Calculus

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Tom M. Apostol Vol.1-part.1-pág.1-455

Transcript of Calculus

Page 1: Calculus
Page 2: Calculus

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O^ob`b nrb kl e^v ^`rboal pl_ob 0/ nrb e^ ab `lkpqfqrfo rk mofjbo `roplab BŠi`ril v FbljbqoŒ^ @k^iŒqf`^-Tklp plpqfbkbk nrb bi `^jfkl sboa^abol m^o^bkqbkabobi BŠi`ril mofk`fmf^ lk rk bpqrafl `ljmibql abi pfpqbj^ ab ilp k•jbolpob^ibp abp^ooliiŠkalil m^pl ^ m^pl ab j^kbo^ iŽdf`^ v ofdrolp^- Nqolp fkpfpqbkbk nrb bi BŠi`ril bp ^kqb qlal rk fkpqorjbkql m^o^ilp fkdbkfbolp u cŒpf`lp:u mlo`lkpfdrfbkqb+ nrb rk `ropl ab_b iibs^o ^ i^p ^mif`^`flkbp abi BŠi`ril ^mbi^kal ^ i^fkqrf`fŽk+ m^o^ abpmr‹p+mlo bi bgbo`f`fl bk i^ obplir`fŽk ab mol_ibj^p+ ^i`^kw^oabpqobw^lmbo^qlof^- Dk ^j_lp mrkqlp ab sfpq^ e^v jr`e^ m^oqb ab o^wŽk-Di BŠi`ril bp rk^ `fbk`f^ abar`qfs^ v rk^ o^j^ ab i^ L^qbjŠqf`^ mro^- @i jfpjlqfbjml bp jrv fjmloq^kqb ob`loa^o nrb bi BŠi`ril qfbkbmolcrka^p o^Œ`bpbk mol,_ibj^p cŒpf`lpv nrb do^k m^oqbab pr mlqbk`f^ v _biibw^abofs^ ab i^ s^ofba^a abprp ^mif`^`flkbp- L^p bp mlpf_ib `lj_fk^o rk abp^ooliil qbŽof`l ofdrolpl `lk rk^p^k^ cloj^`fŽk q‹`kf`^+ v bpqbif_ol obmobpbkqrk fkqbkql ab bpq^_ib`bork pbkpf_ibbnrfif_ofl bkqobi^p alp qbkabk`f^p- @rknrb pb qo^qbbi BŠi`ril `ljl `fbk`f^ abar`,qfs^+kl mlo bpl pb ^_^kalk^k i^p&mif`^`flkbp ^ mol_ibj^p cŒpf`lp-K^p abjlp,qo^`flkbp ab qlalp ilp qblobj^p fjmloq^kqbp pb `lkpfabo^k `ljl rk^ m^oqbbpbk`f^ibk bi abp^ooliil ab i^p fab^p j^qbjŠqf`^p+ v `lk cob`rbk`f^ s^k mob`bafa^pab rk^afp`rpfŽk dblj‹qof`^ l fkqrfqfs^ m^o^a^o ^i bpqraf^kqb rk^ sfpfŽk jŠp mbkbqo^kqbabi mlonr‹ ab i^ abjlpqo^`fŽk- @rknrb bpq^pafp`rpflkbp fkqrfqfs^p mrbabk pboprcf`fbkqbpm^o^bi ib`qlo nrb kl bpq‹ fkqbobp^al bk ilp abq^iibpab i^ abjlpqo^`fŽk+q^j_f‹k pb fk`irvb i^ abjlpqo^`fŽk `ljmibq^ m^o^ ^nrbiilp nrb mobcfbo^krk^bumlpf`fŽk jŠp ofdrolp^-

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qbloŒ ab i^ fkqbdo^`fŽkbp buqobj^a^jbkqb pbk`fii^ bk bpqb`^pl- Lfbkqo^p bi bpqr,af^kqb ^mobkabi^p molmfba^abpab i^ fkqbdo^im^o^crk`flkbp bp`^ilk^a^p+ ^anrfbobbumbofbk`f^bk bi rpl ab i^ klq^`fŽk prj^`fŽk v ^i jfpjl qfbjml pb c^jfif^ofw^`lk bi pfj_lifpjl ab i^ fkqbdo^i-Cb bpq^j^kbo^ pb s^k `lkpqorvbkal ilp mbia^•lpm^o^ nrb i^ qo^kpf`fŽkab crk`flkbp bp`^ilk^a^p ^ lqo^p crk`f`kbp jŠp dbkbo^ibpm^obw` cŠ`fi v k^qro^i-

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K^ pbdrka^ baf`fŽk afcfbobab i^ mofjbo^ bk jr`elp ^pmb`qlp-Rb e^ ^•^afalbi „idb_o^ ifkb^i: ilp qblobj^p abi s^ilo jbafl v i^p ^mif`^`flkbp abi BŠi`ril pbe^k fkqolar`fal bk ilp mofjbolp `^mŒqrilp+v pb e^ ^•^afal _rbk k•jbol abkrbslp v pbk`fiilp bgbo`f`flp- Tk^ fkpmb``fŽk abi Œkaf`bobsbi^ nrb bi if_ol pb e^afsfafal bk `^mŒqrilpab jbklo buqbkpfŽk+abp^ooliiŠkalpb `^a^ rkl pl_ob rk`lk`bmql fjmloq^kqb- U^of^p pb``flkbp e^k pfal bp`ofq^pab krbsl v oblod^kfw^a^pm^o^ molmlo`flk^o rk^ jbglo crka^jbkq^`fŽk v jbglo^o i^ cirfabw ab i^p fab^p-

@i fdr^i nrb bk i^ mofjbo^ baf`fŽk+ `^a^ `lk`bmql krbsl fjmloq^kqb sfbkbmob`bafal ab rk^ fkqolar``fŽk efpqŽof`^+nrb abp`of_b pr abp^ooliil abpab rk^mofjbo^ kl`fŽk cŒpf` fkqrfqfs^ e^pq^ pr clojri^`fŽk j^qbjŠqf`^ mob`fp^-Di bpqr,af^kqb abp`r_ob bk m^oqbilp bpcrbowlpabi m^p^al v ilp qofrkclp ab ilp elj_obpnrb jŠp e^k `lkqof_rfal ^i qbj^- Cb bpqb jlal bi bpqraf^kqb pb `lksfboqb bkm^oqf`fm^kqb`qfsl bk i^ bslir`fŽk ab i^p fab^p v kl nrba^ `ljl jbol l_pbos^alom^pfsl ab ilp obpriq^alp-

K^ pbdrka^ baf`fŽk+`ljl i^ mofjbo^+bpqŠafsfafa^ bk alp sli•jbkbp- K^palpqbo`bo^pm^oqbpmofjbo^p abi Ulirjbk 0 qo^q^kabi BŠi`ril `lk crk`flkbp ab rk^s^of^_ib+ fk`irvbkal i^p pbofbpv rk^ fkqolar``fŽk ^ i^p b`r^`flkbp afcbobk`f^ibp-K^ •iqfj^ qbo`bo^m^oqbabi Ulirjbk 0 fkqolar`b bi „idb_o^ ifkb^i `lk ^mif`^`flkbp^ i^ FbljbqoŒ^ v ^i @kŠifpfp-Fo^k m^oqbab bpqlp qbj^p pb ^mlv^ pŽifa^jbkqb bkbi `Ši`ril ab bgbjmilp nrb firpqo^k i^ qbloŒ dbkbo^i- Diil molmlo`flk^ rk^ jbw`i^ab „idb_o^ v ab @kŠifpfpv `lkqof_rvb ^ mobm^o^obi `^jfkl m^o^ i^ qo^kpf`fŽkabi BŠi`ril `lk rk^ s^of^_ib ^i BŠi`ril `lk s^of^p s^of^_ibp+nrb pb qo^q^bk biUlirjbk Hi- Tk abp^ooliil jŠp ^jmifl ab „idb_o^ ifkb^i pb e^oŠ kb`bp^ofl bk i^pbdrka^ baf`fŽk abi Ulirjbk 00-

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Page 14: Calculus

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Page 20: Calculus

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Page 21: Calculus

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Page 22: Calculus

1 Fiomj_p^^d‡i

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Page 23: Calculus

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Cbpab @onrŒjbabp+ bi abp^ooliil abi j‹qlal ab bue^r`fŽk qrsl nrb bpmbo^o`^pf 07 pfdilp+ e^pq^ nrb bi rpl ab pŒj_lilp v q‹`kf`^p ^idb_o^f`^p pb efwl mob,`fpl bk ilp bpqraflp j^qbjŠqf`lp- Di „idb_o^ bibjbkq^i nrb elv aŒ^ bp c^jfif^o^ i^ j^vloŒ^ ab ilp ^irjklp ab ilp •iqfjlp `roplp ab bkpb•^kw^ pb`rka^of^+ bo^qlq^ijbkqb abp`lkl`fa^ bk qfbjmlp ab @onrŒjbabp+ il nrb e^`Œ^ fjmlpf_ib buqbk,abo bi j‹qlal ^ `r^inrfbo `i^pb ab obdflkbp+ pfk mlpbbo j^kbo^ ^ab`r^a^ abmlabo bumobp^o ilp i^odlp `Ši`rilp bk cloj^ pfjmifcf`^a^-

EHFTQ@ 0-1 Bg h„oj_j _` `sc\p^d‡i \kgd^\_j \ pi\ m`bd‡i n`hd^dm^pg\m,

Tk `^j_fl ibkql mbol obslir`flk^ofl+ bk bi abp^ooliil ab i^p klq^`flkbp j^,qbjŠqf`^p+ bjmbwŽ bk bi pfdil WUH C-B- Di bkdloolpl pfpqbj^ ab krjbo^`fŽkolj^kl crb abpmi^w^al do^ar^ijbkqb mlo ilp `^o^`qbobp ^oŠ_fdlp rqfifw^alp elvaŒ^: ilp pfdklp * v , crbolk fkqolar`falp mlo mofjbo^ sbw+ v pb bjmbw^olk ^ob`lkl`bo i^p sbkq^g^p ab i^ klq^`fŽk ab`fj^i- Cro^kqb bpqb jfpjl mboŒlal+ ilp_ofii^kqbp obpriq^alp ab ilp j^qbjŠqf`lp fq^if^klp S^oq^dif^+ B^oa^kl v Eboo^ofnrb afbolk plir`flkbp ^idb_o^f`^p ^ i^p b`r^`flkbp `•_f`^ v `rŠoqf`^+ bpqfjriŽbi abp^ooliil ab i^ L^qbjŠqf`^ v ^kfjŽ ^ i^ ^`bmq^`fŽk abi ibkdr^gb ^idb_o^f`lkrbsl v prmboflo- Blk i^ fkqolar``fŽk jrv buqbkafa^ ab ilp _fbk bibdfalp pŒj,_lilp ^idb_o^f`lp+ obsfsfŽ bi fkqbo‹p mlo bi ^kqfdrl j‹qlal ab bue^r`fŽk v bkbi pfdil WUH abp`r_ofbolk j•iqfmibp obpriq^alp m^o`f^ibp+ ilp nrb `ljl B^s^,ifbof+ Slof`biif+ Ql_bos^i+ Eboj^q+ O^p`^i v V^iifp crbolk mflkbolp-

Page 24: Calculus

3 Fiomj_p^^d‡i

Fo^ar^ijbkqb+ bi j‹qlal ab bue^r`fŽk crb qo^kpclojŠkalpb bk il nrb elvpb `lkl`b `ljl BŠi`ril fkqbdo^i+krbs^ v mlqbkqbafp`fmifk^ nrb qfbkb krjbol,pŒpfj^p ^mif`^`flkbp kl pŽil bk mol_ibj^p obi^qfslp ^ Šob^p v sli•jbkbp+ pfklq^j_f‹k bk mol_ibj^p ab lqo^p `fbk`f^p- Dpqbj‹qlal+ nrb j^kqfbkb ^idrkl abilp `^o^`qbobp lofdfk^ibp abi j‹qlal ab bue^r`fŽk+ ob`f_fŽ pr j^vlo fjmriplbk bi pfdil WUHH+ab_fal ^ ilp bpcrbowlpab Hp^^`Mbtqlk '0531,0616( v FlqqcofbaKbf_kfw'0535,0605(+ v pr abp^ooliil `lkqfkrŽ aro^kqb bi pfdil WHW+e^pq^ nrb@rdrpqfk,Klrfp B^r`ev '0678,0746( v Aboke^oa Qfbj^kk '0715,0755( ib afbolkrk^ _^pb j^qbjŠqf`^ cfojb- Olpqboflobp^cfk^jfbkqlp v buqbkpflkbp ab i^ qbloŒ^e^k iibd^al e^pq^ i^ L^qbjŠqf`^ `lkqbjmloŠkb^-

0 0-2 :Y Zna\Q\ QRRdUNbPVp[]N_NRYm_RNQRb[ `RTZR[a\ QR]N_mO\YN

@kqbpab mol`babo ^i bpqrafl pfpqbjŠqf`l abi BŠi`ril fkqbdo^i+pboŠ fkpqor`,qfsl ^mif`^o bi j‹qlal ab bue^r`fŽk afob`q^jbkqb ^ rk^ ab i^p cfdro^p m^oqf`r,i^obp qo^q^a^pmlo bi jfpjl @onrŒjbabp- K^ obdfŽk bk `rbpqfŽk bpqŠmobpbkq^a^bk i^ cfdro^ 0-2 v mrbab abp`of_fopb`ljl pfdrb9 Rf pb bifdb rk mrkql ^o_fqo^oflab i^ _^pb ab i^ cfdro^ v pb abpfdk^ mlo s pr afpq^k`f^ ^ /+ i^ afpq^k`f^ sboqf`^iab bpqbmrkql ^ i^ `ros^ bp s%*Dk m^oqf`ri^o+pf i^ ilkdfqra ab i^ _^pb bp ] i^^iqro^ ab i^ cfdro^ bp ]0Š K^ afpq^k`f^ sboqf`^i ab r ^ i^ `ros^ pb abkljfk^ ~loab,k^a^‚ ab s, K^ `ros^ ^pŒabp`ofq^pb abkljfk^ k\mƒ]jg\ v i^ obdfŽk ifjfq^a^ mlobii^ v mlo ilp alp pbdjbkqlp ob`qfiŒkblp+pb ii^j^ n`bh`ioj k\m\]‡gd^j,

K'''''''''''''''''

l s @molufj^`fŽk mlo abcb`ql

EHFTQ@ 0-2 O_ag_hni j[l[\ƒfc]i EHFTQ@ 0-3

@molufj^`fŽk mlo bu`bpl

Page 25: Calculus

Bg h„oj_j _` `sc\p^d‡i k\m\ `g ƒm`\ _` pi n`bh`ioj _` k\mƒ]jg\ 4-

Dpq^cfdro^ mrbab bk`boo^opbbk rk ob`qŠkdril ab _^pb ] v ^iqro^ ]!* `ljlpb sb bk i^ cfdro^ 0-2- N_pbos^kal i^ cfdro^ m^ob`b k^qro^i ^cfoj^o nrb bi Šob^abi pbdjbkql m^o^_Žif`l bp jbklo nrb i^ jfq^a abi Šob^ abi ob`qŠkdril- @onrŒ,jbabp efwl bi plomobkabkqbabp`r_ofjfbkql ab nrb bi Šob^ abi pbdjbkql m^o^,_Žif`l bp bu^`q^jbkqb pi o`m^djab i^ abi ob`qŠkdril: bp ab`fo+ > <]1-1* alkab= abpfdk^ bi Šob^ abi pbdjbkql m^o^_Žif`l- Rb sboŠ ^ `lkqfkr^`fŽk `Žjl pbiibd^ ^ bpqb obpriq^al-

Rb e^`b klqŠo nrb bi pbdjbkql m^o^_Žif`l af_rg^al bk i^ cfdro^ 0-2 kl bpqŠbibdfal bu^`q^jbkqb q^i `ljl il af_rgŽ @onrŒjbabp v nrb ilp abq^iibp nrb

]!„ob^ abi ob`qŠkdril < ,- f0i

f] ,,, ] < i]T T

l \ /\i i

EHFTQ@ 0-4 @ƒg^pgj _`g ƒm`\ _` pi n`bh`ioj k\m\]‡gd^j,

pfdrbk kl plk bu^`q^jbkqb ilp rqfifw^alp mlo ‹i- Rfk bj_^odl+ i^p d_`\n bpbk`f^ibpplk i^p ab @onrŒjbabp: il nrb ^nrŒ pb bumlkb mrbab `lkpfabo^opb `ljl bi j‹,qlal ab bue^r`fŽk bumrbpql `lk i^ klq^`fŽk jlabok^-

Di j‹qlal `lkpfpqb pfjmibjbkqb bk il pfdrfbkqb9 pb afsfab i^ cfdro^ bk rk`fboql k•jbol ab _^ka^p v pb l_qfbkbk alp ^molufj^`flkbp ab i^ obdfŽk+rk^mlo abcb`ql v lqo^ mlo bu`bpl+ rqfifw^kal alp `lkgrkqlp ab ob`qŠkdrilp `ljlpb fkaf`^ bk i^ cfdro^ 0-3- 'Rb rqfifw^k ob`qŠkdrilp jbglo nrb mliŒdlklp ^o_fqo^oflpm^o^ pfjmifcf`^o ilp `Ši`rilp-( Di Šob^ abi pbdjbkql m^o^_Žif`l bp j^vlo nrb biŠob^ qlq^i ab ilp ob`qŠkdrilp fkqboflobp mbol jbklo nrb i^ ab ilp ob`qŠkdrilpbuqboflobp-

Rf `^a^ _^ka^ pb pr_afsfab ^ pr sbw+pb l_qfbkb rk^ krbs^ ^molufj^`fŽk`lk j^vlo k•jbol ab _^ka^p+ i^ obrkfŽk ab i^p Šob^p ab ilp ob`qŠkdrilp fkqb,

Page 26: Calculus

5 Fiomj_p^^d‡i

oflobp ^m`^`* jfbkqo^p nrb bi qlq^i ab i^p Šob^p ab ilp ob`qŠkdrilp buqboflobp_`^m`^`, @onrŒjbabp sfl nrb pb mlaŒ^ ildo^o bi Šob^ `lk bi do^al ab ^molufj^`fŽkabpb^al pfk jŠp nrb qlj^o rk k•jbol prcf`fbkqb ab _^ka^p-

Di `Ši`ril bcb`qfsl bk bpqb `^pl pb ob^ifw^ `ljl pb fkaf`^ ^ `lkqfkr^`fŽk-Blk l_gbql ab pfjmifcf`^o pb pr_afsfab i^ _^pb bk i k\mo`n fdr^ibp+ `^a^ rk^ abilkdfqra \` i 's‹^pb cfd- 0-4(- Klp mrkqlp ab pr_afsfpfŽk `loobpmlkabk ^ ilp pf,drfbkqbp s^ilobp ab r7

k+y +0] *1] * ,,, *&i + i(_ +i] < ] ,i i i i i

K^ bumobpfŽk dbkbo^i ab rk mrkql ab i^ pr_afsfpfŽk bp s <f] Ei* alkab f qlj^ilp s^ilobp pr`bpfslp f < N+ 0+ 1+ 2+ -- + i* Dk `^a^ mrkql f] Fi pb `lkpqorvb biob`qŠkdril buqboflo ab ^iqro^ &f]gi'0 `ljl pb fkaf`^ bk i^ cfdro^ 0-4- Di Šob^ab bpqb ob`qŠkdril bp bi molar`ql ab i^ _^pb mlo i^ ^iqro^ v bp fdr^i ^9

Rf pb abpfdk^ mlo P9 i^ prj^ ab i^p Šob^p ab qlalp ilp ob`qŠkdrilp buqboflobp+mrbpql nrb bi Šob^ abi ob`qŠkdril f+ndhj bp %\1Ei1'f0 pb qfbkb i^ cŽojri^9

'0-0 (

Cb cloj^ ^kŠild^ pb l_qfbkb i^ cŽojri^ m^o^ i^ prj^ my ab qlalp ilp ob`qŠkdrilpfkqboflobp9

'0-1(

K^ cloj^ ab bpq^p prj^p bp ab do^k fjmloq^k`f^ m^o^ pr `Ši`ril- MŽqbpbnrb bi c^`qlo nrb jriqfmif`^ ^ ]! Ei1 bk i^ b`r^`fŽk '0-0( bp i^ prj^ ab ilp `r^,ao^alp ab ilp i mofjbolp k•jbolp k^qro^ibp9

01 * 11 * --- * i/ †

'Di c^`qlo `loobpmlkafbkqb bk i^ b`r^`fŽk '0-1( bp ^kŠildl p^isl nrb i^ prj^qfbkb •kf`^jbkqb i + 0 prj^kalp-( O^o^ s^ilobp do^kabp ab i i^ l_qbk`fŽk abbpq^ prj^ mlo ^af`fŽk afob`q^ ab prp prj^kalp bp mbp^a^+ mbol ^cloqrk^a^,

Page 27: Calculus

Bg h„oj_j _` `sc\p^d‡i k\m\ `g ƒm`\ _` pi n`bh`ioj _` k\mƒ]jg\ 5

jbkqb bufpqb rk^ fkqbobp^kqb fabkqfa^a nrb e^`b mlpf_ib l_qbkbo bpq^ prj^ mlork `^jfkl jŠp pfjmib+ v bp i^ pfdrfbkqb9

'0-2(2 1

01 11 1 i i i) )!%)i <,*,*,-

2 1 5

Dpq^ fabkqfa^a bp sŠifa^ m^o^ qlal bkqbol i 19 0 X mrbab abjlpqo^opb abi pfdrfbk,qb jlal9 Rb m^oqbab i^ cŽojri^ &f) g'1;f6)1C)1f) 0 X pb mlkb bk i^ cloj^

1f0 * 1f * 0 < &f* 0(2 , f1,

G^`fbkal f < 0+ 1+ --- + i + 0+ l_qbkbjlp i^p i + 0 cŽojri^p

2 - 01 * 2 - 0 * 0 < 12 , 02

2 - 11 * 2 - 1 * 0 < 22 , 12

0%h* 0(1 * 0%h* 0( * 0 < h0 * %h* 0(2-

@i prj^o bpq^p cŽojri^p+ qlalp ilp q‹ojfklp abi pbdrkal jfbj_ol pb obar`bkbu`bmql alp v pb l_qfbkb

2ZO* 11 * --- * %h* 0(1\ * 2Z0 * 1* --- * %h* 0(\ * %h* 0( < h0 * O-

K^ pbdrka^ prj^ abi mofjbo jfbj_ol bp i^ prj^ ab ilp q‹ojfklp ab rk^ mol,dobpfŽk ^ofqj‹qf`^ `rvl s^ilo bp ph%h* 0(- Olo q^kql i^ •iqfj^ fdr^ia^a klp a^

'i@(2 1

01 * 11 * --- * &i + 0(1 < --, \ --, * 9 - -2 1 5

Rrj^kal i% ^ ilp alp jfbj_olp+ l_qbkbjlp '0-2(-K^p bumobpflkbp bu^`q^p a^a^p bk ilp pbdrkalp jfbj_olp ab '0-2( v '0-3( kl

plk kb`bp^of^p m^o^ bi l_gbql nrb ^nrŒ pb mbopfdrb+ mbol pfosbk m^o^ abar`focŠ`fijbkqb i^p alp _`ndbp\g_\_`n nrb fkqbobp^k

'H-4( i1

01 * 11 * --- * &i + 0(1 ; , ; 01 * 11 * --- * i0

2

nrb plk sŠifa^p m^o^ qlal bkqbol i 19 0- Dpq^p abpfdr^ia^abp mrbabk abar`fopbcŠ`fijbkqb `ljl `lkpb`rbk`f^p ab '0-2( X 'H-3(+ l afob`q^jbkqb mlo fkar``fŽk-'U‹^pb i^ Rb``fŽk 0 3-0-(

Page 28: Calculus

7 Fiomj_p^^d‡i

Lriqfmif`^kal ^j_^p abpfdr^ia^abp bk '0-4( mlo _^-i\ v e^`fbkal rpl ab'0-0( X '0-1( pb qfbkb9

'0-5(

m^o^`^a^ i* v l_pbosŠkalpb nrb pb mobpbkq mlo mofjbo^ sbw bi k•jbol ]! .2-K^p abpfdr^ia^abp bk '0-5( bumobp^knrb m^o^ `^a^ i bi k•jbol _^ .2 bpqŠ`lj,mobkafal bkqobn,9v P• Obol ^elo^ bp cŠ`fi mol_^o nrb ]%.2 bp bi ˆid^j k•jbolnrb dlw^ ab bpq^ molmfba^a: bp ab`fo+ nrb pf > bp rk k•jbol nrb sbofcf`^ i^pabpfdr^ia^abp

'0-6(

m^o^ `^a^ bkqbol mlpfqfsl i* e^ ab pbo kb`bp^of^jbkqb > < ]\ 1, Olo bpq^ o^wŽkabargl @onrŒjbabp nrb bi Šob^ abi pbdjbkql m^o^_Žif`l bp ]\ 1,

O^o^ mol_^o nrb > < ]\ 1 pb rqfifw^k rk^ sbw jŠp i^p abpfdr^ia^abp '0-4(-Rrj^kal h0 ^ ilp alp jfbj_olp ab i^ abpfdr^ia^a ab i^ fwnrfboa^ bk '0-4( pbl_qfbkb9

Lriqfmif`^kal mlo ]\ k^ v rqfifw^kal '0-0( pb qfbkb

'0-7(

@kŠild^jbkqb+ obpq^kal i0 ab ilp alp jfbj_olp ab i^ abpfdr^ia^a ab i^ abob`e^bk '0-4( v jriqfmif`^kal mlo ]! -i\ pb iibd^ ^ i^ abpfdr^ia^a9

'0-8(

Olo q^kql+`^a^ k•jbol > nrb p^qfpc^d '0-6( e^ ab p^qfpc^`boq^j_f‹k9

'KiN(

m^o^ `^a^ bkqbol i 9880- @elo^ _fbk+ e^v pŽil qobpmlpf_fifa^abp9

> ; ]0

, "

Page 29: Calculus

Be`m^d^djn 8

Rf pb morb_^ nrb i^p alp mofjbo^p `lkar`bk ^ rk^ `lkqo^af``fŽk e^_oŠ ab pbo> < ]! d2+ v^ nrb+ ^i bpqfil ab Reboil`h Glijbp+ pb ^dlq^k ^pŒqlalp i^p mlpf_fif,a^abp-

RrmŽkd^pb nrb i^ abpfdr^ia^a > = ]! d2 crbo^ `fboq^- Cb i^ pbdrka^ abpf,dr^ia^a bk '0-0/( pb l_qfbkb9

'H-00(]1 ]1

7''4'- i

nrb > + ]%d2 bp mlpfqfsl+ pb mrbabk afsfafo> + ]! .2 v jriqfmif`^kal abpmr‹p mlo i pb

m^o^ `^a^ bkqbol i ƒ 0- Orbpql^j_lp jfbj_olp ab '0-00( mlol_qfbkb i^ abpfdr^ia^a

m^o^ `^a^ i, Obol bpq^ abpfdr^ia^a bp bsfabkqbjbkqb c^ip^ m^o^ i<]1e&>+]1e1',Olo q^kql+ i^ abpfdr^ia^a >< ]%d2 `lkar`b ^ rk^ `lkqo^af``fŽk- Cb cloj^ ^kŠ,ild^ pb abjrbpqo^ nrb i^ abpfdr^ia^a > ; ]! d2 `lkar`b ^ pr sbw ^ rk^ `lkqo^,af``fŽk v mlo q^kql ab_b pbo > < ]1e1 `ljl pb e^ ^cfoj^al ^kqbp-

)0 )&, :WR_PVPV\`

0- '^( Llafcf`^o i^ obdfŽk bk i^ cfdro^ 0-2 qlj^kal `ljl loabk^a^ m^o^ `^a^ r bi s^ilo /r0

bk sbw ab s0Š Cf_rg^o i^ krbs^ cfdro^- RŒd^kpbbk bpqb `^pl ilp m^plp mofk`fm^ibp abi^m^oq^al ^kqboflo v `ljmŠobkpb ^j_lp+ bpqraf^kal i^ obmbo`rpfŽk abi `^j_fl bk bi `Ši,`ril ab =+ Dcb`q•bpb il jfpjl pf i^ loabk^a^ bk `^a^ s bp9'_( 1s/) 'b( s/) 'a( 0s/ * )$ 'b( \s/ * `-

1- LlafcŒnrbpb i^ obdfŽk bk i^ cfdro^ 0-2 qlj^kal `ljl loabk^a^ m^o^ `^a^ s* s1 bk sbwab s0Š Cf_•gbpb i^ krbs^ cfdro^-'^( ‰pbpb rk^ `lkpqor``fŽk ^kŠild^ ^ i^ af_rg^a^ bk i^ cfdro^ 0-4 v mor‹_bpb nrb i^pprj^p fkqboflo v buqboflo P9 v p+bpqŠk a^a^p mlo

'_( TqfiŒ`bkpb i^p abpfdr^ia^abp 'nrb mrbabk mol_^opb mlo fkar``fŽk `ljmibq^: s‹^pbRb``fŽk 0 3-1(-

'0-01( i.q2 * 12 * --- * %h * 0(2 ; , ; 02 * 12 * --- * h0

3

m^o^ abjlpqo^o nrb n9 ; ]2-2 ; P9 m^o^ `^a^ i* v mol_^o nrb ]2-2 bp bi ˆid^j k•jbol`ljmobkafal bkqob p+v P9 m^o^ `^a^ i*'b( ƒPr‹ k•jbol prpqfqrvb ^ _3.3 pf i^ loabk^a^ bk `^a^ s bp \s1 * `>

Page 30: Calculus

0/ Fiomj_p^^d‡i

2- K^p abpfdr^ia^abp '0-4( v '0-01( plk `^plp m^oqf`ri^obpab i^p abpfdr^ia^abp jŠp dbkbo^ibp

'0-02(iQ%+

gf * 0f * --- * &i + g'f ; ,, ; gf * 0f * --- * ife * 0

nrb plk sŠifa^p m^o^ `^a^ bkqbol i x 0 X `^a^ bkqbol f ƒ 0- Rrmrbpq^p sŠifa^p '0-02(dbkbo^iŒ`bkpbilp obpriq^alp abi bgbo`f`fl 1-

) )&- 6[mYV`VP_oaVP\QRYZna\Q\ QR6_^boZRQR`

Lbaf^kqb `Ši`rilp ^kŠildlp ^ ilp ob^ifw^alp bk bi ^m^oq^al 0 0-2+@onrŒjbabpiibdŽ ^ i^ `lk`irpfŽk ab nrb bi Šob^ abi pbdjbkql m^o^_Žif`l `lkpfabo^al bp ]1-1,Dpqb eb`el pb ^`bmqŽ `ljl rk qblobj^ j^qbjŠqf`l+ e^pq^ nrb m^p^alp rklp1/// ^•lp pb mbkpŽnrb ab_Œ^k^k^ifw^opbilp obpriq^alp abpab rk mrkql ab sfpq^jŠp `oŒqf`l-O^o^ `ljmobkabo mlo nr‹ er_l nrfbk mrpl bk ara^ i^ s^ifabw ab i^p`lk`irpflkbp ab @onrŒjbabp+bp kb`bp^ofl qbkbomobpbkqbilp `^j_flp fjmloq^kqbpnrb e^k qbkfal ird^o bk i^ ob`fbkqbefpqlof^ ab i^ L^qbjŠqf`^-

B^a^ o^j^ abi `lkl`fjfbkql bp rk `lkgrkql ab fab^p abp`ofq^p mlo jbaflab m^i^_o^p v pŒj_lilp+ v kl pb mrbabk `ljmobkabo bpq^pfab^p pfk rk `lkl`f,jfbkql bu^`ql ab i^p m^i^_o^pv pŒj_lilp nrb pb rqfifw^k-Bfboq^po^j^p abi `lkl,`fjfbkql `lkl`fa^p mlo ndno`h\n _`_p^odqjn* pb afpqfkdrbk ab lqo^p mlonrb bkbii^p pb bifdb \ kmdjmdrk k•jbol ab `lk`bmqlp ~kl abcfkfalp‚ v qlal lqol `lk`bmqlbk bi pfpqbj^ pb abcfkb ^ m^oqfoab ^nr‹iilp- Bfboq^pobi^`flkbp bkqobbpqlp`lk`bmqlpkl abcfkfalp pb qlj^k `ljl \sdjh\n l kjnopg\_jn v lqo^p obi^`flkbp nrb mrbabkabar`fopb ab bpqlp ^uflj^p pb abkljfk^k o`jm`h\n, Di bgbjmil jŠp c^jfif^o abpfpqbj^ abar`qfsl bp i^ FbljbqoŒ^ bibjbkq^i br`ifaf^k^+ nrb e^ pfal bpqraf^a^mlo qla^ mboplk^ `riq^ abpab i^ ‹ml`^ ab i^ Fob`f^ ^kqfdr^-

Di bpmŒofqrab i^ L^qbjŠqf`^ dofbd^+pfdrfbkal bi j‹qlal ab mlpqri^alp vqblobj^p `ljl bk i^ FbljbqoŒ^ ab ilp Bg`h`iojn ab Dr`ifabp+ aljfkŽ bi mbk,p^jfbkql ab ilp j^qbjŠqf`lp e^pq^ i^ ‹ml`^ abi Qbk^`fjfbkql- Tk^ c^pb krbs^v sfdlolp^ bk bi abp^ooliil ab i^ L^qbjŠqf`^ bjmbwŽ `lk i^ ^m^of`fŽkabi „idb_o^bk bi pfdil WUH: X ilp 2// ^•lp nrb pfdrfbolk crbolk qbpqfdlpab do^k `^kqfa^a ababp`r_ofjfbkqlp fjmloq^kqbp- Di o^wlk^jfbkql iŽdf`l mob`fpl abi j‹qlal abar`qfsl`lk bi rpl ab ^uflj^p+ abcfkf`flkbp v qblobj^p+ bpqrsl j^kfcfbpq^jbkqb ^rpbkqbbk bpqbmboŒlal-Klp j^qbjŠqf`lp ab ilp pfdilp WUH+WUHHX WUHHHob`rooŒ^k rk^jbw`i^ `roflp^ ab o^wlk^jfbkql abar`qfsl `lj_fk^al `lk i^ fkqrf`fŽk v i^ mro^`lkgbqro^: v kl bp buqo^•l nrb ^idrklp ab prp obpriq^alp pb e^v^ sfpql mlpqboflo,jbkqb nrb bo^k fk`loob`qlp- Ml l_pq^kqb+rk k•jbol plomobkabkqbab abp`r_of,jfbkqlp fjmloq^kqbp l`roobk bk bpqbmboŒlal+v rk^ do^k m^oqbab bpq^l_o^ e^ pl,_obsfsfal i^ morb_^ ab i^ Gfpqlof^+obmobpbkq^kalrk qof_rql ^ i^ abpqobw vq^ibkql ab ^nrbiilp `fbkqŒcf`lp-

Page 31: Calculus

>iƒgdndn ^m…od^j_`g h„oj_j _` >mlp…h`_`n 00

Br^kal bjmbwŽ ^ afpjfkrfo bi `^ra^i ab krbslp abp`r_ofjfbkqlp+ ^m^ob`fŽrk krbsl mboŒlal ab ^kŠifpfp `oŒqf`l: v ml`l ^ ml`l+ ilp j^qbjŠqf`lp pb sfbolkl_ifd^alp ^ slisbo ^ i^p fab^p `iŠpf`^p abi j‹qlal abar`qfsl+ ^i fkqbkq^o mlkbocrka^jbkqlp cfojbp ^ i^ krbs^ L^qbjŠqf`^ - Dpq^ c^pb abi abp^ooliil+ nrb bjmbwŽ^ mofk`fmflp abi pfdil WHWv e^ `lkqfkr^al e^pq^ bi jljbkql mobpbkqb+e^ ^i`^kw^alrk do^al ab ^_pqo^``fŽk v mrobw^ iŽdf`^ nrb e^ prmbo^al qla^p i^p qo^af`flkbp ab i^`fbk`f^ dofbd^- @ i^ sbw+ e^- molmlo`flk^al rk^ `ljmobkpfŽk jŠp `i^o^ ab ilpcrka^jbkqlp kl pŽil abi BŠi`ril+ pfkl ab qla^p i^p o^j^p ab i^ L^qbjŠqf`^-

G^v jr`e^p cloj^p ab bpqor`qro^o bi BŠi`ril `ljl pfpqbj^ abar`qfsl- Tk^j^kbo^ mlpf_ib+ bp qlj^o ilp k•jbolp ob^ibp `ljl `lk`bmqlp kl abcfkfalp l mofjf,qfslp- @idrk^p ab i^p obdi^p nrb ofdbk i^p lmbo^`flkbp `lk ilp k•jbolp ob^ibpmrbabk qlj^opb `ljl ^uflj^p- Dpqb pfpqbj^ ab ^uflj^p pb e^ fk`irfal bk i^m^oqb 2 ab bpq^ fkqolar``fŽk- Mrbslp `lk`bmqlp+ q^ibp `ljl dio`bm\g* g…hdo`* jiod+ipd_\_* _`mdq\_\* mrbabk abcfkfopb ^ m^oqfoab ilp k•jbolp ob^ibp-K^p molmfba^abpab bpqlp `lk`bmqlp pb abar`bk `ljl qblobj^p ^ m^oqfoab ilp ^uflj^p-

Blkpfabo^kal bi BŠi`ril `ljl rk^ m^oqb abi pfpqbj^ abar`qfsl+ bi obpriq^alab @onrŒjbabp m^o^ bi Šob^ abi pbdjbkql m^o^_Žif`l kl mrbab ^`bmq^opb `ljlrk qblobj^ pf kl pb a^ mobsf^jbkqb rk^ abcfkf`fŽk p^qfpc^`qlof^ ab Šob^- Ml bp`i^ol nrb @onrŒjbabp er_fbo^ clojri^al ^idrk^ sbw rk^ abcfkf`fŽk mob`fp^ abil nrb ‹i bkqbkaŒ^ mlo Šob^- O^ob`b e^_bo qlj^al `ljl `lksbkfl nrb `^a^ obdfŽkqfbkb rk^ Šob^ ^pl`f^a^ ^ bii^- Blk bpq^ efmŽqbpfp pb l`rm^ ab `^i`ri^o Šob^p abobdflkbp m^oqf`ri^obp- Dk prp `Ši`rilp rqfifw^ molmfba^abp abi Šob^ nrb kl pbmrbabk mol_^o jfbkqo^p kl pb mob`fpb nr‹ n` `iod`i_` mlo Šob^- Olo bgbjmil+prmlkb nrb pf rk^ obdfŽk bp fkqboflo ^ lqo^+ bi Šob^ ab i^ obdfŽk jbklo kl mrbabbu`babo ^ i^ ab i^ obdfŽk j^vlo: v q^j_f‹k nrb pf rk^ obdfŽk pb abp`ljmlkbbk alp l jŠp m^oqbp+i^ prj^ ab i^p Šob^p ab `^a^ m^oqb bp fdr^i ^i Šob^ ab qla^i^ obdfŽk- Sla^p bpq^p molmfba^abp pb ^qof_rvbk ^i Šob^+v bk qla^ abcfkf`fŽk nrb pba‹ abi Šob^ bpq^p molmfba^abp e^k ab mlabo abar`fopb `ljl qblobj^p- Dp sbolpŒjfinrb bi jfpjl @onrŒjbabp qlj^o^ bi Šob^ `ljl `lk`bmql mofjfqfsl+ rqfifw^kali^p molmfba^abp jbk`flk^a^p `ljl \sdjh\n,

@`qr^ijbkqb pb `lkpfabo^ i^ l_o^ ab @onrŒjbabp `ljl fjmloq^kqb jŠp nrbmlo `^i`ri^o Šob^p ab cfdro^p m^oqf`ri^obp+ mlonrb e^ prdbofal rk `^jfkl o^wlk^_ibm^o^ _`adidm bi `lk`bmql ab Šob^ m^o^ cfdro^p jŠp l jbklp \m]dom\md\n,@ pr sbw+bi j‹qlal ab @onrŒjbabp prdfbob rk j‹qlal m^o^ abcfkfo rk `lk`bmql jr`el jŠpdbkbo^i+ nrb bp i^ dio`bm\g9 v ^ pr sbw+ i^ fkqbdo^i kl pŽil pb rqfifw^ m^o^ abcfkfov `^i`ri^o Šob^p+ pfkl q^j_f‹k m^o^ bpq^_ib`bo `lk`bmqlp `ljl ilkdfqra ab rk^o`l+ slirjbk+ qo^_^gl v lqolp-

@kqf`fmŠkalpb ^ crqrolp abp^ooliilp+ v rqfifw^kal i^ qbojfklildŒ^ abi BŠi`rilfkqbdo^i+ bi obpriq^al abi `Ši`ril bcb`qr^al bk i^ Rb``fŽk 0 0-2+ m^o^ bi pbdjbkqlm^o^_Žif`l+ pb bumobp^ cob`rbkqbjbkqb `ljl pfdrb9

~K^ fkqbdo^i ab s0 ab N ^ ] bp ]1-1,~

Page 32: Calculus

01 Fiomj_p^^d‡i

v pb bp`of_b pfj_Žif`^jbkqb9

&]s0_s < c1,

9J 2

Di pŒj_lil ` 'rk^ R ^i^od^a^( pb ii^j^ ndbij dio`bm\g v crb fkqolar`fal mloKbf_kfw bk 0564- Di mol`bpl nrb abqbojfk^ bi k•jbol ]1-1 pb abkljfk^ dio`bm\+^d‡i, Klp k•jbolp N v ] nrb ^cb`q^k ^i pfdkl fkqbdo^i pb abkljfk^k g…hdo`n_` dio`+bm\^d4i, Di pfdkl `a s0 _s pb e^ ab `lkpfabo^o `ljl rk qlal- Rr abcfkf`fŽk ab_ba^opb ab i^ jfpj^ j^kbo^ nrb bi af``flk^ofl abp`of_b i^ m^i^_o^ ~L^oqb‚ pfk e^`boobcbobk`f^ ^ ~j^o‚ kf ^ ~qb‚-

Di pŒj_lil ab Kbf_kfw m^o^ i^ fkqbdo^i crb ^`bmq^al molkql mlo jr`elp j^qb,jŠqf`lp+ mlonrb sbŒ^k bk i^ fkqbdo^`fŽk rk qfml ab ~mol`bpl ab prj^`fŽk‚ nrbmbojfqŒ^ prj^o fkcfkfq^p ~`^kqfa^abp fkcfkfq^jbkqb mbnrb•^p‚- Olo bgbjmil+ bk bi`^pl abi pbdjbkql m^o^_Žif`l+ bi Šob^ pb `lk`b_Œ^ `ljl i^ ~prj^‚ ab rk^ fkcfkfa^aab ob`qŠkdrilp fkcfkfq^jbkqb mbnrb•lp ab ^iqro^ s0 v _^pb _s, Di pfdkl fkqbdo^iobmobpbkq^ bi mol`bpl ab prj^`fŽk ab qlalp bpqlp ob`qŠkdrilp- Dpq^ cloj^ abo^wlk^o bp jrv prdbpqfs^ v •qfi cob`rbkqbjbkqb- Cbpab bi mrkql ab sfpq^ iŽdf`l+^alib`b abi abcb`ql ab kl mlabo ^qof_rfo rk pfdkfcf`^al bu^`ql ^i `lk`bmql ~`^k,qfa^a fkcfkfq^jbkqb mbnrb•^‚- @`qr^ijbkqb pb p^_b `Žjl fkqolar`fo i^ fkqbdo^ijbaf^kqb bi k•jbol ob^i+ pfk rqfifw^o `lk`bmqlp jfpqboflplp b fkbumif`^_ibp+ `ljl~fkcfkfqbpfj^i‚- Dpq^ abcfkf`fŽk pb a^oŠ bk bi `^mŒqril 0-

0 0-5 K^ fkqolar``fŽk ^i BŠi`ril nrb pb rqfifw^bk bpqbif_ol

Tk^ bumlpf`fŽk ofdrolp^ v `ljmibq^ q^kql abi BŠi`ril fkqbdo^i `ljl abi afcb,obk`f^i+ abmbkab bpbk`f^ijbkqb ab rk bpqrafl `rfa^alpl abi pfpqbj^ ab ilp k•jbolpob^ibp- Di bpqrafl bk pŒab bpqb pfpqbj^ iibs^al ^ `^_l bk pr qlq^ifa^a+ bp rk qbj^jrv fkqbobp^kqb mbol rk q^kql i^odl+ ab cloj^ nrb obnrfbob rk mbnrb•l slirjbkm^o^ pr `ljmibq^ bumlpf`fŽk- Di j‹qlal pbdrfal bk bpqb if_ol bp bjmbw^o `lk ilpk•jbolp ob^ibp `ljl `g`h`iojn kmdhdodqjnv qlj^o pfjmibjbkqb ^idrk^p ab prpmolmfba^abp crka^jbkq^ibp `ljl \sdjh\n, Dpqlp ^uflj^p v ^idrklp ab ilp qblobj^pjŠp pbk`fiilp nrb mrbabk abar`fopb ab biilp pb afp`rqfoŠk bk i^ m^oqb 2 ab bpqb`^mŒqril- Lr`e^p ab i^p molmfba^abp ab ilp k•jbolp ob^ibp nrb pb e^k qlj^al`ljl ^uflj^p plk mol_^_ibjbkqb c^jfif^obp ^i ib`qlo+ mlo prp bpqraflp ab „idb_o^bibjbkq^i- Rfk bj_^odl+ e^v ^idrk^p molmfba^abp ab ilp k•jbolp ob^ibp nrb kl pbprbibk qbkbo bk `rbkq^ bk bi „idb_o^ bibjbkq^i+ mbol nrb grbd^k rk m^mbi fjmlo,q^kqb bk bi BŠi`ril- Dpq^p molmfba^abp plk `lkpb`rbk`f^ abi ii^j^al \sdjh\ _`g`som`hj npk`mdjm'`lkl`fal q^j_f‹k mlo \sdjh\ _` g\ ^jiodipd_\_' nrb pb bpqraf^oŠ^nrŒ `lk abq^iib- Di ib`qlo mrbab m^o^o pr ^qbk`fŽk bk f^ m^oqb 2 ^kqbp ab bkqo^obk bi `rboml crka^jbkq^i abi qbuql+ l _fbk abg^o i^ ib`qro^ ab bpq^ j^qbof^ m^o^jŠp ^abi^kqb `r^kal pb bk`rbkqob `lk ^nrbii^p m^oqbpab i^ qbloŒ^bk i^p nrb pb

Page 33: Calculus

Fiomj_p^^d‡i \ g\ o`jm…\_` ^jiepiojn 02

rqfifw^k molmfba^abp abi buqobjl prmboflo- K^p j^qbof^p bk bi qbuql nrb abmbka^kabi ^uflj^ abi buqobjl prmboflo pb pb•^i^oŠk `i^o^jbkqb-

O^o^ abp^oolii^o bi BŠi`ril `ljl rk^ qbloŒ^ j^qbjŠqf`^ `ljmibq^+ pboŒ^kb`bp^ofl bumlkbo+ grkql ^i pfpqbj^ ab ^uflj^p abi k•jbol ob^i+ rk `lkgrkqlab ~j‹qlalp ab abjlpqo^`fŽk‚ nrb mbojfqfbo^k abar`fo ilp qblobj^p ^ m^oqfoabilp ^uflj^p- B^a^ ^cfoj^`fŽk bk i^ qbloŒ^ qbkaoŒ^ nrb pbo grpqfcf`^a^ l `ljl~rk^ ibv bpq^_ib`fa^‚ 'bp ab`fo+ rk ^uflj^+ rk^ abcfkf`fŽk l rk qblobj^ mobsf^,jbkqb mol_^al(+ l `ljl bi obpriq^al ab ^mif`^o ^ ibvbp bpq^_ib`fa^p rkl ab ilpj‹qlalp ab abjlpqo^`fŽk ^`bmq^alp- Tk moldo^j^ ab bpq^ k^qro^ibw^ obpriq^oŒ^buqobj^a^jbkqb i^odl v qo^_^glpl+ v ^vra^oŒ^ jrv ml`l ^ i^ `ljmobkpfŽk ab i^j^qbof^ mlo bi mofk`fmf^kqb- @cloqrk^a^jbkqb kl bp kb`bp^ofl mol`babo ab bpq^cloj^ m^o^ iibd^o ^ rk^ _rbk^ `ljmobkpfŽk v j^kbgl abi BŠi`ril- Dk bpqb if_ol pbfkqolar`bk i^p `rbpqflkbp mobp`fkafbkal ab rk cloj^ifpjl bu^dbo^al v pb e^`b^jmifl rpl abi o^wlk^jfbkql dblj‹qof`l `r^kal pb `obb `lksbkfbkqb: mbol ^ijfpjl qfbjml+ pb mol`ro^ nrb i^ bumlpf`fŽk ab i^p j^qbof^p dl`b ab i^ mob`fpfŽk v`i^ofa^a molmf^p ab i^ `fbk`f^ jlabok^- Slalp ilp qblobj^p fjmloq^kqbp ab i^qbloŒ^bk `rbpqfŽk+ bpqŠk bumiŒ`fq^jbkqb bumrbpqlp v ofdrolp^jbkqb abjlpqo^alp-

O^o^ bsfq^o fkqboorjmfo i^ pr`bpfŽk ab fab^p+ ^idrk^p ab i^p abjlpqo^`flkbp^m^ob`bk bk pb``flkbp pbm^o^a^p pb•^i^a^p `lk ^pqbofp`l- Olo i^ jfpj^ o^wŽk+^idrklp ab ilp `^mŒqrilp s^k ^`ljm^•^alp ab pb``flkbp prmibjbkq^of^p bk i^p`r^ibp pb qo^q^k `lk abq^iib ^idrklp qbj^p fjmloq^kqbp obi^`flk^alp `lk bi BŠi`ril-@idrklp ab biilp bpqŠk q^j_f‹k pb•^i^alp `lk ^pqbofp`l m^o^ fkaf`^o nrb mrbabkljfqfopb l mlpmlkbopb pfk nrb pb fkqboorjm^ i^ `lkqfkrfa^a ab i^ bumlpf`fŽk-:Y nrb pb qljbk jŠp l jbklp bk `lkpfabo^`fŽk ilp ^m^oq^alp `lk ^pqbofp`l+ ab,mbkab bk m^oqb ab i^ mobm^o^`fŽk abi ib`qlo v bk m^oqb ab pr fkqbo‹p- K^ mboplk^nrb abpbb rk `ropl `ljmibql ab BŠi`ril q^kql bk qbloŒ^ `ljl bk i^ moŠ`qf`^+qbkaoŠ nrb ibbo qla^ i^ j^qbof^- Di nrb pb fkqbobpb mofjbo^jbkqb mlo i^p fab^p_Špf`^p v i^ moŠ`qf`^+ mlaoŠ prmofjfo ilp ^m^oq^alp `lk ^pqbofp`l-

06BD: &&$# 5RQKMSVRU7GC<8@CLM >6 DMRT_JLMKRQOWQVRU

= *&) =[a_\QbPPVp[N YNFR\_oNQRP\[Wb[a\`

Dk bi bpqrafl ab `r^inrfbo o^j^ ab i^ L^qbjŠqf`^+ pb^ @kŠifpfp+ „idb_o^ lFbljbqoŒ^+ obpriq^ •qfi bjmib^o i^ klq^`fŽk u i^ qbojfklildŒ^ ab i^ SbloŒ^ ab `lk,grkqlp- Dpq^ qbloŒ^+nrb crb abp^oolii^a^ mlo Allib u B^kqlo 'p( ^ cfkbp abi pfdil WHW+

e^ qbkfal rk^ molcrka^ fkcirbk`f^ bk bi abp^ooliil ab i^ L^qbjŠqf`^ bk bi pf,

'p( Fblodb Allib '0704,0753( crb rk iŽdf`l,j^qbjŠqf`l fkdi‹p- Rr if_ol+ Fiq`nodb\^d‡i _` g\ng`t`n _`g k`in\hd`ioj* mr_if`^al bk 0743+ pb•^i^ i^ `ob^`fŽk abi mofjbo pfpqbj^ moŠ`qf`l abKŽdf`^ pfj_Žif`^- Fblodb E- K- O- B^kqlo '0734,0807( u pr bp`rbi^ `ob^olk i^ jlabok^ SbloŒ^ab `lkgrkqlp bk bi mboŒlal 0763,0784-

Page 34: Calculus

03 Fiomj_p^^d‡i

dil uu- G^ rkfcf`^al jr`e^p fab^p ^m^obkqbjbkqb fk`lkbu^p v e^ `lkqof_rfal ^obar`fo do^k k•jbol ab `lk`bmqlp j^qbjŠqf`lp ^ prp crka^jbkqlp iŽdf`lp mlo rkj‹qlal bibd^kqbv pfpqbjŠqf`l- Tk bpqrafl ofdrolpl ab i^ SbloŒ^ab `lkgrkqlp ob,nrbofoŒ rk^ ^jmif^ afp`rpfŽk nrb `lkpfabo^jlp crbo^ abi ^i`^k`b ab bpqbif_ol-Olo cloqrk^+ i^p kl`flkbp _Špf`^p plk bk k•jbol obar`fal+ v bp mlpf_ib abp^oolii^ork `lkl`fjfbkql moŠ`qf`l ab ilp j‹qlalp b fab^p ab i^ SbloŒ^ ab `lkgrkqlp ^qo^s‹p ab rk^ afp`rpfŽk fkcloj^i- Dk ob^ifa^a- kl s^jlp ^ e^`bo rk^ afp`rpfŽkab i^ jlabok^ SbloŒ^ab `lkgrkqlp+ pfkl mob`fp^oi^ qbojfklildŒ^ nrb ab_bobjlp^mif`^o ^ i^p fab^p jŠp l jbklp c^jfif^obp-

Dk L^qbjŠqf`^p+ i^ m^i^_o^ ~`lkgrkql‚ pb bjmib^ m^o^ obmobpbkq^ork^ `l,ib``fŽk ab l_gbqlp `lkpfabo^a^ `ljl rk^ pli^ bkqfa^a- K^p `lib``flkbp abpfdk^a^p`lk klj_obp q^ibp `ljl ~ob_^•l‚+ ~qof_r‚+ ~jr`ebarj_ob‚+ ~bnrfml‚ v ~bib`,qlo^al‚ plk qla^p bgbjmilp ab `lkgrkql- Klp l_gbqlp nrb `lkpqfqrvbk i^ `lib``fŽkpb ii^j^k `g`h`iojn l hd`h]mjn abi `lkgrkql+ v ab biilp pb af`b nrb k`mo`i`^`il nrb bpqŠk^jio`id_jn bk bi `lkgrkql- @ pr sbw+pb af`b nrb bi `lkgrkql ^jiod`i`l bpqŠ^jhkp`noj _` prp bibjbkqlp-

Mlp l`rm^objlp mofk`fm^ijbkqbab `lkgrkqlp ab bkqbpj^qbjŠqf`lp9 `lkgrkqlpab k•jbolp+ ab `ros^p+ ab cfdro^p dblj‹qof`^p+ bq`- Dk jr`e^p ^mif`^`flkbp `lk,sfbkb `lkpfabo^o `lkgrkqlp bk ilp nrb kl pb prmlkb k^a^ ^`bo`^ ab i^ k^qro^ibw^ab prp bibjbkqlp- S^ibp `lkgrkqlp pb ii^j^k ^_pqo^`qlp-K^ SbloŒ^ab `lkgrkqlp^_pqo^`qlpe^ pfal abp^oolii^a^ m^o^ qo^q^o lk q^ibp `lib``flkbp ab l_gbqlp ^o_f,qo^oflp+v mob`fp^jbkqb ^ bp^ dbkbo^ifa^a pb ab_b bi do^k ^i`^k`b ab q^i qbloŒ^-

) *&*&A\aNPV\[R`]N_N QR`VT[N_P\[Wb[a\`

Bloofbkqbjbkqb ilp `lkgrkqlp pb abpfdk^k `lk ibqo^pj^v•p`ri^p9 =) >) a+--- +t+ U) Y9 v ilp bibjbkqlp `lk jfk•p`ri^p9 \* \) `* ,,, *s* u+w-Tqfifw^jlp i^ klq^`fŽk

B9?

m^o^fkaf`^o nrb ~u bp rk bibjbkql ab O| l nrb ~u mboqbkb`b O|) Rf r kl mboqb,kb`b ^ P bplof_fjlp s n# P, Br^kal `lksbkd^+ abpfde^objlp `lkgrkqlp bp`of_fbkalilp bibjbkqlp bkqob `lo`ebqbp: mlo bgbjmil+ bi `lkgrkql ab ilp bkqbolp mlpfqfslpm^obpjbklobp nrb 0/ pb bumobp^ lk bi pŒj_lil x1+ 3+ 5+ 7y jfbkqo^p nrb bi aboj_jn ilp bkqbolp mlpfqfslp pb obmobpbkq^lk x0+ 1+ 2+ --- y: ilp qobpmrkqlp pfdkf,cf`^k ~v ^pŒpr`bpfs^jbkqb‚- Klp mrkqlp prpmbkpfslp q^k pŽil pb rqfifw^k `r^kalbi pfdkfcf`^al ab ~v ^pŒpr`bpfs^jbkqb‚ pb^ `i^ol- Di j‹qlal ab `fq^o ilp bibjbkqlpab rk `lkgrkql bkqob`lo`ebqbp pb ii^j^ cob`rbkqbjbkqb i^ ijo\^d‡i `i gdno\,

Di mofjbo `lk`bmql crka^jbkq^i nrb obi^`flk^ rk `lkgrkql `lk lqol bp i^dbp\g_\_ ab `lkgrkqlp9

CDEHMHBHˆM CD HFT@KC@C CD BNMITMSNR- P` _d^` lp` _jn ^jiepiojn > v ?nji dbp\g`n &j d_„iod^jn' nd^jino\i `s\^o\h`io` _` gjn hdnhjn `g`h`iojn* `i ^ptj

Page 35: Calculus

Pp]^jiepiojn 04

^\nj `n^md]dm`hjn > <?, Pd pij _` gjn ^jiepiojn ^jiod`i` \gbˆi `g`h`ioj lp` ij`noƒ `i `g jomj* _`^dhjn lp` gjn ^jiepiojn nji _dnodiojn u `n^md]dhjn >"?,

DIDLOKN 0- Cb ^`rboal `lk bpq^ abcfkf`fŽk+ ilp alp `lkgrkqlp x1+ 3+ 5+ 7yX x1+ 7+ 3+ 5y plk fdr^ibp+ v^ nrb ^j_lp `lkpq^k ab ilp `r^qol bibjbkqlp 1+ 3+ 5+X 7- Cb bpqb jlal+ `r^kal rp^jlp i^ klq^`fŽk bk ifpq^ m^o^ bumobp^o rk `lkgrkql+bi loabk bk nrb ^m^ob`bk ilp bibjbkqlp bp fkafcbobkqb-

DIDLOKN 1- Klp `lkgrkqlp x1+ 3+ 5+ 7y X x1+ 1+ 3+ 3+ 5+ 7y plk fdr^ibp ^mbp^o ab nrb bk bi pbdrkal `lkgrkql ilp bibjbkqlp 1 v 3 bpqŠk `fq^alp alp sb`bp-@j_lp `lkgrkqlp `lkqfbkbk ilp `r^qol bibjbkqlp 1+ 3+ 5+ 7 X kl lqolp+ ^pŒnrbi^ abcfkf`fŽk bufdb nrb `lkpfabobjlp fdr^ibp bplp `lkgrkqlp- Dpqb bgbjmil mlkbab j^kfcfbpql nrb kl ab_bjlp bufdfo nrb ilp bibjbkqlp `fq^alp bk i^ klq^`fŽk bkifpq^ pb^k qlalp afpqfkqlp- @kŠild^jbkqb bi `lkgrkql ab ibqo^p bk i^ m^i^_o^Jdnndnndkkd bp fa‹kqf`l ^i `lkgrkql vJ* c)n* kw nrb `lkpq^ ab i^p `r^qol ibqo^pafpqfkq^p L+ c)n* v j+

) *&+ EbOP\[Wb[a\`

@ m^oqfo ab rk `lkgrkql a^al mlabjlp cloj^o krbslp `lkgrkqlp+ ii^j^alpnp]^jiepiojn ab ^nr‹i- Olo bgbjmil+ bi `lkgrkql ab ilp bkqbolp mlpfqfslp jbklobpnrb 0/ v afsfpf_ibp mlo 3 'nrb bp bi `lkgrkql x3+ 7y( bp rk pr_`lkgrkql ab ilpbkqbolp mlpfqfslp m^obpjbklobp nrb 0/- Dk dbkbo^i+ a^objlp i^ abcfkf`fŽk pfdrfbkqb9

CDEHMHBHˆM CD RTABNMITMSN- P` _d^` lp` pi ^jiepioj > `n pi np]^jiepioj_`g ^jiepioj ?* u `n^md]dhjn

>m99?*

^p\i_j oj_j `g`h`ioj _` > k`mo`i`^` o\h]d„i \ ?, A`^dhjn o\h]d„i lp` > `noƒ^jio`id_j `i ? j lp` ? ^jiod`i` \ >, Bg n…h]jgj o:: n` podgdu\ k\m\ m`km`n`io\mg\ m`g\^d‡i _` di^gpnd‡i _` ^jiepiojn,

K^ obi^`fŽk = o:: > kl bu`irvb i^ mlpf_fifa^a ab nrb > o:: =+ Dk ob^ifa^a+ ml,abjlp qbkbo i^p alp obi^`flkbp = o:: > v > o:: = mbol bpql pb mobpbkq^ q^k pŽil pf= u > qfbkbk ilp jfpjlp bibjbkqlp- Dk lqo^p m^i^_o^p+

> <? nd u n‡gj nd > o:: ? W ? o:: > ,

Dpqb qblobj^ bp `lkpb`rbk`f^ fkjbaf^q^ ab i^p abcfkf`flkbp ^kqboflobp ab fdr^ia^ab fk`irpfŽk- Rf > o:: Ambol > ), ?* ab`fjlp nrb > bp rk np]^jiepioj kmjkdj ab ?9fkaf`^jlp bpql bp`of_fbkal = b >+

Dk qla^p krbpqo^p ^mif`^`flkbp l`roofoŠ nrb qbkaobjlp cfg^al ab ^kqbj^kl rk`fboql `lkgrkql R+ v pŽil klp fkqbobp^oŠk pr_`lkgrkqlp ab ^nr‹i- Di `lkgrkql crk,

Page 36: Calculus

05 Fiomj_p^^d‡i

a^jbkq^i R mrbab s^of^o ab rk^ ^mif`^`fŽk ^ lqo^: v pboŠ `lkpfabo^al `ljl bi^jiepioj pidq`mn\g ab `^a^ qbloŒ m^oqf`ri^o-K^ klq^`fŽk

ur Gr C R X r p^qfpc^`bMw

abpfdk^oŠbi `lkgrkql ab qlalp ilp bibjbkqlp s ab R nrb p^qfpc^`bki^ molmfba^a O-Br^kal bi `lkgrkql rkfsbop^i ^i nrb klp obcfo^jlp pb pl_obkqfbkab+ ljfqfobjlpbi `fq^oil ^_obsf^kal i^ klq^`fŽk mlkfbkal ur Gr p^qfpc^`bLv+ Dpql pb ibb ~bi `lk,grkql ab qlalp ilp r nrb p^qfpc^`bkL|+ Klp `lkgrkqlp obmobpbkq^alpab bpqbjlalnrba^k `^o^`qbofw^alpmlo rk^ kmjkd`_\_ _`adid_jm\, Olo bgbjmil+ bi `lkgrkql abqlalp ilp k•jbolp ob^ibp mlpfqfslp mlaoŒ abpfdk^opbmlo vs Gs<Lw9 bi `lkgrkqlrkfsbop^i R bk bpqb`^pl pb pl_obkqfbkab nrb bp bi `lkgrkql ab qlalp ilp k•jbolpob^ibp- Cbi jfpjl jlal+ bi `lkgrkql ab qlalp ilp k•jbolp m^obp mlpfqfslpx1+ 3+ 5+ --- y mrbab abpfdk^opb`lk ur G r bkqbol m^omlpfqfsly- M^qro^ijbkqb+ i^ibqo^s mrbab obbjmi^w^opbmlo lqol pfdkl ^ab`r^al- @pŒ+pb mrbab bp`of_fo

ur Er; Ny < us Gs = Ny < vo Go = Ny

bq`‹qbo^-Orbab l`roofo nrb rk `lkgrkql kl `lkqbkd^ bibjbkqlp- Tk q^i `lkgrkql pb

ii^j^ ^jiepioj q\^…j*v pb obmobpbkqjbaf^kqb bi pŒj_lil /- Blkpfabobjlp bi /`ljl pr_`lkgrkql ab `r^inrfbo `lkgrkql- G^v nrfbk fj^dfk^ rk `lkgrkql `ljlrk ob`fmfbkqb'q^i `ljl rk^ _lip^ l rk^ `^g^( nrb `lkqfbkb `fboqlp l_gbqlp+ prpbibjbkqlp- Dkqlk`bp+ bi `lkgrkql s^`Œl pboŒrk ob`fmfbkqbs^`Œl-

O^o^bsfq^oafcf`riq^abp v `lkcrpflkbp+ ab_bjlp afpqfkdrfo bkqobbi bibjbkql sv bi `lkgrkql urv `rvl •kf`l bibjbkql bp r+ 'Tk^ `^g^ `lk rk plj_obol abkqol+bp`lk`bmqr^ijbkqb afpqfkql abi plj_obol `lkpfabo^al plil-( Dk m^oqf`ri^o bi `lk,grkql s^`Œl / kl bp il jfpjl nrb bi `lkgrkql x / y- Dk ob^ifa^a bi `lkgrkql s^`Œll kl `lkqfbkb bibjbkqlp+ jfbkqo^p nrb bi `lkgrkql x/y `lkqfbkb rk bibjbk,ql+ / 'Tk^ _lip^ nrb `lkqfbkb rk^ _lip^ s^`Œ^kl bpqŠs^`Œ^-(Klp `lkgrkqlp nrb`lkqfbkbk rk plil bibjbkql pb ii^j^k ^jiepiojn _` pi `g`h`ioj,

Blk cob`rbk`f^ klp ^vra^jlp ab af^do^j^p m^o^e^`bo fkqrfqfs^pi^p obi^`flkbpbkqob`lkgrkqlp- Olo bgbjmil+ mlabjlp mbkp^onrb bi `lkgrkql rkfsbop^i R bp rk^obdfŽkbk bi mi^kl+v `^a^ rkl ab prp bibjbkqlp rk mrkq•l Klp pr_`lkgrkqlp ab Rmrbabk fj^dfk^opb `ljl `lib``flkbp ab mrkqlp fkqboflobp^ R- Olo bgbjmil+ bk i^cfdro^ 0-5'_( i^ mlo`fŽk plj_ob^a^ bp rk pr_`lkgrkql ab = v q^j_f‹k ab >+K^p ^vra^p doŠcf`^pab bpqbqfml pb ii^j^k _d\bm\h\n _` S`ii v pb rqfifw^k m^o^`ljmol_^o i^ s^ifabw ab `fboqlp qblobj^p ab i^ SbloŒ^ab `lkgrkqlp l m^o^prdbofoj‹qlalp ab abjlpqo^`fŽk ab ilp jfpjlp- M^qro^ijbkqb+ q^ibp abjlpqo^`flkbp pb_^p^k bk i^p abcfkf`flkbp v `lk`bmqlp v pr s^ifabw abmbkaboŠab rk o^wlk^jfbkql`loob`ql v kl mob`fp^jbkqb ab ilp af^do^j^p-

Page 37: Calculus

O`pidji`n* dio`mn`^^dji`n* ^jhkg`h`iojn

) *&, DRb[V\[R`$V[aR_`RPPV\[R`$P\Z]YRZR[a\`

06

@ m^oqfoab alp `lkgrkqlp a^alp = v >) pfbjmob mlabjlp cloj^o rk krbsl`lkgrkql ii^j^al m`pid‡i ab > v ?, Dpqbkrbsl `lkgrkql pb obmobpbkq^lk bipŒj_lil

B^( > r ?

= S > 'pb ibb y= obrkfŽk >|&

l‚ > h ? Bb( = h > < /

EHFTQ@ 0-5 O`pidji`n ` dio`mn`^^dji`n,

v pb abcfkb `ljl bi `lkgrkql ab ilp bibjbkqlp nrb mboqbkb`bk^ = l ^ > l ^^j_lp- Dp ab`fo+ = S > bp bi `lkgrkql ab qlalp ilp bibjbkqlp nrb mboqbkb`bkmlo il jbklp ^ rkl ab ilp `lkgrkqlp =) >+ Dk i^ cfdro^ H-5'^( i^ m^oqbplj_ob^a^obmobpbkq= S >+

@kŠild^jbkqb+ i^ dio`mn`^^d‡i ab > v ? nrb pb obmobpbkq^lk bi pŒj_lil

= i > 'pb ibb9 y= fkqbopb``fŽk>|&

pb abcfkb `ljl bi `lkgrkql ab ilp bibjbkqlp `ljrkbp ^ = v ^ >+ Dk i^ cfdro^ H-5'_(pb obmobpbkqi^ fkqbopb``fŽkab = v >+ Dk i^ cfdro^ H-5'`( pb sb nrb i^ fkqbopb`,`fŽk ab = u > bp bi `lkgrkql /+ mrbpql nrb = u > kl qfbkbk bibjbkqlp `ljrkbp-Clp `lkgrkqlp > u ? pb ii^j^k _dnepiojn pf > i ?; /-

C^alp alp `lkgrkqlp > v ?* pb abcfkb i^ _da`m`i^d\ > + ? 'nrb q^j_f‹k pbii^j^ ^jhkg`h`ioj _` ? m`g\odqj\ >' `ljl bi `lkgrkql ab ilp bibjbkqlp ab >nrb kl mboqbkb`bk^ >+ @pŒmrbp+pbd•k i^ abcfkf`fŽk

= * > < ur Gr D = X r nD>v +

Dk i^ cfdro^ H-5'_( i^ mlo`fŽk kl plj_ob^a^ ab > obmobpbkq> + ?9 i^ kl plj,_ob^a^ ab > obmobpbkq> * =+

K^p lmbo^`flkbp ab obrkfŽk b fkqbopb``fŽkmlpbbk jr`e^p ^k^ildŒ^pcloj^ibp`lk i^ ^af`fŽk v jriqfmif`^`fŽk loafk^of^p ab k•jbolp ob^ibp-Olo bgbjmil+ mrbpql

Page 38: Calculus

07 Fiomj_p^^d‡i

nrb kl bufpqb`rbpqfŽk ab loabk bk i^p abcfkf`flkbp ab obrkfŽk b fkqbopb``fŽk+pbabar`b nrb = S >:> S = X nrb = h >:> h=+ Dp ab`fo+ i^ obrkfŽk v i^ fk,qbopb``fŽkplk lmbo^`flkbp ^jihpo\odq\n, @pfjfpjl af`e^p abcfkf`flkbp bpqŠka^a^pab q^i jlal nrb i^p lmbo^`flkbp plk \nj^d\odq\n8

%=T >& T B < = T %>T B( v %=h >& h B < = h %>h B( -

Dpqlpv lqolp qblobj^p obi^qfslp ^i ~Šidb_o^ab `lkgrkqlp‚ pb `fq^k `ljl Dgbo`f`flpbk i^ Rb``fŽk 01-4- Tkl ab ilp jbglobp j‹qlalp m^o^nrb bi ib`qlo pb c^jfif^of`b`lk i^ qbojfklildŒ^ v i^p klq^`flkbp ^kqbp fkqolar`fa^p bp abar`fo i^p abjlpqo^,`flkbp ab `^a^ rk^ ab bpq^pibvbpcloj^ibp- Tk^ jrbpqo^ abi qfml ab o^wlk^jfbkqlnrb pb kb`bpfq^ ^m^ob`bfkjbaf^q^jbkqb abpmr‹p ab ilp Dgbo`f`flp-

K^p lmbo^`flkbp ab obrkfŽk b fkqbopb``fŽkmrbabk buqbkabopb^ `lib``flkbpcfkfq^pl fkcfkfq^pab `lkgrkqlp+ ab i^ j^kbo^ pfdrfbkqb9Rb^ +$ B rk^ `i^pb 'p( kls^`Œ^ ab `lkgrkqlp- K^ obrkfŽk ab qlalp ilp `lkgrkqlp ab +$ B pb abcfkb `ljl bi`lkgrkql ab qlalp ^nrbiilp bibjbkqlp nrb mboqbkb`bkmlo 0/ jbklp ^ rkl ab ilp`lkgrkqlp ab +$ B)v pb obmobpbkq^lk bi pŒj_lil

T@->`C

Rf bbbp rk^ `lib``fŽk cfkfq^ab `lkgrkqlp+ pb^ mlo bgbjmil bb<u=f; =0* ŠŠŠ *

=iw* bp`of_fjlp

i

S = < S =e < @i T =/ T --- T =i Š>`C hzi

@kŠild^jbkqb+ i^ fkqbopb``fŽkab qlalp ilp `lkgrkqlp ab +$ Bpb abcfkb `ljl bi`lkgrkql ab ^nrbiilp bibjbkqlp nrb mboqbkb`bk^ qlalp ilp `lkgrkqlp ab-& E: pbobmobpbkq^lk bi pŒj_lil

@i fdr^i nrb ^kqbp+m^o^ `lib``flkbp cfkfq^pab `lkgrkqlp bp`of_fjlp9

i

j = < j =f < @i h =0 h +++h=+7_ _ T

'p( O^o^ pfjmifcf`^o bi ibkdr^gb ii^j^jlp ^g\n` ^ rk^ `lib``fŽk ab `lkgrkqlp- O^o^ obmobpbkq^o`i^pbp bjmib^jlp ibqo^p j^v•p`ri^p `ropfs^p- K^ qbojfklildŒ^ v i^ klq^`fŽk rpr^ibp ab i^SbloŒ^ ab `lkgrkqlp pb ^mif`^+ k^qro^ijbkqb+ ^ i^p `i^pbp- @pŒ+mlo bgbjmil+ > C y pfdkfcf`^ nrb> bp rkl ab ilp `lkgrkqlp ab i^ `i^pbz+ v a R: W Fpfdkfcf`^ nrb qlal `lkgrkql ab a mboqbkb`b^ z+ v ^pŒpr`bpfs^jbkqb-

Page 39: Calculus

Be`m^d^djn 08

K^ obrkfŽk v i^ fkqbopb``fŽk pb e^k abcfkfal ab j^kbo^ nrb i^p ibvbp ^pl`f^qf,s^p pb p^qfpc^`bk fkjbaf^q^jbkqb- Dk `lkpb`rbk`f^ kl bufpqfoŠ^j_fd•ba^a `r^kalbp`of_fjlp =f q =0 S --- S =i l =f j =0 j --- j =i ,

) *&- :WR_PVPV\`

i- Tqfifw^o i^ klq^`fŽk bk ifpq^ m^o^ obmobpbkq^oilp pfdrfbkqbp `lkgrkqlp ab k•jbolp ob^ibp-

> < vs Gs/* 0 < Ny -

? < ur G%r * 0(1 < Ny -

b < vs Gs * 7 < 8y -

@ < ur Gr0 * /r/ * r < 1y -

A < ur G%r * 7(1 < 81y-

B < ur G%r/ * .3r&/ < 061y ‘

1- O^o^ ilp `lkgrkqlp abi Dgbo`f`fl 0+ l_p‹osbpb nrb ? R: >, Bfq^o qla^p i^p obi^`flkbp abfk`irpfŽk R: nrb plk sŠifa^p bkqob ilp `lkgrkqlp =) >) a+@) A) B+

2- Rb^k = < wHy+ > < w0+ 1y- Cfp`rqfo i^ s^ifabw ab i^p ^cfoj^`flkbp pfdrfbkqbp 'mol_^onrb rk^p plk `fboq^p v bumif`^o mlo nr‹ i^p lqo^p plk c^ip^p(-

']( > a ?,

'_( = R: >+'b( = C >+

'a( 0 A=+'b( GQ?-'b( 0 b >+

3- Qbplisbo bi Dgbo`f`fl 2 pf = < x0y X > < xx0y+ 0y-4- C^al bi `lkgrkql R < x0+ 1+ 2+ 3y- Dumobp^o qlalp ilp pr_`lkgrkqlp ab R- G^v bk qlq^i

05+ pf `lkq^jlp / v R-5- C^alp ilp `r^qol `lkgrkqlp pfdrfbkqbp

= < xH+1y+ ? < xxiy+x1yy+ b < xxiy+xi+1yy+ A < xxiy+x1y+xi+ 1yy+

afp`rqfo i^ s^ifabw ab i^p ^cfoj^`flkbp pfdrfbkqbp 'mol_^o nrb rk^p plk `Œboq^pu bumif`^omlo nr‹ i^p lqo^p kl il plk(-

']( = < >+'_( = b: >+

'b( = b B-

'`( > C A-

'b( = b @+'b( > b B-

'c( >]@+

'd( ?B A,

'f( = C @+

6- Cbjlpqo^o i^p molmfba^abp pfdrfbkqbp ab i^ fdr^ia^a ab `lkgrkqlp-

']( v\* \w < v\w,

'_( v\* \v < u\) \w,

'b( v\w < u\) by pf u pŽil pf \ < \ < `-

Cbjlpqo^o bi `lkgrkql ab obi^`flkbp ab ilp Dgbo`f`flp 7 ^i 08- '@i cfk^i ab bpq^Rb``fŽk pba^k bgbjmilp ab bpq^p abjlpqo^`ŒŽkbp(-

6, I`t`n ^jihpo\odq\n9 > r ? < ? r >* > i ? < ? i >,

Page 40: Calculus

*( Fiomj_p^^d‡i

7, I`t`n \nj^d\odq\n8 > S &? S B( < %= S ?' S B+ = l) &? l+ B( < %= mq?' k B-

/., I`t`n _dnomd]podq\n8= j &? S B( < %=j ?' S %=j @'*

= S %>k B( < %=S >& k %=S ?&+

00- = o= :=) = h= :=)

./+ = R = q >) = j > R =+

.0+ = S / < =) = k / < /-

.1+ = S %=k >& < =) = k %= S >& < =+

04- Rf > R B u ? R B+ bkqlk`bp > S ? R B-

05- Rf B R = X B R >) bkqlk`bp B R = k >+

06- '^( Oc= b > v >_ B+mol_^o nrb = b B-

'_( Pd> R ? X ? R B+mol_^o nrb > R B-

'b( ƒPr‹ mrbab ^cfoj^opb pf = b > v > RB>

'a( Rf s D = X = R: >) ƒbp `fboql kb`bp^of^jbkqb nrb s D ><

'b( Rf s C = X = C >) ƒbp `fboql kb`bp^of^jbkqb nrb s C ><

.5+ = * %>j B( < %= * A( q %= * ?&+

08- Rb^ z rk^ `i^pb ab `lkgrkqlp- Dkqlk`bp

> * S = < j %>* =&>B,%C >B,%C

v > * j = < S %>* =&+>B,%C >B,C

1/- '^( Cbjlpqo^o nrb rk^ ab i^p alp cŽojri^p pfdrfbkqbp bp pfbjmob `loob`q^ v i^ lqo^ ^idr,k^p sb`bp bp c^ip^9

'f( = * %> * B( < %= * >& S B+

'ff( = * %>q B( < %= * >& * B-

'_( Dpq^_ib`bo rk^ `lkaf`fŽk kb`bp^of^ v prcf`fbkqb ^af`flk^i m^o^ nrb i^ cŽojri^ nrbpb^ fk`loob`q^ pb^ pfbjmob sŠifa^-

A`hjnom\^d‡i _` g\ g`t ^jihpo\odq\ > S ?;? S >, Rb^k U;> S ?*X :> T =+ O^o^ abjlpqo^o nrb W<X pb abjrbpqo^ nrb W R: X b X R W- Rr,mŽkd^pbnrb s D W- Dkqlk`bp s bpqŠmlo il jbklp bk = l bk >+ Krbdl+ s bpqŠmloil jbklp bk ? l bk >9 ab jlal nrb s C X- @pŒ+mrbp+qlal bibjbkql ab W bpqŠq^j_f‹k bk X+ `lk il nrb W R: X- @kŠild^jbkqb+ bk`lkqo^jlp nrb X R W+abjlal nrb W<X-

A`hjnom\^d‡i _` > j ? p: >, Rf s D > j ?* s bpqŠpfjriqŠkb^jbkqb bk >u bk >+ Dk m^oqf`ri^o+s C =+ @pŒ+mrbp+qlal bibjbkql ab = j > bpqŠq^j_f‹kbk >9 mlo il q^kql+> j ? R: >,

Page 41: Calculus

Fiomj_p^^d‡i 10

N]npa ..E+* Sj _kjfqjpk ^_ ]teki]o l]n] _f oeopai]^_ jˆianko l_[f_m

H2-0 Hkqolar``fŽk

G^v jr`elp j‹qlalp m^o^ fkqolar`fo bi pfpqbj^ ab ilp k•jbolp ob^ibp- Tkj‹qlal `loofbkqb bp bi ab bjmbw^o`lk ilp bkqbolpmlpfqfslp 0+1+2+ --- v rqfifw^oilp`ljl _^pb m^o^ `lkpqorfo rk pfpqbj^ jŠp ^jmifl nrb qbkd^ i^p molmfba^abpabpb^a^p- Aobsbjbkqb+ i^ fab^ ab bpqbj‹qlal bp qlj^o ilp bkqbolp mlpfqfslp `ljl_^pb m^o^ cloj^o rk pfpqbj^ jŠp ^jmifl+ nrb bp bi ab ilp k•jbolp m\^dji\g`nmlpfqfslp '`l`fbkqbp ab bkqbolp mlpfqfslp(- Klp k•jbolp o^`flk^ibp mlpfqfslp pbrqfifw^k ^ pr sbw `ljl _^pb m^o^ `lkpqorfo ilp dmm\^dji\g`nmlpfqfslp 'k•jbolpob^ibp `ljl U1 v 4P nrb kl plk o^`flk^ibp(- Di m^pl cfk^i bp i^ fkqolar``fŽk abilp k•jbolp ob^ibpkbd^qfslp v bi `bol- K^ m^oqbjŠp afcŒ`fiabi mol`bpl qlq^i bp bim^pl ab ilp k•jbolp o^`flk^ibp ^ ilp k•jbolp foo^`flk^ibp-

@rknrb i^ kb`bpfa^a abi k•jbol foo^`flk^i pb e^_Œ^ mobpbkq^al v^ ^ ilpj^qbjŠqf`lp ab i^ ^kqfdr^ Fob`f^ bk prp bpqraflp dblj‹qof`lp+ kl pb fkqolargbolkj‹qlalp p^qfpc^`qloflpab `lkpqor``fŽk ab ilp k•jbolp ob^ibp^ m^oqfoab ilp o^`fl,k^ibp e^pq^ bkqo^al bi pfdil WHW-Dk bpq^‹ml`^ pb mbocfi^olkqobpqbloŒ^pafpqfkq^pmlo J^oi Vbfbopqo^pp'0704,0786(+ Fblod B^kqlo '0734,0807( v Qf`e^oa Cbab,hfka '0720,0805(- Dk 0778+ bi j^qbjŠqf`l fq^if^kl Ffrpbmmb Ob^k^ '0747,0821(afl `fk`l ^uflj^p m^o^ ilp bkqbolp mlpfqfslp nrb pb rqfifw^olk `ljl mrkql abm^oqfa^m^o^i^ `lkpqor``fŽk qlq^i- Tk^ bumlpf`fŽk abq^ii^a^ ab bpq^`lkpqor``fŽkbjmbw^kal mlo ilp ^uflj^p ab Ob^k^ v rqfifw^kal bi j‹qlal ab Cbabhfka m^o^fkqolar`fo bi k•jbol foo^`flk^i+ pb bk`rbkqo^ bk bi if_ol ab D- K^ka^r+ Cpi_\+h`iojn _`g >iƒgdndn'Mrbs^ Xloh+ Bebipb^ Or_ifpefkd Bl-+ 0840(-

Di mrkql ab sfpq^ ^almq^al ^nrŒkl bp `lkpqor`qfsl- Rb fkf`f^ bi mol`bpl bkrk mrkql _^pq^kqb ^s^kw^al+ `lkpfabo^kal ilp k•jbolp ob^ibp `ljl `lk`bmqlpmofjfqfslp nrb p^qfpc^`bk^ rk `fboql k•jbol ab molmfba^abpnrb pb qlj^k `ljl^uflj^p: bp ab`fo+pb prmlkb nrb bufpqbk`fboqlp l_gbqlp+ii^j^alp k•jbolp ob^ibp+nrb p^qfpc^`bk ilp 0/ ^uflj^p bkrk`f^alp bk i^p `fk`l Rb``flkbp nrb pfdrbk-Sla^p i^p molmfba^abpab ilp k•jbolp ob^ibpnrb pb rqfifw^oŠkbk bpqbif_ol+ l bpqŠkbkqob ilp ^uflj^p l pb mrbabk abar`fo ab biilp- Br^kal ilp k•jbolp ob^ibp pbabcfkbk jbaf^kqb rk mol`bpl `lkpqor`qfsl+ i^p molmfba^abpnrb pb qlj^k `ljl^uflj^p qbkaoŠk nrb abjlpqo^opb `ljl qblobj^p-

Lfbkqo^p kl pb afd^ 0/ `lkqo^ofl+ i^p ibqo^p\* \) `* ,,, s* v+ w nrb ^m^ob`bkbk ilp ^uflj^p obmobpbkq^kk•jbolp ob^ibp `r^ibpnrfbo^- Klp ^uflj^p pb ^dorm^kbk cloj^ k^qro^i bk qobpdormlp+nrb plk+ \sdjh\n _` ^p`mkj* \sdjh\n _` jm_`iu \sdjh\ _`g `som`hj npk`mdjm'ii^j^al q^j_f‹k \sdjh\ _` ^jiodipd_\_ l \sdjh\_` ^jhkg`odop_',

Page 42: Calculus

++ / iomj_p^^d‡i

) +&* 6dV\ZN` QRPbR_]\

Irkql `lk bi `lkgrkql ab ilp k•jbolp ob^ibp pb prmlkb i^ bufpqbk`f^ ab alplmbo^`flkbp ii^j^a^p \_d^d‡i u hpgodkgd^\^d‡i*q^ibp nrb m^o^ `^a^ m^o ab k•jbolpob^ibp s b v pb mrbab cloj^o i^ nph\ ab s b v+ nrb bp lqol k•jbol ob^i abpfdk^almlo s)t v bi kmj_p^oj ab s mlo v abpfdk^al mlo st l T$ v- K^ prj^ s)t v bimolar`ql st bpqŠk rkŒsl`^jbkqb abqbojfk^alp mlo s b v- @ ilp pfdklp * v - klpb ibp ^pfdk^ lqol pfdkfcf`^al bpmb`f^i nrb bi mob`fp^al bk ilp ^uflj^p-

@WHNL@ 0- OQNOHDC@C BNMLTS@SHU@- s)t;t)s* st;ts,

@WHNL@ 1- OQNOHDC@C @RNBH@SHU@- s)&t)u';&s)t')u* s&tu'< &st'u,

@WHNL@ 2- OQNOHDC@C CHRSQHATSHU@- s&t)u';st)su,

@WHNL@ 3- DWHRSDMBH@CD DKDLDMSNR MDTSQNR- Bsdno`i _jn iˆh`mjn m`\+g`n _dnodiojn*lp` n` di_d^\i kjm N v 0 o\g`n lp` k\m\ ^\_\ iˆh`mj m`\gs n` od`i`8L)s;s)L;s v G-9U;U% 0;s,

@WHNL@ 4- DWHRSDMBH@CD MDF@SHUNR- M\m\ ^\_\ iˆh`mj m`\g s `sdno` piiˆh`mj m`\g v o\g lp` s)t;t)s;L,

@WHNL@ 5- DWHRSDMBH@CDK QDB†OQNBN- M\m\ ^\_\ iˆh`mj m`\gs ;/; N `sdno`pi iˆh`mj m`\gv o\g lp` st <ts < 0-

Kjo\8 Klp k•jbolp N v 0 ab ilp ^uflj^p 4 v 5 plk ilp jfpjlp nrb ilp abi ^uflj^ 3-

Cb ilp ^uflj^p ^kqboflobp pb mrbab abar`fo qla^p i^p ibvbp rpr^ibp abi „idb_o^bibjbkq^i- K^p jŠp fjmloq^kqbp ab bii^p pb ob`ldbk ^ `lkqfkr^`fŽk `ljl qblobj^p-Dk qlalp bpqlp qblobj^p i^p ibqo^p \* \) `* ^) obmobpbkq^k k•jbolp ob^ibp `r^ibp,nrfbo^-

SDNQDL@ 0-0- KDX CD RHLOKHEHB@BHˆM O@Q@ K@ RTL@- Rf \)];\)^*`ioji^`n ] < ^, &Bi k\mod^pg\mnoj kmp`]\ lp` `g iˆh`mj M _`g \sdjh\ 3 `n ˆid^j,'

SDNQDL@ 0-1- ONRHAHKHC@CCD K@ RTRSQ@BBHˆM- A\_jn \ v ] `sdno` pij vn‡gj pi s o\g lp` \ * s <], Bno` s n` _`ndbi\ kjm ] + \, Bi k\mod^pg\mN , \ n` `n+^md]`ndhkg`h`io` +\ v n` _`ijhdi\ `g i`b\odqj _` \,

SDNQDL@ 0-2- ] + \ < ] * &+\',

SDNQDL@ i@- +&+\' < \,

SDNQDL@ 0-4- \&] + b( < \] + \^,

SDNQDL@ 0-5- N• \ < \% /< N-

Page 43: Calculus

>sdjh\n _` ^p`mkj 12

SDNQDL@ 0-6- KDX CD RHLOKHEHB@BHˆM O@Q@ K@ LTKSHOKHB@BHˆM- Rf\] < \^ u \ " N+ `ioji^`n ] < `- &Bi k\mod^pg\mnoj _`hp`nom\ lp` `g iˆh`mj 0 _`g\sdjh\ 3 `n ˆid^j,'

SDNQDL@ 0-7- ONRHAHKHC@CCD K@ CHUHRHˆM- A\_jn \ u ] ^ji \ ;/; N+`sdno`

pij v n‡gj pi s o\g lp` \s < ], I\ s n` _`ndbi\ kjm ] - \ l z v n` _`ijhdi\ ^j^d`io`[

_` ] v \, Bi k\mod^pg\m0.\ n` `n^md]`o\h]d„i \8% v n` _`ndbi\ m`^…kmj^j_` \,

SDNQDL@ 0-8- Rf \ x N+ ioji^`n ]Z\ < ] , \+g,

SDNQDL@ 0-0/- Rf \ x N+ ioji^`n &\+.&*. < \,

SDNQDL@ 0-00- Rf \];L `ioji^`n l \;L l ];L,

SDNQDL@ H-01- &+\'] < +&\]' v &+\'&+]' < \],

RCMPCK? H-02- &\-]' * &^-_'< &\_* ]^'-&]_' nd ]9„ N X _9„ N-

SDNQDL@ H-03- &\-]'&^-_'< &\^'-&]_' nd ]9„ N X _ x N-

SDNQDL@ H-04- &\e]'e&^-_'< &\_'e&]^' pf ] x N+ ` x N+ X _ x N-

O^o^ mlkbo ab j^kfcfbpql `Žjl bpqlp qblobj^p mrbabk l_qbkbopb `ljl `lk,pb`rbk`f^ ab ilp ^uflj^p+ pb a^k i^p abjlpqo^`flkbp ab 0-0 e^pq^ 0-3+ X pboŒ fkp,qor`qfsl m^o^ bi ib`qlo qo^q^o ab abjlpqo^o ilp obpq^kqbp-

A`hjnom\^d‡i _` 0-0- C^al \)];\)^, Dk sfoqra abi ^uflj^ 4+ pb mrbabbibdfo t ab j^kbo^ nrb t)\;L* `lk il `r^i t)&\)]';t)&\)^'* v ^mif`^kali^ molmfba^a ^pl`f^qfs^ &t)\')];&t)\')^* l pb^+L)];L)^, Obol bk sfoqraabi ^uflj^ 3+ pb qfbkb L)];] v L)^;^* l pb^+ ];^, N_p‹osbpb nrb bpqb qblob,j^ abjrbpqo^ nrb bufpqb rk plil k•jbol ob^i nrb qfbkb i^ molmfba^a abi M bk bi^uflj^ 3- Dk bcb`ql+ pf N v N& qrsfbo^k ^j_lp bpq^ molmfba^a+ bkqlk`bp+/*/&</ v /*/</: mlo q^kql+ /*/&</*/ v mlo i^ ibv ab pfjmifcf`^`fŽk /</&-

A`hjnom\^d‡i _` 0-1- C^alp \ v ] pb bifdb t ab j^kbo^ nrb \)t;Lv pb^ s;t)], Dkqlk`bp+ \)s;\)&t)]';&\)t')];L)];], Olo q^kql+ e^vmlo il jbklp rk^ s q^i nrb \ * s < \+ Obol bk sfoqra abi qblobj^ 0-0+ e^v ^ ilprjl rk^- Krbdl e^v pi\ u n‡gj pi\ s bk bpq^p `lkaf`flkbp-

A`hjnom\^d‡i _` 0-2- Rb^ s;]+\ v pb^ t;])& +\', Rb qo^q^ ab mol_^onrb s;t, Olo abcfkf`fŽk ab ]+\* s)\;] v

t * \ < X]* &+\'Z * \ < ] * X&+\' * \Z < ] * N < ],

Page 44: Calculus

13 Fiomj_p^^d‡i

Olo q^kql+s)\;t)\* X bk sfoqra ab 0-0+s;t,

A`hjnom\^d‡i _` 0-3- Rb qfbkb \)& +\';L mlo abcfkf`fŽk ab +\, Obol bpq^fdr^ia^a af`b nrb \ bp bi lmrbpql ab &+\'* bp ab`fo+ nrb \; +&+\' `ljl pb^cfoj^ bk bi qblobj^-

g) +&+ :WR_PVPV\`

0- Cbjlpqo^o ilp qblobj^p abi 0-4 ^i 0-04+rqfifw^kal ilp ^uflj^p 0 ^i 5 v ilp qblobj^p0-0 ^i 0-3-

Dk ilp bgbo`f`flp abi 1 ^i 0/+ abjlpqo^o i^p ^cfoj^`flkbp fkaf`^a^p+ l bpq^_ib`bo i^p fdr^i,a^abp a^a^p- @miŒnrbkpbilp ^uflj^p 0 ^i 5 v ilp qblobj^p abi 0-0 ^i 0-04-

1- ,N < N-2- 0,0 < 0-3- Di `bol kl qfbkb ob`Œmol`l-4- , %[ * \& < , [*\+5- , %[ * ]' < , [ * ]*4+ %[ * \& * %\ * _&< [*_+7- Rf [ :‹ N X \ :‹ N+bkqlk`bp %[\&*f < ^,0 n~,8- , %[,\& < ' , [c\& < [,% * \& pf \ :‹ N-

.-+ %[,\& * %_,^& < %[^ * \_&,%\^& pf_ :‹ N X ^8€ N-

) +&, 6dV\ZN` QR\_QR[

Dpqbdorml ab ^uflj^p pb obcfbob rk `lk`bmql mlo bi nrb pb bpq^_ib`b rk^jm_`i\^d‡i bkqobilp k•jbolp ob^ibp-Rbd•k bpq^loabk^`fŽk pb mrbab ab`fafo pf WQk•jbol ob^i bp j^vlo l jbklo nrb lqol- Rb fkqolar`bk ^nrŒi^p molmfba^abpabloabk+ `ljl rk `lkgrkql ab ^uflj^p obcbobkqbp i krbsl `lk`bmql mofjfqfsl abkjndodqj* m^o^ abcfkfo abpmr‹p ilp `lk`bmqlp ab h\tjm lp` v h`ijm lp` ^ m^oqfoabi ab kjndodqj,

Rrmlkaobjlp nrb bufpqbrk `fboql pr_`lkgrkql Q* b Q+ii^j^al `lkgrkql abk•jbolp kjndodqjn*nrb p^qfpc^`bkilp qobp uflj^p ab loabk pfdrfbkqbp9

@WHNL@ 6- Rf s ` u k`mo`i`^`i \ Q*+ gj hdnhj j^pmm` \ s)t u st,

@WHNL@ 7- M\m\ oj_j m`\g s " N+j U D Q* j +s D Q*+ k`mj ij \h]jn,

@WHNL@ 8- M ma8P*-

@elo^ pb mrbabk abcfkfoilp pŒj_lilp ;+ =+y+v z ii^j^alp obpmb`qfs^jbkqbh`ijm lp`* h\tjm lp`* dbp\g j h`ijm lp`* b dbp\g j h\tjm lp`* ab i^ j^kbo^pfdrfbkqb9

s:t pfdkfcf`^ nrb t+s bp mlpfqfsl-

Page 45: Calculus

>sdjh\n _` jm_`i +.

t<s pfdkfcf`^ nrb s:t,

s83t pfdkfcf`^ nrb l s:t l s;t,

t/s pfdkfcf`^ nrb s83t,

Olo 0/ q^kql+ pb qfbkb s = N pf v pŽil pf s bp ONRHqHUN-Rf s ; N pb af`bnrb s bp i`b\odqj9 pf s 1 M pb af`b nrb s bp ij i`b\odqj, Di m^o ab abpfdr^ia^,abp pfjriqŠkb^p s ; t* t ; w pb bp`of_bk cob`rbkqbjbkqb bk i^ cloj^ jŠp _obsb s :: t ; w: fkqbomobq^`flkbp ^kŠild^p pb a^k ^ i^p abpfdr^ia^abp `ljmrbpq^ps 94 t ; u* s ; t 94 w+U 94 V 94 u*

Cb ilp ^uflj^p ab loabk pb mrbabk abar`fo qla^p i^p obdi^p rpr^ibp ab `Ši`ril`lk& abpfdr^ia^abp+ i^p jŠp fjmloq^kqbp ab i^p `r^ibp pb a^k ^ `lkqfkr^`fŽk `ljlqblobj^p-

SDNQDL@ 0-05- OQNOHDC@C CD SQHBNSNL†@- M\m\ \ t ] iˆh`mjn m`\g`n^p\g`nlpd`m\ n` q`mdad^\ pi\ u n‡gj pi\ _` g\n om`nm`g\^dji`n \ ; ]* ] ; \* \ < ]*

SDNQDL@ 0-06- OQNOHDC@C SQ@MRHSHU@- Pd \ ; ] t ] ; b+`n \ ; `-

SDNQDL@ 0-07- Rf \ ; ] `n \ * b ; ] * `-

SDNQDL@ 0-08- Rf \ ; ] t b = N `n \^ ; ]`,

SDNQDL@ 0-1/- Rf \ ;/; N `n \0 = N-

SDNQDL@ 0-10- 0 = N-

SDNQDL@ 0-11- Pd \ ; ] t ` ; L*`n \` = ]`,

SDNQDL@ 0-12- Pd \ ; ]* `n +\ = , ]* Bi k\mod^pg\mnd\ ; L*`n + \ = N-

SDNQDL@ 0-13- Pd \] = N `ioji^`n \ t ] nji j \h]jn kjndodqjn j \h]jni`b\odqjn,

SDNQDL@ 0-14- Pd \ ; ` t ] ; _*`ioji``n \ * ] ; ` * _,

S^j_f‹k ^nrŒ pb abjlpqo^oŠk pŽil ^idrklp ab bpqlp qblobj^p+ `ljl bgbjmilab `Žjl pb mol`bab bk i^ abjlpqo^`fŽk- Klp abjŠp pb abg^k `ljl bgbo`f`fl ^iib`qlo-

A`hjnom\^d‡i _` 0-05- Rb^ s < ] + \, Rf s < N+ bkqlk`bp ] + \ < \+] < M v+ mlo q^kql+ bk sfoqra abi ^uflj^ 8 kl mrbab pbo kf \ = ] kf ] = \,

Page 46: Calculus

15 Fiomj_p^^d‡i

Rf s ;/; N+bi ^uflj^ 7 ^cfoj^ nrb l s = N l s ; N+ mbol kl ^j_lp: mlo `lkpf,drfbkqb+ l bp \ ; ] l bp ] ; \* mbol kl ^j_lp- Olo q^kql pb sbofcf`^ rk^ v pŽilrk^ ab i^p qobp obi^`flkbp \ < ]* \ ; ]* ] ; \,

A`hjnom\^d‡i _` 0-06- Rf \ ; ]t ] ; ^* bkqlk`bp ] + \ = N v ` + ] = N-Dk sfoqra abi ^uflj^ 6 pb mrbab prj^o l_qbkf‹kalpb %\ * [& * %] * \& = l-Dp ab`fo+ ` + \ = N+v mlo q^kql+\ ; ^,

A`hjnom\^d‡i _` 0-07- Rb^ s < \ * `* v < ] * ^, Dkqlk`bp v , s < ] + \,Obol ] + \< N+mlo q^kql+ \ ; ], Cb alkab v , s = N+il nrb pfdkfcf`^ s ; v-

A`hjnom\^d‡i _` 0-08- Rf [ ; ] bkqlk`bp ] + \ = N- Rf ^< N bk sfoqraabi ^uflj^ 6+ pb mrbab jriqfmif`^o _ mlo %\ * [& l_qbkf‹kalpb %\ * [& _ = l-Obol &] + \'^ < ]` + \^* mlo q^kql+ ]` + \^ ; N v bpql pfdkfcf`^ ]` = \^ `ljlpb nrboŒ^ abjlpqo^o-

A`hjnom\^d‡i _` 0-1/- Rf \< N+ bk sfoqra abi ^uflj^ 6 \+ \ = l- Rf\ ; N+bkqlk`bp , \ = N v+ mlo q^kql+ ', \'} &+ \' = N bk sfoqra abi ^uflj^ 6-Dk ^j_lp `^plp pb qfbkb \0 = N-

A`hjnom\^d‡i _` 0-10- @mif`^kal bi qblobj^ 0-1/ ^i `^pl \ < 0-

") 2-4 Dgbo`f`flp

0- Cbjlpqo^o ilp qblobj^p 0-11 ^i 0-14 rqfifw^kal ilp qblobj^p ^kqboflobp v ilp ^uflj^pabi 0 ^i 8-

Dk ilp bgbo`f`flp abi 1 ^i 0/ abjlpqo^o i^p molmlpf`flkbp v bpq^_ib`bo i^p abpfdr^ia^abpa^a^p- Rb mrbabk rqfifw^o ilp ^uflj^p abi 0 ^i 8 v ilp qblobj^p abi 0- 0 ^i 0-14-

1- Ml bufpqb kfkd•k k•jbol ob^i q^i nrb s0 * 0 < N-2- K^ prj^ ab alp k•jbolp kbd^qfslp bp rk k•jbol kbd^qfsl-3- Rf \ = N: q^j_f‹k g-\< N: pf \ ; N+bkqlk`bp g-\ ; N-4- Rf N ; \ ; ]* bkqlk`bp+ N ; ]+f ; ^,i-5- Rf \ 9# ] X ] ,o `* bp \ 9# ^,6- Rf \ 9# ] X ] n8 ^ v \ < ^* bkqlk`bp ] < ^,7- O^o^ k•jbolp ob^ibp \ u ] `r^ibpnrfbo^+ pb qfbkb \0 * ]0 x N- Rf \] x /+ bkqlk`bp

bp \0 * ]0 = N-8- Ml bufpqb kfkd•k k•jbol ob^i \ q^i nrb s 9# \ m^o^ qlal ob^i s,

0/- Rf s qfbkb i^ molmfba^a nrb N 9# s ; c m^o^ ^\_\ k•jbol ob^i mlpfqfsl c* bkqlk`bp s < N-

0 2-5 M•jbolp bkqbolpv o^`flk^ibp

G^v `fboqlp pr_`lkgrkqlp ab Q nrb pb afpqfkdrbk mlonrb qfbkbk molmfba^abpbpmb`f^ibp ab nrb kl dlw^k qlalp ilp k•jbolp ob^ibp- Dk bpq^ Rb``fŽk pb afp`rqfoŠkalp ab bpqlp pr_`lkgrkqlp+ ilp iˆh`mjn `io`mjn v ilp iˆh`mjn m\^dji\g`n,

Page 47: Calculus

K ˆh`mjn `io`mjn t m\^dji\g`n +0

O^o^ fkqolar`fo ilp bkqbolp ONRHqfUNRpb bjmfbw^ `lk bi k•jbol 0+ `rv^bufpqbk`f^ nrba^ ^pbdro^a^ mlo bi ^uflj^ 3- Di k•jbol 0 * 0 pb obmobpbkqmlo1+ bi 1 * 0 mlo 2+ v ^pŒpr`bpfs^jbkqb- Klp k•jbolp 0+1+ 2+ --- +l_qbkfalp abbpqbjlal mlo i^ ^af`fŽk obmbqfa abi 0 plk qlalp mlpfqfslp+v pb ii^j^k `io`mjnkjndodqjn, Dk ofdlo+bpq^abp`ofm`fŽkab ilp bkqbolp mlpfqfslp kl bp abi qlal mob`fp^mrbp kl ebjlp bumif`^al `lk abq^iib il nrb bkqbkabjlp mlo ~v ^pŒpr`bpfs^jbkqb‚l mlo ~^af`fŽk obmbqfa^abi 0•- Rf _fbk i^ pfdkfcf`^`fŽk fkqrfqfs^ mrbab m^ob`bo`i^o^+bk rk bpqrafl `rfa^alpl abi pfpqbj^ ab ilp k•jbolp ob^ibpbp kb`bp^ofl a^ork^ abcfkf`fŽk jŠp mob`fp^ab ilp bkqbolp mlpfqfslp- G^v s^oflp jlalp ab e^`boil-Tk j‹qlal `lkpfpqb bk fkqolar`fo mofjbol i^ kl`fŽk ab ^jiepioj di_p^odqj,

CDEHMHBHˆM CD BNMITMSN HMCTBSHUN- Ri ^jiepioj _` iˆh`mjn m`\g`n n` _`+ijhdi\ ^jiepioj di_p^odqj nd od`i` g\n kmjkd`_\_`n ndbpd`io`n8

]( Bg iˆh`mj 0 k`mo`i`^` \g ^jiepioj,_( M\m\ oj_j s `i `g ^jiepioj* `g iˆh`mj s * 0 k`mo`i`^` o\h]d„i \g

^jiepioj,

Olo bgbjmil+ Q bp rk `lkgrkql fkar`qfsl- S^j_f‹k il bp bi `lkgrkql Q*- Cbcfkfob,jlp ilp bkqbolp mlpfqfslp `ljl ^nrbiilp k•jbolp ob^ibp nrb mboqbkb`bk^ qlal`lkgrkql fkar`qfsl-

CDEHMHBHˆM CD DMSDQNR ONRHSHUNR- Ri iˆh`mj m`\g n` gg\h\ `io`mj kjndodqjnd k`mo`i`^` \ oj_j ^jiepioj di_p^odqj,

Rb^ O bi `lkgrkql ab qlalp ilp bkqbolp mlpfqfslp- Dp rk `lkgrkql fkar`qfslv^ nrb ^( `lkqfbkb bi 0+v _( `lkqfbkb ^ r * 0 pfbjmob nrb `lkqbkd^ r+ Orbpql nrbilp bibjbkqlp ab O mboqbkb`bk^ qlal `lkgrkql fkar`qfsl+ klp obcbofobjlp ^ O`ljl bi h`ijm `lkgrkql fkar`qfsl- Dpq^molmfba^a abi `lkgrkql O `lkpqfqrvb i^_^pb iŽdf`^ m^o^rk qfml ab o^wlk^jfbkql nrb ilp j^qbjŠqf`lp abkljfk^k _`hjn+om\^d‡i kjm di_p^^d‡i* nrb pb bumlkb `lk abq^iib bk i^ m^oqb3 ab bpq^Hkqolar``fŽk-

Klp lmrbpqlp ab ilp bkqbolp mlpfqfslp pb ii^j^k `io`mjn i`b\odqjn, Klp bkqbolpmlpfqfslp grkql `lk ilp bkqbolp kbd^qfslp v bi N '`bol(+ `lkpqfqrvbk rk `lkgrkql Ynrb pb ii^j^ pfjmibjbkqb ^jiepioj _` gjn `io`mjn,

Dk rk bpqrafl `ljmibql abi pfpqbj^ ab ilp k•jbolp ob^ibp+pboŒkb`bp^ofl ^iiibd^o ^nrŒabjlpqo^o `fboqlp qblobj^p ^`bo`^ ab ilp bkqbolp-Olo bgbjmil+ i^ prj^+i^ afcbobk`f^ l bi molar`ql ab alp bkqbolp bp rk bkqbol+mbol bi `l`fbkqb ab alpbkqbolp kl bp kb`bp^of^jbkqb bkqbol- Rfk bj_^odl+ kl bkqo^objlp bk ilp abq^iibpab q^ibp abjlpqo^`flkbp-

Klp `l`fbkqbp ab bkqbolp \-] 'pfbkal ] ;/; N( pb ii^j^k iˆh`mjn m\^dji\g`n,Di `lkgrkql ab ilp k•jbolp o^`flk^ibp+ obmobpbkq^almlo A$ `lkqfbkb ^ Y `ljlpr_`lkgrkql- Di ib`qlo ab_boŒ^ ljmol_^o nrb O p^qfpc^`bqlalp ilp ^uflj^p ab`rboml v ab loabk- Olo bpq^o^wŽkpb af`b nrb bi `lkgrkql ab ilp k•jbolp o^`fl,

Page 48: Calculus

+1 Fiomj_p^^d‡i

k^ibp bp rk ^p`mkj jm_`i\_j, Klp k•jbolp ob^ibpnrb kl mboqbkb`bk P pb ii^j^kdmm\^dji\g`n,

H 2-6 Hkqbomobq^`fŽkdblj‹qof`^ ab ilp k•jbolp ob^ibp ljl mrkqlp ab rk^ ob`q^

Rfk ara^ nrb bi ib`qlo ab_b bpq^oc^jfif^ofw^al `lk i^ obmobpbkq^`fŽkab ilpk•jbolp ob^ibp mlo jbafl ab ilp mrkqlp ab rk^ ob`q^- Rb bifdb rk mrkql m^o^obmobpbkq^obi N v lqol ^ i^ abob`e^ abi N m^o^obmobpbkq^obi 0+`ljl pb fkaf`^bk i^ cfdro^ 0-6- Dpq^bib``fŽk abqbojfk^ i^ bp`^i^- Rf pb ^almq^ rk `lkgrkql ab^uflj^p ^molmf^alp m^o^ i^ FbljbqoŒ^ br`iŒab^+`^a^ k•jbol ob^i `loobpmlkab^ rkl v pŽil rk mrkql ab i^ ob`q^ v+ob`Œmol`^jbkqb+^a^ mrkql ab i^ ob`q^ ^ rkk•jbol ob^i v pŽil rkl- Olo bpq^o^wŽki^ ob`q^ pb abkljfk^ cob`rbkqbjbkqb m`^o\m`\g l `e` m`\g* v bp `lpqrj_ob rqfifw^oi^p m^i^_o^p iˆh`mj m`\g v kpioj `ljlpfkŽkfjlp- Olo bpl pb af`b jr`e^p sb`bp bi kpioj s bk sbw abi mrkql `loobp,mlkafbkqb ^i k•jbol ob^i s,

K^ obi^`fŽk ab loabk bkqobilp k•jbolp ob^ibpqfbkbrk^ fkqbomobq^`fŽkdblj‹,qof`^pfjmib- Rf s ; v+ bi mrkql s bpqŠ i^ fwnrfboa^ abi mrkql v+ `ljl pb sb bk i^cfdro^ 0-6- Klp k•jbolp mlpfqfslp bpqŠk ^ i^ abob`e^ abi N v ilp kbd^qfslp ^ i^fwnrfboa^ abi N- Rf \ ; \) rk mrkql s p^qfpc^`bi^p abpfdr^ia^abp \ ; s ; ]*pf v pŽil pf r bpqŠbkqob [ u ]*

Dpq^ mlpf_fifa^a ab obmobpbkq^odblj‹qof`^jbkqb ilp k•jbolp ob^ibp bp rk^rufif^o mlabolpl+ mrbp mbojfqb abp`r_ofo v `ljmobkabo jbglo `fboq^pmolmfba^abpab ilp k•jbolp ob^ibp-@rknrb bi ib`qlo ab_b l_pbos^o nrb qla^p i^p molmfba^abpab ilp &k•jbolp ob^ibp nrb pb e^k a^al `ljl qblobj^p ab_bk abar`fopb ab ilp^uflj^p pfk kfkdrk^ obcbobk`f^dblj‹qof`^+ bpql kl mobgrwd nrb kl ab_^ e^`bopbrpl ab i^ FbljbqoŒ^ bk bi bpqrafl ab i^p molmfba^abpab ilp k•jbolp ob^ibp-Olo bi`lkqo^ofl+ i^ FbljbqoŒ^ prdfbob ^ jbkral bi j‹qlal ab abjlpqo^`fŽk m^o^ rkqblobj^ m^oqf`ri^o+u ^idrk^p sb`bp rk ^odrjbkql dblj‹qof`l bp jŠp prdbpqfslnrb i^ abjlpqo^`fŽk mro^jbkqb \i\g…od^\ 'abmbkafbkqb bu`irpfs^jbkqb ab ilp ^ufl,j^p abi k•jbol ob^i(- Dk bpqbif_ol+ pb rqfifw^`lk cob`rbk`f^ i^ fkqrf`fŽk dblj‹,

G G G GN H s t

EHFTQ@ 0-6 Kˆh`mjn m`\g`n m`km`n`io\_jn b`jh„omd^\h`io` `i pi\ g…i`\

qof`^ m^o^^`i^o^o abqbojfk^a^p `rbpqflkbp l m^o^fkar`fo ^ afp`rpflkbp ab lqo^p-Ml l_pq^kqb+i^p abjlpqo^`flkbp ab qlalp ilp qblobj^p fjmloq^kqbp pb mobpbkq^kbkcloj^ ^k^iŒqf`^-

H 2-7 Blq^ prmbofloab rk `lkgrkql- bibjbkql jŠufjl+ buqobjl prmboflo

Klp krbsb ^uflj^p bumrbpqlp e^pq^ ^elo^ `lkqfbkbk qla^p i^p molmfba^abpab ilp k•jbolp ob^ibpbpqraf^alp loafk^of^jbkqb bk „idb_o^ bibjbkq^i- G^v lqol

Page 49: Calculus

@jo\ npk`mdjm_` pi ^jiepioj 18

^uflj^ ab fjmloq^k`f^ crka^jbkq^i bk bi BŠi`ril nrb ab loafk^ofl kl pb bpqraf^bk ilp `roplp ab @idb_o^bibjbkq^i- Dpqb^uflj^ 'r lqol bnrfs^ibkqb( bp kb`bp^oflm^o^ bpq^_ib`boi^ bufpqbk`f^ abi k•jbol foo^`flk^i-

Dk @idb_o^ bibjbkq^i pb mobpbkq^kk•jbolp foo^`flk^ibp `r^kal pb qo^q^abobplisbo `fboq^p b`r^`flkbp `r^aoŠqf`^p- Olo bgbjmil+ pb abpb^ qbkbo rk k•jbolob^i r q^i nrb r0 < 1- @ m^oqfoab ilp krbsb ^uflj^p ^kqboflobpkl pb mrbab mol_^onrb bufpq^rk s bk bi pfpqbj^ ab ilp k•jbolp ob^ibpnrb sbofcfnrb q^i b`r^`fŽk+ v^nrb bpqlp krbsb ^uflj^p plk p^qfpcb`elp q^j_f‹k mlo ilp k•jbolp o^`flk^ibp v kle^v kfkd•k k•jbol o^`flk^i `rvl `r^ao^al pb^ 1- 'Dk bi Dgbo`f`fl 00 ab i^ Rb`,`fŽk 02-01 pb bp_lw^ rk^ abjlpqo^`fŽk ab bpq^^cfoj^`fŽk-( Di ^uflj^ 0/ mbojfqbfkqolar`fo k•jbolp foo^`flk^ibp bk bi pfpqbj^ ab ilp k•jbolp ob^ibp- Rb sboŠq^j_f‹k nrb ^qof_rvb ^i `lkgrkql ab ilp k•jbolp ob^ibprk^ molmfba^a ab `lkqf,krfa^a nrb bp bpmb`f^ijbkqb fjmloq^kqb bk bi bpqrafl abi BŠi`ril-

@kqbpab bumlkbo bi ^uflj^ 0/+ `lksfbkb fkqolar`fo ^idrk^ qbojfklildŒ^v klq^`fŽk bpmb`f^ibp-Rb^ R rk `lkgrkql kl s^`Œl ab k•jbolp ob^ibpv prmlkd^jlpnrb bufpqbrk k•jbol ? q^i nrb

sx?

m^o^qlal s ab R- Dkqlk`bp pb af`b nrb R bpqŠ\^jo\_j npk`mdjmh`io` mlo ?, Di k•,jbol ? pb abkljfk^ rk^ ^jo\ npk`mdjmm^o^R- Cb`fjlp pi\ `lq^ prmboflo ab_fal^ nrb qlal k•jbol j^vlo nrb ? q^j_f‹k bp rk^ `lq^ prmboflo-Rf rk^ `lq^ prmb,oflo ? mboqbkb`bq^j_f‹k ^R+ bkqlk`bp ? pb ii^j^ bi `g`h`ioj hƒsdhj ab R- @ ilprjl mrbab bufpqfork > nrb pb^bibjbkql jŠufjl- Rf bufpqb+pbbp`of_b

> < j^uR-

@pŒnrb+ ? < j^u R pf ? D R X s x ? m^o^qlal s ab R- Tk `lkgrkql pfk `lq^ pr,mboflo pb af`b nrb bp ij \^jo\_j npk`mdjmh`io`,

Klp bgbjmilp nrb pfdrbk firpqo^k bi pfdkfcf`^al ab bpq^pabkljfk^`flkbp-

DIDLOKN 0- Rb^ R bi `lkgrkql ab qlalp ilp k•jbolp ob^ibpmlpfqfslp- Dp rk`lkgrkql kl ^`lq^al prmboflojbkqb- Ml qfbkb`lq^p prmboflobpkf bibjbkql jŠufjl-

DIDLOKN 1- Rb^ R bi `lkgrkql ab qlalp ilp k•jbolp ob^ibp s q^ibp nrbN -z s x 0- Dpqb lkgrkql bpqŠ^`lq^al prmboflojbkqb mlo bi 0- Rr bibjbkql jŠuf,jl bp bi 0-

DIDLOKN 2- Rb^ P bi `lkgrkql ab qlalp ilp k•jbolp ob^ibp s q^ibp nrbN 9999::s ; 0- Dp m^ob`fal ^i `lkgrkql abi bgbjmil 1 p^isl nrb bi mrkql 0 kl bpqŠfk`irfal- Dpqb`lkgrkql bpqŠ^`lq^al prmboflojbkqb mlo bi 0 mbol kl qfbkbbibjbk,ql jŠufjl-

Page 50: Calculus

2/ / iomj_p^^d‡i

@idrklp `lkgrkqlp+ m^ob`falp ^i abi bgbjmil 2+bpqŠk^`lq^alp prmboflojbkqbmbol kl qfbkbkjŠufjl- O^o^biilp bufpqbrk `lk`bmql nrb prpqfqrvb^i abi jŠufjl-Dpqbpb ii^j^ `som`hj npk`mdjm abi `lkgrkql v pb abcfkb `ljl pfdrb9

CDEHMHBHˆM CD DWSQDLN RTODQHNQ- Ri iˆh`mj ? n` _`ijhdi\ `som`hj np+k`mdjm _` pi ^jiepioj ij q\^…j R nd ? od`i` g\n _jn kmjkd`_\_`n ndbpd`io`n8

]( ? `n pi\ ^jo\ npk`mdjm _` R-_( Kdibˆi iˆh`mj h`ijm lp` ? `n ^jo\ npk`mdjm k\m\ R-

Rf R qfbkbjŠufjl+ ‹pqbbp q^j_f‹k buqobjl prmbofloab R- Obol pf R kl mlpbbjŠufjl+ mrbab qbkbobuqobjl prmboflo-Dk bi bgbjmil 2 mob`babkqb+bi k•jbol 0bp buqobjl prmboflom^o^Q pf _fbk Q kl qfbkbjŠufjl- 'Ubo cfdro^ 0-7-(

`lq^p prmboflobp ab Q

.0z...............

buqobjl prmboflo ab Q

rN

R.

`lq^p prmboflobp ab R

.-y0!!

rN

Q.

jŠufjl ab R

^( R qfbkb jŠufjl9j^uR<i

_( Q kl qfbkb jŠufjl+ mbol pŒbuqobjl prmboflo9 prm Q < 0

EHFTQ@ 0-7 @jo\n npk`mdjm`n*hƒsdhj u `som`hj npk`mdjm,

SDNQDL@ 0-15- Ajn iˆh`mjn _dnodiojn ij kp`_`i n`m `som`hjn npk`mdjm`nk\m\ `g hdnhj ^jiepioj,

A`hjnom\^d‡i, Rb^k ? v b alp buqobjlp prmboflobp m^o^ rk `lkgrkql R-K^ molmfba^a _( fjmif`^ nrb b z ? mrbpql nrb ? bp buqobjl prmboflo: ^kŠild^,jbkqb+ ? ƒb v^ nrb b bp buqobjl prmboflo-Krbdl+ qbkbjlp ? < B-

Dpqbqblobj^ klp bumobp nrb pf bufpqbbuqobjl prmboflom^o^rk `lkgrkql R+e^v njg\h`io` rklv mrbab ab`fopb `g buqobjl prmboflo-

Blk cob`rbk`f^ pb bjmib^ bi q‹ojfkl npkm`hj ab rk `lkgrkql bk sbw abbuqobjl prmboflo rqfifw^kal i^ ^_obsf^qro^ npk* bp`of_fbkal bkqlk`bp9

? < prmR-

) +&1 6dV\ZN QRYRda_RZ\`b]R_V\_ NdV\ZNQRP\Z]YRaVabQ!

Olabjlp ^elo^ bpq^_ib`bobi ^uflj^ abi buqobjl prmboflom^o^bi pfpqbj^ abk•jbolp ob^ibp-

Page 51: Calculus

>sdjh\ _`g `som`hj npk`mdjm 20

@WHNL@ 0/- Qj_j ^jiepioj ij q\^…j R _` iˆh`mjn m`\g`n \^jo\_j npk`mdjm+h`io` kjn`` `som`hj npk`mdjm9 `noj `n* `sdno` pi iˆh`mj m`\g ? o\g lp` ? < prm R-

Hkpfpq^jlp rk^ sbw jŠp bk nrb bi buqobjl prmboflo ab R kl mboqbkb`b kb`b,p^of^jbkqb ^ R- Dk ob^ifa^a prm R mboqbkb`b ^ R pf v pŽil pf R mlpbb jŠufjl+ bk`rvl `^pl j^u R < prm R-

K^p abcfkf`flkbp ab ^jo\ dia`mdjm*\^jo\_j dia`mdjmh`io`* h…idhj* pb clojri^kbk cloj^ m^ob`fa^- Di ib`qlo ab_boŒ^ e^`boil `ljl bgbo`f`fl- Rf R qfbkb jŒkfjl+pb bumobp^ mlkfbkal jfk R-

Tk k•jbol I pb ii^j^ `som`hj dia`mdjm 'l …iadhj' ab R pf ^( I bp rk^ `lq^fkcboflo m^o^ R+v _( kfkd•k k•jbol j^vlo nrb H bp `lq^ fkcboflo m^o^ R- Di buqob,jl fkcboflo ab R+`r^kal bufpqb+bp •kf`l v pb abpfdk^ mlo fkc R- Rf R mlpbb jŒkfjl+bkqlk`bp jfk R < fkc R-

Blk bi ^uflj^ 0/+ pb mrbab abjlpqo^o bi pfdrfbkqb

RCMPCK? 0-16- Qj_j ^jiepioj ij q\^…j R \^jo\_j dia`mdjmh`io` kjn`` `som`+

hj dia`mdjm9`noj `n* `sdno` pi iˆh`mj m`\g I o\g lp` I < fkc R-

A`hjnom\^d‡i, Rb^ , R bi `lkgrkql ab ilp k•jbolp lmrbpqlp ab ilp ab R-Dkqlk`bp ,R bp kl s^`Œl v ^`lq^al prmboflojbkqb- Di ^uflj^ 0/ klp af`b nrbbufpqb rk k•jbol ? nrb bp buqobjl prmboflo ab +P, Dp cŠ`fi sbo nrb , ? < fkc R-

Blkpfabobjlp rk^ sbw jŠp ilp bgbjmilp ab i^ Rb``fŽk ^kqboflo- Dk bi bgbj,mil 0+ bi `lkgrkql ab qlalp ilp k•jbolp ob^ibp mlpfqfslp+ qfbkb bi N `ljl buqobjlfkcboflo- Dpb `lkgrkql kl qfbkb jŒkfjl- Dk ilp bgbjmilp 1 v 2+ bi N bp bi jŒkfjl-

Dk qlalp bplp bgbjmilp obpriq^ cŠ`fi ab`fafo pf bi `lkgrkql R bp l kl ^`lq^alprmboflo l fkcboflojbkqb+ v q^j_f‹k bp cŠ`fi abqbojfk^o ilp k•jbolp prm R b fkc R-Di bgbjmil pfdrfbkqb jrbpqo^ nrb ^sbofdr^o i^ bufpqbk`f^ ab i^p `lq^p prmboflo lfkcboflo mrbab obpriq^o afcŒ`fi-

DIDLOKN 3- Rb^ R bi `lkgrkql ab qlalp ilp k•jbolp ab i^ cloj^ 'i * g-i'i*alkab i < 0+1+ 2+ ---- Rf+mlo bgbjmil+ e^`bjlp i < 0+1+ X 2+ bk`lkqo^jlp nrbilp k•jbolp 1+K v -z-zmboqbkb`bk ^ R- Slal k•jbol abi `lkgrkql bp j^vlo nrb 0+`lk il nrb bi `lkgrkql bpqŠ ^`lq^al fkcboflojbkqb v mlo q^kql qfbkb rk buqobjlfkcboflo- Blk rk mbnrb•l bpcrbowl mlabjlp mol_^o nrb 1 bp bi jbklo bibjbkqlab R ab jlal nrb fkc R < jfk R < 1- S^j_f‹k bi `lkgrkql R bpqŠ ^`lq^al prmb,oflojbkqb+ ^rknrb kl bp q^k cŠ`fi abjlpqo^oil- '{Hkq‹kqbpb ( Tk^ sbw p^_fal nrb RbpqŠ ^`lq^al prmboflojbkqb+ bi ^uflj^ 0/ klp ^pbdro^ i^ bufpqbk`f^ abi buqobjlprmboflo ab R- Dk bpqb`^pl kl obpriq^ cŠ`fi abqbojfk^o bi s^ilo abi buqobjl prmbofloab R ^ m^oqfoab i^ abcfkf`fŽk ab bpqb `lkgrkql- Dk rk moŽufjl `^mŒqril sbobjlpnrb bi prm R bp rk k•jbol foo^`flk^i ^molufj^a^jbkqb fdr^i ^ 1+607- Dp rk k•,jbol fjmloq^kqb bk BŠi`ril ii^j^al k•jbol ab Dribo l k•jbol `,

Page 52: Calculus

,+ Fiomj_p^^dƒi

) +&)( ?N ]_\]VRQNQN_^bVZRQVN[NQRYV`aRZNQRY\` [qZR_\` _RNYR`

Dpq^Rb``fŽk `lkqfbkb ^idrk^p molmfba^abp fjmloq^kqbp abi pfpqbj^ ab ilpk•jbolp ob^ibp nrb plk `lkpb`rbk`f^ abi ^uflj^ abi buqobjl prmboflo-

RCMPCK? 0-17- Bg ^jiepioj O _` gjn `io`mjn kjndodqjn 0+1+ 2+ -+ - ij `noƒ\^jo\_j npk`mdjmh`io`,

A`hjnom\^dƒi, RrmŽkd^pb O ^`lq^al prmboflojbkqb- Cbjlpqo^objlp nrbbpql klp `lkar`b ^ rk^ `lkqo^af``fŽk- Orbpql nrb O kl bp s^`Œl+bi ^uflj^ 0/klp af`b nrb O qfbkbbuqobjl prmboflo+pb^ ‹pqb ], Di k•jbol ] + 0+pfbkal jbklonrb \) kl mrbab pbo`lq^ prmboflo ab O- Krbdl+ bufpqbrk jŒkfjl bkqbol mlpfqfsl iq^i nrb i = ] + 0- O^o^ bpqbi qbkbjlp i * 0 = ], Orbpql nrb i * 0 mboqbkb`b^ O+bpql `lkqo^af`b bi nrb $\ pb^ rk^ `lq^ prmboflo m^o^ O-

Bljl `loli^oflp abi qblobj^ 0-17+ pb l_qfbkbk fkjbaf^q^jbkqb i^p `lkpb,`rbk`f^p pfdrfbkqbp9

RCMPCK? 0-18- M\m\ ^\_\ m`\g s `sdno` pi `io`mj kjndodqj i o\g lp` i = s,

A`hjnom\^dƒi* Rf kl crbo^ ^pŒ+s pboŒrk^ `lq^ prmboflo ab O+bk `lkqo^,af``fŽk `lk bi qblobj^ 0-17-

RCMPCK? 0-2/- Pd s = N ` t `n pi iˆh`mj m`\g \m]dom\mdj*sdno` pi `io`mjkjndodqj i o\g lp` is = s+

A`hjnom\^dƒi, @mif`^obi qblobj^ 0-18 `^j_f^kal s mlo tZ~,

K^ molmfba^a abp`ofq^ bk bi qblobj^ 0-2/+ pb abkljfk^ cob`rbkqbjbkqbkmjkd`_\_ \mlpdh`_d\i\ abi pfpqbj^ ab ilp k•jbolp ob^ibp-Fblj‹qof`^jbkqb pfdkf,cf`^ nrb `^a^ pbdjbkql+ q^k i^odl `ljl pb nrfbo^+ mrbab pbo ob`r_fboql mlo rkk•jbol cfkfql ab pbdjbkqlp ab ilkdfqra mlpfqfs^ a^a^+ q^k mbnrb•^ `ljl pbnrfbo^- Dk lqo^p m^i^_o^p+rk^ obdi^ `loq^ mrbab jbafo afpq^k`f^p q^k i^od^p `ljlpb nrfbo^ `lil`Škali^ `lkpb`rqfs^jbkqb- @onrŒjbabp+`lkpfabo^kal ‹pq^ `ljlrk^ molmfba^a crka^jbkq^i ab i^ iŒkb ob`q^+i^ `lkpfaboŽ `ljl rkl ab ilp ^ufl,j^p ab i^ FbljbqoŒ^- Dk ilp pfdilp WHW v WW pb e^k `lkpqorfal dbljbqoŒ^p kl^onrfjbaf^k^p bk i^p nrb pb mobp`fkabab bpqb^uflj^-

@ m^oqfoab i^ molmfba^a ^onrfjbaf^k^+ mlabjlp abjlpqo^o bi qblobj^ pf,drfbkqb nrb klp pboŠ•qfi bk BŠi`ril fkqbdo^i-

RCMPCK? 0-20- Pd om`niˆh`mjn m`\g`n \* s* ` t n\odna\^`i g\n _`ndbp\g_\_`n

%.+.1& \ 8899s 8899\ * yk\m\ oj_j `io`mj i x 0+`ioji^`n s < \,

Page 53: Calculus

Mmjkd`_\_`n api_\h`io\g`n _`g `som`hj npk`mdjm 11

A`hjnom\^d‡i, Rf s = \* bi qblobj^ 0-2/ klp af`b nrb bufpqb rk bkqbolmlpfqfsl i nrb p^qfpc^`b i&s + \' = v+ bk `lkqo^af``fŽk `lk '0-03(- Krbdl klmrbab pbo s ; \* `lk il nrb ab_boŠ pbo s < \,

0 2-00 Oolmfba^abp crka^jbkq^ibp abi buqobjl prmboflo

Dk bpq^Rb``fŽk pb `lkpfabo^k qobpmolmfba^abp crka^jbkq^ibp ab ilp buqobjlpprmboflo b fkcboflo nrb pb rqfifw^oŠk bk il pr`bpfsl- K^ mofjbo^ ab bii^p bpq^_ib`bnrb qlal `lkgrkql ab k•jbolp `lk buqobjl prmboflo `lkqfbkb k•jbolp q^k moŽuf,jlp `ljl pb nrfbo^ ^ af`el buqobjl: abi jfpjl jlal+ rk `lkgrkql `lk buqobjlfkcboflo `lkqfbkb k•jbolp q^k moŽufjlp ^ ‹i `ljl pb nrfbo^-

SDNQDL@ 0-21- P`\ c pi iˆh`mj kjndodqj _\_j v R pi ^jiepioj _` iˆh`+mjn m`\g`n,

]( Rf R od`i` `som`hj npk`mdjm*k\m\ pi ^d`moj s _` R n` od`i`

s< prm R , b+

_( Rf R od`i` `som`hj dia`mdjm*k\m\ pi ^d`moj s _` R n` od`i`

s ; fkc R * b+

A`hjnom\^d‡i _` ^(- Rf bp s x prm R , c m^o^ qlal s ab R+ bkqlk`bpprm R , c pboŒ rk^ `lq^ prmboflo ab R jbklo nrb pr buqobjl prmboflo- Olo `lk,pfdrfbkqb ab_b pbo s = prm R , c mlo il jbklp m^o^ rk s ab R- Dpql abjrbpqo^ ^(-K^ abjlpqo^`fŽk ab _( bp m^ob`fa^-

SDNQDL@ 0-22- OQNOHDC@C @CHSHU@- A\_jn _jn np]^jiepiojn ij q\^…jn

> u ? _` P+ n`\ a `g ^jiepioj

b< u[ * ] G[ C >* ] C ?w ,

]( Rf > u ? kjn``i `som`hj npk`mdjm*`ioji^`n a od`i` `som`hj npk`mdjm* u

prm b < prm > * prm ? ,

_( Rf > V ? od`i`i `som`hj dia`mdjm*`ioji^`n b od`i` `som`hj dia`mdjm*`

fkc b< fkc > * fkc ? ,

A`hjnom\^d‡i, Rrmlkd^jlp nrb > u ? qbkd^k buqobjl prmboflo- Rf ` D a+bkqlk`bp _ < [ * ]* alkab [ C > X ] C A- Olo `lkpfdrfbkqb _ w prm > * prm ?9

Page 54: Calculus

23 / iomj_p^^d‡i

ab jlal nrb prm = * prm > bp rk^ `lq^ prmboflo ab B- Dpql abjrbpqo^ nrb b qfbkbbuqobjl prmboflo v nrb

prm b z prm > * prm ? ,

Rb^ ^elo^ h rk bkqbol mlpfqfsl `r^inrfbo^- Rbd•k bi qblobj^ 0-21 '`lk b < ..h&bufpqbk rk \ bk > v rk ] bk ? q^ibp nrb

0\< prm = * ‚) 0] = npk? + ‚$

Rrj^kal bpq^p abpfdr^ia^abp+ pb l_qfbkb

1\ * ] = prm > * prm ? + ‚) l

1 1prm > * prm ? ; [ * ] * † x prm b * † %

mrbpql nrb \ * ] x prm B- Olo `lkpfdrfbkqb ebjlp abjlpqo^al nrb

1prm b z prm = * prm > ; prm b * †

m^o^ qlal bkqbol i x 0- Dk sfoqra abi qblobj^ 0-20+ ab_b pbo prm b < prm > (prm >+ Dpql abjrbpqo^ ^(+ v i^ abjlpqo^`fŽk ab _( bp m^ob`fa^-

RCMPCK? 0-23- A\_jn _jn np]^jiepiojn ij q\^…jn R t Q _` Q o\g`n lp`

k\m\ oj_j p _` R t oj_j o _` Q, Bioji^`n R od`i` `som`hj npk`mdjm* t Q `som`hjdia`mdjm*t n` q`mdad^\

prm Rz fkc Q,

A`hjnom\^d‡i, B^a^ o ab Q bp `lq^ prmboflo m^o^ R- Olo `lkpfdrfbkqb R qfbkbbuqobjl prmboflo nrb p^qfpc^`b i^ abpfdr^ia^a prm R z o m^o^ qlal o ab P+Krbdlprm R bp rk^ `lq^ fkcboflo m^o^ P) `lk il `r^i P qfbkb buqobjl fkcboflo nrb klmrbab pbo jbklo nrb prm R- Cf`el ab lqol jlal+ pb qfbkb prm R z fkc Q*^jhjpb ^cfojŽ-

") +&)* :WR_PVPV\`

0- Rf s b v plk k•jbolp ob^ibp `r^ibpnrfbo^+ s ; v+ abjlpqo^o nrb bufpqb mlo il jbklp rkk•jbol ob^i w q^i nrb s ; w ; v-

Page 55: Calculus

Bsdno`i^d\ _` m\…^`n^p\_m\_\n _` gjn iˆh`mjn m`\g`n ij i`b\odqjn 24

1- Rf s bp rk k•jbol ob^i ^o_fqo^ofl+mol_^o nrb bufpqbk bkqbolp h v i q^ibp nrb h ; s ; i,2- Rf s = N+abjlpqo^o nrb bufpqb rk bkqbol mlpfqfsl i q^i nrb g-i ; s,3- Rf s bp rk k•jbol ob^i ^o_fqo^ofl+ abjlpqo^o nrb bufpqb rk bkqbol i •kf`l nrb sbofcf`^

i^p abpfdr^ia^abp i 889s ; i * 0- Dpqb i pb abkljfk^ i^ k\mo` `io`m\ ab s v pb abpfdk^mlo WrY+ Olo bgbjmil+ Z4\ < 4+Yph< 1+ n,j < , 2-

4- Rf s bp rk k•jbol ob^i ^o_fqo^ofl+ mol_^o nrb bufpqb rk bkqbol •kf`l i nrb p^qfpc^`b i^abpfdr^ia^a i n9 s ; i * 0-

5- Rf s b v plk k•jbolp ob^ibp ^o_fqo^oflp+s ; v+ mol_^o nrb bufpqb mlo il jbklp rk k•,jbol o^`flk^i l q^i nrb r ; l ; v v abar`fo ab biil nrb bufpqbk fkcfkfqlp- Dpq^ molmfba^apb bumobp^ af`fbkal nrb bi `lkgrkql ab ilp k•jbolp o^`flk^ibp bp _`inj bk bi pfpqbj^ab ilp k•jbolp ob^ibp-

6- Rf s bp o^`flk^i+ s x N+b v bp foo^`flk^i+ abjlpqo^o nrb s * v+ s + v+ st* s-t* tZ~ plkqlalp foo^`flk^ibp-

7- ƒK^ prj^ l bi molar`ql ab alp k•jbolp foo^`flk^ibp bp pfbjmob foo^`flk^i>8- Rf s b v plk k•jbolp ob^ibp `r^ibpnrfbo^+ s ; v+ abjlpqo^o nrb bufpqb mlo il jbklp rk

k•jbol foo^`flk^i w q^i nrb s ; w ; v v abar`fo nrb bufpqbk fkcfkfqlp-0/- Tk bkqbol i pb ii^j^ k\m pf i < 0h m^o^ rk `fboql bkqbol h* b dhk\m pf i * 0 bp m^o-

Cbjlpqo^o i^p ^cfoj^`flkbp pfdrfbkqbp9^( Tk bkqbol kl mrbab pbo ^ i^ sbw m^o b fjm^o-_( Slal bkqbol bp m^o l bp fjm^o-b( K^ prj^ l bi molar`ql ab alp bkqbolp m^obp bp m^o- ƒPr‹ pb mrbab ab`fo ^`bo`^ abi^ prj^ l abi molar`ql ab alp bkqbolp fjm^obp>a( Rf i0 bp m^o+q^j_f‹k il bp i, Rf \0 < 0]0* pfbkal \ v ] bkqbolp+ bkqlk`bp \ v ] plk^j_lp m^obp-b( Slal k•jbol o^`flk^i mrbab bumobp^opbbk i^ cloj^ \g ]* alkab \ u ] plk bkqbolp+rkl ab ilp `r^ibp mlo il jbklp bp fjm^o-

00- Cbjlpqo^o nrb kl bufpqb k•jbol o^`flk^i `rvl `r^ao^al pb^ 1-

XFi_d^\^d‡i, Tqfifw^o bi o^wlk^jfbkql ab obar``fŽk ^i ^_proal- RrmŽkd^pb &\-]'0 < 1+pfbkal \ v \ bkqbolp+ rkl ab biilp mlo il jbklp fjm^o- Tqfifw^o m^oqbpabi Dgbo`f`fl 0/-\

01- K^ molmfba^a ^onrfjbaf^k^ abi pfpqbj^ ab k•jbolp ob^ibp pb abargl `ljl `lkpb`rbk`f^abi ^uflj^ abi buqobjl prmboflo- Cbjlpqo^o nrb bi `lkgrkql ab ilp k•jbolp o^`flk^ibpp^qfpc^`b i^ molmfba^a ^onrfjbaf^k^ mbol kl i^ abi buqobjl prmboflo- Dpql abjrbpqo^nrb i^ molmfba^a ^onrfjbaf^k^ kl fjmif`^ bi ^uflj^ abi buqobjl prmboflo-

y&G+&)+ :dV`aR[PVNQR_NoPRPbNQ_NQNQRY\` [qZR_\` _RNYR`[\ [RTNaVc\`

Rb e^ sfpql ^kqboflojbkqb nrb i^ b`r^`fŽk s0 < 1 kl qfbkb plir`fŽk bkqob ilpk•jbolp o^`flk^ibp- Blk ^rufifl abi ^uflj^ 0} pb mrbab abjlpqo^o nrb i^ b`r^,`fŽk s0 < \ qfbkb plir`fŽk bkqob ilp k•jbolp m`\g`n pf \ 91N- S^i s pb abkljfk^m\…u^p\_m\_\ _` \,

Dk mofjbo ird^o+ pfk qbkbo bk `rbkq^ bi ^uflj^ 0/+ pb mrbabk e^`bo i^ppfdrfbkqbp `lkpfabo^`flkbp- Klp k•jbolp kbd^qfslp kl mrbabk qbkbo o^Œ`bp`r^ao^,a^p+ mrbp pf s0 < \* ^i pbo \ rk `r^ao^al e^ ab pbo kl kbd^qfsl 'bk sfoqra abi qbl,obj^ 0-1/(- @abjŠp+ pf \ < /+ s < N bpi^ •kf`^ o^Œwr^ao^a^ 'mlo bi: qblobj^ )&))"&&rmŽkd^pb+mrbp+ \ = N- Rf s0 < \ bkqlk`bp s ;/; } X &[U'0 < \* mlo q^kql+ s v

Page 56: Calculus

25 Fiomj_p^^d‡i

pr lmrbpql plk ^j_lp o^Œ`bp r^ao^a^p- Obol \ gj nphj qfbkb _jn* mlonrb pfs0 < \ b v1 < \* bkqlk`bp s0 < v1 X &s * v( &s + v( < N+X bk sfoqra abi qblob,j^ 0-00+l s < v l s < , v- Olo q^kql+pf \ qfbkbo^Œ`bpr^ao^a^p+ qfbkb `s\^o\+h`io` _jn,

K^ bufpqbk`f^ab rk^ o^Œwr^ao^a^ mlo il jbklp pb abar`foŠ mlpqboflojbkqbab rk qblobj^ fjmloq^kqb ab BŠi`ril+ `lkl`fal mlo bi qblobj^ abi s^ilo fkqbo,jbafl m^o^ i^p crk`flkbp `lkqfkr^p+ mbol bp fkpqor`qfsl sbo `ljl i^ bufpqbk`f^ab i^ o^Œwr^ao^a^ pb mrbab mol_^o afob`q^jbkqb ^ m^oqfoabi ^uflj^ 0/-

RCMPCK? 0-24- @\_\ iˆh`mj m`\g ij i`b\odqj \ od`i` pi\ m\…u p\_m\_\ iji`b\odq\ ˆid^\,

Kjo\8 Rf \ x N+pr o^Œwr^ao^a^ kl kbd^qfs^ pb fkaf`^oŠ mlo ^0.1 l mlo ,U::- Rf\ = N+i^ o^Œwr^ao^a^ kbd^qfs^ bp , ^0.1 l , -v^-

A`hjnom\^d‡i, Rf+\ < N+bkqlk`bp N bp i^ •kf`^ o^Œwr^ao^a^- RrmŽkd^pbmrbp nrb \ = N-Rb^R bi `lkgrkql ab qlalp ilp k•jbolp ob^ibpmlpfqfslp s q^ibpnrbs0 8999\, Orbpql nrb '0 * \'0 = \* bi k•jbol '0 * \' bp rk^ `lq^ prmboflo ab R-Obol+ R bp kl s^`Œl+ mrbp \eL * \' mboqbkb`b^ R: bk bcb`ql \0 8999\ N * \'0v mlo q^kql \0eL * \'08999\, Dk sfoqra abi ^uflj^ 0/+ R qfbkb rk buqobjlprmboflo nrb pb abpfdk^ mlo ], MŽqbpbnrb ]}x \eL * \' X mlo q^kql ] = N-DufpqbkpŽil qobpmlpf_fifa^abp9 ]0 = \* ]0 ; \* ]0 < \,

RrmŽkd^pb ]! = \ u pb^ ` < ] + &]0 + \'e&0]' < G_ * \e]', Dkqlk`bpM ; _ ; \ u `1 < \0 + %\0 + [& * %\/ * [&/,%1\/& < [ * %\/ * [&/,%1\/& = [ ++

Olo q^kql+`%= s0 m^o^`^a^ s bk R+bp ab`fo+ ` = s m^o^`^a^ s bk R: irbdl `bp rk^ `lq^ prmboflo ab R+u mrbpql nrb _ ; ] pb qfbkbrk^ `lkqo^af``fŽk `lk bieb`el ab pbo ] bi `som`hj npk`mdjm ab R- Olo q^kql+ i^ abpfdr^ia^a ]0 = \ bpfjmlpf_ib-

RrmŽkd^pb]% ; \, Orbpql nrb ] = N pb mrbab bibdfo rk k•jbol mlpfqfsl `q^i nrb _ ; \ u q^i nrb _ ; %[ * \0&d%0\&+ Rb qfbkb bkqlk`bp9

%\ * ?&/ < \/ * ]%/\ * `( ; \/ * 0\] ; \/ * %[ * \/& < [+

Dp ab`fo+ ] * ` mboqbkb`b R- Bljl ] * ` = ]* bpq^ abpfdr^ia^a bpqŠbk`lkqo^af``fŽk `lk nrb \ pb^ rk^ `lq^ prmboflo ab R- Olo q^kql+ i^ abpfdr^ia^a]0 ; [ bp fjmlpf_ib u pŽil nrba^ `ljl mlpf_ib ]0 < [+

)0 2-03 Q^Œ`bpab loabk prmboflo-Olqbk`f^po^`flk^ibp

Di ^uflj^ abi buqobjl prmboflo pb mrbab rqfifw^o q^j_f‹k m^o^ mol_^o i^bufpqbk`f^ ab o^Œ`bpab loabk prmboflo- Olo bgbjmil+ pf i bp rk bkqbol mlpfqfsl

Page 57: Calculus

O`km`n`io\^d‡i _` gjn iˆh`mjn m`\g`n kjm h`_dj _` _`^dh\g`n 15

dhk\m* m^o^ `^a^ ob^i s bufpqbrk k•jbol ob^i t* v rkl plil q^i nrb vN < s,Dpq^ t pb abkljfk^ m\…ui+ndh\ ab s v pb fkaf`^ mlo9

'H-04( l t;x,

Rf i bp k\m* i^ pfqr^`fŽk bp rk ml`l afpqfkq^-Dk bpqb`^pl+ pf s bp kbd^qfsl+ klbufpqbrk k•jbol ob^i s q^i nrb s| < s* mrbpql nrb s| ƒ N m^o^ `^a^ k•jbolob^i t, Rfk bj_^odl+ pf r bp mlpfqfsl+ pb mrbab mol_^o nrb bufpqbrk k•jbolmlpfqfsl u pŽil rkl q^i nrb u• < s, Dpqbu pb abkljfk^ i^ m\…ui+ndh\ kjndodq\ab s v pb fkaf`^ mlo ilp pŒj_lilp '0-04(- Orbpql nrb i bp m^o+%Zs&h < sh v+mlo q^kql+ `^a^ s = N qfbkb alp o^Œ`bpi+ndh\n ob^ibp+t u,u- Rfk bj_^odl+ ilppŒj_lilp Tfc| v Z0!: pb obpbos^k m^o^i^ o^Œwi+ndh\ mlpfqfs^-Ml bumlkbjlp ^nrŒi^p abjlpqo^`flkbp ab bpq^p^cfoj^`flkbp mlonrb pb abar`foŠk jŠp ^abi^kqb `ljl`lkpb`rbk`f^ abi qblobj^ abi s^ilo fkqbojbafl m^o^ i^p crk`flkbp `lkqfkr^p 'sboRb``fŽk 2-0/(-

Rf mbp rk k•jbol o^`flk^i mlpfqfsl+ pb^ m< h-i* alkab h v i plk bkqbolpmlpfqfslp+pb abcfkb s%`ljl &sh'g-i* bp ab`fo `ljl o^Œwi+ndh\ ab u!+ pfbjmob nrb‹pq^ bufpq^-Rf s 6!< N+pb abcfkb s+%< g-s% `lk q^i nrb s! bpq‹ abcfkfa^- O^oqfbkalab bp^pabcfkf`flkbp+bp cŠ`fi `ljmol_^o nrb i^p ibvbprpr^ibp ab ilp bumlkbkqbpplksŠifa^p m^o^ bumlkbkqbpo^`flk^ibp9 s%8s} < r$({) 'u!(! < u&!+u 'tu(! < smtm,

") +&)- DR]_R`R[aNPVp[QRY\` [qZR_\` _RNYR`]\_ ZRQV\QRQRPVZNYR`

Tk k•jbol ob^i ab i^ cloj^

'0-05(

alkab ^l bp rk bkqbol kl kbd^qfsl+ u ]h= [y+ +++ ) [y plk bkqbolp nrb p^qfpc^`bkN R [x R8+ pb bp`of_b `loofbkqbjbkqb bk i^ cloj^ jŠp _obsb pfdrfbkqb9

Rb af`b nrb ‹pq^ bp i^ m`km`n`io\^d‡i _`^dh\g adido\ ab m,Olo bgbjmil9

0 4, < , < /+41 0/ &

0 1,<, </+/14/ 0/1

18 1 4, < 6 * , * ,, < 6+14 -3 0/ 0/1

M•jbolp ob^ibp ab bpq^ `i^pb plk kb`bp^of^jbkqb o^`flk^ibp v qlalp biilp plk abi^ cloj^ l < [,fK| alkab [ bp rk bkqbol- Rfk bj_^odl+ kl qlalp ilp k•jbolpo^`flk^ibp mrbabk bumobp^opbmlo jbafl ab rk^ obmobpbkq^`fŽkab`fj^i cfkfq^-Olo bgbjmil+ pf ppb mrafbo^ bumobp^o pŒ+pb qbkaoŒp< \g 0/‚ l 0[ < 0/‚ m^o^

Page 58: Calculus

27 Fiomj_p^^d‡i

^id•k bkqbol \, Obol bpql bp fjmlpf_ib+ mrbpql nrb 2 kl bp afsfplo ab kfkdrk^mlqbk`f^ ab 0/-

Ml l_pq^kqb+`r^inrfbo k•jbol ob^i s = N mrbab ^molufj^opb `lk rk booloq^k mbnrb•l `ljl pb abpbb mlo jbafl ab rk^ prj^ ab i^ cloj^ '0-05( pf pbqlj^ i prcf`fbkqbjbkqb do^kab- K^ o^wŽk ab biil mrbab sbopb jbaf^kqb bi pf,drfbkqb ^odrjbkql dblj‹qof`l9 pf s kl bp bkqbol+s bpqŠ`ljmobkafal bkqob alpbkqbolp `lkpb`rqfslp+ bp ab`fo+ \• ; s ; \j * 0- Di pbdjbkql nrb rkb \j v\j( 0 mrbab pr_afsfafopb bk afbwm^oqbpfdr^ibp- Rf s kl `lfk`fab `lk rkl ab bpqlpmrkqlp ab pr_afsfpfŽk+ s ab_b bpq^o ljmobkafal bkqobalp ab biilp- Dpql a^ ird^o^ rk m^o ab abpfdr^ia^abp ab i^ cloj^

( [f ; ; * [f * 0\ + s \ +++l 0/ l 0/&

alkab [f bp rk bkqbol '/949 [f 9498(- Rb afsfab ^elo^+ bi pbdjbkql nrb rkb[f [f * 0

[} * , v Mi * ,,, bk afbw m^oqbpfdr^ibp '`^a^ rk^ ab ilkdfqra 0/,1( v

0/ 0/pb `lkqfk•^ bi mol`bpl- Rf abpmr‹p ab rk k•jbol cfkfql ab pr_afsfpflkbp+ rkl abilp mrkqlp `lfk`fab `lk s* s bp rk k•jbol ab i^ cloj^ '0-05(- Rf kl bp ^pŒ+bimol`bpl pb `lkqfk•^ fkabcfkfa^jbkqb v pb bkdbkao^ rk `lkgrkql ab fkcfkfqlpbkqbolp\y* \^0* \1* ,,, Dk bpqb ^pl pb af`b nrb s qfbkbi^ obmobpbkq^`fŽkab`fj^i fkcfkfq^

Cbpmr‹p ab i pr_afsfpflkbp+ s p^qfpc^`bi^p abpfdr^ia^abp

^i Qh ^i [ * 0\j * , * --- * , ; s ; Q * , * --- * [i [[ ,0/ iNk l 0/ fl,

i^p `r^ibp a^k alp ^molufj^`flkbp ab s* rk^ mlo bu`bpl v lqo^ mlo abcb`ql+mlo jbafl ab ab`fj^ibp cfkfqlp nrb afcfbobkbk iN,kl Olo q^kql+pb mrbab ildo^ork do^al ab ^molufj^`fŽk abpb^al pfk jŠp nrb qlj^o h prcf`fbkqbjbkqb do^kab-

Rf s < 0+bp cŠ`fi `ljmol_^o nrb \j < N X \9 < 2 m^o^ `^a^ i x 0+X mloq^kql i^ ^molufj^`fŽk ab`fj^i `loobpmlkafbkqb bp9

0< /+222 ----

B^a^ k•jbol foo^`flk^i qfbkb rk^ obmobpbkq^`fŽkab`fj^i fkcfkfq^-Olo bgbjmil+pf s < U1 pb mrbabk `^i`ri^o mlo q^kqbl q^kql aŒdfqlp ljl pb abpbbk ab pr^molufj^`fŽ]k ab`fj^i- Orbp U1 bpqŠ ljmobkafal bkqob0+3X 0+4v^ nrb 'i+3(1 ;

Page 59: Calculus

O`km`n`io\^d‡i _` gjn iˆh`mjn m`\g`n kjm h`_dj _` _`^dh\g`n 17

1 ; '0+4(1- @kŠild^jbkqb+ bibs^kal ^i `r^ao^al v `ljm^o^kal `lk 1 pb l_qfbkbki^p pfdrfbkqbp ^molufj^`flkbp pr`bpfs^p9

0+30 ; r&1; 0+31+ 0+303 ; r&1; 0+304+ 0+3031 ; r&1; 0+3032 -

N_p‹osbpb nrb bi mol`bpl ^kqboflo bkdbkao^ rk^ pr`bpfŽk ab fkqbos^ilp abilkdfqra 0/,! 0/,1

+ 0/,2+ ‘‘‘ + `^a^ rkl `lkqbkfal bk bi ^kqboflo v `lkqbkfbkal

`^a^ rkl bi mrkql s, Dpql bp rk bgbjmil abi ii^j^al bk`^gb ab fkqbos^ilp+ `lk,`bmql nrb pb rqfifw^ ^idrk^p sb`bp `ljl _^pb m^o^ `lkpqorfo ilp k•jbolp foo^,`flk^ibp ^ m^oqfoab ilp o^`flk^ibp-

Orbpql nrb bk bpqb if_ol pb e^oŠ ml`l rpl ab ilp ab`fj^ibp+ kl pb bpqraf^oŠkprp molmfba^abp `lk qlal abq^iib+ v pŽil pb sboŠ `Žjl pb mrbabk abcfkfo ^k^iŒqf`^,jbkqb bumobpflkbp ab`fj^ibp+ `lk ^rufifl abi ^uflj^ abi buqobjl prmboflo-

Rf s bp rk k•jbol ob^i mlpfqfsl a^al+ pb^ \• bi j^vlo bkqbol f: s, Slj^,al [+) pb^ [f bi j^vlo bkqbol q^i nrb9

[f\j * , ; s,

*)'

Dk dbkbo^i+ abqbojfk^alp \z* \g%,,, * \i+g* pb^ \9 bi j^vlo bkqbol q^i nrb

'H-06(

Rb^ P bi `lkgrkql ab qlalp ilp k•jbolp9

'0-i7(

l_qbkfalp ab bpq^ cloj^ m^o^ i < N+ 0+ 1+ - ! - Orbpql nrb P bp kl s^`Œl v ^`l,q^al prmboflojbkqb+ qfbkb rk buqobjl prmboflo nrb bp cŠ`fi `ljmol_^o nrb `lfk,`fab `lk s, Klp bkqbolp \j* \g%\0* ŠŠŠ ^pŒl_qbkfalp pb mrbabk rqfifw^o m^o^ abcfkfork^ bumobpfŽk ab`fj^i ab s* mlkfbkal9

alkab bi aŒdfql \9 nrb l`rm^ bi ird^o i bp bi j^vlo bkqbol nrb p^qfpc^`b '0-06(-Olo q^kql+ pb mrbab bp`of_fo9

p< /+014/// ----

Rf bk '0-06( pb prpqfqrvb bi pfdkl ab abpfdr^ia^a f: mlo ;&+pb l_qfbkb rk^abcfkf`fŽk ab i^ bumobpfŽk ab`fj^i ^idl afpqfkq^- Di buqobjl prmboflo ab qlalp

Page 60: Calculus

3/ Fiomj_p^^d‡i

ilp k•jbolp ab i^ cloj^ '0-07( bp q^j_f‹k s9 pfk bj_^odl+ ilp bkqbolp^l+ \g%\0* ŠŠŠkl e^k ab pbo kb`bp^of^jbkqb ilp jfpjlp nrb p^qfpc^`bk'0-06(- Olo bgbjmil+ pfpb ^mif`^ i^ pbdrka^ abcfkf`fŽk ^ r < f+pb bk`rbkqo^ ^l < /+ [f < 0+ [/ < 1+^p < 3+ X [i < 8 m^o^ `^a^ h w 3- Dpql `lkar`b ^ i^ obmobpbkq^`fŽkab`fj^ifkcfkfq^

0< /+013888 ----

Di nrb alp k•jbolp ob^ibp mrba^k qbkbo alp obmobpbkq^`flkbpab`fj^ibpafpqfkq^pbp rk pfjmib bgbjmil abi eb`el ab nrb alp `lkgrkqlp afpqfkqlpab k•jbolpob^ibp mrbabk qbkbo rk jfpjl buqobjl prmboflo-

L[ln_ ER+* Eh^o]]cƒh g[n_g•nc][+ mcg\ifim mog[nilcim u]o_mncih_m l_f[]cih[^[m

0 3-0 Dgbjmil ab abjlpqo^`fŽk mlo fkar``fŽk j^qbjŠqf`^

Orbpql nrb prj^kal 0 ^i bkqbol e pb l_qfbkb e * 0 nrb bp j^vlo nrb e) klbufpqbkfkd•k bkqbol nrb pb^ bi h\tjm _` oj_jn, Rfk bj_^odl+ m^oqfbkalabi k•jb,ol 0+pb mrbabk ^i`^kw^o qlalp ilp bkqbolp mlpfqfslp+abpmr‹p ab rk k•jbol cfkfqlab m^plp+m^p^kal pr`bpfs^jbkqb ab h^ f * 0- Dpq^bp i^ _^pb ab rk j‹qlal abo^wlk^jfbkql nrb ilp j^qbjŠqf`lp ii^j^k _`hjnom\^d‡i kjm di_p^^d‡i, Rb firpqo^oŠi^ ^mif`^`fŽk ab bpqbj‹qlal+ abjlpqo^kal bi m^o ab abpfdr^ia^abp rp^a^p bk bi^m^oq^al 0 0-2 m^o^bi `Ši`ril abi Šob^abi pbdjbkql m^o^_Žif`l+bp ab`fo9

'0-08(k2

01 * 11 * --- * 'j , 0(1 ; , ; 01 * 11 * --- * i0Š

2

Dk mofjbo ird^o+ pb `lkpfabo^ i^ abpfdr^ia^a ab i^ fwnrfboa^+cŽojri^ nrb ^_ob,sf^a^jbkqb pb fkaf`^oŠ mlo =%h&'^cfoj^`fŽk obcbofa^^ h&+DpcŠ`fi `ljmol_^o bpq^^cfoj^`fŽk afob`q^jbkqb m^o^ilp mofjbolp s^ilobp ab i, Orbp m^o^ i < 0+1+ 2+mlo bgbjmil+ pb qfbkb9

02

=%f&7/; ,+2

=%/&701 ; 12

+

2=%0&701 * 11 ; 2

2

+

2

prmrbpql nrb pb fkqbomobqnrb i^ prj^ abi mofjbo jfbj_ol bp N `r^kal h < 0-

Page 61: Calculus

Bg kmdi^dkdj _` g\ di_p^^d‡i h\o`hƒod^\ 30

Rb qo^q ab mol_^o nrb =%h&bp `fboq^m^o^`^a^ bkqbol mlpfqfsl h+ Rb mol`bab`ljl pfdrb9 pb prmlkb nrb i^ ^cfoj^`fŽk pb e^ mol_^al m^o^rk s^ilo m^oqf`ri^oab hc pb^ h < f, Dp ab`fo+pb prmlkb nrb pb e^ mol_^al

=%e&701 * 11 * --- * %e* 0(1 ; z2

m^o^ f W cfgl- M\mod`i_j _` `ggj pb e^ ab abar`fo nrb pb sbofcf`^ m^o^ f * )$bp ab`fo9

=%e * 0(9 01 * 11 * --- * f0 ; %e* 0(22

Rrj^kal h1 ^ ilp alp jfbj_olp ab =%e&'nrb pb prmlkb nrb pb e^ mol_^al( pbl_qfbkb i^ abpfdr^ia^a

1 1 f0 e1

f00 *1 * --- * ;,* -,

v m^o^ abar`fo `ljl `lkpb`rbk`f^ ab bii^ i^ =%e * 0(+_^pq^ abjlpqo^o

f0 * a ; &f * 0(22 2

Obol bpql bp `lkpb`rbk`f^ fkjbaf^q^ ab i^ fdr^ia^a9

%e* 0(2

2f1 * 1f

0 * 1f * 0< f1

* f0 * f * -,,,

Olo q^kql+pb e^ abjlpqo^al =%e * 0( `ljl `lkpb`rbk`f^ ab =%e&+Orbpql nrb=%0( pb e^ `ljmol_^al afob`q^jbkqb+ pb pfdrb nrb q^j_f‹k =%/& bp `fboql- R^,_fbkal nrb =%/& bp `fboql+pb pfdrb nrb q^j_f‹k il bp =%0&) v ^pŒpr`bpfs^jbkqb-Orbpql nrb `^a^ bkqbol mrbab ^i`^kw^opbmlo bpqbmol`bpl+ =%h& bp `fboql m^o^qlal s^ilo ab i, Dpql abjrbpqo^ i^ abpfdr^ia^a ^ i^ fwnrfboa^bk '0-08(- K^ abpfdr^i,a^a ^ i^ &abob`e mrbab abjlpqo^opb abi jfpjl jlal-

) ,&* :Y ]_V[PV]V\QRYNV[QbPPVp[ZNaRZmaVPN

Qbcibuflkb bi ib`qlo pl_ob bi `nlp`h\ ab i^ abjlpqo^`fŽk ^kqboflo-Oofjbol pb`ljmorb_^ i^ ^cfoj^`fŽk =%h& m^o^h < 0- Krbdl pb morb_^ nrb mci^ ^cfoj^`fŽk bp`fboq^m^o^rk bkqbol m^oqf`ri^o+o\h]d„i bp `fboq^m^o^bi bkqbol pfdrfbkqb-Cb bpqlpb `lk`irvb nrb i^ ^cfoj^`fŽk bp `fboq^m^o^qlalp ilp bkqbolpmlpfqfslp-

Page 62: Calculus

31 / iomj_p^^dƒi

K^ fab^ ab fkar``fŽk pb mrbab firpqo^oab jr`e^p j^kbo^p ij j^qbjŠqf`^p-Olo bgbjmil+ prmŽkd^pb rk^ cfi^ abplia^alp ab miljl krjbo^alp `lkpb`rqfs^,jbkqb+ u afpmrbpqlpab q^i cloj^ nrb pf rkl ab biilp `^b+ mlo bgbjmil+ bi pb•^i^al`lk bi pŒj_lil e) `el`^ `lk bi pfdrfbkqb pb•^i^al `lk bi pŒj_lil e * 0- Dk pb,drfa^ pb fkqrvb il nrb l`roofoŠ pf bi plia^al k•jbol 0 pb e^`b `^bo e^`f^ ^qoŠp-S^j_f‹k bp `i^ol nrb pf crbo^ bjmrg^al e^`f^ ^qoŠprk plia^al nrb kl crbo^ bimofjbol+ mlo bgbjmil+ bi pb•^i^al `lk id* qlalp ilp plia^alp mlpqboflobp\ „g`^boŒ^k-Dpqbbgbjmil firpqo^ rk^ ifdbo^ dbkbo^ifw^`fŽkabi j‹qlal ab fkar``fŽknrb mrbab bumobp^opbbk i^ cloj^ pfdrfbkqb-

J„oj_j _` _`hjnom\^dƒi kjm di_p^^d‡i, Rb^ >&i' rk^ ^cfoj^`fŽk nrb `lk,qfbkb bi bkqbol i, Rb mrbab `lk`irfo nrb >&i' bp sboa^abol m^o^ `^a^ i ,x i* pfbp,mlpf_ib9

^( Ool_^o nrb =%hg' bp `fboq^-_( Ool_^o+nrb prmrbpq^=%e&sboa^abo^+pfbkal e rk bkqbol ^o_fqo^oflmbol

cfg^al z jG= nrb =%e * 0( bp sboa^abo^-

Dk i^ moŠ`qf`^+kH bp dbkbo^ijbkqb -fdr^i ^ 0- K^ grpqfcf`^`fŽkab bpqbj‹qlalab abjlpqo^`fŽk bp bi pfdrfbkqb qblobj^ obi^qfsl ^ k•jbolp ob^ibp-

SDNQDL@ 0-25- OQHMBHOHNCD HMCTBBHˆM L@SDL„SHB@- P`\ R pi ^jiepioj_` `io`mjn kjndodqjn lp` od`i`i g\n _jn kmjkd`_\_`n ndbpd`io`n8

^( Bg iˆh`mj 0 k`mo`i`^` \g ^jiepioj R-_( Pd pi `io`mj f k`mo`i`^` \ R+o\h]d„i f * 0 k`mo`i`^` \ R-

Bioji^`n oj_j `io`mj kjndodqj k`mo`i`^` \g ^jiepioj R-

A`hjnom\^dƒi* K^p molmfba^abp^( v _( klp af`bk nrb R bp rk `lkgrkql fk,ar`qfsl- 'U‹^pb i^ Rb``fŽk 0 2-5-( Olo `lkpfdrfbkqb R `lkqfbkb `r^inrfbo bkqbolmlpfqfsl-

Br^kal pb bcb`q•^ i^ abjlpqo^`fŽk ab rk^ ^cfoj^`f5k =%h&m^o^qlal h w 0mlo fkar``fŽk j^qbjŠqf`^+ pb ^mif`^ bi qblobj^ 0-25 ^i `lkgrkql R cloj^al mloqlalp ilp bkqbolp m^o^ilp `r^ibp i^ ^cfoj^`fŽk =%h& bp `fboq^- Rf pb abpb^ mol_^onrb >&i' bp `fboq^ pŽil m^o^ qlal i x id* bkqlk`bp pb ^mif`^ bi qblobj^ 0-20 ^i`lkgrkql ab ilp k•jbolp i m^o^ilp `r^ibp bp `fboq^>&i * i. * 0(-

") ,&+ :Y ]_V[PV]V\QRObR[N\_QR[NPVp[

G^v lqo^ molmfba^a fjmloq^kqb ab ilp bkqbolp mlpfqfslp+ii^j^a^ mofk`fmflab_rbk^ loabk^`fŽk+ nrb pb rqfifw^ q^j_f‹k `ljl _^pb m^o^ abjlpqo^`flkbp mlofkar``fŽk- Orbab clojri^opb `ljl pfdrb-

Page 63: Calculus

Bg kmdi^dkdj _` ]p`i\ jm_`i\^d‡i 21

SDNQDL@ 0-26- OQHMBHOHN CD ATDM@ NQCDM@BHˆM- Qj_j ^jiepioj ij q\+^…j_` `io`mjn kjndodqjn ^jiod`i` pij lp` `n `g h`ijm,

N_p‹osbpb nrb bi mofk`fmfl ab _rbk^ loabk^`fŽk pb obcfbob^ `lkgrkqlp abbkqbolp kjndodqjn, Di qblobj^ kl bp `fboql m^o^ `r^inrfbo `lkgrkql ab bkqbolp-Olo bgbjmil+ bi `lkgrkql ab qlalp ilp bkqbolp kl qfbkbrkl nrb pb^ bi jbklo-

Di mofk`fmfl ab _rbk^ loabk^`fŽk mrbab abar`fopb ^ m^oqfoabi mofk`fmfl abfkar``fŽk- Dpql pb abjrbpqo^ bk i^ Rb``fŽk 03-4- Blk`irfjlp bpq^Rb``fŽk `lk rkbgbjmil bk bi nrb pb jrbpqo^ `Žjl pb mrbab ^mif`^obi mofk`fmflab _rbk^ loabk^,`fŽk m^o^ mol_^o qblobj^p obcbobkqbp bkqbolp mlpfqfslp-

Rb^ =%h& i^ pfdrfbkqb fdr^ia^a9

1 1 1 i1 i0

i=%h&70 * 1 * --- * h < , * , * ,-

2 1 5

Rb l_pbos^ nrb =%0( bp `fboq^+mrbpql nrb

Rb qfbkbk+mrbp+alp mlpf_fifa^abp- N _fbk9'f( =%h&bp `fboq^ m^o^`^a^ bkqbol mlpfqfsl h) l'ff( bufpqbmlo il jbklp rk bkqbol mlpfqfsl h m^o^bi nrb =%h&bp c^ip^-Rb qo^q^ab mol_^o nrb pf pb prmlkb i^ 'ff(+ pb iibd^ ^ rk^ `lkqo^af``fŽk-

Dk sfoqra abi mofk`fmfl ab _rbk^ loabk^`fŽk bufpqfoŠ rk bkqbol mlpfqfsl e)nrb pboŠbi h`ijm bkqbol mlpfqfsl m^o^bi `r^i >&i' bp c^ip^- f e^ ab pbo j^vlonrb 0 mrbpql nrb pb e^ sfpql nrb =%0( bp sboa^abo^- Olo q^kql+ i^ fdr^ia^a e^ab pbo `fboq^ m^o^ e ,0 v^ nrb e bp bi jbklo bkqbol m^o^ bi `r^i =%e&bpc^ip^-Rb mrbab+ mrbp+bp`of_fo9

=%e Z 0(9 01 * 11 * --+* %eZ 0(1 < %e* 0(2 * %e* 0(1 * e * 0 -215

Rrj^kal f0 ^ ilp alp jfbj_olp v pfjmifcf`^kal bi ab i^ abob`e^+ pb qfbkb9

e1 e0 e01 * 11 * --- * f0 < , * , * , -

2 1 5

Obol bpq^ fdr^ia^a morb_^ nrb =%e& bp `fboq^: `lk il `r^i pb iibd^ ^ rk^ `lk,qo^af``fŽk mrbp e bo^ rk bkqbol m^o^ bi `r^i =%e&bo^ c^ip^- Bljl i^ molmlpf,`fŽk 'ff( `lkar`b ^ rk^ `lkqo^af``fŽk+ pb sbofcf`^ i^ 'f(+ il nrb morb_^ nrb i^fabkqfa^a bk `rbpqfŽk bp sŠifa^ m^o^ qlalp ilp s^ilobp ab i ƒ 0- Tk^ `lkpb`rbk,`f^ ab bpq^fabkqfa^a bp i^ abpfdr^ia^a ab i^ abob`e^ bk '0-08(-

Page 64: Calculus

-- Fiomj_p^^d‡i

Sla^ abjlpqo^`fŽk bk i^ nrb+ `ljl ^nrŒ+pb e^`b rpl abi mofk`fmflab _rbk^loabk^`fŽk+ pb mrbab prpqfqrfomlo rk^ abjlpqo^`fŽk mlo fkar``fŽk- Olo prmrbpql+nrb pb mlaŒ^e^_bo eb`el i^ abjlpqo^`fŽk bk i^ cloj^ ^`lpqrj_o^a^+ `ljmol,_^kal i^ =%0(+m^o^abpmr‹p m^p^oab i^ =%e&^ i^ =%e * 0(-

) ,&, :WR_PVPV\`

0- Cbjlpqo^o mlo fkar``fŽk i^p cŽojri^p pfdrfbkqbp9

'^( i * 1 * 2 * * h < h%h * 0(.1-

'_( 0 * 2 * 4 * * &0i + 0( < i0Š

'b( 02 * 12 * 22 * * h0 < N * 1 * 2 * --- * h&/+

'a( I2 * 12 * --- * %h * 0(2; h2-2 ; 02* 12 * --- * h1Š

1- N_p‹osbpb nrb

0 < 0 +

0 , 3 < ,N * 1( +

0,3*8<0*1*2+

0 , 3 * 8 , 05 < ,N * 1 * 2 * 3(-

Hka•w`^pb i^ ibv dbkbo^i v abjr‹pqobpb mlo fkar``fŽk-2- N_p‹osbpb nrb

i*f<1,K

i*q*q<1,f+

0 *f *0 *f <1 ,i-

ika•w`^pb i^ ibv dbkbo^i v abjr‹pqobpb mlo fkar``fŽk-3- N_p‹osbpb nrb

h,p<p+M , p(M , f( < K

M , p(M , p(M , 0( < 0•

Hka•w`^pb i^ ibv dbkbo^i v abjr‹pqobpb mlo fkar``fŽk-4- G^ii^o i^ ibv dbkbo^i nrb pfjmifcf`^ bi molar`ql

v abjr‹pqobpb mlo fkar``fŽk

Page 65: Calculus

A`hjnom\^d‡i _`g kmdi^dkdj _` ]p`i\ jm_`i\^d‡i 34

5- Rb^ =%h&i^ molmlpf`fŽk9 0 * 1 * --- * h < c%/h * 0(1-^( Ool_^o nrb pf =%e&bp `fboq^ m^o^ rk bkqbol e) =%e * 0( q^j_f‹k bp `fboq^-_( BofqŒnrbpbi^ molmlpf`fŽk ~ab i^ fkar``fŽk pb pfdrb nrb =%h& bp `fboq^ m^o^ qlal g;+b( So^kpcŽojbpb =%h& `^j_f^kal i^ fdr^ia^a mlo rk^ abpfdr^ia^a nrb bp `fboq^ m^o^qlal bkqbol mlpfqfsl i,

6- Rb^ iF bi jbklo bkqbol mlpfqfsl i m^o^ bi nrb i^ abpfdr^ia^a '0 * s'i< 0 * is )is/

bp `fboq^ m^o^ qlal s = N- B^i`ri^o kH&X abjlpqo^o nrb i^ abpfdr^ia^a bp `fboq^ m^o^qlalp ilp bkqbolp i x kf -

7- C^alp k•jbolp ob^ibp mlpfqfslp [f ) [/ ) ^^ +--- +q^ibp nrb [+} w ][i[f m^o^ qlal h 87771+alkab ` bp rk k•jbol mlpfqfsl cfgl+ miŒnrbpbbi j‹qlal ab fkar``fŽk m^o^ abjlpqo^o nrb\,9 x ^i`k,im^o^ `^a^ i <8 0-

8- Cbjr‹pqobpb mlo fkar``fŽk i^ molmlpf`fŽk pfdrfbkqb9 C^al rk pbdjbkql ab ilkdfqrarkfa^a+ bi pbdjbkql ab ilkdfqra vo+:pb mrbab `lkpqorfo `lk obdi^ v `ljmŠp m^o^ `^a^bkqbol mlpfqfsl i,

0/- Rb^ ] rk bkqbol mlpfqfsl- Cbjlpqo^o mlo fkar``fŽk i^ molmlpf`fŽk pfdrfbkqb9 O^o^ `^a^bkqbol h 8777N bufpqbk bkqbolp kl kbd^qfslp l u l q^ibp nrb9

i < k\ * m* Lxm:],

00- Rb^k i u _ bkqbolp- Rb af`b nrb _ bp rk afsfplo ab i pf i < ^_ m^o^ ^id•k bkqbol ^,Tk bkqbol i pb abkljfk^ kmdhj pf i = 0 v ilp •kf`lp afsfplobp ab i plk 0 v i, Cbjlp,qo^o mlo fkar``fŽk nrb `^a^ bkqbol i = 0 bp l mofjl l molar`ql ab mofjlp-

01- DumiŒnrbpbbi boolo bk i^ pfdrfbkqb ~abjlpqo^`fŽk‚ mlo fkar``fŽk-

Mmjkjnd^d‡i, C^al rk `lkgrkql ab i kf•^p or_f^p+ pf mlo 0/ jbklp rk^ ab i^pkf•^p qfbkb lglp ^wribp+bkqlk`bp i^p i kf•^p qfbkbk lglp ^wribp-

zA`hjnom\^dƒi~ , K^ molmlpf`fŽk bp bsfabkqbjbkqb `fboq^ pf i < 0- Di m^pl ab f^ f * 0 pb mrbab firpqo^o m^p^kal ab i < 2 ^ i < 3- RrmŽkd^pb m^o^ biil nrb i^ mol,mlpf`fŽk bp `fboq^ m^o^ i < 2 X pb^k ./% .0%.1%.2% `r^qol kf•^p or_f^p q^ibp nrb rk^ab bii^p+mlo il jbklp+ qbkd^ lglp ^wribp+mlo bgbjmil+ i^ -.$ Slj^kal }0+.0% .1%`lk,grkq^jbkqb v e^`fbkal rpl ab i^ molmlpf`fŽk `fboq^ m^o^ i < 2+obpriq^ nrb q^j_f‹k .0 v.1 qfbkbk lglp ^wribp- Qbmfqfbkal bi mol`bpl `lk ./% .0 X .2%pb bk`rbkqo^ fdr^ijbkqbnrb z2qfbkb lglp ^wribp- Dp ab`fo+ i^p `r^qol qfbkbk lglp ^wribp- Tk o^wlk^jfbkql ^kŠ,ildl mbojfqb bi m^pl ab f ^ f * 0 bk dbkbo^i-

@jmjg\mdj, Sla^p i^p kf•^p or_f^p qfbkbk lglp ^wribp-

A`hjnom\^d‡i, Orbpql nrb bcb`qfs^jbkqb bufpqb rk^ kf•^ or_f^ `lk lglp ^wribp+pbmrbab ^mif`^o bi obpriq^al mob`babkqb ^i `lkgrkql cloj^al mlo qla^p i^p kf•^p or_f^p-

Kjo\8 Dpqb bgbjmil bp ab_fal ^ F- OŽiv^+nrfbk prdfbob nrb bi ib`qlo `ljmorb_bbumbofjbkq^ijbkqb i^ s^ifabw ab i^ molmlpf`fŽk-

") ,&- 9RZ\`a_NPVp[QRY]_V[PV]V\QRObR[N\_QR[NPVp[

Dk bpq Rb``fŽk pb abar`b bi mofk`fmflab _rbk^ loabk^`fŽk abi ab fkar``fŽk-Rb^ Q rk^ `lib``fŽk kl s^`Œ^ab bkqbolp mlpfqfslp- Prbobjlp abjlpqo^o nrb

Page 66: Calculus

-/ Fiomj_p^^d‡i

P qfbkb rk k•jbol nrb bp bi jbklo+ bpql bp+nrb e^v bk P rk bkqbol mlpfqfsl o,9q^i nrb pl 99:: o m^o^ qlal o ab Q,

Rrmlkd^jlp nrb kl crbo^ ^pŒ-Cbjlpqo^objlp nrb bpql klp `lkar`b ^ rk^`lkqo^af``fŽk- Di bkqbol 0 kl mrbab mboqbkb`bo P 'ab lqol jlal ‹i pboŒ bijbklo k•jbol ab P&+Cbpfdkbjlp `lk R i^ `lib``fŽk ab qlalp ilp bkqbolp mlpf,qfslp i q^ibpnrb i ; o m^o^qlal o ab Q, Olo q^kql 0 mboqbkb`b R mlonrb 0 ; om^o^ qlal o ab Q, Rbdrfa^jbkqb+ pb^ f rk bkqbol mlpfqfsl ab R- Dkqlk`bp f ; om^o^ qlal o ab Q, Cbjlpqo^objlp nrb f * 0 q^j_f‹k bp ab R- Rf kl crbo^ ^pŒ+bkqlk`bp m^o^ rk `fboql n) ab P qbkaoŒ^jlp nc7788e * 0- Orbpql nrb P kl mlpbbk•jbol jŒkfjl+ e^v rk bkqbol o0 bk Q q^i nrb o0 ; od%X mlo q^kql o0 ; f * 0-Obol bpql pfdkfcf`^ nrb o0 8899 e) bk `lkqo^af``fŽk `lk bi eb`el ab nrb e ; o m^o^qlal o ab P+ Olo q^kql e * 0 mboqbkb`b^ R- Rbd•k bi mofk`fmfl ab fkar``fŽk+R `lkqfbkb qlalp ilp bkqbolp mlpfqfslp- Orbpql nrb Q bp kl s^`Œl+bufpqbrk bkqbolmlpfqfsl o bk P+Obol bpqbo ab_b pbo q^j_f‹k ab R 'v^ nrb R `lkqfbkb qlalp ilpbkqbolp mlpfqfslp(- Cb i^ abcfkf`fŽk ab R obpriq^ nrb o ; o*il `r^i bp ^_proal- Olo`lkpfdrfbkqb+ i^ efmŽqbpfpab nrb Q kl mlpbb rk k•jbol jŒkfjl klp iibs^ ^ rk^`lkqo^af``fŽk- Qbpriq^ mrbp nrb Q ab_b qbkbo rk k•jbol jŒkfjl+ v ^ pr sbwbpql morb_^ nrb bi mofk`fmfl ab _rbk^ loabk^`fŽk bp rk^ `lkpb`rbk`f^ abi abfkar``fŽk-

) ,&. :Y `oZO\Y\ `bZNa\_V\

Dk bi `Ši`ril abi Šob^ ab rk pbdjbkql m^o^_Žif`l+^m^ob`bi^ prj^

'0-1/( 01 * 11 * 21 * --- * i0 Š

N_p‹osbpb nrb bi q‹ojfkl dbkbo^i ab bpq^ prj^ bp ab i^ cloj^ f0 v pb l_qfbkb`^a^ rkl ab ilp prj^kalp a^kal ^ f ilp s^ilobp 0+1+2+ --- +i, Dufpqbrk pŒj_liljrv •qfi v `lksbkfbkqb nrb mbojfqb bp`of_fo prj^p bk cloj^ ^_obsf^a^ abkljf,k^al n…h]jgj nph\ojmdj v nrb `lkpfpqb bk i^ ibqo^dofbd^ I, Tqfifw^kal bi pŒj_lilprj^qlofl pb mrbab bp`of_foi^ prj^ '0-1/( `ljl pfdrb9

DpqbpŒj_lil pb ibb9 ~Rrj^ ab f0 abpab 0 e^pq^ h|) Di `lksbkfl bp nrb ilp k•,jbolp nrb ^m^ob`bk bk`fj^ v ab_^gl ab H fkaf`^k bi ob`loofal ab ilp s^ilobpab f, K^ ibqo^ f pb `lkpfabo^ `ljl bi …i_d^`_` nph\^d‡i, Dp bsfabkqb nrb klbp kb`bp^ofl rqfifw^omob`fp^jbkqb i^ ibqo^e) pfkl nrb pb mrbab qlj^o bk pr ird^olqo^ ibqo^ `r^inrfbo^- Olo bgbjmil+ bk sbw ab Ixxg f0 pb mrbab bp`of_foIxg d%*I5xgm*I98%x/h!* bq`-+qla^p i^p `r^ibp plk afpqfkq^pklq^`flkbp m^o^ rk^ jfpj^`lp^- K^p ibqo^pd*d) f* h* bq`-+nrb pb rqfifw^k ^i bcb`ql+ pb abkljfk^k …i_d^`n,Ml pboŒ^`boq^al rqfifw^oi^ ibqo^i m^o^bi Œkaf`bbk bpqbbgbjmil m^oqf`ri^o+mrbpi fkaf`^ v^ bi k•jbol ab q‹ojfklp-

Page 67: Calculus

Bg n…h]jgj nph\ojmdj -0

LŠp dbkbo^i+ `r^kal pb abpb^ cloj^o i^ prj^ ab `fboqlp k•jbolp ob^ibp6> 6'" $$$ " [y7

'0-10 (

rqfifw^kal bi pŒj_lil prj^qlofl pb bp`of_b ^_obsf^a^jbkqb9

'H-11(

Olo bgbjmil9

3

x \f < \g * \0 * \1 * \2 *FRG

4

w rx < Wi * T/ * T0 * T1 * T2 +f<i

@idrk^p sb`bp bp `lksbkfbkqb bjmbw^o i^ prj^`fŽk mlo bi N l mlo ^id•ks^ilo abi Œkaf`b afcbobkqb ab 0- Olo bgbjmil+ pb qfbkb9

3

x s* < Tj * Wi * T/ * T0 * T1 )fzN

4

x i1 < 12 * 22 * 32 * 42-

i;0

Nqo^p cloj^p ab rqfifw^o bi pŒj_lil ab prj^`fŽk+ pb fkaf`^k ^ `lkqfkr^`fŽk9

3

w rh(. < T * r/ * r0 * r1 * r2)Z4B

5

z 1e,0 < 0 * 1 * 11 * 12 * 13 * 14

-ERG

O^o^ mlkbo ab j^kfcfbpql rk^ sbw jŠp nrb i^ bib``fŽk abi Œkaf`b `^ob`b ab fj,mloq^k`f^+ pb l_pbos^ nrb i^ •iqfj^ prj^ pb mrbab bp`of_fo bk `^a^ rk^ ab i^pcloj^p pfdrfbkqbp-

544 5

z1n,0 < I 0l < I03+

h < I 04+

nzi lwK hwK hzi

Page 68: Calculus

-1 / iomj_p^^d‡i

Kjo\8 Cbpab rk mrkql ab sfpq^ bpqof`q^jbkqb iŽdf`l+ ilp pŒj_lilp bk '0-10( v'0-11( kl pb bk`rbkqo^k bkqob ilp ^uflj^p abi pfpqbj^ ab k•jbolp ob^ibp+v mlo il q^kqlabpab rk mrkql ab sfpq^ ofdrolpl+ pb qbkaoŒ^knrb abcfkfo bpqlp krbslp pŒj_lilp ^ m^oqfoab ilp pŒj_lilp mofjfqfslp abi pfpqbj^ `lkpfabo^al- Dpql pb `lkpfdrb jbaf^kqb rk^ _`ad+id^d‡i kjm di_p^^d‡i* i^ `r^i+ `ljl i^ abjlpqo^`fŽk mlo fkar``fŽk+ `lkpq^ ab alp m^oqbp9

^( Rb abcfkb

_( Rrmrbpq^ abcfkfa^ Jyyh[e m^o^ rk h w 0 cfgl+ pb abcfkb9

Olo bgbjmil+ pb mrbab qlj^o i < 0 bk _( v e^`bo rpl ab ^( m^o^ l_qbkbo9

1 0

H[e < H[e * [/ < [f * [/ +hzi h<i

Cbcfkfa^ Kz<i[e ) pb mrbab ^mif`^o lqo^ sbw _( `lk h < 1 m^o^ l_qbkbo

2 1

I [e < I [e * [0 < %[f * [/& * [0 +hzi h<i

Dk sfoqra ab i^ molmfba^a ^pl`f^qfs^ ab i^ ^af`fŽk '^uflj^ 1(+ i^ prj^ %[f * [/& * [0bp i^ jfpj^ nrb [f * %[/ * [0& v+ mlo q^kql+ pb mrbabk prmofjfo ilp m^o‹kqbpfppfk mbifdolab `lkcrpfŽk v bp`of_fo pfjmibjbkqb [f * [/ * [0 m^o^Jyyh[e + @kŠild^jbkqb9

- ,

H[e < H[e * [1 < %[f * [/ * [0& * [1 †h<i h<i

Dk bpqb `^pl pb mrbab _`hjnom\m nrb i^ prj^ %[f * [/ * [0& * [1 bp i^ jfpj^ nrb%[f * [/& * %[0 * [1& X nrb [f * %[/ * [0 * [1&) v mlo q^kql pb mrbabk prmofjfo ilpm^o‹kqbpfp q^j_f‹k pfk mbifdol ab ^j_fd•ba^a v bp`of_fo9

3H[e < [f * [/ * [0 * [1 +Q5R

Oolpfdrfbkal ^pŒ+pb bk`rbkqo^ nrb ^( v _( pfjriqŠkb^jbkqb a^k rk^ abcfkf`fŽk `lj,mibq^ abi pŒj_lil bp`ofql bk '0-11(- Rb `lkpfabo^ nrb i^ klq^`fŽk '0-10( bp jŠp _fbklqo^ cloj^ ab bp`of_fo '0-11(- S^i klq^`fŽk bpqŠ grpqfcf`^a^ mlo i^ ibv ^pl`f^qfs^ dbkbo^iab i^ ^af`fŽk+ nrb ^nrŒ kl pb bkrk`f^oŠ `lk jŠp abq^iib kf abjlpqo^oŠ-MŽqbpb nrb i^ _`adid^d‡i kjm di_p^^d‡i v i^ _`hjnom\^d‡i kjm di_p^^d‡i bk`fboo^k i^jfpj^ fab^ crka^jbkq^i- Tk^ abcfkf`fŽk mlo fkar``fŽk pb abkljfk^ q^j_f‹k _`adid^d‡ikjm m`^pmm`i^d\,

Page 69: Calculus

Bd`m^d^djn -2

) ,&/ :WR_PVPV\`

0- G^ii^o ilp s^ilobp krj‹of`lp ab i^p prj^p pfdrfbkqbp9

3

'^( I e)e:f

2

'b( H00Q( )K5:

4

'b( H'1f * 0(+Œ<N

3

'a( 1 ii*I5G

. 0

'b(1 e%e * 0(!e:f

1- Dpq^_ib`bo i^p pfdrfbkqbp molmfba^abp abi pŒj_lil prj^qlofl-

i i i

'^( 1 %[e * ]e& < 1 [e * 1 ]ee:f h<i h<i

'molmfba^a ^afqfs^(-

i i

'_( 1 %][e& < b1 [eh<i e:f

'molmfba^a eljld‹kb^(-

i'b( 1 %[e * [e*f& < [i + ^l

h<i'molmfba^a qbibp`Žmf`^(-

TqfiŒ`bkpbi^p molmfba^abpa^a^p bk bi Dgbo`f`fl 1+ pfbjmob nrb pb^ mlpf_ib+ m^o^ abar`foi^p cŽojri^p bk ilp Dgbo`f`flp abi 2 ^i 7-

i

1, H0 < h+ 'Di pbkqfal ab bpq^ prj^ bpJyyh\fg* `r^kal [e < 0-(h<i

h

2, I&0f + 0( < i0Šf;g

XFi_d^\^d4i, 0f + 0 < h1 , &f + 0(1-\

XFi_d^\^d4i, ‰pbpbbi Dgbo`f`fl 2 v bi 3\-

x i1 i0 i4, I,*, f

0< &2* 1! * ': -

f;g

Xgi_d^\^d4i, f1 + &f + 0(2 < 1f0 + 1f * 0-[

i i2 i1 i0

6 !&! e0 < , * , * ,%I,*, 3 1 3&f;g

}i 0 ] si)/

0& N! ?9 4 %%%%0 , U

Q5>

pf s9* *0- Kjo\8 Olo abcfkf`fŽk UL < 0-

Page 70: Calculus

4/ / iomj_p^^d‡i

XFi_d^\^d‡i, @miŒnrbpbbi bgbo`f`fl 1 ^ 'i , s' Kz<l sf,Z

_( ƒBrŠi bp i^ prj^ `r^kal s < 0>

8- Cbjlpqo^o mlo fkar``fŽk+ nrb i^ prj^ $zi ' *f&e%/e* 0( bp molmlo`flk^i ^ i* u e^ii^oi^ `lkpq^kqb ab molmlo`flk^ifa^a-

0/- ^( C^o rk^ abcfkf`fŽk o^wlk^_ib abi pŒj_lil HV$ $7[e +

_( Cbjlpqo^o mlo fkar``fŽk nrb m^o^ i x 0 pb qfbkb

00- Cbqbojfk^o pf `^a^ rk^ ab i^p fdr^ia^abp pfdrfbkqbp bp `fboq^ l c^ip^- Dk `^a^ `^pl o^,wlk^o i^ ab`fpfŽk-

0// 0//

'^( J i2 < J i2Š

;.0 ;.:

0// 88

'a( J %c* 0(1 < Id0Š8.: V4B

0//

'_( J 1 < 1//-0</

0// 0//

'b( J '1 * e& < 1 * J e+Q5> Q5>

01- Hkar`fo v abjlpqo^o rk^ obdi^ dbkbo^i nrb pfjmifcfnrb i^ prj^

i 0

If&f * 0(&f;g

02- Cbjlpqo^o nrb 1's:*0 , Rw&; y ; 1'z , Si + 0( pf i x 0- Tqi.o`bpb irb,

dl bpqb obpriq^al m^o^ abjlpqo^o nrb

H 01v&: , 1 ; 5z ; 1v&:::, 0

pf h x 1- Dk m^oqf`ri^o+ `r^kal h < 0/5+ i^ prj^ bpqŠ `ljmobkafa^ bkqob0887 u 0888-

) ,&0 HNY\_NO`\Yba\e QR`VTbNYQNQa_VN[TbYN_

Dp cob`rbkqb bk bi `Ši`ril bi qbkbo nrb lmbo^o `lk abpfdr^ia^abp- Rlk abm^oqf`ri^o fjmloq^k`f^ i^p nrb pb obi^`flk^k `lk i^ kl`fŽk ab q\gjm \]njgpoj,

Page 71: Calculus

S\gjm \]njgpoj u _`ndbp\g_\_ omd\ibpg\m 40

Rf s bp rk k•jbol ob^i+ bi s^ilo ^_plirql ab s bp rk k•jbol ob^i kl kbd^qfslnrb pb abpfdk^ mlo Gthv pb abcfkb `ljl pfdrb9

Yt[ < vU'[

pf s x N+

pf s x l-

N_p‹osbpb nrb ,Gth y s x GtG-Rf ilp k•jbolp ob^ibp bpqŠk obmobpbkq^alp dblj‹,qof`^jbkqb bk bi bgb ob^i+ bi k•jbol Gthpb abkljfk^ afpq^k`f^ ab s ^ N- Rf \ = Nu pf rk mrkql s bpqŠ pfqr^al bkqob +\ v \ bkqlk`bp GthbpqŠ jŠp moŽufjl ^ Mnrb \, K^ bumobpfŽk ^k^iŒqf`^ab bpqb eb`el+ bpqŠ a^a^ mlo bi pfdrfbkqb qblobj^9

RCMPCK? 0-27- Rf \ x N+ `n Gthy \ pf u n‡gj pf +\ x s x \,

A`hjnom\^d‡i, G^v nrb mol_^o alp `rbpqflkbp9 mofjbol+ nrb i^ abpfdr^ia^aGthy \ fjmif`^ i^p alp abpfdr^ia^abp , \ x s x \ v ob`Œmol`^jbkqb+ nrb+ \ x s x \ fjmif`^ Gth49 \,

Rrmrbpql Gth49 [ pb qfbkb q^j_f‹k , [ 49, GtG-Obol l r < Gthl r < , Erfv+ mlo q^kql+ , [ w * Gth49r 49Er 49 [) 0/ `r^i morb_^ i^ mofjbo^ m^oqb abiqblobj^-

O^o^ mol_^o bi ob`Œmol`l+ prmŽkd^pb , \ 49s 49 \, Rf s x N pb qfbkbEr < r 49[8 pf mlo bi `lkqo^ofl bp r w N+ bkqlk`bp Gth< , r27 [+ Dk ^j_lp`^plp pb qfbkb Gthy \* il nrb abjrbpqo^ bi qblobj^-

K^ cfdro^ 0-8- firpqo^ bi+pfdkfcf`^al dblj‹qof`l ab bpqb qblobj^-

9R \ bk bpqbfkqbos^il

'H l \

EHFTQ@ 0-8 Pdbidad^\_j b`jh„omd^j _`g o`jm`h\ /,16,

Blkpb`rbk`f^ abi qblobj^ 0-27+ bp rk^ abpfdr^ia^a fjmloq^kqb nrb bumobp^nrb bi s^ilo ^_plirql ab i^ prj^ ab alp k•jbolp ob^ibp kl mrbab bu`babo ^ i^prj^ ab prp s^lobp ^_plirqlp-

RCMPCK? 0-28- M\m\ s ` u iˆh`mjn m`\g`n ^p\g`nlpd`m\ n` od`i`

Gt* uh49 Yt[ * Esf +

Page 72: Calculus

41 Fiomj_p^^d‡i

Kjo\8 Dpq^ molmfba^a pb abkljfk^ _`ndbp\g_\_ omd\ibpg\m* mrbp `r^kal pb dbkb,o^ifw^ ^ sb`qlobp+ fkaf`^ nrb i^ ilkdfqra ab `^a^ i^al ab rk qofŠkdril bp jbklo l fdr^inrb i^ prj^ ab i^p ilkdfqrabp ab ilp lqolp alp-

A`hjnom\^d‡i, Orbpql nrb s < Gthl s < , Gth+pb qfbkb , Gth9999::s 888899GtG,@kŠild^jbkqb , Guh9999::u <9:Guh-Rrj^kal ^j_^p abpfdr^ia^abp pb qfbkb9

*%Erf* Guh(99999:r * s777778Yt[ * Esf)

u mlo q^kql+ bk sfoqra abi qblobj^ 0-27 pb `lk`irvb nrb9 Gt * uh 9999::Gth* Guh-Slj^kal s < \ + `* b v < ` + ]* bp s * v < \ + ] X i^ abpfdr^ia^a

qof^kdri^o qlj^ i^ cloj^9

g\ + ]g 888899g\ + a[ * h^ , a[ -

Dpq^cloj^ ab i^ abpfdr^ia^a qof^kdri^o pb rqfifw^ cob`rbkqbjbkqb bk i^ moŠ`qf`^-Olo fkar``fŽk j^qbjŠqf`^+ pb mrbab buqbkabo i^ abpfdr^ia^a qof^kdri^o q^i

`ljl pfdrb9

SDNQDL@ 0-3/- Pd \y* \0* ŠŠŠ * \z nji iˆh`mjn m`\g`n ^p\g`nlpd`m\

A`hjnom\^d‡i, O^o^ i < 0 i^ abpfdr^ia^a bp qofsf^i v m^o^ i < 1 bp i^abpfdr^ia^a qof^kdri^o- Rrmrbpq^ `fboq^ m^o^ i k•jbolp ob^ibp+m^o^ i * 0 k•,jbolp ob^ibp \g%\0* ŠŠŠ * \i)g pb qfbkb9

Olo q^kql+bi qblobj^ bp `fboql m^o^ h * 0 k•jbolp pf il bp m^o^ h8 irbdl+ bksfoqra abi mofk`fmfl ab fkar``fŽk+ bp `fboql m^o^ qlal bkqbol mlpfqfsl i,

Di qblobj^ nrb pfdrb `lkpfpqb bk rk^ abpfdr^ia^a fjmloq^kqb nrb pb rqfifw^oŠjŠp ^abi^kqb bk „idb_o^ sb`qlof^i-

SDNQDL@ 0-30- CDRHFT@KC@C CD B@TBGX,RBGV@QY- Pd \,* ,,, * ^! V]+*, , , Š ]9 nji iˆh`mjn m`\g`n ^p\g`nlpd`m\* n` od`i`

'0-12(

Bg ndbij _` dbp\g_\_ `n qƒgd_j nd v n‡gj nd c\t pi iˆh`mj m`\g s o\g lp`\ns * ]*< N k\m\ ^\_\ q\gjm _` f < 0+ 1+ ---- i,

Page 73: Calculus

?e`m^d^djn 42

A`hjnom\^d‡i, O^o^ qlal ob^i s pb qfbkb HYzi&\fs * ]‰0x N mlonrb rk^

prj^ ab `r^ao^alp krk`^ bp kbd^qfs^- Dpql pb mrbab mlkbo bk i^ cloj^

'0-13(

alkabh

> ;F\x*F5G

>s/ * 0?s * B z N +

Prbobjlp abjlpqo^o nrb >/77788 =?+ Rf = < N+`^a^ [e < N+`lk il nrb > < NX bi obpriq^al bp qofsf^i-Rf = ", N+mlabjlp `ljmibq^o bi `r^ao^al u bp`of_fo

&?'0 =? Z ?0

>s/ * 0?s * B < > s * > * >

Di pbdrkal jfbj_ol ^i`^kw^ pr s^ilo jŒkfjl `r^kal r < , ?F>, Olkfbkals ;+?e> bk '0-13(+ l_qbkbjlp ?0

88899>@, Dpql abjrbpqo^ '0-12(- Di ib`qlo ab_b`ljmol_^o nrb bi pfdkl ab fdr^ia^a bp sŠifal pf u pŽil pf bufpqbrk r q^i nrb\j8 * ]d, < N m^o^ `^a^ f,

0 ,3-8 Dgbo`f`flp

0- Ool_^o `^a^ rk^ ab i^p pfdrfbkqbp molmfba^abp abi s^ilo ^_plirql-

'^( Erf < N pf u pŽil pf r < N-

'_( E*rf < GtG-'b( Er * tg < Gu, u\-'a( Erf0 < r0

Š

'b( Erf < pr0Š

'b( Gtuh< xrffsf+

'c( cr,sf < Erf,fsf pfs $! N-'e( Er * uhP Gth* Guh-'f( Erf * Esf O Gt , ue•'g( Efrf * GuhhP Gt , uh-

1- B^a^ abpfdr^ia^a &\d'%ab i^p bp`ofq^p ^ `lkqfkr^`fŽk+ bnrfs^ib bu^`q^jbkqb ^ rk^ abp,fdr^ia^a %\d&+ Olo bgbjmil+ Erf ; 2 pf v pŽil pf ,2 ; r ; 2 v mlo q^kql &\g' bp bnrfs^,ibkqb ^ %\/&+ Cbqbojfk^o qlalp ilp m^obp bnrfs^ibkqbp-

%[f& Erf ; 2- %\/' 3 ; r ; 5-%[0' Gt, 00 ; 2- %\0' ,2 ; r ; 2-%[[& 02 , /rf ; 0- '_^( r; 1 l r ; ,0-%[1& 00 * /rf 99p: 0- %\1& r; /+%[m& Er * 00 = 1- %\3' ,1 ; r ; 3-%[m& Gt* 10 z 4- '_p( *p0mr77O8*. l 0 Or77O8 r2-%[4& 04 , t,hh ; 0- %\4& 0 ; r ; 1-%[m& Er * 40 ; Er * 00- '_p( r O*4 l r w 2-%[6& Er0 + 10 OE+ %\6& f, ; r ; p•%[fK& r ; r0 + 01 ; 1r+ %\gL' ,0 pr O N-

Page 74: Calculus

43 Fiomj_p^^d‡i

2- Cb`fafo pf `^a^ rk^ ab i^p pfdrfbkqbp ^cfoj^`flkbp bp `fboq^ l c^ip^- Dk `^a^ `^pl o^wlk^oi^ ab`fpfŽk-

'^( s ; 4 fjmif`^ Zu\ ; 4-'_( Zu , 40 -' 1 fjmif`^ 2 ; s ; 6-

'b( 00 * 1sg x 0 fjmif`^ s ƒ ,z-'a( Ml bufpqb k•jbol ob^i s m^o^ bi nrb Zu , 00 < Zu , 10-'b( O^o^ qlal s< N bufpqb rk v= N q^i nrb 01u * uh< 4-

3- Cbjlpqo^o nrb bi pfdkl ab fdr^ia^a bp sŠifal bk i^ abpfdr^ia^a ab B^r`ev,R`et^ow pfv pŽil pf bufpqb rk k•jbol ob^i s q^i nrb [er * ]e </ m^o^ qlal f < 0+1+--- +i*

") ,&)( :WR_PVPV\`cN_V\`_RSR_R[aR`NYZna\Q\ QRV[QbPPVp[

Dk bpqb ^m^oq^al pb ob•kbk rk `lkgrkql ab obpriq^alp afsboplp `rv^p abjlpqo^`flkbpplk _rbklp bgbo`f`flp ab ^mif`^`fŽk abi j‹qlal ab fkar``fŽk- @idrklp ab bpqlp bgbo`f`flpmrbabk pbosfo ab _^pb ab afp`rpfŽk v bpqrafl bkqob ilp ^irjklp v bi molcbplo-

@j`ad^d`io` a\^ojmd\g v ]dijhd\g, Di pŒj_lil i 'nrb pb ibb i a\^ojmd\g' pb mrbab abcfkfomlo fkar``fŽk `ljl pfdrb9 N < 0+i < &i + 0( i pf i ƒ 0-

N_p‹osbpb nrb i < 0 - 1 - 2 --- i,

Rf N z f -p i bi ^j`ad^d`io` ]dijhd\g E( pb abcfkb mlo

&i' i f ;f &i+f' %

Kjo\8 @idrk^p sb`bp pb bp`of_b i@f bk sbw ab E( - Dpqlp k•jbolp ^m^ob`bk `ljl`lbcf`fbkqbp bk i^ cŽojri^ ab i^ mlqbk`f^ abi _fkljfl- 'U‹^pb bi Dgbo`f`fl 3 pfdrfbkqb-(

0- B^i`•ibkpb ilp s^ilobp ab ilp pfdrfbkqbp `lbcf`fbkqbp _fkljf^ibp9

'^( 'f(+ '_( 'j+ 'b( 'B+ 'a( 'B+ 'b( 'i+ 'b( 'd(-

1- '^( Cbjlpqo^o nrb9 E( < &ix f'%

'_( R^_fbkal nrb ' x{{( < ;:( `^i`ri^o T(

'a( R^_fbkal nrb _@< &fx2' `^i`ri^o e+

'a( ƒDufpqb rk f q^i nrb _7&< '.z2(>

2- Cbjlpqo^o nrb %hd8f&< %ewf&* F(- Dpq^ molmfba^a pb abkljfk^ a‡mhpg\ \_d+odq\ ab ilp `lbcf`fbkqbp `lj_fk^qloflp l g`t _`g omdƒibpgj _` M\n^\g v molmlo`flk^ rkj‹qlal oŠmfal m^o^ `^i`ri^o pr`bpfs^jbkqb ilp `lbcf`fbkqbp _fkljf^ibp- @ `lkqfkr^`fŽk pba^ bi qofŠkdril ab O^p`^i m^o^ i x 5-

Page 75: Calculus

Be`m^d^djn q\mdjn m`a`m`io`n \g h„oj_j _` di_p^^d‡i 44

10 2

0 3 50 4 0/

0 5 04 1/

2 03 0

0/ 4 004 5 0

3- Cbjr‹pqobpb mlo fkar``fŽk i^ cŽojri^ ab i^ mlqbk`f^ abi _fkljfl9

u rqfiŒ`bpbbi qblobj^ m^o^ abar`fo i^p cŽojri^p9

v pf i = N-

P…h]jgj kmj_p^oj, Di molar`ql ab i k•jbolp ob^ibp

pŒj_lil SHzz0[e$ nrb pb mrbab abcfkfo mlo fkar``fŽk-

cloj^ ab bp`of_fo bpqb molar`ql- N_p‹osbpb nrb9

]h= \0* 9 ŠŠ * \i pb fkaf`^ mlo bi

Di pŒj_lil [.[/$ ++ [i bp lqo^

4- C^o rk^ abcfkf`fŽk mlo fkar``fŽk abi molar`ql SH:<0\fŠ

Cbjlpqo^o mlo fkar``fŽk i^p pfdrfbkqbp molmfba^abp ab ilp molar`qlp9

5- j %[e]e& < 'SH[e& 'j ]e& 'molmfba^a jriqfmif`^qfs^(-

Tk `^pl fjmloq^kqb bp i^ obi^`fŽk9 SH9<0%][e& < _h SHh<0[e$

6- pf `^a^ [e8+` N 'molmfba^a qbibp`Žmf`^(-

7- Rf s ::‹ 0+ abjlpqo^o nrb9 ,,,,0 +s ,

ƒBrŠi bp bi s^ilo abi molar`ql `r^kal s < 0>

Page 76: Calculus

45 Fiomj_p^^d‡i

8- Rf \f ; ]f m^o^ `^a^ s^ilo ab e < 0+1+--- +i * bp cŠ`fi abjlpqo^o mlo fkar``fŽk

Hz<0\f ; Hz<0]f,Cfp`rqfo i^ abpfdr^ia^a `loobpmlkafbkqb m^o^ molar`qlp9

! !QF\f :i]f%f;/ f;/

>gbpi\n _`ndbp\g_\_`n ijo\]g`n

0/- Rf s 9~ 0+ abjlpqo^o mlo fkar``fŽk nrb s! = s m^o^ `^a^ i x 1- Rf N ; s ; 0+ abjlp,qo^o nrb s! ; s m^o^ `^a^ i x 1-

00- CbqbojŒkbkpbqlalp ilp bkqbolp mlpfqfslp h m^o^ilp `r^ibp 1! ; h 01- '^( Blk bi qblobj^ abi _fkljfl abjlpqo^o nrb m^o^ i bkqbol mlpfqfsl pb qfbkb

'_( Rf i = 0+^miŒnrbpbi^ m^oqb'^( v bi Dgbo`f`fl 00 m^o^ abar`fo i^p abpfdr^ia^abp

' 0(! z 01 ; 0 * y ; 0 * I* e ; 2-Q'R

02- '^( Rb^ j rk bkqbol mlpfqfsl- Cbjlpqo^o nrb9

XFi_d^\^d‡i, @miŒnrbpbi^ molmfba^a qbibp`Žmf`^m^o^ i^p prj^p-\

'_( Rf k X i plk bkqbolp mlpfqfslp+abjlpqo^o nrb

%h * 0(0=*0, h.;(.hM ; j * 0 ; %h* /'M 9

@mif`^kal bi ^m^oq^al '^(

'b( Cbjr‹pqobpb mlo fkar``fŽk nrb9

!,0 0=*0!•fM :[i[ ,w,e:f k * 0 e*f

Di ^m^oq^al '_( ^vra^oŠ ^ m^p^obk i^ fkar``fŽk ab i ^ i * 0-03- Rb^k \g!%! \! i k•jbolp ob^ibp+qlalp abi jfpjl pfdkl v qlalp j^vlobp nrb , 0-

@mif`^obi j‹qlal ab fkar``fŽk m^o^ abjlpqo^o nrb9

Page 77: Calculus

Be`m^d^djn q\mdjn m`a`m`io`n \g h„oj_j _` di_p^^d‡i .0

Dk m^oqf`ri^o+ `r^kal \g < \0 < --- < \i < s* alkab s = , 0+ pb qo^kpcloj^ bk9

'0-14( 'i * u(! ƒ 0 * is &_`ndbp\g_\_ _` ?`mijpggd',

Ool_^o nrb pf i = 0 bi pfdkl ab fdr^ia^a pb mobpbkq^bk '0-14( pŽil m^o^ s < N-04- Rf i x 1+ abjlpqo^o nrb i%Xi! x ' (Z pfbkal f i^ m^oqbbkqbo^ ab i-0,05- Klp k•jbolp 0+1+ 2+ 4+ 7+ 02+10+ --- q^ibp nrb `^a^ rkl abpmr‹p abi pbdrkal bp i^ prj^

ab ilp alp ^kqboflobp+pb abkljfk^k iˆh`mjn _` Md]ji\^^d, Rb mrbabk abcfkfo mlo fkar`,`fŽk `ljl pfdrb9

pf i ƒ 1-Cbjlpqo^o nrb

m^o^ `^a^ i ƒ i-

A`ndbp\g_\_`n lp` m`g\^dji\i _dnodiojn odkjn _` kmjh`_djn,Rb^k Ug% U0 * ŠŠŠ * Uh i k•jbolp ob^ibp mlpfqfslp- Rf k bp rk bkqbol kl kril+ i^ h`_d\ _`

kjo`i^d\n k+„ndh\n J• pb abcfkb `bokl pfdrb9

Di k•jbol IE pb abkljfk^ h`_d\ \mdoh„od^\*J/ h`_d\ ^p\_mƒod^\* v J[f h`_d\ \m+h‡id^\,

06- Rf k = N abjlpqo^o nrb I} ; I0M `r^kal Ug% U0* ŠŠŠ * Ui kl plk qlalp fdr^ibp-

XFi_d^\^d‡i, @mif`^o i^ abpfdr^ia^a ab B^r`ev,R`et^ow `lk \f < s$ v ]f < )&I

07- @miŒnrbpbbi obpriq^al abi Dgbo`f`fl 06 m^o^ abjlpqo^o nrb

pf \0 * \0 * ^0 < 7 t\< N+ \ = N+b = N-08- Rb^k [f!$! Mh h k•jbolp ob^ibp mlpfqfslp `rvl molar`ql bp fdr^i ^ 0- Cbjlpqo^o

nrb [f * --- * [i ƒ h v nrb bi pfdkl ab fdr^ia^a pb mobpbkq^ pŽil `r^kal `^a^\f

< 0-

XFi_d^\^d‡i, Blkpfa‹obkpb alp `^plp9 ^( `^a^ \f < 0: _( kl qlal \f < 0- Ool`‹a^pbmlo fkar``fŽk- Dk bi `^pl _( l_p‹osbpb nrb pf \/\0%! \i) < 0+bkqlk`bp mlo il jbklprk c^`qlo+ mlo bgbjmil \g %bp j^vlo nrb 0+v mlo il jbklp rk c^`qlo+ pb^ \i+og*`n jbklonrb i- GŠd^pb ]. < \g\i) X ^miŒnrbpbi^ efmŽqbpfpab fkar``fŽk ^i molar`ql ]g\0 ,,, \ij

qbkfbkal bk `rbkq^ nrb &\g + /'&\i+og + 0( ; N-\

Page 78: Calculus

36 Fiomj_p^^d‡i

1/- K^ h`_d\ b`jh„omd^\ F ab i k•jbolp ob^ibp mlpfqfslp Ug%ŠŠŠ *si bpqŠ abcfkfa^ mlo i^cŽojri^ F < &UFU0%,, si'g-i,

'^( CbpŒdkbpb`lk J• i^ h`_d\ _` kjo`i^d\n k+„ndh\n, Cbjlpqo^o nrb F94 IE X nrbF < &rbhpŽil `r^kal Ug < U0 < --- < sh†

'_( Rb^k k v l bkqbolp+ l ; N ; k, @ m^oqfo ab '^( abar`fo nrb I} ; C ; I} pfUg* U* ,,, * U! kl plk qlalp fdr^ibp-

10- @miŒnrbkpbilp obpriq^alp abi Dgbo`f`fl 1/ m^o^ mol_^o i^ pfdrfbkqb molmlpf`fŽk9 Rf \* ]*v b plk k•jbolp ob^ibp v mlpfqfslp q^ibp nrb \]` < 7+ bkqlk`bp \ * ] * b ƒ 5 X

[\ * [_ * \_ ƒ 01-11- Rf Ug%ŠŠ, * rh plk k•jbolp mlpfqfslp v pf Ue < f`re$ abjlpqo^o nrb

12- Rf [) \) v b plk mlpfqfslp v pf RH [ * \ * b < 0+abjlpqo^o nrb '0 , [&%f * \&%f * b( ƒƒ 5[\_+

Page 79: Calculus

0

<?C 5?>57@D?C

67< 5[<5E<? :>D79B3<

Dk bpqb `^mŒqril pb bumlkb i^ abcfkf`fŽk ab fkqbdo^i v ^idrk^p ab prp molmfb,a^abp crka^jbkq^ibp- O^o^ bkqbkabo i^ abcfkf`fŽk+ bp kb`bp^ofl qbkbo `lkl`fjfbkqlabi `lk`bmql ab crk`fŽk: mlo bifl pb abaf`^k ^idrk^p ab i^p pb``flkbp nrb pfdrbk^ i^ bumif`^`fŽk ab bpqb `lk`bmql v ab lqolp obi^`flk^alp `lk ‹i-

)&) ?N` VQRNOm`VPNQRYN<R\ZRa_oNPN_aR`VN[N

Bljl pb e^ af`el ^kqboflojbkqb+ rk^ ab i^p ^mif`^`flkbp ab i^ fkqbdo^i bp i^clojri^`fŽk abi `lk`bmql ab Šob^- Noafk^of^jbkqb kl pb e^_i^ abi Šob^ bk pŒ+pfklabi Šob^ _` \gbj* il nrb fkaf`^ nrb pb m^oqbab `fboqlp l_gbqlp 'obdflkbp mlifdlk^ibp+obdflkbp `fo`ri^obp+ pbdjbkqlp m^o^_Žif`lp+ bq`-(+ `rv^p Šob^p pb abpb^k jbafo-Rf pb abpb^ iibd^o ^ rk^ abcfkf`fŽk ab Šob^ ^mif`^_ib ^ `i^pbp afpqfkq^p ab l_gbqlp+mofjbo^jbkqb pb ab_boŠ bk`lkqo^o rk `^jfkl bcb`qfsl m^o^ abp`of_fo bpqlp l_gbqlp-

Di j‹qlal jŠp pfjmib ab mob`fp^o af`elp l_gbqlp crb bi ab af_rg^oilp+ q^i`ljl ef`fbolk ilp dofbdlp- Tk `^jfkl jr`el jbglo crb prdbofal mlo Qbk‹ Cbp,`^oqbp '0485,054/( ^i bpq^_ib`bo bk 0526 i^ _^pb ab i^ FbljbqoŒ^ ^k^iŒqf`^-K^ `lirjk^ sboqb_o^i ab i^ dbljbqoŒ^ ab Cbp`^oqbp '`lkl`fa^ ^`qr^ijbkqb mloD`jh`om…\ ^\mo`nd\i\ l D`jh`om…\ \i\g…od^\' bp i^ fab^ ab obmobpbkq^o mrkqlpmlo k•jbolp- Di j‹qlal pbdrfal m^o^ mrkqlp abi mi^kl bp bi pfdrfbkqb9

Rb bifdbk alp ob`q^p mbombkaf`ri^obp ab obcbobk`f^ 'ii^j^a^p `e`n ^jjm_`i\+_jn'* rkl elofwlkq^i 'ii^j^al bgb ab i^p u(+ u bi lqol sboqf`^i 'bi bgb ab i^p s&+Rr mrkql ab fkqbopb``fŽk+ pb fkaf`^ mlo N u pb abkljfk^ jmdb`i, Dk bi bgb s ^i^ abob`e^ abi N pb bifdb `lksbkfbkqbjbkqb rk mrkql+ u pr afpq^k`f^ ^i N pbabkljfk^ _dno\i^d\ pid_\_, K^p afpq^k`f^p sboqf`^ibp `loobpmlkafbkqbp ^i bgb abi^p v pb jfabk `lk i^ jfpj^ afpq^k`f^ rkfa^a- Dkqlk`bp+ ^ `^a^ mrkql abi mi^kl'ii^j^al ^idrk^p sb`bp mi^kl rs& pb ib ^pfdk^ rk m^o ab k•jbolp+ ii^j^alpprp ^jjm_`i\_\n, Dpq^p `lloabk^a^p obmobpbkq^k i^p afpq^k`f^p abi mrkql ^ ilp

.2

Page 80: Calculus

/) Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

bgbp-Dk i^ cfdro^ 0-0 pb a^k ^idrklp bgbjmilp- Di mrkql ab `lloabk^a^p '2+1(bpqŠ pfqr^al qobprkfa^abp ^ i^ abob`e^ abi bgb u u alp rkfa^abp bk`fj^ abibgb s, Di k•jbol 2 bp i^ `lloabk^a^ s abi mrkql+ v bi 1 i^ `lloabk^a^ v- Klpmrkqlp ^ i^ fwnrfboa^ abi bgb v qfbkbk i^ `lloabk^a^ s kbd^qfs^+ ilp pfqr^alpab_^gl abi bgb s* i^ `lloabk^a^ v kbd^qfs^- K^ `lloabk^a^ s ab rk mrkql pbabkljfk^ q^j_f‹k pr \]n^dn\* v i^ v pr jm_`i\_\,

@i a^o rk m^o ab k•jbolp q^i `ljl %[)\& obmobpbkq^kqbab rk mrkql+ pb`lksfbkb nrb i^ ^_p`fp^ \* l `lloabk^a^ s* pb bp`of_b bk mofjbo ird^o- Olo bpql+bi m^o &\*]' pb `lkpfabo^ `ljl rk k\m jm_`i\_j, Dp `i^ol nrb alp m^obploab,k^alp %[) \& v %])^& obmobpbkq^kbi jfpjl k•jbol pf v pŽil pf [ < _ v \ < ^+ Ork,qlp %[)\& q^ibp nrb [ v \ plk ^j_lp mlpfqfslp pb af`b nrb bpqŠk pfqr^alp bk bikmdh`m^p\_m\io`9 pf \ ; N v ] = N bpqŠkbk bi n`bpi_j ^p\_m\io`9 pf \ ; N v] ; N bpqŠkbk bi o`m^`m p\_m\io`9 u pf \ = N u ] ; N bpqŠkbk bi ^p\moj ^p\_m\i+o`, K^ cfdro^ 0-0 mobpbkq rk mrkql bk `^a^ `r^ao^kqb-

O^o^ mrkqlp abi bpm^`fl pb mol`bab ab cloj^ ^kŠild^- Rb qlj^k qobpob`q^pbk bi bpm^`fl mbombkaf`ri^obpbkqob pŒv nrb pb `loqbk bk rk mrkql 'bi lofdbk(-Dpq^pob`q^p abqbojfk^k qobp mi^klp mbombkaf`ri^obpalp ^ alp+ v `^a^ mrkqlabi bpm^`fl mrbab pbo abqbojfk^al a^kal qobpk•jbolp+ `lk ilp pfdklp ^ab`r^,alp+ nrb obmobpbkq^kprp afpq^k`f^p ^ bpqlp mi^klp- Cb i^ FbljbqoŒ^ `^oqbpf^k^qofafjbkpflk^i pb e^_i^oŠ `lk jŠp abq^iib jŠp ^abi^kqb- Cb jljbkql fkqbobp^i^ FbljbqoŒ^ ^k^iŒqf` mi^k^-

Tk^ cfdro^ dblj‹qof`^+ q^i `ljl rk^ `ros^ mi^k^+bp rk `lkgrkql ab mrkqlpnrb p^qfpc^`bk^ rk^ l jŠp `lkaf`flkbp- So^ar`fbkal bpq^p`lkaf`flkbp bk bumob,

afa v

3

2

1 ,,,,,,,,,y'2+1(GGGGGG

' ,1+0(Š-,,,,,,G ZG

bgb s

,4 ,3 ,20 ,1 ,[ NG ,ZGG

3 '+GG

9 ,2G 3

',2+ ,3( ‘-,,,,,,,,<,,

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,,,,,,,,,,,,, --‘'3+,2(

EHFTQ@ 0-0

t

EHFTQ@ 0-1 I\ ^dm^pia`m`i^d\m`km`n`io\_\ kjm g\ `^p\^d‡i

^\mo`nd\i\ s0 * t0 < m0Š

Page 81: Calculus

Cpi^dji`n, F_`\n b`i`m\g`n t `e`hkgjn 50

pflkbp ^k^iŒqf`^p bk i^p `lloabk^a^p s b t* pb l_qfbkbk rk^ l jŠp b`r^`flkbpnrb `^o^`qbofw^k i^ `ros^ bk `rbpqfŽk- Olo bgbjmil+ `lkpfa‹obpb nrb i^ `ros^ bprk^ `fo`rkcbobk`f^ ab o^afl m `lk `bkqol bk bi lofdbk+ `ljl pb fkaf`^ bk i^cfdro^ 0-1- Rb^ M rk mrkql ^o_fqo^ofl ab bpq^ `fo`rkcbobk`f^+ v prmŽkd^pb nrb Mqfbkb ab `lloabk^a^p %r) s&+ Dkqlk`bp+ bi pbdjbkqŽ KL bp i^ efmlqbkrp^ ab rkqofŠkdril ob`qŠkdril `rvlp `^qbqlp qfbkbk ab ilkdfqra Gth+b [uh v mlo q^kql bksfoqra abi qblobj^ ab OfqŠdlo^p9

s0 * t0 <+1-

Dpq^ b`r^`flk pb abkljfk^ i^ `^p\^d‡i ^\mo`nd\i\ ab i^ `fo`rkcbobk`f^+ v pb p^,qfpc^`b mlo i^p `lloabk^a^p ab qlalp ilp mrkqlp ab i^ `fo`rkcbobk`f^ v pŽilmlo bii^p+ ab j^kbo^ nrb bpq^ b`r^`fŽk `^o^`qbofw^ `ljmibq^jbkqb i^ `fo`rkcbobk,`f^- Dpqb bgbjmil jrbpqo^ `Žjl pb ^mif`^ i^ FbljbqoŒ^ ^k^iŒqf`^ m^o^ obar`fo mol,mlpf`flkbp dblj‹qof`^p pl_ob mrkqlp ^ molmlpf`flkbp ^k^iŒqf`^p`lk k•jbolp ob^ibp-

Cro^kqb qlal pr abp^ooliil efpqŽof`l+ bi BŠi`ril u i^ FbljbqoŒ^ ^k^iŒqf`^e^k bpq^al Œkqfj^jbkqb ifd^alp- Cbp`r_ofjfbkqlp bk rkl ab biilp e^k a^al ird^o^ moldobplp bk bi lqol- Dk bpqb if_ol pb foŠk qo^q^kal `lkgrkq^jbkqb `ljl bk prabp^ooliil efpqŽof`l+ mbol pfk lisfa^o nrb b molmŽpfql fkf`f^i bp fkqolar`fo biBŠi`ril afcbobk`f^i b fkqbdo^i- Klp `lk`bmqlp ab FbljbqoŒ^ ^k^iŒqf`^obnrbofalp m^o^biil+ pb foŠk bumlkfbkal `lkclojb pb s^v^k kb`bpfq^kal- Cb jljbkql+ pŽil pbobnrfbobk ml`lp `lk`bmqlp jrv bibjbkq^ibp ab FbljbqoŒ^ ^k^iŒqf`^ mi^k^ m^o^`ljmobkabo ilp orafjbkqlp ab BŠi`ril- O^o^ buqbkabo bi ^i`^k`b u i^p ^mif`^`flkbpabi BŠi`ril pb kb`bpfq^ rk bpqrafl jŠp molcrkal ab i^ FbljbqoŒ^ ^k^iŒqf`^+nrb pbe^oŠ bk ilp `^mŒqrilp 4 v 5 rp^kal ilp j‹qlalp ab BŠi`ril sb`qlof^i- Lfbkqo^pq^kql+ il nrb pb kb`bpfq^ ab FbljbqoŒ^ ^k^iŒqf`^ bp bpq^o rk ml`l c^jfif^ofw^al bkbi af_rgl ab i^p doŠcf`^p ab i^p crk`flkbp-

0+1 Erk`flkbp- Hab^p dbkbo^ibp u bgbjmilp

Dk afsboplp `^jmlp ab i^ ^`qfsfa^a erj^k^+ pb mobpbkq^k obi^`flkbp nrbbufpqbk bkqob rk `lkgrkql ab rklp l_gbqlp u lqol `lkgrkql ab lqolp l_gbqlp-FoŠcf`^p- `^oqldo^j^p+ `ros^p+ q^_i^p+ cŽojri^p+ bk`rbpq^p bk i^ lmfkfŽk m•_if,`^+ bq`- plk c^jfif^obp ^ qlal ^nrbi nrb ibb ilp mbofŽaf`lp- Dk ob^ifa^a pb qo^q^ab mrolp ^oqfcf`flp rp^alp m^o^ abp`of_fo obi^`flkbp bpmb`f^ibp bk cloj^ `r^kqfq^qf,s^- Klp j^qbjŠqf`lp `lkpfabo^k `ljl api^dji`n ^idrklp qfmlp ab bpq^p obi^`flkbp-Dk bpq^ Rb``fŽk- pb a^k rk^p fab^p dbkbo^ibp abi `lk`bmql ab crk`fŽk- Dk i^ Rb`,`fŽk 0-2 lcob`bjlp rk^ abcfkf`fŽk ofdrolp^ ab crk`fŽk-

DIDLOKN i- K^ crbow^ C kb`bp^of^ m^o^ bpqfo^o rk jrbiib ab ^`bol rk^ilkdfqra s ^ m^oqfoab pr ilkdfqra kloj^i+ bp molmlo`flk^i ^ s, Dp ab`fo E < `s*alkab ` bp rk k•jbol fkabmbkafbkqb ab s* nrb bp i^ `lkpq^kqb abi jrbiib- Dpq^

Page 82: Calculus

51 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

cŽojri^ abp`r_fboq^ mlo Ql_boq Gllhb ^ jbaf^alp abi pfdil WUHH pb abkljfk^i^ g`t _` Ejjf` v pb af`b nrb bumobp^ i^ crbow^ bk crk`fŽk abi ^i^od^jfbkql-

DIDLOKN 1- Rb af`b nrb bi slirjbk ab rk `r_l bp crk`fŽk ab i^ ilkdfqra abprp ^ofpq^p- Rf i^p ^ofpq^p qfbkbk ab ilkdfqra u+ bi slirjbk bpqŠ a^al mlo i^ cŽojr,i^ S < u!-

DIDLOKN I- Tk iˆh`mj kmdhj bp qlal,bkqbol i = 0 nrb kl mrbab bumob,p^opb bk i^ cloj^ i < \]* alkab \ v ] plk bkqbolp mlpfqfslp ^j_lp jbklobp nrb i,Klp mofjbolp k•jbolp mofjlp plk9 1+2+ 4+ 6+ 00+ 02+ 06+ 08- C^al rk k•jbolob^i s = N bp mlpf_ib `lkq^o bi k•jbol ab k•jbolp mofjlp jbklobp nrb s, Dpqbk•jbol pb af`b nrb bp rk^ crk`fŽk ab s* pf _fbk kl pb `lkl`b rk^ cŽojri^ ^idb,_o^f`^ pbk`fii^ m^o^ `^i`ri^oil 'pfk kb`bpfa^a ab `lkq^oilp( `r^kal pb `lkl`b s,

K^ m^i^_o^ ~crk`fŽk‚ crb fkqolar`fa^ bk L^qbjŠqf`^p mlo Kbf_kfw+ nrb rqf,ifw^_^ bpqb q‹ojfkl m^o^ abpfdk^o `fboql qfml ab cŽojri^p j^qbjŠqf`^p- LŠpq^oab pb sfl nrb i^ fab^ ab crk`fŽk ab Kbf_kfw qbkŒ^rk ^i`^k`b jrv obar`fal+v mlpqboflojbkqb bi pfdkfcf`^al ab i^ m^i^_o^ crk`fŽk crb bumbofjbkq^kal dbkb,o^ifw^`flkbp moldobpfs^p- @`qr^ijbkqb+ i^ abcfkf`fŽk ab crk`fŽk bp bpbk`f^ijbkqbi^ pfdrfbkqb9 C^alp alp `lkgrkqlp ab l_gbqlp+ bi `lkgrkql W v bi `lkgrkql V*rk^ api^d‡i bp rk^ ibv nrb ^pl`f^ ^ `^a^ l_gbql ab W rkl v pŽil rk l_gbql bk V,Di `lkgrkql W pb abkljfk^ bi _jhdidj ab i^ crk`fŽk- Klp l_gbqlp ab V* ^pl,`f^alp `lk ilp l_gbqlp bk W cloj^k lqol `lkgrkql abkljfk^al bi m`^jmmd_j ab i^crk`fŽk- obpqb mrbab pbo qlal bi `lkgrkql U) mbol kl bp kb`bp^ofl-(

Kbqo^p ab ilp ^ic^_bqlp bpm^•li v dofbdl+ pb rqfifw^k cob`rbkqbjbkqb m^o^abpfdk^o crk`flkbp- Dk m^oqf`ri^o pb rp^k jr`el i^p ibqo^p `) d+ b) F+ D) v ok,Rf a bp rk^ crk`fŽk a^a^ u s bp rk l_gbql ab pr aljfkfl+ i^ klq^`fŽk a&s' pbrqfifw^ m^o^ abpfdk^o bi l_gbql nrb bk bi ob`loofal `loobpmlkab ^ s* bk i^ crk`fŽkd) v pb abkljfk^ bi q\gjm _` g\ api^d‡i a bk s l i^ dh\b`i _` s kjm a, Di pŒj_lila&s' pb ibb+ za ab s~,

K^ fab^ ab crk`fŽk pb mrbab firpqo^o bpnrbjŠqf`^jbkqb ab jr`e^p j^,kbo^p- Olo bgbjmil+ bk i^ cfdro^ 0-2'^( ilp `lkgrkqlp W b X plk pbkalp `lk,grkqlp ab mrkqlp+ u rk^ cib`e^ fkaf`^ `Žjl pb ^m^ob^ rk mrkql ^o_fqo^ofl sab W `lk pr mrkql fj^dbk `%r& ab X- Nqol bpnrbj^ bp bi ab i^ cfdro^ 0-2'_(alkab i^ crk`fŽk ` pb fj^dfk^ `ljl rk^ jŠnrfk^ bk i^ `r^i ilp l_gbqlp abi`lkgrkql W pb qo^kpcloj^k m^o^ molar`fo l_gbqlp abi `lkgrkql X- Br^kal rkl_gbql r bp qo^kpcloj^al mlo i^ jŠnrfk^+ bi obpriq^al cfk^i bp bi l_gbql `%r&+

Dk BŠi`ril bibjbkq^i qfbkb fkqbo‹p `lkpfabo^o bk mofjbo ird^o+ ^nrbii^pcrk`flkbp bk i^p nrb bi aljfkfl v bi ob`loofal plk `lkgrkqlp ab k•jbolp ob^ibp-Dpq^p crk`flkbp pb ii^j^k api^dji`n _` q\md\]g` m`\g l jŠp _obsbjbkqb api^dji`nm`\g`n u pb mrbabk obmobpbkq^o dblj‹qof`^jbkqb jbaf^kqb rk^ doŠcf`^ bk bimi^kl s v- Rb obmobpbkq^ bi aljfkfl W bk bi bgb s* v ^ m^oqfo ab `^a^ mrkql s

Page 83: Calculus

Cpi^dji`n, F_`\n b`i`m\g`n u `e`hkgjn 52

s

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EHFTQ@ 0-2 O`km`n`io\^d‡i `nlp`hƒod^\ _`g ^ji^`koj _` api^d‡i,

ab W pb obmobpbkq^bi mrkql &s*u( alkab u << a&s', K^ qlq^ifa^a ab mrkqlp &s*u(pb abkljfk^ i^ bmƒad^\ ab i^ crk`fŽk-

@ `lkqfkr^`fŽk `lkpfabo^jlp lqolp bgbjmilp ab crk`flkbp ob^ibp-

DIDLOKN 3- I\ api^d‡i d_`iod_\_, Rrmlkd^jlp nrb a&s' < s m^o^ qlalob^i s, Dpq^ crk`fŽk `lk cob`rbk`f^ pb abkljfk^ i^ api^d‡i d_`iod_\_, Rr aljfkflbp bi bgb ob^i+ bpql bp+ bi `lkgrkql ab qlalp ilp k•jbolp ob^ibp- O^o^ `^a^ mrkql%r) u( ab i^ doŠcf`^ bp r < u- K^ doŠcf`^ bp rk^ ob`q^ nrb cloj^ Škdrilp fdr^ibp`lk ilp bgbp `lloabk^alp 's‹^pb cfdro^ 0-3(- Di ob`loofal ab ` bp bi `lkgrkql abqlalp ilp k•jbolp ob^ibp-

DIDLOKN 4- I\ api^d‡i q\gjm \]njgpoj, Blkpfabobjlp i^ crk`fŽk nrb^pfdk^ ^ `^a^ k•jbol ob^i r bi k•jbol kl kbd^qfsl Erf+ Tk^ m^oqb ab pr doŠcf`^bpqŠ obmobpbkq^a^ bk i^ cfdro^ 0-4- Cbpfdk^kal bpq^ crk`fŽk `lk i^ ibqo^ n:+ pb

t

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u

EHFTQ@ 0-3 Dmƒad^\ _` g\api^d‡i d_`iod_\_ y&s' < s*

EHFTQ@ 0-4 Cpi^d‡iq\gjm \]njgpoj %/%&s' < Yt[+

Page 84: Calculus

53 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

qfbkb 9j%r& < Gthm^o^ qlal ob^i r+ Olo bgbjmil+ 9j%-& < N+9j%/& < 1+ ^k&,2( < 2-Blk bpq^ klq^`fŽk bumobp^jlp ^idrk^p molmfba^abp ab ilp s^ilobp ^_plirqlp-

'^( 9j_*r& < 9j%r&+ 'a( 9jW9j%r&Y< 9j%r& +

'b( 9j%r& < pc r0 Š

'b( 9j%r * s& w 9j%r& * 9j%s& 'abpfdr^ia^a qof^kdri^o(-

DIDLOKN 5- I\ api^d‡i iˆh`mj kmdhj, O^o^ `r^inrfbo s = N+ pb^ 5Q!&s'bi k•jbol ab k•jbolp mofjlp jbklobp l fdr^ibp ^ s* Di aljfkfl ab PP bp bi`lkgrkql ab ilp k•jbolp ob^ibp mlpfqfslp- Rr ob`loofal bp bi `lkgrkql ab ilp bkqbolpkl kbd^qfslp xN+ 0+ 1+ -- +y- Dk i^ cfdro^ 0-5 pb obmobpbkq^ rk^ mlo`fŽk ab i^doŠcf`^ ab PP+

t

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1 2 4 6 00 02

i i i i G 0 5 61/

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2 5 7 3/ 21/

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4 01/ 0/ 25177//

EHFTQ@ 0-5 I\ api^d‡i iˆh`mj kmdhj, EHFTQ@ 0-6 I\ api^d‡ia\^ojmd\g,

'Dk ilp bgbp s b v pb e^k rp^al bp`^i^p afpqfkq^p-( Br^kal s `ob`b+ bi s^ilo ab i^crk`fŽk QQ&U' mboj^kb`b `lkpq^kqb e^pq^ nrb s ^i`^kw^ rk k•jbol mofjl+ bk `rvlmrkql bi s^ilo ab i^ crk`fŽk mobpbkq^rk p^iql fdr^i^ 0- Olo `lkpfdrfbkqb i^ doŠcf`^ab PP `lkpfpqb bk pbdjbkqlp ab ob`q^ elofwlkq^ibp- …pqbbp rk bgbjmil ab rk^ `i^pbab crk`flkbp ii^j^a^p api^dji`n `n^\gji\_\n9 ‹pq^p abpbjmb•^k rk m^mbi crka^,jbkq^i bk i^ qbloŒ^ab i^ fkqbdo^i-

DIDLOKN 6- I\ api^d‡i a\^ojmd\g, O^o^ qlal bkqbol mlpfqfsl i* pb abcfkba&i' `ljl i < 0 - 1& --- i, Dk bpqb bgbjmil+ bi aljfkfl ab a bp bi `lkgrkql abilp bkqbolp mlpfqfslp- Klp s^ilobp ab i^ crk`fŽk `ob`bk `lk q^kq^ o^mfabw nrb bpjŠp `lksbkfbkqb mobpbkq^oi^ crk`fŽk bk cloj^ q^_ri^o nrb jbaf^kqb rk^ doŠcf`^-K^ cfdro^ 0-6 bp rk^ q^_i^ ab m^obp &i*iF' m^o^ i < 0+1+ --- + 0/-

Page 85: Calculus

Cpi^dji`n, A`adid^d‡i ajmh\g ^jhj ^jiepioj _` k\m`n jm_`i\_jn 43

Di ib`qlo ab_boŒ^ l_pbos^o alp o^pdlp `^o^`qboŒpqf`lp nrb qfbkbk bk `lj•kqlalp ilp bgbjmilp ^kqboflobp-

'0( O^o^ `^a^ s abi aljfkfl W bufpqb rk^ v pŽil rk^ fj^dbk v bjm^obg^a^`lk ^nrbi s^ilo m^oqf`ri^o ab s,

'1( B^a^ crk`fŽk bkdbkao^ rk `lkgrkql ab m^obp %r) s&) pfbkal r rk bibjbk,ql dbk‹of`l abi aljfkfl W+ b v bp bi bibjbkql •kf`l ab V nrb `loobpmlkab ^ s,

Dk i^ j^vlo m^oqb ab ilp bgbjmilp ^kqboflobp+ pb mobpbkq^olk ilp m^obp%r) u( dblj‹qof`^jbkqb `ljl mrkqlp pl_ob rk^ doŠcf`^- Dk bi bgbjmil 6 ilpmobpbkq^jlp `ljl m^obp `loobpmlkafbkqbp bk rk^ q^_i^- Dk `^a^ `^pl+ `lkl`boi^ crk`fŽk bp `lkl`bo+ bk rk^ r lqo^ cloj^+ oj_jn ilp m^obp &s*u( nrb bkdbkao^-Dpq^ pbk`fii^ l_pbos^`fŽk bp bi lofdbk ab i^ abcfkf`fŽk abi `lk`bmql ab crk`fŽknrb pb bumlkb bk i^ pb``fŽk pfdrfbkqb-

) 0-2 Erk`flkbp- Cbcfkf`fŽk cloj^i `ljl `lkgrkql ab m^obp loabk^alp

Dk i^ Rb``fŽk ^kqboflo+ rk^ crk`fŽk pb abp`of_fŽ `ljl rk^ `loobpmlkabk`f^nrb ^pl`f^ ^ `^a^ l_gbql ab rk `lkgrkql W rkl v pŽil rk l_gbql ab rk `lkgrk,ql V, K^p m^i^_o^p ~`loobpmlkabk`f^‚ v ~^pl`f^ ^‚ mrbab nrb kl qbkd^k i^jfpj^ pfdkfcf`^`fŽk m^o^ qlal bi jrkal+ ab j^kbo^ nrb obclojri^objlp bi `lk,`bmql mlo rk `^jfkl afcbobkqb+_^pŠkalil bk bi `lk`bmql ab `lkgrkql- Mb`bpfq^jlpmofjbol i^ kl`fŽk ab k\m jm_`i\_j ab l_gbqlp-

Dk i^ abcfkf`fŽk ab fdr^ia^a ab `lkgrkqlp+ kl pb jbk`flk^ bi jm_`i bk binrb ^m^ob`bk ilp bibjbkqlp- @pŒnrb+ ilp `lkgrkqlp x1+ Ry X xR+ 1y plk fdr^ibpmlonrb `lkpq^k bu^`q^jbkqb ab ilp jfpjlp bibjbkqlp- Dk `fboq^p l`^pflkbp biloabk `n fjmloq^kqb- Olo bgbjmil+ bk FbljbqoŒ^ ^k^iŒqf`^ mi^k^ i^p `lloabk^a^p%r)u( ab rk mrkql obmobpbkq^k rk m^o loabk^al ab k•jbolp- Di mrkql ab `lloab,k^a^p '1+ 4( kl bp bi jfpjl nrb bi ab `lloabk^a^p '4+ 1(+ pf _fbk ilp ^jiepiojnx1+ 4y X xR+ 1y plk fdr^ibp- Cbi jfpjl jlal+ pf qbkbjlp rk m^o ab l_gbqlp \ v ]'kl kb`bp^of^jbkqb afpqfkqlp( v abpb^jlp afpqfkdrfo rkl ab ilp l_gbqlp+ mlobgbjmil \* `ljl bi kmdh`m bibjbkql u bi lqol+ ]* `ljl bi n`bpi_j* bk`boo^jlp ilpl_gbqlp bk rk m^o‹kqbpfp %[) \&+ Kl `lkpfabo^jlp `ljl rk m^o loabk^al- Cb`fjlpnrb alp m^obploabk^alp %[) \& v %]) ^& plk fdr^ibp pf u pŽil pf prp mofjbolp bibjbk,qlp plk fdr^ibp v prp pbdrkalp bibjbkqlp plk fdr^ibp- Dpql bp+ pb qfbkb

%[) \& < %]) ^& pf v pŽil pf [ < _ v ] < _ ,

U^jlp ^elo^ ^ bpq^_ib`bo i^ abcfkf`fŽk ab crk`fŽk-

CDEHMHBHˆM CD ETMBHˆM- Ri\ api^d‡i a `n pi ^jiepioj _` k\m`n jm_`i\_jn&s*u( idibpij _` gjn ^p\g`n od`i` `g hdnhj kmdh`m `g`h`ioj,

Rf ` bp rk^ crk`fŽk+ bi `lkgrkql ab qlalp ilp bibjbkqlp s nrb ^m^ob`bk `ljlmofjbolp bibjbkqlp ab m^obp &s*u( ab a pb ii^j^ bi _jhdidj ab a, Di `lkgrkql abilp pbdrkalp bibjbkqlp v pb abkljfk^ m`^jmmd_j ab a* l `lkgrkql ab q\gjm`n ab a,

Page 86: Calculus

// Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

Hkqrfqfs^jbkqb+rk^ crk`fŽk mrbab fj^dfk^opb `ljl rk^ q^_i^ nrb `lkpq^ab alp `lirjk^p- B^a^ bkqo^a^ bk i^ q^_i^ bp rk m^oloabk^al %r) s&8 i^ `lirjk^ab i^ r bp bi aljfkfl ab `) v i^ ab i^p v+ bi ob`loofal- Rf alp bkqo^a^p %r) v( v%r)w( ^m^ob`bk bk i^ q^_i^ `lk bi jfpjl s^ilo ab r) m^o^ nrb obmobpbkqbrk^crk`fŽk bp kb`bp^ofl nrb u < w- Cf`el ab lqol jlal+ rk^ crk`fŽk kl mrbabqlj^o alp s^ilobp afpqfkqlpbk rk mrkql a^al s, Olo il q^kql+m^o^qlal s bk bialjfkl ab ` bufpqbbu^`q^jbkqb rk u q^i nrb %r) u( D `+ X^ nrb bpqbu bpqŠabqbo,jfk^al `lk rkf`fa^a rk^ sbw pb `lkl`b s* mlabjlp fkqolar`fo m^o^ ‹i rk pŒj,_lil bpmb`f^i-Dp `lpqrj_ob bp`of_fo

t < y&s'

bk ird^o ab %r) u( C ` m^o^fkaf`^o nrb bi m^o%r) u( mboqbkb`b i `lkgrkql `+Nqo^ j^kbo^ ab abp`of_fo rk^ crk`fŽk ` bpmb`fcf`^kal ilp m^obpnrb `lk,

qfbkb+v nrb bp rpr^ijbkqb mobcbof_ib+lkpfpqb bk abp`of_fo bi aljfkfl ab g*virbdl+ m^o^`^a^ s abi aljfkfl+ abp`of_fo `Žjl pb l_qfbkb bi s^ilo ab i^ crk`fŽk`%r&+Dk obi^`fŽk `lk bpql+pb qfbkbbi qblobj^ pfdrfbkqb`rv^ abjlpqo^`fŽk abg^jlp`ljl bgbo`f`fl m^o^bi ib`qlo-

RCMPCK? 0-0- Ajn api^dji`n a v d nji dbp\g`n pf v n‡gj pf']( a v d od`i`i `g hdnhj _jhdidj* v'_( a&s' < b&s' k\m\ oj_j s _`g _jhdidj _` a,

Blksfbkb a^opb `rbkq^ nrb ilp l_gbqlp r v `%r& nrb ^m^ob`bk bk ilp m^obploabk^alp &s*a&s''ab rk^ crk`fŽk kl qfbkbkmlonr‹ pbok•jbolp pfkl nrb mrbabkpbo l_gbqlp ab `r^inrfbo `i^pb- Dk l`^pflkbp e^objlp rpl ab bpq^ fab^ dbkbo^i+mbol bk i^ j^vloŒ^ ab ilp `^plp klp fkqbobp^oŠkcrk`flkbp ob^ibp+bpql bp+crk`fl,kbp `rvl aljfkfl v ob`loofal pb^k pr_`lkgrkqlp ab i^ ob`q^ ob^i-

@idrk^p ab i^p crk`flkbp nrb ^m^ob`bkbk BŠi`ril pb abp`of_bk bk ilp bgbj,milp pfdrfbkqbp-

)&, @m` RWRZ]Y\QRSb[PV\[R`_RNYR`

/, Cpi^dji`n ^jino\io`n, Tk^ crk`fŽk `rvl ob`loofal `lkpq^ ab rk plilk•jbol pb ii^j^ crk`fŽk `lkpq^kqb- Dk i^ cfdro^ 0-7 pb jrbpqo^ rk bgbjmil+ bk i^nrb `%r& < 2 m^o^qlal r ob^i- K^ doŠcf`^bp rk^ ob`q^ elofwlkq^i nrb `loq^ ^i bgbu bk bi mrkql 'N+ 2(-

0, Cpi^dji`n gdi`\g`n, Tk^ crk`fŽk d abcfkfa^ m^o^ qlal ob^i s jbaf^kqbrk^ cŽojri^ ab i^ cloj^

a%r&< [r * ]

Page 87: Calculus

Jƒn `e`hkgjn _` api^dji`n m`\g`n /0

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EHFTQ@ 0-7 Cpi^d‡i ^jino\io`a&s' < 2-

EHFTQ@ 0-8 Cpi^d‡i gdi`\ga%r&< /r * 0-

EHFTQ@ 0-0/ Mjgdijhdj^p\_mƒod^j a&s' < s0Š

pb ii^j^ crk`fŽk ifkb^i mlonrb pr doŠcf`^ bp rk^ ob`q^- Di k•jbol ] bp i^ loabk^,a^ bk bi lofdbk: bp i^ `lloabk^a^ t abi mrkql 'N+ \& bk bi nrb i^ ob`q^ `loq^ ^ibgb s+ Di k•jbol [ bp i^ mbkafbkqb ab i^ ob`q^- Tk bgbjmil+ a%r&< r) bpqŠ af_r,g^al bk i^ cfdro^ 0-3- Dk i^ cfdro^ 0-8 pb jrbpqo^ lqol a%r&< /r * 0-

1, Cpi^dji`n kjo`i^d\g`n, O^o^ rk bkqbol mlpfqfsl i* pb^ a i^ crk`fŽk ab,cfkfa^ mlo `%r& < u! m^o^ qlal ob^i r+ Br^kal h < 0+‹pq^ bp i^ crk`fŽk fabkqfa^a+obmobpbkq^a^ bk i^ cfdro^ 0-3- O^o^ i < 1 i^ doŠcf`^ bp rk^ m^oŠ_li^+ m^oqb ab i^`r^i pb sb bk i^ cfdro^ 0-0/- O^o^ i < 2+ i^ doŠcf`^ bp rk^ `•_f`^ u bpqŠ obmobpbkq^,a^ bk i^ cfdro^ 0-00 'mŠd- 58(-

2, Cpi^dji`n kjgdi‡hd^\n, Tk^ crk`fŽk mlifkŽjf`^ M bp i^ abcfkfa^ m^o^qlal ob^i s mlo rk^ b`r^`fŽk ab i^ cloj^

i

M&s' < @j * `0u * --- * ^isi < F@fUf,

Q_>

Klp k•jbolp Bl+ _f$ +++) `9 plk ilp ^j`ad^d`io`n abi mlifkljfl+ v bi bkqbol klkbd^qfsl i bp pr bm\_j 'pf `9 ;/; N(- Prba^k fk`irfa^p bk bpqb qfml ab crk`flkbp+ i^pcrk`flkbp `lkpq^kqbp u i^p mlqbk`f^ibp- Klp mlifkljflp ab do^alp 1+ 2 X 3 pb ab,kljfk^k mlifkljflp ^p\_mƒod^jn* ^ˆ]d^jn v ^pƒmod^jn obpmb`qfs^jbkqb- K^ cfdr,o^ 01 mobpbkq^ rk^ m^oqb ab i^ doŠcf`^ ab rk^ crk`fŽk mlifkŽjf`^ `rŠoqf`^ Ma^a^ mlo L%r& <er$ * /r0

Š

Page 88: Calculus

57 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

3, I\ ^dm^pia`m`i^d\, Ulis^jlp ^ i^ b`r^`fŽk `^oqbpf^k^ ab i^ `fo`rkcbobk,`f^+ s0 * v1 < o1 v obplisŠjlpi^ obpmb`ql ^ v- Dufpqbk alp plir`flkbp a^a^p mlo

t < Uo1 , r/ v

'Qb`rboab bi ib`qlo nrb pf \ = /+ bi pŒj_lil X:- obmobpbkq^i^ o^Œwr^ao^a^ ml,pfqfs^ ab \, K^ o^Œwr^ao^a^ kbd^qfs^ bp , p`%c+&Gr_l rk qfbjml bk nrb ilp j^qb,jŠqf`lp ab`Œ^k nrb v bo^ rk^ crk`fŽk ]d+q\g`io` ab s a^a^ mlo v < ,*, sf o1 z s/†

Ml l_pq^kqb+ jlabok^jbkqb kl pb ^ajfqb i^ ~_f,s^ibk`f^‚ `ljl molmfba^a ab i^pcrk`flkbp- K^ abcfkf`fŽk ab crk`fŽk bufdb nrb ^ `^a^ s mboqbkb`fbkqb ^i aljfkfl+`loobpmlkab rkl v pŽil rk s^ilo ab v bk bi ob`loofal- Fblj‹qof`^jbkqb+ bpqlpfdkfcf`^ nrb i^p ob`q^p sboqf`^ibp `loq^k i^ doŠcf`^ bk rk plil mrkql- Olo `lkpf,drfbkqb m^o^ e^`bo e^`bo `ljm^qf_ib bi ^kqboflo bgbjmil `lk bi `lk`bmql qbŽof`l+ab`fjlp nrb i^p alp plir`flkbp m^o^ u abcfkbk _jn crk`flkbp+ a u d+ pfbkal

u b&s' < ,Tn1 , s/a&s' < qm0+ s0

m^o^ `^a^ s nrb p^qfpc^`b , o z s x o- B^a^ rk^ ab bpq^p crk`flkbp qfbkb `ljlaljfkfl bi fkqbos^il `ljmobkafal bkqob ,n v n-Rf Gth= m*kl bufpqb s^ilo ob^iab v q^i nrb s0 * v1 < m%*v ab`fjlp nrb i^p crk`flkbp a v d ij `noƒi _`adid_\nm^o^ q^i r+ Orbpql nrb `%r& bp i^ o^Œwr^ao^a^ kl kbd^qfs^ ab o1

, r0* i^ doŠcf`^

ab ` bp i^ pbjf`fo`rkcbobk`f^ obmobpbkq^a^ bk i^ cfdro^ 0-02- Klp s^ilobp ab i^crk`fŽk d plk z /+ v mlo q^kql i^ doŠcf`^ ab d bp i^ pbjf`fo`rkcbobk`f^ fkcbofloaf_rg^a^ bk i^ cfdro^ 0-02-

4, Pph\n* kmj_p^ojn u ^j^d`io`n _` api^dji`n, Rb^k a u d alp crk`flkbpob^ibp nrb qfbkbk bi jfpjl aljfkfl @+Rb mrbabk `lkpqorfo krbs^p crk`flkbp ^m^oqfoab ` v d mlo ^af`fŽk+ jriqfmif`^`fŽk l afsfpfŽk ab prp s^ilobp- K^ crk`fŽk pabcfkfa^ mlo

p&s' < a&s' * b&s' pf s C @

pb abkljfk^ nph\ ab ` u d u pb obmobpbkq^mlo ` * d- Cbi jfpjl jlal+ bi kmj+_p^oj q < ` +d u bi ^j^d`io` t < ` , d bpqŠk abcfkfalp mlo i^p cŽojri^p

q&s' < a&s'b&s' pf s C @) r&s' < a&s'-b&s' pf s D A X b&s' x N-

Blk bi `lkgrkql ab ilp Dgbo`f`flp nrb pfdrbk pb fkqbkq^ a^o ^i ib`qlo `fboq^pliqro^ bk bi j^kbgl ab i^ klq^`fŽk bjmib^a^ m^o^ i^p crk`flkbp-

Page 89: Calculus

t

EHFTQ@ 0-00 Mjgdijhdj^ˆ]d^j M&s'< s1Š

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EHFTQ@ 0-02 Dmƒad^\n_` g\n_jn api^dji`n

a&s' < rn1 , s/)

a%r&< ]so1 , r/†

0- Rb^ `%r& << r * 0 m^o^ qlal ob^i r+ B^i`ri^o9 a&0'* a& ,1(+ +a&0'* ah* / a&0'*

a`\ * ]'*a&\' * a&]'*a&\'a&]',1- Rb^k `%r& < 0 * r X a%r& << 0 , r m^o^ qlal ob^i r+ B^i`ri^o9 a&0' * b&0'*

a&0' + b&0'*a&0f&0'* a&0'ab&0'*a Xb&0'Z*bXa&0'Z*a&\' * b&+\'*a&o'b& +o',

2- Rb^ j%r& < Zu , 20 * Gt , 00 m^o^ qlal ob^i r+ B^i`ri^o9 k8%K&)n:'i(+ k8%/&) jwd&)

j%,0(+ j%* 1(- Cbqbojfk^o qlalp ilp s^ilobp ab o m^o^ ilp nrb ,`%n* 1( < j%n&+

1+ O_[ `%r&< r0 m^o^ qlal ob^i r+ B^i`ri^o `^a^ rk^ ab i^p cŽojri^p pfdrfbkqbp- Dk `^a^`^pl mob`fp^o ilp `lkgrkqlp ab k•jbolp ob^ibp s* v+ o*bq`-+ m^o^ ilp nrb i^ cŽojri^a^a^ bp sŠifa^-

'^( a&+s' < a&s', 'a( a&0t' < 2a&q','_( a&t' + a&s' < &t + s'&t * s', 'b( a&o0' < a&o'0,

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4- Rb^ a%r&< v3 , r0 m^o^ Gthy 1- Bljmol_^o `^a^ rk^ ab i^p cŽojri^p pfdrfbkqbp bfkaf`^o m^o^ nr‹ s^ilobp ab s* v+ n* v o plk sŠifa^p

'^( b&+s' < b&s',

'_( a%/s& < 1UH<6-

'0( u3b1<0'b( a o < Wnc †

'a( b&\ + 1( < v3^ , \/†

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Page 90: Calculus

0) Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

5- Rb^ ` i^ crk`fŽk abcfkfa^ `ljl pfdrb9 `%r& < 0 m^o^ N 999:r 77780: `%r& < 1 m^o^ 0 ; r w 1-K^ crk`fŽk kl bpqŠ abcfkfa^ pf s ; N l pf s = 1-'^( So^w^o i^ doŠcf`^ ab `+'_( Olkbo a%r&< `%/r&+ Cbp`of_fo bi aljfkfl ab d v af_rg^o pr doŠcf`^-'`( Olkbo b%r& < `%r * 1(- Cbp`of_fo bi aljfkfl ab b u af_rg^o pr doŠcf`^-'a( Olkbo e%r& < `%/r& * `%r * 1(- Cbp`of_fo bi aljfkfl ab e v af_rg^o pr doŠcf`^-

6- K^p doŠcf`^p+ab ilp alp mlifkljflp a%r&< r v `%r& < r1 pb `loq^k bk qobp mrkqlp- Cf_rg^ork^ m^oqb prcf`fbkqb ab prp doŠcf`^p m^o^ sbo `Žjl pb `loq^k-

7- K^p doŠcf`^p ab ilp alp mlifkljflp `r^aoŠqf`lp `%r& :r0*/ v a%r& :/r0(1r * 0 pb`loq^k bk alp mrkqlp- Cf_rg^o i^p mlo`flkbp ab prp doŠcf`^p `ljmobkafa^p bkqob prpfkqbopb``flkbp-

8- Dpqb bgbo`f`fl abp^oolii^ `fboq^p molmfba^abp crka^jbkq^ibp ab ilp mlifkljflp- Rb^`%r& < ƒz<l ?eTe rk mlifkljfl ab do^al h+ Cbjlpqo^o `^a^ rkl ab ilp pfdrfbkqbp^m^oq^alp9'^( Rf h w 0 X `%K&< /+ `%r& < ra%r&) pfbkal d rk mlifkljfl ab do^al h * 0-'_( O^o^ `^a^ ob^i [) i^ crk`fŽk j a^a^ mlo j%r& < `%r * [& bp rk mlifkljfl ab do^al h+'`( Rf h w 0 X `%[& < N m^o^ rk `fboql s^ilo ob^i [) bkqlk`bp `%r& < %r * [&b%r&) pfbkalb rk mlifkljfl ab do^al h * 0- WEh^c][]cƒh7 Blkpfa‹obpb j%r& < `%r * [&+Y'a( Rf `%r& < N m^o^ h ---J- 0 s^ilobp ob^ibp ab r afpqfkqlp+ qlalp ilp `lbcf`fbkqbp ]

fplk

`bol u `%r& < N m^o^ qlal ob^i r+

'b( Rb^ a%r&< •f%;j \erf rk mlifkljfl ab do^al g) pfbkal g w h+ Rf a%r& < `%r& m^o^h * 0 s^ilobp ob^ibp ab s afpqfkqlp+ bkqlk`bp h < i* \f < ^f m^o^ `^a^ s^ilo ab e) va%r&< `%r& m^o^ qlal ob^i r+

0/- Dk `^a^ `^pl+ e^ii^o qlalp ilp mlifkljflp m ab do^al 999:1 nrb p^qfpc^`bk i^p `lkaf`fl,kbp a^a^p-'^( j%K&< jK& < j%/& < 0- '`( j%K&< jK& < H-'_( j%K&< jK& < i+m'1( < 1- 'a( j%K&< jK&+

00- Dk `^a^ `^pl+ e^ii^o qlalp ilp mlifkljflp m ab do^al 999:1 nrb m^o^ qlal ob^i s p^qfp,c^`bk i^p `lkaf`flkbp nrb pb a^k-

'^( j%r& < jK * r&+ 'b( j%/r& < /j%r&+'_( j%r& < jK * r&+ 'a( j%0r& < j%r * 2(-

01- Cbjlpqo^o nrb i^p bumobpflkbp pfdrfbkqbp plk mlifkljflp mlkf‹kali^p bk i^ cloj^ƒzl [ere m^o^ rk s^ilo ab g `lksbkfbkqb- Dk `^a^ `^pl h bp bkqbol mlpfqfsl-

0 i)g i

'^( N * r&/h+ '_( , r ) r :‹ H- 'b( SHN * T/e&+0 , B Q_>

)&. :Y P\[PR]a\ QRm_RNP\Z\ Sb[PVp[QRP\[Wb[a\

Br^kal rk j^qbjŠqf`l fkqbkq^ abp^oolii^o rk^ qbloŒ dbkbo^i nrb ^_^onrbjr`elp `lk`bmqlp afpqfkqlp+mol`ro^ ^fpi^o molmfba^abp`ljrkbp nrb m^ob`bk pbocrka^jbkq^ibp m^o^ `^a^ rk^ ab i^p ^mif`^`flkbp m^oqf`ri^obp nrb `lkpfabo^-Tqfifw^ bkqlk`bp bp^p molmfba^abp `ljl mfbao^p crka^jbkq^ibp ab pr qbloŒ^-Dr`ifabp pfdrfŽ bpqb j‹qlal ^i abp^oolii^o i^ FbljbqoŒ^ bibjbkq^i `ljl rk pfp,qbj^ abar`qfsl _^p^al bk rk `lkgrkql ab ^uflj^p- Mlplqolp ebjlp rqfifw^al bijfpjl mol`bpl bk i^ fkqolar``fŽk ^ufljŠqf`^ abi pfpqbj^ ab k•jbolp ob^ibp+u ilrp^objlp rk^ sbw jŠp bk krbpqo^ afp`rpfŽk abi `lk`bmql ab Šob^-

Page 91: Calculus

Bg ^ji^`koj _` ƒm`\ ^jhj api^d‡i _` ^jiepioj 60

Br^kal ^pfdk^jlp rk Šob^ ^ rk^ obdfŽk mi^k^+^pl`f^jlp rk k•jbol ^ rk`lkgrkql R abi mi^kl- Cbpab bi mrkql ab sfpq^ mro^jbkqb j^qbjŠqf`l+ bpql pfdkfcf`^nrb pb qfbkbrk^ crk`fŽk [ 'crk`fŽk Šob^( nrb ^pfdk^ rk k•jbol ob^i [%O& 'bi Šob^ab R( ^ `^a^ `lkgrkql R ab rk^ `fboq^ `lib``fŽk ab `lkgrkqlp a^a^- Tk^ crk`fŽkab bpq^ k^qro^ibw^+ rvl aljfkfl bp rk^ `lib``fŽk ab `lkgrkqlp v `rvlp s^ilobpplk k•jbolp ob^ibp+pb ii^j^ api^d‡i _` ^jiepioj, Di mol_ibj^ _Špf`l bp bpqb9C^al rk `lkgrkql mi^kl R+ƒnr‹ Šob^ [%O& ^pfdk^objlp ^ R>

Mrbpqol j‹qlal m^o^ ^_loa^o bpqb mol_ibj^ `lkpfpqb bk m^oqfoab `fboq^pmolmfba^abp nrb bi Šob^ ab_fbo^ qbkbo v qlj^oi^p `ljl \sdjh\n m^o^ bi Šob^-Br^inrfbo crk`fŽk ab `lkgrkql nrb p^qfpc^d^bplp ^uflj^p pb ii^j^oŠ crk`fŽkŠob^- Dp kb`bp^ofl abjlpqo^o nrb bufpqbob^ijbkqb+rk^ crk`fŽk Šob^-@nrŒkl fk,qbkq^objlp &e^`boil- Dk `^j_fl+ prmlkbjlp i^ bufpqbk`f^ ab af`e^ crk`fŽk Šob^v abar`fjlp krbs^p molmfba^abp^ m^oqfoab ilp ^uflj^p)-

@kqbp ab bpq^_ib`bo ilp ^uflj^p m^o^ bi Šob^+ e^objlp ^idrk^p l_pbo,s^`flkbp ^`bo`^ ab ilp `lkgrkqlp abi mi^kl ^ ilp nrb pb mrbab ^pfdk^o Šob^-Dpqlppb ii^j^oŠk `lkgrkqlp h`_d]g`n9 i^ `lib``fŽk ab qlalp ilp `lkgrkqlp jbaf_ibp pbabpfdk^oŠ mloqdd, Klp ^uflj^p `lkqfbkbk i^ prcf`fbkqb fkcloj^`fŽk ^`bo`^ ab ilp`lkgrkqlp ab qddm^o^ mbojfqfoklp abjlpqo^o nrb qla^p i^p cfdro^p dblj‹qof`^pnrb ^m^ob`bk bk i^p ^mif`^`flkbp rpr^ibp abi `Ši`ril bpqŠkbk qddv nrb prp Šob^pmrbabk `^i`ri^opb mlo fkqbdo^`fŽk-

Tkl ab ilp ^uflj^p '^uflj^ 4( bpq^_ib`b nrb qlal ob`qŠkdril bp jbaf_ib vnrb pr Šob^ bp bi molar`ql ab i^p ilkdfqrabp ab prp i^alp- K^ m^i^_o^ ~ob`qŠk,dril‚ pb rp^ ^nrŒm^o^obcbofopb `r^inrfbo `lkgrkql `lkdorbkqb )&! ^ rk `lkgrkqlab i^ cloj^

u%r)s& GN pr pb) N pU w ev)

pfbkal c x N X f x N- Klp k•jbolp c v f plk i^p ilkdfqrabp ab ilp i^alp abi ob`,qŠkdril- Blkpfabo^jlp rk pbdjbkql l rk mrkql `ljl rk `^pl m^oqf`ri^o ab rkob`qŠkdril prmlkfbkal nrb c l f 'l ^j_lp( pb^k `bol-

@ m^oqfoab ob`qŠkdrilp mlabjlp `lkpqorfo `lkgrkqlp jŠp `ljmif`^alp-Di `lkgrkql af_rg^al bk i^ cfdro^ 0-03 bp i^ obrkfŽk ab rk^ `lib``fŽk cfkfq^ abob`qŠkdrilp `lk prp _^pbp bk bi bgbs* v pb ii^j^ m`bd‡i `n^\gji\_\, Klp ^uflj^pfjmif`^k nrb `^a^ obdfŽk bp`^ilk^a^ bp jbaf_ib v nrb pr Šob^ bp i^ prj^ ab i^pŠob^p ab ilp ob`qŠkdrilp `ljmlkbkqbp-

‘ Tk^ `lkpqor``fŽk bibjbkq^i ab rk^ crk`fŽk Šob^ pb bk`rbkqo^ bk ilp `^mŒqrilp03 v 11ab Datfk D- Llfpb+ Bg`h`io\mt D`jh`omt Cmjh >i >_q\i^`_ Po\i_kjdio* @aafplk,VbpibvOr_ifpefkd Bl-+ 0852-

Š) K^ `lkdorbk`f^ pb rp^ ^nrŒ bk bi jfpjl pbkqfal nrb bk i^ FbljbqoŒ^ br`ifaf^k^bibjbkq^i- Clp `lkgrkqlp plk `lkdorbkqbp pf prp mrkqlp mrbabk mlkbopb bk `loobpmlkabk`f^rkl ^ rkl ab jlal nrb i^p afpq^k`f^p pb `lkpbosbk- Dpql bp+pf ^ alp mrkqlp j v k bk rk`lkgrkql `loobpmlkabk k% v l% bk bi lqol+ i^ afpq^k`f^ ab k ^ l ab_b pbo fdr^i ^ i^ abk% ^ l%9 pfbkal bpql `fboql m^o^ rk m^o k* l `r^inrfbo^-

Page 92: Calculus

61 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

QbdfŽkbp`^ilk^a^ '^( Blkgrkql ab '_( QbdfŽkbp`^, 'b(QbdfŽkbpb^,loabk^a^p ilk^a^ fkqboflo ilk^a^ buqboflo

EHFTQ@ 0-03 EHFTQ@ 0-04 @jiepioj _` jm_`i\_\n `i^`mm\_jkjm _jn m`bdji`n `n^\gji\_\n,

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Dumlkd^jlp ^elo^ ilp jbk`flk^alp ^uflj^p-

CDEHMHBHˆM @WHNL„SHB@ CD „QD@- Ppkjib\hjn lp` `sdno` pi\ ^g\n` FE_`^jiepiojn _`g kg\ij h`_d]g`n t pi\ api^d‡i _` ^jiepioj \* ^ptj _jhdidj `n,> *^ji g\n kmjkd`_\_`n ndbpd`io`n8

/, Mmjkd`_\_ _` ij i`b\odqd_\_, M\m\ ^\_\ ^jiepioj R _`,>* n` od`i`[%O&w N-

0, Mmjkd`_\_ \_dodq\, PdR V Q k`mo`i`^`i \,GF* o\h]d„i k`mo`i`^`i \,GF*R S Q V R i Q* t n` od`i`

\&P S Q' < \&P' * \&Q' + \&Ph Q' ,

1, Mmjkd`_\_ _` g\ _da`m`i^d\, Pd R V Q k`mo`i`^`i \,> nd`i_j R z Q*`ioji^`n Q + R `noƒ `i,>* t n` od`i` \&Q+ R( < \&Q' + \&P',

2, Fiq\md\i^d\ kjm ^jibmp`i^d\, Pd pi ^jiepioj R k`mo`i`^` \,> t Q `n^jibmp`io` \ R+o\h]d„i Q k`mo`i`^` \ ,GFv o`i`hjn \&P' < \&Q',

3, Bg`^^d‡i _` `n^\g\, Qj_j m`^oƒibpgj O k`mo`i`^` \ ,GF,Pd gjn g\_jn _`O od`i`i gjibdop_`n c t f* `ioji^`n \&O' < cf,

4, Mmjkd`_\_ _` `sc\p^d‡i, P`\ O pi ^jiepioj lp` kp`_` `i^`mm\mn` `i+om`_jn m`bdji`n R t Q*_` hj_j lp`

'i-i(

Page 93: Calculus

Be`m^d^djn 62

Rf `sdno` pij u n‡gj pi iˆh`mj ` lp` n\odna\^` g\n _`ndbp\g_\_`n

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Page 94: Calculus

0- Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

2- Cbjlpqo^o nrb qlal qo^mbwlfab v qlal m^o^ibildo^jl bp jbaf_ib v abar`fo i^p cŽojri^prpr^ibp m^o^ `^i`ri^o pr Šob^-

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Page 95: Calculus

M\mod^dji`n t api^dji`n `n^\gji\_\n 0.

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Page 96: Calculus

0/ Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

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Page 97: Calculus

Pph\ u kmj_p^oj _` api^dji`n `n^\gji\_\n 66

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Page 98: Calculus

01 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

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Page 99: Calculus

A`adid^d‡i _` dio`bm\g k\m\ api^dji`n `n^\gji\_\n 68

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dril ob`qŠkdril-'_( Cba•w`^pb bpqb obpriq^al ^k^iŒqf`^jbkqb ab i^ j^kbo^ pfdrfbkqb- B^j_f^kal bi Œkaf`b

ab prj^`fŽk+ l_p‹osbpb nrb Gy9eWh[d\Y < Gy9eW[%\ * k(g_\-@miŒnrbkpb irbdl ilp Dgbo,

`f`flp 3'^( v '_( ^i `lo`ebqb ab i^ abob`e^-7- Rb^ R rk `lkgrkql ab mrkqlp bk i^ ob`q^ ob^i- K^ api^d‡i ^\m\^o`m…nod^\ab R zp+mlo ab,

cfkf`fŽk+ i^ crk`fŽk Tf; %r& < 0 m^o^ qlal r ab R+ v Tm %r& < N m^o^ ^nrbiilp mrkqlp nrbkl mboqbkb`bk ^ R- Rb^ ` rk^ crk`fŽk bp`^ilk^a^ nrb qlj^ bi s^ilo `lkpq^kqb @f bk bih,&pfjl pr_fkqbos^il . f ab rk^ `fboq^ m^oqf`fŽk ab rk fkqbos^il W[) \Y+ Cbjlpqo^o nrbm^o^ `^a^ s ab i^ obrkfŽk // q /0 S --- q Fi pb qfbkb

i

a&s' <G@fUy,&U',Q_R

Dpq^ molmfba^a pb bumobp^ af`fbkal nrb qla^ crk`fŽk bp`^ilk^a^ bp rk^ `lj_fk^`fŽkifkb^i ab crk`flkbp `^o^`qboŒpqf`^pab fkqbos^ilp-

)&)* Cbcfkf`fŽk ab fkqbdo^i m^o^ crk`flkbp bp`^ilk^a^p

Dk bpq^ Rb``fŽk pb fkqolar`b i^ abcfkf`fŽk ab fkqbdo^i m^o^ crk`flkbp bp,`^ilk^a^p- Dpq^ abcfkf`fŽk pb e^ ab `lkpqorfo ab j^kbo^ nrb pf rk^ crk`fŽk bpkl kbd^qfs^+ pr fkqbdo^i pb^ rk k•jbol nrb pb ^a^mqb ^ i^ fab^ fkqrfqfs^ ab 0/nrb ~pboŒ^‚ bi Šob^ abi ob`fkql ab loabk^a^p `loobpmlkafbkqb-

Rb^ p rk^ crk`fŽk bp`^ilk^a^ abcfkfa^ bk W[) \Y) u pb^ L < vU[) Ug% ŠŠŠ * rhv

i^ m^oqf`fŽk ab W[) \Y q^i nrb p bp `lkpq^kqb bk ilp pr_fkqbos^ilp ^_fboqlp ab L+

Page 100: Calculus

1) Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

Rb abpfdk^ mlo Oe bi s^ilo `lkpq^kqb nrb qlj^ p bk bi pr_ fkqbos^il ^_fboqlh,‹pfjl+ ab j^kbo^ nrb9

P&s' < Pf pf f < 0+1+ --- +i ,

CDEHMHBHˆM CD HMSDFQ@K CD ETMBHNMDR DRB@KNM@C@R- I\ dio`bm\g _` p_` \ \ ]* lp` n` _`ndbi\ kjm `g n…h]jgj 8 n&s'_s* n` _`adi` h`_d\io` g\ ndbpd`io``ƒlgof[7

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] h

n&s' _s < Pf Š &sf + Uf+/' Š"H h<i

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`lkpq^kqb+ mlo i^ ilkdfqra ab fkqbos^il h,pfjl `loobpmlkafbkqb+ cloj^kal bi mol,ar`ql Pf% &Uf + Uf+g' u pb prj^k irbdl qlalp ilp molar`qlp l_qbkfalp-

N_p‹osbpb nrb ilp s^ilobp ab i^ crk`fŽk bk ilp buqobjlp ab ilp fkqbos^ilpkl pb qlj^k bk `rbkq^ v^ nrb kl ^m^ob`bk bk bi pbdrkal jfbj_ol ab '0-2(- Dkm^oqf`ri^o+ pf p bp `lkpq^kqb bk bi fkqbos^il ^_fboql %[) \&) bp ab`fo+ m%r&< _ pf[ ; r ; \) pb qfbkb bkqlk`bp9

G] i

n&s' _s < b &se * Te*f& < ^&] + \' *\ h<i

fkabmbkafbkqb ab `r^ibp pb^k ilp s^ilobp m%[&v m%\&+Rf _ = M v m%r&< _ m^o^qlal r abi fkqbos^il `boo^al W[)\Y) bi `lkgrkql ab loabk^a^p ab p bp rk ob`qŠk,dril ab _^pb \ * [ v ^iqro^ `: i^ fkqbdo^i ab p bp ]%\ * [&) bi Šob^ ab bpqbob`qŠkdril- B^j_f^kal bi s^ilo ab p bk rkl ab ilp mrkqlp \ l ] l bk ^j_lp+`^j_f^ bi `lkgrkql ab loabk^a^p mbol kl pb ^iqbo^ i^ fkqbdo^i abp l bi Šob^ abpr `lkgrkql ab loabk^a^p- Olo bgbjmil+ ilp alp `lkgrkqlp ab loabk^a^p ab i^cfdro^ 0-11 qfbkbk Šob^p fdr^ibp-

t t t

2

Š

1 2 l 1 2s

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jm_`i\_\n,

EHFTQ@ 0-12 @jiepioj _` jm_`i\_\n_` pi\ api^d‡i `n^\gji\_\,

Page 101: Calculus

Mmjkd`_\_`n _` g\ dio`bm\g _` pi\ api^d‡i `n^\gji\_\ 70

Di `lkgrkql ab loabk^a^p ab `r^inrfbo crk`fŽk bp`^ilk^a^ p `lkpq^ ab rkk•jbol cfkfql ab ob`qŠkdrilp+ rkl mlo `^a^ fkqbos^il bk nrb i^ crk`fŽk bp `lkp,q^kqb: bi `lkgrkql ab loabk^a^p mrbab q^j_f‹k mlpbbo l `^ob`bo ab `fboqlp pbd,jbkqlp sboqf`^ibp+ abmbkafbkal ab i^ j^kbo^ `Žjl i^ crk`fŽk bpqŠ abcfkfa^ bk ilpmrkqlp ab pr_afsfpfŽk- K^ fkqbdo^i ab p bp fdr^i ^ i^ prj^ ab i^p Šob^p ab `^a^rkl ab ilp ob`qŠkdrilp+ mobp`fkafbkal ab ilp s^ilobp nrb qlj^ p bk ilp mrkqlp abafsfpfŽk- Dpql bpqŠ ab ^`rboal `lk bi eb`el ab nrb ilp pbdjbkqlp sboqf`^ibp qfbkbkŠob^ `bol v kl `lkqof_rvbk bk k^a^ ^i Šob^ abi `lkgrkql ab loabk^a^p- Dk i^ cfdr,o^ 0-12+ i^ crk`fŽk bp`^ilk^a^ p qlj^ ilp s^ilobp `lkpq^kqbp 1+ 0+v z bk ilp fkqbo,s^ilp ^_fboqlp '0+ 1(+ '1+ 4(+ X '4+ 5( obpmb`qfs^jbkqb- Rr fkqbdo^i bp fdr^i ^

pm%r&r < 1• '1 , 0( * 0 - '4 , 1( * y-'5 , 4( < y3b •

Cb_b l_pbos^opb nrb i^ cŽojri^ m^o^ i^ fkqbdo^i bk '0-2( kl abmbkab ab i^bib``fŽk ab i^ m^oqf`fŽk M jfbkqo^p p pb^ `lkpq^kqb bk ilp pr_fkqbos^ilp ^_fboqlpab L+ Olo bgbjmil+ pf pb+prpqfqrvb L mlo rk^ m^oqf`fŽk jŠp cfk^ O&^•^afbkal rkkrbsl mrkql ab afsfpfŽk o* pfbkal u+ ; o ; u-- Dkqlk`bp bi mofjbo q‹ojfkl abipbdrkal jfbj_ol ab '0-2( pb obbjmi^w^ mlo ilp alp q‹ojfklp n*%'p , sj' Xp-9 &Ug + q(+X ilp obpq^kqbp q‹ojfklp kl `^j_f^k- Orbpql nrb

RH- %f* ul( * RH- &Ug+ f&< RH- &Ug+ sj'*

bi s^ilo ab qla^ i^ prj^ kl `^j_f^- Olabjlp m^p^o ab M ^ `r^inrfbo m^oqf`fŽkjŠp cfk^ kd ^•^afbkal ilp krbslp mrkqlp ab pr_afsfpfŽk rkl qo^p lqol- Dk `^a^m^pl+ i^ prj^ ab '0-2( kl `^j_f^+ ab jlal nrb i^ fkqbdo^i bp i^ jfpj^ m^o^ qlalpilp ^cfk^jfbkqlp ab L+

)&)+ C_\]VRQNQRQRYNV[aRT_NYQRb[N Sb[PVp[R`PNY\[NQN

Dk bpq^ Rb``fŽk pb a^k rk^p molmfba^abp crka^jbkq^ibp ^ i^p nrb p^qfpc^`bi^ fkqbdo^i ab rk^ crk`fŽk bp`^ilk^a^- K^ j^vlo m^oqb ab bpq^p molmfba^abpm^ob`bk l_sf^p `r^kal pb fkqbomobq^k dblj‹qof`^jbkqb v ^idrk^p ab bii^p fk`ir,pl qofsf^ibp- Sla^p bpq^p molmfba^abp plk sŠifa^p m^o^ fkqbdo^ibp ab crk`flkbpjŠp dbkbo^ibp+ v bpq^_ib`fa^p m^o^ bi `^pl ab crk`flkbp bp`^ilk^a^p+ i^p abjlp,qo^`flkbp m^o^ bi `^pl dbkbo^i pboŠk pfjmibp `lkpb`rbk`f^p- K^p molmfba^abp pba^k `ljl qblobj^p v bk `^a^ `^pl pb a^ rk^ fkqbomobq^`fŽk dblj‹qof`^ mlo jbaflab i^p Šob^p- K^p abjlpqo^`flkbp ^k^iŒqf`^pab ilp jfpjlp pb a^k bk i^ Rb``fŽk 0-04-

K^ mofjbo^ molmfba^a bpq^_ib`b nrb i^ fkqbdo^i ab rk^ prj^ ab alp crk`fl,kbp bp`^ilk^a^p bp fdr^i ^ i^ prj^ ab i^p fkqbdo^ibp- …pq^pb `lkl`b `ljl i^ mol,mfba^a \_dodq\ v pb obmobpbkq^bk i^ cfdro^ 0-13-

Page 102: Calculus

1+ Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

n

\ ] \ ] \ ]

EHFTQ@ 0-13- I\ kmjkd`_\_ \_dodq\ _` g\, dio`bm\g,

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F7&n&s'* o&s'Z_s <F7n&s' _s * F7o&s'_s ,

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0n

p

\ ] \ ]

EHFTQ@ 0-14 I\ kmjkd`_\_ cjhjb„i`\ _` g\ Fio`bm\g &^ji ` < 1(-

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Page 103: Calculus

Mmjkd`_\_`n _` g\ dio`bm\g _` pi\ api^d‡i `n^\gji\_\ 72

Di qblobj^ nrb pfdrb bp rk qblobj^ ab ^jhk\m\^d‡i v bumobp^ nrb pf rk^crk`fŽk bp`^ilk^a^ qfbkb bk qlal bi fkqbos^il W[) ] I s^ilobp j^vlobp nrb lqo^+ prfkqbdo^i bk bpqb fkqbos^il q^j_f‹k bp j^vlo-

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`7n&s' _s ; `7o&s'_s ,

K^ fkqbomobq^`fŽk dblj‹qof`^ ab bpqb qblobj^ fkaf`^+ nrb pf rk `lkgrkql ab loab,k^a^p bpqŠ `lkqbkfal bk lqol+ bi Šob^ ab i^ obdfŽk jbklo kl mrbab bu`babo ^ i^ abi^ j^vlo-

K^p molmfba^abp nrb pb e^k a^al e^pq^ ^elo^ pb obcfbobk qla^p bii^p ^ crk,`flkbp bp`^ilk^a^p abcfkfa^p bk rk fkqbos^il `lj•k- K^ fkqbdo^i qfbkb lqo^p mol,mfba^abp fjmloq^kqbp nrb obi^`flk^k fkqbdo^ibp abcfkfa^p bk fkqbos^ilp afpqfkqlp-Dkqob bii^p pb qfbkb bi pfdrfbkqb

SDNQDL@ 0-5- @CHSiUHC@C QDRODBSN @K HMSDQU@KN CD HMSDFQ@BHˆM-

F7n&s' _s * F7n&s'_s < `7n&s'_s pf \:^:],

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Di qblobj^ nrb pfdrb bumobp^ i^ diq\md\i^d\ am`io` \ pi\ om\ng\^d‡i, Rf bi`lkgrkql ab loabk^a^p pb ~qo^pi^a^‚ rk^ `^kqfa^a b+ bi `lkgrkql ab loabk^a^pobpriq^kqb bp bi ab lqo^ crk`fŽk bp`^ilk^a^ o nrb pb abar`b ab i^ p mlo jbafl abi^ b`r^`fŽk9 o&s' < n&s + `', Rf p bpqŠ abcfkfa^ bk X\* ] I+ o bpq^oŠ abcfkfa^ bk

p

,EHFTQ@ 0-15 >_dodqd_\_ ^ji m`nk`^oj

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W[ * b+ ] * bI+ v prp `lkgrkqlp ab loabk^a^p qfbkbk i^ jfpj^ Šob^- Dpq^ molmfb,a^a pb bumobp^ ^k^iŒqf`^jbkqb `ljl pfdrb9

Page 104: Calculus

73 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

SDNQDL@ 0-6- HMU@QH@MBH@EQDMSD @ TM@ SQ@RK@BHˆM-

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k\m\ oj_j m`\g ^ ,

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K^ molmfba^a eljld‹kb^ fkaf`^ `rŠi bp i^ s^of^`fŽk nrb bumbofjbkq^ rk^fkqbdo^i `r^kal pb bcb`q•^ rk `^j_fl ab bp`^i^ bk bi bgb v- Di qblobj^ nrb pba^ ^ `lkqfkr^`fŽk pb obcfbob rk `^j_fl ab bp`^i^ bk bi bgbs, Rf n bp rk^ crk`fŽkbp`^ilk^a^ abcfkfa^ bk bi fkqbos^il W[)\Y X pb ^iqbo^ i^ bp`^i^ bk i^ afob``fŽkelofwlkq^i+ jriqfmif`^kal `^a^ `lloabk^a^ r mlo rk c^`qlo f = N+ i^ krbs^doŠcf`^ bp i^ ab lqo^ crk`fŽk bp`^ilk^a^ o abcfkfa^ bk bi fkqbos^il We[)ebY X ob,i^`flk^a^ `lk n mlo jbafl ab i^ b`r^`fŽk9

pf f\ x s x f],

K^ cfdro^ 0-17 mobpbkq rk bgbjmil `lk f < 1+ X pb sb nrb i^ cfdro^ ^iqbo^a^qfbkb Šob^ al_ib nrb i^ ab i^ cfdro^ lofdfk^i- Dk dbkbo^i+pf pb `lkpfabo^ rk c^`qlo

n

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'0-3( E7n&s'_s < , `7n&s'_s pf [ ; \+

Page 105: Calculus

Lom\n ijo\^dji`n k\m\ g\n dio`bm\g`n 74

Rb abcfkb q^j_f‹k9

b!p' u( _s < N +Š \

abcfkf`fŽk prdbofa^ ^i e^`bo \ < ] bk '0-3(- Dpqlp `lksbkflp mbojfqbk ^cfoj^onrb bi qblobj^ 0-5 bp sŠifal kl pŽil pf ` bpqŠbkqob \ u \) pfkl m^o^ `r^inrfboloabk^`fŽk ab \* ]* `- Di qblobj^ 0-5 pb bp`of_b jr`e^p sb`bp bk i^ cloj^

pn&s' _s * qn&s' _s * qn&s'_s < N-

@kŠild^jbkqb pb mrbab buqbkabo bi `^jml ab s^ifabw abi qblobj^ 0-7+ ^i `^plbk nrb f pb^ kbd^qfsl- Dk m^oqf`ri^o+r^kal f < , 0+bi qblobj^ 0-7 v i^ fdr^i,a^a '0-3( klp a^k

F] m+\\ n&s'_s < K_ p' +s' _s ,

t t

p %r&< p' *r&

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Ki^j^objlp ^ ‹pq^+kmjkd`_\_ _` m`ag`sd‡i ab i^ fkqbdo^i+v^ nrb i^ doŠcf`^ ab i^crk`fŽk n a^a^ mlo n%r&< p' *r& pb l_qfbkb ab i^ crk`fŽk p mlo obcibufŽkobpmb`ql^i bgbv- Dk i^ cfdro^ 0-18 pb obmobpbkqrk bgbjmil-

)&), Ba_N`[\aNPV\[R`]N_NYN V[aRT_NYR`

K^ ibqo^s lp` \k\m`^` `i `g n…h]jgj `wn&s' _s kl grbd^ kfkd•k m^mbibpbk,`f^i bk i^ abcfkf`fŽk ab Šob^-Br^inrfbo lqol pŒj_lil ^ab`r^al pbosfoŠbu^`q^jbkqbfdr^i- Rb rp^k cob`rbkqbjbkqb m^o^biil i^p ibqo^po*p* q* Y+ v bk sbw ab `wn&s' _spb mrbab bp`of_fo `wn&o'_ n) Px n`p' _p* bq`-+pfbkal `lkpfabo^a^p qla^p `ljl kl,q^`flkbp afsbop^p m^o^rk^ jfpj^ `lp^- Klp pŒj_lilp s* o*p* bq`-+nrb pb rqfifw^k

Page 106: Calculus

75 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

bk bpqbpbkqfal+ pb abkljfk^k ~s^of^_ibp ^m^obkqbp‚-Rlk ^kŠild^p ^ ilp Œkaf`bp^m^obkqbprp^alp bk ilp prj^qloflp-

@idrklp ^rqlobp ab if_olp ab BŠi`ril qfbkbk qbkabk`f^ ^ prmofjfo pfjriqŠ,kb^jbkqb i^ s^of^_ib ^m^obkqbv bi pŒj_lil _ v bp`of_fo m^o^ i^ fkqbdo^i pfjmib,jbkqb `wp- Tk^ o^wŽkm^o^ rqfifw^obpqbpŒj_lil ^_obsf^al+ bp nrb bumobp^`lkjŠp crbow^nrb i^ fkqbdo^iabmbkab pli^jbkqb ab i^ api^d‡i p v abi dio`mq\gj X\* ]Z,@pŒ+idrk^p cŽojri^p qlj^k rk^ cloj^ jŠp pfjmib `lk bpq^klq^`fŽk- Olo bgbj,mil+i^ molmfba^a ^afqfs^ pb bumobp^9w&n) n&< `wp * `wn+Rfk bj_^odl+ obpriq^jŠp `ljmif`^al bp`of_fo^idrk^p cŽojri^p+ `ljl mlo bgbjmil+ i^p ab ilp qblobj^p0-6 v 0-7 `lk i^ klq^`fŽk ^_obsf^a^- LŠp fjmloq^kqbp plk+ pfk bj_^odl+ i^p sbkq^,g^p moŠ`qf`^pnrb mobpbkq^i^ klq^`fŽk lofdfk^i ab Kbf_kfw `ljl pb sboŠ jŠp^abi^kqb- Di pŒj_lil _s* nrb bk bpqbjljbkql `^pf m^ob`b prmbocirl+obpriq^ pbork fkpqorjbkql jrv •qfi bk i^ moŠ`qf`^ab `Ši`rilp `lk fkqbdo^ibp-

)&)- :WR_PVPV\`

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'b( bhW/rY^r+

'b(byhW*rY ^r+

1- C^o rk bgbjmil ab crk`fŽk bp`^ilk^a^ p abcfkfa^ bk bi fkqbos^il `boo^al Z/+4\+ nrbqbkd^ i^p pfdrfbkqbp molmfba^abp9 Pxm%r&r < 4+ Pxm%r&r < 1-

2- Ool_^o nrb `wWrYr * `wW*rY ^r < [*\+

3- '^( Rf h bp rk bkqbol mlpfqfsl+ abjlpqo^o nrb Px WnYn < h%h* 0(.1-

'_( Rf `%r& < Px WnYn m^o^ r w N+af_rg^o i^ doŠcf`^ ab ` pl_ob bi fkqbos^il Z/+3\-

4- '^( Cbjlpqo^o nrb `wWn/Yn < 4 , U1 , sf'_( B^i`ri^o Px1 Xo0Z _o*

5- '^( Rf h bp rk bkqbol mlpfqfsl abjlpqo^o nrb Px WnY/n < h%h* i('1k , 0(.5-

'_( Rf a&s' < `wXoZ0 _o m^o^ s x N+af_rg^o i^ doŠcf`^ ab ` bk bi fkqbos^il Z/+2\-

'b( G^ii^o qlalp ilp s^ilobp ab r = N m^o^ ilp nrb `wXoZ0 _o < /%r * 0(-

6- '^( B^i`ri^o `wWRnY_o,

'_( Rf h bp rk bkqbol mlpfqfsl+ abjlpqo^o nrb `7$Y/[_o < h%h* .&%1h* 0(.5-

Page 107: Calculus

Be`m^d^djn 10

7- Oor‹_bpb nrb i^ molmfba^a ab qo^pi^`fŽk 'qblobj^ 0-6( pb mrbab bumobp^o bk i^ cloj^pfdrfbkqb9

G])@ G]a&s' _s < a&s * `' _s,3%4 3

8- Ool_^o nrb i^ molmfba^a pfdrfbkqb bp bnrfs^ibkqb ^i qblobj^ 0-7-

a%f] a%]a&s' _s < f a&fs' _s ,e[ [

0/- C^al rk bkqbol mlpfqfsl j+ Tk^ crk`fŽk bp`^ilk^a^ p bpqŠ abcfkfa^ bk bi fkqbos^il ZN+jY`ljl pfdrb9 n&s' < &[g'ii pf s bpqŠ bk bi fkqbos^il i x s ; i * )$ pfbkal

i < N+&"'" $$$ "A # 0: n&k' < N- OŽkd^pb a&k' < c n&s' _s,

'^( B^i`ri^o .'2(+ .'3( u -&/&1~,'_( ƒO^o^ nr‹ s^ilo 'l s^ilobp( ab j bp Fa&k'Y < 6>

00- Rf bk ird^o ab abcfkfo i^ fkqbdo^i ab rk^ crk`fŽk bp`^ilk^a^ rqfifw^kal i^ cŽojri^ '0-2(pb qlj^o^ `ljl abcfkf`fŽk9

G] in&s' _s < z nx , &se * Te*f& )

\ h<i

pb qbkaoŒ rk^ krbs^ qbloŒ^ ab i^ fkqbdo^`fŽk afpqfkq^ ab i^ a^a^- ƒBrŠibp ab i^p pf,drfbkqbp molmfba^abp pbdrfoŒ^k pfbkal sŠifa^p bk i^ krbs^ qbloŒ^>

'^( Fwn * 09n <pn,

'_( 09&n* q( < 09n * 09o,

%] G*]'b( H\ ? † O < ? \ O+

G*])@ G]'a( n&s' _s < n&s * `' _s,\)^ \

'b( Rf m%r&; n%r& m^o^ `^a^ r bk W[)\Y) bkqlk`bp I9m ; F7n+

01- Qbplisbo bi Dgbo`f`fl 00 rqfifw^kal i^ abcfkf`fŽk

b_ h

m%r&r < z Oe † %rw* WY]i( -Š \ h<i

Dk ilp Dgbo`f`flp nrb pfdrbk pb mfabk i^p abjlpqo^`flkbp ^k^iŒqf`^p ab i^p molmfb,a^abp ab i^ fkqbdo^i a^a^p bk i^ Rb``fŽk 0-02- K^p abjlpqo^`flkbp ab ilp qblobj^p 0-2v 0-7 pb mobpbkq^k `ljl bgbjmil- O^o^ i^p lqo^p pb a^oŠk fkaf`^`flkbp-

A`hjnom\^d‡i _`g o`jm`h\ 0-29 P8` , n&s' _s < b P8n&s' _s m^o^ `^a^ k•jbol ob^i b-

Rb^ L < uri) Wi&--- +rh. rk^ m^oqf`fŽk ab W[) \Y q^i nrb p bp `lkpq^kqb bk ilp pr_fk,

Page 108: Calculus

11 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

qbos^ilp ^_fb,oqlp ab L+ Rb^ m%r&< Oe

Dkqlk`bp+ _ +m%r&< _ +Pf pf Uf+g 9cfkf`fŽk ab fkqbdo^i pb qfbkb9

pf TeZf ; r ; Te %e< 0+1+--- +h&+

s ; Te) v+ mlo q^kql+ bk sfoqra ab i^ ab,

A`hjnom\^d‡i _`g o`jm`h\ /,68

pf e = N-

Rb^ L:uri) TE) +!) riw rk^ m^oqf`fŽk abi fkqbos^il W[)\Y q^i nrb p bp `lkpq^kqb bkilp pr_ fkqbos^ilp ^_fboqlp ab L+ RrmŽkd^pb nrb m%r&< Pd pf Ud[/ ; r ; Ud, Rb^ n%r&:: n&s-f' pf f\ ; s ; f], Dkqlk`bp o&s' < Pd pf s mboqbkb`b ^i fkqbos^il ^_fboqlofs,,~* fs'9 mlo q^kql+ M%< vfsj* fs,* ,,, * fsiw bp rk^ m^oqf`fŽk ab Xf\* f]Z v o bp`lkpq^kqb bk ilp pr_fkqbos^ilp ^_fboqlp ab L$+ Olo q^kql+ o bp rk^ crk`fŽk bp`^ilk^a^`rv^ fkqbdo^i bp9

bS! ! Gc) $&n%r&r < xPd% %erx * erE' < e xPd% &Ud+ Ud+g' < e m%r&r +Šf\ f<i d;g \

02- Cbjlpqo^o bi qblobj^ 0-1 'molmfba^a ^afqfs^(- XFi_d^\^d‡i8 @miŒnrbpbi^ molmfba^a ^af,

qfs^ m^o^ prj^p9 ww:f%[P$ * ]e& < zzzi [e * zzzi ]e +Y

03- Cbjlpqo^o bi qblobj^ 0-3 'molmfba^a ifkb^i(- XFi_d^\^d‡i8 @miŒnrbpbi^ molmfba^a ^af,qfs^ v i^ molmfba^a eljld‹kb^-\

04- Cbjlpqo^o bi qblobj^ 0-4 'qblobj^ ab `ljm^o^`fŽk(+ XFi_d^\^d‡i8 @miŒnrbpbi^ molmfb,

a^a `loobpmlkafbkqb m^o^ prj^p9 y<h[e ; yyh]e pf [e ; ]e m^o^ f < 0+1+ --- +h+Y

05- Cbjlpqo^o bi qblobj^ 0-05 '^afqfsfa^a `lk obpmb`ql ^i fkqbos^il(- XFi_d^\^d‡i8 Rf NGbprk^ m^oqf`fŽk ab W[) `( v L) rk^ m^oqf`fŽk ab W]) \Y) ilp mrkqlp ab LE) grkql `lk ilp abL) cloj^k rk^ m^oqf`fŽk ab W[) \Y+Y

06- Cbjlpqo^o bi qblobj^ 0-6 'fks^of^k`f^ cobkqb ^ rk^ qo^pi^`fŽk(- XFi_d^\^d‡i8 Rf

L < vsj* Wi + --- *s**w bp rk^ m^oqf`fŽk ab X\*\Y ) bkqlk`bp L$ < vsj * B+Tf * B+ ‘‘‘ +

r! * `y bp rk^ m^oqf`fŽk ab W[ * B&+] * _Y+Y

0-05 K^ fkqbdo^iab crk`flkbp jŠp dbkbo^ibp

K^ fkqbdo^iPx m%r&r pb e^ abcfkfal m^o^ rk^ crk`fŽk bp`^ilk^a^- Dk bpqb^m^oq^al pb a^oŠ rk^ abcfkf`fŽk ^mif`^_ib ^ crk`flkbp jŠp dbkbo^ibp n+K^ abcf,kf`fŽk pb `lkpqorfoŠ ab q^i j^kbo^ nrb i^ fkqbdo^i obpriq^kqbdl`b ab qla^p i^pmolmfba^abpa^a^p bk bi ^m^oq^al 0-02-

Page 109: Calculus

H[ chn_al[f ^_ `oh]cih_m g•m a_h_l[f_m 12

. .+@molufj^`fŽk mlo bu`bpl

**,`]]]]]]]] +-,4+@molufj^`fŽk mlo abcb`ql

\ ]

EHFTQ@ 0-2/ >kmjsdh\^d‡i kjm `s^`nj v kjm _`a`^oj _` pi\ api^d‡i a kjm h`_dj _`api^dji`n `n^\gji\_\n,

Di j‹qal bpqŠfkpmfo^al bk bi ab @onrŒjbabp nrb pb bumrpl bk i^ Rb``fŽk 0 0-2-K^ fab^ bp pfjmibjbkqb ‹pq^9 pb bjmfbw^ mlo ^molufj^o mlo abcb`ql v mlo bu`bpli^ crk`fŽk ` jbaf^kqb crk`flkbp bp`^ilk^a^p+ `ljl pb prdfbob bk i^ cfdro^ 0-2/-O^o^ biil pb prmlkb nrb pb bifdb rk^ crk`fŽk bp`^ilk^a^ ^o_fqo^of^+abpfdk^a^ mlop+ `rv^ doŠcf`^ bpqŠ mlo ab_^gl ab i^ ab d*v rk^ crk`fŽk bp`^ilk^a^ ^o_fqo^of^+abpfdk^a^ mlo o*`rv^ doŠcf`^ bpqŠ mlo bk`fj^ ab i^ ab `+ Rf ^elo^ pb `lkpfabo^ bi`lkgrkql ab qlalp ilp k•jbolp Px m%r&r) v Pxn%r&r l_qbkfalp bifdfbkal p v nab qla^p i^p j^kbo^p mlpf_ibp+ pb qfbkb bk sfoqrA abi qblobj^ ab i^ `ljm^o^`fŽk9

.7 m%r&r ; 09n%r& r

Rf i^ fkqbdo^i ab `) e^ ab `rjmifo q^j_f‹k bi qblobj^ ab i^ `ljm^o^`fŽk+ e^ abpbo rk k•jbol `ljmobkafal bkqob `wm%r&r v `wn%r&r m^o^ `^a^ m^o p v n abcrk`flkbp ab ^molufj^`fŽk- Rf bufpqb rk h„g_li „hc]i `lk bpq^ molmfba^a+ m^ob`biŽdf`l qlj^o bpqb k•jbol `ljl abcfkf`fŽk ab fkqbdo^i ab `+

Tk^ pli^ `lp^ mrbab afcf`riq^o bpqb mol`bpl+ v bpq^ afcf`riq^a pb mobpbkq^jrv ^i mofk`fmfl- Cbpdo^`f^a^jbkqb kl bp mlpf_ib ^molufj^o qla^ crk`fŽk pr,mboflojbkqb b fkcboflojbkqb mlo gkbafl ab crk`flkbp bp`^ilk^a^p- Olo bgbjmil+i^ crk`fŽk ` a^a^ mlo i^p b`r^`flkbp9

0a&s' < ,

spf r s : N+ a`L' < k+

bpqŠ abcfkfa^ m^o^ qlal k•jbol ob^i r) mbol bk kfkd•k fkqbos^il W[)\Y nrb `lk,qbkd^ bi lofdbk pb mrbab `lkqlokb^o ` jbaf^kqb crk`flkbp bp`^ilk^a^p- Dpql bpab_fal ^ nrb bk i^ molufjfa^a abi lofdbk+ ` qfbkb s^ilobp ^o_fqo^of^jbkqb do^k,abp+ l af`el ab lqol jlal+ ` kl bpqŠ ^`lq^a^ bk bi bkqlokl abi lofdbk 's‹^pb

Page 110: Calculus

2) Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

cfdro^ 0-20(- Olo biil+ ^i qo^q^oab abcfkfo i^ fkqbdo^i+bp mob`fpl obpqofkdfopb i^pcrk`flkbp nrb plk \^jo\_\n bk X\* ]Z bp ab`fo+ ^ ^nrbii^p crk`flkbp a m^o^ i^p`r^ibp bufpqbrk k•jbol L = N q^i nrb9

'0-4( +Jxa&s' x J

m^o^ `^a^ r bk W[) \Y+ Fblj‹qof`^jbkqb+ i^ doŠcf`^ ab q^ibp crk`flkbp bpqŠ pf,qr^a^ bkqobi^p doŠcf`^pab alp crk`flkbp bp`^ilk^a^p p v o nrb qlj^k ilp s^il,

t

t

J +++++++++++++++++++++++++n%r&< J

s l s

+ J **********************m%r&< , J

EHFTQ@ 0-20 Cpi^d‡i ij \^jo\_\ EHFTQ@ 0-21 Cpi^d‡i \^jo\_\,

obp , L X L obpmb`qfs^jbkqb's‹^pb cfdro^ 0-21(- Dk bpqb`^pl pb af`b nrb ` bpqŠ^`lq^a^ mlo L- K^p alp abpfdr^ia^abp bk '0-4( pb mrbabk bp`of_fo bk i^ cloj^9

Ga&s' GpI+

Qbprbiql bpqbmrkql+ pb mrbab ob^ifw^obi mi^k abp`ofql ^kqbp v clojri^o i^abcfkf`fŽk ab fkqbdo^i-

CDEHMHBHˆM CD HMSDFQ@K CD TM@ ETMBHˆM @BNS@C@- P`\ a pi\ api^d‡i_`adid_\ v \^jo\_\ `i X\* ]Z, P`\i p v o api^dji`n `n^\gji\_\n \m]dom\md\n_`ad+id_\n `i X\* ]Z o\g`n lp`

'0-5( n&s' x a&s' x o&s'

k\m\ ^\_\ s `i X\* ],Z Pd `sdno` pi iˆh`mj /* u n‡gj pij* o\g lp`

'0-6( .7 n&s'_s x / x 09o&s'_s

Page 111: Calculus

Fio`bm\g`n npk`mdjm ` dia`mdjm 80

k\m\ ^\_\ k\m _` api^dji`n `n^\gji\_\n p t o lp` q`mdadlp`i g\n '0-5(+ `ioji^`n`no` iˆh`mj F n` _`ijhdi\ g\ dio`bm\g _` a _`n_` \ \ ] t n` di_d^\ kjm `g n…h]j+gj `wa&s' _s l `w`{@p\i_j F `sdno` n` _d^` lp` a `n dio`bm\]g` `i X\* ]Z,

Rf \ ; ] pb abcfkb F a&s' _s < , `wa&s' _s prmrbpq^a fkqbdo^_ibbk X\* ]Z,S^j_f‹k pb abcfkb `wa&s' _s < N-Rf a bp fkqbdo^_ibbk X\* ]Z* pb af`b nrb i^ fkqb,do^i `wa&s' _s bufpqb-K^ crk`fŽk a pb abkljfk^ dio`bm\i_j* ilp k•jbolp \ v ]ilp g…hdo`n_` dio`bm\^d‡i* v bi fkqbos^il X\* ]Z bi dio`mq\gj _` dio`bm\^d‡i,

)&)/ =[aRT_NYR``b]R_V\_R V[SR_V\_

Rrmlkd^jlp i^ crk`fŽk ` ^`lq^a^ bk W[)\Y+ Rf p v o plk crk`flkbp bp`^ilk^,a^p nrb p^qfpc^`bk'0-5(+ pb af`b nrb p bp fkcboflo ^ o,v nrb o bp prmboflo ^ d*vbp`of_fjlp p z ` w o,

Rb^ P bi `lkgrkql ab qlalp ilp k•jbolp `wn&s' _s l_qbkfalp ^i qlj^o `ljlp qla^p i^p crk`flkbp bp`^ilk^a^p fkcboflobp^ o,v pb^ P bi `lkgrkql ab qlalp ilpk•jbolp `wo&s'_s ^i qlj^o `ljl o qla^p i^p crk`flkbp bp`^ilk^a^p prmboflobp^ a,Dpql bp+

p< wpn&s'_s- p xaw* Q; xI9o&s'_s Fax ow,

Klp alp `lkgrkqlp P v Q plk kl s^`Œlp mrbpql nrb ` bp ^`lq^a^- @pfjfpjl+`wn&s' _s x `wo&s'_s pf p z a x o*ab jlal nrb qlal k•jbol ab P bp jbklo nrb`r^inrfbo^ ab Q, Olo `lkpfdrfbkqb+ pbd•k bi qblobj^ 0-23+P qfbkbbuqobjl prmboflo+v Q buqobjl fkcboflo+nrb p^qfpc^`bki^p abpfdr^ia^abp

qn&s' _s x prm R z fkc P:788qo&s'_s

m^o^ qla^p i^p p v q nrb p^qfpc^`bk p <9:: ` <9:: o, Dpql abjrbpqo^ nrb q^kql prmP`ljl fkc P p^qfpc^`bk'0-6(- Olo il q^kql+ ` bp fkqbdo^_ibbk W[)\Y pf v pŽil pfprm P < fkc Q, bk `rvl `^pl pb qfbkb

oa&s'_s < prmR < ch`P+

Di k•jbol prmP pb ii^j^ dio`bm\g dia`mdjmab a v pb obmobpbkqmlo .'0(- Di k•,jbol Q pb ii^j^ dio`bm\g npk`mdjm ab a v pb obmobpbkqmlo g`-w, @pŒnrb qbkbjlp

g&a' < fkc qH9o&s'_s Ga <9:: ow,

Di o^wlk^jfbkql mob`babkqbabjrbpqo^ bi qblobj^ pfdrfbkqb-

Page 112: Calculus

2+ Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

RCMPCK? 0-8- Qj_\ api^d‡i a \^jo\_\ `i X\* ]Z od`i` pi\ dio`bm\g dia`mdjm`Q& t pi\ dio`bm\g npk`mdjm -&a' lp` n\odna\^`i g\n _`ndbp\g_\_`n

.7 n&s' _s x .'b( y .'b( y F7o&s'_s

k\m\ oj_\n g\n api^dji`n p t o o\g`n lp` p z a x o, I\ api^d‡i a `n dio`bm\]g` `iX\* ]Z nd t n‡gj nd npn dio`bm\g`n npk`mdjm ` dia`mdjm nji dbp\g`n* `i ^ptj ^\njn` od`i`

F7a&s' _s < .'b( < /&a' ,

)&)0 :Y m_RNQRb[ P\[Wb[a\ QR\_QR[NQNRd]_R`NQNP\Z\ b[N V[aRT_NY

Di `lk`bmql ab Šob^ pb fkqolargl ^ufljŠqf`^jbkqb bk i^ Rb``fŽk 0-5 `ljlrk^ crk`fŽk ab `lkgrkql nrb qfbkb `fboq^pmolmfba^abp-@ m^oqfoab bp^p molmfb,a^abp pb abjlpqoŽ nrb bi Šob^ ab rk `lkgrkql ab loabk^a^p ab rk^ crk`fŽk bp`^,ilk^a^ kl kbd^qfs^ bp fdr^i ^ i^ fkqbdo^i ab i^ crk`fŽk- @elo^ abjlpqo^jlp nrbbpl q^j_f‹k bp `fboql m^o^ `r^inrfbo crk`f5k fkqbdo^_ibkl kbd^qfs^- Qb`r‹oabpbnrb bi `lkgrkql ab loabk^a^p ab rk^ crk`f5k kl kbd^qfs^ ` pl_ob rk fkqbos^ilW[)\Y bp bi `lkgrkql ab qlalp ilp mrkqlp %r)s& nrb p^qfpc^`bki^p abpfdr^ia^abpM y V x a&s'* \ x s x ],

RCMPCK? 0-0/- P`\ a pi\ api^d‡i ij i`b\odq\* dio`bm\]g` `i pi dio`mq\gjX\* ]Z* t n`\ O `g ^jiepioj _` jm_`i\_\n _` a nj]m` X\* ]Z, Bioji^`n O `n h`_d]g`t np ƒm`\ `n dbp\g \ g\ dio`bm\g Pxa&s' _s,

A`hjnom\^d‡i, Rb^k R u Q alp obdflkbp bp`^ilk^a^p nrb p^qfpc^`bkR z O y P+Dufpqbkalp crk`flkbp bp`^ilk^a^p p u o nrb p^qfpc^`bkp z ` w o bkW[)\Y) q^ibp nrb

[%O&< F7m%r&r v \&Q' < F7o&s'_s ,

Orbpql nrb a bp fkqbdo^_ib bk X\* ]Z* bi k•jbol / < P8a&s' _s bp bi •kf`l nrbp^qfpc^`bi^p abpfdr^ia^abp

F7n&s' _s x . w F7o&s'_s

m^o^ qla^p i^p crk`flkbp bp`^ilk^a^p n v o nrb `rjmibk p z ` 77788o, Olo `lkpf,drfbkqb ‹pb bp q^j_f‹k bi •kf`l k•jbol nrb p^qfpc^`b[%O&w . 77788[%P&m^o^qla^pi^p obdflkbp bp`^ilk^a^p R u Q q^ibpnrb R o O o Q, Rbd•k i^ molmfba^aab bue^r,`fŽk+bpql abjrbpqo^ nrb O bp jbaf_ib v nrb [%M&< g,

Page 113: Calculus

L]n`mq\^dji`n m`g\odq\n \ g\ o`jm…\ t o„^id^\ _` g\ dio`bm\^d‡i 71

Rb^k P bi `lkgrkql ab loabk^a^p abi qblobj^ 0-0/+ v P& bi `lkgrkql nrbnrba^ pf pb nrfq^k ab O ilp mrkqlp ab i^ doŠcf`^ab `+ Dpql bp+

M$< v&s*s& G\ x s x \ ) M y U ; y&s'w ,

Di o^wlk^jfbkql rqfifw^al m^o^ abjlpqo^o bi qblobj^ 0-0/ q^j_f‹k abjrbpqo^nrb P&bp jbaf_ib v nrb \&L%' < \&N', Olo `lkpfdrfbkqb+ pbd•k i^ molmfba^a abi^ afcbobk`f^ obi^qfs^ ^i Šob^+bi `lkgrkql P , P& bp jbaf_ib v

\&N + N%' < \&N' + \&N%'< N-

k pb^+ebjlp abjlpqo^al bi pfdrfbkqb qblobj^-

RCMPCK? 0-00- P`\ a pi\ api^d‡i ij i`b\odq\* dio`bm\]g` `i pi dio`mq\gjX\* ]Z, I\ bmƒad^\ _` a* `noj `n* `g ^jiepioj

v&s*s& G\ x s x \) U <y&s'w*

`n h`_d]g` t od`i` ƒm`\ dbp\g \ N-

0-08 N_pbos^`flkbp obi^qfs^p^ i^ qbloŒ u q‹`kf`^ ab i^ fkqbdo^`fŽk

Tk^ sbw pb e^ iibd^al ^nrŒ pb mobpbkq^kalp `rbpqflkbp crka^jbkq^ibp9'0( ƒNp„ api^dji`n \^jo\_\n nji dio`bm\]g`n= '1( Ppkp`noj lp` pi\ api^d‡i ``n dio`bm\]g`* •^‡hj n` ^\g^pg\ g\ dio`bm\g _` a=

K^ mofjbo^ `rbpqfŽk pb bpqraf^ bk i^ ~SbloŒ^ab i^ fkqbdo^`fŽk‚ v i^ pbdrka^_^gl bi qŒqrilab ~S‹`kf`^ ab i^ fkqbdo^`fŽk‚- Tk^ obpmrbpq^ ljmibq^ ^ i^ `rbp,qfŽk '0( `loobpmlkab ^ rk kfsbi prmboflo ^ i^ ab rk `ropl mobifjfk^o v kl pbbpqraf^oŠ bk bpqb if_ol- Dk `^j_fl+ a^objlp obpmrbpq^pm^o`f^ibp nrb q^k pŽilobnrfbobk fab^p bibjbkq^ibp-

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K^ afp`rpfŽk ab i^ ~S‹`kf`^ ab i^ fkqbdo^`fŽk‚ `ljfbkw^ bk i^ Rb``fŽk 0-12+alkab pb `^i`ri^ i^ fkqbdo^i`a sL _s* `r^kal mbp bkqbol mlpfqfsl- Krbdl pbabp^ool,ii^k i^p molmfba^abpdbkbo^ibpab i^ fkqbdo^i+q^ibp`ljl i^ ifkb^ifa^a v i^ ^afqfsf,a^a+ v e^`bjlp sbo bk nr‹ cloj^ bp^p molmfba^abpklp ^vra^k ^ ^jmif^o krbp,qolp `lkl`fjfbkqlp bk fkqbdo^ibpab crk`flkbp bpmb`Œcf`^p-

Page 114: Calculus

83 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

)&*( ;b[PV\[R` Z\[pa\[N` e Z\[pa\[N` N a_\f\`& 9RSV[VPV\[R`e RWRZ]Y\`

Tk^ crk`fŽk a pb af`b nrb bp ^m`^d`io` `i pi ^jiepioj R pf a&s' 88899a&t' m^o^`^a^ m^o ab mrkqlp s b v ab R `lk s ; v- Rf pb sbofcf`^ YNabpfdr^ia^a bpqof`q^a&s' ; a&t' m^o^ qlal s ; u bk R pb af`b nrb YNcrk`fŽk bp ^m`^d`io` `i n`iod_j

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EHFTQ@ 0-22 Cpi^dji`n hji‡oji\n,

GGGGr

[ ]

Cb`ob`fbkqb bkpbkqfal bpqof`ql

`nomd^oj`i R- @kŠild^jbkqb+ rk^ crk`fŽk pb af`b _`^m`^d`io` `i R pf a&s' 99888a&t'm^o^ qlal s ; v bk R- Rf a&s' = `%s&m^o^qlal s ; v bk R i^ crk`fŽk pb abkl,jfk^ _`^m`^d`io` `i n`iod_j `nomd^oj i R- Tk^ crk`fŽk pb abkljfk^ hji‡oji\ `i Rpf bp `ob`fbkqb bk R l ab`ob`fbkqb bk R- Jji‡oji\ `i n`iod_j `nomd^ojpfdkfcf`^nrb Z+l bp bpqof`q^jbkqb`ob`fbkqb bk R l bp bpqof`q^jbkqbab`ob`fbkqb bk R- Dk db,kbo^i+bi `lkgrkql R nrb pb `lkpfabo^+ l bp rk fkqbos^il ^_fboql l rk fkqbos^il`boo^al- Dk i^ cfdro^ 0-22 pb a^k bgbjmilp-

EHFTQ@ 0-23 Cpi^dji`n hji‡oji\n \ omjujn,

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Page 115: Calculus

gio`bm\]dgd_\_ _` api^dji`n hji‡oji\n \^jo\_\n 84

DIDLOKN 0- I\n api^dji`n kjo`i^d\g`n, Rf k bp rk bkqbol mlpfqfsl+ pbqfbkb i^ abpfdr^ia^a9

pf N z s ; s)

nrb pb mrbab abjlpqo^o cŠ`fijbkqb mlo fkar``fŽk- Dpql fjmif`^ nrb i^ crk`fŽk `abcfkfa^ m^o^ qlal k•jbol ob^i s mlo i^ b`r^`fŽk a&s' < US bp `ob`fbkqb bk pbk,qfal bpqof`ql bk bi bgb ob^i kl kbd^qfsl- K^ jfpj^ crk`fŽk bp jlkŽqlk^ bkpbkqfal bpqof`ql bk bi bgb ob^i kbd^qfsl 'bp ab`ob`fbkqb pf j bp m^o v `ob`fbkqbpf k bp fjm^o(- Olo q^kql+ ` bp jlkŽqlk^ ^ qolwlp bk `^a^ fkqbos^il cfkfql-

DIDLOKN 1- I\ api^d‡i m\…u p\_m\_\, Rb^ a&s' < Ty m^o^ s x N- Dpq^crk`fŽk bp bpqof`q^jbkqb `ob`fbkqb bk bi bgb ob^i kl kbd^qfsl- Dk bcb`ql+ pfN 9p::s ; v+ qbkbjlp

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b&s' < Sm/ * s/ pf +m x s x m

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)&*) =[aRT_NOVYVQNQQRSb[PV\[R`Z\[pa\[N` NP\aNQN`

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SDNQDL@ 0-01- Pd a `n hji‡oji\ `i pi dio`mq\gj ^`mm\_j X\* ]Z* a `n dio`+bm\]g` `i X\* ]Z,

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Rb^ i rk bkqbol mlpfqfsl u `lkpqorv^jlp alp crk`flkbp bp`^ilk^a^p ab ^moluf,j^`fŽk n9 u o9 abi jlal pfdrfbkqb9 Rb^ M < wt-+Ug% ŠŠŠ * siw rk^ m^oqf`fŽk abW[) \Y bk h pr_fkqbos^ilp fdr^ibp+ bpql bp+ pr_fkqbos^ilp XUf+g% UfZ q^ibp nrb

Page 116: Calculus

2/ Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

s* + s* [ 0 < n\ * \' - i m^o^ `^a^ s^ilo ab e+ Cbcfk^jlp ^elo^ n9 u o9mlo i^pcŽojri^p

Dk ilp mrkqlp ab afsfpfŽk+pb abcfkbk n9 u o9ab jlal nrb pb j^kqbkd^k i^p obi^,`flkbp Pi&U' 949%r& 949nh%r&bk qlal W[) \Y+ Dk i^ cfdro^ 0-24'^( pb jrbpqo^ rkbgbjmil- Blk bpq^bib``fŽk ab crk`flkbp bp`^ilk^a^p+ qbkbjlp

pfbkal i^ •iqfj^ fdr^ia^a rk^ `lkpb`rbk`f^ ab i^ molmfba^a qbibp`Žmf`^ab i^pprj^p cfkfq^p-Dpq^•iqfj^ obi^`fŽk qfbkb rk^ fkqbomobq^`fŽkdblj‹qof`^ jrv pbk,`fii^- K^ afcbobk`f^ P8oi + P8Pi bp fdr^i ^ i^ prj^ ab i^p Šob^pab ilp ob`qŠkdrilpplj_ob^alp ab i^ cfdro^ 0+24'^(- Cbpifw^kal bplp ob`qŠkdrilp e^`f^ i^ abob`e^`ljl pb fkaf`^ bk i^ cfdro^ 0-24'_(+ sbjlp nrb `ljmibq^k rk ob`qŠkdril ab _^pb

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Ulis^jlp ^ bp`of_fo i^ obi^`fŽk ^kqboflo bk i^ cloj^

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Page 117: Calculus

@ƒg^pgj _` g\ dio`bm\g _` pi\ api^d‡i hji‡oji\ \^jo\_\ 75

K^p fkqbdo^ibp prmboflo b fkcboflo ab ` p^qfpc^`bk i^p abpfdr^ia^abp

Lriqfmif`^kal i^p mofjbo^p abpfdr^ia^abp mlo ',0( X prj^kal bi obpriq^al ^ i^ppbdrka^p+ l_qbkbjlp

.Q&* `Q&Q E7oi + E7Pi ,

Tqfifw^kal '0-7( X i^ obi^`fŽk `Q& md%`&)l_qbkbjlp

bl ; .Q&* `Q&R,+ i

m^o^ qlal bkqbol i x 0-&Olo `lkpfdrfbkqb+ pbd•k bi qblobj^ H-20+ ab_b pbo0'.( < 0'.(- Dpql abjrbpqo^ nrb ` bp fkqbdo^_ib bk W[)\Y+

)&** 8mYPbY\QRYNV[aRT_NYQRb[N b[PVp[ Z\[pa\[N NP\aNQN

K^ abjlpqo^`fŽk abi qblobj^ 0-01 kl pli^jbkqb morb_^ nrb i^ fkqbdo^i abrk^ crk`fŽk `ob`fbkqb ^`lq^a^ `sdno`* pfkl nrb q^j_f‹k prdfbob rk j‹qlal m^o^`^i`ri^o bi s^ilo ab i^ fkqbdo^i- …pqbpb bumlkb bk bi qblobj^ pfdrfbkqb-

RCMPCK? 0-02- Ppkjib\hjn a ^m`^d`io` `i pi dio`mq\gj ^`mm\_j X\* ]Z,P`\ s, < \ * f&] + \'-i k\m\ f < N+ 0+ --- + i, Rf / `n pi iˆh`mj ^p\glpd`m\lp` n\odna\^` g\n _`ndbp\g_\_`n

'0-8(

k\m\ oj_j `io`mj i x 0+`ioji^`n / < `wa&s'_s,

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Page 118: Calculus

21 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

m^o^ i x 0- Obol i^ fkqbdo^i Pxa&s'_sn\odna\^` i^p jfpj^p abpfdr^ia^abp nrb g,Tqfifw^kal i^ fdr^ia^a '0-7( sbjlp nrb

l z ./ + aa&U' _s Gy:m^o^qlal i x 0- Olo `lkpfdrfbkqb+ pbd•k bi qblobj^ 0-20+qbkbjlp / <`wa&s'_s*`ljl pb ^cfojŽ-

Blk ^kŠildl o^wlk^jfbkql pb `lkpfdrb abjlpqo^o bi `loobpmlkafbkqb qblobj^m^o^crk`flkbp ab`ob`fbkqbp-

SDNQDL@ 0-03- Ppkjib\hjn a _`^m`^d`io` `i X\*]Z, P`\ sf;^)f&]+\'-ik\m\ f;L* 0+--- +i, Pd/ `n pi iˆh`mj ^p\glpd`m\ lp` n\odna\^` g\n _`ndbp\g_\_`n

] 8 [ x-&Uf' x / x ] 8 [ xa&Uf'F5G Q5>

k\m\ oj_j `io`mj i x 0+`ioji^`n / < Pxa&s'_s,

0-12 BŠi`ril ab YNfkqbdo^iPx rL ^r pfbkal k bkqbol mlpfqfsl

O^o^ jlpqo^o `ljl pb rqfifw^ bi qblobj^ 0-02 `^i`ri^objlp i^ fkqbdo^i`a sM _s pfbkal _ = N u k rk bkqbol mlpfqfsl `r^inrfbo^- K^ fkqbdo^ibufpqbmlonrbbi fkqbdo^kal bp `ob`fbkqb v ^`lq^al bk ZN+\Y+

SDNQDL@ 0-04- Pd k `n pi `io`mj kjndodqj v ] = N+o`i`hjn

A`hjnom\^d‡i, Bljbk`bjlp `lk i^p abpfdr^ia^abp

sŠifa^p m^o^qlal bkqbol i x 0 X qlal bkqbol k x 0- Dpq^pabpfdr^ia^abp mrbabk

Page 119: Calculus

Mmjkd`_\_`n api_\h`io\g`n _` g\ dio`bm\g 77

abjlpqo^opb mlo fkar``fŽk- 'Dk bi Dgbo`f`fl 02 ab i^ Rb``fŽk 03-0/ pb fkaf`^ rk^abjlpqo^`fŽk-( K^ jriqfmif`^`fŽk ab bp^p abpfdr^ia^abp mlo ]M

)/ -iM(g klp a^

] j‚,h &f]'! ]!)/ ] Fi &f]'!, , ;,, ;, ,-i i k)/ i i

hzi hzi

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Olo `lkpfdrfbkqb+ i^p abpfdr^ia^abp '0-8( abi qblobj^ )&)+ pb p^qfpc^`bkmlkfbkal` &s' < s!* \ < /+ v bkqlk`bp . < ]M)g-&k * 0(- Qbpriq^ mrbp nrb ab sM_s :: ]M)/ ,%j * 0(-

)&*, C_\]VRQNQR`Sb[QNZR[aNYR`QRYNV[aRT_NY

@ m^oqfoab i^ abcfkf`fŽk ab fkqbdo^i+bp mlpf_ib abar`fo i^p molmfba^abppf,drfbkqbp+nrb pb abjrbpqo^k bk i^ Rb``fŽk 0-16-

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>_`hƒn* n` od`i`8

oX^oa&s'* ^0b&s'Z _s < Bi oa&s' _s * B1 ob&s' _s ,

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SDNQDL@ 0-06- @CiSiUHC@C QDRODBSN @K HMSDQU@KN CD HMSDFQ@BHˆM- Pd`sdno`i _jn _` g\n om`ndio`bm\g`n ndbpd`io`n* o\h]d„i `sdno` g\ o`m^`m\ v n` od`i`8

ny&s' _s * cb a&s' _s < g`a&s' _s ,\ ;] \

Page 120: Calculus

0// Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

Kjo\8 Dk m^oqf`ri^o+ pf a`n jlkŽqlk^ bk X\*]Z v q^j_f‹k bk X]*^Z* bufpqbk i^palp fkqbdo^ibpGxa b Gxa%lk il nrb q^j_f‹k bufpqbGxa v bp fdr^i ^ i^ prj^ ab ^nr‹ii^p-

SDNQDL@ 0-07- HMU@QH@MBH@EQDMSD @ TM@ SQ@RK@BHˆM- Pd ` `n dio`bm\]g``i X\* ]Z* k\m\ ^\_\ iˆh`mj m`\g ` n` od`i`8

G] ZGba&s' _s < a&s + b( _s ,\ }\)^

SDNQDL@ 0-08- CHK@S@BHˆM N BNMSQ@BBHˆM CDK HMSDQU@KN CD HMSDFQ@BHˆM-

Pd ` `n dio`bm\]g` `i X\* ]Z k\m\ ^\_\ iˆh`mj m`\g f ;C M n` od`i`8

g] 0df] &s'a&s' _s < ,9 a + _s,[ f e[ f

Kjo\8 Dk ilp alp qblobj^p 0-07 v 0-08 i^ bufpqbk`f^ ab rk^ ab i^p fkqbdo^ibp fjmif`^i^ bufpqbk`f^ ab i^ lqo^- Br^kal f < , 0+ bi qblobj^ 0-08 pb ii^j^ kmjkd`_\_ _` m`+`f_rcƒh+

SDNQDL@ 0-1/- SDNQDL@ CD BNLO@Q@BHˆM- Pd ` v d nji \h]\n dio`bm\]g`n`i X\* ]Z W nd b&s' x a&s' k\m\ ^\_\ s `i X\* ]Z n` od`i`8

nd'u( _s 8899na&s' _s ,

Tk `^pl m^oqf`ri^o fjmloq^kqb abi qblobj^ 0-1/ pb qfbkb `r^kal a%r&< Nm^o^ `^a^ r+ Dk bpqb `^pl bi qblobj^ af`b nrb pf `%r& w N bk qlal bi fkqbos^ilX\* ]Z* bkqlk`bp Gxa&s' _s x N- Cf`el ab lqo^ j^kbo^+ rk^ crk`fŽk kl kbd^qfs^qfbkb fkqbdo^i kl kbd^qfs^- S^j_f‹k pb mrbab abjlpqo^o nrb pf pb qfbkb i^ abpf,dr^ia^a bk n`iod_j `nomd^oja%r& ; `%r&) m^o^ qlal r bk X\* ]Z* q^j_f‹k pb sbof,cf`^ i^ jfpj^ abpfdr^ia^a bk pbkqfal bpqof`ql m^o^ i^p fkqbdo^ibp+ mbol i^ abjlp,qo^`fŽk kl bp cŠ`fi-

Dk bi `^mŒqril 4 pb a^oŠk s^oflp j‹qlalp m^o^ `^i`ri^o bi s^ilo ab rk^fkqbdo^i pfk kb`bpfa^a ab ^mif`^o bk `^a^ `^pl i^ abcfkf`fŽk- Rfk bj_^odl+ bpqlpj‹qlalp pŽil plk ^mif`^_ibp ^ rk k•jbol obar`fal ab crk`flkbp v m^o^ i^ j^vlom^oqb ab crk`flkbp fkqbdo^_ibp+ bi s^ilo krj‹of`l ab pr fkqbdo^i pŽil mrbab pbol_qbkfal ^molufj^a^jbkqb: il nrb pb `lkpfdrb ^molufj^kal bi fkqbdo^kal pr,mboflo b fkcboflojbkqb jbaf^kqb crk`flkbp bp`^ilk^a^p+ l mlo jbafl ab lqo^pcrk`flkbp pfjmibp `rv^ fkqbdo^i pb mrbab `^i`ri^o bu^`q^jbkqb- Di qblobj^ ab`ljm^o^`fŽk pb rqfifw^ m^o^ l_qbkbo ^molufj^`flkbp ab i^ fkqbdo^i ab i^ crk,`fŽk bk `rbpqfŽk- Dpq^ fab^ pb ^k^ifw^oŠ `lk jŠp `rfa^al bk bi `^mŒqril 6-

Page 121: Calculus

Fio`bm\^d‡i _` kjgdijhdjn 0/0

)&*- =[aRT_NPVp[QR]\YV[\ZV\`

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'0-0/(

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c*] I] I] & \&E$(.sm _s < , &+s'!%_s< '\0(0&*0 sm _s < , +l l l k)g

il `r^i morb_^ i^ s^ifabw ab '0-0/( m^o^ ] kbd^qfsl- K^ molmfba^a ^afqfs^Px sL _s < c s! _s + absj _s klp `lkar`b ^ i^ cŽojri^ jŠp dbkbo^i

sŠifa^ m^o^qlalp ilp s^ilobp ob^ibpab \ v \) v qlal bkqbol k x N-@idrk^p sb`bp bi pŒj_lil

M&s' 09

pb bjmib^ m^o^abpfdk^o i^ afcbobk`f^ L%\& * L_[&+ Cb bpqbjlal i^ cŽojri^ ^k,qboflo mrbab bp`of_fopb^pŒ9

Dpq^ cŽojri^ v i^ molmfba^a ab ifkb^ifa^a+ klp mbojfqbk fkqbdo^o `r^inrfbomlifkljfl- Olo bgbjmil+ m^o^`^i`ri^o i^ fkqbdo^ii&s0

+ 1s * 4( _s* e^ii^jlp i^fkqbdo^iab `^a^ q‹ojfkl v prj^jlp irbdl ilp obpriq^alp- @pŒmrbp+qbkbjlp

a1 a1 a1 a1 U1 1 u102 02

&s/* 1s * 4( _s < s/ _s + 2 s _s * 4 _s < , , 2 , * 3s :

0 0 0 0 2010 0

< 22

, b ] 2 21

, 01 * 4 2

0, 00< 15 ] 01 * 0/ < 1/-

2 1 0 2 2

Page 122: Calculus

0/1 Ijn ^+ji^`kojn _`g ^ƒg^pgj dio`bm\g

Blk j^vlo dbkbo^ifa^a+ m^o^ `^i`ri^o i^ fkqbdo^i ab `r^inrfbo mlifkljfl fkqb,do^jlp q‹ojfkl ^ q‹ojfkl9

S^j_f‹k mlabjlp fkqbdo^ocrk`flkbp jŠp `ljmif`^a^p+ abpal_iŠkali^p bks^oflp mlifkljflp- Olo bgbjmil+ `lkpfabobjlp i^ fkqbdo^i`wEr%/r * /0 ^r+ Cb_fal^i pfdkl ab s^ilo ^_plirql+ bi fkqbdo^kal kl bp rk mlifkljfl- Ml l_pq^kqb+ lkpf,abo^kal bi pfdkl ab r%/r * N+mlabjlp abp`ljmlkbo bi fkqbos^il Z/+0\ bk alppr_fkqbos^ilp+bk `^a^ rkl ab ilp `r^ibp bi fkqbdo^kal bp rk mlifkljfl- Br^kal ss^oŒ ab N ^i+ bi molar`ql r%/r * 0( `^j_f^ ab pfdkl bk bi mrkql r < f: bp kbd^,qfsl pf N ; s ; pv mlpfqfsl pf p; s ; 0- Olo il q^kql+ bk sfoqra ab i^ mol,mfba^a ^afqfs^ bp`of_fjlp

Hg 00.1 IHEr%/r * 0(0 ^r < , r%/r * 0( ^r * r%/r * 0( ^rl l 0.1

//-0 GF< 'u , /r0' ^r * %/r0

+ r& ^rl 0.1

< 'p , eZ(* '01 , p(< d,

0-15 Dgbo`f`flp

B^i`ri^o `^a^ rk^ ab i^p fkqbdo^ibp pfdrfbkqbp-

0- `7s0_s,

0, a1 s0_s,',

2- I92s1_s,

3- m02U1_s,

3, `73o2 _o*

4, ag 3o2_o,,0

5, `7&3s2 + 2s0& _s*

6, a/ &3s2 + 2s1' _s,,0

8- nh&o0* 0( _o*

0/- I9&1s0+ 2s * 1( _s*

//, `7,0 &6o1 * 4o0 + 0/ * 4( _F,

01- q1&p + g'&p + 1( _pj

02- nh&s * /'0_s,

/2, `8. vs * 0(1 _s,

/3, `7&s + /'&1s + 0( _s*

05- I9F&s+ /'v1s + 0(0 _s,

Page 123: Calculus

Be`m^d^djn )(+

06- I9&0s + 4(2 _s,

.5+ Gx1 &s0 + 2(2 _s,

08- I9s/&s + 4(3 _s,

1/- G9&s * 3(0/ _s* XFi_d^\^d‡i8 Sblobj^ 0-07-\

10- G^ii^o qlalp ilp s^ilobp ab ` m^o^ ilp nrb

'^( I9s&g + s' _s < N+ '_( I9ys&g + s'/ _s < N-

11- B^i`ri^o `^a^ rk^ ab i^p fkqbdo^ibp pfdrfbkqbp- Cf_•gbpb i^ doŠcf`^ ` bk `^a^ `^pl-

'^( a8 y&s' _s vU0 pf M Q s C:$

alkab y&s' < 1 ] spf 0 Q s Q 1-

alkab {'ui z nH , s

pf M Q s Q _+

'_( I9y&s' _spf _otQh:^++

0 , b

` bp rk k•jbol ob^i cfg^al+ N ; ` ; 0-

12- G^ii^o rk mlifkljfl `r^aoŠqf`l M m^o^ bi `r^i M&L' < M&F' < M X `wM&s' _s < 0-

13- G^ii^o rk mlifkljfl `•_f`l M m^o^ bi `r^i M&L' < M&,1( < N+ M&g' < 04+ X 2 ax0M&s' _s < 3-

Be`m^d^djn jko\odqjn,

14- Rb^ ` rk^ crk`fŽk `rvl aljfkfl `lkqfbkb , s pfbjmob nrb `lkqfbkb s, Rb af`b nrb `bp rk^ crk`fŽk k\m pf `% * r& < `%r& v rk^ crk`fŽk dhk\m pf `%* s' < , `%r& m^o^qlal s bk bi aljfkfl ab `+Rf ` bp fkqbdo^_ib bk ZN+\Y abjlpqo^o nrb9

'^( o-&s' _s < 1a8y&s' _s pf ` bp m^o:

'_( a-&s' _s < N pf ` bp fjm^o-

15- Olo jbafl ab ilp qblobj^p 0-07 u 0-08 abar`fo i^ cŽojri^

I9y&s' _s < &] + \' G8yX\ * &] + \'sZ _s,

16- Klp qblobj^p 0-07 u 0-08 prdfbobk rk^ dbkbo^ifw^`fŽk ab i^ fkqbdo^i axy&>s * ?' _s,N_qbkbo bp^ cŽojri^ v abjlpqo^oi^ `lk bi ^rufifl ab ilp `fq^alp qblobj^p- Cfp`rqfoq^j_f‹k bi `^pl > < N-

Page 124: Calculus

0/3 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

17- Lbaf^kqb ilp qblobj^p 0-07 u 0-08 abjlpqo^o i^ cŽojri^

`] n9\ a&^ + s' _s < ^+] a&s' _s ,

0-16 Cbjlpqo^`flkbp ab i^p molmfba^abp crka^jbkq^ibp ab i^ fkqbdo^i

Dpq^ Rb``fŽk `lkqfbkb i^p abjlpqo^`flkbp ab i^p molmfba^abp _Špf`^p ab i^ fkqb,do^i nrb pb `fq^olk bk ilp qblobj^p abi 0-05 ^i 0-1/ ab i^ Rb``fŽk 0-13- Tp^jlpobmbqfa^jbkqb bi eb`el ab nrb qla^ crk`fŽk ` ^`lq^a^ bk rk fkqbos^il W[)\Yqfbkb fkqbdo^i fkcboflo `Q&b fkqbdo^i prmboflo FQ&a^a^p mlo

/&a' < prm S9p Gp z `v ) F%a&< fkc xc o G w ow*

bk alkab n v o plk crk`flkbp bp`^ilk^a^p ^o_fqo^of^p fkcboflobp v prmboflobp ^ o,obpmb`qfs^jbkqb- R^_bjlp+ mlo bi qblobj^ 0-8+ nrb ` bp fkqbdo^_ib pf v pŽil pfX&a' )&g&a' bk `rvl `^pl bi s^ilo ab i^ fkqbdo^i ab ` bp bi s^ilo `lj•k ab i^p fk,qbdo^ibp prmboflo b fkcboflo-

A`hjnom\^d‡i _` g\ kmjkd`_\_ _` gdi`\gd_\_ &Q`jm`h\ 0-05(- Cbp`ljmlkd^,jlp bp^ molmfba^a bk alp m^oqbp9

'@( n&a* d( < n * nd +

'@( ba; ` aa,

O^o^ abjlpqo^o '@(+ mlkd^jlp W%.&< `w` b W%a&:`wa+ Cbjlpqo^objlp nrba&a * c( < /&a * c( < X&a' * W%a&+

Rb^k Pg X P0 crk`flkbp bp`^ilk^a^p `r^ibpnrfbo^ fkcboflobp ^ ` v d+obpmb`qf,s^jbkqb- Orbpql nrb ` v d plk fkqbdo^_ibp+ pb qfbkb

F&a' < prm GnRHGRHw`v)Olo i^ molmfba^a ^afqfs^ abi buqobjl prmboflo 'qblobj^ 0-22(+ q^j_f‹k pb qfbkb

'0-00 ( -&a' * .'d( < prm GnRH* nR1 GRH z E)R1 z dy -

Page 125: Calculus

A`hjnom\^dji`n _` g\n kmjkd`_\_`n api_\h`io\g`n _` g\ dio`bm\g /.3

Obol pf R0 z ` v R1 z d+ bkqlk`bp i^ prj^ p ;P/ * R1 bp rk^ crk`fŽk bp`^ilk^a^fkcboflo ^ ` * d+ v qbkbjlp

a] RH* a] R1 < oR z WQ* d( -\ \ \

Olo il q^kql+bi k•jbol Y'b* d( bp rk^ `lq^ prmboflom^o^bi `lkgrkql nrb ^m^ob`bbk bi pbdrkal jfbj_ol ab '0-00(- Dpq^`lq^ prmboflo kl mrbab pbo jbklo nrb bibuqobjl prmboflo abi `lkgrkql+ ab j^kbo^ nrb

'0-01( /&a' * F&b'x g&a* b' ,

Cbi jfpjl jlal+ pf e^`bjlp rpl ab i^p obi^`flkbp

`Q& < fkc 'n oF E`w oF' *

alkab n)X o0 obmobpbkq^kcrk`flkbp bp`^ilk^a^p ^o_fqo^of^pprmboflobp ^ ` v d+obpmb`qfs^jbkqb+l_qbkbjlp i^ abpfdr^ia^a

'0-02( G&a* b' x g&a' * F&b',

K^p abpfdr^ia^abp '0-01( X '0-02( grkq^p abjrbpqo^k nrb X&a * d( < 0'b * d( <: a&a' * F&b', Olo `lkpfdrfbkqb a * *b bp fkqbdo^_ibv i^ obi^`fŽk '@( bp `fboq^-

K^ obi^`fŽk 'A( bp qofsf^i pf ` < N- Rf ` = N+l_pbosbjlp nrb qla^ crk`fŽkbp`^ilk^a^ R0 < `a bp ab i^ cloj^ R0 < `n* pfbkal p rk^ crk`fŽk bp`^ilk^a^ fkcb,oflo ^ a, @kŠild^jbkqb+ `r^inrfbo crk`fŽk bp`^ilk^a^ n)prmboflo^ `a bp ab i^ cloj^n)< `o* pfbkal o rk^ crk`fŽk bp`^ilk^a^ prmboflo ^ `+ Sbkbjlp mlo il q^kql

X&` ' < prm 'nRHi RH z `a' < prm 'b nR GR xa' < `aR'

v

g&` ' < fkc'c ogY `ax oF' < fkcib no Fax o' < ^-R'}

Krbdl g&`a' < g&`a' < `g&a', @nrŒebjlp rqfifw^al i^p molmfba^abppfdrfbkqbpabibuqobjl prmboflo v abi buqobjl fkcboflo9

'0-03( prm u_r Gr C =v < _ prmur Gr C=v ) fkc u_r Gr D >w < _ fkc ur Gr D >w*

nrb plk sŠifa^p pf ` = N- Dpql abjrbpqo^ 'A( pf ` = N-

Page 126: Calculus

0/5 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

Rf _ ; N+i^ abjlpqo^`fŽk ab 'A( bp _Špf`^jbkqb i^ jfpj^+ bu`bmql nrb qla^crk`fŽk bp`^ilk^a^ PF fkcboflo ^ `a bp ab i^ cloj^ PF < `o* pfbkal o rk^ crk`fŽkbp`^ilk^a^ npk`mdjm ^ a u qla^ crk`fŽk bp`^ilk^a^ n)prmboflo ^ ^a bp ab i^ cloj^n)< `n* pfbkal n rk^ crk`fŽk bp`^ilk^a^ dia`mdjm^ a, @pfjfpjl+ bk ird^o ab '0-03(rqfifw^jlp i^p obi^`flkbp

prm u_r Gr D =v < b fkcur Gr D =v ) fkc u_r Gr D =v < b prm ur Gr D =v)

nrb plk `fboq^ppf ` ; N- Sbkbjlp mrbp

g&`a' << prm xcPF GRH z `a' < prm 'b c o Ga x o' < b fkc 'c o Ga x o' < ^FR' ,

@kŠild^jbkqb+ bk`lkqo^jlp y&`a' < ^F&a', Olo `lkpfdrfbkqb 'A( bp `fboq^ m^o^`r^inrfbo s^ilo ob^i ab `,

A`hjnom\^d‡i _` g\ \_dodqd_\_ m`nk`^oj \g dio`mq\gj _` dio`bm\^d‡i &Q`jm`+h\ 0-06(- Rrmlkd^jlp nrb \ ; ] ; `* v nrb i^p alp fkqbdo^ibpff ` b ab ` bufp,qbk- Cbpfdkbjlp `lk `9f& b e&a' i^p fkqbdo^ibpprmboflo b fkcboflo ab a bk bi fkqbo,s^il W[) b\- Cbjlpqo^objlp nrb

'0-04( `Q&< dQ&< b * p+

Rf m bp rk^ crk`fŽk bp`^ilk^a^ `r^inrfbo^ fkcboflo ^ ` bk W[) bI+ pb qfbkb

X! n < I] n * I+b R ‘aH H ]

Qb`Œmol`^jbkqb+pf PF u P0 plk crk`flkbp bp`^ilk^a^p fkcboflobp ^ ` bk W[)] I uW\) _ I obpmb`qfs^jbkqb+i^ crk`fŽk m nrb `lfk`fab `lk PF bk W[) \ I v `lk P0 bkW\) `I bp rk^ crk`fŽk bp`^ilk^a^ fkcboflo ^ ` bk W[) `I m^o^i^ nrb

bR < nPF * qP0 Š

Olo `lkpfdrfbkqb+ bk sfoqra ab i^ molmfba^a ^afqfs^ abi buqobjl prmboflo 'qblob,j^ 0-22(+qbkbjlp

`Q&< prmGnP GP x G' < prm00!PF GRH z G' * prm GpP0/ P0 x G' < nG * pG,

Page 127: Calculus

A`hjnom\^dji`n _` g\n kmjkd`_\_`n api_\h`io\g`n _` g\ dio`bm\g /.5

@kŠild^jbkqb+ bk`lkqo^jlp

f%`&< bb* .!`)

0/ nrb abjrbpqo^ '0-04( `r^kal \ ; ] ; ^, K^ abjlpqo^`fŽk bp m^ob`fa^ m^o^`r^inrfbo lqo^ afpmlpf`fŽk ab ilp mrkqlp \* \) ^,

A`hjnom\^d‡i _` g\ kmjkd`_\_ _` om\ng\^d‡i &Q`jm`h\ 0-07(- Rb^ d i^ crk,`fŽk abcfkfa^ bk bi fkqbos^il W[ * _) ] * b\ mlo i^ b`r^`fŽk a%T& < `%r * _&+Cb,pfdkbjlp mlo %a&b f%a&i^p fkqbdo^ibp prmboflo b fkcboflo ab d bk bi fkqbos^ilW[ * _) ] * b\- Cbjlpqo^objlp nrb

'0-05( &b' < g&b'< na&s' _s ,

Rb^ p `r^inrfbo crk`fŽk bp`^ilk^a^ fkcboflo ^ d bk bi fkqbos^il W[ * _) ] * _Y+Dkqlk`bp i^ crk`fŽk RH abcfkfa^ bk X\* ]G mlo i^ b`r^`fŽk ny&s' < n&s* `' bp rk^crk`fŽk bp`^ilk^a^ fkcboflo ^ ` bk W[) \Y+ @abjŠp+ qla^ crk`fŽk bp`^ilk^a^ RHfk,cboflo ^ ` bk W[) \F qfbkb bpq^ cloj^ m^o^ rk `fboq^ p fkcboflo ^ a+ S^j_f‹k+ mlo i^molmfba^a ab qo^pi^`fŽk m^o^ i^p fkqbdo^ibp ab i^p crk`flkbp bp`^ilk^a^p+ qbkbjlp

g])@ g] g]n&s' _s < n&s* b( _s < PF&U' _s ,3%4 3 3

Olo `lkpfdrfbkqb pb qfbkb

&b' < prm &/8^^p Gp 999::d( < prm 'pRHGRH999::& < pa&s' _s ,

@kŠild^jbkqb+ bk`lkqo^jlp f%a&< `wy&s' _s* nrb morb_^ '0-05(-

A`hjnom\^d‡i _` g\ kmjkd`_\_ _` _dg\o\^d‡i t ^jiom\^^d‡i &Q`jm`h\ /,/7',Rrmlkd^jlp f = N u abcfk^jlp d bk bi fkqbos^il Xf\* f]G m^o^ i^ fdr^ia^ab&s' < a&s-f', Cbpfdkbjlp mlo 'd( b g&b' i^p fkqbdo^ibp fkcboflo u prmboflo ab bbk Xf\* f]G, Cbjlpqo^objlp nrb

'0-06( &b' < g&b'< f oa&s' _s ,

Rb^ p `r^inrfbo crk`fŽk bp`^ilk^a^ fkcboflo ^ d bk Xf\* f]G, Dkqlk`bp i^ crk`fŽkabcfkfa^ bk X\* ]G mlo i^ fdr^ia^a RH&s' < n&fs' bp rk^ crk`fŽk bp`^ilk^a^

Page 128: Calculus

0/7 Ijn ^ji^`kojn _`g ^ƒg^pgj dio`bm\g

fkcboflo ^ ` bk W[)\Y+ @abjŠp+ qla^ crk`fŽk bp`^ilk^a^ RHfkcboflo ^ ` bk W[)\Yqfbkb bpq^ cloj^- S^j_f‹k+ bk sfoqra ab i^ molmfba^a ab afi^q^`fŽk m^o^ i^p fkqb,do^ibp ab crk`flkbp bp`^ilk^a^p+ qbkbjlp

F+f] g] m]n&s' _s < f n&fs' _s < f I PF&U' _s ,

=6 6 6

Olo `lkpfdrfbkqb

Z&b' < prm 'p^P Gn 8899d( < prm %ebPF GPF 8899a' < e ba&s' _s ,

@kŠild^jbkqb+ bk`lkqo^jlp g&b'< fP8a&s' _s* nrb abjrbpqo^ '0-06( pf f = l-Di jfpjl qfml ab abjlpqo^`fŽk mrbab rqfifw^opb pf g` ; l-

A`hjnom\^d‡i _`g o`jm`h\ _` ^jhk\m\^d‡i &Q`jm`h\ 0-1/(- Rrmlkd^jlpd 99:: bk bi fkqbos^il W[)\Y+ Rb^ p `r^inrfbo crk`fŽk bp`^ilk^a^ fkcboflo ^ a+vpb^ o `r^inrfbo crk`fŽk bp`^ilk^a^ prmboflo ^ `+Rb qfbkb bkqlk`bp `wn 8899`wo*X mloq^kql bi qblobj^ 0-23 klp a^

bd < prm 'bn Gn 8899d( 99::fkc 'bo Gnn8o' < bm,

Dpql abjrbpqo^ nrb P8 b 8899P8a* `ljl abpbŠ_^jlp-

Page 129: Calculus

1

6?<GA6E 6C?=868=BA:E

67 <3

=AF:<D68=kA

1-0 Hkqolar``fŽk

Dk i^ Rb``fŽk 0-07 pb bumobpŽbi Šob^ ab rk `lkgrkql ab loabk^a^p ab rk^crk`fŽk kl kbd^qfs^ `ljl rk^ fkqbdo^i-Dk bpqb`^mŒqrilabjlpqo^objlp nrb q^j,_f‹k pb mrbabk bumobp^ojbaf^kqb fkqbdo^ibpi^p Šob^pab obdflkbp jŠp dbkbo^ibp-@pfjfpjl afp`rqfobjlp lqo^p ^mif`^`flkbp ab i^ fkqbdo^i ^ `lk`bmqlp q^ibp `ljlslirjbk+ qo^_^gl+v moljbaflp- Krbdl+ ^i cfk^i abi `^mŒqril+bpqraf^objlp i^pmolmfba^abp ab i^p crk`flkbp abcfkfa^p jbaf^kqb fkqbdo^ibp-

1-1 Di Šob^ ab rk^ obdfŽk `ljmobkafa^ bkqob alp doŠcf`^pbumobp^a^`ljl rk^fkqbdo^i

Rf alp crk`flkbp o u d bpqŠkobi^`flk^a^p mlo i^ abpfdr^ia^a.bu( z a%r& m^o^qlal r bk rk fkqbos^il W[)\Y) mlkbjlp o x d bk W[)\Y+ Dk i^ cfdro^ 1-0 pb sbkalp bgbjmilp- Rf ox d bk W[)\Y) bi `lkgrkql R `lkpq^ ab qlalp ilp mrkqlp %r)s&nrb p^qfpc^`bki^p abpfdr^ia^abp

x%r&w t x a%r&)

pb abkljfk^ obdfŽkbkqobi^p doŠcf`^pab ov d- Di pfdrfbkqbqblobj^ klp af`b `Žjlpb bumobp bi Šob^ ab R `ljl rk^ fkqbdo^i-

0/8

Page 130: Calculus

&&% >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

\ ]

'^( '_(

EHFTQ@ 1-0 Bg ƒm`\ `iom` _jn bmƒad^\nskm`n\_\ ^jhj pi\ dio`bm\g8

[%O&< `7Wa%r&* x%r&Yr)

RCMPCK? 1-0- Ppkjib\hjn lp` a v d nji dio`bm\]g`n v lp` n\odna\^`ia x c `i X\* ] I- I\ m`bd‡i R `iom` npn bmƒad^\n `n h`_d]g` v np ƒm`\ \&P' qd`i`_\_\ kjm g\ dio`bm\g

'1-0( [%O&< F7Wa%r&* x%r&Y r +

A`hjnom\^d‡i, Rrmlkd^jlp mofjbol nrb a v d plk kl kbd^qfs^p+`ljl pbjrbpqo^ bk i^ cfdro^ 1-0'^(- Rb^k C v F ilp pfdrfbkqbp`lkgrkqlp9

C < u%r)s& G[ w r w ]* N z V ; x%r&v) F < u%r)s& G[ w r w ]* N z V x a%r&v+

Dpql bp+F bp bi `lkgrkql ab loabk^a^p ab a) u B bi ab `) jbklp i^ doŠcf`^ ab `+K^ obdfŽk R bp i^ afcbobk`f^ F , B+ Rbd•k ilp qblobj^p )&)( v 0-ii+E X F plk^j_lp jbaf_ibp- Orbpql nrb B R F+ i^ afcbobk`f^ R < F , B bp g^j_f‹k jbaf,_ib+ v pb qfbkb

[%O&< [%C&* [%B&< F7a%r& r * F7x%r&^r < `7Wa%r&* x%r&Yr +

Dpql abjrbpqo^ '1-0( `r^kal ` v d plk kl kbd^qfs^p-Blkpfabobjlp ^elo^ bi `^pl dbkbo^i `r^kal ` w d bk g\*\Y) mbol kl plk kb,

`bp^of^jbkqb kl kbd^qfs^p- Dk i^ cfdro^ 1-0'_( pb jrbpqo^ rk bgbjmil- Dpqb`^pl

Page 131: Calculus

Be`hkgjn m`np`gojn 000

il mlabjlp obar`fo ^i ^kqboflo qo^pi^a^kal i^ obdfŽk e^`f^ ^oof_^ e^pq^ nrbnrbab pfqr^a^ mlo bk`fj^ abi bgb s, Dpql bp+ bibdfjlp rk k•jbol mlpfqfsl ` pr,cf`fbkqbjbkqb do^kab nrb ^pbdrob nrb N 999:x%r&* _ 7778a%r&* _ m^o^ qlal r bkX\* ]Z, Olo il v^ abjlpqo^al+ i^ krbs^ obdfŽk Q bkqob i^p doŠcf`^p ab ` * ` v d * `bp jbaf_ib+ v pr Šob^ sfbkb a^a^ mlo i^ fkqbdo^i

[%P& < F7W%a%r&* b( , %`%r&* _&Y r < F7Wa%r&* `%r&Y ^r +

Obol pfbkal P `lkdorbkqb ^ R+ ‹pq^ bp q^j_f‹k jbaf_ib v qbkbjlp

[%O&< [%P& < nWa%r&* `%r&Y ^r +‘ ^

Dpql `ljmibq^ i^ abjlpqo^`fŽk-

1-2 Dgbjmilp obprbiqlp

DIDLOKN 0- B^i`ri^o bi Šob^ ab i^ obdflk R pfqr^a^ bkqob i^p doŠcf`^p ab` v d pl_ob bi fkqbos^il ZN+1\ pfbkal `%r& < r%r * 1( W a%r&< r,/+

Pjgp^d‡i, K^p alp doŠcf`^p bpqŠk af_rg^a^p bk i^ cfdro^ 1-1- K^ mlo`flkplj_ob^a^ obmobpbkq^R- X^ nrb `7778d bk bi fkqbos^il ZN+1\+ e^`bjlp rpl abiqblobj^ 1-0 m^o^ bp`of_fo

01 01'4 ( 4 11

12

6[%O&< Wa%r&* `%r&Y ^r < , t , t1 ^r <, , , , <, -l /1 11 2 2

DIDLOKN 1- B^i`ri^o bi Šob^ ab i^ obdfŽk R bkqob i^p doŠcf`^p ab ` v d bk bifkqbos^il Z,0+1\ pf `%r&< r v a%r&< r1-2,

Pjgp^d‡i, K^ obdfŽk R bpqŠ obmobpbkq^a^ bk i^ cfdro^ 1-2- @elo^ kl bp on dbk qlal bi fkqbos^il Y,0+1\- Ml l_pq^kqb+ qbkbjlp ` 7778d bk bi pr_fkqbos^ilZ,0+ N\ X d 999: bk bi pr_ fkqbos^il Z/+1\- @mif`^kal bi qblobj^ 1-0 ^ `^a^ pr_,fkqbos^il+ qbkbjlp

[%O&<ciZd'W( , `%r&Y^r * F7W`%r&* a%r&Y r :

0 ']0(3 ' ]0(1 11 0 13 12<,,,,*,,*,,,,<,-

3 3 1 1 33 05

Page 132: Calculus

001 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

EHFTQ@ 1-1 Ad_gjfi 0- EHFTQ@ 1-2 Ad_gjfi /+

Dk bgbjmilp m^ob`falp ^ ‹pqb+bk ilp nrb bi fkqbos^il W[)\Y mrbab abp`ljml,kbopb bk rk k•jbol cfkfql ab pr_ fkqbos^ilp bk `^a^ rkl ab ilp `r^ibp `w d ld z `) i^ cŽojri^ '1-0( abi qblobj^ 1-0 ^almq^ i^ cloj^

\&P' < F7Gc't( , a&s'/ _s ,

DIDLOKN 2- >m`\ _` pi _dn^j ^dm^pg\m, Tk afp`l `fo`ri^o ab o^afl o bp bi`lkgrkql ab qlalp ilp m•kqlp fkqboflobp^ rk^ `fo`rkcbobk`f^ ab o^afl o u ab ilpmrkqlp ab i^ jfpj^- S^i afp`l bp `lkdorbkqb ^ i^ obdfŽk `ljmobkafa^ bkqobi^pdoŠcf`^pab i^p alp crk`flkbp ` v d abcfkfa^p bk bi fkqbos^il Z, o+o\ mlo i^p cŽo,jri^p

b&s' < Uo1, s/ v a&s' < +qm/* s/

B^a^ crk`fŽk bp ^`lq^a^ u jlkŽqlk^ bk W*l)o\ ab jlal nrb `^a^ rk^ bp fkqbdo^,_ib bk Z, o+o\- Di qblobj^ 1-0 klp af`b nrb i^ obdfŽk bk bpqb`^pl bp jbaf_ib v

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Be`hkgjn m`np`gojn 002

nrb pr Šob^ bp o+mXb&s'+ a&s'Z_s, Cbpfdkbjlp `lk >&m'bi Šob^ abi afp`l- Cb,jlpqo^objlp nrb

=%l&< l0=%f&+

Dpql bp+`g ƒm`\ _` pi _dn^j _` m\_dj m`n dbp\g \g kmj_p^oj _`g ƒm`\ _` pi _dn^jpid_\_ 'afp`l ab o^afl 0( kjm m%*

X^ nrb b&s' + a&s'< 0b&s'*bi qblobj^ 1-0 klp a^

Dk m^oqf`ri^o+ r^kal l < 0+pb qfbkb i^ cŽojri^

>&g' < 1 IH T 0 , s/ _s ,,0

B^j_f^kal i^ bp`^i^ bk bi bgb s* v rqfifw^kal bi qblobj^ 0-08 `lk e < g-m*pbl_qfbkb

>&m'< 1 cob&s' _s < 0mbhb&ms'_s < 0mbhT m0 + &ms'0 _s ;

; /l0 IH T 0 , r0 ^r < m0=%.&+,0

Dpql abjrbpqo^ nrb =%l&< l0=%f&) `ljl pb ^cfojŽ-

CDEHMHBHˆM- P` _`adi` `g iˆh`mj oo^jhj `g ƒm`\ _` pi _dn^j pid_\_,

K^ cŽojri^ nrb pb ^`^_^ ab abjlpqo^o bpq^_ib`b nrb =%l&< zm!*

Di bgbjmil ^kqboflo firpqo^ bi `ljmloq^jfbkql abi Šob^ cobkqb^ i^ afi^q^`fŽkl `lkqo^``fŽk ab i^p obdflkbp mi^k^p- Rrmlkd^jlp nrb P bp rk `lkgrkql a^al abmrkqlp abi mi^kl v `lkpfabobjlp rk krbsl `lkgrkql ab mrkqlp l_qbkfal jriqfmif,`^kal i^p `lloabk^a^p ab `^a^ mrkql ab P mlo rk c^`qlo `lkpq^kqb f = N-Cbpfdkb,jlp &bpqblkgrkql mlo fP v afd^jlp nrb bp pbjbg^kqb ^ P, Di mol`bpl nrb molar`bfP ^ m^oqfoab P qfbkbbi klj_ob dbk‹of`l ab om\inajmh\^d‡i kjm n`h`e\iu\, B^a^mrkql pb abpmi^w^ il i^odl ab rk^ ob`q^ nrb m^p^ mlo bi lofdbk e^pq^ rk^ afp,q^k`f^ ab ‹pqb fdr^i ^i molar`ql ab i^ afpq^k`f^ lofdfk^i mlo f, Rf f = 0+i^ qo^kp,cloj^`fŽk bp rk^ `sk\ind‡i ^ m^oqfoabi lofdbk+ r cjhjo`^d\ ab `bkqol bi lofdbkv o^wŽkj^vlo nrb i^ rkfa^a+ v+pf N ; f ; 0+pb qfbkb rk^ ^jiom\^^d‡i e^`f^ bilofdbk+r cjhjo`^d\ ab `bkqol bi lofdbk v o^wŽkjbklo nrb i^ rkfa^a-

Page 134: Calculus

003 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d4i

Olo bgbjmil+ pf R bp i^ obdfŽk ifjfq^a^ mlo rk^ `fo`rkcbobk`f^ rkfa^a `lk`bkqol bk bi lofdbk+ fP bp rk^ obdfŽk`fo`ri^o `lk`‹kqof`^ ab o^afl f, Dk bi bgbj,mil 2 pb abjlpqoŽ nrb m^o^obdflkbp `fo`ri^obp+bi Šob^ ab fP bp fdr^i ^i molar`qlabi Šob^ab R mlo f!, U^jlp ^elo^ ^ abjlpqo^o nrb bpq^molmfba^a abi Šob^bp sŠ,ifa^ m^o^`r^inrfbo `lkgrkql ab loabk^a^p-

DIDLOKN 3- @jhkjmo\hd`ioj _`g ƒm`\ _` pi ^jiepioj _` jm_`i\_\n am`io`\ pi\ om\inajmh\^d4i mlo n`h`e\iu\, Rb^ a kl kbd^qfs^ b fkqbdo^_ibbk X\* ]Z XRpr `lkgrkql ab loabk^a^p- Dk i^ cfdro^ 1-3'^( pb obmobpbkqrk bgbjmil- Rf ^mif`^,jlp rk^ qo^kpcloj^`fŽk mlo pbjbg^kw^ `lk rk c^`qlo mlpfqfsl f* fP bp bi `lkgrkqlab loabk^a^p ab rk^ krbs^ crk`fŽk d pl_ob bi fkqbos^il Xf\* f]Z ZU‹^pb i^ cfdr,o^ 1-3'_(-\ Tk mrkql %r)u( bpqŠpfqr^al bk i^ doŠcf`^ ab d pf u pŽil pf bi mrkql&s-f* t-f' bpqŠ bk i^ doŠcf`^ ab a, Krbdl t-f < a&s-f'* ab jlal nrbv < fa&s-f', Cf`el ab lqol jlal+ i^ krbs^ crk`fŽk d bpqŠifd^a^ ^ a mlo i^ cŽo,jri^

b&s' < fa&s-f'

\ ] QH f]

']( '_(

EHFTQ@ 1-3 Bg ƒm`\ _` fP `n `g kmj_p^oj _` g\ _` R kjm f0Š

m^o^`^a^ s ab Xf\* f]Z, Olo `lkpfdrfbkqb+ bi Šob^ ab fP sfbkb a^a^ mlo

df] df] n\&fP' < b&s' _s < f y&s-f' _s < f0 y&s' _s *x x \

alkab bk bi •iqfjl m^pl pb e^ rp^al i^ molmfba^a ab afi^q^`fŽk m^o^i^p fkqbdo^ibp'qblobj^ 0-8(- Orbpql nrb `wa&s' _s < \&P'* bpql abjrbpqo^ nrb \&fP' < f0\&P',Dk lqo^p m^i^_o^p+bi Šob^ ab fP bp bi molar`ql ab i^ ab R mlo f!,

DIDLOKN 4- @ƒg^pgj _` g\ dio`bm\g F T/-0 _s, K^ fkqbdo^iobpmb`qlabi Šob^

bp `ljl rk^ bpm^a^ab alp cfilp- Rf _fbk ab loafk^ofl pb rp^ i^ fkqbdo^im^o^`^i`r,i^o Šob^p+^idrk^p sb`bp mlabjlp rqfifw^o krbpqol `lkl`fjfbkql abi Šob^ m^o^

Page 135: Calculus

Be`hkgjn m`np`gojn 004

`^i`ri^o fkqbdo^ibp-Bljl bgbjmil `^i`ri^jlp bi s^ilo ab i^ fkqbdo^i`wT/-0 ^r)

pfbkal \ = N- 'K^ fkqbdo^ibufpqbv^ nrb bi fkqbdo^kal bp `ob`fbkqb v ^`lq^al bkZN+[Y+&

K^ cfdro^ 1-4 obmobpbkqi^ doŠcf`^ ab i^ crk`fŽk n a^a^ mlo n%r&< W0.1 bk

bi fkqbos^il ZN+[Y+ Rr `lkgrkql ab loabk^a^p R qfbkb rk Šob^ a^a^ mlo

\&P' < qU/-0 _s ,

B^i`ribjlp ^elo^ bi Šob^ mlo lqol mol`bafjfbkql- N_pbos^jlp pfjmibjbkqb bki^ cfdro^ 1-4 i^ obdfŽk R v i^ obdfŽk plj_ob^a^ P) nrb grkq^p `ljmibq^k bi ob`,qŠkdril ab _^pb [ v ^iqro^ \% !, Olo q^kql+[%O&* [%P& < [1

-0* ab jlal nrb

\&P' < \1-0 + \&Q' ,

Obol P bp bi `lkgrkql ab loabk^a^p ab rk^ crk`fŽk d abcfkfa^ pl_ob bi fkqbos^ilZN+ i

.1\ abi bgb s jbaf^kqb i^ b`r^`fŽk a%s& < d,@pŒmrbp+qbkbjlp

`\g-% a\g-%[%P&< k b&t' _t < -k t0 _t < [1

-0*

ab jlal nrb [%O&< [1-0 + [1-0 < d\1-0, Dpql abjrbpqo^ nrb

t

&\*\!%'

G'u+s&GGGGGG &!EZU< u ;a&s'

GGGGGGG

p

sl \

DGESP? 1-4 @ƒg^pgj _` g\ dio`bm\g ab U/-0 _s,

Page 136: Calculus

005 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d4i

LŠp dbkbo^i+ pf \ = N v ] = N+mlabjlp rp^o i^ molmfba^a ^afqfs^ ab i^ fkqbdo^im^o^ l_qbkbo i^ cŽojri^

%]

G\

WG.1_s < d&]1

-0

+ \1-0' Š

Di o^wlk^jfbkql ^kqboflo mrbab q^j_f‹k rp^opb m^o^ `^i`ri^o i^ fkqbdo^i`7T.,h _s* pf i bp rk bkqbol mlpfqfsl- Dpq^_ib`bjlp bi obpriq^al bk cloj^ abqblobj^-

RCMPCK? 1-1- M\m\ \ = N+ ] = N v i `io`mj kjndodqj* n` od`i`

'1-1( f] ]g)g-i + \g)g-is% ! _s < ,,,,,,,,

\ 0 z +)T

K^ abjlpqo^`fŽk bp q^k m^ob`fa^ ^ i^ abi bgbjmil 4 nrb abg^jlp ilp abq^iibp^i ib`qlo-

*&, :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 03+ `^i`ri^o bi Šob^ ab i^ obdfŽk R bkqob i^p doŠcf`^p ab ` u dm^o^ bi fkqbos^il W[) \Y nrb bk `^a^ `^pl pb bpmb`fcf`^- G^`bo rk af_rgl ab i^p alp doŠcf`^pu plj_ob^o R-

0- a&s' < 3 , s0*

0, a&s' < 3 , s0*

1, a&s' < s0 * s/)

2, a&s' < s + s0*

3, a&s' < WH.2+

3+a&s' < WH.2+

5, a&s' < WH.2+

6, a&s' < sg-/)

7, a&s' < s0*

/., a&s' < s&s0 + 0(+00- a&s' < Yt[+./+ a&s' < Gt, 00+.0+ a&s' < 1IuI+.1+ a&s' < Gth* Gt , 00+

a%r&< N+b&s' < 7 , 0s/)

b&s' < s0 * 0+b&s' < +s*a%r&< WH.1+

a%r&< WH.1+

a%r&< WH.1+

b&s' < s/)

a%r&< r * 0+

b&s' < s*b&s' ;s0 +/*x&s' < s0 + 0s*a%r&< 0 , 2u2+a%r&< N+

\ < ,1+[ < ,1+\ < ,0+\ ;.*[ < N+[ < 0+

\ < N+\ < N+

\ < ,0+

\ < ,0+\ < ,0+\ ;.*

\ < +S1-1*\ < ,0+-

] < 1-] < 1-];g,

] < 1-];/,] ;0,] < 1-] < 1-

] < 'i * /(.1-];S0,];/,] ;0,

] < f-] ;0,

04- K^p doŠcf`^p ab `%r& < r0 u a%r&< _r1* pfbkal _; N+pb `loq^k bk ilp mrkqlp 'N+N( X'i.b+ g-^0', Cbqbojfk^o ` ab jlal nrb i^ obdfŽk ifjfq^a^ bkqob bp^p doŠcf`^p v pl_ob bifkqbos^il ZN+0.b\ qbkd^ Šob^p-

05- Rb^k `%r& < r * r0* a%r&< [r+ Cbqbojfk^o [ m^o^ nrb i^ obdfŽk pfqr^a^ mlo bk`fj^ab i^ doŠcf`^ ab d v mlo ab_^gl ab ` qbkd^ Šob^-q-

Page 137: Calculus

I\n api^dji`n omdbjijh„omd^\n 006

06- Gbjlp abcfkfal 60& `ljl bi Šob^ ab rk afp`l `fo`ri^o rkfa^a- Dk bi bgbjmil 2 ab i^ Rb`,

`fŽk 1-2+ pb e^ abjlpqo^al nrb 6R < 1 IH z _s* G^`bo rpl ab i^p molmfba^abp ab,0

i^ fkqbdo^i m^o^ `^i`ri^o i^ pfdrfbkqb bk crk`fŽk ab $EP7

'b( c1 %r * 2(X3 , r/ ^r+

07- B^i`ri^o i^p Šob^p ab ilp alab`Šdlklp obdri^obp fkp`ofql v `fo`rkp`ofql bk rk afp`l

`fo`ri^o rkfa^a v abar`fo abi obpriq^al i^p abpfdr^ia^abp 2 ; 6R ; 01'1 , v2(-08- Rb^ B i^ `fo`rkcbobk`f^ rkfa^a+ `rv^ b`r^`fŽk `^oqbpf^k^ bp s0 * t0 < 0- Rb^ B bi `lk,

grkql ab mrkqlp l_qbkfal jriqfmif`^kal i^ `lloabk^a^ r ab `^a^ mrkql %r)s& ab bmlo rk c^`qlo `lkpq^kqb \ = N v i^ `lloabk^a^ t mlo rk c^`qlo `lkpq^kqb ] = l- Di `lk,grkql B pb abkljfk^ bifmpb- 'Br^kal \ < ]* i^ bifmpb bp lqo^ `fo`rkcbobk`f^-(^( Cbjlpqo^o nrb `^a^ mrkql %r)s& ab B p^qfpc^`b i^ b`r^`fŽk `^oqbpf^k^ %r,[&/ (* 'v._(1 < 0-_( Tqfifw^o i^p molmfba^abp ab i^ fkqbdo^i m^o^ abjlpqo^o nrb i^ obdfŽk ifjfq^a^ mlobp^ bifmpb bp jbaf_ib v nrb pr Šob^ bp oh]*

1/- Di Dgbo`f`fl 08 bp rk^ dbkbo^ifw^`fŽk abi bgbjmil 2 ab i^ Rb``fŽk 1-2- Dpq^_ib`bo v ab,jlpqo^o rk^ dbkbo^ifw^`fŽk `loobpmlkafbkqb ^i bgbjmil 3 ab i^ Rb``fŽk 1-2-

10- Blk rk o^wlk^jfbkql m^ob`fal ^i abi bgbjmil 4 ab i^ Rb``fŽk 1-2 abjlpqo^o bi qbl,obj^ 1-1-

*&- ?N` Sb[PV\[R`a_VT\[\Zna_VPN`

@kqbpab fkqolar`fo jŠp ^mif`^`flkbp ab i^ fkqbdo^`f5k+e^objlp rk^ _obsbafdobpf5k m^o^ `ljbkq^o i^p crk`flkbp qofdlklj‹qof`^p- Rrmlkbjlp nrb bi ib`qloqfbkb ^id•k `lkl`fjfbkql ab i^p molmfba^abpab i^p pbfpcrk`flkbp qofdlklj‹qof`^ppbkl+ `lpbkl+ q^kdbkqb+lq^kdbkqb+pb`^kqb v `lpb`^kqb: v prp fksbop^p ^o`l pbkl+^o`l `lpbkl+ ^o`l q^kdbkqb+bq`- Dpq^pcrk`flkbp pb afp`rqbk bk ilp `roplp ab Sof,dlkljbqoŒ^ bk obi^`f5k `lk mol_ibj^p afsboplp nrb obi^`flk^k ilp i^alp v ilpŠkdrilp ab ilp qofŠkdrilp-

K^p crk`flkbp qofdlklj‹qof`^p plk fjmloq^kqbp bk BŠi`ril+ kl p50/ mlo probi^`f5k `lk ilp i^alp v ilp Škdrilp ab rk qofŠkdril+pfkl jŠp _fbk mlo i^p molmfb,a^abp nrb mlpbbk `ljl api^dji`n, K^p pbfp crk`flkbp qofdlklj‹qof`^p qfbkbk bk`lj•k rk^ molmfba^a fjmloq^kqb ii^j^a^ mboflaf`fa^a-

Tk^ crk`f5k a bp k`md‡_d^\ `lk mboŒlalk " N pf pr aljfkfl `lkqfbkb s * kpfbjmob nrb `lkqbkd^ r v pf `%r * m( < `%r& m^o^qlal r abi aljfkfl ab `+ K^pcrk`flkbp pbkl v `lpbkl plk mbof5af`^pab mboŒlal160&+ pfbkal 60&bi Šob^ab rk afp`l`fo`ri^o rkfa^a- Lr`elp mol_ibj^p bk EŒpf` b HkdbkfboŒqo^q^kcbk5jbklp mbof5,af`lp 'q^ibp `ljl sf_o^`flkbp+ jlsfjfbkql mi^kbq^oflv ab lka^p( u i^p crk`flkbppbkl v `lpbkl `lkpqfqrvbk i^ _^pb m^o^bi ^kŠifpfpj^qbjŠqf`l ab q^ibpmol_ibj^p-

K^p crk`flkbp pbkl v `lpbkl mrbabk fkqolar`fopb ab s^of^p j^kbo^p- Olobgbjmil+ e^v abcfkf`flkbp dblj‹qof`^p nrb obi^`flk^k i^p crk`flkbp pbkl v `lpbkl

Page 138: Calculus

007 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

^ ilp Škdrilp+ v e^v lqo^p ab `^oŠ`qbo^k^iŒqf`lnrb fkqolar`bk bp^p crk`flkbp pfkobcbobk`f^^idrk^ ^ i^ FbljbqoŒ^- Tk^p v lqo^p plk bnrfs^ibkqbp+bk bi pbkqfal abnrb qla^p bii^p `lkar`bk ^ i^p jfpj^p crk`flkbp-

Cb loafk^ofl+ `r^kal qo^_^g^jlp `lk pbklp v `lpbklp kl klp fjmloq^k q^kqlprp abcfkf`flkbp `ljl i^p molmfba^abpnrb mrbabk abar`fopb ^ m^oqfoab prp abcf,kf`flkbp- @idrk^p ab bp^p molmfba^abp+fjmloq^kqbp bk BŠi`ril+ pb `fq^k pbdrfa^,jbkqb- Bloofbkqbjbkqb+ abpfdk^jlp ilp s^ilobp ab i^p crk`flkbp pbkl v `lpbklab s mlkfbkal pbks* `lp s* obpmb`qfs^jbkqb-

OQNOHDC@CDR ETMC@LDMS@KDR CDK RDMN X CDK BNRDMN-

/, Ajhdidj _` _`adid^d‡i, I\n api^dji`n n`ij t ^jn`ij `noƒi _`adid_\n `ioj_\ g\ m`^o\ m`\g,

+(

,(

S\gjm`n `nk`^d\g`n, Q`i`hjn `lp N < pbk Œ6i& < 0+`lp 60&< ,0-

@jn`ij _` pi\ _da`m`i^d\, M\m\ s ` v ^p\g`nlpd`m\* o`i`hjn

'1-2( ^jn&t + s' < ^jnt^jns )n`itn`is,

3- A`ndbp\g_\_`n api_\h`io\g`n, M\m\ N ; s ; q 60&+ o`i`hjn

'1-3(pbks .

N ; `lp s ; ,, ; ,, -s `lp s

@ m^oqfoab bp^p `r^qol molmfba^abpmlabjlp abar`fo qla^p i^p molmfba^abpabi pbkl v abi `lpbkl nrb qfbkbk fjmloq^k`f^ bk BŠi`ril- Dpql prdfbob nrb mlab,jlp fkqolar`fo i^p crk`flkbp qofdlklj‹qof`^p ^ufljŠqf`^jbkqb- Dpql bp+mlaoŒ^jlpqlj^o i^p molmfba^abp0 ^ 3 `ljl ^uflj^p abi pbkl u abi `lpbkl u abar`fo qla^pi^p abjŠp molmfba^abp`ljl qblobj^p- O^o^qo^_^g^o loob`q^jbkqb kl ab_b afp`r,qfopbpl_ob rk^ qbloŒ s^`Œ^+bp kb`bp^ofl mol_^o nrb bufpqbkcrk`flkbp nrb p^qfp,c^`bk i^p molmfba^abp^kqboflobp-Olo bi jljbkql m^p^objlp ab i^odl bpqbmol_ib,j^- Oofjbol prmlkbjlp nrb bufpqbk crk`flkbp nrb p^qfpc^`bkbpq^pmolmfba^abpcrka^jbkq^ibp v jlpqo^objlp `Žjl mrbabk abar`fopb i^p abjŠp molmfba^abp-Krbdl+ bk i^ Rb``fŽk 1-6+fkaf`^jlp rk j‹qlal dblj‹qof`l m^o^abcfkfobi pbkl vbi `lpbkl `ljl crk`flkbp `lk i^p molmfba^abpabpb^a^p- Dk bi `^mŒqril 00 q^j,_f‹k bp_lw^jlp rk j‹qlal m^o^ abcfkfo bi pbkl v bi `lpbkl-

SDNQDL@ 1-2- Rf _jn api^dji`n pbk v `lp n\odna\^`i g\n kmjkd`_\_`n 0 \ 2*n\odna\^`i o\h]d„i g\n ndbpd`io`n8

'^( I\ d_`iod_\_ kdo\b‡md^\*pbk! s * `lp! s < 0+ k\m\ oj_j s,'_( S\gjm`n `nk`^d\g`n* pbk N < `lp p60&< pbk 60&< N-

Page 139: Calculus

I\n api^dji`n omdbjijh„omd^\n 008

'a( Bg ^jn`ij `n api^d‡i k\m v `g n`ij `n api^d‡i dhk\m, Bnoj `n* k\m\oj_j s o`i`hjn

`lp %*r&< `lp r) pbk %*r&< ,pbku-

'a( @j+m`g\^dji`n, M\m\ oj_j s* n` od`i`

pbk/06 * r& < `lp r) blp ' 06 * r& < ,pbku-

'b( M`mdj_d^d_\_,M\m\ oj_j s*n` od`i` pbk&s* 106(< pbk s* `lp &s* 106(<< `lp s,

'b( C‡mhpg\n _` \_d^d‡i, M\m\ s ` t ^p\g`nlpd`m\* n` od`i`

`lp &s* t'; `lp s `lp t ,pbk s pbkt *

pbk&s* t' < pbks `lp t * `lp s n`it ,

'c( C‡mhpg\n _` _da`m`i^d\n, M\m\ oj_jn gjn q\gjm`n \ v ]* n` od`i`

\+] \)]pbk [ * pbk ] < 1 pbk,, `lp ,, +

1 1

\+] \)]`lp [ * `lp \ < ,1pbk,, pbk ,, -

+ +

'e( Jjijoji…\, Bi `g dio`mq\gj ZN+q6S I+ `g n`ij `n `nomd^o\h`io` ^m`^d`io` v`g ^jn`ij `nomd^o\h`io` _`^m`^d`io`,

A`hjnom\^d‡i, K^ m^oqb'^( pb abar`b fkjbaf^q^jbkqb pf qlj^jlp s < ubk '1-2( u rp^jlp i^ obi^`fŽk blp N < 0- K^ molmfba^a '^( obpriq^ ab i^ '^( qlj^k,al s < N+s <e6S+ U < 6S X rqfifw^kal i^ obi^`fŽk pbk q6S < 0- Prb bi `lpbkl bpm^oobpriq^ q^j_f‹k ab '1-2(e^`fbkal v < N-@ `lkqfkr^`fŽk abar`fjlp i^ cŽojri^

'1-4( blp 'i6S , r& < pbkr )

e^`fbkal v < p6R bk '1-2(- O^oqfbkal ab bpql v ab '1-2(+ bk`lkqo^jlp nrb bi pbklbp fjm^o+ mrbpql nrb

n`i&+s' < blp 'f * s' < BNR Z06, 'f , u(I <

< blp 06 lp 'z , s' * pbk06pbk'z , s' < +n`is ,

Page 140: Calculus

01/ >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

Dpql abjrbpqo^ '`(- O^o^ mol_^o 'a(+ rqfifw^jlp lqo^ sbw '1-4(+ obbjmi^w^kal mof,jbol s mlo 060! * s u irbdl s mlo +s, Di rpl obfqbo^al ab 'a( klp a^ bkqlk`bp i^pobi^`flkbp ab mboflaf`fa^a 'b(-

O^o^ abjlpqo^o i^p cŽojri^p ab ^af`fŽk m^o^ bi `lpbkl+ _^pq^ obbjmi^w^or mlo +sbk '1-2( u qbkbobk `rbkq^ i^ m^ofa^a l fjm^ofa^a- Tqfifw^kal i^ m^oqb'a( v i^ cŽojri^ ab ^af`fŽk m^o^bi `lpbkl pb l_qfbkb

pbk&s * t' < ,`lp &s * t * y( < ,`lp u `lp &V* y( * pbk s pbk&V* y( <

< `lp s pbkt * pbks `lp t ,

- Dpql abjrbpqo^ 'c(- O^o^ abar`fo i^p cŽojri^p ab afcbobk`f^p 'd(+ obbjmi^w^jlpmofjbol v mlo ,vbk i^ cŽojri^ ab ^af`fŽk m^o^pbk %r * v( l_qbkfbkal

pbk&s+ t' ;n`is^jnt + ^jnsn`it,

Qbpq^kal ‹pq^ ab i^ cŽojri^ m^o^pbk %r * v( u e^`fbkal 0/ jfpjl m^o^i^ crk`fŽk`lpbkl+ iibd^jlp ^

pbk&s* t' ,pbk'u , t' < 0n`it^jns*

`lp %r* s& * `lp %r* s& < ,1 pbkvpbk r +

G^`fbkal r < %[ * \&,/) v < %[ * \&,/) bk`lkqo^jlp nrb bp^p pb `lksfboqbk bki^p cŽojri^p ab afcbobk`f^p 'd(-

K^p molmfba^abpab i^ '^( ^ i^ 'd( pb e^k abar`fal pŽil `lk i^p 0+ 1 X 2-K^ molmfba^a 3 pb rp^ m^o^abjlpqo^o 'e(- K^p abpfdr^ia^abp '1-3( morb_^k nrb`lp s u pbks plk mlpfqfs^p pf N; s ; 60!- Cbpmr‹p ab bpql+pf N ; ] ; \ ; i6i!+

ilp k•jbolp %[ * \&,/ v %[ * \&,/ bpqŠkbk bi fkqbos^il '/+060!(+ v i^p cŽojri^pab afcbobk`f^p 'd( morb_^k nrb pbk \ = pbk ] v `lp \ ; `lp ], Dpql `ljmibq^ i^abjlpqo^`fŽk abi qblobj^ 1-2-

Dk bi moŽufjl `lkgrkql ab Dgbo`f`flp 'mŠdfk^018( pb `lkpfabo^k jŠp molmfb,a^abp ab i^p crk`flkbp pbkl v `lpbkl- Lbk`flk^jlp+ bk m^oqf`ri^o+alp cŽojri^pnrb `lk cob`rbk`f^ pb rp^k bk BŠi`ril- Rlk i^p ii^j^a^p abi ƒibpgj _j]g` l a‡m+hpg\n _` _pkgd^\^d‡i, Sbkbjlp

pbk0s < 1 pbku `lp s* `lp 0s < `lp! s ,pbk1 s < 0 , wpbk,s,

Dpqlp plk+ k^qro^ijbkqb+ pfjmibp `^plp m^oqf`ri^obpab i^p cŽojri^p ab ^af`fŽkl_qbkfalp e^`fbkal v < s, K^ pbdrka^ cŽojri^ m^o^`lp 160! obpriq^ ab i^ mofjbo^`lk i^ fabkqfa^a mfq^dŽof`^-Dpq^q^j_f‹k abjrbpqo^ nrb Z`lpsg R: 0 X Zpbku\ R: 0m^o^ qlal s,

Page 141: Calculus

C‡mhpg\n _` dio`bm\^d‡i k\m\ `g n`ij v `g ^jn`ij 010

1-5 EŽojri^p ab fkqbdo^`fŽkm^o^bi pbkl v bi `lpbkl

K^p molmfba^abpab jlklqlkŒ^ ab i^ m^oqb'e( abi qblobj^ 1-2+grkql `lk i^p`loobi^`flkbp v i^ mboflaf`fa^a+ abjrbpqo^k nrb i^p crk`flkbp pbkl v `lpbkl plkjlkŽqlk^p ^ qolwlp bk `r^inrfbo fkqbos^il- Olo `lkpfdrfbkqb+ jbaf^kqb bi rpl ob,mbqfal abi qblobj^ 0-01+sbjlp nrb bi pbkl u bi `lpbkl+ plk fkqbdo^_ibpbk `r^i,nrfbo fkqbos^il cfkfql- B^i`ri^objlp prp fkqbdo^ibp mif`^kal bi qblobj^ 0-03- Dpqb`Ši`ril rqfifw^alp abpfdr^ia^abp nrb klplqolp bkrk`f^jlp `ljl rk qblobj^9

RCMPCK? 1-3- Pd N ; ym8e60!+ X i x 0+ o`i`hjn

\ x f\ \ x f\'1-5( , J `lp , ; pbk\ ; , J `lp , -

i h<i i i f;L i

A`hjnom\^d‡i, K^p abpfdr^ia^abp '1-5( pboŠkabar`fa^p ab i^ fabkqfa^a

i

'1-6( 1 pbkquH`lp fs < pbk&i )x's x pbkiu +h<i

sŠifa^ m^o^i x 0 X qlal ob^i s, O^o^ abjlpqo^o '1-6(+ rqfifw^jlp i^p cŽojri^p abafcbobk`f^p 'd( abi qblobj^ 1-2 m^o^mlkbo

1 pbkiu `lp fs < pbk&f * i(u , pbk&f + g's,

G^`fbkal e < 0+ 1+ --- + i v prj^kal bp^p fdr^ia^abp+ bk`lkqo^jlp nrb bk i^prj^ abi pbdrkal jfbj_ol pb obar`bk rklp q‹ojfklp `lk lqolp l_qbkf‹kalpb '1-6(-

Rfiu kl bp rk j•iqfmil bkqbol ab 60! mlabjlp afsfafo ^j_lp jfbj_olp ab '1-6(mlo 1 pbk zs obpriq^kal

Gje pbk&i * g's + pbkiu`lp s < ,,,,,,,, -1 pbkv)r

FRG +

Qbbjmi^w^kal i mlo i + 0 X prj^kal 0 ^ ^j_lp jfbj_olp q^j_f‹k l_qbkbjlp

Gj,0f pbk'k , q(u * pbkqu

`lp u < ,,,,,,,, -1 pbkqu

f;L

Page 142: Calculus

011 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

Dp^p alp cŽojri^p plk sŠifa^p pf s 6&< Vgh ) pfbkal h bkqbol- Slj^kal s < \gi*alkab N ; [ w eoobk`lkqo^jlp nrb bi m^oab abpfdr^ia^abp '1-5( bp bnrfs^ibkqb ^ipfdrfbkqb

pbk%h* B9 , pbk'z([ h /h

i 1 pbkA9(pbk%h,B 9 * pbk 'z(

: m_h[ ;9 h 100

00 1 pbk '19(

Dpqbm^o+ pr sbw+bp bnrfs^ibkqb ^i m^o

'1-7(&\'pbk ,

pbk '00 * c&7 * pbk 'z( ; 0i pbk\ ; pbk&i + 0( 9 * pbk'-z( -i 100 '19( 1 00 100

Olo `lkpfdrfbkqb+ abjlpqo^o '1-5( bnrfs^ib ^ abjlpqo^o '1-7(- Cbjlpqo^objlp nrbpb qfbkb

'1-8( pbkcpbk%/h * 0(/ , pbk K ; , pbk /..- ; pbk%/h * .&- * pbk K

L

m^o^N ; 0i6 x ePl+ Br^kal 7 < \-&0i' '1-8( pb obar`b ^ '1-7(-

O^o^ abjlpqo^o i^ abpfdr^ia^a ab i^ m^oqbfwnrfboa^ ab '1-8(+ rp^jlp i^ c5o,jri^ ab ^af`fŽk m^o^ bi pbkl mlkfbkal

%/+.-& m_h%/h * 0(/ <pbk1kN`lpN * `lp1kNpbkN ; pbk /hK pbkN *pbkN> "

e^_fbkal rp^al q^j_f‹k i^p abpfdr^ia^abp

HI ; pbkc`lp q ++K $

l ; `lp /hK w 0 + pbk L< N+

pfbkal qla^p sŠifa^p v^ nrb N ; 0i6 x dQm,K^ abpfdr^ia^a '1-0/( bnrfs^ib ^ i^m^oqbfwnrfboa^ ab '1-8(-

O^o^ abjlpqo^o i^ abpfdr^ia^a ab i^ m^oqbabob`e^ ab '1-8(+ rqfifw^jlp krb,s^jbkqb i^ cŽojri^ ab ^af`fŽk m^o^bi pbkl mlkfbkal

pbk%/h * .&- < pbk/hK `lp _ * `lp /h_ pbk_ +

Page 143: Calculus

C‡mhpg\n _` dio`bm\^d‡i k\m\ `g n`ij v `g ^jn`ij 012

Rrj^kal pbk '( ^ ^j_lp jfbj_olp+ l_qbkbjlp

'1-00( pbk &0i + /'. * pbk 7 < pbk 0i6 '`lp 7 * pbk 7 0 , `lp 0iL' ,pbk 0i6

Obol v^ nrb qbkbjlp

, `lp 0i6 1pbk1 iL pbk i6pbk1kN < 1 pbkkb `lp iL < `lp iL%

bi pbdrkal jfbj_ol ab '1-00( bp fdr^i ^

0 j& L * Ln`iiL' 1 '(`lp'(`lpiL *pbk7pbkkN <pbk i `lp pbk ,, < pbk i ++++++++++^jniL ^jni&'

/ K `lp %h* .&-<pbk i ,

`lp iL

Olo `lkpfdrfbkqb+ m^o^ `ljmibq^o i^ abjlpqo^`fŽk ab '1-8(+ kb`bpfq^jlp q^k pŽilabjlpqo^o nrb

'1-01( ^[j[nx&i[+[/x'[L= p]bk]N

`lp h%& 6

Obol qbkbjlp

`lp iL < `lp &i + 0(7 `lp 7 , pbk'k , /'. pbk 7 ;7; `lp %h* 0(7 `lp 7 ; `lp %h* .&- * )

pbk 7

bk alkab lqo^ sbw ebjlp rqfifw^al i^ abpfdr^ia^a crka^jbkq^i `lp %F; %F.'pbk %F&+

Dpq^ •iqfj^ obi^`fŽk fjmif`^ '1-01(+ `lk il nrb pb `ljmibq^ i^ abjlpqo^`fŽk abiqblobj^ 1-3-

RCMPCK? 1-4- Pd _jn api^dji`n pbk u `lp n\odna\^`i g\n kmjkd`_\_`n api_\+h`io\g`n _` g\ 0 \ g\ 3+k\m\ oj_j \ m`\g n` od`i`

'1-03(

aj\ `lp u _s < pbk \ *

Gj\ pbk u _s < 0 , `lp \ ,

'1-02(

Page 144: Calculus

013 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

A`hjnom\^d‡i, Oofjbol pb abjrbpqo^ '1-02(+ v irbdl rp^jlp '1-02( m^o^abar`fo '1-03(- Rrmlkd^jlp nrb N ; \ Rn.P+ X^ nrb bi `lpbkl bp ab`ob`fbkqbbk ZN+[F) mlabjlp ^mif`^o bi qblobj^ 0-03 v i^p abpfdr^ia^abp abi qblobj^ 1-3l_qbkfbkal '1-02(- K^ cŽojri^ bp sŠifa^ q^j_f‹k m^o^ \ < N+ v^ nrb ^j_lpjfbj_olp plk `bol- Orbabk ^elo^ rqfifw^opbi^p molmfba^abp ab i^ fkqbdo^i m^o^^jmif^o pr s^ifabw ^ qlalp ilp s^ilobp ob^ibp \,

Olo bgbjmil+ pf +o/QR\ RN+bkqlk`bp N R, \ Rp6S+ W i^ molmfba^a abobcibuf5kklp a^

`+\ o oN `lp T ^r < , N `lp %*r& ^r < , N `lp T ^r < ,pbk %*[&< pbk[ +

@pŒ+mrbp+'1-02( bp sŠifa^ bk bi fkqbos^il W*n.P) n.PY+ Rrmlkd^jlp ^elo^ nrbq0SR[ R .P+ Dkqlk`bp *n.P R[ * .P R i0S + ab jlal nrb

a,\ a,ga-0 g\ n9`lp U ^r < `lp U ^r * `lp U ^r <pbk q0S * `lp %r * /Q' ^r :l l .`,/ *E`,/

c\+E`

< 0 , `lp T _s < 0 ,pbk &\ + .P&* pbk' *n.P& < pbk^-*E`,/

Blk biil obpriq^ nrb '1-02( bp sŠifa^ m^o^qlal \ bk bi fkqbos^il W*n.P) .PY+ Obolbpqbfkqbos^il qfbkb ilkdfqra /.P) `lk 0/ nrb i^ cŽojri^ '1-02( bp sŠifa^ m^o^qlal\ mrbpql nrb ^j_lp jfbj_olp plk mbofŽaf`lp obpmb`ql\ `lk mboŒlal/.P+

Rbdrfa^jbkqb rp^jlp '1-02( m^o^ abar`fo '1-03(- @kqb qlal abjlpqo^jlpnrb '1-03( bp sŠifa^ `r^kal \ < /Q .1- @mif`^kal pr`bpfs^jbkqb+ i^ molmfba^a abqo^pi^`f5k+i^ `l,obi^`f5k pbk %r* n.P& < `lp r) v i^ molmfba^a ab obcibuf5k+bk,`lkqo^jlp

/.`,/ GL & ' GL /.`

,/

m_hr^r: pbk r($ ++ ^r: ]imr^r: ]im%*r&^r+l ,E`,/ 1 ,E`,/ l

G^`fbkal rpl ab i^ obi^`f5k `lp %*r& < `lp r v i^ fdr^ia^a '1-02(+ pb l_qfbkb

`+.`,0l pbku ^r < 0-

Olo `lkpfdrfbkqb+ m^o^`r^inrfbo [ ob^i+mlabjlp bp`of_fo

%[m_hU ^r < '0c.1pbkU ^r (`[ pbkU ^r < 0 * %[*E`,/m_h&U * & --( ^r :Il Il .`,/ Il 1

< 0 * c[*E`,/]im T ^r < 0 * pbk &\ + f( < 0 , `lp \ ,Dpql abjrbpqo^ nrb i^ fdr^ia^a '1-02( fjmif`^ '1-03(-

Page 145: Calculus

C‡mhpg\n _` dio`bm\^d‡i k\m\ `g n`ij t `g ^jn`ij 014

DSDLOKN 0- Tp^kal '1-02( u '1-03( grkql `lk i^ molmfba^a ^afqfs^

nz't( _s < nz't( _s + E7y&s' _s*

iibd^jlp ^ i^p cŽojri^p ab fkqbdo^`fŽkjŠp dbkbo^ibp

nlp s _s < pbk ] + pbk \

v

bpbku _s < '0 , `lp ]' + '0 , `lp \' < ,'`lp ] + `lp \',aH

Rf krbs^jbkqb rqfifw^jlp bi pŒj_lil bpmb`f^i x%r&Gym^o^ fkaf`^o i^ afcbobk`f^`%\& * `_[&) mlabjlp bp`of_fobp^p cŽojri^p ab fkqbdo^`fŽkbk i^ cloj^

nlp s _s < pbk s e9 u npbk s _s < ,`lp s 09-

DSDLOKN 1- Blk ilp obpriq^alp abi bgbjmil 0 u i^ molmfba^aab afi^q^`fŽk

G] 0 e@]y&s' _s < , y&s-`' _s *

\ 5 43

l_qbkbjlp i^p cŽojri^p pfdrfbkqbp+sŠifa^p m^o^ _ ;gj+ i7

f] 0 e@] 0`lp `s _s < , `lp s _s < , &n`i`] + pbk`\' *

\ b ^\ b

u

I] 0 e@] 0

pbk`s _s < , pbks _s < , , '`lp `] + `lp `\',\ b KJ b

DSDLOKN 2- K^ fabkqfa^a `lp 0s; 0 ,1pbk1 s fjmif`^pbk! s < 'i,`lp 0s'`lk 0/ nrb+ ^ m^oqfoabi bgbjmil 1+ l_qbkbjlp

c\ 0c\[ 0pbk! s _s < , '0 , `lp 0s' _s < , , , pbk1^ -

l 1 l 1 3

Page 146: Calculus

015 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

Orbpql nrb pbk! s * `lp! s < 0+bk`lkqo^jlp q^j_f‹k

g\ g\ g\ [ 0^jn0s_s; &/+n`i0s'_s;\+ n`i0s_s;+)+n`i0\,l l l 1 3

1-6 Cbp`ofm`fŽk dblj‹qof`^ ab i^p crk`flkbp pbkl v `lpbkl

Dk bpq^Rb``fŽk fkaf`^jlp rk j‹qlal dblj‹qof`l m^o^ abcfkfo i^p crk`flkbppbkl v `lpbkl+ v a^jlp rk^ fkqbomobq^`fŽkdblj‹qof`^ ab i^p molmfba^abpcrka^,jbkq^ibp `fq^a^p bk i^ Rb``fŽk 1-4-

Blkpfabobjlp rk^ `fo`rkcbobk`f^ ab o^afl l v `bkqol bk bi lofdbk- Cbpfdkb,jlp bi mrkql %l)N( mlo =) v pb^ L `r^inrfbo lqol mrkql ab i^ `fo`rkcbobk`f^-Klp alp pbdjbkqlp ob`qfiŒkblpL> v LM abqbojfk^k rk^ cfdro^ dblj‹qof`^ ii^j^a^Škdril nrb obmobpbkq^jlp `lk bi pŒj_lil H>LM, Tk bgbjmil pb obmobpbkqbk i^cfdro^ 1-5- Prbobjlp ^pfdk^o ^ bpqb Škdril rk k•jbol ob^i kl kbd^qfsl s nrbmrbab rp^opb `ljl jbafa^ ab pr j^dkfqra- Di j‹qlal jŠp `loofbkqb m^o^e^`boilbp qlj^o rk^ `fo`rkcbobk`f^ ab o^afl 0 s ii^j^o T0 i^ilkdfqra&abi ^o`l >M*_`n^mdoj

alp sb`bp bi Šob^ abi pb`qlo

EHFTQ@ 1-5 Ri ƒibpgj I >LM _`s m\_d\i`n,

EHFTQ@ 16- A`n^mdk^d‡i b`jh„omd^\ _`pbk s v `lp s*

bk bi pbkqfal `lkqo^ofl ^i ab i^p ^drg^p abi obilg ab = ^ L) v ab`fo nrb i^ jbafa^ab I >LM bp s o^af^kbp- Cbpab rk mrkql ab sfpq^ iŽdf`l+ bpql kl bp p^qfpc^`qloflmlo bi jljbkql mrbp kl pb e^ mob`fp^al bi `lk`bmql ab ilkdfqra ab ^o`l- …pqbpboŠafp`rqfal bk bi `^mŒqril03- Orbpql nrb i^ kl`fŽk ab Šob^e^ pfal v^ afp`rqfa^+mobcbofjlp rqfifw^obi Šob^ abi pb`qlo `fo`ri^o >LM bk ird^o ab i^ ilkdfqra abi^o`l >M ^jhj jbafa^ ab i^ j^dkfqra ab I >LM, Rb pl_obkqfbkab nrb bi pb`qlo

Page 147: Calculus

A`n^mdk^d‡i b`jh„omd^\ _` g\n api^dji`n n`ij v ^jn`ij 016

>LM bp i^ mlo`fŽk jŠp mbnrb•^ abi afp`l `fo`ri^o `r^kal M bpqŠ mlo bk`fj^ abibgb ob^i v i^ j^vlo `r^kal M bpqŠ mlo ab_^gl abi bgb ob^i-

LŠp ^abi^kqb+ `r^kal pb e^v^ afp`rqfal i^ ilkdfqra abi ^o`l+ sbobjlp nrbbi ^o`l >M qfbkb rk^ ilkdfqra bu^`q^jbkqb al_ib abi Šob^ abi pb`qlo >LM, Olo `lk,pfdrfbkqb+ m^o^ `lkpbdrfo i^ jfpj^ bp`^i^ ab jbafa^ ab Škdrilp mlo ilp alp j‹ql,alp+ rp^objlp bi _j]g` abi Šob^ abi pb`qlo >LM `ljl jbafa^ abi Škdril I >LM,Ml l_pq^kqb+ m^o^ l_qbkbo rk^ jbafa^ fkabmbkafbkqb ab i^ rkfa^a ab afpq^k`f^bk krbpqol pfpqbj^ `lloabk^al+ abcfkfobjlp i^ jbafa^ ab I >LM `ljl bi _j]g`_`g ƒm`\ _`g n`^ojm >LM _dqd_d_\ kjm `g ^p\_m\_j _`g m\_dj, Dpq^ o^wŽk kl s^oŒ^pf afi^q^jlp l `lkqo^bjlp bi `Œo`ril+ v mlo q^kql kl pb mfboab dbkbo^ifa^a ^i obp,qofkdfo krbpqo^p `lkpfabo^`flkbp ^i `Œo`ril rkfa^a- K^ rkfa^a ab jbafa^ ^pŒl_qb,kfa^ pb ii^j^ m\_dƒi, @pŒnrb+ ab`fjlp nrb i^ jbafa^ ab rk Škdril I >LM bps o^af^kbp pf s-0 bp bi Šob^ abi pb`qlo >LM abqbojfk^al bk bi afp`l `fo`ri^orkfa^a-

X^ ebjlp fkqolar`fal bi pŒj_lil 6R m^o^ abpfdk^o bi Šob^ ab rk afp`l `fo`ri^orkfa^a- Br^kal M < ',0+ N(+ bi pb`qlo >LM bp rk pbjf`Œo`ril ab Šob^ p6S+ abjlal nrb pr_qfbkab rk Škdril ab 6R o^af^kbp- Di afp`l `ljmibql bp rk pb`qlo ab16S o^af^kbp- Rf fkf`f^ijbkqb M bpqŠ bk 'i+ N( W pb abpmi^w^ rk^ sbw ^iobabalo ab i^`fo`rkcbobk`f^ bk pbkqfal `lkqo^ofl ^i ab i^p ^drg^p abi obilg+ bi Šob^ abi pb`qlo>LM `ob`b ab N ^ 6S+ qlj^kal qlalp ilp s^ilobp abi fkqbos^il ZN+6S\ bu^`q^jbkqbrk^ sbw- Dpq^ molmfba^a+ nrb dblj‹qof`^jbkqb bp ^`bmq^_ib+ mrbab abjlpqo^opbbumobp^kal bi Šob^ `ljl rk^ fkqbdo^i+ mbol kl bumlkaobjlp i^ abjlpqo^`fŽk-

Di pfdrfbkqb m^pl bp abcfkfo bi pbkl u bi `lpbkl ab rk Škdril- Dk ob^ifa^a+mobcbofjlp e^_i^o abi pbkl v abi `lpbkl ab rk iˆh`mj jbglo nrb ab rk Škdril+ab jlal nrb bi pbkl u bi `lpbkl pboŠk api^dji`n abcfkfa^p pl_ob i^ ob`q^ ob^i-Ool`babjlp `ljl pfdrb9 Blkpfabo^jlp rk k•jbol s q^i nrb N ; s ; 16S v pb^ Mbi mrkql ab i^ `fo`rkcbobk`f^ rkfa^a q^i nrb bi Šob^ abi pb`qlo >LM pb^ fdr^i ^r,/+ Rb^k %[) \& i^p `lloabk^a^p ab L+ Dk i^ cfdro^ 1-6 pb obmobpbkq^rk bgbjmil-Klp k•jbolp \ v ] bpqŠk `ljmibq^jbkqb abqbojfk^alp mlo s, Cbcfk^jlp bi pbklv bi `lpbkl ab s `ljl pfdrb9

`lp r < [) pbk s < \+

Cf`el ab lqol jlal+ `lp s bp i^ ^_p`fp^ ab M u pbk s bp pr loabk^a^-

Olo bgbjmil+ `r^kal s < 6S+ qbkbjlp M < ', 0+N( ab jlal nrb `lp 6S < , 0v pbk 6S < N- @kŠild^jbkqb+ `r^kal s < z6S qbkbjlp M < 'N+ 0( v mlo q^kql`lp z6S < N W pbk z6S < 0- Dpqb mol`bafjfbkql a^ bi pbkl v bi `lpbkl `ljl crk,`flkbp abcfkfa^p bk bi fkqbos^il ^_fboql 'N+ 16S(- Rb buqfbkabk i^p abcfkf`flkbp ^qlal bi bgb ob^i mlo jbafl ab i^p fdr^ia^abp pfdrfbkqbp9

pbk N < N+ `lp /< 0+ pbk %r * 16S( < pbk r) `lp %r * 16S( < `lp T †

Page 148: Calculus

017 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^dƒi

K^p lqo^p `r^qol crk`flkbp qofdlklj‹qof`^p pb abcfkbk ^elo^ bk crk`fŽk abi pbklv abi `lpbkl jbaf^kqb i^p `lkl`fa^p cŽojri^p+

pbkuq^k r <,,

`lp s%`lp s

`lqu <,,pbku&

0pb`u <,,

`lp s%- 0`p`u < ,,-

pbku

Dpq^pcrk`flkbp bpqŠabcfkfa^p m^o^qlal ob^i s p^isl bk `fboqlp mrkqlp ^fpi^alp bkilp nrb ilp abkljfk^alobp mrbabk pbo `bol- R^qfpc^`bki^ molmfba^a ab mboflaf`f,a^a `%r * 0/5' < `%r&+K^ q^kdbkqbv i^ `lq^kdbkqb qfbkbkbi mboŒlaljbklo /5,

@ `lkqfkr^`fŽk a^oklp ilp o^wlk^jfbkqlp qofdlklj‹qof`lp m^o^ fkaf`^o `Žjlbp^p abcfkf`flkbp klp iibs^k ^ i^p molmfba^abp crka^jbkq^ibp `fq^a^p bk i^ Rb`,`fŽk 1-4- K^p molmfba^abp 0 v 1 e^k pfal v^ qbkfa^p bk `rbkq^ ^i abcfkfo bi pbklv bi `lpbkl- K^ fabkqfa^a mfq^dŽof` obpriq^ bsfabkqb ^kqb i^ cfdro^ 1-6- Di pbdjbk,ql ob`qfiŒkblKL bp i^ efmlqbkrp^ ab rk qofŠkdril `rvlp `^qbqlp qfbkbk ilkdfqrabpiblp u\ v Zpbkt[+ Olo q^kql+ bi qblobj^ ab OfqŠdlo^pm^o^ qofŠkdrilp ob`qŠkdrilpfjmif`^ i^ fabkqfa^a `lp! s * pbk!s < 0-

Nqo^ sbw rqfifw^oklpbi qblobj^ ab OfqŠdlo^pm^o^a^o rk^ abjlpqo^`fŽk dbl,j‹qof`^ ab i^ cŽojri^ '1-2( m^o^`lp 'u , r&+ Efg‹jlklp bk ilp qofŠkdrilp ob`qŠk,drilp L=K v L>K af_rg^alp bk i^ cfdro^ 1-7- Dk bi qofŠkdril L=K) i^ ilkdfqra abii^al =K bp Zpbku , pbk sg* bi s^ilo ^_plirql ab i^ afcbobk`f^ ab i^p loabk^a^pab N v L+ Cbi jfpjl jlal+ =L qfbkb ilkdfqra Z`lps + `lp uh-Rf ^ obmobpbkqi^ilkdfqra ab i^ efmlqbkrp^ LK) qbkboklp+pbd•k bi qblobj^ ab OfqŠdlo^p+

_0 < 'pbkt + pbkU'0 * '`lp s + `lp t'0,

Olo lqo^ m^oqb+bk bi qofŠkdril ob`qŠkdril L>K bi `^qbql >L qfbkb ilkdfqra00 , `lp 'u , s'- vi^ abi `^qbql >K bp Zpbk'u , t(h- Olo `lkpfdrfbkqb+ bi qblobj^ab OfqŠdlo^pklp a^

_0 < Y0, `lp %s* U'Z0 *pbk1 &V+ r& +

Hdr^i^kal i^p alp bumobpflkbpab _0 v abpmbg^kal `lp 'u , r&) pb l_qfbkb i^ cŽo,jri^ '1-2( m^o^`lp 'u , r&+

Efk^ijbkqb+ i^p abjlpqo^`flkbp dblj‹qof`^p ab i^p abpfdr^ia^abp crka^jbk,q^ibp ab i^ molmfba^a 3 mrbabk a^opb pl_ob i^ cfdro^ 1-8- Bljm^o^oklp q^k pŽilbi Šob^ abi pb`qlo K=L `lk i^ ab ilp qofŠkdrilp KKL v K=>+ Rbd•k i^ abcfkf`fŽka^a^ ab jbafa^ ^kdri^o+ bi Šob^ abi pb`qlo K=L bp ps, Di qofŠkdril K=> qfbkb_^pb 0 v ^iqro^ b) mlo bgbjmil- Olo i^ pbjbg^kw^ ab qofŠkdrilp+ pb bk`rbkqo^e.q < 'pbk u(.'`lp s'* `lk il nrb bi Šob^ abi qofŠkdril K=> bphc;En`i s'-'`lp r&+ Olo `lkpfdrfbkqb+ i^ `ljm^o^`fŽk ab i^p Šob^pklp a^ i^p abpfdr^ia^abp

0 0 0 pbku, pbks `lp s ; , s ; , ,, -1 1 1 `lp s

Page 149: Calculus

Be`m^d^djn 018

N;&^jnt*n`it' >

c < pbku^jns

pbk s

^jns

i N >

EHFTQ@ 1-7 A`hjnom\^d‡i b`jh„omd^\ _`g\ a‡mhpg\ `lp 'u , s',

EHFTQ@ 1-8 A`hjnom\^d‡i b`jh„omd^\ _`g\n _`ndbp\g_\_`n

n`is 0N ; `lp s ; ,, ; ,, -

s `lp s

Cfsfafbkal mloz pbks v qlj^kal ilp ob`Œmol`lp+l_qbkbjlp i^p abpfdr^ia^abp crk,a^jbkq^ibp '1-3(-

Qb`loa^jlp ^00b`qlo rk^ sbw jŠp+ nrb `lk il nrb bk bpq^Rb``fŽk pb `ljbkq^klp molmlkbjlp a^o rk^ fkqbomobq^`fŽkdblj‹qof`^ abi pbkl u abi `lpbkl u abprp molmfba^abpcrka^jbkq^ibp- Dk i^ Rb``fŽk 00-00+pb lcob`b rk bpqrafl ^k^iŒqf`lab bp^p crk`flkbp bk bi nrb kl pb rqfifw^ i^ FbljbqoŒ^-

Dk jr`elp j^kr^ibp ab L^qbjŠqf`^p ^m^ob`bk q^_i^p ab s^ilobp ab pbkl+`lpbkl+ q^kdbkqbv `lq^kdbkqb- Dk i^ cfdro^ 1-0/ 'mŠd- 021( pb e^k af_rg^al i^pdoŠcf`^pab i^p pbfpo^wlkbp qofdlklj‹qof`^p `ljl ^m^ob`bkbk rk fkqbos^il ab rkmboŒlalab ^jmifqra- Qb`roofbkal ^ i^ mboflaf`fa^a pb l_qfbkb bk `^a^ `^pl biobpql ab i^ doŠcf`^-

*&0 :WR_PVPV\`

Dk bpqb `lkgrkql ab Dgbo`f`flp+ pb mrbabk bjmib^o i^p molmfba^abp abi pbkl v abi `lpbkl`fq^a^p bk i^p Rb`b`flkbp ab i^ 1-4 ^ i^ 1-6-0- '^( Cbjlpqo^o nrb pbk i\o < N m^o^ qlal bkqbol i v nrb bplp plk ilp •kf`lp s^ilobp ab s

m^o^ ilp nrb pbk s < N-'_( G^ii^o qlalp ilp s^ilobp ob^ibp s q^ibp nrb `lp s < N-

1- G^ii^o qlalp ilp ob^ibp r q^ibp nrb '^( pbk r < 0: '_( blp r < 0: 'b( pbk r < ,0:'a( `lp s < , 0-

2- Cbjlpqo^o nrb pbk %r * &60&(< , pbk r v blp %r * &60&(< , blp r m^o^ qlal r+3- Cbjlpqo^o nrb pbk 0r < 2 pbk r * 3 pbk! r u blp 0r < blp r * 3 pbk= r blp r m^o^ qlal

ob^i r+ Cbjlpqo^o q^j_f‹k nrb blp 0r < 3 `lp, r * 2 blp r)

4- '^( Cbjlpqo^o nrb pbk i6S < h+`lp d < isf XFi_d^\^d‡i8 G^`bo rpl abi Dgbo`f`fl 3-\

Page 150: Calculus

02/ >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

'_( Cbjlpqo^o nrb /Q < is.2+ @LP /Q < p-'`( Cbjlpqo^o nrb pbk o/Q < BNR o/Q < qs.i-

5- Cbjlpqo^o nrb q^k %r * u( < 'q^k r * q^k u(.'h * q^k r q^k u( m^o^ qlal m^o ab s^ilobpr) u q^ibp nrb q^k r q^k u !! , 0- N_qbkbo i^p `loobpmlkafbkqbp cŽojri^p m^o^ q^k %r * u(v ]in%r * s&+

6- G^ii^o alp k•jbolp = v > q^ibp nrb 2 pbk'u * 0&00!( < = pbk r * > `lp r m^o^ qlal r7- Cbjlpqo^o nrb pf B v \ plk k•jbolp ob^ibp a^alp+ bufpqbk alp k•jbolp ob^ibp = v >

q^ibp nrb B pbk %r * [& < = pbk r * > `lp r m^o^ qlal r+8- Cbjlpqo^o nrb pf = v > plk k•jbolp ob^ibp a^alp+ bufpqbk alp k•jbolp B v \* pfbkal

8 1&9N+ q^ibp nrb i^ cŽojri^ abi Dgbo`f`fl 7 bp sŠifa^-0/- Cbqbojfk^o B v [) pfbkal B = N+ q^ibp nrb B pbk %r * [& < , 1 pbk r * 1 `lp r m^o^

qlal s,00- Cbjlpqo^o nrb pf = v > plk k•jbolp ob^ibp a^alp+ bufpqbk alp k•jbolp B v \* pfbkal

B 1&9N+ q^ibp nrb B `lp %r * [& < = pbk r * > `lp r+ Cbqbojfk^o B v [ pf = < > < 0-01- G^ii^o qlalp ilp k•jbolp ob^ibp s q^ibp nrb pbk s < `lp s,02- G^ii^o qlalp ilp k•jbolp ob^ibp q^ibp nrb pbk s + `lp s < 0-03- Cbjlpqo^o nrb i^p fabkqfa^abp pfdrfbkqbp plk sŠifa^p m^o^ qlalp ilp m^obp s b v9

'^( /]imr]ims < ]im%r * s& * ]im%r * s&+'_( 1pbk r m_hs < `lp %r * s& * `lp %r * s&+'`( /m_hr ]ims < pbk %r * s& * pbk %r * s&+

04- Rf c !! N+ abjlpqo^o nrb i^p fabkqfa^abp pfdrfbkqbp plk sŠifa^p m^o^ qlal s8

pbk %r * b& * pbk r Z pbk %b,/& % w&b * b,/ `lp r * 1 &

]ZiZmZ%ZrZ(Z{ZbZ&Z*Z]ZiZmZr< ] p]b]k]'e].1](pbk %r *y( -b b,/ +

Dpq^p cŽojri^p pb rqfifw^k bk BŠi`ril afcbobk`f^i-05- Cbjlpqo^o pf plk l kl `fboq^p i^p pfdrfbkqbp ^cfoj^`flkbp-

'^( O^o^ qlal r !! N+ pb qfbkb pbk /r !! 1 pbk r+'_( O^o^ `r^inrfbo r) bufpqb rk u q^i nrb blp %r * u( < blp r * `lp u-'b( Dufpqb rk r q^i nrb pbk %r * u( < pbk r * pbk u m^o^ qlal u-'a( Dufpqb rk v !! N q^i nrb k pbk s _s < pbk v-

06- B^i`ri^o i^ fkqbdo^i Pxpbk s _s m^o^ `^a^ rkl ab ilp pfdrfbkqbp s^ilobp ab \ v ]* Dk `^a^`^pl fkqbomobq^obi obpriq^al dblj‹qof`^jbkqb bk crk`fŽk abi Šob^-

']( [ < N+ ] < /Q-4,'_( \ < N+] < /Q-2,'b( \ < N+] < /Q-1,'a( [ < N+] < /Q-0,

B^i`ri^o i^p fkqbdo^ibp ab ilp Dgbo`f`flp abi

'b( [ < N+] < /Q,

'b( \ < N+] < 0/Q,%a&[:*E)\:.+'d( \ < +/Q-4* ] < /Q-2,

07 ^i 16-

.5+ F8%r * pbk r& ^r+

'!.108- Il %r/ * blp r& ^r+

'!.11/- Il 'pbk r * `lp r& ^r+

'!.110- Il Zpbkw , blp -b\_s*

Page 151: Calculus

Be`m^d^djn 020

11- b'p * `lp o' _o,

12- I9Gp* `lp o\ _o,

13- p!Gp* `lp od_o* pf N z s x PP+

14- 491

'q1 * pbk o' _o,

x!.115- Il pbk 0s _s,

I!-1 s

16- l `lp 1 _s,

17- Cbjlpqo^o i^p pfdrfbkqbp cŽojri^p ab fkqbdo^`fŽk+ sŠifa^p m^o^ ] x l9

:B 0Il `lp &\ * ]o' ^n < ] Zpbk&\ * ]s' ,pbk \Z*

<s 0

l pbk &\ * ]o' _o < , ] Z`lp &\ * ]s' + `lp \Z ,

18- '^( G^`bo rpl ab i^ fabkqfa^a pbk 2. < 2 pbk o + 3 pbk! o m^o^ abar`fo i^ cŽojri^ ab fk,qbdo^`fŽk

I9pbk2 o _o <e, f'1 *pbk1 r& `lp r+

'_( Cbar`fo i^ fabkqfa^a `lp 2. < 3 `lp! o + 2 `lp o v rqfifwŠkali^ m^o^ abjlpqo^o nrb

IU `lp! o _o < f'1 * `lp! r& pbk r+

2/- Rf rk^ crk`fŽk ` bp mbofŽaf`^ ab mboŒlal j = N b fkqbdo^_ib bk ZN+jY) abjlpqo^o nrbPD y&s' _s < Px)k y&s' _s m^o^ qlal \,

20- '^( Cbjlpqo^o nrb F! pbk is _s <k! `lp is _s < N m^o^ qlalp ilp bkqbolp i9„ N-'_( Tp^kal i^ m^oqb '^( v i^p cŽojri^p ab ^af`fŽk m^o^ pbkl v `lpbkl+ bpq^_ib`bo i^p pf,drfbkqbp cŽojri^p+ sŠifa^p m^o^ ilp bkqbolp h v i* q^ibp nrb h0 x i09

x1! '1! x1!Il pbk is `lp hs _s < Il pbk is pbk hs _s < Il `lp is `lp hs _s < N +

x_ x1!Il pbk1 is _s < Il `lp&&is _s < QQ * pf i9„ N-

Dpq^p cŽojri^p plk i^p obi^`flkbp ab loqldlk^ifa^a m^o^ bi pbkl v bi `lpbkl-21- @ m^oqfo ab i^ fabkqfa^a

s s s1 pbk 1 `lp fs < pbk &0f * 0( 1 , pbk &0f + 0( 1

u ab i^p molmfba^abp qbibp`Žmf`^p ab i^p prj^p cfkfq^p abjlpqo^o nrb pf r w /g %gbkqbol(+ pb qfbkb

z ] pbk ois `lp o&i * /'sJ `lphu , 0 ‘fxg n`i8gs

Page 152: Calculus

021 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

t s

ks

+(T(

t t

t < q^ks

32 V < ^jos

1

0

s l --- s0&!0,010GG0GGGGG

t t

-

,t;n`^s

+t;^n^s

sN

s---& N p! I--- -Š-0 +(T(

,,0,0 &ej&,0+*

G'+ '+G

00 ', ',GH '- G G '-G H GG H 0G G G

EHFTQ@ 1-0/ Dmƒad^\n _` g\n api^dji`n omdbjijh„omd^\n ^jmm`nkji_d`io`n \ pi dio`mq\gj_` pi k`m…j_j,

Page 153: Calculus

@jjm_`i\_\n kjg\m`n 022

22- Rf s"+0 g4P %g bkqbol(+ abjlpqo^o nrb

z ] pbk gis pbk g`i * i(uJ---pbk er * 0 Š

h<i pbk dFT

23- Rb e^`b obcbobk`f^ ^ i^ cfdro^ 1-6- Olo `ljm^o^`fŽk abi Šob^ abi qofŠkdril D>M `lki^ abi pb`qlo `fo`ri^o D>M* abjlpqo^o nrb pbk s ; s pf N ; s ; 6S- Tp^kal bkqlk`bp bieb`el ab nrb pbk ', r& < , pbk r) abjlpqo^o nrb Yoajht[ ; Gthpf /; Gth; e6S-

1-8 8\\_QR[NQN` ]\YN_R`

G^pq^ ^elo^ ebjlp pfqr^al mrkqlp bk bi mi^kl `lk `lloabk^a^p ob`q^kdri^obp-S^j_f‹k mlabjlp pfqr^oilp `lk `lloabk^a^p mli^obp-Rb e^`b abi jlal pfdrfbkqb-Rb^ M rk mrkql afpqfkql abi lofdbk- Rrmlkd^jlp nrb bi pbdjbkql ab ob`q^ nrb rkbM ^i lofdbk qfbkbilkdfqra m = N u cloj^ rk Škdril '( `lk bi bgbs mlpfqfsl- U‹^pbi^ cfdro^ 1-00- Klp alp k•jbolp m u '( pb ii^j^k `lloabk^a^p mli^obp ab L+ DpqŠkobi^`flk^a^p `lk i^p ob`q^kdri^obp %r) u( mlo i^p fdr^ia^abp

'1-04( s < m`lp '(+ t < opbkN-

t

s < m`jn &'s

j < y2

K < y3

K < y5

t

L < 'u+ s&

st < o pbkc

EHFTQ@ 1-00 @jjm_`i\_\n kjg\m`n, EHFTQ@ 1-01 @pmq\ i ajmh\ _` j^cj ^pt\

`^p\^d‡i kjg\m `n m< y-

Page 154: Calculus

023 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

Di k•jbol mlpfqfsl o pb ii^j^ _dno\i^d\ m\_d\g l m\_dj q`^ojm ab M*v L bp rk ƒibp+gj kjg\m l \mbph`ioj, Cb`fjlp pi Škdril mli^o v kl `g Škdril mli^o mlonrb pf Lp^qfpc^`b '1-04(+ q^j_f‹k il e^`b '( * /h.P `r^inrfbo^ nrb pb^ bi bkqbol hw Blksb,kfjlp bk ii^j^o `lloabk^a^p mli^obp ab M ^ qlalp ilp m^obp ab k•jbolp ob^ibp%l)K& pf p^qfpc^`bk '1-04( pfbkal o= N- Cb bpqb jlal+ rk mrkql a^al mlpbb jŠpab rk m^o ab `lloabk^a^p mli^obp- K^ afpq^k`f^ o^af^i o bpqŠ abqbojfk^a^ `lkrkf`fa^a+ o < T s0 * t0* mbol bi Škdril mli^o K nrba^ abqbojfk^al p^isl j•iqf,milp bkqbolp ab /.P+

Br^kal L bp bi lofdbk+ i^p b`r^`flkbp '1-04( pb p^qfpc^`bk `lk m< N X `r^i,nrfbo K+ Olo bpq^ o^wŽk ^pfdk^jlp ^i lofdbk i^ afpq^k`f^ o^af^i m < N+X `lksbkfjlpbk nrb ^p\glpd`m k•jbol ob^i L mrbab rp^opb `ljl Škdril mli^o-

Rb^ ` rk^ crk`fŽk kl kbd^qfs^ abcfkfa^ bk rk fkqbos^il W[) \Y+ Di `lkgrkqlab qlalp ilp mrkqlp ab `lloabk^a^p mli^obp %l)%d&nrb p^qfpc^d^k l < a`L' bp i^doŠcf`^ ab a bk `lloabk^a^p mli^obp- K^ b`r^`fŽk l < a`L' pb ii^j^ b`r^`fŽk mli^oab bp^ doŠcf`^- O^o^ `fboq^p `ros^p+ i^p b`r^`flkbp mli^obp mrbabk pbo jŠp pbk`f,ii^p v ab rpl jŠp c^slo^_ib nrb i^p b`r^`flkbp `^oqbpf^k^p- Olo bgbjmil+ i^ `fo,`rkcbobk`f^ ab b`r^`fŽk `^oqbpf^k^ s0 * v1 < 3 qfbkb i^ pbk`fii^ b`r^`fŽk mli^ol < 1- K^p b`r^`flkbp '1-04( fkaf`^k `Žjl mrbab m^p^opbab `lloabk^a^p `^oqbpf^,k^p ^ mli^obp-

DIDLOKN- K^ cfdro^ 1-01 klp jrbpqo^ rk^ `ros^ `lk bi ^pmb`ql ab rk l`el`rv^ b`r^`flk `^oqbpf^k^ bp %r0 * t0'1 < tx, Tqfifw^kal '1-04(+ bk`lkqo^jlps0 * v1 < m!*ab jlal nrb i^p `lloabk^a^p mli^obp ab ilp mrkqlp ab bp^ `ros^p^qfpc^`bk i^ b`r^`fŽk l4 < l0 pbk! K) l o1 < Zpbk‹ \+ l < v{pb•Ni- Ml bp afcŒ`fiaf_rg^o bpq^ `ros^ ^ m^oqfo ab i^ b`r^`fŽk mli^o- Olo bgbjmil+ bk bi fkqbos^ilN z ` x .P,/) pbk K `ob`b ab N ^ )$ `lk il nrb l q^j_f‹k `ob`b ab N ^ 0- Rfqr^kalrklp ml`lp mrkqlp `rv^p `lloabk^a^p pb^k cŠ`fibp ab `^i`ri^o+ mlo bgbjmil+ ilpnrb `loobpmlkabk ^ ` < .P,3) .P,1 X .P,0) `^pf af_rg^jlp i^ mlo`fŽk ab i^ `ros^pfqr^a^ bk bi mofjbo `r^ao^kqb- Di obpql pb i^ `ros^ pb l_qfbkb qbkfbkal bk `rbkq^i^ pfjbqoŒ^ ab i^ b`r^`fŽk `^oqbpf^k^+ l i^ pfjbqoŒ^v i^ mboflaf`fa^a ab Zpbk-.+ RboŒ^rk qo^_^gl jŠp afcŒ`fi af_rg^o bpq^ `ros^ ^ m^oqfoab pr b`r^`fŽk `^oqbpf^k^ &Rli^,jbkqb-

*&)( ?N V[aRT_NY]N_NRYm_RNR[ P\\_QR[NQN]\YN_R`

Rb^ o rk^ crk`fŽk kl kbd^qfs^ abcfkfa^ bk rk fkqbos^il W[) \Y) pfbkalN z ] + [ w /.P+ Di `lkgrkql ab qlalp ilp mrkqlp ab `lloabk^a^p mli^obp- %l) %d&

nrb p^qfpc^`bk i^p abpfdr^ia^abp

Kwlwd%_&) [w_w\)

pb abkljfk^ ^jiepioj m\_d\g ab n pl_ob X\* ]Z, K^ obdfŽk plj_ob^a^ ab i^ cfdr,o^ 1-02 bp rk bgbjmil- Rf ` bp `lkpq^kqb bk W[) \Y) pr `lkgrkql o^af^i bp rk pb`qlo

Page 155: Calculus

I\ dio`bm\g k\m\ `g ƒm`\ `i ^jjm_`i\_\n kjg\m`n

EHFTQ@ 1-02 Bg ^jiepioj m\_d\g _` `^jmm`nkji_d`io` \ pi dio`mq\gj X\*]Z,

024

EHFTQ@ 1-03 Bg ^jiepioj m\_d\g _` pi\api^d‡i `n^\gji\_\ R `n pi\ m`pid‡i _`n`^ojm`n ^dm^pg\m`n,Pp ƒm`\ `n FyP0&.' _L,

`fo`ri^o nrb pr_qfbkab rk Škdril ab ] + [ o^af^kqbp- K^ cfdro^ 1-03- jrbpqo^ bi`lkgrkql o^af^i R ab rk^ crk`fŽk bp`^ilk^a^ p- Dk `^a^ rkl ab ilp i pr_fkqbos^,ilp ^_fboqlp &Lf+g* .f' ab W[) \Y bk bi nrb p bp `lkpq^kqb+ ii^jbjlp mlo bgbjmil7'7( < nn* i^ doŠcf`^ ab p bk `lloabk^a^p mli^obp bp rk ^o`l ab `fo`rkcbobk`f^ abo^afl nn*v pr `lkgrkql o^af^i bp rk pb`qlo `fo`ri^o nrb pr_qfbkab rk Škdril abLf + .f+g o^af^kbp-- Cb_fal ^ i^ cloj^ `ljl ebjlp abcfkfal i^ jbafa^ ^kdri^o+bi Šob^ ab bpqb pb`qlo bp n%Ke* -eZE&O8+ Orbpql nrb ] + [ 9#: /4P) `ljl bplp pb`,qlobp kl qfbkbk m^oqb`lj•k rklp `lk lqolp+ mlo i^ ^afqfsfa^a+ bi Šob^ abi `lkgrk,ql o^af^i ab p `loobpmlkafbkqb ^i fkqbos^il `ljmibql W[)\Y sfbkb a^al mlo

alkab R1'/( obmobpbkq^bi `r^ao^al ab m%K&+@pŒmrbp+ m^o^ i^p crk`flkbp bp`^ilk^,a^p+ bi Šob^ abi `lkgrkql o^af^i e^ pfal bumobp^a^ `ljl rk^ fkqbdo^i- U^jlp^elo^ ^ abjlpqo^o nrb bpq^ cŽojri^ fkqbdo^i ^ajfqb j^vlo dbkbo^ifa^a-

RCMPCK? 1-5- A`ndbi`hjn kjm O `g ^jiepioj m\_d\g _` pi\ api^d‡i ij i`+b\odq\ a `i pi dio`mq\gj X\* ]Z* nd`i_j M 9#: ] + \ x 166!+ W npkjib\hjn lp` O `nh`_d]g` b + Pdn`n dio`bm\]g` `i X\* ]Z `g ƒm`\ _` O qd`i` _\_\ kjm g\ dio`bm\g

[%N& < en/Y.' ^K +

A`hjnom\^d‡i, Difg^jlp alp crk`flkbp bp`^ilk^a^p p v o nrb p^qfpc^d^k

N 9#: 5%-&9#:x%K&9#: f%K&

Page 156: Calculus

025 =faoh[m [jfc][]cih_m ^_ f[ chn_al[]cƒh

m^o^qlal K bk W[) \Y) v abpfdkbjlp mlo R v P prp `lkgrkqlp o^af^ibp+obpmb`qfs^,jbkqb- X^ nrb p 999::. 77788o bk W[) \Y) ilp `lkgrkqlp o^af^ibp p^qfpc^`bki^p obi^`fl,kbp ab fk`irpfŽk R ;9: O ;9: Q, Krbdl+ mlo i^ molmfba^a ab jlklqlkŒ^ abi Šob^+pbqfbkb [%O& 77788[%N& 77788[?P&i Obol R v P plk `lkgrkqlp o^af^ibp ab crk`flkbp bp`^il,k^a^p+ mlo il nrb [%O&< F9O/%-&^K v [%P& < FyW/%-& K+ Olo `lkpfdrfbkqb pbqfbkbk i^p abpfdr^ia^abp

m^o^ qla^p i^p crk`flkbp p v o nrb p^qfpc^d^kp 999::a88899& bk W[) \Y+ Obol Q1 v o0

plk crk`flkbp bp`^ilk^a^p ^o_fqo^of^pnrb p^qfpc^`bk R1 999::.088899'1 bk W[)\Y) irbdl+

v^ nrb mbp fkqbdo^_ib+ab_b pbo /[%N& < P8Mz&'^K+ Dpql abjrbpqo^ bi qblobj^-

Kjo\8 Orbab abjlpqo^opb nrb i^ jbkpro^_fifa^a ab O bp rk^ `lkpb`rbk`f^ ab i^efm5qbpfp ab nrb c1 pb^ fkqbdo^_ib+ mbol kl abp^oolii^objlp i^ abjlpqo^`fŽk-

DIDLOKN- O^o^ `^i`ri^o bi Šob^ abi `lkgrkql o^af^i N fkqboflo ^ i^ `ros^bk cloj^ ab l`el af_rg^a^ bk i^ cfdro^ 1-01+ `^i`ri^jlp bi Šob^ ab i^ mlo`fŽkpfqr^a^ bk bi mofjbo `r^ao^kqb v jriqfmif`^jlp irbdl mlo `r^qol- O^o^bpq^`ros^+pb qfbkb l_K& < Zpbk/0 v+ v^ nrb pbk K w N m^o^N 999::K 77788oo.1+bk`lkqo^jlp

06Z.1 06Z.1 '- oo(

[%N& < 3 l cd/%-&^K < 1 l pbk'( ^K < 1 `lpN , `lpf < 1-

*&)) :WR_PVPV\`

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 0 ^i 3+ abjlpqo^o nrb bi `lkgrkql ab mrkqlp `rv^p`lloabk^a^p ob`q^kdri^obp %r)u( p^qfpc^`bk i^ b`r^`f5k `^oqbpf^k^ a^a^+ bp fdr^i ^i ab ilpmrkqlp `rv^p `lloabk^a^p mli^obp %l)5& p^qfpc^`bk i^ `loobpmlkafbkqb b`r^`f5k mli^o-

0- %r * 0(1 * t0 < 0: l < 1 blp K) blp K = N-

/+ s0 * t0 + U < r&s0 * t09 m< 0 * blp K+

1, %r0 * t0'0 < r0 + t0*a 8899r09 l < U blp 0.* blp 0. ƒ N-

1+ %r0 * t0'0 :!r0 + u10: l < Uiblp 1/0-

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 4 ^i 04+ qo^w^oi^ doŠcf`^ ab ` bk `lloabk^a^p mli^obpv `^i`ri^o bi Šob^ abi `lkgrkql o^af^i ab ` bk bi fkqbos^il nrb pb `fq^- Rb prmlkaoŠ nrb `^a^`lkgrkql bp jbaf_ib-

3, Bnkdm\g _` >mlp…h`_`n8a&L' < L* N99::L 88991oo8

4, @dm^pia`m`i^d\ o\ib`io` \g `e` v9 a`L' < 1 blp L* + i .199:: L R:‘‘./+5, Ajn ^dm^pia`m`i^d\n o\ib`io`n \g `d` v9 a`L' < 1 iblp /0+ M Q:L Q:1om*

6, @dm^pia`m`i^d\ o\ib`io` \g `d` s8 a`L' < 3 pbk L* N Q: L Q: i*7, Ajn ^dm^pia`m`i^d\n o\ib`io`n \g `e` s8 a`L' < 3 Zpbk/0+ N99::L R:1oo*

Page 157: Calculus

>kgd^\^d‡i _` g\ dio`bm\^d‡i \g ^ƒg^pgj _` qjgˆh`i`n 026

/., M„o\gj _` mjn\8 {BN( < pbk 0.* T pK w 06.1-//, Ojn\ _` ^p\omj cje\n8 {BN( < Hpbk1/0+ N z L x 106-

/0, L^cj \kg\no\_j8 {BN( < Ui`lp /0+ N z K w 106-

/1, Qm„]jg _` ^p\omj cje\n8 {BN( < Ui`lp 1/0+ N z K w 106-

/2, @\m_djd_`8 {BN( < 0 * `lp K) N z K w 106-/3, @\m\^jg8 y&L' < 1 * `lp L* N z L x 106-

1-01 @mif`^`fŽkab i^ fkqbdo^`fŽk i `Ši`ril ab sli•jbkbp

Dk i^ Rb``fŽk 0-5 pb fkqolargl bi `lk`bmql ab Šob^ `ljl crk`fŽk ab `lkgrkqlnrb p^qfpc^`b fboq^pmolmfba^abpnrb qlj^jlp `ljl ^uflj^p m^o^bi Šob^-Krbdl+bk i^p Rb``flkbp 0-07 v 1-1+ abjlpqo^jlp nrb i^p Šob^pab jr`e^p obdflkbp ml,aŒ^k`^i`ri^opb mlo fkqbdo^`fŽk-Di jfpjl `^jfkl mrbab rqfifw^opb^i qo^q^oabi`lk`bmql ab slirjbk-

Rrmlkd^jlp nrb bufpqbk`fboqlp `lkgrkqlp R ab mrkqlp bk bi bpm^`fl ab qobpafjbkpflkbp+ nrb ii^j^jlp ^jiepiojn h`_d]g`n* v rk^ crk`fŽk ab `lkgrkql H$)ii^,j^a^ api^d‡i qjgph`i* nrb ^pfdk^ ^ `^a^ `lkgrkql jbaf_ib R rk k•jbol p_O&)ii^j^al slirjbk ab R- Tqfifw^jlp bi pŒj_lil a m^o^abpfdk^o i^ `i^pb ab qlalpilp `lkgrkqlp jbaf_ibp bk bi bpm^`fl ab qobpafjbkpflkbp+ v ^ `^a^ `lkgrkql R aba il ii^j^jlp n‡gd_j,

Bljl bk bi `^pl abi Šob^+bkrk`f^jlp rk^p molmfba^abp nrb abpb^oŒ^jlpnrb qrsfbo^ bi slirjbk v i^p qlj^jlp `ljl ^uflj^p m^o^bi jfpjl- K^ bib``fŽkab ilp ^uflj^p klp mbojfqb abjlpqo^o nrb ilp sli•jbkbp ab jr`elp pŽifalpmrbabk `^i`ri^opb mlo fkqbdo^`fŽk- Klp qobpmofjbolp ^uflj^p+ m^ob`falp ^ ilp`loobpmlkafbkqbpm^o^bi Šob^+pb obcfbobk i^p molmfba^abpab kl kbd^qfsfa^a+ ^af,qfsfa^a+ v ab i^ afcbobk`f^- Dk ird^o ab rk ^uflj^ ab fks^of^k`f^ cobkqb ^ i^`lkdorbk`f^+ rqfifw^jlp lqol ab qfml afpqfkql+ii^j^al kmdi^dkdj _` @\q\gd`md,Dpqb^pfdk^ sli•jbkbp fdr^ibp ^ pŽifalp `lkdorbkqbp v q^j_f‹k ^ `fboqlp pŽifalp nrb+kl pfbkal `lkdorbkqbp+ qfbkbk pb``flkbp ab Šob^pfdr^ibp ^i pbo`loq^alp mlo mi^klpmbombkaf`ri^obp^ rk^ ob`q^ a^a^- Blk j^vlo mob`fpfŽk+prmlkd^jlp nrb R v Hpb^k rk pŽifal v rk^ ob`q^ a^alp- Rf E bp rk^ mi^kl mbombkaf`ri^o^ H) i^ fkqbo,pb``fŽk E j R pb ii^j^ pb``fŽk mbombkaf`ri^o^ H+Rf qla^ pb``fŽk mbombkaf`ri^o^ I bp rk `lkgrkql jbaf_ib bk pr molmfl mi^kl+ R pb ii^j^ rk n‡gd_j _` @\q\gd`md,Di mofk`fmflab B^s^ifbof ^pfdk^ sli•jbkbp fdr^ibp ^ alp pŽifalp ab B^s^ifbof+R vP) pf [%Ok E( << [%P k E(m^o^qlal mi^kl E mbombkaf`ri^o ^ rk^ ob`q^ a^a^ H+

Di mofk`fmfl ab B^s^ifbof mrbab fkqbomobq^opbfkqrfqfs^jbkqb `ljl pfdrb-Hj^dfk‹jlklp rk pŽifal ab B^s^ifbof `ljl rk^ mfi^ l jlkqŽk ab iŠjfk^p j^qb,of^ibp abid^a^p+ mlo bgbjmil ab k^fmbp+pfbkal `^a^ iŠjfk^ mbombkaf`ri^o ^ rk^ob`q^ a^a^ H+Rf abpifw^jlp `^a^ iŠjfk^ bk pr molmfl mi^kl mlabjlp `^j_f^o i^cloj^ abi pŽifal mbol kl pr slirjbk-

Page 158: Calculus

027 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

Di ^uflj^ pfdrfbkqb bpq^_ib`b nrb bi slirjbk ab rk m^o^ibibmŒmbalob`q^k,dri^o bp bi molar`ql ab i^p ilkdfqrabp ab prp ^ofpq^p-Tk m^o^ibibmŒmbalob`q^kdri^obp `r^inrfbo `lkgrkql `lkdorbkqb ^ rk `lkgrkql ab i^ cloj^-

'1-05(

Tqfifw^objlp i^ m^i^_o^jŠp `loq^ ~`^g^‚ bk ird^o ab ~m^o^ibibmŒmbalob`q^kdri^o‚-Klp k•jbolp kl kbd^qfslp \* \) ` ab '1-05( plk i^p ilkdfqrabp ab i^p ^ofpq^pab i^`^g^-

Hk`irfjlp+ mlo •iqfjl+ rk ^uflj^ nrb bpq^_ib`b nrb qlal `lkgrkql `lksbulbp jbaf_ib- Tk `lkgrkql pb ii^j^ ^jiq`sj pf+m^o^qlal m^oab mrkqlp M u P abi`lkgrkql+ bi pbdjbkql ab ob`q^ nrb ilp rkb mboqbkb`bq^j_f‹k ^i `lkgrkql- Dpqb^uflj^+ grkql `lk i^p molmfba^abpab ^afqfsfa^a u ab i^ afcbobk`f^+^pbdro^k nrbqlalp ilp pŽifalp bibjbkq^ibp nrb pb mobpbkq^kbk i^p ^mif`^`flkbp abi BŠi`ril plkjbaf_ibp-

Klp ^uflj^p m^o^bi slirjbk mrbabk ^elo^ bpq^_ib`bopbabi pfdrfbkqb jlal-

CDEHMHBHˆM @WHNL„SHB@ CD UNKTLDM- Ppkjib\hjn lp` `sdno` pi\ ^g\n` ^_` n‡gd_jn t pi\ api^d‡i _` ^jiepioj q* ^ptj _jhdidj `n ^) ^ji g\n kmjkd`_\_`nndbpd`io`n8

/, Mmjkd`_\_ _` ij i`b\odqd_\_, M\m\ ^\_\ ^jiepioj R _` ,7a n` od`i`p%O&w N-

0, >_dodqd_\_,Pd R V Q k`mo`i`^`i \,7a * R S Q V R k Q o\h]d„i k`mo`i`+

^`i \ ,7a*t n` od`i` q&PS Q' < q&P'* q&Q' + q&Pk Q',1, Mmjkd`_\_ _` g\ _da`m`i^d\, Pd R V Q k`mo`i`^`i \,7a nd`i_j R p:9 Q*

Q + R k`mo`i`^` \ ,7a t n` od`i` q&Q+ R( < q&Q' + q&P',2, Mmdi^dkdj _` @\q\gd`md, Pd R V Q nji _jn n‡gd_jn _` @\q\gd`md k`mo`i`+

^d`io`n \ ,7a o\g`n lp` \&P k D( y \&Q k D( k\m\ oj_j kg\ij D k`mk`i+_d^pg\m \ pi\ m`^o\ _\_\* `ioji^`n q&P' x q&Q',

3, Bg`^^d‡i _` `n^\g\, Qj_\ ^\e\ ? k`mo`i`^` \ _, Pd gjn g\_jn l \mdno\n _`? od`i`i gjibdop_`n \* ] t `* n` od`i` lp` q&?' < \]`,

4, Qj_j ^jiepioj ^jiq`sj k`mo`i`^` \ m,)

Di ^uflj^ 2 ^pbdro^ nrb bi `lkgrkql s^`Œl / mboqbkb`b ,7a v qfbkbslirjbk`bol- Orbpql nrb p%P* R( z /+ bi ^uflj^ 2 q^j_f‹k fjmif`^ i^ pfdrfbkqbmolmfb,a^a ab jlklqlkŒ^9

q&P' x q&Q'* m^o^`lkgrkqlp P v Q ab-8c q^ibpnrb R p:9 Q,

K^ molmfba^a ab jlklqlkŒ^+ ^ pr sbw+klp jrbpqo^ nrb qlal `lkgrkql mi^kl ^`l,q^al P ab ,7a qfbkb slirjbk `bol- Tk `lkgrkql mi^kl pb ii^j^ \^jo\_j pf bp rkpr_`lkgrkql ab rk `fboql `r^ao^al bk bi mi^kl- Rf `lkpfabo^jlp rk^ `^g^ > ab

Page 159: Calculus

>kgd^\^d‡i _` g\ dio`bm\^d‡i \g ^ƒg^pgj _` qjgˆh`i`n 028

^iqro^ ` nrb qbkd^ bpb `r^ao^al `ljl _^pb+ bkqlk`bp R o9> ab jlal nrb qbkbjlp‚'R( z p%>&< [$]) pfbkal [ i^ ilkdfqra ab `^a^ i^al abi `r^ao^al ab i^ _^pb-Rf crbpb q`P' = N+ mlaoŒ^ qlj^opb ` ab jlal nrb ` ; q&P'-\0

* bk `lkqo^af``fŽk`lk i^ abpfdr^ia^a q`P' x \%^*Dpql abjrbpqo^ nrb ‚'R( kl mrbab pbo mlpfqfs^+`lk il nrb ‚'R( < N+ `ljl pb ^cfojŽ-

N_p‹osbpb nrb bi mofk`fmfl ab B^s^ifbof e^ pfal bpq^_ib`fal bk cloj^ ababpfdr^ia^abp- Rf \&P j C'; \ &Qo< C' m^o^ qlal mi^kl C mbombkaf`ri^o ^ rk^ob`q^ a^a^+ mlabjlp ^mif`^o bi ^uflj^ 3 alp sb`bp m^o^ abar`fo ‚'R( z p%P& uq&Q' x q`P'* u pb qfbkb mlo q^kql q&Q' < q`P',

@ `lkqfkr^`fŽk abjlpqo^jlp nrb bi slirjbk ab rk pŽifal `fiŒkaof`l bpfdr^i ^i Šob^ ab pr _^pb jriqfmif`^a^ mlo pr ^iqro^- Olo pŽifal `fiŒkaof`l bkqbk,abjlp rk `lkgrkql `lkdorbkqb ^ rk `lkgrkql R ab i^ cloj^

p< u%r)s) t&, %r)s& A ?* [ w w z ]w*

pfbkal > rk `lkgrkql jbaf_ib mi^kl v ^`lq^al K^p Šob^p ab i^p pb``flkbp ab Rmbombkaf`ri^obp ^i bgb w abqbojfk^k rk^ crk`fŽk \n* nrb bp bi Šob^ ab i^ pb``fŽk+v nrb qlj^ bi s^ilo `lkpq^kqb \&?' bk bi fkqbos^il \ x w z ]* X bi s^ilo M crbo^ab ‹i- Rb abkljfk^oŠ crk`fŽk Šob^ pb``flk^\ ^ i^ crk`fŽk \n,

Rb^ ^elo^ Q rk^ `^g^ `rv^ crk`fŽk Šob^ pb``flk^i ‰mpb^ fdr^i ^ \n, Di^uflj^ 4 klp af`b nrb q&Q' < \&?'&] + \'* pfbkal \&?' bi Šob^ ab i^ _^pb ab Q*v \ * [ bp pr ^iqro^- Di mofk`fmfl ab B^s^ifbof bpq^_ib`b nrb p_O& < p%P&) abjlal nrb bi slirjbk ab R bp fdr^i ^i Šob^ ab pr _^pb- [%>&) jriqfmif`^a^ mlo pr^iqro^+ ] + \, N_p‹osbpb nrb \&?'&] + \' bp i^ fkqbdo^i ab i^ crk`fŽk \n bk bifkqbos^il W[) \Y+ Cf`el ab lqol jlal+ bi slirjbk ab rk pŽifal `fiŒkaof`l ob`qlbp fdr^i ^ i^ fkqbdo^i ab pr crk`fŽk Šob^ ab i^ pb``fŽk-

q`P' < b \xu' _u ,$[

Olabjlp buqbkabo bpq^ cŽojri^ ^ pŽifalp ab B^s^ifbof jŠp dbkbo^ibp- Rb^ Ork pŽifal ab B^s^ifbof `lk pb``flkbp jbaf_ibp mbombkaf`ri^obp ^ rk^ ob`q^ a^a^I, Blkpfabobjlp rk bgb ab `lloabk^a^p `lfk`fabkqb `lk I 'ii^j^al bgb o&) vpb^ [N%o& bi Šob^ ab i^ pb``fŽk molar`fa^ mlo rk mi^kl mbombkaf`ri^o ^ H bk bimrkql p, Di slirjbk ab O mrbab `^i`ri^opb `lk bi qblobj^ pfdrfbkqb-

SDNQDL@ 1-6- P`\ O pi n‡gd_j _` @\q\gd`md _` ^ ^pt\ api^d‡i ƒm`\ n`^+^dji\g ‰i*n`\ dio`bm\]g` `i pi dio`mq\gj X\* ]Z X ipg\ ap`m\ _`g hdnhj, Bi o\g`n^ji_d^dji`n `g qjgph`i _` O `n dbp\g \ g\ dio`bm\g _`g ƒm`\ n`^^dji\g8

p_N& < a \g &p' _p ,$[

Page 160: Calculus

03/ =faoh[m [jfc][]cih_m ^_ f[ chn_al[]cƒh

@_gimnl[]cƒh+ Difg^jlp crk`flkbp bp`^ilk^a^p p v n q^ibp nrb m8728[O 9#: n

bk W[) \Y X abcfk^jlp p v o `ljl kri^p crbo^ ab W[) \Y+ O^o^ `^a^ pr_fkqbos^ilab W[) \Y bk bi nrb p pb^ `lkpq^kqb+mlabjlp fj^dfk^o rk pŽifal `fiŒkaof`l 'mlobgbjmil+ rk `fifkaol `fo`ri^o ob`ql( `lkpqorfal ab jlal nrb pr Šob^ pb``flk^ibk bpqb pr_ fkqbos^il qbkd^ bi jfpjl s^ilo `lkpq^kqb nrb p- K^ obrkfŽk ab bplp`fifkaolp pl_ob ilp fkqbos^ilp bk ilp nrb p bp `lkpq^kqb bp rk pŽifal R `rvl sl,irjbk p%O&bp+mlo i^ ^afqfsfa^a+ fdr^i ^ i^ fkqbdo^iPxm%o&oi Cbi jfpjl jlal+bufpqb rk pŽifal P) rk^ obrkfŽk ab `fifkaolp+ `rvl slirjbk p%P&< `wn%o&oi

Obol [m%o&< m%o&w [N%o&w n%o&< [P%o& m^o^ qlal o ab W[) \Y) ab jlal nrbbi mofk`fmfl ab B^s^ifbof fjmif`^ nrb p%O&w p%N& :94:p%P&+Dk lqo^p m^i^_o^p+p%N& p^qfpc^`bi^p abpfdr^ia^abp

nn&p'_p x q&O' x no&p'_p

m^o^ qla^p i^p crk`flkbp bp`^ilk^a^p p v o nrb p^qfpc^`bkp :94:[m :94:n bk W[) \Y+

Orbpql nrb [m bp fkqbdo^_ibbk W[) \Y) obpriq^ nrb p%N&< `7[m%o& oiDIDLOKN- Rifog_h ^_ oh mƒfc^i ^_ l_pifo]cƒh+ Rb^ ` rk^ crk`fŽk kl

kbd^qfs^ b fkqbdo^_ibbk rk fkqbos^il W[) \Y+ Rf bi `lkgrkql ab loabk^a^p ab bp^crk`fŽk dfo^ ^iobabalo abi bgbs* bkdbkao^ rk pŽifal ab obslir`fŽk- B^a^ pb``fŽkabqbojfk^a^ mlo rk mi^kl mbombkaf`ri^o ^i bgb s bp rk afp`l `fo`ri^o- Di Šob^abi afp`l `fo`ri^o `loobpmlkafbkqb ^i mrkql s bp QmaYs' * pfbkal m&s' bi `r^ao^alab `%r&+Olo `lkpfdrfbkqb+ pbd•k bi qblobj^ 1-6+ bi slirjbk abi pŽifal 'pf bi pŽ,ifal mboqbkb`b `( bp fdr^i ^ i^ fkqbdo^i`7Qmm&s'_s* pf i^ fkqbdo^i bufpqb-Dkm^oqf`ri^o+pf a&s' < rn1 , s0 m^o^ +m 9#: s 9#: m* bi `lkgrkql ab loabk^a^p ab abp rk afp`l pbjf`fo`ri^o ab o^afl m v bi pŽifal bkdbkao^al bp rk^ bpcbo^ab o^,afl o- K^ bpcbo^bp `lksbu^- Rr slirjbk bp fdr^i ^

coQma0&U'_s < ooco&m0+ s0' _s < 0Qmo&m0+ s0' _s < Qmm1

Blk j^vlo dbkbo^ifa^a+ prmlkd^jlp nrb afpmlkbjlp ab alp crk`flkbp klkbd^qfs^p ` v d nrb plk fkqbdo^_ibpbk rk fkqbos^il W[) \Y X nrb p^qfpc^`bk` w dbk W[) \Y+ Br^kal i^ obdfŽkbkqobprp doŠcf`^pdfo^ ^iobabalo abi bgb r) bkdbkao^rk pŽifal ab obslir`fŽk q^i nrb `^a^ pb``fŽk molar`fa^ mlo rk mi^kl mbombkaf`r,i^o ^i bgb s bk bi mrkql s bp rk^ `lolk^ `fo`ri^o 'rk^ obdfŽk ifjfq^a^ mlo alp`fo`rkcbobk`f^p `lk`‹kqof`^p( `lk Šob^ Qmb0&s'+5ma&s', Olo `lkpfdrfbkqb+ nd d1 , mbp fkqbdo^_ib+bi slirjbk ab af`el pŽifal 'pf q^i pŽifal mboqbkb`b `( sfbkb a^almlo i^ fkqbdo^i I] 1 1

QmXb %r&* ` %r&Yr +\

1-02 Dgbo`f`flp

0- @mif`^o i^ fkqbdo^`fŽk m^o^ `^i`ri^o bi slirjbk ab rk `lkl `fo`ri^o ob`ql bkdbkao^ale^`fbkal dfo^o ^iobabalo abi bgb r i^ doŠcf`^ ab i^ crk`fŽk ` a^a^ mlo `%r&< r] bk bi

Page 161: Calculus

>kgd^\^d‡i _` g\ dio`bm\^d‡i \g ^ji^`koj _` om\]\dj ),)

fkqbos^il N z s x ], Cbjlpqo^o nrb bi obpriq^al bp bi molar`ql ab rk qbo`fl abi Šob^ab i^ _^pb mlo i^ ^iqro^ abi `lkl-Dk `^a^ rkl ab ilp Dgbo`f`flp abi 1 ^i 6+ `^i`ri^o bi slirjbk abi pŽifal bkdbkao^al ^i

dfo^o bi `lkgrkql ab loabk^a^p ab i^ crk`fŽk ` pl_ob bi fkqbos^il fkaf`^al- Cf_rg^o `^a^ rklab ilp `lkgrkqlp ab loabk^a^p-0, y&s' < Ty+ N 9,99:s 8+8890- 4- y&s' < pbk s* /9,99: s 8+889/Q,1, y&s' < sg-2* N 9,99:s 8+8890- 5- y&s' < `lp s* /9,99:s 8+889/Q/0,

1+y&s' < s/) *v7*778 U 8+8891- 6- y&s' < pbk s * `lp s* /9,99: s 8+889/Q,

Dk `^a^ rkl ab ilp Dgbo`f`flp 7 ^i 00+ af_rg^o i^ obdfŽk bkqob i^p doŠcf`^p ab ` v d v`^i`ri^o bi slirjbk abi pŽifal l_qbkfal ^i dfo^o af`e^ obdfŽk ^iobabalo abi bgb s,

5+ y&s' < Ty+ b&s' < 0+ N 9,99:s 8+8890-

6+y&s' < Ty+ b&s' < s0* N 9,99:U p0-.-+ y&s' < pbku+ b&s' < `lp s* N 9,99:s 8+889/Q-2,

..+ y&s' < U3 , s/) b&s' < 0+ N 9,99:s 8+889Uf01- Cf_rg^o i^p doŠcf`^p ab y&s' < +.y v b&s' < sg0 bk bi fkqbos^il Z/+1\- G^ii^o rk k•,

jbol o* 0 ; o ; 1+ ab jlal nrb `r^kal i^ obdfŽk bkqob i^p doŠcf`^p ab ` v d pl_ob bifkqbos^il ZN+nY dfo^ ^iobabalo abi bgb s* bkdbkao^ rk pŽifal ab obslir`fŽk `rvl slirjbkbp fdr^i ^ 4Pn! -1,

02- ƒPr‹ slirjbk ab j^qbof^i pb nrfq^ ab rk^ bpcbo^ ab o^afl 1o `r^kal pb ^qo^sfbp^ `lkrk q^i^aol+ cloj^kal rk ^drgbol `bkqo^al ab o^afl o>

03- Tk pbosfiibqbol pb l_qfbkb mo^`qf`^kal rk ^drgbol `fiŒkaof`l bk rk^ bpcbo^ ab jlal nrbbi bgb ab ^nr‹i m^pb mlo bi `bkqol ab ‹pq^- Rf i^ ilkdfqra abi ^drgbol bp 0c* abjlpqo^onrb bi slirjbk abi pbosfiibqbol bp /Q\c1* pfbkal \ rk k•jbol o^`flk^i-

04- Tk pŽifal qfbkb rk^ _^pb `fo`ri^o ab o^afl 1- B^a^ pb``fŽk molar`fa^ mlo rk mi^klmbombkaf`ri^o ^ rk afŠjbqol cfgl bp rk qofŠkdril bnrfiŠqbol- B^i`ri^o bi slirjbk abipŽifal- ‘

05- K^p pb``flkbp qo^kpsbop^ibp ab rk pŽifal mlo mi^klp mbombkaf`ri^obp ^i bgb s plk `r^,ao^alp `lk `bkqolp bk af`el bgb- Rf ^i `loq^o mlo bi mi^kl mbombkaf`ri^o bk bi mrkql ab^_p`fp^ s* pb l_qfbkb rk `r^ao^al `rvl i^al bp 0s0* pb qo^q^ ab e^ii^o bi slirjbk abipŽifal bkqob s < N X s < \, Cf_rg^o rk bpnrbj^-

06- G^ii^o bi slirjbk ab rk pŽifal `rv^ pb``fŽk qo^kpsbop^i mlo rk mi^kl mbombkaf`ri^o^i bgb s qfbkb ab Šob^ \s~ * ]s * b m^o^ `^a^ s abi fkqbos^il N z s x c, Dumobp^o bislirjbk bk crk`fŽk ab i^p Šob^p ?

/* L X ?0 ab i^p pb``flkbp qo^kpsbop^ibp `loobpmlk,

afbkqbp ^ s < N+s < c-0 v s < c* obpmb`qfs^jbkqb- K^ cŽojri^ nrb obpriq^ pb `lkl`bmlo a‡mhpg\ _`g kmdnh\ojd_`,

07- Cf_rg^o rk bpnrbj^ ab i^ obdfŽk abi mi^kl rs cloj^a^ mlo qlalp ilp mrkqlp %r)u( nrbp^qfpc^`bk i^p abpfdr^ia^abp pfjriqŠkb^p N 9,99:s 8+8891+ s/ 7*778t 8+889i-B^i`ri^o bi slirjbk abi pŽifal l_qbkfal e^`fbkal dfo^o bpq^ obdfŽk9 ^( ^iobabalo abibgb r8 _( ^iobabalo abi bgb v: `( ^iobabalo ab i^ sboqf`^i nrb m^p^ mlo '1+ N(: a( ab i^ el,ofwlkq^i nrb m^p^ mlo 'N+0(-

*&), 6]YVPNPVp[QRYNV[aRT_NPVp[NYP\[PR]a\ QRa_NONW\

G^pq^ ^nrŒkrbpqo^p ^mif`^`flkbp ab i^ fkqbdo^`fŽke^k pfal ^ ilp `lk`bmqlpdblj‹qof`lp ab Šob^ u slirjbk- U^jlp ^elo^ ^ `ljbkq^o rk^ ^mif`^`fŽk ^i `lk,`bmql cŒpf`lab nl[\[di+

Page 162: Calculus

031 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

So^_^gl bp rk^ jbafa^ ab i^ bkbodŒ^lkprjfa^ mlo rk^ crbow^ ^i jlsbork^ m^oqŒ`riab rk mrkql ^ lqol- Dk bpq^pb``fŽk `lkpfabo^jlp bi `^pl jŠp pbk,`fiil+ bi jlsfjfbkql ob`qfiŒkbl-Dpql bp+prmlkbjlp nrb bi jlsfjfbkql pb bcb`q•^^ il i^odl ab rk^ ob`q^ 'nrb pb qlj^ `ljl bgbr& abpab rk mrkql r < [) e^pq^ lqolr < ]* X q^j_f‹k nrb i^ crbow^ ^`q•^ ^ il i^odl ab bpq^ ob`q^- @ajfqfjlp nrb\ ; ] l ] ; \, Rrmlkbjlp ^abjŠp nrb i^ crbow^ nrb ^`q•^ pl_ob i^ m^oqŒ`ri^bp rk^ crk`fŽk ab i^ mlpf`fŽk- Rf i^ m^oqŒ`ri bpqŠbk r) abpfdk^jlp mlo `%r& i^crbow^ nrb ^`q•^ bk bii^+ pfbkal `%r& = N pf ^`q•^ bk i^ afob``fŽk mlpfqfs^ abibgb r) v `%r& ; N pf 0/ e^`b bk pbkqfal `lkqo^ofl- Br^kal i^ crbow^ bp `lkpq^kqb+mlo bgbjmil `%r& < _ m^o^qlal r bkqob[ v \) abcfkfjlp bi qo^_^gl bcb`qr^al mlo` `ljl bi k•jbol ?$ %\ * [&7 i^ crbow^ jriqfmif`^a^ mlo bi abpmi^w^jfbkql-Di qo^_^gl mrbab pbo mlpfqfsl l kbd^qfsl-

Rf i^ crbow^bpqŠjbafa^ bk _di\n v i^ afpq^k`f^ bk ^`io…h`omjn 'pfpqbj^ ^bn'*bi qo^_^gl pb jfab bk _di\n mlo ^`io…h`omj, Tk^ _di\+^`io…h`omj ab qo^_^gl pbii^j^ `mbj Rf i^ crbow^pb jfab bk i`rojin v i^ afpq^k`f^ bk h`omjn 'pfpqbj^ hfn'*bi qo^_^gl pb bumobp bk i`roji kjm h`omj, Tk i`roji+h`omj ab qo^_^gl pb ii^j^djpg`, Tk kbtqlk bnrfs^ib ^ 0/! afk^p+v rk glrib ^ 0/6 bodl Rf i^ crbow^pb jfabbk if_o^p v i^ afpq^k`f^ bk mfbp+jbafjlp bi qo^_^gl bk gd]m\n+kd`,

DIDLOKN- Tk^ mfbao^ab 2 if_o^p ab mbpl pb i^kw^ e^`f^ ^oof_^ ^ il i^odlab rk^ ob`q^+e^pq^ rk^ ^iqro^ ab 04 mfbpv srbisb ^i prbil- Slj^jlp bi bgb s^ il i^odl ab i^ qo^vb`qlof^ v lofbkq^al mlpfqfs^jbkqb e^`f^ ^oof_^- K^ crbow^`lkpq^kqb ab i^ do^sba^a ^`q•^ e^`f^ ^_^gl+ ab jlal nrb `%r& < , 2 if_o^p m^o^`^a^ s* Nz s x 04- Di qo^_^gl bcb`qr^al mlo i^ do^sba^a ^i jlsbo i^ mfbao^abpab+ mlo bgbjmil+ s < 5 mfbpe^pq^ s < 04 mfbpbp , 2&'04 , 5( < , 16 if,_o^p,mfb-Br^kal i^ jfpj^ mfbao^`^b abpab s < 04 mfbpe^pq^ s < 5 mfbp+bi qo^,_^gl bcb`qr^al mlo i^ do^sba^a bp , 2'5 , 04( < 16 if_o^p,mfb-

Rrmlkd^jlp ^elo^ nrb i^ crbow^kl pb^ `lkpq^kqb pfkl nrb pb^ rk^ crk`fŽkab i^ mlpf`fŽk abcfkfa^ bk bi fkqbos^il nrb rkb \ v \+ ƒBŽjl abcfkfjlp bi qo^_^glob^ifw^al mlo n i jlsbo rk^ m^oqŒ`ri abpab [ e^pq^ ]= Kl e^objlp `ljl m^o^bi Šob^ v bi slirjbk- Dpq^_ib`bjlp `fboq^p molmfba^abp nrb sfbkbk fjmrbpq^pmlo bufdbk`f^p cŒpf`^p-Rb abjrbpqo^ irbdl nrb m^o^`r^inrfbo abcfkf`fŽk ab qo^_^gl`lk bp^p molmfba^abp+bi qo^_^gl ob^ifw^al mlo rk^ crk`fŽk crbow^ fkqbdo^_ib `bp fdr^i ^ i^ fkqbdo^i`w%r&^r+

OQNOHDC@CDR ETMC@LDMS@KDR CDK SQ@A@IN- A`ndbi`hjn ^ji r\&a'* `gom\]\ej m`\gdu\_j kjm pi\ api^d‡i ap`mu\ a \g hjq`m pi\ k\mo…^pg\_`n_` \ c\no\ ],Q\g om\]\ej od`i` g\n kmjkd`_\_`n ndbpd`io`n8

/, Mmjkd`_\_ \_dodq\, Rf \ ; ` ; ]* Tx&a' < Tx&a' * Tx&a',0, Mmjkd`_\_ hji‡oji\, Pd a x c `i X\* ]Z* Tx&a' x T8&b', Bnoj `n* pi\

ap`mu\ h\tjm m`\gdu\ pi om\]\dj h\tjm,

Page 163: Calculus

>kgd^\^d‡i _` g\ dio`bm\^d‡i \g ^ji^`koj _` om\]\ej 032

1, C‡mhpg\ `g`h`io\g, Rf a `n ^jino\io`* kjm `e`hkgj a&s' < ` k\m\ oj_j s`i `g dio`mq\gj \]d`moj &\* ]'* T8R' < ^ 8&] + \',

K^ molmfba^a ^afqfs^ mrbab buqbkabopbmlo fkar``fŽk ^ `r^inrfbo k•jbolcfkfql ab fkqbos^ilp-

Dpql bp+pf \ < Tj ; s* ; --- ; Ti < ]* pb qfbkb

i

Tx&a' < ƒTf*

fxg

pfbkal Se bi qo^_^gl ob^ifw^al mlo ` abpab Te*f ^ sd, Dk m^oqf`ri^o+pf i^ crbow^bp rk^ crk`fŽk bp`^ilk^a^ p nrb qlj^ rk s^ilo `lkpq^kqb Oe bk bi fkqbos^il ^_fbo,ql %Te*p Te&) i^ molmfba^a 2 bpq^_ib`b nrb Se < Oe$ %Te * Te*f&) `lk il nrb

@pŒmrbp+m^o^ crk`flkbp bp`^ilk^a^p+ bi qo^_^gl pb bumobp `ljl rk^ fkqbdo^i-Dp cŠ`fi abjlpqo^o nrb bpql bp `fboql bk `^plp jŠp dbkbo^ibp-

RCMPCK? 1-7- Ppkjib\hjn lp` `g om\]\ej n` c\ _`adid_j k\m\ pi\ ^g\n`_` api^dji`n ap`mu\ a _` hj_j lp` n\odna\b\ g\n kmjkd`_\_`n 0+ 1+ X 2- Bg om\]\ej`a`^op\_j `ioji^`n kjm ,pi\ api^d‡i ap`mu\ dio`bm\]g` a \g hjq`m pi\ k\mo…^pg\_`n_` \ c\no\ ] `n dbp\g \ g\ dio`bm\g _` o,

FE)

Tx&a' < \ a&s' _s ,

A`hjnom\^d‡i, Rb^k p u o alp crk`flkbp bp`^ilk^a^p nrb p^qfpc^`bkmR:` R: o bk W[) \Y+ K^ molmfba^a jlkŽqlk^ abi qo^_^gl bpq^_ib`b nrbT8&n' x T8R' R: T9&o', Obol T8&n' < Izn&s' v T8&o' < Px o&s'_s* ab jlal

nrb bi k•jbol T8R' p^qfpc^`bi^p abpfdr^ia^abp

G] ], G]\ n&s'_s R: T\&a' x \ o&s'_s

m^o^ qla^p i^p crk`flkbp bp`^ilk^a^p p u o nrb p^qfpc^`bkp R: ` R: o bk W[) \Y+Orbpql nrb ` bp fkqbdo^_ibbk X\* ]Z* obpriq^ nrb T8R' < Gxa&s' _s,

Kjo\8 Lr`elp ^rqlobp abcfkbk pfjmibjbkqb bi qo^_^gl `ljl i^ fkqbdo^i ab i^ crk,`fŽk crbow^- K^ ^kqboflo afp`rpfŽk mrbab `lkpfabo^opb `ljl rk^ grpqfcf`^`fŽk ab q^i ab,cfkf`fŽk-

Page 164: Calculus

),, >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^dƒi

DIDLOKN- Qm\]\dj i`^`n\mdj k\m\ `nodm\mpi hp`gg`, Rrmlkd^jlp nrb i^crbow^ `%r&kb`bp^of^ m^o^bpqfo^ork jrbiib ab ^`bol rk^ ilkdfqra r jŠp ^iiŠ abpr ilkdfqra k^qro^i bp molmlo`flk^i ^ s &I`t _` Ejjf`', Blilnrbjlp bi bgb s^ il i^odl abi bgbabi jrbiib- Rf i^ crbow^ ab qo^``fŽk ^`q•^ bk i^ afob``fŽk mlpf,qfs^ abi bgb+qbkbjlp `%r&< _r) bk alkab i^ `lkpq^kqb ab qo^``fŽk _ bp mlpfqfs^-'Di s^ilo ab _ mrbab abqbojfk^opb pf `lkl`bjlp i^ crbow^ `%r& m^o^ rk- s^ilom^oqf`ri^o ab s :B N-( Di qo^_^gl mob`fpl m^o^ bpqfo^obi jrbiib rk^ ilkdfqra \ bpPx`%r& r < Px _r ^r < ][0,/) nrb bp rk k•jbol molmlo`flk^i ^i `r^ao^al abiabpmi^w^jfbkql-

Dk bi Ulirjbk 00 v jbaf^kqb i^p fkqbdo^ibpab iŒkb pb bpqraf^ bi qo^_^glm^o^jlsfjfbkqlp ^ 0/ i^odl ab `ros^p-

*&)- :WR_PVPV\`

Dk ilp Dgbo`f`flp 0 v 1 pb prmlkb nrb i^ crbow^ nrb ^`q•^ pl_ob bi obploqb l_bab`b i^ibv ab Gllhb-

0- Rf rk^ crbow^ ab 0/ if_o^p ^i^od^ rk jrbiib biŠpqf`l 0 mrid^a^+ ƒnr‹ qo^_^gl pb ob^ifw^ ^i^i^od^o bi jrbiib 0 mfb>

1- Tk jrbiib qfbkb kloj^ijbkqb i^ ilkdfqra ab 0 jbqol- Tk^ crbow^ ab 0// kbtqlkp0/ `ljmofjb e^pq^ /+8 j- ƒBrŠkqlp glribp ab qo^_^gl pb mob`fp^k m^o^ `ljmofjfoil e^pq^i^ jfq^a ab pr ilkdfqra kloj^i> ƒBrŠi bp i^ ilkdfqra abi jrbiib `r^kal v^ pb e^k ob^,ifw^al 1/ glribp ab qo^_^gl>

2- Tk^ m^oqŒ`ri^ pb jrbsb ^ il i^odl abi bgb s jbaf^kqb rk^ crbow^ fjmriplo^d%r&< 0r0 * 1r kbtqlkp- &B^i`ri^o `rŠkqlp glribp ab qo^_^gl pb ob^ifw^k `lk bp^ crbow^ m^o^qo^pi^a^o i^ m^oqŒ`ri^ ^( abpab s < N e^pq^ s < 6 j: _( abpab s < 1 j e^pq^ s < 6 j-

3- Tk^ m^oqŒ`ri^pb jrbsb ^ il i^odl abi bgb s jbaf^kqb rk^ crbow^ fjmriplo^ a^a^ mlod%r&< [r| * \r afk^p- B^i`ri^o [ v \ ab jlal nrb pb mob`fpbk 8// bodp ab qo^_^gl m^o^abpmi^w^o i^ m^oqŒ`ri^ 0/ `j ^ m^oqfo abi lofdbk+ pf i^ crbow^ bp ab 54 afk^p `r^kals < 4 `j-

4- Tk `^_ib ab 4/ mfbp ab ilkdfqra v 3 if_o^p ab mbpl mlo mfb mbkab ab rk qlokl- B^i`ri^obi qo^_^gl ob^ifw^al ^i bkolii^o 14 mfbp ab `^_ib- Ml `lkpfabo^o jŠp crbow^p nrb i^do^sba^a-

5- Qbplisbo bi Dgbo`f`fl 4 pf pb `rbid^ rk mbpl ab 4/ if_o^p bk bi buqobjl abi `^_ib-6- Tk mbpl ab 04/ if_o^p pb cfg^ bk rk buqobjl ab rk^ `^abk^ `rvl mbpl bp ab 1 if_o^p

mlo mfb- Hkf`f^ijbkqb bi mbpl pb prpmbkab `lk 0/ mfbp ab `^abk^ pl_ob bi _loab ab rkbafcf`fl ab 0// mfbp ab ^iqro^- Blkpfabo^kal pŽil i^ crbow^ ab i^ do^sba^a+ `^i`ri^o biqo^_^gl ob^ifw^al `r^kal bi mbpl pb _^g^ e^pq^ rk^ mlpf`fŽk ab 0/ mfbp pl_ob bi prbil-

7- Dk bi bgbo`f`fl 6+ prmlkbo nrb i^ `^abk^ pŽil qfbkb 5/ mfbp ab ilkdfqra v nrb bi mbplv i^ `^abk^ pb abg^k `^bo ^i prbil+ m^oqfbkal ab i^ jfpj^ mlpf`fŽk fkf`f^i nrb ^kqbp-B^i`ri^o bi qo^_^gl ob^ifw^al mlo i^ crbow^ ab i^ do^sba^a `r^kal bi mbpl ^i`^kw^ bi prbil-

8- Rb^ R%k&bi sliq^gb kb`bp^ofl m^o^ pfqr^o rk^ `^od^ k bk i^p mi^`^p ab rk `lkabkp^alo-Di qo^_^gl kb`bp^ofl m^o^ `^od^o rk `lkabkp^alo abpab l < \ e^pq^ l < ] pb abcfkb jb,af^kqb i^ fkqbdo^i `wR%k&k+ Rf bi sliq^gb bp molmlo`flk^i ^ i^ `^od^+ abjlpqo^o nrb biqo^_^gl ob^ifw^al m^o^ pfqr^o rk^ `^od^ O bk rk `lkabkp^alo abp`^od^al bp cMR%M&+

Page 165: Calculus

R[fil g_^ci ^_ oh[ `oh]cƒh 034

*&). HNY\_ZRQV\QRb[N Sb[PVp[

Dk bi qo^_^gl `fbkqŒcf`lbp kb`bp^ofl `lk cob`rbk`f^ ob^ifw^os^of^p jbaf`fl,kbp bk `lkaf`flkbp pbjbg^kqbp v `^i`ri^o irbdl bi jlig_^ci l g_^c[ `lk i^ fab^ab obprjfo ilp a^qlp- Dufpqbkjr`elp qfmlp•qfibp ab moljbaflp+ bi jŠp `loofbkqbbp i^ g_^c[ [lcng€nc][+ Rf [! [0* ŠŠŠ * [h plk h k•jbolp ob^ibp+pr jbaf^ ^ofqj‹,qf`^ ddbpqŠabcfkfa^ mlo i^ fdr^ia^a

'1-06(

Rf ilp k•jbolp [e plk ilp s^ilobp ab rk^ crk`fŽk ` bk h mrkqlp afpqfkqlp+mlobgbjmil [e < `%Te&)bi k•jbol

bp i^ jbaf^ ^ofqj‹qf`^ ab ilp s^ilobp `%r)&)+++) `%rh&+Olabjlp buqbkabobpqb`lk,`bmql ^i `Ši`ril ab rk s^ilo jbafl kl pŽil m^o^ rk k•jbol cfkfql ab s^ilobp ab`%r&pfkl m^o^ qlalp ilp s^ilobp ab `%r&^i ob`loobo r rk fkqbos^il- K^ abcfkf`fŽknrb pfdrb klp pfosb m^o^ biil-

CDEHMHBHˆM CDK U@KNQ LDCHN CD TM@ ETMBHˆM DM TM HMSDQU@KN- Pd ` `nchn_al[\f_ _h oh chn_lp[fi W[) \Y) ^_`chcgim =%`&)p[fil g_^ci ^_ ` _h W[) \Y)g_^c[hn_ f[ `ƒlgof[

'1-07( 0 G>&a' < ,, a&s' _s ,] + [ \

Br^kal ` bp kl kbd^qfs^+bpq^ cŽojri^ qfbkb rk^ fkqbomobq^`fŽkdblj‹qof`^pbk`fii^- Orbpq^ bk i^ cloj^ &] + \'>&a' <P8a&s' _s* bpq^_ib`b nrb bi ob`qŠk,dril ab ^iqro^ =%`& u _^pb W[)\Y qfbkb i^ jfpj^ Šob^ nrb bi `lkgrkql ab loab,k^a^p ab ` pl_ob W[)\Y+

Olabjlp ^elo^ abjlpqo^o nrb i^ cŽojri^ '1-07( bp bk ob^ifa^a rk^ buqbkpfŽkabi `lk`bmql ab jbaf^ ^ofqj‹qf`^- Rb^ ` rk^ crk`fŽk bp`^ilk^a^ nrb bp `lkp,q^kqb bk `^a^ rkl ab ilp pr_fkqbos^ilp ab X\* \Y) l_qbkfalp ^i afsfafoil bk im^oqbpfdr^ibp- Dk m^oqf`ri^o+pb^ s8 < \ * e%\ * \'-i m^o^e < /+ 0+1+ --- + i*u prmlkd^jlp nrb `%r&< `%r^) pf Te*f ; r ; sd*Dkqlk`bp pboŠ Te * Te*f :: %\ * \da i* `lk il nrb pb qfbkb

0 G] 0 i \ * [ 0 i>&a' < ,, a&s' _s < ,, &!a&sf' ++ < , &!a&sf' Š

]+\ \ ]+\H i iHhzi h<i

Page 166: Calculus

035 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

@pŒmrbp+m^o^ crk`flkbp bp`^ilk^a^p+ bi moljbafl =%`& `lfk`fab `lk i^ jbaf^^ofqj‹qf`^ ab ilp s^ilobp a&sg'*,,, * a&si' qlj^alp bk ilp fkqbos^ilp bk ilp nrb i^crk`fŽk bp `lkpq^kqb-

Blk cob`rbk`f^ pb rqfifw^k jbaf^p ^ofqj‹qf`^p mlkabo^a^p bk ird^o ab i^pjbaf^p ^ofqj‹qf`^p loafk^of^p '1-06(- Rf Ui+ U1+ ‘‘‘ + Th plk h k•jbolp kl kbd^,qfslp 'ii^j^alp k`njn'* kl qlalp `bolp+ i^ jbaf^ ^ofqj‹qf`^ mlkabo^a^ \ ab\g%\0* ŠŠŠ * \i* pb abcfkb jbaf^kqb i^ cŽojri^

i

•rf\f' FRG[:***+

h

•Tfhzi

Br^kal ilp mbplp plk qlalp fdr^ibp+bpqbs^ilo `lfk`fab `lk i^ jbaf^ ^ofqj‹qf`^loafk^of^- K^ buqbkpfŽkab bpqb`lk`bmql ^ i^p crk`flkbp fkqbdo^_ibpsfbkb a^a^mlo i^ cŽojri^

br&s'a&s' _s>&a' ;,\ ] %er&s' _s

pfbkal V rk^ crk`fŽk mbpl kl kbd^qfs^ q^i nrb `wq%r&^r w N-K^p jbaf^p mlkabo^a^p plk jrv rqfifw^a^pbk EŒpf` b HkdbkfboŒ^-Olo bgbj,

mil+ `lkpfabobjlp rk^ s^ofii^ ob`q^ ab ilkdfqra \ v eb`e^ `lk rk j^qbof^i ababkpfa^a s^of^_ib- Blilnrbjlp i^ s^ofii^ ^ 0/ i^odl abi bgb r mlpfqfsl `lk rkbuqobjl bk bi lofdbk N+v abpfdkbjlp `lk g%r& i^ j^p^ ab i^ mlo`fŽk ab s^ofii^ab ilkdfqra r) jbafa^ abpab N- Rf g%r& < Iz k&o' _o m^o^rk^ `fboq^ crk`fŽk fkqb,do^_ib k nrb pb ii^j^ _`ind_\_ _` h\n\ ab i^ s^ofii^- Tk^ s^ofii^ pidajmh` qfbkbrk^ abkpfa^a ab j^p^ `lkpq^kqb- K^ fkqbdo^i-bysk&s' _s pb abkljfk^ bi kmdh`mhjh`ioj ab i^ s^ofii^ bk qlokl ab N+v bi ^`iomj _` bm\q`_\_ bp bi mrkql&rv^`lloabk^a^ s bp

'1-08(

[ aUk&s' _sT:w***

ok&s' _s

…pqbbp rk bgbjmil ab jbaf^ mlkabo^a^- Gbjlp moljbaf^al i^ crk`fŽk afpq^k`f^`%r& < r `lk i^ abkpfa^a ab j^p^ j `ljl crk`fŽk mbpl-

K^ fkqbdo^i`ws0k&s' _s pb ii^j^ n`bpi_j hjh`ioj* l hjh`ioj _` di`m^d\*ab i^ s^ofii^ bk qlokl ab N+v bi k•jbol mlpfqfsl m a^al mlo i^ cŽojri^

as/k&s' _sm0; zlz ]

Gj\k&s' _s

Page 167: Calculus

Be`m^d^djn 036

bp bi m\_dj _` bdmj ab i^ s^ofii^- Dk bpqb `^pl+ i^ crk`fŽk moljbaf^a^ bp bi `r^,ao^al ab i^ crk`fŽk afpq^k`f^+ x%r&< r0

* `lk i^ j^p^ ab abkpfa^a j `ljl crk`fŽkmbpl-

Lbaf^p mlkabo^a^p m^ob`fa^p ^ ‹pq^p q^j_f‹k pb mobpbkq^k bk bi BŠi`ril abmol_^_fifa^abp bk bi `r^i ilp `lk`bmqlp ab `nk`m\iu\ u q\md\iu\ grbd^k bi jfpjlm^mbi nrb bi `bkqol ab do^sba^a v bi jljbkql ab fkbo`f^-

*&)/ :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 0/+ `^i`ri^o bi moljbafl =%`& m^o^ i^ crk`fŽk a^a^ ` bk bifkqbos^il `loobpmlkafbkqb-

0- W%r& < r0* [ ; r ; ], 4, W%r& < `lp r) * 4Pu/ w r w 4Pu/+

5, W%r& < pbk /r) N z r w 4P,/+

5+ W%r& < pbk r `lp r) N z r w 4P,1+

6+W%r& <pbk1 r) N z r w 4Pu/+

.-+ W%r& < `lp! r) N z r w 4P+

/+ W%r& < r0 * r1* N z r ; H-

1, X&s' < UF-0* N z U x 3-2, X&s' < UF-1* 0 z s x 7-3, W%r& < pbk r) N z r w 4P,/+

00- '^( Rf E%r&< r0 m^o^ /94 r 94 ^+ e^ii^o rk k•jbol _ nrb p^qfpc^d^ • ; _ ; [ v q^inrb E%_&pb^ fdr^i ^i moljbafl ab ` bk ZN+[Y+'_( Qbplisbo i^ m^oqb '^( pf `%r& < u!+ pfbkal h rk bkqbol mlpfqfsl `r^inrfbo^-

01- Rb^ `%r& < r0 m^o^ } 94 r 94 0- Di s^ilo jbafl ab Ebk Z/+0\ bp p-G^ii^o rk^ crk`fŽkmbpl kl kbd^qfs^ q q^i nrb i^ jbaf^ mlkabo^a^ ab ` bk Z/+0\+ abcfkfa^ mlo '1-08( pb^'^( : '_( :: 'b( z-

02- Rb^ =%d& bi moljbafl ab ` bk bi fkqbos^il W[) \Y+ Cbjlpqo^o nrb qfbkb i^p molmfba^abppfdrfbkqbp9']( Llijc_^[^ [^cncp[7 =%d * c( < =%`& * =%a&+'_( Llijc_^[^ bigia€h_[7 =%_`& < _=%`& pf b bp rk k•jbol ob^i `r^inrfbo^-'`( Llijc_^[^ gihƒnih[7 =%`&72 =%a& pf ` R d bk W[) \Y+

03- ƒBrŠibp ab i^p molmfba^abp `fq^a^p bk bi Dgbo`f`fl 02 plk sŠifa^p m^o^ i^p jbaf^p mlkab,o^a^p abcfkfa^p mlo '1-08(>

04- Cbpfdkbjlp mlo =w%d&bi moljbafl ab ` bk bi fkqbos^il W[) \Y+'^( Rf \ ; ` ; \) abjlpqo^o nrb bufpqb rk k•jbol o nrb p^qfpc^`b /; o ; 0 q^i nrb=w%u&< n=w%`&* 'i , n&=w%`&+@pŒmrbp+ =w%`& bp rk^ jbaf^ ^ofqj‹qf`^ mlkabo^a^ ab

>x&v' v >x&a',

'_( Cbjlpqo^o nrb bi obpriq^al ab i^ m^oqb '^( q^j_f‹k bp sŠifal m^o^ jbaf^p mlkabo^,a^p `ljl i^p abcfkfa^p mlo '1-08(-

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 05 ^i 10 pb e^`b obcbobk`f^ ^ rk^ s^ofii^ ab ilkdfqra Hpfqr^a^ bk bi bgb s `lk rk buqobjl bk bi lofdbk- Blk i^ abkpfa^a ab j^p^ k nrb pb `fq^ bk`^a^ `^pl+ `^i`ri^o '^( bi `bkqol ab do^sba^a ab i^ s^ofii^+ '_( bi jljbkql ab fkbo`f^bk qlokl ^i lofdbk+ v 'b( bi o^afl ab dfol-

.3+ j%r& < 0 m^o^ M y r w H+H H

.4+ j%r& < 0 m^o^ M wr y1& j%r& < 1 m^o^ /9rwH+

.5+ j%r& < r m^o^ KwrwH+

H H H.6+ j%r& :r m^o^ Kwrw/$ j%r& < , m^o^ /wrwH+

1

Page 168: Calculus

037 >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

0., j%r& < r0 m^o^ N z r w I,H

10- j%r& < r0 m^o^ N z r w 1! &<,

j%r& <,3

Hm^o^ !1 y s x H+

11- Cbqbojfk^o rk^ abkpfa^a ab j^p^ k ab jlal nrb bi `bkqol ab do^sba^a ab rk^ s^ofii^ab ilkdfqra I nrbab pfqr^al ^ rk^ afpq^k`f^ I-2 ab rk buqobjl ab i^ s^ofii^-

12- Dk rk `fo`rfql bi‹`qof`l+ bi sliq^gb _%n&bk bi qfbjml o sfbkb a^al mlo i^ cŽojri^_%n&< 2 pbk /n+ B^i`ri^o9 '^( bi sliq^gb jbafl bk bi fkqbos^il ab qfbjml ZN+$fP,/Y8 '_( i^jbaf^ `r^aoŠqf`^ abi sliq^gb: bpql bp+ i^ o^Œwr^ao^a^ abi moljbafl ab i^ crk`fŽk `0

bk bi fkqbos^il W-)$fP,/Y+13- Dk rk `fo`rfql bi‹`qof`l+ bi sliq^gb _%n&v i^ fkqbkpfa^a ab i^ `loofbkqb c%n&sfbkbk a^alp

mlo i^p cŽojri^p _%n&< 05/ pbk o*c%n&< 1 pbk %n* $fP,3&+K^ mlqbk`f^ jbaf^ pb abcfkbmlo i^ cŽojri^

xdQ_%n&c%n&^n )@ l

pfbkal Q bi mboŒlal abi sliq^gb v ab i^ fkqbkpfa^a- Cbqbojfk^o Q v `^i`ri^o i^ mlqbk`f^jbaf^-

*&)0 ?N V[aRT_NYP\Z\ b[PVp[ QRYYoZVaRb]R_V\_&=[aRT_NYR`V[QRSV[VQN`

Rrmlkbjlp bk bpq^pb``fŽk nrb ` bp rk^ crk`fŽk q^i nrb i^ fkqbdo^iIz `Q&_obufpqbm^o^`^a^ r abi fkqbos^il W[)\Y+ L^kqbkaobjlp [ u ` cfglpu bpqraf^objlpbpq^fkqbdo^i`ljl rk^ crk`fŽk ab r+ Cbpfdk^jlp bi s^ilo ab i^ fkqbdo^i`lk =%r&)`lk 0/ nrb

'1-1/( >&s' < E7a&o' _o pf [ w u z \+

Tk^ b`r^`fŽk `ljl ‹pq^ klp mbojfqb `lkpqorfo rk^ krbs^ 'rk`fŽk = ^ m^oqfoabrk^ crk`fŽk a^a^ a9 bi s^ilo ab = bk `^a^ mrkql ab W[)\Y bp bi abqbojfk^al mlo'1-1/(- @idrk^p sb`bp bpq^crk`fŽk > pb af`b nrb bp rk^ dio`bm\g di_`adid_\ ab a vpb l_qfbkb ^ m^oqfoab a mlo fkqbdo^`fŽk-Cb`fjlp pi\ fkqbdo^ifkabcfkfa^ u kl g\fkqbdo^ifkabcfkfa^ mlonrb = q^j_f‹k abmbkab abi iŒjfqb fkcboflo \, U^ilobp afp,qfkqlp ab \ klp `lkar`foŠk ^ crk`flkbp = afpqfkq^p-Rf rqfifw^jlp rk krbsl iŒjfqbfkcboflo+mlo bgbjmil `* v abcfkfjlp lqo^ fkqbdo^ifkabcfkfa^ B jbaf^kqb i^ b`r^`fŽk

C&s' < n8a&o' _o *

i^ molmfba^a ^afqfs^ klp af`b bkqlk`bp nrb

>&s' + C&s' < E7a&o' _o + n8a&o' _o < a\! a&o' _o *

Page 169: Calculus

I\ dio`bm\g ^jhj api^d‡i _`g g…hdo npk`mdjm, Fio`bm\g`n di_`adid_\n /27

v mlo q^kql i^ afcbobk`f^ >&s' + C&s' bp di_`k`i_d`io` ab s, Olo q^kql alp fkqb,do^ibpfkabcfkfa^p `r^ibpnrfbo^ ab i^ jfpj^ crk`fŽk afcfbobkq^k pŽil bk rk^ `lkp,q^kqb'0^ `lkpq^kqb abmbkab ab i^ bib``fŽk ab [ v ]&+

Br^kal pb `lkl`b rk^ fkqbdo^ifkabcfkfa^ ab Z+bi s^ilo ab rk^ fkqbdo^i`ljl`wQ&_o mrbab `^i`ri^opb mlo pfjmib pr_pqo^``fŽk-Olo bgbjmil+ pf i bp rk bkqbol klkbd^qfsl+ qbkbjlp i^ cŽojri^ abi qblobj^ 0-04+

v i^ molmfba^a ^afqfs^ fjmif`^ nrb

f] d] d\ \i)/ i)go! _o < o! _o [ o! _o < , [\ N N i)g

Dk dbkbo^i+pf B%r&< `wnQ& n) pb qfbkb

'1-10( G<&o'_o < na&o'_o + na&o'_o < C&]' + C&\' ,

Tk^ bib``fŽk afpqfkq ab _ ^iqbo^pli^jbkqb B%r&bk rk^ `lkpq^kqb: bpql kl `^j_f^i^ afcbobk`f^ B%\& * B%[&)ab_fal ^ nrb i^ `lkpq^kqb abp^m^ob`bbk i^ pr_pqo^``fŽk-

Rf rqfifw^jlp bi pŒj_lil bpmb`f^i

m^o^ abpfdk^o i^ afcbobk`f^ B%\& * B%[&) i^ fdr^ia^a '1-10( mrbab mlkbopb bk i^cloj^

F7a&s' _s < C&s'gx < C&]' + B%[&+

Dufpqb+k^qro^ijbkqb+ rk^ obi^`fŽk dblj‹qof`^ jrv pfjmib bkqob rk^ crk,`fŽk ` v prp fkqbdo^ibpfkabcfkfa^p- Dk i^ cfdro^ 1+04'^( pb obmobpbkqrk bgbjmilbk bi nrb ` bp rk^ crk`fŽk kl kbd^qfs^ v bi k•jbol =%r& bp fdr^i ^i Šob^ ab i^obdfŽkplj_ob^a^ pfqr^a^ mlo ab_^gl ab i^ doŠcf`^ab ` abpab \ e^pq^ s, Rf ` qlj^s^ilobp mlpfqfslp v kbd^qfslp+ `ljl bk i^ cfdro^ 1-04'_(+ i^ fkqbdo^i =%r& a^ i^prj^ ab i^p Šob^pab i^p obdflkbp pfqr^a^p mlo bk`fj^ abi bgb s afpjfkrfa^ bki^ prj^ ab i^p Šob^ppfqr^a^p mlo ab_^gl abi jfpjl bgb-

Lr`e^p ab i^p crk`flkbp nrb ^m^ob`bk bk afsbop^p o^j^p ab i^ `fbk`f^ pbmobpbkq^kbu^`q^jbkqb bk bpq^ cloj^+ `ljl fkqbdo^ibpfkabcfkfa^p ab lqo^p crk,`flkbp- …pq bp rk^ ab i^p o^wlkbp mlo i^p nrb rk^ do^k m^oqbabi BŠi`ril bpqŠabaf`^a^ ^i bpqrafl ab i^p fkqbdo^ibpfkabcfkfa^p-

Page 170: Calculus

04/ >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

@ sb`bp rk^ molmfba^am^oqf`ri^oab o fjmif`^ rk^ `loobpmlkafbkqb molmfba^aab i^ fkqbdo^i fkabcfkfa^- Olo bgbjmil+ pf n bp kl kbd^qfs^ bk W[)\Y) i^ fkqbdo^ifkabcfkfa^ = bp `ob`fbkqb+mrbpql nrb pb qfbkb

=%s& * =%r& < GS a&o' _o + GU a&o' _o < ba&o' _o x N+\ \ s

ga&o'_o < Rrj^ ^idb_o^f`^ ab i^p Šob^p

\ s

']( '^(

EHFTQ@ 1-04 Fio`mkm`o\^d‡i b`jh„omd^\ _` g\ dio`bm\g di_`adid_\,

a %r&* a %s&

a%r&aw 3C$

a%r& 1

r B%C s r B%C s1 1

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Page 171: Calculus

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Page 172: Calculus

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Page 173: Calculus

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Page 174: Calculus

)-, >gbpi\n \kgd^\^dji`n _` g\ dio`bm\^d‡i

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Page 175: Calculus

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Page 176: Calculus

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Page 177: Calculus

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Page 178: Calculus

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Dkqlokl J0%j&

, jDkqlokl J0%j&

EHFTQ@ 1-2 Arcmn_ ifj `%r& 77777=) j_li['V

hi m_ ^c]_ h[^[ ^_ ` _h m-

EHFTQ@ 2-2 ` _mn• ^_`chc^[ _h l vifj `%r& 77777`%j&) ^_ g[h_l[ ko_ ` _ms+k ]ihncho[ _h m-

J0%j& pb e^ obmobpbkq^albk bi bgb s, Di ob`qŠkdril plj_ob^al `lkpq^ ab qlalpilp mrkqlp %r)u( m^o^ilp `r^ibp r D J0%j& b u D J.%=&+K^ abcfkf`fŽk ab iŒjfqb^pbdro^ nrb qla^ i^ doŠcf`^ab o `loobpmlkafbkqb ^i fkqbos^il K0%j& bpqŠpfqr^a^ bkbpb ob`qŠkdril+ p^isl m^o^ bi jfpjl mrkql j+

K^ abcfkf`fŽk ab iŒjfqb q^j_f‹k pb mrbab clojri^o jbaf^kqb ilp l[^cim abilp bkqloklp J.%=& v J0%j&+ Dp `lpqrj_ob abpfdk^o bi o^afl ab J.%=& mlo ’ 'ibqo^dofbd^ €jmcfih& v bi ab Jt%j& mlo Ž 'ibqo^ dofbd^ ^_fn[&+Cb`fo nrb ,%r& C J.%=&bp bnrfs^ibkqb ^ i^ abpfdr^ia^a E,%r&* =f ; ’+ v mlkbo nrb r C J0%j&) r ;/; j)bp il jfpjl nrb bp`of_fo N ; Gt , jE ; ˆ- Olo il q^kql+ i^ abcfkf`fŽk ab iŒjfqbmrbab q^j_f‹k bumobp^opb pŒ9

Af m•g\ifi ifj `%r&< = mcahc`c][ ko_ j[l[ ni^i ’ = N+ _rcmn_oh n4 = Nn[f ko_ !)*j

'2-1( E,%r&* =f ; _ mc_gjl_ ko_ N ; Yt , kd ; o4-

N_pbosbjlp nrb i^p qobpfdr^ia^abp+

gdha&s' < >*!&'V

ifj &a&s' + =& < N+ ifj Fa&s' + =f < N+!&,!

Page 179: Calculus

A`adid^d‡i _` g…hdo`_` pi\ api^d‡i 048

plk bnrfs^ibkqbp- Dpq^ bnrfs^ibk`f^ pb e^`b j^kfcfbpq^ q^k molkql `ljl bp`of,_^jlp `^a^ rk^ ab bp^p fdr^ia^abp bk i^ qbojfklildŒ^ ab ’ v -%0+/&+

@i `lkpfabo^o iŒjfqbp `r^kal s x j) `lksfbkb ^ sb`bp abpfdk^o i^ afcbobk`f^s + k `lk bi krbsl pŒj_lil b) v e^`bo irbdl nrb b w N- Dpql fjmif`^ q^k pŽilrk `^j_fl ab klq^`fŽk+ mlonrb+ `ljl pb `ljmorb_^ cŠ`fijbkqb+ i^p alp fdr^ia^,abp pfdrfbkqbp plk bnrfs^ibkqbp9

r*$j

gdha&k * c' < > ,N'+*

gdha&s' < > *

DIDLOKN 0- I…hdo` _` pi\ api^d‡i ^jino\io`, Rb^ a&s' < ` m^o^ qlal s,Dp cŠ`fi abjlpqo^o nrb m^o^ qlal j) qbkbjlp ifj `%r& < ]+ Dk bcb`ql+ a^al rk

3`3'+V

bkqlokl J)%]&) i^ obi^`f‹k '2-0( pb p^qfpc^`b m^o^ `r^inrfbo J,j& mlonrb `%r& < bm^o^ qlal r) `r^inrfbo^ nrb pb^ J)%]&+ Blk i^ klq^`fŽk ab ilp iŒjfqbp+bp`of_fjlp

ifj` < `-

DIDLOKN 1- I…hdo` _` g\ api^d‡i d_`iod_\_, @elo^ bp a&s' < s m^o^ qlals, Olabjlp mol_^o jrv pfjmibjbkqb nrb ifj a&s' < k, O^o^ `r^inrfbo bkqlokl

3`3'+V

J)%j& pb qlj^ J,j& < J)%j&+ Dkqlk`bp i^ obi^`fŽk '2-0( pb ob^ifw^ qofsf^ijbkqb-Blk i^ klq^`fŽk ab iŒjfqb+bp`of_fjlp

fcg t < j+3`3'+V

Klp iŒjfqbp ~i^qbo^ibp‚ mrbabk abcfkfopb bk cloj^ m^ob`fa^- Olo bgbjmil+ pfa&s' x > `r^kal s x k `lk s^ilobp j^vlobp nrb k* ab`fjlp nrb > bp bi g…hdo`kjm g\ _`m`^c\ ab a bk k* b fkaf`^jlp bpql mlkfbkal

gdha&s' < >,?'%<%

Dk i^ qbojfklildŒ^ ab ilp bkqloklp bpql pfdkfcf`^ nrb m^o^ qlal bkqlokl J)%=&)bufpqb ^id•k bkqlokl J0%j& q^i nrb

'2-2( y&s'D J)%=& pfbjmob nrb u D J/%j& u u= j+

Klp iŒjfqbp ^ i^ fwnrfboa^+ nrb pb fkaf`^k mlkfbkal s x j*) pb abcfkbk abi jfpjljlal obpqofkdfbkal s ^ s^ilobp jbklobp nrb j+

Rf ` qfbkb iŒjfqb = bk j) q^j_f‹k qfbkb iŒjfqb ^ i^ abob`e^ u iŒjfqb ^ i^ fwnrfbo,a^ ab j) pfbkal ^j_lp fdr^ibp ^ >, Obol rk^ crk`fŽk mrbab qbkbo bi iŒjfqb ^ i^abob`e^ ab j afpqfkql abi iŒjfqb ^ i^ fwnrfboa^+ `ljl pb sb bk bi bgbjmil pfdrfbkqb-

Page 180: Calculus

05/ Boh]cih_m ]ihncho[m

DIDLOKN 2- Rb^ `%r&< Zui m^o^ qlal r) v pb^ j rk bkqbol `r^inrfbo^-O^o^s^ilobp ab r moŽufjlp ^ j) r ; j) qbkbjlp `%r&< j * 0+X m^o^s^ilobp abr moŽufjlp ^ j) r = j) bp `%r&< j+ Ubjlp+ mlo `lkpfdrfbkqb+ nrb

ifj a&s' < j * 0 X ifj a&s' < j+G%#;"% YX&&&`*>#

Dk rk bgbjmil `ljl ‹pqb+bk bi nrb ilp iŒjfqbp^ i^ fwnrfboa^ v ^ i^ abob`e^plk afpqfkqlp+bi iŒjfqbab ` bk j hi _rcmn_+

DIDLOKN 3- Rb^ `%r&< .,r0 pf r ;/; N+ X `%K&< N- K^ doŠcf`^ ab ` bk i^pmolufjfa^abp abi lofdbk bpqŠobmobpbkq^a bk i^ cfdro^ 2-0'_(- Dk bpqbbgbjmil+ `qlj^ s^ilobp q^k do^kabp `ljl nrbo^jlp bk i^p molufjfa^abp ab M ab jlal nrbkl qfbkb iŒjfqb ^ i^ fwnrfboa^ kf iŒjfqb ^ i^ abob`e^ abi lofdbk- O^o^ abjlpqo^oofdrolp^jbkqb nrb kl bufpqbk•jbol ob^i = q^i nrb ifj `%r&< =) mlabjlp o^,

1T&%#N!V%P%

wlk^o ^pŒ9Rrmlkd^jlp nrb bufpqfbo^rk q^i =) prmlkdŠjlpib = w N- Difg^jlprk bkqlokl J.%=& ab ilkdfqra 0- Dk bi fkqbos^il N ; r ; f,%= * 1(+ qbkbjlp`%r&< 0.f1 = %= *1(1 = = * 1+ ab jlal nrb `%r&kl mrbab bpq^obk bi bkqlo,kl J.%=&+@pŒmrbp+qlal bkqlokl J%K&`lkqfbkb mrkqlp r; M m^o^ilp nrb `%r&bp buqboflo^ J.%=&) `lk 0/ nrb '2-2( kl pb `rjmib m^o^bpqbJ.%=& bibdfal- Krbdl` kl qfbkb iŒjfqb ^ i^ abob`e^ bk N-

DIDLOKN 4- Rb^ `%r&< 0 pf r ;/; N+ X `%K&< N- Dpq^crk`fŽk qlj^ bi s^ilo`lkpq^kqb 0 m^o^qlal s p^isl bk N+ alkab qfbkbbi s^ilo N- Klp iŒjfqbp i^ abob`e^v ^ i^ fwnrfboa^ plk 0 bk qlal mrkql j) `lk 0/ nrb bi iŒjfqbab `%r&) r^kal r qfbk,ab ^ j) bufpqbv bp fdr^i ^ 0- N_p‹osbpb nrb bi iŒjfqbab / bp 0 bk bi mrkql N+ bkq^kql nrb .%-&< l-

2-2 9RSV[VPVp[QRP\[aV[bVQNQQRb[N Sb[PVp[

Dk i^ abcfkf`fŽk ab iŒjfqb kl pb e^`b jbk`fŽk abi `ljmloq^jfbkql ab / bkbi mrkql j+ K^ clojri^`fŽk '2-0( pb obcfbob ^nrbiilp mrkqlp s ;/; j mboqbkb`fbkqbp^i bkqlokl J0%j&) `lk 0/ nrb kl bp kb`bp^ofl nrb ` bpq‹abcfkfa^ bk j+ @abjŠp+ fk,`irpl pf ` bpqŠabcfkfa^ bk j) pr s^ilo ^iiŒkl bp kb`bp^of^jbkqb fdr^i ^i iŒjfqb=+Ml l_pq^kqb+pf l`roob nrb ` bpqŠabcfkfa^ bk j v nrb `%j&< =) pb af`b bkqlk`bpnrb i^ crk`fŽk ` bp `lkqfkr^ bk j+ Cf`el ab lqol jlal+ qbkbjlp i^ pfdrfbkqb ab,cfkf`fŽk-

CDEHMHBHˆM CD BNMSHMTHC@C CD TM@ ETMBHˆM DM TM OTMSN- O_ ^c]_ ko_ oh[`oh]cƒh ` _m]ihncho[ _h oh johni j mc

]( ` _mn•^_`chc^[ _h j) v_( ifj `%r&< `%j&+-+-----!

Page 181: Calculus

@_`chc]cƒh ^_ ]ihnchoc^[^ ^_ oh[ `oh]cƒh 050

Dpq^ abcfkf`fŽk q^j_f‹k mrbab clojri^opb `lk bkqloklp- Tk^ crk`fŽk ` bp`lkqfkr^ bk j pf m^o^ qlal bkqlokl J) W`%j&Ybufpqb rk bkqlokl J0%j& q^i nrb

'2-3( a&s' C J/Xa&k'Z pfbjmob nrb s C J0&k'}

Orbpql nrb `%j& mboqbkb`b pfbjmob ^ JE WF%j&Y) kl pb mob`fp^ i^ `lkaf`fŽks"+ k bk '2-3(- Dpmb`fcf`^kal ilp o^aflp ab ilp bkqloklp+ i^ abcfkf`fŽk ab `lk,qfkrfa^a mrbab a^opb `ljl pfdrb9

Tk^ crk`fŽk ` bp `lkqfkr^ bk k pf m^o^ qlal ’ = N bufpqb rk z = N q^i nrb

Fa&s' + a&k' G; ’ pfbjmob nrb Fs + kd ; y-

Dk i^ cfdro^ 2-2 pb obmobpbkq^dblj‹qof`^jbkqb i^ abcfkf`fŽk ab `lkqfkrfa^a-Dp^ cfdro^ bp m^ob`fa^ ^ i^ 2-1 p^isl nrb bi s^ilo iŒjfqb =) bp fdr^i ^i s^ilo `%j&

`lk il nrb qla^ i^ doŠcf`^ ab ` `loobpmlkafbkqb ^ K0%j& bpqŠ bk bi ob`qŠkdril plj,_ob^al-

DIDLOKN 0- H[m `oh]cih_m ]ihmn[hn_m mih mc_gjl_ ]ihncho[m+ Rf `%r& < _

m^o^ qlal s* bkqlk`bp

ifjc'u( < ifok b < b < a&k'

m^o^ qlal j) `lk il `r^i ` bp `lkqfkr^ m^o^ qlal s,

DIDLOKN 1- H[ `oh]cƒh c^_hnc^[^ _m ]ihncho[ j[l[ ni^i s* Rf `%r& < r

m^o^ qlal s* qbkbjlp

ifokc'u( < ifok u < k < a&k'?'(< (t-p

m^o^ qlal j) irbdl bp `lkqfkr^ m^o^ qlal s^ilo ab s,

DIDLOKN 2- Rb^ `%r& < WrY m^o^ qlal r+ Dpq^ crk`fŽk bp `lkqfkr^ bk qlalmrkql k nrb kl pb^ bkqbol- O^o^ s^ilobp bkqbolp ab s bp afp`lkqfkr^+ v^ nrb biiŒjfqb ab ` kl bufpqb+v^ nrb plk afpqfkqlp ilp iŒjfqbp ^ i^ abob`e^ v ^ i^ fwnrfboa^-Tk^ afp`lkqfkrfa^a ab bpqb qfml+ bk i^ nrb bufpqbk ilp iŒjfqbp ^ i^ abob`e^ v ^ i^fwnrfboa^ mbol plk afpqfkqlp+ pb ii^j^ ^cm]ihnchoc^[^ ^_ m[fni+ Rfk bj_^odl+ v^nrb bi iŒjfqb ^ i^ abob`e^ bp fdr^i ^ `%j& bk `^a^ bkqbol j) ab`fjlp nrb ` bp ]ih*ncho[ jil f[ ^_l_]b[ bk j+

Page 182: Calculus

051 Boh]cih_m ]ihncho[m

DIDLOKN 3- K^ crk`fŽk ` m^o^i^ nrb `%r& < 0. r0 m^o^r ;C+ N+`%K&< N+bpafp`lkqfkr^ bk N- ZUbo cfdro^ 2-0'_(-\ Cb`fjlp nrb bufpqbrk^ ^cm]ihnchoc^[^

ch`chcn[ bk N mlonrb i^ crk`fŽk qlj^ s^ilobp q^k do^kabp `ljl nrbo^jlp bk i^pmolufjfa^abp ab N-

DIDLOKN 4- Rb^ `%r&: 0 m^o^U;C+L* c'N(<N- Dpq crk`fŽk bp `lkqfkr^ bk qlalmrkql s bu`bmql bk N- Dp afp`lkqfkr^ bk N mlonrb a&L' kl bp fdr^i ^i iŒjfqb ab`%r& `r^kal r w N- Dk bpqbbgbjmil i^ afp`lkqfkrfa^a mlaoŒ bsfq^opbifjfq^kali^ crk`fŽk bk N m^o^qbkbobi s^ilo 0 bk sbw ab N-Olo bpq^o^wŽk+rk^ afp`lkqfkrf,a^a ab bpqbqfml pb ii^j^ _pcn[\f_+ N_p‹osbpb nrb i^p afp`lkqfkrfa^abp ab p^iql+kl mrbabk bsfq^opb`^j_f^kal q^k pŽil bi s^ilo ab i^ crk`fŽk ` bk rk mrkql-

2-3 Sblobj^p crka^jbkq^ibp pl_ob iŒjfqbp-Nqolp bgbjmilp ab crk`flkbp `lkqfkr^p

Di `Ši`ril `lk iŒjfqbpmrbab pfjmifcf`^opb`lk cob`rbk`f^ `lk bi qblobj^ pf,drfbkqb nrb molmlo`flk^ rk^p obdi^p_Špf`^p m^o^lmbo^o`lk iŒjfqbp-

SDNQDL@ 2-0- O_[h ` t d ^im `oh]cih_m n[f_m ko_

ifj a&s' < =)rwj

fcga%r& < >+sxk

O_ nc_h_ _hnih]_m

'f( ifj Xa&s' * b&s'Z < > * ?*B''%+$

'ff( ifj Xa&s' + b&s'Z < > + ?*B''%+$

'fff( ifokg&'uy9a%r&< > , ? *!&,,*0(

'fs( gdha&s'eb&s' < >e? pf > "< N-

Jin[7 Tk `^pl m^oqf`ri^o fjmloq^kqb ab 'fff( pb mobpbkq^ r^kal ` bp `lkpq^kqb+bpab`fo `%r& ::= m^o^ qlal r+ Dk bpqb`^pl+ 'fff( pb bp`of_b ifj = +a%r& << = +>+

B_V

K^ abjlpqo^`fŽk abi qblobj^ 2-0 kl bp afcŒ`fi+mbol bp ^idl i^od^+mlo 0/ nrbi^ ebjlp `lil`^al bk lqo^ pb``fŽk 'Rb``fŽk 2-4(- @nrŒ`ljbkq^jlp ^idrk^p`lkpb`rbk`f^p pbk`fii^p abi qblobj^-

N_pbosbjlp mofjbol nrb i^p ^cfoj^`flkbp abi qblobj^ mrbabk bp`of_fopbbkcloj^ rk ml`l afpqfkq^-Olo bgbjmil+ 'f( mrbab mlkbopb`ljl pfdrb9

0fj Xa&s' * b&s'Z < ifj a&s' * ifj b&s' ,[_V [_V [_V

Page 183: Calculus

P_il_g[m `oh^[g_hn[f_m mi\l_ f•gcn_m 052

Dpql klp af`b nrb bi iŒjfqbab rk^ prj^ bp i^ prj^ ab ilp iŒjfqbp-Dp `lpqrj_ob fkaf`^o mlo ` * d+` * d+ ` &dv ` , d i^p crk`flkbp `rvlp s^ilobp

m^o^`^a^ s plk9

`%r& * a%r&) `%r& * a%r&) `%r&+a%r&) u `%r&,c %r&)

obpmb`qfs^jbkqb- Dpq^p crk`flkbp pb abkljfk^k mog[) ^c`_l_h]c[) jli^o]ni v]i]c_hn_ ab ` v d- Rb bkqfbkab nrb bi `l`fbkqb ` , d pŽil bpqŠabcfkfal bk ilp mrkqlpbk ilp nrb a%r&:.: N- Di pfdrfbkqb `loli^ofl ^i qblobj^ 2-0+ bpqŠclojri^al `lkbpq^qbojfklildŒ^ v klq^`fŽk v pb obcfbob crk`flkbp `lkqfkr^p-

SDNQDL@ 2-1- O_[h ` t a ^im `oh]cih_m ]ihncho[m _h oh johni j+ H[ mog[

` * a) f[ ^c`_l_h]c[ ` * a) t _f jli^o]ni oa mih n[g\c€h ]ihncho[m _h j+ Rfa%j& :.: N+ n[g\c€h _f ]i]c_hn_ ` , a _m]ihncho[+

@_gimnl[]cƒh+ Orbpql nrb ` v d plk `lkqfkr^p bk j) pb qfbkbifj `%r& < `%j&Tw.F

v ifj a%r&< a%j&+ @mif`^kal i^p cŽojri^p m^o^ ilp iŒjfqbp+a^a^p bk bi qblob,Tw.F

j^ 2-0 `r^kal = < `%j& v > < a%j&) pb abar`b bi qblobj^ 2-1-Rb e^ sfpql nrb i^ crk`fŽk fa‹kqf`^ v i^ `lkpq^kqb plk `lkqfkr^p m^o^ `r^i,

nrfbo s^ilo ab s, Olo jbafl ab bpqlpbgbjmilp v bi qblobj^ 2-1+pb mrbabk `lkpqorfolqolp jr`elp ab crk`flkbp `lkqfkr^p-

DIDLOKN 0- ?ihnchoc^[^ ^_ jifchigcim+ Rf pb qlj^ `%r& < a%r& < r) abi^ `lkqfkrfa^a abi molar`ql pb abar`b i^ `lkqfkrfa^a bk `^a^ mrkql ab i^ crk`fŽk`rvl s^ilo bk `^a^ s bp s0

Š Olo fkar``fŽk pb morb_^+nrb m^o^`^a^ k•jbol ob^i `v `^a^ bkqbol h i^ crk`fŽk ` m^o^i^ `r^i `%r& < bu! bp `lkqfkr^ m^o^qlal r+ Bljli^ prj^ ab alp crk`flkbp `lkqfkr^p bp ^ pr sbw `lkqfkr^+ mlo fkar``fŽk pb morb_^nrb q^j_f‹k bp `lkqfkr^ i^ prj^ ab rk k•jbol cfkfql ab crk`flkbp `lkqfkr^p-Olo q^kql+qlal mlifkljfl j%r& < 1z<l ]ere_m crk`fŽk `lkqfkr^ bk qlalp ilp mrk,qlp-

DIDLOKN 1- ?ihnchoc^[^ ^_ `oh]cih_m l[]cih[f_m+ Di `l`fbkqb ab alp mlif,kljflp pb ii^j^ `oh]cƒh l[]cih[f+ Rf l bp rk^ crk`fŽk o^`flk^i+ pb qfbkb9

l%r& < j%r& )k%r&

alkab j v k plk mlifkljflp- K^ crk`fŽk l bpqŠabcfkfa^ m^o^qlal k•jbol ob^i rq^i nrb k%r&:.: N- Bljl bi `l`fbkqb ab crk`flkbp `lkqfkr^p bp `lkqfkrl+ i^ crk`fŽko^`flk^i bp `lkqfkr^ bk qlalp ilp mrkqlp bk nrb bpqŠabcfkfa^- Tk bgbjmil pbk`fiilbp l%r& < f,r pf r :.: N- Dpq^crk`fŽk bp `lkqfkr^ m^o^qlal s^ilo ab r p^isl bks < N bk nrb kl bpqŠabcfkfa^-

Page 184: Calculus

053 Boh]cih_m ]ihncho[m

Di qblobj^ nrb pfdrb abjrbpqo^ nrb pf rk^ crk`fŽk b bpqŠfkqbo`^i^a^ bkqoblqo^p alp crk`flkbp nrb qfbkbkbi jfpjl iŒjfqb`r^kal s [[ j) a qfbkbq^j_f‹k bpqbiŒjfqb `r^kal s [[ j+

SDNQDL@ 2-2- OQHMBHOHNCD HMSDQB@K@BHˆM- Oojiha[gim ko_ `%r&wa%r&wb%r&j[l[ ni^i r :.: j _h oh ]c_lni _hnilhi J%j&+Oojiha[gim n[g\c€h ko_

ifj g&'u(< ifj c&s' < \,1V%#S @(((BA

O_ nc_h__hnih]_m ifj a%r&< [+!)!!j

@_gimnl[]cƒh+ Rb^k C%r&< a%r&* `%r&) u D%r&< b%r&* `%r&+K^p abpf,dr^ia^abp ` wb x c fjmif`^k N z b + ` wc + `) l

M y C%r&w D%r&

m^o^qlal r :.: j bk J%j&+ O^o^ abjlpqo^o bi qblobj^+ _^pq^ mol_^o nrb C%r&w N`r^kal r w j) a^al nrb D%r&w N `r^kal r w j+

Rb^ M0'N( rk bkqlokl `r^inrfbo^ ab N- Orbpql nrb D%r&w N `r^kal r w j)bufpqbrk bkqlokl J0%j& q^i nrb

E&s' D M0'N( pfbjmob nrb s D K/%j& v rwj+

Olabjlp prmlkbo nrb J0%j& R: J%j&+ Dkqlk`bp i^ abpfdr^ia^a Nz F z D bpq^,_ib`b nrb C%r&kl bpqŠjŠp ibglp ab N nrb D%r& pf r bpqŠbk J0%j&) r :.: j+ Olo`lkpfdrfbkqb C%r& D M0'N( m^o^q^i s^ilo r) v mlo q^kql C%r&w N `r^kal r w m-Dpql abjrbpqo^ bi qblobj^- K^ jfpj^ abjlpqo^`fŽk bp sŠifa^ pf qlalp ilp iŒjfqbpplk iŒjfqbp^ rk i^al-

Di mofk`fmfl ab fkqbo`^i^`fŽk bp •qfi bk i^ moŠ`qf`^mlonrb ^ jbkral bp ml,pf_ib bk`lkqo^o crk`flkbp ab fkqbo`^i^`fŽk ` u c jŠp j^kbg^_ibp nrb a+ U^jlp ^rqfifw^obpqbmofk`fmflm^o^abjlpqo^o nrb qla^ fkqbdo^ifkabcfkfa^ bp rk^ crk`fŽk`lkqfkr^-

SDNQDL@ 2-3- BNMSHMTHC@C CD K@R HMSDFQ@KDR HMCDEHMHC@R- Oojiha[gimko_ ` _mchn_al[\f_ _h W[)rY j[l[ ni^i r _h W[)\Y) v m_[

>&s' < F7a&o' _o ,

Ahnih]_m f[ chn_al[f ch^_`chc^[ = _m]ihncho[ _h ][^[ johni ^_ W[)\Y+ %Ahfim_rnl_gim ^_f chn_lp[fi n_h_gim ]ihnchoc^[^ [ oh f[^i+&

Page 185: Calculus

Q`jm`h\n api_\h`io\g`n nj]m` g…hdo`n 054

A`hjnom\^d‡i, Difg^jlp k bk X\*]Z, G^v nrb abjlpqo^o nrb >&s' x >&k'`r^kal s x j+ Sbkbjlp

'2-4( =%r&* =%j&< oa&o' ^n +&0(

Orbpql nrb ` bpqŠ ^`lq^a^ bk W[)\Y) bufpqb rk^ `lkpq^kqb L = N q^i nrb,K y a&o'x K m^o^qlal o bk X\*]Z, Rf s< k* fkqbdo^jlp bp^p abpfdr^ia^abpbk bi fkqbos^il Wj) rY l_qbkfbkal

*I%r * j& w =%r&* =%j&w I%r * j&+

Rf s ; k* l_qbkbjlp i^p jfpj^p abpfdr^ia^abp `lk s + k prpqfqrfa^ mlo k + s,Olo `lkpfdrfbkqb+ bk rkl r lqol `^pl mlabjlp e^`bo nrb r w k v ^mif`^o bi mofk,`fmfl ab fkqbo`^i^`fŽk bk`lkqo^kal nrb =%r&w =%j&+Dpql morb_^ bi qblobj^- Rfj bp rk buqobjl ab W[) \Y) qbkbjlp nrb e^`bo nrb r w j abpab bi fkqboflo abifkqbos^il+ `lk il nrb ilp iŒjfqbpplk ^ rk i^al-

DIDLOKN 2- @jiodipd_\_ _`g n`ij v _`g ^jn`ij, Orbpql nrb i^ crk`fŽk

pbkl bp rk^ fkqbdo^ifkabcfkfa^+pbks < F7`lp o_o*bi qblobj^ ^kqboflo klp af`b

nrb i^ crk`fŽk pbkl bp `lkqfkr^ m^o^qlal s, Cbi jfpjl jlal+ bi `lpbkl bp crk`fŽk

`lkqfkr^ m^o^ qlal s v^ nrb `lp s < 0 , F7pbk o_o, K^ `lkqfkrfa^a ab bp^p

crk`flkbp q^j_f‹k pb mrbab abar`fo pfk rqfifw^obi eb`el ab nrb pb^k fkqbdo^ibpfkabcfkfa^p-Dk bi DIbo`f`fl 15 ab i^ Rb``fŽk 2-5 pb bp_lw^ lqo^ abjlpqo^`fŽk-

DIDLOKN 3- Dk bpqb bgbjmil abjlpqo^jlp rk^ fjmloq^kqb cŽojri^ pl_obiŒjfqbp9

'2-5( ifj pbks < 0 +d%\ d

nrb irbdl kb`bpfq^objlp bk BŠi`ril afcbobk`f^i- Orbpql nrb bi abkljfk^alo abi`l`fbkqb 'pbk r&,r qfbkabe^`f^ N`r^kal r zN+ kl mlabjlp bjmib^o bi qblobj^ abi`l`fbkqb ab iŒjfqbpm^o^ abar`fo '2-5(- Dk `^j_fl+ mlabjlp rqfifw^obi mofk`fmflab fkqbo`^i^`fŽk- Dk i^ Rb``fŽk 1-4 sfjlp nrb

Npbks 0;,;,+

s `lp s

Page 186: Calculus

055 Cpi^dji`n ^jiodip\n

sŠifa^ m^o^ N ; s ; iHS- S^j_f‹k bp sŠifa^ m^o^ , 00S ; s ; N v^ nrb`lp ' , r& < `lp r v pbk ' , r& <, pbk r) `lk il nrb i^ ^kqboflo al_ib abpfdr^ia^a bpsŠifa^ m^o^ qlal r :.: N bk bi bkqlokl J%K8 g/Q', Br^kal r w N+ bk`lkqo^jlp`lp r w 0 mrbp bi `lpbkl bp `lkqfkrl bk N+v mlo q^kql i.'`lp r& w 0- Olo `lkpf,drfbkqb+ pbd•k bi mofk`fmfl ab fkqbo`^i^`fŽk+ abar`fjlp '2-5(- Rf abcfkfjlp`%r& < 'pbk r&,r m^o^ r :.: N+`_K& < 0+ bkqlk`bp ` bp `lkqfkr^ m^o^ qlal r+ RrdoŠcf`^ pb obmobpbkqbk i^ cfdro^ 2-3-

t

gdmia&s' < G< a&L';R:

EHFTQ@ 2-3 `%r& < 'pbk r&,r mcr w N+ `%K&< 0- Amn[ `oh]cƒh _m ]ihncho[ j[l[ ni^i r)

DIDLOKN 4-^dji\g kjndodqj,

@jiodipd_\_ _` a&s' < s% k\m\ s = N+ nd`i_j n pi iˆh`mj m\+Di qblobj^ 1-1 klp a^ i^ cŽojri^ ab fkqbdo^`fŽk

.$! sg)g-io% !_o < ,,,+

l 0 * g-i

sŠifa^ m^o^qlal s = N v qlal bkqbol i x 0- Blk ilp qblobj^p 2-3 v 2-0+bk`lk,qo^jlp nrb i^ crk`fŽk = a^a^ mlo =%r&+< rf(.,h bp `lkqfkr^ bk qlalp ilp mrkqlpm = N- @elo^+pb^ a%r& < r$ ! < =%r&,r m^o^r =/- Bljl d bp rk `l`fbkqb ab alpcrk`flkbp `lkqfkr^p+ q^j_f‹k il pboŠ m^o^ qlalp ilp mrkqlp m = N- LŠp dbkb,o^i+ pf `%r& < rh-i* alkab g bp rk bkqbol mlpfqfsl+bkqlk`bp ` bp rk molar`ql abcrk`flkbp `lkqfkr^p u bp+mlo q^kql+ lkqfkr^ bk qlalp ilp mrkqlp m = N- Dpql bpq^,_ib`b i^ `lkqfkrfa^a ab i^ crk`fŽk mlqbk`f^ o,‹pfj^+ `%r&< r$) `r^kal o bp `r^i,nrfbo k•jbol o^`flk^i mlpfqfsl+bk qlalp ilp mrkqlpm = N-Dkm < Nqbkbjlp `lk,qfkrfa^a ^ i^ abob`e^-

K^ `lkqfkrfa^a ab i^ crk`fŽk mlqbk`f^ o,‹pfj^ m^o^o o^`flk^i mrbab q^j_f‹kabar`fopb pfk rqfifw^ofkqbdo^ibp-Dk i^ Rb``fŽk 2-02 pb a^ lqo^ abjlpqo^`fŽk-

Page 187: Calculus

A`hjnom\^dji`n _` gjn o`jm`h\n api_\h`io\g`n nj]m` g…hdo`n /45

+&- 9RZ\`a_NPV\[R`QRY\` aR\_RZN Sb[QNZR[aNYR``\O_R YoZVaR`

Dk bpq^ pb``fŽk abjlpqo^jlp bi qblobj^ 2-0 nrb a^ i^p obdi^p crka^jbkq^ibpm^o^ `^i`ri^o iŒjfqbp ab prj^p+ molar`qlp+ u `l`fbkqbp- Klp ob`roplp ^idb_o^f`lpmofk`fm^ibp nrb pb rqfifw^k bk i^ abjlpqo^`fŽk plk i^p alp molmfba^abp ab ilps^ilobp ^_plirqlp nrb pb jbk`flk^olk bk i^p Rb``flkbp 0 3-7 X 03-8- '0( i^ abpf,dr^ia^a qof^kdri^o+ nrb ^cfoj^ nrb h] * ^h y h]h* E\f m^o^ `r^ibpnrfbo^ [ v ]ob^ibp+ u '1( i^ fdr^ia^a h] \f < h]hh^hnrb bpq^_ib`b nrb bi s^ilo ^_plirql ab rkmolar`ql bp bi molar`ql ab s^ilobp ^_plirqlp-

A`hjnom\^dji`n _` 'f( ` 'ff(- Orbpql nrb i^p alp fdr^ia^abp9

ifjc'u( < = v ifj Xa&s' + =Y < N

plk `ljmibq^jbkqb bnrfs^ibkqbp+ u `ljl pb qfbkb

a&s' * b&s' + %= * >& < Xa&s' + =Y * Xb&s' + >Y )

_^pq^ abjlpqo^o i^p fdr^ia^abp 'f( b 'ff( abi qblobj^ `r^kal ilp iŒjfqbp ab =u ? plk ^j_lp `bol-

RrmŽkd^pb mrbp+ nrb `%r&w N X a%r& w N `r^kal r w j+ Rb abjlpqo^oŠ bkmofjbo ird^o nrb `%r& * a%r&w N `r^kal r w j+ O^o^ biil pb qfbkb nrb mol_^onrb m^o^ `^a^ ’ = N bufpqb rk j = N q^i nrb

'2-6( Fa&s' * c't(0 ; ’ pfbjmob nrb N ; Zu , le ; i+

Rb^ ’ a^al- Orbpql nrb `%r& w N `r^kal r w j) bufpq^ rk .* = N q^i nrb

'2-7( Gb't(0; 9--1

pfbjmob nrb ; Zu , kd ; .. +

@kŠild^jbkqb+ mrbpql nrb a%r& w N `r^kal r ,6 j bufpqb rk K0 = N q^i nrb9

'2-8( Gc't(. ; z pfbjmob nrb N ; Zu , kd ; K/ †

Rf pb fkaf`^ mlo j bi jbklo ab ilp alp k•jbolp -/ u K0* bkqlk`bp+ ^j_^p fdr^i,a^abp '2-7( v '2-8( plk sŠifa^p pf N ; Gt, le ; j* v mlo q^kql+ bk sfoqra ab i^abpfdr^ia^a qof^kdri^o+ pb qfbkb9

Fa&s' * b&s'/ x Fa&s'- * .d'u(0 ; 1 * 1 < ’ -

+ +

Page 188: Calculus

057 Cpi^dji`n ^jiodip\n

Dpql abjrbpqo^ '2-6( nrb+ ^ pr sbw+abjrbpqo^ 'f(- K^ abjlpqo^`fŽk ab 'ff( bp `lj,mibq^jbkqb ^kŠild^+ p^isl nrb bk bi •iqfjl m^pl pb bjmib^ i^ abpfdr^ia^aE,%r&* a%r& 7777880.'t(0 * Gc't(h-

A`hjnom\^d‡i _` 'fff(- RrmŽkd^pbnrb pb e^ abjlpqo^al 'fff( bk bi `^pl m^o,qf`ri^o bk nrb rkl ab ilp iŒjfqbpbp N- Dkqlk`bp bi `^pl dbkbo^i obpriq^ cŠ`fijbkqbab bpqb`^pl m^oqf`ri^o+ ljl pb abar`b ab i^ pfdrfbkqb fdr^ia^a9

a&s'b&s' + >? < a&s' Xb&s' + ?Z * ?Xa&s' + >Z ,

Di `^pl m^oqf`ri^o fjmif`^ nrb `^a^ q‹ojfkl abi pbdrkal jfbj_ol qfbka^ ^ N`r^kal r w m v bk sfoqra ab i^ molmfba^a 'f( i^ prj^ ab ilp alp q‹ojfklp qfbkabq^j_f‹k ^ N- Olo q^kql+ _^pq^ pŽil mol_^o 'fff( bk bi `^pl bk nrb rkl ab ilpiŒjfqbp+mlo bgbjmil >) pb^ N-

RrmŽkd^pbnrb ,%r& w = W a%r&w N `r^kal r w7 m- Rb qo^q^ab mol_^o nrb,%r&$ a%r&w N `r^kal r w m- O^o^ biil pb e^ ab sbo nrb a^al rk k•jbol mlpf,qfsl ` * bufpqbrk j = N q^i nrb

'2-0/( Fa&s'b&s'G ; ’ pfbjmob nrb N ; Zu, kd ; j ,

Orbpql nrb ,%r& w > `r^kal r w m+bufpqbrk ./ q^i nrb

'2-00( Fa&s' + >g ; 0 pfbjmob nrb N ; Zu, le ; .. †

O^o^ q^i r) qbkbjlp 0.'u(0 < E,%r&* = * ?h 9999::E,%r&* ?h * E=E; 0 * G?G-v mloq^kql

'2-01( Fa&s'b&s'G < Fa&s'ggb&s'/ ; 'i * G?G(Gc't(h-

X^ nrb a%r&w N `r^kal r w m+m^o^ qlal ’ = N bufpqbrk K0 q^i nrb

'2-02(`

Gc't(0 ; 0 * G?G pfbjmob nrb N ; Zu, le ; K/)

Olo `lkpfdrfbkqb+ pf ii^j^jlp j ^i jbklo ab ilp alp k•jbolp -/ u K0 bkqlk`bp i^palp abpfdr^ia^abp '2-01( v '2-02( plk sŠifa^p pfbjmob nrb N ; Er * le ; j* vm^o^ q^i s^ilo ab s abar`fjlp '2-0/(+ 0/ nrb `ljmibq^ i^ abjlpqo^`fŽk ab 'fff(-

A`hjnom\^d‡i _` 'fs(- Orbpql nrb bi `l`fbkqb ,%r&. a%r& bp bi molar`ql ab,%r&E> mlo >Ea%r& _^pq^ abjlpqo^o nrb >E a%r&w 0 `r^kal r w m v irbdl ^mif,`^o 'fff(- Rb^ b%r& < a%r&,>) mlo il nrb b%r&w 0 `r^kal r w m+u pb nrfbob ab,jlpqo^o nrb .,b%r& z•0 `r^kal r w m-

Page 189: Calculus

Be`m^d^djn 058

C^al ` = N+ pb qo^q^ ab sbo pf bufpqb rk K = N q^i nrb

'2-03(0]0] , 0 H ; ’ pfbjmob nrb N ; Zu, le ; _ -

b%r&

K^ afcbobk`f^ pb mrbab bp`of_fo `ljl pfdrb9

'2-04( 0]0] 0 0] Eb%r&* 00b%r&* * Gd't(0 -

Orbpql nrb b%r&w 0 `r^kal r w j pb mrbab bibdfo rk -;- q^i nrb ^j_^p abpf,dr^ia^abp9

'2-05(’

Eb%r&* 00 ;,1

u0

Eb%r&* 00 ;,1

pb p^qfpc^d^k pfbjmob nrb N ; Er * kd ; j, K^ pbdrka^ ab bpq^p abpfdr^ia^abpfjmif`^ b%r&= p u mlo q^kql .,fb%r&. < f,b%r& ; 1 m^o^ q^ibp s^ilobp ab r+Djmib^kal bpqb obpriq^al bk '2-04( grkql `lk i^ mofjbo^ abpfdr^ia^a '2-05(+ l_qb,kbjlp '2-03(- Dpql `ljmibq^ i^ abjlpqo^`fŽk ab 'fs(-

2-5 Dgbo`f`flp

Dk ilp Dgbo`f`flp abi 0 ^i 0/+ `^i`ri^o ilp iŒjfqbpv bumif`^o `rŠibp e^k pfal ilp qblobj^prqfifw^alp bk `^a^ `^pl-

00- ifj!1&

rw/T

s0 + \0

7- ifj 1 1 1&rw[T * \s)\

\ &{5N-

03s1 * 11- ifj 64 6 ‘

rwK T * 1

s0+ 3

2-ifj,,-rw/ s + 1

0s/* 1s * 0

3-ifj,,,,,s + 0

bi * c'0 + '1

4- ifj cN_>

8- ifj q^k o,oxj

0/- ifj 'pbk /. * .0 `lp 2.&+nwK

+ Erf00- ifj ,-

sxL) s

r/ * [/

5- ifj 1 1 &rwK T * 0\s * \

\ &{5N-

- Gth01- ifj , -

s -/, s,r/

02- ifj ,-sxL) s

,r/03-ifj ,-

sxL+ s

s0+ \0

6- ifj 1 1&\+L U * 0\s * \

s &{5N-

Page 190: Calculus

06/ Cpi^dji`n ^jiodip\n

Tqfifw^o i^ obi^`fŽk ifj 'pbk r&,r m^o^ bpq^_ib`bo i^p fdr^ia^abp ab ilp Dgbo`f`flp abi04 ^i 1/- n,,*M

pbk /r04- ifj ,, < 1-

(&''%* B

pbk 3s + pbk 1s)0& YVZ%%%%%% 2 *&

s

q^k /r05- ifj,- , <1-

-+,,*/ pbk T

pbk 3s06- ifj ,, < 4-

++,,*/ pbk('

n`is + pbk \08- ifj ,,,,, < `lp^-

(&''%) U +\

0 , `lp s<q•1/-0fj

s0!+,l

10- Cbjlpqo^o nrb ifjK'%:

0,y,,,,,, < -[0

XFi_d^\^d4i8 'h , u&:('h* u&:(< 0, o+Y

11- Tk^ crk`fŽk EbpqŠ abcfkfa^ `ljl pfdrb9

xpbks

a&s' :\s * ]

pf s x `+

pf s< ^ *

pfbkal \* ]* ` `lkpq^kqbp- Rf ] X ` bpqŠk a^alp+ e^ii^o qlalp ilp s^ilobp ab \ 'pf bufpqb^idrkl( m^o^ ilp nrb Ebp `lkqfkr^ bk bi mrkql s < `-

12- Qbplisbo bi Dgbo`f`fl 11 pf Epb abcfkb ab bpqb jlal9

x1 `lp s

%r&:`\s0 * ]

pf s x `+

pf s< `-

13- ƒDk nr‹ mrkql plk crk`flkbp `lkqfkr^p i^ q^kdbkqb v i^ `lq^kdbkqb>14- Rb^ E%r&< %nar&,r pf r w N- Dp_lw^o i^ doŠcf`^ ab E `loobpmlkafbkqb ^ ilp fkqbos^ilp

pbjf^_fboqlp Z, $EP)N( X 'N+ $EPY+ƒPr‹ ib l`roob ^ E%r& r^kal r w N> ƒOrbab abcf,kfopb .%-&ab jlal nrb E pb e^d^ `lkqfkr^ bk N>

15- DpqbDgbo`f`fl lcob`b lqo^ abjlpqo^`fŽk ab i^ `lkqfkrfa^a ab i^p crk`flkbp pbkl v `lpbkl-^( K^ abpfdr^ia^a Zpbku\ ; Zu\+sŠifa^ m^o^ /; Gth; x$EP)crb abjlpqo^a^ bk bi Dgbo,`f`fl 23 ab i^ Rb``fŽk 1-7- Tqfifw^oi^ m^o^ abjlpqo^o nrb i^ crk`fŽk pbkl bp `lkqfkr^ bk N-_( G^`bo rpl ab i^ m^oqb ^( v ab i^ fabkqfa^a `lp /r < 0 , 1 pbk, r m^o^ abjlpqo^o i^`lkqfkrfa^a abi `lpbkl bk N-`( Tqfifw^o i^p cŽojri^p ab ^af`fŽk m^o^ pbk %r * b& v `lp %r * b& m^o^ abjlpqo^o nrb i^pcrk`flkbp pbkl u `lpbkl plk `lkqfkr^p bk `r^inrfbo s^ilo s ob^i-

16- K^ cfdro^ 2-4 jrbpqo^ rk^ mlo`fŽk ab i^ doŠcf`^ ab i^ crk`fŽk Eabcfkfa^ `ljl pfdrb9

0a&s' < pbk,

spf s94 N-

O^o^ r < 0.'k&HS(+pfbkal h bkqbol+ qbkbjlp pbk %f,r& < pbk %h$EP&< N- Dkqob alp ab bplpmrkqlp+ i^ crk`fŽk ^p`fbkab e^pq^ * 0 v _^g^ lqo^ sbw e^pq^ N l _fbk abp`fbkab ^ , 0X srbisb ^ pr_fo ^ N- Olo `lkpfdrfbkqb+ bkqob `r^inrfbo^ ab bplp mrkqlp u bi lofdbk+ i^`ros^ mobpbkq^fkcfkfq^plp`fi^`flkbp- Dpql prdfbob nrb ilp s^ilobp ab i^ crk`fŽk kl qfbkabk^ kfkd•k s^ilo cfgl `r^kal s x N- Cbjlpqo^o nrb kl bufpqb kfkd•k s^ilo ob^i = q^i

Page 191: Calculus

Bd`m^d^djn 060

EHFTQ@ 2-4 `%r&< pbk %f,r& mcU!! N- Amn[ `oh]cƒh _m ^cm]ihncho[ _h N [ohko_ m_^_`ch[ `%K&+

nrb `%r&w = `r^kal r w N- Dpql abjrbpqo^ nrb kl bp mlpf_ib abcfkfo `%K&ab j^kbo^nrb ` pb^ `lkqfkr^ bk N-

WEh^c][]cƒh7 Rrmlkbo nrb bufpq^ rk q^i = v l_qbkbo rk^ `lkqo^af``fŽk-\

17- O^o^ U!! N+pb^ `%r& < Wf,rY) abpfdk^kal mlo WnYbi j^vlo bkqbol 99:n+So^w^o i^ doŠcf`^ab . m^o^ ilp fkqbos^ilp Z,1+ *cY v Zf+1\- ƒPr‹ ib l`roob ^ E%rf `r^kal r w N

qlj^kal s^ilobp mlpfqfslp> ƒv qlj^kal s^ilobp kbd^qfslp> ƒOrbab abcfkfopb .%-& m^o^nrb . pb^ `lkqfkr^ bk N>

18- G^`bo il jfpjl nrb bk bi Dgbo`f`fl 17+ `r^kal E%r&< ',0('0.!&\ m^o^ r!! N-

2/- Kl jfpjl nrb bk bi Dgbo`f`fl 17+ `r^kal E%r&< r%* 0(Z0.!&\ m^o^ r !! N-

20- C^o rk Dgbjmil ab -rk^ crk`fŽk `lkqfkr^ bk rk mrkql ab rk fkqbos^il v afp`lkqfkr^ bkilp abjŠp mrkqlp abi fkqbos^il+ l mol_^o nrb kl bufpqb rk^ q^i crk`fŽk-

20- C^o rk bgbjmil ab rk^ crk`fŽk `lkqfkr^ bk rk mrkql ab rk fkqbos^il v afp`lkqfkr^ bkilp abjŠp mrkqlp abi fkqbos^il+ l mol_^o nrb kl bufpqb rk^ q^i crk`fŽk-

21- Rb^ E%r&< r pbk %f,r& pf U!! N- Cbcfkfo .%-& ab j^kbo^ nrb . pb^ `lkqfkr^ bk N-

22- Rb^ / rk^ crk`fŽk q^i nrb c,%o& * g&q'g889hq, re m^o^ qlalp ilp s^ilobp p v q ab rk fkqbo,s^il W[) \Y+^( Ool_^o nrb . bp `lkqfkr^ bk `^a^ mrkql ab W[) \Y+_( Rrmlkfbkal nrb . pb^ fkqbdo^_ib bk W[) \Y) abjlpqo^o nrb

G&] G &c + \'0

G-&s' _s + &] + \'a&\' x ,1, -

b( LŠp dbkbo^i- Cbjlpqo^o nrb m^o^ `r^inrfbo _ ab W[) \Y) pb qfbkb

Gaa&U' ^r * %b* D'a&@' x %bw \'0 ,

Page 192: Calculus

.4/ Boh]cih_m ]ihncho[m

2-6 ;b[PV\[R` P\Z]bR`aN` e P\[aV[bVQNQ

@ m^oqfoab rk^p crk`flkbp a^a^p mlabjlp `lkpqorfo krbs^p crk`flkbp mlo^af`fŽk+ pr_qo^``fŽk+ jriqfmif`^`fŽk u afsfpfŽk- Dk bpq^ pb``fŽk bumlkbjlp rkkrbsl mol`bafjfbkql m^o^ `lkpqorfo crk`flkbp jbaf^kqb rk^ lmbo^`fŽk `lkl`fa^mlo bi klj_ob ab `ljmlpf`fŽk- U^jlp ^ sboil bk rk bgbjmil-

Rb^ `%r& < pbk 't!(+ O^o^ `^i`ri^o `%r&) mofjbol bibs^jlp r ^i `r^ao^al virbdl qlj^jlp bi pbkl ab r0

Š @pŒmrbp+ %r& pb l_qfbkb `lj_fk^kal lqo^p alp crk,`flkbp+ i^ crk`fŽk bibs^`fŽk ^i `r^ao^al v i^ crk`fŽk pbkl- Rf mlkbjlp p%r& < r0

v o%r& < pbk r) mlabjlp bumobp^o%r& bk crk`fŽk ab o v ab p bp`of_fbkal

_#[$ << XEY#[$G(

Cb`fjlp nrb ` obpriq^ ab i^ `ljmlpf`fŽk ab p v q 'bk bpqbloabk(- Rf `ljmlkbjlpp v o bk bi loabk fksbopl+ l_qbkbjlp rk obpriq^al afpqfkql+pWo%r&Y< 'pbk U'0,

Dpql bp+m^o^`^i`ri^o pWo%r&Y)qlj^jlp mofjbol bi pbkl ab r v irbdl bi `r^ao^alabi pbk s,

Olabjlp ^elo^ `ljbkq^o bpqb mol`bpl `lk j^vlo dbkbo^ifa^a- Rb^k p v qalp crk`flkbp a^a^p `r^ibpnrfbo^- K^ `ljmrbpq^ l i^ `ljmlpf`fŽk ab p v q 'bkbpqbloabk( pb abcfkb `ljl i^ crk`fŽk ` m^o^i^ `r^i

`%r& < oWpur&Y 'pb ibb+yo ab p ab u‚(-

Dp ab`fo+ m^o^ `^i`ri^o bi s^ilo ab ` bk r mofjbol pb `^i`ri^ p%r& v irbdl pb`^i`ri^ o bk bi mrkql p%r&+M^qro^ijbkqb nrb m^o^nrb bpqb`Ši`ril qbkd^ pbkqfal+bp kb`bp^ofl nrb ilp s^ilobp ab p%r& bkqobk bk bi aljfkfl ab i^ crk`fŽk o) v `bpq^oŠpŽil abcfkfa^ bk ^nrbiilp mrkqlp r m^o^ ilp `r^ibp p%r& bpqŠ bk bi al,jfkfl ab p,

Olo bgbjmil+ pf o%r& < Ss v p%r&< 0 , r0* i^ `ljmrbpq^ ` bpqŠa^a^ mlo

`%r& < y- N_p‹osbpb nrb p%r& bpqŠ abcfkfa^ m^o^ qlal k•jbol ob^i r)

jfbkqo^p nrb o bpqŠ abcfkfa^ pŽil m^o^ r88777N- Olo q^kql+ i^ `ljmrbpq^ ` bpqŠabcfkfa^ pŽil m^o^ ^nrbii^p s q^ibp nrb 0 , s0

99888N-Eloj^ijbkqb+ `%r& pb l_qfbkb prpqfqrvbkal r mlo p%r& bk i^ bumobpfŽko%r&+

Olo bpq^ o^wŽk i^ crk`fŽk ` pb fkaf`^ ^idrk^p sb`bp mlo ` < o%p& 'nrb pb ibbyo ab p|&+ Nqo^ klq^`fŽk bjmib^a^ m^o^ fkaf`^o `ljmlpf`fŽk bp9 ` < o l p 'nrbpb ibb o `Œo`ril p& v nrb qfbkb rk^ ^k^ildŒ^ `lk i^ klq^`fŽk ab molar`ql o +p+

Dk bcb`ql+ pb sboŠ ^ `lkqfkr^`fŽk nrb i^ lmbo^`fŽk ab `ljmlpf`fŽk qfbkb ^idr,k^p ab i^p molmfba^abpab i^ jriqfmif`^`fŽk-

K^ `ljmrbpq^ ab qobpl jŠp crk`flkbp pb mrbab e^ii^o `ljmlkfbkal alp+bi obpriq^al `lk i^ qbo`bo^u ^pŒpr`bpfs^jbkqb- @pŒ+i^ crk`fŽk ` a^a^ mlo9

a&s' << `lp Xn`i&s/&Y

Page 193: Calculus

Boh]cih_m ]igjo_mn[m v ]ihnchoc^[^ 062

bp i^ `ljmlpf`fŽk ` < o l %p l q& alkab9

o%r&< `lp r) p%r&<pbku+ v q%r&< r0Š

N_p‹osbpb nrb i^ jfpj^ ` pb mrbab l_qbkbo `ljmlkfbkal o v p$ mofjbol v i^`ljmrbpq^ o l p `lk t+ bp ab`fo ` < %ol- p& l r, Dk bpqbbgbjmil pb `rjmib i^ f_s

[mi]c[ncp[ ab i^ `ljmlpf`fŽk nrb bk cloj^ dbkbo^i bp9

'2-06( p l &ql r' < &pjq' l T

`r^ibpnrfbo^ nrb pb^k i^p crk`flkbp p* q* t pfbjmob nrb qbkd^ pbkqfal cloj^oi^p `ljmrbpq^p nrb ^m^ob`bk bk i^ fdr^ia^a- Di ib`qlo sboŠ nrb i^ abjlpqo^`fŽkab '2-06( bp rk bgbo`f`fl fkjbaf^ql-

Rb l_pbos^oŠ nrb i^ f_s ]ihgon[ncp[ o l p < p l o) kl bp pfbjmob sŠifa^ bki^ `ljmlpf`fŽk- Olo bgbjmil+ pf o%r& < pbk r v p%r& < r0

* i^ `ljmrbpq^ ` < o l p

bpqŠa^a^ mlo `%r& < pbk r0 Znrb pfdkfcf`^ pbk %r0'Z jfbkqo^p nrb i^ `ljmlpf`fŽk

d < p l o bpqŠa^a^ mlo a%r& < pbk! r Znrb pfdkfcf`^ 'pbk U'0Z,Cbjlpqo^objlp ^elo^ rk qblobj^ nrb klp af`b nrb i^ molmfba^a ab i^ `lkqf,

krfa^a pb `lkpbos^ bk i^ lmbo^`fŽk ab `ljmlpf`fŽk- Blk j^vlo mob`fpfŽk+qbkbjlpbi pfdrfbkqb

RCMPCK? 2-4- Oojihc_h^i ko_ p _m]ihncho[ _h j u ko_ o _m]ihncho[ _h k)

mc_h^i k < p%j&) f[ `oh]cƒh ]igjo_mn[ ` < o l p _m ]ihncho[ _h j+

@_gimnl[]cƒh+ Orbpql nrb o bp `lkqfkr^ bk k) m^o^qlal bkqlokl J.Wo%k&Y

bufpqbrk bkqlokl Jck& q^i nrb

'2-07( o%s&C J.Wo%k&Y pfbjmob nrb s C JI& +

Obol k < p%j& v p bp `lkqfkr^ bk j) ab jlal nrb m^o^ bi bkqlokl J0%k& bufpqblqol bkqlokl J ^Bm( q^i nrb

'2-08( S&U' D J0%k& pfbjmob nrb r D Jcj& +

Rf mlkbjlp u < p%r& v `lj_fk^jlp '2-07( `lk '2-08(+ bk`lkqo^jlp nrb m^o^qlal bkqlokl J.%oWpuj&Y&_rcmn_ rk bkqlokl J1%j& q^i nrb

QWp%r&YC J.%oWp%j&Y&pfbjmob nrb r C J0%j&)

l+ af`el ab lqol jlal+ mrbpql nrb `%r& < oWp%r&Y)

`%r& C J.W`%j&Y pfbjmob nrb r C Jcj& +

Dpql pfdkfcf`^nrb ` bp `lkqfkr^ bk j) `ljl pb ^cfojŽ-

Page 194: Calculus

)/, Cpi^dji`n ^jiodip\n

DIDLOKN 0- Rb^ `%r& < pbk r0Š Dp i^ `ljmlpf`fŽk ab alp crk`flkbp `lkqf,

kr^p m^o^qlal s^ilo ab i^ s^of^_ib mlo il nrb ` bp `lkqfkr^ m^o^qlal s,

DIDLOKN 1- O_[,_r& < z < oWp%r&Y)pfbkal o%r&< y+ p%r&: .*r0Š

K^ crk`fŽk p bp `lkqfkr^ pfbjmob+mbol o pŽil il bp m^o^mrkqlp r ƒ N- Krbdl ` bp`lkqfkr^ bk ^nrbiilp s^ilobp r m^o^ilp `r^ibp p%r&ƒ N+bpql bp+bk qlalp ilp mrkqlpnrb p^qfpc^`bkr0

x 0-

+&0 :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 0/+ i^p crk`flkbp ` v d bpqŠk abcfkfa^p mlo i^p cŽojri^p a^a^p-Rf kl pb af`b il `lkqo^ofl+ ilp aljfkflp ab ` v d `lkpfpqbk bk qlalp ilp k•jbolp ob^ibp- Olk,d^jlp b%r& < `Wa%r&Ypfbjmob nrb a%r& bpq‹ bk bi aljfkfl ab `+ Dk `^a^ `^pl+ mob`fp^o bi al,jfkfl ab b v a^o rk^ l jŠp cŽojri^p m^o^ i^ abqbojfk^`fŽk ab b%r&+

0- X&s' < s0 + 0s* b&s' < s * 0-

1- X&s' ;s * 0+ b&s' < s0 + 0s,

1, X&s' < z pf s x N+ b&s' < s0Š

3- X&s' < z pf s zN+ b&s' < +s0Š

3, X&s' < s0* b&s' <z pf s zN-

4, X&s' < +s0* b&s' <z pf s zN-

5,y&s' < pbk s* b&s' <z pf s x N-

7- X&s' <z pf s zN+ b&s' <pbk s,

7, X&s' < RT pf s< L* b&s' < s * RT pf s< N-

/., X&s' < Us * Us pf s< L* b&s' < s * RT pf s< N-

B^i`ri^o ilp iŒjfqbp bk ilp Dgbo`f`flp abi 00 ^i 1/ u bumif`^o nr‹ qblobj^p pb ^mif`^k bk`^a^ `^pl-

01- ifok u&h* Ss,

m_h%r/ * 0(05- ifj i&

!+,0 s+0

06- ifj s pbk,-s

s0 * 700-ifj z3&

f%%*? '

pbk 'q^k o'02-hei,,,-

n*K pbk o

pbk '`lp r&),& YVZ %%%%&

!+,!.1 `lp spbk 'o , 6S(

)-& hei,,,,-o+%! o + 6S

0 , `lp 0s07-ifj,,,,

[,

z,z08- ifj ,,,,,,, -

s

0 , TG, 2s0

1/- ifj ,,,,,[0

Page 195: Calculus

Q`jm`h\ _` ?jgu\ij k\m\ g\n api^dji`n ^jiodip\n 064

10- Rb^k ` v d alp crk`flkbp abcfkfa^p `ljl pfdrb9

r * Erfy&s' < ,,, m^o^ qlal s*

1a%r&< `r

gs,m^o^ s ; N+

m^o^ s 19 N

G^ii^o rk^ cŽojri^ 'l cŽojri^p( m^o^ bi `Ši`ril ab i^ crk`fŽk `ljmrbpq^ b%r& < `Wa%r&Y+ƒO^o^ nr‹ s^ilobp ab s bp `lkqfkr^ c=

11- Qbplisbo bi Dgbo`f`fl 10 `r^kal ` v d pb abcfkbk abi jlal pfdrfbkqb9

y&s' < wypf Erf777780 +

pf Zu\ = 0+ x1 , s0

a%r& < 1pf Erf777781+

pf Erf = 1-

12- Qbplisbo bi Dgbo`f`fl 10 `r^kal b%r& < aW`%r&Y+

2-8 Sblobj^ ab Aliw^kl m^o^i^p crk`flkbp `lkqfkr^p

Dk bi obpqlab bpqb ^mŒqrilpb afp`rqfoŠk ^idrk^p molmfba^abpab i^p crk`flkbp`lkqfkr^p nrb pb rp^k `lk cob`rbk`f^- Lr`e^p ab bii^p ^m^ob`bk `ljl qofsf^ibp`r^kal pb fkqbomobq^kdblj‹qof`^jbkqb+ mlo 0/ nrb ^idrklp pb fk`ifk^k ^ ^`bm,q^oi^p`ljl bsfabkqbp-Rfk bj_^odl+ bp fjmloq^kqb mlkbo ab j^kfcfbpql nrb bpq^pmolmfba^abpkl qfbkbk bk pŒrk^ bsfabk`f^ prmboflo ^ i^ jfpj^ abcfkf`fŽk ab `lk,qfkrfa^a u nrb mlo q^kql e^k ab pboabjlpqo^a^p pf pb nrfbob ^mif`^oi^p`lk `fboq^dbkbo^ifa^a- K^p abjlpqo^`flkbp ab bpq^pmolmfba^abpprbibk e^`bo rpl abi ^uflj^abi buqobjl prmbofloabi pfpqbj^ ab ilp k•jbolp ob^ibp-

Abok^oal Aliw^kl '0670,0737(+ p^`boalqb `^qŽif`l nrb efwl ^mloq^`flkbpfjmloq^kqbp ^ i^p L^qbjŠqf`^p bk i^ mofjbo^ jfq^a abi pfdil WHW+crb rkl abilp mofjbolp bk ob`lkl`bo nrb jr`e^p ab i^p molmfba^abppl_ob crk`flkbp `lk,qfkr^p nrb m^ob`Œ^kl_sf^p obnrboŒ^krk^ abjlpqo^`fŽk- Rrp abjlpqo^`flkbpobcbobkqbp `lkqfkrfa^a crbolk mr_if`^a^p bk 074/ bk pr fjmloq^kqb l_o^ mŽpqrj^M\m\_je\n _`g diadidoj, Tkl ab prp obpriq^alp `lkl`fal mlo bi o`jm`h\ _` ?jgu\ijpb mlkb ab j^kfcfbpql bk i^ cfdro^ 2-5 alkab pb jrbpqo^ i^ doŠcf`^ab rk^ crk`fŽk`lkqfkr^ `+ K^ doŠcf`^bpqŠmlo ab_^gl abi bgb s bk bi mrkql \ u mlo bk`fj^ abibgbs bk bi mrkql \+ Di qblobj^ ab Aliw^kl ^cfoj^ nrb i^ `ros^ e^ ab `loq^o ^i bgb^idrk^ sbw bkqob \ v \+ Dpq^molmfba^a pb mrbab bkrk`f^o ofdrolp^jbkqb `ljlpfdrb9

SDNQDL@ 2-5- SDNQDL@ CD ANKY@MN- P`\ a ^jiodip\ `i ^\_\ kpioj _`gdio`mq\gj ^`mm\_j X\* ]Z W npkjib\hjn lp` a`\' u a&]' od`i`i ndbijn jkp`nojn, Bsdn+o` `ioji^`n kjm gj h`ijn pi ` `i `g dio`mq\gj \]d`moj &\*]' o\g lp` a&^' < l-

A^p^objlp krbpqo^ abjlpqo^`fŽk abi qblobj^ ab Aliw^kl bk i^ pfdrfbkqbmolmfba^aab i^p crk`flkbp `lkqfkr^p nrb bpq^_ib`bjlp ^nrŒ`ljl SB qblobj^-

Page 196: Calculus

065 Boh]cih_m ]ihncho[m

SDNQDL@ 2-6- BNMRDQU@BHˆM CDK RHFMN CD K@R ETMBHNMDR BNMSHMT@R-

O_[ ` ]ihncho[ _h _ v mojiha[gim ko_ `%]&:.: N- Arcmn__hnih]_m oh chn_lp[fi%_* 04+ _ * 04( _h _f ko_ ` nc_h__f gcmgi mcahi ko_ `%]&+

@_gimnl[]c3h ^_f n_il_g[ 2-6- RrmŽkd^pb`%]&= N- Dk sfoqra ab i^ `lkqf,krfa^a+ m^o^`^a^ ` = N bufpqbrk 04 = N q^i nrb9

'2-1/( a&^' + ’ ; a&s' ; a&^' * ’ pfbjmob nrb ` + 04 ; u ; ` * 04 ‘

Slj^kal bi 04 `loobpmlkafbkqb ^ _ < `%]&,/ 'bpq^ ’ bp jimcncp[& bkqlk`bp '2-1/(pb qo^kpcloj^ bk

oa&^' :a&s' ; da&^' pfbjmob nrb ` + 04 ; u ; ` * a-

[

Rg`'

a&_& ***********

` + / `

da&`'

EHFTQ@ 2-5 P_il_g[ ^_ >ift[hi+ EHFTQ@ 2-6 =ko• `%r&= M j[l[ r jlƒrcgi[ _ jo_m `%]& = N-

'U‹^pb cfd- 2-6(- Cb ^nrŒ pb abar`b nrb `%r&= N bk bpqbfkqbos^il u mlo q^kql+`%r&v `%]& qfbkbk bi jfpjl pfdkl- Rf `%]&; N pb qlj^ 04 `loobpmlkafbkqb ^

’ < ,0 `%]& v pb iibd^ ^ i^ jfpj^ `lk`irpfŽk-

Kjo\8 Rf bufpqb `lkqfkrfa^a ^ rk i^al ab `* bkqlk`bp bufpqb bi `loobpmlkafbkqbfkqbos^il rkfi^qbo^i W_) _ * 5( l 'b , 5+bi bk bi `r^i ` qfbkb bi jfpjl pfdkl nrb `%_&+

@_gimnl[]cƒh ^_f n_il_g[ ^_ >ift[hi+ O^o^ cfg^ofab^p+prmŽkd^pb`%[&; Nv a&]' = N q^i `ljl pb e^ eb`el bk i^ cfdro^ 2-5- Orbab e^_bo jr`elp s^ilobpab r bkqob[ v \ m^o^ilp `r^ibp `%r&< N- Rb qo^q^^nrŒab bk`lkqo^o ohi v bpqlpb e^oŠ abqbojfk^kal bi j^vlo r m^o^ bi `r^i `%r&< N- O^o^ biil+ pb^ R bi`lkgrkql ab qlalp ilp mrkqlp abi fkqbos^il W[)\Y m^o^ ilp `r^ibp %r&QN- G^vmlo 0/ jbklp rk mrkql bk R mrbpql nrb `%[& ; N- Olo q^kql+R bp rk `lkgrkqlkl s^`Œl- R bpqŠ^`lq^al prmboflojbkqb mrbpql nrb qlalp ilp mrkqlp ab R bpqŠkbk W[)\Y) v mrbpql nrb qlal `lkgrkql kl s^`Œl ab k•jbolp ob^ibp nrb bpqŠ^`l,

Page 197: Calculus

P_il_g[ ^_f p[fil chn_lg_^ci j[l[ `oh]cih_m ]ihncho[m 066

q^al prmboflojbkqb qfbkb rk _rnl_gi moj_lcil) ^ ‹pqb pb ib ii^j^ ]+ Rb qo^q^ababjlpqo^o nrb `%]& < N-

G^v pŽil qobp mlpf_fifa^abp9 `%]& = N+ `%]& ; N+ v `%]& < N- Rf `%]& = Ne^v rk fkqbos^il 'b , K) b * j' l 'b , j* bi pfb < \) q^i nrb `%r& bp mlpfqfsl pfs bpqŠbk bpqbfkqbos^il- Olo q^kql+ kfkd•k mrkql ab R mrbab bpq^o^ i^ abob,`e^ ab b , K) X mlo q^kql b , K bp rk^ `lq^ prmboflo abi `lkgrkql R- Obol_ * L ; _ u _ bp bi _rnl_gi prmboflo ab R- Olo q^kql i^ abpfdr^ia^a `%]& = N bpfjmlpf_ib- Rf `%]& ; N e^v rk fkqbos^il ?_ * L* _ * L' l W_)_ * i& pf _ < [) bkbi `r^i ` bp kbd^qfs^ v mlo q^kql `%r& ; N m^o^^id•k r = _) `lkqo^ bi eb`el abnrb _ bp rk^ `lq^ prmboflo ab R- Olo q^kql `%]& ; N q^j_f‹k bp fjmlpf_ib v nrb,a^ pŽil i^ mlpf_fifa^a `%]& < N- @abjŠp [ ; _ ; ] mrbpql nrb `_[& ; N t`%\& = l- Blk il `r^i nrba^ abjlpqo^al bi qblobj^ ab Aliw^kl-

2-0/ Sblobj^ abi s^ilo fkqbojbafl m^o^ crk`flkbp `lkqfkr^p

Blkpb`rbk`f^ fkjbaf^q^ abi qblobj^ ab Aliw^kl bp bi n_il_g[ ^_f p[filchn_lg_^ci m^o^ crk`flkbp `lkqfkr^p 's‹^pb cfdro^ 2-7(-

RCMPCK? 2-7- O_[ ]ihncho[ _h ][^[ johni ^_ oh chn_lp[fi W[) \Y+ Oc Tf9 T0 mih ^im johnim ]o[f_mkoc_l[ ^_ W[) \Y n[f_m ko_ %rf& :.:* `%r0'* i^ crk`fŽk `

nig[ ni^im fim p[fil_m ]igjl_h^c^im _hnl_ `%T&. t `%r0' jil fi g_him oh[ p_t _h_f chn_lp[fi %TE$r0',

@_gimnl[]cƒh+ RrmŽkd^pb %rF' ; %r0' v pb^ e rk s^ilo `r^inrfbo^ `lj,mobkafal bkqob`%rf& X `%r/&+ Rb^ d rk^ crk`fŽk abcfkfa^ bk WTf; r/Y `ljl pfdrb9

b&s' ;a&s' + e)

+a&s' < f

y +

\ Š--+,,,,,,!!,,,,, ----Š----,,,,,-JG,,

\ TE T/ ]

\8`%[&*$

\

EHFTQ@ 2-7 Q`jm`h\ _`g q\gjm dio`mh`_dj, EHFTQ@ 2-8 Be`hkgj `i `g lp` ij `n\kgd^\]g` `g o`jm`h\ _` ?jgu\ij,

d bp `lkqfkr^ bk `^a^ mrkql ab ZWi&U0Z X pb qfbkb9

b&sg' <a&sg' + e ; N + b&U0'<a&s0' + e = N -

Page 198: Calculus

067 Cpi^dji`n ^jiodip\n

@mif`^kal bi qblobj^ ab Aliw^kl ^ d pb qfbkba%]&< N m^o^^id•k _ bkqobUg v U0%

il `r^i pfdkfcf` F&`' < e) nrba^kal ^pŒabjlpqo^al bi qblobj^-

Kjo\8 S^kql bk bi qblobj^ ab Aliw^kl `ljl bk bi qblobj^ abi s^ilo fkqbojbafl pbprmlkb nrb ` bp `lkqfkr^ bk qlalp ilp mrkqlp abi fkqbos^il W[)\Y fk`irfalp ilp buqobjlp [ v \+O^o^ bkqbkabo mlo nr‹ bp kb`bp^of^ i^ `lkqfkrfa^a bk ilp buqobjlp \ v ] pb `lkpfabo^ i^`ros^ ab i^ cfdro^ 2-8- …pq bp `lkqfkr^ bk qlalp ilp mrkqlp ab W[) \Y bu`bmql bk [+ @ mbp^oab pbo `%[& kbd^qfs^ v `%\& mlpfqfs^ kl bufpqb kfkd•k r bk W[) \Y m^o^ bi `r^i `%r& < l-

Efk^ijbkqb pb a^ bk bpq^Rb``fŽk rk^ ^mif`^`fŽk abi qblobj^ abi s^ilo fkqbo,jbafl bk i^ `r^i pb abjrbpqo^ nrb `^a^ k•jbol ob^i mlpfqfsl qfbkbrk^ o^Œwk,pfj^+0/ `r^i v^ pb e^_Œ^fkaf`^al bk i^ Rb``fŽk 02-03- Di bkrk`f^al mob`fpl ab bpq^molmfba^a bp bi pfdrfbkqb9

RCMPCK? 2-8- Rf i `n pi `io`mj kjndodqj t pf \ = N+ sdno` pi `io`mj mlpf,odqj t n‡gj pij ] o\g lp`%]! < \,

A`hjnom\^d‡i, Rb^ rk k•jbol b = 0 v q^i nrb N ; \ ; b v `lkpfa‹obpb i^crk`fŽk F abcfkfa^ bk bi fkqbos^il ZN+b\ mlo F&s' < u!+ Dpq^crk`fŽk bp `lkqfkr^bk ZN+b\ v bk ilp buqobjlp pb qfbkb /&.';. v F&`';`iŠ Orbpql nrb L:\:`:`!bi k•jbol [ a^al bpqŠ ljmobkafal bkqobilp s^ilobp ab i^ crk`fŽk .%-& v E%_&+Oloq^kql+bk sfoqra abi qblobj^ abi s^ilo fkqbojbafl+ pb qfbkbE%r&< [ m^o^^id•k r

bk ZN+b\+ pb^ s < \+ Dpql abjrbpqo^ i^ bufpqbk`f^ ab mlo 0/ jbklp rk mlpfqfsl] q^i nrb ]! < \, Ml mrbab e^_bo jŠp nrb rkl mrbpql nrb Ebp `ob`fbkqb bk pbk,qfal bpqof`qlbk ZN+_Y) `lk 0/ `r^i nrba^ abjlpqo^al bi qblobj^-

2-00 Dgbo`f`flp

0- Rb^ ` rk mlifkljfl ab do^al h bk r) `%r& < EwwK?eTe) q^i nrb bi mofjbol v bi •iqfjl`lbcf`fbkqbp Ak v ]i qfbkbk pfdklp lmrbpqlp- Cbjlpqo^o nrb `%r& < N mlo il jbklp m^o^rk s^ilo mlpfqfsl ab s*

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Page 199: Calculus

Af jli]_mi ^_ chp_lmcƒh 068

_ bk ZN+0\ m^o^ bi `r^i `%]& < `- S^i mrkql pb ii^j^ rk johni `cdi ab `+ Di obpriq^al abbpqbDgbo`f`fl bp rk `^pl m^oqf`ri^o abi n_il_g[ ^_f johni `cdi ^_ >liq_l+ WEh^c][]cƒh7@mif`^o bi qblobj^ ab Aliw^kl ^ a%r& < `%r& * r+Y

5- C^a^ rk^ crk`fŽk ` ab s^ilobp ob^ibp `lkqfkr^ bk bi fkqbos^il `boo^al W[) \Y+ Rrmlkfbkalnrb `%[& 94 [ v nrb `%\& w \) abjlpqo^o nrb ` qfbkb rk mrkql cfgl bk W[) \Y+ 'U‹^pb Dgbo,`f`fl 4-(

+&)* :Y ]_\PR`\ QRV[cR_`Vp[

Dk bpq Rb``fŽk pb bpqraf^ lqol j‹qlal fjmloq^kqb nrb pb rp^ `lk cob`rbk`f^m^o^`lkpqorfo krbs^p crk`flkbp ^ m^oqfoab lqo^p crk`flkbp a^a^p- @kqbpab abp,`of_fo bi j‹qlal `lk abq^iib+a^jlp rk pbk`fiil bgbjmil-

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a%s&< s pfdkfcf`^nrb s < y&s' ,

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Page 200: Calculus

07/ Cpi^dji`n ^jiodip\n

Di mol`bpl ab fksbopfŽk mrbab ^mif`^opb^ `r^inrfbo crk`fŽk Enrb qbkd^ i^molmfba^a ab nrb m^o^`^a^ v bk bi ob`loofal ab /* bufpqbrk plil s bk bi aljfkflab F q^i nrb F&s' < v- Dk m^oqf`ri^o+rk^ crk`fŽk `lkqfkr^ v bpqof`q^jbkqbjlkŽ,qlk^ bk rk fkqbos^il W[) \Y qfbkbbp^ molmfba^a-Dk i^ cfdro^ 2-0/ pb jrbpqo^ rkbgbjmil- Rb^k _ < E%[&) < E%\&+Di qblobj^ abi s^ilo fkqbojbafl m^o^ i^p crk,`flkbp `lkqfkr^p klp af`b nrb bk bi fkqbos^il W[) \Y) Eqlj^ qlal s^ilo `ljmobk,afal bkqob ` u ^+ @abjŠp+ kl mrbab qlj^o alp sb`bp bi jfpjl s^ilo mlonrba&U*' :.: `%r0' pfbjmob nrb U* :.: r0Š Olo `lkpfdrfbkqb+ qla^ crk`fŽk `lkqfkr^ bpqof`,q^jbkqb jlkŽqlk^ qfbkb fksbop^-

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Page 201: Calculus

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Page 202: Calculus

071

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Page 203: Calculus

Be`m^d^djn 072

O^o^ bk`^g^o bpqb obpriq^al `lk i^ kl`fŽk ab fksbop^ q^i `ljl pb bumrpl ^kqboflo,jbkqb+ pb mrbab `lkpfabo^o nrb i^ b`r^`fŽk u < s0 kl abcfkb rk^ crk`fŽk ` pfkl_jn crk`flkbp .0 u .1 alkab9

pf /9Q s 9Q` v pf , ` 9Rs 9RM -

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Page 204: Calculus

/62 Cpi^dji`n ^jiodip\n

2-05 Sblobj^ ab ilp s^ilobp buqobjlp m^o^ crk`flkbp `lkqfkr^p

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Dk i^ cfdro^ 2-02 '_(+ R bp bi fkqbos^il `boo^al Z/+1\ X a&s'< g-s pf s = N+.%-&< 0- Dk bpqbbgbjmil+ ` mlpbb rk jŒkfjl ^_plirql bk r < 1+ mbol kl qfbkbjŠufjl ^_plirql- Ml mlpbb jŠufjl ab_fal ^ rk^ afp`lkqfkrfa^a bk rk mrkqlab R-

Page 205: Calculus

Q`jm`h\ _` gjn q\gjm`n `som`hjn k\m\ Xpi^dji`n ^jiodip\n .52

Prbobjlp abjlpqo^o nrb pf R bp rk fkqbos^il `boo^al v Ebp `lkqfkr^ bkqlal R+bkqlk`bp Emlpbb rk jŠufjl ^_plirql v rk jŒkfjl ^_plirql bk R- Dpqbobpriq^al+ `lkl`fal `ljl bi qblobj^ abi jŠufjl 'jŒkfjl( m^o^crk`flkbp `lkqf,kr^p+ pb abar`foŠ `ljl rk^ pbk`fii^ `lkpb`rbk`f^ abi pfdrfbkqb qblobj^-

SDNQDL@ 2-00- SDNQDL@ CD @BNS@BHˆM O@Q@ ETMBHNMDR BNMSHMT@R- O_[E^jiodip\ `i pi dio`mq\gj ^`mm\_j X\*]Z, Bioji^`n E`n \^jo\_\ `i X\*]Z, Bnoj`n* `sdno` pi iˆh`mj a y k o\g lp` F-&s'/x a k\m\ oj_j s `i X\*]Z,

A`hjnom\^d‡i, Q^wlk^jlp mlo obar``fŽk ^i ^_proal l `lkqo^af``fŽk+ rqfif,w^kal rk^ q‹`kf`^ ii^j^a^ j‹qlal ab _fm^oqf`fŽk-Rrmlkd^jlp nrb Eij `n \^j+o\_\ bk X\*]Z, Rb^ b bi mrkql jbafl ab X\*]Z, X^ nrb Ekl bp ^`lq^a^ bkW[) \Y q^jml`l il bpqŠ mlo il jbklp bk rkl ab ilp pr_fkqbos^ilp W[) b\ lW_)\Y+ Rb^ Z^i+\fY ^nrbii^ jfq^a ab W[)\Y bk i^ nrb Ekl bpqŠ ^`lq^a^- Rf Ekl bp ^`lq^a^ bk ^j_^p jfq^abp+ pb^ Z^i+]fY i^ jfq^a fwnrfboa^ ab W[)b\- Blk,qfkrbjlp bi mol`bpl ab _fm^oqf`fŽkobfqbo^a^jbkqb+abpfdk^kal `lk W[i)g* ]i)/Z

i^ jfq^a ab X\i* ]iZ bk i^ `r^i Ekl bp ^`lq^a^+ `lk bi `lksbkfl ab bibdfoi^ jfq^afwnrfboa^ pf Ekl bp ^`lq^a^ bk ^j_^p jfq^abp- Bljl i^ ilkdfqra ab `^a^ fkqbo,s^il bp i^ jfq^a ab pr mob`babkqb+l_pbos^jlp nrb i^ ilkdfqra ab W[i* ]iZ* bp&]+\'-0h

Cbpfdkbjlp `lk > bi `lkgrkql ab ilp buqobjlp fwnrfboalp [) [f) [/) +++ ) ^pŒl_qbkfalp+ u pb^ HW bi buqobjl prmboflo ab =+ S^i mrkql HW bpqŠpfqr^al bk W[)\Y+Olo i^ `lkqfkrfa^a ab Ebk HW+ bufpqbrk fkqbos^il ab i^ cloj^ 'HW , 'g+ HW* '4( bkbi nrb

'2-14( E,%r&* .'GV(G; 0 -

Rf HW < \ bpqbfkqbos^il qfbkb i^ cloj^ X\* \ * 'g(+ v pf HW < ] qfbkbi^ cloj^%\ * 'g+\Y+ K^ abpfdr^ia^a '2-14( fjmif`^

E,%r&G; 0 * G.'GV(G+ab jlal nrb Ebp ^`lq^a^ mlo 0* G.'GV(Gbk bpb fkqbos^il- Rfk bj_^odl+ bi fkqbo,s^il W[i* ]*iZ bpqŠ`lkqbkfal bk 'HW , 'g+ HW * '4( `r^kal h bp il _^pq^kqbdo^kabm^o^ nrb &]+ \'-0i ; 'f- Olo `lkpfdrfbkqb ` q^j_f‹k bp ^`lq^a^ bk X\i* ]ŠŠZ*bk`lkqo^af``fŽk `lk bi eb`el ab nrb ` bpqŠ^`lq^a^ bk W[i* \ ††Y+ Dpq^`lkqo^af``fŽk`ljmibq^ i^ abjlpqo^`fŽk-

Rf ` bp ^`lq^a^ bk W[) \Y) bi `lkgrkql ab qlalp ilp s^ilobp E%r&bpqŠ^`lq^alprmboflob fkcboflojbkqb- Olo `lkpfdrfbkqb+ bpqb`lkgrkql qfbkbrk buqobjl prmboflou rk buqobjl fkcboflo nrb abpfdk^jlp mlo prmEu mlo fkc .) obpmb`qfs^jbkqb-Dpqlbp+bp`of_fjlp

prmc< prmva&s' G\ x s x \v) fkcc< fkc va&s' G\ x s x \v+

O^o^ `r^inrfbo crk`fŽk ^`lq^a^ qbkbjlp fkca x a&s' x prmE m^o^ qlal s bk

Page 206: Calculus

075 *Cpi^dji`n ^jiodip\n

W[)\Y+ @ `lkqfkr^`fŽk abjlpqo^jlp nrb rk^ crk`fŽk `lkqfkr^ ^i`^kw^ ilp s^,ilobp fkc ` v prm ` bk ^id•k mrkql ab W[)\Y+

SDNQDL@ 2-01- SDNQDL@ CDK L„WHLN 'L†MHLN( O@Q@ ETMBHNMDR BNMSH,

i?o- Pd a `n ^jiodip\ `i pi dio`mq\gj ^`mm\_j X\* ]Z* `sdno`i kpiojn ` v _ `iI\* ]Z o\g`n lp`

a&^' < npka v a&_' < ejbb

A`hjnom\^d‡i, A^pq^mol_^o nrb a ^i`^kw^ pr buqobjl prmboflo bk g\* ]Z,O^o^ bi buqobjl fkcboflo _^pq^ qbkbo bk `rbkq^ nrb bi buqobjl fkcboflo ab ` bpbi buqobjl prmboflo ab , `+

Rb^ L < prm`+ Rrmlkaobjlp nrb kl bufpqb rk r bk W[) \Y m^o^ bi nrba&s' < L X pb iibd^oŠ ^ rk^ `lkqo^af``fŽk- Rb^ b&s' < L , a&s', O^o^ qlal s bkX\* ]Z pboŠ bkqlk`bp b&s' = N `lk 0/ nrb i^ crk`fŽk ob`Œmol`^g-b bp `lkqfkr^bk X\* ]Z* mlkd^jlp g-b&s' ; @ m^o^ qlal s bk X\* ]Z* pfbkal @< N- Dpqlfjmif`^ nrb L , a&s' = g-@* `lk 0/ nrb a&s' ; L , g-@ m^o^qlal s ab X\* ]Z,Dpql bpqŠbk `lkqo^af``fŽk `lk bi eb`el ab nrb L bp i^ jbklo `lq^ prmboflo aba bk X\* ]Z, Olo `lkpfdrfbkqb+ a&s' < L m^o^rk s mlo 0/ jbklp bk X\* ]Z,

Jin[7 Dpqb qblobj^ abjrbpqo^ nrb pf Ebp `lkqfkr^ bk W[)\Y) bi prm Ebp pr jŠuf,jl ^_plirql+ v bi fkc Ebp pr jŒkfjl ^_plirql- Krbdl+ bk sfoqra abi qblobj^ abi s^ilofkqbojbafl+ bi ob`loofal ab F bp bi fkqbos^il `boo^al xfkc /* prm b[-

2-06 Sblobj^ ab i^ `lkqfkrfa^a rkfclojb

Rb^ ` rk^ crk`fŽk ab s^ilobp ob^ibpv `lkqfkr^ bk rk fkqbos^il `boo^al W[)\YX pb^k J&a' u h&a' ilp s^ilobp jŠufjl u jŒkfjl obpmb`qfs^jbkqb ab a bk X\* ]Z,@ i^ afcbobk`f^

J&a' + h&a'

i^ ii^j^objlp jn^dg\^d‡i ab a bk bi fkqbos^il X\* ]Z, Rb mlaoŒ rqfifw^oi^ m^i^_o^`so`ind‡i* bk ird^o ab jn^dg\^d‡i* v^ nrb bpq^m^i^_o^ qfbkb bi fk`lksbkfbkqb abprdbofo crk`flkbp lkari^kqbp- Dk qbuqlp ^kqfdrlp pb bjmib^ n\gopn*bnrfs^ibkqb i^,qfkl ab _ofk`l l p^iql: mbol klplqolp `lkpbos^objlp bi klj_ob ab lp`fi^`fŽk+ mlopbo pr rpl jrv dbkbo^ifw^al- N_pbosbjlp nrb i^ lp`fi^`fŽk ab ` bk `r^inrfbopr_fkqbos^il ab W[)\Y kl mrbab prmbo^oi^ ab ` bk W[)\Y+

Cbjlpqo^objlp pbdrfa^jbkqb nrb bi fkqbos^il W[)\Y mrbab pr_afsfafopb abjlal nrb i^ lp`fi^`fŽk ab ` bk `^a^ pr_ fkqbos^il pb^ q^k mbnrb•^ `ljl pb pbnrfbo^- Dpq^molmfba^a+bk cloj^ mob`fp^+bp i^ nrb a^ bi pfdrfbkqb qblobj^+ nrbbnrfs^ib ^i nrb loafk^of^jbkqb pb abkljfk^ o`jm`h\ _` g\ ^jiodipd_\_ pidajmh`,

SDNQDL@ 2-02- P`\ a ^jiodip\ `i pi dio`mq\gj ^`mm\_j X\* ]Z, M\m\ oj_j’ = N `sdno` pi\ k\mod^d‡i _` X\* ]Z* `i pi iˆh`mj adidoj _` np]dio`mq\gjn* o\glp` g\ jn^dg\^d‡i _` ` `i oj_j np] dio`mq\gj `n h`ijm lp` `*

Page 207: Calculus

Q`jm`h\ _` dio`bm\]dgd_\_ k\m\ api^dji`n ^jiodip\n 076

A`hjnom\^d‡i, Q^wlk^objlp mlo `lkqo^af``fŽk+ rqfifw^kal bi j‹qlal ab _f,+m^oqf`flkbp pr`bpfs^p- Rrmlkd^jlp nrb bi qblobj^ bp a\gnj, Dpql bp+ nrb m^o^ rk

`fboql D+ mlo bgbjmil m^o^ D < DN& bi fkqbos^il W[) \Y kl mrbab pbo pr_afsfafalbk rk k•jbol cfkfql ab pr_fkqbos^ilp bk `^a^ rkl ab ilp `r^ibp i^ lp`fi^`fŽk ab` pb^ jbklo nrb Dl‘ Rb^ ` bi mrkql jbafl ab X\* \Y+ Dkqlk`bp m^o^ bpb DN bi qblob,j^ bp c^ipl bk rkl mlo il jbklp ab ilp alp pr_fkqbos^ilp W[)`\ l Zb+\Y+ 'Rf biqblobj^ crbpb `fboql bk ^j_lp pr_fkqbos^ilp+ q^j_f‹k il pboŒ^bk bi fkqbos^il`ljmibql X\* \Y+& Rb^ X\! %0[ ^nrbii^ jfq^a ab X\* \Y bk i^ nrb bi qblobj^ bpc^ipl m^o^ DN& Rf bp c^ipl bk ^j_^p jfq^abp+ pb^ X\y, ]yZ i^ jfq^a fwnrfboa^X\* `Z, Qbfqbo^jlp bi mol`bpl ab _fm^oqf`fŽk+ abpfdk^kal mlo X\!)/< ]i)_ ^nrbii^jfq^a ab X\i* ]iZ bk i^ nrb bi qblobj^ bp c^ipl m^o^ DN& qbkfbkal bk `rbkq^ nrbbibdfjlp i^ jfq^a fwnrfboa^ pf bi qblobj^ bp c^ipl bk ^j_^p jfq^abp ab `[!) \))Y+MŽqbpb nrb i^ lp`fi^`fŽk ab ` bk `^a^ pr_ fkqbos^il ab W[i* ]iZ ^pŒ`lkpqorfal bpmlo il jbklp Dl-

Ki^jbjlp > ^i `lkgrkql ab buqobjlp fwnrfboalp \* \y* \u* ŠŠ, * `lkpqorfalp`ljl pb fkaf`Ž+ u pb^ j8 i^ jŒkfj^ `lq^ prmboflo ab =+ Dpqb mrkql \ bpqŠ pfqr^albk X\* \Y+ Olo i^ `lkqfkrfa^a ab ` bk ?f+) bufpqb rk fkqbos^il '~ , -) j8 * i& bkbi nrb i^ lp`fi^`fŽk ab ` bp jbklo nrb DN& 'Rf j8 < \* bpb fkqbos^il bp X\* \ * K&) upf @g, < \++ bp %\ * j* \ I-( Rfk bj_^odl+ bi fkqbos^il W[i* \iZ bpqŠ abkqol ab'| , i) 'I- * i& `r^kal i bp il _^pq^kqb do^kab m^o^ nrb %\ * \'-0h ; i) `lk ilnrb i^ lp`fi^`fŽk ab ` bk W[i* \iZ bp q^j_f‹k jbklo nrb DN& il nrb bpqŠ bk `lkqo^,af``fŽk `lk bi eb`el ab nrb i^ lp`fi^`fŽk ab ` bp mlo il jbklp DN bk W[! ) \)F+ Dpq^`lkqo^af``fŽk `ljmibq^ i^ abjlpqo^`fŽk abi qblobj^ 2-02-

+&)0 FR\_RZNQRV[aRT_NOVYVQNQ]N_N Sb[PV\[R` P\[aV[bN`

Di qblobj^ ab i^ `lkqfkrfa^a rkfclojb mrbab rqfifw^opb m^o^ abjlpqo^o nrbrk^ crk`fŽk `lkqfkr^ bk W[)\Y bp fkqbdo^_ib bk W[)\Y+

SDNQDL@ 2-03- HMSDFQ@AHKHC@C CD ETMBHNMDR BNMSHMT@R- Pd pi\ api^d‡ia `n ^jiodip\ `i oj_jn gjn kpiojn _` pi dio`mq\gj ^`mm\_j X\* ]Z* `n dio`bm\]g` `iX\* \Y+

A`hjnom\^d‡i, Di qblobj^ 2-00 abjrbpqo^ nrb a bp ^`lq^a^ bk X\* ]Z*`lk 0/ nrb ` qfbkb rk^ fkqbdo^i prmboflo+ ~Q&)v rk^ fkqbdo^i fkcboflo `Q&+Cbjlp,o^objlp nrb `Q&< cQ&+

Difg^jlp rk bkqbol J ƒ 0 X pb^ C < f,J+ Dk sfoqra abi qblobj^ ab i^ `lk,qfkrfa^a rkfclojb+ bibdfal bpqb C bufpqb rk^ m^oqf`fŽk M < urj* Tx) +++ ) riw abW[) \Y bk h pr_fkqbos^ilp q^ibp nrb i^ lp`fi^`fŽk ab ` bk `r^inrfbo pr_ fkqbos^il bpjbklo nrb D- Cbpfdkbjlp mlo J f&e' X hf% &) obpmb`qfs^jbkqb+ bi jŠufjl v bijŒkfjl ^_plirqlp ab ` bk bi f+„ndhj pr_ fkqbos^il WTe*x) TeY+ Sbkbjlp bkqlk`bp

Jf&e' + h-^&e' ; C

Page 208: Calculus

077 Cpi^dji`n ^jiodip\n

m^o^ `^a^ f < 0+1+ --- + i, Rb^k n9 v oh alp crk`flkbp bp`^ilk^a^p abcfkfa^p bkW[)\Y `ljl pfdrb9

Sbkbjlp bkqlk`bp ni&s' xa&s' x oi&swk\m\ qlal s ab X\* \Y+ Sbkbjlp q^j_f‹k

v

K^ afcbobk`f^ ab bp^p alp fkqbdo^ibpbp

Orbpql nrb C < f,J) bpq^abpfdr^ia^a mrbab bp`of_fopbbk i^ cloj^

'2-15( f] g] ] + [oi+ Pi ; ,,l\ \ J

Olo lqo^ m^oqb+i^p fkqbdo^ibpprmboflo b fkcboflo ab ` p^qfpc^`bki^p abpfdr^ia^abp

b F7Pi x fQ&w F7o9,

Lriqfmif`^kal bi mofjbo `lkgrkql ab abpfdr^ia^abp mlo ',0( v prj^kal bi obpri,q^al ^i pbdrkal `lkgrkql l_qbkbjlp

+ G] G]FQ&* `Q& w i oi + \ Pi ,

G^`fbkal rpl ab '2-15( u i^ obi^`fŽk -&a~ x g&a'* qbkbjlp

l pfQ&*,Q& ; ] x [

m^o^ qlal bkqbol K x 0- Olo `lkpfdrfbkqb+ pbd•k bi qblobj^ 0-20+ ab_b pbo-&a' < a&a', Dpql abjrbpqo^ nrb ` bp fkqbdo^_ibbk W[)] Z,

Page 209: Calculus

P_il_g[m ^_f p[fil g_^ci j[l[ `oh]cih_m ]ihncho[m 078

+&)1 FR\_RZN`QRYcNY\_ZRQV\]N_NSb[PV\[R`P\[aV[bN`

Dk i^ Rb``fŽk 1-05 pb abcfkfŽ bi s^ilo moljbafl =%`&ab rk^ crk`fŽk ` pl_obrk fkqbos^il W[)\Y `ljl bi `l`fbkqb P8`%r& r,% \ * [&+ Br^kal ` bp `lkqfkr^+mlabjlp abjlpqo^o nrb bpqb s^ilo moljbafl bp fdr^i ^i s^ilo ab ` bk rk `fboqlmrkql ab W[) \Y+

SDNQDL@ 2-04- SDNQDL@ CDK U@KNQ LDCHN O@Q@ HMSDFQ@KDR- Oc ` `n]ihncho[ _h W[)\Y) j[l[ oh ]c_lni _ ^_ W[)\Y n_h_gim

m]a&s' _ r < a&`'&] + [& +

Š \

@_gimnl[]cƒh+ Qbmobpbkq^jlp mlo g v L+ obpmb`qfs^jbkqb+ ilp s^ilobpjŠufjl v jŒkfjl ab ` bk X\* \Y+ Dkqlk`bp h x a&s' x I m^o^ qlal s ab X\* \Y+Hkqbdo^kal bp^p abpfdr^ia^abp v afsfafbkal mlo ] + \* bk`lkqo^jlp nrbh x >&a' x J* pfbkal >&a' < Gxa&s' _s&] + \', Obol ^elo^ bi qblobj^ abi s^ilofkqbojbafl klp af`b nrb =%`&< `%]&m^o^ rk `fboql _ ab W[)\Y+ Dpql `ljmibq^ i^abjlpqo^`fŽk-

O^o^ s^ilobp jbaflp mlkabo^alp e^v rk obpriq^al ^kŠildl-

SDNQDL@ 2-05- SDNQDL@ CDK U@KNQ LDCHN ONMCDQ@CN O@Q@ HMSDFQ@KDR-

Oojiha[gim ko_ ` u d mih ]ihncho[m _h W[)\Y+ Oc d hi ][g\c[ hoh][ ^_ mcahi_h W[)\Y _hnih]_m) j[l[ oh ]c_lni _ ^_ W[)\Y) n_h_gim

'2-16( Wˆ Wba&s'b&s' _s < a&`' b&s' _s ,x \ Š \

@_gimnl[]cƒh+ Ml `^j_f^kal krk`^ ab pfdkl bk W[)\Y) d bp pfbjmob klkbd^qfs^ l pfbjmob kl mlpfqfs^ bk W[) \Y+ Rrmlkd^jlp nrb d bp kl kbd^qfs^ bkX\* \Y+ Dkqlk`bp mlabjlp o^wlk^o `ljl bk i^ abjlpqo^`fŽk abi qblobj^ 2-04+bu`bmql nrb fkqbdo^jlp i^p abpfdr^ia^abp ga%r& w `%r&a%r&w Ia%r& l_qbkfbkal

'2-17( h m]b&s' _s x m] &s'b&s' _s x J m]b&s' _s,Š i Š %. † \

Rf F)wa%r&r: N+bp^ abpfdr^ia^a abjrbpqo^ nrb P89`%r&a%r&^r< N- Dk bpqb `^pl+i^ b`r^`fŽk '2-16( pb p^qfpc^`b m^o^ `r^inrfbo ` v^ nrb ^j_lp jfbj_olp plk `bol-Cb lqol jlal+ i^ fkqbdo^i ab d bp mlpfqfs^+ v mlabjlp afsfafo mlo bpq^ fkqbdo^ibk '2-17( v ^mif`^o `ljl ^kqbp bi qblobj^ abi s^ilo fkqbojbafl m^o^ `ljmibq^o i^abjlpqo^`fŽk- Rf d bp kl mlpfqfs^+ ^mif`^jlp bi jfpjl o^wlk^jfbkql +b,

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08/ Cpi^dji`n ^jiodip\n

Di qblobj^ abi s^ilo jbafl mlkabo^al klp iibs^ ^idrk^p sb`bp ^ bpqfj^`flkbp•qfibp m^o^i^ fkqbdo^iab rk molar`ql ab alp crk`flkbp v bpmb`f^ijbkqb pf i^ fkqb,do^i ab rkl ab ilp c^`qlobp bp cŠ`fi ab `^i`ri^o- Dk ilp moŽufjlp Dgbo`f`flp pba^k bgbjmilp-

2-1/ Dgbo`f`flp

0- Blk bi qblobj^ 2-05 bpq^_ib`bo i^p abpfdr^ia^abp pfdrfbkqbp9

0 Z0 T6 0

++ x +++_s;,l0NU1 -/ z ,0/

1- Sbkfbkal bk `rbkq^ nrb TG, s0 < '0 , U0' x v mlo jbafl abi qblobj^ 2-05l_qbkbo i^p abpfdr^ia^abp

00 g/-/ 00=,; %x_s:++13 , l U 0 , s} + 13 2 -

2- Tqfifw^o i^ fabkqfa^a 0 * s4 < 'i * s0&%f * s0 * s2' v bi qblobj^ 2-05 m^o^ abjlp,

qo^o nrb m^o^ \ = N+qbkbjlp

f\ _s \0 \2

9 **9[**(*, N 0 * s0 + 2• 4&

SŽjbpb \ < 0.0/ X `^i`ri^o bi s^ilo ab i^ fkqbdo^i `lk pbfp `fco^p ab`fj^ibp-3- Tk^ ab i^p pfdrfbkqbp ^cfoj^`flkbp bp fk`loob`q^- Dumif`^o mlo nr‹ bp c^ip^-

^( K^ fkqbdo^i F: &n`io' o _o = N ab_fal ^ nrb Iz:&n`io' o _o = F: Hpbkqi q_o,

_( K^ fkqbdo^i [ F: 'pbk L o _o < N mlonrb+ pbd•k bi qblobj^ 2-05+ m^o^ rk `fboql b`ljmobkafal bkqob 160! u 360! qbkbjlp

g]pbk o 0 g] `lp &05Q' + BNR &25Q'+ _o < , pbk o_o < ,,,,,,, < N -

1r o ` 1r ?

4- Rf i bp rk bkqbol mlpfqfsl+ rqfifw^o bi qblobj^ 2-05 m^o^ abjlpqo^o nrb

Fq%&i)/Gp &[g'ipbk &o/&_o < ,, +

,-+ @r ip

alkab u::9::9y _ w U%h * 0(6S ‘

5- RrmŽkd^pb nrb ` bp `lkqfkr^ bk W[) \Y+ Rf-oz `%r&^r < N+abjlpqo^o nrb `%]& < N mloil jbklp m^o^ rk ] ab W[) \Y+

6- RrmŽkd^pb nrb ` bp fkqbdo^_ib v kl kbd^qfs^ bk W[) \Y+ Rf-oz `%r&^r < N+abjlpqo^o nrb`%r& < N bk `^a^ mrkql ab `lkqfkrfa^a ab `+ WEh^c][]cƒh7 Rf `%]&= N bk rk mrkql ab`lkqfkrfa^a _) bufpqb rk bkqlokl ab ] bk bi `r^i `%r& = y`%]&+Y

7- RrmŽkd^pb nrb ` bp `lkqfkr^ bk W[) \Y X nrb -oz`%r&a%r& r < N+m^o^ qla^ crk`fŽk dnrb pb^ `lkqfkr^ bk W[) \Y+ Cbjlpqo^o nrb `%r& < N m^o^ qlal r bk W[) \Y+

Page 211: Calculus

3

5[<5E<? 6:87B7>5:3<

,&) =[a_\QbPPVp[UV`ap_VPN

Mbtqlk v Kbf_kfw+fkabmbkafbkqbjbkqb rkl abi lqol+ crbolk bk do^k m^oqbilp obpmlkp^_ibp abi abp^ooliil ab i^p fab^p _Špf`^p abi BŠi`ril fkqbdo^i e^pq^iibd^o ^ `lkpbdrfo nrb mol_ibj^p+ bk pr qfbjml foobplir_ibp+mrafbo^k pboil mloilp krbslp j‹qlalp v ab cloj^ `^pf orqfk^of^- Rr j^vlo ildol crb bpbk`f^ijbkqbbi eb`el ab mlabo crkafo bk rkl bi BŠi`ril fkqbdo^iv i^ pbdrka^ o^j^ fjmlo,q^kqbabi BŠi`ril9 bi BŠi`ril afcbobk`f^i-

K^ fab^ `bkqo^i abi BŠi`ril afcbobk`f^i bp i^ kl`fŽk ab _`mdq\_\, Hdr^i nrbi^ fkqbdo^i+i^ abofs^a^ crb lofdfk^a^ mlo rk mol_ibj^ ab FbljbqoŒ^9 bi mol,_ibj^ ab e^ii^o i^ q^kdbkqbbk rk mrkql ^ rk^ `ros^- Rfk bj_^odl+ ^ afcbobk`f^ab i^ fkqbdo^i+i^ abofs^a^ ^m^ob`b jrv q^oab bk i^ efpqlof^ ab i^ L^qbjŠqf`^-Dpqb`lk`bmql kl pb clojriŽ e^pq^ bi pfdil WUHH+`r^kal bi j^qbjŠqf`l co^k`‹pOfboobab Eboj^q+ qo^qŽab abqbojfk^o ilp jŠufjlp v jŒkfjlp ab `fboq^pcrk`flkbp-

K^ fab^ ab Eboj^q+ _Špf`^jbkqb jrv pfjmib+ mrbab `ljmobkabopb `lk ^rufiflab i^ cfdro^ 3-0- Rb prmlkb nrb bk `^a^ rkl ab prp mrkqlp+bpq^`ros^ qfbkb rk^afob``fŽk abcfkfa^ nrb mrbab sbkfo a^a^ mlo i^ q^kdbkqb-B^a^ rk^ ab bpq^pq^k,dbkqbppb e^ fkaf`^al bk i^ cfdro^ mlo rk^ ob`q^ ab qo^wlp-Eboj^q l_pbosŽ nrbbk ^nrbiilp mrkqlp bk nrb i^ `ros^ qfbkbrk jŠufjl l rk jŒkfjl `ljl ilp ab i^cfdro^+ab ^_p`fp^p s! v WH= i^ q^kdbkqbe^ ab pboelofwlkq^i- Olo q^kql+bi mol_ibj^ab il`^ifw^o bpqlp s^ilobp buqobjlp pb obar`b ^i ab i^ il`^ifw^`fŽk ab i^p q^kdbkqbpelofwlkq^ibp-

Dpql `lkar`b ^ i^ `rbpqfŽk jŠp dbkbo^i ab i^ abqbojfk^`fŽk ab i^ afob``fŽkab i^ q^kdbkqb bk rk kpioj \m]dom\mdj ab i^ `ros^- Di fkqbkql ab obplisbo bpqbmol_ibj^ crb il nrb `lkargl ^ Eboj^q ^ abp`r_ofo ^idrk^p ab i^p fab^p orafjbk,q^of^pobcbobkqbp i^ kl`fŽk ab abofs^a^-

@ mofjbo^ sfpq^ m^ob`b nrb kl e^_oŠ `lkbufŽk bkqobbi mol_ibj^ ab e^ii^obi Šob^ ab rk^ obdfŽk ifjfq^a^ mlo rk^ `ros^ v bi ab e^ii^o i^ q^kdbkqbbk rkmrkql ab rk^ `ros^- Di mofjbol nrb abp`r_ofŽ nrb bpq^palp fab^p+bk ^m^ofbk`f^

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/70 @ƒg^pgj _da`m`i^d\g

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pfk `lkbufŽk+ bpq^_^k Œkqfj^jbkqb ifd^a^p+ crb bi j^bpqol ab Mbtqlk+ Hp^^`A^oolt '052/,0566(- Rfk bj_^odl+ Mbtqlkv Kbf_kfwcrbolk ilp mofjbolp nrb`ljmobkafbolk i^ sboa^abo^ fjmloq^k`f^ ab bpq^obi^`fŽk v i^ bumilq^olk bk cloj^q^i nrb fk^rdro^olk rk^ bq^m^pfk mob`babkqbbk bi abp^ooliil ab i^ L^qbjŠqf`^-

@rknrb i^ abofs^a^ pb fkqolargl fkf`f^ijbkqb m^o^ bi bpqrafl abi mol_ibj^ab i^ q^kdbkqb+molkql pb sfl nrb molmlo`flk^_^ q^j_f‹k rk fkpqorjbkql m^o^bi `Ši`ril ab q`gj^d_\_`n v+ bk dbkbo^i m^o^ bi bpqrafl ab i^ q\md\^d‡i ab rk^crk`fŽk- Dk bi ^m^oq^al pfdrfbkqb pb `lkpfabo^oŠ rk mol_ibj^ m^oqf`ri^o nrb pbobcfbob i `Ši`ril ab rk^ sbil`fa^a- K^ plir`fŽk ab bpqbmol_ibj^ `lkqfbkb qla^pi^p `^o^`qboŒpqf`^pbpbk`f^ibp abi `lk`bmql ab abofs^a^+ v pr ^kŠifpfp`lkar`b ^ i^abcfkf`fŽk dbkbo^i nrb pb a^ bk bi ^m^oq^al 3-2-

3-1 Tk mol_ibj^ obi^qfsl ^ sbil`fa^a

Rb^ rk molvb`qfi i^kw^al sboqf`^ijbkqb abpab bi prbil ^ rk^ sbil`fa^a ab34 j mlo pbdrkal- Oobp`fkafbkal abi olw^jfbkql+ pb prmlkb nrb pli^jbkqb ^`q•^i^ do^sba^a+ mlo il nrb bi molvb`qfipb jrbsb bk iŒkb ob`q^- Rb^ `%n&i^ ^iqro^ bkjbqolp nrb ^i`^kw^ bi molvb`qfi o pbdrkalp abpmr‹p abi i^kw^jfbkql- Rf i^ crbow^ab i^ do^sba^a kl ^`qr^o^ bk ‹i+ bi molvb`qfi`lkqfkr^oŒ^pr_fbkal ^ sbil`fa^a `lkp,q^kqb+ob`loofbkal rk^ afpq^k`f^ ab 34 j `^a^ pbdrkal+ v bk bi qfbjml o pb qbkaoŒ^`%n&< 12n+ Obol ^ `^rp^ ab i^ do^sba^a+ bi molvb`qfis^ obq^oaŠkalpbe^pq^ nrb prsbil`fa^a iibd^ ^ s^ibo `bol+ v ^ m^oqfoab bpqbjljbkql `^b ^i prbil- Dumbofbk`f^pcŒpf`^pfkaf`^k nrb jfbkqo^p bi molvb`qfi bpqŠbk jlsfjfbkql pr ^iqro^ `Q& sfbkba^a^ ^molufj^a^jbkqb mlo i^ cŽojri^

%1+.& `Q& < 12n * 2n0Š

Page 213: Calculus

Ri kmj]g`h\ m`g\odqj \ q`gj^d_\_ 082

Di q‹ojfkl , 2n0 bp ab_fal ^ i^ fkqirbk`f^ ab i^ do^sba^a- N_p‹osbpb nrb `%n&< N`r^kal o < N X o < 8: l pb^+nrb bi molvb`qfiobdobp^^ i^ qfboo abpmr‹p ab 8 pb,drkalp+ mlo 0/ nrb i^ cŽojri^ 3-0 pŽil bp sŠifa^ m^o^N z o x 8-

Di mol_ibj^ ^ `lkpfabo^o bp bi pfdrfbkqb9 A`o`mhdi\m g\ q`gj^d_\_ _`g kmj+t`^odg `i ^\_\ dino\io` _` np hjqdhd`ioj, O^o^ mlabo `ljmobkabo bpqbmol_ibj^+e^v nrb mob`fp^o0/ nrb n` `iod`i_` mlo sbil`fa^a bk `^a^ fkpq^kqb-O^o^ biil+pb fkqolar`b i^ kl`fŽk ab q`gj^d_\_ h`_d\ _pm\io` pi dio`mq\gj _` od`hkj* bpab`fo+abpab bi fkpq^kqbo ^i o * b) abcfkf‹kali^ `ljl bi `l`fbkqb9

afcbobk`f^ ab afpq^k`f^p bk bi fkqbos^il ab qfbjml

fkqbos^il ab qfbjml`Q * b&* `%n&

b

Dpqb`l`fbkqb+ ii^j^al ^j^d`io` di^m`h`io\g* bp rk k•jbol nrb pb mrbab `^i`ri^opfbjmob nrb o v o * c mboqbkbw`^k j_lp ^i fkqbos^il H?$8\- Di k•jbol c mrbabpbo mlpfqfsl l kbd^qfsl+ mbol kl `bol- Rb abg^oŠ cfgl o v pb bpqraf^oŠ il nrb ibl`roob ^i `l`fbkqb fk`objbkq^i+ `r^kal pb a^k ^ b s^ilobp `^a^ sbw jbklobpbk s^ilo ^_plirql-

Olo bgbjmil+ `lkpfa‹obpb bi fkpq^kqbo < 1- ?N afpq^k`f^ ob`loofa^ abpmr‹pab 1 pbdrkalp bp9

`%/& <8/ , 1/ < 6/-

Dk bi qfbjml o < 1 * b i^ afpq^k`f^ ob`loofa^ bp9

`%/ * b& < 34'1 * b& * 4'1 * b&/ < 6/ * /2b * 2b/+

Olo q^kql+i^ sbil`fa^a jbaf^ bk bi fkqbos^il bkqobo < 1 X o < 1 * c bp

`%/ * b& * `%/&

b/2b * 2b

/< 14 ] 2b+

c

Slj^kal s^ilobp ab c `^a^ sbw jŠp mbnrb•lp bk s^ilo ^_plirql+ bpq^sbil`fa^ajbaf^+ pb ^`bo`^ jŠp v jŠp ^ 14- Olo bgbjmil+ pf c < /+0 i^ sbil`fa^a jbaf^bp 13+4: pf c < /+//0+ bp 13+884: pf c < /+////0+ pb l_qfbkb bi s^ilo 13+88884+v`r^kal c < , /+////0 pb l_qfbkb 14+////4- Kl fjmloq^kqb bp nrb pb mrbab l_qb,kbo i^ sbil`fa^a jbaf^ q^k moŽufj^ ^ 14 `ljl pb abpbb+pfk jŠp nrb qlj^o Gdhprcf`fbkqbjbkqb mbnrb•l- Rb abp`of_b bpqbeb`el af`fbkal nrb i^ sbil`fa^a jb,af^ od`i_` \g g…hdo`14 ^p\i_j c od`i_` \ ^`mj, O^ob`b k^qro^i ii^j^o ^i s^ilo abbpqbiŒjfqbi^ q`gj^d_\_ dino\ioƒi`\ bk bi fkpq^kqbo < 1-

Page 214: Calculus

083 @ƒg^pgj _da`m`i^d\g

Klp jfpjlp `Ši`rilp pb mrbabk bcb`qr^o m^o^ `r^inrfbo lqol fkpq^kqb- K^sbil`fa^a jbaf^ bk rk fkqbos^il ^o_fqo^ofl bkqob o v o * b bpqŠ a^al mlo bi `l,`fbkqb9

a&o* c' + a&o'

c

Z34xq * c'+ 3&o* c'0Z + X23o + 3o0Z < * 34 ] /.o [ 3c,

c

Br^kal c qfbkab ^ `bol+ i^ bumobpfŽk ab i^ abob`e^ qfbkab ^i iŒjfqb 34 , /.onrb abcfkb i^ q`gj^d_\_ dino\ioƒi`\ bk bi fkpq^kqb o, Cbpfdk^kal i^ sbil`fa^a fkp,q^kqŠkb^ mlo p%n&pb qfbkb

' 3-1( p%n&< 34 , .-n+

K^ cŽojri^ '3-0( abi bpm^`fl `%n&)abcfkb rk^ crk`fŽk ` nrb fkaf`^ i^ ^iqro^^ nrb pb bk`rbkqo^ bi molvb`qfi bk `^a^ fkpq^kqb ab pr jlsfjfbkql: ` pb abkl,jfk^ api^d‡i kjnd^d‡i l g`t _` `nk\^djn, Rr aljfkfl bp bi fkqbos^il `boo^al ZN+8\

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Page 215: Calculus

A`mdq\_\ _` pi\ api^d‡i 084

v pr doŠcf`^ bp i^ ab i^ cfdro^ 3-1'^(- ZK^ bp`^i^ pl_ob bi bgb sboqf`^i bk ^j_^pcfdro^p 3-1'^( v '_( e^ pfal jlafcf`^a^\- K^ cŽojri^ '3-1( ab i^ sbil`fa^a p%n&

abcfkb rk^ krbs^ crk`fŽk q nrb fkaf`^ i^ o^mfabw `lk nrb pb jrbsb bi molvb`qfibk `^a^ fkpq^kqb ab pr jlsfjfbkql+ pb abkljfk^ api^d‡i q`gj^d_\_ v pr doŠcf`^bp i^ ab i^ cfdro^ 3-1'_(- @i `ob`bo o ab N ^ 8+ p%n&ab`ob`b `lkpq^kqbjbkqb abq'M( < 34 ^ p%6&< , 34- O^o^ e^ii^o bi fkpq^kqb n bk bi `r^i p%n&< N pb ob,prbisb i^ b`r^`fŽk 34 < /.o l_qbkf‹kalpb o < 8.1- Olo q^kql+ bk bi mrkql `bk,qo^i abi jlsfjfbkql i^ fkcirbk`f^ ab i^ do^sba^a obar`b i^ sbil`fa^a ^ `bol vbi molvb`qfi nrba^ fkpq^kqŠkb^jbkqb cfgl- K^ ^iqro^ bk bpqb fkpq^kqb bp `%6,/& :< 0/0+14- Rf o 8~ 8.1+ i^ sbil`fa^a bp kbd^qfs^ v i^ ^iqro^ ab`ob`b-

Di mol`bpl mlo bi `r^i pb l_qfbkb p%n& m^oqfoabi `l`fbkqb fk`objbkq^i pbabkljfk^ ~e^ii^o bi iŒjfqb `r^kal c qfbkab ^ `bol‚+ v pb bumobp^ pfj_Žif`^jbkqb`ljl pfdrb9

'3-2( p%n&< Ecga%n * b& * a&o' ,c+i c

Dpq^ bumobpflk rp^a^ m^o^ abcfkfo i^ sbil`fa^a+ bk bi bgbjmil ^kqboflo+ qfbkbrk pbkqfal jŠp ^jmifl v mbojfqb abcfkfo i^ sbil`fa^a bk jlsfjfbkqlp ^ il i^odlab rk^ iŒkb^ ob`q^+ `r^kal pb `lklw`^ i^ crk`fŽk ab mlpf`fŽk `) v pfbjmob nrbbi `l`fbkqb fk`objbkq^i qfbka^ ^ rk iŒjfqb `r^kal c qfbkab ^ `bol-

,&+ 9R_VcNQNQRb[N Sb[PVp[

Di bgbjmil bumrbpql bk i^ Rb``fŽk ^kqboflo pb•^i^ bi `^jfkl m^o^ fkqol,ar`fo bi `lk`bmql ab abofs^a^- Rb^ ` rk^ crk`fŽk abcfkfa^ mlo il jbklp+ bk rkfkqbos^il ^_fboql %[) \& abi bgb r+ Rb bifdb rk mrkql r bk bpqb fkqbos^il v pb cloj^bi `l`fbkqb ab afcbobk`f^p

a&s * b& * a&s'

c

alkab bi k•jbol c mrbab pbo mlpfqfsl l kbd^qfsl 'mbol kl `bol(+ v q^i nrb s * cmboqbkbw`^ q^j_f‹k ^ %[) \&+ Di krjbo^alo ab bpqb `l`fbkqb jfab i^ s^of^`fŽkab i^ crk`fŽk `r^kal s s^oŒ^ ab s ^ s * c, Di `l`fbkqb obmobpbkq^i^ q\md\^d‡ih`_d\ ab ` bk bi &fkqbos^il nrb rkb s ^ s * c,

Rbdrfa^jbkqb pb e^`b qbkabo c ^ `bol u pb bpqraf^ il nrb ib l`roob ^ bpb `l,`fbkqb- Rf qfbkab e^`f^ rk `fboql s^ilo `ljl iŒjfqb 'u pboŠ bi jfpjl+ q^kql pf cqfbkab ^ `bol `lk s^ilobp mlpfqfslp `ljl kbd^qfslp(+ bkqlk`bp bpb iŒjfqb pb abkl,jfk^ abofs^a^ ab a bk s u pb fkaf`^ mlo bi pŒj_lil a%&s''pb ibb za mofj^ ab s~',Blk il nrb i^ abcfkf`fŽk cloj^i ab `$%r&mrbab bpq^_ib`bopb abi pfdrfbkqb jlal9

Page 216: Calculus

085 @ƒg^pgj _da`m`i^d\g

CDEHMHBHˆMCD CDQHU@C@- I\ _`mdq\_\ a&s' `noƒ _`adid_\ kjm g\ dbp\g_\_

'3-3( a%&s' < Fdha&s * c' + a&s' *b*(K c

^ji o\g lp` `g g…hdo` sdno\, Bg iˆh`mj a&s' o\h]d„i n` _`ijhdi\ ^j`ad^d`io` _`q\md\^d‡i _` a `i s,

Bljm^o^kal '3-3( `lk '3-2( pb sb nrb bi `lk`bmql ab sbil`fa^a fkpq^kqŠkb^bp pfjmibjbkqb rk bgbjmil abi `lk`bmql ab abofs^a^- K^ sbil`fa^a pQ& bp fdr^i^ i^ abofs^a^ aR' `r^kal ` bp i^ ibv ab bpm^`flp: il nrb cob`rbkqbjbkqb pbbumobp^af`fbkal+ nrb i^ sbil`fa^a bp i^ obi^`fŽk bkqob i^ s^of^`fŽk abi bpm^`flv i^ abi qfbjml- Dk bi bgbjmil abp^oolii^al bk i^ Rb``fŽk 3-1 i^ ibv ab bpm^`flp bpqŠa^a^ mlo i^ b`r^`fŽk

a&o'< 23o + 3o0

v pr abofs^a^ ` bp rk^ krbs^ crk`fŽk 'sbil`fa^a( a^a^ mlo

aR' < 34 , gLo,

Dk dbkbo^i+bi mol`bpl ab m^pl ^i iŒjfqb mlo bi nrb pb l_qfbkb a&s' ^ m^oqfoab `%r&) ^_ob rk `^jfkl m^o^ l_qbkbo rk^ krbs^ crk`fŽk ` ^ m^oqfoab rk^crk`fŽk a^a^ a, Dpqbmol`bpl pb abkljfk^ _`mdq\^d‡i* v ` bp i^ kmdh`m\ _`mdq\_\ab `+Rf ` ^ pr sbw bpqŠabcfkfa^ bk rk fkqbos^il ^_fboql pb mrbab q^j_f‹k `^i`ri^onp mofjbo^ abofs^a^+ fkaf`^a^ mlo a! v nrb bp i^ n`bpi_\ _`mdq\_\ ab a, @kŠild^,jbkqb+ i^ abofs^a^ k,pfj^ ab a* nrb pb fkaf`^ mlo a&i'*pb abcfkb `ljl i^ abofs^a^mofjbo^ ab a&i+/', Blksbkaobjlp bk nrb o%!< a* bpql bp+i^ abofs^a^ ab loabk`bol bp i^ jfpj^ crk`fŽk-

Dk bi `^pl abi jlsfjfbkql ob`qfiŒkbl+i^ mofjbo^ abofs^a^ ab i^ sbil`fa^a'pbdrka^ abofs^a^ abi bpm^`fl( pb abkljfk^ \^`g`m\^d‡i, Olo bgbjmil+ m^o^`^i`ri^oi^ ^`bibo^`fŽk bk bi bgbjmil ab i^ Rb``fŽk 3-1 pb mrbab rqfifw^oi^ b`r^`fŽk '3-1(m^o^cloj^o bi `l`fbkqb ab afcbobk`f^p

pQ * b& * pQ&c

X23+gLR)c'Z+X23+/.oZ < +gLc ;+/.,c c

Bljl bpqb`l`fbkqb kl s^oŒ^ i qbkabo c ^ N+pb mrbab `lkpfabo^o nrb od`i_` ^,0/ 'mrbpql nrb bp ,0/ `r^kal b bpqŠmoŽufjl ^ N(- Rb `lk`irvb mrbp+nrbi^ ^`bibo^`fŽk bk bpqbmol_ibj^ bp `lkpq^kqb b fdr^i ^ ,0/+ il nrb fkaf`^ nrb0&0 sbil`fa^a ab`ob`b ^ rk^ o^wŽkab 0/ jbqolp mlo pbdrkal `^a^ pbdrkal- Dk8 pbdrkalp bi ab`ob`fjfbkql qlq^i ab i^ sbil`fa^a bp 8&0/ < 8/ j mlo pbdrk,

Page 217: Calculus

Be`hkgjn _` _`mdq\_\n 086

al+ nrb bpqŠab ^`rboal `lk bi eb`el nrb aro^kqb ilp 8 pbdrkalp ab jlsfjfbkql i^sbil`fa^a `^j_f^ ab p_K& < 34 ^ p%6& < ,34-

,&, :WRZ]Y\`QRQR_VcNQN`

DIDLOKN 0- A`mdq\_\ _` pi\ api^d‡i ^jino\io`, Rrmlkd^jlp nrb a bp rk^crk`fŽk `lkpq^kqb+pb^ mlo bgbjmil `%r& < _ m^o^qlal r+ Di `l`fbkqb ab afcbobk,`f^p bp

a&s * c' + a&s' < ^ + b < M -c c

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DIDLOKN 1- A`mdq\_\ _` pi\ api^d‡i gdi`\g, Rb^ a rk^ crk`fŽk ifkb^i+mlobgbjmil `%r& < gr * \ m^o^ qlal ob^i r+ Rf b :.: N+qbkbjlp

a&s * c' + a&s' < h&s * c' * ] + &hs * ]' < hc < h,

c c c

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a%&s'< h m^o^ `^a^ u-

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a&s * c' + a&s'

b

O^o^ bpqraf^o bpqb`l`fbkqb ^i qbkaboc ^ `bol+ mlabjlp mol`babo ab alp j^kbo^p+l mlo i^ abp`ljmlpf`fŽk c^`qlof^i abi krjbo^alo `lkpfabo^al `ljl afcbobk`f^ abalp mlqbk`f^p k,pfj^p l ^mif`^kal bi qblobj^ abi _fkljfl m^o^ bi abp^ooliil ab%r * c'i, Rbdrfobjlp bi mofjbo j‹qlal v abg^objlp bi pbdrkal `ljl Dgbo`f`flm^o^bi ib`qlo- 'Ubo Dgbo`f`fl 28 ab i^ Rb``fŽk 3-5(

Page 218: Calculus

087 @ƒg^pgj _da`m`i^d\g

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' * c'i i j,h

T * T < &!&s * cgsi+/+fc x ,

Q_>

Dk i^ prj^ e^v i q‹ojfklp- Br^kal b qfbkab ^ N+&s* b&eqfbkab ^ s!* bi h,‹pfjlq‹ojfkl qfbkab ^ TeTi+.

*e < si+.

) v mlo q^kql i^ prj^ ab ilp i q‹ojfklp qfbkab ^is!8!* Cb bpql obpriq^ nrb

F &'u(< isi+/ m^o^qlal s,

DIDLOKN 3- A`mdq\_\ _` g\ api^d‡i n`ij, Rb^ n&s' < pbk s, Di `l`fbkqb abafcbobk`f^p bp

m%r* b& * m%r& pbk'u * b& * pbk r<

c c

O^o^ qo-^kpcloj^oil ab jlal nrb e^d^ mlpf_ib `^i`ri^o bi iŒjfqb`r^kal c x N+rqfifw^jlp i^ fabkqfa^a qofdlklj‹qof`^

q+s t)spbks * pbk s < 1pbk,&,, `lp ,,+ +

mlkfbkal v < s * b+Dpql `lkar`b ^ i^ cŽojri^

pbk%r * b& * pbkr m_h%bd/& % b&,,,,,,, < ,,, `lp s * , -b bd/ /

‘ Dpq^fabkqfa^a bp rk^ `lkpb`rbk`f^ fkjbaf^q^ ab i^ molmfba^a qbibp`Žmf`^ab i^p prj^pcfkfq^p-Dk bcb`ql+pf pb jriqfmif`^ `^a^ q‹ojfkl ab •i^ prj^ mlo %[ * \& pb bk`rbkqo^9

I'G I'G%[ * \& H [e\h*.*e < H%[e(f\h*%e(ff * [e\h*e& < ]! , \h+

Q5> Q5>

Page 219: Calculus

Ad_gjfim ^_ ^_lcp[^[m 088

Br^kal b w N+bi c^`qlo `lp %r * eb& w `lp r mlo i^ `lkqfkrfa^a abi `lpbkl- @pf,jfpjl+ i^ cŽojri^

i& pbku 0Gi,,< +x-+O /

bpq^_ib`fa^ bk i^ Rb``fŽk 2-3+ abjrbpqo^ nrb

'3-4( m_h%bd/&,,* 0 m^o^ qlal b ,,* N-bd/

Olo il q^kql bi `l`fbkqb ab afcbobk`f^pqfbkb`ljl iŒjfqb`lp r `r^kal b w N-Cf`elab lqol jlal+ m$%r&< `lp r m^o^qlal r8 i^ abofs^a^ ab i^ crk`fŽk pbkl bp i^ crk,`fŽk `lpbkl-

DIDLOKN 4- @_lcp[^[ ^_ f[ `oh]cƒh ]im_hi+ Rb^ ]%r&< `lp r+ Cbjlpqo^ob,jlp nrb ]$%r&< , Hpbkr8 bpql bp+i^ abofs^a^ ab i^ crk`fŽk `lpbkl bp jbklp i^crk`fŽk pbkl- O^oq^jlp ab i^ fabkqfa^a

q+s t)s`lp s * `lp u < ,1 pbk&,, pbk,,+ +

u mlkd^jlp t < s * c, Dpql klp `lkar`b ^ i^ cŽojri^

`lp 'u * b& * `lp u < \ m_h%bd/&pbk'u * y( -b bd/ /

K^ `lkqfkrfa^a abi pbkl abjrbpqo^ nrb pbk %r * eb& w pbk r `r^kal b w N: ^ m^o,qfo ab '3-4( l_qbkbjlp ]$%r&< , pbkr+

DIDLOKN 5- @_lcp[^[ ^_ f[ `oh]cƒh l[•t h*mcg[+Rf h bp rk bkqbol mlpfqfsl+pb^ `%r&< uh

.j m^o^r = N- Di `l`fbkqb ab afcbobk`f^pm^o^` bp

y&s * b& * y&s'

c

Olkd^jlp o < %r* op| v S < rf,hi Sbkbjlp bkqlk`bp oh < r * b v pi < r)`lk il nrb c < pi + pi* u bi `l`fbkqb ab afcbobk`f^p qlj^ i^ cloj^

y&s * b& * y&s' p + q &c pi [ qi pm8%* pi+0q * --+* pqi+0 * qi+/ Š

Page 220: Calculus

+)) @ƒg^pgj_da`m`i^d\g

K^ `lkqfkrfa^a ab i^ crk`fŽk o^Œwk,pfj^ morb_^ nrb p x q `r^kal b w N- Olo`lkpfdrfbkqb `^a^ q‹ojfkl abi abkljfk^alo abi jfbj_ol ab i^ abob`e^ qfbkb iŒ,jfqb U!,i `r^kal c x N- Dk qlq^i e^v i q‹ojfklp+ `lk il nrb bi `l`fbkqb ab afcb,obk`f^p qfbkb`ljl iŒjfqb s0

,! Fi, Orbpql nrb q < s% !*bpql abjrbpqo^ nrb

a%&s'< - Wi.k,i-

i

DIDLOKN 6- @jiodipd_\_ _` g\n api^dji`n lp` \_hdo`i _`mdq\_\, Rf rk^crk`fŽk ` qfbkbabofs^a^ bk rk mrkql s* bp q^j_f‹k `lkqfkr^ bk s, O^o^abjlpqo^o,il+ bjmib^jlp i^ fabkqfa^a

a&s * c' ;a&s' * c& &U * c• +a&U~'

nrb bp sŠifa^ m^o^b :B N- Rf e^`bjlp nrb b w N+bi `l`fbkqb ab afcbobk`f^p abipbdrkal jfbj_ol qfbkab ^ d$%r&v+mrbpql nrb bpqb`l`fbkqb bpqŠjriqfmif`^al mlork c^`qlo nrb qfbkab e^`f^ N+bi pbdrkal q‹ojfkl abi pbdrkal jfbj_ol qfbkab ^k• e%&s'< N-Dpql abjrbpqo^ nrb a&s * b& w a&s' `r^kal b w N+Xmlo q^kql nrb `bp `lkqfkr^ bk s,

Dpqbbgbjmil molmlo`flk^ rk krbsl mol`bafjfbkql m^o^mol_^o i^ `lkqfkrfa^aab i^p crk`flkbp- B^a^ sbw nrb bpq^_ib`bjlp i^ bufpqbk`f^ ab rk^ abofs^a^ d$%r&)bpq^_ib`bjlp q^j_f‹k+ ^i jfpjl qfbjml+ i^ `lkqfkrfa^a ab ` bk s, Cb_boŒ l_pbo,s^opb+kl l_pq^kqb+nrb bi ob`Œmol`lkl bp `fboql- K^ `lkqfkrfa^a bk s kl fjmif`^kb`bp^of^jbkqb i^ bufpqbk`f^ab i^ abofs^a^ e%&s', Olo bgbjmil+ `r^kal a&s' < Yt[+bi mrkql s < N bp ab `lkqfkrfa^a ab a Zmrbpqlnrb a&s' x N `r^kal s x N\ mbolkl bufpqb abofs^a^ bk N- 'U‹^pb i^ cfdro^ 3-2-( Di `l`fbkqb ab afcbobk`f^pW`%K* b& * `%-&Y,.E bp fdr^i ^ Ebf,b+ Dpqbs^ib * 0pf b = N v ,0 pf b ; N+v mlo `lkpfdrfbkqb kl qfbkbiŒjfqb`r^kal b w N-

s

EHFTQ@ 3-2 I\ api^d4i `n ^jiodip\ `i M k`mj /%&.'ij `sdno`

Page 221: Calculus

€gb`]m\ _` g\n _`mdq\_\n 1/0

,&- hYTRO_NQRYN QR_VcNQN`

Kl jfpjl nrb ilp qblobj^p obi^qfslp ^ ilp iŒjfqbp ab i^ Rb``fŽk 2-3 klp bk,pb•^k ^ `^i`ri^o bi iŒjfqb ab i^ prj^+ afcbobk`f^+ molar`ql v `l`fbkqb ab alp crk,`flkbp+ bi qblobj^ pfdrfbkqb klp molmlo`flk^ rk `lkgrkql ab obdi^p m^o^ bi `Ši`rilab abofs^a^p-

RCMPCK? 3-0- P`\i a v c _jn api^dji`n _`adid_\n `i pi dio`mq\gj ^jhˆi,Bi ^\_\ kpioj `i lp` a u d od`i`i _`mdq\_\n* o\h]d„i g\n od`i`i g\ nph\ a * d+g\ _da`m`i^d\ a + d+ `g kmj_p^oj nd u `g ^j^d`io` a- d- &M\m\a- d c\t lp` \†\_dmo\h]d„i lp` d c\ _` n`m _dnodio\ _` ^`mj `i `g kpioj ^jind_`m\_j', I\n _`mdq\_\n_` `no\n api^dji`n `noƒi _\_\n kjm g\n ndbpd`io`n a‡mhpg\n8

'f( &a * a&$< a% * a$ )

'ee(&a + a&$< a% + a$ )

'fff( &a%a&$< a , a$ * a + $$

'fs( 'z(& < b%a%na% d& `i kpiojn s _ji_` b&s'( N-

@kqbp ab abjlpqo^o bpqlp qblobj^p+ bp fkqbobp^kqb a^o ^idrk^p ab prp `lk,pb`rbk`f^p- Tk `^pl m^oqf`ri^o ab 'fff( pb qfbkb `r^kal rk^ ab i^p alp crk`flkbpbp `lkpq^kqb+ mlo bgbjmil+ b&s' < ` m^o^ qlal s^ilo ab s, Dk bpqb `^pl+ 'fff( pbqo^kpcloj^ bk9 '` 9a'% < A& .$8 bp ab`fo+ i^ abofs^a^ abi molar`ql ab rk^ crk,`fŽk mlo rk^ `lkpq^kqb bp bi molar`ql ab i^ abofs^a^ ab i^ crk`fŽk mlo i^ `lkp,q^kqb- Blj_fk^kal bpq^ molmfba^a `lk i^ ab i^ abofs^a^ ab rk^ prj^ Zmolmfb,a^a 'f(\ pb qfbkb+nrb m^o^ `^a^ m^o ab `lkpq^kqbp A0 X ^0* bp9

Dpq^ molmfba^a pb abkljfk^ kmjkd`_\_ gdi`\g ab i^ abofs^a^+ v bp ^kŠild^ ^ i^molmfba^a ifkb^i ab i^ fkqbdo^i-

@mif`^kal bi j‹qlal ab fkar``fŽk pb mrbab buqbkabo i^ molmfba^a ifkb^i ^rk k•jbol `r^inrfbo^ cfkfql ab prj^kalp9

alkab `0+ ]0* ]1* ŠŠŠ * ]y plk `lkpq^kqbp v ap a0* Š,Š * cy plk crk`flkbp `rv^p abof,s^a^p plk n*$)t$) +++) `h$+

Page 222: Calculus

1/1 ?•f]ofi ^c`_l_h]c[f

B^a^ cŽojri^ obcbobkqb abofs^a^p pb mrbab bp`of_fo ab alp j^kbo^p+ l`ljl rk^ fdr^ia^a bkqob alp `oh]cih_m) l `ljl rk^ fdr^ia^a bkqob h„g_lim+K^p molmfba^abpabi qblobj^ 3-0 q^i `ljl pb e^k bp`ofql ^kqbp+plk fdr^ia^abpnrb `lkqfbkbk crk`flkbp- Olo bgbjmil+ i^ molmfba^a 'f( fkaf`^ nrb i^ abofs^a^ab i^ crk`fŽk ` * d bp i^ prj^ ab alp crk`flkbp `$ v b%,Br^kal pb `lkpfabo^kilp s^ilobp ab bpq^pcrk`flkbp bk rk mrkql s* pb l_qfbkbk cŽojri^p bkqobk•jb,olp: ^pŒi^ cŽojri^ 'f( fjmif`^

%d* b'%&s' < a%&s' * b%&s',

U^jlp ^elo^ ^ abjlpqo^o bi qblobj^ 3-0-

@_gimnl[]cƒh ^_ 'f(- Rb^ r rk mrkql bk bi nrb bufpqbk ^j_^p abofs^a^p`$%r&v a$%r&+Di `l`fbkqb ab afcbobk`f^p m^o^` * d bp

Xa&s * c' * b&s * c'Z + Xa&s' * b&s'Z ;a&s * c' + a&s' * b&s * c' + b&s' ,

c c c

Br^kal c x N+bi mofjbo `l`fbkqb abi pbdrkal jfbj_ol qfbkab ^ `$%r&v bi pbdrkal^ a$%r&v mlo q^kql i^ prj^ qfbkab ^ `$%r&* a$%r&+K^ abjlpqo^`fŽk ab 'ff( bp^kŠild^-

@_gimnl[]cƒh ^_ 'fff(- Di `l`fbkqb ab afcbobk`f^pm^o^bi molar`ql ` +d bp9

'3-5(a&s * c'b&s * c' + a&s'b&s'

c

O^o^ bpqraf^o bpqb `l`fbkqb `r^kal c x N pb prj^ v obpq^ ^i krjbo^alo rkq‹ojfkl `lksbkfbkqb m^o^nrb pb mrba^ bp`of_fo'3-5( `ljl i^ prj^ ab alp q‹ojf,klp bk ilp nrb ^m^obw`^kilp `l`fbkqbp ab afcbobk`f^p ab ` v d- Rrj^kal v obp,q^kal a%r&%r * b&) '3-5( pb `lksfboqb bk

a&s * c'b&s * c' + a&s'b&s' &' a&s * c' + a&s' a& c' b&s * c' + b&s'+++++c+++++< c s +++c+++* s * +++c+++

Br^kal b w N bi mofjbo q‹ojfkl abi pbdrkal jfbj_ol qfbkab ^ a%r&`$%r&)vmrbpql nrb `%r * b& w `%r&)bi pbdrkal q‹ojfkl qfbkab ^ `%r&a$%r&)il nrb abjrbp,qo^ 'fff(-

@_gimnl[]cƒh ^_ 'fs(- Tk `^pl m^oqf`ri^oab 'fs( pb qfbkb `r^kal `%r&< 0m^o^qlal r+ Dk bpqb`^pl `$%r&< N X 'fs(- pb obar`b ^ i^ cŽojri^

'3-6(

Page 223: Calculus

€gb`]m\ _` g\n _`mdq\_\n 1/2

prmlkfbkal nrb a%r&", N- @ m^oqfoab bpqb `^pl m^oqf`ri^o+pbmrbab abar`fo i^ cŽo,jri^ dbkbo^i 'fs( bp`of_fbkal ` , d `ljl molar`ql v ^mif`^kal 'fff(+ `lk il `r^i pbqfbkb9

Olo q^kql+ nrba^ pli^jbkqb mlo mol_^o '3-6(- Di `l`fbkqb ab afcbobk`f^p ab g-b bp9

'3-7(WfFa%r* b&Y* WfFa%r&Y

c

a%r * b& * a%r&

c

0 0

a%r& a%r * b&

Br^kal c x N+ bi mofjbo `l`fbkqb ab i^ abob`e^ qfbkab ^ a$%r&v bi qbo`bo c^`qloqfbkab ^ .da%r&+Rb obnrfbob i^ `lkqfkrfa^a ab d bk r v^ nrb pb e^`b rpl abi eb`elnrb a%r* b& w a%r&`r^kal b w N- Olo q^kql+ bi `l`fbkqb bk '3-7( qfbkab ^*a$%r&da%T&/)il nrb abjrbpqo^ '3-6(-

Jin[7 O^o^ mlabo bp`of_fo '3-7( bp kb`bp^ofl prmlkbo nrb a%r * b& )++N m^o^qlal bprcf`fbkqbjbkqb mbnrb•l- Dpql bp `lkpb`rbk`f^ abi qblobj^ 2-6-

Di bjmibl abi qblobj^ 3-0 qbkfbkal bk `rbkq^ ilp bgbjmilp bumrbpqlp bk i^Rb``fŽk 3-3+ klp mbojfqb abar`fo krbslp bgbjmilp ab abofs^`fŽk-

DIDLOKN 0- Mjgdijhdjn, Dk bi bgbjmil 2 ab i^ Rb``fŽk 3-3 pb sfl nrb pf`%r&< r!) alkab h bp rk bkqbol mlpfqfsl+ bkqlk`bp `$%r&< ism8%*Orbab pbo fkqb,obp^kqb m^o^ bi ib`qlo bk`lkqo^o ab krbsl bpqb obpriq^al ^ m^oqfoabi `^pl m^oqf`ri^oi < 0+^mif`^kal bi j‹qlal ab fkar``fŽk grkq^jbkqb `lk i^ cŽojri^ ab abofs^`fŽkab rk molar`ql-

Blj_fk^kal bpqb obpriq^al `lk i^ molmfba^a ifkb^i+ pb mrbab abofs^o `r^i,nrfbo mlifkljfl prj^kal i^p abofs^a^p ab `^a^ rkl ab ilp q‹ojfklp: bp ab`fo+ pf

i

a&s' < F@fUf*

Q5>

abofs^kal q‹ojfkl ^ q‹ojfkl pb qfbkb

h

a%&s'< I f@fUf+/Š

Q_>

N_p‹osbpb nrb i^ abofs^a^ ab rk mlifkljfl ab do^al i bp rk mlifkljfl ab do^alh*f+ Olo bgbjmil+ pf `%r&:/r1(2r/*4r(5) bkqlk`bp `$%r&:3r/( .-r*4+

DIDLOKN 1- Cpi^dji`n m\^dji\g`n, Rf mbp bi `l`fbkqb ab alp mlifkljflp+ bpab`fo+ l%r&< j%r&,k%r&)i^ abofs^a^ l$%r&pb mrbab `^i`ri^o mlo jbafl ab i^ cŽojr,

Page 224: Calculus

+)- ?•f]ofi ^c`_l_h]c[f

i^ abi `l`fbkqb 'fs( abi qblobj^ 3-0- K^ abofs^a^ bufpqb m^o^ qlal s bk bi nrbk%r&", N- N_p‹osbpb nrb i^ crk`fŽk l$ ^pŒabcfkfa^ bp ^ pr sbw rk^ crk`fŽk o^`fl,k^i- Dk m^oqf`ri^o+ pf l%r&< f,rh alkab g bp rk bkqbol mlpfqfsl u r ", N pb qfbkb9

Dp`of_fbkal bpqb obpriq^al bk i^ cloj^9 m%&s'< , hs8~8% pb l_qfbkb rk^ buqbk,pfŽk ^ bumlkbkqbp kbd^qfslp ab i^ cŽojri^ a^a^ m^o^ i^ abofs^`fŽk ab mlqbk`f^pk,pfj^p m^o^ i mlpfqfsl-

DIDLOKN 2- Lin_h]c[m ^_ _rjih_hn_ `l[]]cih[lci+ Rb^ `%r&< rm m^o^ r = N+pfbkal l rk k•jbol o^`flk^i- X^ ebjlp abjlpqo^al i^ cŽojri^ ab abofs^`fŽk

'3-8( a%&s'< msm+/

m^o^ l < i.k+ pfbkal h rk bkqbol mlpfqfsl- U^jlp ^elo^ ^ buqbkaboi^ ^ qla^p i^pmlqbk`f^p ab bumlkbkqb o^`flk^i- K^ cŽojri^ m^o^ i^ abofs^`fŽk ab rk molar`qlabjrbpqo^ nrb i^ fdr^ia^a '3-8( q^j_f‹k bp sŠifa^ m^o^ m< 0-i v+ mlo fkar``fŽk+m^o^ m< h-i* pfbkal h `r^inrfbo bkqbol mlpfqfsl- 'Di o^wlk^jfbkql mlo fkar`,`fŽk il e^`bjlp pl_ob g+& Olo q^kql i^ fdr^ia^a '3-8( bp sŠifa^ m^o^ qlal l o^`fl,k^i mlpfqfsl- K^ cŽojri^ m^o^ abofs^o rk `l`fbkqb klp morb_^ nrb '3-8( q^j_f‹k bpsŠifa^ m^o^ l o^`flk^i kbd^qfsl- @pŒmrbp+ pf `%r&< T0

-1

* qbkbjlp `$%r&< eT*g-1

Š

Rf `%r&< T*g-0* bkqlk`bp `$%r&< ,0T*1

-0

Š Dk `^a^ `^pl+ bp mob`fpl nrb r = N-

,&. :WR_PVPV\`

0- Rf `%r& < 1 * r * r0* `^i`ri^o c&'/(+c&'0(+c&'i(+c&',0N(-

1- Rf `%r& < en* s/* 0s* bk`lkqo^o qlalp ilp s^ilobp ab r m^o^ ilp nrb '^( a%&s' :

< N: &]'a%&s'< ,1: &^'a%&s'< 0/-Dk ilp Dgbo`f`flp abi 2 ^i 01+l_qbkbo rk^ cŽojri^ m^o^ $%r& pf `%r& bp i^ nrb pb fkaf`^-

1, a&s' < s0 * 1s * 1-0

5, a&s' < ,, * s3 `lp s,s0 * 0

04 a&s' < s * h! s x ,0-

s6, a&s' < ,,0 ‘ s x 0-

['

G6+a&s' < 1 -* `lpu

sx * 1s * 1.-+ a&s' < ]-3 1 0 -y, * s (

1+a&s' < u3 * pbks,

3, a&s' < s2 pbks,

Page 225: Calculus

Be`m^d^djn 1/4

1 ,pbk s..+ a&s' < 1 -

+ ^jns./+ a&s'

sn`i s<,,,

0 * s/$

02- Rb prmlkb nrb i^ ^iqro^ FR' ab rk molvb`qfi+o pbdrkalp abpmr‹p ab e^_bo pfal i^kw^ale^`f^ ^oof_^ ^ m^oqfoabi prbil `lk rk^ sbil`fa^a fkf`f^i ab Sj jbqolp mlo pbdrkal+ bpqŠa^a^ mlo i^ cŽojri^9

'^( @miŒnrbpbbi j‹qlal abp`ofql bk i^ Rb``fŽk 3-1 m^o^ mol_^o nrb i^ sbil`fa^a jbaf^abi molvb`qfi aro^kqb bi fkqbos^il ab qfbjml ab G G* c bp qj + 0/0 , 3c jbqolp mlopbdrkal+ v nrb i^ sbil`fa^a fkpq^kqŠkb^bk bi fkpq^kqbHbp Ri * 0/0 jbqolp mlo pbdrkal-'_( B^i`•ibpb 'bk crk`fŽk ab qjg bi qfbjml kb`bp^ofl m^o^ nrb i^ sbil`fa^a pb ^krib-'b( ƒBrŠi bp i^ sbil`fa^a ab obdobpl ^ i^ Sfboo^>'a( ƒBrŠi ab_b pbo i^ sbil`fa^a fkf`f^i abi molvb`qfi m^o^ nrb obdobpb^ i^ Sfboo^ ^i `^_lab 0 pbdrkal> ƒv ^i `^_l ab 0/ pbdrkalp> ƒv ^i `^_l ab Q pbdrkalp>'b( Oor‹_bpb nrb bi molvb`qfi pb jrbsb `lk ^`bibo^`fŽk `lkpq^kqb-'c( A•pnrbpb rk bgbjmil ab lqo^ cŽojri^ m^o^ i^ ^iqro^ nrb a‹ ird^o ^ rk^ ^`bibo^`fŽk`lkpq^kqb ab , 0/ jbqolp mlo pbdrkal `^a^ pbdrkal-

03- ƒBrŠi bp bi `lbcf`fbkqb ab s^of^`fŽk abi slirjbk ab rk `r_l `lk obpmb`ql ^ i^ ilkdfqraab `^a^ i^al>

04- ^( Di Šob^ ab rk `Œo`ril ab o^afl l bp $E.!l0 v pr `fo`rkcbobk`f^ bp /$..!l+ Cbjlpqo^o nrb bi`lbcf`fbkqb ab s^of^`fŽk abi Šob^ obpmb`ql ^i o^afl bp fdr^i ^ i^ `fo`rkcbobk`f^-_( Di slirjbk ab rk^ bpcbo^ ab o^afl l bp 1$..!l0-1 v pr Šob^ bp 1$..!l0Š Cbjlpqo^o nrbbi `lbcf`fbkqb ab s^of^`fŽk abi slirjbk obpmb`ql ^i o^afl bp fdr^i ^i Šob^-

Dk ilp Dgbo`f`flp abi 05 ^i 12+ l_qbkbo rk^ cŽojri^ m^o^ d$%r&pf E%r&bp i^ nrb pb fkaf`^

.3+ a&s' <z+0

.4+ a&s' < ++8m;+*i*su

.5+ a&s' < U1-0*

s< N- 0., a&s' < Ug-0 * sg-1 * sg-2* s = N-

s< N- /.+ a&s' < U+g-0 * U+g-1 * s+g-2* s = N-

y//+ a&s' < 0 * s % s = N-s< N-

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Page 226: Calculus

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Page 227: Calculus

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Page 228: Calculus

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Page 229: Calculus

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Page 230: Calculus

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Page 231: Calculus

Lom\n ijo\^dji`n k\m\ g\n _`mdq\_\n 100

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101 @ƒg^pgj _da`m`i^d\g

2- Rb^ `%r& < r * pbk r m^o^ qlal r+ G^ii^o qlalp ilp mrkqlp r m^o^ ilp nrb i^ doŠcf`^ ab `bk %r)`%r| qfbkb mbkafbkqb `bol-

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4- G^ii^o s^ilobp ab i^p `lkpq^kqbp \* ] v ` m^o^ ilp `r^ibp i^p doŠcf`^p ab ilp alp mlifkljflp`%r& < r/ * [r * \ v a%r&< r1 + _ pb `loqbk bk bi mrkql '0+ 1( X qbkd^k i^ jfpj^ q^k,dbkqb bk af`el mrkql-

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7- Cf_rg^o i^ doŠcf`^ ab i^ `•_f`^ `%r& < r * r1 bk bi fkqbos^il `boo^al , 1 z r w 1-G^ii^o i^p `lkpq^kqbp h u ] ab jlal nrb i^ ob`q^ u < hs * ] pb^ q^kdbkqb ^ i^ doŠcf`^ab ` bk bi mrkql ' , 0+N(- Tk^ pbdrka^ ob`q^ nrb m^p^ mlo ', 0+N( bp q^j_f‹k q^kdbkqb^ i^ doŠcf`^ ab ` bk bi mrkql %[) _&+Cbqbojfk^o i^p `lloabk^a^p [ v _+

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O`bg\ _` g\ ^\_`i\ k\m\ g\ _`mdq\^d‡i _` api^dji`n ^jhkp`no\n 0/1

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Page 234: Calculus

103 ?•f]ofi ^c`_l_h]c[f

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Page 235: Calculus

O`bg\ _` g\ ^\_`i\ k\m\ g\ _`mdq\^dƒi _` api^dji`n ^jhkp`no\n 104

Di pbdrkal jfbj_ol ab '3-02( pboŒ bi `l`fbkqb ab afcbobk`f^p `rvl iŒjfqb abcfkbo$%s&)pf bk bi abkljfk^alo bk sbw ab b ^m^ob`fbo^ e+ Rf e :.: N pb `ljmibq^cŠ`fijbkqb i^ abjlpqo^`fŽk jriqfmif`^kal bi krjbo^alo v bi abkljfk^alo mlo fv bi pbdrkal jfbj_ol ab '3-02( qlj^ i^ cloj^9

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'3-05( o%s* o' + o%s&< oXb&o'* o$%s&Y+

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Page 236: Calculus

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=jfc][]cih_m ^_ f[ l_af[ ^_ f[ ][^_h[ 106

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Di bgbjmil mob`babkqb `loobpmlkab ^ ilp ii^j^alp mol_ibj^p pl_ob ]i_`c]c_h*n_m^_ p[lc[]cƒh fca[^im+ N_p‹osbpb nrb kl e^ pfal kb`bp^ofl bumobp^o l bk crk`fŽkab n m^o^ abqbojfk^o i^ abofs^a^ ^l , ^n+ Dpqb eb`el bp bi nrb e^`b nrb i^ obdi^ab i^ `^abk^ pb^ bpmb`f^ijbkqb •qfi bk mol_ibj^p pl_ob ]i_`c]c_hn_m ^_ p[lc[]cƒh

fca[^im+Klp alp bgbjmilp nrb pfdrbk jrbpqo^k `Žjl mrbab rqfifw^opb i^ obdi^ ab

i^ `^abk^ m^o^ l_qbkbo krbs^p cŽojri^p ab abofs^`fŽk-

DIDLOKN 1- C^a^ `%r& < pbk 'u!( `^i`ri^o `$%r&+

Oifo]cƒh+ K^ crk`fŽk ` bp rk^ `ljmlpf`fŽk+ `%r& < o Wp%r&Y)alkab p%r&< r0

v o%r& < pbk r+ O^o^ ^mif`^o i^ obdi^ ab i^ `^abk^ pb kb`bpfq^ abqbojfk^o o$Wp%r&Y:

: o$%r0', Orbpql nrb o$%r&< `lp r pb qfbkb o$%r0

' < `lp 'u!(+ v+ mlo q^kql+ '3-00(a^9

`$%r& < `lp %r/& † p$%r&< `lp %r/& † /r +

Rb mrbab q^j_f‹k obplisbo bi mol_ibj^ ^mif`^kal i^ klq^`fŽk ab Kbf_kfw- Rf pbbp`of_b t < r0 v X < `%r&) bkqlk`bp w < pbk t v ^t,^r < `$%r&+K^ obdi^ ab i^`^abk^ pb bumobp^

_u _u_t 1, < , , < '`lp s&%/r& < `lp %r&{ /r)_s _t _s

nrb `lfk`fab `lk bi obpriq^al l_qbkfal ^kqboflojbkqb k\m\a%&s',

DIDLOKN 2- Rf `%r& < Wp%r&Yhalkab h bp rk bkqbol mlpfqfsl+ `^i`ri^o `%r&bk crk`fŽk ab p%r& u p,_r&+

Oifo]cƒh+ K^ crk`fŽk ` bp rk^ `ljmlpf`fŽk+ `%r&:oWp%r&Y) alkab o%r&:r!+Orbpql nrb o$%r&< h rhwXm_ qfbkb o$Wp%r&Y< hWp%r&Yh*f)si^ obdi^ ab i^ `^abk^ a^9

`$%r& < hWp%r&Yh*Ep$%r&+

Page 238: Calculus

107 @ƒg^pgj _da`m`i^d\g

Rf pb ljfqb i^ obcbobk`f^^ s v pb bp`of_b `ljl rk^ fdr^ia^a bkqobcrk`flkbp+ pbl_qfbkb i^ fjmloq^kqb cŽojri^9

&qi'%< iqi+/q%

nrb fkaf`^ `Žjl pb abofs^ i^ mlqbk`f^ i+ndh\ ab q `r^kal q%bufpqb-K^ cŽojri^ bpq^j_f‹k sŠifa^ m^o^ i^p mlqbk`f^p m\^dji\g`n pf s! v qi+. bpqŠk abcfkfa^p- O^o^obplisbo bi mol_ibj^ jbaf^kqb i^ klq^`fŽk ab Kbf_kfwpb mrbab bp`of_fo t < p%r&

v t < `%r&+ Dkqlk`bp t < sh) ^t,^r < `$%r&v i^ obdi^ ab i^ `^abk^ a^9

_u _u _t k,i &' ( Z ' (\k,i &' (_s < _t _s < it q s < i q s q s *

nrb `lfk`fab `lk i^ mofjbo^ plir`fŽk-

DIDLOKN 3- K^ b`r^`fŽk r0 * d< l0 obmobpbkq rk^ `fo`rkcbobk`f^ abo^afl l v `bkqol bk bi lofdbk- Qbplisfbkal bpq^b`r^`fŽk obpmb`ql^ t bk crk`fŽkab s* pb l_qfbkbk alp plir`flkbp nrb pfosbk m^o^abcfkfoalp crk`flkbp ` v d a^a^pbk bi fkqbos^il Z, m*o\ mlo i^p cŽojri^p

y&s'< qm0+ s0 v a%r&< *Rl0 + r0Š

'K^ doŠcf`^ab ` bp i^ pbjf`fo`rkcbobk`f^ prmboflo+v i^ ab d i^ pbjf`fo`rkcbobk`f^fkcboflo-( Rb qo^q^ab `^i`ri^o i^p abofs^a^p ab ` v d jbaf^kqb i^ obdi^ ab i^ `^,abk^- O^o^` pb ^mif`^ bi obpriq^al abi bgbjmil 2 `lk p%r&< l0 + r0 v h < pv pbl_qfbkb9

'3-1/( a%&s'< Em0 + W1(,0.1' +0s' < +sSm0 [ t1

pfbjmob nrb `%r& ;C N- Di jfpjl j‹qlal ^mif`^al ^ d a^

'N

`#[$

'3-10( b%&s'< \ ,u ,uSm/ Z s/ b&s'

pfbjmob nrb a%r&:B N- N_p‹osbpb nrb pf pb fkaf`^ mlo t v^ pb^ `%r& l a%r&) ^j_^pcŽojri^p '3-1/( v '3-10( mrbabk `lj_fk^opb bk rk^ pli^+ nrb bp9

'3-11('[

s$ < , pf s :8n+N-V

Nqo^ ^mif`^`fŽk •qfi ab i^ obdi^ ab i^ `^abk^+ pb bk`rbkqo^ bk bi j‹qlalab i^ _`mdq\^d‡i dhkg…^do\, O^o^ bumif`^o bi j‹qlal v mlkbo ab j^kfcfbpql prpsbkq^g^p pb _rp`^oŠ ab krbsl bi obpriq^al abi bgbjmil 3 mlo rk `^jfkl jŠppbk`fiil-

Page 239: Calculus

Be`m^d^djn 108

DIDLOKN 4- A`mdq\^d‡i dhkg…^do\, K^ cŽojri^ '3-11( pb mrbab abar`fo af,ob`q^jbkqb ab i^ b`r^`fŽk r0 * v1 < l0 pfk kb`bpfa^a ab obplisboi^ obpmb`ql^ v-Qb`loa^kal nrb v bp rk^ crk`fŽk ab rWv < `%r& l t < a%r&Y pb mrbabk abofs^o^j_lp jfbj_olp ab i^ b`r^`fŽk s0 * t0 < m! u pb qfbkb9

%1+/0& /r * 1(&(! < N-

'Di q‹ojfkl 0tt% bp bi obpriq^al ab abofs^o v1 q^i `ljl pb e^ bumif`^al bk bi bgbj,mil 2-( Qbplisfbkal i^ b`r^`fŽk '3-12( obpmb`ql^ t% pb l_qfbkb '3-11(-

K^ b`r^`fŽk s0 * v1 < +1 pb af`b nrb abcfkb t dhkg…^do\h`io` `ljl crk`fŽkab s 'bk bpqb`^pl abcfkb _jn crk`flkbp( u bi mol`bpl mlo bi `r^i '3-12( pb l_qfbkb^ m^oqfoab bpq^b`r^`fŽk pb abkljfk^ _`mdq\^d‡i dhkg…^do\, Di obpriq^al cfk^i bpsŠifal m^o^i^p alp crk`flkbp ` v d ^pŒabcfkfa^p-N_p‹osbpb nrb bk bi mrkql 'u+ v(ab i^ `fo`rkcbobk`f^ `lk s " M b u " M i^ mbkafbkqbab i^ q^kdbkqbbp , s-t*jfbkqo^p nrb bi o^afl nrb rkb bi `bkqol `lk bi mrkql %r) s& qfbkb mlo mbkafbkqbtZ~, Di molar`ql ab ^j_^p mbkafbkqbpbp , 0+bp ab`fo+i^ q^kdbkqbbp mbombkaf`r,i^o ^i o^afl-

,&)* :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 03+abqbojfk^o i^ abofs^a^ `$%r&+Dk `^a^ `^pl pb pl_obkqfbkabnrb r qlj^ pŽil ilp s^ilobp m^o^ ilp nrb `%r& qfbkb pbkqfal-

H-a&s' < `lp 0s + 1 pbks,s s

5+ a&s' < q^k 1! , `lq 19 -

0, a&s' < ++.h* s0Š

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pbk s.1+ a&s' < Gs * Ss * v:-

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H * g-s % 0 * F-a&s' ,

Page 240: Calculus

11/ @ƒg^pgj_da`m`i^d\g

06- K^ pfdrfbkqb q^_i^ ab s^ilobp pb `^i`riŽ m^o^ rk m^o ab crk`flkbp ` v d v prp abofs^a^p`$ v b%, Blkpqorfo i^ `loobpmlkafbkqb q^_i^ m^o^ i^p alp crk`flkbp `ljmrbpq^p c v fa^a^p mlo b%r&< `Wa%r&Y)e%r&< aW`%r&Y+

s a&s' a%&s' b&s' b%&s'

N 0 . 1 '.0 2 ,1 N 01 N 1 2 0

2 1 3 0 '/

07- Tk^ crk`fŽk ` v prp alp mofjbo^p abofs^a^p pb e^k q^_ri^al `ljl ^ `lkqfkr^`fŽk pbfkaf`^- Olkbo a%r&< `%r0' v `lkpqorfo rk^ q^_i^ m^o^ d v prp alp mofjbo^p abofs^a^pm^o^ s < N+0+ 1-

s a&s' a%&s' a!&s'

N N 0 10 0 0 01 2 1 0

3 5 2 N

08- Cbqbojfk^o i^ abofs^a^ a$%r&bk crk`fŽk ab `$%r&pf9

']( b&s' < a&s0'9

'_( b&s' < a&n`i0 s' * a&^jn0 s'9

'b( b&s' < aXa&s''9'a( b&s' < avaXa&s'Gw,

?i_`c]c_hn_m^_ p[lc[]cƒh fca[^im u ^_lcp[]cƒh cgjf•]cn[+

1/- B^a^ ^ofpq^ab rk `r_l pb afi^q^ ^ o^wŽkab 0 `j mlo pbdrkal- ƒBrŠi bp i^ o^wŽkab s^,of^`fŽk abi slirjbk `r^kal i^ ilkdfqra ab `^a^ ^ofpq^bp '^( 4 `j+ '_( 0/ `j+ 'b( s `j>

10- Tk ^sfŽk pb abpmi^w^ bk srbil elofwlkq^i+ ^ 7 hfiŽjbqolp ab ^iqro^- 'Dk bpqbDgbo,`f`fl pb prmlkb i^ Sfboo^ ii^k^-( K^ orq^ ab srbil m^p^ mlo bk`fj^ ab rk mrkql L abiprbil- K^ afpq^k`f^ bkqob bi ^sfŽk v bi mrkql L afpjfkrvb ^ o^wŽkab 3 hfiŽjbqolp mlojfkrql bk bi fkpq^kqbbk bi nrb bpq^ afpq^k`f^ bp ab 0/ hfiŽjbqolp- B^i`ri^o i^ sbil`fa^aabi ^sfŽk bk hfiŽjbqolp mlo elo^-

11- Dk `^jml ab _^pb_^ii bp rk `r^ao^al `rvl i^al qfbkb 8/ mfbpab ilkdfqra- Tk^ mbilq^bp i^kw^a^ mlo bi _^qb^alo ^ il i^odl ab rk^ iŒkb^nrb m^p^ mlo i^ qbo`bo^ _^pb `lk rk^sbil`fa^a `lkpq^kqb ab 0// mfbpmlo pbdrkal- ƒBrŠi bp i^ o^mfabw`lk nrb s^oŒ^i^ afp,q^k`f^ ab i^ mbilq^ ^ i^ mofjbo^ _^pb+ '^( `r^kal i^ mbilq^ pb bk`rbkqo^ ^ jfq^a ab`^jfkl ab i^ qbo`bo^ _^pb+ '_( `r^kal i^ mbilq^ ^i`^kw^ i^ qbo`bo^ _^pb-

12- Tk _^o`l k^sbd^ m^o^ibi^jbkqb ^ rk^ `lpq^ ob`q^+^ rk^ sbil`fa^a ab 01 jfii^p mlo elo^v ^ rk^ afpq^k`f^ ab 3 jfii^p- ƒBrŠi bp pr sbil`fa^a ab ^molufj^`fŽk ^ rk c^ol ab i^`lpq^ bk bi fkpq^kqb bk nrb afpqb mob`fp^jbkqb 4 jfii^p abi c^ol>

13- Tk ob`fmfbkqbqfbkb cloj^ ab `lkl `fo`ri^o- K^ ^iqro^ bp 0/ j v bi o^afl ab i^ _^pb 3 j-Rb fkqolar`b ^dr^ bk bi ob`fmfbkqb^ rk^ sbil`fa^a `lkpq^kqb ab 4 j2 mlo jfkrql+ ƒ`lknr‹ sbil`fa^a pb bibs^ bi kfsbi abi ^dr^ `r^kal i^ molcrkafa^a abi ^dr^ bp ab 4 j+ pf'^( bi s‹oqf`b abi `lkl bpqŠe^`f^ ^oof_^+'_( ibi s‹oqf`b abi `lkl bpqŠe^`f^ ^_^gl>

Page 241: Calculus

>kgd^\^dji`n _` g\ _`mdq\^d‡i \ g\ _`o`mhdi\^d‡i _` `som`hjn _` api^dji`n 00/

14- Tk abmŽpfql ab ^dr^ qfbkb i^ cloj^ ab rk `lkl `fo`ri^o ob`ql `lk pr s‹oqf`b e^`f^^_^gl- Rr ^iqro^ bp ab 0/ j v bi o^afl ab i^ _^pb ab 04 j- Di ^dr^ p^ib mlo bi clkalab jlal `lkpq^kqb ^ o^wŽk ab 0 j! mlo pbdrkal- Rb sfboqb ^dr^ bk bi abmŽpfql ^ o^wŽkab ` j! mlo pbdrkal- B^i`ri^o ` ab jlal nrb bi kfsbi abi ^dr^ ^p`fbka^ ^ o^wŽk ab 3 jmlo pbdrkal bk bi fkpq^kqb bk nrb bi ^dr^ ^i`^k`b i^ ^iqro^ ab 7 j-

15- Di ^dr^ bkqo^ bk rk q^knrb ebjfpc‹of`l ab 0/ j ab o^afl 'i^ m^oqb mi^k^ e^`f^ ^oof_^(-Dk rk fkpq^kqb a^al+ pb^ c i^ ^iqro^ abi ^dr^ jbafa^ abpab bi clkal+ m bi o^afl ab i^prmbocf`fb if_ob abi ^dr^+ v S bi slirjbk abi ^dr^ bk bi q^knrb- B^i`ri^o _S - _c bk bifkpq^kqb bk nrb c < 4 j- Rf bi ^dr^ bkqo^ ^ o^wŽk `lkpq^kqb ab 4U&2 j! mlo pbdrkal+`^i`ri^o _m- _o* bi `lbcf`fbkqb ab s^of^`fŽk ab m*bk bi fkpq^kqb o bk nrb c < 4 j-

16- Tk qofŠkdril ob`qŠkdril s^of^_ib =>_ bk bi mi^kl rs qfbkb pr Škdril ob`ql bk bi s‹o,qf`b O* rk s‹oqf`b > cfgl bk bi lofdbk+ v bi qbo`bo s‹oqf`b a pl_ob i^ m^oŠ_li^ u < 0 * 265 s0Š

Di s‹oqf`b ? m^oqb abi mrkql 'N+ 0( bk bi qfbjml o < N X pb abpmi^w^ e^`f^ ^oof_^ pfdrfbk,al bi bgb v ^ rk^ sbil`fa^a `lkpq^kqb ab 1 `ok.pbd- ƒBlk nr‹ o^mfabw `ob`b bi Šob^ abiqofŠkdril `r^kal o < 6.1 pbdrkalp>

17- Di o^afl ab rk `fifkaol `fo`ri^o ob`ql ^rjbkq^ `lk rk `lbcf`fbkqb ab s^of^`fŽk `lkp,q^kqbl Rr ^iqro^ bp rk^ crk`fŽk ifkb^i abi o^afl v ^rjbkq^ qobp sb`bp jŠp oŠmfa^jbkqbnrb ‹pqb- Br^kal bi o^afl bp 0 j pr ^iqro^ bp 5 j- Br^kal bi o^afl bp 5 j+ bi slirjbk`ob`b e o^wŽk ab 0 j! mlo pbdrkal- Br^kal bi o^afl bp 25 j+ bi slirjbk ^rjbkq^ ^o^wŽk ab i j! mlo pbdrkal+ pfbkal i bkqbol- B^i`ri^o i,

18- Tk^ m^oqŒ`ri^ bpqŠ l_ifd^a^ ^ jlsbopb ^ il i^odl ab rk^ m^oŠ_li^ `rv^ b`r^`fŽk bpu < s0Š '^( ƒDk nr‹ mrkql ab i^ `ros^ s^oŒ^k i^ ^_p`fp^ v i^ loabk^a^ `lk bi jfpjl`lbcf`fbkqb ab s^of^`fŽk> '_( Dk`lkqo^o bpq^ o^wŽk pf bi jlsfjfbkql bp q^i nrb bk bifkpq^kqb o*bp s < pbk o b v < pbk, o,

2/- K^ b`r^`fŽk s1 * v2 < 0 abcfkb rk^ l jŠp crk`flkbp v ab s, '^( Rrmrbpql nrb bufpqbi^ abofs^a^ v&v pfk obplisbo i^ b`r^`fŽk obpmb`ql ^ v+ abjlpqo^o nrb v&p^qfpc^`b ^ i^b`r^`fŽk s0 )t0t% < N- '_( Rrmrbpql nrb bufpqb i^ pbdrka^ abofs^a^ t!* abjlpqo^o nrbu! < , /rs*2 pfbjmob nrb u y N-

20- Rf N ; r ; 4 i^ b`r^`fŽk r/g0( vH.1< 4 abcfkb v `ljl crk`fŽk ab r+ Rfk obplisboi^ obp,mb`ql ^ u abjlpqo^o nrb u& qfbkb pfdkl `lkpq^kqb- 'Rb prmlkb i^ bufpqbk`f^ ab s$+&

21- K^ b`r^`fŽk 0r0 * 1t0 < 01 abcfkb fjmiŒ`fq^jbkqb alp crk`flkbp v ab r pf Gtey 1- Rr,mrbpql nrb i^ pbdrka^ abofs^a^ v! bufpqb+abjlpqo^o nrb sbofcf`^ i^ b`r^`fŽk 2t1t! < , 8-

22- K^ b`r^`fŽk r pbk rs * /r0 < N- abcfkb fjmiŒ`fq^jbkqb u `ljl crk`fŽk ab T+ Rrmlkfbkalnrb i^ abofs^a^ v&bufpqb+abjlpqo^o nrb p^qfpc^`b i^ b`r^`fŽk t%s/ `lp st * st `lp st (* pbk st * 3u < N-

23- Rf v < s% alkab m bp rk k•jbol o^`flk^i9 m< h-i* pb qfbkb v&< s!* Rrmrbpq^ i^ bufp,qbk`f^ ab i^ abofs^a^ t%*abar`fo i^ cŽojri^ v&< lr7| ^mif`^kal i^ abofs^`fŽkifjmif`fq^ v

i^ cŽojri^ `loobpmlkafbkqb m^o^ bumlkbkqbp bkqbolp-

,&)+ 6]YVPNPV\[RQRYNQR_VcNPVp[N YNQRaR_ZV[NPVp[QR&ilp Rda_RZ\`QRYN`

Sb[PV\[R`

K^ abofs^`fŽk mrbab rqfifw^opbbk i^ il`^ifw^`fŽk ab ilp j^ufjlp v ojjjlpab i^p crk`flkbp- Dk ob^ifa^a+ bk BŠi`ril e^v alp pfdkfcf`^alp ab i^ m^i^_o^ ~jŠ,ufjl‚+ v pb afpqfkdrbk jbaf^kqb ilp ^agbqfslp \]njgpoj u m`g\odqj,Di `lk`bmql abjŠufjl ^_plirql pb fkqolargl bk bi `^mŒqril2- Qb`loabjlp nrb pb af`b nrb rk^

Page 242: Calculus

+++ @ƒg^pgj_da`m`i^d\g

crk`fŽk ` ab s^ilobp ob^ibp qfbkb rk jŠufjl ^_plirql bk rk `lkgrkql R pf bufpqbmlo il jbklp rk mrkql ` bk P q^i nrb

a&s' xa&^' m^o^qlal s bk R-

Di `lk`bmql ab jŠufjl obi^qfsl pb abcfkb ^pŒ9

CDEHMHBHˆM CD L„WHLN QDK@SHUN- Ri\ api^d‡i d,_`adid_\ `i pi ^jidpi+oj P* od`i` pi hƒsdhj m`g\odqj `i pi kpioj ` _` P nd `sdno` pi ^d`moj dio`mq\gj\]d`moj / lp` ^jiod`i` ` o\g lp`

a&s' x a&^' k\m\ oj_j s ndop\_j `i / †&GP,

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Page 243: Calculus

>kgd^\^dji`n _` g\ _`mdq\^d‡i \ g\ _`o`mhdi\^d‡i _` `som`hjn _` api^dji`n 001

l pi h…idhj m`g\odqj `i pi kpioj ` dio`mdjm \ g, Pd g\ _`mdq\_\ g%&^'sdno`* `n

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Rrmlkd^jlp M%]& = N- Rbd•k i^ molmfba^a ab `lkpbos^`fŽk abi pfdkl ab i^pcrk`flkbp `lkqfkr^p+ bufpqb rk fkqbos^il nrb `lkqfbkb ^ _ bk bi nrb M%r& bp mlpf,qfs^- Olo q^kql bi krjbo^alo abi `l`fbkqb M%r& qfbkb bi jfpjl pfdkl nrb bi abkl,jfk^alo m^o^ qlal r ;d; _ bk bpb fkqbos^il- Cf`el ab lqol jlal- `%r& = `%]& `r^kalr = `+ u `%r& ; _ `r^kal r ; ]+ Dpql `lkqo^af`b i^ efmŽqbpfp ab nrb ` qfbkb rkbuqobjl bk ]+ Krbdl+ i^ abpfdr^ia^a M%]& = M bp fjmlpf_ib- Dk cloj^ m^ob`fa^ pbabjrbpqo^ nrb kl mrbab pbo M%_&; N- Olo `lkpfdrfbkqb M%]& < N- `ljl pb ^cfojŽ-Orbpql nrb M%]&< f$%]&)bpql abjrbpqo^ bi qblobj^-

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++- @ƒg^pgj _da`m`i^d\g

kb`bp^of^jbkqb ab_b ^kri^opb bk rk buqobjl+ pf ‹pqb pb mobpbkq bk bi fkqbofloab rk fkqbos^il-

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Page 245: Calculus

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Page 246: Calculus

++/ @ƒg^pgj _da`m`i^d\g

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Page 247: Calculus

Be`m^d^djn ++0

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Page 248: Calculus

++1 @ƒg^pgj_da`m`i^d\g

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^( Zpbk+b, pbkg,\ -999:Er * tg,

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Page 249: Calculus

=jfc][]cih_m ^_f n_il_g[ ^_f p[fil g_^ci ++2

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'3-15( a&t' + a&s' < e%&^'&t+ r&) alkab s ; ` ; t ,

Orbpql nrb `$%]&b s * r plk mlpfqfslp+ il jfpjl ib l`roob ^ `%s&* `%r&)v bpqlpfdkfcf`^ `%r&; `%s&) ljl pb ^cfojŽ- Dpql abjrbpqo^ ^(+ u i^ abjlpqo^`fŽk ab _(bp m^ob`fa^- O^o^ abjlpqo^o _&) rqfifw^jlp i^ fdr^ia^a '3-15( e^`fbkal r < [+X^ nrb `$%]&< /+ qbkbjlp `%s&< `%[& m^o^ qlal s bk W[)\Y) `lk il nrb ` bp`lkpq^kqb bk W[) \Y+

Di qblobj^ 3-6 mlabjlp bjmib^oil m^o^ abjlpqo^o nrb pb mobpbkq^ rk buqob,jl pfbjmob nrb i^ abofs^a^ `^j_f^ ab pfdkl-

RCMPCK? 3-7- Oojiha[gim ` ]ihncho[ _h oh chn_lp[fi ]_ll[^i W[)\Y U ko__rcmn_f[ ^_lcp[^[ f$ _h ni^i johni ^_f chn_lp[fi [\c_lni %[) \&) _r]_jni [][mi _hoh johni ]+

]( Rf f$%r&_mjimcncp[ j[l[ ni^i r ; _ t h_a[ncp[ j[l[ ni^i r = _) ` nc_h_oh g•rcgi l_f[ncpi _h ]+

_( Rf+jil inl[ j[ln_) `$%r&_mh_a[ncp[ j[l[ ni^i r ; _ s jimcncp[ j[l[ ni^ir = _) ` nc_h_oh g•hcgi l_f[ncpi _h ]+

@_gimnl[]cƒh+ Dk bi `^pl ^(+ bi qblobj^ 3-6 ^( klp af`b nrb ` bp bpqof`q^,jbkqb `ob`fbkqb bk W[) _Y u bpqof`q^jbkqb ab`ob`fbkqb bk W_)\Y+ Krbdl `%r&; `%]&

m^o^ qlal r :.: _ bk %[) \&) `lk il nrb ` qfbkb rk jŠufjl obi^qfsl bk ]+

Z\ ` ] ` ]

^( LŠufjl obi^qfsl bk b _( LŒkfjl obi^qfsl bk `-

EHFTQ@ 3-01 Ijn `som`hjn n` km`n`io\i ^p\i_j g\ _`mdq\_\ ^\h]d\ _` ndbij,

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12/ @ƒg^pgj_da`m`i^d\g

Dpql abjrbpqo^ ^( v i^ abjlpqo^`fŽk ab _( bp `ljmibq^jbkqb ^kŠild^- Klp alp`^plp pb e^k obmobpbkq^albk i^ cfdro^ 3-01-

3-06 Bofqboflab i^ abofs^a^ pbdrka^ m^o^ilp buqobjlp

Rf rk^ crk`fŽk ` bp `lkqfkr^ bk rk fkqbos^il `boo^al W[)\Y) bi qblobj^ abilp s^ilobp buqobjlp klp af`b nrb qfbkb rk jŠufjl ^_plirql v rk jŒkfjl ^_pl,irql bk ^id•k mrkql ab W[) \Y+ Rf ` qfbkbabofs^a^ bk `^a^ mrkql fkqboflo+bkqlk,`bp ilp •kf`lp mrkqlp bk ilp nrb mrbabk mobpbkq^opbilp buqobjlp plk9

0( bk ilp buqobjlp abi fkqbos^il \ v ]91( bk ^nrbiilp mrkqlp fkqboflobps bk ilp nrb a%&s'< N-

Klp mrkqlp abi qfml 1( pb ii^j^k `lk cob`rbk`f^ kpiojn ^m…od^jnab `+O^o^ ab`fafopf bk rk mrkql `oŒqf`l bufpqbrk jŠufjl l rk jŒkfjl 'l kf rkl kf lqol(+ kb`bpf,q^jlp jŠp fkcloj^`fŽk ^`bo`^ ab i^ crk`fŽk `+Noafk^of^jbkqb bi `ljmloq^jfbkqlab ` bk rk mrkql `oŒqf`lmrbab abqbojfk^opb ^ m^oqfoabi pfdkl ^idb_o^f`l ab i^abofs^a^ bk i^p molufjfa^abp ab ^, Di qblobj^ nrb pfdrb e^`b sbo nrb rk bpqraflabi pfdkl ab i^ abofs^a^ pbdrka^ bk i^p `bo`^kŒ^pab ` mrbab q^j_f‹k pboklp abrqfifa^a-

SDNQDL@ 3-8- BQHSDQHN CD K@ CDQHU@C@ RDFTMC@ O@Q@ DWSQDLNR DM TM

OTMSN BQ†SHBN- P`\ ` pi kpioj ^m…od^j_` ` `i pi dio`mq\gj \]d`moj &\*]'9 `noj`n* npkjib\hjn \ ; ` ; ] t lp` a%&^'< N- Ppkjib\hjn o\h]d„i lp` `sdno\ g\_`mdq\_\ n`bpi_\ a! `i &\*]', Q`i`hjn `ioji^`n8

]( Pd a! `n i`b\odq\ `i &\*]'* a od`i` pi hƒsdhj m`g\odqj`i ^,_( Pdp! `n kjndodq\ `i &\*]' a od`i` pi h…idhj m`g\odqj`i ^,

Klp alp `^plp bpqŠkobmobpbkq^alpbk i^ cfdro^ 3-01-

A`hjnom\^d‡i, Blkpfabobjlp bi `^pl ^(+ a! ; N bk &\*]', Rbd•k bi qblob,j^ 3-6 '^mif`^al ^ b&(+i^ crk`fŽk `$bp bpqof`q^jbkqb ab`ob`fbkqb bk %[)\&+ Obola%&^'< N+ `lk il nrb a%^j_f^ pr pfdkl ab mlpfqfsl ^ kbd^qfsl bk `* `ljl jrbpqo^i^ cfdro^ 3-01 ^(- Krbdl+ pbd•k bi qblobj^ 3-7+ ` qfbkb rk jŠufjl obi^qfsl bk ^,K^ abjlpqo^`fŽk bk bi `^pl _( bp `ljmibq^jbkqb ^kŠild^-

Rf `! bp `lkqfkr^ bk _) u pf `!%]& ;/; N+ bufpqfoŠrk bkqlokl ab _ bk bi `r^ia! qbkaoŠ bi jfpjl pfdkl nrb a!&^', Olo `lkpfdrfbkqb+ pf a%&^'< N+ i^ crk`fŽk `qfbkb rk jŠufjl obi^qfsl bk _ pf !%]& bp kbd^qfs^+v rk jŒkfjl obi^qfsl pf `!%]&bp mlpfqfs^- Dpqb `ofqbofl _^pq^ m^o^ jr`elp bgbjmilp nrb pb mobpbkq^kbk i^moŠ`qf`^-

Di pfdkl ab i^ abofs^a^ pbdrka^ q^j_f‹k bpqŠobi^`flk^al `lk i^ `lk`^sfa^al i^ `lksbufa^a ab `+Di pfdrfbkqb qblobj^ abjrbpqo^ nrb i^ crk`fŽk bp `lksbu^bk ilp fkqbos^ilp bk ilp nrb `! bp mlpfqfs^+ ljl pb sb bk i^ cfdro^ 3-01 ^(+ Dk i^

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Qm\u\_j _` ^pmq\n 120

cfdro^ 3-01 ^(+ Ebp `Žk`^s^ v^ nrb `! bp kbd^qfs^- A^pq^ afp`rqfo q^k pŽil bi `^plab i^ `lksbufa^a+ v^ nrb pf Ebp `lksbu^+ , Ebp `Žk`^s^-

SDNQDL@ 3-0/- BQHSDQHN CD K@ CDQHU@C@ O@Q@ K@ BNMUDWHC@C- Ppkji+b\hjn E^jiodip\ `i X\* ]Z t lp` o`ib\ _`mdq\_\ `i `g dio`mq\gj \]d`moj &\*]',Pd `$`n ^m`^d`io` `i &\*]' `ioji^`n E`n ^jiq`s\ `i X\* ]Z, Bi k\mod^pg\m*E`n ^ji+q`s\ nd `! `sdno` t `n ij i`b\odq\ `i &\*]',

A`hjnom\^d‡i, Blkpfabobjlp s ; t bk X\* ]Z v mlkd^jlp w < ETU (* 'i , FU'U* alkab N ; GV ; 0- Prbobjlp abjlpqo^o nrb a`u' Q FUa&t' * '0 , GV(

a&s', Orbpql nrb a`u' < FUa&u'* 'i , FU'a&u'*bpql bp il jfpjl nrb abjlpqo^o nrb

'0 , FU'Xa&u' + a&s'Z Q FUXa&t' + a`u'Z ,

Rbd•k bi qblobj^ abi s^ilo jbafl '^mif`^al alp sb`bp(+ bufpqbk mrkqlp ` v ^ nrbp^qfpc^`bk s ; ` ; w v w ; _ ; t q^ibp nrb

a`u' + a&s' <a%&^'&u + s'* v a&t' + a`u' < a%&_'&t + t& +

Orbpql nrb `$bp `ob`fbkqb+ qbkbjlp a%&^' na%&_', @pfjfpjl+ qbkbjlp '0 , HW('Y ,* r& < GV'u , t&) ab jlal nrb mlabjlp bp`of_fo

'i , FU'Xa&u' + a&s'Z < '0 , FU'a%&@'&u+ s' Q FUa%&_'&t+ w( < FUXa&t' + a`u'Z *

il nrb abjrbpqo^ i^ abpfdr^ia^a bufdfa^ mlo i^ `lksbufa^a-

,&)0 F_NfNQ\ QR Pb_cN`

K^ fkcloj^`fŽk obrkfa^ bk ilp qblobj^p ab i^p •iqfj^p pb``flkbp bp `lk cob,`rbk`f^ •qfi bk bi qo^w^al ab `ros^p- @i af_rg^o i^ doŠcf`^ ab rk^ crk`fŽk .) ab_babqbojfk^opb mofjbo^jbkqb bi aljfkfl ab EZbi `lkgrkql ab s^ilobp ab s m^o^ilp `r^ibp bpqŠ abcfkfa^ E%r&Yv+ pf bp cŠ`fi e^`boil+ ab_boŒ^bk`lkqo^opb bi ob`loofalab E 'bi `lkgrkql ab s^ilobp ^i`^kw^alp mlo a', Tk `lkl`fjfbkql abi aljfkfl vabi ob`loofal klp a^ rk^ fab^ ab i^ ^jmifqra ab i^ `ros^ t < E%r&)v^ nrb mob`fp^rk^ mlo`fŽk abi mi^kl rs bk i^ nrb bpqŠ pfqr^a^ i^ `ros^- Rbdrfa^jbkqb bp ^`lkpb,g^_ib pfqr^o ilp mrkqlp 'pf bufpqbk( bk ilp nrb i^ `ros^ `loq^ ^ ilp bgbp `lloabk^alp-K^ fkqbopb``fŽk `lk bi bgb t bp bi mrkql 'N+`%K| prmlkfbkal nrb N mboqbkb`b ^ialjfkfl ab /* v i^p fkqbopb``flkbp `lk bi bgb ab i^p r plk ilp mrkqlp %r) N( m^o^ilp nrb `%r& < N- K^ abqbojfk^`fŽk ab i^p fkqbopb``flkbp `lk bi bgb r mrbab pbo+bk i^ moŠ`qf`^+ jrv afcŒ`fi+v mlabjlp `lkqbkq^oklp `lk s^ilobp ^molufj^alp-

Cb_boŒ^jlp q^j_f‹k abqbojfk^o ilp fkqbos^ilp bk ilp nrb ` bp jlkŽqlk^ bu^,jfk^kal bi pfdkl ab .! u abqbojfk^o ilp fkqbos^ilp ab `lksbufa^a u `lk`^sfa^a

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121 ?•f]ofi ^c`_l_h]c[f

bpqraf^kal bi pfdkl ab `!+ Dpmb`f^i`rfa^al ab_boŠ mlkbopb bk ilp mrkqlp bk ilpnrb i^ doŠcf`^ qfbkb q^kdbkqbpelofwlkq^ibp-

DIDLOKN 0- H[ al•`c][ ^_ v < `%r&)mc_h^i `%r&< r * f,r j[l[ r w N-Dk bpqb`^pl+ kl bufpqbkfkqbopb``flkbp`lk ilp bgbp-K^p alp mofjbo^p abofs^,

a^p bpqŠka^a^p mlo i^p cŽojri^p

a%&s'< i , i.u1+ a!&s' < 0-s1Š

t

s t

G,r&[

Gr&[

1

EHFTQ@ 3-02 Cl•`c][ ^_x%r&< r * f,r+ EHFTQ@ 3-03 Cl•`c][ ^_ x%r&< .,%r0 * 0(-

K^ mofjbo^ abofs^a^ bp mlpfqfs^ pf r0 = 0+kbd^qfs^ pf r0 ; 0+X `bol pf r0 < 0-Krbdl bufpqbrk jŒkfjl obi^qfsl bk s < 0 Xrk jŠufjl obi^qfsl bk s < , 0- O^o^r = /+ i^ abofs^a^ pbdrka^ bp mlpfqfs^ ab j^kbo^ nrb i^ mofjbo^ abofs^a^ bp bp,qof`q^jbkqb `ob`fbkqb- O^o^ s ; N+i^ abofs^a^ pbdrka^ bp kbd^qfs^+v mlo q^kqli^ abofs^a^ mofjbo^ pboŠbpqof`q^jbkqbab`ob`fbkqb-O^o^r moŽufjl ^ /+ bi q‹ojfklr bp mbnrb•l `ljm^o^al ^ f,r) v i^ `ros^ pb `ljmloq^ `ljl i^ doŠcf`^ab v < f,r+'Ubo cfdro^ 3-02-( Olo lqo^ m^oqb+m^o^s^ilobp do^kabp ab r 'mlpfqfslp l kbd^qf,slp(+ bi q‹ojfkl f,r bp mbnrb•l `ljm^o^al `lk r) v i^ `ros^ bp jrv m^ob`fa^ ^i^ ob`q^ v < s, Dk bpqbbgbjmil+ i^ crk`fŽk bp fjm^o+ a&+s' < +a&s'* `lk il`r^i i^ doŠcf`^bp pfj‹qof`^ obpmb`ql^i lofdbk-

Dk bi bgbjmil ^kqboflo+i^ ob`q^ u < s bp rk^ ^pŒkqlqab i^ `ros^- Dk dbkb,o^i+rk^ ob`q^ kl sboqf`^i ab b`r^`fŽk u < gr * \ pb ii^j^ [m•hnin[ ab i^ doŠcf`^ab u < `%r&pf i^ afcbobk`f^ `%r&* %gr * \& qfbkab ^ } `r^kal r qlj^ s^ilobpq^k do^kabp `ljl pb nrfbo^ mlpfqfslp l kbd^qfslp- Tk^ ob`q^ sboqf`^i+s < \* pb

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Ad_l]c]cim 122

ii^j^ [m•hnin[ p_lnc][f pf E`%r&ciibd^ ^ pbo q^k do^kab `ljl pb nrfbo^ `r^kals x \ mlo i^ abob`e^ l mlo i^ fwnrfboa^- Dk bi bgbjmil ^kqboflo+ bi bgb t bp rk^^pŒkqlq^sboqf`^i-

DIDLOKN 1- Cl•`c][ ^_ t < x%r&) ih^_ x%r&< .u%r0* 0(-Dpq^ bp rk^ crk`fŽk m^o+mlpfqfs^ m^o^ qlal s* u bi bgb s bp rk ^pŒkqlq^elof,

wlkq^i- K^ abofs^a^ mofjbo^ sfbkb a^a^ mlo

+0sa%&s' < &s0 * 0(1&

ab jlal nrb `$%r&; N pf r = N+`$%r&= N pf r ; N+u `$%r&< N `r^kal r < N-Olo `lkpfdrfbkqb i^ crk`fŽk `ob`b mlo bk`fj^ abi bgb s kbd^qfsl+ ab`ob`b bk i^m^oqb mlpfqfs^ abi bgb s* v qfbkb rk jŠufjl obi^qfsl bk s < N- Cbofs^kal lqo^sbw+ bk`lkqo^jlp nrb

a!&s' < &s0 * 0(Z,1(, &+0s'0&s0 * /'&0s' < 0&1s

0+ 0(-

%r0 * 0(3 %r0 * 0(2

@pŒnrb `!%r& = N pf 0r0 = 0+X `!%r& ; N pf 0r0 ; 0- Krbdl+ i^ abofs^a^ mofjbo^`ob`b `r^kal s0 = f v ab`ob`b `r^kal s%; f-Dpq^ fkcloj^`fŽk _^pq^ m^o^ af_r,g^o i^ `ros^ ab i^ cfdro^ 3-03- Klp alp mrkqlp ab i^ doŠcf`^ `loobpmlkafbkqbp ^r0 < e+bk ilp nrb i^ abofs^a^ pbdrka^ `^j_f^ pr pfdkl+ pb ii^j^k johnim ^_ch`f_rcƒh+

,&)1 :WR_PVPV\`

Dk ilp pfdrfbkqbp Dgbo`f`flp+ [& e^ii^o qlalp ilp mrkqlp r q^ibp nrb $%r&< N: _( bu^jfk^obi pfdkl ab & v abqbojfk^o ^nrbiilp fkqbos^ilp bk ilp nrb zbp jlkŽqlk^: b( bu^jfk^o bipfdkl ab l u abqbojfk^o ^nrbiilp fkqbos^ilp bk ilp nrb & bp jlkŽqlk^: a( `lkpqorfo rk _l`bqlab i^ doŠcf`^ ab z- Dk `^a^ `^pl+ i^ crk`fŽk& bpqŠ abcfkfa^ m^o^ qlalp ilp s m^o^ ilp `r^ibpqfbkb pbkqfal x%r&+

g, y&s' < s0 + 1s * 1-

0, y&s' < s0 * 2s,1, a&s' < &s + g'0&U* 1(-

2, a&s' < s0* 4s0 * 7s * 4-

3, x%r&< 1 * %r* 0(3-4, y&s' < /-s0,

5, x%r&< r * 0.u1‘

06, y&s' < &s [ g'&s + 2( -

7, y&s' < s-L * s0',/., y&s' < &s0 + 2'-&s0 + 8(-..+ y&s' <pbk1 s,

./+ y&s' < s + pbk s,/1, y&s' < s * `lp s,

/2, v&s' < ds0 * g0 `lp 0s,

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123 @ƒg^pgj _da`m`i^d\g

,&*( :WRZ]Y\`_R`bRYa\QR]_\OYRZN QRRda_RZ\`

Lr`elp mol_ibj^p ab buqobjlp bk L^qbjŠqf`^p mro^p v ^mif`^a^p mrbabk ob,plisbopb pfpqbjŠqf`^jbkqb jbaf^kqb bi rpl abi BŠi`ril afcbobk`f^i- Dk ob^ifa^a+ilporafjbkqlp abi BŠi`ril afcbobk`f^i crbolk bk mofk`fmfl abp^oolii^alp `r^kalEboj^q fkqbkqŽbk`lkqo^o j‹qlalp dbkbo^ibpm^o^abqbojfk^o jŠufjlp v jŒkfjlp-Dk bpq Rb``fŽk obplisbobjlp ^idrklp bgbjmilp v a^objlp ^i ib`qlo i^ lmloqrkfa^aab obplisbo lqolp bk i^ Rb``fŽk 3-10-

Elojri^jlp mofjbol alp mofk`fmflppbk`fiilp nrb mrbabk rp^opb m^o^obplisbodo^k k•jbol ab mol_ibj^p ab buqobjlp-

DIDLOKN 0- Mmdi^dkdj _`g kmj_p^oj hƒsdhj ^ji nph\ ^jino\io`, C^al rkk•jbol mlpfqfsl O+ Cbjlpqo^o nrb bkqobqlalp ilp m^obpab k•jbolp mlpfqfslp sb t q^ibpnrb s * t < P* bi molar`ql st bp bi j^vlo `r^kal s < t < zP,

A`hjnom\^d‡i, Rf s * t < P* V < P + s v bi molar`ql st bp fdr^i ^s&P + s' < sP + s0, Olkd^jlp a&s' < sP + s0, Dpqbmlifkljfl `r^aoŠqf`l qfbkb`ljl abofs^a^ mofjbo^ a%&s'< P + 0s nrb bp mlpfqfs^ m^o^ s ; yP X kbd^qfs^m^o s<xP, Olo q^kql bi jŠufjl ab st pb mobpbkq^r^kal s; P* t;P+s; 3Dpql q^j_f‹k pb mrbab abjlpqo^o pfk rqfifw^obi BŠi`ril- Olkd^jlp pfjmibjbkqba&s';dP0[&U+x P'0 u l_pbosbjlp nrb a&s' bp jŠufjl `r^kal s; P,

DIDLOKN 1- Mmdi^dkdj _` g\ nph\ h…idh\* ^ji kmj_p^oj ^jino\io`, C^al rkk•jbol mlpfqfsl L+ Cbjlpqo^o nrb bkqob qlalp ilp m^obpab k•jbolp mlpfqfslps b t q^ibp nrb st < M*bi nrb e^`b i^ prj^ s * v jŒkfj^ bp s < t < qf-

A`hjnom\^d‡i, Sbkbjlp nrb abqbojfk^o bi jŒkfjl ab i^ crk`fŽk a&s' ;; s * MZs m^o^s = N- K^ mofjbo^ abofs^a^ bp a%&s'< 0 , M-s0, Dpq^bp kbd^qfs^ m^o^ s0 ; M v mlpfqfs^ m^o^ s0 = M* ab j^kbo^ nrb a&s' qfbkb pr jŒkfjlbk s < qf-Krbdl+ i^ prj^ s * t bp jŒkfj^ `r^kal s < t < RL+

DIDLOKN 2- Dkqobqlalp ilp ob`qŠkdrilp ab mboŒjbqola^al+ bi `r^ao^al bpbi ab j^vlo Šob^-

A`hjnom\^dƒi, Tqfifw^jlp bi obpriq^al abi bgbjmil 0- Rb^k s b t ilp i^alpab rk ob`qŠkdril `r^inrfbo^- Rf bi mboŒjbqolbpqŠcfg^al+bkqlk`bp s * t bp `lkp,q^kqb+lk il nrb bi Šob^st qfbkbj^vlo s^ilo `r^kal s < t, Krbdl+ bi ob`qŠkdriljŠufjl bp bi `r^ao^al-

DIDLOKN 3- K^ jbaf^ dblj‹qof`^ ab alp k•jbolp mlpfqfslp kl bu`bab ^ prjbaf^ ^ofqj‹qf`^- Dpql bp+p[\ w w%[ * \&+

Page 255: Calculus

Be`hkgjn m`np`gojn _` kmj]g`h\n _` `som`hjn 124

A`hjnom\^d‡i, C^alp \ = N+ ] = N+ pb^ M < \]j Dkqob qlalp ilp ml,pfqfslp s b t pfbkal st < M* i^ prj^ s * t bp i^ jbklo `r^kal s < t < pL+Dp ab`fo+ pf rs < L) bkqlk`bp r * t ƒ pL * pL < 1 pL+ Dk m^oqf`ri^o+[ * \ ƒ 1 pL < 1 q99]* lk il nrb q99]x 0%[ * \&+ K^ fdr^ia^a pb mobpbkq pfv pŽil pf \ < \+

DIDLOKN 4- Tk _ilnrb ab mbpl T bp jlsfal ^ il i^odl ab rk mi^kl mlork^ crbow^ nrb cloj^ rk Škdril %d lk i^ ob`q^ ab i^ afob``fŽk abi jlsfjfbkql+pfbkal N z ` x /Q* `ljl pb sb bk i^ cfdro^ 3-04- Rrmlkd^jlp nrb i^ obpfpqbk`f^mlo cof``fŽk bp molmlo`flk^i ^ i^ crbow^ kloj^i `lk i^ nrb bi _ilnrb mobpflk^mbombkaf`ri^ojbkqb `lkqo^ bi mi^kl- G^ii^o bi Škdril %dm^o^bi nrb i^ crbow^ abmolmripfŽk kb`bp^of^ m^o^ sbk`bo i^ cof``fŽk pb^ il jŠp mbnrb•^ mlpf_ib-

Pjgp^d‡i, Rb^ C&L' i^ crbow^ ab molmripfŽk-…pq qfbkb rk `ljmlkbkqb sbo,qf`^i e^`f^ ^oof_^ nrb bp B%K&pbk K) ab jlal nrb i^ crbow^ kloj^i ab mobpfŽk`lkqo^ bi mi^kl bp J < S * B%K&pbk `, K^ crbow^ ab cof``fŽk bp jJ) alkab mbp rk^ `lkpq^kqb ii^j^a^ `lbcf`fbkqb ab cof``fŽk- Di `ljmlkbkqb elofwlkq^i ab i^crbow^ ab molmripfŽkbp B%_& lp K+Br^kal ‹pq^ pb fdr^i^ ^ i^ crbow^ ab cof``fŽk+iibd^jlp ^ B%_& lp _ < mZS * B%K&pbk_Y ab i^ nrb bk`lkqo^jlp

B%_&< jS`lp ` * mpbk`

O^o^ e^`bo jŒkfj^ C&`'* e^objlp jŠufjl bi abkljfk^alo b&`' < `lp .) mpbkNbk bi fkqbos^il N z ` x f.P+ Dk ilp buqobjlp+ qbkbjlp a%K&< 0 W a% .P&< m-Dk bifkqboflo abi fkqbos^il+ qbkbjlp

a$%_&< *m_h_ * m`lp _)

ab j^kbo^ nrb d qfbkb rk mrkql `oŒqf`lbk _ < bu+ pfbkal pbk l` < m`lp bu- Dpqla^ d'bu( < `lp bu * m1 lp bu < 'i * k0' `lp bu- Olabjlp bumobp^o lp bu bk crk`fŽkab m-Orbpql nrb m1 lp>~ < pbk1 bu < 0 , `lp! bu+ bk`lkqo^jlp '0 * k0' `lp! HW < 0+`lk 0/ nrb `lp HW < i.UH * j/+ @pŒmrbp a%[& < y- X^ nrb d'bu( bu`bab^ a%K&v ^ a%c.P&)bi jŠufjl ab d pb mobpbkq bk bi mrkql `oŒqf`l-Krbdl i^ crbow^jŒkfj^ mbafa^ bp

B%ET&< kT < kT ,c'hV( sS*:1

DIDLOKN 5- G^ii^o i^ jbklo afpq^k`f^ ab rk mrkql a^al 'N+ \& abi bgb t^ i^ m^oŠ_li^ r0 < 2t, 'Di k•jbol ] mrbab qbkbo`r^inrfbo s^ilo ob^i-(

Page 256: Calculus

125 ?•f]ofi ^c`_l_h]c[f

tCz&'

Erbow^ ab cof`f5kCz&' `lp '(

%K)\&

Erbow^ kloj^i J < S* By%&pbk'(s

EHFTQ@ 3-04 Ad_gjfi 2+ EHFTQ@ 3-05 Ad_gjfi 3+

Oifo]cƒh+ K^ m^oŠ_li^ bpqŠaf_rg^a^ bk i^ cfdro^ 3-05- K^ `^kqfa^a nrb e^vnrb e^`bo jŒkfj^ bp i^ afpq^k`f^ ^) pfbkal

`lk i^ obpqof``fŽkr0 < 1s+ @kqbi^ cfdro^ obpriq^ bsfabkqb nrb `r^kal \ bp h_a[*odqj i^ afpq^k`f^ jŒkfj^ bp G^h-Br^kal bi mrkql 'N+ \& pb abpmi^w e^`f^ ^oof_^pfdrfbkal bi bgbt* bi jŒkfjl bp ] e^pq^ nrb bi mrkql ^i`^kw^ rk^ `fboq^ mlpf`fŽkbpmb`f^i+mlo bk`fj^ ab i^ `r^i bi jŒkfjl bp ; ], U^jlp ^elo^ ^ abqbojfk^obp^ mlpf`fŽk bpmb`f^i-

@kqbqlal+ l_pbosbjlp nrb bi mrkql %r)s& nrb jfkfjfw^ ^ q^j_f‹k jfkfjf,w^ _!* 'Dpq^ l_pbos^`fŽk klp mbojfqb bsfq^o i^ abofs^`fŽk ab i^p o^Œ`bpr^ao^a^p-(Rbdrfa^jbkqb+ mlabjlp bumobp^o_0 bk crk`fŽk •kf`^jbkqb ab s l q^j_f‹k bkcrk`fŽk ab t v abg^jlp `ljl bgbo`f`fl• m^o^ bi ib`qlo abp^oolii^o ilp `Ši`rilp`r^kal z pb bumobp &bkcrk`fŽk ab s,

Olo q^kql i^ crk`fŽk ` nrb e^v nrb e^`bo jŒkfj^ sfbkb a^a^ mlo i^ cŽojri^

Rf _fbk `%s&bpqŠabcfkfa^ m^o^qlal s^ilo ob^i s) i^ k^qro^ibw^abi mol_ibj^ bufdbnrb _rpnrbjlp bi jŒkfjl q^k pŽil bkqob^nrbiilp s^ilobp ab s q^ibpnrb s w N-K^ abofs^a^ bp `$%s&< 3 (/%s * \& nrb bp `bol pŽil `r^kal s < \ * 1- Br^kal] ; 1+ bpql klp iibs^ ^ rk mrkql `oŒqf`lv kbd^qfsl nrb ab_b bu`irfopb mlo i^obpqof``fŽkt x N- Dp ab`fo+ pf _ ; 1+ bi jŒkfjl kl pb mobpbkq bk rk mrkql `oŒ,qf`l- Dk bcb`ql+`r^kal \ ; 1+ sbjlp nrb `$%s&= N `r^kal s w N+X mlo q^kql `bp bpqof`q^jbkqb `ob`fbkqb m^o^ t x N- Olo `lkpfdrfbkqb bi jŒkfjl ^_plirql pbmobpbkq bk bi buqobjl t < N- Di `loobpmlkafbkqb jŒkfjl _ bp S]0 < G^h-

Page 257: Calculus

Be`m^d^djn +,0

Rf ] ƒ 1+ bufpqbrk mrkql `oŒqf`libdŒqfjl bk t;] + 1- Orbpql nrb !&t';0m^o^ qlal v+ i^ abofs^a^ `$ bp `ob`fbkqb+v mlo q^kql bi h…idhj \]njgpoj ab ` pbmobpbkq bk bpqb mrkql `oŒqf`l-Di jŒkfjl ^ bp sf 1%\ * 1( * 3 < 1W^<0-Blk bpql ebjlp abjlpqo^al nrb i^ afpq^k`f^ jŒkfj^ bp G^hpf ] ; 1 u bp1 W^<0 pf ] ƒ 1- 'Di s^ilo ] < 1 bp bi s^ilo m^oqf`ri^o^kqbp `fq^al-(

,&*) :WR_PVPV\`

0- Cbjlpqo^o nrb bkqob qlalp ilp ob`qŠkdrilp ab Šob^ a^a^+ bi `r^ao^al bp bi ab mboŒjbqoljŒkfjl-

1- Tk do^kgbol qfbkb H jbqolp ab ^i^j_ob m^o^ `bo`^o rk qboobkl ab m^pql ob`q^kdri^o ^av^,`bkqb ^ rk jrol ab mfbao^- ƒPr‹ afjbkpflkbp a^oŠk bi Šob^ jŠufj^ ^i qboobkl `bo`^al>

2- Tk do^kgbol nrfbob `bo`^o rk qboobkl ab m^pql ob`q^kdri^o ab Šob^ > ^av^`bkqb ^ rkjrol ab mfbao^- ƒPr‹ afjbkpflkbp bufdbk i^ jŒkfj^ `^kqfa^a ab ^i^j_ob ab `bo`^>

3- C^al R = N- Ool_^o nrb bkqob qlalp ilp k•jbolp mlpfqfslp s b u q^ibp nrb s * u < R+i^ prj^ s0 * v1 bp jŒkfj^ `r^kal s < v-

4- C^al N = N- Ool_^o nrb bkqob qlalp ilp k•jbolp mlpfqfslp s b v q^ibp nrb s0 * v1 < N)i^ prj^ s * v bp jŠufj^ `r^kal s < v-

5- B^a^ i^al ab rk `r^ao^al qfbkb rk^ ilkdfqra H+Cbjlpqo^o nrb bkqob qlalp ilp `r^ao^,alp fkp`ofqlp bk bi `r^ao^al a^al+ bi ab Šob^ jŒkfj^ qfbkb i^alp ab ilkdfqra iKs&1-

6- B^a^ i^al ab rk `r^ao^al qfbkb rk^ ilkdfqra H+G^ii^o bi q^j^•l abi `r^ao^al abjŠufj^ Šob^ nrb mrbab `fo`rkp`of_fopb ^i `r^ao^al a^al-

7- Cbjlpqo^o nrb bkqob qlalp ilp ob`qŠkdrilp nrb mrbabk fkp`of_fopb bk rk `Œo`ril a^al+bi `r^ao^al qfbkb bi Šob^ jŠufj^-

8- Cbjlpqo^o nrb bkqob qlalp ilp ob`qŠkdrilp ab Šob^ a^a^+ bi `r^ao^al qfbkb bi `Œo`ril`fo`rkp`ofql jŒkfjl-

0/- C^a^ rk^ bpcbo^ ab o^afl O, G^ii^o bi o^afl l v i^ ^iqro^ c abi `fifkaol `fo`ri^o ob`ql abj^vlo prmbocf`fb i^qbo^i /$fPlb nrb mrbab fkp`of_fopb bk i^ bpcbo^

00- Dkqob qlalp ilp `fifkaolp `fo`ri^obp ob`qlp ab Šob^ i^qbo^i a^a^+ abjlpqo^o nrb i^ jbklobpcbo^ `fo`rkp`ofq^ qfbkb bi o^afl fdr^i ^i o^afl abi `fifkaol jriqfmif`^al mlos&1 --

01- C^al Sj `lkl `fo`ri^o ob`ql ab o^afl N v ^iqro^ G- G^ii^o bi o^afl v i^ ^iqro^ abi `f,ifkaol `fo`ri^o ob`ql ab j^vlo Šob^ i^qbo^i nrb mrbab fkp`of_fopb bk bi `lkl-

02- G^ii^o i^p afjbkpflkbp abi `fifkaol `fo`ri^o ob`ql ab jŠufjl slirjbk nrb mrbab fkp,`of_fopb bk rk `lkl `fo`ri^o ob`ql ab o^afl N v ^iqro^ G-

03- C^a^ rk^ bpcbo^ ab o^afl O, B^i`ri^o+ bk crk`fŽk ab O* bi o^afl m v i^ ^iqro^ c abi`lkl `fo`ri^o ob`ql ab j^vlo slirjbk nrb mrbab fkp`of_fopb bk bp^ bpcbo^-

04- G^ii^o bi ob`qŠkdril ab j^vlo Šob^ nrb mrbab fkp`of_fopb bk rk pbjf`Œo`ril+ qbkfbkali^ _^pb fkcboflo bk bi afŠjbqol-

05- G^ii^o bi qo^mb`fl ab j^vlo Šob^ nrb mrbab fkp`of_fopb bk rk pbjf`Œo`ril+ qbkfbkal i^_^pb fkcboflo bk bi afŠjbqol-

06- Tk^ `^g^ ^_fboq^ bpqŠ `lkpqorfa^ `lk rk ob`qŠkdril ab `^oqŽk nrfq^kal `r^ao^alp fdr^,ibp bk `^a^ bpnrfk^ v al_i^kal e^`f^ ^oof_^ ilp _loabp- G^ii^o i^p afjbkpflkbp ab i^`^g^ ab j^vlo slirjbk nrb mrbab `lkpqorfopb ab q^i jlal pf bi ob`qŠkdril qfbkb `ljli^alp ^( 0/ v 0/: _( 01 X 07-

07- Rf \ v \ plk ilp `^qbqlp ab rk qofŠkdril ob`qŠkdril `rv^ efmlqbkrp^ bp 0+e^ii^o bi j^vlos^ilo ab 0\ * ],

08- Tk `^jfŽk e^ ab ob`loobo 2// hj bk rk^ `^oobqbo^ ii^k^ ^ sbil`fa^a `lkpq^kqb ab s hjmlo elo^- K^p ibvbp ab `fo`ri^`fŽk mobp`of_bk 24 9,p: s 9,p: 44- Rb prmlkb nrb bi `^o_ro^kqb

Page 258: Calculus

016 @ƒg^pgj_da`m`i^d\g

`rbpq^ ^ 2 mq^p-ifqol v nrb bi `lkprjl bp ab 0/ * s0-/0. ifqolp mlo elo^- Rf bi `lk,ar`qlo `l_o^ M mbpbq^pmlo elo^ v pf l_bab`b qla^p i^p ibvbp ab qoŠcf`l+abqbojfk^o `rŠi bpi^ sbil`fa^a jŠp b`lkŽjf`^ v bi `lpqb abi sf^gb pf M < N+M < 1/+ M < 3/ X M < 5/-

1/- Tk `fifkaol pb e^ l_qbkfal e^`fbkal dfo^o rk ob`qŠkdril ^iobabalo abi bgb s* q^i nrbpr _^pb bpqŠ bk bi bgb s* v qlal bi ob`qŠkdril bpqŠ `lkqbkfal bk i^ obdfŽk `ljmobkafa^bkqob i^ `ros^ v < r,%r0 * 0( v bi bgb r) G^ii^o bi `fifkaol ab slirjbk 0/ j^vlo mlpf_ib-

10- Rb al_i^ rk^ mŠdfk^ ab j^kbo^ nrb i^ bpnrfk^ abob`e^ fkcboflo iibdrb ^ `lfk`fafo `lkbi i^al fwnrfboal ab i^ jfpj^ 's‹^pb cfd- 3-06(- Rf i^ ^k`ero^ ab i^ mŠdfk^ bp 04+13 `j+e^ii^o i^ ilkdfqra jŒkfj^ abi mifbdrb- ƒBrŠi bp bi Škdril nrb cloj^ bpqb mifbdrb jŒkfjl`lk bi i^al abob`el ab i^ mŠdfk^> Rb prmlkb i^ mŠdfk^ prcf`fbkqbjbkqb i^od^ m^o^ bsfq^onrb bi mifbdl ^i`^k`b i^ `^_b`bo^ ab i^ mŠdfk^-

\\\\\\\\\\ --H

EHFTQ@ 3-06 Bd`m^d^dj 0/, EHFTQ@ 3-07 Bd`m^d^dj 00,

11- '^( Tk qofŠkdril fpŽp`bibp bpqŠ fkp`ofql bk rk^ `fo`rkcbobk`f^ ab o^afl o `ljl pb fkaf`^bk i^ cfdro^ 3-07- Rrmlkfbkal bi Škdril 1eV bk bi s‹oqf`b+ `ljmobkafal bkqob N v c&iS+

e^ii^o bi s^ilo jbafl v bi s^ilo jbklo abi mboŒjbqol abi qofŠkdril- C^o qlalp ilp abq^iibpabi o^wlk^jfbkql pbdrfal-'_( ƒBrŠi bp bi o^afl abi jbklo afp`l `fo`ri^o prcf`fbkqbjbkqb do^kab m^o^ `r_ofo oj_jqofŠkdril fpŽp`bibp ab mboŒjbqol a^al I= C^o qlalp ilp abq^iibp abi o^wlk^jfbkql-

12- Tk^ sbkq^k^ qfbkb cloj^ ab ob`qŠkdril qbojfk^al mlo rk pbjf`Œo`ril ab afŠjbqol fdr^i^ i^ _^pb abi ob`qŠkdril- K^ mlo`fŽk ob`q^kdri^o e^ ab pbo ab `ofpq^i qo^kpm^obkqbv i^m^oqb `fo`ri^o e^ ab pbo ab `ofpq^ibp ab `lilo nrb ^ajfqb pŽil i^ jfq^a ab irw ,mlo jbqol`r^ao^al nrb bi `ofpq^i qo^kpm^obkqb-Di mboŒjbqol qlq^i ab i^ sbkq^k^ e^ ab qbkbo ilkdfqracfg^ L+ G^ii^o+ bk crk`fŽk ab L) i^p afjbkpflkbp ab i^ sbkq^k^ nrb abg^ m^p^o i^ j^vlo`^kqfa^a mlpf_ib ab irw-

13- Tk qolwl ab j^abo^ ab 01 aj ab i^odl qfbkb cloj^ ab rk qolk`l ab `lkl `fo`ri^o ob`qlab afŠjbqolp 3 aj v '3 * b& aj bk prp _^pbp+ alkab b w N- Cbqbojfk^o bk crk`fŽk abc bi slirjbk abi j^vlo `fifkaol `fo`ri^o ob`ql nrb pb mrbab `loq^o ab bpqb qolwl abj^abo^+ ab j^kbo^ nrb pr bgb `lfk`fa^ `lk bi abi qolk`l ab `lkl-

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A`mdq\_\n k\m^d\g`n 128

14- C^alp k k•jbolp ob^ibp ]h+ --- + \ij Cbjlpqo^o nrb i^ prj^ Jnyh%r * \*'0 bp jŒkfj^`r^kal s bp i^ jbaf^ ^ofqj‹qf`^ ab ]h+ --- + \ij

15- Rf r = N+pb^ x%r&< 2r0 * =r*3* pfbkal = rk^ `lkpq^kqb mlpfqfs^- G^ii^o bi jbklo s^iloab = q^i nrb x%r&87713 m^o^ qlal r = N-

16- O^o^ `^a^ o ob^i+ pb^ x%r&< , r1 * o0r) v abpfdkbjlp `lk g%n& bi jŒkfjl ab x%r&bkbi fkqbos^il M9o r 9o 0- Cbqbojfk^o bi s^ilo ab g%n& m^o^ `^a^ o abi fkqbos^il, 0 9po 9p 0- Qb`r‹oabpb nrb m^o^ ^idrklp s^ilobp ab o bi jŒkfjl ab x%r&mrbab mobpbk,q^opb bk ilp buqobjlp abi fkqbos^il N 9ps 9p 0-

17- R^_bjlp nrb rk k•jbol s bpqŠ bk rk fkqbos^il \ 9ps 9p]* pfbkal \ = N- Prbobjlp ^mol,ufj^o s mlo jbafl ab lqol k•jbol o bk X\* \Y ab j^kbo^ nrb bi boolo obi^qfsl-io , sg-s*pb^ il jbklo mlpf_ib- Cbpfdkbjlp mlo I%n& bi jŠufjl s^ilo ab En* rf,r `r^kal r s^oŒ^ab \ ^ ]* ^( Cbjlpqo^o nrb bpb jŠufjl pb mobpbkq^ bk rkl ab ilp buqobjlp s < \ ls < ], ]' Cbjlpqo^o nrb J&o' bp jŒkfjl `r^kal o bp i^ jbaf^ ^ojŽkf`^ ab \ v ]* bpqlbp+ `r^kal g-o < f'i.^ * g-]',

",&** 9R_VcNQN`]N_PVNYR`

Dk bpq^ Rb``fŽk pb bumlkb bi `lk`bmql ab abofs^a^ m^o`f^i v pb fkf`f^ ^iib`qlo bk pr klq^`fŽk u pr qbojfklildŒ^- Ml rqfifw^objlp ilp obpriq^alp ab bpq^Rb``fŽk bk kfkdrk^ lqo^ m^oqb ab bpqb Ulirjbk )$ `lk il nrb bpqb qbj^ mrbabljfqfopb l mlpmlkbopb pfk m‹oafa^ ab `lkqfkrfa^a-

Dk bi `^mŒqril 0 pb abcfkfŽ rk^ crk`fŽk `ljl rk^ `loobpmlkabk`f^ nrb^pl`f^ ^ `^a^ l_gbql ab rk `lkgrkql W rk l_gbql v pŽil rkl ab lqol `lkgrkql X+abkljfkŠkalpb ^i `lkgrkql s _jhdidj ab i^ crk`fŽk- G^pq^ ^elo^ pb e^k `lk,pfabo^al crk`flkbp `rvl aljfkfl bo^ rk `lkgrkql ab mrkqlp abi bgb ab i^p T+

Dpq^p crk`flkbp plk i^p ii^j^a^p `lj•kjbkqb api^dji`n _` pi\ q\md\]g` m`\g,Ml bp afcŒ`fi buqbkabo jr`e^p ab i^p fab^p abi BŠi`ril ^ crk`flkbp ab alp l jŠps^of^_ibp ob^ibp-

Tk^ api^d‡i m`\g _` _jn q\md\]g`n m`\g`n bp rk^ crk`fŽk `rvl aljfkfl W bprk `lkgrkql ab mrkqlp abi mi^kl rs+ Rf pb fkaf`^ mlo ` af`e^ crk`fŽk+ pr s^ilobk bi mrkql %r) s& bp rk k•jbol ob^i nrb pb abpfdk^ mlo `%r) s&+ Dp cŠ`fi fj^dfk^o`Žjl rk^ crk`fŽk ab bpq^ `i^pb mrbab mobpbkq^opbbk rk mol_ibj^ cŒpf`l cf`qf`fl-Olo bgbjmil+ pb^ rk^ mi^`^ ab jbq^i ifp^ bk cloj^ ab afp`l `fo`ri^o ab o^afl3 `j nrb bpq‹ pfqr^a^ bk bi mi^kl rs `lk bi `bkqol bk bi lofdbk+ v nrb pb `^,ifbkqb ab q^i j^kbo^ nrb i^ qbjmbo^qro^ bk `^a^ rkl ab prp mrkqlp 'u+ s& bp05 , r0

+ c do^alp `bkqŒdo^alp- Rf pb fkaf`^ mlo `%r) s& i^ qbjmbo^qro^ bk bimrkql %r) s&) bkqlk`bp ` bp rk^ crk`fŽk ab alp s^of^_ibp abcfkfa^ mlo

'3-16( a&s*s& < 05 , t1 , b-

Di aljfkfl ab bpq^ Zrk`fŽk bp bi `lkgrkql ab qlalp ilp mrkqlp %r) s& `rv^ afp,q^k`f^ ^i lofdbk kl bp prmboflo ^ 3- Cbi qblobj^ ab OfqŠdlo^p pb abar`b nrb

Page 260: Calculus

+-) ?•f]ofi ^c`_l_h]c[f

qlalp ilp mrkqlp %r)u( pfqr^alp ^ afpq^k`f^ l abi lofdbk+ p^qfpc^`bki^ b`r^`fŽk

%1+/5& r0 * t0 < l0Š

Olo q^kql+bi aljfkfl ab i^ crk`fŽk bpq^oŠcloj^al mlo qlalp ilp mrkqlp %r) u(nrb p^qfpc^`bki^ abpfdr^ia^a r0 * v1 z 05- N_p‹osbpb nrb bk i^ `fo`rkcbobk`f^'3-17( i^ qbjmbo^qro^ pboŠ`%r)u( < 05 , l!+ Dp ab`fo+ i^ crk`fŽk ` bp `lkpq^kqbbk `^a^ `fo`rkcbobk`f^ `lk `bkqol bk bi lofdbk 's‹^pb cfdro^ 3-08(-

G^v alp j‹qlalp •qfibp m^o^ l_qbkbo rk^ obmobpbkq^`fŽkdblj‹qof`^ abrk^ crk`fŽk ab alp s^of^_ibp- Tkl bp mlo jbafl ab rk^ moj_l`c]c_ bk bi bpm^`fl-O^o^ `lkpqorfo bpq^ prmbocf`fbpb fkqolar`b rk qbo`bo bgb `lloabk^al 'ii^j^albgb t&) nrb m^p^ mlo bi lofdbk v bp mbombkaf`ri^o ^i mi^kl st, Dk i^ m^o^ibi^^i bgb u nrb m^p^ mlo bi mrkql rs) v ^ m^oqfoab bpqbmrkql+ pb qlj^ rk^ `llo,

tu

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EHFTQ@ 3-08 I\ o`hk`m\opm\ `n ^jino\io``i ^\_\ ^dm^pia`m`i^d\ ^ji ^`iomj `i `g

jmdb`i,

EHFTQ@ 3-1/ Ppk`mad^d`m`km`n`io\_\ kjmg\ `^p\^d‡i u < 05, s0 [ v1

abk^a^ t) fdr^i ^ i^ nrb a^ i^ b`r^`fŽk t < `%r)t'* l_qbkf‹kalpb bi mrkql %r)v+v(+

Di ird^o ab qlalp bpqlp mrkqlp bp i^ prmbocf`fbnrb obmobpbkqi^ crk`fŽk-K^ prmbocf`fb `loobpmlkafbkqb ^i bgbjmil ^kqboflojbkqb bumrbpql bpqŠ af,

_rg^a^ bk i^ cfdro^ 3-1/- Rf pb pfq•^ rk qbojŽjbqol bk rk mrkql %r)u( ab i^mi^`^+ bi qlmb ab i^ `lirjk^ ab jbo`rofl ql`^oŒ^^ i^ prmbocf`fbmob`fp^jbkqbbk bi mrkql %r)v+w( alkab t < `%r)s&) rk^ sbw bibdfa^ i^ rkfa^a pl_ob bi bgb t^ab`r^a^jbkqb-

Nqol qfml ab fj^dbk dblj‹qof`^ ab rk^ crk`fŽk ab alp s^of^_ibp pb mrbabaf_rg^o `ljmibq^jbkqb bk bi mi^kl rs+ Dp bi j‹qlal ab i^p f•h_[m ^_ hcp_f nrb

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A`mdq\_\n k\m^d\g`n 130

s

s ^vy

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y

'^( u < rs '_( Bros^p ab kfsbi9 rs < _

EHFTQ@ 3-10 ']( Ppk`mad^d`pt\ `^p\^d‡i `n u;st, '^( @pmq\n_` idq`g ^jmm`nkji_d`io`n\ st < ^jino\io`,

pb rp^ bk i^ `lkcb``fŽk ab j^m^p m^o^ obmobpbkq^ork qboobkl qofafjbkpflk^ibk rk af_rgl _fafjbkpflk^i- Rb prmlkb nrb i^ prmbocf`fb^kqbp abcfkfa^ pb e^`loq^al mlo s^oflp mi^klp elofwlkq^ibp 'm^o^ibilp ^i mi^kl rs&) mlo il nrb i^pfkqbopb``flkbp `lk i^ prmbocf`fbpboŠk rk^p `ros^p cloj^a^p mlo ^nrbiilp mrkqlp%r) v+ w( `rv^ ^iqro^ w bp `lkpq^kqb- Oolvb`q^kal bpq^p`ros^p bk bi mi^kl rs

pb l_qfbkb rk^ c^jfif^ ab ^pmq\n _` idq`g, B^a^ `ros^ ab kfsbi bpqŠ cloj^a^mlo qlalp v pŽil ilp mrkqlp %r) s& `rv^p `lloabk^a^p p^qfpc^`bk i^ b`r^`fŽk`%r) u( < _8 alkab _ bp i^ ^iqro^ `lkpq^kqb m^o^ ^nrbii^ `ros^ m^oqf`ri^o- Dkbi bgbjmil ^kqbp jbk`flk^al+ i^p iŒkb^pab kfsbi plk `fo`rkcbobk`f^p `lk`‹kqof,`^p nrb obmobpbkq^ki^p `ros^p ab qbjmbo^qro^ `lkpq^kqb+ l dnjo`mh\n* `ljl pbaf_rg^k bk rk j^m^ jbqbloliŽdf`l- Nqol bgbjmil ab rk^ prmbocf`fbv prp `ros^pab kfsbi pb mobpbkq bk i^ cfdro^ 3-10- K^ b`r^`fŽk bk bpqb`^pl bp w< rs+ K^prmbocf`fbab cloj^ ab ~pfii^ ab jlkq^o‚ pb `lkl`b `lk bi klj_ob ab k\m\]j+gjd_` cdk`m]‡gd^j,

K^p iŒkb^pab kfsbi bk ilp j^m^p qlmldoŠcf`lp pb af_rg^k cob`rbkqbjbkqbm^o^`^a^ 14 j ab ^iqro^- Br^kal bpqŠkaf_rg^a^p jrv grkq^p+i^ ^iqro^ `^j_f^oŠmfa^jbkqb ^i m^p^oab rk^ iŒkb ab kfsbi ^ i^ pfdrfbkqb: bpql l`roob bk i^ moluf,jfa^a ab rk jlkqb bp`^om^al- Br^kal i^p iŒkb^pab kfsbi bpqŠk_^pq^kqbafpq^k,`f^a^p i^ ^iqro^ s^oŒ abpm^`fl- Rb mrbab qbkbork^ fab^ ab il bp`^om^al ab rkqboobkl`lkpfabo^kal il bpm^`f^a^pnrb pb mobpbkq^kprp iŒkb^pab kfsbi- Rfk bj_^o,dl+ m^o^ ildo^o rk^ fkcloj^`fŽk mob`fp^pl_ob bi `lbcf`fbkqb ab s^of^`fŽk ab i^^iqro^+ pb e^ ab abcfkfo i^ prmbocf`fbmlo jbafl ab rk^ crk`fŽk ^ i^ nrb pb ibmrba^k ^mif`^o ilp `lk`bmqlp abi BŠi`ril afcbobk`f^i-

Page 262: Calculus

131 @ƒg^pgj_da`m`i^d\g

w

. Oi^kl alkab t < Xl

Rrmbocf`fb ab.& b`r^`fŽk u < a&s*t'

V

s

EHFTQ@ 3-11 @pmq\ _` dio`mn`^^d‡i _` pi\ npk`mad^d` w < y&s*t' v pi kg\ij v < Xl+

K^ o^wŽk`lk nrb s^oŒ i^ ^iqro^ bk rk mrkql %rj* Xl( abmbkab ab i^ afob`,`fŽk abi jlsfjfbkql ^ m^oqfoab bpqbmrkql- O^o^ j^vlo pfjmif`fa^a pb `lkpŒ,abo^oŠk ^elo^ mob`fp^jbkqb i^p alp afob``flkbp m^o^ibi^p^ ilp bgbpr b v- RrmŽk,d^pb nrb pb qo^q^ ab rk^ prmbocf`fbabcfkfa^ mlo rk^ b`r^`fŽk ab i^ cloj^w< `%r)u( v pb `loq^ bpq^prmbocf`fbmlo rk mi^kl mbombkaf`ri^o^i bgbX q^i `ljlpb fkaf`^ bk i^ cfdro^ 3-11- Dpqbmi^kl bpqŠcloj^al mlo qlalp ilp mrkqlp %r)v+w(abi bpm^`fl m^o^ ilp `r^ibp i^ `lloabk^a^ X bp `lkpq^kqb+X < Xl+ 'K^ b`r^`fŽkt < Xl pb abkljfk^ b`r^`fŽk abi mi^kl(- K^ fkqbopb``fŽk ab bpqb mi^kl `lk i^prmbocf`fbbp rk^ `ros^ mi^k^+`rvlp mrkqlp p^qfpc^`bki^ b`r^`fŽk w< `%r)Xl(-

Dk bpq^ `ros^+ i^ ^iqro^ w< `%r)Xl( bp crk`fŽk pŽil ab s*RrmŽkd^pb ^elo^ nrb pb m^p^ abi mrkql 'u+ Xl( ^i mrkql 'u+ * b) Xl(- Di

`^j_fl ab ^iqro^ `loobpmlkafbkqb bp `%rj * b) Xl( , `%rj* Xl(- Dpql prdfbob i^cloj^`fŽk abi `l`fbkqb ab afcbobk`f^p-

'3-18( a&sj * c* Vj' + a&sj* Vj'

c

m^o^ abpmr‹p e^`bo qbkabo b w N- Rf bpqb `l`fbkqb qfbkab ^ rk iŒjfqb abcfkfal`r^kal c x N+bpqbiŒjfqbpb abkljfk^ g\ _`mdq\_\ k\m^d\g _` a ^ji m`nk`^oj \ s`i `g kpioj &sj*vl(& O^o^ abpfdk^o i^ abofs^a^ m^o`f^i+e^v s^oflp pŒj_lilp pfbkal^idrklp ab ilp jŠp `loofbkqbp9

j-`si) Ui&

U[

Page 263: Calculus

A`mdq\_\n k\m^d\g`n 132

Di pr_Œkaf`b 0 bk i^p alp •iqfj^p klq^`flkbp pb obcfbob i eb`el ab nrb pŽil i^mofjbo^ `lloabk^a^ s^oŒ^ r^kal pb cloj^ bi `l`fbkqb ab afcbobk`f^p bk '3-18(-@pŒpb qfbkb

a& ' 0! a&sj * c*tj' + a&sj *Vj'

0 Vl *Vj < fj -cxL c

@kŠild^jbkqb+ pb abcfkb i^ _`mdq\_\ k\m^d\g m`nk`^oj \ v bk &sj* Xl( mlo

c9' ( he{'ul+ Vj * f' + y&s l +Vj'1 ul+ Vj < fj +

ewK e

pfbkal i^p klq^`flkbp `loobpmlkafbkqbp

Rf pb bp`of_b w< `%r)s&) q^j_f‹k pb rp^k ilp pŒj_lilp \ug\s u \ug\t m^o^abpfdk^o i^p abofs^a^p m^o`f^ibp-

K^ abofs^`fŽk m^o`f^i kl bp rk `lk`bmql krbsl- Rf pb `lkpfabo^ lqo^ crk,`fŽk d ab rk^ s^of^_ib abcfkfa^ mlo i^ b`r^`fŽk

b&s' < a&s* Vj' *

bkqlk`bp i^ abofs^a^ loafk^of^ a$%rj' bp bu^`q^jbkqb 0/ jfpjl nrb i^ abofs^a^m^o`f^i `x%rj*Vj'% Fblj‹qof`^jbkqb+ i^ abofs^a^ m^o`f^i `x%r)Xl( obmobpbkq i^mbkafbkqb ab i^ q^kdbkqb bk rk mrkql ab i^ `ros^ pb•^i^a^ bk i^ cfdro^ 3-11-Cb i^ jfpj^ j^kbo^+ `r^kal s bp `lkpq^kqb+ bp ab`fo s < sj* i^ b`r^`fŽkw< `%rj* u( abcfkb i^ `ros^ fkqbopb``fŽkab i^ prmbocf`fb lk bi mi^kl `rv^ b`r^,`fŽk bp r < Uj% K^ abofs^a^ m^o`f^i `)%rj* u( a^ i^ mbkafbkqb ab i^ q^kdbkqb ^af`e^ `ros^- Cb bpq^p`lkpfabo^`flkbp pb abar`b nrb m^o^ `^i`ri^o i^ abofs^a^m^o`f^i ab `%r)u( obpmb`ql ^ r) pb mrbab `lkpfabo^o u `ljl pf crbo^ `lkpq^kqbv ^mif`^o i^p obdi^p loafk^of^p abi BŠi`ril afcbobk`f^i- @pŒ+mlo bgbjmil+ pf`%r)v( < 05 , s0

+ v1 pb qfbkb on~*v( < , 0s, @kŠild^jbkqb+ pf pb prmlkb s

cfgl pb bk`rbkqo^ Js* u( < , /s+Nqol bgbjmil bp i^ crk`fŽk a^a^ mlo9

'3-2/( a&s* t' < sn`i V * t0 `lp st

Rrp abofs^a^p m^o`f^ibpplk9

ag&U*t' ;n`it + t1n`ist* a0&U*t' < u `lp t + st0n`i st * 0t `lp st ,

Page 264: Calculus

133 @ƒg^pgj _da`m`i^d\g

K^ abofs^`fŽk m^o`f^i bp rk mol`bpl nrb a^ ird^o ^ krbs^p crk`flkbpc < \--\s v .1 < \ag\t ^ m^oqfoab rk^ crk`fŽk a^a^- Orbpql nrb n)v a0 plk^ pr sbw crk`flkbp ab alp s^of^_ibp+ pb mrbabk `lkpfabo^o npn abofs^a^p m^o,`f^ibp- Dpq^ppb abkljfk^k abofs^a^p m^o`f^ibpab n`bpi_j jm_`i ab a v pb fkaf`^k`ljl pfdrb9

N_p‹osbpb nrb a/,0 pfdkfcf`^ '.0(1+ l pb^ i^ abofs^a^ m^o`f^i ab .0 `lk obpmb`ql^ t,Dk i^ l,klq^`fŽk pb fkaf`^ bi loabk ab abofs^`fŽk bp`of_fbkal+

x [ ,B,+&\`&\t\s + \t \s ,

Dpq^abofs^a^ kl pfbjmob `lfk`fab `lk i^ abofs^a^ m^o`f^i nrb obpriq^ ^i fksbo,qfobi loabk ab abofs^`fŽk9

+ I [ &\`&\s \t ‚r \t

Rfk bj_^odl+ i^ fdr^ia^a ab bpq^palp abofs^a^p m^o`f^ibpqfbkb ird^o bk `fboq^p`lkaf`flkbp nrb pb sbofcf`^k mlo i^ j^vloŒ^ ab crk`flkbp nrb ^m^ob`bk bk i^moŠ`qf`^-Dk bi Ulirjbk 00 pb afp`rqfoŠk bpq^p`lkaf`flkbp-

G^`fbkal obcbobk`f^^i bgbjmil '3-16( pb sb nrb prp abofs^a^p m^o`f^ibpabpbdrkal loabk bpqŠka^a^p mlo i^p cŽojri^p pfdrfbkqbp9

b)f%T)s& < ,1+ cds*t' < cds*t' < N+

O^o^ bi bgbjmil '3-2/( pb l_qfbkb9

a/,/&U*t' < [t2 BNR st *-/,0&U*t' < `lp t + st1 `lp st + 1t0n`i st * 0*/&U*t' < `lp t + st1 `lp st + t0 n`ist + 0t0n`i st < -/*0&s*t' *c*0&U*t' < +sn`it + s0t0^jnst + 0stn`ist + 0stn`ist * 1 `lp st

< ,upbkv , U0t0`lp st + 2st pbkst * 1 `lp st,

Tk bpqrafl jŠp abq^ii^al ab i^p abofs^a^p m^o`f^ibppb sboŠ bk bi Ulirjbk 00-

Page 265: Calculus

Be`m^d^djn 134

",&*+ :WR_PVPV\`

Dk ilp Dgbo`f`flp 0 ^i 7+ `^i`ri^o qla^p i^p abofs^a^p m^o`f^ibp ab mofjbol v pbdrkal loabk-Bljmol_^o bk `^a^ `^pl nrb i^p abofs^a^p m^o`f^ibp a/*0&U* u( u a0,*&U* u( plk fdr^ibp-

.+ x%r) s& < r1 * s1 * 1T/s/+

/+ x%r) s& < r pbk %r * s&+

s2- x%r) s& < rs *,

t

3- Xo~*s& < ST/ * s/+

%s,{5 N(-

3, Xo~9s& < pbk &U0s1',

3+`nr)s& < pbk iblp %/r * 0s&Y+

s * t5, x%r) s& < ,, %r ,{5 s&+

s + ts

7- Xo~*s& < ,,,ST/ * s/

%r)s& ,{5 'N+ N(-

8- Cbjlpqo^o nrb r%it% ir& * s%>tY is& < /t pf '^( t < %r * /s&/) '_( t < %r1 * s1&.,/+

0/- Rf Wnr)s& < rs%%r/ * s/&/ m^o^ %r)s& ,{5 'N+ N(+ abjlpqo^o nrb

Page 266: Calculus
Page 267: Calculus

4

D:?68=kA

:AFD: =AF:<D68=kA K 9:D=H68=kA

-&) ?N QR_VcNQNQRb[N V[aRT_NYV[QRSV[VQN&C_VZR_aR\_RZNSb[QNZR[aNYQRYPmYPbY\

Dk bpq^ Rb``fŽk pb bpqraf^oŠ i^ fjmloq^kqb `lkbuflk bufpqbkqb bkqob fkqb,do^`fŽk v afcbobk`f^`fŽk- :Y qfml ab obi^`fŽk bkqob bpqlp alp mol`bplp bp bk `fboq^cloj^ pbjbg^kqb ^i nrb e^v bkqob ~bibs^o ^i `r^ao^al‚ v ~buqo^bo i^ o^Œwr^,ao^a^‚- Rf pb bibs^ ^i `r^ao^al rk k•jbol mlpfqfsl v irbdl pb _rp`^ i^ o^Œw`r^ao^a^ mlpfqfs^ abi obpriq^al+ pb srbisb ^i k•jbol lofdfk^i- @kŠild^jbkqb+ pfpb `^i`ri^ i^ fkqbdo^i ab rk^ crk`fŽk `lkqfkr^ ` pb l_qfbkb rk^ krbs^ crk`fŽk 'i^fkqbdo^i fkabcfkfa^ ab `&nrb abpmr‹p ab abofs^a^ obmolar`b i^ crk`fŽk lofdfk^i `+Olo bgbjmil+ pf a&s' < s0

* rk^ fkqbdo^i fkabcfkfa^ > ab a nrba^ abcfkfa^ mlo9

:s :B _ ƒ

>&s' < a&o' _o < o/ _o < , , , +b b 2 2

alkab ` bp rk^ `lkpq^kqb- Cbofs^kal pb qfbkb9 >%&s'< s0 < a&s', Dpqb bgbjmilfirpqo^ rk obpriq^al dbkbo^i ii^j^al bi mofjbo qblobj^ crka^jbkq^i abi BŠi`rilnrb pb mrbab bkrk`f^o `ljl pfdrb9

SDNQDL@ 4-0- OQHLDQ SDNQDL@ ETMC@LDMS@K CDK B„KBTKN- P`\ a pi\api^d‡i dio`bm\]g` `i X\* sZ k\m\ ^\_\ s _` X\* ]Z, P`\ ` o\g lp` \ x ` x ] W_`adi\hjn pi\ ip`q\ api^d‡i > _`g ndbpd`io` hj_j8

>&s' < oa&o' _o nd

Bsdno` `ioji^`n g\ _`mdq\_\ >%&s' i ^\_\ kpioj s _`g dio`mq\gj \]d`moj &\*]' `i `glp` a `n ^jiodip\* v k\m\ o\g s o`i`hjn

'4-0( >%&s'< y&s' ,

136

Page 268: Calculus

+-1 O`g\^d‡i `iom` dio`bm\^d‡i u _`mdq\^d‡i

C^jlp mofjbol rk^ grpqfcf`^`fŽkdblj‹qof`^ nrb prdfbob bi mlonr‹ bi qblob,j^ ab_b pbo `fboql: irbdl a^jlp rk^ abjlpqo^`fŽk ^k^iŒqf`^-

Fio`mkm`o\^d‡i b`jh„omd^\, K^ cfdro^ 4-0 jrbpqo^ i^ doŠcf`^ab rk^ crk`fŽk `bk rk fkqbos^il W[)\Y+ Dk i^ cfdro^+b bp mlpfqfsl u

`L8)c n a%!%! a&o'_o < b a&o'_o + b a&o' _o < >&s * c' + >&s' ,

Di bgbjmil bp bi ab rk^ crk`fŽk `lkqfkr^ bk qlal bi fkqbos^il Wr)r * bY+ Olo`lkpfdrfbkqb+ mlo bi qblobj^ abi s^ilo jbafl m^o^fkqbdo^ibp+qbkbjlp

>&s * c' + >&s' < ca&u'* alkab r w w999::s * c ,

Krbdl+ obpriq^

'4-1( >&s * cx + >&s' < a&u' *

M s v s * c ]

EHFTQ@ 4-0 Fio`mkm`o\^d‡ib`jh„omd^\ _`g kmdh`mo`jm`h\ api_\h`io\g _`g @ƒg^pgj,

v+ mrbpql nrb r w wz r * b) bk`lkqo^jlp nrb `%t&w `%r&`r^kal b w N `lks^ilobp mlpfqfslp- Rf b w N `lk s^ilobp kbd^qfslp+ pb o^wlk^ bk cloj^ m^ob`fa^-Olo `lkpfdrfbkqb+ =$%r&bufpqbu bp fdr^i ^ `%r&+

Dpqbo^wlk^jfbkql prmlkb nrb i^ crk`fŽk ` bp `lkqfkr^ bk rk `fboql `iojmijabi mrkql s, Ml l_pq^kqb+i^ efmŽqbpfpabi qblobj^ pb obcfbobq^k pŽil ^ i^ `lkqf,krfa^a ab ` bk rk njgj kpioj s, Olo `lkpfdrfbkqb+ m^o^abjlpqo^o bi qblobj^ _^glbpq^efmŽqbpfphƒn _„]dg rqfifw^jlp rk j‹qlal afpqfkql-

Page 269: Calculus

I\ _`mdq\_\ _` pi\ dio`bm\gdi_`adid_\ 138

A`hjnom\^d‡i \i\g…od^\, Rb^ s rk mrkql bk bi nrb ` bp `lkqfkr^ u prmrbpq^s cfg^+pb cloj^ bi `l`fbkqb9

>&s * c' + >&s'

c

O^o^ abjlpqo^o bi qblobj^ pb e^ ab mol_^o nrb bpqb `l`fbkqb qfbkab ^ n%r&r^kalc x N- Di krjbo^alo bp9

GU)c GU GU)c

>&s * c' + >&s' < b a&o' _o + b a&o' _o < s a&o' _o ,

Rf bk i^ •iqfj^ fkqbdo^i pb bp`of_b `%n&< `%r& * W`%n&* `%r&Y obpriq^9

`U)c aU)c>&s * c' + >&s' < s a&s' _o * s Xa&o' + a&s'Z _o ;

aU)c

; ca&s' * s XG&o' + a&s'Z _o *

ab alkab

'4-2( >&s * c' + >&s' < a&s' * 0- GU)cXG&o' + a&s'Z _o ,c c s

Olo q^kql+ m^o^ `ljmibq^o i^ abjlpqo^`fŽk ab '4-0( bp kb`bp^ofl abjlpqo^o nrb

. c!ifj , XG&o' + a&s~' _o < N-ezN c s

Dk bpq^ m^oqbab i^ abjlpqo^`fŽk bp alkab pb e^`b rpl ab i^ `lkqfkrfa^a ab ` bk s,Rf pb abpfdk^ mlo C%b& bi •iqfjl q‹ojfkl abi pbdrkal jfbj_ol ab '4-2(+

pb qo^q^ ab abjlpqo^o nrb C%b&w N `r^kal b w N- @mif`^kal i^ abcfkf`fŽk abiŒjfqb+pb e^ ab mol_^o nrb m^o^ `^a^ ’ = N bufpqb rk Ž = N q^i nrb

'4-3( C%b& ; ’ pfbjmob nrb N ; b ; Ž-

Dk sfoqra ab i^ `lkqfkrfa^a ab ` bk s* a^al rk ’ bufpqb rk k•jbol mlpfqfsl ˆ

q^i nrb9

'4-4( Fa&o' + a&s' G; q’

Page 270: Calculus

14/ O`g\^d‡i `iom` dio`bm\^d‡i t _`mdq\^d‡i

pfbjmob nrb9

'4-5(

Rf pb bifdb b ab j^kbo^ nrb N ; b ; 7+ bkqlk`bp `^a^ o bk bi fkqbos^il %r)r * bYp^qfpc^`b '4-5( u mlo q^kql '4-4( pb sbofcf`^ m^o^ `^a^ o ab bpqb fkqbos^il- @mif,`^kal i^ molmfba^a FPx)cb&o' _og x Px)cgb&o'/ _o* `r^kal b&o' < E%n&* E%r&)ab i^abpfdr^ia^a bk '4-4( pb m^p^ ^ i^ obi^`fŽk9

GaU)c G aU)c aU)cs &G&o'+ a&s'Z _o x s Fa&o' + a&s'/ _o x s iD _o < ieD ; c` ,

Cfsfafbkal mlo c pb sb nrb '4-3( pb sbofcf`^ m^o^ /; c ; _- Rf c ; N+ rk o^,wlk^jfbkql ^kŠildl abjrbpqo^ nrb '4-3( pb sbofcf`^ pfbjmob nrb /; Gdh; 04+il nrb `ljmibq^ i^ abjlpqo^`fŽk-

4-1 Sblobj^ ab i^ abofs^a^ kri^

Rf rk^ crk`fŽk . bp `lkpq^kqb bk rk fkqbos^il %[) \&) pr abofs^a^ bp kri^ bkqlal bi fkqbos^il %[) \&+ X^ ebjlp abjlpqo^al bpqb eb`el `ljl rk^ `lkpb`rbk`f^fkjbaf^q^ ab i^ abcfkf`fŽk ab abofs^a^- S^j_f‹k pb abjlpqoŽ+ `ljl m^oqb `( abiqblobj^ 3-6+ bi ob`Œmol`l ab bp^ ^cfoj^`fŽk nrb ^nrŒ pb mobpbkq^ `ljl qblobj^fkabmbkafbkqb-

SDNQDL@ 4-1- SDNQDL@ CD K@ CDQHU@C@ MTK@- Rf E$%r&< N k\m\ ^\_\ r`i pi dio`mq\gj \]d`moj /* `n a ^jino\io` `i g,

Dpqb qblobj^+ `r^kal pb rqfifw^ `lj_fk^al `lk bi mofjbo qblobj^ crka^,jbkq^i abi BŠi`ril+ klp `lkar`b ^i pbdrkal qblobj^ crka^jbkq^i nrb pb bpqraf^bk i^ Rb``fŽk pfdrfbkqb-

4-2 Erk`flkbp mofjfqfs^p v pbdrkal qblobj^ crka^jbkq^i abi `Ši`ril

CDEHMHBHˆM CD ETMBHˆM OQHLHSHU@- Ri\ api^d‡i O pb gg\h\ kmdhdodq\&j \iod_`mdq\_\' _` pi\ api^d‡i a `i pi dio`mq\gj \]d`moj / nd g\ _`mdq\_\ _` O`n .) `noj `n* pf L$%r&< E%r&k\m\ oj_j r `i g,

Olo bgbjmil+ i^ crk`fŽk pbkl bp rk^ mofjfqfs^ abi `lpbkl bk qlal fkqbos^ilmlonrb i^ abofs^a^ abi pbkl bp bi `lpbkl- Cb`fjlp pi\ mofjfqfs^ v kl g\ mofjf,

Page 271: Calculus

Cpi^dji`n kmdhdodq\n t n`bpi_j o`jm`h\ api_\h`io\g _`g ^ƒg^pgj 03/

odq\*mlonrb pf L bp rk^ mofjfqfs^ ab ` q^j_f‹k il bp L * e m^o^`r^inrfbo `lkp,q^kqbe+ Qb`Œmol`^jbkqb+alp mofjfqfs^p `r^ibpnrfbo^ L v O ab i^ jfpj^ crk`fŽk` pŽil mrbabk afcbofobk rk^ `lkpq^kqb mlonrb pr afcbobk`f^ M + O qfbkb i^ ab,ofs^a^

M%&s'+ N%&s'< y&s' + y&s' < N

m^o^qla^ s bk / v mlo q^kql+pbd•k bi qblobj^ 4-1+ M + O bp `lkpq^kqb bk g,Di mofjbo qblobj^ crka^jbkq^i abi BŠi`ril klp af`b nrb mlabjlp pfbjmob

`lkpqorfo rk^ mofjfqfs^ ab rk^ crk`fŽk `lkqfkr^ mlo fkqbdo^`fŽk-Br^kal `lj_f,k^jlp bpql `lk bi eb`el ab nrb alp mofjfqfs^p ab i^ jfpj^ crk`fŽk q^k pŽilafcfbobkbk rk^ `lkpq^kqb+l_qbkbjlp bi pbdrkal qblobj^ crka^jbkq^i abi BŠi`ril-

SDNQDL@ 4-2- RDFTMCN SDNQDL@ ETMC@LDMS@K CDK B„KBTKN- Ppkjib\+hjn a ^jiodip\ `i pi dio`mq\gj \]d`moj /* t n`\ M pi\ kmdhdodq\ ^p\glpd`m\ _` ``i g,Bioji^`n* k\m\ ^\_\ ` t ^\_\ s `i /* o`i`hjn

'4-6( M&s' < M&^'* oe`o' _o ,

A`hjnom\^d‡i, Olkd^jlp >&s' < `wa&o'_o, Orbpql nrb a bp `lkqfkr^ bk `^a^r ab .) bi mofjbo qblobj^ crka^jbkq^i klp af`b nrb =$%r&< `%r&m^o^qlal r ab f+Dp ab`fo+> bp rk^ mofjfqfs^ ab ` bk g, Orbpql nrb alp mofjfqfs^p ab ` mrbabk af,cbofoq^k pŽil bk rk^ `lkpq^kqb+ab_b pbo =%r& * L%r& < e m^o^ rk^ `fboq^ `lkp,q^kqbf, Br^kal s < `* bpq^cŽojri^ fjmif`^ +M&^' < f* v^ nrb >&^' < N- Olo`lkpfdrfbkqb+ >&s' + M&s' < +M&^'* ab 0/ nrb l_qbkbjlp '4-6(-

Di qblobj^ 4-2 klp fkaf`^ `Žjl bk`lkqo^o rk^ mofjfqfs^ M ab rk^ crk`fŽk`lkqfkr^ `+ Hkqbdo^kal` abpab rk mrkql cfgl ` ^ rk mrkql ^o_fqo^ofls v prj^kali^ `lkpq^kqb L%]& l_qbkbjlp L%r&+ Obol i^ fjmloq^k`f^ ob^i abi qblobj^ o^af`^bk nrb mlkfbkal i^ b`r^`fŽk '4-6( bk i^ cloj^

'4-7( oe`o' _o < M&s' + M&^',

pb sb nrb mlabjlp `^i`ri^o bi s^ilo ab rk^ fkqbdo^ijbaf^kqb rk^ pfjmib pr_pqo^`,`fŽk pf `lkl`bjlp rk^ mofjfqfs^ L+ Di mol_ibj^ ab `^i`ri^o rk^ fkqbdo^i pb e^qo^kpcloj^al bk lqol mol_ibj^+ bi ab e^ii^o i^ mofjfqfs^ M ab `+ Dk i^ moŠ`qf`^+bi pbdrkal mol_ibj^ bp jŠp cŠ`fi ab ^_loa^o nrb bi mofjbol- B^a^ cŽojri^ ababofs^`fŽk molmlo`flk^ ab j^kbo^ fkjbaf^q^ rk bgbjmil ab rk^ mofjfqfs^ ab rk^`fboq^crk`fŽk `) ab alkab obpriq^ rk^ cŽojri^ ab fkqbdo^`fŽkm^o^af`e^ crk`fŽk-

Cb i^p cŽojri^p ab abofs^`fŽk ^kqbpbpqraf^a^p+u `ljl `lkpb`rbk`f^ abi pb,drkal qblobj^ crka^jbkq^i+ pb mrbabk abar`fo i^p pfdrfbkqbpcŽojri^p ab fkqb,do^`fŽk

Page 272: Calculus

141 N_f[]c3h _hnl_ chn_al[]c3h u ^_lcp[]c3h

DIDLOKN 0- Ehn_al[]c3h ^_ jin_h]c[m l[]cih[f_m+ K^ cŽojri^ ab fkqbdo^`fŽk

'4-8( %h< N+ )$*$ &&&"

pb abjlpqoŽ afob`q^jbkqb bk i^ Rb``fŽk 0-12 ^ m^oqfoab i^ abcfkf`fŽk ab fkqbdo^i-@mif`^kal bi- pbdrkal qblobj^ crka^jbkq^i+ mrbab e^ii^opb ab krbsl bpqb obpri,q^al v ^abjŠp dbkbo^ifw^oil m^o^ bumlkbkqbp o^`flk^ibp- Dk mofjbo ird^o pb l_,pbos^ nrb i^ crk`fŽk L abcfkfa^ mlo -

'4-0/(si(/

L%r&:**i * 0

qfbkb `ljl abofs^a^ L$%r&< r! m^o^ `^a^ h bkqbol kl kbd^qfsl- Cb bpq^ fdr^ia^asŠifa^ m^o^ qlal k•jbol ob^i s* ^mif`^kal '4-7( pb qfbkb

G] ]i)/ + [i+Erh ^r < L%\& * L%[& < ,,,

! i * 0

m^o^ `r^inrfbo fkqbos^il W[)\Y+ Dpq^ cŽojri^+ abjlpqo^a^ m^o^ qlal bkqbol h w N`lkpbos^ pr s^ifabw m^o^ qlal bkqbol kbd^qfsl bu`bmql i < , 0+ nrb pb bu`irvbmrbpql nrb bk bi abkljfk^alo ^m^ob`b i * 0- O^o^ abjlpqo^o '4-8( m^o^ i kbd^,qfsl+ _^pq^ mol_^o nrb '4-0/( fjmif`^ L$%r&< r! `r^kal h bp kbd^qfsl v :.: * )$0/ `r^i bp cŠ`fi ab sbofcf`^o abofs^kal M `ljl crk`fŽk o^`flk^i- G^v nrb qbkbo bk`rbkq^ nrb pf h bp kbd^qfsl+ kf L%r& kf L$%r&bpqŠk abcfkfa^p m^o^ r < N+ v ^i ^mif,`^o '4-8( m^o^ h kbd^qfsl pb ab_bk bu`irfo ^nrbiilp fkqbos^ilp W[) \Y nrb `lkqfbkbkbi mrkql s < N-

Di obpriq^al abi bgbjmil 2 ab i^ Rb``fŽk 3-4+ mbojfqb buqbkabo '4-8( ^ qlalpilp bumlkbkqbp l[]cih[f_m 'bu`bmql , 0( pfbjmob nrb bi fkqbdo^kal bpq‹ abcfkfalbk qlalp ilp mrkqlp abi fkqbos^il X\* ]Z bk `lkpfabo^`fŽk- Olo bgbjmil+ pf L:\:]u i < , pb qfbkb9

G] 0 G] U/-0/

]+ _s < s+g 0_s < ,0 < /%s$\ * s[&+

\ vHW \ 1z !

Dk bi `^mŒqril pfdrfbkqb pb abcfkfoŠ rk^ crk`fŽk mlqbk`f^i dbkbo^i ` q^i nrb`%r& < T? m^o^ ][^[ _rjih_hn_ l_[f ]+ Rb sboŠ nrb bpq^ crk`fŽk qfbkb mlo abofs^a^`$%r&< ?T@

+/ v mlo mofjfqfs^ L%r& < T?(.,%] * 0( pf _ :.: )$ 0/ nrb mbojfqfoŠ

buqbkabo i^ '4-8( ^ qlal bumlkbkqb ob^i bu`bmql , 0-N_p‹osbpb nrb L$%r&< f,r kl mrbab l_qbkbopb mlo abofs^`fŽk ab kfkdrk^

crk`fŽk ab i^ cloj^ L%r& < r!+ Ml l_pq^kqb- bufpqb rk^ crk`fŽk L `rs^ abof,

Page 273: Calculus

Mmjkd`_\_`n _` pi\ api^d‡i _`_p^d_\n _` kmjkd`_\_`n _` np _`mdq\_\ 031

s^a^ bp M%&s'< g-s, Tk^ q^i crk`fŽk bp bsfabkqbjbkqb rk^ fkqbdo^i fkabcfkfa^ abi^ jfpj^: mlo bgbjmil9

EB.L%r& < , ^n

0 opf u= N-

Dpq^ fkqbdo^i bufpqb+mrbpql nrb bi fkqbdo^kal bp jlkŽqlkl- K^ crk`fŽk ^pŒabcfkfa^pb ii^j^ gjb\mdohj 'jŠp `lk`obq^jbkqb+ gjb\mdohj i\opm\g', Rrp molmfba^abp pbabp^oolii^oŠk ab cloj^ pfpqbjŠqf`^ bk bi `^mŒqril 5-

DIDLOKN 1- Fio`bm\^d‡i _` n`ij v ^jn`ij, Orbpql nrb i^ abofs^a^ abi pbklbp bi `lpbkl v i^ abi `lpbkl jbklp bi pbkl+ bi pbdrkal qblobj^ crka^jbkq^i a^i^p cŽojri^p pfdrfbkqbp9

c_ `lp s _s < pbk s i_ < pbk ] + pbk \ *Š \ \

c_ pbku _s < &+^jns' i_ < `lp \ + `lp ] ,Š \ \

Dpq^p cŽojri^p pb `lkl`Œ^k v^+ mrbp pb abjlpqo^olk bk bi `^mŒqril 1 ^ m^oqfo abi^ abcfkf`fŽk ab fkqbdo^i-

Rb l_qfbkbk lqo^p cŽojri^p ab fkqbdo^`fŽk ^ m^oqfo ab ilp bgbjmilp 0 v 1qlj^kal prj^p cfkfq^p ab q‹ojfklp ab i^ cloj^ @u!+ ? pbk s* b `lp s* alkab=) >) b plk `lkpq^kqbp-

4-3 Oolmfba^abpab rk^ crk`fŽk abar`fa^p ab molmfba^abpab pr abofs^a^

Rf rk^ crk`fŽk ` qfbkb abofs^a^ `lkqfkr^ .& bk rk fkqbos^il ^_fboql g*bi pb,drkal qblobj^ crka^jbkq^i ^cfoj^ nrb

'4-0 p( a&s' < a&^' * oa%&o'_o

`r^ibpnrfbo^ nrb pb^k s v ` bk g,Dpq^ cŽojri^+ nrb bumobp^ ` bk crk`fŽk ab prabofs^a^ f$)klp mbojfqb abar`fo molmfba^abp ab rk^ crk`fŽk ^ m^oqfoab molmfba^,abp ab pr abofs^a^- @rknrb i^p molmfba^abp pfdrfbkqbp crbolk v^ afp`rqfa^p bk bi`^mŒqril 3+ mrbab pbo ab fkqbo‹p sbo nrb mrbabk abar`fopb `ljl pbk`fii^p `lkpb,`rbk`f^p ab i^ fdr^ia^a '4-00(-

Rrmlkd^jlp nrb .& bp `lkqfkr^ v kl kbd^qfs^ bk g, Rf s = `* bkqlk`bpIz-%&o'_o ƒ N+W mlo q^kql a&s' ƒ a&^', Dp ab`fo+ pf i^ abofs^a^ bp `lkqfkr^ v klkbd^qfs^ bk g*i^ crk`fŽk bp `ob`fbkqb bk g,

Page 274: Calculus

143 O`g\^d‡i `iom` dio`bm\^d‡i v _`mdq\^d‡i

Dk bi qblobj^ 1-8 pb abjlpqoŽ nrb i^ fkqbdo^i fkabcfkfa^ ab rk^ crk`fŽk`ob`fbkqb bp `lksbu^- Olo `lkpfdrfbkqb+ pf `$bp `lkqfkr^ v `ob`fbkqb bk .) i^ fdr^i,a^a '4-00( abjrbpqo^ nrb ` bp `lksbu^ bk g,@kŠild^jbkqb+ ` bp `Žk`^s^ bk ilpfkqbos^ilp bk ilp nrb `$bp `lkqfkr^ u ab`ob`fbkqb-

-&- :WR_PVPV\`

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 0 ^i HC+ bk`lkqo^o rk^ mofjfqfs^ ab a9 bp ab`fo+ bk`lklqo^o rk^ crk`fŽk L q^i nrb L$%r&: `%r& v ^mif`^o bi pbdrkal qblobj^ crka^jbkq^i m^o^ `^i`r,i^o Iz `%r&^r+

s :‹ N-

3+X&s' < Ue * T^90s0+4s)5

5, a&s' < 1z

6, a&s' < /T/-1 + s+g 1*

6+e&s' < 2 pbk s * /r3Š

s< N-0- evs' < 3s0+

0, d%r&< 2s2+ i1u- s< N-

0+ a&m8'< &s * .&%r0* 1(-

s2)s+12, `%r&< **r*1++

s< N-

s< N- .-+ a&s' < T1,0 * 4 `lp s,

00- Cbjlpqo^o nrb kl bufpqb kfkd•k mlifkljfl -z `rv^ abofs^a^ bpq‹ a^a^ mlo i^ cŽojri^`$%r&< f,r+

01- Cbjlpqo^o nrb F Zo\_o < xsgs m^o^ qlal k•jbol ob^i s*02- Cbjlpqo^o nrb

Es /T/

k %n* oF'0 _o < R%r * Gth( m^o^ qlal r ob^i-

03- Tk^ crk`fŽk ` bp `lkqfkr^ m^o^ `r^inrfbo s v p^qfpc^`b i^ b`r^`fŽk

&U a&o' _o < , * s0 * upbk 0s * P BNR 0s‘ l

m^o^ qlal s, B^i`ri^o av /Q' v %& /Q',

04- Dk`lkqo^o rk^ crk`fŽk ` v rk s^ilo ab i^ `lkpq^kqb `* q^i nrb9

IU a&o' _o < `lp s + p m^o^ qlal s ob^i-

05- Dk`lkqo^o rk^ crk`fŽk ` v rk s^ilo ab i^ `lkpq^kqb `* q^i nrb9

a`U oa&o'_o < pbk s + s `lp s + nT/ m^o^ qlal s ob^i-

Page 275: Calculus

Be`m^d^djn 144

06- Dufpqb rk^ crk`fŽk . abcfkfa^ v `lkqfkr^ m^o^ qlal k•jbol ob^i s nrb p^qfpc^`b rk^b`r^`fŽk ab i^ cloj^9

l! 00U

/4U

/6

Il a&o' _o < ! o%/&o'_o * 7 * 8! * b+

alkab b bp rk^ `lkpq^kqb- Dk`lkqo^o rk^ cŽojri^ bumiŒ`fq^m^o^ ,%r& u e^ii^o bi s^iloab i^ `lkpq^kqb `-

07- Tk^ crk`fŽk . bpqŠ abcfkfa^ m^o^ qlal ob^i s mlo i^ cŽojri^

`&U' < 2 f! 0 * pbk o* l 1 * n/ _o ,

Rfk fkqbkq^o bi `Ši`ril ab bpq^ fkqbdo^i+e^ii^o rk mlifkljfl `r^aoŠqf`l j%r&:[(\r(]r0

q^i nrb j%K&:,%K&) j$%K&:,$%K&) v j!%K&:,!%K&+08- C^a^ rk^ crk`fŽk d+ `lkqfkr^ m^o^ qlal s* q^i nrb d'0( < 4 bG d'q( _d < 1- OŽkd^pb

` &s' < y-by&s + o'0b&o' ^n) abjlpqo^o nrb

a%&s' < s A b&o' _o + n!ob&o'_o *- l Il

u `^i`ri^o .!'0( u `!$%.&+

1/- Rfk `^i`ri^o i^p pfdrfbkqbp fkqbdo^ibp fkabcfkfa^p+ e^ii^o i^ abofs^a^ `$%r&bk `^a^ `^pl pf,%r& bp fdr^i ^

10- Rfk `^i`ri^o i^ fkqbdo^i+ `^i`ri^o `$%r& pf . bpqŠ abcfkfa^ mlo i^ cŽojri^

.$! n3a&s' < ,0 ,3 _o ,++2 * o

11- Dk `^a^ `^pl+ `^i`ri^o .'1( pf . bp `lkqfkr^ v p^qfpc^`b i^ cŽojri^ a^a^ m^o^ qlal s ;!99 N-

'^( F7a&o' _o < s/&g * u( -

`!)'^( l a&o' _o < r0&g * r& +

`a&!'

'`( l o0_o < s0&g * s' *

o++&r*u(

a&o' _o < s,‘ l

'a(

12- K^ _^pb ab rk pŽifal bp bi `lkgrkql ab loabk^a^p ab rk^ crk`fŽk kl kbd^qfs^ . bk bifkqbos^il ZN+[Y+ Sla^p i^p pb``flkbp mbombkaf`ri^obp ^ bpb fkqbos^il plk `r^ao^alp- Dislirjbk abi pŽifal bp

\1 + 0\ `lp \ * '1 , \0' pbk \

m^o^ qlal [ ;!99 N- Rrmlkfbkal nrb . bp `lkqfkr^ bk ZN+[Y) `^i`ri^o ,%[&+

Page 276: Calculus

034 O`g\^d‡i `iom` dio`bm\^d‡i u _`mdq\^d‡i

13- Tk jb`^kfpjl fjmrip^ rk^ m^oqŒ`ri^^ 0/ i^odl ab rk^ ob`q^- DpqŠ `lk`b_fal ab j^kbo^nrb i^ mlpf`fŽk ab i^ m^oqŒ`ri^bk bi fkpq^kqb o ^ m^oqfoabi mrkql fkf`f^i N bk i^ ob`q^bpqŠ a^al mlo i^ cŽojri^ `%n&< -0 * 1q pbk o,Di jb`^kfpjl qo^_^g^ mbocb`q^jbkqb e^pq^bi fkpq^kqb 0 < 6R bk prodb rk^ ^sboŒ^fkbpmbo^a^- @ m^oqfoab bpb jljbkql i^ m^oqŒ`r,i^ pb jrbsb `lk sbil`fa^a `lkpq^kqb '0^ sbil`fa^a ^anrfofa^ bk bi fkpq^kqb 0 < 5Q', B^i,`ri^o 9 ^( pr sbil`fa^a bk bi fkpq^kqb 0 < 6S: _( pr ^`bibo^`fŽk bk bi fkpq^kqb o < 6S:

`( pr ^`bibo^`fŽk bk bi fkpq^kqb o < 5Q9 a( pr mlpf`fŽk ^ m^oqfo ab N bk bi fkpq^kqb0 < 5Q, b( G^ii^o bi fkpq^kqb 0 = 6R bk bi nrb i^ m^oqŒ`ri^ srbisb ^i mrkql fkf`f^i N+l _fbk abjlpqo^o nrb krk`^ obdobp^ ^ N-

14- Tk^ m^oqŒ`ri^pb abpmi^w^ ^ 0/ i^odl ab rk^ ob`q^- Rr mlpf`flk bk bi fkpq^kqb o bp `%n&+

Br^kal N z o}x 0+ i^ mlpf`fŽk sfbkb a^a^ mlo i^ fkqbdo^i

000 * 1 pbk 5QU BNR 5QU

Y'0( < 0 1 ^r )l )s

'Ml fkqbkq^o bi `Ši`ril ab bpq^ fkqbdo^0-( O^o^ o „8 0+ i^ m^oqŒ`ri^pb jrbsb `lk ^`bibo^,`fŽk `lkpq^kqb '0^ ^`bibo^`fŽk ^anrfofa^ bk bi fkpq^kqb o < 0(- B^i`ri^o9 ^i pr ^`bibo^,`fŽk bk bi fkpq^kqb o < 1: _( pr sbil`fa^a `r^kal o < 0: b( pr sbil`fa^a `r^kal 0 = 0:

a( i^ afcbobk`f^ a&o' + a&0( `r^kal o = 0-

15- Dk `^a^ rkl ab ilp `^plp pfdrfbkqbp bk`lkqo^o rk^ crk`fŽk ` '`lk pbdrka^ abofs^a^ '`lkqfkr^( nrb p^qfpc^d^ ^ qla^p i^p `lkaf`flkbp fkaf`^a^p+ l _fbk bumif`^o mlo nr‹ kl bpmlpf_ib bk`lkqo^o rk^ q^i crk`fŽk-'^( d!%r& = N m^o^ `^a^ r) .&'/( < 0+ .$%.&< N-'_( d!%r& = N m^o^ `^a^ r) .&'/( < 0+ .$%.&< 2-'b( %r& = N m^o^ `^a^ r) .$%-&< 0+ `%r&w 0// m^o^ `^a^ mlpfqfsl r+

'a( `!%r& = N m^o^ `^a^ r) .$%-&< 0+ `%r&w 0// m^o^ `^a^ kbd^qfsl r+

16- Tk^ m^oqŒ`ri^pb jrbsb ^ 0/ i^odl ab rk^ ob`q^+pfbkal pr mlpf`fŽk bk bi fkpq^kqb o, `%nf+O^oqb `lk rk^ sbil`fa^a fkf`f^i .$%-&< N X qfbkb rk^ ^`bibo^`fŽk `lkqfkr^ a!&/' >95 m^o^qlal o bk bi fkqbos^il N z o x 0- Cbjlpqo^o nrb i^ sbil`fa^a bp jG( >92 m^o^ qlal o bkrk `fboql fkqbos^il W[) _\+ alkab N ,o [ ; ] ,o 0- pfbkal ] + [ < -

17- C^a^ rk^ crk`fŽk ` q^i nrb i^ fkqbdo^i =%r& < Px`%n&n bufpq^ m^o^ `^a^ r bk rk fkqbo,s^il W[) \Y+ Rb^ b rk mrkql abi fkqbos^il ^_fboql %[) \&+ Blkpfabo^o i^p pfdrfbkqbp ^cfoj^,`flkbp obi^qfs^p ^ ` v >8

^( ` bp `lkqfkr^ bk b-_( ` bp afp`lkqfkr^ bk b-b( ` bp `ob`fbkqb bk %[) \&+a( f$%_&bufpqb-b( f$ bp `lkqfkr^ bk ^,

[& > bp `lkqfkr^ bk b-.0& = bp afp`lkqfkr^ bk b-u( = bp `lksbu^ bk %[)\&+Hi( =$%_&bufpqb-b( =$ bp `lkqfkr^ bk b-

Page 277: Calculus

I\ ijo\^d‡i _` I`d]idu k\m\ g\n kmdhdodq\n 146

HW x2 t

^Dk rk^ q^_i^ `ljl i^ af_rg^a^ ^nrŒ+mlkbo

rk^ Q bk bi `r^ao^al `loobpmlkafbkqb pf i^^cfoj^`fŽk pb•^i^a^ `lk ibqo^ i^qfk^ fjmif`^pfbjmob i^ pb•^i^a^ `lk ibqo^ dofbd^- Cbg^o ilpabjŠp `r^ao^alp bk _i^k`l- Olo bgbjmil+ pf i&

fjmif`^ HW(+j^o`^objlp `lk rk^ P bi `r^ao^alab i^ bpnrfk^ prmboflo fwnrfboa^+ bq`‘---

^

b

a

b

-&. ?N [\aNPVp[ QR ?RVO[Vf]N_N YN`]_VZVaVcN`

Ulis^jlp ^elo^ ^ bpqraf^o i^ obi^`fŽk bkqob fkqbdo^`fŽk v abofs^`fŽk- Oofjbol`ljbkqbjlp rk ml`l i^ klq^`fŽk fkqolar`fa^ mlo Kbf_kfw-

Gbjlp abcfkfal rk^ mofjfqfs^ O ab rk^ crk`fŽk ` `ljl `r^inrfbo crk`fŽkm^o^ i^ nrb L$%r&< `%r&+ Rf ` bp `lkqfkr^ bk rk fkqbos^il+ rk^ mofjfqfs^ sfbkba^a^ mlo rk^ cŽojri^ ab i^ cloj^

M&s' < na&o' _o *Š `

v qla^p i^p abjŠp mkojks^p mrbabk afcbofo ab bp^ q^k pŽil bk rk `lkpq^kqb-Kbf_kfw rpŽ bi pŒj_lil Pa&s' _s m^o^ abpfdk^o rk^ mofjfqfs^ dbkbo^i ab `+ Blkbpq^ klq^`fŽk+ rk^ fdr^ia^a `ljl

'4-01( aa&s' _s < M&s' * b

pb `lkpfabo^ `ljl lqo^ cloj^ ab bp`of_fo L$%r&< `%r&+ Olo bgbjmil+ v^ nrb i^abofs^a^ abi pbkl bp bi `lpbkl+ mlabjlp bp`of_fo

'4-02( H`lp s _s < pbk s * b -

@kŠild^jbkqb+ v^ nrb i^ abofs^a^ ab rh(.,%h * 0( bp u!+ mlabjlp bp`of_fo

'4-03( Fsi)g

s%o_s <,, * a+i * 0

Page 278: Calculus

147 N_f[]cƒh _hnl_ chn_al[]cƒh v ^_lcp[]cƒh

m^o^ `r^inrfbo mlqbk`f^ o^`flk^i `lk q^i nrb i <.< , 0- Di pŒj_lil b obmobpbkq^rk^ `lkpq^kqb ^o_fqo^of^ ab jlal nrb `^a^ rk^ ab i^p fdr^ia^abp '4-02( v '4-03(bp bk ob^ifa^a rk^ ^cfoj^`fŽk bk qlokl ^ rk `lkgrkql `ljmibql ab crk`flkbp-

@ mbp^o ab i^ pbjbg^kw^ ^m^obkqb+bi pŒj_lil ``%r&^r bp `lk`bmqr^ijbkqbafpqfkql abi pŒj_lil ab fkqbdo^`fŽk `w%r& r+ Klp alp e^k pfal lofdfk^alp mlomol`bplp `ljmibq^jbkqb afpqfkqlp9 i^ afcbobk`f^`fŽk v i^ fkqbdo^`fŽk- Rfk bj,_^odl+ `ljl bpqlp mol`bplp bpqŠk obi^`flk^alp mlo ilp qblobj^p crka^jbkq^ibpabi BŠi`ril+ e^v obi^`flkbp bkqob ^j_lp pŒj_lilp-

Di mofjbo qblobj^ crka^jbkq^i fkaf`^ nrb `^a^ fkqbdo^i fkabcfkfa^ ab `bp q^j_f‹k rk^ mofjfqfs^ ab `+ Olo il `r^i+ bk '4-01( pb mrbab prpqfqrfo L%r& mlo`8a&o' _o alkab ` bp rk `fboql iŒjfqb fkcboflo v obpriq^9

'4-04( I a&s' _s < oa&o' _o * a-

Dpql fkaf`^ nrb pb mrbab `lkpfabo^o bi pŒj_lil aa&s' _s `ljl obmobpbkq^kqbab rk^ fkqbdo^i fkabcfkfa^ ab Z+jŠp rk^ `lkpq^kqb-

Di pbdrkal qblobj^ crka^jbkq^i+ bumobp^ nrb m^o^ `^a^ mofjfqfs^ L ab `v `^a^ `lkpq^kqb a+pb qfbkb9

F7a&s' _s < XM&s' * ah09-

Rf pb prpqfqrvb M&s' * b mlo aa&s' _s* bpq^ cŽojri^ pb mrbab bp`of_fo bk i^ cloj^9

'4-05( F7a&s' _s < Ha&s' _s 09-

K^p alp cŽojri^p '4-04( v '4-05( mrbabk `lkpfabo^opb `ljl rk^ bumobpflkpfj_Žif`^ ab ilp qblobj^p mofjbol v pbdrkal crka^jbkq^ibp abi BŠi`ril-

Cb_fal ^ rk^ i^od^ qo^af`fŽk+ jr`elp qo^q^alp ab BŠi`ril `lkpfabo^k bipŒj_lil aa&s' _s `ljl obmobpbkq^kqb ab rk^ !fkqbdo^i fkabcfkfa^! v kl ab rk^crk`fŽk mofjfqfs^ l ^kqfabofs^a^- Dpql bpqŠ grpqfcf`^al+ bk m^oqb+mlo i^ b`r^,`fŽk '4-04( nrb af`b nrb bi pŒj_lil aa&s' _s bp+ ^abjŠp ab rk^ `lkpq^kqb ^af,qfs^ a+ rk^ fkqbdo^i fkabcfkfa^ ab `+ Olo i^ jfpj^ o^wŽk+jr`elp clojri^oflp abL^qbjŠqf`^ `lkqfbkbk buqbkp^p ifpq^p ab cŽojri^p ii^j^a^p ~q^_i^p ab fkqbdo^ibpfkabcfkfa^p‚ pfbkal bk ob^ifa^a q^_i^p ab crk`flkbp mofjfqfs^p- O^o^ afpqfkdrfo bipŒj_lil aa&s' ^r ab I99`%r& r bi •iqfjl pb abkljfk^ fkqbdo^i ^_`chc^[+ Orbpqlnrb bi pbdrkal qblobj^ crka^jbkq^i obar`b bi mol_ibj^ ab i^ fkqbdo^`fŽk ^iab _rp`^o mofjfqfs^p+ i^ bumobpfŽk ~q‹`kf`^ ab fkqbdo^`fŽk‚ pb obcfbob ^i bpqrafl abrk j‹qlal pfpqbjŠqf`l m^o^ e^ii^o mofjfqfs^p- Dpq^ qbojfklildŒ^ pb bk`rbkqo^jr`eŒpfjl bk i^ ifqbo^qro^ j^qbjŠqf`^ u pb ^almq^oŠ q^j_f‹k bk bpqb if_ol- @pŒ+

Page 279: Calculus

Ehn_al[]cƒh jil momncno]cƒh 148

mlo bgbjmil+ `r^kal pb mfab i^ ~fkqbdo^i‚ aa&s' _s pb e^ ab bkqbkabo nrb il nrbpb abpb^ bp i^ mofjfqfs^ jŠp dbkbo^i ab `+

Oofk`fm^ijbkqb pb pfdrbk qobp q‹`kf`^p bk i^ `lkpqor``fŽk ab q^_i^p ab fkqb,do^ibp fkabcfkfa^p+ nrb e^ ab `lkl`bo qlal bi nrb abpbb j^kbg^o Šdfijbkqb bifkpqorjbkql abi BŠi`ril- Rlk 0( chn_al[]cƒh jil momncno]cƒh'nrb pb bumlkaoŠ bkbi ^m^oq^al nrb pfdrb(+ j‹qlal _^p^al bk i^ obdi^ ab i^ `^abk^: 1( chn_al[]cƒhjil j[ln_m) j‹qlal _^p^al bk i^ cŽojri^ ab afcbobk`f^`fŽk ab rk molar`ql 'nrbpb bumlkaoŠ bk bi ^m^oq^al 4-8(: v 2(- fkqbdo^`fŽk mlo ^_m]igjimc]cƒh _h `l[]*

]cih_m mcgjf_m) nrb bp rk^ q‹`kf`^ ^idb_o^f`^ nrb pb afp`rqfoŠ ^i cfk^i abi `^mŒ,qril 5- Dpq^p q‹`kf`^p kl pŽil bumif`^k `Žjl pb e^k `lkpqorfal i^p q^_i^p abfkqbdo^ibp fkabcfkfa^p+ pfkl nrb q^j_f‹k bkpb•^k ^ qo^kpcloj^o `fboq^p fkqbdo^ibp+obar`f‹kali^p ^ lqo^p _Špf`^p nrb pb bk`rbkqo^k bk i^p q^_i^p-

4-6 =[aRT_NPVp[]\_ `b`aVabPV\[

Rb^ O i^ `ljmlpf`fŽk ab alp crk`flkbp L v d+ bp ab`fo M%r& < LWa%r&Y

m^o^ qlal s bk rk `fboql fkqbos^il g, Rf `lkl`bjlp i^ abofs^a^ ab M* pb^L$%r&< `%r&) i^ obdi^ ab i^ `^abk^ klp af`b nrb i^ abofs^a^ ab P sfbkb a^a^ mloi^ cŽojri^ M$%r&< L$Wa%r&Ya$%r&+Orbpql nrb L$ < `) bpql klp ^pbdro^ nrbM$%r&< `Wa%r&Ya$%r&+Dk lqo^p m^i^_o^p

'R-06( M%&s'<a&s' fjmif`^ N%&s'<eXb&s'Zb%&s',

Blk i^ klq^`fŽk ab Kbf_kfw+bpq^ ^cfoj^`fŽk mrbab bp`of_fopb abi jlal pfdrfbkqb9

Rf qbkbjlp i^ cŽojri^ ab fkqbdo^`fŽk

'R-07( Ha&s' _s < M&s' * a+

qbkbjlp q^j_f‹k i^ cŽojri^ jŠp dbkbo^i

' R+08( HaXb&s'Zb%&s' _s < MXb&s'Z * b -

Olo bgbjmil+ pf `%r& < `lp r) bk i^ '4-07( ab_boŠ mlkbopb L%r& < pbk r) abjlal nrb '4-08( pb `lksfboqb bk

'R-1/( H`lp a%r&+a$%r& r < pbk a%r&* b -

Dk m^oqf`ri^o+ pf a%r&< rX pb l_qfbkb

I `lp r1Š 0r0 ^r < pbk r1 * b +

Page 280: Calculus

+/) O`g\^d‡i `iom` dio`bm\^d‡i v _`mdq\^d‡i

obpriq^al nrb pb `ljmorb_^ afob`q^jbkqb v `lk c^`fifa^a mrbpql nrb i^ abofs^a^ab pbkr\ bp 0r0 `lp u!+

N_pbosbjlp ^elo^ nrb i^ cŽojri^ dbkbo^i '4-08( bpqŠobi^`flk^a^ ^ i^ '4-07(mlo rk pbk`fiil mol`bpl jb`Škf`l- Rrmlkd^jlp nrb bk '4-08( prpqfqrfjlp a%r& mlork krbsl pŒj_lil p v obbjmi^`bjlp b%&s'mlo _p-_s* pbd•k i^ klq^`fŽk ab Kbf_kfwm^o^i^p abofs^a^p- Dkqlk`bp i^ '4-08( pb qo^kpcloj^ bk

` _pa&p'+_s < M&p'* a-

J[

@i iibd^o ^nrŒrkl bpqŠcrboqbjbkqb qbkq^al ab obbjmi^w^oi^ `lj_fk^`fŽk _p _sJ[

mlo _pj Rf il e^`bjlp+ i^ •iqfj^ cŽojri^ qlj^ bi ^pmb`ql

'4-10( I a&p' _p < M&p'* a-

N_p‹osbpb nrb bpq^cŽojri^ qfbkb bu^`q^jbkqb i^ jfpj^ cloj^ nrb '4-07(+ p^islnrb bk qla^p m^oqbpbk sbw abi pŒj_lil s ^m^ob`bbi pŒj_lil Q+ Dp ab`fo+ `^a^cŽojri^ ab fkqbdo^`fŽkq^i `ljl '4-07( mrbab a^o ird^o ^ lqo^ jŠp dbkbo^i pfkjŠp nrb e^`bo rk^ pfjmib prpqfqr`fŽk ab ,pŒj_lilp- Rb prpqfqrvb s bk '4-07( mlork krbsl pŒj_lil p m^o^l_qbkbo '4-10(+ v abpmr‹p pb `lkpfabo^ nrb p obmobpbkq^rk^ krbs^ crk`fŽk ab r) q^i `ljl o < a%r&+Qbbjmi^w^jlp bkqlk`bp bi pŒj_lil^o mlo i^ `lj_fk^`fŽk a$%r&r) v i^ fdr^ia^a '4-10( pb obar`b ^ i^ cŽojri^ dbkb,o^i '4-08(-

Olo bgbjmil+ pf prpqfqrfjlp s mlo p bk i^ cŽojri^ ` `lp s _s < pbk s * a+l_qbkbjlp

n`lp p _p < pbkp * a-

Dk bpq^•iqfj^ cŽojri^+ p pb mrbab obbjmi^w^o mlo b&s' v _p mlo b%&s'_s* v ob,priq^ rk^ cŽojri^ `loob`q^ ab fkqbdo^`fŽk'4-1/(-

Br^kal bpqbmol`bpl jb`Škf`l pb rp^ \ g\ diq`mn\* `lkar`b ^i ii^j^al j‹qlalab dio`bm\^d‡i kjm npnodop^d‡i, Di l_gbql ab bpqbj‹qlal bp qo^kpcloj^o rk^ fk,qbdo^i `lk rk fkqbdo^kal `ljmif`^al+ q^i `ljl a1s0 `lp s1 _s* bk rk^ fkqbdo^ijŠp pbk`fii^+`ljl i^ ` `lp p _pj Di j‹qlal bp ^mif`^_ib pfbjmob nrb i^ fkqbdo^ilofdfk^i mrbab bp`of_fopbbk i^ cloj^

HdWa%r&Ya$%r&^r )

v^ nrb i^ prpqfqr`fŽk

o < a%r&) ^o < a$%r&r )

Page 281: Calculus

Fio`bm\^d‡i kjm npnodop^d‡i 150

i^ qo^kpcloj^ bk E`%o& oi Rf pb p^_b bcb`qr^o bpq^ fkqbdo^`fŽk+l_qbkbjlp rk^mofjfqfs^+ ii^j‹jlpi^ L%o&) v i^ fkqbdo^i lofdfk^i pb l_qfbkb prpqfqrvbkal o mloa%r& bk i^ cŽojri^ ab L%o&+

Di ib`qlo mrbab `ljmol_^o nrb kl ebjlp ^qof_rfal pfdkfcf`^al ^idrkl ^ ilppŒj_lilp ^r v ^o `ljl q^ibp- Rb rqfifw^k `ljl fkpqorjbkqlp mro^jbkqb clo,j^ibp nrb klp ^vra^k ^ qo^q^oi^p lmbo^`flkbp j^qbjŠqf`^p bk cloj^ jb`Škf`^-B^a^ sbw nrb rqfifw^jlp bi j‹qlal+ bpq^jlp bk ob^ifa^a ^mif`^kal i^ ^cfoj^,`fŽk '4-06(-

Di ‹ufql ab bpqbj‹qlal abmbkab ab i^ e^_fifa^a bk abqbojfk^o i^ m^oqbabfkqbdo^kal nrb pb e^ ab pr_pqfqrfomlo bi pŒj_lil p* v bpq^e^_fifa^a pb ^anrfbob`lk i^ bumbofbk`f^ nrb pb ildo^ obplisfbkal `^plp m^oqf`ri^obp- Klp bgbjmilpbpmb`f^ijbkqb pbib``flk^alp nrb pb a^k ^ `lkqfkr^`fŽk bkpb•^k i^ j^kbo^ ab^mif`^o bpqbj‹qlal bk i^ moŠ`qf`^-

DIDLOKN 0- Hkqbdo^oEr7g`lp r2 ^r+

Pjgp^d‡i, Rb qo^q^ab bk`lkqo^o ` u d ^ab`r^a^jbkqb m^o^ mlabo bp`of_for1 `lp r2 bk i^ cloj^ `Wa%r&Ya$%r&+Orbpql nrb `lp r2 bp rk^ crk`fŽk `ljmrbpq^+pb mrbab qlj^o `%r& < `lp r v a%r& < r2

* v ab bpq^ j^kbo^ `lp r2 pb bumobp^bk i^ cloj^ dWa%r&Y+Blk bpq^ bib``fŽk ab d bp a$%r&< 1r1 v mlo q^kqldWa%r&Ya$%r&< '`lp r2&%1+-&+Di c^`qlo 3 nrb ^m^ob`bab jŠp+ pb mrbab fkqolar`focŠ`fijbkqb jriqfmif`^kal v afsfafbkal bi fkqbdo^kal mlo 3- @pŒpb qfbkb9

r1 `lp r2 < f'`lp r2&%1r1' < QWa%r&Ya$%r&+

G^`fbkal ^elo^ i^ prpqfqr`fŽk o < a%r&< r2* ^o < a$%r& r < 1r1 ^r) pb qfbkb

I s1 `lp s

2_s < I a&p' _p < I `lp p _p < i&pbkp * b-

Rrpqfqrvbkal p mlo s1 bk bi obpriq^al cfk^i+pb l_qfbkb i^ cŽojri^9

Hs0 `lp s1 _s < 0pbks1 * b+

nrb pb mrbab `ljmol_^o afob`q^jbkqb mlo abofs^`fŽk-Br^kal pb qfbkbrk ml`l ab moŠ`qf`^^idrklp ab ilp m^plp pb bcb`q•^k jbk,

q^ijbkqb+ v bi `Ši`ril pb ob^ifw^ab j^kbo^ _obsb `ljl pfdrb9

Rb^ o < u!9 bkqlk`bp+ ^o < 1r7F ^r) v pb l_qfbkb9

H-[990;9/4:'3 _ s < * H'`lp :'3('3:'2 _s' < pH`lp p _p < qpbk p * a < qpbk:'3 * _-

Page 282: Calculus

151 O`g\^d‡i `iom` dio`bm\^d‡i s _`mdq\^d‡i

N_p‹osbpb nrb bi j‹qlal pb mrbab ^mif`^o bk bpqbbgbjmil+ mlonrb bi bumlkbkqbabi c^`qlo s1 bp bi ab s bk `lp s2 afpjfkrfal bk rk^ rkfa^a-

DIDLOKN 1- Hkqbdo^i` `lp! sn`is _s,

Pjgp^d‡i, Rb^p < `lp s, bkqlk`bp _p < +n`is _s* u pb qfbkb

a `lp! sn`is _s < , c'BNR s'0&+n`is _s' < , ap/_R < ] z2 * B <] BNz2 s ,o`-

S^j_f‹k ^nrŒ pb `ljmorb_^ cŠ`fijbkqb bi obpriq^al cfk^i mlo abofs^`fŽk-

DIDLOKN 2- Hkqbdo^oGn`xxx _s,

Pjgp^d‡i, Rb^ p;qx ;UF-0* bkqlk`bp _p < oi.1 _s l pb^ _s-S9;0 _pjOlo q^kql+

`_AB G (Wsu _s < 1 pbk r _p < , 1 `lp p * B < , 1 `lp s s * B -

DIDLOKN 3- a s_sHkqbdo^o - z -

si*u,

Pjgp^d‡i, Rb^ p < 0 * s0* bkqlk`bp _p < 0s _s* bp ab`fo s _s < _p* v

pb l_qfbkb9

Di j‹qlal ab prpqfqr`fŽk bp fdr^ijbkqb ^mif`^_ib ^ i^p fkqbdo^ibpabcfkfa^p-Olo bgbjmil+ m^o^ `^i`ri^o i^ fkqbdo^i abcfkfa^ Pb-0 `lp! s pbks _s pb abqbojfk^mofjbol i^ fkqbdo^i fkabcfkfa^+`ljl pb efwl bk bi bgbjmil 1+ v irbdl e^`fbkalrpl abi pbdrkal qblobj^ crka^jbkq^i pb mrbab bp`of_fo9

H!/0 0 0! /0 0'6q& (`lp! u pbks _s < , , `lp! T < , , `lp! , , `lp! Ml 2 l 2 1

0

2

@idrk^p sb`bp fkqbobp^ ^mif`^o bi pbdrkal qblobj^ crka^jbkq^i ^ i^ fkqbdo^ibumobp^a^bk crk`fŽk ab o8 pfk bj_^odl+ bk bpqb`^pl pb e^k ab fkqolar`fo mob,

Page 283: Calculus

Fio`bm\^d‡i kjm npnodop^d‡i 152

`fp^jbkqb rklp krbslp iŒjfqbp ab fkqbdo^`fŽk- Dk mofjbo ird^o pb mobpbkq^oŠrk bgbjmil bk bi nrb pb mlkd^ ab j^kfcfbpql bi j‹qlal nrb pb pfdrb+ v abpmr‹ppb grpqfcf`^oŠ bi mol`bpl `lk rk qblobj^ dbkbo^i-

DIDLOKN 4- b 0 0 /0 %r * 0( ^r^ `r ^o -

0Ss0)0s)1

Pjgp^d‡i, Rb^ p < s0 * 0s * 2- Dkqlk`bp _p < &0s* 1( _s v mlo q^kql+

&s * 0( _s 0 _p

T s0 * 0s * 2 < 1! rq -O^o^ l_qbkbo ilp krbslp iŒjfqbp ab fkqbdo^`fŽk pb qfbkb bk `rbkq^ nrb p < 00 pfs < 1 X nrb p < 07 pf s < 2+ X bk `lkpb`rbk`f^ bp9

&1 &s * 0( _s < 0 a/6R+/-0 _p < TS .07 < UH7] U0f -I 1 U s0 * 0s * 2 1 00 00

@i jfpjl obpriq^al pb iibd^ `r^kal pb bumobp^ qlal bk crk`fŽk ab s,

20 &s * 0( _s < T s0 * 0s * 202

< Uf7 , Uif -1 Us0 * 0s * 2 1

Cbjlpqo^jlp ^elo^ rk qblobj^ dbkbo^i nrb grpqfcf`^ bi mol`bpl pbdrfal bkbi bgbjmil 4-

SDNQDL@ 4-3- SDNQDL@ CD RTRSHSTBHˆM O@Q@ HMSDFQ@KDR- Ppkjib\hjnlp` d od`i` pi\ _`mdq\_\ ^jiodip\ d&`i pi dio`mq\gj \]d`moj g,P`\% `g ^jiepioj_` q\gjm`n lp` ojh\ d `i / t npkjib\hjn lp` ` `n ^jiodip\ `i X*Bioji^`n k\m\^\_\ s t ^\_\ ` `i /* o`i`hjn

'4-11( oeXb&o'Zb%&o'_o < F7%ww&a&p' _p ,

A`hjnom\^d‡i, Rb^ \ < b&^' u abcfk^jlp alp krbs^p crk`flkbp M u M abipfdrfbkqb jlal9

M&s' < nua&p' _p

Š \pf s BG* N&s' < oeXb&o'Zb%&o'_o pf s Bg,

Page 284: Calculus

+/- N_f[]cƒh _hnl_ chn_al[]cƒh v ^_lcp[]cƒh

Orbpql nrb O v O plk fkqbdo^ibpfkabcfkfa^p ab crk`flkbp `lkqfkr^p+ qfbkbk abof,s^a^p a^a^p mlo i^p cŽojri^p

M%&s'< a&s'* N%&s'< aXb&s'Zb%&s',

Ki^jbjlp ^elo^ N ^ i^ crk`fŽk `ljmrbpq^+ N%r&< LWa%r&Y+Blk i^ obdi^ abi^ `^abk^+ bk`lkqo^jlp

O%&s'< M%Xb&s'Zb%&s'< aXb&s'Zb%&s'< N%&s',

@mif`^kal alp sb`bp bi pbdrkal qblobj^ crka^jbkq^i+ l_qbkbjlp

aQE!)& ap`!*'a&p' _p < M%&p'_p < MXb&s'Z+ MXb&^'Z< O&s' + O&^' *

rib( rib(

v

naXb&o'Zb%&o'_o < nN%&o'_o < nO%&o'_o < O&s' + O&^' ,

Dpql abjrbpqo^ nrb i^p alp fkqbdo^ibp'4-11( plk fdr^ibp-

4-7 Dgbo`f`flp

Di( ilp bgbo`f`flp abi 0 ^i 1/+ ^mif`^o bi j‹qlal ab prpqfqr`fŽk m^o^ `^i`ri^o i^p fkqbdo^ibp-

0- `U1W*0 ^r+ 7, `7-2`lp 1uU&3 , pbk 0s _s,

a pbk s_s0/- '2 * BNR U'0 Š

` n`is_s00- -z-

,U `lp! U

07pbk vq&:!*i_s01- - n,:,,: -

2 +qs)g

/1, `si+/ pbk si _s* i !! N-

` r•^r03- ‘ n+,,z&

,sH ,ub

/3, ao&/ )o'/-2_o,

.3+ G&U0 * .&*1-0^r+

0}asx_s,

1}ax_s,

`/-1 s_s3-- z-

+0-1U 1 , 1s

` &s * 0( _s3, %r/ * /r * 1(2 ‘

3+ an`i1 s ^r+

5, ` t%t * /'g-1 _u*

`@LPU_s7- - 2 ‘pbk U

Page 285: Calculus

Bd`m^d^djn 154

/5, Fs0&6s1 * 05'0-1_s,

` &n`is * `lp s' _s/6, &n`is [ `lp U'/-1 ,

Gs_s

08- ,z<<<<9999:9999<<<9<<<,UH * s0 * U'i * s0'1

` &U0* 0 , 0U'/-3 _s1/- ) &

'[

10- Cbar`fo i^p cŽojri^p ab ilp qblobj^p 0-07 v 0-08 mlo jbafl abi j‹qlal ab prpqfqr`fŽk-11- Rb^

d%!nLB%r)[& < ' 1 1( ^n)

l o)\l

alkab \< N v m v l plk bkqbolp mlpfqfslp- Cbjlpqo^o nrb C&s* \' < \M)/+0NC&s \* /',12- Cbjlpqo^o nrb

He _o ..,$! _o

/)o0: /)o0

&! 0

pf s< N-

13- Cbjlpqo^o nrb

F8 sh&g + u(! _s < F8 si&g + s'h _s ,

pf h v i plk bkqbolp mlpfqfslp14- Cbjlpqo^o nrb

Fx-0 ax-0l `lp&!u pbk&!s _s < 0+h l `lp>&s _s,

pf h bp rk bkqbol mlpfqfsl-

15- '^( Cbjlpqo^o nrb

n 5mdxu.'pbk s' _s < , qbpbk s' _s ,J + J

XFi_d^\^d‡i8 p < om+ sZ,

'_( @mif`^o '^( m^o^ abar`fo i^ cŽojri^9

cws pbku c/ _s**** ^r < 6n ,,,

l 0 * `lp! s l 0 * s0 Š

16- Cbjlpqo^o nrb F '0 , s0'i+g-0 _s < Iz.1 `lp1k Q _p pf i bp rk bkqbol mlpfqfsl- XFi_d^\+^d‡i8 s < pbk p,Z K^ fkqbdo^i abi pbdrkal jfbj_ol pb mrbab `^i`ri^o mlo bi j‹qlal abfkqbdo^`fŽk mlo m^oqbpnrb pb bumlkaoŠ bk i^ Rb``fŽk pfdrfbkqb-

Page 286: Calculus

+// O`g\^dƒi `iom` dio`bm\^d4i t _`mdq\^d4i

-&1 =[aRT_NPVp[]\_ ]N_aR`

Rb abjlpqoŽ bk bi `^mŒqril3 nrb i^ abofs^a^ ab rk molar`ql ab alp crk,`flkbp ` v d bpqŠa^a^ mlo i^ cŽojri^9

c%&s'<a&s'b%&s'* e%&s'b&s'*

alkab b%T& < `%r&+a%r&+So^ar`fbkal bpql ^ i^ klq^`fŽk ab Kbf_kfw m^o^ mof,jfqfs^p pb qfbkb ` a&s'b %&s'_s * ` a%&s'b&s'_s; a&s'b&s' * a+ nrb pb bp`of_brpr^ijbkqb bk i^ cloj^

'4-12( ` a&s'b%&s'_s <a&s'b&s' + ` a%&s'b&s'_s * b -

Dpq^ fdr^ia^a+ `lkl`fa^ mlo cŽojri^ ab dio`bm\^d4i kjm k\mo`n*a^ ird^o ^ rk^krbs^ q‹`kf`^ ab fkqbdo^`fŽk-

O^o^ `^i`ri^o rk^ fkqbdo^i+mlo bgbjmil ` f&s' _s* ^mif`^kal '4-12(+ pb e^kab bk`lkqo^o alp crk`flkbp ` v d ab j^kbo^ nrb e%r& pb mrba^ bp`of_fo bk i^cloj^ `%r&a$%r&+Rf bpql bp mlpf_ib+^mif`^kal '4-12( pb qbkaoŠ9

` f&s' _s < a&s'b&s' + ` b&s'a%&s'_s * b+

obar`f‹kalpb bi `Ši`ril ab i^ fkqbdo^i a^a^ ^i ab i^ ` b&s'e%&s'_s, O^o^ nrbbi j‹qlal pb^ bcf`^wpb e^k ab bibdfo ` v d ^ab`r^a^jbkqb+ ab j^kbo^ nrb bpq^•iqfj^ fkqbdo^i mrba^ `^i`ri^opb `lk jŠp c^`fifa^a nrb i^ lofdfk^i- @idrk^psb`bp+ obfqbo^kal i^ ^mif`^`fŽk ab '4-12( pb iibd^ ^ rk^ fkqbdo^i ab jŠp cŠ`fi`Ši`ril l nrb pb bk`rbkqo^ bk i^ q^_i^- Klp bgbjmilp obprbiqlp ^ `lkqfkr^`fŽkmlkbk ab j^kfcfbpql i^p sbkq^g^p ab bpqbj‹qlal- Dk bi `^pl ab fkqbdo^ibpabcf,kfa^p+ i^ cŽojri^ '4-12( pb qo^kpcloj^ bk

E7a&s'b%&s'_s < a&]'b&]' + a&\'b&\' + E7a%&s'b&s'_s ,

Olkfbkal p < a&s' v q < b&s' pb qfbkb_p, < a%&s'_s, v _q < b%&s'_s v i^cŽojri^ ab fkqbdo^`fŽk mlo m^oqbpqlj^ rk^ cloj^ ^_obsf^a^ nrb m^ob`b jŠpcŠ`fi ab ob`loa^o9

'4-13( ` p _q < pq + ` q _p * b -

Page 287: Calculus

Fio`bm\^d‡i kjm k\mo`n 156

DIDLOKN 0- Hkqbdo^o Fs `lp s _s,

Pjgp^d‡i, Rb bifdb y&s' < s v b%&s'< `lp s* ab alkab a%&s'< 0 X b&s' :< pbk s v bk sfoqra ab '4-1/( pb qfbkb9

'4-14( I s `lp s _s < s n`is + I pbk s _s * a < upbk s * `lp s * _-

N_p‹osbpb nrb bk bpqb `^pl+ i^ pbdrka^ fkqbdo^i v^ bp `lkl`fa^-

O^o^ bcb`qr^o bi jfpjl `Ši`ril rqfifw^kal i^ klq^`fŽk ^_obsf^a^ ab '4-13( pbbp`of_b9

p < u+ _q < `lp s _s*

_p < _s* q < Hlp u _s < pbk s *

I s `lp s _s < pq + I q _p < s pbk s + I pbk s _s * b < s pbk s * `lp s * b -

Rf pb er_fbo^ bibdfal p < `lp s u _q < s _s* ab alkab _p < , pbk s _sv q < dU0* v '4-13(+ obpriq^oŒ^9

I u`lp u _s <qu1BNR U + pI s0

& ,pbku( _s * b < qu1 `lpu * pI s0 n`is _s) B-

Bljl i^ •iqfj^ fkqbdo^i nrb ^m^ob`b kl e^ pfal qla^sŒ^ `^i`ri^a^+ bpq^ bib``fŽkab p* q kl bp •qfi m^o^ bi `Ši`ril ab i^ fkqbdo^i a^a^- N_p‹osbpb+ pfk bj_^odl+nrb bpq^ •iqfj^ b`r^`fŽk mlaoŒ^ obplisbopb obpmb`ql ^ Fs0 pbk s _s v ^mif`^o'4-14( `lk il `r^i pb l_qbkaoŒ^9

Hs0 pbk s _s < 0s pbk s * 1 `lp s + s0 `lp s * b -

DIDLOKN 1- Hkqbdo^o Fs0 `lp s _s,

Pjgp^d‡i, Rb^ p < s0 v _q < `lp s _s* bkqlk`bp _p < 0s_s v q :F `lp s _s < pbk s* `lk il `r^i pb qfbkb9

&3,04' ` s/^jns_s < Hp _q < pq + ` q_p * b < u1pbku , 1 ` sn`is_s * `-

Page 288: Calculus

+/1 O`g\^d‡i `iom` dio`bm\^d‡i t _`mdq\^d‡i

Dpq^•iqfj^ fkqbdo^ipb mrbab `^i`ri^o ^mif`^kal ab krbsl bi j‹qlal ab fkqbdo^,`fŽk mlo m^oqbp-Orbpql nrb bp ^kŠildl ^i bgbjmil 0+ pb mrbab bp`of_fo afob`q^,jbkqb bi obpriq^al

Es pbku _s < +s `lp s * pbks * _-

Rrpqfqrvbkal bk '4-15( v ^dorm^kal i^p alp `lkpq^kqbp ^o_fqo^of^pbk rk^+ pbqfbkb9

Es0 `lp s _s < s0 pbks * 0s `lp s + 1 pbks * b-

DIDLOKN 2- @idrk^p sb`bp bi j‹qlal c^ii^ mlonrb `lkar`b ab krbsl ^ i^fkqbdo^i lofdfk^i- Olo bgbjmil+ ^i fkqbkq^o `^i`ri^o mlo m^oqbp i^ fkqbdo^ias+/ _s, Rf pb e^`b p < s v _q < s+0_s* bkqlk`bp Xs88_s < ap _q, Blk bpq^bib``fŽk ab p v q pb qfbkb _p < _s v q < , t,0 ab j^kbo^ nrb '4-13( a^9

'4-16( Eu,0 _s < Ep _q < pq + Eq _p * a < ,0 * Eu,0 _s * a+

v pb srbisb ^i mrkql ab m^oqfa^-Olo lqo^ m^oqb+i^ pfqr^`fŽk kl jbglo^ pf pb fk,qbkq^ p < sh v _q < s+h

*/ _s,

Dpqbbgbjmil pb rp^ `lk cob`rbk`f^ m^o^ bsfabk`f^o i^ fjmloq^k`f^ ab kllisfa^o i^ `lkpq^kqb ^o_fqo^of^B- Rf bk i^ cŽojri^ '4-16( kl pb er_fbo^ bp`ofqli^ a+pb er_fbo^ iibd^al ^ i^ b`r^`fŽk as+. _s < , 0 * as+. _s nrb pb rqfifw^^idrk^p sb`bp m^o^ a^o rk^ ^m^obkqbabjlpqo^`fŽk ab nrb N < , 0-

Bljl ^mif`^`fŽk abi j‹qlal ab fkqbdo^`fŽk mlo m^oqbp+pb l_qfbkb lqo^ sbo,pfŽk abi qblobj^ abi s^ilo jbafl mlkabo^al m^o^ fkqbdo^ibp'qblobj^ 2-05(-

SDNQDL@ 4-4- RDFTMCN SDNQDL@ CDK U@KNQ LDCHN O@Q@ HMSDFQ@KDR-

Ppkjib\hjn lp` d `n ^jiodip\ `i X\*]Z* t lp` ` od`i` _`mdq\_\ ^jiodip\ t lp`ipi^\ ^\h]d\ _` ndbij `i X\*]Z, Bioji^`n* k\m\ pi ^d`moj` _` X\*75" o`i`hjn

'4-17( .7a&s'b&s'_s < a&\' E7b&s'_s * a&]'nb&s'_s ,

A`hjnom\^d‡i, Rb^ D&s'< Iz b&o'_o, Bljl nrb d bp `lkqfkr^+ qbkbjlpD%&s'< b&s', Olo `lkpfdrfbkqb+ i^ fkqbdo^`fŽk mlo m^oqbpklp a^

'4-18( .7a&s'b&s'_s < oa&s'D%&s'_s< a&]'D&]'+ oe%&s'D&s'_s *

Page 289: Calculus

Be`m^d^djn +/2

mrbpql nrb C%[& < N- Rbd•k bi qblobj^ abi s^ilo jbafl mlkabo^al+ pb qfbkb

ba%&s'D&s' _s < D&`' ba%&s' _s < D&`'Xa&]' + a`\'Z

m^o^ rk `fboql _ bk W[) \Y+ Olo `lkpfdrfbkqb '4+18(+ pb `lksfboqb bk

`7a&s'b&s' _s < a&]'D&]' + D&`'Xa&]' + a`\'Z < a&\'D&`' * a&]'XD&]' + D&`'Z ,

Dpql abjrbpqo^ '4-17( v^ nrb C%]&< `wa%r& r v Cc\& * C%]&< `wa%r& r)

-&aB :WR_PVPV\`

Blk bi j‹qlal ab fkqbdo^`fŽk mlo m^oqbp ^i`ri^o i^p fkqbdo^ibpab ilp Dgbo`f`flp 0 ^i 5-

0- I s pbks _s, 3- ` u2pbk s _s,

1- I s/ n`is _s, .( cpbk s `lp s _s,

2- GU0`lp s _s, 5- I s pbks `lp s _s,

6- Blk i^ fkqbdo^`fŽk mlo m^oqbpabar`fo i^ cŽojri^

Ipbk1 s _s < +n`is `lp s * I `lp&&s _s ,

Dk i^ pbdrka^ fkqbdo^i+mlkbo `lp&&s < 0 , pbk1 s v ^pŒabar`fo i^ cŽojri^

Ipbk1 s _s < ds + qpbk 0s,

7- Hkqbdo^kal mlo m^oqbpabar`fo i^ cŽojri^

Ipbkj s _s < ,pbkj,0 s `lp s * &i + 0( Gn`ii+/ s `lp1 s _s *

Dk i^ pbdrka^ fkqbdo^i+mlkbo `lp,u < 0 , pbk, s v `lk bpl abar`fo i^ cŽojri^ ob`roobkqb

` n`ii+/s BNR s i + GGpbk! s _s < , i * +i+ pbkj

,1 s _s,

8- Blk ilp obpriq^alp ab ilp Dgbo`f`flp 6 v 7 abjlpqo^o nrb

'!.1'^( Il pbk1 s _s < z-

Page 290: Calculus

+0) O`g\^d4i `iom` dio`bm\^d4i v _`mdq\^d4i

0!.1 2 0!.1 1/Q

'^( l pbk3 s _s < { l pbk1 s _s < &05-

0!.1 - 4 0!.1 3/Q'`( pbk5 s _s < , pbk3 s _s < , -

l 5 l 21

0/- Blk ilp obpriq^alp ab ilp Dgbo`f`flp 6 v 7 abar`fo i^p pfdrfbkqbp cŽojri^p-

'^( Ipbk2 s _s < , `lp s * .1 `lp 2u-

'_( Gn`i2 s _s < fu , n`i0s * g0 pbk3u-

'`( Gn`i<s _s < +ds * 3[ `lp 1s + ej `lp 3s,

00- Blk i^ fkqbdo^`fŽk mlo m^oqbpv ilp obpriq^alp ab ilp Dgbo`f`flp 6 v 0/ abar`fo i^p pf,drfbkqbp cŽojri^p-

'^( Gupbk! s _s < s0 + s pbk0s + p lp 0s,

'_g Gupbk! s _s < n`is + g\ pbk1s + s `lp s * gjy9s `lp 1s,

'`( Gs0 pbk1 s _s < en* 'p , s0' pbk 0s + s `lp 0s,

01- Hkqbdo^kal mlo m^oqbpabar`fo i^ cŽojri^ ob`roobkqb

a^jni+gsn`is i+oa

`lp! s _s < i * +i+ ^jni+0s _s ,

02- Tqfifw^obi obpriq^al abi Dgbo`f`fl 01 m^o^ l_qbkbo i^ cŽojri^ pfdrfbkqb-

'^( F`lp1 s _s < qW * pbk1u-

'_( G@jn1s_s < pbku * k,pbk2u-

'`( G^jn2s_s ;gs * pbk1u )in`i2s,

03- Hkqbdo^kal mlo m^oqbpabjlpqo^o nrb

ax_s;sx)ac_s,

Olkbo s0 < s0 + 0 * 0 bk i^ pbdrka^ fkqbdo^i v abar`fo i^ cŽojri^

ax_U;ox)oaq]_U,

Page 291: Calculus

Be`m^d^djn */)

04- ^( Tp^o i^ fkqbdo^`fŽk mlo m^oqbpm^o^ abar`fo i^ cŽojri^

_( Tqfifw^o i^ m^oqb^( m^o^ `^i`ri^o ab %[0+ U0'3-0 ^r+

05- '^( Rf Fi&s' < `woi&o0* \0'+/-0 _o* ^mif`^o bi j‹qlal ab fkqbdo^`fŽk mlo m^oqbpm^o^abjlpqo^o nrb

pf i x 1-

'_( @mif`^kal '^( abjlpqo^o nrb `wU3&U0 * 4(,001 _s < 057.4 , 3/U4.2-

06- B^i`ri^o i^ fkqbdo^i a8/ o1&2* o1'+/F0 _o* p^_fbkal nrb a8/ '3 * o1'/-0 _o < 00+24- Cb,

g^o bi obpriq^al bk crk`fŽk ab U2 v U2i-

07- Hkqbdo^kal mlo m^oqbpabar`fo i^ cŽojri^

`n`ii)/s 0 n`iis i an`ii+/s+++_s <,,, ,, +++_s,`lpj*i U g `lp>&U g `lpj,0 U

@mif`^o i^ cŽojri^ m^o^ fkqbdo^o ` q^k! s _s v ` q^k s _s,

08- Hkqbdo^kal mlo m^oqbpabar`fo i^ cŽojri^

c`lpj*i U _s < \ y `lpjU [ xa `lpj,0 U _s,

pbkj*0 s i pbk! s i pbkj,0 s

Tqfifw^o i^ cŽojri^ m^o^ fkqbdo^o ` `lq! s _s X ` `lo&T ^r)

1/- ^( G^ii^o rk bkqbol i q^i nrb i F se!&0s' _s < ab ov!&o'_o,

_( B^i`ri^o j se!&0s' _s* p^_fbkal nrb `_K& < 0+ `%/& < 2+ X `$%/& < 4-10- ^( Rf ^-<! bp `lkqfkr^ u kl kri^ bk X\* \Y) u pf bufpqb rk^ `lkpq^kqb h< N q^i nrb

^-<%&o'x h m^o^ qlal o bk X\*]Z* rp^o bi qblobj^ 4-4 m^o^ abjlpqo^o nrb

Gbpbk ^-<&o'_o Gy y -

XFi_d^\^d‡i8 Lriqfmif`^o u afsfafo bi fkqbdo^kal mlo +j$%n&+Y

_( Rf \ = N+ abjlpqo^o nrb GHypbk &o0'_og x 0-\ m^o^ qlal s = \,

Page 292: Calculus

+0+ O`g\^d‡i `iom` dio`bm\^d‡i v _`mdq\^d‡i

"-&)) :WR_PVPV\`QR_R]N`\

0- Rb^ ` rk mlifkljfl q^i nrb `%K&< 0 X pb^ a%r&< r!`%r&+ B^i`ri^o d'N(+ d&'N(+--- + g&'/(-

1- G^ii^o rk mlifkljfl N ab do^al 9R4 q^i nrb N'M( < 0+ L%.& <1+ N&'M( < N!'M( <; L$%.&< l%.& < N-

2- Rf `%r& < ]im r v a%r&< pbk r) abjlpqo^o nrb

a&i'&s' < `lp &s * nh.P& v b&i'&s' < pbk &s * nh.P&+

3- Rf b%r& < `%r&a%r&)abjlpqo^o nrb i^ abofs^a^ k,‹pfj^ ab b sfbkb a^a^ mlo i^ cŽojri^

c&i'&s' < &x'a&f'&s'b&i+f'&s'*

Q5>

bk alkab 'z( obmobpbkq^bi `lbcf`fbkqb _fkljf^i- Dpq^ bp i^ ii^j^a^ a‡mhpg\ _` I`d]idu,4- C^a^p alp crk`flkbp ` v d `rv^p abofs^a^p f$ v d&p^qfpc^`bk i^p b`r^`flkbp

'4-2/( W$%r&:a%r&) a$%r&< *`%r&) a&L' < N+ a%K&< 0+

m^o^ qlal s bk rk `fboql fkqbos^il ^_fboql F nrb `lkqfbkb bi N- Olo bgbjmil+ bp^p b`r^,`flkbp pb p^qfpc^`bk `r^kal `%r& < pbk r v a%r&< `lp r)^( Cbjlpqo^o nrb a0&U' * b0&U' < 0 m^o^ qlal s ab Z+_( Rb^k B v F lqol m^o ab crk`flkbp nrb p^qfpc^d^k '4-2/(- Cbjlpqo^o nrb B%r&< `%r& vC%r& < a%r&) m^o^ qlal r ab F+XFi_d^\^d‡i8 Blkpfabo^o b%r&< WB%r&* `%r&Y * XD&s' + a%r&Y +Yb( ƒPr‹ jŠp pb mrbab ab`fo ^`bo`^ ab i^p crk`flkbp ` v b nrb p^qfpc^`bk 'R-2/(>

5- Tk^ crk`fŽk Z+abcfkfa^ m^o^ qlal k•jbol ob^i mlpfqfsl+ p^qfpc^`b i^ b`r^`fŽk `%r0' < r1

m^o^ `^a^ s = N- Cbqbojfk^o .$%1&+6- Tk^ crk`fŽk d abcfkfa^ m^o^ qlal k•jbol ob^i mlpfqfsl p^qfpc^`b i^p alp `lkaf`flkbp pf,

drfbkqbp9 a%.&< 0 v a$%r0' < r1 m^o^ qlal r = N- B^i`ri^o a%1&+

7- Cbjlpqo^o nrb

0-!&pbko _++ o zN

l o * 0m^o^ qlal s x N-

sl

EHFTQ@ 4-1 Be`m^d^dj 7,

Page 293: Calculus

Be`m^d^djn_` m`k\nj +0,

8- Rb^k B0

v B1

alp `ros^p nrb m^p^k mlo bi lofdbk q^i `ljl pb fkaf`^ bk i^ cfdro^ 4-1-Tk^ `ros^ B pb af`b nrb ~_fpb`^ bk Šob^‚ i^ obdfŽk bkqob B

0v @

0* pf m^o^ `^a^ mrkql M

ab B i^p alp obdflkbp = v > plj_ob^a^p bk i^ cfdro^+ qfbkbk i^ jfpj^ Šob^- Cbqbojfk^oi^ `ros^ prmboflo @

0* p^_fbkal nrb i^ `ros^ _fpb`qofw B qfbkb ab b`r^`fŽk v < s0 v nrb

i^ `ros^ fkcboflo B0 qfbkb ab b`r^`fŽk v <ps0Š

0/- Tk^ crk`fŽk ` bpqŠ abcfkfa^ m^o^ qlal s `ljl pfdrb9

^

[,

a&s' < Npf s bp o^`flk^i+

pf s bp foo^`flk^i-

OŽkd^pb N&c' < a&c'-c pf c x N- ^( Cbjlpqo^o nrb M%b&w N `r^kal b w N- _( Cb,jlpqo^o nrb ` qfbkb abofs^a^ bk N+v `^i`ri^o x&'N(-

Dk ilp bgbo`f`flp 00 ^i 1/+ `^i`ri^o i^p fkqbdo^ibp a^a^p- Hkqbkq^oi^ pfjmifcf`^`fŽk ab ilp`Ši`rilp rqfifw^kal bi j‹qlal ab prpqfqr`fŽk l i^ fkqbdo^`fŽk mlo m^oqbp`r^kal pb^ mlpf_ib-

00- H'1 * 1s' pbk 3s _s,

/0, Gsx_s,

/1, a0U&U0 + 0(8 _s,

&/ /r * 203- Il %3r * 6(^ ^r)

.2+ G+/'h * sP'P ^r)

/4, G8*.&/ + s'0{_s,

06- '[,1 pbk z _s,I0 B

.5+ Gn`i x ^r+

.6+ Fs pbk s0 `lp s0 ^r+

0., FUH * 2 `lp! s n`i0s ^r+

10- Cbjlpqo^o nrb bi s^ilo ab i^ fkqbdo^i F 042rP%r0 * 0(,3 ^r bp /i m^o^ rk `fboql bk,qbol i,

11- Cbqbojfk^o rk m^o ab k•jbolp [ v \ m^o^ ilp `r^ibp F %[r * \&%r0 * 0r * 1(,1 ^r :< 2.1-

12- Rb^ Fi < Px&g + s0'i _s, Cbjlpqo^o nrb &0i * F'Fi < 0i Fi+/* X rqfifw^o bpq^ obi^`fŽk

/0* g\*/2* W Fn}13- Rb^ B%g)h&< d nh%. * o'i ^n) g = N+h = N- Cbjlpqo^o nrb

%g * .&B%g)h& * hB%g * 0+h * 0( < rg(f%f * t(!+

Tqfifw^o bpqb obpriq^al m^o^ `^i`ri^o B%fK) /&+

14- Rb^ X`i' < Px-2 q^k! s _s alkab i x 0- Cbjlpqo^o nrb

'^( a&i * 0( :a&i',

Page 294: Calculus

163 O`g\^d‡i `iom` dio`bm\^d‡i t _`mdq\^d‡i

0'_( a&i' * a&i + 1( < i [ 0 pf i = 1-

0 0'`( ,,0 ; 0a&i' ; ,,0 pf i = 1-

i) i+

15- B^i`ri^o .%-&) p^_fbkal nrb .%$.P&< 1 W nrb b@a&s' * e!&s'Zn`is _s < 4-16- Cbpfdk^o mlo > bi s^ilo ab i^ fkqbdo^i

'! `lp sIl %r * 1(1 ^r +

v `^i`ri^o i^ pfdrfbkqb fkqbdo^i bk crk`fŽk ab >8

0++.1pbk s `lp s++++_s,

l s * 0

K^p cŽojri^p ab ilp Dgbo`f`flp 17 ^i 22 ^m^ob`bk bk q^_i^p ab fkqbdo^ibp-Bljmol_^o `^a^rk^ ab bii^p mlo `r^inrfbo lqol j‹qlal-

06, ax _s <1v^ )]s )\a , x] * `-s sq\ * AU

07, aUagt\U * ] _s < 1 &sag&\s* ]'1-0 + i]asag+gt\s * ] _s' * @&i :l.f , (-\&0i * 2(

1., a x_s < &0h8 g']&sht\ * ]s + h\a x_U' * A &h9j-d,p(•

20- a _s < , z , &0i + 1'\a _s * B &i9j-d.&+sagt\s * ] &i + i(_uci,0 &0i + 0'] uci,0X^u * ]

c

`lpj T `lpj,0 T h + ic`lpj,1 s10, ++_s < *,, +++_s * A &h 9j-di',

pbk! s &h + i'n`iag+/s h+i pbk! s

c

`lpj U `lpjG s h+i * 0a `lp>&s11, ++_s < , ,yyy, , ,,,, +++_s * A &i9j-d.&+

pbk! s &i + i(pbkci,0 s i + 0 pbkci,1 s

23- ^( Dk`lkqo^o rk mlifkljfl L%r& q^i nrb L$%r&* 0L%r& < 3 , 2r * 0r0Š Cbjlpqo^o nrbbufpqbrk^ pli^ plir`fŽk-_( Rf M%r& bp rk mlifkljfl a^al+ abjlpqo^o nrb bufpqb rkl v pŽil rk mlifkljfl L%r&q^i nrb L$%r&* 0L%r& < M%r&+

24- Tk^ pr`bpfŽk ab mlifkljflp 'ii^j^alp kjgdijhdjn _`%?`mijp-gd' pb abcfkb mlo fkar``fŽk`ljl pfdrb9

Li%r& < 0: u FwL`f%T& r !! M pf i x 0-

Page 295: Calculus

Be`m^d^djn _` m`k\nj */-

^( Cbqbojfk^o cŽojri^p bumiŒ`fq^pm^o^ Lfr&) L/%r&) +++) L2%r&+_( Cbjlpqo^o+ mlo fkar``fŽk+ nrb Lh%r& bp rk mlifkljfl bk r ab do^al i* pfbkal bi q‹o,jfkl ab j^vlo do^al u!-`( Cbjlpqo^o nrb Mmo&L'< Mi&g' pf i 98881-a( Cbjlpqo^o nrb Lh%r * 0( , Lh%r&< hrh*. pf h ƒ 0-b( Cbjlpqo^o nrb m^o^ i 888891 qbkbjlp

hzi df L %e& L %K&

0 i + M & ' _ + k*i , k*in , iU U + h&

J i (KRG

c( Cbjlpqo^o nrb Lh%f * r& < %Zf&hjh%r& pf h ƒ 0-

d( Cbjlpqo^o nrb L0i(f%-& < N v O1j,0'p( < N pfi ƒ 0-

25- Rrmlkfbkal nrb 0c!'u(0 9999:g m^o^ `^a^ r bk bi fkqbos^il ZN+[F) v nrb ` qlj^ pr j^vlos^ilo bk rk mrkql fkqboflo ab bpqb fkqbos^il+ abjlpqo^o nrb 00&'/(0* E$%[&d77778[gi Orbab pr,mlkbopb nrb .!pb^ `lkqfkr^ bk ZN+[F+

Page 296: Calculus
Page 297: Calculus

5

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Prfbk cfgb pr ^qbk`fŽk bk obi^`flkbp `r^kqfq^qfs^p+ l bpqraf^ molmfba^abpab crk`flkbp `lkl`fa^p+ l qo^q^ab abp`r_ofo molmfba^abp ab rk^ crk`fŽk abp,`lkl`fa^- Di `lk`bmql ab crk`fŽk bp q^k buqbkpl v q^k dbkbo^i nrb kl bp plo,mobkabkqbbk`lkqo^o rk^ fkjbkp^ s^ofba^a ab crk`flkbp nrb pb mobpbkq^kbk i^k^qro^ibw^-Kl nrb n…n plomobkabkqbbp nrb rk `loql k•jbol ab crk`flkbp bpmb,`f^ibp ofg^k rk^ jriqfqra ab cbkŽjbklp k^qro^ibp qlq^ijbkqb afcbobkqbp-Dk bpqb`^mŒqril pb bpqraf^oŠk ^idrk^p ab bpq^pcrk`flkbp+ bk mofjbo ird^o i^ crk`fŽkild^oŒqjf`^ v pr fksbop^ 'i^ crk`fŽk bumlkbk`f^i( v irbdl i^p crk`flkbp fksbop^pab i^p crk`flkbp qofdlklj‹qof`^p- Slal ^nrbi nrb bpqrafb L^qbjŠqf`^+ v^ pb^`ljl rk^ afp`fmifk^ ^_pqo^`q^+l `ljl fkpqorjbkql bk lqolp aljfkflp `fbkqŒcf`lp+bk`lkqo^oŠ fkafpmbkp^_ib rk `lkl`fjfbkql qbŽof`l v moŠ`qf`l ab bpq^pcrk`flkbpv prp molmfba^abp-

Ool_^_ibjbkqb bi ib`qlo e^_oŠ qbkfal l`^pfŽk ab qo^_^g^o`lk ild^ofqjlpab _^pb 0/ bk „idb_o^ bibjbkq^i l SofdlkljbqoŒ^- K^ abcfkf`fŽk a^a^ `loofbk,qbjbkqb bk „idb_o^ bibjbkq^i bp i^ pfdrfbkqb- Rf s = N+bi ild^ofqjl ab s bk_^pb 0/+ fkaf`^al mlo 0/dHls bp rk rk k•jbol ob^i p q^i nrb i N! < s, Rf s < .-R

b v < i N!+ pb qfbkb9 rs < gLp)q* fdr^ia^a nrb mlo jbafl ab ild^ofqjlp pbbumobp^ab i^ cloj^ pfdrfbkqb9

'5-0( iNdiN %rs&< iNdiN r * iNdilX+

Dpq^molmfba^a crka^jbkq^i+ e^`b nrb ilp ild^ofqjlp pb^k m^oqf`ri^ojbkqb ^mif`^,_ibp ^ ilp `Ši`rilp nrb `lkqfbkbk jriqfmif`^`flkbp- Dp moŠ`qf`l rp^o bi k•jbol 0/`ljl _^pb v^ nrb ilp k•jbolp ob^ibppb bp`of_bk `Žjla^jbkqb bk bi pfpqbj^ ab,`fj^i+ v ^idrklp k•jbolp fjmloq^kqbp q^ibp`ljl /+/0+ /+0+ 0+ 0/+ 0//+ 0///+ ---qfbkbk mlo ild^ofqjlp ilp bkqbolp , 1+ ,0+ N+0+1+ 2+ --- +obpmb`qfs^jbkqb-

+00

Page 298: Calculus

167 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\n diq`mn\n

Rfk bj_^odl+ kl bp mob`fpl qlj^o `ljl _^pb bi k•jbol 0/: `r^inrfbo lqolbkqbol v mlpfqfsl ] ", 0 q^j_f‹k mrbab qlj^opb `ljl _^pb: ^pŒ9

%3+/& p < ild+ s pfdkfcf`^ s < \Q)

u i^ molmfba^a crka^jbkq^i '5-i( pb bumobp^ nrŒ9

'5-2( ild+ &st' < ild+s * ild+V ,

Du^jfk^kal i^ abcfkf`fŽk '5-1( abpab rk mrkql ab sfpq^ `oŒqf`l+pb ib bk,`rbkqo^k ^idrklp c^iilp iŽdf`lp- Dk mofjbo ird^o+ m^o^ bkqbkabo '5-1( bp mob`fplp^_bo nr‹ pfdkfcf`^ ]!, Br^kal p bp rk `io`mj l rk iˆh`mj m\^dji\g '`l`fbkqbab alp bkqbolp(bp cŠ`fi ab abcfkfo+mbol kl l`roob il jfpjl `r^kal p bp dmm\^dji\g,Olo bgbjmil+ ƒ`Žjl pb abcfkfoŠ0/s&1> @rknrb pb iibdrb ^ l_qbkbo rk^ abcfkf`fŽkp^qfpc^`qlof^m^o^ ]!* pb mobpbkq^klqo^p afcf`riq^abp e^pq^ mlabo iibd^o ^ `lk,pfabo^o '5-1( `ljl rk^ _rbk^ abcfkf`fŽk ab ild^ofqjl9 pb e^_oŠ ab abjlpqo^onrb m^o^ `^a^ s = N+bufpqbrk p q^i nrb s < ]!9 X ^abjŠp nrb i^ ibv ab ilpbumlkbkqb ]m]! < ]R(q* pb sbofcf`^ m^o^qlalp ilp bumlkbkqbpob^ibpp v q,

Rb mrbabk sbk`bo qla^p bpq^p afcf`riq^abp v iibd^o ^ rk^ abcfkf`fŽk p^qfp,c^`qlof^ ab ild^ofqjl mlo bpqbj‹qlal mbol bi mol`bpl bp i^odl v mbp^al- @clo,qrk^a^jbkqb+ bi bpqrafl ab ilp ild^ofqjlp pb mrbab iibs^o ^ `^_l mlo rk `^,jfkl `ljmibq^jbkqb afpqfkql nrb bp jr`el jŠp pfjmib v jrbpqo^ bi mlabo vi^ bibd^k`f^ ab ilp j‹qlalp ab `Ši`ril9 kmdh`mj pb fkqolar`b bi ild^ofqjl+ uirbdl pb rp^k ilp ild^ofqjlp m^o^ abcfkfo ]!*

.&* 9RSV[VPVp[QRYY\TN_VaZ\[Nab_NYP\Z\ V[aRT_NY

Di ild^ofqjl bp rk bgbjmil ab rk `lk`bmql j^qbjŠqf`l nrb mrbab pbo abcf,kfal mlo jr`elp `^jfklp afpqfkqlp- Br^kal rk j^qbjŠqf`l fkqbkq^ clojri^ork^ abcfkf`fŽk ab rk `lk`bmql+ bk dbkbo^i qfbkb bk pr mbkp^jfbkql rk^ pbofbab molmfba^abp nrb ‹i abpb^ nrb qbkd^ bpqb `lk`bmql- Du^jfk^kal bpq^pmol,mfba^abp bp `lkar`fal cob`rbkqbjbkqb ^ rk^ cŽojri^ l mol`bpl pfjmib nrbpfosb `ljl abcfkf`fŽk u ab i^ `r^i prodbk bpq^pmolmfba^abp`ljl abar``flkbpiŽdf`^p- Rb sboŠ ^ `lkqfkr^`fŽk `Žjl jbaf^kqb bpqb mol`bpl pb mrbab iibd^o ^i^ abcfkf`fŽk ab ild^ofqjl-

Tk^ ab i^p molmfba^abpnrb pb abpb^ nrb qbkd^ bi ild^ofqjl bp nrb bi ild^,ofqjl ab rk molar`ql pb^ fdr^i ^ G]prj^ ab ilp ild^ofqjlp ab `^a^ rkl abilp c^`qlobp- Dpq^molmfba^a pb `lkpfabo^oŠ bk pŒjfpj^ v pb sboŠ ^ aŽkab pbmrbab iibd^o ^ m^oqfoab bii^- Rf pb prmlkb bi ild^ofqjl `ljl rk^ crk`fŽk f+pb abpb^ nrb bpq^crk`fŽk qbkd^ i^ molmfba^a bumobp^a^mlo i^ cŽojri^

' 5-3( a&st' <a&s' * a&t'

alkab s* v+ st mboqbkb`bk i aljfkfl ab i^ crk`fŽk Z-

Page 299: Calculus

@_`chc]cƒh^_f fia[lcngi h[nol[f ]igi chn_al[f +02

Tk^ b`r^`fŽk q^i `ljl '5-3( nrb bumobp^ rk^ obi^`fŽk bkqob ilp s^ilobp abrk^ crk`fŽk bk alp l jŠp mrkqlp+ pb abkljfk^ rk^ _]o[]cƒh `oh]cih[f+ Lr`elpmol_ibj^p j^qbjŠqf`lp pb obar`bk ^ obplisbo rk^ b`r^`fŽk crk`flk^i bk i^ nrbrk^ plir`fŽk bp rk^ crk`fŽk nrb i^ p^qfpc^d^- Noafk^of^jbkqb rk^ b`r^`fŽk abbpq^ `i^pb qfbkb jr`e^p plir`flkbp afpqfkq^p v bk dbkbo^i bp jrv afcŒ`fi bk`lk,qo^oi^p qla^p- Dp jŠp cŠ`fi _rp`^o pŽil ^nrbii^p plir`flkbp nrb qfbkbk ^idrk^ lqo^molmfba^a+ q^i `ljl `lkqfkrfa^a l afcbobk`f^_fifa^a+ v dbkbo^ijbkqb ‹pq^p plki^p •kf`^p plir`flkbp nrb fkqbobp^k- Dpqb `ofqbofl bp bi nrb pb ^almq^oŠ bk i^obplir`fŽk ab '5-3( _rp`Škalpb pli^jbkqb i^p plir`flkbp afcbobk`f^_ibp- Rfk bj,_^odl bp fkqbobp^kqb sbo nr‹ `lkpb`rbk`f^p pb mrbabk abar`fo ab '5-3( pfk fj,mlkbo ^ ` kfkdrk^ lqo^ obpqof``fŽk-

Tk^ plir`fŽk ab '5-3( bp i^ crk`fŽk nrb bp `bol bk qlal bi bgb ob^i: v ^ab,jŠp+ bp i^ •kf`^ plir`fŽk ab '5-3( nrb bpqŠ abcfkfa^ m^o^ qlalp ilp k•jbolp ob^,ibp- Dk bcb`ql9 pb^ Erk^ crk`fŽk nrb p^qfpc^d^ '5-3(+ pf N mboqbkb`b ^i aljfkflab E pb mrbab mlkbo t < M bk '5-3( l_qbkf‹kalpb .%-&< E%r&* .%-& 0/ nrbfjmif`^ nrb E%r&< N m^o^ `^a^ r bk bi aljfkfl ab f+ Cf`el ab lqo^ cloj^+ pf Nmboqbkb`b ^i aljfkfl ab .) Ee^ ab pbo fa‹kqf`^jbkqb kri^- Olo q^kql+ rk^ plir,`fŽk ab '5-3( kl fa‹kqf`^jbkqb kri^ kl mrbab bpq^o abcfkfa^ bk N-

Rf Ebp rk^ plir`fŽk ab '5-3( u bi aljfkfl ab E`lkqfbkb bi mrkql )$ pb mrbabmlkbo r < t < 0 bk '5-3( v pb l_qfbkb .%.&< /`%.&) ab alkab

a&g' < N-

Rf ^j_lp 0 v , 0 mboqbkb`bk ^i aljfkfl ab Epb mrbab qlj^o s < , 0 b t < , 0ab alkab pb abar`b `%.&< /`%,0( bp ab`fo `%,0( < M-Rf ^elo^+ r) *r) .v , 0 mboqbkb`bk ^i aljfkfl ab `) pb mrbab mlkbo t < , 0 bk '5-3( l_qbkf‹k,alpb E%* r& < E%,0( * E%r&)v mrbpql nrb E%* 0( < M pb qfbkb

a&+s' <a&s' ,

bp ab`fo+ qla^ plir`fŽk ab '5-3( bp kb`bp^of^jbkqb rk^ crk`fŽk j[l+RrmŽkd^pb ^elo^+ nrb ` qfbkb rk^ abofs^a^ `$%r&bk `^a^ r ;/; N- Cbg^kal

t cfgl bk '5-3( v abofs^kal obpmb`ql ^ s '^mif`^kal bk bi mofjbo jfbj_ol i^ obdi^ab i^ `^abk^( pb qfbkb9

ta%&st'<a%&s',

Rf s < 0+ ab bpq^ b`r^`fŽk pb abar`b ta%&t' < a%&/' v+ mlo q^kql+ pb qfbkb9

e%&t' <0&'0( m^o^ `^a^ t jyo8, N-V

Page 300: Calculus

/5- Boh]cƒh fia[lcngi) `oh]cƒh _rjih_h]c[f u `oh]cih_m nlcaihig€nlc][m chp_lm[m

Dk bpq^ b`r^`fŽk pb sb nrb i^ abofs^a^ mbp jlkŽqlk^ v mlo q^kql fkqb,do^_ib bk `^a^ fkqbos^il `boo^al nrb kl `lkqbkd^ bi lofdbk- @abjŠp+ f$bp `lk,qfkr^ bk `^a^ rkl ab bpqlp fkqbos^ilp v pb mrbab ^mif`^o bi pbdrkal qblobj^crka^jbkq^i abi BŠi`ril bp`of_fbkal

a%! a%!/y&s' x y&`'< /%&o'_o <0&'0( , _o ,b b o

Rf s = N+bpq^ b`r^`fŽk bp sŠifa^ m^o^ `^a^ mlpfqfsl ` = N+ v pf bp s ; N bpsŠifa^ m^o^ `^a^ ` kbd^qfsl- Orbpql nrb a&g' < N+bifdfbkal ` < 0 pb qfbkb

x%r&<0&'0('&! _o98 o

pf s< N-

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pf s:L,

Dpq^p alp cŽojri^p m^o^ `%r& mrbabk obrkfopb bk rk^ nrb bp sŠifa^ q^kql pf rbp mlpfqfs^ `ljl kbd^qfs^+ ^ p^_bo9

'5-4( c&&!00x%r&<0&'0( , _o

0 opf rwK+

Dk `lkpb`rbk`f^+ pf bufpqb rk^ plir`fŽk ab '5-3( nrb qfbkb rk^ abofs^a^ bk `^a^mrkql s :.: N bpq^ plir`fŽk e^ ab sbkfo a^a^ kb`bp^of^jbkqb mlo i^ cŽojri^ fkqbdo^iab '5-4(- Rf c&'i(< N+ bkqlk`bp '5-4( fjmif`^ nrb `%r& < N m^o^ `^a^ r :.: N Xbpq^ plir`fŽk `lfk`fab `lk i^ fa‹kqf`^jbkqb kri^- Olo q^kql+ pf ` kl bp fa‹kqf`^,jbkqb kri^ e^ ab pbo .$%0( :.: N+bk `rvl `^pl pb mrbabk afsfafo ^j_lp jfbj_olpab '5-4( mlo c&'i(l_qbkf‹kalpb

'5-5( `%*!//a%r&< , _o

0 opf u z N+

alkab a%r& < `%r&,`$%f&+K^ crk`fŽk a bp q^j_f‹k rk^ plir`fŽk ab '5-3(+ mrbpqlnrb pf F bp plir`fŽk q^j_f‹k il bp `g, Dpql abjrbpqo^ nrb pf '5-3( qfbkb rk^plir`fŽk nrb kl bp i^ fa‹kqf`^jbkqb kri^+ v pf bpq^ crk`fŽk bp abofs^_ib bk qlalpilp mrkqlp+ bu`bmql bk bi lofdbk+ bkqlk`bp i^ crk`fŽk b a^a^ mlo '5-5( bp rk^plir`fŽk+ v oj_\n i^p plir`flkbp mrbabk l_qbkbopb ab ‹pq^ jriqfmif`^kal b mlork^ `lkpq^kqb `lksbkfbkqb-

Page 301: Calculus

A`adid^d‡i _` gjb\mdohj, Mmjkd`_\_`n api_\h`io\g`n 170

Rb e^ ab l_pbos^o nrb bpqbo^wlk^jfbkql kl abjrbpqo^ qla^sŒ^nrb i^ crk`fŽkd ab '5-5( n`\ rk^ plir`fŽk+ mrbpql nrb pb e^ abar`fal '5-5( bk i^ efmŽqbpfpabnrb bufpqŒmlo il jbklp rk^ plir`fŽk kl fa‹kqf`^jbkqb kri^- K^ cŽojri^ '5-5(prdfbob rk `^jfkl m^o^ `lkpqorfo rk^ q^i plir`fŽk+ nrb pb l_qfbkb lmbo^kal bkpbkqfal `lkqo^ofl- Dp ab`fo+ jbaf^kqb '5-5( pb abcfkb i^ crk`fŽk d u irbdl pb `lj,morb_^ nrb bpq^crk`fŽk p^qfpc^`b'5-3(- Dpqbo^wlk^jfbkql fkar`foŒ^^ qlj^o `ljlabcfkf`fŽk ab ild^ofqjl+ i^ crk`fŽk d a^a^ bk '5-5(+ u bkqlk`bp alp k•jbolp afp,qfkqlp qbkaoŒ^krk jfpjl ild^ofqjl+ mrbpql nrb i^ crk`fŽk d qbkaoŒi^ molmfba^a9a%r& < d' , r&+ Dk ^qbk`fŽk ^ `lkpfabo^`flkbp nrb mlpqboflojbkqb pb e^oŠk+ bpmobcbof_ibabcfkfo bi ild^ofqjl ab j^kbo^ nrb alp k•jbolp afpqfkqlpkl qbkd^k bijfpjl ild^ofqjl+ il `r^i pb ildo^ abcfkfbkal bi ild^ofqjl pŽil m^o^ ilp k•jbolpmlpfqfslp- Olo q^kql+pb qlj^oŠ i^ pfdrfbkqb abcfkf`fŽk-

5-2 Cbcfkf`fŽk ab ild^ofqjl- Oolmfba^abp crka^jbkq^ibp

CDEHMHBHˆM- Pd s `n pi iˆh`mj m`\g kjndodqj* _`adidhjn `g gjb\mdohj i\opm\g_` s* _`ndbi\_j kmjqdndji\gh`io` kjm I&s'* ^jhj g\ dio`bm\g

'5-6( `B.I&s' < , _o,

0 o

Br^kal s = 0+ I&s' mrbab fkqbomobq^opbdblj‹qof`^jbkqb `ljl bi Šob^ ab i^ ob,dfŽk plj_ob^a^ ab i^ cfdro^ 5-0-

RCMPCK? 5-0- I\ api^d‡i gjb\mdohj od`i` g\n kmjkd`_\_`n ndbpd`io`n8

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H%.&< N-0

I%&s' < , k\m\ oj_j s = N-s

I&\]' < I&\' * I&]' k\m\ oj_j \< N+ \ = N-

A`hjnom\^d‡i, K^ m^oqb ( pb abar`b fkjbaf^q^jbkqb ab i^ abcfkf`fŽk- O^o^abjlpqo^o _(+ l_pbosbjlp pfjmibjbkqb nrb H bp rk^ fkqbdo^i fkabcfkfa^ ab rk^crk`fŽk `lkqfkr^ u ^mifnrbjlp bi mofjbo qblobj^ crka^jbkq^i abi BŠi`ril- K^molmfba^a `( bp `lkpb`rbk`f^ ab i^ molmfba^a ^afqfs^ ab i^ fkqbdo^i-Dp`of_^jlp

`D] _o a\ _o a\] _o a\] _oI&\]' < , < , * , < I&\' * *n$

0 o go \ o \

Dk i^ •iqfj^ fkqbdo^i ebjlp eb`el i^ prpqfqr`fŽk p < od\* _p < _oZ\* v bk`lk,qo^jlp nrb i^ fkqbdo^i pb obar`b ^ H%\&)il nrb abjrbpqo^ `(-

Page 302: Calculus

060 %Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

t s

k s

t < H%r&

s

EHFTQ@ 5-0 Fio`mkm`o\^d4i_`g gjb\mdohj^jhj pi ƒm`\,

EHFTQ@ 5-1 Dmƒad^\_`g gjb\mdohj i\opm\g,

5-3 FoŠcf`^ abi ild^ofqjl k^qro^i

K^ doŠcf`^ab i^ crk`fŽk ild^ofqjl qfbkbbi ^pmb`ql nrb pb ^mob`f^bk i^ cfdr,o^ 5-1- Lr`e^p molmfba^abpab bpq^`ros^ mrbabk l_qbkbopb+pfk bcb`qr^o kfkd•k`Ši`ril+ pfjmibjbkqb obcfof‹kalpb ^ i^p molmfba^abpabi qblobj^ 5-0- Olo bgbjmil+^ m^oqfoab _( sbjlp nrb H qfbkb abofs^a^ mlpfqfs^ pfbjmob ab jlal nrb bpbpqof`q^jbkqb `ob`fbkqb bk qlal fkqbos^il- Orbpql nrb H%0( < N+i^ doŠcf`^ bpqŠpfqr^a^ mlo bk`fj^ abi bgbs pf s = 0 v mlo ab_^gl pf N ; s ; 0- K^ `ros^ qfbkbmbkafbkqb0 `r^kal s < 0- O^o^ s = 0+i^ mbkafbkqbab`ob`b do^ar^ijbkqb e^`f^`bol `r^kal s `ob`b fkabcfkfa^jbkqb- O^o^ s^ilobp mbnrb•lp ab s* i^ mbkafbkqbbp do^kab v+ ^abjŠp+ qfbkab e^`f^ fkcfkfql `r^kal r ab`ob`b e^`f^ `bol- K^ abof,s^a^ pbdrka^ bp I!&s' < , /-s0 nrb bp kbd^qfs^ m^o^ qlal s* mlo 0/ nrb I bprk^ crk`fŽk `Žk`^s^-

5-4 Blkpb`rbk`f^p ab i^ b`r^`fŽk crk`flk^i I&\]' < I&\' * I&]'

Bljl i^ doŠcf`^abi ild^ofqjl s^ ^p`bkafbkal `r^kal s qfbkab ^ fkcfkfql+pbmrbab plpmb`e^o nrb ilp s^ilobp ab H kl qfbkbk `lq^ prmboflo- Dk bcb`ql+i^ crk,`fŽk kl bpqŠ^`lq^a^ prmboflojbkqb: bpql bp+m^o^ qlal k•jbol mlpfqfsl L 'mlodo^kab nrb pb^( bufpqbks^ilobp ab s q^ibpnrb

'5-7( I&s' = J,

Page 303: Calculus

@jin`^p`i^d\n _` g\ `^p\^d‡i api^dji\g I&\]' <I&\' * I&]' 061

Dpql mlabjlp abar`foil ab i^ b`r^`fŽk crk`flk^i- Br^kal [ < ]* qbkbjlpI&\0

' < 0I&\', Tqfifw^kal i^ b`r^`fŽk crk`flk^i rk^ sbw jŠp mlkfbkal ] < \%*l_qbkbjlp H%[1

' < 0H%[&+Olo fkar``fŽk bk`lkqo^jlp i^ cŽojri^ dbkbo^i

I&\!' < hH%[&

m^o^ `r^inrfbo bkqbol i x 0- Br^kal \ < 1+ pb l_qfbkb K'1cf( < iI&0'* v mlo

q^kql obpriq^

'5-8( I&0!' = KK

`r^kal i =,,I&0'

Dpql abjrbpqo^ i^ ^cfoj^`fŽk '5-7(- Slj^kal ] < g-\ bk i^ b`r^`fŽk crk`flk^i+bk`lkqo^jlp I&g-\' < , I&\', Dk m^oqf`ri^o+ `r^kal \ < 10.+ e^_fbkal bibdfali `ljl bk '5+8(+ pb qfbkb

H%.+*&$< *H%/!& ; *I)1!

il nrb fkaf`^ nrb q^jml`l bufpqb `lq^ fkcboflo m^o^ ilp s^ilobp ab i^ crk`fŽk-Efk^ijbkqb l_pbos^jlp nrb i^ doŠcf`^ `loq^ ^ `^a^ ob`q^ elofwlkq^i pŽil rk^

sbw- Dp ab`fo+ a^al rk k•jbol ob^i \m]dom\mdj] 'mlpfqfsl+ kbd^qfsl l kril(+ bufpqbpij u n‡gj pi \ = N q^i nrb

'5-0/( I&\' < c,

O^o^ abjlpqo^oil pb mrbab o^wlk^o `ljl pfdrb9 Rf ] = N+ bibdfjlp rk bkqbol`r^inrfbo^ i = ]- I&0', Dkqlk`bp+ bk sfoqra ab '5-8(+ I&0i' = ], Rbdrfa^jbkqbbu^jfk^jlp i^ crk`fŽk H bk bi fkqbos^il `boo^al Z0+10.\- Rr s^ilo bk bi buqobjlfwnrfboal bp H%0( < N+ X bk bi buqobjl abob`el bp H%/i', Orbpql nrbN ; ] ; I&0i

'* bi qblobj^ abi s^ilo fkqbojbafl m^o^ crk`flkbp `lkqfkr^p 'qblob,j^ 2-7 ab i^ Rb``fŽk 2-0/( ^pbdro^ i^ bufpqbk`f^ mlo il jbklp ab rk \ q^i nrbI&\' < ], Ml mrbab bufpqfo lqol s^ilo \% q^i nrb I&\%' < ] mlonrb bpql pfdkfcf`^oŒ^I&\' < I&\%' m^o^ \ ;/; \%*v bpql `lkqo^af`b i^ molmfba^a ab `ob`fjfbkql abi il,d^ofqjl- Olo `lkpfdrfbkqb i^ molmlpf`fŽk '5-0/( e^ pfal abjlpqo^a^ m^o^ ] = N-K^ abjlpqo^`fŽk m^o^ \ kbd^qfsl bp `lkpb`rbk`f^ ab ‹p^ pf rqfifw^jlp i^ fdr^ia^aH%f,[& < , H%[&+Dp ab`fo+ ebjlp abjlpqo^al bi pfdrfbkqb

RCMPCK? 5-1- M\m\ ^\_\ iˆh`mj m`\g ] `sdno` `s\^o\h`io` pi iˆh`mj m`\gkjndodqj s ^ptj gjb\mdohj* I&\'* `n dbp\g \ ],

Dk m^oqf`ri^o+ bufpqb rk •kf`l k•jbol `rvl ild^ofqjl k^qro^i bp fdr^i ^ 0-Dpqb k•jbol+ ^i fdr^i nrb PP) pb bk`rbkqo^ q^k obmbqfa^jbkqb bk cŽojri^p j^qb,

Page 304: Calculus

/51 Boh]cƒh fia[lcngi) `oh]cƒh _rjih_h]c[f s `oh]cih_m nlcaihig€nlc][m chp_lm[m

jŠqf`^p nrb bp fkbsfq^_ib bi ^almq^o m^o^ ‹i rk pŒj_lil bpmb`f^i- Kblk^oalDribo '06/6,0672(+ m^ob`b nrb crb bi mofjbol nrb ob`lkl`fŽ i^ fjmloq^k`f^ abbpqbk•jbol u jlabpq^jbkqb il abpfdkŽ mlo `* klq^`fŽk nrb bk pbdrfa^ pb efwlrpr^i-

CDEHMHBHˆM- @_mcah[gim jil _ _f h„g_li j[l[ _f ko_

%3+..& H%_&< 0 -

Dk bi `^mŒqril6 l_qbkaobjlp cŽojri^p bumiŒ`fq^pnrb mbojfqbk `^i`ri^o i^bumobpfŽkab`fj^i ab ` `lk bi do^al ab ^molufj^`fŽk nrb pb abpbb-Rr s^ilo `loob`,ql `lk afbw `fco^p ab`fj^ibp bp 1+6071707174- @pfjfpjl bk bi `^mŒqril6 pb ab,jlpqo^oŠ nrb ` bp foo^`flk^i-

Klp ild^ofqjlp k^qro^ibp pb abkljfk^k q^j_f‹k fia[lcngim h_j_lc[him bkelklo ^ pr fksbkqlo+ Ir^k Mbmbo'044/,0506(- Dp cob`rbkqb bk i^ moŠ`qf`^rqfifw^oilp pŒj_lilp Hkr l ild r bk sbw ab H%r& m^o^abpfdk^o bi ild^ofqjl ab r+

5-5 Kld^ofqjlp obcbofalp^ rk^ _^pb mlpfqfs^]8yo„ 0

Dk i^ Rb``fŽk 5-1 pb e^ sfpql nrb i^ crk`fŽk ` jŠp dbkbo^i abofs^_ib bk bibgb ob^i+nrb p^qfpc^`bi^ b`r^`fŽk crk`flk^i `%rs& < `%r& * `%s&bpqŠa^a^ mlo i^cŽojri^9

'5-01( y&s' < ` ild s *

alkab _ bp rk^ `lkpq^kqb- O^o^ `^a^ _ bpq^ `%r& pb abkljfk^oŠ bi ild^ofqjl abs ^pl`f^al ^ `* u `ljl bp bsfabkqb+pr s^ilo kl pboŠkb`bp^of^jbkqb bi jfpjl nrbbi ild^ofqjl k^qro^i ab s, Rf ` < N+ bp fa‹kqf`^jbkqb kril u bpqb`^pl `^ob`b abfkqbo‹p-Rf ` :.: N pb fkaf`^oŠ ab lqo^ cloj^ i^ abmbkabk`f^ ab ` u _ fkqolar`fbkalbi `lk`bmql ab \[m_ ab ild^ofqjlp-

Cb '5-01( pb abar`b nrb `r^kal _ :.: N bufpqbrk k•jbol ob^i •kf`l \ = Nq^i nrb `%\& < 0- Dpq^ \ bpqŠ obi^`flk^a^ `lk _ mlo jbafl ab i^ fdr^ia^a`ild \ < 0: `ljl \ :.: 0 bp _ < 0.0/d \) X '5-01( pb bumobp bk i^ cloj^

a&s' < ild s ,ild ]

O^o^ bpq^bib``fŽk ab _ pb af`b nrb `%r& bp bi fia[lcngi ^_ r _h \[m_ \ v pb bp,`of_b ild+ r bk sbw ab `%r&+

Page 305: Calculus

Ijb\mdohjn m`a`md_jn \ pi\ ]\n` kjndodq\ ] ", 0 174

CDEHMHBHˆM- Pd ] = /+ ] ;/; 0+ X nd s = N+ `g gjb\mdohj _` s `i ]\n` ] `n`g iˆh`mj

ild sild s <,,

] ild ]%

_ji_` gjn gjb\mdohjn _`g n`bpi_j hd`h]mj nji gjb\mdohjn i\opm\g`n,

N_p‹osbpb nrb ild+ ] < 0- Rf ] < ` pb qfbkb ild+ s < i^d s* bp ab`fo+ ilp il,d^ofqjlp k^qro^ibp plk ilp nrb qfbkbk ab _^pb `, Orbpql nrb ilp ild^ofqjlp ab]\n` ` plk ilp jŠp cob`rbkqbjbkqb rp^alp bk L^qbjŠqf`^+ i^ m^i^_o^ ild^ofqjlfkaf`^ `^pf pfbjmob bi ild^ofqjl i\opm\g, LŠp q^oab+bk i^ Rb``fŽk 5-04 pb abcfkfoŠ]! ab q^i j^kbo^ nrb i^ b`r^`fŽk ]! < s pfdkfcf`^oŠbu^`q^jbkqb 0/ jfpjl nrbi^ p <ild+ s,

Orbpql nrb ilp ild^ofqjlp ab _^pb ] pb l_qfbkbk ab ilp ild^ofqjlp k^qr,o^ibp jriqfmif`^kal mlo i^ `lkpq^kqb i.ild \) i^ doŠcf`^ ab i^ b`r^`fŽk v < ild+ spb mrbab l_qbkbo ab i^ ab u < ild s jriqfmif`^kal qla^p prp loabk^a^p mlo rkjfpjl c^`qlo- Rf ] = 0 bpqb c^`qlo bp mlpfqfsl u pf ] ; 0 bp kbd^qfsl- Dk i^

t t

g:]:`

s l0[[ L:]:+

,, G `++++ ] <,

`

, ; \ ; G`

']( ] = G '_( /; ] ; H

EHFTQ@ 5-2 Dmƒad^\ _` u < gjb* s k\m\ q\mdjn q\gjm`n _` ],

cfdro^ 5-2'^( pb sbk bgbjmilp `lk ] = 0- Rf ] ; 0 pb l_pbos^ nrb g-] = 0 vild \ < , i^d %.,\&) ab j^kbo^ nrb i^ doŠcf`^ ab v < ild+ s pb mrbab l_qbkboab i^ ab t < ildi._u mlo pfjbqoŒ obpmb`ql i bgbs, Klp bgbjmilp ab i^ cfdro^ 5-2'_(pb e^k l_qbkfal ab bpq^cloj^ ^ m^oqfoab ilp ab i^ cfdro^ 5-2'^(-

Page 306: Calculus

064 Cpi^dji gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\n diq`mn\n

5-6 ;p_ZbYN`QRQR_VcNPVp[R V[aRT_NPVp[R[ YN` bR V[aR_cVR[R[Y\TN_VaZ\`

Orbpql nrb i^ abofs^a^ abi ild^ofqjl sfbkb a^a^ mlo i^ cŽojri^ C i^d s < g-sm^o^s = N+pb qfbkbi^ cŽojri^ ab fkqbdo^`fŽk

I z_s < i^d s * b -

@•k jŠp dbkbo^i+pf p < a&s'* pfbkal a rk^ crk`fŽk `lk abofs^a^ `lkqfkr^+ pbqfbkb

'5-02( ` _p,:: < i^d p * b l ` a%&s'_s < gjba&s' * _-

a&s'

G^v nrb qbkbo`rfa^al ^i rqfifw^o'5-02( v^ nrb bi ild^ofqjl kl bpqŠabcfkfalm^o^k•jbolp kbd^qfslp- Olo q^kql+i^p cŽojri^p ab fkqbdo^`fŽk'5-02( plk sŠifa^pq^k pŽil pf o) l `%r&bp mlpfqfs^-

@cloqrk^a^jbkqb+ bp cŠ`fi buqbkabo bi `^jml ab s^ifabw ab bpq^pcŽojri^pab j^kbo^ nrb mrbabk ^mif`^opbm^o^ crk`flkbp nrb pb^k mlpfqfs^p l kbd^qfs^p'mbol ij ^`mj', Rb fkqolar`b pfjmibjbkqb rk^ krbs^ crk`fŽk Ij abcfkfa^ m^o^qlalp ilp k•jbolp ob^ibps ;/; N mlo i^ b`r^`fŽk9

'5-03( HG!&hhIj&s' < i^d Zu\ < , _o *

0 o

abcfkf`fŽk prdbofa^ mlo i^ b`r^`fŽk '5-5( ab i^ Rb``fŽk 5-1- K^ doŠcf`^ ab Ij bppfj‹qof`^ obpmb`ql^i bgbv q^i `ljl pb sb bk i^ cfdro^ 5-3- K^ m^oqb i^ abob`e^abi bgbv bp bu^`q^jbkqb i^ jfpj^ nrb i^ `ros^ ild^oŒqjf`^ ab i^ cfdro^ 5-1-

Orbpql nrb i^d Ersf < i^d 'Gthhuh(< i^d Gth* i^d Guh+i^ crk`fŽk Ij p^qfpc^`bq^j_f‹k i^ b`r^`fŽk crk`flk^i _Špf`^ '5-3(: bp ab`fo+ pb qfbkb9

m^o^ s b v ob^ibp `r^ibpnrfbo^ afpqfkqlp ab `bol- O^o^ s = N pb qfbkbI9 &s' :: g-s v^ nrb Ij&s' m^o^s mlpfqfsl bp il jfpjl nrb i^d s, K^ cŽojri^ ab i^ abof,s^a^ s^ib q^j_f‹k m^o^ s ; N mrbpql nrb bk bpqb`^pl Ij&s' < I& + s' v mloq^kql Ij&s' < , I%&+s' < ,0.' +s' < g-s, Cb ^nrŒobpriq^

'5-04(+ 0

Ij&s' < , m^o^ qlal s^ilo ob^i s x N-r

Page 307: Calculus

C‡mhpg\n _` _`mdq\^d‡i ` dio`bm\^d‡i `i g\n lp` dio`mqd`i`i gjb\mdohjn 065

t

s

EHFTQ@ 5-3 Dmƒad^\ _` g\ api^d‡i Ij%

Olo q^kql+ pf bk i^p cŽojri^p ab fkqbdo^`fŽk mob`babkqbppb mlkb Ij bk sbw abH) pb mrbab buqbkabo pr ^i`^k`b ^ crk`flkbp nrb qlj^k s^ilobp q^kql kbd^qfslp`ljl mlpfqfslp- Olo bgbjmil '5-02( pb mrbab dbkbo^ifw^o ljl pfdrb9

'5-05( a^Q,:9 < i^d hqh* a+ `a%&U' _s < i^d Gb't(0* _-

a&s'

Dsfabkqbjbkqb+ `r^kal pb ^mifnrb '5-05( grkql `lk bi pbdrkal qblobj^ crka^,jbkq^i abi BŠi`ril m^o^ `^i`ri^o rk^ fkqbdo^i fkabcfkfa^ kl pb mrbabk qlj^ofkqbos^ilp nrb fk`irv^k mrkqlp bk ilp nrb o l `%r& pb^k `bol-

DIDLOKN 0- Hkqbdo^o q^k s _s*

Pjgp^d‡i, K^ fkqbdo^i qfbkb i^ cloj^ , ` _pZp* pfbkal p < `lp s* _p :pbks _s, Olo `lkpfdrfbkqb pb qfbkb

ao\i s _s < , a xp < ,i^d hqh* a < ,i^d iblp u\ * a+

cŽojri^ nrb bp sŠifa^ bk `r^inrfbo fkqbos^il bk bi nrb `lp s "N N-Klp alp bgbjmilp nrb pfdrbk plk ^mif`^`fŽk abi j‹qlal ab fkqbdo^`fŽkmlo

m^oqbp-

DIDLOKN 1- Hkqbdo^o i^d s _s,

Page 308: Calculus

066 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\ndiq`mn\n

Pjgp^d‡i, Rb^ p < ild s* _q < _s, Dkqlk`bp _p < _sZs* q < s* v l_qb,kbjlp

` ild u _s <` p _q < pq + ` q _p < u ild u , ` u z _s < u ild s + s * b-

DIDLOKN 2- Hkqbdo^o pbk 'ild s' _s,

Pjgp^d‡i, Rb^ p < n`i 'ild s'* g%< s, Dkqlk`bp _p < `lp 'ild s'&g-s' _s*v bk`lkqo^jlp

Ipbk 'Hld u( _s <Ep _q < pq + Eq _p < u pbk'Hld u( , Elp 'Hld s' _s,

Dk i^ •iqfj^ fkqbdo^ifkqbdo^jlp mlo m^oqbprk^ sbw jŠp+ l_qbkfbkal

Elp 'Hld u( _s < u `lp 'Hld u( * Epbk 'Hld u( _s ,

Blj_fk^kal ‹pq^ `lk i^ fdr^ia^a ^kqboflo+bk`lkqo^jlp nrb

Epbk 'Hld u( _s < fu pbk'Hld u( , qu `lp 'Hld u( * a+

v

Elp 'Hld u( _s < u pbk'Hld u( * qu `lp 'Hld u( * b -

.&0 9R_VcNPVp[Y\TN_oaZVPN

@elo^ pb bumlkaoŠ rk^ q‹`kf`^ `lkl`fa^ mlo _`mdq\^d‡i gjb\m…ohd^\nrb ^jbkral bp rk ^rufif^o mlabolpl bk bi `Ši`ril ab abofs^a^p- Di j‹qlal crbabp^oolii^al bk 0586 mlo Hle^kk Aboklriif '0556,0637( v pr crka^jbkql bp rk^eŠ_fi ^mif`^`fŽk ab i^ obdi^ ab i^ `^abk^-

RrmŽkd^pbnrb pb cloj^ i^ crk`fŽk `ljmrbpq^ ab Kl `lk rk^ crk`fŽk abofs^,_ib `r^inrfbo^ `%r&8 bp ab`fo+

b&s' < IjXa&s'Z < ild Fa&s' G

m^o^ qlal s q^i nrb a&s'%/;N- K^ obdi^ ab i^ `^abk^ ^mif`^a^ grkql `lk '5-04(`lkar`b ^ i^ cŽojri^

'5-06( b%&s'< I•Xa&s'Z , a%&s' < dww8+

Page 309: Calculus

Be`m^d^djn +12

Rf i^ abofs^a^ a$%r&pb mrbab `^i`ri^o ab ^idrk^ lqo^ cloj^+ bkqlk`bp pb mrbabl_qbkbo $%r& ^ m^oqfo ab '5-06( pfk jŠp nrb jriqfmif`^o a$%r&mlo x%r&+Dpqbj‹qlal bp •qfi bk i^ moŠ`qf`^ mlonrb jr`e^p sb`bp a$%r&bp jŠp cŠ`fi ab `^i`ri^onrb d$%r&+Dk m^oqf`ri^o+ bpql bp `fboql `r^kal zbp bi molar`ql M `l`fbkqb ab s^of^pcrk`flkbp pfjmibp- Di bgbjmil nrb pfdrb bp qŒmf`l-

DIDLOKN- B^i`ri^o d$%r&pf a&s' < s0 `lp s&g * U2'+5,

Pjgp^d‡i, Rb qlj^ bi ild^ofqjl abi s^ilo ^_plirql ab y&s' u irbdl pb abofs^-Rb^ mrbp

a%r&< i^d E`%r&. < i^d r0 * i^d Z`lp u\ * i^d '0 * U2'+5 :

< 1 i^d Zu\* i^d Z`lp u\ , 6 i^d '0 * s2',

Cbofs^kal pb qfbkb9

b%&s'< a%&s' < y \ pbk u ] 06s1

a&s' V `lp u i * u

Lriqfmif`^kal m^o x%r&pb l_qfbkb9

a%&s'< 0s `^p u'0 * t3b

06s3 `^p u'0 * U2'6 Š

5-8 :WR_PVPV\`

0- ^( G^ii^o qlalp ilp s^ilobp ab ` q^ibpnrb i^d s < ` * Px_f _o m^o^ qlal s = l-_( Rb^ `%r&< i^d Z'0 * ui.N , ui\ pf r; N- Rf [ v \ plk k•jbolp a^alp+ pfbkal[\ !$! * 0+ e^ii^o qlalp ilp r q^ibp nrb `%rf < `%[& * `%\&+

1- Dk `^a^ `^pl+ e^ii^o rk s ob^i nrb p^qfpc^d^ i^ fdr^ia^a a^a^-'^( i^d '0 * s' < i^d '0 , s', 'b( 1 i^d s < s ild 1+ s &0! 1-

'_( ild '0 * r& < i * ild '0 , r&+ 'a( i^d 'z * s:*f( < 0-2- Rb^ `%rf < 'i^d rcY| pf r; N- Cbp`of_fo ilp fkqbos^ilp bk ilp nrb ` bp `ob`fbkqb+ ab`ob`fbk,

qb+`lksbu^ v `Žk`^s^- Dp_lw^o i^ doŠcf`^ ab `+

Dk ilp Dgbo`f`flp 3 ^i 04+ e^ii^o i^ abofs^a^ `$%r&+Dk `^a^ `^pl+ i^ crk`fŽk ` pb prmlkbabcfkfa^ m^o^ qlal r ob^i m^o^ ilp nrb i^ cŽojri^ a^a^ m^o^ `%rf qfbkb pbkqfal-

2, a&s' < g\b '0 * s/&+

3, a&s' < ildz-

4, y&s' < g\bx*

5, a&s' < i^d 'i^d s',

6, a&s' < g\b&s0 ild s',

s, ' 06+a&s' < pi^d ,1,0 -

B (

Page 310: Calculus

07. Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\ndiq`mn\n

/., a&s' < &s * UH )s0'!

00- a&s' < r9t* , ild '0 * r9t* (-/0,a&s' ;sgjb&s *z]z

0 z )sS].0+ a&s' < 0SN] ild z ] sS] ,

/2, a&s' < uZpbk'ild s' + `lp 'Hld s~',

.2+ a&s' < il%+ `,

Dk ilp Dgbo`f`flp 05 ^i 15+ `^i`ri^o i^p fkqbdo^ibp-

.3+ a081U%06- cild1 s _s,

.5+ Ps ild s ^r)

.6+ P s ild1 T ^r)

c.1+/ _o

1/- ,0,&l * o

0/, P `lq s _s,

//+ Fs! ild &\s' ^r+

01, P s0.-n s _s,

02, a+g_[U+,r iar

00,/,- ild '0 , n&14- 0 _o*

l , o

a ild Yt[15- ,,+<!&!,,,,,!!! ^r)

rRE * ild Hui

16- Cbar`fo i^ cŽojri^ ob`roobkqb

` sh)/ ild! s i `sh ild! s _s < ,,,, , ,, sh ild!,i T _s

h)g h)g

v rqfifw^oi^ m^o^ fkqbdo^o Ps0 ild2 T _s,

17- ^( Rf r; N+pb^ `%r&< r * 0 , ild r) a%r&< ild r * 0 * f,r+ Du^jfk^o ilp pfdklp ab`$ v d&m^o^ abjlpqo^o nrb i^p abpfdr^ia^abp

00 , , ; ild s ; s + 0

s

plk sŠifa^p m^o^ s< N+s :‹-0- Br^kal s < 0+ pb `lksfboqbk bk fdr^ia^abp-_( So^w^o i^p doŠcf`^p ab i^p crk`flkbp = v > abcfkfa^p mlo i^p fdr^ia^abp =%r&< r ,,pu >%r&< 0 , f,r m^o^ r = N+ b fkqbomobq^odblj‹qof`^jbkqb i^p abpfdr^ia^abp ab i^m^oqb ^(-

18- Cbjlpqo^o nrb

- ild '0 * r&hGi,,,,< 0!)**(i B

`lk ilp alp j‹qlalp pfdrfbkqbp9 ^( rqfifw^kal i^ abcfkf`fŽk ab i^ abofs^a^ H$%.&8_( rp^k,al bi obpriq^al abi Dgbo`f`fl 17-

2/ -- Rf [ = N+e^`bo rpl ab i^ b`r^`fŽk crk`flk^i m^o^ abjlpqo^o nrb ild %[l& < l ild [ m^o^qlal k•jbol o^`flk^i m*

Page 311: Calculus

Mjgdijhdjn _` \kmjsdh\^d‡i k\m\ `g gjb\mdohj 180

20- Rb^ L < v\M) \y* \/) ††† ) \iw rk^ m^oqf`fŽk abi fkqbos^il Wf)rY alkab r; .+

'^( Hkqbdo^kal crk`flkbp bp`^ilk^a^p nrb plk `lkpq^kqbp bk ilp pr_fkqbos^ilp ^_fboqlpab L abar`fo i^p pfdrfbkqbp abpfdr^ia^abp9

d&\fx \f

[/' ; ild s ; d&\f

8 \f[/',

h<i f h<i h,i

'_( Hkqbomobq^odblj‹qof`^jbkqb jbaf^kqb Šob^p i^p abpfdr^ia^abp ab '^(-'`( Dpmb`f^ifw^oi^ m^oqf`fŽk m^o^ abjlpqo^o nrb+ m^o^ `^a^ bkqbol i = 0 bp9

i i j,0019f ; ild i ; 19f ,f;0 h<i

21- Cbjlpqo^o i^p pfdrfbkqbp cŽojri^p ab `^j_flp ab _^pb ab ild^ofqjlp-

ild+ s'^( hkc+T < ,0 ] %

jb\'^( 0/c^ r < ild+ [ fia[ r8

22- R^_fbkal nrb ild+ 0/ < 1+2/1474+ `lk pbfp `fco^p ab`fj^ibp bu^`q^p+ `^i`ri^o ild-+ `^mif`^kal rk^ ab i^p cŽojri^p abi Dgbo`f`fl 21- ƒBrŠkq^p `fco^p ab`fj^ibp bu^`q^p pbmrbab ^pbdro^o nrb pb e^k l_qbkfal bk bi obpriq^al>Jin[7 Tk^ q^_i^ `^i`ri^a^- `lk pbfp `fco^p ab`fj^ibp a^ bi s^ilo ild-+ _ < /+323183-

23- Tk^ crk`fŽk /* `lkqfkr^ bk bi bgb ob^i mlpfqfsl+ qfbkb i^ molmfba^a ab nrb `r^ibpnrfbo^nrb pb^k s = N b u = N+i^ fkqbdo^i

F7fEa&o' _o

bp fkabmbkafbkqb ab s 'u mlo q^kql abmbkab pŽil ab s&+Rf `%/& < 1+ `^i`ri^o bi s^ilo abi^ fkqbdo^i =%r& < Rfa&o' ^n m^o^ qlal r = N-

24- Tk^ crk`fŽk /* `lkqfkr^ bk bi bgb ob^i mlpfqfsl+ qfbkb i^ molmfba^a ab nrb

F!$fE F$! FfE0 a&o' _o < V 0 a&o' _o * s 0 a&o' _o

m^o^ qlal s< N v qlal v = N- Rf iN( < 2+ `^i`ri^o F&s' m^o^ `^a^ s ~ N-25- K^ _^pb ab rk pŽifal bp bi `lkgrkql ab loabk^a^p ab rk^ crk`fŽk E`lkqfkr^ bk bi fkqbo,

s^il Z0+[Y+ Sla^p i^p pb``flkbp mbombkaf`ri^obp ^i fkqbos^il Z0-[Y plk `r^ao^alp- Di sl,irjbk abi pŽifal bp d\1gjb0 \ + c^2ild \ * a,y\1 + a,mm^o^ qlal \ x 0- B^i`ri^o E%[&+

.&)( C\YV[\ZV\` QR N]_\dVZNPVp[]N_N RYY\TN_VaZ\

Dk bpq^ Rb``fŽk abjlpqo^objlp nrb i^ crk`fŽk ild^ofqjl mrbab ^molufj^opbmlo `fboqlp mlifkljflp nrb mrbabk rp^opb m^o^ `^i`ri^o ild^ofqjlp `lk bi do^al ab^molufj^`fŽk nrb pb abpbb-

Page 312: Calculus

070 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\ndiq`mn\n

O^o^ pfjmifcf`^o i^p cŽojri^p obpriq^kqbp+mofjbol obbjmi^w^jlp s mlo 0 , sbk i^ fkqbdo^inrb abcfkb bi ild^ofqjl m^o^l_qbkbo

HH,^9_oild'i , r& < ,+

0 o

sŠifa^ pf s ; 0- Di `^j_fl ab s^of^_ib o < 0 , p qo^kpcloj^ ^nrbii^ fdr^ia^a bki^ pfdrfbkqb

h^9_p Œifa 0,ild '0 , r& < ,,+ s^ 0 ^ m^o^ r ; -/0 , p

Rbdrfa^jbkqb ^molufj^jlp bi fkqbdo^kal i.N , o& m^o^ mlifkljflp nrb irbdlfkqbdo^jlp m^o^ l_qbkbo i^p `loobpmlkafbkqbp ^molufj^`flkbp m^o^ bi ild^ofqjl-Bljl mofjbo bgbjmil jlpqo^jlp rk^ pbk`fii^ ^molufj^`fŽk ifkb^i m^o^ bi fkqb,do^kal-

@ m^oqfoab i^ fabkqfa^a ^idb_o^f`^ 0 , o0 < N , r(N * o&) l_qbkbjlp i^cŽojri^

0 p0

++;g)p)++*+'X +'X

sŠifa^ m^o^ `r^inrfbo ob^i p ;C 0- Hkqbdo^kal ‹pq^ bkqob N u s* pfbkal s ; 0+qb,kbjlp

'5-07(

'5-08(s

0 i^9 p0

,ild '0 , r& < r * , * ,, ^o +0 jg+p

K^ doŠcf`^ abi mlifkljfl `r^aoŠqf`l M&s'< s )ds0 nrb ^m^ob`b bk bi pbdrkaljfbj_ol ab '5-08( bpqŠ obmobpbkq^a^bk i^ cfdro^ 5-4 grkql `lk i^ `ros^t < , ild N , s', N_p‹osbpb nrb m^o^ s moŽufjl ^ `bol bi mlifkljfl M&s'bprk^ _rbk^ ^molufj^`fŽk ab , ild N , r&+ Dk bi qblobj^ nrb pfdrb+ rqfifw^jlprk mlifkljfl ab do^al h * 0 m^o^ ^molufj^o i.N , o&) v `lk biil l_qbkbo rkmlifkljfl ab do^al h nrb ^molufjb ild '0 , r&+

RCMPCK? 5-2- P`\ L8 `g kjgdijhdj _` bm\_j i _\_j kjm

s0 s1 si •i sfMi@s' < s * , * , * --- * , < ]-

1 2 i ff;/

Bioji^`n* k\m\ oj_j s ; 0 t oj_j i x 0+n` od`i`

'5-1/( c$!pi,ild'i , s' < Mi&s' * ,, _p,

/0 , p

Page 313: Calculus

Mjgdijhdjn _` \kmjsdh\^d‡i k\m\ `g gjb\mdohj 182

t

+++E+

0+E+

EGG

FFE

EE

F

EF

FE

EE

s

y-Š--yy-Š--y+xx t < , ild '0 , r&

EHFTQ@ 5-4 Mjgdijhdj ^p\_mƒod^j _` \kmjsdh\^d‡i k\m\ g\ ^pmq\ v < , ild 'i , s',

A`hjnom\^d‡i, @ m^oqfoab i^ fabkqfa^a ^idb_o^f`^

0 , pi < '0 , r('i * R * R0 * --- * Ri+/'*

l_qbkbjlp i^ cŽojri^

0 - h

,\ < 0 * p * Q/ * --- * Qh*. * ZQZ )

0 , X 0 , X

sŠifa^ m^o^p ;/; 0- HkqbdoŠkali^bkqobN v s* pfbkal s ; )$ l_qbkbjlp '5-1/(-Olabjlp mlkbo '5-1/( bk i^ cloj^

'5-10(

pfbkal A))%r&i^ fkqbdo^i+

Di s^ilo ab Ah%r& obmobpbkqbi boolo `ljbqfal ^i ^molufj^o ,ild '0 , r& `lk bimlifkljfl Lh%r&+ O^o^ rqfifw^o '5-10( bk ilp `Ši`rilp+ kb`bpfq^jlp `lkl`bo pf bi

Page 314: Calculus

0729 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\n diq`mn\n

boolo bp mlpfqfsl l kbd^qfsl v il do^kab nrb mrbab pbo-Di moŽufjl qblobj^ klpaf`b nrb m^o^s^ilobp ab s mbnrb•lp v mlpfqfslp bi boolo Bi&s' bp mlpfqfsl+ mbolm^o^s kbd^qfs^ bi boolo qfbkbbi jfpjl pfdkl nrb ' , 0(!*0+ pfbkal i bi do^al abimlifkljfl ab ^molufj^`fŽk- Di qblobj^ q^j_f‹k molmlo`flk^ `lq^p prmboflo b fk,cboflo abi boolo-

RCMPCK? 5-3- Pd N ; s ; 0+ o`i`hjn g\n _`ndbp\g_\_`n

'5-11(si(. 0 si(.**9A%r&9****+k*i, i ,i,uk*i

Pd s ; N+ `g `mmjmBi&s' od`i` `g hdnhj ndbij lp` &+ 0(j*0+ v n` od`i`

GGi)//; &[g'i)/B &s' ; [s[ ,i ,k*i

'5-12(

A`hjnom\^d‡i, Rrmlkd^jlp nrb N ; s ; 0- Dk i^ fkqbdo^inrb abcfkb Bi&s'qbkbjlp N 99:::p 88999s* `lk il nrb 0 , s 889990 , p 889990+X mlo q^kql bi fkqbdo^kalp^qfpc^`bi^p abpfdr^ia^abp

pi:x:x,,i,r,i,u

Hkqbdo^kalbpq^pabpfdr^ia^abp+ l_qbkbjlp '5-11(-O^o^ abjlpqo^o '5-12(+ prmlkd^jlp s ; N v pb^ o < , s < Gth-Cb bpqbjlal

o = N v qbkbjlp

f+o pi go &[q'i /Go b!Bi&s' < Bi@*n&< ,, _p < , ,, _q < ',0(!* ,, _q,

N h,q jg)q jg)q

Dpql abjrbpqo^ nrb Bi&s' qfbkbbi jfpjl pfdkl nrb ' , 0(j*0- @pfjfpjl+ qbkbjlp

fo qi Go oi)/ Gthj*0&+g'i)/Bi@s' < ,, _q88999qi_q < ,, < ,,+

jg)q l k*i i)g

0/ `r^i `ljmibq^ i^ abjlpqo^`fŽk ab '5-12(-Di qblobj^ nrb pfdrb klp a^ rk^ cŽojri^ jrv •qfi m^o^ ilp `Ši`rilp `lk

ild^ofqjlp-

RCMPCK? 5-4- Pd N ; s ; 0 t nd h x 0+ n` od`i`

0 * s &s0T

/g*.

&ild ,, < 1 r * , * ---*,,, *,Ng%r& )0 , s 2 0h + 0

Page 315: Calculus

Mjgdijhdjn _` \kmjsdh\^d‡i k\m\ `g gjb\mdohj 184

`i _ji_` `g o„mhdij _` `mmjm*Oh&s'* n\odna\^` g\n _`ndbp\g_\_`n

'5-13(V1i*0 1 , V V

1i*0

,, ; Ng%r& R ,, ,,, -0h * 0 0 , t 0h * 0

A`hjnom\^d‡i, K^ fdr^ia^a '5-10( bp sŠifa^ m^o^ `r^inrfbo s ob^i q^i nrbs ; 0- Rf obbjmi^w^jlp s mlo , s bk '5-10(+ j^kqbkfbkal s = , 0+l_qbkbjlpi^ cŽojri^

'5-14( ,ild'i * u( < Mi&+s' * Bi&+s',

Rf , 0 ; s ; 0+plk sŠifa^p '5-10( u '5-14(- Qbpq^kal '5-14( ab '5-10(+ bk`lk,qo^jlp

'5-15( ild 0 * u < MGs' + Mi&+s' * Bi&s' + Bi&+s',0 , u

Dk i^ afcbobk`f^ Mi&s' + Mi&+ s'* i^p mlqbk`f^p m^obpab s abp^m^ob`bk u i^pmlqbk`f^p fjm^obp pb armif`^k- Olo `lkpfdrfbkqb+ pf i bp m^o+mlo bgbjmil i < 0h*qbkbjlp

M/g%r&* M

/g% *r& < 0&U * t

2 * --- * \t\1

]

i

\,

0

]( +

0 /g * 0

u i^ fdr^ia^a '5-15( pb qo^kpcloj^ bk

0 * s &s0V

1i,0

(i^d ,, < 1 u * , * --- * ,,, * O %r&0 , u 2 0h [ 0 j- +

bk alkab Oh&s' < B0h&s' + B0h&+ s', Dpq^ cŽojri^ bp sŠifa^ pf s bpqŠbk bifkqbos^il ^_fboql , 0 ; s ; 0-L^kqbkd^jlp ^elo^ s bk bi fkqbos^il N ; s ; 0-Dkqlk`bp i^ bpqfj^`fŽk abi qblobj^ 5-3 klp a^

V1i*0 0 V1i*0

,,, ; A0 %r&; ,, ,,,0h * 0 , i , 0 , t 0h * 0

u;+H**

.: +B/g% *r& R0h * 0-

Rrj^kal bpq^pabpfdr^ia^abp+ l_qbkbjlp i^p '5-13(+ v^ nrb 0 * i.N , r& :< '1 , r&,K * r&+

Page 316: Calculus

074 Cpi^d4i gjb\mdohj* api^d4i `skji`i^d\g u api^dji`n omdbjijh„omd^\ndiq`mn\n

DIDLOKN- Slj^kal h < 1 X s < 0+qbkbjlp 'i * s'-&g + s' < 1+ X ob•priq^ i^ cŽojri^

alkab

Cb ^nrŒ obpriq^k i^p abpfdr^ia^abp /+5810 ; ild 1 ; /+5824 `lk jrv ml`lp`Ši`rilp-

.&)) :WR_PVPV\`

0- Tqfifw^o bi qblobj^ 5-4 mlkfbkal s < qv h < 4 m^o^ `^i`ri^o ^molufj^`flkbp ab ild 1-Blkpbos^o krbsb `fco^p ab`fj^ibp bki ilp `Ši`rilp v l_qbkbo i^p abpfdr^ia^abp /+582035/ ;ild 1 ; /+5820365-

1- Rf r <q+ pboŠ 'i * r&,%f * r& < I- @pŒmrbp+ bi qblobj^ 5-4 klp mbojfqb `^i`ri^o i^d 2bk crk`fŽk ab ild 1- Slj^o s < pv h < 4 bk bi qblobj^ 5-4 v bjmib^o ilp obpriq^alpabi Dgbo`f`fl 0 m^o^ l_qbkbo i^p abpfdr^ia^abp 0+/87500 ; i^d 2 ; 0+/87506-

Kjo\8 Orbpql nrb i^d 1 ; i^d ` ; i^d 2+obpriq^ nrb 1 ; ` ; 2-2- Tp^o bi qblobj^ 5-4 `lk s < pm^o^ `^i`ri^o i^d 4 bk crk`fŽk ab i^d 1- Dibdfo bi do^al

abi mlifkljfl ab ^molufj^`fŽk il _^pq^kqb bibs^al m^o^ l_qbkbo i^p abpfdr^ia^abp0+5/8324 ; ild 4 ; 0+5/8327-

3- @mif`^o bi qblobj^ 5-4 `lk s < ,Ho m^o^ `^i`ri^o ild 6 bk crk`fŽk ab ild 4- Dibdfo bi do^alabi mlifkljfl ab ^molufj^`fŽk il _^pq^kqb bibs^al m^o^ l_qbkbo i^p abpfdr^ia^abp0+8348/6 ; ild 6 ; 0+834800-

4- Blk ilp obpriq^alp ab ilp Dgbo`f`flp 0 ^i 3 `^i`ri^o rk^ mbnrb•^ q^_i^ bk i^ nrb ^m^obw,`^k ild i m^o^ i < 1+ 2+ --- + 0/- Tqfifw^o q^kq^p `fco^p ab`fj^ibp ^jmm`^o\n `r^kq^p pb^ ml,pf_ib ^ m^oqfoab i^p abpfdr^ia^abp ab ilp Dgbo`f`flp abi 0 ^i 3-

.&)* ?N Sb[PVp[Rd]\[R[PVNY

Di qblobj^ 5-1 abjrbpqo^ nrb m^o^ qlal s ob^i bufpqbrkl v rk plil v q^inrb H%s&< s, Olo `lkpfdrfbkqb mlabjlp ^mif`^o bi mol`bpl ab fksbopfŽk m^o^ab,cfkfo u `ljl crk`fŽk ab s, K^ crk`fŽk fksbop^ obpriq^kqbpb abkljfk^ api^d4i`skji`i^d\g* l \iodgjb\mdohj* v pb obmobpbkqmlo B,

CDEHMHBHˆM- M\m\ ^p\glpd`m s m`\g*_`adidhjn B&s' ^jhj \lp`g iˆh`mj v^ptj gjb\mdohj `n s, Bnoj `n* v < B&s' ndbidad^\I&t' < s,

Di aljfkfl ab B bp qlal bi bgbob^i: pr ob`loofal bp bi `lkgrkql ab k•jbolpob^ibp mlpfqfsl- K^ doŠcf`^ab A) nrb pb obmobpbkqbk i^ cfdro^ 5-5+pb l_qfbkb abi^ doŠcf`^ abi ild^ofqjl jbaf^kqb rk^ pfjbqoŒ^obpmb`ql i^ ob`q^ v < s, Orbpql

Page 317: Calculus

I\ api^d‡i `skji`i^d\g 186

t

%%%%`&`&&&&&%ŠŠŠŠŠŠŠŠ++ !+t;I&s'

# `&&&:&

:&

. '0+/(E

EEE

s

EHFTQ@ 5-5 I\ bmƒad^\ _` g\ api^d‡i `skji`i^d\g n` j]od`i` _` g\ _`g gjb\mdohj kjm pi\ndh`om…\m`nk`^oj \ g\ m`^o\ u < s,

nrb I XB plk fksbop^prk^ ab lqo^+pb qfbkb

HWA%r&Y< r m^o^qlal r u AWH%s&Y< s m^o^qlal t = N-

B^a^ molmfba^a abi ild^ofqjl mrbab qo^ar`fopb bk rk^ molmfba^a ab i^ buml,kbk`f^i- Olo bgbjmil+ mrbpql nrb bi ild^ofqjl bp bpqof`q^jbkqb`ob`fbkqbv `lkqfkrlbk bi bgbob^i mlpfqfsl+pb abar`b abi qblobj^ 2-0/ nrb i^ bumlkbk`f^i bp bpqof`q^,jbkqb `ob`fbkqb v `lkqfkr^ bk qlal bi bgb ob^i- K^ o‹mif`^ abi qblobj^ 5-0 bp bipfdrfbkqb

RCMPCK? 5-5- I\ api^d‡i `skji`i^d\g od`i` g\n kmjkd`_\_`n ndbpd`io`n8

]( A%K&< 0+ A%f& < `,_( B%&s'< B&s' k\m\ oj_j s,b( B&\ * ]' < B&\'B&]' k\m\ oj_j \ t oj_j ],

A`hjnom\^d‡i, K^ m^oqb ^( pb abar`b ab i^p fdr^ia^abp I&0( < N XH%_&< 0- Rbdrfa^jbkqb abjlpqo^jlp ]&) nrb bp i^ b`r^`fŽk crk`flk^i m^o^ i^bumlkbk`f^i- Rrmlkd^jlp nrb \ v ] plk a^a^p v mlkd^jlp

r < A%[&) t < A%\&) b < H%rs&+

Page 318: Calculus

/65 Boh]cƒh fia[lcngi) `oh]cƒh _rjih_h]c[f u `oh]cih_m nlcaihig€nlc][m chp_lm[m

Sbkbjlp bkqlk`bp

H%r&< [) H%s&< \) A%]&< rs+

Obol _ < H%rs&< H%r&* H%s&< [ * \+ Dpql bp+ _ < [ * \+ Olo q^kql+A%]&< A%[ * \&+ Olo lqo^ m^oqb+ A%]&< rs < A%[&A%\&) ab jlal nrbA%[ * \& < A%[&A%\&)il nrb abjrbpqo^ b(-

K^ ^mif`^`fŽk ab i^ b`r^`fŽk crk`flk^i klp ^vra^ ^ abjlpqo^o _(+ Di `l`fbkqbab afcbobk`f^p m^o^ i^ abofs^a^ A$%r&bp

A%r* b&* A%r&< A%r&A%b&* A%r&< A%r&A%b&* 0 -c c c

Olo il q^kql+ m^o^ mol_^o _( e^v nrb abjlpqo^o nrb

'5-16( ifj A%b&* 0 < 0 -Hq---‘l c

Dumobp^objlp bi `l`fbkqb '5-16( bk crk`fŽk abi ild^ofqjl- Olkd^jlp e:A%b&*.+Dkqlk`bp e * 0 < A%b& lk il nrb Hce * 0( < b v bi `l`fbkqb bp fdr^i ^

'5-17(A%b&* 0

c

e

I&f * 0(

Br^kal b w N bp A%b&w 0+ mrbp i^ crk`fŽk bumlkbk`f^i bp `lkqfkr^ bk 0- Orbpqlnrb e < A%b&* 0+ qbkbjlp e w N `r^kal b ,p N- Obol

H%e* 0( < H%e* 0( , H%.&,,* 09'0( < 0 `r^kal•h ,,* N -f f

Sbkfbkal bk `rbkq^ '5-17(+ bpql abjrbpqo^ '5-16( il `r^i+ ^ pr sbw+ abjrbpqo^ _(

5-02 Dumlkbk`f^ibpbumobp^a^pljl mlqbk`f^pab `

K^ b`r^`fŽk crk`flk^i A%[ * \& < A%[&A%\&qfbkb jr`e^p `lkpb`rbk`f^pfkqbobp^kqbp- Olo bgbjmil+ mlabjlp rqfifw^oi^ m^o^ abjlpqo^o nrb

'5-18( B&m'< `l

m^o^ qlal k•jbol o^`flk^i o-

Slj^jlp mofjbol ] < , [ bk i^ b`r^`fŽk crk`flk^i l_qbkfbkal

A%[&A%*[& < A%K&< 0 +

Page 319: Calculus

@_`chc]cƒh^_ _! j[l[ r l_[f ]o[fkoc_l[ +22

v mlo q^kql A%* [& < f,A%[& m^o^ qlal [ ob^i- Slj^kal \ < [) \ < /[) +++)\ < h[ bk i^ b`r^`fŽk crk`flk^i l_qbkbjlp+ pr`bpfs^jbkqb+ A%/[& < A%[&/)A%0[& < A%[&0) v+ bk dbkbo^i+

'5-2/( A%h[&< B&\'i

m^o^ qlal h bkqbol mlpfqfsl- Dk m^oqf`ri^o+ r^kal [ < )$ l_qbkbjlp

B&i' < `i*

jfbkqo^p nrb m^o^ [ < f,h) pb l_qfbkb A%f&< A%f,h&h+Orbpql nrb A%f,h& = N+biil fjmif`^

B&x' < `% !8

Olo `lkpfdrfbkqb+ pf mlkbjlp \ < g-h bk '5-2/( v ^mif`^jlp '5-20(+ bk`lkqo^jlp

'5-20(

m^o^h v i bkqbolp mlpfqfslp `r^ibpnrfbo^- Cf`el ab lqol jlal+ ebjlp abjlpqo^,al '5-18( m^o^ `^a^ k•jbol o o^`flk^i mlpfqfsl- Bljl A%* o( < f,A%l& < _7!)q^j_f‹k bp sŠifa^ m^o^qlal o o^`flk^i kbd^qfsl-

.&), 9RSV[VPVp[QR 5! ]N_N ? _RNYPbNY^bVR_N

Dk bi ^m^oq^al ^kqboflopb e^ jli\[^i nrb _! < A%r&`r^kal r bp rk l[]cih[f`r^inrfbo^- @elo^ pb ^_`chcl• _! m^o^r foo^`flk^i mlo

'5-21( _T < A%r& m^o^`^a^ r ob^i-

K^ jŠufj^ grpqfcf`^`fŽknrb pb mrbab a^o ab bpq^ abcfkf`fŽk bp nrb `lk bii^ i^ibv ab ilp bumlkbkqbp

'5-22(

bp sŠifa^ m^o^ qlalp ilp k•jbolp ob^ibp [ v \+ Br^kal pb qlj^ i^ abcfkf`fŽk'5-21(+ i^ abjlpqo^`fŽk ab '5-22( bp qofsf^i mrbpql nrb '5-22( kl bp jŠp nrb i^jfpj^ ^cfoj^`fŽk ab i^ b`r^`fŽk crk`flk^i-

K^ klq^`fŽk `s m^o^ A%r& bp rk^ ab i^p `lj•kjbkqb rp^a^p m^o^ i^ buml,kbk`f^i- Dk ^idrk^ l`^pfŽk pb bp`of_b bum'u( bk sbw ab `s mofk`fm^ijbkqb `r^kal

Page 320: Calculus

1.. Cpi^d‡i gjb\mdohj*api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\ndiq`mn\n

^m^ob`bk bumobpflkbp `ljmif`^a^p bk bi bumlkbkqb- Dk bpqb`^mŒqrilpb pbdrfoŠrqfifw^kal ^idrk^p sb`bp A%r&)mbol jŠp q^oab pb rp^oŠ pfbjmob _!+

Rb e^ abcfkfal i^ crk`fŽk bumlkbk`f^i ab j^kbo^ nrb i^p alp b`r^`flkbp

t < `! X s < i^d t

pfdkfcfnrbk bu^`q^jbkqb il jfpjl- Dk bi moŽufjl ^m^oq^al pb abcfkfoŠk mlqbk,`f^p jŠp dbkbo^ibp ab j^kbo^ nrb i^p alp b`r^`flkbp v < \! v s < ild+ v pb^kbnrfs^ibkqbp-

.&)- 9RSV[VPVp[QR>= ]N_N \ = M v s _RNY

Cbpmr‹p ab e^_bo abcfkfal `! m^o^ s ob^i `r^inrfbo^+ kl e^v kfkdrk^ afcf,`riq^a m^o^ a^o rk^ abcfkf`fŽk ab \! m^o^ `^a^ \ = N- Tk j‹qlal bp abcfkfo

\! `ljl bi k•jbol u q^i nrb ild+ t < s9 `i^ol nrb bpqb j‹qlal kl pfosb m^o^\ < 0 mrbpql- nrb bi ild^ofqjl ab _^pb 0 kl bpqŠ abcfkfal- Nqol jlal bp abcfkfo\! mlo i^ cŽojri^9

'5-23( [T < aX0kc]Š

Di pbdrkal j‹qlal bp mobcbof_ib+mlonrb bk mofjbo ird^o bp sŠifal m^o^ qlalmlpfqfsl \ 'fk`irfal \ < 0(+ X bk pbdrkal ird^o mlonrb `lk biil bp jŠp cŠ`fimol_^o i^p pfdrfbkqbp molmfba^abp ab bumlkbk`f^ibp9

i^d \! < s i^d \ , &\]'U < \s]sŠ

[T[.f < [r(.c † 'V(0H < %[.f&T < [T.E †

Pd\ ;C 0+`ioji^`n v < \s ndv n‡gj nds < ild+ v-

K^p abjlpqo^`flkbp ab bpq^p molmfba^abp pb abg^k `ljl bgbo`f`fl ^i ib`qlo-Cb i^ jfpj^ j^kbo^ nrb i^ doŠcf`^ ab i^ crk`fŽk bumlkbk`f^i pb l_qfbkb

ab i^ abi ild^ofqjl mlo pfjbqoŒ^ obpmb`ql ^ i^ ob`q^ s < v+ i^ doŠcf`^ ab v < \!pb mrbab l_qbkbo ab i^ ab u < ild+ s mlo pfjbqoŒ^ obpmb`ql ^ i^ jfpj^ ob`q^:bk i^ cfdro^ 5-6 pb a^k bgbjmilp ab biil- K^p `ros^p bk i^p cfdro^p 5-6 '^( v '_(pb e^k l_qbkfal ab i^p ab 5-2 '^( v '_( obpmb`qfs^jbkqb mlo pfjbqoŒ^- K^ doŠcf`^`loobpmlkafbkqb ^ \ < 0 bp k^qro^ijbkqb i^ elofwlkq^i u < 0-

.&). ;p_ZbYN`QRQR_VcNPVp[R V[aRT_NPVp[R[ YN` bRV[aR_cVR[R[Rd]\[R[PVNYR`

Tk^ ab i^p molmfba^abp jŠp klq^_ibp ab i^ crk`fŽk bumlkbk`f^i bp i^ cŽojri^

%3+02& A$%r&< A%r&)

Page 321: Calculus

C‡mhpg\n _` _`mdq\^d‡i ` dio`bm\^d‡i `i g\n lp` dio`mqd`i`i `skji`i^d\g`n 1./

nrb klp af`b nrb bpq^crk`fŽk bp pr molmf^abofs^a^- Rf i^ ^mif`^oklp grkql `lk i^obdi^ ab i^ `^abk^+ mlaboklp l_qbkbo cŽojri^p ab abofs^`fŽk m^o^crk`flkbp buml,kbk`f^ibp `lk _^pb mlpfqfs^ [ `r^inrfbo^-

Rrmlkd^jlp `%r& < [! m^o^ r = l- Rbd•k i^ abcfkf`fŽk ab \!* mlaboklp bp,`of_fo

x%r&< _rfia[ < A%ri^d [& 8

irbdl+ bk sfoqra ab i^ obdi^ ab i^ `^abk^+ bk`lkqo^oklp

'5-25( a%&s'< B%&si^d \' , i^d \ < B&s i^d \' , i^d \ < \! ild \ ,

Cf`el ab lqol jlal+ i^ abofs^`fŽk ab \! jriqfmif`^ pfjmibjbkqb \! mlo bi c^`qlo`lkpq^kqb ild \* pfbkal bpqbc^`qlo 0 `r^kal \ < `,

t

/: \: `

'^( \< 0

s s

EHFTQ@ 5-6 Dmƒad^\ _` u < \! k\m\ q\mdjn q\gjm`n _` \,

'_( N ; \ ; H

Dsfabkqbjbkqb+ bpq^pcŽojri^p ab abofs^`fŽk `lkar`bk ^rqljŠqf`^jbkqb ^i^p cŽojri^p ab fkqbdo^`fŽk `loobpmlkafbkqbp- Olo bgbjmil '5-24( a^ `lokl ob,priq^al

'5-26( F `U _s < `!% * b+

Page 322: Calculus

1.0 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g u api^dji`n omdbjijh„omd^\n diq`mn\n

bk q^kql nrb '5-25( `lkar`b ^ i^ c5ojri^ jŠp dbkbo^i

a\!%_s;x *Bild [

Dpq^mrbab ^•k dbkbo^ifw^opbmlo bi j‹qlal abprpqfqr`f5k- Rrpqfqrfjlp r bk '5-26(v '5-27( mlo p l_qbkfbkal

%[= N+[ "9 0( -'5-27(

'5-28(- a ]S\m _p < ,, * A

ild [%[ = N+[ "9 0( +

bk alkab p obmobpbkq^r^inrfbo crk`f5k `lk abofs^a^ `lkqfkr^- Rf bp`of_fjlpp < a&s'*u _p < a%&s'_s* i^p c5ojri^p '5-28( pb `lksfboqbk bk

` _`%t&`$%r& r < _`]!)& * A + a[`%!)&

[`?r&`$%r& ^r < ,, * B+ild [

pfbkal i^ pbdrka^ fkqbdo^isŠifa^ m^o^\ = N+\ :.: .)

DIDLOKN 0- Hkqbdo^oPU0`!%6_s,

Pjgp^d„i, Rb^ p < s%*Dkqlk`bp _p < 1s0 _s* u pb l_qfbkb

a0xDIDLOKN 1- Hkqbdo^o- TV _s ,

Pjgp^d‡i, Rb^ p < TV ;sGd, Dkqlk`bp _p < ds+Gd_s < q_s-SU, Krbdlqbkbjlp

a U: ` &g_U' ` 1S 1i*U:>9,_s < 1 1U: , , < 1 1q _p < 1 , * B < ,, * B -z 1 !*FU ild 1 ild 1

DIDLOKN 2- Hkqbdo^oP `lp s `0 ŠŠŠs _s,

Pjgp^d‡i, Rf p < 1 pbk s* pboŠ_p < 1 `lp s _s* v pb l_qfbkb mlo q^kql

Page 323: Calculus

C‡mhpg\n_` _`mdq\^d‡i ` dio`bm\^d‡i `i g\n lp` dio`mqd`i`i `skji`i^d\g`n 1.1

DIDLOKN 3- Hkqbdo^o `Upbks _s,

Pjgp^d‡i, Olkd^jlp p < `!* _q < pbks _s, Dkqlk`bp _p < `%_s* q :< , `lp s* v bk`lkqo^jlp

&4,2.' ` bWpbku_s < ` p _q < Q@ * ` q _p < [`s `lp s * ` `U`lp s _s * _-

K^ fkqbdo^iO `! `lp s _s pb qo^q^abi jfpjl jlal- Slj^jlp p < `s* _q;%^jn s _s*_p < `%_s*q < pbks* v l_qbkbjlp

'5-30( ` `! `lp s _s < `! pbku , ` `! pbku _s * b -

Rrpqfqrv‹kalil bk '5-3/(+ mlabjlp abpmbg^oa`s pbk s _s v obrkfbkal i^p `lkpq^k,qbp ^o_fqo^of^ppb l_qfbkb

` _T`! pbku _s < !1 'pbku , `lp s' * b-

N_p‹osbpb nrb mlabjlp ^mif`^o bpqbobpriq^al bk '5-30( m^o^l_qbkbo q^j_f‹k

F KB`U^jns_s < !1 '`lp s * pbks' * _-

DIDLOKN 4- Hkqbdo^oF w +0 * _T

Pjgp^d‡i, Tk^ j^kbo^ ab qo^q^obpqbbgbjmil bp mlkbo bi fkqbdo^kal bk i^pfdrfbkqb cloj^9

,,<,,,0 * `U `+U* 0

Rbdrfa^jbkqb e^`bjlp p < `+\8* 0+ `lk il nrb _p < , `+U_s* v iibd^jlp ^

F[`[+T

Z_U < *F [+[`+[s[_[s< *F _p < ,ild hqh* b < ,ild '0 * `+U' * `-`+s * 0 `+U * 0 p

Page 324: Calculus

1.2 Cpi^d‡i gjb\mdohj*api^d‡i `skji`i^d\g u api^dji`n omdbjijh„omd^\ndiq`mn\n

Di obpriq^al mrbab mlkbopb ab lqo^ cloj^ pf jlafcf`^jlp bi ild^ofqjl- Olobgbjmil+

0 `!,ild '0 * `+W

' < ild ,,, < ild ,, <0 * `+u `W * 0

< ild 'b!( , ild 'b! * 0( < s + ild '0 * `W',

Nqol jlal ab obplisbo bi bgbjmil `lkpfpqb bk mlkbo

0 `x,,<0,,,-g)`u g)`u

Dkqlk`bp pb qfbkb

`w < s *`w _s < s *` _p *0 * `! 0 * `U p

alkab p < 0 * `!, Dk`lkqo^jlp ^pŒ

a ^r,, < s + ild '0 * `W

' * b+0 * `!

nrb bp rk^ ab i^p cloj^p l_qbkfa^p ^kqbp-

.&)/ :WR_PVPV\`

Dk ilp Dgbo`f`flp 0 ^i 01+e^ii^o i^ abofs^a^ `$%r&+Dk `^a^ `^pl i^ crk`fŽk ` pb i^ prmlkbabcfkfa^ m^o^ qlal r ob^i m^o^ bi nrb i^ bumobpfŽknrb pb a^ ab `%r&qbkd^ pbkqfal-

0- a&s' < b2!&,0- 6- a&s' < /s! Znrb pfdkfcf`^ 1'w!(\-0, a&s' < `\f+ 7- a&s' < ` ŠŠŠŠY-

1, a&s' < `+u!, 8- a&s' < `@jn!u-L' (1+a&s' < ` w- 0/- a&s' < a0/c w-

3, a&s' < a0.w‘ 00- a&s' < ``u Znrb pfdkfcf`^ `:`V&Y+

3+a&s' < 1w‘ 01- a&s' < ``%uZnrb pfdkfcf`^ bum &`&`W~'Z,

B^i`ri^o i^p fkqbdo^ibpfkabcfkfa^p ab ilp Dgbo`f`flp 02 ^i 07-

02- Fs b&!_s, 05- Fm `+/t _s,

.1+ Fs`+u_s, 06- F`q%9_s,

04- Fm b&!_s, 07- Fobw&_s,

Page 325: Calculus

Bd`m^d^djn +(-

08- Cbqbojfk^o qla^p i^p `lkpq^kqbp \ v ] q^ibp nrb 8aU < ] * Px`o _o*1/- Rb^k = < -b`\! `lp \r ^r v > < Hb^!&pbk\r ^r) alkab [ v \ plk `lkpq^kqbp+ kl pfjri,

qŠkb^jbkqb kri^p-Hkqbdo^kal mlo m^oqbpabjlpqo^o nrb

\> + ]? < `\! `lp ]s * B0 + \? * ]> < `\!n`i]s * @0*

pfbkal Bi

v B1

`lkpq^kqbp ^o_fqo^of^p-Cbpmbg^o= v > m^o^ l_qbkbo i^p pfdrfbkqbp cŽojri^pab fkqbdo^`fŽk9

G\! + `\!&\ `lp ]s * ] pbk ]s'

`^jn ]s _s + 1 ]0 * @*\ (

F`\!&\ pbk ]s + ] `lp ]s'

`\! pbk ]s _s < ,,,,,,,,, * @[0 % ]0 Š

Dk ilp Dgbo`f`flp abi 10 ^i 23+ e^ii^o i^ abofs^a^ `$%r&+Ah`^a^ `^pl+ i^ crk`fŽk ` pb pr,mlkb abcfkfa^ m^o^ s^ilobp ob^ibp ab r m^o^ ilp nrb i^ cŽojri^ a^a^ ab `%r&qfbkb pbkqfal-K^ abofs^a^ ild^oŒqjf`^ mrbab pfjmifcf`^o bi qo^_^gl bk ^idrklp `^plp-

0/, y&s' < s!,

00, y&s' < 'i * s'&g * `!%',

`%! + `+!01, y&s' < ,, -

`! * z~

02, y&s' < s\\ * \!\ * \\sŠ

03, y&s' < ild Zild 'ild s'Z,

/3+ y&s' < ild &`%!* y(-

06, y&s' < 'ild s'!,

07, y&s' < sgjb!,

'ild s'!1., y&s'< ,hk!!

B _

20- y&s' < 'pfk s'!jn! * '`lp s' ŠŠŠŠ!

10, x%r&< Ug-!,

s0&1 + U'F-1

00+x%r&< 'i ] r&%0* U'0-1%

05, x%r&< a,h

12, y&s' < SH&s + \d']d,<->

24- Rb^ x%r&< r!) alkab r = N u l bp rk k•jbol ob^i `r^inrfbo^- K^ cŽojri^ `$%r&< lrl*g

pb abjlpqoŽ m^o^ m m\^dji\g,'^( Ool_^o nrb bpq^ cŽojri^ bp sŠifa^ q^j_f‹k m^o^ m ob^i `r^inrfbo^- XFi_d^\^d‡i8 Dp,`of_fo sm < `m ild s,Z

'_( Cfp`•q^pb _^gl nr‹ `lkaf`flkbp bi obpriq^al ab '^( pb ^mif`^ m^o^ s x N-25- @mif`^o i^ abcfkf`fŽk \! < b!&ild^ m^o^ abar`fo i^p pfdrfbkqbp molmfba^abp ab i^ bu,

mlkbk`f^i dbkbo^i9'^( ild \T < s ild \,'_( &\]'! < \!%]!,'b( \!\V < \!)V,

'a( &\!'V < &\V'! < \!V,

'b( Pd \ x 0+ `ioji^`n v < ^! nd v n‡gj nd s < ild+ t,

Page 326: Calculus

+)- Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\n diq`mn\n

26- Rb^ `%r&< n%[$!* [*!$& pf [ = N- Ool_^o nrb-

a&s * t' * a&s + t' < 0a&s'a&t',

27- Rb^ `%r& < _?$!alkab _ bp rk^ `lkpq^kqb- Ool_^o nrb .&'/( < _ u ^mif`^o bpqb obpriq^alm^o^ abjlpqo^o i^ pfdrfbkqb obi^`f5k9

`?$! * 0

ifj ,,, ;^,y--,-j T

28- Rb^ ` rk^ crk`f5k abcfkfa^ bk qlal bi bgb ob^i+`lk abofs^a^ .& nrb p^qfpc^`b i^ b`r^`f5k9

a%&s'< ^a&s' m^o^ qlal s*

alkab ` bp rk^ `lkpq^kqb- Ool_^o nrb bufpqb rk^ `lkpq^kqb G q^i nrb a&s' < H`^!* m^o^ `^,a^ s*

WEh^c][]cƒh7 GŠd^pb a%r&< `%r&_*?$! u `lkpfa‹obpb a$%r&+Y3/- Rb^ ` rk^ crk`f5k abcfkfa^ bk qlal bi bgb ob^i- Rrm5kd^pb ^abjŠp nrb . p^qfpc^`b H^b`r^,

`f5k crk`flk^i9

'f( a&s * t' < a&s'a&t' m^o^ qlal s b v-

'^( @mif`^kal p50/ i^ b`r^`fŽk crk`flk^i abjlpqo^o nrb a&L' bp N 5 0- Cbjlpqo^o q^jl_f‹k nrb pf .%-&{{0 N bkqlk`bp `%r&{{0 N m^o^ ni^i r+ RrmŽkd^pb+ ^abjŠp ab 'f(+ nrb ,$%r&bufpqb m^o^ qlal s* u abjr‹pqobkpb i^p pfdrfbkqbp molmfba^abp9'_( a%&s'a&t';a%&t'&s' m^o^ qlal s b v-

'b( Dufpqb rk^ `lkpq^kqb _ q^i nrb `$%r&< ]`%r& m^o^ qlal r+'a( E%r&< _@%! pf .%-&{{0 N- WEh^c][]cƒh7 U‹^pb Dgbo`f`fl 28-

30- '^( Rb^ `%r&< _! * 0 , r m^o^ qlal r+ Cbjlpqo^o nrb ,)%r&w N pf r w N X `$%r&w Npf s ,o N-G^`fbkal rpl ab bpqb eb`el abar`fo i^p abpfdr^ia^abp

`%!= 0 )s*

sŠifa^p m^o^ qlal s = N- 'Br^kal s < N+pb `lksfboqbk bk fdr^ia^abp-(Hkqbdo^obpq^p abpfdr^ia^abp m^o^ abar`fo i^p pfdrfbkqbp+ qla^p sŠifa^p m^o^ s = N9

[0

'^( `%! = 0 * s * 1e&[/

_*!$9E*r(/c$

s0 u2`+%!= 0 , s * , ,,

1 2 &

s0 u2'b( `%! = 0 * s * 1f * 2 &

'ag Dkrk`f^o i^ dbkbo^ifw^`f5k prdbofa^ v abjr‹pqobpb-31- Rf i bp rk bkqbol mlpfqfsl v pf s = N+abjlpqo^o nrb

v nrb & s'+i`!%: .*88 pf s :ij

Difdfbkal i bk cloj^ ^ab`r^a^+ abar`fo nrb 1+4 ; ` ; 1+88-

Page 327: Calculus

Cpi^dji`n cdk`m]‡gd^\n 2/6

32- Rb^ `%r)u( < rV alkab r = N- Cbjlpqo^o nrb

.&)0 ;b[PV\[R` UV]R_OpYVPN`

jX,< tss*/U[ u

jXjt < sV ildu-

Eob`rbkqbjbkqb bk @kŠifpfppb mobpbkq^k`fboq^p `lj_fk^`flkbp ab crk`fl,kbp bumlkbk`f^ibp nrb jbob`bk nrb pb ibp a‹ klj_obp bpmb`f^ibpu nrb pb bpqr,afbk `ljl bgbjmilp ab krbs^p crk`flkbp- Dpq^p`lj_fk^`flkbp pb abkljfk^kn`ij cdk`m]‡gd^j 'pbke(+ ^jn`ij cdk`m]‡gd^j '`lpe(+ o\ib`io` cdk`m]‡gd^\ 'q^ke(+bq`‹qbo^+v pb abcfkbk `ljl pfdrb9

_T Z _*Tpbkeu < ,,,+

1

0`p`eu < ,,+

pbkeu

t

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Page 328: Calculus

1.6 Cpi^d‡i gjb\mdohj*api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

.&)1 Dgbo`Œ`Œ`p

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Page 329: Calculus

Fiq`mn\n _` g\n api^dji`n omdbjijh„omd^\n ,)2

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Page 330: Calculus

20/ Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

t t

s

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a&s' < pbks6S 6S

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Page 331: Calculus

/ iq`mn\n _` g\n api^dji`n omdbjijh„omd^\n 200

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sŠifa^ m^o^ , 0 ; s ; 0-

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'5-35( ` _s-{,,,,,9 < ^o`pbks * B -UH , s0

Hkqbdo^kal mlo m^oqbppb l_qfbkb lqo^ krbs^ cŽojri^ ab fkqbdo^`fŽk

`^o`pbku _s < s ^o`pbku , `sQ < If ^o`pbk u * y * `-Di `lpbkl v i^ q^kdbkqb pb,Hksfboqbkab cloj^ ^kŠild^- O^o^ bi `lpbkl pb

^`lpqrj_o^ bibdfo bi fkqbos^il Z/+60!\ m^o^ e^`bo i^ fksbopfŽk 's‹^pb cfd- 5-00(-K^ crk`fŽk fksbop^ obpriq^kqb+ii^j^a^ ^o`l `lpbkl+ pb abcfkb `ljl pfdrb9

p < ^o``lp q fj mif`^ q < `lp p u

Dk i^ cfdro^ 5-01 bpqŠobmobpbkq^a i^ doŠcf`^ ab i^ crk`fŽk ^o``lp-O^o^ fksboqfoi^ q^kdbkqbpb bifdb bi fkqbos^il ^_fboql ',q&6S+ y&6S('s‹^pb cfdr,

o^ 5-02( v pb abcfkb bi ^o`l q^kdbkqb`ljl pfdrb9

p < ^obq^kq fjimif`^ q < q^k p u

Dk i^ cfdro^ 5-03 pb e^ af_rg^al rk^ m^oqbab i^ doŠcf`^ ab i^ crk`fŽk ^o`lq^kdbkqb-

Page 332: Calculus

1/0 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g s api^dji`n omdbjijh„omd^\n diq`mn\n

t

s s

t

l

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'5-36( ,0C ^o``lp r < -z,,,,,,:++

u 0 , s/

sŠifa^ m^o^ , 0 ; s ; 0 v

0'5-37( C ^o`q^k s8988,,1&

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sŠifa^ m^o^ qlal bi k•jbol ob^i s,

t t

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Page 333: Calculus

Fiq`mn\n _` g\n api^dji`n omdbjijh„omd^\n 202

'5-36( pb mrbab qo^ar`fo bk i^ pfdrfbkqb cŽojri^ ab fkqbdo^`fŽk9

h&!0x _o < ,'^o``lp s + ^o``lp N( < z , ^o``lp s

l 0 , o0 0

pf , 0 ; s ; 0- Bljm^o^kal '5-38( `lk '5-34( pb abar`b i^ obi^`fŽk fk,, ^o``lp s < ^ob pbk s '0/ nrb q^j_f‹k pb mrbab abar`fo ab i^ `lkl`fa^ fabkqfa^apbk %n.P * s& < `lp t* bp`of_fbkal t < ^o``lp r&+ Blk i^ klq^`fŽk ab Kbf_kfwm^o^ fkqbdo^ibp abcfkfa^p pb mrbab bp`of_fo '5-38( `ljl pfdrb9

'5-38(

'5-4/( a _s-z,,,,:+< ,^o``lp r * _-&T 0 ] s0

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'5-40( /%!_o,, < ^o`q^kul 0 * o0

l a ^r,,1 < ^o`q^k r * b -g)s

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a\m^^jns_U < s ^o``lp s * a x < u^o``lpu , z * a+

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g)s

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'5-41(/Q

m^o^ qlal r ob^i+^o``lq s < , , ^obq^k s1

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pf Gtf y 0+^o`pb` s < ^o``lp ,s

'5-43(0

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Page 334: Calculus

1/2 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

.&** :WR_PVPV\`

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,0f+@ o``lp s < 9o-<<<,{{

rG*Tc

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0 * s0

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Cbar`fo i^p cŽojri^p ab fkqbdo^`fŽkab ilp Dgbo`f`flp5 ^i 0/-

pf Hui = 0-

pf Erf = 0-

4, P ^o`blq s _s < s ^o``lq s * ild '0 * s0' * B-

5, P ^o`pb`r ^r < r ^o`pb`r * [9{ild Er * yG * `-

6, a ^o``p` s _s < s ^o``p` s * [9Hild Fs * y Z* B-

7, P '^o`pbkuc _s < u'^o`pbkU'0 + 0s * 1z ^o`pbks * B-

`\m^n`is ii,zH ^o`pbku b/., ++_s < ild , ,, * -[! [ [

00- '^( Cbjlpqo^o nrb C ' ^obblqs + ^obq^kz( < N m^o^qlal s x N-

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Dk ilp Dgbo`f`flp01 ^i 14+bk`lkqo^oi^ abofs^a^ `$%r&+Rb prmlkb nrb bk `^a^ `^pl i^crk`fŽk ` bpqŠabcfkfa^m^o^qlalp ilp s^ilobp ob^ibpr m^o^ilp `r^ibp i^ cŽojri^ `%r& qfbkbpbkqfal-

s./+ a&s' < ^o`pbkX -

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0.1+ a&s' < ^o`blp, -

s

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Page 335: Calculus

?d`m^d^djn (&)

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0 , s0

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f*r

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15- Ool_^o nrb _t-_s < &s* t'-&s + u( pf ^o`qd &t-s' < ild qs0 * t0,

16- B^i`ri^o _0t-_s0 pf v < '^o`pbk u(zm^o^ Hui; 017- Rb^ y&s' < ^o`q^k s + s ) ds\, Du^jfk^o bi pfdkl ab .$+m^o^ abjlpqo^o nrb

[0

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Page 336: Calculus

0.3 Boh]cƒh fia[lcngi) `oh]cƒh _rjih_h]c[f t `oh]cih_m nlcaihig€nlc][m chp_lm[m

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,,, < ^o`q^k s * b -0* s

0

Rb a^oŠ ^ `lkqfkr^`fŽk rk j‹qlal m^o^ `^i`ri^o i^ fkqbdo^i ab rk^ crk`fŽk o^,`flk^i `r^inrfbo^ v pb sboŠ nrb bi obpriq^al mrbab bumobp^opbpfbjmob mlo jbaflab- mlifkljflp+ crk`flkbp o^`flk^ibp+ ^o`l q^kdbkqbpv ild^ofqjlp-

K^ fab^ _Špf`^ abi j‹qlal `lkpfpqb bk abp`ljmlkbo rk^ co^``fŽk bk prj^ab co^``flkbp pfjmibp nrb mrbabk fkqbdo^opbmlo i^p q‹`kf`^p a^a^p ^kqboflojbkqb-Rb bumlkaoŠ i^ j^kbo^ dbkbo^i ab mol`babo mlo jbafl ab rk k•jbol ab bgbj,milp pbk`fiilp nrb fkaf`^k qlalp ilp m^plp bpbk`f^ibp abi j‹qlal-

DIDLOKN 0- Dk bpqb bgbjmil pb bjmfbw^ `lk alp co^``flkbp pfjmibpf,%r * 0( X f,%r * 2( `rv^p fkqbdo^ibppb `lkl`bk+ v pb sb nr‹ l`roob `r^kalpb-cloj^ rk^ `lj_fk^`fŽk ifkb^i ab bpq^pco^``flkbp- Olo bgbjmil+ pf pb qlj^alp sb`bp i^ mofjbo^ co^``fŽk+jŠp qobpsb`bp i^ pbdrka^+ pb qfbkb

1 2 0&s* 2( * 1&s+ 0( 3s * 2+s,,,0 * +s,*,2 < +&+s+++/+'&+s+)+1'+< s0 * 0s + 2

Kbvbkal ^elo^ bpq^ cŽojri^ ab abob`e^ ^ fwnrfboa^+af`b nrb i^ crk`fŽk o^`fl,k^i l a^a^ mlo l%r&< %2r * 0&,%r0 * /r * 2( pb bumobp^ ljl rk^ `lj_fk^`fŽkifkb^i ab f,%r * 0( v f,%r * 2(- Olo q^kql+ pb mrbab bp`of_fo i^ fkqbdo^i ab lbp`of_fbkal9

a 3s* 2 _s < 0ax * 1ax < 10/d Zu, 00 * 2 ild Zu* 2[* `-s/ * 0s + 2 s + 0 s * 2

DIDLOKN 1- Di bgbjmil ^kqboflo prdfbob rk mol`bafjfbkql m^o^ `^i`ri^ofkqbdo^ibpab i^ cloj^ E%[r * \&,%r0 * /r * 0&^r+ Olo bgbjmil+ m^o^ `^i`ri^oE%/r * 2&,%r0 * /r * 0&^r pb qo^q^ab bumobp^obi fkqbdo^kal `ljl `lj_fk^,`fŽk ifkb^i ab f,%r * 0( v f,%r * 2( bp`of_fbkal

'5-44( 0s)3 > ?<,,*,,s0 * 0s + 2 s + 0 s * 2

Page 337: Calculus

. hn_al[]cƒh jil `l[]]cih_m mcgjf_m 206

alkab = X > plk `lkpq^kqbp nrb pb e^k ab abqbojfk^o- Rf pb mrbabk bk`lkqo^o =u ? ab j^kbo^ nrb i^ b`r^`fŽk '5-44( pb^ rk^ fabkqfa^a+ bkqlk`bp i^ fkqbdo^i abi^ co^``fŽk abi mofjbo jfbj_ol bp fdr^i ^ i^ prj^ ab i^p fkqbdo^ibp ab i^p co^``fl,kbp abi pbdrkal jfbj_ol- O^o^ e^ii^o = v > pb jriqfmif`^k ^j_lp jfbj,_olp ab '5-44( mlo %r * i('u * 2( m^o^ nrfq^o ilp abkljfk^alobp- Blk il `r^ipb qfbkb

'5-45( =%r * 2( * >%r * 0( < /r * 4-

O^o^ abqbojfk^o = v > ^ m^oqfo ab bpq^ fdr^ia^a e^v alp j‹qlalp `lj•kjbkqbrp^alp- Tkl `lkpfpqb bk fdr^i^o ilp `lbcf`fbkqbp ab i^p mlqbk`f^p fdr^ibp ab s bk'5-45(- Dpql `lkar`b ^ i^p b`r^`flkbp = * > < 1 X 0= * > < 4- Qbplisfbkalbpqb m^o ab b`r^`flkbp pfjriqŠkb^p+ pb l_qfbkb = <p+> < i• Di lqol j‹qlal`lkpfpqb bk a^o ^ s bk '5-45( alp s^ilobp afpqfkqlp `lk il `r^i pb l_qfbkb lqolm^o ab b`r^`flkbp bk = u >+ Dk bpqb `^pl m^oqf`ri^o+ i^ mobpbk`f^ ab ilp c^`qlobpr * 0 X r * 2 prdfbob bi qlj^o ilp s^ilobp r < 0 X r < , 2- Olkfbkal r < 0bk '5-45( bi `lbcf`fbkqb ab ? pb ^kri^ v pb qfbkb 2> < 6+ l pb^ > < p-@kŠild^,jbkqb pb mrbab ^kri^o bi `lbcf`fbkqb ab = mlkfbkal r < , 2+ `lk il `r^i , 1> :< , 0+ l pb^ ? < p- Dk ^j_lp `^plp pb e^k e^ii^al ilp s^ilobp nrb p^qfpc^,`bk '5-44(+ ab j^kbo^ nrb pb qfbkb9

` /r * 4 ^r < W`w * 0-`w < >--ild Zu , 00 * F ild Zu* 20 * `-r0 * /r * 2 3 r * 0 3 r * 2 3 3

Dp `i^ol+ nrb bi j‹qlal bumrbpql bk bi bgbjmil 1+ pb ^mif`^ q^j_f‹k ^ fkqb,do^ibp ab i^ cloj^ ``%r&,a%r& ^r bk i^p nrb ` bp rk mlifkljfl ifkb^i v d rk mlif,kljfl `r^aoŠqf`l nrb pb mrbab abp`ljmlkbo bk molar`ql ab c^`qlobp ifkb^ibp`lk `lbcf`fbkqbp ob^ibp a%r& < %r * r)&%r * r0', Dk bpqb `^pl+ bi `l`fbkqb pb mrbabbumobp^o `ljl rk^ `lj_fk^`fŽk ifkb^i ab 0. %r * u+( v 0. %r * r0' X i^ fkqb,do^`fŽk ab `%r&,a%r& `lkar`b ^ i^ `lj_fk^`fŽk `loobpmlkafbkqb ab ilp q‹ojfklpild^oŒqjf`lp i^d Gt , VGGX i^d Gt , u10•

Klp bgbjmilp mob`babkqbp pb obcfbobk ^ crk`flkbp o^`flk^ibp ` , d bk i^p nrbbi do^al abi krjbo^alo bp jbklo nrb bi abi abkljfk^alo- Tk^ crk`fŽk o^`flk^i`lk bpq^ molmfba^a pb abkljfk^ rk^ crk`fŽk o^`flk^i jlijc[+ Rf ` , d bp cgjlijc[)bp ab`fo+ bi do^al ab ` kl bp jbklo nrb bi do^al ab d+ pb mrbab bumobp^oooz`ljlprj^ ab rk mlifkljfl v rk^ crk`fŽk o^`flk^i molmf^- Dk bcb`ql+ _^pq^ pfjmib,jbkqb afsfafo ` mlo d m^o^ l_qbkbo9

y&s' < N&s' * O&s'*a%r& a%r&

Page 338: Calculus

1/6 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\n diq`mn\n

alkab P v O plk mlifkljflp 'ii^j^alp ^j^d`io` v m`noj*obpmb`qfs^jbkqb( ab j^,kbo^ nrb bi obpql bp ab do^al jbklo nrb a+ Olo bgbjmil9

T0 * 1s < u * 1 * iNu * 5

s0 + 0s + 2 s0+ 0s + 2

Olo q^kql+^i bpqraf^o i^ q‹`kf`^ ab fkqbdo^`fŽk+kl pb nrfq^ dbkbo^ifa^a ifjfqŠkalpb^ i^p crk`flkbp o^`flk^ibp kmjkd\n u mlo q^kql bk il pr`bpfsl pb `lkpfabo^oŠaa&s'-b&s' _s alkab a bp ab do^al jbklo nrb d-

Tk qblobj^ dbkbo^i ab „idb_o^ af`b nrb qla^ crk`fŽk o^`flk^i pb mrbabbumobp^oljl prj^ cfkfq^ab co^``flkbp ab i^ cloj^9

=

%r * [&eu

?s * b&s0 * ]s * ^'h%

alkab f u h plk bkqbolp mlpfqfslp u >* O*a+ \* ]* ` `lkpq^kqbp `lk i^ `lkaf`fŽk]! + 2^ ; P- Dpq^`lkaf`fŽk fkaf`^ nrb bi mlifkljfl s0 * ]s * ` kl pb mrbababp`ljmlkbo bk c^`qlobp ifkb^ibp `lk `lbcf`fbkqbp ob^ibp+nrb bp il jfpjl nrbab`fo nrb i^ b`r^`fŽk `r^aoŠqf`^ s0 * ]s * ` < N kl qfbkb o^Œ`bpob^ibp- Tkmlifkljfl ab bpq^cloj^ pb af`b nrb bp dmm`_p^d]g`bk bi `^jml ob^i- Br^kal rk^crk`fŽk o^`flk^i pb bumobp^ab i^ j^kbo^ fkaf`^a^ pb af`b nrb pb e^ abp`lj,mrbpql bk am\^^dji`n ndhkg`n, Olo q^kql+bi mol_ibj^ ab fkqbdo^obpq^ crk`fŽk e^nrba^al obar`fal ^i ab fkqbdo^oprp co^```flkbp pfjmibp+ 0/ nrb pb ildo^ cŠ`fijbkqb `lk i^p q‹`kf`^p nrb pb bumlkbk bk ilp bgbjmilp nrb pfdrbk-

@nrŒkl pb qo^q^oŠab mol_^o nrb i^ abp`ljmlpf`fŽk bk co^``flkbp pfjmibpbufpqbpfbjmob+pfkl nrb pb sboŠ 'mlo jbafl ab bgbjmilp( `Žjl pb l_qfbkbk i^pco^``flkbp pfjmibp bk mol_ibj^p `lk`obqlp- Dk `^a^ `^pl+ `r^kal prog^+i^ abp,`ljmlpŒ`Œ‹k bk co^``flkbp m^o`f^ibppb mlaoŠ bcb`qr^o afob`q^jbkqb-

Dp `lksbkfbkqb afp`rqfo mlo pbm^o^al ilp `^plp+ pbd•k pb^ i^ cloj^ bk nrbpb abp`ljmlkd^ bi abkljfk^alo abi `l`fbkqb a&s'-b&s' bk molar`ql ab c^`qlobp-

@>PL 0- Bg _`ijhdi\_jm `n pi kmj_p^oj _` a\^ojm`n gdi`\g`n _dnodiojn, Rr,mŽkd^pba%r& abp`ljmrbpql bk h c^`qlobp ifkb^ibp+bp ab`fo9

a%r& < %r * rf&%r * r/& +++ %r * rh& †

Rb l_pbos^ nrb rk^ `lj_fk^`fŽk ifkb^i ab i^ cloj^

>g >T,,* ---*,,s + Wi T * sh

pb obar`b ^ rk^ •kf`^ co^``fŽk `lk bi `lj•k abkljfk^alo a%r& pfbkal bi krjb,o^alo ab bpq^co^``fŽk rk mlifkljfl ab do^al jbklo nrb i nrb `lkqfbkb i^p =+

Page 339: Calculus

Fio`bm\^d‡i kjm am\^^dji`n ndhkg`n 208

Olo q^kql+ pf pb mrbabk bk`lkqo^o i^p = ab j^kbo^ nrb bpqb krjbo^alo pb^ fdr^i^ `%r& pb qfbkb i^ abp`ljmlpf`fŽk

a&s' >g >i,<,,* ---*,,a%r& t , Wi W , ri%

v i^ fkqbdo^i ab a&s'-b&s' pboŠ fdr^i ^H99&H@ei^d Fs + sdd,Dk bi bgbjmil nrb pfdrb pbobplisboŠ rk `^pl m^o^ i < 2-

DIDLOKN 2- a /T/ * 3s + 0Hkqbdo^o ,,,,, _s ,

s\)s0+0s

Pjgp^d‡i, Orbpql nrb s1 * s0+ 0s < s&s + 0'&s * 1( bi abkljfk^alo bp

bi molar`ql ab c^`qlobp ifkb^ibp afpqfkqlp u pb qo^q^ ab e^ii^o @i&>0* >1* ab j^kbo^nrb9

0s/ * 4u , ) < >g * ,--9 K * ,,-9f-K-u^ * u1

, 1u W W , 0 W * 1

Prfq^kal abkljfk^alobp pb qfbkb

/r/ * 2r * 0 < =E%r * .&%r * 1( * =/r%r * 1( * =[r%r * 0(-

O^o^ V < N pb qfbkb ,1@i < ,0+ bp ab`fo+ @i < i- O^o^ r < 0 pb l_qfbkb91>0 < 5+ l pb^+=0 < 1+ X m^o^ s < ,1 obpriq^ ‹,q+< ,2+ l pb^+ >1 < , i- Oloq^kql pb qfbkb9

a0U0* 4u , ) _s < a _s * 0ax + 0ax :u^ * s0 [ 0s 1 s s + 0 1 s * 1

< hhkc Yt[ * 1hkc Yt , 00, phkc Yt * 10* b

@>PL 1- Bg _`ijhdi\_jm `n pi kmj_p^oj _` a\^ojm`n gdi`\g`n \gbpijn _` gjn^p\g`n n` m`kdo`i, Rb firpqo^ bpqb `^pl `lk rk bgbjmil-

DIDLOKN 3- H U0 * 0s * 2Hkqbdo^o ,,,,,, ^r +

%r * .&%r * 0(1

Pjgp^d‡i, Rb e^k ab bk`lkqo^o >c >0* >1 ab j^kbo^ nrb

'5-46( u1 * /r * 2 < ,,,5, * y * @^ -

&s + /'&s * i> V , 0 s * 0 'u * 0(1

Page 340: Calculus

10. Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

Rlk kb`bp^of^pi^p alp co^``flkbp >_&s * 0( v >n-&s * 0(1+ pŒljl >*-&s + 0(^ cfk ab `lkpbdrfo rk mlifkljfl ab do^al alp bk bi krjbo^alo v qbkbo q^kq^pb`r^`flkbp `ljl `lkpq^kqbp `r^kal pb qo^qbab abqbojfk^o i^p >, Prfq^kal abkl,jfk^alobp pb qfbkb9

%3+25& r0 * /r * 2 < =x%r * 0(1 * =0%r * .&%r * 0( * =1%r * 0( -

Rrpqfqrvbkal s < 0 pb qfbkb 2>* < 5 l pb^ @+< - Rf s < , 0 pb l_qfbkb+ 0>n < 1 X @p < , 0- Rb kb`bpfq^ lqo^ b`r^`fŽk m^o^ abqbojfk^o >0Š Orbpqlnrb kl bp mlpf_ib lqo^ bib``fŽk ab s nrb ^krib ^id•k c^`qlo+pb mol`ro^ qlj^o sab j^kbo^ nrb ilp `Ši`rilp pb^k il jŠp pbk`fiilp mlpf_ibp- Olo bgbjmil+ e^`fbkals < N pb iibd^ ^ i^ b`r^`fŽk 2 < >*+ >0 + @p ab 0/ nrb obpriq^ >0: * p-Nqol j‹qlal bp abofs^o ^j_lp jfbj_olp ab '5-47( v irbdl prpqfqrfork^ r `lksb,kfbkqb- Cbofs^kal bk '5-47( pb l_qfbkb i^ b`r^`fŽk

v e^`fbkal s < , 0+pb bk`rbkqo^9 N < , 0>0 * @p bp ab`fo >0 < g>n < , p+`ljl ^kqbp-G^ii^a^p i^p > nrb p^qfpc^`bk'5-46( pb qfbkb9

` T/ * /r * 2 ^r * w `w * `w ` ^r *%r * f&%r * 0(1 , 1 r * 0 1 r * 0 , %r * 0(1 ,

200< ,ild Zu, 00 , ,ild Zu* 00 * ,, * B-1 1 [ % 0

Rf+bk bi mofjbo jfbj_ol ab '5-46( er_fbo^ e^_fal bi c^`qlo %r * i (! bksbw ab %r * 0(1 pb er_fbo^ qbkfal nrb ^•^afo bk bi pbdrkal jfbj_ol bi q‹ojfkl>2,%r * i (!- LŠp dbkbo^i+pf rk c^`qlo ifkb^i ^m^ob`bk sb`bp bk bi abkljfk^alo+m^o^bpqbc^`qlo pb e^ ab qlj^o rk^ prj^ ab k q‹ojfklp+ bp ab`fo9

'5-48(

alkab i^p = plk `lkpq^kqbp- O^o^ `^a^ c^`qlo ifkb^i obmbqfalpb e^ ab qlj^o rk^prj^ ab bpqbqfml-

@>PL 2- Bg _`ijhdi\_jm ^jiod`i` a\^ojm`n ^p\_mƒod^jn dmm`_p^d]g`nidibp+ij _` gjn ^p\g`n n` m`kdo`,

DIDLOKN 4- Hkqbdo^o F 0T/ * /r * 1 ^r +u2 , 0

Page 341: Calculus

. hn_al[]cƒh jil `l[]]cih_m mcgjf_m 210

Pjgp^d‡i, Di abkljfk^alo pb mrbab abp`ljmlkbo bk bi molar`ql s0 * 0 <; %r * .&%r0 * r * 0(+ alkab r0 * r * 0 bp foobar`f_ib+ v pb qfbkb rk^ abp`ljml,pf`fŽk ab i^ cloj^9

1s/ * 1u , 1 < y * ?s * b

u2, 0 u , 0 u1 * u * 0

Dk i^ co^``fŽk ab abkljfk^alo s0 * s * 0 pb mlkb `ljl krjbo^alo rk mlifkljflab mofjbo do^al ?s * B^ cfk ab qbkbo q^kq^p b`r^`flkbp `ljl `lkpq^kqbp `r^kalpb abqbojfk^k =) >) B- Prfq^kal abkljfk^alobp v obplisfbkal obpmb`ql ^ =) >)X B pb qfbkb9 = < 0+ > < 1+ B < 2- Olo q^kql+ pb mrbab bp`of_fo9

` 0T/ * 1u , 1 _s :`w (` 1u * 2 _s,u2 ] 0 u , 0 u1 * u * 0

K^ mofjbo^ fkqbdo^i abi pbdrkal jfbj_ol bp ild Er * 00- O^o^ `^i`ri^o i^ pbdrka^fkqbdo^i+ pb bp`of_b9

` 1u * 2 _s <` 1u * 0 _s * ` 1 _su1 * u * 0 u1 * u * 0 u1 * u * 0

` _s< ild %r

/ * u * 0( * 1 'u * V * -

G^`fbkal p < s * pv \ < Uf i^ •iqfj^ fkqbdo^i bp9

` _p 1 p 3- m8 1u * 01 < ,^o`q^k , < ,r 2 ^obq^k z-

p/ * '&0--1 '&0-- '&0-- 2 r 2

Olo q^kql pb qfbkb9

I 0T/ * 1u , 1 _s < ild Zu ] 00 * ild 'u 1 * u * 0( * yU2 ^o`q^k 1z :: 0 * bu2 ] 0 2 s 2

@>PL 3- Bg _`ijhdi\_jm ^jiod`i` a\^ojm`n ^p\_mƒod^jn dmm`_p^d]g`n\gbpijn_` gjn ^p\g`n `noƒi m`k`od_jn, K^ pfqr^`fŽk ^nrŒ bp ^kŠild^ ^ i^ abi `^pl 1- @ajf,qfjlp nrb bp mlpf_ib rk^ abp`ljmlpf`fŽk ab x%r&,a%r& bk co^``flkbp pfjmibp+ bkmofjbo ird^o bk rk^ prj^ ab i^ cloj^ '5-48( mlo `^a^ c^`qlo ifkb^i+ q^i `ljlpb afgl ^kqboflojbkqb: u bk pbdrkal ird^o+ pf rk c^`qlo `r^aoŠqf`l foobar`f_ib

Page 342: Calculus

100 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\ndiq`mn\n

pb obmfqbg sb`bp+ pb ^ajfqb nrb pb mrbab abp`ljmlkbo bk rk^ prj^ ab j q‹o,jfklp+ ab i^ cloj^

alkab `^a^ krjbo^alo bp ifkb^i-

DIDLOKN 5- Hkqbdo^o

Pjgp^d‡i, Rb bp`of_b

s% + T0 * 0s/

* s * 1 > ?s * B As * B,,,,,,,,, < ,, * *&s + /'&s0 * 1(1 s + 0 s0 * 1 &s0 * 1(1

Prfq^kal abkljfk^alobp u obplisfbkal obpmb`ql^ =) >) a+ C+ u A pb qfbkb-

> < 0+ >: f+ _< ,h+ @< ,0+ A: N-

Olo q^kql+obpriq^+

F T1 * s0 * 0s

/* s * 1 _s < Fw (F fu , q_s *F s _s :

&s + /'&s0 * 1(1 2 s + 0 s0 * 1 &s0 * 1(1

0 F _s 0F 0s _s 0 F _s 0 F 0s _s< 2! s + 0 * 2 s0 * 1 , &2 s0 * 1 , 1 &s0 * 1(1 <

0 0 U1 s< ,Hld Zu, 00* , ild %r0 * 1( , , ^o`q^k -n::*, , / L+

0 0)+++)@,1 s0 * 1

Klp bgbjmilp mob`babkqbpplk jlabilp qŒmf`lpab ilp nrb pb mobpbkq^kbkdbkbo^i- Di mol_ibj^ ab i^ fkqbdo^`fŽkab crk`flkbp o^`flk^ibp molmf^ppb obar`b^i `Ši`ril ab fkqbdo^ibpab i^ cloj^

E _s

%r* \'i% E s_s

%r/ * \r * ]&g$v F _s

&s0 * ]s * ^'h ,

Page 343: Calculus

/ io`bm\g`n lp` kp`_`i om\inajmh\mn` `i dio`bm\g`n _` api^dji`n m\^dji\g`n 101

K^ mofjbo^ fkqbdo^i bp ild Er * [f pf h < 0 X %r * [&!*i,%f * h& pf h ; .+O^o^ `^i`ri^o i^p lqo^p alp pb bumobp^i^ cloj^ `r^aoŠqf`^ `ljl prj^ ab alp`r^ao^alp9

alkab p < s )]-0 v j8 < /!-2^ + ]0Š 'Dpql bp mlpf_ib mrbpql nrb 2^ + ]0 ;

= N- K^ prpqfqr`fŽk p < s * ]-0 obar`b bi mol_ibj^ ^i ab `^i`ri^o

'5-5/( u

K^ mofjbo^ bp 0ild &p0* nt1( pf h < 0 X 0&p0 * nt1(!,j.'i , h' pf h = 0- Rfh < 0 i^ pbdrka^ fkqbdo^ibk '5-5/( sfbkb a^a^ mlo i^ cŽojri^9

m_p 0 p

,1,,1 < , ^o`q^k , * b -Š p)ms nt nt

Di `^pl h = 0 pb obar`b ^i `^pl h < 0 ^mif`^kal obfqbo^a^jbkqb i^ cŽojri^ab ob`roobk`f^9

nrb pb l_qfbkb mlo fkqbdo^`fŽkmlo m^oqbp-Cb 0/ af`el pb abar`b nrb qla^ crk`fŽko^`flk^i mrbab pbo fkqbdo^a^ mlo jbafl ab mlifkljflp+ crk`flkbp o^`flk^ibp+^o`lpq^kdbkqbpv ild^ofqjlp9

.&*, =[aRT_NYR`^bR ]bRQR[a_N[`S\_ZN_`RR[ V[aRT_NYR`QRSb[PV\[R`_NPV\[NYR`

Tk^ crk`fŽk ab alp s^of^_ibp abcfkfa^ mlo rk^ b`r^`fŽk ab i^ cloj^

! `W

L%r) s& < ƒ ƒ[hŠirhs!

j<N i;L

pb abkljfk^ kjgdijhdj _` _jn q\md\]g`n, Di `l`fbkqb ab alp ab bpqlp mlifkl,jflp pb abkljfk^ api^d‡i m\^dji\g _` _jn q\md\]g`n, Hkqbdo^ibpab i^ cloj^9aO`n`i s* `lp s' _s alkab O bp rk^ crk`fŽk o^`flk^i ab alp s^of^_ibp pb mrbabobar`fo jbaf^kqb i^ prpqfqr`fŽk p < q^k ps ^ fkqbdo^ibpab i^ cloj^ am&p'_palkab m bp rk^ crk`fŽk o^`flk^i ab rk^ s^of^_ib- K^ •iqfj^ fkqbdo^i pb mrbab

Page 344: Calculus

102 Cpi^d‡i gjb\mdohj*api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

`^i`ri^o jbaf^kqb i^p q‹`kf`^p nrb pb ^`^_^k ab abp`of_fo-Rb firpqo^ bi j‹qlal`lk rk bgbjmil m^oqf`ri^o-

DIDLOKN 0- Hkqbdo^oFZE*^T+pbkU * `lp s

Pjgp^d‡i, K^ prpqfqr`fŽk p < q^k ps a^

s < 1 ^o`q^kp *1

_s< ++_p0 * p0

%

s s 1 q^kds 0ppbks < 1pbk,`lp, < ,,z<,,

1 1 pb`! qu 0 * p/$

`lp T < 1 ':NR1 z ] 0 < ]1 ]] 0 < ]1 ]] 0 < 0 , p/

1 pb`! qu 0 * p0 0 * p0%

v

0p * 0 , p,pbks * `lp s < 1

0 * p

Olo q^kql pb qfbkb9

a _s + ,1 o _p + +0G _ppbku * `lp s + ! p0 + 0p + 0 , &p+ \'&p + ]' *

alkab \ < 0 * / u ] < 0 , /- Di j‹qlal ab co^``flkbp pfjmibp `lkar`b ^

` _p + ]0 Z']0 ]0( _p%o* [&%o * ]' + [ * ] o * [ * o * ]

u mrbpql nrb+ [ * ] < 1 r&1 pb l_qfbkb9

'5 50(` _s r&1 Gp + ] G s&1 Gq^k s + 0 * s&1{- ,,,,<,Hld ,, *B<,Hld ,,,,,, )@,pbku * `lp s 1 p + \ 1 q^kqu , 0 , r&1

Di •iqfjl obpriq^al pb mrbab pfjmifcf`^o rqfifw^kal fabkqfa^abp qofdlklj‹qof`^p^ab`r^a^p- Dk mofjbo ird^o pb l_pbos^ nrb / , 0 < q^k q0S ab j^kbo^ nrb bi

Page 345: Calculus

Fio`bm\g`n lp` kp`_`i om\inajmh\mn` `i dio`bm\g`n _` api^dji`n m\^dji\g`n 103

krjbo^alo ab i^ •iqfj^ co^``fŽk bp q^k s * q^k i6S- Di abkljfk^alo pb mrbabbp`of_fobk i^ cloj^9

Slok^kal ild^ofqjlp bk i^ cloj^ fkaf`^a^ bk '5-50( v `lj_fk^kal bi q‹ojfkl, U1 ild ' s1 * 0( `lk rk^ `lkpq^kqb ^o_fqo^of^+pb mrbab bp`of_fo '5-50( bki^ cloj^9

H _s T1 G &s 6S( G< , ild q^k , * , * b-pbks * `lp s 1 1 7

Dk rk^ Rb``fŽk ^kqboflo pb abargl i^ cŽojri^ ab fkqbdo^`fŽk

I _s-n,,,:: < ^o`pbku&T 0 , s/

`ljl `lkpb`rbk`f^ ab i^ cŽojri^ m^o^abofs^o ^olpbk s, K^ mobpbk`f abi ^o`pbk sprdfbob nrb q^j_f‹k mlaoŒ^^i`ri^opb bpq^fkqbdo^ijbaf^kqb i^ prpqfqr`fŽk qofdlkl,j‹qof`^ o < ^o`pbk s, Sbkbjlp bkqlk`bp

s < pbkq+ _s < `lp o_o* x;Sg+n`i0o;^jno*

v bk`lkqo^jlp nrb

I _s < I `lp o_o < G_o < o < ^o`pbku -x ^jno

Dpq^bp rk^ _rbk^ prpqfqr`fŽk pf bi fkqbdo^kal `lkqfbkb rR<V1- Dk dbkbo^i+`r^inrfbo fkqbdo^i ab i^ cloj^ aO&s* S\0 + s0' _s* bk alkab O obmobpbkqrk^crk`fŽk ab alp s^of^_ibp+pb mrbab qo^kpcloj^o jbaf^kqb i^ prpqfqr`fŽk

s < \n`io* _s < \ `lp o_o*

bk rk^ fkqbdo^i ab i^ cloj^ aO&\ pbk o*\ `lp o'\ `lp o_o, Dpq^+ pr sbw+pb mrb,ab pfbjmob fkqbdo^omlo jbafl ab rkl ab ilp j‹qlalp ^kqbp bumrbpqlp-

DIDLOKN 1- Hkqbdo^o I s_s

3 , s0 * T 3 , s0 Š

Page 346: Calculus

104 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g v api^dji`n omdbjijh„omd^\n diq`mn\n

Pjgp^d‡i, Rb^ s < 1 pbk o* _s < 1 `lp o _o* sf 3 , s/ << 1 `lp o*u bk`lk,qo^jlp nrb

` s _s ` 3 pbko`lp o _o ` pbk o_o

3 , s0 * sf 3 , s0 < 3 `lp! o * 1 `lp o < `lp o * p<

< ,ild Gp* `lp qf * a < ,ild 'i * T 3 , s/& * _-

Di jfpjl j‹qlal pfosb m^o^fkqbdo^ibpab i^ cloj^

I O&s*S\0 + &`s * _'0' _s 9

pb rqfifw^ i^ prpqfqr`fŽk qofdlklj‹qof`^ _r * ^ < [ pbk o,Dk cloj^ m^ob`fa^ pb obprbisbk fkqbdo^ibpabi qfml

jbaf^kqb i^ prpqfqr`fŽk `s * _ < \ q^k o* _s < \ pb`! o _o, O^o^ fkqbdo^ibpab i^cloj^

I O&s*S&`s * _'0 + \0' _s *

pb bjmib^ i^ prpqfqr`fŽk `s * _ < \ pb` o* ` _s < \ pb` oq^k o_o, Dk rkl r lqol`^pl+ bi krbsl fkqbdo^kal pb `lksfboqb bk rk^ crk`fŽk o^`flk^i ab pbk o u `lp o,

.&*- :WR_PVPV\`

B^i`ri^o i^p pfdrfbkqbp fkqbdo^ibp9

` 0s)1g, &s [ 0'&s * 4( _s,

0, a&s * /'&Us8x'&U * 2(

1, ` u2 ZT1_8) 1&

aUx)0U+4

3- l]l 1 1 _s,(i:! )s + s

` 6U1)53, %r * 0('1u * 0(2 _s,

a2U/)U)g

5- u2 ] i ^r+

` x_s6- z * 3s0 * 3&

`B%,6, +0++_s,s )s

Page 347: Calculus

Bd`m^d^djn ,+0

6+ ` U&U0_

8 0(1 -

/., ` &U* /'&U xU0'0&U* 2(2 -

..+ ` &UU9z(1-

./+ ` U-x u &

` s0_s02- 1 5&

U )s+

` &s * 1( _s/2, s0 [ 2s * 3 -

` ^r

/3, &s0[ 2s * 2'&s0 + 2s * 4( -

` &s + 2( _s05- u2 * 1s0 * 0s%

.4+ a&U0 x 0(1&

bs * 007-‘ U1 [ 0 _s,

` s2)/.6+ r%r0 * 0(1 ^r)

0., ` 3 _U0

2&B ' B

/.+ ` U&x0+88' ^r+

00, a U-x 0 -

01, a u3 _8 0 -

` s0_s13- ,1,,,,1-

&U * 0s * 1(

` 2U3 + 003, &s3 * U * 0(1 _s,

04, ` _s ,1 pbk U + `lp U * 4

05, ` _s0 * \ `lp U

/5+ ` ^r0 * \^jns

` pbk1 T

18- 0 1 _s,*pbk U

` _s

2/- ^1pbk1 T * ]/ `lp1 T

1/, ` _s 1&\n`is * ] `lp s'

0!.1 pbk s _s21- ,,,,,,-

l 0 * `lp s * n`is

11, ` U2 , s0_s,

01+ `+,ZrZZ ^r)s2 +s/

'N ; [ ; 0(-

&\ = 0(-

%[\ w N(-

%[ w N(-

13, FR

1 s+ s0

_s,

Fqs0 )s

14, s _s,

15, F qs0 * 4 _s,

16, F+;;;s;;_U,qs/)s)/

17, `+w+sT* )s

`S0 + s +s/

3/- 1 ^r)s

XFi_d^\^d‡i8 Dk bi Dgbo`f`fl 3/+ jriqfmif`^o krjbo^alo v abkljfk^alo mlo

U1 , s + s/+Y

Page 348: Calculus

106 Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g u api^dji`n omdbjijh„omd^\ndiq`mn\n

.&*. :WR_PVPV\`QR_R]N`\

0- Rb^ `%r& < Iz 'ild n&,%n* 0( ^n pf r = N- B^i`ri^o `%r& * `%f,r&+ Bljl `ljmol_^`fŽk pb

sbofcf`^oŠ9 a&0' * cj < qild1 1+1- Dk`lkqo^o rk^ crk`fŽk `) `lkqfkr^ m^o^ qlal s 'v kl `lkpq^kqbjbkqb kri^(+ q^i nrb

d%!pbk oW/%r&< a&o'1 _o

l )^jno

2- Hkq‹kqbpb `^i`ri^o F_!$,r ^r ^mif`^kal bi j‹qlal ab fkqbdo^`fŽk mlo m^oqbp-

3- Hkqbdo^oIc10/d 'bBlp!&( ^r)4- Tk^ crk`fŽk ` bpqŠ abcfkfa^ mlo i^ b`r^`fŽk

G 2s )0`%r& < r%r * H('u * 1( pf s<L

'^( Dk`lkqo^o i^ mbkafbkqb ab i^ doŠcf`^ ab ` bk bi mrkql bk nrb s < 0-'_( K^ obdfŽk abi mi^kl `ljmobkafa^ bkqob i^ doŠcf`^ v bi fkqbos^il Z0+3\ pb e^`b dfo^o^iobabalo abi bgb s bkdbkao^kal rk pŽifal ab obslir`fŽk- B^i`ri^o bpq^ fkqbdo^i v mol_^onrb pr s^ilo bp %5Qild '14.7(-

5- Tk^ crk`fŽk E bpqŠ abcfkfa^- mlo i^ pfdrfbkqb fkqbdo^i fkabcfkfa^9

d!%`o

B%r&< , ^n0 o

pf s< N-

'^( ƒO^o^ nr‹ s^ilobp ab r bp `fboql nrb ild r 94 B%r&<'_( Cbjlpqo^o nrb If _n,%n* [& ^n < _*9FWB%r* [& * B%f * [&Y+'b( Cb cloj^ ^kŠild^+ bumobp^oi^p pfdrfbkqbp fkqbdo^ibp bk crk`fŽk ab E-

d!%„%o+_o

0 o % d!%`o

!/^n)0 o

6- Dk `^a^ `^pl+ a^o rk bgbjmil ab rk^ crk`fŽk `lkqfkr^ ` nrb p^qfpc^d^ i^p `lkaf`flkbpcfg^a^p m^o^ qlal ob^i s* l _fbk bumif`^o mlo nr‹ rk^ q^i crk`fŽk kl bufpqb9

'^( im~_o < `!*

'_( FwF%n&n < 0 , 1!+1- Z1!+1pfdkfcf`^ 1'!+1(-\

'b( FQ%n&_o < W/%r&* 0-

7- Rf `%r * u( < `%r& `%s& m^o^ qlal r b u u pf `%r& < 0 * ra%r& alkab a%r&z&i `r^kalr w N abjlpqo^o nrb '^( `$%r&bufpqb m^o^ `^a^ r) v '_( x%r&< _!)

8- C^a^ rk^ crk`fŽk a nrb qfbkb rk^ abofs^a^ a$%r&m^o^ `^a^ ob^i r v nrb p^qfpc^`b i^ppfdrfbkqbp b`r^`flkbp9

a$%K&< 1 v a%r * s& < _ffa%r&* _!$a%s&m^o^ qlal r v qlal s+

Page 349: Calculus

Be`m^d^djn _` m`k\nj ,+2

'^( Ool_^o nrb a%/r& < /_!a%r& v bk`lkqo^o rk^ cŽojri^ ^kŠild^ j[l[a%0r&+'_( Fbkbo^ifw^o '^( m^o^ a^o rk^ cŽojri^ nrb obi^`flkb a%hr& `lk a%r& sŠifa^ m^o^qlal bkqbol mlpfqfsl i, Ool_^o bi obpriq^al mlo fkar``fŽk-'`( Ool_^o nrb d'N( < N X e^ii^o bi iŒjfqb ab a%b&,b `r^kal b w N'a( Dufpqb rk^ `lkpq^kqb B q^i nrb a$%r&< a%r&* ?_$ m^o^ qlal r+ Cbjlpqo^oil u e^ii^obi s^ilo ab B- XFi_d^\^d‡i8 @mif`^o i^ abcfkf`fŽk ab i^ abofs^a^ ab b%&s',Z

0/- Tk^ crk`fŽk mbofŽaf`^ ab mboŒlal [ p^qfpc^`b `%r * [& < `%r& m^o^ qlal r ab pr aljfkfl-ƒPr‹ pb mrbab ab`fo ab rk^ crk`fŽk nrb bp abofs^_ib m^o^ qlal s^ilo ab s u nrb p^qfp,c^`b rk^ b`r^`fŽk ab i^ cloj^9

a&s * \' < ]a&s'

m^o^ qlal s* alkab \ v \ plk `lkpq^kqbp mlpfqfs^p& 00- @miŒnrbpbi^ abofs^`fŽk ild^oŒqjf`^ m^o^ abar`fo i^p cŽojri^p ab abofs^`fŽk ab mol,

ar`qlp v `l`fbkqbp+ ab i^p cŽojri^p `loobpmlkafbkqbp ab prj^p v afcbobk`f^p-01- Rb^ > < F `o-R * 0( _o, Dumobp^o ilp s^ilobp ab i^p pfdrfbkqbp fkqbdo^ibp+mlo jbafl

ab i^ fkqbdo^i >8

f\ `+o

'^( 0 _o,[*f o + \ +

X/ `o'`( Il &o* 0(1 _o*

GE o`o%

'_( ,1,0 _o*l o (

'a( a`o ild '0 * o' _o*

02- Rb^ j%r& < @i * @FU * @0U0 X pb^ W%r&< _Tj%r&+

'^( Ool_^o nrb l_K&) abofs^a^ k,pfj^ ab ` bk bi mrkql N+bp ?j * h?. * h%h * 0(`1&'_( Qbplisbo bi jfpjl mol_ibj^ `r^kal m bp rk mlifkljfl ab do^al 2-'`( Fbkbo^ifw^oil ^ rk mlifkljfl ab do^al h,

03- Rb^ `%r& < r pbk [r+ Ool_^o nrb a&0ig&s' < &[/'i&\0is pbk^u , 0i\0i+/ blp \s',04- Cbjlpqo^o nrb9

XFi_d^\^d‡i8 /-&f * h * 0( < j of)h _o,Z

05- Rb^ B%r&< ba&o' ^n+ Cbqbojfk^o rk^ cŽojri^ 'l cŽojri^p( m^o^ `^i`ri^o B%r& m^o^qlal ob^i s* pf ` bpqŠ abcfkfa^ `ljl pfdrb9

']( a&o' < &o* 0qI(1-

w0 , o/

'_( a&o' < 0 ] Enf

'b( a&o' < `+/ogŠ

pf Gphy 0+

pf Gph= 0- 'a( a&o' < bi jŠufjl ab 0 v o/†

06- Tk pŽifal ab obslir`fŽk bpqŠ bkdbkao^al mlo i^ olq^`fŽk ab i^ doŠcf`^ s:`%r& m^o^ ZN+[Y^iobabalo abi bgb s, Rf m^o^ `^a^ \ = N bi slirjbk bp \0 * \* e^ii^o i^ crk`fŽk `+

Page 350: Calculus

22/ Cpi^d‡i gjb\mdohj* api^d‡i `skji`i^d\g t api^dji`n omdbjijh„omd^\ndiq`mn\n

i_- Rb^ `%r&:_*0+ m^o^ qlal r) Rb abpfdk^ mlo O%n&bi `lkgrkql ab loabk^a^p ab ` bk bifkqbos^il ZN+nF) pfbkal n = N- Rb^ =%n&bi Šob^ ab O%n&)R%n&bi slirjbk abi pŽifal l_qb,kfal mlo i^ olq^`fŽk ab O%n&bk qlokl ^i bgbs* v S%n&bi slirjbk abi pŽifal l_qbkfal mloi^ olq^`fŽk ab O%n&bk qlokl ^i bgbv- B^i`ri^o9 ^( =%n&8_( R%n&8b( S%n&8a( ifj R%n&f= 'q(-

nwK

08- Rb^ ^ rk k•jbol q^i nrb pbke ` < -'Ml fkqbkq^o bi `Ši`ril ab `-( Dk `^a^ `^pl e^ii^oqlalp ^nrbiilp s 'pf bufpqbk ^idrklp( nrb p^qfpc^`bk i^ b`r^`fŽk a^a^- Dumobp^oi^ obp,mrbpq^ bk crk`fŽk ab 0/d 1 X 0/d 2-

'^( ild'b&! * q`/t * 0( < ^, '_( ild'b&! , Ub1w , 0( < ^,

1/- Cbqbojfk^o pf `^a^ rk^ ab i^p molmlpf`flkbp pfdrfbkqbp bp `fboq^ l c^ip^- Ool_^o i^p `fboq^p-

'^( 10/d 4 < Rild 1-

i

'`( 19f+/-0 ; 1U:: m^o^ qlal i x 0-

f;g

'a( i * pbke s x `lpe s m^o^ qlal s,

Dk ilp Dgbo`f`flp 10 ^i 13+ bpq^_ib`bo `^a^ abpfdr^ia^a bu^jfk^kal bi pfdkl ab i^ abof,s^a^ ab rk^ crk`fŽk ^ab`r^a^-

110- , s ; pbk s ; s

nn

om

pf L:s:W%

00, s 8 0; ild' i * T ; yu2

01, s + 5! ; pbk s ; s pf

13- 'ui&* t]'g-] ; %T; * t\'g-\

14- Cbjlpqo^o nrb

pf s< N-

s< N-

pf s = N+t = N+ v N ; [ ; \+

'^( Px`8%o_o < `+!%&`%!+ 0 , s',

'_( H$!`+oo0_o < 0 `+!%&`%!+ i , s + x',

'`( o`+oo1_o < 2 `+%!&b&!, 0 , s + x x x',

'a( Dkrk`f^o i^ dbkbo^ifw^`fŽk prdbofa^ v abjlpqo^oi^ mlo fkar``fŽk-15- Rf [) \) ^i&\

f) plk a^alp+ u [\ :‹ /+ mol_^o nrb bufpqbk `lkpq^kqbp =) >) a q^ibp nrb9

`\/ pbk s * ]/ `lp s++++]+++_s < >s * ? ild i^ pbk s * ] `lp u\ * `-\n`is * ^jns

XFi_d^\^d‡i8 Ool_^o nrb bufpqbk > v ? q^ibp nrb9

[f pbks * ]. BNR T < @'^pbk s * ] `lp s' * ?&\ lp s + ] pbks',Z

Page 351: Calculus

Be`m^d^djn_` m`k\nj 220

16- Dk`lkqo^o bk `^a^ `^pl rk^ crk`fŽk /* nrb p^qfpc^d^i^p `lkaf`flkbp a^a^p9

'^( X%&s/&< g-s'_( Z&'pbk1r& < `lp! U

'b( Z&'pbkr& < `lp! r

m^o^s< K)m^o^ qlal s*m^o^ qlal s*

f%f&< 0-f%f&< 0-f%f&< 0-

'a( Z&'ild r& < wym^o^ N ; s x )$

m^o^ s< &"/&.' ;.,

17- Tk^ crk`fŽk+ ii^j^a^ bi gjb\mdohj dio`bm\g v nrb pb obmobpbkq mlo Kf+pb abcfkb `ljlpfdrb9

Es_oHc%r&< ,

1 ild opf s x 1-

Dpq^ crk`fŽk ^m^ob`b bk i^ SbloŒ^ ^k^iŒqf`^ab k•jbolp+ alkab pb abjrbpqo^ nrb Id&s'bp rk^ ^molufj^`fŽk jrv _rbk^ m^o^ bi k•jbol ab mofoklpvo s, Cbar`fo i^p pfdrfbkqbpmolmfba^abp ab Id&s',

s H! _o 1'^( Hc%r&< ild r * I1 ild1 o + ild 1 -

T j,h f s mLg _o'_( Hc%r&< ild r *ƒildh*i r * h I1 ildk*i o * `i*

FRG

alkab b- bp rk^ `lkpq^kqb 'abmbkafbkqb ab h&+G^ii^o bpq^ `lkpq^kqb-'b( Ool_^o nrb bufpqbrk^ `lkpq^kqb ] q^i nrb GNdW `o-o _o `, Id&s'* v e^ii^o bi s^ilo ab ],'a( Dumobp^o i^ fkqbdo^i `$7F0o F&o+ 0( _o mlo jbafl abi ild^ofqjl fkqbdo^i alkab` < 0 * pild 1-

'b( Rb^ F&s < a3 Id&`0U+2' + `0 Id&`0U+0' pf s = 2- Ool_^o nrb9

b1&!a%&s'< 1 2 1 -

s + s (

18- Rb^ F&s' < ildgo\ pf s ; N- Cbjlpqo^o nrb F qfbkb fksbop^+ v abpfdk^o bpq^ fksbop^ mlo d-ƒBrŠi bp bi aljfkfl ab d> G^ii^o rk^ cŽojri^ m^o^ `^i`ri^o a%s&m^o^ `^a^ u bk bi al,jfkfl ab d- Cf_rg^o i^ doŠcf`^ ab d-

2/- Rb^ F&s' <P3&g* o1'+g-0F_o pf s x N- 'Ml fkqbkq^o bi `Ši`ril ab bpq^ fkqbdo^i-(^( Cbjlpqo^o nrb Ebp bpqof`q^jbkqb `ob`fbkqb bk bi bgb ob^i kl kbd^qfsl-_( Cbpfdk^o mlo d i^ fksbop^ ab `+ Cbjlpqo^o nrb i^ abofs^a^ pbdrka^ ab d bp molmlo,`flk^i ^ d1 Zbpql bp+a!%s&< ]a0%s& m^o^ `^a^ v bk bi aljfkfl ab d\ v e^ii^o i^ `lkpq^k,qb ab molmlo`flk^ifa^a-

Page 352: Calculus
Page 353: Calculus

6

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Klp mlifkljflp cfdro^k bkqobi^p crk`flkbp jŠp pbk`fii^p nrb pb bpqraf^k bk@kŠifpfp-Rlk ^ab`r^a^p m^o^ qo^_^g^obk `Ši`rilp krj‹of`lp mlonrb prp s^ilobppb mrbabk l_qbkbo bcb`qr^kal rk k•jbol cfkfql ab jriqfmif`^`flkbp u ^af`flkbp-Dk bi `^mŒqril5 pb sfl nrb i^ crk`fŽk ild^ofqjl mrbab ^molufj^opb mlo mlif,kljflp il nrb klp mbojfqb `^i`ri^o ild^ofqjlp `lk i^ mob`fpfŽknrb pb abpbb-Dk bpqb`^mŒqrilabjlpqo^objlp nrb jr`e^p lqo^p crk`flkbp+ q^ibp `ljl i^ buml,kbk`f^i v i^p qofdlklj‹qof`^p+ mrbabk q^j_f‹k ^molufj^opb mlo mlifkljflp- Rf i^afcbobk`f^ bkqob rk^ crk`fŽk u pr ^molufj^`fŽk mlifkŽjf`^ bp prcf`fbkqbjbkqbmbnrb•^+ bkqlk`bp mlabjlp+ ^ bcb`qlp moŠ`qf`lp+`^i`ri^o `lk bi mlifkljfl bkird^o ab e^`boil `lk i^ crk`fŽk lofdfk^i-

Dufpqbkjr`e^p j^kbo^p ab ^molufj^o rk^ crk`fŽk a^a^ ` mlo mlifkljflp+abmbkafbkal abi rpl nrb pb e^ ab e^`bo ab i^ ^molufj^`fŽk- Dk bpqb`^mŒqrilklpfkqbobp^oŠl_qbkbo rk mlifkljfl nrb `lfk`fa^ `lk ` v ^idrk^p ab prp abofs^a^p bkrk mrkql a^al- Djmbw^jlp krbpqol `ljbkq^ofl `lk rk bgbjmil pbk`fiil-

Rrmlkd^jlp nrb ` bp i^ crk`fŽk bumlkbk`f^i+ `%r& < _!+ Dk bi mrkql r < N+i^ crk`fŽk ` v qla^p prp abofs^a^p s^ibk 0- Di mlifkljfl ab mofjbo do^al

a%r&< 0 * u

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Rf ^molufj^jlp ` mlo rk mlifkljfl ab pbdrkal do^al M nrb `lfk`fa^ `lk `v prp alp mofjbo^p abofs^a^p bk N+mlabjlp bpmbo^ork^ jbglo ^molufj^`fŽk ab` nrb `lk i^ crk`fŽk ifkb^i d+mlo il jbklp bk i^p molufjfa^abp ab 'N+ 0(- Di ml,ifkljfl

M%r&< 0 * u * qu1

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Page 354: Calculus

223 >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

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'6-0(i re u1 u!

L%r& < &!, < 0 * u * , * --- * ,xf + T F5:

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Page 355: Calculus

Mjgdijhdjn _` Q\tgjm `ib`i_m\_jn kjm pi\ api^d‡i 224

6-1 C\YV[\ZV\` QRFNeY\_R[TR[Q_NQ\]\_ b[N Sb[PVp[

Rrmlkd^jlp nrb ` qfbkbabofs^a^p e^pq^ bi loabk i bk bi mrkql s < N+pfbkali x 0+b fkqbkqbjlp bk`lkqo^o rk mlifkljfl M nrb `lfk`fa^ `lk ` v prp i mofjbo^pabofs^a^p bk N- Cb_bk p^qfpc^`bopbi * 0 `lkaf`flkbp+ ^ p^_bo

'6-1( M&L' < a&L'* M%&L'<.$%-&) j9h&%i& <x_h&%i&)

^pŒnrb bkp^v^jlp rk mlifkljfl ab do^al i* mlo bgbjmil

'6-2(

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'6-3(e&f'&L'

b ,,,f + e

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M&L' < a&L'* L$%K&< .$%-&)

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Page 356: Calculus

225 =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

Dk i^ jfpj^ cloj^+ mlabjlp abjlpqo^o nrb bufpqbrk mlifkljfl v rkl pŽilab do^al z i nrb `lfk`fab `lk ` u prp i mofjbo^p abofs^a^p bk bi mrkql s < \,Dk bcb`ql+bk ird^o ab '6-2(+ mlabjlp bp`of_fo L loabk^al pbd•k i^p mlqbk`f^p abs + \ v mol`babo `ljl ^kqbp-Rf `^i`ri^jlp i^p abofs^a^p bk bi mrkql \ bk ird^oab N+iibd^jlp ^i mlifkljfl

'6-4(h e&f'& '

L%r& < &!]] [ %r Z [&e+H+ e Q5>

Dpqbbp bi •kf`l mlifkljfl ab do^al z h nrb p^qfpc^`bi^p `lkaf`flkbp

L%[&< x%[&) k%&\'<a%&\'*

v pb ib ii^j^ jifchigci ^_ P[sfil bk elklo abi j^qbjŠqf`l Aollh S^vilo '0574,0620(- Blk j^vlo mob`fpfŽk+ab`fjlp nrb bi mlifkljfl '6-4( bp bi jifchigci ^_

P[sfil ^_ al[^i h a_h_l[^i jil ` _h _f johni [+Blksfbkb qbkbork^ klq^`fŽk nrb fkafnrb i^ abmbkabk`f^ abi mlifkljfl ab

S^vilo L obpmb`ql ab ` v h+ Hkaf`^objlp bp^ abmbkabk`f^ bp`of_fbkal L < Ph`

l L < Ph%`&+Di pŒj_lil Py pb abkljfk^ ij_l[^il ^_ P[sfil ab do^al h+ Br^kalbpqblmbo^alo pb ^mif`^ ^ rk^ crk`fŽk d*molar`b rk^ krbs^ crk`fŽk Ph`) bi mlif,kljfl ab S^vilo ab do^al i, Di s^ilo ab bpq^ crk`fŽk bk s pb obmobpbkq^lkPh`%r& l mlo PhW`%r&Y+Rf nrbobjlp fkaf`^o i^ abmbkabk`f^ -obpmb`qlab [) bp`of_f,jlp Ph`%r8[& bk ird^o ab Ph`%r&+

DIDLOKN 0- Br^kal ` bp i^ crk`fŽk bumlkbk`f^i+ `%r& < A%r& < `s* qbkbjlpA%,9&%r&< `s m^o^ qlal e) ^pŒnrb A%e&%K&< _K < 0+ X bi mlifkljfl ab S^viloab do^al h dbkbo^al mlo B bk N bp bi a^al mlo i^ cŽojri^

i se s/ u!Q B&s' < Q &`U' < &!, < 0 * s * , * --- * , -

h h H+e 1 h Q5>

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iPhA%r8 0( < &! - 9-- %r * .U+

I,f Q5>

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Page 357: Calculus

@ƒg^pgj ^ji kjgdijhdjn _` Q\tgjm ,,0

@pŒmrbp q^k pŽil ^m^ob`bk mlqbk`f^p fjm^obp ab r bk ilp mlifkljflp ab S^vilodbkbo^alp mlo i^ crk`fŽk pbkl+ bk N- Di mlifkljfl ab S^vilo ab do^al 0i * 0qfbkb i^ cloj^

[- [/ _ B,(_

Q/h(E%m_hr& < T * 2 * 4 ,6 * --- * ',0(! &0i * 0(

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,%&0 s2 s4 Š s0Š

Q, '`lp r& < 0, , * , , , * --- * ',0( ,-1- 1 3 5 %/h&

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6-2 8mYPbY\P\[ ]\YV[\ZV\` QRFNeY\_

Rf rk^ crk`fŽk Eqfbkb abofs^a^p ab loabk i bk rk mrkql \* mlabjlp pfbjmobcloj^o pr mlifkljfl ab S^vilo Qig mlo jbafl ab i^ cŽojri^

i a&fg& 'Qia&s' < 1 ,{I, &s + \'! ,

FR:

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SDNQDL@ 6-1- Bg jk`m\_jm _` Q\tgjm P8 od`i` g\n kmjkd`_\_`n ndbpd`io`n8]( Idi`\gd_\_, Rf ?f V ^, nji ^jino\io`n*

]' A`mdq\^d‡i, I\ _`mdq\_\ _` pi kjgdijhdj _` Q\tgjm _` E n pi kjgdijhdj_` Q\tgjm _` -%9 n _`^dm*n` od`i`

Page 358: Calculus

227 >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

b( Fio`bm\^d‡i, Ri\ dio`bm\g di_`adid_\ _` pi kjgdijhdj _` Q\tgjm _` a `npi kjgdijhdj _` Q\tgjm _` pi\ dio`bm\g di_`adid_\ _` a, Ad^cj ^ji h\+tjm km`^dnd‡i* nd b&s' < `7a&o'_o* n` od`i` `ioji^`n

A`hjnom\^d‡i, B^a^ molmlpf`fŽk ^(+ _( l b( bp rk^ b`r^`fŽk nrb ifd^ alpmlifkljflp abi jfpjl do^al- O^o^ abjlpqo^o `^a^ rk^ ab bii^p l_pbosbjlp pfj,mibjbkqb nrb bi mlifkljfl nrb ^m^ob`b bk bi mofjbo jfbj_ol qfbkb bi jfpjls^ilo v i^p jfpj^p abofs^a^p bk bi mrkql \ nrb bi nrb ^m^ob`b bk bi pbdrkaljfbj_ol- Dkqlk`bp _^pq^ nrb ob`loabjlp i^ molmfba^a ab rkf`fa^a abi qblob,j^ 6-0- N_p‹osbpb nrb i^ abofs^`fŽk ab rk mlifkljfl ob_^g^ pr do^al+ bk q^kqlnrb i^ fkqbdo^`fŽkil ^rjbkq^-

Di qblobj^ pfdrfbkqb klp af`b il nrb pr`bab `r^kal prpqfqrfjlp s mlo `s bkrk mlifkljfl ab S^vilo-

SDNQDL@ 6-2- OQNOHDC@C CD RTRSHSTBHˆM- Rb^ b&s' < a&^s'* nd`i_j `pi\ ^jino\io`, P` od`i` `ioji^`n

Bi k\mod^pg\m*^p\i_j \ < /+ o`i`hjn Qib&s' < Qia&^s',

A`hjnom\^d‡i, X^ nrb b&s' < a&^s'* i^ obdi^ ab i^ `^abk^ klp a^

b%&s'< ^a%&^s'* b!&s' < ^0e!&^s'* ---+

Olo q^kql l_qbkbjlp

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I e I e Q5> Q5>

DIDLOKNR- Qbbjmi^w^kal s mlo , s bk bi mlifkljfl ab S^vilo `loobpmlk,afbkqb ^ `!* bk`lkqo^jlp nrb

s0 s1 u!P 'b,N9( < 0 , s * , , , * ---* ']i(k , -i 1 2 i

Page 359: Calculus

?•f]ofi ]ih jifchigcim ^_ P[sfil 228

Orbpql nrb `lpe s < d`%!* dm%!*mlabjlp rqfifw^oi^ molmfba^a ab ifkb^ifa^a m^o^l_qbkbo

K^ abofs^`fŽk klp a^

s1 s3 U0i+/P0i[/&n`ics' < s * , * , * --- * ,,,

2 4 %/h * 0(

Di qblobj^ nrb pfdrb bp q^j_f‹k •qfi bk i^ pfjmifcf`^`fŽk ab `Ši`rilp `lk mlifkl,jflp ab S^vilo-

SDNQDL@ 6-3- O_[ M9 oh jifchigci ^_ al[^i h ƒ 0- O_[h ` v d ^im `oh]ci*h_m]ih ^_lcp[^[m ^_ il^_h h _h N v mojiha[gim ko_

'6-5(

_h ^ih^_ a%r&w N ]o[h^i r w N- Af jifchigci L8 _m_f jifchigci ^_ P[sfil a_*h_l[^i jil ` _h N-

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DIDLOKNR- O^oqfbkal ab i^ fabkqfa^a ^idb_o^f`^

'6-6(0 i)g

,, < 0 * s * s/ * --- * u! * [s[[/+s /+s%

sŠifa^ m^o^ qlal r " 0+ sbjlp nrb '6-5( pb p^qfpc^`b `lk `%r&< 0.'0 z r&Lh%r&< 0 * r * --- * r!) v a%r&< r,%. * r&+ Orbpql nrb a%r&w N `r^kals x N+bi qblobj^ 6-3 klp af`b nrb

Hkqbdo^kalbpq^obi^`fŽk `lkpbdrfjlp bpqblqol mlifkljfl ab S^vilo

s0 s1 si%/

Q Z,i^d '0 , r&Y < r * , * , * --- * ,, -i)g 1 2 i * 0

Page 360: Calculus

23/ >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

Dk '6-6( mlabjlp obbjmi^w^o s mlo , s0 m^o^ `lkpbdrfo

@mif`^kal bi qblobj^ 6-3 rk^ sbw jŠp+ bk`lkqo^jlp nrb

Hkqbdo^kal bpq^ obi^`fŽk iibd^jlp ^ i^ cŽojri^

i , U0f(/

Q0i)g '^o`q^k r& < H%Z.&e0f p 9f;L (

6-3 Dgbo`f`flp

0- So^w^o i^p doŠcf`^p ab ilp mlifkljflp ab S^vilo P1 'pbk r& < r * r1-1 v S4'pbk u( :

; r * r1-1 * r3-3 , Olkbo bpmb`f^i &^qbk`fŽk bk ilp mrkqlp bk ilp nrb i^p `ros^p `lo,q^k ^i bgb r) Bljm^o^o bpq^p doŠcf`^p `lk i^ ab %r&< pbk r+

1- G^`bo 0/ jfpjl nrb bk bi Dgbo`f`fl 0 m^o^ ilp mlifkljflp ab S^vilo Q0

'`lp u(+ Q2

'`lp r&)u %r&< `lp r+

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2, Qi&g x s' <c%Z.&e s!,Q5>

3, Q0i)g&, x s0' <dU0f(f+Q5>

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i 00f+/

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Q5+

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Page 361: Calculus

C‡mhpg\ _` Q\tgjm ^ji m`noj 230

6-4 EŽojri^ ab S^vilo `lk obpql

Ulis^jlp ^elo^ ^ rk^ afp`rpfŽk abi boolo bk i^ ^molufj^`fŽk ab rk^ crk,`fŽk a jbaf^kqb pr mlifkljfl ab S^vilo Qia bk rk mrkql \, Di boolo pb abcfkb `ljli^ afcbobk`f^ Bi&s' < a&s' + Qia&s', @pŒnrb+ pf a qfbkb abofs^a^ ab loabk i bk \*mlabjlp bp`of_fo

'6-7(

…pq pb abkljfk^ a‡mhpg\ _` Q\tgjm ^ji m`noj Bi&s'9 bp •qfi `r^kal mlabjlp bpqf,j^o i^ j^dkfqra ab Bi&s', Dumobp^objlp bi boolo `ljl rk^ fkqbdo^iv bkqlk`bppb bpqfj^ i^ j^dkfqra ab i^ fkqbdo^i-Blkpfabobjlp mofjbol bi boolo nrb ^m^ob`b`lk rk^ ^molufj^`fŽk ifkb^i-

RCMPCK? 6-4- Ppkjib\hjn lp` a od`i` _`mdq\_\ n`bpi_\ a! ^jiodip\ `i pi^d`moj `iojmij _` \, Bioji^`n* k\m\ oj_j s `i `n` `iojmij* n` od`i`

a&s' <a`\' * e%&\'&s + \' * B/&s' *

`i _ji_`

B/&s' < F7&s + o'a!&o' _o ,

A`hjnom\^d‡i, Rbd•k i^ abcfkf`fŽk abi boolo mlabjlp bp`of_fo

By&s' < a&s' + a`\' + a%&\'&s + \';v! a%&o'_o + a%&\' GU _o < GU XG%&o'+ a%&\'Z_o,\ \ \

K^ •iqfj^ fkqbdo^imrbab mlkbopb bk i^ cloj^ Px p _q* alkab p < a%&o'+ a%&\'*v q < o + s, @pfjfpjl _pg_o < a!&o' v _q-_o < 0+ `lk il nrb i^ cŽojri^ abfkqbdo^`fŽkmlo m^oqbpklp a^

B/&s' < np _q < pq HW , FU &o+ s'a!&o' _o < FU 'u , o'a!&o' _o *\ \ \ \,

mrbpql nrb p < N `r^kal o < \* v q8< N `r^kal o < s, Dpql abjrbpqo^ bi qbl,obj^

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Page 362: Calculus

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RCMPCK? 6-5- Oojiha[gim ko_ ` n_ha[ ^_lcp[^[ ]ihncho[ ^_ il^_h h * 0_h oh ]c_lni chn_lp[fi ko_ ]ihn_ha[ [+ Ahnih]_m) j[l[ ni^i r _h _mn_chn_lp[fi) n_*h_gim f[ `ƒlgof[ ^_ P[sfil

j a&f'& 'a&s' < &!,! [\ &s + \o * Bds' *

+7 e Q5>

mc_h^i

Bds' < 0--aU &s + ooa&i)/'&o' _o ,i \

@_gimnl[]cƒh+ Di qblobj^ pb abjrbpqo^ mlo fkar``fŽk obpmb`ql^ h+ Kl eb-jlp v^ abjlpqo^al m^o^i < 0- Rrmlkd^jlp ^elo^ nrb bp `fboql m^o^rk `fboql iv 0/ s^jlp ^ abjlpqo^o m^o^ i * 0- Dp`of_^jlp i^ cŽojri^ ab S^vilo '6-7( `lki * 0 v `lk i v obpq^kal l_qbkbjlp

[ [ki)/f&\' [ \ k*0

Bi)/&s' + Bi&s' 'k*i( &s '

Tqfifw^jlp i^ bumobpfŽkfkqbdo^iab Ah%r&v l_pbos^kal nrb %r * [&h(f,%h * 0( <<< Iz%r * o(! _o pb l_qfbkb

0 :B a&i)/'& $LBBi)/&s' < , &s + ooki)g'&o' _o + \ &s + U _o :

i \ i \

< 0-- aU&U [ o'iXa&i)/'&o' + ki)/'&\'Z _o ,T \

K^ •iqfj^ fkqbdo^i mrbab bp`of_fopbbk i^ cloj^ `7o ^i) alkab o << ` %h(f&%n&** `%h(f&%[&v p < , %r * n&!n*f,%h* 0(- Hkqbdo^kalmlom^oqbpv qbkfbkal bk `rbk,

q^ nrb p < N `r^kal o < \* v nrb q < N `r^kal o < s* bk`lkqo^jlp nrb

0 LB 0 <B 0 LBBiE&s' < , p _q < , , q _p < ,,, &s + oo)V&i)0'&o' _o ,i \ i \ &i * 0( \

Dpql `ljmibq^ bi m^pl fkar`qfsl ab i ^ i * 0+`lk il nrb bi qblobj^ bp sŠifalm^o^ qlal i x 0-

6-5 Dpqfj^`fŽk abi boolobk i^ cŽojri^ ab S^vilo

Orbpql nrb bi boolo A!%r& bk &HcŽojri^ ab S^vilo e^ pfal bumobp^al bkcloj^ ab fkqbdo^inrb ^cb`q^ ^ i^ abofs^a^ ab loabk i * 0 ab ` kb`bpfq^jlp ^idrk^

Page 363: Calculus

Bnodh\^d‡i _`g „mmjmi g\ a‡mhpg\ _` Q\tgjm 232

fkcloj^`fŽk jŠp ^`bo`^ l7| ^kqbpab mlabo bpqfj^o i^ j^dkfqra ab Ah%r&) `ljlpb bumif`^ bk bi qblobj^ nrb pfdrb-

RCMPCK? 6-6- Pd g\ _`mdq\_\ &i * g'+„ndh\ _` a n\odna\^` g\n _`ndbp\g_\_`n

&5,7' h xy:/L)g'&o' x J

k\m\ oj_j o `i pi ^d`moj dio`mq\gj lp` ^jio`ib\ \* `ioji^`n k\m\ oj_j s `i `no`dio`mq\gj o`i`hjn g\ `nodh\^d‡i ndbpd`io`

'6-0/(&s + \'/L)/ &s + \'/L)g

h :B&s':Jx+x&i * 0( , /. + &i * 0(

pf s< \*

t

'6-00(& '/.)/ & '/.)/

h \ + s ; &[g'i)/B &s' ; J \ + s%h* 0( , h * %h* 0(

pf s ; \,

A`hjnom\^d‡i, Rrmlkd^jlp nrb s = \, Dkqlk`bp i^ fkqbdo^i m^o^ Bi&s' pbbuqfbkab ^i fkqbos^il W[)rY+ O^o^ `^a^ o bk bpqbfkqbos^il qbkbjlp &'u, ih = N+`lk il nrb i^p abpfdr^ia^abp '6-8( klp a^k

h &s + o'i ; &s + o'i y:/L).&o' ; J &s + o'/Li + i + i

Hkqbdo^kalbkqob \ u s* bk`lkqo^jlp nrb

'6-01( ha%! Ja%!+ &s + o'i _o 888899Bi&s' 888899+ &s + o'i _o ,i \ i \

K^ prpqfqr`fŽk p < s + o*_p < , _o klp a^

F$! G!*+\ & '/.)/&s + o'i _o < Ri _p < s + \ *\ N i)g

`lk 0/ nrb '6-01( pb obar`b ^ '6-0/(-Rf s ; \* i^ fkqbdo^`fŽkpb bcb`q•^ bk Xs*\Z, O^o^ `^a^ o bk bpqbfkqbos^il

qbkbjlp o 9988s* `lk 0/ nrb ',0 &h%r* n&h< %n* r&h8877 N- Olo `lkpfdrfbkqb+ ml,abjlp jriqfmif`^o i^p abpfdr^ia^abp '6-8( mlo bi c^`qlo kl kbd^qfsl &+g'/L&s+o'di b fkqbdo^jlp bkqob s u \ l_qbkfbkal '6-0i(-

Page 364: Calculus

233 =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

DIDLOKN 0- Rf `%r& < _7FFu [ < N+ qbkbjlp i^ cŽojri^

Orbpql nrb `9!(.&%r&< _!) i^ abofs^a^ l| bp jlkŽqlk^ `ob`fbkqb bk `r^inrfbofkqbos^il+ u mlo q^kql p^qfpc^`b i^p abpfdr^ia^abp _] x `%!(.&%n&w _@ bk qlal fkqbo,s^il ab i^ cloj^ W\) b\- Dk rk q^i fkqbos^il+ i^p abpfdr^ia^abp obi^qfs^p ^ Ah%r&abi qblobj^ 6-6 pb p^qfpc^`bk m^o^ h < `\ v L < `!, Dk m^oqf`ri^o+ `r^kal ] < N+qbkbjlp

W!*i W!*i,,, ; A 'u( ; bB,,

%h* 0( , ! , %h* 0( pf M ; s x `-

Olabjlp rqfifw^o bpq^p bpqfj^`flkbp m^o^ `^i`ri^o bi k•jbol ` ab Dribo- Rbqlj^ ] < N+` < 0+ s < 0+ X qbkfbkal bk `rbkq^ nrb ` ; 2 l_qbkbjlp

'6-02(! 0

b < ƒ, * A))%f&)f;L e

alkab %hw 0( z A))%f&; %hw 0(

Dpql mbojfqb bi `Ši`ril ab ` `lk bi do^al ab ^molufj^`fŽk nrb pb abpbb- Olo bgbj,mil+ pf abpb^jlp bi s^ilo ab ` `lk pfbqb `fco^p ab`fj^ibp bu^`q^p+ bibdfjlp rkh q^i nrb 0,%h * 0( ; iiN,p- Oolkql sbobjlp nrb h < 01 bp prcf`fbkqb- Blk_^pq^kqb o^mfabw pb mrbab `^i`ri^o rk^ q^_i^ ab s^ilobp ab f,h ab_fal ^ nrb f,h mrbab `^i`ri^opb afsfafbkal f,%h * 0( mlo h+ K^ pfdrfbkqb q^_i^ m^o^ 2 z h w 01`lkqfbkb bplp k•jbolp obalkab^alp e^pq^ krbsb ab`fj^ibp- Di ~obalkabl‚ bpqŠbk `^a^ `^pl fkaf`^al mlo rk jŠp l rk jbklp il nrb klp fkaf`^ pf i^ `loob``fŽkbp mlo bu`bpl l mlo abcb`ql- 'Dk `r^inrfbo `^pl+ bi boolo bp jbklo nrb jbaf^rkfa^a abi •iqfjl loabk ab`fj^i `lkpfabo^al-(

h h h T

2 /+055555 556 , 7 /+/// /13 7/1 ,3 /+/30 555556 , 8 /+/// //1 645 ,4 /-//7 222 222 * 0/ /+/// /// 165 ,5 /+//0 277 778 , 00 /+/// /// /14 *6 /+/// 087302 , 01 /+/// /// //1 *

Page 365: Calculus

Bnodh\^d‡i _`g `mmjm`i g\ a‡mhpg\ _` Q\tgjm 234

Klp q‹ojfklp `loobpmlkafbkqbp ^ i < N+ 0+ 1 qfbkbk prj^ -Rrj^kal bpqbk•jbol ^ ilp s^ilobp ab i^ q^_i^ 'm^o^ i z01(+ l_qbkbjlp rk qlq^i ab 1+60717072/-Rf qbkbjlp bk `rbkq^ ilp obalkablp+ bi s^ilo `a`^odqj ab bpq^ prj^ mrbab pbojbklo nrb ^nrbi s^ilo+ afcfofbkal ^ il prjl bk pab rk^ rkfa^a abi •iqfjl loabkab`fj^i `lkpbos^al 'ab_fal ^ ilp pfbqb pfdklp jbklp(+ l mrbab bu`babo ^ il prjlbk eab rkfa^a abi •iqfjl loabk ab`fj^i 'ab_fal ^ ilp qobp pfdklp jbklp(- Ki^jb,jlp ^ i^ prj^ p- Dkqlk`bp qlal il nrb mlabjlp ^pbdro^o jbaf^kqb bpqb `Ši`rilbp i^ abpfdr^ia^a 1+607170715 ; p ; 1+607170721- K^ bpqfj^`fŽk abi booloD+1'i( klp a^ /+///////// z A^f& ; /+////////0- Orbpql nrb ` < p * D+1'i(+bpqb `Ši`ril klp iibs^ ^ i^p abpfdr^ia^abp pfdrfbkqbp m^o^ _7

1+607170715 ; ` ; 1+607170722-

Dpql klp af`b nrb bi s^ilo ab `* `lk pfbqb`fco^p ab`fj^ibp+ bp ` < 1+6071707+ l nrbbi s^ilo ab ` obalkab^al ^ l`el ab`fj^ibp+ bp ` < 1+60717072-

DIDLOKN 1- Fmm\^dji\gd_\_ _` `, Olabjlp rqfifw^o i^ ^kqboflo bpqfj^`fŽk abiboolo Ai& 0( m^o^ abjlpqo^o nrb ` bp foo^`flk^i- Djmbw^jlp bp`of_fbkal i^p abp,fdr^ia^abp '6-02( abi jlal pfdrfbkqb9

0 :`+ v%g,: 2 -%h* 0( , J+-zf %h* 0(

Q_>

Lriqfmif`^kal mlo i * pb l_qfbkb

0 ‚&! i% 2 2++:i `+ ,,-9;,,;,i * 0 , e i * 0 , 3

Q_>

'6-03(

pf i ƒ2- O^o^ `^a^ i i^ prj^ obpmb`ql ^ f bp rk bkqbol- Rf ` crbo^ o^`flk^i+ ml,aoŒ^jlp bibdfo i il _^pq^kqb do^kab m^o^ nrb q^j_f‹k i ` crbpb bkqbol- Obol bk,qlk`bp '6-03( klp afoŒ^nrb i^ afcbobk`f^ ab bplp alp bkqbolp pboŒ rk k•jbol bkqbol kl j^vlo nrb +0/ `r^i bp fjmlpf_ib- Olo q^kql ` kl mrbab pbo o^`flk^i-

K^p ^molufj^`flkbp mlo mlifkljflp klp mbojfqbk `lk cob`rbk`f^ l_qbkbo s^,ilobp krj‹of`lp ^molufj^alp ab fkqbdo^ibp &nrb kl pb mrbabk `^i`ri^o afob`q^,jbkqb jbaf^kqb crk`flkbp bibjbkq^ibp- Tk bgbjmil c^jlpl bp i^ fkqbdo^i

Es 0

y&s' < l `8%z,

nrb pb mobpbkq^ bk i^ qbloŒ^ab mol_^_fifa^abp v bk jr`elp mol_ibj^p ab EŒpf`^-Dp p^_fal nrb i^ crk`fŽk a ^pŒabcfkfa^ kl bp rk^ api^d‡i `g`h`io\g, Dp ab`fo+

Page 366: Calculus

235 >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

` kl mrbab l_qbkbopb ^ m^oqfoab mlifkljflp+ bumlkbk`f^ibp+ild^ofqjlp+ crk`flkbpqofdlklj‹qof`^p l qofdlklj‹qof`^p fksbop^p jbaf^kqb rk k•jbol cfkfql ab m^plp bkilp nrb fkqbosbkd^k i^p lmbo^`flkbp ab ^af`fŽk+pr_pqo^``fŽk+jriqfmif`^`fŽk+ afsf,pfŽk+l `ljmlpf`fŽk- Nqolp bgbjmilp nrb `lk cob`rbk`f^ pb mobpbkq^kq^kql bkqbloŒ^ljl bk i^ moŠ`qf`^plk i^p fkqbdo^ibp

'Dk i^ mofjbo^ ab ‹pq^p+pb pl_obkqfbkab nrb bi `l`fbkqb 'pbk L-o pb obbjmi^w^mlo 0 `r^kal o < N- Dk i^ qbo`bo^+f bp rk^ `lkpq^kqb+L:f:/,' Sbojfk^jlp bpq^Rb``fŽk `lk rk bgbjmil nrb e^`b sbo `Žjl i^ cŽojri^ ab S^vilo mrbab rp^opbm^o^l_qbkbo rk^ _rbk^ bpqfj^`fŽk ab i^ fkqbdo^i G>-0lg}_o,

DIDLOKN 2- K^ cŽojri^ ab S^vilo m^o^`%`lk i < 3 klp a^

'6-04(s0 s1 s2

bY < 0 * r * , * , * , * A %r&+1 2 3 3

Rrmlkd^jlp ^elo^ nrb s x N- Dk `r^inrfbo fkqbos^il ab i^ cloj^ Z , b+N\ qbkb,jlp `+` x `! x 0+ ab jlal nrb mlabjlp rqfifw^oi^p abpfdr^ia^abp '6-00( abiqblobj^ 6-6 `ljl g < `+` u L < 0 m^o^ bp`of_fo

.: %*.&2A 't( ; %ZT&23 , 4 pf s ; N-

Cf`el ab lqol jlal+ pf s ; N+ bkqlk`bp B2&s' bp kbd^qfs^ v z s3-3 , Qbbjmi^,w^kal s mlo , o0 bk '6-04(+ qbkbjlp

'6-05(

bk alkab , p0/. R z A+%* o0' ; N- Rf N po x q-bk`lkqo^jlp nrb o

/.- R z

z 'f(il.4 ; /+/////8- @pŒnrb+ pf fkqbdo^jlp '6-05( bkqobN u p+l_qbkbjlp

IH.1- 0 0 0 0 0`8%_o < , , ,, * ,,, , ,,, * ,,, , '(+

l 12-12 4•14•1 6•16•2 8&18&3

alkab N ; '( ; /+/////34- Qbalkab^kal e^pq^ `r^qol ab`fj^ibp+ bk`lkqo^jlpPx-0mo0 _o < /+3502-

Page 367: Calculus

Lom\n ajmh\n _` g\ a‡mhpg\ _` Q\tgjm ^ji m`noj 236

)6-6 Ba_N`S\_ZN` QRYNSp_ZbYNQRFNeY\_P\[ _R`a\

Gbjlp bumobp^al bi boolo bk i^ cŽojri^ ab S^vilo `ljl rk^ fkqbdo^i

S^j_f‹k mrbab bumobp^opbbk jr`e^p lqo^p cloj^p- Orbpql nrb bi c^`qlo %r * n&!abi fkqbdo^kal krk`^ `^j_f^ ab pfdkl bk bi fkqbos^il ab fkqbdo^`fŽk+v nrb y:i)g' bp`lkqfkr^ bk bpqbfkqbos^il+bi qblobj^ abi s^ilo jbafl mlkabo^al m^o^ fkqbdo^ibp'qblobj^ 2-05( klp a^

Es FU & 'i)g&s + o'w&i)gg&o' _o < e&i)gg&^' 'u ] o'i _o < e&i)gg&^' W , \ *

b \ i * 0

bpq^kal ` bk bi fkqbos^il `boo^al nrb rkb \ `lk s, Olo `lkpfdrfbkqb+ bi boolomrbab bp`of_fopbbk i^ cloj^

`&i)/'& ,'C %r&< b %r Z [&h(f +

h &i * 0(

Dpq^bp i^ ii^j^a^ cloj^ ab K^do^kdbabi obpql-Rb m^ob`b^ ilp ^kqboflobpq‹ojf,klp ab i^ cŽojri^ ab S^vilo+ p^isl nrb i^ abofs^a^ `%h(f&%]&bpqŠ`^i`ri^a^ bk rk`fboql mrkql b abp`lkl`fal u kl bk bi mrkql \, Di mrkql b abmbkab ab s v ab i*q^kql `ljl ab `+

Blk rk o^wlk^jfbkql ab qfml afpqfkql+mlabjlp mobp`fkafoab i^ `lkqfkrfa^aab a&i)/' v abar`fo i^ cŽojri^ ab K^do^kdbv lqo^p cloj^p abi obpql _^gl efmŽqbpfpjŠp a‹_fibp- Rrmlkd^jlp nrb l7| bufpqbbk rk `fboql fkqbos^il ^_fboql %b)e& nrb`lkqbkd^ bi mrkql \* v nrb a&i' pb^ `lkqfkr^ bk bi fkqbos^il `boo^al Wb)eY+ Difg^,jlp `r^inrfbo s :.: \ bk Wb) eY+ O^o^ pfjmifcf`^o+ ^ajfq^jlp nrb s = \, L^k,qbkd^jlp r cfgl v abcfk^jlp rk^ krbs^ crk`fŽk B bk bi fkqbos^il W[) rY abi pf,drfbkqb jlal9

i xfe&%&C&o' < a&o' * y \\ o %r * o'f ,

K e Q_+

N_pbosbjlp nrb B%r&< `%r& v B%[&< Ph`%r8 [&) `lk il nrb B%r& * B%[&< Ah%r&+

K^ crk`fŽk E bp `lkqfkr^ bk bi fkqbos^il `boo^al W[) rY u bp abofs^_ib bk bifkqbos^il ^_fboql %[)r&+ Rf `^i`ri^jlp B$%n&)qbkfbkal bk `rbkq^ nrb `^a^ q‹ojfkl

Page 368: Calculus

,-1 =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

ab i^ prj^ nrb abcfkb B%n&bp rk molar`ql+ bk`lkqo^jlp nrb qlalp ilp q‹ojfklppb obar`bk bu`bmql rkl+ v klp nrba^ i^ fdr^ia^a

C%vow< %r * o'i y&i)/'vo' ,T

Rb^ ^elo^ F `r^inrfbo crk`fŽk `lkqfkr^ bk W[)rY v abofs^_ib bk %[)r&+ Olabjlpbkqlk`bp ^mif`^o i^ cŽojri^ abi s^ilo jbafl ab B^r`ev 'qblobj^ 3-5( v bp`of_fo

C$u]vWBurv* Bu[vY< B$u]vWCur&* C%[&Y+

m^o^rk `fboql _ bk bi fkqbos^il ^_fboql %[) r&+ Rf F&kl bp `bol bk %[) r&) pb l_qfbkbi^ pfdrfbkqb cŽojri^ m^o^ bi boolo Ah%r&7

C%v^wAhur& < , WCurv* Cu[&Y +

C$u]&

Olabjlp bumobp^obi boolo ab s^of^p j^kbo^p jbaf^kqb bib``flkbp afpqfkq^pab i^crk`fŽk F- Olo bgbjmil+ qlj^kal C%n&< %r * p(j*p+ l_qbkbjlp i^ cloj^ abK^do^kdb

x&i)g'v@'B %r&< *w*ur * [n(.

i 'z* 0( &

Slj^kal C%n&< s + o*l_qbkbjlp lqo^ cŽojri^+ ii^j^a^ cloj^ ab B^r`ev abiobpql+

alkab \ ; ` ; s ,

y&i)/F&@'Bds' < *&s + ^ovs + [&)

T(alkab \ ; ` ; s ,

Rf C%n&< %r * yO+ pfbkal j w 0+ l_qbkbjlp i^ cŽojri^

x&i)/'v 'Bivs' < *` &s+ ^o)g+!&s [ \'! *

T(Valkab \ ; ` ; s ,

6-7 :WR_PVPV\`

Dk ilp Dgbo`f`flp 0+1 X 2 pb a^k bgbjmilp ab cŽojri^p ab S^vilo `lk obpql- Dk `^a^ `^plabjlpqo^o nrb bi boolo p^qfpc^`b i^p abpfdr^ia^abp nrb pb a^k-

Ji '\h(g,0t1g,0

pbk r <ƒ &0f [ 0( * B/h%r&)h<i

zth1kG

GAch%r&. w %/h * 0( -

Page 369: Calculus

Be`m^d^djn ,-2

i &[/'f U0f

1- `lp r <‚ %/e& * A/h(.%r&)Q5>

i+g &,0'fU0f)/

2- ^o`q^k r <‚ 0f * 0 * A/h%r&)Q5>

3- ^( N_qbkbo bi k•jbol o < Xi4 , 2 `ljl ^molufj^`fŽk ab i^ o^Œwkl kri^ ab i^ b`r^,`fŽk r0 + pbk r rqfifw^kal bi mlifkljfl ab S^vilo ab qbo`bo do^al nrb ^molufj^ pbk r+_( Cbjlpqo^o nrb i^ ^molufj^`fŽk ab i^ m^oqb ^( p^qfpc^`b i^ abpfdr^ia^a

0Zpbko , o10 ; ,

1//&

a^al nrb Xi4 , 2 ; /+8- Cb`fo pf i^ afcbobk`f^ 'pbk o , l0' bp mlpfqfs^ l kbd^qfs^- C^oilp abq^iibp abi o^wlk^jfbkql pbdrfal-

4- ^y Tqfifw^o bi mlifkljfl ab S^vilo ab qbo`bo do^al nrb ^molufj^ ^o`q^k s m^o^ l_qbkbobi k•jbol o < ;U10 , 2(.1 `ljl ^molufj^`fŽk ab i^ o^Œwkl kri^ ab i^ b`r^`fŽk^o`q^k s < s0Š

_( C^al nrb U10 ; 3+5 v nrb 105 < 54425+ abjlpqo^o nrb i^ ^molufj^`fŽk ab i^ m^o,qb ^( p^qfpc^`b i^ abpfdr^ia^a

6-m0

+ ^o`q^k nh; 0//&

Cb`fo pf i^ afcbobk`f^ %lw* ^o`q^k o( bp mlpfqfs^ l kbd^qfs^- C^o ilp abq^iibp abi o^wlk^,jfbkql pbdrfal-

'h0 * s1.[ `

5- Cbjlpqo^o nrb Il 0 * r3- ^r * 0 * 20& alkab

XF-0 06- Cbjlpqo^o nrb /+382837 ; Il 0 * th ^r ; /+38&2847-

7- ^( Rf Nz- r w 0+abjlpqo^o nrb pbk r < r * r1-1 * l%r&) alkab Wl%r&W7O8:d'3-3 ,_( Tqfifw^o i^ bpqfj^`fŽk ab i^ m^oqb ^( m^o^ bk`lkqo^o rk s^ilo ^molufj^al ab i^ fkqb,do^i aa0-0 pbk %r0' ^r+ C^o rk^ bpqfj^`fŽk abi boolo-

8- Tqfifw^o ilp qobp mofjbolp q‹ojfklp kl krilp ab i^ cŽojri^ ab S^vilo ab pbk s m^o^ bk,`lkqo^o rk s^ilo ^molufj^al ab i^ fkqbdo^i F 'pbk sdZ~ _s v a^o rk^ bpqfj^`fŽk abi boolo-ZRb pl_obkqfbkab nrb bi `l`fbkqb 'pbk rfcr bp fdr^i ^ 0 `r^kal r < N-\

0/- Dk bpqb Dgbo`f`fl pb bumlkb rk j‹qlal m^o^ `^i`ri^o /Q* rp^kal i^ cŽojri^ ab S^viloab ^o`q^k s a^a^ bk bi Dgbo`f`fl 2- Rb _^p^ bk nrb /Q bp moŽufjl ^ 2+1+ab jlal nrbd/Q bp moŽufjl ^ /+7 Ž p+v bpqb s^ilo bp moŽufjl ^ 3 ^obq^k p-Olkbo l` < ^o`q^k p+aG < 3/_ , d/Q,^( Tqfifw^o i^ fabkqfa^a q^k'@ * >& < 'q^k = * 7(.'0, q^k = q^k >& mlkfbkal =:>:i]v irbdl = < > < 1/` m^o^ e^ii^o q^k 1/` < Z=,1 v q^k 3/` < pby&Tqfifw^o bkqlk`bp i^ fabk,qfa^a rk^ sbw jŠp `lk = < 300+ > < , d/Q l_qbkfbkal q^k aG 0 e,- Dpql lofdfk^ i^ pfdrfbk,qb fabkqfa^a klq^_ib abp`r_fboq^ bk 06/5 mlo Zlek L^`efk '057/,0640(9

/Q < 05 ^o`q^k p, 3 ^o`q^k 0 e,-

Page 370: Calculus

24/ >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

_( Tqfifw^o bi mlifkljfl ab S^vilo Shh'^obq^k r& blk r < em^o^ abjlpqo^o nrb

2+047217823 ; 05 ^obq^ke; 2+047217861-

b( Tqfifw^o bi mlifkljfl ab S^vilo P.^obq^k r& blk r < 1e m^o^ abjlpqo^o nrb

,/+/056252/8 ; ,3 ^obq^kj ; ,/+/056252//-

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6-8 Nqo^pl_pbos^`flkbp pl_ob bi boolobk i^ cŽojri^ ab S^vilo- K^ klq^`fŽk j

Rf` qfbkb abofs^a^ ab loabk %h* 0( `lkqfkr^ bk rk `fboql fkqbos^il nrb `lk,qbkd^ rk mrkql \* mlabjlp bp`of_fo i^ cŽojri^ ab S^vilo bk i^ cloj^

'6-06(i z;gh' (

a&s' < &!]] \ &s + \V * Bi&s' ,I, f F5:

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pfbkal L = l- Krbdl+ pbd•k bi qblobj^ 6-6+ qbkbjlp i^ bpqfj^`fŽk ab boolo

H 00/*0

GBi&s' Gy J s + \%h* 0(

m^o^ `^a^ s bk X\ + b+\ * bi Rf j^kqbkbjlp s :.: \ v afsfafjlp bp^ abpfdr^i,a^a mlo Gt , \g/L

* bk`lkqo^jlp nrb

.:/ A))%r& 0; K Yt , \g ,+ &s + \'! + &i * 0(

Rf ^elo^ e^`bjlp nrb r w [) sbjlp nrb Ah%r&,%r * [&! w N- Dpql il bumobp^jlpaf`fbkal nrb bi boolo Ah%r& bp ab loabk fkcboflo ^ %r * [&! `r^kal r w \

Page 371: Calculus

Knl[m i\m_lp[]cih_m mi\l_ _f _llil _h f[ `ƒlgof[ ^_ P[sfil+ H[ hin[]cƒh*i 02.

Dp ab`fo+ bk i^p `lkaf`flkbp bpq^_ib`fa^p+ %r& mrbab ^molufj^opb bk rk bk,qlokl ab [ mlo rk mlifkljfl bk %r * [& ab do^al h) u bi boolo bk bpq^^molufj^,`fŽk bp ab loabk fkcboflo ^ %r * [&h `r^kal r w [+

Dk 08/8 D- K^ka^r+ 'p( fkqolargl rk^ klq^`fŽk bpmb`f^i jrv ^molmf^a^`r^kal pb rqfifw^bk `lkbufŽk `lk i^ cŽojri^ ab S^vilo- Dpq^bp i^ klq^`fŽk j 'i^klq^`fŽk l jfk•p`ri^( v pb abcfkb `ljl pfdrb-

CDEHMHBHˆM- Oojiha[gim a%r& ;/; M j[l[ ni^i r ;/; [ _h oh ]c_lni chn_lp[fi

ko_ ]ihn_ha[ [+ H[ hin[]cƒh

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mcahc`c][ ko_

ifj 'u( < N-r**;[ a%r&

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moŽufjl ^ [) `%r& bp mbnrb•l `ljm^o^al `lk a%r&+

DIDLOKN 0- `%r& < i%0( `r^kal r w [ pfdkfcf`^nrb `%r& w N `r^kal r w [+

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Tk^ fdr^ia^a ab i^ cloj^ `%r& < b%r& * i%a%r| pfdkfcf`^nrb `%r& * b%r& :i%a%r| l+ af`el ab lqol jlal+ W`%r&* b%r&Y,a%r& w N `r^kal r w [+

DIDLOKN 2-

`r^kal s+L,

pbk s + sSbkbjlp pbk r < r * i%r& mlonrb ++s++

pbk s,,,s

izN

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i e&fg& 'e&s' < 19z &s + \'! * jzs + \V' `r^kal s ,,,* \ *

FR:

'p( Dajrkal K^ka^r '0766,0827( crb rk- c^jlpl j^qbjŠqf`l ^ibjŠk nrb efwl fjmloq^kqbp`lkqof_r`flkbp ^ i^ L^qbjŠqf`^- Dp `lkl`fal mlo prp if_olp ab @kŠifpfp v ab SbloŒ^ ab k•,

jbolp+ Gbklp ab `i^ofa^a-

Page 372: Calculus

241 >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

pfbjmob nrb i^ abofs^a^ l| pb^ `lkqfkr^ bk rk `fboql fkqbos^il nrb `lkqbkd^ bimrkql \, Dpql bumobp^+_obsbjbkqb+ bi eb`el ab nrb bi q‹ojfkl ab boolo bp mbnrb•l`ljm^o^al `lk %r * [&! `r^kal r bp moŽufjl ^ [+ Dk m^oqf`ri^o+ab i^ afp`rpfŽkeb`e^ bk ^kqboflobp pb```flkbp+ qbkbjlp ilp bgbjmilp pfdrfbkqbp ab cŽojri^ abS^vilo bumobp^a^p`lk i^ klq^`fŽk l9

0,, < 0 * u * u1 * --- * r5. * i%rh

& `r^kal u ,, N-+'[ (

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1 2 3 i

t+ s5.

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1 i

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2 4 6 %/h * 0(

u1 u3 u5 s05.

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2 4 6 0i + 0

Dk ilp `Ši`rilp obi^qfslp ^ ^molufj^`flkbp ab S^vilo+ `lk cob`rbk`f^ pb e^`bkb`bp^ofl `lj_fk^o s^oflp q‹ojfklp nrb `lkqfbkbk bi pŒj_lil,l- Dk bi qblobj^nrb pfdrb pb a^k rk^p ml`^p obdi^p pbk`fii^p m^o^ bi j^kbgl ab ilp pŒj_lilp,l-K^ j^vlo m^oqbab i^p pfqr^`flkbp nrb bk i^ moŠ`qf`^mrbabk prodfo mrbabk ob,plisbopb `lk bp^p obdi^p-

SDNQDL@ 6-7- „KFDAQ@ CD RŒLANKNR,N- @p\i_j s x \* o`i`hjn8

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Page 373: Calculus

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ag&U' € a0&U' < ag&U' € a0&U' *

b&s' b&s' b&s'

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O^o^ abjlpqo^o b(+m^oqfjlp ab i^ fabkqfa^a ^idb_o^f`^

0 p++;/+p)p++g)p g)p

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0 * a%r&

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'* qbkbjlp

`lp s10

2 < 0 * s0 * j&s0' `r^kal s x N-0 , qu * i%r & 1

Olo `lkpfdrfbkqb+ obpriq^

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+ +-

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s0 s1

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Page 374: Calculus

243 =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

u ^pŒpb l_qfbkb

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bk alkab o < , r,/ * r0,0 * i%r0', Obol `r^kal o ,* N+ qbkbjlp _! ;

< 0 * o * dp80* i%o0'* `lk il nrb l_qbkbjlp

s s0 0' U s0 (1 1 U ggs0 1

b! < 0 , , * , * j&s0' * , \ , * , * j&s0

' * j&s ' <0 , , * , * j&s ',1 2 1 1 2 1 13

Rf ^mif`^jlp bpq^fdr^ia^a ^ '6-07(+ l_qbkbjlp i^ cŽojri^ abpb^a^-

6-0/ @mif`^`flkbp ^ i^p cloj^p fkabqbojfk^a^p

X^ ebjlp bumif`^al `Žjl pb rqfifw^k i^p ^molufj^`flkbp mlo mlifkljflp bkbi `Ši`ril ab s^ilobp ab crk`flkbp- S^j_f‹k pb mrbabk rqfifw^obk bi `Ši`ril abiŒjfqbp-Kl s^jlp ^ bumif`^o `lk ^idrklp bgbjmilp-

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, \! [ ]U

di,,,['U [

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`%< 0 * o * j&o' `r^kal o ,,,* N

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Page 375: Calculus

=jfc][]cih_m [ f[m W[lg[m ch^_n_lgch[^[m ,..

@nrŒebjlp rqfifw^al bi eb`el ab nrb i%r ild [& < i%r& t i%r ild \& < i%r&+Rf ^elo^ obpq^jlp u l_pbos^jlp nrb i%r& * i%r&< i%r&) bk`lkqo^jlp \! + ]! :: r%.ia [*Eia \&(i%r&+ Cfsfafbkal mlo r u qbkfbkal bk `rbkq^ nrb i%r&,r:i%f&)l_qbkbjlp

\! + ]! \ \,,, < ild , * -%.&,,,*ild , `r^kal s ,,,*N-s ] ]

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s s 1

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s

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Page 376: Calculus

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ifj '0 * s& \ggg < b! -!!")

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6-00 Dgbo`f`flp

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pbk^u5-ifj,

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Page 377: Calculus

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Page 378: Calculus

247 =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

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Page 379: Calculus

O`bg\ _` I%Ejkdo\g k\m\ g\ ajmh\ di_`o`mhdi\_\ L-L 248

`sdno` t od`i` `g q\gjm o\g ^jhj I* `ioji^`n `g g…hdo`

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Dk bpqb `^pl a&s' < pbk s u b&s' < s, Di `l`fbkqb ab abofs^a^p bp a%&s'-b%&s':< '`lp s'-g u qfbkab ^ 0 `r^kal s x N- Rbd•k bi qblobj^ 6-8 bi iŒjfqb '6-12( q^j,_f‹k bufpqb v bp fdr^i ^ 0-

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h-s + o\i,s0j

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Page 380: Calculus

-25/ =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

bi `l`fbkqb '6-13( pb qo^kpcloj^ bk

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/+ ` he `&` + g's

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fkc^if_ib-

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Page 381: Calculus

O`bg\ _` I%Ejkdo\g k\m\ g\ ajmh\ di_`o`mhdi\_\ L-L 250

Di mofjbo m^pl bp `loob`ql mbol bi pbdrkal kl- Di `l`fbkqb %3r * /&,%/r * 0( klbp fkabqbojfk^al `r^kal s x 0- Di iŒjfqb `loob`ql+ 3+ pb l_qfbkb prpqfqrvbkal smlo 0 bk %3r * /&%/r * 0(-

DIDLOKN 5- @idrk^p sb`bp pb mrbab obar`fo bi qo^_^gl e^`fbkal rk `^j_flab s^of^_ib- Olo bgbjmil+ pb mlaoŒ^ ^mif`^o afob`q^jbkqb i^ obdi^ ab K&GŽmfq^im^o^`^i`ri^o bi iŒjfqb

0+ TVfj -.,+

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mbol pb mrbab bsfq^o i^ abofs^`fŽk ab i^ o^Œwr^ao^a^ bp`of_fbkal o < v9:9v l_,pbos^kal nrb9

Uf o, 0 0ifj , < ifj ,, < 00j ,, < , , +

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Rb qo^q^ ^elo^ ab abjlpqo^o bi qblobj^ 6+8-

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C&s' <a&s' pf s89d8,\* B%[&< N+

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Page 382: Calculus

,/+ =jlircg[]cƒh ^_ `oh]cih_m jil jifchigcim

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6-02 Dgbo`f`flp

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ifj 'u,2pbk 1s * \s+/ * ]' < N>!+zl

Page 383: Calculus

Ijn n…h]jgjn * // t + ^j, 252

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]s + pbk s l U \ * o

05- Tk ^o`l ab `fo`rkcbobk`f^ ab o^afl 0 pr_qfbkab rk Škdril ab s o^af^kbp+ N ; s ; n.P)`ljl pb sb bk i^ cfdro^ 6-1- Di mrkql a bp i^ fkqbopb``fŽk ab i^p alp q^kdbkqbp bk > u>+ Rb^ P%r& bi Šob^ abi qofŠkdril =>_ u O%r&bi Šob^ ab i^ obdfŽk plj_ob^a^- B^i`ri^o9

lEHFTQ@6-1 Dgbo`q`ql i_-

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Page 384: Calculus

253 >kmjsdh\^dƒi _` api^dji`n kjm kjgdijhdjn

bj_^odl+ ^i pŒj_lil * // kl pb ib ab_b ^qof_rfo kfkd•k pfdkfcf`^al kjm „g hdnhj*v pb a^oŠk abcfkf`flkbp mob`fp^pab s^of^p molmlpf`flkbp nrb `lkqfbkbk bpqbpŒj_lil-

Tk^ ab bii^p bp i^ pfdrfbkqb9

ifj a&sw < =)2)2&&&&&r#((

nrb pb ibb ~bi iŒjfqbab `%r&) `r^kal r qfbkab ^ jŠp fkcfkfql+bp =|+ K^ fab^ nrbpb nrfbob bumobp^o lk biil bp nrb ilp s^ilobp ab i^ crk`fŽk `%r& mrbabk pbo q^kmoŽufjlp ^ > `ljl pb nrfbo^ qlj^kal s prcf`fbkqbjbkqb do^kab- O^o^ nrb bpq^molmlpf`fŽk qbkd^ rk pbkqfal j^qbjŠqf`l mob`fpl pb e^ ab bumif`^o nrb pfdkfcf`^~q^k moŽufjl `ljl pb nrfbo^‚ v ~prcf`fbkqbjbkqb do^kab‚+ il `r^i pb ildo^ `lki^ pfdrfbkqb abcfkf`fŽk9

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Br^kal fkqbobp^`lkl`bo bi `ljmloq^jfbkql ab `%r& m^o^ s^ilobp ab r i`jb\odqjn do^kabp+pb fkqolar`b bi pŒj_lil , // 'jbklp fkcfkfql( u pb bp`of_b9

ifj a&sw < =0$"$./!

Page 385: Calculus

Him m•g\ifim * // s * // 254

nrb pfdkfcf`^9 O^o^ `^a^ C = N bufpqb rk I = N q^i nrb

G.'t( , =f ; C pfbjmob nrb s ; *I+

Rf C bpqŠ abcfkfa^ mlo '6-14( bp cŠ`fi `ljmol_^o nrb i^p alp fdr^ia^abp

ifj x%r&< > u EcgB%n&< =`!!K*;'E"'JJ

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RCMPCK? 6-0/- Oojiha[gim ko_ ` u c nc_h_h ^_lcp[^[m `$%r&u a$%r&j[l[

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u Hfj a%r&< N+N'%%JJ

s ko_ a$%r&<0&/ N j[l[ r = I+ Rf `$%r&,a$%r&nc_h^_ [ oh f•gcn_ ]o[h^i r ,* * //+

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'6-15( ifj a%&s' < Hu,*ll d&'u(

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j^ i^ cloj^ fkabqbojfk^a^ N.N `r^kal o x /* pb `lkpfabo^ bi `l`fbkqb ab i^pabofs^a^p B$%n&,C$%n&+@mif`^kal i^ obdi^ ab i^ `^abk^+ pb qfbkb9

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v D%&o'< ,0 c&%.&+n0 n

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G^v bsfabkqbjbkqb rk qblobj^ ^kŠildl ^i 6-0/ `r^kal pb `lkpfabo^ biiŒjfqb m^o^ s ,,,=, , ll-

Page 386: Calculus

,// >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

/&)- ?oZVaRV[SV[Va\`

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'6-16( gdha&s' < * //

!)*([

l+ q^j_f‹k+

'6-17( a&s' ++* // `r^kal s ++\

m^o^ fkaf`^o nrb a&s' pb mrbab e^`bo q^k do^kab `ljl pb nrfbo^ qlj^kal sprcf`fbkqbjbkqb moŽufjl ^ \, Di pfdkfcf`^al mob`fpl ab bpqb pŒj_lil bpqŠ a^almlo i^ pfdrfbkqb abcfkf`fŽk9

CDEHMHBHˆM- Ijn n…h]jgjn `i '6-16( t '6-17( ndbidad^\i lp` \ ^\_\ iˆh`mjkjndodqj K &o\i bm\i_` ^jhj n` lpd`m\' ^jmm`nkji_` jomj iˆh`mj kjndodqj 04&lp` _`k`i_` _` K( o\g lp`

a&s' = I nd`hkm` lp` M ; Yt , ]h ; 04 ‘

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gdha&s' < * // +s+)\)

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gdha&s' < * // +?'%3'

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Page 387: Calculus

I…hdo`n diadidojn

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ifj x%r&< * //d%##\\

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Di ib`qlo kl qbkaoŠ afcf`riq^a m^o^ clojri^o abcfkf`flkbp ^kŠild^p m^o^ ilppŒj_lilp

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'6-18( ifj i^d s < * // -d%#\\

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Page 388: Calculus

257 >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

Cb i^ obi^`fŽk nrb bufpqb bkqob i^ crk`fŽk ild^oŒqjf`^ v i^ bumlkbk`f^i bp cŠ`fimol_^o nrb9

'6-2/( v 'l ifj `+s < N(-9q9,**`l;''7;$

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ifj 0-- < N -9q9,*`u9{s\,

ifj s%< *// v

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ifj `+/-s: * //

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u,,l*

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6-05 Bljmloq^jfbkql ab ild s v `! m^o^s^ilobp do^kabp ab s

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,* *// bk ^j_lp `^plp `r^kal s m+m\ 'l t,* € //(- Dk bpqb `^pl+ ab`fjlp nrbbi `l`fbkqb `%r&, a%r& ^almq^ i^ cloj^ fkabqbojfk^a^ //.//- Dufpqbk s^of^p buqbk,pflkbp ab i^ obdi^ ab K&GŽmfq^inrb ^ jbkral klp ^vra^k m^o^ abqbojfk^o bi`ljmloq^jfbkql ab rk `l`fbkqb `r^kal ^almq^ i^ cloj^ fkabqbojfk^a^ //.//-

Ml l_pq^kqb+ kl bumlkaobjlp bp^p buqbkpflkbp mlonrb jr`elp ab ilp bgbjmilpnrb pb mobpbkq^k bk i^ moŠ`qf`^ mrbabk qo^q^opb ^mif`^kal bi qblobj^ nrb pfdrbv nrb abp`of_b bi `ljmloq^jfbkql abi ild^ofqjl v ab i^ bumlkbk`f^i m^o^ s^il,obp do^kabp ab s,

RCMPCK? 6-00- Pd \ = N v ] = N+ n` od`i`

'6-20( ifj 'Hld U'] < Nu,*ll u!

v

'6-21(

Page 389: Calculus

@jhkjmo\hd`ioj _` g\b s u `s k\m\ q\gjm`n bm\i_`n _` s 147

A`hjnom\^d‡i, Oofjbol abjlpqo^jlp '6-20( X irbdl rqfifw^jlp bi obpriq^alm^o^ abar`fo '6-21(- Orbab e^`bopb rk^ abjlpqo^`fŽk pbk`fii^ ab '6-20( m^oqfbkalab i^ abcfkf`fŽk ab ild^ofqjl `ljl fkqbdo^i- Rf b = N u o x )$ qbkbjlp n*f w n_*f+

Krbdl+ pf r = 0+ mlabjlp bp`of_fo

Gs 0 GU ` 0 `

N ; i^d u < , _o x m8%_o < y ; y -iq 0 ` B

Olo `lkpfdrfbkqb+ pb qfbkb

'Hld u(! T.y`+\N ; ,, ; ,, m^o^ qlal ` = N -

s\ ^]

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Dpql abjrbpqo^ '6-20(- O^o^ abjlpqo^o '6-21(+ e^`bjlp bi `^j_fl ab s^of^_ibn < _!) Dkqlk`bp r < i^d n) v mlo q^kql r\,_[T < %fian&\,n[

† Obol o x *// `r^k,al u ,,,* * //+ `lk il `r^i '6-21( bp `lkpb`rbk`f^ ab '6-20(-

Blk rk^ k^qro^i buqbkpfŽk ab i^ klq^`fŽk,l+ mlabjlp bp`of_fo i^p molmlpf`fl,kbp pl_ob iŒjfqbp nrb ^`^_^jlp ab abjlpqo^o bk i^ cloj^

'i^d U'] < i%r\' `r^kal r ,,,* * // +

v

Cf`el ab lqol jlal+ mlo do^kab nrb pb^ ] u mlo mbnrb•l nrb pb^ [ '^j_lp ml,pfqfslp(+ 'ild U'] qfbkab ^ fkcfkfql jŠp ibkq^jbkqb nrb s!, @pfjfpjl+ s] qfbkab ^fkcfkfql jŠp ibkq^jbkqb nrb `%!8

DIDLOKN 0- Dk bi bgbjmil 3 ab i^ Rb``fŽk 6-01 pb abjlpqoŽ nrb bi `ljmlo,q^jfbkql ab _*f,T , r m^o^ r moŽufjl ^ N kl mlaŒ^ pbo ab`fafal jbaf^kqb rk k•,jbol `r^inrfbo^ ab ^mif`^`flkbp ab i^ obdi^ ab K&GŽmfq^im^o^ bi `^pl N.N- Mll_pq^kqb+ pf bp`of_fjlp o < 0. s* bpqb `l`fbkqb pb qo^kpcloj^ bk o- `o u ^almq^ i^cloj^ fkabqbojfk^a^ //.// `r^kal om+~*//- Di qblobj^ 6-00 klp af`b nrb

ifj d;Lo+<)jj `o ,

Olo q^kql+ _*f,rcr w N `r^kal r w /* l+ bk lqo^p m^i^_o^p+ _*f,T < i%r& `r^k,al s x /*-

@abjŠp ab N.N b //.// bufpqbk lqo^p cloj^p fkabqbojfk^a^p- @idrk^p abbp^p+obmobpbkq^a^p `lk ilp pŒj_lilp N& //+ /•+ b //•+ pb firpqo^k `lk ilp bgbjmilp

Page 390: Calculus

26/ >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

nrb pb a^k ^ `lkqfkr^`fŽk- Dk bgbjmilp m^ob`falp ^ bplp+ qo^kpcloj^`flkbp ^idb,_o^f`^p klp mbojfqbk ^ jbkral obar`fo ^ rk^ cloj^+ fkabqbojfk^a^ abi qfml N.N l//.// nrb mrbab pboobprbiq^`lk i^ obdi^ ab K&GŽmfq^i+mlomlifkljflp ab ^moluf,j^`fŽk+ l mlo jbafl abi qblobj^ 6-00-

DIDLOKN 1- 'N - '/(- Cbjlpqo^o nrb ifj+ --‘l* u! i^d s < N m^o^ `^a^ s^ilocfgl ab %T = N-

Pjgp^d‡i, Olkfbkal n < i.u+ bk`lkqo^jlp nrb s! i^d s < , 'i^d n&, '1 v+ bksfoqra ab '6-20(+ qfbkab ^ M `r^kal ',* * ^`*

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DIDLOKN 3- '//•(- Cbjlpqo^o nrb ifj !+!&*b` r.,F& < 0-

Pjgp^d‡i, Olkbo o < i.u v ^mif`^o bi obpriq^al abi bgbjmil 2-

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Page 391: Calculus

Be`m^d^djn 260

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Page 392: Calculus

,0+ >kmjsdh\^d‡i _` api^dji`n kjm kjgdijhdjn

16- Cbjlpqo^o nrb 'i * r&$ < 0 * _r * i%r& `r^kal r w N- @mif`^o bpqb obpriq^al m^o^`^i`ri^o bi iŒjfqb ab

17- O^o^ rk `fboql s^ilo ab b+ bi iŒjfqb

ifj u%r3* 69u3 * 1(! , rv*2M#((

bp cfkfql v kl kril- Cbqbojfk^o bpb b v `^i`ri^o bi s^ilo abi iŒjfqb-18- Rb^k a%r&< r_t$ v `%r& < Pxa%n&%n* f,n& _o* B^i`ri^o bi iŒjfqb ab ,))%r&,a!%r& `r^kal

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ab ,)%r&,a$%r& `r^kal r ,* * // bp cfkfql v kl kril- Cbqbojfk^o b v `^i`ri^o bi s^ilo abiiŒjfqb-

20- Rb^ `%r& < _*fE!)$ pf T8€ N+X `%K&< l-^( Cbjlpqo^o nrb m^o^ qlal g; N+ %r&,r! w N `r^kal r w N-_( Cbjlpqo^o nrb m^o^ r :‹ N i^ abofs^a^ k,pfj^ ab ` bp ab i^ cloj^ `%$&%r&< `%r&L%f,r&)

pfbkal Lnn& rk mlifkljfl bk o,b( Cbjlpqo^o nrb a&L' < N m^o^ qlal i x 0- Dpql abjrbpqo^ nrb qlal mlifkljfl ab

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^kr^i+ ^`rjriŠkalpb ilp fkqbobpbpg sb`bp mlo ^•l: bp ab`fo pb prmlkb bi ^•l afsfafalbk h m^oqbpfdr^ibp v `^a^ h+ndhj ab ^•l bi fkqbo‹p molar`fal pb fk`lomlo^ ^i `^mfq^i-^( Cbjlpqo^o nrb bi `^mfq^i qlq^i l_qbkfal ^i `^_l ab k ^•lp bp L%f * l,g& †††† Rf l vi pb j^kqfbkbk cfglp+bp^ `^kqfa^a qfbkab ^ M`mj `r^kal h ,* * `b+ Dpqbeb`el a^ lofdbk^ i^ pfdrfbkqb abcfkf`fŽk9 Cb`fjlp nrb rk^ `^kqfa^a ab afkbol bpqŠ fjmrbpq^ ^ fkqbo‹p`lkqfkrl ^i l mlo rkl ^kr^i pf i^ `^kqfa^a `%n&abpmr‹p ab n ^•lp bp `%K&_l

$) pfbkal n`r^inrfbo k•jbol ob^i kl kbd^qfsl- B^i`ri^o ^molufj^a^jbkqb bi qfbjml kb`bp^ofl m^o^nrb rk^ `^kqfa^a ab afkbol pb armifnrb `lil`Škali^ bk pr _^k`l ^i 5 $ ^kr^i ^ fkqbo‹p`ljmrbpql+ ^`rjriŠkalpb ilp fkqbobpbp_( bk cloj^ `lkqfkr^+ `( mlo qofjbpqobp-

Page 393: Calculus

7

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Rb mobpbkq^krk^ do^k s^ofba^a ab mol_ibj^p+ bk ilp `r^ibp pb abpb^ abqbo,jfk^o rk bibjbkql s^of^_ib ^ m^oqfoab pr `lbcf`fbkqb ab s^of^`fŽk- Olo bgbjmil+pb nrfbob abqbojfk^o i^ mlpf`fŽk ab rk^ m^oqŒ`ri jŽsfi `lkl`fbkal pr sbil`fa^al ^`bibo^`fŽk: l _fbk+ a^a^ rk^ prpq^k`f^ o^af^`qfs^ nrb pb abpfkqbdo^+`lk`lbcf`fbkqb ab s^of^`fŽk `lkl`fal+ pb qo^q^ab abqbojfk^o i^ `^kqfa^a ab prpq^k`f^obj^kbkqb abpmr‹p ab rk qfbjml a^al- Dk bgbjmilp `ljl ‹pqlp+pb qo^q^ab ab,qbojfk^o rk^ api^d‡i _`n^jij^d_\ jbaf^kqb a^qlp obi^`flk^alp mlo rk^ b`r^`fŽknrb `lkqfbkb mlo 0/ jbklp rk^ ab i^p abofs^a^p ab i^ crk`fŽk abp`lkl`fa^- Dpq^pb`r^`flkbp pb bkljfk^k `^p\^dji`n _da`m`i^d\g`n v pr bpqrafl `lkpqfqrvb rk^ abi^p o^j^p ab i^ L^qbjŠqf`^ nrb qfbkbk jŠp ^mif`^`flkbp-

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'7-0( a%&s'<a&s'

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262

Page 394: Calculus

263 giom^_p^^dƒi \ g\n `^p\^dji`n _da`m`i^d\g`n

i^ Lb`Škf`^ ab cirfalp- ^pŒ`ljl bk lqolp `^mŒqrilp ab i^ EŒpf`^ j^qbjŠqf`^-Cf`e^ b`r^`fŽk ^ajfqb qfmlp afpqfkqlp ab plir`flkbp bkqob i^p `r^ibp `noƒia&s* s&;; s * /s) a&s*s& < `U `lp s) v a&s*s& < ild &s/ * a'}

Di bpqrafl ab i^p b`r^`flkbp afcbobk`f^ibp bp rk^ m^oqb ab i^ L^qbjŠqf`^nrb+ nrfwŠ jŠp nrb `r^inrfbo lqo^+ e^ pfal afob`q^jbkqb fkpmfo^a^ mlo i^ Lb`Š,kf`^+ i^ @pqolkljŒ^ v i^ EŒpf`^j^qbjŠqf`^- Rr efpqlof^ bjmbwŽ bk bi pfdil WUHH

`r^kal Mbtqlk+ Kbf_kfw v ilp Aboklriif obplisfbolk ^idrk^p b`r^`flkbp afcbobk,`f^ibp pbk`fii^p nrb pb mobpbkq^olk bk mol_ibj^p ab FbljbqoŒ^ v Lb`Škf`^- Dplpmofjbolp abp`r_ofjfbkqlp+ nrb `ljbkw^olk ^iobabalo ab 058/+ iibs^olk do^ar^i,jbkqb ^i abp^ooliil ab rk^ bpmb`fb ab ~_lip^ ab qor`lp‚ m^o^ obplisbo `fboqlp qfmlpm^oqf`ri^obp ab b`r^`flkbp afcbobk`f^ibp- Rf _fbk bplp qor`lp plk ^mif`^_ibp obi^qf,s^jbkqb bk ml`lp `^plp+ klp mbojfqbk obplisbo jr`e^p b`r^`flkbp afcbobk`f^ibpnrb pb mobpbkq^k bk Lb`Škf`^ u FbljbqoŒ^+ ab jlal nrb pr bpqrafl bp ab fjmlo,q^k`f^ moŠ`qf`^- @idrklp ab bplp j‹qlalp bpmb`f^ibp v ^idrklp ab ilp mol_ibj^p nrb`lk biilp mlabjlp obplisbo pboŠk bumrbpqlp e^`f^ bi cfk^i ab bpqb `^mŒqril-

K^ bumbofbk`f^ e^ abjlpqo^al nrb bp afcŒ`fi l_qbkbo qbloŒ^pj^qbjŠqf`^p abdo^k dbkbo^ifa^a ^`bo`^ ab i^p plir`flkbp ab i^p b`r^`flkbp afcbobk`f^ibp+ p^islm^o^ rklp ml`lp qfmlp- Dkqob ‹pq^p mlabjlp `fq^o i^p ii^j^a^p b`r^`flkbp afcbobk,`f^ibp gdi`\g`n nrb pb mobpbkq^k bk rk^ do^k s^ofba^a ab mol_ibj^p `fbkqŒcf`lp-Klp qfmlp jŠp pbk`fiilp ab b`r^`flkbp afcbobk`f^ibp ifkb^ibp v ^idrk^p ab prp ^mif,`^`flkbp pb `ljbkq^k bk bpqb `^mŒqril ab fkqolar``fŽk- Dk bi Ulirjbk 00 pb e^`brk bpqrafl jŠp `ljmibql ab i^p b`r^`flkbp ifkb^ibp-

7-1 SbojfklildŒ^ v klq^`fŽk

Br^kal pb qo^_^g^ `lk rk^ b`r^`fŽk afcbobk`f^i q^i `ljl '7-0( pb ^`lpqrj,_o^ bp`of_fo t bk sbw ab `%r&)t% bk sbw ab n$%r&v i^p abofs^a^p ab loabk prmb,oflo pb fkaf`^k mlo t!* t!%* bq`- S^j_f‹k pb rqfifw^k lqo^p ibqo^p bk ird^o ab tq^ibp `ljl p* @) w+bq`- Rb bkqfbkab mlo jm_`i ab rk^ b`r^`fŽk afcbobk`f^i biab i^ abofs^a^ ab j^vlo loabk nrb ^m^ob`b bk i^&b`r^`fŽk- @pŒ'7-0( bp rk^b`r^`fŽk ab mofjbo loabk nrb mrbab bp`of_fopb t% < t, K^ b`r^`fŽk afcbobk`f^it% < s1t * pbk 'uv!( bp ab pbdrkal loabk-

Dk bpqb `^mŒqril pb `lkpfabo^oŠk+ bk mofjbo ird^o+ i^p b`r^`flkbp ab mofjboloabk bk i^p nrb pb mrbab abpmbg^opbi^ t%*u nrb pb bp`of_bk ab i^ j^kbo^ pfdrfbkqb9

%5+/& s$ < a&s*s&)

bk alkab i^ bumobpfŽk `%r) s& abi pbdrkal jfbj_ol qfbkb afsbop^p cloj^p m^oqf`ri^,obp- Tk^ crk`fŽk abofs^_ib t < V&s' bp rk^ njgp^d‡i ab '7-1( bk rk fkqbos^il /pf i^ crk`fŽk X v pr abofs^a^ X&p^qfpc^`b i^ obi^`fŽk

V%&s'<aXs* V&s''

Page 395: Calculus

Q`mhdijgjb…\ v ijo\^d‡i 264

m^o^ qlal s bk g,Di `^pl jŠp pbk`fiil pb mobpbkq^ `r^kal y&s*u( bp fkabmbkafbk,qb ab v- Dk q^i `^pl+ '7-1( pb `lksfboqb bk

'7-2( t% < M%r&)

bk alkab O pb prmlkb nrb bp rk^ crk`fŽk a^a^ abcfkfa^ bk rk `fboql fkqbos^il g,Qbplisbo i^ b`r^`fŽk afcbobk`f^i '7-2( pfdkfcf`^ bk`lkqo^o rk^ mofjfqfs^ ab P-

Di pbdrkal qblobj^ crka^jbkq^i abi BŠi`ril klp af`b `Žjl e^`boil `r^kal O bp

`lkqfkr^ bk rk^ fkqbos^il ^_fboql g,Rbk`fii^jbkqb+ pb fkqbdo^ O u pb ^dobd^ rk^`lkpq^kqb `r^inrfbo^- @pŒ+qla^ plir`fŽk ab '7-2( nrba^ fk`irfa^ bk i^ cŽojri^

'7-3( t < HN&u( ^ u * b +

pfbkal b rk^ `lkpq^kqb `r^inrfbo^ 'ii^j^a^ `loofbkqbjbkqb `lkpq^kqb ^o_fqo^of^ abfkqbdo^`fŽk(- K^ b`r^`fŽk afcbobk`f^i '7-2( mlpbb rk^ fkcfkfa^a ab plir`flkbp+ rk^m^o^ `^a^ s^ilo ab B-

Rf kl bp mlpf_ib `^i`ri^o i^ fkqbdo^i '7-3( mlo jbafl ab crk`flkbp bibjbkq^ibp+

q^ibp `ljl mlifkljflp+ crk`flkbp o^`flk^ibp+ crk`flkbp qofdlklj‹qof`^p v qofdlklj‹,qof`^p fksbop^p+ ild^ofqjlp+ u bumlkbk`f^ibp+ pb `lkpfabo^ nrb i^ b`r^`fŽk afcbobk`f^ie^ pfal obprbiq^ pf i^ plir`fŽk mrbab bumobp^opb jbaf^kqb fkqbdo^ibp ab crk`flkbp`lkl`fa^p- Dk i^ moŠ`qf`^+ bufpqbk s^oflp j‹qlalp m^o^ `^i`ri^o ^molufj^a^jbkqbfkqbdo^ibp nrb klp iibs^k ^ rk^ •qfi fkcloj^`fŽk ^`bo`^ ab i^ plir`fŽk- LŠnrfk^p`^i`ri^alo^p ^rqljŠqf`^p ab ^iq^ sbil`fa^a pb e^k afpb•^al mbkp^kal bk bpqb qf,

ml ab mol_ibj^p-

DIDLOKN- Jjqdhd`ioj gdi`\g _`o`mhdi\_j kjm g\ q`gj^d_\_, Rrmlkd^jlpnrb rk^ m^oqŒ`ri^ pb jrbsb ^ il i^odl ab rk^ ob`q^ ab j^kbo^ nrb pr sbil`fa^abk bi fkpq^kqb o bp 1 pbk o, Cbqbojfk^o pr mlpf`fŽk bk bpb fkpq^kqb o,

Pjgp^d‡i, Rf s%n& obmobpbkq^ i^ mlpf`fŽk bk bi fkpq^kqb c) jbafa^ ^ m^oqfoabi mrkql fkf`f^i+ i^ abofs^a^ U$%n&obmobpbkq^i^ sbil`fa^a bk bi fkpq^kqb o, Sbkb,jlp+ mrbp+pbd•k bi bkrk`f^al

t%&o'< 0n`io,

Hkqbdo^kal+ bk`lkqo^jlp nrb

V&o' < 1 Ipbk o _o * a < ,1 `lp o * _-

Dpql bp qlal `r^kql mlabjlp abar`fo ^`bo`^ ab U%n& m^oqfo•kf`^jbkqb abi `l,kl`fjfbkql ab i^ sbil`fa^a: ^idl jŠp ab fkcloj^`fŽk bp kb`bp^of^ m^o^ cfg^o i^

crk`fŽk ab mlpf`fŽk- Olabjlp abqbojfk^o b pf `lkl`bjlp bi s^ilo ab V bk rk

Page 396: Calculus

265 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

`fboql fkpq^kqb- Olo bgbjmil+ pf U%K&<< N+bkqlk`bp B < 1 X i^ crk`fŽk mlpf`fŽk bpU%n&< 1 , 1 `lp 0- Obol pf U%K&< 1+ bkqlk`bp B < 3 X i^ crk`fŽk mlpf`fŽk pb,oŠ U%n&< 3 , 1 `lp n+

Dk `fboqlp ^pmb`qlp bi bgbjmil nrb ^`^_^jlp ab obplisbo bp qŒmf`lab il nrbbk dbkbo^i l`roob- Dk ^id•k jljbkql abi mol`bpl ab obplir`fŽk ab rk^ b`r^`fŽkafcbobk`f^i ab mofjbo loabk+ pb obnrfbob rk^- fkqbdo^`fŽk m^o^ bifjfk^o i^ abofs^a^v&v bk bpb jljbkql ^m^ob`b rk^ `lkpq^kqb ^o_fqo^of^ B- Di jlal mlo bi `r^i i^`lkpq^kqb B bkqo^ bk i^ plir`fŽk abmbkaboŠ ab i^ k^qro^ibw^ ab i^ b`r^`fŽk afcb,obk`f^i a^a^- Orbab ^m^ob`bo `ljl rk^ `lkpq^kqb ^afqfs^+ `ljl bk i^ b`r^`fŽk'7-3( mbol bp jŠp cŠ`fi nrb ^m^obw`^ bk ^idrk^ lqo^ cloj^+ Olo bgbjmil+ `r^kalobplisbjlp i^ b`r^`fŽk t% < t ab i^ Rb``fŽk 7-2+ bk`lkqo^jlp nrb qla^ plir`fŽkqfbkb i^ cloj^ t < ?_! )

Dk jr`elp mol_ibj^p bp kb`bp^ofl pbib``flk^o bkqob qla^p i^p plir`flkbp i^nrb qfbkb rk s^ilo ^pfdk^al bk rk `fboql mrkql- Di s^ilo ^pfdk^al pb abkljfk^^ji_d^d‡i did^d\g*u bi mol_ibj^ ab abqbojfk^o rk^ q^i plir`fŽk bp rk kmj]g`h\ _`q\gjm`n did^d\g`n, Dpq^ qbojfklildŒ^ pb lofdfkŽ bk i^ Lb`Škf`^+ bk alkab+ `ljl bkbi bgbjmil ^kqboflo+ bi s^ilo ^pfdk^al obmobpbkq^i^ mlpf`fŽk bk rk `fboql fkpq^kqbfkf`f^i-

Bljbkw^objlp krbpqol bpqrafl ab i^p b`r^`flkbp afcbobk`f^ibp `lk rk `^plm^oqf`ri^o fjmloq^kqb-

0&+ :PbNPVp[QVSR_R[PVNYQR]_VZR_\_QR[ ]N_NYNSb[PVp[Rd]\[R[PVNY

K^ crk`fŽk bumlkbk`f^i bp fdr^i ^ pr molmf^ abofs^a^+ v il jfpjl bp sŠifalm^o^ `r^inrfbo molar`ql ab rk^ crk`fŽk bumlkbk`f^i mlo rk^ `lkpq^kqb- Dp cŠ`fiabjlpqo^o nrb ‹p^p plk i^p •kf`^p crk`flkbp nrb p^qfpc^`bk bp^ molmfba^a bk qlalbi bgb ob^i-

RCMPCK? 7-0- Pd B `n pi iˆh`mj m`\g _\_j* `sdno` pi\ t n‡gj pi\ api^d‡i alp` n\odna\^` g\ `^p\^d‡i _da`m`i^d\g

a%&s'<a&s'

k\m\ oj_j s m`\g t lp` n\odna\^` o\h]d„i g\ ^ji_d^d‡i did^d\g a&L'< B- Bno\ api+^d‡i qd`i` _\_\ kjm g\ a‡mhpg\

a&s' < @`!,

A`hjnom\^d‡i, Dp cŠ`fi `ljmol_^o nrb i^ crk`fŽk a&s' < @`! p^qfpc^`b i^b`r^`fŽk afcbobk`f^i v i^ `lkaf`fŽk fkf`f^i a^a^p- Sbkbjlp nrb abjlpqo^o ^elo^nrb ‹pq^ bp i^ ˆid^\ plir`fŽk-

Rb^ t < a%r& rk^ plir`fŽk `r^inrfbo^ ab bpqb mol_ibj^ ab s^ilobp fkf`f^ibp9

a$%r&< a%r& m^o^ qlal r d'N( < _-

Page 397: Calculus

B^p\^dji`n _da`m`i^d\g`n gdi`\g`n _` kmdh`m jm_`i ,00

Prbobjlp abjlpqo^o nrb b&s' < @`*w9l nrb b&s'`+T < B- Blkpfabobjlp i^ crk`fŽkc&s' < b&s'`+U v abjlpqobjlp nrb pr abofs^a^ pfbjmob bp `bol- K^ abofs^a^ ab csfbkb a^a^ mlo

b$%r&< a$%r&_*T* a%r&_*T < _*TWa$%r&* a%r&Y< N-

Krbdl+ pbd•k bi qblobj^ ab i^ abofs^a^ kri^+ b bp `lkpq^kqb- Obol d'N( < B mlo ilnrb b%K&< c'M(aM < B- Olo q^kql+ qbkbjlp b%r& < B m^o^ qlal r il `r^i pfdkfcf,`^ nrb a%r& < ?_!) `ljl pb abpb^_^ abjlpqo^o-

Di qblobj^ 7-0 bp rk bgbjmil ab qblobj^ ab bufpqbk`f^ v rkf`fa^a- Mlp af`bnrb bi mol_ibj^ ab s^ilobp fkf`f^ibp a^al od`i` rk^ plir`fŽk 'bufpqbk`f^( v nrbqfbkb pi\ pli^ plir`fŽk 'rkf`fa^a(- Di l_gbql ab do^k m^oqb ab i^ fksbpqfd^`fŽk bki^ qbloŒ^ ab i^p b`r^`flkbp afcbobk`f^ibp bp abp`r_ofo qblobj^p ab bufpqbk`f^ vrkf`fa^a m^o^ `i^pbp ^jmif^p ab b`r^`flkbp-

Rbdrfa^jbkqb `ljbkq^jlp rk qfml fjmloq^kqb nrb fk`irvb i^ b`r^`fŽk afcb,obk`f^i v&< M%r& v i^ b`r^`fŽk v&< v `ljl `^pl m^oqf`ri^o-

0&, :PbNPV\[R`QVSR_R[PVNYR`YV[RNYR`QR]_VZR_\_QR[

Tk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^

'7-4( t%* L%r&s< M%r&)

bk alkab M v O plk crk`flkbp a^a^p+ pb abkljfk^ b`r^`fŽk afcbobk`f^i gdi`\g _`kmdh`m jm_`i, Klp q‹ojfklp nrb `lkqfbkbk i^ crk`fŽk fk`Ždkfq^ v v pr abofs^a^ v&^m^ob`bk `ljl rk^ `lj_fk^`fŽk ifkb^i ab u b t%, K^p crk`flkbp M v P pb prmlkbk`lkqfkr^p bk rk `fboql fkqbos^il ^_fboql g, U^jlp ^ _rp`^o qla^p i^p plir`flkbpv abcfkfa^p bk g,

Blkpfabobjlp mofjbol bi `^pl m^oqf`ri^o bk bi nrb bi pbdrkal jfbj_ol+M%r& bp fa‹kqf`^jbkqb kril- K^ b`r^`fŽk

'7-5( t% * L%r&s< M

pb ii^j^ b`r^`fŽk cjhjb„i`\ l m`_p^d_\ `loobpmlkafbkqb ^ i^ '7-4(- Qbplisbobjlpi^ b`r^`fŽk eljld‹kb^ v irbdl rqfifw^objlp bi obpriq^al m^o^ obplisbo i^ b`r^`fŽkkl eljld‹kb^ '7-4(-

Rf v kl bp kri^ bk &" i^ b`r^`fŽk '7-5( bp bnrfs^ibkqb ^ i^ b`r^`fŽk

%5+4& t% < *L%r&V

Dpql bp+qla^ v kl kri^ nrb p^qfpc^d^ '7-5( p^qfpc^`b q^j_f‹k '7-6( v ob`Œmol`^jbkqb-

Page 398: Calculus

,01 Ehnli^o]]cƒh [ f[m _]o[]cih_m ^c`_l_h]c[f_m

Rrmlkd^jlp ^elo^ nrb v bp rk^ crk`fŽk mlpfqfs^nrb p^qfpc^`b'7-6(- Orbpql nrb bi`l`fbkqb v&et bp i^ abofs^a^ ab ild t* i^ b`r^`fŽk '7-6( pb `lksfboqb bk @ild v <: *L%r&) ab i^ nrb obpriq^ ild t < , `L%r& r * a+ `lk il `r^i qbkbjlp

'7-7( s < _*=?r&) alkab =%r& < G L%r& ^r * `-

Dp ab`fo+ pf bufpqbrk^ plir`fŽk mlpfqfs^ ab '7-5(+ kb`bp^of^jbkqb ab_b qbkbo i^cloj^ '7-7( m^o^ rk `fboql s^ilo ab B- Qbpriq^ ^elo^ cŠ`fi `ljmol_^o nrb qla^crk`fŽk '7-7( bp rk^ plir`fŽk ab i^ b`r^`fŽk eljld‹kb^ '7-5(- Dk bcb`ql+qbkbjlp

s$ < Z_*=?r&=$%r& < ZL%r&_*=%r& < *L%r&s +

@pŒnrb+ ebjlp bk`lkqo^al qla^p i^p plir`flkbp mlpfqfs^pab '7-5(- Blk biil+ ob,priq^ pbk`fiil bumobp^oqla^p i^p plir`flkbp- Dpq^_ib`bjlp bi obpriq^al `ljl rkqblobj^ ab bufpqbk`f^ v rkf`fa^a-

RCMPCK? 7-1- Oojiha[gim O ]ihncho[ _h oh chn_lp[fi [\c_lni g, Afcd[gim

oh johni ]o[fkoc_l[ [ _h g u m_[ \ oh h„g_li l_[f ]o[fkoc_l[+ Arcmn_ _hnih]_m

oh[ v mƒfi oh[ `oh]cƒh v < `%r& ko_ m[ncm`[]_ _f jli\f_g[ ^_ p[fil_m chc]c[f_m

'7-8( s$ * L%r&s< N+ ]ih a&\' < ] *

_h _f chn_lp[fi g,Amn[ `oh]cƒh pc_h_ ^[^[ jil f[ `ƒlgof[

'7-0/( `%r& < \_*=?r& ) ^ih^_ =%r& < oL%n& n +

@_gimnl[]cƒh+ Rb^ ` i^ crk`fŽk abcfkfa^ mlo '7-0/(- Dkqlk`bp =%[& < N `lkil nrb `%[& < \_! < \+ K^ abofs^`fŽk klp e^`b sbo nrb ` p^qfpc^`bi^ b`r^`fŽk af,cbobk`f^i '7-8(+ mlo il nrb ` bp rk^ plir`fŽk abi mol_ibj^ ab s^ilobp fkf`f^ibp-Sbkbjlp ^elo^ nrb mol_^o nrb bp i^ •kf`^ plir`fŽk-

Rb^ d rk^ plir`fŽk `r^inrfbo^- Prbobjlp abjlpqo^o nrb a%r& < \_*=%F$&l nrba%r&_>

&!' < \+ Olo q^kql bp k^qro^i fkqolar`fo b%r& < a%r&wR&+K^ abofs^a^ ab b

sfbkb a^a^ mlo

'7-00( b$%r&< a$%r&_=?T&* a%r&_=%r&=$%r&< _=%r&Wa$%r&* L%r&a%r&Y+

Orbpql nrb d p^qfpc^`bi^ b`r^`fŽk afcbobk`f^i '7+8(+qbkbjlp a$%r&* L%r&a%r&< Nbk qlal g*irbdl b$%r&< N m^o^qlal r ab g,Dpql pfdkfcf`^nrb b bp `lkpq^kqb bk g,Rb qfbkb mrbp+ b%r& < b%[& < a%[&w%[&< a%[& < \+ Cf`el ab lqo^ j^kbo^a%r&w%T&< \) ab j^kbo^ nrb a%r& < \_*=%r&) il `r^i abjrbpqo^ nrb d < `+ Dpql`ljmibq^ i^ abjlpqo^`fŽk-

Page 399: Calculus

B^p\^dji`n _da`m`i^d\g`n gdi`\g`n _` kmdh`m jm_`i 268

K^ •iqfj^ m^oqbab i^ abjlpqo^`fŽk ^kqboflo prdfbob rk j‹qlal m^o^obplisboi^ b`r^`fŽk afcbobk`f^i kl eljld‹kb^ '7-4(- Rrmlkd^jlp nrb d bp rk^ crk`fŽknrb p^qfpc^d^ '7-4( v mlkd^jlp c&s' < b&s'`>&U' bk alkab+ `ljl ^kqbp+=%r& < Iz L%n& n) Dkqlk`bp i^ b`r^`fŽk '7-00( q^j_f‹k bp sŠifa^+ mbol v^ nrb dp^qfpc^`b'7-4(+ i^ cŽojri^ klp a^ m^o^b$%r&)

c%&s' < `>&U'N&s' ,

Qb`loa^kal ^elo^ bi pbdrkal qblobj^ crka^jbkq^i bp`of_fjlp

c&s' < c&\' * o`>/a'N&o' _o ,$[

Olo q^kql+v^ nrb c&\' < b&\'* qla^ plir`fŽk d ab '7-4( qfbkb i^ cloj^

'7-01( b&s' < `+>&s'c&s' < b&\'`+>&s' * `+>&s' oN&o'`>&o' _o ,

Qb`Œmol`^jbkqb+mlo abofs^`fŽk afob`q^ ab '7-01(+ bp cŠ`fi `ljmol_^o nrb `^a^rk^ ab bp^pd bp plir`fŽk ab '7-4(+ `lk il nrb ebjlp bk`lkqo^al oj_\n i^p plir,`flkbp- N_qbkbjlp+ mrbp+bi obpriq^al pfdrfbkqb-

RCMPCK? 7-2- Ppkjib\hjn lp` M v O nji ^jiodip\n `i pi dio`mq\gj \]d`m+oj g, Bgde\hjn pi kpioj \ ^p\glpd`m\ `i g v n`\ ] ^p\glpd`m iˆh`mj m`\g, Bsdno``ioji^`n pi\ api^d‡i v pi\ njg\ v < a&s' lp` n\odna\^` `g kmj]g`h\ _` q\gjm`ndid^d\g`n

s$ * L%r&s < M%r&+ ^ji &\' < ]

`i `g dio`mq\gj g,Bno\ api^d‡i qd`i` _\_\ kjm g\ a‡mhpg\

a&s' < ]`+>&s' * `+>&sg nN&o'`>&o' _o *$[

`i _ji_` >&s' < `wM&o'_o*

G^pq^ ^elo^ i^ m^i^_o^~fkqbos^il‚ pfdkfcf`^_^rk fkqbos^il ^`lq^al ab i^ clo,j^ &\*]'*X\* ]Z* X\* ]' l &\*]Z* pfbkal \ ; ], Dp `lksbkfbkqb q^j_f‹k `lkpfabo^ofkqbos^ilp kl ^`lq^alp- Rb obmobpbkq^kjbaf^kqb ilp pŒj_lilp %[) * //(+ ', //+ [&)W[) *//( u ',//+ [Y) v pb abcfkbk abi pfdrfbkqbjlal9

%[) * bk( < ur Hu= [v ) ', //+ [& < ur Gr ; [v)

W[)*&//( < wt Yt = [v) ', //+ \g < vs Gs x \w ,

Page 400: Calculus

27/ Fiomj_p^^dƒi \ g\n `^p\^dji`n _da`m`i^d\g`n

@abjŠp+ `lksfbkb obcbofopbi `lkgrkql ab oj_jn ilp k•jbolp ob^ibp`fqŠkalil `ljlbi fkqbos^il ', //+ * '/(- @pŒmrbp+`r^kal afp`rq^jlp rk^ b`r^`fŽk afcbobk`f^il pr plir`fŽk bk rk fkqbos^il E)pb pl_obkqbkaboŠ nrb Ebp rkl ab ilp krbsb qfmlpabp`ofqlp-

DIDLOKN- G^ii^o qla^p i^p plir`flkbp ab i^ b`r^`fŽk afcbobk`f^i ab mofjboloabk r` * '0 , r&s < `! bk bi fkqbos^il 'N+* '/(-

Oifo]c•h+ Oofjbol pb mlkb i^ b`r^`fŽk bk i^ cloj^ v&* L%r&s< M%r&af,sfafbkal mlo s, Dpql klp a^

'0 ( `

0.g

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`lk 0/ `r^i L%r&< f,r * 0 X M%r&< _0!$,r+ Orbpql nrb L v O plk `lkqfkr^p bkbi fkqbos^il 'N+* '/(+ bufpqbrk^ plir`fŽk •kf`^ t < `%r& nrb p^qfpc^`b`r^inrfbo`lkaf`fŽk fkf`f^i a^a^ ab i^ cloj^ `_[& < ], Dumobp^objlp qla^p i^p plir`flkbp bkcrk`fŽk abi s^ilo fkf`f^i bk bi mrkql \ < 0- Dp ab`fo+ a^al `r^inrfbo k•jbolob^i \) abqbojfk^objlp qla^p i^p plir`flkbp m^o^i^p nrb `%.&< \+

B^i`ri^jlp mofjbol

=%r& <dLgk&o'_o <d%!'z, 0( _o < i^d r * &s + 0( -

Sbkbjlp mlo q^kql `+>'v( < `!%+g+Fjbs< `U+g-s* v `>&o' < o`+8%*`lk il nrbbi qblobj^ 7-2 klp af`b nrb i^ plir`fŽk sfbkb a^a^ mlo i^ cŽojri^

u,i !%+gFU10 `!%+g `!%F%!y&s' < ] [` + * y y o`g+o_o < ] + * ,+, `g _o ;

s sgo T Wi

`!*+g `U `!%+g `0s b!&,q,i: ] + * , &`U+ `' < ] x * , , , -

s s -t s s

Dpql il mlabjlp q^j_f‹k bp`of_fobk i^ cloj^

`/$! * ]_A#?$2 %%%%$

s

pfbkal B < ]`8%+ `, Kl nrb klp a^ qla^p i^p plir`flkbp bk bi fkqbos^il 'N+* '/(-Orbab pbo fkqbobp^kqbbpqraf^o bi `ljmloq^jfbkql ab i^p plir`flkbp `r^kal

s x N-Rf ^molufj^jlp i^ bumlkbk`f^i jbaf^kqb pr mlifkljfl ifkb^i ab S^vilo+ bk,

Page 401: Calculus

Be`m^d^djn 270

`lkqo^jlp nrb _/T < 0 * /r * i%r& v `s < 0 * r * i%r& `r^kal r w N+ `lkil nrb qbkbjlp

y&s' < '0 * B( * '1 * @'s * j&s' < 0 * B * '1 * B( * -%.& +u u

Olo `lkpfdrfbkqb+ pŽil i^ plir`fŽk `loobpmlkafbkqb ^ b < , 0 qfbkab ^ rk iŒjfqbcfkfql `r^kal r w N+pfbkal bpb iŒjfqb 0-

0&- :WR_PVPV\`

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 0 ^i 4+ obplisbo bi mol_ibj^ ab s^ilobp fkf`f^ibp bk bifkqbos^il nrb pb fkaf`^-

0- s$ * 1t < `/T bk ', //+ * //(+ `lk t < N `r^kal s < N-

0, st% + 0t < s3 bk 'N+ * //(+ `lk t < 0 `r^kal s < 0-

1, t% * t q^k s < pbk 0s bk %*w.P) w.P&)`lk t < 1 `r^kal s < N-

2, s$ * st < s0 bk ', //+ * //(+ `lk t < N `r^kal s < N-

^r4- , * s < `/n bk ', //+ * //(+ `lk s < 0 `r^kal o < l-

_o

5- G^ii^o qla^p i^p plir`flkbp ab v&pbk s * v `lp s < 0 bk bi fkqbos^il 'N+$fP&+Cbjlpqo^onrb rk^ bu^`q^jbkqb ab bpq^p plir`flkbp qfbkb iŒjfqb cfkfql `r^kal s x N+X lqo^ il qfbkbq^j_f‹k cfkfql `r^kal s x $fP+

6- G^ii^o qla^p i^p plir`flkbp ab r%r * i(v&* v < r%r * f&/_+s%bk bi fkqbos^il ', 0+ N(-Ool_^o nrb qla^p i^p plir`flkbp qfbkabk ^ N `r^kal s x 0+ X nrb q^k pŽil rk^ ab bii^pqfbkb iŒjfqb cfkfql `r^kal s x N-

7- G^ii^o qla^p i^p plir`flkbp ab v&* v `lq s < 1 `lp s bk bi fkqbos^il 'N+$fP&+Ool_^o nrbbu^`q^jbkqb rk^ ab bp^p q^j_f‹k bp plir`fŽk bk ', //+ * //(-

8- G^ii^o qla^p i^p plir`flkbp ab %r */&%r * 1't% * 1v < %r * f&%r * 1( bk `^a^ rkl abilp fkqbos^ilp pfdrfbkqbp9 ^( ', //+1(: _( '1+2(: b '2+ *//(- Cbjlpqo^o nrb qla^p i^pplir`flkbp qfbkabk ^ rk iŒjfqb cfkfql `r^kal s x 1+ X kfkdrk^ qfbkb iŒjfqb cfkfql `r^kalsx1,

0/- Olkd^jlp m%r&< 'pbk r&,r pf r w N+ X m_K&< 0- Cbcfk^jlp P%r& < Pxm%n&n+ Cbjlp,qo^o nrb i^ crk`fŽk `%r& < rP%r& p^qfpc^`b i^ b`r^`fŽk afcbobk`f^i rs$ * u < r pbk r bkbi fkqbos^il ', //+ * //( v e^ii^o qla^p i^p plir`flkbp bk bpb fkqbos^il- Cbjlpqo^o nrbi^ b`r^`fŽk afcbobk`f^i kl qfbkb plir`fŽk nrb p^qfpc^d^ i^ `lkaf`fŽk fkf`f^i a`L' < 0+ Xbumif`^o mlo nr‹ bpql kl `lkqo^af`b bi qblobj^ 7-2-

00- Ool_^o nrb bufpqb rk^ pli^ crk`fŽk o,`lkqfkr^ bk bi bgb ob^i mlpfqfs^+ q^i nrb

.cU

X&s' < 0 * , X&o'_o[ 0

m^o^ qlal s = N u e^ii^o bpq^ crk`fŽk-01- K^ crk`fŽk ` abcfkfa^ mlo i^ b`r^`fŽk

X&s' < s`&/+*s%'-0+ s`+s%-0 `wm0`o%-0_o

Page 402: Calculus

,1+ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

m^o^s = N qfbkbi^p molmfba^abpab nrb 0( bp `lkqfkr^ bk bi bgbob^i mlpfqfsl+v 1( p^lqfpc^`bi^ b`r^`fŽk

a&s' < 0 , s F7a&o' _o

m^o^qlal s = N-G^ii^o qla^p crk`flkbp `lk bp^palp molmfba^abp-

A]o[]cƒh ^_ >_lhiocffc+ Tk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^ s$* L%r&s < M%r&sh)alkab i kl bp N kf 0+pb ii^j^ b`r^`fŽk ab Aboklriif- Dpq^b`r^`fŽk kl bp ifkb^i ab_fal^ i^ mobpbk`f ab u!+ Di bgbo`f`fl pfdrfbkqbjrbpqo^ nrb pfbjmob mrbab qo^kpcloj^opb bkrk^ b`r^`fŽk ifkb^i ab mofjbo loabk `lk rk^ krbs^ crk`fŽk fk`Ždkfq^ q* alkab q < v!+f < 0 , k-

02- Rb^ e rk^ `lkpq^kqb kl kri^- Rrmlkd^jlp nrb O u O plk `lkqfkr^p bk rk fkqbos^il 0-Rf [ D / X pf ] bp rk k•jbol ob^i `r^inrfbo^+ pb^ p < a%r&i^ •kf`^ plir`fŽk abi mol,_ibj^ ab s^ilobp fkf`f^ibp i$ * eL%r&p < eM%r& bk .) `lk a%[&< \+ Rf h8€ 0 Xf < 0 , h) abjlpqo^o nrb rk^ crk`fŽk v < `%r& kl fa‹kqf`^jbkqb kri^ bk /* bp rk^plir`fŽk abi mol_ibj^ ab s^ilobp fkf`f^ibp

s$ * L%r&s < M%r&sh bk .) `lk a&\'x%< ]

pf v pŽil pf i^ mlqbk`f^ h,‹pfj^ ab ` bp fdr^i ^ d bk f+Dk `^a^ rkl ab ilp Dgbo`f`flp 03 ^i 06+obplisbo bi mol_ibj^ ab s^ilobp fkf`f^ibp bk bi

fkqbos^il nrb pb `fq^-

.1+ s$ * 1s < /_!$sf,/ bk ' , //+ * //(+ `lk t < 1 `r^kal r < N-

.2+ s$ * U < Zs/%T/ * r * 0( bk ', //+ *! //(+ `lk s < 0 `r^kal r < N-.3+ rf$ * /s < 1r0sf,/ bk ', //+ * //(+ `lk s < N `r^kal r < 0-.4+ rf$ * s < v1u1ild r bk 'N+* //(+ `lk s < `r^kal s < 0-

.5+ /rsf$ * 'i * T&s/ < b&!bk 'N+* //(+ `lk '^( s < v: `r^kal r < 0: '_( V < ,v:`r^kal r < 0: %]& rk iŒjfqbcfkfql `r^kal r ,* N-

08- Tk^ b`r^`fŽk ab i^ cloj^ v&* L%r&s* M%T&s/< N%r& pb ii^j^ _]o[]cƒh ^_ Nc]][nc+'Ml pb `lkl`b j‹qlal m^o^obplisbo i^ b`r^`fŽk dbkbo^i ab Qf``^qf-( Cbjlpqo^o nrb pfrbp rk^ plir`fŽk `lkl`fa^ ab bp^ b`r^`fŽk+ bufpqbkbkqlk`bp lqo^p plir`flkbp ab i^cloj^ v < p * g-q* pfbkal q rk^ crk`fŽk nrb p^qfpc^`brk^ b`r^`fŽk ifkb^i ab mofjboloabk-

1/- K^ b`r^`fŽk ab Qf``^qf v&* v * v1 < 1 qfbkbalp plir`flkbp `lkpq^kqbp-O^oqfoab `^a^rk^ ab bp^p v rqfifw^obi Dgbo`f`fl 08 m^o^e^ii^o lqo^p plir`flkbp abi jlal pfdrfbkqb9^( Rf , 1 99o] ; 0+ e^ii^o rk^ plir`fŽk bk ', //+ * //( m^o^ i^ nrb u< ] `r^kals < N-_( Rf ] x 0 l ] ; , 1+e^ii^o rk^ plir`fŽk bk bi fkqbos^il ', //+ * //( m^o^i^nrb O < ] `r^kal s ;.,

0&. 6YTb[\` ]_\OYRZN So`VP\` bR P\[QbPR[ N RPbNPV\[RQVSR_R[PVNYR`QR]_V%ZR_ \_QR[

Dk bpq Rb``fŽk afp`rqfobjlp s^oflp mol_ibj^p cŒpf`lpnrb mrbabk pboclojr,i^alp j^qbjŠqf`^jbkqb `ljl b`r^`flkbp afcbobk`f^ibp-Dk `^a^ `^pl+ i^ b`r^`fŽk

Page 403: Calculus

Mmj]g`h\n a…nd^jnlp` ^ji_p^`i \ `^p\^dji`n _da`m`i^d\g`n _` kmdh`m jm_`i 161

afcbobk`f^i obmobpbkqrk^ pfjmifcf`^`fŽk fab^ifw^a^ abi mol_ibj^ cŒpf`lv pb ii^j^hj_`gj h\o`hƒod^j _`g kmj]g`h\, K^ b`r^`fŽk afcbobk`f^i pb mobpbkq^ ljl rk^qo^ar``fŽk ab rk^ `fboq^ ibv cŒpf`^+q^i `ljl i^ pbdrka^ ibv abi jlsfjfbkql abMbtqlk+ i^ ibv ab i^ ~`lkpbos^`fŽk‚+ bq`- Mrbpqol molmŽpfql nrŒ kl bp grpqfcf`^oi^ bib``fŽk abi jlabil j^qbjŠqf`l pfkl jŠp _fbk abar`fo `lkpb`rbk`f^p iŽdf`^pabi jfpjl- B^a^ jlabil bp pli^jbkqb rk^ ^molufj^`fŽk ab i^ ob^ifa^a+ v prgrpqfcf`^`fŽkmboqbkb`bmolmf^jbkqb ^ i^ `fbk`f^ ^ i^ nrb bi mol_ibj^ `loobpmlkab-Rf,i^ fkqrf`fŽk l i^ bsfabk`f^ bumbofjbkq^i `lk`rboa^k `lk ilp obpriq^alp abar,`falp j^qbjŠqf`^jbkqb+ ^mob`f^jlp nrb bi jlabil klp obpriq^ •qfi- Rf kl bp ^pŒ+fkqbkq^jlp bk`lkqo^o rk jlabil jŠp `lksbkfbkqb-

DIDLOKN 0- A`ndio`bm\^d‡i m\_d\^odq\, @rknrb ilp afpqfkqlp bibjbkqlp o^,af^`qfslp mobpbkq^kafcbobk`f^pklq^_ibp bk prp `lbcf`fbkqbp ab abpfkqbdo^`fŽk+qla^pi^p prpq^k`f^p qfbkbk i^ molmfba^a `lj•k ab nrb i^ sbil`fa^a ab abp`ljmlpf`fŽkab rk^ abqbojfk^a^ prpq^k`f^ bk `^a^ fkpq^kqbbp molmlo`flk^i ^ i^ `^kqfa^a abprpq^k`f^ bufpqbkqbbk ^nrbi fkpq^kqb-Rf pb abpfdk^ mlo t < `%n&i^ `^kqfa^a abprpq^k`f^ o^af^`qfs^ bufpqbkqbbk bi fkpq^kqbn) i^ abofs^a^ t% < d$%n&obmobpbkqi^sbil`fa^a ab `^j_fl ab t bk bi fkpq^kqbo v i^ ibv ab abp`ljmlpf`fŽk bumobp^9

s$< *es)

alkab f bp rk^ `lkpq^kqb mlpfqfs^ 'ii^j^a^ ^jino\io` _` _`ndio`bm\^d‡i' `rvl s^iloabmbkab abi bibjbkql m^oqf`ri^o nrb pb bpqŠabp`ljmlkfbkal- Di pfdkl jbklp bpab_fal ^ nrb t ab`ob`b `r^kal o `ob`b+v mlo q^kql t% bp pfbjmob kbd^qfsl- K^ b`r^,`fŽk afcbobk`f^i t% < , ft bp bi jlabil j^qbjŠqf`l rqfifw^al m^o^mol_ibj^p ob,i^qfslp ^ abpfkqbdo^`fŽko^af^`qfs^- Sla^ plir`fŽk v < `%n&ab bpq^b`r^`fŽk afcb,obk`f^i qfbkb i^ cloj^

'7-02( aR' < a&L'`+en

Olo `lkpfdrfbkqb+ m^o^abqbojfk^o i^ `^kqfa^a mobpbkqbbk bi fkpq^kqbo*kb`bpfq^jlp`lkl`bo i^ `^kqfa^a fkf`f^i `_K& v bi s^ilo ab i^ `lkpq^kqb ab abpfkqbdo^`fŽke+

Dp fkqbobp^kqbsbo nr‹ fkcloj^`fŽk pb mrbab abar`fo ab '7-02(+ pfk `lkl`bobu^`q^jbkqb bi s^ilo ab `_K& l ab e+ Dk mofjbo ird^o pb l_pbos^ nrb m^o^kfkd•ks^ilo cfkfql abi qfbjml o pb ^kri^ `%n&mrbpql nrb i^ bumlkbk`f^i `+fo bp pfbjmobmlpfqfs^: mlo q^kql+kl pb mrbab e^_i^o ab ~qfbjml qlq^i ab sfa^‚ ab rk^ prpq^k`f^o^af^`qfs^- Rfk bj_^odl+ bp mlpf_ib abqbojfk^o bi qfbjml kb`bp^ofl m^o^ nrb pbabpfkqbdobrk^ am\^^d‡i ab i^ jrbpqo^- Eob`rbkqbjbkqb pb bifdb i^ co^``fŽk i+ v biqfbjml P bk bi `r^i `%P&,`%K&< z pb abkljfk^ pc^[ g_^c[ ab i^ prpq^k`f^+nrb bpbi qfbjml kb`bp^ofl m^o^ nrb i^ j^p^ ab i^ prpq^k`f^ o^af^`qfs^ pb obarw`^ ^ i^

Page 404: Calculus

273 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

jfq^a- Dpqbs^ilo ab Q pb mrbab abqbojfk^o obplisfbkal i^ b`r^`fŽk `+fQ < qobp,mb`ql Q, Slj^kal ild^ofqjlp pb qfbkb +fQ < ,ild 1 Ž Q < 'ild 0'-f, Orbpqlnrb bp9

a&o * P& a&L'`+f&o)Q' +qm 0; ;` <,

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EHFTQ@ 7-0 A`ndio`bm\^d4i m\_d\^odq\^ji qd^g\h`_d\ Q,

pb sb nrb i^ sfa^ jbaf^ bp i^ jfpj^ `r^inrfbo^ nrb pb^ i^ jrbpqo^ ab rk j^qbof^io^af^`qfsl a^al- K^ cfdro^ 7-0 a^ rk^ fab^ dbkbo^i ab i^ cloj^ ab rk^ `ros^ ababpfkqbdo^`fŽko^af^`qfs^-

DIDLOKN 1- @\…_\ _` pi ^p`mkj `i pi h`_dj m`ndno`io`, Tk `rboml bkobmlpl ab j^p^ h bp i^kw^al ^ do^k ^iqro^ bk i^ ^qjŽpcbo^ qboobpqob-Rrmrbpql nrb`^b bk iŒkb ob`q^ v nrb i^p •kf`^p crbow^pnrb ^`q•^k pl_ob ‹i plk i^ ab i^ do^,sba^a qboobpqob%ga) alkab d bp i^ ^`bibo^`fŽk ab,i^ do^sba^a+ prmrbpq^`lkpq^kqb(v rk^ crbow^obpfpqbkqb'ab_fa^ ^ i^ obpfpqbk`f abi ^fob( nrb bp molmlo`flk^i ^ prsbil`fa^a+ pb qo^q^ab bpqraf^o bi jlsfjfbkql obpriq^kqb-

Rb^ n < a&o'i^ afpq^k`f^ ob`loofa^ mlobi jŽsfi bk bi fkpq^kqbo v pb^ q < n%:: a&o' pr sbil`fa^a- Cb i^ efmŽqbpfpab nrb m^oqbabi obmlpl pb abar`b .$%-&< N-

G^v alp crbow^pnrb ^`q•^k pl_ob bi `rboml+ rk^ abp`bkabkqb ga ab_fa^ ^ prmbpl v lqo^ ^p`bkabkqb , fq 'ab_fa^ ^ i^ obpfpqbk`f abi ^fob( alkab f bp rk^

Page 405: Calculus

Mmj]g`h\n a…nd^jnlp` ^ji_p^`i \ `^p\^dji`n _da`m`i^d\g`n _` kmdh`m jm_`i 163

`lkpq^kqb mlpfqfs^- K^ pbdrka^ ibv ab Mbtqlk af`b nrb i^ prj^ ab i^p crbow^pnrb ^`q•^k bk rk `rboml bk `^a^ fkpq^kqbbp fdr^i ^i molar`ql ab pr j^p^ g

mlo pr ^`bibo^`fŽk- Rf pb fkaf`^ mlo [ i^ ^`bibo^`fŽk bk bi fkpq^kqbo*bkqlk`bp\ < q%< n! v i^ ibv ab Mbtqlk a^ i^ b`r^`fŽk

h\ < hb + fq,

…pq pb mrbab `lkpfabo^o `ljl rk^ b`r^`fŽk afcbobk`f^i ab pbdrkal loabk pf pb`lkpfabo^ i^ crk`fŽk ab abpmi^w^jfbkql n l ab mofjbo loabk pf pb `lkpfabo^ i^crk`fŽk sbil`fa^a q, Bljl b`r^`fŽk ab mofjbo loabk bk q* bp ifkb^i v mrbab bp,`of_fopbbk i^ cloj^

G Qq )+q;b,

g

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'7-03(

N_p‹osbpb nrb q x hba f `r^kal o,* * ^j, Rf abofs^jlp i^ b`r^`fŽk '7-03(+ bk,`lkqo^jlp nrb i^ ^`bibo^`fŽk bk qlal fkpq^kqbbp \ < b`+fo/hŠ @pfjfpjl \ x N`r^kal o,* * ^j, Hkqbomobq^alcŒpf`^jbkqb+bpql pfdkfcf`^ nrb i^ obpfpqbk`f^abi^fob qfbkab ^ bnrfif_o^o i^ crbow^ab i^ do^sba^a-

Orbpql nrb q < n%*i^ b`r^`fŽk '7-03( bp ^ pr sbw rk^ b`r^`fŽk afcbobk`f^i bki^ crk`fŽk ab abpmi^w^jfbkql n* nrb mrbab fkqbdo^opbafob`q^jbkqb obpriq^kal9

ga ag/

p < e o * h1 `+fo-h * b-

Orbpql nrb p < N `r^kal o < N pb qfbkb b < , ag! df0 obpriq^kal i^ b`r^`fŽkabi jlsfjfbkql9

1

R < ga o * ag %_*en,h [ 0( -Q Q/

Rf i^ sbil`fa^a fkf`f^i bp Rj `r^kal o < N+ i^ cŽojri^ '7-03( m^o^i^ sbil`fa^abk bi qfbjml o pb e^ ab prpqfqrfomlo

ga '0 +fo-h' * +fo-hR * * * ` qj`,' Q

Page 406: Calculus

275 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Dp fkqbobp^kqbklq^o nrb m^o^oj_\ sbil`fa^a fkf`f^i 'mlpfqfs^+kbd^qfs^ l `bol( i^sbil`fa^a iŒjfqb`r^kal o `ob`b fkabcfkfa^jbkqb bp ga` e) k•jbol fkabmbkafbkqbab qj, Di ib`qlo ab_b _rp`^o i^ bumif`^`fŽk ab bpqbeb`el bk o^wlkbp ab `^oŠ`qbocŒpf`l-

DIDLOKN 2- Ri kmj]g`h\ nj]m` `iamd\hd`ioj, Di `lbcf`fbkqb ab s^k^`flkab i^ qbjmbo^qro^ ab rk `rboml bp molmlo`flk^i ^ i^ afcbobk`f^ bkqobpr qbjmbo^,qro^ v i^ abi jbafl ^j_fbkqb- &I`t _` `iamd\hd`ioj _` K`roji,' Rf t < a&o'bpi^ qbjmbo^qro^ 'abp`lkl`fa^( abi `rboml bk bi fkpq^kqbo v I%n&abpfdk^ i^ qbjmbo^,qro^ '`lkl`fa^( abi jbafl ^j_fbkqb+ i^ ibv ab Mbtqlk `lkar`b ^ i^ b`r^`fŽk af,cbobk`f^i

'7-04( s$ < *eWs * I%n&Y l s$ * es < e I%n&)

pfbkal e rk^ `lkpq^kqb mlpfqfs^-Dpq^b`r^`fŽk ifkb^i ab mofjbo loabk bp bi jl,abil j^qbjŠqf`l nrb rp^jlp m^o^ilp mol_ibj^p ab bkcof^jfbkql- K^ •kf`^ plir`fŽkab i^ b`r^`fŽk nrb p^qfpc^`bi^ `lkaf`fŽk fkf`f^i `%[& < ] sfbkb a^a^ mlo i^cŽojri^

'7-05( a&o' < ]`8! * `8! `7fJ&p'`eQ _p *

Blkpfabobjlp ^elo^ rk `^pl m^oqf`ri^obk bi nrb bi `rboml m^p^ab 1//• ^ 0//•bk 3/ jfkrqlp ^i pbo prjbodfal bk rk jbafl `rv^ qbjmbo^qro^ pb j^kqfbkb `lkp,q^kqb+pb^ mlo bgbjmil J&o' < 0/•+ Rf jbafjlp o bk jfkrqlp v a&o'bk do^alp+ qbkb,jlp a&L' < 1// X i^ b`r^`fŽk '7-05( klp a^

'7-06( , age&o' < 0..`+en * /.f`+ef l `eo _p :

: 0..`+ef * 0/'0 , `+en&< 0/ * /7.`+en†

Olabjlp `^i`ri^o e ^ m^oqfoab i^ fkcloj^`fŽk ab nrb `%1-&< 0//- Olkfbkal bk&6,/5' o < 3/+ bk`lkqo^jlp 8/ < /7.`+1in

) `lk 0/ nrb +2.f < ild '8/.08/(+e < fl'0ld 08 , ild 8(-

Rbdrfa^jbkqb+ `^i`ri^jlp bi qfbjml nrb kb`bpfq^ bpqbjfpjl j^qbof^i m^o^bkcof^opbab 1//• ^ 0//• pf i^ qbjmbo^qro^ abi jbafl pb j^kqfbkb ^ R•- Dkqlk`bpi^ b`r^`fŽk '7-05( bp sŠifa^ `lk i^ jfpj^ `lkpq^kqb e mbol `lk I%o& < R- Dk ir,d^o ab '7-06(+ mlkbjlp i^ cŽojri^

a&o' < 4 * /73`+en†

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Mmj]g`h\n a…nd^jnlp` ^ji_p^`i \ `^p\^dji`n _da`m`i^d\g`n_` kmdh`mjm_`i 165

O^o^ bk`lkqo^o bi fkpq^kqb n m^o^ bi `r^i `%n&< 0//+ mlkbjlp 84 < /73`+fo* `lk

il nrb +fo< ild '84.084( < ild /8.28(+ X mlo q^kql

o < 0'ild 28 ] i^d 08( < 3/ i^d 28 , i^d 08f i^d 08 , i^d 8

Dk rk^ q^_i^ ab ild^ofqjlp `lk `r^qol `fco^p ab`fj^ibp+ bk`lkqo^jlp ild 28 << 2+5525+ i^d 08 < 1+8333+ W i^d 8 < 1+0861 `lk il nrb+ ^molufj^kal+ bk`lkqo^-jlp o < 3/'/+608(.'/+636( < 27+4 jfkrqlp-

K^ b`r^`fŽk afcbobk`f^i '7-04( bumobp^ nrb i^ sbil`fa^a ab bkcof^jfbkqlab`ob`b `lkpfabo^_ibjbkqb `r^kal i^ qbjmbo^qro^ abi `rboml qfbkab ^ ^`bo`^opb ^i^ qbjmbo^qro^ abi jbafl- Bljl bgbjmil+ pb mrbab _rp`^o bi qfbjml kb`bp^oflm^o^ bkcof^o i^ jfpj^ prpq^k`f^ ab 0/// ^ 0// `lk bi jbafl `lkpq^kqbjbkqb^ 3., Di `Ši`ril `lkar`b ^ ild '4.84( < , fo* l

n < 0i^d 08 < 3/ i^d 08 < 3/'1-833( < 047 jfkrqlp-f i^d 08 , i^d 8 /-636

N_p‹osbpb nrb bi abp`bkpl ab qbjmbo^qro^ ab 0/// ^ 0// kb`bpfq^ rk qfbjml nrbbu`bab ^ `r^qol sb`bp bi qfbjml kb`bp^ofl m^o^ m^p^o ab 1//

/^ 0///-

DIDLOKN 3- Ri kmj]g`h\ _` _dnjgp^d‡i, Tk abmŽpfql `lkqfbkb 0// 0 abrk^ afplir`fŽk p^ifk^ `rv^ `lk`bkqo^`fŽk bp 1+4 d ab p^i mlo ifqol- Tk^ afplir`fŽk`lkqbkfbkal 1 d ab p^i mlo ifqol bkqo^ bk bi abmŽpfql ^ o^wŽk ab 4 0 mlo jfkrqlv i^ jbw`i^ 'nrb pb e^`b rkfclojb mlo bi jlsfjfbkql( p^ib ^ i^ jfpj^ sbil`fa^a-Dk`lkqo^o i^ `^kqfa^a ab p^i nrb e^v bk `^a^ fkpq^kqb bk bi abmŽpfql-

Rb^ v < `Q& bi k•jbol ab do^jlp ab p^i nrb e^v bk bi abmŽpfql o jfkrqlpabpmr‹p ab e^_bo `ljbkw^al i^ jbw`i^- G^v alp c^`qlobp nrb molar`bk i^ s^of^,`fŽk ab v+ i^ afplir`fŽk nrb ^dobd^ p^i ^ o^wŽk ab 0/ d mlo jfkrql v i^ jbw`i^nrb p^ib nrb afpjfkrvb i^ `^kqfa^a ab p^i ^ o^wŽk ab 2%s.0//( do^jlp mlo jfkrql-'K^ co^``fŽk t-gLL obmobpbkq^i^ `lk`bkqo^`fŽk bk bi qfbjml n+&Olo q^kql i^ b`r^,`fŽk afcbobk`f^i bp9

t%< 0/ , -jt l t%* djt < 0/-

Dpq^ b`r^`flk ifkb^i bp bi jlabil j^qbjŠqf`l m^o^ krbpqol mol_ibj^- X^ nrbv < 14/ `r^kal o < N+ i^ •kf`^ plir`fŽk sfbkb a^a^ mlo i^ cŽojri^

'7-07( s < 14/b,p.1/ * b,p.

1/ F70/bq.1/ _p < 1// * 3.`+o-/-

Dpq^ b`r^`fŽk jrbpqo^ nrb v = 1// m^o^ qlal o v nrb v z 1// `r^kal o `ob`bfkabcfkfa^jbkqb- Krbdl bi jŒkfjl ab `lkqbkfal ab p^i bp 1// d+ il nrb q^j_f‹k

Page 408: Calculus

277 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

er_fbo^ mlafal abar`fopb abi bkrk`f^al abi mol_ibj^- Dk i^ b`r^`fŽk '7-07( pbmrbab abpmbg^o o bk crk`fŽk ab t l_qbkf‹kalpb9

o < 1/ i^d ' 4/ (-u , 1//

Dpq^ b`r^`flk mbojfqb&bk`lkqo^o bi qfbjml bk bi nrb i^ p^i `lkqbkfa^ pb^ rk^abqbojfk^a^ `^kqfa^a t* pfbjmob nrb 1// ; t ; 14/-

DIDLOKN 4- @dm^pdojn`g„^omd^jn, Dk i^ cfdro^ 7-1'^(+ ^m^ob`b rk^ crbo,w^ bib`qoljlqofw+ rk^ obpfpqbk`f^+ v rk^ ^rqlfkar``fŽk `lkb`q^a^p bk pbofb-K^ crbow^ bib`qoljlqofw molar`b rk sliq^gb nrb lofdfk^ rk^ `loofbkqb bi‹`qof`^nrb ob`loob bi `fo`rfql- Rf bi ib`qlo kl bpqŠc^jfif^ofw^al `lk ilp `fo`rfqlp bi‹`qof`lp+kl ab_b mobl`rm^opb- O^o^ krbpqol l_gbql+ qlal il nrb mob`fp^jlp `lkl`bo ^`bo`^abi `fo`rfql bp nrb bi sliq^gb+ abpfdk^al mlo R%n&)v i^ fkqbkpfa^a ab i^ `loofbkqb+abpfdk^a^ mlo F&o'*plk crk`flkbp abi qfbjml o ifd^a^p mlo rk^ b`r^`fŽk afcbobk`f^iab i^ cloj^

'7-08( IF%&o'* OF&o'< S&o',

@nrŒ I X O pb prmlkbk `lkpq^kqbp mlpfqfs^p- Rb ii^j^k obpmb`qfs^jbkqb+ i^di_p^o\i^d\ v i^ m`ndno`i^d\ abi `fo`rfql- K^ b`r^`fŽk afcbobk`f^i bp rk^ clojri^,`fŽk j^qbjŠqf`^ ab rk^ ibv ab `lkpbos^`fŽk+ ii^j^a^ g`t _`g qjgo\e` _` Hdm^ccjaa*v pfosb `ljl jlabil j^qbjŠqf`l m^o^ bi `fo`rfql-

@nrbiilp ib`qlobp kl sbop^alp bk `fo`rfqlp mrbabk bk`lkqo^o •qfi fj^dfk^onrb i^ `loofbkqb bp `ljl bi ^dr^ nrb `fo`ri^ mlo rk qr_l- K^ crbow^ bib`qoljlqofw'loafk^of^jbkqb _^qboŒ^l dbkbo^alo( bp ^kŠild^ ^ rk^ _lj_^ nrb e^`b cirfo bi^dr^: i^ obpfpqbk`f^ pb m^ob`b ^ i^ cof``fŽk bk bi qr_l+ nrb qfbkab ^ lmlkbopb ^icirgl ab `loofbkqb: u i^ fkar`q^k`f^ bp rk^ fkcirbk`f^ bpq^_fifw^alo^ nrb qfbkab ^fjmbafo `^j_flp _orp`lp bk i^ `loofbkqb ab_falp ^ s^of^`flkbp p•_fq^p bk bi sliq^gb-

Di qfml `loofbkqb ab mobdrkq^p obi^qfs^p ^ q^ibp `fo`rfqlp bp bpqb9 Rf pb ^mif`^bk bi `fo`rfql rk `fboql sliq^gb R%n&)ƒ`rŠi bp i^ fkqbkpfa^a obpriq^kqb E%n&<K^ pl,ir`fŽk pb `lkpfdrb jbaf^kqb rk^ b`r^`fŽk afcbobk`f^i ifkb^i ab mofjbo loabk-Rf .%-& obmobpbkq^i^ fkqbkpfa^a fkf`f^i bk bi fkpq^kqb o < N+ i^ b`r^`fŽk qfbkb i^plir`fŽk

F&o'< F&L'`+Oo-I* `+OogIbRl& `Os-I _s ,

Tk `^pl m^oqf`ri^o fjmloq^kqb pb mobpbkq^ `r^kal bi sliq^gb ^mif`^al bp `lkp,q^kqb+mlo bgbjmil R%n&< A m^o^ qlal o, Dk bpqb `^pl+ i^ fkqbdo^`fŽk obpriq^ cŠ`fiv klp `lkar`b ^ i^ cŽojri^

F&o'< y * '0'/( ] x'`+Oo-I

Page 409: Calculus

Mmj]g`h\n a…nd^jnlp` ^ji_p^`i \ `^p\^dji`n _da`m`i^d\g`n _` kmdh`m jm_`i 167

. &o'

Hkar`qlo

Hkqbkpfa^a Bab i^ `loofbkqb . 'N( = Fc

Hkqbkpfa^aab i^ `loofbkqb / 'N( < nc

O

yBN

Erbow^bib`qoljlqofw

. %K&Hkqbkpfa^a Bab i^ `loofbkqb . '/(; O

Qbpfpqbk`f^

']( '^(

EHFTQ@ 7-1 ]( Ad\bm\h\ k\m\ pi ^dm^pdojndhkg` `i n`md`,_( Fio`ind_\_ m`npgo\io \g \kgd^\mpi qjgo\e` ^jino\io` B,

Dpql abjrbpqo^ nrb i^ k^qro^ibw^ ab i^ plir`fŽk abmbkab ab i^ obi^`fŽk bkqobi^fkqbkpfa^a fkf`f^i 0'/( v bi `l`fbkqb B-O, Rf 0'/( < B- O* bi q‹ojfkl bumlkbk`f^ikl ^m^ob`bv i^ fkqbkpfa^a bp `lkpq^kqb+g&o'< B-O, Rf 0'/( = B-O* bi `lbcf`fbkqbabi q‹ojfkl bumlkbk`f^i bp mlpfqfsl u i^ fkqbkpfa^a ab`ob`b e^`f^ bi s^ilo iŒjfqbB-O `r^kal n*( *//- Rf 0'/( ; B-O* i^ fkqbkpfa^a `ob`b e^`f^ bi s^ilo iŒjf,qb B-O, K^ `lkpq^kqb B-O`n i^ ^jhkji`io` `no\^dji\md\ ab i^ fkqbkpfa^a+u biq‹ojfkl bumlkbk`f^i Xg&L'+ B-OZ`+Nn

,H bp i^ ^jhkji`io` q\md\]g` ab i^ jfpj^-

U‹^kpb bgbjmilp bk i^ cfdro^ 7-1'_(-Klp bgbjmilp mob`babkqbpe^`bk sbo bi mlabo rkfcf`^alo u i^ rqfifa^a moŠ`qf`^

ab i^p b`r^`flkbp afcbobk`f^ibp-Lrbpqo^k `Žjl afsboplp qfmlpab mol_ibj^p cŒpf`lpmrbabk `lkar`fo bu^`q^jbkqb ^i jfpjl qfml ab b`r^`fŽk afcbobk`f^i-

K^ b`r^`fŽk afcbobk`f^i '7-08( bp ab bpmb`f^ifkqbo‹pab_fal ^ nrb prdfbob i^mlpf_fifa^a ab ^`ljbqbo i^ plir`fŽk ab rk^ ^jmif^ s^ofba^a ab mol_ibj^p cŒpf`lprp^kal jbaflp bi‹`qof`lp- Olo bgbjmil+ prmlkd^jlp rk mol_ibj^ cŒpf`lnrb klp`lkarw`^ ^ rk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^

s$ * \t < M)

pfbkal \ rk^ `lkpq^kqb mlpfqfs^ u P rk^ crk`fŽk `lkl`fa^- Olabjlp fkqbkq^oi^`lkpqor``fŽk ab rk `fo`rfql bi‹`qof`l `lk rk^ fkar`q^k`f^ I v rk^ obpfpqbk`f^O

Page 410: Calculus

28/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

ab j^kbo^ nrb pb^ O- I < \ u bkqlk`bp ^mif`^o rk sliq^gb IN bk bi `fo`rfql-SbkaoŒ^jlp bkqlk`bp rk `fo`rfql bi‹`qof`l `lk bi jfpjl jlabil j^qbjŠqf`l nrbbi mol_ibj^ cŒpf`l-@pŒpb mlaoŒ^kl_qbkbo a^qlp krj‹of`lp ab i^ plir`fŽk abimol_ibj^ cŒpf`l lk jbaf`flkbp ab i^ fkqbkpfa^a bk bi `fo`rfql bi‹`qof`l- Dpq^fab^pb e^ mrbpql bk moŠ`qf`^v e^ `lkar`fal ^i abp^ooliil ab ilp ^\g^pg\_jm`n \i\g‡+bd^jn,

7-6 Dgbo`f`flp

Dk ilp Dgbo`f`flp nrb pfdrbk+ rqfifw^o rk^ b`r^`f5k afcbobk`f^i ab mofjbo loabk ^ab`r^a^`ljl jlabil j^qbjŠqf`l abi mol_ibj^-

0- K^ sfa^ jbaf^ abi o^afl bp ^molufj^a^jbkqb 05// ^•lp- Dk`lkqo^o nr‹ mlo`bkq^gb abrk^ `^kqfa^a a^a^ ab o^afl pb e^ abpfkqbdo^al bk 0// ^•lp-

1- Rf rk^ `bm^ ab _^`qbof^p ^rjbkq^ bk cloj^ molmlo`flk^i ^ i^ `^kqfa^a mobpbkqbu pf i^ml_i^`f5k pb armif`^ bk rk^ elo^+ ƒbk `rŠkql ^rjbkq^oŠ ^i `^_l ab 1 elo^p>

2- Rb^ u < `Q& i^ `^kqfa^a ab rk^ prpq^k`f^ nrb bufpqb bk bi fkpq^kqb o, Rrmlkd^jlp nrbpb abpfkqbdo^ bk cloj^ molmlo`flk^i ^ i^ `^kqfa^a mobpbkqb-Rf i bp rk bkqbol mlpfqfsl+ bik•jbol P m^o^ bi `r^i `%P&< `%K&,h bp i^ sfa^ k,‹pfj^ ab i^ prpq^k`f^-^( Cbjlpqo^o nrb af`el s^ilo Q `n `g hdnhj k\m\ oj_\ hp`nom\ _` pi h\o`md\g _`o`m+hdi\_j* v `^i`ri^o Q bk crk`f5k ab i v ab i^ `lkpq^kqb ab abpfkqbdo^`f5k f,_( Rf \ v \ plk a^alp+ mol_^o nrb ` mrbab bumobp^opbbk i^ cloj^

a&o' < a&\'r:o<y&]'g+T&o'

v abqbojfk^o q%n&+Dpql morb_^ nrb i^ `^kqfa^a mobpbkqbbk bi fkpq^kqb n bp rk^ jbaf^dblj‹qof`^ mlkabo^a^ ab i^p `^kqfa^abp bufpqbkqbp bk alp fkpq^kqbp o < \ v o < ],

3- Tk elj_ob molsfpql ab rk m^o^`^Œa^ppb i^kw^ abpab do^k ^iqro^- Di mbpl `lkgrkql abielj_ob v bi m^o^`^Œa^pbp 87 hd- Rb^ p%n&i^ sbil`fa^a 'bk jbqolp mlo pbdrkal( o pbdrkalpabpmr‹p abi i^kw^jfbkql- Cro^kqb ilp 0/ mofjbolp pbdrkalp ^kqbp ab ^_ofopb bi m^o^`^Œ,a^p pb prmlkb nrb i^ obpfpqbk`f^ abi ^fob bp /+0p%n&hd- Cbpmr‹p+ rk^ sbw ^_fboql bi m^o^,`^Œa^p+i^ obpfpqbk`f^ abi ^fob bp /p%n&hd- Rb prmlkb nrb i^ ^`bibo^`f5k ab i^ do^sba^a bp8+7 j.pd1 v pb qo^q^ ab e^ii^o c5ojri^p bumiŒ`fq^pm^o^ i^ sbil`fa^a p%n&v bi qfbjml n+'Orbab rqfifw^opb i^ ^molufj^`f5k `+P-2 < 26.017 bk ilp `Ši`rilp-(

4- Sbkfbkal bk `rbkq^ bi bgbjmil 1 ab i^ Rb``f5k 7-5+ u rqfifw^kal i^ obdi^ ab i^ `^abk^m^o^ bp`of_fo

_q _n _q _q*:**:p*_o _o _n _n

abjlpqo^o nrb i^ b`r^`f5k afcbobk`f^i abi bgbjmil mrbab bumobp^opbabi jlal pfdrfbkqb9

_n ]q_q;x

bk alkab ] < g` f u _ < bmiaf9 Hkqbdo^obp^ b`r^`f5k m^o^ bumobp^o n bk crk`f5k ab q,Bljm^o^o bi obpriq^al `lk i^p c5jri^p m^o^ q v n abar`fa^p bk bi bgbjmil-

Page 411: Calculus

Bd`m^d^djn 280

5- Llafcf`^o bi bgbjmil 1 ab i^ Rb``fŽk 7-5 prmlkfbkal nrb i^ obpfpqbk`f^ abi ^fob bp mol,mlo`flk^i ^ q0Š Cbjlpqo^o nrb i^ b`r^`fŽk afcbobk`f^i mrbab mlkbopb bk `^a^ rk^ ab i^pcloj^p pfdrfbkqbp9

_n h q_q < e `1 , q/ 8

_o h_q < e `1 , q/$

bk alkab b < Uhb-f, Hkqbdo^o ^a^ rk^ ab bi0^p v l_qbkbo i^p pfdrfbkqbp cŽojri^p m^o^ m8

`]o [ `+]o

q < b ,,,, < b q^ke ]o *`]o * `+]o

bk alkab ] < Sfb-h, Cbqbojfk^o ilp s^ilobp iŒjfqb ab S `r^kal ox * ll +6- Tk `rboml bk rk^ e^_fq^`fŽk ^ 5/• E pb bkcoŒ^ab 1//• E ^ 01/• E bk jbaf^ elo^-

'^( Ool_^o nrb pr qbjmbo^qro^ abpmr‹p ab o jfkrqlp bp 5/ * /2.`+en alkabf < '0ld 6 , ild 2(.2/-'_( Ool_^o nrb bi qfbjml kb`bp^ofl m^o^ ^i`^kw^o i^ qbjmbo^qro^ QLC bpqŠ a^al mlo i^cŽojri^ o< ZHld03/ , ild %P* 3-&Y,e alkab 5/ ; P w 1//-'`( G^i0^o bi fkpq^kqb bk nrb i^ qbjmbo^qro^ bp 8/• E-'a( G^i0^o rk^ cŽojri^ nrb bumobpb bk crk`fŽk ab o i^ qbjmbo^qro^ abi `rboml `r^kali^ qbjmbo^qro^ ab i^ e^_fq^`fŽk kl pb j^kqbkd^ `lkpq^kqb+ pfkl nrb afpjfkrv^ ^ o^wŽkab 0• E `^a^ 0/ jfkrqlp- Rb prmlkaoŠ nrb i^ qbjmbo^qro^ ab i^ e^_fq^`fŽk bp 5/• E`r^kal i^ abi `rboml bp 1//• E-

7- Tk qbojŽjbqol pb j^kqbkŒ^ dr^oa^al bk rk^ e^_fq^`fŽk `rv^ qbjmbo^qro^ bo^ 64• E-Bfk`l jfkrqlp abpmr‹p ab e^_boil p^`^al ab i^ e^_fq^`fŽk bi qbojŽjbqol j^o`^ 54• E-Nqolp `fk`l jfkrqlp abpmr‹p j^o`^ 5/• E- B^i`ri^o i^ qbjmbo^qro^ buqboflo-

8- Tk q^knrb `lkqfbkb 267+42 i ab rk^ afplir`fŽk p^ifk^ l_qbkfa^ ^i afplisbo 11+57 hd abp^i- Olo rk^ bkqo^a^ cirvb ^dr^ ^i q^knrb ^ o^wŽk ab 00+25 i mlo jfkrql j^kqbkf‹kalpbi^ `lk`bkqo^`fŽk rkfclojb mlo jbafl ab ^dfq^alobp- ƒBrŠkq^ p^i e^_oŠ bk bi q^knrb ^i`^_l ab rk^ elo^ pf mlo rk abp^d•b p^ib afplir`fŽk ^ o^wŽk ab 6+46 i mlo jfkrql>

0/- K^p `lkaf`flkbp plk i^p abi Dgbo`f`fl ^kqboflo- Di clkal abi q^knrb bpqŠ `r_fboql `lkrk^ jbw`i^ ab p^i v j^qbof^i fkplir_ib+ v pb prmlkb nrb i^ p^i pb afprbisb `lk rk^ sbil,`fa^a molmlo`flk^i ^ i^ afcbobk`f^ bkqob i^ `lk`bkqo^`fŽk ab i^ plir`fŽk v i^ ab rk^afplir`fŽk p^qro^a^ '251 do^jlp mlo ifqol( v nrb pf bi ^dr^ crbo^ mro^ pb afplisboŒ^342+5 d ab p^i mlo jfkrql- ƒBrŠkq^ p^i e^_oŠ bk i^ plir`fŽk `r^kal e^v^ qo^kp`roofali^ elo^>

00- Blkpfabobjlp rk `fo`rfql bi‹`qof`l m^ob`fal ^i abi Dgbo`f`fl 4 ab i^ Rb``fŽk 7-5- Rrmlk,d^jlp nrb i^ crbow^ bib`qoljlqofw bp rk dbkbo^alo ab `loofbkqb ^iqbok^ nrb molar`b rksliq^gb R%n&< A pbk ro* alkab A v r plk `lkpq^kqbp mlpfqfs^p- Rf .%-&< N+ abjlpqo^o nrbi^ fkqbkpfa^a qfbkb i^ bumobpfŽk

9 9Z<.%.& < +;++;+++[+[+[++,8+[n`i&ro+ |( * ++++`+Oo-IT O0 * r0I0 O0 * r0I0 %

bk alkab `s pŽil abmbkab ab r* I v O, Cbjlpqo^o nrb \8 < N `r^kal I < N-01- Dk bi bgbjmil 4 ab i^ Rb``fŽk 7-5+ prmlkbo nrb bi sliq^gb bp rk^ crk`fŽk bp`^ilk^a^ ab,

cfkfa^ ^pŒ9A%n&< A pf \ x o 9l9: \) pfbkal \ = N: A%n&< N m^o^ `r^inrfbo lqol s^ilo ab o,

Page 412: Calculus

,2+ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Rf .%-&< N abjlpqo^o nrb i^ fkqbkpfa^a sfbkb a^a^ mlo i^p cŽojri^p pfdrfbkqbp9E%n&< M pf o 4 [8

AF&o'< , 'i , `+O&o+\'-I'

O

Apf \ x o x ]9 F&o'< N `+OogI &`O]gI + `O\gI' pf o x ],

G^`bo rk bpnrbj^ fkaf`^kal i^ k^qro^ibw^ ab i^ doŠcf`^ ab g,

@m`^dhd`ioj _` g\ kj]g\^d‡i, Dk bi bpqrafl abi `ob`fjfbkql ab rk^ ml_i^`fŽk 'nrb mrbabpbo erj^kl+ ^kfj^i l _^`qbof^kl(+ i^ crk`fŽk nrb `rbkq^ bi k•jbol s ab fkafsfarlp mob,pbkqbpbk bi fkpq^kqb o bp kb`bp^of^jbkqb rk^ api^d‡i bp`^ilk^a^ nrb pli^jbkqb qlj^ s^ilobpbkqbolp- Olo `lkpfdrfbkqb bi sboa^abol ^j`ad^d`io` _` ^m`^dhd`ioj _sa _o bp `bol 'pf o bpqŠ`lkqbkfal bk rk fkqbos^il ^_fboql alkab s bp `lkpq^kqb(+ l _fbk i^ abofs^a^ _sa _o kl bufpqb'`r^kal s p^iq^ ab rk bkqbol ^ lqol(- Ml l_pq^kqb+ pb mrbabk l_qbkbo fkcloj^`flkbp •qfibp pfpb prmlkb nrb i^ ml_i^`fŽk s bp rk^ crk`fŽk `lkqfkr^ ab o `lk abofs^a^ `lkqfkr^ _s-_o bk`^a^ fkpq^kqb- Dk i^ efmŽqbpfp ^kqboflo+ pb mlpqri^k i^p ~ibvbp ab `ob`fjfbkql ab i^ ml_i^,`fŽk‚+ nrb abmbkabk ab c^`qlobp abi jbafl ^j_fbkqb nrb mrbabk bpqfjri^o l obq^oa^o bi `ob`f,jfbkql-

Olo bgbjmil+ pf bi jbafl ^j_fbkqb qfbkb rk bcb`ql mbnrb•l l kril+ m^ob`b k^qro^i prmlkbonrb i^ sbil`fa^a ab `ob`fjfbkql bp molmlo`flk^i ^i qlq^i ab i^ ml_i^`fŽk+ v bkqlk`bp i^ ibvab `ob`fjfbkql qlj^oŠ i^ cloj^9

'7-1/(_s+ ;fs_o Š

alkab f bp rk^ `lkpq^kqb nrb abmbkab ab i^ k^qro^ibw^ ab i^ ml_i^`fŽk- Orbab l`roofo bkabqbojfk^a^p `lkaf`flkbp nrb bi c^`qlo f s^oŒb`lk bi qfbjml+ v i^ ibv ab `ob`fjfbkql '7-1/(mrbab dbkbo^ifw^opb `ljl pfdrb9

'7-10(J[_o < f&o's,

Rf+mlo ^idrk^ o^wŽk+i^ ml_i^`fŽk kl mrbab bu`babo ^ `fboql j^uojl L 'mlo bgbjmil+mlo ^dlq^opb ilp ^ifjbkqlp(+ m^ob`b k^qro^i prmlkbo i^ sbil`fa^a ab `ob`fjfbkql molmlo`flk^i^ ^j_lp s v L , s pfjriqŠkb^jbkqb- Rb qfbkb mrbp rk pbdrkal qfml ab ibv ab `ob`fjfbkql-

'7-11(J[, < fs&J + s'*_o

alkab+ `ljl bk '7-10(+ f mrbab pbo `lkpq^kqb+ l jŠp dbkbo^ijbkqb f mrbab s^of^o `lk biqfbjml- Lbglo^p qb`kliŽdf`^p mrbabk e^`bo nrb bi s^ilo ab L `obw`^ l ab`obw`^ m^ri^qfk^,jbkqb v mlo q^kql pb mrbab dbkbo^ifw^o '7-11( prmlkfbkal nrb L s^oŒ^`lk bi qfbjml-

02- Dumobp^o s bk crk`fŽk ab o m^o^ `^a^ rk^ ab i^p ~ibvbp ab `ob`fjfbkql‚ bk '7-1/( v'7-11( '`lk f v L ^j_^p `lkpq^kqbp(- Ool_^o nrb bi obpriq^al ab '7-11( pb mrbab bumob,p^o `ljl pfdrb9

'7-12( Kr:*****

0 * _*[f n*ni& )

alkab HW bp rk^ `lkpq^kqb u oj bp bi qfbjml bk bi nrb s < I./+

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Be`m^d^djn 282

03- Blkpfa‹obpb i^ ibv ab `ob`fjfbkql bk i^ cŽojri^ 'R-12( abi Dgbo`f`fl 02 v prmŽkd^pb nrbe^`fbkal bi `bkpl bk qobp fkqbos^ilp ab qfbjml fdr^ibp o! o0* o1* ilp k•jbolp e^k pfals s s Cbjlpqo^o nrb pb qfbkbk a^qlp prcf`fbkqbp m^o^ abqbojfk^o L v nrb bk bcb`ql

i& 1& 2&

pb qfbkb9

'7-13(U1&U0 + Ug' + UF&U1 + s0'

J < T/ ***********rw * UFU1

04- Cbar`fo i^ cŽojri^ nrb dbkbo^ifw^ 'R-12( abi Dgbo`f`fl 02 m^o^ i^ ibv ab `ob`fjfbkql 'R-11(`r^kal f kl bp kb`bp^of^jbkqb `lkpq^kqb- Dumobp^o bi obpriq^al `lk obi^`fŽk ^i qfbjmlo! m^o^ bi `r^i s < L .1-

05- Di Bbkprp Arob^r a^ ilp pfdrfbkqbp a^qlp 'bk jfiilkbp( ab ml_i^`fŽk bk ilp Dpq^alpTkfalp bk fkqbos^ilp ab 0/ ^•lp abpab 068/ ^ 084/: 2+8+4+2+6+1+8+5+01+8+06+ 12+ 20+28+ 4/+ 52+ 65+ 81+ iNR+ 011+ 024+ 04/-'^( @mif`^kal i^ b`r^`fŽk 'R-13( abqbojfk^o bi s^ilo ab L ^ _^pb ab ilp a^qlp abi `bkplm^o^ 068/+ iR4/ v 080/-'_( Kl jfpjl nrb bk '^( m^o^ ilp ^•lp 080/+ 082/ X 084/-'b( O^oqfbkal ab ilp `Ši`rilp eb`elp bk '^( v '_( ƒpb mrbab `lkpfabo^o `ljl ^`bmq^,_ib l kl i^ ibv ab `ob`fjfbkql 'R-12( m^o^ i^ ml_i^`fŽk ab ilp Dpq^alp Tkfalp>

06- '^( Cf_•gbpb i^ doŠcf`^ ab ild s `ljl crk`fŽk ab o*alkab s obmobpbkq^ ilp a^qlp abi`bkpl a^alp bk bi Dgbo`f`fl 05- Tqfifw^o bpq^ doŠcf`^ m^o^ abjlpqo^o nrb i^ ibv ab `ob`f,jfbkql 'R-1/( pb p^qfpc^`Œ^ lk jr`e^ ^molufj^`fŽk abpab 068/ ^ 080/- Cbqbojfk^o rks^ilo jbafl o^wlk^_ib ab f m^o^ bpqb mboŒlal-'_( CbqbojŒkbpb rk s^ilo jbafl o^wlk^_ib ab f m^o^ bi mboŒlal abpab 081/ ^ 084/: pr,mŽkd^pb nrb i^ ibv ab `ob`fjfbkql 'R-1/( bp sŠifa^ m^o^ bpq^ e) v mobab`fo i^ ml_i^`fŽk abilp Dpq^alp Tkfalp m^o^ ilp ^•lp 1/// v 1/4/-

iR- K^ mobpbk`f^ ab qlufk^p bk rk `fboql jbafl abpqorvb rk `riqfsl ab _^`qbof^p+ pfbkal bi`l`fbkqb afcbobk`f^i ab abpqor``fŽk molmlo`flk^i ^i k•jbol ab _^`qbof^p v ^ i^ `^kqfa^aab qlufk^p mobpbkqbpbk bi `riqfsl- Rf kl er_fbo^ qlufk^p i^p _^`qbof^p `ob`boŒ^k `lk

s s s

']( '_( 'b(

s s s

'`( 'b( #:

EHFTQ@ R-2 Dgbo`f`fl iR-

Page 414: Calculus

283 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

rk^ sbil`fa^a molmlo`flk^i ^ i^ `^kqfa^a qlq^i ab _^`qbof^pbufpqbkqb-Rb^ s bi k•jbolab _^`qbof^psfsfbkqbpbk bi fkpq^kqbo, RrmŽkd^pbnrb i^ `^kqfa^a ab qlufk^p `ob`b `lksbil`fa^a `lkpq^kqbv nrb i^ molar``fŽk ab qlufk^p bjmfbw^bk bi fkpq^kqbo < l- Dpq^,_ib`bo rk^ b`r^`fŽk afcbobk`f^im^o^s, Qbplisbo i^ b`r^`fŽk afcbobk`f^i-Tk^ ab i^p`ros^p ab i^ cfdro^ 7-2 bp i^ nrb obmobpbkqjbglo bi `ljmloq^jfbkql dbkbo^i ab s`ljl crk`fŽk ab o,Cb`fo `rŠi bp i^ bibdfa^ u bumif`^obi mlonr‹-

0&0 :PbNPV\[R`YV[RNYR`QR`RTb[Q\ \_QR[ P\[ P\RSVPVR[aR`P\[`aN[aR`

Tk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^

s! * L.%r&s$* L0%r&s< N%r&

pb abkljfk^ `^p\^d‡i gdi`\g _` n`bpi_j jm_`i, K^p crk`flkbp M*v M0 nrb jriqf,mif`^k i^ crk`fŽk fk`Ždkfq^ t v pr abofs^a^ t% plk ilp ^j`ad^d`io`n ab i^ b`r^`fŽk-

O^o^ i^p b`r^`flkbp ifkb^ibp ab mofjbo loabk+ afjlp rk qblobj^ ab bufpqbk`f^v rkf`fa^a v abqbojfk^jlp qla^p i^p plir`flkbp jbaf^kqb rk^ cŽojri^- Rf _fbkbufpqbrk qblobj^ ab bufpqbk`f^v rkf`fa^a m^o^i^ b`r^`fŽk dbkbo^i ifkb^i ab pb,drkal loabk+ kl e^v rk^ cŽojri^ nrb klp a‹ qla^p i^p plir`flkbp+ p^isl bk ^idrklp`^plp m^oqf`ri^obp-Dk bi Ulirjbk 00 pb bumlkb rk bpqrafl ab i^ b`r^`fŽk ifkb^idbkbo^i ab pbdrkal loabk- @nrŒpŽil qo^q^jlp bi `^pl bk bi nrb ilp `lbcf`fbkqbpLg v L0 plk `lkpq^kqbp-Br^kal bi pbdrkal jfbj_ol N%r& bp fa‹kqf`^jbkqb kril+ i^b`r^`fŽk pb ii^j^ cjhjb„i`\,

K^ b`r^`fŽk ifkb^i eljld‹kb^ `lk `lbcf`fbkqbp `lkpq^kqbp crb i^ mofjbo^ b`r^,`fŽk afcbobk`f^i ab rk qfml dbkbo^i nrb pb obplisfŽ `ljmibq^jbkqb- Dk 0632+Dribomr_if`Ž rk^ mofjbo^ plir`fŽk- @m^oqbab pr fkqbo‹pefpqŽof`l+bpq^b`r^`fŽk pb mob,pbkq^bk rk^ do^k s^ofba^a ab mol_ibj^p ab ^mif`^`fŽk+ab j^kbo^ nrb pr bpqraflbp ab fjmloq^k`f^ moŠ`qf`^-@abjŠp+ mlabjlp a^o cŽojri^p m^o^ qla^p i^p plir,`flkbp-

Blkpfabobjlp rk^ b`r^`fŽk ifkb^i eljld‹kb^ `lk `lbcf`fbkqbp `lkpq^kqbp nrbbp`of_fjlp ^pŒ9

s! * \t% * \s < M Š

Arp`^jlp plir`flkbp bk qlal bi bgbob^i ', //+ * //(- Tk^ plir`fŽk bp i^ crk`fŽk`lkpq^kqb t < N- Dpq^pb ii^j^ i^ plir`fŽk omdqd\g,Mlp fkqbobp^e^ii^o plir`flkbpkl qofsf^ibp+v `ljbkw^jlp krbpqol bpqrafl `lk ^idrklp `^plp m^oqf`ri^obpm^o^ilpnrb mrbabk bk`lkqo^opb plir`flkbp kl qofsf^ibp+mlo pfjmib fkpmb``fŽk- Dk qlalpbplp `^plp+ bi `lbcf`fbkqb ab t%bp kril+ v i^ b`r^`fŽk qfbkbi^ cloj^ v! * ]t < N-Ubobjlp nrb obplisbo bpq^b`r^`fŽk m^oqf`ri^obnrfs^ib ^ obplisbo bi `^pl dbkbo^i-

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Bsdno`i^d\ _` njgp^dji`n _` g\ `^p\^d‡i t! * ]t < M 284

7-8 Dufpqbk`f^ab plir`flkbp ab YNb`r^`fŽk t! * ]t < N

DIDLOKN0- I\ `^p\^d‡i t! < N- Dk bpqb`^pl plk krilp ilp alp `lbcf`fbk,qbp \ v \) W mlabjlp abqbojfk^o qla^p i^p plir`flkbp `lk c^`fifa^a- Rrmlkd^jlpnrb t bp rk^ crk`fŽk `r^inrfbo^ nrb p^qfpc^d t! < N bk ', //+ * '/(- Dkqlk`bppr abofs^a^ v&bp `lkpq^kqb+mlkd^jlp v&< `0‘ Hkqbdo^kalbpq^obi^`fŽk+bk`lkqo^,jlp nrb u bp kb`bp^of^jbkqb ab i^ cloj^

bk alkab A0 W @0 plk `lkpq^kqbp-Qb`Œmol`^jbkqb+m^o^`r^inrfbo m^oab `lkpq^kqbpA0 X ^0* bi mlifkljfl ab mofjbo do^al v < ^/s * ?0 p^qfpc^`bv! < N+ `lk 0/ nrbebjlp e^ii^al qla^p i^p plir`flkbp m^o^bpqb`^pl-

Rbdrfa^jbkqb prmlkbjlp nrb ] ", M W qo^q^jlp mlo pbm^o^al ilp `^plp ] ; Mu ] = N-

DIDLOKN1- B^p\^d‡i v! * ]t < N+ nd`i_j ] ; l- X^ nrb ] ; N+ mlab,jlp bp`of_fo \ < , O+ pfbkal e = N+v i^ b`r^`fŽk afcbobk`f^i qlj^ i^ cloj^

Tk^ plir`fŽk fkjbaf^q^ bp v < `%8!*v lqo^ t < `+er† @ m^oqfoab bii^p mlabjlp

l_qbkbo lqo^p plir`flkbp `lkpqorvbkal `lj_fk^`flkbp ifkb^ibp ab i^ cloj^

pfbkal B0 W B1 `lkpq^kqbp ^o_fqo^of^p-Dk bi qblobj^ 7-5 pb abjlpqo^oŠ nrb oj_\ni^p plir`flkbp nrba^k fk`irfa^p bk bpq^ cŽojri^-

DIDLOKN2- B^p\^d‡i v! * ]t < N+ nd`i_j ] = N- @nrŒmlabjlp bp`of_fo] < f!* alkab f = /+ v i^ b`r^`fŽk afcbobk`f^i qlj^ i^ cloj^

Nqo^ sbw l_qbkbjlp plir`flkbp ab jlal fkjbaf^ql- Tk^ plir`fŽk bp t < `lp fs* vlqo^ t < pbk er+ @ m^oqfoab bii^p ildo^jlp lqo^p plir`flkbp cloj^kal `lj_fk^,`flkbp ifkb^ibp+

bk alkab ^* v @0 plk `lkpq^kqbp `r^ibpnrfbo^- Di qblobj^ 7-5 abjlpqo^oŠ nrb bpq^cŽojri^ fk`irvb qla^p i^p plir`flkbp-

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,2/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

7-0/ Qbar``fŽk ab i^ b`r^`fŽk dbkbo^i ^i `^pl m^oqf`ri^o t! * ]t ;.,

Di mol_ibj^ ab obplisbo rk^ b`r^`fŽk ifkb^i ab pbdrkal loabk `lk `lbcf`fbk,qbp`lkpq^kqbp mrbab obar`fopb ^i ab obplisbo ilp `^plp m^oqf`ri^obpnrb ^`^_^jlpab sbo- Dufpqbrk j‹qlal m^o^e^`bo il nrb pb ^mif`^ q^j_f‹k ^ b`r^`flkbp jŠpdbkbo^ibp-K^ fab^ bp `lkpfabo^o qobpcrk`flkbp v+ p* v q q^ibp nrb v < QR+ Cbof,s^kal l_qbkbjlp v&< RS% * p%q*b v! < pq! * 0p%q%* p!q, Dumobpbjlp ^elo^i^ `lj_fk^`fŽk v! * \t% * ]t bk crk`fŽk ab p v q, N_qbkbjlp

'7-14( s! * \t% * \s < pq! * 0p%q%* p!q * \&pq%* p%q'* ]pq :

; &q!* \q%* ]q'p * &0q%* \q'p% * qp!,

Difg^jlp pbdrfa^jbkqb q m^o^ nrb bi `lbcf`fbkqb ab p%pb^ `bol- Dpql bufdb nrbq%< , \q-0* `lk il `r^i mlabjlp bibdfo q < `! !, O^o^ bpq^ q qbkbjlpj! < , \q%-0 < \0q-2* u bi `lbcf`fbkqb ab p bk '7-14( pb `lksfboqb bk

\0q \0q 2] [ \0

q! * \q%* ]q < , , , * ]q < ,,, q,313

@pŒmrbp+i^ b`r^`fŽk '7-14( pb obar`b ^

&2] + [

0't! * \t% * ]t < p! * 3 p q,

Orbpql nrb q < `+\u-0* i^ crk`fŽk q krk`^ bp `bol+ `lk il `r^i v p^qfpc^`bi^ b`r^,`fŽk afcbobk`f^i v! * \t% * ]t < N pf v pŽil pf p p^qfpc^`bp! * f' 2] + \0'p < N-@pŒmrbp+ebjlp abjlpqo^al bi pfdrfbkqb qblobj^-

RCMPCK? 7-3- P`\i u t p _jn api^dji`n o\g`n lp` t < pm\!-0, Bioji^`n*`i `g dio`mq\gj &+ //+ * L@ '* v n\odna\^` g\ `^p\^d‡i _da`m`i^d\g v! * \t% * ]t < Nnd u n‡gj nd p n\odna\^` g\ `^p\^d‡i _da`m`i^d\g

00 * 2] + [0 Mp ++2+p;,

Dpqb qblobj^ obar`b bi bpqrafl ab i^ b`r^`fŽk v! * \t% * ]t < M ^i `^plm^oqf`ri^o v! * ]t < N- Gbjlp bk`lkqo^al plir`flkbp kl qofsf^ibpab bpq^b`r^,`fŽk mbol+p^isl m^o^ bi `^pl ] < N+ kl ebjlp abjlpqo^al nrb pb e^k e^ii^aloj_\n i^p plir`flkbp-

Page 417: Calculus

Q`jm`h\ _` pid^d_\_ k\m\ g\ `^p\^d‡i v! * ]t < M 286

0&)) FR\_RZNQRb[VPVQNQ]N_NYNRPbNPVp[t! * \p .)

Di mol_ibj^ ab abqbojfk^o qla^p i^p plir`flkbp ab i^ b`r^`fŽk v! * ]t < Nmrbab obplisbopb `lk bi pfdrfbkqb o`jm`h\ _` pid^d_\_,

RCMPCK? R-4- Ppkjib\hjn _jn api^dji`n a t d lp` n\odna\b\i g\ `^p\^d‡i_da`m`i^d\g v! * ]t < N `i &+ //+ * //(- Ppkjib\hjn o\h]d„i lp` n\odna\^`i g\n^ji_d^dji`n did^d\g`n

a`L' < b&L'* .$%-&< a$%K&+

Bioji^`n `n a&s' < b&s' k\m\ oj_j s,

A`hjnom\^d‡i, Rb^ c&s' < a&s' + b&s', Prbobjlp mol_^o nrb c&s' < Mm^o^ qlal r+ G^objlp bpql bumobp^kal b bk crk`fŽk ab prp ^molufj^`flkbp mlomlifkljflp ab S^vilo-

N_pbos^jlp mofjbol nrb c bp q^j_f‹k rk^ plir`fŽk ab i^ b`r^`fŽk afcbobk,`f^i v! * \s < N X p^qfpc^`bi^p `lkaf`flkbp fkf`f^ibp b%K&< N+ b$%K&< N- Sla^crk`fŽk u nrb p^qfpc^d i^ b`r^`fŽk afcbobk`f^i qfbkbabofs^a^p ab `r^inrfbo loabkbk ', //+ * //( v mrbabk `^i`ri^opb mlo abofs^`fŽk obfqbo^a^ab i^ b`r^`fŽk af,cbobk`f^i- Olo bgbjmil+ mrbpql nrb v! < , ]t* qbkbjlp t%! < , ]t%* bt&2'< ] ]t! < ]%~, Olo fkar``fŽk bk`lkqo^jlp nrb i^p abofs^a^p ab loabk m^osfbkbk a^a^p mlo

bk q^kql nrb i^p ab loabk fjm^o plk tg0i+g'< ',0 'i+/]i+/t%, Orbpql nrb c&L' vb$%K&plk ^j_^p N+ obpriq^ nrb qla^p i^p abofs^a^p b%h&%i&plk kri^p- Olo `lkpf,drfbkqb+`^a^ mlifkljfl ab S^vilo bkdbkao^al mlo c bk bi mrkql s < N qfbkbqlalpprp `lbcf`fbkqbp krilp-

@mifnrbjlp ^elo^ i^ cŽojri^ ab S^vilo `lk obpql 'qblobj^ 6-5(+rp^kal rkmlifkljfl ab ^molufj^`fŽk ab do^al fjm^o 0i + 0+ W bk`lkqo^jlp nrb

b%r&< A0i[/%r& †

alkab A/!*.%r& bp bi q‹ojfkl ab boolo bk i^ cŽojri^ ab S^vilo- O^o^`ljmibq^o i^abjlpqo^`fŽk+ e^`bjlp m^qbkqbnrb bi boolo mrbab e^`bopb q^k mbnrb•l `ljl pbnrfbo^ qlj^kal i prcf`fbkqbjbkqb do^kab-

Tqfifw^jlp bi qblobj^ 6-6 m^o^ bpqfj^o i^ j^dkfqra abi q‹ojfkl ab boolo-O^o^ biil kb`bpfq^jlp bpqfj^o i^ j^dkfqra ab i^ abofs^a^ b%/h&+ Blkpfabobjlp`r^inrfbo fkqbos^il `boo^al cfkfql X+^* ^Z* pfbkal ` = l- X^ nrb b bp `lkqfkr^bk bpb fkqbos^il+bp ^`lq^a^ bk ‹i+ pb^ mlo bgbjmil fb%r& R L bk Z, _) ]Y+ X^ nrb

Page 418: Calculus

287 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

c&!g!'&s'< &+g'!]!c&s'* qbkbjlp i^ bpqfj^`fŽk Fc&0!'&s'/ x Jg]y! bk W*]) I- Diqblobj^ 6-6 klp a^ GCs,p't(0 y Jg]X!s0i,%/h& `lk il nrb+ bk bi fkqbos^ilW*])bi+qbkbjlp i^ bpqfj^`fŽk

K G^hjs/h K G^hj^/h J>/4.

/; Gd't(0 ; ,,, ; ,,, < ,* * %/h& * %/h& %/h& $

pfbkal > < ES,0`, Cbjlpqo^jlp ^elo^ nrb >h-h qfbkab e^`f^ N `r^kalh x * ll- Dpql bp l_sfl pf N z = w 0- Rf = = 0+mlabjlp bp`of_fo

'7-15(

>h > > > > > >f & > Yh+f

h ;g%0!%f%f)g!%9x f f)oG %

pfbkal f ; h, Rf bibdfjlp m^o^ f bi j^vlo bkqbol z >* bkqlk`bp = ; f * 0 vbi •iqfjl c^`qlo qfbkab ^ N `r^kal h x * ^`, Krbdl = h - h qfbkab ^ N `r^kalg w //+ `lk il nrb i^ abpfdr^ia^a '7-15( morb_^ nrb b%r&< N m^o^ qlal r bkW*_) b\- Obol+v^ nrb _ bp ^o_fqo^ofl+pb abar`b nrb b%r&< N m^o^ qlal r ob^i-Dpql `ljmibq^ i^ abjlpqo^`fŽk-

Kjo\8 Di qblobj^ 7-4 klp af`b nrb alp plir`flkbp ab i^ b`r^`fŽk afcbobk`f^iv! * ]t < N nrb qfbkbk bi jfpjl s^ilo v i^ jfpj^ abofs^a^ bk N ab_bk `lfk`fafo m^o^qlal s* K^ bib``fŽk abi mrkql N kl bp bpbk`f^i- Di jfpjl o^wlk^jfbkql jrbpqo^ nrb biqblobj^ q^j_f‹k bp `fboql pf bi mrkql N pb obbjmi^w^ mlo rk mrkql ` `r^inrfbo^- Dk i^abjlpqo^`fŽk ^kqboflo+ _^pq^ rqfifw^o ^molufj^`flkbp mlifkŽjf`^p ab S^vilo bk ` bkird^o ab e^`boil bk N-

7-01 Rlir`fŽk `ljmibq^ ab i^ b`r^`fŽk t! * ]t < N

Di qblobj^ ab rkf`fa^a klp mbojfqb `^o^`qbofw^oqla^p i^p plir`flkbp ab i^b`r^`fŽk afcbobk`f^i t! * ]t < N-

RCMPCK? 7-5- P`\i ] pi iˆh`mj m`\g t _jn api^dji`n Sq V Q0 `i', //+ * '/( _`adid_\n ^jhj ndbp`8

]( Pd ] << N+ o.%r&< 0+ R0&U' << r+_( Pd ] ; N+ kji`hjn ] < , f0 V _`adidhjn pg&s' < `f8o* p0&s' < `+fdgd,b( Pd ] = N+kji`hjn ] < f0 t _`adidhjn pg&s' < `lp fs* R0&U' < pbk fs,

Bioji^`n oj_\ njgp^d‡i _` g\ `^p\^d‡i _da`m`i^d\g t! * ]t < M `i &+ //+ * '/(

od`i` g\ ajmh\

'7-16(

nd`i_j bi V @0 ^jino\io`n,A`hjnom\^d‡i, Dk i^ Rb``fŽk 7-8 abjlpqo^jlp nrb m^o^`^a^ m^oBi X B1 i^

crk`fŽk t a^a^ bk '7-16( bp rk^ plir`fŽk ab i^ b`r^`fŽk t! * ]t < N- Ool_^jlp

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Pjgp^d‡i ^jhkg`o\ _` g\ `^p\^d‡i t! * \t% * ]t < M 288

^elo^ nrb qla^p i^p plir`flkbp qfbkbkbpq^cloj^- Di `^pl ] < N crb bpq^_ib`fal bki^ Rb``fŽk 7-8+ab jlal nrb mlabjlp prmlkbo nrb \ :B* N-

K^ fab^ ab i^ abjlpqo^`fŽk bp bpq^9Rb^ t < `%r& rk^ plir`fŽk `r^inrfbo^ abv! * ]t < N- Rf mlabjlp abjlpqo^o nrb bufpqbk`lkpq^kqbp `9 X B1 nrb p^qfpc^`bkbi m^oab b`r^`flkbp

'7-17(

bkqlk`bp ` v BiTi * @0x plk plir`flkbp ab i^ b`r^`fŽk afcbobk`f^i v! * ]t < Nnrb qfbkbkbi jfpjl s^ilo u i^ jfpj^ abofs^a^ bk N-Rbd•k bi qblobj^ ab rkf`fa^a+obpriq^ nrb ` < BiTi * @0R0Š

Dk bi `^pl _(+ qbkbjlp pg&s';`!!%* p0&s' <z<, `lk 0/ nrb pg&.';R0&.'; 0v o7%K&< e) o$/%-&< *e+ Blk biil i^p b`r^`flkbp '7-17( pb `lksfboqbk bkAi * ?0 < a`L'* v A0 , A1 < a%&L'-f, Sfbkbk i^ plir`fŽk Ai < g-&L' * oa%&L'af*?0 < Œ.'N(, oa%&L'af,

Dk bi `^pl `(+ qbkbjlp pg&s' < `lp fs* p0&s' < pbk fs* u ^pŒbp p.&L' < 0+p0&.' < N+p%&L'< N+p%0&.'< f* v i^p plir`flkbp plk Bi < a`L'* v B1 < a&L'ef,Orbpql nrb pfbjmob bufpqbkA0 v ?0 nrb p^qfpc^`bk'7-17(+ i^ abjlpqo^`fŽk bp `lj,mibq^-

7-02 Rlir`fŽk `ljmibq^ ab i^ b`r^`fŽk t! * \t% * ]t < N

Di qblobj^ 7-3 klp af`b nrb t p^qfpc^`bi^ b`r^`fŽk afcbobk`f^i v! * \t%* ]t <Npf u pŽil pf p p^qfpc^`bp! * G2] + \0'p < N+pfbkal t < `+\8^-0p,Rbd•k bi qblob,j^ 7-5 p^_bjlp nrb i^ k^qro^ibw^ab `^a^ plir`fŽk p abmbkab abi pfdkl ^idb_o^f,`l abi `lbcf`fbkqb ab p* bpql bp+abi pfdkl ab 2] + \0 l ab \0 + 2], @i k•jbol\0 + 2] ib ii^j^jlp _dn^mdhdi\io` ab i^ b`r^`fŽk afcbobk`f^i t! * \t% * ]t < NX 0/ abpfdk^jlp mlo ^+ Br^kal `lj_fk^jlp ilp obpriq^alp ab ilp- qblobj^p 7-3v 7-5 l_qbkbjlp bi pfdrfbkqb-

RCMPCK? 7-6- P`\ _ < \0 + 2] `g _dn^mdhdi\io` _` g\ `^p\^d‡i _da`m`i^d\ggdi`\g t! * \t% * ]t < N- Qj_\ njgp^d‡i _` `no\ `^p\^d‡i `i &+`l+ * '/( od`i`g\ ajmh\

'7-18(

`i _ji_` Bh V @0 nji ^jino\io`n* t g\n api^dji`n Th V R0 n` _`o`mhdi\i n`bˆi `gndbij \gb`]m\d^j _`g _dn^mdhdi\io` _`g hj_j ndbpd`io`8

]( Pd _ < N+ py&s' < 0 X p0&s' < s,_( Pd _ 8| N+ p/&s' < `!8^ u p0&s' < z~*nd`i_j f < pT`-b( Pd _ ; N+pg&s' < `lp fs*t*pds' < pbk fs* nd`i_j f < fs +_,

Page 420: Calculus

-)) Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Kjo\8 Dk bi `^pl _(+ bk bi nrb bi afp`ofjfk^kqb _ bp mlpfqfsl+ i^ plir`fŽk u bk'7-18( bp rk^ `lj_fk^`fŽk ifkb^i ab alp crk`flkbp bumlkbk`f^ibp+

bk alkab

\ +\ * R^&0< ,!1 * e < 1 &

\ +\ +*F_&1< , 1 , e < 1 -

Klp alp k•jbolp &0 v &1 qfbkbk `ljl prj^ &0* o1< , \ v `ljl molar`ql&iod< /&\0 + ^& < \+ Olo `lkpfdrfbkqb+ plk i^p o^Œ`bpab i^ b`r^`fŽk `r^aoŠqf`^

*0)[*)];L,

Dpq^ bp i^ abkljfk^a^ `^p\^d‡i ^\m\^o`m…nod^\ pl`f^a^ ^ i^ b`r^`fŽk afcbobk`f^i

s! * [s$ * \s < N ‘

Di k•jbol _ < \0 + 2] q^j_f‹k pb ii^j^ afp`ofjfk^kqb ab bpq^ b`r^`flk `r^aoŠqf`^pr pfdkl ^idb_o^f`l abqbojfk^ i^ k^qro^ibw^ ab i^p o^Œ`bp-Rf _ x /+ i^ b`r^`fŽk `r^aoŠ,qf`^ qfbkb o^Œ`bpob^ibp a^a^p mlo ' , \ € S_'-0, Rf _ ; /+ kl qfbkb o^Œ`bpob^ibp mbolab_b qbkboi^p ^jhkg`e\n o

0v m

0Š K^ abcfkf`fŽk ab i^ crk`fŽk bumlkbk`f^i mrbab ^jmif^opb

ab j^kbo^ nrb `lx!) u `l0% qbkd^k pfdkfcf`^al `r^kal o0

u o1

plk k•jbolp `ljmibglp-Dpq^ ^jmif^`fŽk+ bumrbpq^ bk bi `^mŒqril 8+ pb e^`b ab jlal nrb i^ `lj_fk^`fŽk ifkb^i'7-18( mrba^ q^j_f‹k bp`of_fopb `ljl rk^ `lj_fk^`fŽk ifkb^i ab `ly! v `l

+!) `r^kalo

0v o

1pb^k `ljmibglp-

Blk`irfjlp bpq^Rb``fŽk e^`fbkal s^of^p l_pbos^`flkbp- Orbpql nrb qla^p i^pplir`flkbp ab i^ b`r^`fŽk afcbobk`f^i t! * \t% * ]t < } bpqŠk `lkqbkfa^p bk i^cŽojri^ '7-18(+ i^ `lj_fk^`fŽk ifkb^i abi pbdrkal jfbj_ol pb ii^j^ ^ jbkral i^njgp^d‡i b`i`m\g ab i^ b`r^`fŽk afcbobk`f^i- Tk^ plir`fŽk `r^inrfbo^ l_qbkfa^m^oqf`ri^ofw^kal ilp s^ilobp ab i^p `lkpq^kqbp ^/ X @0 pb abkljfk^ rk^ njgp^d‡ik\mod^pg\m,

Olo bgbjmil+ qlj^kal ?/ < 0+?0 < N+virbdl ?/ < N+?0 < 0+l_qbkbjlp i^palp plir`flkbp m^oqf`ri^obp

Dpq^palp plir`flkbp plk ab bpmb`f^ifjmloq^k`f^ mlonrb i^p `lj_fk^`flkbp ifkb^ibp`lkpqorfa^p `lk bii^p klp a^k qla^p i^p plir`flkbp- Tk m^o ab plir`flkbp `r^ibp,nrfbo^ `lk bpq^molmfba^abp rk^ ]\n` abi `lkgrkql ab qla^p i^p plir`flkbp-

Tk^ b`r^`fŽk afcbobk`f^i qfbkb pfbjmob jŠp ab rk^ _^pb- Olo bgbjmil+ i^b`r^`fŽk v! < 8v qfbkb i^ _^pb RE < _

0i]) R/ < _*O!$+ Obol q^j_f‹k qfbkb i^ _^pb

Ty < `lpe 0r ) T0 < pbke 0r+ Dk bcb`ql+ mrbpql nrb %,$uF&< T/ * T0 X

Page 421: Calculus

Be`m^d^djn 3/0

_*0

uFF <--: VG , T/) qla^ `lj_fk^`fŽk ifkb^i ab _0T v _w0uFF bp q^j_f‹k rk^ `lj_fk^,

`fŽk ifkb^i ab VG X T0Š Krbdl+ bi m^o TF * T0 bp lqo^ _^pb-Orbab abjlpqo^opb nrb qlal m^o ab plir`flkbp Sy u S0 ab rk^ b`r^`fŽk afcb,

obk`f^i v! * \t% * ]t < N pboŠ rk^ _^pb pf i^ o^wŽk |* SF kl bp `lkpq^kqb-Rf _fbk ^nrŒ kl s^jlp ^ kb`bpfq^o bpq^ molmfba^a+ i^ jbk`flk^jlp mlonrb qfbkbfjmloq^k`f^ bk i^ qbloŒ^ab i^p b`r^`flkbp ifkb^ibp ab pbdrkal loabk `lk `lbcf,`fbkqbp kl `lkpq^kqbp- Dk bi Dgbo`f`fl 12 ab i^ Rb``fŽk 7-03 pb bp_lw^ rk^ ab,jlpqo^`fŽk-

0&), :WR_PVPV\`

G^ii^o qla^p i^p plir`flkbp ab i^p pfdrfbkqbp b`r^`flkbp afcbobk`f^ibp bk ', //+ * '/(-

0- s! * 1s < N-

/+ ` * 1s < N-0+` * 1s$ < N-1+ s! * 1s$ < N-2+ s! * /s$ * 0s < N-

3+ s! * /s$ * 0s < N-4+` * /s$ * /s < N-

5+` * /s$ * 2s < N-6+` * /s$ * s < N-

.-+ s! * /s$ * s < N-

Dk ilp Dgbo`f`flp 00 ^i 03+ e^ii^o i^ plir`fŽk m^oqf`ri^o nrb p^qfpc^d^ i^p `lkaf`flkbpfkf`f^ibp a^a^p-

..+ /s! * 0s$ < N+ `lk s < H b s$ < H `r^kal s < N-

./+ s! * /2s < N+ `lk s < ,0 b s$ < N `r^kal s < 2-

.0+ s! * 1s$ * U < N+ `lk s < 1 b s$ < ,0 `r^kal s < 0-

.1+ ` * 1s$ * 2s < N+ `lk s < 1 b s$ < ` `r^kal s < N-

04- K^ doŠcf`^ ab rk^ plir`fŽk p ab i^ b`r^`fŽk afcbobk`f^i v! , 3v&* 18v < N `loq^ i^doŠcf`^ ab rk^ plir`fŽk q ab i^ b`r^`fŽk v! * 3v&* 02v < N bk bi lofdbk- K^p alp `ro,s^p qfbkbk i^ jfpj^ mbkafbkqb bk bi lofdbk- Cbqbojfk^o p v q pf Q$-.P& < 0-

05- K^ doŠcf`^ ab rk^ plir`fŽk p ab i^ b`r^`fŽk afcbobk`f^i v! , 2v&, 3v < N `loq^ i^ doŠcf`^ab rk^ plir`fŽk q ab i^ b`r^`fŽk v! * 3v&, 4v < N bk bi lofdbk- Cbqbojfk^o p v qpf i^p alp `ros^p qfbkbk mbkafbkqbp fdr^ibp bk bi lofdbk v pf

, S&U'2 300j ,, <,

sx) // o%r& 5 -

06- G^ii^o qlalp ilp s^ilobp ab i^ `lkpq^kqb f q^ibp nrb i^ b`r^`fŽk afcbobk`f^i v! * ft < Nqbkd^ rk^ plir`fŽk kl qofsf^i t < ,e%T&m^o^ i^ `r^i ,e%K& < ,e%f& < N- O^o^ `^a^ s^ilo^ajfpf_ib ab e) abqbojfk^o i^ `loobpmlkafbkqb plir`fŽk t <-f %r&+Blkpfa‹obkpb ilp s^il,obp mlpfqfslp v kbd^qfslp ab e+

07- Rf %[)\& bp rk mrkql a^al abi mi^kl v pf g bp rk k•jbol ob^i a^al+ abjlpqo^o nrbi^ b`r^`fŽk afcbobk`f^i v! * f0t < N qfbkb bu^`q^jbkqb rk^ plir`fŽk `rv^ doŠcf`^ m^p^mlo %[)\& v qfbkb bk ‹i i^ mbkafbkqb g+ Cfp`rqfo q^j_f‹k bi `^pl e < N-

08- ^( Rb^k %[x) ]f& v %[/) ]/& alp mrkqlp bk bi mi^kl q^ibp nrb [f * [/ :‹ gl) pfbkal hbkqbol- Cbjlpqo^o nrb bufpqb bu^`q^jbkqb rk^ plir`fŽk ab i^ b`r^`fŽk afcbobk`f^iv! * v < N `rv^ doŠcf`^ m^p^ mlo bplp alp mrkqlp-_( K^ molmlpf`fŽk ab i^ m^oqb^(+ ƒbp `fboq^ pfbjmob pf [f * [/ bp rk j•iqfmil ab $fP<

Page 422: Calculus

3/1 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

b( Fbkbo^ifw^o bi obpriq^al ab i^ m^oqb [& m^o^ i^ b`r^`fŽk u! * f0t < N- Cfp`rqfo q^jl_Œ‹k bi `^pl f < N-

1/- Dk `^a^ `^pl+ e^ii^o rk^ b`r^`fŽk afcbobk`f^i ifkb^i ab pbdrkal loabk nrb pb p^qfpc^d^m^o^ r

hv p0Š

'^( of%T&< `!* Q/%T&< _*t+'_( !i'W( < _/t) Q/%T&< r_/!$+'b( Qf%T&< _*t,/ blp r) o/%r& < _*t,/ pbk r+'a( Qf%T&< m_h%/r * 0(+Q/%T&< pbk %/r * 1(-'b( Qf%T&< blpe r) Q/%T&< pbke s*

Slihmec[hi+ C^a^p i^p crk`flkbp p* u p,% i^ crk`fŽk S abcfkfa^ mlo S%r& ;; Qf%T&Qw%r&* Q/%T&Qw%r&bp pr Slihmec[hi) abkljfk^`fŽk rp^a^ bk ^qbk`fŽk ^ I- L- G-Volkphf '0667,0742( nrb crb nrfbk i^ fkqolargl- Klp Dgbo`f`flp nrb pfdrbk pb obcfbobk ^molmfba^abp abi Volkphf^kl-

10- ^( Rf bi Volkphf^kl S%r&{bib rh

&uo0

bp kril m^o^ qlal s^ilo ab r bk rk fkqbos^il^_fboql g*abjlpqo^o nrb bi `l`fbkqb Q

/,Qf bp `lkpq^kqb bk g, Cf`el ab lqol jlal+ pf

Q/,Qf kl bp `lkpq^kqb bk H) bkqlk`bp S%]& o5 N mlo il jbklp m^o^ rk _ ab f+_( Cbjlpqo^o nrb i^ abofs^a^ abi Volkphf^kl bp T%< QEQ)! * p*po,

11- Rb^ T bi Volkphf^kl ab alp plir`flkbp rh

&p0

ab i^ b`r^`fŽk afcbobk`f^i v! * \t%)]t;L*pfbkal \ v ] `lkpq^kqbp-^( Cbjlpqo^o nrb S p^qfpc^`b i^ b`r^`fŽk ab mofjbo loabk S$ * [S < N X mlo q^kqlS%r& < S%K&_*{{+ Dpq^ cŽojri^ morb_^ nrb pf V'N( o5 N+bkqlk`bp S%r&l3- m^o^ qlal r+_( Rrmlkfbkal nrb rh kl bp fa‹kqf`^jbkqb kri^+ abjlpqo^o nrb V'N( < N pf v pŽil pfQ/,Qf bp `lkpq^kqb-

12- Rb^k qg v q0 alp plir`flkbp `r^ibpnrfbo^ ab i^ b`r^`fŽk afcbobk`f^i v! * \t% * ]t < Nq^ibp nrb q

0- q

gkl bp `lkpq^kqb-

^( Rb^ v < x%r&rk^ plir`fŽk ab i^ b`r^`fŽk afcbobk`f^i- Tqfifw^o molmfba^abp abi Volkp,hf^kl m^o^ abjlpqo^o+ nrb bufpqbk `lkpq^kqbp ^

fv ^, q^ibp nrb

_( Cbjlpqo^o nrb qla^ plir`fŽk bp ab i^ cloj^ v < ]g pg * @0S0Š Cf`el ab lqol jlal+cloj^k rk^ _^pb m^o^ bi `lkgrkql ab qla^p i^p plir`flkbp-

7-04 D`r^`flkbp ifkb^ibp kl eljld‹kb^p ab pbdrkal loabk `lk `lbcf`fbkqbp`lkpq^kqbp

Ulis^jlp ^ afp`rqfo i^p b`r^`flkbp kl eljld‹kb^p ab i^ cloj^

'7-2/( s! * \t% * \s < N )

bk i^p nrb ilp `lbcf`fbkqbp [ v \ plk `lkpq^kqbp v bi pbdrkal jfbj_ol N bp rk^crk`fŽk `r^inrfbo^ `lkqfkr^ bk ',//+ * '/(- K^ afp`rpfŽk mrbab pfjmifcf`^opbjbaf^kqb bi rpl ab rk lmbo^alo- O^o^ `r^inrfbo crk`fŽk ` `lk abofs^a^p l v .!)

Page 423: Calculus

D` - gdi`\g`n ij cjhjb„i`\n _` n`bpi_j jm_`i ^ji ^j`ad^d`io`n ^jino\io`n 2.1

mlabjlp abcfkfo rk lmbo^alo H nrb qo^kpcloj^ ` bk lqo^ crk`fŽk H%`&abcfkfa^mlo i^ b`r^`fŽk

I&e' <a! * \g%* ]G ,

Lbaf^kqb bpqb lmbo^alo+ i^ b`r^`fŽk afcbobk`f^i '7-2/( pb bp`of_b bk i^ cloj^pbk`fii^

H%s&< N+

Dp cŠ`fi `ljmol_^o nrb I&t* * V0' < I&t*' * I&V0'* v nrb I&^t' < ^I&t'm^o^ `r^inrfbo `lkpq^kqb `- Olo `lkpfdrfbkqb+ m^o^ `r^inrfbo m^o ab `lkpq^kqbpAh W ?0* qbkbjlp

Dpq^bp i^ ii^j^a^ kmjkd`_\_ _` gdi`\gd_\_ abi lmbo^alo I,Rrmlkd^jlp ^elo^ nrb Xi b V0 plk alp plir`flkbp `r^ibpnrfbo^ ab i^ b`r^`fŽk

I&t' < O, Orbpql nrb I&t*' < I&V0' < O* i^ ifkb^ifa^a klp a^

mlo il nrb V0 + Xi bp rk^ plir`fŽk ab i^ b`r^`fŽk eljld‹kb^ H%s&< N- Olo ilq^kql+ pboŠV0 + Xi < Bi Ui * @0S0* bk alkab Bi Ui * @0S0 bp i^ plir`fŽk dbkbo^i abi^ b`r^`fŽk eljld‹kb^+ l pb^

V0 < A0T0 * @0S0 * VF Š

Dpq^b`r^`fŽk ab_b p^qfpc^`bopbm^o^oj_j m^oab plir`flkbp Xi b V0 ab i^ b`r^`fŽkkl eljld‹kb^ H%s&< O, Olo `lkpfdrfbkqb+ pf mlabjlp abqbojfk^o rk^ njgp^d‡ik\mod^pg\mXi ab i^ b`r^`fŽk kl eljld‹kb^+ oj_\n i^p plir`flkbp bpqŠk`lkqbkfa^pbk i^ cŽojri^

'7-20(

pfbkal Ai v A1 `lkpq^kqbp ^o_fqo^of^p-B^a^ rk^ ab q^ibpt bp bsfabkqbjbkqb rk^ pl,ir`fŽk ab I&t' < Q mlonrb I&@*S* * @0S0 * Xi( < I&@*S* * @0S0' * I&t*' :< N * N < N+ X^ nrb qla^p i^p plir`flkbp ab H%s&< N pb bk`rbkqo^k bk '7-20(+i^ bumobpfŽkBiUi * @0S0 * Xi pb ii^j^ njgp^d‡i b`i`m\g ab '7-2/(- @pŒnrb+ eb,jlR abjlpqo^al bi qblobj^ nrb pfdrb-

RCMPCK? 7-7- Pd t* `n pi\ njgp^d‡i k\mod^pg\m_` g\ `^p\^d‡i ij cjhjb„i`\H%s&< O* g\ njgp^d‡i b`i`m\g n` j]od`i` nph\i_j \ Xi g\ njgp^d‡i b`i`m\g _` g\^jmm`nkji_d`io` `^p\^d‡i cjhjb„i`\ H%s&< N-

Page 424: Calculus

-)- Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Di qblobj^ 7-6 klp af`b `Žjl pb bk`rbkqo^ i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽkeljld‹kb^ I&t' < N- Dp^ qfbkb i^ cloj^ t < @/S/ * @0S Š,* bk alkab

'7-21(

abqbojfkŠkalpb i^p crk`flkbp Q/ X Q0 mlo jbafl abi afp`ofjfk^kqb ab i^ b`r^`fŽk+`ljl pb bumif`Žbk bi qblobj^ 7-6- @elo^ abjlpqo^jlp nrb SF v S0 mrbabk rp^opbm^o^`lkpqorfo rk^ plir`fŽk m^oqf`ri^oVg ab i^ b`r^`fŽk kl eljld‹kb^ H%s&< N+

Dk i^ `lkpqor``fŽk fkqbosfbkbrk^ crk`fŽk S abcfkfa^ mlo i^ fdr^ia^a

Dpq^pb ii^j^ bi rmjinfd\ij ab UH u S09 ^idrk^p ab prp molmfba^abppb bumrpfbolkbk ilp Dgbo`f`flp 10 v 11 ab i^ Rb``fŽk 7-03- Mb`bpfq^objlp i^ molmfba^a ab nrbS%r& krk`^ bp `bol- Dpq^mrbab abjlpqo^opb jbaf^kqb ilp j‹qlalp fkaf`^alp bkilp Dgbo`f`flp l mrbab `ljmol_^opb afob`q^jbkqb v bk m^oqf`ri^om^o^i^p crk`flkbpSF u S0 a^a^p bk '7-21(-

RCMPCK? 7-8- P`\i SF t S0 g\n njgp^dji`n _` g\ `^p\^d‡i I&t' < N _\_\n `i&6,10'*nd`i_j I&t' < t! * \t% * ]t, P`\ T `g rmjinfd\ij _` SF t S0Š Bioji^`ng\ `^p\^d‡i ij cjhjb„i`\ H%s&< O od`i` pi\ njgp^d‡i k\mod^pg\mVg _\_\ kjmg\ a‡mhpg\

nd`i_j

'7-22( LN%r&

o/%r&< , p0%r&** ^r )S%r& ` N%r&

n0%r&< p/%r&** ^r +S%r&

A`hjnom\^d‡i, Hkqbkqbjlp e^ii^o crk`flkbp odv o0 q^ibpnrb i^ `lj_fk^`fŽkXi < odSF * o0S0 p^qfpc^d i^ b`r^`fŽk I&Vg' < O, Sbkbjlp

Br^kal cloj^jlp i^ `lj_fk^`fŽk ifkb^i I&V/' < t9 * \tx * ]V/* ilp q‹ojfklpnrb `lkqfbkbk od v o0 abp^m^ob`bk ab_fal ^ nrb I&q/' < I&q0' < N- Klp q‹ojf,klp obpq^kqbpklp a^k i^ obi^`fŽk

Page 425: Calculus

B^n, gdi`\g`n ij cjhjb„i`\n _` n`bpi_j jm_`i ^ji ^j`ad^d`io`n ^jino\io`n 2.3

Prbobjlp bibdfo o*u o0 ab jlal nrb H%s)&< N+ Dpql mlabjlp `lkpbdrfoil pf bib,dfj^p n)v o0 ab jlal nrb

v ovqv * oxqx < O ,

Rb qo^q^ ab rk m^o ab b`r^`flkbp ^idb_o^f`^p `lk i^p fk`Ždkfq^p nwu ox, Di abqbojf,k^kqb abi pfpqbj^ bp bi Volkphf^kl ab q* u q0Š Orbpql nrb ‹pqb krk`^ bp `bol+ bipfpqbj^ qfbkb rk^ plir`fŽk a^a^ mlo

u nw< pxNdS+

Hkqbdo^kal bp^p obi^`flkbp+ l_qbkbjlp i^p cŽojri^p '7-22(+ `ljmibq^kal ^pŒi^abjlpqo^`fŽk-

Di j‹qlal mlo bi nrb ebjlp l_qbkfal i^ plir`fŽk t* pb ii^j^ ^ sb`bp ab n\md\+^dƒi _` ^jino\io`n, Erb rqfifw^al mofjbol mlo Zle^kk Aboklriif bk 0586 m^o^ obplisbob`r^`flkbp ifkb^ibp ab mofjbo loabk+ u irbdl mlo K^do^kdb bk 0663 m^o^ b`r^,`flkbp ifkb^ibp ab pbdrkal loabk-

Kjo\8 Orbpql nrb i^p crk`flkbp o*u o0 abi qblobj^ 7-8 pb bumobp^k `ljl fkqbdo^ibpfkabcfkfa^p+ `^a^ rk^ ab bii^p bpqŠ abqbojfk^a^ p^isl rk^ `lkpq^kqb ^afqfs^- Rf prj^jlprk^ `lkpq^kqb ^* ^ o* u rk^ `lkpq^kqb ^

0^ o

0`^j_f^jlp i^ crk`fŽk X+ bk rk^ krbs^

crk`fŽk X1 < X+ * @*S* )@0S0Š Dk sfoqra ab i^ ifkb^ifa^a- qbkbjlp

il `r^i klp af`b nrb i^ krbs^ crk`fŽk X1 q^j_f‹k bp rk^ plir`fŽk m^oqf`ri^o ab i^ b`r^,`fŽk kl eljld‹kb^-

DIDLOKN 0- G^ii^o i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk t! * t < qd s bk&+i-0* i-0',

Pjgp^d‡i, K^p crk`flkbp q* u R0 ab i^p fdr^ia^abp '7-21( sfbkbk a^a^p mlo

px%r&< `lp r )

Rr Volkphf^kl bp S%r& < px%r&p8%r&* R/%T&Rw%r&< `lp! r * pbk! r < 0- Oloq^kql+ ab '7-22( l_qbkbjlp

oy&s'< , I pbku q^k u _s < pbk u , ild Zpb` u * q^k u\ +

u

o/&s' < H`lp u q^k u _s < ` pbk u _s < ,`lp u -

Page 426: Calculus

-)/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

@pŒnrb+ rk^ plir`fŽk m^oqf`ri^o ab i^ b`r^`fŽk kl eljld‹kb^ bp

(&0 < o/&U'R/&U' * o0&U'S0&U' < pbk r `lp r * BNR r ild Zpb`r * q^k u\ ,pbk r `lp r ;

< ,`lp s ild Zpb`s * q^k u\ -

Rbd•k bi qblobj^ 7-7+ pr plir`fŽk dbkbo^i bp

t < @/ BNR r * ]0m_hr * `lp r ild Zpb r * q^k u\-

@rknrb bi qblobj^ 7-8 molmlo`flk^ rk j‹qlal dbkbo^i m^o^ abqbojfk^o rk^plir`fŽk m^oqf`ri^o ab H%s&< N) e^v j‹qlalp bpmb`f^ibp nrb `lk cob`rbk`f^ plkab ^mif`^`fŽk jŠp cŠ`fi `r^kal i^ crk`fŽk O qfbkb `fboq^p cloj^p m^oqf`ri^obp- Dk i^Rb``fŽk pfdrfbkqb bumlkbjlp rk j‹qlal ^ab`r^al pf O bp rk mlifkljfl l bi mol,ar`ql ab rk mlifkljfl mlo rk^ bumlkbk`f^i-

7-05 L‹qlalp m^oqf`ri^obp m^o^ i^ abqbojfk^`fŽk ab rk^ plir`fŽk m^oqf`ri^oab i^ b`r^`fŽk kl eljld‹kb^ t! * \t% * ]t < O

@>PL 0- Bg n`bpi_j hd`h]mj O `n pi kjgdijhdj _` bm\_j i, Rf ] ," N+pfbjmob mlabjlp bk`lkqo^o rk mlifkljfl ab do^al i nrb p^qfpc^`b i^ b`r^`fŽk-Dkp^vbjlp rk mlifkljfl ab i^ cloj^

i

c&U' <F\fsfQ5>

`lk `lbcf`fbkqbp fkabqbojfk^alp- Rrpqfqrvbkal bk i^ b`r^`fŽk afcbobk`f^i H%s&< N

b fdr^i^kal ilp `lbcf`fbkqbp ab i^p mlqbk`f^p pbjbg^kqbp ab s* mlabjlp abqbojf,k^o [8 )[h*f) +++) [f +[} bk cloj^ pr`bpfs^- @ `lkqfkr^`fŽk ^mif`^jlp bi j‹qlal^ rk bgbjmil-

DIDLOKN 0- G^ii^o i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk v! * v < u!+

Pjgp^d‡i, K^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk eljld‹kb^ v! * v < N sfbkba^a^ mlo u < B0 BNR s * ?0 pbk s, @ bii^ qbkbjlp nrb prj^o rk^ plir`fŽk m^oqf`r,i^o ab i^ b`r^`fŽk kl eljld‹kb^- Orbpql nrb bi pbdrkal jfbj_ol bp rk mlifkljfl`•_f`l v nrb bi `lbcf`fbkqb ab u kl bp kril+ fkqbkq^jlp bk`lkqo^o rk^ plir`fŽkm^oqf`ri^o ab i^ cloj^ UE%r&< =r1 * >r0 * ?r * @+ Cbofs^kal alp sb`bp+ bk,`lkqo^jlp nrb s!%r& < 3=r * />+ K^ b`r^`fŽk afcbobk`f^i `lkar`b ^ i^ obi^`fŽk

%3=r * />& * %=r1 * >r0( _r * @&< r1Š

Page 427: Calculus

Oifo]cƒh j[lnc]of[l ^_ f[ _]o[]cƒh hi bigia€h_[ s! * [s$ * \s < N 1-4

Habkqfcf`^kal `lbcf`fbkqbpab i^p mlqbk`f^p^kŠild^p ab s* l_qbkbjlp = < 0+> < N+B < ,5+X C < N+ab jlal nrb rk^ plir`fŽk m^oqf`ri^obp sx%r&< r0 * 3r+ @pŒnrb+ i^ plir`fŽk dbkbo^i bp

t < B0 BNR s * @0 pbks * s1 + 4s ,

Orbab pbo fkqbobp^kqbi^ `ljm^o^`fŽk ab bpqb j‹qlal `lk bi ab s^of^`fŽkab `lkpq^kqbp- K^p fdr^ia^abp '7-22( klp a^k

o/&s' < , ` u2pbk s _s < +&1s1+ 5(pbk s * &s1

+ 4s' `lp s

v

o/&s' < ` s0 `lp s _s < &1s/* 5( `lp s * &s0

* 4s'n`is,

Br^kal cloj^jlp i^ `lj_fk^`fŽk nx.$x * o0R0* bk`lkqo^jlp i^ plir`fŽk m^oqf`ri^osx%r&< r0 * 3r) `ljl ^kqbp- Dk bpqb `^pl+ bi rpl abi j‹qlal ab s^of^`fŽk ab`lkpq^kqbp bufdb bi `Ši`ril ab i^p fkqbdo^ibp`r1 pbk r ^r b `r0 `lp r ^r+ Blk bij‹qlal ab ilp `lbcf`fbkqbp fkabqbojfk^alp+ kl pb mob`fp^i^ fkqbdo^`fŽk-

Rf bi `lbcf`fbkqb \ bp `bol+ i^ b`r^`fŽk s! * [s$ < N kl mrbab p^qfpc^`bopb`lk rk mlifkljfl ab do^al i* mbol pŒlk rkl ab do^al i * 0 pf \ :.: N- Rf plkkrilp [ v ]* i^ b`r^`fŽk bp t! < O9 pr plir`fŽk dbkbo^i bp rk mlifkljfl ab do^ali * 1 l_qbkfal `lk alp fkqbdo^`flkbp pr`bpfs^p-

?=OK 1- Af m_aoh^i gc_g\li nc_h_ f[ `ilg[ N%r& < jur&_g7FF)mc_h^i m ohjifchigci ^_ al[^i h) s g oh[ ]ihmn[hn_+

Dk bpqb`^pl bi `^j_fl ab s^of^_ib s < o%r&_g7FFqo^kpcloj^ i^ b`r^`fŽk afcb,obk`f^i s! * [s$ * \s < N bk rk^ krbs^ b`r^`fŽk

o! * %/g * [&o$ * %g0 * [g * \&o < L +

Dpq^bp abi qfml afp`rqfal bk bi `^pl 0+ab jlal nrb pfbjmob qfbkbrk mlifkljflplir`fŽk Qx+ Krbdl+ i^ b`r^`fŽk lofdfk^i qfbkb rk^ plir`fŽk m^oqf`ri^oab i^ cloj^sx < of%r&_g7FF)alkab S0 bp rk mlifkljfl- Rf g0 * [g * ] :.: N+bi do^al ab S0 bpbi jfpjl nrb bi do^al ab m- Rf g0 * [g * ] < Nmbol /g * [ :.: N+bi do^alab TH bp rk^ rkfa^a j^vlo nrb bi ab m- Rf g0 * [g * \ < N X /g * [ < N+bido^al ab S0 bp ab alp rkfa^abp j^vlo nrb bi ab m-

DIDLOKN 1- G^ii^o rk^ plir`fŽk m^oqf`ri^o ab i^ b`r^`fŽk t! * t < s`17FF+

Oifo]cƒh+ Di `^j_fl ab s^of^_ib s < o_0FFklp `lkar`b ^ i^ krbs^ b`r^`fŽko! * 3o$ * iNr < r+ Dkp^v^kal of%r& < =r * >) bk`lkqo^jlp i^ plir`fŽk m^o,qf`ri^o ox%r&< %2r * 2(.4/+ `lk 0/ nrb rk^ plir`fŽk m^oqf`ri^o ab i^ b`r^`fŽklofdfk^i bp XH < _17FF%2r* 2(.4/-

Page 428: Calculus

3/7 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Di j‹qlal ab ilp `lbcf`fbkqbp fkabqbojfk^alp q^j_f‹k mrbab rp^opb pf O qfbkbi^ cloj^ N%r& < j%r&_g!) `lp EU+T) l N%r& < j%r&_gT pbk yr) pfbkal j rk mlifkljflv g v bu `lkpq^kqbp- Dk ^j_lp `^plp+ bufpqbpfbjmob rk^ plir`fŽk m^oqf`ri^oab i^cloj^ Uf%T& < _g{wWk%r&]imfU+r * l%r&m_h_rrY) bk alkab k v l plk mlifkljflp-

0&)/ :WR_PVPV\`

G^ii^o i^ plir`fŽk dbkbo^i ab `^a^ rk^ ab i^p b`r^`flkbp afcbobk`f^ibp ab ilp Dgbo`f`flp0 ^i 06- Rf i^ plir`fŽk kl bp sŠifa^ bk qlal bi bgb sboqf`^i+ a^o rk fkqbos^il bk bi nrb pb^sŠifa^-0- s! * &$< s,0, s! * s$ < s0Š

1, s! * t% < s/ * 0s,2, t! + 0t%* 1t < s0†

3, s! * 4(! * 2t < s/ * 0s * 0-3+ s! * s$ * 3s < 0s1 * 3s0 + 5s * 1-4+ s! * 1s < `0!,

6, t! *, 2t < `+/!+

06- v! * 5v&* 8(&< x%r&)alkab x%r&< 0jŠp s*

07- Rf f bp rk^ `lkpq^kqb kl kri^+ abjlpqo^o nrb i^ b`r^`fŽk v! , f0'% < N%r& qfbkb rk^ pl,ir`fŽk m^oqf`ri^o Xi a^a^ mlo

7, t! * s$ * 0t < `!,iN- s! * s$ * /s < `0!,00- t! * s$ * 0t < _T * `0!,/0, s! * 1(! * s < s * 0s `!*

.0+ s! * 1(! * s < `+!-s/†

.1+ s! * s < `lq1 s,

.2+ s! * U < 1./ * `U',

.3+ s! * s$ * /s < `U-L * `U',m^o^ 0 z r w 1+ X x%r&< N m^o^ qlalp ilp ab,

0 a!V/ < , O&o'pbke f&s + o' _o ,f l

G^ii^o i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk v! , 8v < `\!,08- Rf f bp rk^ `lkpq^kqb kl kri^+ abjlpqo^o nrb i^ b`r^`fŽk v! )f/t < O&s' qfbkb rk^ pl,

ir`fŽk m^oqf`ri^o Xi a^a^ mlo

..!V/ < , O&o'pbk f&s + o' _o,e l

G^ii^o i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk v! * 8v < pbk 0r+

Dk `^a^ rkl ab ilp Dgbo`f`flp 1/

0., s! * s < pbk s,/.+ s! * s < `lp s,00, t! * 2t < 1s `lp s,

^i 14+ abqbojfk^o i^ plir`fŽk dbkbo^i-

01, s! )2t < 1spbk s,02, t! + 1t%< 1b1!pbk s,03, s! * s < `/r `lp 1s,

0&)0 :WRZ]Y\`QR]_\OYRZN So`VP\` bR P\[QbPR[ N RPbNPV\[RYV[RNYR`QRpb,Tb[Q\ \_QR[ P\[ P\RSVPVR[aR`P\[`aN[aR`

DIDLOKN 0- Jjqdhd`ioj \mh‡id^j ndhkg`, Rrmlkd^jlp nrb rk^ m^oqŒ`ri^bpqŠl_ifd^a^ ^ jlsbopb bk rk^ ob`q^ `lk pr ^`bibo^`fŽk afofdfa^ e^`f^ rk mrkql

Page 429: Calculus

Be`hkgjn _` kmj]g`h\n a…nd^jn -)2

cfgl ab i^ ob`q^ v molmlo`flk^i ^ i^ afpq^k`f^ ^ bpb mrkql cfgl- Rf qlj^jlp bi lofdbkbk bi mrkql cfgl v bp t i^ afpq^k`f^ bk bi fkpq^kqb s* bkqlk`bp i^ ^`bibo^`fŽk v! ab_bpbo kbd^qfs^ `r^kal s bp mlpfqfs^+ v mlpfqfs^ `r^kal s bp kbd^qfs^- Olo `lkpfdrfbk,qb mlabjlp bp`of_fo v! < , f0t* l

t! * f0t < N+

pfbkal f0 rk^ `lkpq^kqb mlpfqfs^- Dpq^ bp i^ b`r^`fŽk afcbobk`f^i abi hjqdhd`ioj\mh‡id^j ndhkg`, Rb bjmib^ ^ jbkral `ljl jlabil j^qbjŠqf`l m^o^ bi jlsf,jfbkql ab rk mrkql bk rk jb`^kfpjl sf_o^kqb q^i `ljl rk^ `rboa^ qbkp^ l rkaf^m^pŽk sf_o^kqb- K^ jfpj^ b`r^`fŽk pb mobpbkq^ bk i^ qbloŒ^ab `fo`rfqlp bi‹`,qof`lp bk alkab pb ii^j^ b`r^`fŽk abi lp`fi^alo ^ojŽkf`l-

Di qblobj^ 7-5 klp af`b nrb qla^p i^p plir`flkbp qfbkbk i^ cloj^

'7-23( t < > n`ifs * ? `lp fs*

bk alkab = u > plk `lkpq^kqbp ^o_fqo^of^p- Olabjlp bumobp^o i^p plir`flkbp bkcrk`fŽk abi pbkl l abi `lpbkl q^k pŽil- Olo bgbjmil+ mlabjlp fkqolar`fo krbs^p`lkpq^kqbp b v 9u+bk alkab

u?

N' < ^o`q^k =$B < S>0 * >0

bkqlk`bp qbkbjlp 'sbo cfdro^ 7-3( > < b`lp /'+ A < b pbk /'+ u i^ b`r^`fŽk '7-23(pb `lksfboqb bk

t < B `lp | pbk uu * Bpbk N' `lp fs < @n`i&fs * N'( ‘

Br^kal i^ plir`fŽk pb bp`of_b bk bpq^ qloj^+ i^p `lkpq^kqbp b v 9uqfbkbk rk^pbk`fii^ fkqbomobq^`fŽk dblj‹qof`^ 'sbo cfdro^ 7-4(- Klp s^ilobp buqobjlp ab v+nrb pb mobpbkq^k `r^kal pbk ne| * u( < ~ 0+ plk ~ B- Br^kal s < /+ bi abp,mi^w^jfbkql fkf`f^i bp b pbk j8 Br^kal s `ob`b+ i^ m^oqŒ`ri^lp`fi^ bkqob ilp s^ilobpbuqobjlp * b u , b `lk mboŒlal 05Q- f, Di Škdril fs * N' pb ii^j^ bi ƒibpgj _`a\n` u bi jfpjl s bp bi s^ilo fkf`f^i abi Škdril ab c^pb-

DIDLOKN 1- Sd]m\^dji`n \hjmodbp\_\n, Rf rk^ m^oqŒ`ri^ prgbq^ ^ rk jl,sfjfbkql ^ojŽkf`l pfjmib p•_fq^jbkqb bp pljbqfa^ ^ rk^ crbow^ buqbok^ molmlo,`flk^i ^ pr sbil`fa^a+ bi krbsl jlsfjfbkql p^qfpc^`b rk^ b`r^`fŽk afcbobk`f^i abi^ cloj^

t! * 0^t% * f0t < N +

Page 430: Calculus

30/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

EHFTQ@ 7-3

t

EHFTQ@ 7-4 Jjqdhd`ioj \mh„id^j ndhkg`,

bk alkab _ v e/ plk `lkpq^kqbp+_ ;/; N+e; N- Rf ` = N+sbobjlp nrb qla^p i^pplir`flkbp qfbkabk ^ `bol `r^kal s ,* * ^`* Dk bpqb`^pl+ pb af`b nrb i^ b`r^`fŽkafcbobk`f^i bp `no\]g`, K^ crbow^buqbok^lofdfk^ rk^ \hjmodbp\^d‡i abi jlsfjfbk,ql- Rf ` ; N+sbobjlp nrb ^idrk^p plir`flkbp qfbkbk s^ilobp q^k do^kabp `ljl pbnrfbo^ `r^kal s ,* * ll- Dk bpqb`^pl+ pb af`b nrb i^ b`r^`fŽk bp di`no\]g`,

Orbpql nrb bi afp`ofjfk^kqb ab i^ b`r^`fŽk bp _ < %/?&/ * 1e/ < 1%]/ * e/&)i^ k^qro^ibw^ab i^p plir`flkbp bpqŠabqbojfk^a^ mlo i^p j^dkfqrabp obi^qfs^pab^0 v f!* Klp qobp ^plp _ < N+_ = N+X _ ; N mrbabk pbo^k^ifw^alp `ljl pfdrb9

^( Adn^mdhdi\io` ^`mj8 `%< f!, Dk bpqb`^pl+ qla^p i^p plir`flkbp qfbkbk i^cloj^

t < `+@!%&>* >r&+

Rf ` = N+qla^p i^p plir`flkbp qfbkabk ^N `r^kal s ,* * ^j* Dpqb ^pl pb `fq^ `ljl\hjmodbp\hd`ioj ^m…od^j,Rf ? ;/; N+`^a^ plir`fŽk `^j_f^oŠ ab pfdkl bu^`q^jbkqbrk^ sbw ab_fal ^i c^`qlo ifkb^i = * >r+ Dk i^ cfdro^ 7-5'^( pb obmobpbkq rkbgbjmil -- Rf ` ; N+ `^a^ plir`fŽk kl qofsf^i qfbkab ^ * // l ^ , // `r^kalr*( )..,

_( Adn^mdhdi\io` kjndodqj8 ^0 = f!, Rbd•k bi qblobj^ 7-6 qla^p i^p plir`fl,kbp qfbkbk i^ cloj^

bk alkab c < yS_ < S^0+ f0

Š Orbpql nrb c/ < ]/ * f!* qbkbjlp c/ * ]0 ; N`lk il nrb &c+ ^'&c * ^' ; N- Olo `lkpfdrfbkqb+ ilp k•jbolp c ,,Š, v c * `qfbkbk pfdklp lmrbpqlp- Rf ` = N+bkqlk`bp c * ` bp mlpfqfsl `lk il nrb c + ` bpkbd^qfsl+ v mlo q^kql ^j_^p bumlkbk`f^ibp _%b*]&FF v _*%b(]&FF qfbkabk ^ `bol `r^kals+)+) ^j, Dk bpqb`^pl+ `fq^al `ljl \hjmodbp\hd`ioj `skji`i^d\g* qla^p i^p plir,

Page 431: Calculus

Be`hkgjn _` kmj]g`h\n a…nd^jn 300

`flkbp qfbkabk ^ N m^o^s x * ^^ , Dk i^ cfdro^ 7-5'^( pb obmobpbkqrk bgbjmil-B^a^ plir`fŽk mrbab `^j_f^o ab pfdkl mlo il jbklp rk^ sbw-

Rf ` ; /+ bkqlk`bp c + ` bp mlpfqfsl mbol c * ` bp kbd^qfsl- @pŒnrb+ ^j_^pbumlkbk`f^ibp _%b*]&r v _*%b(]&r qfbkabk ^ * // `r^kal s x * ol+ `lk 0/ nrb krb,s^jbkqb bufpqbkplir`flkbp `lk s^ilobp ^_plirqlp q^k do^kabp `ljl pb nrfbo^-

b( Adn^mdhdi\io` i`b\odqj8 `1 ; O- Dk bpqb`^pl+ qla^p i^p plir`flkbp qfbkbki^ cloj^

t < @`+@Un`i&cs* `s'*

bk alkab c < is <a < Sf/ * ^/† Rf `< N+qla^ plir`fŽk kl qofsf^i lp`fi^+mbol i^ ^jmifqra ab i^ lp`fi^`fŽk qfbkab e^`f^ `bol `r^kal s x * j~* Dpqb`^pl pbii^j^ \hjmodbp\hd`ioj jn^dg\io` v pb obmobpbkqbk i^ cfdro^ 7-5'_(- Rf ` ; N+qla^pi^p plir`flkbp kl qofsf^ibpqlj^k s^ilobp mlpfqfslp v kbd^qfslp q^k do^kabp `ljlpb nrfbo^ `r^kal s x * ol-

@jloqfdr^jfbkql `oŒqf`l

@jloqfdr^jfbkql lp`fi^kqb

&0• ! &&&&&&&&%&`&%

^( Cfp`ofjfk^kqb N l mlpfqfsl _( Cfp`ofjfk^kqb kbd^qfsl

EHFTQ@ 7-5 Sd]m\^dji`n \hjmodbp\_\n lp` n` km`n`io\i `jhj njgp^dji`n _` u! * 0`t% (( f0t < N+ ^ji ` = N+ v _dn^mdhdi\io` 2&^0+ f0',

DIDLOKN 2- @dm^pdojng„^omd^jn, Rf fkqbo`^i^jlp rk^ `^m^`fa^a 'mlo bgbj,mil+ rk `lkabkp^alo( bk bi `fo`rfql bi‹`qof`l abi bgbjmil 4 ab i^ Rb``fŽk 7-5+ i^b`r^`fŽk afcbobk`f^i nrb pfosb ab jlabil m^o^ bpqb`fo`rfql sfbkb a^a^ mlo

IFG&o'* OF&o'* y a g&o'_o < S&o'*

Page 432: Calculus

301 Ehnli^o]]cƒh [ f[m _]o[]cih_m ^c`_l_h]c[f_m

pfbkal B rk^ `lkpq^kqb mlpfqfs^ ii^j^a^ ][j[]c^[^+ K^ abofs^`fŽk ab bpq^ b`r^,`fŽk a^ rk^ b`r^`fŽk ifkb^i ab pbdrkal loabk ab i^ cloj^

HE!%n&* NE$%n&* 0-E%n&< R$%n&+A

Rf bi sliq^gb ^mif`^al R%n&bp `lkpq^kqb+bi pbdrkal jfbj_ol bp `bol u i^ b`r^`fŽkqlj^ i^ cloj^

f!%n&* y f$%n&* \0 E%n&< M -I I@

Dpqbbp bi jfpjl qfml ab b`r^`fŽk ^k^ifw^al bk bi bgbjmil 1+p^isl nrb 1` bpqŠobbjmi^w^al mlo O- I* X f0 mlo /-&I@', Dk bpqb`^pl+ bi `lbcf`fbkqb ` bp mlpfqfslv i^ b`r^`fŽk bp pfbjmob bpq^_ib-Cf`el ab lqol jlal+ i^ fkqbkpfa^a f%n&pfbjmobqfbkab ^ N `r^kal o ,,,* * ^j* K^ qbojfklildŒ^ abi bgbjmil 1 q^j_f‹k pb rqfifw^^nrŒ- Rb af`b nrb i^ `loofbkqb bp ^jloqfdr^a^ `oŒqf`^jbkqb`r^kal bi afp`ofjf,k^kqb bp `bol %?N0 < 1H&) bumlkbk`f^ijbkqb `r^kal bi afp`ofjfk^kqb bp mlpfqfsl%?N0 = 1H&) u lp`fi^kqb `r^kal bi afp`ofjfk^kqb bp kbd^qfsl %?N0 ; 1H&+

DIDLOKN 3- Iipcgc_hni ^_ oh ]ib_n_ ]ih g[m[ p[lc[\f_+ Tk `lebqb bpfjmrip^al jbaf^kqb i^ fdkf`fŽk abi `^o_ro^kqb bk rk^ `Šj^o^ ab `lj_rpqfŽk+ ^imbojfqfoi^ bumripfŽk e^`f^ ^qoŠpab ilp molar`qlp ab i^ `lj_rpqfŽk- Rrmlkd^jlpnrb bi `lebqb m^oqbabi obmlpl v pb jrbsb sboqf`^ijbkqb e^`f^ ^oof_^ ^ il i^odlab rk^ ob`q^- Cbpfdkbjlp i^ ^iqro^ abi `lebqb bk bi fkpq^kqbo mlo l%n&)i^ j^p^abi `lebqb 'fk`irfal bi `^o_ro^kqb( mlo g%n&)v i^ sbil`fa^a ab i^ j^qbof^ bumrip^,a^+ `lk obi^`fŽk ^i `lebqb+ mlo ]%n&+Dk ^rpbk`f^ ab crbow^pbuqbok^p+i^ b`r^`fŽk

'7-24( g%n&l!%n&< g$%n&]%n&

pb rqfifw^`ljl jlabil j^qbjŠqf`l m^o^afp`rqfo bi jlsfjfbkql- Di mofjbo jfbj,_ol+ g%n&l!%n&)bp bi molar`ql ab i^ j^p^ abi `lebqb mlo pr ^`bibo^`fŽk- Di pbdrkaljfbj_ol+ g$%n&]%n&)bp i^ crbow^ab ^`bibo^`fŽk bk bi `lebqb jlqfs^a^ mlo bi bj,mrgb abp^oolii^al mlo bi jb`^kfpjl ab fjmripfŽk- Dk ilp bgbjmilp nrb ^nrŒ pb`lkpfabo^k+ g%n&v ]%n&plk `lkl`falp l mrbabk bumobp^opbbk crk`fŽk ab l%n&l prabofs^a^ l$%n&'0^ sbil`fa^a abi `lebqb(- K^ b`r^`fŽk '7-24( pb `lksfboqb bk rk^b`r^`fŽk afcbobk`f^i ab pbdrkal loabk obpmb`ql^ i^ crk`fŽk mlpf`fŽk l+

Rf bpqŠkq^j_f‹k mobpbkqbpcrbow^pbuqbok^p+q^ibp`ljl i^ do^sba^a+ bkqlk`bpbk ird^o ab '7-24(+ rp^jlp i^ b`r^`fŽk

'7-25( g%n&l!%n&< g$%n&]%n&* B%n&)

bk a^kav B%n&obmobpbkqi^ prj^ ab qla^p i^p crbow^pbuqbok^pnrb ^`q•^k pl_obbi `lebqb bk bi fkpq^kqbo,

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AF_gjfim ^_ jli\f_g[m `•mc]im 302

@kqbp ab `lkpfabo^o bgbjmilp m^oqf`ri^obp+e^objlp rk o^wlk^jfbkql nrbmrbab pbosfo m^o^ grpqfcf`^oi^ b`r^`fŽk '7-24(- @ q^i cfk `lkpfabo^jlp mofjbolrk `lebqb nrb nrbj^ pr `^o_ro^kqb `lk fkqbojfqbk`f^p+bk cloj^ m^ob`fa^ ^ i^p_^i^p ab rk ^oj^ ab crbdl- Dk m^oqf`ri^o+ lkpfabo^jlp rk fkqbos^il ab qfbjmlXo*o * bY) bk alkab b bp rk k•jbol mlpfqfsl mbnrb•l: prmlkd^jlp nrb rk^`fboq^ `^kqfa^a ab j^qbof^ ab molmripfŽkbp bumbifa^ bk bi qfbjml o*v nrb kl bpbumrip^a^ `^kqfa^a ^idrk^ bk bi fkqbos^il pbjf^_fboql %n)o * bY+ Blk bpqlp pr,mrbpqlp+l_qbkbjlp rk^ cŽojri^ `rvl iŒjfqb+r^kal c ,6 N+bp i^ b`r^`fŽk '7-24(-

Hkjbaf^q^jbkqb ^kqbp ab i^ bumripfŽk ab j^qbof^ bk bi fkpq^kqbo*bi `lebqbqfbkb rk^ j^p^ g%n& v rk^ sbil`fa^a p%n&+@i cfk^i abi fkqbos^il Wn)n * bY) bi`lebqb qfbkb j^p^ g%n * b& v sbil`fa^a p%n* b&+ K^ j^p^ ab i^ j^qbof^ bumri,p^a^ bp g%n&* g%n * b&) u pr sbil`fa^a aro^kqb bi fkqbos^il bpp%n&* ]%n&)mrbpqlnrb ]%n&bp i^ sbil`fa^a ab bumripfŽk `lk obi^`fŽk ^i `lebqb- Irpq^jbkqb ^kqbp abi^ bumripfŽk ab j^qbof^ molmriplo^ bk bi fkpq^kqbo*bi `lebqb bp rk pfpqbj^ `lkjljbkql g%n&p%n&+Dk bi fkpq^kqbn * b) bpqb pfpqbj^ `lkpq^ ab alp m^oqbp+rk`lebqb `lk jljbkql g%n * b&p%n* b& v i^ j^qbof^ bumrip^a^ `lk jljbkqlWg%n&* g%n * b&YWp%n&* ]%n&Y+K^ ibv ab `lkpbos^`fŽk ab jljbkqlp bpq^_ib`bnrb bi jljbkql abi krbsl pfpqbj^ ab_b pbofdr^i ^i abi ^kqfdrl- Olo `lkpfdrfbkqb+qbkbjlp

g%n&p%n&< g%n * b&p%n* b& * Wg%n&* g%n * b&YWp%n&* ]%n&Y)

ab i^ nrb l_qbkbjlp

g%n * b&Wp%n* b& * p%n&Y< Wg%n* b& * g%n&Y]%n&+

Cfsfafbkal mlo c v e^`fbkal nrb c ,6 N+bk`lkqo^jlp nrb

g%n&p$%n&< g$%n&]%n&)

nrb bp bnrfs^ibkqb ^ i^ b`r^`fŽk '7-24(-Blkpfabobjlp rk `^pl m^oqf`ri^o bk bi nrb bi `lebqb m^oqbabi obmlpl `lk

rk mbpl fkf`f^i ab s hfilp 'fk`irvbkal ] hfilp ab `^o_ro^kqb( v nrb pb jrbsbsboqf`^ijbkqb e^`f^ ^oof_^ pfdrfbkal rk^ ob`q^- Rrmlkd^jlp nrb bi `^o_ro^kqbpb `lkprjb bk cloj^ `lkpq^kqb ^ o^wŽkab f hfilp mlo pbdrkal v nrb ilp molar`qlpab i^ `lj_rpqfŽk plk abp`^od^alp afob`q^jbkqb e^`f^ ^qoŠp`lk rk^ sbil`fa^a`lkpq^kqb ab ` jbqolp mlo pbdrkal `lk obi^`fŽk ^i `lebqb- Rrmlkd^jlp nrb i^•kf`^ crbow^ buqbok^ nrb ^`q•^ pl_ob bi `lebqb bp i^ ^qo^``fŽk qboobpqob-Prbob,jlp p^_bo ^ nr‹ ^iqro^ iibd^oŠ bi `lebqb ^kqbp ab nrb qlal pr `^o_ro^kqb pb`lkprj^-

Page 434: Calculus

303 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Orbpql nrb qlal bi `lj_rpqf_ib pb `lkprjb `r^kal fo < ]* obpqofkdfjlp o^i fkqbos^il N9R::o x ]gf* K^ •kf`^ crbow^ buqbok^ nrb ^`q•^ pl_ob bi `lebqb bp*g%n&a)i^ sbil`fa^a ]%n&< *]) `lk il nrb i^ b`r^`fŽk '7-25( pb qo^kpcloj^ bk

h&o'm!&o'< +h%&o'` + h&o'b ,

Di mbpl abi `lebqb bk bi fkpq^kqb o bp s , fo* X pr j^p^ h&o' bp 's , fo'-b9irbdl qbkbjlp g$%n&< , f - d u i^ b`r^`fŽk ^kqboflo qlj^ i^ cloj^

!%& g$%n& e_m o ;+++`+b;++++b,S#> M ' fo

Hkqbdo^kal+ u rqfifw^kal i^ `lkaf`fŽk fkf`f^i o&'N(< N+ bk`lkqo^jlp

t , fom%&o'< +` ild ,, , bo ,

s

Hkqbdo^kal krbs^jbkqb u bjmib^kal i^ `lkaf`fŽk fkf`f^i l_K& < N+ l_qbkbjlp i^obi^`fŽk

`&r+fo' r+fo 01m&o'< y,yGkc ,, , , bo * `o,e t 1

Slal bi `lj_rpqf_ib pb e^ `lkprjfal `r^kal o < \ , e+ Dk bpqb fkpq^kqb i^ ^iqro^ bp

'7-26( m&x'< `&r + ]' ild t , ] [ b]0 * `] ,

f f t 1 f/ f

Dpq^ cŽojri^ bp sŠifa^ pf ] ; t- O^o^ `fboqlp `lebqbp+ bi mbpl abi `lebqb s^`Œlab `^o_ro^kqb bp mbnrb•l obpmb`ql ^i mbpl ab af`el `^o_ro^kqb+ v bp fkqbobp^kqb`lkpfabo^o bi `^pl iŒjfqb ] < t- Ml mlabjlp mlkbo ] < t bk '7-26( mlo i^ mob,pbk`f^ abi q‹ojfkl ild 't , \&,q+ Ml l_pq^kqb+ pf e^`bjlp nrb \ w t+ bi mofjboq‹ojfkl ab '7-26( bp rk^ cloj^ fkabqbojfk^a^ `lk iŒjfqb N- Olo `lkpfdrfbkqb+`r^kal ] x t+ bi s^ilo iŒjfqb abi pbdrkal jfbj_ol ab '7-26( bp

0 br/ `r q 1< , , , * , < , , bQ * `Q

0 f/ f 1 ‘

bk alkab Q < t . f bp bi qfbjml kb`bp^ofl m^o^ nrb qlal bi mbpl t pb `lkprj^-

7-08 Dgbo`f`flp

Dk ilp Dgbo`f`flp abi 0 ^i 4+ pb prmlkb nrb rk^ m^oqŒ`ri^ pb jrbsb `lk jlsfjfbkql^ojŽkf`l pfjmib+ ab ^`rboal `lk i^ b`r^`fŽk t < b n`i&fs * GVy K^ q`gj^d_\_ ab i^ m^oqŒ`r,

Page 435: Calculus

Be`m^d^djn 304

i^ pb abcfkb `ljl i^ abofs^a^ t%, H[ `l_]o_h]c[ abi jlsfjfbkql bp bi ob`Œmol`l abi mboŒlal-'OboŒlal < /$4c*de8 cob`rbk`f^ < e,/4P+& K^ cob`rbk`f^ obmobpbkq^bi k•jbol ab `f`ilp aba_pˆ],alp bk i^ rkfa^a ab qfbjml+ `lk q^i nrb f = l-

0- G^ii^o i^ ^jmifqra a pf i^ cob`rbk`f^ bp .,4P u ilp s^ilobp fkf`f^ibp ab u b u& '`r^kals < N( plk 1 v 3+ obpmb`qfs^jbkqb-

1- G^ii^o i^ sbil`fa^a `r^kal v bp `bol+ p^_fbkal nrb i^ ^jmifqra bp 6 v i^ cob`rbk`f^ 0/-2- Cbjlpqo^o nrb i^ b`r^`fŽk abi jlsfjfbkql mrbab q^j_f‹k bp`of_fopb abi jlal pfdrfbkqb9

t < > BNR &-g-U * %F&+

G^ii^o b`r^`flkbp nrb obi^`flkbk i^p `lkpq^kqbp =) h* %F)u a+ e) lq-

3- G^ii^o i^ b`r^`fŽk abi jlsfjfbkql p^_fbkal nrb v < 2 b v&< N X nrb bi mboŒlal bp p-4- G^ii^o i^ ^jmifqra abi jlsfjfbkql pf bi mboŒlal bp 16R u i^ sbil`fa^a bp ~ Sj `r^kal

v < Wk&5- Tk^ m^oqŒ`ri^ bpqŠ pljbqfa^ ^ rk jlsfjfbkql ^ojŽkf`l pfjmib- Hkf`f^ijbkqb pr abp,

mi^w^jfbkql bp 0+pr sbil`fa^a 1 v pr ^`bibo^`fŽk , 01- B^i`ri^o pr abpmi^w^jfbkql v pr^`bibo^`fŽk `r^kal i^ sbil`fa^a bpU 7-

6- O^o^ rk `fboql k•jbol mlpfqfsl f* i^ b`r^`fŽk afcbobk`f^i abi jlsfjfbkql ^ojŽkf`lpfjmib u! * f0t < N qfbkb plir`flkbp ab i^ cloj^ u < `%r& `lk `_K& < `%0&< N X`%r&; N m^o^ qlal r bk bi fkqbos^il ^_fboql N ; r ; 2- B^i`ri^o f u e^ii^o qla^p i^pplir`flkbp-

7- K^ fkqbkpfa^a G&-' ab i^ `loofbkqb nrb `fo`ri^ bk bi fkpq^kqb . bk rk `fo`rfql bi‹`qof`ll_bab`b ^ i^ b`r^`fŽk afcbobk`f^i l%n&* .'p( < C%n&)bk alkab F'.( bp rk^ crk`fŽk bp`^,ilk^a^ a^a^ mlo C%n&< 0 pf N z o x /4P) C%n&< N m^o^ qlalp ilp abjŠp s^ilobp ab o,Cbqbojfk^o i^ plir`fŽk nrb p^qfpc^`b i^p `lkaf`flkbp fkf`f^ibp 0'/( < N+0&'/( < 0-

8- K^ fkqbkpfa^a F%n&ab i^ `loofbkqb nrb `fo`ri^ bk bi fkpq^kqb o bk rk `fo`rfql bi‹`qof`ll_bab`b ^ i^ b`r^`fŽk afcbobk`f^i

0!'q( * OF%&o'* gR' < pbk ro *

pfbkal O X r `lkpq^kqbp mlpfqfs^p- K^ plir`fŽk mrbab bumobp^opb bk i^ cloj^ 0'0( <: B%n&* = pbk&ro * u (+alkab B%n&w N `r^kal o x * //+ u = u GV plk `lkpq^kqbp nrbabmbkabk ab O v r* `lk > = N- Rf bufpqb rk s^ilo ab r nrb e^d^ > q^k do^kab `ljlpb^ mlpf_ib+ bkqlk`bp S,%/4P& pb ii^j^ `l_]o_h]c[ ^_ l_mih[h]c[ abi `fo`rfql-^( G^ii^o qla^p i^p cob`rbk`f^p ab obplk^k`f^ `r^kal N < 0-_( G^ii^o ^nrbiilp s^ilobp ab O m^o^ ilp `r^ibp bi `fo`rfql qbkaoŠ rk^ cob`rbk`f^ ab ob,plk^k`f^-

0/- Tk^ k^sb bpm^`f^i obdobp^ ^ i^ Sfboo^- Rrmlkd^jlp nrb i^ •kf`^ crbow^ buqbok^ nrb ^`q•^pl_ob bii^ bp i^ do^sba^a+ v nrb `^b pfdrfbkal rk^ ob`q^ afofdfa^ e^`f^ bi `bkqol ab i^Sfboo^- Di bcb`ql ab i^ do^sba^a bp m^o`f^ijbkqb `lkqo^oobpq^al bk`bkafbkal rk `lebqbnrb ^`q•^ `ljl cobkl- Di `lj_rpqf_ib ab bpqb `lebqb bp `lkprjfal bk cloj^ `lkpq^kqb^ o^wŽk ab f hfilp mlo pbdrkal v bi j^qbof^i bumrip^al qfbkb rk^ sbil`fa^a `lkpq^kqb ab^ jbqolp mlo pbdrkal `lk obi^`fŽk ^i `lebqb- Dk`lkqo^o rk^ cŽojri^ m^o^ i^ afpq^k`f^ob`loofa^ mlo i^ k^sb bpm^`f^i bk pr `^Œa^ bk bi fkpq^kqb o pf m^oqbabi obmlpl bk o<N `lkrk mbpl fkf`f^i ab f hfilp-

00- Tk `lebqb ab t hfilp ab mbpl fkf`f^i m^oqb abi obmlpl bk bi bpm^`fl if_ob 'pfk crbow^pbuqbok^p( v pb jrbsb ^ il i^odl ab rk^ qo^vb`qlof^ ob`qfiŒkb^-Di `^o_ro^kqb pb `lkprjb^ i^ o^wŽk `lkpq^kqb ab f hfilp mlo pbdrkal u ilp molar`qlp ab i^ `lj_rpqfŽk plk abp,`^od^alp e^`f^ ^qoŠp ^ i^ sbil`fa^a `lkpq^kqb ab ^ jbqolp mlo pbdrkal `lk obi^`fŽk ^i`lebqb- G^ii^o i^ afpq^k`f^ ob`loofa^ bk bi fkpq^kqb o,

01- Qbplisbo bi Dgbo`f`fl )) pf i^ sbil`fa^a fkf`f^i abi `lebqb bp Ul u ilp molar`qlp ab i^`lj_rpqfŽk plk nrbj^alp `lk sbil`fa^a q^i nrb i^ j^qbof^ bumrip^a^ nrbab bk obmlplbk bi bpm^`fl-

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305 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

0&*( BO`R_cNPV\[R_RYNaVcN`N YN RPbNPV\[RQVSR_R[PVNYR`[\ YV[RNYR`

Orbpql nrb i^p b`r^`flkbp afcbobk`f^ibp ifkb^ibp ab pbdrkal loabk `lk `lbcf,`fbkqbp `lkpq^kqbp pb mobpbkq^kbk q^k ^jmif^ s^ofba^a ab mol_ibj^p `fbkqŒcf`lp+bp ob^ijbkqb sbkq^glpl nrb afpmlkd^jlp ab j‹qlalp pfpqbjŠqf`lp m^o^obplisboi^p-Lr`e^p b`r^`flkbp kl ifkb^ibp prodbk bpmlkqŠkb^jbkqb bk mol_ibj^p cŒpf`lpvdblj‹qof`lp+ mbol kl bufpqbrk^ qbloŒ^ljmibq^ `ljm^o^_ib ^ i^ ab i^p b`r^`flkbpifkb^ibp- Dk i^ fkqolar``fŽk ab bpqb`^mŒqriljbk`flk^oklp rk^ ~_lip^ ab qor`lp‚nrb e^ pfal abp^oolii^a^ m^o^ qo^q^ojr`elp `^plp m^oqf`ri^obpab b`r^`flkbp klifkb^ibp- Sbojfk^oklp bpqb`^mŒqril`lk i^ afp`rpfŽk ab ^idrklp ab bplp ~qor`lp‚ v^idrklp ab ilp mol_ibj^p nrb obplisboklp `lk pr ^vra^- RŽil `lkpfabo^objlp b`r^,`flkbp ab mofjbo loabk bk i^p nrb mrbab abpmbg^opbi^ abofs^a^ v&v pb mrbabkbumobp^obk i^ cloj^

'7-27( s$ <`nr)s& +

Qb`loabjlp nrb rk^ plir`fŽk ab '7-27( bk rk fkqbos^il Ebp `r^inrfbo crk,`fŽk+v < U%r&)abofs^_ib bk E v nrb p^qfpc^`bi^ obi^`fŽk U$%r&< `Wr) U%r&Ym^o^qlal s ab g,Dk bi `^pl ifkb^i+ abjlpqo^oklp rk qblobj^ ab bufpqbk`f^ v rkf`fa^anrb klp af`b nrb bufpqbrk^ v pŽil rk^ plir`fŽk nrb p^qfpc^`brk^ `lkaf`fŽk fkf`f^i^pfdk^a^- @abjŠp+ afpmlkboklp ab rk^ cŽojri^ m^o^abqbojfk^o bp^ plir`fŽk-

Dpql kl pr`bab bk bi `^pl dbkbo^i- Tk^ b`r^`fŽk kl ifkb^i mrbab ij qbkbopl,ir`fŽk nrb p^qfpc^d rk^ `lkaf`fŽk fkf`f^i a^a^+ l mrbab qbkbohƒn _` pi\, Olobgbjmil+ i^ b`r^`fŽk &t%'0+ st% * v * 0 < N kl qfbkb plir`fŽk `lk u < N`r^kal s < N+v^ nrb bpql bufdfoŒnrb '0(1 < ,0 `r^kal s < N- Olo lqo^ m^oqb+i^ b`r^`fŽk v&< 0t0-1 qfbkb alp plir`flkbp afpqfkq^p+U)%r&< N b U0%r&:r1* nrbp^qfpc^`bki^ `lkaf`fŽk fkf`f^i v < N `r^kal r < N-

@pŒmrbp+bi bpqrafl ab i^p b`r^`flkbp kl ifkb^ibp lcob`b jŠp afcf`riq^abp ^ `^rp^ab i^ kl bufpqbk`f^ l kl rkf`fa^a ab i^p plir`flkbp- S^j_f‹k+ fk`irpl `r^kalbufpqbkplir`flkbp+ mrbab nrb kl pb^ mlpf_ib abqbojfk^oi^p bumiŒ`fq^jbkqbbk crk,`fŽk ab crk`flkbp pbk`fii^p-@ sb`bp mlaboklp bifjfk^o i^ abofs^a^ v&ab i^ b`r^`fŽkafcbobk`f^i v iibd^o ^ rk^ obi^`fŽk ab i^ cloj^

B%r)s&< M

nrb pb p^qfpc^`bm^o^ ^idrk^p+ l nrfwŠp qla^p+ i^p plir`flkbp- Rf bpq^ b`r^`flkmrbab obplisbopb obpmb`ql^ u bk crk`fŽk ab s* iHbd^jlp ^ rk^ cŽojri^ bumiŒ`fq^m^o^ i^ plir`fŽk- Blk j^vlo cob`rbk`f^+ kl l_pq^kqb+i^ b`r^`fŽk bp abj^pf^al`ljmif`^a^ m^o^ abpmbg^obk bii^ v- Olo bgbjmil+ bk rk^ Rb``fŽk mlpqboflobpqr,af^objlp i^ b`r^`fŽk afcbobk`f^i

& \'[t <,,+

t)s

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@pmq\n dio`bm\g`n u ^\hkjn _dm`^^dji\g`n 306

v bk`lkqo^objlp nrb qla^ plir`fŽk p^qfpc^`b kb`bp^of^jbkqb i^ obi^`fŽk

'7-28( -i^d %r/ * f&* ^o`q^k ƒ9 * B < M0 s

m^o^ rk^ `fboq^ `lkpq^kqb B- RboŒ fk•qfi fkqbkq^o abpmbg^o bk bpq^ b`r^`flk i^ vbk crk`fŽk ab s, Dk rk `^pl m^ob`fal ^ ‹pqb+ ab`fjlp nrb i^ obi^`fŽk '7-28( bprk^ a‡mhpg\ dhkg…^do\ m^o^ i^p plir`flkbp- Blj•kjbkqb pb af`b nrb i^ b`r^`fŽkafcbobk`f^i e^ pfal ~obprbiq^‚ l ~fkqbdo^a^‚ `r^kal pb iibd^ ^ rk^ cŽojri^ fj,miŒ`fq abi qfml B%r)u( < N bk i^ nrb kl ^m^ob`bk abofs^a^p ab i^ crk`fŽk fk`Žd,kfq^- @ sb`bp bp^ cŽojri^ obsbi^ fkcloj^`fŽk •qfi ^`bo`^ ab i^p plir`flkbp- Olo lqo^m^oqb+bi ib`qlo `ljmol_^oŠ nrb q^i obi^`fŽk fjmiŒ`fq^ mrbab pbo jbklp •qfi nrbi^ jfpj^ b`r^`fŽk afcbobk`f^i m^o^ bpqraf^o molmfba^abp ab i^p plir`flkbp-

Dk i^ pfdrfbkqb Rb``fŽk jlpqo^jlp `Žjl mrbab l_qbkbopb+ `lk cob`rbk`f^+ fk,cloj^`fŽk `r^ifq^qfs^ ^`bo`^ ab i^p plir`flkbp afob`q^jbkqb ^ m^oqfoab i^ b`r^`fŽkafcbobk`f^i pfk rk `lkl`fjfbkql ab cŽojri^p bumiŒ`fq^pl fjmiŒ`fq^p ab i^p plir,`flkbp-

0&*) 8b_cN` V[aRT_NYR`e PNZ]\` QV_RPPV\[NYR`

Blkpfabobjlp rk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk+ q^i `ljl v&< `%r) s&)

v prmlkd^jlp nrb ^idrk^ ab i^p plir`flkbp p^qfpc^`bk rk^ obi^`fŽk fjmiŒ`fq^ abi^ cloj^

'7-3/( B%r)t* B( < N+

pfbkal B rk^ `lkpq^kqb- Rf fkqolar`fjlp rk pfpqbj^ ab `lloabk^a^p ob`q^kdri^ov pb•^i^jlp qlalp ilp mrkqlp %r)u( `rv^p `lloabk^a^p p^qfpc^`bk '7-3/( m^o^ rk`fboql s^ilo ab B+ l_qbkbjlp rk^ `ros^ ii^j^a^ ^pmq\ dio`bm\g ab i^ b`r^`fŽkafcbobk`f^i- Noafk^of^jbkqb s^ilobp afpqfkqlp ab B a^k `ros^p fkqbdo^ibp afpqfk,q^p+ mbol qla^p bii^p qfbkbk rk^ molmfba^a dblj‹qof`^ `lj•k- K^ b`r^`fŽk afcb,obk`f^i v&< `%r)v( ifd^ i^ mbkafbkqb v&bk `^a^ mrkql %r)s& ab i^ `ros^ ^ i^p`lloabk^a^p s b v- @i qlj^o B qlalp ilp s^ilobp+ i^ `lib``fŽk ab `ros^p fkqbdo^ibpl_qbkfa^ pb ii^j^ c^jfif^ ab `ros^p nrb abmbkab ab pi njgj k\mƒh`omj,

Olo bgbjmil+ `r^kal i^ b`r^`fŽk afcbobk`f^i bp v&< 2+ i^ fkqbdo^`fŽk klp a^u < 0r * B+ v i^p `ros^p fkqbdo^ibp cloj^k rk^ c^jfif^ ab ob`q^p+qla^p `lk mbk,afbkqb fdr^i ^ 2- K^ `lkpq^kqb B obmobpbkq^ bi pbdjbkql nrb `^a^ rk^ ab bp^pob`q^p fkqbo`bmq^ pl_ob bi bgb t,

Rf i^ b`r^`fŽk afcbobk`f^i bp v&< s* i^ fkqbdo^`fŽk molar`b v < ds0 * B+ vi^p `ros^p fkqbdo^ibp cloj^k rk^ c^jfif^ ab m^oŠ_li^p `ljl pb sb bk i^ cfdro^ 7-6-Nqo^ sbw+ i^ `lkpq^kqb B klp fkaf`^ i^ fkqbopb``fŽk ab `^a^ `ros^ `lk bi bgb v-

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307 / iomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

t t

EHFTQ@ 7-6 @pmq\n dio`bm\g`n _` g\ `^p\^d‡i_da`m`i^d\g t% < s,

EHFTQ@ 7-7 @pmq\n dio`bm\g`n _` g\ `^p\^d‡i_da`m`i^d\g v&< v-

K^ cfdro^ 7-7 obmobpbkq i^ c^jfif^ ab `ros^p bumlkbk`f^ibp+ v < @`!* nrb plk`ros^p fkqbdo^ibpab i^ b`r^`fŽk afcbobk`f^i v&< v- Tk^ sbw jŠp+ B obmobpbkqbipbdjbkql fkqbo`bmq^alpl_ob bi bgb t, Dk bpqb`^pl+ B bp q^j_f‹k fdr^i ^ i^ mbk,afbkqb ab i^ `ros^ bk bi mrkql bk nrb `loq^ ^i bgbv-

Dk i^ cfdro^ 7-8 pb e^ obmobpbkq^alrk^ c^jfif^ ab ob`q^pkl m^o^ibi^p-Rlki^p `ros^p fkqbdo^ibpab i^ b`r^`fŽk

'7-30(

u i^ b`r^`fŽk

'7-31( t < `s ,iB1-

a^ rk^ c^jfif^ ab plir`flkbp nrb abmbkab ab rk plil m^oŠjbqol-Dpq^ bp rk^ c^jfif^ nrb mlpbb rk^ `iqjgq`io`* bpql bp+rk^ `ros^ nrb qfbkb i^molmfba^a ab nrb bk `^a^ rkl ab prp mrkqlp bp q^kdbkqb^ rk^ `ros^ ab i^ c^,

Page 439: Calculus

@pmq\n dio`bm\g`n v ^\hkjn _dm`^^dji\g`n 308

t

EHFTQ@ 7-8 @pmq\n dio`bm\g`n _` g\ `^p\^d‡i

_t 0 &_t'0_da`m`i^d\g t < s _s ,!3 _s ,

t

s s,( l

EHFTQ@ 7-0/ Pjgp^d‡i _` g\ `^p\^d‡i'7-3i( lp` ij k`mo`i`^` \ g\ a\hdgd\ _`

g\ `^p\^d‡i &6,20'

jfif^- ')( @nrŒ i^ bkslisbkqb bp u < s0 u pr doŠcf`^ bp i^ `ros^ ab qo^wlp ab i^cfdro^ 7-8- K^ bkslisbkqb ab rk^ c^jfif^ ab `ros^p fkqbdo^ibp bp ^ pr sbw rk^`ros^ fkqbdo^i ab_fal ^ nrb i^ mbkafbkqb v i^p `lloabk^a^p bk rk mrkql ab i^bkslisbkqb `lfk`fabk `lk i^p ab rk^ ab i^p `ros^p fkqbdo^ibp ab i^ c^jfif^- Dk bpqbbgbjmil+ bp cŠ`fi `ljmol_^o afob`q^jbkqb nrb v < s0 bp rk^ plir`fŽk ab '7-30(-N_p‹osbpb nrb bpq^ plir`fŽk m^oqf`ri^o kl mboqbkb`b ^ i^ c^jfif^ '7-31(- Orbabkl_qbkbopb lqo^p plir`flkbp nrb kl mboqbkb`bk ^ i^ c^jfif^ rkfbkal mlo`flkbp ab`ros^p ab i^ c^jfif^ `lk mlo`flkbp ab i^ bkslisbkqb- Dk i^ cfdro^ 7-0/ pb jrbpqo^rk bgbjmil- K^ ob`q^ q^kdbkqb bk = obpriq^ ab qlj^o B < , 1 bk '7-31( v i^q^kdbkqb bk > ^_ B < y-K^ plir`fŽk obpriq^kqb+ u < `%r&)sfbkb a^a^ ^pŒ9

v

+0U + 0

x%r&< wT/ ZZ J

1 05

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Dpq^ crk`fŽk ^ajfqb abofs^a^ v p^qfpc^`b i^ b`r^`fŽk afcbobk`f^i '7-30( m^o^ qlals^ilo ob^i ab s, Qbpriq^ bsfabkqb nrb mlo bi jfpjl mol`bafjfbkql mlaoŒ^k `lkp,qorfopb fkcfkfa^a ab bgbjmilp m^ob`falp- Dpqb bgbjmil e^`b sbo nrb mrbab kl pbocŠ`fi a^o qla^p i^p plir`flkbp mlpf_ibp ab rk^ b`r^`fŽk afcbobk`f^i-

Dp mlpf_ib ^ sb`bp bk`lkqo^o rk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk nrbpb p^qfpc^d^ mlo qla^p i^p `ros^p ab rk^ c^jfif^ ab rk plil m^oŠjbqol- Ub^jlp alpbgbjmilp-

')( u ob`Œmol`^jbkqb- `^a^ `ros^ ab i^ c^jfif^ bp q^kdbkqb ^ i^ bkslisbkqb-

Page 440: Calculus

31/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

DIDLOKN 0- G^ii^o rk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk ^ i^ nrb p^,qfpc^`bkqla^p i^p `fo`rkcbobk`f^p `lk `bkqol bk bi lofdbk-

Pjgp^d‡i, Tk^ `fo`rkcbobk`f^ `lk `bkqol bk bi lofdbk v o^afl B qfbkb mlob`r^`fŽk s0 * t0 < @0

Š Br^kal B s^ qlj^kal qlalp ilp s^ilobp mlpfqfslp+l_qb,kbjlp qla^p i^p `fo`rkcbobk`f^p `lk `bkqol bk bi lofdbk- O^o^ bk`lkqo^o rk^ b`r^,`fŽk afcbobk`f^i ab mofjbo loabk nrb qbkd^ bp^p`fo`rkcbobk`f^p `ljl `ros^p fkqb,do^ibp+_^pq^ abofs^o i^ b`r^`fŽk `^oqbpf^k^l_qbkfbkal 0s * 0tt% < N- @pŒmrbp+`^a^ `fo`rkcbobk`f^ p^qfpc^`bi^ b`r^`fŽk afcbobk`f^i t%< , s-t,

DIDLOKN 1- G^ii^o rk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk m^o^i^ c^jfif^ab qla^p i^p `fo`rkcbobk`f^p nrb m^p^kmlo bi lofdbk v nrb qfbkbkprp `bkqolp pl_obbi bgbs,

Pjgp^d‡i, Rf bi `bkqol ab rk^ `fo`rkcbobk`f^ bpqŠbk &@*N( v m^p^ mlo bilofdbk+bi qblobj^ ab OfqŠdlo^pklp af`b nrb `^a^ mrkql %r)s& ab i^ `fo`rkcbobk`f^p^qfpc^`bi^ b`r^`fŽk `^oqbpf^k^ %r * ?&/ * t0 < ?0

* nrb mrbab bp`of_fopb`ljlpfdrb

'7-32( s0 * t0 + 0@s < M

O^o^bk`lkqo^o i^ b`r^`fŽk afcbobk`f^i nrb qbkd^ bp^p`fo`rkcbobk`f^p `ljl `ros^pfkqbdo^ibp+abofs^jlp '7-32( l_qbkfbkal 0s * 0tt% + 1B < N+l

'7-33( s * tt% < `-

Orbpql nrb bpq^b`r^`fŽk `lkqfbkb a+pb p^qfpc^`b•kf`^jbkqb m^o^i^ `fo`rkcbobk,`f^ '7-32( `loobpmlkafbkqb ^i jfpjl B- O^o^ l_qbkbo rk^ b`r^`fŽk afcbobk`f^i nrbpb p^qfpc^d m^o^qla^p i^p `ros^p '7-32(+ ab_bjlp bifjfk^o B- OlaoŒ^jlp abofs^o'7-33( v l_qbkaoŒ^jlp 0 * tt! * &t%'0< N- Dpq^bp rk^ b`r^`fŽk afcbobk`f^i abpbdrkal loabk nrb pb p^qfpc^`bm^o^qla^p i^p `ros^p '7-32(- Olabjlp l_qbkbo rk^b`r^`fŽk ab mofjbo loabk bifjfk^kal B ^idb_o^f`^jbkqb bkqob '7-32( v '7-33(-Rrpqfqrvbkal s * Vd mlo b bk '7-32(+ l_qbkbjlp s0 * t0 + 0s&s * tt%'* b`r^,`fŽk ab mofjbo loabk nrb mlabjlp mlkbo bk i^ cloj^ t% < %s/ * r/&,%/rs&+

K^ cfdro^ 7-00 obmobpbkqil nrb pb ii^j^ rk ^\hkj _dm`^^dji\g ab rk^ b`r^,`fŽk afcbobk`f^i- Dp pfjmibjbkqb rk `lkgrkql ab mbnrb•lp pbdjbkqlp ob`qfiŒkblpq^kdbkqbp s^of^p `ros^p fkqbdo^ibp-Di bgbjmil m^oqf`ri^oobmobpbkq^albk i^ cfdr,o^ 7-00 bp rk `^jml afob``flk^i ab i^ b`r^`fŽk v&< t,

Orbab `lkpqorfopb rk `^jml afob``flk^i pfk obplisbo i^ b`r^`fŽk afcbobk`f^i-Rb bifdb rk mrkql+ mlo bgbjmil &\*]'* v pb `^i`ri^ bi k•jbol a`\* ]' l_qbkfal mloprpqfqr`fŽk bk bi pbdrkal jfbj_ol ab i^ b`r^`fŽk afcbobk`f^i t% < cn|)t', Rf bufp,qb rk^ `ros^ fkqbdo^inrb m^pbmlo bpb mrkql+ pr mbkafbkqbbk ‹i ab_b pbo fdr^i

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^ `_[) \&+ Olo `lkpfdrfbkqb+ pf af_rg^jlp rk qo^wl ob`qfiŒkblmlo bi `fq^al mrkql%[) \& u nrb qbkd^ bp^ mbkafbkqb+af`el mbnrb•l pbdjbkql ab ob`q^ cloj^oŠ m^oqbab rk `^jml afob``flk^i ab i^ b`r^`fŽk afcbobk`f^i- Cf_rg^kal s^oflp ab bplpqo^wlp+mlabjlp `lkpbdrfo rk^ fab^ `i^o^ abi `ljmloq^jfbkql dbkbo^i ab i^p`ros^p fkqbdo^ibp-@ sb`bp q^i fkcloj^`fŽk `r^ifq^qfs^ mrbab pbo i^ nrb pb mob`fpb-@asf‹oq^pb nrb mrkqlp afpqfkqlp 'N+\& bk bi bgb t lofdfk^k `ros^p fkqbdo^ibpafp,qfkq^p-Dpql klp a^ rk^ o^wŽkdblj‹qof`^ ab i^ ^m^of`fŽkab rk^ `lkpq^kqb ^o_fqo^of^^i fkqbdo^ork^ b`r^`fŽk ab mofjbo loabk-

0&** :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 01+ bk`lkqo^o rk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk nrbqbkd^ `ljl `ros^p fkqbdo^ibp i^ c^jfif^ ab `ros^p a^a^-.+ 0s * 1t < B- 5- s0 * t0 * 0@t < 0-0, V < @`+/!+ 6- t < @&s+ g'`!,1, s0 + t0 < B- 7- t2&U * 1( < B'u , 1(-1+ st < B- 8- t < B `lp s,3, t0 < @s, 0/- ^o`q^k t * ^o`pbk s < B-

00- Sla^p i^p `fo`rkcbobk`f^p nrb m^p^k mlo ilp mrkqlp '0+N( X ', 0+N(-01- Sla^p i^p `fo`rkcbobk`f^p nrb m^p^k mlo ilp mrkqlp '0+ 0( X ', 0+ , 0(-

Dk i^ `lkpqor``fŽk abi `^jml afob``flk^i ab rk^ b`r^`fŽk afcbobk`f^i+ ^ sb`bp bi qo^_^glmrbab ^_obsf^opb `lkpfabo^_ibjbkqb pf pfqr^jlp mofjbol ^nrbiilp mrkqlp bk ilp nrb i^ mbk,

Page 442: Calculus

311 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

afbkqbu&qbkd^ rk s^ilo `lkpq^kqb B- O^o^ `^a^ a+ bplp mrkqlp bpqŠkbk rk^ `ros^ ii^j^a^dnj^gdi\,02- Cf_rg^o i^p fpl`ifk^p+ `loobpmlkafbkqbp^ i^p mbkafbkqbp lkpq^kqbph+0+ v 1 m^o^ i^

b`r^`fŽk afcbobk`f^iv&< s0 * t0, Blk i^ ^vra^ ab i^p fpl`ifk^p+`lkpqorfo rk `^jml af,ob``flk^i m^o^i^ b`r^`fŽk b fkqbkq^oi^ abqbojfk^`fŽk ab i^ cloj^ ab i^ `ros^ fkqbdo^inrb m^p^mlo bi lofdbk-

03- Cbjlpqo^o nrb i^p fpl`ifk^p ab i^ b`r^`fŽk afcbobk`f^i v&< s * v cloj^k rk^ c^jfif^ab ob`q^pabmbkafbkqbab rk m^oŠjbqol- So^w^oi^p fpl`ifk^p `loobpmlkafbkqbp^ i^p mbk,afbkqbp`lkpq^kqbpN+€q+€ 0+ € + € 1- Blk i^ ^vra^ ab i^p fpl`ifk^p+ `lkpqorfo rk`^jml afob```flk^i u bp_lw^oi^ `ros^ fkqbdo^inrb m^p^mlo bi lofdbk- Tk^ ab i^p `ros^pfkqbdo^ibpbp q^j_f‹k rk^ fpl`ifk^: e^ii^oi^-

04- So^w^os^of^p fpl`ifk^p u `lkpqorfo rk `^jml afob``flk^i m^o^i^ b`r^`fŽk-

_t &_t'0V :r* * ,^r ^r

Rf pb af_rg^ `lk `rfa^al bi `^jml afob``flk^i+ pb mrbab abqbojfk^o rk^ c^jfif^ ab pl,ir`flkbp abmbkafbkqbab rk m^oŠjbqol ^ m^oqfoabi ^pmb`ql abi `^jml afob``flk^i-

7-12 D`r^`flkbp pbm^o^_ibpab mofjbo loabk

Tk^ b`r^`fŽk afcbobk`f^i ab mofjbo loabk ab i^ cloj^ u&< `%r)u( bk i^ nrbbi pbdrkal jfbj_ol `%r)u( pb bp`fkab bk bi molar`ql ab alp c^`qlobp+rkl abmbk,afbkqb ab s q^k pŽil v bi lqol •kf`^jbkqb ab v+ pb abkljfk^ `^p\^d‡i n`k\m\]g`,Dgbjmilp mrbabk pbo v&< s1

* v&< v+ v&< pbkv ild s* v&< s-ob v+ bq`- @pŒrk^b`r^`fŽk pbm^o^_ibmrbab bumobp^opbbk i^ cloj^

t%< M%r&N%s&)

alkab O v Q plk crk`flkbp a^a^p- Br^kal N%s&:.: N+mlabjlp afsfafo mlo N%s&vmlkbo i^ b`r^`fŽk bk i^ cloj^

=%s&s$< N&s'*

pfbkal =%s&< f,N%s&+ Di qblobj^ nrb pfdrb klp fkaf`^ `Žjl bk`lkqo^o rk^cŽojri^ fjmiŒ`fq nrb pb p^qfpc^d m^o^`r^inrfbo plir`fŽk ab rk^ q^i b`r^`fŽk-

RCMPCK? 7-0/- P`\ u < V&s' pi\ njgp^d‡i ^p\glpd`m\ _` g\ `^p\^d‡i _da`+m`i^d\g n`k\m\]g`

%5+12& =%s&s$< M%r&

o\g lp` X&n`\ ^jiodip\ `i pi dio`mq\gj \]d`moj g,Ppkjib\hjn lp` P u g\ api^d‡i^jhkp`no\ > l W nji \h]\n ^jiodip\n `i g,P`\ E ^p\glpd`m kmdhdodq\_` >* `noj

Page 443: Calculus

B^p\^dji`n n`k\m\]g`n _` kmdh`mjm_`i 312

`n* ^p\glpd`m api^d‡i o\g lp` F&< >, Bioji^`n g\ njgp^d‡i X n\odna\^` g\ a‡mhpg\dhkg…^do\,

'7-35( C%s&< G M%r&^r * b

k\m\ pi ^d`mojq\gjm _` B- O`^…kmj^\h`io`* ndt n\odna\^` '7-35( `ioji^`n t `n pi\njgp^d‡i _` &6,23',

A`hjnom\^d‡i, Orbpql nrb X bp rk^ plir`fŽk ab '7-34(+ ab_b pbo

%5+14& =WU%r&Ys$%r&< M%r&

m^o^`^a^ r ab g, X^ nrb F&< >*bp^ b`r^`fŽk pb `lksfboqb bk

C$WU%r&YU$%r&< M%r&+

Obol+pbd•k i^ obdi^ ab i^ `^abk^+ bi mofjbo jfbj_ol bp i^ abofs^a^ ab i^ crk`fŽk`ljmrbpq^ F l X- Olo `lkpfdrfbkqb F l X bp rk^ mofjfqfs^ ab P+ 0/ `r^i pfdkfcf`^nrb

'7-37( CWU%r&Y< G M%r&^r * b

m^o^rk `fboql s^ilo ab B- Dpq^bp i^ obi^`fŽk '7-35(- Qb`Œmol`^jbkqb+pf t < U%r&p^qfpc^`b'7-35(+ i^ abofs^`fŽk klp a^ '7-36(+ 0/ nrb abjrbpqo^ nrb X bp rk^ pl,ir`fŽk ab i^ b`r^`fŽk afcbobk`f^i '7-34(-

Kjo\8 K^ cŽojri^ fjmiŒ`fq^ '7-35( q^j_f‹k mrbab bumobp^opb bk crk`fŽk ab >,@ m^oqfoab '7-36( qbkbjlp

H=WU%r&YU$%r&r < G M%r& r * _-

Rf e^`bjlp i^ prpqfqr`fŽk+ s < U%r&) ^s < U$%r& r bk i^ fkqbdo^i ab i^ fwnrfboa^+ i^b`r^`fŽk pb qo^kpcloj^ bk

'7-38( G= %s&s < GM%r& r * `-

Orbpql nrb i^ fkqbdo^i fkabcfkfa^ ` =%s&^s obmobpbkq^ rk^ mofjfqfs^ `r^inrfbo^ ab =)i^ b`r^`fŽk '7-38( bp lqo^ j^kbo^ ab bp`of_fo '7-35(-

Dk i^ moŠ`qf`^+ i^ cŽojri^ '7-38( pb l_qfbkb afob`q^jbkqb ab '7-34( mlo rk mol`bpljb`Škf`l- Dk i^ b`r^`fŽk afcbobk`f^i '7-34( mlkbjlp i^ abofs^a^ v&bk i^ cloj^ ^s,^ru i^ `lkpfabo^jlp `ljl Tk `l`fbkqb l_qbkfbkal i^ obi^`fŽk =%s&^s < M%r&^r+ Olkbjlpirbdl ilp pfdklp ab fkqbdo^`fŽk bk ^j_lp jfbj_olp ab bp^ b`r^`fŽk v prj^jlp i^ `lkp,

Page 444: Calculus

-+- Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

q^kqb b l_qbkfbkal '7-38(- Di qblobj^ 7-0/ molmlo`flk^ i^ grpqfcf`^`fŽk ab bpqb mol`bpljb`Škf`l+ bi `r^i bp lqol bgbjmil ab i^ bcf`^`f^ ab i^ klq^`fŽk ab Kbf_kfw-

DIDLOKN- K^ b`r^`fŽk kl ifkb^iuv& * v < v1 bp pbm^o^_ibv^ nrb mrbabbp`of_fopbbk i^ cloj^

'7-4/(t% 0

< ,t&t + 0( r

`lk q^inrb bu'u , N :.: N X s :.: N- Dk bpqb`^pl i^p alp crk`flkbp u < N b t < 0plk bsfabkqbjbkqb plir`flkbp ab st% * v < t0, K^p obpq^kqbpplir`flkbp+ pf bufp,qbk+p^qfpc^`bk'7-4/( v+ mlo q^kql+pbd•k bi qblobj^ 7-0/ q^j_f‹k p^qfpc^`bk

` _t *` _s * Hs%s * 0( t

m^o^ rk `fboql s^ilo ab i^ `lkpq^kqb G+ Orbpql nrb bi fkqbdo^kal abi mofjbojfbj_ol bp g-&t + M , g-t* `r^kal fkqbdo^jlp+ bk`lkqo^jlp nrb

i^d Es * 00 , i^d Guh< i^d Zu\* G+

Dpql klp a^ G'u, k.uh < Erf_G l 'u , M.W< ?r m^o^ rk `fboql s^ilo ab B-Cbpmbg^kal v+ l_qbkbjlp i^ cŽojri^ bumiŒ`fq^

'7-40( 0s:**+

0, Bu

Di qblobj^ 7-0/ klp af`b nrb m^o^rk s^ilo `r^inrfbo^ ab B nrb bifg^jlp+ bp^ tbp rk^ plir`fŽk: mlo `lkpfdrfbkqb bk bpqbbgbjmil ebjlp abqbojfk^al qla^p i^pplir`flkbp9 i^p crk`flkbp `lkpq^kqbp v < M b v < 0 X qla^p i^p crk`flkbp abcfkf,a^p mlo '7-4/- N_p‹osbpb nrb m^o^B < N pb l_qfbkb i^ plir`fŽk `lkpq^kqb v < 0-

0&*, :WR_PVPV\`

Dk ilp Dgbo`f`flp abi 0 ^i 01+ prmlkbo nrb bufpqbk plir`flkbp u bk`lkqo^o rk^ cŽojri^fjmiŒ`fq^^ i^ nrb p^qfpc^d^ki^p plir`flkbp-

/, t% < Ud-t0,1- q^k s `lp t < +t%q^kt,1, %r * f&s$ * s/ < N-1+s$ < %s* f&%s* 1(-

2+sws$:r+3+ %r * f&s$ < rs+

6- 'i , T/&.,/s$ * 0 * s/ < N-

6, rs%f * r/&s$ * 'i * s/& < N-6+ %r/

* 1&s$< s+.-+ rss$ < 0 * r/ * s/ * T/s/+

..+ ss$ < `\8)0q pbk s,

./+ r^r * s ^s < rs%r^s * s^r&+

Page 445: Calculus

B^p\^dji`n cjhjb„i`\n _` kmdh`m jm_`i 314

Dk ilp Dgbo`f`flp 02 ^i 05 bk`lkqo^o crk`flkbp `+ `lkqfkr^p bk qlal bi bgb ob^i+ nrbp^qfpc^d^k i^p `lkaf`flkbp a^a^p- Br^kal pb^ cŠ`fi e^ii^oi^p qla^p+ e^`boil: bk qlal `^pl-e^ii^o bi j^vlo k•jbol mlpf_ib-

/1, a&s' < 1 * PR@L _o,/2, a&s'a&s' < 3s* a`L' < 0-/3, &%&s'* 0s``%r& < N+ a`L' < l-.3+ a0&s' * Xe%&U'Z0< 0- Jin[7 `%r& < , 0 bp rk^ plir`fŽk-

06- Tk^ crk`fŽk ` kl kbd^qfs^+ `lkqfkr^ bk qlal bi bgb ob^i+ qfbkb i^ molmfba^a abnrb pr `lkgrkql ab loabk^a^p `loobpmlkafbkqb ^ rk fkqbos^il ^o_fqo^ofl qfbkb Šob^ mol,mlo`flk^i ^ i^ ilkdfqra abi fkqbos^il- G^ii^o `+

07- Qbplisbo bi Dgbo`f`fl 06 pf bi Šob^ bp molmlo`flk^i ^ i^ afcbobk`f^ ab ilp s^ilobp ab i^crk`fŽk bk ilp buqobjlp abi fkqbos^il-

08- Qbplisbo bi Dgbo`f`fl 07 prpqfqrvbkal ~afcbobk`f^‚ mlo ~prj^‚-1/- Qbplisbo bi Dgbo`f`fl 07 prpqfqrvbkal ~afcbobk`f^‚ mlo ~molar`ql‚-

0&*- :PbNPV\[R`U\Z\Tn[RN` QR]_VZR_\_QR[

Blkpfabobjlp ^elo^ rk qfml bpmb`f^i ab b`r^`fŽk ab mofjbo loabk

'7-41( s$ < a&s*t'*

bk i^ nrb bi pbdrkal jfbj_ol qfbkb rk^ molmfba^a `lkl`fa^ mlo cjhjb`i`d_\_*i^ `r^i `lkpfpqb bk

'7-42( Wnnr)ns&< Wnr)s&

m^o^ qlal s* t* v qlal o ;/; N- Dp ab`fo+ prpqfqrvbkal s mlo os b t mlo ot kl `^j_f^bi s^ilo ab ed~*s&+ K^p b`r^`flkbp ab i^ cloj^ '7-41( nrb qfbkbk bpq^ molmfba^a pbabkljfk^k cjhjb„i`\n '^idrk^p sb`bp q^j_f‹k pb ii^j^k cjhjb„i`\n _` bm\_j^`mj', Olo bgbjmil9

* t [ s * &s0 * t0'1 * U s0 * t0

t < ,, + t < ,,,+ t < ,pbk +t * s st t s0+g

@mif`^kal '7-42( `lk o < g-s* i^ b`r^`fŽk afcbobk`f^i bk '7-41( pb qo^kpcloj^ bk9

t% < i^d s + Hldu +

'7-43(

K^ mobpbk`f^ abi `l`fbkqb t-s bk bi pbdrkal jfbj_ol+ prdfbob i^ fab^ ab fkqol,ar`fo rk^ krbs^ crk`fŽk fk`Ždkfq^ q `lk q < tZz, Dkqlk`bp u < qs* d:: q%s* q* v bpq^ prpqfqr`fŽk qo^kpcloj^ '7-43( bk9

_qq%s * q < a&g*q' l s + < a&g*q' + q,

_s

Page 446: Calculus

-+/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

K^ •iqfj^ b`r^`fŽk bp pbm^o^_ibab mofjbo loabk `lk i^ fk`Ždkfq^ q, Olabjlprqfifw^o bi qblobj^ 7-0/ v l_qbkbo rk^ cŽojri^ fjmiŒ`fq^ m^o^ q v obbjmi^w^obkqlk`bp q mlo s,r `lk il nrb pb l_qfbkb rk^ cŽojri^ fjmiŒ`fq^m^o^s+

DIDLOKN- Qbplisbo i^ b`r^`fŽk afcbobk`f^i s$ < %s* r&,%s * r&+

Pjgp^d‡i, K^ b`r^`fŽk pb bp`of_b bk i^ cloj^

G t-s + )C < -

sYr * 0

K^ prpqfqr`fŽk q < t-s i^ qo^kpcloj^ bk

_q q + 0 0 * q0

r*:***p:***+_s q * 0 q * 0

@mif`^kal bi qblobj^ 7-0/ pb l_qfbkb

` q ` 0 ` _s++ _q * ,, _q < , , * b-0 * q0 0 * q0 s

K^ fkqbdo^`fŽk a^

pild '0 * q/& * ^o`q^k q < ,ild Zu\* _-

Qbbjmi^w^kal q mlo tZ~* qbkbjlp

qild &s0 * s0' + qild s0 * ^o`q^k9 99< ,ild Zu\* a+

s

v mrbpql nrb ild s0 < 1 ild Gtz+obpriq^

qild %r/ * f&* ^obq^k9 99< _-s

K^p plir`flkbp ab rk^ b`r^`fŽk eljld‹kb^ t% < `%r) t' qfbkbk ^idrk^p mol,mfba^abp dblj‹qof`^p fkqbobp^kqbp-@kqb qlal+ cŠ`fijbkqb pb abjrbpqo^ nrb i^pob`q^pnrb m^p^kmlo bi lofdbk plk fpl`ifk^p ab i^ b`r^`fŽk- Qb`loabjlp nrb rk^fpl`ifk^ ab s$ < `%r) s& bp rk^ `ros^ ^ il i^odl ab i^ `r^i i^ mbkafbkqbs$ bp `lkp,q^kqb-Dpq^molmfba^a pb l_pbos^ bk i^ cfdro^ 7-01 nrb jrbpqo^ rk `^jml afob`,`flk^i ab i^ b`r^`fŽk afcbobk`f^i s$ < , /s,r+

Page 447: Calculus

B^p\^dji`n cjhjb„i`\n _` kmdh`m jm_`i 316

K^ fpl`ifk^ `loobpmlkafbkqb ^ i^ mbkafbkqb` qfbkb ab b`r^`fŽk +0t-s;^*jt;+o^s*t mlo q^kql bp rk^ ob`q^ nrb m^p^mlo bi lofdbk v ab mbkafbkqb, p`-O^o^ abjlpqo^o i^ molmfba^a bk dbkbo^i `lkpfa‹obpb i^ ob`q^ ab mbkafbkqbh nrbm^p mlobi lofdbk+`rv^ b`r^`fŽk bpu <gr m^o^qlal %r)u( v bk m^oqf`ri^o+bi mrkql%.+g& mboqbkb`b i^ ob`q^- RrmŽkd^pb^elo^+mlo o^wŽkab pfjmif`fa^a+ nrb mlo

`^a^ mrkql ab i^ ob`q^ v < hs m^p^ rk^ `ros^ fkqbdo^i- K^ mbkafbkqb ab i^`ros^ fkqbdo^i nrb m^p^ mlo bi mrkql %[)\& ab bpq^ ob`q^ bp `%[) \& < `%[)g[&+Rf^"N pb mrbab rqfifw^oi^molmfba^aab eljldbkbfa^a '7-42( bp`of_fbkal `%[)g[&:

t

,,,s

EHFTQ@ 7-01 @\hkj _dm`^^dji\g _` g\ `^p\^d‡i _da`m`i^d\g v&< , 0t-s, I\n dnj^gdi\nnji m`^o\n lp` k\n\i kjm `g jmdb`i,

d%.) g&+ Dpab`fo+pf %[) \& 99E, 'N+N(i^ `ros^ fkqbdo^inrb m^p^ mlo %[) \& qfbkb i^jfpj^ mbkafbkqbnrb i^ `ros^ fkqbdo^i nrb m^p^ mlo '0+ g&+ Olo q^kql+ i^ ob`q^

Page 448: Calculus

317 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

t < hs bp rk^ fpl`ifk^+ `ljl pb e^_Œ^ af`el- 'Rb mlaoŒ^ mol_^o q^j_f‹k nrb‹pq^p plk i^p •kf`^p fpl`ifk^p ab rk^ b`r^`fŽk eljld‹kb^-(

Dpq^ molmfba^a& ab i^p fpl`ifk^p prdfbob rk^ molmfba^a ab i^p `ros^p fkqb,do^ibp `lkl`fa^ mlo diq\md\i^d\ m`nk`^oj \ g\n om\inajmh\^dji`n kjm n`h`e\iu\,Qb`r‹oabpb nrb rk^ pbjbg^kw^ qo^kpcloj^ rk `lkgrkql R bk rk krbsl `lkgrkqlfP l_qbkfal jriqfmif`^kal i^p `lloabk^a^p ab `^a^ mrkql ab R mlo rk c^`qlo`lkpq^kqb e = l- B^a^ ob`q^ nrb m^p^ mlo bi lofdbk pb qo^kpcloj^ bk pŒjfpj^mlo rk^ pbjbg^kw^- Olo q^kql+ i^p fpl`ifk^p ab i^p b`r^`flkbp afcbobk`f^ibp eljl,d‹kb^p nrba^k fks^of^kqbp mlo qo^kpcloj^`flkbp ab pbjbg^kw^+ v bk `lkpb`rbk`f^il jfpjl ab_b l`roofo ^i `^jml afob``flk^i- Dpql prdfbob nrb i^p qo^kpcloj^`flkbpmlo pbjbg^kw^ qo^kpcloj^k `ros^p fkqbdo^ibp bk `ros^p fkqbdo^ibp- O^o^ abjlpqo^oil^k^iŒqf`^jbkqb prmŽkd^pb nrb R bp rk^ `ros^ fkqbdo^i abcfkfa^ mlo rk^ cŽojri^bumiŒ`fq^9

'7-44( t < B%r&+

Cb`fo nrb R bp rk^ `ros^ fkqbdo^i ab t% < `%r)t' pfdkfcf`^ nrb9

'7-45( B$%r&< `n|8B%r|

m^o^ `^a^ r bk `lkpfabo^`fŽk- Rb^ ^elo^ %r) s& rk mrkql `r^inrfbo^ ab fP, Orbpqlnrb bi mrkql %r,e) s,e& mboqbkb`b ^ R+ prp `lloabk^a^p p^qfpc^`bk '7-44(+ abalkab s,e < B%r,e& l s < eB%r,e&+ Dp ab`fo+ i^ `ros^ eO bpqŠ abcfkfa^ mlo i^b`r^`fŽk t < C%r&)alkab C%r& < eB%r,e&+ N_p‹osbpb nrb i^ abofs^a^ ab F bpqŠ

a^a^ mlo9

D%&s'< fC%&x' , c< C%&x' ,

O^o^ mol_^o nrb fP bp rk^ `ros^ fkqbdo^i ab t% < `%r) t' _^pq^ mol_^o nrbC$%r&< `%r) C%r| l il nrb bp il jfpjl+ nrb

'7-46(

Obol+ pf pb prpqfqrvb s mlo s-f bk i^ b`r^`fŽk '7-45( v pb ^mif`^ i^ molmfba^a abeljldbkbfa^a `lk o < f pb l_qfbkb9

il `r^i abjrbpqo^ '7-46(: bp ab`fo+ fP bp rk^ `ros^ fkqbdo^i pfbjmob nrb il pb^ P,Tk bgbjmil pbk`fiil bk bi nrb bpq^ molmfba^a dblj‹qof`^ bp bsfabkqb bp i^ b`r^,

Page 449: Calculus

>gbpijn kmj]g`h\n a…nd^jnt b`jh„omd^jn -+2

`flk eljld‹kb^ t%< , s-t* `rv^p `ros^p fkqbdo^ibp cloj^k rk^ c^jfif^ ab `fo,`rkcbobk`f^p `lk`‹kqof`^p `lk rk m^oŠjbqol+ a^a^ mlo i^ b`r^`fŽk s0 * t0 < B-

Rb mrbab abjlpqo^o q^j_f‹k nrb pf i^p `ros^p fkqbdo^ibp ab rk^ b`r^`fŽkab mofjbo loabk plk fks^of^kqbp mlo i^p qo^kpcloj^`flkbp ab pbjbg^kw^+ i^ b`r^,`fŽk afcbobk`f^i bp kb`bp^of^jbkqb eljld‹kb^-

7-15 Dgbo`f`flp

0- Ool_^o nrb i^ prpqfqr`fŽk v < r f i qo^kpcloj^ i^ b`r^`fŽk eljld‹kb^ v&< `%r) v( bk rk^b`r^`fŽk ab mofjbo loabk bk q bk i^ nrb i^p s^of^_ibp bpqŠk pbm^o^a^p-@idrk^p sb`bpbpq^ prpqfqr`fŽk `lkar`b ^ fkqbdo^ibpnrb plk jŠp cŠ`fibp ab `^i`ri^o nrb i^p l_qbkfa^pmlo i^ prpqfqr`fŽk v < sq bpqraf^a^ bk bi qbuql-

Qbplisbo i^p b`r^`flkbp afcbobk`f^ibp ab ilp Dgbo`f`flp 1 ^i 00-

'[0, s$ <z-

t

1, s$ < 0 * y-s

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3, &0a + s0't% * 1st < N-

3+ sg% < t + T s0 * t0,

5, s0t% * st * 1c < N-

6, t0 * &s0 + st * a't% < N-

* t&s0 * st * t0'

7, V < s&s0 * 1st * t0'%

.-+ s$ < g* upbk g,s s

00- s&t * 2s't% * t&s * 2t' < N-

7-16 @idrklp mol_ibj^p cŒpf`lpv dblj‹qof`lp nrb `lkar`bk ^ b`r^`flkbp abmofjbo loabk

Rbdrfa^jbkqb `ljbkq^jlp ^idrklp bgbjmilp ab mol_ibj^p cŒpf`lp v dblj‹,qof`lp nrb `lkar`bk ^ b`r^`flkbp afcbobk`f^ibp ab mofjbo loabk nrb plk eljl,d‹kb^p l pbm^o^_ibp-

Qm\t`^ojmd\n jmojbji\g`n, Cb`fjlp nrb alp `ros^p pb `loq^k jmojbji\gh`io`bk rk mrkql pf prp q^kdbkqbp `lk ‹i plk mbombkaf`ri^obp- Tk^ `ros^ nrb `loq^loqldlk^ijbkqb ^ qla^p i^p `ros^p ab rk^ c^jfif^ pb ii^j^ qo^vb`qlof^ loqldlk^iab i^ c^jfif^- K^ cfdro^ 7-02 jrbpqo^ ^idrklp bgbjmilp- Klp mol_ibj^p obi^qfslp^ qo^vb`qlof^p loqldlk^ibp plk fjmloq^kqbp bk L^qbjŠqf`^ mro^ v ^mif`^a^-Olo bgbjmil+ bk i^ qbloŒ^ab cirgl ab cirfalp+ alp c^jfif^p loqldlk^ibp ab `ros^ppb ii^j^k g…i`\n `lpdkjo`i^d\g`n v g…i`\n _` ^jmmd`io` obpmb`qfs^jbkqb- Dk i^ qbloŒ^abi `^ilo+ pb ii^j^k g…i`\n dnjo`mh\n v g…i`\n _` agpej,

Page 450: Calculus

32/ Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Rrmlkd^jlp a^a^ rk^ c^jfif^ ab `ros^p nrb p^qfpc^`bkrk^ b`r^`fŽk afcbobk,`f^i ab mofjbo loabk+ mlo bgbjmil

'7-47( s$ <Wnr)s& +

Di k•jbol `%r)s& bp i^ mbkafbkqb ab rk^ `ros^ fkqbdo^i nrb m^p^ mlo 'u+ s&+

K^ mbkafbkqbab `^a^ qo^vb`qlof^loqldlk^i bk bpb mrkql bp bi ob`Œmol`lkbd^qfsl* nf`%r)s&) ab jlal nrb i^p qo^vb`qlof^ploqldlk^ibp p^qfpc^`bki^ b`r^`fŽk af,cbobk`f^i

'7-48(G Gs:***

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Rf '7-47( bp pbm^o^_ib+bkqlk`bp '7-48( bp q^j_f‹k pbm^o^_ib-Rf '7-47( bp eljl,d‹kb^+ q^j_f‹k '7-48( il bp-

DIDLOKN 0- G^ii^o i^p qo^vb`qlof^ploqldlk^ibp ab i^ c^jfif^ ab qla^p i^p`fo`rkcbobk`f^p nrb m^p^kmlo bi lofdbk v qfbkbk prp `bkqolp bk bi bgbs,

Pjgp^d‡i, Dk bi bgbjmil 1 ab i^ Rb``fŽk 7-10 pb bk`lkqoŽ nrb bpq^c^jfif^qfbkb mlo b`r^`fŽk `^oqbpf^k^ r/ * v1 , /?r < M v nrb p^qfpc^`bi^ b`r^`fŽk af,cbobk`f^i v&< %s/* r0&,%/rs&+ Qbbjmi^w^kal bi pbdrkal jfbj_ol mlo pr ob`Œ,mol`l kbd^qfsl+ bk`lkqo^jlp nrb i^p qo^vb`qlof^ploqldlk^ibp p^qfpc^`bki^ b`r^`fŽkafcbobk`f^i

G 0stt < -

s0 +g

Dpq^b`r^`fŽk eljld‹kb^ mrbab fkqbdo^opb lk i^ prpqfqr`fŽk t < RT) v pb iibd^^ i^ c^jfif^ ab `ros^p fkqbdo^ibp

r0 * t0 + 0@t < M Š

Dpq^bp rk^ c^jfif^ ab `fo`rkcbobk`f^p nrb m^p^kmlo bi lofdbk v qfbkbkprp `bkqolpbk bi bgbt, @idrk^p ab bp^p`fo`rkcbobk`f^p bpqŠkaf_rg^a^p bk i^ cfdro^ 7-02-

Mmj]g`h\n _` k`mn`^p^d‡i, Tk mrkql P bpqŠprgbql ^ jlsbopb ^ il i^odl abrk^ `ros^ mi^k^ BhŠ Nqol mrkql M bk bi jfpjl mi^kl ~mbopfdrb‚ bi mrkql P-

Dpql bp+M pb jrbsb ab q^i j^kbo^ nrb pr afob``fŽk ab jlsfjfbkql bpqŠpfbjmoblofbkq^a^ e^`f^ P- Di mrkql M mlo q^kql abp`of_b lqo^ `ros^ B1 ii^j^a^ ^pmq\ _`k`mn`^p^d‡i, Dk bi bgbjmil obmobpbkq^albk i^ cfdro^ 7-03+ i^ `ros^ @*bp bi bgbv-Dk rk mol_ibj^ ab mbopb`r`fŽk pb abpb^ bk`lkqo^o i^ `ros^ B1 `r^kal pb `lkl`b

Page 451: Calculus

>gbpijn kmj]g`h\n a…nd^jnv b`jh„omd^jn 320

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i^ `ros^ b v `fboq^ fkcloj^`fŽk ^af`flk^i obi^qfs^ ^ O v P+ mlo bgbjmil+ rk^ ob,i^`fŽk bkqobprp mlpf`flkbp l prp sbil`fa^abp-

Br^kal ab`fjlp nrb i^ afob``fŽk ab jlsfjfbkql ab O bpqŠpfbjmob lofbkq^a^e^`f^ P+ nrbobjlp pfdkfcf`^onrb i^ q^kdbkqb^ B1 bk O m^p^mlo P- Olo il q^kql+pf abpfdk^jlp `lk %r)u( i^p `lloabk^a^p ob`q^kdri^obpab O bk rk fkpq^kqba^al+ u

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K^ fkcloj^`fŽk ^af`flk^i loafk^of^jbkqb klp mbojfqb `lkpfabo^o W b X `ljl crk,`flkbp `lkl`fa^p ab s b v+ bk `rvl `^pl i^ b`r^`fŽk '7-5/( pb `lksfboqb bk rk^b`r^`fŽk afcbobk`f^i ab mofjbo loabk `lk i^ fk`Ždkfq^ v- Blkpfabobjlp ^elo^ rkbgbjmil bk bi nrb bp^ b`r^`fŽk bp pbm^o^_ib-

DIDLOKN 1- Tk mrkql P pb jrbsb pl_ob rk^ ob`q^ Bh+v rk mrkql O mbopf,drb ^ O ab j^kbo^ nrb i^ afpq^k`f^ ab O ^ O qbkd^ rk s^ilo `lkpq^kqb f = N-Rf M kl bpqŠfkf`f^ijbkqb pl_ob ah&e^ii^o i^ `ros^ ab mbopb`r`fŽk-

Page 452: Calculus

321 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

Pjgp^d‡i, Sljbjlp B+ `ljl bgb t v pfqrbjlp M fkf`f^ijbkqb bk bi mrkql%e)N(- Orbpql nrb i^ afpq^k`f^ ab L ^ P bp e) ab_b pbo'W , U'0 * %U* t'0 < e!+Obol W < M pl_ob Bi&`lk il nrb qbkbjlp V + t; qf0 + s0* u i^ b`r^`fŽk af,cbobk`f^i '7-5/( pb `lksfboqb bk

Sf0 [ s0

s$:****'[

HkqbdoŠkali^ `lk i^ ^vra^ ab i^ prpqfqr`fŽk r < e `lp o v rqfifw^kal bi eb`el abnrb u < N `r^kal r < e+ l_qbkbjlp i^ obi^`fŽk

e * T e0+ s0

x - 1 1V < f ild ,,,,, , s f + r +

r

K^ `ros^ ab mbopb`r`fŽk bk bpqb bgbjmil pb ii^j^ om\^omdu9bpqŠ af_rg^a^ bk i^cfdro^ 7-03-

P\gd_\ _` pi g…lpd_j kjm pi jmdad^dj, Rb prmlkb nrb rk abmŽpfql'kl kb`b,p^of^jbkqb `fiŒkaof`l( `lkqfbkb rk iŒnrfal- Di iŒnrfal p^ib abi abmŽpfql ^ qo^s‹pab rk lofcf`fl ab _loabp mbocb`q^jbkqb mrifjbkq^alp- Rf kl er_fbo^ olw^jfbkql'v mlo q^kql kl er_fbo^ m‹oafa^ ab bkbodŒ^(i^ sbil`fa^a ab p^ifa^ pboŒfdr^i ^ptas jbqolp mlo pbdrkal+ alkab v bumobp i^ ^iqro^ 'bk jbqolp( abpab _f lofcf`fl^ i^ prmbocf`fbif_ob abi iŒnrfal- ')( 'U‹^pb cfd- 7-04-( Rf >j bp bi Šob^ 'bk jbqolp`r^ao^alp( abi lofcf`fl+ bkqlk`bp >jS0bt obmobpbkqbi k•jbol ab jbqolp `•_f`lpmlo pbdrkal ab iŒnrfal nrb p^ib mlo bi lofcf`fl- @ `^rp^ abi olw^jfbkql+ bi `eloolpb `lkqo^b rk ml`l+ v i^ sbil`fa^a ab abp`^od^ bp ^molufj^a^jbkqb ^>jS0bt*alkab ` bp rk k•jbol abqbojfk^al bumbofjbkq^ijbkqb u nrb pb abkljfk^ ^j`ad+^d`io` _` _`n^\mb\, Noafk^of^jbkqb+ m^o^ lofcf`flp mrifalp bi s^ilo ^molufj^alab ` bp /+5/- G^`fbkal rpl ab bpqlp a^qlp v qlj^kal d < 8+7 pb bk`rbkqo^ nrb i^sbil`fa^a ab p^ifa^ bp ^molufj^a^jbkqb fdr^i ^ 1+54qV jbqolp mlo pbdrkal vnrb i^ sbil`fa^a ab abp`^od^ bp 1+54 =j qV jbqolp `•_f`lp mlo pbdrkal-

Rb^ R%s&bi slirjbk ab iŒnrfal `lkqbkfal bk bi abmŽpfql `r^kal i^ ^iqro^abi iŒnrfal bp t, Rf bi Šob^ ab i^ pb``fŽk ob`q^ abi abmŽpfql^ i^ ^iqro^ p bp >&p'*pb qfbkb S&t' < `w>&p' _p* ab alkab obpriq^ _S -_t < >&t', Cb il af`el bk bimŠoo^cl^kqboflo obpriq^ nrb bi `lbcf`fbkqb ab s^of^`fŽk abi slirjbk `lk obpmb`ql^i qfbjml bp _S - _o < 0*43>j ru jbqolp `•_f`lp mlo pbdrkal+ alkab bi pfdkljbklp pb ab_b ^ nrb bi slirjbk ab`ob`b- @mif`^kal i^ obdi^ab i^ `^abk^+ obpriq^9

_ S [ _ S _ V [ >& ' _ V_o + _t _o + V _o %

')( Rf rk^ m^oqŒ`ri ab j^p^ g `^b if_objbkqb ^ il i^odl ab rk^ afpq^k`f^ u u ^i`^kw^ rk^sbil`fa^a q* pr bkbodŒ^ fk‹qf`^ dhq0 e^ ab pbo fdr^i ^ i^ bkbodŒ mlqbk`f^i hbt 'bi qo^_^gl

ob^ifw^al m^o^ bibs^oi^ ^ i^ ^iqro^ s&+Qbplisfbkal obpmb`ql ^ q pb qfbkb q < S0bt,

Page 453: Calculus

>gbpijn kmj]g`h\n a…nd^jnt b`jh„omd^jn 322

Blj_fk^kal ‹pq^ `lk i^ b`r^`fŽk _S - _o < , 1+54@l ru pb l_qfbkb i^ b`r^`fŽkafcbobk`f^i

>&t' x < +0*43>jSV *

Dpq^ b`r^`fŽk pbm^o^_ib pb rqfifw^ `ljl jlabil j^qbjŠqf`l m^o^ mol_ibj^pobi^qfslp ^ i^ p^ifa^ ab cirfalp mlo rk lofcf`fl+ K^ ^iqro^ t ab i^ prmbocf`fb bpqŠifd^a^ ^i qfbjml o mlo rk^ b`r^`fŽk ab i^ cloj^

'7-50( E=%p& `x_t;+0*43>j _o)@,XEs

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Gt

y

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DIDLOKN 2- Blkpfa‹obpb ^elo^ bi `^pl m^oqf`ri^o bk nrb =%s& bp `lkpq^kqb+bp ab`fo =%s& < = m^o^ qlal s) u nrb bi kfsbi abi iŒnrfal e^ _^g^al ab 0/ jbqolp

Page 454: Calculus

323 Fiomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

^ 8 jbqolp bk 0/ jfkrqlp '5// pbdrkalp(- Dpqlpa^qlp pb mrbabk fkqolar`fo bk ilpiŒjfqbpab i^p fkqbdo^`flkbp+u ab i^ b`r^`fŽk afcbobk`f^i '7-50( pb abar`b9

c7_t a4.., ,,<- < f _o*0/ UX N

alkab f < 0*43>j- >, Cb i^ b`r^`fŽk ^kqboflo pb mrbab abqbojfk^o f mrbp

UHN, U8 < 4..f

hl

@elo^ pb mrbab `^i`ri^o bi qfbjml kb`bp^ofl m^o^ nrb bi kfsbi abp`fbka^ ab rks^ilo ^ lqol a^al- Olo bgbjmil+ pf bk bi fkpq^kqbn)bi kfsbi bp 6 jbqolp v bk bifkpq^kqbo0 bp 0 jbqol 'qH v o0 jbafalp bk jfkrqlp( bkqlk`bp e^ ab pbo9

H-h_t .3-n/

, , < f _o*6 Uv 5/cH

ab alkab obpriq^9

iN'U6 , i('UV * 2( < '0/('0 534( 5 051 <0/,8 & '+ (

< 0/0+2jfkrqlp-

7-17 Dgbo`f`flp QRobm^pl

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 0 ^i 0/ bk`lkqo^o i^p qo^vb`qlof^p loqldlk^ibp ab i^pc^jfif^p ab `ros^p a^a^p.+ /r * 1t < B- 4- r/t < B-0, st < B- 5- t < @`+/r†

1, s0 * t0 * 0@t < 0- 6- s0 + t0 < B-3- v1 < @s, 7- t < B `lp s,8- Sla^p i^p `fo`rkcbobk`f^p nrb m^p^k mlo ilp mrkqlp N+ N( X ', 0+ N(-

0/- Sla^p i^p `fo`rkcbobk`f^p nrb m^p^k mlo ilp mrkqlp N+ 0( v ', 0+ , 0(-00- Tk mrkql O pb jrbsb e^`f^ ^oof_^ pfdrfbkal bi bgb v mlpfqfsl- Tk mrkql L) fkf`f^ijbkqb

bk '0+ N(+ mbopfdrb O ab j^kbo^ nrb pr afpq^k`f^ ^i bgb v bp pab i^ afpq^k`f^ ^ O abpabbi lofdbk- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab i^ `ros^ ab mbopb`r`fŽk-

01- Qbplisbo bi Dgbo`f`fl 00 `r^kal i^ co^``fŽk hpb obbjmi^w^ mlo rk k•jbol mlpfqfsl `r^i,nrfbo^ e+

02- Tk^ `ros^ ab b`r^`fŽk `^oqbpf^k^ v < x%r&m^p^ mlo bi lofdbk- So^w^kal ob`q^p m^o^ibi^p^ ilp bgbp mlo rk mrkql ^o_fqo^ofl ab i^ `ros^ pb cloj^ rk ob`qŠkdril `lk alp i^alppl_ob ilp bgbp-K^ `ros^ afsfab `r^inrfbo^ ab bplp ob`qŠkdrilp bk alp obdflkbp = v >) rk^ab i^p `r^ibp qfbkb rk Šob^ fdr^i ^ i sb`bp i^ lqo^- G^ii^o i^ crk`fŽk `+

Page 455: Calculus

Be`m^d^djn _` m`k\nj 324

03- Qbplisbo bi Dgbo`f`fl 02 pf i^p alp obdflkbp = v > qfbkbk i^ molmfba^a ab nrb+ `r^kaldfo^k ^iobabalo abi bgb s* abp`of_bk pŽifalp rkl ab ilp `r^ibp qfbkb rk slirjbk isb`bp j^vlo nrb bi abi lqol-

04- K^ doŠcf`^ ab rk^ crk`fŽk ` abofs^_ib v kl kbd^qfs^ m^p^ mlo bi lofdbk v mlo bi mrkql%.)/,4P&+ Rf+ m^o^ qlal s< N+bi `lkgrkql ab loabk^a^p ab ` pfqr^al mlo bk`fj^ abifkqbos^il ZN+ rY abp`of_b rk pŽifal ab slirjbk r0`%r& `r^kal dfo^ ^iobabalo abi bgb r)e^ii^o i^ crk`fŽk `+

05- Tk^ crk`fŽk ` abofs^_ib v kl kbd^qfs^ bpqŠ abcfkfa^ bk bi fkqbos^il `boo^alZN+ 0\ pfbkal`%.&< N- O^o^ `^a^ [) N ; [ ; 0+ i^ ob`q^ s < [ `loq^ bi `lkgrkql ab loabk^a^p ab `bk alp obdflkbp nrb qfbkbk Šob^p = v > obpmb`qfs^jbkqb+ pfbkal = bi Šob^ ab i^ obdfŽkab i^ m^oqbfwnrfboa^- Rf = * > < /`%[& * 0[ * \) pfbkal \ rk^ `lkpq^kqb fkabmbkafbkqbab \* e^ii^o i^ crk`fŽk ` v i^ `lkpq^kqb ],

06- K^ doŠcf`^ ab rk^ crk`fŽk ` m^p^ mlo ilp alp mrkqlp Ol < 'N+ 0( X Md< '0+ N(- O^o^ qlalmrkql L < %r)u( ab i^ doŠcf`^+ i^ `ros^ bpqŠ pfqr^a^ mlo bk`fj^ ab i^ `rboa^ LiL) v biŠob^ =%r& ab i^ obdfŽk `ljmobkafa^ bkqob i^ `ros^ v i^ `rboa^ LLi bp fdr^i ^ r1Š Cbqbojf,k^o i^ crk`fŽk `+

07- Tk abmŽpfql `lk m^obabp sboqf`^ibp qfbkb rk^ pb``fŽk `r^ao^a^ ab 3 jbqolp `r^ao^alpab Šob^- Di ^dr^ p^ib abi abmŽpfql mlo rk lofcf`fl ab b`bkqŒjbqolp `r^ao^alp- Rf bi kfsbiabi ^dr^ bpqŠ fkf`f^ijbkqb 1 jbqolp mlo bk`fj^ abi lofcf`fl+ e^ii^o bi qfbjml kb`bp^oflm^o^ nrb abp`fbka^ 0 jbqol-

08- Tk abmŽpfql qfbkb i^ cloj^ ab rk `lkl `fo`ri^o ob`ql `lk pr s‹oqf`b e^`f^ ^oof_^- G^ii^obi qfbjml kb`bp^ofl m^o^ s^`f^oil+ abi iŒnrfal nrb il iibk^+ mlo rk lofcf`fl mo^`qf`^al bkpr _^pb- Dumobp^o bi obpriq^al bk crk`fŽk ab i^p afjbkpflkbp abi `lkl v abi Šob^ @labi lofcf`fl-

1/- K^ b`r^`fŽk rs! * v&* '0 , r&s < N qfbkb rk^ plir`fŽk ab i^ cloj^ v < `!8!* alkab gbp `lkpq^kqb- Cbqbojfk^o bpq^ plir`fŽk bumiŒ`fq^jbkqb-

10- Qbplisbo i^ b`r^`fŽk afcbobk`f^i %r * s0& * 3rs/s$ < N e^`fbkal rk `^j_fl ab s^of^_ib^ab`r^al nrb i^ `lksfboq^ bk ifkb^i-

11- Qbplisbo i^ b`r^`fŽk afcbobk`f^i '0 * t0`0%'t% * v < N fkqolar`fbkal bi `^j_fl ab s^of^,_ibp ab i^ cloj^ v < p`!! alkab h bp rk^ `lkpq^kqb v p bp rk^ krbs^ crk`fŽk fk`Ždkfq^-

12- '^( C^a^ rk^ crk`fŽk ` nrb p^qfpc^`b i^p obi^`flkbp

/ $%r&:`C& pf s< K) aL' < 1+

pf bp v < `%r&) mol_^o nrb v p^qfpc^`b rk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^9

r/s! * [rs$ * \s < N +

alkab \ v ] plk `lkpq^kqbp- Cbqbojfk^o \ v ],

'_( Dk`lkqo^o rk^ plir`fŽk ab i^ cloj^9 a&s' < Bu!-13- '^( Rb^ p rk^ plir`fŽk kl kri^ ab i^ b`r^`fŽk ab pbdrkal loabk9

s! * L%r&s$* M%r&s< N ‘

Ool_^o nrb i^ prpqfqr`fŽk v < pq qo^kpcloj^ i^ b`r^`fŽk

s! * L%r&s$* M%r&s< N%r&

bk rk^ b`r^`fŽk ifkb^i ab mofjbo loabk bk p$+

Page 456: Calculus

325 giomj_p^^d‡i \ g\n `^p\^dji`n _da`m`i^d\g`n

'_( N_qbkbo mlo q^kqbl rk^ plir`fŽk kl kri^ ab i^ b`r^`fŽk v! , 2t%* s0&t%+ 2t' < NX ^mif`^o bi j‹qlal a^al bk '^( m^o^ bk`lkqo^o rk^ plir`fŽk ab

s! * 1s$ * r0%s$* 1s&< /r_*%FF0,[

q^i nrb v < N b v&< 3 pf s < N-14- BfbkqŒcf`lpabi @g^u @qljf`p Vlohp ^fpi^olk rk do^jl ab rk krbsl bibjbkql o^af^`qfsl

ii^j^al abqboflorj- Rb bk`lkqoŽ nrb pb abpfkqbdo^_^ `lk rk^ sbil`fa^a molmlo`flk^i ^i`r^ao^al ab i^ `^kqfa^a bufpqbkqb- Cbpmr‹p ab rk ^•l nrba^_^ qdo^jl-'^( Dpq^_ib`bo v obplisbo i^ b`r^`fŽk afcbobk`f^i nrb a^ i^ j^p^ abi abqboflorj nrb nrba^bk bi fkpq^kqb o,'_( B^i`ri^o i^ `lkpq^kqb ab abpfkqbdo^`fŽk bk rkfa^abp ab dok! v ^•l,i-

15- RrmŽkd^pb nrb bk bi bgbo`f`fl mob`babkqb pb prpqfqrvb i^ m^i^_o^ ^p\_m\_j mlo m\…u^p\_m\_\* nrba^kal ilp lqolp a^qlp ilp jfpjlp- Ool_^o nrb bk bpqb `^pl i^ &prpq^k`f^ pbabpfkqbdo^oŒ^qlq^ijbkqb bk rk qfbjml cfkfql v e^ii^o bpqb qfbjml-

16- @i mofk`fmf^o i^ cfb_ob abi lol+ i^ ml_i^`fŽk ab Blvlqb Fri`e+ @ofwlk^+ bo^ 254- @ m^o,qfo ab bpqb jljbkql+ i^ ml_i^`fŽk er_fbo^ fal `ob`fbkal+ nrba^kal `^a^ ^•l jriqfmif`^a^mlo `* p^isl bi do^k k•jbol ab jrboqbp ~^``fabkq^ibp‚ nrb bp ab rk^ sŒ`qfj^ af^of^ mlo`^a^ 0// `fra^a^klp- Qbplisfbkal i^ b`r^`fŽk afcbobk`f^i ^ab`r^a^ abqbojfk^o `ljlcrk`flkbp abi qfbjml+ '^( i^ ml_i^`fŽk nrb qbkaoŒ^Blvlqb Fri`e o ^•lp abpmr‹p abi `l,jfbkwl ab i^ cfb_ob abi lol+ v '_( bi k•jbol qlq^i ab jrboqbp ^i `^_l ab o ^•lp-

17- ƒBlk nr‹ sbil`fa^a qbkaoŒ^nrb pbo i^kw^al e^`f^ ^oof_^ rk `lebqb m^o^ nrb kl sli,sfbo^ krk`^ ^ i^ qfboo^> 'RŽil pb qfbkb bk `rbkq^ i^ crbow^ ab i^ do^sba^a qboobpqob-(

18- Rb^ v < E%r&i^ plir`fŽk ab i^ b`r^`fŽk afcbobk`f^i

* 0t0 * st < 1t0 * 4

nrb p^qfpc^`b i^ `lkaf`fŽk fkf`f^i .%-& < N- 'Ml fkqbkq^o i^ obplir`fŽk ab bp^ b`r^`fŽk-(^( K^ b`r^`fŽk afcbobk`f^i jrbpqo^ nrb .!%-& < N- Cfp`rqfo pf E qfbkb rk jŠufjl l rkjŒkfjl obi^qfslp bk N l _fbk kf rkl kf lqol-_( N_p‹osbpb nrb E$%r&w N m^o^ `^a^ r w N X nrb `$%r&w em^o^ `^a^ r w -i^I - Cbqbo,jfk^o alp k•jbolp mlpfqfslp [ v ] q^ibp nrb E%r&= [r * \ m^o^ `^a^ r w -hh-b( Cbjlpqo^o nrb U-t0 x N `r^kal s x * ll + Cbq^ii^o bi o^wlk^jfbkql-a( Cbjlpqo^o nrb t-s qfbkab ^ rk iŒjfqb cfkfql `r^kal s x * // v abqbojfk^oil-

2/- C^a^ rk^ crk`fŽk Enrb p^qfpc^d^ i^ b`r^`fŽk afcbobk`f^i

rd!_r& * 0rW $_r'Z0 < 0 , _*%FF

m^o^ qlal s ob^i- 'Ml obplisbo i^ b`r^`fŽk-(^( Rf Eqfbkb rk buqobjl bk rk mrkql ` z N+ abjlpqo^o nrb q^i buqobjl bp rk jŒkfjl-_( Rf Eqfbkb rk buqobjl bk N ƒbp rk jŠufjl l rk jŒkfjl> Irpqfcf`^o i^ `lk`irpfŽk-b( Rf `_K& < .$%-& < N+ e^ii^o i^ jbklo `lkpq^kqb > q^i nrb y&s' 9p:>s0% m^o^ qlal s x l-

Page 457: Calculus

8

Al@:DBE 8B@C?:>BE

1&) =[a_\QbPPVp[UV`ap_VPN

K^ b`r^`fŽk `r^aoŠqf`^ s0 * 0 < N kl qfbkbplir`fŽk bk bi pfpqbj^ ab ilp k•,jbolp ob^ibpmlonrb kl bufpqbrk k•jbol ob^i `rvl `r^ao^al pb^ , 0- G^k pfalfkqolar`falp krbslp qfmlp ab k•jbolp+ ii^j^alp iˆh`mjn ^jhkg`ejn* m^o^ `lk,pbdrfo i^p plir`flkbp ab q^ibpb`r^`flkbp- Dk bpqb_obsb `^mŒqrilqo^q^jlp ab ilpk•jbolp `ljmibglp v sbjlp nrb plk fjmloq^kqbp bk i^ obplir`fŽk ab b`r^`flkbp^idb_o^f`^p v bk `Ši`ril afcbobk`f^i b fkqbdo^i-

X^ bk bi pfdil WUH pb fkqolargl bi pŒj_lil sf< m^o^ mol`ro^opb i^p plir,`flkbp ab i^ b`r^`fŽk s0 * 0 < N- DpqbpŒj_lil+ jŠp q^oab obmobpbkq^al`lk i^ibqo^d*pb `lkpfaboŽ `ljl rk k•jbol cf`qf`fl l fj^dfk^ofl nrb ab_Œ qo^q^opb idb,_o^f`^jbkqb `ljl `r^inrfbo k•jbol ob^i+p^isl nrb pr `r^ao^al bo^ , 0- @pŒ+mlo bgbjmil+ bi mlifkljfl `r^aoŠqf`l s0 * 0 crb c^`qlofw^al bp`of_fbkalr0 * 0 < r0

+ e1< %r * c&%r* f(+ u i^p plir`flkbp ab i^ b`r^`fŽk r0 * 0 < Npb afbolk `ljl s < ,*,f+ pfk mobl`rm^opb abi pfdkfcf`^al l s^ifabw ab q^ibp cŽo,jri^p- Dumobpflkbpq^ibp`ljl 1 * 2f pb ii^j^olk k•jbolp `ljmibglp+ v pb rqfif,w^olk ab jlal mro^jbkqb cloj^i `^pf 2// ^•lp ^kqbp ab nrb crbo^k abp`ofqlpab rk^ j^kbo^ nrb mrbab pbo`lkpfabo^a^ `ljl p^qfpc^`qlof bk i^ ^`qr^ifa^a-

@ mofk`fmflp abi pfdil WHW+J^oi Eofbaof`e F^rpp '0666,0744( v Vfiif^jQlt^k G^jfiqlk '07/4,0754( fkabmbkafbkqbjbkqb v `^pf ^i jfpjl qfbjml mol,mrpfbolk i^ fab^ ab abcfkfoilp k•jbolp `ljmibglp `ljl m^obploabk^alp %[)\& abk•jbolp ob^ibpalq^alp ab `fboq^pmolmfba^abpbpmb`f^ibp-Dpq^fab^ elv pb ^`bmq^qlq^ijbkqb v i^ bumlkbjlp bk i^ Rb``fŽk pfdrfbkqb-

1&*&9RSV[VPV\[R`e ]_\]VRQNQR`

CDEHMHBfˆM- Rf \ t ] nji iˆh`mj m`\g`n* `g k\m &\*]' n` gg\h\ iˆh`mj^jhkg`ej* nd g\ dbp\g_\_* g\ \_d^d‡i u g\ hpgodkgd^\^d‡i _` k\m`n n` _`adi`i _`g

hj_j ndbpd`io`8

326

Page 458: Calculus

327 Kˆh`mjn ^jhkg`ejn

]( Fbp\g_\_8 &\*]' < &^*_' ndbidad^\ \ < ` u ] < _,_( Pph\8 &\*]' * &^*_' < &\ * `* ] * _',b( Mmj_p^oj8 &\*]'&^*_' < &\^ + ]_* \_ * ]^',

K^ abcfkf`fŽk ab fdr^ia^a klp af`b nrb bi m^o%[)\& pb `lkpfabo^ `ljl rk m^ojm_`i\_j, @pŒ+bi k•jbol `ljmibgl '1+ 2( kl bp fdr^i ^i '2+ 1(- Klp k•jbolp \ v ]pb ii^j^k ^jhkji`io`n ab &\*]', Di mofjbo `ljmlkbkqb+ \* bp i^ k\mo` m`\g abik•jbol `ljmibgl: bi pbdrkal `ljmlkbkqb+ ]* pb ii^j^ k\mo` dh\bdi\md\,

N_p‹osbpb nrb bi pŒj_lil d < U ,0 kl ^m^ob`bbk bpq^abcfkf`fŽk-Cbkqol abml`l fkqolar`fobjlp d `ljl rk k•jbol `ljmibgl m^oqf`ri^o`lk qla^p i^p molmfb,a^abp ^idb_o^f`^p nrb ^qof_rŒ^k i pŒj_lil cf`qf`fl T ,0 ilp ^kqfdrlp j^qbjŠ,qf`lp- Ml l_pq^kqb+^kqbp ab bpql+ afp`rqfobjlp i^p molmfba^abp _Špf`^p ab i^plmbo^`flkbp nrb ^`^_^jlp ab abcfkfo-

RCMPCK? 8-0- I\n jk`m\^dji`n _` \_d^d‡i u hpgodkgd^\^d‡i _` iˆh`mjn^jhkg`ejn n\odna\^`i g\n g`t`n ^jihpo\odq\* \nj^d\odq\ v _dnomd]podq\, Bnoj `n* nds* v+ u* nji iˆh`mjn ^jhkg`ejn ^p\g`nlpd`m\* o`i`hjn

I`t ^jihpo\odq\8 s * v < v * s* st < ts,I`t \nj^d\odq\8 s * 'u * v( < &s * u( * u* s&tu' < &st'u,I`t _dnomd]podq\8 s&t * v( < st * su,

A`hjnom\^d‡i, Sla^p bp^p ibvbp pb sbofcf`^k `lk c^`fifa^a afob`q^jbkqb ^m^oqfoab i^ abcfkf`fŽk ab prj^ v molar`ql- Olo bgbjmil+ m^o^ abjlpqo^o i^ ibv^pl`f^qfs^ m^o^ i^ jriqfmif`^`fŽk+ bp`of_fjlp s < &Ug%U0'* X < 'XH +V0'*w < ' w++W0' X l_pbos^jlp nrb

U&tW' < 'Wi&U/'&VFWF + V0W0 * VFW0 * V0WF'

: &UgV/WF + V0W0' + U0&VFW0* X1YH(+UF&VFW0* V0WF' * UgVFWF + V0W0'' :

: &vUFVF + U0V0'WF + &UFV0 * U0VF'W0* &UFV0 * U0VF'WF * &UFVF + U0V0'W0' :

< 'z0X0 , U0V0* UFV0 * U/Vg'&Wg* W0' < &sV'W,

Cb-j^kbo^ pbjbg^kqb pb abjrbpqo^k i^p ibvbp `lkjrq^qfs^ v afpqof_rqfs^-

Di qblobj^ 8-0 abjrbpqo^ nrb bi `lkgrkql ab qlalp ilp k•jbolp `ljmibglpp^qfpc^`bilp qobpmofjbolp ^uflj^p abi pfpqbj^ ab ilp k•jbolp ob^ibp+`ljl pbafbolk bk i^ Rb``fŽk 0 2+1-@elo^ abjlpqo^objlp nrb ilp ^uflj^p 3+ 4 X 5 q^j,_f‹k pb p^qfpc^`bk-

Orbpql nrb 'N+ N( * %[) \& m^o^qlalp ilp `ljmibglp %[) \&) bi k•jbol `lj,mibgl 'N+ N( bp bi bibjbkql kbrqol m^o^i^ ^af`fŽk- Rb ib ii^j^ bi k•jbol `ljmibgl

Page 459: Calculus

A`adid^dji`n v kmjkd`_\_`n 328

`bol- @kŠild^jbkqb+ bi `ljmibgl '0+ N( bp bi bibjbkql kbrqol m^o^ i^ jriqfmif`^,`fŽk mlonrb

%[)\&%.) N( < %[)\&

m^o^ qlal %[) \&+ @pŒnrb+ bi ^uflj^ 3 pb p^qfpc^`b `lk 'N+ N( `ljl kbrqol l fa‹k,qf`l m^o^ i^ ^af`fŽk v 'i + N( `ljl kbrqol m^o^ i^ jriqfmif`^`fŽk-

O^o^ sbofcf`^o bi ^uflj^ 4+ l_pbosbjlp pli^jbkqb nrb ', [) * \& * %[) \& :< 'N+ N(+ ^pŒnrb %*[) +]' bp bi lmrbpql ab %[)]', Dp`of_fjlp *%[) ]' bk ir,d^o ab %*[)+]',

Efk^ijbkqb+ abjlpqo^jlp nrb `^a^ k•jbol `ljmibgl kl kril qfbkb rk ob`Œ,mol`l obpmb`ql ^i bibjbkql kbrqol 'i+ N(- Dpql bp+pf %[) \& :.: 'N+ N(+bufpqb rk k•,jbol `ljmibgl %_)^& q^i nrb

%[) \&%]) ^& < '0+ N(-

Dk bcb`ql+ bpq^ b`r^`fŽk bnrfs^ib ^i m^o ab b`r^`flkbp

\` + ]_ < 0+ \_ * ]` < N+

nrb qfbkbk i^ plir`fŽk •kf`^

'8-0(

K^ `lkaf`fŽk %[) \& :.: 'N+ N( ^pbdro^ nrb [0 * \0 :.: N+ mlo il nrb bi ob`Œ,mol`l bpqŠ _fbk abcfkfal- Dp`of_fjlp %[)\p< l f,%[) \& m^o^ obmobpbkq^o biob`Œmol`l ab %[) \&+ @pŒmrbp+ qbkbjlp

'8-1( pf %[) \& <.< 'N+ N( -

K^ afp`rpfŽk ^kqboflo jrbpqo^ nrb bi `lkgrkql ab ilp k•jbolp `ljmibglpp^qfpc^`b ilp pbfp ^uflj^p abi pfpqbj^ ab ilp k•jbolp ob^ibp- Olo `lkpfdrfbkqb+qla^p i^p ibvbp abi „idb_o^ nrb pb abar`bk abi `lkgrkql ab ^uflj^p q^j_f‹k plksŠifa^p m^o^ ilp k•jbolp `ljmibglp- Dk m^oqf`ri^o+ ilp qblobj^p abi H-0 ^i 0-04 abi^ Rb``fŽk 02-1 plk qlalp sŠifalp m^o^ ilp k•jbolp `ljmibglp il jfpjl nrb m^o^ilp k•jbolp ob^ibp- Di qblobj^ 0-7 klp af`b nrb bufpqbk ilp `l`fbkqbp ab k•jbolp`ljmibglp- Dpql bp+ pf %[) \& v %_) & plk alp k•jbolp `ljmibglp pfbkal %[) \& :.::.: 'N- N(+ bufpqb bu^`q^jbkqb rk k•jbol `ljmibgl %r) u( q^i nrb %[) \&%r) u( <: %_) &+ Dk bcb`ql+ qbkbjlp %r)u( < %_) &%[)]q8%*

Page 460: Calculus

22. Kˆh`mjn ^jhkg`ejn

8-2 Klp k•jbolp `ljmibglp `ljl b[N buqbkpfŽkab ilp k•jbolp ob^ibp

Cbpfdkbjlp `lk B bi `lkgrkql ab ilp k•jbolp `ljmibglp- Blkpfabobjlp bipr_`lkgrkql B+ ab B `lkpqfqrfal mlo ilp k•jbolp `ljmibglp ab i^ cloj^ %[)K&)bpql bp+ilp nrb qfbkbkkri^ i^ m^oqbfj^dfk^of^- K^ prj^ l bi molar`ql ab alp bib,jbkqlp ab Bl q^j_f‹k mboqbkb`b Bl- Dk bcb`ql+qbkbjlp

'8-2( %[)N( * %\) N( < %[* \) K& v &\*L'&]*N( < &\]*N( -

Dpql abjrbpqo^ nrb mlabjlp prj^o l jriqfmif`^o alp k•jbolp `ljmibglp ab B+prj^kal l jriqfmif`^kal pŽil i^p m^oqbpob^ibp-N il nrb bp il jfpjl+ obpmb`ql^i^ ^af`fŽk v ^ i^ jriqfmif```fŽk+ ilp k•jbolp ab B+pb `ljmloq^k abi jfpjl jlalnrb pf crbo^k k•jbolp ob^ibp-Kl jfpjl bp sŠifal m^o^ i^ prpqo^``fŽk v i^ afsf,pfŽk+v^ nrb *%[)N( < %*[)N( v &]*L'+g;&]+! N( pf ] ;/; N- Olo bpqbjlqfsl+kloj^ijbkqb kl e^`bjlp afpqfk`fŽk bkqobbi k•jbol ob^i r v bi k•jbol `ljmibgl&s*N( `rv^ m^oqbob^i bp s9 `lksbkfjlp bk fabkqfcf`^os v &s*N(+ v bp`of_fjlpr < %r)N(- Dk m^oqf`ri^o+bp`of_fjlp N < 'N+N(+ 0 < '0+ N(+ ,0 < ',0+ N(+ X^pŒpr`bpfs^jbkqb- Cb bpqbjlal+ mlabjlp fj^dfk^o bi pfpqbj^ ab ilp k•jbolp`ljmibglp `ljl rk^ buqbkpfŽkabi ab ilp k•jbolp ob^ibp-

K^ obi^`fŽk bkqobB+ v bi pfpqbj^ ab ilp k•jbolp ob^ibpmrbab bpq^_ib`bopbbkcloj^ ifdbo^jbkqb afpqfkq^-Rb^ Q bi `lkgrkql ab ilp k•jbolp ob^ibp+u abpfdkb,jlp `lk ni^ crk`fŽk nrb ^mif`^ `^a^ k•jbol ob^i r bk bi k•jbol `ljmibgl %r)K&+Dpql bp+pf s C Q+mlkbjlp

a&s' < &s*N( -

K^ crk`fŽk . ^pŒabcfkfa^ qfbkb`ljl aljfkfl Q v `ljl ob`loofal Bl+ v ^mif`^ bib,jbkqlp afpqfkqlp ab Q bk bibjbkqlp afpqfkqlp ab Bl- Olo sbofcf`^o bpq^pmolmfba^,abp+pb af`b nrb obpq^_ib`brk^ ^jmmm`nkji_`i^d\ pij \ pij bkqobQ v B!- K^p lmb,o^`flkbp ab ^af`fŽk v jriqfmif`^`fŽk pb `lkpbos^k ^ qo^s‹p ab bpq^`loobpmlkabk`f^-Dpql bp+qbkbjlp

a`\ * ]' <a`\' * a&]' u a&\]' < a&\'e&]' *

pfbkal bpq^pfdr^ia^abp q^k pŽil rk^ krbs^ clojri^`fŽk ab '8-2(- Orbpql nrb Qp^qfpc^`bilp pbfp^uflj^p+ il jfpjl ib l`roob ^ B!- Klp alp `rbomlp ab k•jbolpQ u Bl pb af`b nrb plk dnjhjmajn9 i^ crk`fŽk onrb ilp ifd^ `ljl ^kqbppb e^ abp,`ofql+ pb abkljfk^ dnjhjmadnhj, Dk il `lk`bokfbkqb ^ i^p lmbo^`flkbp ab ^af`fŽku jriqfmif`^`fŽk+ kl e^`bjlp afpqfk`flkbp bkqob `rbomlp fpljloclp- Olo bpqlfabkqfcf`^jlp bi k•jbol ob^i r `lk bi k•jbol `ljmibgl %r)N(- Di pfpqbj^ abk•jbolp `ljmibglp B bp rk^ `so`ind‡i abi ab ilp k•jbolp ob^ibp Q mlonrb `lk,qfbkbrk pr_`lkgrkql B+ fpljlocl ^ Q-

Page 461: Calculus

H[ ohc^[^ cg[ach[lc[ d 330

Di `rboml B^ mrbab pbo il^_h[^i ab jlal nrb pb p^qfpc^d^k ilp qobp ^uflj^pab loabk ab i^ Rb``fŽk 0 2-3- Dk bcb`ql+ abcfkfjlp %r) N( `ljl mlpfqfsl pf v pŽil pfr = N- Dp jrv pbk`fiil `ljmol_^o nrb ilp ^uflj^p 6+ 7 X 8 pb p^qfpc^`bk+ mloil nrb B+ bp rk `rboml loabk^al- Di fpljlocfpjl ` ^kqbp abp`ofql `lkpbos^ q^j_f‹kbi loabk v^ nrb ^mif`^ ilp bibjbkqlp mlpfqfslp ab Q pl_ob ilp bibjbkqlp mlpfqfslpab Bl-

1&, ?N b[VQNQVZNTV[N_VN<

Klp k•jbolp `ljmibglp qfbkbk ^idrk^p molmfba^abp ab i^p nrb kl dlw^kilp k•jbolp ob^ibp- Olo bgbjmil+ i^ b`r^`fŽk `r^aoŠqf`^ r0 * 0 < N+ nrb kl qfbkbplir`fŽk bk bi `^jml ob^i+ mrbab ^elo^ obplisbopb `lk bi rpl ab ilp k•jbolp `lj,mibglp- Dk bcb`ql+ bi k•jbol `ljmibgl 'N+ 0( bp rk^ plir`fŽk+ mrbpql nrb qbkbjlp

'N+ 0(1 < 'N+ 0('/+ 0( < 'N& N , 0 - 0+N& 0 * 0 - N( < ',0+ N( < ,0 -

Di k•jbol `ljmibgl 'N+ 0( il obmobpbkq^jlp `lk i^ ibqo^ d v pb ii^j^ ohc^[^ cg[*ach[lc[+ Sfbkb i^ molmfba^a ab nrb pr `r^ao^al bp , 0+e1< , 0- Di ib`qlo mrbab`ljmol_^o cŠ`fijbkqb nrb ', f(1 < , 0+ ^pŒnrb r < , f bp lqo^ plir`fŽk ab i^b`r^`fŽk r0 * 0 < N-

@elo^ mlabjlp obi^`flk^o i^ fab^ ab m^o loabk^al `lk i^ klq^`fŽk bjmib^a^mlo ilp ^kqfdrlp j^qbjŠqf`lp- N_pbosbjlp mofjbol nrb i^ abcfkf`fŽk ab jriqfmif,`^`fŽk ab k•jbolp `ljmibglp klp a^ %\) /('/+ 0( < 'N+ \&) v qbkbjlp mlo q^kql

%[)\& < %[)N( * 'N+ \& < %[)N( * %\)/('/+0( -

Olo `lkpfdrfbkqb+ pf bp`of_fjlp [ < %[)N(+ \ < %\) N(+ b d < 'N+ 0(+ `lkpbdrfjlp%[) \& < [ * \c+ Dp ab`fo+ ebjlp abjlpqo^al bi pfdrfbkqb

RCMPCK? 8-1- Pi^i h„g_li ]igjf_di %[)\& jo_^_ _rjl_m[lm_ _h f[ `ilg[%[)\& < [ * \c+

K^ sbkq^g^ ab bpq^ klq^`fŽk `lkpfpqb bk nrb klp ^vra^ bk bi j^kbgl ^idb_o^f`lab i^p cŽojri^p bk i^p nrb fkqbosfbkb i^ ^af`fŽk v i^ jriqfmif`^`fŽk- Olo bgbjmil+pf jriqfmif`^jlp [ * \c mlo _ * ^c) rqfifw^kal i^p ibvbp afpqof_rqfs^ v ^pl`f^qfs^+v obbjmi^w^kal e1mlo ,0+bk`lkqo^jlp nrb

%[* \c&%]* ^c&< [] * \^ * %[^ * \]&c)

Page 462: Calculus

331 Kˆh`mjn ^jhkg`ejn

nrb+ k^qro^ijbkqb+ bpqŠab ^`rboal `lk i^ abcfkf`fŽk ab jriqfmif`^`fŽk- @kŠild^,jbkqb+ m^o^`^i`ri^o bi ob`Œmol`lab rk k•jbol `ljmibgl kl kril [ * \c) mlabjlpbp`of_fo

0 \ + ]d \ + ]d \ ]d

\ * ]d < &\ * ]d'@\ + ]d' < \0 * ]0 < \0 * ]0+ \0 * …&

Dpq^cŽojri^ bpqŠab ^`rboal `lk i^ a^a^ bk '8-1(-Blk i^ fkqolar``fŽk ab ilp k•jbolp `ljmibglp+ ebjlp d^k^al jŠp nrb il nrb

prmlkb mlabo obplisbo i^ pbk`fii^ b`r^`fŽk `r^aoŠqf`^ s0 * 0 < N- Blkpfabobjlp+mlo bgbjmil+ i^ b`r^`fŽk `r^aoŠqf`^ \s% * ]s * ` < N+bk i^ nrb \* ]* ` plkob^ibp v [ ;/; N- Bljmibq^kal bi `r^ao^al+ mlabjlp bp`of_fo bpq^ b`r^`fŽk bk i^cloj^

%T * z(1 * 2\^ + ]

/< M -

0\ 2\/

Rf 2\^ + ]! x N+i^ b`r^`fŽk qfbkb i^p o^Œ`bpob^ibp ', ] ~ T ]/* 2\^'-&0\', Rf

1[] * \0 = N+bi mofjbo jfbj_ol bp mlpfqfsl m^o^qlal s^ilo ob^i ab r v i^ b`r^`fŽk kl qfbkbo^Œ`bpob^ibp-Dk bpqb`^pl+ kl l_pq^kqb+bufpqbkalp o^Œ`bpljmibg^p+a^a^p mlo i^p cŽojri^p

u] , S2\^ + ]/

l0 < , , , z0\ 0\

'8-3(] , S2\^ + ]/

m. < , , * G,,,,0\ 0\

Dk 0688+ F^rpp abjlpqoŽ nrb qla^ b`r^`fŽk ^idb_o^f`^ ab i^ cloj^

bk i^ nrb \j* \g% ,,, * \9 plk k•jbolp ob^ibp `r^ibpnrfbo^+ `lk \,9 ;/; N+qfbkb rk^plir`fŽk `ljmibg^ pf i x 0- @abjŠp+ fk`irpl bk bi `^pl bk nrb ilp `lbcf`fbkqbp\,9 ]h= ŠŠŠ + \z pb^k `ljmibglp+ bufpqbrk^ plir`fŽk `ljmibg^- Dpqbeb`el pb `lkl`b`lk bi klj_ob ab Q`jm`h\ api_\h`io\g _`g ƒgb`]m\, ')( Dk ‹i pb abjrbpqo^ nrbkl bp kb`bp^ofl `lkpqorfo k•jbolp jŠp dbkbo^ibpnrb ilp `ljmibglp m^o^ obplisbob`r^`flkbp ^idb_o^f`^p `lk `lbcf`fbkqbp `ljmibglp-

')( Dk `r^inrfbo if_ol ab qbloŒ^ab crk`flkbp ab s^of^_ib `ljmibg^ mrbab bk`lkqo^opb rk^abjlpqo^`fŽk abi qblobj^ crka^jbkq^i abi Šidb_o^+Olo bgbjmil+ s‹^pb J- Jklmm+ Qc`jmt ./Cpi^odjin* Clsbo Or_if`^qflkp+ Mbt Xloh+ 0834+l D- Gfiib+ >i\gtod^ Cpi^odji Qc`jmt* Uli- )$Ai^pabii Or_ifpefkd Bl-+ 0848- Orbab sbopb rk^ abjlpqo^`fŽk jŠp bibjbkq^i bk N- R`eobfbov D- Rmbokbo+Fiomj_p^odji oj Jj_`mi >gb`]m\ \i_ J\omds Qc`jmt* Bebipb^ Or_ifpefkd Blok,m^kv Mbt Xloh! 0840-

Page 463: Calculus

Fio`mkm`o\^d‡i b`jh„omd^\, J‡_pgj t \mbph`ioj --,

1&- =[aR_]_RaNPVp[TR\Zna_VPN&@pQbY\e N_TbZR[a\

Orbpql nrb rk k•jbol `ljmibgl %r) s& bp WQm^oloabk^al ab k•jbolp ob^,ibp+ mrbab obmobpbkq^opbdblj‹qof`^jbkqb jbaf^kqb rk mrkql abi mi^kl+ l mlork^ cib`e^ l sb`qlo dblj‹qof`l nrb rk^ bi lofdbk `lk bi mrkql %r) s&) `ljl jrbp,qo^ i^ cfdro^ 8-0- Olo biil+ ^i mi^kl rs) pb ib ii^j^ ^ jbkral mi^kl `ljmibgl-Di bgb s bp bi bgb ob^i: bi bgb t bp bi bgb fj^dfk^ofl- Noafk^of^jbkqb i^p m^i^_o^piˆh`mj ^jhkg`ej u kpioj pb rp^k fkafpqfkq^jbkqb- @pŒnrb afobjlp bi mrkql wbk ird^o ab ab`fo bi mrkql `loobpmlkafbkqb ^i k•jbol `ljmibgl u*

K^p lmbo^`flkbp ab ^af`fŽk u prpqo^``fŽk ab k•jbolp `ljmibglp qfbkbk rk^pbk`fii^ fkqbomobq^`fŽkdblj‹qof`^- Rf alp k•jbolp `ljmibglp 1+ u Y1 pb obmobpbk,q^k jbaf^kqb cib`e^p nrb rkbk bi lofdbk `lk w+v W0* obpmb`qfs^jbkqb+bkqlk`bp i^prj^ u* * Y1 bpqŠabqbojfk^a^ mlo i^ g`t _`g k\m\g`gƒbm\hj, K^ cib`e^ nrb rkbbi lofdbk `lk bi mrkql w+ * Y1 bp rk^ af^dlk^i abi m^o^ibildo^jl abqbojfk^almlo N+ w+v W0* `ljl pb sb bk i^ cfdro^ 8-1- K^ lqo^ af^dlk^i bpqŠobi^`flk^a^ `lk i^afcbobk`f^ ab w+v V/$ K^ cib`e^ ab w+^ Y1 bp m^o^ibi^v ab fdr^i ilkdfqra nrb i^nrb rkb N `lk Y1 , V)8 i^ cib`e^ bk bi pbkqfal lmrbpql+ ab Y1 ^ w+ pb obi^`flk^ abijfpjl jlal `lk w+, W0%

Rf %r) s& ", 'N+ N(+ mlabjlp bumobp^or b s bk `lloabk^a^p mli^obp

r < l `lp `*v l_qbkbjlp

t < opbk ‹ +

s * dt < m&^jn ` * dn`i`','8-4(

t

)%r)s&< r * csGG+G

8+t < mpbk `+GG

l

EHFTQ@ 8-0 O`km`n`io\^d‡i b`jh„omd^\ _`giˆh`mj ^jhkg`ej s * dt,

s

EHFTQ@ 8-1 >_d^d‡i v npnom\^^d‡i _` iˆh`+mjn ^jhkg`ejn m`km`n`io\_jn b`jh„omd^\h`i+

o` h`_d\io` g\ g`t _`g k\m\g`g‡bm\hj,

Page 464: Calculus

--- Kˆh`mjn ^jhkg`ejn

Di k•jbol mlpfqfsl l) nrb obmobpbkq^i^ afpq^k`f^ ab %r) u( ^i lofdbk+ pb ii^j^ gi*

_pgj l q\gjm \]njgpoj ab r * dt v pb obmobpbkq^`lk Er * dtg, @pŒmrbp+ qbkbjlp

Yt * dtg < Us0 * t0 ,

Di Škdril mli^o 7 bp rk \mbph`ioj ab s * dt, Cb`fjlp pi ^odrjbkql u kl `g^odrjbkql mlonrb m^o^ rk mrkql a^al %r) u( bi Škdril 7 bpqŠ abqbojfk^al p^islj•iqfmilp ab 16S- @idrk^p sb`bp bp mobcbof_ib ^pfdk^o rk ^odrjbkql •kf`l ^ rkk•jbol `ljmibgl- Dpql mrbab e^`bopb ifjfq^kal %d s^of^o bk rk fkqbos^il pbjf,^_fboql ab ilkdfqra 16S- Klp fkqbos^ilp ZN+16S( X ',6S+ 6S\ plk ilp jŠp `lj•kjbkqbrp^alp- Tqfifw^objlp bi fkqbos^il ',6S+ 6S\ X ii^j^objlp ^i `loobpmlkafbkqb %dbi\mbph`ioj kmdi^dk\g ab r * dt9 bpqb s^ilo ab %dil obmobpbkq^jlp `lk [la%r * dt',@pŒmrbp+ pf s * dt ;/; N X m< Er * dtg* abcfkfjlp [la%r * dt' `ljl bi •kf`l k•,jbol ob^i %dnrb p^qfpc^`b i^p `lkaf`flkbp

s < m`jn 5) t < opbk 7+ ,6S ; 7 z 6S-

@i k•jbol `ljmibgl `bol+ ib ^pfdk^jlp bi jŽaril N v `lksbkfjlp bk nrb `r^i,nrfbo k•jbol ob^i 7 mrbab rp^opb `ljl ^odrjbkql-

Orbpql nrb bi s^ilo ^_plirql ab rk k•jbol `ljmibgl u bp pfjmibjbkqb i^ilkdfqra ab rk pbdjbkql+ kl ab_b plomobkaboklp nrb dl`b ab i^p molmfba^abp abilp s^ilobp ^_plirqlp ab ilp k•jbolp ob^ibp- Olo bgbjmil+ qbkbjlp

Yv[ = N pf u !-8+N+ u

Fblj‹qof`^jbkqb+ bi s^ilo ^_plirql Gv+, u0/ obmobpbkq^ i^ afpq^k`f^ bkqob ilpmrkqlp w+v W0 abi mi^kl `ljmibgl- K^ abpfdr^ia^a qof^kdri^o

q^j_f‹k bp sŠifa^- @abjŠp+ qbkbjlp i^p cŽojri^p pfdrfbkqbp m^o^ ilp s^ilobp ^_pl,irqlp ab molar`qlp v `l`fbkqbp ab k•jbolp `ljmibglp9

'8-5(

u

GWF G< i< -0W0 GY10

Rf bp`of_fjlp WF < \ * \c v W0 < @ * ^c) l_qbkbjlp '8-5( `ljl `lkpb`rbk`f^fkjbaf^q^ ab i^ fabkqfa^a

Page 465: Calculus

Bd`m^d^djn --.

K^ cŽojri^ m^o^\wIw10pb abar`b ab '8-5( pf bp`of_fjlp w+ ljl rk molar`ql+

Rf w< s * dt* bi ^jhkg`ej ^jidpb\_j ab w bp bi k•jbol `ljmibgl u ;s+dt,Fblj‹qof`^jbkqb+ oobmobpbkqbi pfj‹qof`l ab wobpmb`ql^i bgbob^i- K^ abcfkf`fŽkab `lkgrd^al fjmif`^ nrb

uo< Gvh1‘

Cbg^jlp+ `ljl bgbo`f`fl+^i ib`qlo i^ `ljmol_^`fŽk ab bp^p molmfba^abp-Rf rk^ b`r^`fŽk `r^aoŠqf`^ `lk `lbcf`fbkqbp ob^ibp kl qfbkb o^Œ`bpob^ibp+prp

o^Œ`bp ljmibg^p+ a^a^p mlo '8-3(+ plk `lkgrd^a^p- Qb`Œmol`^jbkqb+pf l) X l)

plk `ljmibglp `lkgrd^alp+ mlo bgbjmil m*< N' * daG v m0 < N' , c%F)pfbkal N' v%F ob^ibp+bkqlk`bp l) X mz plk o^Œ`bpab rk^ b`r^`fŽk `r^aoŠqf`^ `lk `lbcf`fbkqbpob^ibp- Dk bcb`ql+ qbkbjlp

v

`lk il nrb

u i^ b`r^`fŽk `r^aoŠqf`^ bk `rbpqfŽk bp

1&. :WR_PVPV\`

0- Dumobp^o ilp k•jbolp `ljmibglp pfdrfbkqbp bk i^ cloj^ \+n+]d,'^( 'i * f(1- 'b( 'i * f(.'i , 1f(-'_( i.f- 'b( f4 * f05‘

'`( 0.'0 * f(- 'c( 0 * c * f1 * f2Š

'a( '1 * 1.&1 + 2., 'e( Lf * f('i * e,7(-

1- B^i`ri^o ilp s^ilobp ^_plirqlp ab ilp k•jbolp `ljmibglp pfdrfbkqbp-']( 0 * d* 'a( H * d* e1Š

'_( 2 * 3f- 'b( f6 * eiN‘

'`( 'i * f(.'H , f(- 'b( 1'0 , f( * 2'1 * L*2- B^i`ri^o bi jŽaril u bi ^odrjbkql mofk`fm^i ab `^a^ rkl ab ilp k•jbolp `ljmibglp

pfdrfbkqbp-

'^( 1f- 'b( ,0-'_( ,2f- 'a( 0-

Page 466: Calculus

--/ J„g_lim ]igjf_dim

'b( ,2 * s2 f- 'd( ',0 , f(2-

'b( N * e(./- 'f( i.N * e(-'c( ',0 * f(2- 'g( i.N * f(1-

3- Dk `^a^ `^pl+ abqbojfk^o qlalp ilp k•jbolp ob^ibp s b v nrb p^qfpc^`bk i^ obi^`fŽk a^a^-']( r * cs < r * cs+ 'a( %r * cs&/ < %r * cs&/+

s * dt'_( s * dt < Zu * euh- 'b( ,,l < s + dt*

T *EU0//

'b( Zu * euh< Zu , euh- 'b( H df < s * dt*Q5>

4- Blkpqorfo rk^ obmobpbkq^`fŽk abi `lkgrkql ab qlalp ilp u abi mi^kl `ljmibgl nrb p^qfp,c^d^k `^a^ rk^ ab i^p `lkaf`flkbp pfdrfbkqbp-']( Yv[ ; 0- 'a( Yv, 00 < Yv* 00-'^( w * V < 0- 'b( Zw, eh< Zw* eh-'_( u + u < f- 'b( v * u < Gv01Š

5- Rb^ ` rk mlifkljfl ab `lbcf`fbkqbp ob^ibp-^( Cbjlpqo^o nrb a&u' < aR' m^o^ qlal `ljmibgl w-_( Tqfifw^o i^ m^oqb ^( m^o^ abar`fo nrb ilp `bolp kl ob^ibp ab ` 'pf bufpqbk( ab_bk mob,pbkq^opb ^ m^obp ab `ljmibglp `lkgrd^alp-

6- Ool_^o nrb kl pb mrbab fkqolar`fo rk^ obi^`fŽk ab loabk bk bi pfpqbj^ ab ilp k•jbolp`ljmibglp ab j^kbo^ nrb pb p^qfpc^d^k ilp qobp ^uflj^p ab loabk ab i^ Rb``fŽk 0 2-3-

XFi_d^\^d‡i8 RrmŽkd^pb nrb pb mrbab fkqolar`fo rk^ q^i loabk^`fŽk b fkqbkq^oab`fafo pf i^ rkfa^a fj^dfk^of^ d bp mlpfqfs^ l kbd^qfs^-\

7- Cbcfkfo i^ pfdrfbkqb ~pbraloabk^`fŽk‚ bkqob ilp k•jbolp `ljmibglp- Rf t < r * dt,ab`fjlp nrb t bp mlpfqfsl pf v pŽil pf r = N- ƒBrŠibp ab ilp ^uflj^p ab loabk ab i^ Rb`,`fŽk 0 2-3 pb p^qfpc^`bk `lk bpq^ abcfkf`fŽk abi k•jbol u mlpfqfsl>

8- Qbplisbo bi Dgbo`f`fl 7 pf i^ pbraloabk^`fŽk pb abcfkb ^pŒ9ab`fjlp nrb u bp mlpfqfslpf u pŽil pf Yv[ = N-

0/- Qbplisbo bi Dgbo`f`fl 7 pf i^ pbraloabk^`f5k pb abcfkb ^pŒ9Rf w< s * dt* ab`fjlp nrbw bp mlpfqfsl pf v p5il pf s = v-

00- Qbmobpbkq^obi `lkgrkql ab qlalp ilp `ljmibglp w nrb p^qfpc^`bk `^a^ rk^ ab i^p `lk,af‹flkbp pfdrfbkqbp-'^( ./t * 20 ; 0-

'^( Gv*hh;hv,hh-01- Rb^ s < %[t * \&,%]t * ^&) alkab [+

'a( Yv , ehy Gv* eh-'a( Yv[ y /0u * 00-

\) ^ v ^ plk ob^ibp- Cbjlpqo^o nrb

T + HU < %[^ * \]&%t * t&,E]t * ^f0 Š

Rf \_ + ]^ = N- mol_^o nrb i^p m^oqbpfj^dfk^of^p ab w u s qfbkbk bi jfpjl pfdkl-

1&/ :d]\[R[PVNYRP\Z]YRWN`

Prbobjlp ^elo^ buqbkabo i^ abcfkf`fŽk ab `! ab jlal nrb qbkd^ pfdkfcf`^al`r^kal s pb obbjmi^`b mlo rk k•jbol `ljmibgl `r^inrfbo^ w- Dpq^ buqbkpfŽk i^

Page 467: Calculus

Bskji`i^d\g`n ^jhkg`ejn --0

abpb^jlp ab jlal nrb i^ ibv ab bumlkbkqbp+ %`! < _!(\) pb^ sŠifa^ m^o^ `r^ibp,nrfbo^ `ljmibglp \ v \+ X+k^qro^ijbkqb+ kb`bpfq^jlp nrb `% `lfk`fa^ `lk i^ buml,kbk`f^i rpr^i `r^kal w pb^ ob^i- Dufpqbks^oflp j‹qlalp bnrfs^ibkqbp m^o^ iibs^o^ `^_l bp^ buqbkpfŽk-@kqbpab bpq^_ib`boi^ abcfkf`fŽk ab `! nrb ebjlp bibdfal+a^objlp rk^ afp`rpfŽk ebroŒpqf` nrb pbosfoŠ `ljl mrkql ab m^oqfa^v jlqfslm^o^ bp^ abcfkf`fŽk-

Rf bp`of_fjlp w< s * fv+bkqlk`bp+ pf i^ ibv ab bumlkbkqbp ab_b pbo sŠifa^m^o^ k•jbolp `ljmibglp+ ab_b pbo

Orbpql nrb `! e^ pfal v^ abcfkfa^ `r^kal s bp ob^i+krbpqol mofjbo l_gbqfsl bpiibd^o ^ rk^ abcfkf`fŽk mi^rpf_ib ab `cU `r^kal v pb^ ob^i- Orbp _fbk+ pf €f ab_bpbo rk k•jbol `ljmibgl+ mlabjlp bp`of_fo

'8-6( `dt < =%s&* c>%s&)

bk alkab = v > plk crk`flkbp ob^ibp nrb ab_bk abqbojfk^opb- Cbofsbjlp ^j_lpjfbj_olp ab i^ b`r^`fŽk '8-6(+ prmlkfbkal nrb = v > pb^k abofs^_ibp+v j^kb,g^kal bi k•jbol `ljmibgl f `ljl pf crbo^ rk k•jbol ob^i- Kibd^jlp bkqlk`bp ^

'8-7( d`%q< =$%s&* c>$%s&+

Cbofs^kal rk^ sbw jŠp+ bk`lkqo^jlp nrb

[`dt < =!%s&* c>!%s&+

Bljm^o^kal bpq^b`r^`fŽk `lk '8-6( sbjlp nrb = v > ab_bk p^qfpc^`boi^p b`r^,`flkbp

=!%s&< *=%s& v >!%s&< *>%s&+

Cf`el ab lqol jlal+ `^a^ rk^ ab i^p crk`flkbp = v > bp rk^ plir`fŽk ab i^ b`r^,`fŽk afcbobk`f^i p! * / < N- Blk 0/ af`el bk bi `^mŒqril7+ mlabjlp ^pbdro^o nrbbpq^b`r^`fŽk qfbkb bu^`q^jbkqb rk^ plir`fŽk `lk s^ilobp fkf`f^ibp ^pfdk^alp .%-&

v .$%-&+Rf mlkbjlp v < N bk '8-6( v '8-7( W ob`loa^jlp nrb `! < 0+bk`lkqo^jlpnrb = v > qfbkbk ilp s^ilobp fkf`f^ibp

=%K& < 0+ =$%K&< N+ u >%K&< N+ >$%K&< 0 -

Rbd•k bi qblobj^ ab rkf`fa^a m^o^b`r^`flkbp afcbobk`f^ibp ab pbdrkal loabk `lk`lbcf`fbkqbp `lkpq^kqbp+ab_b pbo

=%s& < `lp s u >%s&;n`it,

Page 468: Calculus

--1 Kˆh`mjn ^jhkg`djn

Dk lqo^p m^i^_o^p+pf `.s ab_b pbo rk k•jbol `ljmibgl `lk i^p molmfba^abpnrb^`^_^jlp ab abp`of_fo+bkqlk`bp ab_bjlp qbkbo_ff, < `lp s * dpbks+ Dpq^afp`r,pfŽk a^ lofdbk ^ i^ abcfkf`fŽk pfdrfbkqb-

CDEHMHBHˆM- Pd w < s * dt* _`adidhjn `W ^jhj `g iˆh`mj ^jhkg`oj _\_jkjm g\ `^p\^d‡i

%6+6& `%< `U&^jn t * cn`it' ,

N_p‹osbpb nrb `V < `s `r^kal t < N: irbdl bpq^ bumlkbk`f^i `lfk`fab `lki^ bumlkbk`f^i loafk^of^ `r^kal wbp ob^i- Tqfifw^objlp ^elo^ bpq^abcfkf`fŽk m^o^abar`fo i^ ibv ab bumlkbkqbp-

RCMPCK? 8-2- Pd\ t ] nji iˆh`mjn ^jhkg`ejn* o`i`hjn

A`hjnom\^d‡i, Dp`of_fbkal \ < s * dt v ] < p * dq*qbkbjlp

_[ < _F$%]imt * cpbkv(+

`lk 0/ nrb

_[_] < _7l_))W]im t `lp p * pbkt pbks * f'`lp t pbks * n`it `lp p&Y+

Tqfif`bjlp ^elo^ i^p cŽojri^p ab ^af`fŽk m^o^`lp %s* p& u pbk %s* p& u i^ ibvab bumlkbkqbpm^o^bumlkbk`f^ibp ob^ibp+ lk il nrb i^ b`r^`fŽk ^kqboflo pb `lk,sfboqbbk

'8-00 ( _[_] < _P(QW]im %s* p& * fpbk'v * p&Y+

Orbpql nrb [ * ] < %r * o& * c%s* q(+ bi pbdrkal jfbj_ol ab '8-00( bp _\(]Š

Dpql abjrbpqo^ '8-0/(-

RCMPCK? 8-3- Qj_j iˆh`mj ^jhkg`oj w;/; N kp`_` `skm`n\mn` `i g\ ajmh\

%6+./& u;m`d7*

`i _ji_` m< Zw\t %F < ^od 'w( * 0i5Q*nd`i_j i pi `io`mj ^p\glpd`m\, Bno\ m`+km`n`io\^d‡i n` _`ijhdi\ ajmh\ kjg\m _` w-

A`hjnom\^d‡i, Rf w< s * dt* i^ obmobpbkq^`fŽk'8+4( klp a^

u < o'`lp _ * cpbk_& )

Page 469: Calculus

Boh]cih_m ]igjf_d[m --2

alkab l < Gvhv %d< ^od %t&* Wiom*pfbkal h rk bkqbol `r^inrfbo^- Obol pf qlj^,j^p r < N b s < f_ bk '8+8(+ l_qbkbjlp i^ cŽojri^

`d6 < `lp ` * opbk `*0/ nrb abjrbpqo^ '8-01(-

K^ obmobpbkq^`fŽk ab ilp k•jbolp `ljmibglp bk i^ cloj^ mli^o '8-01( bpjrv •qfi bk i^ jriqfmif`^`fŽk v afsfpfŽk ab k•jbolp `ljmibglp- Olo bgbjmil+ pfW/ < m,`! v W0 < m0`d9f;) qbkbjlp

'8-02(

Olo `lkpfdrfbkqb bi molar`ql ab ilp jŽarilp+ l+ly+ bp bi jŽaril abi molar`ql W/W0*

ab ^`rboal `lk i^ b`r^`fŽk '8-5(+ v i^ prj^ ab ilp ^odrjbkqlp+ ` * N.+ bp rk^odrjbkql abi molar`ql W/W0%

Br^kal u < m`!* i^ ^mif`^`fŽk obfqbo^a^ ab '8-02( klp a^ i^ cŽojri^

sŠifa^ m^o^ `r^inrfbo bkqbol i kl kbd^qfsl- S^j_f‹k bp sŠifa^ m^o^ s^ilobp kbd^,qfslp ab h pf abcfkfjlp w!! `ljl %t*.&g `r^kal h bp bkqbol mlpfqfsl-

@kŠild^jbkqb+ qbkbjlp

`lk 0/ nrb bi jŽaril ab W/-W0 bp mGm0X i^ afcbobk`f^ ` + l. bp rk ^odrjbkqlab W'W0%

1&0 ;b[PV\[R` P\Z]YRWN`

Tk^ crk`fŽk o `rvlp s^ilobp plk k•jbolp `ljmibglp pb abkljfk^ crk`fŽk`ljmibg^- Rf bi aljfkfl ab o bp rk `lkgrkql ab k•jbolp ob^ibp+ o pb ii^j^ crk,`fŽk `ljmibg^ ab s^of^_ib ob^i- Rf bi aljfkfl bp rk `lkgrkql ab k•jbolp `lj,mibglp+o bp rk^ crk`fŽk `ljmibg^ ab s^of^_ib `ljmibg^- Tk bgbjmil bp i^ crk`fŽkbumlkbk`f^i+ abcfkfa^ mlo i^ b`r^`fŽk

a`u' < `%

m^o^ qlal `ljmibgl u, Lr`e^p ab i^p crk`flkbp bibjbkq^ibp abi BŠi`ril+ q^ibp `ljli^ bumlkbk`f^i+ bi ild^ofqjl+ v i^p crk`flkbp qofdlklj‹qof`^p+ mrbabk buqbkabopb v

Page 470: Calculus

34/ Kˆh`mjn ^jhkg`ejn

`lksboqfopb bk crk`flkbp ab s^of^_ib `ljmibg^- 'Ubo Dgbo`f`flp 8 v 0/ ab i^ Rb`,`fŽk 8-0/-( Dk bpqbj^o`l jŠp ^jmifl `lk cob`rbk`f^ ^m^ob`bk krbs^p molmfba^,abp b fkqboobi^`flkbp-Olo bgbjmil+ i^ crk`fŽk bumlkbk`f^i `ljmibg^ bp mbofŽaf`^-Dk bcb`ql+pf w< s * dt v pf i bp rk bkqbol `r^inrfbo^+ qbkbjlp

_t(/hllc < _TW]im %s * 1jo( * fpbk'v * 1jo(\ < _T%]ims * fpbkv( < _• +

Ubjlp ^pŒnrb a`u * 0iQmd'< a`u'* mlo il nrb a qfbkb bi mboŒlal0Qmd,Dpq^mol,mfba^a ab i^ crk`fŽk bumlkbk`f^i pŽil pb mlkb ab j^kfcfbpql `r^kal bpqraf^jlpi^ bumlkbk`f^i `ljl crk`fŽk ab rk^ s^of^_ib `ljmibg^-

Di mofjbo bpqrafl pfpqbjŠqf`l abi BŠi`ril afcbobk`f^i b fkqbdo^i`lk crk`flkbpab s^of^_ib `ljmibg^ crb eb`el mlo B^r`ev ^ mofk`fmflpabi pfdil WHW-Cbpab bk,qlk`bp i^ qbloŒ pb e^ abp^oolii^al v `lksboqfal bk rk^ ab i^p o^j^p jŠp fjmlo,q^kqbpb fkqbobp^kqbpab i^ L^qbjŠqf`^- Rb e^ `lksboqfal bk rk fkpqorjbkql fkafp,mbkp^_ibm^o^cŒpf`lpb fkdbkfbolp v qfbkb `lkbuflkbp `^pf `lk `r^inrfbo o^j^ abi^ L^qbjŠqf`^ mro^- @nrŒkl pb bumlkaoŠ bpq^qbloŒ^:q^k pŽil e^objlp rk bpqraflbibjbkq^i abi `Ši`ril `lk crk`flkbp `ljmibg^p ab s^of^_ib ob^i-

Rrmlkd^jlp nrb ` bp rk^ crk`fŽk `ljmibg^ abcfkfa^ bk rk `fboql fkqbos^il Eab k•jbolp ob^ibp-O^o^ `^a^ s ab H) i^ crk`fŽk a&s' qlj^ rk s^ilo `ljmibgl+ ^pŒnrb mlabjlp bp`of_fo

y&s' < p&s' * dq&s'*

pfbkal p&s' v q&s' ob^ibp- Dpq^ b`r^`fŽk abqbojfk^ alp crk`flkbp ob^ibp p v qii^j^a^p m^oqbpob^i b fj^dfk^of^ ab ` obpmb`qfs^jbkqb: bp`of_fjlp i^ fdr^ia^a abcloj^ _obsb mlkfbkal ` < p * dq, Klp `lk`bmqlp ab `lkqfkrfa^a+ abofs^`fŽk bfkqbdo^`fŽkab ` mrbabk abcfkfopb^ qo^s‹p ab ilp `lk`bmqlp ^kŠildlp m^o^ p v q*`ljl pb fkaf`^ bk i^ pfdrfbkqbabcfkf`fŽk-

CDEHMHBHˆM- Pd a < p * dq* _`^dhjn lp` a `n ^jiodip\ `i pi kpioj \ ndg\n api^dji`n p v q nji \h]\n ^jiodip\n `i `n` kpioj, I\ _`mdq\_\ _` a n` _`adi`kjm g\ dbp\g_\_

a%&s'< p-`s' * dq-`s'

nd`hkm` lp` \h]\n _`mdq\_\n p-`s' u qx&s' `sdno\i, >iƒgjb\h`io`* _`adidhjn g\dio`bm\g _` a*kjm g\ dbp\g_\_

qa&s' _s < pp&s' _s * dpq&s' _s

^ji o\g lp` `sdno\i g\n dio`bm\g`n _`g n`bpi_j hd`h]mj,@ i^ sfpq^ ab bpq^abcfkf`fŽk+kl bp plomobkabkqbbk`lkqo^o nrb jr`elp ab

ilp qblobj^p abi BŠi`ril afcbobk`f^i u abi fkqbdo^ipb^k sŠifalp q^j_f‹k m^o^crk,

Page 471: Calculus

Be`hkgjn _` a‡mhpg\n _` _`mdq\^d‡i ` dio`bm\^d‡i 340

`flkbp `ljmibg^p- Olo bgbjmil+ i^p obdi^p ab abofs^`fŽk ab prj^p+ molar`qlp+ v`l`fbkqbp 'qblobj^ 3-0( plk sŠifa^p m^o^ crk`flkbp `ljmibg^p- Di mofjbol u bipbdrkal qblobj^p crka^jbkq^ibp abi BŠi`ril 'qblobj^p 4-0 u 4-2( ^pŒ`ljl biqblobj^ ab i^ abofs^a^ kri^ 'qblobj^ 4-1( q^j_f‹k pb `rjmibk `lk crk`flkbp`ljlibg^p+ O^o^ firpqo^o `lk nr‹ c^`fifa^a mrbabk abjlpqo^opb bplp qblobj^p+ `lk,pfabobjlp bi qblobj^ ab i^ abofs^a^ kri^9

Pd a%&s'< M k\m\ oj_j s _` pi dio`mq\gj \]d`moj >"`ioji^`n a `n ^jino\io``i g,

A`hjnom\^d‡i, Olkd^jlp a < p * dq, Orbpql nrb a%< p%* dq%*i^ efmŽ,qbpfp`$< N bk f pfdkfcf`^ nrb p%u q%plk ^j_^p kri^p bk f+Krbdl+ pbd•k bi qblob,j^ 4-1+ p u q plk ^j_^p `lkpq^kqbp bk g,Olo q^kql ` bp `lkpq^kqb bk g,

1&1 :WRZ]Y\`QRSp_ZbYNQRQR_VcNPVp[R V[aRT_NPVp[

Dk bpq^ Rb``fŽk afp`rqfjlp rk bgbjmil fjmloq^kqb ab crk`fŽk `ljmibg^ abs^of^_ib ob^i+ i^ crk`fŽk ` abcfkfa^ m^o^ qlal s^ilo ob^i r mlo i^ b`r^`fŽk

y&s' < `os*

alkab o bp rk k•jbol `ljmibgl cfgl- Br^kal o bp ob^i+ i^ abofs^a^ ab bp^ crk`fŽksfbkb a^a^ mlo i^ cŽojri^ `$%r&< o„!, Cbjlpqo^jlp ^elo^ nrb bpq^ cŽojri^ bpq^j_f‹k sŠifa^ m^o^ o `ljmibgl

RCMPCK? 8-4- Pd a&s' < `o\8k\m\ oj_j s m`\g u pi o ^jhkg`ej adej*a%&s'< o`%!,

A`hjnom\^d‡i, Olkd^jlp o < bu * d&1*pfbkal bu v a1 ob^ibp- Cb i^ abcfkf`fŽkab bumlkbk`f^i `ljmibg^ obpriq^

y&s' < `o\8 < `\s)d-gs < `\s `lp ?s * d`\Un`i 6s ,

Olo `lkpfdrfbkqb+ i^p m^oqbpob^i b fj^dfk^of^ ab ` sfbkbk a^a^p mlo

&7,/2' p&s' < `\s `lp aGs u q&s' < `\sn`i &1s ,

Dp^p crk`flkbp plk abofs^_ibp m^o^ qlal s^ilo ab s u prp abofs^a^p plk

p%&s' < `s`\s `lp a1s + a1`\sn`i &1s * q%&s' < `s`\s n`ia1s * a1`\s `lp aGs Š

Orbpql nrb a%&s'< p%&s'* dq%&s'*qbkbjlp

a%&s' < `s`\s&^jn aGs * en`ia1s' * da1`\s&^jn aGs * d n`iaGs' :

< 'bu * daG'`&\)d-dgs< o`os,

Dpql `ljmibq^ i^ abjlpqo^`fŽk-

Page 472: Calculus

341 J„g_lim ]igjf_dim

Di qblobj^ 8-4 qfbkb ^idrk^p `lkpb`rbk`f^p fkqbobp^kqbp-Olo bgbjmil+ pf^almq^jlp i^ klq^`fŽk ab Kbf_kfwm^o^i^p fkqbdo^ibpfkabcfkfa^p+mlabjlp mlkbobi qblobj^ 8-4 bk i^ cloj^

'8-0 4(

`r^kal q ;/; N- Rf mlkbjlp q < k_ * daG b fdr^i^jlp i^p m^oqbpob^i b fj^dfk^of^bk i^ b`r^`fŽk '8-04(+ l_qbkbjlp i^p cŽojri^p ab fkqbdo^`fŽk

u

G ~&! `0 ^ `z!%&j^`lp aGs * %FpbkxIu(` `lp s s < 1

k_ * aG0

G ~&! O ^ * `z!%&j^pbkcIu , uF`lp %Fr&` pbk{Iu s + 1 &

k_ * vG0

nrb plk sŠifa^p pf k_ v uF kl plk `bol-Nqo^ `lkpb`rbk`f^ abi qblobj^ 8-4 bp i^ `lkbufŽk bkqob bumlkbk`f^ibp `lj,

mibg^pv b`r^`flkbp afcbobk`f^ibp ifkb^ibp ab pbdrkal loabk `lk `lbcf`fbkqbp `lkp,q^kqbp-

RCMPCK? 8-5- ?ihmc^_l_gim f[ _]o[]cƒh ^c`_l_h]c[f

%6+.3& s! * \t% * \s < N+

_h f[ ko_ [ s \ mih ]ihmn[hn_ml_[f_m+H[m j[ln_m l_[f _ cg[ach[lc[ ^_ f[ `oh]cƒh `^_`chc^[ _h %* //+ * '/( jil f[ _]o[]cƒh `%r&< _.r mih mifo]cih_m ^_ f[ _]o[]cƒh^c`_l_h]c[f '8-05( pf t mƒfi pf p_moh[ l[•t ^_ f[ _]o[]cƒh ][l[]n_l•mnc][

o/ * \o * ] < N-

@_gimnl[]cƒh+ Rb^ H%s&< v! * [s$ * \s+ Orbpql nrb d$%r&< n_,r) q^j_f‹kqbkbjlp `!%r& < o%`!*lk il ko_H%`& < _.rS * [n * \&+ Obol _b7 krk`^ bp`bol v^ nrb _nr_*n7FF< _! < 0- Krbdl+ H%`&< N pf v pŽil pf o0 * [n * \ < N-Obol pf bp`of_fjlp ` < o * d~*bk`lkqo^jlp H%`&< H%o&* cH%p&)v mlo q^kqlH%`&< N pf v pŽil pf H%o&< N X H%p&< N- Dpql `ljmibq^ i^ abjlpqo^`fŽk-

Kjo\8 Rf o < \ * yM* i^p m^oqbpob^i b fj^dfk^of^ ab ` sfbkbk a^a^p mlo '8-03(-Rf i^ b`r^`fŽk `^o^`qboŒpqf`qfbkb alp o^Œ`bpafpqfkq^p+ob^ibp l `ljmibg^p+ i^ `lj_fk^`fŽkifkb^i

bp i^ plir`fŽk dbkbo^i ab i^ b`r^`fŽk afcbobk`f^i- Dpql bpqŠab ^`rboal `lk ilp obpriq^alpabjlpqo^alp bk bi qblobj^ 7-6-

Page 473: Calculus

Be`m^d^djn 342

:[ ilp Dgbo`f`flp nrb pfdrbk pb afp`rqbk lqolp bgbjmilp ab crk`flkbp `lj,mibg^p-

1&)( :WR_PVPV\`

0- Dumobp^o `^a^ rkl ab ilp pfdrfbkqbp k•jbolp `ljmibglp bk i^ cloj^ \ * ]d,

']( `mmd-0Š

'_( 0`+mmd-0Š

'b( 1`mmd,

'a( [`+mmdŠ

'b( d * `0mmdŠ

'b&(mmd-2,

'c( `mmd-2[ `+mmd-2Š

0 , `mmd-0

'd( 0 * `mmd-0Š

1- Dk `^a^ `^pl+ e^ii^o qlalp ilp s^ilobp ab s b v nrb p^qfpc^`bk i^ obi^`fŽk a^a^-

']( s * dt < s` !*

'_( s * dt < t`cT†

'b( _s(cs < ,0-

0 * d ,'`( ,0 ,- < s`%!*

,0

2- ^( Ool_^o nrb `% :‹ N m^o^ qlal `ljmibgl w-_( G^ii^o qlalp ilp `ljmibglp w m^o^ ilp nrb `%< 0-

3- ^( Rf '( bp ob^i+ abjlpqo^o nrb

_cK * z8!`lpb <,,,,

1v

_cK Z _*cK

pbk _ < ,,,,0d

_( Tp^o i^p cŽojri^p abi ^m^oq^al ^( m^o^ abar`fo i^p fabkqfa^abp

`lp! _ < ql * `lp /_&) pbk! _ < ql , `lp /_& +

4- ^( Cbjlpqo^o bi Q`jm`h\ _` Iicpl_)

'`lp _ * dpbk _&h < `lp h‚ * dpbk h_ )

sŠifal m^o^ qlal s^ilo ob^i ` v qlal bkqbol i mlpfqfsl-_( G^`bo i < 2 bk i^ m^oqb ^( v abar`fo i^p fabkqfa^abp qofdlklj‹qof`^p-

pbk 0_ < 2 `lp! _ pbk _ ,pbk2 _) `lp 0_ < `lp! _ * 2 `lp bpbk1 _ +

5- Cbjlpqo^o nrb qla^ prj^ qofdlklj‹qof`^ ab i^ cloj^

i

Oh%r&< q^l * 1 &\f `lp er * \} m_her&hzi

Page 474: Calculus

343 J„g_lim ]igjf_dim

mrbab bumobp^opb`ljl prj^ ab bumlkbk`f^ibp `ljmibg^p+

i

Pi&U' < 1 ?e_ce!$)fx+i

alkab ?e < c%[e * 8\e& m^o^ e < 0+ 1+ --- + h+ Cbqbojfk^o i^p `loobpmlkafbkqbp cŽojr,i^p m^o^ ^8z,

6- Rf h v i plk bkqbolp+ abjlpqo^o nrb

&0R `di!* `+dh!*_s < xN9J D_

_( Tqfifw^o bi ^m^oq^al ^( m^o^ abar`fo i^p obi^`flkbp ab loqldlk^ifa^a m^o^ bi pbkl ubi `lpbkl %g v h plk bkqbolp+ g0 :‹ h0y'8

RH h ", i *

pf h < i ,

a0! a0R G0!l pbk is `lp hs _s < l pbk is pbk hs _s < l `lp is `lp hs _s < N +

n a0Rl pbk1 is _s < l `lp! is _s < oo pf h"*K+

7- C^al rk k•jbol `ljmibgl w :‹ N- Olkbo w < m`%!*bk alkab .; ^od'w(- Rb^k Yi ;O`%\*bk alkab O < m.,h u- HW< L-i* u D < „oc.h) pfbkal i bkqbol mlpfqfsl-^( Cbjlpqo^o nrb u8< w: bpql bp+Vf bp rk^ o^Œwk,‹pfj^ ab w-_( Cbjlpqo^o nrb w qfbkb bu^`q^jbkqb i o^Œ`bpk,‹pfj^p afpqfkq^p+

u nrb bpqŠk pfqr^a^p pl_ob rk^ `fo`rkcbobk`f^ ab&o^afl O `ljl ilp s‹oqf`bp ab rk ml,iŒdlkl obdri^o-b( Cbqbojfk^o i^p qobp o^Œ`bp`•_f`^p ab f-a( Cbqbojfk^o i^p `r^qol o^Œ`bp`r^oq^p ab f-b( Cbqbojfk^o i^p `r^qol o^Œ`bp`r^oq^p ab , d,

8- K^p abcfkf`flkbp ab i^p crk`flkbp pbkl v `lpbkl mrbabk ^jmif^opb ^i mi^kl `ljmibgl`ljl pfdrb9

_ct * _*ct

`lp W : w pbk W < ++u+9++

Br^kal w bp ob^i+ bp^p cŽojri^p `lfk`fabk `lk i^p crk`flkbp pbkl v `lpbkl loafk^of^p-'Ubo Dgbo`f`fl&3-( Tqfifw^o bp^p cŽojri^p m^o^ abar`fo i^p pfdrfbkqbp molmfba^abp abi pbklv abi `lpbkl `ljmibglp- @nrŒp* q v w obmobpbkq^kk•jbolp `ljmibglp+ pfbkal w < s * dt*

'^( pbk %o * p& < pbk r`lp p * `lp om_hp+

'_( `lp %o * p& < `lp o `lp p ,pbk rpbk p+

'`( pbk! W * `lp, W < 0-'a( `lp %cs&< `lpev+ pbk %cs&< :pbkev-'b( `lp w < `lp s `lpe s * cpbk s pbke s+'N pbk w < pbk s `lpe t * c`lp s pbke t,

Page 475: Calculus

Be`m^d^djn ,--

0/- Rf u bp rk k•jbol `ljmibgl kl kril+ abcfkfjlp Kld u* bi ild^ofqjl `ljmibgl ab u* mloi^ b`r^`fŽk

K^d u < i^d Zw\* eod'w(-

Br^kal u bp ob^i v mlpfqfsl+ bpq^ cŽojri^ `lfk`fab `lk bi ild^ofqjl loafk^ofl- Djmib^obp^ cŽojri^ m^o^ abar`fo i^p molmfba^abp pfdrfbkqbp ab ilp ild^ofqjlp `ljmibglp-

'^( K^d ',0( < id* K^d 'f( < oof.1-

'_( K^d &WgW0'< K^d Wy * K^d W0 * 0immd*'b( K^d &Wy-W0'< K^d Yi , K^d W0 * Wi+i c)'a( `Ijb w < u,

alkab i bp rk bkqbolalkab i bp rk bkqbol

00- Rf t v u plk k•jbolp `ljmibglp+ u o &N+ abcfkfjlp u! mlo jbafl ab i^ b`r^`fŽk

LI 4 52 /37 u*

bk alkab Kld u bpqŠ abcfkfal `ljl bk bi Dgbo`f`fl 0/-^( B^i`ri^o 0e+fe+X ' ,0(e-

_( Cbjlpqo^o nrb u\u] < u\)] pf \* \ v u plk `ljmibglp- ! </-

b( N_p‹osbpb nrb i^ b`r^`fŽk

kl pb p^qfpc^`b `r^kal u* < W0 < , 0 X t < d, ƒBrŠibp plk i^p `lkaf`flkbp nrb ab_bk`rjmifo tx u W0 m^o^ ^pbdro^o nrb i^ b`r^`fŽk '8-06( bp sŠifa^ m^o^ qlal `ljmibgl s>

Dk ilp Dgbo`f`flp abi 01 ^i 04+ H obmobpbkq^ bi lmbo^alo ifkb^i abcfkfal mlo H%s&;< v! * \t% * ]t* pfbkal \ v ] `lkpq^kqbp ob^ibp-

01- Cbjlpqo^o nrb pf N bp rk^ crk`fŽk `ljmibg^+ mlo bgbjmil N%r&< L%r&* cM%r&)bkqlk,`bp rk^ crk`fŽk `ljmibg^ `%r&< o%r&* cp%r&p^qfpc^`b i^ b`r^`fŽk afcbobk`f^i H%s&< N%r&bk rk fkqbos^il . pf v pŽil pf o v p p^qfpc^`bk i^p b`r^`flkbp H%o&< L%r& v H%p&< M%r&bk g,

02- Rf = bp `ljmibgl v r bp ob^i+ abjlpqo^o nrb i^ b`r^`fŽk afcbobk`f^i H%s&< >`d^SU qfbkbplir`flkbp `ljmibg^p ab i^ cloj^ v < ?`%n+m* `lk q^i nrb ] !%!r/ l \r !!&/- Dumobp^o bik•jbol `ljmibgl > bk crk`fŽk ab \* \) =) X j~*

03- Rrmlkbo nrb b bp ob^i u ] !%!~(1- Tp^o ilp obpriq^alp abi Dgbo`f`fl 02 m^o^ abjlpqo^o nrbi^ b`r^`fŽk afcbobk`f^i H%s&< b`lp qr qfbkb rk^ plir`fŽk m^oqf`ri^o ab i^ cloj^u < = `lp %qr * /9(+ alkab

> vHZ

q^k HW < ]+++0

Š+ r

04- Rrmlkbo nrb b bp ob^i u \ !$! r/† Cbjlpqo^o nrb i^ b`r^`flk afcbobk`f^i H%s&< b pbktuqfbkb rk^ plir`fŽk m^oqf`ri^o ab i^ cloj^ v < > pbk %qr * /9( v bumobp^o > v l9 bkcrk`fŽk ab \* ]* b v r,

Page 476: Calculus