Calculus

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Tom M. Apostol Vol.1-part.2-pág.456-813

Transcript of Calculus

Page 1: Calculus
Page 2: Calculus

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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-

Mm‡gjbj \ g\ n`bpi_\ `_d^d‡i

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-

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Mm‡gjbj HW

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Page 17: Calculus

WWHH gi_d^` \i\g…od^j

05-06 Rfpqbj^p ab b`r^`flkbp ifkb^ibp05 -07 S‹`kf`^p ab `Ši`ril05-08 Hksbop^p ab j^qof`bp `r^ao^a^p05-1/ Dgbo`f`flp05-10 Dgbo`f`flp s^oflp pl_ob j^qof`bpRlir`flkbp ^ ilp bgbo`f`flp†kaf`b ^ic^_‹qf`l

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Page 18: Calculus

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Page 19: Calculus

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Dk bpqb`^mŒqrilpb bpqraf^k rklp mol_ibj^p bk m^oqbmi^kqb^alp e^`b rklp13// ^•lp+ `r^kal bi cfiŽplcl dofbdl YbkŽk ab Dib^ '384,324 ^- ab B-( mob`fmfqŽrk^ `ofpfp bk i^ L^qbjŠqf`^ ^kqfdr^ bpq^_ib`fbkal ^idrk^p m^o^alg^p fkdbkflp^p-Tk^ ab bii^p+ii^j^a^ cob`rbkqbjbkqb i^ k\m\_jd\ _`g ^jmm`_jm pb mrbab bumlkboab i^ j^kbo^ pfdrfbkqb9

Tk `loobalo kl mrbab ^i`^kw^o krk`^ i^ jbq^ mlonrb pfbjmob e^ ab ob`l,ooboi^ jfq^a ab rk^ afpq^k`f^ ^kqbp ab ob`loobo i^ afpq^k`f^ qlq^i- Dp ab`fo+`r^kal e^v^ ob`loofal i^ mofjbo^ jfq^a+ qbkaoŠ nrb `loobo i^ lqo^ jfq^a-Br^kal e^v^ ob`loofal i^ jfq^a ab ‹pq^+ ib nrba^oŠ qla^sŒ^ i^ `r^oq^m^oqb+r^kal e^v^ `loofal i^ jfq^a ab bpq^ `r^oq^ m^oqb+ib nrba^oŠ i^l`q^s^ m^oqbv ^pŒpr`bpfs^ b di_`adid_\h`io`,YbkŽk mbkpŽ+bsfabkqbjbkqb+ bk rk^ pfqr^`fŽk fab^i bk i^ nrb bi `loobalo

bp rk^ m^oqŒ`ri l mrkql nrb pb jrbsb ab rk buqobjl ^ lqol ab rk pbdjbkqlab ob`q^- O^o^ ^k^ifw^o bi o^wlk^jfbkql ab YbkŽk `lk jŠp abq^iib pb prmlkbnrb bi `loobalo m^oqbabi mrkql j^o`^al `lk 0 bk i^ cfdro^ 0/-0 v `loob e^`f^i^ jbq^ j^o`^a^ `lk N- K^p mlpf`flkbp fkaf`^a^p mlo -z+ +e+---+bq`-+pb•^i^ki^ co^``fŽk ab `^oobo^ nrb pb e^ ab `loobo qla^sŒ^ `r^kal pb ^i`^kw^ bi mrkqlj^o`^al- Dpq^pco^``flkbp '`^a^ rk^ ab i^p `r^ibp bp i^ jfq^a ab i^ ^kqboflo(pr_afsfabk qlal bi qo^vb`ql bk rk k•jbol fkabcfkfal ab m^oqbp ^a^ sbw jŠpmbnrb•^p- Orbpql nrb m^o^ ob`loobo mlo pbm^o^al `^a^ rk^ ab bpq^pm^oqbppbkb`bpfq^ rk^ `^kqfa^a mlpfqfs^ ab qfbjml+ m^ob`b k^qro^i ^cfoj^o nrb bi qfbjmlkb`bp^ofl m^o^ bi qo^vb`ql qlq^i e^ ab pbo i^ prj^ qlq^i ab qla^p bpq^p`^kqf,a^abp ab qfbjml- Cb`fo nrb bi `loobalo krk`^ mrbab ^i`^kw^o i^ jbq^ bnrfs^ib ^ab`fo nrb krk`^ iibd^ ^ bii^ bk rk qfbjml cfkfql: l+ af`el ab lqol jlal+ nrb i^prj^ ab rk k•jbol fkcfkfql ab fkqbos^ilp mlpfqfslp ab qfbjml kl mrbab pbocfkfq^-

K^ ^cfoj^`fŽk ab YbkŽk ab nrb rk k•jbol fifjfq^al ab `^kqfa^abp mlpf,qfs^p kl mrbab qbkbo rk^ prj^ cfkfq^+crb `lkqo^af`e^ 1/// ^•lp jŠp q^oab

346

Page 20: Calculus

347 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

`lk i^ `ob^`fŽk ab i^ qbloŒ ab i^p pbofbpfkcfkfq^p-Dk ilp pfdilp WUHH v WUHHH

^idrklp j^qbjŠqf`lp bjmbw^olk ^ mbkp^onrb bo^ mlpf_ib buqbkabo i^ fab^ abprj^ loafk^of^ ab `lkgrkqlp adidojn ab k•jbolp ^ `lkgrkqlp diadidojn* ab j^,kbo^ nrb bk ^idrklp `^plp i^ ~prj^‚ ab rk `lkgrkql ab fkcfkfqlp k•jbolp pb^cfkfq^- O^o^ sbo `Žjl pb mrbab e^`bo bpq^ buqbkpfŽk u qbkbo rk^ fab^ ab i^pafcf`riq^abp nrb mrbabk mobpbkq^opbm^o^ biil+ `lksfbkb ^k^ifw^o i^ m^o^alg^ abYbkŽk `lk jŠp abq^iib-

RrmŽkd^pb nrb bi `loobalo ^kqbp jbk`flk^al+ `loob ^ q`gj^d_\_ `lkpq^kqbt nrb kb`bpfq^ Q jfkrqlp m^o^ i^ mofjbo^ jfq^a abi ob`loofal- O^o^ bi pfdrfbkqb`r^oql ab ob`loofal kb`bpfq^oŠQ.1 jfkrqlp+ m^o^ bi l`q^sl pfdrfbkqb Q- 3 jfkr,qlp u bk dbkbo^i m^o^i^ m^oqb ljmobkafa^ bkqob /-0i u /-0i

)/ kb`bpfq^oŠQ-0i

jfkrqlp- K^ ~prj^‚ ab qlalp bpqlp fkqbos^ilp pb mrbab fkaf`^o pfj_Žif`^jbkqbbp`of_fbkal i^ pfdrfbkqb bumobpfŽk9

' KN-i(@ @ @Q) + * , * --- * , * ----1 3 0T

Dpqbbp rk bgbjmil ab i^p ii^j^a^p n`md`ndiadido\n u bi mol_ibj^ ^nrŒ bpqŠbkab`fafo pf bp mlpf_ib bk`lkqo^o rk^ cloj^ k^qro^i ab ^pfdk^oib rk k•jbol nrbpb mrba^ ii^j^o nph\ ab i^ pbofb-

K^ bumbofbk`f^cŒpf` af`b nrb bi `loobalo nrb `loob ^ sbil`fa^a `lkpq^kqb^i`^kw^oŠ pr jbq^ bk rk qfbjml al_ib abi nrb kb`bpfq^_^ m^o^^i`^kw^o pr mrkql

l d

EHFTQ@ 0/-0 I\ k\m\_je\ _` W`i4i,

jbafl- Orbpql nrb kb`bpfq^ Q jfkrqlp m^o^ i^ jfq^a abi ob`loofal+ e^_Œ^ abbjmib^o 0Q jfkrqlp m^o^ bi ob`loofal `ljmibql- Dpqbo^wlk^jfbkql prdfbob nrbpb ab_b ^pfdk^o i^ ~prj^‚ 0Q ^ i^ pbofbbk '0/-0( bpmbo^kal nrb i^ fdr^ia^a

'0/-1(P P P

Q * , * , * --- * , * --- < 0Q1 3 0i

mrba^ pbo ~sŠifa^‚ bk ^id•k pbkqfal-

Page 21: Calculus

I\ k\m\_je\ _` W`i‡i 348

K^ qbloŒ^ab i^p pbofbp fkcfkfq^p mob`fp^ `Žjl pb e^ ab fkqbomobq^obpq^ fdr^i,a^a- K^ fab^ bp i^ pfdrfbkqb9 Oofjbol pb prj^k rk iˆh`mj adidoj ab q‹ojfklp+ilp i mofjbolp+ fkaf`^kal bpq^ prj^ mlo n!, @pŒpb qfbkb9

'0/-2( Q Q Qp ;Q)+)+)}jj)+,! 1 3 +!'*

v bpq^ prj^ pb abkljfk^ nph\ k\m^d\g k,pfj^- Rb bpqraf^ abpmr‹p bi `ljmlo,q^jfbkql ab n9 `r^kal i qlj^ s^ilobp `^a^ sbw jŠp do^kabp- Dk m^oqf`ri^o pbqo^q^ ab abqbojfk^o pf i^p prj^p m^o`f^ibp n9 qfbkabk ^ rk iŒjfqb cfkfql `r^kali `ob`b fkabcfkfa^jbkqb-

Dk bpqb `^pl bp cŠ`fi sbo nrb bi s^ilo iŒjfqb ab i^p prj^p m^o`f^ibp bp 0Q,

Dk bcb`ql+ `^i`ri^kal ^idrk^p ab bpq^p prj^p m^o`f^ibp pb qfbkb9

RH< P)Q 1

R1 < Q)!0 <!1 Q*Q Q 5

R2 < P * !1 * z< 3 P)

Q Q Q /3R3 < P * !1 * z* 7! < 7 P+

Rb l_pbos^ nrb bpqlp obpriq^alp pb mrbabk bumobp^o`ljl pfdrb9

RH< '1 , f&P) P0 < '1 , &P) P1 < '1 , &P) P2 < '1 , f&P+

il `r^i `lkar`b ^ mbkp^o bk rk^ cŽojri^ dbkbo^i ab i^ cloj^9

'0/-3( n9 < '1 , 0Hf&Q m^o^ qlal bkqbol mlpfqfsl i,

K^ cŽojri^ 'i/-3( pb `ljmorb_^ cŠ`fijbkqb mlo fkar``fŽk- Orbpql nrb 0.1!,0 ,* N`r^kal i `ob`b fkabcfkfa^jbkqb+ obpriq^ p! ,* 0Q, Olo q^kql+ i^ fdr^ia^a '0/-1(bp ~`fboq^‚ pf pb fkqbomobq^nrb 0Q bp bi g…hdo`ab i^p prj^p m^o`f^ibp n,9 Dpqbmol`bpl ab iŒjfqb m^ob`b fks^ifa^o i^ l_gb`fŽk ab YbkŽk nrb i^ prj^ ab rkk•jbol fkcfkfql ab fkqbos^ilp ab qfbjml kl mrbab pbo krk`^ cfkfq^-

@elo^ a^objlp rk ^odrjbkql nrb molmlo`flk^ rk ^mlvl `lkpfabo^_ib ^imrkql ab sfpq^ ab YbkŽk- Rrmlkd^jlp nrb bk bi ^kqboflo ^kŠifpfp ab i^ m^o^alg^abi `loobalo pb e^`b rk mbnrb•l mbol fjmloq^kqb `^j_fl- Dk sbw ab `lkpf,abo^o i^ sbil`fa^a `lkpq^kqb+ prmŽkd^pb nrb ab`ob`b do^ar^ijbkqb ab j^kbo^nrb kb`bpfq^ Q jfkrqlp m^o^ fo ab 0 ^ 0.1+ Q-0 m^o^ fo ab 0.1 ^ 0.3+ Q-1 jfkr,qlp m^o^ fo ab 0.3 ^ 0.7+ u bk dbkbo^i Q-i jfkrqlp m^o^ fo ab 0.1!,0 ^ 0.1!-

Page 22: Calculus

-/) Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

Di ~qfbjml qlq^i‚ nrb kb`bpfq^oŠ m^o^ i^ `^oobo^+ sbkaoŠ ^elo^ obmobpbkq^al mloi^ pfdrfbkqb pbofb fkcfkfq^9

'i/-4(

Dk bpqb `^pl+ i^ bumbofbk`f^ cŒpf`^ kl prdfbob kfkdrk^ ~prj^‚ l_sf^ l k^qro^im^o^ ^pfdk^o ^ af`e^ pbofb v mlo q^kql bpqb bgbjmil e^v nrb bpqraf^oil abpabrk mrkql ab sfpq^ `ljmibq^jbkqb j^qbjŠqf`l-

Hdr^i nrb ^kqbp+ pb fkqolar`bk i^p prj^p m^o`f^ibp Pi9 bp ab`fo9

'0/-5( Q Q Qp+< Q * !1 * 2! * --- * ::,-

v pb qo^q^ ab sbo nr‹ l`roob ^ n9 `r^kal i `ob`b fkabcfkfa^jbkqb- Dpq^ prj^m^o`f^i kl bp q^k cŠ`fi ab bpqraf^o `ljl i^ ab '0/-2(+ mrbp kl bufpqb rk^ cŽojri^^kŠild^ ^ i^ '0/-3( nrb pfjmifcfnrb i^ bumobpfŽk abi pbdrkal jfbj_ol ab '0/-5(-Rfk bj_^odl+ mlo `ljm^o^`fŽk ab bpq^p prj^p m^o`f^ibp `lk rk^ fkqbdo^i ^mol,mf^a^ pb mrbab sbo nrb qlj^k s^ilobp q^k do^kabp `ljl pb nrfbo^-

Dk i^ cfdro^ 0/-1 pb sb m^oqb ab i^ efm‹o_li^ n%r&< f,r m^o^ r = N- 'Dk bibgb u pb e^ jlafcf`^al i^ bp`^i^-( Klp ob`qŠkdrilp af_rg^alp+ qfbkbk rk Šob^ qlq^ifdr^i ^ i^ prj^

'0/-6(

Di Šob^ ab i^ obdflk abqbojfk^a^ mlo i^ efm‹o_li^ v bi fkqbos^il Z0+ i * 0\ bpax)gU+g ^r < ild %h* 0(+ v mrbpql nrb bpq^ Šob^ kl mrbab bu`babo i^ prj^ ab

i^p Šob^p ab ilp ob`qŠkdrilp+ pb qfbkb i^ abpfdr^ia^a

'0/-7( 0 0 00 * , * , * --- * , = ild %h* 0(1 2 i m8 * Š

Lriqfmif`^kal ^j_lp jfbj_olp mlo Q pb l_qfbkb m8ƒ Q ild %h* 0(- Dp ab`fo+pf i^ sbil`fa^a abi `loobalo ab`ob`b q^i `ljl pb e^ fkaf`^al ^kqboflojbkqb+ biqfbjml kb`bp^ofl m^o^ ^i`^kw^o bi mrkql /-0i bp mlo il jbklp Q ild %h* 0( jf,krqlp- Orbpql nrb ild %h * 0( ^i `ob`bo h qlj^ s^ilobp q^k do^kabp `ljl pbnrfbo^+ pb `rjmib bk bpqb `^pl i^ m^o^alg^ ab YbkŽk+ bp ab`fo nrb bi `loobalokl ^i`^kw^oŠ i^ jbq^ bk rk qfbjml cfkfql-

K^ qbloŒ^ dbkbo^i ab pbofbp fkcfkfq^p e^`b rk^ afpqfk`fŽk bkqob pbofbp `ljl'0/-0( `rv^p prj^p m^o`f^ibp qfbkabk ^ rk iŒjfqb cfkfql+ v pbofbp `ljl '0/-4( `rv^poqi]o m^o`f^ibp kl qfbkbk iŒjfqb cfkfql- K^p mofjbo^p pb abkljfk^k ^jiq`mb`io`n

Page 23: Calculus

I\ k\m\_je\ _` W`i‡i 350

t

K^ prj^ ab i^p 0Šob^p ab ilp ob`qŠkdrilp bp 0 * , * - - * --- * - -

1 2 i

Di Šob^ ab i) Y

i^ obdfŽk plj_ob^a^ bp / r *X^r <Hld %h* 0(

sl 1 2 i k*0

EHFTQ@ 0/-1 Pdbidad^\_j b`jh„omd^j _` g\ _`ndbp\g_\_ H * /-0 * --- * Efh ƒ i^d &i * 0(-

v i^p pbdrka^p _dq`mb`io`n, Klp mofjbolp fksbpqfd^alobp bk bpqb aljfkfl mlkŒ^kml`^ l kfkdrk^ ^qbk`fŽk bk i^p `rbpqflkbp ab `lksbodbk`f^ v afsbodbk`f^- So^,q^_^k i^p pbofbpfkcfkfq^p`ljl pf crbo^k prj^p loafk^of^p cfkfq^p+prgbq^p ^ i^pibvbp rpr^ibp abi „idb_o^ pfk qbkbo bk `rbkq^ nrb bpq^pibvbp kl mrbabk buqbk,abopb rkfsbop^ijbkqb ^ i^p pbofbpfkcfkfq^p-Olo bpl kl bp plomobkabkqb nrb pbe^v^ sfpql- jŠp q^oab nrb ^idrklp ab ilp mofjbolp obpriq^alp l_qbkfalp crbo^kfk`loob`qlp- @cloqrk^a^jbkqb+ jr`elp ab ^nrbiilp mflkbolp qbkŒ^krk^ fkqrf,`fŽk v abpqobw^ml`l cob`rbkqbp+nrb ibp bsfq^_^ iibd^o ^ `lk`irpflkbp c^ip^p+^rknrb biilp kl mrafbo^k grpqfcf`^o prp j‹qlalp- Dkqob ilp mofjbolp j^qbjŠ,qf`lp nrb pb l`rm^olk ab i^p pbofbpl`rm^ rk ird^o mobbjfkbkqb Kblk^oa Dribo-Dribo abp`r_oŒ^cŽojri^ qo^pcŽojri^+ ^ `r^i jŠp fkqbobp^kqb+v ^ i^ sbw rqfifw^_^i^p pbofbpfkcfkfq^p`ljl `lk`bmql rkfcf`^alo ab afsbop^p o^j^p ab i^ L^qbjŠ,qf`^ nrb e^pq^ bkqlk`bp bpq^_^k pfk obi^`flk^o- K^ do^k `^kqfa^a ab qo^_^glpab Dribo nrb e^k pl_obsfsfal ^i m^pl abi qfbjml bp rk qof_rql ^ pr klq^_fiŒpfjlfkpqfkql ab il j^qbjŠqf`^jbkqb `loob`ql-

K^ buqbkpfŽkabi rpl ab i^p pbofbpfkcfkfq^pbjmbwŽ jŠp q^oab bk bi pfdil WUHH-

`bo`^ ab `fk`rbkq^ ^•lp abpmr‹p abi k^`fjfbkql ab Dribo+ `lfk`fafbkal `lk bimofk`fmfl abi abp^ooliil abi BŠi`ril fkqbdo^i Mf`eli^p Lbo`^qlo '051/,0576( vVfiif^j Aolrk`hbo '051/,0573( abp`r_ofbolk bk 0557 rk^ pbofb fkcfkfq^ m^o^

Page 24: Calculus

351 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

bi ild^ofqjl ^i fkqbkq^o `^i`ri^o bi Šob^ ab rk pbdjbkql efmbo_Žif`l- Ol`l abp,mr‹p+ Mbtqlk abp`r_ofŽ i^ n`md` ]di‡hd^\, Dpqlp abp`r_ofjfbkqlp `lkpqfqrvbkrk mrkql crka^jbkq^i ab i^ efpqlof^ ab i^ L^qbjŠqf`^- Tk `^pl m^oqf`ri^oab i^ pbofb _fkŽjf`^ bp bi `lkl`fal o`jm`h\ _`g ]dijhdj nrb ^cfoj^ nrb9

'0 * so < &W'se)

Q5>

alkab r bp rk k•jbol ob^i ^o_fqo^ofl+ h rk bkqbol kl kbd^qfsl+ v F( bp bi`lbcf`fbkqb _fkljf^i- Mbtqlk bk`lkqoŽ nrb bpq^ cŽojri^+ sŠifa^ m^o^ s^ilobp _h*o`mjn ab i pb mlaŒ^ buqbkabo ^ bumlkbkqbp m`\g`n `r^ibpnrfbo^+ prpqfqrvbkal i^prj^ cfkfq^ abi pbdrkal jfbj_ol+ mlo rk^ pbofb cfkfq^ `lksbkfbkqb+ pf _fbk klil abjlpqoŽ- Dcb`qfs^jbkqb+ bk rk bpqrafl `rfa^alpl ab i^ pbofb _fkljf^i prodbk^idrk^p `rbpqflkbp _^pq^kqb abif`^a^p ab `lksbodbk`f^ ^ i^p nrb kl pb mlaŒ^obpmlkabo bk i^ ‹ml`^ ab Mbtqlk-

Ol`l abpmr‹p ab i^ jrboqb ab Dribo bk 0672+ bi `^ra^i ab krbslp abp`r,_ofjfbkqlp bjmbwŽ ^ afpjfkrfo v bi mboŒlal cloj^i bk i^ efpqlof^ ab i^p pbofbpiibdŽ ^ pr q‹ojfkl- Tk krbsl mboŒlal+ v jŠp `oŒqf`l+ bjmbwŽ bk 0701 `r^kalF^rpp mr_if`Ž i^ `‹ib_ob jbjlof^ nrb `lkqbkŒ^+ mlo mofjbo^ sbw bk i^ efpqlof^+rk bpqrafl ofdrolpl ab i^ `lksbodbk`f^ ab ^idrk^p pbofbp fkcfkfq^p- Ol`lp ^•lpjŠp q^oab+ B^r`ev fkqolargl rk^ abcfkf`fŽk ^k^iŒqf`^ abi `lk`bmql ab iŒjfqb bkpr qo^q^al @pmnj_` >iƒgdndn\gb`]m\d^j 'mr_if`^al bk 0710(+ v bumrpl ilp crka^,jbkqlp ab i^ qbloŒ^jlabok^ ab `lksbodbk`f^ v afsbodbk`f^- Dk i^ Rb``fŽk nrbpfdrb pb bumlkaoŠk ilp orafjbkqlp ab bpq^ qbloŒ^-

)(&* EbPR`V\[R`

Dk bi ibkdr^gb `loofbkqb i^p m^i^_o^p ~pbofb‚ v ~pr`bpflk‚ plk pjljj^p vpb rqfifw^k m^o^ abpfdk^o rk `lkgrkql ab `lp^p l pr`bplp afpmrbpqlp bk rk loabk-Dk L^qbjŠqf`^+ bpq^p m^i^_o^p qfbkbk rk pfdkfcf`^al q‹`kf`l bpmb`f^i- K^ m^i^,_o^ ~pr`bpfŽk‚ qfbkb rk pbkqfal ^kŠildl ^i abi ibkdr^gb `loofbkqb+ mrbp `lk bii^pb nrfbob fkaf`^o rk `lkgrkql ab l_gbqlp mrbpqlp bk loabk+ mbol i^ m^i^_o^~pbofb‚ pb rp^ bk rk pbkqfal `ljmibq^jbkqb afpqfkql- @nrŒ pb bpqraf^oŠ bi `lk,`bmql ab pr`bpfŽk abg^kal bi ab pbofb m^o^ abcfkfoil jŠp q^oab bk bi ^m^oq^al 0/-4-

Rf ^ `^a^ bkqbol mlpfqfsl i bpqŠ ^pl`f^al rk k•jbol ob^i \i* bkqlk`bp pbaf`b nrb bi `lkgrkql loabk^al

abcfkb rk^ pr`bpfŽk fkcfkfq^- B^a^ q‹ojfkl ab i^ pr`bpflk qfbkb ^pfdk^al rkbkqbol mlpfqfsl+ ab j^kbo^ nrb pb mrbab e^_i^o abi kmdh`m o„mhdij ^i&abi n`+bpi_j o„mhdij \0 X bk dbkbo^i abi o„mhdij i+ndhj \ij B^a^ q‹ojfkl \z qfbkb rkpfdrfbkqb \iE X mlo q^kql kl e^v rk ~•iqfjl‚ q‹ojfkl-

Page 25: Calculus

Pp^`ndji`n 352

Klp bgbjmilp jŠp `loofbkqbp ab pr`bpflkbp pb mrbabk `lkpqorfo a^kal ^idrk^obdi^ l cŽojri^ nrb abcfk^ bi q‹ojfkl k,pfjl- @pŒ+mlo bgbjmil+ i^ cŽojri^\9 < //i abcfkb i^ pr`bpfŽk `rvlp `fk`l mofjbolp q‹ojfklp plk9

@idrk^p sb`bp pb kb`bpfq^k alp l jŠp cŽojri^p+ mlo bgbjmil9

[0i+/ < 0+

pfbkal bk bpqb`^pl ilp mofjbolp q‹ojfklp9

0+1+0+7+0+07+0+21+0

Nqo^ cloj^ `loofbkqb ab abcfkfo rk^ pr`bpfŽk bp jbaf^kqb rk `lkgrkql ab fkp,qor``flkbp nrb fkaf`^k `Žjl pb l_qfbkb rk q‹ojfkl ^ m^oqfoab ilp ^kqboflobp-@pŒ+pb qfbkbmlo bgbjmil9

Dpqbj‹qlal m^oqf`ri^o pb `lkl`b mlo cŽojri^ ab m`^pmm`i^d\ v abcfkb rk^ pr`b,pfŽk c^jlp^ ii^j^a^ ab Cd]ji\^^d, ) Klp mofjbolp q‹ojfklp plk9

0+ 0+1+ 2+ 4+ 7+ 02+10+ 23-

Dk qla^ pr`bpfŽk il bpbk`f^i bp nrb bufpqbrk^ crk`fŽk ` abcfkfa^ bk ilpbkqbolp mlpfqfslp+ q^i nrb `%h& bp bi q‹ojfkl k,pfjl ab i^ pr`bpfŽk m^o^ `^a^i < 0+1+ 2+ --- Dcb`qfs^jbkqb+ ‹pqb bp bi `^jfkl jŠp `lksbkfbkqb m^o^ bpq^_ib,`bo rk^ abcfkf`fŽk q‹`kf`^ ab pr`bpfŽk-

CDEHMHBHˆM- Ri\ api^d‡i a ^ptj _jhdidj `n `g ^jiepioj _` oj_jn gjn `io`+mjn kjndodqjn 0+1+ 2+ ! - n` _`ijhdi\ np^`nd‡i diadido\, Bg q\gjm a&i' _` g\ api+^d‡i n` _`ijhdi\ `g o„mhdij i+ndhj _` g\ np^`nd‡i,

Di m`^jmmd_j ab i^ crk`fŽk 'bp ab`fo+ bi `lkgrkql ab s^ilobp ab i^ crk`fŽk(pb mlkb jr`e^p sb`bp ab j^kfcfbpql+ bp`of_fbkal ilp q‹ojfklp bk loabk- @pŒ9

H'h'*a&0'*a&1'% ,,, *a&i'* ,, * ,

) Ef_lk^``f+ `lkl`fal q^j_f‹k mlo Kblk^oal ab Ofp^ '0064,014/(+ bk`lkqoŽ bpq^ pr`bpfŽk ^iqo^q^ork mol_ibj^ obi^qfsl ^ ilp mol`bplp ebobafq^oflp bk ilp `lkbglp-

Page 26: Calculus

353 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

Olo o^wlkbp ab _obsba^a pb rqfifw^ i^ klq^`fŽk v-&i'w m^o^ fkaf`^o i^ pr`bpflk`rvl q‹ojfkl k,pfjl bk E%h&+Blk jr`e^ cob`rbk`f^ i^ abmbkabk`f^ ab h pb fkaf`^rqfifw^kal pr_Œkaf`bp+ v pb bp`of_b \,* Pi* sz* Ri* N klq^`fŽk ^kŠild^ bk sbw abn_h&+

K^ `rbpqfŽk nrb pb nrfbob `lkpfabo^o+ bp ab`fafo pf ilp q‹ojfklp E%h&qfbkabkl kl ^ rk iŒjfqb cfkfql `r^kal i `ob`b fkabcfkfa^jbkqb- O^o^ biil+ pb mob`fp^buqbkabo bi `lk`bmql ab iŒjfqb ^ i^p pr`bpflkbp+ il nrb pb ildo^ `lk i^ abcfkf`fŽkpfdrfbkqb-

CDEHMHBHˆM- Ri\ np^`nd‡i vo&i'w od`i` g…hdo I nd*k\m\ ^\_\ iˆh`mj kjndod+qj ’+ `sdno` jomj iˆh`mj kjndodqj M &lp` `i b`i`m\g _`k`i_` _` `Z o\g lp`

E,%h&* HE; ’ k\m\ oj_j h ƒ K,

Bi `no` ^\nj* _`^dhjn lp` g\ np^`nd‡i vo`i'w ^jiq`mb` c\^d\ I t `n^md]dhjn

Fdha&i' < I* l a&i' ,* H l i+)jj,

Ri\ np^`nd‡i lp` ij ^jiq`mb` n` gg\h\ _dq`mb`io`,

Dk bpq^ abcfkf`fŽk ilp s^ilobp ab i^ crk`fŽk E%h&v bi iŒjfqb H mrbabk pbok•jbolp ob^ibp l `ljmibglp- Rf Ev H plk `ljmibglp+ mrbabk abp`ljmlkbopb bkpr m^oqb ob^i b fj^dfk^of^+ pb^k ‹pq^p F < p * dq v I < \ * d], Dkqlk`bp qbkb,jlp F&i' + I < p&i' + \ * dXq&i' + ]Z, K^p abpfdr^ia^abp

fo%h&* ]h Q E,%h&* HE v Ep%h&* \f Q E,%h&* JG

morb_^k nrb i^ obi^`fŽk F&i' x I fjmif`^ nrb p&i' x \ v q`i' x ] `r^kali x ll + Qb`Œmol`^jbkqb+ i^ abpfdr^ia^a

E,%h&* HE Q fo%h&* ]h * Ep%h&* ^h

abjrbpqo^ nrb i^p alp obi^`flkbp p&i' x \ v qvi' x ] fjmif`^k F&i' x I `r^kali x ^j * Cf`el ab lqol jlal+ rk^ pr`bpfŽk `ljmibg^ E`lksbodb pf v pŽil pf i^m^oqbob^i p v i^ m^oqbfj^dfk^of^ q `lksbodbk pbm^o^a^jbkqb+ bk `rvl `^pl qbkbjlp

gdha&i' < ifj p&i' * fifj q&i',

Dp `i^ol+ nrb qla^ crk`fŽk abcfkfa^ m^o^ qlalp ilp k•jbolp s ob^ibp v mlpf,qfslp mrbab pbosfo m^o^ `lkpqorfo rk^ pr`bpfŽk obpqofkdfbkal r ^ qlj^o pŽil s^il,obp `io`mjn, Dpql bumif`^ i^ do^k ^k^ildŒ^ bkqob i^ abcfkf`fŽk nrb pb ^`^_^ ab a^o

Page 27: Calculus

Oo]_mcih_m gihƒnih[m ^_ h„g_lim l_[f_m 354

v i^ Rb``fŽk 6-03 m^o^ crk`flkbp jŠp dbkbo^ibp- K^ ^k^ildŒ^ pb mobpbkq^ q^j_f‹kbk ilp f•gcn_m ch`chcnim u pb abg^ ^i ib`qlo bi abcfkfo ilp pŒj_lilp

gdha&i' < * // v ifjc'k( < , //

`ljl pb efwl bk i^ Rb``fŽk 6-04 `r^kal ` bp ab s^ilobp ob^ibp- Rf ` bp `ljmibg^+bp`of_fjlp `%h&w // `r^kal h w // pf E`%h&.w * ^j ,

K^ co^pb ~pr`bpfŽk `lksbodbkqb‚ pb bjmib^ pŽil m^o^ pr`bpflkbp `rvl iŒjfqbbp `chcni+ Rr`bpflkbp `lk iŒjfqb * // l , // pb af`b nrb plk afsbodbkqbp- K^pcŽojri^p nrb pfdrbk abcfkbk ^idrk^p pr`bpflkbp-

a&i' < %ZE&h )h.f$

a&i' < pbk 1& a&i' < '\G(j'0 *y(+ a&i' < `!di-0 ,

K^p obdi^p _Špf`^p m^o^ iŒjfqbp ab prj^p+ molar`qlp+ bq`-+ plk sŠifa^p q^j_f‹km^o^ iŒjfqbp ab pr`bpflkbp `lksbodbkqbp- Di ib`q`o kl bk`lkqo^oŠ afcf`riq^a bk i^clojri^`fŽk ab af`elp qblobj^p+ v prp abjlpqo^`flkbp pb e^`bk bk cloj^ ^kŠild^^ i^p ab i^ Rb``fŽk 2-4-

K^ `lksbodbk`f^ l afsbodbk`f^ ab jr`e^p pr`bpflkbp pb mrbabk abqbojfk^orqfifw^kal molmfba^abp ab crk`flkbp `lkl`fa^p nrb bpqŠk abcfkfa^p m^o^ qlal smlpfqfsl- Rb a^k ^ `lkqfkr^`fŽk ^idrklp bgbjmilp fjmloq^kqbp `rvlp iŒjfqbp pbmrbabk bk`lkqo^o afob`q^jbk-b l rqfifw^kal ^idrklp ab ilp obpriq^alp abar`falpbk bi `^mŒqril 6-

'0/-8( Hfj 0-, < N pf l`= N-i+ NB( i\,

ifj u! < N RH Zu\ ; i-ixjj

'0/-0/(

'0/-00('ild i'!

ifj ,,, < N m^o^ qlal \ = N+\ = N -i+ // i]

'0/-01( ifj i% ! < 0 -

'0/-02( ifj '0 * x'i < _[k,l` i

m^o^ qlal ob^i [ +

0/-2 Rr`bpflkbp jlkŽqlk^p ab k•jbolp ob^ibp

Tk^ pr`bpfŽk un%h&vpb af`b nrb bp ]l_]c_hn_ pf

a&i' xa&i * 0( m^o^ qlal i ƒ H-

Page 28: Calculus

-// Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Dpql pb fkaf`^ _obsbjbkqb bp`of_fbkal `%h&8!+ Olo lqo^ m^oqb+pf pb qfbkb9

a&i' xa&i * 0( m^o^qlal i x 0+

pb af`b nrb i^ pr`bpfŽk bp _`^m`^d`io` v pb bp`of_ba&i'%<G,,Tk^ pr`bpfŽk pb ii^j^jlkŽqlk^ `r^kal bp `ob`fbkqb l ab`ob`fbkqb-

Dp `Žjlal qo^_^g^o`lk pr`bpflkbp jlkŽqlk^p mrbp pr `lksbodbk`f^ l af,sbodbk`f^ pb mrbab abqbojfk^o cŠ`fijbkqb- Dk bcb`ql+ pb qfbkb bi pbk`fiil `ofqboflpfdrfbkqb-

RCMPCK? 0/-0- Ri\ np^`nd‡i hji‡oji\ ^jiq`mb` ndt n‡gj nd `n \^jo\_\,

Kjo\8 Tk^ pr`bpfŽk va&i'wpb af`b nrb bpqŠ\^jo\_\ pf bufpqbrk k•jbol mlRfqfslLq^i nrb E`%h&Gy L m^o^ qlal h+ Tk^ pr`bpfŽk nrb kl bpqŠ ^`lq^a^ pb abkljfk^ hi\^jo\_\

A`hjnom\^d‡i, Dp `i^ol nrb rk^ pr`bpfŽk kl ^`lq^a^ kl mrbab `lksbodbo+mlo q^kql pb e^ ab abjlpqo^o pli^jbkqb nrb rk^ pr`bpfŽk jlkŽqlk^ u ^`lq^a^`lksbodb-

Rrmrbpql a&i'-! pb^ K bi buqobjl prmboflo abi `lkgrkql ab s^ilobp ab i^crk`fŽk- 'Orbpql nrb i^ pr`bpfŽk bpqŠ ^`lq^a^+ bk sfoqra abi ^uflj^ 0/ abi`lkgrkql ab ilp k•jbolp ob^ibp+qfbkb rk buqobjl prmboflo-( Dkqlk`bp E%h&77787H

H * b H

` r ` r h- - ----- H.%.& .%/& .%0& .%1& F&K' F&i'

EHFTQ@ 0/-2 Ri\ np^`nd‡i ^m`^d`io` \^jo\_\ ^jiq`mb` c\^d\ np `som`hj npk`mdjm,

m^o^ qlal i* v pb qo^q^ab mol_^o nrb i^ pr`bpfŽk `lksbodb e^`f^ I,Rb^ ’ rk k•jbol mlpfqfsl ^o_fqo^ofl-Bljl H * ’ kl mrbab pbo rk^ `lq^

prmboflo ab oj_jn ilp k•jbolp F&i' e^ ab pbo I + b ; F&K'm^o^ ^id•k K 'bpqbK abmbkab ab D(- Orbpql nrb a&i'-! *pf i x K bp F&K'x F&i'*mlo q^kql+I + ’ ;9 `%h&w H m^o^qlal h w J) `ljl pb sb bk i^ cfdro^ 0/-2- Cb bpq^pabpfdr^ia^,abp pb abar`b9

L88898I + a&i' ; ’ m^o^ qlal i x K

il nrb pfdkfcf`^ nrb i^ pr`bpfŽk `lksbodb e^`f^ H) `ljl pb nrboŒ^abjlpqo^o-Rfa&i'%<G,i^ abjlpqo^`fŽk bp ^kŠild^+ pfbkal bk bpqb`^pl bi iŒjfqbbi buqobjl

fkcboflo ab ilp s^ilobp ab i^ crk`fŽk-

Page 29: Calculus

Bd`m^d^djn 356

)(&, :WR_PVPV\`

Dk ilp Dgbo`f`flp 0 ^i 11+ pb abcfkb rk^ pr`bpflk un%h&vmlo i^ cŽojri^ a^a^- Dk `^a^`^pl+ '^( abqbojfk^o pf i^ pr`bpfŽk `lksbodb l afsbodb+ u '_( e^ii^o bi iŒjfqb ab `^a^ pr`b,pfŽk `lksbodbkqb- Dk ^idrklp `^plp mrbab pbo •qfi prpqfqrfo bi bkqbol i mlo rk k•jbolob^i mlpfqfsl ^o_fqo^ofl s v bpqraf^o i^ crk`fŽk ab s obpriq^kqb mlo ilp j‹qlalp abi `^mŒ,qril 6- Rb mrbabk ^mif`^o i^p cŽojri^p '0/-8( ^ '0/-02( a^a^p ^i cfk^i abi ^m^oq^al 0/-1-

i i * G0- g`i' < i * H , +i+ ,

i/ i/ * 00, g`i' < ::,9z99,z, +i+ ,

01 g`i' < [1iZ(Z% ZZ0['[i[

, 1i)g * ' [0'i)g ,

.0+ g`i' < y , Ty-

i5Q1, ,%h&< `lp 1! -

i0 * 1i + 12, g`i' < 3i0 Š

.1+ -&i' < i\!9 alkab X\g ; 0-

i3 g`i' <,, 0i%

/3, g`i' < gjbii* \< 0-i

0/////k.3+ g`i' < 0 * i0

.4+,%h& <'0 *yn4, g`i' < G* ' \G(j-

G* '\G(j5, g`i' < ,,, , ,

i

6, g`i' < '\G(ji

7, e8&i' < 10.!-

G* '\G(j* +

i i5Q.5+ g`i' < 0 * ,,`lp,-

i * H 1

.6+ g`i' <' G* yn0., g`i' < `+mmdi-0Š

010- g`i' < , `+mmdi-0Š

i.-+ g`i' < i&+F'i,

h0-1 pbk %h &&&$ 6#;$ 2 % &

i * 000, g`i' < i`+mmdi-0Š

K^p pr`bpflkbp u[iw ab ilp Dgbo`f`flp abi 12 ^i 17 plk qla^p `lksbodbkqbp- Olo q^kql+m^o^ `^a^ D = N mobcfg^al bufpqb rk bkqbol J 'nrb abmbkab ab D(+ q^i nrb g\i + JG; D

pf h878;7K pfbkal I < ifjkzNI \ij Cbqbojfk^o bk `^a^ `^pl bi s^ilo ab K nrb `loobpmlkab^ ilp pfdrfbkqbp s^ilobp ab D9 D < 0: /+0: /+/0: /+//0: /+///0-

G01, \i < ,-

i

0/3+ \i <+-

i,

i/1+ \i < ,,h&

i)' ]H(k*i

/2+ \i < ,,,i

0i05, \h < i1 * 0 -

06, \9 < ']i(k'9lo

Page 30: Calculus

-/1 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

18- Cbjlpqo^o nrb rk^ pr`bpfŽk kl mrbab `lksbodbo e^`f^ alp iŒjfqbp afpqfkqlp-2/- Rrmrbpql nrb ifok-+z // [} < N-@mif`^kal i^ abcfkf`fŽk ab iŒjfqb abjlpqo^o nrb

ifj ++zll \9 < N-20- Rf ifj++zll [i < > X ifj++zll ]i ;? ^mif`^kal i^ abcfkf`fŽk ab iŒjfqb abjlpqo^o nrb

ifj++zll &\i * ]i' < > * ? u ifj **xjj&`\i' < `> alkab ` bp rk^ `lkpq^kqb-21- O^oqfbkal ab ilp obpriq^alp l_qbkfalp bk 2/ v 20 abjlpqo^o nrb pf ifjkzll \h; = bp

ifj++]-ll \x < >0, Cbpmr‹p+ e^`fbkal rpl ab i^ fabkqfa^a 0\i]i < &\i * ]i'0 + \/ * ]0

abjlpqo^o nrb gdhixjj&\i]i' < >? pf ifjkzll \i ;> v Hfjkzll ]i < ?, i i

22- Rf \ bp rk k•jbol ob^i v i rk bkqbol kl kbd^qfsl+ bi `lbcf`fbkqb _fkljf^i 'z( bpqŠ abcf,kfal mlo9

&\' < \&\ + /'&\ + 1( --- '] , i * 0(i * ,T(

'^( Rf \ < , p+mol_^o nrb9

4

, 05& &\' 133 < 017&

'_( Rb^ \,• < ' ]H(k &+x-0', Ool_^o nrb \! = N v nrb \i(. ; \i ,

23- Rb^ ` rk^ crk`fŽk ob^i jlkŽqlk^ `ob`fbkqb v ^`lq^a^ bk bi fkqbos^il Z/+0\- Cbcfk^jlpalp pr`bpflkbp vPiw u voiw abi pfdrfbkqb jlal9

FE FE c'0( , a`L'^( Cbjlpqo^o nrb p! z l X&s' _s x `! X nrb N z jX&s' _s + Oh w + i , ,

_( Cbjlpqo^o nrb i^p pr`bpflkbp um))vv Q))v`lksbodbk ^j_^p e^`f^ bi iŒjfqb &gX&U' r +-il

b( Dpq^_ib`bo u abjlpqo^o rk obpriq^al `loobpmlkafbkqb ^i fkqbos^il Z^+ _\-24- Tqfifw^o bi bgbo`f`fl 23 m^o^ bpq^_ib`bo i^p pfdrfbkqbp obi^`flkbp9

0 i &f 1 0'^( ifj , ƒ,(< 2 -

$n.ZZ T8 ; 9.: ;

i +'_( ifj ƒ,, < ild 1-

kzNN i * efx/

h!! i oo

'b( ifj I xf0 <,-kzNN i * 3

Q_R

'a( ifj z 0 ‘ mkzNN 5- r&i0 * f0 < ild 'i * U 1(-

i 0 f! 0'b( ifj ‚ ,pbk, < ,-

h**K? X4Y; ; *2

y 0 f! 0'b( ifj H,pbk1, < ,-

kzNN fxg i i /

Page 31: Calculus

P`md`n diadido\n -/2

)(&- ER_VRV[SV[VaN`

@ m^oqfo ab rk^ pr`bpflk ab k•jbolp ob^ibp+ pb mrbab cloj^o rk^ ip`q\pr`bpfŽk prj^kal ilp q‹ojfklp pr`bpfs^jbkqb- @pŒ+pf i^ pr`bpfŽk a^a^ qfbkbilp q‹ojfklp9

NY$3' " II$ " 3; & ((( &

pb cloj^ i^ pr`bpfŽk ab i^p ~prj^p m^o`f^ibp‚9

RH< ^i&

v ^pŒpr`bpfs^jbkqb+ bpq^kal abcfkfa^ i^ prj^ m^o`f^i ab ilp i mofjbolp q‹ojfklp`ljl pfdrb9

'0/-03(h

n,9< ^i * \0 * --- * \h < \fŠ

F5G

K^ pr`bpfŽk vPiw ab i^p prj^p m^o`f^ibp pb ii^j^ n`md` diadido\ l pfjmibjbkqbn`md`*v pb fkaf`^ q^j_f‹k mlo ilp pŒj_lilp pfdrfbkqbp9

'0/-04( ^i * \0 * \1 * ---+ ^i * \0 * --- * \i * --- +

Olo bgbjmil+ i^ pbofb 9yhg-f obmobpbkq^i^ pr`bpfŽk xp-ym^o^ i^ `r^i9

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Page 32: Calculus

-0) Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

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Page 33: Calculus

Mmjkd`_\_ _` gdi`\gd_\_ _` g\n n`md`n ^jiq`mb`io`n 360

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Page 34: Calculus

361 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Di qblobj^ 0/-1 qfbkb rk `loli^ofl fkqbobp^kqbnrb pb rp^ `lk cob`rbk`f^m^o^ bpq^_ib`bo i^ afsbodbk`f^ ab rk^ pbofb-

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Page 35: Calculus

P`md`n o`g`n^‡kd^\n 362

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Page 36: Calculus

-0- Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

nrb pb mrbab l_qbkbo prmofjfbkal ilp m^o‹kqbpfpu pfjmifcf`^kal- Rf pb e^`b ^elo^ i^jfpj^ lmbo^`fŽk bk i^ pbofb fkcfkfq^9

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Page 37: Calculus

P`md` b`jh„omd^\ 364

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Page 38: Calculus

365 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

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Page 39: Calculus

Be`m^d^djn 366

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Page 40: Calculus

-01 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

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Page 41: Calculus

Be`m^d^djn ji `skm`ndji`n _`^dh\g`n -02

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Page 42: Calculus

-1) Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

alkab N ,o [+ ,o 8 m^o^ `^a^ f x i- Di k•jbol r bpqŠ obi^`flk^al `lk ilp aŒdfqlp [i$ [f$[0* %ŠŠ mlo jbafl ab i^p abpfdr^ia^abp9

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Page 43: Calculus

@mdo`mdjn_` ^jiq`mb`i^d\ 370

i x ^j , Dk ^idrklp `^plp m^oqf`ri^obp+ ljl mlo bgbjmil i^ pbofb dblj‹qof`^+i^p prj^p m^o`f^ibpn9 pb mrbabk pfjmifcf`^o e^pq^ bi mrkql ab mlabo abqbojfk^ocŠ`fijbkqb pr `ljmloq^jfbkql `r^kal i x ^`* Rfk bj_^odl+ bk i^ j^vloŒ^ abilp `^plp bpq^ cloj^ pfjmifcf`^a^ m^o^ n9 kl bufpqbv afcŒ`fijbkqb pb mrbab bpq^,_ib`bo i^ `lksbodbk`f^ l afsbodbk`f^ mlo bi j‹qlal fkaf`^al- X^ ^i mofk`fmfl+ilp nrb fksbpqfd^_^k bk bpqb`^jml+ jrv bpmb`f^ijbkqb B^r`ev v prp `lkqbjmloŠ,kblp+ pb afbolk `rbkq^ ab bpq^ afcf`riq^a+ v afbolk rklp ~`ofqboflp ab `lksbodbk,`f^‚+ `lk ilp nrb biraŒ^k i^ kb`bpfa^a ab rk `lkl`fjfbkql bumiŒ`fqlab i^p prj^pm^o`f^ibp-@idrklp ab bpqlp `ofqboflp+ilp jŠp pbk`fiilp v jŠp •qfibp pb bpqraf^oŠkbk bpqb^m^oq^al+mbol ^kqbp pb e^oŠk ^idrk^p l_pbos^`flkbp dbkbo^ibp ^`bo`^ abi^ k^qro^ibw^ ab bpqlp `ofqboflp-

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Page 44: Calculus

371 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Dpql bp rk bgbjmil ab rk `ofqbofl nrb bp abi qfml 'ff( v kl abi qfml 'f(- K^`lkaf`fŽk '0/-26( ij bp prcf`fbkqbm^o^i^ `lksbodbk`f^ ab rk^ pbofb-Olo bgbjmil+`r^kal \i < g-i i^ `lkaf`fŽk \9 ,* N pb p^qfpc^`bv pfk bj_^odl i^ pbofb Gg-iafsbodb- K^ sboa^abo^ rqfifa^a ab bpqb `ofqbofl bp nrb a^ rk^ `lkaf`fŽk npad+^d`io` m^o^ i^ _dq`mb`i^d\, Dp ab`fo+ pf bi q‹ojfkl \i ab i^ pbofbG-- ij qfbkab^ `bol+ bkqlk`bp i^ pbofb e^ ab pbo afsbodbkqb- Dpq^ molmlpf`fŽk bp iŽdf`^jbkqbbnrfs^ibkqb ^i qblobj^ 0/,5-

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Page 45: Calculus

@mdo`mdjn_` ^jhk\m\^d‡i k\m\ n`md`n _` o„mhdijn ij i`b\odqjn 372

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Bioji^`n -1 \9 ^jiq`mb` pf v n‡gj pf -1 ] 9 ^jiq`mb`,

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CDEHMHBHˆM- P` _d^` lp` _jn np^`ndji`n v\iw u v]iw _` iˆh`mjn ^jhkg`ejnnji \ndio‡od^\h`io` dbp\g`n pf

ifj Mh < 0-k,b `` \ 8

Page 46: Calculus

-1- Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Dpq^obi^`fŽk pb fkaf`^ pfj_Žif`^jbkqb bp`of_fbkal

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SDNQDL@ 0/-0/- Ajn n`md`nG\,9 u G]i _` o„mhdijn kjndodqjn u \ndio‡od^\+h`io` dbp\g`n j \h]\n ^jiq`mb`i j \h]\n _dq`mb`i,

DIDLOKN 1- K@ ETMBHˆM YDS@ CD QHDL@MM- Dk bi bgbjmil 0 abi ^m^oq^al0/-6 pb mol_Ž nrb G.,%h0 * h& bp rk^ pbofbqbibp`Žmf`^`lksbodbkqb- Tqfifw^kal‹pq^ `ljl pbofbab `ljm^o^`fŽk pb pfdrb nrb FF-i0 bp `lksbodbkqb+ mrbpql nrb., h0

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Page 47: Calculus

Bg ^mdo`mdjdio`bm\g 374

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Page 48: Calculus

-1/ Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Rf m 9= 0+ bi q‹ojfkl hE*O w N `r^kal h w // v mlo q^kql voiw `lksbodb- Dk sfoqraabi `ofqbofl fkqbdo^i+bpql fjmif`^ i^ `lksbodbk`f^ ab i^ pbofbm^o^ p = 0-

Rf n ; 0+bkqlk`bp oŠŠx // v i^ pbofbafsbodb- Di `^pl m^oqf`ri^o n < 0 '0^n`md`\mh‡id^\' pb bpqrafŽ v^ bk i^ pb``fŽk 0/-4- Rr afsbodbk`f^ bo^ v^ `lkl`fa^ab Kbf_kfw-

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Page 49: Calculus

@mdo`mdjn_` g\ m\…ut _`g ^j^d`io` k\m\ n`md`n_` o„mhdijn ij i`b\odqjn 265

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Page 50: Calculus

-11 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

i^ ^mif`^`fŽk afob`q^ abi `ofqbofl ab `ljm^o^`fŽk 'qblobj^ 0/-7( bumobp^nrb \,9 `lksbodb- K^p abpfdr^ia^abp bk '0/-31( plk bnrfs^ibkqbp ^

'0/-32(

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cloj^ moŠ`qf`^pfk e^`bo obcbobk`f^^i k•jbol s,

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Page 51: Calculus

@mdo`mdjn_` g\ m\…uv _`g ^j^d`io` k\m\ n`md`n_` o„mhdijn ij i`b\odqjn 267

Tk^ ^mif`^`fŽk ifdbo^jbkqb afpqfkq^ abi `ofqbofl ab `ljm^o^`fŽk `lkar`b ^i`ofqbofl abi `l`fbkqb-

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Efk^ijbkqb+ '`( pb morb_^ rqfifw^kal ilp jfpjlp bgbjmilp Prb bk bi qbl,obj^ 0/ 01-

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Page 52: Calculus

-2) Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

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Page 53: Calculus

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XFi_d^\^d‡i, @mif`^o bi Dgbo`f`fl 04 `lk ]i)g < i,Z

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\i)g = a&i',<0,,*,\• i i6

m^o^ i x J)

bk alkab E,%h&.,o L m^o^ qlal h) bkqlk`bp G[i `lksbodb pf = = 0 u afsbodb pf = ,o 0-

Page 54: Calculus

270 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

XFi_d^\^d‡i, Rf > :l l 0+ rqfifw^o bi Dgbo`f`fl 05- Rf > < 0+ ^mif`^o bi Dgbo`f`fl 04 `lk]i(. < h ild h+Y

07- @mif`^o bi `ofqbofl ab F^rpp 'abi Dgbo`f`fl 06( m^o^ abjlpqo^o nrb i^ pbofb

z '0 •2 - 4 %/h * g''fJ--- 1 -3 - 5 %/h&i;g

`lksbodb pf e = 1 v afsbodb pf e w 1- Dk bpqb bgbjmil bi `ofqbofl abi `l`fbkqb kl pfosb-

)(&)/ ER_VRNYaR_[NQN`

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'0/-34(//

1 &[/'i+/\i < \g + \0 * ^p , \2 * --- * &[/'i+/\i * !&+i;g

alkab `^a^ \9 = N bp mlpfqfs^-Dgbjmilp ab pbofbp^iqbok^a^p bo^k `lkl`falp ab ilp mofjbolp fksbpqfd^al,

obp- Rb e^ `fq^al v^ i^ pbofbild^oŒqjf`^

s0W

R s2 u!ild '0 * r& < r * &1* !2 , &3* --- * %Z.&h*. i * ----

Bljl pb abjlpqo^oŠ jŠp q^oab+bpq^ pbofb `lksbodb v pr prj^ bp ild 'i * r&m^o^ ,0 ; s x i- O^o^ s mlpfqfsl bp rk^ pbofb ^iqbok^a^- Dk m^oqf`ri^o+pfs < 0 pb l_qfbkb i^ cŽojri^9

'0/-35( 0 0 0 &[/'i+/ild 1 < 0 , , * , , , * --- * ,, * ---

1 2 3 i %

nrb af`b nrb i^ prj^ ab i^ pbofb^ojŽkf`^ ^iqbok^a^ bp i^d 1- Dpqbobpriq^al bp abbpmb`f^i fkqbo‹p qbkfbkal bk `rbkq^ nrb i^ pbofb ^ojŽkf`^ 1 ggi afsbodb-

Hkqfj^jbkqb obi^`flk^a^ `lk '0/-35( bp i^ fkqbobp^kqbcŽojri^

'0/-36( 4P 0 0 0 '\0(!,0,<0,,*,,,*+ --* ,,* ---3 2 4 6 0i + 0

Page 55: Calculus

P`md`n\go`mi\_\n 382

abp`r_fboq^ mlo I^jbp Fobdlov bk 0560- Kbf_kfwbk`lkqoŽ ab krbsl bpq^cŽojri^bk 0562 `^i`ri^kal bi Šob^ abi `Œo`ril rkfa^a-

@j_^p pbofbp'0/-35( v '0/-36( plk pbofbp iqbok^a^pab i^ cloj^ '0/-34( bk i^pnrb u[iw ab`ob`b jlkŽqlk^jbkqb e^`f^ `bol- Kbf_kfwl_pbosŽ+ bk 06/4+ nrb bpq^pfjmib molmfba^a ab \9 fjmif`^ i^ `lksbodbk`f^ ab oj_\ pbofb^iqbok^a^-

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'0/-37( k\m\ ^\_\ x 0 -

K^p abpfdr^ia^abp bk '0/-37( a^k rk^ j^kbo^ ab bpqfj^o bi boolo nrb pb `ljbqb^i ^molufj^o i^ prj^ R mlo rk^ prj^ m^o`f^i n+, K^ mofjbo^ abpfdr^ia^a bu,mobp^nrb bi boolo R , n9 qfbkb bi pfdkl ', 0(!+ nrb bp bi abi mofjbo q‹ojfklabpmob`f^al+ ', g'i\i * h&K^ pbdrka^ abpfdr^ia^a ^cfoj^ nrb bi s^ilo ^_plirql abbpqbboolo bp jbklo nrb bi abi mofjbo q‹ojfkl abpmob`f^al+

n,9 i m^o +Eh+i fjm^o

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A`hjnom\^d‡i, K^ fab^ ab i^ abjlpqo^`fŽk ab i^ obdi^ ab Kbf_kfwbp jrvpfjmib u bpqŠ obmobpbkq^a bk i^ cfdro^ 0/-4- K^p prj^p m^o`f^ibp P0i '`lk rkk•jbol m^oab q‹ojfklp( cloj^k rk^ pr`bpfŽk `ob`fbkqb mrbpql nrb P0i)0 + P0i :: \0i)F + \0i)0 = N-@kŠild^jbkqb i^p prj^p m^o`f^ibpP0i+/ cloj^k rk^ pr`bpfŽkab`ob`fbkqb- @j_^p pr`bpflkbp bpqŠk^`lq^a^p fkcboflojbkqb mlo Q1 X prmboflojbkqbmlo RH&Olo q^kql+`^a^ pr`bpfŽk vP0iw X vP0i+gw pfbkal jlkŽqlk^ v ^`lq^a^+ `lk,sbodb e^`f^ rk iŒjfqb+bp ab`fo P0i x R&X P0i+/ x O!+ Obol R&< R! mrbpql nrb9

R&, R! < ifj P0i + ifj P0i+/ < ifj &P0i + P0i+g' < ifj &+\0i' < N-i ,,,Šjj i ,,,Š//

Hkaf`^kal bpqb iŒjfqb `lj•k mlo R+ bp `i^ol nrb i^ pbofb `lksbodb v qfbkb mloiŒjfqb R-

O^o^ abar`fo i^p abpfdr^ia^abp bk '0/-37( pb o^wlk^ `ljl pfdrb9 Orbpql nrbP0i5%t P0i+g%<: bp9

u R z P0i)g ; P0i+/ m^o^qlal i ƒ 0 -

Page 56: Calculus

-2- Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

Olo q^kql+pb qfbkbk i^p abpfdr^ia^abp9

v

nrb `lkpfabo^a^p `lkgrkq^jbkqb `lkar`bk ^ '0/-37(- Blk il `r^i pb qfbkb i^abjlpqo^`fŽk `ljmibq^-

DIDLOKN 0- Orbpql nrb F-i!%, v g-i x N `r^kal i x //+ i^ `lksbodbk`f^ab i^ pbofb^ojŽkf`^ ^iqbok^a^ i , p* x, * --! bp rk^ `lkpb`rbk`f^ fkjb,af^q^ ab i^ obdi^ ab Kbf_kfw-K^ prj^ ab bpq^pbofbpb `^i`riŽ v^ bk bi bgbjmil 3-

DIDLOKN 1- K^ pbofb^iqbok^a^z %Z.&h 'ild h&,h `lksbodb- O^o^ abjlpqo^oil^mif`^kal i^ obdi^ ab Kbf_kfwpb e^ ab mol_^o nrb 'ild h&,h w N `r^kal h w //

u nrb 'ild i'gi!%,, Kl mofjbol pb abar`b ab i^ b`r^`fŽk 'i/-00( ab i^ Rb``fŽk0/-1- O^o^ mol_^o il pbdrkal pb l_pbos^ nrb i^ crk`fŽk ` m^o^ i^ `r^i

a&s' < ild s `r^kal s = Ns

qfbkb i^ abofs^a^ `$%r&< 'i , ild r&,+l+ Br^kal r = _ ‹pq^ bp kbd^qfs^ v ` bpjlkŽqlk^ ab`ob`fbkqb- Dk m^oqf`ri^o `%h* 0( ; `%h&m^o^ h ‚ 2-

DIDLOKN 2- Bljl `lkpb`rbk`f^ ab i^ obdi^ ab Kbf_kfwpb mrbab abar`fo q^j,_f‹k rk iŒjfqb fjmloq^kqb- Rb^9

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\\ < p+ [ < '^ ^r3 I1 T $

--- +

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(.

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K^ prj^ m^o`f^i %/h * 0( pb mrbab bumobp^oljl pfdrb9

.0 _s 0 a\ _s 0 di _s 0P0i+/ < 0 , , * , , , * --- * ,, , , * , <

0 T 1 1 T i + 0 h*. T i

0 0 Fi _s i 0< 0 * , * --- * , , , < 0 * , * --- * , , ild i ,0 i 0 T 1 i

Page 57: Calculus

P`md`n\go`mi\_\n 384

Orbpql nrb m/h*. ,* b `r^kal i ,* // pb l_qfbkb i^ pfdrfbkqb cŽojri^9

'0/-38( ifj '0 * 0* --- * y , ild i' < _-i+\j , 1

Di k•jbol b abcfkfal mlo bpqbiŒjfqb pb abkljfk^ ^jino\io` _` Bpg`m 'fkaf`^a^^idrk^p sb`bp mlo t', Hdr^i nrb 6R u `* bpqbk•jbol ^m^ob`bbk jr`e^p cŽojri^p^k^iŒqf`^p-Rr s^ilo `lk afbw `fco^p ab`fj^ibp bu^`q^p bp9 /+4661045538- Tk mol,_ibj^ fkqbobp^kqb+qla^sŒ^pfk obplisbo+bp ^sbofdr^o pf i^ `lkpq^kqb ab Dribo bp o^,`flk^i l foo^`flk^i-

K^ obi^`fŽk '0/-38( q^j_f‹k mrbab bumobp^opbljl pfdrb9

'0/-4/(i 0‚ f < ild i * b * -%.& `r^kal i x // -

f;g

Cb bpql pb abar`b nrb i^ o^wŽk'0 * e* --- * i.k(.ild h ,* 0 `r^kal h ,* ;‚H+

ab jlal nrb i^p prj^p m^o`f^ibpab i^ pbofb^ojŽkf`^ plk ^pfkqŽqf`^jbkqb fdr^ibp ^ild i, Dpql bp+qbkbjlp

T 0‚ e!$*$ild i `r^kal i x // -

Q_R

K^ obi^`fŽk '0/-4/( kl pŽil bumif`^ mlo nr‹ i^ pbofb ^ojŽkf`^ afsbodb+ pfkl nrbq^j_f‹k klp molmlo`flk^ rk^ fab^ `lk`obq^ abi `ob`fjfbkql ab prp prj^p m^o,`f^ibp- Dk bi moŽufjl bgbjmil rqfifw^jlp bp^ obi^`fŽk m^o^abjlpqo^o nrb i^ pbofb^ojŽkf`^ ^iqbok^a^ qfbkb prj^ fdr^i ^ ild 1-

DIDLOKN 3- Rb^ n,9 < 91:<<0 ', i(h,i . f, R^_bjlp nrb Ph qfbkab ^ rk iŒ,jfqb `r^kal g ,* //+ v s^jlp ^elo^ ^ abjlpqo^o nrb bpb iŒjfqbbp ild 1- Br^kalh bp m^o+pb^ h < 0i* mlabjlp pbm^o^oilp q‹ojfklp mlpfqfslp v kbd^qfslp l_qb,kfbkal

@mif`^kal '0/-4/( ^ `^a^ prj^ abi •iqfjl jfbj_ol ^ i^ abob`e^+ l_qbkbjlp

P0i < 'Hld 0i * b * -%.| * 'Hld i * b * -%.| < ild 1 * -%.&)

`lk 0/ nrb n0i ,* ild 1 `r^kal i ,* ^j * Dpql abjrbpqo^ nrb i^ &prj^ ab i^ pbofb^ojŽkf`^ ^iqbok^a^ bp ild 1-

Page 58: Calculus

385 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

)(&)0 8\[cR_TR[PVNP\[QVPV\[NYe NO`\YbaN

@ mbp^oab pbo i^ pbofb ^ojŽkf`^ ^iqbok^a^ I &+ 0(!,0 Fi `lksbodbkqb+ i^pbofbnrb pb l_qfbkb prpqfqrvbkal `^a^ q‹ojfkl mlo pr s^ilo ^_plirql bp afsbodbkqb-Dpql morb_^ nrb bk dbkbo^i i^ `lksbodbk`f^ ab H[/. kl fjmif`^ i^ `lksbodbk`f^H h]++.-Dk pbkqfal `lkqo^ofl pb qfbkb bi pfdrfbkqb qblobj^9

RCMPCK? 0/-04- PdHg\/.g jiq`mb`* o\h]d„i ^jiq`mb` H\9 t o`i`hjn

'0/-40(

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Orbpql nrb ]i bp N Ž 10]++0qbkbjlp N z \ 8 w 1G]!h+v mlo q^kql H h]+-Gal,jfk^ H],9 , Olo `lkpfdrfbkqb Hc/. `lksbodb v+`ljl v^ pb e^ af`el+ bpql fjmif`^ i^`lksbodbk`f^ ab H \,9 ,

Rrmlkd^jlp ^elo^ nrb ilp q‹ojfklp [8 plk `ljmibglp+ mlkd^jlp [y < Ri * djzpfbkal Qh v Qh ob^ibp-Orbpql nrb Eohf w g\ig*i^ `lksbodbk`f^ ab I g\ig fjmif`^ i^`lksbodbk`f^ ab Hypig* ^ pr sbw fjmif`^ i^ `lksbodbk`f^ ab HQh v^ nrb Qh bpob^i- @kŠild^jbkqb+ I p! `lksbodb- Dk sfoqra ab i^ ifkb^ifa^a+ i^ pbofbH&pi * yRi' `lksbodb-

O^o^ abjlpqo^o '0/-40(+ l_pbosbjlp nrb FIx;g \fg x Ix;g g\fg* u e^d^jlpirbdl nrb i x // ‘

CDEHMHBHˆM- Ri\ n`md` H\9 n` gg\h\ \]njgpo\h`io` ^jiq`mb`io` nd Hg\ig^jiq`mb`, Bn ^ji_d^dji\gh`io` ^jiq`mb`io` nd H\! ^jiq`mb` t `i ^\h]dj Hi^+!_dq`mb`,

RfH\! v H]9 plk ^_plirq^jbkqb `lksbodbkqbp+ 0/ jfpjl ib l`roob ^ i^ pbofbI %w[h* %F\i' `r^ibpnrfbo^ nrb pb^k z v %F+Dpql pb abar`b fkjbaf^q^jbkqb ab i^p

abpfdr^ia^abp= = = )) ))

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Page 59: Calculus

@mdo`mdjn_` ^jiq`mb`i^d\ _` Admd^cg`ot >]`g -20

^sbofdr^o i^ `lksbodbk`f^ \]njgpo\ ab rk^ pbofb ab q‹ojfklp `ljmibglp `r^ibp,nrfbo^- Dk bpq^Rb``fŽk bumlkbjlp alp `ofqboflp nrb pb rqfifw^k ^ jbkral m^o^abqbojfk^o i^ `lksbodbk`f^ bk bi `^pl bk nrb rk^ pbofb kl `lksbog^ ^_plirq^,jbkqb- @j_lp `ofqboflp e^`bk rpl ab rk^ fabkqfa^a ^idb_o^f`^ ii^j^a^ a‡mhpg\_` nph\^d‡i k\m^d\g _` >]`g* bk jbjlof^ abi j^qbjŠqf`l klorbdl Mfbip Gbkofh@_bi '07/1,0718(- Cf`e^ cŽojri^ bp m^ob`fa^ ^ i^ ab i^ fkqbdo^`fŽk mlo m^oqbpvmrbab bkrk`f^opb `ljl pfdrb-

SDNQDL@ 0/-05- EˆQLTK@ CD RTL@BHˆM O@QBH@K CD @ADK- P`\i v\iw tv ]iw _jn np^`ndji`n _` iˆh`mjn ^jhkg`ejn* t gg\h`hjn

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nrb klp a^ '0/-41(-

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A`hjnom\^d‡i, Tp^kal i^ klq^`fŽk abi qblobj^ 0/-05+ bufpqbrk L = N q^inrb F>i x L m^o^ qlal i, Olo `lkpfdrfbkqb >i]i)F w N `r^kal i x ^`9 O^o^bpq^_ib`bo i^ `lksbodbk`f^ ab 1- \i]i* qbkbjlp nrb abjlpqo^o q^k pŽil nrb i^pbofb 1- >f&]f + ]f)F& bp `lksbodbkqb- Orbpql nrb ]i%! * qbkbjlp i^ abpfdr^ia^a

Page 60: Calculus

-21 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

Obol i^ pbofb &]9+ \e(o' bp rk^ pbofbqbibp`Žmf`^ lksbodbkqb nrb aljfk^

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SDNQDL@ 0/-07- BQHSDQHN CD @ADK- P`\i \9 pi\ n`md` ^jiq`mb`io` _`o„mhdijn ^jhkg`ejn v v]iw pi\ np^`nd‡i hji‡oji\ ^jiq`mb`io` _` o„mhdijn m`\+g`n, Bioji^`n g\ n`md` \,]• ^jiq`mb`,

A`hjnom\^d‡i, Tqfifw^jlp lqo^ sbw i^ klq^`fŽk abi qblobj^ 0/-05- K^ `lk,sbodbk`f^ ab \9 fjmif`^ i^ ab i^ pr`bpfŽk u= ††v X mlo q^kql i^ ab i^ pr`bpfŽkv>i]i(o w, @pfjfpjl+ v>iw bp rk^ pr`bpfŽk ^`lq^a^- Di obpql ab i^ abjlpqo^`fŽkbp pbjbg^kqb ^ i^ abi `ofqbofl ab Cfof`eibq-

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i u! ] 0wTe:T*Z*+

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Page 61: Calculus

Be`m^d^djn 388

Olkfbkal r < _/c`c bk bpq^cŽojri^+ alkab cIbp ob^i mbol kl j•iqfmil bkqbol ab 06+

bk`lkqo^jlp

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Page 62: Calculus

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Page 63: Calculus

O`jm_`i\^d‡i _` n`md`n 4/0

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Page 64: Calculus

4/1 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

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Page 65: Calculus

O`jm_`i\^d‡i _` n`md`n 4/2

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Page 66: Calculus

4/3 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

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Page 67: Calculus

O`jm_`i\^d‡i _` n`md`n 4/4

_( Rf J \9 `n \]njgpo\h`io` ^jiq`mb`io`* g\n _jn n`md`nJ \x u I\9 ^jiq`m+b`i* u o`i`hjn

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Page 68: Calculus

4/5 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

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Page 69: Calculus

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04- G^ii^o qlalp ilp bkqbolp \ x 0 m^o^ ilp nrb i^ pbofb F88g%h &0,%[h& `lksbodb-05- Rb^k id ; i

0; k^ ; --- ^nrbiilp bkqbolp mlpfqfslp nrb kl `lkqfbkbk bi N bk pr obmob,

pbkq^`fŽk ab`fj^i- @pŒmlo bgbjmil+ id < 0+ i0 < 1+ --- + i7 < 8+ kilN < 00+ --- +i/6

< 08+ i/7

< 10+ bq`- Cbjlpqo^o nrb i^ pbofb ab ilp ob`Œmol`lpF98ygdif `lksbodb uqfbkb rk^ prj^ jbklo nrb 8/-

XFi_d^\^d‡i8 Djmib^o i^ pbofb 8 H9<l'8. 0/(j nrb aljfk^ i^ pbofb bk bpqrafl-\

06- Rf [ bp rk k•jbol ob^i ^o_fqo^ofl+ pb^ mh%[& < 0\ * u\ * ---* i\, Cbqbojfk^o bi pf,drfbkqb iŒjfqb9

, ni&\ * 0(di,,,, -

hwKK hOh%[&

'Blkpfabo^o ilp s^ilobp mlpfqfslp v kbd^qfslp ab \ ^pŒ`ljl \ < N-(07- ^( Rf j v k p5k bkqbolp cfglp+j w k w 0+ abjlpqo^o nrb

_( K^ pbofb nrb pfdrb bp rk^ obloabk^a^ ab i^ pbofb ^ojŽkf`^ ^iqbok^a^ bk i^ nrb ^m^ob,`bk ^iqbok^qfs^jbkqb qobp q‹ojfklp mlpfqfslp pbdrfalp ab alp q‹ojfklp kbd^qfslp9

0 *h*p,p, ,*p*, *he,e,p* * * , \ ----

Cbjlpqo^o nrb i^ pbofb `lksbodb u nrb pr prj^ bp ild 1 * pild h

XFi_d^\^d‡i8 Blkpfabo^o i^ prj^ m^o`f^i R4+ v rqfifw^o i^ m^oqb ^(-\

b( Qbloabk^o i^ pbofb ^ojŽkf`^ ^iqbok^a^+ bp`of_fbkal ^iqbok^qfs^jbkqb k q‹ojfklp mlpfqf,slp pbdrfalp ab l q‹ojfklp kbd^qfslp- @mif`^o bkqlk`bp i^ m^oqb ^( m^o^ abjlpqo^o nrbbpq^ pbofb obloabk^a^ `lksbodb v qfbkb `ljl prj^ ild 1 * pfia%j,k&+

Page 70: Calculus

3.6 Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

)(&*+ =[aRT_NYR`VZ]_\]VN`

Di `lk`bmql ab fkqbdo^iPxa&s' _s pb fkqolargl bk bi `^mŒqril0 `lk i^p obpqof`,`flkbp ab nrb i^ crk`fŽk a bpqrsfbo^ _`adid_\ v \^jo\_\ bk rk dio`mq\gj adidojW[)\Y+ Di l_gbql ab i^ qbloŒ ab i^ fkqbdo^`fŽk pb mrbab buqbkabo pf pb ^cilg^kbpq^pobpqof``flkbp-

O^o^ bjmbw^o+ pb mrbab bpqraf^o bi `ljmloq^jfbkql ab Pxa&s' _s `r^kal] x * _k+ Dpql `lkar`b ^ i^ kl`fŽk ab dio`bm\g diadido\ l q^j_f‹k ii^j^a^ dio`+bm\g dhkmjkd\ _` kmdh`m\%nk`^d` u nrb pb fkaf`^ mlo bi pŒj_lil P8 a&s' _s, Nqo^buqbkpfŽk pb l_qfbkb `r^kal pb qlj^ bi fkqbos^il W[) \Y cfkfql+mbol ` kl bpqŠ^`lq^a^ bk bi fkqbos^il- K^ krbs^ fkqbdo^i ^pŒl_qbkfa^ '`lk rk m^pl ^ iŒjfqb^ab`r^al( pb abkljfk^ dio`bm\g dhkmjkd\ _` n`bpi_\ `nk`^d`, O^o^ afpqfkdrfoi^p fkqbdo^ibpabi `^mŒqril0 ab i^p fkqbdo^ibpfjmolmf^p+^ i^p mofjbo^p pb ibp prbibii^j^o fkqbdo^ibp~molmf^p‚-

Lr`e^p crk`flkbp fjmloq^kqbp bk bi @kŠifpfppb mobpbkq^k`ljl fkqbdo^ibpfjmolmf^p ab rk^ r lqo^ `i^pb+v pb e^`b rk bpqrafl abq^ii^al ab bpq^pcrk`flkbpbk `roplp prmboflobpab BŠi`ril- @nrŒpb `lkpfabo^oŠk pŽil ilp ^pmb`qlpjŠp bib,jbkq^ibp ab i^ qbloŒ^+v pŽil pb a^oŠk ^idrk^p abcfkf`flkbp+qblobj^p v bgbjmilp-

Dp bsfabkqb nrb i^p abcfkf`flkbp `loobpmlkafbkqbp ^ i^p fkqbdo^ibpfjmolmf^pqfbkbkrk do^k m^ob`fal `lk i^p ab i^p pbofbpfkcfkfq^p-Olo q^kql+kl bp plomobkabkqbnrb jr`elp ab ilp qblobj^p bibjbkq^ibp pl_ob pbofbpqbkd^k pr o‹mif`^ bk i^ qbloŒ^ab i^p fkqbdo^ibpfjmolmf^p-

Rf i^ fkqbdo^imolmf^ Pxa&s' _s bufpqbm^o^ `^a^ ] 9888\* pb mrbab abcfkfo rk^krbs^ crk`fŽk E`ljl pfdrb9

X&]' < n^

a&s' _s m^o^ `^a^ ] 9888\ *&^

K^ crk`fŽk g ^pŒabcfkfa^ pb abkljfk^ dio`bm\g diadido\ l dio`bm\g dhkmjkd\ abmofjbo^ bpmb`fbu pb fkaf`^ mlo jbafl abi pŒj_lil P8 a&s' _s, K^ fkqbdo^ipb af`bnrb bp ^jiq`mb`io` pf bi iŒjfqb

'0/-50( ifj F&]' < ifj n_a&s' _s

]+))j88 _,*ll ‘--\

bufpqbv bp cfkfql- Dk `^pl `lkqo^ofl pb af`b nrb i^ fkqbdo^iP8a&s' _s bp _dq`m+b`io`, Rf bi iŒjfqbbk '0/-50( bufpqbu bp fdr^i ^ >* pb af`b nrb bi k•jbol > bp biq\gjm ab i^ fkqbdo^iv pb bp`of_b9

b!a&s' _s < > ,

Dpq^ abcfkf`fŽk bp ^kŠild^ ^ i^ a^a^ m^o^ i^p pbofbpfkcfkfq^p-Klp s^ilobpab i^ crk`fŽk F&]' grbd^k bi jfpjl m^mbinrb i^p ~prj^p m^o`f^ibp‚+mlo il nrb

Page 71: Calculus

Fio`bm\g`ndhkmjkd\n 4/8

pb i^p ii^j^oŠ ~fkqbdo^ibpm^o`f^ibp‚- N_p‹osbpb nrb bi pŒj_lil I9a&s' pb rp^q^kql m^o^ i^ fkqbdo^i `ljl m^o^ bi s^ilo ab i^ fkqbdo^i `r^kal ‹pq^ `lksbodb-'BljmŠobpb `lk i^ l_pbos^`fŽk eb`e^ e^`f^ bi cfk^i ab i^ Rb``fŽk 0/-4-(

DIDLOKN 0- K^ fkqbdo^ifjmolmf^ `wr*m ^r afsbodb pf n 490 X `lksbodb pfn = 0- O^o^ abjlpqo^oil+ l_p‹osbpb nrb9

%

\.*P + )

X&]' <E7u! _s < 0 , p

ild ]

pf n8yg8,0+

pf p < 0 -

Olo q^kql+Fd]' qfbkab ^ rk iŒjfqbcfkfql pf v pŽil pf p = )$ bk `rvl `^pl bi iŒjfqbbp9

hll 0s+n_s < ,, -

0 R , 0

Di `ljmloq^jfbkql ab bpq^fkqbdo^ibp ^kŠildl ^i ab i^ pbofb`loobpmlkafbkqb ^ i^

crk`fŽk wbq^+m_m&< G&9hi8z,

DIDLOKN 1- K^ fkqbdo^i`7pbk r ^r afsbodb mrbpql nrb9

F&]' < F7pbks _s < 0 , `lp ]*

u kl qfbkab ^ kfkd•k iŒjfqb`r^kal ] ,,* * ^j*

K^p fkqbdo^ibpfkcfkfq^pab i^ cloj^ czll `%r&^r pb abcfkbk ab j^kbo^ ^kŠil,d^- @abjŠp+ pf I+9-ll `%r& r v P8`%r& r plk \h]\n ^jiq`mb`io`n m^o^ rk `* pbaf`b nrb i^ fkqbdo^i Rzll `%r& r bp `lksbodbkqb v pr s^ilo pb abcfkb `ljl i^ prj^9

'0/-51( H7a&s' _s < ojja&s' _s * 0// a&s' _s ,

'Dp cŠ`fi abjlpqo^o nrb bp fkabmbkafbkqb ab i^ bib``fŽk ab ]+& K^ fkqbdo^iRzll `%r&^r pb af`b nrb afsbodb pf rk^ mlo il jbklp ab i^p fkqbdo^ibpabi pbdrkaljfbj_ol ab '0/-51( afsbodb-

Page 72: Calculus

3/. Pp^`ndji`n* n`md`n*dio`bm\g`n dhkmjkd\n

DIDLOKN 2- K^ fkqbdo^i axjj ` ,^iwi _s `lksbodb pf \ = N- Dk bcb`ql+ pf] = N+ pb qfbkb9

&] `+\/ug _s < &] `+\u _s < `+\]

+ 0z - Il Il +\ \

Olo q^kql+ `7n]0vh `lksbodb v pr s^ilo bp nYy+Olo lqo^ m^oqb+pf ] =/+ pbqfbkb9

`r^kal ] x // ‘

Olo q^kql+axjj `+\/ug _s q^j_f‹k `lksbodb v pr s^ilo bp g-\, Dp ab`fo+ pb qfbkbbll o]0vh _s < 0-\, N_p‹osbpb+ pfk bj_^odl+ nrb i^ fkqbdo^i axjj `+[t _s _d+q`mb` mrbpql nrb axjj `+\u _s afsbodb-

Bljl bk bi `^pl ab i^p pbofbp+pb qfbkbk q^j_f‹k bk bi `^pl ab i^p fkqbdo^ibpfjmolmf^p afsboplp `ofqboflpab `lksbodbk`f^- Di jŠp pbk`fiil ab biilp pb obcfbobfkqbdo^kalp mlpfqfslp-

SDNQDL@ 0/-12- Pd g\ dio`bm\g kmjkd\ a8a&s' _s `sdno` k\m\ ^\_\ ] x \ t nda&s' x N k\m\ oj_j s x N+ `ioji^`n a8 a&s' _s ^jiq`mb` pf- t n‡gj pf `sdno` pi\^jino\io` L = N o\g lp`8

ba&s' _s x J k\m\ ^\_\ ] x \ ,

Dpqbqblobj^ bp i^ _^pb abi pfdrfbkqb `ofqbofl ab `ljm^o^`fŽk-

SDNQDL@ 0/-13- Pd g\ dio`bm\g kmjkd\ ax a&s' _s `sdno` k\m\ ^\_\ ] x \ t ndN xa&s' x b&s' k\m\ oj_j s x \* _ji_` a8 b&s' _s ^jiq`mb`* `ioji^`n a8 a&s' _so\h]d„i ^jiq`mb` t

KNN a&s' _s x Kll b&s' _s ,

Jin[+ Rb af`b nrb i^ fkqbdo^i n8a%r&r ^igch[ i^ fkqbdo^iP8 %r& r+

SDNQDL@ 0/-14- BQHSDQHN CD BNLO@Q@BHˆM CDK KŒLHSD- Pd g\n _jn dio`+bm\g`n kmjkd\n axa&s' _s t ax b&s' _s `sdno`i k\m\ ^\_\ ] x \* nd`i_j a&s' x N Vb&s' = M k\m\ oj_j s ƒ \* t pf

Hfj a&s' < `*b,b,q,ll a%r&

`ioji^`n g\n dio`bm\g`n o8a&s'_s*t*a8 b&s' _sj ^jiq`mb`i \h]\n j _dq`mb`i \h]\n,

'0/-52( _ji_` ^8y„ M +

Page 73: Calculus

Fio`bm\g`n dhkmjkd\n 400

Kjo\, Rf bi iŒjfqb bk '0/-52( bp /+ pŽil pb mrbab `lk`irfo nrb i^ `lksbodbk`f^ ab`7a%r& r fjmif`^ i^ `lksbodbk`f^ ab P8x%r& r+

K^p abjlpqo^`flkbp ab ilp qblobj^p abi 0/-12 ^i 0/-14 plk ^kŠild^p ^ ilp abilp `loobpmlkafbkqbp ab pbofbp v pb abg^k `ljl bgbo`f`flp-

DIDLOKN 3- O^o^ `^a^ ob^i n*i^ fkqbdo^i b:+z:s% _s `lksbodb+ `ljl obpriq^ab i^ `ljm^o^`fŽk `lk `8)s+/ _s mrbpql nrb `+!%s%-s+/,* N `r^kal s ,* * \n*

K^p fkqbdo^ibp fjmolmf^p ab pbdrka^ bpmb`fb pb mrbabk fkqolar`fo `ljlpfdrb9 Rb prmlkb nrb ` bpqŠ abcfkfa^ bk bi fkqbos^il pbjf ^_fboql %[)\Y+ v nrb i^fkqbdo^i by%n&n bufpqb m^o^ `^a^ s nrb p^qfpc^`b \ ; s 4: \+ Rb abcfkb bkqlk`bprk^ krbs^ crk`fŽk Eab i^ j^kbo^ pfdrfbkqb9

F&s' < E7a&o' _o pf z: s m8|)

K^ crk`fŽk F ^pŒabcfkfa^ pb abkljfk^ dio`bm\g dhkmjkd\ _` n`bpi_\ `nk`^d` vpb fkaf`^ mlo bi pŒj_lil ax)a&o' _o, Rb af`b nrb i^ fkqbdo^i ^jiq`mb` pf bi iŒjfqb

'0/-53( ifj•I'u( < ifj G] a&o' _os+)\) s+)\) s

bufpqb u bp cfkfql- Rf bpql kl l`roob pb af`b nrb i^ fkqbdo^i ax)a&o' _o _dq`mb`,Rf bi iŒjfqb bk '0/-53( bufpqb v bp fdr^i^ >* bi k•jbol > pb af`b nrb bp bi q\gjmab i^ fkqbdo^i u pb bp`of_b9

G] a&o' _o << > ,\)

DIDLOKN 4- Rb^ a&o' < o! pf o = N-Rf ] = N v s = N pb qfbkb9

v

]g+% [ sg+P

F%r&<E7o+P_o < 0 , p

ild \ * i^d u

pf R9E, 0 +

pf p < 0 -

Br^kal s x N *+ F&s' qfbkab ^ rk iŒjfqb cfkfql pf u pŽil pf p ; 0- Olo q^kql+ i^fkqbdo^i ab) o+P _o `lksbodb pf n ; 0 u afsbodb pf p 1 0-

Dpqbbgbjmil pb mlaoŒ^qo^q^oab lqo^ cloj^- Rf pb e^`b i^ prpqfqr`fŽk o < g-p*_o < +p+/o) pb l_qfbkb9

`] cigWo! _o < pn

+/ _p ,

$! † fd\

Page 74: Calculus

401 Oo]_mcih_m)m_lc_m)chn_al[f_mcgjlijc[m

Br^kal s ,* /*+ g-s ,* *// v mlo q^kql ab) o+P_o < a%&a]QP+/ _p* pfbjmob nrb i^•iqfj^ fkqbdo^i `lksbog^- Dk sfoqra abi bgbjmil 0+ ‹pq^ `lksbodb pf v pŽil pfn + 1 ; , 0+bp ab`fo+ p ; 0-

Di bgbjmil ^kqboflo firpqo^ rk eb`el dblj‹qof`l klq^_ib- Blkpfa‹obpb i^crk`fŽk ` abcfkfa^ mlo `%r&<U+1-2 m^o^ N ; r w 0- K^ fkqbdo^i F* `%r& r `lk,sbodb+mbol i^ fkqbdo^i F* /Qa0&s' _s afsbodb- Fblj‹qof`^jbkqb+ bpql pfdkfcf`^ nrbbi `lkgrkql abi mi^kl ifjfq^al mlo bi bgb r) t < `%r&)r < N X r < 0 qfbkb Šob^cfkfq^+mbol nrb bi pŽifal l_qbkfal mlo pr olq^`fŽk ^iobabalo abi bgb s qfbkbslirjbk fkcfkfql-

K^p fkqbdo^ibpfjmolmf^p ab i^ cloj^ `w*a&o' _o pb abcfkbk ab j^kbo^ ^kŠ,ild^- Rf i^p alp fkqbdo^ibp ax)a&o' _o X `7*a&g' _o plk [g\[m ]ihp_la_hn_m) pbbp`of_b

G9a&o' _o < o)a&o' _o * oa&o'_o,

Kjo\, @idrklp ^rqlobp bp`of_bk P8 bk sbw ab Px)}

K^ abcfkf`fŽk pb mrbab buqbkabo 'ab j^kbo^ l_sf^( ^ rk k•jbol cfkfql `r^i,nrfbo^ ab prj^kalp- Olo bgbjmil+ pf ` kl bpqŠ abcfkfa^ bk alp mrkqlp ` ; ^fkqboflobp^ rk fkqbos^il W[)\Y pb af`b nrb i^ fkqbdo^i fjmolmf^ axa&o' _o `lk,sbodb u qfbkb bi s^ilo Px+a&o'_o * axxa&g' _o * i)a&g' _d* pfbjmob nrb `^a^rk^ ab bpq^pfkqbdo^ibp`lksbog^- @•k jŠp+ pb mrbabk `lkpfabo^o `lj_fk^`flkbp~jfuq^p‚ q^ibp `ljl ax)a&o' _o * a8 a&o' _o nrb pb bp`of_b nu,i~ _o l `lj_fk^,`flkbp ~jfuq^p‚ ab i^ cloj^ Pxxa&o' _o * ‚)a&g' _o * ax a&g' _o nrb pb bp`of_bpfjmibjbkqb `7a&o' _o,

DIDLOKN 5- H[ `oh]cƒh a[gg[+ Rf p = N+i^ fkqbdo^i`}))( `m%o%8!^n `nksbo,db- Dpq^fkqbdo^ipb e^ ab fkqbomobq^oljl rk^ prj^+ ab i^ cloj^9

'0/-54(

K^ pbdrka^ fkqbdo^i `lksbodb m^o^ qlal ob^i 4+ bk sfoqra abi bgbjmil 3- O^o^bpqraf^o i^ mofjbo^ fkqbdo^ipb mlkb o < g-p v pb l_pbos^ nrb9

Obol Iz mg-pp+P+g _p `lksbodb m^o^n =N mlo `ljm^o^`fŽk `lk `wp+n+/_pj Oloq^kql+ i^ fkqbdo^i j* `+ooP

+/ _o `lksbodb m^o^ 4 = N- Br^kal n = N i^ prkf^bk '0/-54( pb fkaf`^ mlo obp(-K^ crk`fŽk i&pŒabcfkfa^ pb ii^j^ `oh]cƒh a[gg[)

Page 75: Calculus

Be`m^d^djn 402

fkqolar`fa^ mlo mofjbo^ sbw mlo Dribo bk 0618- Sfbkb i^ molmfba^a fj,mloq^kqb nrb n'j * 0( < h `r^kal h bp rk bkqbol `r^inrfbo^ ƒN- 'O^o^ i^abjlpqo^`fŽk+ s‹^pb Dgbo`f`fl 08 ab i^ Rb``fŽk 0/-13-(

Klp `ofqboflp ab `lksbodbk`f^ a^alp bk ilp qblobj^p 0/-12 ^i 0/-14 qfbkbk^kŠildlp m^o^ i^p fkqbdo^ibpfjmolmf^p ab pbdrka^ bpmb`fb-Di ib`qlo kl qbkaoŠafcf`riq^a bk clojri^o ‹i jfpjl q^ibp `ofqboflp-

)(&*, :WR_PVPV\`

Dk `^a^ rkl ab ilp Dgbo`f`flp abi 0 ^i 0/ bpqraf^o i^ `lksbodbk`f^ ab i^ fkqbdo^i fj,molmf^ `loobpmlkafbkqb+

c7: s0- - . _s*

l su3*0 `./-ar

5- - m _s,.) SU

5, a/+ ild s _s,/* f*r

7- c`l ZTZ_s,]`l `lpe s

6+ a/+ Z^r +

.) SU ildu

0/- 0// &g_U['n,1 r iar

b`l 1

1- ,// _*T ^r+

b`l 0

2- - . _s,l&su2*0

b`l 0

3- - ., ^r+l q`T

bll `+S,*

4- - m _s,.) SU

00- O^o^ rk `fboql s^ilo ob^i a+ i^ fkqbdo^i

hNN' ?r 0(1 s0 * 0 , 0s * 0 _s

`lksbodb- Cbqbojfk^o a u `^i`ri^o i^ fkqbdo^i-01- O^o^ rk `fboql s^ilo ob^i a+i^ fkqbdo^i

`lksbodb- Cbqbojfk^o a u `^i`ri^o i^ fkqbdo^i-

02- O^o^ rk `fboql s^ilo ob^i b i^ fkqbdo^i

h`l'0 B (+++++++ _sl UH * 0s/ s * 0

`lksbodb- Cbqbojfk^o a u `^i`ri^o i^ fkqbdo^i-

Page 76: Calculus

403 Pp^`ndji`n* n`md`n*dio`bm\g`ndhkmjkd\n

03- G^ii^o 0/p s^ilobp ab \ v \ q^ibp nrb

0..&0U0 * ]s * \ [' [/ s&0s * [& 0 _s + 0 -

04- ƒO^o^ nr‹ s^ilobp ab i^p `lkpq^kqbp \ u^ bufpqb u bp fdr^i ^ 0 bi iŒjfqb pfdrfbkqb>

, FGgWi * \s/ * ]sej 1 ^r )

Ii,*ll ,Ii U * s * 0

05- ^( Cbjlpqo^o nrb

&d+c_s '0 _U'ifj , * ,

c+L) ,0 U Š c U5) v nrb ifj gcpbk s _s < N -

e,*ll +c

_( Cb`fo pf `lksbodbk l afsbodbk i^p pfdrfbkqbp fkqbdo^ibp fjmolmf^p-

06- ^( Cbjlpqo^o nrb i^ fkqbdo^i G* 'pbk rfEr ^r `lksbodb-

_( Ool_^o nrb ifj!+]l* ud {'`lp o'-o0 _o < 0-

b( Cb`fafo pf i^ fkqbdo^i F* '`lp o'-o0 _o `lksbodb l afsbodb-

07- ^( Rf ` bp jlkŽqlk^ ab`ob`fbkqb m^o^ qlal r87770 X pf `%r&w N `r^kal r w * //+

abjlpqo^o nrb i^ fkqbdo^i ay%! `%r&^r v i^ pbofb `%h&plk ^j_^p `lksbodbkqbp l ^j_^pafsbodbkqbp-

WEh^c][]cƒh7 Qb`r‹oabpb i^ abjlpqo^`fŽk abi `ofqbofl ab i^ Œkqbdo^i-i

_( C^o rk bgbjmil ab rk^ crk`fŽk ` kl jlkŽqlk^ m^o^ i^ `r^i i^ pbofb `%h&`lksbog^v i^ fkqbdo^i Ic&%r&^r afsbog^-

08- Rb^ obp(< Px oP+/`+o _o m^o^ p = N 'crk`fŽk d^jj^(- @mif`^o i^ fkqbdo^`fŽk mlo m^oqbpm^o^ mol_^o nqb obp* 0( < pc&'p(+Cbpmr‹p abjlpqo^o mlo fkar``fŽk nrb H&'k* 0( < h m^o^ i bkqbol mlpfqfsl-

B^a^ rkl ab ilp Dgbo`f`flp 1/ ^i 14 `lkqfbkb rk^ molmlpf`fŽk+ kl kb`bp^of^jbkqb `fboq^+pl_ob rk^ crk`fŽk ` abcfkfa^ &O^o qlal s 98880- Dk `^a^ rkl ab biilp+ i obmobpbkq^rk bkqbolmlpfqfsl+ v Fi i^ fkqbdo^i Io x%r&^r nrb pb prmlkb nrb bufpqb pfbjmob- Cb `^a^ rk^ a^o i^abjlpqo^`fŽk l rk `lkqo^bgbjmil-

1/- Rf ` bp jlkŽqlk^ ab`ob`fbkqb v pf ifj i+`+j8 Fi bufpqb+ bkqlk`bp i^ fkqbdo^i Ic y&s' _s`lksbodb-

10- Rf fcg!)Zii`%r& < N X ifjk]olik < =) bkqlk`bp Pa `%r&^r `lksbodb v pr s^ilo bp =+11- Rf i^ pr`bpfŽk xHky`lksbodb+ i^ fkqbdo^i Ou! `%r&^r `lksbodb-12- Rf ` bp mlpfqfs^ v pf fcgh*(i]fh < =) bkqlk`bp Pm`%r&^r `lksbodb v s^ib =+

Page 77: Calculus

Be`m^d^djn -)-

13- Rrmrbpql nrb d$%r&bufpqb m^o^ `^a^ r 19 0 X nrb bufpqb rk^ `lkpq^kqb L = N q^i nrb0q.'u(0 99: L m^o^ qlal r 19 0: pf gdhi[<!g* < = i^ fkqbdo^i Pm`%r&^r `lksbodb v pr s^ilobp >,

14- Rf O`$`%r&^r `lksbodb+ bkqlk`bp gdh!*[!a&U' < N-

Page 78: Calculus
Page 79: Calculus

))

EG8:E=BA:E K E:D=:E 9: ;GA8=BA:E

00-0 Blksbodbk`f^ mrkqr^i ab pr`bpflkbp ab crk`flkbp

Dk bi `^mŒqril 0/ ebjlp `lkpfabo^al pr`bpflkbp `rvlp q‹ojfklp bo^k k•jbolpob^ibp l `ljmibglp- @elo^ nrbobjlp `lkpfabo^o pr`bpflkbp vaiw `rvlp q‹ojfklppb^k api^dji`n ob^ibp l `ljmibg^p nrb qbkd^k rk aljfkfl `lj•k bk i^ ob`q^ ob^il bk bi mi^kl `ljmibgl- O^o^ `^a^ s abi aljfkfl+ mlabjlp `lkpqorfo lqo^ pr`b,pfŽk u`h%r&v ab k•jbolp `rvlp q‹ojfklp plk ilp `loobpmlkafbkqbp s^ilobp ab i^pcrk`flkbp- Cbpfdkbjlp `lk R bi `lkgrkql ab mrkqlp s m^o^ ilp nrb bpq^ pr`bpfŽk`lksbodb- K^ crk`fŽk ` abcfkfa^ bk R mlo i^ fdr^ia^a

a&s' < ifj ai&s' pf s A R+

pb ii^j^ i^ api^d‡i g…hdo`ab i^ pr`bpfŽk vaiw* v ab`fjlp nrb i^ pr`bpfŽk vaiw^jiq`mb` kpiop\gh`io` e^`f^ a bk bi `lkgrkql R-

Di bpqrafl ab q^ibp pr`bpflkbp bpqŠ bk mofk`fmfl obi^`flk^al `lk bi qfml abmobdrkq^ pfdrfbkqb9 Rf `^a^ q‹ojfkl ab rk^ pr`bpfŽk vaiw qfbkb rk^ `fboq^ molmfb,a^a+ `ljl+ mlo bgbjmil+ i^ `lkqfkrfa^a+ i^ abofs^_fifa^a l i^ fkqbdo^_fifa^a+ ƒbknr‹ `lkaf`flkbp pb `lkpbos^ bp^ molmfba^a bk i^ crk`fŽk iŒjfqb> Olo bgbjmil+ pf`^a^ crk`fŽk `h bp `lkqfkr^ bk rk mrkql s* ƒil bp q^j_f‹k i^ crk`fŽk iŒjfqb `<Di bgbjmil nrb pfdrb abjrbpqo^ nrb+ bk dbkbo^i+ kl il bp-

DIDLOKN 0- Pp^`nd‡i _` api^dji`n ^jiodip\n ^ji api^d‡i g…hdo _dn^jiodip\,Rb^ `h%r&< u! pf N z r w 0- Dk i^ cfdro^ 00-0 pb e^k obmobpbkq^al ^idrklp q‹o,jfklp- K^ pr`bpfŽk vaiw `lksbodb mrkqr^ijbkqb bk bi fkqbos^il `boo^al ZN+0\+ vpr crk`fŽk iŒjfqb ` sfbkb a^a^ mlo i^ cŽojri^

a&s' < ifj u! < xN!zNN 0

pf N z s ; 0+

pf s < 0 -

&406

Page 80: Calculus

407 Oo]_mcih_m t m_lc_m^_ `oh]cih_m

N_p‹osbpb nrb i^ crk`fŽk iŒjfqb` bp afp`lkqfkr^ bk 0+pf _fbk `^a^ q‹ojfkl ab i^pr`bpfŽk bp `lkqfkr^ bk qlal bi fkqbos^il ZN+0\-

DIDLOKN 1- Oo]_mcƒh j[l[ f[ ko_ ifj `7 Eh%r&r :.: `\ ifj Eh%r& r+ Rb^h*(?`F [$h ++++†ii

`h%r&< hr%f * r0&h m^o^N z r w 0- Dk bpqbbgbjmil+ i^ pr`bpfŽk u`hv `lksbodbmrkqr^ijbkqb e^`f^ rk^ crk`fŽk iŒjfqbonrb bp `bol bk qlal mrkql abi fkqbos^il`boo^al ZN+0\- Dk i^ cfdro^ 00-1 pb e^k obmobpbkq^alilp mofjbolp q‹ojfklp abi^ pr`bpfŽk- K^ fkqbdo^iab Fi bk bi fkqbos^il ZN+0\ sfbkb a^a^ mlo

d/ dg i '0 , s0'i)gY/ i` &s'_s < i s&g+ s0o_s < , ,,,,, < ,,,-l i l 1 h * 0 l /%h * 0(

Olo `lkpfdrfbkqb qbkbjlp ifj D`h%r& ^r < 0+ mbol d ifj `h%r&^r < N- Cf`elj\kk j\nk

ab lqol jlal+ bi iŒjfqbab i^p fkqbdo^ibpkl bp fdr^i ^ i^ fkqbdo^iabi iŒjfqb-Dpqb

'0+ 0(

ss

EHFTQ@ 00-0 Pp^`nd‡i _` api^dji`n ^jiodip\n^ji Xpi^dƒi g…hdo_dn^jiodip\,

EHFTQ@ 00-1 Pp^`nd‡i _` api^dji`n k\m\g\ lp` d*x N `i il+ 0\ k`mjEdi ,* 0 ^p\i_j i x ll-

bgbjmil morb_^ nrb i^p alp lmbo^`flkbp ab ~m^pl ^i iŒjfqb‚ b ~fkqbdo^`fŽk‚ klpfbjmob plk fkqbo`^j_f^_ibp- 'Ubo q^j_f‹k ilp bgbo`f`flp 06 u 07 ab i^ pb``fŽk00-6(

Fblodb F- Rqlhbp '0708,08/2(+ Oefiifm K- s- Rbfabi '0710,0785(+ v J^oiVbfbopqo^ppcrbolk ilp mofjbolp bk `ljmol_^o nrb pb kb`bpfq^_^ ^idrk^ `lkaf`fŽk

Page 81: Calculus

?ihp_la_h]c[ ohc`ilg_ ^_ mo]_mcih_m_ `oh]cih_m 408

^af`flk^i m^o^ grpqfcf`^o bi fkqbo`^j_fl ab bp^p lmbo^`flkbp- Dk 0737+ Rqlhbp uRbfabi 'fkabmbkafbkqbjbkqb v `^pf ^i jfpjl qfbjml( fkqolargbolk rk `lk`bmql^elo^ ii^j^al ]ihp_la_h]c[ ohc`ilg_ v abjlpqo^olk nrb m^o^ rk^ pr`bpfŽk rkfclo,jbjbkqb `lksbodbkqb i^p lmbo^`flkbp ab m^pl ^i iŒjfqb b fkqbdo^`fŽk mrbabk fkqbo,`^j_f^opb- LŠp q^oab Vbfbopqo^pp abjlpqoŽ nrb bi `lk`bmql bp ab do^k fjmloq^k,`f^ bk @kŠifpfp prmboflo- Dk i^ Rb``fŽk moŽufj^ fkqolar`fjlp bi `lk`bmql v abjlp,qo^jlp pr obi^`fŽk `lk i^ `lkqfkrfa^a u i^ fkqbdo^`fŽk-

))&* Blksbodbk`f^ rkfclojb QR`bPR`V\[R`QRSb[PV\[R`

Rb^ vaiw rk^ pr`bpfŽk nrb `lksbodb mrkqr^ijbkqb bk rk `lkgrkql R e^`f^ rk^crk`fŽk iŒjfqb `+ Rbd•k i^ abcfkf`fŽk ab iŒjfqb+bpl pfdkfcf`^ nrb m^o^ `^a^ s ab Rv m^o^ `^a^ ’ = N bufpqb rk bkqbol J) nrb abmbkab ab s v ab b+ q^i nrbE`h%r&* x%r&.; ’ `lk q^i nrb h w J+ Rf bi jfpjl J pfosb m^o^ ni^im ilp mrkqlp rab R+ bkqlk`bp i^ `lksbodbk`f^ pb ii^j^ ohc`ilg_ bk R- Dpql bp+ qbkbjlp i^ pf,drfbkqb

CDEHMHBHˆM- Qh[ mo]_mcƒh _ `oh]cih_m vaiw m_ff[g[ ohc`ilg_g_hn_ ]ih*p_la_hn_ b[]c[ ` _h oh ]ihdohni Q pf j[l[ ni^i C = M _rcmn_oh J %^_j_h^c_hn_n[hmƒfi ^_ D( n[f ko_ h ƒ J cgjfc][

E`h%r&* `%r&.; C j[l[ ni^i r ^_ R-

Arjl_m[gim mcg\ƒfc][g_hn_ _mi _m]lc\c_h^i

d9x ` ohc`ilg_g_hn_ _h R-

EHFTQ@ 00-2 Pdbidad^\_j b`jh„omd^j _` g\ ^jiq`mb`i^d\ pidajmh`, Rf i 19 K* oj_\ g\ bmƒad^\_` ^\_\ ~| `noƒ ndop\_\ \ _dno\i^d\ h`ijm lp` C _` g\ bmƒad^\ _` g\ api^d‡i gdhdo` `+

Br^kal i^p crk`flkbp ai plk ab s^ilobp ob^ibp+ bufpqb rk^ fkqbomobq^`fŽk dbl,j‹qof`^ pbk`fii^ ab i^ `lksbodbk`f^ rkfclojb- K^ abpfdr^ia^a `h%r&* `%r& ; C bpbnrfs^ibkqb ^i m^o ab abpfdr^ia^abp

a&s' + ’ :ai&s' :a&s' * ’-

Page 82: Calculus

41/ Pp^`ndji`n v n`md`n_` api^dji`n

Rf ‹pq^pplk `fboq^pm^o^qlal i $8<Ju qlal s ab R+bkqlk`bp qla^ i^ doŠcf`^ab dz`loobpmlkafbkqb ^ R bpqŠbk rk^ _^ka^ ab ^iqro^ 1iN pfj‹qof`^jbkqb pfqr^a^ obp,mb`ql ab i^ doŠcf`^ab d,`ljl pb fkaf`^ bk i^ cfdro^ 00-2-

))&+ 8\[cR_TR[PVNb[VS\_ZRe P\[aV[bVQNQ

Cbjlpqo^jlp ^ `lkqfkr^`fŽk nrb i^ `lksbodbk`f^ rkfclojb qo^kpjfqb i^ `lk,qfkrfa^a ab ilp q‹ojfklp ab i^ pr`bpfŽk vaiw ^ i^ crk`fŽk iŒjfqb`+

RCMPCK? 00-0- Pd cy w a pidajmh`h`io` `i pi dio`mq\gj R u ^\_\ api^d‡iai `n ^jiodip\ `i ^\_\ kpioj k _` R+g\ api^d‡i g…hdoa o\h]d„i `n ^jiodip\ `i k,

@_gimcl[]c•h+ Ool_^objlp nrb m^o^ qlal C = N bufpqb rk bkqlokl J%j&q^i nrb Fa&s' + a&k' G; C pfbjmob nrb s C K&k' h R- Rf C = N bpqŠa^al+ bufpqbrk bkqbol J q^i nrb i x K fjmif`^

Fai&s' + a&s'/ ; y m^o^qlal s ab R-2

Orbpql nrb aK bp `lkqfkr^ bk j) bufpqbrk bkqlokl J%j& q^i nrb

Gyr't( , aK&M' G; z m^o^qlal r ab K&k' i R -

Olo il q^kql+m^o^qlal s ab J%j& h R+ qbkbjlp

Fa&s' + a&k' G< Fa&s' + a**&s' * a**&s' + a**&k' * aq&k' + a&k' G99::

8899Fa&s' + aq&s' G* Gbr't( , aK&M'G* Faq&k' + a&k' G-

Orbpql nrb `^a^ q‹ojfkl abi pbdrkal jfbj_ol bp ; iN.2+ bk`lkqo^jlpFa&s' + a&k' G; D+ il `r^i `ljmibq^ i^ abjlpqo^`fŽk-

Di qblobj^ ^kqboflo qfbkbrk^ ^mif`^`fŽk fjmloq^kqb ^ i^p pbofbpab crk`flkbp-Rf ilp s^ilobp ab i^p crk`flkbp ai&s' plk prj^p m^o`f^ibpab lqo^p crk`flkbp+ mlobgbjmil+

i

ai&s' < I pf&s' *Q_R

v pf gz x ` mrkqr^ijbkqb bk R+qbkbjlp bkqlk`bp

;HM

a&s' < ifj ai&s' <Hpgs'h*K?& ,f;g

Page 83: Calculus

@jiq`mb`i^d\ pidajmh` ` dio`bm\^d‡i 410

m^o^ `^a^ s ab R- Dk bpqb `^pl+ pb af`b nrb i^ pbofb EfEf lksbodb mrkqr^ijbkqbe^`f^ i^ crk`fŽk prj^ `+ Rf d8x ` rkfclojbjbkqb bk R+ab`fjlp nrb i^ pbofb Gg-f

`lksbodb rkfclojbjbkqb e^`f^ `+ Rf `^a^ q‹ojfkl p*bp rk^ crk`fŽk `lkqfkr^ bk rkmrkql k ab R+`^a^ prj^ m^o`f^i ai q^j_f‹k bp `lkqfkr^ bk k `lk il nrb+ bk sfoqraabi qblobj^ 00-0+ l_qbkbjlp bi pfdrfbkqb `loli^ofl-

RCMPCK? 00-1- Rf pi\ n`md` _` api^dji`n GEEe ^jiq`mb` piddjmh„h`io`c\^d\ g\ api^d‡i nph\ a `i pi ^jiepioj R+u pf ^\_\ o„mhdij p* `n ^jiodipj `i pikpioj k _` R+ g\ nph\Z o\h]d„i `n ^jiodip\ `i k,

Kjo\8 S^j_f‹k mlabjlp bumobp^o pfj_Žif`^jbkqb bpqb obpriq^al bp`of_fbkal

JA AC

ifj GEEe%T&<Gifj .Ex)+%r&+T**$;*dF 9.: f;g T*!$F&

Dumobp^jlp bpql af`fbkal nrb m^o^ rk^ pbofb rkfclojbjbkqb `lksbodbkqb mlabjlp fkqbo,`^j_f^o bi pŒj_lil ab m^pl ^i iŒjfqb `lk bi ab prj^`fŽk+ l nrb mlabjlp m^p^o ^iiŒjfqb q‹ojfkl ^ q‹ojfkl-

))&, 8\[cR_TR[PVNb[VS\_ZRR V[aRT_NPVp[

Di pfdrfbkqb qblobj^ abjrbpqo^ nrb i^ `lksbodbk`f^ rkfclojb klp mbojfqbfkqbo`^j_f^o bi pŒj_lil ab fkqbdo^`fŽk `lk bi ab m^pl ^i iŒjfqb-

RCMPCK? 00-2- Ppkjib\hjn lp` Xz x a pidajmh`h`io` `i pi dio`mq\gjX\* ]Z* v lp` ^\_\ api^d‡i ai `n ^jiodip\ `i X\* ]Z, A`adi\hjn pi\ ip`q\ np^`nd‡ivbiw h`_d\io`

pf r D W[) \Y )

u kjib\hjn

b&s' < oa&o'_o ,

Bioji^`n bi x d pidajmh`h`io` `i X\* ]Z, Pdh]‡gd^\h`io`* o`i`hjn

ifj nai&o'_o < I!&ifj ai&o' _o ,k,b `l \ \ ^ ,p `l

A`hjnom\^d‡i, K^ abjlpqo^`fŽk bp jrv pbk`fii^- C^al C = N+bufpqb rk bk,qbol J q^i nrb i x J fjmif`^

ai&o' + a&o' ; \C\

I'Hm^o^ qlal o ab W[) \Y +

Page 84: Calculus

.++ Pp^`ndji`n v n`md`n _` api^dji`n

Krbdl+ pf r D W[)\Y u pfk z J) qbkbjlp

Ea††%r&* a%r&G< G&URŠŠ&o'+ a&o~_o G888899G]/aŠŠ&o'+ a&o'/_o ; &][‹[ _o < b +, G\ \ G\ ] + [

`lk 0/ nrb d-- z d rkfclojbjbkqb bk W•)\Y+

Nqo^ sbw+ ljl `loli^ofl+ qbkbjlp rk obpriq^al ^kŠildl m^o^i^p pbofbp:

RCMPCK? 00-3- Ppkjib\hjn lp` pi\ n`md` _` api^dji`n HQf ^jiq`mb` pid+ajmh`h`io` c\^d\ g\ api^d‡i nph\ a `i pi dio`mq\gj X\* \Y) nd`i_j ^\_\ Rf ^ji+odip\ `i X\* \Y+ Pd s C X\* \Y) _`adidhjn

v b&s' <FUa&o'_o ,

Bioji^`n d-- z d pidajmh`h`io` `i X\* \Y+ Ad^cj _` jomj hj_j* o`i`hjn

zzfGFUpf&o'_o <FU xx HGpdo' _o

j

A`hjnom\^d‡i, @mifnrbjlp bi qblobj^ 00-2 ^ i^ pr`bpfŽk ab prj^p m^o`f^ibpvFiw a^a^ mlo

,+ j

.††&o'< I pf&o'*f;/

v l_pbosbjlp nrb Pxa,,&o'_o < If;/ Px pf&o'_o,

Blk cob`rbk`f^ bi qblobj^ 00-3 pb bumobp af`fbkal nrb rk^ pbofbrkfclojb,jbkqb `lksbodbkqb mrbab fkqbdo^opbq‹ojfkl ^ q‹ojfkl-

00+-4 G[N `lkaf`fŽk prcf`fbkqb ]N_N i^ `lksbodbk`f^ rkfclojb

Vbfbopqo^ppfkaf`Ž rk `ofqbofl m^o^ mol_^o nrb `fboq^p pbofbpplk rkfclojb,jbkqb `lksbodbkqbp- Di `ofqbofl bp ^mif`^_ib pfbjmob nrb i^ pbofba^a^ mrba^ pboaljfk^a^ mlo rk^ pbofbkrj‹of`^ ab q‹ojfklp mlpfqfslp-

Page 85: Calculus

Ri\ ^ji_d^d‡i npad^d`io` k\m\ g\ ^jiq`mb`i^d\ pidajmh` 301

SDNQDL@ 00-4- BQHSDQHN L CD VDHDQRSQ@RR- A\_\ pi\ n`md` _` api^dji`nJ Qh lp` ^jiq`mb` kpiop\gh`io` c\^d\ pi\ api^d‡i a `i pi ^jiepioj R- Pd `sdno`pi\ n`md` iph„md^\ ^jiq`mb`io` _` o„mhdijn kjndodqjn J J i o\g lp`

l 49 gpi&s'd 49J9 k\m\ oj_j i ƒ 0 V oj_j s _` R+

`ioji^`n g\ n`md` K Qh ^jiq`mb` pidajmh`h`io` `i R-

A`hjnom\^d‡i, Di `ofqbofl ab `ljm^o^`fŽk morb_^ nrb i^ pbofb I pi&s' `lk,sbodb ^_plirq^jbkqb m^o^ `^a^ s ab R- O^o^ `^a^ s ab R+qbkbjlp

Orbpql nrb i^ pbofb HJ-^ `lksbodb+ m^o^ `^a^ ’ = N bufpqbrk bkqbol K q^i nrbi ƒ K fjmif`^

NB

J J-^ ; p -fxi)g

Dpql morb_^ nrb

m^o^qlal i ƒ J Xqlal s ab R- Olo il q^kql+i^ pbofbK Qh `lksbodb rkfclojbjbkqbe^`f^ ` bk R-

K^ abofs^`fŽk q‹ojfkl ^ q‹ojfkl ab rk^ pbofbcrk`flk^i `r^inrfbo^ bp ^prkqljŠp abif`^al+ bk `r^kql ^ i^ `lkpbos^`fŽk ab molmfba^abp+nrb i^ fkqbdo^`fŽkq‹o,jfkl ^ q‹ojfkl- Olo bgbjmil+ i^ pbofb Hwf'pbk is'-i0 `lksbodb m^o^qlal s^iloab s v^ nrb bp aljfk^a^ mlo K /-i0

Š @abjŠp+ i^ `lksbodbk`f^ bp rkfclojb bkqlal bi bgb ob^i- Ml l_pq^kqb+i^ pbofbl_qbkfa^ abofs^kal q‹ojfkl ^ q‹ojfkl bpH '`lp isoa i* v ‹pq^ _dq`mb` `r^kal s < N- Dpqbbgbjmil abjrbpqo^ nrb i^ abof,s^`fŽk q‹ojfkl ^ q‹ojfkl mrbab abpqorfo i^ `lksbodbk`f^+ ^rk `r^kal i^ pbofblofdfk^i pb^ rkfclojbjbkqb `lksbodbkqb- Olo `lkpfdrfbkqb+bi mol_ibj^ ab grpqf,cf`^o bi fkqbo`^j_fl ab i^p lmbo^`flkbp ab abofs^`fŽk v prj^`fŽk bp+bk dbkbo^i+jŠp pbofl nrb bk bi `^pl ab i^ fkqbdo^`fŽk-Blk bpqbbgbjmil bi ib`qlo mrbab `lj,mol_^o nrb i^p j^kfmri^`flkbp `loofbkqbp `lk prj^p cfkfq^pkl pfbjmob mrbabkbcb`qr^opb `lk pbofbp+fk`irpl bk bi `^pl bk nrb i^p pbofbpab nrb pb qo^qbpb^krkfclojbjbkqb `lksbodbkqbp- @ `lkqfkr^`fŽk kl obcbofjlp ^ rk^p pbofbpab crk,`flkbp ab qfml bpmb`f^i+ii^j^a^p pbofbpab mlqbk`f^p+nrb mrbabk j^kbg^opb bkjr`e^p l`^pflkbp `ljl pf crbo^k prj^p cfkfq^p-

Page 86: Calculus

413 Pp^`ndji`n t n`md`n_` api^dji`n

00-5 Rbofbpab mlqbk`f^p- BŒo`ril ab `lksbodbk`f^

Tk^ pbofbab i^ cloj^

\i&u + \o < ^l * \y&u+ \' * --- * \i&u + ^(! * ---q0-<l

pb ii^j^ pbofbab mlqbk`f^p ab u + \, Klp k•jbolp u* \* v ilp `lbcf`fbkqbp \9 plk`ljmibglp- Blk `^a^ pbofbab mlqbk`f^p bpqŠ^pl`f^al rk `Œo`ril+ ii^j^al ^…m^pgj_` ^jiq`mb`i^d\* q^i nrb i^ pbofb`lksbodb ^_plirq^jbkqb m^o^ qlal u fkqboflo ^ijfpjl+ v afsbodb m^o^ qlal u buqboflo-Di `bkqol abi `Œo`ril bp \ v pr o^afl m pb

QbdfŽk abafsbodbk`f^

EHFTQ@ 00-3 @…m^pgj_` ^jiq`mb`i^d\ _` pi\ n`md`_` kjo`i^d\n,

ii^j^ m\_dj _` ^jiq`mb`i^d\, 'Ubo cfdro^ 00-3-( Dk `^plp buqobjlp+ bi `Œo`rilmrbab obar`fopb ^ Gz plil mrkql \* bk `rvl `^pl m< N+N mrbab `lkpfpqfo bk qlalbi mi^kl `ljmibgl+ bk `rvl `^pl ab`fjlp nrb l < * ]_) K^ bufpqbk`f^abi `Œo`rilab `lksbodbk`f^ pb abjrbpqo^ bk bi qblobj^ 00-6-

Di `ljmloq^jfbkql ab i^ pbofbbk ilp mrkqlp colkqbo^ abi `Œo`ril kl mrbabmobab`fopb-Blk bgbjmilp pb sb nrb mrbab e^_bo `lksbodbk`f^ bk kfkdrkl+ bk^idrkl+ l bk qlalp ilp mrkqlp colkqbo^-

O^o^ do^k m^oqbab i^p pbofbpab mlqbk`f^p nrb bk i^ moŠ`qf`^pb mobpbkq^k+bio^afl ab `lksbodbk`f^ mrbab abqbojfk^opb jbaf^kqb bi `ofqbofl abi `l`fbkqb l biab i^ o^Œw+ljl bk ilp bgbjmilp nrb pfdrbk-

DIDLOKN 0- O^o^ e^ii^o bi o^afl ab `lksbodbk`f^ ab i^ pbofb ab mlqbk`f^pGui-i * ^mif`^jlp bi `ofqbofl abi `l`fbkqb- Rf u ;/; N+i^ o^wŽkab q‹ojfklp `lkpb,

`rqfslp qfbkb `ljl s^ilo ^_plirql

Gui)g h G Yv[

%h* 0( ui < h * 0 -

Page 87: Calculus

P`md`n _` kjo`i^d\n, @…m^pgj_` ^jiq`mb`i^d\ 414

Orbpql nrb bpqb `l`fbkqb qfbkab e^`f^ N `r^kal i x //+ iibd^jlp ^ i^ `lk`irpfŽkab nrb i^ pbofb `lksbodb ^_plirq^jbkqb m^o^ qlal `ljmibgl w ;/; N- S^j_f‹k `lk,sbodb m^o^ w < N+`lk il nrb bi o^afl ab `lksbodbk`f^ bp * ^`*

Orbpql nrb bi q‹ojfkl dbkbo^i ab rk^ pbofb `lksbodbkqb ab_b qbkabo ^ N+bi ob,priq^al abi bgbjmil ^kqboflo morb_^ nrb

]Thei,</[%\\ ;

m^o^ qlal w `ljmibgl- Dpql bp+h ~`ob`b jŠp oŠmfa^jbkqb‚ nrb i^ mlqbk`f^ k,pfj^ab `r^inrfbo k•jbol `ljmibgl w cfgl `r^kal i x ^j*

DIDLOKN 1- O^o^ ^sbofdr^o i^ `lksbodbk`f^ ab i^ pbofb i/1iui* rqfifw^jlpbi `ofqbofl ab i^ o^Œw-Sbkbjlp

&i/1i .wik(i.k < 2 Zw\i/-i ,,* 2 Zw\ r^kal i ,,* // +

v^ nrb i0-i < &i!-i'0 v i% ! x 0 `r^kal i x ll- Olo `lkpfdrfbkqb+ i^ pbofb `lk,sbodb ^_plirq^jbkqb pf Zw\; eu afsbod` pf Zw\= - Di o^afl ab `lksbodbk`f^ bp p-Dpq^ pbofb ab mlqbk`f^p afsbod` bk qlal mrkql colkqbo^ ab_fal ^ nrb+ pf Zw\< e+bi q‹ojfkl dbkbo^i qfbkb s^ilo ^_plirql i!,

DIDLOKN 2- O^o^ `^a^ rk^ ab i^p pbofbp uiai v uiai/) bi `ofqbofl abi`l`fbkqb klp af`b nrb bi o^afl ab `lksbodbk`f^ bp 0- K^ mofjbo^ afs`odb bk bimrkql colkqbo^ u < 0 mbol `lksbodb bk qlalp ilp abjŠp mrkqlp colkqbo^ 'sboRb``fŽk 0/-08(- K^ pbdrka^ pbofb `lksbodb bk qlal mrkql colkqbo^ mrbpql nrb bpaljfk^a^ mlo I 0. i!,

Sbojfk^jlp bpq^Rb``fŽk abjlpqo^kal nrb qla^ pbofb ab mlqbk`f^p mlp`b `Œo`r,il ab `lksbodbk`f^- K^ abjlpqo^`fŽk pb ^mlv^ `k bi qblobj^ pfdrfbkqb-

SDNQDL@ 00-5- Pd g\ n`md` _` kjo`i^d\n H\iui ^jiq`mb` `i pi kpioj w ;/; L*kjm `e`hkgj k\m\ w < YH&n` od`i`8

]( I\ n`md` ^jiq`mb` \]njgpo\h`io` k\m\ oj_j X nd`i_j Gvh; GXGG-_( I\ n`md` ^jiq`mb` pidajmh`h`io` `i oj_j _dn^j ^dm^pg\m_` ^`iomj `i N

u m\_dj O ; GX00-

A`hjnom\^d‡i, Orbpql nrb \iux `lksbodb+ pr q‹ojfkl dbkbo^i qfbkab e^`f^N `r^kal i x ^j* Dk m^oqf`ri^o+ g\iux/ ; i^ m^oqfoab rk `fboql i 1 K, Rb^ R rk`Œo`ril ab o^afl O* pfbkal N ; O ; HYHH-Pd Y D R X i 1 K* qbkbjlp Zw\9999::O v

alkab n < Gy h•

Page 88: Calculus

415 Pp^`ndji`n t n`md`n_` api^dji`n

Orbpql nrb N ; o ; 0+i^ pbofbG\iui bp aljfk^a^ mlo i^ pbofbdblj‹qof`^ `lk,sbodbkqbGo!, Dk sfoqra abi `ofqbofl L ab Vbfbopqo^pp+i^ pbofbG\iui `lksbodbrkfclojbjbkqb bk R- Dpql morb_^ ^(+ Di o^wlk^jfbkql morb_^ q^j_f‹k nrb i^pbofb G\iui `lksbodb ^_plirq^jbkqb m^o^ `^a^ w ab R- Obol v^ nrb `^a^ u q^inrb EtW ; HYii bpqŠbk rk `fboql `Œo`ril R ab o^afl O ; HYii+bpql morb_^ q^j_f‹ki^ m^oqb (-

SDNQDL@ 00-6- DWHRSDMBH@CD TM B†QBTKN CD BNMUDQFDMBH@- Pd g\ n`md`_` kjo`i^d\n G\iui ^jiq`mb` kjm gj h`ijn k\m\ pi w ;/; N+ kjm `e`hkgj k\m\Y < Yi+ X _dq`mb`kjm gj h`ijn k\m\ pi u*kjm `e`hkgj k\m\ Y < W0* `sdno` pi iˆ+h`mj m`\gkjndodqj mo\g lp` g\ n`md` jiq`mb` \]njgpo\h`io` nd Gvh; mt _dq`mb`ndGvh= m,

A`hjnom\^d‡i, Cbpfdkbjlp `lk > bi `lkgrkql ab qlalp ilp k•jbolp mlpfqfslpGv.m^o^ilp nrb i^ pbofbab mlqbk`f^p G\iui `lksbodb- Di `lkgrkql > kl bp s^`Œlv^ nrb+ mlo efmŽqbpfp+lkqfbkb GXhe-@pfjfpjl+ kfkd•k k•jbol ab = mrbab pboj^,vlo nrb 0zH'ab_fal ^i qblobj^ 00-5(- Krbdl+ GX1ebp rk^ `lq^ prmbofloab =+ Orbpqlnrb > bp rk `lkgrkql kl s^`Œl ab k•jbolp mlpfqfslp ^`lq^al prmboflojbkqb+ qfbkbbuqobjl prmboflo nrb abpfdk^jlp `lk m, Dp bsfabkqb nrb m< N v^ nrb m< GXhh-Dk sfoqra abi qblobj^ 00-5 kfkd•k k•jbol ab > mrbab prmbo^o^ m, Olo `lkpf,drfbkqb+i^ pbofbafsbodb pf Zw\= l+ Obol bp cŠ`fi abjlpqo^o nrb i^ pbofb`lksbodb\]njgpo\h`io` pf Gvh; m, Rf Gvh; m*bufpqbrk k•jbol mlpfqfsl ubk > q^i nrbGvh; s ; m,Rbd•k bi qblobj^ 00-5+i^ pbofbG\iui `lksbodb ^_plirq^jbkqb- Dpql`ljmibq^ i^ abjlpqo^`fŽk-

Bljl bp k^qro^i+bufpqbrk qblobj^ ^kŠildl m^o^pbofbpab mlqbk`f^p ab w , \nrb mrbab abar`fopb abi `^pl nrb ^`^_^jlp ab qo^q^o+fkqolar`fbkal bi `^j_fl abs^of^_ib Y < w , \, Di `Œo`ril ab `lksbodbk`f^ qfbkb pr `bkqol bk \* `ljl pb sbbk i^ cfdro^ 00-3-

00-6 Dgbo`f`flp

Dk ilp Dgbo`f`flp abi 0 ^i 05+abqbojfk^o bi o^afl ab `lksbodbk`f^ l ab i^p pbofbpab ml,qbk`f^p nrb pb a^k- Dk ilp Dgbo`f`flp abi 0 ^i 0/+ ^sbofdr^o i^ `lksbodbk`f^ bk ilp mrkqlpcolkqbo^ pf m bp cfkfql-

0-c87+T'U

// !

1- 19%h90(1&!!,N l

// %t* 2(!

2- 19%h* 0(1! -i;j

y ',0(!11!w1!3- `8 /hz

!5*

Page 89: Calculus

Bd`m^d^djn .+0

//

4- ‚ Z0 , &[0'iZui,

k<i//

5- z i ui,H iih:f

ƒNN'i& 2 - 4 --- %/h * 0((2 i00- 1 3 u ,

- •5••• %/h&.f:.

// ' Gn01- ƒ 0 * xGi w!+

T5R

w %ZE&h%t* i(k

5, I i/ * Gi;j

//

7- ƒ \h/uh) >4H4+(

//

02- ‚ 'pbk `kgw!+ \< N-

i;j//

03- ƒ'pbke \itu!* \< N-

i;j//ƒ w!

04- \i * n88 \< N+ ] = N-.f:.

\< N+] = N-

06- Rf Fi %r&< hr_*is0 m^o^ h < 0+ 1+ --- + v r ob^i+ abjlpqo^o nrb

ifj &Fai&s'_s :‹ 'Gifj ai&s' _s,k]NN >\ >\ hZK?8

Dpqb bgbjmil abjrbpqo^ nrb i^p lmbo^`flkbp ab fkqbdo^`fŽk v ab m^pl ^i iŒjfqb kl pfbj,mob mrbabk fkqbo`^j_f^opb-

07- Rb^ Eh%r&<'pbk isqoi* v m^o^ `^a^ r ob^i cfgl pb^ E%r&< ifjkzll Eh%r&+Cbjlpqo^o nrb

ifj ax&L' :‹ Z&'N(-hwKK

Dpqb bgbjmil morb_^ nrb i^p lmbo^`flkbp ab abofs^`fŽk v ab m^pl ^i iŒjfqb kl pfbjmobmrbabk fkqbo`^j_f^opb-

08- Cbjlpqo^o nrb i^ pbofb H9&<i'pbk is'-i0 `lksbodb m^o^ qlal ob^i s* v abpfdkbjlp prprj^ `lk E%r&+Cbjlpqo^o nrb E bp `lkqfkr^ bk ZN+&60!\+ X rqfifw^o bi qblobj^ 00-3 m^o^abjlpqo^o nrb

oa&s' _s < 1c&0i x 0(2 -J I5G

1/- Rb p^_b nrb

pf N99::s 8899/.P †

Page 90: Calculus

417 Pp^`ndji`n u n`md`n _` api^dji`n

Tqfifw^o bpq^ cŽojri^ v bi qblobj^ 00-3 m^o^ abar`fo i^p pfdrfbkqbp cŽojri^p

// &[/'i)/ /Q1

'_( H%/h * 0(2 < 21 -Hizi

))&0 C_\]VRQNQRQRYN Sb[PV\[R` _R]_R`R[aNQN`]\_ `R_VR_RNYRQR]\aR[PVN`

Dk bpq Rb``fŽk klp ifjfq^jlp ^ pbofbpm`\g`n _` kjo`i^d\n* bpql bp ^ pbofbpab i^cloj^ \i&u + \'i bk i^p nrb w+\ v ilp `lbcf`fbkqbp \9 plk qlalp k•jbolp ob^ibp-Dp`of_fobjlp s bk ird^o ab u* Di `Œo`ril ab `lksbodbk`f^ ifjfq^ bk bi bgbob^i rkfkqbos^il &\ + o+\ * o( pfj‹qof`l obpmb`ql^i mrkql \9 ^ q^i fkqbos^il 0/ ii^j^jlpdio`mq\gj _` ^jiq`mb`i^d\ ab i^ pbofbob^i ab mlqbk`f^p \i&s + \'i, Di k•jbol opb ii^j^ o^afl ab `lksbodbk`f^- 'U‹^pb cfdro^ 00-4(-

Cfsbodbk`f^Blksbodbk`f^ ^_plirq^

Cfsbodbk`f^

H'W \

EHFTQ@ 00-4 Fio`mq\gj _` ^jiq`mb`i^d\ _` pi\ n`md`m`\g _` kjo`i^d\n,

B^a^ pbofbob^i ab mlqbk`f^p abcfkb rk^ crk`fŽk prj^ `rvl s^ilo bk `^a^ sabi fkqbos^il ab `lksbodbk`f^ sfbkb a^al mlo

//

y&s' < H\i@s + \'! ,hwK

Rb af`b nrb i^ pbofbm`km`n`io\ g\ api^d‡i a bk bi fkqbos^il ab `lksbodbk`f^+v pb i^ abkljfk^ bi _`n\mmjggj `i n`md` ab a pbd•k i^p mlqbk`f^p ab \,

Dufpqbkalp mol_ibj^p _Špf`lp ^`bo`^ ab ilp abp^ooliilp bk pbofbab mlqbk`f^pnrb ^nrŒklp fkqbobp^k9

0( C^a^ i^ pbofb+e^ii^o molmfba^abpab i^ crk`fŽk prj^ `+1( C^a^ rk^ crk`fŽk o,sbo pf mrbab pbol kl obmobpbkq^amlo rk^ pbofbab

mlqbk`f^p- Qbpriq^ nrb pŽil ^idrk^p crk`flkbp bpmb`f^ibpmlpbbk abp^ooliil bk pbofbab mlqbk`f^p- Rfk bj_^odl Ni^ i^pb ab q^ibpcrk`flkbp `lkqfbkb i^ j^vlo m^oqbabilp bgbjmilp nrb pb mobpbkq^kbk i^ moŠ`qf`^+v mlo q^kql pr bpqrafl bp ab do^kfjmloq^k`f^- Ulis^jlp ^elo^ ^ i^ afp`rpfŽk ab i^ `rbpqfŽk 0(-

Di qblobj^ 00-5 klp af`b nrb i^ pbofbab mlqbk`f^p `lksbodb ^_plirq^jbkqbm^o^ `^a^ r abi fkqbos^il ^_fboql %[ * o+[ * l&) v nrb `lksbodb rkfclojbjbkqb

Page 91: Calculus

Mmjkd`_\_`n _` g\n api^dji`n m`km`n`io\_\n kjm n`md`n m`\g`n _` kjo`i^d\n 307

bk qlal pr_fkqbos^il `boo^al W[ * N) [ * NY) alkab N ; N ; l+ Orbpql nrb `^a^q‹ojfkl ab i^ pbofb ab mlqbk`f^p bp rk^ crk`fŽk `lkqfkr^ bk qlal bi bgb ob^i+obpriq^abi qblobj^ 00-1 nrb i^ crk`fŽk prj^ ` bp `lkqfkr^ bk qlal pr_fkqbos^il `boo^alX\ + O* \ * OZ* u mlo q^kql bk bi fkqbos^il ^_fboql &\ + m*\ * O', @pfjfpjl+bi qblobj^ 00-3 klp af`b nrb mlabjlp fkqbdo^o i^ pbofb ab mlqbk`f^p q‹ojfkl ^ q‹o,jfkl bk qlal pr_fkqbos^il `boo^al W[ * N) [ * NY+ Dpq^p molmfba^abp ab i^p crk,`flkbp obmobpbkq^a^p mlo pbofbp ab mlqbk`f^p nrba^k `lk`obq^a^p bk bi qblobj^

pfdrfbkqb-

RCMPCK? 00-7- Rf pi\ api^d‡i a `noƒ m`km`n`io\_\ kjm g\ n`md` _` kjo`i^d\n

'00-0(NB

a&s' < G\**&s+ \'ih:K

`i pi dio`mq\gj \]d`moj &\ + m*\ * m'* `n ^jiodip\ `i `n` dio`mq\gj* t np dio`bm\g`i ^p\glpd`m np] dio`mq\gj ^`mm\_j kp`_` ^\g^pg\mn` dio`bm\i_j g\ n`md` o„mhdij \o„mhdij, Bi k\mod^pg\m*k\m\ oj_j s _` &\ + m*\ * m'* o`i`hjn

Esa&o'_o < y \ia%U&o+ \'i _o < y y 'u , \'i)g\ I* \ I*i)/

I5: IR:

Di qblobj^ 00-7 q^j_f‹k abjrbpqo^ nrb bi o^afl ab `lksbodbk`f^ ab i^ pbofbfkqbdo^a^ bp mlo il jbklp fdr^i ^i ab i^ pbofb lofdfk^i- Cbjlpqo^objlp ^elo^ nrb^j_^p pbofbp qfbkbk bu^`q^jbkqb bi jfpjl o^afl ab `lksbodbk`f^- Cbjlpqobjlpmofjbol nrb rk^ pbofb ab mlqbk`f^p mrbab abofs^opb q‹ojfkl ^ q‹ojfkl bk bi fkqbofloab pr fkqbos^il ab `lksbodbk`f^-

RCMPCK? 00-8- P`\ a g\ api^d‡i m`km`n`io\_\ kjm g\ n`md` '00-0( `i `g dio`m+q\gj _` ^jiq`mb`i^d\ &\ + m*\ * m', Bioji^`n o`i`hjn8

^( I\ n`md` _`mdq\_\ HHz<0i\i&s + \'i+g od`i` o\h]d„i m\_dj _` ^jiq`m+b`i^d\ m,

_( I\ _`mdq\_\ o%&s'sdno` k\m\ ^\_\ s _`g dio`mq\gj _` ^jiq`mb`i^d\ t qd`i``skm`n\_\ kjm

a%&s'<Gi\i&s + ^o,hŠ

h:f

A`hjnom\^d‡i, O^o^ pfjmifcf`^o+ prmlkd^jlp bk i^ abjlpqo^`fŽk \ < N-Cbjlpqobjlp mofjbol nrb i^ pbofb abofs^a^ `lksbodb ^_plirq^jbkqb bk bi fkqbos^il

Page 92: Calculus

42/ Pp^`ndji`n t n`md`n_` api^dji`n

%*l) l&+Difg^jlp `r^inrfbo r mlpfqfsl q^i nrb N ; r ; l) v pb^ c rk k•jbol ml,pfqfsl q^i nrb N ; s ; s * b ; m, Dkqlk`bp i^p pbofbp_` a&s' v ab a&s* b& plk^_plirq^jbkqb `lksbodbkqbp- Krbdl+ mlabjlp bp`of_fo

'00-1( y&s * c' + y&s' [ x &s * c'i [ sic + J-+[i c

h:K

K^ pbofbabi pbdrkal jfbj_ol bp ^_plirq^jbkqb `lksbodbkqb v^ nrb bp rk^ `lj,_fk^`fŽk ifkb^i ab pbofbp`lksbodbkqbp- @ `lkqfkr^`fŽk ^mifnrbjlp bi qblobj^ abis^ilo jbafl m^o^bp`of_fo

&s * c'i + u! < ci^x+/ *

alkab s ; `8 ; s * c, Krbdl+ i^ pbofb '00-1( bp fa‹kqf`^ ^ &i pbofb

'00-2(//

w$h[ ]i+.x i ii;/

nrb ab_b pbo ^_plirq^jbkqb `lksbodbkqb+ mrbpql nrb i^ ab i^ fdr^ia^a 00-1 il bp-K^ pbofb'00-2( kl bp-v^ rk^ pbofbab mlqbk`f^p+mbol aljfk^ i^ pbofbab mlqbk`f^p1 i\,s!8%* `lk il nrb bpq^•iqfj^ pbofbab_b pbo^_plirq^jbkqb `lksbodbkqb m^o^bpb s^ilo ab s, Dpql abjrbpqo^ nrb bi o^afl ab `lksbodbk`f^ ab i^ pbofbabofs^a^1 i\*,s!+g bp mlo il jbklp fdr^i ^ m,Olo lqo^ m^oqb+bi o^afl ab `lksbodbk`f^ abi^ pbofbabofs^a^ kl mrbab bu`babo ^ m mlonrb bpq^pbofbabofs^a^ aljfk^ i^ lof,dfk^i 1 \is!, Dpql morb_^ i^ m^oqb (-

O^o^ abjlpqo^o i^ m^oqb_(+ pb^ d i^ crk`fŽk prj^ ab i^ pbofbabofs^a^:

//

b&s' < 1 i\hs!+/i;/

@mif`^kal bi qblobj^ 00-7 ^ d+mlabjlp fkqbdo^oq‹ojfkl ^ q‹ojfkl bk bi fkqbos^ilab `lksbodbk`f^ l_qbkfbkal

Il9 //

b&o'_o <1 \*9si <y&s' + \j ,l i;/

Orbpql nrb d bp `lkqfkr^+ bi mofjbo qblobj^ crka^jbkq^i abi BŠi`ril klp af`b nrb`$%r&bufpqbv bp fdr^i ^ a%r& m^o^`^a^ r abi fkqbos^il ab `lksbodbk`f^- Dpql ab,jrbpqo^ _(-

Kjo\o Orbpql nrb qla^ pbofb ab mlqbk`f^p 1 [))%r * [&{ mrbab l_qbkbopb abofs^kal prpbofb fkqbdo^a^+1 [+%r * [&{(E,%h* 0(+ bi qblobj^ 00-8 klp af`b nrb i^p alp pbofbpqfbkbk bi jfpjl o^afl ab `lksbodbk`f^-

Page 93: Calculus

Mmjkd`_\_`n _` g\n api^dji`n m`km`n`io\_\n kjm n`md`n m`\g`n _` kjo`i^d\n 31/

Klp qblobj^p 00-7 v 00-8 grpqfcf`^k i^p j^kfmri^`flkbp cloj^ibp ab i^Rb``fŽk 0/-7 bk alkab l_qrsfjlp s^oflp abp^ooliilp bk pbofbab mlqbk`f^prqfifw^kali^ abofs^`fŽk v i^ fkqbdo^`fŽkq‹ojfkl ^ q‹ojfkl ab i^ pbofbdblj‹qof`^- Dk m^o,qf`ri^o+bpqlp qblobj^p bpq^_ib`bk i^ s^ifabw ab ilp abp^ooliilp

r & 0'isi)gild 'i * r& < !&!,,,!,,

H i)/h:K

u Jll&[g'is0i)/

^o`q^k s < +/h * .

hwK

pfbjmob nrb s bpq‹ bk bi fkqbos^il ^_fboql 0 ; s ; 0-Bljl rk^ `lkpb`rbk`f^ jŠp abi qblobj^ 00-8+ l_qbkbjlp nrb i^ crk`fŽk

prj^ ab rk^ pbofbab mlqbk`f^p qfbkb abofs^a^p ab oj_j loabk v nrb mrbabk pbo`^i`ri^a^p mlo abofs^`fŽk obfqbo^a^q‹ojfkl ^ q‹ojfkl ab i^ pbofb ab mlqbk`f^p-Rf F&s' < H\i&s + \'i v abofs^jlp bpq^cŽojri^ f sb`bp v mlkbjlp irbdl s < \bk bi obpriq^al+ bk`lkqo^jlp nrb x9e&%[&< fd\• `lk il `r^i bi `lbcf`fbkqb h,‹pfjl[e sfbkb a^al mlo i^ cŽojri^

afg&\'[ * ** m^o^ f < 0+1+2+----

! f + e

Dpq^cŽojri^ q^j_f‹k bp sŠifa^ m^o^ f < N pf fkqbomobq^jlpF&L'&\'ljl a`\', @pŒmrbp+bi abp^ooliil ab . bk pbofbab mlqbk`f^p qfbkb i^ cloj^,--%..+1& a&s' < xaf'&\' &s [ \'f,

H e Q5>

Dpq^molmfba^a mrbab clojri^opb `ljl rk o`jm`h\ _` pid^d_\_ m^o^ilp abp^ool,iilp bk pbofbab mlqbk`f^p-

RCMPCK? 00-0/- Pd _jn n`md`n _` kjo`i^d\n I \i&s + \'! v I ]**&s+ \'!od`i`i g\ hdnh\ api^d‡i nph\ . `i pi ^d`moj `iojmij _`g kpioj \* `ioji^`n g\n_jn n`md`nnji dbp\g`n o„mhdij \ o„mhdij9 `i m`\gd_\_* o`i`hjn \9 < ]9 < F&i'&\'-i k\m\ ^\_\ i 1 N-

K^ fdr^ia^a '00-3( abjrbpqo^ q^j_f‹k nrb i^p prj^p m^o`f^ibpab rk^ pbofbab mlqbk`f^p plk pbk`fii^jbkqb ilp mlifkljflp ab S^vilo ab i^ crk`fŽk prj^ bk bimrkql \, Dk lqo^p m^i^_o^p+pf rk^ crk`fŽk ` bp obmobpbkq^_ibmlo rk^ pbofbabmlqbk`f^p bk rk fkqbos^il %[ * l) [ * l&) bkqlk`bp i^ pr`bpfŽk ab mlifkljflp abS^vilo vQia&s9 \'w bkdbkao^a^ bk \ mlo a `lksbodb mrkqr^ijbkqb bk bpb fkqbos^ile^`f^ i^ crk`fŽk prj^ g,@abjŠp+ i^ `lksbodbk`f^ bp rkfclojb bk qlal pr_ fkqbos^il`boo^al abi fkqbos^il ab `lksbodbk`f^-

Page 94: Calculus

421 Pp^`ndji`n t n`md`n _` api^dji`n

00-8 Rbofbab S^vilo dbkbo^a^ mlo rk^ crk`fŽk

Ulis^jlp ^elo^ ^i pbdrkal mol_ibj^ nrb prodfŽ ^i `ljbkw^o i^ Rb``fŽk mob,`babkqb- Dpql bp+a^a^ rk^ crk`fŽk o,^sbofdr^o pf bp l kl abp^oolii^_ib bk pbofbab mlqbk`f^p bk rk `fboql fkqbos^il ^_fboql bk qlokl ^i mrkql \,

R^_bjlp mlo il nrb pb ^`^_^ ab abjlpqo^o nrb q^i crk`fŽk ab_b qbkbokb`b,p^of^jbkqb abofs^a^p ab qlal loabk bk rk `fboql fkqbos^il ^_fboql bk qlokl ^imrkql \ v nrb ilp `lbcf`fbkqbp ab pr abp^ooliil bk pbofbab mlqbk`f^p plk ilp a^alpmlo i^ fdr^ia^a '00-3(- Rrmlkd^jlp+ bkqlk`bp+ nrb m^oqfjlp ab rk^ crk`fŽk `nrb qbkd^ abofs^a^p ab `r^inrfbo loabk bk rk fkqbos^il ^_fboql bk qlokl ^i mrk,ql \, Cfobjlp nrb rk^ q^i crk`fŽk bp diadido\h`io` abofs^_ib bk bpb fkqbos^il+ Ol,abjlp bkqlk`bp ajmh\m i^ pbofbab mlqbk`f^p

// d%ef%[&'00-4( z ,, &s+ \g}

H e Q_>

Dpq^ pb ii^j^ n`md` _` Q\tgjm b`i`m\_\ kjm a `i \, Elojribjlp ^elo^ alp mob,drkq^p9 ƒBlksbodb bp^ pbofbm^o^ `r^inrfbo lqol r afpqfkql abi r < [< Rf bp ^pŒ+ƒpr prj^ bp fdr^i ^ `%r&< @rknrb plomobka^+i^ `lkqbpq^`fŽk ^ bp^p mobdrkq^pbp+bk dbkbo^i+~kl‚- K^ pbofbmrbab pbo l kl `lksbodbkqb m^o^s ;/; \ v+ pf il bp+prprj^ mrbab l kl `lfk`fafo `lk `%r&+Dk bi Dgbo`f`fl 13 ab i^ Rb``fŽk 00-02 pb a^rk bgbjmil ab rk^ pbofbnrb `lksbodb e^`f^ rk^ prj^ afpqfkq^ab `%r&+

Tk^ `lkaf`fŽk kb`bp^of^ v prcf`fbkqb m^o^ mlabo `lkqbpq^o ^cfoj^qfs^jbkqbi^p alp mobdrkq^p+mrbab `lkpbdrfopb rp^kal i^ cŽojri^ ab S^vilo `lk obpql+nrba^ rk abp^ooliil adidoj ab i^ cloj^

wx9e&%[& e B &,'%..+3& e&s' < H++y9y+&s+ \' * i u -

f;L

K^ prj^ cfkfq^ bp bi mlifkljfl ab S^vilo ab do^al i dbkbo^al mlo ` bk \* vAh%r& bp bi boolo `ljbqfal ^i ^molufj^o ` `lk pr mlifkljfl ab S^vilo- Rf e^`bjlp00 ,* // bk '00-5(+ sbjlp nrb i^ pbofbab mlqbk`f^p '00-4( `lksbodboŠ e^`f^ `%r& pfv pŽil pf bi q‹ojfkl ab boolo qfbkab ^ N- Dk i^ Rb``fŽk nrb pfdrb e^`bjlp i^ afp`r,pfŽk ab rk^ `lkaf`fŽk prcf`fbkqbm^o^nrb bi q‹ojfkl ab boolo qfbka^ ^ N-

00-0/ Blkaf`fŽk prcf`fbkqb m^o^ i^ `lksbodbk`f^ ab rk^ pbofbab S^vilo

Dk bi qblobj^ 6-5 pb abjlpqoŽ nrb bi q‹ojfkl ab boolo bk i^ cŽojri^ ab S^vilomrbab bumobp^opbljl rk^ fkqbdo^i

%..+4& Bi&s' < 0, &!&s+ o'iy:i)gg&o' _oh G\ ,

Page 95: Calculus

A`n\mmjggjn `i n`md` _` kjo`i^d\n _` api^dji`n `skji`i^d\g t omdbjijh„omd^\n 311

bk `r^inrfbo fkqbos^il bk qlokl ^i mrkql \ bk bi nrb l7| pb^ `lkqfkr^- Olo `lk,pfdrfbkqb+pf ` bp fkabcfkfa^jbkqb abofs^_ib+qbkbjlp pfbjmob bp^ obmobpbkq^`fŽkabiboolo+`lk il nrb i^ pbofbab S^vilo `lksbodb e^`f^ `%r& pf v pŽil pf i^ fkqbdo^iqfbkab e^`f^ N `r^kal i x ^`*

K^ fkqbdo^ipb mrbab mlkbo bk rk^ cloj^ ^idl jŠp •qfi mlo jbafl ab rk `^j,_fl ab s^of^_ib- Dp`of_^jlp

o < s * &\ + s'p * _o < +&s + \' _p *

v l_pbosbjlp nrb p s^oΠab 0 ^ N `r^kal o s^oΠab \ ^ s, Olo q^kql+i^ fkqbdo^i'00-6( pb `lksfboqb bk

'00-7(%T \'i)g//Bi&s' < , p%/&i)g'Xs * &\ + s'pZ _p ,

h l

Dpq^cloj^ abi boolo klp mbojfqb a^o i^ pfdrfbkqb `lkaf`fŽk prcf`fbkqbm^o^i^ `lk,sbodbk`f^ ab rk^ pbofbab S^vilo-

RCMPCK? 00-00- Rf a `n diadido\h`io` _`mdq\]g` `i pi dio`mq\gj \]d`mojF < &\ + m*\ * m'* t nd `sdno` pi\ ^jino\io` kjndodq\ > o\g lp`

'00-8( Glj('t(0 y >i k\m\ i < 0+1+2+--- +

v oj_j s _` I* `ioji^`n g\ n`md` _` Q\tgjm b`i`m\_\ kjm a `i \ ^jiq`mb` c\^d\ a&s'k\m\ ^\_\ s _` g,

A`hjnom\^d‡i, Tqfifw^kal i^ abpfdr^ia^a '00-8( ab i^ cŽojri^ fkqbdo^i'00-7(+l_qbkbjlp i^ bpqfj^`fŽk

Zu \gi)/ // Zu \gi)/ >i)/ ?h)/M ; hC 't(0 ; , >i)/ pi _p < , < ,,,

+ i + i N &i)/' &i)g' %

bk alkab ? < >gs + ]h- Obol m^o^qlal ?* ?! Fi qfbkab ^ N `r^kal i x //+ ^pŒnrb Ah%r&w N m^o^`^a^ r ab f+

00-00 Cbp^ooliilp bk pbofbab mlqbk`f^pab i^p crk`flkbp bumlkbk`f^i v qofdlkl,j‹qof`^p

K^p crk`flkbp pbkl v `lpbkl v qla^p prp abofs^a^p bpqŠk^`lq^a^p mlo bi k•,jbol 0 bk qlal bi bgb ob^i- Olo `lkpfdrfbkqb+ i^ abpfdr^ia^a '00-8( bp sŠifa^ `lk

Page 96: Calculus

423 Oo]_mcih_mv m_lc_m_ `oh]cih_m

= < 0 pf `ur& < pbk r l pf `ur& < `lp r) v qbkbjlp ilp abp^ooliilp bk pbofb

s1 s3 s% U0i+/

pbk s < s + + * , , , * --- * ']0(k,i ,,, * --- +2 4 6 &0i + 0(

s/ s1 s3 s/h

`lp T < 0 , , * , , , * --- * %Z.&h * * ---+1 3 5 %/h&

sŠifalp m^o^qlal r ob^i- O^o^i^ crk`fŽk bumlkbk`f^i+ `ur& < _!) qbkbjlp l&ur& :z m^o^qlal r) ^pŒnrb bk `r^inrfbo fkqbos^il cfkfql %*l) o( qbkbjlp ``EF 949 mj Olo`lkpfdrfbkqb+ '00-8( pb p^qfpc^`bm^o^= < `!* Orbpql nrb o bp `r^inrfbo^+ bpql ab,jrbpqo^ nrb bi pfdrfbkqbabp^ooliil bk pbofbab mlqbk`f^pbp sŠifal m^o^qlal s ob^i9

s0 si

``EF;g)s)+)}}})+)}}},1 h

Klp ^kqboflobp abp^ooliilp bk pbofbpab mlqbk`f^p abi pbkl v abi `lpbkl pbmrbabk qlj^o `ljl mrkql ab m^oqfa^ ab rk bpqrafl `ljmibq^jbkqb ^k^iŒqf`lab i^p crk`flkbp qofdlklj‹qof`^p- Slj^kal bpq^ppbofbp ljl ^_`chc]cih_m abi pbklv abi `lpbkl+ bp mlpf_ib abar`fo pŽil ab bii^p qla^p i^p molmfba^abp^k^iŒqf`^pv^idb_o^f`^p ab i^p crk`flkbp qofdlklj‹qof`^p- Olo bgbjmil+ ab i^p pbofbppb ab,ar`bk fkjbaf^q^jbkqb i^p cŽojri^p9

pbk N < N+ `lp N < 0+ pbk %*r& < , pbk r) `lp ', r&: `lp r)

@ pbk r < `lp r) @ `lp r < , pbk r+

K^p cŽojri^p ab ^af`fŽk pb mrbabk l_qbkbo mlo jbafl abi pfdrfbkqb ^oqfcf`fl9pb^k p v q krbs^p crk`flkbp abcfkfa^p mlo i^p b`r^`flkbp9

our& < pbk %r* [& * pbk r `lp [ * `lp r pbk [)

pur& < `lp %r* [& * `lp r `lp [ * pbk r pbk [)

alkab [ bp rk k•jbol ob^i cfgl+v pb^ `ur& < XRvU'Z0 * Wpur|&/+Dkqlk`bp bp cŠ`fi`ljmol_^o nrb o$ur&< pur& u p$ur&< , our& u mlo q^kql `$ur&< N m^o^qlal r+Cb ^nrŒ obpriq^ nrb `ur& bp rk^ `lkpq^kqb+ v mrbpql nrb `uK&< N+e^ ab pbo`ur& < N m^o^qlal r+ Dpql fjmif`^ our& :pur& < N m^o^qlal u+l ab lqo^ cloj^9

pbk %r* [& < pbk r `lp [ * `lp r pbk [)

`lp %r * [& < `lp u`lp [ * pbk r pbk [+

Page 97: Calculus

Q`jm`h\ _` ?`mino`di 424

Di k•jbol 6S+ pb mrbab fkqolar`fo `ljl bi jbklo k•jbol mlpfqfsl q^i nrbpbk r < N 'pb mrbab mol_^o nrb bufpqb rk^ q^i r& v abpmr‹p pb mrbab mol_^onrb bi pbkl v bi `lpbkl plk mbofŽaf`^p`lk mboŒlal16S+ nrb pbk 'q6S( < 0+W nrb`lp ' 6S( < N- Klp abq^iibp+nrb kl pb bumlkaoŠk ^nrŒ+pb mrbabk bk`lkqo^o bkbi if_ol Qc`jmt \i_ >kkgd^\odji ./ Fiedido` P`md`n*mlo J- Jklmm 'Mrbs^ Xloh9G^ckbo+ 0840(-

r ))&)* FR\_RZN QR 7RZ`aRV[

Di qblobj^ ))&)) abjrbpqo^ nrb i^ pbofb ab S^vilo ab rk^ crk`fŽk ` `lk,sbodb pf i^ abofs^a^ k,pfj^ a&i' kl `ob`b jŠp oŠmfa^jbkqb ab i^ mlqbk`f^ k,pfj^ab rk `fboql k•jbol mlpfqfsl- Nqo^ `lkaf`fŽk prcf`fbkqbm^o^ i^ `lksbodbk`f^ crbclojri^a^ mlo bi j^qbjŠqf`l orpl RbodbfM- Abokpqbfk'077/,(-

SDNQDL@ 01-01- SDNQDL@ CD ADQMRSDHM- Pd` u oj_\n npn _`mdq\_\n nji iji`b\odq\n `i pi dio`mq\gj ^`mm\_j ZN+mG*noj `n* nd

a&s' ƒ M v

k\m\ ^\_\ s `i ZN+mGv ^\_\ i < 0+ 1+ 2+ --- + `ioji^`n* nd N z s ; m*g\ n`md`_` Q\tgjm

^jiq`mb` c\^d\ a&s',

A`hjnom\^d‡i, Di obpriq^al bp qofsf^i m^o^ s < N+ ^pŒnrb prmlkd^jlp nrbM ; s ; m,Tqfif`bjlp i^ cŽojri^ ab S^vilo `lk obpql m^o^bp`of_fo

'00-0/(h a&f'-L'

a&s' < !! ][, sf * Bi&s' ,J-- e Q5>

Cbjlpqo^objlp nrb bi q‹ojfkl ab boolo p^qfpc^`bi^p abpfdr^ia^abp

'00-00(

Dpql+ ^ pr sbw+morb_^ nrb Ah%r&w N `r^kal i x // mrbpql nrb bi `l`fbkqb%r,l&h(f w N `r^kal N ; r ; l+

Page 98: Calculus

425 Oo]_mcih_m v m_lc_m^_ `oh]cih_m

O^o^ abjlpqo^o '00-00(+ rqfifw^jlp i^ cloj^ fkqbdo^iabi boolo nrb, pb a^ bk'00-7( `lk [ < N9

si)/ d/

Aw%r&< ,, ohd%h(ff%r * ro& ^o +i% l

Dpq cŽojri^ bp sŠifa^ m^o^`^a^ r abi fkqbos^il `boo^al ZN+lY+ Rf r :B N+mlkbjlp

B %r& 0 c/

Bh%r&< ZhZ < , ohd%h(ff%r * ro& ^o +rh(. h l

K^ crk`fŽk l| bp jlkŽqlk^ `ob`fbkqbbk bi fkqbos^il ZN+o\ v^ nrb pr abofs^a^ bpkl kbd^qfs^- Olo `lkpfdrfbkqb+ qbkbjlp

d%h(f&%r * ro& < jhD&Wr%. * o&Yw jh(ffWl%. * o&Y

pf N z o w 0+ il `r^i &fjmif`^ nrb Bh%r&w Bh%l& pf N ; u z o- Cf`el ab lqoljlal+ qbkbjlp Ai%r&,r!(. w A))%l&,li(. l

'00-01( %r&h(.

A7h%r&w l Ah%l&+

Olkfbkal r < o bk i^ fdr^ia^a '00-0/(+ sbjlp nrb Ah%l&w `%l& mlonrb `^a^ q‹o,jfkl ab i^ prj^ bp kl kbd^qfsl- @mif`^kal bpql bk '00-01(+ l_qbkbjlp '00-00( il`r^i `ljmibq^ i^ abjlpqo^`fŽk-

))&)+ :WR_PVPV\`

O^o^ `^a^ rk^ ab i^p pbofbpab mlqbk`f^p ab ilp bgbo`f`flp 0 ^i 0/ abqbojfk^o bi `lkgrkqlab qlalp ilp s^ilobp ob^ibp s m^o^ ilp nrb `lksbodb v `^i`ri^o pr prj^- Klp abp^ooliilp bkpbofb ab mlqbk`f^p v^ sfpqlp mrbabk rqfifw^opb`r^kal `lksbkd^-

NB

0- ƒ']i(kt1k-

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Page 99: Calculus

Be`m^d^djn .,0

x si

8- J---- %h* 2( -h:K

w%r * 0(!0/- J---- %h* 1( -

T_>

B^a^ rk^ ab i^p crk`flkbp ab ilp bgbo`f`flp abi 00 ^i 10 qfbkb rk^ obmobpbkq^`fŽk bkpbofb ab mlqbk`f^p ab s, Rrmrbpq^ i^ bufpqbk`f^ abi abp^ooliil+ `ljmol_^o nrb ilp `lbcf`fbkqbpqfbkbk i^ cloj^ a^a^+ v abjlpqo^o nrb i^ pbofb `lksbodb m^o^ ilp s^ilobp ab s fkaf`^alp-Br^kal `lksbkd^ mrbabk rqfifw^opb ilp abp^ooliilp v^ sfpqlp-

‚NN'ild \'i00- \T < ,,+ , u!+

T(h:K

H6> 'qlal s', XFi_d^\^d‡i8 \! < _!EKa[+Y

// U0i(/

/0, pbke u <•&0i * 0( 'qlal s',T_>

// 00i+/

02 pbk&&r < !! %Z.&h(. ** r/h 'qlal r&+ XFi_d^\^d‡i8 `lp /r < 0 , 1 pbk! r+Y- J---- %/h& h:.

0 y sh

03- 1 ] s < J---- /i)/T_>

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01 , 3s Kll ' %ZE&h&08- ,,,,, < 0 * ,, sh

5 , 3s + s/ 4hT_>

'ft[ ; 0(-

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1/- 1 0 , ‘ {, ‚ pbk 2 ss * s *s 2 hwK

XFi_d^\^d‡i8 s1 + 0 < &s + /'&s0 * s * 0(-[

'Gth ; 0(-

Page 100: Calculus

427 Pp^`ndji`n t n`md`n _` api^dji`n

s 0 1//

' 0 , &[/'i'&(0 ,,,,, < , i * ,,,, si

z - '0 , r&%. * r/& 1 00<0 1

11- Cbqbojfk^ob0 `lb•`Œbkqb`pabi-abp^ooliil bk pbofb ab mlqbk`f^p pbk &0s * f6S( <H9&<l\isi,

12- Rb^ E%r&< '1 * ('1(4.-1 Cbqbojfk^o ilp `lbcf`fbkqbp \j* \g ,,, * \2 ab i^ pbofb ab S^vilo db•kbo^a^ mlo . bk N-

13- Rb^ E%r&:_*.,!! pf r !+ N+X 0'/( < l-^( Cbjlpqo^o nrb . qfbkb abofs^a^p ab qlal loabk bk qlal bi bgb ob^i-_( Cbjlpqo^o nrb y:i' < N m^o^ qlal i x 0- Dpqbbgbjmil morb_^ nrb i^ pbofb ab S^vilodbkbo^a^ mlo . bk qlokl ^i mrkql N `lksbodb bk qlal bi bgb ob^i+ mbol nrb obmobpbkqznjg\h`io` bk bi lofdbk-

'Gth; 0(-

))&), ER_VRQR]\aR[PVN e RPbNPV\[RQVSR_R[PVNYR`

@ sb`bp i^p pbofbpab mlqbk`f^pklp mbojfqbk l_qbkbo plir`flkbp ab b`r^`flkbpafcbobk`f^ibp`r^kal c^ii^k ilp lqolp j‹qlalp- Dk bi Ulirjbk 00 pb a^ rk^ afp`r,pfŽk pfpqbjŠqf`^abi bjmibl ab i^p pbofbpab mlqbk`f^pbk i^ qbloŒab i^p b`r^`flkbpafcbobk`f^ibpifkb^ibp ab pbdrkal loabk- @nrŒfirpqo^jlp `lk rk bgbjmil ^idrk^ abi^p fab^p v q‹`kf`^p obi^qfs^p^ bpqb^prkql-

Blkpfa‹obpb i^ b`r^`fŽk afcbobk`f^i ab pbdrkal loabk9

'00-02( '0 , s0&s! < */s+

Rrmlkd^jlp nrb bufpqbrk^ plir`fŽk+ pb^ v < `%r&)nrb pb mrbab bumobp^omlojbafl ab rk^ pbofbab mlqbk`f^pbk rk bkqlokl abi lofdbk+bp ab`fo9

'00-03(

Dk mofjbo ird^o pb e^k ab abqbojfk^o ilp `lbcf`fbkqbp \j* ]h= \0* ŠŠŠ

Tk^ j^kbo^ ab mol`babo bp i^ pfdrfbkqb9 Cbofs^kal '00-03( alp sb`bp+ pbl_qfbkb9

//

t! <Gh%h* f&[hrh*0 ,

i;0

Lriqfmif`^kal mlo 0 , s08 pb bk`rbkqo^9

'00-04()) ))

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Page 101: Calculus

P`md`n_` kjo`i^d\n v `^p\^dji`n _da`m`i^d\g`n 428

Rrpqfqrvbkal `^a^ rk^ ab i^p pbofbp '00-03( X '00-04( bk i^ b`r^`fŽk afcbobk`f^i pbl_qfbkb rk^ b`r^`fŽk nrb `lkqfbkb alp pbofbp ab mlqbk`f^p+ sŠifa^ bk rk bkqloklabi lofdbk- Dk sfoqra abi qblobj^ ab i^ rkf`fa^a+ bpq^p pbofbp ab mlqbk`f^p e^kab pbo fdr^ibp q‹ojfkl ^ q‹ojfkl+ v pbmrbabk fdr^i^o ilp `lbcf`fbkqbp ab si l_qb,kf‹kalpb i^p obi^`flkbp9

%h* /&%h* f&[i)0 + h%h* f&[i < */[i

l il nrb bp il jfpjl9

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Page 106: Calculus

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K^ fab^ ab bjmib^o rk k•jbol m^o^pfqr^ork mrkql bk rk^ ob`q^ crb `lkl`fa^mlo ilp ^kqfdrlp dofbdlp- Dk 0526 Cbp`^oqbpbuqbkafŽ bpq^fab^+rqfifw^kal rk j[lab k•jbolp &\g%\0' m^o^ pfqr^o rk mrkql bk bi mi^kl+ u rk^ o`mi\ ab k•jbolp&\g%\0 * \n' m^o^ pfqr^o rk mrkql bk bi bpm^`fl- Dk bi pfdil WHWilp j^qbj-Šqf`lp@- B^vibv '0710,0784( v G- F- Fo^ppj^kk '07/8,0766( mol_^olk nrb kl bo^ kb,`bp^ofl abqbkbopbbk i^p qbok^p ab k•jbolp- Rb mrbab q^j_f‹k `lkpfabo^o rk^^p\o`mi\ ab k•jbolp &\g%\0 * \n *\2' l+ jŠp dbkbo^i+rk^ h*jf[ ab k•jbolp ob^ibp

&\g%\0 * Š Š Š * \h&

Page 109: Calculus

Bg `nk\^dj q`^ojmd\g_` g\n i+kg\n _` iˆh`mjn m`\g`n 436

m^o^ qlal bkqbol i ƒ 0- Tk^ q^i i+kg\ pb ii^j^ kpioj i+_dh`indji\g l q`^ojm i+_dh`indji\g* pfbkal ilp k•jbolp \g *\9 *,,, ,\9 i^p ^jjm_`i\_\n l ^jhkji`io`nabi sb`qlo- Di `lkgrkql ab qlalp ilp sb`qlobp i+_dh`indji\g`n pb ii^j^ `nk\^djq`^ojmd\g_` i+kg\n*l pfjmibjbkqb i+`nk\^dj, Kl abpfdk^jlp `lk SiŠ

Orbab mobdrkq^obi ib`qlo bi jlqfsl mlo bi `r^i bpq^jlp fkqbobp^,alpbk bpm^,`flp ab afjbkpfŽk j^vlo nrb qobp-Tk^ `lkqbpq^`fŽk bp nrb jr`elp mol_ibj^pnrb prmlkbk bi bpqrafl ab pfpqbj^p ab rk k•jbol do^kab ab b`r^`flkbp pb ^k^if,w^k `lk j^vlo c^`fifa^a fkqolar`fbkal sb`qlobp bk rk k,bpm^`fl `lksbkfbkqb vobbjmi^w^kal qla^p ^nrbii^p b`r^`flkbp mlo rk^ pli^ b`r^`fŽk sb`qlof^i- Nqo^sbkq^g^ bp i^ ab nrb mlabjlp qo^q^oab rk^ sbw+jr`e^p molmfba^abp`ljrkbp ^ilp bpm^`flp ab rk^+ alp+ qobpl jŠp afjbkpflkbp+ bpql bp+molmfba^abpfkabmbk,afbkqbpab i^ afjbkpflk^ifa^a abi bpm^`fl- Dpql bpqŠab ^`rboal `lk bi bpmŒofqrabi^ jlabok^ j^qbjŠqf`^ nrb c^`fifq^ bi abp^ooliil ab j‹qlalp ab ^jmif^p pŒkqbpfpm^o^^q^`^o afpqfkqlpmol_ibj^p ab j^kbo^ pfjriqŠkb^-

Cbpdo^`f^a^jbkqb+ i^p obmobpbkq^`flkbpdblj‹qof`^p nrb plk rk^ do^k ^vra^bk i^ firpqo^`fŽkv grpqfcf`^`fŽkab `lk`bmqlp pl_ob sb`qlobp+`r^kal i < 0+1+X 2+kl mrbabk rqfifw^opb r^kal i = 2: mlo biil+ bi bpqrafl abi @idb_o^sb`qlof^i bkbpm^`flpab jŠp ab qobpafjbkpflkbp ab_b e^`bopb mlo bkqbol `lk jbaflp ^k^iŒqf`lp-

Dk bpqb`^mŒqrilabpfdk^jlp loafk^of^jbkqb ilp sb`qlobp `lk ibqo^pj^v•p`r,i^p >* ?* a+ --- +u ilp `ljmlkbkqbp `lk i^p `loobpmlkafbkqbpjfk•p`ri^p \* ]* `* ,,,@pŒ+bp`of_fjlp

O^o^`lksboqfo Sh bk rk^ bpqor`qro^ ^idb_o^f`^+fkqolar`fjlp i^ dbp\g_\_ ab sb`ql,obp v alp lmbo^`flkbp i^ \_d^d‡i ab sb`qlobp v i^ hpgodkgd^\^d‡i kjm `n^\g\m`n,K^ m^i^_o^ ~bp`^i^o‚ pb rqfifw^ ^nrŒ`ljl pfkŽkfjl ab ~k•jbol ob^i‚-

CDEHMHBHˆM- Ajn q`^ojm`n > v ? _` Si nji dbp\g`n nd`hkm` lp` ^jdi^d_`inpn ^jhkji`io`n, Bnoj `n*pf > < &\g%\0 * ŠŠŠ * \i' v ? < '_h Š]0 * ŠŠŠ * ]i'* g\ `^p\+^d‡i q`^ojmd\g> < ? od`i` `s\^o\h`io` `g hdnhj ndbidad^\_j lp` g\n i `^p\^dji`n`n^\g\m`n

I\ nph\ > * ? n` _`adi` ^jhj `g q`^ojm j]o`id_j nph\i_j gjn ^jhkji`io`n ^j+mm`nkji_d`io`n8

Rf ` `n pi `n^\g\m*_`adidhjn ^> j >^ ^jhj `g q`^ojm j]o`id_j hpgodkgd^\i_j ^\_\^jhkji`io` _` > kjm `8

Page 110: Calculus

.-1 €gb`]m\ q`^ojmd\g

O^oqfbkal ab bpq^abcfkf`fŽk bp cŠ`fi `ljmol_^o i^p pfdrfbkqbpmolmfba^abpabbp^p lmbo^`flkbp9

RCMPCK? 01-0- I\ \_d^d‡i _` q`^ojm`n `n ^jihpo\odq\,

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t \nj^d\odq\*

= * %>* B( < %=* >& * B-

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v n\odna\^` g\n _jn g`t`n _dnomd]podq\n

^&>* ?' < ^> * ^?* u &^* _'> < ^> * _> ,

K^p abjlpqo^`flkbp ab bp^p molmfba^abp plk `lkpb`rbk`f^ fkjbaf^q^ ab i^abcfkf`fŽk u pb abg^k `ljl bgbo`f`fl m^o^ bi ib`qlo-

Di sb`qlo `lk qlalp ilp `ljmlkbkqbp N pb ii^j^ q`^ojm ^`mj v pb obmobpbkq^`lk N- Sfbkb i^ molmfba^a ab nrb > * N < > m^o^ qlal sb`qlo >9 af`el ablqol jlal+ N bp rk bibjbkql kbrqol l fa‹kqf`l m^o^ i^ ^af`fŽk ab sb`qlobp- Disb`qlo ', g'> nrb q^j_f‹k pb obmobpbkq^lk , > pb ii^j^ bi jkp`noj ab >, S^j,_f‹k bp`of_fjlp > + ? bk ird^o ab > * ', ?' Xil ii^j^jlp _da`m`i^d\ ab > u ?,K^ b`r^`fŽk %=* >& * > < = abjrbpqo^ nrb i^ prpqo^``fŽk bp i^ lmbo^`fŽk fk,sbop^ ab i^ ^af`fŽk- N_p‹osbpb nrb L> < N X nrb g> < >,

Di ib`qlo e^_oŠ l_pbos^al i^ pbjbg^kw^ bkqob ilp sb`qlobp bk bi bpm^`fl ab1 afjbkpflkbp u ilp k•jbolp `ljmibglp- @j_lp pb abcfkbk `ljl m^obploabk^alpab k•jbolp ob^ibp u ^j_lp pb prj^k ab i^ jfpj^ j^kbo^- @pŒmrbp+bk il nrbe^`b obcbobk`f^^ i^ ^af`fŽk+ ilp k•jbolp `ljmibglp u ilp sb`qlobp _fafjbkpflk^ibp^idb_o^f`^jbkqb plk fkafpqfkdrf_ibp- RŽil pb afcbobk`f^k ^i fkqolar`fo i^ jriqf,mf`^`fŽk-

K^ jriqfmif`^`fŽk a^ ^i pfpqbj^ ab k•jbolp `ljmibglp bi `lkgrkql ab molmfb,a^abp nrb q^j_f‹k mlpbbk ilp k•jbolp ob^ibp-Orbab abjlpqo^opb '^rknrb obpriq^afcŒ`fi(nrb bu`bmql m^o^ i < 0 X i < 1+ kl bp mlpf_ib fkqolar`fo i^ jriqfmif`^,`fŽk bk Ri ab jlal nrb pb p^qfc^d^kqla^p i^p molmfba^abp-Rfk bj_^odl mrbabkfkqolar`fopb bk Si `fboqlp molar`qlp nrb kl p^qfpc^`bk oj_\n i^p molmfba^abp-Olo bgbjmil+ bk i^ pb``fŽk 01-4 e^_i^objlp abi kmj_p^oj dio`mdjmab alp sb`qlobpbk SiŠ Di obpriq^al ab bpq jriqfmif`^`fŽk bp rk bp`^i^o+kl bp rk sb`qlo- Dk i^ pb`,`fŽk 02-8 pb qo^q^oŠabi kmj_p^oj q`^ojmd\g, Dpq^jriqfmif`^`fŽk pŽil bp ^mif`^_ib

Page 111: Calculus

Fio`mkm`o\^d‡ib`jh„omd^\ k\m\ i 4 2 438

bk bi bpm^`fl S\, Di obpriq^al pfbjmob bp rk sb`qlo+ mbol bpqb molar`ql kl bp`lkjrq^qfsl-

)*&+ =[aR_]_RaNPVp[TR\Zna_VPN]N_N ?+++,+1

Rf _fbk i^p abcfkf`flkbp ^kqboflobpplk `ljmibq^jbkqb fkabmbkafbkqbpab i^FbljbqoŒ^+ilp sb`qlobp u i^p lmbo^`flkbp `lk biilp qfbkbk rk^ fkqbobp^kqbfkqbo,mobq^`fŽkdblj‹qof`^ m^o^bpm^`flp ab afjbkpfŽk qobpl jbklo nrb qobp-G^objlpobmobpbkq^`flkbpm^o^alp afjbkpflkbp u ob`ljbka^jlp ^i ib`qlo nrb pb i^p e^d^m^o^bpm^`flp ab qobpu ab rk^ afjbkpflkbp-

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EHFTQ@ 01-0 Bg q`^ojm b`jh„omd^j,>E _`g kpioj > \g ?,

, ,EHFTQ@ 01-1 >? X @A nji `lpdq\g`io`nkjmlp` ? + > < C , B-

Tk m^o ab mrkqlp > u ? pb gg\h\ q`^ojm b`jh„omd^j pf rkl ab ilp mrkqlp+mlo bgbjmil >* bp bi kpioj did^d\gu bi lqol+ ?* bp kpioj `som`hj, Qbmobpbkq^jlprk sb`qlo dblj‹qof`l `lk rk^ cib`e^ ab = ^ >) `ljl sbjlp bk i^ cfdro^ 01-0+vbjmib^jlp i^ klq^`fŽk =>+

Klp sb`qlobp dblj‹qof`lp plk bpmb`f^ijbkqb •qfibp m^o^ obmobpbkq^ofboq^pj^dkfqrabp cŒpf`^pq^ibp`ljl crbow^p+abpmi^w^jfbkqlp+sbil`fa^abp+ u ^`bibo^`fl,kbp+nrb mlpbbk j^dkfqra u afob``fŽk- K^ ilkdfqra ab i^ cib`e^ bp rk^ jbafa^ab i^ j^dkfqra v i^ mrkq^ ab i^ cib`e^ fkaf`^ i^ afob``fŽk nrb pb mob`fp^-

Page 112: Calculus

44/ >gb`]m\ q`^ojmd\g

Rrmlkd^jlp nrb fkqolar`fjlp rk pfpqbj^ `lloabk^al `lk lofdbk N- K^ cfdr,

o^ 01-1 jrbpqo^ alp sb`qlobp di v ?@ q^ibpnrb > * = < C , B- Dk crk`fŽk abilp `ljmlkbkqbp+ bpql pfdkfcf`^ nrb

v

Bljm^o^kal ilp qofŠkdrilp fdr^ibp ab i^ cfdro^ 01-1+sbjlp nrb i^p alp cib`e^p

nrb obmobpbkq^k=> v ?@ qfbkbk i^ jfpj^ ilkdfqra+ plk m^o^ibilp+b fkaf`^k i^jfpj^ afob``fŽk- Ki^j^jlp ^ q^ibp sb`qlobp dblj‹qof`lp `lpdq\g`io`n, Dpql bp+, ,ab`fjlp => bp bnrfs^ibkqb ^ ?@ pfbjmob nrb

'01-0( >*=:@*?+

N_p‹osbpb nrb ilp `r^qol mrkqlp =) >) B+ @ plk s‹oqf`bp ab rk m^o^ibildo^jl-'Ubo cfdro^ 01-2-( K^ b`r^`fŽk '01-0( q^j_f‹k pb mrbab bp`of_fo bk i^ cloj^> * A < ? * B il nrb klp af`b nrb ilp q„mod^`njkp`nojn _`g k\m\g`gjbm\hj od`+i`i g\ hdnh\ nph\, Dk m^oqf`ri^o+pf rkl ab ilp s‹oqf`bp+mlo bgbjmil >* bp bilofdbk N+ `ljl bk i^ cfdro^ 01-3+ bi sb`qlo dblj‹qof`l nrb rkb N ^i s‹oqf`blmrbpql A `loobpmlkab ^i sb`qlo prj^ A < ? * B- Dpql pb bumobp af`fbkal nrbi^ ^af`fŽk ab sb`qlobp `loobpmlkab dblj‹qof`^jbkqb ^ i^ ^af`fŽk ab sb`qlobpdblj‹qof`lp mlo jbafl ab i^ ibv abi m^o^ibildo^jl- K^ fjmloq^k`f^ ab ilp sb`qlobpbk i^ cŒpf` molsfbkb abi eb`el klq^_ib ab nrb jr`e^p j^dkfqrabp cŒpf`^p'q^ibp`ljl crbow^p+sbil`fa^abp v ^`bibo^`flkbp( pb `lj_fk^k mlo jbafl ab i^ ibv abim^o^ibildo^jl-

@

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nph\8 > * A < ? * B-

EHFTQ@ 01-3 I\ \_d^d4i _` q`^ojm`n dio`mkm`o\_\b`jh„omd^\h`io` ^ji g\ g`t _`g k\m\g`gjbm\hj,

Page 113: Calculus

Bd`m^d^djn 440

@ cfk ab pfjmifcf`^o i^ klq^`fŽk+ rqfifw^objlp bi jfpjl pŒj_lil m^o^ abpfdk^ork mrkql ab Ri '`r^kal i x 2( X bi sb`qlo dblj‹qof`l nrb rkb bi lofdbk ^ bpb

,* ,*mrkql- @pŒmrbp+ bp`of_fjlp = bk ird^o ab @=) > bk ird^o ab @>) bq`- S^j_f‹kbp`of_fjlp ^idrk^p sb`bp > bk ird^o ab `r^inrfbo sb`qlo dblj‹qof`l bnrfs^ibkqb ^

@=+Olo bgbjmil+ i^ cfdro^ 01-4 obmobpbkq^dblj‹qof`^jbkqb i^ prpqo^``fŽk ab sb`,qlobp- Clp sb`qlobp dblj‹qof`lp bpqŠk abpfdk^alp `lk > * =) X plk bnrfs^ibkqbp-Sfbkbk i^ jfpj^ ilkdfqra u i^ jfpj^ afob``fŽk-

K^ cfdro^ 01-5 obmobpbkq^ dblj‹qof`^jbkqb i^ jriqfmif`^`fŽk mlo bp`^i^obp-Rf > < ]=) bi sb`qlo dblj‹qof`l > qfbkb `ljl ilkdfqra bi molar`ql ab hahmlo i^ilkdfqra ab >9 qfbkb i^ jfpj^ afob``fŽk nrb > pf _ bp mlpfqfsl+ u i^ afob``fŽklmrbpq^ pf ` bp kbd^qfsl-

> +?

EHFTQ@ 01-4 Pdbidad^\_j b`jh„omd^j_` g\ npnom\^^d‡i _` q`^ojm`n,

-7

EHFTQ@ 01-5 Jpgodkgd^\^d‡i _` q`^ojm`n,kjm `n^\g\m`n,

K^ fkqbomobq^`fŽk dblj‹qof`^ ab ilp sb`qlobp bk S i m^o^ i x 2 prdfbob rk^j^kbo^ ab abcfkfo bi m^o^ibifpjl bk rk bpm^`fl ab afjbkpfŽk i `r^inrfbo^-

CDEHMHBHˆM- Ajn q`^ojm`n > u ? _` S i od`i`i g\ hdnh\ _dm`^^d‡i nd ? < ^>k\m\ pi ^d`moj `n^\g\m kjndodqj `* u g\ _dm`^^d‡i jkp`no\ nd ? < ^> k\m\ pi ^d`moj` i`b\odqj, P` gg\h\i k\m\g`gjn nd ? < ^> k\m\ pi ^d`moj ` ij ipgj,

N_p‹osbpb nrb bpq^ abcfkf`fŽk mbojfqb `lkpfabo^o nrb qlal sb`qlo qfbkb i^jfpj^ afob``fŽk nrb ‹i jfpjl ,molmfba^a nrb pbdro^jbkqb kb`bpfq^objlp- S^j,_f‹k pb l_pbos^ nrb bpq^ abcfkf`fŽk ^pfdk^ ^i sb`qlo `bol i^p pfdrfbkqbp molmfba^,abp9 Di sb`qlo `bol bp bi •kf`l nrb qfbkb i^ afob``fŽk ab pr lmrbpql u mlo q^kqlbi •kf`l sb`qlo nrb qfbkb i^ afob``fŽk lmrbpq^ ^ pŒjfpjl- Di sb`qlo `bol bp bi•kf`l sb`qlo m^o^ibil ^i sb`qlo `bol-

)*&, :WR_PVPV\`

0- Rb^k > < '0+ 2+ 5(+ ? < '3+ ,2+2(+ X B < '1+ 0+4( qobp sb`qlobp ab S\% Cbqbojfk^o ilp`ljmlkbkqbp ab `^a^ rkl ab ilp sb`qlobp9 ^( > * ?9 _( > + ?9 b( > * ? + B:a( 5> + 0? + 1@9 a( 0> * ? + 1@,

Page 114: Calculus

..+ >gb`]m\ q`^ojmd\g

1- Cf_rg^o ilp sb`qlobp dblj‹qof`lp nrb rkbk bi lofdbk ^ ilp mrkqlp = < '1+0( X A < '0+ 2(-Dk i^ jfpj^ cfdro^+qo^w^obi sb`qlo dblj‹qof`l nrb rkb bi lofdbk ^i mrkql B < > * o?m^o^ `^a^ rkl ab ilp pfdrfbkqbp s^ilobp ab o8o < p: o < : o < : o < 0: o < 1:o < ,0: o < - ,1-

2- Qbplisbo bi bgbo`f`fl 1 pf B < o> * ?, ,3- Rb^k > < '1+ 0(+? < '0+ 2(+ X B < s> * t?* bk alkab s b u plk bp`^i^obp-

^( So^w^o bi sb`qlo nrb rkb bi lofdbk ^ B m^o^ `^a^ rkl ab ilp pfdrfbkqbp m^obpab s^,ilobp ab s b v9 T < V < f: s < z+t < : s < +t < f: s < 1+V < ,0: s < 2+

V < +09s < ,p+s ; 9s < *E)s ;0,_( ƒPr‹ `lkgrkql bp bi ab ilp mrkqlp B l_qbkfalp `r^kal s b u qlj^k qlalp ilp s^il,obp ob^ibp q^ibp nrb s * v < 0> 'G^`bo rk^ `lkgbqro^ v jlpqo^o bi ird^o dblj‹qof`l bki^ cfdro^- Ml e^`bo i^ abjlpqo^`fŽk-(b( C^o rk^ fab^ abi `lkgrkql ab qlalp ilp mrkqlp B l_qbkfalp ^i s^of^o fkabmbkafbkqb,jbkqb s b v bk ilp fkqbos^ilp Nz s x 0+Nz V x 0+X e^`bo rk^ obmobpbkq^`fŽkab bpb`lkgrkql-a( ƒPr‹ `lkgrkql bp bi ab qlalp ilp mrkqlp B l_qbkfalp pf s s^oŒ^ bk bi fkqbos^ilNz s x 0 b v ob`loob qlalp ilp k•jbolp ob^ibp>b( ƒPr‹ `lkgrkql obpriq^ pf s b v ob`loobk ^j_lp qlalp ilp k•jbolp ob^ibp>

4- Rb^k = < '1+ 0( X > < '0+ 2(- Cbjlpqo^o nrb qlal sb`qlo B < %]g

%]0

' ab R0

mrbabbumobp^opbbk i^ cloj^ B < s> * t?, Dumobp^os b v bk crk`fŽk ab `

hv ^

5- Rb^k > < '0+ 0+0(+? < 'N+ 0+0(+X B < '0+ 0+N( qobpsb`qlobp ab U^ v A;s>)t?)u@*alkab s u w plk bp`^i^obp-^( Cbqbojfk^o &ilp `ljmlkbkqbp ab C-_( Rf C < N abjlpqo^o nrb s < v < w< N-`( G^ii^o s* v+ w q^ibp nrb C < '0+ 1+ 2(-

6- Rb^k > < '0+ 0+ 0(+ ? < 'N+ 0+ 0( X B < '1+ 0+ 0( qobp sb`qlobp ab S\% v C < s> )t? * t?) bk alkab r) v+ w plk bp`^i^obp-^( Cbqbojfk^o ilp `ljmlkbkqbp ab C-_( G^ii^o s* v+ v u* kl qlalp krilp+ q^ibp nrb C < N-b( Cbjlpqo^o nrb kfkdrk^ bib``fŽk ab s* v+ w e^`b C < '0+ 1+ 2(-

7- Rb^k = < '0+ 0+ 0+ N(+ > < 'N+ 0+ 0+ 0(+ B < '0+ 0+ N+ N( qobp sb`qlobp ab R2

% vC < s> * t? * wB pfbkal s* v+ w bp`^i^obp-^( Cbqbojfk^o ilp `ljmlkbkqbp ab C-_( Rf C :-) abjlpqo^o nrb s < v < w< N-b( G^ii^o s* v+ w q^ibp nrb C < '0+ 4+ 2+ 3(-a( Cbjlpqo^o nrb kfkdrk^ bib``fŽk ab s* v+ w e^`b C < '0+ 1+ 2+ 3(-

8- Dk S, abjlpqo^o nrb alp sb`qlobp m^o^ibilp ^ rk jfpjl sb`qlo plk m^o^ibilp bkqob pŒ-0/- C^alp `r^qol sb`qlobp kl krilp >* ?* B+ C ab S, q^ibp nrb B < > * ? v > bp m^o^ibil

^ C- Cbjlpqo^o nrb B bp m^o^ibil ^ C pf v pŽil pf ? bp m^o^ibil ^ C-00- ^( Cbjlpqo^o+ m^o^ ilp sb`qlobp ab R+)i^p molmfba^abp ab i^ ^af`fŽk v ab i^ jriqfmif,

`^`fŽk mlo bp`^i^obp a^a^p bk bi qblobj^ 01-0-_( Lbaf^kqb sb`qlobp dblj‹qof`lp bk bi mi^kl+ obmobpbkq^obi pfdkfcf`^al dblj‹qof`l abi^p alp ibvbp afpqof_rqfs^p &` * _'> < ^> * _> v `&> * ?' < ^> * `?,

01- Rf rk `r^aofiŠqbol L>?@ ab S0 bp rk m^o^ibiŽdo^jl nrb qfbkb > v B `ljl s‹oqf`bplmrbpqlp+ abjlpqo^o nrb > )d&@ + >' ; ?, ƒPr‹ qblobj^ obi^qfsl ^ ilp m^o^ibildo^,jlp mrbab abar`fopb ab bpq^ fdr^ia^a>

)*&- C_\QbPa\ R`PNYN_

Hkqolarw`^jlp ^elo^ rk krbsl qfml ab jriqfmif`^`f5k ii^j^a^ molar`ql bp,`^i^o l fkqboflo ab alp sb`qlobp bk S D+

Page 115: Calculus

Mmj_p^oj `n^\g\m "..,

CDEHMHBHˆM- Pd > < &\**,,, *\i' u ? < o]* *,,, *]i' nji _jn q`^ojm`n _`Sh† np kmj_p^oj `n^\g\m n` m`km`n`io\ ^ji > , ? W n` _`adi` ^ji g\ dbp\g_\_

!=$ > < I [e\e{

fxg

@pŒmrbp+m^o^ `^i`ri^o = +> jriqfmif`^jlp ilp `ljmlkbkqbp `loobpmlkafbk,qbpab = v > X prj^jlp irbdl qlalp ilp molar`qlp- Dpq^jriqfmif`^`fŽk qfbkb i^pmolmfba^abp^idb_o^f`^p pfdrfbkqbp-

SDNQDL@ 01-1- M\m\ oj_jn gjn q`^ojm`n >* ?* B _` s+ t oj_jn gjn `n^\g\m`n`* n` od`i`i g\n kmjkd`_\_`n ndbpd`io`n8

'^( =$ > < >$ ='_( > , &? * B( < > 8? * > , B'b( ]%=$ >& < %]=&+> < = +%]>&'a( = += = N nd =8xc7N

'b( =$ = < M nd = < N-A`hjnom\^d‡i, K^p qobpmofjbo^p molmfba^abpplk cŠ`fibp `lkpb`rbk`f^p ab

i^ abcfkf`fŽk v pb abg^k `ljl bgbo`f`flp- O^o^ abjlpqo^o i^p alp •iqfj^p+ rp^jlpi^ obi^`fŽk > , > < -K \x, Orbpql nrb `^a^ q‹ojfkl bp kl kbd^qfsl+ i^ prj^ bpkl kbd^qfs^- @abjŠp+ i^ prj^ bp `bol pf v pŽil&pf `^a^ q‹ojfkl ab i^ prj^ bp`bol v bpql q^k pŽil mrbab l`roofo pf > < N-

Di molar`ql bp`^i^o qfbkb rk^ fkqbomobq^`fŽkdblj‹qof`^ fkqbobp^kqbnrb pbsboŠ bk i^ pb``fŽk 01-8- @kqbpab afp`rqfoi^+kl l_pq^kqb+jbk`flk^jlp rk^ abp,fdr^ia^a fjmloq^kqb obi^qfs^ ^ molar`qlp bp`^i^obpnrb bp crka^jbkq^i bk „idb_o^sb`qlof^i-

&g`t^jihpo\odq\'*&g`t_dnomd]podq\'*&cjhjb`i`d_\_'*&kjndodqd_\_'*

SDNQDL@ 01-2- CDRHFT@KC@C CD B@TBGX,RBGV@QY- Pd> V ? nji q`^ojm`n_` S i* o`i`hjn

'01-1( %=+>&/ w %=+=&%>+>&+

>_`hƒn* `g ndbij _` _`ndbp\g_\_ `n `g qƒgd_j ndt n‡gj ndpij _` gjn q`^ojm`n`n `g kmj_p^oj _`g jomj kjm pi `n^\g\m,

A`hjnom\^d‡i, Dumobp^kal `^a^ &rkl ab ilp alp jfbj_olp ab '01-1( bkcrk`fŽk ab ilp `ljmlkbkqbp+ l_qbkbjlp

nrb bp i^ abpfdr^ia^a v^ abjlpqo^a^ bk bi qblobj^ H-30-

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443 €gb`]m\ q`^ojmd\g

Oobpbkq^objlp lqo^ abjlpqo^`fŽk ab '01-1( nrb kl rqfifw^ ilp `ljmlkbkqbp-S^i abjlpqo^`fŽk bp fkqbobp^kqbmlonrb e^`b sbo nrb i^ abpfdr^ia^a ab B^r`ev,R`et^qw bp rk^ `lkpb`rbk`f^ ab i^p `fk`l molmfba^abpabi molar`ql bp`^i^o nrbpb `fq^k bk bi qblobj^ 01-1 v kl abmbkab ab i^ abcfkf`fŽk nrb pb rqfifwŽm^o^ab,ar`fo bp^p molmfba^abp-

O^o^ iibs^o ^ `^_l bpq^-abjlpqo^`fŽk+ l_pbosbjlp mofjbol nrb '01-1( bpqofsf^i pf = l > bp bi sb`qlo `bol- Olo q^kql+mlabjlp prmlkbo nrb = u > plk^j_lp kl krilp- Rb^ b bi sb`qlo

b < r= * s>) alkab s < >$ > u s:={>+

K^p molmfba^abp a( v b( fjmif`^k nrb @8b z N- Br^kal bumobp^jlp bpql bkcrk`fŽk ab s b u+obpriq^ '01-1(- O^o^ bumobp^o@8 a bk crk`fŽk ab s b u+rqfifw^,jlp i^p molmfba^abp^(+_( u b( l_qbkfbkal

_• a < %r= * s>& +%r= * s>& < r/%= +=& * /rs%=$ >& * s/%> +>& +

Tqfifw^kal i^p abcfkf`flkbp ab s b u v i^ abpfdr^ia^a @8 a y /+ pb iibd^ ^

%>$>&/%=+=& * /%= +>&/%>+>& * %=+>&/%>+>& w N-

K^ molmfba^a a( fjmif`^ nrb >$ > = } mrbpql nrb > :.:-) `lk il nrb mlabjlpafsfafo mlo %>$>& l_qbkfbkal

%>$>&%=+=& * %=+>&/ w N+

nrb `lfk`fab `lk '01-1(- Dpql q^j_f‹k abjrbpqo^ nrb bi pfdkl fdr^i bp sŠifal bk'01-1( pf u pŽil pf a < M+ Obol a < k pf v pŽil pf s> < t?, @ pr sbw+bpq^fdr^i,a^a pb sbofcf`^ pf u pŽil pf rkl ab ilp sb`qlobp bp bi molar`ql abi lqol mlo rkbp`^i^o-

K^ abpfdr^ia^a ab B^r`ev,R`et^ow qfbkb fjmloq^kqbp ^mif`^`flkbp ^ i^p mol,mfba^abp ab i^ gjibdop_ l ijmh\ ab rk sb`qlo+ `lk`bmql nrb bumlkbjlp pbdrfa^,jbkqb-

01-5 Klkdfqra l kloj^ ab rk sb`qlo

K^ cfdro^ 01-6 jrbpqo^ bi sb`qlo dblj‹qof`l nrb rkb bi lofdbk ^i mrkql> < %[f$^+{(bk bi mi^kl- @ m^oqfoabi qblobj^ ab OfqŠdlo^pbk`lkqo^jlp nrb i^ilkdfqra ab > sfbkb a^a^ mlo i^ cŽojri^

ilkdfqra ab = < S\x * \x,

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Ijibdop_ j ijmh\ _` pi q`^ojm 444

>

=

EHFTQ@ 01-6 Bi S0 g\ gjibdop_

_` > `n T \d * \x ,EHFTQ@ 01-7 Bi S!, g\ gjibdop_ _` > `n

U^f * [w * [w+

Dk i^ cfdro^ 01-7 pb jrbpqo^ bi af_rgl `loobpmlkafbkqb bk S\, @mif`^kal biqblobj^ ab OfqŠdlo^p alp sb`bp+ bk`lkqo^jlp nrb i^ ilkdfqra ab rk sb`qlo dbl,j‹qof`l > bk S! sfbkb a^a^ mlo

ilkdfqra ab > < T \d * \x * \x ,

N_p‹osbpb nrb bk rkl r lqol `^pl i^ ilkdfqra ab = sfbkb a^a^ mlo %=+= (0.1+ i^o^Œwr^ao^a^ abi molar`ql bp`^i^o ab > mlo pŒjfpjl- Dpq^ cŽojri^ prdfbob rkj‹qlal m^o^ fkqolar`fo bi `lk`bmql ab ilkdfqra bk SiŠ

CDEHMHBHˆM- Pd > `n pi q`^ojm `i Si* np gjibdop_ j ijmh\ n` _`ndbi\ ^jiGG?GGu n` _`adi` h`_d\io` g\ dbp\g_\_

GG?00 < &> , >'/-0 ,

K^p molmfba^abp crka^jbkq^ibp abi molar`ql bp`^i^o `lkar`bk ^ i^p `loobp,mlkafbkqbp molmfba^abp ab i^ kloj^-

RCMPCK? 01-3- Pd > `n pi q`^ojm _` Si u ` pi `n^\g\m* o`i`hjn g\n nd+bpd`io`n kmjkd`_\_`n8

^( GG?zG= N nd > ;/; N &kjndodqd_\_'*^( h?GG < M nd = < M+a( Gh_?00< G_hhh?h0 &cjhjb`i`d_\_',

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445 €gb`]m\q`^ojmd\g

A`hjnom\^d‡i, K^p molmfba^abp^( v _( plk `lkpb`rbk`f^ fkjbaf^q^ ab i^pmolmfba^abpa( v b( abi qblobj^ 01-1- O^o^ abjlpqo^o b( rqfifw^jlp i^ molmfba^aab eljldbkbfa^a abi molar`ql bp`^i^o l_qbkfbkal

Ef]= 00< %_=$ _=&f,/ < %_/= +=&.,/ < %_/&f /%= +=&.,/ < Ya[ GG?GG-

K^ abpfdr^ia^a ab B^r`ev,R`et^ow q^j_f‹k pb mrbab bumobp^obk crk`f5kab i^ kloj^- Dii^ bpq^_ib`b nrb

'01-2(

Slj^kal i^ o^Œwr^ao^a^ mlpfqfs^ ab `^a^ jfbj_ol+ mlabjlp q^j_f‹k bp`of_foi^ abpfdr^ia^a ab B^r`ev,R`et^ow bk i^ cloj^ bnrfs^ibkqb

'01-3( f=$ @Gy GG?GGph@GG•

Tqfifw^objlp ^elo^ i^ abpfdr^ia^a ab B^r`ev,R`et^ow m^o^abar`fo i^ abpfdr^ia^aqof^kdri^o-

SDNQDL@ 01-4- CDRHFT@KC@C SQH@MFTK@Q- Pd > X ? nji q`^ojm`n _` Si*

o`i`hjn

GG?* @GGy GG?00* GG@GG-

>_`hƒn* `g ndbij dbp\g `n qƒgd_j ndu n‡gj ndpij _` gjn q`^ojm`n `n `g kmj_p^oj _`gjomj kjm pi `n^\g\m,

A`hjnom\^d‡i, O^o^ bsfq^o i^p o^Œ`bp r^ao^a^p+ bp`of_fjlp i^ abpfdr^ia^aqof^kdri^o bk i^ cloj^ bnrfs^ibkqb

'01-4(

Di mofjbo jfbj_ol ab '01-4( bp

jfbkqo^p nrb bi pbdrkal jfbj_ol bp

'GG? G0* GG@GG(1< GG?001* 100? 00YG@GG* GG@G01-

Bljm^o^kal bp^p alp c5ojri^p+ sbjlp nrb '01-4( bp sŠifa^ pf v p5il pf pbqfbkb

'01-5( = +> w GG?0GGG@GG-

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Lmojbji\gd_\_ _` q`^ojm`n 446

Obol =$ > w =$ > `lk il nrb '01-5( obpriq^ ab i^ abpfdr^ia^a ab B^r`ev,R`et^ow+ bk i^ cloj^ '01-3(- Dpql morb_^ nrb i^ abpfdr^ia^a qof^kdri^o bp `lkpb,`rbk`f^ ab i^ abpfdr^ia^a ab B^r`ev,R`et^ow-

K^ molmlpf`fŽk ob`Œmol` q^j_‹k bp `fboq^- Dpql bp+pf i^ abpfdr^ia^a qof^k,dri^o bp `fboq^ q^j_f‹k il bp '01-5( m^o^ > v m^o^ +>* ab il nrb l_qbkbjlp'01-2(- @pŒmrbp+i^ abpfdr^ia^a qof^kdri^o v i^ ab B^r`ev,R`et^ow plk iŽdf`^,jbkqb bnrfs^ibkqbp- @abjŠp+ bi pfdkl ab fdr^ia^a s^ib bk rk^ pf v pŽil pf s^ibbk i^ lqo^+`lk il nrb pb `ljmibq^ i^ abjlpqo^`fŽk abi qblobj^ 01-4-

Dk i^ cfdro^ 01-8 pb obmobpbkq dblj‹qof`^jbkqb i^ abpfdr^ia^a qof^kdri^o-Blk biil pb mrbab ^cfoj^o nrb i^ ilkdfqra ab rk i^al ab rk qofŠkdril kl prmbo^i^ prj^ ab i^p ilkdfqrabp ab ilp lqolp alp i^alp-

01-6 Noqldlk^ifa^a ab sb`qlobp

@ il i^odl ab i^ abjlpqo^`fŽk ab i^ abpfdr^ia^a qof^kdri^o 'qblobj^ 01-4(+pbl_qrsl i^ cŽojri^

'01-6( GG?* @G01< GG?001* GG@GG1* 0> , ?

7%8

,,,,,,,,,,,EEEEEEE

E

,EE>EEEEE+

E++>

EHFTQ@ 01-8 Pdbidad^\_j b`jh„omd^j _`g\ _`ndbp\g_\_ omd\ibpg\m8

GG?* @GG999::GG?0G* GG@GG-

EHFTQ@ 01-0T Ajn q`^ojm`n k`mjk`i_d^pg\m`n n\ode\^`i g\ d_`iod_\_kdo\b‡md^\8

GG?* @G01< GG?001* GG@G01‘

nrb bp sŠifa^ m^o^ alp sb`qlobp `r^ibpnrfbo^ = v > ab R+++K^ cfdro^ 01-0/jrbpqo^ alp sb`qlobp dblj‹qof`lp mbombkaf`ri^obpbk bi mi^kl- Eloj^k rk qofŠk,

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447 >gb`]m\ q`^ojmd\g

dril ob`qŠkdril `rvlp `^qbqlp qfbkbk ilkdfqrabp GG?GGv GG@GGv `rv^ efmlqbkrp^ qfbkbilkdfqra G?* @GG-Di qblobj^ ab OfqŠdlo^pbpq^_ib`b nrb

GG?* >E./ < GG?001 * GG@G01-

Bljm^o^kal bpqbobpriq^al `lk '01-6(+ sbjlp nrb = +> < N-Cf`el ab lqol jlal+bi molar`ql bp`^i^o ab alp sb`qlobp mbombkaf`ri^obpabi mi^kl bp `bol- Dpq^mol,mfba^a a^ lofdbk ^ i^ abcfkf`fŽk ab sb`qlobp mbombkaf`ri^obpbk Ri,

CDEHMHBHˆM- Ajn q`^ojm`n > v ? _` Si nji k`mk`i_d^pg\m`n p jmojbji\g`n nd=$> < N-

K^ fdr^ia^a '01-6( jrbpqo^ nrb alp sb`qlobp = v > ab Rh plk loqldlk^ibppf u pŽil pf GG?* >S < GG?U* EE>S+ Dpq^bp i^ fabkqfa^a ab OfqŠdlo^pbk Sh†

)*&0 :WR_PVPV\`

0- Rb^k > < '0+ 1+ 2+ 3(+ 7 < ',0+ 1+ ,2+ N( X b < 'N+ 0+N+ 0( qobp sb`qlobp ab S2Š

B^i`ri^o `^a^ rkl ab ilp pfdrfbkqbp molar`qlp9'^( = +>8 '_( >$ B: '`( = +B: 'a( = l %> * ?&8 'b( %= * >& +B-

1- C^alp qobp sb`qlobp > < '1+ 3+ ,6(+ 7 < '1+ 5+ 2(+X b < '2+ 3+ ,4(- Dk `^a^ rk^ abi^p bumobpflkbp pfdrfbkqbp pb mrbabk fkqolar`fo m^o‹kqbpfp ab rk^ pli^ j^kbo^ m^o^ l_qb,kbo rk^ bumobpfŽk nrb qbkd^ pbkqfal- Hkqolar`fo af`elp m^o‹kqbpfp v bcb`qr^o i^p lmb,o^`flkbp-

'^( = +>?8 '_( = +> * B: 'b( = * > l B: 'a( =>{ B: 'b( =F> l B-

2- Cbjlpqo^o pf bp l kl `fboq^ i^ molmlpf`fŽk pfdrfbkqb obcbobkqb ^ sb`qlobp bk Si8 Rf> /7< >} a u > m %.*bp 7 < B-

3- Cbjlpqo^o pf bp l kl `fboq^ i^ molmlpf`fŽk pfdrfbkqb nrb pb obcfbob ^ sb`qlobp bk Si8

Rf > l 7 m^o^ qlal 7+ bp > ;.,4- Rf = < '1+0+,0( X 7 < '0+,0+1(+ e^ii^o rk sb`qlo kl kril B ab R) q^i nrb = l B <

< 7 lB < N-5- Rf > < '0+,1+2( X 7 < '2+0+1(+e^ii^o ilp bp`^i^obp s b v q^ibp nrb b < s> * v7 bp rk

sb`qlo kl kril u nrb B /7 < N-6- Rf > < '1+ ,0+ 1( X 7 < '0+ 1+ ,1(+ e^ii^o alp sb`qlobp b v C ab S nrb p^qfpc^d^k

qla^p i^p `lkaf`flkbp pfdrfbkqbp9 > < b * C+ 7 C < N+B m^o^ibil ^ >+7- Rf = < '0+ 1+ 2+ 3+ 4( X 7 < %.)c*d* *x*'*e^ii^o alp sb`qlobp a u C ab R nrb

p^qfpc^d^k qla^p i^p `lkaf`flkbp pfdrfbkqbp9 7 < b * /@) C• = < N+ b m^o^ibil 4^ =+8- Rb^k > < '1+,0+ 4(+ 7 < ',0+ ,1+ 2(+ X b < '0+,0+ 0( qobp sb`qlobp ab S l B^i`ri^o

i^ kloj^ ab `^a^ rkl ab ilp pfdrfbkqbp sb`qlobp9 ^'^( = * >8 '_( = * >8 'b( = * > * B: 'a( = * > * `-

0/- Dk `^a^ `^pl e^ii^o rk sb`qlo 7 ab S0 q^i nrb, 7 - > < N X 00700< GG?GGpf9

'^( = < '0+0(: '^( = < '0+,0(: '`( = < '1+,2(: '`( = < %[) \&+

00- Rb^k > < '0+ ,1+ 2( X 7 < '2+ 0+1( alp sb`qlobp ab S\% Dk `^a^ `^pl+ e^ii^o rk sb`,qlo B ab ilkdfqra 0 m^o^ibil ^9'^( = * >8 '_( = * >8 'b( = * />8 'a( = * />8 'b( /= * >+

01- C^alp ilp sb`qlobp ab R1% = < '3+ 0+ ,2(+ 7 < '0+ 1+ 1(+ B < N+ 1+ ,1(+ @ < '1+0+1(+X B <'1+ ,1+ ,0(- Cbqbojfk^o qlalp ilp m^obp loqldlk^ibp-

02- G^ii^o qlalp ilp sb`qlobp ab S 1 nrb qfbkbk i^ jfpj^ ilkdfqra nrb > v ib plk loqldlk^ibp pf9

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Mmjt`^^dji`n, €ibpgj _` _jn q`^ojm`n `i `g `nk\^dj _` i _dh`indji`n 337

'^( > <'0+1(: '^( > <'0+ ,1(: '`( > <'1+ ,0(: '`( > <',1+0(-

03- Rf > < '1+,0+0( X ? < '2+,3+ ,3(+ e^ii^o rk mrkql b abi bpm^`fl ab 2 afjbkpflkbpq^i nrb =) >) X b plk ilp s‹oqf`bp ab rk qofŠkdril ob`qŠkdril-

04- Rf = < '0+ ,0+1( X > < '1+ 0+,0(+e^ii^o rk- sb`qlo kl kril a ab R) loqldlk^i ^ = u ^ >+05- Rb^k = < '0+ 1( X > < '2+ 3( alp sb`qlobp ab R0Š G^ii^o ilp sb`qlobp L u O ab R

0q^ibp nrb = < L * P+ pfbkal O m^o^ibil ^ >) v P loqldlk^i ^ >+

06- Qbplisbo bi bgbo`f`fl 05 pf ilp sb`qlobp mboqbkb`bk ^ R+$pfbkal = < '0+ 1+ 2+ 3( X? < '0+ 0+ 0+ 0(-

07- C^alp ilp sb`qlobp > < '1+ ,0+ 0(+? < '0+1+,0(+ X b < '0+0+,1( ab S1Š G^ii^o ilp

sb`qlobp C ab i^ cloj^ s? * uA loqldlk^ibp ^ > u ab ilkdfqra 0-08- Cbjlpqo^o nrb m^o^ alp sb`qlobp = v > ab R) pb qfbkb i^ fabkqfa^a

GG?*@001 ,GG? ,@001 ;2> %?*

X mlo q^kql >} ? < N pf v pŽil pf GG?* @GG< GG?, @GG-Hkqbomobq^obpqb obpriq^al dbl,j‹qof`^jbkqb bk S 1: i^p af^dlk^ibp ab rk m^o^ibildo^jl plk fdr^ibp pf v pŽil pf bi m^o^,ibildo^jl bp rk ob`qŠkdril-

1/- Cbjlpqo^o nrb m^o^ alp sb`qlobp `r^ibpnrfbo^ > v ? ab S* pb qfbkb

G?* @G01* GG?, @G01< 1 GG?001* 1 GG@G01‘

ƒPr‹ qblobj^ dblj‹qof`l ^`bo`^ ab ilp i^alp v af^dlk^ibp ab rk m^o^ibildo^jl pbmrbab abar`fo ab bp^ fabkqfa^a>

10- Di qblobj^ dblj‹qof`l nrb pfdrb prdfbob rk^ fabkqfa^a sb`qlof^i obi^qfs^ ^ qobp sb`qlobp=) > X B- Cb`fo `rŠi bp u abjlpqo^o nrb bp sŠifa^ m^o^ ilp sb`qlobp ab S*, S^i fabkqfa^amolmlo`flk^ rk^ abjlpqo^`fŽk abi qblobj^ `lk j‹qlalp sb`qlof^ibp-

~K^ prj^ ab ilp `r^ao^alp ab ilp i^alp ab rk `r^aofiŠqbol `r^inrfbo^ prmbo^ ^ i^prj^ ab ilp `r^ao^alp ab i^p af^dlk^ibp bk `r^qol sb`bp bi `r^ao^al ab i^ ilkdfqraabi pbdjbkql ob`qfiŒkbl nrb rkb ilp mrkqlp jbaflp ab i^p af^dlk^ibp-‚

11- Tk sb`qlo = ab R) qfbkb ilkdfqra 5- Tk sb`qlo > ab Ri qfbkb i^ molmfba^a ab nrb m^o^qlal m^o ab bp`^i^obp s b v ilp sb`qlobp s>9+ t? X 2t> + 7s? plk loqldlk^ibp-B^i`ri^o i^p ilkdfqrabp ab > u ab /= * 0>+

12- C^alp alp sb`qlobp > < '0+1+2+3+ 4( X ? < /+ h+H+h+p( ab Sj% G^ii^o alp sb`qlobpB v C nrb p^qfpc^d^k i^p qobp `lkaf`flkbp pfdrfbkqbp9 B bp m^o^ibil ^ =) C bp loqldlk^i^ =) v > < B * C-

13- C^alp bk R) alp sb`qlobp = v > kl krilp v kl m^o^ibilp+ abjlpqo^o nrb bufpqbk sb`ql,obp B v C bk R) nrb p^qfpc^`bk i^p qobp`lkaf`flkbp abi bgbo`f`fl 12 v bumobp^o B v C bkcrk`fŽk ab = u >+

14- Cbjlpqo^o pf bp l kl `fboq^ `^a^ rk^ ab i^p molmlpf`flkbp pfdrfbkqbp obi^qfs^p ^ sb`,qlobp bk S*8^( Rf > bp loqldlk^i ^ >) GG?* r>.. w GG?GGm^o^ qlal k•jbol ob^i r+_( Rf GG?(r>EE w GG?GGm^o^ qlal k•jbol ob^i r) = bp loqldlk^i ^ >+

)*&1 C_\eRPPV\[R`&h[TbY\ QRQ\` cRPa\_R` R[ RYR`]NPV\ QR i QVZR[`V\[R`

Di molar`ql bp`^i^o ab alp sb`qlobp bk S0 qfbkb rk^ fkqbomobq^`fŽk dblj‹qof`^fkqbobp^kqb- K^ cfdro^ 01-00 ^( jrbpqo^ alp sb`qlobp dblj‹qof`lp kl krilp = v >nrb cloj^k rk Škdril %d+Dk bpqb bgbjmil+ qbkbjlp N ; %d; %5Q, K^ cfdro^01-00 _( jrbpqo^ bi jfpjl sb`qlo = u alp sb`qlobp mbombkaf`ri^obp `rv^ prj^bp >, Tkl ab biilp+ o?* bp bi molar`ql ab ? mlo rk bp`^i^o nrb ii^j^jlp i^ kmj+t`^^d‡i _` > nj]m` ?, Dk bpqb bgbjmil+ o bp mlpfqfsl v^ nrb N ; %d; e6S-

Page 122: Calculus

45/

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f> <molvb``f5kab@ pl_obA

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Olabjlp rqfifw^o ilp molar`qlp bp`^i^obp m^o^ bumobp^o o bk crk`fŽk ab = v?, Olkbjlp mofjbol o? 9-z--b < > X qlj^o irbdl bi molar`ql bp`^i^o ab `^a^

jfbj_ol mlo ? l_qbkfbkal

o? , ? * @8? < > , ? ,

Obol @8? < N+ ab_fal ^ nrb b pb af_rgŽ mbombkaf`ri^o ^ ?, Olo `lkpfdrfbkqbn‚ 7> < = +>) `lk il nrb qbkbjlp

'01-7( =$> =$>n:**:**)>$ > EE>EE0

Olo lqo^ m^oqb+bi bp`^i^o o lofdfk^ rk^ pbk`fii^ obi^`fŽk m^o^ bi Škdril K+ Cb i^cfdro^ 01-00 _(+ sbjlp nrb

`lp ` < 00 o?// < qI \GG?GG GG?G &

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'01-8(

l

=$>`lp b< ,,,

GG?GGGG@GG

> , ? < GG?00GG@GG`lp `,

Cf`el ab lqol jlal+ bi molar`ql bp`^i^o ;ib alp sb`qlobp kl krilp = v > ab R0

bp fdr^i ^i molar`ql ab qobp k•jbolp9 i^ ilkdfqra ab =) i^ ab >) v bi `lpbkl abiŠkdril nrb cloj^k-

Page 123: Calculus

Ijn q`^ojm`n ^jjm_`i\_jn pido\mdjn 450

K^ fdr^ia^a '01-8( prdfbob rk^ j^kbo^ ab abcfkfo bi `lk`bmql ab Škdril bkRiŠ K^ abpfdr^ia^a ab B^r`ev,R`et^ow+ bumobp^a^bk i^ cloj^ '01-3(+ jrbpqo^nrb bi `l`fbkqb abi pbdrkal jfbj_ol ab '01-8( qfbkb s^ilo ^_plirql z 0 m^o^alp sb`qlobp `r^ibpnrfbo^ ab S i, N ab lqol jlal+ qbkbjlp

=$>,0 ; ,, ; 0-,GG?GGGG@GG,

Olo il q^kql+bufpqbrk plil k•jbol ob^i %dbk bi fkqbos^il N z %dw 6R q^i nrb '01-8(bp `fboq^-Cbcfkfjlp bi Škdril bkqob= v > `ljl bpb k•jbol %d+K^ afp`rpfŽk ^kqb,oflo pb obprjb bk i^ abcfkf`fŽk pfdrfbkqb-

CDEHMHBHˆM- P`\i > v ? _jn q`^ojm`n _` Si* ^ji ? ;/; N- Bg q`^ojm o?*nd`i_j

=$>n:**

>$>$

n` gg\h\ g\ kmjt`^^d‡i _` > nj]m` ?, Pd > X ? nji \h]jn ij ipgjn* `g ƒibpgj %d

lp` ajmh\i n` _`adi` h`_d\io` g\ dbp\g_\_,

=$>%d< ^o``lp ,,,

GG?GGGG@GG

Kjo\8 K^ crk`fŽk ^ob `lpbkl obpqofkdb '( ^i fkqbos^il M 994'x994 /Q, N_p‹osbpb q^j,_f‹k nrb '( < q0S `r^kal ={ > < N-

01-0/ Klp sb`qlobp `lloabk^alp rkfq^oflp

Dk bi `^mŒqril8 ^mobkafjlp nrb qlal k•jbol `ljmibgl %[)\& mrbab bumob,p^opbbk i^ cloj^ [ * ]d*bk alkab d obmobpbkqbi k•jbol `ljmibgl 'N+ 0(- @kŠ,ild^jbkqb+ qlal sb`qlo %[) \& ab R0 mrbab bumobp^opbbk i^ cloj^

%[)\& < [%f) N( * \%K) 0( -

Klp alp sb`qlobp '0+ N( X 'N+ 0( nrb jriqfmif`^k ilp `ljmlkbkqbp \ v ] pb abkljf,k^k q`^ojm`n ^jjm_`i\_jn pido\mdjn,Hkqolar`fjlp ^elo^ bi `lk`bmql ^kŠildl bk SiŠ

CDEHMHBHˆM- Bi Si* gjn i q`^ojm`n Bg< '0+ N +--- +N(+ B0 < 'N+ 0+ N+ --- +N(+,,, *Bi < 'N+ N+ -- N+ 0( n` gg\h\i q`^ojm`n ^jjm_`i\_jn pido\mdjn, Bg f+„ndhj^jhkji`io` _` B9 `n 0 u oj_jn gjn _`hƒn ^jhkji`io`n nji ^`mj,

Page 124: Calculus

451 >gb`]m\ q`^ojmd\g

K^ abkljfk^`fŽk ab ~sb`qlo rkfq^ofl‚ mol`bab abi eb`el ab nrb `^a^sb`qlo Bf qfbkb ilkdfqra 0- N_p‹osbpb nrb bplp sb`qlobp plk loqldlk^ibp bkqob pŒ+bpql bp+ bi molar`ql bp`^i^o ab alp `r^ibpnrfbo^ ab biilp bp `bol+

Bf Š B* < M pf f x d+

RCMPCK? 01-5- Qj_j q`^ojm W < 'u-+ --- + si' _` Si kp`_` `skm`n\mn` `ig\ ajmh\

i

W < t0C0 * --- * rhAh <GreAe{FRG

>_`hƒn* `no\ m`km`n`io\^d‡i `n ˆid^\, Bnoj `n* nd

vi

`ioji^`n s* < Ue k\m\ ^\_\ q\gjm f < 0+1+ --- + i,

A`hjnom\^d‡i, K^ mofjbo^ molmlpf`fŽk obpriq^ fkjbaf^q^jbkqb ab i^ abcf,kf`fŽk ab ^af`fŽk v jriqfmif`^`fŽk mlo bp`^i^obp- K^ rkf`fa^a bp `lkpb`rbk`f^ abi^ abcfkf`fŽk ab fdr^ia^a ab sb`qlobp-

Tk^ prj^ abi qfml Gy>y pb ii^j^ ^jh]di\^d‡i gdi`\g ab ilp sb`qlobp=f) +++)=iŠ Di qblobj^ 01-5 klp af`b nrb qlal sb`qlo ab Ri mrbab bumobp^opb`ljl `lj_fk^`fŽk ifkb^i ab ilp sb`qlobp `lloabk^alp rkfq^oflp- Dumobp^jlp bpqlaf`fbkal nrb ilp sb`qlobp `lloabk^alp rkfq^oflp Ch=--- +Bi dbkbo^k bi bpm^`fl Si,S^j_f‹k ab`fjlp nrb dbkbo^k S i `lk pid^d_\_ mlonrb i^ obmobpbkq^`fŽk ab rksb`qlo `ljl `lj_fk^`fŽk ifkb^i ab Af) +++) Bi bp •kf`^- Nqolp `lkgrkqlp ab sb`,qlobp afpqfkqlp ab ilp Bg* ,,, *Bi q^j_f‹k dbkbo^k S i `lk rkf`fa^a+ v bk i^ pb`,`fŽk 01-01 bpqraf^objlp q^ibp `lkgrkqlp-

Dk R0 ilp sb`qlobp `lloabk^alp rkfq^oflp Af v A0 cob`rbkqbjbkqb pb abpfd,k^k+ obpmb`qfs^jbkqb+ `lk ilp pŒj_lilp d u gbk kbdofq^ `ropfs^- Dk U^ pb rqfifw^kilp pŒj_lilp o,dv f bk ird^o ab Af) B0* B\, @idrk^p sb`bp pb `lil`^ rk^ cib`e^ lrk^ _^oo^ bk`fj^ abi pŒj_lil+ mlo bgbjmil . l g, Di pfdkfcf`^al dblj‹qof`labi qblobj^ 01-5 bpqŠ obmobpbkq^al bk i^ cfdro^ 01-01 m^o^ i < 2-

Br^kal ilp sb`qlobp pb bumobp^k `ljl `lj_fk^`flkbp ifkb^ibp ab ilp sb`,qlobp `lloabk^alp rkfq^oflp+ ilp `Ši`rilp ^idb_o^f`lp `lk ^nrbiilp mrbabk bcb`qr^o,pb `lk i^p prj^p GreAe ab ^`rboal `lk i^p obdi^p rpr^ibp abi „idb_o^- Blkpf,abo^kal ilp `lbcf`fbkqbp ab ilp sb`qlobp `lloabk^alp rkfq^oflp+ mrbabk l_qbkbopbilp `ljmlkbkqbp bk `r^inrfbo jljbkql abi `Ši`ril- Olo bgbjmil+ m^o^ prj^oalp sb`qlobp+ > < %[f; +++ ) [i' u ? < &]p% ,, * ]i'* bp`of_fjlp

Page 125: Calculus

Be`m^d^djn 452

EHFTQ@ 01-01 Ri q`^ojm @ _` U^ `skm`n\_j ^jhj ^jh]di\^d‡i gdi`\g _` c)d)f,

u ^mif`^jlp i^ molmfba^a ab ifkb^ifa^a ab prj^p cfkfq^p l_qbkfbkal

i i i

> * ? < G\!B! * G]!B! < G&\! * ]!'B** ,!<0 !z0 !z0

Di `lbcf`fbkqb ab B! bk bi pbdrkal jfbj_ol bp bi h,‹pfjl `ljmlkbkqb ab i^ prj^=+ * >+

)*&)) :WR_PVPV\`

0- Cbqbojfk^o i^ molvb``fŽk ab @ pl_ob ? pf @ < 'i+ 1+ 2( X ? < 'i+ 1+ 1(-1- Cbqbojfk^o i^ molvb``fŽk ab @ pl_ob ? pf @ < '3+ 2+ 1+ 0( X ? < '0+0+0+0(-2- ^( Rb^k @ < '5+ 2+ ,1( X \* ]* ^ ilp Škdrilp nrb @ cloj^ `lk ilp sb`qlobp `lloabk^alp

rkfq^oflp c)d)e) obpmb`qfs^jbkqb- B^i`ri^o `lp \* `lp \ v `lp ^, Dpqlp pb ii^j^k ilp`lpbklp afob`qlobp ab @- _( G^ii^o qlalp ilp sb`qlobp ab Rx ab ilkdfqra 0 m^o^ibilp ^ =+

3- Cbjlpqo^o nrb bi Škdril nrb cloj^k @ < '0+ 1+ 0( X N < '1+ 0+ ,0( bp bi al_ib abi nrbcloj^k b < '0+3+ 0( X @ < '1+ 4+ 4(-

4- Cbqbojfk^o sb`qlof^i jbkqb ilp `lpbklp ab ilp Škdrilp abi qofŠkdril bk bi bpm^al ab2 afjbkpflkbp `rvlp s‹oqf`bp plk ilp mrkqlp '1+ ,0+ 0(+'0+ ,2+ ,4(+ X '2+ ,3+ ,3(-

5- Sobp sb`qlobp >* O* b ab U^ p^qfpc^`bk i^p molmfba^abp pfdrfbkqbp9

eh?GG< GGAGG< 4+ he@h0< 0+ GG?, ? * AGG< GG?* ? * AGG-

Rf bi Škdril nrb cloj^k = u > bp +!.7! e^ii^o bi nrb cloj^k > u B-

Page 126: Calculus

*+) >gb`]m\ q`^ojmd\g

6- C^alp qobpsb`qlobp kl krilp =) >) b ab R!+ RrmŽkd^pb nrb bi Škdril nrb cloj^k = v bbp fdr^i ^i nrb cloj^k > v B- Cbjlpqo^o nrb b bp loqldlk^i ^i sb`qlo GG@GG?, GG?GG@-

7- Rb^ '( bi Škdril nrb cloj^k ilp alp sb`qlobp pfdrfbkqbp ab S Š,8 > < '0+ 0+ --- + 0( v? < '0+ 1+ --- + h&+G^ii^o bi s^ilo iŒjfqb ab '( `r^kal h$*( ]] +

8- Qbplisbo bi bgbo`f`fl 7 pf = < '1+ 3+ 5+ --- + /h& v > < '0+ 2+ 4+ --- + /h * 0(+0/- C^alp alp sb`qlobp = < '`lp '(+ ,pbk '( v > < 'pbk '(+mlp K& ab R0Š

^( Cbjlpqo^o nrb = v > plk sb`qlobp loqldlk^ibp ab ilkdfqra 0- G^`bo rk af_rgl bkbi nrb = v > clojbk rk Škdril %F < 6S.5-_( G^ii^o qlalp ilp sb`qlobp %r)s& ab R0 q^ibp nrb %r)s& :r= * s>+ @pbdro^opb abnrb pb `lkpfabo^k qlalp ilp mlpf_ibp s^ilobp ab '(-

00- Cbjlpqo^o sb`qlof^ijbkqb nrb i^p af^dlk^ibp ab rk olj_l plk mbombkaf`ri^obp-01- Eloj^kal bi molar`ql bp`^i^o ab ilp alp sb`qlobp '`lp [) pbk [& v '`lp \) pbk \&) abar`fo

i^ fabkqfa^a qofdlklj‹qof`^ `lp %[ * \& < `lp [ `lp \ * pbk [ pbk \+

02- Rf %F bp bi Škdril nrb cloj^k ilp sb`qlobp kl krilp = v > ab R!$ abjlpqo^o nrb

GG?, >EE0 < EE=001* GG@G01, 1 GG?GGGG@GG`lp %F+

Hkqbomobq^aldblj‹qof`^jbkqb bk S0* bpqb bp bi qblobj^ abi `lpbkl ab i^ SofdlkljbqoŒ^-03- Rrmlkd^jlp nrb bk ird^o ab abcfkfo bi molar`ql bp`^i^o ab alp sb`qlobp > < %[! $†† ) [))&

v > < %\! +++ +\ ††& mlo i^ cŽojri^ ={> < 09:<0[e\e) rp^jlp i^ abcfkf`fŽk pfdrfbkqb9

h

> , ? < 09g\f]fY ,F5G

ƒBrŠibp ab i^p molmfba^abp abi qblobj^ 01-1 plk sŠifa^p `lk bpq^ abcfkf`fŽk> ƒDp sŠ,ifa^ i^ abpfdr^ia^a ab B^r`ev,R`et^ow>

04- Rrmlkd^jlp nrb bk S/ abcfkfjlp bi molar`ql bp`^i^o ab alp sb`qlobp > < %[f$ [/& v? < '_

h+]0' `lk i^ cŽojri^

Cbjlpqo^o nrb plk sŠifa^p qla^p i^p molmfba^abp abi qblobj^ 01-1 `lk bpq^ abcfkf`f5k-ƒDp sŠifa^ i^ abpfdr^ia^a ab B^r`ev,R`et^ow>

05- Qbplisbo bi bgbo`f`fl 04 pf bi molar`ql bp`^i^o ab alp sb`qlobp = < %[! [0* [[& v? < '_

!]0* ]\' ab S\ pb abcfkb jbaf^kqb i^ cŽojri^ >} ? < 1^i_i * \0]0 * ^^_^

* ^i_^ * ^^_i -06- Rrmlkd^jlp nrb bk ird^o ab abcfkfo i^ kloj^ ab rk sb`qlo > < %[ ) +++ ) [))& `lk i^

c5ojri^ &>}@(i.z rp^jlp i^ abcfkf`fŽk pfdrfbkqb9

^( Cbjlpqo^o nrb bpq^ abcfkf`fŽk ab kloj^ p^qfpc^`b qla^p i^p molmfba^abp ab ilp qbl,obj^p 01-3 v 01-4-_( Tp^o bpq^ abcfkf`fŽk bk S0 u obmobpbkq^obk rk^ cfdro^ bi `lkgrkql ab qlalp ilpmrkqlp 'u+ u( ab kloj^ 0-

Page 127: Calculus

Biqjgq`io` gdi`\g _` pi ^jiepioj adidoj _` q`^ojm`n 454

b( ƒBrŠibp ab i^p molmfba^abp ab ilp qblobj^p 01-3 v 01-4 pboŒ^ksŠifa^p pf rpŠo^jlp i^abcfkf`fŽk

07- Rrmlkd^jlp nrb i^ kloj^ ab rk sb`qlo > < %[f$+++)[i' pb abcfkfbo^ `lk i^ cŽojri^

GG?GG< j^u F[ef)f9e9h

alkab bi pŒj_lil abi pbdrkal jfbj_ol obmobpbkq^ bi jŠufjl ab ilp i k•jbolpH]hh+g\0/* ,,, * g\ig}^( ƒBrŠibp ab i^p molmfba^abp ab ilp qblobj^p 01-3 u 01-4 plk sŠifa^p `lk bp^ abcfkf`fŽk>_( Tp^o bp^ abcfkf`fŽk ab kloj^ bk S 1 X obmobpbkq^obk rk^ cfdro^ bi `lkgrkql abqlalp ilp mrkqlp %r)u( ab kloj^ 0-

08- Rf > < %[f$ +++† [i' bp, rk sb`qlo ab S Š, abcfkfo alp kloj^p abi jlal pfdrfbkqb9

i

GG?000< Y g\fgfx/

v GG?001 < j^u g\fg,f98e98h

Cbjlpqo^o nrb GG?G01z GG?GGy GG?GGH‘Hkqbomobq^odblj‹qof`^jbkqb bpq^ abpfdr^ia^a bkbi mi^kl-

1/- Rf = X > plk alp mrkqlp bk bi bpm^`fl ab i afjbkpflkbp+ i^ afpq^k`f^ ab = ^ > pb abpfd,k^ `lk ^%=) >& X pb abcfkb `lk i^ fdr^ia^a ^%=) >& < GG?, >EE+Cbjlpqo^o nrb i^ afpq^k,`f^ qfbkb i^p molmfba^abp pfdrfbkqbp9

'^( ^%=)>& < ^%>)=&+ '_( ^%=)>& < N pf u p50/ pf = < >+'b( ^%=)@(y ^%=)A( * ^%?)>&+

)*&)* :[c\YcR[aR YV[RNYQRb[ P\[Wb[a\ SV[Va\QRcRPa\_R`

Rb^ R < u=x) +++) @a WQ`lkgrkql kl s^`Œl nrb `lkpq^ ab e sb`qlobp ab Rh)

alkab f* bi k•jbol ab sb`qlobp+mrbab pbojbklo+ fdr^il j^vlo nrb i* i^ afjbkpfŽkabi bpm^`fl- Rf rk sb`qlo Wab Si mrbab obmobpbkq^opb`ljl rk^ `lj_fk^`fŽk ifkb^iab @i+ --- + =e) mlo bgbjmil

f

U < u^d>d*'4)

pb af`b nrb R b`i`m\ ^i sb`qlo W-

CDEHMHBHˆM- Bg ^jiepioj _` aj_jn gjn q`^ojm`n b`i`m\_jn kjm R n` _`ijhdi\`iqjgq`io` gdi`\g _` R v n` _`ndbi\ kjm I&P',

Page 128: Calculus

.// >gb`]m\ q`^ojmd\g

Cf`el ab lqol jlal+ i^ bkslisbkqb ifkb^i ab P bp pfjmibjbkqb bi `lkgrkql abqla^p i^p mlpf_ibp`lj_fk^`flkbp ifkb^ibp ab sb`qlobp bk P, N_p‹osbpb nrb i^p `lj,_fk^`flkbp ifkb^ibp ab sb`qlobp ab I&P' mboqbkb`bkq^j_f‹k ^ I&P', Cb`fjlp nrbP b`i`m\ oj_j `g `nk\^dj Si pf I&P' < SiŠ

CHCKNJM 0- Rb^ P < x@iy&Dkqlk`bp I&P' `lkpq^ ab qlalp ilp molar`qlp ab=f mlo bp`^i^obp-

CHCKNJM 1- Slal `lkgrkql P < x@i+--- -@a dbkbo^ bi sb`qlo kril v^ nrbN < L>y * --- )L>e† Dpq^ obmobpbkq^`fŽk+bk i^ nrb qlalp ilp `lbcf`fbkqbpBi&-‘‘ +?e plk `bol+ pb ii^j^ m`km`n`io\^d‡i omdqd\gabi sb`qlo kril- Rfk bj_^odl+mrbabk bufpqfo lj_fk^`flkbp ifkb^ibp kl qofsf^ibpnrb obmobpbkqbkN- Olo bgbjmil+prmlkd^jlp nrb rkl ab ilp sb`qlobp ab P bp bi molar`ql ab lqol mlo rk bp`^i^o+pb^ >0 < 0>y, Sbkbjlp bkqlk`bp jr`e^p obmobpbkq^`flkbpkl qofsf^ibpab N+ mlobgbjmil

l < 0o>. * o>/ * L>0 * --- * L>E_)

pfbkal o `r^inrfbo bp`^i^o kl kril-Mlp fkqbobp^kbk bpmb`f^i ilp `lkgrkqlp P nrb dbkbo^k ilp sb`qlobp ab rk^

pli^ j^kbo^-

CDEHM(BHˆM- Ri ^jiepioj P < u=b$ ++) @a _` q`^ojm`n _` Sh b`i`m\ \ V^ji pid^d_\_ ndP b`i`m\ \ V t nd

'01-0/(ib

W < I^d>dcw.

tib

T < I_d>d8.:

dhkgd^\ `* < _d k\m\ oj_j f -

Dk i^p alp prj^p nrb ^m^ob`bkbk '01-0/(+ pb pl_obbkqfbkab nrb ilp sb`qlobp@e --- +>f bpqŠkbp`ofqlpbk bi jfpjl loabk ^pŒljl q^j_f‹k nrb i^ fjmif`^`fŽk'01-0/( bp sŠifa^ m^o^ rk^ loabk^`fŽk mobcfg^a mbol ^o_fqo^of^ab ilp sb`qlobp>c ,,, *>f,

RCMPCK? 01-6- Ri ^jiepioj P b`i`m\ pi q`^ojm ^p\glpd`m\ _` I&P' ^ji pid+^d_\_ ndv n‡gj ndP b`i`m\ ^ji pid^d_\_ `g q`^ojm ^`mj,

A`hjnom\^d‡i, Rf P dbkbo^ `r^inrfbo sb`qlo ab I&P' `lk rkf`fa^a+ bsfabk,qbjbkqb dbkbo^ N `lk rkf`fa^a- O^o^ abjlpqo^o bi ob`Œmol`l+prmlkd^jlp nrb Rdbkbo^ N `lk rkf`fa^a v bifg^jlp `r^inrfbo sb`qlo W ab H%O&+Rrmlkd^jlp nrbR dbkbo^ W ab alp j^kbo^p+ mlo bgbjmil

ib

U < I^d>dc:.

uib

V < I_d>d,c:.

Page 129: Calculus

Fi_`k`i_`i^d\ gdi`\g 456

Qbpq^kal+ bk`lkqo^jlp nrb N < 1zz0 @@d9+ _d'>d , Obol mrbpql nrb R dbkbo^ N `lkrkf`fa^a+ ab_b pbo @d+ _d < M m^o^ qlal c)^pŒnrb R dbkbo^ W `lk rkf`fa^a-

)*&)+ =[QR]R[QR[PVNYV[RNY

Di qblobj^ 01-6 abjrbpqo^ i^ fjmloq^k`f^ ab ilp `lkgrkqlp nrb dbkbo^k `lkrkf`fa^a bi sb`qlo `bol- S^ibp `lkgrkqlp pb afpqfkdrbk `lk rk klj_ob bpmb`f^i-

CDEHMHBHˆM- Ri ^jiepioj R < v> * ,,, * @a lp` `ib`i_m\ ^ji pid^d_\_ `gq`^ojm ^`mj n` _`ijhdi\ ^jiepioj _` q`^ojm`n gdi`\gh`io` di_`k`i_d`io`, A` ij n`m\n…R `n pi ^jiepioj gdi`\gh`io` _`k`i_d`io`,

Cf`el ab lqol jlal+ i^ di_`k`i_`i^d\ pfdkfcf`^ nrb R bkdbkao^ N •kf`^,jbkqb `lk i^ obmobpbkq^`fŽk qofsf^i9

f

1 ^d>d < M fjmif`^ qlal `• < M -f<i

K^ _`k`i_`i^d\ pfdkfcf`^ nrb R bkdbkao^ N bk ^idrk^ cloj^ kl qofsf^i- Dpql bp+

m^o^ rklp `fboqlp bp`^i^obp b! --- + `n* qbkbjlp

f

1 ^d>9< N mbol kl qlal `* bp `bol-c:f

Rf _fbk i^ abmbkabk`f^ b fkabmbkabk`f^ plk molmfba^abp ab ilp ^jiepiojn absb`qlobp+ bp `loofbkqb ^mif`^o q^j_f‹k bp^p abkljfk^`flkbp ^ ilp jfpjlp sb`qlobp-Olo bgbjmil+ ilp sb`qlobp ab rk `lkgrkql ifkb^ijbkqb fkabmbkafbkqb pb ii^j^k ^jbkral sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp- Blksbkfjlp q^j_f‹k bk ii^j^o `lk,grkql ifkb^ijbkqb fkabmbkafbkqb ^i `lkgrkql s^`Œl-

Klp bgbjmilp nrb pfdrbk mrbabk pbosfo m^o^ molcrkafw^o bk ilp `lk`bmqlp ababmbkabk`f^ b fkabmbkabk`f^-

DIDLOKN 0- Rf rk pr_`lkgrkql Q ab rk `lkgrkql R bp abmbkafbkqb+ bi jfpjlR bp abmbkafbkqb+ mlonrb pf Q dbkbo^ N bk cloj^ kl qofsf^i+ 0/ jfpjl e^`b R- Dpqlbp iŽdf`^jbkqb bnrfs^ibkqb ^ i^ molmlpf`fŽk ab nrb qlal pr_`lkgrkql ab rk`lkgrkql fkabmbkafbkqb bp fkabmbkafbkqb-

DIDLOKN 1- Klp i sb`qlobp `lloabk^alp rkfq^oflp B! ,,, * Bi ab Si dbkbo^kN `lk rkf`fa^a ^pŒmrbp plk ifkb^ijbkqb fkabmbkafbkqbp-

DIDLOKN 2- Br^inrfbo `lkgrkql nrb `lkqbkd^ bi sb`qlo `bol bp abmbkafbkqb-Olo bgbjmil+ pf =) < N+ qbkbjlp i^ obmobpbkq^`fŽk kl qofsf^i N < .=) * K=0 (--- * K=e+

Page 130: Calculus

457 =fa_\l[ q`^ojmd\g

DIDLOKN 3- Di `lkgrkql R < uc)d)8* gy ab sb`qlobp ab R0 bp ifkb^ijbkqbabmbkafbkqb mlonrb qbkbjlp i^ obmobpbkq^`fŽkkl qofsf^i abi sb`qlo `bol

l < c* f * %*E&%c* d&+

Dk bpqbbgbjmil bi pr_`lkgrkql Q < vd*ew bpifkb^ijbkqb fkabmbkafbkqb-Di qbo`bosb`qlo+ d* g+bp ab i^ bkslisbkqb ifkb^i ab P+Di qblobj^ pfdrfbkqb abjrbpqo^ nrbpf ^agrkq^jlp ^ dv f `r^inrfbo sb`qlo ab i^ bkslisbkqb ifkb^i ab P) l_qbkbjlp rk`lkgrkql abmbkafbkqb-

SDNQDL@ 01-7- P`\ R < x@i&--- +@a pi ^jiepioj gdi`\gh`io` di_`k`i_d`i+o` _` f q`^ojm`n _` Si* v n`\ I&P' g\ `iqjgq`io` gdi`\g _` R- Qj_j ^jiepioj _`f * 0 q`^ojm`n _` I&P' `n gdi`\gh`io` _`k`i_d`io`,

A`hjnom\^d‡i, K^ abjlpqo^`fŽk pb e^`b mlo fkar``fŽk bk e) k•jbol ab sb`,qlobp ab R- Rrmlkd^jlp mofjbol f < 0- Dkqlk`bp+ mlo efmŽqbpfp+R `lkpq^ ab rksb`qlo+ q^i `ljl @i+pfbkal @i z N v^ nrb R bp fkabmbkafbkqb-Sljbjlp ^elo^alp sb`qlobp afpqfkqlp `r^ibpnrfbo^ ?/ v ?0 ab I&P', Dkqlk`bp `^a^ rkl bp bi mol,ar`ql ab @i mlo rk bp`^i^o+mlo bgbjmil >g ;^/=g v >0 < ^0=x) kl pfbkal ?g X@0 ilp alp krilp- Lriqfmif`^kal >) mlo @0 X>0 mlo @g X obpq^kal+bk`lkqo^jlp nrb

Dpq^bp rk^ obmobpbkq^`fŽkkl qofsf^i ab N ^pŒnrb >) v >0 plk abmbkafbkqbp-Dpqlabjrbpqo^ bi qblobj^ `r^kal f < 0-

Rrmlkd^jlp ^elo^ nrb bi qblobj^ bi{sŠifal m^o^f + 0 Xs^jlp ^ abjlpqo^onrb q^j_f‹k il bp m^o^ e+ Sljbjlp `r^inrfbo `lkgrkql ab e * 0 sb`qlobp abH%O&)pb^ P < v?g* ?0* ŠŠŠ * ?E/w% Prbobjlp abjlpqo^o nrb P bp ifkb^ijbkqb ab,mbkafbkqb-Orbpql nrb `^a^ >c bp ab H%O&)mlabjlp bp`of_fo

'01-00(e

n*< ƒ \dd>dc:.

k\m\ `^a^ d < 0+1+ --- +f * 0- Du^jfkbjlp qlalp ilp bp`^i^obp \,* nrb jriqfmif,`^k @i v bp`fka^jlp i^ abjlpqo^`fŽk bk alp `^plp pbd•k pb^k qlalp ilp bp`^i^obpkrilp l kl-

@>PL 0- ^fi < M k\m\* oj_j d < 0+1+ --- +f * 0- Dk bpqb`^pl i^ prj^ ab'01-00( kl fk`irvb @i ^pŒnrb `^a^ >x ab P mboqbkb`b i^ bkslisbkqb ifkb^i abi`lkgrkql P%< v>0* ŠŠŠ * >_, Obol P%bp ifkb^ijbkqb fkabmbkafbkqb v `lkpq^ abf + 0 sb`qlobp- Rbd•k i^ efmŽqbpfpab fkar``fŽk+ bi qblobj^ bp sŠifal m^o^ f + 0`lk il nrb bi `lkgrkql P bp abmbkafbkqb-Dpql abjrbpqo^ bi qblobj^ bk bi B^pl 0-

Page 131: Calculus

Fi_`k`i_`i^d\ gdi`\g ./2

@>PL 1- Kj oj_jn gjn `n^\g\m`n \c nji ^`mj, Rrmlkd^jlp nrb ^0i " N- 'Rfbp mob`fpl+mlabjlp slisbo ^ krjbo^o ilp D m^o^nrb pb^ ^pŒ-(Slj^kal e< 0 bki^ fdr^ia^a '01-00( v jriqfmif`^kal ^j_lp jfbj_olp mlo ^,* pfbkal ^o :\yya\i* iibd^jlp ^

f

]x>. < [cf=f * K ]xMxd=d +ex0

Cb bpq^fdr^ia^a obpqbjlp i^ '01-00( v l_qbkbjlp

]x>. * >x < J %]x[.d * [cd&=d )0<1

m^o^ d < 1+ --- + e * 0- Dpq^ b`r^`fŽk bumobp^ `^a^ rkl ab ilp e sb`qlobp`fDh, ?d `ljl `lj_fk^`fŽk ifkb^i ab f + 0 sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp>0* ŠŠŠ * >f, Rbd•k i^ efmŽqbpfpab fkar``fŽk+ ilp f sb`qlobp ^,?* + ?, ab_bk pboabmbkafbkqbp-Olo il q^kql+m^o^-rklp `fboqlp bp`^i^obpo0* ŠŠŠ * nc*y+kl qlalp krilp+qbkbjlp

(%*I nx%]x>.* >x& < M +z<1

ab i^ nrb obpriq^

Obol ‹pq^ bp rk^ `lj_fk^`fŽk ifkb^i kl qofsf^i ab Di +--- +>e(f nrb obmobpbkqbisb`qlo `bol+ `lk il `r^i ilp sb`qlobp Di&--- +DTi ab_bk pbo abmbkafbkqbp-Dpql`ljmibq^ i^ abjlpqo^`fŽk-

@ `lkqfkr^`fŽk abjlpqo^jlp nrb bi `lk`bmql ab loqldlk^ifa^a bpqŠŒkqfj^,jbkqb obi^`flk^al `lk i^ fkabmbkabk`f^ ifkb^i-

CDEHMHBHˆM- Ri ^jiepioj _` q`^ojm`n R < v>g<,,, *@a _` Sdgn` _`ijhdi\jmojbji\g nd>d Š >y < M nd`hkm` lp` e" c+Ad^cj _` jomj hj_j* _jn q`^ojm`n _dn+odiojn ^p\g`nlpd`m\ _` pi ^jiepioj jmojbji\gnji k`mk`i_d^pg\m`n,

RCMPCK? 01-8- @p\glpd`m^jidpioj jmojbji\g R < v>g* ,,, * @a _` q`^oj+m`nij ipgjn _` S i `n gdi`\gh`io` di_`k`i_d`io`, >_`hƒn* ndR b`i`m\ pi q`^ojm W+n`\ `no`

'01-01(g`

T < I`y>y*f<i

Page 132: Calculus

46/ €gb`]m\ q`^ojmd\g

`ioji^`n gjn a\^ojm`n `n^\g\m`n `h+ ŠŠŠ,^y qd`i`i _\_jn kjm g\n a‡mhpg\n

T$ =d

&/0,/1' ^d < ,, k\m\ d < 0+1+--- +f,>9%>e

A`hjnom\^d‡i, Cbjlpqobjlp mofjbol nrb R bp ifkb^ijbkqb fkabmbkafbkqb-Rrmlkd^jlp nrb Gyyh`,>* < N- Lriqfmif`^kal bp`^i^ojbkqb `^a^ jfbj_ol mlo @iv qbkfbkal bk `rbkq^ nrb =f +=c:K m^o^ `^a^ d ;/; 0+bk`lkqo^jlp `i'@G Š=f& < N-Obol '@i& =f& ;/; N mrbpql nrb @i ;/; N+ `lk il nrb BG< N- Qbmfqfbkal bpqbo^wl,k^jfbkql `lk =f prpqfqrfal mlo =x) bk`lkqo^jlp nrb `^a^ ?x < ^- Olo `lkpfdrfbkqbR dbkbo^ N `lk rkf`fa^a mlo il nrb R bp ifkb^ijbkqb fkabmbkafbkqb-

Rrmlkd^jlp ^elo^ nrb R dbkbo^ T `ljl bk '01-01(- Eloj^kal bi molar`qlbp`^i^o ab T mlo =x `ljl ^kqbp+bk`lkqo^jlp nrb ]x%=x$=x& < W&=x ab alkabl_qbkbjlp '01-02(-

Rf qlalp ilp sb`qlobp @i+ --- +>e abi qblobj^ 01-8 qfbkbkkloj^ 0+i^ cŽojri^m^o^ilp jriqfmif`^alobp pb pfjmifcf`^ nrba^kal bk i^ cloj^

^e;U}>9,

Tk `lkgrkql ab sb`qlobp loqldlk^ibp u=x) +++) @a `^a^ rkl ab ilp `r^ibp qfbkbkloj^ 0+ pb abkljfk^ `lkgrkql jmojijmh\g, Klp sb`qlobp `lloabk^alp rkfq^oflpDi+--- +Bi plk rk bgbjmil ab `lkgrkql loqlkloj^i-

)*&), 7N`R`

Dp `loofbkqb bpqraf^o `lkgrkqlp ab sb`qlobp nrb dbkbobk `r^inrfbo sb`qlo abT+ `lk rkf`fa^a- S^ibp `lkgrkqlp pb ii^j^k ]\n`n ab Sh†

CDEHMHBHˆM- Ri ^jiepioj R < u=b$ ++) @a _` q`^ojm`n _` Sh `n pi\ ]\n`k\m\ Si ndR b`i`m\ oj_j q`^ojm _` s+ ^ji pid^d_\_, Pd*\_`hƒn* R `n jmojbji\g*`ioji^`n R n` _`ijhdi\ ]\n` jmojbji\g,

@pŒmrbp+rk^ _^pb bp rk `lkgrkql ifkb^ijbkqb fkabmbkafbkqbnrb dbkbo^ qlalbi bpm^`fl Sh† Di `lkgrkql ab sb`qlobp `lloabk^alp rkfq^oflp bp rk bgbjmil ab_^pb- Dpqb`^pl m^oqf`ri^o ab _^pb bp q^j_f‹k rk^ _^pb loqldlk^i- Cbjlpqobjlp^elo^ nrb qla^ _^pb `lkqfbkb bi jfpjl k•jbol ab bibjbkqlp-

RCMPCK? 01-0/- Bi pi `nk\^dj q`^ojmd\g_\_j Si* g\n ]\n`n od`i`i g\n kmj+kd`_\_`n ndbpd`io`n8

^( Qj_\ ]\n` ^jiod`i` `s\^o\h`io` i q`^ojm`n,_( @p\glpd`m^jiepioj _` q`^ojm`n gdi`\gh`io` di_`k`i_d`io`n `n pi np]^ji+

epioj _` pi\ ^d`mo\]\n`,b( @p\glpd`m^jiepioj _` i q`^ojm`n gdi`\gh`io` di_`k`i_d`io`n `n pi\ ]\n`,

Page 133: Calculus

Be`m^d^djn 460

A`hjnom\^d‡i, Klp sb`qlobp `lloabk^alp rkfq^oflp Dil ‘‘‘ + Bi `lkpqfqrvbkrk^ _^pb m^o^Si, Rf abjlpqo^jlp nrb alp _^pbp `r^ibpnrfbo^ `lkqfbkbk bi jfpjlk•jbol ab sb`qlobp l_qbkbjlp ^(-

Rb^k R u Q alp _^pbp+qbkfbkal R+f sb`qlobp v Q nrb qbkd^ l sb`qlobp- Rfl 8~ e) bkqlk`bp P `lkqfbkb mlo il jbklp e * 0 sb`qlobp ab H%O&)v^ nrb H%O&:Ri,

Olo `lkpfdrfbkqb+ bk sfoqra abi qblobj^ 01-7+Q ab_b pboifkb^ijbkqb abmbkafbkqb+bk `lkqo^af``fŽk `lk i^ efmŽqbpfpab nrb Q bp rk^ _^pb- Dpql pfdkfcf`^ nrb klmrbab pbo m= f* mlo q^kql pboŠmx f, @mif`^kal bi jfpjl o^wlk^jfbkql fkqbo,`^j_f^kal R v P) bk`lkqo^jlp nrb e w l+ Krbdl+ l < e `lk il nrb i^ m^oqb^(nrba^ abjlpqo^a^-

O^o^mol_^o _(+ pb^ R < x@i+--- +@a `r^inrfbo `lkgrkql ab sb`qlobp ifkb^i,jbkqb fkabmbkafbkqbab RiŠ Oc H%O&< Ri* R bp rk^ _^pb- Rf kl bp ^pŒ+bufpqbrk`fboql sb`qlo W ab Ri nrb kl mboqbkb`b H%O&+@agrkqbjlp bpqbsb`qlo ^ R v `lk,pfabobjlp bi krbsl `lkgrkql R&< x@i+ + >f* Tv+ Rf bpqb`lkgrkql crbo^ ab,mbkafbkqb+bufpqfoŒ^krklp bp`^i^obp`{+ +BGi&kl qlalp krilp+ q^ibpnrb

/:

1 ^d>d* @f)/U < M -d;g

Obol @f)o ;/; N mrbpql nrb ?h= --- + >f plk fkabmbkafbkqbp-Krbdl+ mlaoŒ^jlp ob,plisbo bp^ b`r^`fŽk obpmb`ql^ W v bk`lkqo^o nrb W C H%O&)bk `lkqo^af``fŽk `lkbi eb`el ab nrb W kl mboqbkb`b H%O&+Olo `lkpfdrfbkqb+ bi `lkgrkql R&bp ifkb^i,jbkqb fkabmbkafbkqbmbol `lkqfbkb f) 0 sb`qlobp- Rf I&P%'< Sh) R&bp rk^ _^pbv+mrbpql nrb R bp rk pr_`lkgrkql ab P%*i^ m^oqb_( nrba^ abjlpqo^a^- Rf R&kl bprk^ _^pb+mlabjlp o^wlk^o `lk R& ljl il ef`fjlp `lk R+l_qbkfbkal rk krbsl`lkgrkql R! nrb `lkqfbkb f * 1 sb`qlobp v bp ifkb^ijbkqb fkabmbkafbkqb-Rf R!bp rk^ _^pb+i^ m^oqb_( bpqŠabjlpqo^a^- Rf kl+ obmbqfjlp bi mol`bpl- Cb_bjlpiibd^o ^ rk^ _^pb ^i `^_l ab rk k•jbol cfkfql ab obmbqf`flkbpabi mol`bpl+ ab lqoljlal l_qbkaoŒ^jlp rk `lkgrkql fkabmbkafbkqb`lk i * 0 sb`qlobp+ bk `lkqo^af`,`fŽk `lk bi qblobj^ 01-7- Olo il q^kql i^ m^oqb_( nrba^ abjlpqo^a^-

Efk^ijbkqb+ rqfifw^kal i^p m^oqbp ( v _( abjlpqo^jlp `(- Rb^ R `r^inrfbo`lkgrkql ifkb^ijbkqb fkabmbkafbkqbnrb `lkpqb ab i sb`qlobp- Rbd•k i^ m^oqb_(+R bp rk pr_`lkgrkql ab rk^ `fboq^ _^pb >+ Obol pbd•k ^( i^ _^pb > qfbkb bu^`q^,jbkqb i bibjbkqlp+ `lk il nrb R < >+

01-04 Dgbo`f`flp

0- Rb^k: u d ilp sb`qlobp `lloabk^alp rkfq^oflp ab S0Š G^ii^o bk `^a^ `^pl bp`^i^obp s b u

q^ibp nrb r%c * d&* ': * d& bp fdr^i ^']( c8 '_( e9 'a( 2: , 3e9 'a( 6: * 3e,

1- Rf > < '0+ 1(+ ? < '1+ ,3(+ X b < '1+ ,2( plk qobp sb`qlobp ab S0

* e^ii^o rklp bp`^,i^obp s b u q^ibp nrb a < s> * s>+ ƒBrŠkqlp m^obp ab bplp bufpqbk>

Page 134: Calculus

.0+ >ob`]m\ q`^ojmd\g

2- Rf > < '1+ ,0+0(+ ? < '0+1+ ,0(+ X b < '1+ ,00+6( plk qobpsb`qlobp ab S\% e^ii^orklp bp`^i^obp s b u q^ibp nrb a < s> * t?,

3- Cbjlpqo^o nrb bi bgbo`f`fl 2 kl qfbkb plir`fŽk pf b pb obbjmi^w^ mlo bi sb`qlo '1+ 00+6(-4- Rb^k > v A alp sb`qlobp kl krilp ab S,

^( Rf = X > plk m^o^ibilp+abjlpqo^o nrb plk ifkb^ijbkqb abmbkafbkqbp-_( Rf = X > kl plk m^o^ibilp+abjlpqo^o nrb plk ifkb^ijbkqb fkabmbkafbkqbp-

5- Rf %[)\& u %_)& plk alp sb`qlobp ab R0% abjlpqo^o nrb plk ifkb^ijbkqb fkabmbkafbkqbp

pf u pŽil pf \_ + ]` :I5 N-6- G^ii^o qlalp ilp k•jbolp ob^ibp n m^o^ ilp `r^ibp ilp alp sb`qlobp '0 * o*0 , n& v

'0 , o*g* n&ab S0

pb^k ifkb^ijbkqb fkabmbkafbkqbp-7- Rb^k I+E+e ilp sb`qlobp `lloabk^alp rkfq^oflp ab R p&Cbjlpqo^o nrb ilp `r^qol sb`qlobp

c)d*e) 8 * f * e plk ifkb^ijbkqb abmbkafbkqbp+mbol nrb qobp`r^ibpnrfbo^ ab biilp plkifkb^ijbkqb fkabmbkafbkqbp-

8- Rb^k: u f ilp sb`qlobp `lloabk^alp rkfq^oflp ab R/

v R < x:+: * ew,^( Cbjlpqo^o nrb R bp ifkb^ijbkqb fkabmbkafbkqb-_( Cbjlpqo^o nrb f mboqbkb`b^ i^ bkslisbkqb ifkb^i ab R-b( Dumobp^o2: , 3g `ljl `lj_fk^`fŽk ifkb^i ab : b : * g-a( Cbjlpqo^o nrb H%O&< R0Š

0/- Blkpfabo^o ilp qobp sb`qlobp > <:+? < : * f v b < c * f * 1f ab S1Š

^( Cbjlpqo^o nrb bi `lkgrkql u=) 4$ax bp ifkb^ijbkqb fkabmbkafbkqb-_( Dumobp^o ^a^ rkl ab ilp sb`qlobp f v f `ljl `lj_fk^`fŽk ifkb^i ab >* ? X B-b( Dumobp^o1f , 2g* 3f `ljl `lj_fk^`fŽk ifkb^i ab >* A+ X b-a( Cbjlpqo^o nrb u=)>)_v bp rk^ _^pb m^o^ Up&

00- Rb^k = < '0+ 1(+A < '1+ ,3(+ b < '1+ ,2(+ X C < '0+ ,1( `r^qol sb`qlobp ab R0Š

Eloj^o qlalp ilp pr_`lkgrkqlp kl s^`Œlp ab u=) A+a+Cy nrb plk ifkb^ijbkqb fkab,mbkafbkqbp-

01- Rb^k > < '0+ 0+ 0+N(+ A < 'N+ 0+ 0+ 0( X b < '0+ 0+N+ N( qobpsb`qlobp ab S1†

^( Cbqbojfk^o pf =) >) b plk ifkb^ijbkqb abmbkafbkqbp l fkabmbkafbkqbp-_( N_qbkbo rk sb`qlo kl kril C q^i nrb =) >) a+C pb^k abmbkafbkqbp-b( N_qbkbo rk sb`qlo A q^i nrb =) >) a+A pb^k fkabmbkafbkqbp-a( Dibdfal B ab i^ m^oqbbi+ bumobp^obi sb`qlo W < '0+ 1+ 2+ 3( `ljl `lj_fk^`fŽkifkb^i ab =) >) a+A+

02- ^( Cbjlpqo^o nrb ilp qobp sb`qlobp pfdrfbkqbp ab Up plk ifkb^ijbkqb fkabmbkafbkqbp9

'U2+ 0+N(+'0+ U2- 0(+'N+0+U2(-_( Cbjlpqo^o nrb ilp qobppfdrfbkqbpplk abmbkafbkqbp9%p$~)0+N(+ '0+U1+ 0(+ 'N+ -0+U1(-`( G^ii^o qlalp ilp k•jbolp ob^ibp o m^o^ ilp `r^ibp ilp qobpsb`qlobp pfdrfbkqbp ab U^plk abmbkafbkqbp9 %n)0+N(+ '0+ o*0(+ 'N+ 0+n&+

03- Blkpfabo^o ilp pfdrfbkqbp `lkgrkqlp ab sb`qlobp ab R3&Dk `^a^ `^pl+ e^ii^o rk pr_`lk,grkql ifkb^ijbkqb fkabmbkafbkqb nrb `lkqbkd^ bi j^vlo k•jbol mlpf_ib ab sb`qlobp-

'^( x'i+/+0+N(+ 'i+ 0+0+0(+ '/+0+/+0(+ '1+N+,0+ N(y-'_( x'0+0+0+0(+'0+,0+0+0(+ '0+,0+ ,0+0(+ 'i+ ,0+ ,0+ ,0(y-'b( x'0+0+0+0(+ '/+0+0+0(+ 'N+/+0+0(+ 'N+N+N+0(y-

04- C^alp qobpsb`qlobp ifkb^ijbkqb fkabmbkafbkqbp =) >) b ab R†+ Cbjlpqo^o pf plk l kl`fboq^p i^p molmlpf`flkbp pfdrfbkqbp-^( = * >) A * a+= * a plk ifkb^ijbkqb fkabmbkafbkqbp-_( = * >) > * a+= * a plk ifkb^ijbkqb fkabmbkafbkqbp-

05- ^i Cbjlpqo^o nrb rk `lkgrkql R ab qobpsb`qlobp ab U^ bp rk^ _^pb m^o^ S pf v pŽilpf pr bkslisbkqb ifkb^i H_m&lkqfbkb ilp qobpsb`qlobp `lloabk^alp rkfq^oflp 9*dv e+_( Dpq^_ib`bo v abjlpqo^o rk^ dbkbo^ifw^`fŽk ab i^ m^oqb^( m^o^ S Š,

Page 135: Calculus

Bg `nk\^dj q`^ojmd\g S i&@' _` i+kg\n _` iˆh`mjn ^jhkg`ejn 351

06- Dk`lkqo^o alp _^pbp m^o^ U^ nrb `lkqbkd^k ilp alp sb`qlobp 'N+ 0+ 0( X '0+ 0+ 0(-07- Dk`lkqo^o alp _^pbp m^o^ S 3 nrb qbkd^k `ljrkbp pŽil ilp alp sb`qlobp 'N+ 0+ 0+ 0( (

'0+ 0+ 0+ 0(-

08- Blkpfabo^o ilp pfdrfbkqbp `lkgrkqlp ab sb`qlobp ab S\8

R < x'i+ 0+0(+'/+0+1(+ '0+ N+,0(y+ P < x'1+ 0+N(+'1+ N+,1(y+ Q < x'i+ 1+2(+'0+ 2+R(y

^( Cbjlpqo^o nrb H%P&R: H%O&+_( Cbqbojfk^o qla^p i^p obi^`flkbp ab fk`irpfŽk nrb bufpqbk bkqob ilp `lkgrkqlp H%O&)H%P&)v H%Q&+

1/- Cbpfdkbjlp `lk = u > alp pr_`lkgrkqlp cfkfqlp ab sb`qlobp bk rk bpm^`fl sb`qlof^i Rij

v `lk H%=&v H%>&prp bkslisbkqbp ifkb^ibp- Ool_^o `^a^ rk^ ab i^p molmlpf`flkbp pf,drfbkqbp-^( Rf = R: >) bkqlk`bp H%=&R: H%>&+_( H%=l) >&R: H%=&h H%>&+b( C^o rk bgbjmil bk bi nrb H%=i A( ,{‹ H%=&i H%>&+

)*&). :Y R`]NPV\cRPa\_VNYRh%?&QR ,mi^pQR[qZR_\` P\Z]YRW\`

Dk i^ pb``fŽk 01-1 pb abcfkfŽ bi bpm^`fl sb`qlof^i Rh `ljl bi `lkgrkql abqla^p i^p k,mi^p ab k•jbolp ob^ibp-K^ fdr^ia^a+ i^ ^af`fŽk ab sb`qlobp+v i^ jri,qfmif`^`fŽkmlo bp`^i^obppb abcfkfbolk bk crk`fŽk ab ilp `ljmlkbkqbp abi pfdrfbk,qb jlal9 Rf > < %[! +++ ) ^++(v ? < '_

!--- +]i'* bkqlk`bp

= < > pfdkfcf`^ \9 < \) m^o^`^a^ c< 0+1+--- +i *

^> < &^\g%,,, *^\i' ,

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Page 136: Calculus

463 >gb`]m\ q`^ojmd\g

mlo pŒjfpjl bp mlpfqfsl+ef`fjlp rpl abi eb`el ab nrb rk^ prj^ ab `r^ao^alp abk•jbolp ob^ibp bp mlpfqfs^- Orbpql nrb rk^ prj^ ab `r^ao^alp ab k•jbolp `lj,mibglpmrbab pbo&kbd^qfs^+qbkbjlp nrb jlafcf`^o i^ abcfkf`fŽk ab = +> pf nrbob,jlp `lkpbos^o i^ molmfba^a ab nrb pb^ mlpfqfsl- O^o^ Rh%?&)rp^jlp i^ pfdrfbkqbabcfkf`fŽk ab molar`ql bp`^i^o-

CDEHMHBHˆM- Rf = < &\ *,,, *\i' u ? < n\+7+++*]i' nji _jn q`^ojm`n _`Rh%?&)_`adidhjn np kmj_p^oj `n^\g\m >%? ^ji g\ a‡mhpg\

h

=$ > < H\f]f*f;g

_ji_` ]f `n `g ^jhkg`ej ^jiepb\_j _` ]~,

N_p‹osbpb nrb bpq^ abcfkf`fŽk bpqŠ ab ^`rboal `lk i^ ^kqbp a^a^ m^o^ Ri

mlonrb \e < \e `r^kal \| bp ob^i- K^p molmfba^abpcrka^jbkq^ibp abi molar`qlbp`^i^o+`loobpmlkafbkqbp ^ i^p abi qblobj^ 01-1+qlj^k ^elo^ i^ pfdrfbkqb cloj^-

RCMPCK? 01-00- M\m\ oj_jn gjn q`^ojm`n >* ?* a _` Rh%?&u oj_jn gjn `n^\+g\m`n^jhkg`ejn `* o`i`hjn

'^( = +> < >$ =)'_( > , &?* @'; >%? * > , B+'b( ]%= +>& < %]=&+> < = +%€>&)'a( > , > = N nd > l{‹ N+'b( > , > < M nd > < N-

Sla^p bp^p molmfba^abp plk pbk`fii^p `lkpb`rbk`f^p ab i^ abcfkf`fŽk v prpabjlpqo^`flkbp pb abg^k `ljl bgbo`f`flp- Di ib`qlo ab_boŒ l_pbos^o nrb ^m^ob`bbi`lkgrd^al bk i^ molmfba^a ^( `r^kal pb fksfboqbbi loabk ab ilp c^`qlobp- @pfjfp,jl+ ^m^ob`bbi `lkgrd^al abi c^`qlo bp`^i^o bk i^ molmfba^a b( `r^kal bi bp`^i^o `m^p^ab rk i^al ^i lqol abi mrkql-

K^ abpfdr^ia^a ab B^r`ev,R`et^ow qlj^ ^elo^ i^ cloj^

'01-03( h? - A01 z %=+=&%>+>& +

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Orbpql nrb bi molar`ql bp`^i^o ab rk sb`qlo mlo pŒjfpjl bp kl kbd^qfsl+ ml,abjlp fkqolar`fo i^ kloj^ ab rk sb`qlo ab Tj'A( jbaf^kqb i^ cŽojri^ rpr^i

GG?GG< %=$>'/-0,

Page 137: Calculus

Be`m^d^djn 464

K^p molmfba^abpcrka^jbkq^ibp ab i^p kloj^p+ `ljl pb bpq^_ib`fbolk bk bi qbl,obj^ 01-3+q^j_f‹k plk sŠifa^p pfk jlafcf`^`fŽk m^o^R))%?&+K^ abpfdr^ia^a qof^k,dri^o+ GG?* @GGy GG?GG* GG@GG+q^j_f‹k s^ib bk R))%?&+

K^ loqldlk^ifa^a ab sb`qlobp bk R))%?&pb abcfkb `lk i^ obi^`fŽk = +> < N-Bljl bk bi `^pl ob^i+alp sb`qlobp = v > ab Rh%?&plk loqldlk^ibp pf v pŽil pfp^qfpc^`bki^ fabkqfa^a mfq^dŽof`^+GG?* @ 01< GG?G01* GG@G01-

Klp `lk`bmqlp ab bkslisbkqb ifkb^i+ fkabmbkabk`f^ ifkb^i+ u _^pb+pb abcfkbkm^o^R)8%?&bu^`q^jbkqb `ljl bk bi `^pl ob^i- Klp qblobj^p abi 01-6 ^i 01-0/ vprp abjlpqo^`flkbp plk qla^p sŠifa^p pfk jlafcf`^`fŽk m^o^ Rh%?&+

)*&)/ :WR_PVPV\`

0- Rb^k = < '0+ c&) > < %c)*c&) X b < %/c)0( qobp sb`qlobp ab R0%?&+B^i`ri^o `^a^ rkl

ab ilp pfdrfbkqbp molar`qlo bp`^i^obp9

%[&=$>8 %\&>$=8

'b( ?} B: 'c( > , B:

'g( %= * c>& +%= * c>&+

1- Rf = < '1+ 0+ *c& X > < %c),0+ /c&) e^ii^o rk sb`qlo kl kril b ab R1%?& loqldlk^i

pfjriqŠkb^jbkqb ^ = v >+2- Cbjlpqo^o nrb m^o^ alp sb`qlobp `r^ibpnrfbo^ = u > ab R+%?&)qbkbjlp i^ fabkqfa^a

'b( %c=&$>8'e( %> * B( - =8

'a( = +%c>&8'f( %= * A( - >8

'b( %c=&+%c>&8

GG?* @G01< GG?001* GG@G01* = +> * = +>+

3- Cbjlpqo^o nrb m^o^ alp sb`qlobp `r^ibpnrfbo^ = v > ab R+%?&)qbkbjlp i^ fabkqfa^a

GG?* > 001, GG?, > 001< /%= +> * = +>&+

4- Cbjlpqo^o nrb m^o^ alp sb`qlobp `r^ibpnrfbo^ = v > ab R+%?&)qbkbjlp i^ fabkqfa^a

GG?* @G01* &h? , @G01< 1 GG?001* 1 h@G01Š

5- ^( Cbjlpqo^o nrb m^o^ alp sb`qlobp `r^ibpnrfbo^ = v > ab R+%?&)i^ prj^ = +? * = +>bp ob^i-_( Rf = X > plk sb`qlobp kl krilp ab R+%?&)abjlpqo^o nrb

=$>(=$>,1;,,,,, ;1, GG?GGGG@GG, -

6- Cbcfkfjlp bi Škdril '( cloj^al mlo alp sb`qlobp kl krilp = v > ab R+%?&jbaf^kqb i^fabkqfa^a

i Z w%=+> * =7.c&, ^obblp GG?G0GG@GG -

Page 138: Calculus

.0/ €gb`]m\ q`^ojmd\g

K^ abpfdr^ia^a abi bgbo`f`fl 5 abjrbpqo^ nrb pfbjmob bufpqb rk •kf`l Škdril '( bk bifkqbos^il `boo^al N z '( z .l nrb p^qfpc^`b bpq^ fdr^ia^a- Cbjlpqo^o nrb

GG?, @G01< GG?001* GG@G01, 1 GG?0GGG@Flp %F+

7- @mif`^o i^ abcfkf`fŽk abi bgbo`f`fl 6 ^i `Ši`ril abi Škdril cloj^al mlo ilp alp sb`qlobppfdrfbkqbp ab RŠ'B(9 = < 'i+ N+c) c)g(+v > < 'g+ g+o,N+c&+

8- ^( Cbjlpqo^o nrb ilp qobpsb`qlobp pfdrfbkqbp cloj^k rk^ _^pb m^o^ Up'B(9 > < 'H+N+N(+? < 'N+ c)-/(+ b < N+ 0+ c&+_( Dumobp^obi sb`qlo '4+ 1 , c)/c& `ljl `lj_fk^`fŽk ifkb^i ab =) >) B-

0/- Cbjlpqo^o nrb i^ _^pb ab ilp sb`qlobp `lloabk^alp rkfq^oflp Ch& Š-- + Cj ab Uj q^j_f‹k`lkpqfqrvbk rk^ _^pb m^o^ Tj'A(-

Page 139: Calculus

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Dk bpqb`^mŒqrilpb qo^q^ab i^p ^mif`^`flkbp abi „idb_o^ sb`qlof^i ^i bpqraflab i^p ob`q^p+ilp mi^klp u i^p pb``flkbp `Žkf`^p- Dk bi `^mŒqril03 bi „idb_o^ sb`,qlof^i pb `lj_fk^ `lk ilp j‹qlalp abi `Ši`ril+ v pb a^k lqo^p ^mif`^`flkbp ^i bpqr,afl ab `ros^p v ^ `fboqlp mol_ibj^p ab Lb`Škf`^-

Di bpqrafl ab i^ FbljbqoŒ^ `ljl pfpqbj^ abar`qfsl+ crb `lk`b_fal mloDr`ifabp ^molufj^a^jbkqb 2// ^•lp ^kqbpab Ibpr`ofpql+ bjmbw^kal `lk rk `lk,grkql ab ^uflj^p l mlpqri^alp nrb abp`of_bk molmfba^abp ab ilp mrkqlp u i^pob`q^p-

Klp `lk`bmqlp ~mrkql‚ v ~ob`q^‚ pb qlj^k `ljl kl`flkbp mofj^of^p v klpb abcfkbk- Rb abcfkbk lqolp `lk`bmqlp ^ m^oqfoab ilp mrkqlp v ob`q^p+v ilp qblob,j^p pb abar`bk pfpqbjŠqf`^jbkqb ^ m^oqfoab ilp ^uflj^p- Dr`ifabp bpq^_ib`fŽafbw^uflj^p `lk ilp nrb fkqbkqŽabar`fo qlalp prp qblobj^p- G^ pfal abjlpqo^al nrbbplp ^uflj^p kl plk ^ab`r^alp m^o^i^ qbloŒ^-Olo bgbjmil+ bk i^ abjlpqo^`fŽk abpr mofjbo qblobj^+ Dr`ifabp e^`b rk^ efmŽqbpfpqŠ`fq^ obi^qfs^ ^ i^ fkqbopb``fŽkab alp `fo`rkcbobk`f^p nrb kl bpqŠ`r_fboq^ mlo prp ^uflj^p- Cbpab bkqlk`bp e^kpfal clojri^a^p lqo^p ifpq^pab ^uflj^p ab ilp nrb obpriq^k qlalp ilp qblobj^p abDr`ifabp- K^ jŠp c^jlp^ bp i^ nrb afl bi j^qbjŠqf`l ^ibjŠk C^sfa Gfi_boq '0751,0832( bk pr l_o^ Dmpi_g\b`i _`m D`jh`omd`* mr_if`^al bk 0788- 'Dufpqbrk^ qo^,ar``fŽk fkdibp^9Qc` Cjpi_\odjin ./ D`jh`omt* Nmbk Blroq Or_ifpefkd Bl-+ 0836-(Dpqbqo^_^gl+abi nrb pb ef`fbolk pfbqbbaf`flkbp ^ibj^k^p bk sfa^ ab Gfi_boq+fk^r,droŽ i^ L^qbjŠqf`^ ^_pqo^`q^abi pfdil uu-

Gfi_boq `ljbkwŽ pr bpqrafl ab i^ FbljbqoŒ^ mi^k^ `lk `fk`l `lk`bmqlp nrbkl abcfkfŽ9kpioj* m`^o\*`i 'obi^`fŽk bkqobrk mrkql v rk^ ob`q^(+ iom` 'obi^`fŽkbkqobrk mrkql v rk m^oab mrkqlp(+v ^jibmp`i^d\ 'obi^`fŽk bkqobm^obpab mrk,qlp(- C^ bkqlk`bp nrfk`b ^uflj^p ^ m^oqfoab ilp `r^ibp abp^oolii^ qla^ i^ Fbljb,qoŒmi^k^ br`ifaf^k^- K^ FbljbqoŒ^ abi bpm^`fl pb _^p^ bk sbfkqf•k ^uflj^p nrbfk`irvbk pbfp`lk`bmqlp nrb kl pb abcfkbk-

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Page 140: Calculus

467 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

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02-1 Qb`q^pbk bi bpm^`fl k,afjbkpflk^i

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CDEHMHBHˆM- P`\ M pi kpioj _\_j t > pi q`^ojm ij ipgj _\_j, Bg ^jiepioj_` oj_jn gjn kpiojn _` g\ ajmh\ M* o>*`i _ji_` o m`^jmm`oj_jn gjn iˆh`mjnm`\g`n*`n pi\ m`^o\ lp` k\n\ kjm M t `n k\m\g`g\ \ >, A`ndbi\hjn `n\ m`^o\ ^jiI&M9>' V `n^md]dhjn

I&M9>' < vM * o> Gom`\gw j* hƒn ]m`q`h`io`* I&M9>' ;vM * o>w,

P` _d^` lp` pi kpioj P `noƒ `i g\ m`^o\ I&M9>' nd P D I&M9>',

Dk bi pŒj_lil H%L8=&) bi mrkql L bp`ofql bk mofjbo ird^o bpqŠbk i^ ob`q^+v^ nrb `loobpmlkab ^ o < N- Di pbdrkal mrkql+ >* pb ii^j^ q`^ojm _` _dm`^^d4iab i^ ob`q^- K^ ob`q^ H%K8=& nrb m^p^ mlo bi lofdbk N bp i^ bkslisbkqb ifkb^iab >9 `lkpq^ ab qlalp ilp molar`qlp ab > mlo bp`^i^obp-K^ ob`q^ mlo M m^o^ibi^^= pb l_qfbkb prj^kal L ^ `^a^ sb`qlo ab i^ bkslisbkqb ifkb^i ab =+

K^ cfdro^ 02-0 jrbpqo^ i^ fkqbomobq^`fŽkdblj‹qof`^ ab bpq^abcfkf`fŽk bk R1Š

B^a^ mrkql M * o> mrbab obmobpbkq^opbmlo bi buqobjl ab rk sb`qlo dblj‹qof`lqo^w^al mlo bi lofdbk- Br^kal o s^oŒ qlj^kal qlalp ilp s^ilobp ob^ibp+bi `loobp,mlkafbkqb mrkql M * o> abp`of_b rk^ ob`q^nrb m^p^mlo M u m^o^ibi^^i sb`qlo >,K^ cfdro^ 02-0 jrbpqo^ ilp mrkqlp `loobpmlkafbkqbp^ ^idrklp s^ilobp ab o bk i^palp ob`q^pH%L8=& u H%K8=&+

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>gbpi\n kmjkd`_\_`n n`i^dgg\n _` g\n m`^o\n 468

H%L8=&

H%K8=&

EHFTQ@ 02-0 I\ m`^o\ I&M9>' kjm M k\m\g`g\ \ > u np m`g\^d‡i b`jh„omd^\ ^ji g\ m`^o\IT9 >' kjm L k\m\g`g\ \ >,

)+&+ @idrk^p molmfba^abppbk`fii^p ab i^p ob`q^p

Oofjbol abjlpqo^jlp nrb bi sb`qlo afob``fŽk > nrb ^m^ob`bbk i^ abcfkf`fŽkab H%L8=& mrbab obbjmi^w^opbmlo `r^inrfbo lqol sb`qlo m^o^ibil ^ =+ 'Qb`loab,jlp nrb alp sb`qlobp > u ? pb ii^j^k m^o^ibilp pf > < ^? m^o^rk `fboql bp`^i^o` kl kril-(

RCMPCK? 02-0- Ajn m`^o\n I&M9>' X I&M9?' lp` k\n\i kjm `g hdnhj kpi+oj M nji dbp\g`n nd v n‡gj nd gjn q`^ojm`n _` _dm`^^d‡i > v ? nji k\m\g`gjn,

A`hjnom\^d‡i, Rrmlkd^jlp mofjbol nrb I&M9>' < I&M9?', Sljbjlp rkmrkql bk I&M9>' afpqfkql ab M*mlo bgbjmil M * >, Dpqbmrkql bpqŠq^j_f‹k bkI&M9?' ab j^kbo^ nrb M * > < M * ^? m^o^ rk `fboql bp`^i^o ^, Krbdl+ qbkb,jlp > < ^? u ` <.<, N v^ nrb > ;-;+., Olo `lkpfdrfbkqb+ > v ? plk m^o^ibilp-

Cbjlpqobjlp ^elo^ bi ob`Œmol`l-Rrmlkd^jlp nrb = u > plk m^o^ibilp+pb^>;^? m^o^rk `fboql ^;-;+L,Rf P bpqŠbk I&M9>'* bkqlk`bp qbkbjlp P < M)o>;; M * o&^?' < M * &^o'?* `lk 0/ lp` P bpqŠ bk I&M9?', Olo `lkpfdrfbkqbI&M9>' Q I&M9?', Cbi jfpjl jlal+ I&M9?' Q I&M9>'* mlo q^kql I&M9>' ;: I&M9?',

@ `lkqfkr^`fŽk abjlpqo^jlp nrb bi mrkql M nrb ^m^ob`bbk i^ abcfkf`fŽk abH%L8=& mrbab obbjmi^w^opb mlo `r^inrfbo lqol mrkql O pfqr^al bk i^ jfpj^ob`q^-

Page 142: Calculus

36. >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

RCMPCK? 02-1- Ajn m`^o\n I&M9>' X I&N9 >' ^ji `g hdnhj q`^ojm _` _d+m`^^d‡i > nji dbp\g`n ndv n‡gj nd P `noƒ `i I&M9>',

A`hjnom\^d‡i, Rrmlkd^jlp nrb I&M9>' < I&N9 >', Orbpql nrb P bpqŠbkI&N9>'* P q^j_f‹k bpqŠbk I&M9>', O^o^ abjlpqo^o bi ob`Œmol`l+prmlkd^jlpnrb P bpqŠ bk I&M9>'* pb^ P < O * ^>, Prbobjlp abjlpqo^o nrb I&M9>' :I&N9 >', Rf W C I&M9>'* bkqlk`bp W < O * o> m^o^ rk `fboql o, ObolO < P , ^>* ^pŒnrb W < P , ^> * o> < P * &o+ ^'>* v mlo q^kql W q^j,_f‹k bpqŠbk I&N9>', Olo il q^kql I&M9>' R I&N9>', @kŠild^jbkqb+ bk`lkqo^,jlp I&N9 >' R I&M9>'* `lk il `r^i i^p alp ob`q^pplk fdr^ibp-

Tkl ab ilp c^jlplp mlpqri^alp ab Dr`ifabp bp bi kjnopg\_j _` g\n k\m\g`g\nnrb bp iŽdf`^jbkqb bnrfs^ibkqb ^ i^ molmlpf`fŽk ab nrb ~mlo rk mrkql a^al bufpqbrk^ v pŽil rk^ ob`q^ m^o^ibi^ ^ rk^ ob`q^ a^a^‚- Cbar`fobjlp bpq^ molmfba^a`ljl rk^ `lkpb`rbk`f^ abi qblobj^ 02-0- Mb`bpfq^jlp mofjbol abcfkfo bi m^o^ib,ifpjl ab ob`q^p-

CDEHMHBHˆM- Ajn m`^o\n I&M9>' W I&N9?' nji k\m\g`g\n nd npn q`^ojm`n _`_dm`^^d‡i> v ? nji k\m\g`gjn,

RCMPCK? 02-2- A\_jn pi\ m`^o\ I v pi kpioj O ij k`mo`i`^d`io` \ I*`sdno` pi\ v n‡gj pi\ m`^o\ I% lp` ^jiod`i` O v `n k\m\g`g\ \ I,

A`hjnom\^d‡i, Rrmlkd^jlp nrb i^ ob`q^ a^a^ qfbkb bi sb`qlo ab afob``fŽk>, Blkpfabobjlp i^ ob`q^ I%< I&N9>', Dpq^ob`q^ `lkqfbkb P v bp m^o^ibi^^ I,Di qblobj^ 02-0 klp af`b nrb ‹pq^ bp i^ •kf`^ ob`q^ `lk bp^palp molmfba^abp-

Kjo\8 K^odl qfbjml ilp j^qbjŠqf`lp plpmb`e^olk nrb bi mlpqri^al ab i^p m^o^ibi^pmlaŒ^abar`fopb ^ m^oqfoab ilp lqolp mlpqri^alp ab Dr`ifabp+ mbol qlalp ilp fkqbkqlp m^o^abjlpqo^oil obpriq^olk fk•qfibp- @ mofk`fmflp abi pfdil WHWilp j^qbjŠqf`lp J^oi E- F^rpp'0666,0744(+ I- Aliv^f '07/1,075/(+ v M- H- Kl_^q`ebsphf '0682,0745( iibd^olk ^ i^ `lk,sf``fŽk ab nrb bi mlpqri^al ab i^p m^o^ibi^pkl mlaŒ^abar`fopb ab ilp lqolp v abp^ool,ii^olk FbljbqoŒ^p kl br`ifaf^k^p+ bpql bp+FbljbqoŒ^p bk i^p nrb kl bp sŠifal bi `fq^almlpqri^al- Di qo^_^gl ab bplp elj_obp fkpmfoŽ^ lqolp j^qbjŠqf`lp v `fbkqŒcf`lpi^ ^j,mif^`fŽk ab prp mrkqlp ab sfpq^ ^`bo`^ ab i^p ~sboa^abp ^`bmq^a^p‚ v ^ ob`e^w^o lqolp^uflj^p Prb aro^kqb pfdilp e^_Œ^k pfal `lkpfabo^alp `ljl p^do^alp-

S^j_f‹k pb abar`b `lk c^`fifa^a i^ pfdrfbkqb molmfba^a ab i^p ob`q^p nrbDr`ifabp bpq^_ib`fŽ `ljl rk ^uflj^-

RCMPCK? 02-3- Ajn kpiojn _dnodiojn _`o`mhdi\i pi\ m`^o\, Bnoj `n* ndO" P+ `sdno` pi\ v n‡gj pi\ m`^o\ lp` ^jiod`i` O v P- Mp`_` _`n^md]dmn`jhj_f ]ihcohni xO * n%M* L&v +

Page 143: Calculus

O`^o\n v api^dji`n q`^ojmd\g`n 470

A`hjnom\^d‡i, Rb^ I i^ ob`q^ nrb m^p^mlo M v bp m^o^ibi^^ N+M* bpql bp+

H < H%L8N + L& < uL * o&N+ L&v+

Dpq^ob`q^ `lkqfbkb ^ M v^ P 'qlj^o o < N m^o^M v o < 0 m^o^M&+Rb^ ^elo^ I%`r^inrfbo ob`q^ nrb `lkqbkd^ M v P- Cbjlpqo^objlp nrb I% < I, Orbpql nrbI% `lkqfbkb M*qbkbjlp I% < I&M9>' m^o^^id•k > ;/; N- Obol q^j_f‹k I% `lkqfbkbP `lk il nrb M * ^> < P m^o^rk `fboql `- Krbdl qbkbjlp P , M < ^>* alkab` ;/; N v^ nrb P ;/; M, Olo `lkpfdrfbkqb P , M bp m^o^ibi^^ > `lk il nrb+ pbd•kbi qblobj^ 02-1+qbkbjlp I% < I&M9>' < I&M9N+M' < I,

DIDLOKN- Di qblobj^ 02-3 klp a^ rk pbk`fiil j‹qlal m^o^ ^sbofdr^o pf rkmrkql P bpqŠbk rk^ ob`q^a^a^ I&M9>', Mlp af`b nrb P bpqŠbk I&M9>' pf v pŽilpf P , M bp m^o^ibil ^ >, Olo bgbjmil+ `lkpfabobjlp i^ ob`q^ I&M9>'* alkabL < '0+ 1+ 2( X = < '1+ ,0+ 4(- O^o^ ^sbofdr^o pf bi mrkql P < '0+ 0+3( bpqŠbkbp^ ob`q^+bu^jfkbjlp P , M < 'N+ ,0+ 0(- Orbpql nrbP , M kl bp bi molar`qlab > mlo rk bp`^i^o+bi mrkql '0+ 0+3( kl bpqŠbk bp^ ob`q^- Olo lqo^ m^oqb+pfP < '4+ N+ 02(+ bk`lkqo^jlp nrb P , L < '3+ ,1+-0/( < /=) ^pŒnrb P bpqŠbk i^ ob`q^-

K^ abmbkabk`f^ ifkb^i ab alp sb`qlobp bk S i* mrbab bumobp^opblk ibkdr^gbdblj‹qof`l-

SDNQDL@ 02-4- Ajn q`^ojm`n > u ? _` Si nji gdi`\gh`io` _`k`i_d`io`n pfu n‡gj pf `noƒi `i g\ hdnh\ m`^o\ lp` k\n\ kjm `g jmdb`i,

A`hjnom\^d‡i, Rf bp `bol rkl ab ilp sb`qlobp > l ?* bi obpriq^al bp qofsf^i-Rf ^j_lp plk kl krilp+ bkqlk`bp > v ? plk abmbkafbkqbppf v pŽil pf ? < o> m^o^rk `fboql bp`^i^o o, Obol ? < o> pf v pŽil pf ? bpqŠbk i^ ob`q^ nrb m^p^ mlo bilofdbk v bp m^o^ibi^^ >,

02-3 Qb`q^p v crk`flkbp sb`qlof^ibp

Di `lk`bmql ab ob`q^ pb mrbab obi^`flk^o ^i ab crk`fŽk- K^ `loobpmlkabk`f^nrb ^pl`f^ ^ `^a^ k•jbol ob^i o bi sb`qlo M * o>* bp rk bgbjmil ab crk`fŽk `rvlaljfkfl bp bi `lkgrkql ab ilp k•jbolp ob^ibpv `rvl ob`loofal bp i^ ob`q^ H%L8=&+Rf abpfdk^jlp i^ crk`fŽk `lk bi pŒj_lil W+bi s^ilo ab i^ crk`fŽk T%n&bk o sfbkba^al mlo i^ b`r^`fŽk

'02-0( U&o'< M * o>,

Ki^j^jlp ^ ‹pq^+crk`fŽk sb`qlof^i ab rk^ s^of^_ib ob^i-K^ `lkpfabo^`fŽk ab bp^ crk`fŽk bp fjmloq^kqb ab_fal ^ nrb+ `ljl sbobjlp

bk bi `^mŒqril03+klp a^ rk j‹qlal k^qro^i m^o^bpqraf^o `ros^p bk cloj^ jŠpdbkbo^i-

Page 144: Calculus

360 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

Di bp`^i^o o ab i^ b`r^`fŽk '02-0( pb abkljfk^ ^ jbkral k\mƒh`omj* v i^b`r^`fŽk '02-0( pb ii^j^ `^p\^d‡i q`^ojmd\g k\m\h„omd^\ l pfjmibjbkqb `^p\^d‡iq`^ojmd\g ab i^ ob`q^-@ sb`bp `lksfbkb fj^dfk^o i^ ob`q^ `ljl i^ qo^vb`qlof^ abrk^ m^oqŒ`rijŽsfi+ bk `rvl `^pl bi m^oŠjbqol o bp bi od`hkj u bi sb`qlo U&o' biq`^ojm kjnd^d‡i

N_pbosbjlp nrb alp mrkqlp U&\' u U&]' ab rk^ ob`q^ a^a^ I&M9>' plkfdr^ibp pf v pŽil pf qbkbjlp M * \> < M * ]> l &\ + ]'> < N- Orbpql nrb> ;/; N+ bpq^•iqfj^ obi^`fŽk bp sŠifa^ pf v pŽil pf [ < ], @pŒnrb+ s^ilobp afpqfk,qlp abi m^oŠjbqol o `lkar`bk ^ mrkqlp afpqfkqlp ab i^ ob`q^-

Blkpfabobjlp ^elo^ qobpmrkqlp afpqfkqlpab rk^ ob`q^ a^a^+ pb^k U&\'* U&]'*v U&^'* pfbkal \ = ], Cb`fjlp nrb U&^' bpqŠ`iom` U&\' u U&]' pf ` bpqŠbkqob\ v ]* bpql bp+pf \ ; ` ; ],

K^ `lkdorbk`f^ mrbab abcfkfopbbk crk`fŽk ab i^p kloj^p- Tk m^oab mrkqlpM*N pb ii^j^ ]ihalo_hn_ ^ lqol m^o M%*N&pf GGN, /00 < GGN&, N&00-K^ kloj^EEL* /00 pb ii^j^ q^j_f‹k afpq^k`f^ bkqob L v N-

Dpql `ljmibq^ i^p abcfkf`flkbp ab ilp `lk`bmqlp kpioj* m`^o\* `i* `iom`* v^jibmp`i^d\ bk krbpqol jlabil ^k^iŒqf`labi bpm^`fl br`iŒabl ab i afjbkpflkbp-Blk`irfjlp bpq^pb``fŽk `lk ^idrk^ lqo^ l_pbos^`fŽk obi^qfs^ ^ i^p b`r^`flkbp m^,o^j‹qof`^p m^o^i^p ob`q^pbk bi bpm^`fl qofafjbkpflk^i-

Rf rk^ ob`q^ m^p^mlo alp mrkqlp afpqfkqlp M u N+ mlabjlp rqfifw^oM + N`ljl sb`qlo ab afob``fŽk = bk i^ b`r^`fŽk 'j-0(: i^ b`r^`fŽk sb`qlof^i ab i^ob`q^ bp bkqlk`bp

T%n&< L * o&N+ L& l U&o'< oN *'0 , o'M,

K^p b`r^`flkbp sb`qlof^ibp pb mrbabk bumobp^oq^j_f‹k bk crk`fŽk ab ilp`ljmlkbkqbp+ Olo bgbjmil+ pf bp`of_fjlp L < %j)k) l&) = < %[) \) ]&) v T%n&:: %r)v+w(+i^ b`r^`fŽk '02-0( bp bnrfs^ibkqb ^ i^p qobpb`r^`flkbp bp`^i^obp

'02-1( s < j * o\* t < l * o]* X < m* o`,

Dpq^pplk i^p `^p\^dji`n `n^\g\m`n k\m\h„omd^\n l pfjmibjbkqb `^p\^dji`n k\m\h„+omd^\nab i^ ob`q^: plk •qfibp bk ilp `Ši`rilp bk ilp nrb fkqbosfbkbkilp `ljmlkbk,qbp-K^ b`r^`fŽk sb`qlof^i bp jŠp pbk`fii^ u jŠp k^qro^i m^o^bpqraf^o-i^p molmfb,a^abp dbkbo^ibpab i^p ob`q^p-

Rf qlalp ilp sb`qlobp plk abi bpm^`fl ab alp afjbkpflkbp+ pb kb`bpfq^k pŽil i^palp mofjbo^p b`r^`flkbp m^o^j‹qof`^p '02-1(- Dk bpqb `^pl+ mlabjlp bifjfk^o obkqobi^p alp b`r^`flkbp m^o^j‹qof`^p v l_qbkbjlp i^ obi^`fŽk

'02-2( \%r * j& * [%s w k&< /+

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O`^o\n v api^dji`n q`^ojmd\g`n 472

nrb pb ii^j^ `^p\^d‡i ^\mo`nd\i\ ab i^ ob`q^- Rf \ ;/; N+^nr‹ii^ mrbab bp`of_fopbbk i^ cloj^

]V + k < , 'u , j& +

\

Di kpioj &k*l' bpqŠbk i^ ob`q^: bi k•jbol ] - \ bp i^ k`i_d`io` ab i^ ob`q^-K^ b`r^`fŽk `^oqbpf^k^ '02-2( mrbab q^j_f‹k bp`of_fopbmlo jbafl ab mol,

ar`qlp bp`^i^obp-Rf mlkbjlp K < &]**[&) U < %r)s&) v j < %j) k&) i^ b`r^`fŽk'02-02( pb `lksfboqb bk

%T* L&{ J: M l T{J:L{J+

Di sb`qlo J bp mbombkaf`ri^o ^i sb`qlo ab afob``fŽk = mrbpql nrb J$ = :: ]\ + \] < N: bi sb`qlo K pb ii^j^ q`^ojm ijmh\g ^ i^ ob`q^- K^ ob`q^ `lkpq^ab qlalp ilp mrkqlp T nrb p^qfpc^`bki^ obi^`fŽk %T * L& +J < N-

Dk i^ cfdro^ 02-1 pb jrbpqo^ bi pfdkfcf`^al dblj‹qof`l ab bp^ obi^`fŽk- Klpmrkqlp L v T bpqŠkbk i^ ob`q^v bi sb`qlo kloj^i J bp loqldlk^i ^i T*L+ K^ cfdr,o^ prdfbob nrb bkqobqlalp ilp mrkqlp U ab i^ ob`q^+bi ab jbklo ilkdfqra zGVGGpbl_qfbkb `r^kal U bk i^ molvb``fŽk ab M pl_ob K, C^jlp ^elo^ rk^ abjlpqo^`fŽk^idb_o^f`^ ab bpqbeb`el-

H&

J Ub`qlo kloj^i

>

k

j s

EHFTQ@ 02-1 O`^o\ `i `g kg\ij st lp` k\n\ kjm M ^ji q`^ojm ijmh\g K, @\_\ kpioj W_` g\ m`^o\ n\odna\^` &U+ M' , K < N-

RCMPCK? 02-5- P`\ I g\ m`^o\ _` S0 ^jindno`io` `i oj_jn gjn kpiojn U lp`n\odna\^`i

T$J:L$J)

Page 146: Calculus

362 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

`no\i_j M `i g\ m`^o\ u nd`i_j K pi q`^ojm ij ipgj ijmh\g \ g\ m`^o\, Mjib\hjn

^: EL$JEGGLGG

Qj_j V _` I od`i` gjibdop_ zGVGGƒ _, >_`hƒn* GGVGG< _ ndv n‡gj ndV `n g\ kmj+t`^^d‡i _` M nj]m` K8

U; oK* L$J_ji_` o < ,,l

J$J

A`hjnom\^d‡i, Rf W B I* qbkbjlp W&K < k, K, Rbd•k i^ abpfdr^ia^a abB^r`ev,R`et^ow+ qbkbjlp

GL +JE < ET$ J Gy 0GVGG GGL0G+

0/ nrb fjmif`^ zGVGG19 EL$JE,EEJEE< ^+ Di pfdkl ab fdr^ia^a bp sŠifal pf v pŽilpf T < nJ m^o^rk `fboql bp`^i^o o*bk `rvl `^pl j+ J < W&J < nJ$ J) `lk 0/`r^i n < j+ J,J$ J+ Dpql `ljmibq^ i^ abjlpqo^`fŽk-

Cbi jfpjl jlal mlabjlp abjlpqo^o nrb pf P bp rk mrkql a^al ab S0 klpfqr^al bk i^ ob`q^H) bkqlk`bp m^o^rk `fboql V ab H bi jbklo s^ilo ab GGV, OGGbp E%L* M&$LG.GGLGG+X bpql l`roob `r^kal V , O bp i^ molvb``fŽk ab L * Opl_ob bi sb`qlo kloj^i J+ Di k•jbol

G'N, P(& LG

GGLGG

pb ii^j^ _dno\i^d\ _`n_` `g kpioj O \ g\ m`^o\ I, Di ib`qlo mlaoŒ^obmobpbkq^obpqlp `lk`bmqlp bk rk^ cfdro^ m^ob`fa^ ^ i^ 02-1-

)+&- :WR_PVPV\`

0- Tk^ ob`q^ H ab R 1 `lkqfbkb ilp alp mrkqlp L < ', 2+0( X O < '0+ 0(- Cbqbojfk^o `rŠibpab ilp pfdrfbkqbp mrkqlp bpqŠk bk H+ ^( 'N+N(: _( '/+0(: b( '0+1(: a( '1+0(: b( ',1+0(-

1- Qbplisbo bi bgbo`f`fl 0 pf L < '1+ , 0( X O < ',3+1(-2- Tk^ ob`q^ H ab R0 `lkqfbkb bi mrkql L < ', 2+ 0+0( X bp m^o^ibi^ ^i sb`qlo '0+ , 1+2(-

Cbqbojfk^o `rŠibp ab ilp pfdrfbkqbp mrkqlp bpqŠk bk I, ^( 'N+N+N(: _( '1+ ,0+3(:`( ',1+ ,0+3(: a( ',3+2+ ,1(: b( '1+ ,8+05(-

3- Tk^ ob`q^ I `lkqfbkb ilp alp mrkqlp M < ',2+0+0( X P < '0+1+6(- Cbqbojfk^o `rŠibpab ilp pfdrfbkqbp mrkqlp bpq9k bk H+ ^( ',6+ /+4(: _( ',6+ N+ ,4(: `( ',00+0+00(:a( ',00+ ,0+00(: b( ',0+ !+3(: b( ',e+p+2(+d( ',0+ p,3(-

4- Dk `^a^ `^pl+ abqbojfk^o pf ilp qobpmrkqlp M* O+ O bpqŠk bk rk^ ob`q^-'^( M < '1+0+0(+P < '3+ 0+,0(+ O < '2+ ,0+0(-'e( M < '1+1+2(+P < ',1+2+ 0(+O < ',5+3+ 0(-'b( M < '1+0+0(+P < ',1+2+0(+ O < '4+ ,0+0(-

Page 147: Calculus

Mg\ijn `i `g `nk\^dj `p^g…_`j i+_dh`indji\g 474

5- Dkqob ilp l`el mrkqlp pfdrfbkqbp =) >) X b bpqŠk bk rk^ ob`q^- Cbqbojfk^o qlalp ilppr_`lkgrkqlp ab qobpl jŠp mrkqlp nrb bpqŠk bk iŒkb ob`q^9 =:%/) 0+ 0(+>:%3) ,0+ 0(+b < ',5+ R+ 0(+ A <',1+2+ 0(+ B < '0+ 0+ 0(+ C < ',3+ 3+ 0(+ F < ',02+ 8+ 0(+F < '03+ ,5+ 0(-

6- Tk^ ob`q^ m^p^ mlo bi mrkql L < '0+ 0+0( X bp m^o^ibi^ ^i sb`qlo = < '0+ 1+2(- Nqo^ ob`q^mlo N < '1+ 0+N( bp m^o^ibi^ ^i sb`qlo > < /+7+ 02(-•Cbjlpqo^o nrb i^p alp na_p]o pb`loq^k v abqbojfk^o pr mrkql ab fkqbopb``fŽk-

7- ^( Cbjlpqo^o nrb alp ob`q^p H%L8=& X H%M8>& ab R! pb `loq^k pf v pŽil pf L * Mmboqbkb`b^ i^ bkslisbkqb ifkb^i ab = v >+_( Cbqbojfk^o pf pb `loq^k l kl i^p alp ob`q^p pfdrfbkqbp ab S**8

H < x'i+ 0+,0( * o&,1+0+ 2(y+ H$< x'2+,3+0( * o&,0+4+ 1(y-

8- Rb^ T%n&< L * n= rk mrkql ^o_fqo^ofl bk i^ ob`q^ H%L8=&) pfbkal L < '0+ 1+ 2( X> < '0+ ,1+1(+ X pb^ N < M+2+ 0(-^( B^i`ri^o GGO, W'q(001+ `r^ao^al ab i^ afpq^k`f^ bkqob N v T%n&+_( Cbjlpqo^o nrb e^v bu^`q^jbkqb rk mrkql W'ql( m^o^ bi nrb i^ afpq^k`f^GGO, V'p(eebp jŒkfj^ v `^i`ri^oi^-b( Cbjlpqo^o nrb N + W'ql( bp loqldlk^i ^ >,

0/- Rb^ N rk mrkql kl pfqr^al bk i^ ob`q^ H%L8=& ab R!+^( Rb^ `%n&< GGO, W'q(001+ alkab T%n&< L * n=i Cbjlpqo^o nrb `%n&bp rk mlifkljfl`r^aoŠqf`l bk o v nrb q^i mlifkljfl ^i`^kw^ pr s^ilo jŒkfjl bk rk plil s^ilo ab o*q^i `ljl o < ojj_( Cbjlpqo^o nrb N + W'ql( bp loqldlk^i ^ =+ +

00- C^a^p alp ob`q^p m^o^ibi^p H%L8=& v H%M8=& ab R!+ Cbjlpqo^o nrb l _fbk H%L8=& ;H%M8=& l i^ fkqbopb``fŽk H%L8=& Œ[H%M8=& bp s^`Œ^-

01- C^a^p alp ob`q^p H%L8=& X H%M8>& ab R! nrb kl plk m^o^ibi^p- Cbjlpqo^o nrb i^ fk,qbopb``fŽk bp s^`Œ^ l `lkpq^ ab rk plil mrkql-

02-5 Oi^klp bk bi bpm^`fl br`ifabl k,afjbkpflk^i

Rb abcfkfŽ rk^ ob`q^ bk bi bpm^`fl k,afjbkpflk^i `ljl rk `lkgrkql ab i^ clo,j^ vM * o>w l_qbkfa^ prj^kal ^ rk mrkql a^al M qlalp ilp sb`qlobp ab i^ bk,slisbkqb ifkb^i ab rk sb`qlo > kl kril- Cb jlal m^ob`fal pb abcfkb rk mi^kl+`lk i^ afcbobk`f^ ab nrb prj^jlp ^ M qlalp ilp sb`qlobp ab i^ bkslisbkqb ifkb^i abalp sb`qlobp = u > ifkb^ijbkqb fkabmbkafbkqbp- O^o^ ^pbdro^oklp nrb Rh `lkqfbkbalp sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp+ prmlkbjlp abpab bi mofk`fmfl nrb i ƒ 1-Lr`e^p ab krbpqo^p ^mif`^`flkbp pb obcbofoŠk ^i `^pl i < 2-

CDEHMHBHˆM- Ri ^jiepioj K _` kpiojn _` Si `n pi kg\ij nd `sdno`i pikpioj M s _jn q`^ojm`n gdi`\gh`io` di_`k`i_d`io`n > s ? o\g`n lp`

J < vM * n> * o? Gn*om`\gw,

Cbpfdk^objlp bi `lkgrkql jŠp _obsbjbkqb bp`of_fbkal L < vk * n> * o?w,B^a^ mrkql ab L ab`fjlp nrb bpqŠ `i bi mi^kl- Dk m^oqf`ri^o+ qlj^kal p < o < /+

Page 148: Calculus

364 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

> * o?

EHFTQ@ 02-2 Mg\ij lp` k\n\ kjm M b`i`m\_j kjm > v ?* W np m`g\^d‡i b`jh„omd^\^ji `g kg\ij lp` k\n\ kjm N v `noƒ b`i`m\_j kjm > v ?,

sbjlp nrb O bpqŠbk bi mi^kl- Di `lkgrkql xO * n> * o?w q^j_f‹k pb ii^j^ mi^klnrb m^p^mlo O dbkbo^al mlo = v >+ Br^kal O bp bi lofdbk+ bi mi^kl bp pfjmib,jbkqb i^ bkslisbkqb ifkb^i ab > u ?, K^ cfdro^ 02-2 jrbpqo^ rk mi^kl ab S0 nrbm^p^ mlo bi lofdbk v dbkbo^al mlo = v >) X rk mi^kl nrb m^p^ mlo rk mrkql Okl kril v dbkbo^al mlo bi jfpjl m^oab sb`qlobp-

@elo^ abar`fobjlp ^idrk^p molmfba^abpab ilp mi^klp ^kŠild^p ^ i^p ab i^pob`q^p a^a^p bk ilp qblobj^p abi 02-0 ^i 02-3- K^ mofjbo^ klp jrbpqo^ nrb ilpsb`qlobp+> u ? ab i^ abcfkf`fŽk abi mi^kl vM * n> * o?w mrbab obbjmi^w^opbmlo`r^inrfbo lqol m^onrb qbkd^ i^ jfpj^ bkslisbkqb ifkb^i-

RCMPCK? 02-6- Ajn kg\ijn L < xO * n> * o?w v L&< xO * pb * HCylp` k\n\i kjm `g hdnhj kpioj O nji dbp\g`n nd t n‡gj nd g\ `iqjgq`io` gdi`\g _`> t ? ^jdi^d_` ^ji g\ _` b t C-

A`hjnom\^d‡i, Rf i^ bkslisbkqb ifkb^i ab > u ? bp fdr^i ^ i^ ab a u C+ bpbsfabkqb nrb L < I$+ Qb`Œmol`^jbkqb+prmlkd^jlp nrb L < I$+ Di mi^kl L`lkqfbkb ^ O * = u ^ O * >+ Orbpql nrb bplp mrkqlp bpqŠk ^j_lp q^j_f‹k bkI$) `^a^ rkl ab ilp&= u > ab_b bpq^obk i^ bkslisbkqb ifkb^i ab a u C- @kŠild^,jbkqb+ a u C bpqŠk^j_lp bk i^ bkslisbkqb ifkb^i ab = u >+ Olo `lkpfdrfbkqb i^bkslisbkqb ifkb^i ab = u > bp fdr^i ^ i^ ab a u C-

Di qblobj^ pfdrfbkqbjrbpqo^ nrb bi mrkql O nrb ^m^ob`bbk i^ abcfkf`fŽk abimi^kl vM)n>)o?w mrbab prpqfqrfopbmlo `r^inrfbo lqol mrkql O abi jfpjl mi^kl-

Page 149: Calculus

Mg\ijn `i `g `nk\^dj `p^g…_`j i+_dh`indji\g .10

RCMPCK? 02-7- Ajn kg\ijn L < vM)n> * o?w t L& < wO * n> * o?wb`i`m\_jn kjm gjn hdnhj q`^ojm`n > t ? ^jdi^d_`i pf t n‡gj nd O `noƒ `i L-

A`hjnom\^d‡i, Rf L < J%* bkqlk`bp O bpqŠ `fboq^jbkqb bk L- O^o^ abjlp,qo^o bi ob`Œmol`l+ prmlkd^jlp nrb P bpqŠ bk L+ pb^ P < M * \> * ]?, Slokb,j^p `r^inrfbo mrkql W ab L- Dkqlk`bp W < M * n> * o? m^o^ rklp `fboqlp bp,`^i^obp p u o, Obol M;N+\>+]?* ab jlal nrb U;N)&n+\'>)&o+]'?,Olo `lkpfdrfbkqb W bpqŠ bk I$) `lk il nrb L R: I$+ Cbi jfpjl jlal+ bk`lk,qo^jlp nrb J% R: L+ ^pŒnrb ilp alp mi^klp plk fdr^ibp-

Di mlpqri^al ab i^p m^o^ibi^p ab Dr`ifabp 'qblobj^ 02-2( qfbkb rk^ cloj^^kŠild^ m^o^ ilp mi^klp- @kqbp ab bpq^_ib`bo bpqb qblobj^ kb`bpfq^jlp abcfkfo bim^o^ibifpjl ab alp mi^klp- K^ abcfkf`fŽk bpqŠ prdbofa^ mlo i^ obmobpbkq^`fŽk dbl,j‹qof`^ ab i^ cfdro^ 02-2-

CDEHMHBHˆM- Ajn kg\ijn L < vM * n> * o?w V L& < wO * pB * oAw njik\m\g`gjn nd g\ `iqjgq`io` gdi`\g _` > s ? `n dbp\g \ g\ _` B s C- A`^dhjn o\h]d„ilp` pi q`^ojm W `n k\m\g`gj \g kg\ij L pf W k`mo`i`^` \ g\ `iqjgq`io` gdi`\g _`= t >+

RCMPCK? 02-8- A\_jn pi kg\ij L t pi kpioj P ij k`mo`i`^d`io` \ L+ `sdn+o` pi kg\ij t n‡gj pij K& lp` ^jiod`i` P t `n k\m\g`gj \ K-

A`hjnom\^d‡i, Rb^ L < vM * n> * o?w X `lkpfabobjlp bi mi^klL& < wO * n> * o?w, Dpqb mi^kl `lkqfbkb O u bp dbkbo^al mlo ilp jfpjlp sb`,qlobp = u > nrb bkdbkao^k L- Olo `lkpfdrfbkqb L& bp m^o^ibil ^ L- Rf L! bp lqolmi^kl nrb m^p^ mlo O m^o^ibil ^ L+ bkqlk`bp

IE < vN * n@ * oAw

bk alkab i^ bkslisbkqb ifkb^i ab B v C bp fdr^i ^ i^ ab = v >+ Rbd•k bi qblobj^02-6+ ab_b pbo I! < I$+ Olo il q^kql L& bp bi •kf`l mi^kl mlo P m^o^ibil ^ L-

Di qblobj^ 02-3 klp af`b nrb alp mrkqlp afpqfkqlp abqbojfk^k rk^ ob`q^-Di qblobj^ nrb pfdrb abjrbpqo^ nrb qobp mrkqlp afpqfkqlp abqbojfk^k rk mi^kl+`lk q^i nrb ilp qobp mrkqlp kl bpq‹k ^ifkb^alp-

RCMPCK? 02-0/- Rf M*P V O nji om`nkpiojn ij ndop\_jn `i g\ hdnh\ m`^o\*`sdno` pi kg\ij J t n‡gj pij lp` ^jiod`i` `njn om`nkpiojn, Q\g kg\ij `noƒ _\_jkjm `g ^jiepioj

' 02-3( I < vM * n&N + M' * ooO + M'w,

Page 150: Calculus

366 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

A`hjnom\^d‡i, Rrmlkd^jlp mofjbol nrb rkl ab ilp qobpmrkqlp+ mlo bgbj,mil L) pb^ bi lofdbk- Dkqlk`bp O u N kl bpqŠkbk rk^ jfpj^ ob`q^ nrb m^pbmlo bilofdbk ab jlal& nrb plk ifkb^ijbkqb fkabmbkafbkqbp-Olo `lkpfdrfbkqb+ bkdbkao^krk mi^kl nrb m^p^ mlo bi lofdbk+ pb^ ‹pqb

J% < vnN * oOw,

Dpqbmi^kl `lkqfbkb ilp qobpmrkqlp N+ O v O,Cbjlpqobjlp ^elo^ nrb K& bp bi •kf`l mi^kl nrb `lkqfbkb bplp qobpmrkqlp-

Br^inrfbo lqol mi^kl nrb m^pbmlo bi lofdbk qfbkb i^ cloj^

J! < vn> * oBw*

pfbkal = X > ifkb^ijbkqb fkabmbkafbkqbp-Rf L! `lkqfbkb P v N) qbkbjlp

'02-4( M < \> * ]?* O < ^> * _B*

m^o^ `fboqlp bp`^i^obp \* \) `9 ^+ Krbdl+ qla^ `lj_fk^`fŽk ifkb^i ab P u N bpq^j_f‹k rk^ `lj_fk^`fŽk ifkb^i ab = u >) ^pŒnrb I$m I!+

O^o^ abjlpqo^o nrb L! Q I$) _^pq^ abjlpqo^o nrb = u > plk `^a^ rkl abbiilp rk^ `lj_fk^`fŽk ifkb^i ab O u O, Lriqfmif`^kal i^ mofjbo^ b`r^`fŽk '02-4(mlo ^ u i^ pbdrka^ mlo \ u obpq^kal+bifjfk^jlp > u pb l_qfbkb

d\_ + ]^'> < _N + ]O,

K^ afcbobk`f^ \_ + ]` kl mrbab pbo`bol+ ab lqol jlal P v O pboŒ^kabmbkafbk,qbp-Olo il q^kql mlabjlp afsfafo mlo \_ + ]` u bumobp^o> `ljl `lj_fk^`fŽkifkb^i ab P u N+ @kŠild^jbkqb+ mlabjlp bumobp^o> `ljl `lj_fk^`fŽk ifkb^i abO u O* `lk il nrb J! R J%, Dpql abjrbpqo^ bi qblobj^ `r^kal rkl ab ilp qobpmrkqlp L) P+ N bp bi lofdbk-

O^o^ abjlpqo^o bi qblobj^ bk bi `^pl dbkbo^i+pb^ L bi `lkgrkql '02-3(+a < P , M u C < O + M, Cbjlpqo^jlp mofjbol nrb a u A plk ifkb^ijbkqbfkabmbkafbkqbp- Rf kl+ qbkaoŒ^jlp C < o` m^o^ ^id•k bp`^i^o o* aŠkalklpN * L < n%M* L&) l N < L * n%M* L&) bk `lkqo^af``fŽk `lk bi eb`el ab nrbM* P+ O kl bpqŠkbk i^ jfpj^ ob`q^- Olo `lkpfdrfbkqb bi `lkgrkql J bp rk mi^klnrb m^p^mlo M u bpqŠdbkbo^al mlo bi m^oab sb`qlobp ifkb^ijbkqb fkabmbkafbkqbpa u A, Dpqbmi^kl `lkqfbkb ilp qobpmrkqlp M*O u O 'qlj^jlp n < 0+o < N m^o^l_qbkbo P+ u n < N+ o < 0 m^o^ l_qbkbo N&+ @elo^ qbkbjlp nrb abjlpqo^o nrb‹pqb bp bi •kf`l mi^kl nrb `lkqfbkb L) O u N

Rb^ J% `r^inrfbo mi^kl nrb `lkqbkd^ M*O u O, X^ nrb J% bp rk mi^kl nrb`lkqfbkb L) qbkbjlp

J% < vM * n> * oBw

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Mg\ijn t api^dji`n q`^ojmd\g`n 478

m^o^ rk `fboql m^o ab sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp = v >+ Rb^ Iw < um= (* o?w bi mi^kl nrb m^p^ mlo bi lofdbk dbkbo^al mlo bi jfpjl m^o > v ?, Dsfabk,qbjbkqb L& `lkqfbkb rk sb`qlo W pf v pŽil pf Iw `lkqfbkb T*L+ Orbpql nrb L&`lkqfbkb O v N) bi mi^kl I w `lkqfbkb b < O , L v C < N * L+ Obol ^`^_^jlpab abjlpqo^o nrb bufpqb rk mi^kl v pŽil rkl nrb `lkqfbkb N+ b v C mrbpql nrbb v C plk ifkb^ijbkqb fkabmbkafbkqbp- Olo `lkpfdrfbkqb Jx < um_ * oAw* abjlal nrb L& < vM * pb * oAw < L- Dpql `ljmibq^ i^ abjlpqo^`fŽk-

Dk bi qblobj^ 02-4 pb abjlpqoŽ nrb alp sb`qlobp ab Si plk ifkb^ijbkqb ab,mbkafbkqbp pf v pŽil pf bpqŠk bk rk^ jfpj^ ob`q^ nrb m^p^ mlo bi lofdbk- Di qblob,j^ nrb pfdrb bp bi `loobpmlkafbkqb ^i `^pl ab qobp sb`qlobp-

RCMPCK? 02-00- Qm`n q`^ojm`n >* ?* b _` Si nji gdi`\gh`io` _`k`i_d`io`nnd t n‡gj nd `noƒi `i pi hdnhj kg\ij lp` k\n\ kjm `g jmdb`i,

A`hjnom\^d‡i, Rrmlkd^jlp nrb >* ?* b plk abmbkafbkqbp- Olabjlp bkqlk,`bp bumobp^o rkl ab ilp sb`qlobp `ljl `lj_fk^`fŽk ifkb^i ab ilp lqolp alp+ pb^b < n> * o?, Rf > X ? plk fkabmbkafbkqbp+ bkdbkao^k rk mi^kl nrb m^p^ mlobi lofdbk u b bpqŠ bk bpqb mi^kl- Rf = X > plk abmbkafbkqbp+ bkqlk`bp =) > X bbpqŠk pfqr^alp bk rk^ jfpj^ ob`q^ nrb m^p^ mlo bi lofdbk+ v mlo q^kql bpqŠk bk`r^inrfbo mi^kl nrb m^pb mlo bi lofdbk nrb `lkqfbkb ilp qobp mrkqlp F+)? X a-

O^o^ abjlpqo^o bi ob`Œmol`l+ prmlkd^jlp nrb =) >) b bpqŠk bk rk jfpjlmi^kl nrb m^p^ mlo bi lofdbk+ pb^ ‹pqb bi mi^kl L- Rf = X > plk abmbkafbkqbp+ bk,qlk`bp =) > X b plk abmbkafbkqbp+ v kl e^v k^a^ jŠp nrb abjlpqo^o- Rf = X >plk fkabmbkafbkqbp+ dbkbo^k rk mi^kl L& nrb m^p^ mlo bi lofdbk- Rbd•k bi qblob,j^ 02-0/+ bufpqb rk mi^kl v pŽil rkl nrb m^p^ mlo N v `lkqfbkb = v >+ Olo `lk,pfdrfbkqb L& < L- Orbpql nrb b bpqŠ bk bpb mi^kl+ ab_b pbo b < n> * o?* `lkil nrb >* > v b plk abmbkafbkqbp-

)+&/ CYN[\` e Sb[PV\[R` cRPa\_VNYR`

K^ `loobpmlkabk`f^ nrb ^pl`f^ ^ `^a^ m^o ab k•jbolp ob^ibp p v o bi sb`qloM * n> * o? bk bi mi^kl L < vM* n> * o?w bp lqol bgbjmil ab crk`fŽk sb`qlof^i-Dk bpqb `^pl+ bi aljfkfl ab i^ crk`fŽk bp bi `lkgrkql ab qlalp ilp m^obp ab k•jb,olp ob^ibp %m)0( v pr ob`loofal bp bi mi^kl L- Rf abpfdk^jlp i^ crk`fŽk mlo W u prps^ilobp mlo W'p+i(+ bkqlk`bp m^o^ `^a^ m^o %m)n& qbkbjlp

'02-5( U&n*o' < M * n> * o?,

Dpq^ crk`fŽk W bp rk^ crk`fŽk sb`qlof^i ab alp s^of^_ibp ob^ibp- Klp bp`^i^obp p v opb ii^j^k m^oŠjbqolp- u i^ b`r^`fŽk '02-5( bp i^ b`r^`fŽk m^o^j‹qof`^ l sb`qlof^i

Page 152: Calculus

37. >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

abi mi^kl- Dpql bp il jfpjl nrb i^ obmobpbkq^`fŽkab rk^ ob`q^ jbaf^kqb rk^crk`fŽk sb`qlof^i ab rk^ s^of^_ib ob^i- K^ mobpbk`f ab alp m^oŠjbqolp bk i^ b`r^,`fŽk '02-5( a^ ^i mi^kl rk^ `r^ifa^a _fafjbkpflk^i- Br^kal `^a^ sb`qlo bpqŠbkS 2 X pb bumobp bk crk`fŽk ab prp `ljmlkbkqbp+ mlo bgbjmil

v T%m)o' < %r)t* w(+

i^ b`r^`fŽk sb`qlof^i '02-5( mrbab obbjmi^w^opbmlo qobpb`r^`flkbp bp`^i^obp+

r < OH* m[f * o]f)

Klp m^oŠjbqolp p v o pfbjmob mrbabk bifjfk^opb bkqobbp^p qobpb`r^`flkbp l_qb,kfbkal rk^ b`r^`fŽk ifkb^i ab i^ cloj^ \s * ]t * ^u << _* ii^j^a^ b`r^`fŽk `^olqbpf^k^abi mi^kl- Olkboklp pbdrfa^jbkqb rk bgbjmil-

DIDLOKN- Rb^ L << vM * n> * o?w* alkab M << '0+ 1+ 2(+ > << '0+ 1+ 0(+X ? << '0+,3+ ,0(- K^ b`r^`fŽk sb`qlof^i `loobpmlkafbkqb bp

U&n*o' < '0+1+2(* p'i+1+0(* o&g*,3+ ,0(-

Cb bpq^ l_qbkboklp i^p qobpb`r^`flkbp m^o^j‹qof`^p

s;g)n)o* t < 1 * 1o , 2o* u;1)n+o,

O^o^ l_qbkbo rk^ b`r^`fŽk `^oqbpf^k^+mlkboklp i^ mofjbo^ v i^ qbo`bo^b`r^`flkbpbk i^ cloj^ s + 0 << p * o*w, 2 << n+o, Rrj^kal v irbdl obpq^kal bp^pb`r^,`flkbp+ bk`lkqo^oklp nrb 1p << s * w , 3+ 0o << s+ w* 1- Rrpqfqrvbkal bk i^pbdrka^ b`r^`fŽk ab i^ v+ l_qbkboklp i^ b`r^`fŽk `^oqbpf^k^ s * v , 2w << ,5-Ulisbobjlp ^ bpqraf^o i^p b`r^`flkbp `^oqbpf^k^pbk i^ pb``fŽk 02-05-

)+&0 :WR_PVPV\`

0- Rb^ L < uL * m= * f>v) alkab L < '0+ 1+ ,2(+ = < '2+ 1+ 0( X > < '0+ N+3(- Cbqbo,jfk^o `rŠibp ab ilp pfdrfbkqbp mrkqlp bpqŠk bk L-

'^( '0+1+N(: '_( '0+1+0(: 'b( '5+3+5(: 'a( '5+5+5(: 'b( '5+5+,4(-1- Klp qobpmrkqlp L < '0+ 0+ ,0(+ O < '2+ 2+ 1( X N < '2+ ,0+ ,1( abqbojfk^o rk mi^kl

L- Cb`fo `rŠibp ab ilp mrkqlp pfdrfbkqbp bpqŠk bk L-'^( '1+1+ p(: '_( '3+ N+,p(: '`( ',2+0+ ,2(: 'a( '2+0+2(: 'b( 'N+N+N(-

2- Cbqbojfk^o i^p b`r^`flkbp bp`^i^obp m^o^j‹qof`^p m^o^ `^a^ rkl ab ilp mi^klp pfdrfbkqbp-^( Di mi^kl nrb m^p^ mlo '0+ 1+ 0( X bpqŠdbkbo^al mlo ilp sb`qlobp 'N+ 0+N( X '0+ 0+3(-_( Di mi^kl nrb m^p^ mlo '0+ 1+ 0(+ 'N+ 0+N( X '0+ 0+3(-

3- Tk mi^kl L qfbkb i^p b`r^`flkbp bp`^i^obp m^o^j‹qof`^p-

s < 0 * n + 1.+ t < 1 * n * 3.+ u;0n)-,

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Mmj_p^oj q`^ojmd\g 480

^( Cbqbojfk^o `rŠibp ab ilp pfdrfbkqbp mrkqlp bpqŠk bk L9 'N+ N+ N(+ '0+ 1+ N(+'1+ ,2+ ,2(-_( G^ii^o ilp sb`qlobp L)= u > q^ibp nrb L < uL * m= * n>v+

4- Rb^ L bi mi^kl abqbojfk^al mlo qobp mrkqlp M* P+ O kl ^ifkb^alp-^( Rf k* l* m plk qobp bp`^i^obp q^ibp nrb k * l * m< 0+ abjlpqo^o nrb kM * lN * mObpqŠ bk L-_( Cbjlpqo^o nrb qlal mrkql ab L bp ab i^ cloj^ kM)lN)mO* alkab k)l)m;g,

5- Cbqbojfk^o i^ b`r^`fŽk ifkb^i `^oqbpf^k^ ab i^ cloj^ [r * ]t * ]t < _ m^o^ `^a^ rklab ilp mi^klp pfdrfbkqbp-^( Oi^kl nrb m^p^ mlo '1+ 2+ 0( X bpqŠ dbkbo^al mlo '2+ 1+ 0( X ',0+ ,1+ ,2(-_( Oi^kl nrb m^p^ mlo '1+ 2+ 0(+ ',1+ ,0+ ,2( X '3+2+ ,0(-b( Oi^kl nrb m^p^ mlo '1+ 2+ 0( X bp m^o^ibil ^i mi^kl nrb m^p^ mlo bi lofdbk v bpqŠdbkbo^al mlo '1+ N+ ,1( X '0+ 0+ 0(-

6- K^ b`r^`fŽk `^oqbpf^k^ ab rk mi^kl L bp 0r * 4v * w < 8-^( Cbqbojfk^o `rŠibp ab ilp pfdrfbkqbp mrkqlp bpqŠk bk L9 'N+ ,1+ ,0(+ ',0+ ,1+1(+'2+ 0+,4(-_( G^ii^o ilp sb`qlobp L) = X > q^ibp nrb L < uL * m= * n>v+

7- Blkpfabobjlp ilp alp& mi^klp K < vM * n> * o?w X K&< wO * pB * oAw* alkabL < '0+0+0(+ = < '1+ ,0+2(+ > < ',0+ N+1(+ P < '1+ 2+0(+ B < '0+ 1+ 2( X C < '2+1+0(-G^ii^o alp mrkqlp afpqfkqlp pfqr^alp bk i^ fkqbopb``fŽk L 'g I$+

8- C^alp rk mi^kl L < uL * m= * n>v) alkab L < '1+ 2+ 0(+= < '0+ 1+ 2( X > < '2+ 1+ 0(+X lqol mi^kl L& `rv^ b`r^`fŽk `^oqbpf^k^ bp s + 1v * w< N-^( Cbqbojfk^o pf K v K& plk m^o^ibilp-_( G^ii^o alp mrkqlp bk i^ fkqbopb``fŽk K& 'g K! pf L! qfbkb i^ b`r^`fŽk `^oqbpf^k^

r * 0t * t < N+

0/- Rb^k H i^ ob`q^ nrb m^p^ mlo '0+ 0+ 0( m^o^ibi^ ^i sb`qlo '1+ ,0+2( X L bi mi^kl nrbm^p^ mlo '0+ 0+ , 1( X dbkbo^al mlo ilp sb`qlobp '1+ 0+ 2( X 'N+ 0+ 0(- Ool_^o nrb bufpqbrk mrkql v pŽil rkl bk i^ fkqbopb``fŽk H 'g L X abqbojfk^oil-

00- Tk^ ob`q^ `lk sb`qlo ab afob``fŽk T bp m^o^ibi^ ^ rk mi^kl L pf T bp m^o^ibil ^ L-Rb^ H i^ ob`q^ nrb m^p^ mlo '0+ 0+ 0( X bp m^o^ibi^ ^i sb`qlo '1+ ,0+ 2(- Cbqbojfk^o pfH bp m^o^ibi^ ^ `^a^ rkl ab ilp mi^klp pfdrfbkqbp-^( Oi^kl nrb m^p^ mlo '0+ 0+ ,1( X dbkbo^al mlo '1+ 0+2( X ]+ 0+ 0(-_( Oi^kl nrb m^p^ mlo '0+ 0+ ,1(+ '2+ 4+ 1( X '1+ 3+ ,0(-b( mf^kl ab b`r^`fŽk `^oqbpf^k^ s * 1v * 2w < ,2-

01- Clp mrkqlp M v P bpqŠk bk rk mi^kl L- Cbjlpqo^o nrb qlal mrkql ab i^ ob`q^ nrbm^p^ mlo L v O mboqbkb`b q^j_f‹k ^ L-

02- C^a^ i^ ob`q^ H nrb m^p^ mlo '0+ 1+ 2( X bp m^o^ibi^ ^i sb`qlo '0+ 0+ 0(+ X a^al rk mrkql'1+ 2+ 4( nrb kl bpqŠ bk I, G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl L nrb m^p^ mlo'1+ 2+ 4( X nrb `lkqfbkb qlalp ilp mrkqlp ab I,

03- C^alp rk^ ob`q^ I v rk mrkql M kl pfqr^al bk I, Cbjlpqo^o nrb bufpqb rk mi^kl vpŽil rkl nrb m^p^ mlo M v `lkqfbkb qlalp ilp mrkqlp ab I,

)+&1 C_\QbPa\cRPa\_VNY

Dk jr`e^p ^mif`^`flkbp abi „idb_o^ sb`qlof^i ^ mol_ibj^p ab FbljbqoŒ^ uab Lb`Škf`^ obpriq^ •qfi afpmlkbo ab rk j‹qlal cŠ`fi ab l_qbkbo rk sb`qlo mbo,mbkaf`ri^o ^ `^a^ rkl ab alp sb`qlobp a^alp = u >+ Dpql pb `lkpfdrb `lk bi mol,ar`ql sb`qlof^i = W > nrb pb abcfkb ^pŒ9

Page 154: Calculus

370 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

CDEHMHBHˆM- P`\i > < '^i& \0* \n' t ? < &]/* ]0* ]\' _jn q`^ojm`n _` S\,Pp kmj_p^oj q`^ojmd\g > U? &`i `no` jm_`i' n` _`adi` ^jhj `g q`^ojm

@ m^oqfoab bpq^abcfkf`fŽk pb abar`bk `lk c^`fifa^a i^p molmfba^abppfdrfbkqbp-

RCMPCK? 02-01- M\m\ oj_jn gjn q`^ojm`n >* ?* B _` U^ V k\m\ oj_j iˆh`+mj m`\g ` o`i`hjn8

]( = V > < *%> t =&_( = V %>* B( < %= V >& * %= V B(b( ]%= W >& < %]=&W >)a( > , &> V ?' < N &jmojbji\gd_\_ m`nk`^oj \ >'*b( ?% &> V ?' < N &jmojbji\gd_\_ m`nk`^oj \ 5&)b( GG?V @G01< GG?G0100@G01

, &>%?'0 &d_`iod_\_ _` I\bm\ib`'*c( > V ? < N nd s n‡gj nd > s ? nji gdi`\gh`io` _`k`i_d`io`n,

&ndh`om…\\go`mi\_\'*&g`t _dnomd]podq\'*

A`hjnom\^d‡i, K^p m^oqbp (+ _( v b( obpriq^k fkjbaf^q^jbkqb ab i^ abcfkf,`fŽk v pb abg^k `ljl bgbo`f`flp m^o^bi ib`qlo- O^o^ abjlpqo^o a(+ l_pbosbjlp nrb

K^ m^oqbb( pb abar`b abi jfpjl jlal+ l mrbab abar`fopb ab ^( v a(- O^o^ ab,jlpqo^o c(+ bp`of_fjlp

u

u `ljmol_^jlp irbdl nrb ilp alp pbdrkalp jfbj_olp `lfk`fabk-K^ molmfba^a c( jrbpqo^ nrb > W ? < N pf v pŽil pf &>%?'0 < FF>TFF?T,

Rbd•k i^ abpfdr^ia^a ab B^r`ev,R`et^ow 'qblobj^ F/+0&) bpl l`roob pf u pŽil pf rklab ilp sb`qlobp bp bi molar`ql abi lqol mlo rk bp`^i^o- Cf`el ab lqol- jlal+= W > < M pf u pŽil pf = u > plk ifkb^ijbkqb abmbkafbkqbp+il nrb abjrbpqo^ d(-

DIDLOKNR- K^pm^oqbp ( v d( abjrbpqo^k nrb = W = < N- Cb i^ abcfkf`fŽkab molar`ql sb`qlof^i bk`lkqo^jlp nrb

crd:e) dre:c) erc:d+

Page 155: Calculus

Mmj_p^oj q`^ojmd\g 482

Di molar`ql sb`qlof^i kl bp ^pl`f^qfsl- Olo bgbjmil+ qbkbjlp

c u %cu g( < c u e < , g pfk bj_^odl %cV c&V f < N V f < N -

Di qblobj^ nrb pfdrb jrbpqo^ alp molmfba^abp crka^jbkq^ibp abi molar`qlsb`qlof^i-

RCMPCK? 02-02- P`\i > t ? _jn q`^ojm`n gdi`\gh`io` di_`k`i_d`io`n `i S\,P` od`i`8

]( Ijn q`^ojm`n >* ?* > V ? nji gdi`\gh`io` di_`k`i_d`io`n,_( Qj_j q`^ojm K _` U^ jmojbji\g \ > t ? ndhpgoƒi`\h`io` `n `g kmj_p^oj

_` pi `n^\g\m kjm > V ?,

A`hjnom\^d‡i, Rb^ B < > W ?, Dkqlk`bp B ;/; N mrbp > v ? plk ifkb^ijbkqbfkabmbkafbkqbp- C^alp ilp bp`^i^obp \* ]* ` q^ibp nrb \> * ]? * ^@ < N+ clojb,jlp bi molar`ql bp`^i^o ab `^a^ jfbj_ol mlo B u qbkfbkal bk `rbkq^ nrb> , B < ?% B < N bk`lkqo^jlp ` < N- Dpql a^ \> * ]? <, N+ `lk 0/ nrb[ < ] < N v^ nrb > u ? plk fkabmbkafbkqbp- Dpql abjrbpqo^ ^(-

Rb^ J rk sb`qlo `r^inrfbo^ loqldlk^i ^ i^ sbw ^ = u ^ >) v pb^ B < = W >+Cbjlpqo^objlp nrb

%J$A> < %J$J&%? ?&+

Dkqlk`bp ab i^ abpfdr^ia^a ab B^r`ev,R`et^ow 'qblobj^ 01-2( obpriq^ nrb J bpbi molar`ql ab B mlo rk bp`^i^o-

Orbpql nrb =) > X B plk ifkb^ijbkqb fkabmbkafbkqbp+ p^_bjlp+ bk sfoqra abiqblobj^ 01- 0/ b(+ nrb dbkbo^k S\, Dk m^oqf`ri^o+ dbkbo^k K* ab jlal nrb ml,abjlp bp`of_fo

K < \> * ]B * ^@

m^o^ `fboqlp bp`^i^obp \* \) ^, Dpql klp a^

J$ J < J$ %[=* \ > * ^@' < _J +B

mrbpql nrb J += < J$ > < N- S^j_f‹k+ v^ nrb B&= < B&> < N+ qbkbjlp

B• K < B• &\> * Fd?* ^@' < ^@%B-

Olo `lkpfdrfbkqb+ &K%K'&@%B( < &^K%@'&@ , B( < &K%@'&^@%B( < &K%@'0*0/nrb `ljmibq^ i^ abjlpqo^`fŽk-

Page 156: Calculus

372 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

Di qblobj^ 02-01 klp c^`fifq^ i^ fkqbomobq^`fŽkdblj‹qof`^ abi molar`ql sb`,qlof^i- Olo i^p molmfba^abpa( u b(+ p^_bjlp nrb = W > bp mbombkaf`ri^o pfjri,qŠkb^jbkqb ^ = u ^ >+ Br^kal bi sb`qlo = V > pb obmobpbkqdblj‹qof`^jbkqbjbaf^kqb rk^ cib`e^+i^ afob``fŽk ab i^ cib`e^ abmbkab ab i^p mlpf`flkbp obi^qfs^p

fH,f

^( Rfpqbj^ `lloabk^al lofbkq^al bkpbkqfal afob`ql-

f

H,ef

> s?

_( Rfpqbj^ `lloabk^al lofbkq^albk pbkqfal obqoŽdo^al-

EHFTQ@ 02-3 Mjnd^dji`n m`g\odq\n_` >* ? W > W ?,

ab ilp qobpsb`qlobp rkfq^oflp `lloabk^alp- Rf c)f X e pb `lil`^k `ljl pb sb bki^ cfdro^ 02-3 ^(+ pb af`b nrb cloj^k rk ndno`h\ ^jjm_`i\_j jmd`io\_j `i n`iod_j_dm`^oj, Dk bpqb`^pl+ i^ afob``fŽk ab > W%?bpqŠabqbojfk^a^ mlo i^ ~obdi^ abi^ j^kl abob`e^‚- Dpql bp+`r^kal = dfo^ e^`f^ > ab jlal nrb ilp abalp ab i^j^kl abob`e^ pb•^ibk bi pbkqfal ab i^ olq^`fŽk+bkqlk`bp bi mrid^o fkaf`^ i^ af,ob``fŽk ab = W > 'prmlkfbkal+ ab ^`rboal `lk il nrb pb afp`rqb+ nrb bi mrid^obpqŠmbombkaf`ri^o^ ilp lqolp abalp(- Dk rk pfpqbj^ `lloabk^al lofbkq^al bk pbk,qfal obqoŽdo^al+ ljl bk i^ cfdro^ 02-3 _(+ i^ afob``fŽk ab = W A pb fksfboqbumrbab abqbojfk^opb `lk rk^ `loobpmlkafbkqb obdi^ ab i^ j^kl fwnrfboa^-

K^ ilkdfqra ab = W > qfbkb rk^ fkqbomobq^`fŽkdblj‹qof`^ fkqbobp^kqb-Rf =X ? plk sb`qlobp kl krilp nrb cloj^k rk^ Škdril `* pfbkal N z ` x 6S+ mlabjlpbp`of_fo= +> < GY?GGYY@eplp ` bk i^ molmfba^a c( abi qblobj^ 02-01 l_qbkfbkal

ab i^ nrb obpriq^

GG?u >EE < GG?00GG@GGpbk‹ -

Orbpql nrb GG@GGpbk ` bp i^ ^iqro^ abi m^o^ibildo^jl abqbojfk^al mlo = u > 'sbocfdro^ 02-4(+sbjlp nrb g\ gjibdop_ _` > V ? `n dbp\g\g ƒm`\ _` `n` k\m\g`gjbm\hj,

Page 157: Calculus

Bg kmj_p^oj q`^ojmd\g `skm`n\_j `i ajmh\ _` _`o`mhdi\io` 373

„ob^ < GG?uA00

EHFTQ@ 02-4 I\ gjibdop_ _` > V ? `n dbp\g \g ƒm`\ _`g k\m\g`gjbm\hj _`o`mhdi\_j kjm > v ?,

)+&)( :Y ]_\QbPa\ cRPa\_VNYRd]_R`NQ\ R[ S\_ZN QRQRaR_ZV[N[aR

K^ cŽojri^ nrb abcfkb bi molar`ql sb`qlof^i mrbab mlkbopb bk cloj^ jŠp`ljm^`q^ `lk i^ ^vra^ ab ilp abqbojfk^kqbp- Rf \* ]* `* _ plk `r^qol k•jbolp+ i^afcbobk`f^ \_ + ]` pb abpfdk^ ^ jbkral `lk bi pŒj_lil

u nrb pb ii^j^ _`o`mhdi\io` 'ab pbdrkal loabk(- Klp k•jbolp \* ]* `* _ plk prp`g`h`iojn* v ab`fjlp nrb bpqŠk `lil`^alp bk alp adg\n \* ] v `* _ v bk alp ^jgph+i\n \* ` u ]* _, N_p‹osbpb nrb rk fkqbo`^j_fl ab i^p alp cfi^p l ab i^p alp `lirj,k^p pŽil `^j_f^ bi pfdkl abi abqbojfk^kqb- Olo bgbjmil+ mrbpql nrb \_ + ]` :; +&]` + \_'* qbkbjlp

Rf bumobp^jlp `^a^ rkl ab ilp `ljmlkbkqbp abi molar`ql sb`qlof^i `ljl rkabqbojfk^kqb& ab loabk alp+ i^ cŽojri^ nrb abcfkb = W > qlj^ i^ cloj^

Dpql mrbab q^j_f‹k bumobp^opbbk crk`fŽk ab ilp sb`qlobp <"d,e `ljl pfdrb9

'H2-6(

Page 158: Calculus

374 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

Klp abqbojfk^kqbp ab qbo`boloabk pb bp`of_bk `lk qobpcfi^pv qobp`lirjk^pv mrbabk abcfkfopbbk crk`fŽk ab ilp abqbojfk^kqbp ab loabk alp mlo i^ cŽojri^

'02-7(

Dpql bp il nrb pb ii^j^ bi ~abp^ooliil‚ ab rk abqbojfk^kqb mlo ilp bibjbkqlpab i^ mofjbo^ cfi^- N_pbosbjlp nrb bi abqbojfk^kqb abi pbdrkal jfbj_ol nrbjriqfmif`^ [f mrbab l_qbkbopb abi abqbojfk^kqb abi mofjbo jfbj_ol prmofjfbkali^ cfi^ v i^ `lirjk^ bk i^p nrb ^m^ob`b[f$ Klp lqolp alp abqbojfk^kqbp abi pbdrk,al jfbj_ol pb l_qfbkbk abi jfpjl jlal-

Dk bi Ulirjbk 00 pb bpqraf^k ilp abqbojfk^kqbp ab loabk j^vlo nrb qobp-Mlp molmlkŒ^jlp •kf`^jbkqb fkqolar`fo ilp abqbojfk^kqbp ab Žoabkbp pbdrkal vqbo`bol m^o^ afpmlkbo ab rk fkpqorjbkql m^o^ bp`of_fo `fboq^pcŽojri^p bk cloj^`ljm^`q^ nrb mbojfq^ ob`loa^oi^p `lk j^vlo c^`fifa^a-

Klp abqbojfk^kqbp qfbkbk mibkl pfdkfcf`^al pf ilp bibjbkqlp ab i^ mofjbo^cfi^ plk sb`qlobp- Olo bgbjmil+ pf bp`of_fjlp bi abqbojfk^kqb

d f e

u ~abp^oolii^jlp‚ pbd•k i^ obdi^ bpq^_ib`fa^ bk '02-7(+ bk`lkqo^jlp nrb bi ob,priq^al `lfk`fab `lk bi pbdrkal jfbj_ol ab '02-6(- Cb lqol jlal+ mlabjlp bp,`of_fo i^ abcfkf`fŽk abi molar`ql sb`qlof^i = W > bk i^ cloj^ `ljm^`q^ pfdrfbkqb9

d f e

> u ? < [f [/ ^^

]R ], ^^

Olo bgbjmil+ m^o^ `^i`ri^o bi molar`ql sb`qlof^i ab > < 1: , 6e * 1f v> < 3: * 0e) bp`of_fjlp

Page 159: Calculus

Be`m^d^djn 486

)+&)) :WR_PVPV\`

0- Rb^k > < +m c * 0f* ? <1: * f , f* b < d)0e * 0f, B^i`ri^o `^a^ rkl ab ilp pfdrfbk,qbp sb`qlobp bk crk`fŽk ab c)g+e7

']( = t >8'^( ? t A:'b( B u =8

'a( =r 'B t =&8

'a( %= t >& t B:'b( = t %> r?&8

'c( %= u A( u >8'd( %= * >& t %= * ?&8

'f( %= u >& u %= r?&+

1- Dk `^a^ `^pl e^ii^o rk sb`qlo ab ilkdfqra 0 bk R 2 loqldlk^i ^ i^ sbw ^ = v ^ >7

']( > < c * f * f*'_( = < 1: , 2g * 1e)'b( > < c * 1g* 1f*

? < 1: * 1e + f9

? < ,:&*4g* 5f9

> < ,2: * 1g, e+

2- Dk `^a^ `^pl rqfifw^o bi molar`ql sb`qlof^i m^o^ `^i`ri^o bi Šob^ abi qofŠkdril ab s‹oqf,`bp =) >) B9

'^( = < 'N+1+1(+'_( > < ',1+2+ 0(+'b( > < 'N+N+N(+

? < '1+ N+,0(+? < '0+ ,2+3(+> < 'N+ 0+0(+

B < '2+3+ N(:B < '0+1+0(:B<'i+N+i(-

3- Rf > < 1: * 3e * 1f* ? <1: * 6g * 2f* v b < 2: * 2: * 4f* bumobp^o bi molar`ql sb`,qlof^i %= * a( u %> * =& bk crk`fŽk ab c)d*e+

4- Cbjlpqo^o nrb GG?W @GG< Keh GG@ Gpf v pŽil pf = v > plk loqldlk^ibp+5- C^alp alp sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp = v- > ab U^&Rb^ b < %> W =& * >+

^( Cbjlpqo^o nrb = bp loqldlk^i ^ > * a-_( Cbjlpqo^o nrb bi Škdril %F nrb cloj^k ? v b p^qfpc^`b f6S ; '( ; 6S-

b( Rf GG@GG< 0 X GG@W ?GG< 1+ `^i`ri^o i^ ilkdfqra ab a-6- Rb^k = v > alp sb`qlobp loqldlk^ibp bk U^&qbkfbkal `^a^ rkl ilkdfqra 0-

^( Cbjlpqo^o nrb =) >) = W > bp rk^ _^pb loqlkloj^i m^o^ U^&_( Rb^ b < %= W >& W =+ Cbjlpqo^o nrb GGAhG< 0-b( So^w^o rk^ cfdro^ nrb jrbpqo^ i^ obi^`fŽk dblj‹qof`^ bkqob =) >) X = W > X rqfifw^obp^ cfdro^ m^o^ l_qbkbo i^p obi^`flkbp

%= u >& u = < >) %= V >& V > < *=+

a( Cbjlpqo^o i^p obi^`flkbp ab i^ m^oqb b( ^idb_o^f`^jbkqb-7- ^( Rf = W > < N X = +> W N+ rkl mlo il jbklp ab ilp sb`qlobp = l > bp kril- Cb,

jlpqo^o bpq^ molmlpf`fŽk v a^o pr fkqbomobq^`fŽk dblj‹qof`^-_( C^al = +‹ N- Rf = W > < = W a v ={ > < ={ a+ abjlpqo^o nrb > < a-

8- Rb^k > < &1:, e * 0f v b < 2: * 2e + f,

^( G^ii^o rk sb`qlo > q^i nrb = W > < a- ƒG^v jŠp ab rk^ plir`fŽk>_( G^ii^o rk sb`qlo > q^i nrb = W > < a u = +> < 0- ƒG^v jŠp ab rk^ plir`fŽk>

0/- C^alp rk sb`qlo kl kril = v rk sb`qlo b loqldlk^i ^ =) ^j_lp bk R +Cbjlpqo^o nrbbufpqb rk plil sb`qlo > q^i nrb = W > < b v = +> < 0- ^

00- Sobp s‹oqf`bp ab rk m^o^ibildo^jl plk ilp mrkqlp = < '0+ N+ 0(+ > < ',0+ 0+ 0(+b < '1+ ,0+1(-

^( G^ii^o qlalp ilp mrkqlp C nrb mrbabk pbo bi `r^oql s‹oqf`b abi m^o^ibildo^jl-_( B^i`ri^o bi Šob^ abi qofŠkdril =>_+

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376 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

01- C^alp alp sb`qlobp kl m^o^ibilp = v > ab R1 pfbkal = +> < 1+ GG?GG< 0+ GG@GG< 3- Rb^B < /%= W >& * 0>+ B^i`ri^o = +%>* ?&) GhAeG+v bi `lpbkl abi Škdril '( nrb cloj^k> v B-

02- C^alp alp sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp = v > ab R2&Cbqbojfk^o pf `^a^ rk^ abi^p pfdrfbkqbp molmlpf`flkbp bp `fboq^ l c^ip^-^( = * >) = * >) = W > plk ifkb^ijbkqb fkabmbkafbkqbp-_( = * >) = (%= W >&) > (%= W >& plk ifkb^ijbkqb fkabmbkafbkqbp-b( =) >) %=* >& W %=* >& plk ifkb^ijbkqb fkabmbkafbkqbp-

03- ^( Cbjlpqo^o nrb qobp sb`qlobp =) >) B+ ab R1 bpqŠk bk rk^ jfpj^ ob`q^ pf u pŽil- pf%>* =& W 'B , =& < N-_( Rf = !+ >) abjlpqo^o nrb i^ ob`q^ nrb m^p^ mlo = v > `lkpfpqb bk bi `lkgrkql abqlalp ilp sb`qlobp L q^ibp nrb %L * =& W %L * >& < N-

04- C^alp alp sb`qlobp loqldlk^ibp =) > ab R1

* ^j_lp ab ilkdfqra 0- Rb^ L rk sb`qlo nrbp^qfpc^`b i^ b`r^`fŽk L W > < =*L+ Cbjlpqo^o `^a^ rk^ ab i^p molmlpf`flkbp-^( M bp loqldlk^i ^ ? X-qfbkb ilkdfqra isf_( L) >) L W > cloj^k rk^ _^pb m^o^ R

a( %L V >& V > < , L+

a( L < i@ , c%= t >&+

)+&)* C_\QbPa\ZVda\

Klp molar`qlp bp`^i^o v sb`qlof^i mrbabk `lj_fk^opb m^o^cloj^o bi kmj_p^ojhdsoj > , ? W B+`rvl pfdkfcf`^al bp > , &?W B( bu`irpfs^jbkqb- Orbpql nrb bpqbbp rk molar`ql bp`^i^o ab alp sb`qlobp+pr s^ilo bp rk bp`^i^o- Olabjlp `^i`ri^obpqb bp`^i^o mlo jbafl ab abqbojfk^kqbp- Dp`of_^jlp = < %[f$ \0 * [0&)A < %\/ * \) ) \1'* B < '`+ +B1 +B^( X bumobpbjlp > W B bk i^ cloj^ '02-6(- Elo,j^kal bi molar`ql bp`^i^o `lk >* l_qbkbjlp

@pŒmrbp+= +> W B bp fdr^i ^i abqbojfk^kqb `rv^p cfi^p plk ilp `ljmlkbkqbp abilp c^`qlobp >* ? X B-

Dk bi qblobj^ 0,2-01 pb bk`lkqoŽ nrb alp sb`qlobp = v > plk ifkb^ijbkqb ab,mbkafbkqbppf v pŽil pf pr molar`ql sb`qlof^i = V > bp bi sb`qlo kril- Di qblobj^pfdrfbkqba^ rk `ofqbofl ^kŠildl `loobpmlkafbkqb m^o^i^ abmbkabk`f^ ifkb^i ab qobpsb`qlobp-

RCMPCK? 02-03- Qm`nq`^ojm`n >* ?* B _` S1 nji gdi`\gh`io` _`k`i_d`io`nndv n‡gj nd

=$> V `< l-

Page 161: Calculus

Mmj_p^oj hdsoj 488

A`hjnom\^d‡i, Rrmlkd^jlp mofjbol nrb >* ?* X b plk abmbkafbkqbp-Rf > X a plk abmbkafbkqbp+bkqlk`bp > W a < k+ u mlo q^kql =$ > W a < N-Rrmlkd^jlp+ pbdrfa^jbkqb+ nrb ? u a plk fkabmbkafbkqbp-Orbpql nrb ilp qobpplkabmbkafbkqbp+bufpqbk rklp bp`^i^obp \* ]* `* kl qlalp krilp+ -q^ibp nrb\> * ]? * ^` < N- Dk bpq^obi^`fŽk ab_b pbo\ ;/; N+ab lqol jlal ? u a pboŒ^kabmbkafbkqbp-Olo `lkpfdrfbkqb+ mlabjlp afsfafo mlo \ u bumobp^o= `ljl rk^`lj_fk^`fŽk ifkb^i ab ? u a+ mlo bgbjmil > < o? * n`, Eloj^kal bi molar`qlbp`^i^o ab `^a^ jfbj_ol mlo ? W a+bk`lkqo^jlp

> , &? t B( < o?%? t B * n`+ ? t B < N+

mrbpql nrb ? u B plk ^j_lp loqldlk^ibp ^ ? W a- Olo q^kql i^ abmbkabk`f^ ab=) > X b fjmif`^ nrb = +> W b < N-

O^o^ abjlpqo^o bi ob`Œmol`l+prmlkd^jlp nrb = +> W a < k- Rf > X a plkabmbkafbkqbp+q^j_f‹k il plk =) > v &B+v bi qblobj^ bpqŠabjlpqo^al- Rrmlkd^,jlp+ mrbp+nrb ? u a plk ifkb^ijbkqb fkabmbkafbkqbp-Dkqlk`bp+pbd•k bi qblobj^02-02+ilp qobpsb`qlobp >) a+u > W a plk ifkb^ijbkqb fkabmbkafbkqbp-Krbdl+ bk,dbkao^k > `lk il nrb mlabjlp bp`of_fo

> < \? * ]@ * ^&? t B(

m^o^ `fboqlp bp`^i^obp \* ]* ^, Eloj^kal bi molar`ql bp`^i^o ab `^a^ jfbj_olmlo > W a u qbkfbkal bk `rbkq^ nrb =$ %>W a( < N+bk`lkqo^jlp ` < N+^pŒnrb = < [> * \_+ Dpql abjrbpqo^ nrb =) > X b plk ifkb^ijbkqb abmbkafbkqbp-

DIDLOKN- O^o^ abqbojfk^o pf ilp qobpsb`qlobp '1+ 2+ ,0(+ '2+ ,6+4(+ X'0+ ,4+1( plk abmbkafbkqbp+cloj^jlp pr molar`ql jfuql+ bumobp^al bk cloj^ ababqbojfk^kqb

1 2,0

2 ,6 4 < 1' , 03 * 14( , 2'5 , 4( , 0',04 * 6( < 16 -

,4 1

Orbpql nrb bi molar`ql jfuql kl bp kril+ ilp qobpsb`qlobp plk ifkb^ijbkqb fkab,mbkafbkqbp

Di molar`ql jfuql qfbkb rk^ fkqbobp^kqbfkqbomobq^`fŽkdblj‹qof`^- K^ cfdr,o^ 02-5 jrbpqo^ rk m^o^ibibmŒmbalabqbojfk^al mlo qobpsb`qlobp dblj‹qof`lp =)>) b kl pfqr^alp bk bi jfpjl mi^kl- Rr ^iqro^ bp 0GA00 `lp 0=+ pfbkal 0= bi Škdrilnrb cloj^k = W > X a- Dk bpq^ cfdro^+`lp 0= bp mlpfqfsl mlonrb N 99::0= ; 6S-Di Šob^ abi m^o^ibildo^jl nrb cloj^ i^ _^pb bp GG?W 700- v ‹pq^ bp q^j_f‹k bi

Page 162: Calculus

4.. >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

Šob^ ab `^a^ pb``fŽk m^o^ibi^ ^ i^ _^pb- Hkqbdo^kal bi Šob^ pb``fŽk bkqob N vGGAhGlp `k* bk`lkqo^jlp nrb bi slirjbk abi m^o^ibibmŒmbalbpEE= u >EE '00AGGlp `k'*bi Šob^ ab i^ _^pb jriqfmif`^a^ mlo i^ ^iqro^- Obol qbkbjlp

GG?u @GG'GGAGG`lp`bm(< %=u >&$ B-

Cf`el ab lqol jlal+ bi molar`ql jfuql = W > +B bp fdr^i ^i slirjbk abi m^o^ib,ibmŒmbalabqbojfk^al mlo >* O* B- Br^kal f6S ; `k x 6S+ `lp `k bp kbd^qfsl v bimolar`ql = W >$ B bp bi s^ilo lmrbpql ^i abi slirjbk- Rf =) >) b bpqŠkbk rkmi^kl nrb m^p^mlo bi lofdbk+ plk ifkb^ijbkqb abmbkafbkqbpv pr molar`ql jfuqlbp kril- Dk bpqb`^pl+ bi m^o^ibibmŒmbalabdbkbo^ v qfbkb slirjbk `bol-

>s?`

@iqro^ < 00B00 `lp &!

EHFTQ@ 02-5 Fio`mkm`o\^d‡ib`jh„omd^\ _`g kmj_p^oj hdsoj ^jhj qjgph`i_` pi k\m\g`g`k…k`_j,

Dpq^ fkqbomobq^`fŽkdblj‹qof`^ abi molar`ql jfuql prdfbob `fboq^p molmfba^,abp ^idb_o^f`^p abi jfpjl- Olo bgbjmil+ rk^ mbojrq^`fŽk `Œ`if`^ ab ilp qobpsb`,qlobp =) >) b abg^ bi molar`ql jfuql fks^of^_ib- Blk bpql nrbobjlp fkaf`^o nrb

'02-8( >s?%@;?s@%>;@s>%?

Tk^ abjlpqo^`fŽk ^idb_o^f`^ ab bp^ molmfba^a pb mrbab sbo bk bi bgbo`f`fl 6 abi^ pb``fŽk 02-03- Dpq^molmfba^a fjmif`^ nrb bi mrkql v bi ^pm^plk fkqbo`^j_f^,_ibp bk rk molar`ql qofmib-Dk bcb`ql+ i^ `lkjrq^qfsfa^a abi molar`ql bp`^i^ofjmif`^ %>W ?&{ = < = +%>W B( v `r^kal bpql pb `lj_fk^ `lk i^ mofjbo^fdr^ia^a ab '02-8(+ bk`lkqo^jlp nrb

'02-0/( >s?}@;>}?s@,

Di molar`ql qofmib= +> W b ^ jbkral pb fkaf`^ `lk bi pŒj_lil W=>_Y pfk fkaf,`^o bi mrkql kf bi ^pm^-Cb_fal ^ i^ fdr^ia^a '02-0/(+ kl e^v ^j_fd•ba^a `lk

Page 163: Calculus

O`bg\ _` @m\h`m k\m\ m`njgq`m pi ndno`h\ _` om`n`^p\^dji`n gdi`\g`n 4./

bpq^ klq^`fŽk+ bi molar`ql abmbkab q^k pŽil abi loabk ab ilp c^`qlobp =) >) Bv kl ab i^p mlpf`flkbp abi mrkql v abi ^pm^-

02-02 Qbdi^ ab Bo^jbo m^o^obplisbo rk pfpqbj^ ab qobpb`r^`flkbp ifkb^ibp

Di molar`ql jfuql mrbab rqfifw^opb m^o^ obplisbo rk pfpqbj^ ab qobp b`r^`fl,kbp ifkb^ibp `lk qobp fk`Ždkfq^p s* v+ w- Rrmlkd^jlp nrb bi pfpqbj^ bpqŠ bp`ofqlbk i^ cloj^

'@-h (

\\s * c\V * B^Y < _\ ,

Rb^ = bi sb`qlo ab `ljmlkbkqbp \* *\0 * \n u abcfk^jlp abi jfpjl jlal >) B+ u @+Dkqlk`bp i^p qobp b`r^`flkbp '02-00( plk bnrfs^ibkqbp ^ i^ •kf`^ b`r^`fŽk sb`,qlof^i

'02-01( s> * s> * wB< A,

Rf jriqfmif`^jlp bp`^i^ojbkqb ilp alp jfbj_olp ab bpq^ b`r^`fŽk mlo ? W B+ ml,mlkfbkal X>?@Z bk ird^o ab > , ? V B+ bk`lkqo^jlp nrb

rW=>?Y * sW>>?Y * tW?>?Y < W@>?Y+

Orbpql nrb X??@Z < X@?@Z< N+ilp `lbcf`fbkqbp ab v v ab w abp^m^ob`bk v l_,qbkbjlp

'02-02( W@>?YT:**

W=>?Ypf W=>?Y <:… N-

Cbi jfpjl jlal iibd^jlp ^ cŽojri^p ^kŠild^p m^o^ v v w- @pŒmrbp qbkbjlp

'02903( W=@?Y

s < W=>?Yu

W=>@YV:**

W=>?Ypf W=>?Y <:… M -

K^ `lkaf`fŽk X>?@Z ;/; N pfdkfcf`^ nrb ilp qobp sb`qlobp >* ?* B plk ifkb^ijbkqbfkabmbkafbkqbp- Dk bpqb `^pl+ '02-01( jrbpqo^ nrb qlal sb`qlo @ bk bi bpm^`flqofafjbkpflk^i bpqŠ dbkbo^al mlo =) >) B v ilp jriqfmif`^alobp s* v+ w bpqŠk abqbo,jfk^alp `lk rkf`fa^a mlo i^p &cŽojri^p '02-02( u '02-03(- Br^kal ilp molar`qlp

Page 164: Calculus

5/1 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

jfuqlp nrb ^m^ob`bk bk bp^p cŽojri^p pb mlkbk `ljl abqbojfk^kqbp+bi obpriq^alpb `lkl`b `lk bi klj_ob ab m`bg\ _` @m\h`m m^o^ i^ obplir`fŽk abi pfpqbj^'02-00(9

B8 @G 78 ?G B8 78 \g @G B8

J, ], ^,F \, J, `+ \, ]0 J,

B? @? 7? ?? B? 7? ?? @? B?s; t; P5?G ^G 78 ?G @G 78 \g @G 78

H, ], 7+ H, ], 7+ H, ], 7+

?? @? 7? ?? @? 7? ?? @? 7?

Rf X>?@Z < N+bkqlk`bp >* ?* B bpqŠkbk rk mi^kl nrb m^p^ mlo bi lofdbkv bi pfpqbj^ kl qfbkb plir`fŽk ^ jbklp nrb @ bpq‹ bk af`el mi^kl- Dk bcb`ql+ilpsb`qlobp >* ?* B plk ifkb^ijbkqb abmbkafbkqbp+ab prboqb nrb bufpqbk bp`^i^obpp* q* t kl qlalp krilp q^ibpnrb p> * q? * tB :-) Rf i^ qbok^ &s*v+ w( p^qfp,c^`b '02-01(+ il jfpjl l`roob `lk i^ qbok^%r * no) v * np) w* nq& m^o^qlal o*v^ nrb qbkbjlp

&s * op'> * &t * oq'? * 'w * or'@;

; s> * t? * u@ * o&p>* q? * r@' < s> * t? * u@,

)+&), :WR_PVPV\`

0- B^i`ri^o bi molar`ql jfuql = +> W b bk `^a^ `^pl-

'^( = < '2+ N+N(+ > < 'N+ 3+ N(+ b < 'N+ N+7(-'_( > < '1+2+,0(+ ? < '2+ ,6+4(+ b < '0+ ,4+1(-

'b( > < '1+0+2(+ ? < ',2+ N+5(+ b < '3+4+,0(-

1- G^ii^o qlalp ilp k•jbolp ob^ibp o m^o^ ilp nrb ilp qobpsb`qlobp M+o, 0(+'p+0+N(+'N+ 0+p(plk ifkb^ijbkqb abmbkafbkqbp-

2- B^i`ri^o bi slirjbk abi m^o^ibibmŒmbalabqbojfk^al mlo ilp sb`qlobp d* d)d * e) e * d,3- Cbjlpqo^o nrb = W > << =$ %> u c&c* = +%>u d&d* = +%>u e&e+

4- Cbjlpqo^o nrb c u %= u c&* g u %= u g( * e u %= u e& < /=+

5- ^( G^ii^o qlalp ilp sb`qlobp \d * ]e * ^f nrb p^qfpc^d^k i^ obi^`fŽk

%[c* ]e * ]e& +e t %3c* 2g * 1e& < 2 -

_( G^ii^o bi sb`qlo \d * ]e * ^f ab jbklo ilkdfqra nrb p^qfpc^d^i^ obi^`fŽk ab ^(-

Page 165: Calculus

Be`m^d^djn 5/2

6- G^`bo rpl ab i^p molmfba^abp ^idb_o^f`^p+ ab ilp molar`qlp bp`^i^o v sb`qlof^i+ m^o^ ab,jlpqo^o i^p pfdrfbkqbp molmfba^abp abi molar`ql jfuql

]( %=* >& +%=* >& t B < N-_( ={ > W B < *>{ = W B- Dpql abjrbpqo^ nrb ^i fksboqfo i^ mlpf`fŽk ab ilp alp mof,jbolp sb`qlobp `^j_f^ bi pfdkl-

XFi_d^\^d‡i8 Tqfifw^o i^ m^oqb ^( v i^p ibvbp afpqof_rqfs^p-\b( ={ > W B < *={ B W >+ Dpql abjrbpqo^ nrb i^ mbojrq^`fŽk ab ilp sb`qlobp pbdrk,al v qbo`bol `^j_f^ bi pfdkl- XFi_d^\^d‡i8 Tqfifw^o i^ pfjbqoŒ^ ^iqbok^a^-\a( ={ > W B < *?$ > W =+ Dpql abjrbpqo^ nrb i^ mbojrq^`fŽk ab ilp sb`qlobp mof,jbol u qbo`bol `^j_f^ bi pfdkl- XFi_d^\^d‡i8 Tqfifw^o _( u `(-\

Hdr^i^kal ilp pbdrkalp jfbj_olp ab _(+ b(+ v a(+ bk`lkqo^jlp nrb

=$>r?:>$?r=:]*= r>)

il nrb abjrbpqo^ nrb rk^ mbojrq^`fŽk `Œ`if`^ ab =) >) B abg^ fks^of^_ib bi molar`qljfuql-

8- Dpqb bgbo`f`fl bp_lw^ rk^ abjlpqo^`fŽk ab i^ fabkqfa^a sb`qlof^i

'02-04( = u %> V B( < %?$=&> * %>$=&? †

nrb ^idrk^p sb`bp pb ii^j^ cŽojri^ ~`^_ jbklp _^`‚- Rb^k > < %\Y) \0* \\'*B < '`-+ ^0% `^i+ abjlpqo^o nrb

eu &? u B( < ^,? + ],@*

Dpql abjrbpqo^ '02-04( bk bi `^pl m^oqf`ri^o = < f- Cbjlpqo^o i^p cŽojri^p `loobpmlkafbkqbp m^o^ > < f v > ;f* v `lj_fk^oi^p irbdl m^o^ l_qbkbo '02-04(-

0/- Tqfifw^o i^ cŽojri^ ~`^_ jbklp _^`‚ abi bgbo`f`fl 8 m^o^ abar`fo i^p pfdrfbkqbp fabkqf,a^abp sb`qlof^ibp-

B^( ?= u >& u BB u @&< B@ u >$ @&? * ?= u >$ ?&@+'_( = u ?> u B( * > u 'B u =& * B u ?= u >& < N-'b( = u %> u B( < %= u >& u B pf v p50/ pf > u BB u =& < N-'a( %= u >& +'B u @&< %>$@&%=+B( , %>$?&%=+@&+

00- Br^qol sb`qlobp =) >) B+ @ ab U^ p^qfpc^`bk i^p obi^`flkbp = WB& > < 4+ = W @{ > < 2+B * @ < e* 1g* e) B , @ < e, e+ B^i`ri^o %=W >& W 'B W @&bk crk`fŽk bk c)d*e+

01- Cbjlpqo^o nrb %= * >& +%> W B( W 'B W =& < %={ > W ?&/+02- Cbjlpqo^o pf bp l kl `fboq^ i^ cŽojri^ = W W= W %= W >&Y +B < , GG?G01= +> W B-03- ^( Cbjlpqo^o nrb bi slirjbk abi qbqo^baol `rvlp s‹oqf`bp plk >* ?* B+ A bp

qE%>* =& +'A , =& t %@* =&W+

_( B^i`ri^o af`el slirjbk `r^kal = < '0+ 0+ 0(+ > < 'N+ /+ 1(+ B < 'N+ 2+ N(+ XA < '3+ /+ N(-

04- ^( Rf > !! B+ abjlpqo^o nrb i^ afpq^k`f^ abpab = ^ i^ ob`q^ nrb m^p^ mlo > v B bp

EE%=* >& t 'A , >&EE,EE>* AGG-

_( B^i`ri^o bp^ afpq^k`f^ `r^kal = < '0+ ,1- ,4(+ > < ',0+0+0( v B < '3+ 4+0(-

Page 166: Calculus

4.2 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

05- K^ cŽojri^ ab Gbolk m^o^ `^i`ri^o bi Šob^ R ab rk qofŠkdril `rvlp i^alp qfbkbk ilk,

dfqrabp \* ]* ` bpR < tn&n + \'&n + ]'&n + `'* pfbkal n < &\ * ] * ^'-0, Dpqb bgbj,

mil bp_lw^ rk^ abjlpqo^`fŽk sb`qlof^i ab bp^ cŽojri^-Rrmlkd^jlp nrb bi qofŠkdril qfbkb ilp s‹oqf`bp bk /+ = X >) pfbkal GG?GG< \*

GG@GG< \) GG@, ?GG< ^,^( Blj_fk^o i^p alp fabkqfa^abp

+0> , ? < GG?, @G01, GG?G01

, GG@G01

m^o^ l_qbkbo i^ cŽojri^

_( Olkbo i^ cŽojri^ ab i^ m^oqb ^( bk i^ cloj^

Q1 < g%+4&\* ] * ^'&\ * ] + ^'&^ + \ * ]'&^ * \ + ]' *

v ab ^eŒabar`fo i^ cŽojri^ ab Gbolk-

Qbplisbo jbaf^kqb i^ obdi^ ab Bo^jbo bi pfpqbj^ ab b`r^`flkbp bk `^a^ rkl abilp bgbo`f`flp 06+ 07 X 08-

/5,s)0t)1u;3* 0s+t)2u;gg* +t)u;1,/6, s * t * 0u < 3+ 1s + t + u < 1+ 0s * 3t * 1u < 2-.6+ s * t < R+ s * u < 1+ U * u < 4-

1/- Rf L < '0+ 0+ 0( X = < '1+ 0+ ,0(+ abjlpqo^o nrb `^a^ mrkql %r) v+ t& ab i^ ob`q^uL * o>g p^qfpc^`b bi pfpqbj^ ab b`r^`flkbp ifkb^ibp s + v * u < 0+ s * v * 2w < 4+0r * v * 6w < 00-

)+&)- HRPa\_R[\_ZNYR`N ]YN[\`

Dk i^ pb``fŽk 02-5 pb abcfkfŽ bi mi^kl `ljl rk `lkgrkql ab i^ cloj^vM * n> * o?w* alkab > v ? plk sb`qlobp ifkb^ijbkqb fkabmbkafbkqbp-@elo^abjlpqo^jlp nrb ilp mi^klp bk Uc mrbabk `lkpfabo^opb ab jlal `ljmibq^jbkqbafpqfkql+`lk bi `lk`bmql ab sb`qlo kloj^i-

CDEHMHBHˆM- P`\ K < vM * n> * o?w `g kg\ij lp` k\n\ kjm M u b`i`m\_jkjm > v ?, Ri q`^ojm K _` U^ `n k`mk`i_d^pg\m \ J*nd `n k`mk`i_d^pg\m \ g\ q`u\ > v \ ?, Pd*\_`hƒn* K `n ij ipgj* `ioji^`n K n` gg\h\ q`^ojm ijmh\g \g kg\ij,

Kjo\8 Rf K} > < K , ? < N+ bkqlk`bp K} &n>* o?' < N+ ab jlal nrb rk sb`qlombombkaf`ri^o ^ i^ sbw ^ = v ^ > bp mbombkaf`ri^o ^ `r^inrfbo sb`qlo ab i^ bkslisbkqbifkb^i ab > v ?, @pfjfpjl+ pf K bp kloj^i ^ rk mi^kl- q^j_f‹k il bp oK m^o^ qlal s^iloob^i o Š,Šl-

Page 167: Calculus

S`^ojm`n ijmh\g`n \ kg\ijn 5/4

RCMPCK? 02-04- A\_j pi kg\ij L < vk * n> * o?w lp` k\n\ kjm M tb`i`m\_j kjm > t ?, P`\ K < > W ?, Q`i`hjn `ioji^`n8

^( K `n pi q`^ojm ijmh\g \ K-_( L `n `g ^jiepioj _` oj_jn gjn q`^ojm`n W _` S 2 lp` n\odna\^`i g\ `^p\^d‡i

'02-05( &U + M'} K < N-

A`hjnom\^d‡i, Orbpql nrb L bp rk mi^kl+ > v ? plk ifkb^ijbkqb fkabmbk,afbkqbp+ ^pŒnrb = W > :.: N- Dpql abjrbpqo^ ^( v^ nrb = W > bp loqldlk^i ^ i^sbw ^ = v ^ >+

O^o^ abjlpqo^o _(+ pb^ L& bi `lkgrkql ab qlalp ilp sb`qlobp W ab S1 nrb p^,qfpc^`bk i^ b`r^`fŽk '02-05(- Rf W C L+ W , L bp i^ bkslisbkqb ifkb^i ab = v >)^pŒnrb W , L bp loqldlk^i ^ J+ Olo `lkpfdrfbkqb W C L& 0/ nrb abjrbpqo^ nrbL ;9: I$+ Qb`Œmol`^jbkqb+ prmlkd^jlp W D I$+ Dkqlk`bp W p^qfpc^`b '02-05(-Orbpql nrb =) >) J plk ifkb^ijbkqb fkabmbkafbkqbp 'qblobj^ 02-02(+ dbkbo^k`r^inrfbo sb`qlo ab S 2 `lk 0/ nrb+ bk m^oqf`ri^o+ qbkbjlp

U + M < n> * o? * pK

m^o^ `fboqlp bp`^i^obp p+o*p, Eloj^kal bi molar`ql bp`^i^o ab `^a^ jfbj_ol mloK* bk`lkqo^jlp p < N+ ^pŒnrb W , M < n> * o?, Dpql abjrbpqo^ nrb W C L-Krbdl+ L& ;9: L+ 0/ nrb `ljmibq^ i^ abjlpqo^`fŽk ab _(-

Di pfdkfcf`^al dblj‹qof`l abi qblobj^ 02-04 pb jrbpqo^ bk i^ cfdro^ 02-6-Klp mrkqlp L v W bpqŠk bk bi mi^kl v bi sb`qlo kloj^i J bp loqldlk^i ^ T*L+Dp^ cfdro^ prdfbob bi qblobj^ pfdrfbkqb

RCMPCK? 02-05- A\_jn pi kg\ij L lp` k\n\ kjm pi kpioj M t pi q`^ojmij ipgj K ijmh\g \ K+ n`\

'02-06(^ < GN&JE+

GGLGG

Bioji^`n oj_j V `i K od`i` gjibdop_ GGVGGƒ _, >_`hƒn* o`i`hjn GzVGG< _ pf tn‡gj pf W `n g\ kmjt`^^d‡i _` M nj]m` K8

u < oK*L$J

_ji_` n < ,,-J$J

A`hjnom\^d‡i, K^ abjlpqo^`fŽk pb abar`b ab i^ abpfdr^ia^a ab B^r`ev,R`et^ow pfdrfbkal bi jfpjl j‹qlal nrb bk bi qblobj^ 02-5+ ^i l_qbkbo bi obpri,q^al ^kŠildl m^o^ i^p ob`q^p bk S0Š

Page 168: Calculus

4.4 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

Blk bi jfpjl o^wlk^jfbkql bk`lkqo^jlp nrb pf O bp rk mrkql kl pfqr^albk L+ bkqlk`bp bkqob qlalp ilp mrkqlp V bk L i^ jbklo ilkdfqra GGV, OGGqfbkbird^o `r^kal W , O bp i^ molvb``fŽk ab L * O pl_ob J+ Dpq^ ilkdfqra jŒkfj^bp E%L* M&$JE,EEJfE v pb ii^j^ i^ _dno\i^d\ _`n_` P \g kg\ij, Di k•jbol _ ab'02-06( bp i^ afpq^k`f^ abpab bi lofdbk ^i mi^kl-

02-05 D`r^`flkbp ifkb^ibp `^oqbpf^k^p m^o^ mi^klp

Klp obpriq^alp ab ilp qblobj^p 02-04 v 02-05 q^j_f‹k mrbabk bumobp^opb bkcrk`fŽk ab ilp `ljmlkbkqbp- Rf bp`of_fjlp J < %[) \) _&) L < &Ug%XH+Yi(+ XW < %T) v+t&) i^ b`r^`fŽk 02-05 qlj^ i^ cloj^

'02-07(

Dpq^ bp i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl v pŽil pb p^qfpc^`b m^o^ ^nrbiilp mrkqlp'u+ v+ w( nrb bpqŠk bk bi mi^kl- Di `lkgrkql ab mrkqlp nrb p^qfpc^`bk '02-07( klpb ^iqbo^ pf jriqfmif`^jlp `^a^ `lbcf`fbkqb \* \) ` mlo rk bp`^i^o kl kril o, Dpqlbnrfs^ib q^k pŽil ^ rk^ bib``fŽk afpqfkq^ ab sb`qlo kloj^i bk '02-05(-

So^kpmlkfbkal ilp q‹ojfklp nrb kl abmbkabk ab s* v+ v w pb l_qfbkb

%.0+.6& \s * \s * ^u < `h+

pfbkal _* < \UF * ]VF * ?VF, Rb af`b nrb rk^ b`r^`fŽk ab bpqb qfml bp gdi`\g bkr)s)V+ @`^_^jlp ab abjlpqo^o nrb qlal mrkql %r)s)t& ab rk mi^kl p^qfpc^`b rk^b`r^`fŽk ifkb^i `^oqbpf^k^ '02-08( bk i^ nrb \* \) ? kl plk qlalp krilp- Qb`Œmol`^,jbkqb+ qla^ b`r^`fŽk ifkb^i `lk bp^ molmfba^a+ obmobpbkq^ rk mi^kl- 'Di ib`qlomrbab `ljmol_^oil `ljl bgbo`f`fl-(

Di k•jbol aH ab i^ b`r^`fŽk '02-08( lofdfk^ rk^ pbk`fii^ obi^`fŽk m^o^ i^afpq^k`f^ ^ abi mi^kl ^i lofdbk- Orbpql nrb `G:j+ J) qbkbjlp G`hh< GN&LG<`GGLGG-Dk m^oqf`ri^o HaHH< _ pf i^ kloj^i K qfbkb ilkdfqra 0- Di mi^kl m^p^ mlo bi lof,dbk pf u pŽil pf ^E < N-

DIDLOKN- K^ b`r^`fŽk `^oqbpf^k^ 0s * 5v * 1u < 5 obmobpbkq^ rk mi^kl`lk sb`qlo kloj^i K < 1: * 5g* 1f, Dp`of_^jlp i^ b`r^`fŽk `^oqbpf^k^ bk i^cloj^

[ s ]'%'%'5*201

bk i^ nrb pb mlkb ab j^kfcfbpql nrb bi mi^kl `loq^ ^ ilp bgbp `lloabk^alp bk ilpmrkqlp '2+ N+N(+ 'N+ 0+ N(+X 'N+ N+1(- Klp k•jbolp 2+ 0+ 1 pb ii^j^k+ obpmb`qfs^,jbkqb+ ilp n`bh`iojn dio`m^`ko\_jn bk ilp bgbp mlo bi mi^kl- Di `lkl`fjfbkql abbplp pbdjbkqlp mbojfqb obmobpbkq^obi mi^kl oŠmfa^jbkqb- Dk i^ cfdro^ 02-7 pbjrbpqo^ rk^ mlo`fŽk abi mi^kl+ Rr afpq^k`f^ _ ^i lofdbk bp _ < 5.GGLGG< 5.6-

Page 169: Calculus

Be`m^d^djn 5/6

u

'N+/+1(

J

'N+ 0- N(t

it

EHFTQ@ 02-6 Mg\ij lp` k\n\ kjm M u U^ji q`^ojm ijmh\g K,

EHFTQ@ 02-7 Mg\ij lp` dio`m^`ko\n`bh`ioj 2+ 0+1-

Clp mi^klp m^o^ibilpqfbkbkrk^ kloj^i J `lj•k- Rf J < %[) \) _&) i^p b`r^,`flkbp `^oqbpf^k^pab alp mi^klp m^o^ibilp mrbabk bp`of_fopb^pŒ9

\s * \s * ^u < ^g * \s * \s * ^u < ^0 *

afcbobk`fŠkalpb q^k pŽil bk ilp pbdrkalp jfbj_olp- Di k•jbol E^g + _0.,EEJEE pbii^j^ afpq^k`f^ bkqobilp alp mi^klp+abcfkf`fŽk prdbofa^ mlo bi qblobj^ 02-05-

Clp mi^klp pb ii^j^k mbombkaf`ri^obppf rk^ kloj^i ^ rkl bp mbombkaf`ri^o^ rk^ kloj^i ^i lqol- N jŠp dbkbo^i+pf i^p kloj^ibp ^ alp mi^klp cloj^k rkŠkdril %F)ab`fjlp nrb %Fbp bi Škdril nrb cloj^k ilp alp mi^klp-

)+&)/ :WR_PVPV\`

0- C^alp ilp sb`qlobp @ < 1: * 1e+ 2f X ? < f * f,^( G^ii^o rk sb`qlo kl kril J mbombkaf`ri^o ^ i^ sbw ^ = v ^ >+_( C^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl nrb m^p^ mlo bi lofdbk v bpqŠ dbkbo^al mlo

@ v >+b( C^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl nrb m^p^ mlo N+ 1+ 2( X bpqŠ dbkbo^a^ mlo

? v >+1- Tk mi^kl qfbkb `ljl b`r^`fŽk `^oqbpf^k^ s * 0t + 0u * 6 < N- G^ii^o9

^( rk sb`qlo kloj^i ab ilkdfqra rkfa^a:_( ilp pbdjbkqlp fkqbo`bmq^alp mlo bi mi^kl bk ilp bgbp:b( i^ afpq^k`f^ abi mi^kl ^i lofdbk:a( bi mrkql P abi mi^kl jŠp moŽufjl ^i lofdbk-

2- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl nrb m^p^ mlo N+ 1+ ,2( X bp m^o^ibil ^i mi^kl1s + v * 0u < 3- ƒBrŠi bp i^ afpq^k`f^ bkqob ilp alp mi^klp>

Page 170: Calculus

4.6 >kgd^\^dji`n+_`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

3- Br^qol mi^klp qfbkbk b`r^`flkbp `^oqbpf^k^p r * 1v , 1w < 4+ 0r * 5v * 2w< 1+/r * v * 1w < ,0+ X r * 1v * w < 6-^( Cbjlpqo^o nrb alp ab biilp plk m^o^ibilp v ilp lqolp alp plk mbombkaf`ri^obp-_( G^ii^o i^ afpq^k`f^ bkqob ilp alp mi^klp m^o^ibilp-

4- Klp qobp mrkqlp '0+ 0+ ,0(+ '2+ 2+ 1(+ X '2+ ,0+ ,1( abqbojfk^k rk mi^kl- G^ii^o ^( rksb`qlo kloj^i ^i mi^kl: _( rk^ b`r^`fŽk `^oqbpf^k^ abi mi^kl: `( i^ afpq^k`f^ abi mi^kl^i lofdbk-

5- G^ii^o rk^ b`r^`fŽk `^oqbpf^k^ abi mi^kl abqbojfk^al mlo '0+ 1+ 2(+ '1+ 2+ 3(+ X',0+ 6+,1(-

6- Cbqbojfk^o bi Škdril cloj^al mlo ilp mi^klp s * u < 0 b u * w < 1-7- Tk^ ob`q^ m^o^ibi^ ^ rk sb`qlo kl kril J pb abkljfk^ mbombkaf`ri^o ^i mi^kl L pf J

bp kloj^i ^ L- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl nrb m^p^ mlo '1+ 2+ ,6(+ p^l_fbkal nrb i^ ob`q^ nrb m^p^ mlo '0+ 1+ 2( X '1+ 3+ 01( bp mbombkaf`ri^o ^ af`el mi^kl-

8- G^ii^o i^ b`r^`fŽk sb`qlof^i m^o^j‹qof`^ ab i^ ob`q^ nrb `lkqfbkb bi mrkql '1+ 0+ ,2(X bp mbombkaf`ri^o ^i mi^kl 2s + 2v * w < 4-

0/- Tk mrkql pb abpmi^w^ bk bi bpm^`fl ab jlal nrb bk bi fkpq^kqb o pr mlpf`fŽk sfbkba^a^ mlo bi sb`qlo TQ& < '0 , n&x* '1 , 0n&d* %/n* .&e+^( Cbjlpqo^o nrb bi mrkql pb jrbsb ^ 0/ i^odl ab rk^ ob`q^- 'K0^j‹jlpi^ H+&_( G^ii^o rk sb`qlo K m^o^ibil ^ I,b( ƒDk nr‹ fkpq^kqb bi mrkql ql`^ bi mi^kl 0s * 2u * 1w * 0 < N>a( G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl m^o^ibil ^i ab i^ m^oqb `( v nrb `lkqbkd^bi mrkql T%0&+b( G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl mbombkaf`ri^o ^ H -nrb `lkqbkd^ bi mrkqlT%/&+

00- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl nrb m^p^ mlo '0+ 0+ 0( pf rk sb`qlo kloj^i Kcloj^ ilp Škdrilp /Q* d/Q* 6S+ `lk c)g+e) obpmb`qfs^jbkqb-

01- B^i`ri^o bi slirjbk abi qbqo^baol `rvlp s‹oqf`bp plk bi lofdbk v ilp mrkqlp bk ilp nrbilp bgbp `lloabk^alp `loq^k bi mi^kl s * 1u * 2w < 5-

02- G^ii^o rk sb`qlo > ab ilkdfqra 0 mbombkaf`ri^o ^ {* 1g, 1f X m^o^ibil ^i mi^kls + v * 4w < 0-

03- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl m^o^ibil ^ ilp alp sb`qlobp {* f v f * f v nrb`loq^ bi bgb s bk '1+ N+N(-

04- G^ii^o qlalp ilp mrkqlp pfqr^alp bk i^ fkqbopb``fŽk ab ilp qobp mi^klp 0r * v * w < 4+0r * v * 4w < 6+ r * v * 0t < 2-

05- Cbjlpqo^o nrb qobp mi^klp `rv^p kloj^ibp plk ifkb^ijbkqb fkabmbkafbkqbp qfbkbk rkplil mrkql `lj•k-

06- Tk^ ob`q^ `lk sb`qlo ab afob``fŽk > bp m^o^ibi^ ^ rk mi^kl L pf > bp m^o^ibil ^ L-Tk^ ob`q^ nrb `lkqfbkb '0+ 1+ 2( bp m^o^ibi^ ^ `^a^ rkl ab ilp mi^klp s * 1v * 2w < 3+/r * 2u * 3w < 4- G^ii^o i^ b`r^`fŽk sb`qlof^i m^o^j‹qof`^ ab bp^ ob`q^-

07- C^a^ rk^ ob`q^ H kl m^o^ibi^ ^i mi^kl L+ abjlpqo^o nrb i^ fkqbopb``fŽk H j L `lk,qfbkb pŽil rk mrkql-

08- ^( Cbjlpqo^o nrb i^ afpq^k`f^ abpab bi mrkql %ri$ Xl+ wl( ^i mi^kl [r * ]t * ]t * _ < N-bp

g\sj * ]tj * ]Vj * `e%[0 * ]0 * @0'/-0

_( G^ii^o bi mrkql M abi mi^kl 3s + .1s * 1w * 8 < N nrb bpq‹ jŠp moŽufjl ^i mrk,ql M:%*/) 04+,6(+

1/- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl m^o^ibil ^i /r * X * 1w * 3 < N pf bi mrkql'2+ 1+ ,0( bnrfafpq^ ab ^j_lp-

Page 171: Calculus

I\n n`^^dji`n ^‡id^\n /)2

10- ^( Rf ilp mrkqlp =) >) B abqbojfk^k rk mi^kl+ abjlpqo^o nrb i^ afpq^k`f^ abpab rkmrkql P ^ af`el mi^kl bp H'P *=&{ %> * =& W 'B , =&f,EE%>* =& W 'B , ?(GG•_( B^i`ri^o bp^ afpq^k`f^ pf P < N+ /+ N(+ = < 'N+ I+ 0(+ > < N+ ,I+ 0( X B < '1+ 2+ 3(-

11- Cbjlpqo^o nrb pf alp mi^klp K v K&kl plk m^o^ibilp+ pr fkqbopb``fŽk K 'Z J% bp rk^ob`q^-

12- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl m^o^ibil ^ f v nrb m^p^ mlo i^ fkqbopb``fŽk abilp mi^klp `rv^p b`r^`flkbp plk r * 1v * 2w < 3 X /r * v * w< 1-

13- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ abi mi^kl m^o^ibil ^i sb`qlo 0c * f * $/e pf `lkqfbkb qlalpilp mrkqlp ab i^ ob`q^ ab fkqbopb``fŽk ab ilp mi^klp&s * v < 2 X 1v * 2w < 3-

)+ &)0 ?N` `RPPV\[RPp[VPN`

Tk^ ob`q^ jŽsfi F nrb `loq^ ^ rk^ ob`q^ cfg^ = bk rk mrkql L) cloj^kal`lk bii^ rk Škdril `lkpq^kqb %F)pfbkal N ; %F; H6S+ bkdbkao^ rk^ prmbocf`fb bk bibpm^`fl qofafjbkpflk^i ii^j^a^ `lkl `fo`ri^o ob`ql- K^ ob`q^ F bp i^ b`i`m\omdu abi`lkl+ > bp bi `e`* u M pr q„mod^`,B^a^ rkl ab ilp `lklp af_rg^alp bk i^ cfdro^ 02-8qfbkb bi bgb sboqf`^i- K^p mlo`flkbp prmboflo b fkcboflo abi `lkl nrb pb rkbk bkbi s‹oqf`b pb ii^j^k cje\n abi `lkl --K^p `ros^p l_qbkfa^p `loq^kal bi `lkl `lk rkmi^kl nrb kl m^pb mlo bi s‹oqf`b pb ii^j^k n`^^dji`n ^‡id^\n* l pfjmibjbkqb ^‡id+^\n, Rf bi mi^kl pb`^kqb bp m^o^ibil ^ rk^ ob`q^ abi `lkl nrb m^pb mlo bi s‹oqf`b+i^ `Žkf`^ bp rk^ k\mƒ]jg\, Dk `r^inrfbo lqol `^pl i^ fkqbopb``fŽk pb ii^j^ `gdkn`

EHFTQ@ 02-8 I\n n`^^dji`n ^‡id^\n,

l cdk„m]jg\* pbd•k nrb bi mi^kl `loqb rk^ elg^ abi `lkl l i^p alp obpmb`qfs^jbkqb-'Ubo cfdro^ 02-8-( K^ efm‹o_li^ `lkpq^ ab alp ~o^j^p‚ rk^ bk `^a^ elg^ abi `lkl-

Lr`elp abp`r_ofjfbkqlp fjmloq^kqbp q^kql bk i^ L^qbjŠqf`^ mro^ `ljlbk i^ ^mif`^a^ e^k qbkfal obi^`fŽk `lk i^p pb```flkbp `Žkf`^p- Di bpqrafl mlo

Page 172: Calculus

4/. >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g \ g\ b`jh`om…\ \i\g…od^\

@mlilkfl ab i^p `Žkf`^p bk bi pfdil GGG^kqbpab I-B- crb rkl ab ilp qo^_^glpjŠpklq^_ibp ab i^ FbljbqoŒ^ `iŠpf`^ dofbd^-Tklp 1/// ^•lp jŠp q^oab+F^ifibl abp,`r_oŒ^nrb rk molvb`qfii^kw^al elofwlkq^ijbkqb abpab il ^iql ab rk^ qloob+ ^b ^i^ qfboo abp`of_fbkal rk^ qo^vb`qlof^m^o^_Žif`^ 'pf pb mobp`fkabab i^ obpfpqbk`f^abi ^fob v pb prmlkb nrb bi jlsfjfbkql qfbkb ird^o pl_ob rk^ m^oqbab i^ prmbo,cf`fb qboobpqobnrb pb prmlkb mi^k^(- Tkl ab ilp jljbkqlp `rj_obp ab i^ efp,qlof^ ab i^ @pqolkljŒ^ qfbkb ird^o ^iobabalo ab 05// `r^kal Jbmibo prdfbobnrb qlalp ilp mi^kbq^ppb jrbsbk bk Žo_fq^pbiŒmqf`^p-Tklp 7/ ^•lp jŠp q^oab+Mbtqlk abjlpqo^_^ nrb i^p Žo_fq^pmi^kbq^of^pbiŒmqf`^pfjmif`^k i^ ibv ab do^,sfq^`fŽk rkfsbop^i bk i^ nrb i^ crbow^ ab ^qo^``fŽk bp molmlo`flk^i ^i fksboplabi `r^ao^al ab i^ afpq^k`f^ bkqob ilp `rbomlp nrb pb ^qo^bk- K^ qbloŒ ab i^do^sfq^`fŽk rkfsbop^i clojri^a^ mlo Mbtqlk pb `lkpfabo^ ^idrk^p sb`bp `ljlbi j^vlo abp`r_ofjfbkql `fbkqŒcf`lnrb pb e^ ob^ifw^al- K^p pb``flkbp `Žkf`^p^m^ob`bk kl pŽil bk i^p Žo_fq^pab ilp mi^kbq^pv p^q‹ifqbp+pfkl q^j_f‹k `ljlqo^vb`qlof^pab m^oqŒ`ri^p qŽjf`^p bibjbkq^ibp- Dpqlp bgbjmilp v jr`elp lqolpjrbpqo^k i^ fjmloq^k`f^ ab i^ qbloŒ ab i^p pb``flkbp `Žkf`^p nrb afcŒ`fijbkqbbp bpqfj^a^ bk qla^ pr fjmloq^k`f^-

G^v lqo^p abcfkf`flkbp ab i^p pb``flkbp `Žkf`^p nrb plk bnrfs^ibkqbp- Dk rk^ab bii^p pb `lkpfabo^k rklp mrkqlp bpmb`f^ibpii^j^alp `i]im v bkqlk`bp i^ bifmpbpb mrbab abcfkfo`ljl bi ird^o ab qlalp ilp mrkqlp abi mi^kl `rv^ prj^ ab afpq^k,

GCfob`qofwGGG

^) * _0 < `lkpq^kqb'bifmpb(

G ) * ^0. < `lkpq^kqb'efm‹o_li^( ^) < _0

'm^oŠ_li^(

EHFTQ@ 02-0/ A`adid^dji`n aj^\g`n _` g\n n`^^dji`n ^‡id^\n,

`f^p aH X _0 ^ alp mrkqlp C* v E- '0lp cl`lp( bp `lkpq^kqb 's‹^pb cfd- 02-0/(-Rf ilp cl`lp `lfk`fabk i^ bifmpbpb obar`b ^ rk^ `fo`rkcbobk`f^- K^ efm‹o_li^ bpbi ird^o ab qlalp ilp mrkqlp m^o^ ilp `r^ibp H ^y * a10 bp `lkpq^kqb v i^ m^oŠ_li^bp bi ird^o ab qlalp ilp mrkqlp q^ibpnrb i^ afpq^k`f^ ^ rk mrkql cfgl E 'ii^j^albi cl`l( bp fdr^i ^ i^ afpq^k`f^ ^ rk^ m`^o\ cfg^ ii^j^a^ afob`qofw'nrb kl qfbkbkfkdrk^ obi^`fŽk `lk i^ afob`qofwab rk `fifkaol(-

Page 173: Calculus

I\n n`^^dji`n ^‡id^\n 500

Dpcbo^QG

El`l B+

EHFTQ@ 02-00 A`hjnom\^d‡i _` A\i_`gdi,

G^v rk o^wlk^jfbkql jrv pfjmib v bibd^kqb nrb morb_^ nrb i^ molmfba^acl`^i ab rk^ bifmpbbp `lkpb`rbk`f^ ab pr abcfkf`fŽk `ljl pb``fŽk ab rk `lkl-Dpq^abjlpqo^`fŽk crb abp`r_fboq^ bk 0711 mlo bi j^qbjŠqf`l _bid^ F- O- C^k,abifk '0683,0736( rqfifw^kal alp bpcbo^pnrb plk q^kdbkqbp^i `lkl v ^i mi^klpb`^kqb q^i `ljl pb fkaf`^ bk i^ cfdro^ 02-00- Dpq^pbpcbo^pplk q^kdbkqbp i `lkl^ 0/ i^odl ab alp m^o^ibilp B0 v ?0Š Rb abjlpqo^oŠ nrb ilp mrkqlp E0 v E1 &ab`lkq^`ql ab i^p bpcbo^p lk bi mi^kl plk mob`fp^jbkqb ilp cl`lp ab i^ bifmpb-

Rb^ M rk mrkql ^o_fqo^ofl ab i^ bifmpb-Di mol_ibj^ bpqŠ bk mol_^o nrb,* ,*

EELBY00 * GGND100 bp `lkpq^kqb+ bp ab`fo+ fkabmbkafbkqb ab i^ bib``fŽk ab L+ O^o^

Page 174: Calculus

4/0 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

biil pb af_rg^ bk bi `lkl i^ ob`q^ nrb s^ ab N ^ M v pb^k =f v =0 prp fkqbopb`,- ,* ,*

`flkbp `lk i^p `fo`rkcbobk`f^p BH v B1 obpmb`qfs^jbkqb- Dkqlk`bp LBE v L=E,* ,*

plk alp q^kdbkqbp^ 9RH abpab L v mlo q^kql E,LBfff < GGN?hhh-@kŠild^jbkqb,* ,*

EELB0.. < EEL=0.. v+ mlo q^kql+pb qfbkb9

,* ,* ,*

Obol EEL=ff. * E,L=0.. < GG?i?100+nrb+bpi^ afpq^k`f^ bkqobi^p alp `fo`rkcbobk`f^pB+ v B1 jbafa^ pl_ob i^ prmbocf`fbabi `lkl- Dpql abjrbpqo^ nrb EH v B0 plkilp cl`lp ab i^ bifmpb+ljl pb e^_Œ^prmrbpql-

Llafcf`^kal ifdbo^jbkqb bpq^ abjlpqo^`fŽk pb qfbkbk i^p `loobpmlkafbkqbpm^o^i^ efm‹o_li^ v i^ m^oŠ_li^- Dk bi `^pl ab i^ efm‹o_li^ pb e^ ab `lkpfabo^ork^ bpcbo^ bk `^a^ elg^ abi `lkl+ v m^o^ i^ m^oŠ_li^ rk^ bpcbo^ q^kdbkqb ^imi^kl pb`^kqb bk bi cl`l- Dpq^bpcbo^bp q^kdbkqb^i `lkl ^ il i^odl ab rk^ `fo,`rkcbobk`f^ pfqr^a^ bk rk mi^kl `rv^ fkqbopb``fŽk `lk bi mi^kl pb`^kqb bp i^afob`qofwab i^ m^oŠ_li^- Blk bpq^pfkaf`^`flkbp bi ib`qlo mrbab mol_^o nrb i^pmolmfba^abpcl`^ibp ab i^ efm‹o_li^ v i^ m^oŠ_li^ pb mrbabk abar`fo ab pr abcf,kf`fŽk `ljl pb``flkbp ab rk `lkl-

02-08 Du`bkqof`fa^aab i^p pb``flkbp `Žkf`^p

Nqo^molmfba^a`^o^`qboŒpqf`ab i^p pb``flkbp `Žkf`^p pb obcfbob rk `lk`bmqlii^j^al bu`bkqof`fa^a- Tk^ pb``fŽk `Žkf`^ mrbab abcfkfopb ljl rk^ `ros^ abp`ofq^mlo rk mrkql nrb pb jrbsb bk rk mi^kl ab j^kbo^ nrb i^ o^wŽkab prp afpq^k`f^p^ rk mrkql cfgl v ^ rk^ ob`q^ cfg^bp `lkpq^kqb- Dpq^ o^wŽk`lkpq^kqb pb ii^j^`s^`iomd^d_\_ ab i^ `ros^ v pb abpfdk^ mlo `, 'Ml ab_b `lkcrkafopb `lk bi k•jbol` ab Dribo-( K^ `ros^ bp rk^ `gdkn` pf N ; ` ; 0+ rk^ k\mƒ]jg\ pf ` < 0+ X rk^cdk„m]jg\ pf ` = 0- Di mrkql cfgl pb ii^j^ aj^j v i^ ob`q^ cfg^_dm`^omdu,

@almq^objlp bpq^ abcfkf`fŽk `ljl _^pb ab krbpqol bpqrafl ab i^p `Žkf`^pv^ nrb mbojfqb qo^q^opfjriqŠkb^jbkqb ilp qobpqfmlpab `Žkf`^p v c^slob`b bi rpl abilp j‹qlalp sb`qlof^ibp- Dk bpq^afpbkpfŽk pb pl_obkqfbkab nrb qlalp ilp mrkqlpv ob`q^pbpqŠkbk bi jfpjl mi^kl-

CDEHMHBHˆM- A\_jn pi\ m`^o\ I* pi kpioj C ij k`mo`i`^d`io` \ I* u piiˆh`mj kjndodqj `, A`ndbi`hjn ^ji _&U*I' g\ _dno\i^d\ _` pi kpioj V \ I, Bg^jiepioj _` oj_jn gjn W lp` n\odna\^`i g\ m`g\^d‡i

'02-1/( GGV, DGG< ` ^cT) H&

`n pi\ ^‡id^\ ^ji `s^`iomd^d_\_ `, I\ ^‡id^\ `n pi\ `gdkn` PF ` ; 0+pi\ k\mƒ]jg\nd` < 0+X pi\ cdk„m]jg\ nd` = 0-

Page 175: Calculus

Bs^`iomd^d_\_ _` g\n n`^^dji`n ^‡id^\n 502

Rf K bp rk sb`qlo kloj^i ^ I v M `r^inrfbo mrkql ab I i^ afpq^k`f^ _&U*I'ab `r^inrfbo mrkql W ^ H sfbkb a^a^ mlo i^ cŽojri^

^%T H& < E%T* L& +JE ++ GGLGG

Br^kal K qfbkb ilkdfqra 0+ bpq^ bumobpfŽkpb pfjmifcf`^ v nrba^ _&U*I' :<H'W , L&{ LG+X i^ b`r^`fŽk crka^jbkq^i '02-1/( ab i^p pb``flkbp `Žkf`^p pbqo^kpcloj^ bk

'02-10( 0GT * DGG< ` E%T* L& +JE+

K^ ob`q^ H pbm^o^bi mi^kl bk alp obdflkbp nrb ii^j^objlp ^o_fqo^of^jbkqb~mlpfqfs^‚ v ~kbd^qfs^‚ pbd•k i^ bib``fŽk ab K, Rf 'W , M' , K = N+ab`fjlp nrbW bpqŠbk bi pbjfmi^kl mlpfqfsl+ v pf 'W , M' , K ; N bk bi pbjf mi^kl kbd^qfsl-O^o^ ilp mrkqlp ab i^ ob`q^ I qbkbjlp 'W , M' , K < N-Dk i^ cfdro^ 02-01 i^ bib`,`fŽk abi sb`qlo kloj^i J fkaf`^ nrb ilp mrkqlp pfqr^alp ^ i^ abob`e^ ab H bpqŠkbk bi pbjf mi^kl mlpfqfsl v ilp ab i^ fwnrfboa^ bk bi kbd^qfsl-

Blilnrbjlp bi cl`l E bk bi pbjfmi^kl kbd^qfsl+ `ljl pb fkaf`^ bk i^ cfdr,o^ 02-01+v bifg^jlp M ab jlal nrb pb^ bi mrkql ab I jŠp moŽufjl ^ E- Dkqlk`bpL * D < ^J) pfbkal G`h< GGN, DGGi^ afpq^k`f^ abi cl`l ^ i^ afob`qofw-Orbpql nrb

HUsb`qlo rkfq^ofl++! kloj^i ^ H

GGV,DGG++.-.

-.-.

!!-. GEl`l C ŠŠ888 )& %%%%%% M < C * _K

w%T * B&+Jw^ * %T * B&{J

Cfob`qofwH

EHFTQ@ 02-01 Ri\ ^‡id^\ _` `s^`iomd^d_\_ ` `n `g ^jiepioj _` oj_jn gjn W lp` n\odna\^`iGGV, DGG< _f%T * B&+J * `e-

E bpqŠbk bi pbjfmi^kl kbd^qfsl+ qbkbjlp 'E , M' , K < , _ ; N+^pŒnrb _ bpmlpfqfsl- Rrpqfqrvbkal M mlo E * _K bk '02-10(+ l_qbkbjlp bi qblobj^ pfdrfbk,qb+nrb pb obmobpbkqbk i^ cfdro^ 02-01-

Page 176: Calculus

4/2 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

RCMPCK? 02-06- P`\ a pi\ ^‡id^\ ^ji `s^`iomd^d_\_ `* aj^j C*u _dm`^omduI \ pi\ _dno\i^d\ _ _` B+PdK `n pi q`^ojm pido\mdj ijmh\g \ I t ndC `noƒ `i `gn`hdkg\ij i`b\odqj _`o`mhdi\_j kjm K* `ioji^`n b `n `g ^jiepioj _` oj_jn gjnkpiojn U lp` n\odna\^`i g\ `^p\^d‡i

'02-11( 00T * DGG < ` E%T* B& +J * ^c +

02-1/ D`r^`flkbp mli^obp ab i^p `Žkf`^p

K^ b`r^`fŽk abi qblobj^ 02-06 mrbab pfjmifcf`^opb pf `lil`^jlp bi cl`l bkrk^ mlpf`fŽk bpmb`f^i-Olo bgbjmil+ pf bi cl`l bpqŠbk bi lofdbk i^ b`r^`fŽk pbqo^kpcloj^ bk

'02-12( GGVGG< ` ET$ J * ^c+

+Dpqcloj^ bp m^oqf`ri^ojbkqb •qfi pf nrbobjlp bumobp^oW bk crk`fŽk ab `lloab,k^a^p mli^obp-Sljbjlp i^ afob`qofwH sboqf`^i+`ljl bk i^ cfdro^ 02-02+v mlkd^,jlp K < d, Rf W qfbkb`lloabk^a^p mli^obpmv `*qbkbjlp HHWHH;m* W&K ;m `lp `*v i^ b`r^`fŽk '02-12( pb `lksfboqb bk

'02-13( l < _ Holp _ * ^c +

Rf W bpqŠ ^ i^ fwnrfboa^ ab i^ afob`qofw+qbkbjlp oblp ` ; ^) ^pŒ nrbHolp ` + ^c < ^ * l `lp ` u '02-13( qlj^ i^ cloj^ l < _%^ * oblp %d&)l+ abp,mbg^kal m*qbkbjlp

'02-14(`_

m;` `lp _ * 0

Rf W bpqŠ^ i^ abob`e^ ab i^ afob`qofw+qbkbjlp oblp ` = ^) ^pŒnrb '02-13( ^almq^i^ cloj^

m < `&m lp ` + ^& )

l_qbkfbkal

m << `_` `lp ` + 0

Orbpql nrb l = N+bpq^•iqfj^ b`r^`fŽk fjmif`^ _ = 0- Cf`el ab lqol jlal+ pŽilm^o^i^ efm‹o_li^ bufpqbkmrkqlp ^ i^ abob`e^ ab i^ afob`qofw-@pŒmrbp+ebjlp ab,jlpqo^al bi pfdrfbkqb qblobj^ nrb pb obmobpbkqbk i^ cfdro^ 02-02-

'02-15(

Page 177: Calculus

Bd`m^d^djn 504

Cfob`qofw Cfob`qofw

GGGGGG

'( G

Cx

^( l `lp %F ; ^ bk i^ bifmpb+i^ m^oŠ_li^ vi^ o^j^ fwnrfboa^ ab i^ efm‹o_li^-

_( m `lp %F = ^ bk i^ o^j^ abob`e^ abi^ efm‹o_li^-

EHFTQ@ 02-02 @‡id^\n ^ji `^p\^d‡i kjg\m m< ah m`lp %F * `e - Bg oj^j C `n `g jmdb`iv `noƒ \ g\ dulpd`m_\ _` g\ _dm`^omdu,

RCMPCK? 02-07- P`\ b pi\ ^‡id^\ ^ji `s^`iomd^d_\_ `* pi aj^j E `i `gjmdb`i* v pi\ _dm`^omduq`mod^\g I \ pi\ _dno\i^d\ _ \ g\ _`m`^c\ _` E- Pd N; `888990+g\ ^‡id^\ b `n pi\ `gdkn` j pi\ k\mƒ]jg\9 oj_j kpioj _` b `noƒ \ g\ dulpd`m_\ _`I u n\odna\^` g\ `^p\^d‡i kjg\m

'02-16(`_

l:`^jn &'* h

Pd ` = 0+g\ ^pmq\ `n pi\ cdk„m]jg\ ^ji pi\ m\h\ \ ^\_\ g\_j _` I, Ijn kpiojn _`g\ m\h\ _` g\ dulpd`m_\ n\odna\^`i '02-16( u gjn _` g\ m\h\ _` g\ _`m`^c\ n\odna\^`i

'02-17( `_l:

`^jn &'+

Dk bi pfdrfbkqb`lkgrkql ab bgbo`f`flp pb afp`rqbk i^p b`r^`flkbp mli^obp`loobp,mlkafbkqbp ^ lqo^p mlpf`flkbp ab i^ afob`qofw-

)+&*) :WR_PVPV\`

0- Cbjlpqo^o nrb i^ b`r^`fŽk '02-11( abi qblobj^ 02-06 ab_b obbjmi^w^opb mlo

GGV, BEE< ` x%T* B&$J * `e

pf B bpqŠbk bi pbjfmi^kl mlpfqfsl abqbojfk^al mlo J+

Page 178: Calculus

4/4 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

1- Rb^ a rk^ `Žkf`^ ab bu`bkqof`fa^a `* rk cl`l bk bi lofdbk+ u afob`qofw sboqf`^i H ^ rk^afpq^k`f^ ^ ^ i^ fwnrfboa^ ab B+^( Cbjlpqo^o nrb pf b bp rk^ bifmpb l rk^ m^oŠ_li^+ qlal mrkql ab b bpqŠ ^ i^ abob`e^ab H v p^qfpc^`b i^ b`r^`fŽk mli^o

`_m;g+`^jnL%

_( Cbjlpqo^o nrb pf b bp rk^ efm‹o_li^+ ilp mrkqlp ab i^ o^j^ ab i^ abob`e^ p^qfpc^`bki^ b`r^`fŽk ab i^ m^oqb^( v ilp ab i^ o^j^ fwnrfboa^ p^qfpc^`bk o < , `_-L * b`lp %F&+

N_p‹osbpb nrb bk bpqb `^pl 0 * ` `lp %F pfbjmob bp kbd^qfs^- -2- Rf rk^ `Žkf`^ qfbkb rk^ afob`qofwelofwlkq^i ^ afpq^k`f^ _ mlo bk`fj^ ab rk cl`l pfqr^al

bk bi lofdbk- abjlpqo^o nrb prp mrkqlp p^qfpc^`bk i^p b`r^`flkbp mli^obp l_qbkfa^p abi^p abi qblobj^ 02-07 obbjmi^w^kal `lp ‹ mlo pbk %F+ƒBrŠibp plk i^p `loobpmlkafbkqbpb`r^`flkbp mli^obp pf i^ afob`qofw bp elofwlkq^i v bpqŠ mlo ab_^gl abi cl`l>

B^a^ rkl ab ilp bgbo`f`flp abi 3 ^i 8 a^ rk^ b`r^`fŽk mli^o ab rk^ `Žkf`^ `lk rkcl`l B bk bi lofdbk u rk^ afob`qofw sboqf`^i ^ i^ abob`e^ ab B+ Dk `^a^ `^pl+ abqbojfk^o i^bu`bkqof`fa^a ` v i^ afpq^k`f^ _ abi cl`l ^ i^ afob`qofw-G^`bo rk af_rgl jlpqo^kal& i^ obi^,`fŽk ab i^ `ros^ ^ pr cl`l u ^ pr afob`qofw-

1,& T <,,,,

0 )xjnL%

05, m < 0 -

,!1! * `lp K

22+ l < 0 * p lp K †

53+ l:***

2*`lpN

37 l:****

- 0 * 1 `lp %F$

38- o < 0 * `lp %F†

Dk `^a^ rkl ab ilp bgbo`f`flp abi 0/ ^i 01+ rk^ `Žkf`^ ab bu`bkqof`fa^a ` qfbkb rk cl`lbk bi lofdbk v rk^ afob`qofw a^a^ mlo pr b`r^`fŽk `^oqbpf^k^- Dk `^a^ `^pl+ `^i`ri^o i^ afp,q^k`f^ ^ abi cl`l ^ i^ afob`qofwu abqbojfk^o i^ b`r^`fŽk mli^o ab i^ `Žkf`^- O^o^ rk^ efm‹o,_li^+ a^o rk^ b`r^`fŽk mli^o m^o^ `^a^ o^j^- G^`bo rk af_rgl jlpqo^kal i^ obi^`fŽk ab i^`ros^ ^ pr cl`l v ^ pr afob`qofw-

.-+ ` < p: afob`qofw9 0r * 1s < 14-ii-b<i: afob`qofw9 2s)1t;03,./+ ` < 1: afob`qofw9 s * t < 0-02- Tk `ljbq^ pb jrbsb pfdrfbkal rk^ Žo_fq^ m^o^_Žif`^ `lk bi Rli bk bi cl`l- Br^kal bi

`ljbq^ bpqŠ ^ 0/7 hfiŽjbqolp abi Rli+ rk sb`qlo nrb rkb bi cl`l ^i `ljbq^ cloj^ rkŠkdril ab 60&.2 `lk bi sb`qlo rkfq^ofl J qo^w^al m-lo bi cl`l mbombkaf`ri^ojbkqb ^ J)bpq^kal bi cl`l bk bi pbjfmi^kl kbd^qfsl abqbojfk^al mlo J+^( G^ii^o i^ b`r^`fŽk mli^o ab i^ Žo_fq^+qlj^kal bi lofdbk `ljl cl`l+ v `^i`ri^kal i^jbklo afpq^k`f^ bkqob bi `ljbq^ v bi Rli-_( Qbplisbo i^ m^oqb^( pf bi cl`l bpqŠ bk bi pbjfmi^kl mlpfqfsl abqbojfk^al mlo J+

T-11 BŽkf`^ppfj‹qof`^p obpmb`ql i lofdbk

- Rb af`b nrb rk `lkgrkql ab mrkqlp bp ndh„omd^jm`nk`^oj \g jmdb`i pf , WbpqŠbk bi `lkgrkql pfbjmob nrb W mboqbkbw`^ ‹i- Cbjlpqo^jlp pbdrfa^jbkqb nrb

Page 179: Calculus

@‡id^\n ndh„omd^\n m`nk`^oj \g jmdb`i 506

bi cl`l ab rk^ bifmpb l ab rk^ efm‹o_li^ mrbab pfbjmob pfqr^opb ab jlal nrb i^`Žkf`^ pb^ pfj‹qof`^ obpmb`ql ^i lofdbk- O^o^ e^`boil bp`of_^jlp i^ b`r^`fŽk crk,a^jbkq^i '02-11( ^pŒ9

'02-18( 00U + DGG< ` F&U+ C'% K + ^c < ` FU%K + C%K + ^c < g`U%K + ]h+

alkab \ < `_ * `C% K, Dibs^kal ^j_lp jfbj_olp ^i `r^ao^al- l_qbkbjlp

%.0+0-& 00VG01, 0C} U * 00D001 < `0&U%K'0 + 0`\U% K * \0,

Rf qfbkb nrb e^_bo pfjbqoŒ^ obpmb`ql ^i lofdbk+ bpq^ b`r^`fŽk ab_b q^j_f‹k p^qfp,c^`bopb `r^kal W pb obbjmi^`b mlo , W+ aŠkalklp

'02-20( GGVG01 * 0C} U * 00D001 < „&U%K'0 * 0`\U% K * \0,

Qbpq^kal '02-20( ab '02-2/(+ qbkbjlp pfjbqoŒ^ pf v pŽil pf

C%U; `\U}K Ž &C+ `\K' , U <9< M -

Dpq^ b`r^`fŽk mrbab p^qfpc^`bopb m^o^ qlal W ab i^ `ros^ pf v pŽil pf B v J bpqŠkifd^alp mlo i^ b`r^`fŽk

'02-21( C; `\K* alkab \ < `_ * `]< K ,

K^ obi^`fŽk C < `\K fjmif`^ C%K < `\* l_qbkfbkal \ < `_ * `%\, Rf ` < 0+bpq^ •iqfj^ b`r^`fŽk kl mrbab p^qfpc^`bopb v^ nrb ^) i^ afpq^k`f^ abi cl`l ^ i^ af,ob`qofw+kl bp kri^- Dpql pfdkfcf`^ nrb m^o^ i^ m^oŠ_li^ kl e^v pfjbqoŒ^ obpmb`ql^i lofdbk- Rf ` ;/; 0+pfbjmob mlabjlp p^qfpc^`bo i^ obi^`fŽk '02-21( qlj^kal

'02-22(`_

[:**+0 , `/

B:wJ+0 , `/

N_p‹osbpb nrb \ = N pf ` ; 0 v \ ; N pf ` = 0- Olkfbkal C < `\K bk '02-2/(l_qbkbjlp bi pfdrfbkqb

RCMPCK? 02-08- P`\ B pi\ ^jid^\ ^ji `s^`iomd^d_\_ ` ;/; 0 V pi aj^j C\ pi\ _dno\i^d\ _ _` pi\ _dm`^omduI, Pd K `n pi q`^ojm pido\mdj ijmh\g \ I tC < `\K* nd`i_j \ < `_-&/ + `0

'* `ioji^`n A `n `g ^jiepioj _` oj_jn gjn kpi+ojn W lp` n\odna\^`i g\ `^p\^d‡i

'02-23(

Page 180: Calculus

4/6 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

Dpq^ b`r^`fŽk bsfabk`f^ i^ pfjbqoŒ^obpmb`ql ^i lofdbk v^ nrb kl `^j_f^`r^kal W pb obbjmi^w^ mlo +s,Cb_fal ^ bpq^pfjbqoŒ^+i^ bifmpbv i^ efm‹o_li^qfbkbk `^a^ rk^ alp cl`lp+ pfj‹qof`^jbkqb `lil`^alp obpmb`ql^i `bkqol+ v alpaf,ob`qof`bp+q^j_f‹k pfj‹qof`^jbkqb `lil`^a^p obpmb`ql^i `bkqol-

K^ b`r^`fŽk '02-23( pb p^qfpc^`b `r^kal W < *(*[J+ Dplp alp mrkqlp pbii^j^k q„mod^`nab i^ `Žkf`^- Di pbdjbkql nrb ilp rkb bp bi `e` h\tjm pf i^ `Žkf`^bp rk^ bifmpb+v bi `e` om\inq`mnjpf bp rk^ efm‹o_li^-

Rb^ J$ rk sb`qlo rkfq^ofl loqldlk^i ^ J+ Rf W < \J$) bkqlk`bp W&J < N+^pŒnrb i^ b`r^`fŽk '02-23( pb p^qfpc^`bm^o^T < ]K%pf v pŽil pf ]0 * _0[0 < _%*

Dpql bufdb ` ; 0+ ]% < \0&/ + `0', Di pbdjbkql nrb rkb ilp mrkqlp W < +)+]K%*

alkab ] < \x pb ii^j^ `e` h`ijm ab i^ bifmpb-

L]n`mq\^d‡i8 Rf mlkbjlp ` < N bk '02-23(+ obpriq^ zGVGG< \, b`r^`fŽk ab rk^`fo`rkcbobk`f^ ab o^afl \ v `bkqol bk bi lofdbk- @ i^ sfpq^ ab '02-22(+ mlabjlp `lkpf,abo^o q^i `fo`rkcbobk`f^ `ljl `^pl iŒjfqb ab rk^ bifmpb bk i^ nrb ` x N X _ x // abjlal nrb _^ w [+

)+&*+ :PbNPV\[R`PN_aR`VN[N`QRYN Pp[VPN`

O^o^ l_qbkbo i^p b`r^`flkbp `^oqbpf^k^pab i^ bifmpbv ab i^ efm‹o_li^+ bp`of,_fjlp '02-23( bk crk`fŽk ab i^p `lloabk^a^p ob`q^kdri^obp ab W- Difg^jlp K < d'0/ nrb pfdkfcf`^ nrb i^p afob`qof`bp plk sboqf`^ibp( v pb^ W < %r) s&+Dkqlk`bpyyUT < s0 * v1+ W&K < s* v '02-23( qlj^ i^ cloj^ s0 * m* `0\0; `0s0 * \%*l s0%f * `0

' * v1 < \0%f * `0'* il nrb klp a^ -

N+ \,,*,,+,,,<0-\0 \0%. Z `0

'

Dpq^b`r^`fŽk `^oqbpf^k^obmobpbkqi^ bifmpb%_; 0( v i^ efm‹o_li^ %_= 0( v pbaf`b nrb bpqŠbk i^ ajmh\ ^\i‡id^\, Klp cl`lp bpqŠk bk ilp mrkqlp o\`* N( v&+\`* N(: i^p afob`qof`bpplk i^p ob`q^psboqf`^ibps < \g` u s < +n\Z `,

Rf ` ; 0+mlkbjlp ] < \n- 0 , `/ u bp`of_fjlp i^ b`r^`fŽk ab i^ bifmpbbki^ cloj^ `^kŽkf`^

'02-24(

'02-25(u1 t0'%'5*([/ ]/

N(+pfbkal ` < \` < S\/* ]/† Dk i^ cfdro^Rrp cl`lp bpqŠk bk %])K&v %*_)

02-03 ^( pb jrbpqo^ rk bgbjmil-

Page 181: Calculus

B^p\^dji`n ^\mo`nd\i\n ^_ g\n ^‡id^\n 508

Rf _ = 0+mlkbjlp ] < h]hT_/ * 0 X bp`of_fjlp i^ b`r^`fŽk ab i^ efm‹o_li^bk i^ cloj^ `^kŽkf`^

'02-26(

Rrp cl`lp bpqŠkbk 0/% mrkqlp &`*N( u &+`* N(+pfbkal ` < g\G`< S\/ * ]/† Dki^ cfdro^ 02-03 _( pb jrbpqo^ rk bgbjmil-

L]n`mq\^d‡i8 Cbpmbg^kal bk '02-26( i^ v bk crk`fŽk ab s* l_qbkbjlp alp plir`flkbp

'02-27(] ,-++

V < |+qs/ +\/†

g\g

O^o^ s^ilobp ab s do^kabp v mlpfqfslp+ bi k•jbol +zs%+\%bp `^pf fdr^i ^ s* ^pŒnrb bi pbdrkaljfbj_ol ab '02-27(+ bp moŽufjl- ^ ~ \r,f[`+ Dp cŠ`fi abjlpqo^o nrb i^ afcbobk`f^ bkqobt*;]sg\g b t*;] !U&u&,^&.H^iqfbkab ^ N `r^kal U+) *//- Dpq^ afcbobk`f^ bp

] , -++ ] s/* &s/

* \/& g\g ] g\g ]IH , 01 < , %r * U r/

* [/& < , ,,,,,,,,,,,, < ; ,,

h]h g\g s * qs/* \/ s * Ss/

* \/ s

t

'^( Difmpb

+ +y * n---- < 0: ]%< \% + `&\% ]%

t

s

'_( Gfm‹o_li^

EHFTQ@ 02-03 @‡id^\n ^ji `s^`iomd^d_\_ ` :{‹ 0+ ndh„omd^\n m`nk`^oj \g jmdb`i, Ijn aj^jn`noƒi `i '~ `* N(+ nd`i_j ` < y\g`, Ijn omdƒibpgjn m`g\^dji\i b`jh„omd^\h`io` \* ]* ^,

Page 182: Calculus

40. >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

ab j^kbo^ nrb VF+V0+). `r^kal s+) *//- Olo `lkpfdrfbkqb+ i^ ob`q^ t;]s-g\g bp rk^^pŒkqlqab i^ efm‹o_li^- K^ ob`q^t;+]sgg\g`n i^ lqo^^pŒkqlq^-Rb af`b nrb i^ efm‹o_li^ qfbkabe^`f^ bp^pob`q^p^pfkq5qf`^jbkqb-Dk i^ cfdro^ 02-03\& pb obmobpbkq^ki^p ^pŒkqlq^p-

K^ b`r^`f5k `^oqbpf^k^ab i^ bifmpbv ab i^ efm‹o_li^ qlj^ afpqfkq^cloj^ pfi^p afob`qof`bpkl plk sboqf`^ibp-Olo bgbjmil+ pf i^p afob`qof`bppb qlj^k elofwlk,q^ibp+mlabjlp `lkpfabo^o J < g bk i^ b`r^`f5k '02-23(- Orbpql nrb W&J :< W&f < t* l_qbkbjlp rk^ b`r^`f5k `^oqbpf^k^ m^ob`fa^ ^ i^ '02-24(+ bu`bmqlnrb s b t bpqŠkmbojrq^a^p- K^ cloj^ `^kŽkf`^ bk bpqb`^pl bp

'02-28(

Rf i^ `5kf`^ pb qo^pi^a^mlo^af`f5k ab rk sb`qlo T} < %rj * Ui& ^ `^a^ rkl abprp mrkqlp+bi `bkqol pboŠ%rj *Vj' bk ird^o abi lofdbk- K^p `loobpmlkafbkqbp b`r^,`flkbp `^oqbpf^k^p mrbabk l_qbkbopb ab '02-24( l '02-28( prpqfqrvbkal s mlos + Tj b V mlo V + Vj,

O^o^ l_qbkbo rk^ b`r^`fŽk `^oqbpf^k^ m^o^ i^ m^oŠ_li^+ `lkpfabobjlp abkrbsl i^ b`r^`fŽk crka^jbkq^i '02-1/( `lk ` < 0- Sljbjlp `ljl afob`qofwi^ob`q^ sboqf`^i r < *_ v pfqrbjlp bi cl`l bk %_)N(- Rf W < %r) s&) qbkbjlpW , E < %r * _) s&) v i^ b`r^`fŽk '02-1/( klp a^ %r * _&/ * v1 < Gt* _./+Dpql pfjmifcf`^ i^ b`r^`fŽk `^kŽkf`^

'02-3/( t0 < 2^s,

Di mrkql jbafl bkqobbi cl`l v i^ afob`qofw'bi lofdbk bk i^ cfdro^ 02-04( pb ii^j^q„mod^ab i^ m^oŠ_li^+v i^ ob`q^ nrb m^p^mlo bi s‹oqf`b v bi cl`l bp bi `e` ab i^m^oŠ_li^- K^ m^oŠ_li^ bp pfj‹qof`^ obpmb`ql^ pr bgb-Rf ` = N+i^ m^oŠ_li^ bpqŠ^i^ abob`e^ abi bgb t* `ljl bk i^ cfdro^ 02-04- Br^kal ` ; N+i^ `ros^ bpqŠ^ i^fwnrfboa^ abi bgbv-

Rf pb bifdbk ilp bgbpab jlal nrb bi cl`l bpq‹ bk bi bgbv bk bi mrkql 'N+_&

v pf i^ ob`q^ elofwlkq^i t < +` pb qlj^ `ljl afob`qofw+i^ cloj^ `^kŽkf`^ ab i^b`r^`fŽk F `^oqbpf^k^ qlj^ i^ cloj^

s0 < 2^t,

Br^kal ` = N i^ m^oŠ_li^ pb ^_ob e^`f^ ^oof_^ `ljl jrbpqo^ i^ cfdro^ 02-05+v `r^kal ` ; N+pb ^_ob e^`f^ ^_^gl-

Rf i^ m^oŠ_li^ ab i^ cfdro^ 02-04 pb qo^pi^a^ ab jlal nrb pr s‹oqf`b bpq‹ bkbi mrkql %rj * Xl(+ i^ `loobpmlkafbkqb b`r^`fŽk bp

%s* VL'0 < 2^&s + sj'%

Page 183: Calculus

Be`m^d^djn 510

t t

Cfob`qofws; + b

s

+))$! %r)s&-++&! G

yy GGG

> G'''''''''' ''''''R'''''Cfob`qofw t < , b

EHFTO& 02-04 I\ k\mƒ]jg\ v1 < 2^s, EHFTQ@ 02-05 I\ k\mƒ]jg\ s0 < 3`v

Di cl`l bpqŠ ^elo^ bk bi mrkql %rj * _) Xl( X i^ afob`qofw bp i^ ob`q^ r < Uj + `-Di bgb ab i^ m^oŠ_li^ bp i^ ob`q^ X < Xl&

@kŠild^jbkqb+ rk^ qo^pi^`fŽk ab i^ m^oŠ_li^ ab i^ cfdro^ 02-05 klp `lkar`b^ i^ b`r^`fŽk

`lk cl`l bk %rj* Xl * ]&+ K^ ob`q^ X < Xl , _ bp pr afob`qofw+i^ ob`q^ r < Tj

pr bgb-Di ib`qlo mrbab bk`lkqo^o bkqobqbkfal abjlpqo^o nrb rk^ m^oŠ_li^ krk`^

qfbkb ^pŒkqlq^p-

)+&*, Dgbo`f`flp

B^a^ rk^ ab i^p b`r^`flkbp bk ilp bgbo`f`flp abi 0 ^i 5 obmobpbkq^rk^ bifmpb-G^ii^o i^p`lloabk^a^p abi `bkqol+ ilp cl`lp v ilp s‹oqf`bp+ v af_rg^o `^a^ `ros^- Cbqbojfk^o q^j_f‹ki^ bu`bkqof`fa^a-

s0, t00- 0// * 25 < 0-

t0 s01- 0// * 25 < H-

%r * 1(1 %s * 2(12- 05 * ,8, < G-

2, 7s0 * 03t0 < 14-

3, 2t0 * 1s0 < G-

%r * 0(1 %s * 1(15- ,,05, * 14 < 0-

Page 184: Calculus

400 >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

Dk `^a^ rkl ab ilp bgbo`f`flp abi 6 ^i 01+ e^ii^o i^ b`r^`fŽk `^oqbpf^k^ 'bk i^ cloj^`^kŽkf`^ ^molmf^a^( m^o^ i^ bifmpbnrb p^qfpc^`b i^p `lkaf`flkbp a^a^p- Cf_rg^o `^a^ `ros^-

6- Bbkqol bk 'N+N(+rkcl`l bk ' +N(+rk s‹oqf`b bk '0+ N(-7- Bbkqol bk ',2+ 3(+ pbjfbgbp ab ilkdfqra 3 v 2+bgbj^vlo m^o^ibil ^i bgb s,8- Kl jfpjl nrb bk bi bgbo`f`fl 7+ p^isl nrb bi bgb j^vlo bp m^o^ibil ^i bgb v-

0/- U‹oqf`bp bk ',0+ 1(+ ',6+ 1(+ bgb jbklo ab ilkdfqra 1-00- U‹oqf`bp bk '2+ ,1(+ '02+ ,1(+ cl`lp bk '3+ ,1(+ '01+ ,1(-01- Bbkqol bk '1: 0(+bgb j^vlo m^o^ibil ^i bgbs* i^ `ros^ m^p^mlo ilp mrkqlp '5+ 0( X '1+ 2(-

B^a^ rk^ ab i^p b`r^`flkbp bk ilp bgbo`f`flp 02 ^i 07 obmobpbkq rk^ efm‹o_li^- G^ii^oi^p `lloabk^a^p abi `bkqol+ ilp cl`lp v ilp s‹oqf`bp- Cf_rg^o `^a^ `ros^ v jlpqo^o i^p mlpf,`flkbp ab i^p ^pŒkqlq^p-B^i`ri^o q^j_f‹k i^ bu`bkqof`fa^a-

s0 t002- 0// , 53 < 0-

t0 s003- 0// , 53 < 0-

%r * 2(104- 3 , %s* 2(1 < 0-

.3+ 6r0 + .3t0 < 033-

/5, 2s/ * 3t/ * 1/ < N-

%r * 0(107- 3

%s* 1(18 < 0-

Dk `^a^ rkl ab ilp bgbo`f`flp 08 ^i 12+ e^ii^o i^ b`r^`fŽk `^oqbpf^k^ 'bk i^ cloj^ `^k‹,kf`^ ^ab`r^a^( m^o^ i^ efm‹o_li^ nrb p^qfpc^`b i^p `lkaf`flkbp a^a^p- Cf_rg^o `^a^ `ros^ ui^p ^pŒkqlq^p-08- Bbkqol bk 'N+ N(+rk cl`l bk '3+ N(+rk s‹oqf`b bk '1+ N(-1/- El`lp bk 'N+ € Ui(+ s‹oqf`bp bk 'N+ € 0(-10- U‹oqf`bp bk '€1+ N(+^pŒkqlq^pv < |0s,11- Bbkqol bk ',0+ 3(+ rk cl`l bk ',0+ 1(+ rk s‹oqf`b bk ',0+ 2(-12- Bbkqol bk '1+ ,2(+ bgb qo^kpsbopl m^o^ibil ^ rkl ab ilp bgbp`lloabk^alp+ i^ `ros^ m^p^

mlo '2+ ,0(v ',0+ N(-13- ƒO^o^ nr‹ s^ilo 'l s^ilobp( ab b i^ ob`q^ 1s + 1v < b pboŠ q^kdbkqb ^ i^ efm‹o_li^

s0 + 1t0 < 0014- K^p ^pŒkqlq^pab rk^ efm‹o_li^ plk i^p ob`q^p /r * v < N X /r * v < N- G^ii^o i^

b`r^`fŽk `^oqbpf^k^ ab i^ `ros^ pf m^p^ mlo bi mrkql '2+ ,4(-B^a^ rk^ ab i^p b`r^`flkbp bk ilp bgbo`f`flp 15 ^i 20 obmobpbkq rk^ m^oŠ_li^- G^ii^o

i^p `lloabk^a^p ab ilp s‹oqf`bp+ i^ b`r^`fŽk ab i^ afob`qofw+v i^ abi bgb- Cf_rg^o `^a^ rk^ab i^p `ros^p-04, t0 < +6s, 18- s0 < 4t,

05, t0 < 1s, 2/- s0 * 6t < N-17- 'u , 0(1 < i1u , 5- 20- %r * 1(1 < 1s * 8-

Dk `^a^ rkl ab ilp bgbo`f`flp abi 21 ^i 26+ e^ii^o i^ b`r^`fŽk `^oqbpf^k^ 'bk i^ cloj^`^kŽkf`^ ^ab`r^a^( m^o^ i^ m^oŠ_li^ nrb p^qfpc^`bi^p `lkaf`flkbp a^a^p v af_rg^o i^ `ros^-21- El`l bk 'N+ ,f(: b`r^`fŽk ab i^ afob`qofw+s < f-22- U‹oqf`b bk 'N+ N(: b`r^`fŽk ab i^ afob`qofw+s < ,1-23- U‹oqf`b bk ',3+ 2(: cl`l bk ',3+ 0(-24- El`l bk '2+ ,0(: b`r^`fŽk ab i^ afob`qofw+s <p-25- Dgb m^o^ibil ^i bgb v: m^p^ mlo 'N+ 0(+'0+N( X '1+ N(-26- Dgb m^o^ibil ^i bgb r8 s‹oqf`b bk '0+ 2(: m^p^ mlo ',0+ ,0(-27- O^oqfbkal ab i^ abcfkf`fŽk cl`^i+ e^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab i^ m^oŠ_li^ `rvl cl`l

bp bi lofdbk v `rv^ afob`qofwbp i^ ob`q^ /r * v < 0/-

Page 185: Calculus

Be`m^d^djn q\mdjn nj]m` ^‡id^\n 512

)+&*- :WR_PVPV\`cN_V\``\O_RPp[VPN`

0- Cbjlpqo^o nrb bi Šob^ ab i^ obdflk ifjfq^a^ mlo i^ bifmpb s0 - \0 * t0- ]0 < 0 bp fdr^i^ \] jriqfmif`^al mlo bi Šob^ ab rk `Œo`ril ab o^afl 0-

L]n`mq\^d‡i8 Dpq^ molmlpf`fŽk mrbab abjlpqo^opb ^ m^oqfo ab i^p molmfba^abp db,kbo^ibp ab i^ fkqbdo^i+pfk ob^ifw^o kfkdrk^ fkqbdo^`fŽk-

1- ^( Cbjlpqo^o nrb bi slirjbk abi pŽifal ab obslir`fŽk bkdbkao^al ^i dfo^o i^ bifmpbr0 - [0 * t0- \0 < 0 ^iobabalo ab pr bgb j^vlo bp fdr^i ^ [\| jriqfmif`^al mlo bi slir,jbk ab rk^ bpcbo^ rkfa^a-

L]n`mq\^d‡i8 Dpq^ molmlpf`fŽk mrbab abjlpqo^opb ^ m^oqfo ab i^p molmfba^abpdbkbo^ibp ab i^ fkqbdo^i+ pfk ob^ifw^o kfkdrk^ fkqbdo^`fŽk-

_( ƒBrŠi bp bi obpriq^al pf i^ bifmpb dfo^ ^iobabalo ab pr bgb jbklo>2- G^ii^o qlalp ilp k•jbolp mlpfqfslp = v >) = = >) q^ibp nrb bi Šob^ ab i^ obdflk ifjf,

q^a^ mlo i^ bifmpb >s0 * ?t0 < 2 bp fdr^i ^ i^ ab i^ obdfŽk ifjfq^a^ mlo i^ bifmpb

%= * >&r/ * %= * >&s/ < 2 -

3- Tk ^o`l m^o^_Žif`l qfbkb rk^ _^pb ab ilkdfqra ] v ^iqro^ c, Cbqbojfk^o bi Šob^ ab i^obdfŽk ifjfq^a^ mlo bi ^o`l v i^ _^pb-

4- K^ obdfŽk ifjfq^a^ mlo i^ m^oŠ_li^ v1 < 5r v i^ ob`q^ r < 1 dfo^ ^iobabalo abi bgb r+G^ii^o bi slirjbk abi pŽifal ab obslir`fŽk ^pŒbkdbkao^al-

5- Clp m^oŠ_li^p qfbkbk mlo b`r^`flkbp v1 < /%r * 0( b v1 < 1%r * 1( X ifjfq^k rk^ ob,dfŽk abi mi^kl O,^( B^i`ri^o mlo fkqbdo^`fŽk bi Šob^ ab O,_( G^ii^o bi slirjbk abi pŽifal ab obslir`fŽk bkdbkao^al ^i dfo^o O ^iobabalo abi bgb s,b( Kl jfpjl nrb bk _(+ mbol dfo^kal O ^iobabalo abi bgb v-

6- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab i^ `Žkf`^ `lkpqfqrfa^ mlo qlalp ilp mrkqlp 'u+ u( `rv^afpq^k`f^ ^i mrkql 'N+ 1( bp i^ jfq^a ab i^ afpq^k`f^ ^ i^ ob`q^ v < 7-

7- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab i^ m^oŠ_li^ `rvl cl`l bpqŠ bk bi lofdbk v `rv^ afob`qofwbp i^ ob`q^ s * v * 0 < N-

8- G^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab rk^ efm‹o_li^ nrb m^p^ mlo bi lofdbk+ v nrb prp^pŒkqlq^pplk i^p ob`q^p v < /r * 0 b v < */r * 2-

0/- ^( O^o^ `^a^ j = /+ i^ b`r^`fŽk jr/ * %j * /&s/ < j/ * /j obmobpbkq^rk^ bifmpb- G^,ii^o 'bk crk`fŽk ab j& i^ bu`bkqof`fa^a v i^p `lloabk^a^p ab ilp cl`lp-_( G^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab i^ efm‹o_li^ nrb qfbkb ilp jfpjlp cl`lp nrb i^

bifmpb ab i^ m^oqb ^( v nrb qfbkb bu`bkqof`fa^a sf00- Dk i^ pb``fŽk 02-11 pb abjlpqoŽ nrb rk^ `Žkf`^ pfj‹qof`^ obpmb`ql ^i lofdbk p^qfpc^`b

i^ b`r^`fŽk GGV, DGG< g`U} K + \g* alkab \ < `_ * `C} K, Tqfifw^o bpq^ obi^`fŽk m^o^abjlpqo^o nrb .GV, DGG* GGV* DGG< 0\ pf i^ `Žkf`^ bp rk^ bifmpb- Dk lqo^p m^i^_o^p+i^ prj^ ab i^p afpq^k`f^p ab `r^inrfbo mrkql ab i^ bifmpb ^ prp cl`lp bp `lkpq^kqb-

01- Sbkfbkal bk `rbkq^ bi bgbo`f`fl 00+ abjlpqo^o nrb m^o^ `^a^ o^j^ ab i^ efm‹o_li^ i^afcbobk`f^ GGV, DGG, GGV* DGGbp `lkpq^kqb-

02- ^( Cbjlpqo^o nrb rk^ qo^kpcloj^`fŽk mlo eljlqb`f^ 'prpqfqr`fŽk ab s mlo os b v mlo ns&qo^kpcloj^ rk^ bifmpb `lk `bkqol bk bi lofdbk bk lqo^ bifmpb `lk i^ jfpj^ bu`bkqof`fa^a-_( Cbjlpqo^o q^j_f‹k bi ob`Œmol`l- Dpql bp+pf alp bifmpbp `lk`‹kqof`^p qfbkbk i^ jfpj^bu`bkqof`fa^a v bgbpj^vlobp pl_ob i^ jfpj^ ob`q^+bpqŠk obi^`flk^a^p mlo rk^ eljlqb`f^-b( Cbjlpqo^o ilp obpriq^alp ^kŠildlp ^ ilp ^( v _( m^o^ i^p efm‹o_li^p-

Page 186: Calculus

-*, >kgd^\^dji`n _`g ƒgb`]m\ q`^ojmd\g\ g\ b`jh`om…\ \i\g…od^\

03- Tqfifw^o i^ b`r^`fŽk `^oqbpf^k^ nrb obmobpbkq^ qla^p i^p `Žkf`^p ab bu`bkqof`fa^a `v `bkqol bk bi lofdbk m^o^ abjlpqo^o nrb q^ibp `Žkf`^p plk `ros^p fkqbdo^ibp ab i^b`r^`fŽk afcbobk`f^i v&< %_0+ f&r,s+

L]n`mq\^d‡i8 Orbpql nrb ‹pq^ bp rk^ b`r^`fŽk afcbobk`f^i eljld‹kb^ 'pb``fŽk7-14(+ bi `lkgrkql ab qla^p bp^p `Žkf`^p ab bu`bkqof`fa^a ` bp fks^of^kqb cobkqb ^ rk^qo^kpcloj^`fŽk mlo eljlqb`f^- 'BljmŠobpb `lk bi bfbo`f`fl 02-(

04- ^( Cbjlpqo^o nrb bi `lkgrkql ab qla^p i^p m^oŠ_li^p bp fks^of^kqb cobkqb ^ rk^ qo^kp,cloj^`fŽk mlo pbjbg^kw^- Dpql bp+ rk^ q^i pbjbg^kw^ qo^kpcloj^ rk^ m^oŠ_li^ bk rk^m^oŠ_li^-_( G^ii^o qla^p i^p m^oŠ_li^p pbjbg^kqbp ^ v < s0Š

05- K^ ob`q^ r * v * 3 < N bp q^kdbkqb ^ i^ m^oŠ_li^ v1 < f3r+ G^ii^o bi mrkql ab `lkq^`ql-06- ^( C^al [ :; N- Rf i^p alp m^oŠ_li^p v1 < 1j%r * [& v r0 < 2lt plk q^kdbkqbp+ abjlp,

qo^o nrb i^ `lloabk^a^ s abi mrkql ab `lkq^`ql bpqŠ abqbojfk^a^ •kf`^jbkqb mlo \,_( G^ii^o rk^ `lkaf`fŽk m^o^ \* m v l nrb bumobpb bi eb`el ab nrb i^p alp m^oŠ_li^ppb^k q^kdbkqbp-

07- Blkpfa‹obpb bi ird^o dblj‹qof`l ab ilp mrkqlp L abi mi^kl m^o^ ilp nrb i^ afpq^k`f^ abM ^i mrkql '1+ 2( bp fdr^i ^ i^ prj^ ab i^p afpq^k`f^p ab M ^ ilp alp bgbp `lloabk^alp-^( Cbjlpqo^o nrb i^ m^oqbab bpb ird^o pfqr^a^ bk bi mofjbo `r^ao^kqb bp m^oqb ab rk^efm‹o_li^- Rfqr^o i^p ^pŒkqlq^pv e^`bo rk af_rgl-_( Dp_lw^o i^ doŠcf`^ abi ird^o+ bk ilp lqolp `r^ao^kqbp-

08- Clp m^oŠ_li^p qfbkbk bi jfpjl mrkql `ljl cl`l v i^ jfpj^ ob`q^ `ljl bgb+mbol prps‹oqf`bp bpqŠk ^ afpqfkql i^al abi cl`l- Cbjlpqo^o nrb i^p m^oŠ_li^p pb `loq^k loqldl,k^ijbkqb 'l pb^ nrb prp q^kdbkqbp plk mbombkaf`ri^obp bk ilp mrkqlp ab fkqbopb``fŽk(-

1/- ^( Cbjlpqo^o nrb i^ b`r^`fŽk `^oqbpf^k^

obmobpbkq^qla^p i^p `Žkf`^p pfj‹qof`^p obpmb`ql ^i lofdbk `lk cl`lp bk 'b+ N( v %*])K&+_( L^kqbkbo cfgl ^ v abpfdk^o `lk R bi `lkgrkql ab qla^p bp^p `Žkf`^p l_qbkfa^p ^iqlj^o [0 qlalp ilp k•jbolp mlpfqfslp 89 ]!+ Cbjlpqo^o nrb qla^ `ros^ ab R p^qfpc^`b i^b`r^`fŽk afcbobk`f^i

%^t'0 ^s

st + * %r0 + t0 + ]0' + + st < N -_s _s

b( Cbjlpqo^o nrb R bp loqldlk^i ^ pŒjfpjl: bpql bp+ bi `lkgrkql ab qla^p i^p qo^vb`,qlof^p loqldlk^ibp ^ i^p `ros^p ab R bp bi jfpjl R- XFi_d^\^d‡i8 Qbbjmi^w^o v&mlo,i.v& bk i^ b`r^`fŽk afcbobk`f^i ab i^ m^oqb \&+Y

10- Cbjlpqo^o nrb bi ird^o ab ilp `bkqolp ab rk^ c^jfif^ ab `fo`rkcbobk`f^p+ nrb m^p^kqla^p mlo rk mrkql a^al v plk q^kdbkqbp ^ rk^ ob`q^ a^a^+ bp rk^ m^oŠ_li^-

11- Cbjlpqo^o nrb bi ird^o ab ilp `bkqolp ab rk^ c^jfif^ ab `fo`rkcbobk`f^p+ nrb plk qla^pq^kdbkqbp 'buqbok^jbkqb( ^ rk^ `fo`rkcbobk`f^ a^a^ v ^ rk^ ob`q^ a^a^+ bp rk^ m^oŠ,_li^- 'Di bgbo`f`fl 10 mrbab `lkpfabo^opb `ljl rk `^pl m^oqf`ri^o-(

12- ^( Tk^ `rboa^ ab ilkdfqra 7H_h pb qo^w^ mbombkaf`ri^o ^i bgb ab i^ m^oŠ_li^ v1 < 2^s,Rb^k M v P ilp mrkqlp bk ilp nrb i^ `rboa^ `loq^ i^ m^oŠ_li^- Cbjlpqo^o nrb bi sb`qlonrb rkb N `lk L bp mbombkaf`ri^o ^i nrb rkb N `lk P-

_( K^ `rboa^ ab rk^ m^oŠ_li^ qo^w^a^ mlo bi cl`l u m^o^ibi^ ^ i^ afob`qofw pb ii^j^

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Be`m^d^djn q\mdjn nj]m` ^‡id^\n 514

g\_j m`^oj l k\mƒh`omj, Cbjlpqo^o nrb i^ ilkdfqra abi i^al ob`ql bp al_ib ab i^ afpq^k`f^abi cl`l ^ i^ afob`qofw+v abjlpqo^o irbdl nrb i^p q^kdbkqbp ^ i^ m^oŠ_li^ bk ilp buqobjlpabi i^al ob`ql `loq^k bi bgb ab i^ m^oŠ_li^ bk i^ fkqbopb``fŽk ab ‹pqb `lk i^ afob`qofw-

13- Clp mrkqlp L u O pb ii^j^k pfj‹qof`lp obpmb`ql ^ rk^ `fo`rkcbobk`f^ pf L u O bpqŠk^ifkb^alp `lk bi `bkqol+ pf bi `bkqol kl bpqŠ bkqob biilp+ u pf bi molar`ql ab prp afp,q^k`f^p ^i `bkqol bp fdr^i ^i `r^ao^al abi o^afl- Rf O abp`of_b i^ ob`q^ s * 1v , 4 < /+e^ii^o bi ird^o abi mrkql L pfj‹qof`l ab P obpmb`ql ^ i^ `fo`rkcbobk`f^ s0 * v1 < 3-

Page 188: Calculus
Page 189: Calculus

),

8h?8G?B 8BA ;GA8=BA:E H:8FBD=6?:E

03-0 Erk`flkbp sb`qlof^ibp ab rk^ s^of^_ib ob^i

Dpqb`^mŒqril `lj_fk^ bi „idb_o^ sb`qlof^i `lk ilp j‹qlalp abi BŠi`ril vabp`of_b ^idrk^p ^mif`^`flkbp ^i bpqrafl ab `ros^p v ^idrklp mol_ibj^p ab Lb`Š,kf`^- Di `lk`bmql ab crk`fŽk sb`qlof^i bp crka^jbkq^i bk bpqbbpqrafl-

CDEHMHBHˆM- Ri\ api^d‡i ^ptj _jhdidj `n pi ^jiepioj _` iˆh`mjn m`\g`nt ^ptj m`^jmmd_j`n pi np]^jiepioj _`g `nk\^dj i+_dh`indji\g Sh n` _`ijhdi\api^d‡i q`^ojmd\g_` pi\ q\md\]g` m`\g,

Gbjlp bk`lkqo^al q^ibp crk`flkbp bk bi `^mŒqril 02- Olo bgbjmil+ i^ ob`q^nrb m^p^mlo rk mrkql L v bp m^o^ibi^^ rk sb`qlo kl kril = bp bi ob`loofal abi^ crk`fŽk sb`qlof^i W a^a^ mlo

r%n&< L * `=

m^o^qlal ob^i o,

K^p crk`flkbp sb`qlof^ibp pb abpfdk^oŠk `lk ibqo^pj^v•p`ri^p `ropfs^p q^ibp`ljl B) F+ W+U) bq`-+l jbaf^kqb ibqo^pjfk•p`ri^p `ropfs^p kbdofq^p a*a) bq`-Di s^ilo ab rk^ crk`fŽk B bk o pb abpfdk^+`loofbkqbjbkqb+ mlo B%n&+Dk ilp bgbj,milp nrb bpqraf^objlp+ bi aljfkfl ab B pboŠrk fkqbos^il nrb mrbab `lkqbkbo rkll alp buqobjlp l nrb mrbab pbo fkcfkfql-

03-1 Nmbo^`flkbp^idb_o^f`^p-Bljmlkbkqbp

K^p lmbo^`flkbp rpr^ibp abi „idb_o^ sb`qlof^i mrbabk ^mif`^opb m^o^ `lj,_fk^o alp crk`flkbp sb`qlof^ibp l rk^ crk`fŽk sb`qlof^i `lk rk^ crk`fŽk ob^i- Rf

516

Page 190: Calculus

/+1 @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

E X F plk crk`flkbp sb`qlof^ibp+v pf p bp rk^ crk`fŽk ob^i+qbkfbkal qla^p rk al,jfkfl `lj•k+ abcfkfjlp krbs^p crk`flkbp E * F+ oB) v B$ F jbaf^kqb

&C* D'&o' < C&o' * D&o' * &pC'&o' < p&o'C&o'* &C%D'&o' < C&o'} D&o' ,

K^ prj^ B * F v bi molar`ql oB plk sb`qlof^ibp+ jfbkqo^p nrb bi molar`ql bp,`^i^o B +F bp ob^i- Rf B%nvX C%nv bpqŠkbk bi bpm^`fl ab 2 afjbkpflkbp+ q^j_f‹kmlabjlp abcfkfo bi molar`ql sb`qlof^i B W F `lk i^ cŽojri^

&C V D'&o' < C&o' t D&o',

K^ lmbo^`fŽk ab `ljmlpf`fŽk mrbab ^mif`^opbm^o^ `lj_fk^o crk`flkbp sb`,qlof^ibp `lk crk`flkbp ob^ibp-Olo bgbjmil+ pf C bp rk^ crk`fŽk sb`qlof^i `rvl al,jfkfl `lkqfbkb bi ob`loofal ab rk^ crk`fŽk ob^i p*i^ crk`fŽk `ljmrbpq^ F < E l pbp rk^ krbs^ crk`fŽk sb`qlof^i abcfkfa^ mlo

C%n&< BWo%n&Y

m^o^ `^a^ o bk bi aljfkfl ab p,Rf rk^ crk`fŽk B qfbkb prp s^ilobp bk R i* `^a^ sb`qlo B%nvqfbkb i `ljml,

kbkqbp+u mlabjlp bp`of_fo

C&o' < &ag&o'*a•&o'!,,, *ai@o~ ,

@pŒmrbp+ ^a^ crk`fŽk sb`qlof^i E lofdfk^ h crk`flkbp ob^ibpo,*,,, *cy `rvlp s^,ilobp bk o plk ilp `ljmlkbkqbp ab B%nv+Hkaf`^jlp bpq^ obi^`fŽk bp`of_fbkalE < '.0& --- + ai'* v ii^j^jlp -f bi h•‹pfjl `ljmlkbkqb ab B+

03-2 KŒjfqbp+abofs^a^p+b fkqbdo^ibp

Klp `lk`bmqlp crka^jbkq^ibp abi BŠi`ril+ q^ibp`ljl iŒjfqb+abofs^a^ b fkqb,do^i+-q^j_f‹k mrbabk buqbkabopb^ i^p crk`flkbp sb`qlof^ibp- Dumobp^jlp pbk`fii^,jbkqb i^ crk`fŽk sb`qlof^i bk crk`fŽk ab prp `ljmlkbkqbp u ob^ifw^jlp i^p lmb,o^`flkbp abi `Ši`ril pl_ob ilp `ljmlkbkqbp-

CDEHMHBHˆM- Pd E < '.0= --- + ai' `n pi\ api^d‡i q`^ojmd\g*_`adidhjn `g g…+hdo`* g\ _`mdq\_\ t g\ dio`bm\gkjm

ifok a&o' < 'ifokc0'q(+ --- +ifokcS.'qy( +n*(j n*$L n*.F

C%&ow< &av&o'*,,, *ax&o~*

bC&o' _o < Q7ag&o'_o* ,,, *E7ado' _o' *

nd`hkm` lp` gjn ^jhkji`io`n _` gjn n`bpi_jn hd`h]mjn o`ib\i n`iod_j,

Page 191: Calculus

I…hdo`n*_`mdq\_\n ` dio`bm\g`n /+2

Cb`fjlp q^j_f‹k nrb C bp ^jiodip\* _`mdq\]g` l dio`bm\]g` bk&rk fkqbos^ilpf `^a^ `ljmlkbkqb ab C qfbkbi^ `loobpmlkafbkqb molmfba^abk bi fkqbos^il-

@ i^ sfpq^ ab bp^pabcfkf`flkbp+kl mrbab plomobkabobk`lkqo^o nrb jr`elp abilp qblobj^p pl_ob iŒjfqbp+ lkqfkrfa^a+ abofs^`fŽk+ b fkqbdo^`fŽk ab crk`flkbpob^ibpq^j_f‹k plk sŠifalp m^o^crk`flkbp sb`qlof^ibp- U^jlp ^ bpq^_ib`bo^idrklpab ilp qblobj^p nrb rqfifw^jlp bk bpqb`^mŒqril-

RCMPCK? 03-0- PdC*F+ X p nji _`mdq\]g`n `i pi dio`mq\gj*gj hdnhj L^pmm`^ji C * E+ pC*v C%E+ Wo`i`hjn

%B* F(&< D&* C$) &pC'%< p%C* pg!* %B$F(& < B${ F * B$ C$+

PdC WE od`i`i gjn q\gjm`n `i S 2 o\h]d„i o`i`hjn

%Bt F(& < D&t F * B t C$+

A`hjnom\^d‡i, O^o^ sbo i^ j^o`e^ ab i^p abjlpqo^`flkbp afp`rqfjlp i^ cŽo,jri^ m^o^ %oB&$+K^p abjlpqo^`flkbp ab i^p lqo^p plk m^ob`fa^p v pb abg^k `ljlbgbo`f`flp m^o^ bi ib`qlo-

Dp`of_fbkal B < '.! --- +ai'* qbkbjlp

pC < &pa * ,,, * pai' * &pC'%< &&pa '%*,,, *&pai'%' ,

Obol i^ abofs^a^ abi h,‹pfjl `ljmlkbkqb ab pC bp &pa_ < p%X9* paf* ^pŒnrbqbkbjlp

&pC'%< p%&ex*,,, *ai' * p&av * ,,, *ax' < p%C * p B$)

Di ib`qlo l_pbos^oŠ nrb i^p cŽojri^p ab abofs^`fŽk abi qblobj^ 03-0 plk^kŠild^p ^ i^p ab i^ abofs^`fŽk ab rk^ prj^ l rk molar`ql ab crk`flkbp ob^ibp-Orbpql nrb bi molar`ql sb`qlof^i kl bp `lkjrq^qfsl+ ab_b rkl mobpq^oqbk`fŽk ^iloabk ab ilp c^`qlobp bk i^ cŽojri^ `loobpmlkafbkqb ^ %B W F(&-

K^ cŽojri^ m^o^i^ abofs^`fŽk B +F klp a^ bi pfdrfbkqb qblobj^ nrb pb rqfif,w^oŠ`lk cob`rbk`f^-

RCMPCK? 03-1- Pd pi\ api^d‡i q`^ojmd\g`n _`mdq\]g` t `n _` gjibdop_ ^jin+o\io` `i pi dio`mq\gj \]d`moj /* `ioji^`n C%E&< N `i g, AF@cj _` jomj hj_j*C%&o'`n k`mk`i_d^pg\m \ C&o'k\m\ ^\_\ o `i g,

A`hjnom\^d‡i, Olkd^jlp b&o'< GGD'p(U< C&o'}C&o', Olo efmŽqbpfp+d bp`lkpq^kqb bk .) v mlo il q^kql d&< N bk f+ Obol v^ nrb d bp rk molar`ql bp`^i^o+qbkbjlp d&< E! C * C%C%< 0C} C%,Olo il q^kql C%C%< N-

Page 192: Calculus

52/ @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

Di qblobj^ nrb pfdrb qo^q^ab i^p crk`flkbp `lkqfkr^p- Rr abjlpqo^`fŽk pbabar`b cŠ`fijbkqb abi qblobj^ 2-4 v 3-1 nrb `lkqfbkbk ilp obpriq^alp ^kŠildlpm^o^ crk`flkbp ob^ibp-

SDNQDL@ 03-2- P`\ L < E l p* _ji_` E `n pi\ api^d‡i q`^ojmd\g v p `npi\ api^d‡i m`\g, Rf p `n ^jiodip\ `i o v E `n ^jiodip\ `i p&o'y+ioji^`n L `n^jiodip\ `i o, Rf g\n _`mdq\_\n p%&o'v C%Xp&o'Zsdno`i* `ioji^`n L%&o'o\h]d„i`sdno` u qd`i` _\_\ kjm g\ m`bg\ _` g\ ^\_`i\*

D-&o' < C%Xp&o'Zp-&o',

Rf rk^ crk`fŽk sb`qlof^i E bp `lkqfkr^ bk rk fkqbos^il `boo^al W[)\Y) -bk,qlk`bp `^a^ `ljmlkbkqb bp `lkqfkr^ v mlo q^kql fkqbdo^_ibbk W[)\Y) ^pŒnrb E bpfkqbdo^_ibbk W[)\Y+ Klp qobpqblobj^p nrb pfdrbk molmlo`flk^k i^p molmfba^abpcrka^jbkq^ibp ab i^ fkqbdo^iab i^p crk`flkbp sb`qlof^ibp- Dk `^a^ `^pl+ iŠp ab,jlpqo^`flkbp pb abar`bk ^i jljbkql ab ilp obpriq^alp ^kŠildlp m^o^i^p fkqbdo^ibpab crk`flkbp ob^ibp-

SDNQDL@ 03-3- KHMD„KHC@C X @CHSHUHC@C- Pd g\n api^dji`n q`^ojmd\g`nEv L nji dio`bm\]g`n`i X\*]Z* o\h]d„i gj `n^,F%* ^0. k\m\ ^p\g`nlpd`m\ A0 X ^0*

v o`i`hjn

>ndhdnhj* k\m\ ^\_\ ` `i, X\*]Z* o`i`hjn

n8C&o' _o <F7C&o' _d * bC&o' _o ,

SDNQDL@ 03-4- OQHLDQ SDNQDL@ ETMC@LDMS@K CDK B„KBTKN-& Ppkjib\+hjn lp` D `n pi\ api^d‡i q`^ojmd\g^jiodip\ `i X\*]Z, Pd ` D X\* ]Z* _`adi\hjng\ dio`bm\gdi_`adid_\ > ^jhj g\ api^d‡i q`^ojmd\g_\_\ kjm

>&s' <F7C&o' _o pf [ R u R \+

Bioji^`n >%&s'sdno`*v n` od`i` >%&s'< C&s' k\m\ ^\_\ s _` &\*]',

SDNQDL@ 03-5- RDFTMCN SDNQDL@ ETMC@LDMS@K CDK B„KBTKN- Ppkjib\+hjn lp` g\ api^d‡i q`^ojmd\gE od`i` _`mdq\_\ ^jiodip\ E&i pi dio`mq\gj \]d`moj g,Bioji^`n* k\m\ ^\_\ `g`^^d‡i _` ` t s `i F) o`i`hjn

C&s' < C&^' * nC%&o'_o ,`

Page 193: Calculus

I…hdo`n* _`mdq\_\n ` dio`bm\g`n 520

Di qblobj^ nrb pfdrb bp rk^ buqbkpfŽk ab i^ molmfba^a ]+f$77B%n&n :+.$77]B%n& n) obbjmi^w^kal i^ jriqfmif`^`fŽk mlo bi bp`^i^o _ mlo bi molar`ql

bp`^i^o mlo rk sb`qlo B-

RCMPCK? 03-6- Pd E < &e/j ,,, * Fi' `n dio`bm\]g` `i X\* ]Z* k\m\ oj_j q`^ojma < &^/j ,,, * ^i' `g kmj_p^oj `n^\g\m _• D `n dio`bm\]g` `i X\* ]Z* V o`i`hjn

a mj B%E&E < '0&_•B%n&nŠ &X Š oF

A`hjnom\^d‡i, Orbpql nrb `^a^ `ljmlkbkqb ab E bp fkqbdo^_ib+ qbkbjlp

@mifnrbjlp ^elo^ bi qblobj^ 03-6 v i^ abpfdr^ia^a ab B^r`ev,R`et^ow vl_qbkbjlp i^ pfdrfbkqb molmfba^a fjmloq^kqb ab i^p fkqbdo^ibp ab crk`flkbp sb`,qlof^ibp-

RCMPCK? 03-7- Pd D V GGDGGnji dio`bm\]g`n `i X\* ]Z o`i`hjn

'03-0( GG-A B%n&E 00x,@GGD'p(00n +

A`hjnom\^d‡i, Olkd^jlp b < -n99C&o'_o, Rf b </+ bkqlk`bp '03-0( obpriq^qofsf^i- Rrmlkd^jlp+ mrbp+ nrb b /:0< l v ^mifnrbjlp bi qblobj^ 03-6 l_qbkfbkal

'03-1(&k mj

GGAGG1< a• a < _[ B%E&n < ha• B%E&n +,*\ Š,p

Orbpql nrb bi molar`ql bp`^i^o b•E %n&bp crk`fŽk ob^i+ qbkbjlp i^ abpfdr^ia^a

'03-2( m8a C&o' _o x,m8naC&o'/ _o x H9GGAhGGGD'p(00_o*

alkab bk bi •iqfjl m^pl pb e^ bjmib^al i^ abpfdr^ia^a ab B^r`ev,R`et^ow+ha•D'p(0y GGAeGGGD'p(GG-Blj_fk^kal '03-1( v '03-2(+ iibd^jlp ^

GGAGG1z GGAGGoGGD'p(00_o ,

Š \

X^ nrb heAeG= /+ mlabjlp afsfafo mlo GAeGl_qbkfbkal '03-0(-

Page 194: Calculus

410 @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

),&, :WR_PVPV\`

B^i`ri^o i^p abofs^a^p B$%n&u B!%n&m^o^ `^a^ rk^ ab i^p crk`flkbp sb`qlof^ibp ab ilpbgbo`f`flp abi 0 ^i 5-

0- C&o' < &o*o0*o1*o2',

0, C&o' < '`lp o*pbk1 o*pbk 0o*q^k o',

1, C&o' < '^o`pbk o*^obblp o',

2, C&o' < 0`o9 * 1`oe,

3, C&o' < `lpe oc* pbke 0oe * `+1of,

04, C&o' < ild 'i * o0'9 * ^obq^k oe * ,0 ,1 f,

)o6- Rb^ E rk^ crk`fŽk sb`qlof^i a^a^ mlo

0o 0 \, o0C&o' < ,, : * ,, d * f

0 * o0 0 * o0 Š

Cbjlpqo^o nrb bi Škdril cloj^al mlo B%n&u B$%n&bp `lkpq^kqb+ bpql bp+ fkabmbkafbkqbab o,B^i`ri^o i^p fkqbdo^ibp ab i^p crk`flkbp sb`qlof^ibp bk ilp bgbo`f`flp abi 7 ^i 00-

d. *7- l &o*Vo*`o'_o,

&*-28- Il 'pbk o*`lp o*q^k o' _o,

01- B^i`ri^o > , ?* pfbkal > < 1: , 2e * f u ? < j &o`0o9* o blpe 0oe * 0o`+0of' _o,02- C^alp rk sb`qlo > kl kril v rk^ crk`fŽk sb`qlof^i E q^i nrb B%n&{> < o m^o^ qlal o*

u q^ibp nrb bi Škdril cloj^al mlo B$%n&u > bp `lkpq^kqb 'fkabmbkafbkqb ab n&+Cbjlp,qo^o nrb B!%n&bp loqldlk^i ^ B$%n&+

03- C^alp ilp sb`qlobp cfglp kl krilp = v >) mlkd^jlp B%n&< `/n= * `+/n>+ Cbjlpqo^o nrbB!%n&qfbkb i^ jfpj^ afob``fŽk nrb B%n&+

04- Rf F < E W B$) `^i`ri^o F& bk crk`fŽk ab E v ab i^p abofs^a^p ab E-05- Rf F < E• E&W B!) abjlpqo^o nrb F& < E• E&W B$$$+06- Cbjlpqo^o nrb fcgn*(j B%n&< = pf v pŽil pf ifj n*L EEB%n&* ?GG< N-07- Cbjlpqo^o nrb rk^ crk`fŽk sb`qlof^i E bp abofs^_ib bk rk fkqbos^il ^_fboql . pf u pŽil pf

m^o^ `^a^ o bk . qbkbjlp

0C&o' < ifj , XC&o* b&* C&o'Z,

.z,*k c

08- Cbjlpqo^o bi qblobj^ ab i^ abofs^a^ kri^ m^o^ i^p crk`flkbp sb`qlof^ibp- Rf B$%n&< Nm^o^ `^a^ n bk rk fkqbos^il ^_fboql .) bufpqb rk sb`qlo b q^i nrb B%n&< b m^o^ qlalo ab g,

1/- C^alp ilp sb`qlobp cfglp = v > X i^ crk`fŽk sb`qlof^i E q^i nrb B!%n&< n= * >) abqbo,jfk^o B%n&pf D'M( < B W B$%K&< A-

10- Tk^ b`r^`fŽk afcbobk`f^i ab i^ cloj^ W&'t( * j%r&U%r&< M%r&)alkab j bp rk^ crk`fŽkob^i f-a^a^+ O rk^ crk`fŽk sb`qlof^i b V rk^ crk`fŽk sb`qlof^i abp`lkl`fa^+ pb ii^j^b`r^`fŽk afcbobk`f^i sb`qlof^i ab mofjbo loabk- Cbjlpqo^o nrb pf j u O plk `lkqfkr^pbk rk fkqbos^il /* bkqlk`bp m^o^ `^a^ \ ab / u `^a^ sb`qlo ? bufpqb rk^ plir`fŽk u pŽil

Page 195: Calculus

>kgd^\^dji`n \ g\n ^pmq\n, Q\ib`i^d\ 522

rk^ U nrb p^qfpc^`b i^ `lkaf`fŽk fkf`f^i U%[&< >) X nrb bp^ plir`fŽk sfbkb a^a^ mloi^ cŽojri^

U%n&< >_*k%n&* _*k%n&wM%r&€%!$&^r )

pfbkal k%r&< bk&o' _o,11- Tk^ crk`fŽk sb`qlof^i E p^qfpc^`b i^ b`r^`fŽk nB$%n&< B%n&* n= m^o^ `^a^ n w N+alkab

= bp rk sb`qlo cfgl- B^i`ri^o E!'i( v B%0&bk crk`fŽk ab =) pf B%.&< /=+12- G^ii^o rk^ crk`fŽk sb`qlof^i E+ `lkqfkr^ bk bi fkqbos^il 'N+ * BW(+ q^i nrb

0 F%!C&s' < s`!%> * , C&o'_o *B 0

m^o^ qlal s = N+pfbkal > rk sb`qlo cfgl kl kril-13- Tk^ crk`fŽk sb`qlof^i E+ nrb krk`^ bp `bol v qfbkb abofs^a^ `lkqfkr^ B$%n&m^o^ qlal o*

pfbjmob bp m^o^ibi^ ^ pr abofs^a^- Cbjlpqo^o nrb bufpqb rk sb`qlo `lkpq^kqb > v rk^crk`fŽk ob^i mlpfqfs^ o q^i nrb B%n&< o%n&=m^o^ qlal n+

),&- 6]YVPNPV\[RN YN Pb_cN`&FN[TR[PVN

Rb^ W rk^ crk`fŽk sb`qlof^i `rvl aljfkfl bp rk fkqbos^il g,Br^kal o ob`l,oobH) ilp `loobpmlkafbkqbps^ilobp ab i^ crk`fŽk T%n&ob`loobk rk `lkgrkql ab mrk,qlp nrb ii^j^jlp bmƒad^\ ab i^ crk`fŽk W- Rf ilp s^ilobp ab i^ crk`fŽk bpqŠkbkbpm^`flp ab 1 Ž 2 afjbkpflkbp+ mlabjlp obmobpbkq^odblj‹qof`^jbkqb i^ doŠcf`^-Olo bgbjmil+ pf U&o' < M * o>* alkab M v > plk sb`qlobp cfglpbk S\* `lk > ;C L*i^ doŠcf`^ab W bp rk^ ob`q^ nrb m^p^mlo L v bp m^o^ibi^^ =+ Tk^ crk`fŽk jŠpdbkbo^i abp`of_foŠ rk^ doŠcf`^ jŠp dbkbo^i+`ljl prdfbob bi bgbjmil ab i^ cfdr,o^ 03-0- Rf W bp `lkqfkr^ bk F*q^i doŠcf`^pb ii^j^ ^pmq\9 l `lk jŠp mob`fpfŽki^`ros^ abp`ofq^mlo W- @ sb`bp ab`fjlp nrb i^ `ros^ bp abp`ofq^k\m\h„omd^\h`io`mlo W- Di fkqbos^il f pb ii^j^ dio`mq\gj k\m\h„omd^j9 `^a^ o ab f pb ii^j^k\mƒh`omj,

K^p molmfba^abpab i^ crk`fŽk W mrbabk rqfifw^opbm^o^ fksbpqfd^omolmfb,a^abp dblj‹qof`^p ab prp doŠcf`^p-Dk m^oqf`ri^o+i^ abofs^a^ W&bpqŠifd^a^ ^i`lk`bmql ab q^kdbk`f^+`ljl bk bi `^pl ab rk^ crk`fŽk ob^i- Elojbjlp bi `l,`fbkqb ab afcbobk`f^p

U& * c' + U&o'

bb fksbpqfdrbjlp pr `ljmloq^jfbkql `r^kal c x N- Dpqb`l`fbkqb bp bi molar`qlabi sb`qlo T%n* b& * T%n&mlo bi bp`^i^o f,b+ Di krjbo^alo+ T%n* b& * T%n&)obmobpbkq^albk i^ cfdro^ 03-1+bp m^o^ibil ^i sb`qlo '03-3(- Rf bumobp^jlp bpqb

'03-3(

Page 196: Calculus

523 @ƒg^pgj^ji Xpi^dji`n q`^ojmd\g`n

`l`fbkqb ab afcbobk`f^p bk crk`f5k ab prp `ljmlkbkqbp u e^`bjlp nrb c x /+ bk,`lkqo^jlp nrb

ifj U&o * c' + U&o' < U%&o'*xz,*M c

u

s

EHFTQ@ 03-0 ?olp[ ^_m]lcn[ jil _fp_]nil T%n&+

s

EHFTQ@ 03-1 Af p_]nil W'q * b& * T%n& _mj[l[f_fi [f WT%n* b& * T%n&Y,b+

prmlkfbkal nrb i^ abofs^a^ T$%n&bufpq^-K^ fkqbomobq^`fŽkdblj‹qof`^ ab bpq^ob,i^`f5kprdfbob i^ pfdrfbkqb abcfkf`f5k-

CDEHMHBH5M- P`\ B pi\ ^pmq\ _`n^mdo\kjm pi\ api^d4i q`^ojmd\g^jiodip\ W-Pd `sdno` g\ _`mdq\_\ U%&o'v ij `n ipg\* g\ m`^o\ lp` k\n\ kjm U&o'v k\m\g`g\ \U&o'n` gg\h\ o\ib`io` \ B `i U&o',Bg q`^ojm U%&o'n` _`ijhdi\ q`^ojm o\ib`io`\ B `i T%n&+

DIDLOKN 0- O`^o\, O^o^ rk^ ob`q^ a^a^ mlo U&o'< M * o>* pfbkal= :.:-) qbkbjlp T$%n&< =) ^pŒnrb i^ ob`q^ q^kdbkqbbk `^a^ mrkql `lfk`fab `lki^ doŠcf`^ab W+molmfba^a nrb `fboq^jbkqb abpbŠ_^jlp-

DIDLOKN 1- @dm^pia`m`i^d\, Rf W abp`of_b rk^ `fo`rkcbobk`f^ ab o^afl \v `bkqol bk L) bkqlk`bp GGV'p(, NGG< \ m^o^`^a^ o, Di sb`qlo T%n&* L pb ii^j^m\_dj q`^ojm9 mrbab obmobpbkq^opbdblj‹qof`^jbkqb mlo rk^ cib`e^ abpab bi `bk,qol ^i mrkql T%n&+Orbpql nrb bi o^afl sb`qlo qfbkb ilkdfqra `lkpq^kqb+bi qblobj^03-1 klp af`b nrb bp mbombkaf`ri^o^ pr abofs^a^ v mlo q^kql mbombkaf`ri^o^ i^ob`q^ q^kdbkqb-@pŒmrbp+m^o^rk^ `fo`rkcbobk`f^+ krbpqo^ abcfkf`f5k ab q^kdbk`f^bpqŠab ^`rboal `lk i^ nrb pb a^ bk i^ FbljbqoŒ^ mi^k^ bibjbkq^i-

Page 197: Calculus

>kgd^\^dji`n \ g\n ^pmq\n, Q\ib`i^d\ 524

DIDLOKN 2- Fiq\md\i^d\ am`io` \ pi ^\h]dj _` k\mƒh`omj, Erk`flkbp afp,qfkq^pmrbabk qbkbo i^ jfpj^ doŠcf`^- Olo bgbjmil+ prmlkd^jlp nrb W bp rk^crk`fŽk sb`qlof^i `lkqfkr^ abcfkfa^ bk rk fkqbos^il . v nrb p bp rk^ crk`fŽk ob^iabofs^_ib `lk p%pfbjmob afpqfkq^ab `bol bk rk fkqbos^il F) v q^i nrb bi ob`loofalab p pb^ g, Dkqlk`bp i^ crk`fŽk V abcfkfa^ bk F mlo i^ b`r^`fŽk

U%n&< TWo%n&Y

bp rk^ crk`fŽk sb`qlof^i `lkqfkr^ nrb qfbkbi^ jfpj^ doŠcf`^nrb W- Clp qrk`flkbpW b V ^pŒobi^`flk^a^p pb ii^j^k `lpdq\g`io`n, Rb af`b ab bii^p nrb molmlo`flk^kobmobpbkq^`flkbpm^o^j‹qof`^p afpqfkq^pab i^ jfpj^ `ros^- @pfjfpjl pb af`bnrb i^ crk`fŽk p abcfkb rk `^j_fl ab m^oŠjbqol-

Klp `lk`bmqlp dblj‹qof`lp jŠp fjmloq^kqbp ^pl`f^alp ^ rk^ `ros^ plk ^nrb,iilp nrb mboj^kb`bk fks^of^kqbpcobkqb^ rk `^j_fl ab m^oŠjbqol- Olo bgbjmil+bp cŠ`fi abjlpqo^o nrb i^ q^kdbkqbbp fks^of^kqb- Rf i^ abofs^a^ T$Wo%n&Ybufpqb+i^obdi^ ab i^ `^abk^ jrbpqo^ nrb U$%n&q^j_f‹k bufpqbu sfbkb a^a^ mlo i^ cŽojri^

s$%n&< U%Xp&o'Zp%&o',

K^ abofs^a^ o$%n&krk`^ bp `bol- Rf T$Wo%n&Ybp kl kri^+ U$%n&q^jml`l bp kri^+ abjlal nrb U$%n&bp m^o^ibil ^ T$Wo%n&Y+Olo `lkpfdrfbkqb ^j_^p obmobpbkq^`flkbpW b V klp `lkar`bk ^ i^ jfpj^ q^kdbkqbbk `^a^ mrkql ab i^ `ros^-

DIDLOKN 3- Mmjkd`_\_`n _` m`ag`sd‡i `i g\n ^‡id^\n, K^p `Žkf`^p qfbkbkmolmfba^abpab obcibufŽkrp^a^p `lk cob`rbk`f^ bk bi afpb•l ab fkpqorjbkqlp Žmqf,

'^(Difmpb

'^(Gfm‹o_li^

S^kdbkqb

EHFTQ@ 03-2 Mmjkd`_\_`n _` m`ag`sd‡i _` g\n ^‡id^\n,

Page 198: Calculus

525- @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

`lp v ^`•pqf`lp- Klp o^vlp irjfklplp nrb m^oqbk ab rkl ab ilp cl`lp ab rk obcib`qlobiŒmqf`l-`lksbodboŠk bk bi lqol cl`l+ `ljl jrbpqo^ i^ cfdro^ 03-2 ^(- Klp o^vlpirjfklplp afofdfalp e^`f^ rkl ab ilp cl`lp ab rk obcib`qlo efmbo_Žif`l `lksbo,dboŠk bk bi lqol cl`l+ `ljl sbjlp bk i^ cfdro^ 03-2 _(- Dk rk obcib`qlo m^o^_Žif`l+ilp o^vlp irjfklplp m^o^ibilp ^i bgb `lksbodbk bk bi cl`l+ `ljl sbjlp bk i^cfdro^ 03-2 b(+O^o^ bpq^_ib`bo bp^p molmfba^abp ab obcibufŽk+ kb`bpfq^jlp abjlp,qo^o nrb bk `^a^ cfdro^ ilp Škdrilp ` plk fdr^ibp- Blkpbdrfobjlp bpql m^o^ i^ bifmpbv m^o^ i^ efm‹o_li^ v abg^jlp ^i ib`qlo i^ abjlpqo^`fŽk m^o^ i^ m^oŠ_li^-

Blilnrbjlp rk cl`l B0 bk bi lofdbk v pb^k !0 v !1 alp sb`qlobp rkfq^oflp`lk i^ jfpj^ afob``fŽk nrb V v V, B0* obpmb`qfs^jbkqb+ pfbkal V rk mrkql^o_fqo^ofl ab i^ `Žkf`^- 'Ubo cfdro^ 03-3-( Rf ^) < GGVGGv ^) < GGV, E100+plki^p afpq^k`f^p cl`^ibp bkqob W s ilp cl`lp B/ v B0* obpmb`qfs^jbkqb+ qbkbjlp

s < ^/!/ u

Blkpfabobjlp ^elo^ U%!/%!0%_/ *X _0 `ljl crk`flkbp abcfkfa^p bk rk `fboqlfkqbos^il ab k•jbolp ob^ibp- Rrp abofs^a^p bpqŠk obi^`flk^a^p mlo i^p b`r^`flkbp

'03-4(

Orbpql nrb !0 v !1 qfbkbk ilkdfqra `lkpq^kqb+ `^a^ rkl bp mbombkaf`ri^o ^ pr ab,ofs^a^+ ^pŒnrb i^p b`r^`flkbp '03-4( klp a^k T!!g < ^8 X T!!0 < ^w+Rrj^kalv obpq^kal bp^p obi^`flkbp+ bk`lkqo^jlp nrb

'03-5(

Dk i^ bifmpb+^E * ^0 bp `lkpq^kqb+ ab jlal nrb ^8 * ^w< N- Dk `^a^ o^j^ abi^ efm‹o_li^ _* + _0 bp `lkpq^kqb+ ^pŒnrb _ z, _x < N- Olo `lkpfdrfbkqb+ i^pb`r^`flkbp '03-5( klp a^k

bk i^ bifmpb+ bk i^ efm‹o_li^-

Rb^ Q < U%.GGV.GGrk sb`qlo rkfq^ofl `lk i^ jfpj^ afob``fŽk nrb U%,Dkqlk`bp Q`nq^kdbkqb ^ i^ `Žkf`^+ v qbkbjlp

bk i^ bifmpb+ Q%!0< Q%!g bk i^ efm‹o_li^-

Rfc(i v `/ plk+ obpmb`qfs^jbkqb+ ilp Škdrilp nrb Q cloj^ `lk !0 v !1& pfbkalN z bi z 60&X-N z `0 x 60&+^nrbii^p alp •iqfj^p b`r^`flkbp jrbpqo^k nrb

bk i^ bifmpb+ bk i^ efm‹o_li^-

Page 199: Calculus

>kgd^\^dji`n \g hjqdhd`ioj ^pmqdg…i`j /,0

'^( K0 < &Gn, K) bk i^ bifmpb '_( K0 < K) bk i^ efm‹o_li^

EHFTQ@ 03-3 A`hjnom\^dji`n _` g\n kmjkd`_\_`n _` m`ag`sd‡i k\m\ g\ `gdkn` v g\ cdk„m]jg\,

Krbdl qbkbjlp L0 < 6S , 'IH bk i^ bifmpb+v &G0 < 'IH bk i^ efm‹o_li^- Dpq^pobi^,`flkbp bkqob ilp Škdrilp 'IH v K0 a^k i^p molmfba^abpab obcibufŽkab i^ bifmpbvab i^ efm‹o_li^-

),&. 6]YVPNPV\[RNYZ\cVZVR[a\ Pb_cVYo[R\&HRPa\_cRY\PVQNQ$cRY\PVQNQe NPRYR_NPVp[

Rrmlkd^jlp nrb rk^ m^oqŒ`ri pb jrbsb bk bi bpm^`fl ab 1 Ž 2 afjbkpflkbpab jlal nrb pr mlpf`fŽk bk bi fkpq^kqbo obcbofa^^ rk `fboql pfpqbj^ `lloabk^alsbkd^ a^al mlo rk sb`qlo T%n&+Br^kal o s^oŒ bk rk fkqbos^il ab qfbjml+ bi `^,jfkl ob`loofal mlo i^ m^oqŒ`ri bp pbk`fii^jbkqb i^ doŠcf`^ ab W- @pŒmrbp+i^crk`fŽk sb`qlof^i W klp pfosb `ljl jlabil j^qbjŠqf`l m^o^ abp`of_fo bi jlsf,jfbkql- @ i^ crk`fŽk sb`qlof^i W i^ ii^j^jlp api^d‡i kjnd^d‡i abi jlsfjfbkql-Klp `lk`bmqlp cŒpf`lpq^ibp`ljl sb`qlo sbil`fa^a+ sbil`fa^a+ u sb`qlo ^`bibo^`fŽkmrbabk abcfkfopbbk crk`fŽk ab i^p abofs^a^p ab i^ crk`fŽk ab mlpf`fŽk- Dk i^ pf,drfbkqb afp`rpfŽk prmlkbjlp nrb i^ crk`fŽk mlpf`fŽk mrbab abofs^opb `r^kq^psb`bp pb^ mob`fpl pfk ab`foil bk `^a^ l`^pfŽk-

CDEHMHBHˆM- @jind_`m`hjn pi hjqdhd`ioj _`n^mdojkjm pi\ api^d‡i q`^oj+md\gW- I\ _`mdq\_\ U%&o'n` gg\h\ q`^ojm q`gj^d_\_ `i `g dino\io` o, I\ gjibdop__`g q`^ojm q`gj^d_\_* GGV&'p(GG+n` gg\h\ q`gj^d_\_, I\ _`mdq\_\ n`bpi_\ U!&o' _`gq`^ojm kjnd^d‡i, n` gg\h\ q`^ojm \^`g`m\^d‡i,

Page 200: Calculus

527 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

Kjo\^d‡i, @ sb`bp i^ crk`fŽk mlpf`fŽk W pb obmobpbkq mlo m*bi sb`qlosbil`fa^a mlo q* i^ sbil`fa^a mlo q v i^ ^`bibo^`fŽk \, @pŒnrb q < m%*q < 00 b\+u \ < q%< m!*

Rf bi sb`qlo sbil`fa^a T$%n&pb obmobpbkqmlo rk sb`qlo dblj‹qof`l ifd^al ^i^ `ros^ bk T%n&)sbjlp nrb bpqŠpfqr^al bk i^ ob`q^ q^kdbkqb-Di rpl ab i^ m^i^_o^~sbil`fa^a‚ m^o^i^ ilkdfqra abi sb`qlo sbil`fa^a pb grpqfcf`^oŠbk i^ pb``fŽk 03-01bk alkab pb abjrbpqo^ nrb i^ sbil`fa^a bp bi `lbcf`fbkqb ab s^of^`fŽk ab i^ ilkdf,qra ab ^o`l ^ il i^odl ab i^ `ros^- Dpql bp il nrb bi sbil`Œjbqol ab rk ^rqljŽsfifkqbkq^jbafo- @pŒmrbp+i^ ilkdfqra abi sb`qlo sbil`fa^a klp af`b i^ o^mfabw lknrb i^ m^oqŒ`ri pb jrbsb bk `^a^ fkpq^kqb+v pr afob``fŽk klp fkaf`^ e^`f^ aŽkabs^- Di sb`qlo sbil`fa^a `^j_f^oŠ pf jlafcf`^jlp i^ sbil`fa^a l i^ afob``fŽk abijlsfjfbkql 'l ^j_lp(- Di sb`qlo ^`bibo^`fŽk bp rk^ jbafa^ ab bpqb `^j_fl-K^ ^`bibo^`fŽk lofdfk^ bi bcb`ql nrb rkl bumbofjbkq^ `r^kal rk ^rqljŽsfi `^j,_f^ pr sbil`fa^a l pr afob``fŽk- @ afcbobk`f^ abi sb`qlo sbil`fa^a+ bi sb`qlo ^`b,ibo^`fŽk kl bpqŠkb`bp^of^jbkqb bk i^ ob`q^ q^kdbkqb-

DIDLOKN 0- Jjqdhd`ioj m`^odg…i`j, Blkpfabobjlp rk jlsfjfbkql `rvlsb`qlo mlpf`fŽk bp

m&o'< M * a&o'> *

alkab L v = plk sb`qlobp cfglp v = :.: M+ Dpqbjlsfjfbkql pb ob^ifw^^ il i^odlab rk^ ob`q^ nrb m^p^mlo L v m^o^ibi^^ =+ Di sb`qlo sbil`fa^a+ i^ sbil`fa^a v bisb`qlo ^`bibo^`fŽk sfbkbk a^a^p mlo

q&o'< a%&o'>* q&o' < Ghr'p(00< Gb&'p(0GG?00+ \&o' < a!&o'> ,

Rf a&o'v a!&o'kl plk `bol+ bi sb`qlo ^`bibo^`fŽk bp m^o^ibil ^i sb`qlo sbil`fa^a-

DIDLOKN 1- Jjqdhd`ioj ^dm^pg\m, Rf rk mrkql 'u+ s& ab S0 bpqŠobmobpbk,q^al mlo prp `lloabk^a^p mli^obp o v '(+qbkbjlp

r < l `lp '(+ s:lm_h%&+

Rf o bp cfgl+mlo bgbjmil o < \* u pf `& mrbab s^of^o bk rk fkqbos^il `r^inrfbo^ ab^jmifqra mlo il jbklp 05Q* bi `loobpmlkafbkqb mrkql %r)u( abp`of_b rk^ `fo`rk,cbobk`f^ ab o^afl [ v `bkqol bk bi lofdbk- Rf `lkpfabo^jlp '( `ljl rk^ crk`fŽkabi qfbjml+ mlo bgbjmil '( < `%n&)qbkbjlp rk jlsfjfbkql a^al mlo i^ crk`fŽk abmlpf`fŽk

m&o'< \ ^jna&o'd * \n`ia&o'e,

Page 201: Calculus

>kgd^\^dji`n \g hjqdhd`ioj ^pmqdg…i`j 528

Di `loobpmlkafbkqb sb`qlo sbil`fa^a bp

q&o'< m%&o'< +\a%&o'n`ia&o'd* \a%&o'jna&o'e*

abi nrb pb abar`b nrb i^ sbil`fa^a bk bi fkpq^kqbo bp

q&o' < Ghr'p(00< \ Gb&'p(G-

Di c^`qlo 0c&'q(0< y_` - _oy pb ii^j^ q`gj^d_\_ \ibpg\m ab i^ m^oqŒ`ri^-Tk `^pl m^oqf`ri^o fjmloq^kqb pb mobpbkq^ r^kal b < ro* alkab r bp rk^

`lkpq^kqb mlpfqfs^-Dk bpqb`^pl+ i^ m^oqŒ`ri m^oqbabi mrkql %[) N( bk bi fkpq^kqbo < N X pb jrbsb bk bi pbkqfal `lkqo^ofl ^i ab i^p ^drg^p abi obilg pfdrfbkal i^`fo`rkcbobk`f^ `lk sbil`fa^a ^kdri^o `lkpq^kqb ^\, K^p cŽojri^p m^o^bi sb`qlo ml,pf`fŽk+sb`qlo sbil`fa^a v sbil`fa^a pb qo^kpcloj^k bk

m&o'< \ `lp ro c* \ pbkd~og+ q&o'< +r\n`i jdo c* r\ `lp ^jo g+ q&o'< \r ,

Di sb`qlo ^`bibo^`fŽk sfbkb a^al mlo

\&o' 9< +r0\ `lp ^jo c* q0\ pbk lfof < +ic&o'*

nrb jrbpqo^ nrb i^ ^`bibo^`fŽk qfbkbpfbjmob afob``fŽk lmrbpq^ ^i sb`qlo mlpf`fŽk-Br^kal pb obmobpbkq^lk rk sb`qlo dblj‹qof`l qo^w^al bk bi ird^o nrb l`rm^ i^m^oqŒ`ri^+bi sb`qlo ^`bibo^`fŽk bpqŠafofdfal e^`f^ bi `bkqol ab i^ `fo`rkcbobk`f^-Cb_fal ^ bpql+i^ ^`bibo^`fŽk pb ii^j^ ^`iom…k`o\* abkljfk^`fŽk molmrbpq^mloMbtqlk-

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Page 202: Calculus

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Page 203: Calculus

Be`m^d^djn 530

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/-+ @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

8- Qbcfof‹kalpb ^i bgbo`f`fl 6+ pb^ p&o' bi sb`qlo rkfq^ofl p&o'< n`irod+ ^jn ^jo Z* Cb,jlpqo^o nrb bufpqbk alp `lkpq^kqbp = v > q^ibp nrb q u \ < =o%,& * >e) v bumobp^o> v ? bk crk`fŽk _` \* ]* v S

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Page 205: Calculus

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Page 206: Calculus

/-- @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

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Page 207: Calculus

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Page 208: Calculus

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pfbkal p&o' rk sb`qlo rkfq^ofl- Olo `lkpfdrfbkqb HHS&'q(00< foi'q(f v bpql abjrbpqo^nrb HHS&'q(00bp rk^ jbafa^ abi `lbcf`fbkqb ab s^of^`fŽk abi Škdril ab fk`ifk^`fŽkabi sb`qlo q^kdbkqb- Br^kal j^%&o'= N+ bi Škdril bp `ob`fbkqb+ v mlo q^kqlR&-' < J%n&+Br^kal j^%&o'; N+ bi Škdril bp ab`ob`fbkqb+ bk bpqb `^pl+ p&o' ;: *J%n&+Dk i^ cfdro^ 03-00 pb obmobpbkq^kilp alp `^plp- N_p‹osbpb nrb bi Šk,dril ab fk`ifk^`fŽk ab p&o' bp l`'q( * 6S v^ nrb

p&o' < ,pbkl`'q(f * `lp j^&o'e <`lp &L@&o'* x'd* pbk &L@&o'* x'e,

),&1 :WR_PVPV\`

Klp bgbo`f`flp abi 0 000 5 -pb obcfbobk ^ ilp jlsfjfbkqlp abp`ofqlp bk ilp bgbo`f`flp abi0 ^i 5+ obpmb`qfs^jbkqb+ ab i^ pb``fŽk 03-6- O^o^ bi s^ilo ab o nrb pb `fq^+ ^( bumobp^o bisb`qlo q^kdbkqb rkfq^ofl P v bi kloj^i mofk`fm^i J bk crk`fŽk ab c)d*e8 _( bumobp^o i^ ^`b,ibo^`fŽk \ `ljl rk^ `lj_fk^`fŽk ifkb^i ab P u J+

0- o < 1- 2- o < T- 4- o < 0-

0, o < QQ, 3- o < QQ, 5- o < QQ

6- Cbjlpqo^o nrb pf bi sb`qlo ^`bibo^`fŽk bp pfbjmob `bol+ bi jlsfjfbkql bp ob`qfiŒkbl-7- Cbjlpqo^o nrb bi `ljmlkbkqb kloj^i abi sb`qlo ^`bibo^`fŽk bp Ghru ]hh.hhrhh-8- O^o^ `^a^ rk^ ab i^p molmlpf`flkbp pfdrfbkqbp obcbobkqbp ^ i^ `ros^ abp`ofq^ mlo rk^

m^oqŒ`ri^ jŽsfi bk bi bpm^`fl ab 2 afjbkpflkbp+ a^o Tk^ abjlpqo^`fŽk l mlkbo rk`lkqo^bgbjmil-

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Be`m^d^djn /-0

^( Rf bf sb`qlo sbil`fa^a bp `lkpq^kqb+ i^ `ros^ bp mi^k^-_( Rf i^ sbil`fa^a bp `lkpq^kqb+ i^ `ros^ bp mi^k^-b( Rf bi sb`qlo ^`bibo^`fŽk bp `lkpq^kqb+ i^- `ros^ bp mi^k^-a( Rf bi sb`qlo sbil`fa^a bp mbombkaf`ri^o ^i sb`qlo ^`bibo^`fŽk+ i^ `ros^ bp mi^k^-

0/- Tk^ m^oqŒ`ri^ ab j^p^ rkfa^a `lk sb`qlo mlpf`fŽk l%n&bk bi fkpq^kqb o pb jrbsb bkbi bpm^`fl _^gl i^ ^``fŽk ab `fboq^p crbow^p-^( Cbjlpqo^o nrb o u \ < N Œjmif`^ o u q < `* pfbkal ` rk sb`qlo `lkp`^kqb-_( Rf o u Te < `* alkab ` bp rk sb`qlo `lkpq^kqb+ abjlpqo^o nrb bi jlsfjfbkql pbob^ifw^ bk rk mi^kl- Blkpfabo^o ` x N X ` < N-`( Rf i^ crbow^ obpriq^kqb nrb ^`q•^ pl_ob i^ m^oqŒ`ri^ bpqŠ pfbjmob afofdfa^ e^`f^ bilofdbk+ abjlpqo^o nrb i^ m^oqŒ`ri^ pb jrbsb bk rk mi^kl-a( ƒDi molar`ql o u q bp kb`bp^of^jbkqb `lkpq^kqb pf rk^ m^oqŒ`ri^ pb jrbsb bk rkmi^kl>

00- Tk^ m^oqŒ`ri^pb jrbsb ^ il i^odl ab rk^ `ros^ ab q^i j^kbo^ nrb bi sb`qlo sbil`fa^acloj^ rk Škdril `lkpq^kqb `lk rk sb`qlo rkfq^ofl a^al ^,^( Rf i^ `ros^ bpqŠ bk rk mi^kl nrb `lkqbkd^ `* abjlpqo^o nrb bi sb`qlo ^`bibo^`fŽk bi=+`bol l bp m^o^ibil ^i sb`qlo sbil`fa^a-_( C^o rk bgbjmil ab rk^ q^i `ros^ 'kl mi^k^( m^o^ i^ nrb bi sb`qlo ^`bibo^`fŽk krk`^bp `bol kf m^o^ibil ^i sb`qlo sbil`fa^a-

01- Tk^ m^oqŒ`ri^ pb jrbsb ^ il i^odl ab i^ bifmpb 1s0 * v1 < 0 `lk sb`qlo ab mlpf`fŽkm&o'<a&o'9* b&o'e,Di jlsfjfbkql bp q^i nrb bi `ljmlkbkqb elofwlkq^i abi sb`qlo sb,il`fa^a bk bi fkpq^kqb o bp *a%n&+^( voa jrbsb i^ m^oqŒ`ri^pl_ob i^ bifmpb bk afob``fŽk ^ c^slo l `lkqo^of^ ^ i^p ^drg^pabi obilg>_( Cbjlpqo^o nrb bi `ljmlkbkqb sboqf`^i abi sb`qlo sbil`fa^a bk bi fkpq^kqb o bp mol,mlo`flk^i ^ `%n&v e^ii^o bi c^`qlo ab molmlo`flk^ifa^a-b( ƒBrŠkql qfbjml pb kb`bpfq^ m^o^ nrb i^ m^oqŒ`ri^ob`loo^ rk^ sbw i^ bifmpb>

02- Tk^ `ros^ mi^k^ b bk bi mofjbo `r^ao^kqb qfbkb mbkafbkqb kbd^qfs^ bk `^a^ rkl ab prpmrkqlp v m^p^ mlo bi mrkql 'i+ 0(- Di sb`qlo mlpf`fŽk o nrb rkb bi lofdbk `lk rk mrkql`r^inrfbo^ %r)u( ab a cloj^ rk Škdril L `lk c)u bi sb`qlo sbil`fa^a cloj^ rk Škdr,il 0= `lk o*pfbkal N ; K ; d/Q*v N ; 9j ; /Q, Rf 2 q^k 9j < 3 `lq `& bk `^a^ mrkql aba+ e^ii^o i^ b`r^`fŽk `^oqbpf^k^ ab a u af_rg^o i^ `ros^-

03- Tk^ ob`q^ mbombkaf`ri^o ^ i^ ob`q^ q^kdbkqb ^ rk^ `ros^ mi^k^ pb ii^j^ ob`q^ kloj^i-Rf bk `^a^ mrkql ab rk^ `fboq^ `ros^ mi^k^ a pb qo^w^k i^ kloj^i u rk^ ob`q^ sboqf`^i+bp^p alp ob`q^p fkqbo`bmq^k pl_ob bi bgb s rk pbdjbkql ab ilkdfqra 1- G^ii^o i^ b`r^`fŽk`^oqbpf^k^ ab bp^ `ros^ pf m^p^ mlo bi mrkql '0+1(- Rlk mlpf_ibp alp plir`flkbp-

04- C^alp alp sb`qlobp cfglp kl krilp = v > nrb cloj^k rk Škdril K) pfbkal N ; K ;9 .P+

Tk jlsfjfbkql `lk sb`qlo ab mlpf`fŽk l%n&bk bi fkpq^kqb o*p^qfpc^`b i^ b`r^`fŽk af,cbobk`f^i

l$%n&< = u l%n&

v i^ `lkaf`fŽk fkf`f^i obN( < >+^( Cbjlpqo^o nrb bi sb`qlo ^`bibo^`fŽk [%n&bp loqldlk^i ^ =+_( Cbjlpqo^o nrb i^ sbil`fa^a bp `lkpq^kqb v `^i`ri^oi^ bk crk`fŽk ab =) > X K+`( Cf_rg^o i^ `ros^+ jlpqo^kal pr obi^`fŽk `lk ilp sb`qlobp = v >+

05- Dpqb bgbo`f`fl abp`of_b `Žjl bi sb`qlo rkfq^ofl q^kdbkqb v i^ kloj^i mofk`fm^i nrba^k^cb`q^alp mlo rk `^j_fl ab m^oŠjbqol- Rrmlkd^jlp nrb rk^ `ros^ b bpqŠ abp`ofq^ mloalp crk`flkbp bnrfs^ibkqbp W b X+ pfbkal U%n&< TWo%n&Y+Cbpfdkbjlp i^ q^kdbkqb rkf,q^of^ `loobpmlkafbkqb ^ W `lk bi pŒj_lil Qu v i^ `loobpmlkafbkqb ^ W mlo Qt,

^( Cbjlpqo^o nrb bk `^a^ mrkql ab b qbkbjlp Ps%n&< PrWo%n&Ypf o bp bpqof`q^jbkqb`ob`fbkqb+ mbol pf o bp bpqof`q^jbkqb ab`ob`fbkqb bkqlk`bp Ps%n&<, PrWo%n&Y+Dk bi

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/-1 @ƒg^pgj, ji api^dji`n q`^ojmd\g`n

mofjbo `^pl+ pb af`b nrb p ^jin`mq\ g\ jmd`io\^d4i9 bk bi pbdrkal `^pl+ p diqd`mo`g\jmd`io\^d‡i,_( Ool_^o nrb ilp `loobpmlkafbkqbp sb`qlobp kloj^ibp mofk`fm^ibp Kz v Kq p^qfpc^`bkJs%n&< JrWo%n&Y bk `^a^ mrkql ab B- Cbar`fo nrb bi mi^kl lp`ri^alo bp fks^of^kqbcobkqb ^ rk `^j_fl ab m^oŠjbqol-

03-0/ Cbcfkf`fŽk ab ilkdfqra ab rk ^o`l

Dk afsbop^p m^oqbp abi BŠi`ril v ab i^ FbljbqoŒ^ ^k^iŒqf`^ pb e^`b obcb,obk`f^ ^ i^ ilkdfqra ab rk ^o`l ab `ros^- @kqbp ab mlabo bpqraf^o i^p molmfb,a^abp ab i^ ilkdfqra ab rk^ `ros^ bp mob`fpl a^o rk^ _`adid^d‡i ab ilkdfqra abrk ^o`l- Di molmŽpfql ab bpqb ^m^oq^al bp clojri^o bpq^ abcfkf`fŽk- Dpq^ iibs^oŠab• j^kbo^ k^qro^i ^ i^ `lkpqor``fŽk ab rk^ crk`fŽk 'ii^j^a^ crk`fŽk ilkdfqraab ^o`l( nrb jfa^ i^ ilkdfqra ab i^ qo^vb`qlof^ abp`ofq^ mlo rk^ m^oqŒ`ri^ jŽsfibk `^a^ fkpq^kqb ab pr jlsfjfbkql- @idrk^p ab i^p molmfba^abp crka^jbkq^ibpab bpq^ crk`fŽk pb bpqraf^k bk i^ pb``fŽk 03-01- Dk m^oqf`ri^o+ pb mol_^oŠ nrbm^o^ i^ j^vlo m^oqb ab i^p `ros^p nrb pb mobpbkq^k bk i^ moŠ`qf`^ bpq^ crk`fŽk pbmrbab bumobp^o `ljl i^ fkqbdo^i ab i^ sbil`fa^a-

O^o^ iibd^o^ rk^ abcfkf`fŽk ab il nrb pb bkqfbkab mlo ilkdfqra ab rk^`ros^+ pb mol`bab `ljl pf pb er_fbo^ ab jbafo bpq^ ilkdfqra `lk rk^ obdi^ do^,ar^a^- Dk mofjbo ird^o pb pb•^i^k bk i^ `ros^ rklp `r^kqlp mrkqlp nrb pbqlj^k `ljl s‹oqf`bp ab rk mliŒdlkl fkp`ofql 'bk i^ cfd- 03-01 pb a^ rk bgbjmil(+irbdl pb jfab i^ ilkdfqra qlq^i ab bpq^ mlifdlk^i `lk i^ obdi^ do^ar^a^ v pb`lkpfabo^ ‹pq^ `ljl rk^ ^molufj^`fŽk ab i^ ilkdfqra ab i^ `ros^- Rb l_pbos^nrb ^idrklp mliŒdlklp ~^molufj^k‚ i^ `ros^ jbglo nrb lqolp- Dk m^oqf`ri^o+ pfpb bjmfbw^ mlo rk mliŒdlkl LE v pb `lkpqorvb rk krbsl mliŒdlkl fkp`ofql L0

^•^afbkal krbslp s‹oqf`bp ^ ilp ab LE bp k^qro^i nrb i^ ilkdfqra ab L0 pb^ j^vlonrb i^ ab LE) q^i `ljl pb sb bk i^ cfdro^ 03-02- Cb i^ jfpj^ j^kbo^ pb mrbabkfo cloj^kal q^kqlp mliŒdlklp `ljl pb nrfbo^ `lk ilkdfqrabp `^a^ sbw j^vlobp-

Olo lqo^ m^oqb+fkqrfqfs^jbkqb pb l_pbos^ nrb i^ ilkdfqra ab `^a^ mliŒdlklfkp`ofql kl ab_b bu`babo ^ i^ ab i^ `ros^ 'mrbpql nrb bi pbdjbkql ab ob`q^ bpi^ qo^vb`qlof^ jŠp `loq^ bkqob alp mrkqlp(- Dp ab`fo+ bi k•jbol nrb pb qlj^mlo abcfkf`fŽk `ljl ilkdfqra ab rk^ `ros^+ e^ ab pbo rk^ ^jo\ npk`mdjmab i^pilkdfqrabp ab qlalp ilp mliŒdlklp fkp`ofqlp- Olo q^kql+ m^ob`b k^qro^i abcfkfo i^ilkdfqra ab rk^ `ros^ `ljl bi `som`hj npk`mdjmab i^p ilkdfqrabp ab qlalp ilpmlpf_ibp mliŒdlklp fkp`ofqlp-

Dk i^ j^vlo m^oqb ab `ros^p nrb pb mobpbkq^k bk i^ moŠ`qf`^+ bpq^ abcfkf`fŽk&bp •qfi v _^pq^ m^o^ ^pfdk^o rk^ ilkdfqra ^ i^ `ros^- Rfk bj_^odl+ bp plomobk,abkqb nrb bufpq^k `fboqlp `^plp m^qliŽdf`lp bk ilp nrb bpq^ abcfkf`fŽk kl bp ^mif,`^_ib- G^v `ros^p m^o^ i^p `r^ibp kl e^v buqobjl prmboflo ab i^p ilkdfqrabpab ilp mliŒdlklp fkp`ofqlp- 'Dk bi bgbo`f`fl 11 ab i^ pb``fŽk 03-02 pb a^ rk bgbjmil(-Olo q^kql+ pb e^`b mob`fpl `i^pfcf`^o i^p `ros^p bk alp `^qbdloŒ^p+i^p nrb qfbkbk

Page 211: Calculus

A`adid^d‡i _` gjibdop_ _` pi \m^j

EHFTQ@ 03-01 @pmq\^ji pi\kjgdbji\g din^mdo\,

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EHFTQ@ 03-02 I\ kjgdbji\g >?@ od`i` h\tjmgjibdop_ lp` g\ kjgdbji\g >@,

rk^ ilkdfqra v i^p nrb kl i^ qfbkbk-K^pmofjbo^p pb abkljfk^k ^pmq\n m`^odad^\]g`nv i^p lqo^p ij m`^odad^\]g`n,

O^o^ clojri^o bpq^pfab^p bk q‹ojfklp ^k^iŒqf`lp+bjmbw^jlp `lk rk^ `ros^bk bi bpm^`fl ab 2 Ž 1 afjbkpflkbp abp`ofq^mlo rk^ crk`fŽk sb`qlof^i m*v `lkpfab,objlp i^ mlo`fŽk ab i^ `ros^ abp`ofq^ mlo l%n& i s^of^o o bk rk fkqbos^il W[)\Y+@i mofk`fmfl+prmlkbjlp q^k pŽil nrb mbp `lkqfkr^ bk bi fkqbos^il m^o^j‹qof`l-Cbpmr‹p ^•^afobjlp lqo^p obpqof``flkbp-

Blkpfabobjlp ^elo^ rk^ m^oqf`fŽkO abi fkqbos^il W[)\Y)

L < voj* od%,,, *oiw * alkab [ < pl ; nc; --- ; o9< \+

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m&oi'*obpmb`qfs^jbkqb- 'Dk i^ cfdro^ 03-03 pb jrbpqo^ rk bgbjmil `lk i < 5-(Klp i^alp ab bp^ mlifdlk^i qfbkbk ilkdfqrabp

Olo `lkpfdrfbkqb+ i^ ilkdfqra ab i^ mlifdlk^i tqg!(+ nrb abpfdk^jlp `lk QQ&M'* bpi^ prj^

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54/ @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

k\m\ oj_\n g\n k\mod^dji`n _` X\*]Z* n` _d^` lp` g\ ^pmq\ `n m`^odad^\]g t g\gjibdop_ _`g \m^j* di_d^\_\ kjm >&\*]'* n` _`adi` ^jhj `g `som`hj npk`mdjm_`oj_jn gjn iˆh`mjn 06R'N(G-Pd ij `sdno` pi K+ g\ ^pmq\ n` _`ijhdi\ ij m`^odad^\]g`,

N_p‹osbpb nrb pf bufpqbrk L nrb p^qfpc^d '03-0/( m^o^ `^a^ m^oqf`fŽkLpb qfbkb9

'03-00( 06S'O(. z >&\*]' x J*

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EHFTQ@ 03-03 M\mod^d‡i_` X\* ]Z `i n`dn np] dio`mq\gjn u g\ kjgdbji\g din^mdo\^jmm`nkji_d`io`,

Dp cŠ`fi mol_^o nrb rk^ `ros^ bp ob`qfcf`^_ib pfbjmob nrb pr sb`qlo sbil,`fa^a q pb^ `lkqfkrl bk bi fkqbos^il m^o^j‹qof`l X\* \Y+ Dk bcb`ql+ bi qblobj^pfdrfbkqb fkaf`^ nrb bk bpqb `^pl pb mrbab qlj^o i^ fkqbdo^i ab i^ sbil`fa^a`ljl rk^ `lq^ prmboflo ab qlalp ilp k•jbolp DL&E+

RCMPCK? 03-0/- P`\ ..&/' `g q`^ojm q`gj^d_\_ _` g\ ^pmq\ ^ji q`^ojm kjnd+^d‡i mdoZt n`\ ^&o' < Ghr'p(00g\ q`gj^d_\_, Pd q `n ^jiodip\ `i X\*]Z* g\ ^pmq\ `nm`^odad^\]g t np gjibdop_ >&\*]' n\odna\^` g\ _`ndbp\g_\_

'03-01( >&\* ]' x X] q&o'_o ,Š \

Page 213: Calculus

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A`hjnom\^d‡i, O^o^ `^a^ m^oqf`fŽk O ab X\* ]Z* qbkbjlp

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G.P%L&Gy `7q&o'_o

m^o^ qla^p i^p m^oqf`flkbp O+v mlo q^kql bi k•jbol `wp%n&_o bp rk^ `lq^ prmbofloabi `lkgrkql ab qlalp ilp k•jbolp 00R'N(G-Dpql abjrbpqo^ nrb i^ `ros^ bp ob`qfcf,`^_ib v ^i jfpjl qfbjml pb sb nrb i^ ilkdfqra =%[) \& kl mrbab bu`babo ^ i^ fkqb,do^i ab i^ sbil`fa^a-

LŠp ^abi^kqb pb abjlpqo^oŠ nrb i^ abpfdr^ia^a '03-01( bp+ bk bcb`ql+ rk^dbp\g_\_, O^o^ biil pb qbkaoŠ nrb e^`bo rpl ab i^ \_dodqd_\_ ab i^ ilkdfqra abrk ^o`l+ molmfba^a nrb pb bpqraf^oŠ bk bi moŽufjl ^m^oq^al-

),&)) 6QVaVcVQNQQR YNY\[TVabQQR N_P\

Rf rk^ `ros^ ob`qfcf`^_ib pb `loq^ bk alp qolwlp+ i^ ilkdfqra ab qla^ i^`ros^ bp i^ prj^ ab i^p ilkdfqrabp ab i^p alp m^oqbp- Dpq^ bp lqo^ ab bpq^p mol,mfba^abp ~fkqrfqfs^jbkqb fkjbaf^q^p‚ v `rv^ abjlpqo^`fŽk kl bp qofsf^i- Dpq^molmfba^a pb abkljfk^ \_dodqd_\_ _` g\ gjibdop_ _`g \m^j v pb mrbab bumobp^o^k^iŒqf`^jbkqb `ljl pfdrb9

RCMPCK? 03-00- @jind_`m`hjn pi\ ^pmq\ m`^odad^\]g` _` gjibdop_ >&\* ]'*_`n^mdo\ kjm pi q`^ojm mR' ^p\i_j o q\m…\ `i pi dio`mq\gj X\* ]Z, Rf \ ; ` ; ]*n`\i A+ v A1 g\n ^pmq\n _`n^mdo\n kjm t&o' ^p\i_j o q\m…\ `i gjn dio`mq\gjn X\* `Zv X`* ]Z* m`nk`^odq\h`io`, Bioji^`n B+ t B1 o\h]d„i nji m`^odad^\]g`n t* nd>&\*_( u >&^*]' m`km`n`io\i npn m`nk`^odq\n gjibdop_`n* o`i`hjn %

=%[) \& < =%[8b( * @'`+ \& +

A`hjnom\^d‡i, Rb^k OH u O1 m^oqf`flkbp ^o_fqo^of^p ab X\* `Z v X`* ]Z obp,mb`qfs^jbkqb- Klp mrkqlp ab O+ v ilp ab O1 `lkgrkq^jbkqb cloj^k rk^ krbs^m^oqf`fŽk L ab W[)\Y m^o^ i^ `r^i pb qfbkb9

'03-02(

Page 214: Calculus

541 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

0/ nrb mrb_^ nrb /5Qvky'/ v /5Q&M0' bpqŠk ^`lq^a^p mlo =%[) \& v mlo q^kql B{u B1 plk ob`qfcf`^_ibp- Cb '03-02( pb abar`b q^j_f‹k9

Rf pb `lkpfabo^ ^elo^ M0 cfgl u ky s^of^_ib bk bi `lkgrkql ab qla^p i^p m^oqf,`flkbp ab W[) ]F) mrbpql nrb bi k•jbol =%[) \& * 06S'O1(0 bp rk^ `lq^ prmboflo abqlalp ilp k•jbolp 06S'O{(0 kl mrbab pbo jbklo nrb pr buqobjl prmboflo nrb bp>&\*`', Olo q^kql pb qfbkb >&\* `' 8899>&\* ]' + 06S'O1(0+ l il nrb bp il jfpjl9

06S'O1(0 99:: >&\* ]' + >&\* ^' ,

Dpql morb_^ nrb >&\* ]' + >&\* `' bp rk buqobjl prmboflo ab qla^p i^p prj^p 5Q&M0&.) u mrbpql nrb kl mrbab pbo jbklo nrb pr buqobjl prmboflo =%]) \& pbqfbkb >&^*]' 8899>&\* ]' + >&\* ^'* l pb^9

'03-03( >&\*^' * >&^*]' R >&\*]' ,

O^o^ abjlpqo^o i^ fdr^ia^a _^pq^ mol_^o i^ abpfdr^ia^a `lkqo^of^- O^o^ biilpb bjmfbw^ mlo rk^ m^oqf`fŽk `r^inrfbo^ L ab W[) \ I- Rf pb ^agrkq^ bi mrkql _ ^ Lpb l_qfbkb rk^ m^oqf`fŽk m*ab W[) _H u rk^ m^oqf`fŽk L0 ab W_)\ H ab j^kbo^ nrb9

Dpql morb_^ nrb =%[) b( * @'`+ \& bp rk buqobjl prmboflo ab qlalp ilp k•jbolp06S'O(H+ u mrbpql nrb ‹pqb kl mrbab pbo jbklo nrb bi buqobjl prmboflo+ e^ ab pbo9

>&\*]' 8899>&\*^' * >&^*]' ,

Dpq^ abpfdr^ia^a+ grkql `lk '03-03(+ fjmif`^ i^ molmfba^a ^afqfs^-

),&)* ;b[PVp[ Y\[TVabQQRN_P\

Rrmlkd^jlp nrb rk^ `ros^ bp bi `^jfkl abp`ofql mlo rk sb`qlo mlpf`fŽk l%n&+Tk^ mobdrkq^ k^qro^i bp bpq^9 ƒBrŠkql e^_oŠ ^s^kw^al i^ m^oqŒ`ri^ jŽsfi ^ ili^odl ab i^ `ros^ bk bi fkpq^kqb o= O^o^ afp`rqfo bpq^ `rbpqfŽk+ fkqolar`fjlp g\ api+^d‡i gjibdop_ _` \m^j p- abcfkfa^ `ljl pfdrb9

m%.&< =%[) .& nd 0= \* n`\' < N-

K^ fdr^ia^a m_[&< N pfdkfcf`^ q^k pŽil nrb bpq^jlp prmlkfbkal nrb bi jlsf,jfbkql `ljfbkw^ `r^kal o < \,

Page 215: Calculus

Cpi^d‡i gjibdop_ _` \m^j 542

Di qblobj^ ab i^ ^afqfsfa^a klp mbojfqb abar`fo ^idrk^p molmfba^abp fj,mloq^kqbp ab n, Olo bgbjmil+ qbkbjlp bi pfdrfbkqb

RCMPCK? 03-01- M\m\ oj_\ ^pmq\ m`^odad^\]g`*g\ api^d‡i gjibdop_ _` \m^j n`n hji‡oji\ ^m`^d`io` `i X\* ]Z, Bnoj `n* o`i`hjn

'03-04(

A`hjnom\^dƒio Rf \ x nE ; n/ w ]* pb qfbkb9

alkab i^ •iqfj^ fdr^ia^a bp `lkpb`rbk`f^ ab i^ ^afqfsfa^a- Orbpql nrb @'o+ S ƒ N+nrba^ abjlpqo^al '03-04(-

@ `lkqfkr^`fŽk pb abjlpqo^oŠ nrb i^ crk`fŽk p qfbkb rk^ abofs^a^ bk `^a^mrkql fkqboflo abi fkqbos^il m^o^j‹qof`l u nrb bpq^ abofs^a^ bp fdr^i ^ i^ sbil`fa^aab i^ m^oqŒ`ri^-

RCMPCK? 03-02- P`\i n g\ api^d‡i gjibdop_ _` \m^j \nj^d\_\ \ pi\ ^pmq\t q&o' g\ q`gj^d_\_ `i `g od`hkj o, Pdq `n ^jiodip\ `i X\* ]Z* `ioji^`n g\ _`mdq\_\n%&o'sdno` k\m\ ^\_\ o _` n\* ]Z V qd`i` _\_\ kjm g\ a‡mhpg\

'03-05( n%&o'< q&o',

A`hjnom\^dƒi* Cbcfk^jlp aR' < Iz q&p' _pj R^_bjlp nrb a%R'< qvo' bksfoqra abi mofjbo qblobj^ crka^jbkq^i abi BŠi`ril- Cbjlpqo^objlp nrb m$%n&<p%n&+

@ q^i cfk cloj^jlp bi `l`fbkqb ab afcbobk`f^p

'03-06( 0H m&o* cx + m&o' ..+

Rrmlkd^jlp mofjbol nrb b = k- Di pbdjbkql ab ob`q^ nrb rkb ilp mrkqlp m&o'um&o* b& mrbab `lkpfabo^opb `ljl rk^ mlifdlk^i nrb ^molufj^ bi ^o`l nrb rkbbplp alp mrkqlp- Olo `lkpfdrfbkqb+ bk sfoqra ab '03-00(+ qbkbjlp

yym&o* c' + m&o'00 99p:>&o*o * c' < n&o* c' + n&o',

@mif`^kal bpqb obpriq^al bk '03-06( grkql `lk i^ abpfdr^ia^a '03-01( abi qblobj^03-0/ l_qbkbjlp

.X m&o* cx + m&o'0[ 99p:n&o* cx + n&o' 99p:z aocS&R' _p ;e&o * cx + e&o',

Page 216: Calculus

543 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

Tk o^wlk^jfbkql ^kŠildl morb_^ nrb bpq^p abpfdr^ia^abp q^j_f‹k plk sŠifa^pm^o^ c ; l- Rf e^`bjlp c x N+bi `l`fbkqb ab afcbobk`f^p ab i^ fwnrfboa^ qfbkab^ Zo&'w(00< q&o' v bi ab i^ abob`e^ qfbkab ^ a%&o'< q&o', Qbpriq^ ab bpql nrb bi`l`fbkqb Xn&o* c' + n&o'Z- c q^j_f‹k qfbkab ^ q&o', Obol bpql pfdkfcf`^ nrb n%&o'bufpqbu bp fdr^i ^ p%n&) ljl pb nrboŒ abjlpqo^o-

Di qblobj^ 03-02 bpqŠab ^`rboal `lk krbpqo^ kl`fŽk fkqrfqfs^ ab sbil`fa^a`ljl i^ afpq^k`f^ ob`loofa^ mlo rkfa^a ab qfbjml aro^kqb bi jlsfjfbkql-

Tqfifw^kal '03-05( grkql `lk bi pbdrkal qblobj^ crka^jbkq^i abi BŠi`ril+mlabjlp `^i`ri^o i^ ilkdfqra abi ^o`l fkqbdo^kal i^ sbil`fa^a- @pŒmrbp+bi `^jfklob`loofal mlo rk^ m^oqŒ`ri aro^kqb rk fkqbos^il ab qfbjml Xo/< o0Z bp

Go0 Go0

n&o0'+ n&o/' < p&'o(_o < q&o'_o ,o* o*

Dk m^oqf`ri^o+ r^kal n)< [ v ny < \) l_qbkbjlp mlo i^ ilkdfqra abi ^o`l i^pfdrfbkqb fkqbdo^i9

>&\* ]' <aq&o' _o,

DIDLOKN 0- Ijibdop_ _` pi \m^j _` ^dm^pia`m`i^d\, O^o^ `^i`ri^o i^ ilk,dfqra ab rk ^o`l ab `fo`rkcbobk`f^ ab o^afl \* mlabjlp fj^dfk^o rk^ m^oqŒ`ri^jŽsfi ^ il i^odl ab i^ `fo`rkcbobk`f^ ab ^`rboal `lk i^ b`r^`fŽk l%n&< [ `lp od(( \ pbk og- Di sb`qlo sbil`fa^a bp q&o'< , \ pbk od * \ `lp od Xi^ sbil`fa^a bpp%n&< [+ Hkqbdo^kali^ sbil`fa^a bk rk fkqbos^il ab ilkdfqra %F)bk`lkqo^jlp nrbi^ ilkdfqra abi ^o`l abp`ofql bp [%F+Cf`el ab lqol jlal+ i^ ilkdfqra ab rk ^o`lab ‹fo`rkcbobk`f^ bp molmlo`flk^i ^i Škdril `loobpmlkafbkqb: i^ `lkpq^kqb ab mol,mlo`flk^ifa^a bp bi o^afl ab i^ `fo`rkcbobk`f^- O^o^ rk^ `fo`rkcbobk`f^ rkfa^aqbkbjlp \ < 0+ X i^ ilkdfqra abi ^o`l bp bu^`q^jbkqb fdr^i ^i Škdril jbafal-

DIDLOKN 1- Ijibdop_ _` g\ bmƒad^\ _` pi\ api^d‡i m`\g, K^ doŠcf`^ab rk^crk`fŽk ob^i ` abcfkfa^ bk rk fkqbos^il W[)\Y mrbab `lkpfabo^opb `ljl rk^ `ros^`lk sb`qlo mlpf`fŽk l%n&a^al mlo

Q&o' < od* e&o'e ,

Di sb`qlo sbil`fa^a `loobpmlkafbkqb bp q&o'< d * e%&o'e*v i^ sbil`fa^a bp

q&o' < 00q&o'00 < UH * Xa%&o~'0,

Page 217: Calculus

Ad_l]c]cim 544

Olo `lkpfdrfbkqb+ i^ ilkdfqra ab i^ doŠcf`^ ab ` bk rk fkqbos^il W[)rY sfbkb a^a^mlo i^ fkqbdo^i

&/2,/6' n&s'<F8q&o'_o <F8UH* Xa%&o'Z0_o ,

),&)+ :WR_PVPV\`

Dk ilp bgbo`f`flp abi 0 ^i 8+ e^ii^o i^ ilkdfqra abi `^jfkl abp`ofql mlo rk^ -m^oqŒ`ri^jŽsfi pl_ob rk^ `ros^ ab ^`rboal `lk i^ b`r^`fŽk a^a^+ aro^kqb bi fkqbos^il ab qfbjmlnrb bk `^a^ `^pl pb bpmb`fcf`^-0- m&o'< \&/ + `lp o'd * \&o + pbk o'e* N z o x /.P) \ = N-0, m&o'< _f `lp od * _f pbk od* N z o x 1-

1, m&o'< ^'`lp o * opbk o'd * ^'pbk o + o `lp o'e* N z o x /.P) \ = N-

b1 b1

2, m&o'< ,,{: `lp! od * !]n`i1 od) M y o x /.P) b1 < \0+ ]0

* M ; ] ; \,

3, m&o'< ^'pbke o + o'd * ^'`lpe o + G(f+ N z o x Q* \ = N-

4, m&o'< pbk od * of * '0 , `lp o'f 'N z o x /.P&+5, m&o'< od * 1o5 * 4o1f 'N z o x 1(-

6, m&o'< od * ild 'pb` o'e * ild 'pb` o * q^k o'f 'N z o x c.P&+7, m&o'< \ `lp ^jo c * \ n`iroe * ]rf &ojx n w nE&$

0/- G^ii^o rk^ fkqbdo^i m^ob`fa^ ^ i^ ab /3-07( m^o^ i^ ilkdfqra ab i^ doŠcf`^ ab rk^ b`r^,`fŽk ab i^ cloj^ r < c'u(+ qbkfbkal c abofs^a^ `lkqfkr^ bk rk fkqbos^il W_) Y+

00- K^ b`r^`fŽk ab rk^ `ros^ bp v1 < s1Š G^ii^o i^ ilkdfqra abi ^o`l nrb rkb N+ ,0( ^N+ 0(-

01- Clp mrkqlp = v > ab rk `Œo`ril rkfa^a ab `bkqol @ abqbojfk^k bk ‹i rk pb`qlo `fo`ri^o=@>+ Ool_^o nrb i^ ilkdfqra abi ^o`l => bp fdr^i ^ alp sb`bp bi Šob^ abi pb`qlo-

02- Dpq^_ib`bo fkqbdo^ibp m^o^ i^p ilkdfqrabp ab i^p `ros^p `rv^p b`r^`flkbp plk^( v < #F) N 99:s 8890: _( s < o * ild n)X < o[+ ild o*0 99:o 889 , Ool_^o nrb i^ pbdrka^ilkdfqra bp bi molar`ql ab i^ mofjbo^ mlo T 1 -

03- ^( Dpq^_ib`bo i^ fkqbdo^i nrb a^ i^ ilkdfqra ab i^ `ros^ v < b `lpe %r,_& abpab r < N^ r < [ %[ = N _ = N(-_( Ool_^o nrb bi molar`ql ab i^ ilkdfqra ab bpq^ `ros^ mlo ` bp fdr^i ^i Šob^ ab i^obdfŽk ifjfq^a^ mlo v < ha `lpe %r,_&) bi bgb r) bi bgb v v i^ ob`q^ r < [+`( B^i`ri^o bpq^ fkqbdo^i v e^ii^o i^ ilkdfqra ab i^ `ros^ `r^kal \ < 1-

04- Cbjlpqo^o nrb i^ ilkdfqra ab i^ `ros^ v < `lpe r nrb rkb ilp mrkqlp 'N+ 0( X %r) `lpe u(bp pbke s pf s = N-

05- Tk^ crk`fŽk kl kbd^qfs^ ` qfbkb i^ molmfba^a ab nrb pr `lkgrkql ab loabk^a^p bk rkfkqbos^il `r^inrfbo^ qfbkb rk Šob^ molmlo`flk^i ^ i^ ilkdfqra abi ^o`l ab i^ doŠcf`^`loobpmlkafbkqb ^i fkqbos^il- G^ii^o `+

06- Tqfifw^kal i^ b`r^`fŽk sb`qlof^i m&-'< \ pbk od* \ `lp odalkab /; \ ; \* mol_^o nrbi^ ilkdfqra H ab rk^ bifmpb bpqŠ a^a^ mlo i^ fkqbdo^i

Page 218: Calculus

545 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

alkab ` < sf \/ * ]/-\, Di k•jbol ` bp i^ bu`bkqof`fa^a ab i^ bifmpb- Dpqb bp rk `^plm^oqf`ri^o ab rk^ fkqbdo^i ab i^ cloj^9

ii^j^a^ chn_al[f _f•jnc][ ^_ m_aoh^[ ]f[m_) alkab N 999:e ; 0- Klp k•jbolp A%e& bpqŠkq^_ri^alp m^o^ s^oflp s^ilobp ab f,

g?, Rf N ; ] ; 2\* pb^ m&o'< \&o+ pbk o'd* \&g + `lp o'e * ] pbk xo f, Ool_^o nrb i^ilkdfqra ab i^ qo^vb`qlof^ abp`ofq^ abpab o < N e^pq^ o < /$EP bp >[A%e&) alkab A%e& qfbkbbi pfdkfcf`^al a^al bk bi bgbo`f`fl 06 v e/ < 0 , %\,1[&/+

08- Tk^ m^oqŒ`ri^ pb jrbsb `lk sb`qlo mlpf`fŽk

m&o'< o> * o0> * 0&/+o'1-0> t >)

alkab = v > plk alp sb`qlobp rkfq^oflp cfglp nrb cloj^k rk Škdril ab $EP,0 o^af^kbp-B^i`ri^o i^ sbil`fa^a ab i^ m^oqŒ`ri^bk bi fkpq^kqb o v e^ii^o `r^kql qfbjml fksfboqb m^o^abpmi^w^opb rk^ afpq^k`f^ ab 01 rkfa^abp ab ilkdfqra ab ^o`l abpab i^ mlpf`fŽk fkf`f^i,l‚-

1/- ^( Br^kal rk `Œo`ril orba^ 'pfk abpifw^jfbkql( ^ 0/ i^odl ab rk ob`q^+ rk mrkql abi^ `fo`rkcbobk`f^ abp`of_b rk^ `ros^ ii^j^a^ ]c]fic^_+ Rf i^ ob`q^ cfg^ bp bi bgb r v pfbi mrkql jŽsfi %r)u( bpqŠ fkf`f^ijbkqb bk bi lofdbk+ abjlpqo^o nrb `r^kal bi `Œo`rildfo^ rk Škdril L qbkbjlp

r < [%L + pbk L' * t < [%. * `lp nc&)

alkab \ bp bi o^afl abi `Œo`ril- Dp^p plk i^p b`r^`flkbp m^o^j‹qof`^p ab i^ `f`ilfab-_( Qbcfof‹kalpb ^ i^ m^oqb ^(+ abjlpqo^o nrb ^sd^r < ]inni u abar`fo nrb i^ ob`q^q^kdbkqb ^ i^ `f`ilfab bk %r)s&cloj^ rk Škdril x&%gQ*K& `lk bi bgb r+ G^`bo rk doŠcf`lv jlpqo^o nrb i^ ob`q^ q^kdbkqb m^p^ mlo bi mrkql jŠp ^iql ab i^ `fo`rkcbobk`f^+

1 - Rb^ b rk^ `ros^ abp`ofq^ mlo alp crk`flkbp bnrfs^ibkqbp W b U) alkab X'qi < W Wo%n&Ym^o^ ` 999:o 8889+ Rf i^ crk`fŽk p nrb abcfkb bi `^j_fl ab m^oŠjbqol qfbkb abofs^a^ `lk,qfkr^ bk W]) ^Y abjlpqo^o nrb

&R&_' &_

Hqh` EET$%o&ff^o:+E] GGW&'p(h `p+

v abar`fo nrb i^ ilkdfqra ab ^o`l ab b bp fks^of^kqb cobkqb ^ rk q^i `^j_fl ab m^oŠ,jbqol-

11- Blkpfabobjlp i^ `ros^ mi^k^ `rv^ b`r^`fŽk sb`qlof^i bp m&o'< od * a&o'e* alkab

a&o' < o `lp 'co( v o 78:-) a`L' < l-

Blkpfabobjlp i^ pfdrfbkqb m^oqf`fŽk abi fkqbos^il ZN+0\9

M < xl+J+0i x 0 &--- +z+ z+ 0 y -

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@pmq\opm\ _` pi\ ^pmq\ 546

Cbjlpqo^o nrb i^ `loobpmlkafbkqb mlifdlk^i fkp`ofq^ $EP%L&qfbkb ilkdfqra

000EM' = 0 * , * , * --- *,

1 2 0i

v abar`fo nrb bp^ `ros^ kl bp ob`qfcf`^_ib-

),&), 8b_cNab_NQRb[N Pb_cN

Dk rk^ ob`q^+bi sb`qlo rkfq^ofl q^kdbkqb Q kl `^j_f^ pr afob``fŽk u mloq^kql P$: N- Rf i^ `ros^ kl bp rk^ iŒkb^ob`q^+i^ abofs^a^ S& jfab i^ qbk,abk`f^ ab i^ q^kdbkqb^ `^j_f^o pr afob``fŽk- Di `lbcf`fbkqb ab s^of^`fŽk l ab,ofs^a^ ab i^ q^kdbkqbrkfq^of^ m`nk`^oj \ g\ gjibdop_ _`g \m^j pb abkljfk^ sb`qlo^pmq\opm\ ab i^ `ros^- Rb abpfdk^ mlo _Q -_n* alkab p obmobpbkqi^ ilkdfqra abi^o`l- K^ obdi^ ab i^ `^abk^ u i^ cŽojri^ m$%n&< p%n&mbojfqbk obi^`flk^o bi sb`qlo`ros^qro^ _Q - _n `lk i^ abofs^a^ Q%obpmb`ql^i qfbjml jbaf^kqb i^ b`r^`fŽk

_Q < _o _Q < \0 Q%&o'< \0 Q&o'_n _n _o p&'o( q&o' ,

Orbpql nrb P%n&< GGR&'p(00J%n&)l_qbkbjlp

'03-08( _Q < 00 P%n&..J%n&)_n q&o'

nrb af`b nrb bi sb`qlo `ros^qro^ qfbkb i^ jfpj^ afob``fŽk nrb i^ kloj^i mofk,`fm^i J%n&+Di c^`qlo bp`^i^o nrb jriqfmif`^ ^ J%n&bk '03-08( bp rk k•jbol klkbd^qfsl ii^j^al ^pmq\opm\ ab i^ `ros^ bk o u pb abpfdk^ mlo H&o'&Hbp i^ ibqo^dofbd^ h^mm^(-@pŒ+i^ `ros^qro^ H&o'abcfkfa^ `ljl i^ gjibdop_ _`g q`^ojm ^pmq\+opm\ bpqŠa^a^ mlo i^ cŽojri^ pfdrfbkqb9

'03-1/( H&o' < 00o'q(00 -, q&o'

DIDLOKN 0- @pmq\opm\ _` pi\ ^dm^pia`m`i^d\, O^o^ rk `Œo`ril ab o^afl \*a^al mlo i^ b`r^`fŽk m&o'< \ `lp od * ] pbk oe* qbkbjlp q&o'< , \ pbk od (( \ `lp oe* q&o' < \* Q&o'< , pbk od * `lp oe* u Q%&o'< ,`lp od + pbk odKrbdl qbkbjlp 0GR&'p(G < 0 ^pŒnrb G%n&< nYy+Dpql morb_^ nrb rk^ `fo`rkcb,obk`f^ qfbkb `ros^qro^ `lkpq^kqb- Di ob`Œmol`lab i^ `ros^qro^ bp bi o^afl ab i^`fo`rkcbobk`f^-

Page 220: Calculus

/.1 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

Br^kal H&o' x N pr fksbopl pb abkljfk^ m\_dj _` ^pmq\opm\ u pb abpfdk^mlo k&o' 'm bp i^ ibqo^ dofbd^ ol(- K^ `fo`rkcbobk`f^ bk bi mi^kl lp`ri^alo abo^afl k&o' v `bkqol pfqr^al pl_ob i^ kloj^i v e^`f^ bi buqobjl abi sb`qlo `ros^qro^pb ii^j^ ^…m^pgjjn^pg\_jm, Rb mrbab abjlpqo^o nrb bi `Œo`ril lp`ri^alo bp i^ ml,pf`fŽk iŒjfqbab i^p `fo`rkcbobk`f^p nrb m^p^kmlo qobpmrkqlp moŽufjlp ab i^ `ros^`r^kal alp ab ilp mrkqlp pb ^molufj^k ^i qbo`bol- Cb_fal ^ bpq^ molmfba^a pbaf`b nrb bi `Œo`ril lp`ri^alo bp bi `Œo`ril nrb ~jbglo pb ^grpq^ ^ i^ `ros^‚ bk`^a^ rkl ab prp mrkqlp-

DIDLOKN 1- @pmq\opm\ _` pi\ ^pmq\ kg\i\, Dk rk^ `ros^ mi^k^ pb e^sfpql nrb 0GQ%&o'00< hat&'p(h+ alkab GV'p(bp bi Škdril ab fk`ifk^`fŽk abi sb`qlo q^kdbk,qb+pb•^i^al bk i^ cfdro^ 03-00- Orbpql nrb FU%&o'9!!_jg_o < &_FU-_n'&_n-_o' ;S&o'_dUW^m)i^ b`r^`fŽk '03-1/( fjmif`^9

Cf`el ab lqo^ cloj^+ i^ `ros^qro^ ab rk^ `ros^ mi^k^ bp bi s^ilo ^_plirql ab i^abofs^a^ ab '0- obpmb`ql i^ ilkdfqra abi ^o`l-Lfab bi `^j_fl ab afob``fŽk obpmb`qbabi `^jfkl ob`loofal ^ il i^odl ab i^ `ros^-

DIDLOKN 2- @pmq\n kg\i\n _` ^pmq\opm\ ^jino\io`, Rf _\a _n bp rk^ `lkp,q^kqbkl kri^9 _\a _n < \* bkqlk`bp HW < \n * ] alkab ] bp rk^ `lkpq^kqb+v mloq^kql P < `lp %[m* ]'d * pbk &\n * ]'e, Hkqbdo^kal pb qfbkb9m< &g-\' pbk&\n* ]'d + &g-\' `lp &\n * ]'e * >* alkab > bprk sb`qlo `lkpq^kqb-Cb^nrŒobpriq^Ghn, ?GG< h.h]h+bp ab`fo+i^ `ros^ bp rk^ `fo`rkcbobk`f^ 'l rk ^o`lab `fo`rkcbobk`f^( ab `bkqol bk bi buqobjl ab = v o^afl h.h]h-Dpql abjrbpqo^ nrbrk^ `ros^ ab `ros^qro^ `lkpq^kqb G ;/; N bp rk^ `fo`rkcbobk`f^ 'l ^o`l ab `fo`rk,cbobk`f^( ab o^afl f,G+

U^jlp ^elo^ ^ abjlpqo^o rk qblobj^ nrb obi^`flk^ i^ `ros^qro^+ i^ sbil`fa^av i^ ^`bibo^`fŽk-

RCMPCK? 03-03- M\m\ pi hjqdhd`ioj ^p\glpd`m\ ^ji q`^ojm q`gj^d_\_ q&o'*q`gj^d_\_ q&o'*q`^ojm \^`g`m\^d‡i \&o'* u ^pmq\opm\ G%n&)o`i`hjn

'03-10( \&o' < q%&o'Q&o'* H&o'q0&o'K&o',

Bno\ a‡mhpg\* \ np q`u* dhkgd^\

'03-11( |'n( < Gh]'p(t r'MGG-q1&o'

Page 221: Calculus

Be`m^d^djn 548

A`hjnom\^d‡i, O^o^ abjlpqo^o '03-10(+ mlkbjlp '03-1/( bk i^ cloj^GGR&'p(00< G%n&R%n&)nrb klp a^ P$%n&< G%n&p%n&J%n&+Rrpqfqrvbkal bpq^ bumobpfŽkab P$%n&bk i^ b`r^`fŽk '03-7(+ l_qbkbjlp '03-10(-

O^o^ abjlpqo^o '03-11(+ clojbjlp bi molar`ql sb`qlof^i \&o' u q&o'* rqfif,w^kal '03-10( m^o^ [%n&v i^ cŽojri^ p%n&< p%n&P%n&m^o^ bi sb`qlo sbil`fa^a- Dpqlklp a^

'03-12( [ t p < p$pP V P * Gp1J V P < Go1J V P

v^ nrb Q W Q < N- Rf `lkpfabo^jlp i^ ilkdfqra ab `^a^ jfbj_ol ab '03-12( ul_pbos^jlp nrb

GGLW RGG< GGLGGGGRGGpbkf6S < 0+

l_qbkbjlp Gh]V ree < GR[) il nrb morb_^ '03-11(-

Dk i^ moŠ`qf`^ obpriq^ jŠp pbk`fiil `^i`ri^o ilp sb`qlobp q v \ 'abofs^kal bisb`qlo mlpf`fŽk l&8 mlo q^kql i^ b`r^`fŽk '03-11( klp molmlo`flk^ rk j‹qlal •qfim^o^ `^i`ri^o i^ `ros^qro^- Dpqb j‹qlal bp loafk^of^jbkqb jŠp pbk`fiil nrb abqbo,jfk^o i^ `ros^qro^ ^ m^oqfoab pr abcfkf`fŽk-

Rf pb qo^q^ ab rk^ iŒkb^ob`q^ pb qfbkb \ u q < N X mlo q^kql i^ `ros^qro^ bp`lkpq^kqbjbkqb `bol- Tk^ `ros^ `lk `ros^qro^ mbnrb•^ bk rk mrkql qfbkb bk bpqbmrkql o^afl ab `ros^qro^ do^kab u bk i^ molufjfa^a ab bpqb mrkql afcfbob ml`lab rk^ iŒkb^ ob`q^- Olo q^kql+ i^ `ros^qro^ bp i^ jbafa^ ab i^ qbkabk`f^ ab rk^`ros^ ^ abpsf^opb ab i^ iŒkb^ ob`q^-

),&)- :WR_PVPV\`

0- Blkpfa‹obkpb i^p `ros^p abp`ofq^p bk ilp bgbo`f`flp abi 0 ^i 5 ab i^ pb``fŽk 03-8 v bk`^a^ `^pl abqbojfk^o i^ `ros^qro^ G%n&m^o^ bi s^ilo fkaf`^al ab n+

1- Tk^ e‹if`b bpqŠ abp`ofq^ mlo i^ crk`fŽk ab mlpf`fŽk m&o'< \ `lp L'o d* \ pbk ^joZ * ]jnof*Cbjlpqo^o nrb qfbkb `ros^qro^ `lkpq^kqb H < [,%[0 * ]0',

2- Clp sb`qlobp rkfq^oflp cfglp = v > cloj^k rk Škdril %F)pfbkal N ; `&; nm*Tk^ m^oqŒ`ri^pb jrbsb pl_ob rk^ `ros^ ^i^_b^a^ ab j^kbo^ nrb pr sb`qlo ab mlpf`fŽk m&o'v bi sb`,qlo sbil`fa^a q&o' bpqŠk obi^`flk^alp mlo i^ b`r^`fŽk q&o' < = V m&o', Rf m`L' < >) ab,jlpqo^o nrb i^ `ros^ qfbkb `ros^qro^ `lkpq^kqb v `^i`ri^oi^ bk crk`fŽk ab K+

3- Tk mrkql pb jrbsb bk bi bpm^`fl pbd•k i^ b`r^`fŽk sb`qlof^i

m&o' < 3 `lp od * 3 pbk od* 3 `lp o e +

^( Ool_^o nrb i^ qo^vb`qlof^ bp rk^ bifmpb v e^ii^o i^ b`r^`fŽk abi mi^kl nrb `lkqfbkbaf`e^ bifmpb-_( Ool_^o nrb bi o^afl ab `ros^qro^ bp k&o' < 1&U w'i ,Œ,pbk!o'1-0,

Page 222: Calculus

//) @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

4- O^o^ i^ `ros^ `rv^ b`r^`fŽk sb`qlof^i bp m&o' < `%Z * `m%Z* s1 o =" abjlpqo^o nrb i^

`ros^qro^ bp H&o' < q!d-&`o* `+o'0,

5- ^( O^o^ rk^ `ros^ mi^k^ abp`ofq^ mlo i^ b`r^`fŽk l%n&< r%n&8* s%n&d)abjlpqo^o nrb i^`ros^qro^ sfbkb a^a^ mlo i^ cŽojri^

Er$%n&s!%n&* s$%n&r!%n&0

G%n&< %Wr$%n&Y/* Ws$%n&Y/v0,/

_( Rf rk^ `ros^ mi^k^ qfbkb i^ b`r^`fŽk `^oqbpf^k^ t < x%r&)abjlpqo^o nrb i^ `ros^,qro^ bk bi mrkql %r) x%r| bp

Ed!%r&0xi * Xa%&U'Z0w1-0Š

6- Rf rk mrkql pb jrbsb ab j^kbo^ nrb ilp sb`qlobp sbil`fa^a v ^`bibo^`fŽk qfbkbk pfbj,mob ilkdfqra `lkpq^kqb+ mol_^o nrb i^ `ros^qro^ bp `lkpq^kqb bk qlalp ilp mrkqlp abi`^jfkl- Dumobp^obpq^ `lkpq^kqb mlo jbafl ab Gh]hhv Hhrhh-

7- Rf alp `ros^p ab b`r^`flkbp `^oqbpf^k^p t < x%r& b v < a%r& plk q^kdbkqbp bk bi mrkql%[) \& v qfbkbk i^ jfpj^ `ros^qro^ bk bpqb mrkql mol_^o nrb EE!%[&.:fa!%[&f+

8- O^o^ `fboqlp s^ilobp ab i^p `lkpq^kqbp \ v \) i^p alp `ros^p ab b`r^`flkbp `^oqbpf^k^pv < [r%\ * r& v %r * /&s < r pb `loq^k pli^jbkqb bk rk mrkql O+ qfbkbk rk^ ob`q^q^kdbkqb `lj•k bk O+ v i^ jfpj^ `ros^qro^ bk O-^( G^ii^o qlalp ilp \ v ] nrb p^qfpc^`bk qla^p bp^p `lkaf`flkbp-_( O^o^ `^a^ bib``fŽk mlpf_ib ab \ u \ nrb p^qfpc^d^k i^p `lkaf`flkbp a^a^p+ `lkpqorfork doŠcf`l ab i^p alp `ros^p- Llpqo^o ab nr‹ j^kbo^ pb `loq^k bk L+

0/- ^( Cbjlpqo^o nrb bk bi s‹oqf`b ab rk^ m^oŠ_li^ bi o^afl ab `ros^qro^ ^i`^kw^ pr s^ilojŒkfjl-_( C^alp alp sb`qlobp rkfq^oflp cfglp = v > nrb cloj^k rk Škdril %d)pfbkal N ; %d; 6S-

K^ `ros^ `lk sb`qlo mlpf`fŽk m&o'< f= * .0> bp rk^ m^oŠ_li^ pfqr^a^ bk bi mi^kldbkbo^al mlo = v >+ Cbqbojfk^o 'bk crk`fŽk ab =) > X &e' bi sb`qlo mlpf`fŽk abi s‹oqf`bab bp^ m^oŠ_li^- Orbab rqfifw^opb i^ molmfba^a ab i^ m^oŠ_li^ bpq^_ib`fa^ bk i^ m^oqb ^(-

00- Tk^ m^oqŒ`ri^pb jrbsb ^ il i^odl ab rk^ `ros^ mi^k^ `lk sbil`fa^a `lkpq^kqb fdr^i^ 4- R^ib abi lofdbk bk bi fkpq^kqb 0 < N `lk sbil`fa^a fkf`f^i 3e* v krk`^ m^p^ ^ i^fwnrfboa^ abi bgb s+ Dk qlal jljbkql i^ `ros^qro^ abi `^jfkl bp E_%n&< /n+ Cbpfdkbjlp`lk [%n& bi Škdril nrb cloj^ bi sb`qlo sbil`fa^a `lk bi bgb s mlpfqfsl bk bi fkpq^kqb o,^( Cbqbojfk^o bumiŒ`fq^jbkqb \&o' `ljl crk`fŽk ab i-_( Cbqbojfk^o bi sb`qlo sbil`fa^a q&o' bk crk`fŽk ab dv g-

01- Tk^ m^oqŒ`ri^pb jrbsb ^ il i^odl ab rk^ `ros^ mi^k^ `lk sbil`fa^a `lkpq^kqb fdr^i^ 1- Di jlsfjfbkql bjmfbw^ bk bi lofdbk `r^kal 0 < N X bi sb`qlo sbil`fa^a fkf`f^ip_K& bp 1:- Rb p^_b nrb bk `^a^ fkpq^kqb i^ `ros^qro^ bp E_%n&< 30- G^ii^o bi sb`qlo sbil,

`fa^a `r^kal 0 < qv&:pf i^ `ros^ k^ bpqŠ krk`^ ab_^gl abi bgb s,

),&). ?\` cRPa\_R cRY\PVQNQe NPRYR_NPVp[R[ P\\_QR[NQN]\YN_R`

@ sb`bp bp jŠp k^qro^i abp`of_fo ilp mrkqlp ab rk^ `ros^ mi^k^ bk `lloabk^,a^p mli^obp nrb bk `lloabk^a^p ob`q^kdri^obp-Orbpql nrb i^p `lloabk^a^p ob`q^k,drH^obp%r)s& bpqŠkifd^a^p ^ i^p mli^obp l u _ mlo i^p b`r^`flkbp

r < l `lp ` * t < l pbk` *

Page 223: Calculus

Ijn q`^ojm`n q`gj^d_\_ v \^`g`m\^d‡i `i ^jjm_`i\_\n kjg\m`n 44/

n< sd * te

G

8 t < mpbk `&) &`&&&&`&`&``&

`& s < oblp `&0 s

t

j s

EHFTQ@ 03-04 @jjm_`i\_\n kjg\m`n, EHFTQ@ 03-05 Ijn q`^ojm`n pido\mdjnp* v ..-$

bi sb`qlo ab mlpf`fŽk m < sd * te nrb rkb bi lofdbk `lk 'u+ u( sfbkb a^al mlo

m < m`lp ` d* mpbk` d < m&lp ` d * pbk ` g( +

pfbkal l < Ghnhh-Dk i^ cfdro^ 03-04 pb ^mob`f^bp^ obi^`fŽk-Di sb`qlo `lp ` d* pbk `&d bp rk sb`qlo ab ilkdfqra rkfa^a nrb qfbkb i^

jfpj^ afob``fŽk nrb m, Dpqbsb`qlo rkfq^ofl pb abpfdk^ `loofbkqbjbkqb mlo p9 X i^b`r^`fŽk ^kqboflo pb bp`of_b ^pŒ9

alkab QQ < BNR `&d* pbk `e,

Blksfbkb q^j_f‹k fkqolar`fo rk sb`qlo rkfq^ofl pj% mbombkaf`ri^o ^ pQ% nrb pbabcfkb `ljl pfdrb9

^QQ ` , `& †Qi < , < ,pbk . * `lp F+

JK

N_pbosbpbnrb qbkbjlp

_pj < ,`lp ` c* pbkbg < *QP

JK

Dk bi bpqrafl ab i^p `ros^p mi^k^p+ilp alp sb`qlobp rkfq^oflp QQ X Qj abpbjmb•^kbi jfpjl m^mbibk `lloabk^a^p mli^obp nrb ilp sb`qlobp rkfq^oflp cv d bk `lloab,

Page 224: Calculus

//+ @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

k^a^p ob`q^kdri^obp-K^ cfdro^ 03-05 jrbpqo^ ilp sb`qlobp rkfq^oflp p9 v p6 ifd^alp^ i^ `ros^+ bk ^idrkl ab prp mrkqlp-

Rrmlkd^jlp ^elo^ nrb i^p `lloabk^a^p mli^obp l v ` plk crk`flkbp ab o*mlo bgbjmil l < `%n&)%&< a%n&+Cbar`fobjlp cŽojri^p m^o^ bumobp^oilp sb`qlobpsbil`fa^a v ^`bibo^`fŽk bk crk`fŽk ab p*v p6Š O^o^bi sb`qlo ab mlpf`fŽk+qbkbjlp

m < mRm< a&o'pm ,

Orbpql nrb ` abmbkab abi m^oŠjbqol o*il jfpjl ib l`roob ^i sb`qlo rkfq^ofl p*v e^v nrb qbkboil bk `rbkq^ `r^kal pb `^i`ri^ bi sb`qlo sbil`fa^a- @pŒmrbp qb,kbjlp

_m _&mpm' _m _p9q < , < ,, < , Qm* m+ ,

_o _o _o _o

Tqfifw^kal i^ obdi^ ab i^ `^abk^+ mlabjlp bumobp^o _p,d_o bk crk`fŽk ab p6

bp`of_fbkal

'03-13(_p* _` _p9 _`+;++;+R6%_o _o _` _o

v i^ b`r^`fŽk m^o^bi sb`qlo sbil`fa^a pb `lksfboqb bk

'03-14(_m _L

q;+p)m+p,_o n _o 6

Klp c^`qlobp bp`^i^obp _m-_o v m_†o_o nrb jriqfmif`^k ^ p9 u p6 pb ii^j^k+obpmb`qfs^jbkqb+^jhkji`io`n m\_d\g u om\inq`mn\g abi sb`qlo sbil`fa^a-

Orbpql nrb p9 v p6 plk sb`qlobp rkfq^oflp loqldlk^ibp+ bk`lkqo^jlp nrb

&_mY0 &_LV

q , q < _m- * m_o F $

`lk il nrb i^ sbil`fa^a q sfbkb a^a^ mlo i^ cŽojri^

G&_m'0 &_`'0q< , * m+ ,_o _o

Cbofs^kal ^j_lp jfbj_olp ab '03-14(+ bk`lkqo^jlp nrb bi sb`qlo ^`bibo^,`fŽk qfbkb i^ bumobpfŽkpfdrfbkqb

\ < &_0mp _m_Rm' * &m_

0` p * _m _` R6 * m_` _R6'

_o0 n * _o _o _o0 7 _o _o _o _o

Page 225: Calculus

Jjqdhd`ioj kg\ij ^ji \^`g`m\^d‡i m\_d\g 552

K^ abofs^a^ _pmg_omrbab bumobp^opbbk crk`fŽk ab Qj mlo jbafl '03-13(- @kŠil,d^jbkqb mlabjlp bumobp^oi^ abofs^a^ ab Qj mlo i^ b`r^`fŽk

_pj _` _pj _`,<,,< ++Rmj

_o _o _` _o

Dpql klp iibs^ ^ i^ pfdrfbkqbcŽojri^ nrb bumobp [ bk crk`fŽk ab prp `ljmlkbkqbpo^af^i u qo^kpsbop^i9

'03-15( \ < &_0m [ m&_`Y0'p * &m_

0` * 1 _m _`'pj,

_o0 _o F$ _o0 _o _o

Br^kal %d< o*i^ `ros^ mrbab abp`of_fopbjbaf^kqb i^ b`r^`fŽk mli^o l < a`L',Dk bpqb`^pl+ i^p cŽojri^p m^o^ i^ sbil`fa^a+ v ilp sb`qlobp sbil`fa^a v ^`bibo^,`fŽk pb pfjmifcf`^k `lkpfabo^_ibjbkqb+ v l_qbkbjlp

_mq < , p* * mpj*JK

),&)/ @\cVZVR[a\]YN[\ P\[ NPRYR_NPVp[_NQVNY

Di sb`qlo ^`bibo^`fŽk pb ii^j^ m\_d\g pf bi `ljmlkbkqb qo^kpsbop^ibk i^ fdr^i,a^a '03-15( bp pfbjmob `bol- Dpqb`ljmlkbkqb bp fdr^i ^

Olo `lkpfdrfbkqb+ bi sb`qlo ^`bibo^`fŽk bp o^af^i pf v pŽil pf l0 _ &eF_o bp `lkpq^kqb-Di jlsfjfbkql mi^kl `lk ^`bibo^`fŽk o^af^i qfbkb rk^ fkqbomobq^`fŽkdblj‹,

qof`^ fkqbobp^kqbbk crk`fŽk abi Šob^- Cbpfdkbjlp `lk =%n&bi Šob^ ab i^ obdfŽk_^oofa^ mlo bi sb`qlo mlpf`fŽk bkqobrk fkpq^kqbo < [ u rk fkpq^kqbmlpqbofloo,Dk i^ cfdro^ 03-06 i^ obdfŽkplj_ob^a^ bp rk bgbjmil- Cbjlpqo^objlp nrb bi `lb,cf`fbkqb ab s^of^`fŽk fkpq^kqŠkbl ab bp^ Šob^ bp bu^`q^jbkqb fdr^i ^ xm0_`g_o,Dpql bp+qbkbjlp

'03-16( >%&o' < 0-m/ __` ,0 n

Cb bpql obpriq^ nrb bi sb`qlo ^`bibo^`fŽk bp o^af^i pf u pŽil pf bi sb`qlo mlpf`fŽk_^oobbi Šob^+ab j^kbo^ nrb bi Šob^_^oofa^ pb^ molmlo`flk^i ^i qfbjml bjmib^al

Page 226: Calculus

//- @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

O^o^ abjlpqo^o '03-16(+ prmlkbjlp nrb bp mlpf_ib bifjfk^o o bkqob i^p alpb`r^`flkbp l < `%n&)%&< a%n&)v bk `lkpb`rbk`f^ bumobp^ol `ljl crk`fŽk ab `*mlkd^jlp m< N%_&+ Dpql pfdkfcf`^ nrb bufpqb rk^ crk`fŽk ob^i N q^i nrbNWa%n&Y< `%n&+Dkqlk`bp i^ obdfŽk plj_ob^a^ ab i^ cfdro^ 03-06 bp bi `lkgrkqlo^af^i ab N bk bi fkqbos^il Wa%[&)a%n&Y+Rbd•k bi qblobj^ 1-5+bi Šob^ ab bp^ ob,dfŽk sfbkb a^a^ mlo i^ fkqbdo^i

m8=%n&< p N0%_& _ +i %[&

Cbofs^kal bpq^ fkqbdo^i+pbd•k bi mofjbo qblobj^ crka^jbkq^i abi BŠi`ril v i^obdi^ ab i^ `^abk^+ bk`lkqo^jlp nrb

>%&o'< z Gv0Xb&o'Zb%&o'< z a0&o'b%&o' < z+1zz-+

il `r^i abjrbpqo^ '03-16(-

),&)0 8\\_QR[NQN`PVYo[Q_VPN`

Rf i^p `lloabk^a^p r b t ab rk mrkql L%r) t* w( abi bpm^`fl pb prpqfqrvbkmlo i^p `lloabk^a^p mli^obp m u K) ilp qobpk•jbolp m,,z* w pb abkljfk^k ^jjm_`+

u

u

t

s

EHFTQ@ 03-06 Bg q`^ojm _` jimn]c•h ]\mm` `gƒm`\ ^ji `g ^j`ad^d`io` _` q\md\^d‡i

EHFTQ@ 03-07 @jjm_`i\_\n ^dg…i_md^\n,

. ^K=$%n&< , m0[

1 _o %

Page 227: Calculus

Be`m^d^djn 554

i\_\n ^dg…i_md^\nabi mrkql M, Di k•jbol kl kbd^qfsl m obmobpbkq^elo^ i^ afp,q^k`f^ abi mrkql M ^i bgb w+q^i `ljl pb fkaf`^ bk i^ cfdro^ 03-07- Klp mrkqlpabi bpm^`fl m^o^ilp nrb m bp `lkpq^kqb bnrfafpq^k abi bgbw v mlo q^kql mboqbkb`bk^ rk `fifkaol `fo`ri^o 'ab ^nrŒbi klj_ob ab `lloabk^a^p ^dg…i_md^\n',

O^o^ bpqraf^o `ros^p \g\]`\_\n bk `lloabk^a^p `fiŒkaof`^p+i^ b`r^`fŽk abio^afl sb`qlo m pb e^ ab prpqfqrfomlo rk^ ab i^ cloj^9

m< mp9 * u&o'f ,

K^p cŽojri^p `loobpmlkafbkqbp m^o^ ilp sb`qlobp sbil`fa^a u ^`bibo^`fŽk pbl_qfbkbk pfk jŠp nrb prj^o ilp q‹ojfklp u%&o'fu u!&o'f* ^ ilp pbdrkalp jfbj_olpab i^p cŽojri^p _fafjbkpflk^ibp bk '03-14( u '03-15(-

),&)1 :WR_PVPV\`

0- Tk^ m^oqŒ`ri^ pb jrbsb bk bi mi^kl ab j^kbo^ nrb pr mlpf`fŽk bk bi fkpq^kqb o qfbkb`lloabk^a^p mli^obp m< o* .; o, G^ii^o i^p cŽojri^p m^o^ ilp sb`qlobp sbil`fa^a q v^`bibo^`fŽk \* v i^ `ros^qro^ G bk rk fkpq^kqb o `r^inrfbo^-

1- Tk^ m^oqŒ`ri^pb jrbsb bk bi bpm^`fl ab j^kbo^ nrb pr mlpf`fŽk bk bi fkpq^kqb n qfbkb`lloabk^a^p `fiŒkaof`^p m< o, .; o, w < o,^( Ool_^o nrb i^ `ros^ bpqŠ pfqr^a^ bk rk `lkl v e^ii^o rk^ b`r^`fŽk `^oqbpf^k^ m^o^bpqb `lkl 'i^ `ros^ pb abkljfk^ c„gd^` ^‡id^\',_( G^ii^o i^p cŽojri^p m^o^ i^ sbil`fa^a q* i^ ^`bibo^`fŽk \ v i^ `ros^qro^ F@ bk bifkpq^kqb o,`( G^ii^o rk^ cŽojri^ m^o^ abqbojfk^o bi Škdril nrb cloj^k i^ q^kdbqb ^ i^ `ros^ v i^dbkbo^qofw abi `lkl bk `^a^ mrkql ab i^:`ros^-

2- Tk^ m^oqŒ`ri^pb jrbsb bk bi bpm^`fl ab j^kbo^ nrb pr mlpf`fŽk bk bi fkpq^kqb o qfbkb`lloabk^a^p `fiŒkaof`^p m< pbk n) %&< o, w < ild pb` o* alkab N z o ; q6S-^( Ool_^o nrb i^ `ros^ bpqŠ pfqr^a^ bk rk `fifkaol ab b`r^`fŽk s0 * 'u \ z(1 <9 i-_( G^ii^o rk^ cŽojri^ 'bk crk`fŽk ab n&m^o^ abqbojfk^o bi Škdril nrb cloj^ bi sb`qlosbil`fa^a `lk e+

3- Rf rk^ `ros^ qfbkb i^ b`r^`fŽk mli^o m < a&L'*_ji_` \ x L x ] x \ * 16S+ abjlpqo^onrb i^ ilkdfqra ab ^o`l bp

4- K^ `ros^ ab b`r^`flk mli^o l < [%. * `lp K&) alkab [ = N v N z K w 16S+ pb ii^j^^\m_djd_`, So^w^o i^ doŠcf`^ ab Hz `^oaflfab m< 3'0 * `lp L' v `^i`ri^o i^ ilkdfqra abpr ^o`l-

5- Tk^ m^oqŒ`ri^pb jrbsb pfdrfbkal rk^ `ros^ mi^k^ `rv^ b`r^`fŽk mli^o bp l < _^6* alkab` bp rk^ `lkpq^kqb v K s^oŒ^ bkqob N v /$EP+^( G^`bo rk doŠcf`l fkaf`^kal i^ cloj^ ab i^ `ros^ m^o^ `^a^ rkl ab ilp pfdrfbkqbps^ilobp ab _7 _ < N- _ < 0+ _ < ,0-_( Cbpfdkbjlp mlo H%_&i^ ilkdfqra abi [l]i ab `ros^ v mlo [%_& bi Šob^ ab i^ obdfŽk_^oofa^ mlo bi sb`qlo ab mlpf`fŽk `r^kal '( s^oŒ^ ab N ^ 16S- B^i`ri^o H%_&v [%_& bkcrk`fŽk ab `,

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/// @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

6- So^w^o i^ `ros^ `rv^ b`r^`fŽk mli^o bp m< pbk, %F) N z %F w 16S+ X jlpqo^o nrb `lkpq^ab alp _r`ibp-^( G^ii^o bi Šob^ ab i^ obdfŽk ifjfq^a^ mlo rk _r`ib ab i^ `ros^-_( B^i`ri^o i^ ilkdfqra ab rk _r`ib ab i^ `ros^-Dk `^a^ rkl ab ilp bgbo`f`flp abi 7 ^i 00+obmobpbkq^oi^ `ros^ mi^k^ `rv^ b`r^`fŽk mli^opb a^ v `^i`ri^o i^ ilkdfqra ab pr ^o`l-

7- o < %F) N z %F w 6S- 0/- o < 0 * `lp %F) N z %F w 6S-

6+ m < `?* N z %F w 6S- 00- m < 0 , `lp %F) N z %F w 16S-

01- Rf rk^ `ros^ qfbkb i^ b`r^`fŽk mli^o m< a& _&) abjlpqo^o nrb pr o^afl ab `ros^qro^ ksfbkb a^al mlo i^ cŽojri^ k < &m0* m%0'1-0-gm0+ mm! * 0m%0/*alkab m%;o&`' v mjj;,o%&`',

02- O^o^ `^a^ rk^ ab i^p `ros^p ab ilp bgbo`f`flp abi 7 ^i 00+ `^i`ri^o bi o^afl ab `ros^qro^m^o^ bi s^ilo fkaf`^al ab `,^( Br^inrfbo '( bk bi bgbo`f`fl 7- b( 7 < 60& bk bi bgbo`f`fl 0/-_( Br^inrfbo '( bk bi bgbo`f`fl 8- a( 7 < z 60& bk bi bgbo`f`fl 00-

03- Cbpfdkbjlp mlo 0= bi Škdril+ N z 0= y 6S+ cloj^al mlo bi sb`qlo ab mlpf`fŽk v bi sb`qlosbil`fa^a ab rk^ `ros^- Rf i^ `ros^ bpqŠ bumobp^a^ bk `lloabk^a^p mli^obp+ abjlpqo^onrb p pbk 0= < o v p `lp 0= < ^l e ^c&) pfbkal p i^ sbil`fa^a-

04- Tk molvb`qfi `lebqb bpqŠ molvb`q^al ab j^kbo^ nrb afpm^o^al pb afofg^ afob`q^jbkqbe^`f^ bi _i^k`l- Cb_fal ^ c^iilp q‹`kf`lp+ pr afob``fŽk bk bi srbil bcb`qfsl cloj^ rkŠkdril cfgl j8 x N `lk i^ afob``fŽk abpab bi molvb`qfi ^i _i^k`l- Cbqbojfk^o i^ qo^vb`,qlof^ `r^kal pb afpm^o^ e^`f^ rk _i^k`l cfgl- Cfp`rqfo i^ cloj^ ab i^ qo^vb`qlof^ ^is^of^o l`- ƒ@i`^kw^oŠ bi molvb`qfi bi _i^k`l> 'Rrmlkbo nrb bi jlsfjfbkql pb ob^ifw^ bkrk mi^kl-(

05- Cb_fal ^ c^iilp jb`Škf`lp+ ilp q‹`kf`lp abi i^kw^jfbkql e^k mboafal bi `lkqoli ab rkmolvb`qfi `lebqb i^kw^al ob`fbkqbjbkqb- Rb p^_b nrb bi molvb`qfi pbdrfoŠ rk `ropl ob`qf,iŒkbl `lk sbil`fa^a `lkpq^kqb+ ab afob``fŽk abp`lkl`fa^- Br^kal bi molvb`qfi bpqŠ ^3 jfii^p ab afpq^k`f^ pb e^ il`^ifw^al rk fkpq^kqb v pb e^ mboafal ab krbsl- Hkjbaf^q^,jbkqb pb i^kw^ rk ^kqfmolvb`qfi `lk sbil`fa^a `lkpq^kqb qofmibnrb i^ abi mofjbol- ƒBrŠie^ ab pbo bi `ropl abi pbdrkal molvb`qfi m^o^ nrb ^i`^k`b ^i mofjbol> 'Rb prmlkb nrb^j_lp molvb`qfibp pb jrbsbk bk bi jfpjl mi^kl-(

06- Ool_^o nrb pf rk^ b`r^`fŽk afcbobk`f^i eljld‹kb^ ab mofjbo loabk ab i^ cloj^u&< `%r)u( pb bp`of_b bk `lloabk^a^p mli^obp+ pb obar`b ^ rk^ b`r^`fŽk pbm^o^_ib-@mif`^o bpqb j‹qlal m^o^ obplisbo u&< 'u , r&,%s * r&+

07- Tk^ m^oqŒ`ri^ 'jŽsfi bk bi bpm^`fl( qfbkb sbil`fa^a a^a^ mlo q < rf u m*alkab rbp rk^ `lkpq^kqb v m bp bi sb`qlo mlpf`fŽk- Ool_^o nrb i^ m^oqŒ`ri^pb jrbsb pl_ob rk^`fo`rkcbobk`f^ `lk sbil`fa^a ^kdri^o `lkpq^kqb r, 'K^ sbil`fa^a ^kdri^o bpqŠ abcfkfa^ mloE^5.^nf) alkab '( bp bi Škdril mli^o bk bi fkpq^kqb n+&

08- Tk^ m^oqŒ`ri^pb jrbsb bk rk mi^kl mbombkaf`ri^o ^i bgb u, Di jlsfjfbkql qfbkb ird^o^ il i^odl ab rk^ `fo`rkcbobk`f^ `lk `bkqol bk bpqb bgb-^( Ool_^o nrb bufpqb rk sb`qlo q%n&m^o^ibil ^i bgb u q^i nrb9

q&o'< r&o' t m&o'*

alkab m&o'v q&o' plk ilp sb`qlobp mlpf`fŽk v sbil`fa^a bk bi fkpq^kqb o, Di sb`qlo r&o'pb ii^j^ q`^ojm q`gj^d_\_ \ibpg\m v pr j^dkfqra r&o' < Ghs'p(00bp i^ q`gj^d_\_ \ibpg\m,_( Di sb`qlo \&o' < r%&o'pb ii^j^ sb`qlo \^`g`m\^d‡i \ibpg\m, Cbjlpqo^o nrb bi sb`qlo^`bibo^`fŽk j&o' Y<q%&o'Zsfbkb a^al mlo i^ cŽojri^

j&o' < Xr&o' , m&o'Zr&o'+ r/&o'm&o'* `u'q( t m&o',

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>kgd^\^dji`n \g hjqdhd`ioj kg\i`o\mdj 556

b( Rf i^ m^oqŒ`ri^ bpqŠ bk bi mi^kl rs v pf i^ sbil`fa^a ^kdri^o q%n& bp `lkpq^kqb+ pb^q%n&< q) abjlpqo^o nrb bi sb`qlo ^`bibo^`fŽk [%n&bp `bkqoŒmbqlv nrb+ [%n&< *q/l%n&+

1/- Rb af`b nrb rk `rboml bpqŠ pljbqfal ^ rk hjqdhd`ioj mdbd_j pf+m^o^ `^a^ m^o ab m^oqŒ`r,i^p k u l bk bi `rboml- i^ afpq^k`f^ Ghn0+'q(, ml&o'00 bp fkabmbkafbkqb ab o* alkab mk&o' uml&o' fkaf`^k ilp sb`qlobp mlpf`fŽk ab j v k bk bi fkpq^kqb o, Ool_^o nrb m^o^ rk `rbomloŒdfal bk bi nrb `^a^ m^oqŒ`ri^ dfo^ ^iobabalo abi bgb w pb qfbkb9 pj%n& :q%n& u lj%n&)alkab q%n&bp i^ jfpj^ m^o^ `^a^ m^oqŒ`ri^+v pj%n&bp i^ sbil`fa^a ab i^ m^oqŒ`ri^j+

),&*( 6]YVPNPV\[RNYZ\cVZVR[a\ ]YN[RaN_V\

Cbpmr‹p abi ^kŠifpfp ab do^k k•jbol ab a^qlp pl_ob jlsfjfbkql mi^kbq^ofl^`rjri^alp e^pq^ bi ^•l 05//+ bi ^pqoŽkljl ^ibjŠk Ile^kkbp Jbmibo '0460,052/(pb molmrpl abp`r_ofo i^p ibvbp j^qbjŠqf`^p nrb ofdbk bi jlsfjfbkql ab ilp mi^,kbq^p- Dkqlk`bp pb `lkl`Œ^k pbfp mi^kbq^p v pbd•k i^ qbloŒ^ab Blm‹okf`l pbprmlkŒ^nrb prp Žo_fq^pbpq^_^k pfqr^a^p bk bpcbo^p`lk`‹kqof`^p ^iobabalo abiRli- Jbmibo fkqbkqŽ abjlpqo^o nrb ilp o^aflp ab bpq^p bpcbo^pbpq^_^k obi^`fl,k^alp `lk ilp `fk`l mlifbaolp obdri^obp ab i^ FbljbqoŒ^- Rb ib l`roofŽ i^ fab^fkdbkflp^ ab nrb bi pfpqbj^ pli^o bpq^_^ `lkpqorfal `ljl rk oljmb`^_bw^p`efkl- Dk bi `bkqol abi pfpqbj^ pfqr^_^ bi Rli+ v abpmr‹p bk pr`bpfŽk `lil`^_^i^p pbfp bpcbo^p`lk`‹kqof`^p nrb mlaŒ^kfkp`of_fopbv `fo`rkp`of_fopb ^ ilp `fk`lmlifbaolp obdri^obp, l`q^baol+ f`lp^baol+ alab`^baol+ qbqo^baol v `r_l+ bk bpqbloabk 'ab abkqol e^`f^ crbo^(+ K^ bpcbo^ jŠp fkqbok^+fkp`ofq^ bk bi l`q^baolobdri^o+ `loobpmlkab ^ i^ Žo_fq^ ab Lbo`rofl- K^ nrb ib pbdrŒ^+fo`rkp`ofq^ ^il`q^baol b fkp`ofq^ ^i f`lp^baol+ `loobpmlkab ^ i^ Žo_fq^ ab Ubkrp- K^ Žo_fq^ab i^ Sfboo^ bpq^_^ bk i^ bpcbo^`fo`rkp`ofq^ ^i f`lp^baol b fkp`ofq^ ^i alab`^baol+v ^pŒpr`bpfs^jbkqb: i^ bpcbo^ jŠp buqbok^+`lkqbkfbkal i^ Žo_fq^ ab I•mfqbo+bpq^oŒ^fo`rkp`ofq^ ^i `r_l- @ mbp^oab nrb bpq^ qbloŒ m^ob`Œ^loob`q^ abkqolab rk 4 $ ab boolo+i^p l_pbos^`flkbp ^pqolkŽjf`^p bcb`qr^a^p bk bpqb mboŒlalpb e^`Œ^k `lk rk q^kql mlo `fbkql ab boolo jr`el jbklo+ v Jbmibo cfk^ijbkqbmbkpŽ bk jlafcf`^o bpq^ qbloŒ^-Cbpmr‹p ab jr`elp bpqraflp mlpqboflobppb ibl`roofŽ nrb ilp a^qlp l_pbos^alp obi^qfslp ^ Žo_fq^p`loobpmlkaŒ^k jŠp ^ qo^,vb`qlof^p `g…kod^\nnrb ^ i^p qo^vb`qlof^p `fo`ri^obp abi pfpqbj^ ab Blm‹okf`l-Efk^ijbkqb+ qo^p bpcrbowl fk`bp^kqb+Jbmibo afl qobpibvbp c^jlp^p+ abp`r_fboq^pbjmŒof`^jbkqb+ nrb bumif`^_^k qlalp ilp cbkŽjbklp ^pqolkŽjf`lp `lkl`falpe^pq^ bkqlk`bp- Rb mrbabk bkrk`f^o `ljl pfdrb9

Mmdh`m\g`t _` H`kg`m, Klp mi^kbq^pabp`of_bk Žo_fq^pbiŒmqf`^pbk rkl ab`rvlp cl`lp bpqŠbi Rli-

P`bpi_\ g`t _` H`kg`m, K^p Šob^p_^oofa^p mlo bi o^afl sb`qlo abpab bi Rli^ rk mi^kbq^ plk molmlo`flk^ibp ^i qfbjml-

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//1 @ƒg^pgj ^ji api^dji`n q`^ojmd\g`n

Q`m^`m\ g`t _` H`kg`m, Di `r^ao^al abi mboŒlal ab rk mi^kbq^ bp molmlo,`flk^i ^i `r_l ab pr afpq^k`f^ jbaf^ ^i Rli-

L]n`mq\^d‡i8 Olo k`m…j_j ab rk mi^kbq^ pb bkqfbkab bi qfbjml kb`bp^ofl m^o^ nrbob`loo^ rk^ sbw i^ Žo_fq^ biŒmqf`^-K^ _dno\i^d\ h`_d\ ^i pli bp i^ jfq^a ab i^ ilkdfqra abibgb j^vlo ab i^ bifmpb-

K^ clojri^`fŽk ab bpq^p ibvbp ^ m^oqfo abi bpqrafl ab q^_i^p ^pqolkŽjf`^pcrb rk eb`el jrv klq^_ib- Bbo`^ ab rklp 4/ ^•lp jŠp q^oab+ Mbtqlk mol_Žnrb i^p qobp ibvbp ab Jbmibo bo^k `lkpb`rbk`f^ ab pr pbdrka^ ibv abi jlsf,jfbkql v ab i^ `‹ib_ob ibv ab i^ do^sfq^`fŽk rkfsbop^i- Dk bpq^ pb``fŽk+ e^`fbkalrpl abi j‹qlal sb`qlof^i+ pb sboŠ `Žjl pb mrbabk abar`fo i^p ibvbp ab Jbmibo abi^p ab Mbtqlk-

Rli

EHFTQ@ 03-08 Bg q`^ojm kjnd^d‡i _`n_` `g Pjg \g kg\i`o\,

Rrmlkd^jlp nrb pb qfbkb rk Rli cfgl ab j^p^ I v rk mi^kbq^ jŽsfi ab j^p^g ^qo^Œalmlo bi Rli `lk rk^ crbow^ B+ 'Oobp`fkafjlp ab i^ fkcirbk`f^ ab lqo^p crbo,w^p-( K^ pbdrka^ ibv abi jlsfjfbkql ab Mbtqlk bpq^_ib`b nrb

'03-17( C < h\ *

alkab \ bp bi sb`qlo ^`bibo^`fŽk abi jlsfjfbkql abi mi^kbq^- Cbpfdkbjlp `lk mbi sb`qlo mlpf`fŽk abpab bi Rli ^i mi^kbq^ 'sbo cfdro^ 03-08(+ pb^k m< Ghnhhv p9 rksb`qlo rkfq^ofl `lk i^ jfpj^ afob``fŽk nrb m* pŒnrb m< mp9, K^ ibv ab i^ do^sf,q^`fŽk rkfsbop^i bpq^_ib`b nrb

C < ,E hJ1 3=&

m

alkab F bp rk^ `lkpq^kqb- Blj_fk^kal ‹pq^ `lk '03-17(+ l_qbkbjlp

'03-18( DJ[ < , ,1, ˆ, *

m

Page 231: Calculus

>kgd^\^dji`n \g hjqdhd`ioj kg\i`o\mdj //2

0/ nrb klp af`b nrb i^ ^`bibo^`fŽk bp m\_d\g, Cbjlpqo^objlp bk pbdrfa^ nrb i^Žo_fq^ bpqŠ bk rk mi^kl- Tk^ sbw p^_fal bpql+ pb abar`b fkjbaf^q^jbkqb ab ilpobpriq^alp ab i^ pb``fŽk 03-06 nrb bi Šob^ _^oofa^ mlo bi sb`qlo mlpf`fŽk bp mol,mlo`flk^i ^i qfbjml-

O^o^ abjlpqo^o nrb bi `^jfkl bpqŠ bk rk mi^kl rqfifw^jlp bi eb`el ab nrbo v \ plk m^o^ibilp- Rf fkqolar`fjlp bi sb`qlo sbil`fa^a q < _m- _o* qbkbjlp

_q _q _m _o W \ < o W , * q W q < o W , * , W q < , 'o W q' ,

_o _o _o _o

Orbpql nrb m W \ < N+ bpl pfdkfcf`^ nrb o W q bp rk sb`qlo `lkpq^kqb+ pb^n W q < `,

Rf ` < N+ bi sb`qlo mlpf`fŽk o bp m^o^ibil ^i q v bi jlsfjfbkql bp ob`qfiŒkbl-Orbpql nrb i^ qo^vb`qlof^ ab rk mi^kbq^ kl bp ob`qfiŒkb^+ab_b pbo ` x N- K^ obi^,`fŽk o W q < ` abjrbpqo^ nrb m 8 ` < N+ ^pŒnrb bi sb`qlo mlpf`fŽk bpqŠ bk rkmi^kl mbombkaf`ri^o ^ `, Orbpql nrb i^ ^`bibo^`fŽk bp o^af^i+ o _^oob bi Šob^ bkrk^ o^wŽk `lkpq^kqb bp ab`fo molmlo`flk^ijbkqb ^i qfbjml- Dpql abjrbpqo^ i^pbdrka^ ibv ab Jbmibo-

Dp cŠ`fi mol_^o nrb bp^ `lkpq^kqb ab molmlo`flk^ifa^a bp bu^`q^jbkqb i^ jf,q^a abi sb`qlo `- Dk bcb`ql+ pf rp^jlp `lloabk^a^p mli^obp v bumobp^jlp i^ sb,il`fa^a bk crk`fŽk ab p* v Qj `ljl bk i^ b`r^`fŽk '03-14(+ bk`lkqo^jlp nrb

'03-2/( , ' &_m _`' l _`` < n V q < omp9 V _o p* * n_o Rj < m8_o p* V pn *

v mlo q^kql Ghahh< Gn1_`-_og, Rbd•k '03-16( bpql bp fdr^i ^ 0 >%&o'F*alkab >%&o'bpi^ sbil`fa^a `lk i^ nrb bi o^afl sb`qlo _^oob bi Šob^ l q`gj^d_\_ \m`jg\m,

Dk i^ cfdro^ 03-1/ pb obmobpbkq^i^ pbdrka^ ibv ab Jbmibo- K^p alp obdflkbpplj_ob^a^p+ nrb plk _^oofa^p mlo bi sb`qlo mlpf`fŽk bk fkqbos^ilp ab qfbjmlfdr^ibp+ qfbkbk Šob^p fdr^ibp-

Cbjlpqo^objlp ^elo^ nrb bi `^jfkl bp rk^ bifmpb- @kqb qlal+ clojbjlp bimolar`ql sb`qlof^i \ W `* rqfifw^kal '03-18( v '03-2/(+ v bk`lkqo^jlp nrb

v^ nrb \ < _q- _o X Qj < _p,Z _L* i^ b`r^`fŽk ^kqboflo obi^qfs^ ^ \ W ` q^j_f‹kmrbab bp`of_fopb `ljl pfdrb9

J J_o &q V `' < _o &@Jp*' ,

Page 232: Calculus

56/ ?•f]ofi ]ih `oh]cih_m p_]nilc[f_m

Hkqbdo^kal l_qbkbjlp

q W b < DJpl * ] *

alkab \ bp lqol sb`qlo `lkpq^kqb- Olabjlp mlkbo bpq^ fdr^ia^a bk i^ cloj^

'03-20( q W b < DJ&pl * `'*

pfbkal DJ` < ], Blj_fk^objlp ‹pq^ `lk '03-2/( m^o^ bifjfk^o q u l_qbkbo rk^bumobpfŽk m^o^ m,@ q^i cfk jriqfmif`^jlp bp`^i^ojbkqb ^j_lp jfbj_olp ab '03-2/(mlo b u ^j_lp jfbj_olp ab '03-20( mlo m*Hdr^i^kal i^p alp bumobpflkbp abi mol,ar`ql jfuql m 8 q W b+ iibd^jlp ^ i^ b`r^`fŽk

'03-21( DJm&/ * `^jn 3• < ^/)

bk i^ nrb _ <hhah&9_ < E,_f,) v ;.= obmobpbkq^bi Škdril cloj^al mlo bi sb`qlo `lkp,q^kqb _ u bi o^afl sb`qlo l+ 'Ubo i^ cfdro^ 03-10-( Rf mlkbjlp ^ < ]0d%CI_&) i^b`r^`fŽk '03-21( pb qo^kpcloj^ bk

'03-22(`_

l:`^jn 3= * 0

l m < `& * m`lp 3• -

No_fq^ Cfob`qofw

`m`jn o-G

yG_ + m`jn o-G G

GGGG

_ +*G

EHFTQ@ 03-1/ P`bpi_\ g`t _` H`kg`m, I\n _jn Cdbpm\ 03-10 I\ m\u‡i m-&_+m`lp ;.• `nm`bdji`n njh]m`\_\n* ]\mmd_\n `i dio`mq\gjn g\ `s^`iomd^d_\_ _` ` < hGahh-

dbp\g`n _` od`hkj* od`i`i g\ hdnh\ ƒm`\,

Rbd•k bi qblobj^ 02-07+ ‹pq^ bp i^ b`r^`fŽk mli^o ab rk^ `lokb^ `lk bu`bkqk`f,a^a _ v rk cl`l bk bi Rli- K^ cfdro^ 03-10 jrbpqo^ i^ afob`qofw qo^w^a^ mbombkaf`r,

Page 233: Calculus

Be`m^d^djn _` m`k\nj 560

i^ojbkqb ^ ` ^ rk^ afpq^k`f^ ^ abi Rli- K^ afpq^k`f^ abpab bi mi^kbq^^ i^ afob`qofwbp _ + m `lp ;c=+ u i^ o^wŽkme&_+ m`jn ;c‚ bp i^ bu`bkqof`fa^a `, K^ `Žkf`^ bprk^ bifmpbpf ` ; 0+ rk^ m^oŠ_li^ pf ` < 0+ X rk^ efm‹o_li^ pf ` = 0- Orbpqlnrb ilp mi^kbq^pob`loobk `^jfklp `boo^alp+ i^ Žo_fq^nrb `lkpfabo^jlp ab_b pbork^ bifmpb-Dpql abjrbpqo^ i^ mofjbo^ ibv ab Jbmibo-

Efk^ijbkqb+ abarw`^jlp i^ qbo`bo^ibv ab Jbmibo- Rrmlkd^jlp nrb i^ bifmpbqfbkbbi bgbj^vlo ab ilkdfqra 0\ v bi bgbjbklo ab ilkdfqra 0], Di Šob^ab i^ bifm,pb pboŠmrbp i\]* Rb^ Q bi qfbjml bjmib^al mlo bi mi^kbq^bk a^o rk^ srbiq^ ^pr Žo_fq^ biŒmqf`^-Orbpql nrb bi sb`qlo mlpf`fŽk _^oob bi Šob^ `lk i^ sbil`fa^a^obli^o x^* qbkbjlp x^Q < i\] * l _fbk Q < 05Q\]e^, Prbobjlp abjlpqo^o nrbP0 bp molmlo`flk^i ^ \!,

Cb i^ pb``fŽk 02-11 abar`fjlp \0 < [0%. * _0'* _^ < [%. * _0

'* ^pŒnrb

^0 < DJ`_ < DJ\&/ + €&)

v qbkbjlp mlo q^kql

Orbpql nrb N bp bi molar`ql ab rk^ `lkpq^kqb mlo \%*bpql abjrbpqo^ i^ qbo`bo^ibv ab Jbmibo-

),&*) :WR_PVPV\`QR_R]N`\

0- Rb^ mbi sb`qlo nrb rkb bi lofdbk ^ rk mrkql ^o_fqo^ofl ab i^ m^oŠ_li^ v1 < s* pb^k N' biŠkdril nrb l cloj^ `lk i^ ob`q^ q^kdbkqb+N z N' z /Q* X ` bi Škdril nrb cloj^ l `lkbi bgb s mlpfqfsl+ N z '( z /Q, Dumobp^o &!bk crk`fŽk ab `,

1- Cbjlpqo^o nrb bi sb`qlo Q :s8 * 0^e bp q^kdbkqb ^ i^ m^oŠ_li^ v1 < 2^s bk bi mrkql%r)s&) v nrb bi sb`qlo J ::/]c * te bp mbombkaf`ri^o ^ P+

XFi_d^\^d‡i8 Dp`of_fo i^ b`r^`fŽk sb`qlof^i ab i^ m^oŠ_li^+ bjmib^kal t `ljlm^oŠjbqol-\

2- Cbjlpqo^o nrb i^ b`r^`iNk ab i^ ob`q^ ab mbkafbkqb h nrb bp q^kdbkqb ^ i^ m^oŠ_li^v1 < 2^s mrbab bp`of_fopb bk i^ cloj^ v z hs * ^gh* ƒBrŠibp plk i^p `lloabk^a^p abimrkql ab `lkq^`ql>

3- ^( Qbplisbo bi bgbo`f`fl 2 m^o^ i^ m^oŠ_li^ 'v , vl(1 < 2^&s + r))&+_( Qbplisbo bi bgbo`f`fl 2 m^o^ i^ m^oŠ_li^ r0 < 1]s) v+ bk dbkbo^i+ m^o^ i^ m^oŠ_li^%r * ri'0 < 1]%s * Xl(&

4- Cbjlpqo^o nrb i^ b`r^`fŽk ab rk^ ob`q^ q^kdbkqb ^ i^ m^oŠ_li^ v1 < 2^s bk bi mrkql'Wi )sx& mrbab bp`of_fopb bk i^ cloj^v v < /]%r * r &+

5- Qbplisbo bi bgbo`f`fl 4 m^o^ `^a^ rk^ ab i^p m^oŠ_li^p `fq^a^p bk bi bgbo`f`fl 3-

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/0+ @ƒg^pgj^ji api^dji`n q`^ojmd\g`n

6- ^( Rb^ L rk mrkql ab i^ m^oŠ_li^ v < s0Š Rb^ P bi mrkql ab fkqbopb``fŽk ab i^ ob`q^kloj^i bk L `lk bi bgb v- ƒBrŠi bp i^ mlpf`fŽk iŒjfqb ab P `r^kal L qfbkab e^`f^ bibgb v>_( Qbplisbo bi jfpjl mol_ibj^ m^o^ i^ `ros^ v < `%r&) pfbkal .&'/( < l-

7- K^ ob`q^ v < _ `loq^ ^ i^ m^oŠ_li^ v < r0 bk alp mrkqlp- G^ii^o bi o^afl ab i^ `fo`rk,cbobk`f^ nrb m^p^ mlo bplp alp mrkqlp v mlo bi s‹oqf`b ab i^ m^oŠ_li^- Di o^afl ab,mbkab ab ^, ƒPr‹ ib l`roob ^i o^afl `r^kal ` x N>

8- Cbjlpqo^o nrb rk mrkql &s +v ( bpqŠ _`iomj* `i l ap`m\ ab i^ bifmpb s0-\0 * t0-]0 < 01 1 l l / 0pbd•k nrb s1-\0 * tj-] pb^ h`ijm* dbp\ l h\tjm nrb -

0/- C^a^ rk^ bifmpb s0-\0 * t0-]0 < 0- Cbjlpqo^o nrb ilp sb`qlobp P v J a^alp mlo

Q V, s,< , _1& * ,::10+

plk+ obpmb`qfs^jbkqb+ o\ib`io` v ijmh\g ^ i^ bifmpb `r^kal pb ^mif`^k bk bi mrkql&s*t', Rf bi Škdril bu`‹kqof`l ab &sj%Xl( bp {.* abjlpqo^o nrb i^ q^kdbkqb bk &sj%Xl(qfbkb i^ b`r^`fŽk `^oqbpf^k^

s t, `lp }/ * , pbk }/ < 0 -\ ]

00- Cbjlpqo^o nrb i^ q^kdbkqb ^ i^ bifmpb s0-\0 * t0-]0 < 0 bk bi mrkql &sj &Xl( qfbkbmlo b`r^`fŽk sjs-\0 * tjt-]0 < 0-

01- Cbjlpqo^o nrb bi molar`ql ab i^p afpq^k`f^p ab ilp cl`lp ab rk^ bifmpb ^ rk^ ob`q^q^kdbkqb `r^inrfbo^ bp `lkpq^kqb+ pfbkal bp^ `lkpq^kqb bi `r^ao^al ab i^ ilkdfqra abipbjfbgb jbklo-

02- Rb qo^w^k alp ob`q^p q^kdbkqbp ^ i^ bifmpb s0 * 2t0 < 7+ m^o^ibi^p ^ i^ ob`q^ s * 1v < 6-G^ii^o ilp mrkqlp ab `lkq^`ql-

03- Tk^ `fo`rkcbobk`f^ m^p^ mlo ilp alp cl`lp ab rk^ bifmpb v bp q^kdbkqb ^ bii^ bk alpmrkqlp- G^ii^o i^ bu`bkqof`fa^a ab i^ bifmpb-

04- Rb^ R rkl ab ilp alp s‹oqf`bp ab rk^ efm‹o_li^ `rvl bgb qo^kpsbopl qfbkb ilkdfqra /[v `rv^ bu`bkqof`fa^a bp 1- Rb^ L rk mrkql pfqr^al bk i^ jfpj^ o^j^ nrb R+ Cbpfdkb,jlp `lk > bi Šob^ ab i^ obdfŽk ifjfq^a^ mlo i^ efm‹o_li^ v mlo bi pbdjbkql ob`qfiŒkblRL) v pb^ m i^ ilkdfqra ab RL+^( Blil`^o ilp bgbp `lloabk^alp bk rk^ mlpf`fŽk `lksbkfbkqb v bp`of_fo i^ b`r^`fŽk abi^ efm‹o_li^-_( Dumobp^o bi Šob^ > `ljl rk^ fkqbdo^i v+ pfk fkqbkq^o bi `Ši`ril ab bp^ fkqbdo^i+ ab,jlpqo^o nrb =l*a qfbkab ^ rk iŒjfqb `r^kal bi mrkql L qfbkab ^ R+ G^ii^o bpb iŒjfqb-

05- Cbjlpqo^o nrb ilp sb`qlobp P < %s,\0'9 * %r,[0&d v J < %r,[0'9 + %s,\0&dplk+ obpmb`,qfs^jbkqb+ q^kdbkqb v kloj^i ^ i^ efm‹o_li^ s0-\0 + t0-]0 < 0 pf pb ^mif`^k bk bi mrkql%r)u( ab i^ `ros^-

06- Cbjlpqo^o nrb i^ ob`q^ q^kdbkqb ^ i^ efm‹o_li^ s0-\0 + t0-]0 < 0 bk bi mrkql &sj &Xl(qfbkb `ljl b`r^`fŽk sjs-p0 + tjt-]0 < 0-

07- K^ ob`q^ kloj^i bk `^a^ mrkql ab rk^ `ros^ v i^ nrb rkb ^nrbi mrkql `lk bi lofdbkcloj^k rk qofŠkdril fpŽp`bibp `rv^ _^pb bpqŠ bk bi bgb s, Cbjlpqo^o nrb i^ `ros^ bprk^ efm‹o_li^-

08- K^ kloj^i bk rk mrkql L ab rk^ `ros^ `loq^ ^i bgb s bk W v ^i bgb v bk X- G^ii^o i^`ros^ pf `^a^ mrkql L bp bi mrkql jbafl abi `loobpmlkafbkqb pbdjbkql ob`qfiŒkbl WXv pf bi mrkql '3+ 4( bpqŠ bk i^ `ros^-

1/- Cbjlpqo^o nrb bi molar`ql ab i^p afpq^k`f^p abpab `r^inrfbo mrkql ab rk^ efm‹o_li^^ prp ^pŒkqlq^pbp `lkpq^kqb-

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Be`m^d^djn_` m`k\nj 562

10- K^ b`r^`flk mli^o ab rk^ `ros^ bp l < `fK&+ G^ii^o ` pf rk ^o`l `r^inrfbo^ nrb rk^alp mrkqlp afpqfkqlp ab i^ `ros^ qfbkb ilkdfqra molmlo`flk^i ^9 ^( ^i Škdril cloj^almlo ilp alp o^aflp sb`qlobp: _( i^ afcbobk`f^ ab i^p afpq^k`f^p abi lofdbk ^ ilp alpmrkqlp: `( bi Šob^ abi pb`qlo cloj^al mlo bi ^o`l v ilp alp o^aflp sb`qlobp-

11- Rf rk^ `ros^ bk bi bpm^`fl ab 2 afjbkpflkbp bpqŠ obmobpbkq^a^ mlo rk^ crk`fŽk sb`ql,of^i l abcfkfa^ bk rk fkqbos^il m^o^j‹qof`l W[) \Y) abjlpqo^o nrb bi molar`ql jfuqll$%n&+l%[& W l%\& bp `bol mlo il jbklp m^o^ rk s^ilo ab n bk %[)\&+ Hkqbomobq^obpqbobpriq^al dblj‹qof`^jbkqb-

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'*

:EC68=BE ?=A:6?:E

)-&) =[a_\QbPPVp[

@ 0/ i^odl ab bpqbif_ol ebjlp bk`lkqo^al jr`elp bgbjmilp ab l_gbqlp j^qb,jŠqf`lp nrb mrbabk prj^opb rklp `lk lqolp v jriqfmif`^opb mlo k•jbolp ob^ibp-@kqbqlal+ ilp k•jbolp ob^ibp plk l_gbqlp ab q^i k^qro^ibw^-Nqolp bgbjmilp plki^p crk`flkbp sb`qlof^ibp+ilp k•jbolp `ljmibglp+ i^p pbofbpv ilp sb`qlobp bk bibpm^`fl i+_dh`indji\g, Dk bpqb`^mŒqrilqo^q^jlp rk `lk`bmql j^qbjŠqf`l dbkbo^i+ii^j^al `nk\^dj gdi`\g* nrb fk`irvb qlalp bplp bgbjmilp v jr`elp lqolp `ljl`^plp m^oqf`ri^obp-

Aobsbjbkqb+ rk bpm^`fl ifkb^i bp rk `lkgrkql ab bibjbkqlp ab k^qro^ibw^`r^inrfbo^ pl_ob bi nrb mrbabk ob^ifw^opbfboq^plmbo^`flkbp ii^j^a^p \_d^d‡i vhpgodkgd^\^d‡i kjm iˆh`mjn, @i abcfkfo rk bpm^`fl ifkb^i kl bpmb`fcf`^jlp i^k^qro^ibw^ ab ilp bibjbkqlp kf ab`fjlp `Žjl pb ob^ifw^k i^p lmbo^`flkbp bkqobbiilp- Dk `^j_fl+ bufdfjlp nrb i^p lmbo^`flkbp qbkd^k `fboq^p molmfba^abp nrbqlj^jlp `ljl ^uflj^p ab rk bpm^`fl ifkb^i- U^jlp ^elo^ ^ e^`bo `lk abq^iib rk^abp`ofm`fŽkab bplp ^uflj^p-

)-&* 9RSV[VPVp[QRR`]NPV\YV[RNY

Rb^ S rk `lkgrkql kl s^`Œl ab l_gbqlp+ii^j^alp `g`h`iojn, Di `lkgrkql Spb ii^j^ bpm^`fl ifkb^i pf p^qfpc^`bilp afbw ^uflj^p pfdrfbkqbp nrb pb bkrk`f^kbk qobpdormlp+

>sdjh\n _` ^g\pnpm\@WHNL@ 0- BK@TRTQ@ QDRODBSN CD K@ @CHBHˆM- > oj_j k\m _` `g`h`iojn

s ` t _` S ^jmm`nkji_` pi `g`h`ioj ˆid^j _` S gg\h\_j nph\ _` s ` t* _`ndbi\_jjil r * t,

564

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565 Bnk\^djn gdi`\g`n

@WHNL@ 1- BK@TRTQ@ QDRODBSN CD K@ LTKSHOKHB@BHˆM ONQ M‰LDQNR QD@,

KDR- > oj_j s _` S u oj_j iˆh`mj m`\g \ ^jmm`nkji_` pi `g`h`ioj _` S gg\h\_jkmj_p^oj _` \ kjm s* _`ndbi\_j kjm \s,

>sdjh\n k\m\ g\ \_d^d‡i

@WHNL@ 2- KDX BNMLTS@SHU@- M\m\ oj_j s v oj_j v _` S* o`i`hjns * u < u * s,

@WHNL@ 3- KDX @RNBH@SHU@- @p\g`nlpd`m\ lp` n`\i s* v+ w _` S* o`i`hjn%r * u( * w < r * 'u * t&+

@WHNL@ 4- DWHRSDMBH@CD DKDLDMSN BDQN- Bsdno` pi `g`h`ioj `i S* _`+ndbi\_j ^ji `g n…h]jgj N+ o\g lp`

s)L;s k\m\ oj_j s _` S,

@WHNL@ 5- DWHRSDMBH@CD NOTDRSNR- M\m\ oj_j s _` S* `g `g`h`ioj &+g'sod`i` g\ kmjkd`_\_

s * ',i(u < N-

>sdjh\n k\m\ g\ hpgodkgd^\^d‡i kjm iˆh`mjn

@WHNL@ 6- KDX @RNBH@SHU@- M\m\ oj_j s _` S v oj_j k\m _` iˆh`mjnm`\g`n \ u ]* o`i`hjn

\&]s' < &\]'s ,

@WHNL@ 7- KDX CHRSQHATSHU@ O@Q@ K@ @CHBHˆM DM S, M\m\ oj_j s v oj_jt _` S u oj_j iˆh`mj m`\g \* o`i`hjn

\&s * t' < \s * \t ,

@WHNL@ 8- KDX CHRSQHATSHU@ O@Q@ K@ @CHBHˆM CD M‰LDQNR- M\m\ oj_js _` S u oj_j k\m _` iˆh`mjn m`\g`n \ u ]* o`i`hjn

&\* ]'s < \s * ]s ,,

@WHNL@ 0/- DWHRSDMBH@CD DKDLDMSN HC…MSHBN- M\m\ oj_j s _` S* o`i`+hjn Fs < s,

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Be`hkgjn _` `nk\^djn gdi`\g`n /00

Klp bpm^`flp ifkb^ibp ^pŒabcfkfalp+pb ii^j^k+ ^ sb`bp+bpm^`flp ifkb^ibp m`\g`nm^o^ obp^iq^obi eb`el ab nrb pb jriqfmif`^k ilp bibjbkqlp ab R mlo k•jbolpob^ibp-Rf bk ilp ^uflj^p 1+ 6+7 X 8 pb obbjmi^w^ iˆh`mj m`\g mlo iˆh`mj ^jh+kg`ej* i^ bpqor`qro^ nrb obpriq^ pb ii^j^ `nk\^dj gdi`\g ^jhkg`ej, @idrk^p sb`bprk bpm^`fl ifkb^i pb ii^j^ q^j_f‹k `nk\^dj q`^ojmd\g gdi`\g l pfjmibjbkqb `nk\^djq`^ojmd\g9 ilp k•jbolp rqfifw^alp `ljl jriqfmif`^alobp pb ii^j^k `n^\g\m`n, Tkbpm^`fl ifkb^i ob^i qfbkb k•jbolp ob^ibp `ljl bp`^i^obp: rk bpm^`fl ifkb^i `lj,mibgl qfbkb `ljl bp`^i^obp k•jbolp `ljmibglp- Rf _fbk `lkpfabo^objlp mofk`fm^i,jbkqb bgbjmilp ab bpm^`flp ifkb^ibp ob^ibp+qlalp ilp qblobj^p plk sŠifalp m^o^bpm^`flp ifkb^ibp `ljmibglp- Br^kal afd^jlp bpm^`fl ifkb^i pfk jŠp+ pb pl_obkqbk,aboŠ nrb bi bpm^`fl mrbab pbo ob^i l `ljmibgl-

)-&+ :WRZ]Y\QRR`]NPV\YV[RNYR`

Rf mob`fp^jlp bi `lkgrkql R v ab`fjlp `Žjl pb prj^k prp bibjbkqlp v `Žjlpb jriqfmif`^k mlo k•jbolp+ l_qbkbjlp rk bgbjmil `lk`obql ab bpm^`fl ifkb^i-Di ib`qlo cŠ`fijbkqb mrbab `ljmol_^o nrb `^a^ rkl ab ilp bgbjmilp pfdrfbkqbpp^qfpc^`bqlalp ilp ^uflj^p m^o^ rk bpm^`fl ifkb^i ob^i-

DIDLOKN 0- Rb^ R < B$ bi `lkgrkql ab qlalp ilp k•jbolp ob^ibp+v pb^ks * u u \s i^ ^af`fŽk u i^ jriqfmif`^`fŽk loafk^of^p ab k•jbolp ob^ibp-

DIDLOKN 1- Rb^ S < b bi `lkgrkql ab qlalp ilp k•jbolp `ljmibglp+ abcf,kfjlp s * v `ljl i^ ^af`fŽk loafk^of^ ab k•jbolp `ljmibglp+ v \s `ljl i^ jri,qfmif`^`fŽkabi k•jbol `ljmibgl s mlo bi k•jbol ob^i \, @rknrb ilp bibjbkqlp abR pb^k k•jbolp `ljmibglp+ ‹pqb bp rk bpm^`fl ifkb^i ob^i mlonrb ilp bp`^i^obpplk ob^ibp-

DIDLOKN 2- P`\ S < SiŠ bi bpm^`fl sb`qlof^i ab qla^p i^p k,mi^p ab k•jb,olp ob^ibp+ lk i^ ^af`fŽk v i^ jriqfmif`^`fŽk mlo bp`^i^obp abcfkfa^p bk i^ cloj^loafk^of^ bk crk`fŽk ab ilp `ljmlkbkqbp-

DIDLOKN 3- Rb^ R bi `lkgrkql ab qlalp ilp sb`qlobp Ri loqldlk^ibp ^ rksb`qlo kl kril a^al J+ Rf i < 1+ bpqbbpm^`fl ifkb^i bp rk^ ob`q^ nrb m^p^mlo N`lk J `ljl sb`qlo kloj^i- Rf s< 2+ bp rk mi^kl nrb m^p^mlo N `lk J `ljlsb`qlo kloj^i-

Klp pfdrfbkqbpbgbjmilp pb ii^j^k `nk\^djn api^dji\g`n, Klp bibjbkqlp ab Splk crk`flkbp sb`qlof^ibp+ `lk i^ prj^ ab alp crk`flkbp ` u d abcfkfa^p bk i^cloj^ loafk^of^9

%.* a&%r&< x%r&* a%r&

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/01 Bnk\^djn gdi`\g`n

m^o^qlal ob^i s bk i^ fkqbopb``fŽkab ilp aljfkflp ab Ev d- K^ jriqfmif`^`fŽk abrk^ crk`fŽk F mlo rk bp`^i^o ob^i \ pb abcfkb ^pŒ9\g bp ^nrbii^ crk`fŽk `rvl s^ilobk `^a^ s abi aljfkfl ab Ebp \g&s', Di bibjbkql `bol bp i^ crk`fŽk `rvlp s^ilobpplk krilp m^o^ qlal s, Di ib`qlo mrbab `ljmol_^o cŠ`fijbkqb nrb `^a^ rkl abilp `lkgrkqlp pfdrfbkqbpbp rk bpm^`fl crk`flk^i-

DIDLOKN 4- Di `lkgrkql ab qla^p i^p crk`flkbp abcfkfa^p bk rk fkqbos^ila^al-

DIDLOKN 5- Di `lkgrkql ab qlalp ilp mlifkljflp-

DIDLOKN 6- Di `lkgrkql ab qlalp ilp mlifkljflp ab do^al z i* pfbkal icfgl- 'Rfbjmob nrb `lkpfabobjlp bpqb`lkgrkql+ pb pl_obkqbkaboŠ nrb pfbjmob bpqŠfk`irfal bi mlifkljfl kril-( Di `lkgrkql ab qlalp ilp mlifkljflp ab do^al dbp\g^ i kl bp rk^ bpm^`fl ifkb^i mlonrb kl pb p^qfpc^`bkilp ^uflj^p ab `i^rpro^- Olobgbjmil+ i^ prj^ ab alp mlifkljflp ab do^al i mrbab kl pbo ab do^al i,

DIDLOKN 7- Di `lkgrkql- ab qla^p i^p crk`flkbp `lkqfkr^p bk rk fkqbos^ila^al- Rf bi fkqbos^il bp W[) \Y) abpfdk^jlp bpqbbpm^`fl `lk ?%[) \&+

DIDLOKN 8- Di `lkgrkql ab qla^p i^p crk`flkbp abofs^_ibp bk rk mrkql a^al-

DIDLOKN 0/- Di `lkgrkql ab qla^p i^p crk`flkbp fkqbdo^_ibpbk rk fkqbos^ila^al-

DIDLOKN 00- Di `lkgrkql ab qla^p i^p crk`flkbp Eabcfkfa^p bk bi mrkql 0pfbkal F&0( < N- Di k•jbol N bp bpbk`f^i bk bpqbbgbjmil- Rf obbjmi^w^jlp N mlork k•jbol kl kril `* sfli^jlp bi ^uflj^ ab `i^rpro^-

DIDLOKN 01- Di `lkgrkql ab qla^p i^p plir`flkbp ab rk^ b`r^`fŽk afcbobk`f^iDkb^i eljld‹kb^ v! * \t% * ]t < N+alkab \ v ] plk `lkpq^kqbp a^a^p- S^j_f‹k^nrŒbp bpbk`f^i bi N- Di `lkgrkql ab plir`flkbp ab rk^ b`r^`fŽk afcbobk`f^i kleljld‹kb^ kl p^qfpc^`bilp ^uflj^p ab `i^rpro^-

Dpqlp bgbjmilp v jr`elp lqolp e^`bk m^qbkqb Žjl bi `lk`bmql ab bpm^`flifkb^i bpqŠbuqbkafal mlo bi „idb_o^+ i^ FbljbqoŒ^ v bi @kŠifpfp-Br^kal pb abar`brk qblobj^ ab ilp ^uflj^p ab rk bpm^`fl ifkb^i+ l_qbkbjlp rk obpriq^al sŠifalm^o^ `^a^ bgbjmil `lk`obql- Tkfcf`^kal s^oflp bgbjmilp ab bpqbjlal+ `lkpbdrf,jlp rk `lkl`fjfbkql jŠp molcrkal bk `^a^ rkl- Dk l`^pflkbp bi `lkl`fjfbkqlab rk abqbojfk^al bgbjmil ^vra^ m^o^ ^kqf`fm^ol fkqbomobq^oobpriq^alp sŠifalpm^o^ lqolp bgbjmilp v mlkb bk bsfabk`f^ obi^`flkbp nrb ab lqol jlal mlaoŒ^km^p^ofk^asboqfa^p-

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@jin`^p`i^d\n `g`h`io\g`n _` gjn \sdjh\n 568

)-&, 8\[`RPbR[PVN RYRZR[aNYR`QRY\` NdV\ZN`

Klp qblobj^p nrb pfdrbk pb abar`bk cŠ`fijbkqb ab ilp ^uflj^p ab rk bpm^`flifkb^i-

SDNQDL@ 04-0- TMHBHC@C CDK DKDLDMSN BDQN- Bi ^p\glpd`m `nk\^dj gdi`\g`sdno` pi `g`h`ioj ^`mj v n‡gj pij,

A`hjnom\^d‡i, Di ^uflj^ 4 klp ^pbdro^ nrb bufpqbmlo 0/ jbklp rk bibjbkql`bol- Rrmlkd^jlp nrb bufpq^kalp+ pb^k /0 v K0* G^`fbkal s << /0 X N < N1 bkbi ^uflj^ 4+ l_qbkbjlp /0 * N1 < /0+ @kŠild^jbkqb+ e^`fbkal r < N1 XN < /0+ bk`lkqo^jlp N1 * /0 < N1+ Obol /0 * N1 < N1 * /0 mlo i^ ibv `lk,jrq^qfs^+ ^pŒnrb /0 < N1+

SDNQDL@ 04-1- TMHBHC@C CD DKDLDMSNR NOTDRSNR- Bi ^p\glpd`m `nk\^djgdi`\g oj_j `g`h`ioj od`i` `s\^o\h`io` pi jkp`noj, Bnoj `n* k\m\ oj_j s `sdno`pi t* v n‡gj pij o\g lp` s * v < N-

A`hjnom\^d‡i, Di ^uflj^ 5 klp af`b nrb `^a^ s qfbkb mlo 0/ jbklp rklmrbpql+ ^ p^_bo ',0 &r+ Rrmlkd^jlp nrb r qbkd^ alp lmrbpqlp+pb^k W0 b V0%Dk,qlk`bp s * X0 < N X s * X1 < N- Rrj^kal X1 ^ ilp alp jfbj_olp ab i^ mofjbo^fdr^ia^a v ^mif`^kal ilp ^uflj^p 4+ 3 X 2+ l_qbkbjlp nrb

U/ * %r * UE&< U/ * M < U/ )

v

U/ * %r* UE&< 'e * r& * UE < N * UE < Uc * N < UE +

Olo `lkpfdrfbkqb W0 < V0* `lk 0/ nrb s qfbkb bu^`q^jbkqb rk lmrbpql+ bi bibjbk,ql %*f&r+

Kjo\^d‡i, Di lmrbpql ab s pb abpfdk^ mlo +s, K^ afcbobk`f^ v , s pb abcfkb`ljl i^ prj^ v * ', t(+

Di qblobj^ pfdrfbkqb jrbpqo^ rk `lkgrkql ab molmfba^abp nrb ofdbk ilp`Ši`rilp ^idb_o^f`lp bibjbkq^ibp bk rk bpm^`fl ifkb^i-

SDNQDL@ 04-2- Bi pi `nk\^dj gdi`\g* _`ndbi`hjn ^ji s ` t _jn `g`h`iojn^p\g`nlpd`m\ v ^ji \ v ] _jn `n^\g\m`n ^p\g`nlpd`m\, Q`i`hjn `ioji^`n g\n kmj+kd`_\_`n ndbpd`io`n8

]( Kr < N-_( \L < N-

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/1) Bnk\^djn gdi`\g`n

a( ', \'s < , &\s' < \&+ t(+

a( Pd \s < N+ j ]d`i \ < M j s < N+ j gjn _jn,a( Pd \s < \t v \ ;/; N+ `ioji^`n s < v-b( Pd \s < ]s u s ;/; N+ `ioji^`n \ < ],c( , &s* u( < ' , s' * ' , u( < , s+t,e( s * s < 0s* s) s )s < 1s* v `i b`i`m\g* 6<0 s < is,

Cbjlpqo^objlp ^(+ _( u b( u abg^jlp `ljl bgbo`f`flp i^p abjlpqo^`flkbp ab i^plqo^p molmfba^abp-

A`hjnom\^d‡i _` ^(- Rb^ w< Ls, Cbpb^jlp abjlpqo^o nrb&w< N- Rr,j^kal w ^ pŒjfpjl u ^mif`^kal bi ^uflj^ 8+ bk`lkqo^jlp nrb

w* w < Ls * Ls < 'N * L's < Ls < u ,

Rrjbjlp ^elo^ +u ^ ^j_lp jfbj_olp v l_qbkbjlp u < N-

A`hjnom\^d‡i ab _(- Rb^ w< \L* prj^o w^ pŒjfpjl+ u ^mif`^obi ^uflj^ 7-

A`hjnom\^d‡i ab b(+ Rb^ w< &+\'s, Rrj^kal w ^ \s u ^mif`^kal bi ^ufl,j^ 8+ bk`lkqo^jlp nrb

u * \s < &+\'s * \s < &+\ * \'s < Ls < L*

^pŒnrb u bp bi lmrbpql ab \s* w< +&\s', @kŠild^jbkqb+ pf prj^jlp \&+s' ^\s v ^mif`^jlp bi ^uflj^ 7 v i^ molmfba^a _(+ bk`lkqo^jlp nrb \&+s' < +&\s',

)-&- :WR_PVPV\`

Dk ilp bgbo`f`flp abi 0 ^i 17+ abqbojfk^o pf `^a^ rkl ab ilp `lkgrkqlp a^alp bp rkbpm^`fl ifkb^i ob^i+ pf i^ ^af`fŽk u jriqfmif`^`fŽk mlo bp`^i^obp ob^ibp bpqŠ abcfkfa^ bki^ cloj^ rpr^i- O^o^ ^nrbiilp bk ilp nrb kl bp ^pŒ+ab`fo `rŠibp plk ilp ^uflj^p nrb kl pb`rjmibk- K^p crk`flkbp ab ilp bgbo`f`flp 0 ^i 06 plk ob^ibp- Dk ilp bgbo`f`flp 2+ 3 X 4+ `^a^crk`fŽk qfbkb rk aljfkfl nrb `lkqfbkb N v 0- Dk ilp bgbo`f`flp 6 ^i 01+ `^a^ aljfkfl `lk,qfbkb qlalp ilp k•jbolp ob^ibp-

0- Sla^p i^p crk`flkbp o^`flk^ibp-1- Sla^p i^p crk`flkbp o^`flk^ibp don,lk bi do^al ab /8# nrb bi do^al ab d 'fk`irvbk,

al .:-&+2- Sla^p i^p E `lk .%-&< .%.&+3- Sla^p i^p E`lk /,%-& :,%.&+4- Sla^p i^p E`lk .%.& < 0 * .%-&+5- Sla^p i^p crk`flkbp bp`^ilk^a^p abcfkfa^p bk ZN+0I-6- Sla^p i^p F bk i^p nrb F&s' x N `r^kal s x * ^j *7- Sla^p i^p crk`flkbp m^obp-8- Sla^p i^p crk`flkbp fjm^obp-

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Pp]`nk\^djn _` pi `nk\^dj gdi`\g 570

0/- Sla^p i^p crk`flkbp ^`lq^a^p-00- Sla^p i^p crk`flkbp `ob`fbkqbp-01- Sla^p i^p crk`flkbp `lk mboŒlal /$EP+02- Sla^p i^p E fkqbdo^_ibp bk Z/+0\ `lk i E%r&^r< N-

03- Sla^p i^p Efkqbdo^_ibp bk Z/+0\ `lk ia&s'_s x N-04- Sla^p i^p E nrb p^qfpc^`bk E%r&< iN , r& m^o^ qlal r)05- Slalp ilp mlifkljflp ab S^vilo ab do^al z i m^o^ rk i cfgl 'fk`irvbkal bi mlifkl,

jfl `bol(-06- Sla^p i^p plir`flkbp ab rk^ b`r^`fŽk afcbobk`f^i ifkb^i eljld‹kb^ ab pbdrkal loabk

v! * L%r&s$* M%r&s < N+pfbkal L v O crk`flkbp a^a^p+ `lkqfkr^p m^o^ qlal s*07- Sla^p i^p pr`bpflkbp ob^ibp ^`lq^a^p-08- Sla^p i^p pr`bpflkbp ob^ibp `lksbodbkqbp-1/- Sla^p i^p pbofbp ob^ibp `lksbodbkqbp-10- Sla^p i^p pbofbp ob^ibp ^_plirq^jbkqb `lksbodbkqbp-11- Slalp ilp sb`qlobp %r)v+t& ab R2 `lk t < N-12- Slalp ilp sb`qlobp %r)v+ t& ab S 2 `lk r < N l v < N-13- Slalp ilp sb`qlobp %r)v+ t& ab Ud `lk v < 2r+14- Slalp ilp sb`qlobp %r)s)t& ab R

1`lk 0r(1s: 0+ w bb Gh+

15- Slalp ilp sb`qlobp %r)s)t& ab Ud nrb plk molar`qlp ab N+ 1+ 2( mlo bp`^i^obp-16- Slalp ilp sb`qlobp 'u+ v+t& ab S 2 `rvlp `ljmlkbkqbp p^qfpc^`bk rk pfpqbj^ ab qobp b`r^,

`flkbp ifkb^ibp ab i^ cloj^

17- Slalp ilp sb`qlobp ab Rj nrb plk `lj_fk^`flkbp ifkb^ibp ab alp sb`qlobp a^alp = v >+18- Rb^ S < Q*+ bi `lkgrkql ab ilp k•jbolp ob^ibp mlpfqfslp- Cbcfk^jlp i^ ~prj^‚ ab alp

bibjbkqlp r b v ab S `ljl pr molar`ql r* v 'bk bi pbkqfal loafk^ofl(+ v abcfk^jlp i^~jriqfmif`^`fŽk‚ ab rk bibjbkql r ab S mlo rk bp`^i^o _ mlkfbkal r|+ Cbjlpqo^o nrbS bp rk bpm^`fl ifkb^i ob^i `lk bi bibjbkql `bol-

2/- Cbjlpqo^o nrb bi ^uflj^ 0/ kl mrbab abar`fopb ab ilp lqolp ^uflj^p-

XFi_d^\^d‡i8 C^o rk bgbjmil ab rk `lkgrkql S `lk lmbo^`flkbp nrb p^qfpc^d^kilp ^uflj^p abi 0 ^i 8 mbol kl biiN-\

20- Rb^ R bi `lkgrkql ab qlalp ilp m^obp loabk^alp %r) )r<& ab k•jbolp ob^ibp- Dk `^a^ `^plabqbojfk^o pf R bp l kl rk bpm^`fl ifkb^i `lk i^p lmbo^`flkbp ab ^af`fŽk v jriqfmif`^,`fŽk mlo bp`^i^obp abcfkfa^p `ljl pb fkaf`^- Rf bi `lkgrkql kl bp rk bpm^`fl ifkb^i+fkaf`^o `rŠibp plk ilp ^uflj^p nrb kl pb `rjmibk-^( 'Wi&T/& * 'Xi %V0' < 'Wi * XH)T/ * U/&) [%rx )T/& < %[rf) L',_( 'Wi&T/& * 'uz)U/& < 'Wi * uz+N(+ [%Tf) T/& < %[rE) [T/&$`( 'Wi&T/& * 'uz+X1( < 'Wi&T/ * U/&) [%rf )T/& < %[rf) [T/&$a( 'Wi&T/& * 'Xi+ X1( < 'HWi* u10+ Guz* X10(+ [%rf) T/& < %E[rff) f[r/.&+

21- Cbjlpqo^o i^p m^oqbpab i^ a( ^ i^ e( abi qblobj^ 04-2-

)-&. EbOR`]NPV\QRb[ R`]NPV\YV[RNY

C^al rk bpm^`fl ifkb^i S pb^ R rk pr_`lkgrkql kl s^`Œl ab S, Rf R bp q^j,_f‹k rk bpm^`fl ifkb^i+ bkqlk`bp R pb ii^j^ np]`nk\^dj ab S, Di qblobj^ nrb pfdrb

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/1+ Bnk\^djn gdi`\g`n

a^ rk pbk`fiil `ofqbofl m^o^ abqbojfk^o pf rk pr_`lkgrkql ab rk bpm^`fl ifkb^ibp l kl rk pr_bpm^`fl-

RCMPCK? 04-3- P`\ R pi np]^jiepioj ij q\^…j _` pi `nk\^dj gdi`\g S,Q\g np]^jiepioj R `n pi np]`nk\^dj ndv n‡gj ndn\odna\^` gjn \sdjh\n _` ^g\pnpm\,

A`hjnom\^d‡i, Rf R bp rk pr_bpm^`fl+ p^qfpc^`bqlalp ilp ^uflj^p ab rkbpm^`fl ifkb^i+v mlo q^kql+bk m^oqf`ri^o+ilp ^uflj^p ab `i^rpro^-

Cbjlpqobjlp ^elo^ nrb pf R p^qfpc^`bilp ^uflj^p ab `i^rpro^+ p^qfpc^`bq^j_f‹k ilp lqolp- K^p ibvbp `lkjrq^qfs^ v ^pl`f^qfs^ m^o^ i^ ^af`fŽk '^uflj^p2 v 3( X ilp ^uflj^p m^o^i^ jriqfmif`^`fŽk mlo bp`^i^obp '^uflj^p abi 6 ^iql(pb p^qfpc^`bk^rqljŠqf`^jbkqb bk R mlonrb plk sŠifalp m^o^ qlalp ilp bibjbkqlpab R+E^iq^ `ljmol_^o ilp ^uflj^p 4 v 5+ i^ bu,fpqbk`f abi bibjbkql `bol bk R+v i^ bufpqbk`f^ab rk lmrbpql m^o^`^a^ bibjbkql ab R-

Rb^ s rk bibjbkql `r^inrfbo^ ab R- 'R qfbkbmlo 0/ jbklp rk bibjbkql v^ nrbkl bp s^`Œl-( Rbd•k bi ^uflj^ 1+\s bpqŠbk R m^o^qlal bp`^i^o \, Slj^kal \ < N+obpriq^ nrb Kr bpqŠbk R- Obol Kr < N+ bk sfoqra abi qblobj^ 04-2 ^(+ `lk 0/`r^i M C R+X pb p^qfpc^`bbi ^uflj^ 4- Slj^kal [ < , 0+sbjlp nrb %*.&rbpqŠbk R- Obol s * &+F's < N v^ nrb s v &+F's bpqŠk^j_lp bk S* ^pŒnrb bi^uflj^ 5 &pbp^qfpc^`bbk R- Olo `lkpfdrfbkqb R bp rk pr_bpm^`fl ab R+

CDEHMHBHˆM- P`\ R pi np]^jiepioj ij q\^…j _` pi `nk\^dj gdi`\g S, Ri`g`h`ioj s _` S _` g\ ajmh\

f

T < z ?xTx){zi

`i _ji_` Wi= ‘‘‘ + Te k`mo`i`^`i oj_jn \ R v Bi= ‘‘‘ + ?e nji `n^\g\m`n*n` _`ijhdi\^jh]di\^d‡i gdi`\g _` `g`h`iojn _` R- Bg ^jiepioj _` oj_\n g\n ^jh]di\^dji`ngdi`\g`n adido\n_` `g`h`iojn _` R n\odna\^` gjn \sdjh\n _` ^g\pnpm\ v kjm o\ioj`n pi np]`nk\^dj _` S, A`^dhjn _` `n` np]`nk\^dj lp` `noƒ b`i`m\_j kjm R+ jo\h]d„i g` gg\h\hjn g\ `iqjgq`io` gdi`\g _` R+v gj _`ndbi\hjn kjm I&P', Pd R`n q\^…j*_`adidhjn I&P' ^jhj vLw*`g ^jiepioj ^jino\ n‡gj _`g `g`h`ioj ^`mj,

Blkgrkqlp afpqfkqlpmrbabk dbkbo^o bi jfpjl pr_bpm^`fl- Olo bgbjmil+ bi bp,m^`fl R0 bpqŠdbkbo^al mlo `^a^ rkl ab ilp pfdrfbkqbp `lkgrkqlp ab sb`qlobp9uc)dv)uc)d)c* dv)uK) c)*c)d) *d) d* gy-Di bpm^`fl ab qlalp ilp mlifkljflp j%n&

ab do^al 9Qi bpqŠdbkbo^al mlo bi `lkgrkql ab i * 0mlifkljflp x0+o*x*,,, *q! y-S^j_f‹k bpqŠdbkbo^al mlo bi `lkgrkql x0+o-0* z.2+ --- +o!-&i * 0(y v mlo

x0+'0 * \n&) '0 * V+ --- +'0 * n&!v+Di bpm^`fl ab qlalp ilp mlifkljflp bpqŠdb,kbo^al mlo bi `lkgrkql fkcfkfql ab ilp mlifkljflp x0+o,x*, , , w,

@i iibd^o ^nrŒ prodbk ab jlal k^qro^i krjbolp^p mobdrkq^p-Olo bgbjmil+ƒnr‹ bpm^`flp mrbabk dbkbo^opbmlo rk k•jbol cfkfqlab bibjbkqlp ƒRf rk bpm^`flbpqŠdbkbo^al mlo rk k•jbol cfkfql ab bibjbkqlp+ `rŠi bp bi jbklo k•jbol abbibjbkqlp kb`bp^oflp> O^o^afp`rqfo bpq^p`rbpqflkbp v lqo^p `lk bii^p obi^`flk^a^p

Page 245: Calculus

@jiepiojn _`k`i_d`io`n ` di_`k`i_d`io`n `i pi `nk\^dj gdi`\g 461

fkqolar`fjlp ilp `lk`bmqlp ab _`k`i_`i^d\* di_`k`i_`i^d\* ]\n`n v _dh`ind‡i,X^ bk bi `^mŒqril 01 bk`lkqo^jlp bp^p fab^p ^i bpqraf^o bi bpm^`fl sb`qlof^i Rh

@elo^ s^jlp ^ buqbkaboi^p^ bpm^`flp ifkb^ibp ab qfml dbkbo^i-

)-&/ 8\[Wb[a\` QR]R[QVR[aR`R V[QR]R[QVR[aR`R[ b[ R`]NPV\YV[RNY

CDEHMHBHˆM- Ri ^jiepioj P _` `g`h`iojn _` pi `nk\^dj gdi`\g S n` gg\h\_`k`i_d`io` pf `sdno` pi ^jiepioj adidoj _` `g`h`iojn _dnodiojn _` P* Vh= ŠŠŠ + sd*t pi ^jmm`nkji_d`io` ^jiepioj _` `n^\g\m`n `h+ ŠŠŠ+ ^d* ij oj_jn ^`mj* o\g`n lp`

f

F@dUd < N-8.:

Bg ^jiepioj P n` gg\h\ di_`k`i_d`io` pf ij `n _`k`i_d`io`, Bi o\g ^\nj* ^p\g`n+lpd`m\ lp` n`\i gjn `g`h`iojn _dnodiojn Tx) +++ ) s* _` P t gjn `n^\g\m`n `h+ ŠŠŠ+ ]c)

f

x @}U} < My p pfzi

dhkgd^\ _f < Bw < --- < ?e < M -

Rf _fbk i^ abmbkabk`f^ u i^ fkabmbkabk`f^ plk molmfba^abpab ilp `lkgrkqlpab bibjbkqlp+ mlabjlp q^j_f‹k ^mif`^o bp^p abkljfk^`flkbp ^ ilp bibjbkqlpjfpjlp- Olo bgbjmil+ ilp bibjbkqlp ab rk `lkgrkql fkabmbkafbkqbpb ii^j^k bib,jbkqlp fkabmbkafbkqbp-

Rf O bp rk `lkgrkql cfkfql+i^ abcfkf`fŽk ^kqboflo bpqŠab ^`rboal `lk i^ a^a^bk bi `^mŒqril01 m^o^bi bpm^`fl RiŠ Ml l_pq^kqb+i^ abcfkf`fŽk a^a^ ^nrŒkl bpqŠobpqofkdfa^ `lkgrkqlp cfkfqlp-

DIDLOKN 0- Rf rk pr_`lkgrkql Q ab rk `lkgrkql P bp abmbkafbkqb+bi jfpjlP `n _`k`i_d`io`, Dpql bp iŽdf`^jbkqb bnrfs^ibkqb ^ i^ ^cfoj^`fŽk ab nrb qlalpr_`lkgrkql ab rk `lkgrkql fkabmbkafbkqbbp fkabmbkafbkqb-

DIDLOKN 1- Rf rk bibjbkql ab O bp bi molar`ql ab lqol mlo rk bp`^i^o+Rbp abmbkafbkqb-

DIDLOKN 2- Rf N D R+bkqlk`bp O bp abmbkafbkqb-

DIDLOKN 3- Di `lkgrkql s^`Œl bp fkabmbkafbkqb-

Dk bi `^mŒqril01 crbolk afp`rqfalp jr`elp bgbjmilp ab `lkgrkqlp abmbkafbk,qbpb fkabmbkafbkqbp-Klp bgbjmilp nrb ^ `lkqfkr^`fŽk pb `ljbkq^k+ firpqo^k bplp`lk`bmqlp bk bpm^`flp crk`flk^ibp- Dk `^a^ `^pl bi bpm^`fl ifkb^i crka^jbkq^i Sbp bi `lkgrkql ab qla^p i^p crk`flkbp ob^ibpabcfkfa^p bk i^ ob`q^ ob^i-

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DIDLOKN 4- Rb^k pg&o'< `lp! o* p0&o' < pbk! o* p\&o' < 0 m^o^ qlal k•jb,ol ob^i o, K^ fabkqfa^a mfq^dŽof` morb_^ nrb p* * Q0 + S^ < N+ ^pŒnrb i^p qobpcrk`flkbp p+*p0* p9 plk abmbkafbkqbp-

DIDLOKN 5- Rb^ Qe%n&< oe m^o^ f < N+ 0+ 1+ --- + v o ob^i- Di `lkgrkqlR < uoj* Sh+ Q0* ŠŠŠ w bp fkabmbkafbkqb-O^o^ abjlpqo^o bpql+_^pq^ abjlpqo^o nrbm^o^ `^a^ i ilp i * 0 mlifkljflp p,* p,* ,,, * Ri plk fkabmbkafbkqbp-Tk^ obi^,`fŽk ab i^ cloj^ I ?eQe < N pfdkfcf`^ nrb

'04-0(

m^o^ qlal ob^i o, Br^kal o < N+ bk`lkqo^jlp nrb Bl < N- Qbmfqfbkal bi mol`bpl+bk`lkqo^jlp nrb `^a^ `lbcf`fbkqb ?e bp `bol-

DIDLOKN 6- Rf ^i&--- + \,9 plk k•jbolp ob^ibp afpqfkqlp+i^p i crk`flkbpbumlkbk`f^ibp

plk fkabmbkafbkqbp-Olabjlp abjlpqo^o bpql mlo fkar``fŽk pl_ob i, Di obpriq^albp qofsf^i `r^kal i < 0- Olo `lkpfdrfbkqb+ prmlkd^jlp nrb bp sŠifa^ m^o^ i + 0crk`flkbp bumlkbk`f^ibp v `lkpfabobjlp ilp bp`^i^obp Ah+ ŠŠŠ+ `9 q^ibp nrb

'04-1(i

-K]e_[er < N-Q_R

Rb^ [y bi j^vlo ab ilp h k•jbolp ^i&--- +\ij Lriqfmif`^kal ^j_lp jfbj_olp ab'04-1( mlo `+\ I!$) l_qbkbjlp

'04-2(i

-K]e_%[e*[I&!$+: N-Q_R

Rf e :.: L+ bi k•jbol [e * [o bp kbd^qfsl- Olo `lkpfdrfbkqb+ `r^kal r w * NB bki^ b`r^`fŽk'04-2(+`^a^ q‹ojfkl `lk e:f:I qfbkab^ `bol v bk`lkqo^jlp nrb BL < N-Rrmofjfbkal bi q‹ojfkl L,‹pfjl ab '04-1( v ^mif`^kal i^ efmŽqbpfpab fkar``fŽk+bk`lkqo^jlp nrb `^a^ rkl ab ilp i + 0 obpq^kqbplbcf`fbkqbp `* bp `bol-

SDNQDL@ 04--4- P`\ R pi ^jiepioj di_`k`i_d`io` lp` ^jino\ _` f `g`h`iojn_` pi `nk\^dj gdi`\g S u n`\ I&P' `g np] `nk\^dj b`i`m\_j %kjmR- Bioji^`n oj_j^jiepioj _` f * 0 `g`h`iojn _` I&P' `n _`k`i_d`io`,

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?\n`n v _dh`ind‡i 574

A`hjnom\^d‡i, Br^kal S < Si* bi qblobj^ 04-4 pb obar`b ^i 01-7- Rf bu^jfk^jlp i^ abjlpqo^`fŽk abi 01-7 bk`lkqo^jlp nrb •kf`^jbkqb pb _^p^ bk bieb`el ab nrb Si bp rk bpm^`fl ifkb^i v kl bk lqo^ molmfba^a m^oqf`ri^o ab SiŠ

Olo `lkpfdrfbkqb i^ abjlpqo^`fŽk a^a^ m^o^ bi qblobj^ 01-7 bp sŠifa^ m^o^ rk

bpm^`fl ifkb^i S `r^inrfbo^-

)-&0 A^pbp u QVZR[`Vp[

CDEHMHBHˆM- Ri ^jiepioj adidoj P _` `g`h`iojn _` pi `nk\^dj gdi`\g S n`gg\h\ ]\n` adido\ _` S nd P `n di_`k`i_d`io` v b`i`m\ S, Bg `nk\^dj S `n _`_dh`ind‡i adido\ nd od`i` pi\ ]\n` adido\, A` jomj hj_j* S `n _` diadido\n _dh`i+

ndji`n,

SDNQDL@ 04-5- P`\ S pi `nk\^dj gdi`\g _` _dh`ind‡i adido\, Bioji^`n oj_\]\n` adido\ _` S od`i` `g hdnhj iˆh`mj _` `g`h`iojn,

A`hjnom\^d‡i, Rb^k P v Q alp _^pbp cfkfq^p ab S, Rrmlkd^jlp nrb P v Q`lkpq^k obpmb`qfs^jbkqb ab f u h bibjbkqlp- Orbpql nrb P bp fkabmbkafbkqb u bk,dbkao^ R) bi qblobj^ 04-4 klp af`b nrb qlal `lkgrkql ab e * 0 bibjbkqlp ab Rbp abmbkafbkqb- Olo `lkpfdrfbkqb+ qlal `lkgrkql ab jŠp ab f bibjbkqlp ab S bpabmbkafbkqb- X^ nrb P bp rk `lkgrkql fkabmbkafbkqb+ ab_b pbo h 8889e+Di jfpjlo^wlk^jfbkql `lk P v Q fkqbo`^j_f^a^p9 morb_^ nrb f 8889h, Olo il q^kql f < h,

CDEHMHBHˆM- Pd pi `nk\^dj gdi`\g S od`i` pi\ ]\n` _` i `g`h`iojn* `g `i+o`mj i n` gg\h\ _dh`ind‡i _` S, Bn^md]dhjn i < afj S,

DIDLOKN 0- Di bpm^`fl Ri qfbkb afjbkpfŽk i, Tk^ _^pb bp bi `lkgrkql abilp h sb`qlobp `lloabk^alp rkfq^oflp-

DIDLOKN 1- Di bpm^`fl ab qlalp ilp mlifkljflp j%n& ab do^al 999:i qfbkbafjbkpfŽk i * 0- Tk^ _^pb bp bi `lkgrkql ab i * 0 mlifkljflp x 0+ o*o!*,,, *oh

v+

Slal mlifkljfl ab do^al f: h bp rk^ `lj_fk^`fŽk ifkb^i ab bplp h * 0 mlif,kljflp-

DIDLOKN 2- Di bpm^`fl ab i^p plir`flkbp ab i^ b`r^`fŽk afcbobk`f^iv! , 0t% + 2v < N qfbkb afjbkpfŽk 1- Tk^ _^pb bpqŠ cloj^a^ mlo i^p alp crk,`flkbp ox%r&< `+s* o0%r&< _!) Sla^ plir`fŽk bp rk^ `lj_fk^`fŽk ifkb^i abbp^p alp9

DIDLOKN 3- Di bpm^`fl ab qlalp ilp mlifkljflp j%n& bp ab fkcfkfq^p afjbk,pflkbp- Di `lkgrkql fkcfkfql x0+ o*o!*,,, w dbkbo^ bpqb bpm^`fl v kfkd•k `lkgrkqladidoj ab mlifkljflp dbkbo^ bi bpm^`fl-

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/1/ Bnk\^djn gdi`\g`n

RCMPCK? 04-6- P`\ S pi `nk\^dj gdi`\g _` _dh`ind‡i adido\ ^ji afj S < i,P` od`i`8

]( @p\glpd`m ^jiepioj _` `g`h`iojn di_`k`i_d`io` _` S `n pi np]^jiepioj_` pi\ ^d`mo\ ]\n` k\m\ S,

_( @p\glpd`m ^jiepioj _` i `g`h`iojn di_`k`i_d`io`n `n pi\ ]\n` k\m\ S,

A`hjnom\^d‡i, K^ abjlpqo^`fŽk ab ^( bp fa‹kqf`^ ^ i^ ab i^ m^oqb _( abiqblobj^ 01-0/- K^ abjlpqo^`fŽk ab _( bp fa‹kqf`^ ^ i^ ab i^ m^oqb b( abi qblobj^01-0/-

Rb^ R rk bpm^`fl ifkb^i ab afjbkpfŽk i v `lkpfabobjlp rk^ _^pb `rvlpbibjbkqlp `g%,,, * `i pb qlj^k bk rk `fboql loabk- Tk^ q^i _^pb loabk^a^ i^ `lk,pfabo^jlp `ljl rk^ k,mi^ &`! ,* , * `i', Rf U D S* mlabjlp bumobp^o s `ljl rk^`lj_fk^`fŽk ifkb^i ab bplp bibjbkqlp _^pb9

'04-3(h

V < ^d`dŠ<-&

Klp `lbcf`fbkqbp bk bpq^ b`r^`flk abqbojfk^k rk^ k,mi^ ab k•jbolp '`h+ ŠŠŠ+ `j(nrb bpqŠ rkŒsl`^jbkqb abqbojfk^a^ mlo s, Dk bcb`ql+ pf qbkbjlp lqo^ obmobpbk,q^`fŽk ab s `ljl `lj_fk^`fŽk ifkb^i ab `g% ,,, * `i* mlo bgbjmil s < z0_,`* *obpq^kal ab '04+3( bk`lkqo^jlp nrb yh'b+ , _d'`d < N- Obol v^ nrb ilp bib,jbkqlp _^pb plk fkabmbkafbkqbp+ bpl fjmif`^ nrb AG< ^) m^o^ `^a^ c)`lk 0/ `r^i'_h+ ŠŠŠ+ _h& < &_! ,,, *_h&+

Klp `ljmlkbkqbp ab i^ k,mH loabk^a^ '`h+ ŠŠŠ+ ^h& abqbojfk^a^ mlo '04-3(pb ii^j^k ^jhkji`io`n _` s m`nk`^oj \ g\ ]\n` jm_`i\_\ &`g%,,, * `i',

)-&1 :WR_PVPV\`

Dk `^a^ rkl ab ilp bgbo`f`flp abi 0 ^i 0/+ R bp bi%r)v+w( ab S 2 `rvlp `ljmlkbkqbp p^qfpc^`bk i^ `lkaf`fŽkrk pr_bpm^`fl ab S 2& Rf 0/ bp+`^i`ri^o afj R-

0- s < N-0, s * t < N-1, s * t * w < N-1+s ;t,3, s < t < u*

`lkgrkql ab qlalp ilp sb`qlobpnrb pb a^- Cbqbojfk^o pf R bp

3+ s < t lo s < u*5, s0 + t0 < N-5+ s * t < 0-7, V < 0s u u < 1s,

.-+ s * t * u < N X s + t + u < N-

Rb^ O- bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ab do^al 9p: i* pfbkal i cfgl- Dk `^a^bgbo`f`fl abi 00 ^i 1/+ pb^ R bi `lkgrkql ab qlalp ilp mlifkljflp . ab m- nrb p^qfpc^`bk i^`lkaf`fŽk a^a^- Cbqbojfk^o pf R bp rk pr_bpm^`fl ab L h| Rf il bp+`^i`ri^o afj R-00- .'/( < N-01- .&'/( < N-

Page 249: Calculus

Mmj_p^ojndio`mdjm`n*nk\^djn `p^g…_`jn,Kjmh\n /10

02- c!'/( < N-.1+ .%-&* .&'/( < M+.2+ .%-&< .%.&+.3+ .%-&< .%/&+/5, Ebp m^o-.5+ Ebp fjm^o-/7, Ebp ab do^al 49f* pfbkal f ; i* l E< N-0., Ebp ab do^al e+ pfbkal e ; i* l E< N-10- Dk bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp j%n&+abp`of_fo bi pr_bpm^`fl bkdbk

ao^al mlo `^a^ rkl ab ilp pfdrfbkqbp lkgrkqlp ab mlifkljflp v abqbojfk^o pr afjbkpfŽk+'^( xi- od*o2w9 '_( vo*q2+oPw9 'b( vo*odw9 'a( xi * o*'i * q(0y-

11- Dk bpqbbgbo`f`fl+I&P' bp bi pr_bpm^`fl dbkbo^al mlo rk pr_`lkgrkql P ab rk bpm^`flifkb^i R+Cbjlpqo^o i^p molmlpf`flkbp ab i^ ^( ^ i^ b(-]( P p9:I&P',_( RfR o9: SR9: Tu pf P bp rk pr_bpm^`fl ab R+ bkqlk`bp H%O&o9: P+Dpq^molmfba^a pbbumobp af`fbkal nrb I&P' bp bi g_hil pr_bpm^`fl ab S nrb `lkqfbkb P,b( Tk pr_`lkgrkql P ab S bp rk pr_bpm^`fl ab S pf u pŽil pf I&P' < P,a( Rf R p9: P p9: R) bkqlk`bp H%O&p9:olc+b( Rf P X Q plk pr_bpm^`flp ab S* q^j_f‹k il bp R c!&[ Q,.& Rf O X P plk pr_`lkgrkqlp ab R) bkqlk`bp H%Oc!&[neO78$H%O&c!&[olc+d( C^o rk bgbjmil bk bi nrb H%O%&S(z H%O&c!&[H%P&+

12- Rb^ U bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibpabcfkfa^p bk i^ ob`q^ ob^i- Cbqbo,jfk^o pf `^a^ rkl ab ilp pfdrfbkqbppr_`lkgrkqlp ab R bp abmbkafbkqbl fkabmbkafbkqb-B^i`ri^o i^ afjbkpfŽk abi pr_bpm^`fl dbkbo^al mlo `^a^ `lkgrkql-'^( xi+oC*b!ii&y+\ x ], 'b( x`lp s* pbksw,'_( voC*soCw, 'c( x`lphu+pbkhsw,

'`( xi+GD+soCw, 'e( xi+`lp 1u+pbkhsw,'a( voC*soC* oqEy- 'f( xpbks* pbk0sw,'b( v`:m*+* `lpe sw, 'g( v`:mlp s* `+gg%pbksw,

13- Rb^k S rk bpm^`fl ifkb^i ab afjbkpfŽk cfkfq^+v P rk pr_bpm^`fl ab S, Cbjlpqo^o `^a^rk^ ab i^p molmlpf`flkbp pfdrfbkqbp-^( P bp ab afjbkpfŽk cfkfq^v afj P 49 afj S,_( afj P < afj S pf v pŽil pf P < S,b( Sla^ _^pb ab P bp m^oqbab rk^ _^pb ab S,a( Tk^ _^pb ab S kl `lkqfbkb kb`bp^of^jbkqb rk^ _^pb ab P,

04-0/ Oolar`qlp fkqboflobp-bpm^`flp br`iŒablp- Mloj^p

Dk i^ FbljbqoŒ^ br`iŒab^ loafk^of^+ ^nrbii^p molmfba^abpnrb `rbkq^k `lki^ mlpf_fifa^a ab jbafo ilkdfqrabp ab pbdjbkqlp ob`qfiŒkblpu Škdrilp cloj^alp mloob`q^p pb ii^j^k molmfba^abph„omd^\n,Dk krbpqol bpqrafl ab REE$abcfkfjlp i^pilkdfqrabp v ilp Škdrilp bk crk`fŽk abi molar`ql bp`^i^o- Prbobjlp ^elo^ buqbk,abo bp^p fab^p ^ bpm^`flp ifkb^ibp jŠp dbkbo^ibp-Oofjbol fkqolar`fobjlp rk^ db,kbo^ifw^`fŽk abi molar`ql bp`^i^o+ nrb ii^j^objlp& kmj_p^oj dio`mdjm*v irbdlabcfkfobjlp i^ ilkdfqra u bi Škdril zk crk`fŽk ab bpqbmolar`ql fkqboflo-

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/11 Bnk\^djn gdi`\g`n

Di molar`ql bp`^i^o s} v ab alp sb`qlobp s < &Ug< ŠŠŠ * si' b v < 'XH+ ‘-‘ Vi'ab Rh pb abcfkfŽ bk bi `^mŒqril 01 mlo i^ cŽojri^

'04-4(i

t 9V < UdV9,45*

Dk rk bpm^`fl ifkb^i dbkbo^i+bp`of_fjlp %r) u( bk ird^o ab r +u m^o^ilp molar`qlpfkqboflobp+v abcfkfjlp bi molar`ql ^ufljŠqf`^jbkqb v kl jbaf^kqb rk^ cŽojri^-Dpql bp+bpq^_ib`bjlp rk^p `fboq^p molmfba^abp nrb nrbobjlp nrb p^qfpc^d^kilpmolar`qlp fkqboflobpv i^p `lkpfabo^jlp `ljl \sdjh\n,

CDEHMHBHˆM- Ri `nk\^dj gdi`\g m`\g S od`i` pi kmj_p^oj dio`mdjmnd \ ^\_\k\m _` `g`h`iojn s ` u _` S ^jmm`nkji_` pi iˆh`mj m`\g ˆid^j &s*u( lp` n\odn+a\^` gjn ndbpd`io`n \sdjh\n ^p\g`nlpd`m\ lp` n`\i s* v+ w _` S v k\m\ oj_jn gjn`n^\g\m`n m`\g`n `-

0( %r)s& < %s)r&1( &s*s * w( < &s*s& * &s*w(2( ]%r)s& < %_r)s&3( %r)r& = N mc r vi9 M

&^jihpo\odqd_\_* l ndh`om…\',&_dnomd]podqd_\_*l gdi`\gd_\_',&\nj^d\odqd_\_*p cjhjb`i`d_\_',&kjndodqd_\_',

Tk bpm^`fl ifkb^i `lk rk molar`ql fkqboflo pb ii^j^ `nk\^dj m`\g `p^g…_`j,

L]n`mq\^d‡i8 G^`fbkal ` < N bk '2(- bk`lkqo^jlp nrb &L, u( < N m^o^ qlal u-

Dk rk bpm^`fl ifkb^i `ljmibgl+ rk molar`ql fkqboflo %r) u( bp rk k•jbol`ljmibgl nrb p^qfpc^`bilp jfpjlp ^uflj^p nrb ilp abi molar`ql fkqboflo ob^i+bu`bmql bi ab i^ pfjbqoŒ nrb pb obbjmi^w^ mlo i^ obi^`fŽk

'0&( 'u+s& < %s)r& )

pfbkal %s)r& bi `ljmibgl `lkgrd^al ab %s)r&+ Dk bi ^uflj^ ab eljldbkbfa^a+ bijriqfmif`^alo bp`^i^o ` mrbab pbo `r^inrfbo k•jbol `ljmibgl- Cbi ^uflj^ ab i^eljldbkbfa^a v '0&(+iibd^jlp ^ i^ obi^`fŽk

%r)_s&< %_s)r& < ]%s)r& < ]%r)s& +

Tk bpm^`fl ifkb^i `ljmibgl `lk rk molar`ql fkqboflopb ii^j^ `nk\^dj `p^g…_`j^jhkg`ej, '@ sb`bp pb rp^ q^j_f‹k i^ abkljfk^`fŽk ab `nk\^dj pido\mdj,' Tkbgbjmil bp bi bpm^`fl sb`qlof^i `ljmibgl Rh%?&_obsbjbkqb afp`rqfal bk i^ pb`l_†‡j 01-05-

@rknrb klp fkqbobp^kmofk`fm^ijbkqb ilp bgbjmilp ab bpm^`flp br`iŒablp ob^,ibp+ilp qblobj^p ab bpqb`^mŒqrilplk sŠifalp m^o^ bpm^`flp br`iŒablp `ljmibglp-

Page 251: Calculus

Mmj_p^ojn dio`mdjm`n*`nk\^djn `p^g…_`jn, Kjmh\n /12

Br^kal ab`fjlp bpm^`fl br`iŒabl pfk jŠp+ bkqbkabobjlp nrb mrbab pbo ob^i l`ljmibgl-

Di ib`qlo ab_fbo^ `ljmol_^o nrb `^a^ bgbjmil nrb pfdrb p^qfpc^`bqlalp ilp^uflj^p abi molar`ql fkqboflo-

DIDLOKN 0- Dk Si pb^ %r)u( < T$ s- bi molar`ql bp`^i^o loafk^ofl ab r b u-

DIDLOKN 1- Rf s < &Ug%s0' b V < 'Xi +V0' plk alp sb`qlobp ab R0* abcfkf,jlp %r) u( jbaf^kqb i^ cŽojri^

Dpqbbgbjmil mlkb ab j^kfcfbpql nrb mrbabk bufpqfojŠp ab rk molar`ql fkqboflobk rk bpm^`fl ifkb^i a^al-

DIDLOKN 2- Rb^ ?%[) \& bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibp `lk,qfkr^p bk rk fkqbos^il W[)\Y+ Cbcfk^jlp rk molar`ql fkqboflo ab alp crk`flkbp` v d `lk&i^ cŽojri^

&G*d( <G8a&o'b&o'_o,

Dpq^cŽojri^ bp ^kŠild^ ^ i^ b`r^`fŽk '04-4( nrb abcfkb bi molar`ql bp`^i^o ab alpsb`qlobp bk RiŠ Klp s^ilobp ab i^p crk`flkbp `%n&u a%n&abpbjmb•^k bi m^mbiabilp `ljmlkbkqbp s* b Xf u i^ fkqbdo^`fŽkbi ab i^ prj^-

DIDLOKN 3- Dk bi bpm^`fl ?%[) \&) abcfkfjlp

&G*d( ;G8 r&o'a&o'b&o'_o*

alkab s bp rk^ crk`fŽk mlpfqfs^cfg ab @&\*]', S^i crk`fŽk pb ii^j^ api^d‡i k`nj,Dk bi bgbjmil 2 qbkbjlp q%n&< 0 m^o^qlal o,

DIDLOKN 4- Dk bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp-abcfkfjlp

%F)d( < aj!! `+oy&o'b&o'_o ,

Cb_fal ^i c^`qlo bumlkbk`f^i+ bpq^ fkqbdo^i fjmolmf^ `lksbodb m^o^ qlal m^o abmlifkljflp ` u d-

Page 252: Calculus

/2) Bnk\^djn gdi`\g`n

SDNQDL@ 04-7- Bi pi `nk\^dj `p^g…_`j S* oj_j kmj_p^oj dio`mdjmn\odna\^`g\ _`ndbp\g_\_ _` @\p^ct+P^cr\mu8

x%r)s&./ Q %r) r&%s)s& k\m\ oj_j s u oj_j t `i S,

>_`hƒn* `g ndbij _` dbp\g_\_ `n qƒgd_j ndv n‡gj nds ` v nji _`k`i_d`io`n,

A`hjnom\^d‡i, Br^kal pb abjlpqoŽ bi obpriq^al ^kŠildl m^o^ ilp sb`qlobpbk s+ 'qblobj^ 01-2(+ pb efwl obp^iq^onrb i^ abjlpqo^`fŽk bo^ rk^ `lkpb`rbk`f^ab i^p molmfba^abpabi molar`ql bp`^i^o `fq^a^p bk bi qblobj^ 01-1 v kl pb efwlabmbkaboab i^ abcfkf`fŽk m^oqf`ri^orqfifw^a^ m^o^abar`fo bp^pmolmfba^abp-Olo ilq^kql+i^ jfpj^ abjlpqo^`fŽk bp sŠifa^ bk `r^inrfbo bpm^`fl br`iŒabl ob^i- Br^kal^mif`^jlp bp^ abjlpqo^`fŽk bk rk bpm^`fl br`iŒabl `ljmibgl+ l_qbkbjlp i^ abpf,dr^ia^a 'u+ t'&t* s' Q &s*s'&t*t'* nrb `lfk`fab `lk i^ abpfdr^ia^a ab B^r`ev,R`et^ow v^ nrb

%r)s&%s)r& < %r)s&%r)s& < x%r)s&./ †

DIDLOKN- @mif`^kal bi qblobj^ 04-7 ^i bpm^`fl ?%[) \& `lk bi molar`qlfkqboflo &a9d( < `wa&o'b&o'_o*bk`lkqo^jlp nrb i^ abpfdr^ia^a ab B^r`ev,R`et^owpb qo^kpcloj^ bk

Di molar`ql fkqboflo mrbab rqfifw^opbm^o^ fkqolar`fo bi `lk`bmql j‹qof`l abilkdfqra bk `r^inrfbo bpm^`fl br`iŒabl-

CDEHMHBHˆM- Bi pi `nk\^dj `p^g…_`j S* `g iˆh`mj ij i`b\odqj Ghthh_`ddid_jkjm g\ `^p\^d‡i

Yt[ < %r)W(0.1

n` _`ijhdi\ ijmh\ _`g `g`h`ioj s,

Br^kal i^ abpfdr^ia^a ab B^r`ev,R`et^ow pb bumobp bk crk`fŽk ab i^p klo,j^p+ qlj^ i^ cloj^

h't+u(0Q Yt[ fEsff +

Orbpql nrb bp mlpf_ib abcfkfo rk molar`ql fkqboflo ab jr`e^p j^kbo^p+ i^kloj^ ab rk bibjbkql abmbkaboŠabi molar`ql fkqboflobibdfal- Dpq^c^iq^ ab rkf,

Page 253: Calculus

Lmojbji\gd_\_ `i pi `nk\^dj `p^g…_`j 580

`fa^a bo^ ab bpmbo^o-Dpqb eb`el bp ^kŠildl ^i ab nrb mlabjlp ^pfdk^o k•jbolpafpqfkqlp ^ i^ jbafa^ ab i^ ilkdfqra ab rk pbdjbkql ob`qfiŒkbl a^al+ pbd•k i^bib``fŽk ab bp`^i^ l rkfa^a ab jbafa^- Di qblobj^ nrb pfdrb a^ i^p molmfba^abpcrka^jbkq^ibp ab i^p kloj^p nrb kl abmbkabk ab i^ bib``fŽk ab molar`ql fkqboflo-

RCMPCK? 04-8- Bi pi `nk\^dj `p^g…_`j* oj_\ ijmh\ od`i` g\n kmjkd`_\_`nndbpd`io`n k\m\ oj_jn gjn `g`h`iojn s ` v+ t oj_jn gjn `n^\g\m`n `9

^( Zu\ < M pf s < N-_( Zu\ = N pf s x M &kjndodqd_\_',b( Zbu\ < Ya[ Ghthh &cjhjb`i`d_\_',a( Zu * uhhQ: Zu\ * Ghuhh &_`ndbp\g_\_ omd\ibpg\m',

Bg ndbij _` dbp\g_\_ `n qƒgd_j `i g\ _`ndbp\g_\_ omd\ibpg\m pf v n‡gj pf s ` v nji_`k`i_d`io`n,

A`hjnom\^d‡i, K^p molmfba^abp ^(+ _( v b( pb abar`bk fkjbaf^q^jbkqb abilp ^uflj^p abi molar`ql fkqboflo- O^o^ abjlpqo^o a( l_pbosbjlp nrb

Efr * sff/ < %r* s) r * s& < 'u+ r& * %s)s& * %r)s& * %s)r& :

< GGth01* hGuh01* 'u+ s& * %r)s& +

K^ prj^ 'u+ u( * 'u+ s& bp ob^i- K^ abpfdr^ia^a ab B^r`ev,R`et^ow morb_^ nrb

E%r)u(0 R: GhthhGhuhhu nrb h't+u(0 R: GhthhGhuhh+^pŒnrb qbkbjlp

Zu* uh01Q: GGthh1* GGuh01* 100thhGhuhh< '00thh* Ghuhh(1-

Dpql abjrbpqo^ a(- Di pfdkl ab fdr^ia^a bk a( bp sŠifal pfbjmob nrb il pb^ bk i^abpfdr^ia^a ab B^r`ev,R`et^ow-

CDEHMHBHˆM- Bi pi `nk\^dj `p^g…_`j m`\g S* `g ƒibpgj ajmh\_j kjm _jn `g`+h`iojn ij ipgjn s ` v n` _`adi` ^jhj `g iˆh`mj ` _`g dio`mq\gj M Q: ` Q: 4P lp`n\odna\^` g\ `^p\^d‡i

'04-5( `lp %d< %r)t' ,Ghthhhhuhh

L]n`mq\^d‡i8 K^ abpfdr^ia^a ab B^r`ev,R`et^ow morb_^ nrb bi `l`fbkqb abi pb,drkal jfbj_ol ab '04-5( bpqŠ bk bi fkqbos^il Y,0+0[+pŒnrb bufpqb pŽil rk '( bkZN+%/QZ `rvl `lpbkl bp fdr^i ^i ab bpqb `l`fbkqb-

)-&)) B_a\T\[NYVQNQR[ b[ R`]NPV\RbPYoQR\

CDEHMHBHˆM- Bi pi `nk\^dj `p^g…_`j S* _jn `g`h`iojn s ` u n` gg\h\i jmoj+bji\g`n pf np kmj_p^oj dio`mdjm `n ^`mj, Ri np]^jidpioj P _` S `n pi ^jidpioj

Page 254: Calculus

/2+ Bnk\^djn gdi`\g`n

jmojbji\g nd &s*u( < M k\m\ oj_j k\m _` `g`h`iojn _dnodiojns ` u _` Q- Ri ^ji+epioj jmojbji\g n` gg\h\ jmojijmh\g nd^\_\ pij _` npn `g`h`iojn od`i` ijmh\ /,

Di bibjbkql `bol bp loqldlk^i ^ qlal bibjbkql ab S9 bp bi •kf`l bibjbkqlloqldlk^i ^ pŒjfpjl- Di pfdrfbkqb qblobj^ abjrbpqo^ rk^ obi^`fŽk bkqobloqldlk^,ifa^a v abmbkabk`f^-

SDNQDL@ 04-0/- Bi pi `nk\^dj `p^g…_`j S* oj_j ^jiepioj jmojbji\g _``g`h`iojn ij ipgjn `n di_`k`i_d`io`, Bi k\mod^pg\m*i pi `nk\^dj `p^g…_`j _`_dh`ind‡i adido\ ^ji afj S < i* oj_j ^jiepioj jmojbji\g lp` ^jino` _` i `g`+h`iojn ij ipgjn `n pi\ ]\n` k\m\ S,

A`hjnom\^d‡i, Rb^ R rk `lkgrkql loqldlk^i ab bibjbkqlp kl krilp ab S*v prmlkd^jlp nrb rk^ `fboq^ `lj_fk^`fŽk ifkb^i cfkfq^ab bibjbkqlp ab R bp `bol+pb^

f

I@dUc < /+8.:

alkab `^a^ rx D R- Eloj^kal bi molar`ql bp`^i^o ab `^a^ jfbj_ol mlo Tf vqbkfbkal bk `rbkq^ nrb &Ug% Ud' < N pf d 6&<0+ bk`lkqo^jlp nrb BH'WH+ r)& < N-Obol %r) ) Ug' 6&<N v^ nrb Ug 6&<M `lk il `r^i A0 < N- Qbmfqfbkal bi o^wlk^jfbkql`^j_f^kal Tf mlo r+) bk`lkqo^jlp nrb `^a^ _x < N-Dpql morb_^ nrb R bp fkabmbk,afbkqb- Rf afj S < i v pf R `lkpq^ ab i bibjbkqlp+ bi qblobj^ 04-6 _( abjrbpqo^nrb R bp rk^ _^pb m^o^R+

DIDLOKN- Dk bi bpm^`fl ifkb^i ob^i B'N+ 166!( `lk bi molar`ql fkqboflo'g+d( < `w!a&s'b&s'_s* pb^ R bi `lkgrkql ab i^p crk`flkbp qofdlklj‹qof`^p xrl+

S!

z1& ‘‘‘ y a^a^p mlo

pj&s' < 0+ R0i+/&U' < `lp is * p/h&s' < pbk is* m^o^ i < 0+1+----

Pd h 6&< i* qbkbjlp i^p obi^`flkbp ab loqldlk^ifa^a

^pŒnrb R bp rk `lkgrkql loqldlk^i- Orbpql nrb kfkd•k bibjbkql ab R bp bi bib,jbkql `bol+ R bp fkabmbkafbkqb-K^ kloj^ ab `^a^ bibjbkql ab O pb `^i`ri^ cŠ`fi,jbkqb- Sbkbjlp uoj * rl( < `w!^r < 166! v+m^o^ h w 0+qbkbjlp

01! 1&R0i+g%R0i+g' < l `lp is _s < 6S+ I\ 1

&R0i* R0i' < l pbk is _s < 6S-

Page 255: Calculus

Lmojbji\gd_\_ `i pi `nk\^dj `p^g…_`j 582

Olo `lkpfdrfbkqb+ Ghqkhh< U106 v 00 Ri 00 < U: m^o^ i ƒ 0- Cfsfafbkal `^a^ Ri mlopr kloj^+ l_qbkbjlp rk `lkgrkql loqlkloj^i x>IN+bmi+o1+&!y alkab>Ik<rj.hhqjhh-@pŒmrbp+ qbkbjlp

0`FFi%r&< - ., +

T 106

`lp is`FF/h*f%T& < s!!: +

n`iis`FF/h%T&< s!!: + m^o^ i ƒ 0 -

Dk i^ pb``fŽk 04-04 abjlpqo^objlp nrb qlal bpm^`fl br`iŒabl ab afjbkpfŽkcfkfq^ qfbkb rk^ _^pb loqldlk^i- Di qblobj^ nrb pfdrb jrbpqo^ `Žjl pb `^i`ri^kilp `ljmlkbkqbp ab rk bibjbkql obi^qfslp ^ rk^ q^i _^pb-

RCMPCK? 04-00- P`\ S pi `nk\^dj `p^g…_`j _` _dh`ind‡i adido\ i* v npkji+b\hjn lp` P < v`g* ,,, * `iw `n pi\ ]\n` jmojbji\g k\m\ S, Pd pi `g`h`ioj s `noƒ`skm`n\_j ^jhj pi\ ^jh]di\^d‡i gdi`\g _` gjn `g`h`iojn _` g\ ]\n`* n`\ „no\

'04-6(i

T < I^d`d*cwf

`ioji^`n npn ^jhkji`io`n m`g\odqjn \ g\ ]\n` jm_`i\_\ &`g%,,, * `i' qd`i`i _\_jnkjm g\n a‡mhpg\n

'04-7(&s*`9' 0 1

A+ < ,,,( k\m\ g< + +--- +i ,+ 'b-+`d

Bi k\mod^pg\m*nd P `n pi\ ]\n` jmojijmh\g* ^\_\ `* qd`i` _\_\ kjm

&/3,7' ]d < %r)_d& †

A`hjnom\^d‡i, Eloj^kal bi molar`ql fkqboflo ab `^a^ jfbj_ol ab '04+6(`lk `,* l_qbkbjlp

h

'u+ bI < •^d&`d*bI < ^g`d) `d&

f<i

mrbpql nrb &`y*`y' < M pf d ;/; {-Dpql fjmif`^ '04-7(+ v `r^kal &`y*`y' < 0+ l_qb,kbjlp '04-8(-

Rf v`g% ,,, * `iw bp rk^ _^pb loqlkloj^i+ i^ b`r^`fŽk '04-6( mrbab bp`of_fopbbk i^ cloj^

'04-0/(i

T < •&s* `d'`d,cwf

Page 256: Calculus

/2- Bnk\^djn gdi`\g`n

Di pfdrfbkqb qblobj^ morb_^ nrb bk rk bpm^`fl br`iŒabl ob^i ab afjbkpfŽkcfkfq^`lk rk^ _^pb loqlkloj^i bi molar`ql fkqbofloab alp bibjbkqlp bp fdr^i ^ i^prj^ ab ilp molar`qlp ab prp `ljmlkbkqbp-

RCMPCK? 04-01- P`\ S pi `nk\^dj `p^g…_`j m`\g _` _dh`ind‡i adido\ i*s npkjib\hjn lp` v`g%,,, * `iw `n pi\ ]\n` jmojijmh\g k\m\ S, M\m\ oj_j k\m _``g`h`iojn s ` u _` S* o`i`hjn

'04-00( !&s*t' < -1&s*`*'&t*b-( &C‡mhpg\ _` M\mn`q\g',f<i

Bi k\mod^pg\m*^p\i_j s < v+ o`i`hjn

'04-01(i

GGth01< -1 &s* `*'0,d;g

A`hjnom\^d‡i, Eloj^kal bi molar`ql fkqboflo ab ^j_lp jfbj_olp ab i^b`r^`fŽk '04-0/( `lk v+ v ^mif`^kal i^ molmfba^a ab ifkb^ifa^a abi molar`ql fkqb,oflo+l_qbkbjlp '04-00(- Br^kal s < v+ i^ b`r^`fŽk '04-00( pb obar`b ^ '04-01(-

L]n`mq\^d‡i8 K^ b`r^`fŽk '04-00( pb abkljfk^ `ljl pb fkaf`^ bk elklo abL- @- O^opbs^i '0665,0725 ^molufj^a^jbkqb( + nrb l_qrsl bpqb qfml ab cŽojri^ bk rkbpm^`fl crk`flk^i bpmb`f^i-

)-&)* :WR_PVPV\`

0- Rb^k r < 'u+ --- )r)& b v < 'X! --- +v+( sb`qlobp ^o_fqo^oflp ab S*, Cbqbojfk^o bk `^a^`^pl pf %r)u( bp rk molar`ql fkqboflo bk S! pf %r)u( bpqŠ abcfkfal mlo i^ cŽojri^ nrb pba^- Dk bi `^pl bk nrb %r)u( kl pb^ rk molar`ql fkqboflo+ ab`fo `rŠibp plk ilp ^uflj^pnrb kl pb p^qfpc^`bk-

i

']( %r)s& < ,0Ud GWeh-dxg &

i '/-0'a( %r)s& < cwf T8U4

'_( %r)s& <0 d UdVd f{i i

'b( %r)s& ;,0Ud,0Ve,cw. dw.

i i i

'b( &s*s& < -1 'u+ * Vd'0 + -1s9 + ,0V5,O5R O5R O5R

1- Rrmlkd^jlp nrb j^kqbkbjlp ilp qobp mofjbolp ^uflj^p abi molar`ql fkqboflo ob^i'pfjbqoŒ^+ifkb^ifa^a v eljldbkbfa^a( mbol obbjmi^w^jlp bi `r^oql ^uflj^ mlo rkl krb,sl '3&(9 'u+ r& < N pf u pŽil pf r < N- Cbjlpqo^o nrb l %r)r& = N m^o^ qlal r {:‹ NN _fbk 'u+ s': N m^o^ qlal s {:‹ N-

Page 257: Calculus

Ad_l]c]cim 584

WEh^c][]cƒh7 Rrmlkbo %r)r& = N m^o^ rk `fboql r w M X %s7s&; N m^o^ rk `fboqlv z N- Dk bi bpm^`fb dbkbo^al mlo xu+vy+e^ii^o rk bibjbkql w z N `lk %t)t& < N-\

Cbjlpqo^o nrb bk ilp bgbo`f`flp abi 2 ^i 6 `^a^ rk^ ab i^p molmlpf`flkbp bp sŠifa^ m^o^qlal m^o ab bibjbkqlp s b v ab rk bpm^`fl br`iŒabl ob^i-

1, %r)s& < N pf u p50/ pf EEr* vii < HHu, sff+1+ %r)s& < N pf v p50/ pf HHu* vi01 < HHuii1* iHvii1-3, %r)t' < N pf v p50/ pf EEr* ^t! x !uHHl]n] qlal b ob^i

3+ %r * s) r * s& < N pf u p50/ pfiiu! < HKsii-

6- Rf s b v plk bibjbkqlp kl krilp nrb cloj^k rk Škdril 7+ bkqlk`bp

Hiu, vQ1 < Tuii1 * HHvii1, 1 EErff.ivii `lp 7 -

7- Dk bi bpm^`fl ifkb^i ob^i B'i+ _&) abcfkfjlp rk molar`ql fkqboflo mlo

%`)a& <`7'ild r&`%r&a%r&r +

^( Rf `%r&< u&:+i`ri^o HHcii-

_( G^ii^o rk mlifkljfl ab mofjbo do^al a%r&< ^ * \r nrb pb^ loqldlk^i ^ i^ crk`f5k`lkpq^kqb `%r&< i-

8- Dk bi bpm^`fl ifkb^i ob^i @&+g* 0(+ pb^ %`)a& ;Pxg a&o'bR'_o, Blkpfabo^o i^p qobp crk,`flkbp p! p0 rc a^a^p mlo

p.&o' < 0 + R\&o' < 0 * o,

Cbjlpqo^o nrb alp ab bii^p plk loqldlk^ibp+ alp cloj^k bkqob pŒrk Škdril %Qm-1*v alpcloj^k bkqob pŒrk Škdril %Qm-4,

0/- Dk bi bpm^`fl ifkb^i L+ ab qlalp ilp mlifkljflp ob^ibp ab do^al z i* abcfkfjlp

^( Cbjlpqo^o nrb 'g+d( bp rk molar`ql fkqboflo m^o^ MŠ,_( B^i`ri^o 'g+d( `r^kal `Q& < o X a%n&< \o * \+b( Rf `un&< o*e^ii^o qlalp ilp mlifkljflp d loqldlk^ibp ^ `+

00- Dk bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp+abcfkfjlp &a*d( < `7$+oy&o'b&o'_o,^( Cbjlpqo^o nrb bp^ fkqbdo^i fjmolmf^ `lksbodb ^_plirq^jbkqb m^o^ qlalp ilp mlifkl,jflp ` u d-_( Rf r+%n&< o%m^o^ k < N+ 0+ 1+ --- + abjlpqo^o nrb 'u+ r ††&< 'j * h& +b( B^i`ri^o %`)d( `r^kal `%n&< %n* 0(1 v a%n&< n0 * 0-

a( G^ii^o qlalp ilp mlifkljflp ab mofjbo do^al a%n&< [ * ]o loqldlk^ibp ^ `Q&< 0* o,01- Dk bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp+ abqbojfk^o pf %`)d( bp l kl rk

molar`ql fkqboflo `r^kal pb abcfkb 'g+d( `lk i^ c5ojri^ nrb pb a^- Dk bi `^pl bk nrb%`)d( kl bp rk molar`ql fkqboflo+ fkaf`^o nr‹ ^uflj^p kl plk obpmbq^alp- Dk b(+ l vd& fkaf`^k abofs^a^p-

Page 258: Calculus

/2/ Bnk\^djn gdi`\g`n

]( &a*b' <X&g'b&g',

^( &a*b' < GF7X&o'b&o'_o f{b( &X*b' <I9 %&o'b%&o'_o,

a( zj < Q7X&o'_o' Q7b&o' _o',

/1, S bpqŠ cloj^al `lk qla^p i^p pr`bpflkbp fkabcfkfa^p ab k•jbolp ob^ibp xu+-y m^o^ ilp`r^ibp i^p pbofbp Hr/ `lksbodbk- Rf r < ur+v b u < vt,w plk alp bibjbkqlp ab R)abcfkfjlp i

//

%r)s& < I rhUh$I5G

^( Cbjlpqo^o nrb bpq^ pbofb `lksbodb ^_plirq^jbkqb-XFi_d^\^d‡i8 Tp^o i^ abpfdr^ia^a ab B^r`ev,R`et^ow m^o^ ^molufj^o i^ prj^

Kz<iXr)+Uhf+Y_( Cbjlpqo^o nrb R bp rk bpm^`fl ifkb^i `lk %r)u( `ljl molar`ql fkqboflo-b( B^i`ri^o %r)u( pfu+ < f,h b u- < f,%h * 0( m^o^ h <8 0-a( B^i`ri^o %r)u( pf u+ < 1• b u- < f,hc m^o^ h 777770-

03- Rb^ S bi `lkgrkql ab qla^p i^p crk`flkbp ob^ibp E`lkqfkr^p bk ZN+* //( v q^ibp nrb i^fkqbdo^i Px _*``/%n&^n`lksbodb- Cbcfk^jlp 'g+d( < Px _*`f%n&a%n&^n+^( Cbjlpqo^o nrb i^ fkqbdo^i nrb a^ 'g+d( `lksbodb ^_plirq^jbkqb m^o^ `^a^ m^o abcrk`flkbp Eu d ab R+

XFi_d^\^d‡i8 @mif`^o i^ abpfdr^ia^a ab B^r`ev,R`et^ow m^o^ ^molufj^o i^ fkqb,do^i FS_*n.E%n&a%n&f^n+Y

_( Cbjlpqo^o nrb R bp rk bpm^`fl ifkb^i `lk '.+ d( `ljl molar`ql fkqboflo-b( B^i`ri^o 'g+d( pf f%n&< _ v a%n&:n{) alkab h < N+0+ 1+ ----

04- Dk rk bpm^`fl br`iŒabl `ljmibgl+ abjlpqo^o nrb bi molar`ql fkqboflo qfbkb i^p pfdrfbkqbpmolmfba^abp m^o^ qlalp ilp bibjbkqlp s* v+ w v qlalp ilp `ljmibglp \ v ],

']( %[r) \s& < [b%r) s&+ '_( %r)[s * \t& < [%r) s& * b%r)t&+05- Cbjlpqo^o nrb bk qlal bpm^`fl br`iŒabl plk sŠifa^p i^p fabkqfa^abp pfdrfbkqbp-

']( Yt * uh01 < GGth01* GGuh01* %r)s& * %s)r&+'^( Ght * uhh1 , Yt , uh01 < /%r)s& * /%s)t(-'b( GGt* uh01 * Zu , uh01 < 1 GGthh1* 1 Ghu001-

06- Cbjlpqo^o nrb bi bpm^`fl ab qla^p i^p crk`flkbp `ljmibg^p `lkqfkr^p bk rk fkqbos^ilW[) \Y pb qo^kpcloj^ bk rk bpm^`fl rkfq^ofl pf abcfkfjlp rk molar`ql fkqboflo mlo i^cŽojri^

&a*b' <I9r&o'X&o'b&o'_o *

alkab s bp rk^ crk`fŽk mlpfqfs^ cfg^+`lkqfkr^ bk W[) \Y+

)-&)+ 8\[`a_bPPVp[ QRP\[Wb[a\` \_a\T\[NYR`&@na\Q\ QR <_NZ%EPUZVQa

-Slal bpm^`fl ifkb^i ab afjbkpfŽk cfkfq^qfbkbrk^ _^pb cfkfq^:Rf bi bpm^`fl bpbr`iŒabl+ mlabjlp `lkpqorfo pfbjmob rk^ _^pb jmojbji\g, Dpqbobpriq^al pb abar,

Page 259: Calculus

@jinomp^^d‡i _` ^jiepiojn jmojbji\g`n, J„oj_j _` Dm\h+P^chd_o 475

boo^ `ljl `lkpb`rbk`f^ ab rk qblobj^ `rv^ abjlpqo^`fŽk bkpb•^ ^ `lkpqorfo`lkgrkqlp loqldlk^ibp bk `r^inrfbo bpm^`fl br`iŒabl+ ab afjbkpfŽk cfkfq^ l abfkcfkfq^p afjbkpflkbp- K^ `lkpqor``fŽk pb ii^j^ h„oj_j _` Dm\h+P^chd_o* bk jb,jlof^ ab I- O- Fo^j '074/,0805( v D- R`ejfaq '0734,0810(-

SDNQDL@ 04-02- SDNQDL@ CD NQSNFNM@KHY@BHˆM- P`\ s** s,* ,,, * pi\ np+^`nd‡i adido\ j di_`adid_\ _` `g`h`iojn _` pi `nk\^dj `p^g…_`j S* t _`ndbi`hjn^ji Ios• ,,, * Uf' `g np]`nk\^dj b`i`m\_j kjm gjn f kmdh`mjn _` `njn `g`h`iojn,Bsdno` pi\ np^`nd‡i ^jmm`nkji_d`io` _` `g`h`iojn Xi&V,* ,,, * _` S lp` od`i` g\nndbpd`io`n kmjkd`_\_`n k\m\ ^\_\ `io`mj f8

^( Bg `g`h`ioj Xh*i `n jmojbji\g \ oj_j `g`h`ioj _`g np]`nk\^dj I &t! ,,, Ue&+

_( Bg np]`nk\^dj b`i`m\_j kjm XH= ‘‘‘ + Ue `n `g hdnhj lp` `g b`i`m\_jkjm WH= ‘‘‘ + Te7

b( I\ np^`noji t! V,* ,,, * `n ˆid^\* n\gqj a\^ojm`n `n^\g\m`n, Bnoj `n* pfsu) sw)++++) `n jom\ np^`nd‡i _` `g`h`iojn _` S lp` n\odna\^`i g\n kmjkd`_\_`n ^(t _(+ `ioji^`n kjm ^\_\ f `sdno` pi `n^\g\m ?e o\g lp` sw< ?eUe$

A`hjnom\^d‡i, Blkpqorv^jlp ilp bibjbkqlp t! V0%,,, * mlo fkar``fŽk- O^o^fkf`f^o bi mol`bpl+ qlj^jlp t* < Tf$ Rrmlkd^jlp ^elo^ nrb ebjlp `lkpqorfal

, t! ,Š, * Ul ab jlal nrb ^( v _( pb p^qfpc^`bk `r^kal e < l+ Cbcfk^jlp Ul() jb,af^kqb i^ b`r^`fŽk

'04-02(n

Vm)g< UE/ + x\…Vd%f<i

alkab ilp bp`^i^obp \g* ,,, * \9 qfbkbk nrb abqbojfk^opb- O^o^ d w m*bi molar`qlfkqboflo ab Ul() `lk Ux sfbkb a^al mlo

m

&Vm)g*t9' < 'WG0+t9' + \d&Vd% t9' < 'uo*i& t9' + \9&t9* t9'*c:.

mrbpql nrb %Ux)Ux& < N pf d :.: z-Rf t9 :.: -) mlabjlp e^`bo Ul(f loqldlk^i ^ Uxqlj^kal

'04-03(

Rf Ux < N+ bkqlk`bp Ul(f bp loqldlk^i ^ Ux m^o^ `r^inrfbo [x nrb pb bifg^+ bk bpqb`^pl bibdfjlp [x < N- @pŒmrbp+ bi bibjbkql Ul(f bpqŠ _fbk abcfkfal v bp loqldlk^i

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/21 Bnk\^djn gdi`\g`n

^ `^a^ rkl ab ilp ^kqboflobpbibjbkqlp X! ---‘ Vm%Olo `lkpfdrfbkqb+ bp loqldlk^i^ qlal bibjbkql abi pr_ bpm^`fl -

H%Uf$+++ ) Ul& +

Dpql abjrbpqo^ ^( `r^kal f < l * 0-O^o^ abjlpqo^o _( `r^kal e < l * 0 +qbkbjlp nrb mol_^o nrb

H%Uf)$! $Ul(f& < H%Tf!$! rl(.&$ a^al nrb H%Uf)$! )Ul& < H%r/*}}}* rl&{Klp l mofjbolp bibjbkqlp Vg%,,, * Xo mboqbkb`bk

H%r.) ††† ) rl&

u mlo q^kql bpqŠkbk bi pr_bpm^`fl jŠp ^jmifl H%r.) +++ ) rl(.&$ Di krbsl bibjbk,ql Ul(f a^al mlo '04-02( bp rk^ afcbobk`f^ ab alp bibjbkqlp ab H%r.) +++ ) rl(.&

^pŒnrb q^j_f‹k bpqŠbk H%r.) +++ ) rm(.&+ Dpql abjrbpqo^ nrb

K^ b`r^`fŽk '04-02( morb_^ nrb rl(. bp i^ prj^ ab alp bibjbkqlp ^_H%Uf ) +++ $Ul(f&`lk il nrb rk o^wlk^jfbkql ^kŠildl a^ i^ fk`irpfŽk bk bi lqol pbkqfal9

Dpql abjrbpqo^ _( `r^kal f < l * 0- Olo il q^kql ^( v _( e^k pfal abjlpqo^alpmlo fkar``fŽk obpmb`qlab e+

Efk^ijbkqb abjlpqo^jlp `( mlo fkar``fŽk obpmb`qlab e+ Di `^pl e < 0 bpqofsf^i-Olo `lkpfdrfbkqb+ prmlkd^jlp nrb b( bp `fboql m^o^e < l u `lkpfabobjlpbi bibjbkql Uw(. + Dk sfoqra ab _(+ bpqbbibjbkql mboqbkb`b

H%Uf) +++ ) Ul(f& )

^pŒnrb mlabjlp bp`of_fo

l(fU8(. < 1 ?cUc< w+* ?l(.Ul(f )

f<i

alkab u9 A H%sE++++ ) Vm', Prbobjlp abjlpqo^o nrb u+< N- Olo i^ molmfba^a^(+Uwf u ]l(fUl(f plk ^j_lp loqldlk^ibp ^ t*+ Olo `lkpfdrfbkqb+ pr afcbobk`f^+ t*)bp loqldlk^i ^ u,, Cf`el ab lqol jlal+ u9 bp loqldlk^i ^ pŒjfpjl+ ^pŒnrbu9 < N- Dpql `ljmibq^ i^ abjlpqo^`fŽk abi qblobj^ ab loqldlk^ifa^a-

Dk i^ `lkpqor``fŽk ^kqboflo+mrbab pr`babo nrb Ul(f < N m^o^^id•k m,Dkqlk,`bp '04-02( morb_^ nrb Ul)g bp rk^ `lj_fk^`fŽk ifkb^i ab X! --- +m*)u mlo q^kql

Page 261: Calculus

@jinomp^^d‡i _` ^jiepiojn jmojbji\g`n, J„oj_j _` Dm\h+P^chd_o 477

ab Ug%ŠŠŠ * s,* ^pŒnrb ilp bibjbkqlp Ug%ŠŠŠ * Um)g plk abmbkafbkqbp-Dk lqo^p m^,i^_o^p+pf ilp f mofjbolp bibjbkqlp U! ŠŠŠ * Uf plk fkabmbkafbkqbp+ilp bibjbkqlp`loobpmlkafbkqbpt! ,,, *Ue plk ij ipgjn, Dk bpqb ^pl ilp `lbcf`fbkqbp\y ab '04-02(sfbkbk a^alp mlo '04-03(+ v i^p cŽojri^p nrb abcfkbk sx) +++ ) Ue pb `lksfboqbk bk

'04-04( X0 < Wi&

m!! &Um)g%Vy' m 0 1 f 0

Vm)g < Um)g + I &, ,' Vd ^o^ m< + +--- + , -fzi t! t*

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Page 262: Calculus

0)) Bnk\^djn gdi`\g`n

pb ii^j^ i^ jlis_]]nih _` s nj]m` t, Dk bi j‹qlal ab Fo^j,R`ejfaq '04-04(+`lkpqorfjlp bi bibjbkql VQ)g obpq^kal ab UQ)g i^ molvb``fŽk ab UQ)g pl_ob `^a^rkl ab ilp ^kqboflobpbibjbkqlp Uf; ††+ ) m*+K^ cfdro^ 04-0 obmobpbkqi^ `lkpqor`,`fŽk dblj‹qof`^ bk bi bpm^`fl sb`qlof^i S1Š

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Page 263: Calculus

@jhkg`h`iojn jmojbji\g`n, Mmjt`^^dji`n 6/0

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Page 264: Calculus

6/1 Bnk\^djn gdi`\g`n

S* _`o`mhdi\m pi `g`h`ioj `i R ^pt\ _dno\i^d\ \ s n`\ gj hƒn k`lp`†\ kjnd]g`,K^ afpq^k`f^ bkqobalp bibjbkqlp s b u pb abcfkb `ljl i^ kloj^ Ght, uhh-

@kqbpab afp`rqfo bpqbmol_ibj^ bk pr cloj^ dbkbo^i+`lkpfabobjlp rk `^plm^oqf`ri^o+obmobpbkq^albk i^ cfdro^ 04-1- @nrŒR bp bi bpm^`fl sb`qlof^i U^ XR bprk pr_bpm^`fl ab afjbkpfŽk alp+ rk mi^kl nrb m^p^mlo bi lofdbk- C^al s ab R)bi mol_ibj^ `lkpfpqb bk bk`lkqo^o+bk bi mi^kl R+bi mrkql p jŠp moŽufjl ^ s,

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Page 265: Calculus

@jhkg`h`iojn jmojbji\g`n, Mmjt`^^dji`n 6/2

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Page 266: Calculus

0)- Bnk\^djn gdi`\g`n

Efk^ijbkqb+ abjlpqobjlp nrb i^ kloj^ ab r sfbkb a^a^ mlo i^ cŽojri^ mfq^,dŽof`^- Sbkbjlp

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Page 267: Calculus

=jlircg[]cƒh ƒjncg[ ^_ _f_g_hnim ^_ oh _mj[]ci _o]f•^_i 4-2

Obol heo, ph01ƒ N+ `lk il nrb Ght, oT ƒ Ght, nT* s^ifbkal bi pfdkl fdr^i pfv pŽil pf p < o8Dpql `ljmibq^ i^ abjlpqo^`fŽk-

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Page 268: Calculus

0)/ Bnk\^djn gdi`\g`n

7788h+ Rf F C B' , 0+ 0(+ abpfdkbjlp `lk Fi i^ molvb``fŽk ab F pl_ob R- Sbkbjlpbkqlk`bp

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Page 269: Calculus

Be`m^d^djn 0)0

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o- Rb^ S bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibp ` `lkqfkr^p bk ZN+* BC( u q^ibpnrb i^ fkqbdo^i I:&_*n`/%n&^n `lksbodb- Cbcfk^jlp %d)d( < Po_*n`%n&a%n&^n) v pb^ Xl+ XH&X1& -‘- + bi `lkgrkql l_qbkfal ^mif`^kal bi j‹qlal ab Fo^j,R`eokfaq ^ UF! UF% s0% ŠŠŠ *

alkab rh%n& < o! m^o^ i 19 N- Cbjlpqo^o nrb si%n& < 0+XH'0( < n * 0+U/%.& < n/ * 1n * 1+t-o' < o1 + 7o0 * 07q , 5-

5- Dk bi bpm^`fl ifkb^i ob^i B'H+ 2( `lk molar`ql fkqboflo %d) d( <F `%r&a%r&^r) pb^`%r& < i.u v abjlpqo^o nrb bi mlifkljfl `lkpq^kqb d jŠp moŽufjl ^ ` bp d < pild 2-B^i`ri^o 9Gc, beh1m^o^ bpqb c-

6- Dk bi bpm^`fl ifkb^i ob^i B'N+ 1( `lk molar`ql fkqboflo %d)d( < F `%r&a%r&^r) pb^`%r& < _! v abjlpqo^o nrb bi mlifkljfl `lkpq^kqb d jŠp moŽufjl ^ ` bp d < -i %_0 + 0(-B^i`ri^o Ghc, `S m^o^ bpqb c- !

7- Dk bi bpm^`fl ifkb^i ob^i ?%*E) 0( `lk molar`ql fkqboflo %d) d( :Fwf`%+$9&a%r&^r) pb^`%r& < _$ u e^ii^o bi mlifkljfl c jŠp moŽufjl ^ `+ B^i`ri^o Ghc, `f70 m^o^ bpqb c-

8- Dk bi bpm^`fl ifkb^i ob^i B'N+ 1&:6( `lk molar`ql fkqboflo %d) d( < -0&4:: %r&a%r&^r) pb^`%r& < r+ Dk bi pr_bpm^`fl dbkbo^al mlo o))%r& < 0+ r+ %r& < `lp r) o/%r& < pbk r) e^ii^obi mlifkljfl qofdlklj‹qof`l jŠp moŽufjl ^ `+

0/- Dk bi bpm^`fl ifkb^i R abi bgbo`f`fl R+mlkbo `%r& < _*T v e^ii^o bi mlifkljfl ab mofjbodo^al jŠp moŽufjl ^ `+

Page 270: Calculus
Page 271: Calculus

'+

FD6AE;BD@68=BA:E?=A:6?:E K @6FD=8:E

).&) F_N[`S\_ZNPV\[RYV[RNYR`

Tkl ab ilp jlaboklp l_gbqfslp abi @kŠifpfp bp rk bpqrafl ^jmifl ab crk`fl,kbp `rvlp aljfkflp v ob`loofalp plk pr_`lkgrkqlp ab bpm^`flp ifkb^ibp- S^ibp crk,`flkbp pb ii^j^k om\inajmh\^dji`n* \kgd^\^dji`n* r jk`m\_jm`n, Dpqb `^mŒqril qo^q^ab ilp bgbjmilp jŠp pbk`fiilp- ii^j^alp qo^kpcloj^`flkbp gdi`\g`n* nrb pb mobpbk,q^k bk qla^p i^p o^j^p ab i^ L^qbjŠqf`^- K^p molmfba^abp ab qo^kpcloj^`flkbpjŠp dbkbo^ibp pb l_qfbkbk ^ jbkral ^molufjŠkali^p jbaf^kqb qo^kpcloj^`fl,kbp ifkb^ibp-

Hkqolar`fjlp mofjbol i^ klq^`fŽk v i^ qbojfklildŒ^ jŠp `loofbkqb obi^qfs^ ^crk`flkbp `r^ibpnrfbo^- Rb^k S v T alp `lkgrkqlp- Di pŒj_lil

P7R**(S

pb rp^oŠ m^o^ fkaf`^o nrb Q bp rk^ crk`fŽk `rvl aljfkfl bp S v `rvlp s^ilobpbpqŠk bk T, O^o^ `^a^ s ab S* bi bibjbkql Q&s' ab T pb ii^j^ dh\b`i _` s\ om\q„n _` Q* v ab`fjlp nrb Q \kgd^\ s `i Q&s', Rf > bp rk pr_`lkgrkql `r^i,nrfbo^ ab S* bi `lkgrkql ab qla^p i^p fjŠdbkbp Q&s' m^o^ s ab > pb ii^j^ g\ dh\+b`i _` > \ om\q„n_` Q v pb obmobpbkq^mlo Q&>', K^ fj^dbk abi aljfkfl S* Q&S'*bp bi ob`loofal ab P+

Rrmlkd^jlp ^elo^ nrb S v T plk bpm^`flp ifkb^ibp nrb qfbkbk bi jfpjl `lk,grkql ab bp`^i^obp+ v abcfk^jlp rk^ qo^kpcloj^`fŽk ifkb^i `ljl pfdrb-

CDEHMHBHˆM- Rf S X T nji _jn `nk\^djn gdi`\g`n* pi\ api^d‡i Q8S x T n`gg\h\ om\inajmh\^d‡i gdi`\g _` S `i T* pf od`i` g\n kmjkd`_\_`n ndbpd`io`n8

]( P%r * u( < P%r&* P+%s&_( P%]r& < ]P%r&

^p\g`nlpd`m\ lp` n`\i s ` u _` S*k\m\ oj_j s _` S v ^p\glpd`m `n^\g\m ^,

6/8

Page 272: Calculus

60/ Qm\inajmh\^dji`n gdi`\g`n t h\omd^`n

Dpql pfdkfcf`^ nrb Q `lkpbos^ i^ ^af`fŽk v i^ jriqfmif`^`fŽk mlo bp`^i^obp-K^p alp molmfba^abpmrbabk `lj_fk^opb bk rk^ cŽojri^ nrb bpq^_ib`b nrb

Q&\s * ]t' < \Q&s' * ]Q&t'

m^o^ qlal s v qlal t ab S v qlalp ilp bp`^i^obp \ v ], Olo fkar``fŽk+ qbkbjlpq^j_f‹k i^ obi^`fŽk jŠp dbkbo^i

m^o^ i bibjbkqlp `r^ibpnrfbo^ s! ,,, *s9 ab R u i bp`^i^obp `r^ibpnrfbo^^i&--- )[iŠ

Di ib`qlo mrbab `ljmol_^o cŠ`fijbkqb nrb ilp bgbjmilp pfdrfbkqbpplk qo^kp,cloj^`flkbp ifkb^ibp-

DIDLOKN 0- Qm\inajmh\^d‡i d_„iod^\, K^ qo^kpcloj^`fŽk Q8S x S* alkabP%r&< r m^o^ qlal r ab R) pb abkljfk^ qo^kpcloj^`fŽk fa‹kqf`^ u pb abpfdk^mlo / l mlo gq*

DIDLOKN 1- Qm\inajmh\^d‡i ^`mj, K^ qo^kpcloj^`fŽk Q8S x S nrb ^mif`^`^a^ bibjbkql ab S bk N pb ii^j^ qo^kpcloj^`fŽk `bol u pb abpfdk^ mlo k-

DIDLOKN 2- Jpgodkgd^\^d‡i kjm pi `n^\g\m adej ^, Sbkbjlp ^nrŒQ8S x S*alkab Q&s' < `s m^o^qlal s ab S, Br^kal ` < 0+pb qo^q^ab i^ qo^kpcloj^`fŽkfa‹kqf`^- Br^kal ` < /+ bp i^ qo^kpcloj^`fŽk `bol-

DIDLOKN 3- B^p\^dji`n gdi`\g`n, Rb^k S < Si v T < S h, C^alp hik•jbolp ob^ibp\p* `lk e< 0+1+--- +h v f < 0+1+ --- +i* abcfk^jlp Q8Si x s-+`ljl pfdrb9 P ^mif`^ `^a^ sb`qlo r < %r! +++)ri' ^_ Ri bk bi sb`qlo s < %sx),,, *Vh' ab S h ab ^`rboal `lk i^p b`r^`flkbp

h

Uc < H [cere m^o^ c < 0+1+--- +g +f;g

DIDLOKN 4- Mmj_p^oj dio`mdjm ^ji pi `g`h`ioj adej, Rb^ S rk bpm^`flbr`iŒabl- O^o^ rk bibjbkql cfgl w ab S* abcfk^jlp Q8S x Q ^pŒ9Rf s C S*P%r&< %r)t&) bi molar`ql fkqboflo ab r mlo w-

DIDLOKN 5- Mmjt`^^d‡i nj]m` pi np]`nk\^dj, Rb^k S rk bpm^`fl br`iŒablv P rk pr_bpm^`fl ab S ab afjbkpfŽk cfkfq^-Cbcfk^jlp Q8S x P ^pŒ9Rf s C S*P%r& bp i^ molvb``fŽk ab r pl_ob O+

Page 273: Calculus

Kˆ^g`j u m`^jmmd_j 600

DIDLOKN 6- Bg jk`m\_jm _`mdq\^d‡i, Rb^ S bi bpm^`fl ifkb^i ab qla^p i^pcrk`flkbp ob^ibp ` abofs^_ibp bk rk fkqbos^il ^_fboql %[) \&+ K^ qo^kpcloj^`fŽkifkb^i nrb ^mif`^ `^a^ crk`fŽk ` ab S bk pr abofs^a^ n$pb ii^j^ lmbo^alo abofs^,`fŽk v pb abpfdk^ mlo A, @pŒmrbp+ qbkbjlp A8 S x S* alkab A&a' < n$m^o‹i

`^a^ ` ab R+

DIDLOKN 7- Bg jk`m\_jm dio`bm\^d‡i, Rb^ S bi bpm^`fl ifkb^i ab qla^p i^pcrk`flkbp ob^ibp `lkqfkr^p bk rk fkqbos^il W[)\Y+ Rf ` C R) abcfk^jlp d < P%h`ljl i^ crk`fŽk S a^a^ mlo

a%r&< a'&o' _o pf [77788r 77788],

Dpq^ qo^kpcloj^`fŽk P pb ii^j^ lmbo^alo fkqbdo^`fŽk-

).&* AqPYR\ e _RP\__VQ\

Dk bpq^ pb``fŽk+ P obmobpbkq^rk^ qo^kpcloj^`fŽk ifkb^i ab rk bpm^`fl ifkb^iS bk rk bpm^`fl ifkb^i T,

RCMPCK? 05-0- Bg ^jiepioj Q&S' &m`^jmmd_j_` Q' `n pi np]`nk\^dj _` T,>_`hƒn* Q \kgd^\ `g `g`h`ioj ^`mj _` S `i `g `g`h`ioj ^`mj _` T,

A`hjnom\^d‡i, O^o^ abjlpqo^o nrb Q&S' bp rk pr_bpm^`fl ab T* q^k pŽilkb`bpfq^jlp `ljmol_^o ilp ^uflj^p ab `i^rpro^- Sljbjlp alp bibjbkqlp `r^ibp,nrfbo^ ab Q&S'* pb^k Q&s' v Qos&+ Dkqlk`bp Q&s' * Q&t' < Q&s * s&) ^pŒnrbQ&s' * Q&t' mboqbkb`b ^ Q&S', @pfjfpjl+ m^o^ `r^inrfbo bp`^i^o ` qbkbjlp^Q&s' < Q&^s'* `lk il nrb ^Q&s' mboqbkb`b ^ Q&S', Olo `lkpfdrfbkqb+ Q&S' bp rkpr_bpm^`fl ab T, Slj^kal ` < N bk i^ obi^`fŽk Q&^s' < ^Q&s'*bk`lkqo^jlp nrbP%[&< [+

CDEHMHBHˆM- Bg ^jiepioj _` oj_jn gjn `g`h`iojn _` S lp` Q \kgd^\ `i \ n`gg\h\ iˆ^g`j _` Q u n` _`ndbi\ kjm K&Q', >n…kp`n* o`i`hjn

K&Q' < vs Gs C S V Q&s' < Ny -

RCMPCK? 05-1- Bg iˆ^g`j _` Q `n pi np]`nk\^dj _` S,

A`hjnom\^d‡i, Rf s b v bpqŠk bk K&Q'* il jfpjl ibp l`roob ^ s *- v v ^ ^sm^o^ qlalp ilp bp`^i^obp `* v^ nrb

Q&s* s& < Q&s' * Q&'%'< M v Q&^s' < ^Q&s' < N-

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601 Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

Klp bgbjmilp nrb pfdrbk abp`of_bk ilp k•`iblp ab i^p qo^kpcloj^`flkbp ifkb^,ibp a^a^p bk i^ pb``fŽk 05-0-

DIDLOKN 0- Qm\inajmh\^d‡i d_„iod^\, Di k•`ibl bp xNy+ bi pr_bpm^`fl`lkpqfqrfal q^k pŽil mlo bi bibjbkql `bol-

DIDLOKN 1- Qm\inajmh\^d‡i ^`mj, Orbpql nrb qlal bibjbkql ab S pb ^mif`^bk `bol+ bi k•`ibl bp bi jfpjl S,

DIDLOKN 2- Jpgodkgd^\^d‡i kjm pi `n^\g\m adej ^, Rf ` ;/; N+ bi k•`ibl pŽil`lkqfbkb bi bibjbkql N- Rf ^ < N+ bi k•`ibl bp R+

DIDLOKN 3- B^p\^dji`n gdi`\g`n, Di k•`ibl bpqŠ `lkpqfqrfal mlo qlalp ilpsb`qlobp 'u-+ --- +si' ab s+ m^o^ ilp `r^ibp

i

G[cere < N m^o^ d < 0+1+--- +g +Q_R

DIDLOKN 4- Mmj_p^oj dio`mdjm kjm pi `g`h`ioj adej w- Di k•`ibl `lkpq^ab qlalp ilp bibjbkqlp ab S loqldlk^ibp ^ w-

DIDLOKN 5- Mmjt`^^d‡i nj]m` pi np]`nk\^dj P, Rf U D R) qbkbjlp i^•kf`^ abp`ljmlpf`fŽk loqldlk^i s < p * p0- 'pbd•k bi qblobj^ 04-04(- Orbpql nrbP%r& < p+ qbkbjlp P%r& < N pf u pŽil pf r < n/,, Olo `lkpfdrfbkqb+ bi k•`ibl &bp3/,* bi `ljmibjbkql loqldlk^i ab P,

DIDLOKN 6- Lk`m\_jm _`mdq\^d‡i, Di k•`ibl bpqŠ cloj^al mlo qla^p i^pcrk`flkbp `lkpq^kqbp bk bi fkqbos^il a^al-

DIDLOKN 7- Lk`m\_jm dio`bm\^d‡i, Di k•`ibl `lkqfbkb pli^jbkqb i^ crk,`fŽk `bol-

05-2 CfjbkpfŽk abi k•`ibl v o^kdl ab i^ qo^kpcloj^`fŽk

S^j_f‹k bk bpq^ pb``fŽk Q obmobpbkq^rk^ qo^kpcloj^`fŽk ab rk bpm^`fl ifkb^iR bk rk bpm^`fl ifkb^i S+ Mlp fkqbobp^ i^ obi^`fŽk bkqob i^p afjbkpflkbp ab R) abik•`ibl J%P&u abi ob`loofal P%R&+Rf R bp ab afjbkpfŽk cfkfq^+bi k•`ibl q^j_f‹kil pboŠ mlo pbo rk pr_bpm^`fl ab R+ Dk bi qblobj^ nrb pfdrb+ abjlpqo^jlp nrbbi ob`loofal P%R& q^j_f‹k bp ab afjbkpfŽk cfkfq^: pr afjbkpfŽk pb ii^j^ l[haiab P+

Page 275: Calculus

Adh`ind‡i _`g iˆ^g`j u m\ibj _` g\ om\inajmh\^d‡i 602

RCMPCK? 05-2- Pd S `n _` _dh`ind‡i adido\* o\h]d„i gj `n l_s&) u o`i`hjn

'05-0( afj J%P&* afj P%R&< afj R+

Ad^cj _` jomj hj_j* g\ _dh`ind‡i _`g iˆ^g`j hƒn `g m\ibj _` pi\ om\inajmh\^d‡igdi`\g `n dbp\g \ g\ _dh`ind‡i _` np _jhdidj,

A`hjnom\^d‡i, Rb^k i < afj S u `!!! `y rk^ _^pb m^o^ K&Q'* alkabe < afj J%P& w i, Rbd•k bi qblobj^ 04-6+ bplp bibjbkqlp cloj^k m^oqb ab rk^`fboq^ _^pb ab R) mlo bgbjmil ab i^ _^pb

'05-1(

alkab f * l < h+ Cbjlpqo^objlp, nrb ilp l bibjbkqlp

'05-2(

cloj^k rk^ _^pb ab Q&S'*abjlpqo^kal ^pŒnrb afj Q&S';m, Orbpql nrb f)m;i*q^j_f‹k bpl abjrbpqo^ '05-0(-

Cbjlpqo^jlp mofjbol nrb ilp l bibjbkqlp ab '05-2( dbkbo^k P%R&+Rfv C P%R&)bp v < P%r&m^o^ rk `fboql r ab R) v mlabjlp bp`of_fo r < ]._. (* ---* ?e(l_e(l+ Krbdl+ qbkbjlp

f)m f f+d+m f)m

s < Q&s' < I^G&`d' <K ^G&`9' * 19 ^dQ&`9'< 19 ^dQ&`d'f:f q<i c:.+7)0 c:e(f

mrbpql nrb P%_.& < --- :P%_e&: N- Dpql abjrbpqo^ nrb ilp bibjbkqlp '05-2(dbkbo^k P%R&+

Cbjlpqobjlp ^elo^ nrb bplp bibjbkqlp plk fkabmbkafbkqbp- Rrmlkd^jlp nrbbufpqfbo^k bp`^i^obp ?e(f) +++ ) ?e(l q^ibp nrb

o`+d+m

/7 ^dQ&`9'< N]dxf)g

Dpql fjmif`^oŒ^ nrb

&

o^+qm 'Q 19 :z, < N

x:e*d 0

Page 276: Calculus

603 Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

mlo 0/ nrb bi bibjbkql r < ]en*)_e(f * --- * ?e(l_e(l pboŒabi k•`ibl J%P&+Rfd,kfcf`^ bpql nrb bufpqfoŒ^kbp`^i^obp?.) ††† ) ^` q^ibpnrb s < ^.`. * --- * `n`n* `lkil nrb qbkaoŒ^jlp

g` F^)Q

V , V < y `,`• + x `,`y < M -fzi :zh*i

Obol `ljl ilp bibjbkqlp '05-1( plk fkabmbkafbkqbp+qlalp ilp bp`^i^obp `* e^k abpbo `bol- Olo `lkpfdrfbkqb+ ilp bibjbkqlp '05-2( plk fkabmbkafbkqbp-

Jin[7 Rf R bp ab afjbkpfŽk fkcfkfq^ mlo 0/ jbklp rkl ab ilp alp J%P& l P%R&bpab afjbkpfŽk fkcfkfq^- Dk bi bgbo`f`fl 2/ ab i^ Rb``fŽk 05-3 pb bp_lw^ rk^ abjlpqo^`fŽkab bpqb eb`el-

).&, :WR_PVPV\`

Dk `^a^ rkl ab ilp bgbo`f`flp abi 0 ^i 0/+ pb abcfkb rk^ qo^kpcloj^`fŽk P7R0

x R0

jbaf^kqb i^ cŽojri^ a^a^ m^o^ P%r)s&) alkab %r)u( bp rk mrkql `r^inrfbo^ ab R0Š Cbqbo,jfk^o bk `^a^ `^pl pf Q bp ifkb^i- Rf Q bp ifkb^i+ ab`fo `rŠibp plk bi k•`ibl v bi ob`loofal+ v`^i`ri^o prp afjbkpflkbp

f+ P%r)s& < %s)r&+/+ P%r)s& < %r) *s&+

0+ P%r)s& < %r) K&+

1+ P%r) s& < %r) r&+

2+ P%r) s& < %r/) s/&+

3+ P%r)s& < %_T)_$f&+

4+ P%r)s& < %r) .&+

5+ P%r)s& < %r * f)s * 0(-6+ P%r) s& < %r * s) r * s&+

.-+ P%r) s& < %/r * s) r * s&+

G^`bo il jfpjl bk `^a^ rkl ab ilp bgbo`f`flp abi 00 ^i 04- Rf i^ qo^kpcloj^`fŽkQ8S 1 z S0 bp i^ nrb pb fkaf`^-..+ P e^`b dfo^o `r^inrfbo ,mrkql bi jfpjl Škdril ;.= ^iobabalo abi lofdbk- Dpql bp+P ^mif`^

rk mrkql ab `lloabk^a^p mli^obp %l)K& bk bi mrkql ab `lloabk^a^p mli^obp %l) -(9,|)alkab ;.= bp cfgl- @abjŠp+ Q ^mif`^ N bk pŒjfpjl-

/0, Q ^mif`^ `^a^ mrkql bk pr pfj‹qof`l obpmb`ql ^ rk^ ob`q^ cfg^ nrb m^p^ mlo bi lofdbk-/1, Q ^mif`^ qlal mrkql bk bi mrkql '0+ 0(-.1+ P ^mif`^ `^a^ mrkql ab `lloabk^a^p mli^obp %l) & bk bi mrkql ab `lloabk^a^p %/l) K&+

@abjŠp+ Q ^mif`^ N bk pŒjfpjl-.2+ P ^mif`^ `^a^ mrkql ab `lloabk^a^p mli^obp 'o+ K& bk bi mrkql ab `lloabk^a^p %l) /-&+

@abjŠp+ P ^mif`^ N bk pŒjfpjl-

G^`bo il jfpjl bk `^a^ rkl ab ilp bgbo`f`flp 05 ^i 12 pf i^ qo^kpcloj^`fŽk Q8S 2 z S 2

bpqŠ abcfkfa^ mlo i^ cŽojri^ nrb pb a^ m^o^ Pn8r) v+ w(+alkab %r) v+ w( bp rk mrkql ^o_fqo^ofl&ab S\%

.3+ P%r) t* w( < 'w+s) r&+

.4+ P%r) s) w( < %r) s) K&+

.5+ P%r) s) t& < %r) /s) 0t&+

.6+ P%r)s) t& < %r) s) 0(-

/-+ P%r)s) t& < %r * E)s * 0+t * 0(-/.+ P%r)s) u' < %r * i+v * 1+u * 2(-//+ P%r) s) t& < %r) s/) W1',

/0+ P%r) s) t& < %r * t) N+ r * s&+

Page 277: Calculus

Be`m^d^djn /)-

Dk `^a^ rkl ab ilp bgbo`f`flp abi 13 ^i 16+ i^ qo^kpcloj^`fŽk P7R w R bp i^ nrb pbfkaf`^- Cbqbojfk^o+ bk `^a^ `^pl+ pf Q bp ifkb^i- Rf il bp+ ab`fo `rŠibp plk bi k•`ibl v biob`loofal v `^i`ri^o prp afjbkpflkbp `r^kal pb^k cfkfq^p-13- Rb^ R bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp j%r& ab do^al z h+ Rf j D R)

k < P%j& pfdkfcf`^ nrb k%r&< j%r * 0( m^o^ qlal ob^i ab r+

14- Rb^ S bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibp abofs^_ibp bk bi fkqbos^il ^_fboql',0+ 0(- Rf ` C R) d < P%`&pfdkfcf`^ nrb a%r&< r`$%r& m^o^ qlal r ab ',0+ 0(-

15- Rb^ R bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibp `lkqfkr^p bk W[) \Y+ Rf ` D R)d < P%`&pfdkfcf`^ nrb

a%r& <G8G&o'pbk %r * o' _o m^o^ [ n8 r 7788] ,

16- Rb^ U bi bpm^`fl ab qla^p i^p crk`flkbp ob^ibp abofs^_ibp alp sb`bp bk rk fkqbos^il^_fboql %[)\&+ Rf X C U+ abcfkfo P%s&< u! * Ls$ * Ou pfbkal L v O alp `lkpq^kqbp-

17- Rb^ U bi bpm^`fl ifkb^i ab qla^p i^p pr`bpflkbp ob^ibp `lksbodbkqbp wt-x+Cbcfkfjlp rk^qo^kpcloj^`fŽk P7RwR ^pŒ9Rf r:ur!v bp rk^ pr`bpfŽk `lksbodbkqb `lk iŒjfqb [)mlkbjlp P%r&< us!v) alkab u! < [ * r8 m^o^ h ƒ 0- Cbjlpqo^o nrb P bp ifkb^i v ab`fo`rŠibp plk bi k•`ibl u bi ob`loofal ab S-

18- Rb^ U bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp `lkqfkr^p bk bi fkqbos^il X+5Q* 5QI- Rb^ Rbi pr_`lkgrkql ab U nrb `lkpq^ ab qla^p i^p crk`flkbp ` nrb p^qfpc^`bk i^p qobpb`r^`flkbp

oG&o'_o < N+ qooG&o'lp o_o < N+ 'oo G&o'pbk o_o < N -† *.P

^( Cbjlpqo^o nrb R bp rk pr_bpm^`fl ab U-_( Cbjlpqo^o nrb R `lkqfbkb i^p crk`flkbp `%r&< `lp hr v `%r&< pbk hr m^o^ `^a^i < 1+ 2+ -- +`( Cbjlpqo^o nrb R bp ab afjbkpfŽk fkcfkfq^-

Rb^ P7U z U i^ qo^kpcloj^`fŽk ifkb^i abcfkfa^ ^pŒ9Rf ` D R) d < P%`&pfdkfcf`^ nrb

a%r& <qooxi * `lp %r * o'wG&o'n )

2/-

a( Cbjlpqo^o nrb P%R&)bi ob`loofal ab P) bp ab afjbkpfŽk cfkfq^ v e^ii^o rk^ _^pbm^o^ P%R&+

b( Cbqbojfk^o bi k•`ibl ab P+

c( G^ii^o qlalp ilp k•jbolp ob^ibp ^ 9%!N X qla^p i^p crk`flkbp ` kl kri^p ab U q^ibp nrbP%`&< _Y+'N_p‹osbpb nrb rk^ q^i Emboqbkb`b ^i ob`loofal ab P+&Rb^ Q8U z T rk^ qo^kpcloj^`fŽk ifkb^i ab rk bpm^`fl ifkb^i U bk rk bpm^`fl ifkb^iS+ Rf R bp ab afjbkpfŽk fkcfkfq^+ abjlpqo^o nrb mlo il jbklp rkl ab ilp alp P%R&l J%P& bp ab afjbkpfŽk fkcfkfq^-

WEh^c][]cƒh7 RrmŽkd^pb afj J%P& < e) afj P%R&< l) pb^ _f$ +++) _f rk^ _^pbm^o^ J%P& v pb^k _f) +++) _! _e(f) +++ `^,9 bibjbkqlp &fkabmbkafbkqbpab U+ pfbkalh; l+ Klp bibjbkqlp P%_e(f&)+++) P%_e)!&plk abmbkafbkqbp v^ nrb h; l) Tqfifw^obpqb eb`el m^o^ l_qbkbo rk^ `lkqo^af``fŽk-\

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5/4 Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

).&- B]R_NPV\[RNYTRO_NVPN`P\[ a_N[`S\_ZNPV\[RYV[RNYR`

K^p crk`flkbp `rvlp s^ilobp mboqbkb`bk ^ rk bpm^`fl ifkb^i a^al S mrbabkprj^opb rk^p `lk lqo^p v mrbabk jriqfmif`^opb mlo bp`^i^obp ab T ab ^`rboal`lk i^ abcfkf`fŽk pfdrfbkqb-

CDEHMHBHˆM- P`\i R9S x T u Q8S x T _jn api^dji`n ^ji pi _jhdidj^jhˆi S t ^ji q\gjm`n k`mo`i`^d`io`n \ pi `nk\^dj gdi`\g T, Rf ` `n pi `n^\g\m^p\glpd`m\ _` T* _`adidhjn g\ nph\ R * Q v `g kmj_p^oj ^Q kjm g\n `^p\^dji`n

' 05-3( 'Q * Q'&s' < P&s' * Q&s' * &^Q'&s' < ^Q&s'

k\m\ oj_j s _` S,

Mlp fkqbobp^ bpmb`f^ijbkqb bi `^pl bk bi nrb S bp q^j_f‹k rk bpm^`fl ifkb^i`lk ilp jfpjlp bp`^i^obp nrb S+ Dk bpqb `^pl abpfdk^jlp `lk 1!' R) S& bi `lk,grkql ab qla^p i^p qo^kpcloj^`flkbp ifkb^ibp ab S bk T,

Rf R X Q plk alp qo^kpcloj^`flkbp ifkb^ibp ab 1!' S* T'* bp rk pbk`fiil bgbo`f,`fl `ljmol_^o nrb R* Q v ^Q q^j_f‹k plk qo^kpcloj^`flkbp ifkb^ibp ab 1&' R) S&+

@•k jŠp- Blk i^p lmbo^`flkbp nrb ^`^_^jlp ab abcfkfo+ bi jfpjl `lkgrkql/!% R) S& pb qo^kpcloj^ bk rk krbsl bpm^`fl ifkb^i- K^ qo^kpcloj^`fŽk `bol pfosbab bibjbkql `bol bk bpb bpm^`fl+ v i^ qo^kpcloj^`fŽk %*f&P bp i^ lmrbpq^ ab P+Rb `ljmorb_^ nrb pb p^qfpc^`bk ilp afbw ^uflj^p ab rk bpm^`fl ifkb^i- Olo `lk,pfdrfbkqb+ qbkbjlp bi pfdrfbkqb-

RCMPCK? 05-3- Bg ^jiepioj /!%R) T' _` oj_\n g\n om\inajmh\^dji`n gdi`\+g`n _` S `i T `n pi `nk\^dj gdi`\g ^ji g\n jk`m\^dji`n _` \_d^d‡i u hpgodkgd^\+^d‡i kjm `n^\g\m`n _`adid_\n `i '05-3(-

Tk^ lmbo^`fŽk ^idb_o^f`^ jŠp fkqbobp^kqb nrb pb bcb`q•^ `lk i^p qo^kpclo,j^`flkbp ifkb^ibp bp i^ ^jhkjnd^d‡i j hpgodkgd^\^d‡i ab qo^kpcloj^`flkbp- Dpq^lmbo^`fŽk kl rqfifw^ i^ bpqor`qro^ ^idb_o^f`^ ab rk bpm^`fl ifkb^i v mrbab abcfkfopb`lk bkqbo^ dbkbo^ifa^a abi pfdrfbkqb jlal9

CDEHMHBfˆM- A\_jn gjn ^jiepiojn Q) S, T, P`\i Q8Q w S pi\ api^d‡i ^ji_jhdidj R v q\gjm`n `i R) v R9R w T jom\ api^d‡i ^ji _jhdidj R v q\gjm`n `iT, I\ ^jhkjnd^d‡i PQ `n g\ api^d‡i PQ8R x T _`adid_\ kjm

&PQ'&s' < PXQ&s'Z k\m\ oj_j s `i R,

@pŒmrbp+ m^o^ ^mif`^o s jbaf^kqb i^ `ljmlpf`fŽk PQ* ^mif`^jlp mofjbol sjbaf^kqb Q*v irbdl ^mif`^jlp Q&s' mlo jbafl ab R- Dpql pb obmobpbkq^bk i^ cfdr,o^ 05-0-

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Lk`m\^dji`n \gb`]m\d^\n ^ji om\inajmh\^dji`n gdi`\g`n 606

PQ8 R+++,T

EHFTQ@ 05-0 Dmƒad^j _` g\ ^jhkjnd^d‡i _` _jn om\inajmh\^dji`n,

K^ `ljmlpf`fŽk ab crk`flkbp ob^ibp pb e^ bk`lkqo^al obmbqfa^psb`bp bkkrbpqol bpqrafl abi BŠi`ril+ v ebjlp sfpql nrb i^ lmbo^`fŽk+bk dbkbo^i+kl bp`lkjrq^qfs^- Ml l_pq^kqb+`ljl bk bi `^pl ab i^p crk`flkbp ob^ibp+i^ `ljmlpf,`fŽk p^qfpc^`bi^ ibv ^pl`f^qfs^-

SDNQDL@ 05-4- Pd Q8R x S* P8S x T, u O8T x W nji om`napi^dji`n* o`+i`hjn

N%OP&< %NO&P

A`hjnom\^d‡i, K^p crk`flkbp O&PQ' v &OP'Q qfbkbk ^j_^p aljfkfl R ts^ilobp bk W- O^o^ `^a^ s ab Q) qbkbjlp

WN%OP&Y%r&< NW%OP&%r&Y< NWOWP%r&YYW W%NO&PF%r&< %NO&WP%r&Y< NWOWP%r&YY)

il nrb abjrbpqo^ Prb N%OP&< %NO&P+

CDEHMHBHˆM- Rb^ Q8S x S pi\ api^d‡i lp` \kgd^\ S `i n…hdnhj, A`adid+hjn di_p^odq\h`io` g\n kjo`i^d\n `io`m\n _` Q ^jhj ndbp`8

QL; f) P! < P$E;| j[l[ h = i-

@nrŒ. obmobpbkqi^ qo^kpcloj^`fŽk fa‹kqf`^- Di ib`qlo mrbab `ljmol_^o nrbi^ ibv ^pl`f^qfs^ fjmif`^ i^ ibv ab bumlkbkqbpQ!%Q!< Q!%8!m^o^ qlalp ilp bkqb,olp 00/ kbd^qfslp h v i,

Di qblobj^ nrb pfdrb morb_^ nrb i^ `ljmlpf`fŽk ab qo^kpcloj^`flkbp gdi`\g`nbp ifkb^i-

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607 Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

RCMPCK? 05-5- Pd R* S* T nji `nk\^djn gdi`\g`n ^ji gjn hdnhjn `n^\g\m`n*v nd Q8RxS v P8SxT nji om\inajmh\^dji`n gdi`\g`n* g\ ^jhkjnd^d‡i PQ8RxT`n gdi`\g,

A`hjnom\^d‡i, O^o^ qlal s v qlal v ab R v qlalp ilp bp`^i^obp \ v ]* qb,kbjlp

&PQ'&\s * ]t' < PXQ&\s * ]t'Z < PX\Q&s' * ]Q&t'Z < \PQ&s' * ]PQ&t' ,

K^ `ljmlpf`fŽk mrbab `lj_fk^opb ,`lk i^p lmbo^`flkbp ^idb_o^f`^p ab ^af`fŽkv jriqfmif`^`fŽk mlo bp`^i^obpbk i&'q*S& iibd^kal ^i pfdrfbkqb

RCMPCK? 05-6- P`\i R* S* T `nk\^djn gdi`\g`n ^ji gjn hdnhjn `n^\g\m`n*npkjib\hjn lp` R v Q k`mo`i`^`i \ 2_%S* T'* v n`\ ` pi `n^\g\m ^p\glpd`m\,

]( M\m\ ^p\glpd`m api^d‡i O ^ji q\gjm`n `i S* o`i`hjn

'Q * Q'O < PO * QO u &^P'O < ^&PO' ,

_( M\m\ ^p\glpd`m om\inajmh\^d‡i gdi`\g O8T x R* o`i`hjn

O&P* Q' < OP * OQ u O&^P' < ^&OP' ,

K^ abjlpqo^`fŽk bp rk^ `lkpb`rbk`f^ fkjbaf^q^ ab i^ abcfkf`fŽk ab `ljml,pf`fŽk v pb abg^ `ljl bgbo`f`fl-

).&. =[cR_`N`

@i bpqraf^o i^p crk`flkbp ob^ibp^mobkafjlp `Žjl `lkpqorfo krbs^p crk`flkbpjbaf^kqb i^ fksbopfŽk ab crk`flkbp jlkŽqlk^p- Prbobjlp ^elo^ buqbkabo bi j‹,qlal ab fksbopfŽk ^ rk^ `i^pb jŠp dbkbo^i ab crk`flkbp-

C^a^ rk^ crk`fŽk P) krbpqol l_gbqfsl bp bk`lkqo^o+pf bp mlpf_ib+lqo^ crk`fŽkR `rv^ `ljmlpf`fŽk `lk Q pb^ i^ qo^kpcloj^`fŽk fa‹kqf`^- Orbpql nrb i^ `ljml,pf`fŽk+bk dbkbo^i+kl bp `lkjrq^qfs^+ qbkbjlp nrb afpqfkdrfo PQ ab QP, Olo ilq^kql fkqolar`fjlp alp qfmlpab fksbop^p nrb ii^j^jlp jsbop^ mlo i^ abob`e^ bfksbop^ mlo i^ fwnrfboa^-

CDEHMHBHˆM- A\_jn _jn ^jiepiojn S u T u pi\ api^d‡i Q8S x T, P` _d^`lp` pi\ api^d‡i P8Q&S' x S `n diq`mn\ _` Q kjm g\ dulpd`m_\ pf PXQ&s'Z < sk\m\ oj_j s _` S, `noj `n* nd

PQ; o,*

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Fiq`mn\n 608

_ji_` gq `n g\ om\inajmh\^d‡i d_„iod^\ nj]m` R+ Ri\ api^d‡i N7P%R&w R n`gg\h\ diq`mn\ _` P kjm g\ _`m`^c\ nd PWN%s&Y< V k\m\ oj_j s _` P%R&)noj `n* nd

QO < FPq&)

_ji_` ,l%R& `n g\ om\inajmh\^d‡i d_„iod^\ nj]m` Q&S',

CHCKNJM- Ri\ api^d‡i ndi diq`mn\ kjm g\ dulpd`m_\ k`mj ^ji _jn diq`mn\nkjm g\ _`m`^c\, Rb^k S < x0+ 1y X T < xNy- Cbcfkfjlp Q8S x T `ljl pfdrb9P%f&< P%/&< N- Dpq^ crk`fŽk qfbkb alp fksbop^p mlo i^ abob`e^ N7S w R XO%8T x S a^a^p mlo

N%K&< 0 + N$%K&< 1-

Ml mrbab qbkbo fksbop^ mlo i^ fwnrfboa^ R v^ nrb biil bufdfoŒ^

0 < OWP%E&Y< O_K& v- 1 < OWP%/&Y< O_K&+

Dpqb pbk`fiil bgbjmil mlkb ab j^kfcfbpql nrb kl qfbkb nrb bufpqfo kb`bp^of^jbkqbfksbop^ mlo i^ fwnrfboa^ v nrb i^ fksbop^ mlo i^ abob`e^ kl qfbkb nrb pbo kb`bp^,of^jbkqb •kf`^-

Sla^ crk`fŽk Q8S x T qfbkb mlo-il jbklp rk^ fksbop^ ^ i^ abob`e^- Dk bcb`,ql+ `^a^ s ab P%R&qfbkb i^ cloj^ s < P%r&m^o^ ^i jbklp rk r ab R+ Rf bibdfjlprkl ab bplp s^ilobp r v abcfkfjlp N%s&< r) bkqlk`bp PWN%s&Y< P%r&< s m^o^`^a^ t ab P%R&) pŒnrb N bp rk^ fksbop^ mlo i^ abob`e^- K^ kl rkf`fa^a mrbab mob,pbkq^opb ab_fal ^ nrb mrbab e^_bo jŠp ab rk s ab S nrb pb ^mifnrb bk rk t abP%R&+Cbkqol ab ml`l abjlpqo^objlp 'qblobj^ 05-8( nrb pf `^a^ s ab P%R&bpi^ fj^dbk ab pi n‡gj s ab S* i^ fksbop^ mlo i^ abob`e^ bp •kf`^-

@kqbpabjlpqo^objlp nrb pf bufpqb fksbop^ mlo i^ fwnrfboa^ bp •kf`^ v+ ^i jfpjlqfbjml+ bp fksbop^ ^ i^ abob`e^-

RCMPCK? 05-7- Ri\ Q8S x T kp`_` o`i`m \ gj hƒn pi\ diq`mn\ kjm g\dulpd`m_\, Rf Q od`i` diq`mn\ kjm g\ dulpd`m_\ R+ `ioji^`n R `n o\h]d„i diq`mn\kjm g\ _`m`^c\,

A`hjnom\^d‡i, Rrmlkd^jlp nrb Q qbkd^ alp fksbop^p mlo i^ fwnrfboa^+O7P%R&w R v O$7P%R&w R+ Difg^jlp `r^inrfbo s bk P%R&+Cbjlpqo^objlp nrbO%s&< O$%s&+Bljl s < P%r&m^o^ rk `fboql r ab R) qbkbjlp

OWP%r&Y< r u O$WP%r&Y< r)

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61/ Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

mrbpql nrb O u O$ plk ^j_^p fksbop^p mlo i^ fwnrfboa^- Olo `lkpfdrfbkqb O%s&< rv O$%s&< r) `lk il nrb O%s&< O$%s&m^o^ qlal v ab P%R&+Olo il q^kql O < O$

il nrb abjrbpqo^ nrb i^p fksbop^p mlo i^ fwnrfboa^ `lfk`fabk-Cbjlpqobjlp ^elo^ nrb qla^ fksbop^ mlo i^ fwnrfboa^ O bp q^j_f‹k fksbop^

mlo i^ abob`e^- Difg^jlp rk bibjbkql `r^inrfbo^ v bk P%R&+Cbjlpqo^objlp nrbPWO%s&Y< s+ Orbpql nrb s C P%R&)qbkbjlp s < P%r& m^o^ rk `fboql r ab R+ ObolO bp fksbop^ mlo i^ fwnrfboa^+ ^pŒnrb

r < OWP%r&Y< O%s&+

@mif`^kal P) iibd^jlp ^ P%r& < PWO%s&Y+Obol v < P%r&) `lk il nrb s < PWO%s&Y)

il `r^i `ljmibq^ i^ abjlpqo^`fŽk-Di qblobj^ nrb pfdrb `^o^`qbofw^ qla^p i^p crk`flkbp nrb qfbkbk fksbop^ mlo

i^ fwnrfboa^-

RCMPCK? 05-8- Ri\ api^d‡i Q8S x T od`i` diq`mn\ kjm g\ dulpd`m_\ nd vn‡gj nd Q \kgd^\ `g`h`iojn _dnodiojn _` S `i `g`h`iojn _dnodiojn _` T9 `noj `n*nd t n‡gj nd*k\m\ ^p\g`nlpd`m\ s ` v _` S*

'05-4( sxt dhkgd^\ P%r& w P%s&+

Kjo\8 K^ `lkaf`fŽk '05-4( bp bnrfs^ibkqb ^ i^ ^cfoj^`fŽk

'05-5( P%r&< P%s& fjmif`^ s ;t,

Tk^ crk`fŽk Q nrb p^qfpc^`b '05-4( l '05-5( m^o^ `r^ibpnrfbo^ r b v ab S pb abkljfk^ ohi\ pij bk S,

A`hjnom\^d‡i, Rrmlkd^jlp nrb P bp i^ fksbop^ mlo i^ fwnrfboa^ ab Q* v nrbP%r&< P%s&+Prbobjlp abjlpqo^o nrb r: u- @mif`^kal O) bk`lkqo^jlp OWP%r&Y:

OWP%s&Y+Orbpql nrb OW4$%r&Y< r v OWP%s&Y< v+ bpql fjmif`^ r < v- Blk biilnrba^ abjlpqo^al nrb rk^ crk`fŽk `lk fksbop^ mlo i^ fwnrfboa^ bp rkl ^ rkl bkpr aljfkfl-

Cbjlpqobjlp ^elo^ bi ob`Œmol`l- Rrmlkd^jlp nrb Q bp rkl ^ rkl bk U-Dk`lkqo^objlp rk^ crk`fŽk P8 Q&U( z U nrb bp fksbop^ ab Q mlo i^ fwnrfboa^-Rf s C P%R&) bkqlk`bp s < P%r& m^o^ rk `fboql r ab U- Dk sfoqra ab '05-5(+ bufp,qb _r[]n[g_hn_ oh r bk U m^o^ bi `r^i s < P%r&+Cbcfk^jlp O%s& ljl bpb r+ Dpql+bp+ abcfk^jlp O bk P%R& `ljl pfdrb9

O%s&< s fjmif`^ nrb P%r&< t,

Sbkbjlp bkqlk`bp OWP%r&Y< r m^o^ `^a^ r ab R) ^pŒnrb OP < gq* Olo `lkpf,drfbkqb+ i^ crk`fŽk P ^pŒabcfkfa^ bp fksbop^ ab Q mlo i^ fwnrfboa^-

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Qm\inajmh\^dji`n gdi`\g`n pij \ pij 610

CDEHMHBHˆM- P`\ Q8S x T pij \ pij `i S, I\ ˆid^\ diq`mn\ _` Q kjm g\dulpd`m_\ &g\^p\g n\]`hjn lp` o\h]d„i `n diq`mn\ kjm g\ _`m`^c\' n` _`ndbi\kjm Q+/, A`^dhjn lp` Q `n diq`mod]g`*v gg\h\hjn \ Q+gg\ diq`mn\ _` Q,

Klp obpriq^alp ab bpq^ pb``fŽk pb obcfbobk ^ crk`flkbp `r^ibpnrfbo^- Rbdrfa^,jbkqb ^mif`^jlp bp^p fab^p ^ i^p qo^kpcloj^`flkbp ifkb^ibp-

).&/ F_N[`S\_ZNPV\[R YV[RNYR`b[\ N b[\

Dk bpq^ pb``fŽk+ R v S obmobpbkq^k bpm^`flp ifkb^ibp `lk ilp jfpjlp bp`^,i^obp+ v Q8S x T bp rk^ qo^kpcloj^`fŽk ifkb^i ab , g%&S*T', K^ ifkb^ifa^a ab Qklp mbojfqb bumobp^o ab s^of^p j^kbo^p i^ molmfba^a ab nrb rk^ qo^kpcloj^`fŽkifkb^i pb^ rkl ^ rkl-

RCMPCK? 05-0/- P`\ Q8S x T pi\ om\inajmh\^d‡i gdi`\g _` , g%&S*T',Pji `lpdq\g`io`n g\n ndbpd`io`n kmjkjnd^dji`n,

]( Q `n pij \ pij `i S,_( Q `n diq`mod]g v np diq`mn\ Q+/8Q&S' x S `n gdi`\g,b( M\m\ oj_j s _` S* Q&s'< N dhkgd^\ s < N- Bnoj `n*`g iˆ^g`j K&Q' ^ji+

od`i` njg\h`io` `g `g`h`ioj ^`mj _` S,

A`hjnom\^d‡i, Cbjlpqo^objlp nrb ^( fjmif`^ _(+ _( fjmif`^ b(+ v b( fjmif,`^ ^(- Rrmlkd^jlp mofjbol nrb ^( bp `fboq^- Q qfbkb bkqlk`bp fksbop^ 'pbd•k biqblobj^ 05-8(+ v qbkbjlp nrb abjlpqo^o nrb Q+/ bp ifkb^i- Sljbjlp alp bibjbk,qlp `r^ibpnrfbo^ p v p ab Q&S', Dkqlk`bp p < Q&s' v p < Q&t' m^o^ ^id•k s v^id•k v ab R+ O^o^ alp bp`^i^obp `r^ibpnrfbo^ \ v \) qbkbjlp

\p * ]p < \Q&s' * ]Q&t' < Q&\s* ]t' *

v^ nrb Q bp ifkb^i- Krbdl+ ^mif`^kal Q+/* qbkbjlp

Q+g&\p* ]p' < \s * ]t < \Q+g&p'* ]Q+g&p'*

^pŒnrb Q+. bp ifkb^i- Olo `lkpfdrfbkqb ^( fjmif`^ _(-Rrmlkd^jlp pbdrfa^jbkqb nrb _( bp `fboq^- Sljbjlp rk s `r^inrfbo^ ab S

m^o^ bi `r^i Q&s' < N- @mif`^kal Q+.) bk`lkqo^jlp nrb s < Q+/&.' < N+ mrbpql

nrb Q+. bp ifkb^i- Olo `lkpfdrfbkqb+ _( fjmif`^ b(-Olo •iqfjl+ prmlkd^jlp `fboq^ b(- Sljbjlp alp bibjbkqlp `r^ibpnrfbo^ p

v q ab S pfbkal Q&p';Q&q', Olo i^ ifkb^ifa^a+ qbkbjlp Q&p+g%'< Q&p'+Q&p';< N+ ^pŒnrb p+p < N- Olo `lkpfdrfbkqb+ Q bp rkl ^ rkl bk S* u nrba^ `ljmib,q^a^ i^ abjlpqo^`fŽk abi qblobj^-

Page 284: Calculus

0++ Qm\inajmh\^dji`n gdi`\g`n t h\omd^`n

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k\m\ Q&S',

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!F^d`d < Nc:f

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Page 285: Calculus

Be`m^d^djn 0+,

Olo •iqfjl+ prmlkd^jlp `fboq^ a(- Cbjlpqo^objlp nrb P%r& < N fjmif`^s < N- Rb^ v`g * ,,, *`iw rk^ _^pb m^o^ R+ Rf s C R) mlabjlp bp`of_fo

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).&0 :WR_PVPV\`

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1- Rb^ R < xN+ 0+ 1y- Cbp`of_fo qla^p i^p crk`flkbp P7R w R m^o^ i^p `r^ibp P%R&< R+Dk qlq^i plk pbfp- CbpŒdkbkpblk PE$P0* ŠŠŠ * P

4X `lkpqorfo rk^ q^_i^ ab jriqfmif`^`fŽk

nrb jrbpqob i^ `ljmlpf`fŽk ab `^a^ m^o- Hkaf`^o `rŠibp plk rkl ^ rkl bk R) v a^o prpfksbop^p-

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x S0

pb abcfkb `lk i^cŽojri^ nrb pb a^ m^o^ P%r)s&) pfbkal %r)s& rk mrkql `r^inrfbo^ ab R

0Š Cbqbojfk^o bk

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1Š 2Dk `^a^ `^pl+

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Page 286: Calculus

613 Qm\inejmh\^dji`n gdi`\g`n u h\omd^`n

Dk ilp bgbo`f`flp abi 11 ^i 14+ R X Q obmobpbkq^k crk`flkbp `lk aljfkfl S v s^ilobpbk S, Dk dbkbo^i PQ ŠŠŠQP, Rf PQ < QP* ab`fjlp nrb R u Q ^jihpo\i,

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ab lqol jlal+ i^ fksbop^ ab PQ bp i^ `ljmlpf`fŽk ab i^p fksbop^p+ qlj^a^p bk loabkfksbopl-

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'Q * Q'0 < Q1 * 0PQ * Q0 u

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17- Rb^ R bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp j%r&+ Rb^k @ bi lmbo^alo abof,s^`fŽk v Q i^ qo^kpcloj^`fŽk ifkb^i nrb ^mif`^ k&s' bk sk%&s',^( Olkbo k&s' < 1 * 1s + s0 * 2s1 v abqbojfk^o i^ fj^dbk ab k ^ qo^s‹p ab `^a^ rk^ab i^p qo^kpcloj^`flkbp pfdrfbkqbp9 A* Q* AQ* QA* AQ + QA* NB1 , A0M,_( Cbqbojfk^o ilp mlifkljflp k ab S m^o^ ilp `r^ibp Q&k' < k,b( Cbqbojfk^o ilp mlifkljflp k ab S m^o^ ilp `r^ibp &AQ + 0A'&k' << N-a( Cbqbojfk^o ilp mlifkljflp k ab S m^o^ ilp `r^ibp &AQ + QA'i&k' < @lnj&+

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l%r& < j%K&)i

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Page 287: Calculus

Qm\inajmh\^dji`n gdi`\g`n ^ji q\gjm`n \ndbi\_jn 614

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Page 288: Calculus

4/3 Pl[hm`ilg[]cih_m fch_[f_m v g[nlc]_m

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Page 289: Calculus

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Page 290: Calculus

617 Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

A`hjnom\^d‡i, @mif`^kal Q ^ `^a^ rkl ab ilp jfbj_olp ab '05-00( u rqf,ifw^kal '05-0/(+ l_qbkbjlp

bk alkab `^a^ uz sfbkb a^a^ mlo '05-02(+ Dpql `ljmibq^ i^ abjlpqo^`fŽk-

G^_fbkal bibdfal rk m^o ab _^pbp %_f$ +++ )_h& v %q) ) +++ ) Th' m^o^ S v T*obpmb`qfs^jbkqb+ qla^ qo^kpcloj^`fŽk ifkb^i Q8S x T qfbkb rk^ obmobpbkq^`fŽkj^qof`f^i %nx^+Qb`Œmol`^jbkqb+ pf afpmlkbjlp ab hi bp`^i^obp `lil`^alp cloj^kalrk^ j^qofw ob`q^kdri^o %nce&v bibdfjlp rk m^o ab _^pbp loabk^a^p m^o^ S v T*bp cŠ`fi abjlpqo^o nrb bufpqb bu^`q^jbkqb rk^ qo^kpcloj^`fŽk ifkb^i Q8S x Tnrb qfbkb bp^ obmobpbkq^`fŽk j^qof`f^i- Cbcfkfjlp P pfjmibjbkqb `lk ilp bibjbkqlp_^pb ab S mlo jbafl ab i^p b`r^`flkbp '05-0/(- Dkqlk`bp+ pbd•k bi qblobj^ 05-01+bufpqb rk^ v pŽil rk^ qo^kpcloj^`fŽk P7R w S `lk bplp s^ilobp ^pfdk^alp- K^ fj^,dbk P%r&ab rk mrkql r ab R sfbkb bkqlk`bp a^a^ mlo i^p b`r^`flkbp '05-01( u'05-02(-

DIDLOKN 0- @jinomp^^d‡i _` pi\ om\inajmh\^d‡i gdi`\g \ k\modm_` pi\ h\+omdu_\_\, Rrmlkd^jlp nrb afpmlkbjlp ab i^ j^qofw 1 W 2-

Difg^jlp i^p _^pbp rpr^ibp ab sb`qlobp `lloabk^alp rkfq^oflp m^o^ S1 u S0Š Dk,qlk`bp i^ j^qofw a^a^ obmobpbkq^rk^ qo^kpcloj^`fŽk ifkb^i Q8S1 x S0 nrb ^mif`^rk sb`qlo `r^inrfbo^ &UF% r0* r8x& ab RE bk bi sb`qlo 'XH+U/& ab R0 ab ^`rboal `lki^p b`r^`flkbp ifkb^ibp

DIDLOKN 1- @jinomp^^d‡i _` pi\ m`km`n`io\^d‡i h\omd^d\g _` pi\ om\inajm+h\^d‡i gdi`\g _\_\, Rb^ S bi bpm^`fl ifkb^i ab qlalp ilp mlifkljflp ob^ibp k&s'ab do^al z 2- Dpqb bpm^`fl qfbkb afjbkpfŽk 3+ u bibdfjlp i^ _^pb '0+ s* s0* u!(-Rb^ @ bi lmbo^alo abofs^`fŽk nrb ^mif`^ `^a^ mlifkljfl j%r& ab R bk pr abof,s^a^ j$%r&+Olabjlp `lkpfabo^o @ `ljl rk^ qo^kpcloj^`fŽk ifkb^i ab R bk S)alkab T bp bi bpm^`fl qofafjbkpflk^i ab qlalp ilp mlifkljflp ob^ibp ab,do^al z 1-Dk T bibdfjlp i^ _^pb '0+ T) u!(+ O^o^ bk`lkqo^o i^ obmobpbkq^`fŽk j^qof`f^i ab Aobi^qfs^ ^ bp^ bib``fŽk ab _^pbp+ qo^kpcloj^jlp 'abofs^jlp( `^a^ bibjbkql _^pb

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O`km`n`io\^d‡i h\omd^d\g_` g\n om\inajmh\^dji`n gdi`\g`n 507

ab S X il bumobp^jlp `ljl rk^ `lj_fk^`fŽk ifkb^i ab ilp bibjbkqlp _^pb ab T,@pŒmrbp+bk`lkqo^jlp nrb

@%f&< M < M * Kr * Kr/) @%r&< 0 < 0 * Kr * Kr0*

@%r0' < /r < M * /r * Kr0 *

Klp `lbcf`fbkqbp ab bplp mlifkljflp abqbojfk^k i^p ^jgphi\n ab i^ obmobpbkq^`fŽkj^qof`f^i ab @+Olo `lkpfdrfbkqb+ i^ obmobpbkq^`fŽkmbafa^ sfbkb a^a^ mlo i^ pf,drfbkqb j^qofw 2 V 3-

O^o^ e^`bo klq^o bi eb`el ab nrb i^ obmobpbkq^`fŽkj^qof`f^i abmbkab kl pli^,jbkqb ab ilp bibjbkqlp _^pb pfkl q^j_f‹k ab pr loabk+ fksfoq^jlp bi loabk ab ilpbibjbkqlp _^pb bk T u rqfif`bjlp+ bk pr ird^o+ i^ _^pb loabk^a^ 'u!+ s* 0(- Dkqlk,`bp ilp bibjbkqlp _^pb ab S pb qo^kpcloj^k bk ilp jfpjlp mlifkljflp l_qbkfalp^kqbp+mbol ilp `ljmlkbkqbp ab ‹pqlp obi^qfslp ^ i^ krbs^ _^pb %r0

* r) 0( ^m^ob`bkbk loabk fksbopl- Olo `lkpfdrfbkqb+ i^ obmobpbkq^`fŽkj^qof`f^i ab A ^elo^ bp

B^i`ribjlp rk^ qbo`bo^ obmobpbkq^`fŽkj^qof`f^i ab @) rp^kal i^ _^pb'0+0 * r) 0 * r * r0

* 0 * r * r0 * u!( m^o^ S* v i^ _^pb %.)r)r0' m^o^ S+

Klp bibjbkqlp _^pb ab S pb qo^kpcloj^k ^pŒ9

@%f&< N+ @%. * r& < 0+ @%. * r * r/& < 0 * /r )

`lk il nrb i^ obmobpbkq^`fŽkj^qof`f^i bk bpqb`^pl bp

Page 292: Calculus

51. Qm\inajmh\^dji`n gdi`\g`n t h\omd^`n

).&)) 8\[`a_bPPVp[ QRb[N _R]_R`R[aNPVp[ZNa_VPVNYR[ S\_ZN QVNT\[NY

X^ nrb bp mlpf_ib l_qbkbo afpqfkq^p obmobpbkq^`flkbp j^qof`f^ibp ab rk^ qo^kp,cloj^`fŽk ifkb^i a^a^ jbaf^kqb i^ bib``fŽk ab _^pbp afpqfkq^p+ m^ob`b k^qro^ifkqbkq^o bibdfo _^pbp ab jlal nrb i^ j^qofw obpriq^kqb qbkd^ rk^ cloj^ il jŠppbk`fii^ mlpf_ib- Di qblobj^ nrb pfdrb morb_^ nrb mlabjlp e^`bo qlalp ilp bib,jbkqlp N bu`bmql ilp ab i^ af^dlk^i nrb s^ abpab bi s‹oqf`b prmboflo fwnrfboal ^ifkcboflo abob`el- @ il i^odl ab bp^ af^dlk^i e^_oŠ rk^ efibo^ ab rklp pbdrfalpab `bolp+ pfbkal bi k•jbol ab rklp fdr^i ^i o^kdl ab i^ qo^kpcloj^`fŽk- Tk^j^qofw &oyf' `lk qlalp ilp bibjbkqlp oyf < N `r^kal d ;/; f pb ii^j^ h\omdu _d\+bji\g, ,

RCMPCK? 05-03- P`\i S u T `nk\^djn gdi`\g`n _` _dh`ind‡i adido\* ^jiafj S < i u afj T < h, Ppkjib\hjn lp` Q C 1 &S*T' t lp` m< afj Q&S'm`km`n`io` `g m\ibj _` Q, Bsdno`i `ioji^`n pi\ ]\n` &`** ,,, *`i' k\m\ S v jom\%q) ) +++ ) qi' k\m\ T o\g`n lp`

'05-03( Q&`d'< Td k\m\ e< 0+1+--- +m*

t

'05-04( Q&`d'< N k\m\ e< m* 0+--- +i ,

Mjm ^jindbpd`io`* g\ h\omdu &odf'_` Q m`g\odq\ \ `n\n ]\n`n od`i` oj_jn gjn `g`h`iojn^`mj `s^`koj gjn m `g`h`iojn _` g\ _d\bji\g lp` q\g`i

no < n00 < --- < nll < 0 -

A`hjnom\^d‡i, Blkpqorfjlp mofjbol rk^ _^pb m^o^ T, Orbpql nrb Q&S'bp rk pr_bpm^`fl ab T `lk afj Q&S' < m*bi bpm^`fl Q&S' qfbkb rk^ _^pb ab mbibjbkqlp bk T* pb^k ‹pqlp r* *,,, *TmŠ Rbd•k bi qblobj^ 04-6+ bplp bibjbkqlpcloj^k rk pr_`lkgrkql ab rk^ `fboq^ _^pb m^o^ T, Olo `lkpfdrfbkqb mlabjlp ^a,grkq^o rklp bibjbkqlp Tl() ) 7++) Tg ab jlal nrb

'05-05(

pb^ rk^ _^pb m^o^ T,Rbdrfa^jbkqb `lkpqorfjlp rk^ _^pb m^o^ R+ B^a^ rkl ab ilp l mofjbolp

bibjbkqlp Tc ab '05-05( bp i^ fj^dbk mlo il jbklp ab rk bibjbkql ab R+Difg^jlp,rkl ab q^ibp bibjbkqlp ab R u Hi^j‹jlpib `,, Dkqlk`bp P%_x&< Sx m^o^ d < 0+1+--- +m ^pŒnrb '05-03( pb p^qfpc^`b- Rb^ ^elo^ e i^ afjbkpfŽk abi k•`ibl J%P&+Rbd•k bi qblobj^ 05-2 qbkbjlp h < e * l+ Orbpql ko_J%P& < e) bi bpm^`fl J%P&

Page 293: Calculus

@jinomp^^d‡i _` pi\ m`km`n`io\^d‡i h\omd^d\g `i ajmh\ _d\bji\g 51/

qfbkbrk^ _^pb nrb `lkpq^ ab f bibjbkqlp ab S nrb abpfdk^jlp mlo bn*q + ‘‘‘ ,`m9~,O^o^ `^a^ rkl ab bplp bibjbkqlp+ i^ b`r^`fŽk '05-04( pb p^qfpc^`b-Olo il q^kql+m^o^`ljmibq^o i^ abjlpqo^`fŽk+ qbkbjlp nrb abjlpqo^o nrb bi `lkgrkql loabk^al

'05-06(

bp rk^ _^pb m^o^ R+ X^ nrb afj R < h < l * e) pŽil qbkbjlp nrb abjlpqo^onrb bplp bibjbkqlp plk fkabmbkafbkqbp-Rrmlkd^jlp nrb rk^ `fboq^ `lj_fk^`fŽkifkb^i ab biilp pb^ `bol+ mlo bgbjmil

'05-07(

@mif`^kal Q Xe^`fbkal rpl ab i^p b`r^`flkbp '05-03(v '05-04(+ bk`lkqo^jlp nrb

o*0q o

GoQ&`o'<< G„oTo<< K+p,h p,h

Obol T/ * ŠŠŠ +Vn plk fkabmbkafbkqbp+v mlo q^kql Bh < --- < `*< N- Olo `lkpf,drfbkqb+ilp m mofjbolp q‹ojfklp ab '05-07( plk `bol+ mlo il `r^i '05-07( pb ob,ar`b ^

m)f

G,`9 < K+Œ<o*i

Obol _l(/ * ŠŠŠ )_l(e plk fkabmbkafbkqbpmrbpql nrb cloj^k rk^ _^pb m^o^ J%P&)v mlo q^kql--nF&<&--- :?l(e: N-Olo `lkpfdrfbkqb+ qlalp ilp _x ab '05-07( plk`bol+ irbdl ilp bibjbkqlp ab '05-06( cloj^k rk^ _^pb m^o^ S, Dpql `ljmibq^ i^abjlpqo^`fŽk-

DIDLOKN- Mlp obcbofjlp ^i bgbjmil 1 ab i^ pb``fŽk 05-0/+ alkab C bp bilmbo^alo abofs^`fŽk nrb ^mif`^ bi bpm^`fl R ab ilp mlifkljflp ab do^al 9#2bk bi bpm^`fl S ab ilp mlifkljflp ab do^al 9#1- Dk bpqbbgbjmil+ bi ob`loofalP%R&< S) ^pŒnrb P qfbkb o^kdl 2- @mif`^kal bi j‹qlal pbdrfal bk bi qblobj^05-03+ bibdfjlp `r^inrfbo _^pb m^o^ T* mlo bgbjmil i^ _^pb '0+ s* s0', Tk `lk,grkql ab mlifkljflp ab R nrb pb ^mif`^ pl_ob bplp bibjbkqlp bp %r) qu1+ .*r0&+

@jmif^jlp bpqb`lkgrkql m^o^ ildo^o rk^ _^pb m^o^ R ^agrkq^kal bi mlifkljfl`lkpq^kqb 0+nrb bp rk^ _^pb m^o^bi k•`ibl ab C- Olo `lkpfdrfbkqb+ pf bjmib^jlpi^ _^pb %r)qu1

+ fo+ 0( m^o^ R v i^ _^pb '0+ r) r0' m^o^ S) i^ `loobpmlkafbkqb

obmobpbkq^`fŽkj^qof`f^i m^o^C qfbkb i^ cloj^ af^dlk^i

Page 294: Calculus

510 Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

).&)* :WR_PVPV\`

Dk qlalp ilp bgbo`f`flp bk ilp nrb pb `lkpfabob bi bpm^`fl sb`qlof^i R+)i^ _^pb nrb pbrqfifw^oŠ pboŠ i^ ab ilp sb`qlobp `lloabk^alp rkfq^oflp pf kl pb af`b il `lkqo^ofl- Dk ilp bgbo,`f`flp obi^qfslp ^ i^ j^qofw ab rk^ qo^kpcloj^`fŽk ifkb^i Q8S x T pfbkal S < T* pf kl pbfkaf`^ il `lkqo^ofl qlj^objlp i^ jfpj^ _^pb bk S v bk T,

0- Cbqbojfk^o i^ j^qofw ab `^a^ rk^ ab i^p pfdrfbkqbp qo^kpcloj^`flkbp ifkb^ibp ab S, bk S,8^( i^ qo^kpcloj^`fŽk fa‹kqf`^+_( i^ qo^kpcloj^`fŽk `bol+b( jriqfmif`^`fŽk mlo rk bp`^i^o cfgl ^,

1- Cbqbojfk^o i^ j^qofw m^o^ `^a^ rk^ ab i^p pfdrfbkqbp molvb``flkbp-

']( Q8 S1 ,=, S0 Š

'_( P7 R1 ,=, R0*

'b( Q8 Up ,=, S0)

alkab Q&UF%U0 *s1' < 'Wi&U0'%alkab Q&UF%U0* U1' < &s0* U1'%alkab Q&sF%U0 *U1* U2 *sn' < &s0* U1* s2'}

2- Tk^ qo^kpcloj^`fŽk ifkb^i Q8S0

x S u ^mif`^ ilp sb`qlobp _^pb c v f `ljl pfdrb9

P%c&< c * g+ PQ& < /c * f -

^( B^i`ri^o P%0c* 1d&v L%0c * 1d&bk crk`fŽk ab c v g-_( Cbqbojfk^o i^ j^qofw ab Q v ab O-b( Qbplisbo i^ m^oqb_( pf i^ _^pb %c)d&pb obbjmi^w^ mlo %_f$ _t&) pfbkal _f < c * g+_/

: 1d * g-3- Tk^ qo^kpcloj^`fŽk ifkb^i P7R

0x R

0pb abcfkb ^pŒ9B^a^ sb`qlo %r)s& pb qo^kpcloj^ bk

pr pfj‹qof`l obpmb`ql ^i bgb v v irbdl pb armif`^ pr ilkdfqra m^o^ l_qbkbo P%r)s&+ Cb,qbojfk^o i^ j^qofw ab Q v i^ ab Q0,

4- Rb^ Q8S1

x S1

rk^ qo^kpcloj^`fŽk ifkb^i q^i nrb

P%e&< /c * 2g* 2e) S'g * e& < c) Q&d* f * f' <f , f ,

^( B^i`ri^o P%c* 1g* 0e&U abqbojfk^o i^ afjbkpfŽk abi k•`ibl v bi o^kdl ab P+_( Cbqbojfk^o i^ j^qofw ab P+

5- O^o^ i^ qo^kpcloj^`fŽk ifkb^i abi bgbo`f`fl 4+ pb `lkpfabo^k i^p alp _^pbp `lfk`fabkqbp`lk %_f$ _t$ _0&) pfbkal _f < '1+ 2+ 4(+ _/ < '0+ /+ N(+ _0 < '/+ 0+,0( X abqbojfk^o i^j^qofw Q obi^qfs^ ^ i^p krbs^p _^pbp-

6- Tk^ qo^kpcloj^`fŽk ifkb^i P7R1 x R0

^mif`^ ilp sb`qlobp _^pb `ljl pfdrb9 P%c&< 'N+N(+S'g( < '0+ 0(+P%e&< '0+ ,0(-^( B^i`ri^o P%1c* f * e&v abqbojfk^o i^ afjbkpfŽk abi k•`ibl v bi o^kdl ab P+_( Cbqbojfk^o i^ j^qofw ab Q,b( Tqfifw^kal i^ _^pb ‰,'*e& bk Ud v i^ 't-+ s1( bk R

0% pfbkal VH < '0+0(+ V < '0+1(+

abqbojfk^o i^ j^qofw&ab Q obi^qfs^ ^ bp^p _^pbp- 1

a( G^ii^o i^p _^pbp %_F) _/) _0& m^o^ S0 v 'Vq+ s1( m^o^ S/ m^o^ i^p `r^ibp i^ j^qofw ab

Q qbkd^ i^ cloj^ af^dlk^i-7- Tk^ qo^kpclzj^`fŽk ifkb^i P7R0 x R

1^mif`^ ilp sb`qlobp _^pb `ljl pfdrb9 P%c&:

%.)-).&) P%d&< ',0+/+0(-^( B^i`ri^o P%/c* 0d&v abqbojfk^o i^ afjbkpfŽk abi k•`ibl v bi o^kdl ab P+_( Cbqbojfk^o i^ j^qofw ab P+b( G^ii^o _^pbp %_))_t& m^o^ S0 v 'tI - V1+ V02( m^o^ S

1m^o^ i^p `r^ibp i^ j^qofw- ab Q

qfbkb cloj^ af^dlk^i-

8- Qbplisbo bi bgbo`f`fl 7 pf P%c&<'0+ /+ 0( u P%d&< '0+ 0+ 0(-

Page 295: Calculus

Bnk\^djn gdi`\g`n _` h\omd^`n 622

0/- Rb^k S u T alp bpm^`flpifkb^ibp+ j_lp ab afjbkpfŽk 1 u `lk i^ jfpj^ _^pb %_f$_/&+Rb^ Q8S ,* T rk^ qo^kpcloj^`fŽk ifkb^i q^i nrb

P%_f * _t& < 0_f * a_t) P%0_f * /_/& < 6bh * /0_/ +

^( B^i`ri^o Q&`/ * _f& u abqbojfk^o i^ afjbkpfŽk abi k•`ibl u bi o^kdl ab Q,_( Cbqbojfk^o i^ j^qofw ab Q obi^qfs^^ i^ _^pb a^a^-b( Tqfifw^om^o^ S i^ _^pb %_f$_/& v e^ii^o rk^ krbs^ _^pb ab i^ cloj^ 'bi * [_/)1b

h* \_

0' m^o^S) m^o^i^ nrb i^ j^qofw ab P qbkd^i^ cloj^ af^dlk^i-

Dk bi bpm^`fl ifkb^i ab qla^p i^p crk`flkbp ob^ibp+^a^ rkl ab ilp pfdrfbkqbp lkgrkqlpbp fkabmbkafbkqbv dbkbo^rk pr_bpm^`fl R ab afjbkpfŽk cfkfq^-Tqfifw^obi `lkgrkql a^al`ljl _^pb m^o^S v pb^ C9S ,* S bi lmbo^alo abofs^`fŽk-Dk `^a^ `^pl+ e^ii^o i^ j^qofwab C v i^ ab C1 obi^qfs^ i^ _^pb nrb pb bifdb-00- 'pbku+blp r&+ 04- ',blp r) pbkr&+01- '0+r) b!&(- 05- 'pbkr) _imr) r pbkr) r_im r&+02- '0+0 * r) 0 * r * b!&(- 06- 'b!&pbku+_T blp r&+03- 'b!&+r_!$&+ 07- 'b1!&pbk0r) _T blp 0r&+08- Dibdfoi^ _^pb '0+ s* s0* WR( bk bi bpm^`fl ifkb^i S ab qlalp&ilp mlifkljflp ob^ibpab

do^al 9p::2- Rb^k C bi lmbo^alo abofs^`fŽk v Q8S ,* S i^ qo^kpcloj^`fŽk ifkb^i nrb^mif`^ j%r& bk rj$%r&+ Blk obi^`fŽk^ i^ _^pb a^a^+ abqbojfk^o i^ j^qofw ab `^a^ rk^ab i^p qo^kpcloj^`flkbp pfdrfbkqbp9^( Q9 _( AQ9 b( QA9 a( QA + AQ9 b( QF9b( QgA0 + A0Q0,

1/- Blk obpmb`ql i bgbo`f`fl 08- Rb^T i^ fj^dbk ab U^ qo^s‹pab QA, G^ii^o _^pbp m^o^S v T m^o &i^pnrb i^ j^qofw QA qbkd^cloj^ af^dlk^i-

).&)+ :`]NPV\` YV[RNYR`QRZNa_VPR`

Gbjlp sfpql `Žjl i^p j^qof`bp pb mobpbkq^kbpmlkqŠkb^jbkqb `ljl obmobpbk,q^`flkbp ab i^p qo^kpcloj^`flkbp ifkb^ibp- S^j_f‹k pb mrbabk `lkpfabo^o i^p j^,qof`bp`ljl bibjbkqlp bufpqbkqbplk fkabmbkabk`f^ ab i^p qo^kpcloj^`flkbp ifkb^,ibp- Bljl q^ibpbibjbkqlp+ cloj^k lqo^ `i^pb ab l_gbqlp j^qbjŠqf`lp nrb mrbabkabcfkfopbmlo jbafl ab i^p lmbo^`flkbp ^idb_o^f`^p nrb mrbabk ob^ifw^opb`lkbiilp- K^ obi^`fŽk `lk i^p qo^kpcloj^`flkbp ifkb^ibp a^ lofdbk ^ bp^p abcfkf`flkbp+mbol q^i obi^`fŽk pboŠmlo bi jljbkql fdklo^a^-

Rb^k j v i alp bkqbo^pmlpfqfslp v pb^ / ŠŠŠ,ŠŠbi `lkgrkql ab qlalp ilp m^obpabbkqbolp %c)g( q^ibpnrb 0 z d ph* 0 z g z i8 Br^inrfbo crk`fŽk = `rvl aljfkflpb^ / ŠŠŠ,ŠŠpb abkljfk^ h\omduj W i, Di s^ilo ab i^ crk`fŽk >&d*g( pb ii^j^ `g`h`i+ni cd ab i^ j^qofw v pb abpfdk^oŠ q^j_f‹k mlo [c8+Noafk^of^jbkqb pb afpmlkbkqlalp ilp s^ilobp ab i^ crk`fŽk bk rk ob`qŠkdril nrb `lkpq^ ab j cfi^p v i `l,irjk^p+ abi jlal pfdrfbkqb

[.. [./ [.h

-/. -// [/h

Page 296: Calculus

623 Qm\inajmh\^dji`n gdi`\g`n t h\omd^`n

Klp bibjbkqlp \p mrbabk pbol_gbqlp ^o_fqo^oflpab k^qro^ibw^`r^inrfbo^- Mloj^i,jbkqb pboŠkk•jbolp ob^ibpl `ljmibglp+ mbol ^ sb`bp `lksfbkb `lkpfabo^o j^qof`bp`rvlp bibjbkqlp plk lqolp l_gbqlp+mlo bgbjmil+ crk`flkbp- S^j_f‹k abpfdk^objlpi^p j^qof`bp jbaf^kqb i^ klq^`fŽk ^_obsf^a^

= < &\d)&!$77x/*1+ l > < %[cd&+

Rf h < i* i^ j^qofw pb ii^j^ ^p\_m\_\, Tk^ j^qofw 0 W i pb ii^j^ h\omdu adg\9rk^ j^qofw h W 0 bp rk^ h\omdu, ^jgphi\,

Clp crk`flkbp plk fdr^ibp pf v pŽil pf qfbkbk bi jfpjl aljfkfl v qlj^k ilpjfpjlp s^ilobp bk `^a^ bibjbkql abi aljfkfl- Orbpql nrb i^p j^qof`bp plk crk,`flkbp+ alp j^qof`bp > < &\dy'v ? < &]yy'plk fdr^ibp pf v pŽil pf qfbkbk bi jfpjlk•jbol ab cfi^p+bi jfpjl k•jbol ab `lirjk^p+ b fdr^ibp bibjbkqlp \yy < ],y m^o^`^a^ m^o %c)d&+

Rrmlkd^jlp ^elo^ nrb ilp bibjbkqlp plk k•jbolp 'ob^ibp l `ljmibglp( vabcfk^jlp i^ ^af`fŽk ab j^qof`bp v i^ jriqfmif`^`fŽk mlo bp`^i^obp pfdrfbkal bijfpjl j‹qlal nrb m^o^crk`flkbp ob^ibpl `ljmibg^p `r^ibpnrfbo^-

CDEHMHBHˆM- Pd > < &\dy's ? < &]yy'nji _jn h\omd^`n h W i s nd,` `n pi`n^\g\m ^p\glpd`m\* _`adidhjn g\n h\omd^`n > * ? t ^> _`g hj_j ndbpd`io`

> * ? < &\9e* ]de' * ^> < &^\9e',

I\ nph\ n‡gj n` _`adi` ^p\i_j > t ? od`i`i `g hdnhj o\h\†j h V i,

DIDLOKN- Rf

> < Y 0+ ,9\ ? < Zz

N ziu,0 N '+

qbkbjlp bkqlk`bp

> * ? < Z9+ ,zi < + - ,9i <'. N

,0\0> : &+g'? :'+ ,1 N ,0 1 ',

Cbcfkfjlp i^ j^qofw N `ljl i^ j^qofw h W i `rvlp bibjbkqlp plk qlalp N-Blk bp^p abcfkf`flkbp+bp fkjbaf^ql bi bgbo`f`fl ab `ljmol_^o nrb bi `lkgrkql abqla^p i^p j^qof`bp h W i bp rk bpm^`fl ifkb^i- Kl abpfdk^jlp `lk Jh*i, Rf ilp bib,jbkqlp plk k•jbolp ob^ibp+bi bpm^`fl Jh*i bp rk bpm^`fl ifkb^i ob^i- Rf plk k•,jbolp `ljmibglp+ Lj i bp rk bpm^`fl ifkb^i `ljmibgl- Dp q^j_f‹k cŠ`fi abjlpqo^o

Page 297: Calculus

Fnjhjmadnhj `iom` om\inajmh\^dji`n gdi`\g`n v h\omd^`n 513

nrb bpqbbpm^`fl bp ab afjbkpfŽk h W i, Dk bcb`ql+ rk^ _^pb m^o^ Jh*i `lkpq^ab hi j^qof`bp nrb qfbkbk rk bibjbkql fdr^i ^ 0 v qlalp ilp abjŠp fdr^ibp ^ N-Olo bgbjmil+ i^p pbfp j^qof`bp

Z0 • /\+} } }

ZN 0 /\+} } }

ZN } 0\+ Z/ } /\+} } } iNN

ZN } /\+/0/ ZN • /\+

N N 0

cloj^k rk^ _^pb m^o^ bi `lkgrkql ab qla^p i^p j^qof`bp 1 W 2-

).&), =`\Z\_SV`Z\ R[a_Ra_N[`S\_ZNPV\[RYV[RNYR`e ZNa_VPR`

Ulis^jlp ^elo^ ^ i^ obi^`fŽk bkqob j^qof`bp v qo^kpcloj^`flkbp ifkb^ibp-Rb^k S v T alp bpm^`flp ifkb^ibp ab afjbkpfŽk cfkfq^`lk afj S < i v afj T < h,Difg^jlp rk^ _^pb %_f) +++ ) _)x& m^o^ S v lqo^ %Sf) +++ ) Sh' m^o^ T, Dk bpq^afp`rpfŽk bp^p_^pbp pb j^kqfbkbk cfg^p-Cbpfdkbjlp`lk /%R) S& bi bpm^`fl ifkb^iab qla^p i^p qo^kpcloj^`flkbp ifkb^ibp ab R bk S+ Rf RC /%R) S&) pb^ g%P&i^j^qofw ab P obi^qfs^^ i^p _^pbpa^a^p- Qb`loabjlp nrb g%P& pb abcfkb `ljl pfdrb-

K^ fj^dbk ab `^a^ bibjbkql _^pb _e pb bumobp^ ljl rk^ `lj_fk^`fŽk ifkb^iab ilp bibjbkqlp _^pb ab T8

'05-i8(h

Q&`e& < GnceSc m^o^ f < 0+1+--- +i ,c:f

Klp jriqfmif`^alobp bp`^i^obp nce plk ilp bibjbkqlp ce ab g%P&+@pŒmrbp+qbkbjlp

'05-1/(

K^ b`r^`fŽk '05-1/( abcfkb rk^ krbs^ crk`fŽk g `rvl aljfkfl bp /%R) S& v`rvlp s^ilobp plk j^qof`bp ab Jh*i, Orbpql nrb qla^ j^qofw h W i bp i^ j^qofwg%P& m^o^ rk^ `fboq^ P ab 1' S* T'* bi ob`loofal ab g bp Jh*i, Di qblobj^ pf,drfbkqb morb_^ nrb i^ qo^kpcloj^`fŽk g7 0&S* T' ,,* Jh*i bp ifkb^i v rkl ^ rklbk /%R) S&+

SDNQDL@ 05-04- SDNQDL@ CD HRNLNQEHRLN- M\m\ ^p\g`nlpd`m\ P v Q _`1' S* T' v oj_jn gjn `n^\g\m`n `* o`i`hjn

g_O * P& < g_O& * g%P& u g%]P& < ]g%P& +

>_`hƒn*g_O& < g%P&

\n…lp` h `n pij \ pij `i 1' S* T',

dhkgd^\ R< Q*

Page 298: Calculus

625 Pl[hm`ilg[]cih_m fch_[f_m s g[nlc]_m

@_gimnl[]cƒh+ K^ j^qofw g%P& bpqŠcloj^a^ `lk ilp c^`qlobp nh ab '05-08(-Cbi jfpjl jlal+ i^ j^qofw g_O& bpqŠ`lkpqfqrfa^ `lk ilp c^`qlobp Oceab i^p b`r^,`flkbp

'05-10(i

P&`f';0ndfTd m^o^ f;/*0* ,,, *i,fzi

Orbpql nrb qbkbjlpi

'R * Q'&`f' < 1 &Pdg%* odf'Tdfzi

ug

&^Q'&`f' < 1 &^odf'Td *c:f

l_qbkbjlp g_O * P& < %Oce* nce&< g_O& * g%P& v g%]P& < %]nce&< ]g%P&+

Dpql abjrbpqo^ nrb g bp ifkb^i+O^o^ abjlpqo^o nrb g bp rkl ^ rkl+ prmlkd^jlp nrb g_O& < g%P&) pfbkal

R < %Oce&v P < %nc^+K^p b`r^`flkbp '05-08( v '05-10( abjrbpqo^k nrb O%_e&:: P%_^ m^o^`^a^ bibjbkql _^pb _m)^pŒnrb O%r&< P%r& m^o^qlal r ab R) u mloq^kql R < Q,

L]n`mq\^d‡i8 K^ crk`fŽk h bp rk dnjhjmadnhj, Dibdfa^p rk^p _^pbp+ h bpq^_ib`brk^ `loobpmlkabk`f^ rkl ^ rkl bkqob bi `lkgrkql ab i^p qo^kpcloj^`flkbp i& %R)S& vbi `lkgrkql Jh ,Š ab i^p j^qof`bp h W i, K^p lmbo^`flkbp ab ^af`fŽk u jriqfmif`^`fŽk mlobp`^i^obp pb `lkpbos^k ^ qo^s‹p ab bp^ `loobpmlkabk`f^- Klp bpm^`flp ifkb^ibp i& %R)S&u Jh ,i pb af`b nrb plk dnjhjmajn, Hk`fabkq^ijbkqb+ bi qblobj^ 05-00 abjrbpqo^ nrb bialjfkfl ab rk^ qo^kpcloj^`fŽk ifkb^i rkl ^ rkl qfbkb i^ af[S0mzz9zkfdr^i ^ pr ob`loofal-Olo `lkpfdrfbkqb+ afj g%&S*T' < afj Jh ,i < hi,

Rf R < S u bibdfjlp i^ jfpj^ _^pb m^o^^j_lp+ i^ j^qofw g%f& `loobpmlk,afbkqb ^ i^ qo^kpcloj^`fŽk fa‹kqf`^ F8S x S bp rk^ j^qofw af^dlk^i `lk ilp bib,jbkqlp ab i^ af^dlk^i fdr^ibp ^ 0 u qlalp ilp abjŠp fdr^ibp ^ N- Dpq^pb ii^j^c^_hnc^[^ l g[nlct ohc^[^ u pb abpfdk^ `lk F l `lk Fi,

).&)- @bYaV]YVPNPVp[QRZNa_VPR`

@idrk^p qo^kpcloj^`flkbp ifkb^ibp mrbabk jriqfmif`^opb mlo jbafl ab i^ `lj,mlpf`fŽk- Cbcfkfobjlp ^elo^ i^ jriqfmif`^`fŽk ab j^qof`bp ab j^kbo^ nrb bi mol,ar`ql ab alp j^qof`bp `loobpmlka^ ^ i^ `ljmlpf`fŽk ab i^p qo^kpcloj^`flkbp ifkb^,ibp nrb biilp obmobpbkq^k-

Qb`loabjlp nrb pf Q8R x S X R9S x T plk qo^kpcloj^`flkbp ifkb^ibp+pr`ljmlpf`fŽk OP7Q w S bp rk^ qo^kpcloj^`fŽk ifkb^i a^a^ mlo

OP%r&< OWP%r&&m^o^qlal r ab Q +

Rrmlkd^jlp nrb Q) R) X S plk ab afjbkpfŽk cfkfq^+mlo bgbjmil

afj R < i * afj S; j+ afj T; h,

Page 299: Calculus

Jpgodkgd^\^d‡i _` h\omd^`n 0,0

Difg^jlp _^pbp m^o^ Q) R) X S+ Blk obi^`fŽk ^ bp^p _^pbp g_O& bp rk^ j^qofwh W k* Q bp rk^ j^qofw k W i* u PQ bp rk^ j^qofw h W i, K^ pf,drfbkqb abcfkf`fŽk ab jriqfmif`^`fŽk ab j^qof`bp klp mbojfqb abar`fo i^ obi^`fŽkh&PQ' < h&P'h&Q', Dpql buqfbkab ^ ilp molar`qlp i^ molmfba^a ab fpljlocfpjl-

CDEHMHBHˆM- P`\i > pi\ h\omdu h V l ^p\glpd`m\* v ? pi\ h\omdu l V i^p\glpd`m\* o\g`n ^jhj

v > < %\${FU{wZf$&.-+0,

Bg kmj_p^oj >? n` _`adi` ^jhj g\ h\omdu h V i b < &@de' ^ptj `g`h`ioj ddqd`i`_\_j kjm

'05-11($L

?cd < G[ce\ed +FRG

L]n`mq\^d‡i8 Di molar`ql >? pŽil bpqŠ abcfkfal pf bi k•jbol ab `lirjk^p ab= bp fdr^i ^i ab cfi^p ab >+

Rf bp`of_fjlp >d m^o^ bumobp^o i^ cfi^ c ab > u ?% m^o^ i^ `lirjk^d ab >) u i^p fj^dfk^jlp `ljl sb`qlobp ab afjbkpfŽk m+i^ prj^ '05-11( bp pfj,mŒbjbkqb bi molar`ql bp`^i^o >y , ?%*Dp ab`fo+ bi bibokbkql-qg ab >? bp bi molar`qlbp`^i^o ab i^ cfi^ d ab > mlo i^ `lirjk^ d ab ?8

>? + &> , ?e'h,i, e c)d:fz

@pŒmrbp+ i^ jriqfmif`^`fŽk ab j^qof`bp mrbab `lkpfabo^opb `ljl rk^ dbkbo^ifw^,`fŽk abi molar`ql bp`^i^o-

DIDLOKN i- Rb^k > < Z 2,0

u ? bp 2 W 1+ bi molar`ql >? bp i^ j^qofw 1 W 1

10\ -'0

Klp bibjbkqlp ab => pb `^i`ri^k ^pŒ

@i&>f < 2• 3 * i+ 4 * 1• N < 06+ = ! ?0 < 2 - 5 * 0 - ' , 0( * 1 - 1 < 10 +

>0% >f < ',0( - 3 * 0 - 4 * N-/< 0+ =0Š ?0 < ',0( - 5 * 0 - ',0( * N&1 < ,6 -

Page 300: Calculus

516 Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

DIDLOKN 1- Rb^

u

@nrŒ= bp 1 W 2 X > bp 2 Wi+ `lk il nrb => bp i^ j^qofw 1 W 0 a^a^ mlo

>? < X>g%?gZ < Z,8\+>/† >f 7

Orbpql nrb >g%?g < 1• ',1( * 0 - 0 * ',2( -1 < ,8 X >0Š ?g < 0 - ',1( ** 1• 0 * 3 - 1 < 7-

DIDLOKN 2- Rf = X > plk alp j^qof`bp `r^ao^a^p abi jfpjl q^j^•l+ bkqlk,`bp => u >= bpqŠk abcfkfa^p- Olo bgbjmil+ pf

v

bk`lkqo^jlp nrb

Z02 7\>?; *

1 ,1 Z,0 0/\?> < -

2 01

Dpqb bgbjmil morb_^ nrb bk dbkbo^i => :B >=+ Rf => < >=) ab`fjlp nrb = u >nji k`mhpo\]g`n l nrb ^jihpo\i,

DIDLOKN 3- Rf Fk bp i^ j^qofw fabkqfa^a k W k*`ioji^`n Fk> < > m^o^ qla^j^qofw >* j W i* v ?g9 < ? m^o^ qla^ j^qofw ?* h W j+ Olo bgbjmil-

Cbjlpqo^jlp pbdrfa^jbkqb nrb i^ j^qofw ab rk^ `ljmlpf`fŽk PQ bp bi mol,ar`ql ab i^p j^qof`bp g_O& u gc$P&+

Page 301: Calculus

Jpgodkgd^\^d‡i _` h\omd^`n 628

SDNQDL@ 05-05- P`\i Q8S ,* S u P8S ,* T _jn om\inajmh\^dji`n gdi`\g`n*_ji_` S* S* T nji `nk\^djn gdi`\g`n _` _dh`ind‡i adido\,Bioji^`n* `g`bd_\n pi\n]\n`n ade\n*g\n h\omd^`n _` Q+ Q W PQ `noƒi m`g\^dji\_\n kjm g\ `^p\^d‡i

h&PQ' < h&P'h&Q' ,

A`hjnom\^d‡i, Rrmlkd^jlp nrb afj S < i* afj S < k* afj T < h, Rb^k'rH +‘‘‘ +r++(rk^ _^pb,m^o^R) 'UH&‘‘‘ &Rk' rk^ _^pb m^o^R) v 'VH+ ‘‘‘ )+q!)&rk^_^pb m^o^ T, Blk obi^`fŽk ^ bp^p _^pbp qbkbjlp

&!h`P' < %Oc,&$7787$.$alkab P&q**'< Gny!ry m^o^ f < 0+1+--- +k*•*.

u10

h&Q' < &od-'8%y8%/%alkab Q&pd'< Fo!dS! m^o^ f < 0+1+--- +i,!'*

Olo `lkpfdrfbkqb+ qbkbjlp

`lk biil bk`lkqo^jlp nrb

X^ ebjlp l_pbos^al nrb i^ jriqfmif`^`fŽk ab j^qof`bp kl pfbjmob p^qfpc^`bi^ ibv `lkjrq^qfs^- Di qblobj^ pfdrfbkqbmorb_^ nrb p^qfpc^`bi^p ibvbp ^pl`f^qfs^v afpqof_rqfs^-

SDNQDL@ 05-06- KDXDR @RNBH@SHU@X CHRSQHATSHU@O@Q@K@ LTKSHOKHB@,BHˆM CD L@SQHBDR- A\_\n g\n h\omd^`n >* ?* B-

^( Pd gjn kmj_p^ojn >&?@'v &>?'@ od`i`i n`iod_j* o`i`hjn

>,&?@'< &>?'@ &g`t \nj^d\odq\',

_( Ppkjib\hjn lp` > t ? n`\i _`g hdnhj o\h\†j, Pd >@ t ?@ od`i`i n`i+od_j*o`i`hjn

&>* ?'@ < >@ * ?@% &g`t _dnomd]podq\kjm g\ _`m`^c\'*

`i o\ioj lp` nd @> t @? od`i`i n`iod_j* o`i`hjn

@&>* ?' < @>* @? &g`t_dnomd]podq\kjm g\ dulpd`m_\',

Page 302: Calculus

0-) Pl[hm`ilg[]cih_m fch_[f_m u g[nlc]_m

@_gimnl[]cƒh+ Dp^pmolmfba^abpmrbabk abar`fopb afob`q^jbkqb ^ m^oqfoabi^ abcfkf`fŽk ab jriqfmif`^`fŽk ab j^qof`bp+ mbol mobcbofjlp o^wlk^o abi pfdrfbkqbjlal- Hkqolarw`^jlp ilp bpm^`flp ifkb^ibp ab afjbkpfŽk cfkfq^Q) R) r* W v i^pqo^kpcloj^`flkbp ifkb^ibp P7Q w R) R9R w S) N7S w W q^ibp nrb cfg^a^prk^p_^pbp+qbkbjlp

= < g%N&) > < g_O&) A < h&Q',

Rbd•k bi qblobj^ 05-05+ bp g%NO&< => X g%OP&< >?+ Cb i^ ibv ^pl`f^qfs^m^o^ i^ `ljmlpf`fŽk+ bk`lkqo^jlp nrb N%OP&< %NO&P+@mif`^kal bi qblobj^05-05 rk^ sbw jŠp ^ bp^ b`r^`fŽk+ l_qbkbjlp g%N&g%OP&< g%NO&g%P&l=%>?& < %=>&?) nrb abjrbpqo^ ^(- K^ abjlpqo^`fŽk ab _( mrbab e^`bopb `lkrk o^wlk^jfbkql m^ob`fal-

CDEHMHBHˆM- Oc= _moh[ g[nlct ]o[^l[^[) ^_`chcgim f[ jin_h]c[ _hn_l[ ^_= jil ch^o]]cƒh ]igi mcao_7

>L < .) >i < >>i+g m^o^i x 0-

).&). :WR_PVPV\`

0- Rf > < Y 0,0

'-

3 ,zi> < Z,9 9H+B < Zz ,z\+ `^i`ri^o > * B+ =>)

4 ,1 i ,2>=) =?) ?=) =%/> * 0?&+

1- Rb^ = < Zz yH-G^A^o qla^p i^p j^qof`bp >) 1 W 1+ q^ibp nrb ^( => < N9 _( >= < N-

2- G^ii^o bk `^a^ `^pl \* \) `* ^ m^o^ nrb pb p^qfpc^d^ i^ b`r^`fŽk a^a^-

J /

2 1

Page 303: Calculus

Be`m^d^djn 630

3- B^i`ri^o bk `^a^ `^pl => * >=+

n 1 1\ Z,: 0

RQ'^( = < 1 0 1+ 85 1

0 1 2 1

y Z: N 9i > w W 7,1\

'_( = 0 ,1 3 -

,0 1 iI ,2 4 00

4- Rf > bp rk j^qofw `r^ao^a^+ abjlpqo^o nrb > i>h < > i)h m^o^ qlalp ilp bkqbolph x N+ i x N-

5- Rb^ @ Z0 iI- Bljmol_^o nrb = 1 << N 0 Z/0 10I X `^i`ri^o > i,

Z

BNR K6- Rb^ > < -

pbk L,pbk /I - ZBNR 0.

Bljmol_^o nrb > 1 <BNR K pbk1/

,pbk1/Iv `^i`ri^o > i,

`lp 0.

0 9\- Bljmol_^o nrb >0 < Yy y y[- B^i`ri^o >1 v >%, Rrmlkbo

/0 //0

rk^ cŽojri^ dbkbo^i m^o^ >i u abjlpqo^oi^ mlo fkar``fŽk-

8- Rb^ > < Z]9 yH-Cbjlpqo^o nrb > 1 < 0> + / X `^i`ri^o > /..,

0/- G^ii^o qla^p i^p j^qof`bp =) 1 W 1+ q^ibp nrb = 1 < N-00- ^( Ool_^o nrb rk^ j^qofw =) 1 W 1+ `lkjrq^& `lk `r^inrfbo j^qofw 1 W 1 pf v pŽil pf

> `lkjrq^ `lk `^a^ rk^ ab i^p `r^qol j^qof`bp

_( G^ii^o qla^p bp^p j^qof`bp =+01- K^ b`r^`fŽk >0 < . pb p^qfpc^`b m^o^ `^a^ rk^ ab i^p j^qof`bp 1 W 1

Page 304: Calculus

0-+ Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

alkab \ X ` plk k•jbolp ob^ibp ^o_fqo^oflp- G^ii^o qla^p i^p j^qof`bp =) 1 W 1+ q^ibpnrb >0 < g,

02- Rf > ; Z 1 ,HI,1 2

=? :> v @= < >+

Z6 5I -X > < 8 7 & e^ii^o j^qof`bp B v C+ 1 W 1+ q^ibp nrb

03- ^( Bljmol_^o nrb i^p fabkqfa^abp ^idb_o^f`^p

%= * >&/ < =/ * /=> * >/ u %= * >&%= * >& < =/ * >/

kl plk `fboq^p m^o^ i^p j^qof`bp 1 W 1+ = < Yy ,,zI v > < Yy yH-_( Llafcf`^o bi pbdrkal jfbj_ol ab bp^p fabkqfa^abp m^o^ l_qbkbo cŽojri^p sŠifa^pm^o^ qla^p i^p j^qof`bp `r^ao^a^p = v >+b( ƒO^o^ nr‹ j^qof`bp = v > plk sŠifa^p i^p fabkqfa^abp bpq^_ib`fa^p bk ^(>

).&)/ EV`aRZNQRRPbNPV\[RYV[RNYR`

Rb^ = < %[cd&rk^ j^qofw g W h ab k•jbolp a^a^+ v pb^k AG+ŠŠŠ+@h lqolph k•jbolp- Tk `lkgrkql ab h b`r^`flkbp ab i^ cloj^

'05-12(i

I\dfsf < `* m^o^ c < )$*$ &&&$h*fx/

pb ii^j^ pfpqbj^ ab h b`r^`flkbp ifkb^ibp `lk i fk`Ždkfq^p- Blkpfabo^jlpWi + ‘-‘ +u+ `ljl fk`Ždkfq^p- Tk^ njgp^d‡i abi pfpqbj^ bp rk^ k,mi^ `r^inrfbo^ abk•jbol &Ug% ŠŠŠ * Ui' m^o^ ilp nrb pb p^qfpc^`bk qla^p i^p b`r^`flkbp- K^ j^qofw >pb ii^j^ h\omdu _` gjn ^j`ad^d`io`n abi pfpqbj^-

Klp pfpqbj^p ifkb^ibp mrbabk bpqraf^opb mlo jbafl ab i^p qo^kpcloj^`flkbpifkb^ibp- Dibdfjlp i^p _^pbp rpr^ibp ab sb`qlobp `lloabk^alp rkfq^oflp bk Ri u Rh,

K^ j^qofw ab ilp `lbcf`fbkqbp > abqbojfk^ rk^ qo^kpcloj^`fŽk ifkb^i+ Q8Si x S h*

nrb ^mif`^ rk sb`qlo ^o_fqo^ofl T < %r) ) +++ ) ri' ab Ri bk bi sb`qlo u < 'XH + --- +

Vh' ab S h a^al mlo i^p b`r^`flkbp ifkb^ibp

i

Vd< I\dfsf m^o^ c: 0+1+ --- +h,!5*

Page 305: Calculus

Pdno`h\n _` `^p\^dji`n gdi`\g`n 521

Rb^ ` < '`0 +‘‘‘ +@h' bi sb`qlo ab S i `rvlp `ljmlkbkqbp plk ilp k•jbolp nrb^m^ob`bk bk bi pfpqbj^ '05-12(- Dpqbpfpqbj^ mrbab bp`of_fopbjŠp pbk`fii^jbkqbmlkfbkal

P%r&< `-

Di pfpqbj^ qfbkb rk^ plir`fŽk pf v pŽil pf ` bpqŠbk bi ob`loofal ab P+Rf rk plils ab Si pb ^mif`^ bk `* bi pfpqbj^ qfbkb rk^ pli^ plir`fŽk- Rf jŠp ab rk s pb^mif`^ bk `* bi pfpqbj^ ^ajfqb jŠp ab rk^ plir`fŽk-

DIDLOKN 0- Ri ndno`h\ ndi njgp^d‡i, Di pfpqbj^ s * t < 0+ s * t < 1kl qfbkb plir`fŽk- K^ prj^ ab alp k•jbolp kl mrbab pbo ^ i^ sbw 0 v 1-

DIDLOKN 1- Ri ndno`h\ ^ji njgp^d‡i ˆid^\, Di pfpqbj^ s * t < 0+r * s < N qfbkb bu^`q^jbkqb rk^ plir`fŽk9 %r)s& < S+p(-

DIDLOKN 2- Ri ndno`h\ ^ji hƒn _` pi\ njgp^d‡i, Di pfpqbj^ s * t < 0+nrb `lkpq^ ab rk^ b`r^`fŽk `lk alp fk`Ždkfq^p+ qfbkb jŠp ab rk^ plir`fŽk- Clpk•jbolp `r^ibpnrfbo^ `rv^ prj^ pb^ 0 a^k rk^ plir`fŽk-

@ `^a^ pfpqbj^ ifkb^i '05-12(+ mlabjlp ^pl`f^o lqol pfpqbj^

i

F\dfsf;L m^o^ d;/*0* ,,, *h*1-2

l_qbkfal obbjmi^w^kal `^a^ `* bk '05-12( mlo N- Dpqbpb ii^j^ bi ndno`h\ cjhj+b„i`j `loobpmlkafbkqb ^i '05-12(+ Rf ` a, N- bi pfpqbj^ '05-12( pb ii^j^ ij cj+hjb„i`j, Tk sb`qlo s ab Si p^qfpc^oŠbi pfpqbj^ eljld‹kbl pf v pŽil pf

P%r&< N:

alkab P bp i^ qo^kpcloj^`fŽk ifkb^i abqbojfk^a^ mlo i^ j^qofw ab ilp `lbcf`fbkqbp-Di pfpqbj^ eljld‹kbl qfbkb pfbjmob i^ plir`fŽk s < N+ mbol mrbab qbkbo lqo^p-Di `lkgrkql ab plir`flkbp abi pfpqbj^ eljld‹kbl bp bi k•`ibl ab Q, Di qblobj^pfdrfbkqb bumlkb i^ obi^`fŽk bkqobi^p plir`flkbp abi pfpqbj^ eljld‹kbl v i^p abipfpqbj^ kl eljld‹kbl-

SDNQDL@ 05-07- Ppkjib\hjn lp` `g ndno`h\ ij cjhjb„i`j '05-12( o`ib\pi\ njgp^d‡i* kjm `e`hkgj ],

]( Pd pi q`^ojm s `n pi\ njgp^d‡i _`g ndno`h\ ij cjhjb„i`j* `ioji^`n `gq`^ojm q < s + ] `n pi\ njgp^d‡i _`g ^jmm`nkji_d`io` ndno`h\ cjhj+b„i`j,

Page 306: Calculus

522 Qm\inajmh\^dji`n gdi`\g`n t h\omd^`n

_( Pd pi q`^ojm q `n pi\ njgp^d‡i _`g ndno`h\ cjhjb„i`j* `g q`^ojm s <q * ]`n pi\ njgp^d‡i _`g ndno`h\ ij cjhjb„i`j,

A`hjnom\^d‡i, Rb^ Q8Sddx S g i^ qo^kpcloj^`fŽk ifkb^i abqbojfk^a^ mlo i^j^qofw ab ilp `lbcf`fbkqbp+`ljl ^kqbp pb e^ af`el- Orbpql nrb ] bp rk^ plir`fŽkabi pfpqbj^ kl eljld‹kbl qbkbjlp P%\& < ]+ Rb^k r v p alp sb`qlobp ab s+ q^ibpnrb q < s+], Dkqlk`bp qbkbjlp

Q&q'< Q&s + ]' < Q&s' + Q&c' < Q&s' + ^,

Olo `lkpfdrfbkqb P%r& < _ pf u pŽil pf Pni& < &N-Dpql abjrbpqo^ ^ i^ sbw ^( u ^(+

Dpqbqblobj^ morb_^ nrb bi mol_ibj^ ab e^ii^o qla^p i^p plir`flkbp ab ropfpqbj^ kl eljld‹kbl pb bp`fkab bk alp m^oqbp90( G^ii^o qla^p i^p plir`flkbpq abi pfpqbj^ eljld‹kbl+ bpql bp+abqbojfk^kal bi k•`ibl ab Q9 v 1( e^ii^o rk^plir`fŽk m^oqf`ri^o ] abi pfpqbj^ kl eljld‹kbl- Rrj^kal ]\ `^a^ rkl ab ilpsb`qlobp q abi k•`ibl P) pb l_qfbkbk qla^p i^p plir`flkbp s < q * \ abi pfpqbj^kl eljld‹kbl-

Rb^ e i^ afjbkpfŽk ab K&Q', Rf mlabjlp bk`lkqo^o e plir`flkbp di_`k`i_d`i+o`n H! ) +++ ) i&h abi pfpqbj^ eljld‹kbl+ bii^p cloj^oŠk rk^ _^pb m^o^K&Q'* v ml,abjlp l_qbkbo `r^inrfbo w9 ab J%P& cloj^kal qla^p i^p `lj_fk^`flkbp ifkb^ibp

alkab n))+++*of plk bp`^i^obp ^o_fqo^oflp-Dpq^`lj_fk^`fŽk ifkb^i pb ii^j^ njgp+^d‡i b`i`m\g _`g ndno`h\ cjhjb„i`j, Rf ] bp rk^ plir`fŽk m^oqf`ri^oabi pfpqbj^ kleljld‹kbl+ bkqlk`bp qla^p i^p plir`flkbp s sfbkbk a^a^p mlo

Dpq^ `lj_fk^`fŽk ifkb^i pb ii^j^ njgp^d‡i b`i`m\g _`g ndno`h\ ij cjhjb„i`j,Di qblobj^ 05-07 mrbab mlkbopb bk bpq^ lqo^ cloj^9

RCMPCK? 05-08- P`\ Q8Si ,* Sh g\ om\inajmh\^d‡i gdi`\g o\g lp` Q&s' < t*_ji_` s < &Ug% ŠŠŠ * Ui'* t < 'Xi + ‘-- *Vh'* b

i

Vd< x\dfUf k\m\ e< 0+1+--- +h ,h<i

Page 307: Calculus

Q„^id^\n _` ^ƒg^pgj 634

P`\ f g\ _dh`ind‡i _`g iˆ^g`j _` Q, Pd UH + ‘‘‘ + Re Pji f njgp^dji`n di_`k`i_d`i+o`n _`g ndno`h\ cjhjb„i`j Q&s'< M+Wnd] `n pi\ njgp^d‡i k\mod^pg\m_`g ndno`h\ij cjhjb„i`j Q&s' < `* `ioji^`n g\ njgp^d‡i b`i`m\g _`g ndno`h\ ij cjhjb„i`j `n

s < \ * 00r0 * --- * ffqf*

Dpqbqblobj^ kl klp af`b `Žjl bk`lkqo^o rk^ plir`fŽk m^oqf`ri^o ] abi pfp,qbj^ kl eljld‹kbl+ kf i^p plir`flkbp -TG +ŠŠŠ+Re abi pfpqbj^ eljld‹kbl- Mlp af`bq^k pŽil il nrb mrbab l_qbkbopb `r^kal bi pfpqbj^ kl eljld‹kbl qbkd^ rk^ pl,ir`fŽk- Di pfdrfbkqb bgbjmil+ ^rknrb jrv pbk`fiil+ firpqo^ bi qblobj^-

DIDLOKN- Di pfpqbj^ r * u < 1 qfbkb `ljl pfpqbj^ eljld‹kbl ^pl`f^al i^b`r^`fŽk r * v < N- Olo `lkpfdrfbkqb+ bi k•`ibl `lkpq^ ab qlalp ilp sb`qlobp abR0 ab i^ cloj^ %n)*n&)pfbkal n^o_fqo^ofl-Orbpql nrb %n)*n& < n%.),0(+ ‹pqb bprk pr_bpm^`fl rkf afjbkpflk^i ab S0 `lk _^pb '0+,0(- Tk^ plir`fŽk m^oqf`ri^oabi pfpqbj^ kl eljld‹kbl bp '/+1(- Olo q^kql+i^ plir`fŽk dbkbo^i abi pfpqbj^ kleljld‹kbl sfbkb a^a^ mlo

%r)s& < '/+1( * 0'0+,0( l s < 0+ t;0+/*

pfbkal o ^o_fqo^ofl-

05-07 S‹`kf`^p ab `Ši`ril

Ulis^jlp ^i mol_ibj^ abi `Ši`ril bcb`qfsl ab i^p plir`flkbp ab rk pfpqbj^ifkb^i kl eljld‹kbl- @rknrb pb e^k abp^oolii^al jr`elp j‹qlalp m^o^ ^q^`^obpqbmol_ibj^+ qlalp bufdbk `Ši`rilp `lkpfabo^_ibp pf bi pfpqbj^ bp ab do^k q^j^,•l- Olo bgbjmil+ m^o^obplisbo rk pfpqbj^ ab afbwb`r^`flkbp `lk bi jfpjl k•jbolab fk`Ždkfq^pmrbabk pbokb`bp^of^ps^of^p elo^p ab `Ši`rilp+ fk`irpl `lk i^ ^vra^ab rk `^i`ri^alo j^kr^i-

U^jlp ^ `ljbkq^o rk j‹qlal jrv rqfifw^al+ nrb pb ii^j^ h„oj_j _` `gdhd+i\^d‡i _` D\pnn+Fjm_\i* nrb bp obi^qfs^jbkqb pbk`fiil v mrbab moldo^j^opb cŠ`fi,jbkqb m^o^ `^i`ri^alobp bib`qoŽkf`lp ab ^iq^ sbir`fa^a- Di j‹qlal `lkpfpqb bk i^^mif`^`fŽk ab qobplmbo^`flkbp crka^jbkq^ibp ^ i^p b`r^`flkbp ifkb^ibp abi pfpqbj^9

0( Fio`m^\h]dj _` _jn `^p\^dji`n,0' Jpgodkgd^\^d‡i _` oj_jn gjn o„mhdijn _` pi\ `^p\^d‡i kjm pi `n^\g\m ij

ipgj,1' Pph\ _` pi\ `^p\^d‡i \ jom\ hpgodkgd^\_\ kjm pi `n^\g\m,

Page 308: Calculus

635 Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

B^a^ sbw nrb bcb`qr^jlp rk^ ab bp^p lmbo^`flkbp bk bi pfpqbj^ l_qbkbjlp rkkrbsl pfpqbj^ `lk i^p jfpj^p plir`flkbp- Clp pfpqbj^p `lk i^p jfpj^p plir`flkbppb ii^j^k `lpdq\g`io`n, Dcb`qr^kal bp^p lmbo^`flkbp rk^ qo^p lqo^ ab jlal pfpqb,jŠqf`l iibd^jlp mlo cfk ^ rk pfpqbj^ bnrfs^ibkqb nrb mrbab obplisbopb ^ pfjmibsfpq^-

Hirpqo^objlp bi j‹qlal `lk ^idrklp bgbjmilp m^oqf`ri^obp- Rb sboŠ bkqlk`bp`Žjl pb ^mif`^ bi j‹qlal bk dbkbo^i-

DIDLOKN 0- Pdno`h\ ^ji njgp^d‡i ˆid^\, Blkpfabobjlp bi pfpqbj^

/r * 3t * 1t < ,2

s + /s!7x! w < 4

s + 1s * 5w < 0/ -

Dpqb pfpqbj^ qfbkb plir`fŽk •kf`^+ s <013+ t;53* w < 20+ nrb l_qbkaobjlp mlo bij‹qlal ab bifjfk^`fŽk ab F^rpp,Hloa^k+ O^o^ bsfq^o qo^_^gl kl `lmf^jlp i^p ib,qo^ps* v+ w kf ilp pfdklp ab fdr^ia^a+ pfkl nrb qo^_^g^objlp `lk i^ h\omdu\hkgd\_\

'05-13( ,z\0/

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'05-14(*+-=64

20

i^ j^qofw ^jmif^a^ abpmr‹p ab rk^ pr`bpfŽk ab lmbo^`flkbp cfi^- Di `loobpmlkafbkqbpfpqbj^ ab b`r^`flkbp bp r < 013+ X < 64+ w < 20+ nrb klp a^ i^ plir`fŽkabpb^a^-

Page 309: Calculus

Q„^id^\n _` ^ƒg^pgj 0-0

Di mofjbo m^pl bp l_qbkbo rk 0 bk bi s‹oqf`b prmboflo fwnrfboal ab i^ j^qofw

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Z

0 ,1 0

1 ,4 3

0 ,3 5,z\-

0/

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'05-15( Yy<yy ,024

4

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Zz ,z ,z 04z\&N ,1 4

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'05-16( Yy,z ,z

N N 0izI-20

@i iibd^o ^nrŒ+bi `loobpmlkafbkqb pfpqbj^ ab b`r^`flkbp sfbkb a^al mlo

s + /s * u < 4

V + 0u < 02

u < 20 -

Page 310: Calculus

0-1 Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

Dpq^p b`r^`flkbp mrbabk obplisbopb pr`bpfs^jbkqb m^oqfbkal ab i^ qbo`bo^ v qo^,_^g^kal e^`f^ ^qoŠp+aŠkalklp

u < 20+ V < 02 * 0u < 02 * 51 < 64+ s < 4 * 0t + u < 4 * 04/ , 20 < 013-

l _fbk+ mlabjlp `lkqfkr^o bi mol`bpl ab F^rpp,Hloa^k `lksfoqfbkal bk `bolpqlalp ilp bibjbkqlp pfqr^alp mlo bk`fj^ ab i^ af^dlk^i ab rklp bk i^ pbdrka^u bk i^ qbo`bo^ `lirjk^p- Lriqfmif`^kal i^ pbdrka^ cfi^ ab '05-16( mlo 1 u pr,j^kal bi obpriq^al ^ i^ mofjbo^+ l_qbkbjlp

Z

0 N

N 0

: :

,2 G 20\,1002 -0 20

Olo •iqfjl+ jriqfmif`^jlp i^ qbo`bo^ cfi^ mlo 2 u prj^jlp bi obpriq^al ^ i^ mofjbo^cfi^+ u irbdl jriqfmif`^jlp i^ qbo`bo^ cfi^ mlo 1 u prj^jlp bi obpriq^al ^ i^ pb,drka^ `lk il nrb iibd^jlp ^ i^ j^qofw '05-14(-

DIDLOKN 1- Pdno`h\ ^ji hƒn _` pi\ njgp^d‡i, Blkpfabobjlp bi pfdrfbkqbpfpqbj^ ab 2 b`r^`flkbp `lk 4 fk`Ždkfq^p9

'05-17(

0s + 3t * 2u * p + q < ,2

s + /s * u + p * q < 4

s + 1s * 4u * 0p + q < 0/ -

K^ `loobpmlkafbkqb j^qofw ^jmif^a^ bp

<3

,4 3 0 ,0 ,2\,1 0 ,0 0 4 -

,3 5 1 ,0 0/

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,05 08

'2 **

,2 3

013\64 -

20

Page 311: Calculus

Q„^id^\n _` ^ƒg^pgj 0-2

Di `loobpmlkafbkqb pfpqbj^ ab b`r^`flkbp mrbab obplisbopb obpmb`ql ^ s* v+ w bkcrk`fŽk ab p v q aŠkalklp

r < 013 * .3o * .6p

t < 64 * 7p + iis

u < 20 * 1p + 2q,

Rf e^`bjlp o < n) X p < o0* pfbkal o*v ny k•jbolp ob^ibp ^o_fqo^oflp+ v abqbojf,jfk^jlp r) v+ w jbaf^kqb bp^p b`r^`flkbp+ bi sb`qlo %r) v+t) o) p& ab R+a^al mlo

%r)t* t) o) p&< '013 * 0500 , 0801+64 * 800 , 0001+20 * 200 , 301+00+ '1(

bp rk^ plir`fŽk m^oqf`ri^o abi pfpqbj^ kl eljld‹kbl '05-17(- Klp alp sb`qlobp

%r)s) t) o) p&< '013+64+20+ N+N( * 00'05+8+2+0+ N( * 01',08+ ,00+ ,3+ /+0(-

Dpq^ b`r^`fŽk klp a^ i^ plir`fŽk dbkbo^i abi pfpqbj^- Di sb`qlo '013+ 64+ 20+ N+N(bp rk^ plir`fŽk m^oqf`ri^o abi pfpqbj^ kl eljld‹kbl '05-17(- Klp alp sb`qlobp'05+8+2+0+ N( X ',08+ ,00+ ,3+ /+0( plk plir`flkbp abi `loobpmlkafbkqb pfp,qbj^ eljld‹kbl- Orbpql nrb plk fkabmbkafbkqbp+ `lkpqfqrvbk rk^ _^pb m^o^ bibpm^`fl ab qla^p i^p plir`flkbp abi pfpqbj^ eljld‹kbl-

DIDLOKN 2- Pdno`h\ ndi njgp^d‡i, Blkpfabobjlp bi pfpqbj^

'05-18(

0s + 3t * 2u < ,2

r * /s * t < 4

u , 1s * 3u < 0/-

Dp fa‹kqf`l ^i abi bgbjmil 0 bu`bmql nrb bk bi `lbcf`fbkqb ab w bk i^ qbo`bo^ b`r^`fŽke^ pfal `^j_f^al bi 5 mlo rk 4- K^ j^qofw ^okme^a^ `loobpmlkafbkqb bp

Z

1 ,4 30 ,1 00 ,3 4

,2\4 -

0/

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'05-2/(0

'+N

hy[-20

Page 312: Calculus

64/ Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

Br^kal i^ •iqfj^ cfi^ pb bumobp^ ljl b`r^`fŽk+ iibd^jlp ^ N < 20- Olo `lkpf,drfbkqb bi pfpqbj^ lofdfk^i kl qfbkb plir`fŽk mrbpql nrb ilp alp pfpqbj^p '05-18(u '05-2/( plk bnrfs^ibkqbp-

Dk `^a^ rkl ab ilp bgbjmilp ^kqboflobp+bi k•jbol ab b`r^`flkbp kl bu`baŒ^^i ab fk`Ždkfq^p- Rf e^v jŠp b`r^`flkbp nrb fk`Ždkfq^p+bi mol`bpl ab F^rpp,Zloa^k mrbab ^•k ^mif`^opb-Olo bgbjmil+ `lkpfabobjlp bi pfpqbj^ abi bgbjmil 0+nrb qfbkb i^ plir`fŽk r < 013+X< 64+ w < 20- Rf ^agrkq^jlp rk^ krbs^ b`r^,`fŽk ^ bpqbpfpqbj^ nrb pb^ p^qfpcb`e^mlo i^ jfpj^ qbok^+mlo bgbjmil+ i^ b`r^,`fŽk /r * 2u * w< 43+ bkqlk`bp bi mol`bpl ab bifjfk^`fŽk klp iibs^ ^ i^ j^qofw^jmif^a^

hyyy0z9\M M 0 20

N N N N

`lk rk^ cfi^ ab `bolp bk i^ m^oqbfkcboflo- Obol pf ^agrkq^jlp rk^ krbs^ b`r^`fŽknrb kl pb p^qfpc^d^ mlo i^ qbok^ '013+ 64+ 20(+ mlo bgbjmil i^ b`r^`fŽks * u * w< 0+ bkqlk`bp bi mol`bpl ab bifjfk^`fŽk klp `lkar`b ^ i^ j^qofw^jmif^a^ ab i^ cloj^

<

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).&)1 =[cR_`N`QRZNa_VPRPbNQ_NQN`

Rb^ > < %[xd&rk^ j^qofw `r^ao^a^ h W h) -q^inrb ?> < &"pfbkal / i^ j^,qofwfabkqfa^a i W i* bkqlk`bp > pb ii^j^ ij ndibpg\m v ? i^ diq`mn\ ab > mlog\ dulpd`m_\,

Dibdfa^ i^ _^pb rpr^i ab ilp sb`qlobp `lloabk^alp rkfq^oflp ab Ri* pb^P7Rj x Ri i^ qo^kpcloj^`fŽk ifkb^i `lk j^qofw g%P&< =+ Sbkbjlp bkqlk`bp bipfdrfbkqb

Page 313: Calculus

giq`mn\n _` h\omd^`n ^p\_m\_\n 640

RCMPCK? 05-1/- I\ h\omdu > `n ij ndibpg\m nd v n‡gj nd Q `n di q`mod]g`, Pd?> < F* `ioji^`n ? < h&Q+/

',

A`hjnom\^d‡i, Rrmlkd^jlp nrb > bp kl pfkdri^o u nrb ?> < g, Cbjlpqo^,objlp nrb Q&s' < N fjmif`^ s < N- C^al s q^i nrb Q&s' < N+ pb^ W i^ j^qofw`lirjk^ i V 0 cloj^a^ ^ m^oqfoab ilp `ljmlkbkqbp ab s, Orbpql nrb Q&s' < N+i^ j^qofw molar`ql =T bp rk^ j^qofw `lirjk^ i V 0 cloj^a^ mlo `bolp+ ^pŒnrb?&>U' bp q^j_f‹k rk^ j^qofw `lirjk^ ab `bolp- Obol ?&>U' < &?>'U < gU < W+mlo il nrb qlal `ljmlkbkqb ab s bp N- Olo `lkpfdrfbkqb+ Q bp fksboqf_ib+ u i^b`r^`fŽk QQ+/ < g fjmif`^ nrb h&Q'h&Q+/

' < g l >h&Q+/' < g, Lriqfmif`^kal

^ i^ fwnrfboa^ mlo ." bk`lkqo^jlp jq H!!( < ?, Qb`Œmol`^jbkqb+ pf Q bp fksboqf_ibbkqlk`bp Q+/Q bp i^ qo^kpcloj^`fŽk fa‹kqf`^ ^pŒnrb h&Q+/'h&Q' bp i^ j^qofwfabkqfa^a- Olo `lkpfdrfbkqb > bp kl pfkdri^o u h&Q+g'> < g,

Sla^p i^p molmfba^abp ab i^p qo^kpcloj^`flkbp ifkb^ibp fksboqf_ibp qfbkbk pr`lkqo^m^oqfa^ m^o^ i^p j^qof`bp kl pfkdri^obp- Dk m^oqf`ri^o+ i^p fksbop^p mlo i^fwnrfboa^ 'pf bufpqbk( plk •kf`^p+ v qla^ fksbop^ mlo i^ f-wnrfboa^ bp q^j_f‹k fk,sbop^ mlo i^ abob`e^- Cf`el ab lqol jlal+ pf > bp kl pfkdri^o v ?> < >"bkqlk,`bp >? < g, Ki^j^jlp ^ ? i^ diq`mn\ ab > X i^ abpfdk^jlp mlo > +h- K^ fksbo,p^ = ,0 q^j_f‹k bp kl pfkdri^o u np fksbop^ bp =+

Rbdrfa^jbkqb abjlpqo^jlp nrb bi mol_ibj^ ab i^ abqbojfk^`fŽk bcb`qfs^ abilp bibjbkqlp ab i^ fksbop^ ab rk^ j^qofw kl pfkdri^o bp bnrfs^ibkqb ^ i^ obplir,`fŽk ab i pfpqbj^p ifkb^ibp kl eljld‹kblp-

Rb^ = < pq+z(kl pfkdri^o v R&9&0:[ < %\xd&pr fksbop^- Klp bibjbkqlp ab =u > +GbpqŠk ifd^alp mlo i^p i0 b`r^`flkbp-

'05-20(i

I [ce\eF < %dcF$f;g

pfbkal %dxd< 0 pf d < c+v %dxd< N pf d ;/; d+ O^o^ `^a^ s^ilo cfgl ab d) mlabjlp`lkpfabo^o '05-20( `ljl rk pfpqbj^ kl eljld‹kbl ab i b`r^`flkbp ifkb^ibp `lki fk`Ždkfq^p ].d) ]/d ) ††† ) \hd+ Orbpql nrb > bp kl pfkdri^o+ `^a^ rkl ab bplppfpqbj^p qfbkb plir`fŽk •kf`^+ i^ `lirjk^ d ab >+ Slalp bplp pfpqbj^p qfbkbk i^jfpj^ j^qofw ab `lbcf`fbkqbp > u afcfbobk q^k pŽil bk prp pbdrkalp jfbj_olp-Olo bgbjmil+ pf > bp rk^ j^qofw 2 W 2+ bufpqbk 8 b`r^`flkbp bk '05-20( nrb mrbabkobmobpbkq^opbljl 2 pfpqbj^p ifkb^ibp nrb qfbkbk i^p pfdrfbkqbp j^qof`bp ^jmif^a^p9

Page 314: Calculus

0.+ Qm\inajmh\^dji`n gdi`\g`n u h\omd^`n

Rf ^mif`^jlp bi mol`bpl ab F^rpp,Iloa^k+ iibd^jlp ^ i^p obpmb`qfs^pj^qof`bp ^j,mif^a^p

Z

0 N N ^S[N 0 N \0/ *

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Dk i^ moŠ`qf`^^molsb`e^jlp bi eb`el ab nrb ilp qobppfpqbj^p qfbkbk i^ jfpj^j^qofw ab `lbcf`fbkqbp u obplisbjlp ilp qobppfpqbj^p ab rk^ sbw qo^_^g^kal `lki^ j^qofw ^jmif^a^&

Z^r \/0 \/1 0 N zi{0/ {00 .01 N 0

\1/ \10 {11 N N

Di mol`bpl ab bifjfk^`fŽk klp iibs^ ^

<RN N _r ]/0

\!Y0 N ]0/ ]00 ]01 Š

N 0 ]1/ ]10 ]11

K^ j^qofw ab i^ m^oqbabob`e^ ab i^ _^oo^ sboqf`^i bp i^ fksbop^ abpb^a^- K^ ab i^fwnrfboa^ bp i^ j^qofw fabkqfa^a 2 W 2-

Ml bp mob`fpl`lkl`bo ab ^kqbj^kl pf> bp kl pfkdri^o- Rf > bp ndibpg\m*mlab,jlp ^•k ^mif`^o bi j‹qlal ab F^rpp,Hloa^k+ mbol l`roob nrb bk bi mol`bpl rklab ilp bibjbkqlp ab i^ af^dlk^i pb `lksfboqb bk `bol+ v kl pboŠmlpf_ib qo^kpcloj^o= bk i^ j^qofw fabkqfa^a-

).&*( :WR_PVPV\`

@mif`^kal bi mol`bpl ab F^rpp,Hloa^k ^ `^a^ rkl ab ilp pfpqbj^p pfdrfbkqbp+ abqbojfk^oi^ plir`fŽk dbkbo^i+ pf bufpqb-

.+ s * t * 0t < 4

/r * t * 1t < 00

+t * u < 2-

0, 1s * 0t * u < 0

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1, 1s * /s * u < 0

3s * 1t * 1u < 15s * 1s * 3u < 2-

2, 1s * /s * u < 0

3s * 1t * 1u < 15s * 1s * 3u < 2

s * t + u < N-

Page 315: Calculus

Be`m^d^djn 642

3, 1s + /s * 3u * p < 0s * t + 1u * 0p < 1

4s * t + 2u * 1p < 6-

4, s * t + 1u * p < 40s + t * u + 0p < 1

5s * t + 5u * 1p < 2-

5, s * t * 0u * 1p * 2q < N0s * 1t * 6u * 0Hp * /2q < N1s * 1t * 4u * .-.E * /3q < N-

5+ r * /s * t * /.. < ,10s * 2t + u}+ 400 < 81r * t * t * 00 < 43s + 1t * 0u * 00 < 2-

8- Cbjlpqo^o nrb bi pfpqbj^ r * u * /t < 1+ /r * u * 0t < 1+ 2r * u * [t < 5+ qfbkb pl,ir`fŽk •kf`^ pf \ 9%!7- G^ii^o qla^p i^p plir`flkbp `r^kal \ < 7-

0/- ^( Cbqbojfk^o qla^p i^p plir`flkbp abi pfpqbj^

3s * /s * 4u * 100 < ,0

s + t * W + 00 < ,1-

_( Cbqbojfk^o qla^p i^p plir`flkbp abi pfpqbj^

3s * 1t + 4u * 100 < , 0

s + t * u + p < ,1

s)t)u 3+

00- Dpqb bgbo`f`fl klp fkaf`^ `Žjl pb abqbojfk^k qla^p i^p j^qof`bp kl pfkdri^obp 1 W 1-Cbjlpqo^o nrb

X\ ]G X _ %8 < &\_ + ]`'g,b _ ,b \

Cbar`fo nrb Z9 9\ bp kl pfkdri^o pf v pŽil pf \_ + ]` 9%!N+bk `rvl `^pl pr fksbop^ bp

0 W^ *\F\_ + ]` ,b \%

Cbqbojfk^o i^ fksbop^ ab `^a^ rk^ ab i^p j^qof`bp ab ilp bgbo`f`flp abi 01 ^i 05-

01- Z 9 2 yH-'* +

02 Yy,: [

03- Z,z zz ,9I

Page 316: Calculus

643 Qm\inajmh\^dji`n gdi`\g`n v h\omd^`n

Yy1 2

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N N

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1 N 1 N N N

N 2 N GN N05-

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N N N N 1 N

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0- Rf rk^ j^qofw `r^ao^a^ qfbkb rk^ `lirjk^ ab `bolp l rk^ cfi^ ab `bolp+ abjlpqo^o nrbbp pfkdri^o-

1- O^o^ `^a^ rk^ ab i^p molmlpf`flkbp pfdrfbkqbp obi^qfs^p ^ j^qof`bp i W i* a^o rk^ ab,jlpqo^`fŽk l rk `lkqo^bgbjmil-^( Rf => * >= < N+ bkqlk`bp =/>0 < >0=/+_( Rf = X > plk kl pfkdri^obp+bkqlk`bp = * > bp kl pfkdri^o-`( Rf > X > plk kl pfkdri^obp+bkqlk`bp => bp kl pfkdri^o-a( Rf =) >) X = * > plk kl pfkdri^obp+bkqlk`bp = * > bp kl pfkdri^o-b( Rf =0 < N+ bkqlk`bp = * . bp kl pfkdri^o-c( Rf bi molar`ql ab e j^qof`bp =f +++=e bp kl pfkdri^o+ `^a^ rk^ ab i^p j^qof`bp =) bp

kl pfkdri^o-

2- Rf = < E yH+e^ii^o rk^ j^qofw kl pfkdri^o L q^i nrb j*f =L < Zz ] ziX

\ fI ,,3- K^ j^qofw > < e ] +alkab d0 < ,0+\ ;T * U4(+ v ] < 'i , U4(+ qfbkb i^

molmfba^a ab nrb =/ < =+ Cbp`of_fo bk cloj^ `ljmibq^ qla^p i^p j^qof`bp =) 1 W 1+`lk bibjbkqlp `ljmibglp q^ibp nrb >0 < =+

4- Rf =/ < =) abjlpqo^o nrb %= * F&e < . * %/e * f&=+5- K^ qbloΠab i^ obi^qfsfa^a rqfifw^rk `lkgrkql ab b`r^`flkbp ab i^ cloj^ r$ < [%r * pn&)

v&< v+ u%< u* o%< \&o+ qs-^0', @nrŒq obmobpbkq i^ sbil`fa^a ab rk l_gbql nrb pb

jrbsb+ ` i^ sbil`fa^a ab i^ irw+ u \ < B.U,:91 , q0* alkab Grh; `- K^ qo^kpcloj^`fŽknrb ^mif`^ bi sb`qlo _f afjbkpflk^i &s*o' bk &s%*o%'pb ii^j^ om\inajmh\^d‡i _` Ijm`iou,Rr j^qofw obi^qfs^ ^ i^p _^pbp rpr^ibp pb abpfdk^ `lk Ivq' u sfbkb a^a^ mlo

I&q' < \X 0 ,isI-+q^+/

Page 317: Calculus

Be`m^d^djnq\mdjn nj]m` h\omd^`n 644

N_p‹osbpb nrb H%p&bp kl pfkdri^o v nrb H%K&< 0- Cbjlpqo^o nrb H%p&H%o&< H%q&)pfbkal s < %o* %$&]0-&Ro* ]0', Dp ab`fo+ bi molar`ql ab alp qo^kpcloj^`flkbp abKlobkqw bp lqo^ qo^kpcloj^`fŽk ab Klobkqw-

6- Rf `^j_f^jlp i^p cfi^p mlo i^p `lirjk^p bk rk^ j^qofw ob`q^kdri^o =) i^ krbs^ j^qofw^pŒl_qbkfa^ pb ii^j^ i^ om\inkp`no\ ab = v pb abpfdk^ mlo =f+ Olo bgbjmil+ pf qbkbjlp

> x X8 8 9I- bkqlk`bp >% + X8 8Z

Cbjlpqo^o nrb i^p qo^kpmrbpq^p qfbkbk i^p molmfba^abp pfdrfbkqbp9

'^( %=E&n< =+ '_( %= * >&E< =n * >E+ 'b( %]=&n< ?=f+'a( %=>&n< >$=$) 'b( %=.l. < %=*.&. pf = bp kl pfkdri^o-

7- Tk^ j^qofw `r^ao^a^ = pb ii^j^ j^qofw loqldlk^i pf ==E < f+ Bljmol_^o ~rb i^ j^qofw

Zblp 7 , pbk 7I

1 W 1 pbk 7 `lp '( bp loqldlk^i m^o^ `^a^ k•jbol ob^i '(- Rf > bp `r^inrfbo

j^qofw loqldlk^i i W i* abjlpqo^o nrb prp cfi^p+`lkpfabo^a^p `ljl sb`qlobp ab Ri* clo,j^k rk `lkgrkql loqldlk^i-

8- O^o^ `^a^ rk^ ab i^p molmlpf`flkbp pfdrfbkqbp ^`bo`^ ab i^p j^qof`bp i W i* a^o rk^abjlpqo^`fŽk l bk pr ird^o rk `lkqo^ bgbjmil-^( Rf = X > plk loqldlk^ibp+ = * > bp loqldlk^i-_( Rf = X > plk loqldlk^ibp+ => bp loqldlk^i-`( Rf = X => plk loqldlk^ibp+ > bp loqldlk^i-

/., J\omd^`n _` E\_\h\m_* ii^j^a^p ^pŒmlo Z^`nrbp G^a^j^oa '0754,0852(+ plk ^nrbii^pj^qof`bp i W i `lk i^p molmfba^abp pfdrfbkqbp9

0- B^a^ abjbkql bp 0 Ž , 0-00- B^a^ cfi^+`lkpfabo^a^ `ljl rk sb`qlo ab Ri* qfbkb ilkdfqra fdr^i ^ Ty -

000- Di molar`ql bp`^i^o ab alp cfi^p afpqfkq^p `r^ibpnrfbo^ bp N-K^p j^qof`bp ab G^a^j^oa pb mobpbkq^k bk `fboqlp mol_ibj^p ab dbljbqoŒ^ v bk i^qbloŒ^ab k•jbolp+ v e^k pfal ^mif`^a^p ob`fbkqbjbkqb bk i^ `lafcf`^`fŽk Žmqfj^ m^o^ i^`ljrkf`^`fŽk bpm^`f^i- @ mbp^o ab pr ^m^obkqb pfjmif`fa^a+ mobpbkq^k jr`elp mol_ibj^ppfk obplisbo- Di mofk`fm^i mol_ibj^ kl obprbiql bk bpqbjljbkql bp bi ab abqbojfk^o qlalpilp s^ilobp ab i m^o^ ilp nrb bufpqb rk^ j^qofw ab G^a^j^oa i W i, Dpqb bgbo`f`fl a^fab^ ab rk^ plir`fŽk m^o`f^i-^( Cbqbojfk^o qla^p i^p j^qof`bp ab G^a^j^oa 1 W 1 'e^v bu^`q^jbkqb 7(-_( Dpq^ m^oqb abi bgbo`f`fl bp_lw^ rk^ abjlpqo^`fŽk pbk`fii^ abi pfdrfbkqb qblobj^9Rf > `n pi\ h\omdu _` E\_\h\m_ i W i* nd`i_j i = 1+ `ioji^`n i `n pi hˆgodkgj _` 2,K^ abjlpqo^`fŽk pb _^p^ bk alp ibj^p jrv pbk`fiilp obi^qfslp ^ ilp sb`qlobp bk bi bpm^`fl ab afjbkpfŽk i, Cbjlpqo^o `^a^ rkl ab bplp ibj^p u ^mif`^oilp ^ i^p cfi^p ab i^j^qofw ab G^a^j^oa m^o^ abjlpqo^o bi qblobj^-

KDL@ 0- Pd W+V* Y nji q`^ojm`n jmojbji\g`n _` Sh) n` od`i`

&U) s&+&U* X( < GGVG01Š

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Page 318: Calculus
Page 319: Calculus

EB?G8=BA:E 6 ?BE :>:D8=8=BE

Fiomj_p^^d‡i

)00-3 Dgbo`f`flp 'mŠd- 8(

0- ']( ]0 '^( ]0

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2- '_( Oh ; e * 0 ; n9

'`( ]0 * ] 'b( g\]0 * ]`

\]f)g'b( ,, * ]`

f * 0

01-4 Dgbo`f`flp 'mŠd- 08(

0- = < x0+,0y+ > < x0y+b < xiy+@ < x1y+A < xi + ,06y+

C < xi+,06+ ,7 * v36+ ,7 , X36y-/+ = p999=) > p999=) > p999>) > p999b+ > p999A) > p999B) b p999=) b p999>) b p999b+ b p999A) b

p999B) @ p999@)A p999A) A p999B) B p999B+ 'Ml qbkbo bk `rbkq^ i^p fk`irpflkbp ~molmf^p‚+(

2- '^( `fboq^ '_( `fboq^ 'b( c^ip^ 'a( `fboq^ 'b( c^ip^ 'c( c^ip^3- '^( `fboq^ '_( `fboq^ 'b( `fboq^ 'a( `fboq^ 'b( c^ip^ 'c( c^ip^

4- /+x0y+ x1y+x2y+x3y+xi+ 1y+ xi+ 2y+xi+ 3y+ x1+2y+x1+3y+x2+3y+ xi+ 1+ 2y+xi+ 1+3y+xi+ 2+3y+

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i

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646

Page 320: Calculus

647 Pjgp^dji`n \ gjn `e`m^d^djn

5- '_( =%f& c^ip^%/h * 0(1

'b( 0 * 1 * --- * i ; 7

6- kH < 2

03-6 Dgbo`f`flp 'mŠd- 38(

0- '^( 0/ '^( 04 'b( 06/ 'a( 177 'b( 25 'b( P-5

7- '^( i * 0

8- `lkpq^kqb < 100- '^( `fboq^ '^( c^ip^ 'b( c^ip^ 'a( c^ip^ 'b( c^ip^ 'b( c^ip^

i01-

i * 0

0 3-8 Dgbo`f`flp 'mŠd- 42(

1- '^i& ]0'* &\0 * ]n'* '^^+ ]5'* &\2* ]gL'* &\n* _^(+&\4* ]n'* &\5*]7'* &\n* ]2'* &\7* ]4'* &\gL% _0(2- '^( c^ip^ '_( `fboq^ 'b( `fboq^ 'a( c^ip^ 'b( c^ip^

") 3-0/ Dgbo`f`flp s^oflp pl_ob i^ fkar``fŽk 'mŠd- 43(

0- '^( 0/ 'H>( 0 'b( 6 'a( 10 'b( 57/ 'b( H1- '_( 06 'b( 8 'a( Ml

l k*i i

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6, 0h

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?[j•nofi .

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/, a&0' ;1*a&+0' < ,0+ +a&0' < ,2+c'q( <0,+ i.c'1( ;o*a&\ )]' ;\ )] * 0+

a`\' * a&]' < \ * ] * 0*a&\'a&]' < \] * \ * ] * G0 , ,a&0' * b&0' < 0*a&0' + b&0' < 2*a&0'b&0' < +1*a&0'ab&0' < +1*,aXb&0'Z < N+

bX,a&0'Z < +0*,a&\' * b& +\' < 1 * 0\*a&o'b& +o' < '0 * ow0

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4- ']( Gthy 1 '_( GHGy H 'a( Gphƒ z 'a( N z \ x 3 'a( Gohy 3'b( Gthy 1+ s n! N

5- '_( xuiN xs x Hy 'b( xu01 xs z3y a( Di aljfkfl bp s^`Œl

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Page 321: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 648

00- '^( k&s' < \s&g + s' * ]* \ v ] ^o_fqo^oflp '_( k&s' < ?) ^ ^o_fqo^ofl'b( k&s' < \s* \ ^o_fqo^ofl 'a( k&s' < ?) ` ^o_fqo^ofl

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Page 322: Calculus

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Page 323: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn

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Page 324: Calculus

0/+ Pjgp^dji`n \ gjn `e`m^d^djn

//, os0 + os * `lp &s0' + `lp s/0, o&s1 + `lp 1s * 0(

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3- N+abcfkfa^ pŽil bk s < N 8- Fs *Z.9:+s = N

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1+*( s0 pf s ƒ N: N pf s ; N

Page 325: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 652

11- 0 pf 0 99::Gth99::W2: N bk ilp abj^p s^ilobp ab s,

01, s/ pf s x N: N pf s ; N

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Page 326: Calculus

653 Pjgp^dji`n \ gjn `e`m^d^djn

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)

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Page 327: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 543

05- '^( A(&v * b' < '0 * b-a'A(a * '0 )&-b'A(b `r^kal ,a&s' v b&s' kl bp N:

A(&a%b' < b0 A(a * a0 A(b9

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Page 328: Calculus

0// Pjgp^dji`n \ gjn `e`m^d^djn

/6, s b%&s' b!&s'

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1

l2

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l0/25

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3-08 Dgbo`f`flp 'mŠd- 122(

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`( `$ `ob`b pf s = N: ab`ob`b pf s ; N-

1, [& €0 _( ` `ob`b pf Gth= 0: ab`ob`b pf Gth; 0 b( mob`b pf r = N:

ab`ob`b pf s ; N3- ^( 0+2 _( a `ob`b pf s ; 0 l pf s = 2: ab`ob`b pf 0 ; s ; 2 b( n$ ob`b

pf s = 1: ab`ob`b pf s ; 1

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b( `$ `ob`b pf s ; N+ l pf s = N

Page 329: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 656

6- ^( 10.2 _( ` `ob`b pf s ; N+ l pf s = 10.2: ab`ob`b pf N ; s ; 10.2

a( l `ob`b pf s ; N+ l pf s<L

7- ^( 1 _( ` `ob`b pf s ; 0+ l pf 0 ; s ; 1: ab`ob`b pf 1 ; s ; 2+ l pf

s< 2

b( l `ob`b pf s ; 0+ l pf s = 2: ab`ob`b pf 0 ; s ; 2

8- ^( €0 _( ` `ob`b pf ysg ; 0: ab`ob`b pf Gth= 0 b( l `ob`b pf +S1 ; s ; N+

l pf s = S19 ab`ob`b pf s ; +S1*l pf N ; s ; U20/- ^( N _( ` `ob`b pf s ; ,2 l pf ,2 ; s ; N: ab`ob`b pf N ; s ; 2+

l pf s = 2 b( l `ob`b pf ysg< 2: ab`ob`b pf Gth; 2

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01- '^( /h.P '_( ` `ob`b m^o^ qlal r'b( l `ob`b pf /h.P ; r ; %/h * .&.P8 ab`ob`b pf %/h * .&.P ; r ; /h.P

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03- '^( N '_( ` `ob`b pf s = N: ab`:b`b pf s ; N 'b( m`ob`b m^o^ qlal s

,&*) Dgbo`f`flp 'mŠd- *+/!

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Page 330: Calculus

0/1 Pjgp^dji`n \ gjn `e`m^d^djn

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Page 331: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 658

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8- d , U2 05- s&s0 * 0(,0.1 * b0 06- wj&6s0 * 16(4.2 * b

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*` 07- x&n`is + blp U'0-1 * b1 08- 1UH * UH*!> * b

))& ,,,< * b 1/- +o&s + 0(1.4 * bq`jns

-&)( :WR_PVPV\` ]mT&*.1!

0- pbk s + s blp s * B

0, 0s pbk s * 1 blp s + s/ blp s * B1, s1n`is * 1s0`jns8+ 4sn`is + 5`lpu * b2, +s1`jns * 1s0n`is * 4s`jns + 5pbku * B

4- fpbk1u * b5- ppbk 0s + qu blp 0s * b

04- '_( &3/Q-10'\3

06- '2U20 * U2 , 00-24(07- q^ks +s9 fq^k2u ,q^ku )s08- ,`lqu +s9 +/`jo0s )^jos )s1/- '^( i < 3 '_( 1

Page 332: Calculus

66/ Pjgp^dji`n \ gjn `e`m^d^djn

)4-00 Dgbo`f`flp ab obm^pl 'mŠd- 161(

.+ aQ^%K& < N pf N O e O i + 0: a9h&%i& < i /+ 3r2 * .2r1 * fKr0 * 0/( ,5, gg6+ U < .3r0,6

0/- '_( Z&'N( < N

00- ,z `lp 2r * 124 m_h2r * nr `lp 2r

01- 0'0 * U0'1-0

02- ,20/.1/03- 26.717004- il'0 * U3'4

05- 0.15454/06- `lp , `lp 0.5+ W./%r * 0(0.1 , 13\ pbk %r * 0(0.3 , 1W%r* 0(2.3 , 3%r * 0(0.3\ `lp %r * 0(0.3

08- ipbk1 s0

1/- ,f'0 * 2`lp&&U'1-0

//+ [ < 8+ \ < I-{12- 0Z&F+ pS+Sb02, g\s/1 * nr./ * i[uhh

15- 2

16-z'z * ]0 ]] >'1 1 6R * 1

23- '^( j%r& < *r/ * r * 0

24- '^( L/%r& < r * p: L0%r& < r0 + r * p: L1%r& < r1 + nr0 * nr8Lcr& < r2 + /r1 * r0 + ŠŠg}j9 L3%r& < r3 + dr2 * dU1 + pt

@\k…opgj 4

'g-8 Dgbo`f`flp 'mŠd- 178(

0- '^( 0 '_( %[ * _(.'i * [\&

` + 0 &`0 ,8,,, 0(11- '^( N '_(

` * 0'b( 3 'a(

1_0

2- `ob`b pf N ; r ; `* ab`ob`b pf r = `9 `lksbu^ pf r = `1-0* `Žk`^s^ pf N ; r ; `1-0

%/r&,%. * r0' 0/-h%r * UH * r/'i

3-UH * r0

4- r,K * r0' 00- 0.Z1'0 * U r * 0(\5- r,%r0 + 3(

01- ild %r * U r0 * 0(6- .,%r ild r&7- %/,r& * .,%r ild r& 02- .,%[ * \r0'

2( r,%r2 *.& 03- 1 pbk 'ild r&

Page 333: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 55/

.2+ *f,%r ild1 r&

05- eild 01 * 1sg * b.4+ s ild10th , 0s ild Fsg * 0s * b.5+ wT/ ild Gth, ds/ * b/7, xs0gjb0gsg + ds0gjb Fsg * ds0 * b1/- 210- ild Zpbk-b\* b

si(f si(f00, h * 0 ild g\sg + %h* 0(1 * b pf h83 ,0: eild1 g\sg * b pf h < ,0

[0

12- 2! 'ild10th , eild Gth* f( * b13- ild Zild sg * b14- ,1

15- f' ,1 * ild Erfbfn * ild Erf * bs.

16-- !3 gjb1gsX + /14U2ild10th * 110s2gjb Fsg + 0z6U2 * b12, 2gjbs24- 2 * 1gjbs,14, \,gjb \

5-06 Dgbo`f`flp 'mŠd- 2/3(

0- 1`1s+/ 5- 0U ild 11- 6s„s0

6- 0/)s0s ild 12- [0s`+s0

7- '`lp s'`!} s`V8U 8- ,'oaj 0U'`!LP0U

3-0V 9 0/- 0

`g-U 00- _T__s

4- ,6 01- ! _T`U`` ``

02- `U&s + 0( * b03- +`+U&s * 0( * b 07- +d&s0 * 0'`+U0 * b04- `U&s0 + 0s * 1( * b 08- ] < `\* \ ^o_fqo^ofl05- +d`+0U&s0 * s * „ * b 10- sU&g * ild s'

06- 0&V 9 + g'`VU * b 11- 0 * 'i * 0s * 0s0'`U0

12- 2&`U * `+U'+013- \\s\\+g * \s\+/\s\ ild \ * \s\\!&gjb \'014- /-Xs ild s ild 'ild s'Z15- `U&g* `0U'+/-0

16- sssssXx * ild s * 'ild U'0G

17- 'ild s'U &ild ild s * HNzs'

Page 334: Calculus

00+ Pjgp^dji`n \ gjn `e`m^d^djn

18- 1u,i*0ld s i^d s

'i^d U'U+g

2/- u0*ild r Wr * 1'i^d T&/ * r i^d r i^d 'i^d r&Y

20- 'pbku(i*B/7W X`\o/ r ,i^d %m_hr&Y* '`lp r&.($$$r Zq^k! r * i^d &^\p r&Y

0/+ r*0)g-T%. * i^d r&

21r * 03r/ * 1r0 * /r2

22- 2'0 ] r&/%0 Z T&/,0%0 * T&2,0

5-08 Dgbo`f`flp 'mŠd- 2/7(

05-06-07-08-1/-

422!3

pbke s < 041&blpe s < yz

,{,X-*+

qf+.

5-11 Dgbo`f`flp 'mŠd- 203(

001- ,,, oe Gth; 1

U3 , s/

02- ‘ . pf Er * 00 ; U1sf * /r * r/

003 ,,,, oe Gth= H

- HuiUu1 , 0

`lp s04-

g`\n uipf s x %e* S/Q* e bkqbol

05-Uz

pf1'0 * r&

rwK

0 * r2

06-0 * r3

/r07-

Hui'0 * r0'

pf r w N

s18- ^o`pbk z * b

s * G2/- ^o`pbk U1 * b

pbk 0s08- pboo&r * `\n2 r pf r w %e* c&.P

G1/-

1'0 * s/&

`lp r * m_hr10- ,,,,, pf e.P ; s ; %e * n&.P

Upbk1u

s11- ,,,, oe N ; Gth; H

WrfTG*r/

12- 0.'0 * s/& pf s x 0

2s13- ,,,,,, oe Gth; 0

Uf , s2&\m``\n U0'1

G14- ,,,,,,,,, pf s = 0

/r U r * 0 ^o`b^p %.,R,w&

05, 0r * '0 * /r/& ^o`pbk r

'0 , T/&/ '0 , U/'3-0

Page 335: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 662

0 s20- , ^obq^k , * b

\ \

10, '\] ^o`q^kg Fws' * a pf \] = k:

\ GTy*trR^RG,,,, ild ] * a pf \] ; k1 h]HT+\] re]h , s rh^ 1 0s+/

22- U6 ^obq^k U6 * b

23- F'G * s0' ^obq^k s + sZ * b

s1 1 * s0 ,++24- , ^obblp s + ++ r&0 , s/ * a

2 8

25-z'H * u1('^obq^k U'0 + s ^obq^k s * y i^d '0 * s0' * b

26- '0 * r& ^obq^k [.9: , [.9: * b 0' r &31- , ^obq^k s + ++27- '^obq^k [.9:(1 * b 1 0 * s

/

28- G^o`pbk r * rX, 0 , r0' * b 32- ^obq^k _T * b%r * 0(a&Gb'&p9hhhs

3/- ,,,<<<<,, * b1[.0 * s0

%r * 0(a&!&'&'9hhhs

30- ‘ .]] * B1s 0 * s0

*_

^obblq `!33- z ild '0 * `+0s' + s * B

`

s -+++23, \ ^o`pbk , , [ \/

* s/ * B[

/%\ * od' Gs + \35- H 0 ^o`pbk ,, * b

]+\ ]+\

36- 0 g] + ^i &] + ^(^o`pbk F7<: * iU'u , \'&] + s' X0s + &\ * ]'Z * b

5-14 Dgbo`f`flp 'mŠd- 215(

0- i^d Gt , 10 * i^d Zu * 40 * b

G'=: * 1(3 G

1- z i^d &s9 0'&s * U * b

2- , 0%r 0] 0( * y i^d G9 9 y G* B

2,xU0 [ s * i^d Gs19s]*01( G* b

2 24- i^d Gt * 00 , &0s * 0(1 * 0s * 0 * a5- 10^d Gt , 00 * i^d %r0 * r * 0( * b4+ s * h ^obq^k s + ,x^obq^k %rd/&* B7- 10^d xrf * i^d Er * 00 * b

Page 336: Calculus

663 Pjgp^dji`n \ gjn `e`m^d^djn

G8- i^d Gth, pi^d %r/ * 0( * /%r/ * 0( * b

6r/ * 2-r * 57 0 G%r * E&%r* 1(05.

.-+ 1%r * /&%r * U * di^d %r * 2(06 * bG

))& ,,0 * f[a Er * GG* bs)01- pi^d Er/

* 00 , f[a Gth* b.0+ r * y i^d Gt , 10 , ) i^d Er * 20 * b

303- i^d Er * 10 , r Z 1 * b

004-1,, , ^o`q^k %r * 1(* b

'[

05- 3 f[a Er * 00 ,z i^d Gth, pi^d Gt * 10 * b

06- ei^d G9 y y0, 0&U8[ 0( * b

07 -i i^d %r * 0(1 b‘ 2 s/)s)/ )

G.6+ f[a Erf * ,1,,0 * b

s (

h G G Er +0//-+ 1r * 1r/ * 7 i^d *r* * b

Gth10- i^d- .,+,, , U * ^o`q^k s * b

sf * s/

//+ cf[a E%r* E&,%r* 0(0 , p•^o`q^k r * b

G r/ * rR/ * : : rR/12- ‘ l f[a +, *- l ^o`q^k ,, * b

3s 1 r/ * rp 1 * H 1s 1 H, r/

/1+ %r/ * /r * 1(,0 * ^o`q^k %r * 0( * b/2+ *r,%r3 * r * 0( * b

0 0 * 2 q^k %r,/&15- U4 ^o`q^k U 4 * b

1 %FE* \ U'16- ‘ . ]] ^o`q^k ,, q^k , * brG , \/ +0 * \ 1

G G[ * `lp r * T [0+ : pbk r G

17- ,,,,,,,9,, i^d ,,,,,,,,, * bS\0

+ 0 i * \ `lp s

07, r * qU1 ^o`q^k %R/ q^k r& * b

Page 337: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn

1., 8] ^obq^k „ q^k s' * B

blp s+)& % %&%%%%% * B

[%[+m_hr * ] blp r&

21- '6S.3( , iild 1

22- iuU2 , s/ * z ^o`pbk ':2( * B

23- ,U2 +s/ * B

,-++ ,m 'U2 [s/ *r.h(24- s 2 , s/

* s 2 ild s * B

25- U s0 * s * iild '1U s0 * s * 0s * 0( * B

26- iusz * •zHld'u * qs/ * 4( * B

27- U s0 * s * 0 , iild &0s * 0 * 0q%+s+0,*,u,*,H( * B

28- ild &0s * 0 * 1U s0 * 0( * B

U1 +s +s0 U1 'U1 +s [s0 U1( &0s * 0(3/- , s * 3!!! ild s + 3!!! , ^o`pbk [,2,

5-15 Dgbo`f`flpab obm^pl 'mŠd-217(

0- a&s' * aL-s' < iNld U'0

0, a&s' < ild U2.'1 * `lp s'

3- 06 03 5Q&2s}* 1(

-& N! ,0y '_( 1 < ,,,,, ^r0 s&s * /'&s * 1(

5- ']( s x 0 'a( C&\s' + C&\'9 C&s' + x * `9 s`% ! + ` + CD'

6- '^( Ml bufpqbq^i crk`fŽk '_( Z/s i^d 1 'b( iu ~ 08- '^( b&1s' < 1`0Ub&s' '_( b&is' < i`&i+g'Ub&s' 'b( 1 'a( B < 1

/., a&s' < ]sd\b&s'* pfbkal d rk^ crk`fŽk mbofŽaf`^ `lk mboŒlal \,

01- '^( +>`+\ '_( i@ 'b( > * 0 , ib 'a( ` i^d 1 , >

02- '_( Bl * i@. * i&i + g'^0 * i&i + /'&i + 1(`2

H +T

'b( Pdk&s' <•<fUf* bkqlk`bp a&i'&j' ;•f &9'@fFR: FR:

05- ']( ds/&s * Gth(

'_( s + s1 oe Gthy 0: s+gs Gth* yy oe3r

Erf = 0

'b( 0 , `+U pf s x N: `U + 0 pf

'`( s pf Erf w 0: gs1 * yy pf2 2 [

s ; N

Gth= 0

00.

%7

Page 338: Calculus

00/ Pjgp^dji`n \ gjn `e`m^d^djn

.4+ a&s' < S&0s * F'-Qm07- '^( U , `+0o' '_( Qm&/ + `+2o' 'b( zSoZi , `+0o&0o* 0(\

08- '^( i^d 2 , 1 i^d 1 '_( Ml bufpqb kfkd•k s ob^i1/- '^( `fboq^ '_( c^ip^ 'b( `fboq^ 'a( c^ip^ pf s ; N

'a( Pl

14- 'a( &s`+ooi _o < i `+s& `U + c7w&Il f;L

16- '^( a&o' < 1Ui , H pf o = N

'_( a&o' < o + o0 * pf N z o x 0'b( aR' < o + do0 * p pf Zo\z 0

'a( a&o' < o pf o x N: a&o' < _f * 0 pf o = N

++ 8%e* 0( 17- '_( `i < ,1 ‚ il f 1 'b( ] < ild 1

e:. d

/6+ a%s&< +`V9 qlalv2/- '_( `lkpq^kqb < e

?[j•nofi 4

6-7 Dgbo`f`flp 'mŠd- 237(

44U1 U17- '_( 561 * N) alkab ENEw 657/ ; 1 - 0/,

3

8- /-8350 * O* alkab GPG; 1 - 0/,3

6-00 Dgbo`f`flp 'mŠd- 245(

0- 0 * s ild 1 * s/ ild1 11- blp 0 * 'blp 0 , pbk .&%r * 0( , i'1pbk 0 * blp E&%r* 0(1 * f'pbk 0 , 2 blp .&%r * 0(2

[1 [2 /1[3 [4

1, s + s0 + + * , , , *,5 1 01/ 7

-(.(5-6-7-8-

2/-22-

6-02 Dgbo`f`flp 'mŠd- 251(

0- [31- ,1

\ < N+ ] < 0+ b < , ,f 0/- 0 04- h 1/- ,1 14- +`-05

\-] 00- 0 05- ,0 10- f gjb\ 15- `+F-0

y 01- ild \-gjb ] 06- ,0 11- H- 16- bi.52 5

,f 02- G 07- G 12- g-` 17- G!2 !5 1

G 03- 08- 13- `0 18- 0!1 1\ < 1: iŒjfqb < ha`L' < N: .+'/( < N: m`L' < 3: iŒjfqb < `0

2- p3- ,f

3, %[,S/( e

Page 339: Calculus

Pjgp^dji`n \ gjn `e`m^o^ojn 000

6- G.Ty7- ,1

8-

0/- ,y

0H- h%h * 0(.101- i'^1 , ]0'-&\0]0'

02- 5 `r^kal s ++,N: 2-5Q `r^kal s ++,5Q-0.1+ \ < ,2: \ < z,.2+ \ < 3: \ < 005- '^( P%r&< q^k vr * hpbk r '_( O%r&< wr * !wpbk r 'b(

/5, oB-I

>oblp fo07-

0f

6-06 Dgbo`f`flp 'mŠd-260(

0- N 7- `-0 04- N 11-

1- 0 8- *// 05- G 12- `%

2- ,y 0/- 0 06- ,0 13- `0-p

3- E,R\ 00- N 07- G 14- 01

4- 0 01- N 08- ` 15- ild 1o3

5- 0 02- p 1/- G 16- 01

6- N 03- N 10- i.b 17- ` < i: iŒjfqb < p18- 0

1

2/- b < 0: iŒjfqb < R021- '_( 00+44 ^•lp 'b( 00+56 ^•lp

@\k…opgj 6

7-4 Dgbo`f`flp 'mŠd-270(

0- t;`1T[`0T G' 0( -5, V <91 H *:9 'B , `+T*&

5+ t < pbk r * ?,m_hr

6+ t < ': < yn&s * s x 1 * `(.-+ V < rP%r& * ?r

..+ v&s' < H * ild s

01- RŽil i^ crk`fŽk a^a^

1, t < 3 blp s + 1 `lp, s

2, V < s0+ 1 * 0`+s0-0

3, s < •%`/n* e`+n

3+ t < %r * ?&,m_hr

.1+ t < &S0`0U * `0s [ `U'0

.2+ t < .,%r/ * r * 1 , _*T&

.3+ t < %r1 + U'0

.4+ t < .,%r/ * r * r/fiar&

; &`U * `0+U'/-0 &`U * `0+U'!/-0 pbke s07- '^( t [0s '_( t < , 0s 'b( t0 < +s+

^`1s * 1 ] * 1 `1s * 1B ] + 01/- '^( t < b 0r 0 pfbkal B < +] + '_( V < 0r B pfbkal B <,,

,` + + 0 ` + ] * 1

Page 340: Calculus

001 Pjgp^dji`n \ gjn `e`m^d^djn

7-6- Dgbo`f`flp 'mŠd- 28/(

0//'0 , 1,0.05( < 3+1 mlo `fbkql

Br^qol sb`bp i^ `^kqfa^a fkf`f^i

0-

1-2- '^( Q < 'h]c i'-f

3- 145'0 , b,H.G( pf

4, q ,,* sf hb-f

'_( q%l& < %\ * u&,%\ * [&

N 99::x 99::0/: 05 * /44`/-*/. pf o ; 0/

6- 'b( 43+4 jfk

7- 44•8- 7+74 Jd

0/- 13+70 Jd02- O^o^ i^ b`r^`fŽk '7-1/(+ s < tka&+'G,hqh: m^o^ i^ b`r^`fŽk '7-11(+ V < Jf

0'a( P < HNhZ0 * '5// , l&e * 'i3//h , f&_*efY

.2+ s < IWf *bum' *IF,,%EE&^ff&F*E05- '^( 1// Lfiilkbp '_( 106 Lfiilkbp06- '^( /+/15 mlo ^•l '_( /+/00 mlo ^•l: 15/ Lfiilkbp 34/ Lfiilkbp/6, _sg_o < er%. * \m'9 r < ri_e%E*)xE$S8 `ros^ 'a(

7-03 Dgbo`f`flp 'mŠd-3/0(

f+ V < ^f`/T * ^/`+/T

/+ t < `hblp /r * ?/ pbk /r

1, t < Bi * ^/`1T

1+ S < B0 * ^/`+‰Š z,

2+ U < _F!%]f \n Y%/r * ]0 pbk [ /r&00- t < 0 , w_*0F!,/

./+ s < +`\n &3s + 04(

\ ] {,{,.0+ U <91 …'k&,,G(*91 a+`.,GG+ alkab \ < 1 , [ 4+ \ < 1 * [ 4

3+ U < ]._! * ]0_*7fF$

5, t < _F7%]/ blp r * @0 pbk-s(5+ t < _F$%]. blp /r * `1pbk /r&7, t < )+)*F$%]

/* ]

0r&

.-+ t < `%&^/ * ^0s'

.1+ S < /_*/F!%_im r * pbk r&/3, wE%T&< x`0G!+!n`i3s9 q&s' < x`+0s+!n`i1s/4, EE%T&< 4&`1T * `+T'-39 yvs' < `! [ `+js

.4+ e < i/4P/8 ad,&s' < Bpbk j,t &i < 0+1+2+ --- (

08- '_( K\ 'b( pf f ,:H`, N i^ `lkaf`fŽk bp [f * C/ ,:H`, iigZ9

1/- '^( s! * s < N

'_( s! * 1.$ * 1s < N'b( ` * s$ * ws < N'a( s! * 1s < N'b( s! * s < N

7-06 Dgbo`f`flp 'mŠd-3/7(

f+ U < ]f_F$ * ]0_*s + r

08 t < ]f_r * ]/ * 0s + s/ * /s0

Page 341: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 557

1, U < ^y`+U * ^0 * gs1

2, t < `G%&^y`lp *d•s * ^ 1 pbk qds' + 176 * xs * (s0 * ,fou^

3, t < ^y`s * ^0`2U * y1 * os * y%s0

4, t < `y`0U * `0`+1G%+ .1 * qu , s0 + os1

5, t < &^y * oU'`0U * `0`+0U

6, t < _x `lp 0s * ^/ pbk1u * d`+/r

7, t < ^y`0G * &`0 * +/+s'`s

.-+ t < ^y` 0G%* ^0`U * o`0s

//, t;^y`*0U)&`/))s'`T)+ `0U

/0, t < &^y * ^0s * +gd+s\'`U* s * 1

/1, V < &`y * `0s + i^d FsY'`+U

/2, t < `{pbk s * &^y * i^d Zbpbs * `lq sg' `lp s + 1/3, V < `y`s * x0`+s * &`U + `+U' h]c '0 * `U' + s`! + 0

.3+ U < &^y * os'`s * `+U * `0`+0U + ,q , d&`U * `+0U' i^d '0 * `U'

x

'bi * ^0s'`+0T pf s ; H \ s = 1+.4+ U :

&\ * ]s'`+1U * y pf H z s 9R1

.5+ U < ^y„s * `0`+1U * ns`1s

.6+ t ;&^y+ xs'`\n1s )&^0 ,ip(pbk2u0., t ;&`y+xs'`\ns )^0n`is

/.+ t < ]x `lp r * %]/ * zu(pbk r

00, t ;^g^\n0s )^0n`i0s )s`\ns * 4pbku

01, t < `y `lp 0s * ^0 pbk 0s * s pbk s + `lp s02, t < `{ * ^0„s +x`0U&1 pbk s * `lp s'03, t < @gpbk tr,h, `1 `lp s * gj`0U&1pbk 1s + `lp 1s'

0&)1 :WR_PVPV\`'mŠd- 303(

i- 1U11- ~ 03„0R

1, > < a+ g < f* x2 < l` , /Q2, U < 2 `\n 2/QU

4- b < %sw* qx'g-0

3+ s < U5+ s! < *./s < ,3U5/QU

5, V < +> pbk 2!& pfbkal > mlpfqfsl

zpbko * 0 , blp o pf N9R o 9R0/Q*6, F&o':

pbk o pf o99880/Q

8- '^( 0.'10SU1( '_( O ; 1

.-+ m&o'< dbo/* ^o * ^&o+ z(ild '0 , 9(

))& m&o'< ^o * `'f , o' ild '0 , z(

Page 342: Calculus

01) Pjgp^dji`n \ gjn `e`m^d^djn

SRj S

/0, l%n&< f ild S Z fo

7-11 Dgbo`f`flp 'mŠd-310(

f+ s$ * he< N/+ s$ * /s < N1, ss$ * s < N

1+ rs$ * s < N2+ /rs$ * s < N

00- %r/ * s/ * H(v&, /rs < N./+ %r/ * /rs * s/ * /&s$ * s/ * /rs * r/ * 1 < N.1+ s * t < ,0 bp ^ i^ sbw rk^ `ros^ fkqbdo^i v rk^ fpl`ifk^.2+ s < ?r * ?/8 bkslisbkqb9 s < ,iu1

4, %r/ * v1 , .&s$ * /rs < N

5, %r * E&s$ * rs < N6, %r/

* 2'/% + V < N6+ s$ * s q^k s < N

0/- T i , r/ s$ * s/ * H < N

7-13 Dgbo`f`flp 'mŠd- 313(

i- v2 < 8ds2 * B1- blp s < Bbi.b!p t,(-(

4-00-

01-02-

03-04-05-

s%? * i^d Gt * 00( < iU * 1 < ?%s * 0&_T

s/(/w:?

%s* &_*0t < _T%_imr * pbk r& * Br/ * 0 < ?%s/ * 0(X&s' < 0`s+.

X&s' < U4u1 * 0X&s' < &,i^d'0 * s/&

W%r& < ~ 0: W%r& < pbk %r * B( : q^j_f‹k+ ^nrbii^p crk`flkbp `lkqfkr^p `rv^p doŠcf`^p

mrbabk l_qbkbopb rkfbkal mlo`flkbp ab i^p `ros^p u < pbk %r * a( `lk mlo`flkbp abi^p ob`q^p u < ~ 0- Tkl ab q^ibp bgbjmilp bp x%r&< , 0 m^o^ r 94 N+x%r&< pbk %r * x/Q'm^o^ /94 r 94 2&06!+ x%r&< 0 m^o^ r w 2&06!

.4+ X&s' < B

.5+ W%r& < =_T,]

.6+ X&s' < N0., a&s' < N

3+ U < ?%r * 0&_T

6- ^obq^k t * ^obpbk s < B7- '0 * s/&%E * r/& < ?r/

6+ t2&U * 1( < ?%r * 1(0/- 0 * s/ < ?r/`r/

7-15 Dgbo`f`flp 'mŠd- 318(

/+ r/ * s/ < B

1, s < r ild x?rf

1+ r/ * s/ < ?r1

2+ s/ < ?%r/ * s/&0

3+ r/ * /?s < ?/) B = N4+ s%?r/ * 0( < s

s7- ^obq^k , * ild Guh< B

t

s T s08- , , ,- * ild , < B

s t s

0/ q^k H< @`T

, 0s

00- %r * s&0 < ?T2s2

Page 343: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 670

7-17 Dgbo`f`flp ab obm^pl 'mŠd- 323(

5, st < A5+ t0 + ild 'pbk1 r& < B6+ &s + @'0 * t0 < B1,0

.-+ s0 * t0 + @&s * t' * 1 < N00- V < , 0s ild s

0- 1s + /s < B

/+ s0 + t0 < A1, s0 * t0 + @s * H < N1+ 0s0 * t0 < B4- 1v1 , s/ < B3+ s/ < T * B

0/0, V < , fsgjbs

.0+ X&s' < Bu!+ l X&s' < ?TE,`f

.1+ X&s' < ^`j, l X&s' < @sg-&0i'

0 s/3, V < : s1 * 1

0/QO/tc08- z pbd+pfbkal O bi o^afl ab i^ _^pb v c i^ ^iqro^ abi `lkl

q

2/4, V < h'h, b9,h(: ] < 0` + 2

/5, V < +4s0 * 3s * G.5+ pl-p pbd

0., V < `!10- E< + s * @U+F-0m^o^ s = N+l E< + s m^o^ qlal s

00, h < ,0: u10/d hnh< o`+0%!* @t0

12- '^( \ < N+] < p '_( X&s' < /TF-0

13- '_( V < ‹&!, `+!*1-1

14- '^( F-&o * 0( do^jlp bk o ^•lp '_( H do^jlp! ^•lp!

15- Zi , q'1 , S0'oZ0 do^jlp bk o ^•lp 1 * U1 ^•lp

16- '^( 143`+0Š43o`fra^a^klp bk q ^•lp '_( 254'0 , `+0*43o' abcrk`flkbp bk q ^•lp

17- 5+85okf.pbd < 14 /45 jf.e18- '^( LŒkfjl obi^qfsl bk N '_( \ < z+] < $ 6 H 'a( e2/- '_( LŒkfjl 'b( p

@\k…opgj 7

8!-5 Dgbo`f`flp 'mŠd-334(

0- '^( 0d '_( +d 'b( p, pe 'a( 07 * d 'b( ,z ) d 'b( H * c

'c( N 'e( 0 * d

1- '^( U1 '_( 4 'b( 'a( 'b( U1 'b( U 542- '^( o ;0*. ;g/Q '_( o ;1*. < + /Q 'b( o < 0+/ ;/Q {'a( o < 0+/ </

'b( o < 1U2+ L < 3/Q-4 'b( o < 0+L < iHS 'c( o < 1U1+ L < o/Q

'e( o < 1U1+L < +o/Q 'f( o < qU1+ L < ,^ 'g( l < h+L < *•/Q3- '^( t < N+s ^o_fqo^ofl '_( s = N+t < N 'b( Slal s v qlal V 'a( s < N+

t ^o_fqo^ofl: l t < N+s ^o_fqo^ofl 'b( s < 0+V < N 'b( s < 0+V < N

Page 344: Calculus

01+ Pjgp^dji`n \ gjn `e`m^d^djn

8-0/ Dgbo`f`flp 'mŠd- 342(

0- '^( '_( ,1f 'b( ,2 'a( 0 'b( 0 * e 'b( 'i * c&,X,/

'd( U1 e %b&*c1- '^( s < N+r ^o_fqo^ofl '_( r < s < N 'b( r < N+V < %/h * 0(6S+ pfbkal h rk

bkqbol `r^inrfbo^ 'a( r < 0+V < .P * /h.P) pfbkal h rk bkqbol `r^inrfbo^2- '_( u < 0i5Qd*pfbkal i rk bkqbol `r^inrfbo^

4, @f < &\[f * d][f' m^o^ f < 0+1+--- +i

0/- 'b( iU2 * if+ , U2 * nc) ,e

'a( \ * ]e* +\ + ]d*+] * \d* ] + \d* pfbkal \ < gG0 * U1 v ] < gG0 + [.1'b( \ + ]d* +\ * ]d* ] * \d* +] + \d*pfbkal \ v ] ilp ab 'a(

00- '^( 0+zm!*_*Q 'b( ,6S ; ^od'wi( * ^od &W0' x oo

.0+ > < =E%\ * r0 * \rd'

@\k…opgj/.

0/-3 Dgbo`f`flp 'mŠd- 356(

0- '^( Blksbodb #@$ N 01- '^( Blksbodb #@$ p1- '^( Blksbodb '_( ,0 02- '^( Blksbodb '_( N2- '^( Cfsbodbkqb 03- '^( Blksbodb #@$ M

3- '^( Blksbodb '_( , 04- '^( Blksbodb #@$ M

4- #?$ Blksbodb #@$ N 05- #?$ Blksbodb #@$ N5- '^( Cfsbodbkqb 06- '^( Blksbodb #@$ `/

6- #?$ Blksbodb #@$ N 07- '^( Cfsbodbkqb

7- '^( Cfsbodbkqb 08- '^( Blksbodb #@$ N8- '^( Blksbodb #@$ 1/- #?$ Cfsbodbkqb

0/- '^( Cfsbodbkqb 10- '^( Blksbodb #@$ N00- #?$ Blksbodb #@$ N 11- '^( Cfsbodbkqb

12- K< /-B13- K< /-B14- K< i.D15- K< VZz

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17-ild D

K<ild '8.0/(

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F5:

prj^ ab 0 ^ i* K^p alp pr`bpflkbp vn,w u vo,w `lksbodbk e^`f^ i^ fkqbdo^i Fadur&^r+

0/-8 Dgbo`f`flp 'mŠd- 366(

11- ']( 0 '_( 0` + 2 'b( ` * 0

Page 345: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 672

12- '_( 413- '^( Ha‹kqf`^ '_( Ml fa‹kqf`^ 'b( Ml fa‹kqf`^ 'a( Ha‹kqf`^

)0/-0/ Dgbo`f`flp pl_ob abp^ooliilp ab`fj^ibp 'mŠd- 368(

0- p1- e 2- yzz+€-1+ g>4- h

0/-03 Dgbo`f`flp 'mŠd- 375(

i- Cfsbodbkqb1- Blksbodbkqb2- Blksbodbkqb3- Blksbodbkqb4- Blksbodbkqb5- Blksbodbkqb

-6- Blksbodbkqb04- Blksbodbkqb m^o^ p = 0:

05- Blksbodbkqb06- Blksbodbkqb07- Blksbodbkqb

7-8-

0/-00-

01-02-03-

afsbodbkqb m^o^ p z 0

BlksbodbkqbCfsbodbkqb

BlksbodbkqbCfsbodbkqbBlksbodbkqbCfsbodbkqbBlksbodbkqb

0/-05

0-

1-,(3-4-5-

02-03-

Dgbo`f`flp 'mŠd-38/(

Blksbodbkqb

BlksbodbkqbBlksbodbkqb

CfsbodbkqbCfsbodbkqb

CfsbodbkqbBlksbodbkqbBlksbodbkqb pf

6- Cfsbodbkqb7- Blksbodbkqb8- Blksbodbkqb

0/- Cfsbodbkqb

00- Blksbodbkqb

01- Cfsbodbkqb

l ; l ; 0+ l `r^kal r < e7y)e bkqbol `r^inrfbo^

0/-1/ Dgbo`f`flp 'mŠd-388(

i- Blkaf`flk^ijbkqb `lksbodbkqb1- Blkaf`flk^ijbkqb `lksbodbkqb2- Cfsbodbkqb m^o^ p 9,9:N: `lkaf`flk^ijbkqb `lksbodbkqb m^o^

`lksbodbkqb m^o^ p = 0

3- @_plirq^jbkqb `lksbodbkqb4- @_plirq^jbkqb `lksbodbkqb5- @_plirq^jbkqb `lksbosbkqb6- Cfsbodbkqb7- Cfsbodbkqb8- Cfsbodbkqb

N ; p z 09 ^_plirq^jbkqb

0/-00-

01-02-03-04-

Blkaf`flk^ijbkqb `lksbodbkqb@_plirq^jbkqb `lksbodbkqbCfsbodbkqb@_plirq^jbkqb `lksbodbkqb@_plirq^jbkqb `lksbodbkqb

Cfsbodbkqb

Page 346: Calculus

01- Pjgp^dji`n \ gjn `d`m^d^djn

05- @_plirq^jbkqb `lksbodbkqb

06- @_plirq^jbkqb `lksbodbkqb07- @_plirq^jbkqb `lksbodbkqb08- Blkaf`flk^ijbkqb `lksbodbkqb

1/- Blkaf`flk^ijbkqb `lksbodbkqb

14-

10- Cfsbodbkqb11- Blkaf`flk^ijbkqb `lksbodbkqb12- Cfsbodbkqb13- Blkaf`flk^ijbkqb `lksbodbkqb

Cfsbodbkqb m^o^ p R N: `lkaf`flk^ijbkqb `lksbodbkqb m^o^N ; p RH:`lksbodbkqb m^o^ p = 0

@_plirq^jbkqb `lksbodbkqb@_plirq^jbkqb `lksbodbkqbCfsbodbkqb@_plirq^jbkqb `lksbodbkqb@_plirq^jbkqb `lksbodbkqb@_plirq^jbkqb `lksbodbkqb

@_plirq^jbkqb `lksbodbkqb]5>Slal u

Slal u nrb p^qfpc^d^ Gvh; 2Slal uSlal u bu`bmql bkqbolp kbd^qfslp

15-16-17-18-2/-20-

21-22-23-

24-25-26-

^_plirq^jbkqb

Slal u !! 0 nrb p^qfpc^d^ ug R 0Etf ; b,i.RR

Qj_juSlal u !! N nrb p^qfpc^d^N R Zw, iH R i

Slal u !! ,0 nrb p^qfpc^d^ g1w* 20 R iSlal t < r * dt `lk r ƒ N

Slal u nrb p^qfpc^d^01 * g-ug = GSlal t nrb p^qfpc^d^ Y1* Eftf = iSlal u !! N

Er * ghR mm-2*f bkqbol `r^inrfbo^

Gt, ghR mm-4*f bkqbol `r^inrfbo^

27-28-3/-30-

31-32-33-34-35-

36-37-

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0- '^( N

'_( Blksbodb pf b oG:bi iŒjfqb bp Npf b ; 0: bi iŒjfqb bp 0 pf b < 0: afsbodb pf b = 01- '^( G '_( bi j^vlo ab ilp alp !k•jbolp \ v ]

2- q^i * \03- N * U4(4- N6- Cfsbodbkqb

7- Blksbodbkqb pf p ; p: afsbodbkqb pf p ƒ p8- Blksbodbkqb

0/- Cfsbodbkqb

00- Cfsbodbkqb03- b Q 2.2+ \ ƒ 2

\ * 006- Br^kal \ ƒ ,0+ bi iŒjfqb bp - `r^kal \ R , K bi iŒjfqb bp N

\ * 1&

)(&*, :WR_PVPV\` ]mT&-)+!

0- Cfsbodbkqb1- Blksbodbkqb2- Blksbodbkqb3- Blksbodbkqb4- Blksbodbkqb

5- Blksbodbkqb6- Blksbodbkqb

7- Blksbodbkqb8- Cfsbodbkqb

0/- Blksbodbkqb pf p = 0: afsbodbkqb -pfp R i

Page 347: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 674

00- b < -z: i^ fkqbdo^i qfbkb bi s^ilo eildf01- b < -z: i^ fkqbdo^i qfbkb bi s^ilo eild z-

2 ]02- b < z&.1:i^ fkqbdo^i qfbkb bi s^ilo U1 ild U 1

/2, \ < ] < 0` + 1

U204- ^<i: \:E*Z4P

05- '_( Cfsbodbk ^j_^p06- 'b( Cfsbodb

@\k…opgj //

))&6 Dgbo`f`flp 'mŠd-415(

0- m < 1: `lksbodbkqb m^o^ 0 u 0 ; 10, m < 1: `lksbodbkqb m^o^ GvhR 1+ u :-‹ 1

1, l < 1: `lksbodbkqb m^o^ Gv* 20 R 1+ t :-‹ ,0

2, m < z: `lksbodbkqb m^o^ GvhR p2+ m < z: `lksbodbkqb m^o^ Gvh; p3+ l < _8 `lksbodbkqb m^o^ Gvh; _

5, m < 0: `lksbodbkqb m^o^ Gv* 00 R 05+ m < * ^`

6+ m < 3: `lksbodbkqb m^o^ Yvh ; 3

.-+ m < 0: `lksbodbkqb m^o^ -ug ; 000- .$:.

./+ m < f,_

.0+ m < * NB pf \ < f5Q* f bkqbol: m < 0 pf \ :-‹ f5Q

.1+ m < `+\

04- l < j^u %[)\&.3+ m < jfk L-\* f,\&

00-02 Dgbo`f`flp 'mŠd-425(

0- Gth; 0: i.N * s/&

1- Gth; 2: 0.'2 , r&2- Gth; 0: s-L + U'0

1+ Fsg; 0: +s-L * U'0

0 ild N * /r&4- Gth; y: 0 * 0s * 0s,

4, +gns:o9 +gjbL+0s'/ [

6- ,1 R s ; 1: , ^o`q^k ,[ 1

7- Slal s9 `+s1

Page 348: Calculus

01/ Pjgp^dji`n \ gjn `e`m^d^djn

8- Slal s9 m0&`T * 0 , s + s0' pf s :‹ N+N pf s < N

`U++F + s0/ Slal u : ,,,oe s :‹ i 9 gnds < 0

+ %r * 0(1 & 1

U1186

11- ,z

01, \j < 3U1+ ^i < N+ \0 < 4U1- \1 < N+ \2 < iis1

))&). Dgbo`f`flp 'mŠd- -,*!

&i * 1'&i + 1(0- \i)0 < &i * 0'&i * 0( \i m^o^ i x N: a&s' < 0 , 1s0

%h* 1&%h* 2( - 0/0, \i)0 < &i * 0'&i * 0( \! m^o^ i x N: a&s' < 0s + 2 s0

2- Slal s3- Slal s

4- Slal r8 [ < g*] ;.5- Slal s9 a&s' < `r/

6- Slal s9 a&s' < `U + s + 07- Slal s9 a&s' < blp 0s

8- Slal s9 a&s' < s * pbke 1s./+ t < 0 * s * s/ * wT0 * ---/1, U < s * ds2 * -2U5 * !!g!aojsgL * ---.1+ U ;ds0 * 001WR * jxjU6 * 776//WHi * ---

/4, V ;@L&/ * y s1i

'9p'1 - 2('4 - 5( --- X&1i + 0( - &1i'Z

* B0

' s * y '2& 3('5& 6(& xx%x99i'%'2.0 * 0(\(

' ƒ`l &[/'is0i' ƒ`l &[/'i)gs0ix/

.4+ t ;^j 0 * ,,,, *`G

,,, \

Š j<01& 3+ -- &0i' kzi 0 •2 --- &0i + 0(

07- ^i < ,0+ \0 < N+ \1 < {: a&s' < &s * g'`+0U

6 pbk s blp s + 0/7, \n < N+ \4 < , ,7& 9a&s' < ,, * 1 pf s x N: a`L' <y: a&/Q'< +0-/Q0

( B B

1/- 'b( U1 < 0+3031024512

10- '_( U2 < 0+621/4/7/6457766

Page 349: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 010

@\k…opgj /0

)*&, :WR_PVPV\` ]mT&--)!

0- '^( '4+/+8( H'_( ',2+5+2( 'b( '2+ ,0+3( 'a( ',6+13+10( 'b( '/+/+ N(

3, s < y'2`0 , ^0'* V < x&0@0 + BH(

5- '^( &s * u*s * t * u*s * t' 'b( s < 1+ V < 0+ u < ,0

6- '^( &s * 0u*s * t * u*s * t * u' '_( Tk bgbjmil9 s < ,1+ V < u < H7- '^( &s)u*s)t)u*s)t*t' 'b( s;+F* t;2* u;0

01- K^p af^dlk^ibp ab rk m^o^ibiŽdo^jl pb `loq^k bk pr mrkql jbafl-

)*&0 :WR_PVPV\` ]mT&--0!

0- '^( ,5 '^( 1 'b( 51- '^( %=$>&? < '10+17+ ,24(

'a( =%>$ B( < '2/+5/+ , 0/4(

'a( } 'b( 3'_( =$ %> * B( < 53 'b( %= * >& +B < 61

'1 3 ,6(

'b( =d%>$A( < , , ,04&04& 04+

4- Tk bgbjmil9 '0+ ,4+ ,2(

5- Tk bgbjmil9 s < ,1+ V < 0

6- B < p',0+ ,1+1(+ A <ˆ'11+ ,0+ HC(

'4 6 0 ,4 ,6(

7- B < HUH+1+2+3+4(+ @ < HS&33& 22&77&44

8- '^( S52 '_( UH3 'b( S31 'a( 40/- '^( '0+ ,0( l ',0+0( '_( '0+0( l ',0+ ,0(

'a( %\)*[& l ' *\) [&

000- ']( Š.]'3+ ,0+4(

s31

0'a( U31',4+,3+,H(

./+ = W >) A v @) A v A) @ W A02- '^( '1+ ,0( X ',1+0( '_( '1+0( X ',1+ ,0(

'`( '0+1( v ',0+ ,1(03- Tk bgbjmil9 B < '7+0+0(04- Tk bgbjmil9 B < '0+ ,4+ ,2(

.3+ L < G,;2+ 3(+ M < 11R' ,3+2(

.4+ L < U+ )$ 0+0(+ M < z',2+ ,0+0+2(

007- ~ U1 '/+0+0(

'b( '2+1( l ',2+ ,1(

0'^( U03 ',1+ ,2+0(

0'b( +.]',0+,4+3(

s31

0'b( :<'0+/+0(

U1

'b( '0+1( v ',0+ ,1(

1/- K^ prj^ ab ilp `r^ao^alp ab ilp i^alp ab rk m^o^ibiŽdo^jl `r^inrfbo^ bp fdr^i ^ i^prj^ ab ilp `r^ao^alp ab i^p af^dlk^ibp-

11- 3: 01U1

Page 350: Calculus

677 Pjgp^dji`n \ gjn `e`m^d^djn

0 ' 6 1 ,4 ,03(12- B < oo'i+ 1+2+3+4(+ @ < RG0/+1&!2&!!3&,4,

13- B < `=) @ < > * `=) pfbkal ` < u= +>&,%=+=&

01-00 Dgbo`f`flp 'mŠd- 452(

0- 080>

0, g?

,( '^(5 2 ,1

!6&6& 6 '5 2 ,1('_( 6&6&6 v '

,5 ,2 z(6 & 6 &6

4- N+Tzy+T 350

4, =+mm-6

5+ !/Q-48- N

0/- _( K^ b`r^`fŽk bp sŠifa^ m^o^ s b s `r^ibpnrfbo^ pf blp _ < 0: pf `lp _ w 0 i^ •kf`^plir`fŽk bp s < t < N

03- Sla^p bu`bmql _(-06- b( Sla^p bu`bmql bi qblobj^ 01-3 ^(-

07- ^( Sla^p

01-04 Dgbo`f`flp 'mŠd-460(

0- '^( s < t < p '^( s < ,p+t < p 'b( s < 3+ V < ,0'a( s < 0+ V < 5

0, s < q- t < e1, s < 2+ V < ,36- Sla^p o x N8- 'b( 6: , 3'f * g(

0/- '_( f < ? + >* f < F_ , ?' 'b( .3> + /2? * 3@'..+ u=v) u>v) u?v) u@v)u=) >v) u=) By+u=) @v)u>) By+u?) @v01- '^( Hkabmbkafbkqb '_( Tk bgbjmil\: @ < = 'b( Tk bgbjmil9 A < 'N+N+/+0(

'a( Difdfbkal B < 'N+N+N+0(+ obkbjlp U < 0> * ? + b * 1B

02- 'b( o < N+U1+,U103- '^( x'H+/+0+ N(+'/+0+ /+0(+ '1+ N+, 0+N(y '_( Di `lkgrkql a^al 'b( Di `lkgrkql a^ab06- x'N+ 0+0(+'0+0+0(+ '/+0+ N(y+ x'N+0+0(+'0+0+0(+ 'N+N+H(y07- x'H+0+0+0(+'/+0+0+0(+ 'N+/+0+0(+ 'N+N+N+i(y+

xbi+ 0+0+0(+'/+0+0+0(+ '/+0+ N+N(+'N+/+0+ N(y

.6+ H%Q&&<H%P&$77H%O&1/- Tk bgbjmil 09 > < vBg! ,, * BmG* ? < vBg * B0* B0 * Bi< ,,, * Bi[g * Bi< Bi * Bgw

01-06 Dgbo`f`flp 'mŠd-464(

0- '^( ,0 , f '_( ,0 * f 'b( 0 , f

'b( 1 , f 'c( ,f 'e( ,0 * 1f 'e('`( ,0 * e 'b(

,2 , 1f 'g( 1f,0 , e

Page 351: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 012

1- Tk bgbjmil9 '0 * c),4 , 2f+0 , 2f(

6, Qmg17, 1> + ? * 1a

@\k…opgj /1

02-4 Dgbo`f`flp 'mŠd-473(

H- '_(+ 'a(+ X 'b(1+ '^( v 'b(2- 'b(+ 'a(+ v 'b(3- '_(+ 'b(+ v 'b(4- '^( Ml '_( Ml 'b( Ml

3+ =) >) a+@)B bpqŠk^ifkb^alp6- Rb `loq^k bk '4+ 8+02(

7- '_( Ml

8- '^( 801 * 6o * 8 '_( iU54

02-7 Dgbo`f`flp 'mŠd-48/(

G- 'b( v 'b(

1- '^(+ '_(+ u 'b(

2- '^( r < 0 * o* t < 1 * p * o* t < 0 * 30'_( s < p * o* t < 0 * p+ u < p * 2o

3- '^( '0+1+ N( X '1+ ,2+ ,2( '_( I < x'i+ 1+N( * p'i+ 0+1( * 0' ,1+3+ 0(y5- '^(+ '_(+ v 'b( s + /s * u < ,26- '^( 'M+,1+ ,0( u ',0+ ,1+1(

'_( J < x'N+ ,1+ ,&90(* p' ,0+ N+2( * o&1*2+5(y

7- Clp bgbjmilp9 ',4+1+5( v ',03+2+06(8- '^( Rf '_(Clp bgbjmilp9 '0+/+ ,0( v ',0+/+0(

0/- ',1+z+ , z&(0H- '^(+ '_( u 'b( Ml

.0+ s + t < ,0

02-00 Dgbo`f`flp 'mŠd-486(

0- '^( ',1+2+ ,0( '_( '3+ ,4+2(

'b( '7+2+ ,6( 'b( '0/+ 00+4('e( ',1+/+3(

01- ']( €- .] ',3+2+ 0( '_(

s 15

2- '^( 01&! '_(zU24 'b(3- 7: * f , 0f

5- '_( `lp ` bp kbd^qfsl 'b( UR8- '^( rk^ plir`fŽk bp ? < ,: , 1f

'b( '3+ ,3+1('c( ',1+,7+,01(

'a( '7+ 0/+3('e( '1+ ,1+ N(

0~ - .\ ',30+ ,07+6(

s 1/43

qU2

0'b( ~ U5 '0+1+0(

'_( c * f , e bp i^ •kf`^ plir`fŽk

Page 352: Calculus

68/ Pjgp^dji`n \ gjn `e`m^d^djn

00- '^( Sobp mlpf_fifa^abp: A < ? * b , > < 'N+N+1(+ A < > * b , ? < '3+ ,1+1(+

@ < = * > * b < ' ,1+1+ N( '_( qU501- ,3: 7U2: ,qU2

02-03 Dgbo`f`flp 'mŠd- 5/1(

0- '^( 85 '_( 16 'b( ,73

1- N+ U1+ ,U12- 15- '^( &0] + 0(: * ]e * ^f* pfbkal_ v b ^o_fqo^oflp

0f- ,2: * 0e * 3f03: '_( 1

04- '_( U1//4.30.4+ s < 0+ U < ,0+ Y < 1.5+ s < 0+ U < ,0+ Y < 1.6+ s < 0+ V < 3+ u < 0

'_( *nc* bf

02-06 Dgbo`f`flp 'mŠd- 5/6(

0- '^( ',6+1+,1( '^( +5s)0t+0u;.1- '^( 'p+z+,0( '_( ,6+ ,†+† 'b( {1, 1s + t * 1w< ,4: 8.U033- '_( G,U54- '^( '0+1+ ,1( '_( s * 0t + 0u < 4 'b( q4,gLs+1t+5u)/5;.6- Clp Škdrilp: %Qm-1 X 0%Qm-1

5+ s * /s * 7u * 44 < M7, U&o' < '1+ 0+z2( * o&2*,2+ 0(

0/- '_( J < '0+ 2+ ,1( 'b( o < 0

'b( s * 1t + 0u * 08 < M

..+ s * S0t * u < 1 * U101- 5

002- +.* '6+,7+ ,2(

s 011

.1+ s + t * u < 104- 'g+N+p(/5, U&o' < '0+1+2( * o&g*,1+0(08- '_( L < ,k'4+ ,03+1(

'b( +5s * /s * 0u < ,8'a( ',z+,T+T&,(

'a( 0s * 1t * 0u * 04 < N

02-10 Dgbo`f`flp 'mŠd-504(

1, m < `_-&/ + ` pbk 7(: m < +`_-&/ * ` pbk 7(1+ ` < 0+ ^ < 13, ` < p+_ < 53+ ` < + ^ < 54+ ` < 1+ ^ < 0

6, ` < 1+ _ < 1

Page 353: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 680

6+ ` < 0+ ^ < 3.-+ ^ < 4+ m < 14.'0/ * 2 `lp %d* 3pbk'g(00- ^ < 4+ m < 14.'4 * 3 `lp %d* 2 pbk %d&

./+ ^ < fU1+ m < 0.'`lp 7 * pbk %d* {U1(+ m < 0.'`lp %d* pbk 7 , {U1(

02- '^( m < 0+4 u 0/7.'0 * `lp 7(: 6+4 W 0/6 Jj '_( m < 4 u 0/6.'0 , `lp 7(:

1-4 V 0/6 Jj

02-13 Dgbo`f`flp 'mŠd- 510(

0- Bbkqolbk'/+/(:cl`lpbk'€7+/(: s‹oqf`bp bk '€HN+N(: `;o1- Bbkqol bk 'N+N(: cl`lp bk 'N+ €7(: s‹oqf`bp bk 'N+ € 0/(: ` < q

2- Bbkqol bk '1+ ,2(: cl`lp bk '1 €U6+ z2(:s‹oqf`bp bk '5+ ,2(+',1+ ,2(: ` <U6.3

3- Bbkqol bk '/+/(: cl`lp bk'€z+N(: s‹oqf`bp bk '€z+N(: ` <q

4- Bbkqol bk 'N+N(: cl`lp bk ' €U2.5+ N(: s‹oqf`bp bk '€U2.2+ N(: ` < f5- Bbkqol bk ',0+ ,1(: cl`lp bkb ,0+0(+ ',0+ ,4(: s‹oqf`bp bk ',0+2(+ ',0+ ,6(: `; e5, 5s/ * 05v1 < 6

%r * U %s* 3(17- 05 *,8, < 0

%r * 2(1 %s* 3(18- 8 * 05 < 0

%r * 3(10/- 8 * %s* 1(1 < 0

%r * 7(1 %s* 1(100- 14 * 8 < 0

%r * 1(1 %s* 0(101- 05 * ,3, < 0

02- Bbkqol bk '/+/(: cl`lp bk'€1U30+/(: s‹oqf`bp bk '€HN+N(: ` <U30.4

03- Bbkqol bk '/+/(: cl`lp bk'N+ €1U30(: s‹oqf`bp bk 'N+ €0/(: ` <U30.4

04- Bbkqolbk',2+2(:cl`i-R bk',2 €U4+2(: s‹oqf`bp bk ',0+2(+',4+2(: ` <U4.1

05- Bbkqol bk 'N+N(: cl`lp bk'€4+/(: s‹oqf`bp bk '€3+/(: ` < 4.306- Bbkqol bk 'N+N(: cl`lp bk'N+ €2(: s‹oqf`bp bk 'N+ €1(: ` <q

07- Bbkqol bk '0+ ,1(: cl`lp bk'0 €U02+ ,1(: s‹oqf`bp bkq'2+ ,1(+',0+ */&8_ <fU02

s0 t008- 3! , 01 < 0

0., t0 + s0 < 0

s/ v110- !3 , 05 < 0

%r * 0(100, %s* 3(1 , 2 < 0

5%s* 2(1 2%r * 1(112- 16 , 16 < 0

Page 354: Calculus

02+ Pjgp^dji`n \ gjn `e`m^d^djn

13- €U[203, 2s0 + t0 < 00

15- U‹oqf`b bk 'N+N(: afob`qofw s < 1: bgbt < N10- U‹oqf`b bk 'N+N(: afob`qofws < ,e: bgbt < N

17- U‹oqf`b bk 'p+0(: afob`qofws < ,z: bgbt < 018- U‹oqf`b bk 'N+N(: afob`qofwt < , z: bgbs < N2/- U‹oqf`b bk 'N+N(: afob`qofw t < 1: bgb s < N

20- U‹oqf`b bk ',1+ ,p(: afob`qofwgv < +,ga9 bgb s < ,1

10, s0 < +t11, t0 < 6s

01+ %r * 3(1 < *5%s * 2(02+ %s * 0(1 < 2%r * c&03+ %r * f,I1 < 1'v * c&15, %s* 2(1 < +6&s + 0(

16, s0 + 2st * 3. * 2.s * 0.t + 0// < N

02-14 Dgbo`f`flp s^oflp pl_ob `Žkf`^p 'mŠd- 512(

0+ ? = N+ = < i'i * S3'?.+ c\b4- 056S5- '^( p '_( 16S 'b( 376S.4/, T0-Y0 * t0-/4 < 06, s0 + 0st * t0 + 0s + 0t < 0

7, t0 + 2s0 + 2t * 2s < N

0/- '^( ` < S0-&k * 1(: cl`lp bk 'T1+N( u ',T1+ N(04- '_( V < @s/) B {{5 N05- '3+7(00- ']( s < p] '^( 0/kl0 < 2\1

.5+ %r * $(1 * %s* q(1 < q

'_( 4s0 + 1g < 3

?[j•nofi .1

03-3 Dgbo`f`flp 'mŠd- 521(

0- m&o' < '0+ 0o*1o0 * 2o1'9 C!&o' < 'N+1+4o*/0o0'

0, m&o' < ' ,pbk o*pbk1q+1 blp 0o*pb`1 o(9 C!&o' < ' ,`lp o*0 `lp 0o*+2 pbk1q+ 1 pb`! qq^k o'1, C&-' < ''0 , /0'+/-0* ,'0 , '1(,0.1(: C!&-' < &/&/ * '1(,&2.1+,.'0 * '1(,2.1(

2, C&o' < &0`o*1`o'9 C!&o' < &0`o*1`o'

3, C%&o'< 'pboceo*1 `lpe 0o* [1`+0n&8 C!&o' < '`lpe .) 3 pbke 0o*b`+0n&

4, C%&o'< &0o-&F* o0'*0.'0 * o/&) +0--&/ * o0~9

C!&o' < ''1 , 0o0'-&F * o0'0* +0o-&F * .0'0* '501 , 1(.'0 * '1(2(7- 'p+z+a,G(

8- '0 ,pT1+ pT1+ild pT1(

Page 355: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 02,

&g)` g)`'

0/- ild ,1, +0 , ild ,1,

00- .*`+0*/+0-`'

01- N/3, D%&o'< C&o' t m&o'

0., C&o' < -yo1> * do0? * o` * A00, m& ' < >* C&1' < x5 * 2 ild 1'>01, C&s' < `U&s * /'> + `>

03-6 Dgbo`f`flp 'mŠd- 530(

0- q&o'< '2 , 1o0'9 * 4oe * '2 * 1o0'f9 \&o' < +4o9 * 5g * 4of9 q&o'< 2U1- N * o0'

0, b'o( < ,pbk od * blp og * `%f9 \&o' < ,blp od +pbk og * `%f9 q&o'< N * `0o'g-0

1, q&o'< 2'blp o + opbko'9 * 1&n`io * o blp 0(g * 2f9 \&o' < ,2'1 pbk o * o blp o'9 )2'1 blp o + opbko'e9 q&o'< &7o0* 14(0.1

o o2, q&o'< N , blp 0(: * n`ioe * 1 blp 1-f9 \&o' < n`io d * blp oe + pbk1-f9 q&o'< 1

3, q&o'< Pod * 4o0e * 1f9 \&o' < 5: * 01qg: q&o' < 4o0 * 24, q&o';9)`jnoe)n`iof9 \&o';+n`ioe)`jnof9 q&o';S0,

7, > < \]jm%*? < \0r1

00- '_( 6„o-`jn0 _04- '^( s&o' < 3 blp 0o* t& ' < 2 pbk 0o '_( s0-/4 * t0/7 < 0

/4, 1Q-2

03-8 Dgbo`f`flp 'mŠd- 535(

0- '^( Q < ,qlU1 ',2: * 3g * 3f'9 K < ,z: , qf '_( \ < 01U1 Q * 4K

N * €!&8* `/!e * `mf1- '^( Q < ,N * `0!'+/-0e * `!L * „!'+/-0f9 K < ,,,,,,,,

'0 * `0!'/-0&/ * 1‹!(i.1'_( \ < N * `0!'+/-0X`0!Q * '0 * 0`0!'/-0KZ

2- '^( P < z: * we8 J < f '_( \ < 3J

3- '^( P <:: J < ,fU1 'g * e&8 '_( \ < U1 J4- '^( Q < 0'1: * 0e * f'9 K < T * 1g , 0f' '_( \ < /0Q * 4K

5- '^( Q < fU1: * qf* df9 K < ,fU1g * fU1 f '_( \ < K8- Blkqo^bgbjmil m^o^ _( v a(9 jlsfjfbkql pl_ob rk^ e‹if`b

00- Tk bgbjmil9 m&o'< 0a`0o blp o _od * 0a`0o pbkq _oZ * `0of9 q&o' cloj^ rk Škdril

`lkpq^kqb `lk e) mbol \&o' krk`^ bp `bol kf m^o^ibil ^ q&o'

01- '^( Blkqo^of^ ^ i^p ^drg^p abi obilg '_( 2 'b( 0/Q-S1/1, s/-1 * t/-2 < 0/2, t0 < 2s9 v1 < c , 2s

04- '_( GG?GGGG@GGpbk '(

03-02 Dgbo`f`flp 'mŠd-544(

0- 6\

1- T1-'… , 0(

Page 356: Calculus

02- Pjgp^dji`n \ gjn `e`m^d^djn

1, 0/Q0\

1+ 1%[1 + \1&,%[\&

3, 0\&blpe bsblpe Q + 0(3+ 0S0/Q

6- 4/

7- U1 ild '0 * X./&

7, /.// Ty &oF+ oj'

.-+ Pxq0 * %a$%sSs

15U02 , 0500- 16

02- '^( Gj/

T 0 * `0s _s1

03- 'b( bpbke,`

+l 'U1 blpe %P,/& * qr99cQ', U 1 \ ild ‘ E

0 *s 1

``' 1'_( 0 1 * '1 _o

.3+ X&s' < hblpe 'z * ^'* N X&s' < e

/7, p%l&< 0 * 0o9 2 rkfa^abp ab qfbjml

),&)- :WR_PVPV\`'mŠd- 548(

0- '0( ,.4 '1( '0 * 0`0mm'g 0&/ * `0mm'+1-0 '2( 154 '3( 0U1 '4( +9wd '5( e0

2- HHAiipbk%d

3- ']( s < u5, H < hG]hh.hhrh01

6+ \ < 0+ \ < 1: pb `loq^k bk 'N+N(0/- U‹oqf`b bk ,i blp %d> * 0blp1'g ?

00- '^( ~'o( < d/Q + 3o0 '_( q&o' < 4 pbk 3o0 d * 4 blp 3o0f

./+ S0d * U1g

),&)1 :WR_PVPV\` ]mT&..-!

0- q&m' < p9 * op`9 \&o' < +opl * 0p`9 H&o' < '1 * o0&%E* o0'+1-01- ']( q&o' < p9 * oR` * e8 \&m' < , op9 * 0p`9 H&o' < &o2 * 3o0 * 6'/-0&0 * o0'+1-0

'_( ^o``lps! 1.'1 * o/&

2- '_( d/Q + o4- 21

TG* `/

5- '_( I&`' < ,,, &`/ll? * 0( pf ` o5 N: I&L' < 0/Q,[%K& < +@ _

`1ll] * 0\&^'< +2+`+ pf ` ", N:

Page 357: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 684

6- '^( 2oo.05 '_( 1 * fU2ild '1 * U2(

5+ 5Q&5Q0 * 0(001 * ild '6S * U 6S1 * 0(

8- U1 &`! + 0(0/- 300- 7

02- '^( &.0 * /'1-0-&.0 * 1( '_( U1 `6 'b( s 1 * U1 'a( U1

04- m< %j`+6lil: _i^k`l bk bi lofdbk+ mrkql ab m^oqfa^ abi `lebqb m< &/&%F < N:\ bp bi Škdril+ N ; \ ; $.P) abqbojfk^al mlo q v +m9 m^o^ N ; \ ; $.P,/ bi `^jfkl bprk^ bpmfo^i m^o^ i^ nrb mx N `r^kal %F `ob`b fkabcfkfa^jbkqb: m^o^ \ < $.P,/ bp rk^`fo`rkcbobk`f^ bk qlokl ^i lofdbk: m^o^ $.P,/ ; \ ; $.P bp rk^ bpmfo^i m^o^ i^ nrb m`ob`b fkabcfkfa^jbkqb `r^kal %F `ob`b fkabcfkfa^jbkqb-

05- SŽjbpb `ljl bgb s mlpfqfsl i^ ob`q^ nrb s^ abpab i^ mlpf`fŽk l_pbos^a^ ^ `r^qol jfii^pab afpq^k`f^ ^ i^ _^pb ab i^kw^jfbkql- Blkqfk•bpb ^ il i^odl ab bpq^ ob`q^ qobp jfii^p'm^o^ bsfq^o i^ mlpf_fifa^a ab nrb bi molvb`qfi srbis^ ^ i^ _^pb( v abpmr‹p pb pfdrb i^bpmfo^i l < k6-SO

06- ild U r0 * t0 * ^obq^k %s,r& < B

03-10 Dgbo`f`flp ab obm^pl 'mŠd-560(

0- q^k N' < q^k _,%/ * q^k! K&

2- %_,g/)/_,g&3- '^( t + Ui < g%r * ri& * ^gh9 q^kdbkqb bk Huƒ* _,g/)si * /_,g&

'_( U * Ui < g%r * ri& * ]h€8 q^kdbkqb bk %ri * /_g) Ui -* _g0'

5- 'Xi , Ui&%U* Ui& < /_%r * Wi , /ri&8 rEU < /Fxr * TEUf8%Tf * ri&%U * Ui& < 1'Xi , Ui&%r * ri& * 'Wi , TK&%Uf* Ui&

6- '^&( 'N+ (

G'_( Dp`of_foP < 'N+]&s~, pf }a!&L' x N bkqlk`bp ]&s' ++)a&L' * ,, `r^kal U ,,* N-

.!%-&Dk `r^inrfbo lqol `^pl ,\%r&. ,,* * // `r^kal r ,,* N-

G* ` G7 +< ,, ,,* , `r^kal b ,,* N- 1 1

02- '1+0(+',1+ ,0(

03- ei1=%)& G

/3, 1s0 + v1 < 1\0Š ,,,,*,+ +2 14\

10- '^( a`L' < f pbk %_* @'* l a`L' < f

'_( a`L' < @`L-Sf0*.) alkab f/ = H

'b( X&L' < &0-f' pb` %_* B(+ l a`L' < 0-f

?[j•nofi .2

04-4 Dgbo`f`flp 'mŠd-57/(

0- pf 1- pf 2- pf 3- pf

Page 358: Calculus

02/ Pjgp^dji`n \ gjn `e`m^d^djn

4- Ml 02- pf 10- pf5- pf 03- Ml 11- pf6- pf 04- pf 12- Ml

7- pf 05- pf 13- pf8- pf 06- pf 14- Ml

0/- pf 07- pf 15- pf00- Ml 08- pf 16- pf

01- pf 1/- pf 17- pf

20- '^( Ml '_( Ml 'b( Ml 'a( Ml

04-80-

1-2-3-

06-07-08-1/- Ml10- '^( afj < 2 '_( afj < 2 'b( afj < 1 'a( afj < 112- '^( pf \ x N X ] x N+bi `lkgrkql bp fkabmbkafbkqb+ afj < 2: pf \ Ž ] bp `bol+ bi `lk,

grkql bp abmbkafbkqb: afj < 1 '_( fkabmbkafbkqb+ afj < 1 'b( pf \ x N+ fkabmbk!afbkqb+ afj < 2: pf \ < N+abmbkafbkqb+ afj < 1 'a( fkabmbkafbkqb: afj < 2'b( abmbkafbkqb: afj < 1 'b( fkabmbkafbkqb afj < 1 'c( fkabmbkafbkqb afj < 1

'e( abmbkafbkqb: afj < 1 'f( fkabmbkafbkqb: afj < 1 'g( fkabmbkafbkqb: afj < 1

04-01 Dgbo`f`flp 'mŠd-583(

0- '^( Ml '_( Ml 'b( Ml 'a( Ml 'b( pf

7- '^( yTa1 * 0 '_( a%r&< ]&r Z a1 9 0(+] ^o_fqo^ofl

%h * .&%/h * 0( h * 0 '& /h * 0(&0/- '_( 4i \ * ,1, ] 'b( b&o' < \ o +Z9+ * \ ^o_fqo^ofl

00- 'b( 32 'a( b&o'< [%. +xo'* [ ^o_fqo^ofl01- '^( Ml '_( Ml 'b( Ml 'a( Ml02- 'b( 0 'a( `/ * 003- 'b( i -0i)g

Dgbo`f`flp 'mŠd- 575(

pf 1 4- pf 8- pf 02- pf! i

pf 1 5- Ml 0/- pf !03- pf ! i

pf 1 6- Ml ii- pf i 04- pf i

pf 1 7- Ml 01- pf i 05- pf i

pf afj < 0 ( fi pf i bp m^o+ Ei% * 0( pf i bp fjm^opf afj < fi pf i bp m^o+ x&i * 0( pf i bp fjm^opf e * 0

04-05 Dgbo`f`flp 'mŠd-6/5(

0- '^( v '_( yU2 '0+0+0(+ e.5 '0+ ,1+0(

1- '^(zU1'0+ 0+/+/(+ Z{U5',H+ 0+1+/(+ z[.2'0+ ,0+0+2(\ G

'_( yU2 '0+0+ /+0(+ ] ., '0+ ,1+5+0(&r31

5- | ,00ld1 2

Page 359: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 686

5, `x + h

26, +E` ,b,0( )+s9 h+5`+xŠ `

8- 6R , 1pbku

0/- p,qu

05-3

0-

1-2-3-4-5-6-7-8-

0/-00-01-02-03-04-

Hq=-

06-

07-

08-

1/-10-

11-12-13-

14-

15-

16-

@\k…opgj /4

Dgbo`f`flp 'mŠd- 603(

Kfkb^i: afjbkpfŽk abi k•`ibl N+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl 0+ o^kdl iKfkb^i: afjbkpfŽk abi k•`ibl 0+ o^kdl iMl ifkb^iMl ifkb^i

Ml ifkb^i

Ml ifkb^iKfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 1

Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 1

Ml ifkb^i

Kfkb^i : afjbkpfŽk abi k•`ibl /+ o^kdl 1Ml ifkb^iKfkb^i: afjbkpfŽk abi k•`ibl N+ o^kdl 2Kfkb^i: afjbkpfŽk abi k•`ibl 0+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl 2Ml ifkb^i

Ml ifkb^iMl ifkb^iM l ifkb^iKfkb^i: afjbkpfŽk abi k•`ibl 0+ o^kdl 1

Kfkb^i: afjbkpfŽk abi k•`ibl /+ o^kdl i * 0

Kfkb^i: afjbkpfŽk abi k•`ibl 0+ o^kdl fkcfkfql

Kfkb^i: afjbkpfŽk abi k•`ibl fkcfkfq^+ o^kdl 1Kfkb^i: afjbkpfŽk abi k•`ibl 1+ o^kdl fkcfkfql

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Page 360: Calculus

021 Pjgp^dji`n \ gjn `e`m^d^djn

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Page 361: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 688

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Page 362: Calculus

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Page 363: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 7/0

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Page 364: Calculus

7/1 Pjgp^dji`n \ gjn `e`m^d^djn

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rk^ plir`fŽk

Page 365: Calculus

Pjgp^dji`n \ gjn `e`m^d^djn 1),

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Page 366: Calculus
Page 367: Calculus

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Page 368: Calculus

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Page 369: Calculus

gi_d^` \ga\]„od^j

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Page 370: Calculus

1)1 gi_d^` \ga\]„od^j

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cŽojri^p ab ^af`fŽk m^o^ pbklp v `lpbklp+ 008ENTQHDQ+INRDOG+045+ 6/4cob`rbk`f^ abi jlsfjfbkql ^ojŽkf`l pfjmib+

304crk`fŽk ^`lq^a^- 8/

`^o^`qboŒpqf`^+68 'Dgbo`f`fl 7(`ljmibg^+ 338`ljmrbpq^+ 061+ 606

`lkqfkrfa^a ab i^+ 062abofs^`fŽk ab i^+ 102+ 52/

`Žk`^s^+ 040+ 12/ab `lkgrkql+ 60`lksbu^+ 040+ 12/`lpbkl+ `lkqfkrfa^a ab i^+ 054+ 060

abofs^`fŽk ab i^+ 088b`r^`fŽk afcbobk`f^i m^o^ i^+ 284fkqbdo^`fŽk ab i^- 012+ 142molmfba^abp ab i^+ 007pbofb ab mlqbk`f^p ab i^+ 423

`lq^kdbkqb+ 017`ob`fbkqb+ 83ab`ob`fbkqb+ 83abcfkfa^ jbaf^kqb rk^ fkqbdo^i+ 037bibjbkq^i+ 234bumlkbk`f^i `ljmibg^+ 337

abcfkf`fŽk ab+ 185abofs^a^ ab i^+ 187fkqbdo^i ab i^+ 2/0pbofb ab mlqbk`f^p m^o^ i^+ 423

c^`qlof^i+ 53

crk`fŽk d^jj^+ 401+ 403 'Dgbo`f`fl 08(efmbo_Žif`^+ 2/6fabkqfa^a+ 52fjm^o+ 0/2 'Dgbo`f`fl 14(fjmiŒ`fq^+108fkqbdo^_ib+ 80fksbop^+ 068+ 2/8ifkb^i+ 55

^ qolwlp+ 041jlkŽqlk^+ 83

^ qolwlp+ 0(3

kl ^`lq^a^+ 78m^o+ 0/2 'Dgbo`f`fl 14(m^oqb bkqbo^+ 67mbofŽaf`^+ 006mlpf`fŽk+ 526+ 550o^`flk^i+ 1/2+ 205,215

fjmolmf^+ 206molmf^+ 207

pbkl bk bi mi^kl `ljmibgl+ 343 'Dgbo- 8(`lkqfkrfa^a ab i^+ 054+ 060b`r^`fŽk afcbobk`f^i m^o^ i^+ 284abofs^`fŽk ab i^+ 087fkqbdo^`fŽk ab i^+ 012+ 142pbofb ab mlqbk`f^p ab !i^+423molmfba^abp ab i^+ 007

q^kdbkqb+ 017sb`qlof^i+ 516wbq^ ab Qfbj^kk- 373

crk`flkbp ab Abppbi+432 'Dgbo`f`fl 0/(`lkqfkr^p+ ^`lq^`fŽk ab+ 073abofs^_ibp+ `lkqfkrfa^a ab+ 1//jlkŽqlk^p ^ qolwlp+ 83mlifkŽjf`^p+ 56

`lkqfkrfa^a ab i^p+ 052abofs^`fŽk ab i^p+ 1/2fkqbdo^`fŽk ab i^p+ 87+ 0/0ab alp s^of^_ibp+ 212

mlqbk`f^ibp+ 56+ 87ob^ibp+ 51qofdlklj‹qof`^p+ 006,022

`ljmibg^p+ 343 'Dgbo`f`fl 8(`lkqfkrfa^a+ 054+ 06/abofs^`fŽk ab+ 087molmfba^abp crka^jbkq^ibp+ 007abp`ofm`fŽk dblj‹qof`^+ 015doŠcf`^p ab i^p+ 021fkqbdo^`fŽk ab+ 012fksbop^p+ 2/8,202pbofbpab mlqbk`f^p m^o^ i^p+ 423

ab s^of^p s^of^_ibp+ 128

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gi_d^` \ga\]„od^j

F

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br`ifaf^k^+ 0/+ 466dbljbqoŒ^p kl ^onrfjbaf^k^p+ 21

kl br`iŒab^p+ 47/FHAAR+INRH@GVHKK@QC+434FQ@L+INQFDM ODCDQRNM+586FQ@RRL@MM+GDQL@MM+435FQDFNQX+I@LDR+365+ 381

G

G@C@L@QC+I@BPTDR+644G@LHKSNM+VHKKH@LQlt@M+ 326+ 434GD@UHRHCD+NKHUDQ+434e‹if`b+ 53/

`fo`ri^o+ 53/GHKADQS+C@UHC+466efm‹o_li^+ 5/8+ 501+ 508GNNJD+ QNADQS+51

0

fabkqfa^a ab K^do^kdb+ 481mfq^dŽof`^+ 007+ 447+ 464+ 6/2

fdr^ia^a ab `lkgrkqlp+ 03ab crk`flkbp+ 55ab k•jbolp `ljmibglp+ 327ab sb`qlobp+ 436+ 462

fkabmbkabk`f^ ifkb^i+ 456+ 572fkqbdo^_fifa^a ab crk`flkbp `lkqfkr^p+ 076

ab rk^ crk`fŽk `lkqfkr^+ 076ab rk^ crk`fŽk jlkŽqlk^+ 84

fkqbdo^`fŽk ab i^ crk`fŽk ild^oŒqjf`^+ 177ab crk`flkbp jlkŽqlk^p+ 86

o^`flk^ibp+ 2 05, 212qofdlklj‹qof`^p+ 012+ 142+ 212

mlo co^``flkbp pfjmibp+ 205,212mlo m^oqbp+155,158ab mlifkljflp+ 87mlo prpqfqr`fŽk+ 148,153

fkqbdo^i+ `ros^+ 306abcfkfa^+ abcfkf`fŽk ab i^+ 8/

molmfba^abp ab i^+ 88+ 0//biŒmqf`^+545 'Dgbo`f`fl 06(ab rk^ crk`fŽk ^`lq^a^+ 8/

1)2

fkqbdo^i ab rk^ crk`fŽk ^`lq^a^ `ljmibg^+ 34/bp`^ilk^a^+ 68sb`qlof^i+ 517

fjmolmf^ afsbodbkqb+ 4/7+ 400ab mofjbo^ bpmb`fb+4/7ab pbdrka^ bpmb`fb+4/7

fkabcfkfa^+ 037+ 053fkcboflo+ 80prmboflo+ 80

fkqbdo^kal+ 80fkqbomobq^`fŽkdblj‹qof`^ ab i^ abofs^a^+ 1/6

ab i^ fkqbdo^i+ 70+ 81+ 0/8fkqbopb``fŽk ab `lkgrkqlp+ 06fkqbos^il ^_fboql+ 63

`boo^al+ 63ab `lksbodbk`f^+ 417

fkqbos^ilp+ 63+ 268fksbop^ mlo i^ abob`e^+ 608+ 640

mlo i^ fwnrfboa^+ 607+ 64/fpl`ifk^p+ 311+ 317fpljlocfpjl+ 33/+ 625fplqboj^p+ 130+ 318

H

I•mfqbo+ 556

I

JDOKDQ+ING@MMDR+50/+ 556

J

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100+ 146+ 161+ 263+ 382KDNM@QCNCD OHR@'Ef_lk^``f(+ 352ibv ^pl`f^qfs^ bk rk bpm^`fl ifkb^i+ 565

m^o^ i^ ^af`fŽk ab k•jbolp+ 11+ 327ab sb`qlobp+ 437

`ljmlpf`fŽk ab crk`flkbp+ 062+ 606jriqfmif`^`fŽk ab k•jbolp+ 11+ 327obrkfŽk b fkqbopb``fŽk ab `lkgrkqlp+ 07

`lkjrq^qfs^ bk Tk bpm^`fl ifkb^i+ 565m^o^ i^ ^af`fŽk ab k•jbolp+ 11+ 327

ab sb`qlobp+ 437molar`qlp bp`^i^obp+ 577

sb`qlof^ibp+ 442

Page 372: Calculus

70/ gi_d^` \ga\]„od^j

ibv+jriqfmif`^`fŽk ab k•jbolp+ 11+327obrkfŽk b fkqbopb``fŽkab `lkgrkqlp+ 07

afpqof_rqfs^ bk rk bpm^`fl ifkb^i+ 556m^o^ molar`qlp bp`^i^obp+442+ 577

sb`qlof^ibp+ 481k•jbolp+ 11+ 327lmbo^`flkbp `lk `lkgrkqlp+ 1/ 'Dgbo,

`f`fl 0/(ab Gllhb+ 51+ 033ab Mbtqlk ab bkcof^jfbkql+ 275

ab i^ do^sfq^`fŽk rkfsbop^i+ 557abi jlsfjfbkql+ 274+ 557

abi m^o^ibildo^jl+- 332+ 44/ibvbp ab `ob`fjfbkql+ 281+ 282

ab Jbmibo+ 556+ 557K&GŽm S@K+ FTHKK@TLD EQ@Mi:NHR @MSNHMD+

247iŒjfqbmlo i^ abob`e^+ 048

ab rk^ crk`fŽk+ 045fkcfkfql+255ab fkqbdo^`fŽk+353mlo i^ fwnrfboa^+ 048qblobj^p pl_ob+ 01+ 80

iŒjfqbpfkcfkfqlp+255,257^ rk i^al+ 048,05/

ifkb^ifa^a ab i^p abofs^a^p+ 1/0ab i^p fkqbdo^ibp+71+ 88ab i^p pbofbp`lksbodbkqbp+ 360ab ilp lmbo^alobp ab S^vilo+ 226

iŒkb^pbnrfmlqbk`f^ibp+ 318KNA@SBGDURJH+MHJNK@HHU@MNUHBG+47/ild^ofqjlp+ `Ši`ril ab+ 183,185

ab _^pb ]* 062` 'kbmbof^klp l k^qro^ibp(+170,173

ilkdfqra ab rk sb`qlo+ 443lqo^p abcfkf`flkbp ab+453 'Dgbo`f`flp 06+07(

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K

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ab `lbcf`fbkqbp+631`lirjk^ 'sb`qlo `lirjk^(+ 615+ 623af^dlk^i+ 62/cfi^ 'sb`qlo cfi^(+ 623+ 635fabkqfa^a+ 625

j^qofw fksbop^+64/kl pfkdri^o+ 64/loqldlk^i+ 644 'Dgbo`f`fl 7(pfkdri^o+ 641rkfa^a+ 625

jŠufjl ^_plirql+ 073ab rk^ crk`fŽk+ 073

obi^qfsl ab rk^ crk`fŽk+ 111jbaf^ ^ofqj‹qf`^+ 46+ 034

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LDQB@SNQ+ MHBGNK@R+350+ 365Lbo`rofl+ 556j‹qlal ab bifjfk^`fŽk ab F^rpp,Iloa^k+ 634

ab bue^r`fŽk+ 3,8ab Fo^j,R`ejfaq+ 586ab ilp jŒkfjlp `r^ao^alp+ 128 'Dgbo`f,

`fl 14(jŒkfjl ^_plirql+ 073

ab rk^ crk`fŽk+ 073obi^qfsl ab rk^ crk`fŽk+ 111

jlabil ^k^iŒqf`l ab i^ dbljbqoŒ^ br`ifaf^k^+467j^qbjŠqf`l+ 272

jŽaril ab Tk k•jbol `ljmibgl+ 333jljbkql+ 035

ab fkbo`f^+ 035jlsfjfbkql ^ il i^odl ab rk^ `ros^+ 526

^ojŽkf`l+ 3/7`fo`ri^o+ 527ab rk `lebqb+ 301mbofŽaf`l+ 3/8+ 304+ 557

jriqfmif`^`fŽk ab crk`flkbp+ 57+ 66+ 052ab j^qof`bp+ 625ab k•jbolp+ 11+ 44+ 327ab qo^kpcloj^`flkbp+ 605ab sb`qlobp mlo bp`^i^obp+437+ 565

L

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528kloj^ ab rk sb`qlo+ 443

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Page 373: Calculus

-i_d^` \ga\]„od^j

kloj^i ^ rk^ `ros^ mi^k^+ 536^ rk mi^kl+ 5/3^ rk^ ob`q^+472

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m^o^ mofjfqfs^p+ 146k•`ibl+ 600k•jbol `ljmibgl+ ^odrjbkql ab rk+ 332

`bol+ 328`lkgrd^al+ 334

` '_^pb ab ilp ild^ofqjlp k^qro^ibp(+`Ši`rilabi+ 233

abcfkf`fŽk abi+ 173foo^`flk^ifa^a abi+ 234

om* Ši`ril abi+ 238 'Dgbo`f`fl 0/(abcfkf`fŽk abi+ 002

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bk rk bpm^`fl br`iŒabl+ 580ab mi^klp+ 5/6ab ob`q^p+1/7abi pbkl v abi `lpbkl+ 020 'Dgbo`f`fl 20(ab sb`qlobp+ 446

m

m^oŠ_li^+ 3+ 56+ 5/8+ 501+ 51/m^o^_lilfab efmbo_Žif`l+ 130m^o^alg^ ab YbkŽk+ 346,351m^o^ibibmŒmbal+027m^o^ibifpjl ab mi^klp+ 472

ab ob`q^p+47/

700

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^cfk^jfbkql ab rk^+ 65O@RB@K+AK@HRD+2OD@MN+FHTRDOOD+ 10mbkafbkqb ab rk^ `ros^+ 1/6mbojrq^`fŽk+ 4/2mbombkaf`ri^ofa^a ab mi^klp+ 5/6

ab ob`q^p+1/7ab sb`qlobp+ 446

6n 'mf(+ Ši`ril ab+ 238 'Dgbo`f`fl 0/(mi^kl lp`ri^alo+ 533mi^klp+ 474,48/mlifkljfl `r^aoŠqf`l+ 56mlifkljflp ab Kbdbkaob+6//

ab S^vilo+ 224,23/mlpqri^alp ab Ob^kl m^o^ ilp k•jbolp bk,

qbolp+10mlqbk`f^ abi _fkljfl+ 44 'Dgbo`f`fl 3(+ 351+

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ab B^s^ifbof+ 026+ 027ab fkqbo`^i^`fŽk m^o^ iŒjfqbp+053

mol_ibj^ ab s^ilobp fkf`f^ibp+ 265mol_ibj^p ab mbopb`r`fŽk+ 32/molar`ql bp`^i^o 'fkqboflo(+442+ 463+ 576

jfuql 'bp`^i^o qofmib(+487sb`qlof^i+ 480

moljbafl+ 46+ 072molmfba^a ^afqfs^ abi Šob^+61

ab i^p abofs^a^p+ 1/0abi buqobjl prmboflo v abi buqobjl fk,

cboflo+ 22ab i^ fkqbdo^i+71+ 72+ 88+ 52/ab i^ ilkdfqra abi ^o`l+ 540

ab ilp moljbaflp+ 036 'Dgbo`f`fl 02(ab i^p pbofbp`lksbodbkqbp+ 360

ab i^p prj^p cfkfq^p+38abi qo^_^gl+ 031abi slirjbk+ 027

^onrfjbaf^k^ ab ilp k•jbolp ob^ibp+21ab bue^r`fŽk abi Šob^+61eljld‹kb^ ab i^p fkqbdo^ibp+Q1

ab i^p pbofbp+360ab i^z prj^p cfkfq^p+38

ab jlklqlkŒ^ abi Šob^+62

Page 374: Calculus

701 gi_d^` \ga\]„od^j

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ab `ros^qro^+ 546ab dfol+ 036

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sb`qlof^i ab rk^+ 470m^o^ibifpjl ab+ 47/mbkafbkqbab rk^+ 1/6+ 472q^kdbkqb+1/6+ 523sb`qlo kloj^i ^ rk^+ 472

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obloabk^`fŽk ab pbofbp+4/0,4/5obmobpbkq^`fŽk_fk^of^+37/

arlab`fj^i+ 37/ab j^qof`bp+ 615

obpql ab B^r`ev bk i^ cŽojri^ ab S^vilo+237

ab K^do^kdb bk i^ cŽojri^ ab S^vilo+ 237bk i^ cŽojri^ ab S^vilo+ 230,237

obrkfŽk ab `lkgrkqlp+ 06

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p

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Page 375: Calculus

gi_d^` \ga\]„od^j

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ab ilp s^ilobp buqobjlp m^o^crk`flkbp `lk,qfkr^p+ 075

qblobj^p ab bufpqbk`f^+266+ 284pl_ob crk`flkbp `lkqfkr^p+ 052+ 062,078crka^jbkq^ibp abi `Ši`ril+ 136+ 140+ 52/

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ab rk^ m^oŠ_li^+ 51/sf_o^`flkbp+ 3/8

^jloqfdr^a^p+ 3/8sfa^ jbaf^+ 272slirjbk+ abcfkf`fŽk ^ufljŠqf`^ abi 027

ab i^ bpcbo^+03/ &ab pŽifalp ab obslir`fŽk+ 03/ab pŽifalp ab pb``fŽk `lkl`fa^+ 028

t

V@KKHR+INGM+ 2VDHDQRSQ@RR+J@QK+ 10+ 408+ 412VQNMRJH+ I- L- GNDMD+ 3/1tolkphf^kl+ 3/1 'Dgbo`f`fl 10(+ 3/3