璋編著國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

微積分講義

黃文璋編著國立高雄大學應用數學系

一百零一年十月

�9>C, x�Î¶��.§�.oÝ�×.ß¸à, /�²°,

�ÈN.�ë�°.5Ë.�à��3��.C{.�., ��5��ýL, ô� u�ODÝ�B�Ñà��5�¬ §�.oÝ¥�Iê, v3.ß^¡Ý.ê�, ù×à6�½¥�Ý��&Æ�T9>C, tÝ��5Ý>C, ñ�.ß�5Ýó.IY, ¬�.h¯.ßE¨�ó., b×�MÝÃF�^¡ÂÕ&Ë®Þ`, �|ó.Ý��¼¤��Í>Cx�ãC�×°¨bÝ��5h, LÍÎApostol :

Calculus, Vol I and Vol II, 2nd ed. (1967, 1969, John Wiley &

Sons)�Courant and John : Introduction to Calculus and Analysis,

Vol I and Vol II (1965, 1974, Springer-Verlag)CJohnson : Calculus

(1971, Allyn and Bacon)�ÍhtÝ»�Á9, ú��Õ²,E&ËÃFÝD¡, ù½��K, ¬- £�ËÑÎ� �\ïòGÝI5�×ÍhtÝ¥¥/��², ôT�Õ\R¼�øo, XÛ�2øiÎô��®ï�æb§, Íh39]«Ý�¨¬Î¼Aß��Íh�F¶yèõOG, ã�-]��.ÆC_4��¡W

�ü�6Ö¯Þ��ß, C?Ê��ôÜÃ�KÑ�Ý�®�r|z��BðèºÆCC_4îÝ*�Yî�ht±Ñ�ÝÌÍ, JÎ�¡ãA-Ò��eÃÆ�ßC³·ý��QRÝ��É�=EÍh/�è��K¼Ñ, b°Î&9OKÎ�s¨Ýþ´��Æ¿�ÎÍ>C�|´·�PÓ¨Ý��

?Z*Ó»101O10`

i

êêê ggg

ÏÏÏ×××aaa ÁÁÁ§§§ 1

1.1 ó�ÝÁ§ . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Á§ÝÃÍP² . . . . . . . . . . . . . . . . . . . . . 10

1.3 �ZóCiø£ . . . . . . . . . . . . . . . . . . . . . 19

1.4 ÐóÝÁ§ . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 Á§�§C=�P . . . . . . . . . . . . . . . . . . . 40

1.6 =�P�×M"D . . . . . . . . . . . . . . . . . . 55

ÏÏÏÞÞÞaaa ��555��555ÝÝÝ��+++ 71

2.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.2 «� . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3 �5Ý�L . . . . . . . . . . . . . . . . . . . . . . . 74

2.4 �5ÝÃÍP²C§¡ . . . . . . . . . . . . . . . . . 89

2.5 ë�ÐóÝ�5 . . . . . . . . . . . . . . . . . . . . . 105

2.6 ��5 . . . . . . . . . . . . . . . . . . . . . . . . 112

2.7 0óÝ�LCÃÍP² . . . . . . . . . . . . . . . . . 118

2.8 )WÐóC2Ðó��5 . . . . . . . . . . . . . . . 138

ÏÏÏëëëaaa ��555��555��nnn;;; 149

3.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

3.2 ��5ÃÍ�§ . . . . . . . . . . . . . . . . . . . . . 151

3.3 �ó�ð° . . . . . . . . . . . . . . . . . . . . . . . 160

3.4 5I�5 . . . . . . . . . . . . . . . . . . . . . . . . 170

iii

ÏÏÏ°°°aaa ��555��TTTààà 181

4.1 ÁÂ��LCíÂ�§ . . . . . . . . . . . . . . . . . 181

4.2 OÁÂC0% . . . . . . . . . . . . . . . . . . . . . . 195

4.3 ��"�P . . . . . . . . . . . . . . . . . . . . . . . 213

4.4 Á§�� . . . . . . . . . . . . . . . . . . . . . . 231

4.5 �5�Tà®Þ . . . . . . . . . . . . . . . . . . . . . 240

ÏÏÏ"""aaa øøø÷÷÷ÐÐÐóóó 257

5.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.2 Eó . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

5.3 ¼óÐó . . . . . . . . . . . . . . . . . . . . . . . . 274

5.4 �QW��<[ . . . . . . . . . . . . . . . . . . . . . 291

5.5 ¼óCEóÐó�×MD¡ . . . . . . . . . . . . . 298

5.6 Ô`ÐóCDë�Ðó . . . . . . . . . . . . . . . . . 317

5.7 �5*» . . . . . . . . . . . . . . . . . . . . . . . . 326

5.8 ÐóÝÍ�P² . . . . . . . . . . . . . . . . . . . . . 342

5.9 Ó� . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

ÏÏÏ000aaa ��555��TTTààà 361

6.1 O«� . . . . . . . . . . . . . . . . . . . . . . . . . . 361

6.2 �� . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

6.3 =�CI»«� . . . . . . . . . . . . . . . . . . . . . 373

6.4 ��5�óÂ�Õ . . . . . . . . . . . . . . . . . . . 382

ÏÏÏÚÚÚaaa óóó��CCCùùùóóó 389

7.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

7.2 ùóÝÃÍP² . . . . . . . . . . . . . . . . . . . . . 395

7.3 Ñ4ùó . . . . . . . . . . . . . . . . . . . . . . . . 402

7.4 øýùó . . . . . . . . . . . . . . . . . . . . . . . . 419

7.5 ��5 . . . . . . . . . . . . . . . . . . . . . . . . . . 438

ÏÏÏâââaaa ÐÐÐóóóóóó��CCCÐÐÐóóóùùùóóó 457

8.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

iv

8.2 @F[e . . . . . . . . . . . . . . . . . . . . . . . . 462

8.3 í8[e . . . . . . . . . . . . . . . . . . . . . . . . 470

8.4 �ùó . . . . . . . . . . . . . . . . . . . . . . . . . . 481

8.5 �ùó�P² . . . . . . . . . . . . . . . . . . . . . . 488

8.6 Í�nyùóÝ�� . . . . . . . . . . . . . . . . . . 503

ÏÏÏÜÜÜaaa 999��ÐÐÐóóóCCCÍÍÍ��555��555 515

9.1 9�Ðó . . . . . . . . . . . . . . . . . . . . . . . . 515

9.2 Á§C=� . . . . . . . . . . . . . . . . . . . . . . . 517

9.3 ]'0ó . . . . . . . . . . . . . . . . . . . . . . . . 525

9.4 �0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

9.5 )WÐóC2Ðó��5 . . . . . . . . . . . . . . . 543

9.6 a�5 . . . . . . . . . . . . . . . . . . . . . . . . . . 554

9.7 ÁÂ . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

9.8 9�Ðó��5 . . . . . . . . . . . . . . . . . . . . . 570

ÏÏÏèèèaaa ¥¥¥��555 581

10.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

10.2 Þ¥�5��L . . . . . . . . . . . . . . . . . . . . . 582

10.3 ¥�5�×MD¡ . . . . . . . . . . . . . . . . . . 591

10.4 Green �§ . . . . . . . . . . . . . . . . . . . . . . . 604

10.5 �ó�ð . . . . . . . . . . . . . . . . . . . . . . . . 608

10.6 {î¥�5 . . . . . . . . . . . . . . . . . . . . . . . 615

10.7 �¥�5 . . . . . . . . . . . . . . . . . . . . . . . . 625

10.8 �+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

ÏÏÏèèè×××aaa ��555]]]��PPP 639

11.1 G� . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

11.2 ×$aP�5]�P . . . . . . . . . . . . . . . . . . 643

11.3 Þ$aP�5]�P . . . . . . . . . . . . . . . . . . 654

11.4 �5ÒÝ�5]�P . . . . . . . . . . . . . . . . . . 668

11.5 ª�5]�P . . . . . . . . . . . . . . . . . . . . . 671

v

11.6 �P×$aP�5]�P . . . . . . . . . . . . . . . 675

11.7 �5]�P�ùó� . . . . . . . . . . . . . . . . . . 678

õõõSSS 685

�ZõS . . . . . . . . . . . . . . . . . . . . . . . . 685

zZõS . . . . . . . . . . . . . . . . . . . . . . . . 693

vi

ÏÏÏ×××aaa

ÁÁÁ§§§

1.1 óóó��ÝÝÝÁÁÁ§§§

ÁÁÁ§§§Î��5ÝÃ�, Ía&Æµ¼"DÁ§, v�ó��óóó��ÝÝÝÁÁÁ§§§&Æ�\-�#ÇÕ�3�.`�.ÕÇ��ó,

»A0.3, &Æá¼¸â?Î13�ùÇ

(1.1) 0.3 = 3 · 1

10+ 3 · 1

102+ · · ·+ 3 · 1

10n+ · · · = 1

3,

9�«Í@µ�àÕÁ§�.Gn4Ý�fùóÝõ

sn =3

10

1− (1/10)n

1− 1/10 �

�&ÆêàÕËÍP², Ï×Í u¯n×à¦�, J“t¡”snÝÂÞ�(1.1)P�ËÍ�r� £�ó�õÝ“Â”×ø; ÏÞÍ “t¡”(1/10)n�Â 0�.h

3

10

1− 0

1− 1/10=

3

10

10

9=

1

3�

X|&Æ�\µb�§Á§ÝB�, Ç¸`EÁ§ÝÃFÎÄè5z½�

1

2 Ï×a Á§

�Äö8ÝÎ, �Kß3¬�Ý�Á§Ý�L��µì, -ðD£2¸à�»A, �."OùÝó.��, bt�2¹óÝÞC, 3?�à�ð��Õ9ì9v|óCÞ�¨ÝÞê�

(i) 12Ý¹ób(1) 1Í (2) 12Í (3) P§9Í;

(ii) 2, 4, 6Ýt�2¹óÎ(1) 12 (2) 2 (3) P§��X], ËÞÝ“ýã�n”/ (3)�9ËæÍ��3�.��¨ÝÞêµÎ�¨Ý��T&�¨3��5ï¢ï�ÎÑ@Ý�~b�A¢�LÁ§÷? &Æ�:9ìÞ»�Ï×» �Êó

�

1,1

2,1

4,1

8,

1

16, · · ·�

hó��¼��, Á§Â«{Î0�E��Ä3;»`�(�222��666�1(ù��FìS)“×M�H, ^ãÍ�, 0t�Ï”�0t�ÏÎ¼NgK�y×F, ¬“t¡”º§�øµ^b1Ý�¨², ë»`�Ý��SSS(V�-G260O) �»��ÕhÜÜÜaaaÕÕÕ

��®¥�3vvviii��¶½“v�RÞ, X´RK, v�êv, |�y��v, J�iø)��PX´z”�39ð��Á§ÝÃF�Qk�, ´2�6`��ÎM&9�b×°ó�&ÆãÌD°-�:�ÍÁ§Â�G�1, 1

2, 1

4, · · ·

×»�¨², 'bó�{an, n ≥ 1}, van = 1/n, Çb

1,1

2,1

3, · · · ,

1

n, · · ·�

n��, hó��#�0�&Æ�?×M1�Aì: �an = 1/n,

��×Ñóε,©�n > 1/ε,J0 < an = 1/n < ε�Æ�Ï[1/ε]+14R,Í�[ · ] t�JóÐó, anÝÂ/�yε�êãy{an, n ≥ 1} �3, X|�¡��¢×Ñóε, ©�nÈ�, -b0 < an < ε��×&�Ýó, u�y�×Ñó, hóÄ 0, Æan�Á§Â 0�&Æ�2àì�Br

limn→∞

an = 0,

T1n��∞`(B n →∞),ó�{an, n ≥ 1}[[[eeeÕ0(converges

to zero), B an → 0�

1.1 ó�ÝÁ§ 3

î�ó��N×4/ Ñ, ¬u¯n×à¦�, |Ðr¼�î, Ç�n → ∞, JÏn4º�¼�#�0�&Æ:Õ×ó��Ñó, ÍÁ§ÂQ 0�3Á§�ðb9×vÝ¨é, Çæ¼N×4í��Ýf�, ÍÁ§Q��¨², AËb§ó�õ ×b§ó, ëb§ó�õ) ×b§ó, ��nÍb§ó�õ�Îb§ó�¬P§9Íb§ó�õ÷? µ�×�Îb§óÝ��!B�, 3Á§��¢¯�íb��sß�.h, �A“�Ûíÿ, h��A”, -��D*�.�Ûíÿ�` ��∞, �h`ºsß%�¯µ�P°��Ý�G�ó��Á§ 0b¿Ë¶°:

n →∞`, an → 0,

T

(1.2) limn→∞

an = 0,

Í�lim limit�¹¶�&Æ¯��ÕîPÝ�L�u&Æµ�ÌDan, Jan�¼��, 3Ï1004�¡ÝN×4/�y1/100, 3Ï1,0004�¡ÝN×4/�y1/1, 000, õv.�4Q^b×4an

Î0, ¬u&ÆÌDÿÈò, �Q�¡ÝN×4an�0Ý-û, �|��&ÆX��Ä9Ë�Õ��)�¼�ß��¢ÛÈò? 9��Õ&Æ

X��? u&Æ�E9Ë�¯��@Ý�L, -�E(1.2)P��ó.î�@Ý1��&Æ�½|¿¢¼�Õ, :��´z½°�3óaî�î�

hó�, ¬óã Iε = (−ε, ε), |0 Í�T�uε = 2, �QXbÝan/a3Iε�; uε = 1/10, J´104�a3Iε�, ��Ï114R/a3Iε��¯ε?�°, ÉA1ε = 1/1, 000, J�Ï1,0014R,

an ô/a3Iε��2, �¡εãÿ9�, ©�ε > 0, Ä�0Õ×t�ÝÑón0, ��1/n0 < ε, J©bG«b§4a1, · · · , an0−1, �a3Iε�, ��Ïn04R(Çn ≥ n0), an/a3Iε��

4 Ï×a Á§

¿àî»��°, &Æ�E×�Ý“ó�{an, n ≥ 1}�Á§ a”,

�×´�@��L�§ø�Î

(1.3) limn→∞

an = a ?

n��`, an�aÝ-ûÞ��, Ç|an − a|��, �9��Õ�÷? �|�×ε > 0, �|an − a|�î&, �.¬&Xb|an − a|/6��, ©�n��`, |an − a|��Ç��ð�1, ©��|0Õ×Ín0,¸ÿn ≥ n0`, |an−a| < εµÈÝ�uÆÿhε�È�,�|�ð×Í�t¡,u�¡�£×Íε > 0,/�ðÕ�0Õ×n0(hn0��εbn, ÇE�!Ýε�0Õ�!Ýn0), ¸ÿn ≥ n0`(ÇnÈ�),

|an − a| < ε, J&Æ-!�(1.3)PWñ�&ÆÅ�Î1n��`, an��ya, Çan − a = 0, �Î1|an −

a|6�y��×Ñó, ©�nÈ��|an − a| ≥ 0QÕGWñ, ¬×&�Ýó��y��×Ñó, µ©b0Ý�X|1h`(1.3)PWñ��×ε > 0, �1(1.3)PWñ, µ60Õ×Ín0, ¸ÿn ≥ n0`

|an − a| < ε,

Çan ∈ (a− ε, a+ ε)�A�ε�ÿ��, ;ðn0-��¬u×à�|ðÕ, µ�ÿ�#å(1.3)PÝ��Ý�&ÆÞî��°¶W×�LAì�

��LLL1.1.'b×ó�{an, n ≥ 1}�E×a ∈ R, u∀ε > 0, D3×n0 ≥ 1, ¸ÿn ≥ n0`, |an − a| < ε, JÌlimn→∞ an = a�

�K�.ïE�L1.1¬��|#å, 3�ìî, 9Î×Í�Õ�|ÝúÞ�h.Í��â°ÍB�: (1)∀ε > 0, (2)D3×n0 ≥ 1,

(3)¸ÿn ≥ n0`, (4)|an − a| < ε��zÍ�Ý.�n;, Î�BÄ×ð` Ý�&ÆÝêÝÎ�¯|an − a| < ε, 3%��µì? ©�n ≥ n0Ç�, EG«Ýa1, · · · , an0−1�|�à§º��n0êÎ%�? ©�0Õ×ÍÇ�, Î�|�½X�Ýε��!Ý�&ÆX�

1.1 ó�ÝÁ§ 5

ºJ�×°�§, à£°�§, �|QÃ&Æ´">2O��KÁ§Â�¬Î�L1.1ÎtqÍÝ, P¨W�§��à, �kO×ó��Á§, �Îÿ¶Iµ°2µ�L�¯�Q3ºà�L1.1`,

Ä6�á¼T��Á§Âb�&Æ:9ìÝ¨Þ»�

»»»1.1.'an = n/(n + 1), n ≥ 1, &ÆJ�limn→∞ an = 1�k¸

(1.4) |an − a| =∣∣∣∣

n

n + 1− 1

∣∣∣∣ =1

n + 1< ε,

Ç�n + 1 > 1/ε, Tn > 1/ε − 1�A�ε ≥ 1, J1/ε − 1 ≤ 0, Æh`∀n ≥ 1, /�¸(1.4)PWñ; u0 < ε < 1Jãn0 = [1/ε]Ç�,

Í�[ · ] t�JóÐó�3�L1.1�Ýn0¬�°×, �:�u0Õ×n0Êà, JN×fn0�ÝJóù/Êà�X|u�Ñε ¢, /ãn0 = [1/ε] + 1�Q�|, Tãn0 = [1/ε] + 10ôP÷�

»»»1.2.'an = (2n2+3)/(n2+2n), n ≥ 1,9ì&ÆJ�limn→∞ an =

2�k¸

|an − 2| =∣∣∣∣2n2 + 3

n2 + 2n− 2

∣∣∣∣ =

∣∣∣∣4n− 3

n2 + 2n

∣∣∣∣

<4n

n2 + 2n=

4

n + 2<

4

n< ε,

©�n > 4/εÇ�, Çãn0 = [4/ε] + 1�

E×ó�{an, n ≥ 1}, uD3×a ∈ R, ¸ÿlimn→∞ an = a,J&Æ1{an, n ≥ 1}[[[eee(convergent), T1{an, n ≥ 1}[eÕa, ÍJÌ sss÷÷÷(divergent)�ñÇ�:�u{an, n ≥ 1}[e�a ∈ R,

J{an, n ≥ 2}ù[e�a, #��*G«b§4, A�Ê{an, n ≥ k},Í�k ×ü�ÑJó, )[e�a�×ó�Ý[e�Í, ¥�ÝÎÍ��III(tail), �&G«b§4�3h&Æ�Eb§CP§��|1��QóÝ/)¬Pî

&, Ç�Qó�|��, ¬��×�Qó, ¸×�Îb§, ©b

6 Ï×a Á§

×Íb§Ýó, Qôµb&�b§Íb§Ýó�/), ôÎb&�¬P§9Íb§Ýó�/)µ�×�b&Ý�X|uan = n,

J{an, n ≥ 1} ×b§ó�, .N×an/ ×b§Â��Ä�Q{an, n ≥ 1}¬&×b&ó�(Ç�D3×ðóK > 0, ¸ÿ|an| ≤K, ∀n ≥ 1)��½n�¦�, anô×à¦�, X�øÄN×Ñó, 9µÎ}¡&ÆºD¡Ýn →∞`, an →∞, Çlimn→∞ an = ∞�ãy∞¬&×@ó, Æh`)1Á§�D3��ylimn→∞ an�D3êÎ%��¤? ôµÎ&Æ��"î∀a ∈

R,

(1.5) limn→∞

an 6= a�

�îPÇlimn→∞ an = a�Wñ, Ç�.�

(1.6) ∀ε > 0, D3× n0 ≥ 1, ¸ÿ n ≥ n0`, |an − a| < ε�

&Æµ�¼:: �1∀ε > 0, · · ·�Wñ, ©�0Õ×ε > 0¸ÿ· · ·�WñÇ��3h· · ·Ç(1.6)P�¡ëÍB��k¸D3×n0 ≥ 1,

44�Wñ, -6E∀n0 ≥ 1, 44�Wñ, 3h44Ç(1.6)P�¡ÞB��t¡kn ≥ n0`, |an− a| < ε, �Wñ, -©�0Õ×n ≥ n0,

¸ÿ|an − a| ≥ ε�À�Aì: E×a ∈ R, uD3×ε > 0, ¸ÿE∀n0 ≥ 1, ÄD3×n ≥ n0, ¸|an − a| ≥ ε, J(1.5)PWñ��u∀a ∈ R, /�ðÕh¯, Jlimn→∞ an-�D3, T1n → ∞`,

an�Á§�D3, Ç{an, n ≥ 1}s÷�

»»»1.3.�Jlimn→∞(−1)n�D3�JJJ��.�an = (−1)n��J�∀a ∈ R, limn→∞(−1)n 6= a��'a ≥0�ãε = 1/2, Jn �ó`,

|an − a| = | − 1− a| = a + 1 > ε,

Ç∀n0 ≥ 1, ÄD3×�ón ≥ n0, ¸ÿ|an − a| ≥ ε�Íg'a < 0�)ãε = 1/2, Jn �ó`,

|an − a| = |1− a| = 1− a > ε,

1.1 ó�ÝÁ§ 7

Ç)b∀n0 ≥ 1, D3×�ón ≥ n0, ¸ÿ|an − a| ≥ ε�ÿJ�

9ì&Æ��óóó��Ý�L�'bó�{an, n ≥ 1}�uan+1 ≥an, ∀n ≥ 1,J{an, n ≥ 1}Ì ��¦¦¦;uan+1 ≤ an,∀n ≥ 1,J{an, n ≥1} Ì ��333��¦C�3ó�ÙÌ��ó��uan+1 > an,

∀n ≥ 1, J{an}Ì �}�¦�!§��L�}�3, C�}��ó��ÐóÝ��ôbv«Ý�L, &Æµ�¶�¼Ý�3Á§�×�ÃÍÝ�� vb&Ýó�Ä[e, �ì�

§�

��§§§1.1.'ó�{an, n ≥ 1} ��vvvbbb&&&, Jlimn→∞ anD3�JJJ��.�'{an, n ≥ 1} �¦, v'|an| ≤ K, ∀n ≥ 1, Í�K ×ü�ÝÑó�.{an, n ≥ 1}ó�Â�/)S = {a1, a2, · · · }bî&, Æã@ó��t�î&2§(�9ìÛ1.1)á, h/)bt�î&D3�&Æ|L�ht�î&��an ≤ L,∀n ≥ 1, )Wñ�&Æ�J�LÇ limn→∞ an�E�×ε > 0, .L − ε¬&S�×î&, ÆÄD3×n0 ≥ 1, ¸

ÿL − ε < an0(Qn0��εbn)�JE∀n ≥ n0, .an ≥ an0 ,

ÆL− ε < an ≤ L�ùÇ∀ε > 0, �0Õ×n0 ≥ 1, ¸ÿ

0 ≤ L− an < ε, ∀n ≥ n0�

�îP�|L− an| < ε, ∀n ≥ n0, ��(3hL− anÄ� �), Æµ�LL = limn→∞ an��y{an, n ≥ 1} �3Ý�µù!§�J�

ÛÛÛ.(ttt��îîî&&&222§§§). 'B @ó�×&èvbî&��/),

JBbttt��îîî&&&(least upper bound, ¹¶ lub)�T1Bbîîî@@@&&&(supermum), |supB��

3h'B @ó�×&è�/)�uK��x ≤ K, ∀x ∈ B, vuk1 < K, JÄb×l ∈ B, ¸ÿl > k1, JÌK B�t�î&(î

8 Ï×a Á§

@&)�t�î&uD3Ä°×, ¬�×�òy�/), �supB ∈B, JB�bt�-ô, Ç supB�supB /∈ B, JB��D3t�-ô�v«2, ô��Lttt��ììì&&&(greatest lower bound, ¹¶ glb)�T1B�ììì@@@&&&(infimum), |infB��»A, EB =

[0, 1)T(0, 1], supB/ 1,¬Gï, supB /∈ B,E¡ïsupB ∈ B�êÞï�inf/ 0�

E×ó�{an, n ≥ 1}, &Æ��limn→∞ an = ∞Climn→∞ an =

−∞��L�

��LLL1.2.u∀k > 0, D3×n0 ≥ 1, ¸ÿn ≥ n0`, an > k, J|limn→∞ an = ∞��; u∀k > 0, D3×n0 ≥ 1, ¸ÿn≥n0 `,

an < −k, J|limn→∞ an = −∞��

ulimn→∞ an = ∞, J&Æ1n → ∞`, ans÷Õ∞, T1Á§ ∞��ú�×g, h`Á§¬�D3�limn→∞ an = −∞ô�v«21��ê;ð&Æ¶limn→∞ an = a`, Ç2âa ×@ó�9ìbËÍ�§1.1 �ñÇÝ.¡�

��§§§1.1.'D3×n0 ≥ 1, ¸ÿó�{an, n ≥ 1}�Ïn04R ��,

vhó� bbb&&&, Jlimn→∞ anD3�

��§§§1.2.'ó�{an, n ≥ 1} �� b&�uhó� �¦,

Jlimn→∞ an = ∞; uhó� �3, Jlimn→∞ an = −∞�

»»»1.4.'an = 10n/n!, n ≥ 1��l�hó�3G104 �¦, �Ï114��3�êan ≥ 0, ∀n ≥ 1, .h{an, n ≥ 1} b&�Æã�§1.1á, limn→∞ anD3�

»»»1.5.'an = n2 + (−1)n, n ≥ 1�.{an, n ≥ 1} �¦�� b&,

Ælimn→∞ an = ∞�

êÞ 9

»»»1.6.'an = (n2 + 1)/n, &Æ�J�limn→∞ an = ∞�´�k

an =n2 + 1

n= n +

1

n> k,

©�n > kÇ��Ç∀k > 0,ãn0 = [k]+1,Jn ≥ n0`, an > k�Æµ�Lá, limn→∞ an = ∞�¨², .ó�{an, n ≥ 1} �¦, Æù�¿à�§1.2, �ÿÕ!øÝ��

»»»1.7.�J(i) limn→∞ 1

nα = 0, α > 0;

(ii) limn→∞ nβ = ∞, β > 0�JJJ��.&Æ©J(i)�∀ε > 0 k¸

∣∣∣∣1

nα− 0

∣∣∣∣ =1

nα< ε,

©�n > (1/ε)1/αÇ��Æ�ãn0 = [(1/ε)1/α] + 1�

E×ó�{an, n ≥ 1}, XÛ“ãÁ§”, µÎX�limn→∞ an�ãÁ§ôµW ×ËºÕ�A�b¨×ó�{bn, n ≥ 1}, &Æ��º®

(1.7) limn→∞

(an + bn) = limn→∞

an + limn→∞

bn

ÎÍWñ? ¼�Î�8��ãÁ§, ��Î�ãÁ§�8�, ÞïÝ�L�8!�3ó.î×��, uÞÞºÕÝg�øð, XÿÝ��¬�×�8��»A, tÝKó©»,

√a + b 6= √

a +√

b�

ð�1, �]�8�ÞºÕ��øð�¬3ÊÝf�ì, 3Êf�ì, (1.7)PÎWñÝ�3ì×;&ÆÞ��Á§Ý×°ÃÍP²�

10 Ï×a Á§

êêê ÞÞÞ 1.1

1. 'an = 2n/(n− 3), n ≥ 4��µ�LJ�limn→∞ an = 2�

2. 'an =√

n/(2 +√

n), n ≥ 1��µ�LJ�limn→∞ an = 1�

3. 'an = (n2 + 3)/(n + 2), n ≥ 1��µ�LJ�limn→∞ an =

∞�4. 'an = (n− 2n2)/(n2 +3), n ≥ 1��µ�LJ�limn→∞ an =

−2�

5. 'an = 1n+1

+ 1n+2

+ · · ·+ 12n

, n ≥ 1��Jlimn→∞ anD3�

6. 'a1 = 1, an+1 =√

3an + 1, n ≥ 1��Jlimn→∞ anD3�

7. 'an =∑n

i=1 i−2, n ≥ 1��Jlimn→∞ anD3�

8. 'an =∑n

i=1(i(i + 1))−1, n ≥ 1��Olimn→∞ an�

9. 'a1 = 3, an+1 = 5/an, n ≥ 1��1�ì�Ý.0ÎÍÑ@:

�y = limn→∞ an, �ÿy = 5/y, .hy =√

5(−√5�))�

1.2 ÁÁÁ§§§ÝÝÝÃÃÃÍÍÍPPP²²²

'b×ó�{an, n ≥ 1}, Jn →∞`, hó��bÞÍÁ§D3�àÌîh��ÎEÝ�.'bÞÁ§a, a′D3,va 6= a′,&Æ�ã�×FÝÑóε ,¸ÿI1 = (a−ε, a+ε)�I2 = (a′−ε, a′+ε)�¥P�JtÝb§4�², XbÝan�a3I1�, ô�a3I2�, �hÛ��¨²,ulimn→∞ an = aD3,Jhó� b&�àÌîôÎEÝ�.tÝb§4, ÉA1a1, a2, · · · , am−1, ÍõÝan/a3(a − 1, a + 1)�(Çãε = 1)�X|�0Õ×b§ �âXban�&ÆÞh��W�Aì, ¬Þ|îÝ�°�@2¶��

1.2 Á§ÝÃÍP² 11

��§§§2.1.'ó�{an, n ≥ 1}�Á§D3, J(i) Á§Â°×;

(ii) hó�b&�JJJ��.�limn→∞ an = a ∈ R�

(i) 'D3¨×@óa′ 6= a, ô��limn→∞ an = a′�ãε = |a −a′|/2�JµÁ§Ý�Lá, D3n1, n2 ≥ 1, ¸ÿ

|an − a| < ε, ∀n ≥ n1,

|an − a′| < ε, ∀n ≥ n2�

Æn ≥ max{n1, n2}`, ãë��P

|a− a′| ≤ |an − a|+ |an − a′| < 2ε = |a− a′|,

hÛ��ÆÿJÁ§Â°×�(ii)ãε = 1,JD3n0 ≥ 1,¸ÿ|an−a| < 1, ∀n ≥ n0�Çn ≥ n0

`, |an| ≤ |a|+ 1��

K = max{|a1|, |a2|, · · · , |an0−1|, |a|+ 1},

J|an| ≤ K, ∀n ≥ 1�ÿJhó�b&�

ãyÁ§ÂÄ°×, Æ3Á§ÂD3Ý�µì, �¡hó�G«Ý4ó�;9�, @�2¸Æ©º3Ø×�Â!�¯�®�v®�º÷¼÷�, hÇó��%%%��(stationary)�9ô�|1� ¢uÁ§ÂD3, &Æ-1hó�[e, .tâhó�ÝÂÎ�|ßéÝ��s÷Ýó�, ��¸ÆÝÂº×à®�, ��ì¼, ��¸ÆÝÂºPc¼Ý��(TPc¼Ý��), ôÎ��ì¼�3^ðà+, ×�lÏ��Ý��[e×F, ôbv«Ý�¤�Ç�O��Ý� ��ß�, ��s÷�9ìÝ�§ÎnyÁ§Ý°JºÕ�

��§§§2.2.'bÞó�{an, n ≥ 1}C{bn, n ≥ 1}, vlimn→∞ an = a,

limn→∞ bn = b/D3�J

12 Ï×a Á§

(i) limn→∞(an + bn) = a + b,

(ii) limn→∞(an − bn) = a− b,

(iii) limn→∞(anbn) = ab,

(iv) limn→∞ an/bn = a/b, ub 6= 0�JJJ��.&Æ©J(iii)C(iv), (i)C(ii)�J�º3êÞ�

(iii) ∀ε > 0, µ�L�0Õn0 ≥ 1, ¸ÿn ≥ n0`,

|an − a| < ε, v |bn − b| < ε�

J.anbn − ab = bn(an − a) + a(bn − b), Æãë��P

|anbn − ab| ≤ |bn||an − a|+ |a||bn − b|< (|a|+ |bn|)ε ≤ (|a|+ K)ε,

h�ã�§2.1á, {bn, n ≥ 1} ×b&ó�, Æ�'|bn| ≤ K, ∀n ≥1, Í�K > 0 Ø×�Â�9ìÝM»Î3J�Á§ÝÄ��ðàÝ*»�E��×ε >

0, &Æ�ãε1 = ε/(|a| + K), Jµî�.0, �0Õ×n′0 ≥ 1, ¸ÿn ≥ n′0`,

|anbn − ab| < (|a|+ K)ε1 = ε�ÆÿJ(iii)�

(iv) &Æ6J�∀ε > 0, D3×n0 ≥ 1, ¸ÿ

(2.1)

∣∣∣∣an

bn

− a

b

∣∣∣∣ < ε, ∀n ≥ n0�

�û(iii), ¿àë��P, �ÿ∣∣∣∣an

bn

− a

b

∣∣∣∣ =

∣∣∣∣anb− abn

bnb

∣∣∣∣ ≤|b||an − a||bnb| +

|a||bn − b||bnb|(2.2)

=1

|bn| |an − a|+ |a||bn||b| |bn − b|�

ãî��P&Æá¼T�J�Ý, .nÈ�¡, |an− a|C|bn− b|/º��¬)b×°Þ;, 1/|bn|º�º��? #�|bn| = 0§�ð?

�Ä9°Î��XÝ�

1.2 Á§ÝÃÍP² 13

.limn→∞ bn = b 6= 0, ÆD3×n1 ≥ 1, ¸ÿn ≥ n1`, |bn − b| <|b|/3(Çãε = |b|/3)��|b| − |bn| < |bn− b|(hù ëëë��PPP), Æÿn ≥ n1`, |bn| > 2|b|/3 > 0�X|&Æ©�Ên ≥ n1, .t¡��n →∞, Æ×��µ§×n ≥ n1, QÎ�|Ý�9ì��Ý�®µÎ¯(2.1)PWñ��∀ε > 0, �0Õ

×n2 ≥ 1, ¸ÿ

|an − a| < |b|3

ε, |bn − b| < |b|23(|a|+ 1)

ε, ∀n ≥ n2�

�ãn0 = max{n1, n2}�Jn ≥ n0`, .

1

|bn| <3

2|b| ,

Æã(2.2)P,∣∣∣∣an

bn

− a

b

∣∣∣∣ ≤3

2|b||b|3

ε +3|a|2|b|2

|b|23(|a|+ 1)

ε

<ε

2+

ε

2= ε�

h�3�|bn−b|Ýî&`,.ab�� 0,Æ5Òã 3(|a|+1)�ÿJ�

9ìÝ.¡ôÎ��QÝ, .©�3î�§�ãbn = α, ∀n ≥ 1,

Ç��

��§§§2.1.'limn→∞ an = aD3�JE∀ðóα,

(i) limn→∞(an + α) = a + α,

(ii) limn→∞ αan = αa�

��§§§2.3.ulimn→∞ an = a, limn→∞ bn = b, van ≤ bn,∀n ≥ 1, Ja ≤b�JJJ��.'a > b,&Æ�0�ë;�ãε = (a−b)/2 > 0�ã�'á,D3×n0 ≥ 1,¸ÿn ≥ n0`, |an−a| < ε,v|bn−b| < ε�Æn ≥ n0`,

bn < b + ε = b +a− b

2= a− a− b

2= a− ε < an,

14 Ï×a Á§

h��'an ≤ bn,∀n ≥ 1, �)�Æa ≤ b�

Íg&Æ:½(Ýôôô^æææ§§§(Squeezing principle), êÌôôô^��§§§Tëëë��»»»��§§§(Sandwich rule)�

��§§§2.4.'bëó�{an, n ≥ 1}, {bn, n ≥ 1}C{cn, n ≥ 1}, an ≤bn ≤ cn, ∀n ≥ 1,vlimn→∞ an = limn→∞ cn = a,Jlimn→∞ bn = a�JJJ��.ã�'á, ∀ε > 0, D3×n0 ≥ 1, ¸ÿ

|an − a| < ε, |cn − a| < ε, ∀n ≥ n0�

Ça− ε < an, cn < a + ε, ∀n ≥ n0�

.an ≤ bn ≤ cn,∀n ≥ 1, Æa− ε < bn < a + ε, ∀n ≥ n0, ùWñ�ÿJ�

9ìÝ.¡, ÍJ�º3êÞ�

��§§§2.2.'{an, n ≥ 1} ×b&ó�, vlimn→∞ bn = 0, J

limn→∞

anbn = 0�

»»»2.1.�O

limn→∞

2n3 + n2 + 4n

n3 + 3n2 + 2n + 1��.´�Þ5�5Ò!t|n3, ÿ

2n3 + n2 + 4n

n3 + 3n2 + 2n + 1=

2 + n−1 + 4n−2

1 + 3n−1 + 2n−2 + n−3�

ã»1.7á, n → ∞`, n−1, n−2, n−3/��0��¿à�§2.2, -ÿ

limn→∞

2n3 + n2 + 4n

n3 + 3n2 + 2n + 1=

2 + 0 + 0

1 + 3 · 0 + 2 · 0 + 0= 2�

1.2 Á§ÝÃÍP² 15

»»»2.2.'|x| < 1, &Æ�J�

limn→∞

xn = 0�

JJJ��.ux = 0, îPQWñ�Íg'0 < x < 1, v�a > 0, ��

x =1

1 + a�

�¿àE∀n ≥ 1, Ca > 0,

(1 + a)n ≥ 1 + na,

ÿn →∞`,

xn =1

(1 + a)n≤ 1

1 + na=

n−1

n−1 + a−→ 0

0 + a= 0�

�yu−1 < x < 0, J0 < |x| < 1, ��J�limn→∞ |x|n = 0,

Ælimn→∞ xn = 0(�êÞ)�

ÛÛÛ.E∀x > −1CJón ≥ 1, (1 + x)n ≥ 1 + nx, h Bernoulli��PPP(Bernoulli inequality)�×©½�µ, �|ó.hû°, TÞÞÞ444PPP��§§§J��

»»»2.3.�On →∞`, n√

n�Á§Â��..n ≥ 1, ��¢×�yT�y1Ýó, �ng]¡)�yT�y1�Æ

n√

n ≥ 1��¿àÕÕÕ��¿¿¿ííí(arithmetic mean)�yT�y¿¿¿¢¢¢¿¿¿ííí(geometric

mean), ÿ

n√

n = (√

n · √n · 1 · 1 · · · 1)1/n

≤ 1

n(√

n +√

n + 1 + 1 + · · ·+ 1)

=1

n(2√

n + n− 2) =2√n

+ 1− 2

n,

16 Ï×a Á§

Í�1 · 1 · · · 1�b(n− 2)Í1�.h&ÆÇÿ

1 ≤ n√

n ≤ 2√n

+ 1− 2

n�

t¡¿àô^�§, .n →∞`, 1√nC 1

n/��0, ÆÿJ

limn→∞

n√

n = 1�

¨×ð�Ý®°Î¿àÞ4P�§��an = n√

n�.an ≥ 1,

Æan = 1 + hn, Í�hn ≥ 0�J

n = ann = (1 + hn)n

≥ 1 + nhn +n(n− 1)

2h2

n ≥n(n− 1)

2h2

n�

ÆEn ≥ 2, b

h2n ≤

2

n− 1,

Ç

hn ≤√

2√n− 1�

.h

1 ≤ an = 1 + hn ≤ 1 +

√2√

n− 1�

�û»1.7�(i)ÝJ�, �ÿîP��1�Æãô^�§ÿJlimn→∞ an = 1�

&Æ:ÕÊ2¿à×°�§, 3OÁ§`��6ã�L�s�

ÛÛÛ.E��n ≥ 1, Ca1, a2, · · · , an > 0,

(a1a2 · · · an)1/n ≤ 1

n(a1 + a2 + · · ·+ an),

v�rWñ, uv°ua1 = a2 = · · · = an�hÇ½(ÝÕÕÕ¿¿¿��PPP, êÌÕÕÕ��¿¿¿¢¢¢¿¿¿íííÂÂÂ��PPP�

êÞ 17

ÛÛÛ.ÞÞÞ444PPP��§§§(Binomial theorem). E∀n ≥ 1,

(a + b)n =n∑

i=0

(n

i

)an−ibi,

Í�(

n

i

)=

n!

i!(n− i)!=

n(n− 1) · · · (n− i + 1)

i! �

�h, &ÆEó�ÝÁ§�Ý×°ÃÍÝ+Û�ì×;&ÆÞBã.0�ZóCiø£, |¸��EÁ§�9×FYê�Á§�)b×°ÞC, A�ó��Ðóó��, &Æ|¡º�D¡��&ÆEë�ÐóC¼ó�Eó?!Y¡, Þ��O?9Á§�

êêê ÞÞÞ 1.2

1. �J�§2.2�(i)C(ii)�

2. 'limn→∞ an = a, �Jlimn→∞ |an| = |a|�ê®ÍYWñÍ?

3. 'limn→∞ |an| = 0, �Jlimn→∞ an = 0�

4. �J�§2.2�

5. 'limn→∞ an = a > 0, �JD3×n0 ≥ 1, ¸ÿan > 0,∀n ≥n0�

6. 'limn→∞ an = a > 0, vlimn→∞ bn = ∞��J(i) limn→∞ an/bn = 0,

(ii) limn→∞ bn/an = ∞,

(iii) limn→∞ anbn = ∞�7. 'bÞ94Pf(x)Cg(x), JElimn→∞ f(n)/g(n)�b¢.¡?

8. 'a1 = 1, an =√

1 + an−1, n ≥ 2��Olimn→∞ an�

18 Ï×a Á§

9. 'a1 = 1, a2 = 2, an+2 = an+1 + an, n ≥ 1��Olimn→∞ an�

10. �Jó�√

2,√√

2,

√√√2, · · ·[e, ¬OÍÁ§Â�

11. �Jó�√

2,√

2 +√

2,

√2 +

√2 +

√2, · · ·[e, ¬OÍÁ§

Â�12. �Olimn→∞(n2 + 3n)/(n3 + 2n2 + 1)�13. �Olimn→∞(2n3 + 4n2 + 5)/(7n2 + n + 6)�14. �Olimn→∞

√1 + n−1�

15. �Olimn→∞√

n(√

n + 1−√n)�16. û»2.3ÝÏÞË®°, �Olimn→∞ n/αn, α > 1�17. �Jlimn→∞(

√n + 1−√n)

√n + 1/2 = 1/2�

18. �Jlimn→∞( 3√

n + 1− 3√

n) = 0�19. �Jlimn→∞ n!/nn = 0�20. �Jlimn→∞

∑ni=1 i/n2 = 1/2�

21. �Jlimn→∞∑n

i=1(n + i)−2 = 0�22. �Jlimn→∞

∑ni=1(n + i)−1/2 = ∞�

23. �Jlimn→∞∑n

i=1(n2 + i)−1/2 = 1�

24. �Olimn→∞ 10n/n!�25. �Olimn→∞(1 + (−1)n)/n�26. �X�limn→∞ n(

√n2 + 2− n)�Â, ¬µ�LJ��

27. �JTÍJE×Ðóf , limn→∞(an−bn) = 0,0llimn→∞ f(an)

−f(bn) = 0�

1.3 �ZóCiø£ 19

1.3 ��ZZZóóóCCCiiiøøø£££

&Æ��Êó�{an, n ≥ 1}, Í�

an =n∑

i=0

1

i!= 1 +

1

1!+

1

2!+ · · ·+ 1

n!,

¬J�ÍÁ§D3�h �§1.1“��vb&�ó�Ä[e”�×�?ÝTà»��´�, �Q{an, n ≥ 1} �¦�Íg

an = 1 + 1 +1

2+

1

2 · 3 +1

2 · 3 · 4 + · · ·+ 1

2 · 3 · · ·n≤ 1 + 1 +

1

2+

1

22+

1

23+ · · ·+ 1

2n−1

= 1 +1− (1/2)n

1− 1/2< 3,

Ç{an, n ≥ 1}|3 ×î&�Æ{an, n ≥ 1} [e�×@ó�ãyhÁ§Â�π×ø3&9ó.Ý2P��¨, X|��¹Ým��¸×Í©½ÝÐr��ó.��ZZZ(Euler, 1707-1783)«{ÎÏ×Í�ºÕhó�¥�PÝó.�, ¬|e¼�î�¡¼eµ�Â 2à, ô�Ì ��ZZZóóó(Euler’s number)�e�π�- Îó.�t¥�ÝËÍø÷ó�ãye n →∞`an�Á§, Æe��î

(3.1) e =∞∑i=0

1

i!�

9ì1�A¢|{an, n ≥ 1}Oe��«Â, ¯@îan[e�eÝ>�Á"�)¢Ã×¿¢ùó, E��n > m,

an = am +1

(m + 1)!+

1

(m + 2)!+ · · ·+ 1

n!

≤ am +1

(m + 1)!(1 +

1

m + 1+

1

(m + 1)2+ · · ·+ 1

(m + 1)n−m)

≤ am +1

(m + 1)!

1

1− 1/(m + 1)= am +

1

m ·m!�

20 Ï×a Á§

ÆEn > m,

(3.2) am < an ≤ am +1

m ·m!�

u�n →∞, JîP0l

(3.3) am < e ≤ am +1

m ·m!,∀m ≥ 1�

ÇE∀m ≥ 1, am�e�-t9 1m·m!�ãym!�½mW�>�Á

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1.3 �ZóCiø£ 21

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22 Ï×a Á§

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1 <(1 + 1

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1.3 �ZóCiø£ 23

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24 Ï×a Á§

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O A C

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√4− C2

n�

êÞ 25

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2, µ��ÿ

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×�Jb

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2n

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√· · · = π,

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êêê ÞÞÞ 1.3

1. �Jlimn→∞ cos2(π/2n) = 1�

2. �J(3.10)P�

26 Ï×a Á§

3. �Oì�Á§Â

limn→∞

√2−

√2 +

√2 +

√· · ·�

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5. �Dn��i�²6Ñn9\��\��OBn�Dn�n;�

6. �JA2n An�Bn�¿¢¿í�

1.4 ÐÐÐóóóÝÝÝÁÁÁ§§§

G«&ÆD¡ó�ÝÁ§, E×ó�{an, n ≥ 1},&Æ;ð©º®n → ∞`, an�Á§, ��º®n → 5`, an�Á§�h.n©��y1, 2, 3, 4, 5,· · · , X|un�#�5, JOa5Ç��¬u�×Ðóf(x), JEyx → 5, x → −3.2, x →∞Tx → −∞`, f(x)��¢Â, 9Ë®Þ-b�LÝ��v&Æ�¿àó�Á§Ý�L¼�LÐÐÐóóóÝÝÝÁÁÁ§§§�'b×Ðóf(x), &Æ�á¼x → a`, f(x)º��¢Â? ´

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ighborhood)�Çc�×Ï½Ä (a, b)Ý�P,Í�a < c < b, �E∀r > 0, (c − r, c + r)Ì c�×�5 rÝEEEÌÌÌÏÏÏ½½½(symmetric

neighborhood of c of radius r)�u(a, b) c�×Ï½,J(a, b)\{c} =

(a, c) ∪ (c, b)Ì c�×��TTTÏÏÏ½½½(deleted neighborhood)�

1.4 ÐóÝÁ§ 27

�p:�cÝN×Ï½Ä�â×EÌÏ½�3D¡ÐóÝÁ§`, &Æð¢ÃÏ½ÝÃF�.hkOlimx→a f(x), ´�f3aÝ×�TÏ½��b�L, �yf(a)bP�LÅÎ^bn;Ý�.&Æ©Î�Ox×à#�a`, f(x)º��¢Â,¬�Î�Of(a)�X|�AO

limx→1

x2 + x− 2

x− 1, lim

x→0

sin x

x,

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�aÝ¼\, ôµÎfa��a, h`|x ↑ aTx → a−��!§bx ↓ a(Tx → a+), ��aÝ�\��a�Çb`ºOlimx→a+

Tlimx→a−, 5½Ì ��ÁÁÁ§§§(right-hand limit)C¼¼¼ÁÁÁ§§§(left-hand

limit), )Ì��ÁÁÁ§§§(one-sided limit)�¨3¼:limx→a f(x)Î%��¤? �:x → a��'bó

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N×º��aÝó�{an, n ≥ 1}, K�Olimn→∞ f(an), §¡îÎ��|Ý�®�&Æ�ûï31.1;, �Ló�ÝÁ§`, ¢ã��Ý/�, ��Á§Ý�L�'|a| < ∞, |b| < ∞�Eylimx→a f(x) = b, f(x) → bÎ&Æt

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28 Ï×a Á§

f(x) → bÝ�×�Ýh-|h� Á§Ý�L�

��LLL4.2.u∀ε > 0, D3×δ > 0, ¸ÿ0 < |x − a| < δ`, |f(x) −b| < ε, JÌx → a`, f(x) → b, v|

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f(x) = b

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1.4 ÐóÝÁ§ 29

½)�9ì&Æ�¿Í»��

»»»4.1.'a > 0 ×ðó, �Jlimx→a

√x =

√a�

JJJ��.&Æ��¾WE∀ε > 0,

(4.2) |√x−√a| < ε�

�îP��y √a− ε <

√x <

√a + ε�

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a− ε�×� Ñ, ¿]¡��Pµ�×�)Wñ(��−3 < 2 < 5Ý��)��ãε1 = min{ε,√a}�ãyε1 ≤ ε,Æu

(4.3) |√x−√a| < ε1,

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√a− ε1 <

√x <

√a + ε1�

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a− (2√

aε1 − ε21) < x < a + (2

√aε1 + ε2

1)�

ãy2√

aε1 − ε21 = ε1(2

√a− ε1) > 0, Æ©�

x ∈ (a− 2√

aε1 + ε21, a) ∪ (a, a + 2

√aε1 + ε2

1)

(h a�×�TÏ½), J(4.3)PWñ, .h(4.2)PWñ�Ah-ÿJ(4.2)PÝ�Qãy0 < 2

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1 < 2√

aε1 + ε21,ÆuË��O�δ, J�

ãδ = 2√

aε1 − ε21, J0 < |x− a| < δ`, |√x−√a| < ε�

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30 Ï×a Á§

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|√x− 4| < 0.01�

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(4.4) |f(x)− 2| < ε�

�

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x− 1√x + 3 + 2

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1.4 ÐóÝÁ§ 31

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|f(x)− 1

2| =

∣∣∣∣1√

x + 3− 1

2

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2√

x + 3

∣∣∣∣

=

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x + 3)

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»»»4.5.'

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{1, ux−1 Jó ,

x, ux−1� Jó�

32 Ï×a Á§

Çux ∈ S = {· · · ,−13,−1

2,−1, 1, 1

2, 1

3, · · · },Jf(x) = 1,ÍJf(x) =

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|f(x0)− b| = b− x0 > b− b/2 = b/2 = ε�.hlimx→0 f(x) 6= b�

(ii) 'b = 0�ãε = 1/2�E∀δ > 0, 3(0, δ)�, ÄD3×x1 ∈S, J

|f(x1)− b| = |f(x1)| = 1 /∈ (−ε, ε)�.hlimx→0 f(x) 6= b�

(iii) 'b < 0�û(i)�ÿlimx→0 f(x) 6= b�

ÛÛÛ.\ï��/�limx→1 f(x)ÎÍD3? uD3, ��Á§Â, ¬J��

»»»4.6.'

f(x) =

{1, ux b§ó,

0, ux P§ó�Jû»4.5�D¡�J�limx→0 f(x)�D3, #�limx→a f(x)�D3, ∀a ∈ R(�êÞÏ1Þ)�

Íg&ÆD¡Í�&ËÁ§, ´�:G«�èÄÝ�Á§C¼Á§�

��LLL4.3.uE∀ε > 0,D3×δ > 0,¸ÿE∀x ∈ (a, a+δ), |f(x)−b| <ε,JÌf3a��ÁÁÁ§§§ b,|limx→a+ f(x) = b��uE∀ε > 0,D

1.4 ÐóÝÁ§ 33

3×δ > 0, ¸ÿE∀x ∈ (a − δ, a), |f(x) − b| < ε, JÌf3a�¼¼¼ÁÁÁ§§§ b, |limx→a− f(x) = b��x → a+Tx → a−ê�5½|x ↓ aTx ↑ a��

»»»4.7.��2

limx→0

|x|x�D3,

¬

limx→0+

|x|x

= 1, v s limx→0−

|x|x

= −1�¯@îbì��§�

��§§§4.2.EÐóf , limx→a f(x) = b, uv°u

(4.5) limx→a+

f(x) = limx→a−

f(x) = b�

JJJ��.'limx→a f(x) = b, J∀ε > 0, D3×δ > 0, ¸ÿux ∈ (a −δ, a)∪ (a, a + δ), J|f(x)− b| < ε�ã�L4.3á, hÇ0�(4.5)PWñ�Íg'(4.5)PWñ�J∀ε > 0, D3×δ1 > 0, ¸ÿ

|f(x)− b| < ε, ∀x ∈ (a− δ1, a),

vD3×δ2 > 0, ¸ÿ

|f(x)− b| < ε , ∀x ∈ (a, a + δ2)�

Æuãδ = min{δ1, δ2}, J

|f(x)− b| < ε, ∀x ∈ (a− δ, a) ∪ (a, a + δ)�

µ�LáhÇ�limx→a f(x) = b�ÿJ�

ã�§4.2á, u×Ðóf , 3ØFa�¼Á§��Á§b×�D3, T4Þï/D3, ¬ÍÂ�!, Jf3a�Á§�D3�

34 Ï×a Á§

»»»4.8.'f(x) = [x], x ∈ R, �t�JóÐó�.EN×Jón,

limx→n+

[x] = n, limx→n−

[x] = n− 1,

Þï��, Ælimx→n[x]�D3��yua� Jó, J

limx→a

[x] = [a]�

»»»4.9.'

f(x) =

{2x + 1, x ≥ 1

x, x < 1�J

limx→1+

f(x) = 3, limx→1−

f(x) = 1,

vlimx→1 f(x)�D3�

b`×Ðóf , 3x�#�a`, ÍÂºPc¼Ý¦��?�@×Fý, &Æbì��L�

��LLL4.4.u∀k > 0, D3×δ > 0, ¸ÿ

f(x) > k, ∀x ∈ (a, a + δ),

JÌf3a��Á§ P§�, v|

limx→a+

f(x) = ∞

��u∀k > 0, D3×δ > 0, ¸ÿ

f(x) < −k, ∀x ∈ (a, a + δ),

JÌf3a��Á§ �P§�, v|

limx→a+

f(x) = −∞

1.4 ÐóÝÁ§ 35

��

!§��L

limx→a−

f(x) = ∞, limx→a−

f(x) = −∞,

Climx→a

f(x) = ∞, limx→a

f(x) = −∞�Q, ãy∞C−∞/&@ó, X|î�¿Ë�µKòyÁ§�D3�

»»»4.10.'f(x) = 1x, x 6= 0�àÆ¼:limx→0+

1x

= ∞, 9ì&Æ¼�Jh¯�∀k > 0, �¸

1

x> k,

�hÇ(¥�xã0Ý��Ä¼)

0 < x <1

k�

Æuãδ = k−1,JE∀x ∈ (0, δ), f(x) > k�µ�LhÇ�limx→0+1x

= ∞�

¨², ù�J�

limx→0−

1

x= −∞�

#�&Æb?×�Ý��

(4.6) limx→a+

1

(x− a)p= ∞, p > 0 ,

(4.7) limx→a−

1

(x− a)p= −∞, p = r/s, Í�r, s ÞÑ�ó�

�y = f(x), ulimx→a+ f(x) = ∞, J3x = a��, f�%��lA%4.1�

36 Ï×a Á§

-

6

Ox

a

y

%4.1.

Çxãa��, �¼�#�a`, f �%�º�¼�#�àax = a�.hàax = a, Ì f�%�Ý×��aaa(asymptote)�?×�2,ulimx→a+ f(x),Climx→a− f(x),Þï�Kb× ∞T

−∞, Jàax = a, Ì f�%�Ý×kkkààà��aaa(vertical asymp-

tote)�&Æ$bì��L�

��LLL4.5.u∀ε > 0, D3×k > 0, ¸ÿ|f(x) − b| < ε, ∀x > k, JÌlimx→∞ f(x) = b�!§��Llimx→−∞ f(x) = b�

»»»4.11.&Æ�J

(4.8) limx→∞

1

x3= 0�

�∀ε > 0, ∣∣∣∣1

x3− 0

∣∣∣∣ < ε

Ç (¥�.x →∞, Æ©�Êx > 0)

0 <1

x3< ε�

�îP��yx > ε−1/3�Æuãk = ε−1/3, Jux > k, |f(x)| <

ε�µ�Lá(4.8)PWñ�

1.4 ÐóÝÁ§ 37

´×�Ý��ÍJ�º3êÞ�

(4.9) limx→∞

1

(x− a)p= 0, a ∈ R, p > 0�

b`x��`, f(x)ô�W��, 9µÎì��L�

��LLL4.6.u∀k > 0, D3×n > 0, ¸ÿf(x) > k, ∀x > n, J|

(4.10) limx→∞

f(x) = ∞

��

!§��L

limx→∞

f(x) = −∞, limx→−∞

f(x) = ∞, limx→−∞

f(x) = −∞�

|î9°µÎ&Æx�ºD¡ÝÁ§�L�Þ9¿Í�L8!f´, \ïT@�º�ºÍæ§�b°®Þ&ÆãÌD°-�:�ÍÁ§, ¬b°´�ÓÝ®Þ, &B×j�ÛÝJ�, Î��|X�ÍÁ§Ý�Á§Î��5ÝÃ�, hÃFu��z½, ?¡�9§¡K�W5§WaÝ�

»»»4.12.'f(x) = x2�E∀k > 0, ãn =√

k, Jx > n =√

k`,

f(x) = x2 > n2 = k, µ�Lálimx→∞ x2 = ∞�

ì�?×�Ý��ÍJ�º3êÞ�

(4.11) limx→∞

(x− a)p = ∞, a ∈ R, p > 0�

�y = f(x), ulimx→∞ f(x) = b, Jáx��`, f(x)�Âº�#�b, vÍ%�º�#�àay = b�Çàay = b f�%�Ý×��a�ùÇu

limx→∞

f(x) = b T limx→−∞

f(x) = b,

38 Ï×a Á§

Jàay = bÌ f�%�Ý×iii¿¿¿��aaa(horizontal asymptote)�

»»»4.13.�0

y =4x2

x2 + 1

�%��.

´��:�%�EÌyy�(�Ðó)�ê.

0 ≤ x2

x2 + 1< 1,

Æ0 ≤ y < 4�êlim

x→∞4x2

x2 + 1= 4,

Æy = 4 i¿��a�Þy;¶

y =4x2

x2 + 1= 4− 4

x2 + 1,

J��:�3x > 0�y �¦�bÝî��P², �Bà0¿ÍF,

�pÿÕì%�

-

6

Ox

y

%4.2.

êÞ 39

êêê ÞÞÞ 1.4

1. �JE»4.6��Ðóf , limx→a f(x)�D3, ∀a ∈ R�

2. Oì�&Á§, Í�[ · ] t�JóÐó�(i) limx→1

2x2−3x+1x−1

, (ii) limh→0(x+h)2−x2

h,

(iii) limx→∞√

x1+x

, (iv) limx→∞(x+1)2

x2+1,

(v) limx→0 tan x, (vi) limx→0 x2 cos 5x,

(vii) limx→01−√1−x2

x2 , (viii) limx→0+|x3|x3 ,

(ix) limx→1−(1− x + [x]− [1− x]), (x) limx↑2[x2 + 1],

(xi) limt→−1+

√1+t

1+t5, (xii)limx→0+

(x+1)2

(x−1)2−1�

3. Oì�&Á§�(i) limx→∞(

√4 + x2 − x), (ii) limx→−∞ 1−x+3x2

1+x2 ,

(iii) limx→∞√

x√

x−6x+5

, (iv) limx→−∞x√−x√1−4x3 ,

(v) limx→∞√

x3+xx

, (vi) limx→∞(x2+1x+1

− x2+2x+2

)�4. µε− δÝ]°J�ì�&Á§�

(i) 'f(x) = c, ∀x ∈ R, Jlimx→a f(x) = c, ∀a ∈ R,

(ii) limx→3

√x + 1 = 2,

(iii) limx→21√2+x

= 12,

(iv) limx→41x

= 14,

(v) limx→−2(x2 + x) = 2�

5. �X�ì�&%��a, ¬0Í%�(i) f(x) = 1

x−2, (ii) f(x) = 2x

x+2,

(iii) f(x) = 1(x−2)2

, (iv) f(x) = 2x(x+2)2

,

(v) f(x) = 2x2

(x+2)2, (vi) f(x) = 2

x2−9,

(vii) f(x) = 2x2

x2+9, (viii) f(x) = x + 1

x,

(ix) f(x) = x2−1x2−4

, (x) f(x) = x2−1x2+4

,

(xi) f(x) = x2+1x2−4

, (xii) f(x) = x2+1x2+4 �

6. �J(4.6)C(4.7)P�

40 Ï×a Á§

7. �J(4.9)P�

8. �J(4.11)P�

9. �JE∀x ∈ R, limm→∞(cos(xπ))2mD3, vÁ§Â= 1, ux Jó; Á§Â= 0, ux� Jó�

10. �JE∀x ∈ R, limn→∞ (limm→∞(cos(n!xπ))2m)D3, vÁ§Â= 1, ux b§ó, Á§Â= 0, ux P§ó�

11. '

f(x) =

{1, ux b§ó,

x, ux P§ó��Jlimx→1 f(x) = 1�

12. '

f(x) =

{1, ux−1 Jó,

1− x, ux−1� Jó��Jlimx→0 f(x) = 1�

13. �J (i) limx→1

√x + 3 6= 3, (ii) limx→2(x + 3)(x2 + 1) 6= 1�

1.5 ÁÁÁ§§§��§§§CCC===��PPP

ÐóÝÁ§�§b×°Îv«ó�Á§�Ý��, J�ô�Kv«�¨², ùb×°ó��^bÝÁ§�§�bÝ9°�§, �¯&ÆO?9ÝÁ§�

��§§§5.1.'limx→a f(x) = b, vlimx→a g(x) = c�J(i) limx→a(f(x) + g(x)) = b + c,

(ii) limx→a(f(x)− g(x)) = b− c,

(iii) limx→a f(x)g(x) = bc,

1.5 Á§�§C=�P 41

(iv) limx→a f(x)/g(x) = b/c, c 6= 0�

��§§§5.2.(ôôô^��§§§). 'D3×δ > 0,¸ÿf(x) ≤ g(x) ≤ h(x),∀x ∈(a− δ, a) ∪ (a, a + δ), v

limx→a

f(x) = limx→a

h(x) = b,

Jlimx→a

g(x) = b�îÞ�§�Ýðóa, b, c/ @ó, |¡&Æ��©½Î�, Á§

ÑT�P§�&Æ5½º|∞, C−∞�î�¼�

��§§§5.3.'limx→a f(x) = 0, vD3×δ > 0 , ¸ÿ|g(x)| ≤ k, ∀x ∈(a− δ, a) ∪ (a, a + δ), Í�k ×ðó�J

limx→a

f(x)g(x) = 0�

»»»5.1.ãyg(x) = sin x ×ø�Ðó, ��2limx→∞ sin x�D3�ãh�:�limx→0 sin(1/x)ù�D3,¬.| sin(1/x)| ≤ 1,∀x 6=0, Æ¿à�§5.3, ÿ

limx→0

x sin1

x= 0�

»»»5.2.O

limx→∞

(x2 + 1

x + 1− x2 + 2

x + 2

)�

��.´��¥�ÝÎ, &Æ��ÞkO�Á§¶W

limx→∞

(x2 + 1

x + 1

)− lim

x→∞

(x2 + 2

x + 2

),

. hÞÁ§/�D3�6�;5ÿ

limx→∞

x2 − x

(x + 1)(x + 2)= lim

x→∞1− x−1

1 + 3x−1 + 2x−2=

1

1= 1,

42 Ï×a Á§

h�àÕx →∞ `, x−1, 3x−1, 2x−2/��0, �¿à�§5.1�

9ì&Æ+Û=�ÝÃF�

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f(a), ux×à#�a`, f(x)ô��#�f(a), J&Æ1f3a

=��3��5s"Ý\�, �I5X�§ÝÐó/ =�, .h£`

E=�ÝËÑ�L¬Î�~�àÕèâtS, 3×°Î§Ý®Þî��¨�=�ÝÐó�9Í�©½Îmmmñññfff(Fourier, 1758-

1830)3#æ.Ý�®, 2¸ÝèÜtS��Ýó.�, EÐóC=�PÝ�L?�T2�µÚ�4QE�9ß¼1, =�9ÍÞÎ��ç�ÄÝ�A=�ëF/ì$; ó×Y�, =�ëg/ÿÑ«�¬��×Í´?Ý�L¬��|�»A, 3WebsterÝCÎîbAìÝ�Õ:

continuity: the condition of being continuous, something that

continuous without a break�continuous: continuing without interruption�&Æ�QP°ãî��ÕÝ�=�Ý�¤�Ç¸3�ZCÎ

�, A¬È�ºh1986O�ÌÝ[�, b===��ÐÐÐóóó: ó.(Þ, ñÐó�¦��Í��ó¦�!`�y

ëï, Û�=�Ðó�t&¯�Bá¼=�ÐóÝ�¤, ÍJî��Õô�|Ý

��y“�”3[�îJã�Õ, �ÄTÎ“��”�E, ¬“��”ôã�Õ�[�3G��Õìê�|1�: »AbÐóy =

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∆x�yë`, ∆yù ë, Jf(x) =�Ðó�\ï��9ìÝ�L5.1f´, [�9ð1��¬��J, vb×ý0, �T&��:�¼�

1.5 Á§�§C=�P 43

|î�KÎ*^ðà��zZCÎîÝ�Õ, ?��1ëyOG,£`ó.�E=�ÝÃFùÎ�ÿWÝ�×àÕ�-1821O,

ÞÞÞ��(Cauchy, 1789-1857)�|ε − δÝ]°��Á§Ý�L�Q¡Ah��=�Ý�L, h�L ó.�X��, ¬;à�*�¬9�ÎBÄ×yõOÝ��Ý�

��LLL5.1.'×Ðóf3ØFab�L, v

(5.1) limx→a

f(x) = f(a),

JÌf3a===��uf3×/)A�N×F/=�, JÌf3A=��uf3Í�L½�N×F/=�, JÌf ×=�Ðó�

»»»5.3.'f(x) = 1/x, Jf3N×x 6= 0/=�, ¬f3x = 0�=�,

.0�3f��L½��'g(x) = 1, ∀x 6= 1,vg(1) = 3�J.limx→1 g(x) = 1 6= g(1),

Æg3x = 1�=��

×Ðóf3ØFa =�, µÎ3a�Á§�D3, vhÁ§ÂµÑ?Îf3a�Âf(a)�.h(5.1)Pê��y

(5.2) limx→a

|f(x)− f(a)| = 0�

&Æ�á

limx→a

x = a�

¿à�§5.1ÿ

limx→a

xn = limx→a

x · x · · · x = a · · · a = an�

#�ub×mg94P

f(x) = bmxm + bm−1xm−1 + · · ·+ b1x + b0,

44 Ï×a Á§

J¿à�§5.1�J�

limx→a

f(x) = f(a),∀a ∈ R,

Çf ×=�Ðó�êãyb§P(Ç5P) Þ94PÝ¤, Æ�§5.1�(iv)0l

��§§§5.4.N×b§P3Í�L½�, / =�Ðó�

?×�PÝ��Aì�

��§§§5.5.'fCgÞÐó3aF/=��Jf + g, f − gCfg/3aF=��v©�g(a) 6= 0, Jf/g3a Fù=��

��§§§5.1.'fCg3/)Aî/ =�Ðó, Jf + g, f − g, fg3Aî/=��ug(x) 6= 0, ∀x ∈ A, Jf/g3Aù =��

»»»5.4.'

f(x) =

{ √2− x, x < 2,

x− 2, x ≥ 2�

Jf3x > 2Cx < 2�/=��yx = 2, .

limx→2+

f(x) = limx→2−

f(x) = 0 = f(2),

Æf3x = 2ù=��Çf ×=�Ðó�

»»»5.5.'

f(x) = [x +1

2]− [x], x ∈ R,

Í�[ · ] t�JóÐó, &Æ0fÝ×I5%�Aì�

1.5 Á§�§C=�P 45

6

-12

1 32

2

1

Ox

f(x)

%5.1.

EN×�ón, f3n/2��¼Á§C�Á§×Í 0×Í 1, Þï��Çf3n/2ÝÁ§�D3, .hf3n/2��=��3Íõ2]J/=��

»»»5.6.'

f(x) =(x− 1)(x− 2)

x− 1,

f3x = 1^b�L, vtÝ3x = 1/=��¬.

limx→1

f(x) = limx→1

(x− 2) = −1,

Æu�f(1) = −1, Jf ×Õ�=��Ðó�

×Ðóf3ØFau�=�, vlimx→a f(x)D3, h`A�¥±�Lf3a�Â, -�¸f3a=�, A»5.6�9Ë�=�Ì ��ÉÉÉ��ÝÝÝ��===��(removable discontinuity)��y»5.5��=�F, &ÆµP°�v«Ý�§Ý, .limx→n/2 f(x)�D3�»5.5��=�F, Ì ®®®��ÝÝÝ��===��(jump discontinuity)�êãy3∀x = n/2, f��Á§D3v�yf(n/2), .h&Æ1f3n/2 ��===��(continuous from the right)�!§��L¼¼¼===��:�uf3ØFa, ÉÎ�=�ê ¼=�, Jf3a =��¨²,

uf(x) = 1/x2, x 6= 0, f(0) = 0,Jf30�=��ãylimx→0 f(x) =

∞, &ÆÌhË�=� PPP§§§ÝÝÝ��===��(infinite discontinuity)��y = f(x), &Æ�3x-y¿«î0�Í%��uf3ØF=�,

J3�FÍ%�Î=#��\Ý�A�f3Ø =�, J3

46 Ï×a Á§

� f�%�-K^b�\�&ÆÀ�×ìÐóÝ�=�ºbì��'f(x)3x = a�=�:

(i) limx→a f(x)D3, ¬��yf(a)�h�µ��Î.f3a^b�L,T4b�L¬f(a) 6= limx→a f(x)�GïAf(x) = (x−a)/(x−a), ¡ïA'a 6= 0, �f(x) = x2, x 6= a, f(a) = 0�9µÎG«X1Ý�É�Ý�=��u¥±�Lf(a)�¸f3a=��

(ii) limx→a+ f(x)�limx→a− f(x)/D3, ¬Þï��(iii) limx→a+ f(x)Tlimx→a− f(x)�D3, ¬f3aÝ!� b§

Â�Í;êÞt¡×Þ ×»�(iv)limx→a+ f(x)Tlimx→a− f(x) = ∞T−∞�Af(x) = 1/(x−

a)2�¡ëË�=�Ì ÍÍÍ²²²Ý(essential), .P°¥±�Lf(a)�Â

¸fW 3a=��.Ä)WÐó, BãÐóÝ)W, &Æ�ÿÕ&P&øÝÐ

ó�ì��§ ny)WÐóÝÁ§, ÍJ�º3êÞ�

��§§§5.6.'limx→a g(x) = b, vf3b=�, J

(5.3) limx→a

f(g(x)) = f(limx→a

g(x)) = f(b)�

»»»5.7.´��pJ�E∀a > 0, CÑJón,

(5.4) limx→a

n√

x = n√

a,

.hf(x) = n√

x3N×x > 0 =��¿àh��C�§5.6-ÿ

limx→2

√2x2 − 7x + 6

x2 − 3x + 2=

√limx→2

2x2 − 7x + 6

x2 − 3x + 2=√

1 = 1�Í�$b×°nyÁ§ P§�Ý��'

limx→a

f(x) = ∞ , limx→a

g(x) = c�J

limx→a

(f(x) + g(x)) = ∞,

limx→a

(f(x)− g(x)) = ∞�

1.5 Á§�§C=�P 47

ê

limx→a

f(x)g(x) =

{∞, uc > 0,

−∞, uc < 0,

c = 0JÚ�µ��(�� Ü×°»�¼D¡), &Æ|¡º�"D�¨²,

limx→a

f(x)/g(x) =

{∞, uc > 0,

−∞, uc < 0�

�b�K�¶Ý, A¼Á§��Á§�limx→∞ f(x) = a �, &Æ�××�Ü&Ë��9°K�pÿûTÑ;¨b�§ÝJ��ÿÕ�

Í;t¡&Æ¼:¿Í´©�ÝÁ§�

»»»5.8.'b×Ðó

f(x) =

{1/q, ux = p/q, p, q !²Jó, q > 0,

0, ux = 0 TP§ó�

�Jf3N×P§óC0 =��JJJ��.ãy3�×��Ýb§óp/qÝ�×Ï½�/bP§óD3,

�f3P§óÝÂ 0, �f3�b§óÝÂb×ü�Ý-²1/q, X|f3b§ó(tÝë�²)�=�Î��Ý�¨'0 < a < 1 ×P§ó, Jf(a) = 0�E3a��×Ï½�Ý

×P§óx, f(x) = 0 = f(a), Æ&Æ©ml�b§óÇ��&ÆÝÃÍ�°Î9øÝ: a P§ó, P°�îW5ó, A��

ÿÕ×|5ó�îÝ�«Â(Çh5ó�a�#�), h5óÝ5��5Ò/6��E×��ÝÑJóq, 5Ò≤ q�Ë5ó, ©bb§9Í(q(q −

1)/2Í), 9°ó�, ©b×Íûat�(��bÞ5ó�a�û,

ÍJaù b§ó)�hó�a�ûÒ|δ��

48 Ï×a Á§

3 (a− δ, a + δ)��×b§óx = p1/q1, .fG�5óK?#�a, .h¬&¸Æ�Ý×Í, Æq1 > q, v

f(x) =1

q1

<1

q�

¨�×ε > 0, ãq �y1/ε�t�ÑJó(Çu1/ε ÑJó,

Jq = 1/ε; ÍJq = [1/ε] + 1), Q¡µG�1�0�δ�JE�×(a − δ, a) ∪ (a, a + δ)�Ýb§óx, 'x = p1/q1, ãG�1�áq1 > q > 1/ε�.h

|f(x)− f(a)| = | 1q1

| = 1

q1

<1

q< ε�

Æµ�Lálimx→a

f(x) = f(a) = 0,

.hf3a=��yua 0T×�3(0,1) ÝP§ó, ûG�J°, �ÿf3a

ù=�, hI5&Æº�\ï��D¡�

31.3;&Æ�0�

(5.5) limn→∞

(1 +1

n)n = e,

Í�Á§Îº½ÑJón = 1, 2, · · · ãÝ�h��.ÂÕÐóÝ��

»»»5.9.�J

(5.6) limx→∞

(1 +1

x)x = e�

JJJ��.N×x ∈ R, ��

(5.7) [x] ≤ x < [x] + 1,

Í�[ · ] t�JóÐó�ã(5.7)Pÿ, E∀x > 0,

1 +1

[x] + 1< 1 +

1

x≤ 1 +

1

[x]�

1.5 Á§�§C=�P 49

ãhêÿ

(1 +1

[x] + 1)[x] < (1 +

1

x)[x] ≤ (1 +

1

x)x

≤ (1 +1

[x])x ≤ (1 +

1

[x])[x]+1�

ã(5.5)Pê�ÿ

limn→∞

(1 +1

n + 1)n = lim

n→∞(1 +

1

n + 1)n+1 · (1 +

1

n + 1)−1 = e,

Clim

n→∞(1 +

1

n)n+1 = lim

n→∞(1 +

1

n)n · (1 +

1

n) = e�

Ælim

x→∞(1 +

1

[x] + 1)[x] = lim

x→∞(1 +

1

[x])[x]+1 = e,

�ãô^�§ÇÿJ(5.6)P�

¿à(5.6)P, ô�Aì2J�

(5.8) limx→−∞

(1 +1

x)x = e�

�u = −x, J

limx→−∞

(1 +1

x)x = lim

u→∞(1− 1

u)−u = lim

u→∞(u− 1

u)−u

= limu→∞

(u

u− 1)−u = lim

u→∞(1 +

1

u− 1)u

= limu→∞

(1 +1

u− 1)u−1(1 +

1

u− 1) = e�

Ç�¡x →∞Tx → −∞, (1 + 1/x)x/��e�(5.6)P�¨×�� (J�º3êÞ)

(5.9) limx→0

(1 + x)1/x = e�

¯@îE∀x ∈ R,

(5.10) limn→∞

(1 +x

n)n = ex,

50 Ï×a Á§

�ÄÍJ�ø�êG&ÆX.Ý, .hº3Ï5.3;�J��¨²,

Ç¸Î(5.6)P, &Æô�'��E¼ób�5ÝÝ�, ��º²¶(1 + 1/x)xÝ�L�3Ï"a&Æº¥±E¼ó��ÛÝ�L�

ëëë��ÐÐÐóóó3��5�6�½¥�Ý��3��5�, ;ð��Îã©©©��(radian measure), �×iø�� 2π�&Æ�:Þ��Ý��:

limx→0

sin x = 0,(5.11)

limx→0

cos x = 1�(5.12)

(5.11)P�¿ày = sin x �Ðó, v(�%5.2)

(5.13) 0 < sin x < x, ∀0 < x <π

2,

�àô^�§Çÿ��y(5.12)P, �¿à(5.13)P, ÿì��P

0 < 1− cos x ≤ 1− cos2 x = sin2 x < x2,∀0 < x <π

2,

�àô^�§, Cy = cos x �Ðó-�ÿÕ�ã(5.11)C(5.12)PÇÿf(x) = sin x, Cg(x) = cos x3x = 0/=��¿à

(5.14) sin(x + h) = sin x cos h + cos x sin h,

C

(5.15) cos(x + h) = cos x cos h− sin x sin h

Þ2P, 5½�h → 0, Çÿf(x) = sin xCg(x) = cos x/ =�Ðó��ytan x = sin x/ cos x, 3cos x 6= 0�, Çx 6= ±π/2,±3π/2,

· · · , /=��Í�ë�Ðó, Acot x, sec x, csc x��=��, ô��¶��9ì&ÆJ��5�, nyë�Ðó×¥�ÝÁ§��

1.5 Á§�§C=�P 51

��§§§5.7.

(5.16) limx→0

sin x

x= 1�

JJJ��.E0 < x < π/2, bì�%��

O A

B

C

D

x

x

sin xtan x

%5.2.

A%5.2b×�5 1�G�, ÇOA = OB = 1, ∠AOB = x,CA

⊥OA� ã0 < x < π/2�Bãf´4OAB, G�OABCà�ë��OACÝ«�, �ÿì��P

1

2sin x <

1

2x <

1

2tan x�

ãhêÿ

cos x <sin x

x< 1,∀0 < x <

π

2�ãylimx→0 cos x = 1, Æãô^�§ÿ

limx→0+

sin x

x= 1�

�y3x = 0�¼Á§, ¿àsin(−x) = − sin x, ux < 0, J−x > 0,

vsin x/x = sin(−x)/(−x), .h

limx→0−

sin x

x= lim

x→0+

sin x

x= 1�

ÆÿJ(5.16)PWñ�

¿à�§5.7, Çÿ

limx→0

tan x

x= lim

x→0

sin x

x

1

cos x= lim

x→0

sin x

xlimx→0

1

cos x= 1

52 Ï×a Á§

(¥��|tWÞÁ§Ý�, Î.ÞÁ§/D3);

limx→0

sin 2x

sin x= lim

x→0

sin 2x

2x

x

sin x· 2

= limx→0

sin 2x

2xlimx→0

1

sin x/x· 2 = 1 · 1 · 2 = 2;

limx→0

1− cos x

x= lim

x→0

1− cos2 x

x(1 + cos x)= lim

x→0

sin2 x

x(1 + cos x)

= limx→0

sin x

x

1

1 + cos x· sin x = 1 · 1

2· 0 = 0�

t¡, ìPô�pÿÕ(J�º3êÞ):

(5.17) limx→0

1− cos x

x2=

1

2�

(5.16)P�Á§Î��(indeterminate form)Ý×Ë�.x →0 `, sin x/x�5��5Ò/��0, .hb0/0Ý�P, P°ñÇÿáÁ§Â ¢�9vÁ§&Æ|¡º��á"D�

êêê ÞÞÞ 1.5

1. SàÊÝ�§C�áÝ��Oì�Á§�(i) limx→a(x

5 + 3x2 + 2x + 1), (ii) limx→1+

√x2−1x2+1

,

(iii) limx→1

√x(x+2)x+1

, (iv) limx→0

√x3+1−1

x2 �

2. Oì�Á§�(i) limt→1

1−t3

2−√t2+3,

(ii) limt→2

√1+√

2+t−√3

t−2,

(iii) limy→∞(√

1 + y −√y),

(iv) limn→−∞( 3√

n3 + n− 3√

n3 + 1),

(v) limm→0

√4+m+m2−2√4+m−m2−2

,

(vi) limx→0x√

9−x+x3−3,

êÞ 53

(vii) limh→0

√1+

3√h+2

h−27,

(viii) limm→0

3√8+m2+m3− 3√8+m3√8+m− 3√8+m2−m3

,

(ix) limx→0

5√1+x5− 5√1+x2

3√1−x3− 3√1−x2,

(x) limt→0

5√1+t5− 3√1+t3

t3 �3. Oì�&Á§�

(i) limx→0sin ax

x, (ii) limx→0

tan 2xsin x

,

(iii) limx→0sin axsin bx

, (iv) limx→0sin 5x−sin 3x

x,

(v) limx→asin x−sin a

x−a, (vi) limx→0

1−cos 2xx2 ,

(vii) limx→0cos 3x−cos x

x2 , (viii) limx→0sin 4x−sin x

sin 2x,

(ix) limx→0x sin x1−cos x

, (x) limh→0cos(x+h)−cos x

h,

(xi) limx→0tan x−sin x

x2 , (xii) limx→0(1

sin x− 1

tan x)�

4. 'f , g5½ mCng94P, f(x) = amxm + · · · + a0, g(x) =

bnxn + · · ·+ b0�O

(i) limx→0f(x)g(x)

, (ii) limx→∞f(x)g(x)

, (iii) limx→−∞f(x)g(x)�

5. �J(5.9)P�

6. Olimx→0(1 + x2)1/x2Climx→0(1 + sin x)1/ sin x�

7. 'acdf 6= 0, O

limx→∞

( limy→∞

ax2 + bxy + cy2

dx2 + exy + fy2)− lim

y→∞( limx→∞

ax2 + bxy + cy2

dx2 + exy + fy2)�

8. �Jì�&Á§�D3�(i) limx→−1

x+2x+1

, (ii) limx→−2|x+2|x+2

,

(iii) limy→0|y|−y

y, (iv) limy→1

|y−1|−y+1|y−1|+y−1

,

(v) limx→0|x|−x|x|3−x3 , (vi) limx→2(x− [x]), [ · ] t�JóÐó�

9. 'limx→∞ f(x) = b, �µ�LJ�limx→0+ f(1/x) = b�

10. (i) 'limx→a f(x) = b, �µ�LJ�limx→a |f(x)| = |b|;(ii) 'f ×=�Ðó, µ�LJ�|f |ù ×=�Ðó�

54 Ï×a Á§

11. ulimx→a |f(x)| = b,�®limx→a f(x)ÎÍÄD3v�ybT−b?

12. ulimx→a f(x)D3, �JÍÁ§ÂÄ°×�

13. '

f(x) =

{sin x, x ≤ c,

ax + b, x > c,

Í�a, b, c ðó�ub, c �á, �X�a�Â, ¸f ×=�Ðó�

14. Eì�Ðó¥�îÞ�

f(x) =

{2 cos x, x ≤ c,

ax2 + b, x > c�

15. 'f(x) = x sin(1/x), x 6= 0, vf(0) = 1��Jf3x = 0 ×�É��=�, ¬¥±�Lf(0)¸f3x = 0=��

16. 'f(x) = tan x/x, x 6= 0�®ÎÍ��Lf(0)�Â, ¸ÿf3x = 0=��

17. D¡ì�Ðó�=�P�(i) f(x) = |x|

x, (ii) f(x) = x+a

x−a,

(iii) f(x) = |x2+1|x+1

, (iv) f(x) = x sin(1/x), x 6= 0, f(0) = 0�18. 'f��L½ [0,∞), v

f(x) =

{p sin(1/q), ux = p/q, (p, q) = 1, q ≥ 1,

x, ux P§óT0�

�®f3x = 0ÎÍ=�?

19. '

f(x) =

{1, ux b§ó,

x, ux P§ó�D¡f�=�P�

1.6 =�P�×M"D 55

20. �f(x) = limn→∞ xn, x ∈ [0, 1]�D¡f�=�P, ¬0f�%��

21. �f(x) = [x] +√

x− [x]�D¡f�=�P, ¬0f�%��

22. 'f(x) = [1/x], x 6= 0, Í�[ · ] t�JóÐó��0f3 [−2,−1/5]�%�, ¬¼�Í�=��

23. 5½Ef1(x) = (−1)[1/x], x 6= 0, Cf2(x) = x(−1)[1/x], x 6= 0,

¥�îÞ�

24. �J�§5.1�

25. �J�§5.2�

26. �J�§5.3�

27. �J�§5.6�

28. �0

f(x) = sin1

x, x 6= 0,

�%��ÎÍ��Lf(0), ¸ÿf3x = 0=�? êf�%��g(x) = sin x�%�b¢8«C8²�?

29. 'an =∑n

i=1 9/10i, n ≥ 1��O

(i) limn→∞[an], (ii) [limn→∞ an],

Í�[ · ] t�JóÐó�

1.6 ===��PPP��×××MMM"""DDD

�h�T&�EÁ§C=��b×°�MÝÝ��&Æá¼3�Ã{Æ`�,µ�BãÁ§ÝÄ�, Õ��K%�Ý«��`a�Cñ�Ý��pppñññ`�EÁ§4^b�ÛÝÃF, ¬ôs"��

56 Ï×a Á§

K§¡��BÄ×yO|îÝ��ýýý000(trial and error), ��|×8-Ý]P, ©à¿�à��Á§Ý�L��5C�5ôK�|Á§ÝÄ�¼�L��.R��.êó.ÝJ�, LÍÎ¿¢.Ý®Þ, �

��'êá¼�OJÝ¯�¡, ¶Iµ°, b`��à�Ah°°�MÜ°�, t¡�ÌÿJ�BÄ0OÝ�.IY, &ÆâyEó.îÝ�ì.�b´züÝÝ��¿«¿¢.3¨�ó.�Ý�4¬��, ¬¸Eèº�ì]«ÝIY, ��×àÎ��.h3��5��#åÁ§, E9BÄ�` C9ß�ÉÝ�þ,

�s"�Ah��Ýε − δÝÁ§�L, ��3y` /��#å,

#�ºà�A, �&|¯�b�KÁ§ÂÎ×¿µ�:�Ý, 9]«ÝàÌQ��´

��¬ÎÂÕ´Þ´Ý�µ, ©?È�ε − δÝ°��Á§ÝÌFu��z, �¬?¡Ý.ê´�|, ô¸� ó.Ý�æè>�¨×·g��y=�,Aî;X�&Æ\bhÃF, A=�Ý�óx ∈

[0, 1],b½yx = 1, 2, 3, · · ·�ãy��5�X�§ÝÐó, ðÎ=�Ðó, TÎ@@@ððð===��ÐÐÐóóó(piecewise continuous function), Af(x)

= [x], X|E=�Ðó�ÿ�¨¿8:�!ñ×è, Á§C=�4KÎ@@@FFFÝÝÝ(pointwise)P², ôµÎ×F×F:ÐóÎÍÁ§D3, ÎÍ=��¬3D¡ØÐó3ØFÝÁ§T=�P`, -6��Ðó3£×F!�ÝÂ�»A, u©�f(0) = 0, JP°ÿálimx→0 f(x) ¢?

3�@ÍaG&Æ��¿Í=�ÐóÝ¥��´�, G«èÄ&ÆðºÂÕ)WÐó, ì��§¼�=�PBÄ)WÝºÕ)º1¹�

��§§§6.1.'f = u◦v�ÞÐóu, v�)WÐó�uv3a=�, u3q =

v(a)=�, Jf3a=��JJJ��..u3q=�, Æ∀ε1 > 0, D3×δ1 > 0, ¸ÿ

(6.1) |u(y)− u(q)| < ε1, ∀|y − q| < δ1�

1.6 =�P�×M"D 57

êv3a=�, vq = v(a), ÆEε2 = δ1 > 0, D3×δ2 > 0, ¸ÿ

(6.2) |v(x)− q| < ε2, ∀|x− a| < δ2�u�y = v(x), Jã(6.1)C(6.2)Pÿ, E∀|x − a| < δ2, -b|u(y) −u(q)| < ε1, Ç|u(v(x)) − u(v(a))| < ε1, �hÇ |f(x) − f(a)| <

ε1�Æµ�Láf3a=��

»»»6.1.'f(x) = sin(3x2 + 5x + 2)�Jf = u ◦ v, Í�u(x) = sin x,

v(x) = 3x2 + 5x + 2, / =�Ðó, Æf ×Õ�=��Ðó�

»»»6.2.'f(x) =√

1− sin2 x�Jf = u ◦ v, Í�u =√

x, v = 1 −sin2 x�v ×Õ�=��Ðó, �u©3x ≥ 0=��.v ≥ 0Wñ, Æf ×Õ�=��Ðó�u; �Êf1(x) =

√1− x2, Jf13

x2 ≤ 1�=��

ny=�Pb�K©�ÝP², àÌîKÎ�QWñÝ, 9ì&ÆB�¬J�¿Í´ÃÍÝP²�Ï×Í�+ÛÝÎBolzano��§§§(Bolzano theorem)�Bolzano (1781-1848) ×�ßy$¸�Fx>ßl, �ÕÎ�\�ºÕ&9ny=�ÐóÝB�, 4:R¼��, ¬u��?Â½2Tà, -6�J�Ý.ï�×�3èÜtSGf, �Eó.b�KQ¤, LÍÎÞ¨��ÛÝÌF, Só.5��´�&Æ�×S§�

SSS§§§6.1.===��ÐÐÐóóó��ÐÐÐrrr111¹¹¹PPP²²²(sign-preserving property of con-

tinuous function). 'f3c=�, vf(c) 6= 0�JD3×δ > 0, ¸ÿf(x)�f(c)�Ðr8!, ∀x ∈ (c− δ, c + δ)�JJJ��.'f(c) > 0�ã=�Ý�Lá, ∀ε > 0, D3×δ > 0, ¸ÿ

(6.3) f(c)− ε < f(x) < f(c) + ε, ∀x ∈ (c− δ, c + δ)�uãε = f(c)/2 > 0, J(6.3)PW

1

2f(c) < f(x) <

3

2f(c), ∀x ∈ (c− δ, c + δ)�

58 Ï×a Á§

.hf(x)�f(c)×ø/ Ñ, ∀x ∈ (c− δ, c + δ)�uf(c) < 0, ãε = −f(c)/2Ç�, ÿJ�

ÛÛÛ.uf3cG �=�(T¼=�), JîS§�; D3×δ > 0, ¸ÿf(x)�f(c)!r, ∀x ∈ [c, c + δ) (Tx ∈ (c− δ, c])�

¿àhS§C@ó�ÙÝt�î&2§(��§1.1�Û), �J��§6.2, C½(Ý�q�§�

��§§§6.2.(Bolzano��§§§). 'Ðóf3T [a, b]îN×F/=�,

vf(a)�f(b) Ðr8D�JD3×c ∈ (a, b), ¸ÿf(c) = 0�JJJ��.'f(a) < 0vf(b) > 0�3(a, b)�1��b&9x�¸f(x) =

0, &Æ©��0�×Í-ÿJÝ��&Æ�0ÝÎt�Ý£×Í��S = {x|a ≤ x ≤ bvf(x) ≤ 0}�S¬&è/), h.f(a) <

0, Æa ∈ S��∀x ∈ S, x < b, .hb S�×î&, Æãt�î&2§ác = sup SD3�&Æ�J�cÇ XO, Çf(c) = 0�

f(c)©bëË��: f(c) > 0, f(c) < 0 Cf(c) = 0�uf(c) > 0,

JãS§6.1á,D3 (c−δ, c+δ)(uc = bJ (c−δ, c]),¸ÿf3h �/ Ñ�Æ3S�Ý-ô/��òy(c − δ, c + δ)�,

ÇS�Ý-ô/�yT�yc− δ, .hc− δù S�×î&��c−δ < c, h�c S�t�î&�)�Æf(c) > 0��Íg,uf(c) < 0,JD3 (c− δ, c+ δ)(uc = aJ [c, c+ δ)),

¸ÿf3h ��Æºb×x > c, ¸ÿf(x) < 0, .h9Íx ∈S, �9ê�c S�t�î&�)�Æf(c) < 0ù��yì°×��©bf(c) = 0Ý�êa < c < b, h.�áf(a) <

0, f(b) > 0��yuf(a) > 0vf(b) < 0, !§�J�

Bolzano�§ÇÎ1, Í%��x�Ý×Ð�Õ¨×Ð, Ä�ºUÄx��àÌî:ÎEÝ, Ðóu�=�, Q�|®Äx�, =�ÝÉ�J��(}¡Ý�§6.3ôÎÃy9Ëæ§)�9ÎBolzano3

1.6 =�P�×M"D 59

�-1817Os�Ý��,&Æ|ì%¼1��&�3�.�X.ÄÝ0]�PÝq, bXÛ�q�§, ¯@îÇ Bolzano�§�

a b

%6.1.

ãBolzano�§, ñÇ�ÿ9ìÝ=�ÐóÝ�� ÂÂÂ��§§§(Inter-

mediate-value theorem)�

��§§§6.3.'f3[a, b]=�, v'D3x1, x2 ∈ [a, b], x1 < x2, ¸ÿf(x1)

6= f(x2)�JEN×+yf(x1)�f(x2) �y, D3×c ∈ (x1, x2), ¸ÿf(c) = y�JJJ��.�´×�P'f(x1) < f(x2)�JE�×y ∈ (f(x1), f(x2)),

�g ×�L3[x1, x2]�Ðó, v

g(x) = f(x)− y�

Jg3[x1, x2]=�, vg(x1) = f(x1) − y < 0, g(x2) = f(x2) − y >

0�Æ¿àBolzano�§, D3×c ∈ (x1, x2), ¸ÿg(c) = 0, ¬hÇf(c) = y, ÿJ�

3îÞ�§�, &Æ/�Of3 ÝÐF6=�, uf3ÐF�=�, J��-�×�EÝ�»A, ã[a, b] = [0, 1], vf(x) =

1,∀x ∈ (0, 1], f(0) = −1�J�:�îÞ�§Ý��/�Wñ�¿à� Â�§�J�×&Æ3@ó�!áÝ��

��§§§6.1.'n ×ÑJó, JEN×a > 0, ]�Pxn = aªb×Ñq�

60 Ï×a Á§

JJJ��.ã×c > 1��0 < a < c, v�

f(x) = xn, x ∈ [0, c]�

Jf3[0, c]=�, f(0) = 0, f(c) = cn�.0 < a < c < cn, Ça+yf(0)�f(c)� �Æã�§6.3á, D3×b ∈ (0, c), ¸ÿf(b) =

a�9-J�ÝD3P�ê.f3[0, c] ×�}�¦Ðó, ��b¨×x, ¸ÿf(x) = xn = a�J±�

ó.�bXÛüüü��FFF��§§§(Fixed-point theorem), 9ìÝ�§ Í×©½Ý�µ�J�¬�p, ©�¿àBolzano�§Ç�, &Æº3êÞ�

��§§§6.4.'f ×�L3[a, b]î�=�Ðó, a < b, v'a ≤ f(x) ≤b,∀x ∈ [a, b]�JD3×c ∈ [a, b]¸ÿf(c) = c�

3��5�, OÁÁÁÂÂÂ(extreme value) ×¥�Ý�Þ, =�Ðó39]«ôb×°��ÂÿD¡�'f ×�L3@óî×/)S�@ÂÐó�uD3×c ∈ S, ¸

ÿ

f(x) ≤ f(c), ∀x ∈ S,

JÌf3Sîb��EEEÁÁÁ��(absolute maximum)�f(c)JÌ f3Sî��EÁ�Â�uD3×d ∈ S, ¸ÿ

f(x) ≥ f(d), ∀x ∈ S,

JÌf3Sîb��EEEÁÁÁ��(absolute minimum)�uf3cb�EÁ�, Jf3S��%�, t{Fsß3x = c, v

{� f(c)�!§�1��EÁ��»A, 'f(x) = sin x, S =

[0, π]�Jf3x = π/2b�EÁ�, 3x = 0Cπb�EÁ��êuf(x) = 1/x, S = (0, 1]�Jf3S��EÁ�sß3x = 1,¬f3S

îP�EÁ��Q�:�f3x = 0�=��

1.6 =�P�×M"D 61

&Æ��J�ÝÎ, uf3×T �=�, Jf3h , ÄÉb�EÁ�vb�EÁ��9µÎ8¥�Ý===��ÐÐÐóóóÝÝÝÁÁÁÂÂÂ��§§§(Extreme-value theorem for continuous function)��Ä&Æ�m�ì��§�

��§§§6.5.(===��ÐÐÐóóó��bbb&&&��§§§, Boundedness theorem for continu-

ous function). 'f3T [a, b]î=�, Jf3[a, b]îb&�JJJ��.&Æ�¿à===��ÞÞÞ555°°°(successive bisection), |DJ°¼J��'f3[a, b]� b&��c [a, b]��F�JfÄ3[a, c]�T

[c, b]�� b&(b��3Þ� /�b&)�'[a1, b1] ¸f�b&Ý£×Í� (ubËÍJó¼\£×Í)�¥�hM»,Ngu3Þ� f/�b&, Jó¼\£×Í, v|[an+1, bn+1]��[an, bn]XóÝº¸f�b&Ý£×Í� ��Q[an, bn]�� (b− a)/2n��A = {a, a1, a2, · · · }, ãyA ⊂ [a, b] ×b&/), ÆA�î

@&D3��α = sup A, Jα ∈ [a, b]�ãyf3α=�, ÆD3×δ > 0, ¸ÿ(Çãε = 1)E∀x ∈ S = (α− δ, α + δ),

(6.4) |f(x)− f(α)| < 1�

¬uα = a,J S�; [a, a+δ);uα = b,JS�; (b−δ, b]�(6.4)Pê0l

(6.5) |f(x)| < 1 + |f(α)|, ∀x ∈ S�

Æf3S�|1+ |f(α)| Í×î&�ãy[am, bm] ⊂ [an, bn],∀m ≥ n,

Æ|�α ∈ [an, bn],∀n ≥ 1�ÆunÈ�, ¸ÿ[an, bn]Ý��(b −a)/2n < δ,J[an, bn] ⊂ (α−δ, α+δ) = S�Æã(6.5)á, f3[an, bn]�b&�h��G�'f3[an, bn]� b&ë;�hë;0�f3[a, b]îb&�J±�

uÐóf3[a, b]b&, J/){f(x)|a ≤ x ≤ b}bî&vbì&,

62 Ï×a Á§

.h�/)bî@&Cì@&, 5½|sup fCinf f��Ç

sup f = sup{f(x)|a ≤ x ≤ b}, inf f = inf{f(x)|a ≤ x ≤ b}�

E×b&Ðóf , -binf f ≤ f(x) ≤ sup f, ∀x ∈ [a, b]�9ìÝ�§¼�[a, b]î�×=�Ðóf , Ä3[a, b]�ºãÂinf fCsup f�

��§§§6.6.(===��ÐÐÐóóó��ÁÁÁÂÂÂ��§§§). 'f3[a, b]î=�, a < b�JD3c, d ∈ [a, b], ¸ÿ

f(c) = sup f v f(d) = inf f�

JJJ��.&Æ©�J�D3c ∈ [a, b], ¸ÿf(c) = sup fÇ��Q¡¿àinf f = sup(−f),-�ÿÕôD3×d ∈ [a, b],¸ÿf(d) = inf f��M = sup f�'�D3×c ∈ [a, b], ¸ÿf(c) = sup f , &Æ�

0�ë;��g(x) = M − f(x)�Jg(x) > 0, ∀x ∈ [a, b]�.h1/gù3[a, b]

=�(¿à�§5.1)�Jã�§6.5, 1/g3[a, b]b&, '1/g(x) < K,

∀x ∈ [a, b], Í�K > 0�ãhêÿ

M − f(x) > 1/K, ∀x ∈ [a, b],

Æ

f(x) < M − 1/K, ∀x ∈ [a, b],

�h�M f3[a, b]�t�î&ë;�ÇD3×c ∈ [a, b], ¸ÿf(c)

= M = sup f�J±�

î��§¼�, uf3[a, b]=�, Jsup fCinf f5½ �EÁ�C�EÁ��ã� Â�§(Ç�§6.3)á, f3[a, b]î�Â½Ç [inf f, sup f ]�Í�$b=�P, 3ãDÐó`)º1¹, &Æ©B�Aì, J

�¬�p, ©�iÍ%¼:-��¡�

1.6 =�P�×M"D 63

��§§§6.7.'f3T [a, b]=�v �}�¦��c = f(a), d =

f(b), v�g f�DÐó, ÇE∀y ∈ [c, d], g(y) = x, Í�x ∈ [a, b]��y = f(x)�J

(i) g3[c, d]�}�¦;

(ii) g3[c, d]=��

t¡,&Æ+Ûííí888===��PPP(uniform continuity), TÌ×××lll===��3|ε − δ¼�L=�`, kl�Ðóf3a=�, E∀ε > 0, &Æ

60×δ > 0, ¸ÿ|f(x)− f(a)| < ε, ∀|x− a| < δ�E×ε > 0, &Æá¼δÝã°¬�°×, ×Ë0Õ×Íδ, f�δ�ÝÑó/Êà��v&Æô�×��µ§×δ�øÄØÂ, .©�0ÕÇ�, ��ÝÑóÎ�b��Êà�ð�1, f3a=�, Î×ËIIIÝÝÝPPP²²²(local property), ©�f3a!�×Ï½(�¡9�)bn�!ñ×è, &Æô©m�Ê�5�ÝεÇ�, .Ehε, u�0Õδ, J�δ)�ÊàfG�ε��ÝεÂ�¨²,δÝóã�¬�εbn, �aôbn��|�,uf3a!

�Ý%�´¿c, Jδ��°; uf3a!�Ý%�´q, Jδµ��°�u3Ø �, δ©�εbn, ��aPn, &Æµ1f ííí888===��(uniformly continuous)�ôµÎu∀ε > 0, D3×δ > 0, ¸ÿE I�Ý�Þx, y, u��|x− y| < ε, -/¸|f(x)− f(y)| < ε, JÌf3I�í8=��.h, &Æ�|1f3ØF=�, ¬Ìf3ØFí8=�QÎ^b�LÝ, Ä6Î3× �ºD¡fÎÍºí8=��uf ×í8=�Ðó, �y = f(x), JEÞx, ©�È#�, X

ETÝyÂ��#�, ��ÞxÝ�HPn�&ÆÜ¿Í»�¼:�

»»»6.3.�f(x) = 2x + 3, I = R�J.E∀ε > 0

|f(x)− f(a)| = 2|x− a| < ε, ©�|x− a| < δ = ε/2,

Æf3Rî í8=��

64 Ï×a Á§

»»»6.4.f(x) = x2, I = [0, 1]�J.E∀x, a ∈ I,

|f(x)− f(a)| = |x2 − a2| = |x + a||x− a| ≤ 2|x− a| < ε,

©�|x− a| < δ < ε/2, Æf3Ií8=��¯@îuÞIðW�×b§Ý , f) í8=��Äf3RîQ&í8=��J�Aì�'f3Rîí8=��

�ε = 1, �'�0Õ×δ > 0, ��í8=�Ýf��¨ã

a =1

δ, x =

1

δ+

δ

2,

J|x− a| = δ/2 < δ, ¬

|f(x)− f(a)| = |x + a||x− a| > 2

δ· δ

2= 1,

�)�hë;0�f3Rî� í8=��

î»×å&Æ, ×ÐóÎÍí8=�, b`�XãÝ bn�

»»»6.5.'f(x) = 1/x, x > 0, I = (0, 1], Jf3Iî=�, ¬� í8=��J�Aì�ãε = 10, v'�0Õ×0 < δ < 1( ¢�Ah§×δÝP�?),

��í8=�Ýf��¨ãx = δ, a = δ/11,J|x−a| = 10δ/11 < δ,

¬|f(x)− f(a)| = 10

δ> 10�

h�í8=�Ý�'�)�A�0f(x) = 1/x, x > 0, �%�, �:�f3x�#�0`,�;

&ð", Ä6§×xÝ�;��1JfÝ�;��¬3x =

1!�, %�µ�¿c, Ç¸xÒ1}G°, fÝ�;¬��9Îf� í8=�Ýæ.��Ä&Æbì��

��§§§6.8.'Ðóf3×T I =�, Jf3I í8=��

Sà�§6.8`, tÝÐó =�, ô�TÝ��X|, Af(x) = 1/x, I = (0, 1]µ��¬E∀0 < a < 1, f3I = [a, 1] í

1.6 =�P�×M"D 65

8=��Qb°Ðó3&T ô��í8=�, �»6.4��§6.8�J�màÕbnËËË��(covering)Ý×ÃÍ�§�&ÆW�vJ�yì�hS§��ÕXÛååå[[[PPP(compactness), �¢�×�/)¡T{��5Ýh(AApostol (1974))�3h'b×°� XxWÝ×/)C, C× I�E∀x ∈ I, uxÄòyC�Ø× ,-ÌC I�×Ë��»A,'C={(−1, 0), (−1

2, 1), (1

3, 2), (3

2, 3)},

JC [0, 2]�×Ë��

SSS§§§6.2.'C ×°� XxWÝ/), v T [a, b]�×Ë��JÄD3CÝ×b§�/C

′ù [a, b]�×Ë��

JJJ��.�

A = {x|x ∈ [a, b], v[a, x]� C�×b§�/XË�},

|�a ∈ A, ÆA 6= ∅�ê�

c = sup A�

�:�c ≤ b�ê.C�ÝN×-ô/ � , ÆD3×(e, f) ∈C, ¸ÿc ∈ (e, f)�ãyc = sup A, ÆÄD3x0 ∈ A, ¸ÿx0 ∈(e, f)�ÍJc 6= sup A�.x0 ∈ A,µA��L,D3C�×b§�/,Ì� B, [a, x0]�

×Ë��Ah×¼C′= B ∪ {(e, f)}

[a, c]�×bbb§§§ËËË��, Æcù3A��9ì&Æ¼:¯@îc = b, .hC

′ [a, b]�×b§Ë�, ÍS§-ÿJÝ�G«�¼�c ≤ b,uc < b,v'd ∈ (c, b)∩(e, f)�J.C

′ [a, d]

×Ë�, Æd ∈ A, ¬hQ�c = sup Aë;�Æc < b�Wñ, Çc =

b�J±�

3îS§�, C-Ì [a, b]Ý×��ËËË��(open covering)�hS§¼�, E×T Ý×�Ë�, ÄD3×b§Ý�Ë��

66 Ï×a Á§

¨3&Æ��J��§6.8��×ε > 0, .f3I=�, Æ∀t ∈ I, D3×rt > 0, ¸ÿf(t) −

ε/2 < f(x) < f(t) + ε/2, ∀x ∈ (t − rt, t + rt) ∩ I��Ut = (t −rt/2, t + rt/2)�J{Ut, t ∈ I} I�×�Ë��.hãS§6.2, D3×b§Ý�Ë�, |

(t1 − r1

2, t1 +

r1

2), (t2 − r2

2, t2 +

r2

2), · · · , (tn − rn

2, tn +

rn

2)

��ãδ = min{r1/2, r2/2, · · · , rn/2}�&ÆÞJ�hδÐ)í8=��ÝmO�E�×c ∈ I,JcòyØ×(ti−ri/2, ti +ri/2), i = 1, · · · , n�ux

∈ (c − δ, c + δ), .δ ≤ ri/2, Æx�c/òy(ti − ri, ti + ri)��hÇ�f(x)�f(c)/3(f(ti)− ε/2, f(ti) + ε/2)��Æ|f(x)− f(c)| <ε�ùÇux ∈ (c− δ, c + δ), -b|f(x)− f(c)| < ε�ãyhf�xPn,Æµ�Lf3Iîí8=��J±�

êêê ÞÞÞ 1.6

1. '

f(x) =x + |x|

2, x ∈ R, g(x) =

{x, x < 0,

x2, x ≥ 0�Oh(x) = f(g(x)), ¬¼�h�=��

2. '

f(x) =

{1, |x| ≤ 1,

0, |x| > 1,g(x) =

{2− x2, |x| ≤ 2,

2, |x| > 2�Oh1(x) = f(g(x))Ch2(x) = g(f(x)), ¬5½¼�h1Ch2�=��

3. 'f(x) =∑n

k=0 ckxk ×ng94P��J

(i) uc0cn < 0, �Jf(x) = 0�Kb×Ñq;

(ii) un �ó, �Jf(x) = 0�Kb×@q�

êÞ 67

4. �¿àBolzano�§, 5½X�ì�&]�PqÝP��(i) 3x4 − 2x3 − 36x2 + 36x− 8 = 0 ;

(ii) 2x4 − 14x2 + 14x− 1 = 0 ;

(iii) x4 + 4x3 + x2 − 6x + 2 = 0�

5. 'n ×Ñ�ó, a < 0��J]�Pxn = aªb×�q�

6. 'f(x) = tan x��® ¢4f(π/4) = 1vf(3π/4) = −1, ¬3 [π/4, 3π/4]�, QP°0Õ×x¸ÿf(x) = 0�

7. �J�§6.4�

8. 'f ×3[a, b]î�=�Ðó, a < b, vf(a) ≤ a, f(b) ≥b��JD3×c ∈ [a, b], ¸ÿf(c) = c�

9. 'f ×=�Ðó, vf(f(x)) = x, ∀x ∈ R��JD3×x ∈R, ¸ÿf(x) = x�

10. (i) 'Ðóf(x)3x = 0=�, vf(x + y) = f(x)f(y), ∀x, y ∈R��Jf3Rî=��(ii)'Ðóf(x)3x = 0=�, vf(x + y) = f(x) + f(y), ∀x, y ∈R��Jf3Rî=��

11. 'f3Rî=�, vE�×b§óx, f(x) = 1��Jf(x) = 1,

∀x ∈ R�

12. 'f3[0, 1]=�, vf©ãb§óÂ�uf(1) = 1, �Jf(x) =

1, ∀x ∈ [0, 1]�

13. �J3�§6.7�, u=�Ý�'; b&, J��×�Wñ�

14. u|f |3/)S�bî&, �Jf3S�b&�

15. 'f3[a, b]=�, ¿àS§6.2, �Jf3[a, b] b&�

68 Ï×a Á§

16. 'f�L3[0, 1]î, v

f(x) =

{x, ux b§ó,

1− x, ux P§ó��J

(i) f©3x = 12=�;

(ii) fÝÂ½ [0, 1]�

17. 'f3[a, b]=��J(i) D3×c ∈ [a, b], ¸ÿf(c) = 1

2(f(a) + f(b));

(ii) E∀0 ≤ λ ≤ 1, D3d ∈ [a, b], ¸ÿf(d) = λf(a) + (1 −λ)f(b)�

18. �Jì�Ðó/ í8=��(i) f(x) = axn, I = [c, d], c < d;

(ii) f(x) =√

x, I = [1, 5];

(iii) f(x) = x2 − x, I = [−1, 1];

(iv) f(x) = 2/x, I = [1,∞)�

19. �Jì�Ðó/� í8=��(i) f(x) = x+1

x−1, I = (1, 2);

(ii) f(x) = x3, I = (0,∞)�

20. �l�ì�Ðó¢ï í8=�, ¢ï�Î�(i) f(x) =

√x, I = [0, 1];

(ii) f(x) = x3+2x+1

, I = [0,∞);

(iii) f(x) =√

1− x2, I = [0, 1];

(iv) f(x) = 3√

1− x2, I = [−1, 1];

(v) f(x) = 1x2−3x+2

, I = (1, 2];

(vi) f(x) = 1x2−3x+2

, I = [1.01, 1.99]�

êÞ 69

21. 'f, g/3Rîí8=��(i) �Jf + gù3Rîí8=�;

(ii) �®fg3RîÎÍí8=�? J�TÍJ��

22. 'Ðóf3 Iîí8=�, �Jf3Iîù=�vb&�

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1974).Mathematical Analysis, 2nd ed. Addison-

Wesley, Reading, Massachusetts.

70 Ï×a Á§

ÏÏÏÞÞÞaaa

��555��555ÝÝÝ��+++

2.1 GGG��

�5��5, ��5�ÝËÍÃÍÝÞC��Ëû9OR, -ëëÏÏ2, bny�5��5Ý"D, �3�Ã{Æ`�/Í�W��¡êcXì¼, àÕè0�èÚtS, ó.��ê��#Ty�5��5Ý"D�¬�5ËÑb�ÙÝs", Û��yèÚtS, ¬v;ðh�ypppñññ(Newton, 1642-1727)C¾¾¾¾¾¾¹¹¹+++(Leibniz,

1646-1716)Þ��Ý�xï�9Í�Ýn" , �5C�5Í¼Î&�3s"Ý, pñC¾¾¹+:�Þï! YºÕ, v|��5ÃÍ�§�ñÞï� Ýn;��h�¡-bÝ��5, ¬v">2s"R¼��ÆÞ�ô�Ì ��5Ýs�ï�Q¹¿�¡, &Æô��1��5µÎMQ�ËÍßXs�,

�Ñ9ËÍßÎ9�2b�É�3pñC¾¾¹+�G, &9I.�, Aðððyyy(Fermat, 1601-1665)�¦¦¦999¯(Galielo, 1564-1643, L�¿FZ.�CÎ§.�),��ééé��(Kepler, 1571-1630, Æ»FZ.�CÎ§.�), åÕ`I.�_O±áIÝ@s, E��5Ý¦Ã-�bÝQ¤��pñÝ�/Barrow (1630-1677), -¿{�s¨Ý�5��5Û! YºÕ�pñÞ��5ÝÃFB�Ý´z½, �¾¾¹+Xx�Ý×°·»ÝÐr, C�ÕÝ]°);à�

71

72 ÏÞa �5��5Ý�+

*��5B�ÆÞßÝúW�¡, 2¸Ý5�Ý&9r½, A��555°°°(calculus of variations) C�5]�Ýs", 3I.Ý&9r½ôb�KTà��ß?�ÝÎ, 4Qpñ�¾¾¹+CÍ�4, ßé��5Ahú�bæÝ�Ì, �ÆQÎ�Þ��5�ÝÃÍÃFÂz(»AÁ§ÝÃF)��ÆXàÕÝ“PPP§§§��ÝÝÝ��”(infinitely

small quantities (Çinfinitesimals)),3�ìîQÎ��×\Ý(�2.7

;)�×àÕèÜtS, �EÁ§��ÛÝ�L, CbÝ=�ÙÝÃF¡, ��5ÝÃF�ÕËÑ�z�ÍaE�5C�5��×°�MÝ+Û, �½¡«¿aÝ", ��ºE��5Ý/�ºXXz½, £`bÄ��¥\Ía, Þ�EÃÍÝÃFb?×MÝÝ��

2.2 «««��

BÄ×ð��Ýs"¡, �¿¢.T�QI.�, X®ßÝ×°àÌÝÃF, ��|�5T�5¼à��5Îà¼à��ABÄàîØFÝ6a, T×àaîØÔ�É�Ý>�, T?×�2, à¼O��;;;£££(rate of change)��y¿«î×\& àÝ ½, -�¢Ã�5¼Oÿ�¿¢CÎ§�$b�KÃF, �|�5¼�î�Í;&Æµ|O«�¼S�5ÝÃF�E×� l´ wÝÎ�, ¸Ý«�Ç lw, h¯@&Æ��.

-“á¼”�µ�#åh¯@?Ý, . u��#å×°2', -�Mp��ó.�ï.ÞßîÎq#�Ý, �ï.ê�ß.8ñT�3�9�>�,&Ë¶É??3“8*bß”¡,�Q�k�“ßÝD3”��-Î��>ÝÏ×Í2'�Î�«�á¼¡, ë��Ý«�Ç ��{��Ý×�(ë��BÊ6v¡�Ú ×Î��)�ê.��×9\��5W×°ë��, 9\�Ý«�, Ç��LW£°ë��«�Ýõ�9\�5Wë��Ý]°¬�°×, �Ä�|J��¡§�5, Í«�KÎ��Ý�u¿«î× ½R, Í\& ×à, J&Æ-��z½h ½

ÎÍb«��uÞh ½w3¿«î×b]}�Ýüî, J5½

2.2 «� 73

Bã�ÕR�Xâ}�ó, C�âR�}�ó, -V¯�£�R�«��RÝ«�-+y�ÞÂ� (9ôÎ×2')�©�Þ}��5��, 0--�¼�� Ý�E3O×\& `aÝ ½�«�, 3ó.î´|û>,

&Æ�Êx-y¿«î× ½R, Í¼\\& àax = a, �\\& àax = b, î]\& ×ÑÝ=�Ðóf(x)�%�, ì]\& x��&ÆÌR 3f�%�ìãa�b(under the graph of f from

a to b), �%2.1�

O a bx

y

R

y = f(x)

%2.1.

kOR�«�, &ÆÞ [a, b]5Wn5, �×��, 3N×5F, ix��ka, Ah-ÞR5Wn�s, �%2.2�

O a bx

y

%2.2.

9°�sÝ«�, ¬�fR�«��|O�?×M2, &Æ�5½ãîCãì, Ç|²#C/#Î�Ý«�, ¼£�9°�sÝ

74 ÏÞa �5��5Ý�+

«��ôµÎ3N×�s�, 5½|f3Í��t�ÂCt�Â {�Î�, �%2.3�

O a bx

y

%2.3.

&Æ�5½O�²#Î�«�õ, C/#Î�«�õ, JRÝ«�, -+yÞï� �àÌî, uÞ[a, b]5Ý�¼�Þ, JRÝ«�@�-��O�Þ

@Ý��Ã{ÆR, ¿àh]°(ÇM¼°), �Õ��KàìÝ«��h�°�1�Ìb�5ÝÕ��î�²#C/#Î�, 5½ETÞ$VÐó, Í�×ÍÝ%�

3fÝ%��î, ×Í3�ì�&Æ�|1Î�îC�ì, |Þ$VÐó¼¿�f�$VÐóÝ“�5”, µ�L £°ETÎ�«�Ýõ�9ø-Þ�5�«�ÝÃF=R¼Ý�\ï�|�?fãa�bÝ�5, -º+yÞ$VÐóÝ�5 �3�5�, $$$VVVÐÐÐóóó6�×¥�Ý��3ì×;&Æ-Þ¢ã$VÐó¼�L�5�ny$VÐóÝ×°�5ÝP², �¢�Apostol (1967)

Chapter 1�

2.3 ��555ÝÝÝ��LLL

3î×;&Æ�|àÌÝ]P, #å×àìX��Ý«�, Q¡|×Á§Â�î«�, 9ì&ÆÞhM»DÄ¼�&Æ��|àÌÝ]P, ¼�î×=�àìX��Ý«��Þ|öÜ��Ý]P, �J�×ó�Î�«�õÝÁ§D3, hÁ§ÂÇ �5!

2.3 �5Ý�L 75

`ôÎ«�Ý�L� Ý¸O×AG, \& `aÝ ½�«�, ´|y�§, &

Æ��'`aÝI5, ×=�ÐóÝ%��'f�L3T [a, b]î, ×=�v&�ÝÐó�32ý¿«î, ãf�%�,

Þàax = aCx = b, Cx�X�� ½|R��, Ç 3fÝ%�ìãa�bÝ ½�EyR&Æ��×óA, Ì Í«««��(area)�«�AÄ6b�9\�«�×lÝP²�ÇA6�yT�yN×�âyR�9\�Ý«�, ��yT�yN×�âR�9\�Ý«��&ÆÞJ�ªb×óA��hP²�3E«�×MD¡G,

&Æ�+Û× �555vvv(partition)ÝÃF�'bn + 1ÍF

a = x0 < x1 < · · · < xn−1 < xn = b,

Þ[a, b]5WnÍ�

[x0, x1], [x1, x2], · · · , [xn−1, xn]�

&Æ-|ÐrP = {x0, x1, · · · , xn}

�î�nÍ� , ¬Ìh [a, b]�×5v��ÄE×5vP , b`Î¼5vFx0, x1, · · · , xn, b`Î¼5v¡ÝnÍ� , ÞïÍ²¬P-², .Gï�X�¡ï, �¡ïù�X�Gï�u×5v�N×� /��, -Ìh ×ÑÑÑ!!!555vvv(regular partition)�»A, '[a, b] = [1, 6], JP = {1, 2, 2.5, π,

√15, 6}, �Þ[1, 6]5v

W[1, 2], [2, 2.5], [2, 5, π], [π,

√15], [

√15, 6]

�5Í� ��P = {1, 2, 3, 4, 5, 6}J [1, 6]�×Ñ!5v, v5W

[1, 2], [2, 3], [3, 4], [4, 5], [5, 6]

�5Í�� :�P = {x0, x1, · · · , xn} ×Ñ!5v, uv°u

x1 − x0 = x2 − x1 = · · · = xn − xn−1 =b− a

n,

76 ÏÞa �5��5Ý�+

ùÇuv°u

xi = a +i(b− a)

n, i = 0, 1, · · · , n�

¨E ½R, kOÍ«��P = {x0, x1, · · · , xn} [a, b]��×5v, .f =�, EN×� [xi−1, xi], i = 1, · · · , n, �3Í�0Õ×ui, ¸ÿf(ui) f3[xi−1, xi]�Á�Â(�Ï×a�§6.6)�Ah&Æ-C�nÍ�]�, Í95½ [xi−1, xi], { f(ui), i =

1, · · · , n�9°�]�Ç�W×9\�(Ì ��]]]999\\\��, rectan-

gular polygon), ¬/#yR�u|I(P )�h/#yR��]9\�Ý«�, J

I(P ) =n∑

i=1

f(ui)(xi − xi−1)�

!§,3N×� [xi−1, xi],0Õ×vi¸ÿf(vi) f3[xi−1, xi]�Á�Â�Q¡5½|[xi−1, xi] 9, f(vi) {, C�nÍ�]��9°�]�ù�W×�]9\�, v²#yR, Í«�

C(P ) =n∑

i=1

f(vi)(xi − xi−1)�

ãyN×/#yR��]9\�, Ä�âyN×²#yR��]9\�, ÆE�Þ[a, b]�5vP1, P2,

I(P1) ≤ C(P2)�

.hu�L,U5½�ì�Þ/):

L = {I(P )|P [a, b] �×5v},U = {C(P )|P [a, b] �×5v},

JL�N×ó/ U�×ì&, vU�N×ó/ L�×î&�ã@ó�Ù�t�î&2§, /)Lbt�î&, /)Ubt�ì&�u�

Al = lub L, Au = glb U,

2.3 �5Ý�L 77

JãG«Ý1�á,

Al ≤ Au�

uAl = Au, J��2, RÝ«�A-�ã A = Al = Au�Ah×¼,E×=�v&�ÝÐóf ,3Í%�ìãa�bÝ ½,Í«�A-�µî�M»¼�LÝ�Q�:�hM»��Á��, ��Kæ��O�×«��ô.Ah, ��¨��5ÃÍ�§Ý¥�, .¿à��§, &Æ-�| #Ý]P¼O�«��nyAl = AuÝJ�, &Æº3}¡, 9ì�Ü×OI(P )CC(P )�»�

»»»3.1.'f(x) = 4−x2, [a, b] = [−2, 2], ½R 3f�%�ìãa�b,

ê'P = {−2,−1/2, 1, 2}�f3� [−2,−1/2], [−1/2, 1]C[1, 2]

�Á�Â5½ f(−2) = 0, f(1) = 3, f(2) = 0, Æ

I(P ) = 0 · (−1

2+ 2) + 3 · (1 +

1

2) + 0 · (2− 1) =

9

2�

�f3ëÍ� �Á�Â5½ f(−1/2) = 15/4, f(0) = 4, f(1) =

3, Æ

C(P ) =15

4(−1

2+ 2) + 4(1 +

1

2) + 3(2− 1) =

117

8 �

¨3&ÆÞf =�Ý�', w´ G�OÎb&Ðó(T îÝ=�ÐóQù b&)�'f ×�L3T [a, b]îÝb&Ðó�E[a, b]��×5

vP = {x0, x1, · · · , xn}, �

S(P ) =n∑

i=1

mi(xi − xi−1),

T (P ) =n∑

i=1

Mi(xi − xi−1),

78 ÏÞa �5��5Ý�+

Í�mi, Mi, 5½ f3� [xi−1, xi]�glbClub��G&Æ�Êf =�Ðó, �T î�=�Ðó, ÄbÁ�Â(vÇ Íglb)CÁ�Â(Ç Ílub)�¬¨3ÝfG b&Ðó, .hÁ�TÁ�Âµ�×�D3Ý�î�S(P )CT (P )5½Ì fny5vP3[a, b]�ìììõõõ(lower sum

)Cîîîõõõ(upper sum)�ê.E∀i = 1, · · · , n,Cx ∈ [xi−1, xi], mi

≤ f(x) ≤ Mi, ÆS(P ) ≤ T (P )�

Ý]-, &Æ|�6CÒ∆(Çδ��¶)¼�î-û, ∆xiÇ xi −xi−1, i = 1, · · · , n�.hS(P )CT (P )�;¶

S(P ) =n∑

i=1

mi∆xi, T (P ) =n∑

i=1

Mi∆xi�

E[a, b]�×5vP , u5�¼�Ø°� �5WóÍ, JÿÕ[a, b]�×±Ý5vP ′, P ′Ì P�×ÞÞÞ555(refinement)�»A, uP = {1, 2, 3, 4}, JP ′ = {1, 1.5, 2, 2.7, 3, 3.2, 4} P�×Þ5��:�P ′ P�×Þ5, uv°uPX�âÝ£°5vF�/), P ′X�âÝ£°5vF�/)Ý�/�9ì ×nyÞ5��Ý��, àÌî��ÎEÝ�h��

¼�Þ5º¸ìõ¦�, ¸îõ3��

SSS§§§3.1.'P ′ P�×Þ5, JS(P ) ≤ S(P ′), vT (P ′) ≤ T (P )�JJJ��.&Æ©6�Ê5vP ′fP9×� Ý�µÇ�( %�?)�'P = {x0, x1, · · · , xn}, ê Ý�-, 'P ′ = {x0, x, x1, · · · , xn}, Í�x0 < x < x1�J

S(P ′) = m(x− x0) + m′(x1 − x0) +n∑

i=2

mi∆xi,

Í�mCm′5½ f3[x0, x]C[x, x1]�glb�.m1 f3[x0, x]∪ [x, x1]

�glb, Æm1 ≤ mvm1 ≤ m′�.h

m1∆x1 = m1((x− x0) + (x1 − x)) ≤ m(x− x0) + m′(x1 − x)�

2.3 �5Ý�L 79

Æÿ

S(P ) =n∑

i=1

mi∆xi = m1∆x1 +n∑

i=2

mi∆xi ≤ S(P ′)�

!§�JT (P ′) ≤ T (P )�J±�

'b[a, b]�Þ5vP1CP2, JhÞ5v�5vF�Ð/P P1

CP2��!Þ5�»A, P1 = {1, 2, 5}, P2 = {1, 3, 4, 5}, JP =

{1, 2, 3, 4, 5} P1�Þ5ù P2�Þ5�¿àh��, �J�ì��§�

��§§§3.1.'f T [a, b]î�×b&Ðó, P1, P2 [a, b]��Þ5v�JS(P1) ≤ T (P2)�JJJ��.'P P1CP2�×�!Þ5�JãS§3.1, S(P1) ≤ S(P )vT (P ) ≤ T (P2)��S(P ) ≤ T (P ), Æÿ

S(P1) ≤ S(P ) ≤ T (P ) ≤ T (P2)�

ÿJ�

¿àS§3.1C�§3.1, êñÇ�ÿì�.¡�

��§§§3.2.'P ′ P�×Þ5, JT (P ′)− S(P ′) ≤ T (P )− S(P )�

E[a, b]î�×b&Ðóf , )�

L = {S(P )|P [a, b]�×5v},(3.1)

U = {T (P )|P [a, b]�×5v}�(3.2)

Jã�§3.1á, L�N×ó/ U�×ì&, �U�N×ó/ L�×î&�Æãt�î&2§, ÿlubLCglbU/D3�.h�bì��L�

80 ÏÞa �5��5Ý�+

��LLL3.1.'f 3[a, b]î�×b&Ðó, L,U��LA(3.1)C(3.2)

P�Jfãa�b�ììì��555(lower integral), |∫ b

a¯

f��, v

∫ b

a¯

f = lub L�

�fãa�b�îîî��555(upper integral), |∫ b

a¯ f��, v

∫ b

a

¯f = glb U�

ãlub��Lá, ∀ε > 0, D3[a, b]�×5vP1, ¸ÿ

0 ≤∫ b

a¯

f − S(P1) < ε,

�qAglb��Lá, D3[a, b]�×5vP2, ¸ÿ

0 ≤ T (P2)−∫ b

a

¯f < ε�

ÆuãP P1CP2�×�!Þ5, J¿àS§3.1, ÿ

(3.3) 0 ≤∫ b

a¯

f − S(P ) < ε,

v

(3.4) 0 ≤ T (P )−∫ b

a

¯f < ε�

ôµÎ∀ε > 0, D3[a, b]�×5vP , ¸ÿ(3.3)C(3.4)P!`Wñ�ãyN×îõT (P ) L�×î&, v

∫ b

a¯

f = lub L, ÆEN×5

vP ,∫ b

a¯

f ≤ T (P )�.h∫ b

s¯

f U�×ì&�ê.∫ b

a¯ f = glb U , ÆE

N×5vP ,

(3.5) S(P ) ≤∫ b

a¯

f ≤∫ b

a

¯f ≤ T (P )�

2.3 �5Ý�L 81

uftÝb&, ù3[a, b]��, 9ì1��BãÊ2óã5vP , ¸ÿS(P )�T (P )��#��EN×ÑJón, �Pn = {x0, x1, · · · , xn} [a, b]�×Ñ!5v,

Ç∆x1 = ∆x2 = · · · = ∆xn = (b− a)/n��'f3[a, b] �¦�J3 [xi−1, xi], f�glb f(xi−1), lub f(xi), i = 1, · · · , n�.h

S(Pn) =n∑

i=1

f(xi−1)∆xi =

(n∑

i=1

f(xi−1)

)b− a

n,

T (Pn) =n∑

i=1

f(xi)∆xi =

(n∑

i=1

f(xi)

)b− a

n �

ãîÞPÿ

T (Pn)− S(Pn) = (f(xn)− f(x0))b− a

n(3.6)

=(b− a)(f(b)− f(a))

n �

�:�E∀ε > 0, ©�nãÿÈ�, J�¸

(3.7) 0 ≤ T (Pn)− S(Pn) < ε�

�yuf �3, ûî�D¡, )�ÿÕ(3.7)P�ÇÿJE�×[a, b]î, ��vb&ÝÐóf , D3×5vP , ¸ÿS(P )��#�T (P )�ê(3.5)P0luf b&Ðó, JEN×[a, b]îÝ5vP ,

(3.8) 0 ≤∫ b

a

¯f −

∫ b

a¯

f ≤ T (P )− S(P )�

Æu��îf ��Ðó, Jãy∀ε > 0, D3[a, b]î�×5vP , ¸ÿT (P )− S(P ) < ε, .h¿à(3.8)PÇÿ

0 ≤∫ b

a

¯f −

∫ b

a¯

f < ε, ∀ε > 0�

ãhñÇÿÕ∫ b

a¯ f − ∫ b

a¯

f = 0�&ÆÞ��W�yì�

82 ÏÞa �5��5Ý�+

��§§§3.3.'f T [a, b]î�×b&v��Ðó, J

(3.9)

∫ b

a¯

f =

∫ b

a

¯f�

»»»3.2.'f(x) = c ×ðó, ∀x ∈ [a, b]�|�EN×5vPn, S(Pn)

= T (Pn) = c(b− a)�Æ∫ b

a¯

c =

∫ b

a

¯c = c(b− a)�

»»»3.3.'f(x) = 4− x, x ∈ [1, 3], Of3[1, 3]�ì�5Cî�5��.ãPn [1, 3]�×Ñ!5v, ÇPn = {x0, x1, · · · , xn}, Í�∆xi =

2/n, v

xi = 1 +2i

n, i = 0, 1, · · · , n�

êf3[xi−1, xi]�glb f(xi) = 4− xi = 3− 2i/n�.h

S(Pn) =n∑

i=1

(3− 2i

n

)2

n=

6n

n− 4

n2

n∑i=1

i

= 6− 4

n2

n(n + 1)

2= 6− 2(n + 1)

n= 4− 2

n�

!§�ÿ(T¿à(3.6)P, �Þf(b)�f(a)øð, .h�f �3)

T (Pn) = 4 +2

n�

�ã(3.5)Pá, E∀n ≥ 1,

4− 2

n= S(Pn) ≤

∫ b

a¯

f ≤∫ b

a

¯f ≤ T (Pn) = 4 +

2

n�

Æ∫ b

a¯

f =∫ b

a¯ f = 4�

»»»3.4.t�JóÐóf(x) = [x]3 [2, 5] b&v�¦, Oì�5Cî�5�

2.3 �5Ý�L 83

JJJ��.'P3n [2, 5]�×3n�5Ñ!5v,ÆN×�5�� 1/n�v.f3GnÍ� �glb/ 2, Íg 3, �Íg 4, Æ

S(P3n) =n∑

i=1

2 · 1

n+

2n∑i=n+1

3 · 1

n+

3n∑i=2n+1

4 · 1

n= 2 + 3 + 4 = 9�

(3.6)Pê��

T (P3n)− S(P3n) =(5− 2)(f(5)− f(2))

3n=

3

n,

ÆT (P3n) = 9 + 3/n�.h9 ≤ ∫ 5

2¯

f ≤ ∫ 5

2¯ f ≤ 9 + 3/n�Æÿ

∫ 5

2¯

f =

∫ 5

2

¯f = 9�

b&9&ÆðÂÕÝ��Ðó, /��(3.9)P, ãhÇSå�ì��L�

��LLL3.2.'Ðóf3T [a, b]îb&, JfÌ 3[a, b]��(inte-

grable), uv°u∫ b

a¯

f =∫ b

a¯ f�uf3[a, b]��, Jfãa�b��5,

|∫ b

af(x)dx��, v�L

(3.10)

∫ b

a

f(x)dx =

∫ b

a¯

f =

∫ b

a

¯f�

Ðóf Ì ��555ÕÕÕ��(integrand)�

ã»3.2á, E×ðóc,

∫ b

a

cdx = c(b− a)�(3.11)

b`&Æ1∫ b

af(x)dxD3, Í��Ç f3[a, b]��ã�§

3.2-á, T îÝb&v��Ðó ��¨², &Æ¯1�

84 ÏÞa �5��5Ý�+

�5ÝÐrAì�Ðr∫ ¾¾¹+y�-1675OXx, .z

ZCÒSZ��ÿ, ��õõõ(sum)ÝÏ×ÍCÒ�.h∫ b

af(x)dx�

�×°�Af(x)dxÝ4�õ, ÑAìõCîõ5½ ×°mi∆xiCMi∆xi�õ�uf(x) ≥ 0, Jf(x)dx��WÎ×{� f(x), v9I dx��]��«��3Ðr

∫ b

af(x)dx�, ËÑm�à¼�

îf�a�bÝ�5, Í@©b∫ b

afÝI5��Ä, &ÆX�ºs¨¾

¾¹+Ý�5Ðr, 3&9ºÕ�, Î×�?Ý�Õ'��.h&Æ;ð)2à|

∫ b

af(x)dx¼�î�5�¬3

∫ b

af(x)dx�, ¬�Î

&àx��, 2àÍ�CÒ/��.h,

∫ b

a

f(x)dx =

∫ b

a

f(t)dt =

∫ b

a

f(u)du = · · ·�

3hÐrx, t, u�, Ì ÌÌÌaaa��óóó(dummy variable), µA!∑n

i=1 ai

�Ýi ×ø�¨², G�

∫ b

af(x)dx/©Ea < b�b�L� Ý]-, &Æ�

∫ a

a

f(x)dx = 0,

vub < a, � ∫ b

a

f(x)dx = −∫ a

b

f(x)dx�

(3.8)PE�×b&Ðó/Wñ, ãhÇÿì��§�

��§§§3.4.b&Ðóf3T [a, b]��,uv°u∀ε > 0,D3[a, b]�×5vP , ¸ÿ

0 ≤ T (P )− S(P ) < ε�9ìÝ�§àÌîôÎEÝ�

��§§§3.5.'Ðóf3T [a, b]î��, Jf3N×[a, b]�TÝ� ù��JJJ��.'[c, d] ⊂ [a, b]�E∀ε > 0, .f��, ã�§3.3á, D3×5

2.3 �5Ý�L 85

vP , ¸ÿ0 ≤ T (P ) − S(P ) < ε��P ′ ×�âc, dÞF�PÝÞ5�.S(P ′) ≥ S(P )vT (P ′) ≤ T (P ), Æ

0 ≤ T (P ′)− S(P ′) ≤ T (P )− S(P ) < ε�

ãP1 P ′�×�/v [c, d]�×5v, J

0 ≤ T (P1)− S(P1) ≤ T (P ′)− S(P ′) < ε�

Æã�§3.3, ÇÿJf3[c, d]��

A!«�, �5bì��P�

��§§§3.6.'f3T [a, b]C[b, c]/��, Jf3[a, c]��, v∫ b

a

f(x)dx +

∫ c

b

f(x)dx =

∫ c

a

f(x)dx�

JJJ��.ã�§3.3, ∀ε > 0, 5½D3[a, b]C[b, c]�5vP1, P2, ¸ÿ

0 ≤ T (P1)− S(P1) <ε

2, 0 ≤ T (P2)− S(P2) <

ε

2�

ãhêÿ

(3.12) 0 ≤ (T (P1) + T (P2))− (S(P1) + S(P2)) < ε�

.S(P1)+S(P2)CT (P1)+T (P2)5½ f3[a, c]�×ìõCîõ,Æ��¿à�§3.3ÇÿJf3[a, c]��Ígã(3.5)P(h`

∫ b

a¯

f =∫ b

a¯f =

∫ b

af(x)dx), ÿ

S(P1) ≤∫ b

a

f(x)dx ≤ T (P1),

S(P2) ≤∫ c

b

f(x)dx ≤ T (P2)�

.h

S(P1) + S(P2) ≤∫ b

a

f(x)dx +

∫ c

b

f(x)dx ≤ T (P1) + (P2)�

86 ÏÞa �5��5Ý�+

.f3[a, c]��, Æ∫ c

af(x)dxù+yÍìõS(P1) + S(P2)Cîõ

T (P1) + T (P2)� �Æã(3.11)C(3.12)PÇÿ∣∣∣∣∫ c

a

f(x)dx−∫ b

a

f(x)dx−∫ c

b

f(x)dx

∣∣∣∣ < ε, ∀ε > 0�

¬hÇ ∫ c

a

f(x)dx =

∫ b

a

f(x)dx +

∫ c

b

f(x)dx�

Í�§J±�

u∫ b

af(x)dxC

∫ c

bf(x)dx/D3, J�pJ�, �¡a�bCb�c�

��,∫ c

af(x)dxùD3, v

∫ b

a

f(x)dx +

∫ c

b

f(x)dx =

∫ c

a

f(x)dx�

êã�§3.2C3.5Çÿì�.¡�

��§§§3.1.'Ðóf3[a, b] b&v@@@ððð��(piecewise monotonic),

Jf3[a, b]��

�×94P3× î, / b&v@ð��(|¡º1�),

sineÐóCcosineÐóùÎ�¯@î, &ÆðÂÕÝÐó�, �9Î@ð��Ý, X|&Æ�ÿÕ�KÐó ��Í;XD¡ÝÐó��, �Xf�(��L3.2) hÐó b

&, v6�L3×(b§Ý)T î�ã&ÆÝ.0Ä�, A5v�ãÁ�, CÁ�Â�, �:�Îm�b×°f�, ��1JìõCîõ/b§, v��!×Â�u � b§, TÐó� b&, J�Kb×f(xi)∆xi = ∞(T−∞), h`Äb×ìõTîõ ∞T(−∞)�b&CT , µ1JÝìõCîõ/b§�¨², &ÆôÿÕ3T î, ×b&v@ð��ÝÐóÄ�

��T îÝ×=�Ðóù��ÝJ�Jº3}¡�Q &T, TÐó&@ð��T&=�ôb��º��, &ÆX�º1

êÞ 87

�(9ÎXÛTheory of integrationÝP�)��yá¼×ÐóÎ��¡, êA¢�ÕÍ�5Â÷(9ÎXÛTechnique of integrationÝP�)? �&Æ@M+Û?9ÝÃÍP²C]°�¡, Þ�¸&Æ�O��KÝ�5�

êêê ÞÞÞ 2.3

1. �f(x) = 1/x, [a, b] = [1/2, 2], P ×Ñ!5v, Þ[a, b]5W6Í�� OI(P )CC(P )�

2. Eì�ÐóC5v, 5½OÍS(P )CT (P )�(i) f(x) = 1− x2, [a, b] = [0, 2], P = {0, 1

2, 1, 3

2, 2};

(ii) f(x) = 2x2, [a, b] = [−1, 1], P = {−1,−12, 0, 1

2, 1};

(iii) f(x) = x3, [a, b] = [−2, 0],

P = {−2,−53,−4

3,−1,−2

3,−1

3, 0};

(iv) f(x) = 1/x, [a, b] = [−4,−1], P = {−4,−3,−2,−1};(v) f(x) = 1/x2, [a, b] = [1, 4], P = {1, 3

2, 2, 5

2, 3, 7

2, 4}�

3. 'f(x) = x, P [a, b]�×5v��J

(i) S(P ) < (b2 − a2)/2 < T (P );

(ii)∫ b

a¯

f =∫ b

a¯ f = (b2 − a2)/2�

4. 'f(x) = x3, �JE�×[a, b]�5vP ,

(i) S(P ) < (b4 − a4)/4 < T (P );

(ii)∫ b

a¯

f =∫ b

a¯ f = (b4 − a4)/4�

5. 'f(x) = 1/x2, P = {x0, x1, · · · , xn} [a, b]�×5v, b > a >

0�(i)¶�S(P )CT (P );

88 ÏÞa �5��5Ý�+

(ii)�J

1

x2i+1

<1

xixi+1

<1

x2i

, ∀i = 0, 1, · · · , n− 1,

.hxi+1 − xi

x2i+1

<1

xi

− 1

xi+1

<xi+1 − xi

x2i

,

ãh�ÿ

S(P ) <1

a− 1

b< T (P );

(iii) �J∫ b

a¯

f =∫ b

a¯ f = 1

a− 1

b�

6. 'f(x) = x3, a < b��J∫ b

a¯

f =

∫ b

a

¯f =

(b4 − a4)

4 �

7. 'f(x) = 1/x3��JEN×[a, b]�5vP ,

(i) S(P ) < 12

(1a2 − 1

b2

)< T (P );

(ii)∫ b

a¯

f =∫ b

a¯ f = 1

2

(1a2 − 1

b2

)�

8. 'Ðóf��L

f(x) =

{0, ux b§ó,

1, ux P§ó�

O∫ 1

0¯

fC∫ 1

0¯ f , ¬¼�f3[0, 1]ÎÍ��

9. 'Ðóf��L

f(x) =

{x, ux b§ó,

0, ux P§ó�

�O∫ 1

0¯

fC∫ 1

0¯ f , ¬¼�f3[0, 1]ÎÍ��

2.4 �5ÝÃÍP²C§¡ 89

10. 'ÞÐófCg3[a, b]/ =�, vf(x) ≤ g(x),∀x ∈ [a, b]��J ∫ b

a¯

f ≤∫ b

a¯

g, v∫ b

a

¯f ≤

∫ b

a

¯g�

11. u|f |3[0, 1]��, �®f3[0, 1]ÎÍùÄ��, uÎJJ�, �ÎJÜ×D»�

2.4 ��555ÝÝÝÃÃÃÍÍÍPPP²²²CCC§§§¡¡¡Í;&Æ+Û×°�5ÝÃÍP²CÃÍ§¡�´��§3.2¼

�, T îÝb&v��ÝÐóÄ ��, �ÍJ�Ä�Çèº×OÍ�5ÂÝ]°�&Æ¶W×�§Aì�

��§§§4.1.'f T [a, b]î�×b&Ðó, �xi = a + i(b − a)/n,

i = 0, 1, · · · , n, Í�n ×ÑJó�(i) uf �¦, vE∀n ≥ 1, B��ì��P

(4.1)b− a

n

n−1∑i=0

f(xi) ≤ B ≤ b− a

n

n∑i=1

f(xi),

JB =∫ b

af(x)dx�

(ii) uf �3, vE∀n ≥ 1, B��ì��P

(4.2)b− a

n

n∑i=1

f(xi) ≤ B ≤ b− a

n

n−1∑i=0

f(xi),

JB =∫ b

af(x)dx�

JJJ��.&Æ©J�(i), (ii)ÝJ�v«�ã�§3.2á, E∀n ≥ 1,

b− a

n

n−1∑i=1

f(xi) ≤∫ b

a¯

f =

∫ b

a

f(x)dx =

∫ b

a

¯f ≤ b− a

n

n∑i=1

f(xi)�

90 ÏÞa �5��5Ý�+

îP=!(4.1)P-ÿ∣∣∣∣B −

∫ b

a

f(x)dx

∣∣∣∣ ≤b− a

n(f(b)− f(a)), ∀n ≥ 1�

ÆB =∫ b

af(x)dx�

tÝ�§3.5, �5$b×°ÃÍÝP², Ç¸�§3.5ô��îW´×�ÝlP(��§4.3)�9°�§©�¿àî×;Ý�5�L, ¢ÃìõCîõ-�ÿÕ, .h&Æ¯�J��tÝ�§4.6�², ÍõÞ¼¿à��5ÃÍ�§, ñÇ�ÿÕ�\ïô��ãàÌ¼:¸Æ ¢Wñ�

��§§§4.2.(aaaPPP). 'ÞÐófCg3T [a, b]��, JE��@óc1�c2, c1f + c2gù��, v

(4.3)

∫ b

a

(c1f(x) + c2g(x))dx = c1

∫ b

a

f(x)dx + c2

∫ b

a

g(x)dx�

î�§��|2.ÂÕnÍÐóÝ�µ, &Æ¯��¶�

��§§§4.3.(��PPP). ì��Þ�5D3, Kº0lÏëÍ�5D3, v

(4.4)

∫ b

a

f(x)dx +

∫ c

b

f(x)dx =

∫ c

a

f(x)dx�

��§§§4.4.(¿¿¿ÉÉÉ��PPP). 'f3T [a, b]��, JE�×@óc,

(4.5)

∫ b

a

f(x)dx =

∫ b+c

a+c

f(x− c)dx�

��§§§4.5.(MMM��;;;��). 'f3T [a, b]��, JE∀k 6= 0,

(4.6)

∫ b

a

f(x)dx =1

k

∫ kb

ka

f(x

k)dx�

2.4 �5ÝÃÍP²C§¡ 91

��§§§4.6.(fff´��§§§). 'f�g/3T [a, b]��, vg(x) ≤ f(x),

∀x ∈ [a, b], J

(4.7)

∫ b

a

g(x)dx ≤∫ b

a

f(x)dx�

��§§§4.1.'f�g/3T [a, b]��, v|g(x)| ≤ f(x),∀x ∈ [a, b],

J ∣∣∣∣∫ b

a

g(x)dx

∣∣∣∣ ≤∫ b

a

|g(x)| dx ≤∫ b

a

f(x)dx�

�§4.6�×ñÇÝ.¡ , uf3[a, b]��vf(x) ≥ 0,∀x ∈[a, b],J

∫ b

af(x)dx ≥ 0�ê�§4.1Í@àÕug3[a, b]��,J|g|ù

3[a, b]��, h��J�º3êÞ�4Q&ÆÞ¼ºD¡&Ë�5*», ��mNgKã�L�s

¼O�5, ¬�O�¼Ý�5Í@ÎÁKó, �9óÝ�5Î“Õ��”Ý(|¡º1�%�§Õ��), �©�O�«Â�×Ðó3Ø× ��, ©Î1Í�5ÂD3, ¬��îh�5Â��@2�î�¼�|ìõCîõT|Í��ÐóÝ�5¼¿�, ÎðàÝ]P, �§4.6-Îh`Ý§¡µA, �à¼ÿÕXkO�5�×î&Cì&�Íg&Æ¼D¡=�Ðó��P�.×T î�=�Ð

ófÄ b&, .hãî×;Ý.0á(��L3.1), f3[a, b]�bì�5Cî�5�9ì&ÆÞJ�,f =�`, Íì�5�î�58�, .hf3[a, b]��´�E[a, b] �×5vP = {x0, x1, · · · , xn}, �

‖P‖ = max{∆x1, ∆x2, · · · , ∆xn},||P ||Ì 5vP�PPPóóó(norm), �5v¡Xÿ� Ý��∆x1

, · · · , ∆xn��t�ï�Æ‖P‖ ≥ ∆xi, ∀i = 1, · · · , n, vD3×i¸ÿ‖P‖ = ∆xi�

��§§§4.7.'f T [a, b]î�×=�Ðó, Jf3[a, b]��JJJ��.ã�§3.3á, u�J�E∀ε > 0, D3[a, b]�×5vP , ¸

92 ÏÞa �5��5Ý�+

ÿ0 ≤ T (P )− S(P ) < ε, Í�§-ÿJÝ�.T îÝ=�Ðó, Ä í8=�(�Ï×a�§6.8),

Æ∀ε > 0, D3×δ > 0, ¸ÿ

(4.8) |f(x)− f(c)| < ε

b− a, ∀x, c ∈ [a, b] v |x− c| < δ�

¨'P = {x0, x1, · · · , xn} [a, b]�×5v, v��||P || < δ�ê�f(ui)Cf(vi)5½�f3[xi−1, xi]�Á�(Çglb)CÁ�Â(Çlub)�J

(4.9) T (P )− S(P ) =n∑

i=1

(f(vi)− f(ui))∆xi�

.ui, vi ∈ [xi−1, xi], Æ|ui − vi| ≤ ∆xi ≤ ||P || < δ�.hã(4.8)Pá

(4.10) 0 ≤ f(vi)− f(ui) <ε

b− a, ∀i = 1, · · · , n�

Þh�á(4.9)P, Çÿ(¥�∑n

i=1 ∆xi = b− a)

0 ≤ T (P )− S(P ) <ε

b− a

n∑i=1

∆xi = ε�

ÿJ�

'×b&Ðóf3T [a, b]��, �g ¨×b&Ðó, vg(x)

= f(x), ∀x ∈ (a, b)�ÇtÝ��3ÐF², ÞÐó�Â/8!, J�p:�ÞÐó3[a, b]�î�58!, ì�5ô8!�.hg3[a, b]ù��, v�5ÂÇ

∫ b

af(x)dx�?×�2Jbì��

��

��§§§4.8.'×b&Ðóf3T [a, b]��, vtÝ3b§ÍF²,

¨×b&Ðóg��g(x) = f(x), ∀x ∈ [a, b], Jg3[a, b]ù��,

v∫ b

ag(x)dx =

∫ b

af(x)dx�

Þ�§4.7�4.8�)Çÿì�.¡�

2.4 �5ÝÃÍP²C§¡ 93

��§§§4.2.'f T [a, b]î�×b&Ðó, v©3b§ÍF�=�,

Jf3[a, b]��

AG, ©3b§ÍF�=�ÝÐó, Ì @ð=�ÝÐó�uÞ�§4.2��“bbb§§§ÍÍÍ”; “��óóóÍÍÍ”, Í��)Wñ, &Æµ�#åh¯@?Ý, J�¯��Ä;ðb�§4.2�B�ÈÝ�Ï×a»5.8Ç ×©3�óÍF�=�ÝÐó�3î×;&Æ�ÿÕ, T î�b&v@ð��ÝÐó, Ä

��, ¨3êÿ@ð=��b&Ðóù��uk�5ÝÐó T îÝ=�Ðó, b&Ýf��QWñ�¬uÐó¬&3JÍ [a, b]î=�, Jb&Ý�'µbÄ�Ý�»A, f(x) = 1/x,

x ∈ (0, 1], f3x = 0�=�, .h�§4.7C�§4.1/�Êà�|¡&ÆºU"��ÐóÝÃF, £`ET Ý�O�w´, .�b°Ðó, Ag(x) = 1/

√x, x ∈ (0, 1], 4)3x = 0�=�, ¬QÎ�

��ÄG�f(x) = 1/x)3(0, 1]��G&Æ�ÊìõCîõ, ÞT [a, b]5v¡, 5½3N×�

ãf�glbmiClubMi, �ÿÕõS(P ) =∑n

i=1 mi∆xiCT (P ) =∑ni=1 Mi∆xi�.huf3[a, b]��, J

(4.11) S(P ) ≤∫ b

a

f(x)dx ≤ T (P )�

9ì&Æ�Êf´×�Ýõ�'Ðóf3[a, b]��, vP = {x0, x1, · · · , xn} [a, b]�×5v�

3N×� [xi−1, xi]��ã×ózi, i = 1, · · · , n, J

R(P ) =n∑

i=1

f(zi)∆xi

Ì f3[a, b]�×Riemannõõõ(Riemann sum)�Riemann èÜtSÆ»×½(Ýó.�, �3Þè°Ï`(�-1850O), s�×Sny5�ÝÃ��¡Z, ´�|õÝÁ§��5��Û�L�

94 ÏÞa �5��5Ý�+

�×ìõTîõ/ ×Riemannõ, v.mi ≤ f(zi) ≤ Mi,∀i =

1, · · · , n, ÆÄb

(4.12) S(P ) ≤ R(P ) ≤ T (P )�

.f3[a, b]��,.h∀ε > 0,D3×5vP ,¸ÿ0 ≤ T (P )−S(P ) <

ε��êb(4.11)C(4.12)P, Æ∀ε > 0, D3×5vP , ¸ÿ∣∣∣∣R(P )−

∫ b

a

f(x)dx

∣∣∣∣ < ε�

Ç©�Ê2óã5vP , �¸Riemannõ��5ÂÁ#��&Æ-Þ|Riemannõ¼¿�

∫ b

af(x)dx, B�WAì��§�

��§§§4.9.'b&Ðóf3T [a, b]��, {Pn, n ≤ 1} ×ó��[a, b]Ý5v, v��

(4.13) limn→∞

||Pn|| = 0�

ê'R(Pn) �×ETPn�Riemannõ, J

(4.14) limn→∞

R(Pn) =

∫ b

a

f(x)dx�

JJJ��.u�J�∀ε > 0, D3×n0 ≥ 1, ¸ÿ

(4.15) 0 ≤ T (Pn)− S(Pn) < ε, ∀n ≥ n0,

J.AGX�S(Pn) ≤ R(Pn) ≤ T (Pn), vS(Pn) ≤ ∫ b

af(x)dx ≤

T (Pn), Æb∣∣∣∣R(Pn)−

∫ b

a

f(x)dx

∣∣∣∣ < ε, ∀n ≥ n0�

�µÁ§Ý�L-ÿJ(4.14)PWñ�.�'f3[a, b]��, ã�§3.3á, ∀ε > 0, D3[a, b]�×5

vP = {x0, x1, · · · , xk}, ¸ÿ

(4.16) 0 ≤ T (P )− S(P ) <ε

2�

2.4 �5ÝÃÍP²C§¡ 95

êÄ�óãP¸ÿk ≥ 2(uk ≤ 1, J¿à�§3.1, ãP�×k ≥ 2Ý�5vÇ�)�.f ×b&Ðó, ÆD3×K > 0, ¸ÿ

−K ≤ f(x) ≤ K, ∀x ∈ [a, b]�

EN×5vPn, Ík − 1ÍT?KÝ� , ��â5vP�ÝFx1,· · · ,xk−1, vPn�yõÝN×� , /�âyPÝØ×� �(\ï��0×%::), êPnÝ� ó, |#Pn��, ¬�6�yT�yk�E5vPn,

(4.17) T (Pn)− S(Pn) =

#Pn∑i=1

(Mi −mi)∆ui,

Í�Mi,mi5½ f3Pn�ÏiÍ� [ui−1, ui] �ÝlubCglb�.|mi|, |Mi| ≤ K, v∆ui ≤ ||Pn||, Æ(4.17)P��N×4/�yT�y2K||Pn||�.h(4.17)P��£°4�, ET�âx1, · · · , xk−1

Ý� Ýõ, �øÄ2K(k− 1)||Pn||�¨², E×�âyPÝ� [xj−1, xj]ÝPn�� [ui−1, ui],

(Mi −mi)∆ui ≤ (M ′j −m′

j)∆xj ,

Í�M ′i ,m

′i5½ f3[xj−1, xj]�ÝlubCglb(h.m′

j ≤ mi,M′j ≥

Miv∆ui ≤ ∆xi)�.h(4.17)P��£°4�, ET��âx1, · · ·, xk−1Ý� Ýõ, �øÄT (P )− S(P )�ãîD¡á,

(4.18) 0 ≤ T (Pn)− S(Pn) ≤ (T (P )− S(P )) + 2K(k − 1)||Pn||�êã(4.13)P��'á, ∀ε > 0, D3×n0 ≥ 1, ¸ÿ

(4.19) ||Pn|| < ε

4K(k − 1), ∀n ≥ n0�

�)(4.16)�(4.18)C(4.19)P, ÿ

0 ≤ T (Pn)− S(Pn) <ε

2+

ε

2= ε, ∀n ≥ n0,

96 ÏÞa �5��5Ý�+

Ç(4.15)PWñ�Í�§ÿJ�

ãî�§ñÇÿ, uPn [a, b]�×n�5Ñ!5v, J(.||Pn||= (b− a)/n → 0, n →∞)

limn→∞

n∑i=1

f(zi)∆x =

∫ b

a

f(x)dx�

îP�¼� ×Riemannõ�Á§, ∆x = (b− a)/n, zi [xi−1, xi]��×F�9ì ×Tà�

»»»4.1.'p ×ÑJó, vb > 0, �O∫ b

0xpdx�

JJJ��..f(x) = xp3[0, b]î=�, Æù��.huã= b/n, zi =

ib/n, i = 1, · · · , n, J¿à�§4.9, ÿ∫ b

0

xpdx = limn→∞

n∑i=1

f(zi)∆x = limn→∞

b

n

n∑i=1

(ib

n

)p

=bp+1

p + 1,

Í�Á§ÂàÕêÞÏ1Þ�ãhêÿ(¿à�§4.3), EN×ÑJóp, C∀b > a ≥ 0,

(4.20)

∫ b

a

xpdx =bp+1 − ap+1

p + 1 �

»»»4.2.'f(x) =∑n

k=0 ckxk ×ng94P�J¿àî»C�§4.2

ÿ∫ b

a

f(x)dx =

∫ b

a

n∑

k=0

ckxkdx =

n∑

k=0

ck

∫ b

a

xkdx =n∑

k=0

ckbk+1 − ak+1

k + 1 �

»»»4.3.�Þ limn→∞

∑ni=0(n + i)−1�îW×�5�

��..n∑

i=0

1

n + i=

1

n+

n∑i=1

1

1 + i/n

1

n=

1

n+

n∑i=1

f(zi)∆x ,

2.4 �5ÝÃÍP²C§¡ 97

Í�f(x) = (1 + x)−1, [a, b] = [0, 1], ∆x = 1/n, zi = i/n, ê1/n → 0,

Æã�§4.9, ÿ

limn→∞

n∑i=0

1

n + i=

∫ 1

0

1

1 + xdx�

�¥�ÝÎ, (1 + x)−13[0, 1]î ��vb&, Æ∫ 1

0(1 + x)−1dxÝ@

D3�

��5�b�9�!ÌÍÝíííÂÂÂ��§§§(Mean-value theorem), 9ìËÍÎny�5Ý�

��§§§4.10.(��555��íííÂÂÂ��§§§). 'f3[a, b]î=�,JD3×c ∈ [a, b],

¸ÿ

(4.21)

∫ b

a

f(x)dx = f(c)(b− a)�

JJJ��.´�a = b`, ã(4.21)P�¼�/ 0, Í�§�QWñ�¨'b > a��MCm5½�f3[a, b]�Á�ÂCÁ�Â, Jm ≤f(x) ≤ M, ∀x ∈ [a, b]�Æ¿à�§4.6Çÿ

m(b− a) =

∫ b

a

mdx ≤∫ b

a

f(x)dx ≤∫ b

a

Mdx = M(b− a),

Ç

m ≤ 1

b− a

∫ b

a

f(x)dx ≤ M�

�¿à=�Ðó�� Â�§(Ï×a�§6.3)ÿ, D3×c ∈ [a, b],

¸ÿ

f(c) =1

b− a

∫ b

a

f(x)dx,

�hÇ(4.21)P�

3î�§�, f =�Ý�'ÎÄ�Ý, �ì»�

98 ÏÞa �5��5Ý�+

»»»4.4.'

f(x) =

{0, 0 ≤ x ≤ 1/2,

1, 1/2 < x ≤ 1�

J∫ 1

0f(x)dx = 1/2 6= f(c)(1− 0), ∀c ∈ [0, 1]�

'f(x) ≥ 0, ∀x ∈ [a, b], �§4.10¼�, 3f%�ì, ãa�bÝ«�, �|×{ f(c), 9) [a, b]��]�Ý«�¼ã��uf(x)�×� Ñ, ô)bv«Ý�Õ, ©�3f(x) ≤ 0�Þ«�Ú �Â�ã(4.21)PÇÿ, E×=�Ðóf ,

(4.22) m(b− a) ≤∫ b

a

f(x)dx ≤ M(b− a) ,

Í�MCm5½�f3[a, b]�Á�ÂCÁ�Â�hPù ×Ef3[a, b]î��5Ý£�, ¸��5�×��Ýî&Cì&��y�§4.10 ¢Ì íÂ�§÷? �Aì2�Õ�ubb§ÍÂf1, f2, · · · , fn, JÍ¿í

f1 + f2 + · · ·+ fn

n ��u�×Ðóf , �EP§9Í�!Ýf(x)ã¿í, Í�x [a, b]��×ó, ×Í�QÝ�° �[a, b]��ãnÍó, ' x1, x2, · · · , xn,

Q¡Of(x1) + f(x2) + · · ·+ f(xn)

n,

��n →∞�h¿íÝÁ§uD3,ô�&ÆA¢ã{xi}bn�¬uã{xi} [a, b]��5F, J

1

n

n∑i=1

f(xi) =1

b− a

n∑i=1

f(xi)∆xi,

Í�∆xi = (b− a)/n, Jn →∞`, h¿í��

1

b− a

∫ b

a

f(x)dx =

∫ b

af(x)dx∫ b

adx�

2.4 �5ÝÃÍP²C§¡ 99

∫ b

af(x)dx/(b− a)-Ì f3[a, b]�¿¿¿íííÂÂÂ(mean value)�(4.22)P

¼�, E×=�Ðóf , h¿íÂ+ym�M� ��(4.21)PJÎ13 [a, b]�, Äb×cD3, ¸ÿf(c)�y�¿íÂ��Ä�§4.10©Î1Jc�D3, ¬Î¼�c ¢Â�¨², ùb×ny�J¿íÝ��, h î�§�.Â�

��§§§4.11.'fCgí [a, b]î�=�Ðó, v'g3[a, b]î� &ÑT� &��JD3×c ∈ [a, b], ¸ÿ

(4.23)

∫ b

a

f(x)g(x)dx = f(c)

∫ b

a

g(x)dx�

JJJ��.'g3[a, b]/&��ûî�§�J�, ã��Pmg(x) ≤f(x)g(x) ≤ Mg(x)�s, �ÿ

(4.24) m

∫ b

a

g(x)dx ≤∫ b

a

f(x)g(x)dx ≤ M

∫ b

a

g(x)dx�

u∫ b

ag(x)dx = 0, J(4.24)P0l

∫ b

af(x)g(x)dx = 0, h`�×c ∈

[a, b] , /�¸(4.23)PWñ�u∫ b

ag(x)dx 6= 0, JÄ Ñ(.'g(x)

≥ 0,∀x ∈ [a, b])�Þ(4.24)PN×4/t|∫ b

ag(x)dx, Q¡��¿

à=�ÐóÝ� Â�§ÇÿJ�

î��JJJíííÂÂÂ��§§§(Weighted mean-value theorem), ð�à¼� ÞÐó¶��5Ý£��©½uÍ�b×ÐóÝ�5´|O`, -�àî�3êÞ��:Õ×°TàÝ»��y %�Ì �J¿í? �´×�P, �'3[a, b]�g ≥ 0vg 6≡ 0(.�'g� &ÑT� &�, �ug ≡ 0, (4.23)PQWñ)�J

g1f1 + g2f2 + · · ·+ gnfn

g1 + g2 + · · ·+ gn

nÍ�f1, · · · , fn��J¿í, Í�gi�fi�J¥��û�§4.9�¡Ý1�, JÿÐóf��J¿í (Í�gÌ JJJ¥¥¥ÐÐÐóóó(weight

function)) ∫ b

af(x)g(x)dx∫ b

ag(x)dx

�

100 ÏÞa �5��5Ý�+

�§4.11¼�, î��J¿íº�yf3[a, b]�ÝØ×Âf(c)�t¡&Æ¼:�5�°×P��T)Bÿ&ÆX�Ý�5Ý�L, ÎÙ�y ÝE¿«î

×°ð�Ý%�, ��×ÊÝ«�Ý�L�&Æ��º�á¼,

tÝ�L3.2, ÎÍ�|Í�]P¼�L�5? Q)�Ìn£°&Æ- �bÝP²� Ý?�@2D¡, &Æ�½¼X�A¢�«�×Í��Ý

�L�´�, «��Ú ×///)))ÐÐÐóóó(set function), ÇEØ×©�Ý/)R, &Æ��×@óA(R), ¬Ì� A�«��.ÐóA��L½ ×/)�/), ÆÌ� /)Ðó�Íg, E£°/),

&Æ��Í“«�”÷? uf ×3[a, b]�b&�&�, v��Ðó, Jã2.3;�D¡á, 3ÐófÝ%�ì, ãa�bÝ ½R, Ä6b×“«�”�9vÝ ½R,Ì ×ÁÁÁ///(ordinate set), ¬v&Æ�OE�×/)Ðó, u�Ì� “«�”, Í�L½6�âXb9vÁ/�&ÆóãÁ/�9I ×�� (half-open interval), AhË8ÏÝ/), A[a, b)C[b, c)�ø/ è/)��L3.2X��5ù ×/)Ðó, Í�L½ XbÝÁ

/�h.E×&��Ðóf , uR 3fÝ%�ì, ãa�bÝ ½, J&Æ�R�ÐóÂ

∫ b

af(x)dx�

¨², 9ìÝ×°P²&Æ- ôÎ«�ÐóAT�bÝ�A��L½|D��

(i) A(R) ≥ 0, ∀R ∈ D�(ii) A(R1) ≤ A(R2),∀R1, R2 ∈ D vR1 ⊂ R2�(iii)'R1, · · · , Rn ∈ D,vRi∩Rj = φ, ∀i 6= j, i, j = 1, · · · , n(R1,

· · · , Rn Ì !!!ÊÊÊ(mutually disjoint))�J

A(R1 ∪R2 ∪ · · · ∪Rn) = A(R1) + A(R2) + · · ·+ A(Rn)�

(iv) 'f ×ãÂk�ðóÐó, vR 3fÝ%�ìãa�bÝ ½, JA(R) = k(b− a)�&Æ��J�ÝÎ, E�×/)ÐóA, uÍ�L½�âXbÁ

/, ¬��î�°ÍP², JÍÐóÂ��L3.2X��58

2.4 �5ÝÃÍP²C§¡ 101

!�ð��, �5 °×��&ÆXm�/)Ðó�h��W�yì�

��§§§4.12.'A ×/)Ðó, Í�L½�âXbÁ/, ¬��G�f�(i)-(iv)�ê'f ×3 [a, b]�b&�&�, v��Ðó,

R 3fÝ%�ìãa�bÝ ½�J

(4.25) A(R) =

∫ b

a

f(x)dx�

JJJ��.EN×[a, b]�5vP , &ÆÞJ�

(4.26) S(P ) ≤ A(R) ≤ T (P )�

�.f ��, ��(4.26)P�°×ÂÇ ∫ b

af(x)dx, Ah(4.25)P-

ÿJÝ�'P = {x0, x1, · · · , xn} [a, b]�×5v, v

S(P ) =n∑

i=1

mi∆xi

8ETÝìõ�E∀i = 1, 2, · · · , n, �Li�9I [xi−1, xi],

{� mi�Î��ãP²(iv), A(Li) = mi∆xi�ê.R1, R2, · · · ,

Rn !Ê, ÆãP²(iii)

A(L1 ∪ L2 ∪ · · · ∪ Ln) = A(L1) + A(L2) + · · ·+ A(Ln) = S(P )�

êL1 ∪ L2 ∪ · · · ∪ Ln R�×�/, ÆãP²(ii)

S(P ) = A(L1 ∪ L2 ∪ · · · ∪ Ln) ≤ A(R)�

!§�JA(R) ≤ T (P ), .h(4.26)PWñ�

102 ÏÞa �5��5Ý�+

êêê ÞÞÞ 2.4

1. 'p ×ÑJó, ¿à

bp − ap = (b− a)(bp−1 + bp−2a + · · ·+ bap−2 + ap−1),

�JEN×ÑJói,

ip <(i + 1)p+1 − ip+1

p + 1< (i + 1)p�

¿àîPÿn−1∑i=1

ip <np+1

p + 1<

n∑i=1

ip ,∀n ≥ 2�

ãh�ÿ

limn→∞

1

np+1

n∑i=1

ip =1

p + 1�

2. �5½Þì�ÞÁ§�îW�5�(i) lim

n→∞∑n

k=1k

n2+k2 ,

(ii) limn→∞

(n

n2+12 + nn2+22 + n

n2+32 + · · ·+ nn2+4n2

)�

3. (i) �J2n∑

i=n+1

1

i=

2n∑m=1

(−1)m−1

m,∀n ≥ 1�

(ii) ¿à(i) Þ

limn→∞

(1

1− 1

2+

1

3− 1

4+ · · ·+ 1

2n− 1− 1

2n

)

�îW×�5�4. �J

(i) 12≤ ∫ 2

11tdt ≤ 1;

(ii) 1− 1x≤ ∫ x

11tdt ≤ x− 1,∀x > 1;

(iii) 12+ 1

3+ · · ·+ 1

n≤ ∫ n

11tdt ≤ 1 + 1

2+ · · ·+ 1

n−1, Í�n ≥ 2

×ÑJó�

êÞ 103

5. ¿à�§4.11, �Jì��P�1

10√

2≤

∫ 1

0

x9

√1 + x

dx ≤ 1

10�

6. ¿à√

1− x2 = (1− x2)/√

1− x2C�§4.11, �Jì��P�

11

24≤

∫ 1/2

0

√1− x2dx ≤ 11

24

√4

3�

7. ¿à1 + x6 = (1 + x2)(1− x2 + x4)C�§4.11, �JE∀a > 0,

1

1 + a6

(a− a3

3+

a5

5

)≤

∫ a

0

1

1 + x2dx ≤ a− a3

3+

a5

5�

8. 'f3[a, b]=�, Í�a < b, v∫ b

af(x)dx = 0��JD3×c ∈

[a, b], ¸ÿf(c) = 0�

9. 'f ×&�Ðó, v3[a, b]��Ju∫ b

af(x)dx = 0, J

3N×f�=�F, f�Â 0�(èî: 'c f�×=�Fvf(c) > 0, JD3c�×Ï½A ⊂ [a, b], ¸ÿf(x) > f(c)/2,

∀x ∈ A)

10. 'f3[a, b]=�, vEN×[a, b]î�=�Ðóg,∫ b

af(x)g(x)dx

= 0��Jf(x) = 0,∀x ∈ [a, b]�

11. 'f3[a, b]��(i) �J|f |3[a, b]ù��;

(ii)∣∣∣∫ b

af(x)dx

∣∣∣ ≤∫ b

a|f(x)|dx�

12. �Oì�&�5�(i)

∫ 1

−1|2x + 1|dx;

(ii)∫ 3

−2[4x− 1]dx, Í�[ · ] t�JóÐó;

(iii)∫ 2

0|x(x− 1)(x− 2)|dx;

(iv)∫ −4

−2(x + 4)10dx�(èî: ¿à�§4.4)

104 ÏÞa �5��5Ý�+

13. �Jë��«��y9¶|{��

14. �J�5 rÝi«� 2∫ r

−r

√r2 − x2dx�

15. �Jì�ny�5ÝCauchy��PPP(Cauchy’s inequality for

integrals)�E�Þ=�ÐófCg,

∫ b

a

f 2(x)dx

∫ b

a

g2(x)dx ≥( ∫ b

a

f(x)g(x)dx)2

�

16. (i) ¿à�5∫ 1000

0x10dx, �O110 + 210 + · · ·+ 100010�×�«

Â;

(ii) ¿à�5∫ 1000

2x−10dx, �O2−10 + 3−10 + · · · + 1000−10�

×�«Â�

17. 'Ðóf3[0, b]��, Í�b > 0�(i) uf �Ðó, �J

∫ b

−bf(x)dx = 2

∫ b

0f(x)dx;

(ii) uf �Ðó, �J∫ b

−bf(x)dx = 0�

18. 'f(x) = x− [x]��Jf3[0, 5]��, ¬OÍ�5Â�

19. '

f(x) =

{sin x/x, x 6= 0,

1, x = 0�

�Jf3[0, 1]��

20. '

f(x) =

{x sin(1/x), x 6= 0,

0, x = 0�

�Jf3[0, 1]��

2.5 ë�ÐóÝ�5 105

21. 'f�L3[0, 1]î, v

f(x) =

{1/q, x = p/q, v (p, q) = 1,

0, x = 0,TP§ó�

�Jf3[0, 1]��

22. 'a0, a1, · · · , an @ó, v��

a0

1+

a1

2+ · · ·+ an

n + 1= 0�

¿à�5�íÂ�§,�J]�Pa0+a1x+a2x2+· · ·+anxn =

0, �Kb×@q�

2.5 ëëë��ÐÐÐóóóÝÝÝ��555

3��5�, ë�ÐóÝ2��¥�, Íæ.¬�©Î¸Æ�)ë��\Ýn;, x�Î¸ÆXÌbÝÐóP²�&Æ�'��3�.`�-�!�sine(Ñ<)�cosine(õ<)�tangent(Ñ6)�cotangent(õ6)�secant(Ñv)Ccosecant(õv)�0Íë�Ðó, C¸ÆÝDÐóarc sine, · · ·�ë�ÐóÝ¥�P²�×,

-Îøøø��PPP(periodicity)�3�AÎ§��C¯��, ðº�§ø�Ý¨é, Aè��Ï�º»C®��º��y"D9v®Þ`, Í�X��ÕÝó., ðñÒ�ÝsineCcosineÐó��3�.`, Eë�ÐóÄb×�ÑÝ�é, Ç2P©½9�Í�ì�¿ÍP², 3��5�Î´ðàÕÝ:

(i) cos 0 = sin(π/2) = 1, sin 0 = cos(π/2) = sin π = 0;

(ii) sin2 x + cos2 x = 1, ∀x ∈ R;

(iii) cos(−x) = cos x, sin(−x) = − sin x, Çcosine �Ðó, sine

�Ðó;

(iv) sin(x + π/2) = cos x, cos(x + π/2) = − sin x, ∀x ∈ R;

(v) sin(x + 2π) = sin x, cos(x + 2π) = cos x, ∀x ∈ R, Çø�/ 2π;

106 ÏÞa �5��5Ý�+

(vi) E∀x, y ∈ R,

cos(x + y) = cos x cos y − sin x sin y,

sin(x + y) = sin x cos y + cos x sin y,

ãhÇÿ¹�2P

sin 2x = 2 sin x cos x, cos 2x = cos2 x− sin2 x = 1− 2 sin2 x;

(vii) E∀x, y ∈ R,

sin x− sin y = 2 sin(x− y

2) cos(

x + y

2),

cos x− cos y = −2 sin(x− y

2) sin(

x + y

2);

(viii) 3[0, π/2] , sineÐó �}�¦, cosineÐó �}�3;

(ix) 0 < cos x < sin x/x < 1/cos x, ∀x ∈ (0, π/2)�

¨b×°2PCP²Þ¼umàÕ&Æ��ãP²(iv)�(v)C(viii)á, sineCcosineÐó/ @ð��ÝÐ

ó, .hã�§3.2, 3�×b§Ý �, sineCcosineÐó/ ��Ðó��yÍ�5Â�¢ã�§4.1ÿÕ��Ä&Æ)màÕì��P�

��§§§5.1.E∀u ∈ [0, π/2]Cn ≥ 1,

(5.1)u

n

n∑

k=1

cos

(ku

n

)< sin u <

u

n

n−1∑

k=0

cos

(ku

n

)�

JJJ��.&Æ�Jì�ë�ùóÝ�P:

2 sin(1

2x)

n∑

k=1

cos(kx) = sin((n +1

2)x)− sin(

1

2x),(5.2)

∀n ≥ 1, x ∈ R�

2.5 ë�ÐóÝ�5 107

´�¿àë�ÐóÝP²(vii)�ÿ

2 sin(1

2x) cos(kx) = sin((k +

1

2)x)− sin((k − 1

2)x)�

ÞîPEk = 1, 2, · · · , n, ¼�&�8�, Çÿ(5.2)P�ux/2� π�Jó¹, J�Þ(5.2)P�N×4/t|2 sin(x/2),

ÿ

(5.3)n∑

k=1

cos(kx) =sin ((n + 1/2)x)− sin(x/2)

2 sin(x/2) �

3îP�|n− 1ã�n, v¼�&�1, ÿ

(5.4)n−1∑

k=0

cos(kx) =sin((n− 1/2)x) + sin(x/2)

2 sin(x/2) �

îÞP©�x 6= 2mπ, Í�m ×Jó, /Wñ�ãx = u/n, Í�0 < u ≤ π/2, Jã(5.3)C(5.4)Pá, (5.1)P�

ì��P��:

u

n

sin((n + 1/2)u/n)− sin(u/(2n))

2 sin(u/(2n))< sin u

<sin((n− 1/2)u/n) + sin(u/(2n))

2 sin(u/(2n)) �(5.5)

�h��Pê��yìP:

sin((n + 1/2)u/n)− sin(u/(2n))(5.6)

<sin(u/(2n))

u/(2n)sin u

< sin((n− 1/2)u/n) + sin(u/(2n))�

�u�J�E∀0 < 2nθ ≤ π/2,

sin((2n + 1)θ)− sin θ <sin θ

θsin(2nθ)(5.7)

< sin((2n− 1)θ) + sin θ

108 ÏÞa �5��5Ý�+

Wñ, J�θ = u/(2n)-ÿÕ(5.6)PÝ�&Æ�J(5.7)P�¼\£Í��P�¿àë�Ðó�P²(vi),

ÿ

sin((2n + 1)θ) = sin(2nθ) cos θ + cos(2nθ) sin θ(5.8)

< sin(2nθ)sin θ

θ+ sin θ,

h�ôàÕì�ë�Ðó�P²(¥�0 < 2nθ < π/2)

cos θ < sin θ/θ, 0 < cos(2nθ) ≤ 1, sin θ > 0�

ñÇ�:�ã(5.8)P-0�(5.7)P�¼\£Í��P�ÍgJ�(5.7)P��\£Í��P�)¿àP²(vi)ÿ,

sin((2n− 1)θ) = sin(2nθ) cos θ − cos(2nθ) sin θ�

îP¼�&�sin θ, ÿ

sin((2n− 1)θ) + sin θ(5.9)

= sin(2nθ)

(cos θ + sin θ

1− cos(2nθ)

sin(2nθ)

)�

ãy1− cos(2nθ)

sin(2nθ)=

2 sin2(nθ)

2 sin(nθ) cos(nθ)=

sin(nθ)

cos(nθ),

(5.9)P��y

sin(2nθ)(

cos θ + sin θsin(nθ)

cos(nθ)

)

= sin(2nθ)cos θ cos(nθ) + sin θ sin(nθ)

cos(nθ)

= sin(2nθ)cos((n− 1)θ)

cos(nθ) �

.h, u�J�

(5.10)cos((n− 1)θ)

cos nθ>

sin θ

θ,

2.5 ë�ÐóÝ�5 109

J(5.7)P�\£Í��P-ÿJÝ�(5.10)��P, ãì�.0-ñÇ�ÿ:

cos(nθ) = cos((n− 1)θ) cos θ − sin((n− 1)θ) sin θ

< cos((n− 1)θ) cos θ < cos((n− 1)θ)θ

sin θ,

Í�àÕP²(ix) cos θ < θ/sin θ�Í�§J±�

bÝ�§5.1, -�J�ì�sineCcosineÐó��52P�

��§§§5.2.E∀u ∈ R, ∫ u

0

cos xdx = sin u,(5.11)

∫ u

0

sin xdx = 1− cos u�(5.12)

JJJ��.&Æ�J(5.11)P�'0 < u ≤ π/2�.cosineÐó3[0, u]� �3, vb(5.1)��P, Æ¿à�§4.1, -ÿÕ(5.11)PE∀u ∈(0, π/2]Wñ�uu = 0, J.(5.11)P�¼��/ 0, Æ(5.11)P)Wñ�uu ∈ [−π/2, 0], J0 ≤ −u ≤ π/2�¿àcosine��ÐóP², ÿ

∫ u

0

cos xdx = −∫ −u

0

cos xdx = − sin(−u) = sin u�

Æ(5.11)PE∀u ∈ [−π/2, π/2]Wñ�uu ∈ [π/2, 3π/2], Ju − π ∈[−π/2, π/2]�.h∫ u

0

cos xdx =

∫ π/2

0

cos xdx +

∫ u

π/2

cos xdx

= sin(π/2) +

∫ u−π

−π/2

cos(x + π)dx = 1−∫ u−π

−π/2

cos xdx

= 1− sin(u− π) + sin(−π/2) = sin u�Æ(5.11)PEu ∈ [−π/2, 3π/2]Wñ�h �� 2π, �.(5.11)

P�¼��, í ø� 2π�ø�Ðó, ÆÿJ(5.11)PE∀u ∈ RWñ�

110 ÏÞa �5��5Ý�+

Íg&Æ¿à(5.11)P¼J�(5.12)P�ã�§4.4�sin(x + π/2) = cos xCcosine �Ðó, ÿ

∫ π/2

0

sin xdx =

∫ 0

−π/2

sin(x + π/2)dx

=

∫ 0

−π/2

cos xdx =

∫ π/2

0

cos xdx = sin(π/2) = 1�

Æ(5.12)PEu = π/2Wñ��E�×u ∈ R,

∫ u

0

sin xdx =

∫ π/2

0

sin xdx +

∫ u

π/2

sin xdx

= 1 +

∫ u−π/2

0

sin(u + π/2)dx = 1 +

∫ u−π/2

0

cos xdx

= 1 + sin(u− π/2) = 1− cos u�

ÇÿJ(5.12)P, J±�

�§5.2èºsineCcosineÐóÝ�52P, ÍJ�4�p, QÎ�3��&Æ.Ý�5C��5ÃÍ�§, Þ��D|2ÿÕhÞ2P�£`��-��º��5ÃÍ�§Ý�æÝ��&Æ�X|)à#.0�hÞ�5, Î Ý¯��Ý�|GXèÄÝ, �5Î�|^b�5, ��}s"Ý�¿à�§4.3, �ÿsineCcosineÐó3×� ��5, Ç

∫ b

a

cos xdx = sin b− sin a = sin x∣∣∣b

a,(5.13)

∫ b

a

sin xdx = −(cos b− cos a) = − cos x∣∣∣b

a,(5.14)

Í�E×Ðóf ,

f(x)∣∣∣b

a= f(b)− f(a)�(5.15)

êÞ 111

hÐr3�5�ð¸à�9ìÝ2P¿à�§4.5-�ÿÕ, &Æº3êÞ�

∫ b

a

cos(kx)dx =1

k(sin(kb)− sin(ka)),(5.16)

∫ b

a

sin(kx)dx = −1

k(cos(kb)− cos(ka))�(5.17)

»»»5.1.O∫ u

0sin2 xdxC

∫ u

0cos2 xdx�

��.´�¿àcos(2x) = 1− 2 sin2 x, �ÿ∫ u

0

sin2 xdx =1

2

∫ u

0

(1− cos(2x))dx

=1

2u− 1

2

∫ u

0

cos(2x)dx =1

2u− 1

4sin(2u),

Í�t¡×�5àÕ(5.16)P, Íg¿àsin2 x + cos2 x = 1, ÿ∫ u

0

cos2 xdx =

∫ u

0

(1− sin2 x)dx

= u−∫ u

0

sin2 xdx =1

2u +

1

4sin(2u)�

êêê ÞÞÞ 2.5

1. �J(5.16)C(5.17)P�

2. �Oì��5�(i)

∫ π/2

0| sin x− cos x|dx, (ii)

∫ π

0|12

+ cos x|dx,

(iii)∫ π/2

0(sin 2x + cos 3x)dx, (iv)

∫ π/2

0(sin2 2x− cos2 3x)dx�

3. (i) �0�sin 3t = 3 sin t− 4 sin3 t, cos 3t = 4 cos3 t− 3 cos t;

(ii) ¿à(i), �O∫ x

0sin3 tdtC

∫ x

0cos3 tdt�

112 ÏÞa �5��5Ý�+

4. �JE×��Ýø�Ðóf , vø�' p,∫ p

0

f(x)dx =

∫ a+p

a

f(x)dx,∀a ∈ R�

5. (i) E�×� 0�Jón, �J∫ 2π

0

sin nxdx =

∫ 2π

0

cos nxdx = 0;

(ii) E�ÞJóm,n, m2 6= n2, �J∫ 2π

0

sin nx cos mxdx =

∫ 2π

0

sin nx sin mxdx

=

∫ 2π

0

cos nx cos mxdx = 0,

∫ 2π

0

sin2 nxdx =

∫ 2π

0

cos2 nxdx = π, n 6= 0�

6. 'x� 2πÝJó¹��Jn∑

k=1

sin(kx) =sin(nx/2) sin ((n + 1)x/2)

sin(x/2) �

2.6 ��555

'Ðóf3 [a, x]��, ∀x ∈ [a, b]�´�&Æá¼, Ðóf3 [a, b]��5, �aCbbn, ùÇ

∫ b

af(u)du ×a, b�Ðó� Ý

?×MÝ�G��5��5 �n;, &Æ�Êf3[a, x]��5, Í�x ∈ [a, b], v�ÐóA ,

(6.1) A(x) =

∫ x

a

f(u)du, x ∈ [a, b]�

ÐóAÌ f�×��555(indefinite integral)��5 ×Ðó, ��5 ×óÂ, Þï�L�!�¨², &Æ�X|1 “×”

2.6 ��5 113

��5, h.A�abn, �!Ýa-�L��!ÝA�uð×�!Ý ì&, ÉA1c, v�

A1(x) =

∫ x

c

f(u)du,

J

A(x)− A1(x) =

∫ x

a

f(u)du−∫ x

c

f(u)du =

∫ c

a

f(u)du,

ÆA(x)−A1(x)�xPn�ð�1, ×Ðó��Þ��5Ý- ×ðó(hðó�aCcbn)�ã×Ðó�×��5, �Õ�Í�×�5�»A, ã»4.1á,

EN×Jón ≥ 1, ∫ x

0

undx =xn+1

n + 1�.h

∫ b

a

undu =

∫ b

0

undu−∫ a

0

undu =bn+1 − an+1

n + 1 �

×��, uáf�×��5A, J

(6.2)

∫ s

c

f(u)du = A(s)− A(c) = A(u)|sc�×ÐófCÍ��5b×��Ý¿¢n;��5A(x)�

3`ay = f(u)�u�� ãu = a�x �“«�”�ÇufºãÑÂC�Â,JA(x)�3u�î]�«�3*u�ì]�«��ôµÎÞu�ì]�«�Ú �Â��%6.1�

A(x) =∫ x

af(u)du

∫ x

af(u)du =«��óõ

f(u)

O a xu

f(u)

O a xu

++

%6.1. ��5�«��¿¢n;

114 ÏÞa �5��5Ý�+

ó.�, b&9Ðó, Î|ØÐó�×��5Ý�P�¨�¿à��5, ô�C��KbàÝ±Ðó�9Î��5�, ��KS»D¡��5Ýx�æ.�×�9ì&Æ¼:��5�×°P²�´�ì��§, ¼��

�5 ×=�Ðó�

��§§§6.1.'b&Ðóf3[a, b]��, vE∀x ∈ [a, b], ÐóA��LA(6.1)P�JA3∀p ∈ [a, b] =�(3p = aTp = bJ ��=�)�JJJ��.E∀p ∈ [a, b], &Æ�J�x → p`, A(x) → A(p)��.f b&, ÆD3×K > 0, ¸ÿ|f(u)| ≤ K, ∀u ∈ [a, b]�.h(¿à�§4.1), ux > p, J

|A(x)− A(p)| = |∫ x

p

f(u)du| ≤∫ x

p

|f(u)|du

≤∫ x

p

Kdu = K(x− p)�

ux < p, J!§�J�|A(x)− A(p)| ≤ K(p− x)�Çb

|A(x)− A(p)| ≤ K|x− p|�îPÇ0l

limx→p

A(x) = A(p)�

Q3î�D¡�, up = aTb, &Æ©�ÿÕ��=��J±�

»»»6.1.31.5;�JÄsineCcosineÐó/ =��u¿à�§6.1,

Cî;J�Ý(5.11)C(5.12)P,ù�ÿsineCcosine/ =�Ðó�

b`á¼ÐófÝØ×P², ù�0�ETÝ��5�×©�P²�»A, uf3[a, b] &�, J.

A(y)− A(x) =

∫ y

x

f(t)dt ≥ 0, ∀a ≤ x ≤ y ≤ b,

2.6 ��5 115

ÆA3[a, b] ×¦Ðó�¿¢îÝ�LÇ , E×&�Ðó, 3Í%�ìãa�xÝ«�, �½x�¦��¦�¨b×P², µ�Î£��|�ã¿¢¼1��&Æ�m�ì

��L�

��LLL6.1.uE∀x, y ∈ [a, b], vα ∈ (0, 1), Ðóg��

(6.3) g(z) ≤ αg(y) + (1− α)g(x),

Í�z = αy+(1−α)x,JÌg3[a, b] ��ÐÐÐóóó(convex function)��u(6.3)P��rDÄ¼, JÌg3[a, b] ��ÐÐÐóóó(concave funct-

ion)�

&Æ¯�1�Aì�Ex < y, uz = αy + (1 − α)x, Jz − x =

α(y − x), ÇzûxÝûÒ [x, y]��α¹�αã0�1, zÇãxÉ�y, �(z, αg(y) + (1− αg(x))Çº½=#(x, g(x)), (y, g(y))�aðã¼?�É��P(6.3)µÎ1, g�%�3G�aðÝì]��y�Ðó, JÎÐó%�3=#að�î]�E×�Ðó,

=#%�î�ËF, JhÞF �%�3haðÝì], �ÐóJDÄ¼�%6.2 α = 1/2Ý��

g(x)+g(y)2

g(x) g(x+y2

) g(y)

x x+y2

y

g(x+y2

)

g(x) g(x)+g(y)2

g(y)

x x+y2

y

(a)�Ðó (b)�Ðó

%6.2. |¿¢%�¼1��Ðó��Ðó

116 ÏÞa �5��5Ý�+

��§§§6.2.'f [a, b]î�×��Ðó��A(x) =∫ x

af(u)du, x ≥

a�(i) 'f ¦Ðó, JA �Ðó;

(ii) 'f 3Ðó, JA �Ðó�JJJ��.&Æ©J(i), .E×3Ðó, −f ¦Ðó, ¿à(i)y−fÇ�ÿÕ(ii)�'f3[a, b]�¦�Ea ≤ x < y ≤ b, C0 < α < 1, �z = αy +

(1− α)x�&Æ�J�

(6.4) A(z) ≤ αA(y) + (1− α)A(x)�

ê.A(z) = αA(z) + (1− α)A(z), Æu�J�

αA(z) + (1− α)A(z) ≤ αA(y) + (1− α)A(x),

T(1− α)(A(z)− A(x)) ≤ α(A(y)− A(z))

Ç��.A(y)− A(z) =∫ y

zf(u)du,A(z)− A(x) =

∫ z

xf(u)du, Ç

�J�

(6.5) (1− α)

∫ z

x

f(u)du ≤ α

∫ y

z

f(u)du�

.f �¦, Æ

f(u) ≤ f(z),∀x ≤ u ≤ z, f(z) ≤ f(u),∀z ≤ u ≤ y�

ãhBÄ�5(¿à�§4.6), ÿ∫ z

x

f(u)du ≤ f(z)(z − x), f(z)(y − z) ≤∫ y

z

f(u)du�

¬z = αy + (1− α)x, ê�;¶ (1− α)(z − x) = α(y − z), Æî�Þ��Pê0l

(1− α)

∫ z

x

f(u)du ≤ (1− α)f(z)(z − x) = αf(z)(y − z)

≤ α

∫ y

z

f(u)du�

êÞ 117

.h(6.5)PWñ, �(i)ôµJ�Ý�

»»»6.2.ãycosineÐó3[0, π] �3,Æsin x =∫ x

0cos udu3[0, π]

�Ðó��3[π, 2π], cosine �¦, Æ3[π, 2π], sine �Ðó�

êêê ÞÞÞ 2.6

1. �Oì�&�5�(i)

∫ 2x

−1(1 + t + t2)dt, (ii)

∫ x2

x(1

2− sin t)dt,

(iii)∫ x2

x(u2 + sin 3u)du, (iv)

∫ x

−π(1 + cos 2u)2du�

2. 'f ×�Ýø�Ðó, ø� 2, ê'f3�× /��,

�g(x) =∫ x

0f(u)du�

(i) �JEN×Jón, g(2n) = 0;

(ii) �Jg �Ðó, v ø�2�ø�Ðó�

3. '�Ðóf ×ø� 2�ø�Ðó�ê'f3�× /��g(x) =

∫ x

0f(u)du�

(i) �Jg �Ðó, vg(x + 2) = g(x) + g(2);

(ii) �|g(1)¼�îg(2)Cg(5);

(iii) �Og(1)�Â, ¸ÿg ø�2�ø�Ðó�

4. 'fCg3�× /��, vf �Ðó, g �Ðó�ê'f(5) = 7, f(0) = 0, g(x) = f(x + 5), f(x) =

∫ x

0g(u)du,∀x ∈

R��J(i) f(x− 5) = −g(x),∀x ∈ R;

(ii)∫ 5

0f(u)du = 7;

(iii)∫ x

0f(u)du = g(0)− g(x)�

5. �f(x) = [x], Í�[ · ] {úÐr��0F (x) =∫ x

0f(u)du,

x ∈ [0, 5], �%��:�4f� =�Ðó, ¬FQ =�Ðó, v@@@ðððaaaPPP(piecewise linear)�

118 ÏÞa �5��5Ý�+

6. 'f(x) = x− [x]− 1/2, ux� Jó, vf(x) = 0, ux ×Jó, Í�[ · ] {úÐr��F (x) =

∫ x

0f(u)du, u ∈ R�

(i) �Jf(x + 1) = f(x), ∀x ∈ R, v0f3x ∈ [0, 1]�%;

(ii) �JF (x) = (x2 − x)/2, ∀x ∈ [0, 1], vF ø� 1�ø�Ðó;

(iii) �|[ · ]¼�îF (x)�

2.7 000óóóÝÝÝ��LLLCCCÃÃÃÍÍÍPPP²²²

3�.Ýó.�&Æ.Ä, u33�`��Ý122¦, J>� 12 ÷ 3 = 4, ÇN�`42¦�9�«Í@Î�Ý×Í�', ÇÎ|�>3��©�Î�>��Ý, í�|h]PO�>��Q, �¶ð^�T_�ÝB��, ��3N×`ÑÝ>�«{�¼8!, ôµÎ��¬&|�>3��3Î§�, &Æô.Ä�ãa�ô¬&|�>aì�E��>Ýº�, &Æµ��©1>�Ý, �Ä6ý�Î£×`ÑÝ>��yA¢OØ×`Ñ�>�÷?

'Ø_�3{>2î�ù, v�f(x)�3` x, �ØF�R_��H�J�` x�` x + h, _��Ýf(x + h)− f(x), .h39ð� �¿í>�

f(x + h)− f(x)

h �

�:��.�O�ÝXÛ>�, Í@Î¿í>��î�h¬�×��y0, uh < 0, Jx + h < x, h`

f(x + h)− f(x)

h=

f(x)− f(x + h)

−h

��` x+h�x,_��¿í>��.h�¡h ÑT�, (f(x + h)

−f(x))/h�3` x!�×` , _��¿í>��|h|�|��

2.7 0óÝ�LCÃÍP² 119

�, ¬�� 0, �u©á3x��Hf(x), ÎP¿í>��u�h → 0, (f(x + h)−f(x))/hº��Õ%�? Q, �Xf�ÎhÁ§�D3��uD3, hÁ§��Ý�L ¢? ¯��ÕÝ, �ÞhÁ§Ú 3` x�� >>>��(instantaneous velocity)�ö�ó.ÝÌF¼:, G�f(x)�×��ûÒ, ��3`

x�ØË�, A{��á�, Cß®��Æ(f(x + h)− f(x))/h,

Ç�3` [x, x + h] �¿í¦�Ý��h → 0, �Á§Ç �`¦��, T1�;£�&Æ�×�LAì�

��LLL7.1.Ðóf3x�000óóó(derivative), |f ′(x)��(\�f prime of

x), Í�L

(7.1) f ′(x) = limh→0

f(x + h)− f(x)

h,

©�î�Á§D3�f ′(x)êÌ f3x��;£(the rate of change

off at x)��u(7.1)P�Á§D3,-Ìf3x��(differentiable)�uf3�L½�NF/��, -1fÎ×��ÐÐÐóóó(differentiable

function),T1f��uf3x=�,Jf ′+(x) = limh→0+(f(x+h)−f(x))/h, f ′−(x) = limh→0− (f(x + h)− f(x))/h, 5½Ì f3x��0óC¼0ó, Þï/Ì��000óóó(one-sided derivative)�êuf ′+(x) = ∞, vf ′−(x) = ∞, JÌf3x�0óf ′(x) = ∞(.∞¬&×@ó, Æh`0ó¬�D3)�!§��Lf ′(x) = −∞�

ñÇ�:�f3x��, uv°uf3x��0óC¼0ó/D3,

vÞï8��'−∞ ≤ a < b ≤ ∞, ufE(a, b)�N×F/��, &Æ-1f3(a, b)��ua 6= −∞, �f3(a, b)��, v3x = a��0óD3, J1f3[a, b)��, !§��Lf3(a, b]T[a, b]��G«9gèÄ, pñC¾¾¹+Ý¥�Q¤, �)�5��5��5�Ýx��°µÎ0óÝÃF�A!�5ÎRÙy¿¢®Þ�ÝO«�, 0óôÎRÙy¿¢.�, AO3¿«î×`aîØF�6aE£�¬��5RÙ�\, ô�èÚtS�, ðykX�Ø°Ðó�Á�CÁ�Â, �bÝ0óÝÃF�

120 ÏÞa �5��5Ý�+

��3�.`�K.Ä666aaa(tangent line)�3iîØF, ��FCiT8=aðkàÝàa, -ÎiÝ×6a��ÄYiîÝ6a, #��×àîÝ6a, µP°Ah�LÝ, &Æ�#å6aÝÃF?Ý�.AiÝ²6Ñ9\��, &ÆÄ�íD|2#å, �Î_~Í�L��'¿«îb×à, v3àîN×F/b×6aD3�%7.1|Ìaà0�3¿ÍFÝ6a�ðy¥�Õ3àîÝÁ�CÁ��, 6a i¿�.hu�0ÁÂ, «{©�0�¢�bi¿6aÇ��

x0 x1

%7.1. à3x0Cx1bi¿6a

¯��º��¡¼Ý��Ý�ðy�ó.�.Âî��°, �Oà3�¢×F�6aE£, Q¡Qµ×à"Dì�, âys"��5.�y:�ì, �5��5Îm�8�Ý�OàìX��Ý«

�, «{�Î�OàîØF�6aE£b¢nÐ�Ï×�s¨ÕhÞïTbÛ6n;ÝÎpñÝ�/Barrow�¬pñC¾¾¹+ ´��ºÕÞï� n;�¥�P, ¬��ñRÍ �n;�hn;(��5ÃÍ�§), -Þó.�s"�x�×èGÝ±S-�4Q0óÎRÙyO6aÝ®Þ, ¬AÍ;×��XD¡Ý, ¡

¼ñÇs¨0óôèº×O>�C&Ë�;£Ý]°�3�×°»��G, !ñ×è(7.1)P�ìP��:

2.7 0óÝ�LCÃÍP² 121

(7.2) f ′(x) = limu→x

f(u)− f(x)

u− x �

b`&Æº|îP¼O0ó�

»»»7.1.�f(x) = x2, �Of ′��.E∀x ∈ R,

limh→0

f(x + h)− f(x)

h= lim

h→0

(x + h)2 − x2

h= lim

h→0(2x + h) = 2x

D3�Æf ′(x) = 2x, ∀x ∈ R, Çf ×��Ðó�

»»»7.2.�f(x) = 1/x, �Of ′��.E∀x 6= 0,

limh→0

f(x + h)− f(x)

h= lim

h→0

(x + h)−1 − x−1

h

= limh→0

−h

hx(x + h)= − 1

x2�

Æf ′(x) = −x−2, ∀x 6= 0�

»»»7.3.�f(x) =√

x, �Of ′�JJJ��.E∀x > 0,

limh→0

f(x + h)− f(x)

h= lim

h→0

√x + h−√x

h

= limh→0

x + h− x

h(√

x +√

x + h)=

1

2√

x�

î�Á§E∀x > 0/Wñ�Æf ′(x) = (2√

x)−1,∀x > 0�

3î»�, 4f��L½ [0,∞), ¬f30��B�5¡Xÿ�f ′ù ×Ðó, ¬�L½�×��f��L½8!��2ý,

f ′ ��L½= f ��L½ \ {f��ÝF}�

122 ÏÞa �5��5Ý�+

»»»7.4.�f(x) = xn, Í�n ×ÑJó, �Of ′��.E∀x ∈ R,

limh→0

f(x + h)− f(x)

h= lim

h→0

(x + h)n − xn

h= nxn−1,

&Æ6¯Ý� Ý�ÕÄ��Æf ′(x) = nxn−1, ∀x ∈ R�

»»»7.5.'f(x) = c, ∀x ∈ R, �Of ′��..f(x + h)− f(x) = c− c = 0, Æf ′(x) = 0,∀x ∈ R�

ÞîÞ»�), ÇÿEN×&�Jón,

(7.3) (xn)′ = nxn−1�

»»»7.6.�f(x) = sin x, �Of ′��.´�

f(x + h)− f(x) = sin(x + h)− sin x = 2 sin(h/2) cos(x + h/2)�

êh → 0`,

sin(h/2)

h=

sin(h/2)

h/2

1

2→ 1

2, cos(x + h/2) → cos x�

Æ

limh→0

f(x + h)− f(x)

h= 2 · 1

2· cos x = cos x�

.hsineÐó�0ó cosineÐó, &Æ�|

(7.4) (sin x)′ = cos x, x ∈ R,

��

»»»7.7.�f(x) = cos x, �Of ′��.ûî», ¿à

cos(x + h)− cos x = −2 sin(h/2) sin(x + h/2),

2.7 0óÝ�LCÃÍP² 123

�pÿÕf ′(x) = − sin x�&Æ|

(7.5) (cos x)′ = − sin x, x ∈ R,

��

�yÍ�ë�ÐóÝ�5&Æº3êÞ�3Ï2.5;�, &Æ�:ÕsineÐó�cosineÐó��5n;Û6�îÞ»¼�Þï��5ùb��Ýn;�uÞËÍ��¿3×R:, �5��5Ý! �YºÕ, �2Qu¨�0óÉQ��6aE£(4&Æ$Î1�A¢ã0óO6a),

àÌî×Ðó3ØFÄ6=�, ��b6aD3, ô��9µÎ9ìÝ��

��§§§7.1.'Ðóf3Fx��, Jf3x=��JJJ��.�bì��P:

f(x + h)− f(x) = hf(x + h)− f(x)

h, h 6= 0�

�h → 0, .(f(x + h)− f(x))/h → f ′(x), Æ

f(x + h)− f(x) → 0 · f ′(x) = 0�

ÿJ�

�Ä=�Ðó�×��»A, �f(x) = |x|, Jf ×Õ�=��Ðó�¬.

(f(0 + h)− f(0))/h = |h|/h,

h → 0`, Á§�D3(�Á§ 1¼Á§ −1), Æ3x = 0��

124 ÏÞa �5��5Ý�+

-

6

f(x) = |x|

x

y

O

%7.2. f(x) = |x|�%�

��Î×Íf=��úÝf��uá×Ðó3ØF��, -áhÐó3�Fù=�Ý�×3ØFx��Ðó, B�5¡Xÿ�Ðó, �×�)3x��, #�ô�×�3x=��0ó ×±ÝÐó, æ¼ÐóbÝP², 0ó�×�ºb, 3êÞ��:Õ×°»��0óô�ã¿¢¼�Õ�(f(a + h)− f(a))/h, �3y = f(x) �

%�î,=#(a, f(a))�(a+h, f(a+h))ÞF�àa(Ì vvvaaa(secant

line))�E£��h → 0, ôµÎ¯a+ h×à#�a, uG�vaE£�Á§D3, JÁ§`ÝvaµÚ 3(a, f(a))�6a, �%7.3�

-

6

O a a + hx

y

%7.3.

uh → 0`, |(f(a + h) − f(a))/h| → ∞, Jh → 0`, vaº�¼�q�h`3(a, f(a))�6aÇ�L kàax = a��yEf(x) = |x|, 3x = 0¬P6aD3�h.3%�î=#(0, 0)�Í

2.7 0óÝ�LCÃÍP² 125

��×F�vaE£ 1, �=#(0, 0)�Í¼��×F�vaE£ −1, ÆvaE£�Á§�D3, .hf3x = 0��&Æ¶×�LAì�

��LLL7.2.3y = f(x)�%�î×F(a, f(a))�6a (i) Ä(a, f(a))vE£ f ′(a)�àa, uf ′(a)D3;

(ii) àax = a, ulimh→0 |(f(x + h)− f(x))/h| = ∞�tÝ(i)T(ii)Ý�µ, %�3(a, f(a))�6a�D3�

�p:�3(i)Ý�µ, 6a]�P

(7.6) y − f(a) = f ′(a)(x− a)��y3(a, f(a))�°°°aaa(normal line), Í�L Ä(a, f(a))v�6akà�àa�Æuf ′(a) 6= 0, J°aE£ −1/f ′(a), v]�P

(7.7) y − f(a) = − 1

f ′(a)(x− a)�

uf ′(a) = 0, J°a kàax = 0, u6a kàax = 0, J°a i¿ay = f(a)�

»»»7.8.�O3f(x) = x2�%�î, 3F(2, 4)�6aC°a��..f ′(x) = 2x, Æf ′(2) = 4�.h6a

y − 4 = 4(x− 2)��°a

y − 4 = −1

4(x− 2)�

A!OÁ§`, b×ny°JºÕÝ�§, 0óùbì�ETÝ�§�

��§§§7.2.'ÞÐóf�g, b8!��L½, v'f�g/3ØFx��Jf + g, f − g, fg, f/g/3x��(Eyf/g, g(x)6� 0)�êh`

126 ÏÞa �5��5Ý�+

(i) (f + g)′ = f ′ + g′,

(ii) (f − g)′ = f ′ − g′,

(iii) (fg)′ = f ′g + fg′,

(iv) (f/g)′ = (gf ′ − fg′)/g2�JJJ��.(i) kOf + g3x�0ó, �¶�ìP�

f(x + h) + g(x + h)− (f(x) + g(x))

h

=f(x + h)− f(x)

h+

g(x + h)− g(x)

h �h → 0`, .fCg/3x��, .hîP��Ë45½��f ′(x)

Cg′(x)�ÆÿJ�(ii) û(i)-�ÿJ�(iii) µ�L�O

f(x + h)g(x + h)− f(x)g(x)

h

h → 0`�Á§�BÄ�×4C3×4g(x)f(x + h), ÿh → 0`,

f(x + h)g(x + h)− f(x)g(x)

h

= g(x)f(x + h)− f(x)

h+ f(x + h)

g(x + h)− g(x)

h→ g(x)f ′(x) + f(x)g′(x)�

h�àÕ��0�=�, X|h → 0`, f(x + h) → f(x)�ÿJ�(iv) &Æ�J(iv)�×©»(Çãf(x) ≡ 1)

(7.8)

(1

g

)′= − g′

g2�

hPuJ�, J¿à(iii), ãìP-ÿJÝ�(

f1

g

)′=

1

gf ′ + f

(1

g

)′=

f ′

g− fg′

g2=

gf ′ − fg′

g2 �

�(7.8)PêãìP-ñÇ:�ÿJ�

(7.9)1/g(x + h)− 1/g(x)

h= −g(x + h)− g(x)

h

1

g(x)

1

g(x + h)�

2.7 0óÝ�LCÃÍP² 127

h�)àÕ.g3x��,X|g3x=�,.hh → 0`, g(x+h) →g(x)�

ug(x) ðóÐó, Ag(x) = c, ∀x ∈ R, J.g′(x) = 0, Æãî�§�(iii)ÿ

(7.10) (cf)′ = cf ′�

h��)(ii), -ÿì�.¡�

��§§§7.1.'f1, f2, · · · , fn/3x��, c1, c2, · · · , cn ðó, J

(7.11)

(n∑

i=1

cifi

)′

=n∑

i=1

cif′i�

!ñ×è, k(7.9)Pb�L, EÈ�Ýh, g(x + h)�� 0��.�ág(x) 6= 0, ÆãÏ×aS§6.1, 1J9�¯^®Þ�êãî�§�(iii), ù�pÿÕì�.¡�

��§§§7.2.'f1, f2, · · · , fn/3x��, J

(7.12) (n∏

i=1

fi)′ =

n∑i=1

(f ′i∏

j 6=i

fj)�

îP�∏

j 6=i fj�tÝfi, Þf1, f2, · · · , fn8¶�

»»»7.9.'b×94Pf(x) =∑n

i=0 cixi��ã(7.3)P(xi)′ = ixi−1,∀i

≥ 0, �ã�§7.1Çÿ

f ′(x) =n∑

i=0

icixi−1 =

n∑i=1

icixi−1,

×n− 1g�94P�

»»»7.10.'f(x) = p(x)/q(x) ×b§P, Í�p(x)�q(x)/ 94P, q(x) 6= 0�Jã�§7.2�(iv)�ÿf ′(x)�©½2, up(x) = 1,

128 ÏÞa �5��5Ý�+

q(x) = xm, Çf(x) = 1/xm,m ≥ 1, x 6= 0, J

f ′(x) =xm · 0−mxm−1

x2m=−m

xm+1= −mx−m−1�

ãîPC(7.3)P-ÿì�´×�ÝEN×Jón,

(7.13) (xn)′ = nxn−1�

Qun ≤ −1, Jx6� 0�

»»»7.11.'f(x) = sin x/(x2 − 4), J

f ′(x) =(x2 − 4) cos x− 2x sin x

(x2 − 4)2 �

�yf ′Ý�L½�f��L½8!, / R \ {2,−2}�

»»»7.12.3(7.13)P�, &ÆÿÕEN×Jón, xn�0ó��yf(x)

= xα, x > 0, Í�α = p/q ×b§ó, Í0ó ¢? �'α Ñ, v'p, q ÞÑJó�J

(7.14)f(x + h)− f(x)

h=

(x + h)p/q − xp/q

h �

�x1/q = ξ, (x + h)1/q = ξ1�ûÏ×a»4.1�®°, �ÿ

(7.15) limx→a

x1/q = a1/q, a ≥ 0�

Ælimh→0(x + h)1/q = x1/q, Çh → 0`, ξ1 → ξ�.h(7.14)PW

f(x + h)− f(x)

h=

ξp1 − ξp

ξq1 − ξq

=ξp−11 + ξp−2

1 ξ + · · ·+ ξp−1

ξq−11 + ξq−2

1 ξ + · · ·+ ξq−1�

ãîPÇÿ(¿àh → 0`, ξ1 → ξ)

limh→0

f(x + h)− f(x)

h=

pξp−1

qξq−1= αξp−q = αxα−1�

2.7 0óÝ�LCÃÍP² 129

�yuα < 0, )b8!Ý��, hI5º3êÞ��ÆÿEN×b§óα(uα < 1, ��ÝP��|D¡, �p, q�Âbn),

(7.16) (xα)′ = αxα−1�

»»»7.13.'y = f(x) = x1/3, x ≥ 0, Jf ′(x) = 13x−2/3, x > 0, f3x =

0��0ó�D3�.h → 0`, |(f(0 + h) − f(0))/h| → ∞, Æ�Lf3(0, 0)�6a kàax = 0��%7.4�

-

6

Ox

yf(x) = x1/3

%7.4. f(x) = x1/3 �%

¨², uf(x) = x3, x ∈ R, f ′(x) = 3x2, f ′(0) = 0, Æf3(0, 0)�6a i¿ay = 0, �°a kàax = 0�\ï��0h`y =

f(x)�%��yuf(x) =

√x, hÐó�L3x ≥ 0�3x = 0, f(x)P¼0

ó, v�0ó ∞(.h�D3)�3x = 0�6aÇ y��t¡'y = f(x) =

3√

x2 = x2/3, x ≥ 0�Jf30��0ó +∞(X|�D3)�A%7.5�:�h`y = f(x)�%�3x =

0�6aù y��EXÛ5ð�LÝÐó, 3=#F�0ó��T�§, �ì»�

»»»7.14.'

f(x) =

{x2 + 2, x ≥ 0,

3 cos x− 1, x < 0�

130 ÏÞa �5��5Ý�+

-

6

Ox

yf(x) = x2/3

%7.5. f(x) = x2/3

Ex > 0, f(x) = x2 + 2, Æf ′(x) = 2x; Ex < 0, f(x) = 3 cos x − 1,

Æf ′(x) = −3 sin x, /P®Þ��yf3x = 0�0ó÷? ��.x ≥ 0`, f(x) = x2 +2, µ| f ′(0) = 2 · 0 = 0�.3x = 0�0ó�30!�f�Â/bn, �30�¼�, f¬&x2 + 2 ÝlP�Æ©?ã�L¼O3x = 0�0ó�´�f(0) = limx→0+ f(x) =

limx→0− f(x) = 2, Æf3x = 0=��uf3x = 0�=�(ÉA13x ≥ 0�, Þf; f(x) = x2 + 3), Jf3x = 0Ä��, £Í®Þ-��XÝ�¨3.f3x = 0=�, X|$6µ�D¡�&Æ5½�Ê3x = 0��0óC¼0ó, ÿ

f ′+(0) = limh→0+

f(h)− f(0)

h= lim

h→0+

h2 + 2− 2

h= 0,

f ′−(0) = limh→0−

f(h)− f(0)

h= lim

h→0−3 cos h− 1− 2

h

= 3 limh→0−

cos h− 1

h= 0,

Þï8�v/ 0, Æf ′(0) = 0D3�Ç

f ′(x) =

{2x, x ≥ 0,

−3 sin x, x < 0�

êf ′) ×Õ�=��Ðó, vûî�D¡�ÿ, f ′tÝ3x = 0²/��

2.7 0óÝ�LCÃÍP² 131

\ïô��J,u};�×ìf ,�f(x) = x2 +2x+2, x ≥ 0,

Íõ��Jf3x 6= 0�/��, �3x = 0)=�¬��ú�×g, E9Ë5ð�LÝÐó, 3ø#F(AÍ»��x =

0)�0ó, �©½º��G«�1ÄEÍ», Îf ′(x) = 2x, “∀x >

0”, �&f ′(x) = 2x, “∀x ≥ 0”��v, ô��.hµñÇì�¡

f ′+(0) = limx→0+

f ′(x) = limx→0+

(2x) = 0

(4QÍ»�9�nÎEÝ)�Ü»��, 'b×Ðóf(x) = 1,

∀x > 0, f(x) = −1, ∀x < 0, vf(0) = 0�.f3x = 0�=�,

Æf3x = 0Q��êf ′(x) = 0, ∀x 6= 0, Ælimx→0+ f ′(x) =

limx→0− f ′(x) = 0�.h

limx→0+

f ′(x) 6= f′+(0) = lim

h→0+

f(h)− f(0)

h= lim

h→0+

1

h= ∞,

v

limx→0−

f ′(x) 6= f′−(0) = lim

h→0−f(h)− f(0)

h= lim

h→0−−1

h= ∞�

�Ä3Ø°f�ì, E×Ðóf , limx→a f ′(x) = f ′(a)ÎºWñÝ, �Ï°a�§1.6�êÞ�$b×°ny5ð�LÐó�0ó,

�º&�Yê�

Í;t¡, &ÆE�5ÝÐr�×°��1��3ó.Ýs"�,Ðr×à6�½¥�Ý��+�−�×�÷

� = �√�e�π�i�, /ÎÐr�b°ÐrAxn, n!, �Þ×ð

��ÝB�, |×��Ý�P¼�î�Í�A∫ b

af(x)dx, ¬�©Î

èø&Æ�5, ÎA¢ÿÕ(Ç∑

f(xi)∆xi�Á§), ¬¯&Æ�Æ��Õ, |O�5Â��Äb`��b¿Í�!ÝÐr, /�8!Ý�L, Ú�!�

µ�2à��5�-b¿ÍÐr¬D�G«2àÝBrf ′ , La-

grange (1736-1812)3èâtSÏXS�hBrú��5¡, ÿÕ×±ÝÐó, �f ′3x�Â, |f ′(x)��u�y = f(x), Jy′ô

132 ÏÞa �5��5Ý�+

��0óf ′(x)�ãyf ′) Ðó, Æ&Æ�EÐóf ′��5, �ÿÕÞÞÞ$$$000óóó(second derivative), Çf ′′ = (f ′)′��f ′′�0ó, -Îë$0ó, Çf ′′′ = (f ′′)′�×��f (4) = (f ′′′)′, · · · , f (n) =

(f (n−1))′, T|y′′, y′′′, y(4), · · · , y(n)��©�XÿÐó)��, -�µ��5, �ÿì×$0ó, 9°ÙÌ{{{$$$000óóó(higher deriva-

tives)�LagrangeÝÐr,�pñX2àÝy, y,· · · ,-²¬��3Î§îÝ>�C�>�, )2àpñÝÐr�¨²$b×°Ðr, A3�-1800O, Arbogast (1759-1803),

|Df�f�0ó, hÐrôûÅ2�¸à�ÐrD-Ì ×��555ºººÕÕÕ(differentiation operation), hÐr×å&ÆDf ×ãfB�5¡, ÿÕÝ±Ðó�{$0óJ|D2f,D3f, · · · , D4f��3x�Â|Df(x), D2f(x) , · · · , ��3DÐrì, ��|Þ|GÝ2P»ðÄ¼�A

D cos x = − sin x, D2 cos x = D(− sin x) = − cos x,

D(f + g) = Df + Dg�

¬�º�, D2 Þg�5, D2f = D(Df)��Î(Df)2�3ó.5�]«Ý\�s"�, ¾¾¹+ÕÎt��º, ��ó

ãÐr�¥�Ýó.��Bð��9�G, �Í�ó.�D¡&ËÐr�8��5�X|�3¨�ó.Ýs"�®ß¥�ÝÅ(, I5æ.Ç Í��C°bÐ�æÝÐr, �9Í�9�Ù�¾¾¹+�¾¾¹+s"�×��G«Xè]Q�!ÝÐr��y = f(x),

�|∆y

∆x�î(f(x + h)− f(x))/h, ¬Ì� ---555¤¤¤(difference quotient), Ç|∆y�(f(x + h)− f(x)), ∆x�h�Ðr∆ Ì ---555ºººÕÕÕ(difference

operator)�3Á§`, Ç�h → 0, -5¤��f ′(x), ¾¾¹+| dy

dx(Tdy/dx)�hÁ§, Ç

dy

dx= lim

∆x→0

∆y

∆x�

2.7 0óÝ�LCÃÍP² 133

¾¾¹+�©2à�!Ý�5Ðr, �E0óÝ�°ô��ß�!��- ËÍ“P§�Ý”�dy�dx(Ì��555ÝÝÝ(differentials))

�¤, Ç dy/dx, �¬Þ0ódy/dxÌ ��555¤¤¤(differential quo-

tient)�¾¾¹+Þdy, dx�W��±Ýó, 4�Î0, ¬Í�EÂQ�y��Ñó�4Q¾¾¹+¬P°Eî�P§�Ý�ÃF, ��×�ß´�

��L, �CÍ�4, Q��ã2ºàhËÐr, s"Í��5Ý§¡�¬b&9ßQ.hÆÿ��5b°��¤È(dy/dxJÍÎ×ÍÐr, �&dyt|dx, ¬b`ê�t�Wdy, dx), ¬��¶£°ºÕ]°ÝÑ@P�àÕèÜtS, Þ�CÍ�ó.�, �@�|Á§ÝÃF¼ã�“P§�Ý”hÃF�¬P¡A¢, ¾¾¹+ÝP§�Ý�ÃF, E&9ßÝ��5Qb8QÃ�9Ë¤îÝ]P, àÌî´|#å, ¬v�´"2ÿÕ@jîÎÑ@Ý��4Qb°¾¾¹+Ý�°, ¡¼�s¨¬�Ñ@, ¬�X2àÝ

ÐrQÂ øF�|Ðrdy/dx¼�î0ó, ��î�ÿÕ0óÝÄ��|¡&Æºµ�s¨, 2àhÐr, b°2P�´D|B�C�ºà�¿à¾¾¹+ÝÐr, bì��î°:

f ′(x) =dy

dx, f ′′(x) =

d2y

dx2, f ′′′(x) =

d3y

dx3, · · · , f (n)(x) =

dny

dxn�u|DÝÐrJ DnyTDnf�ubÞ��Ðóu = f(x), v = g(x),

J

d

dx(u + v) =

du

dx+

dv

dx,

d

dx(uv) = u

dv

dx+ v

du

dx�

�b`ô| dfdx¼�îf ′�uy = f(x), ��î0ó3ØFa�Â, ô

�2àdy

dx

∣∣∣∣x=a

, Tdf

dx

∣∣∣∣x=a�

t¡&Æ1�, ¾¾¹+Þ$0óÐrÝã¼, Íõ{$0óÝÐrô�v«2ÿÕ�

134 ÏÞa �5��5Ý�+

¾¾¹+ÞÞ$0ó, Ú Þ$-5¤�Á§��x1 = x + h,

x2 = x+2h�XÛÞ$-5¤, Ç ×$-5¤�×$-5¤�Ç

1

h(y2 − y1

h− y1 − y

h) =

1

h2(y2 − 2y1 + y),

Í�y = f(x), y1 = f(x1), vy2 = f(x2)�u|∆x�h, ∆y1 =

y2 − y1, v∆y = y1 − y, J

y2 − 2y1 + y = ∆y1 −∆y = ∆(∆y) = ∆2y�

Çy2 − 2y1 + y�Ú y�-5(Ç∆y)Ý-5, ÇÞ$-5�.h¿àî�Ðr, Þ$-5¤�¶ ∆2y/(∆x)2, Í�5Ò ∆xÝ¿],

�5� y�Þ$-5, Ë�Ý“2”��L�!�Þ$0ó.h��î

f ′′(x) = lim∆x→0

∆2y

(∆x)2�

îP- ¾¾¹+|d2y/dx2�Þ$0óÝã¼�!ñ×è, ∆∆ = ∆2�-5Ý-5, ÇÞ$-5�êÞ$-5

¤ÝÁ§, - Þ$0ó, h¯Í@$6J��.&ÆÎÞÞ$0ó�L , ×$0ó�×$-5¤ÝÁ§, Ç

f ′′(x) = limh→0

f ′(x + h)− f ′(x)

h= lim

h→0

∆f ′

h �

©�Þ$0ó=�, hÞ�L-��Ä.&Æ©Î�¯��E¾¾¹+ÝÐrbÍÃF, .h-¯�J��

êêê ÞÞÞ 2.7

1. �Oì�&Ðó�0ó, Cf ′��L½, �¿à�áÝ�§�

êÞ 135

(i) f(x) = x3 + sin x,

(ii) f(x) = x3/2 sin x,

(iii) f(x) = (x + 1)−1, x 6= −1,

(iv) f(x) = (x2 + 1)−1 + x5 cos x,

(v) f(x) = (2 + cos x)−1,

(vi) f(x) = (x2 + x sin x)/(x3 + cos x),

(vii) f(x) =√

x/(1 + x2),

(viii) f(x) = x/(1 +√

x),

(ix) f(x) = (1 + 2x−1)(2 + x−2),

(x) f(x) = (x2 − x−2)2�2. µ�L�Oì�&Ðó�0ó, ¬��f ′��L½�

(i) f(x) =√

3x− 2, (ii) f(x) = (2x− 5)−1/2,

(iii) f(x) = sin(x2), (iv) f(x) = cos(x3)�3. ��J

(tan x)′ = sec2 x, (sec x)′ = tan x sec x,

(cot x)′ = − csc2 x, (csc x)′ = − cot x csc x,

Q9°P�WñÝ�Xf�Î, N×P��Ýx�¸¼�/b�L�

4. �Oì�Ðó��5, Í�Ðó/�L3¸Íb�L��(i) f(x) = tan x sec x, (ii) f(x) = sin x/x,

(iii) f(x) = x tan x, (iv) f(x) = (x + sin x)−1,(v) f(x) = (x + cos x)−1,

(vi) f(x) = (x2 + cos x)/(2x2) + sin x�5. 'f(x) = x2 − 3x + 2��O%�3x = −2�6aC°a�

6. 'f1(x) = 3x2, f2(x) = 2x3 + 1��JÞï�%�3F(1, 3)86, ÇÞï3�Fb8!Ý6a�

7. 'f(x) = x2, x > 0, = 0, x ≤ 0��Oy = f(x)�%�3(0, 0)

�6a�

136 ÏÞa �5��5Ý�+

8. �¿à2P

1 + x + · · ·+ xn = (xn+1 − 1)/(x− 1), x 6= 1,

Bã�55½O�ì�&õ�2P�(i) 1 + 2x + 3x2 + · · ·+ nxn−1,

(ii) 12x + 22x2 + 32x3 + · · ·+ n2xn�

9. �Oì�&Ðó�0ó, ¬��f ′��L½�(i) f(x) = |x2 − 4|, (ii) f(x) = |x3 − 1|,(iii) f(x) = (|x| − |x + 1|)3, (iv) f(x) = x3|x + 1|,(v) f(x) = [x],(vi) f(x) = x2, x ≥ 0, = −x2, x < 0�

10. �¾\ì�&Ðó3x = 0ÎÍ��, u��JO�0ó, ¬D¡h`f ′3x = 0ÎÍ=�, ÎÍ��(i) f(x) = x|x|, (ii) f(x) = x2|x|,(iii) f(x) = x3|x|, (iv) f(x) = x2/3,

(v) f(x) = |x|3/2, (vi) f(x) = x|x|3/2�11. �f(x) = x + sin x, �O��f ′(x) = 0�XbÝx�

12. '

f(x) =

{x2, x ≤ c,

ax + b, x > c,

Í�a, b, c ðó��O¸f ′(c)D3�f�(|c¼�îa, b)�

13. �5½EAìÝf1, f2, ¥�îÞ�

f1(x) =

{1/|x|, |x| > c,

a + bx2, |x| ≤ c;f2(x) =

{sin x, x ≤ c,

ax + b, x < c�

14. �f(x) = (1−√x)/(1 +√

x), x > 0, �ODf , D2f , D3f�

êÞ 137

15. 'f ′(a)D3, �¾\ì�¢ï Ñ@, ¬��Ê§ã�(i) f ′(a) = limh→0

f(a)−f(a−h)h

,

(ii) f ′(a) = limh→0f(a+2h)−f(a+h)

h,

(iii) f ′(a) = limh→0f(a+h)−f(a−h)

2h,

(iv) f ′(a) = limh→0f(a+3h)−f(a)

3h �16. E×Ðóf , �L

D∗f(x) = limh→0

f 2(x + h)− f 2(x)

h,

Í�f 2(x) = f(x) · f(x)��ÞÐóf , g,

(i) �OD∗(f + g), D∗(f − g), D∗(fg), D∗(f/g);

(ii) �|fCDf¼�îD∗f ;

(iii) �®£ËÐóf , ��D∗f = Df?

17. �OÐóf , ¸ÿf ′(x) = |x|, x ∈ R�

18. �

f1(x) =

{q−1, x = p/q, (p, q) = 1,

0, x = 0 TP§ó;

f2(x) =

{q−2, x = p/q, (p, q) = 1,

0, x = 0 TP§ó��5½D¡f1, f23x = 0��P�

19. �J(7.15)PWñ�

20. �J(7.16)PEα < 0Wñ�

21. 'f ×��Ðó�E×a ∈ R, �

g(x) =

{f(x), x > a,

f(a) + f ′(a)(x− a), x ≤ a�

�Jg) ×��Ðó�

138 ÏÞa �5��5Ý�+

22. 'Ðóf��limt→∞ f(t) = c, Í�c ×ðó, Çf|y = c ×��a��®limt→∞ f ′(t)ÎÍÄ 0? J�TÍJ��

2.8 )))WWWÐÐÐóóóCCC222ÐÐÐóóó��555E¿Í��5Ðó, &Æ��§Í°JºÕ��5�¬E

)WÐó(&9Ðó/|)WÐóÝ�P�¨), ��ÝAsin(x2)$�ã�Là#OÍ0ó(�î×;�êÞ), ´�ÓÝ§�ð? Í;&Æ-Þ�×ny)WÐó�5��§, Ì ===ÅÅÅ!!!JJJ(chain

rule)�bÝh!J, Þ��»�2¦�&ÆX��5ÝÐó��×»��

»»»8.1.'f(x) = x2, g(x) = x2 + 1, J(f ◦ g)(x) = (x2 + 1)2 =

x4 + 2x2 + 1, .h(f ◦ g)′(x) = 4x3 + 4x = 4x(x2 + 1)�

&Æ×:Õ3î»�, ��Þf ◦ g;�, Q¡¿à|GbÝ��Õ�f ◦ g�0ó�¬b`f ◦ g¬P°;�, AG�sin(x2)T√

x2 + sin x�Esin(x2), &Æá¼sin xCx2�0ó; E√

x2 + sin x,

&Æôá¼√

xCx2 + sin x�0ó, 9vÝ»�, =Å!JKÊà�

��§§§8.1.'f = u ◦ v, vv′(x)Cu′(y)/D3, Í�y = v(x)�Jf ′(x)

D3, v

(8.1) f ′(x) = u′(y) · v′(x) = u′(v(x)) · v′(x)�JJJ��.�ξ = v(x + h)− y, v'ξ 6= 0�Jv(x + h) = y + ξ, v

f(x + h)− f(x)

h=

u(v(x + h))− u(v(x))

h(8.2)

=u(y + ξ)− u(y)

h=

u(y + ξ)− u(y)

v(x + h)− v(x)

v(x + h)− v(x)

h

=u(y + ξ)− u(y)

ξ

v(x + h)− v(x)

h �

2.8 )WÐóC2Ðó��5 139

h → 0`, ξ → 0(h.v3x��, Æv3x=�), .h

u(y + ξ)− u(y)

ξ→ u′(y),

v(x + h)− v(x)

h→ v′(x),

ÆÿJ(8.1)P�¬.b��EP§&9Ýh, ¸ÿξ = 0, h`(8.2)P�Wñ��

µb®ÞÝ(u©bb§Íh¸ÿξ = 0, J^®Þ, %�?)�h`&Æ6¯�ÑÑî�®°��

(8.3) g(t) =u(y + t)− u(y)

t− u′(y), t 6= 0�

ãîPêÿ

(8.4) u(y + t)− u(y) = t(g(t) + u′(y))�

4(8.3)P©Et 6= 0�b�L, ¬�:�(8.4)PEXbt/Wñ, ©��×b§Ýg(0)�ê.t → 0`, g(t) → u′(y)− u′(y) = 0, Æ&Æ�g(0) = 0, AhgW ×30=��Ðó�Þξ�á(8.4)P��t, v¿à(8.2)Pÿ

f(x + h)− f(x)

h=

u(y + ξ)− u(y)

h(8.5)

=ξ

h(g(ξ) + u′(y))�

îPEξ = 0)Wñ��h → 0, .ξ/h → v′(x), vg(ξ) → g(0) =

0(h.h → 0`, ξ → 0, �g30=�), Æã(8.5)P)ÿh`

f(x + h)− f(x)

h→ v′(x)u′(y),

Í�§J±�

(8.1)Pê�|ì��Ý2P�î�

(8.6) (u(v))′ = u′(v) · v′�

140 ÏÞa �5��5Ý�+

T¿à¾¾¹+ÝBr, �z = u(v), y = v(x), J

(8.7)dz

dx=

dz

dy

dy

dx�

»»»8.2.�f(x) = x√

x2 + 4, J

f ′(x) = (x)′√

x2 + 4 + x(√

x2 + 4)′�

.g(x) =√

x2 + 4 = u(v(x)), Í�u(x) =√

x, v(x) = x2 + 4,

�u′(x) = 1/(2√

x), v′(x) = 2x, Æ

(√

x2 + 4)′ =1

2√

x2 + 4· 2x =

x√x2 + 4

�

.hf ′(x) =

√x2 + 4 +

x√x2 + 4

�

»»»8.3.�f(x) = sin(x2)�.(sin x)′ = cos x, (x2)′ = 2x, �sin(x2) hÞÐó�)W, Æ

f ′(x) = cos(x2) · 2x�

u�y = x2, z = f(x), Jz = sin y, .h¿à(8.7)P

dz

dx=

dz

dy

dy

dx= cos y · 2x = cos(x2) · 2x�

»»»8.4.�f(x) = (v(x))n, Í�n ×b§ó, v'v′(x)D3�ãyf = u(v), Í�u(x) = x, vu′(x) = nxn−1, Æ

f ′(x) = n(v(x))n−1v′(x)�

E×)WÐó, A��ÿ!Y, Q�|à#¶�Í0ó��à�@2Þu, v��¶��»A, EÐóg(x) =

√x2 + 4, A¢ÿÕÐ

óÂ? �×x, �ÿx2 + 4, ��]ÿ√

x2 + 4�¬O0ó`ÎÞG�

2.8 )WÐóC2Ðó��5 141

M»DÄ¼��E√

x�5, Íõ(Çx2 + 4)î¹��, �√

x�0ó 1/(2

√x), Æ�ÿ1/(2

√x2 + 4), Ígx2 + 4�0ó 2x�hÞï

8¶Çÿ2x/(2√

x2 + 4)�¨², ¾¾¹+Ý2P(8.7)Pôèº&Æ×ÿÕ)WÐó�0

óÝÄ��z = u(v), y = v(x)�&Æ�Odz/dx, �B×ª+y,

Ç�Odz/dy, ¬h&&ÆÝêÝ, ��¶îdy/dx �ÎEx��5�dz/dyCdy/dxÞÐr, î×;��ÕÄ¬&dzt|dy, Cdyt|dxÝ�¤�¬3(8.7)P��, u�WdyÎ�“V5”Ý, -ÿ¼�dz/dx�9øÝÞ��V5ÝÞ4V*Ý�°, Ý@�QÃ&ÆÝ�h2P�/��§8.1ô��|�.Â�ëÍ#��b§ÍÐó�)WÝ

�µ�»A, 'f = u ◦ v ◦ w, v'w′(x), v′(y), u′(z)/D3, Í�y = w(x), z = v(y)�Jf ′(x)D3, v

(8.8) f ′(x) = u′(z)v′(y)w′(x)�

T¶W

(8.9) (u(v(w)))′ = u′(v(w))v′(w)w′�

T�ξ = f(x), Jbì�¾¾¹+ÝBr:

(8.10)dξ

dx=

dξ

dz

dz

dy

dy

dx�

Íg&Æ¼:2Ðó��5��I5&Æ�D¡ÄÝÐó,

/��@2|×]�P�î��Aãy = x3 + 1, �L�×Ðóf(x) = x3 + 1, f�%�ôÇ î�]�P�%��¬¬&N×Ðó/�Ah�@2�L��AEì�x, y�]�P

(8.11) x3 − x = y3 − y2 + 24,

T

(8.12) y2 = sin(x2 + y2),

142 ÏÞa �5��5Ý�+

µ�Î��|2��|x¼�îy(T|y¼�îx)��Ä)b��D3×Ðóf , ¸ÿ

x3 − x = f 3(x)− f 2(x) + 24,

EN×3f��L½ÝxWñ(Ey2 = sin(x2 + y2)ô�bv«Ý��)�h`&Æ-1�]�P2â2�L�×Ðó�2Ðó��5??��Bã��Ðó�ÿ, Í�ÝÄ�-Ì

� 2Ðó��5�&Æ�×°»�, ��-�Ý�hÄ��

»»»8.5.'x, y��(8.11)P, v'y = f(x)�JÞ(8.11)PN×4Ex

�5, ÿ

3x2 − 1 = 3y2 dy

dx− 2y

dy

dx,

.h��

(8.13)dy

dx=

3x2 − 1

3y2 − 2y�

h�àÕ¿à)WÐó��5,

dy3

dx=

dy3

dy

dy

dx= 3y2 dy

dx,

dy2

dx=

dy2

dy

dy

dx= 2y

dy

dx�

ê3(8.11)PXà��%�î, Ä(3, 1)9F�6a ¢? ã(8.13)P,

h`6aE£ dy

dx=

3 · 32 − 1

3 · 12 − 2 · 1 = 26,

ÆÄ(3, 1)�6a

(y − 1) = 26(x− 3)�

»»»8.6.'x, y��x2 + y2 = 16��:�h ×�5 4�iÝ]�P�×Íi¬�Î×Ðó%�, ¬î�iy =

√16− x2Cì�

i= −√16− x2 5½�L�Ðó�¿à2Ðó��5ÿ

2x + 2ydy

dx= 0,

2.8 )WÐóC2Ðó��5 143

Æ

(8.14)dy

dx= −x

y�

3x = 2`dy/dx ¢? ã(8.14)Pá,�á¼y�Â��ÿÕdy/dx��EN×x(tÝx = 4T−4), bËÍy�ÍET�u��y =

√12,

Jdy/dx = −2/√

12; u��y = −√12, Jdy/dx = 2/√

12�

¿à2Ðó��5, ô�ODÐó�0ó�'y = f(x) ×1−1Ðó, vf ′(x)D3�|g�f�DÐó, ÇEyòyØ×/),

x = g(y), v

(8.15) f ′(g(y))g′(y) = 1,

T

(8.16) g′(y) =1

f ′(g(y))=

1

f ′(x),

©�f ′(x) 6= 0�ã(8.16)Pá, 3ØË�Lì, ×Ðó�0ó, �ÍDÐó�0ó! ÅÅÅóóó(reciprocal)�QÂÿ¥�ÝÎ, (8.16)P�.0¬��Û,.&Æ�'gEy��¯@îuf��,JE©�¸f ′(g(x)) 6= 0�x, g′(x)/D3�J�¬�Hp, &Æ�yÍ;t¡�

»»»8.7.�y = f(x) = x3 ×1 − 1Ðó, Æf�DÐógD3, vx =

g(y) = y1/3�Jã(8.16)P

g′(y) =1

f ′(x)=

1

3x2=

1

3y2/3,

�à#ãg(y) = y1/3O0óXÿ8!�

!ñ×è, (8.16)Pu|¾¾¹+ÝÐr�îÞ?z½, Ç

(8.17)dx

dy=

1dydx

�

144 ÏÞa �5��5Ý�+

�ú�×g, dx/dyTdy/dxJÍÎ×ÍÐr, X|(8.17)PWñ, ¬&Î.��Ã5ó;��y¼��¬3�5�, b�9u2à¾¾¹+Ðr¡, �óºÕî-×lÝ2P�9µÎ&Æ3î×;èÄÝ, ¾¾¹+Ðr�ß�ª��

»»»8.8.�y = f(x) = xq, x ≥ 0, Í�q ≥ 1 ×ÑJó, J

f ′(x) = qxq−1�

ê�f�DÐó x = g(y) = yq−1

�ã(8.16)Pÿ

dg

dy=

1

f ′(x)=

1

qxq−1=

1

qy(q−1)/q=

1

qyq−1−1�

.huÞx�yøð(&Æ´êY|x�Ðó��ó), -ÿ

(xq−1

)′ = q−1xq−1−1�

�yxp/q�0ó ¢? .xp/q = (xq−1)p, ¿à)WÐó��5, ÿ

(xp/q)′ = p(xq−1

)p−1(xq−1

)′ = px(p−1)/qq−1xq−1−1 = (p/q)xp/q−1,

h��(7.16)P×l�

¿àDÐó��52P$�O�ADë�Ðó�0ó, &Æº3Ï"a�D¡�t¡&ÆJ�DÐó��52P�

��§§§8.2.'f ×3T [a, b]��}�¦v=��Ðó,ê�g f

�DÐó�uEØx ∈ (a, b), f ′(x)D3v� 0, Jg′(y)D3v� 0, Í�y = f(x), vg′(y)��(8.16)P�JJJ��.'f ′(x)D3v� 0, Í�x ∈ (a, b)��y = f(x), &ÆÞJ�

(8.18) limk→0

g(y + k)− g(y)

k=

1

f ′(x)�

êÞ 145

�h = g(y + k)− g(y), x = g(y), J

h = g(y + k)− x,

v

x + h = g(y + k)�ÞîPË�ãÐóf , ÿ

y + k = f(x + h),

.hk = f(x+h)−f(x)�.g �}�¦(hãyf �}�¦),Æuk 6= 0, Jh = g(y + k)− g(y) 6= 0�.huk 6= 0, -b

g(y + k)− g(y)

k=

h

f(x + h)− f(x)(8.19)

=1

(f(x + h)− f(x))/h�

.g3y =�(�Ï×a�§6.7),Æk → 0`, h = g(y+k)−g(y) →0�Çk → 0`, h → 0�Æ3(8.19)P�, �k → 0, Jt¼�→g′(y), �t��→ 1/f ′(x)�J±�

êêê ÞÞÞ 2.8

1. �Oì�Ðó��5�(i) f(x) = (2x + x2)3/2, (ii) f(x) = (x− x−2)1/2,

(iii) f(x) = (1 +√

x) 3√

x2 + x, (iv) f(x) = 3√

x + sin(x2),

(v) f(x) =√

x +√

x +√

x, (vi) f(x) = sinn x cos(nx),

(vii) f(x) = sin(sin(sin x)), (viii) f(x) = sin2 x sin(x3),

(ix) f(x) = x2√

1 + x2/(1 +√

x),

(x) f(x) = sin(cos2 x) · cos(sin2 x)�

2. 'f(x) = 1+x6, g(x) = x3, h(x) = x+x−1��y = f(g(h(x))),

�Ody/dx�

146 ÏÞa �5��5Ý�+

3. �f(x) = (1 + x−1)−1, x 6= 0, g(x) = (1 + f−1(x))−1��Of ′(x)Cg′(x)�

4. 5½Eì�Ðó, �Og′(x)�(i) g(x) = f(x2), (ii) g(x) = f(sin2 x) + f(cos2 x),

(iii) g(x) = f(f(x)), (iv) g(x) = f(f(f(x)))�

5. �y = x√

x2 + 1, �O3Í%�îyx = 0�6a�

6. 'f, g/Ex��, �0�ì�¾¾¹+Ý¶��52P�

dn

dxn(f(x)g(x)) =

n∑i=0

(n

i

)f (i)(x)g(n−i)(x)�

7. �¿à2Ðó�5°, Ody/dx�(i) x2 + y2 − r2 = 0, r ×ðó,

(ii) x− y3 − 3xy2 = 0,

(iii) (x + y)(x− y)−1 = y−1�

8. Eì�&]�PX�L��y = f(x),�5½Of ′(x)Cf ′′(x)�(i) x = (y5 + y + 1)/(y2 + y + 1),

(ii) x = (1−√y)/(1 +√

y),

(iii) y +√

xy = x2,

(iv) x2y2 + xy = 2�

9. (i) 'y = 2x3 − x, �Jy′′ = xy′′′;

(ii) 'y =√

4x2 + 1, �Jyy′ = 4x;

(iii) 'y =√

x2 + ax, �Jx2 + y2 = 2xyy′;

(iv) 'y =√

ax2 + bx, �Jy = −b2/(4y3)�

10. �JÞ`a3y = 2x + x4y3C2y + 3x + y5 = x3y, 3æF�6a!8kà�

êÞ 147

11. �3ì�&%�î, OÄX��ÝF�6a�(i) xy = 4, (−2,−2), (ii) x + x2y2 − y = 1, (1, 1),

(iii)x2 + xy + y2 = 3, (1, 1), (iv) x3 + y3 = 6xy, (3, 3)�

12. E0 < x < 5, ]�Px1/2 + y1/2 = 5�L�y ×x�Ðó��Bã��y, �Jy′/!r(�'y′D3)�

13. ]�P3x2 + 4y2 = 12�L�y Þx�2Ðó, Í�|x| ≤2�'y′′D3, �J4y3y′′ = −9�

14. ]�Px sin xy + 2x2 = 0�L�y ×x�Ðó�'y′D3, �Jy′x2 cos xy + xy cos xy + sin xy + 4x = 0�

15. ]�Px3 + y3 = 1�L��©×Íy x�Ðó�(i) 'y′D3, �Bã��y, �Jx2 + y2y′ = 0;

(ii) 'y′′D3, �Jy′′ = −2xy−5, ©�y 6= 0�

16. �

fn(x) =

{xn sin(1/x), x 6= 0,

0, x = 0��5½D¡f1, f2, f3Cf4�(i)Ý=�P, (ii)��P, (iii)0ó�=�PC��P�

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1967).Calculus, Vol I, 2nd ed. John Wiley &

Sons, New York, New York.

148 ÏÞa �5��5Ý�+

ÏÏÏëëëaaa

��555��555��nnn;;;

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.��s"ìÝ®`, hs"É��ÆÞ�, ô�âcy�ÆÞ��èÚtSÝ�ö, b�Kþ�ÝI.�, �Æ4�3.�&�, ¬Qlæy;�¦¿¯C�é�Ýó.�®�¿à;*C_�(��êGÐ,îÝ-¿), 9°I.�1¹Û6Ý#Ç��î�O6aC«�-Î�Æq�·¶Ý®Þ�pñC¾¾¹+Ý��Q¤, -Î�z½2:�¼, hÞ®ÞbÛ6Ýn=�3�ÆW�,

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149

150 Ïëa �5��5�n;

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Mathematica”(�Qó.Ýï.æ§, �-1687O�W¬�Ì, Îà¼�Õ�QI.ÝÎP)�, 4b&9��Î|��5Ý]°ÿÕ,

¬pñ�K|�Î¿¢Ý]P¼�¾, 3�h�¿{:��5ÝÅ��Äpñ|øøøóóó°°°(fluxions, pñÌ0ó øøøóóó, �3�-

1671O¶WMethods Fluxionum et Serierum Infinitarum×h, ¬àÕ�-1736O��Ì), s�Ý×°ny��5Ý½®�¡, ×°�Ý�Ìï, -��¾¾¹+ÝBß, Ê¡��5~bÎ s�Ý�pñÝ�Ìï¼�¾¾¹+P��¹¿�¡, 3`£±§¡ÝÃ��Ìn, ×6T�sÝ�(ì, ËÍß}ñv!`xC�h§¡, Í@Î��QÝ×�¯�ãyI.&Ä5ú��ÉP®�s�Ý8�J, Ë]ÝÊÜ¯@îE3I.Ý@~�, �<��ß5²@~W�, WñÝ×ûrø�3èÚ�èâtS , 3ó.5�]«, ��6Xºì¼Ýz

½C�ÛÝ�O, «{K��¥Ú�àÆ(T1àÌ)??ã�J��`Ý�£¬��<ßÆE£°±]°è��¢²¶�ûÅÝ�°Î, Þ��5|´züÝ]P¼�¾, �¬�m��v��5?`��5Ýs", Îßé3Kó¿�&ð8|Ýó.�W�, .h�^bsß�¥Ýý0��v£¿�ó.�ÝàÌÁAÞ, X- ÎEÝ��, ;ðôÎÑ@Ý��î�Æ@~Ý®Þ??Î¼�y�Q¨é, �|@jÝóA¼l��5�XÿÝ��Ñ@P, .h��5�¦�×�?ÝqÃ, �^b0á H�

3.2 ��5ÃÍ�§ 151

¬3°»�¬ú(1789-1799)�¡, 2¸Ý>�¿Ó;�?9ß�á.êÝ��, E��5è�ÑÑ, µW ×�Ñ��cÝ�®�¬àÕèÜtS, ��5�|±«ê�¨�`�*^, ��5�¬W �¯.ßÿÕ�Ûó.IY�×��, ¬v��!òy¿¢.T�ó., �s"W 5�., 9°�VKÎ�£°�xïXÎ�]ÕÝ�ÍaÞ+Û��5ÃÍ�§CËÍx�Ý�5]°, 9°Þ�

�»�2è�&Æ�5Ý�æ�

3.2 ��555ÃÃÃÍÍÍ��§§§

Í;+Û&Æ�9gèÕÝ��5ÃÍ�§��5�©bh�§Î�Ì ÃÍ�§, A!�ó�©b×ÍÃÍ�§, ��h�§�}©P�Í�§�5ËI5, &ÆW� Þ�§��5ÃÍ�§, �ñR�5��5Ýn;�ãhn;�:�,

�5��5v«ËÍ! �YÝºÕ, A¿]C�]�Þ×Ñó¿]¡, �ãÍÑ¿]q, -ÿ/æó(Çux > 0, J

√x2 = x, &Æ�

à�Þx¿], à#�ÿx2�¿]q x)�!ø2, uÞ×=�Ðó�5, ÿÕ×±ÝÐó( æ¼Ðó��5), �Þh±Ðó�5, �ÿ/æ¼ÝÐó�Aãf(x) = x2, J

A(x) =

∫ x

c

f(t)dt =1

3x3 − 1

3c3

f�×��5�B�5¡ÿA′(x) = x2 = f(x)�9ìJ ×�Ý��

��§§§2.1.(��555ÃÃÃÍÍÍ��§§§ÝÝÝÏÏÏ×××III555). 'E∀x ∈ [a, b], f3[a, x]��×s ∈ [a, b], �Lì�Ðó

(2.1) A(x) =

∫ x

s

f(t)dt, x ∈ [a, b]�

152 Ïëa �5��5�n;

JE∀x ∈ (a, b), ©�f3x=�, A-3x��, v

(2.2) A′(x) = f(x)�

JJJ��.'x ∈ [a, b] f�×=�F, &Æ��J�

(2.3) limh→0

A(x + h)− A(x)

h= f(x)�

ã(2.1)P, v.f(t) = f(x) + (f(t)− f(x)), Æ

A(x + h)− A(x) =

∫ x+h

s

f(t)dt−∫ x

s

f(t)dt =

∫ x+h

x

f(t)dt

=

∫ x+h

x

f(x)dt +

∫ x+h

x

(f(t)− f(x))dt

= hf(x) +

∫ x+h

x

(f(t)− f(x))dt�

.h

(2.4)A(x + h)− A(x)

h= f(x) +

1

h

∫ x+h

x

(f(t)− f(x))dt�

Æu�J�îPt�£4h → 0`, ù��0, (2.3)P-ÿJÝ�.f3x=�, Æ∀ε > 0, D3×δ > 0, ¸ÿ

|f(t)− f(x)| < 1

2ε, ∀|t− x| < δ�

óãh, ��0 < h < δ, JE∀t ∈ [x, x + h], |t − x| < δWñ,

Æ|f(t) − f(x)| < ε/2�ê¿àN×Ðó, Í�5¡Ý�EÂ, �yT�yÍ�EÂ��5(ÏÞa�§4.1), ÿ

∣∣∣∣∫ x+h

x

(f(t)− f(x))dt

∣∣∣∣ ≤∫ x+h

x

|f(t)− f(x)|dt

≤∫ x+h

x

ε

2dt =

1

2hε�

3.2 ��5ÃÍ�§ 153

ãhÇÿ∀ε > 0, D3×δ > 0, ¸ÿ0 < h < δ`,∣∣∣∣1

h

∫ x+h

x

(f(t)− f(x))dt

∣∣∣∣ ≤1

2ε < ε�

u−δ < h < 0, àv«ÝD¡)�ÿÕîPWñ�Í�§J±�

î��§¼�, 3ÊÝf�ì(Çf3x�=�), ukÞ×Ðó�5¡Xÿ�±Ðó��5, J�6Q×��5Ä�, ±Ðó3x�0óÇ æ¼Ðó3x�Â�\ï�p��Ü�×°3x�=�vh`(2.2)P�Wñ�Ðóf�uf3x�×Ï½ =�(�§2.1©�'f3x=�), Jî�J�

��;&9��Êh > 0, v'f3[x, x + h]=��ã�5�íÂ�§(ÏÞa�§4.10), D3×z ∈ [x, x + h], ¸ÿ

A(x + h)− A(x) =

∫ x+h

x

f(t)dt = hf(z)�Æ

limh→0+

A(x + h)− A(x)

h= lim

h→0+f(z) = f(x)�

�yh < 0Ý�µ!§�J�'f ≥ 0, J�5��«�, \ï��0×%(�'f=�), -�

Ý�G�Ä�Ý¿¢�L�Íg&ÆD¡��5ÃÍ�§ÝÏÞI5�´�uf (a, b)��×ðóÐó, Jf ′(x) = 0, ∀x ∈ (a, b)�D

�, uf ′(x) = 0, �3x�f�6aE£ 0, Ç3x��6a ×i¿a�.h, u∀x ∈ (a, b), f ′(x) = 0, JàÌîf3(a, b) ×ðó�h��¿à&ÆY3¡«ny�5ÝíÂ��(�4.1;)-ñÇ�ÿ�&Æ-�#å?Ý, Ç3(a, b)�×Ðóf�0ó 0, uv°uf3(a, b)� ðó�#½&Æ�DDD000óóó(antiderivative) ��L�D0óêÌæææÐÐÐóóó(primitive function)�ãC«î�:�ËÍ(Þ&�Ý��

��LLL2.1.ÐóFu��F ′(x) = f(x),∀x ∈ (a, b),-Ì Ðóf3(a, b)

�D0ó�

154 Ïëa �5��5�n;

»A, sineÐó cosineÐó3�× �×D0ó(.(sin x)′ =

cos x)�×Ðó�D0ó¬�°×, h.u0Õf�×D0óF ,

JF + C, Í�C ×ðó, ù f�×D0ó�D�, uF , P/ f�D0ó, J.F ′(x) − P ′(x) = f(x) − f(x) = 0, ∀x ∈ (a, b),

ÆF −P3(a, b)îù ×ðó�ôµÎ!×Ðó�ÞD0óÝ- ×ðó��5ÃÍ�§ÝÏ×I5×å&Æ, E×=�Ðó, Bã�5,

�ÿÍ×D0ó�h��î�ÞD0ó�- ×ðó, -ÿì��§�

��§§§2.2.(��555ÃÃÃÍÍÍ��§§§ÝÝÝÏÏÏÞÞÞIII555). 'f3� (a, b)î=�,

vF f3(a, b)î�×D0ó�JE∀s, x ∈ (a, b),

(2.5) F (x) = F (s) +

∫ x

s

f(t)dt�

JJJ��.�A(x) =∫ x

sf(t)dt�ãy�'f3(a, b)=�, �§2.1¼�,

A′(x) = f(x), ∀x ∈ (a, b)�ùÇA f3(a, b)î�×D0ó�ê.ÞD0ó�- ×ðó, ÆD3×ðóC, ¸ÿA(x) − F (x) =

C��x = s, .A(s) = 0, ÆÿC = −F (s)�.h(2.5)PWñ�

�§2.2×å&Æ, E=�Ðóf�×D0óF , K�Bã�ó�×s, Þfãs�5�x, Q¡�îF (s)-ÿÕF (x)�¬9��Îh�§t�Ý��uÞ(2.5)P;¶ ìP, -�:�h�§Ý�æÝ�

(2.6)

∫ x

s

f(t)dt = F (x)− F (s) = F (t)∣∣∣x

s�

Çf3[s, x]î��5, u�0Õf�×D0óF , -ñÇ�ÿÝ�O�5Í¼Î×��ÝÜÝ�®, ¬¨3h®ÞQ»ðWOD0ó�×��, ¡ï´Gï�|9Ý��¡à%�]P, ©��0�f�×D0ó, Jf��5-�XÝ�ô.hN×Í�52P-ET×�52P��AÄ�3�.`�, N×94PÝ¶�2P,

3.2 ��5ÃÍ�§ 155

-ET×5�.PÝ2P�uNg�5K�ãî×a��L½W,

JËÎ�s05�¬�5Î&Æ´�ôNÝ, &Æ×'´º�§&Ë�5ÝºÕ�9ì��¿Í»��

»»»2.1.En ×&�Jó, 3î×a&Æ�à#ã�LÕ�

(2.7)

∫ b

a

xndx =bn+1 − an+1

n + 1 �

¬En� JóÞA¢? �§2.2×å&Æ, �|�D/×ì, :£ÍÐó�5¡�ÿxn��3î×a»7.12, &Æô�J�, EN×b§ón, (xn)′ = nxn−1, �hÇ(xn/n)′ = xn−1, ©�n 6= 0�ÆEN×b§ón 6= −1, (

xn+1

n + 1

)′= xn�

Çxn+1/(n + 1) xn�×D0ó, Í�n 6= −1 b§ó�.h�§2.20l(2.7)PEn 6= −1 b§óùWñ�

�yn = −1Tn �×@óÝ�µ, &Æ��ÕÏ"a.Ý¼óCEó¡, �bð°��¯@î(2.7)P, EN×n 6= −1�@ó/Wñ��n = −1Ý�µ�¨²�§�

»»»2.2.32.5;&Æ�ðÝ×j�G, �O�sineÐóCcosineÐóÝ�5�¿à(sin x)′ = cos xC(cos x)′ = − sin x, Çásin x cos x�×D0ó, − cos x sin x�×D0ó��§2.2-ñÇ0l

∫ b

a

cos xdx = sin x∣∣∣b

a= sin b− sin a,

∫ b

a

sin xdx = (− cos x)∣∣∣b

a= cos b− cos a

Í;t¡&ÆED0óC��5Ý��, º��×°1��

156 Ïëa �5��5�n;

�§2.1¼�×ÐófÝ��5, »A∫ x

sf(t)dt, ì�®Þ�

�: ��f , O×ÐóF , ��

(2.8) F ′(x) = f(x)�

h®Þ�O&Æ�×�5ÝºÕ�3ó.�ðº�¨9ËYºÕÝ®Þ�� Ý�9v®Þ, ??S±ÝÃF(»A, Ý�a + x = b�SJó, Ý�ax = b�Sb§ó, Í�a, b Þ�Qó)�ã�(2.8)P&ÆðS±Ðó, hF|¡º1��(2.8)P�×ÐóF , -Ì f�D0ó��(2.8)P, T1

0f�×D0ó, y:�ì��5��ÎË/¯�¬��2.1QJ�Ý, N×f��5, Ä f�×D0ó��Äh��¬Î��X, 0�f�XbD0ó(Ç0�(2.8)P�Xb�) Ý®Þ�f��5©Î×ÍD0ó, ��$bÍ�fÝD0ó��§2.2/�Ý9I5Ý®Þ�Çf��×D0óF , f�×��5�î×ðó�ôµÎ1

(2.9) F (x) = C +

∫ x

s

f(t)dt�

&Æ��º��3îP�, ðóC�|6¯�h.Bã;��5�ì§s, Xÿ�D0ó�æ¼ÝD0ó-×Íðó�ùÇ�×D0óBÊ2óã�5ì§¡, �¶W×��5�¬;ð, u6¯ðóC, QP°ÿÕXbD0ó�»A, uf(x) ≡ 0,

J��5∫ x

sf(t)dt ≡ 0, ¬�×ðó/ f�D0ó�¨×»

'f(x) = x, J��5 x2/2 − s2/2, x2/23�×&�Ýó, ¬x2/2 + 1ù f�×D0ó�.hu��î�XbD0ó,

J(2.9)P��ðóC, ��6¯�ãyD0óC��5bG�n;, &Æ-�.Â��5

ÝÃF, |-��âXbÝD0ó�E�×C +∫ x

sf(t)dt, &Æ

KÞÌ� ��5, Ah×¼-�à� ½D0óC��5Ý��5CD0óÍ¼ÎËÍ�!ÝÃF, bÝ��5ÃÍ�§, hÞÃF-)� ×Ý��y

∫ b

af(x)dxJÌ ��555(definite

integral)�

3.2 ��5ÃÍ�§ 157

*¡, &Æº|(9ôÎ¾¾¹+X2àÝ)

(2.10) F (x) =

∫f(x)dx

¼��f�×D0ó�ÇBÊ2óãðóCCs¡,

F (x) = C +

∫ x

s

f(u)du�

(2.10)P��î]°, µÎÞ�5Ýî�ì§sCx/6�, ¬|x �5Ý�ó��}2ý, �5Ý�óCî§/2àxÎ�?Ý¶°�&Æ�©½º�, Ðr

∫f(x)dx©Î��f�Ø×D0ó�

(2.10)PÝ�L�d

dxF (x) = f(x)

��LÎ��8!Ý�'P f�×D0ó, ¿à(2.10)P�Br-ÿ ∫

f(x)dx = P (x) + C,

Í�C ×ðó, Ì �5ðó�»A, .(sin x)′ = cos x, X|�¶W ∫

cos xdx = sin x + C��

∫ b

a

cos xdx = (sin x + C)∣∣∣b

a= (sin b + C)− (sin a + C)

= sin b− sin a = sin x∣∣∣b

a�

ôµÎ3O��5`, ðóC×��-�6¯��×�Jb

(2.11)

∫ b

a

f(x)dx =

∫f(x)dx

∣∣∣b

a�

¨², ¾¾¹+ÝÐr, $bì�-¿�� Ý�-'f(x) =

F ′(x) ×=�Ðó�J�§2.2�|ìP�î�

F (b)− F (a) =

∫ b

a

F ′(x)dx =

∫ b

a

dF (x)

dxdx =

∫ b

a

dF (x),

158 Ïëa �5��5�n;

Í�3t¡×�P�, �AÞdxV5*�ÿ��5¼O��5, A��5Ý�5P�, âb�3�5

Õ��L½�ÝF, -�9�º��»A, 3O∫

1

x2dx

`, .x−2 ÝD0ó −x−1 + C, Æ

(2.12)

∫1

x2dx = −1

x+ C�

.x−1Ý�L½��â09F, X|îP�Wñ, 6x 6= 0��îPÍ@ ×��5, ôµÎD0óÝ2P��¼��ËÐó�8�, �L½ô�8!, Ç/ R \ {0}�.h, u¿à��5Ý2P�Oì��5 ∫ 3

−1

1

x2dx

µ��Ý�.�5P��â0, ¬x = 0��¸(2.12)PWñ�E�I5Ý�52P, &Æð�©½¼�¸ÍWñ��óP��×]«Î Ý��, ×]«- ��K§�Ðó�b�L, 2P�b�L�

êêê ÞÞÞ 3.2

1. �|OD0óÝ]°, Oì��5�(i)

∫ 5

1(4x4 + 2x)dx, (ii)

∫ 1

0(x + 1)(x3 − 2)dx,

(iii)∫ 4

2x4+x−3

x3 dx, (iv)∫ 3

0(1 +

√x)2dx,

(v)∫ 3

1(√

2x +√

x3)dx, (vi)∫ 4

12x2−6x+7

2√

xdx,

(vii)∫ 8

1(2x1/3 − x−1/3)dx, (viii)

∫ b

a(3 sin x + cos 2x)dx,

(ix)∫ 6

3

√y − 2dy, (x)

∫ 1

0(z + 1)−1/2dz�

2. (i)�JD(√

2x + 1) = (2x + 1)−1/2, ¬O∫ 4

0(2x + 1)−1/2dx;

(ii)�JD(√

1 + 2x2) = 2x/√

1 + 2x2,¬O∫ 2

0x/√

1 + 2x2dx�

êÞ 159

3. �J�D3×94Pf , ��f ′(x) = x−1�

4. �O

(i)∫ x

0|t|dt, (ii)

∫ x

0(t + |t|)2dt,

(iii)∫ 4

−2(|x− 1|+ |x + 1|)dx, (iv)

∫ 2

0max{3x, 4− x2}dx�

5. �O=�Ðóf , ¸Í��∫ x

0

f(t)dt = −1

2+ x2 + x sin 2x +

1

2cos 2x, ∀x ∈ R�

6. �O=�ÐófCðóc, ¸ÿ∫ x

c

tf(t)dt = cos x− 1

2, ∀x ∈ R�

7. �O=�ÐófCðóc, ¸ÿ

∫ x

0

f(t)dt =

∫ 1

x

t2f(t)dt +1

8x16 +

1

9x18 + c�

8. 'g ×=�Ðó, ��g(1) = 5v∫ 1

0g(t)dt = 2��

f(x) =1

2

∫ x

0

(x− t)2g(t)dt�

�J

f ′(x) = x

∫ x

0

g(t)dt−∫ x

0

tg(t)dt�

¬AhOf ′′(1)Cf ′′′(1)�

9. 'g3x��, f3u = g(x)=��

F (x) =

∫ g(x)

s

f(t)dt,

�OdF (x)/dx�

160 Ïëa �5��5�n;

10. 'g1, g23x/��, f3u1 = g1(x)Cu2 = g2(x) /=��

F (x) =

∫ g2(x)

g1(x)

f(t)dt,

�OdF (x)/dx�

11. ¿àîÞ, �Oì�&ÐóEx��5�(i) f(x) =

∫ x2

0(1 + t2)−2dt, (ii) f(x) =

∫ x3

x2 sin(1 + t2)dt,

(iii) f(x) =∫ cos x

sin x

√t + sin tdt,

(iv) f(x) =∫ sin

√x√

x

√2 + cos tdt�

12. �Oì�&��5�(i)

∫x sin x2dx, (ii)

∫t2√

1 + t3dt�

13. 'f3[a, b]=�, �O¸ìPWñ�f��

D(

∫ x

a

f(t)dt) =

∫ x

a

Df(t)dt�

14. 'f ′(x)3x ≥ 1�=�, a > 1 ×ðó, [ · ]�t�JóÐó��J(i)

∫ a

1[x]f ′(x)dx = [a]f(a)− (f(1) + · · ·+ f([a]));

(ii)∫ a

1[x2]f ′(x)dx = [a2]f(a)−(f(1)+f(

√2)+· · ·+f(

√[a2]))�

3.3 ��óóó��ððð°°°

3î×;Ý��5ÃÍ�§×å&Æ, O�5Ý®Þ, ��»ð O��5Ý®Þ�XÛ�5Ý*», -Î¼�×O��5�b�ÙÝ]°�&9>Ihº��×°��5Ý�(Ì ðððààà��555��), Þ×

°ð�ÐóÝ�52P��Í�bëË*»Îf´¥�Ý�Ï

3.3 �ó�ð° 161

×Ë ��óóó��ððð°°°(change of variable, TÌintegration by substitu-

tion), ÏÞË 555III��555°°°(integration by parts), ÏëË III555555PPP°°°(integration by partial fractions)�9¿Ëx�Ý�5*», tÝÜÃ�5��ñ, ¬ð�à¼Þ×°XkOÝ�5, »ð �5��bÝÃÍ�P, .hO�Í�5�Í;-�D¡�ó�ð°��ó�ð°Îã�5Ý=Å!J¼Ý�3ÊÝf�ì(gb×

=�Ý0ó(9ËÐóÌ ===��, continuously differentiable, ¥�¬&¼gÉ=�ê��)CF (g(x)) =�, ×�5Ýf�), &Æb

(3.1) DF (g(x)) = F ′(g(x))g′(x),

.h ∫F ′(g(x))g′(x)dx = F (g(x)) + C�

u�F ′ = f , Çÿ

(3.2)

∫f(g(x))g′(x)dx = F (g(x)) + C,

Í�F f�×D0ó�u��

u = g(x), du = g′(x)dx,

J(3.2)P�;¶ ì�´ðàÝ�P:

(3.3)

∫f(g(x))g′(x)dx =

∫f(u)du

∣∣∣u=g(x)�

îPÇÌ �52P��ó�ð, Í�kaì¶×u = g(x)��5¡u�|g(x)ã��.F f�×D0ó, Æÿ

(3.4)

∫f(u)du

∣∣∣u=g(x)

= (F (u) + C)∣∣∣u=g(x)

= F (g(x)) + C�

uÎ��5, J�p:�(3.3)P�¶WìP, J�Jº3}¡�

(3.5)

∫ b

a

f(g(x))g′(x)dx =

∫ g(b)

g(a)

f(u)du�

162 Ïëa �5��5�n;

&Æ¯��Õ×ì�3(3.3)P�¼�, &Æ®Ý�ó�ð, Ç�u = g(x), Q¡��du g�0ó¶îdx, Ç�du = g′(x)dx��3(3.5)P�, æ¼x�P� ãa�b, .u = g(x), ÆuÝP� ãg(a)�g(b)��Ä�ú�×g,

∫f(u)du©Î×Ðr, ��5,

Í�Ýdu�}¼:, Í@^%��L�&Æ�u = g(x), Q¡|du¼ã�g′(x)dx, ©Î×ËÐrîÝ'�, QÃ&ÆÞó.ÝºÕ»ðW^_Ý]P�9Î¾¾¹+Ðr�×g�¨Í-¿��&Æ�X|�ÿÕ(3.3)P, XµAÝµÎ(3.1)PÝ�5!J�u2à¾¾¹+ÝÐr, J(3.1)PW

(3.6)

∫f(g)dg =

∫f(u)du�

�.u = g(x), Æ

g′(x)dx =du

dxdx

uÞîP��“5�”C“5Ò”�dx��¡, µâ?ÿÕdu, �A�mÍ��J, -�ÿÕ(3.3)P��ó�ð°�ÍW�, ÐÚ&ÆÎÍ�Ñ@2X��5Õ��,

£×I5Ä6|uã��b`�©×Ë�ðí�W��9ì�¿Í»��

»»»3.1.�O∫

2x(x2 + 1)3dx��.�u = x2 + 1, Jdu = 2xdx, v

∫2x(x2 + 1)3dx =

∫u3du

∣∣∣u=x2+1

= (1

4u4 + C)

∣∣∣u=x2+1

=1

4(x2 + 1)4 + C�

»»»3.2.�O∫ 3

1x√

x2 − 1dx��.�u = x2 − 1Jdu = 2xdx�êx = 1`, u = 0; x = 3`, u =

8�Æ

3.3 �ó�ð° 163

∫ 3

1

x√

x2 − 1dx =1

2

∫ 8

0

√udu =

1

2(2

3u3/2)

∣∣∣8

0

=1

3(83/2 − 03/2) =

16√

2

3 �

3î»�, uÞ�5Õ�; √

x2 − 1, :R¼«{´�|, ¬QP°2àG��ð�.xdx = du/2, 2 ×ðó, î»��5Õ�þz2, &Æ�|�ð¡, �t|2, ¬QP°3�5², t|×x�Í»}¡, &Æº|Í��ð, ¼OÍ�5�¨², ô��O�x

√x2 − 1�D0ó

1

2

2

3u3/2 =

1

4(x2 − 1)3/2,

��á�5Ýî�ì§3C1, Xÿ�n)8!�

»»»3.3.�O∫

u2

(u3+1)2du�

��.�y = u3 + 1, Jdy = 3u2du, v∫

u2

(u3 + 1)2du =

1

3

∫1

y2dy =

−1

3

1

y+ C = − 1

3(u3 + 1)+ C�

»»»3.4.�O∫

x3 cos x4dx��.�u = x4, Jdu = 4x3dx, v∫

x3 cos x4dx =1

4

∫cos udu =

1

4sin u + C =

1

4sin x4 + C�

»»»3.5.�O∫ 3

2x+1√

x2+2x+3dx�

��.�u = x2 + 2x + 3, Jdu = 2(x + 1)dx, v∫ 3

2

x + 1√x2 + 2x + 3

dx =1

2

∫ 18

11

1√udu =

√u∣∣∣18

11=√

18−√

11�

Qô��O��5∫

x + 1√x2 + 2x + 3

dx =√

x2 + 2x + 3 + C,

164 Ïëa �5��5�n;

J∫ 3

2

x + 1√x2 + 2x + 3

dx =√

x2 + 2x + 3∣∣∣3

2=√

18−√

11�

»»»3.6.'φ′D3v=�, J∫

φ(x)φ′(x)dx =

∫φ(x)dφ(x) =

1

2φ2(x) + C�

×�Jb ∫φn(x)φ′(x)dx =

1

n + 1φn+1(x) + C�

»A, ∫sinn x cos xdx =

1

n + 1sinn+1 x + C�

!§, uÐóf(x)3x ∈ [−1, 1] =�, J∫ b

a

f(sin x) cos xdx =

∫ sin b

sin a

f(x)dx�

uãa = 0, b = 2π, J�ó�ðu = sin x, ¬&×1−1Ý�ð�h`∫ 2π

0

f(sin x) cos xdx =

∫ 0

0

f(u)du = 0�

¨3&Æ�×(3.5)P�J��

��§§§3.1.'g�0óg′3� (r, s)î=�, �J�g3(r, s)�Â½,

ê'f3J=��JE∀a, b ∈ (r, s),

(3.7)

∫ b

a

f(g(x))g′(x)dx =

∫ g(b)

g(a)

f(u)du�

JJJ��.�η = g(a), v�LÞ±ÐóP,QAì:

P (ξ) =

∫ ξ

η

f(u)du, ξ ∈ J,

Q(ξ) =

∫ ξ

η

f(g(x))g′(x)dx, ξ ∈ J�

3.3 �ó�ð° 165

ã��5ÃÍ�§(�§2.1),

P ′(ξ) = f(ξ), Q′(ξ) = f(g(ξ))g′(ξ)�

Æã�5�=Å!J, ÿ

(P (g(ξ)))′ = f(g(ξ))g′(ξ) = Q′(ξ)�

�ã��5ÃÍ�§(�§2.2),

∫ g(b)

g(a)

f(u)du =

∫ g(b)

g(a)

P ′(u)du = P (g(b))− P (g(a)),

v∫ b

a

f(g(x))g′(x)dx =

∫ b

a

Q′(x)dx =

∫ b

a

(P (g(x)))′dx

= P (g(b))− P (g(a))�

Æ(3.7)PWñ�

BÄ|îÝD¡, ��E�ó�ð°, T�b×�MÝÝ��3^£¡�, �^�ó ôb�ó�ð�4Í�L, ��5�©Î Ý�ÕîÝ]-, ��Ý�ð�!, ¬� ÝºÕÄ�QÎv«Ý��ó�ð, Î��^£¡�×¥�ÝÞC�¯@î, &9^£¡�ÝÞC, ??�3��5�0ÕÍæ��9ì&ÆE�ó�ð, ��×°1��5ÃÍ�§, Þ�5Ý®Þ» OD0ó�49E�5¼

1, �Î×��, ¬b`)�|:�×Ðó�D0ó�»A, EG�»3.1�»3.5, Í�b¿Í×¿��:��ÍD0ó�¬BÄÊ2�ð¡, �5Õ�-»ðW´��Ý�P, ��¿à!áÝ2PÕ��5�Q�ó�ð°¬&0�, �9`Î¿àh°)��;, 9`-©?¨Õ¸H�3¸à�ó�ð°`, u�5Õ�bA(3.2)P¼��f(g(x))g′(x)Ý�P, J�z½2��u = g(x), Q¡'°O�f�D0óÇ�, g¬m� ×1−1ÝÐó�¬b`×�

166 Ïëa �5��5�n;

��5Õ�Îf(x), ÇO∫ b

af(x)dx, ��Ý�ðÎ�u = g(x), 9`

µb°Þ;�º�Ý�u3[a, b]�, g′(x)/� ÑT� �, Çh`g ×ãx�u��

}��Ý�ð�.hDÐóx = g−1(u)D3, v

(3.8)

∫ b

a

f(x)dx =

∫ g(b)

g(a)

f(g−1(u))(g−1(u))′du�

¬ug � �}��Ý�ð, -��®ßý0Ý, �ì»�

»»»3.7.�O∫ 2

−1x2dx�

��.h�5Â�� 13x3|2−1 = 3�¬u�x2 = u, ��x =

√u,

dx = 1/(2√

u), vx = −1`u = 1, x = 2`, u = 4, Jÿì�ý0��

∫ 2

−1

x2dx =

∫ 4

1

u1

2√

udu =

1

2

∫ 4

1

u1/2du =1

2

2

3u3/2

∣∣∣4

1=

7

3�

ý0sß3¢�? �x2 = u, ��x =√

uTx = −√u, ÚxÝP��Ç��Þæ�5;¶WÞ4�5Ýõ��∫ 2

−1

x2dx =

∫ 0

−1

x2dx +

∫ 2

0

x2dx =

∫ 0

−1

u(− 1

2√

u)du +

∫ 2

0

u(1

2√

u)du

= −1

2

∫ 0

−1

u1/2du +1

2

∫ u

0

u1/2du = −1

3(0− (−1)) +

1

3(8− 0) = 3,

-ÿÑ@Ý��Ý�

X|Eg� �}��Ðó`, .h`g��b�©×ÍDÐó,

-��Þ�5 tW¿Í� , v¸g3N×� gbDÐó��yu�5Õ� f(g(x))g′(x)Ý�P, Ç¸g� �}��Ðó, J.h`�à��DÐóx = g−1(u), )ï|?Ý®°ô^®Þ�»A3»3.1�, u; O

∫ 3

−22x(x2 + 1)3dx, Jµ»3.1XÿD

0ó, �5ÂT 14(9 + 1)4 − 1

4(4 + 1)4 = 1

4(104 − 54)�¬Aì�:

3.3 �ó�ð° 167

�, utWËÍ�5, )ÿ8!�n�∫ 3

−2

2x(x2 + 1)3dx =

∫ 0

−2

2x(x2 + 1)3dx +

∫ 3

0

2x(x2 + 1)3dx

=

∫ 1

5

2(−√u− 1)u3 −1

2√

u− 1du +

∫ 10

1

2√

u− 1u3 1

2√

u− 1du

=

∫ 1

5

u3du +

∫ 10

1

u3du =1

4u4

∣∣∣1

5+

1

4u4

∣∣∣10

1

=1

4(1− 54) +

1

4(104 − 1) =

1

4(104 − 54)�

¨², b`�5Õ�©bf(g(x))Ý�P, ÇO∫

f(g(x))dx�h`u�0Õ×Ðóh, ¸ÿ

(3.9) f(g(x)) = h(g(x))g′(x),

J-�à�ó�ð�Ç�u = g(x), �b

(3.10)

∫f(g(x))dx =

∫h(g(x))g′(x)dx =

∫h(u)du�

�uu = g(x)b×=�v� ëÝ0óg′(x), J(3.9)PWñ�h.9`DÐóx = g−1(u)D3, vb×=�Ý0ó

(3.11)dg−1(u)

du=

dx

du=

1

g′(x)�

ãîP�:� ¢��'g′(x) 6= 0��LÐóh

(3.12) h(u) = f(u)dg−1(u)

du= f(u)/g′(x),

Í�t��PWñÎàÕ(3.11)P�J|u = g(x)�áîP-ÿ(3.9)P�.h3Ê2f�ì, (3.12)P�|Wñ, v

∫f(g(x))dx =

∫h(u)du =

∫f(u)

dg−1(u)

dudu(3.13)

=

∫f(u)

dx

dudu,

168 Ïëa �5��5�n;

¾¾¹+ÝÐrê��î�Í-¿��.ï�6BÝÎ, 3|uã�g(x)¡, æ¼Ý�5

∫f(g(x))dx 6=

∫f(u)du�

ð�1, ¬�Î©Þg(x); u��, �Î�Þduô¶îdx/du�f´(3.2)Pt¼Ct�4-��¡��yuO��5, �5�î�ì§ô�ET�ð�

»»»3.8.�O∫ 4

11/√

xdx��.�u =

√x, Jx = u2, dx = 2udu, v

∫ 4

1

1√xdx =

∫ 2

1

1

u2udu =

∫ 2

1

2du = (2u)∣∣∣2

1= 2�

¨², u�u = 1/x, J

∫ 1

1/2

sin(1

x)dx =

∫ 1

2

sin u−1

u2du =

∫ 2

1

sin u

u2du�

»»»3.9.�O∫

x√

2− xdx��.�2− x = u, J

∫x√

2− xdx =

∫−(2− u)

√udu =

2

5u5/2 − 4

3u3/2 + C

=2

5(2− x)5/2 − 4

3(2− x)3/2 + C�

êêê ÞÞÞ 3.3

1− 27Þ, �O&�5�

êÞ 169

1.∫

x√

1 + 3xdx� 2.∫

(x + 2)√

4x + 5dx�3.

∫x2√

x + 1dx� 4.∫

sin3 xdx�5.

∫x(x− 1)1/3dx� 6.

∫cos xsin3 x

dx�7.

∫sin x

(3+cos x)2dx� 8.

∫sin x√cos3 x

dx�9.

∫x5√1−x6 dx� 10.

∫x2(2x3 + 3)2/3dx�

11.∫

sin x+cos x√sin x−cos x

dx� 12.∫

x3√x2+1−1

dx�13.

∫ (x2−2x+1)1/5

1−xdx� 14.

∫ √1+√

x√x

dx�15.

∫ √u√

1 + u√

udu� 16.∫

xn−1 sin xndx�17.

∫ 2

−19x2(1 + 3x3)2dx� 18.

∫ −5

0

√1− 3udu�

19.∫ 2

−2x√

1+8x2 dx� 20.∫ 3

0x3√1+x

dx�21.

∫ 1

01√

1+√

xdx� 22.

∫ 2

11x2

√1− 1

xdx�

23.∫ π/4

0cos 2x

√4− sin 2xdx� 24.

∫ 8

3sin√

x+1√x+1

dx�25.

∫ 2

1x−1√

xdx� 26.

∫ √x√

1+xdx�

27.∫

x√1+x2+(1+x2)3/2

dx�28. �J ∫ 1

x

1

1 + t2dt =

∫ 1/x

1

1

1 + t2dt, ∀x > 0�

29. �JE�ÞÑJóm,n,∫ 1

0

xm(1− x)ndx =

∫ 1

0

xn(1− x)mdx�

30. �J(�¿àx = sin u��ð)EN×ÑJón,∫ 1

0

(1− x2)n−1/2dx =

∫ π/2

0

cos2n udu�

31. (i) �J(�¿àu = π − x��ð)∫ π

0

xf(sin x)dx =π

2

∫ π

0

f(sin x)dx�

(ii) �¿à(i)0�∫ π

0

x sin x

1 + cos2 xdx = π

∫ 1

0

1

1 + x2dx�

170 Ïëa �5��5�n;

32. �JE�×ÑJóm,

∫ π/2

0

cosm x sinm xdx = 2−m

∫ π/2

0

cosm xdx�

33. �

F (x, a) =

∫ x

0

tp

(t2 + a2)qdt,

Í�a > 0, p, q ÞÑJó��J

F (x, a) = ap+1−2qF (x/a, 1)�

34. �K =∫ 1

−1dy, ñÇ:�K = 2�¬u�y = x5/2, JÿK =

52

∫ 1

1x3/2dx = 0��ÕÍ�æ �

3.4 555III��555

3î×a&Æ�ÿì�ÞÐó�¶�Ý�52P�

(4.1) D(f(x)g(x)) = f(x)g′(x) + g(x)f ′(x)�

uOîPË�ÝD0ó, -ÿ

(4.2) f(x)g(x) =

∫f(x)g′(x)dx +

∫g(x)f ′(x)dx + C�

T¶W

(4.3)

∫f(x)g′(x)dx = f(x)g(x)−

∫g(x)f ′(x)dx + C�

hP-Ì 5I�5�2P, ¸èº×±Ý�5*»��9hÞ(4.3)P��ðóC6¯*��Ä.(4.3)P�¼��&b×D0ó,�×ÐóÝD0ó¬�°×(-×Íðó),.hðóCÎÄ�Ý,

ÍJ(4.3)P�¼��×�8��Äb` Ý�-, &Æ??6�C��yuÎO��5, (4.3)PJW

(4.4)

∫ b

a

f(x)g′(x)dx = f(x)g(x)∣∣∣b

a−

∫ b

a

g(x)f ′(x)dx�

3.4 5I�5 171

u�u = f(x), v = g(x), v2à¾¾¹+ÝÐr, Çdu = f ′(x)dx,

dv = g′(x)dx, h`(4.3)PW ì�´|B7Ý�P�

(4.5)

∫udv = uv −

∫vdu�

5I�5Ý2P�ÞXkOÝ�5, »; ´|OÝ�P�»A, kO

∫h(x)dx, ��0�ÞÐófCg, ��h(x) = f(x)g′(x),

v∫

g(x)f ′(x)dx´|O��J∫

h(x)dx = f(x)g(x)−∫

g(x)f ′(x)dx + C�

b`�BÄ�©×gÝ5I�5ÝM», ��;�×Í´|�5Ý�P�.h¶Wfg′��b�9�!Ý¶°, A¢0�ÊÝfCg,

Îm�×°B�Ýá��QÐófCgôm�×°f�, A��Cgf ′ô��9ì�¿Í»��

»»»4.1.�O∫

x cos xdx��.ãf(x) = x, g′(x) = cos x, Jf ′(x) = 1, g(x) = sin x�ã(4.2)P

∫x cos xdx = x sin x−

∫sin xdx + C(4.6)

= x sin x + cos x + C�

3îP�,∫

sin xdxÎ×Í&Æ!áÝ�5, êE∫

sin xdx, &Æ�à¶Wcos x + C, .ÞðóÝõ) ×ðó, ��bðóCÝ��Ä�B�, (4.6)P�ÝËÍC¬�×�¼!×ðó, ¬;ð Ý��, &Æ¬�ðàÍ�CÒ�

3î»�, uãf(x) = cos x, g′(x) = x, Jf ′(x) = − sin x,

g(x) = x2/2, .h

(4.7)

∫x cos xdx =

1

2x2 cos x +

∫1

2x2 sin xdx,

172 Ïëa �5��5�n;

ÿÕ×?�ÓÝ�5∫

x2 sin xdx�.hu�^óãÑ@ÝfCg, Î��¼Ý��Äb`fCgb�©×Ëó°, &Æ|¡º1��¨², ¿à(4.5)P, î»ô�Aì.0�∫

x cos xdx =

∫xd sin x = x sin x−

∫sin xdx + C

= x sin x + cos x + C�ê3(4.7)P�, ¿à(4.6)P, �ÿ

∫x2 sin xdx = 2

∫x cos xdx− x2 cos x

= 2x sin x + 2 cos x− x2 cos x + C�

»»»4.2.�O∫

x3√1+x2 dx�

��.�u = x2, dv =

x√1 + x2

dx,

J¿à�ó�ð(6�ðó)

v =

∫x√

1 + x2dx =

√1 + x2�

Æ∫

x3

√1 + x2

dx =

∫udv = x2

√1 + x2 −

∫ √1 + x2dx2 + C

= x2√

1 + x2 −∫

2x√

1 + x2dx + C

= x2√

1 + x2 − 2

3(1 + x2)3/2 + C�

¨², \ïô��|�ó�ð°(�u = 1 + x2)OG��5, ºÿÕ8!�n�

»»»4.3.�O∫

x2 cos xdx��.�u = x2, v = sin x, J

(4.8)

∫x2 cos xdx =

∫udv = x2 sin x− 2

∫x sin xdx + C�

3.4 5I�5 173

�û»4.1, ÿ∫

x sin x = −x cos x + sin x + C�

Þh��á(4.8)P, v¿àËðóõ) ×ðó, ÿ∫

x2 cos xdx = x2 sin x + 2x cos x− 2 sin x + C�

»»»4.4.�JE∀n ≥ 2,

∫secn xdx(4.9)

=1

n− 1(secn−2 x tan x + (n− 2)

∫secn−2 xdx) + C�

JJJ��.´�, .(tan x)′ = sec2 x, Æãu = secn−2 x, v = tan x, J∫

secn xdx =

∫udv = secn−2 x tan x−

∫tan xd secn−2 x + C

= secn−2 x tan x− (n− 2)

∫secn−3 x sec x tan x tan xdx + C

= secn−2 x tan x− (n− 2)

∫secn−2 x tan2 xdx + C,

Í�àÕ(sec x)′ = sec x tan x�êtan2 x = sec2 x− 1, Æ∫

secn−2 x tan2 xdx =

∫secn xdx−

∫secn−2 xdx�

.h, ãîÞP, B;�-ÿ(4.9)P�

3(4.9)P�, &ÆÞsec x�ng]Ý�5, ;W×n − 2g]Ý�5, 9Î�5�ðbÝ]P, (4.9)P-Î×ËLLL]]]222PPP(recursive

formula)�E(4.9)Pun �ó, J��¼�un �ó, Jt¡ºÿÕ

∫sec xdx, h�5��&Æ.ÕEóÐóÝ�5��

��

174 Ïëa �5��5�n;

êÞ�$�b×°L]2P�ÉA1, E∀n ≥ 1,

(4.10)

∫sinn xdx = − 1

nsinn−1 x cos x +

n− 1

n

∫sinn−2 xdx�

9ì¼:, A¢ãîP¼ÿÕ×|P§Ý¶�¼�îiø£πÝ]°�ã(4.10)Pÿ

(4.11)

∫ π/2

0

sinn xdx =n− 1

n

∫ π/2

0

sinn−2 xdx, ∀n ≥ 1�

D«2¸àhL]2P, ÿ

∫ π/2

0

sin2m xdx =2m− 1

2m

2m− 3

2m− 2· · · 1

2·∫ π/2

0

dx, ∀m ≥ 1,

C∫ π/2

0

sin2m+1 xdx =2m

2m + 1

2m− 2

2m− 1· · · 2

3·∫ π/2

0

sin xdx, ∀m ≥ 1�

.h

∫ π/2

0

sin2m xdx =2m− 1

2m

2m− 3

2m− 2· · · 1

2· π

2,∀m ≥ 1,(4.12)

∫ π/2

0

sin2m+1 xdx =2m

2m + 1

2m− 2

2m− 1· · · 2

3,∀m ≥ 1�(4.13)

îÞP¼�&8t, ÿ(4.14)

π

2=

2 · 21 · 3

4 · 43 · 5

6 · 65 · 7 · · ·

2m · 2m(2m− 1)(2m + 1)

∫ π/2

0sin2m xdx

∫ π/2

0sin2m+1 xdx

�

îP��Þ�5Ý¤, m → ∞`, ��1, h�ãì�.0ÿá�E0 < x < π/2, .0 < sin x < 1, v

0 < sin2m+1 x < sin2m x < sin2m−1 x,

3.4 5I�5 175

Æ

0 <

∫ π/2

0

sin2m+1 xdx ≤∫ π/2

0

sin2m xdx ≤∫ π/2

0

sin2m−1 xdx�

ÞîPN×4&t|∫ π/2

0sin2m+1 xdx, v.ã(4.11)P�ÿ

∫ π/2

0sin2m−1 xdx

∫ π/2

0sin2m+1 xdx

=2m + 1

2m= 1 +

1

2m,

Æb

1 ≤∫ π/2

0sin2m xdx

∫ π/2

0sin2m+1 xdx

≤ 1 +1

2m�

ãhÇÿG�\��¨3(4.14)P�, �m →∞, Çÿ

(4.15)π

2= lim

m→∞2

1

2

3

4

3

4

5

6

5

6

7· · · 2m

2m− 1

2m

2m + 1�

hP Wallis (1616-1703)Xÿ, ×Oπ��Ý2P�4|¡&Æº:Õ, $b&9�!ÝOπ Ý2P, ¬(4.15)P��ÎtÚSßÝ×Í�Q, (4.15)PEOπ Â��, ¬�Î�bà, h.(4.15)P��[e�X�ã(4.15)Pô�0�×�î

√πÝ]°, ôq b¶�´�.

limm→∞ 2m/(2m + 1) = 1, Æ(4.15)P��y

limm→∞

22 · 42 · · · (2m− 2)2

32 · 52 · · · (2m− 1)22m =

π

2�

ÞË��¿], vÞ5�5Ò!¶|2 · 4 · · · (2m− 2), ÿ√

π

2= lim

m→∞2 · 4 · · · (2m− 2)

3 · 5 · · · (2m− 1)

√2m = lim

m→∞22 · 42 · · · (2m− 2)2

(2m− 1)!

√2m

= limm→∞

22 · 42 · · · (2m)2

(2m)!

√2m

2m= lim

m→∞(22 · 12)(22 · 22) · · · (22 ·m2)

(2m)!√

2m�

ãhÇÿ

(4.16) limm→∞

(m!)222m

(2m)!√

m=√

π�

176 Ïëa �5��5�n;

5I�5 ×bàÝ�Ì, ¬ô¬&�;�P �»A, ukO

∫x−1dx, �u = x, v = −x−1, J

(4.17)

∫x−1dx =

∫udv = −1 +

∫x−1dx + C,

��ÿÕ×ÍaOÝ�5∫

x−1dx�¯@î, Ç¸�u = xn, v =

−x−n/n, )Î��;Ý�3(4.17)P�, u6�ðóC, Jÿ0 = −1(��

∫x−1dx)��)

§Ý��X|ðóCÎÄ�Ý�QuED0óÝ�ÝÈDS,

á¼(4.17)P�¼��∫

x−1dx5½��x−1�Ø×D0ó, .h�×�8�, µ�ºbæWÝ�t¡&Æ�×�5��JíÂ�§(�ÏÞa�§4.11)Ý¨×

ÌÍ, h ×5I�5°ÝTà�

��§§§4.1.'g3 [a, b]=�, f ′ =�v3[a, b] /!r�JD3×c ∈ [a, b], ¸ÿ

(4.18)

∫ b

a

f(x)g(x)dx = f(a)

∫ c

a

g(x)dx + f(b)

∫ b

c

g(x)dx�

JJJ��.�G(x) =∫ x

ag(t)dt�.g =�, Æã��5ÃÍ�§, ÿ

G′(x) = g(x),∀x ∈ [a, b]��¿à5I�5,∫ b

a

f(x)g(x)dx =

∫ b

a

f(x)G′(x)dx(4.19)

= f(b)G(b)−∫ b

a

f ′(x)G(x)dx,

Í�àÕG(a) = 0, Æf(b)G(a) = 0��ãÏÞa�§4.11, ÿD3×c ∈ [a, b], ¸ÿ

∫ b

a

f ′(x)G(x)dx = G(c)

∫ b

a

f ′(x)dx = G(c)(f(b)− f(a))�

ÞîP�á(4.19)P, ÿ∫ b

a

f(x)g(x)dx = f(b)G(b)−G(c)(f(b)− f(a))

= f(a)G(c) + f(b)(G(b)−G(c))�

êÞ 177

.G(c) =∫ c

ag(x)dx, G(b)−G(c) =

∫ b

cg(x)dx,ÆîPÇ(4.18)P�ÿ

J�

êêê ÞÞÞ 3.4

1− 8Þ/2à5I�5Ý]°�

1. �O(i)∫

x3 cos xdx, (ii)∫

x sin x cos xdx�

2. �J∫

sin2 xdx = x/2− sin 2x/4�¿àh��CîÞ, �J

(i)∫ π/2

0sin2 xdx = π

4;

(ii)∫ π/2

0sin4 xdx = 3

4

∫ π/2

0sin2 xdx = 3π

16;

(iii)∫ π/2

0sin6 xdx = 5

6

∫ π/2

0sin4 xdx = 5π

32�

3. �J

(i)∫

sin3 xdx = −34cos x + 1

12cos 3x + C;

(ii)∫

sin4 xdx = 38x− 1

4sin 2x + 1

32sin 4x + C;

(iii)∫

sin5 xdx = −58x + 5

48cos 3x− 1

80cos 5x + C�

4. �J

(i)∫

x sin2 xdx = 14x2 − 1

4x sin 2x− 1

8cos 2x + C;

(ii)∫

x sin3 xdx = 34sin x− 1

36sin 3x− 3

4x cos x+ 1

12x cos 3x+C;

(iii)∫

x2 sin2 xdx = 16x3 + (1

8− 1

4x2) sin 2x− 1

4x cos 2x + C�

5. �0�E∀n ≥ 2,

∫sinn xdx = −sinn−1 x cos x

n+

n− 1

n

∫sinn−2 xdx�

6. �JE∀n ≥ 2,

∫cosn xdx =

cosn−1 x sin x

n+

n− 1

n

∫cosn−2 xdx�

178 Ïëa �5��5�n;

7. �JE∀n ≥ 1,∫

(a2 − x2)ndx =x(a2 − x2)n

2n + 1+

2a2n

2n + 1

∫(a2 − x2)n−1dx�

8. �JE∀m,n ≥ 1,∫

sinn+1 x

cosm+1 xdx =

1

m

sinn x

cosm x− n

m

∫sinn−1 x

cosm−1 xdx,

∫cosm+1 x

sinn+1 xdx = − 1

n

cosm x

sinn x− m

n

∫cosm−1 x

sinn−1 xdx,

∫sinn x cosm xdx =

sinn+1 x cosm−1 x

n + m

+m− 1

n + m

∫sinn x cosm−2 xdx�

9ì&Þ�2à�¢�áÝ�5*»�9.

∫(2 + 3x) sin 5xdx� 10.

∫x√

1 + x2dx�11.

∫ 1

−2x(x2 − 1)9dx� 12.

∫ 1

02x+3

(6x+7)3dx�

13.∫

x4(1 + x5)5dx� 14.∫ 1

0x4(1− x)20dx�

15.∫ 2

1x−2 sin( 1

x)dx� 16.

∫sin( 4

√x− 1)dx�

17.∫

x sin x2 cos x2dx� 18.∫ √

1 + 3 cos2 x sin 2xdx�19. �J

∫ √a + bx

xdx = 2

√a + bx + a

∫1

x√

a + bxdx�

20. 'n 6= −3/2, �J∫

xn√

ax + bdx

=2

a(2n + 3)(xn(ax + b)3/2 − nb

∫xn−1

√ax + bdx)�

21. 'm 6= −1/2, �J∫

xm

√a + bx

dx =2

(2m + 1)b(xm

√a + bx−ma

∫xm−1

√a + bx

dx)�

êÞ 179

22. 'n 6= 1, a, b 6= 0��J∫

1

xn√

ax + bdx

= −√

ax + b

(n− 1)bxn−1− (2n− 3)a

(2n− 2)b

∫1

xn−1√

ax + bdx�

23. �J ∫ x

0

sin t

t + 1dt ≥ 0,∀x ≥ 0�

24. �f(n) =∫ π/4

0tann xdx, n ≥ 1��J

(i) f(n + 1) < f(n);

(ii) f(n) + f(n− 2) = 1n−1

, n > 2;

(iii) 1n+1

< 2f(n) < 1n−1

, n > 2�

25. ��f(π) = 2, C∫ π

0(f(x) + f ′′(x)) sin xdx = 5��Of(0)�

Â�

26. �

A =

∫ π

0

cos x

(x + 2)2dx�

�|A�îì��5

∫ π/2

0

sin x cos x

x + 1dx�

27. (i) �Jªb×94PP (x), ��

P ′(x)− 3P (x) = 4− 5x + 3x2, ∀x ∈ R;

(ii) ��×94PQ(x), �Jªb×94P��

P ′(x)− 3P (x) = Q(x), ∀x ∈ R�

180 Ïëa �5��5�n;

28. (i) �J

∫ π/2

0

cos4 x√2− sin 2x

dx =

∫ π/2

0

sin4 x√2− sin 2x

dx;

(ii) ¿à(i), �O∫ π/2

0cos4 x/(

√2− sin 2x)dx�Â�

¢¢¢��ZZZ¤¤¤

1. |Ï�(1994). Leibniz A¢��5? ó.Fê��,

Ï18àÏ3�, 3-14�

ÏÏÏ°°°aaa

��555��TTTààà

4.1 ÁÁÁÂÂÂ��LLLCCCíííÂÂÂ��§§§

�5t�ÝTà�×, -Îà¼ÜÃO×ÐóÝÁ�ÂTÁ�Â�&9Tà®Þ�ÝOt·�(optimum solution), ð�»ð ,

O×ÐóÝÁ�ÂTÁ�ÂÝ®Þ�uÞ0óÚ ×Ðó��`�;£, J�à¼O�AÎ§îÝ>�C�>��Á�ÂbËË, ×ËÎ&Æ31.6;D¡ÄÝ�EÁ�Â�3×

/)S�, uD3×c ∈ S, ¸ÿ

(1.1) f(x) ≤ f(c), ∀x ∈ S,

JÌf3cb�EÁ�Âf(c)��EÁ�Âù�v«2�L�uf(c)

f3S��EÁ�Â, B S�×�/), vc ∈ B, J�Qf(c)ù f3B��EÁ�Â�Ï×a�§6.6ô¼�, T îÝ=�Ðó, Äb�EÁ�ÂC�EÁ�Â�¨×ËÁÂÎ8EÁÂ, Í�LAì�

��LLL1.1.'f ×�L3/)S��@ÂÐó, ê'c ∈ S�uD3×�âc�� I, ¸ÿ

f(x) ≤ f(c), ∀x ∈ I ∩ S,

181

182 Ï°a �5�Tà

JÌf3cb888EEEÁÁÁ��ÂÂÂ(relative maximum)f(c)�

!§��L888EEEÁÁÁ��ÂÂÂ(relative minimum)�b`&Æ©18EÁ�T8EÁ�, �6¯“Â”�8EÁ�

�8EÁ�, )Ì888EEEÁÁÁÂÂÂ(relative extreme value, Trelative ex-

tremum, extremum��ó extrema)�u×ó Ðóf3S��×8EÁÂ, JÌhó f�×ÁÂ�3�L1.1�, c ÁÂsß�,

�f(c) ×ÁÂ��EÁ�C�EÁ�, J)Ì�EÁÂ��:�N×�EÁÂ, / ×8EÁÂ�8EÁ��, t�ï-Î�EÁ�, 8EÁ��, t�ï-Î�EÁ��8EÁ�êÌIIIÁÁÁ��(local maximum), �EÁ�êÌ ��IIIÁÁÁ��(global maxi-

mum)�h.N×8EÁ�, Ç ×Ï½�Ý�EÁ��.h, b`&ÆºTÙ21Á�(TÁ�), . ;ð¬�p5ï, X¼ÝÎ8EÁ�(TÁ�), T�EÁ�(TÁ�)�uÐóf3cb×8EÁ�, Jh8EÁ�ù f3c �×Ï½Ý

�EÁ��N×�EÁ�QôÎ×8EÁ��%1.1 ×°��Ý�µ�

-

6

-

6�EÁ�

?

π2 π

�EÁ�I

�EÁ�µ

x

y

O

�EÁ�R

8EÁ�

?

�EÁ�6 8EÁ�

6

f(x) = sin x, 0 ≤ x ≤ π f(x) = x(1− x)2, −1/2 ≤ x ≤ 2

−12

13

1 2x

y

O

%1.1. Ðó�ÁÂ

4.1 ÁÂ��LCíÂ�§ 183

��§§§1.1.'f�L3×� (a, b), vf3c ∈ (a, b)b8EÁÂ�uf ′(c)D3, Jf ′(c) = 0�JJJ��.3(a, b)î�LÐóQ

Q(x) =

{(f(x)− f(c))/(x− c), x 6= c,

f ′(c), x = c�

.f ′(c)D3, Æx → c`, Q(x) → f ′(c) = Q(c), .hQ3c =��u�J�Q(c)= 0, Jf ′(c)= 0-ÿJÝ�&ÆÞàDJ°, Ç5½0�Q(c) > 0, �Q(c) < 0/�)�'Q(c) > 0, JãÏ×aS§6.1, D3c�×Ï½, ¸ÿQ 3h

Ï½�/ Ñ�Ç3hÏ½�, Ex 6= c, Q(x)�5��5Ò!r�Æ3hÏ½�, f(x) > f(c), ∀x > c, f(x) < f(c), ∀x < c�h�f3cbÁÂ�)�ÆQ(c)�� Ñ�!§�JQ(c) < 0�)�.hQ(c) = 0, Çf ′(c) = 0�ÿJ�

î��§, Î3yÁÂsß�x = c�0óD3, vc ×/F(Ç� \&F)��'ìÝ��.h9ìÝ.¡, -��ºWñ�

��§§§1.1.'f3cbÁÂ, JÄbì�ëË�µ�×sß:

(i) f ′(c)D3v 0,

(ii) f ′(c)�D3,

(iii) c \&F�

.N×�EÁÂù ×8EÁÂ, �§1.1ôÊà�EÁÂÝ�µ��Ä, �§1.1�Y�Ë, Çb��(i)�(ii) T(iii)�, b×Wñ, ¬f3cQPÁÂ�»A, uf(x) = x3, f ′(x) = 3x2, Æf ′(0) =

0�¬f ×�¦Ðó, .hf30PÁÂ, �%1.2��§1.1¼�0ó�D3�,b��sß3ÁÂ�»A,'f(x) =

|x|, Jf3x = 0��, ¬f30Qb×8EÁ�(ôÎ�EÁ�), �%1.3�

184 Ï°a �5�Tà

-

6 f(x) = x3

x

y

O

%1.2. f ′(0) = 0¬f30PÁÂ

-

6

O

f(x) = |x|

x

y

%1.3. f ′(0)�D3, ¬f30bÁÂ

9ì ×��Ý»��

»»»1.1.�f(x) = x−1�Jf��L½ (−∞,0)∪(0,∞)�Í%� Ô`a,��3�.`��B!�Ý��Qf3�L½�,P8EÁÂ(ùP�EÁÂ)�¬f3(0, 1], b�EÁ�(ù 8EÁ�),

sß3x = 1; f3(0, 1)PÁÂ; 3[−2,−1]��EÁ�(3x = −2),

C�EÁ�(3x = −1)/D3; 3[−2, 0)©b�EÁ��

'c f��L½�Ý×ÍF, f ′(c)�D3, Tf ′(c) = 0, JcÌ f�×ÛÛÛ&&&FFF(critical point)�»A, uf(x) = |x|, J0 ×Û&

4.1 ÁÂ��LCíÂ�§ 185

F; uf(x) = x3, J0ù ×Û&F��Äb°hÎÌ(c, f(c)) ×Û&F, b°hJ©3f ′(c) = 0`, �Ìc ×Û&F�3�ì×�§�G, &Æ�E0óÝÑT�, ºEÐó®ßÝÅ(�×°D¡�uÐóf3ØFc=�, J3x�#�c`, f(x)ÝÂ�f(c)ôº�

#��uf3x = c��, ºb¢.¡÷? ��Ä=�, .hbn=�PÝ��, h`Ef/Êà�¬��f=��úÿ9, .h&ÆEfôTá¼ÿ?9��'f ′(c) > 0, J

limh→0

f(c + h)− f(c)

h= f ′(c) > 0�

.h©�hÈ�, Jf(c + h)− f(c)

h> 0�

ÇhÈ�`, f(c + h)− f(c)�h!r�ÆEÈ�Ýh, f(c + h)> f(c),

h > 0; f(c + h)< f(c), h < 0�ôµÎD3c�×Ï½, ¸ÿ3hÏ½�, f �¦�D�, uf ′(c) < 0, JD3c�×Ï½, ¸ÿ3hÏ½�, f �3�¨3&Æ�|�ì��§Ý�

��§§§1.2.'f3T [a, b]=�, vf ′(a)f ′(b)< 0, Jf3(a, b)�bÛ&F�JJJ��.'f ′(a) > 0, f ′(b) < 0�ãÏ×a�§6.6, =�Ðóf 3T [a, b]�, Äb�EÁ��¬.f ′(a) > 0, .h3aÝ!�f �¦, Æ�EÁ��sß3a�!§�EÁ�ù�sß3b�Æ3� (a, b)�, fb×�EÁ��Çÿ�§1.1�(i)T(ii)Wñ�.hf3(a, b)�bÛ&F��yuf ′(a) < 0vf ′(b) > 0, !§�J�

»»»1.2.�Jf(x) = x4−2x3+3x− 1, 3(−1, 2)bÛ&F�JJJ��..f ×94P, Æf3[−1, 2]=��êf ′(x) = 4x3 − 6x2 + 3,

Æf ′(−1) = −7 < 0, f ′(2) = 11 > 0�Æã�§1.2ÇÿJ�

186 Ï°a �5�Tà

Íg&Æ:Rolle��§§§(Rolle’s theorem), 9Î°»ó.�Rolle

(1652-1719)3�-1690OJ�Ý�

��§§§1.3.'Ðóf3T [a, b]=�, v'f(a) = f(b)�Jf3� (a, b)��Kb×Û&F�JJJ��.uf(x) = f(a), ∀x ∈ [a, b], Jf3[a, b] ×ðóÐó�.hf ′(x) = 0, ∀x ∈ (a, b)�Ç∀x ∈ (a, b)/ f�Û&F�uD3×x0 ∈ (a, b), ¸ÿf(x0) 6= f(a), .T îÝ=�Ðó, Äb�EÁ�, C�EÁ�, Æuf(x0) > f(a), Jf3(a, b)�D3×�EÁ�; uf(x0) < f(a), Jf3(a, b)�D3×�EÁ��Æã�§1.1á, D3×c ∈ (a, b), ¸ÿf ′(c)�D3, Tf ′(c) = 0�ÿJ�

9ì ×ñÇÝ.¡�

��§§§1.2.'Ðóf3T [a, b]=�, 3� (a, b) ��, ê'f(a)

= f(b)�J�Kb×c ∈ (a, b), ¸ÿf ′(c) = 0�

3�§1.2��'ì, y(a, b)�Äb×Fc, ¸ÿ3c�6a i¿, �%1.4�

a c b

f(a) f(b)

A B

f ′(c) = 0

%1.4. Rolle�§�¿¢�î

»»»1.3.'f(x) = x2 − 4x, x ∈ [−1, 5]�.f(5) = f(−1) = 5, Æf3(−1, 5)�b×Û&F��p:�f ′(2) = 0, Ç2 ×Û&F�

4.1 ÁÂ��LCíÂ�§ 187

»»»1.4.'f(x) = 1 − |1 − x|, x ∈ [0, 2]�ãyf(0) = f(2) = 0,

Æf3(0, 2)b×Û&F, hF�QÇx = 1, f�×��ÝF�

A!3�5�, �5�ùbíííÂÂÂ��§§§(Mean-value theorem for

derivatives)�h�§:R¼¬�R¿, ¬Qb��Ý;¨, &Æº�1��3Rolle�§�, �'f(a) = f(b), �3E∀x ∈ (a, b), f ′(x) D3

ì, &ÆÿÕf�%�3(a, b)�, ÄbØ×FÍ6a i¿�ð�1, 3Ø×F�6a, ¿�(a, f(a))�(b, f(b))�=a�uf(a) 6=f(b)ºA¢? ¯��ÕÝ, ºb×v«Ý��, ÇD3×Fc ∈(a, b), ¸ÿf�%�3c�6a, ¿�(a, f(a))�(b, f(b))�=a�Þa¿�, ê�ÍE£8!�&ÆB�¬J�h��Aì�

��§§§1.4.(��555ÝÝÝíííÂÂÂ��§§§). 'f3T [a, b]=�, 3� (a, b)

��J�Kb×c ∈ (a, b), ¸ÿ

(1.2) f(b)− f(a) = f ′(c)(b− a)�

JJJ��.&Æ�¿àRolle�§, .hm�×Ðó, 3 �ÞÐFb8!ÝÐóÂ��

h(x) = f(x)(b− a)− x(f(b)− f(a))�

J�Qh(a) = h(b) = bf(a)−af(b),vh3(a, b)��Æ¿àRolle�§ÿ,D3×c ∈ (a, b),¸ÿh′(c) = 0�.h′(x)= f ′(x)(b−a)−(f(b)−f(a)), .hÇÿJ(1.2)Wñ�

ê(1.2)P�;¶

(1.3)f(b)− f(a)

b− a= f ′(c)�

ãîP&Æ�?Ý�íÂ�§Ý�L��'b×Ô�, 3×àaî?GÉ�, f(t)��` t�Ô�X�ÝûÒ�J(1.3)P�¼

188 Ï°a �5�Tà

�, �3` [a, b] , Ô��¿í>�, êf ′(t)�3`Ñt��`>��(1.3)PÇ�3(a, b)�ÄD3Ø×`Ñ, ¸ÿ3�`Ñ��`>�, �y3[a, b] �¿í>��¯@î, f ′ù3(a, b)=�`,

h��ã=�ÐóÝ� Â�§(Ï×a�§6.3), ñÇ�ÿ( %�?)�

%1.5 ��îíÂ�§�¿¢�L�

a

(c, f(c))

c b

A

B

ba c2

B

A

c1

(a) (b)

%1.5. íÂ�§�¿¢�L

¨², íÂ�§©Î1Jc�D3, ¬Î�¼�c3¢��Eb°Ðóf , ??cÂ¬�|O��ÄÍ�§Ý¥�P, 3ycÝD3, ¿àhP², �ÿÕ¨×°&ÆX��Ý��, �¬�m�á¼c�@6ÝÂ�ê�èø&�ÝÎ, 3¸àÍ�§`, uf¬&3∀x ∈ (a, b)��, -�×�ÊàÝ�»A, f(x) = |x| ×=�Ðó, vtÝ3x = 0²/=��¬¬�D3×c ∈ (−1, 1), ¸ÿf ′(c)= (f(1)−f(−1))/(1−(−1))= 0�9ìÝ�§(Cauchy’s mean-value formula) íÂ�§�×�.

Â�

��§§§1.5.(ÞÞÞ��íííÂÂÂ��§§§). 'ÞÐóf�gí3T [a, b]=�, 3� (a, b)��J�Kb×c ∈ (a, b), ¸ÿ

(1.4) f ′(c)(g(b)− g(a)) = g′(c)(f(b)− f(a))�

4.1 ÁÂ��LCíÂ�§ 189

JJJ��.�

h(x) = f(x)(g(b)− g(a))− g(x)(f(b)− f(a))�

Jh(a) = h(b)= f(a)g(b)−g(a)f(b)�.hãRolle�§, D3×c ∈(a, b), ¸ÿh′(c) = 0�BãÞhEx�5, Çÿ(1.4)PWñ�

9ì&Æ�¿Í»��

»»»1.5.�f(x) = 2x2 − x + 1, x ∈ [0, 1]�Jf ′(x) = 4x− 1�ê

f(1)− f(0)

1− 0=

2− 1

1= 1�

Æ�f ′(c) = 4c− 1 = 1, ÿc = 1/2�

»»»1.6.'f(x) = x4 − 7x3 + 2x2 + 4, x ∈ [0, 1]�Jf ′(x) = 4x3 −21x2 + 4x�ê

f(1)− f(0)

1− 0=

0− 4

1= −4�

ãíÂ�§, b×c ∈ (0, 1), ��4c3 − 21c2 + 4c = −4�¬c¬�|��

»»»1.7.'p > 1, x > 1��J

(1.5) p(x− 1) < xp − 1 < pxp−1(x− 1)�

JJJ��.�f(t) = tp, t ∈ [1, x], x > 1�Jf ′(t)= ptp−1�ãíÂ�§,

D3×c ∈ (1, x), ¸ÿ

f ′(c) = pcp−1 =xp − 1

x− 1,

Çxp−1= p(x−1)cp−1�.c ∈ (1, x),vp > 1,Æ1< cp−1< xp−1�¿àh��PñÇ:�(1.5)PWñ�

190 Ï°a �5�Tà

»»»1.8.�J

5 +5

52≤√

26 ≤ 5 +1

10�JJJ��.�f(x) =

√x, x ∈ [25, 26]�Jf ′(x) = 1/(2

√x)�ãíÂ�

§, D3×c ∈ (25, 26), ¸ÿ

f ′(c) =1

2√

c=

√26−√25

26− 25=√

26−√

25,

T √26 =

√25 +

1

2√

c= 5 +

1

2√

c�

.c ∈ (25, 26),

5

52<

√26

52=

1

2√

26<

1

2√

c<

1

2√

25=

1

10�

J±�

�Aî», E√

26X�Ý×î�ì§, 3£�Õ^�Hs¾Ý`�bÍÄ��*^�Õ�ÌAh]-, 9Ë��PÝàHQ^£��Ý�Íg&Æ¼:×33.2;, &Æ-�àÄÝ��J��Çu×

T [a, b]îÝ=�Ðóf , ��f ′(x) = 0, ∀x ∈ (a, b), Jf3(a, b)

×ðó�àÌîh��ÎEÝ�.3ØF�0ó 0, �3�F�6aE£ 0, Ç6a¿�x�, �u`a3N×F�6a/¿�x�, h`aÄ ×¿�x��àa�ü�×t ∈ (a, b), JE∀x ∈ (a, b), vx > t, .f3[t, x] =�v��, .híÂ�§Êà,

vD3×c ∈ (t, x), ¸ÿ

f(x)− f(t) = f ′(c)(x− t) = 0�

Æÿf(x) = f(t)�!§ux ∈ (a, b), vx < t, ù�ÿf(x) =

f(t)�.hf3(a, b) ×ðó�&Æá¼×Ðóu��Ä=�, ¿àíÂ�§��ÿ´=�P

?×MÝ£G�'Ðóf3T [a, b]��, ê'f ′(x)3[a, b]b

4.1 ÁÂ��LCíÂ�§ 191

&(uf ′(x)3[a, b] =�,Jf ′(x)3[a, b]Äb&),Ç'D3×ðóM

> 0, ¸ÿ|f ′(x)| ≤ M , ∀x ∈ [a, b]�E��Þx1, x2 ∈ (a, b), ãíÂ�§á, D3×ξ ∈ (x1, x2), ¸ÿ

|f(x2)− f(x1)| = |f ′(ξ)(x2 − x1)| ≤ M(x2 − x1)�

.hE∀ε > 0, ©�ãδ = ε/M , J|x2 − x1| ≤ δ`, |f(x2) −f(x1)| ≤ ε�»A, uf(x) = x2, x ∈ [−a, a]�.

|f ′(x)| = |2x| ≤ 2a,

Æ|x2 − x| ≤ ε/2a`, |f(x2)− f(x1)| ≤ ε�3h, &ÆÌ×Ðó��Lipschitzfff��(Lipschitz condition,

Lipschitz (1832-1903) Æ»ó.�),T1 Lipschitz===��(Lipsch-

itz continuous), uD3×ðóM , ¸ÿE�Þ�L½�Ýx1, x2,

|f(x2)− f(x1)| ≤ M |x2 − x1|�uLipschitzf�Wñ, Jì�¤

f(x2)− f(x1)

x2 − x1

��EÂ, ù|M Íî&��:�×3T îb×=�0óÝÐóf , Ä Lipschitz=��¬Ç¸×¬&3N×F/��ÝÐó,

ôb�� Lipschitz=��»Af(x) = |x|�¨×]«,¬&N×=�Ðó,/ Lipschitz=��»A,ãf(x)

= x1/3, Jf(x)− f(0)

x− 0= x−2/3,

x��`, f¬&b&, Æf3[0, x]� Lipschitz=��Qf ′(x)

= x−2/3/3, x�#�0`,ô� b&(G«¼�uf ′ b&, f- Lipschitz=�)�Lipschitz=�ÝÐó, XxWÝ/), �ây=�ÐóÝ/), v�â0óÎ=�ÝÐó�/)�ð�1,

Lipschitz=�, Î×f=�P�ú, �f=��3Ýf��3ó.��Lipschitz=�, ×v¥�ÝÐó�

192 Ï°a �5�Tà

Í;t¡&Æ¼:íÂ�§�¨×Tà�32.8;�êÞ, &Æ�O

f(x) =

{x2 sin(1/x), x 6= 0,

0, x = 0,

�0ó�h ×Õ��, ¬0ó�×�=��»�¯@î, .

f ′(x) =

{2x sin(1/x)− cos(1/x), x 6= 0,

0, x = 0,

Æf ′(x)tÝ3x = 0²/=��vlimx→0+ f ′(x)Climx→0− f ′(x)/�D3�9ì ×¾½0ó�=�PÝ��

��§§§1.6.'Ðóf3a�×Ï½N�=�, vf ′(x)D3, ∀x 6= a�ê'

limx→a

f ′(x) = b

D3�Jf ′(a)D3v�yb, Çh`f ′3a=��JJJ��.¿àíÂ�§, ÿE∀u ∈ N , vu 6= a, D3×ξ+ya�u , ¸ÿ

f(u)− f(a)

u− a= f ′(ξ)�

�u → a, Jξ → a, �ã�'áf ′(ξ) → b�.hu → a`, îP¼��Á§D3, v�yb�¬hÇ�f ′(a)D3v�yb�ÿJ�

&��¿àî��§, ¥�ÏÞa»7.14�9ì¨×8nÝ��, ù�v«2ÿÕ, J�º3êÞ�

��§§§1.7.'f3[a, b]=�, 3(a, b)��, vlimx→a+ f ′(x) = ∞�J

limx→a+

f(x)− f(a)

x− a= ∞�

êÞ 193

êêê ÞÞÞ 4.1

1. �¿àÞ�íÂ�§, J�E∀x > 0, D3×z ∈ (0, x), ¸ÿ

sin x− x

x3= −1

6cos z�

2. �Eì�&ÐóOÁÂ, ¬¼�¢ï 8EÁÂ, ¢ï �EÁÂ�(i) f(x) = 4− x, x ∈ [−2, 4];

(ii) f(x) = x2 − 2x + 2, x ∈ R;

(iii) f(x) = x3 + x2 + x− 4, x ∈ [−1, 1];

(iv) f(x) = x4 − x3, x ∈ [0, 1];

(v) f(x) = x2 + 4x−2, x ∈ [−2, 2];

(vi) f(x) = (x− 1)/(x2 + 3), x ∈ [−4, 2]�3. Eì�&ÐóC , ��JíÂ�§ÎÍWñ�

(i) f(x) = (x− 1)/x, x ∈ [1, 3];

(ii) f(x) = 1 + x2, x ≥ 0, = 1− x2, x < 0, x ∈ [−1, 1];

(iii) f(x) = (3− x2)/2, x ≤ 1, = x−1, x ≥ 1, x ∈ [0, 2]�4. �J

(i) 3 + 128≤ 3√

28 ≤ 3 + 127

;

(ii) 2 + 2165

≤ 5√

33 ≤ 2 + 180�

5. 'f ×Þg94P��J3y = f(x)�%�î,=#(a, f(a))

C(b, f(b))�àa, Ä¿�3x = (a + b)/2�6a�

6. 'f(x) = 1− x2/3��Õ %�4f(1) = f(−1), vf ′(x)3[−1, 1]�/� ë, ¬hQ�ÀDRolle�§�

7. �¿àRolle�§J�,E∀b ∈ R,]�Px3−3x+b = 03[−1, 1]

�, t9©b×q�

8. �Jx2 = x sin x + cos xªbÞ@q�

194 Ï°a �5�Tà

9. �Ju

xn + a1xn−1 + a2x

n−2 + · · ·+ an−1x = 0

b×Ñqx = r, J

nxn−1 + (n− 1)a1xn−2 + (n− 2)a2x

n−3 + · · ·+ an−1 = 0

b×�yr�Ñq�

10. �JE]�Pxn + ax + b = 0, un �ót9bÞ@q; un

�ót9bë@q�

11. �Jx > sin x, ∀x > 0; x < tan x, ∀x ∈ (0, π/2)�

12. �J

(i) | sin x− sin y| ≤ |x− y|;(ii) nyn−1(x − y)≤ xn − yn≤ nxn−1(x − y), 0 < y ≤ x, n =

1, 2, · · ·�13. 'f3[a, b]��, vEØ×c ∈ (a, b), f ′(x) ≤ 0, ∀a ≤ x < c,

f ′(x) ≥ 0, ∀c < x ≤ b��Jf(x) ≥ f(c), ∀x ∈ [a, b]�

14. �JíÂ�§�;¶ : 'f3[x, x + h]=�, 3(x, x + h)��, Í�h > 0�JD3×θ ∈ (0, 1), ¸ÿ

f(x + h) = f(x) + hf ′(x + θh)�

5½Ef(x) = x2, f(x) = x3, X�θ(|xCh��)�ü�x,

x 6= 0, 5½EG�ÞÐó, Oh → 0`θ �Á§�

15. 'Ðóf3T [a, b]=�, 3� (a, b)Þg��ê'(a, f(a))�(b, f(b))�=a, øy = f(x)�%�y(c, f(c)),

Í�c ∈ (a, b)��JD3×ξ ∈ (a, b), ¸ÿf ′′(ξ) = 0�

16. 'Ðóf3@óî��Juf(0) = 0v|f ′(x)| ≤ |f(x)|,∀x ∈ R, Jf(x) = 0,∀x ∈ R�

4.2 OÁÂC0% 195

17. uD3c�×�TÏ½D,C×ðóM > 0(M��cbn),¸ÿ

|f(x)− f(c)| < M |x− c|α,∀x ∈ D,

JÌf3c��αggg��Lipschitzfff��(Lipschitz condition of or-

der α)�'f3c��αg�Lipschitzf��Juα > 0, Jf

3c=�, uα > 1, Jf3c��

18. 'f�L3[a, b], uD3×ðóM > 0, ¸ÿ

|f(x)− f(y)| < M |x− y|α,∀x, y ∈ [a, b],

JÌf3[a, b]��αggg��ííí888Lipschitzfff��(uniform Lipschitz

condition of order α)��Juα > 1, Jf3[a, b] ×ðó�

19. �J�§1.7�

4.2 OOOÁÁÁÂÂÂCCC000%%%

31.3;&Æ��L��Ðó, ¿à0ó�¾½Ðó��P, �ì�§�

��§§§2.1.'Ðóf3× I=��(i) uE∀x ∈ I, vx� I�ÐF, f ′(x) > 0, Jf3I � �}�

¦;

(ii) uE∀x ∈ I, vx� I�ÐF, f ′(x) < 0, Jf3I� �}�3�JJJ��.&Æ©J(i), (ii)�J�v«�'x1, x2 ∈ I, vx1 < x2�JãíÂ�§, D3×c ∈ (x1, x2), ¸

ÿf(x2)− f(x1) = f ′(c)(x2 − x1)�

196 Ï°a �5�Tà

.x2 > x1, vã�'f ′(c) > 0, Æf(x2) > f(x1)�ÿJ�

3î�§�,u(i)��f�f ′(x) > 0,; f ′(x) ≥ 0,Jÿf3I� �¦; u(ii)��f�f ′(x) < 0; f ′(x) ≤ 0, Jÿf3I� �3�¨², '×Ðóf3 I�=�, v3I�tÝb��3ÐF

², /PÛ&F�Jã�§1.2, E∀x ∈ I, vx� ÐF, f ′(x) > 0,

Tf ′(x) < 0�h��î�§2.1, -ÿì��§�

��§§§2.2.'Ðóf3× I=�, v3I �tÝb��3ÐF², /PÛ&F, Jf3I� �}��

E×=�Ðóf , ×Ë&Æ0�¸XbÝÛ&F, Jf 3£° ��, -K�X��»A, 'a < b, f�Þ8µÝÛ&F,

vf3[a, b]=��Jã�§2.2ÿ:

(i) uf(a) < f(b), Jf3[a, b]�}�¦;

(ii) uf(a) > f(b), Jf3[a, b]�}�3�\ïÎÍ:�, %��Êf(a) = f(b)Ý�µ? ¯@î

ãRolle�§, ñÇ:�h`f(a) 6= f(b)�ê�§2.2, ôÊà � b§Ý�µ, ã9ìÝ»��:��

»»»2.1.�0�Ðóf(x) = x3 − 3x + 1 �� .´�f�0ó f ′(x) = 3x2−3,.hx = 1, −1 f�Û&F��ì��

x −2 −1 1 2

f(x) −1 3 −1 3

�:�f(−2) < f(−1), f(−1) > f(1), f(1) < f(2)�êf 3(−∞,

−1)C(1,∞)�/PÛ&F, vf(−2) < f(−1), f(2) > f(1), Æã�§2.2, f3(−∞,−1] � �}�¦, 3[1,∞) � �}�¦�.f(−1) > f(1), Æf3[−1, 1]� �}�3�

4.2 OÁÂC0% 197

3î»�, &ÆX�Ýf��¦C�3ÝP��E×Ðó, t�Ý�Í� Ý, �Äy0�Í%��&ÆX�ºD¡, A¢�¸%�´Þ@2�0��Äã»2.1�Ý�¡, -�ÿf�%��VA%2.1�

-

6(−1, 3)

(2, 3)

(1,−1)−3 −1 O 1 3

x

y

−1

−3

1

3

%2.1. f(x) = x3 − 3x + 1�%�

�:×»�

»»»2.2.'f(x) = x + 4/x2��O¸f �� , ¬0f�%��.f�%�A%2.2�

-

6

(2, 3)

O 2 4x−2−4

2

4

6

y

−2

−4

%2.2. f(x) = x + 4/x2�%�

´�f��L½ R\{0},v ×=�Ðó�êf ′(x) = 1−8x−3,

x 6= 0��:�f ′(x) = 0uv°ux = 2,Ç2 f°×�Û&F�Æã�§2.2, f3(−∞, 0), (0, 2], [2,∞)/ �}��ê.f ′(−1) >

0, f ′(1) < 0, f ′(3) > 0, Æã�§2.1á, f3(−∞, 0)� �}�¦,

3(0, 2]� �}�3, 3[2,∞) � �}�¦�

198 Ï°a �5�Tà

êlimx→0 f(x) = ∞, Æx = 0 ×kà��a�ê|x|��`,

f(x)�#�x,Çlimx→−∞(f(x)−x)= limx→∞(f(x)−x)= 0,.hy =

x ×E��a(ulimx→∞(f(x)− (mx+ b)) = 0,Tlimx→−∞(f(x)−(mx + b)) = 0, Jàay = mx + bÌ y = f(x)�%�Ý×E��a)�ã%2.1�:�, uf(x) = x3 − 3x + 1, Jf3x = −1b×8EÁ

�, 3x = 1b×8EÁ�, vÁ��Á�Â5½ 3C−1��yf

¬P�EÁÂ�ã%2.2�:�, uf(x) = x + 4/x2, J3x = 1b×8EÁ�, vf¬P8EÁ�C�EÁÂ�×��, �¿à�§2.1 ¼OÁÂ�&ÆÞ��B�Aì�

��§§§2.3.'=�Ðóf3� (a, b), tÝ��3c², /��(i) uf ′(x) > 0, ∀x < c, f ′(x) < 0, ∀x > c, Jf3c b×8EÁ

�;

(ii) uf ′(x) < 0, ∀x < c, f ′(x) > 0, ∀x > c, Jf3c b×8EÁ��JJJ��.3(i)Ý�µ, ã�§2.1á, f3(a, c)�}�¦, v3(c, b) �}�3�.hE∀x ∈ (a, b), vx 6= c, f(x) < f(c)�Æf3cb×8EÁ��!§�J(ii)�

�§2.3�|%2.3¼1�, E×��Ðó, 0ó;�Ñ�r�, µºbÁÂ®ß�

f ′(x)>0 f ′(x)<0 f ′(x)<0 f ′(x)>0

a c b a c b

(a)3cbÁ� (b)3cbÁ�

%2.3. ÁÂsß�!��%�

4.2 OÁÂC0% 199

X|kOÐóf�ÁÂ, ÍM»Aì�(i) 0�f�XbÛ&FC\&F, |A�9°F�/)�(ii) 'c ∈ A, 0�×c�Ï½(a, b), ¸ÿf3[a, b] ∩ D =�, Í�D f��L½, v[a, b] ∩ A = {c}, Ç3[a, b]�, c f°×�Û&FT\&F�(iii)5½�Õf(a)�f(b)Cf(c)��'c � \&F�

(1) uf(a) < f(c), vf(b) < f(c), Jf3cb×8EÁ�;

(2) uf(a) > f(c), vf(b) > f(c), Jf3cb×8EÁ��(3) uî�ÞË�µ/Îsß, Jf3cPÁ��

�yuc \&F,c ¼\&F©�f´f(b)�f(c)��;c �\&F©�f´f(b)�f(c)��Ç��î�M»(iii), �|ì�(iii)′ ã��

(iii)′5½�Õf ′(a)Cf ′(b)�(1) uf ′(a) > 0, vf ′(b) < 0, Jf3cb8EÁ�;

(2) uf ′(a) < 0, vf ′(b) > 0, Jf3cb8EÁ�;

(3) uî�Þ�µ/Îsß, Jf3cPÁ��yuc \&F, c ¼\&F©�:f ′(b)�Ñ�; c �\&F©�:f ′(a)�Ñ��9ì�¿Í»��

»»»2.3.'f(x) = x3�Jf ′(x) = 3x2, 0 f°×�Û&F�.f ′(x)

> 0, ∀x 6= 0, Æ0¬&f�ÁÂ�¯@îf ×�}�¦Ðó, ¬PÁÂD3�

»»»2.4.'f(x) = x3 + 3x2 − 1��Of�8EÁÂ��.´�f ′(x) = 3x2 + 6x = 3x(x + 2), .h−2, 0 f�Û&F�êf ′(x) > 0, ∀x < −2, f ′(x) < 0, ∀ − 2 < x < 0, f ′(x) > 0, ∀x >

0, Æã�§2.3, f3−2b×8EÁ�, f30b×8EÁ��BO�f(−2) = 3, f(0) = −1, �ÿf�%�A%2.4�

»»»2.5.'f(x) = x3 +3x−1��Of��aC8EÁÂ¬0Í%��.x → 0+`, f(x) → ∞, x → 0−`, f(x) → −∞, Æx = 0 k

200 Ï°a �5�Tà

à��a�Ígf ′(x) = 3x2− 3x−2 = 3(x4− 1)/x2�ÿ−1, 1 f�Û&F�êf ′(x) > 0, ∀x < −1, f ′(x) < 0, ∀0 < x < 1, f ′(x) < 0,

∀0 < x < 1, f ′(x) > 0, ∀x > 1�Æf3−1b8EÁ�, 31b8EÁ��vf(−1) = −4Cf(1) = 4 5½ 8EÁ�ÂC8EÁ�Â�f�%�A%2.5�

-

6(−2, 3)

(0,−1)−1−2−3−4 1 2 3

1

2

3

−1

x

y

O

%2.4. f(x) = x3 + 3x2 − 1�%�

-

6

−40

10

40

1 2 3−1−2−3x

y

O

%2.5. f(x) = x3 + 3x−1�%�

3ÏÞa�L6.1, &Æ��ÐóC�ÐóÝ�L�&Æ��×8nÝ�LAì�

��LLL2.1.Ðóf�%�, Ì 3(c, f(c)) îîî��(concave upward),

uf ′(c)D3, vD3c�×�TÏ½D, ¸ÿf3D��%�/3Ä(c, f(c))�6aÝî]�!§��Lììì��(concave downward)�

4.2 OÁÂC0% 201

î�Cì��%��lA%2.6�

O cx

y

(c, f(c))

O cx

y

(c, f(c))

(a) 3(c, f(c)) î� (b) 3(c, f(c)) ì�

%2.6. î�Cì��%�

�:�E×��Ðóf , u3Ø ½� �(�)Ðó, Jf �%�3h ½� î�(ì�)�Ý�Ðó��P, �¸&Æ�Þ@20%�9ì ×¾½�§�

��§§§2.4.'Ðóf3c�Ø×Ï½��(i) uf ′′(c) > 0, Jf�%�3(c, f(c)) î�;

(ii) uf ′′(c) < 0, Jf�%�3(c, f(c)) ì��JJJ��.'f ′′(c) > 0, Çf ′(x)3c�×Ï½� ¦Ðó�JD3c�×Ï½N , ¸ÿE∀x ∈ N , f ′(x) < f ′(c), ux < c; f ′(x) > f ′(c),

ux > c�êãíÂ�§(�§1.4)ÿ, D3×ξ+yx�c� , ¸ÿ

f(x)− f(c)− f ′(c)(x− c) = f ′(ξ)(x− c)− f ′(c)(x− c)

= (f ′(ξ)− f ′(c))(x− c)�

�g(x) = f(x) − f(c) − f ′(c)(x − c)�E∀x ∈ N , ux > c, Jc <

ξ < x, .hf ′(ξ) > f ′(c), vg(x) > 0; ux < c, Jx < ξ < c,

.hf ′(ξ) < f ′(c), h`)bg(x) > 0�ÇÿE∀x ∈ Nvx 6= c,

g(x) > 0�.y = f(c)+f ′(c)(x− c) Ä(c, f(c))�6a, (x, f(c)+f ′(c)(x−

c)) 6aî×F, �(x, f(x)) f�%�îÝ×F, uf(x) > f(c) +

202 Ï°a �5�Tà

f ′(c)(x−c),�f3x�%�36aÝî]�.huE∀x ∈ N ,vx 6=c, g(x) > 0, f�%�3(c, f(c)) î��ÿJ(i)�!§�J(ii)�

9ì�§ �§2.4�×.¡, h ¿àÞÞÞ$$$000óóó¼¼¼¾¾¾½½½ÁÁÁÂÂÂ(second derivative test for extrema)�

��§§§2.5.'f ′(c) = 0, vf3c�×Ï½��(i) uf ′′(c) < 0, Jf3cb×8EÁ�;

(ii) uf ′′(c) > 0, Jf3cb×8EÁ��JJJ��..f ′(c) = 0, Æf�%�3cb×i¿6a�uf ′′(c) < 0,

Jã�§2.4, f�%�3c�!� ì�, Ç3c!�f(x) < f(c),

Æf3cb×8EÁ��ÿJ(i)��y(ii)ù!§�J�

uf ′(c) = 0, vf ′′(c) = 0, Jî�§-�Êà¼¾½ÁÂ�9`��¿à�§2.3, 9Î´ÃÍÝ�§, �à�'Þ$0óD3�Í;t¡º�D¡f ′′(c) = 0Ý�µ�ãÞ$0ó�Ñ�, ù�¾½Ðó��P��ì�§�

��§§§2.6.'f3T [a, b]=�, 3� (a, b) ��uf ′3(a, b)

�¦, Jf3[a, b] �Ðó�©½2, uf ′′3(a, b)�D3v&�,

Jf �Ðó�JJJ��.3[a, b]�ãx < y��z = αy + (1− α)x, Í�0 < α < 1�&Æ6J�

f(z) ≤ αf(y) + (1− α)f(x)�ê.f(z) = αf(z) + (1− α)f(z), ÆÇ��y�J�

(1− α)(f(z)− f(x)) ≤ α(f(y)− f(z))�

ãíÂ�§, D3ξ ∈ (x, z), η ∈ (z, y), ¸ÿ

f(z)− f(x) = f ′(ξ)(z − x), v f(y)− f(z) = f ′(η)(y − z)�

4.2 OÁÂC0% 203

.f ′ �¦, vξ < η, Æf ′(ξ) ≤ f ′(η)�ê��J(1 − α)(z − x) =

α(y − z)�Æÿ

(1− α)(f(z)− f(x)) = (1− α)f ′(ξ)(z − x) ≤ αf ′(η)(y − z)

= α(f(y)− f(z))�ÿJf �Ðó��yuf ′′ ≥ 0, Jf ′ �¦, Æt¡×I5ôÿJ�

&Æ�b×(Þ�+Û, ÇDDD`FFF(point of inflection)TÌjjjFFF�

��LLL2.2.'b×Ðóf , uD3c�×Ï½(a, b)¸ÿf ′′(x) > 0, ∀x ∈(a, c), f ′′(x) < 0, ∀x ∈ (c, b) (TDÄ¼f ′′(x) < 0, ∀x ∈ (a, c),

f ′′(x) > 0, ∀x ∈ (c, b)), J(c, f(c))Ì f%��×D`F, TÌf3c

b×D`F�

uf3cb×D`F, vf ′′(c)D3, JÄbf ′′(c) = 0�æ.Aì�'(a, b) A�L2.2�, X�c�×Ï½, v�g = f ′�.f ′′(x)

D3, Æg′(x)D3, ∀x ∈ (a, b)�.hg′3∀x ∈ [ξ, η]=�, Í�∀a <

ξ < c < η < b, vg′(ξ)g′(η) < 0�Æã�§1.2, g3(ξ, η)�bÛ&F�ãyE∀x ∈ (a, c) ∪ (c, b), g′(x)D3v� 0, ÆhÛ&FÄ c�v.g′(c) = f ′′(c)D3, ÆhÛ&F� g′�D3ÝF,

Çg′(c) = f ′′(c) = 0�ùÇuf 3D`Fc�Þ$0óD3, Jhc f ′�Û&F, Çf ′′(c) = 0�¬¬&N×f ′�Û&F, /º¸fbD`F�»A, 'f(x) = x4,

Jf ′(x) = 4x3, f ′′(x) = 12x2�.f ′′(0) = 0, Æ0 f ′ �×Û&F�¬f ′(x) > 0, ∀x 6= 0, Æf30PD`F�

»»»2.6.'f(x) = x3 − 3x2, Of�8EÁÂCD`F, ¬0Í%��.´�f ′(x) = 3x2 − 6x = 3x(x − 2), f ′′(x) = 6x − 6 = 6(x −1)�Æ0C2 f�Û&F�ê.f ′′(0) = −6 < 0, f ′′(2) = 6 >

0, Æf30 b8EÁ�Âf(0) = 0, f32b8EÁ�Âf(2) =

204 Ï°a �5�Tà

−4�.f ′′(x) = 0�°×� x = 1, Æx = 1 °×b��¸fbD`F��.f ′′(x) > 0, ∀x > 1, f ′′(x) < 0, ∀x < 1, Æã�L2.2,

f 31bD`F�f�%�A%2.7�

-

6

−2 −1 1 2 3 4

−2

−4

2

x

y

O

(1,−2)

%2.7. f(x) = x3 − 3x2�%�

3��5�, x�Î�§Ðó, D¡ÐóÝ&ËP²�G«�èÄE×Ðó, t�Ý�¸Ý, �Äy0�Í%��X|9ì&Æà�0%Ý×°M»�

1. ´�X��aaa�kà��aÞ%�5âWóÍI5, �W&�}ñÝ¿Í ½; i¿��aJ�:�“tâ”(Çx →∞Tx →−∞), %�Ý�l{�; ubE��aù0�, �îtâxºì��àa�

2. X�ÁÂ�9ÎÐóÝIt{�Ct±��3ÁÂ!�,

%��l��%2.8�

x0 x0

(a) Á�Â (b) Á�Â%2.8. ÁÂ!�ÐóÝ%�

4.2 OÁÂC0% 205

tÝ\&F², 3ÁÂÝ0ó 0, T�D3(ÇÛ&F), hã%2.8 �:��×��, u3ØFÐó�0óD3, J3�F!�%�º´¿â; u0ó�D3, ��=�(Af(x) = [x], 3x = 1),

T�0óC¼0ó/D3, ¬�8�(h`3�F%�´JÞ,

Af(x) = |x|, 3x = 0), Tb×��0ó��∞T−∞ (h`3�Fkàx��àa, �Ú Í“6a”, Af(x) = x1/3, 3x = 0)�X|kX�ÁÂ, �0�Û&F�A�3Ø×Û&Fc, ÐófÞ

g��, vf ′′(c) 6= 0, Jµ�§2.5¾½Á�TÁ��uf ′′(c) = 0,

J�¿à�§2.3¼¾½�&Æ|¡ôº�D¡f ′′(c) = 0Ý�µ�3. X�D`F�ã%2.8�:�, 3Á�Â�¼��%�?

ì;¨, uP�;, ºP§;¨ì��D`FµÎ;�%�ÝlV�3D`F!�Ý%�, �lbì�°v�

( a )

( c )

( b )

( d )

f ′′ < 0

f ′′ > 0

f ′′ > 0

f ′′ < 0

f ′′ > 0

f ′′ < 0

f ′′ < 0

f ′′ > 0

%2.9. D`F!�ÐóÝ%�

Í�(a)C(b)v, f ′ > 0, ÇÐó �¦; (c)C(d)v, f ′ < 0, ÇÐó �3�f ′′ > 0, �f ′ �¦, Ç%��E£�¼��; f ′′ < 0

JD��f ′′ > 0, A�Î�¦Ðó, �%�º�¼�q(.E£��); A�Î�3Ðó, �%�º�¼�¿c(.E£ã�Â��¦

206 Ï°a �5�Tà

�, X|�EÂ��)�f ′′ < 0ô�bv«2D¡�u3Ø×FÝ¼��, f ′′Ðr8D, J�F D`F(�L2.2)��ã�L2.2�¡ÝD¡á, u3ØD`Fc, f ′′(c)D3, JÄbf ′′(c) = 0�9ôèº&Æ×Í0D`FÝ]P, �0�º¸Þ$0ó 0ÝF, �¾½3�FÎÍºbD`F�ã%2.9, ��:�D`FÝD3, Î��QÝ�A3%2.9 (a)�, u%��AF&D`F, J3AF�¼�, %�4�¼�¦{, ¬�T�¼�¿c, 3AF¾Õt{�, uP�;, ÄÝAF-�?ìÝ��AF-�A c%�?ì, %�-»�?ìÝ�T,¸Í)Qî>, vî>>�¦"(.f ′′ > 0)�!§��ÕÍ�ëË�µ�ã%2.9ô�:�, E×=�Ðó, 3Þ8ÏÝ&\&�ÁÂ

(hÞ8ÏÝÁÂÄ�× Á�, × Á�, %�?), Äªb×D`FD3�X�D`F, Þ�¸%�iÿ´Þ@�4. O�x�Cy��^û�Çuby = f(x), 0�º¸y = 0�x,

Ç %�;Äx��, CO�f(0), Ç%�;Äy ��ô�à0�3×°xÂ�y�Â, Ah%�º?Þ@�B�×ÍæJ, 3ØF��T¼�, uP�;(AÛ&F�D

`F, T��a), J%�µ×à5T�ì�(?îT?ì)�»A, 'y = x2, h ×Õ��ÝÐó�3x = 0b×Á�, X|3x = 0!�, %�A%2.3 (b)�ê3x > 0�, P�¢�;, X|%��\î>�∞; 3x < 0 �, ôP�¢�;, X|�½x��, %�ô×àî>�∞�9ì��¿Í»��XÛ0%, -�â0��a�Û&F

CD`F�

»»»2.7.�0f(x) = x4 − 4x3 + 10�%��.´�f ′(x) = 4x3−12x2 = 4x2(x−3),X|0, 3 Û&F�êf ′′(x)

= 12x2 − 24x = 12x(x− 2), X|3x = 0, 2,��bD`F�.f ′′(3) > 0, Æ3x = 3b8EÁ�Âf(3) = 81 − 108 +

10 = −17�¬f ′′(0) = 0, ÆÞ$0ó�¾½ÁÂ°´[�.

4.2 OÁÂC0% 207

3x = 0�¼��×Ï½, f ′/ �, Æ3x = 0PÁÂ�ê3x =

0�¼��×Ï½, f ′′Ðr8D�3x = 2�!�, f ′′�Ðrù8D�Æ(0, f(0))C(2, f(2))/ D`F, �f(0) = 10, f(2) = −6�%�¬P��a(h 94P), x → ∞`f(x) → ∞, x →

−∞`, f(x) → −∞�x = 0`, f(0) = 10, �x��^û ��x4 − 4x3 + 10 = 0�xÂ, ¬�|��, ¬�ã�q��(ÏÞa�§6.2) 0��«Â�f�%�A%2.10�

-

6

−1 O 1 2 3 4

5

15

10

−5

−10

−15

x

y

(2,−6)

(3,−17)

%2.10. f(x) = x4 − 4x3 + 10 �%�

»»»2.8.�0f(x) = x + 1/x�%��.f��L½ R\{0}�x → 0+`, f(x) →∞, x → 0−`, f(x) →−∞, Æx = 0 kà��a�êx → ∞ `, (f(x) − x) → 0, x →−∞`, (f(x)− x) → 0, Æy = x E��a�.f ′(x) = 1 − 1/x2, f ′′(x) = 2/x3�Æ3x = 1,−1bÛ&

F�.f ′′(1) = 2 > 0, Æ3x = 1bÁ�Âf(1) = 2; f ′′(−1) =

−2 < 0, Æ3x = −1 bÁ�Âf(−1) = 2�qA|îÝD¡, ñÇ�ÿÕ9ì%2.11�&��¥�Õ, &Æ¬Î�JÎÍbD`F�h.3x > 0�,

f/��, �©b×ÁÂ, vP��a, Æ�ºbD`F�3x < 0

��µùv«�

208 Ï°a �5�Tà

-

6

−4

−2

2

4

−4 −2 2 4x

y

O

y=x+1/x

y=x

y=1/x

%2.11. f(x) = x + 1/x�%�

»»»2.9.�0f(x) = 1/(1 + x2)�%��.´�x →∞`, f(x) → 0; x → −∞`, f(x) → 0�Æx � i¿��a�.f(x) > 0, ∀x ∈ R, Æ%�/3x�î]�êPkà��a�.f(x) = f(−x), Æ%�EÌyy��B�5�ÿ

f ′(x) =−2x

(1 + x2)2, f ′′(x) =

2(3x2 − 1)

(1 + x2)3 �

.h3x = 0b×Û&F�ê.f ′′(0) = −2 < 0, Æ3x = 0 b×8EÁ��f ′′(x) = 0, ÿx = 1/

√3, Tx = −1/

√3�f ′′�Ñ�rA

ì:

x x < −1/√

3 −1/√

3 < x < 1/√

3 x > 1/√

3

f ′′ + − +

Æ3x = 1/√

3C−1/√

3, /bD`F, vf(1/√

3) = f(−1/√

3) =

3/4�f�%�A%2.12�uPD`F, 3x > 0�, %�º×à?ì;¨�¬.3x =

1/√

3bD`F, X|%�-»Ä¼, �¼�¿c, 3x��, %�¿{Îi¿Ý(Ç|x� ��a)�D�, u�:�3x = 0 bÁ

4.2 OÁÂC0% 209

-

6

− 1√3

1√3

−1 1 2x

y

O

%2.12. f(x) = 1/(1 + x2) �%�

�, v|x� ��a, Já3x > 0�ÝØFÄbD`F�D`FÝD3�1Î&ð5{�QÝ�

»»»2.10.�0f(x) = x4 − 2x3�%��.´�h 94P, ÆP��a�êf ′(x) = 4x3 − 6x2 = 4x2(x −3/2), f ′′(x) = 12x(x− 1)��f ′(x) = 0ÿx = 0Tx = 3/2�.f ′′(0)

= 0, ÆãÞ$0óP°¾½3x = 0ÎÍ ÁÂ�ê3x = 0!�,

f ′(x)/ �, Æ3x = 0PÁÂ�êf ′′(3/2) > 0, Æ3x = 3/2bÁ�Âf(3/2) = −27/16�&Æ��f ′�Ñ�rAì�

x x < 0 0 < x < 3/2 x > 3/2

f ′ − − +

�f ′′(x) = 0ÿx = 0T1, hÞxÂ ��bD`F��f ′′ �Ñ�rAì�

x x < 0 0 < x < 1 x > 1

f ′′ + − +

Æ3x = 0Cx = 1/bD`F,vf(0) = 0, f(1) = −1�êf(2) = 0,

Æ%�;Äx = 0Cx = 2��f�%�A%2.13�

210 Ï°a �5�Tà

-

6

1 2x

y

O

%2.13. f(x) = x4 − 2x3�%�

Í;t¡&Æ¼:|Þ$0ó¾½ÁÂ`, uf ′(c) = 0, vf ′′(c)

= 0�A¢? &Æ��×�§2.5�.Â��'Ðóf , 3c�×Ï½(a, b) n + 1$��, vf (n+1)3c =

�, ê'f (k)(c) = 0, ∀1 ≤ k ≤ n�JD«¿àÞ�íÂ�§, ÿE∀x ∈ (a, b),

f(x)− f(c)

(x− c)n+1=

f ′(x1)

(n + 1)(x1 − c)n=

f ′(x1)− f ′(c)(n + 1)(x1 − c)n

=f ′′(x2)

(n + 1)n(x2 − c)n−1=

f ′′(x2)− f ′′(c)(n + 1)n(x2 − c)n−1

...

= sf (n)(xn)− f (n)(c)

(n + 1)!(xn − c)=

f (n+1)(xn+1)

(n + 1)!,

Í�x1+yx�c , x2+yx1�c , õv.�ãîPÇÿ

(2.1) f(x) = f(c) +f (n+1)(xn+1)

(n + 1)!(x− c)n+1,

Í�xn+1+yx�c �EyfbAîÝ�î°, &Æì×;º��áD¡��'f (n+1)(c) 6= 0�J©�x�cÈ#�, f (n+1)(xn+1)�f (n+1)(c)

!r�.h¿à(2.1)P, Çÿ×¾½f3cÎÍ Á�TÁ�ÂÝãJAì:

êÞ 211

(i) un �ó, vf (n+1)(c) > 0, Jf3x = cbÁ��(ii) un �ó, vf (n+1)(c) < 0, Jf3x = cbÁ��(iii) un �ó, Jf3x = cPÁÂ�

»»»2.11.(i) 'f(x) = x3�Jf ′(0) = f ′′(0) = 0, vf ′′′(0) = 6 >

0�ETyG�ãJ, n = 2 �ó, Æf3x = 0PÁÂ�(ii) f(x) = x4�Jf ′(0) = f ′′(0) = f ′′′(0) = 0, vf (4)(0) = 24 >

0�ETyG�ãJ, n = 3 �ó, vf (4)(0) > 0, Æf3x = 0bÁ��

êêê ÞÞÞ 4.2

1. �Eì�&Ðó, O�f �� , ¬0�y = f(x)�%��(i) f(x) = x2 − 2x + 8, (ii) (x− 2)3(x + 1)2,

(iii) f(x) =√

x +√

x + 1, (iv) f(x) = −x3 + 3x− 5,

(iv) f(x) = x + 1/x, (vi) f(x) = x + 5/(2x + 3),

(vii) f(x) = (x− 1)1/3 + 12(x + 1)2/3�

2. �Oì�&Ðó�ÁÂ�(i) f(x) = x2

4+ 4

x, (ii) f(x) = x3 + 3

x,

(iii) f(x) = 1√x

+√

x9

, (iv) f(x) = x3

x2+1,

(v) f(x) = x3 + 3x2 − 9x + 10,

(vi) f(x) = x3 + 4x2 − 3x− 9�

3. �0ì�&Ðó�%�, ¬¼��a�ÁÂCD`F(u0��Þ@Â, �O��óÏ×�)�(i) f(x) = x3 − x, (ii) f(x) = 2x3 − 3x2,

(iii) f(x) = (x− 1)2(x + 2), (iv) f(x) = x + 1/x2,

(v) f(x) = x/(1 + x2), (vi) f(x) = x− sin x,

(vii) f(x) = x+2x2+2x+4

, (viii) f(x) = x1/3(x− 4),

212 Ï°a �5�Tà

(ix) f(x) = 1(x−1)(x−3)

, (x) f(x) = x√

x + 3,

(xi) f(x) = (x2 − 4)/(x2 − 9), (xii) f(x) = x + 1/(x− 1),

(xiii) f(x) = (2x3 + x2 − 1)/(x2 − 1),

(xiv) f(x) = x3 − 6x2 + 9x + 5�

4. �®uÐóf3[a, b]C[b, c]�/ �¦, Í�a < b < c, JfÎÍÄ3[a, c]��¦�

5. 'ÞÐófCg,/3[a, b]��¦��®f+gCfgÎÍù3[a, b]

��¦�

6. �0f(x) = (x − a1)2 + (x − a2)

2 + · · · + (x − an)2�%�, Í�a1, · · · , an ðó�

7. �Oì�&Ðó, 3X� ��EÁÂ�(i) f(x) = 1− |1− x|, x ∈ [−1, 1],

(ii) f(x) = (x− 1)x1/3, x ∈ [1/2, 2],

(iii) f(x) = x2 + 4/x, x ∈ [1, 5],

(iv) f(x) = x3 − 12x + 3, x ∈ [−5, 3],

(v) f(x) = (x− 2)/(x + 2), x ∈ [0, 4],

(vi) f(x) = x−√2 sin x, x ∈ [0, π],

(vii) f(x) = 3x5 − 25x3 + 60x, x ∈ [−1, 3]�

8. �¾½f(x) = sin x− x + x3/6 3x = 0ÎÍbÁÂ, ÎÍbD`F�

9. �¾½f(x) = cos x + x2/2 3x = 0ÎÍbÁÂ, ÎÍbD`F�

10. �0f(x) = x + sin x�%�, ¬��a, ÁÂCD`F�

11. �O=�Ðóf Cðóc, ¸ÿ∫ x

c

f(t)dt = sin x− x cos x− 1

2x2, ∀x ∈ R�

4.3 ��"�P 213

4.3 ��"""��PPP

&Æ�+Û�0ÝÃF�33.7;, &Æ�|AìÝ]P�L0ó: 'b×Ðóy = f(x),

J

f ′(x) = limh→0

f(x + h)− f(x)

h= lim

∆x→0

∆y

∆x,

Í�∆y = f(x + h) − f(x), ∆x = h�E×ü�Ýx, &Æ�LÐóε

(3.1) ε = ε(h) =f(x + h)− f(x)

h− f ′(x) =

∆y

∆x− f ′(x)�

J�Qlimh→0

ε(h) = 0�¥�∆y��óãxÉ��x + h`, T�óyÝ;�(TÌ¦¦¦��(increment))��ã(3.1)P, �ÿ

(3.2) ∆y = f ′(x)∆x + ε∆x�

Ç∆y��î ËÍ��õ, Ï×Í �f ′(x)WÑf�f ′(x)∆x, ÏÞÍ ε∆x, h4©�∆xãÿÈ�, J�∆x�fÂ(Çε)�|��&Æ-Þf ′(x)∆x Ì y3x��0, ¬|dy��, Ç

(3.3) dy = df(x) = f ′(x)∆x�

E�×��Ðóf , C×ü�Ýx, Í�0 ×h = ∆x�aPÐó�»A, EÐóy = x2,

dy = d(x2) = 2x∆x = 2xh�

uÐó y = x, .Í0ó ðó1, Æÿ

(3.4) dx = ∆x = h�

.h(3.3)PW

(3.5) dy = df(x) = f ′(x)dx�

214 Ï°a �5�Tà

(3.2)Pô�;¶

(3.6) ∆y = f ′(x)dx + εdx = dy + εdx�

ÆT�óÝ¦�∆y, ��0dy�- εdx, h4;ð� 0�»A,

uy = x2, Jdy = 2xdx, v

∆y = (x + dx)2 − x2 = 2xdx + (dx)2 = dy + εdx,

Æh`ε = dx�A3.7;X�, 2à¾¾¹+Ý�5Ðrdy/dx, öÜ©Î.0

ó ∆y/∆x∆x → 0`�Á§�¼�dy/dx JÍÎ×ÍÐr,

�&dy t|dx�¬uµ&Æî�E�0Ý�L, �ÿ(3.5)P,

dy/dx @jî-��dyt|dx�h�dy�dx, �B^b“P§�Ý”��LÝ(�(3.4)C(3.5)ÞP), �ÎÞ5½ h = ∆x �aPÐó�u∆x��,Jdy�dxô��dy�dx�¤,º�yf�0óf ′(x), ô�� Ý, .©Î.(3.5)P;¶��!§, &Æô��L{$�0�Ç�

d2y = f ′′(x)(∆x)2 = f ′′(x)(dx)2,

d3y = f ′′′(x)(dx)3,

��Ah×¼�¾¾¹+E{$0óÝÐr, ôº×l(Þ(dx)2¶dx2, (dx)3¶Wdx3�)�

(3.2)Pô�;¶

(3.7) f(x + h) = f(x) + ∆y = f(x) + hf ′(x) + εh,

Çuü�x, f(x + h)uÚ ×h�Ðó, ��î ×hÝaPÐóf(x) + hf ′(x), �î×0-εh�h0-©�hÈ�, �h8f-��|aPÐóf(x)+hf ′(x) = f(x)+dy¼¿�f(x+h),ôµÎ6¯εh, Ç|dyã�∆y�Í¿¢�L , |3x�6af(x + h) =

f(x) + hf ′(x), ¼ã�`ay = f(x)�©�hÈ�, Í0--�ºH�, �%3.1�

4.3 ��"�P 215

-

6

y

x x + dx

¾ -dx = h

?

6

6dy?

∆y

O

y

%3.1. �0dy�¦�∆y

ã%3.1�:�, 3x!�Ý×ÍFx + dx, ÍÐóÂÍ¼T f(x + h) = f(x) + ∆y, ¬f�� ×�ÓÝÐó, ∆y�×�?O�aPÐóÎ×&ð��ÝÐó, ÐóÂ��|O��©�h��, y = f(x)�%��Äx�6a-²��, X|3x!�ØF�ÐóÂ, �|36aîETÝyã�, Çf(x) + dy�9øÝ£�4CW0-∆y − dy, ¬Q�¸�Õ�|&9��y0-9�, Î&Æ.&ÝP�? Ú�!�µ��×��, u�0-��, h-�ãÿ��9ì&Æ¿àíÂ�§, ¼£�0-εhÝ��ãíÂ�§á, E∀h > 0, D3×ξ+yx�x + h , ¸ÿ

f(x + h)− f(x) = hf ′(ξ),

Æ

ε =f(x + h)− f(x)

h− f ′(x) = f ′(ξ)− f ′(x)�

uf3xÝ!� Þg��, J�×g¿àíÂ�§�ÿ

ε = f ′(ξ)− f ′(x) = (ξ − x)f ′′(η),

Í�η +yξ�x ÝØó, Æηù+yx�x + h �uD3×ðóM > 0, ¸ÿf ′′3T [x, x + h](h h > 0Ý�µ, uh < 0J ; [x + h, x])��EÂ|M Í×î&, J

|ε| = |(ξ − x)f ′′(η)| ≤ hM�

216 Ï°a �5�Tà

Æf(x + h)�f(x) + hf ′(x)�-εh, Í�EÂ�øÄMh2�hÈ�`, Mh2�hf ′(x)8fº��, t&f ′(x) 0(�(3.7)P)�AG«�1�ÄÝ, 3×� , |×aPÐó¼¿�×Ðó, 3@jTà`�¥��¬Ah, Ç¸3´{�Ýó.5��, ô�¥��&Æºµ�D¡9Ë¿�, }¡ôºJ�, ¯@îG�0-, �£�ÿ?Þ@, Ç|εh| ≤ Mh2/2�&Æ�Þ|aPÐó, ¼¿�ÐóÝ�°�|.Â�94P�

1Î5��, XÂÕÝÐó�t��Ý�E×94PÐóy = f(x),

��×x, �¬��|�Õ�ÐóÂf(x), bn�5C�5ÝºÕ,

�1ôÎt�|Ý�9ì&ÆÞJ�, &9Ðó/�|×ÊÝ94P¼¿��'f ×3x = 0, ng��ÝÐó, n ≥ 1�&Æ�0×9

4PP , h94P3x = 0�f�n$0ó/8!(3ØË�Lì,

�P�f3x = 0!�È#�)�ÇP��ì�n + 1Íf�

(3.8) P (0) = f(0), P ′(0) = f ′(0), · · · , P (n)(0) = f (n)(0)�

Æ&Æ�×ng94P, Ç�

(3.9) P (x) = c0 + c1x + c2x2 + · · ·+ cnx

n�

3(3.9)P�, �x = 0, ÇÿP (0) = c0, Æc0 = f(0)�ÍgÞ(3.9)P¼��5½Ex�5, ��x = 0, ÿP ′(0) = c1, Æc1 =

f ′(0)�õhv., �ÿ

(3.10) ck =f (k)(0)

k!, k = 0, 1, · · · , n,

Í�f (0) = f�Æu×gó�øÄn�94P��(3.8)P, JÍ;óÄ6��(3.10)P(uf (n)(0) 6= 0, JP�gó n)�D�, u×94P, Í;ó��(3.10)P, ôÄ��(3.8)P(h94P�;ób��yn)�.h&ÆÇJ�Ýì��

4.3 ��"�P 217

��§§§3.1.'f ×3x = 0, ng��Ðó�JªD3×gó�øÄn�94PP , ��(3.8)P�f�, vP

(3.11) P (x) =n∑

k=0

f (k)(0)

k!xk�

!§�J, ªD3×gó�øÄn�94PP , �f3x = a�n$0ó/8!�¯@î, ©�ÞP¶W×x− a�¶�, ¬AG�.0,

J�ÿ

(3.12) P (x) =n∑

k=0

f (k)(a)

k!(x− a)k�

h °×Ýgó�øÄn�94P, ��

(3.13) P (a) = f(a), P ′(a) = f ′(a), · · · , P (n)(a) = f (n)(a)�

&Æ-Þ(3.12)P��94PÌ f3a�ng��999444PPP(Taylor

polynomial, Brook Taylor (1685-1731) z»ó.�, ÎpñÝ.ß, 3�-1715O�ÌMethodus Incrementorum Directa et Inversa,

èºÞ×Ðó"�WùóÝ]°, ×tS¡, {úCÞ�ÞÍ]°�Û;), ¬|Pn�h94P�

»»»3.1.�OsineÐó3π/2�4g��94P��.�f(x) = sin x�J

f(x) = sin x, f(π/2) = 1,

f ′(x) = cos x, f ′(π/2) = 0,

f ′′(x) = − sin x, f ′′(π/2) = −1,

f ′′′(x) = − cos x, f ′′′(π/2) = 0,

f (4)(x) = sin x, f (4)(π/2) = 1�

ÆP4(x) = 1− 1

2(x− π

2)2 +

1

24(x− π

2)4�

218 Ï°a �5�Tà

�uãa = 0,.×��,bf (2k+1)(0) = (−1)k, f (2k)(0) = 0,Æ

P2n−1(x) = P2n(x) = x− x3

3!+

x5

5!− x7

7!+ · · ·+ (−1)n−1 x2n−1

(2n− 1)!�

»»»3.2.�OcosineÐó30�2ng��94P��.�f(x) = cos x, .f (2k)(0) = (−1)kvf (2k+1)(0) = 0, ∀k ≥ 0, Æ

P2n(x) = 1− x2

2!+

x4

4!− x6

6!+ · · ·+ (−1)n x2n

(2n)!�

ì��§, �à¼�;O��94PÝ�Õ�

��§§§3.2.'Qn ×ng94P, n ≥ 1�ê'f�g Þ3x = 0, ng��Ðó, v

(3.14) f(x) = Qn(x) + xng(x),

Í�g(0) = 0�JQn f30�ng��94P�JJJ��.�h(x) = f(x) − Qn(x) = xng(x)�J|�h(0) = 0vh30�´n$0ó/ 0, .hQn��(3.8)P�f��Æã�§3.1áQn =

Pn�

»»»3.3.Bãt°�ÿì��P

1

1− x= 1 + x + x2 + · · ·+ xn +

xn+1

1− x, x 6= 1�

Æ(3.14)PWñ, Í�

f(x) =1

1− x, Qn(x) = 1 + x + · · ·+ xn, g(x) =

x

1− x�

êg(0) = 0�Æ�§3.2¼�f30�ng��94P 1 + x + · · · +xn�

E×ng��ÝÐóf , &Æ+ÛÝf3a�ng��94PPn�Pn�f3aF�Â8!,v´n$0óù8!(�(3.13)P)�¬Pn~b

4.3 ��"�P 219

�f�×�8!�u�Rn(x) = f(x)− Pn(x), J

f(x) = Pn(x) + Rn(x),

T

(3.15) f(x) =n∑

k=0

f (k)(a)

k!(x− a)k + Rn(x)�

Rn(x)Ì f3a�Ïngõõõ444(the nth remainder term of f at a)�(3.15)PÌ |Rn(x) õ4�fÝ��222PPP(Taylor’s formula), ôÌ f�×��"�2P, T�Ì��"P(Taylor’s expansion)�A�&Æ�£�õ4Rn(x)��, J(3.15)P�´bà�&Æ�ÞRn(x)|×�5¼�î, Q¡�£�h�5��&Æ��×n = 1Ý��

��§§§3.3.'Ðóf3a�Ø×Ï½B, b×=�ÝÞ$0ó�JE∀x ∈ B,

(3.16) f(x) = f(a) + f ′(a)(x− a) + R1(x),

Í�

(3.17) R1(x) =

∫ x

a

(x− t)f ′′(t)dt�

JJJ��.ã(3.15)Pÿ

R1(x) = f(x)− f(a)− f ′(a)(x− a) =

∫ x

a

(f ′(t)− f ′(a))dt

=

∫ x

a

(f ′(t)− f ′(a))d(t− x)

= (f ′(t)− f ′(a))(t− x)|xa −∫ x

a

(t− x)f ′′(t)dt

=

∫ x

a

(x− t)f ′′(t)dt,

Í�Ïë�Ï°Í�P, ÛàÕ5I�5�ÿJ�

220 Ï°a �5�Tà

9ì E×�Ýn��

��§§§3.4.'Ðóf3a�Ø×Ï½B, b×=�ÝÏn + 1$0ó�JE∀x ∈ B, &Æbì��2P

(3.18) f(x) =n∑

k=0

f (k)(a)

k!(x− a)k + Rn(x),

Í�

(3.19) Rn(x) =1

n!

∫ x

a

(x− t)nf (n+1)(t)dt�

JJJ��.&ÆÞEnàó.hû°¼J, �n = 1Çî×�§�¨'Í�§En�yØJómWñ, &Æ�J�En = m + 1ùWñ�5½En = m, Cn = m + 1¶�(3.15)P¬83, ÿ

Rm+1(x) = Rm(x)− f (m+1)(a)

(m + 1)!(x− a)m+1�

ÞRm|(3.19)P��5ã�(.�'(3.19)PEn = mWñ), v¿à

(x− a)m+1

m + 1=

∫ x

a

(x− t)mdt,

ÿ

Rm+1(x) =1

m!

∫ x

a

(x− t)mf (m+1)(t)dt− f (m+1)(a)

m!

∫ x

a

(x− t)mdt

=1

m!

∫ x

a

(x− t)m(f (m+1)(t)− f (m+1)(a))dt�

�ÞîPt¡×�5¶W∫ x

audv, Í�

u = f (m+1)(t)− f (m+1)(a), v = −(x− t)m+1

m + 1 �

Q¡¿à5I�5, �ÿ(.t = a`u = 0, t = x`v = 0)

Rm+1(x) =1

m!

∫ x

a

udv = − 1

m!

∫ x

a

vdu

=1

(m + 1)!

∫ x

a

(x− t)m+1f (m+2)(t)dt�

4.3 ��"�P 221

Æn = m + 1`, Í�§Wñ�ãó.hû°á, Í�§E∀n ≥ 1Wñ�J±�

ãy��2P�Ý0-Rn(x), ��îW×nyf�Ïn + 1$0óÝ�5, Æuáf (n+1)�×î�ì&, J�ÿ×Rn(x)�×î�ì&��ì�§�

��§§§3.5.'f�Ïn + 1$0ó, 3a�Ø×Ï½B� =�, v��E∀t ∈ B,

(3.20) m ≤ f (n+1)(t) ≤ M,

Í�m, M Þðó�JE∀x ∈ B,

(3.21) m(x− a)n+1

(n + 1)!≤ Rn(x) ≤ M

(x− a)n+1

(n + 1)!, u x > a,

v

(3.22) m(a− x)n+1

(n + 1)!≤ (−1)n+1Rn(x) ≤ M

(a− x)n+1

(n + 1)!, u x < a�

JJJ��.´�'x > a, JRn(x)Î×3[a, x]��5(�(3.19)P)�E∀t ∈ [a, x], .(x− t)n ≥ 0, Æã(3.20)Pÿ

m(x− t)n

n!≤ (x− t)n

n!f (n+1)(t) ≤ M

(x− t)n

n! �

ÞîPN×4Etãa�x�5, ÿ

m

n!

∫ x

a

(x− t)ndt ≤ Rn(x) ≤ M

n!

∫ x

a

(x− t)ndt�

�Þ ∫ x

a

(x− t)ndt =

∫ x−a

0

undu =(x− a)n+1

n + 1

�áîP, -ÿJ(3.21)P�

222 Ï°a �5�Tà

ux < a, JRn(x)Î3[x, a]��5�E∀t ∈ [x, a], (−1)n(x −t)n = (t − x)n ≥ 0�ÆuÞ(3.20)P�N×4, &¶|(−1)n(x −t)n/n!, )î¹��Pn;��ãx�a�5Çÿ(3.22)P�

»»»3.4.uf(x) = sin xva = 0, Jbf(x) = P2n(x) + R2n(x), v

sin x = x− x3

3!+

x5

5!− x7

7!+ · · ·+ (−1)n−1 x2n−1

(2n− 1)!+ R2n(x)�

.f (2n+1)(t)��ycos tT− cos t, Æ|f (2n+1)(t)| ≤ 1�ÇM�ã 1, m�ã −1�ÆE∀x > 0, ã(3.21)Pÿ

|R2n(x)| ≤ x2n+1

(2n + 1)!�

b&9:«��ÝÐó, ¬P°��¼, ôµÎP°|×��Ðó(elementary function, �A94P�b§P�ë�Ðó�Dë�Ðó�¼ó�EóC9°ÐóÝ°JºÕT)W, -Ì ��ÐÐÐóóó)¼�î�»A,

∫ 1

0

sin(x2)dx,

∫ 1

0

sin x

xdx,

ãysin x/x3x = 0 P�L, u3x = 0&Æ|1ã�sin 0/0, JhÐó3x = 0ù=�, .h3[0, 1] Ý�5-b�LÝ(|¡uÂÕv«�µ�Ðó, &ÆôºAh�§)�¢ã94P¼¿�Ðó, -�à¼O9°P°|��Ðó¼�îÝ�5��«Â�

»»»3.5.�O∫ 1

0sin x2dx��«Â�ãî»á(ãn = 4)

sin x = x− x3

3!+

x5

5!− x7

7!+ R8(x)�

.f (9)(x) = sin x, Æ3x ∈ [0, 1], 0 ≤ f (9)(x) ≤ sin 1 ≤ 1�.h

0 ≤ R8(x) ≤ x9

9!�

4.3 ��"�P 223

Æ

sin x2 = x2 − x6

3!+

x10

5!− x14

7!+ R8(x

2),

v

0 ≤ R8(x2) ≤ x18

9! �.

0 ≤∫ 1

0

R8(x2)dx ≤

∫ 1

0

1

9!x18dx =

1

19 · 9!,

Æ ∫ 1

0

sin x2dx =1

3− 1

7 · 3!+

1

11 · 5!− 1

15 · 7!+ θ,

Í�

0 ≤ θ ≤ 1

19 · 9!,

0-�1�&ð��

��"P�Ýõ4, 4��3(3.19)P, ¬$b×°Í�Ý�î°�´�.(3.19)P��5Õ��Ý(x − t)n, 3�5 �, /Î�r, vê�'f (n+1)3h � =�, ÆãÏÞa�§4.11�5��JíÂ�§, ÿ

∫ x

a

(x− t)nf (n+1)(t)dt = f (n+1)(ξ)

∫ x

a

(x− t)ndt

= f (n+1)(ξ)(x− a)n+1

n + 1,

Í�ξ Ø+y|a, x ÐF�T �ÝØF(¥�, a�x�×�£×Í�)�.hõ4�¶W

(3.23) Rn(x) =f (n+1)(ξ)

(n + 1)!(x− a)n+1,

�

(3.24) f(x) =n∑

k=0

f (k)(a)

k!(x− a)k +

f (n+1)(ξ)

(n + 1)!(x− a)n+1�

224 Ï°a �5�Tà

(3.23)PÌ Lagrange�P�õ4(Lagrange’s form of the remain-

der)�Rn¶W9Ë�P, :R¼��2P�ÝG«Ý4�8«(�(3.18)P), ©�Äf (n+1)��á3ØFξ�&a�êξQ�a,

xCfbn��Ä, Ê2¿àÞ�íÂ�§(�Ía�§1.5), �|�m�

'f (n+1) =�,-�ÞRn¶W(3.23)P,J�Í@�H�Ó,b·¶Ý\ï�¢�Apostol (1967) pp. 283-284, ��¬�¨×õ4Ý�î°(Ì ÞÞÞ��PPP)�¨², ua = 0, Jÿ

(3.25) f(x) =n∑

k=0

f (k)(0)

k!xk +

f (n+1)(ξ)

(n + 1)!xn+1,

Í�ξ+y0�x �h©½�PÝ��"PÌ� Maclaurin222PPP(Maclaurin’s formula, Maclaurin (1698-1746) Á}(Scotch)

Ýó.�, ��!×`��3�-1742O, ��¡¼|�(Cú(Ý2P��Ä\y�25O, h2P-�3StirlingÝ×S½®��¨)�Í;t¡&Æ+ÛXÛo-Br(o-notation, \�the little-oh

notation)�'Ðóf3a�Ø×Ï½Bb×=�ÝÏn + 1$0ó, h`

f(x) =n∑

k=0

f (k)(a)

k!(x− a)k + Rn(x) ,∀x ∈ B�

¨'x ∈ [a− c, a + c] ⊂ B, Í�c > 0, Jãf (n+1)3hT �) =�, ÿf (n+1)3hT � b&(Ï×a�§6.4)�ÇD3×ðóM > 0, ¸ÿ

|f (n+1)(t)| ≤ M, ∀t ∈ [a− c, a + c]�

Æ(3.19)�õ4��

|Rn(x)| ≤ M|x− a|n+1

(n + 1)!, ∀x ∈ [a− c, a + c]�

4.3 ��"�P 225

.h

0 ≤∣∣∣∣

Rn(x)

(x− a)n

∣∣∣∣ ≤M

(n + 1)!|x− a|,(3.26)

∀x ∈ [a− c, a) ∪ (a, a + c]�

Æu�x → a, JRn(x)/(x − a)n → 0�&ÆÌh x → a`,

Rn(x)Ý��$$$(order)±y(x−a)n(\�Rn(x) is of smaller order than

(x− a)n as x → a)�ôµÎ3G�f�ì, x�#�a`, f(x)�|×x − aÝng

94P¼¿�, v0-Ý�$±y(x − a)n�&Æ�|9ø�, »A, x → 0`, x2 → 0vx3 → 0�¬x3#�0Ý>�"Äx2(x =

0.1`, x2 = 0.01�x3 = 0.001G�y0.01)�4!ø��0, &Æ)�|f´Í>��x2�2x2�Q>�Î“!×�ù”Ý", x2êfx"�X|Rn(x)Ý�$�y(x−a)n,x → a�ÇÎ1x → a`,

Rn(x)?0þÝ>�"Ä(x − a)n�uRn(x)�¶W×x − aÝ¶�,

JRn(x)Ýg]µÄ6�yn�QRn�×�Î×x− aÝ¶��ny�$, 35.8;&Æº��×°�� Ý]-, &ÆS

Landau (1877-1938)3�-1909O+ÛÝo-Br�

��LLL3.1.'xòyaÝØÏ½, g(x) 6= 0, ©�x 6= a�J

(3.27) f(x) = o(g(x)), x → a,

�

(3.28) limx→a

f(x)

g(x)= 0�

3î��L�, aô�|Î∞T−∞�©�xÈ�`, g(x) 6= 0, v

limx→∞

f(x)

g(x)= 0,

-�¶Wf(x) = o(g(x)), x →∞�

226 Ï°a �5�Tà

Ðrf(x) = o(g(x))\�f(x) is little-oh of g(x), Tf(x) is of

smaller order than g(x), Í�� x�#�a`, �g(x)8ff(x)��

»»»3.6.(i) x2 = o(x), x → 0;

(ii) sin2 x = o(x), x → 0;

(iii) f(x) = o(1), x → a, uv°uf(x) → 0, x → a;

(iv) f(x) = o(xn), x → a, uv°uf(x)/xn → 0, x → a�

&Æôðb�AìPÝ¶°:

f(x) = h(x) + o(g(x)), x → a,

hÇf(x)− h(x)

g(x)→ 0, x → a�

ôµÎf(x)− h(x) = o(g(x)), x → a�»A, .

sin x− x

x=

sin x

x− 1 → 0, x → 0,

Æsin x = x + o(x)�

»»»3.7..x →∞ `, x2/x3 → 0, Æx2 = o(x3), x →∞�

u.o-BrSá��"P�, v¿àG«�¼�ÝRn(x)/(x −a)n → 0, Jf(x)�¶W

(3.29) f(x) =n∑

k=0

f (k)(a)

k!(x− a)k + o((x− a)n), x → a,

©�f (n+1)3�âaÝØT =��ã(3.29)P�:�, x�#�a`, f(x)«y×x− aÝng94P,v0-�(x− a)n8f��

4.3 ��"�P 227

9ì µA&Æ�G�§ÄÝ��"P;

1

1− x= 1 + x + x2 + · · ·+ xn + o(xn), x → 0�

sin x = x− x3

3!+

x5

5!− x7

7!+ · · ·+ (−1)n−1 x2n−1

(2n− 1)!+ o(x2n),

x → 0�cos x = 1− x2

2!+

x4

4!− x6

6!+ · · ·+ (−1)n x2n

(2n)!+ o(x2n+1),

x → 0��º�ÝÎ, 'b×Ðóg, Jo(g(x))¬&×©�ÝÐó, �ÎØ

×�g(x)8f, ��ÝÐó�X|x2 = o(x), vx3 = o(x), ¬��.hÿÕx2 = x3�b`&Æº´¯�2 5, �¶Wx2 = o1(x),

x3 = o2(x)�¬;ð©��z½o©Î×Br, o(x)bÍ©�Ý�¤,

J-�6�o1, o2�Ý��ýýý (subscript) 1C2�Eyo-Br, &Æ��×°ð�ÝbnÍºÕÝ��

��§§§3.6.x → a`,

(i) o(g(x)) + o(g(x)) = o(g(x));

(ii) o(g(x))− o(g(x)) = o(g(x));

(iii) o(cg(x)) = o(g(x)), uc 6= 0;

(iv) f(x) · o(g(x)) = o(f(x)g(x));

(v) o(o(g(x))) = o(g(x));

(vi) 11+g(x)

= 1− g(x) + o(g(x)), ux → a`, g(x) → 0�

JJJ��.�J(i)�Ç'f1(x) = o(g(x)), f2(x) = o(g(x))�Jx →a`,

f1(x) + f2(x)

g(x)=

f1(x)

g(x)+

f2(x)

g(x)→ 0 + 0 = 0,

ÆÿJf1(x) + f2(x) = o(g(x))�(ii)−(v)�J�v«(i), º�\ï��W�gJ(vi)��¶�

ì��P:1

1 + u= 1− u + u · u

1 + u�

228 Ï°a �5�Tà

�|u = g(x)�áîP�.x → a`, g(x) → 0, Æ

g(x)

1 + g(x)→ 0, x → a�

.h(vi)ÿJ�

»»»3.8.�Jtan x = x + 13x3 + o(x3), x → 0�

JJJ��..

cos x = 1− 1

2x2 + o(x3), x → 0,

Æ¿à�§3.6�(vi), ãg(x) = −12x2 + o(x3), ÿ

1

cos x=

1

1− 12x2 + o(x3)

= 1 +1

2x2 − o(x3) + o(−1

2x2 + o(x3))

= 1 +1

2x2 + o(x2), x → 0�

h�àÕx → 0`,

−o(x3)

x2= −o(x3)

x3· x → 0,

Æ−o(x3) = o(x2), v

o(−12x2 + o(x3))

x2=

o(−12x2 + o(x3))

−12x2 + o(x3)

−12x2 + o(x3)

x2

→ 0 · (−1

2+ 0) = 0,

ãhÿ o(−12x2 +o(x3)) = o(x2),êo(x2)+o(x2) = o(x2)�.hx →

0`,

tan x =sin x

cos x= (x− 1

6x3 + o(x4))(1 +

1

2x2 + o(x2))

= x− 1

3x3 + o(x3),

Í�t¡×�rWñ, º�\ï��J�

êÞ 229

êêê ÞÞÞ 4.3

31 − 5Þ, �EX�ÝÐóf�nCa, ¶�f3a�ng��94PPn(x)�

1. f(x) = cos x, n = 5, a = π/2�

2. f(x) = sec x, n = 3, a = π/5�

3. f(x) =√

1 + x, n = 5, a = 0�

4. f(x) = 1√1−x

, n = 4, a = 0�

5. f(x) = 1(1−x)2

, n = 4, a = 0�

36 − 10Þ, �Tn(f(x))�Ðóf(x)30�ng��94P��J&Þ��P�

6. Tn( 11+x

) =∑n

k=0(−1)kxk�

7. T2n+1(x

1−x2 ) =∑n

k=0 x2k+1�

8. Tn( 12−x

) =∑n

k=0xk

2k+1�

9. Tn((1 + x)α) =∑n

k=0

(αk

)xk, Í�α ×b§ó, v

(α

k

)=

α(α− 1) · · · (α− k + 1)

k! �

10. T2n(sin2 x) =∑n

k=1(−1)k−1 22k−1

(2k)!x2k�(èî: cos 2x = 1 −

2 sin2 x)

311− 13Þ, �EX�Ýn¶�&Ðó�Maclaurin2P�

11. f(x) = sin x, n = 6�

12. f(x) = tan x, n = 4�

230 Ï°a �5�Tà

13. f(x) = 1√1−x

, n = 4�

14. �J

sin x =n∑

k=1

(−1)k−1x2k−1

(2k − 1)!+R2n(x), Í�|R2n(x)| ≤ |x|2n+1

(2n + 1)!�

15. �J

cos x =n∑

k=0

(−1)kx2k

(2k)!+R2n+1(x), Í�|R2n+1(x)| ≤ |x|2n+2

(2n + 2)!�

16. �JMaclaurin2P��"P��, ÇÍ��×í�0�¨×�

17. �JD3×c ∈ (0, 1), ¸ÿ∫ 1

0

1 + x30

1 + x60dx = 1 +

c

31�

18. �J

0.493948 <

∫ 1/2

0

1

1 + x4dx < 0.493958�

19. �|sin x = x − x3/3! + R4(x), O∫ √2/2

0sin x2dx�×�«Â,

¬��0-�P��

20. �|sin x = x − x3/3! + x5/5! + R6(x), O∫ 1

0sin x/xdx�×�

«Â, ¬��0-�P�(AG3x = 0)|1ã�sin x/x)�

21. �JD3×ëg94PP (x), ¸ÿ

x cos x = P (x) + o((x− 1)3), x → 1�

22. �0�×t±gÝ94PP (x), ¸ÿ

sin(x− x2) = P (x) + o(x6), x → 0�

4.4 Á§�� 231

4.4 ÁÁÁ§§§��

3�§Á§®Þ`, &ÆðºÂÕXÛ��(�Ï×aR-bÝ)�»A,O

limx→0

f(x)

g(x),

�limx→0 f(x) = limx→0 g(x) = 0�b`9Ë�µ, )�D|�X�»A, Olimx→0 x3/x2, �p:�Á§Â 0�¬b`µ�Î£��|Ý�»A, Olimx→0(x − tan x)/(x − sin x)�Q�b×°Í��PÝ��Alimx→af(x) = ∞, vlimx→ag(x) = ∞,

Olimx→a f(x)/g(x); limx→∞ f(x) = limx→∞ g(x) = 0, Olimx→∞f(x)/g(x); Tlimx→∞ f(x) = ∞, vlimx→∞ g(x) = ∞, Olimx→∞f(x)/g(x)��b×°��, Alimx→0 f(x) = 0, vlimx→0 g(x) =

∞, Olimx→0 f(x)g(x); Tlimx→0 f(x) = ∞, vlimx→0 g(x) = ∞,

Olimx→0(f(x) − g(x))�×��, Á§��, bì�¿Ë�P:

∞−∞, 0 · ∞,0

0,∞∞ , ∞0, 1∞, 00�

&Æ1�×ì9°ÐrÝ�¤�'bÞÐófCg, C×ðóa,

a ∈ R = R ∪ {∞,−∞}�î�&Ë�P5½�

x → a` �O(i) f(x) →∞, g(x) →∞, limx→a(f(x)− g(x)),

(ii) f(x) → 0, g(x) →∞, limx→a f(x) · g(x),

(iii) f(x) → 0, g(x) → 0, limx→a f(x)/g(x),

(iv) f(x) →∞, g(x) →∞, limx→a f(x)/g(x),

(v) f(x) →∞, g(x) → 0, limx→a(f(x))g(x),

(vi) f(x) → 1, g(x) →∞, limx→a(f(x))g(x),

(vii) f(x) → 0, g(x) → 0, limx→a(f(x))g(x)�

Í�b°�PÎ��Ý, ÃÍî©b×Ë0/0Ý�P, Í�¿Ë/�; 0/0Ý�P��Ä.∞/∞Ý�Pôð�¨, &Æ;ðµ�©½2.∞/∞Ý�P, ; 0/0Ý�P�ð�1, &Æx

232 Ï°a �5�Tà

��Ê0/0, C∞/∞ÞË�P�¨², a�|Î×@ó, ô�|Î∞T−∞�a @ó`, ô��Ê��Á§(Çx → a+Tx →a−) ��êtÝG«ÚË��, Q�b×°Á§Îny−∞Ý��A3(iv)�, f(x) → ∞, vg(x) → −∞,

Olimx→a f(x)/g(x)�9ì&ÆS×ËBã�5¼O��Á§Ýbà]°, Ç111ÄÄÄ¾¾¾!!!JJJ(L’Hospital’s rule)�L’Hosptial (1661-

1704)b`¶WL’Hoptial, �Î°»ó.��3�-1696O, ¶�aªîÏ×Í��5>Ih�hhb�9�!ÝÌÍ, ��5Ý�Â øF, �hQ¤��¬�h�I5Ý/�, �Àh½(Ý1Ä¾!J, C]Í@KÙ�y1Ä¾�/�×Johann Bernoulli

(1667-1748, êÌJohn Bernoulli)�1Ä¾!JÝÃÍ�°, ÎãÞÐó�0óÝ¤f ′(x)/g′(x)�Á

§, ¼Of(x)/g(x)�Á§� %�hÞÁ§ºbn;÷? 3�×�JÝJ��G, &Æ¯�1�Aì�'fCg��f(a) = g(a) =

0�JE∀x 6= a, ©�g(x) 6= 0, ìPWñ:

f(x)

g(x)=

f(x)− f(a)

g(x)− g(a)=

(f(x)− f(a))/(x− a)

(g(x)− g(a))/(x− a)�

uf ′(a)Cg′(a)D3, vg′(a) 6= 0, Jx → a, îP�t��, ��f ′(a)/g′(a)�.hx → a`, f(x)/g(x) → f ′(a)/g′(a)��Ä31Ä¾!J�, ¯@î¬�m�Ef�g, Cf ′�g′3a F

��¢�'��©m�'x → a`, f(x)Cg(x)/��0 (3hÎ¼E0/0Ý��), vf ′(x)/g′(x)��×b§ÝÁ§Â�1Ä¾!J-Î1, h`f(x)/g(x)ù��!×Á§Â�&ÆB�¬J�h��Aì�

��§§§4.1.'ÐófCg, 3� (a, b)��, a < b, v'

(4.1) limx→a+

f(x) = limx→a+

g(x) = 0�

ê'g′(x) 6= 0, ∀x ∈ (a, b), v

(4.2) limx→a+

f ′(x)

g′(x)= L

4.4 Á§�� 233

D3�J

(4.3) limx→a+

f(x)

g(x)= L�

JJJ��.&ÆÞ¿àÞ�íÂ�§(�§1.5), 3×|a ÐF�T �¬.fCgb��3aP�L, .h&Æ�LÞ±ÐóAì��

F (x) =

{f(x), x 6= a,

0, x = a;G(x) =

{g(x), x 6= a,

0, x = a�JFCG/3a=��êã�'CF�G��L, E∀x ∈ (a, b), ÐóFCG, /3T [a, x]=�, v3� (a, x) ��X|EÐóF�G, C [a, x], ¿àÞ�íÂ�§, ÿD3×c ∈ (a, x), ¸ÿ

(F (x)− F (a))G′(c) = (G(x)−G(a))F ′(c)�

.F (a) = G(a) = 0, ÆîPW

(4.4) f(x)g′(c) = g(x)f ′(c)�

êg′(c) 6= 0(��'g′3(a, b)�/� 0),vg(x)�� 0 (ÍJ.G(x)

= G(a) = 0,ãRolle�§á,D3×x1 ∈ (a, x)¸ÿG′(x1) = g′(x1) =

0, h�g′3(a, b)�í� 0�))�.hBË�!t|g(x)g′(c),

(4.4)PW f(x)

g(x)=

f ′(c)g′(c)�

x → a+, c → a+ (h.c ∈ (a, x)),�êã�'limx→a+ f ′(x)/g′(x)

= LD3, Æx → a+`, îP��L, Ç(4.3)PWñ�

î��§ÎjE�Á§��p�|ÑÑf�, �ÿÕ×ny¼Á§(Çx → a−)TË�Á§(Çx → a)Ý��9ì�¿Í»��

»»»4.1.�O

limx→π

sin x

x− π�

234 Ï°a �5�Tà

��..limx→π sin x = limx→π(x− π) = 0, ¿à1Ä¾!Jÿ

limx→π

sin x

x− π= lim

x→π

cos x

1= −1�

»»»4.2.�Olimx→0

x− tan x

x− sin x��.h ×0/0��, f(x) = x− tan x, g(x) = x− sin x��

f ′(x)

g′(x)=

1− sec2 x

1− cos x

) ×0/0��îP��y

1− 1/ cos2 x

1− cos x= − 1− cos2 x

cos2 x(1− cos x)= −1 + cos x

cos2 x→ −2,

x → 0�ÆXkO�Á§ −2�

3î»�, u�B�;, J(1 − sec2 x)/(1 − cos x) = −(1 +

cos x)/ cos2 x�&×��, Æ��¿à1Ä¾!J, ÍJ5�5Ò&��5, ÿx → 0`,

− sin x

2 cos x sin x= − 1

2 cos x→ −1

2

�ý0��©�)Î��, 1Ä¾!J-��¸à, ¬�@�Î��ê3�ÕÄ��, Ê2��5��5Ò��!4, ð��;ºÕ�

»»»4.3.�O

limx→1

3x2 − 2x− 1

x2 − x ��.¿à1Ä¾!J, ÿXkO�Á§

limx→1

6x− 2

2x− 1= lim

x→1

6

2= 3�

4.4 Á§�� 235

�QÏ×Í�PÎýÝ, .limx→1(6x − 2)/(2x − 1)�&×��¯@îh`��Þx|1�á, �ÿÁ§(6− 2)/(0− 1) = 4�

»»»4.4.�O

limx→0

sin x− x

x3 ��.&Æ©��ÕÄ�Aì:

limx→0

sin x− x

x3= lim

x→0

cos x− 1

3x2= lim

x→0

− sin x

6x= lim

x→0

− cos x

6= −1

6�

3î»�, EÏÞÍ�r��ÝÁ§, &Æô�¿àÏ×aÝ(5.16)P, Ç

(4.5) limx→0

sin x

x= 1,

�à#ÿÕÁ§Â −1/6�¬Âÿ¥�ÝÎ, &ÆÎÍ�¿à1Ä¾!J¼J�(4.5)P? 3Ï×a�§5.7�, &ÆðÝ×j�G�J�(4.5)P, ¬uà1Ä¾!J, �"-�ÿ

limx→0

sin x

x= lim

x→0

cos x

1= 1�

h�Ýn"Î,¸à1Ä¾!J,Ä6àÕsineÐó��5,¬sineÐó��5, Q�àÕ(4.5)P(�ÏÞa»7.6)�3ó.î&ÆÎ��9øÇ�J��ÇãA0�B, ¬A�Wñ, QÿàÕB�1Ä¾!J��.Â�´�:nyx → ∞`, f(x)/g(x)ÝÁ

§�u�t = 1/x, J

limx→∞

f(x)

g(x)= lim

t→0+

f(1/t)

g(1/t)= lim

t→0+

f1(t)

g1(t),

Í�f1(t) = f(1/t), g1(t) = g(1/t), t 6= 0�.hã�§4.1�ÿì��

��§§§4.2.'f ′(x)Cg′(x)/D3, ∀x > M , Í�M ×ü�ÝÑó�ê'

limx→∞

f(x) = limx→∞

g(x) = 0,

236 Ï°a �5�Tà

vg′(x) 6= 0, ∀x > M�¨ulimx→∞ f ′(x)/g′(x)D3,J�ÿlimx→∞f(x)/g(x)ùD3, vÞÁ§Â8��JJJ��.�L = limx→∞ f ′(x)/g′(x)�v�f1(t) = f(1/t), g1(t) = g(1/t),

t 6= 0�J

(4.6)f(x)

g(x)=

f1(t)

g1(t),

Í�t = 1/x��Qx → ∞, uv°ut → 0+�.t → 0+`,

f1(t)/g1(t) ×0/0��, Æ�ãf ′1(t)/g′1(t)�Á§¼Of1(t)/

g1(t) �Á§�ã=Å!J

f ′1(t) = − 1

t2f ′(

1

t), g′1(t) = − 1

t2g′(

1

t)�

êã�'g1(t) 6= 0, ∀0 < t < 1/M�ãîP�ÿ, x = 1/tvx >

M`,f ′1(t)g′1(t)

=f ′(1/t)g′(1/t)

=f ′(x)

g′(x)�

.hux → ∞`, f ′(x)/g′(x) → L, Jt → 0+ `, f ′1(t)/g′1(t) →

L��AGX�, 1Ä¾!Jê0l

L = limt→0+

f ′1(t)g′1(t)

= limt→0+

f1(t)

g1(t)�

¬ã(4.6)P, f1(t)/g1(t) = f(x)/g(x), J±�

Q&Æô�¿�2¶�×x → −∞`, v«�§4.2Ý��êx → a(Ta+, a−, a ô�|Î∞T−∞)`, uf(x) →∞vg(x) = ∞, h`&Æ1f(x)/g(x)b×∞/∞��, J)bETÝ1Ä¾!J, &Æµ�W�Ý�ÃÍîE0/0T∞/∞Ý��, 3Êf�ì,

(4.7) limx→a

f(x)

g(x)= lim

x→a

f ′(x)

g′(x)�

ê3¸à1Ä¾!J`, uîP��Á§ ∞(T−∞), J¼�Á§ù ∞(T−∞)�

4.4 Á§�� 237

»»»4.5.�O

limθ→π/2−

sec θ

tan θ�

��.h ×∞/∞��¿à1Ä¾!J, ÿ

limθ→π/2−

sec θ

tan θ= lim

θ→π/2−sec θ tan θ

sec2 θ= lim

θ→π/2−tan θ

sec θ�

u�à×g1Ä¾!J, ÿ

limθ→π/2−

tan θ

sec θ= lim

θ→π/2−sec2 θ

sec θ tan θ= lim

θ→π/2−sec θ

tan θ�

�Q�P°ãh°O�Á§Â�¬.tan θ/ sec θ = sin θ, Æÿθ →π/2− `, Á§Â 1�

î»�î1Ä¾!J¬&0�, 3@jTà�, b`��°�;�

»»»4.6.�O

limx→∞

x2 + 5x

3x2 + x + 2��.Bã1Ä¾!J�O�Á§ 1/3, h�&Æ|GXá8!�

»»»4.7.�O

limx→0+

(1

sin x− 1

x)�

��.h ×∞−∞��Bã;5, �»; ×0/0��

limx→0+

(1

sin x− 1

x) = lim

x→0+

x− sin x

x sin x= lim

x→0+

1− cos x

sin x + x cos x

= limx→0+

sin x

2 cos x− x sin x=

0

2− 0= 0�

¿à1Ä¾!J`, �bÊÆP, �`º�ÎÍ ��

»»»4.8.�O

limx→∞

x4(cos1

x− 1 +

1

2x2)�

238 Ï°a �5�Tà

��.h ×∞ · 0 ��u�t = 1/x, J

limx→∞

cos(1/x)− 1 + 1/(2x2)

1/x4= lim

t→0+

cos t− 1 + t2/2

t4

= limt→0+

− sin t + t

4t3= lim

t→0+

− cos t + 1

12t2= lim

t→0+

sin t

24t=

1

24�

�ÕÏ"a&Æ+Û¿Í±Ðó¡, ��Þ?°�;�∞0,

1∞C00Ý��, ôº3£`�D¡�!ñ×è, \ïÎÍ��;, ¢9ëËÁ§ù ��? &Æ©1�00Ý��, ÍõËËº�\ï� �D¡�Ey(f(x))g(x), &Æá¼, ug(x) ≡0vf(x)�ºãÂ0, J(f(x))g(x) ≡ 1; �uf(x) ≡ 0vg(x)�ºãÂ0, J(f(x))g(x) ≡ 0�¨², uf(x) ≡ c1 6= 0 ×ðó, vg(x) →0, J(f(x))g(x) → 1; ug(x) ≡ c2 6= 0 ×ðó, vf(x) → 0,

J(f(x))g(x) → 0�¬¨3Îf�gK3��, vÞïí?0#�,

9`(f(x))g(x)º��¢Â, µºb&Ë��, 6ÚÍ½�µ��¨², b`ô�¿àî×;Ý��"P, ¼O��ÝÁ§�»A, E»4.4, &Æ�à1Ä¾!J, O�Á§ −1/6�¬¿àìPô�!øÿÕhÂ�

sin x− x

x3=

x− x3/6 + o(x4)− x

x3=−x3/6 + o(x4)

x3 �

»»»4.9.�O

limx→0

1

x(cot x− 1

x)�

��.´�ã»3.8, x → 0`,

cot x =1

tan x=

1

x + 13x3 + o(x3)

=1

x

1

1 + 13x2 + o(x2)

=1

x(1− 1

3x2 + o(x2)) =

1

x− 1

3x + o(x)�

Æx → 0`,

1

x(cot x− 1

x) = −1

3+ o(1) → −1

3�

êÞ 239

êêê ÞÞÞ 4.4

1 − 18Þ, �¿à1Ä¾!J, 19 − 24Þ, J¿à��"POÁ§�

1. limx→0tan x

x � 2. limx→0tan x−xx−sin x�

3. limx→π/2sec2 3xsec2 x� 4. limx→0

cot 3xcot 2x�

5. limu→0tan 2uu sec u� 6. limx→0+

x−sin x(x sin x)3/2�

7. limx→0x cot x−1

x2 � 8. limx→π/2tan 3xtan x�

9. limx→π/2tan x−5sec x+4� 10. limx→0+

1√x( 1

sin x− 1

x)�

11. limx→∞ x1/4 sin(1/√

x)� 12. limx→0sin x−x−x3/6

x5 �13. limx→2

3x2+2x−16x2−x−2 � 14. limx→∞(x2 −√x4 − x2 + 1)�

15. limx→a+

√x−√a+

√x−a√

x2−a2 � 16. limt→1ntn+1−(n+1)tn+1

(t−1)2 �17. limx→1

Pnk=1 xk−n

x−1 � 18. limx→1+( 1x−1

− 1√x−1

)�19. limx→0

1−cos x2

x2 sin x2� 20. limx→01−cos2 xx tan x �

21. limx→03 tan 4x−12 tan x3 sin 4x−12 sin x� 22. limx→0

cos(sin x)−cos xx4 �

23. limx→0tan 2xsin 3x� 24. limx→0

1−cos 2x−2x2

x4 �25. �

f(x) =sin 4x sin 3x

x sin 2x ��5½Olimx→0 f(x)Climx→π/2 f(x)�

26. �OðóaCb, ¸ÿ

limx→0

(x−3 sin(3x) + ax−2 + b) = 0�

27. �OðóaCb, ¸ÿ

limx→0

1

bx− sin x

∫ x

0

t2dt√a + t

= 1�

28. �EÑJón, k, D¡¢`ì�Á§D3, uD3¬OÍÁ§Â�

limx→0+

(1

sinn x− 1

xk

)�

240 Ï°a �5�Tà

29. A%'b×�5 1, �� x(©�)�i=, C 3A�BÞF�6aÝøF�

O A

C

B

�T (x)�ë��ABC�«�, S(x)�YÅ�«��OT (x)�S(x)Climx→0+ T (x)/S(x)�

4.5 ��555��TTTààà®®®ÞÞÞG«¿;XD¡ÝKÎny�5Ý&ËTà��h&Æ�1�

Ìn�5ÝÃÍ�Ì�EXa�XÝ×@j®Þ, &ÆµÎ�¿àÊÝ�á��, Q¡Þ�®Þ��

»»»5.1.àÝn��100cm3�iÖ�ª, à%�M��¸C]t6��.'iÖlÝ�5 r, { h�Jã�á�f�, ÿ

(5.1) πr2h = 100�kC]t6, Ç�«�t��«� îìÞi«��«�,

Ç

f(r) = 2πr2 + 2πrh = 2πr2 + 2πr · 100

πr2

= 2πr2 +200

r, 0 < r < ∞,

4.5 �5�Tà®Þ 241

h�ã(5.1)P, ÿh = 100/(πr2)�á�&Æ-Î�0�r, ¸f(r)t�, Ç0�EÁ�Â�B�5¡, ÿ

f ′(r) = 4πr − 200

r2=

4(πr3 − 50)

r2,

f ′′(r) = 4π +400

r2> 0�

�f ′(r) = 0, ÿr = r0 = (50/π)1/3`bÛ&F�êf ′′(r) Ñ, Æh 8EÁ�Â�ãy�EÁ�Â, ��sß3Û&F, Tf��ÝF��

limr→0+

f(r) = ∞, limr→∞

f(r) = ∞,

/�ºÎÁ��Æ3r0 = (50/π)1/3`b�EÁ��h`{

h = 100/(πr2) = 2(50/π)1/3 = 2r0�Ç{�yà5`, C]t6�

»»»5.2.(i) �óãÞ&�ó, ¸Íõ 1, v¿]õt�;

(ii) �óãÞ&�ó, ¸Íõ 1, v¿]õt��.'Þó xCy, �áx + y = 1�ÍÞ(i)-Î�¸x2 + y2t�,

(ii) �¸x2 + y2t��.y = 1 − x, Æx2 + y2= x2 + (1 − x)2=

2x2 − 2x + 1��f(x) = 2x2−2x+1,Jf ′(x) = 4x−2, f ′′(x) = 4 > 0��f ′(x)

= 0, ÿx = 1/2 °×�Û&F�ê.f ′′(1/2) > 0, Æ3x = 1/2bÁ��ê\&Fx = 0T1, f(0) = f(1) = 1 > f(1/2) = 1/2�Æ3x = 1/2(h`x = y)b�EÁ�, 3x = 0Tx = 1/b�

EÁ��Âÿ¥�ÝÎ, uÍ»; xCy/ Ñó, Jx = 0Cx =

1/�3�L½�, Æh`P�EÁ�Â�

»»»5.3.(i) óãÞ&�ó, ¸Íõ 1, �¶�t�;

(ii) óãÞ&�ó, ¸Íõ 1, �¶�t��.Í»®°v«îÞ�(i)�� x = y = 1/2, (ii)�� x = 0,

y = 1 Tx = 1, y = 0��u; óãÞÑó, J(ii)P��

242 Ï°a �5�Tà

»»»5.4.3x-y¿«îb×eÎax2 = 4y, 3y�îb×�F(0, b)��OeÎa�(0, b), t#�ÝF��.(0, b)�eÎaî�×F�ûÒ, d =

√x2 + (y − b)2, Í�x2 =

4y�ub ≤ 0, J�QeÎaî(0, 0)�(0, b)t�, vûÒ −b�¨'b > 0�´�Od�Á�, �Od2�Á�Î��Ý��

d2 = x2 + (y − b)2 = 4y + (y − b)2 = y2 + y(4− 2b) + b2, y ≥ 0�

�f(y) = y2 + y(4− 2b) + b2, y ≥ 0, J

f ′(y) = 2y + 4− 2b,

f ′′(y) = 2 > 0�

�f ′(y) = 0, ÿy = b− 2 °×�Û&F�ub < 2, .b − 2 < 0, h`b − 2� �(.y > 0)�Çb < 2`,

PÛ&F��l�\&F, y = 0`, f(0) = b2, �limy→∞ f(y) =

∞�Æ3y = 0b�EÁ��¯@îub < 2, Jf ′(y) > 0, ∀y ≥0�Çh`f �¦, ÆÁ�sß3f�L½�¼�ÐFy = 0�h`Á�Â

√f(0) = b�

ub ≥ 2, .f ′′(y) > 0, ∀y ≥ 0, Æ3Û&Fy = b − 2, b8EÁ�, vÁ�Â

f(b− 2) = (b− 2)2 + (b− 2)(4− 2b) + b2 = 4(b− 1)�

ÇtyûÒ √

f(b− 2) = 2√

b− 1��¡Aì: b < 2`, eÎaî(0, 0)t#�(0, b), vûÒ |b|�

b ≥ 2`, eÎaî(2√

b− 2, b − 2)�(−2√

b− 2, b − 2)/ t#�(0, b) �F, vûÒ 2

√b− 1�

»»»5.5.ØlTàó.�, kæ»��×I_��2¥�æðÝÕ°Î, N�ÃÍð5000-, Nbx(¶�69}8x-�¨², _ ð��Nß1000-, NøÄ40ß, Nß�K}ø�ßó6¹�3�8�(»: 45ß`, NßVþ1000 − 6(45 − 40) = 970)�»��t9©

4.5 �5�Tà®Þ 243

�ð60�_��®_�9K`, Nß¿í�HtK?

��.'bxß, JÀ�ð

g(x) =

{5000 + 8x2 + x(1000− 6(x− 40)), 40 ≤ x ≤ 60,

5000 + 8x2 + 1000x, 0 < x < 40�

�¿í�H

f(x) =g(x)

x=

{5000

x+ 2x + 1240, 40 ≤ x ≤ 60,

5000x

+ 8x + 100, 1 ≤ x < 40�

�f ′(x) = 0, ÿx = 50, 40 ≤ x ≤ 60; x = 25, u1 ≤ x < 40��

f(50) = 100 + 100 + 1210 = 1440,

f(25) = 200 + 200 + 1000 = 1400�

êEÐF1, 40C60, 5½b

f(1) = 5000 + 8 + 1000 = 6008,

f(40) = 125 + 100 + 1240 = 1465,

f(60) = 5000/60 + 120 + 1240 = 14431

3�

Æáf(25) t�Â, Ç_� 25ß`, Nß¿í�HtD�

»»»5.6.��ë��×\�C«�, �OÍø�t�ï��.'4ABC�, AB��2a��.«�ù��, ÆABî�{h

ü��Þhë��Hy2ý¿«î, ABw3x�î, v|æF Í�F,cFC�2ý (x, h)�µÞ�á�Ox,¸ÿAC +BC +2at�, ÇAC + BCt��

f(x) = AC+BC =√

(x + a)2 + h2+√

(a− x)2 + b2, −a ≤ x ≤ a�

244 Ï°a �5�Tà

J

f ′(x) =x + a√

(x + a)2 + h2+

x− a√(x− a)2 + h2

,

f ′′(x) =−(x + a)2

√((x + a)2 + h2)3

+1√

(x + a)2 + h2

+−(x− a)2

√((x− a)2 + h2)3

+1√

(x− a)2 + h2

=h2

√((x + a)2 + h2)3

+h2

√((x− a)2 + h2)3�

�f ′(x) = 0, ¬B;�ÿ4ah2x = 0, Æx = 0 °×�Û&F�.

f ′′(0) =2h2

(a2 + h2)2/3> 0,

Æ3x = 0, bÁ�Âf(0) = 2√

a2 + h2 �êf(a) = f(−a) =√4a2 + h2 + h > 2

√a2 + h2 = f(0), Æ3x = 0b�EÁ�Â�Ç

h`4ABC �Të��

-

6

A−a O x a

B

h

C(x, h)

x

y

¨², Í»u; ��×\�Cø�, Ç�J��Të��ù «�t�ï�

»»»5.7.'¿«îbA�BÞF3×àa�!×��3hàaîO×F, ¸hF�A, BÞF�ûÒõt��.A%, Ç�3x�î0×FP , ¸ÿPA + PBt��

4.5 �5�Tà®Þ 245

�

f(x) =√

x2 + h2 +√

(a− x)2 + h21, 0 ≤ x ≤ a�

J

f ′(x) =x√

x2 + h2+

x− a√(a− x)2 + h2

1

,

f ′′(x) =h2

√(x2 + h2)3

+h2

1√((a− x)2 + h2

1)3

> 0�

�f ′(x) = 0, ÿ

x√x2 + h2

=a− x√

(x− a)2 + h21

,

�hÇcos α = cos β, ùÇα = β�êf ′′(x) Ñ, Æα = β`,

fbÁ��.©b×Á�, hÄ �EÁ�(��f�%�Ý�Ï)�

-

6

O

A

h

α βh1

B

y

x-x¾ P

¾ a -

Í»��.î�DDD æææFFF(optical law of reflection)bn��.îb×¥�Ýðððyyytttyyy` æææ§§§(Fermat’s principle of least

time)�Ç3Ø°X��Ýf�ì, �aãA�BX�Ý5, µXm�tyÝ` ��3hty` �tyûÒ!L, .hãABaî×F�B, Já �α�yD �β`, X��` ty�

3î»�, uA�B3àa�²�, J�QP AB�àa�øF�î»Qô�à¿¢]°¼�, 8*��|G-!�Ý��

246 Ï°a �5�Tà

yì», u�¢Ã��5, -�Î��|Ý�

»»»5.8.'¿«îÞFA�B3×àa�²�(A%)�'3A9��B£�>�5½ c1Cc2�OãA�BXm` ty�5��.��2, Í»Ç�3x�0×FP , ¸ÿºÞaðAPCPB�ty` ��

f(x) =1

c1

√h2 + x2 +

1

c2

√h2

1 + (a− x)2, 0 ≤ x ≤ a�

�f ′(x) = 0, ÿ

1

c1

x√h2 + x2

=1

c2

a− x√h2

1 + (a− x)2

`, fbÁ�(v �EÁ�)��îPê��y

sin α

c1

=sin β

c2�

�J�3x�îªb×F��îP�

-

6

O

A

h

y

xPx

α

β

B

h1¾ a -

Í»)�à�.îÝty` Ýæ§¼1��'3ÞË+²�(Aè��i), �a>�5½ c1, c2�JãÏ×Ë+²�ÝAF�ÏÞË+²�ÝBF, �aX�Ý5m��îP(hÇSnell’s

law of refraction)�

4.5 �5�Tà®Þ 247

¨², bXÛ888nnn>>>£££(related rates)Ý®Þ�

»»»5.9.'b×i�Ý[�,{ 42M,9�5 52M,|N�`3ñ]2MÝ>£¥iáÍ��h(t)�3` ti�{�, Oh = 2`,

dh/dt�

4

5

��.Í»ÇO3i{�h = 2`, h��`¦�>��V (t)�3` t, [�i��, ãÞ�á

(5.2)dV

dt= 3�

.�ã��dV/dt, �Odh/dt, X|Ì� 8n>£��r(t)�3` t`i«��5, JãÞ�á,

r(t)

h(t)=

4

5�

ê

(5.3) V (t) =1

3πr2h =

1

3π(

5

4h)2h =

25

48πh3�

�)(5.2)C(5.3)P, ÿ

3 =dV

dt=

25

483πh2dh

dt�

.hh = 2`,dh

dt= 3

48

25π

1

12=

12

25π�

»»»5.10.Øßð3*\Pe, *Òi«30Î�PÕe`, [PaÝ>� NJ2Î��Oa� 50Î`, eºi«�>��

248 Ï°a �5�Tà

��.ãÞ�áds/dt = −2, kOs = 50`dx/dt�.

x2 + 302 = s2,

A x

30s

Æ

2xdx

dt= 2s

ds

dt= −4s,

vdx

dt= −2s

x�

s = 50`, x = 40, Æh`dx/dt = −100/40 = −2.5, Ç�`>� NJ−2.5Î�>� �, .eûAFÝûÒ�3�

»»»5.11.'×²^�{� 7ó,i¿�>²�>� N5Ö10ó�Øß32«AFÌ?²^, O²^ûhßi¿ûÒ 24ó`, Ì?��;��.A%, �ádx/dt = −10, kOx = 24`, dθ/dt�¿àtan θ =

x/7, ÿd tan θ

dt= sec2 θ

dθ

dt=

1

7

dx

dt= −10

7�

.hdθ

dt= −10

7cos2 θ�

x = 24`, s =√

72 + 242 = 25�h`cos θ = 7/25, Æ

dθ

dt= −10

7(

7

25)2 = − 70

605�

4.5 �5�Tà®Þ 249

θ

x

7s

A

3�9Tà®Þ�, ðºÂÕm�]�PÝqÝ`Î�'b×Ðóf , &Æ�0�£°º��f(x) = 0Ýx�A�fÎ×gPTÞgP, Q^®Þ, ëgPC°gPôb2P�, ©Î´�Ó�¬Îfô�×�Î94P, Ç¸Î��ÝëgP, £°�ÓÝ2P, b`ô��ÿ�¯&Æá¼, qÝÂ��~b ¢�9ì&Æ-èº×0qÝ�«�Ý]°, Ì� pppñññ°°°(Newton’s method)�'f ×��Ðó, &Æ�0f(x) = 0�q��0×#�qÝ

ó, |x0 ��, h�¿à�q�§(Ï×a�§6.2)�k0×?#�qÝÂ, ®Äy = f(x)�%�îÝF(x0, f(x0))�6a, ¬øx�yx1�6a�]�P

y = f(x0) + f ′(x0)(x− x0)�

�y = 0-O�x^û, Ç

x1 = x0 − f(x0)

f ′(x0)�

Q¡�x1�s, ¥�î�M», µ�ÿx2, x3, · · · , à�Þ@� &ÆX��Ä,uR�Âx00ÿ�?,µ�ÿÕÝx1, x2,· · · ,��Òq�G�×��, ��¹x0#�y = f(x)�ÁÂ�êxn+1�xnbì�n;, 9Îpñ°ÝL]2P, J�Jº3ê

Þ�

(5.4) xn+1 = xn − f(xn)

f ′(xn), n = 0, 1, 2, · · ·�

}¡&ÆºJ�, 3ÊÝf�ì, n →∞`, xnº��f(x) = 0

�×q(¬v[eÿ�")�&Æ�:%5.1�

250 Ï°a �5�Tà

-

6

x2r x1 x0O

y

x

y = f(x)

(x0, f(x0))

%5.1. pñ°�M»

'r f(x) = 0�×q, A%5.1, u3r!�, f�%� î�,

Jx0!ã3rÝ��, Ahx1, x2,· · · , �º�¼�#�r�Í�Ý��, ô��|ã%�¾\x0, ã3¢�´·�'3r�×Ï½B�, f ′′=�, vf ′(x), f ′′(x)� 0�.f(r) = 0,

ã(3.24)��"Pÿ(ãa = xn, x = r)

0 = f(r)− f(xn) + f ′(xn)(r − xn) +f ′′(c)

2(r − xn)2,

Í�c+yr�xn �ÞîPN×4&t|f ′(xn), ÿ

f(xn)

f ′(xn)+ (r − xn) +

f ′′(c)2f ′(xn)

(r − xn)2 = 0,

ãîPêÿ

(5.5) xn+1 − r = xn − f(xn)

f ′(xn)− r =

f ′′(c)2f ′(xn)

(r − xn)2,

Í�Ï×�PÎã(5.4)P�¼�f ′�f ′′ÝÑ��µ�b°Ëà), &Æ©D¡Í�×Ë, Íõë

Ë�µÝD¡v«��'3B�f ′(x) > 0, f ′′(x) > 0,Jã(5.5)Pár < xn+1�ur <

xnùWñ,J.f �¦, f(xn) > f(r) = 0,Æã(5.4)P,êÿxn+1 <

4.5 �5�Tà®Þ 251

xn�.hx1 > x2 > · · · > xn > xn+1 > r ×��vb&Ýó�, .hlimn→∞ xnD3, |r′�hÁ§Â��3(5.4)P�Ë��n → ∞,

ÿr = r′ = lim

n→∞xn�

ÇÿJãpñ°ÿÕÝó�{xn, x ≥ 1}, Ý@º��fÝ×Íq�Íg:pñ°OqÝ0-�´�ãíÂ�§ÿ(¿àf(r) = 0)

r − xn =f(r)− f(xn)

f ′(c)=−f(xn)

f ′(c),

Í�c+yr�xn �Æu|f ′(c)| ≥ M , Í�M ×ðó, J

(5.6) |r − xn| ≤ |f(xn)|M �

.nÈ�`, xn�#�r, .hf(xn)�#�f(r) = 0, ÆîPnÈ�`, �à¼®0-�×�?Ý£�Â�gã(5.5)Pÿ

xn+1 − r =f ′′(c)

2f ′(xn)(r − xn)2�

.hu|f ′(xn)| ≥ M1, |f ′′(c)| ≤ M2, Í�M1, M2 Þðó, Jÿ

(5.7) |xn+1 − r| ≤ M2

2M1

|xn − r|2�

ÆuM2/M1��, ãîP�:�xn��rÝ>�8"�îPù�� 0-�£��

»»»5.12.�O2x3 + x2 − x + 1 = 0�×�«q��.�f(x) = 2x3 + x2− x + 1, f ′(x) = 6x2 + 2x− 1�.h(5.4)PW

(5.8) xn+1 = xn − 2x3n + x2

n − xn + 1

6x2n + 2xn − 1 �

252 Ï°a �5�Tà

.f(−1) = 1 > 0, f(−2) = −9 < 0, Æ3(−2,−1)�b×q�ãy|1| < | − 9|, Æ�?qT´#�−1�ãx0 = −1.2, �¿à(5.8)P�ÿ�5.1�

n xn f(xn) f ′(xn) f(xn)/f ′(xn) xn+1

1 −1.20000 0.18400 5.24000 0.03511 −1.23511

2 −1.23511 −0.00711 5.68276 −0.00136 −1.23375

3 −1.23375 0.00001 5.66533 0.00000 −1.23375

4 −1.23375

�5.1

ã�5.1�:�, x3Cx4��óG5��8!, &Æ�V��cÝ, Ç−1.23375 ×�«q, v0-�y10−5�

uxn+1−xn�¼��,-�ÿÕ�«q�uxn+1−xn��0,

Jpñ°´[, �ì»�

»»»5.13.�¿àpñ°, O√

3 ��«Â��.√

3Ç x2− 3 = 0�Ñq�.1.72 = 2.89 < 3 < 1.82 = 3.24�ãx0 = 1.8�êf(x) = x2 − 3, f ′(x) = 2x,

xn+1 = xn − x2n − 3

2xn

=1

2(xn +

3

xn

)�

.h

x1 = 1.73333,

x2 = 1.73205,

x3 = 1.73205�

ãyx2�x3��óG5�8!, Æá√

3��óÏ5�Ý�«Â 1.73205�uk?Þ@, x1Ý�ó�ó69ã¿��

4.5 �5�Tà®Þ 253

3îÞ»�, &Æ:ÕÇ¸�¿à(5.6)T(5.7)�0-2P, ¸àpñ°¿g¡, -ôV¯�:�0-Ý��9ìÞ», Ç¿à0-2P�

»»»5.14.�x0 = 3��O√

7��«Â�(i) àpñ°�ëg¬£�0-;

(ii) u�0-�y1020, ®m|pñ°�¿g?

��.(i) �f(x) = x2 − 7, f ′(x) = 2x, f ′′(x) = 2�

xn+1 = xn − f(xn)

f ′(xn)= xn − x2

n − 7

2xn

=x2

n + 7

2xn�

µ��ÿ

x1 =8

3, x2 =

127

48, x3 =

32257

12192.= 2.6457513123�

.r < c < x3, �r > 2, Æ|f ′(c)| > 4�.hã(5.6)P

|r − x3| ≤ |f(x3)|4

< 1.7 · 10−9�

�¬:�©|pñ°�ëg, -�8Þ@, vá

2.6457513106 < r < 2.6457513124�

(ii) ã(5.7)Pÿ

|xn − r| ≤(

M2

2M1

)2n−1

|x0 − r|2n

�

�p:�√

7 > 2.6, Æ|f ′(xn)| = |2xn| > 5.2, êf ′′(c) = 2, ÆM2 =

2�¨², |x0 − r| ≤ 0.4��(

0.4

5.2

)2n−1

· 0.4 < 10−20,

�ÿn = 5Ç��

254 Ï°a �5�Tà

»»»5.15.�Ox3 + x− 1 = 0�q��óÏë��.�f(x) = x3 + x − 1�.f(0) = −1�f(1) = 1Ðr8D, Æ3(0, 1) b×q�êf ′(x) = 3x2 + 1 > 0, Æfb°×Ýq3(0, 1)

�ãx0 = 1, Jã

xn+1 = xn − f(xn)

f ′(xn)= xn − x2

n + xn − 1

3x2n + 1

=2x3

n + 1

3x2n + 1

,

µ��ÿx1 = 0.75, x2 = 5986

, x3 = 523407767077

.= 0.682�ãM1 = 1�:

�x3�Þ@��óÏë��

»»»5.16.'f(x) = x1/3, J�Qx = 0 f(x) = 0�°×q�¬.f ′(x) = 1

3x−2/3, Æu2àpñ°, J

xn+1 = xn − f(xn)

f ′(xn)= xn − x

1/3n

13x−2/3n

= xn − 3xn = −2xn�

.h

xn+1 = (−2)xn = (−2)2xn−1 = · · · = (−2)n+1x0�

ÆA�x0óãÝ�Î0, Jxn�º��0�

êêê ÞÞÞ 4.5

1. �`ay = x3îã×FP , v'ÄP�6aø`ay¨×FQ��J3Q�6aE£, 3P�6aE£�4¹�

2. 'f(x) = ax2 + bx + c, Í�a, b, c @ó�ê'|f(x)| ≤ 1,

∀|x| ≤ 1��J|f ′(x)| ≤ 4, ∀|x| ≤ 1�

3. 'b×ü», 9 Ñ]�, XàC]�«� C, �O9\�C»{, ¸��t��

êÞ 255

4. b×ª´12¦, ��*×séÃÚ'éÝ�ì*52¦�*Ý�Ã��'i9Ú'ðà, 2«Ú'ðàÝ5/3¹��®º%�aÚ'tBz?

5. �JiÝ/#ë��, |Ñë��Ý«�t��

6. �JiÝ²6ë��, |Ñë��Ýø�ty�

7. �OeÎay = x2ît#�(3, 0)ÝF�

8. �J(32, 3

4), `ay = x2 − xît#�(1, 1)ÝF�

9. �3Yix2/a2 + y2/b2 = 1, a > b, î0×F, ¸ÿhF�(c, 0)�ûÒty�

10. ��×\� 12%ÝÑ]�ü�, ^�°Í�(ù Ñ]�),

àW×P��]�»��®h�]��t�� ¢?

11. 'b×9�5 6%v{ 10%�Ñi��O/#yhiÖ�Ýt��, ¬O�h`iÖ�9�5�{�

12. 'bàa¼=#A, BËFCB, CËF, ê'ABkàBC�*k3ABî0×FP , ¬Ñ�±¼PC, �¼APômÑ��uÑ��¼��±, N2¦WÍ 1f2, ê'AB 402¦, BC 202¦��0�P , ¸��Àðàt±�

13. 'b×�5 6ÎÝ¦Ï_¦, |NJ1/10ñ]ÎÝ>��O_¦Ý�5L3�>��

14. ØV��20Î, Eê3�', ¬'V�ÝcÐ|NJ1.5ÎÝ>�âì��OV�cÐû2«16Î`, V�9IÒ��'�>��

15. 'Øß�{6Î, |NJ4ÎÝ>�'{ 14ÎÝÕu��,

�θ�hß�I�u�Ý=a�uÖ �ô��OhßÒuÖ12Î`, θÝ�;£�

256 Ï°a �5�Tà

16. �J(5.4)P�

17. �|pñ°Ox3 + x2 + x = 2�×�«q¬Þ@��óÏ3�,

ãx0 = 1�

18. �|pñ°Ox3 + 2x2 − 2x = 5�×�«q, ãx0 = 1.5�

19. '

f(x) =

{ √x, x ≥ 0,

−√−x, x < 0��Ju|pñ°Of(x) = 0 �q, Jÿx0 = −x1 = x2 =

−x3 = · · ·�20. (i) �Jx3 − x− 1 = 0, ªb×@q;

(ii) �|pñ°, O(i)��]�P��«q��ó5�Þ@�

21. �Ox3 + x + 1 = 0Ýq��óÏë��

22. �Jx3 − 4x + 1 = 0ªbë@q, ¬Ohëq��«Â��ó5�Þ@�

23. �O2x − cot x = 0+y0�1 �×�«q, ¬Þ@��óÏ3��

24. kO√

13��«Â�(i) �x0 = 4, �àpñ°Ëg, ¬£�0-;

(ii) u�0-�y10−10, �®�àpñ°¿g?

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1967). Calculus, Vol I, 2nd ed. John Wiley &

Sons, New York, New York.

ÏÏÏ"""aaa

øøø÷÷÷ÐÐÐóóó

5.1 GGG��

��.R-.ÝÐó, EÐóÝ�L, &Æ¬Î��9Ý§×, X|�|b&P&øÝÐó�b°�P��ÝÐó, 3ó.�6�½¥�Ý��A94P�5P��9°ÅÎ�|§�Ý, .�Î��Ý¯Î??�bà��Äb×°�©�ÝÐó, Qô�à¼à�&9�!Ý¨é�Ía-Î�+Û×°9vÝÐó�A!ø÷ó,¸ÆP°|+�−�×�÷C√ �,XÛ�ó]°¼�î, 9ËÐóÌ øøø÷÷÷ÐÐÐóóó(transcendental function)�?�@2ý, E×Ðóf , uD3×u�94Pp(u) =

∑nk=0 ak(x)uk, Í�;

óa0(x), a1(x), · · · , an(x), / x�@;ó94P, v�� 0, ¸ÿEN×f�L½�Ýx, p(f(x)) = 0, Jf Ì ��óóóÐÐÐóóó(algebraic

function), �Î�óÐó-Ì ø÷Ðó�»A, f(x) =√

x,

Cg(x) = 3√

x + 1/ �óÐó, .�Æ5½��f 2(x) − x =

0Cg3(x)− x− 1 = 0�Çp(u)5½ u2 − xCu3 − (x + 1)�Ía�D¡Ýø÷Ðó, x�ÎEEEóóóÐÐÐóóó(logarithmic func-

tion)�¼¼¼óóóÐÐÐóóó(exponential function), �bë�Ðó(3G«Ýa;�D¡�K), CDDDëëë��ÐÐÐóóó(inverses of the trigonometric func-

tions)�9°Ðó3ó.Ýr½�, �¡Î3hé]«TTà]«,

KÎ��þKÝ�Ì��Kny¸ÆÝb¶P², ?ð�ß �

257

258 Ï"a ø÷Ðó

��á_~�E��5��, bÝ9°Ðó, �U"&ÆÝ�5�æ, ¯O��5, �W×4?9�9zÝ�®�

5.2 EEEóóó

3Ïëa»2.1, EN×��y−1Ýb§ón, ¿à��5ÃÍ�§, ��

∫ b

axndx, Ç

∫ b

a

xndx =bn+1 − an+1

n + 1 �

�yn = −1`, îP�QP°Wñ�.Eb > a > 0, f(x) =

x−1 ×=�Ðó, Æf3[a, b] ��, vÍ�×Riemannõ/º��

∫ b

ax−1dx�¿àh��, �q = n

√b/a, ¬|¿¢ùóa,

aq, aq2, · · · , aqn−1, aqn = b, Þ[a, b]5Wn5, v�xi = aqi, i =

1, · · · , n�JRiemannõ

Rn =n∑

i=1

f(xi)∆xi =n∑

i=1

(aqi)−1(aqi − aqi−1)

=n∑

i=1

(aqi)−1aqi(1− 1

q) = n(1− n

√a/b) −−−−→n →∞

∫ b

a

x−1dx�

�Än →∞`, n(1− n√

a/b)�Á§ ¢,êGQôP°O��¬&Æá¼Eb > a > 0,

∫ b

ax−1dxÎD3Ý�&Æ-|x−1�×��

5, SEóÐólog x, v�

(2.1) log x =

∫ x

1

1

udu, x > 0�

?�@2ý, log x x��QQQEEEóóó(natural logarithm), ux > 1, -��Ô`aÐóf(u) = u−1, 3%�ìã1�xÝ«��ãyy = u−1, u > 0, ×ÑÝ=�Ðó, Ðólog x, EXb

Ýx > 0/b�L, ¬vÎ×=�, v�}��¦�Ðó��y

5.2 Eó 259

3(2.1)P��, �5ì§ã 1, ©Î Ý]-�Ahº¸

(2.2) log 1 = 0,

vlog x > 0, ∀x > 1, log x < 0, ∀0 < x < 1��u−13 [a, b],

a < b, ��5J

(2.3)

∫ b

a

1

udu = log b− log a�

h�5�Ô`aÐóu−1, 3%�ìãa�bÝ«��Ah&Æ-SÝ×3ó.�, ��ë�Ðó8!ÁÌÝ±Ð

ó�3×MD¡G, &Æ�:hÐób£°ÃÍP²�

��§§§2.1.EóÐóbì�P²:

(i) log 1 = 0;

(ii) (log x)′ = 1/x, x > 0;

(iii) log(xy) = log x + log y, ∀x, y > 0�JJJ��.(i)�BJ�Ý, (ii)J¿à��5ÃÍ�§ñÇ�ÿ�¨J�(iii)�´�bìP

log(xy) =

∫ xy

1

1

udu =

∫ x

1

1

udu +

∫ xy

x

1

udu�

�¿à�ó�ð, �u/x = t, Jdu = xdt, v∫ xy

x

1

udu =

∫ y

1

1

xtxdt =

∫ y

1

1

tdt�

ÿJ(iii)�

ãî�§�(iii), ãy = 1/x, Jÿ

0 = log 1 = log x + log1

x�

Æb

(2.4) log1

x= − log x�

260 Ï"a ø÷Ðó

×�Jb

logy

x= log y + log

1

x= log y − log x�

êD«2¿à�§2.1�(iii), ÿE∀xi > 0, i = 1, · · · , n,

(2.5) log(x1 · · ·xn) =n∑

i=1

log xi�

©½2, E��ÑJón,

(2.6) log(xn) = n log x�

îPEn = 0 )Wñ, h.x0 = 1�un ×�Jó, J−n > 0, v

log(xn) = log(1

x−n) = − log(x−n) = −(−n) log x = n log x�

¨'α = p/q ×b§ó, Í�p, q Jó�J(xα)q = xp, v

log xα =1

qlog(xα)q =

1

qlog xp =

p

qlog x = α log x�

Æÿ

(2.7) log xα = α log x,

E∀x > 0, Cb§óαWñ�

31.3;&Æ�ÿÕ|(1 + 1/n)nn → ∞`�Á§, ¼Sðóe�¯@îe��

(2.8) log e = 1,

Ç3`ay = 1/x�ì, ã1�eÝ«� 1�&Æ¼: ¢(2.8)PWñ�(2.8)P e�×©P,ôµÎ&Æô��Le, ��log x = 1�°×Ý@ó�.log x =�Ðó, Æ(¿àÏ×a�§5.6)

log e = log( limn→∞

(1 +1

n)n) = lim

n→∞log(1 +

1

n)n = lim

n→∞n log(1 +

1

n)�

5.2 Eó 261

�¿à�5�íÂ�§, ÿ

log(1 +1

n) =

∫ 1+ 1n

1

1

udu =

1

ξn

1

n,

Í�ξn ∈ (1, 1 + 1/n), �nbn��2limn→∞ ξn = 1, Æ

log e = limn→∞

n1

ξn

1

n= 1�

Íg¼:EóÐó�%��y = log x, x > 0�J

dy

dx=

1

x> 0, ∀x > 0,

d2y

dx2= − 1

x2< 0, ∀x < 0�

Æ%� �¦vì�, Ç%�W�´cX�¯@î, hcXøÄ&ÆÝ��35.5;, &ÆºÞlog x�xn8f, £`��-º´bÃF�.E∀x > 0, y′Cy′′/D3v� 0, ÆPÁÂCD`F�êG«�èÄx > 1`, log x > 0, x < 1`, log x < 0, �ã(2.6)P, -ÿ(¥�h ��Ðó)

limx→∞

log x = ∞, limx→0+

log x = −∞�

.h%�Pi¿��a, �x = 0 kà��a�%2.1 y = log x

�%��

-

6

1 2 e 3x

O

y

1y = log x

%2.1. �QEóÐó�%�

262 Ï"a ø÷Ðó

�.`�&ÆÍ@-.ÄEó,×��ÎD¡|10 999 (base)ÝðàEó�£`;ðÎ9ø�LÝ: 'x > 0, J|log10 x�x3|10 9�Eó, vÍÂu ��10u = x�@ó�.hlog10 10 = 1,

ux = 10u, vy = 10v, J.xy = 10u+v, Æ

(2.9) log10(xy) = u + v = log10 x + log10 y�

h��§2.1�(iii)f´�ãyb(2.9)P, ¸Eó3�§¶°`]-�K�»A

log10(2.7× 1019) = 19 + log10 2.7,

log10(2.7× 10−19) = −19 + log10 2.7,

log10(3.28× 105) · (6.79× 107) = 5 + 7 + log10 3.28 + log10 6.79�

æ¼��T��Ýó, BãEó¡, W ÊÝ��Ëó8¶,

J�¢ããEó��;�Õ�3£�Õ�Ì�s¾Ý`�, EóÝ@Î�bàÝ�ßÆ�|EóÝæ§, s��ÕM, ¬��Eó��*^�Õ�ÌûÅvb[, Eó3�Õî��u|?¥�, ¬¸)Îó.î×Á¥�ÝÐó�|10 9ÝEó, �X|Ì “ðàEó”, Î.3@ó�Ù�,

&Æ2àè�, .hðàEób&9�ÕîÝ]-�¯@î, EóÝ9¬�m� 10, �¢Ñób 6= 1, /�2à 9�Ç

(2.10) u = logb x ⇐⇒ x = bu�

�(2.9)P�EóÝÃÍP²W

(2.11) logb(xy) = logb x + logb y�

ËÍ�!Ý9ÝEó, ôbì�n;�'a, b > 0, v/� 1, J

(2.12) loga x =logb x

logb a�

|î9°, �VÎ&Æ�.`�XD¡ÝEóÞC�

5.2 Eó 263

�}2ý, |(2.10)P¼�LEó, �ìî¬&£��Û�´�, kÝ�(2.10)P, &Æ��á¼buÝ�L�'u > 0, u Jó,

Jbu = b · b · · · b, �=¶uÍb�uu = p/q ×b§ó, Jbu =q√

bp��uu < 0, bu = 1/b−u��yb0J� 1�Æu b§ó`,

buÝ�L�Õz½�¬u� b§ó`, bu Ý�L, µ��|ßéÝ�ÉA110

√2Î%��¤? A¢OÍÂ? buÝ�L�P°�

z, ?¢µ��@�N×x > 0, ªb×u ��x = bu, Ah��Llogb x = u��v(2.11)P�WñêÃybu+v = bu · bv, u, v ∈ R, �u, v� b§ó`, 9ô�á�A¢J��X|�.`�|î�]PSEó, �� X#å, @3Î

Ãy�.ßEó.Ý�ò�È, ^bs¨Í�Ý��QG��ü�Î�|�R¼, �Äº�ðøa�(2.1)P, -Î2à×��!Ý]P, ¼�LEó, É��ê��5Ý�æC8Y�

9ì1� ¢|(2.1)P��5, ¼�L�QEó�}¡&Æôº¼�, uE¼ó¥±Ê2�L¡, J�Î�|¼ó¼�LEó�´�, 3ó.�&ÆE"DÐóÝP², ð��·¶�ÉA1:

ë�Ðób�9YúÝP²�êAÐó�ø�P��PC��P��Þ¼��ub^ºÑ^£¡, N×�^�ó, ôbÍ&�©�ÝP²�9×°P²�, b×°ÎXÛ©ÇÝP²�ÇØ×P² Ø×(T×v)ÐóX}©ÌbÝ, :ÕhP²Wñ, &Æ-�X�Î£×(v)Ðó�9�AßÝ¼ò ßÝ©Ç×��EóÐóÝP²�, tÚS&ÆÝ,��ÎÞÑóÝ¶��Eó,

�yhÞóEó�õ(Ç(2.11)P)�'b×Ðóf , &Æ�Tf��

(2.13) f(xy) = f(x) + f(y), x, y ∈ D,

Í�D f��L½�9Ëny×Ðó3ËÍ|îÝF�]�P,

Ì ×½½½ÐÐÐ]]]��PPP(functional equation)�&9ó.îÝ®Þ, ð�»ð �×½Ð]�P, ôµÎ0��½Ð]�P�XbÝÐó�;ð�×½Ð]�P¬&|¯, b`º�9�×°f�, A�

264 Ï"a ø÷Ðó

'Ðó =�T��9Ý×°9vÝ�', �¸&Æ3�×½Ð]�PÝÄ��, ��×°Ä�ÝºÕ, AãÁ§T�5��Q, ó.î3�×®Þ`, &Æ;ð�Tf��3�?, Ç�m�Ý�'��, ��Q��ú�?, Ç��0�t×�Ý��×ó.�§Ý.Â, ??�æf�Ýw´T��Ý?×��|:�f(x) = 0, ∀x ∈ D, (2.13)P�×��¯@î9Î

0 ∈ D`, (2.13)P�°×��J�Aì: '0 ∈ Dvf (2.13)P�×��ãy = 0, ÿ

f(0) = f(x) + f(0), ∀x ∈ D,

.hf(x) = 0, ∀x ∈ D�Ç©�0 ∈ D, Jf(x) = 0, ∀x ∈ D�¬0 6∈ D`, (2.13)PtÝf(x) ≡ 0²ÎÍbÍ��? �×]�P, �0�ÍXb�, u©Î:�bØ°�, �®¬Î�@��@�tÝ9°�², �ô^bÍ��¨'0 6∈ D�u1 ∈ D, J�x = y = 1, ÿ

f(1) = 2f(1),

.hf(1) = 0�

u1, −1/3D�, ãx = y = −1, ÿ

f(1) = 2f(−1),

.hf(−1) = 0�

ux,−x, 1,−1 ∈ D, ãy = −1, ÿ

f(−x) = f(−1) + f(x),

.hf(−x) = f(x)�

5.2 Eó 265

Ç��(2.13)P�f ×�Ðó��'E∀x 6= 0, f ′(x)/D3�Þ(2.13)P¼��Ex�5(ü�y),

ÿyf ′(xy) = f ′(x)�

3îP��x = 1, ÿyf ′(y) = f ′(1), .h

(2.14) f ′(y) =f ′(1)

y, ∀y 6= 0�

ãîPêÿ, 3N×��â0�T , f ′ ��, .h3� f ′

��ê.(2.14)Pù0lf ′3N×G� =�, .h��5ÃÍ�§Êà, v

(2.15) f(x)− f(c) =

∫ x

c

f ′(t)dt = f ′(1)

∫ x

c

1

tdt�

ux > 0, (2.15)PE∀c > 0Wñ�ux < 0, (2.15)PE∀c < 0Wñ�.f(1) = 0, Æ3(2.15)P�, �c = 1, ÿ

f(x) = f ′(1)

∫ x

1

1

tdt, x > 0�

�¿àf(x) = f(−x), ÿ

f(x) = f ′(1)

∫ −x

1

1

tdt, x < 0�

îÞPÇ0l

(2.16) f(x) = f ′(1)

∫ |x|

1

1

tdt, x 6= 0�

ÇÿJuf ′(x)D3∀x 6= 0(hf�Í@�ê°, �Ä Ý�-, v©Î��Ý�EóÝ�L®ßÝ�^,&Æ)�h�'),J(2.13)P��A(2.16)P�.f ′(1) ×ðó, (2.16)P�;¶

(2.17) f(x) = k

∫ |x|

1

1

tdt, x 6= 0,

266 Ï"a ø÷Ðó

Í�k = f ′(1)�uk = 0, J(2.17)P0lf(x) = 0, ∀x 6= 0, h��f(x) ≡ 0×l�uf ′(1) 6= 0, Jf(x) 6= 0, ∀x 6= 0�Ç3tÝx = 0�², f ′(x)D3, vf� 0�ì, &ÆJ�(2.13)P��Ä (2.17)PÝ�P, Í�k 6= 0 ×ðó��Ä6º�ÝÎ, |îÝD¡, ©ÎÿÕu(2.13)P3×°©�

Ýf�ìb�, J�Ä6b(2.17)PÝ�P��y(2.13)PÎÍËb�÷? E×A(2.17)P�Ðóf , ��J��(2.13)P, Æ(2.13)PÝ@b��|îÝ.0, Î&Æ|(2.1)P, ¼�LEóÐóÝ�^�4

Q(2.17)P��, �L3x 6= 0�, u©�Êx > 0, JÐóW 1− 1�êk ×ðó, Ý�-, &Æã 1�ã|îÝD¡á, 3fyÑó��Ýf�ì, ��f(xy) = f(x) +

f(y)ÝÐó

(2.18) f(x) = k log x,

Í�k ×ðó�k = 0Ý�µ&Æ�4t, .h`f 0�uk 6=0, �Aì¢ãS�!9ÝEó, ��î�f�kÝn;�ã(2.18)Pá,uk 6= 0,JD3×b > 0,¸ÿf(b) = 1,Çk log b =

1�Æb 6= 1, vk = 1/ log b, Ç(2.18)P�;¶

f(x) = log x/ log b�

&Æ�×�LAì�

��LLL2.1.'b > 0,vb 6= 1�E∀x > 0, x3|b 9�Eó,|logb x��, v

(2.19) logb x =log x

log b�

ã(2.19)PC�§2.1�(ii), ñÇÿ

(2.20)d

dxlogb x =

1

x log b�

5.2 Eó 267

ê�:�logb b = 1, vub = e, Jloge x = log x, Æ�QEó, Ç |e 9�Eó�ãy|b 9ÝEó �QEót|×ðó,

.hãy = log xÝ%�, ©�N×Á2ýt|.�log b, -�ÿÕy = logb xÝ%��b > 1, .log b > 0, h.� Ñ, b < 1, h.� ��&Æ�×°%�Aì�

-

6

-

6

O O1 1x x

yy

1<b<e

b=e

b>e

1e<b<1 b= 1

e

0<b< 1e

(a) b > 1 (b) b < 1

%2.2. y = logb x �%�

�QEóêÌNapierian logarithm, Î ÝSF¸Ýs�ïNaiper(1550-1617, �ÎÍ¼õó.�, Á}Merchiston2]Ý2x), �3�-1614O, ��Ï×ÍEóÝóÂ�(�Ý�20OÝ�G), vÎ|e 9Ý�QEó�b`º|ln x¼©½�î�QEó, �Ä.ó.�ÂÕÝEó, ;ðÎ�QEó, X|´9Ý`¡,

µ|log x�î�QEó, ¬vð6�“�Q”ÞC, �©ÌEó�Íg¼:A¢¿àEóÐó¼O�5�.D log x = 1/x, x > 0, Æb

(2.21)

∫1

xdx = log x + C�

Í;×��&Æ-¼�, tÝ−1�²Ýb§ón, xnÝ�5/��bÝ(2.21)P, EXbb§ón, xnÝ��5/á¼Ý��yn� b§óÝ�µ, &Æ}¡�D¡�ã(2.21)PñÇ�ÿ,

268 Ï"a ø÷Ðó

uf =��, J

(2.22)

∫f ′(x)

f(x)dx = log f(x) + C�

Q, ¯ñÑÊÆÕ.Eó©�L3Ñó, .hîP©Ef(x) > 0�b[�uf(x)� ÑÞA¢?

�:9ìÝD¡�´�ux 6= 0, J∫ |x|

1

1

tdt = log |x|�

¨², ux > 0,d

dxlog |x| = d

dxlog x =

1

x,

ux < 0,d

dxlog |x| = d

dxlog(−x) =

−1

−x=

1

x�Çÿ

(2.23)d

dxlog |x| = 1

x�.h�¡x ÑT�,

(2.24)

∫1

xdx = log |x|+ C,

ãhÇÿ©�f(x) 6= 0, vf3x =��,

(2.25)

∫f ′(x)

f(x)dx = log |f(x)|+ C�

Ah×¼, ÇÞ(2.21)PC(2.22)P, U"Õ(2.24)PC(2.25)P�Q3¿à(2.24)P(T(2.25)P)O��5`, �5 ��âx =

0(Tf(x) = 0)�

»»»2.1.�O∫

tan xdx��.�×¿:�£×ÐóÝ0ó�ytan x, E�.ï¬��|��Ä, u¶Wtan x = sin x/ cos x, v¥�Õ(cos x)′ = − sin x, J

∫tan xdx =

∫ −f ′(x)

f(x)dx,

5.2 Eó 269

Í�f(x) = cos x�Æã(2.25)Pÿ

∫tan xdx = − log | cos x|+ C�

îPE¸cos x 6= 0�x/Wñ�

»»»2.2.�O∫

sec xdx��.�u = sec x + tan x, J

du = (sec x tan x + sec2 x)dx = sec x(sec x + tan x)dx = u sec xdx�

.h

∫sec xdx =

∫1

udu = log |u|+ C = log | sec x + tan x|+ C�

Q¯��º- î�®°«{Î3/�n, ÍJA¢�:��u = sec x + tan x�9ì&Æèº¨×´�QÝ®°��u =

sin x, Jdu = cos xdx, v

∫sec xdx =

∫cos x

cos2 xdx =

∫cos x

1− sin2 xdx =

∫1

1− u2du

=1

2

∫(

1

1− u+

1

1 + u)du =

1

2log

∣∣∣∣1 + u

1− u

∣∣∣∣ + C

=1

2log

1 + sin x

1− sin x+ C =

1

2log

(1 + sin x)2

cos2 x+ C

= log

∣∣∣∣1 + sin x

cos x

∣∣∣∣+C=log | sec x + tan x|+ C�

»»»2.3.�O∫

x/(x2 − 1)dx��.

∫x

(x2 − 1)dx =

1

2

∫2x

x2 − 1dx =

1

2log |x2 − 1|+ C�

270 Ï"a ø÷Ðó

»»»2.4.�O∫ −1

−3x2/(3x− 1)dx�

��.�u = 3x− 1, Jdu = 3dx, vx = (u + 1)/3�.h∫ −1

−3

x2

3x− 1dx =

1

27

∫ −4

−10

(u + 1)2

udu =

1

27

∫ −4

−10

(u + 2 +1

u)du

=1

27(1

2u2 + 2u + log |u|)

∣∣∣−4

−10= − 1

27(30 + log

5

2)�

»»»2.5.�O∫

log xdx��.¿à5I�5ÿ

∫log xdx = x log x−

∫xd log x = x log x−

∫x · 1

xdx

= x log x−∫

1dx = x log x− x + C�

»»»2.6.�O∫

sin(log x)dx��.¿à5I�5ÿ∫

sin(log x)dx = x sin(log x)−∫

xd(sin(log x))

= x sin(log x)−∫

x cos(log x) · 1

xdx = x sin(log x)−

∫cos(log x)dx

= x sin(log x)− x cos(log x)−∫

sin(log x)dx,

Í�t¡×�PWñ, ÎE∫

cos(log x)dx��×g5I�5�Þ

∫sin(log x)dxÉÕ¼�)¿, ÿ

∫sin(log x)dx =

1

2x sin(log x)− 1

2x cos(log x) + C�

Ahôÿ(.ÅóÏÞ�r��b×∫

cos(log x)dx)∫

cos(log x)dx =1

2x sin(log x) +

1

2x cos(log x) + C�

!ñ×è, 3&ÆÞÞ∫

sin(log x)dx)¿, ¢ºÿÕðóC? Íæ. ÞD0ó��º-×ðó, ��8��

5.2 Eó 271

t¡¼:, A¢¿àEó¼�;�5Ý�Õ�h Johann

Bernoulli3�-1697OXs"�¼Ý�´�ã(2.25)Pÿ

(2.26)d

dxlog |f(x)| = f ′(x)

f(x)�

�g(x) = log |f(x)|, ug′(x)´f ′(x)�|O, J(2.26)P0l

(2.27) f ′(x) = g′(x)f(x)�

h°©½Îf ×°��ÝÐó�¶�`, t bà�9ì�¿Í»��

»»»2.7.�OD log√

(x2 + 1)3/(x2 − 1)��.��;Eóÿ

log

√(x2 + 1)3

x2 − 1=

3

2log(x2 + 1)− 1

2log(x2 − 1)�

.h

D log

√(x2 + 1)3

x2 − 1=

3

2

2x

x2 + 1− 1

2

2x

x2 − 1=

3x

x2 + 1− x

x2 − 1

=2x3 − 4x

x4 − 1 �

»»»2.8.�f(x) = (x + cos x)3(x2 + sin x)−4, �Of ′(x)��.�g(x) = log |f(x)|, J

g(x) = 3 log |x + cos x| − 4 log |x2 + sin x|�.h

g′(x) =f ′(x)

f(x)=

3(1− sin x)

x + cos x− 4(2x + cos x)

x2 + sin x �Æ

f ′(x) = 3(1− sin x)(x + cos x)2(x2 + sin x)−4

−4(2x + cos x)(x + cos x)3(x2 + sin x)−5�

272 Ï"a ø÷Ðó

êêê ÞÞÞ 5.2

1. �0ì�&ÐóÝ%��(i) y = log(−x), x < 0, (ii) y = log |x|, x 6= 0,

(iii) y = log(1 + x), x > −1, (iv) y = log(1− x), x < 1,

(v) y = log x/x, x > 0�2. �Jlogb x = (logb a)(loga x) = loga x/ loga b�

3. �5½|EóÐó�î

(i)∫ x

−11tdt, x < 0, (ii)

∫ x

01

1+tdt, x > 0�

4. �JE∀m ≥ 2,

1

2+

1

3+ · · ·+ 1

m< log m < 1 +

1

2+ · · ·+ 1

m− 1�

5. (i) �O��log x = c +∫ x

et−1dt, ∀x > 0, �ðóc�

(ii) �f(x) = log((1 + x)/(1− x)), −1 < x < 1�ua, b Þðóvab 6= −1, �O��f(x) = f(a) + f(b) �Xbx�

6. �Jlog u + u = 0ªb×@ó��

7. ��ì�&]�P�(i) log(1 + x) = log(1− x),

(ii) 2 log x = x log 2,

(iii) log(1 + x) = 1 + log(1− x),

(iv) log(√

x +√

x + 1) = 1�8. �J

(i) 1− x−1 < log x < x− 1, ∀x > 0, x 6= 1;

(ii) x/(1 + x) < log(1 + x) < x, ∀x > 0,

(iii) x− x2/2 < log(1 + x) < x, ∀x > 0;

(iv)∑2n

i=1(−1)i+1xi/i < log(1 + x)

<∑2n+1

i=1 (−1)i+1xi/i, ∀x > 0�

êÞ 273

9. �Oì�&Ðó�0ó�(i) y = log

√3 + 2x2, (ii) y = log(1 +

√x + 1)−2,

(iii) y = log(log x), (iv) y = logx e,

(v) y = log(x2 log x3), (vi) y = (x3 + 4x)5(2x2 + cos x)−2,

(vii) y = x sin(log x)− x2 cos(log x),

(viii) y = (x3 − 1)4(√

x + 1)3/(x2 + 1)1/3�10. �Oì�&�5�

(i)∫

12+3x

dx, (ii)∫ 3

0x

2x2+3dx,

(iii)∫

1x log x

dx, (iv)∫

log xx√

1+log xdx,

(v)∫

log xx

dx, (vi)∫

1x log x log(log x)

dx,

(vii)∫

log2 xdx, (viii)∫ 1−e2

0log(1−t)

1−tdt,

(ix)∫

x log2 xdx, (x)∫

xn log(ax)dx,

(xi)∫

x2 log2 xdx, (xii)∫

cot xdx�11. �Jì�L]2P, Í�m 6= −1�

∫xm logn xdx =

xm+1 logn x

m + 1− n

m + 1

∫xm logn−1 xdx,

¬¿àh2P, O�∫

x3 log3 xdx�

12. �5½¿à1Ä¾!JCÏ8Þ(ii), J�

limx→0+

log(1 + x)

x= 1�

13. 'n, r ÑJó��O

limr→∞

(1

r + 1+

1

r + 2+ · · ·+ 1

r + nr)�

14. (i) ¿à»2.5, �JE∀n ≥ 2,

log 1 + log 2 + · · ·+ log(n− 1)

< n log n− n + 1 < log 2 + log 3 + · · ·+ log n;

274 Ï"a ø÷Ðó

(ii) ¿à(i), �J

e(n + 1

e)n+1 > n! > e(

n

e)n�

37.3;�êÞ, &Æº��n!�£��

15. '=�Ðóf(x)�L3x > 0,��E∀x, y > 0,∫ xy

xf(t)dt�xP

n��f(2) = 2, �OÐóA(x) =∫ x

1f(t)dt, x > 0�

16. '=�Ðóf(x)�L3x > 0, ��∫ xy

1

f(t)dt = y

∫ x

1

f(t)dt + x

∫ y

1

f(t)dt, ∀x, y > 0�

�f(1) = 3, �Of(x), x > 0�

17. �f(x) = (log(x + 1)− log x)/ log2 x, x ≥ 2��Jf ×�3Ðó, vlimx→∞ f(x) = 0�

18. �JÄ�D3Þ94Pf(x)Cg(x), ¸ÿ

log x = f(x)/g(x), ∀x > 0�

19. ¿àlog2 3.= 1.58, ó.hû°CíÂ�§, �JE∀n ≥ 3,

1

2+

1

3+ · · ·+ 1

n>

1

2log2 n�

5.3 ¼¼¼óóóÐÐÐóóó

3î×;&Æ¼�, log x ×�}��¦ÝÐó, v��limx→0+ log x = −∞, limx→∞ log x = ∞�ÆE∀y ∈ R, ªD3×x > 0, ¸ÿlog x = y�Çlog x ×ã(0,∞)Ì�R�1 − 1, vÌWÝÐó�.hy = log x�DÐóD3, &Æ|x = E(y)��Ð

5.3 ¼óÐó 275

óE��L½ R, Â½ (0,∞), v) =�C�}�¦, �Ï×a�§6.7)�.y = log x�x = E(y), hÞPX��x�y �n;Î×

øÝ(hÞPÌ ��), ÇkáE(y)�Â ¢, &Æm�0ÕEóÂ y�x, JE(y) = x�ê.EóÐó 1 − 1Ðó, vÍÂ½ R, Æ∀y ∈ R, log x = yªb×��¨EN×b§óα,

ã(2.7)C(2.8)P, ÿ

log eα = α log e = α�

Æ

E(α) = eα�

ÇEN×b§óα, E(α)�Â eÝαg]�uα = m/n,Í�m,n ÞJóvn 6= 0,JeαÝ�L n

√em�uα� b§ó�A¢? .Ð

óEÝ�L½ @ó/), �&ÆêG©Eb§ó, ��ÐóE�Â�uα P§ó, ×Ít�QÝ�LeαÂÝð°Î�

(3.1) eα = E(α),

ê.E ×=�Ðó, ÆE�×b§ó�{αn, n ≥ 1}, ©�limn→∞αn = α, J

(3.2) eα = E(α) = limn→∞

E(αn) = limn→∞

eαn�

Çeα {eαn , n ≥ 1}�Á§, hèº×α P§ó`, Oeα�M»�h¡E�×@óα, &Æ-ÞE(α)¶Weα��ex, x ∈ R, -Ì ¼óÐó, h ×�L3@ó, ��}�¦v=�ÝÐó, ¬ãÑÂ�b` Ý¸P��Î£�{Ñ, ôÞex¶Wexp{x}�.3x-y¿«î, Þ! DÐó�Ðó%�, EÌyàay = x�Æã%2.1Çÿì�¼óÐó�%��Ä}¡&Æºà#��Í%��

276 Ï"a ø÷Ðó

-

6

−2 −1 O 1 2x

123

e

y

%3.1. ¼óÐó�%�

ãyy = log x�x = ey ��, .hìPWñ:

(3.3) elog x = x,

Çlog x ��ey = x��y�u�b¼óÐó, �àh¼�L9 e�Eó�D�, ùb

(3.4) log ey = y,

Çey ��Í�QEó y�ó, ôµÎlog x = y �x�ã��5ÝÌF, �|×��ÝÐóy = 1/x��5, ¼�L

�QEó, Q¡|�QEóÝDÐó, ¼�LeÝg], Î´�|Ý�à9Ë]P, Ðólog xCex��PC=�P, -ñÇWñ, ��m©½ÝJ��Ey

e = limn→∞

(1 +1

n)n,

�b×´×�Ý�P, ÇÞex�î ì�Á§

(3.5) ex = limn→∞

(1 +x

n)n�

h ¨×OexÝM»�kJîP, ©��J�ó�{sn, n ≥ 1}�Á§ xÇ�, Í�

sn = log(1 +x

n)n�

5.3 ¼óÐó 277

h.un →∞`, sn → x, J.ex ×=�Ðó, Æ

esn → ex,

�êã(3.3)P,

esn = (1 +x

n)n,

Æ(3.5)PWñ�¨J�limn→∞ sn = x�ûïî;log e = 1�J�, ´�b

sn = n log(1 +x

n) = n

∫ 1+x/n

1

1

udu�

�ã�5�íÂ�§á, D3×ξn ∈ (1, 1 + x/n), ¸ÿ

sn = n1

ξn

(1 + x/n− 1) =x

ξn�

�n →∞`, ξn → 1, ÆÿJlimn→∞ sn = x, .h(3.5)PWñ�Íg&Æ¼:��×ÑóÝg], A¢�L�ÕêG c, &Æ

©áb§g]Ý�L�@�'x > 0, vα ×b§ó�ã(2.7)Pá

log xα = α log x�

îPê�;¶

(3.6) xα = eα log x�

�hP��, EN×@óα/b�L�.hE∀x > 0Cα ∈ R, &Æ|(3.7)P¼�Lxα:

(3.7) xα = eα log x, x > 0, α ∈ R�

ãh�LôñÇ:�, E∀α ∈ R, f(x) = xα, x > 0, ×=�Ðó,

vE�×Ñóa, C�×@ób, ab = eb log ab�LÝ�×ÑóÝg], & b§ó`, �A¢�LÝ®Þô�XÝ, ¬QÎàEó¼�L�

278 Ï"a ø÷Ðó

ã(3.7)Pê¿à(3.4)P, Çÿ(2.7)P�×.Â:

(3.8) log xα = α log x,∀x > 0, α ∈ R�

�x = eβ, Çlog x = β, J(3.7)PW

(3.9) (eβ)α = eαβ,∀α, β ∈ R�

?×�Ý�� , E∀x > 0,

(xα)β = (eα log x)β = eαβ log x = (elog x)αβ = xαβ,

Ç

(3.10) (xα)β = xαβ,∀x > 0, α, β ∈ R�

¨², ùb¶�2P:

(3.11) xαxβ = xα+β,∀x > 0, α, β ∈ R�

kJîP, ©��J�

(3.12) log(xαxβ) = log(xα+β)

Ç��¿à�§2.1�(iii)C(3.8)Pÿ

log(xαxβ) = log xα + log xβ = α log x + β log x

= (α + β) log x = log(xα+β)�

Æ(3.12)PÿJ�¿à¼óCEó, �O�×°&Æ|GÕ��Ý�5C�5, �

ìÞ»�

»»»3.1.3Ïëa»2.1&Æ��EN×� −1�b§óα, ÿÕ

(3.13)

∫ b

a

uαdu =bα+1 − aα+1

α + 1 �

5.3 ¼óÐó 279

�yα = −1, î×;�D¡ÄÝ, -0lEó�α P§ó`, u�O��5

φ(x) =

∫ x

1

uαdu, x > 0

(¥�uα ×u�=�ÐóÆ��), J-�ÿÕuα3�×��â0� [a, b]îÝ��5��'x > 1�ã(3.7)P

uα = eα log u,

Í�E∀u ∈ [1, x], log u ≥ 0�'β, γ Þ/��y−1�b§ó, v��

β ≤ α ≤ γ�9Ëβ, γbP§9Í, &Æ�ã×E�J

β log u ≤ α log u ≤ γ log u�

.¼óÐó ¦Ðó, îPê0l

eβ log u ≤ eα log u ≤ eγ log u,

ùÇuβ ≤ uα ≤ uγ�

.h ∫ x

1

uβdu ≤ φ(x) ≤∫ x

1

uγdu,

�ã(3.13)P, îP-0l

(3.14)1

β + 1(xβ+1 − 1) ≤ φ(x) ≤ 1

γ + 1(xγ+1 − 1)�

¨ãb§ó�{βn, n ≥ 1}C{γn, n ≥ 1}, ��limn→∞ βn = limn→∞γn = α�Jã¼óÐó�=�P, ÿ

xβn+1 = e(βn+1) log x, � xγn+1 = e(γn+1) log x

280 Ï"a ø÷Ðó

/��e(α+1) log x = xα+1�.hã(3.14)P, Cô^�§ÿ

φ(x) =1

α + 1(xα+1 − 1)�

u0 < x ≤ 1ù!§�J�ÇJ�E∀a, b > 0, C@óα 6= −1,

(3.15)

∫ b

a

uαdu =1

α + 1(bα+1 − aα+1)�

êα > 0 `, îP�.Â�Ea, b �Þ@ó/Wñ�

»»»3.2.3ÏÞa»7.12, &Æ�ÿE�×b§óα,

(3.16)d

dxxα = αxα−1�

3h&Æ�Þî�2P, .Â��@óα�´�ã(3.15)Pÿ

∫ x

a

uβdu =1

β + 1(xβ+1 − aβ+1), β 6= −1, a, x > 0�

�ã��5ÃÍ�§, ÿ

xβ =d

dx(

1

β + 1(xβ+1 − aβ+1)) =

1

β + 1

d

dxxβ+1�

ãhÇÿ(|αã�β + 1), E∀α 6= 0(Çβ 6= −1),

d

dxxα = αxα−1�

ê.α = 0`, xα = 1, ÆîPEα = 0ùWñ�

ãîÞ»á, E�×@óα,

(3.17) (xα)′ = αxα−1�

¬

(3.18)

∫xαdx =

1

α + 1xα+1 + C

5.3 ¼óÐó 281

©Eα 6= −1�Wñ�Exα, ©�α 6= 0, �5º¸g]K1, �5º¸g]91�h.α = 0`, xα = x0 = 1, �1Ý�5 0, ¬&x−1�Æ

∫x−1dx �ºÎxÝØ×g], ô^b�×xÝg]��

5 x−1�5?blog xÞh�µ�R¼,

∫x−1dx = log |x| + C��(log x)′

= x−1�EóÐóÝ�¨, �¬Þh�5C�5Ýþý�î, Í�9YúÝP², ?ð�&Æ/æPM�

Íg&Æ�×��§2.1¿�Ý�§�

��§§§3.1.¼óÐóbì�P²:

(i) e0 = 1, e1 = e;

(ii) (ex)′ = ex, ∀x ∈ R;

(iii) ea+b = eaeb, ∀a, b ∈ R�JJJ��.(i) .log 1 = 0vlog e = 1, Æ(i)Wñ�gJ(ii)�.y = ex�x = log y! DÐó, Æ

dex

dx=

dy

dx=

1

dx/dy=

1

d log y/dy=

1

1/y= y = ex,

ÿJ�t¡3(3.11)P�, ãx = e, α = a, β = b, ÇÿJ(iii)Wñ�

î�§¼�,

(3.19) (ex)′ = ex�

9Î¼óÐót©�Ý×ÍP², Ç¸Ý0ó) ¸Í��

»»»3.3.3îÞ», &ÆBããÁ§C��5ÃÍ�§, ÿÕ(xα)′ =

αxα−1�9ì:, A¢¿à¼óÐó��5=Å!J, à#ÿh��G«�ÿ

xα = eα log x, x > 0, α ∈ R�

282 Ï"a ø÷Ðó

u�φ(x) = log x, ψ(u) = αu, g(y) = ey, Jxα = g(ψ(φ(x))�ê

φ′(x) =1

x, ψ′(u) = α, g′(y) = ey,

Æ

d

dx(xα) = g′(y)ψ′(u)φ′(x) = ey · α · 1

x

= eα log x · α · 1

x= α

xα

x�

.h

(3.20)d

dx(xα) = αxα−1�

h°�Q´îÞ»�, 6|Þb§ó�¼¿�α�|9Ý�bÝ(3.20)P, �ã��5ÃÍ�§, Çÿ

(3.21)

∫xαdx =

xα+1

α + 1+ C, α 6= −1�

9ø�Ý�ÿÕ(3.20)P, �ÿÕ(3.21)P�Ä�, ��»3.1C3.2,

�ÿ(3.21)P�ÿ(3.20)PÝÄ�f´, h��K�

Íg&Æ¼:|��×Ñó 9Ý¼óÐó�

»»»3.4.&Æ��Lex, ∀x ∈ R, ô�LÝxα = eα log x, ∀x > 0, α ∈R��y9�ÎeÝ¼óÐó, ô�p�L�E∀a > 0, ×Í�LaxÝ]°Î�ax = y, Í�y��

(3.22) loga y = x�

�Äh]PEa = 1µ�ÊàÝ, .&Æ¬��L9Î1ÝEó�¨×Ë]PÎfï(3.7)P, �

(3.23) ax = ex log a, x ∈ R�

5.3 ¼óÐó 283

&Æ´�K9Ë]P, ×]«Î.h°E∀a > 0/Êà; vê�ñÇ:�ax, x ∈ R, ×=�Ðó; ¨×]«Î2àh°, EJ�ì�¼óÝP²´�|:

log ax = x log a,(3.24)

(ab)x = axbx,(3.25)

axay = ax+y,(3.26)

(ax)y = (ay)x = axy,(3.27)

y = ax, uv°ux = loga y,Í� a 6= 1�(3.28)

î�"Í��J�º3êÞ�¿à=Å!J, &Æ�ÿ�§3.1

�×.¡�

��§§§3.1.E∀a > 0,

(3.29)dax

dx= ax log a�

JJJ��.�f(x) = ax, ¿à(3.23)P, f�;¶

f(x) = ex log a�

Æã=Å!J, ÿ

f ′(x) = ex log a d

dx(x log a) = ex log a log a = ax log a�

�§3.1¯@î´�§3.1�(ii)×��.log e = 1, Æa = e`,

Çÿ(3.19)P�ê(3.29)P¼�, E��×a > 0, y = ax �0ó, �¸Í�WÑf�¨², a 6= 1, u|(3.23)P¼�Lax, Jù�A�§3.1�(ii)�J°, v¿à(2.20)P, ÿax�0óAì�

dax

dx=

1

d loga y/dy= y log a = ax log a�

284 Ï"a ø÷Ðó

ã(3.19)C(3.29)P, ¿à��5ÃÍ�§, Çÿì��52P�∫

exdx = ex + C,(3.30)∫

axdx =ax

log a+ C, a > 0 v a 6= 1�(3.31)

¿à�ó�ð, îÞPê0l?×�Ýny¼óÝ�52P�ÇEN×=��ÝÐóf ,

∫ef(x)f ′(x)dx = ef(x) + C,(3.32)

∫af(x)f ′(x)dx =

af(x)

log a+ C, a > 0, a 6= 1�(3.33)

»»»3.5.�5½Of(x) = xxCg(x) = xxx�0ó�

��.´�

log f(x) = log xx = x log x�ÞîPË�5½Ex�5, ÿ

f ′(x)

f(x)= log x + x · 1

x= 1 + log x,

Æ

f ′(x) = f(x)(1 + log x) = xx(1 + log x)�!§,

log g(x) = log xxx

= xx log x,

.h

g′(x)

g(x)= (xx)′ log x + xx · 1

x= xx(1 + log x) log x + xx−1�

Æ

g′(x) = xxx

(xx(1 + log x) log x + xx−1)�

5.3 ¼óÐó 285

Eyî», b¿�6©½º�Ý�´�'bÞ��ÝÐóξCη,

CÞðóa > 0, b�J

d

dx(η(x))b = (η(x))b−1η′(x), ã(3.17)P,

d

dxaξ(x) = aξ(x)ξ′(x) log a, ã(3.29) P�

¬xx¬�òyîËvÐó�×, �Îb(η(x))ξ(x)Ý�P, Ç9�¼ó/ Ðó�»3.5-èºÝ×O9v�PÝÐó�0óÝ]°�¨², xxxÝºÕ��A¢? ¯@î

xxx

= x(xx),

ôµÎ¼óI5ãî?ìÕ�×��

x(xx) 6= (xx)x�

îP��yxx2, �xxx¬�8!�x = 3 `, îP¼� 327, �

� 39, Þï��yx = 1T2`, îP¼��8��

»»»3.6.�O∫

x2ex3dx�

��.�u = x3, Jdu = 3x2dx�.h∫

x2ex3

dx =1

3

∫eudu =

1

3eu + C =

1

3ex3

+ C�

»»»3.7.�O∫

cos xe2 sin xdx��.�u = 2 sin x, Jdu = 2 cos xdx�.h

∫cos xe2 sin xdx =

1

2

∫eudu =

1

2eu + C =

1

2e2 sin x + C�

»»»3.8.�O∫

2√

x√xdx�

��.�u =√

x, Jdu = 12x−1/2dx�.h

∫2√

x

√x

dx = 2

∫2udu = 2

2u

log 2+ C =

2√

x+1

log 2+ C�

286 Ï"a ø÷Ðó

»»»3.9.�O∫

ex sin xdx��.¿à5I�5,

∫ex sin xdx = −

∫exd cos x = −ex cos x +

∫cos xdex(3.34)

= −ex cos x +

∫ex cos xdx + C�

��×Å∫

ex cos xdx, �ÿ

(3.35)

∫ex cos xdx = ex sin x−

∫ex sin xdx + C�

Þh(3.35)P�á(3.34)P, ÿ∫

ex sin xdx = −ex cos x + ex sin x−∫

ex sin xdx + C,

ãhÇ��∫

ex sin xdx =1

2ex sin x− 1

2ex cos x + C�

�Ä�¥�ÝÎ, î�£°ðóC, ��¬&!øÝðó�¬;ð Ý��, &Æ�©½ 5C1, C2, · · ·�

»»»3.10.�O∫

11+ex dx�

��.y:�ì, Í»«P��R��Aì2;¶�5Õ�:

1

1 + ex=

e−x

e−x + 1�

u�u = e−x + 1, Jdu = −e−xdx, .h∫

e−x

e−x + 1dx = −

∫1

udu = − log |u|+ C = − log(1 + e−x) + C�

3t¡×�P&Æ6��EÂ, h.1 + e−x Ñ�Íg�Þ�5Õ�;¶

1

1 + ex= 1− ex

1 + ex,

5.3 ¼óÐó 287

u�u = 1 + ex, J∫

1

1 + exdx = x−

∫ex

1 + exdx = x−

∫1

udu

= x− log |u|+ C = x− log(1 + ex) + C�

ËgXÿÝ�n:R¼«{�!, ¯@î©�BÄÊÝ;�, Þï-8!Ý�Ç

− log(1 + e−x) = log1

1 + e−x= log

ex

ex + 1= log ex − log(1 + ex) = x− log(1 + ex)�

Í;&Æ|EóÐóÝDÐó¼�L¼ó, Ç¸(3.5)PÝWñ,

ôÎàÕEóÐóÝP²�b°hJÎ��L¼ó, �|¼óÐóÝDÐó, ¼�LEó��ÆÝ]°, µÎ�SÐrE(x), Í�

E(x) = limn→∞

(1 +x

n)n, x ≥ 0�

àv«J�

e = limn→∞

(1 +1

n)n

Ý]°, J�E∀x > 0, ó�{an(x), n ≥ 1} ��vb&, Í�

an(x) = (1 +x

n)n�

.hlimn→∞ an(x)D3, �E(x)ôµb�LÝ�ux < 0, J�L

E(x) =1

E(−x)�

Ah×¼, E∀x ∈ R, E(x)/b�@Ý�L, hÐó-Ì ¼óÐó�#ì¼-�J�EN×b§ór,

(3.36) E(r) = er,

Q¡ÎEN×@óùWñ�.h

(3.37) ex = limn→∞

(1 +x

n)n,∀x ∈ R�

288 Ï"a ø÷Ðó

Ç9ø�L�¼Ý¼óÐó, �&Æ�G.0�¼ÝÎ×øÝ�9ì&Æ¼: ¢(3.36)PWñ�¼óÐóÝ�9P²4/�

à#¿à(3.37)P¼O, ¬h�©Î¯��Ý�, b¨×�L¼óÐóÝ]P, &Æ¬��H9�G3ny9Í]PÝD¡�G«�èÄ, 3��5�, ��LEó��L¼ó, Î×Ë´?Ý]P�'p ×ÑJó, J

((1 +p−1

n)n)p = (1 +

1

pn)np�

�n →∞, J.pn →∞, ÆîP0l

(E(p−1))p = e,

Ë\�pg], ÿE(p−1) = e1/p�

Çÿn →∞`,

(3.38) (1 +1

pn)n → e1/p�

�'b×ÑJóq, J

(3.39) ((1 +1

pn)n)q = (1 +

1

pn)nq�

�m = nq,JîP��ê�y(1+q/(pm))m��n →∞,Jm →∞,

.h(3.39)P0l(v¿à(3.38)P)

(e1/p)q = E(q/p)�

ÆEN×Ñb§ór, &ÆJ�ÝE(r) = er�ur = 0, J

E(0) = 1 = e0�

ur < 0, J

E(r) =1

E(−r)=

1

e−r= er�

5.3 ¼óÐó 289

ÆÿJ(3.36)PEN×b§órWñ�t¡&Æ¼:¼óÐó�%��

»»»3.11.�0y = ex, x ∈ R, �%��.´�

limn→∞

en = ∞,

Ælimx→∞ ex = ∞, limn→−∞ ex = limx→∞(1/ex) = 0, .hx � ×i¿��a�êy′ = ex > 0, Æ%� �¦, PÛ&F�êy′′ = ex > 0, Æ

ùPD`F�x = 0 `, y = 1, x = 1 `, y = e, �ÿy�%�A%3.1�

»»»3.12.�0y = e−x2, x ∈ R, �%��

��.h ×�Ðólimx→∞ e−x2= limx→−∞ e−x2

= 0, Æx � i¿��a�êy′ = −2xe−x2

, y′′ = −2e−x2+ 4x2e−x2

= 4e−x2(x2 − 1/2)�5

½�y′ = 0Cy′′ = 0, ÿ3x = 0bÛ&F, 3x = 1/√

2T−1/√

2��bD`F�B»�PÝ®°, ÿ3x = 0bÁ�Â1, 3x =

1/√

2C−1/√

2bD`F�y�%�A%3.2�

-

6

xO

y

y = e−x2

%3.2. y = e−x2�%�

3Tàó.�, ©½Î3^£¡TÙ�.�, y = e−x2 ×¥�ÝÐó, &Æ|¡º�D¡�\ï��Ï°a%2.12f´�Þï�%�«{bF, ¬37.5;, &Æº¼�¯@îÞï b×��Ý-²�

290 Ï"a ø÷Ðó

êêê ÞÞÞ 5.3

1. �Oì�&ÐóÝ0ó�(i) f(x) = e

√x, (ii) f(x) = 2x2

,

(iii) f(x) = ecos2 x, (iv) f(x) = eex,

(v) f(x) = log(ex + 1), (vi) f(x) = (1 + e−3x2)3,

(vii) f(x) = log√

ex + 1, (viii) f(x) = ex(e2x − x)−2�

2. �Oì�&�5�(i)

∫xexdx, (ii)

∫x2e−xdx,

(iii)∫

xe−xdx, (iv)∫

e√

xdx,

(v)∫

x3e−x2dx, (vi)

∫ex/(1− ex)dx,

(vii)∫

ex+exdx, (vii)

∫(ex + e−x + 2)dx,

(ix)∫

log xx

dx, (x)∫

1x log x log(log x)

dx�

3. �Oì�&Ðó�0ó�(i) f(x) = (log x)x/xlog x, (ii) f(x) = xlog x,

(vi) f(x) = log(ex +√

1 + e2x), (iv) f(x) = (log x)x,

(v) f(x) = log(log(log x)), (iii) f(x) = x1/x,

(vii) f(x) = (sin x)cos x + (cos x)sin x, (v) f(x) = axa+ aax

�

4. �0ì�&Ðó�%��(i) f(x) = x log x, x > 0, (ii) f(x) = x2e−x,

(iii) f(x) = eex, (vi)f(x) = (e2x − 1)/(e2x + 1),

(v) f(x) = ee−x, (vi) f(x) = e−ex

,

(vii) f(x) = e−e−x, (viii) f(x) = log log x, x > 1�

5. �J(3.24)�(3.28)�"P�

6. �OÐóf(x) = e−1/x2, x 6= 0, �ÁÂ�

7. �f(x) = 2x/(1 + e2x)��JEN×ÑJón, f (n)(0)/ Jó�

5.4 �QW��<[ 291

8. ¿àíÂ�§, �J

ex ≥ 1 + x, e−x ≥ 1− x, ∀x ≥ 0�ãh, ¿à�5, �JEN×ÑJón, Cx ≥ 0,

ex ≥ 1 + x +x2

2!+ · · ·+ xn

n!,

1− x +x2

2!− x3

3!+ · · ·+ x2n

(2n)!≥ e−x

≥ 1− x +x2

2!− x3

3!+ · · ·+ x2n

(2n)!− x2n+1

(2n + 1)!�

9. 'a, b Þðóv�!` 0��

A =

∫eax cos bxdx, B =

∫eax sin bxdx�

¿à5I�5, �J

aA− bB = eax cos bx + C1,

aB + bA = eax sin bx + C2,

Í�C1, C2 Þðó��A,B, �ÿì��52P�∫eax cos bxdx =

eax(a cos bx + b sin bx)

a2 + b2+ C,

∫eax sin bxdx =

eax(a sin bx− b cos bx)

a2 + b2+ C�

ÛÛÛ.ù�Bã5I�5, 5½ÿÕî�ÞP��

5.4 ��QQQWWW��<<<[[[¼óÐó, ð�à¼à��Q&&9bnW�C<[Ýÿ

P�ãÍ;X�Ý×°»�, �:�¼óÐóÝ¥�P, ChÐó�®ßÎ��QÝ�

292 Ï"a ø÷Ðó

(A) ã×�5]�¼S�¼óÐó×]�P�uâb0ó-Ì ��555]]]��PPP(differential equa-

tion)�&Æ�Aì2ã×��ÝP²¼S�¼óÐó�

��§§§4.1.'Ðóy = f(x)��ì�]�P:

(4.1) y′ = αy,

Í�α ×ðó�J(4.1)P��

(4.2) y = f(x) = ceαx,

Í�c ×ðó�JJJ��.´�E�×ðóc, |�Jy = ceαx��(4.1)P�D�, &Æ6J�9Î°×��(4.1)PÝÐó�'y (4.1)P�×�, �u =

ye−αx�J

u′ = y′e−αx − αye−αx = e−αx(y′ − αy) = 0,

h�àÕãyy (4.1)P�×�, Æy′ = αy�.×Ðó�0óu 0, JhÐó ×ðó�Æu = c, vy = ceαx, Í�c ×ðó�

î��§��, ¼óÐó�×©P: 9Î°×0ó�æÐóWÑf(y′/y = α)�Ðó�×�Ðó�5¡, �æÐóº��!(��94P�ë�Ðó�Eó�), ¬¼óÐó�5¡, �æÐóÃÍîÎ8!Ý(©-×ðó¹)�(4.1)PW ¼óÐó�×¥�ÝP²�

(B) �¿®Þ�'D×��Ýþ3X�v2�¿�>,¿£ NO100α%�u

NO�>×g, JxO¡�Í¿õ

(1 + α)x�

5.4 �QW��<[ 293

uN`�>×g, JxO¡�Í¿õ

(1 + α/12)12x�

uN^�>×g, ×O|365^�, JxO¡�Í¿õ

(1 + α/365)365x�

u×O�>ng, JxO¡�Í¿õ

(1 + α/n)nx�

¯n×à¦�, §¡îÎ��Ý, Ç3�¼�yÝ` /-�¿×g, Jn → ∞`, xO¡�Í¿õ��eαx(�(3.5)P, C¿àux

×u�=�Ðó)�h�x�mÎÑJó, �|Î�×Ñó��\2�¿, �&ÆÝ�ÆÎ¿>ºf×O�>×g¦�&

9��Ü×»��α = 0.1, vD×��Ýþ�u×O�>×g, J×O¡�Í¿õ 1.1�¬u�\�¿, J×O¡�Í¿õ e0.1 .

= 1.1051709, ¬�f1.1�9K�ub×�X��Ì¸O¿£ 9%, ¬N^�>×g, ¨×�X� O¿£ 10%, ¬×O�>×g�JÏ×�X�×O¡ÝÍ¿õ, �ye0.09 .

= 1.0941742 <

1.10�ÇÏÞ�X�, )ÎEDV´b¿�u�BÄ9°f´, ;ð&Æº0| �\�¿, µº¸¿>�»�2¦��b°ßÎÝW�, Í@ôÎv«�\2�¿, .h3` t�ß

Î�ó�f(t), ;ðbì��P

(4.3) f(t) = ceαt,

Í�|�c = f(0)�3` t = 0�ó��4QßÎÝó�TÎJó, ¬ó��`, (4.3)P)Î×Í�?Ý�î°, ©b°�Ý0-��u3` t�ó�, b(4.3)��P, J3` t��;£f ′(t) = αceαt�.h

(4.4) f ′(t) = αf(t),

ùÇ3�×`Ñ��;£, �¨b��f ×ðó�

294 Ï"a ø÷Ðó

¨×]«, 3(A)��:ÕÝ, u(4.4)PWñ, Jf(t) = ceαt, Í�c ×ðó�X|hÞÿPÎ��Ý: �;£�¨b��f ×ðó, CW�Î|�\�¿Ý]P�b°w PÎ², Í<[��;ôÎ�¨b�WÑf, Ç(4.4)P

��α �Â, .h3` t��ùb(4.3)PÝ�P�

»»»4.1.'Øw PÎ²��<� 1600O(ÇB1600O¡Í²�3�)�u×��b150¸, �OtO¡�õ�, C�®BÄ9ò²�W 30 ¸?

��.�f(t)�3` t�²�, Çf(t) = 150eαt�ã�'

f(1600) = 150e1600α = 150/2,

.h1600α = − log 2, Tα = − log 2/1600�Æt O¡

f(t) = 150eαt = 150e−t log 2/1600 = 150(1

2)t/1600�

Íg'f(t) = 30, Çeαt = 1/5�.h

t = − log 5

α=

1600 log 5

log 2.= 3715�

»»»4.2.'X��O¿£ 5%, v�\�¿��®B9ò¡, Í¿õ�¾æ¼Ý3¹?

��.ÇOt�Â��

e0.05t = 3,

Tt = 20 log 3.= 21.9722, ÇV21.97O¡�

»»»4.3.�'ØÔ��3>��>�WÑf�ê'R�>� 1002M/J, vB2J¡>�402M/J��®

(i) 5J¡�Ô��>� ¢?

(ii) �` a, a > 0, ��9KûÒ�

5.4 �QW��<[ 295

��.�v(t)�3` tÔ��>�, Jµ�'dv/dt = −kv, Í�k >

0 ×ðó�.hv(t) = ce−kt�

êv(0) = c = 100, Æ

v(t) = 100e−kt�

êã�'v(2) = 40 = 100e−2k, Æk = 12log 2.5, v

v(t) = 100e−t2

log 2.5 = 100et2

log 25 = 100(

2

5)t/2�

(i) v(5) = 100(2

5)5/2�

(ii) ûÒ =

∫ a

0

v(t)dt =

∫ a

0

100et2

log(2/5)dt

=100

12log(2

5)e

t2

log(2/5)∣∣∣a

0=

200

log(2/5)

((2

5)a/2 − 1

)�

»»»4.4.'ØÞÁ×��b1000Í, vB2�`¡W 8000Í��®Bt�`¡bu�?

��.µÞ�f(t) = 1000eαt,

êf(2) = 1000e2α = 8000, Æ2α = log 8�.h

f(t) = 1000eαt = 1000 · 8t/2�

Ø°]�P�¼óÐón;Û6, &Æ��ì��§�

��§§§4.2.'g [0,∞)î�×=�Ðó, v��

(4.5) g(x + y) = g(x) + g(y), ∀x, y ≥ 0�

J

(4.6) g(x) = g(1)x, ∀x ≥ 0�

296 Ï"a ø÷Ðó

JJJ��.'D3×x0 > 0, ¸ÿg(x0) 6= g(1)x0��

h(x) = g(x)− g(1)x,

Jhù��h(x + y) = h(x) + h(y), ∀x, y ≥ 0,

vh(x0) 6= 0�êãh(nx) = nh(x), Í�n ×ÑJó, �ÿ

limn→∞

|h(nx0)| = ∞�¬hÛ��, .�Aì2J�h ×b&Ðó�ãyh(1) = 0, Æh(x + 1) = h(x), ∀x ≥ 0, Çh ×ø�Ð

ó��hê [0, 1]î�×=�Ðó, .h3[0, 1]�Ä b&�ø�Pê0lh3[0,∞) b&�hë;0lx0�D3, Ç(4.6)PWñ�J±�

9ì î��§�×ñÇÝ.¡�

��§§§4.1.'f [0,∞)î�×=�Ðó, v��

(4.7) f(x + y) = f(x)f(y), ∀x, y ≥ 0�

JE∀x ≥ 0, f(x) = 0, T

f(x) = fx(1)�

3�§4.1�, &Æ¬Î�'f��êÞ�b×ÞÎ3f��Ý�'ì, �(4.7)P, ��f´Í�°�

êêê ÞÞÞ 5.4

1. 'f ×�L3Rî�Ðó, v��

f(x + y) = f(x)f(y), ∀x, y ∈ R�

êÞ 297

(i) �Jf(0) = 0T1, vuf(0) 6= 0, Jf(x) 6= 0, ∀x ∈ R;

(ii)�Juf ′(x)D3, ∀x ∈ R,Jf ′(x)f(y) = f ′(y)f(x), ∀x, y ∈R; D3×ðóc, ¸ÿf ′(x) = cf(x), ∀x ∈ R, uf(0) 6= 0,

Jf(x) = ecx�

2. 'f ×ãÑÂ�=�Ðó, v��

f(√

x2 + y2) = f(x)f(y), ∀x, y ∈ R�

�Jf(x) = fx2(1)�

3. 'Øw PÎ²��<� 1600O��® â9ò¡, Í²�W æ¼�1/3?

4. 'Øw PÎ²3K1/10m10O��®m9òhÎ²W æ¼�1/5?

5. 'ØÞÁuÇò3Fº¦�×¹��®9ò�¦�50 ¹?

6. 'ØÞÁuÇò2FÍó��ã5000¦�12000��®�9ò,

Íó��¾100000?

7. 'ØX�|�¿NO�>×g��®O¿£ ¢, �¸D�8O¡, Í¿õ�¹?

8. ØßD�3×N��¿×gÝX�, O¿£ 8%��®¢`ÍD��¹?

9. Øß5½&D×0-3A, BÞ�X��AX�O¿£ 4%,

vN��¿×g, BX�O¿£ 4.125%, vNO�¿×g��®BÄJóO¡, £×�X�´b¿?

298 Ï"a ø÷Ðó

5.5 ¼¼¼óóóCCCEEEóóóÐÐÐóóó��×××MMMDDD¡¡¡

&Æ�:¼óCEóÐóÝ��"P�´�ã

1

1− u= 1 + u + u2 + · · ·+ un−1 +

un

1− u, u 6= 1,

Ë�Eu�5, ã0�x, ÿ

− log(1− x) = x +1

2x2 +

1

3x3 + · · ·+ 1

nxn +

∫ x

0

un

1− udu,

Ç

(5.1) − log(1− x) = Pn(x) + Rn(x), x < 1�

Í�Ïng��94P

Pn(x) = x +1

2x2 +

1

3x3 + · · ·+ 1

nxn,

õ4

Rn(x) =

∫ x

0

un

1− udu�

ì�§�à¼£�õ4Ý��

��§§§5.1.u0 < x < 1, J

(5.2)xn+1

n + 1≤ Rn(x) ≤ 1

1− x

xn+1

n + 1�

ux < 0, JRn(x)�(−1)n+1!r, v

(5.3) 0 < (−1)n+1Rn(x) ≤ |x|n+1

n + 1�

JJJ��.�'0 < x < 1�.Rn(x)Ý�5P� 0 ≤ u ≤ x, .h1 −x ≤ 1− u ≤ 1�Æ�5Õ��

un ≤ un

1− u≤ un

1− x�

5.5 ¼óCEóÐó�×MD¡ 299

5½Eî��P�N×4ã0��x, Çÿ(5.2)P�Íg'x < 0, v�t = −x = |x|�Jt > 0, v

Rn(x) = Rn(−t) =

∫ −t

0

un

1− udu = −

∫ t

0

(−v)n

1 + vdv

= (−1)n+1

∫ t

0

vn

1 + vdv�

.hRn(x)�(−1)n+1!r�ê

(−1)n+1Rn(x) =

∫ t

0

vn

1 + vdv ≤

∫ t

0

vndv =tn+1

n + 1=|x|n+1

n + 1,

ÿJ(5.3)P�

ì��§EOEóÂ, QÃ��

��§§§5.2.'0 < x < 1vm ≥ 1, J

(5.4) log(1 + x

1− x) = 2(x +

x3

3+ · · ·+ x2m−1

2m− 1) + Em(x),

Í�

(5.5)x2m+1

2m + 1≤ Em(x) ≤ 2− x

1− x

x2m+1

2m + 1�

JJJ��.ã(5.1)Pÿ(|−xã�x)

(5.6) − log(1 + x) = Pn(−x) + Rn(−x), x > −1�

u−1 < x < 1, J(5.1)�(5.6)P!`Wñ�ãhÞPÇÿ(¼��&83)

(5.7) log(1 + x

1− x) = Pn(x)− Pn(−x) + Rn(x)−Rn(−x)�

3Pn(x)−Pn(−x)�, x��g]4/��,��g]4W 2¹�.hun �ó, ÉA1n = 2m, J

P2m(x)− P2m(−x) = 2(x +x3

3+ · · ·+ x2m−1

2m− 1),

300 Ï"a ø÷Ðó

v(5.7)PW

(5.8) log(1 + x

1− x) = 2(x +

x3

3+ · · ·+ x2m−1

2m− 1) + Em(x),

Í�

(5.9) Em(x) = R2m(x)−R2m(−x)�(5.8)PE−1 < x < 1Wñ�¨u§×0 < x < 1, Jã�§5.1

(5.10)x2m+1

2m + 1≤ R2m(x) ≤ 1

1− x

x2m+1

2m + 1,

v

(5.11) 0 < −R2m(−x) ≤ x2m+1

2m + 1�

Þ(5.10)�(5.11)Þ��P, ET48�, v¿à

1 +1

1− x=

2− x

1− x,

ÇÿJ(5.5)P�

»»»5.1.�Olog 2��«Â��.ãx = 1/3, m = 2, J(1 + x)/(1− x) = 2�ã(5.4)Pÿ

log 2 = 2(1

3+

1

81) + E2(

1

3),

Í�1

1215=

1

5(1

3)5 ≤ E2(

1

3) ≤ 1

2(1

3)5 =

1

486�.h4©àÕ��Ý�Õ, ��ÿ

0.6921 < log 2 < 0.6935�

»»»5.2.'f(x) = ex�.f (n)(x) = ex, ∀n ≥ 0, vf (n)(0) = e0 = 1,

Æf30�ng��94P

Pn(x) = 1 + x +x2

2!+ · · ·+ xn

n!=

n∑

k=0

xk

k!�


�õ4

Rn(x) =1

n!

∫ x

0

(x− t)netdt�Çex��"P

(5.12) ex = Pn(x) + Rn(x)�u0 < x ≤ c, J1 < et ≤ ec, ∀t ∈ [0, x]�.h

xn+1

(n + 1)!≤ Rn(x) ≤ ec xn+1

(n + 1)!, 0 < x ≤ c�

uãx = c = 1, v¿àe < 3, J

(5.13) e =n∑

k=0

1

k!+ Rn(1),

v1

(n + 1)!< Rn(1) <

3

(n + 1)!�

¿àh��O�e�Â��Þ@�»A, n = 12`, �ÿ

2.718281826 < e < 2.718281833,

��ó7�Þ@�¯@î, 31.3;, &Æ�ãó�ÝÁ§0�

(5.14) e =∞∑i=0

1

i!�

�yA¢ã(5.13)PÿÕ(5.14)P, &Æº3ÏÜa"D�ùó�[eP`�J��

AÏ°a»3.5, ãex��"Pô�à¼£��5Â�

»»»5.3.ãn = 4, Jex��"P

(5.15) ex = 1 + x +x2

2!+

x3

3!+

x4

4!+ R4(x)�

302 Ï"a ø÷Ðó

'x ≤ 0, J3 [−c, 0], e−c ≤ ex ≤ 1, ÆãÏ°a��§3.5,

.m = e−cvM = 1, Æ

0 < (−1)5R4(x) ≤ (−x)5

5!, x < 0�

ùÇux < 0, Jx5

5!≤ R4(x) < 0�

3(5.15)P, |−t2ã�áx, Jÿ

(5.16) e−t2 = 1− t2 +t4

2!− t6

3!+

t8

4!+ R4(−t2),

Í�

−t10

5!≤ R4(−t2) < 0�

u0 ≤ t ≤ 1/2, J

t10

5!≤ (

1

2)10 1

5!< 0.000009�

Þ(5.15)P, ¼��&�ã0�1/2�5, ÿ

∫ 1/2

0

e−t2dt =1

2− 1

3 · 23+

1

5 · 25 · 2!− 1

7 · 27 · 3!+

1

9 · 29 · 4!− θ,

Í�0 < θ ≤ 0.0000045�ãîPÿ∫ 1/2

0

e−t2dt.= 0.4613�

u¿ào-Br, ã|îÝ��ÿ

log(1 + x) = x− x2

2+

x3

3− x4

4+ · · ·+ (−1)n−1xn

n(5.17)

+o(xn), x → 0,

ex = 1 + x +x2

2!+ · · ·+ xn

n!+ o(xn), x → 0�(5.18)


»»»5.4.�J

(5.19) (1 + x)1/x = e(1− x

2+

11x2

24+ o(x2)), x → 0�

JJJ��.´�(1 + x)1/x = elog(1+x)/x�

�ã(5.17)P

log(1 + x) = x− x2

2+

x3

3+ o(x3), x → 0,

.hlog(1 + x)

x= 1− x

2+

x2

3+ o(x2), x → 0�

v

(5.20) (1 + x)1/x = exp(1− x

2+

x2

3+ o(x2)) = e · eu, x → 0,

Í�

(5.21) u = −x

2+

x2

3+ o(x2)�

êu → 0`,

(5.22) eu = 1 + u +u2

2+ o(u2),

Þ(5.21)P�á(5.22)P, ÿ

eu =1− x

2+

x2

3+ o(x2) +

1

2(−x

2+

x2

3+ o(x2))2 + o(x2)(5.23)

=1− x

2+

11x2

24+ o(x2)�

t¡Þ(5.23)P�á(5.20)P��-ÿJ(5.19)P�

9ì&Æ¼:f

limn→∞

(1 +1

n)n = e,

304 Ï"a ø÷Ðó

?×�Ý��&Æ�àÕõ4 Lagrange�P�log(1 + x)Ý��"P, h�¿àÏ°a�(3.25)PÿÕ�Ç

log(1 + x)(5.24)

= x− 1

2x2 + · · ·+ (−1)n−1xn

n+

(−1)n

(n + 1)(1 + ξ)nxn+1, x > −1,

Í�ξ+y0�x� �

��§§§5.3.'bó�{an, n ≥ 1}C{bn, n ≥ 1}, ��limn→∞ an = 0,

vlimn→∞ anbn = c�J

(5.25) limn→∞

(1 + an)bn = ec�

JJJ��..limn→∞ an = 0, ÆD3ÑJón0, ¸ÿn ≥ n0`, |an| ≤1/2�E∀n ≥ n0, ¿à(5.24)Pÿ

log(1 + an) = an − 1

2(1 + ξ)−2a2

n,

Í�ξ�anbn, ��|ξ| ≤ |an| ≤ 1/2�Æn →∞`,

bn log(1 + an) = anbn − 1

2(1 + ξ)−2a2

nbn → c�

ÿJ(5.25)P�

¿à�§5.3, ñÇ�ÿ

limn→∞

(1 +1

n+

1

n2)n = e�

êûî��§�J�, ù�ÿ×ny=��óÝ��

��§§§5.1.'limx→a f(x) = 0, vlimx→a f(x)g(x) = c, Í�a ∈ R ∪{∞,−∞}�J

(5.26) limx→a

(1 + f(x))g(x) = ec�


(5.26)P�×©» (�Ï×a(5.8)P):

limx→∞

(1 +1

x)x = e,

ãhêÿ

(5.27) limx→0

(1 + x)1/x = e�

�(5.19)PÇ ×´(5.27)P?Þ@Ý£�, ��x�#�0`, (1 +

x)1/x �e�fÂÝ�«Â�

»»»5.5.�'|ÆC^ÆC, v'NÍCºÆýÝ^£ p, 0 <

p < 1�|Xn�ÆnÍC¡À�ýÝCó, JXn ¢? &Æá¼Q�×�, Xnb�� 0, 1, · · · , n�ã4�à)�Ý��á, u|P (Xn = k)�Xnº�yk�^£, J

P (Xn = k) =

(n

k

)pk(1− p)n−k, k = 0, 1, · · · , n�

'p��, vn��, ¬np�«yλ, Í�λ > 0, JEN×ü�Ýk,

P (Xn = k)��«Â ¢?

´�ÞP (Xn = k);¶

P (Xn = k) =1

k!

n(n− 1) · · · (n− k + 1)

nk(np)k(1− p)n(1− p)−k�

.n��`(¥�kü�)

n(n− 1) · · · (n− k + 1)

nk

.= 1, (np)k .

= λk,

(1− p)n .= e−λ, (1− p)−k .

= 1,

Æn��`,

P (Xn = k).=

1

k!· 1 · λk · e−λ(1− p)−k =

e−λλk

k! �

9�QÎ×´��ÝEP (Xn = k)��«�î°�

306 Ï"a ø÷Ðó

EóÐóC¼ó��¦T�3, b&9b¶ÝP², 9ì ×»�

»»»5.6.'

f(x) =1

log(1 + 1/x)− x, x ≥ 1�

�Jf ×¦Ðó�JJJ��.�

g(y) =1

log(1 + y)− 1

y, 0 < y ≤ 1,

Í»Ç��yJ�g y��3Ðó�.

g′(y) =−1

(1 + y) log2(1 + y)+

1

y2=

(1 + y) log2(1 + y)− y2

y2(1 + y) log2(1 + y),

vy2(1 + y) log2(1 + y) > 0, Æu�J�

h(y) = (1 + y) log2(1 + y)− y2 < 0,∀0 < y ≤ 1,

Ç��hæ�30¬P�L, ¬&Æ��h(0) = 0, Jh3y ≥ 0 ��(\ï�J�)�¨u�J�

h′(y) = log2(1 + y) + 2 log(1 + y)− 2y < 0, ∀0 < y ≤ 1,

Ç�(h ×�5f�)��ê.h′(0) = 0(hI5º�\ï��W, ¥�¬&h(0) = 0, h

′(0)- 0), Æu�J�E∀0 < y ≤ 1,

h′′(y) =2 log(1 + y)

1 + y+

2

1 + y− 2 =

2(log(1 + y)− y)

1 + y< 0

Ç�(hù ×�5f�)�¬9ÎEÝ, .log(1 + y) < y, ∀y >

0�J±�

35.2;&Æ�èÄEóÐóW�cX�8D2, ¼óÐóW�Á">�ì��§�î, EóÐóW�>�´��xÑÝ¶�/X,

�¼óÐó�W�"Ä��x�¶��


��§§§5.4.'a > 0, b > 0�J

limx→∞

logb x

xa= 0,(5.28)

limx→∞

xb

eax= 0�(5.29)

JJJ��.&Æ�J(5.28)P, Q¡¿à(5.28)P¼J(5.29)P�´�b

t−1 ≤ tc−1,∀c > 0, t ≥ 1�.hux > 1, J

0 < log x =

∫ x

1

1

tdt ≤

∫ x

1

tc−1dt =xc − 1

c<

xc

c�

ãîPÇÿ

0 <logb x

xa<

xbc−a

cb,∀x > 1, c > 0�

uãc = a/(2b), Jx → ∞`, xbc−a = x−a/2 → 0�.h(5.28)PÿJ�kJ(5.29)P, �t = ex�Jx = log t, v

xb

eax=

logb t

ta �

¬x →∞`, t →∞, Æ(5.28)PÇ0l(5.29)P�

(5.29)P�×ñÇÝ.¡ , E�×94PfCðóa > 0,

limx→∞

f(x)

eax= 0�

¿ào-Br, î�§��î

logb x = o(xa), x →∞,(5.30)

xb = o(eax), x →∞�(5.31)

308 Ï"a ø÷Ðó

�¡b9�Ca9�, ©�Þï/ Ñ, Jlogb x��∞Ý>�, XÄxa, �xb��∞Ý>�, êXÄeax�

»»»5.7.�O

limx→0+

e−1/x

x �

��.h ×0/0��l�D«¿à1Ä¾!Jÿ

limx→0+

e−1/x

x= lim

x→0+

e−1/x(1/x2)

1= lim

x→0+

e−1/x

x2

...

= limx→0+

e−1/x

n!xn+1�

�Q�¡�5¿g, ��Ý�PP°�´, Æ1Ä¾!J´[�u�t = 1/x, J¿à�§5.3ÿ

limx→0+

e−1/x

x= lim

t→∞e−t

1/t= lim

t→∞t

et= 0�

Ç

e−1/x = o(x), x → 0 +�34.4;&Æ�èÄ$b×°��Á§Î��, Ç00, �∞0

Ý�P�9°BÊ2»ð¡,/�; 0/0T∞/∞Ý��bnEóT¼óÝ��, ôð�¢Ã�§5.3¼�§�

»»»5.8.�J

(5.32) limx→0+

xα log x = 0,∀α > 0�

JJJ��.�t = 1/x, J

xα log x = − log t

tα �.x → 0+`, t →∞, Æ¿à(5.28)PÇÿJ(5.32)P�


î»��|.Â�ÇE��α, β > 0,

(5.33) limx→0+

xα(log x)β = 0�

»»»5.9.�Jlim

x→0+xx = 1�

JJJ��.h ×00Ý��´�

xx = ex log x�

�ãî»álim

x→0+x log x = 0�

ê¼óÐó =�Ðó, Æ

limx→0+

xx = elimx→0+ x log x = e0 = 1�

»»»5.10.�Jlim

x→∞x1/x = 1�

JJJ��.h ×∞0Ý��t = 1/x,Jx1/x = 1/tt�êx →∞`,

t → 0+, .h¿à»5.9ÇÿJÍ»�

»»»5.11.�O

limx→0+

√x

1− e2√

x��.AìÇÿ:

limx→0+

√x

1− e2√

x= lim

t→0+

t

1− e2t= lim

t→0+

1

−2e2t= −1

2�

Í;t¡&Æ�×b¶Ý��Ex > 0, �

S0(x) = x, S1(x) = xx, S2(x) = xxx

, · · · ,

310 Ï"a ø÷Ðó

×��

Sn(x) = xSn−1(x) = xx···x

, n ≥ 2�&Æ��º�?E∀x > 1, limn→∞ Sn(x) = ∞, vE∀x < 1,

limn→∞ Sn(x) = 0�¯@îE∀0 < x ≤ e1/e, limn→∞ Sn(x)[e,

limn→∞ Sn(e1/e) = e, vEx > e1/e, n → ∞`, Sn(x)�Á§�D3��¢�Bromwich (1949)�

êêê ÞÞÞ 5.5

1. �Oì�&Á§Â, Í�a, b �y0�ðó�(1) limx→0

log(1+x)e2x−1 � (2) limx→0

sin xarctan x�

(3) limx→0ax−1bx−1

, a, b 6= 1� (4) limx→1log x

x2+x−2�(5) limx→0

log(1+x)−x1−cos x � (6) limx→0

x(ex+1)−2(ex−1)x3 �

(7) limx→1sin(π/(2x))·log x

(x3+5)(x−1) � (8) limx→0ebx2−cos x

x2 �(9) limx→0

ax−asin x

x3 � (10) limx→1 x1/(1−x)�(11) limx→0(x + e2x)1/x� (12) limx→0

(1+x)1/x−ex �

(13) limx→0

((1+x)1/x

e

)1/x

� (14) limx→0x−(1+x) log(1+x)

x2 �(15) limx→0(

1x− 1

ex−1)� (16) limx→1(

1log x

− 1x−1

)�(17) limx→0(

ax+bx

2)1/x� (18) limx→∞x(a1/x− b1/x)�

(19) limx→∞ x((1 + 1x)x − e)� (20) limx→∞ xe−x2 ∫ x

0et2dt�

(21) limx→0ax−bx

x � (22) limx→0(1 + x + x2)cot x�(23) limx→0(sin x/x)1/x2

� (24) limx→∞(sin x/x)1/x2

�2. �Oì�&Á§Â, Í�a, b �y0�ðó�

(1) limx→0(2−x)ex−x−2

x3 � (2) limx→0log(cos ax)log(cos bx)�

(3) limx→1+xx−x

(1−x)+log x� (4) limx→0e−1/x2

x1000 �(5) limx→∞

log(a+bex)√a+bx2 � (6) limx→π

log | sin x|log | sin 2x|�

(7) limx→ 12−

log(1−2x)tan(πx) � (8) limx→∞ ax

xb , a > 1�(9) limx→1− log x log(1− x)� (10) limx→0+ x(xx−1)�(11) limx→0+(xxx − 1)� (12) limx→0−(1− 2x)sin x�(13) limx→0+ x1/ log x� (14) limx→0+(cot x)sin x�

êÞ 311

(15) limx→π/4(tan x)tan 2x� (16) limx→0+(log 1x)x�

(17) limx→0+ xe/(1+log x)� (18) limx→1(2− x)tan(πx/2)�(19) limt→∞

R t+1t e−x2

/xadx

e−t2/ta+1 �(20) limx→0(

1log(x+

√1+x2)

− 1log(1+x)

)�

3. ¿àíÂ�§, �Jì��P�(i) 1

1+x< log(1 + x)− log x < 1

x, x > 0,

(ii) 1− x/y < log(y/x) ≤ y/x− 1, ∀0 < x < y�

4. 'f(x) = (x2)x, x 6= 0, f(0) = 1��Jf3x = 0=��

5. �¾½f(x) = ex − x − x2/2 − x3/6, 3x = 0ÎÍbÁ�TÁ��

6. 'f(x) = (1 + 1/x)x��Of ′(1)Climx→∞ f ′(x)�

7. 'f(x) = ecx, Í�c ×ðó�¿àf ′(0) = c, �J

limx→0

ecx − 1

x= c�

8. �On, ¸ÿx → ∞`, xne−x4 ∫ x

0et4dt�Á§D3, ¬OhÁ

§Â�

9. 'b > a > 0, �J

limt→0

(

∫ 1

0

(bx + a(1− x))tdx)1/t = e−1(bb/aa)1/(b−a)�

10. 'n ×ÑJóvx > 0��J

(1 +x

n)n < ex,∀x > 0,v ex < (1− x

n)−n,∀x < n�

Ê2óãn, �J2.5 < e < 2.99�

11. E∀a > 0, �JD3×c > 0, ¸ÿlog x < xa, ∀x > c�

312 Ï"a ø÷Ðó

12. Ea > 1, CÑJón��JD3×c > 0, ¸ÿxn < ax, ∀x >

c�

13. (i) ãx = 1/3Cm = 5, ¿à�§5.2, �J

0.6931460 < log 2 < 0.6931476;

(ii) ãx = 1/5, J(1 + x)/(1 − x) = 3/2�¿à(i)C�§5.2,

ãm = 5, �J

1.098611 < log 3 < 1.098617�

14. �Oðóa, ¸ÿx → 0`, x−2(eax − ex − x)�Á§D3, ¬OhÁ§Â�

15. 'Ðóf�ë$0óD3v=�, ¬��

limx→0

(1 + x +f(x)

x)1/x = e3�

�Of(0), f ′(0), f ′′(0)Climx→0(1 + f(x)/x)1/x�(èî: ¿àulimx→0 g(x) = A, Jg(x) = A + o(1), x → 0)

16. '

f(t) =E

R(1− e−Rt/L),

Í�E, R, L ÑÝðó��OlimR→0+ f(t)�

17. �Oc Â, ¸ÿlim

x→∞(x + c

x− c)x = 4�

18. (i) �JE∀c ∈ R,

(1 + x)c = 1 + cx + o(x), x → 0;

(ii) ¿à(i), �O

limx→∞

((x4 + x2)1/2 − x2)�

êÞ 313

19. OcÂ, ¸ÿlim

x→∞((x5 + 7x4 + 2)c − x)

D3, ¬Oh`�Á§Â�

20. E∀a 6= 1, �

f(x) = (ax − 1

x(a− 1))1/x, x 6= 0�

�J

limx→∞

f(x) =

{a, a > 1,

1, 0 < a < 1;

limx→−∞

f(x) =

{1, a > 1,

a, 0 < a < 1�

21. �Oì�ÞÁ§Â�(i) limx→∞(ex/2 + x2((1 + 1/x)x − e));

(ii) limx→∞ x((1 + 1/x)x − e log(1 + 1/x)x)�

22. �f(x) =∫ x

1g(t)(t + 1/t)dt, Í�g(t) = tet2��O

limx→∞

f ′′(x)

g′′(x)�

23. �f(x) =∫ x

0e2t(3t2 + 1)1/2dt, g(x) = xce2x��Oc �Â, ¸

ÿlimx→∞ f ′(x)/g′(x)D3v� 0, ¬Oh`�Á§Â�

24. �0f(x) = log x/x, x > 0,�%��EÑJón,X�(√

n)√

n+1

�(√

n + 1)√

n¢ï´�; X�eπ�πe¢ï´��

25. �Þ789, 798

, 879, 897

, 978, 987�0ó, ¶ï��4��

26. �0�xx3= 3�XbÑ@ó��

27. �0f(x) = x2e−x + 1�%��

314 Ï"a ø÷Ðó

28. �¿àpñ°, �x− ex = 0��óÏ6��

29. �J

1− 1

2+

1

3− 1

4+ · · ·+ 1

2n− 1− 1

2n=

1

n + 1+ · · ·+ 1

2n�

¬¿àîP, |×�5¼�î∑n

i=1(−1)i−1i−1��«Â�

30. �J

(A

A + B/n)n

En�3, Í�n ≥ 1 Jó, A, B > 0 Þðó�

31. 'f(x + y) = f(x)f(y), ∀x, y ∈ R, vf(x) = 1 + xg(x), Í�limx→0 g(x) = 1��J(i) f ′(x)D3, ∀x ∈ R;

(ii) f(x) = ex�

32. '

f(x) =

∫ x

1

log t

t + 1dt, x > 0�

�Of(x) + f(1/x), ¬�Jf(2) + f(1/2) = log2 2/2�

33. �JE∀x, y > 0, 0 < a < b,

(xb + yb)1/b < (xa + ya)1/a�

34. �f(x) = log x− x1/k, Í�k ×ÑJó�(i) �Of�Á�Â;

(ii) �Jf(x) = 0ªbÞ@q, |rk, sk��, v

0 < rk < kk < sk < k2k;

(iii) �JukÈ�, Je < rk < e1+δ, ∀δ > 0;

(iv) �Jlimk→∞ rk = e, limk→∞ sk = ∞�

êÞ 315

35. �Ot�Ýa, Ct�Ýb, ��EN×ÑJón,

(1 +1

n)n+a ≤ e ≤ (1 +

1

n)n+b�

36. 'f ×=�Ðó, v

f(x) =

∫ x

0

f(t)dt, ∀x ≥ 0�

�Jf(x) = 0, ∀x ≥ 0�(Û. ÍÞù��¨33.4;�êÞ, &Æ¨3�Ì´9Ý, T�?D|2��)

37. 'f(x) = e−1/x2, x 6= 0, vf(0) = 0�

(i) �Jf ×=�Ðó;

(ii) �JE∀m > 0, limx→0 f(x)/xm = 0;

(iii) �JE∀x 6= 0, f (n)(x) = f(x)P (1/x), Í�P (t) ×t�94P;

(iv) �Jf (n)(0) = 0, ∀n ≥ 1�ÍÞ�îf30��ng��94P/ 0�

38. 'a, r Þü��ÑÝðó, vr > 1�EN×ÑJók, �nk

��(a + n)k ≤ rnk�t�ÝÑJón��Jlimk→∞ nk/kD3, vOhÁ§Â�

39. �Jex ×ø÷Ðó�(èî: �àDJ°�Ç'ex ×�óÐó, v��

an(x)enx + an−1(x)e(n−1)x + · · ·+ a1(x)ex + a0(x) = 0,

Í�a0(x), a1(x), · · · , an(x)/ x�94P, Q¡¿àlimx→∞e−xxn = 0, ∀n ≥ 1, �0�ë;�)

40. �JN×�óÐó�DÐó, ) ×�óÐó, .h¿àîÞÇÿlog xù ø÷Ðó�

316 Ï"a ø÷Ðó

41. �

f(x) =

∫ x

1

et

tdt, x > 0�

(i) �O��log x ≤ f(x)�XbÝx�(ii) �J

∫ x

1

et

t + adt = e−a(f(x + a)− f(1 + a)),Í�a ×ðó�

(iii) A(ii), �|f¼�îì�&�5:

∫ x

1

eat

tdt,

∫ x

1

et

t2dt,

∫ x

1

e1/tdt�

42. EN×ÑJón, �

an =

∫ x

0

e−ttndt�

�Oa1, a2, a3, ¬��×�Ýan, �|ó.hû°J��

43. �5½O=�Ðóf�gCh, ��E∀x ∈ R,

∫ x

0

f(t)dt = ex,

∫ x2

0

g(t)dt = 1− 2x2

,

∫ x

0

h(t)dt = h2(x) + 1,

uP�ù1�æ.�

44. �

A =

∫ 1

0

et

t + 1dt�

�5½|A�îì��5:

(i)∫ a

a−1e−t

t−a−1dt, (ii)

∫ 1

0tet2

t2+1dt,

(iii)∫ 1

0et

(t+1)2dt, (iv)

∫ 1

0et log(1 + t)dt�

5.6 Ô`ÐóCDë�Ðó 317

5.6 ÔÔÔ`ÐÐÐóóóCCCDDDëëë��ÐÐÐóóó

b×°¼óÐóÝ)W, 35�C��îàH��, .hE9°©½Ý)W&Æ�|ú(�9°ÐóÙÌÔÔÔ`ÐÐÐóóó (hyper-

bolic functions), 5½ hyperbolic sine(�Ìsinh), hyperbolic co-

sine(cosh), hyperbolic tangent(tanh)��Í�L

sinh x =ex − e−x

2, cosh x =

ex + e−x

2,

tanh x =sinh x

cosh x=

ex − e−x

ex + e−x, coth x =

1

tanh x,

sechx =1

cosh x, cschx =

1

sinh x�

9°ÐóQ«ë�Ðóm�8�, ¬¸Æ�ë�Ðób×°v«ÝP², hã¸ÆÝ�L]Pô�:��ê.

cosh2 x− sinh2 x = 1,

Æu�u = cosh x, v = sinh x, Ju2 − v2 = 1, Í%�ª ×Ô`a,

9Îú( Ô`ÐóÝæ.�Ô`Ðó�Ô`aÝn;, -�9µAë�Ðó�iÝn;×ø(sin2 x + cos2 x = 1)�&Æ��Ô`ÐóÝ×°ÃÍP², J�Jº3êÞ�

1. cosh2 x− sinh2 x = 1�2. sinh(−x) = − sinh x�3. cosh(−x) = cosh x�4. tanh(−x) = − tanh x�5. sinh(x + y) = sinh x cosh y + cosh x sinh y�6. cosh(x + y) = cosh x cosh y + sinh x sinh y�7. sinh(2x) = 2 sinh x cosh x�8. cosh(2x) = cosh2 x + sinh2 x�9. cosh x + sinh x = ex�10. cosh x− sinh x = e−x�

318 Ï"a ø÷Ðó

11.(cosh x + sinh x)n = cosh(nx) + sinh(nx), n ×Jó�12.2 sinh2(x/2) = cosh x− 1�13.2 cosh2(x/2) = cosh x + 1�14. tanh2 x + sech 2x = 1�15. coth2 x− csch 2x = 1�16.D sinh x = cosh x�17.D cosh x = sinh x�18.D tanh x = sech 2x�19.D coth x = −csch 2x�20.Dsechx = −sechx tanh x�21.Dcsch x = −cschx coth x�

î�9°2P�, b°�ë�ÐóÝ2Pv«, b°J�P��Íg&ÆD¡Dë�Ðó, 9Î�5.��þKÝÐó�´

�:sineÐó�kDÐóD3, Ä6hÐó3Ø ��Q9Ë �9, A[−π/2, π/2], [π/2, 3π/2], [−3π/2,−π/2]

�/Î�µ&Æ?¡ÝàH��, �Ã×Í9vÝ /��Ä;ð&ÆÃã[−π/2, π/2], ¬�L×±ÐófAì:

f(x) = sin x, x ∈ [−π/2, π/2]�

9ø�LÝÐó �}�¦, ¬EN×[−1, 1] Ý@ó, /�ãÂ�Çf �[−π/2, π/2]Ì�[−1, 1]Ý1−1vÌWÝÐó�Æb×°×ÝDÐóg, ã[−1, 1]Ì�[−π/2, π/2], ��y = f(x), x =

g(y)�hÐógÌ DDDÑÑÑ<<<(inverse sineTarc sine)Ðó, |arcsinTsin−1��u2àsin−1�©½�T, sin−1 x sin xÝDÐó, �&1/ sin x�Æ

u = arcsin v, v ∈ [−1, 1]

�v = sin u, u ∈ [−π/2, π/2]�


ãÏÞa(8.16)P�ÿarc sineÐó�0ó�.

f ′(x) = cos x,

Æg′(y) =

1

f ′(x)=

1

cos x=

1√1− sin2 x

=1√

1− y2�

¬kîPWñ, y��y1T−1�Æÿ(Þx�yøð)

(6.1) D arcsin x =1√

1− x2, −1 < x < 1�

ãîPñÇÿì��52P:

(6.2)

∫ x

0

1√1− t2

dt = arcsin x, −1 < x < 1�

!ñ×è, &Æb�9�!Ý]P¼�Lë�Ðó�tð�ÝµÎ��!�Ý, |¿¢Ý]P¼�L�»A, E×E\ 1 Ýà�ë��, ub×��©� x, JÍE\��Ç sin x�¬&Æô�Bã(6.2)P, ��Larc sineÐó(ÑA!&Æ|×�5¼�LEóÐó)�Q¡ÞsineÐó�L arc sineÐóÝDÐó, �cosineÐó sineÐóÝ0ó, õv.�9Î��à��Ý]°¼�Lë�Ðó�ã(6.1)Pêÿì��5Ý2P:

(6.3)

∫1√

1− x2dx = arcsin x + C, −1 < x < 1�

Q&Æ§�k¸îPWñ, x6òy(−1, 1)�hÑAarcsine ÐóÎ�L3[−1, 1] , ¬©3(−1, 1) ��ÍgD¡cosineCtangentÐóÝDÐó�EycosineÐó, &Æ

;ðã�L½ [0, π], 3h �, cosine �[0, π]Ì�[−1, 1]

Ý1−1vÌWÝÐó, .hDÐó, ÇDDDõõõ<<<(inverse cosineTarc

cosine)ÐóD3, |arccosTcos−1��Æ

u = arccos v, v ∈ [−1, 1]

320 Ï"a ø÷Ðó

�v = cos u, u ∈ [0, π]�

�ytangentÐó, ã�L½ (−π/2, π/2), J��LDDDÑÑÑ666(inverse tangentTarc tangent)Ðó, |arctanTtan−1��Æ

u = arctan v, v ∈ (−∞,∞)

�v = tan v, v ∈ (−π/2, π/2)�

A!ÿÕ(6.1)P, �ÿì��52P:

(6.4) D arccos x =−1√1− x2

,−1 < x < 1,

C

(6.5) D arctan x =1

1 + x2, x ∈ R�

ã(6.4)Pêÿì��

(6.6)

∫ x

0

1√1− t2

dt = − arccos t∣∣∣x

0=

π

2− arccos x�

f´(6.2)�(6.6)P, Çÿ

(6.7) arcsin x + arccos x =π

2�

îPù�ãsin(π/2 − y) = cos y, �y = arccos x�ÿ�êã(6.4)Pô0lì��5Ý2P:

(6.8)

∫1√

1− x2dx = − arccos x + C, −1 < x < 1�

uf´(6.3)C(6.8)ÞP, �ÿarcsin x + arccos x ×ðó��x =

0ñÇÿhðó π/2�ã(6.5)Pôÿ

(6.9)

∫ x

0

1

1 + t2dt = arctan x, x ∈ R,


C

(6.10)

∫1

1 + x2dx = arctan x + C, x ∈ R�

u¿à5I�5C(6.1)P, �ÿ∫

arcsin xdx = x arcsin x−∫

x√1− x2

dx

= x arcsin x +√

1− x2 + C�

!§b∫

arccos xdx = x arccos x−√

1− x2 + C,∫

arctan xdx = x arctan x− 1

2log(1 + x2) + C�

!§, ô��Lcotangent, secantCcosecantÐó�DÐó�ê&Æ�arccotxãÂ3(0, π), arcsecxãÂ3(0, π/2) ∪ (π, 3π/2)�ãW9Ë�©ÝÂ½Î Ý�5îÝ]-��arccscxãÂ3(−π,

−π/2) ∪ (0, π/2)�ãhÇÿ(ûOarcsin x�0óÝM», &�ã�ÕÄ��, �:� ¢�Þarcsecx�Â½, ã (0, π/2) ∪ (π, 3π/2))

(6.11) Darcsecx =1

x√

x2 − 1, |x| > 1,

C

(6.12)

∫1

x√

x2 − 1dx = arcsecx + C�

¨², ùb

Darccotx = − 1

1 + x2, x ∈ R,(6.13)

Darccscx = − 1

x√

x2 − 1, |x| > 1�(6.14)

ãî�9°Dë�ÐóÝ0ó2P, �p:� ¢3�5�, umàÕDë�Ðó, &Æ;ð(ô©m�)2àarc sine, arc tangent

Carc secantÐó-ÈÝ�&Æ��ëÍðàÝ�52P�

322 Ï"a ø÷Ðó

»»»6.1.E∀a 6= 0, �J∫

1√a2 − x2

dx = arcsinx

|a| + C, |x| < |a|,(6.15)

∫1

x2 + a2dx =

1

aarctan

x

a+ C, x ∈ R,(6.16)

∫1

x√

x2 − a2dx =

1

aarcsec

x

a+ C, |x| > |a|�(6.17)

JJJ��.¿à=Å!J, ÿ

D arcsinx

|a| =1√

1− x2/a2

1

|a| =1√

a2 − x2,

Æ(6.15)PÿJ�!§�J(6.16)C(6.17)P�

ë�ÐóCDë�Ðó, / ¥�Ýø÷Ðó�bÝ9ËvÐó, ��»�2è{&Æ�5Ý�æ�b¶ÝÎ, Dë�, ÐóÝ0ó/��Îø÷Ðó, � �óÐó�ë�ÐóÝ0ó, J) ë�Ðó�A!Eó ¼óÝDÐó, ¼óÐóÝ0ó) ø÷Ðó, �EóÐóÝ0óµW �óÐóÝ�

»»»6.2.�OD arctan(1 + x2), CD arctan(1/x)��.¿à=Å!J, ÿ

D arctan(1 + x2) =1

1 + (1 + x2)2· 2x =

2x

x4 + 2x2 + 2�

Íg

D arctan(1/x) =1

1 + (1/x)2· (− 1

x2) = − 1

1 + x2�

3î»�, .

D arctan(1/x) = −D arctan x,

Æarctan(1/x) = − arctan x + C,

êÞ 323

Í�C ×ðó�ê.arctan 1 = π/4, ÆC = π/2, Çÿ

arctan x + arctan(1/x) =π

2,∀x 6= 0�

»»»6.3.�O∫

x√4−x4 dx�

��.�u = x2, Jdu = 2xdx�.h∫

x√4− x4

dx =

∫1

2

1√4− u2

du =1

2arcsin(

u

2) + C

=1

2arcsin(

x2

2) + C�

»»»6.4.�ODarcsec√

x��.¿à=Å!Jÿ

Darcsec√

x =1√

x√

(√

x)2 − 1

1

2√

x=

1

2x√

x− 1, x > 1�

ãhê�ÿ∫

1

2x√

x− 1dx = arcsec

√x + C, x > 1�

êêê ÞÞÞ 5.6

1. �JÔ`Ðó�P²1−21�

2. �J(6.4)�(6.5)�(6.11)�(6.13)C(6.14)�"P�

3. �Oì�&Ðó��5�(1) f(x) = arcsin(2x)� (2) f(x) = arccos(1−x

2)�

(3) f(x) = arccos( 1x)� (4) f(x) = arcsec(4x)�

(5) f(x) = arctan(x2 + 1)� (6) f(x) = arcsin(ex)�(7) f(x) = arcsin(sin x)� (8) f(x) = arctan(

√x2 − 1)�

(9) f(x) = arctan(e2x)e2x � (10) f(x) = arcsin 1−x2

1+x2�(11) f(x) = log(arctan x)� (12) f(x) = arcsec

√x2 − 1�

324 Ï"a ø÷Ðó

(13) f(x) =√

arcsin 3x� (14) f(x) = arctan(tan2 x)�(15) f(x) = log(arccos 1√

x)� (16) f(x) = (arccos(x2))−2�

(17) f(x) = arctan(x +√

1 + x2)�(18) f(x) = arctan(x +

√1 + x2)�

4. �Jì�&�52P�(i)

∫arccotxdx = xarccotx + 1

2log(1 + x2) + C,

(ii)∫

arcsecxdx = xarcsecx− log |x +√

x2 − 1|+ C,

(iii)∫

arccscxdx = xarccscx + log |x +√

x2 − 1|+ C,

(iv)∫

(arcsin x)2dx = x(arcsin x)2 − 2x + 2√

1− x2 arcsin x

+C,

(v)∫

arcsin xx2 dx = log |1−

√1−x2

x| − arcsin x

x+ C�

5. �J

arctan

(x + 1

x− 1

)+ arctan x = C,

Í�C ×ðó, ¬OCÂ�

6. (i) �JD(arccotx− arctan x−1) = 0, ∀x 6= 0;

(ii) �J�D3×ðóC, ¸ÿarccotx− arctan x−1 = C, ∀x 6=0�¬�Õh��ÎÍ)§�

7. 'arctan(y/x) = log√

x2 + y2��Jdy/dx = (x + y)/(x −y)�

8. 'y = arcsin x/√

1− x2, |x| < 1��Od2y/dx2�

9. 'y = sin(a arctan x), a ×ðó��J

(1 + x2)2 d2y

dx2+ 2x(1 + x2)

dy

dx+ a2y = 0�

10. 'y = sin(a arcsin x), a ×ðó��J

(1− x2)2 d2y

dx2− x

dy

dx+ a2y = 0�

êÞ 325

11. ¿àíÂ�§, �J

| arctan x− arctan y| ≤ |x− y|, x, y ∈ R�

12. �Olimn→∞∑n

k=1n

n2+k2�Â�

13. 'a ×ðó, �O

limx→0

arcsin ax

x �

14. �OÐóf(x) = sin x + 12sin 2x�ÁÂCD`F, ¬0Í%�

15. �OÐóf(x) = x + sin x�ÁÂCD`F, ¬0Í%�

16. �5½OÐóf(x) = cos x cosh xCg(x) = x + cos x�ÁÂ�

17. �Oì�&�5�1.

∫ 1/2

01√

1−x2 dx� 2.∫ 3

√3√

31

x2+9dx�

3.∫ √2

2/√

31

x√

x2−1dx� 4.

∫ex

e2x+1dx�

5.∫ −3

√2

−61

x√

x2−9dx� 6.

∫1

x√

1−log2 xdx�

7.∫

cos x1+sin2 x

dx� 8.∫

1√1−2x−x2 dx�

9.∫

1a+bx2 dx, ab 6= 0� 10.

∫1

x2−x+2dx�

11.∫

x arctan xdx� 12.∫

x2 arccos xdx�13.

∫x(arctan x)2dx� 14.

∫arctan

√xdx�

15.∫

arctan x1+x2 dx� 16.

∫1√

e2x−1dx�

17.∫

arctan√

x√x(1+x)

dx� 18.∫ √

1− x2dx�19.

∫xearctan x

(1+x2)3/2 dx� 20.∫

earctan x

(1+x2)3/2 dx�21.

∫x2

(1+x2)2dx� 22.

∫ arccot(ex)ex dx�

23.∫

(a+xa−x

)1/2dx, a > 0� 24.∫ √

(x− a)(x− b)dx, b 6= a�25.

∫1√

(x−a)(b−x)dx, b 6= a� 26.

∫tan2 x

3dx�

326 Ï"a ø÷Ðó

5.7 ��555***»»»G«èÄ, ¢Ãø÷Ðó��»�2è{&Æ�5Ý�æ�Í

;&Æ-�+Û×°�5Ý]°�Q9°Î¿ËýãÝ�5]°, ??&ÆÂÕ×�5, ��àÕ�©×Ë]°, Ty�Ý¿Í]°¡, ��¼�

(A) ë�Hð°'D3×Ë�ó�b§Ðóf�u�5Õ�(i) bf(x,

√a2 − (cx + d)2)Ý�P, J??�cx + d = a sin t;

(ii) bf(x,√

a2 + (cx + d)2)Ý�P, J??�cx + d = a tan t;

(iii) bf(x,√

(cx + d)2 − a2)Ý�P, J??�cx + d = a sec t�BÄ9Ë�ð¡, �5Õ�??»ð sin tTcos t Ýb§Ðó�

»»»7.1.�O ∫1√

1− x2dx�

��.h�53î×;�O�Ä, arcsin x + C�¨�x = sin t,

Jdx = cos tdt, v√

1− x2 =√

1− sin2 t = cos t�.h∫1√

1− x2dx =

∫1

cos tcos tdt =

∫dt = t + C = arcsin x + C�

3î»�,√

1− sin2 tÎÍT¶W| cos t|? .cos t�×� Ñ�¯@î, 3�x = sin t`, &Æ�§×t ∈ [−π/2, π/2], 3hP�/,

cos tÄ &�, .h�à¶W| cos t|�

»»»7.2.�O ∫x

4− x2 +√

4− x2dx�

��.�x = 2 sin t, Jdx = 2 cos tdt,√

4− x2 = 2 cos t�.h∫x

4− x2 +√

4− x2dx =

∫4 sin t cos t

4 cos2 t + 2 cos tdt =

∫sin t

cos t + 1/2dt

= − log |12

+ cos t|+ C − log(1 +√

4− x2) + C ′,

5.7 �5*» 327

Í�C ′ = C + log 2�

»»»7.3.'a > 0��O∫

x2

√x2 + a2

dx�

��.�x = a tan t, dx = a sec2 tdt, t ∈ (−π/2, π/2)�J√

x2 + a2 =√a2(1 + tan2 t) = a sec t�h�)àÕut ∈ (−π/2, π/2), Jsec t Ñ�.h

∫x2

√x2 + a2

dx =

∫a2 tan2 t

a sec ta sec2 tdt = a2

∫tan2 t sec tdt

= a2

∫(sec2 t− 1) sec tdt = a2

∫sec3 tdt− a2

∫sec tdt�

.�G��∫

sec tdt = log | sec t+tan t|+C,9ìO∫

sec3 tdt�ãÏëa(4.9)P

∫sec3 tdt =

1

2(sec t tan t +

∫sec tdt) + C,

Æ

(7.1)

∫sec3 tdt =

1

2(sec t tan t + log | sec t + tan t|) + C�

.h∫

x2

√x2 + a2

dx =a2

2(sec t tan t− log | sec t + tan t|) + C

=x

2

√x2 + a2 − a2

2log

∣∣∣∣∣

√x2 + a2

a+

x

a

∣∣∣∣∣ + C

=x

2

√x2 + a2 − a2

2log |

√x2 + a2 + x|+ C ′,

Í�C ′ = C + a2 log a/2�

328 Ï"a ø÷Ðó

î»µÎ×ÍXÛ��555222PPP(integration formula)�»A, kO∫

x2

√3x2 + 2

dx,

�Þî��5;

1√3

∫x2

√x2 + (

√2/3)2

dx,

J:�©��a =√

2/3, -�¢Ã»7.3�2P, ÿÕkO��5

1√3(x

2

√x2 + 2/3− 1

2

2

3log |

√x2 + 2/3 + x|) + C�

»»»7.4.�O ∫1

a2 sin2 x + b2 cos2 xdx�

��.�u = (a/b) tan x, du = (a/b) sec2 xdx, .h∫

1

a2 sin2 x + b2 cos2 xdx =

1

b2

∫1

(a2/b2) tan2 x + 1

1

cos2 xdx

=1

b2

∫1

u2 + 1

b

adu =

1

abarctan u + C =

1

abarctan(

a

btan x) + C�

(B) b§P��5XÛb§PÇÞ94P�¤�b§PÝ0ó) ×b§ó, ¬

b§PÝ�5µ�×�Îb§PÝ�»A,∫

1

xdx = log |x|+ C,

T ∫1

1 + x2dx = arctan x + C,

/&b§P�9ì&ÆD¡×�b§PÝ�5�b§P�1Î×vð�, ¬v��ÝÐó�5º2, 9vÐó/��¼, �v�|94P�b§P�EóTarc tangentÐó�î�

5.7 �5*» 329

ÃÍÝ�°Î9øÝ: �.×b§P¶WIII555555PPP(partial frac-

tions)�õ, Q¡¿à×�Ý�5*»×4×4��¼�

»»»7.5.�O ∫x2 + 4x− 1

x3 − xdx�

��.´�x2 + 4x− 1

x3 − x=

1

x+

2

x− 1− 2

x + 1�.h

∫x2 + 4x− 1

x3 − xdx =

∫1

xdx +

∫2

x− 1−

∫2

x + 1dx

= log |x|+ 2 log |x− 1| − 2 log |x + 1|+ C = log

∣∣∣∣x(x− 1)2

(x + 1)2

∣∣∣∣ + C�

u�O ∫2x4 + 3x3 − x2 + x− 1

x3 − xdx�

�ã�t°ÿÕ

2x4 + 3x3 − x2 + x− 1

x3 − x= 2x + 3 +

x2 + 4x− 1

x3 − x �

.h∫

2x4 + 3x3 − x2 + x− 1

x3 − xdx = x2 + 3x + log

∣∣∣∣x(x− 1)2

(x + 1)2

∣∣∣∣ + C�

î»�î, u�5Õ� ×�5P(Ç5�gó�f5Ò±), J�; ñ5P, ôµÎÞb§Pf/g¶W

f(x)

g(x)= q(x) +

r(x)

g(x),

Í�q(x)Cr(x)/ 94P,vr(x)�gó±yg(x)�gó�u×��f(x)�gó-±yg(x)�gó, Jq(x) = 0vr(x) = f(x)�94Pg(x)ÝI5Q^®Þ, Í�5) ×94P, Æ&Æ©m�ÊË5P(Ç5�gó±y5Ògó�b§P)��5Ç��

330 Ï"a ø÷Ðó

ã�ó�Ý��á, N×@;ó�94P/��îW×°@;ó�×gPCÞgP�¶��»A,

x3 − x = x(x− 1)(x + 1),

x3 − 8 = (x + 2)(x2 − 2x + 4),

x4 + 4x2 + 4 = (x2 + 2)2�

.h'b×b§Pf(x)/g(x), vf(x)�gó±yg(x)(Ç Ë5P),

J�5�g(x)W×°×gPCÞgP�¶��Q¡Þf(x)/g(x)

, ¶Wb§Íì�9Ë�PÝ5P�õ:

A

(x + a)kC

Bx + C

(x2 + bx + c)m,

Í�k, m ÑJó, A�B�C�a�b�c @ðóvb2−4c < 0�f�b2 − 4c < 0, �ÞgPx2 + bx + c ��3@ó�5��×b§P, ¶Wî�9°5P�õ¡, -ÌÞÍ5�WI55P�.hb§PÝ�5®Þ, -»ðWÍI55P��5��3�.�.I55P`, ��¬�á¼b¢àH, A*s¨æ¼3�5`àÿî�&Æ�}�1�×ìnyI55P�u(x + a)r g(x)�×.

P, v(x + a)r+1� g(x)�.P, Jf(x)/g(x)¶WI55P¡, -br4

A1

x + a+

A2

(x + a)2+ · · ·+ Ar

(x + a)r,

Í�A1, · · · , Ar ðó¬�� 0�©�g(x)bØ×9Ë×gPÝ.�, Jf(x)/g(x)¶WI55P¡, -b×Í9Ëõ�u(x2 + bx +

c)s g(x)�×.P, b2 − 4c < 0, v(x2 + bx + c)s+1� g(x)�.P,

Jf(x)/g(x)¶WI55P¡, -bs4

B1x + C1

x2 + bx + c+

B2x + C2

(x2 + bx + c)2+ · · ·+ Bsx + Cs

(x2 + bx + c)s,

Í�Bi, Ci, i = 1, · · · , s, ðó, ¬�� 0�©�g(x)bØ×9ËÞgPÝ.�, Jf(x)/g(x)¶WI55P¡, -b×Í9Ëõ�

5.7 �5*» 331

&Æ�aJ�N×b§P, /�|AîÝ5�WI55P��Äãì�×°»�, ��:�A¢5�×b§P�9ìÞ» 5Ò�¶W×°Þ�!Ý×gP�¶��

»»»7.6.�O ∫1

x2 − a2dx�

��.´��¶W1

x2 − a2=

A

x− a+

B

x + a�B;5¡ÿ

A(x + a) + B(x− a) = 1�5½|x = aCx = −a�áîP, -��

A =1

2a, B = − 1

2a�

.h1

x2 − a2=

1

2a(

1

x− a− 1

x + a),

v∫

1

x2 − a2dx =

1

2a(

∫1

x− adx−

∫1

x + adx)

=1

2a(log |x− a| − log |x + a|) + C =

1

2alog

∣∣∣∣x− a

x + a

∣∣∣∣ + C�

»»»7.7.�O ∫2x2 + 5x− 1

x3 + x2 − 2xdx�

��..x3 + x2 − 2x = x(x− 1)(x + 2), Æ

2x2 + 5x− 1

x3 + x2 − 2x=

A1

x+

A2

x− 1+

A3

x + 2�

.h

2x2 + 5x− 1 = A1(x− 1)(x + 2) + A2x(x + 2) + A3x(x− 1)�

332 Ï"a ø÷Ðó

5½|x = 0, −1C−2�áîP, ��

A1 =1

2, A2 = 2, A3 = −1

2�.h∫

2x2 + 5x− 1

x3 + x2 − 2xdx =

1

2

∫1

xdx + 2

∫1

x− 1dx− 1

2

∫1

x + 2dx

=1

2log |x|+ 2 log |x− 1| − 1

2log |x + 2|+ C�

ì» 5Ò�¶W×°×gP�¶�, vg]b�y1ï�

»»»7.8.�O ∫x2 + 2x + 3

(x− 1)(x + 1)2dx�

��.�OA1, A2, A3, ¸ÿ

x2 + 2x + 3

(x− 1)(x + 1)2=

A1

x− 1+

A2

x + 1+

A3

(x + 1)2�

B;5¡ÿ

x2 + 2x + 3 = A1(x + 1)2 + A2(x− 1)(x + 1) + A3(x− 1)��x = 1, ÿA1 = 3/2; �x = −1, ÿA3 = −1; �x = 0, ¬ÞA1,

A3�á, ��A2 = −1/2�.h∫

x2 + 2x + 3

(x− 1)(x + 1)2dx=

3

2

∫1

x− 1dx− 1

2

∫1

x + 1dx−

∫1

(x + 1)2dx

=1

2log |x− 1| − 1

2log |x + 1|+ 1

x + 1+ C�

ì» 5Òâb��5�ÝÞgP�¶�, N4�gó/ 1g�

»»»7.9.�O ∫x2 + x + 1

(2x + 1)(x2 + 1)dx�

5.7 �5*» 333

��.�OA, B, C, ¸ÿ

x2 + x + 1

(2x + 1)(x2 + 1)=

A

2x + 1+

Bx + C

x2 + 1 �

B;5¡ÿ

x2 + x + 1 = (A + 2B)x2 + (B + 2C)x + (A + C)�

ãyÞ94P8�, ETx!g�;óÄ68�, .h

A + 2B = 1, B + 2C = 1, A + C = 1�

��A = 3/5, B = 1/5, C = 2/5�.h∫

x2 + x + 1

(2x + 1)(x2 + 1)dx

=3

5

∫1

2x + 1dx +

1

5

∫x

x2 + 1dx +

2

5

∫1

x2 + 1dx

=3

10log |2x + 1|+ 1

10log(x2 + 1) +

2

5arctan x + C�

ì» 5ÒâbÞgPvg]b�y1ï�

»»»7.10.�O ∫x4 − x3 + 2x2 − x + 2

(x− 1)(x2 + 2)2dx�

��.�¶Wì�I55P

x4 − x3 + 2x2 − x + 2

(x− 1)(x2 + 2)2=

13

x− 1+

23x− 1

3

x2 + 2+

−x

(x2 + 2)2�

.h∫

x4 − x3 + 2x2 − x + 2

(x− 1)(x2 + 2)2dx

=

∫ 13

x− 1dx +

∫ 23x− 1

3

x2 + 2dx−

∫x

(x2 + 2)2dx

=1

3log |x− 1|+ 1

3

∫2x

x2 + 2− 1

3

∫1

x2 + 2dx− 1

2

∫2x

(x2 + 2)2dx

334 Ï"a ø÷Ðó

=1

3log |x− 1|+ 1

3log(x2 + 2)−

√2

6arctan

x√2

+1

2

1

x2 + 2+ C�

(C) �; b§P��53èÚ�èâtS��5s"Ý��, ó.�ðlæy0�Í

�5b“��@@@ÝÝÝÝÝÝ��PPP”(closed form, Ç�|��Ðó�î)�Ðó��5*»�¼�?¡, E�5Ý/�ô�¼�Ý��âyó.�s¨, Ç¸£°“�@Ý”Ðó(A��Ðó), �N×/��¼, �¬��ô�m��.h ÝO�5�Ý¥y&Ë�5*»Ýr#, �@�¼Qì¼�¬�Îb×°¥�ÝW�ºì¼, ÇXbb§PÍ�5b�@Ý�P�h��µÎ3(B)�, XD¡Ýb§P��5, &Æ�¢×°»�¼1��9ì&Æ��×°J§�3(B)�, &Æ¼�, E×Ë5P, �Þ¸; ×°

A

(x + a)kC

Bx + C

(x2 + bx + c)m

�õ, Í�b2 < 4c�u�x + a = u, J∫

A

(x + a)kdx = A

∫1

ukdu�

�u�v = (2x + b)/√

4c− b2, J∫

Bx + C

(x2 + bx + c)mdx =

∫B(x + b/2) + C −Bb/2

((x + b/2)2 + c− b2/4)mdx

= (4

4c− b2)m

∫B′v − C ′

(v2 + 1)m

√4c− b2

2dv

= K1

∫v

(v2 + 1)mdv + K2

∫1

(v2 + 1)mdv,

Í�

B′ = B

√4c− b2

2, C ′ = C −Bb/2,

K1 = (4

4c− b2)mB′

√4c− b2

2, K2 = −(

4

4c− b2)mC ′

√4c− b2

2 �

5.7 �5*» 335

X|Eb§PÝ�5®Þ, -»ð ©m5½�Êì��ÐóÝ�5:

1

xn,

1

(x2 + 1)n,

x

(x2 + 1)n�

&Æµ�¼:�´�∫

1

xndx =

{− 1

(n−1)xn−1 + C, n > 1,

log |x|+ C, n = 1�

X|1/xn��5) ��Ðó�Ígu�ξ = x2 + 1, J

∫x

(x2 + 1)ndx =

1

2

∫1

ξndξ =

{− 1

2(n−1)(x2+1)n−1 , n > 1,12log(x2 + 1), n = 1�

t¡¼O

In =

∫1

(x2 + 1)ndx�

'n > 1, J

1

(x2 + 1)n=

1

(x2 + 1)n−1− x2

(x2 + 1)n,

v∫

1

(x2 + 1)ndx =

∫1

(x2 + 1)n−1dx−

∫x2

(x2 + 1)ndx�

EîPt�×4¿à5I�5, ÿ∫

x2

(x2 + 1)ndx = − 1

2(n− 1)

∫xd

1

(x2 + 1)n−1

= − 1

2(n− 1)

x

(x2 + 1)n−1+

1

2(n− 1)

∫1

(x2 − 1)n−1dx�

Æÿì�L]2P:∫

1

(x2 + 1)ndx =

1

2(n− 1)

x

(x2 + 1)n−1(7.2)

+2n− 3

2(n− 1)

∫1

(x2 + 1)n−1dx�

336 Ï"a ø÷Ðó

un− 1)�y1, J¥�î�M»�t¡ÿ∫

1

x2 + 1dx = arctan x + C�

ÆÿIn�|b§PCarctan x¼�î�ã(7.2)PêÿE∀n > 1Cα 6=0,

∫1

(x2 + α2)ndx =

1

2α2(n− 1)

x

(x2 + α2)n−1

+2n− 3

2α2(n− 1)

∫1

(x2 + α2)n−1dx,

� ∫1

x2 + α2dx =

1

αarctan

x

α+ C�

ãî�D¡á, Eb§P, &Æ@@�|��Ðó¼�îÍ�5�Qu�ÿ!Y,Å�×�&�; 1/xn, 1/(x2+1)nTx/(x2+

1)nÝ�P�Ç¸E

A

(ax + b)n,

Bx

(ax2 + bx + c)n,

C

(ax2 + bx + c)n

ô�pO��5�b×°Ðó��5�»; b§P��5�»A, 'b×Þ�

ó�b§Pf , �kO∫

f(sin x, cos x)dx�u�

t = tanx

2,

J

x = 2 arctan t, dx =2

1 + t2dt,

sinx

2=

t√1 + t2

, cosx

2=

1√1 + t2

,

sin x = 2 sinx

2cos

x

2=

2t

1 + t2,

cos x = 2 cos2 x

2− 1 =

1− t2

1 + t2�

5.7 �5*» 337

.h∫

f(sin x, cos x)dx =

∫f(

2t

1 + t2,1− t2

1 + t2) · 2

1 + t2dt�

ÇÞæ�5»; ×b§P��5, �b§PÝ�5ê�B�XÝ�

»»»7.11.�O ∫1

sin x + cos xdx�

��.Aî�tan(x/2) = t, J

sin x + cos x =1 + 2t− t2

1 + t2,

v∫

1

sin x + cos xdx =

∫1 + t2

1 + 2t− t22

1 + t2dt = −2

∫1

t2 − 2t− 1dt

= − 1√2

∫(

1

t− 1−√2− 1

t− 1 +√

2)dt

= − 1√2

log

∣∣∣∣∣t− 1−√2

t− 1 +√

2

∣∣∣∣∣ + C

= − 1√2

log

∣∣∣∣∣tan(x/2)− 1−√2

tan(x/2)− 1 +√

2

∣∣∣∣∣ + C�

¨²,E∫

R(x,√

a2 − (cx + d)2)dx,∫

R(x,√

a2 + (cx + d)2)dx,

C∫

R(x,√

(cx + d)2 − a2)dx, BÄë�Hð°¡(�(A)), /�ÞkO��5, » O×nysin uCcos u�b§P, Q¡��t =

tan(u/2), -» O×b§P��5�9ì ×°ô�»ð Ob§PÝ�5�»�

»»»7.12.�O ∫1√

x + 2 3√

xdx�

338 Ï"a ø÷Ðó

��.�x = u6, J(6�� Ý�Õ)

∫1√

x + 2 3√

xdx = 6

∫u5

u3 + 2u2du

= 2u3 − 6u2 + 24u− 48 log |u + 2|+ C

= 2√

x + 6 3√

x + 24 6√

x− 48 log( 6√

x + 2) + C�

»»»7.13.�O ∫ √1 + x−√1− x√1 + x +

√1− x

dx�

��.�b§;�5Õ��5Òÿ∫ √

1 + x−√1− x√1 + x +

√1− x

dx =

∫2− 2

√1− x2

2xdx

=

∫1−√1− x2

xdx�

��1− x2 = u2, îPt¡×�5, ê�y∫

1−√1− x2

x2xdx =

∫1− u

1− u2(−u)du = −

∫u

1 + udu

= −u + log |1 + u|+ C = −√

1− x2 + log(1 +√

1− x2) + C�

»»»7.14.�O ∫ex + 1

e2x − ex + 2dx�

��.�u = ex, Jdu = exdx, v∫

ex + 1

e2x − ex + 2dx =

∫u + 1

u2 − u + 2

1

udu,

W ×b§P��5�6��ÕÄ�, ÿîP��5�y

1

2log |u| − 1

4log(u2 − u + 2) +

5

2√

7arctan(

2u− 1√7

) + C

=1

2x− 1

4log(e2x − ex + 2) +

5

2√

7arctan(

2ex − 1√7

) + C�

5.7 �5*» 339

�Ä�º�ÝÎ, »ðWb§P¡, Î1J��¼, ¬.�; I55P, b`�Õ)Î8�Ó�ÆubÍ�]P��|2O��5, ¬�à©2Þ�5Õ�; ×b§P��ì»�

»»»7.15.�O ∫sin3 x

cos x + 2dx�

��.�u = cos x, J∫

sin3 x

cos x + 2dx =

∫1− cos2 x

cos x + 2sin xdx = −

∫1− u2

u + 2du

=

∫(u− 2 +

2

u + 2)du =

1

2u2 − 2u + 3 log |u + 2|+ C

=1

2cos2 x− 2 cos x + 3 log(cos x + 2) + C�

¬u|u = tan(x/2)�á, Jæ�5W ∫

16u3

(u2 + 3)(u2 + 1)3du�

4Î×b§P��5, ¬�p��ÕÄ�º�Ó&9�

¨², 3O ∫1

a2 sin2 x + b2 cos2 xdx,

u�t = tan x, º´�t = tan(x/2)��&9�×��, u�5Õ� sin2 x, cos2 x, Tsin x cos x�b§P, ;ð��t = tan x�h.

cos2 x =1

1 + tan2 x,

sin2 x = 1− cos2 x =tan2 x

1 + tan2 x,

sin x cos x = tan x cos2 x =tan x

1 + tan2 x�

ê??ë�ÐóÝ�5, º´b§PÝ�5 &ÆXK��»A,

4Q∫

xn(1− x2)n/2dx �; ×b§P��5, ¬u�x = sin u, J»ð O

∫sinn ucosm+1 udu, �h�¿àL]O�(�êÞ4.4)�

340 Ï"a ø÷Ðó

À�, O�5, ¬P×�Ýt·]°, B�á�QÎtx�Ý,

©��Õ�æ?, 4b`��ÿ2àt�-Ý]°, ¬)��Ñ@�n�9ì ×»��O ∫

1

a cos x + b sin xdx, a2 + b2 > 0�

&Æ4�¿àt = tan(x/2)��ð(�»7.11), ¬u�

A =√

a2 + b2, sin θ =a

A, cos θ =

b

A,

J�5W

1

A

∫1

sin(x + θ)dx =

1

Alog | tan(

x + θ

2)|+ C�

êêê ÞÞÞ 5.7

1. �Oì�&�5�(1)

∫ √25− x2dx� (2)

∫ √9x2 − 4dx�

(3)∫ √

4−x2

xdx� (4)

∫ √x2 − 4dx�

(5)∫

x√

9x2 − 4dx� (6)∫

1x√

x2+9dx�

(7)∫

1(x2+9)2

dx� (8)∫

1(x2−4)2

dx�(9)

∫1

(x2−4)2dx� (10)

∫ √x2−a2

xdx�

(11)∫ √

x2−a2

x2 dx� (12)∫ √

a2−x2

x2 dx�(13)

∫x2√

a2 − x2dx� (14)∫ √

x2 + a2dx�(15)

∫ √a2+x2

x2 dx� (16)∫

x2√x2−a2 dx�

(17)∫

x2√a2−x2 dx� (18)

∫1

x√

a2−x2 dx�(19)

∫1

x√

a2+x2 dx� (20)∫

1x√

x2−a2 dx�(21)

∫ √a2 − x2dx� (22)

∫1

a2 sin2 x−b2 cos2 xdx�

(23)∫

1(x2−4x+5)2� (24)

∫(x + 3)2

√x2 + 6x + 8dx�

(25)∫

x√3−x2 dx� (26)

∫ √3−x2

xdx�

(27)∫ √

x2+xx

dx� (28)∫

x√x2+x+1

dx�(29)

∫1√

x2+xdx� (30)

∫ √2−x−x2

x2 dx�

êÞ 341

2. �Oì�&�5�(1)

∫x+1x2−x

dx� (2)∫

xx2−5x+6

dx�(3)

∫x3

x2−2x−3dx� (4)

∫6x2+1

2−x−6x2 dx�(5)

∫3x−1

4x2−4x+1dx� (6)

∫1

4x2+12x+9dx�

(7)∫

x2+1x3+x2−2x

dx� (8)∫

4x2−3x(x+2)(x2+1)

dx�(9)

∫x2

x4−16dx� (10)

∫1

x3−x2 dx�(11)

∫x3+1x3−4x

dx� (12)∫

x3+1x3−1

dx�(13)

∫2x2+1(x−2)3

dx� (14)∫

x2+x+1(x+1)3

dx�(15)

∫2x3+x2+5x+4

x4+8x2+16dx� (16)

∫x4+x3+18x2+10x+8

(x2+9)3dx�

(17)∫

3x+1(x2−4)2

dx� (18)∫

x3+1(4x2−1)2

dx�(19)

∫1

x2(x−1)dx� (20)

∫1

x(x2+1)dx�

(21)∫

1x4−1

dx� (22)∫

x4

x4+5x2+4dx�

(23)∫

1(x+1)(x+2)2(x+3)3

dx� (24)∫

1(x2−4x+4)(x2−4x+5)

dx�(25)

∫x4+1

x(x2+1)2dx� (26)

∫1

x4+1dx�

(27)∫

x2

(x2+2x+2)2dx� (28)

∫4x5−1

(x5+x+1)2dx�

(29)∫

12 sin x−cos x+5

dx� (30)∫

11+a cos x

dx, 0 < a < 1�(31)

∫1

1+a cos xdx, a > 1� (32)

∫sin2 x

1+sin2 xdx�

(33)∫ π/2

0sin x

1+cos x+sin xdx� (34)

∫1

(a sin x+b cos x)2dx, a 6= 0�

(35)∫

1a2 sin2 x+b2 cos2 x

dx, ab 6= 0�3. '×=�Ðóf� 0, v��

f 2(x) =

∫ x

0

f(t)sin t

2 + cos tdt�

�Of�

4. 'f ×b§P, a, b 6= 0, p, q Jó, q > 0��t = (ax +

b)1/q��J∫f(x)(ax + b)p/qdx =

∫f(

1

a(tq − b))tp

q

atq−1dt�

5. ¿àîÞ, �O

(i)∫

x2−1(2x+3)3/2 dx, (ii)

∫x√

x+3x2+4

dx�

342 Ï"a ø÷Ðó

6. �f ×Ë�ó�b§P�(i) �u = tanh(x/2), �J

∫f(cosh x, sinh x)dx =

∫f(

1 + u2

1− u2,

2u

1− u2)

2

1− u2du�

(ii) �t = ex, �J∫

f(cosh x, sinh x)dx =

∫f(

u2 + 1

2u,u2 − 1

2u)1

udu�

7. �5½¿àîÞ��ËË�ð, O

(i)∫

12+cosh x

dx, (ii)∫

1sinh x+cosh x

dx�

8. 'f ×Ë�ó�b§P, a, b, α, β, n ≥ 1 ðó��

t = (ax + b

αx + β)1/n,

�J∫

f(x, (ax + b

αx + β)1/n)dx

=

∫f(−βtn + b

αtn − a, t)

aβ − bα

(atn − a)2ntn−1dt�

9. ¿àîÞ, �O

(i)∫

x+1x+2

(x+3x+4

)1/3dx, (ii)∫

x√

5−x√x+2

dx�

5.8 ÐÐÐóóóÝÝÝÍÍÍ��PPP²²²

Í;&ÆEÐóÝ��ØË]PÝf´, ¬D¡×°©��b¶ÝÐó�

5.8 ÐóÝÍ�P² 343

(A) Ðó��f´f´ÞÐóÍ�ó��`, ÍÐóÂÝ��3��5�Î�

¥�Ý, 9µÎXÛ��ÝÝÝ��$$$(order of magnitude)�34.3;&Æ�+ÛÝhÃF, Í;��×°��E∀α > 0, x → ∞, Jxα, logα x, ex, eαx/��∞�¬¸

Æ��∞ Ý>�Q�8!�»A, .x → ∞`, x3/x2 → ∞,

Æx3��∞Ý>�"Äx2�&Æ-Ìx3fx2, b´{Ý�$��∞�!§©�α > β > 0, Jxαfxβ, b´{Ý�$��∞�×��, ux →∞`, |f(x)| → ∞, |g(x)| → ∞, v|f(x)/g(x)|

→ ∞, JÌf(x) ��∞Ý�${yg(x)�D�, u|f(x)/g(x)| →0, x → ∞, JÌf(x)��∞Ý�$±yg(x)��ux → ∞`,

|f(x)/g(x)|��×� 0 �ðó, T×à+yËü�Ñó , JÌhÞÐó, b8!Ý�$��∞�»A, 'f(x) = ax3 + bx2 + c, a 6= 0, Jf�g(x) = x3, b8!Ý

�$��∞, �ffh(x) = x2, b´{Ý�$��∞�'bÞÐófCg, vf��∞Ý�$�yg, Jf + g�fb8!

Ý�$�4!øÎ��∞, &Æ|�$�{±f´Í>��"X�h

ÑA!3/)�, P§/), ô�|�óÝC��óÝ, f´ÍÃó��35.3;, !øÎ��0, &Æf´Í�$�{±�

(B) ¼óCEóÝ�$�35.5;, &Æ�ÿE∀a > 0,

limx→∞

log x

xa= 0 C lim

x→∞xa

ex= 0�

x > 1`, E!×x, a��xa-��; D�a��xa-��¬log x

W�Ý>�Xy�×xa, �exW�Ý>�"Ä�×xa�3�Õ^I.�Ý�Õ°, bXÛ94P` C¼ó` , µÎà�Xm` Ý��¨×]«, log xÝ>�, QêXÕp|�, �y�×xa�¼ó�Eó��ËÍÁÐ, ÞXbxÝ¶�ô3Í �

344 Ï"a ø÷Ðó

ã|îD¡á, ¼óÐó��∞Ý�$, {y�×xÝ¶�; �EóÐó��∞Ý�$, ±y�×xÝ¶��ãhê�ÿÕ�9�${y¼óÐó, T�$±y¼óÐóÝÐó�ÉA, eexÝ�$,

{y¼óÐó, �log(log x)Ý�$, ±yEóÐó�©�xÈ�, Ðóx�log x�log(log x)�log(log(log x))�ô\K

��Õ��, ¬>�Q×Íf×ÍX�»A, ux = 10100, 9Î×Í��Ýó, ¬log xV 230, �log(log x)V© 5.4�&Æ�à�×Ðóf��∞Ý>�, b`�|1|xÝØg],

A|xÝag]��∞, Í�a > 0�h�x →∞`f(x)/xa → c,

Í�c ×��y0�ðó�ôµÎG«X1Ý, f�xab8!Ý�$�¬|¼óÝ>��∞, µÎ�${y�×xa, ùÇ�D3×a > 0, ¸f|xÝag]��∞�¨×]«, ufÝ�${yxa�±yxa+1, JÎÍD3×0 < ε < 1, ¸ÿfÝ�$�xa+ε8!, ùÇfÎ|xÝa + εÝg]��∞? �nÎÍ�Ý�»A,

f(x) = xa log x, Jx →∞`,

f(x)

xa→∞,

xa+1

f(x)→∞,

ÆfÝ�${yxa�±yxa+1�¬¬�D3×0 < ε < 1, ¸ÿx →∞, f(x)/xa+ε��0T∞�¨², ËÐóô�×��f´Í�$�{±�»A, �h(x) =

f(x)/g(x), Í�

f(x) = x2 sin2 x + x + 1,

g(x) = x2 cos2 x + x,

Jx →∞`, f(x) →∞vg(x) →∞�¬uEN×Jón,

h(nπ) =nπ + 1

(nπ)2 + nπ→ 0,

h((n +1

2)π) = (n +

1

2)π + 1 +

1

(n + 12)π→∞�

Æx → ∞`, h(x)�Á§É� 0ô� ∞�êh(x)ô�º×à+yËü�Ñó �.hP°f´f�g�$�{±�

5.8 ÐóÝÍ�P² 345

4Q&ÆE�$Ý�L, ¬��|f´�ÞÐó, �Ä9�Î×�þ´�.E�Aî�ÝfCg, &Æ;ð�nTÍ�$Ý��, .Ç¸á¼Í×�Â, E¨×Ðó¬��èºbàÝ£G�

(C) 3�×FÝ�$'EØ×ξ ∈ R, x → ξ`, f(x) →∞vg(x) →∞�J&Æô

�f´Í�$��©�ÞG«x →∞`��$Ýf´, }�ÑÑÇ��»A, .x → ξ`,

e|x−ξ|−1 →∞, |x− ξ|−α →∞, ∀α > 0,

v

(8.1)e|x−ξ|−1

|x− ξ|−α→∞,

Æx → ξ`, e|x−ξ|−1��∞Ý�$, {y|x − ξ|−α, ∀α > 0�!§x → ξ`, .

(8.2)log |x− ξ||x− ξ|α → 0,

Æx → ξ `, log |x− ξ|��0Ý�$, ±y|x− ξ|−α, ∀α > 0�

(D) ��0��$tÝf´��∞Ý>�, A!35.3;, ô�f´��0Ý>

��»A, .x → 0`,

e−|x|−1 → 0, |x|α → 0, ∀α > 0,

v

(8.3)e−|x|

−1

|x|α → 0,

346 Ï"a ø÷Ðó

Æx → 0`, e−|x|−1��0Ý�$, {y��×xÝ¶��!§x →

0`, .1/ log |x| → 0, v

(8.4)|x|α

1/ log |x| → 0,

Æx → 0`, 1/ log |x|��0Ý�$, ±y�×xÝ¶��

(E) O-BrCo-Br34.3;&Æ�So-Br, E�ÞÐóf(x)Cg(x), f = o(g)Ç

�f��$±yg, ôµÎf/g → 0�hBr�Êà&9�!Ý�µ��âÐó��0T∞, C¢óx��∞T�×ξ ∈ R�ÐroÎÙ��$�&Æ��×°¨bÝ��Aì�

xα = o(xβ), ∀α < β, x →∞,

log x = o(xα), ∀α > 0, x →∞,

e−x = o(x−α), ∀α > 0, x →∞,

e−x−1

= o(xα), x → 0+,

log |x| = o(x−1), x → 0,

¨², &Æ$bO-Br�f = O(g)�f(x)��$t9�g(x)8!�E∀a ∈ R ∪ {∞,−∞},

f(x) = O(g(x)), x → a,

uv°uD3×ðóK, Ca�×�TÏ½N(ua = ∞, JN =

(c,∞), Í�c Ø@ó), ¸ÿ

|f(x)| ≤ K|g(x)|, ∀x ∈ N�&Æ\�f is big O of g at a�»A

sin x = O(x), x → 0,

x3 + x = O(x3), x →∞,√

10x− 1 = O(√

x), x →∞,

ex − 1 = O(x), x → 0�

5.8 ÐóÝÍ�P² 347

?×�2, 3f = O(g)�Ðr�, ô��6�'x��×a, ©�f�g�fÂ b&Ç��»A,

log x = O(x), x > 1,

x = O(sin x), |x| < π/2�

¿àO-Br, b`�?Þ@2à�0-�»A,

f(x + h) = f(x) + hf ′(x) + o(h), h → 0,

�;

f(x + h) = f(x) + hf ′(x) + O(h2), h → 0�

Tbcos x = 1 + O(x2), ∀x ∈ R�

êx ô�|ó�ã��»A

1√1 + 4n2

=1

2n+ O(

1

n2), n →∞�

(F) ×°©�ÝÐó��5��O�ÛP, ??¯×°�.ïqpÊT�9�

«b×°�ÿ�AhÝ§ã�4Q��5�Ý×°ÃÍÃF, A=�PC¿âP, KÙ�yàÌ, ¬QÄ6ÞÍ�@;, Ah3�ìî�b�L��9ø×¼, -SÝ�ÛÝ�L, �´�Ýæ¼àÌÝ©P�»A, kÞ=�PÝÃF�Û;, -�ÿ�m�8��ÝhéÃF, ��DTæ�`a=;ÝÃF; ��JÎ×Ífæ¼ÿWÝ`a¿âÝÃF, §×?9, C?héÝÃF��A9vP��¹Ýþ´, º¯×°Ä�E�ìIY�ÈÝß, TÎ1E�ª8YÝ�ì��·¶Ýß, �×��µP°#å��5Ý£°ÃÍÃFC.0]P�?��1Í��b�K¬��|Ý�Õ�&Æ�T��@�Èá��5Ý£×�¤��Ù��9ì&Æ¢¿Í�P��ÝÐó, ¯��}�Ý�b`ºb×

°&Æï��ÕÝ¨ésß�

348 Ï"a ø÷Ðó

»»»8.1.'f(x) = e−1/x2, Í%�A%8.1�

-

6

1O

1

x

y

y = e−1/x2

%8.1.

hÐó&Æ35.5;�êÞ�#ÇÄ, ¸3x = 0ÍP�L, ¬u�f(0) = 0, Jf3x = 0=��ê

f ′(x) =2

x3e−1/x2

, x 6= 0,

�

f ′(0) = limh→0

e−1/h2 − 0

h= 0�

!§�O�

f (n)(0) = 0,∀n ≥ 2�

.hf30�N×$0ó/D3v�y0��Ï8.5;, &ÆºÝ�hÐóÝ©�P�

»»»8.2.'f(x) = e−1/x�Ex > 0, hÐó�î»��ÐóÝ� ��«, Çx → 0+`, f(x) → 0, vf (n)(x) → 0, ∀n ≥ 1�u�Lf(0) = 0, Jf

(n)+ (0) = 0, ∀n ≥ 1�

f�%�A%8.2�¬ux → 0−, J�x → 0+��µ--��h`f(x)CXbf (n)(x), /��∞, vf30 �¼0ó�D3�Eb§ÐóCë�ÐóÎ�º9øÝ�uy = a ×kà��a, Jx → a+Tx → a−`, f(x)Ä��∞T−∞�

5.8 ÐóÝÍ�P² 349

-

6

1

y = e−1/x

y = e−1/x

O

1

x

y

%8.2.

»»»8.3.'f(x) = tanh(1/x)�ãy

f(x) =e1/x − e−1/x

e1/x + e−1/x,

hÐó3x = 0P�L�ê

limx→0+

f(x) = 1 6= −1 = limx→0−

f(x)�

Æf3x = 0b×®�Ý�=��¬.E∀x 6= 0

f ′(x) = − 4

x2(e1/x + e−1/x)2,

Ælim

x→0+f ′(x) = lim

x→0−f ′(x) = 0�

f�%�A%8.3�

-

6

x

y

O

−1

1y = tanh 1

x

%8.3.

350 Ï"a ø÷Ðó

»»»8.4.'f(x) = x tanh(1/x)�ãy

f(x) = xe1/x − e−1/x

e1/x + e−1/x,

bÝx9Í.�, î»�f3x = 0��=�P-��*Ý�.

limx→0

f(x) = 0,

Æ��Lf(0) = 0, ¸f3x = 0=��¬

f ′(x) =e1/x − e−1/x

e1/x + e−1/x− 1

x(

2

e1/x + e−1/x)2,

Æf3x = 0b×®�Ý�=��f�%�3x = 04=�, ¬´JÞ, Æ0ó�D3��Ä3x = 0��0óC¼0ó/D3, v5½�y1C−1�

f�%�A%8.4�

-

6

1

y = x tanh 1x

x

y

O 1−1

%8.4.

»»»8.5.'f(x) = x sin(1/x), f(0) = 0�f�%�A%8.5�9ÍÐó&Æ|G#ÇÄ�×�Ðó3×b§ �, Îãb§ð��ÝI5XàW�¬30!�, f�\®�, �Äf)Î×=�Ðó�¨²,

f ′(x) = sin1

x− 1

xcos

1

x, x 6= 0,

�:�x → 0`, f ′(x)3∞�−∞ M��êf ′(0)�D3, v3x =

0�¼0óC�0óô/�D3�

êÞ 351

-

6

O

y

x1/2π 1/π 2/π

y = x sin 1x

%8.5.

êêê ÞÞÞ 5.8

1. �J(8.1)�(8.4)P�°ÍÁ§��

2. E�×a ∈ R ∪ {∞,−∞}, �J(i) O(f) + O(g) = O(f + g);

(ii) O(f) ·O(g) = O(fg);

(iii) O(f) · o(g) = o(fg);

3. 'f(x) = O(g(x)), x → a, Í�a ∈ R��J∫ x

0

f(u)du = O(

∫ x

0

g(u)du), x → a�

4. �J

(i) sin x = x + O(x3), x → 0;

(ii) cos x = 1− x2/2 + O(x4), x → 0;

(iii) ex = 1 + x + O(x2), x → 0;

(iv) log(1 + x) = x + O(x2), x → 0;

(v) (1+b/x)x = eb(1−b2/(2x)+O(x−2)), x →∞,Í�b ∈ R�

352 Ï"a ø÷Ðó

5.9 ÓÓÓ��

�h, ��5ÝÃÍÞC, &Æ�K�l#ÇÝ�A!�V�,

&Æ�ÞÚx�ñR¼�|¡��Ý�®, µÎc"�.ÂTTà��5�, XD¡Ý®Þ9�Î�Ðóbn�ÐóÝ�LÎ�

´êÝ: bËÍ/)A�B, ãA�BÝ×ET, ©��A �N×-ô©ET×ÍB�Ý-ô, JhET-Ì ×Ðó�X|&Æ�|b&P&øÝÐó�9Í�94Ðó�b§Ðó�ë�Ðó�Dë�Ðó�¼óCEó, |C¸ÆÝ°JºÕC)WÐóÎtÃÍÝÐó, X|&Æ�ÙÌ� ��Ðó�N×��Ðó/�OÍ�5(�Ä��3b°2]��), Í0

ó) ×��Ðó�X|&ÆE��ÐóÝ�5ºÕ�1��ßé��y�5, 4 �5ÝYºÕ, ×��, 3��5�6�Ý��Q´¥�, vÍ�Õ, �Qô´�5ÌÃ;P�ãyb��5ÃÍ�§, N×�52PF ′(x) = f(x), -ETAìÝ×�52P

∫f(x)dx = F (x) + C�

A5.7 ;X�, XÛ��×�5�@Ý�P, µÎ0�×��ÐóF

��îP��y£°Î�|b�@Ý�P��5ÝÐó, &Æ�-�9Ká¼Ý�¬4Q&Æá¼N×=�Ðó/��, Ç¸&9�P�|ÝÐó, QP°��@Ý�P��5�tÝ|GXèÄÝ, ×°�P��Ý��Ðó��5, �A

∫ex

xdx,

∫ √a0 + a1x + · · ·+ anxndx, n ≥ 3,

/Î»��3èÜtS, ó.��J�î��5, /P�@Ý�P�5?�5.ÝêÝ, ¬&µÎ.Ðó�@Ý��¼��%Þ

N×Ðó��5�@2�î�¼, ©Î.êó.ÈòÝß, ×Ë�QÝ�?�9�£°��÷â&N, '×ÍÍ{)Ã;Ýß,

�^x��Î·¶, Î×øÝ¼§�N×=�Ðó, &Æ/�|

5.9 Ó� 353

ÍRiemannõ, ¼¿�Í3�×T ��5�.h¬�Î1, ×Ðó��5×��|×��Ðó¼�î, �§��¼�hA!Esin 0.2�log 3�e

√2, ¸ÆÝÂ~bÎ9K÷? �Ãô©�|×ó

�¼¿�Ý��Þ×ÐóÝ�5, |��Ðó¼�î, tx�Ýæ.ÛÎÃy&ÆE��ÐóÝP²´Ý�, C�O�ÍóÂ´�|�×ÐóÝ�5, ��|&ÆX!áÝÐó¼�î, ×Í�XÝ

ð°µÎS±Ðó, ÇEh�5ú(�&Æ|G-9ø�Ä, »A, �

log x =

∫ x

1

1

tdt,

.��î�t−1 Ý�5�Ah×¼SÝEóÐó, �&Æô0�Ý�KnyhÐó�P²�ë�Ðó&Æô�v«2S�)©�¢Ãb§Ðó, �Bã�5CãDÐóÝÄ��ÉA1(ù¢�5.6 ;), �|

arctan x =

∫ x

0

1

1 + t2dt

�arc tangentÐó��L, Q¡ãDÐó, ÿÕtangentÐó�AhÍ�"Íë�Ðó-�µ�ÿÕÝ�Q, u�|

arcsin x =

∫ x

0

1√1− t2

dt

arcsineÐó��L, ô�µ�ÿÕ0Íë�Ðó�2à9Ë]P¼�Lë�Ðó, µ��ñÒÝÄ�àÌÝ¿¢�L]P�Q9ø×¼, &Æµm�F�G��0�, Ä�ã¿¢]P-�D|ÿ�Ý×°ë�ÐóÝP²�î�9ËE|��ÐóÝ�5¼�L±ÐóÝ"D, E25

�.Ýs"QÃ��(�ìÛ)�¨×]«, tÝ�s��K�5Ý*», ô�ñðà�5�,Ah×¼, ��Þ£°b�@Ý�P��5, BÄ×°M», ;W�5�î�bÝ�5�P, �O��5�

ÛÛÛ.35.5;�êÞ�, &Æ:Õ4∫ x

1et/tdtP°�W��Ðó, ¬

u�f(x)�h�5, J∫ x

1eat/tdt,

∫ x

1et/t2dtC

∫ x

1e1/tdt/�|f(x)�

354 Ï"a ø÷Ðó

î�9v»��9, 9ì ¨×´¥�Ý, ÇYYYiii��555(elliptic

integrals)�XÛYi�5, Ç�5Õ� ×ëgT°g94PÝ¿]q�

b§Ðó�9Í�©½¥�ÝÎ

(9.1) u(s) =

∫ s

0

1√(1− x2)(1− k2x2)

dx�

Yi�5ú(Ýã¼ h�5®ß�OYiÝ=�(�6.3;)�u(s)

�DÐós(u)ô�¥��k = 0`, .u(s) = arcsin x, Æs(u) =

sin u�s(u)Î×ËJacobianYiÐó, ;ð|snu��, |�îhÐó sineÐó�.Â�u(s) YiÐó�×æ�, hÐó3��Ðó¡��¥�, v3Î§îTàô�Â�b°�5BÄ×°��Ý�ð, -�»ðWYi�5, 4Qæ¼

Ý�P:R¼�Yi�5��×ø�»A, u�u = cos(x/2), J∫

1√cos α− cos x

dx = −k√

2

∫1√

(1− u2)(1− k2u2)du,(9.2)

k =1

cos(α/2);

u�u = sin x, J∫

1√cos 2x

dx =

∫1√

(1− u2)(1− 2u2)du;

u�u = sin x, J∫

1√1− k2 sin2 x

dx =

∫1√

(1− u2)(1− k2u2)du�

&ÆôE�5��5Ýn;, �×°��1��5Î×´�5?ÃÍÝºÕ, ¬v�5¡Xÿ�Ðó, )òy&Æ�áÝÐóP��¬Î=�ÐóÄ��, 4Q�×��îW��Ðó, ��QÎ×´úÝf�, �G&Æ�ÂÄ&9=�Ðó, 3Ø°F��, Í@�WeierstrassÝ`�R, -C��K=�ÐóQÕ��Ý»��9Ë»�&Æº3Ïâa�D¡�

5.9 Ó� 355

ÀÀ¼1, �5��5¬�Î£��|�5¢ï�ºÕ´ÃÍ�×��ÎÆÿ�5´p, ¬�Ø°ÌF¼1, �5Qê´ÃÍ�3Ïâa&Æºw´�5Ý�L, |U��5ÐóÝr½�¨², &ÆD¡ÐóÝ&Ë� , �âÁÂ�D`FC��

a��'ßÝ�¨(T�æ�Wµ) Î�|�;Ý, �f(t)�3` t

Øß��¨�fb`�¦b`�3, ��ß×ß�R��¨3b&ËÉ��8°, �Æð|“��¯¯Ý��í3RþFî”¼ÚS��Þ��XÕ�ÆÝ°î�XÛRþF, ��Wt = 0, Tt

�H�`, ��í3RþF, ��è{f3t = 0!��Â��Äã%9.1�:�, E£��Î´¥�Ý�4Q×��g(t) > f(t), ¬¡¼f(t)�Îø÷g(t)�ÐófÝE£fgÝE£�!

Ot

g(t)

f(t)

%9.1.

Qfô�×�Î×à�¦, 3¾Õ{)`(Á�Â), -��ìªÝ��§cµ�ìª, µÄ6�bD`F, | cÍìªÝ�T,

�¸f��î>�×`Ý?ìª, ¬PH�n;, ¬&Æ�èø� ÝÎ, Äæ“�¹H\¾Õ�EÁ�Â”�b°ß��Î3èâÏ¼�, �î×X?�.`, ¾ÕÁ�Â, �h-P©��¨, Íf(t)

�%�b��A%9.2�ã%9.2�:�, 3t = t0, fbÁ�Â, �¡-z��W, �?��aê��×ËÁ�ÂsßÝ, f-��3,

9`µmxCD`F, -»�T, 3�ìª>�, ¸¾Õ�9(Á�Â)¡, ��î>�×à?îÝßßÎ��sÝ, bRb�, b{�(î>)b�>(ìª), ©�Î¼)b�T(Á�Âº��¨), ßþ

356 Ï"a ø÷Ðó

)b�;(i¿��a��H\�¨), £µ�|Ý�Ý�ÐóÝ9°P², E&ÆÝßßÌ, ºb�K@��!ñ×è, 4Ðóð�J¼à�&ËÿP, »A, |f(t)�î3

` tØË¯Î�Â�¬9�î3` t, �¯Îb×�Â, 9ÎXÛXXX��ÿÿÿPPP(deterministic model)�f´×�(Qô´�Ó)ÝÿP ��^ÿÿÿPPP(stochastic model)�ôµÎ3` t, �¯Î�×�b×�Â, �Î��^ÝÝÝ(random)�9Î×?)§ÝÿP�Ü×��Ý»�¼:, EØ×©½Ý` t, b1

2Ý^£f(t) = 1, 1

3Ý^

£f(t) = 2, 16Ý^£f(t) = 3�A�.µ�Ì?ÕÝf(t)à0�¼,

)ÿ×Ðó%�, ¬ð×gÌ?, ��ºÿÕ×��!ÝÐó, 9µÎ�^Ý�¤�9Ë�^ÝÿP, Þ¼&�3^£¡, T�^Ä�Ý��º.Õ�

Ot

t0

f(t)

%9.2.

Ía+ÛÝ¿Í¥�Ýø÷Ðó�´�Î! DÐóÝEó�¼ó, 9ËÍÐób�9�úÝP², &Æ�D¡�9�°»ó.�Laplace �1Ä“EóÝs�, �;Ý�Õ, ¸FZ.�Ý.ú¦�×¹”�ãh�:�EóÝs�, Xèº�ÕîÝ-¿�=²óÝÍóô�Eóbn:

(9.3) φ(n) =n

log n+ O(

n

log2 n),

Í�φ(n)��øÄn�²óÝÍó�nyEóC¼óÐóÝ×°ßÆ, �¢�R)�(1984)×h�, bnEóC¼óÝ¿SZa�3

5.9 Ó� 357

^£¡�, &9¥�Ý555µµµÐÐÐóóó(distribution function)�¼óÐóbn, �9»»»ððð(transform)ô¢Ã¼óÐó�Þ¼&�ºX�#ÇÕ�êãy¼óÐó, ð�à��QW�ÝÿP, ÍD3�1Î&ð

�QÝ, 5{F§�ÿ�®ßÝ�9�«êb×Ín"Ýó3Yg, Çe9Íø÷ó�¨×Í¥�Ýø÷Ðó ë�Ðó, 9Îø�ÐóÝ��E

" tÝßß�>ÝW�², êb�AÏ¦º»��p�Ç��Ý¨é�ë�Ðó39�, -6�Á¥�Ý��ø÷óπ, êÎÍ�Ýü÷, ¸�eÝ��8ñT��ë�Ðó�¼óÐó , Í@n;Û6, |¡&Æº1��&Æ3�ªZª(½¡, ðºb“\�ßh��Í ß”ÝÕ

��3E¼óCë�Ðó�Ý�¡, &Æô�ÿ�ØY��QÝwú, ±�&Æ9ËÍÐó��ì]«Îè{&Æ�5Ý�æ, ��î]«, JÎè{&Æ�ªó.Ý`æ, ¸&ÆÌ»y"õ9°Ðó, 8YP²Ý�E��

!!!��: ðððààà��555��

1.∫

xndx = 1n+1

xn+1 + C, n 6= −1�

2.∫

1xdx = log |x|+ C�

3.∫

sin xdx = − cos x + C�

4.∫

cos xdx = sin x + C�

5.∫

sec2 xdx = tan x + C�

6.∫

csc2 xdx = − cot x + C�

7.∫

sec x tan xdx = sec x + C�

8.∫

csc x cot xdx = − csc x + C�

358 Ï"a ø÷Ðó

9.∫

exdx = ex + C�

10.∫

1√a2−x2 dx = arcsin x

|a| + C, a 6= 0�

11.∫

1a2+x2 dx = 1

aarctan x

a+ C, a 6= 0�

12.∫

1x√

x2−a2 dx = 1aarcsecx

a+ C�

13.∫

1x√

ax+bdx = 1√

blog

∣∣∣√

ax+b−√

b√ax+b+

√b

∣∣∣ + C, a 6= 0, b > 0�

14.∫

1x√

ax+bdx = 2√−b

arctan√

ax+b−b

+ C, a 6= 0, b < 0�

15.∫

1xn√

ax+bdx = − 1

b(n−1)

√ax+b

xn−1 − (2n−3)a(2n−2)b

∫1

xn−1√

ax+bdx, ab 6= 0,

n 6= 1�

16.∫ √

ax+bx

dx = 2√

ax + b + b∫

1x√

ax+bdx, a 6= 0�

17.∫

1x2−a2 dx = 1

2alog

∣∣x−ax+a

∣∣ + C, a 6= 0�

18.∫ √

x2 ± a2dx = x2

√x2 ± a2 ± a2

2log |x +

√x2 ± a2|+ C�

19.∫

1√x2±a2 dx = log |x +

√x2 ± a2|+ C, a 6= 0�

20.∫ √

a2 − x2dx = x2

√a2 − x2 + a2

2arcsin x

a+ C, a 6= 0�

21.∫

1(x2+a2)n dx = 1

2(n−1)a2

{x

(x2+a2)n−1 + (2n− 3)∫

1(x2+a2)n−1 dx

},

a 6= 0, n 6= −1�

22.∫

x sin xdx = sin x− x cos x + C�

23.∫

xn sin xdx = −xn cos x+nxn−1 sin x−n(n−1)∫

xn−2 sin xdx,

n ≥ 2�

24.∫

x cos xdx = cos x + x sin x + C�

25.∫

xn cos xdx = xn sin x + nxn−1 cos x−n(n− 1)∫

xn−2 cos xdx,

n ≥ 2�

5.9 Ó� 359

26.∫

sinm x cosn xdx

=

{1

m+n(− sinm−1 x cosn+1 x + (m− 1)

∫sinm−2 x cosn xdx)

1m+n

(sinm+1 x cosn−1 x + (n− 1)∫

sinm x cosn−2 xdx),

m + n 6= 0�

27.∫

sinn xdx = − 1n

sinn−1 x cos x + n−1n

∫sinn−2 xdx, n ≥ 2�

28.∫

sin2 xdx = −12sin x cos x + x

2+ C�

29.∫

cosn xdx = 1n

sin x cosn−1 x + n−1n

∫cosn−2 xdx, n ≥ 2�

30.∫

cos2 xdx = 12sin x cos x + x

2+ C�

31.∫

sin2 x cos2 xdx = −14sin x cos3 x + 1

8sin x cos x + x

8+ C�

32.∫

tan xdx = log | sec x|+ C�

33.∫

tan2 xdx = tan x− x + C�

34.∫

tann xdx = 1n−1

tann−1 x− ∫tann−2 xdx, n ≥ 2�

35.∫

cot xdx = log | sin x|+ C�

36.∫

sec xdx = log | sec x + tan x|+ C�

37.∫

secn xdx = 1n−1

(secn−2 x tan x + (n− 2)∫

secn−2 xdx),

n ≥ 2�

38.∫

xeaxdx = 1a2 (ax− 1)eax + C, a 6= 0�

39.∫

xneaxdx = xn

aeax − n

a

∫xn−1eaxdx, a 6= 0, n ≥ 1�

40.∫

eax sin bxdx = 1a2+b2

(a sin bx− b cos bx)eax + C, a 6= 0�

41.∫

eax cos bxdx = 1a2+b2

(a cos bx + b sin bx)eax + C, a 6= 0�

42.∫

log xdx = x log x− x + C�

360 Ï"a ø÷Ðó

43.∫

xm logn xdx = 1m+1

(xm+1 logn x− n∫

xm logn−1 xdx),

m 6= −1�

44.∫

logn xdx = x logn x− n∫

logn−1 xdx�

45.∫

xn log xdx = xn+1

n+1(log x− 1

n+1) + C, n 6= −1�

46.∫

logn xx

dx = 1n+1

logn+1 x + C, n 6= −1�

47.∫

1x log x

dx = log(log x) + C�

48∫

arcsin xdx = x arcsin x +√

1− x2 + C�

49.∫

xn arcsin xdx = 1n+1

xn+1 arcsin x− ∫xn+1√1−x2 dx, n ≥ −1�

50.∫

arctan xdx = x arctan x− 12log(x2 + 1) + C�

51.∫

xn arctan xdx = 1n+1

(xn+1 arctan x− ∫xn+1

x2+1dx), n ≥ −1�

52.∫

arcsecxdx = xarcsecx− log |x +√

x2 − 1|+ C�

¢¢¢��ZZZ¤¤¤

1. R)�(1984). �ó.�I.`�ã, ¬��

2. Bromwich, T. J. I’A. (1991). An Introduction to the Theory

of Infinite Series, 3rd ed. Chelsea Publishing Company, New

York, New York.

ÏÏÏ000aaa

��555��TTTààà

6.1 OOO«««��

�5ÝÏ×ÍTà-Îà¼O«��ãÏÞa�§4.12á, 3×%�ì�«��|ì�]P¼�LÍ«��

��LLL1.1.'Ðóf3[a, b]=�vãÑÂ�|R�f�%�ì, ãa�b

Ý ½�JR�«�A(R)��L

A(R) =

∫ b

a

f(x)dx�

»»»1.1.�f(x) = sin x, �O3f�%�ì, ã0�π/2�«��.µ�LXkO�«�

∫ π/2

0

sin xdx = − cos x∣∣∣π/2

0= 1�

'f3 [a, b]=�,¬ã�Â�J∫ b

af(x)dx < 0,vãx�, f�

%�Cx = a, x = bÞàaX�� ½R�«�-�L

A(R) = −∫ b

a

f(x)dx�

361

362 Ï0a �5�Tà

?×�2, uf�g/3[a, b]=�, vf(x) ≥ g(x), ∀x ∈ [a, b], J+yf�g�%� ãa�bÝ ½R, Í«��L

A(R) =

∫ b

a

(f(x)− g(x))dx�

»»»1.2.�O+yÞÐóf(x) = 2x − x2, �g(x) = x − 2 �%� Ý«��.�OhÞÐó%��øF��

x− 2 = 2x− x2�

��x = 2Tx = −1�ÆÞ%�8øy(2, 0)�(−1,−3)�f �%� ×eÎa, g�%� ×àa, �kO%1.1�YÅI5Ý«��

-

6

(2,0)f(x) = 2x− x2

g(x) = x− 2

ROx

y

(−1,−3)

%1.1.

µ�LR�«�

A(R) =

∫ 2

−1

((2x− x2)− (x− 2))dx

=

∫ 2

−1

(−x2 + x + 2)dx = (−1

3x3 +

1

2x2 + 2x)

∣∣∣2

−1=

9

2�

»»»1.3.�O�5 r�i«��

6.1 O«� 363

��.Ç�Ox2 + y2 = r2�%�/Ý«��ê�Qh«� Ï×é§�«�Ý4¹�Æ

i«� = 4

∫ r

0

√r2 − x2dx = 4

∫ π/2

0

r2 cos2 θdθ

= 4r2

∫ π/2

0

1

2(cos 2θ + 1)dθ = 2r2(sin 2θ/2 + θ)

∣∣∣π/2

0= πr2�

»»»1.4.�O3f(x) = e−x�%�ì, ã0�bÝ«�, Í�b > 0��.

«� =

∫ b

0

e−xdx = 1− e−b�

»»»1.5.�O]�Py2 = 4x2 − x4 �%�X�� ½Ý«��.&Æ��%A%1.2�

-

6

−2 2O

Rx

y

%1.2.

�I%�Î+yx = −2�x = 2 �ãEÌP, ©mO�R �«�,

�¶|4Ç��

A(R) =

∫ 2

0

√4x2 − x4dx

∫ 2

0

x√

4− x2dx

= −1

2

∫ 0

4

√udu− 1

2

2

3u

32

∣∣∣0

4=

8

3,

Í�àÕ�ó�ð, �u = 4− x2�Æ«� 32/3�

364 Ï0a �5�Tà

»»»1.6.A%1.3, i¿ay = c, 3Ï×é§øày = 2x − 3x3yËF��Oc�Â, ¸ÞYÅIÝ«�8!�

-

6

Ox

y

y = c

y = 2x− 3x3

b

%1.3.

��.�(b, c)�àay = c, �à�ÏÞÍøF�&Æ-Î�Oc�Â, ¸ÿ ∫ b

0

(c− (2x− 3x3))dx = 0�

ãîPÇÿ

cb− b2 +3

4b4 = 0�

ê.(b, c)3àî, Æÿ

c = 2b− 3b3�

�îPÇÿb = 2/3, c = 4/9�

êêê ÞÞÞ 6.1

1. �Oì�&]�P�%�X�� ½Ý«�, ¬0Í�%�(1) y = x3, y = 0, x = 1, x = 3�(2) y =

√x, y = −√x, x = 4�

6.2 �� 365

(3) y = x2 − 4, y = 4− x2�(4) y = x2, y = 1�(5) x2y = 4, 3x + y − 7 = 0�(6) y2 = 4x, x = 1�(7) y = −x2 + 4x− 3, 3(0,−3)C(4,−3)�6a�(8) y2 = a2x6 − x8�

2. �Oã`ay = a cosh(x/a), x�Càax = a, x = −aX�� ½�«��

3. �Oã`ay = a tanh(x/a), x�Càax = m, x = nX�� ½�«��

4. �¿à�5, J�EN×ÑJón,

2

3n√

n <√

1 +√

2 + · · ·+√n <

4n + 3

6

√n�

6.2 ��

¿à�5, ô�Oè �Ø° ½Ý��A!«�, è �Ø°©½Ý ½��, &Æ�áA¢O��»A��´�{��& a�bCc��]�, Í�� abc; Ö�Ý�� 9«�¶|{�3h×ñ�CÌ ×ÖÖÖ��(cylinder), u¸ãì�%�X��:

ËÍ5½a3Þ¿�Ý¿«Ý��R1CR2(R1CR25½Ì C�î�ì9), �yC��«, R1CR2�\&îETFÝ=aðXàW, vN×=a/kàR1CR2Xò�¿«, hÞ¿«ÝûÒÌ C�{, �%2.1��y´×�Ý ½��÷?

'b×ñ�S�×¿«u�S8ø, Íø/ ×¿«îÝ ½,

Ì� ^«««(cross section, TÌ^½½½)�'S�Ø×ü�àakà�

366 Ï0a �5�Tà

Xb^«Ý«� �á, vÍ�; =��ùÇ'b×2ýàaL,

¸ÿñ�S+y5½kàLîËFa�b �¿« , vBÄ[a, b]�×FxvkàL�¿«, ES�^«�A(x) �á(�%2.2), ê'ÐóA(x)3[a, b]=��©bE��î�f�Ýñ�S, &Æ��LÍ��

S

R1

R2

%2.1.

A(a) A(x) A(b)

a x bL

%2.2.

'P = {x0, x1, · · · , xk} [a, b]�×5v, ãzi ∈ [xi−1, xi], i =

1, · · · , k, Q¡5½C�{� ∆xi = xi − xi−1, ^«� A(zi)�iÖ�J|ì�Riemannõõõ

R(P ) =k∑

i=1

A(zi)∆xi

6.2 �� 367

�S��V (S)�×�«Â�%2.3 ×k = 3�»�

a = x0 z1 x1 z2 x2 z3 x3 = b x

A(z1) A(z2) A(z3)

%2.3.

óã×ó�[a, b]�5v{Pn, n ≥ 1}, ¸ÿ

limn→∞

||Pn|| = 0,

J&Æ-�LV (S)

V (S) = limn→∞

R(Pn),

Í�R(Pn)�ETPn�×Riemannõ��ãRiemann�§(ÏÞa�§4.9)á, îP��Á§D3v�y

∫ b

aA(x)dx�áîX�, &

Æ-|∫ b

aA(x)dx, ��G«Xà�f��ñ�S Ý��

»»»2.1.�O�5 r�¦SÝ��.A%2.4, 3x− y− z2ý�, Þ¦Tw3æF, &Æ��3x-y ¿«�7Å�N×kàx��¿«, �¦�^«�

A(x) = πy2,

Í�x2 + y2 = r2, x ∈ [−r, r]�Æ

A(x) = π(r2 − x2),

v

V (S) =

∫ r

−r

A(x)dx =

∫ r

−r

π(r2 − x2)dx = π(r2x− 1

3x3)

∣∣∣r

−r=

4

3πr3�

368 Ï0a �5�Tà

−r x rx

y

(x, y)

A(x)

y

%2.4.

ãî»:�¿à�5, &Æ��D|2O�¦��Ã{Æu)3t, Ä��(�

»»»2.2.�S�Þ+yx = 0, x = 4, x�CeÎay2 = x � ½,

�x�I»Xÿ�ñ�, �%2.5��OS��

O x 4x

y

y2 = x

y

%2.5.

��..^«� ×�5y�i, Í«�

A(x) = πy2 = πx,

Æ

V (S) =

∫ 4

0

πxdx = 8π�

6.2 �� 369

»»»2.3.'bÞ{�øÄr,9�5 r�ÞiÖ�,Í�T�kà�OÍ8øI5S��, �%2.6�

z

x

yx2 + y2 = r2

R

z2 + x2 = r2

R

yz

(x, 0, 0) - (x, y, 0)

(r, 0, 0)

%2.6.

��.´��:�S��×kàx��y�Tz��^«/ Ñ]�,

%2.6©��1/4Ý8øI5�A%2.6�YÅI5Ý«�

y2 = yz = z2 = r2 − x2�

Æ^«� A(x) = 4(r2 − x2), x ∈ [−r, r],

v

V (S) =

∫ r

−r

4(r2 − x2)dx = 4(r2x− 1

3x3)

∣∣∣r

−r=

16

3r3�

'f(x), x ∈ [a, b], Í�0 ≤ a < b, ×=�Ðó, ê'f(x) ≥ 0,

∀x ∈ [a, b]�uÞf �%�ì, ãa�bÝ ½, |R��, �x�I»,

JãG«D¡á, Xÿñ�S��

(2.1) V (S) =

∫ b

a

πf 2(x)dx�

370 Ï0a �5�Tà

¬uÞR�y�I», Þÿ×�èÝñ�, |Q��, JQ��V (Q)

¢? (2.1)P�Q�ÊàÝ�(2.1)P�ã¼, ÞS5W�9�ß(Ç{��)ÝÖ��yQ, &Æ�ãõ��, �5W�9�è� û��ÝÖ��ôµÎ'Pn = {x0, x1, · · · , xn} [a, b]�×5v��Qi�3f�%�ì, ãxi−1�xi � ½, �y�I»Xÿ�III»»»��J

(2.2) Q =n⋃

i=1

Qi,

êπmi(x

2i − x2

i−1) ≤ V (Qi) ≤ πMi(x2i − x2

i−1),

Í�mi, Mi5½�f 3[xi−1, xi]�Á�CÁ�Â�ãî��Pÿ

mi ≤ V (Qi)

π(x2i − x2

i−1)≤ Mi�

�ã=�Ðó�� Â�§(Ï×a�§6.3)ÿ, D3×ξi ∈ (xi−1,

xi), ¸ÿ

(2.3)V (Qi)

π(x2i − x2

i−1)= f(ξi)�

.hu�∆xi = xi − xi−1, J

V (Q) =n∑

i=1

V (Qi) =n∑

i=1

π(x2i − x2

i−1)f(ξi)(2.4)

=n∑

i=1

π(xi + xi−1)f(ξi)∆xi

=n∑

i=1

2πξif(ξi)∆xi +n∑

i=1

π(xi − 2ξi + xi−1)f(ξi)∆xi�

'{Pn} ×ó�[a, b]�5v, v��limn→∞ ||Pn|| = 0�Jn →∞ `, (2.4)Pt¡×�r��Ï×4, ��

∫ b

a2πxf(x)dx��y

ÏÞ4õ, .|xi − 2ξi + xi−1| ≤ 2∆xi,

6.2 �� 371

Æ

∣∣∣n∑

i=1

π(xi − 2ξi + xi−1)f(ξi)∆xi

∣∣∣≤ 2πn∑

i=1

|f(ξi)|∆x2i

≤ 2π||Pn||n∑

i=1

|f(ξi)|∆xi → 0 ·∫ b

a

|f(x)|dx = 0�

áîX�, &Æ-�L

(2.5) V (Q) = 2π

∫ b

a

xf(x)dx�

»»»2.4.�¿à(2.5)P, O�5 r�¦SÝ��.�

f(x) =√

r2 − x2, x ∈ [0, r]�Þf�%�ì, ã0�rÝ ½�y�I», Xÿñ��Ý2¹Ç ¦Ý��Æ

V (S) = 2 · 2π∫ r

0

x√

r2 − x2dx = 4π · (−1

3)(r2 − x2)

32

∣∣∣r

0

= −4π

3(0− r3) =

4

3πr3,

h�»2.1 Xÿ8!�

»»»2.5.�Of(x) = log x�%�ì, ãx = 1�x = 3� ½, �y�I», Xÿñ�S��.ã(2.5)Pÿ

V (S) = 2π

∫ 3

1

x log xdx = 2π · 1

2x2(log x− 1

2)∣∣∣3

1

= π(9(log 3− 1

2)− (log 1− 1

2)) = π(9 log 3− 4),

h�∫

x log xdx�à#O, T¿àÏ"a�ðà�5��

372 Ï0a �5�Tà

êêê ÞÞÞ 6.2

1. �0ì�&]�PX�� ½Ý%�, ¬O& ½�x�I»Xÿñ��(i) y = x2, y = 0, x = 2;

(ii) y = log x, y = 0, x = 2, x = 5;

(iii) y =√

4 + x, x = 0, y = 0;

(iv) y = ex, y = 0, x = 1, x = 4;

(v) y = 1/x, y = 0, x = 1, x = 3;

(vi) y = sin x, y = 0, x = π/2�

2. �O9�5 r, { h�Ñi�Ý��

3. 'b×&��=�Ðóf(x)�E∀a > 0, uf�%�ìã0�a

Ý ½, Ex�I», Xÿñ�� a2 + a, OhÐóf(x),

x > 0�

4. 'f(x) = e−2x, x ∈ R��A(t)�f�%�ìã0 �tÝ«�,

V (t)�f�%�ì, ã0�tÝ ½�x�I»Xÿñ��,

W (t)�fã0�tÝ%��y�I»Xÿñ��OA(t),

V (t),W (t)Climt→0 V (t)/A(t)�

5. '

f(x) =

√4x + 2

x(x + 1)(x + 2), x > 0�

�Jf�%�ì, ãx = 1�x = 4� ½, �x�I»Xÿñ�� π log(25/8)�

6. �Oã`ay = sec x, x�, x = π/4Cx = −π/4�� ½,

�x �I», Xÿñ��

7. �Oã`ay =√

x�y = x, 3x ∈ [0, 2] � ½, �y�I»,

Xÿñ��

6.3 =�CI»«� 373

8. �Oy = ex�%�ì, xã1�5 Ý ½,�y�I»,Xÿñ��

9. �Oy = cos x�%�ì, xã14π�1

2π � ½, �y�I», X

ÿñ��

6.3 ===��CCCIII»»»«««��

&Æá¼u×Ðóf =�, JÍ%�-� \; uf ��, JÍ%�ºÈ“¿â”, ��ºbv«f(x) = |x|, Í%�b×´JÞÝF3x = 0�uf =��, JÍ�;£f ′(x)) =�, ��ºbv«

f(x) =

{x2 sin(1/x), x 6= 0,

0, x = 0,

4f ′(x)D3, ∀x ∈ R, ¬x�ê�0, f ′(x)�º�#�f ′(0), �Q�\2®��uf ′ù=�, ��Í%�º´©Îf =�`“?¿â”�&Æ-Ìf =��(Çf ′ =�)`, f ×¿âÐó, T�Ìf ¿¿¿âââ(smooth)�G«Ë;�D¡ÝO«�C��Ý®Þ�E¿«î×à,

&Æô�OÍ%�î, +yØËF �à��, ¬Ì� ===��(arc length)� Ý�;Ðr, |P (x)�àî×F(x, f(x))��uP (c)CP (d) àîÞ8²F,|

↪→P (c)P (d)�hÞF �=(arc)�

Ç↪→

P (c)P (d)�3àî, ãFP (c)��P (d)X�B�I5�'f ×�L3T [a, b]îÝÐó, ×Í£�

↪→P (a)P (b)�=

�Ý]°, |/#5a��õ¼£��'P = {x0, x1, · · · , xn}�[a, b]�×5v,|IP�nÍaðXàW�5aP (x0)P (x1) · · ·P (xn)

�IP��||IP |��, �Q

|IP | =n∑

i=1

P (xi−1)P (xi),

374 Ï0a �5�Tà

Í�P (xi−1)P (xi)�=#P (xi−1)�P (xi)�að��¿àûÒ2P, êÿ

(3.1) |IP | =n∑

i=1

√(xi − xi−1)2 + (f(xi)− f(xi−1))2�

A�f ×��Ðó, JíÂ�§Êà, Ç3(xi−1, xi)�D3×Fzi,

¸ÿf(xi)− f(xi−1) = (xi − xi−1)f

′(zi) = f ′(zi)∆xi,

Í�∆xi = xi − xi−1�.h

(3.2) |IP | =n∑

i=1

√1 + f ′2(zi)∆xi�

x

y

O

P (a)

P (b)

P (x1)

P (x2)

P (x3)

%3.1.

.ËF Ý=a, |àaty, X||IP |�yT�y↪→

P (a)P (b)Ý��vu5vÝ�¼�Þ, &Æï�|IP |�G�=��#��ÇuP1, P2, · · · ×ó�[a, b]�5v, ��limn→∞ ||Pn|| = 0,

J×Í)§Ý�?Î|limn→∞ |IPn|�↪→

P (a)P (b)�=�, ©�hÁ§D3�ãy(3.2)P�Ðó

√1 + f ′23[a, b]�×Riemannõ, �©�hÐ

ó3[a, b]��, JãÏÞa�§4.9á, limn→∞ ||Pn|| = 0 `,

(3.3) limn→∞

|IPn| =∫ b

a

√1 + f ′2(x)dx�

6.3 =�CI»«� 375

�√

1 + f ′23[a, b]��Ý×Í�5f� f ′3[a, b]=�, ôµÎf ×¿âÐó�qA|îD¡, &Æ�ì��L�

��LLL3.1.'f T [a, b]�×¿âÐó, J↪→

P (a)P (b)�=�

(3.4) L =

∫ b

a

√1 + f ′2(x)dx�

»»»3.1.�O3Ðóf(x) = x3/2�%�î, ãx = 0�x = 4�=��..f ′(x) = 3

2x1/2 =�, .h

L =

∫ 4

0

√1 +

9

4xdx =

4

9

3

2(1 +

9

4x)3/2

∣∣∣4

0=

8

27(10√

10− 1)�

»»»3.2.�OeÎay = 4kx2, k > 0, +yx ∈ [a, b]�=��..dy/dx = 8kx =�, Æ

L =

∫ b

a

√1 + 64k2x2dx = 8k

∫ b

a

√x2 + (

1

8k)2dx�

ãÏ"aðà�5��2P18, ÿ

L = 8k(x

2

√x2 +

1

64k2+

1

2

1

64k2log(x +

√x2 +

1

64k2)∣∣∣b

a�

»»»3.3.�Oix2 + y2 = k2, k > 0, �ø��.�y =

√k2 − x2, J

dy

dx=

−x√k2 − x2

,

Æî�i�=�

∫ k

−k

√1 +

x2

k2 − x2dx = k

∫ k

−k

1√k2 − x2

dx = k arcsinx

k

∣∣∣k

−k

= k(arcsin 1− arcsin(−1)) = kπ�

376 Ï0a �5�Tà

.hiø� 2kπ�

Ey�ÎÐó%�Ýà,&Æð|¢¢¢óóó°°°(parameter method)¼à��Ç|

(3.5) x = x(t), y = y(t), t ∈ I,

¼�îhà, Í�tÌ ¢ó�;ð�'x(t)�y(t)/ =�Ðó,

vb8!Ý�L½�Q|GÝÐó%�ù�|

x = x(t) = t, y = y(t) = f(t)

Ý¢ó°¼�î�¨², Aix2 + y2 = r2, Í%�¬&×Ðó%�,

x = r cos t, y = r sin t, t ∈ [0, 2π),

|¢ó°��î�Þ;&Æ�9D¡Ý, �¢�Apostol (1967)

Chapter 14, uI = [a, b], vx(t)�y(t)/ =��, J(3.5)Pà�Ýà�=��L

(3.6) L =

∫ b

a

√(x′(t))2 + (y′(t))2dt�

�p:�u Ðó%�x = t, y = f(t),J(3.6)P-W (3.4)P�Æ|(3.6)P¼�Là�=� �L3.1�×.Â�

»»»3.4.�|¢ó°¥�»3.3��.�

x = x(t) = k cos t, y = y(t) = k sin t, t ∈ [0, 2π)�

Jã(3.6)P,

iø� =

∫ 2π

0

√(−k sin t)2 + (k cos t)2dt =

∫ 2π

0

kdt = 2kπ�

h�»3.2Xÿ8!�

6.3 =�CI»«� 377

»»»3.5.�JYi

x = x(t) = a sin t, y = y(t) = b cos t, t ∈ [0, 2π),

Í�0 < b < a, �ø� ì��5

(3.7) 4a

∫ 2π

0

√1− e2 sin2 tdt,

Í�e =√

a2 − b2/a�JJJ��.ãEÌP, ©6O�3Ï×é§�=��¶|4Ç��Ï×é§�=�

∫ π/2

0

√(x′(t))2 + (y′(t))2dt =

∫ π/2

0

√a2 cos2 t + b2 sin2 tdt

=

∫ π/2

0

√a2 − (a2 − b2) sin2 tdt

= a

∫ π/2

0

√1− ((a2 − b2)/a2) sin2 tdt

= a

∫ π/2

0

√1− e2 sin2 tdt,

ÿJ�(3.7)P��5-Î×ËYi�5(�5.9;)�

t¡, |¢ó°¼O=�, ô�D|2.Â�3îè (#�nîè )�36.1;&ÆXO, KÎbn¿«î%�Ý«��y`«Ý«

�A¢O÷? ÉA1×�5 r�¦Ý�«� ¢? 'b×=�Ðóy = f(x), x ∈ [a, b], Þf�%�Ex�I», ÿÕ×I»`«�3ÊÝf�ì, hI»`«Ý«�Î�|O�Ý�&Æ�:A¢Oi��«��

»»»3.6.'b×Ñi�, ®Þ¿�9�¿«, Þi�^�Þ�55½ r1Cr2�i�E«´' l(�%3.2)�J^��E««� πl(r1 + r2)�

378 Ï0a �5�Tà

JJJ��.´�ÞG�E«6�wHy¿«î, Jÿ×A%3.3b8!c�ÞG�� ÝYÅI5��θ�G��ô�, Þi=�55½ kCk + l�êa1 = 2πr1 = (k + l)θ, a2 = 2πr2 = kθ�%3.3�YÅI5�«

�, ÞG�«��-�Ç

π(k + l)2(θ/2π)− πk2(θ/2π) =θ

2((k + l)2 − k2) =

θl

2(2k + l)

=l

2((k + l)θ + kθ) =

l

2(2πr1 + 2πr2) = πl(r1 + r2)�

ÿJ�

r1

lr2

%3.2.

a1

la2

k

θ

%3.3.

¿à»3.6, &Æ-bOI»««�Ý2PÝ�

6.3 =�CI»«� 379

»»»3.7.'f(x) ≥ 0, ∀x ∈ [a, b], ×=��ÝÐó�Þf�%��x�I», |S�Xÿ�I»«�&Æ�OS �«�A(S)��Pn = {x0, x1, · · · , xn} [a, b]�×5v, &Æ|(xi−1, f(xi−1))

�(xi, f(xi))�=að,�x�I»,ÿÕ×Aî»��i�Ý^«�«�, �f3[xi, xi−1]�%��x�I»XÿI»«�«�Ý�«Â�Jãî»á,

(3.8)

A(S).=

n∑i=1

π(f(xi−1) + f(xi))√

(xi − xi−1)2 + (f(xi)− f(xi−1))2�

x

y

O xixi−1

�ãíÂ�§á, D3×ξi ∈ (xi−1, xi), ¸ÿ

f(xi)− f(xi−1) = (xi − xi−1)f′(ξi)�

ÞîP�á(3.8)P, ÿ

(3.9) A(S).=

n∑i=1

π(f(xi−1) + f(xi))√

1 + (f ′(ξi))2∆xi,

Í�∆xi = xi − xi−1��{Pn}�×ó�[a, b]�5v, ��limn→∞||Pn|| = 0�(3.9)P��, Í@�Î×Riemannõ, ¬�J�(3h¯�, �Ä¢�(2.5)P�.0Ä��pÿÕ), n → ∞`, ¸��

2π

∫ b

a

f(x)√

1 + (f ′(x))2dx�

380 Ï0a �5�Tà

&Æ-�L

(3.10) A(S) = 2π

∫ b

a

f(x)√

1 + (f ′(x))2dx�

uf(x)�×� &�, J

(3.11) A(S) = 2π

∫ b

a

|f(x)|√

1 + (f ′(x))2dx�

»»»3.8.�Of(x) = x3, x ∈ [0, 2], �%��x�I», XÿI»«S�«��.ã(3.10)Pá

A(S) = 2π

∫ 2

0

x3√

1 + 9x4dx = 2π1

54(1 + 9x4)3/2

∣∣∣2

0

=π

27((145)3/2 − 1)�

»»»3.9.�O�5 r�¦Ý�«��.Þf(x) =

√r2 − x2, x ∈ [−r, r],�%��x�I»Çÿ¦«S�.

(3.12) f ′(x) =−x√

r2 − x2, x ∈ (−r, r),

v

f(x)√

1 + (f ′(x))2 =√

r2 − x2√

1 + x2/(r2 − x2) = r�

Æã(3.10)P

(3.13) A(S) = 2π

∫ r

−r

f(x)√

1 + (f ′(x))2dx = 4πr2�

\ïÎÍ�:�, 3(3.12)P�, x��yrT−r, hE(3.13)P��5ÎÍº®ß®Þ?

êÞ 381

êêê ÞÞÞ 6.3

1. �Oày = 4− 2x3/2, ã(0, 4)�(4,−12) �=��

2. �Oày = x3/6 + 1/(2x), ã(1, 2/3)�(3, 14/3) �=��

3. �Oà5y3 = x2, 3ix2 + y2 = 6/�=��

4. �O/Yax2/3 + y2/3 = a2/3�=��

5. �O3ày2 = x3, ãæF�6aE£�x��ô� π/4��=��

6. �O3ày = log(x2 − 1)�%�î, ãx = 2�x = 3�=��(èî: 1

x2−1= 1

x−1− 1

x+1)

7. �Oày =√

x− x2 + arcsin√

x�=��

8. �Oày = arcsin x±√1− x2�=��

9. �O3ey = (ex + 1)/(ex − 1)�%�î, xãa�b �=�, Í�0 < a < b�

10. �O3y = (4x2/9 + 1)3/2�%�î, ã(0, 1)�(2, 125/27) �=��

11. �O3y = x2/4 − log x/2�%�î, ãx = 1�x = 2 �=��

12. �5½O×Ô�º½ì�&à�×µX�B�ûÒ�(i) x(t) = a(1− cos t), y(t) = a(t− sin t), t ∈ [0, 2π], a > 0;

(ii) x(t) = et cos t, y(t) = et sin t, t ∈ [0, 2];

(iii) x(t) = a(cos t + t sin t), y(t) = a(sin t − t cos t), t∈[0, 2π],

a > 0;

13. �J3y = cosh x�%�î,ã(0, 1)�(x, cosh x)�=� sinh x,

Í�x > 0�

382 Ï0a �5�Tà

14. �Oy = sin x, x ∈ [0, π], �%�, Ex�I»XÿI»«�«��

15. �Oy = x3/2, x ∈ [0, 1], �%�, Ex�I»XÿI»«�«��

6.4 ��555��óóóÂÂÂ��ÕÕÕ

u�áf�D0ó, J-P°¿à��5ÃÍ�§O��5∫ b

af(x)dx�Â��Ä&ÆQ�Oh��5��«Â�Í��Þ@

��]°�×-Î¿àÏÞa�§4.9, |Riemannõ¼¿��&Æ�B�h�§Aì�'f 3T [a, b]�=�Ðó, {Pn, n ≥ 1} ×ó��[a, b]Ý

5v, v��

limn→∞

||Pn|| = 0�

ê'R(Pn) �×ETPn�Riemannõ, J

limn→∞

R(Pn) =

∫ b

a

f(x)dx�

ÆE∀ε > 0, D3×ÑJók, ¸ÿn ≥ k`,

R(Pn) ∈ (

∫ b

a

f(x)dx− ε,

∫ b

a

f(x)dx + ε)�

Ç©�n ≥ k, R(Pn) ∫ b

af(x)dx�×�«Â, v0-�yε�

3@jTà`, �ãP1, P2, · · · [a, b]�×Ñ!5vó��uPn

= {x0, x1, · · · , xn}, J�|

R(Pn) =n∑

i=1

f(xi−1)∆x,

6.4 ��5�óÂ�Õ 383

T

R1(Pn) =n∑

i=1

f(xi)∆x,

�ÏnÍRiemannõ, Í�∆x = (b − a)/n�©�nÈ�, R(Pn)�R1(Pn)/�#�

∫ b

af(x)dx�&Æô�|Þï�Õ�¿í¼£�∫ b

af(x)dx, ;ðh ×??Ý£��

1

2(R(Pn) + R1(Pn)) =

∆x

2(

n∑i=1

f(xi−1) +n∑

i=1

f(xi))

=∆x

2(f(x0) +

n−1∑i=1

2f(xi) + f(xn))�

.huPn = {x0, x1, · · · , xn} [a, b]�×Ñ!5v, J(4.1)∫ b

a

f(x)dx.=

b− a

2n(f(x0) + 2f(x1) + · · ·+ 2f(xn−1) + f(xn))�

î��52P-Ì VVV��°°°(trapezoidal rule)�h(Ì�ã¼Aì�'f(x) ≥ 0, x ∈ [a, b], JA%4.1�:�, (4.1)P�� nÍ{,

/ ∆x = (b− a)/n�V�Ý«�õ�

-

6

x0 x1 x2 x3 x4

f(x0)f(x1)f(x2)f(x3)f(x4)

a bx

y

O

%4.1.

384 Ï0a �5�Tà

»»»4.1.'f(x) =√

x2 + 1��¿àV�°O3f�%�ì, ã0�3

�«��.Ç�O

∫ 3

0

√x2 + 1dx�ãÏ"aðà�5��2P18á

∫ 3

0

√x2 + 1dx =

x

2(√

x2 + 1 +1

2log(x +

√x2 + 1)

∣∣∣3

0

=3√

10

2+

1

2(log 3 +

√10)

.= 5.6526�

g¿àV�°, ãn = 6, J

P6 = {0, 0.5, 1, 1.5, 2, 2.5, 3}�

f 39°5vF��ó2��«ÂAì:

x 0 .5 1 1.5 2 2.5 3

f(x) 1 1.12 1.41 1.80 2.24 2.69 3.16

�ã(4.1)Pÿ

∫ 3

0

√x2 + 1dx

.=

1

4(1 + 2.24 + 2.82 + 3.60 + 4.48 + 5.38 + 3.16)

= 5.67�

hÂ�G�µD0óÿÕÝÂ&ð#��

Íg¼:V�°�×.Â, h°Ù�ySimpson (1710-1761), Æ

Ì� Simpson°°°(Simpson’s rule), ×´V�°?Þ@Ý£��5Ý]°�'f ×3[a, b]î=��Ðó, V�°Î|×að¼¿�f�%

�, �Simpson°Î|×eÎa¼¿�f�%�, X|êÌeeeÎÎÎaaa°°°(parabolic rule)�&Æ�:eÎaÝ×°P²��ëÍ��a�8²F(−∆x, y0), (0, y1)C(∆x, y2), Jªb×eÎa

y = a2x2 + a1x + a0

6.4 ��5�óÂ�Õ 385

;ÄhëF, Í�;óa0, a1, a2��ì�ë]�P�

y0 = a2(−∆x)2 + a1(−∆x) + a0,(4.2)

y1 = a0,(4.3)

y2 = a2(∆x)2 + a1∆x + a0�(4.4)

�3heÎaì, ã−∆x�∆x�«�(�%4.2)

∫ ∆x

−∆x

(a2x2 + a1x + a0)dx = (

a2x3

2+

a1x2

2+ a0x)

∣∣∣∆x

−∆x

=∆x

3(2a2(∆x)2 + 6a0)�

ã(4.2)−(4.4)P, �J�

y0 + 4y1 + y2 = 2a2(∆x)2 + 6a0�

hÇ%4.2�eÎaìÝ«��

-

6

x

y

O−∆x ∆x

y0

y1 y2

%4.2.

uÞeÎa¿É, 'b×eÎay = a0x2 + a1x + a0;Ä(x0, y0),

(x1, y1)C(x2, y2)�ëF, �%4.3, Í�∆x = x2 − x1 = x1 − x0�J�QeÎaìÝ«�)8!, Ç

∫ x2

x0

(a2x2 + a1x + a0)dx =

∆x

3(y0 + 4y1 + y2)�

386 Ï0a �5�Tà

'Pn = {x0, x1, · · · , xn} [a, b]�×Ñ!5v, Í�n ×�ó,

v�∆x = (b− a)/n�J∫ b

a

f(x)dx

=

∫ x2

x0

f(x)dx +

∫ x4

x2

f(x)dx + · · ·+∫ xn−2

xn−4

f(x)dx +

∫ xn

xn−2

f(x)dx�

Í�N×�5∫ xi+2

xif(x)dx/�|×;ÄëF(xi, f(xi)), (xi+1,

f(xi+1))C(xi+2, f(xi+2))�eÎaìÝ«�¼¿��êãG«�ÿÝ��á:

-

6

x

y

O x0 x1 x2

y0y1 y2∆x ∆x

%4.3.∫ x2

x0

f(x)dx.=

∆x

3(f(x0) + 4f(x1) + f(x2)),

∫ x4

x2

f(x)dx.=

∆x

3(f(x2) + 4f(x3) + f(x4)),

...∫ xn−2

xn−4

f(x)dx.=

∆x

3(f(xn−4) + 4f(xn−3) + f(xn−2)),

∫ xn

xn−2

f(x)dx.=

∆x

3(f(xn−2) + 4f(xn−1) + f(xn))�

Þ9°P�¼��&5½8�, Çÿ∫ b

a

f(x)dx.=

b− a

3n(f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + · · ·

+2f(xn−2) + 4f(xn−1) + f(xn))�(4.5)

6.4 ��5�óÂ�Õ 387

hÇSimpson°E�5�¿��

»»»4.2.�¿àSimpson°, £�∫ 1

0

√1− x2dx, ãn = 4�

��.´�P4 = {0, 0.25, 0.5, 0.75, 1}�

√1− x239°5vF��ó3��«ÂAì:

x 0 14

12

34

1√1− x2 1 .968 .866 .661 0

.hãSimpson°∫ 1

0

√1− x2dx

.=

1

12(1 + 4 · .968 + 2 · .866 + 4 · .661 + 0)

.= 0.771�

¯@î(�»1.3)

∫ 1

0

√1− x2dx = π/4

.= 0.785�

unãÿ?�, J0-Qº?��

»»»4.3.�Of(x) = 1/x�%�î, ã(1, 1)�(5, 1/5) �=�Ý�«Â��..f ′(x) = −1/x2, Æ=�

∫ 5

1

√1 + 1/x4dx =

∫ 5

1

√1 + x4/x2dx�

ãn = 4, P4 = {1, 2, 3, 4, 5}, �ÿî��5��«Â

1

3(√

2 + 4 ·√

17

4+ 2 ·

√82

9+ 4 ·

√257

16+

√626

25)

.=

1

3(1.414 + 4.124 + 2.014 + 4.008 + 1.001)

.= 4.187�

@j=�JV 4.08�

388 Ï0a �5�Tà

êêê ÞÞÞ 6.4

1. µV�°, �Oì�&�5��«Â�(i)

∫ 3

11/xdx, n = 8; (ii)

∫ −1

−41/xdx, n = 6;

(iii) 4∫ 1

0

√1− x2dx, n = 4; (iv)

∫ 1

04/(1 + x2)dx, n = 4;

(v)∫ 3

−1

√4 + x3dx, n = 4�

2. µV�°,ãn = 4,�OYix2+16y2 = 25,ã(3, 1)�(5, 0) �=�Ý�«Â�

3. �µSimpson°, ¥�Ï1Þ�

4. �¿àSimpson°, ãn = 6, O/Yax2/3 + y2/3 = a2/3X� ½�«�Ý�«Â�

5. �¿àSimpson°, ãn = 4, O5y = x5/2�%�î, ãx =

0�x = 1 �=�Ý�«Â�

6. �¿àSimpson°, ãn = 4, Oy2 = x5/25�%�î, ãx = 0

�x = 2 �=��

ÏÏÏÚÚÚaaa

óóó��CCCùùùóóó

7.1 GGG��

&Æ�D¡Äó�, ùóô9gèÕ�¯@î8*\��.R,

��-�!�ùó, LÍÎ�A�-ùóC�fùó��'b×ó�a1, a2, · · · , an, · · ·�Jµ�OÍI5õ, �ÿó

�{sn, n ≥ 1}, Í�

s1 = a1, s2 = a1 + a2, · · · , sn = a1 + · · ·+ an =n∑

i=1

ai�

n → ∞, -ÿÕõ∑∞

i=1 ai, hÌ ×PPP§§§ùùùóóó(infinite series), T©�Ìùùùóóó(series), .t&©½Î�, ÍJ;ð&Æ©�ÊP§94Ý@óùó�D�, 'b×ùó

∑∞i=1 ai, JÍI5õs1 = a1,

s2 = a1 + a2, · · · , ùxW×ó��u

limn→∞

sn = A,

Í�A ×@ó, JÌùó∑∞

i=1 ai[e, vAÌ h[eùó�õ, ÍJÌ s÷�×s÷ùó�õ¬�D3�3

∑∞i=1 ai�,

a1, a2, · · ·Ì ùóÝ4, �an Ì Ïn4T×××��444(general term)�

389

390 ÏÚa ó�Cùó

3��5s"Ý��, ó.�¬Îº�ùóÝ[eCs÷Ý®Þ��Æ.ùóÚ �b§ÍóÝõv«, | æ�Ý�óºÕ)Wñ�»A,

a1 + a2 + · · ·+ a2n = (a1 + a2) + (a3 + a4) + · · ·+ (a2n−1 + a2n),

a1 + a2 + a3 = a1 + a3 + a2�

Ý�Õ]-, b`&Æºb9vºÕ�¬EP§ÍóÝõ, 9ËºÕQ�×�)Wñ�\�ó.�X�ÿÝ×°| ÎEÝ��,

Í�b°¡¼Q�J�ÎýÝ�5?, A|GXèÄÝ, £°��5Ý�xï, �Æ9�íÌbÁAÞÝàÆ, �vW°»ú, .h4Q�Æ��ÿ�J�, Xb�ÆÿÕÝ��, ¬�Æ¬ÎÙìH9Ýý�3£°�xï�, qÂÿ×èÝÎ�Z��¬×Í#×Í, s¨Ý&9YúÝó.2P, ¬¿àP§ùó, Þó.�&9�8�Ýr½J)R¼��K�ZÝW��;ÄaªÝl��øFì¼, K�h�y�²{´ðÝàÆ, á¼%��, ó.îÎÑ@Ý�ùóÝûÅ�D¡, Û��èÚtS¡f, V\y�Z�ßÝ

"èOG, ¬��5.\�Ýs"!M�Mercater (1620-1687)CBrouncker (1620-1684), y�-1668O, ÝOÔ`a%�ìØ ½Ý«�, s¨Ý|×ùó¼�îEóÝ2P�hs¨�Ú ó.ªîÝ×¦�*�¨², ãÞ4P�§ÿEN×ÑJón,

(1 + x)n =n∑

i=0

(n

i

)xi�

pñs¨, î�2P�.ÂÕn �×@ó, ©�ÞîP��; ×ÊÝP§ùó��Ä�¬Î��¢J��3ì×a&ÆºD¡h®Þ, '`��:�, 9Í��ÕùóÝ[e®Þ, �£��ÎpñÝ`��/�Ý�3�Z�?(�-1783O)¡�ò, ùóÝ@~#�-<[ì

¼�àÕ�-1812O, {ús�×½(Ý@~ ×, Í��âaªî, Ï×gEØ°©½Ýùó�[eP, �JC�ÛÝ�

7.1 G� 391

§�¿O¡, Þ�3�1821O�ÌÝ½®�, +ÛÁ§ÃFÝ��L�Ah×¼,¨�ó.�[eCs÷ÝÃ��Õ¦�ì¼�Ía-Î�EùóCÍ8n®Þ�×°�MD¡�tÝÏ×a�ÄÝó�[e��LC�§, 3h&Æ��èº

×¾\ó�[eTs÷Ýbà��´�u×ó�{an, n ≥ 1},[eÕ×b§ÝÂa, Jn��`, an�a�-ûÞ��.h,

um�n/��,Jam�an�-ûù��ôµÎ©�4óÈ�,J�Ë4��#��ÑP21, &Æbì��§�

��§§§1.1.'{an, n ≥ 1}[e�JE∀ε > 0, D3×n0 ≥ 1, ¸ÿ

|am − an| < ε, ∀m,n ≥ n0�

JJJ��.�lim

n→∞an = a�

J∀ε > 0, D3×n0 ≥ 1, ¸ÿ

|an − a| < ε/2, ∀n ≥ n0�

ÆEm, n ≥ n0,

|am − an| ≤ |am − a|+ |an − a| < ε/2 + ε/2 = ε�

J±�

ãî��§ÇS�9ìÝ�L�

��LLL1.1.ó�{an, n ≥ 1}Ì ×ÞÞÞ��óóó��(Cauchy sequence), uÍ��ì�ÞÞÞ��fff��(Cauchy condition):

∀ε > 0, D3×n0 ≥ 1, ¸ÿ|am − an| < ε, ∀m,n ≥ n0�

�§1.1¼�, N×[eó�Ä ×Þ�ó��ÍYùË, �ì��§�J�3h¯�, �¢�Apostol (1974) Theorem 4.8�


��§§§1.2.N×Þ�ó�Ä[e�

�§1.2, ;ðÎà¼E×¯��á¼Á§Â ¢Ýó�, J�Í[e�ã�§1.1C1.2á, ×ó�[e, uv°uhó� Þ�ó��hÌ ÞÞÞ��[[[eee¾¾¾½½½°°°(Cauchy’s convergence criterion)�

»»»1.1.�

an = 1− 1

2+

1

3− 1

4+ · · ·+ (−1)n−1 1

n, n ≥ 1�

k¾\hó�ÎÍ[e, 4Á§Â¬�|:�(&Æ}¡º�D¡Í®Þ), ¬E∀m > n ≥ n0,

|am − an| = | 1

n + 1− 1

n + 2+ · · ·+ (−1)m−n−1 1

m| < 1

n≤ 1

n0

(�êÞ), Æ

|am − an| < ε, ∀m > n ≥ n0 > 1/ε�

.h{an, n ≥ 1} ×Þ�ó��ã�§1.2á, limn→∞ anD3�

»»»1.2.'ó�{an, n ≥ 1}, ��

|an+2 − an+1| ≤ 1

2|an+1 − an|, ∀n ≥ 1�

9ìJ�limn→∞ anD3��bn = |an+1 − an|, J

0 ≤ bn+1 ≤ bn/2,

Æãó.hû°, �ÿ0 ≤ bn+1 ≤ b1/2n, ∀n ≥ 1�.hlimn→∞ bn =

0�êE∀m > n,

am − an =m−1∑

k=n

(ak+1 − ak),

7.1 G� 393

.h

|am − an| ≤m−1∑

k=n

bk ≤ bn(1 +1

2+ · · ·+ 1

2m−1−n) < 2bn�

ÆÿJ{an, n ≥ 1} ×Þ�ó�, .hlimn→∞ anD3�

b`ô�¿àô^�§, ¼Oó��Á§Â�

»»»1.3.�5½Oì�ëÁ§Â�(i) limn→∞( 1√

n2+12 + 1√n2+22 + · · ·+ 1√

n2+n2 ),

(ii) limn→∞( 1√n2+1

+ 1√n2+2

+ · · ·+ 1√n2+n

),

(iii) limn→∞( 1√n2+1

+ 1√n2+2

+ · · ·+ 1√n2+n2 )�

��.(i) kO�Á§

limn→∞

1

n

n∑i=1

1√1 + (i/n)2

=

∫ 1

0

1√1 + x2

dx

= log(x +√

1 + x2)∣∣∣1

0= log(1 +

√2)�

(ii) .

1√n2 + n

≤ 1√n2 + i

≤ 1√n2 + 1

, i = 1, 2, · · · , n,

Æn√

n2 + n≤

n∑i=1

1√n2 + i

≤ n√n2 + 1

�

n →∞, îPËÐ�Á§Âí 1�ÆkO�Á§Â 1�(iii) .

1√n2 + i

≥ 1√n2 + n2

=1√2n

, i = 1, 2, · · · , n2,

Æn2∑i=1

1√n2 + i

≥ n2

√2n

=n√2�

.hXkO�Á§ ∞�


êêê ÞÞÞ 7.1

1. �Jì�&�P�(i) limn→∞ 1

n

∑ni=1(

in)2 = 1

3,

(ii) limn→∞∑n

i=11

n+i= log 2,

(iii) limn→∞∑n

i=1n

n2+i2= π

4,

(iv) limn→∞∑n

i=11n

sin iπn

= 2π,

(v) limn→∞∑n

i=11n

sin2 iπn

= 12,

(vi) limn→∞1+ n√e+

n√e2+···+ n√

en−1

n= e− 1�

2. 'bó�{an, n ≥ 1}, an5½Aì��5½¾\&ó�ÎÍ[e, u[e¬O�Á§Â�(i) an = n(−1)n

, (ii) an = nan, |a| < 1, (iii) an = 1 + (−1)n,

(iv) an = (−1)n

n+ 1+(−1)n

2, (v) an = n2/3 sin(n!)

n+1 �

3. '|an| < 2, v|an+2 − an+1| ≤ 18|a2

n+1 − a2n|, ∀n ≥ 1��

J{an, n ≥ 1}[e�4. '×&�ó�{an, n ≥ 1}, ��

(2− an)an+1 = 1�

�Jlimn→∞ an = 1�

5. �an = (n + 1)c − nc, Í�c ×@ó��X�¸limn→∞ anD3�cÂ, ¬Oh`�limn→∞ an�

6. (i) '0 < x < 1, �Olimn→∞(1 + xn)1/n;

(ii) 'a > b > 0, �Olimn→∞(an + bn)1/n�

7. 'ó�{an, n ≥ 1}��an+1 = (an + an−1)/2, n ≥ 2�(i) 'limn→∞ anD3��|a1Ca2�hÁ§�

7.2 ùóÝÃÍP² 395

(ii) �JE��a1Ca2, limn→∞ anD3�(èî: 5½�Êó�{a2n, n ≥ 1}C{a2n−1, n ≥ 1})

8. 'b×ó�{an, n ≥ 1}, a1 = 1, an+1 =√

1 + an, n ≥ 1��Jhó�[e¬OÍÁ§Â�

9. 'b×ó�{an, n ≥ 0}, a0 = 1, a1 = 1, a−1n+2 = a−1

n+1 + a−1n ,

n ≥ 0��Jhó�[e¬OÍÁ§Â�

10. �JE∀m > n ≥ 1,

0 <1

n + 1− 1

n + 2+ · · ·+ (−1)m−n−1 1

m<

1

n + 1�

11. '∑∞

i=1 ai[e, �Jn → ∞`, Ïn4¡�õ4Rn → 0, Í�

Rn = an+1 + an+2 + · · ·�

7.2 ùùùóóóÝÝÝÃÃÃÍÍÍPPP²²²

×ùóubì��P, -Ì ×2f r �¿¿¿¢¢¢ùùùóóó(geometric

series):

∞∑i=1

ari−1 = a + ar + ar2 + · · ·+ arn−1 + · · ·�

ã

1− rn = (1− r)(1 + r + r2 + · · ·+ rn−1),

�ÿ´n4�I5õ

sn = a

n∑i=1

ri−1 =a(1− rn)

1− r=

a

1− r− arn

1− r, r 6= 1�


.lim

n→∞rn = 0, ∀|r| < 1,

Ælim

n→∞sn =

a

1− r− a

1− rlim

n→∞rn =

a

1− r, ∀|r| < 1�

ÇJ�E∀|r| < 1, ¿¢ùó[e, võ a/(1− r)�Ç

(2.1)∞∑i=1

ari−1 =a

1− r, |r| < 1�

\ïô��|Þ�[e¾½°, ¼J�NÍ|r| < 1 Ý¿¢ùó/[e�¿à(2.1)P, �ÞØ°5P"�Wùó�»A, uãa = 1, r =

−x2, Jÿ

1

1 + x2=

1

1− (−x2)= 1 + (−x2) + (−x2)2 + · · ·= 1− x2 + x4 − x6 + · · ·=

∞∑i=1

(−1)i−1x2i−2 =∞∑i=0

(−1)ix2i�

î�ùóEx2 < 1, T|x| < 1[e�N×ùóÝI5õxW×ó��D�, EN×ó�{sn, n ≥ 1},

©��

a1 = s1, a2 = s2 − s1, · · · , an = sn − sn−1, · · · ,

Jsn = a1 + · · ·+ an�

Ç{sn, n ≥ 1} [eùó∑∞i=1 ai�I5õó��

'×ó�{sn, n ≥ 1}�Á§D3�u�*G«b§4, ÉA1m − 14, �ÿÕó�sm, sm+1, · · · , sm+n, · · · , J)b8!ÝÁ§�Ç

limn→∞

sn = limn→∞

sm+n, ∀m ≥ 1�


©½2, usn ×[eùó∑∞

i=1 ai�´n 4ÝI5õ, J

an = sn − sn−1,

vlim

n→∞an = lim

n→∞sn − lim

n→∞sn−1 = 0�

ã|îD¡Çÿì��§�

��§§§2.1.'ùó∑∞

i=1 ai[e, Jlimn→∞ an = 0�

�§2.1�Y�Ë, Çlimn→∞ an = 0`, ùó∑∞

i=1 ai�×�[e��Ä�§2.1ð�J¼¾½×ùóÎÍs÷�Çulimn→∞ an

6= 0, J∑∞

i=1 ais÷�»A, E×¿¢ùó∑∞

i=1 ari−1, u|r| ≥ 1, J.

limn→∞

rn 6= 0,

Æ|r| ≥ 1`, î�¿¢ùós÷�

»»»2.1.ùó∞∑i=1

1

i= 1 +

1

2+ · · ·+ 1

n+ · · ·

Ì ��õõõùùùóóó(harmonic series)�.E∀n ≥ 1,

s2n − sn =1

n + 1+

1

n + 2+ · · ·+ 1

2n

>1

2n+

1

2n+ · · ·+ 1

2n=

1

2,

Æã�§1.1á, ó�{sn, n ≥ 1}s÷�.hùó∑∞

i=1 1/is÷, 4Qn →∞`, hùóÝ×�4an = 1/n → 0�

»»»2.2.�Êùó

1 +1√2

+1√3

+ · · ·+ 1√n

+ · · · =∞∑i=1

1√i�


n →∞`, ×�4an = 1/√

n → 0�¬´n4õ

sn >1√n

+ · · ·+ 1√n

=n√n

=√

n →∞�

Æhùós÷�

�ÄQ¬&N×s÷ùó, ÍI5õÄ��∞T−∞�»A, Eùó

1− 1 + 1− 1 + 1− 1 + · · · ,

Í�Ïn4�I5õsn b§, vøý½ 1T0, Æhùós÷�A!�5

∫ b

af(x)dx�Ýx, 3

∑∞i=1 ai�Ýi ù Ìa�ó, �

|ðWj, k, l��êb`ùóõº�i = 0TØ×Jó��×��, EN×Jók ≥ 0,

∑∞i=k ai�

∑∞i=1 biÎ×øÝ, Í�bi =

ak+i−1�u�º��, Ý�-, b`|∑

ai��∑∞

i=k ai�EN×k ≥ 1, ãy

∞∑i=1

ai =k−1∑i=1

ai +∞∑

i=k

ai,

�b§Íõ∑k−1

i=1 aiÄ b§, Æ∑∞

i=1 ai�∑∞

i=k ai!`[eTs÷�h� Ý-¿, k = 1, �L

∑0i=1 ai = 0�ôµÎvAó�,

×ùóÝ[eTs÷, �å�áTJ�×°b§4ÝÅ(�9Î %�b`|

∑ai�¶°��

∑∞i=k ai(�î×ð), ��H�´Î�Ï

¿4��Íg, &Æ:ùóÝaPP²�´�Eb§ÍóÝõ, b9ìÞ

��¬¥�ÝP²:

(2.2)n∑

i=1

(ai + bi) =n∑

i=1

ai +n∑

i=1

bi,

C

(2.3)n∑

i=1

(cai) = c

n∑i=1

ai,


Í�c ×ðó�îÞP²)¿¡, Ç b§õÝaPP²: E�Þα, β ∈ R,

(2.4)n∑

i=1

(αai + βbi) = α

n∑i=1

ai + β

n∑i=1

bi�

î��, �.Â�ùóÝ�µ�ì��§¼�, EÞ[eùó)baPÝP²�

��§§§2.2.'∑∞

i=1 aiC∑∞

i=1 bi Þ[eùó, vα, β Þ@ó�Jùó

∑(αai + βbi)ù[e, vÍõ��ìP:

(2.5)∞∑i=1

(αai + βbi) = α

∞∑i=1

ai + β

∞∑i=1

bi�

JJJ��.ã(2.4)P, E∀n ≥ 1,

n∑i=1

(αai + βbi) = α

n∑i=1

ai + β

n∑i=1

bi�

�n → ∞, JîP��Ï×4, ��α∑∞

i=1 ai, �ÏÞ4��β

∑∞i=1 bi�Æn → ∞`, îP¼�, ��G�ÞÁ§Â�õ,

Ç(2.5)PWñ�

��§§§2.3.u∑

ai[ev∑

bis÷, J∑

(ai + bi)s÷�JJJ��..bi = (ai+bi)−ai,�

∑ai[e,Æ�§2.2¼�,u

∑(ai+bi)[

e, J∑

biù[e�Æ∑

bis÷`,∑

(ai + bi)��[e�

»»»2.3..∑

1/is÷�∑

1/2i[e, Æ∑

(1/i + 1/2i)s÷�

�yu∑

aiC∑

bi/s÷, h`∑

(ai + bi)ºA¢? �n �×�,b`[eb`s÷�»A,uai = bi = 1,J

∑ai,

∑biC

∑(ai+

bi)/s÷; �uai = 1, bi = −1, J∑

ai�∑

bi)/s÷, ¬∑

(ai +

bi)[e�


Íg&Æ¼:¥¥¥PPPùùùóóó(telescoping series)�ub×ùó∑∞

i=1 ai,

Í�

ai = bi − bi+1, i ≥ 1,

JÌh ¥Pùó�&Æbì��§�

��§§§2.4.'ai = bi − bi+1, i ≥ 1�J∑

ai[e, uv°uó�{bi}[e, vh`

∞∑i=1

ai = b1 − limi→∞

bi�

JJJ��.�sn =∑n

i=1 ai, J

sn = (b1 − b2) + (b2 − b3) + · · ·+ (bn − bn+1) = b1 − bn+1�

Æó�{sn}�{bn}, !`[eTs÷�v[e`,

∞∑i=1

ai = limn→∞

sn = b1 − limn→∞

bn+1 = b1 − limi→∞

bi�

»»»2.4.'an = (n2 + n)−1�J

an =1

n− 1

n + 1,

v∑∞

n=1 an ×¥Pùó�Æ∞∑

n=1

an = 1− limn→∞

1

n= 1�

»»»2.5.'

an =1

(n + x)(n + x + 1)(n + x + 2)

=1

2(

1

(n + x)(n + x + 1)− 1

(n + x + 1)(n + x + 2)), n ≥ 1,

êÞ 401

Í�x� ×�Jó�J∑∞

n=1 an ×¥Pùó, v

∞∑n=1

an =1

2(x + 1)(x + 2)�

»»»2.6..log

n

n + 1= log n− log(n + 1),

vlimn→∞ log n = ∞, Æ∑∞

n=1 log(n/(n + 1))s÷�

»»»2.7.�J

(2.6) π = 4∞∑i=1

arccot(2i2)�

JJJ��.´�E∀n ≥ 1, ¿àarctan u + arctan v = arctan((u + v)/(1−uv)), Carctan(−u) = − arctan u, �ÿ

n∑i=1

arccot(2i2) =n∑

i=1

arctan(1

2i2) =

n∑i=1

arctan((2i + 1)− (2i− 1)

(2i + 1)(2i− 1) + 1)

= −n∑

i=1

(arctan(2i− 1)− arctan(2i + 1)

= arctan 1− arctan(2n + 1)�

�n →∞Çÿ(2.6)P�

&9ùó&Æ©�¾\Í[eTs÷, �yõµ�×��|O��¥Pùóu[eJÍõ�O�, ©�limn→∞ bn�O��¨²,

¿¢ùóôÎ�O�ÍõÝùó�

êêê ÞÞÞ 7.2

1. �5½J�ì�ùó[e, ¬O�Íõ�(1)

∑∞n=1

1(2n−1)(2n+1)� (2)

∑∞n=1

n(n+1)(n+2)(n+3)�


(3)∑∞

n=21

n2−1� (4)∑∞

n=12n+1

n2(n+1)2�(5)

∑∞n=1

2n+3n

6n � (6)∑∞

n=12n+n2+n

2n+1n(n+1)�(7)

∑∞n=1

√n+1−√n√

n2+n � (8)∑∞

n=1(−1)n−1(2n+1)

n(n+1) �(9)

∑∞n=1

n(n+1)!� (10)

∑∞n=1

n2−n−1n! �

(11)∑∞

n=1(n−1)!(n+1)!� (12)

∑∞n=1

1n(n+1)(n+2)�

(13)∑∞

n=2log((1+1/n)n(1+n))

(log nn)(log(n+1)n+1)� (14)∑∞

n=1 arctan 1n2+n+1�

(15)∑∞

k=16k

(3k+1−2k+1)(3k−2k)� (16)∑∞

n=2 log(1− 1/n2)�

7.3 ÑÑÑ444ùùùóóó

'b×ùó∑

ai, uai≥ 0, ∀i≥ 1, JÌh ×ÑÑÑ444ùùùóóó(positive

term series, Tnonnegative term series)�E×Ñ4ùó, .I5õ{sn, n ≥ 1}, ×�¦ó�, ¿àÏ×a�§1.1, -ÿ9ìÝ��

��§§§3.1.'∑

ai ×Ñ4ùó�J∑

ai[e, uv°uI5õó�bî&�

31.3;, &Æ�¿à�§3.1, J�ùó∑∞

i=0 1/i![e�§¡î¼1, k¾½×ùó�e÷P, 6l�ÍI5õó

�{sn, n ≥ 1}, ÎÍÁ§D3�¬Aî×;èÄÝ, �9`Îsn

¬P��Ý�P, .hôµ�|:�n → ∞`, sn�Á§ÂÎÍD3�\�Ý"Dï�AÞ��ß, -º�Õb9ÍpÞD3, .�s"�×°¾½ùóe÷PÝ]°, ��àBãOI5õ¼¾½�ùó�e÷PÝ¾½°bë�v: (i) �5f�, (ii) Ä�f�,

(iii) ��f��'C nyùó∑

ai�Ø×f�, Jî�ëv5½�Aì1�:

(i) uCWñ, J∑

ai[e;

(ii) u∑

ai[e, JCWñ;

7.3 Ñ4ùó 403

(iii)∑

ai[e, uv°uCWñ�»A, '

∑ai ×¥Pùó, ai = bi − bi+1, Jlimi→∞ biD3,

∑

ai[e�×�5f��limn→∞ an = 0 ∑

ai[e�×Ä�f��êu�á

∑ai ×�fùó, J2f�y1, hùó[eÝ

��f��EÄ�f�, �à¼¾½ùós÷, Çulimn→∞ an 6= 0,

Jhùós÷�

Í;�D¡nyÑ4ùó�eee÷÷÷PPPÝ¾½°�

��§§§3.2.fff´lll��°°°(comparison test). '∑

aiC∑

bi ÞÑ4ùó�uD3×ÑÝðóc, ¸ÿ

(3.1) ai ≤ cbi, ∀i ≥ 1,

J∑

bi[e, 0l∑

ai[e�JJJ��.�sn = a1 + · · ·+an, tn = b1 + · · ·+ bn, n ≥ 1�J(3.1)PWñ,

0lsn ≤ ctn�u∑

bi[e, JD3×�y0 �ðóM , ¸ÿtn =∑ni=1 bi ≤ M , .hsn ≤ cM�Æã�§3.1á,

∑ai[e�J±�

3�§3.2�f�ì, ×Í��Ý�� , u∑

ai s÷, J∑

bis÷��uD3×c > 0, ¸ÿ(3.1)PWñ, &Æ-1ùó

∑aiååå×××

yyy∑

bi(∑

ai is dominated by∑

bi), T1∑

biYg∑

ai�ê�QuD3×n0 ≥ 1, ¸ÿ(3.1)PWñ, ∀i ≥ n0, J�§3.2��)Wñ�#�, u{an}C{bn}Î�Ø4�� &�, J�§3.2 ��)Wñ�×ùó�e÷P, �åG«b§4ÝÅ(�Í;Í�Ý�§ô�bv«Ý.Â�

��§§§3.3.ÁÁÁ§§§fff´lll��°°°(limit comparison test). 'an, bn > 0,

∀n ≥ 1, v

(3.2) limn→∞

an

bn

= 1�

J∑

ai[e, uv°u∑

bi[e�


JJJ��.ã(3.2)P, ÇÿD3×n0 ≥ 1, ¸ÿn ≥ n0`, 1/2 < an/bn <

3/2�.hbn < 2an, van < 3bn/2, ∀n ≥ n0�Æã�§3.2, ÇÿJÍ�§�

ÛÛÛ3.1. uD3×ðóc > 0, ¸ÿ

limn→∞

an

cbn

= 1,

ùÇ

(3.3) limn→∞

an

bn

= c

`, ã�§3.3á, h`∑

ai�∑

cbi !`[eTs÷�Æ(3.3)PWñ`,

∑ai [e,uv°u

∑bi[e�¨²,uc = 0,J|�h`

©�ÿÕ�¡:∑

bi[e, 0l∑

ai[e, �∑

ais÷, 0l∑

bis÷�t¡, uc = ∞, J

∑bis÷, 0l

∑ais÷,

∑ai[e, 0

l∑

bi[e�

Íg��×°�L�

��LLL3.1.'bÞó�{an}�{bn}, ��

limn→∞

an

bn

= 1,

JÌ{an}�{bn}��«««888��(asymptotically equal), v|

an ∼ bn, n →∞

��(\�an is asymptotically equal to bn)�

ã�§3.3Çÿì�.¡�

��§§§3.1.'bÞÑ4ùó∑

anC∑

bn, an, bn > 0, ∀n ≥ 1, vD3×c > 0, ¸ÿan ∼ cbn, n → ∞�J

∑an�

∑bn!`[e, Ts

÷�

7.3 Ñ4ùó 405

»»»3.1.3»2.4, &Æ�J�∑

(n2 + n)−1 ×[eÝ¥Pùó�.1

n2∼ 1

n2 + n, n →∞,

Æã�§3.1á,∑

1/n2[e�êE∀s > 2,∑

1/nså×y∑

1/n2�Æ

∑1/ns[e, ∀s ≥ 2�}¡&ÆºJ�, ¯@î

∑1/ns[e,

∀s > 1�u�

(3.4) ζ(s) =∞∑

n=1

1

ns, s > 1,

-�L�½(ÝRiemann zetaÐÐÐóóó(Riemann zeta-function, �ÌzetaÐÐÐóóó)��Zs¨&9nyζ(s)�YúÝ2P, »A,

ζ(2) =∞∑

n=1

1

n2=

π2

6,

h2PêGãy�Ì��, &Æº�8.6;�.0�

»»»3.2.ãy∑

1/ns÷, Æu×Ñ4ùóÝ×�4, �1/n�«8�,

Jhùós÷�»A,

∞∑n=1

1√n(n + 1)

�∞∑

n=1

sin1

n

/s÷�¨², Bã�

∑1/n2f´á,

∑∞n=1 n/

√n5 + 1[e�!§

∑∞n=1

(n + 2)/√

n3 + ns÷, �∑∞

n=1(2n + 1)/√

n6 + n2[e�

»»»3.3.�D¡ùó∑∞

n=1 n2−n sin(n−1)�e÷P��..

limn→∞

sin(n−1)

n−1= 1,

ÆD3×n0 ≥ 1, ¸ÿn ≥ n0`, 0 < n sin(n−1) < 2�.hn ≥n0`,

0 <n

2nsin(n−1) <

2

2n�


.¿¢ùó∑∞

n=1 2/2n[e, Æã�§3.2,∑∞

n=1 n2−n sin(n−1)[e�

»»»3.4.�D¡ùó∑∞

i=2 1/ log i�e÷P��..

1

log i>

1

i,

�∑

i−1s÷, Æ∑∞

i=2 1/ log is÷�&�Tô�:�, ùó�4ó,

%��ã2��

»»»3.5.�D¡ì�Þùó�e÷P�(i)

∑∞n=1(2 + cos n)/3n;

(ii)∑∞

n=1(n− 1)/(2n2)��.(i) E∀n ≥ 1,

0 ≤ 2 + cos n

3n≤ 3

3n=

1

3n−1,

�∑

1/3n−1[e, Ææùó[e�(ii) E∀n ≥ 2,

0 <1

4n≤ n

4n2≤ 2n− 2

4n2≤ n− 1

2n2,

�ùó∑

1/(4n)s÷, Ææùós÷�

k�b[2¸àf´l�°, &Æµ6á¼?9ùó[eTs÷�¿¢ùóCzetaÐóãy�P��, ðJ¼à�f´Ýùó�¿à9ìÞ�3�-1837O,J�Ý��555lll��°°°(integral test),

�¾½?9ùóÝe÷P�

��§§§3.4.'f ×�L3[1,∞)��3ÝÑÐó�E∀n ≥ 1, �

sn =n∑

i=1

f(i), tn =

∫ n

1

f(x)dx�

7.3 Ñ4ùó 407

Jó�{sn}C{tn}!`[eTs÷�JJJ��.ã%3.1, �:�

n∑i=2

f(i) ≤∫ n

1

f(x)dx ≤n−1∑i=1

f(i),

�hÇsn − f(1) ≤ tn ≤ sn−1�

ãy{sn}�{tn}, / ��¦, ãî��P, �:�Þó�!`bî&, T!`Pî&��¦�ó�ÎÍs÷, µÚÍÎÍbî&D3�ÆÿJ{sn}�{tn}, !`[eTs÷�

-

6

O 1 2 · · · nx

f(2)

f(n)

y

∑ni=2 f(i) ≤ ∫ n

1f(x)dx

-

6

O 1 2 · · · nx

f(1)

f(n− 1)

y

∫ n

1f(x)dx ≤ ∑n−1

i=1 f(i)

%3.1.

»»»3.6.E∀p > 0, ùó∑∞

n=1 1/npÌ ×pùùùóóó(p series)�3»3.1,

�J�p ≥ 2`, hùó[e�9ì¼:, A¢¿à�5l�°, J�hùó[e, uv°up > 1�ã

f(x) =1

xp, p > 0�

J

sn =n∑

i=1

f(i) =n∑

i=1

1

ip,

tn =

∫ n

1

f(x)dx =

∫ n

1

1

xpdx

{n1−p−1

1−p, p 6= 1,

log n, p = 1�


up > 1, Jn →∞`, n1−p → 0, Æ{tn}[e�ã�5l�°á, h`

∑1/npù[e�up ≤ 1, Jn →∞ `, tn →∞, Æ{tn}s÷, .

h∑

1/npùs÷�

»»»3.7.�¾½∑∞

k=1 k/ek�e÷P��.ã

f(x) =x

ex= xe−x,

Jf(x) > 0, vf ′(x) = e−x(1 − x) < 0, ∀x > 1�Æf3[1,∞) Ñv�3�ê

∫ n

1

xe−xdx = 2e−1 − (n + 1)e−n → 2e−1, n →∞�

ÆX�Ýùó[e�¨², ô�¿àÛ3.1, ÞX�Ýùó�[eÝ¿¢ùó

∑e−k/2

8f, �ÿÕ!øÝ�¡�

»»»3.8..

tn =

∫ n

2

1

x(log x)pdx =

{(log n)1−p−(log 2)1−p

1−p, p 6= 1,

log(log n)− log(log 2), p = 1,

Æ{tn}[euv°up > 1�.h

∞∑n=2

1

n(log n)p

[e, uv°up > 1�

&Æá¼∑

1/ns÷, �©�nÝg]f1�, ÉA11 + ε, Í�ε�|Î×��Ýü�Ñó, J

∑ 1

n1+ε

7.3 Ñ4ùó 409

[e�î»¼�, ¯@î5Ò�“?�°”�h.4∞∑

n=2

1

n log n

s÷, �∞∑

n=2

1

n(log n)1+ε

[e, Í�ε� ×��ü�Ñó��u�

an =1

n(log n)1+ε, bn =

1

n1+ε,

J

limn→∞

an

bn

= limn→∞

nε

(log n)1+ε= ∞�

EÞÑ4ùó∑

an�∑

bn, vlimn→∞ an/bn = ∞, Jã∑

bn[e,

¬P°@�∑

anÎÍ[e�Í»Ç¼�, 4ùó∑

anf∑

bn“��9”, ¬

∑an )b��[e�9Ë»�Í@�9, Aãan = 1/n2,

bn = 1/n3, Jlimn→∞ an/bn = ∞, ¬∑

an�∑

bn /[e�nyÑ4ùó, $bËËðàÝl�[eÝ]°�9ÎÞ�¢

ã�¿¢ùó∑

xn f´�s"�¼Ý�´�'×ùó

∑an�ÏØ4R, ��

(3.5) 0 ≤ an ≤ xn,

Í�0 < x < 1�Jãf´l�°á,∑

an[e��(3.5)Pê��y

(3.6) 0 ≤ a1/nn ≤ x�

ãhÇÿì�qqqPPPlll��°°°(root test)�

��§§§3.5.'×Ñ4ùó∑

an, ��

limn→∞

a1/nn = R�


(i) uR < 1, Jùó[e;

(ii) uR > 1, Jùós÷;

(iii) uR = 1, Jh°´[�JJJ��.(i) 'R < 1, vó×x��R < x < 1�JD3×n0 ≥ 1, ¸ÿn ≥ n0`(3.6)PWñ�Æãf´l�°á, ùó[e�

(ii) ´�R > 1, 0lbP§9Ían > 1�Æn → ∞`, an��0, .hùós÷�

(iii) 'bÞùó∑

1/nC∑

1/n2, Gïs÷¡ï[e, ¬Þï�RÂ/ 1�ÆR = 1`, ùób`[eb`s÷�

»»»3.9.�5½D¡ì�&ùó

(i)∑∞

n=31

(log n)n , (ii)∑∞

n=1

(n

n+1

)n2

, (iii)∑∞

n=1n2

2n

�e÷P��.(i) .n →∞`,

a1/nn =

1

log n→ 0,

Æùó[e�(ii) .n →∞`,

a1/nn = (

n

n + 1)n =

1

(1 + 1/n)n→ 1

e< 1,

Æùó[e�(iii) .n →∞`,

a1/nn =

(n2

2n

)1/n

=n2/n

2→ 1

2< 1,

Æùó[e�

¨×v«Ýl�° fffÂÂÂlll��°°°(ratio test)�

��§§§3.6.'b×ùó∑

an, �ÏØ4��an > 0, v��

limn→∞

an+1

an

= L�

7.3 Ñ4ùó 411

(i) uL < 1, Jùó[e;

(ii) uL > 1, Jùós÷;

(iii) uL = 1, Jh°´[�JJJ��.(i) 'L < 1, ¬ã×x��L < x < 1�JD3×n0 ≥ 1, ¸ÿan+1/an < x, ∀n ≥ n0�.h

an+1

xn+1<

an

xn, ∀n ≥ n0�

ÆEn ≥ n0, ó�{an/xn} �3�©½2, n ≥ n0,

an

xn≤ an0

xn0�

ùÇan ≤ cxn,

Í�c = an0/xn0�Æ

∑anå×y[eùó

∑cxn�ÿJ(i)�

(ii) L > 1, D3×n0 ≥ 1, ¸ÿan+1 > an, ∀n ≥ n0�Æn →∞`, an��0, .h

∑ans÷�

(iii) )ãÞùó∑

1/nC∑

1/n2 », -�ÿJ�

E×ùó∑

an, Ç¸an+1/an < 1, ∀n ≥ 1, ô�1Jhùó[e�.h`limn→∞ an+1/an)b�� 1�»A, uan = 1/n,

Jan+1/an = n/(n + 1) < 1, ∀n ≥ 1, ¬∑

ans÷��ÄunÈ�`,

an+1 > an > 0, J∑

anÄs÷, .h`an��0�

»»»3.10.�D¡ì�&ùóÝe÷P�(i)

∑∞n=1

nn

n!;

(ii)∑∞

n=1n34n

n!;

(iii) 1 + 12

197+ 2!

32 (197)2+ 3!

43 (197)3+ · · ·�

��.(i) .n →∞`,

an+1

an

=(n + 1)n+1

(n + 1)!

n!

nn=

(n + 1

n

)n

= (1 +1

n)n → e > 1,

Æùós÷�


(ii) .n →∞`,

an+1

an

=(n + 1)34n+1

(n + 1)!

n!

n34n=

4(n + 1)2

n3→ 0,

Æùó[e�(iii) ´�×�4

an =(n− 1)!

nn−1(19

7)n−1, n ≥ 1�

�n →∞ `,

an+1

an

=nn

(n + 1)n

19

7→ 1

e

19

7< 1,

h.19/7.= 2.714, �e > 2.718�

3î»(i)�, u; D¡ùó∑

n!/nn, J|�hùó[e(.h`limn→∞ an+1/an = 1/e < 1)�ãhÇÿ

(3.7) limn→∞

n!

nn= 0�

ÇEn��`, nn�W�"Än!�u|o−Br�î, Ç

n! = o(nn), n →∞�Q(3.7)P, ù�¿àô^�§yì��Pà#ÿÕ;

0 <n!

nn=

1

n

2

n

3

n· · · n

n

n<

1

n�

»»»3.11.�D¡ì�&ùóÝe÷P, Í�k ×ðó�(i)

∑∞n=1 (log n)k/2n,

(ii)∑∞

n=1 en/nk��.(i) .n →∞ `,

an+1

an

=(log(n + 1))k

2n+1

2n

(log n)k=

1

2

(log(n + 1)

log n

)k

→ 1

2< 1,

7.3 Ñ4ùó 413

Æùó[e�(ii) .n →∞`,

an+1

an

= e(n

n + 1)k → e > 1,

Æùós÷�

&ÆÂÕÝ»��, qPl�°CfÂl�°, ??!`ÊàT!`´[�´[`µ�:ÎÍbÍ�]°�¾½�×ùó�[eTs÷Î@�Ý, Ø×]°´[, ��îhùó�e÷P��X��3êÞ�, &Æº�+Û¨²ËÍl�°��Ä×��,

fÂl�°, ´qPl�°?à�h.;ð�ÕfÂ, ´�ng]�|�¬ÎqPl�°, ÊàP´Â�?�@21, ãfÂl�°J�ùó[e`, ãqPl�°, ;ðù�ÿÕùó[e; �qPl�°´[`, fÂl�°, ;ðù´[�9]«ÝD¡�¢�Rudin (1964) Chapter 3�9ì-�×fÂl�°´[, ¬qPl�°W�Ý»��

»»»3.12.'b×ùó

∞∑n=1

2(−1)n−n =1

22+

1

21+

1

24+

1

23+

1

26+

1

25+ · · ·�

Ja2n

a2n−1

= 2,a2n+1

a2n

=1

8,

Ælimn→∞ an+1/an�D3, .hfÂl�°´[�¬n →∞`,

n√

an = 2((−1)n−n)/n → 2−1 < 1,

ÆãqPl�°áhùó[e�

Í;t¡&Æ��×b¶Ý»��


»»»3.13.'b×�}�¦�ÑJóó�{an, n ≥ 1}�»A, {an, n ≥1} = {1, 2, 3, 7, 10, 12, · · · }��un�a1, a2, · · · , an�t�2¹ó��J

∑∞n=1 1/un[e�

JJJ��.´�, N×ÑJóu��×.óv, Ä��v ≤ √u, Tv = u/v1,

Í�v1 ≤√

u�.hu�.ó, �øÄ2√

uÍ�¯@î2√

u, ©bu ×¿]ó`, � Jó��3h�µ(u ¿]ó), &Æ3�Õu�.ó`,

√uÕÝËg�Æu�.óÍó, �y2

√u�

¨.0< a1< a2< · · · , �un a1, a2, · · · , an�t�2¹ó, Æa1,

a2, · · · , an/ un8²Ý.ó�.hn < 2√

un, Ç1/un < 4/n2��4∑

1/n2 ×[eùó�ÆÿJ∑∞

n=1 1/un[e�

êêê ÞÞÞ 7.3

1. �¾\ì�&ùó�e÷P�(1)

∑∞n=1

13n−1+2� (2)

∑∞k=1 sin( k

k2+1)�

(3)∑∞

k=1k

k2+103� (4)∑∞

k=2 sin( 1k2+k

)�(5)

∑∞i=3

1i√2� (6)

∑∞k=4

√k

k2−4�(7)

∑∞k=3

log kk2 � (8)

∑∞n=1 n3e−n�

(9)∑∞

r=1arctan rr2+1 � (10)

∑∞m=2

m+1(m+2)2m�

(11)∑∞

n=1n+12n � (12)

∑∞n=1

n2

2n�(13)

∑∞n=1

| sin nx|n2 � (14)

∑∞n=1

2+(−1)n

2n �(15)

∑∞n=1

n!+3n

(n+2)!� (16)∑∞

n=2log n

n√

n+1�(17)

∑∞n=1

1√n(n+10)� (18)

∑∞n=1

1+√

n(n+1)3−1�

(19)∑∞

n=21

(log n)s� (20)∑∞

n=1|an|10n , |an| < 10�

(21)∑∞

n=1n cos2(nπ/3)

2n � (22)∑∞

n=11

n√n�(23)

∑∞n=1

12log n� (24)

∑∞n=1

13log n�

(25)∑∞

n=1 n sin( 1n2 )� (26)

∑∞n=1(

2n2n+1

− 2n−12n

)�(27)

∑∞n=1

1(n!)1/n� (28)

∑∞n=2

1(log n)1/n�

(29)∑∞

n=1log nn1+a , a > 0� (30)

∑∞n=1

(log n)2

n1+a , a > 0�(31)

∑∞k=2

√k

k2−sin2(100k)� (32)∑∞

n=21

(log n)log n�(33)

∑∞n=1

1n1+1/n� (34)

∑∞n=3

1(log n)log log n�

êÞ 415

(35)∑∞

n=1

∫ 1/n

0

√x

1+x2 dx� (36)∑∞

n=1

√2n−1 log(4n+1)

n(n+1) �(37)

∑∞n=1

∫ n+1

ne−

√xdx� (38)

∑∞n=3

1n log n(log log n)s�

(39)∑∞

n=21

log(n!)� (40)∑∞

n=1(1− cos 1n)�

(41)∑∞

n=11·3·5···(2n−1)2·4·6···(2n) � (42)

∑∞n=1(

√1 + n2 − n)�

(43)∑∞

n=1 ns(√

n + 1− 2√

n +√

n− 1)�2. �¾\ì�&ùó�e÷P�

(1)∑∞

n=1(n!)2

(2n)!� (2)∑∞

n=1(n!)2

2n2 �(3)

∑∞n=1

2nn!nn � (4)

∑∞n=1

3nn!nn �

(5)∑∞

n=1n!3n� (6)

∑∞n=1

n!22n�

(7)∑∞

n=11

(log n)1/n� (8)∑∞

n=1(n1/n − 1)n�

(9)∑∞

n=1 e−n2

� (10)∑∞

n=1(1n− e−n2

)�(11)

∑∞n=1

(1000)n

n! � (12)∑∞

n=1nn+1/n

(n+1/n)n�(13)

∑∞n=1

n3(√

2+(−1)n)n

3n � (14)∑∞

n=1 rn| sin nx|, r > 0�(15)

∑∞n=1

2n(n!)2

(2n)! � (16)∑∞

n=1 n2e−n2

�(17)

∑∞n=1

1nn� (18)

∑∞n=1(

2n+53n−2

)n�(19)

∑∞n=1 n3( n

2n−1)n� (20)

∑∞n=1(

n2

3sin( 2n+1

n3+n2 ))n�

(21)∑∞

n=1n!nn

(2n)!� (22)∑∞

n=1(2n)!

3n2 �(23)

∑∞n=1

n!

2n2� (24)∑∞

n=1n

4nn!�(25)

∑∞n=1

4n

n4n!� (26)∑∞

n=1(1− 1n)n2

�(27)

∑∞n=1 2n sin 1

3n� (28)∑∞

n=1cos2 n

(√

n+1)3�

3. 'b×ùó∑∞

n=1 1/n1+θ(n)vlimn→∞ θ(n) = LD3��JuL >

0, Jùó[e; uL < 0, Jùós÷�¬D¡L = 0 Ý�µ�

4. �O

limn→∞

1√n

(1√1

+1√2

+ · · ·+ 1√n

)�

5. �JE∀a ∈ R,

−π

2<

∞∑n=1

a

n2 + a2<

π

2�


6. Eùó∑n

i=1 log i, ÿû�5l�°ÝJ°, �J

(n

e)ne < n! < n(

n

e)ne�

î��PEn!�×Õ¯Ý£��×Íf´?Ý£�ÎStir

-ling222PPP(Stirling’s formula):

(n

e)n√

2πn < n! < (n

e)n√

2πn(1 +1

4n),

.hn! ∼ (

n

e)n√

2πn�

7. �Juan ≥ 0, ∀n ≥ 1, v∑

an[e, JE∀p ≥ 1,∑

apnù[

e�

8. '∑

u2iC

∑v2

i/[e�(i) �J

∑ni=1(ui − vi)

2 ≤ 2∑n

i=1 u2i + 2

∑ni=1 v2

i , ∀n ≥ 1;

(ii) ¿à(i), �JE∀p ≥ 2,∑

(ui − vi)p[e�

9. �Juan ≥ 0, ∀n ≥ 1, vó�{nan}b&, J∑

a2n[e�

10. '{an, n ≥ 1} ×�3�0�ó�, v∑

an[e�(i) �OSk =

∑kn=1 n(an − an+1), k ≥ 1;

(ii) �J∑

n(an − an+1)[e�

11. �Julimn→∞ n2an = a > 0, J∑

an[e�

12. �Juan > 0, ∀n ≥ 1, v∑

an[e, J∑

(1/an)s÷�

13. 'D3×n0 ≥ 1, ¸ÿan, bn > 0, ∀n ≥ n0��cn = bn −bn+1an+1/an�(i) �JuD3×r > 0, ¸ÿcn > r, ∀n ≥ n0, J

∑an [

e�(èî: J�∑n

k=n0ak ≤ an0bn0/r)

(ii) 'cn ≤ 0, ∀n ≥ n0, v∑

1/bns÷��J∑

ans÷�(èî: J�

∑1/bnå×y

∑an)

êÞ 417

14. 'an > 0, ∀n ≥ 1��JuD3×r > 0C×n0 ≥ 1, ¸ÿ

an+1

an

≤ 1− 1 + r

n,∀n ≥ n0,

J∑

an[e; �u

an+1

an

≥ 1− 1

n, ∀n ≥ n0,

J∑

ans÷(èî: 3îÞ�ãbn+1 = n)�hÇ Raabelll��°°°(Raabe’s test)�

15. 'an > 0, ∀n ≥ 1, vD3×n0 ≥ 1, ×s > 1, C×M > 0, ¸ÿ

an+1

an

= 1− A

n+

f(n)

ns,∀n ≥ n0,

Í�|f(n)| ≤ M , ∀n ≥ 1��JuA > 1J∑

an[e, uA ≤1J

∑ans÷�hÇ{{{úúúlll��°°°(Gauss’ test)�(èî: uA 6=

1, ¿àîÞ; uA = 1¿àÏ12Þ, vãbn+1 = n log n)

16. ¿à{úl�°, �Jùó

∞∑n=1

(1 · 3 · 5 · · · (2n− 1)

2 · 3 · 5 · · · (2n)

)k

[euv°, uk > 2(EhùófÂl�°´[)�

17. 'n1 ≤ n2 ≤ · · · ×ó��ÑJó, vN×ni/&²ó,

(ni, nj) = 1, ∀i 6= j��®∑∞

i=1 1/ni[eÍ?

18. �Bn(x) = 1x + 2x + · · ·+ nx, x > 0�

�J∞∑

n=2

Bn(logn 2)

(n log2 n)2

[e�


19. (i) ¿àtan θ = cot θ − 2 cot 2θ, �J

n∑

k=1

1

2ktan(

x

2k) =

1

2ncot(

x

2n)− cot x, x 6= 0;

(ii) �J

∞∑

k=1

1

2ktan(

x

2k) =

1

x− cot x, x 6= 0�

20. 'n �ó`an = 1/n, n �ó`an = 1/n2��¾½∑∞

n=1 an

�e÷P�

21. �J∑∞

n=1(√

na + 1−√na), a > 2`[e�

22. �J∑∞

n=2(log(n+1)− log n)/ log2 n[e�(èî: ¿à»3.8)

23. �Jùó∞∑

n=1

1 · 2 · 3 · · ·n(α + 1)(α + 2) · (α + n)

[e, uv°uα > 1�

24. �J∑∞

n=1(1− 1/√

n)n[e�

25. '{an}C{bn} Þó�, vean = an + ebn , n ≥ 1�(i) �JE�×n ≥ 1, uan > 0, Jbn > 0;

(ii) uan > 0, ∀n ≥ 1, v∑

an[e, �J∑

(bn/an)[e�

26. �O�Xb¸∑∞

n=1(n!)c/(3n)![e�c�

27. �O�Xb¸∑∞

n=1(n!)3/(an)![e�ÑJóa�

28. 'n1 < n2 < n3 < · · · óC��â0�ÑJóó�, Çn1 = 1,

n2 = 2, · · · , n9 = 9, n10 = 11, · · · , n18 = 19, n19 = 21, · · ·��J

∑∞k=1 1/nk[e, vÍõ�y90�(èî: J�ùóå×

y9∑∞

n=0(9/10)n)

7.4 øýùó 419

29. 'a ×@ó, �sn(a) = 1a + 2a + · · ·+ na��O

limn→∞

sn(a + 1)

nsn(a) �

30. �5½Eì�Þùó¾½Íe÷P, ¬1�fÂl�°CqPl�°/´[�(i)

∑∞n=1

(5+(−1)n

2

)−n

, (ii)∑∞

n=1

(5+(−1)n

2

)n

�

7.4 øøøýýýùùùóóó

3î×;&ÆD¡ÝÑ4ùó, Í;&ÆD¡Ñ4C�4/bÝùó�´�:øøøýýýùùùóóó(alternating series), Ç×Ñ��4øýÝùó, Í�P

∞∑n=1

(−1)n−1an = a1 − a2 + a3 + · · ·+ (−1)n−1an + · · · ,

Í�an > 0�Q∑∞

n=1(−1)nanù ×øýùó�¾¾¹+3�-1705Os¨, 0l×øýùó[eÝ��P

²�ãhÇÿì��§, Ì ¾¾¾¾¾¾¹¹¹+++!!!JJJ(Leibniz rule)�

��§§§4.1.'{an} ×�3�0�ó��Jøýùó∑∞

n=1(−1)n−1an

[e�u|S �ùóõ, J�Ïn4�I5õsn��

(4.1) 0 < (−1)n(S − sn) < an+1, ∀n ≥ 1�

JJJ��.ãy{an} �3, Æ��ó4�I5õ{s2n, n ≥ 1} �¦ó�, ��ó4�I5õ{s2n−1, n ≥ 1} �3ó��Þó�/|s2 ì&, |s1 î&�.��v b&Ýó�Ä[e, ÆD3S ′CS ′′, ¸ÿ

limn→∞

s2n = S ′, limn→∞

s2n−1 = S ′′�


ê

S ′−S ′′ = limn→∞

s2n− limn→∞

s2n−1 = limn→∞

(s2n−s2n−1) = limn→∞

(−a2n) = 0,

ÆS ′ = S ′′�u|S�h�!Á§Â, J�Qó�{sn}[e�S�Íg&Æ0�(4.1)P�.{s2n}�¦, v{s2n−1}�3, Æ

s2n < s2n+2 ≤ S, S ≤ s2n+1 < s2n−1, ∀n ≥ 1�

.h

0 < S − s2n ≤ s2n+1 − s2n = a2n+1,

v

0 < s2n−1 − S ≤ s2n−1 − s2n = a2n�ãî�Þ��P, Çÿ(4.1)P�J±�

3î��§�, u�Ø4��, {an}��3v[e�0, Jøýùó

∑∞n=1(−1)n−1an)[e�Qh`(4.1)Pµ�×�WñÝ�

»»»4.1.ãyó�{1/n, n ≥ 1}�3�0, Æøýùó1 − 12

+ 13−

14

+ · · ·[e�hùóõ}¡&ÆºO��Í»��î, Ç¸

∑anC

∑bn/s÷, ¬

∑(an − bn) Qb��[e, 3han =

1/(2n− 1), bn = 1/(2n), n ≥ 1�

»»»4.2.�

f(x) =log x

x, x > 0,

Jx > e`, f ′(x) = (1 − log x)/x2 < 0�Æx > e`f �3�.huã

an =log n

n,

Jn > 3`, an �3, vlimn→∞ an = 0�Æÿøýùó∑

(−1)n

log n/n[e�

7.4 øýùó 421

»»»4.3.�Jùó

1− 1

1!+

1

2!− 1

3!+ · · ·+ (−1)n 1

n!+ · · ·

[e, ¬£�ÍõS��óÏ3��..ó�{1/n!}�3�0, Æùó[e�ê.

1

7!< 0.0002,

ÆS

.= 1− 1 +

1

2− 1

6+

1

24− 1

120+

1

720.= 0.368�

¯@îhùóõ e−1, Æe−1 .= 0.368�

»»»4.4.�D¡ùó

1− 2

3+

3

5− · · ·+ (−1)n−1 n

2n− 1+ · · ·

�e÷P��.4×�4

n

2n− 1=

1

2(1 +

1

2n− 1)

�3, ¬

limn→∞

n

2n− 1=

1

26= 0,

Æhùós÷�

»»»4.5.�D¡ùó∞∑i=1

(−1)i i

i2 + 1

�e÷P��.�f(x) = x/(x2 + 1), Jf ′(x) = (1 − x2)/(x2 + 1)2 < 0, ∀x >

1�Æf3 [1,∞) �3�.hó�{i/(i2 + 1), i ≥ 1}ù�3�êlimn→∞ n/(n2 + 1) = 0, Ææùó[e�


»»»4.6.�Êùó

1√2− 1

− 1√2 + 1

+1√

3− 1− 1√

3 + 1+ · · ·

+1√

n− 1− 1√

n + 1+ · · ·�

4Qùó�×�4��0, ¬×�4¬&�3, Æ�§4.1�Êà�êG2n4�õ

n+1∑

k=2

(1√

k − 1− 1√

k + 1) =

n+1∑

k=2

2

k − 1= 2

n∑

k=1

1

k�

Æn →∞`, I5õ��∞, .hæùós÷�

�Ä�¥�ÝÎ, �§4.1©Îèº×¾\øýùóÎÍ[e��5f��Ç¸��§4.1�Ýf�, øýùó

∑∞n=1(−1)n−1an

, )b��[e�9Í�an��0, QÎÄ�Ý, ¬{an}�3µ�ÎÄ�Ý��}¡»4.11�

»»»4.7.¿à¾¾¹+!J, �0�×¥�ÝÁ§��

a1 = 1, a2 =

∫ 2

1

1

xdx, a3 =

1

2, a4 =

∫ 3

2

1

xdx, · · · ,

×�4

a2n−1 =1

n, a2n =

∫ n+1

n

1

xdx, n ≥ 1�

|�an ↓ 0, Æøýùó∑∞

n=1(−1)n−1an[e�u|γ�hùóõ,

��Ïn4�I5õ sn, J

s2n−1 = 1 +1

2+ · · ·+ 1

n−

∫ n

1

1

xdx = 1 +

1

2+ · · ·+ 1

n− log n�

.n →∞`, s2n−1 → γ, Æÿ

(4.2) limn→∞

(1 +1

2+ · · ·+ 1

n− log n) = γ�

7.4 øýùó 423

ðóγÌ ��ZZZðððóóó(Euler’s constant)�A!πCe, 35��, γ ×¥�Ýðó, ÍÂV 0.5772156649�¬�uπCe, &ÆP°|×��Ý2P¼�îγ, #�γ b§óTP§ó, �*)Îá�

(4.2)Pê��î

(4.3)n∑

i=1

1

i= log n + γ + o(1), n →∞�

ãhÇÿ

limn→∞

(1 +1

2+ · · ·+ 1

n)/ log n = 1�

.hêbn∑

i=1

1

i∼ log n, n →∞�

ã(4.3)P,�¬��:� ¢�õùó∑

1/is÷,ô�:�∑

1/i

W��>��¿à(4.3)Pô�O�×°ùóÝõ�

»»»4.8.�

sm =m∑

i=1

(−1)i−1 1

i= 1− 1

2+

1

3− 1

4+ · · ·+ (−1)m−1 1

m�

3»4.1�¼�limm→∞ smD3, 9ì¼OÍõ�´�b

s2n =n∑

i=1

1

2i− 1−

n∑i=1

1

2i= (

2n∑i=1

1

i−

n∑i=1

1

2i)−

n∑i=1

1

2i

=2n∑i=1

1

i−

n∑i=1

1

i�

¿à(4.3)P, ÿ

s2n = (log 2n + γ + o(1))− (log n + γ + o(1))

= log 2 + o(1)�Æn →∞`, s2n → log 2�.hsn → log 2�ÇÿJ

1− 1

2+

1

3− 1

4+ · · ·+ (−1)m−1 1

m+ · · · = log 2�


¨², ô�AìO�s2n → log 2(ù�êÞ2.4Ï3Þ):

s2n =2n∑

i=n+1

1

i=

n∑i=1

1

n + i=

n∑i=1

1

n

1

1 + i/n→

∫ 1

0

1

1 + xdx = log 2�

»»»4.9.�¾\ùó

1

3+

1

3√

3+

1

3√

3 3√

3+ · · ·+ 1

3√

3 3√

3 · · · n√

3+ · · ·

�e÷P��.ùó�×�4

an = (31+ 12+···+ 1

n )−1�

�.n →∞`,

1 +1

2+ · · ·+ 1

n− log n → γ,

Æu�

bn =1

3log n=

1

nlog 3,

J

limn→∞

an

bn

=1

3γ> 0�

�.log 3 > 1, Æ∑

bn[e, .h∑

an [e�\ïô��|qPl�°, TfÂl�°, l�î�

∑anÎÍ[

e�.an+1/anCa1/nn n → ∞ `, /��1, ÆhÞ°3h/´

[�

Íg&ÆD¡��EEE[[[eee(absolute convergence)Cfff��[[[eee (con-

ditional convergence)�4Qøýùó

∑(−1)n−1/n[e, ¬uÞN×4ã�EÂ, Q

ÿÕ×s÷ùó∑

1/n�ôµÎ×��,∑

an[e, �×�0l

∑ |an|[e��Äu∑ |an|[e, Q�0�

∑an[e, �ì�

§�

7.4 øýùó 425

��§§§4.2.'∑ |an|[e, J

∑anù[e, v

(4.4) |∞∑

n=1

an| ≤∞∑

n=1

|an|�

JJJ��.�bn = an + |an|, u�J�∑

bn[e, J.an = bn− |an|, Æã�§2.2Çÿ

∑an[e�

.E∀n ≥ 1, bn = 0T2|an|, Æ

0 ≤ bn ≤ 2|an|,

.h∑ |an|Yg

∑bn, Æ

∑bn[e, .h

∑an [e�

�y(4.4)P, ã

|n∑

i=1

ai| ≤n∑

i=1

|ai|, ∀n ≥ 1,

��n →∞Çÿ�

��LLL4.1.u∑ |an|[e, Jùó

∑anÌ �E[e; u

∑ |an|s÷,

�∑

an[e, J∑

anÌ f�[e�

��§§§4.3.u∑

anC∑

bn/ �E[e, JE∀α, β ∈ R,∑

(αan +

βbn)ù �E[e�JJJ��.E∀n ≥ 1,

n∑i=1

|αai + βbi| ≤ |α|n∑

i=1

|ai|+ |β|∞∑i=1

|bi|

≤ |α|∞∑i=1

|ai|+ |β|∞∑i=1

|bi| < ∞�

ÆI5õ∑n

i=1 |αai+βbi| b&, ∀n ≥ 1�.hùó∑ |αan+βbn|[

e�ÿJ�

»»»4.10.ùó∑

(−1)n/n2�

∑(−1)n(2/3)nC

∑∞n=0(−1)n/n!/ �E

[e, �∑∞

n=2(−1)n/ log n f�[e�


»»»4.11.�Êøýùó

2− 1

22+

2

32− 1

42+

2

52− 1

62+ · · · =

∞∑n=1

(−1)n−1an�

Ç�ó4 a2n−1 = 2/(2n− 1)2, �ó4 a2n = 1/(2n)2�.∞∑

n=1

2

(2n− 1)2C

∞∑n=1

1

(2n)2

/ [eùó, Æã�§4.3á,

∞∑n=1

(2

(2n + 1)2− 1

(2n)2)

ù�E[e�ÇÿJæøýùó[e, 4{an}&�3: a2n+1 > a2n,

∀n ≥ 2�

3î×;Xè�, l�Ñ4ùó�e÷PÝ]°, QK�à¼l�×ùóÎÍ�E[e�9ì�èºËÍ, E£°��E[e�ùóÝe÷P�¾½°�hÞ¾½°/6àÕì�AbelIII555õõõ222PPP(Abel partial summation formula)�

��§§§4.4.'{an}C{bn} Þó�, v�An =∑n

i=1 ai�J

(4.5)n∑

i=1

aibi = Anbn+1 +n∑

i=1

Ai(bi − bi+1)�

JJJ��.u�A0 = 0, Jai = Ai − Ai−1, i = 1, · · · , n�Æn∑

i=1

aibi =n∑

i=1

(Ai − Ai−1)bi =n∑

i=1

Aibi −n∑

i=1

Aibi+1 + Anbn+1�

ÿJ(4.5)P�

Abel2PÝÞß, v«5I�5�ôµÎE(4.5)P¼��õ, ;|��õ�îAnbn+1¼ã�, �Xf�QÎ(4.5)P��õ,

7.4 øýùó 427

�´|ßé�ã(4.5)P�:�, n → ∞`, uùó∑n

i=1 Ai(bi −bi+1)Có�{Anbn+1}/[e, J

∑aibi[e�9ìËÍ�§Ç��

G�ùóCó�[e��5f�, .hù ∑

aibi[e�×�5f��

��§§§4.5.(Dirichletlll��°°°(Dirichlet’s test)). 'ùó∑

an �I5õ ×b&ó�, ê'{bn} ×�3�0Ýó��J

∑anbn[e�

JJJ��.'An��LA3�§4.4�, ã�'áD3×M > 0, ¸ÿ|An| ≤ M , ∀n ≥ 1�.hn → ∞`Anbn+1 → 0�Æu�J�

∑Ai(bi − bi+1)[e, JÍ�§ÿJ�.{bn}�3, Æ

|Ai(bi − bi+1)| ≤ M(bi − bi+1)�

�∑

(bi − bi+1) ×[eÝ¥Pùó, vYg|Ai(bi − bi+1)|�Æ∑Ai(bi − bi+1) �E[e, .hù[e�J±�

\ïÎÍ�:��§4.1 î��§�×©»�

��§§§4.6.(Abellll��°°°(Abel’s test)). '∑

an[e, v{bn} ×��[eó�, J

∑anbn[e�

JJJ��.)2à�§4.4�ÝÐr�ã�'∑

an[e, Æó�{An}[e, .hó�{Anbn+1}ù[e�{An}[e, ê0l{An}b&�¨'limn→∞ bn = c, J

∑anbn =

∑an(bn − c) + c

∑an�

.{bn − c} ×�3�0�ó�,Æ�§4.5�f�/��(|bn − cã�bn)�ãhÿ

∑an(bn − c)[e, .�

∑anbn[e�

ksìDirichletl�°��, &Æm�9á¼×°I5õ b&Ýùó�QN×[eùó/bhP²�¯@î, 3�§4.5C4.6�, an/� �ó�3h, ×�óó�{an}[e, uv


°uÍ@Ió�CÌIó�/[e, v{an}Á§ Í@Ió��ÌIó��Á§õ��×v¥�Ýs÷ùó, ¬I5õ b&Ýùó, ¿¢ùó

∑xn, Í�x ×�ó, v|x| 6= 1�9ì&

Æ�×hùóI5õ�×î&Ý�§�ãh�§ô�0�ÏÞa�(5.3)P�´�¥�Õ, u|x| = 1, Jx = eiθ, Í�θ ×@ó,

i =√−1, �eiθ = cos θ + i sin θ�

��§§§4.7.'θ ∈ R, v� 2π�Jó¹�JE∀n ≥ 1,

(4.6)n∑

k=1

eikθ =sin(nθ/2)

sin(θ/2)ei(n+1)θ/2,

v

(4.7) |n∑

k=1

eikθ| ≤ 1

| sin(θ/2)|�

JJJ��.θ� 2π�Jó¹, eiθ 6= 1, h`¿à�fùóõÝ2P, �AìÿJ(4.6)P:

n∑

k=1

eikθ =eiθ(1−einθ)

1− eiθ

=einθ/2 − e−inθ/2

eiθ/2 − e−iθ/2ei(n+1)θ/2 =

sin(nθ/2)

sin(θ/2)ei(n+1)θ/2�

ê.| sin(nθ/2)| ≤ 1v|ei(n+1)θ/2| = 1, Æã(4.6)PÇÿ(4.7)P�

'{bn} ×�3�0�@ó�, vãan = xn, Í�x ×�ó,

v|x| = 1, x 6= 1�Jã�§4.5C4.7á∑∞

n=1 bnxn[e��:��§4.1 h��x = −1�×©»�¨ãx = eiθ, Í�θ ∈ R, vθ� 2π�Jó¹��Êùó∑∞

n=1 bnxn, JãG«�D¡á, Í@I∑∞

n=1 bn cos nθ, �ÌI∑∞

n=1

bn sin nθ/[e�©½2, uãbn = 1/nα, α > 0, Jì�ùó/[e:

∞∑n=1

einθ

nα,

∞∑n=1

cos nθ

nα,

∞∑n=1

sin nθ

nα �

7.4 øýùó 429

êα > 1, î�&ùó, ./å×y∑

1/nα, Æ/�E[e�t&©½Î�, &Æ�ÊÝùó, Í&4í @ó�!ñ×è, .3(4.6)PË�@ICÌI6&�8�, Æÿ

(4.8)n∑

k=1

cos kθ =sin(nθ/2) cos((n + 1)θ/2)

sin(θ/2),

C

(4.9)n∑

k=1

sin(kθ) =sin(nθ/2) sin((n + 1)θ/2)

sin(θ/2) �

�p:�(4.8)P, �ÏÞa(5.3)PÎ8!Ý�Í;t¡&Æ¼D¡ùùùóóó��¥¥¥444(rearrangements of series)�3

7.1;&ÆèÄ, Eb§Íó, ;�8�Ý5�¬�;�Íõ�3�-1833O, Þ�s¨EP§ùóh��µ�×�)WñÝ�»A,

3»4.8, &ÆJ�Ý

(4.10) 1− 1

2+

1

3− 1

4+ · · · = log 2�

¬uÞhùó¥4, �¶ËÑ4�¶×�4, Jÿì�±ùó:

(4.11) 1 +1

3− 1

2+

1

5+

1

7− 1

4+

1

9+

1

11− 1

6+ · · ·�

h±ùó�æøý�õùóÝ&4, b1−1vÌWÝn;�¬�Aì2J�ÍõQ� log 2��tn�(4.11)P�ùó�Ïn4ÝI5õ�JEt3m, .�â2mÍ

Ñ4CmÍ�4, Æ¿à(4.3)P, ÿ

t3m =2m∑

k=1

1

2k − 1−

m∑

k=1

1

2k= (

4m∑

k=1

1

k−

2m∑

k=1

1

2k)− 1

2

m∑

k=1

1

k

=4m∑

k=1

1

k− 1

2

2m∑

k=1

1

k− 1

2

m∑

k=1

1

k

= (log 4m + γ + o(1))− 1

2(log 2m + γ + o(1))− 1

2(log m + γ + o(1))

=3

2log 2 + o(1)�


Ælim

m→∞t3m =

3

2log 2�

ê.t3m+1 = t3m + 1/(4m + 1), t3m−1 = t3m − 1/(2m), Æ

limm→∞

t3m+1 = limm→∞

t3m−1 = limm→∞

t3m =3

2log 2�

Çÿ

limn→∞

tn =3

2log 2�

ãî»�:�Þ×[eùó¥4¡, ��ÿÕ×�!Ýõ�¯@î9ì&ÆÞJ�h¨é©bEf�[eÝùó�ºsß�ôµÎ×�E[eÝùó4B¥4ô�º;�Íõ�

��§§§4.8.'∑

an ×�E[e�ùó, Íõ S�JN×∑

an�¥4, ) �E[e, vÍõ S�JJJ��.'

∑bn

∑an�×¥4, ÇD3×ãÑJó, Ì�ÑJó

�1− 1vÌW�Ðóf , ¸ÿbn = af(n)�.Ñ4ùó∑ |bn| �I5

õ, |∑ |an| Íî&, Æ

∑ |bn|[e, Ç∑

bn ×�E[eùó�Íg&ÆJ�

∑bn = S��

Bn =n∑

k=1

bk, An =n∑

k=1

ak, A∗n =

n∑

k=1

|ak|, S∗ =∞∑

k=1

|ak|�

J.n →∞`, An → S, A∗n → S∗, Æ∀ε > 0, D3×n0 ≥ 1, ¸ÿ

|An0 − S| < ε

2, |A∗

n0− S∗| < ε

2�

Ehn0ã×n1, ¸ÿ

(4.12) {1, 2, · · · , n0} ⊆ {f(1), f(2), · · · , f(n1)}�.f�Â½ ÑJó�/), Æî�n1×�0ÿÕ�Jn 6= n1,

|Bn − S| = |Bn − An0 + An0 − S|(4.13)

≤ |Bn − An0|+ |An0 − S| ≤ |Bn − An0|+ε

2�

7.4 øýùó 431

¬êb

|Bn − An0| = |n∑

k=1

bk −n0∑

k=1

ak| = |n∑

k=1

af(k) −n0∑

k=1

ak|�

ã(4.12)P, îPt��EÂ�a1, · · · , an0º��Æ

(4.14) |Bn − An0| ≤ |an0+1|+ |an0+2|+ · · · = |A∗n0− S∗| < ε

2�

Þ(4.14)P�á(4.13)P, ÿ

|Bn − S| < ε

2+

ε

2= ε, ∀n ≥ n1�

ÇÿJ∑

bn [e�S�

3î��§�,∑

an�E[eÛ Ä��Riemmans¨, E×f�[eÝùó, ©�BÊÝ¥4, �¯Íõ[e��×��Ý@ó�Riemann�J�,àÕf�[eùó�×©P,ÇÄbP§9ÍÑ4, CP§9Í�4(ÍJµÎ�E[eÝ)�u�

a+n =

an + |an|2

, a−n =an − |an|

2,

Ç

a+n =

{an, uan ≥ 0,

0, uan < 0,a−n =

{0, uan ≥ 0,

an, uan < 0,

J∑

a+n�

∑a−n , 5½

∑an�Ñ4C�4I5, �v

an = a+n + a−n�

hÞùó�∑

anbì�n;�

��§§§4.9.'b×ùó∑

an�(i) u

∑anf�[e, J

∑a+

n�∑

a−n/s÷;

(ii) u∑

an�E[e, J∑

a+n�

∑a−n/[e, v

(A)∑∞

n=1 an =∑∞

n=1 a+n +

∑∞n=1 a−n�


JJJ��.(i) '∑

anf�[e�J∑

12an[ev

∑12|an|s÷�Æã�

§2.3á∑

a+n�

∑a−n/s÷�

(ii) '∑

an�E[e, J∑

12anC

∑12|an| /[e�Æ�ã�

§2.3á,∑

a+n�

∑a−n/[e�ê.an = a+

n + a−n , Æ(A)Wñ�

»A,Eùó∑

(−1)n−1/n,hùó f�[e,vETÝa+2n−1 =

1/(2n−1), a−2n−1 = 0, a+2n = 0, a−2n = −1/(2n)��Q

∑a+

nC∑

a−n/s÷�¬E

∑(−1)n−1/n2, hùó �E[e, |�ETÝ

∑a+

nC∑a−n/[e�t¡&ÆJ�Riemann¥¥¥444��§§§(Riemann’s Rearrangement

Theorem)�

��§§§4.10.'∑

an ×f�[eùó, S ×��@ó�JD3

∑an�×¥4

∑bn[e�S�

JJJ��.'a+nCa−n��LAG, ãî×�§á,

∑a+

nC∑

a−n/s÷�9ì1�A¢¥4

∑an�.

∑a+

n ×Ñ4ùó, vI5õ��∞(¥�hùós÷), ÆuãÈ9Ý4ó, J�¸ÍI5õ�yS�'p1 Xm�4ó�J

p1∑n=1

a+n > S, v

q∑n=1

a+n ≤ S, ∀q < p1�

Íg.∑

a−n�N×4/ &Ñ, vI5õ��−∞(hùós÷),

Æu3∑p1

n=1 a+n�îÈ9Ýa−n 4, J�¸Íõ�yS�'m�n14,

Çp1∑

n=1

a+n +

n1∑n=1

a−n < S, vp1∑

n=1

a+n +

m∑n=1

a−n ≤ S, ∀m < n1�

¥�î�M», ��îÈ9ÝÑ4a+n , ¸ÍõøÄS, ��î

È9Ý�4a−n¸Íõ�yS�×àµ�ì�, -ÿ∑

an�×¥4

∑bn��Ïp14R,

∑bn�I5õ�S�-t9 Ø×an(��

G�¥4ÝÄ�), ¬n → ∞`an → 0(h.∑

an f�[e),

Æ∑

bn�I5õ��S�ÇÿJùó∑

bn[e�S�J±�

êÞ 433

êêê ÞÞÞ 7.4

1. �¾\ì�&ùó�e÷P�(1)

∑∞n=1(−1)n−1 1√

n� (2)∑∞

n=2(−1)n−1 2log(n+1)�

(3)∑∞

n=1(−1)n−1 n+13n � (4)

∑∞n=1(−1)n n

n2+2�(5)

∑∞k=1(−1)k 1

2k−1� (6)∑∞

k=1(−1)k−1 k2k�

(7)∑∞

i=2(−1)i−1 ilog i� (8)

∑∞i=2(−1)i 2i−1

5i+1�(9)

∑∞i=1(−1)i−1 log i

i � (10)∑∞

i=1(−1)i−1 sin 1i�

(11)∑∞

i=2(−1)i+1 log2 ii � (12)

∑∞i=2(−1)i+1 logp i√

i, p ≥ 1�

(13)∑∞

i=1(−1)i+1arccot i� (14)∑∞

i=1(−1)i+1i arctan 1i2�

2. �Oì�ùóõ��óÏë�Þ@�(1)

∑∞n=1(−1)n−1 1

22(n−1)� (2)∑∞

n=2(−1)n−1 1(2n−2)!�

(3)∑∞

n=1(−1)n−1 1(2n−1)3� (4)

∑∞n=2(−1)n−1 1

n·3n�3. �¾\ì�&ùó�e÷P, u[e¬¼�Î�E[eTf�[e�(1)

∑∞n=1(−1)n−1 1

(2n−1)!� (2)∑∞

n=1(−2)n

n! �(3)

∑∞n=1(−1)n+1 n!

9n� (4)∑∞

k=1(−1)k+1

3√k �

(5)∑∞

n=1(−1)n (n+100)n3 � (6)

∑∞k=1(−1)k k100

(k+2)!�(7)

∑∞k=1(−1)k( 1

k)1/k� (8)

∑∞k=1(−1)k k!

100k�(9)

∑∞k=2(−1)k 4k+1

7k2−1� (10)∑∞

k=1(−1)k+1 3k

k32k+3�(11)

∑∞k=1(−1)k k77k+3

23k � (12)∑∞

k=1(−1)k k!1·3·5·(2k−1)�

(13)∑∞

k=2(−1)k 1k log2 k� (14)

∑∞k=2(−1)k sin 10k√

k3 �(15)

∑∞n=1(−1)n(n−1)/2 1

2n� (16)∑∞

n=1(−1)n(2n+103n+1

)n�(17)

∑∞n=2(−1)n 1√

n+(−1)n� (18)∑∞

n=1(−1)n n2

n2+1�(19)

∑∞n=1(−1)n 1

log(en+e−n)� (20)∑∞

n=1(−1)n 1n log2(n+1)�

(21)∑∞

n=1(−1)n n37

(n+1)!� (22)∑∞

n=1(−1)n∫ n+1

ne−x

xdx�

(23)∑∞

n=1 sin(log n)� (24)∑∞

n=1 log(n sin 1n)�

(25)∑∞

n=1(−1)n(1− n sin 1n)� (26)

∑∞n=1(−1)n(1− cos 1

n)�

(27)∑∞

n=1(−1)n arctan 12n+1� (28)

∑∞n=1 log(1 + 1

| sin n|)�


(29)∑∞

n=2 sin(nπ + 1log n

)� (30)∑∞

n=11·4·7···(3n−2)2·4·6···(2n) �

(31)∑∞

n=2(−1)n 1(n+(−1)n)s� (32)

∑∞n=1(−1)n(n−1)/2 n100

2n �(33)

∑∞n=1(−1)n sin(1/n)

n � (34)∑∞

n=1(−1)n tan( 1n)�

(35)∑∞

n=1(−1)n arctan n√n � (36)

∑∞n=1(−1)n(e− (1 + 1

n)n)�

(37)∑∞

n=11·3·5···(2n−1)3·6·9···(3n) � (38)

∑∞n=1(−1)n 1

n(1+1/2+···+1/n)�(39)

∑∞n=1(−1)n(π

2− arctan(log n))�

(40)∑∞

n=1(−1)n+1(a1/n − 1), a > 0�4. �5½X�¸ì�&ùó[e�@óxÝ/)�

(1)∑∞

n=1 nnxn� (2)∑∞

n=1(−1)n x3n

n!�(3)

∑∞n=1

x3n

3n� (4)∑∞

n=1xn

nn�(5)

∑∞n=1(−1)n 1

x+n� (6)∑∞

n=1xn√

nlog 2n+1

n �(7)

∑∞n=1(1 + 1

5n+1)n2

x17n� (8)∑∞

n=0(x−1)n

(n+1)!�(9)

∑∞n=1

(−1)n(x−1)n

n � (10)∑∞

n=1(2x+3)n

n log(n+1)�(11)

∑∞n=1

(−1)n

2n−1

(1−x1+x

)n

� (12)∑∞

n=1

(x

2x+1

)n

�(13)

∑∞n=1

nn+1

(x

2x+1

)n

� (14)∑∞

n=11

(1+x2)n�(15)

∑∞n=1(−1)n 2n sin2n x

n � (16)∑∞

n=12n sinn x

n �5. �B(n)�32��ÑJónÝ1�Íó�»A, B(6) =

B(1102) = 2, B(15) = B(11112) = 4, Í�6 = 1102, �32

�ì, 6��î° 110�(i) �JB(2n) = B(n), B(2n + 1) = B(2n) + 1 = B(n) + 1�(ii) �

S =∞∑

n=1

B(n)

n(n + 1),

�JS = 2 log 2�

6. 'b×øýùó∑∞

n=1(−1)nan, Í�

a2n−1 =1

n, a2n =

1

n2, n ≥ 1�

�Qlimn→∞ an = 0va2n ≤ a2n−1, ∀n ≥ 1��Jhùó�[e, ¬�Õ ¢¾¾¹+!J3h�Êà�

êÞ 435

7. u∑∞

n=1(−1)n−1(an − bn) ×øýÝ�3ùó, ¬v[e��®h`

∑∞n=1(−1)n−1anÎÍ[e? J�TÍJ��

8. ¿àStirling2P(�7.3;�êÞ), EkD¡ì�ùó�e÷PC, ÎÍ�E[e�

∞∑n=1

(−1)n

(1 · 3 · 5 · · · (2n− 1)

2 · 4 · 6 · · · (2n)

)k

�

9. �O¸ì�ùó[e�rÝXb��Â�

1− 1

2r+

1

3− 1

4r+ · · ·+ 1

2n− 1− 1

(2n)r+ · · ·�

(èî: ¿à�5l�°ÝJ�, ÿÕ´2n4C2n− 14I5õ�î�ì&)

10. �O�¸ì�ùó[e�rÝXb��Â�

1 +1

3r− 1

2r+

1

5r+

1

7r− 1

4r+ · · ·�

11. �5½O�¸ì�ùó(i) [e, (ii) �E[e, �ðóaCb�a

1− b

2+

a

3− b

4+ · · ·+ a

2n− 1− b

2n+ · · ·�

12. �®

(i) u∑∞

n=1 an [e, ÎÍ0l∑∞

n=1(an + an+1)[e?

(ii) u∑∞

n=1(an + an+1)[e, ÎÍ0l∑∞

n=1 an[e?

(iii) u∑∞

n=1(|an|+ |an+1|)[e, ÎÍ0l∑∞

n=1 |an|[e?

13. '∑

an[e, van > an+1 > 0, ∀n ≥ 1��Jlimn→∞ nan =

0�

14. 'an > 0, v∑

an[e��J∑

a−1n s÷�


15. '∑ |an|[e��J

∑a2

nù[e�¬Ü×D»1�ÍY�Ë�

16. 'an ≥ 0, v∑

an[e��JE∀p > 1/2,∑√

ann−p[e�

¬Ü×»1�p = 1/2`, G��×�Wñ�

17. �JTÍJ9ì&B�:

(i) u∑

an�E[e, J∑

a2n/(1 + a2

n)ùQ�(ii) u

∑an�E[e, van 6= −1, ∀n ≥ 1, J

∑an/(1 + an) ù

Q�(iii) uan > 0, ∀n ≥ 1, v

∑ans÷, J

∑a2

nùQ;

(iv) uan > 0, ∀n ≥ 1, v∑

a2n[e, J

∑an/nùQ�

18. �Ju∑∞

n=1 an�E[e, J∑∞

n=1((n + 1)/n)anùQ�

19. �Ju∑

an[e, J∑

an/nùQ�

20. 'b{an}C{bn}Þó�, an ≥ an+1 ≥ 0, ∀n ≥ 1, vbn ≥ 0,

∀n ≥ 1�ê'

limn→∞

an = 0, limn→∞

an/bn = 1�

.∑∞

n=1(−1)nan[e,«��?∑∞

n=1(−1)nbnô[e��JTÍJh�?�

21. '∑∞

n=1 a2n�

∑∞n=1 b2

n/[e��J∑∞

n=1 anbn�E[e�

22. 'an > 0, ∀n ≥ 1, v∑∞

n=1 ans÷��Jì�ùó[e:

∞∑n=1

an

(1 + a1)(1 + a2) · · · (1 + an)�

23. ¿àAbell�°, �Ju∑∞

n=1 nan[e, J∑∞

n=1 anùQ�Û:

ÍÞuan ≥ 0, ∀n ≥ 1, -��|�J��

êÞ 437

24. �¿àDirichletl�°,D¡∑∞

n=1 sin nθ/n�e÷P, θ ∈ R�

25. �D¡ì�Þùó�e÷P:

(i)∞∑

n=1

sin nθ/ log n, θ ∈ R;

(ii)1

2 · 1 +2

3 · 3 −3

4 · 2 +4

5 · 3 +5

6 · 7 −6

7 · 4 + · · ·

+3n− 2

(3n− 1)(4n− 3)+

3n− 1

(3n)(4n− 1)− 3n

(3n + 1)(2n)+ · · ·�

(èî: ¿àAbell�°)

26. '∑∞

r=1 ar[e, vSn = a1 + a2 + · · ·+ an��J

limn→∞

1

n

n∑i=1

Si =∞∑

r=1

ar�

¿àG��O

limn→∞

1

n(1

2+

2

3+

3

4+ · · ·+ n

n + 1)�

27. 'p > q ≥ 1 Þü�ÑJó��J

limn→∞

pn∑

k=qn

1

k= log

p

q�

28. (i) ÞøýÝ�õùó¥4, vµëÑ4�Ë�4, Ç1 + 13

+15− 1

2− 1

4+ 1

7+ 1

9+ 1

11− 1

6− 1

8+ · · ·��Jhùó[e, võ

log 2 + 12log 3

2�(èî: �ÊI5õó�s5n, ¬¿àîÞ)

(ii) �O1− 12− 1

4+ 1

3− 1

6− 1

8+ · · · + 1

2n−1− 1

4n−2− 1

4n+ · · ·

�õ�(iii) �.Â(i) C(ii), ��×�Ý��


7.5 ��555

3ÏÞa�L�5∫ b

af(x)dx`,&Æ��×°§×�ÇÐófÄ

6b&, v [a, b] b§�µA£`Ý5vCOîõ�ìõÝÄ�, hÞf�Û Ä6�.h£`�A|ìÝ�5, /^b�L:

∫ ∞

1

1

x3dx,

∫ 1

−1

sin x

xdx,

∫ ∞

0

e−x2

dx,

∫ ∞

0

arctan xdx�

Í;&Æ¼:A¢w´h§×, |U��ÐóÝr½�9µÎÍ;X�+ÛÝ��555(improper integral, êÌÂÂÂLLL��555T��ÑÑÑððð��555, �GÝ�5-Ì ÑÑÑððð��555(proper integral))�´�, 'E∀b ≥ a, �5

∫ b

af(x)dxD3�u

limb→∞

∫ b

a

f(x)dx

D3vb§, JÌ∫∞

af(x)dx[e, ÍJÌ s÷, v¶W

(5.1)

∫ ∞

a

f(x)dx = limb→∞

∫ b

a

f(x)dx,

¬Ìh ÏÏÏ×××lll��555(improper integral of the first kind), ×PPP§§§��555(infinite integral)�êXÛ

∫∞a

f(x)dx D3, �∫∞

af(x)dx[

eÝ�¤Î8!Ý��:�î�ny��5Ý�L, v«P§ùóÝ�L�»A,∫ b

af(x)dx�Ì III555��555(partial integral), v«ùóÝI5õ�35��b&9¥�ÝÐó, Î|��5Ý�P�¨, Þ¼&�

3{��5�Ý��, º��á"D9vÐó, Í;©Î�×°�MÝ+Û�

»»»5.1.�Ê��5∫∞

1x−sdx�´�E∀b > 1,

∫ b

1

x−sdx =

{b1−s−1

1−s, s 6= 1,

log b, s = 1�

7.5 ��5 439

�:�b →∞`, î��5[e, uv°us > 1, vh`∫ ∞

1

x−sdx =1

s− 1�

�Q∫∞1

x−sdxÝe÷� , v«pùó∑∞

n=1 n−p�

»»»5.2..b →∞`,

∫ b

0

sin xdx = 1− cos b

�Á§¬�D3, Æ∫∞

0sin xdxs÷�

EÏ×l��5, Í[e�Í, x�:Í�I�h.∫ b

af(x)dx

D3, ∀b ≥ a, Æ∫∞

af(x)dxD3, uv°uD3×c > a, ¸ÿ∫∞

cf(x)dxD3�êuf(x)�D0óF (x)D3, b`ºbì�¶°:

∫ ∞

a

f(x)dx = F (x)∣∣∣∞

a= lim

x→∞F (x)− F (a)�

Íg, P§�5∫ b

−∞ f(x)dxô�v«2�L, Ç

(5.2)

∫ b

−∞f(x)dx = lim

a→−∞

∫ b

a

f(x)dx�

�uD3×c ∈ R(¥�cÂ¬�°×), ¸ÿ∫ c

−∞ f(x)dx C∫∞

cf(x)dx

/D3, JÌ∫∞−∞ f(x)dx[e, vÍÂ�L

(5.3)

∫ ∞

−∞f(x)dx =

∫ c

−∞f(x)dx +

∫ ∞

c

f(x)dx�

uîP��Þ�5, �Kb×�D3, JÌ∫∞−∞ f(x)dx s÷�Â

ÿ¥�ÝÎ∫∞−∞ f(x)dx, ¬�×��ylimb→∞

∫ b

−bf(x)dx�»A,∫∞

−∞ sin xdx s÷, ¬limb→∞∫ b

−bsin xdx = 0; Tf(x) = xù ×

»�¬u∫∞−∞ f(x)dxD3, JÍÂ�y

limb→∞

∫ b

−b

f(x)dx


(J�º3êÞ)�î�EEEÌÌÌÝÝÝÁÁÁ§§§(symmetric limit),Ì ∫∞−∞ f(x)

dx �ÞÞÞ��xxxÂÂÂ(Cauchy principal value)�

»»»5.3.�Ê��5∫∞−∞ e−k|x|dx, Í�k > 0�9ì&ÆJ�¸[

e�´�E∀b > 0,

∫ b

0

e−k|x|dx =

∫ b

0

e−kxdx =e−kb − 1

−k→ 1

k, b →∞�

Æ∫∞0

e−k|x|dx[e, vÍÂ 1/k�¨², E∀a < 0,

∫ 0

a

e−kxdx =

∫ 0

a

ekxdx =

∫ −a

0

e−k|x|dx → 1

k, a → −∞�

.h∫∞−∞ e−k|x|dx = 2/k�

»»»5.4.�Ê��5∫∞−∞ 1/(1 + x2)dx�´�

∫ ∞

0

1

1 + x2dx = lim

b→∞

∫ b

0

1

1 + x2dx = lim

b→∞arctan x

∣∣∣b

0

= limb→∞

arctan b =π

2�

!§ ∫ 0

−∞

1

1 + x2dx =

π

2�

Æ∫∞−∞ 1/(1 + x2)dx = π[e�

»»»5.5.�Sn =∑n2

j=1n

n2+j2 , n ≥ 1��Olimn→∞ Sn��..

n

n2 + j2=

1

n

1

1 + (j/n)2,

Æ ∫ (j+1)/n

j/n

1

1 + x2dx <

n

n2 + j2<

∫ j/n

(j−1)/n

1

1 + x2dx�

7.5 ��5 441

.h ∫ (n2+1)/n

1/n

1

1 + x2dx < Sn <

∫ n

0

1

1 + x2dx�

3îP�n →∞, v¿à»5.4Ý��, Çÿlimn→∞ Sn = π/2�

A!3ùó�, E��5ùb×°ny[eÝl�°�9ì ×E�5Õ� Ñ`��Ýl�°�

��§§§5.1.'f(x) ≥ 0, ∀x ≥ a, ê'E∀b ≥ a,∫ b

af(x)dx/D3�J∫∞

af(x)dx[e, uv°uD3×M > 0, ¸ÿ

∫ b

a

f(x)dx ≤ M, ∀b ≥ a�

î��§��§3.1ET, ôÎàÕ��vb&Ýó�Ä[eh×��¿àh�§ñÇ�ÿì��§3.2ETÝ�§, ôÌ fff´lll��°°°�

��§§§5.2.'E∀x ≥ a, 0 ≤ f(x) ≤ g(x)�ê'∫ b

af(x)dxD3, ∀b ≥

a, v∫∞

ag(x)dx[e�J

∫∞a

f(x)dx[e, v∫ ∞

a

f(x)dx ≤∫ ∞

a

g(x)dx�

3î�§�, &ÆÌ∫∞

ag(x)dxYg

∫∞a

f(x)dx�ãyf(x) ≥ 0,

Æ∫ b

af(x)dx ×b �¦Ðó�ê.f(x) ≤ g(x), Æ

∫ b

af(x)dx ≤∫ b

ag(x)dx ≤ ∫∞

ag(x)dx < ∞�Æ�§5.2ù ×Ë��vb&Ý

ó�Ä[e�Tà�9ìÝ�§, ùÌ ÁÁÁ§§§fff´lll��°°°�

��§§§5.3.'E∀x ≥ a, f(x) ≥ 0, g(x) > 0�ê'E∀b ≥ a,∫ b

af(x)dx

C∫ b

ag(x)dx/D3�u

(5.4) limx→∞

f(x)

g(x)= c, c 6= 0,


J∫∞

af(x)dxC

∫∞a

g(x)dx,!`[eTs÷�uc = 0,J∫∞

ag(x)dx

[e, 0l∫∞

af(x)dx[e��uc = ∞, J

∫∞a

g(x)dxs÷, 0l

∫∞a

f(x)dx s÷�

î�ë�§�J�, ãy5½v«�§3.1�3.3�J�, Æº�\ï� ��W�¨², &Æbì��, J�Jº3êÞ�

��§§§5.4.(i)E∀a ∈ R ∪ {−∞}, u∫∞a|f(x)|dx[e, J

∫∞a

f(x)dx[e, v ∣∣∣

∫ ∞

a

f(x)dx∣∣∣≤

∫ ∞

a

|f(x)|dx;

(ii) uf(x) ≥ 0, ∀x ≥ 0, f(x) ≤ 0, ∀x < 0, J∫∞−∞ f(x)[e, u

v°u∫∞−∞ |f(x)|dx [e�

3}¡»5.11&ÆÞ:Õ,u∫∞

af(x)dx[e,J

∫∞a|f(x)|dx�×

�[e�Æ�§5.4 ×Âÿº�Cf´Ý��, ô��§4.2Eï�

»»»5.6.¿à»5.1C�§5.3, ÿ∫∞1

x/(3x2 + 4x + 5)dxs÷, �∫∞

1x2

/(x4 + 5x + 6)dx[e�

»»»5.7.�Ê∫∞0

e−x2/2dx�.x > 2`x2/2 > x, Æ

0 < e−x2/2 < e−x, ∀x > 2�

� ∫ ∞

2

e−xdx = limb→∞

∫ b

2

e−xdx = limb→∞

(e−2 − e−b) = e−2

D3, Æã�§5.2á,∫∞

1e−x2/2dxD3�ê0 ≤ x ≤ 1`, 0 <

e−x2/2 ≤ 1�Æ∫ 1

0e−x2/2dxQD3�.h

∫ ∞

0

e−x2/2dx =

∫ 2

0

e−x2/2dx +

∫ ∞

2

e−x2/2dx

7.5 ��5 443

D3�¨², ô�¿à0 < e−x2/2 < e−x+1, ∀x ≥ 0, v∫∞0

e−x+1dx =

e, �ÿ∫∞0

e−x2/2dxD3�3Ïèa&ÆºJ�

(5.5)

∫ ∞

0

e−x2/2dx =

√π

2�

9Î×Íb¶Ý��, .��5∫

e−x2/2dx¬&��Ðó, ��5

∫∞0

e−x2/2dxQ��@2O��Íg, .e−x2/2 ×�Ðó, Æ

∫ ∞

−∞e−x2/2dx = 2

√π

2=√

2π,

.h ∫ ∞

−∞

1√2π

e−x2/2dx = 1�

×Ðógu��g(x) ≥ 0, ∀x ∈ R, v∫∞−∞ g(x)dx = 1(Ç3g�

%�ì, �x �î] � ½Ý«� 1), JgÌ ×^£££ÛÛÛ��ÐÐÐóóó(probability density function)�Æu�

(5.6) f(x) =1√2π

e−x2/2, x ∈ R,

Jf ×^£Û�Ðó�E×[eÝ��5, �ó�ð)Êà�u�x = (t− µ)/σ, Jÿ

∫ ∞

−∞

1√2πσ

e−(t−µ)2/(2σ2)dt = 1�

ÆE∀µ ∈ R, σ > 0,

(5.7) f(x) =1√2πσ

e−(x−µ)2/(2σ2), x ∈ R,

) ×^£Û�Ðó�!ñ×è, Æ»Ý10y¸üJÎ|{{{úúú ß, ß¼�b×(5.7)P�ÐóCÍ%��000---§§§¡¡¡Î{úE^£¡Ýx�Q¤�{ús¨(5.7)P�L��^£Û�Ðó, 30-§¡�, 6�½×Á¥�Ý��


»»»5.8.E∀s ∈ R, .

limx→∞

e−xxs

x−2= 0,

v∫∞1

x−2dx[e, Æ∫∞1

e−xxsdx[e�

&Æ�+Û¨×Ë��5�'×Ðóf , �L3(a, b], vE∀x ∈(a, b],

∫ b

xf(t)dtD3�J

∫ b

a+f(t)dtÌ ÏÏÏÞÞÞlll��555(improper in-

tegral of the second kind), vu

limx→a+

∫ b

x

f(t)dt

D3vb§, JÌh��5[e, ÍJÌ s÷�h`¬�L∫ b

a+

f(t)dt = limx→a+

∫ b

x

f(t)dt�

»»»5.9.E∀b > x > 0,

∫ b

x

t−sdt =

{b1−s−x1−s

1−s, s 6= 1,

log b− log x, s = 1�

x → 0+î��5D3, uv°us < 1�Æ∫ b

0+t−sdt D3, uv

°us < 1�¨², Bãt = 1/u��ð, �ÿ

∫ b

x

t−sdt =

∫ 1/x

1/b

us−2du�

x → 0+, 1/x →∞, .h

∫ b

0+

t−sdt =

∫ ∞

1/b

us−2du,

©�îP��5D3��ã»5.1á, îP��5D3, uv°us− 2 < −1, Çs < 1�

7.5 ��5 445

»»»5.10.�O∫ 1

0+x log xdx�

��.ãðà�5�, �ÿ∫ 1

0+

x log xdx = limt→0+

∫ 1

t

x log xdx = limt→0+

1

4(2x2 log x− x2)

∣∣∣1

t

= limt→0+

1

4(−2t2 log t + t2 − 1) = −1

4,

Í�àÕlimt→0+ t2 log t = 0(�¿à1Ä¾!JJ�)�

»5.9�î, ÏÞl��5�; Ï×l��5, X|&Æ¬�m©½��, nyÏÞl��5Ýl�°�&Æô�v«2�L��5

∫ b−a

f(t)dt�êu∫ c

a+f(t)dt�∫ b−

cf(t)dt/[e, J�L

∫ b−

a+

f(t)dt =

∫ c

a+

f(t)dt +

∫ b−

c

f(t)dt�

b` Ý�-, ??|∫ b

af(t)dtã�

∫ b−a+

f(t)dt, T∫ b

a+f(t)dt, T∫ b−

af(t)dt�»A3

∫ b

0t−2dt�, &Æá¼hÇ�

∫ b

0+t−2dt, .t��

0��5Ý�L�|�}�.Â�»A, uf3c, d/P�L, Í

�a < c < d < b, Ju∫ c−

af(t)dt,

∫ d−c+

f(t)dt,∫ b

d+f(t)dt/[e, -

Ì∫ b

af(t)dt[e, v|hë�5�õ, �

∫ b

af(t)dt��L�¨²,

ô�bËvl��5��)�»A, �Ê∫ b

a+f(t)dt +

∫∞b

f(t)dt, v|

∫∞a+

f(t)dt, T∫∞

af(t)dt�h�)lÝ��5�

»»»5.11.9ì&Æ¼¾½∫∞

0sin x/xdx�e÷P, h�5Ì

Dirichlet��555(Dirichlet integral), � DirichletX"D�.limx→0+ sin x/x = 1, Æu�f(x) = sin x/x, x > 0, f(0) = 1,

Jf3[0,∞] =��Æ∫∞

0sin x/x dx�Ú Ï×l��5, vh

�5[e, uv°u∫∞

1sin x/xdx [e�E∀b ≥ 1, ¿à5I�5

ÿ ∫ b

1

sin x

xdx =

− cos x

x

∣∣∣∣b

1

−∫ b

1

cos x

x2dx�


.b →∞`,

−cos x

x

∣∣∣b

1= cos 1− cos b

b→ cos 1,

v. ∣∣∣cos x

x2

∣∣∣ ≤ 1

x2,

�∫∞1

1/x2dx[e,Æã�§5.4á,∫∞1

cos x/x2dxD3,.h∫∞

0sin x

/xdx[e�¯@î�J�∫∞0

sin x/xdx = π/2(�Courant and John

(1965) pp.589-591TÍhÏèa)�!ñ×è, .f(x) ×=�Ðó, Æu�5 �î§� P§�, �Î×b§ÝÑó, Jh�5[[[eee�¬uÞ�5Õ�ã�EÂ, Jÿ×s÷Ý��5

∫∞0| sin x|/xdx(J�º3êÞÏ26Þ, ù�»5.18)�\ï��

0sin x/x, x > 0, �%, |Ý�∫∞

0sin x/xdxC

∫∞0| sin x|/xdx�¿

¢�L�3h,

∫∞0

sin x2dx(Í[eP�ì»), C∫∞0

cos x2dx(Í[ePº3êÞÏ18Þ)Ì Fresnel��555(Fresnel integral), �¨y��aaaÝÝÝ�� æææ§§§(theory of diffraction of light)�ãFresnel�5á, Ç¸limx→∞ f(x) 6= 0,

∫∞0

f(x)dx)b��[e�9Î�ùó[e�!�(ù�êÞÏ23Þ)�¯@î, Ç¸f(x)� b&,

∫∞0

f(x)dx)��[e�»A, �Ê

∫∞0

2u cos u4du�u= 4√

nπ, n = 0, 1, 2, · · · ,�5Õ�W 2 4

√nπ cos nπ = 2 4

√nπ, T−2 4

√nπ, Æ�5Õ�� b

&�¬u�u2 = x, Jh�5W ∫∞

0cos x2dx [[[eee�

»»»5.12.ã»5.2á∫∞

0sin xdxs÷, 9ìJ�

∫∞0

sin x2dx[e�Aî», &Æ©mJ�

∫∞1

sin x2dx[eÇ��E∀b ≥ 1, �t =

x2, Jÿ ∫ b

1

sin x2dx =1

2

∫ b2

1

sin t√t

dt�

ûî», )¿à5I�5, �J�∫∞1

sin t/√

tdt[e(�êÞÏ18Þ)

�J±�

#½&Æ+Û×3Tàó.�Á¥�ÝgammaÐÐÐóóó�

7.5 ��5 447

»»»5.13.'α > 0, 9ìJ�∫∞0

e−ttα−1dt [e��Þh�5;¶ ∫ 1

0

e−ttα−1dt +

∫ ∞

1

e−ttα−1dt�

Í�ÏÞÍ�5�3»5.8�á¼[e��yÏ×Í�5, �t =

1/u, J ∫ 1

0

e−ttα−1dt =

∫ ∞

1

e−1/uu−α−1du�

¬©��∫∞1

u−α−1duf´, -á∫∞1

e−1/uu−α−1du[e, ∀α > 0�.h

∫ 1

0e−t tα−1dt[e, ∀α > 0�Æα > 0`,

∫∞0

e−ttα−1dt[e�&Æ-�L×±Ðó

(5.8) Γ(α) =

∫ ∞

0

e−ttα−1dt, α > 0�

Γ Ì gammaÐó, �Z3�-1729OXS, hÐób×°b¶ÝP²�AE∀α > 0,

Γ(α + 1) =

∫ ∞

0

e−ttαdt =

∫ ∞

0

tαde−t

= −tαe−t∣∣∣∞

0+

∫ ∞

0

αe−ttα−1dt = αΓ(α),

Ç

(5.9) Γ(α + 1) = αΓ(α), ∀α > 0�

ê

(5.10) Γ(1) =

∫ ∞

0

e−tdt = 1�

ã(5.9)C(5.10)ÞPÇÿ, EN×ÑJón,

(5.11) Γ(n + 1) = n!�

ê¿à(5.5)P, �ÿ(J�º3êÞ)

(5.12) Γ(1

2) =

√π�


�¿à(5.9)P, Çÿ

Γ(3

2) = Γ(

1

2+ 1) =

1

2Γ(

1

2) =

√π

2 �

!§EN×ÑJón, /�O�Γ(n/2)�

ã(5.8)Pÿ∫ ∞

0

e−ttα−1

Γ(α)dt = 1, ∀α > 0�

u�ó�ð, �t = βx, Í�β > 0 ×ðó, Jÿ∫ ∞

0

e−βx(βx)α−1

Γ(α)βdx =

∫ ∞

0

βαxα−1e−βx

Γ(α)dx = 1�

Æu�

(5.13) f(x) =

{βαxα−1e−βx

Γ(α), x > 0,

0, x ≤ 0,

Í�α, β ÞÑÝðó, Jf(x) > 0, ∀x > 0, v∫ ∞

0

f(x)dx = 1�

(5.13)P�L�Ðófù ×¥�Ý^£Û�Ðó�¨², ã»5.4á, u�

(5.14) h(x) =1

π(1 + x2), x ∈ R,

J. ∫ ∞

−∞

1

π(1 + x2)dx = 1,

Æh(x)ù ×^£Û�Ðó��¿à�ó�ð, �x = (t− θ)/a, ÿ∫ ∞

−∞

a

π(a2 + (t− θ)2)dt = 1�

7.5 ��5 449

ÆE∀a > 0, θ ∈ R,

(5.15) f(x) =a

π(a2 + (x− θ)2), x ∈ R,

h ×ð�Ý^£Û�Ðó�35.3;&Æ�¼�, 4f1(x) = 1/(1 + x2)�f2(x) = e−x2Ý%

�:R¼bF, Í@Þï Qb��Ý-²�¯@î�pl�E∀n ≥ 1,

∫∞0

xnf1(x)dx/s÷, �∫∞0

xnf2(x)dx/[e�4x →∞`, f1(x)�f2(x)/�3�0, ¬f2(x)��0Ý>�´f1(x)"�9(Ì?f2(x)/f1(x) �Á§Ç�á), Æf2(x)¶î�×xn, n ≥ 1, Í�5)[e��f1(x)��0Ý>�´X, ©�¶î×xn, n ≥ 1,

Í�5-s÷Ý�Íg¼:Lalpace»»»ððð(Laplace transform)�'×Ðóg(x)�L

3[0,∞) î, vg(x) ≥ 0, ∀x ≥ 0�J

(5.16) ψ(u) =

∫ ∞

0

e−uxg(x)dx, u > 0,

Ì g�Laplace»ð, ©�î��5D3�h ó.�×¥�Ý»ð�uf��LA(5.13)P, Jf�Lalpace»ð

ψ(u) =

∫ ∞

0

e−ux βαxα−1e−βx

Γ(α)dx = βα

∫ α

0

xα−1e−(β+u)x

Γ(α)dx

=βα

(β + u)α

∫ ∞

0

(β + u)αxα−1e−(β+u)x

Γ(α)dx =

(β

β + u

)α

,

h�àÕt¡×�5�Ý�5Õ�ùb(5.13)P��fÝ�P, ©Îβ ; β + u, ÆÍ�5) 1�

»»»5.14.'b×Ðóf(x) = 1/x2, x ≥ 1�JhÐó%��=� ∫ ∞

1

√1 + (−2/x3)2dx =

∫ ∞

1

√x6 + 4

x3dx = ∞,

h.limx→∞√

x6 + 4/x3 = 1 6= 0�¬f�%�ì, ã1�∞�«� ∫ ∞

1

1

x2dx = −1

x

∣∣∣∣∞

1

= 1,


Û×b§Â�¨², �g(x) = 1/x, x ≥ 1, Jg�%�ì, ã1�∞�«�

∫ ∞

1

1

xdx = ∞�

¬g�%��x�I»X��

π

∫ ∞

1

1

x2dx = π,

×b§Â�î�9°��, y:�ìº�ßb°�#��Ä.¸Æ��Õ

�!îÝ��«�T��, X|[es÷ , ¬PÄQÝn;�

Bã�ó�ð, ×��5b`�»ð ×Ñð�5�»A, u�x = sin u, J

∫ 1

0

1√1− x2

dx =

∫ π/2

0

du =π

2�

¨×]«, ×=�ÐóÝ�5, ôb��»ðW×��5�9Ë�µsß3u�u = φ(x), v3�5 ÝÐF, 0óφ′(x) 0,

Ædx/du P§��Í;t¡&Æ�¼:¿Í»�, |¸\ïE��5?Ý��

»»»5.15.�O ∫ 3

1

1√(x− 1)(3− x)

dx�

��.ãy�5Õ�, 3�5 ÝËÍÐF/P�L, Æ�ÞXkO��5;¶

limε,δ→0+

[∫ a

1+ε

1√(x− 1)(3− x)

dx +

∫ 3−δ

a

1√(x− 1)(3− x)

dx

],

Í�1 < a < 3�.∫1√

(x− 1)(3− x)dx =

∫1√

1− (x− 2)2dx = arcsin(x− 2) + C,

7.5 ��5 451

Æ∫ 3

1

1√(x− 1)(3− x)

dx = limε,δ→0+

(arcsin(1− δ)− arcsin(ε− 1))

=π

2+

π

2= π�

»»»5.16.�O ∫ ∞

1

1

ex+1 + e3−xdx�

��.9Î×Ï×l��5��y = x− 1, J∫

1

ex+1 + e3−xdx =

1

e2

∫1

ex−1 + e1−xdx =

1

e2

∫ey

e2y + 1dy

=1

e2arctan ey + C =

1

e2arctan ex−1 + C�

Æ∫ ∞

1

1

ex+1 + e3−xdx = lim

b→∞

∫ b

1

1

ex+1 + e3−xdx

= limb→∞

1

e2(arctan eb−1 − arctan e0) =

1

e2(π

2− π

4) =

π

4e2�

»»»5.17.'f ×��Ðó, vf(1) = 1, ê

(5.17) f ′(x) =1

x2 + f 2(x), ∀x ≥ 1�

�Jlimx→∞ f(x)D3, vÁ§Â�y1 + π/4�JJJ��.ã(5.17)Páf ′(x) > 0, ∀x ≥ 1, Æ3x ≥ 1�, f �}�¦�.hf(t) > f(1) = 1, ∀t > 1, v

f ′(t) =1

t2 + f 2(t)<

1

t2 + 1, ∀t > 1�

ê.(5.17)P�� ×=�Ðó, Æf ′ù ×=�Ðó�.hE∀x > 1, f ′(t)3[1, x]��, v

f(x) = 1+

∫ x

1

f ′(t)dt < 1+

∫ x

1

1

t2 + 1dt < 1+

∫ ∞

1

1

t2 + 1dt = 1+

π

4�


.f(x)3x ≥ 1, ��vb&, Ælimx→∞ f(x)D3, v�y1 +

π/4�J±�

»»»5.18.'Ðóf(x)�L3x ≥ 0, ÍÐó%�A%5.1�

-

6

xO

y

1 2 3 4 5 6

112 1

3

−1

−12

−13

%5.1.

×��f(2n + 1/2) = 1/(n + 1), vf(2n + 3/2) = −1/(n +

1), n ≥ 0, v%�ã×°�Të��XàW, ¬EÌyx��J∫∞0

f(x)dx[e, vÍÂ 0�¬∫∞

0|f(x)|dxs÷�uÑ;f , ¸Í

Ðó%��Ý�Të��, 9I��µ� 1, 1, 1/2, 1/2, 1/3,

1/3,· · · , �ë��{î¹ 1, �ÿ×Ðóg, Jhù ×limx→∞g(x) 6= 0, ¬

∫∞0

g(x)dx[e�»�

êêê ÞÞÞ 7.5

1. �l�ì�&��5�e÷P, u[e¬OÍÂ�(1)

∫∞1

1x√

xdx� (2)

∫ 4

−∞1

(5−x)2dx�

(3)∫ 3

−∞1√7−x

dx� (4)∫∞

0x

1+x2 dx�(5)

∫ 4

31√x−3

dx� (6)∫ 0

−21√

4−x2 dx�(7)

∫∞0

1√ex dx� (8)

∫ 1

013√x

dx�(9)

∫ 4

01

x√

xdx� (10)

∫∞1

11+x2 dx�

(11)∫∞

31

x2−2xdx� (12)

∫ π/2

π/4sec xdx�

(13)∫ π/2

01

1−sin xdx� (14)

∫∞−1

x

ex2 dx�(15)

∫∞2

1x2−1

dx� (16)∫∞

0e−√

x√x

dx�

êÞ 453

(17)∫∞

0e−x cos xdx� (18)

∫ 1

0x4 log xdx�

(19)∫∞−∞

1ex+e−x dx� (20)

∫∞0

cos x√1+x3 dx�

2. �l�ì�&��5�e÷P�(1)

∫ 1

01√sin x

dx� (2)∫ 1

01√

tan xdx�

(3)∫∞

0x√

x4+1dx� (4)

∫∞0

1√x3+1

dx�(5)

∫∞0

log x√x

dx� (6)∫ 1

0log x1−x

dx�(7)

∫∞−∞

xcosh x

dx� (8)∫ 1

01√

x log xdx�

(9)∫∞

21

x(log x)2dx� (10)

∫∞0

x1+x6 sin2 x

dx�

3. �Ju∫∞−∞ f(x)dx[e, JÍÂ�ylimb→∞

∫ b

−bf(x)dx�

4. ¿à(5.5)P, �JΓ(1/2) =√

π�

5. (i) ¿à(5.5)P, �O∫∞

0x2e−x2

dx;

(ii) 'f��LA(5.7)P, �J∫∞−∞(x− µ)2f(x)dx = σ2�

6. 'f��LA(5.13)P��J∫∞0

xf(x)dx = α/β , v∫∞

0(x −

α/β)2f(x)dx = α/β2�

7. �0�ì�gammaÐó�×°Í��P�(i) Γ(α) = 2

∫∞0

t2α−1e−t2dt,

(ii) Γ(α) =∫ 1

0(log(1/t))α−1dt,

(iii) Γ(α) = cα∫∞

0tα−1e−ctdt, c > 0,

(iv) Γ(α) =∫∞−∞ eαte−et

dt�

8. �5½Eì�ÞÐó, OÍLaplace»ð�(i) f(x) = λe−λx, x ≥ 0, λ > 0;

(ii) f(x) = sin2 x, x ≥ 0�9. �J

∫ 4

0

1√x(x + 4)

dx =π

4,

∫ ∞

4

1√x(x + 4)

dx =π

4�


10. �5½Oì�&�5�(i)

∫∞1

log xx2 dx, (ii)

∫ e

0x2 log xdx�

(iii)∫ π/2

1( 1

x2 − csc x cot x)dx,

(iv)∫ π/2

0(sec x tan x− sec2 x)dx,

11. �Oa, b�Â, ¸∫∞0

xa/(1 + xb)dx[e�

12. �5½O¸ì�&��5[e�pÝ/)�

(i)∫ 1

0xp log xdx, (ii)

∫ 1

0xp log2 xdx, (iii)

∫∞1

log x/xpdx�

13. (i) ¿à2x/π < sin x ≤ 1, ∀x ∈ (0, π/2], �J��5∫ π/2

0log sin xdxD3;

(ii) ¿à

∫ π/2

0

log sin xdx =

∫ π/2

0

log cos xdx =

∫ π

π/2

log sin xdx,

C∫ π/2

0

log sin 2xdx =

∫ π/2

0

log sin xdx+

∫ π/2

0

log cos xdx+

∫ π/2

0

log 2dx,

�J∫ π/2

0log sin 2xdx = −(π log 2)/2�

14. �J ∫ ∞

0

1

1 + x4dx =

π√

2

4 �

15. �J

limλ→∞

∫ ∞

0

1

1 + λx4dx = 0�

16. �5½Oðóc�Â¸ì�&�5[e, ¬Oh`&�5Â�(i)

∫∞2

( cxx2+1

− 12x+1

)dx,

(ii)∫∞

1( x

2x2+2c− c

x+1)dx,

(iii)∫∞

0( 1√

1+2x2 − cx+1

)dx�

êÞ 455

17. �5½Oðóa, b�Â, ¸ì�Þ�PWñ�(i)

∫∞1

(2x2+bx+ax(2x+a)

− 1)dx = 1,

(ii) limp→∞∫ p

−px3+ax2+bx

x2+x+1dx = 1�

18. (i) �J∫∞1

sin x/√

xdx[e,

(ii) �Jlimx→0+ x∫ 1

xcos t/t2dt = 1,

(iii) �¾½∫ 1

0cos t/t2dt�e÷P,

(iv) �J∫∞0

cos x2dx [e�

19. �J∫∞0

sin2(π(x + x−1))dxs÷�

20. �¾½∫∞0

sin t/(1 + t)dt�e÷P�

21. �5½O¸ì�Þ�5[e�sÝP��(i)

∫∞0

xs−1/(1 + x)dx, (ii)∫∞0

sin x/xsdx�

22. (i) 'f(x) ×��3�Ðó, vlimx→∞ f(x) = 0��J

∫∞1

f(x)dx�ùó∑∞

n=1 f(n)!`[eTs÷;

(ii) �Ü×&��Ðóf , ¸ÿ∑∞

n=1 f(n)[e, ¬∫∞

1f(x)dx

s÷�

23. (i) uf(x)3x ≥ 0� ×�3ÝÑÐó, v∫∞0

f(x)dx[e, �Jlimx→∞ f(x) = 0;

(ii) ?×�2, �Julimx→∞ f(x)D3, v∫∞0

f(x)dx [e,

Jlimx→∞ f(x) = 0�

24. 'f3[0, 1]=��J

limx→∞

x

∫ 1

x

f(z)

z2dz = f(0)�

25. 'Ðóh��LA(5.14)P�O∫∞−∞ xh(x)dx�

26. �J∫∞0| sin x|/xdxs÷�


27. (i) 'a > 0 ×ðó��J

limh→0+

∫ a

−a

h

h2 + x2dx = π;

(ii) 'f(x)3x ∈ [−1, 1]=��J

limh→0

∫ 1

−1

h

h2 + x2f(x)dx = πf(0)�

28. �J∞∑

k=n

1

k2 log k∼

∫ ∞

n

1

x2 log xdx ∼ 1

n log n�

29. �J�§5.4�

30. �J»5.18�∫∞

0f(x)dxC

∫∞0

g(x)dx[e,¬∫∞0|f(x)|dxs÷�

31. �Oì�&Á§�(i) limt→∞

R∞t e−x2

dx

e−t2/t, (ii) limt→∞

R t+1t e−x2

/xadxR∞t e−x2

/xadx, a > 0�

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1974). Mathematical Analysis, 2nd ed. Addison-


2. Courant, R. and John, F. (1965). Introduction to Calculus and

Analysis, Vol I. Springer-Verlag, New York, New York.

3. Rudin, W. (1964). Principles of Mathematical Analysis, 2nd ed.

McGraw-Hill Book Company,New York, New York.

ÏÏÏâââaaa

ÐÐÐóóóóóó��CCCÐÐÐóóóùùùóóó

8.1 GGG��

&Æá¼u|x| < 1, J¿¿¿¢¢¢ùùùóóó∑∞

n=0 xn[e�(1− x)−1�Ç

(1.1) 1 + x + x2 + · · ·+ xn + · · · = 1

1− x, |x| < 1�

�u|x| ≥ 1, Jî�¿¢ùós÷�3î×a, &Æ4"Dùó�e÷P�¬�I5�&Æ¾�[eÝùó,ÍõQP°O��¿¢ùó, ÎKó&Æ�|×��Ý2P, ¼�îÍõ�[eùó�hùóÝ¥�P , �ã¸�s, �ÿÕ&9b¶Ýùó�õ�»A, 3(1.1)P�, u|x2ã�x, Jÿ×©b�g4Ýùó:

(1.2) 1 + x2 + x4 + · · ·+ x2n + · · · = 1

1− x2, |x| < 1�

�yukO©â�g4Ýùóõ, �3(1.2)P�¼��&¶|x, Jÿ

(1.3) x + x3 + x5 + · · ·+ x2n+1 + · · · = x

1− x2, |x| < 1�

u3(1.1)P�|−xã�x, Jÿ

(1.4) 1− x + x2 + · · ·+ (−1)nxn + · · · = 1

1 + x, |x| < 1�

457

458 Ïâa Ðóó�CÐóùó

3(1.4)P�u|x2ã�x, Jÿ

(1.5) 1−x2+x4−x6+· · ·+(−1)nx2n+· · · = 1

1 + x2, |x| < 1�

ÞîPË�!¶|x, êÿ(1.6)

x− x3 + x5− x7 + · · ·+ (−1)nx2n+1 + · · · = x

1 + x2, |x| < 1�

Q��ÿÕ�98nÝùó�A3(1.2)P�, |2xã�x, Jÿ×3|x| < 1/2[eÝùó�î�9°ùó,K�Ú P§gÝ94P(polynomials of infinite

order), TÌ��ùùùóóó(power series), ôµÎ¸Æ/b∞∑

n=0

anxn

Ý�P, Í�a0, a1, · · · , Ì h�ùó�;ó�Ey94P, �¡ÎOÍÐóÂ, T�5��5ÝºÕK&ð��pñ�ßs¨&9ÐóK��W�ùó, �vùóÝ;ób8-Ý!�, 9�1Î��5s"��Ý×¥�Wµ�tÝ(1.1)−(1.6)P£°², ´½(Ý»�$b

ex = 1 + x +1

2!x2 + · · ·+ 1

n!xn + · · · ,(1.7)

sin x = x− 1

3!x3 +

1

5!x5 − · · ·+ (−1)n

(2n + 1)!x2n+1 + · · · ,(1.8)

cos x = x− 1

2!x2 +

1

4!x4 − · · ·+ (−1)n

(2n)!x2n + · · ·�(1.9)

î�ë�ùóÝ�î°, /E∀x ∈ RWñ�}¡&ÆºJ�, E(1.1)−(1.9)P, uÞN×�r�¼��5½

Ex �5T�5, �P)Wñ�ôµÎE�ùóÝ�5C�5, &Æ�AE(b§g�)94Pv«Ý�§�»A, Þ(1.1)P¼��5½Ex�5, Jÿ

(1.10) 1 + 2x + 3x2 + · · ·+ nxn−1 + · · · = 1

(1− x)2, |x| < 1�

8.1 G� 459

�uÞ(1.4)P¼��5½Ex�5, Jÿ

log(1 + x) = x− 1

2x2 +

1

3x3 − 1

4x4 + · · ·+ (−1)n

n + 1xn+1(1.11)

+ · · · , |x| < 1�îPÇÞEóô�îW�ùó, 9ÎMercator (1620-1687)�Brouncker (1620-1684) Ý�ÕÔ`a%�ìÝ«�,y�-1668Os¨Ý�4Q(1.1)P©E|x| < 1Wñ, �Ä(1.11)PEx = 1 ùWñ�Þx = 1�á(1.11)P, Jÿ

(1.12) log 2 = 1− 1

2+

1

3− 1

4+ · · · ,

9Î3��5s"Ý��, �ó.�� ÌcÝ�P�×�¨², uÞ(1.5)P¼��Ex�5, Jÿ

arctan x = x− 1

3x3 +

1

5x5 − 1

7x7 + · · ·+ (−1)n

2n + 1x2n+1(1.13)

+ · · · , |x| < 1�9ÎGregory (1638-1675)y�-1671Os¨Ý, hPE|x| ≤ 1Wñ�.arctan 1 = π/4, Æã(1.13)Pÿ

π

4= 1− 1

3+

1

5− 1

7+ · · ·

(Ì Leibniz-Gregory series, h.¾¾¹+.�Õ��iÝ«�,

y�-1673O¥±s¨h��), T

(1.14) π = 4(1− 1

3+

1

5− 1

7+ · · · )�

A!Elog 2Ý"P, îPôÎ×½(Ý�P�&Æ:ÕbÝ��5, &Q�|×��Ýùó, �îiiiøøø£££π��Ä, (1.14)P¬&Î�Õπ t?Ý2P, .(1.14)P��ù

ó[e�X�ôµÎ�Õ�94, ��ÿÕπÝG¿��9ì&Æèº¿Í�´"[e�πÝ2Pº&�¢��´�, �α =

arctan(1/2), β = arctan(1/3), Jã

tan(α + β) =tan α + tan β

1− tan α tan β=

12

+ 13

1− 16

= 1 = tan(π

4),


ÿπ

4= α + β = arctan

1

2+ arctan

1

3

= (1

2− 1

3 · 23+

1

5 · 25− · · · ) + (

1

3− 1

3 · 33+

1

5 · 35− · · · )�

¨², .(¿à.arctan u+ arctan v= arctan((u + v)/(1 − uv)), Æ2 arctan u = arctan(2u/(1− u2)))

4 arctan1

5= 2 · 2 arctan

1

5= 2 arctan

5

12= arctan

120

119,

varctan

120

119− arctan

1

239= arctan 1 =

π

4,

.hπ

4= 4 arctan

1

5− arctan

1

239�Æÿ

π = 16(1

5− 1

3 · 53+

1

5 · 55− · · · )

− 4(1

239− 1

3 · 2393+

1

5 · 2395− · · · ),

9�QÎ×[e?"ÝÝùó�3�-1706O, Machin(1680-1751)

¿àarctan(1/5)ÝG704, Carctan(1/239)ÝG304, ÿÕπÝ�óG100��°»��Ý×ÍÝhÍ��µπÂÕ�ó×y0��y %��ÕπÂ?R�¦Ë(1985)×ZèÕËÍ§ã: (1) Ý�ÕπÂ, &Æ.ºÝ�9àé\�Õ, Cl��óêÝ]°; (2)Oπ

ÂÎ?�±é\, CIY�PßõÝ?]°�t¡, %�&9ð�Ý��Ðó/��îW�ùó÷(A(1.7)−

(1.9)P)? ¯@î9¬�H�M, ãÏ°a�§3.4Ý��2Pá, �×30Ý×Ï½(n + 1)g��ÝÐó, /�|×gó�øÄnÝ��94P¼¿��3(1.7)−(1.9)P�, £°�ùóÝI5õ, Ç ��94P�×Ðóf30Ý×Ï½��$0ó/D3, JEN×ÑJón, ��2P×å&Æ, f�¶W

(1.15) f(x) =n∑

k=0

akxk + Rn(x),

êÞ 461

Í�b§õ∑n

k=0 akxk ×gó�øÄn��94P, �Rn(x)�

9Ë¿�Ý0-�u3(1.15)P�, ��x, �¯n��∞, J��94P, ��×�ùó

∑∞k=0 akx

k, Í�

ak = f (k)(0)/k!�

uEØ°x, n →∞`, 0-4Rn(x) → 0, JE9Ëx, 3(1.15)P�, �n →∞, -ÿ

f(x) = limn→∞

n∑

k=0

akxk + lim

n→∞Rn(x) =

∞∑

k=0

akxk�

ùÇG��ùó[e�f(x)�u×x�º¸limn→∞ Rn(x) = 0, JI5õ

∑nk=0 akx

k, -�º��f(x)�»A, &Æb(1 − x)−1 =

1+x+x2+ · · ·+xn+Rn(x),Í�Rn(x) = xn+1/(1−x)�E∀|x| < 1,

n → ∞ `, Rn(x) → 0, Æ(1.1)PWñ�b`¬�Î£��|¾\Rn(x)ÎÍ��0�3£°�µì, n →∞, Rn(x)º��0,

9Î&Æb·¶Ý�Þ�3î×a&ÆD¡Ý×�ùóÝ[eP,

£°��Þ�ÜÃ&Æ, ¼¾½�ùóÝ[eP�f�ùó?×�Ý, µÎXÛÐÐÐóóóùùùóóó(series of functions), Ç

×ùóÝN×4 ×Ðó��h8nÝµÎÐÐÐóóóóóó��(sequence of

functions), Ía-�"DÐóó�CÐóùó�

êêê ÞÞÞ 8.1

1. �mBãJ�N×M»�)°P, �0�ì�&2P(/E|x| < 1Wñ)�

(i)∑∞

n=1 nxn = x(1−x)2

; (ii)∑∞

n=1 n2xn = x2+x(1−x)3

;

(iii)∑∞

n=1 n3xn = x3+4x2+x(1−x)4

; (iv)∑∞

n=1 n4xn = x4+11x3+11x2+x(1−x)5

;

(v)∑∞

n=1xn

n= log 1

1−x; (vi)

∑∞n=1

x2n−1

2n−1= 1

2log 1+x

1−x;

(vii)∑∞

n=0(n + 1)xn = 1(1−x)2

; (viii)∑∞

n=0(n+1)(n+2)

2!xn = 1

(1−x)3;

(ix)∑∞

n=0(n+1)(n+2)(n+3)

3!xn = 1

(1−x)4�


2. ãîÞ�(i)−(iv), &Æ��E∀k ≥ 1, D3×kgÝ94PPk(x), ¸ÿ

∞∑n=1

nkxn =Pk(x)

(1− x)k+1,

Í�Pk(x)�t±g4 x, t{g4 xk��¿àó.hû°J�h¯@, ¬�mJ�N×nyùó�ºÕÝ)°P�

3. ãÏ1Þ�(vii)−(ix), &Æ��E∀k ≥ 1,

∞∑n=0

(n + k

k

)xn =

1

(1− x)k+1�

�Jh¯@, ¬�mJ�N×nyùóºÕÝ)°P�

4. ¿à(1.7)P, �mJ�N×nyùóºÕÝ)°P, �O

(i)∑∞

n=2n−1n!

; (ii)∑∞

n=2n+1n!

; (iii)∑∞

n=2(n−1)(n+1)

n! �

5. ¿à(1.7)P, �mJ�N×nyùó�ºÕÝ)°P, �J

(i)∑∞

n=1n2xn

n!= (x2 + x)ex;

(ii)∑∞

n=1n3

n!= ke, Í�k ×ÑJó, ¬O�k�

8.2 @@@FFF[[[eee

&ÆD¡ÄÐó�[e, ÇOlimx→a h(x), ôD¡Äó��[e,

ÇOlimn→∞ an�Í;D¡×��î�ÞË[eÝ�µ�'b×ó��Ðó{fn}, v'9°Ðób8!Ý�L½�EN

×�L½�Ýx, �ÿ×ó�{fn(x)}��S �¸hó�[eÝx�/)�J×�L3SîÝÐóf , Í�

(2.1) f(x) = limn→∞

fn(x), ∀x ∈ S,

8.2 @F[e 463

Ì ó�{fn}�ÁÁÁ§§§ÐÐÐóóó(limit function)�&Æ¬Ìó�{fn}3S

î@@@FFF[[[eee(converges pointwise)�f�¬|fn → f(@F[e),

Tfn −→p f��ãyðóù ×©½ÝÐó, ÆÐóó�[e�ÃF, ðó

ó�[e�ÃFÝ.Â�ê.E×ü�Ýx, {fn(x)} ×ðóó�,

ÆÍÁ§, �µOðóó��Á§Ý]°¼O�ãÞ�[el�°á, Ç¸�áÁ§Ðóf , &Æ)�¾�Ðó

ó�{fn}ÎÍ[e�ôµÎ{fn}[e�×Á§Ðó, uv°uD3×/)S, vE∀x ∈ S C�×ε > 0, D3×n0 ≥1, ¸ÿE∀m,

n ≥ n0,

|fm(x)− fn(x)| < ε�¨², ¯@î&Æ|G-�D¡ÄÐóó��Á§�»A, 3Ï"a(3.5)P�Þex�î

ex = limn→∞

(1 +x

n)n�

h�fn(x) = (1 + x/n)n ×x�ng94P�!ø35.3;,&Æô�BèÄE×P§óα, �0×��α �b§ó�{αn}, v�L

xα = limn→∞

xαn�

N×Ðó-b×%��ÍET, G�ÞÐóó��[e, ×å&ÆexCxα�%�, �5½Ú (1 + x/n)nCxαn�%��Á§��ÄÁ§Ðó�%� Ðó%��Á§, h¯@ÎàÕ�-èÜtS�f, � ó.�X§��

|G&Æ1Ä, &ÆEÐóÝP²ð��·¶�×]«��Î. @jTàîÝm�, ×]«Î.D¡ÐóÝP², ðñ�&Æ��Ý�¶�EÐóó�, &Æx��D¡ì�®Þ: uó�{fn}��N×fn, /bØ×P², A=��T��, JÁ§Ðóf , ÎÍùÌbh©P? ÉA1, uN×fn/3x=�, JfÎÍù3x=�? 9ì&Æ��¿Í»��


»»»2.1.&Æ�×=�Ðóó�, ¬Á§Ðó�=��»�E∀n ≥ 1,

�fn(x) = xn, x ∈ [0, 1], JN×fn/ =�Ðó�êÁ§Ðóf

f(x) = limn→∞

xn =

{0, 0 ≤ x < 1,

1, x = 1,

3x = 1 �=��h»�î

(2.2) limx→a

limn→∞

fn(x) = limn→∞

limx→a

fn(x)

�×�Wñ�ã%2.1�¯�Ý� ¢n → ∞`, fn�Á§3x =

1�=��&��:��½nÝ¦�, fnÝ%�tÝ3x = 1², �¼�ì�x�, ¬fn(1) = 1, ∀n ≥ 1�n →∞ `, fn�Á§fÝ%�,

3x = 1-\*Ý�Æf3x = 1�=��

-

6

%2.1. =�Ðóó�ÍÁ§ÐóQ�=�

x

y

O 1

(1,1)

»»»2.2.&Æ�×

(2.3) limn→∞

∫ b

a

fn(x)dx =

∫ b

a

limn→∞

fn(x)dx

�Wñ�»�E∀n ≥ 1, �

fn(x) = nx(1− x2)n, x ∈ [0, 1]�

8.2 @F[e 465

J.fn(0) = fn(1) = 0, ∀n ≥ 1, vE∀x ∈ (0, 1), 0 < (1 − x2) < 1,

Æ(¿àÏ"a�§5.4)

f(x) = limn→∞

fn(x) = 0, ∀x ∈ [0, 1]�

&Æ�×°fn(x)�%�y%2.2�.∫ 1

0

fn(x)dx = n

∫ 1

0

x(1− x2)ndx = −n

2

(1− x2)n+1

n + 1

∣∣∣∣1

0

=n

2(n + 1),

Æ

limn→∞

∫ 1

0

fn(x)dx = limn→∞

n

2(n + 1)=

1

2�

¬ ∫ 1

0

limn→∞

fn(x)dx =

∫ 1

0

0dx = 0,

Þï��h»�î�5�Á§, �×��yÁ§��5��5Î×��ÕÁ§ÝºÕ, Aî», ËbnÁ§ÝºÕ??��øð�

-

6

%2.2. Ðóó�Á§��5��y�5�Á§

x

y

O 1

n = 3

n = 2

n = 1

¨uÞfn;

fn(x) = n2x(1− x2)n, x ∈ [0, 1],


J

limn→∞

∫ 1

0


n2

2n + 2= ∞,

¬.limn→∞ fn(x) = 0, ∀x ∈ [0, 1], Æ∫ 1

0limn→∞ fn(x)dx, �limn→∞∫ 1

0fn(x)dx)�8�, vÞï-²��

»»»2.3.&Æ�×��Ðóó�{fn}, Á§ÐóD3, ¬{f ′n}s÷�»�E∀n ≥ 1, �

fn(x) = sin nx/√

n, x ∈ R�

Jlim

n→∞fn(x) = 0, ∀x ∈ R�

¬f ′n(x) =

√n cos nx,

ÆE�×x ∈ R, limn→∞ f ′n(x)/�D3�%2.3�×°fn(x)�%��

-

6

%2.3. Ðó�Á§D3¬0óó�s÷

x

y

O π

n = 1

n = 2

n = 3

n = 8?

6n = 16

0óù ×OÁ§ÝÄ�, î»�î

(2.4) limn→∞

(d

dxfn(x)) =

d

dx( limn→∞

fn(x))

�×�Wñ�&Æ��¿ÍËÁ§ºÕ, �×��øð�»�

8.2 @F[e 467

»»»2.4.'b×ÞÞÞ¥¥¥óóó��(double sequence){amn,m ≥ 1, n ≥ 1}, Í�

amn =m

m + n�

J.EN×ü�Ýn,

limm→∞

amn = 1,

Ælim

n→∞lim

m→∞amn = 1�

¬EN×ü�Ým,

limn→∞

amn = 0,

Ælim

m→∞lim

n→∞amn = 0�

h»�î

(2.5) limn→∞

limm→∞

amn = limm→∞

limn→∞

amn

�×�Wñ�

»»»2.5.�î»8nÝ×Í��,

(2.6)∞∑

n=1

∞∑m=1

amn =∞∑

m=1

∞∑n=1

amn

ô�×�Wñ(¥�×ùóÝI5õ�W×ó�)�'b×P§ÝÎÎÎppp(matrix), Í(m,n)�HÝ-ô amnvhÎp

0 12

14

18

116

132

· · ·−1

20 1

214

18

116

· · ·−1

4−1

20 1

214

18

· · ·−1

8−1

4−1

20 1

214

· · ·...

......

......

.... . .

�

J∞∑

n=1

amn = 2−m + 2−m−1 + · · · = 2−m+1, m ≥ 1,


Æ∞∑

m=1

∞∑n=1

amn = (1 + 2−1 + 2−2 + · · · ) = 2�

!§∞∑

n=1

∞∑m=1

amn =∞∑

n=1

(−2−n+1) = −2�

Æ(2.6)P�Wñ�E×b§ÝÎp, u�OÍXb-ôÝõ, �Þ��(column) 8�

T�Þ��(row)8�, XÿÀõÎ��Ý�¬E×P§ÝÎp, øðhÞ8�Ý5�, ��ºÿÕ×8²ÝÀõ�9vÝ»��9, \ï�÷ÿûÍ», ��C�¿Í�

»»»2.6.E∀n ≥ 0,�

un(x) =x2

(1 + x2)n, x ∈ R�

JN×un/ =�Ðó��

f(x) =∞∑

n=0

un(x) =∞∑

n=0

x2

(1 + x2)n�

.un(0) = 0, ∀n ≥ 0, Æf(0) = 0�Ex 6= 0, îPt��, ×[e�¿¢ùó, vÍõ 1 + x2�Æ

f(x) =

{0, x = 0,

1 + x2, x 6= 0�

Í»�»2.1v«, 4E∀k ≥ 0,∑k

n=0 un(x) =�, ¬P§Í=�Ðó�õ, Q�×�) ×=�Ðó�

»»»2.7.E∀m ≥ 1, �

fm(x) = limn→∞

(cos(m!πx))2n�

êÞ 469

|áum!x ×Jó, Jfm(x) = 1, ÍJfm(x) = 0�¨�

f(x) = limm→∞

fm(x),

ux� ×b§ó, Jm!xÄ� Jó, ∀m ≥ 1, .hfm(x) = 0,

∀m ≥ 1, Æf(x) = 0�ux ×b§ó, �'x = p/q, Í�p, q ÞJó, q > 0, J©�m ≥ q, m!x ×Jó, Æf(x) = 1�.h

limm→∞

limn→∞

(cos(m!πx))2n =

{0, ux� b§ó,

1, ux b§ó�

hÁ§Ðó ×Õ��=��Ðó, 4æ¼N×(cos(m!πx))2n/ =�Ðó�

ãî«9°»�á, uÞÁ§Ä�D|2øð, ��ºÿÕý0Ý��Stokes (1819-1903), Seidel (1821-1896)CWeierstrass�ß,

Ît\�ºÕøðÞÁ§ºÕ, Îm�×°Ü²f�Ýó.��3�-1848O, StokesCSeidel, ¿{!`¬5½2+Û×°¡¼�Ì ííí888[[[eee(uniform convergence, TÌ×××lll[[[eee)ÝÃF��ÆJ�E×í8[eÝó�,�5CÁ§ÞºÕÎ�øðÝ�}¡Weierstrass J�, í8[eÝÃF, 3{��¥��3ì×;&ÆÞJ�hÃF,¬¼�¸�=�PC��Ýn;�

êêê ÞÞÞ 8.2

1. �fn(x) = x2n/(1+x2n), x ∈ R, n ≥ 1��J(2.2)PÎÍWñ�

2. �5½Eì�Ðóó�{fn}, �J∫ 1

0

limn→∞


∫ 1

0

fn(x)dx

ÎÍWñ�


(i) fn(x) = n2x(1− x)n, x ∈ [0, 1];

(ii) fn(x) = nxe−nx, x ∈ [0, 1];

(iii) fn(x) = nxe−nx2, x ∈ [0, 1];

(iv)

fn(x) =

2n2x, 0 ≤ x ≤ 12n

,

n− 2n2(x− 12n

), 12n≤ x ≤ 1

n,

0, 1n≤ x ≤ 1;

(v)

fn(x) =

2n3x, 0 ≤ x ≤ 12n

,

n2 − 2n3(x− 12n

), 12n≤ x ≤ 1

n,

0, 1n≤ x ≤ 1�

3. �fn(x) = x/(1 + n2x2), x ∈ [−1, 1], n ≥ 1��J(2.4)PÎÍWñ�

4. �fn(x) = sin nx/n, x ∈ R, ¬�f(x) = limn→∞ fn(x)��J

limn→∞

f ′n(0) 6= f ′(0)�

8.3 ííí888[[[eee

'{fn} ×3/)Sî@F[e�fÝÐóó��ãÁ§Ý�Lá, hÇ�E∀x ∈ SCE∀ε > 0, D3×n0 ≥ 1, ¸ÿ|fn(x) −f(x)| < ε, ∀n ≥ n0�¬î�n0;ðº�xCεbn�ôµÎE�!ÝxTε, X0ÕÝn0��!�¬uE∀x ∈ S, �0Õ×8!Ýn0,

J9Ë[e, -Ì 3Sîííí888[[[eee(��1.6 ;Ðó�í8=�8f´)�&Æ¶×�LAì�

8.3 í8[e 471

��LLL3.1.×ó��Ðó{fn}, u��E∀ε > 0, D3×n0 ≥ 1(n0©�εbn), ¸ÿn ≥ n0`,

(3.1) |fn(x)− f(x)| < ε, ∀x ∈ S,

JÌ{fn}3Sîí8[e�f({fn} converges uniformly to f on S),

¬|

fn → f(í8[e)

��T¶Wfn −→u f�

í8[e�|b×��Ý¿¢�Õ�´�|fn(x)− f(x)| < ε, �

f(x)− ε < fn(x) < f(x) + ε

��uîPE∀n ≥ n0, Cx ∈ SWñ, JXbfn3S�Ý%�, -+y×´� 2ε, |f �T�Ýñ��, �%3.1�\ïô��,

%�u� í8[e, J{fn}�f , -^bA%3.1Ýn;�

f + ε

fn

f = limn→∞ fn

f − ε

%3.1. í8[e�¿¢�L

9ì ×��Ý»��

»»»3.1.�fn(x) = (sin n2x)/n, x ∈ R, n ≥ 1�ãf(x) = 0, ∀x ∈R�J.

|fn(x)− f(x)| = |sin n2x

n| ≤ 1

n,


Æ∀ε ≥ 0,©�n ≥ n0 = [1/ε]+1,J|fn(x)−f(x)| < ε�Æ{fn}3R

îí8[eÕ0�

ì��§¼�í8[e, �Þ=�PFL�Á§Ðó�

��§§§3.1.'{fn}3Sîí8[e�f�uN×fn3Ø×p ∈ S/=�,

Jf3pù=��JJJ��.&ÆÞJ�E∀ε >0, D3×p�Ï½N(p), ¸ÿ

|f(x)− f(p)| < ε, ∀x ∈ N(p) ∩ S�

ãí8[eÝ�', ∀ε > 0, D3×n0 ≥ 1, ¸ÿn ≥ n0 `,

|fn(x)− f(x)| < ε

3, ∀x ∈ S�

ê.fn03p=�, ÆD3×p�Ï½N(p), ¸ÿ

|fn0(x)− fn0(p)| < ε

3, ∀x ∈ N(p) ∩ S�

ÆE∀x ∈ N(p) ∩ S,

|f(x)− f(p)| = |f(x)− fn0(x) + fn0(x)− fn0(p) + fn0(p)− f(p)|≤ |f(x)− fn0(x)|+ |fn0(x)− fn0(p)|+ |fn0(p)− f(p)|<

ε

3+

ε

3+

ε

3= ε�

ÿJ�

î��§ô�Tà�Ðóùó�E∀n ≥ 1, 'Ðófn(x) ×Ðóùó

∑∞k=1 uk(x)�I5õ, Ç

fn(x) =n∑

k=1

uk(x)�

u3Sîfn −→p f , J

f(x) = limn→∞

fn(x) =∞∑

k=1

uk(x),∀x ∈ S�

8.3 í8[e 473

h`Ìùó∑

uk@F[e�õÐóf�u3Sîfn −→u f , JÌùó

∑ukí8[e�f�¨uN×uk/3ØFp ∈ S=�,JN×I5

õfn, ù3p=��Æã�§3.1, Çÿì�.¡�

��§§§3.1.'Ðóùó∑

uk, 3Sîí8[e�õÐóf , v'N×uk/3ØFp ∈ S=�, Jfù3p=��

�§3.1��, �|ì�Ðr¼�î:

limx→p

∞∑

k=1

uk(x) =∞∑

k=1

limx→p

uk(x)�

h�E×í8[eÝùó, &Æ�øðÁ§�õÞºÕ�ê!ñ×è, í8[e©ÎFL=�P�×�5f�, ¬&Ä

�f��»A, 3»2.2�, &Æb×ó�&í8[e�=�Ðó,

¬ÍÁ§Ðóù ×=�Ðó�¨², í8[eº1¹???ÝÝÝ�� (good behavior)�&ûÝ� , �ì»�

»»»3.2.�

fn(x) =

{1/n, ux b§ó,

0, ux P§ó�J�Q{fn}3Rîí8[e�f(x) ≡ 0�¬4N×fn Õ��=��Ðó, Á§Ðóf , Q ×Õ�=�ÝÐó�

Íg&Æ¼:í8[e, ô0��5�Á§ÞºÕ�øð�

��§§§3.2.'3[a, b]îfn −→u f , v'N×fn/3[a, b]=��E∀x ∈[a, b], �

gn(x) =

∫ x

a

fn(t)dt,

g(x) =

∫ x

a

f(t)dt�


J3[a, b]î, gn −→u g�©½2gn −→p g, Ç∀x ∈ [a, b],

limn→∞

∫ x

a

fn(t)dt =

∫ x

a

limn→∞

fn(t)dt�

JJJ��.�´×�P, 'b > a�.fn −→u f , Æ∀ε > 0, D3×n0 ≥ 1,

¸ÿn ≥ n0`,

|fn(t)− f(t)| < ε

b− a, ∀t ∈ [a, b]�

ÆE∀x ∈ [a, b], ©�n ≥ n0,

|gn(x)− g(x)| = |∫ x

a

(fn(t)− f(t))dt| ≤∫ x

a

|fn(t)− f(t)|dt

<

∫ x

a

ε

b− adt ≤ ε,

ÆÿJgn −→u g�

�Äí8[e, ¬&�5�Á§�øð�Ä�f�, �ì»�

»»»3.3.'fn��LA3»2.1��ãyh ×=�Ðóó�, Qb×�=�ÝÁ§Ðó�Æã�§3.1á, &3[0, 1]í8[e�¬n →∞`,

∫ 1

0

fn(x)dx =

∫ 1

0

xndx =1

n + 1→ 0�

Ælimn→∞∫ 1

0fn(x)dx =

∫ 1

0limn→∞ fn(x)dx =

∫ 1

0f(x)dx = 0�

\ï�÷�J: 3î»�, 4{fn}3 [0, 1]�í8[e, ¬3N×��â1�[0, 1]Ý� î, Qí8[e�&Æ)bì�.¡�

8.3 í8[e 475

��§§§3.2.'Ðóùó∑

uk3[a, b]îí8[e�õÐóf , v'N×uk/3[a, b]=��E∀x ∈ [a, b], �

gn(x) =n∑

k=1

∫ x

a

uk(t)dt,

g(x) =

∫ x

a

f(t)dt�

J3[a, b] îgn −→u g, v

limn→∞

n∑

k=1

∫ x

a

uk(t)dt =

∫ x

a

limn→∞

n∑

k=1

uk(t)dt,

ùÇ∞∑

k=1

∫ x

a

uk(t)dt =

∫ x

a

∞∑

k=1

uk(t)dt�

JJJ��.�fn(t) =∑n

k=1 uk(t), J

∫ x

a

fn(t)dt =n∑

k=1

∫ x

a

uk(t)dt,

�à�§3.2ÇÿJ�

�§3.2 ¼�í8[eÝùó, Íõ��5 @4�5�õ�A¢¾½í8[e÷? Weierstrassèº×bàÝ¾½°(Ì

Weierstrass M -test for Uniform Convergence),h°EX��ÝÐóùó, uD3×Yg¸Ý[eÑ4ùó-Êà�

��§§§3.3.(Weierstrass M -lll��°°°). 'Ðóùó∑

un, 3Sî@F[e�f�uD3×[eÝÑ4ùó

∑Mn, ¸ÿ

0 ≤ |un(x)| ≤ Mn, ∀n ≥ 1, ∀x ∈ S,

J∑

un3Sîí8[e�f�


JJJ��.´�ãf´l�°á, 3Sî�∑

un(x)�E[e�f�E∀x ∈S,

|f(x)−n∑

k=1

uk(x)| = |∞∑

k=n+1

uk(x)| ≤∞∑

k=n+1

|uk(x)| ≤∞∑

k=n+1

Mk�

.∑

Mk[e, Æ∀ε > 0, D3×n0 ≥ 1, ¸ÿn ≥ n0 `,

∞∑

k=n+1

Mk < ε�

�)G�Þ��P, ÿ

|f(x)−n∑

k=1

uk(x)| < ε, ∀n ≥ n0, x ∈ S�

µ�LhÇ�∑

un3Sîí8[e�f�

»A, ã�§3.3á, E∀a ∈ R,∑∞

n=1 ex/n23S = (−∞, a)í8[e�¬E

∑∞n=1 log x/n2, x > 0, �§3.3-�ÊàÝ��y×

ùó�@4�5Ýõ, ÎÍº�yõ��5÷? ×��, Þï¬�8�, Ç¸Îí8[eÝ�µ�»A, .| sin nx/n2| ≤ 1/n2,

Æ∑∞

n=1 sin nx/n2[e��ã�§3.3á, h í8[e�¬hùó@4�5Ýõ

∑cos nx/n, 3x = 0s÷�h»�î, Ç¸Îí

8[e, @4�5ô��ºÓû[eP�Æ×��, �JõC�5��øð, f�JõC�5��øðpÿ9�¨², 3»2.3

�, .|fn(x)| ≤ 1/√

n, Æ3Rîfn −→u f(x) ≡ 0, ¬E�×x ∈ R,

{f ′n(x)}/�[e��Ä�Ý�, &9Eb§ÍõWñÝºÕ,

EP§Íõµ�×�ºWñÝ�38.5 ;&ÆJº:Õ, &9�ùóÝºÕ, ð�ÞhùóÚ b§Íõ×��y¢`(2.4)PWñ, |Cnyí8[e�×MÝD¡, �¢�Apostol (1974)

Chapter 9, TRudin (1964) Chapter 7, h�©Î�MÝ+Û��Ä3�@Í;G, &Æ5½�ó�Cùó�í8[eÝÞÞÞ��fff��(Cauchy condition)�

8.3 í8[e 477

��§§§3.4.'{fn} ×ó��L3Sî�Ðó�JD3×Ðóf , ¸ÿfn3Sîí8[e�f , uv°uì�f�(Ì Þ�f�)Wñ:

∀ε > 0, D3×n0 ≥ 1, ¸ÿE∀m,n ≥ n0,

(3.2) |fm(x)− fn(x)| < ε, ∀x ∈ S�

JJJ��.�J�Ä�P,Ç�'fn3Sîí8[e�f�Jµ�L, ∀ε >

0, D3×n0 ≥ 1, ¸ÿn ≥ n0 `,

|fn(x)− f(x)| < ε/2, ∀x ∈ S�

ÆE∀m ≥ n0,

|fm(x)− f(x)| < ε/2, ∀x ∈ S�

îÞ��PÇ0l(3.2)PWñ�gJ�5P�Ç�'Þ�f�Wñ�JE∀x ∈ S, ó�{fn(x)}

[e��f(x) = lim

n→∞fn(x), x ∈ S�

&Æ6J�{fn}3Sîí8[e�f�ã�'(3.2)P, E×��Ýε > 0, D3×n0 ≥ 1, ¸ÿn ≥ n0`,

|fn(x)− fn+k(x)| < ε

2, ∀k ≥ 1, ∀x ∈ S�

.h

limk→∞

|fn(x)− fn+k(x)| = |fn(x)− f(x)| ≤ ε/2, ∀x ∈ S�

Æun ≥ n0, -b

|fn(x)− f(x)| < ε, ∀x ∈ S�

�hÇ�{fn} 3Sîí8[e�f�

9ì ×ñÇÝ.¡�


��§§§3.3.∑

fn(x)3Sîí8[e, uv°u∀ε > 0, D3×n0 ≥ 1,

¸ÿn ≥ n0`,

(3.3) |n+m∑

k=n+1

fk(x)| < ε, ∀m ≥ 1, ∀x ∈ S�

êêê ÞÞÞ 8.3

1. �D¡ì�&Ðóó�{fn}�í8[eP�(i) fn(x) = xn − x2n, x ∈ [0, 1];

(ii) fn(x) = sin(x/n), x ∈ [−a, a], a > 0;

(iii) fn(x) =√

x2 + n−2, x ∈ R;

(iv) fn(x) = nx/(1 + nx), x ∈ [0, 1];

(v) fn(x) = xn/(1 + xn), x ∈ (1− a, 1 + a), 0 < a < 1�

2. �D¡ì�&ùóÐó�í8[eP�(i)

∑∞n=1(sin nx)/

√n4 + x4, x ∈ R;

(ii)∑∞

n=1 2n sin(1/(4nx)), x > 0;

(iii)∑∞

n=1(−1)n(x + n)n/nn+1, x ∈ [0, 1]�

3. uD3×ðóM > 0, ¸ÿ|fn(x)| ≤ M , ∀n ≥ 1, ∀x ∈S, JÌ{fn}3Sîííí888bbb&&&(uniformly bounded)��JuN×fn3Sîb&, v{fn}3Sîí8[e�f , J{fn}3Sî í8b&�

4. '3Sî{fn}í8[e�f , {gn}í8[e�g��J{fn +

gn}3Sîí8[e�f + g�

5. 'fn(x) = 1/(nx + 1), 0 < x < 1, n ≥ 1��J{fn}3(0, 1)@F[e, ¬¬&í8[e�

6. 'fn(x) = x/(nx + 1), 0 < x < 1, n ≥ 1��J{fn}3(0, 1)í8[e�

êÞ 479

7. �fn(x) = xn,Jã»2.2á, {fn}3[0, 1]@F[e,¬&í8[e�'g ×3[0, 1]î�=�Ðó, vg(1) = 0, �J{g(x)xn}3[0, 1]í8[e�

8. '{fn}3Sîí8[e�f ,vN×fn3Sî=��ê'{xn} ×3Sî[e�x�ó��J

limn→∞

fn(xn) = f(x)�

9. �fn(x) = ncx(1− x2)n, x ∈ R, n ≥ 1�(i) �JE∀c ∈ R, {fn}3[0, 1]@F[e;

(ii) �X�¸{fn}3[0, 1]í8[e�cÂ;

(iii) �¼�¸limn→∞∫ 1

0fn(x)dx =

∫ 1

0limn→∞ fn(x)dx Wñ

�cÂ�

10. �fn(x) = x/(1 + nx2), x ∈ R, n ≥ 1��5½O{fn}C{f ′n}�Á§ÐófCg�(i) �Jf ′(x)D3, ∀x ∈ R, ¬f ′(0) 6= g(0), ¬O�¸f ′(x) =

g(x)�Xbx;

(ii) �O�¸fní8[e�f�RÝXb� ;

(iii) �O�¸f ′ní8[e�g�RÝXb� �

11. �fn(x) = e−n2x2/n, x ∈ R, n ≥ 1��J

(i) {fn}3Rîí8[e�0;

(ii) {f ′n}3Rî@F[e�0;

(iii) {f ′n}3�×�â0�RÝ� �í8[e�

12. '{fn} ×�L3[0, 1]�=�Ðóó�, v{fn}3[0, 1]îí8[e�f��JTÍJ

limn→∞

∫ 1−1/n

0

fn(x)dx =

∫ 1

0

f(x)dx�


13. �fn(x) = 1/(1 + n2x2), 0 ≤ x ≤ 1, n ≥ 1�(i) �J{fn}3[0, 1]@F[e, ¬¬&í8[e;

(ii) �®limn→∞∫ 1

0fn(x)dx =

∫ 1

0limn→∞ fn(x)dxÎÍWñ?

14. �JE∀α > 1/2,∑∞

n=1 x(nα(1 + nx2))−13N×R �b§Ý� í8[e�ê®G�ùó3Rî, ÎÍí8[e�

15. �Ju∑∞

n=1 |an|[e, J∑∞

n=1 an sin nx, �∑∞

n=1 an cos nx3Rî/í8[e�

16. �

f(x) =∞∑

n=1

1/(1 + n2x)�

(i) �O¸î�ùó�E[e�xÝ/);

(ii) �®3£° , î�ùóí8[e?

(iii) �®3£° , î�ùó�í8[e?

(iv) �®hùó[e`, fÎÍ=�?

(v) �®fÎÍb&?

17. �

fn(x) =

0, x < 1(n + 1),

sin2(π/x), 1/(n + 1) ≤ x ≤ 1/n,

0, x > 1/n�

(i) �J{fn}[e�×=�Ðó, ¬&í8[e;

(ii) �J∑

fn(x)�E[e, ∀x ∈ R, ¬¬&í8[e�

18. �Jùó∞∑

n=1

(−1)n(x2 + n)/n

3N×b& í8[e, ¬E�×x ∈ R/&�E[e�

8.4 �ùó 481

19. �JE∀x ∈ R,∑∞

n=1 sin nx/n2/[e, ¬|f(x) �Íõ��Jf3[0, π]=�, ¬¿à�§3.2, �J

∫ π

0

f(x)dx = 2∞∑

n=1

1

(2n− 1)3�

20. �¿àì�2P∞∑

n=1

cos nx

n2=

x2

4− πx

2+

π2

6, ∀x ∈ [0, 2π],

C�§3.2, 0�

(i)∑∞

n=11n2 = π2

6; (ii)

∑∞n=1

(−1)n+1

(2n−1)3= π3

32�

8.4 ��ùùùóóó38.1 ;, &ÆèÕ�ùóÝ¥�P�h ×©½ÝÐóùó,

¸Ìn�K�?ÝP²�Í;&Æ-D¡9Ëùó�´�u×P§ùó, bì��P

∞∑n=0

an(x− a)n,

-Ì ×(x− a) ��ùó�&Æ��ì�Þ�§�

��§§§4.1.'∑

anxnEØx = x1 6= 0[e�J

(i) hùóEN×��|x| < |x1|�x�E[e;

(ii) hùó3N× [−r, r] í8[e, Í�0 < r < |x1|�JJJ��..

∑anxn

1[e, Æn →∞`, anxn1 → 0�©½2, D3×n1 ≥

1, ¸ÿn ≥ n1`, |anxn1 | < 1�EØ×0 < r < |x1|, �S = [−r, r]�

ux ∈ Svn ≥ n1, J|x| ≤ r, v

|anxn| = |anxn

1 ||x

x1

|n < | xx1

|n ≤ | r

x1

|n = tn,


Í�t = |r/x1|�.0 < t < 1, ùó∑

anxnå×y[eÝ¿¢ùó

∑tn��ãWeierstrass M -l�°, ÿ

∑anx

n3Sîí8[e�Æ(ii)ÿJ�ãî�D¡, ù�J�

∑anxn, E∀x ∈ S�E[e�ê.N×

��|x| < |x1|�x, Äa3Ø×S�, Í�5r < |x1|�ÆÿJ(i)�

ì�§¼�, E×x(Ça = 0)��ùó, Í[eP� ×|0 �TÝ �

��§§§4.2.'ùó∑

anxn3Øx = x1 6= 0[e, v3Øx = x2s

÷�JD3×r > 0, ¸ÿhùó3|x| < r�E[e, v3|x| > rs÷�JJJ��.�

A = {|x||∑

anxn [e}�

ã�'|x1| ∈ A, ÆA&è/)�êã�§4.1á, �y|x2|�ó/�3A�, Æ|x2| A�×î&�ãt�î&2§á, Ab×t�î&,

|r��Qr > 0, h.r ≥ |x1|�ê.r A�t�î&, ÆA��-ô/��øÄr�Æ

∑anx

n, E∀|x| > rs÷�ÍgE∀|x| < r,

D3×Ñób ∈ A, ¸ÿ|x| < b < r(ÍJr� t�î&), �ã�§4.1á,

∑anx

n�E[e�J±�

�§4.2 ¼�, E×�ùó∑

anxn, ¸Í[eÝP�, ÄÎ×|0 �T� (ô��©b09×F)�E×�ÝÐóùó, Q�×�Ah�»A, �Êùó

∑sin x/n, Í[eP� {nπ|n =

0,±1,±2, · · · }, �Q� × ��yub×(x − a)��ùó, ôñÇ�¶�ETÝ�§4.1C4.2, h`[e Î|a �T��Ä�§4.2¬Î�×å&Æùó

∑anx

n3|x| = rÎÍ[e�Ex =

rT−r, ��}¾\ùó�e÷P�ãyb�§4.2, &ÆÌr ∑anx

n�[[[eee��555(radius of convergence), �∑

anxn [eÝP

�, Ì [[[eee (interval of convergence), �� (−r, r), [−r, r),

(−r, r] T[−r, r]��:�[e�5, ù��L ¸∑

anxn [e�

8.4 �ùó 483

Xbx Ý/)�t�î&��ùó3Í[e Ý/F[e, ¬3� ÝÐFµ�×�[eÝ�êrô�� 0, Ç

∑anxn ©b

x = 0�[e,h`ùóõ a0; rô�� ∞,h`E∀x ∈ R,/¸ùó[e, .h[e (−∞,∞)�¨², ã�§4.1á,

∑anx

n

3N× [−b, b]í8[e, Í�0 < b < r�&9&Æ;ðÂÕÝ�ùó, ??�¢ÃfÂl�°, TqPl

�°, OÍ[e�5�

»»»4.1.�Oì�&ùó�[e�5r, C[e �(i)

∑xn/n!;

(ii)∑

(−1)n−1xn/n;

(iii)∑

nxn;

(iv)∑

(x− 2)n/(3nn2)��.(i) .

limn→∞

∣∣∣∣xn+1/(n + 1)!

xn/n!

∣∣∣∣ = limn→∞

|x|n + 1

= 0, ∀x ∈ R,

ÆhùóE∀x ∈ R/[e�.hr = ∞, v[e R�(ii) .

limn→∞

∣∣∣∣(−1)nxn+1/(n + 1)

(−1)n−1xn/n

∣∣∣∣ = limn→∞

n

n + 1|x| = |x|,

Æhùó3|x| < 1[e, 3|x| > 1s÷�ê�Qx = 1`, hùó[e, x = −1`ùós÷�.hr = 1, v[e (−1, 1]�

(iii) .

limn→∞

∣∣∣∣(n + 1)xn+1

nxn

∣∣∣∣ = limn→∞

n + 1

n|x| = |x|,

Ær = 1�ê�Qùó3x = 1C−1/s÷�Æ[e (−1, 1)�(iv) .

limn→∞

∣∣∣∣(x− 2)n+1

3n+1(n + 1)· 3nn2

(x− 2)n

∣∣∣∣ = limn→∞

n2

3(n + 1)2|x− 2| = |x− 2|

3,


.hùó3|x − 2|/3 < 1[e, Ç3|x − 2| < 3[e, Ær = 3�ê|x− 2| = 3, ùóW

∑1/n2, Æ[e�.hùó3|x− 2| ≤ 3 [

e, Ç[e [−1, 5]�

ãî»�(i)á,∑

xn/n! E∀x ∈ R[e, ÆÍ×�4��0, Ç

(4.1) limn→∞

xn

n!= 0, ∀x ∈ R�

ôµÎn!´�×ü�@óxÝng], W��", h Á§�×¥�Ý��×�ùóÝ;ó, u�×°;�b`¬�Å(Í[e�5�»

A, uE∑

anxn�N×4, !¶|×&ëðó, Q�Å(Íe÷

P�×Í©»Î, uÞ∑

anxn�N×4/t|x, �ÿ∑

anxn−1, T

/¶|x, �ÿÕ∑

anxn+1, /�æùób8!�[e �¨²,

ùbì��§�

��§§§4.3.'{cn, n ≥ 0} ×ó��Ñó, v��

(4.2) limn→∞

n√

cn = 1�

J∑

anxn�

∑cnanxn, b8!�[e�5�

JJJ��.�rCr′5½�∑

anxnC∑

cnanxn �[e�5��'0 < r <

∞, vãx��0 < |x| < r, J(r − |x|)/|x| > 0��ã×ε > 0, ��ε < (r − |x|)/|x|, ãhêÿ

(1 + ε)|x| < r�

ê.�'(4.2)PWñ, ÆD3×n0 ≥ 1, ¸ÿE∀n ≥ n0,

1− ε < n√

cn < 1 + ε�

.h|cnx

n| = ( n√

cn|x|)n < ((1 + ε)|x|)n, ∀n ≥ n0,

Æùó∑∞

n=n0|cnanx

n|å×y∑∞n=n0

|an|((1+ε)|x|)n��(1+ε)|x| <r, Í�r

∑anx

n�[e�5, Æ∑∞

n=n0|an|((1 + ε)|x|)n [e, .

8.4 �ùó 485

h∑∞

n=n0|cnanx

n|ù[e��î´b§Ýn0 4, &Æ-�J�E∀|x| < r,

∑∞n=0 cnanx

n�E[e�Æ∑

cnanxn�[e�5r′ ≥r�Íg, .

∞∑n=0

anxn =

∞∑n=0

1

cn

(cnanxn),

v

limn→∞

n√

1/cn = 1,

Æuãùó∑

cnanxn�s, ûG�D¡, �ÿùó∑

anxn �[e�5r ≥ r′�h��P�G«�ÿÝr′ ≥ r �), ÇÿJr = r′�ur = ∞, J}�ÑÑG�J°, �ÿr′ = ∞�urTr′b×

Ñ, JãG�D¡�ÿr = r′�ÆurTr′b× 0, J¨×�º Ñ,

ùÇÞï/ 0�Æ�¡£×Ë�µ, /br = r′�J±�

»»»4.2..

limn→∞

n√

n = limn→∞

n√

1/n = 1,

Æ�Æî-²��Ýëùó∑∞

n=1 xn,∑∞

n=1 nxn�∑∞

n=1 xn/n, b8!Ý[e�5, v/ 1�¬[e µ� (−1, 1), (−1, 1),

[−1, 1)¬�¼8!�\ï�p:�, E∀k ∈ R,∑∞

n=1 nkxn�[e�5/ 1�

»»»4.3.ãcn = n, �ÿ∑

anxn�

∑nanx

nb82!Ý[e�5�.h

∑anx

n�∑

nanxn−1b8!Ý[e�5, ¡ï Gï@4�5Ý

õ�

t¡, &Æô��Ê´×�Ý�ùó∑

an(z − a)n, Í�z, a,

an/ �ó�JÍ;Ý��)Wñ, ©Î[e , �; [[[eeeiii(circle of convergence)�Çu[e�5r, Jùó3|a iTr �5Ýi/[e, 3i²s÷��y3|z − a| = r)�¨²D¡, Ú�µ�!ºb�!Ý�¡�


êêê ÞÞÞ 8.4

1. �O¸ì�&ùó[e�xÝ/)�

(1)∑∞

n=0 xn/√

n + 1� (2)∑∞

n=0(−1)nxn�(3)

∑∞n=0 x2n/n!� (4)

∑∞n=1 nxn/2n�

(5)∑∞

n=0(3x)n/2n+1� (6)∑∞

n=0 n!xn/10n�(7)

∑∞n=1(−1)nn2xn� (8)

∑∞n=1 n(x− 1)n−1/3n�

(9)∑∞

n=1(n!)3xn/(2n)!� (10)∑∞

n=1 xn/ log(n + 1)�(11)

∑∞n=1 n!xn/nn� (12)

∑∞n=2(−1)nxn/(n log2 n)�

(13)∑∞

n=0(2x + 1)n/3n� (14)∑∞

n=0 an2xn, x < a < 1�

(15)∑∞

n=1 3√

nxn/n� (16)∑∞

n=0(−1)nxn/(n + 1)2�(17)

∑∞n=1 xn2

� (18)∑∞

n=1(1− (−2)n)xn�(19)

∑∞n=1(log x)n� (20)

∑∞n=0(−1)n22nx2n/(2n)�

(21)∑∞

n=1(1 + xn)−1� (22)∑∞

n=0(1 + 2 + · · ·+ 2n)xn�(23)

∑∞n=1 sin(x/2n)� (24)

∑∞n=1(x/2)n(2n)!/(n!)2�

(25)∑∞

n=3 xnn−(1+2(log log n)/ log n)�(26)

∑∞n=1(−1)n+1x2n−1/(2n− 1)!�

(27)∑∞

n=1(−1)n−1x2n−1/(n + 1)�(28)

∑∞n=0(x + 2)n/((n + 1)2n)�

(29)∑∞

n=0(−1)n+1(x + 1)2n/((n + 1)25n)�(30)

∑∞n=1(−1)n(2x− 1)n/n!�

(31)∑∞

n=1(−1)n(x + 1)n/(n2 + 1)�(32)

∑∞n=0(x− 2)n/(2n

√n + 1)�

(33)∑∞

n=1 nxn/((n + 1)(n + 2)2n)�

2. �Oì�&�ùó�[e�5�

êÞ 487

(1)∑∞

n=1 nnxn/n!� (2)∑∞

n=1(n!)3x3n/(3n)!�(3)

∑∞n=1(1 + 1/n)n2

xn� (4)∑∞

n=1(sin an)xn, a > 0�(5)

∑∞n=0 n!x2n� (6)

∑∞n=0(2n)!xn!�

(7)∑∞

n=1(1·3···(2n−1)2·4···(2n)

)3xn� (8)∑∞

n=12·4···(2n)

1·3···(2n−1)x4n�

(9)∑∞

n=0(sinh an)xn, a > 0� (10)∑∞

n=1 eΣni=1i−1

xn�(11)

∑∞n=1(a

n + bn + cn)xn, a, b, c > 0�(12)

∑∞n=1 xn/(an + bn), a, b > 0�

(13)∑∞

n=1(an

n+ bn

n2 )xn, a, b > 0�

(14)∑∞

n=1(1 + an

+ bn2 )

n2xn, a, b > 0�

3. un �ó, �an = 2−n, un �ó, �an = 2−n+1��Jh`

∑anx

n�[e (−2, 2)�¥�limn→∞ an+1/an¬�D3�

4. '∑

anxn�[e�5 r��J

(i) uD3×ðóM > 0, ¸ÿ|an| ≤ M , ∀n ≥ 0, Jr ≥ 1;

(ii) uD3×ðóM ≥ N > 0, ¸ÿN ≤ |an| ≤ M , ∀n ≥ 0,

Jr = 1�

5. 'limn→∞ n√

an = r > 0��Jh`∑

anxn�[e�5 1/r�

6. �Ju∑

anxn�[e�5 r, J

∑anx2n�[e�5

√r�

7. 'p, q ÞÑJó��O∞∑

n=0

(n + p)!

n!(n + q)!xn

�[e�5�

8. �O¸∑∞

n=1(xn/(1 + x2n))[e�xÝ/)�

9. �JtÝ|x| = 1²,∑∞

n=0(x2n

/(1−x2n+1))[e,v|x| < 1`,

ùóõ x/(1− x), |x| > 1`, ùóõ −(x− 1)−1�


8.5 ��ùùùóóó��PPP²²²�×�ùó

∑∞n=0 an(x − a)n, EN×òyÍ[e Ýx, �L

Ðóf

f(x) =∞∑

n=0

an(x− a)n�

&ÆÌùó∑

an(x − a)n, fnya��ùùùóóó"""PPP(power-series ex-

pansion of f about a), TÌ f��ùó�î°�ã8.1;á,

∑∞n=0 xn f(x) = 1/(1− x)ny0��ùó"P�ù

Ç1

1− x=

∞∑n=0

xn, |x| < 1�

�ÄîP¼�E|x| > 1)b�L, ¬��©E|x| < 1�b�L�ôµÎ1|x| > 1`, 1/(1− x)¬Pny0 ��ùó"P�

E�ùó"P, &Æ��ºEì�®Þ�Õ·¶:

(i) �×ùóD¡ÍõÐóf�P²;

(ii) D¡¢`×Ðób�ùó"P�&Æs¨©b×°©�ÝÐó, �b�ùó"P��Ä�9ó

&Æð�ÝÐó, ÅÎK�"�W�ùó�X|E�ùóÝ"D,

-�¥��&Æ�:®Þ(i)�

u∑

an(x − a)n�[e�5 r, Jã�§4.1á, h�ùó3 (a − r, a + r) �E[e, 3[a − b, a + b]í8[e, Í�0 < b <

r�.�ùó�N×4an(x − a)n/ @óî�=�Ðó, Æã�§3.1á, E∀0 < b < r, õÐófù3N×[a − b, a + b]=��.hf3(a− r, a + r)=��¨², ã�§3.2á, &Æ�Þh�ùó3N× [a− b, a + b], 0 < b < r, @4�5�&ÆE£°��î �ùó�ÐóÝP², W�yì�

��§§§5.1.'D3×a ∈ R, Cr > 0, ¸ÿE∀x ∈ (a − r, a + r), Ðófb�ùó"PAì:

8.5 �ùó�P² 489

(5.1) f(x) =∞∑

n=0

an(x− a)n�

J(i) f3(a− r, a + r)=�;

(ii) f3N×�ây(a− r, a + r)�T ��5, � �ùó@4�5�õ�©½2, E∀x ∈ (a− r, a + r),

(5.2)

∫ x

a

f(t)dt =∞∑

n=0

an

∫ x

a

(t− a)ndt =∞∑

n=0

an

n + 1(x− a)n+1�

ã(5.2)Pá, f��5)b�ùó�î°��ã�§4.3á, hÞùób8!Ý[e�5�9ì ×ny@4�5Ý�§�

��§§§5.2.'f3 (a − r, a + r), ��îW(5.1)P, Í�r��ùó�[e�5�J

(i) �5ùó∑∞

n=1 nan(x− a)n−1�[e�5) r;

(ii) E∀x ∈ (a− r, a + r), f ′(x)D3, v

f ′(x) =∞∑

n=1

nan(x− a)n−1�

JJJ��.(i) �J��3»4.3�ÍgJ�(ii)��g ��5ùó�õ, Ç

g(x) =∞∑

n=1

nan(x− a)n−1�

EÐóg, ã�§5.1á, �3(a− r, a + r)�@4�5, v∫ x

a

g(t)dt =∞∑

n=1

(an(t− a)n|xa) =∞∑

n=1

an(x− a)n = f(x)− a0�

.g =�, ã��5ÃÍ�§ÝÏ×I5á, E∀x ∈ (a− r, a + r),

f ′(x)D3v�yg(x)�ÿJ(ii)�


38.3;, &Æ�¼�E×�ÝÐóùó, Ç¸í8[e, @4�5Ýõ, ô�×��yõ��5, ¬E�ùóQWñ�ãyb�§5.1C5.2, 38.1;&ÆBã@4�5T�5, �ÿÕ±Ý�ùó�ºÕ, Û/)°�©½2, &Æbì�"P:

log(1 + x) =∞∑

n=0

(−1)n

n + 1xn+1, |x| < 1,(5.3)

log(1− x) = −∞∑

n=0

1

n + 1xn+1, |x| < 1,(5.4)

arctan x =∞∑

n=0

(−1)n

2n + 1x2n+1, |x| < 1�(5.5)

9°"Pô�à¼O�Aarctan 0.5, Tlog 0.7 ��«Â�ã(5.3)C(5.4)P, ÿ

(5.6)1

2log(

1 + x

1− x) =

∞∑n=0

1

2n + 1x2n+1, |x| < 1�

xã−1É��1`, (1 + x)/(1 − x)ãÂ3XbÑóî�Æ©�Ê2óãxÂ, &Æ�ã(5.6)PÿÕ�×Ñó�EóÂ��§5.2�¨×.¡ , ×�ùó�õÐó, Í��$0ó/D

3, v�Bã@4�5�ÿ�uf(x) =∑∞

n=0 an(x − a)n, JB�5k g¡, v�x = a, ÿ

f (k)(a) = k!ak,

Æxk �;ó

ak =f (k)(a)

k!, k ≥ 1�

�ya0 = f (0)(a) = f(a)�Æf��ùó"P

(5.7) f(x) =∞∑

k=0

f (k)(a)

k!(x− a)k�

hP²Ç �ùó"P�°°°×××PPP��§§§(Uniqueness theorem)�

8.5 �ùó�P² 491

��§§§5.3.'Þ�ùó∑

an(x− a)n�∑

bn(x− a)n3a�Ø×Ï½�,

b8!�õÐóf�JhÞùó@48�, Ç

an = bn = f (n)(a)/n!, ∀n ≥ 0�

ã(5.7)P, ù�:�×�ùó�I5õ, Ç ÍõÐó3a��"P�ð�1, uÐóf3 (a − r, a + r), ��W×�ùó, Jf3a��94Pó�, 3(a − r, a + r)�@F[e�õÐóf�Lbï, 3(a − r, a + r)��×TÝ� , h í8[e�#½&Æ¼:, Í;×��Xè�ÝÏÞÍ®Þ, Ç��×Ðóf , ¢`�3aF�Ø×Ï½, b�ùó"P?

G«�JÄ,9ËÐóÄ3a�Ø×Ï½ÝN×$0ó/D3(.hlog x�x1/330/P°"�, .Þï3x = 0�0ó/�D3), vhÐó��ùó"PA(5.7)PX��¨'b×3a �Ø×Ï½Ý��$0ó/D3ÝÐóf , 9ËÐóÌ 3� PPP§§§ggg��(infinitely differentiable)�J&Æ�¶�ì��ùó

(5.8)∞∑

k=0

f (k)(a)

k!(x− a)k�

hùóÌ ãf3aX®ß��ùùùóóó(Taylor’s series generated by

f at a)�&ÆñÑº®: tÝ3x = a², hùóÎÍº[e? A�ÎÝ�, ÍõÎÍ�yf(x)? �ß#²ÝÎ, ×��, hÞ®ÞÝ�n/ Í��tÝ3x = a ², hùó�×�[e, Ç¸[e,

Íõô�×��yf(x)�36.5 ;�êÞ, &Æ�:Õ, uf(x) =

e−1/x2, x 6= 0, vf(0) = 0, h ×P§g��ÝÐó, vE∀k ≥ 0,

f (k)(x) 3x = 0 =��¬.f (k)(0) = 0, ∀k ≥ 0, Æf30 ��ùó"P�õ 0 6= f(x)�9ËÐóÍ@�9�g(x) = e−1/x2

sin(1/x),

x 6= 0, g(0) = 0, ù ×»�9ì&Æ�¸hÞ®Þ��n, / ù��f��´�ãÏ°a(3.15)P��2Pá

(5.9) f(x) =n∑

k=0

f (k)(a)

k!(x− a)k + Rn(x),


�r��b§4Ýõ f , 3a�ng��94P, Rn(x) õ4, Ç|��94P¼¿�f�0-�u3(5.9)P�, �n → ∞, J�:�(5.8)P��ùóº[e�f(x), uv°u0-4Rn(x)��0�9ì�§5.4C5.5, ��¸h0-4��0 Ý�5f��3Ï°a(3.19)P, &ÆÞRn(x)�îW×�5, Ç

(5.10) Rn(x) =1

n!

∫ x

a

(x− t)nf (n+1)(t)dt

©�f (n+1)3a�Ø×Ï½=�, vxòyhÏ½, îP-Wñ�ÆufP§g��, E∀n ≥ 1, &Æ/�Þ0-�îWîP, .h��ùó[e�f(x), uv°u(5.10)P��, n →∞ `��0�¿à�ó�ð, �t = x + (a− x)u, JRn�;¶

(5.11) Rn(x) =(x− a)n+1

n!

∫ 1

0

unf (n+1)(x + (a− x)u)du�

&Æbì��§�

��§§§5.4.'Ðóf3(a−r, a+r)P§g��,v'D3×ðóA > 0,

¸ÿ

(5.12) |f (n)(x)| ≤ An, ∀n ≥ 1, ∀x ∈ (a− r, a + r)�

JE∀x ∈ (a− r, a + r), f3aX®ß��ùó[e�f(x)�JJJ��.ã(5.11)Pv¿à(5.12)Pÿ

0 ≤ |Rn(x)| ≤ |x− a|n+1

n!An+1

∫ 1

0

undu =|x− a|n+1an+1

(n + 1)!=

Bn+1

(n + 1)!,

Í�B = A|x − a|�.E∀B ∈ R, n → ∞`, Bn+1/(n + 1) →0(�(4.1)P), ÆÿJE∀x ∈ (a− r, a + r), Rn(x) → 0�

»»»5.1.�f(x) = sin x, g(x) = cos x�.

|f (n)(x)| ≤ 1, |g(n)(x)| ≤ 1, ∀n ≥ 1, x ∈ R,

8.5 �ùó�P² 493

Æ�¡Ef(x)Tg(x), (5.12)PWñ, Í�A = 1�.h

sin x = x− 1

3!x3 +

1

5!x5 − · · ·+ (−1)n

(2n + 1)!x2n+1 + · · · ,(5.13)

cos x = 1− 1

2!x2 +

1

4!x4 − · · ·+ (−1)n

(2n)!x2n + · · · ,(5.14)

∀x ∈ R�¿à�§5.1C5.2, ãî�Þ"P, ñÇ�:�

(sin x)′ = cos x, (cos x)′ = − sin x,∫ x

0

sin tdt = 1− cos x,

∫ x

0

cos tdt = sin x�

»»»5.2.�f(x) = ex�´�f (n)(x) = ex, ∀n ≥ 1, x ∈ R�êEx3�×b§ (−r, r)�, Í�r > 0, ex ≤ er, �er > 1�.h(5.12)PWñ, Í�A = er�.r � ��×Ñó, ÇÿJE∀x ∈ R, ìPWñ:

(5.15) ex = 1 + x +1

2!x2 + · · ·+ 1

n!xn + · · ·�

ãh"Pô�:�

(ex)′ = ex,

∫ x

0

etdt = ex − 1�

»»»5.3.�Oì�ùóõ�

1− 1

4+

1

7− 1

10+ · · ·+ (−1)n

3n + 1+ · · ·�

��.&Æ�J�XkO�ùóõ, �;¶ ×��5, Ç

(5.16)∞∑

n=0

(−1)n

3n + 1=

∫ 1

0

1

1 + t3dt�

´�E∀t 6= −1, Ck ≥ 1,

1

1 + t3=

k−1∑n=0

(−1)nt3n +(−1)kt3k

1 + t3 �


.h

∫ 1

0

1

1 + t3dt−

k−1∑n=0

(−1)n

∫ 1

0

t3ndt = (−1)k

∫ 1

0

t3k

1 + t3dt�

ãhêÿ

∣∣∣∫ 1

0

1

1 + t3dt−

k−1∑n=0

(−1)n

3n + 1

∣∣∣=∣∣∣∫ 1

0

t3k

1 + t3dt

∣∣∣≤∫ 1

0

t3kdt =1

3k + 1�

��k →∞, Çÿ(5.16)P�ê∫ 1

0

1

1 + t3dt =

1

3(

∫ 1

0

1

1 + tdt +

∫ 1

0

2− t

1− t + t2dt)

=1

3(log(1 + t)− 1

2log(1− t + t2) +

√3 arctan

2t− 1√3

)∣∣∣1

0

=1

3(log 2 +

√3(

π

6+

π

6)) =

1

3(log 2 +

π√3),

hÇ XkO�ùóõ�

\ï��:, �§5.1ÎÍÊàÍ»?

»»»5.4.�f0(x) = ex, vfn+1(x) = xf ′n(x), ∀n ≥ 0��J

∞∑n=0

fn(1)

n!= ee�

JJJ��..

f0(x) =∞∑

k=0

xk

k!,

ê�'f1(x) = xf ′0(x), Æ

f1(x) =∞∑

k=0

kxk

k! �

8.5 �ùó�P² 495

.h¿àó.hû°, ÿ

fn(x) =∞∑

k=0

knxk

k! �

Æ∞∑

n=0

fn(1)

n!=

∞∑n=0

∞∑

k=0

kn

k!n!=

∞∑

k=0

1

k!

∞∑n=0

kn

n!=

∞∑

k=0

1

k!ek = ee�

h�.kn/(k!n!) Ñ, Æ8�Ý5��øð�ÿJ�

�§5.4¼�, u×Ðóf�n$0óf (n), W�Ý>��yØÑó�ng], Jf��ùó[e�f�9ì ¨×¸��ùó, [e�fÝ�5f�, 9ÎÆ»ó.�Bernstein (1880-1968)X��

��§§§5.5.'3ØT [0, r], f (n)(x) ≥ 0, ∀n ≥ 0, ∀x ∈ [0, r]�JE∀x ∈ [0, r), ��ùó

∞∑

k=0

f (k)(0)

k!xk

[e�f(x)�JJJ��.ux = 0, Í�§QWñ�¨'0 < x < r�)¿à��2P, E∀n ≥ 0, Þf¶W

(5.17) f(x) =n∑

k=0

f (k)(0)

k!xk + Rn(x)�

&ÆÞJ�

(5.18) 0 ≤ Rn(x) ≤ (x

r)n+1f(r)�

J.n →∞`, (x/r)n+1 → 0, Æ-ÿÕRn(x) → 0�)�ÞRn(x);¶W(5.11)P��P, ©Îh�a = 0, Ç

Rn(x) =xn+1

n!

∫ 1

0

unf (n+1)(x− xu)du�


îPE∀x ∈ [0, r]Wñ�ur 6= 0, �

Fn(x) =Rn(x)

xn+1=

1

n!

∫ 1

0

unf (n+1)(x− xu)du�

ã�'.f (n+1)�0óf (n+2) &�, Æf (n+1)3[0, r] �¦�.hE∀u ∈ [0, 1],

f (n+1)(x− xu) = f (n+1)(x(1− u)) ≤ f (n+1)(r(1− u)),

ÆÿFn(x) ≤ Fn(r), ∀0 < x ≤ r�

ãhêÿ

(5.19)Rn(x)

xn+1≤ Rn(r)

rn+1≤ f(r)

rn+1,

Í��P, àÕ3(5.17)P�, �x = r�.(5.17)P�r��ÝõN×4, / &�, ÆRn(x) ≤ f(r)�ãhÿ(5.18)PWñ�J±�

Íg&Æ¼:ÞÞÞ444PPPùùùóóó(binomial series)�ãÞ4P�§, EN×ÑJóm, &Æb

(a + b)m = am +

(m

1

)am−1b + · · ·+

(m

k

)am−kbk + · · ·+ bm�

(a+ b)m�(a+b) · (a+b) · · · (a+b)=¶m4,.hî�2PÍ@ÎàÕ4�à)Ý*»�¬um� ÑJó÷? &Æ�ãa = 1, b = x,

¬�Ê(1 + x)m�Juá(1 + x)m�"P, ×�Ý

(a + b)m = am(1 + b/a)m

-ô�¶�, ©�a 6= 0�m ÑJó`,

(1 + x)m = 1 +

(m

1

)x +

(m

2

)x2 + · · ·+

(m

k

)xk + · · ·+ xm�

8.5 �ùó�P² 497

um� ÑJó, &Æ)�Þ(1 + x)m�îW×ùó, ©Îh`ùó-�ºâcÝ�&ÆºÿÕ×XÛÝÞ4Pùó, vÍ�P (|cã�m):

(5.20)∞∑

n=0

anxn = 1 +

(c

1

)x +

(c

2

)x2 + · · ·+

(c

n

)xn + · · · ,

h�an =(

cn

), �

(5.21)

(c

n

)=

c(c− 1) · · · (c− n + 1)

n!, n ≥ 1�

¥�c ×&�Jó`, ã(5.21)Pÿ

(5.22)

(c

n

)=

c!

n!(c− n)!, 0 ≤ n ≤ c,

h�&ÆÄ��.`�4�à)�, Xÿ8!�Æ(5.21)PÇ.Âc

� &�Jó`,(

cn

)Ý�L, �(5.22)P��r��, .bc!, Æ©

Ec &�Jó`�Wñ�êã(5.21)P�:�, c ×&�Jó`, E∀n > c,

(cn

)= 0, h`(5.20)PW ×b§Ýùó�th�µ

²(Çc� &�Jó), (5.20)P/ ×P§ùó�&Æ�¼:(5.20)P�ùóÝ[e�5�ãfÂl�°, .

limn→∞

∣∣∣an+1xn+1

anxn

∣∣∣= limn→∞

∣∣∣c− n

n + 1

∣∣∣|x| = |x|�

Æ|x| < 1, ùó[e; |x| > 1, ùós÷, Ç[e�5 1�E∀c ∈ R, (5.20)P�Þ4Pùó, �L×ÐófAì:

(5.23) f(x) =∞∑

n=0

anxn, |x| < 1,

uc ×&�Jó, ãÞ4P�§, fÇW

(5.24) f(x) = (1 + x)c�

9ìJ�, ¯@îE∀c ∈ R, f/bî��P�


u�J�

(5.25) D(f(x)(1 + x)−c) = 0,

Jf(x)(1 + x)−c = K,

Í�K ×ðó�ê.f(0) = 1, ÆK = 1�Ah×¼(5.24)PE∀c ∈ R-WñÝ�&Æ��J�(5.25)P�Ç�J�

D(f(x)(1 + x)−c) = f ′(x)(1 + x)−c − cf(x)(1 + x)−c−1 = 0�

ÞîPN×4&¶|(1 + x)1+c, ÿ

(5.26) f ′(x)(1 + x)− cf(x) = 0�

9ìJ�(5.23)P�L��f , ��(5.26)P�ã�§5.2, E∀|x| < 1,

f ′(x) =∞∑

n=1

nanxn−1 =

∞∑n=0

(n + 1)an+1xn,

Æ

xf ′(x) =∞∑

n=1

nanxn =∞∑

n=0

nanxn�

.h

f ′(x)(1 + x) =∞∑

n=0

((n + 1)an+1 + nan)xn�

ê

(n + 1)an+1 + nan

=(n + 1)c(c− 1) · · · (c− n)

(n + 1)!+

nc(c− 1) · · · (c− n + 1)

n!

=c(c− 1) · · · (c− n + 1)

n!(c− n + n) = can,

8.5 �ùó�P² 499

Æ

f ′(x)(1 + x) =∞∑

n=0

canxn = cf(x)�

AhÇJ�(5.26)PWñ, .h(5.25)PWñ, Æ(5.24)PWñ�J±�

&ÆÞ��W�yì, hÇ.ÂÝÞÞÞ444PPP��§§§�

��§§§5.6.E∀c ∈ R, |x| < 1,

(1 + x)c =∞∑

n=0

(c

n

)xn(5.27)

= 1 +

(c

1

)x +

(c

2

)x2 + · · ·+

(c

n

)xn + · · ·�

ÞÞ4P�§.ÂÕÊà��g], �1Îpñ3ó.Ý�9s¨�, 8ÂÿkªÝ×�, 9Î�\�@~ÝW��ÄA!èâtSÝÍ�ó.��êY, pñ¬ÎEh�§, �×ËÑÝJ��pñ:ÕWallis�Oπ2P(�3.4;), @s��×°v«Ý�Õ, âys¨Þ4Pùó�àÕèÜtS, �b�ß��ÝJ��¨�ê&Æ"�(1 + x)c�&xc, h.¡ï3x = 0¬&Xb0ó/D3, t&c ×&�Jó, �h`xc� ×�ùóÝ�¨², &ÆÎ| #ÝW°J�(1 + x)c�"PA(5.27)P, ù�à#0��Ç�f(x) = (1 + x)c, JÁ|:�

f (k)(0) =

(c

k

), ∀k ≥ 0�

Æu�J�n → ∞`, õ4Rn(x)��0, Jã�§5.5, -ÿÕ(5.27)PWñÝ��Äh®Þ4��p, ¬ô¬&A&Æ�GÿÕsin x, cos x, ex��õ4, ��0×��|�X|�I5Ý>Ih�º9ø #ÝJ�(5.27)P�b·¶Ý\ï, �¢�Courant

and John (1965) pp. 456-457,nylimn→∞ Rn(x) = 0�à#J��


»»»5.5.�O√

1 + x�"P��..

√1 + x = (1 + x)1/2, ÆE∀|x| < 1,

√1 + x

= 1 +1

2x +

12(1

2− 1)

2!x2 + · · ·+

12(1

2− 1) · · · (1

2− n + 1)

n!xn + · · ·

= 1 +1

2x− 1

22 · 2!x2 + · · ·+ (−1)n+11 · 3 · · · (2n− 3)

2nn!xn + · · ·�

»»»5.6.�Olog(x +√

1 + x2)�"P��.�

f(x) = log(x +√

1 + x2),

J

f ′(x) =1 + x/

√1 + x2

x +√

1 + x2=

1√1 + x2�

ÆkOf�"P, &Æ©m�O(1 + t2)−1/2�"P, Q¡�@4�5Ç��ã�§5.6ÿ, ©�t2 < 1,

(1+t2)−1/2 = 1−1

2t2+

1 · 322 · 2!

t4+· · ·+(−1)n 1 · 3 · · · (2n− 1)

2n · n!t2n+· · ·�

Æã�§5.1, E∀|x| < 1,

log(x +√

1 + x2) =

∫ x

0

(1 + t2)−1/2dt

= x− 1

2 · 3x3 +1 · 3

22 · 5 · 2!x5 + · · ·

+(−1)n 1 · 3 · · · (2n− 1)

2n · (2n + 1) · n!x2n+1 + · · ·�

êêê ÞÞÞ 8.5

1. �Oì�&ùó[eÝP�, ¬OÍõ��¿à�GXÿny�ùó"PÝ��

êÞ 501

(1)∑∞

n=0(−1)nx2n� (2)∑∞

n=0xn

3n+1�(3)

∑∞n=0 nxn� (4)

∑∞n=0(−1)nnxn�

(5)∑∞

n=0(−2)n n+2n+1

xn� (6)∑∞

n=12nxn

n �(7)

∑∞n=0

(−1)n

2n+1(x

2)2n� (8)

∑∞n=0

(−1)nx3n

n! �(9)

∑∞n=0

xn

(n+3)!� (10)∑∞

n=0(x−1)n

(n+2)!�(11)

∑∞n=2

x3n

2n� (12)∑∞

n=1(−1)n+1(x2n−1 + x2n)�(13)

∑∞n=0

x4n

2n+1� (14)∑∞

n=1xn

n(n+1)�(15)

∑∞n=1

x4n−3

4n−3� (16)∑∞

n=0 ne−nx�2. �'ì�&Ðó��ùó"PD3��JÍ�ùóbX�Ý�P, vùó3X�Ýx�P� [e�

(1) ax =∑∞

n=0(log a)n

n!xn, a > 0, x ∈ R�(èî: ax = ex log a)

(2) sinh x =∑∞

n=0x2n+1

(2n+1)!, x ∈ R�

(3) sin2 x =∑∞

n=1(−1)n+1 22n−1

(2n)!x2n, x ∈ R�

(èî: cos 2x = 1− 2 sin2 x)

(4) 12−x

=∑∞

n=0xn

2n+1 , |x| < 2�(5) e−x2

=∑∞

n=0(−1)nx2n

n!, x ∈ R�

(6) sin3 x = 34

∑∞n=1(−1)n+1 32n−1

(2n+1)!x2n+1, x ∈ R�

(7) log√

1+x1−x

=∑∞

n=0x2n+1

2n+1, |x| < 1�

(èî: ¿àlog(1 + x)Clog(1− x)�"P)

(8) x1+x−2x2 = 1

3

∑∞n=1(1− (−2)n)xn, |x| < 1

2�(èî: Þ¼�¶WI55P)

(9) 1x2+x+1

= 2√3

∑∞n=0 sin(2π(n+1)

3)xn, |x| < 1�

(èî: x3 − 1 = (x− 1)(x2 + x + 1))

(10) 12−5x6−5x−x2 =

∑∞n=0(1 + (−1)n

6n )xn, |x| < 1�

3. �O3sin(2x + π/4)��ùó"P∑∞

n=0 anxn��;óa98�

4. �O∫ 1/2

0x/(1 + x3)dx��óÏ4��

5. �O∫ x

0log(1 + t)/tdt�"P�


6. �O∫ x

0arctan t/tdt�"P�

7. �

S(x) =x2

1 · 2 +x3

2 · 3 + · · ·+ xn

(n− 1) · n + · · · ,

�OS ′(x), �¿àlog(1 − x)�"P, |Oî�ùóõS(x), ¬��[eÝP��

8. �BãOÞg0óS′′(x), |ÿì�ùóõS(x), ¬��[e

ÝP��

S(x) =x2

1 · 2 −x4

3 · 4 + · · ·+ (−1)n−1x2n

(2n− 1) · 2n + · · ·�

9. �5½Oì�&Ðó��ùó"P, ¬O&ùó�[e�5�(1) log(1 + x + x2)� (2) log(1 + 3x + 2x2)�(3) log(1− x− 2x2)� (4) arctan(2x/(1− x2))�(5) arctan(2x3/(1 + 3x2))�

10. �5½Oì�&Ðó��ùó"P, ¬O&ùó�[e�5�(1)

√1− x� (2) 3

√1 + x2�

(3) x(1− x2)−1/2� (4) (1 + 2x)−3�(5) x(4− x)3/2� (6)

√2 + x�

(7) arcsec(1/x)� (8) log(√

1 + x2 − x)�(9) (1 + x + x2)−1� (10) (1− x− 2x2)−1�

11. �5½Oarcsin xC(arcsin x)2��ùó"P�

12. �J(ÍÞù�Ïè×a»2.6)

arcsin x√1− x2

= x +2

3x3 +

2 · 43 · 5x5 +

2 · 4 · 63 · 5 · 7x7 + · · ·

=∞∑

n=0

22n(n!)2x2n+1

(2n + 1)! �

8.6 Í�nyùóÝ�� 503

13. û»5.3, �J(h {úXJ�)E∀a, b > 0,

1

a− 1

a + b+

1

a + 2b− 1

a + 3b+ · · · =

∫ 1

0

ta−1

1 + tbdt�

14. �

u = 1 +x3

3!+

x6

6!+ · · · , v =

x

1!+

x4

4!+

x7

7!+ · · · ,

w =x2

2!+

x5

5!+

x8

8!+ · · · , f(x) = u3 + v3 + w3 − 3uvw�

�Jf ′(x) = 0, ∀x ∈ R, ¬Of(x)�

8.6 ÍÍÍ��nnnyyyùùùóóóÝÝÝ��

ùóÝD¡, ÎKó¿ÍÌbòaªÝÞC�×�tÝ��5�², 3�Kó.Ýr½�, ô6�½¥�Ý��P§ùó, ÎÞ×óTÐó|×P§ÝÄ�¼�îÝ×Ë]P,

¬¬&°×Ý]P�PPP§§§ÝÝÝ¶¶¶��(infinite product) ÎÍ�Ý]P�×�»AWallis¶�, Ç

(6.1)π

2=

2

1

2

3

4

3

4

5

6

5

6

7· · · ,

ÛÞπ/2�îWP§Ý¶��×��, XÛP§Ý¶�

∞∏i=1

ai = a1 · a2 · a3 · · · ,

�I5¶�ó�a1, a1a2, a1a2a3, · · ·

�Á§, ©�hÁ§D3�.ha1, a2, a3, · · · , Qô�|ÎØ×¢óx�Ðó�9ì ×b¶Ý»�:

(6.2) sin x = x(1− x2

π2)(1− x2

(2π)2)(1− x2

(3π)2) · · ·�


hYúÝ2P, Í¥�P¬�±y

sin x = x− x3

3!+

x5

5!− · · ·�

9ùÎ�ZÝ¥�Q¤�×�kÝ�(6.2)P, &Æ�:×ì´��Ý94P�'f(x) = a0 +

a1x + · · ·+ anxn ×ng94P, a0 6= 0, v'f(x) = 0bnÍ8²Ýqx1, · · · , xn, Jã�óÃÍ�§á, f(x) �Aì5�W×°×gPÝ¶�:

f(x) = an(x− x1) · · · (x− xn),

QÍ��x1, · · · , xnb��Î�ó�ãîPêÿ

f(x) = C(1− x

x1

)(1− x

x2

) · · · (1− x

xn

),

Í��x = 0 Ç�ÿC = a0�¨uf(x)� ×94P, � ×´�ÓÝÐó, &Æ)�®,

ÎÍ�Þf 5�W×gPÝ¶��×��, Q�×�ðÿÕ�»A, .E�×x/�º¸ex = 0, .hexP°Ah5��Zs¨, EsineÐó, 9Ë5�QÎ��Ý�ôµÎb(6.2)P, hPE∀x ∈ RWñ, J��¢�Courant and John (1965) pp.602-

603�ã(6.2)P, �:�sin x = 0�q 0, ±π, ±2π, · · ·�u|x =

π/2�á(6.2)P, Jÿ

(6.3) sinπ

2= 1 =

π

2(1− 1

22 · 12)(1− 1

22 · 22)(1− 1

22 · 32) · · ·�

ê¿à

1− 1

22 ·m2=

(2m− 1)(2m + 1)

2m · 2m , m ≥ 1,

Jã(6.3)PÇÿ(6.1)P�&Æ�:�7.3;��LÄÝzetaÐó�×8n®Þ��ß�#

ÝÎ, Ðó

ζ(s) =∞∑

n=1

1

ns, s > 1,

8.6 Í�nyùóÝ�� 505

�²ób×¥�Ýn;�E²óp1 = 2, p2 = 3, p3 = 5, · · · , Cs ≥1, .

0 <1

psi

< 1,

Æã¿¢ùó1

1− 1/psi

= 1 +1

psi

+1

p2si

+1

p3si

+ · · ·�Ei = 1, 2, 3, · · · , ÞîP¼��5½8¶, ��Ñ9øÝ8¶ÎÍ)°, J¼�ÿ

limn→∞

n∏i=1

1

1− 1/psi�

�y��, .N×�y1�Jó, b°×�².ó¶��î°, Æ��

1 +1

2s+

1

3s+ · · · = ζ(s)�

.hzetaÐó��îWì�¶�

(6.4) ζ(s) = (1

1− 1/2s)(

1

1− 1/3s)(

1

1− 1/5s) · · · , s > 1�

ã(6.4)Pù�0�²óbP§9Í�h.u©bb§9Í²ó, |p1, p2, · · · , pr��, J(6.4)P��, ©Îb§Íó8¶, ÆÇ¸Es = 1, ù ×b§Â, ¬&Æ�áζ(1) = ∞, hë;0�²óbP§9Í�î�J��p�|�Û;�Qh°f�¿¦Æ|DJ°,E

²óbP§9ÝJ�, àÕ?9Ý�ÌC´�Ó��Ä.ÍàÕ��!Ý]°, X|)qÂÿ×.�¿à�ùó"P, tÝ�|OÐó�óÂ², ô�¢hU"×°

ÐóÝ�L½��ó�»A, ã

(6.5) ex = 1 + x +1

2!x2 + · · ·+ 1

n!xn + · · · , x ∈ R,

¼óÐóÝ�L½Í¼Î@ó, EN×�óa + ib, Í�i =√−1,

a, b ∈ R, &Æ�|îP¼�Lea+ib�ê¿à¼óÝP²

(6.6) ea+ib = eaeib,


&Æ©��Leib-�|Ý�Þx = ib�áexÝ"PvÞ@I�ÌI5�, ÿ

eib = 1 + ib +1

2!(ib)2 + · · ·+ 1

n!(ib)n + · · ·

= 1− 1

2!b2 +

1

4!b4 + · · ·+ (−1)n b2n

(2n)!+ · · ·

+i(b− 1

3!b3 + · · ·+ (−1)n b2n+1

(2n + 1)!+ · · · )

= cos b + i sin b,

9µÎ��ZZZ222PPP(Euler’s formula)�ãhñÇÿÕ

eiπ + 1 = 0�9Î×Í�b¶vÎ-ÝP�, ¸�â-�9Îó.�t¥�Ý5Íó0, 1, e, π, iCÞÐr+, =�¼óCë�Ðó, Î�Q&t»úÝËvÐó, Bã�ùó, ��ñRÍ Ýn;��Ä�}2ý, |9øÝ]°.0�

(6.7) eix = cos x + i sin x, x ∈ R,

¬�Î��Û�h.ex�"P, Î3x @óÝ�'ì�ÿ, .h(6.5)P, Ex = ib)Wñ, Îm�J�Ý��Ä�Zô�Îàî�Ý]°, Bã�ùó¼J��&ÆB��Ý“J°”Aì, Q��Û, ¬QÎÎlèâtSÝ]°�´�, &�3�.`��.ÄDe Moivre222PPP(De Moivre

(1667- 1754),z»ó.�): EN×Jón,

(6.8) (cos θ + i sin θ)n = cos nθ + i sin nθ�u|θ = x/n�áîP, Jÿ

(6.9) cos x + i sin x = (cosx

n+ i sin

x

n)n�

E×ü�Ýx, n��`, cos(x/n)�cos 0 = 1�-²��; ê.

x/n → 0 `,sin(x/n)

x/n→ 1,

8.6 Í�nyùóÝ�� 507

Æn��`, sin(x/n)�«yx/n�X|ì�Á§2PTÎ)§Ý:

(6.10) cos x + i sin x = limn→∞

(1 + ix/n)n�

4Q35.3;, &ÆXÿÕÝex = limn→∞(1 + x/n)n, ÎE∀x ∈ R,

¬�Z�'hP, |ix�áx)Wñ�.h(6.10)P�� eix,

Æ(6.7)PWñ�kE�Z2P�×�ÛÝó.�L, m�¢Ã��ÐóÝ§¡,

9ÎèÜtSó.Ý¥�Wµ�×�Q$b×°Í�Ý®Þ(A�ùó�[eTs÷), ù2¸��Ðó¡Ýs"�¨², &Æô�|�ùó, |��Ý]°¼�L�AsineÐ

óCcosineÐó��ë�Ðó�&9P², /�|9°"PÿÕ�tÝ3î×;:ÄÝ, �ÿÕ�52P², Í�A

sin 0 = 0, cos 0 = 1, sin(−x) = − sin x, cos(−x) = cos x,

KñÇ�ÿ��°2Pô�Aì�ÿ��u, v ÞÐó, �L

u(x) = sin(x + a)− sin x cos a− cos x sin a,

v(x) = cos(x + a)− cos x cos a + sin x sin a,

Í�a ×ü�@ó�v�

f(x) = u2(x) + v2(x)�

J|J(¿à�ÿÝ�52P)

u′(x) = v(x), v′(x) = −u(x),

Æf ′(x) = 0, ∀x ∈ R�

.hf ×ðó�ê.f(0) = 0, Æÿf(x) = 0, ∀x ∈ R�.hu(x) = v(x) = 0, ∀x ∈ R�ÇJ�

sin(x + a) = sin x cos a + cos x sin a,

cos(x + a) = cos x cos a− sin x sin a,


�yπ, ��L ��sin x = 0�t�Ñó(hx�J�D3), Ah×¼, �J�sineCcosine ø�2π�ø�Ðó, vsin(π/2) = 1,

cos(π/2) = 0�9°Þ;&Æ�a9D¡, �¢�Knopp (1951)×h�&Æ�:×¢ãùó�s÷, ¼¾\×ó��×�4��0Ý

��×»��¿àStirling2P(�7.3;êÞ), ñÇ�ÿu

an =(2n)!

4n(n!)2,

Jn →∞`, an ∼ 1/√

n → 0, v

an+1

an

=2n + 1

2n + 2< 1, ∀n ≥ 1,

Æøýùó∑

(−1)nan f�[e�¬u�¿àStirling2P, kJ�limn→∞ an = 0¬��|�9ì&Æèº×��Ý¾½]°�

��§§§6.1.'{an} ×�}�3�Ñ4ó��bn = 1 − an+1/an,

n ≥ 1�Jlimn→∞ an = 0, uv°u∑

bns÷�JJJ��.´�&Æ¥�Õ, t&limn→∞ bn = 0, ÍJ

∑bn�

∑log(1 −

bn)/s÷�¨×]«, ulimn→∞ bn = 0, J¿à1Ä¾!J�ÿ

limn→∞

bn

− log(1− bn)= 1�

ÆãÁ§f´l�°á,∑

bns÷,uv°u∑

log(1−bn)s÷�h�.0 < bn < 1, Æ− log(1− bn) > 0, ∀n ≥ 1�¨.

an+1 = an(1− bn), ∀n ≥ 1,

Æan = a1(1− b1)(1− b2) · · · (1− bn−1), ∀n ≥ 2,

.hlimn→∞ an = 0, uv°ulimn→∞ log an = −∞, uv°u

limn→∞

(log a1 +n∑

i=1

log(1− bi)) = −∞,

8.6 Í�nyùóÝ�� 509

uv°u∑

log(1 − bn)s÷��J�∑

log(1 − bn)s÷, uv°u

∑bns÷�J±�

/Õ�§6.1�GÝøýùó∑

(−1)nan�.bn= 1 − an+1/an=

1/(2n + 2), ∀n ≥ 1, ��2∑

bns÷, Æã�§6.1ÿlimn→∞ an =

0, �Ä¿àStirling2P�¨², ¿à�§6.1, ô�Á�|2ÿÕ�A

∞∑n=0

(−1)n nn

enn!,

CÞ4Pùó∞∑

n=0

(c

n

), c > −1,

[e�&Æ��×°nyùóÝb¶��, J�/¯��b·¶Ý

\ï�¢�8nÝh°�9ì ×nyË�ùó8¶Ý��

��§§§6.2.'bÞ�ùó

f(x) =∞∑

n=0

anxn, |x| < r1,

g(x) =∞∑

n=0

bnxn, |x| < r2�

JE∀|x| < min{r1, r2},

f(x)g(x) =∞∑

n=0

cnxn,

Í�

(6.11) cn =n∑

k=0

akbn−k�


J��Apostol (1973) Theorem 9.24��:�Þ�ùó8¶,

Ìbv«94P8¶Ý��&Æá¼=�Ðó�×��, ¯@î�b?ÁÐÝ��

��§§§6.3.D3×3RîÕ��=�Ðó�

9ËÐó�9, ¬¬&&Æð�ÝÐó, 9ì�×»��

φ(x) =

{x, 0 ≤ x ≤ 1,

2− x, 1 < x ≤ 2,

�U"φ��L�Xb@ó, ¸φ ×ø� 2�Ðó, Çφ ��

φ(x + 2) = φ(x)�

Jφ3Rî=��Íg�

(6.12) f(x) =∞∑

n=0

(3

4)nφ(4nx)�

.0 ≤ φ ≤ 1, ã�§3.3á, (6.12)PX�L�ùó, 3Rîí8[e�Æã�§3.1ÿ, f3Rî=��yf3RîÕ��, ÍJ��Rudin (1964) pp.141-142,

3h¯��

&ÆèÄ9g, 94PÎt��ÝÐó�EN×T îÝ=�Ðó, �|×ó��94P¼í8¿��9ÎWeierstrassJ�Ý½(Ý��&ÆW�yì, �¢�Rudin (1964) pp.146-148�A�×Ðób�ùó"P, J�ãÍ�ùó�I5õ( 94P), ¼í8¿�(��§4.1)�Ðó��Äh�§Ý©�P, 3y©�E=�Ðóf , -�0Õ94P¼¿�, Q9Ë94PÝÁ§, µ�×�ÎfÝ�ùóÝ(.f¬�×�b�ùó"P)�

8.6 Í�nyùóÝ�� 511

��§§§6.4.'f [a, b]î�×=�Ðó, JE∀ε > 0, D3×94PP (��εbn), ¸ÿ

(6.13) |f(x)− P (x)| < ε, ∀x ∈ [a, b]�

Íg&Æ:AbelÁÁÁ§§§��§§§(Abel’s limit theorem)�´�u|x| <1, J

(6.14) log(1− x) = −∞∑

n=1

xn

n�

&Æá¼îP��Ex = −1ù[e,Ç øýùó∑∞

n=1(−1)n−1/n�¬&ÆÎÍù�Þx = −1�á(6.14)P�¼�, �ÿÕ

log 2 =∞∑

n=1

(−1)n−1

n?

ì��§¼��nÎù�Ý(h��3Ïâaù�èÄ)�J�J�Apostol (1973) Theorem 9.31(Abel’s limit theorem)�

��§§§6.5.(AbelÁÁÁ§§§��§§§). '

(6.15) f(x) =∞∑

n=0

anxn, |x| < r�

uî�ùóEx = rù[e, Jlimx→r− f(x)D3, v

limx→r−

f(x) =∞∑

n=0

anrn�

�§6.5bì�Ý.¡, ÍJ�ô�àÕ�§6.2, ��Apostol

(1973) Theorem 9.32�

��§§§6.1.'∑∞

n=0 an,C∑∞

n=0 bn Þ[eùó,�cn��LA(6.11)P

�J©�∑∞

n=0 cn[e,

(6.16)∞∑

n=0

cn = (∞∑

n=0

an)(∞∑

n=0

bn)�


37.3;&Æ��×�Zs¨bnπÝ2P, Ç

(6.17)∞∑

n=1

1

n2=

π2

6�

h2Pb&9�!ÝJ°(�êÞ8.3), Þ¼&�ub^ºÑê´{�Ýó.�, A��Ðó¡��-º.Õ�9ì&Æèº×©m¿à�*X.Ý�ÌÝÃÍJ°�´�¿à�E[eùóÝP², ìPWñ:

∞∑n=1

1

n2=

∞∑n=0

1

(2n + 1)2+

∞∑n=1

1

(2n)2,

Æu�J�

(6.18)∞∑

n=0

1

(2n + 1)2=

π2

8

ÇÿJ(6.17)P�ã�§5.6, �ÿ(1 − x2)−1/2�"P��¿à�§5.1�ÿ(ù�

êÞ8.5)

(6.19) arcsin x = x +∞∑

n=1

1 · 3 · 5 · · · (2n− 1)

2 · 4 · 6 · · · (2n)· x2n+1

2n + 1, |x| < 1�

¿àRaabel�°(�êÞ7.3), �J�(6.19)P��ùóx =

1 `[e(hI5�J�º�&�)�.hã�§3.3 Weierstrass

M -l�°á, (6.19)P��ùó3[−1, 1]í8[e��ã�§6.5á,

(6.19)PEx = 1T−1ù/Wñ�¨|x = sin t�á(6.19)P�Ë�,

ÿ(6.20)

t = sin t +∞∑

n=1

1 · 3 · 5 · · · (2n− 1)

2 · 4 · 6 · · · (2n)· sin2n+1 t

2n + 1, t ∈ [−π

2,π

2]�

Þ(6.20)P�¼��5½ã0�π/2�5, v¿à�§5.1, ÿ(6.21)

π2

8= 1 +

∞∑n=1

1

2n + 1

1 · 3 · 5 · · · (2n− 1)

2 · 4 · 6 · · · (2n)

∫ π/2

0

sin2n+1 tdt�

8.6 Í�nyùóÝ�� 513

Ïëa(4.13)P, ê��

(6.22)

∫ π/2

0

sin2n+1 tdt =2 · 4 · 6 · · · (2n)

3 · 5 · 7 · · · (2n + 1), ∀n ≥ 1�

Þ(6.22)P�á(6.21)PÇÿ(6.18)P�J±�

t¡&Æ¼:Tauber��§§§(Tauber’s theorem)�×��, �§6.5 �Y�§¬�Wñ�ÇufbA(6.15)P��P, J

∑anr

n

s÷`, f(r−)ôb��D3�»A, ãan = (−1)n, Jf(x) =

1/(1 + x), ∀|x| < 1�êx → 1−`, f(x) → 1/2�¬∑

(−1)nQs÷�3�-1897O, Tauber (V1866-1947)J�, ©�E;ó{an}�×°§×, JAbel�§�Y�§Wñ�Í¡b&99×vÝ��,

¬KÌ Tauberian Theorems�9ì ×t��Ý�µ, b`Ì TauberÏÏÏ×××��§§§(Tauber’s first theorem), J��Apostol (1973)

Theorem 9.33�

��§§§6.6.�f(x) =∑∞

n=0 anxn, |x| < 1, v'limn→∞ nan = 0�

ulimx→1− f(x) = S, J∑∞

n=0 an[e, võù S�

t¡&Æ�×nyP§¶��[eÝ��, J��Apostol

(1974) Theorems 8.52C8.55�

��§§§6.7.(i)'an > 0, ∀n ≥ 1,J∏∞

n=1(1+an)[e,uv°u∑

an[e�

(ii) '1 6= an ≥ 0, ∀n ≥ 1, J∏∞

n=1(1− an) [evÁ§� 0, uv°u

∑an[e�

¢¢¢��ZZZ¤¤¤

1. R�¦Ë(1985). 3.14159 · · · _Õ9? I.`�Ï16àÏ7�,

529-532�




3. Rudin, W. (1964). Principles of Mathematical Analysis, 2nd

ed. McGraw-Hill Book Company, New York, New York.

4. Knopp, K. (1951). Theory and Application of Infinite Series.

Hafner, New York, New York.


Analysis, Vol I. Springer-Verlag, New York, New York.

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999��ÐÐÐóóóCCCÍÍÍ��555��555

9.1 999��ÐÐÐóóó

�h&ÆXD¡Ý, 9� ��óÝ@ÂÐó, Af(x), T1×îÝÐó�&Æô�bãnîè Rn, Ì�mîè Rm�'�ÂÝ'''��óóóÐÐÐóóó(vector-valued function of a vector variable)�3hE∀n ≥ 1, Rn = {(a1, a2, · · · , an)|ai ∈ R, i = 1, · · · , n}��um =

1, JÌh @Â�'��óÐó, T�Ìööö��(scalar)Ðó, b`ôÌh 999��óóóÐÐÐóóó(T999��ÐÐÐóóó)�;ð�|f(x, y),Tg(x, y, z),5½�ËÍ�ó, CëÍ�ó�@ÂÐó�»A, �

u = x + y,

JE∀x, y ∈ R, b×u�ÍET; �

v = log(1− x2 − y2 − z2),

E∀x, y, z ∈ R, vx2 + y2 + z2 < 1, b×v�ÍET�×��,

uz = f(x, y), x, yÌ ��óóó(independent variables), zÌ TTT��óóó(dependent variable)�Íax�D¡9�Ðó, �y'�ÂÐóÝD¡, �¢�Apostol (1969) Chapter 8, &Æ©º3Ï9.6;#ÇÕ�

515

516 ÏÜa 9�ÐóCÍ�5��5

E9�Ðó, &Æ�GXD¡Ý, ny��ÐóÝÁ§�=��0óC�5ÝÃF, &9)Êà�¬ôb×°|GX^b±ÝÃF��Äb×ÍæJÎ: u×EË�óÐóWñÝ�§, ??�D|2.ÂÕëÍ|îÝ�ó, �ÍJ��à��¢@²îÝ;��.h, &9`Î&Æ©D¡Ë�óÝÐó, h`ô´||¿¢%�¼1��yëÍ|îÝ�ó�Ðó, uÂÿD¡, &ÆôºÇC�'bËÍnîÝF, x = (x1, · · · , xn), y = (y1, · · · , yn), Í///

��(inner product)��L

(1.1) x · y =n∑

k=1

xkyk�

�||x|| = (x · x)1/2 = (∑n

k=1 x2k)

1/2�x�Pó(norm), T1x��||x−y||�x, yÞF�ûÒ�E×a ∈ RnC×r > 0, Rn��

||x− a|| < r

�XbFÝ/), Ì ×|a �T, �5 r��Ýnî¦(n-ball),

¬|B(a; r)��un = 1, hÇ ×|a �T�� ; un =

2, hÇ×|a iT�i8; un = 3, hÇ×|a �T�@T¦�'S Rn�×�/, va ∈ S�uD3×|a �T��Ýnî

¦, ¸ÿh¦�âyS�, JaÌ S�×///FFF(interior point)�S�Xb/F�/)|intS��×�âa��/)(open set), ôÌ a�×Ï½�3h, uS =intS, JSÌ ×�/)�¨², 'S ⊂ Rn, JE×a ∈ Rn, uD3×|a �T��

Ýnî¦, ¸h¦��âSÝ�¢F, JaÌ S�×²²²FFF(exterior

point)�S�Xb²F�/)|extS��×FuÉ� S�/F, ô� S�²F, -Ì S�\\\&&&FFF(boundary point)�&Æ¬�©½ 5FC'��Ðr�'x = (x1, · · · , xn) ×n

îÝF, v = (v1, · · · , vn) ×nî'�, Jx + v = (x1 + v1, · · · , xn +

vn)�êEÞnî'�uCv, Í/��L)A(1.1)P�ê�J�

(1.2) u · v = ||u|| · ||v|| cos θ,

9.2 Á§C=� 517

Í�θ u�v�ô��ã(1.2)Pêÿ

(1.3) |u · v| ≤ ||u|| · ||v||�

îPô�¿àÞ��P¼J�, hI5=!(1.2)P/º�&�� Yê�t¡, uA1, A2 R�Þ�/, JÍÎÎÎ��ÉÉÉ¶¶¶��(Cartesian prod-

uct)��L

A1 × A2 = {(a1, a2)| a1 ∈ A1, a2 ∈ A2}�

uA1, A2, A3 R�ë�/, J

A1 × A2 × A3 = {(a1, a2, a3)| a1 ∈ A1, a2 ∈ A2, a3 ∈ A3},

õv.�.hR2 = R×R, R3 = R×R×R�

9.2 ÁÁÁ§§§CCC===��

��ó�Á§ÝÃF, ��|2.Â�9�Ðó�'f : S →R, Í�S Rn�×�/, Çf ×|S �L½�n�óÝÐó�JEa = (a1, · · · , an) ∈ Rn, �|f(a)�f(a1, · · · , an)�E×b ∈ R,

(2.1) limx→a

f(x) = b,

T¶Wx → a`, f(x) → b, �

(2.2) lim||x−a||→0

|f(x)− b| = 0�

(2.2)Pê��y

(2.3) lim||h||→0

|f(a + h)− b| = 0�

518 ÏÜa 9�ÐóCÍ�5��5

(2.1)Pô�|Ï½¼1��Ç(2.1)PWñ, uv°uEN×b�Ï½N , D3×B(a; r), ¸ÿ©�x ∈ B(a; r) ∩ S \ {a}, Jf(x) ∈N�ux = (x1, · · · , xn), a = (a1, · · · , an), J||x − a|| = ((x1 −a1)

2 + · · ·+ (xn − an)2)1/2�Æ||x− a|| → 0, uv°uxk − ak → 0,

∀k = 1, · · · , n�ER2�ÝF, u|(x, y)�x, (u, v)�a, J(2.1)P�¶W

lim(x,y)→(u,v)

f(x, y) = b�

�yER3�ÝF, u|(x, y, z)�x, (u, v, w)�a, J(2.1)P�¶W

lim(x,y,z)→(u,v,w)

f(x, y, z) = b�

×Ðóf , u��

(2.4) limx→a

f(x) = f(a),

JÌ3a=��uf3∀a ∈ S=�, JÌf3S=��

»»»2.1.'f(x, y) = x, x, y ∈ R, �Jf ×=�Ðó�JJJ��.&Æ6J�, E∀a = (u, v),

(2.5) limx→a

f(x) = f(a)�

�∀ε > 0,(x, y) ∈ B(a; ε)`,.||(x, y)−(u, v)|| < ε,0l|x−u| <ε, Æ|f(x)− f(a)| = |x− u| < ε�.hµ�Lá(2.5)PWñ�

ãyî�9�Ðó�Á§C=�Ý�L, Í²î��Ðó�Ý�LÎ×øÝ, .h�|ï�, &9|GbÝ, nyÁ§C=�Ý��, 39î�ôbETÝ��

��§§§2.1.'fCg Þn�óÐó,vlimx→a f(x) = b, limx→a g(x) =

c�J(i) limx→a(f(x) + g(x)) = b + c;

9.2 Á§C=� 519

(ii) limx→a λf(x) = λb, ∀λ ∈ R;

(iii) limx→a f(x) · g(x) = bc;

(iv) limx→a f(x)/g(x) = b/c, c 6= 0�

��§§§2.2.'fCg Þn�óÐó, v/3aF=��Jf + g, f −gCfg, /3aF=�, v©�g(a) 6= 0, Jf/g3aFù=��

×Ðóu ×°axk11 xk2

2 · · ·xknn �õ, Í�a ∈ R, k1, · · · , kn &

�Jó, -Ì ×n�ó�94Ðó(TÌ94P)�»A,

f(x, y) = x2 − 3xy + y3 + 5y2 + 7

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y)Clog(cos2(x2 + y2)), 3Íb�L�/=��ÆÏ×ÍÐó3JÍx-y¿«/=�, ÏÞÍÐó3(x, y) 6= (0, 0)�=�, ÏëÍÐó3x + y 6= 0�=�, Ï°ÍÐó3x2 + y2� π/2��ó¹�=��î�9¿Í»��î, E×Þ�óÐó, Í�=�FÝ/), ��â×°âÒÝF, ×f`aT×°`a�9ì&Æ�×Þ�óÐó, EN×�ó/=�, ¬uÚ ×Þ�óÝÐó, J� =��

»»»2.2.�

f(x, y) =

{xy

x2+y2 , (x, y) 6= (0, 0),

0, (x, y) = (0, 0)�ü�y = y0, uy0 = 0, Jf(x, 0) = 0, ∀x ∈ R, Æf(x, 0) ×=�Ðó;uy0 6= 0,Jf(x, y0) = xy0/(x

2 +y20) ×5Ò� 0�b§P,Æ

) =�Ðó�.hEN×ü�Ýy, f(x, y) ×x�=�Ðó�!§, uü�x, f(x, y)) ×yÝ=�Ðó�¬uy = x, v(x, y) 6= (0, 0), J

f(x, y) =x2

2x2=

1

2,

520 ÏÜa 9�ÐóCÍ�5��5

Æu(x, y)º½àay = x��(0, 0), Jf(x, y) → 1/2 6= f(0, 0) =

0�.huÚ ×Þ�óÝÐó, Jf3(0, 0)�=��¯@î,

u(x, y)º½àay = mx��(0, 0), Jf(x, y) → m/(m2 + 1)�3Í»�, 4

limy→0

(limx→0

f(x, y)) = limy→0

f(0, y) = 0,

vlimx→0

(limy→0

f(x, y)) = limx→0

f(x, 0) = 0,

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mx��(0, 0), J

f(x, y) =mx3

x4 + m2x2→ 0�

ôµÎ(x, y)º½�×y = mx�àa��(0, 0)`, f(x, y)/��0�¬h¬�1J(x, y) → (0, 0)`, f(x, y)�Á§D3�¯@î, u(x, y)º½eÎay = x2��(0, 0), Jf(x, y) → 1/2 6=0�.h(x, y) → (0, 0)`, f(x, y)�Á§�D3�

»»»2.4.�f(x, y) =

xy

|x|+ |y| , (x, y) 6= (0, 0)�

�Olim(x,y)→(0,0) f(x, y)�

êÞ 521

��..|x|≤√

x2 + y2, |y| ≤√

x2 + y2, v|x|+|y|≥√

x2 + y2, ∀(x, y)∈R2, ÆE∀(x, y) 6= (0, 0),

|f(x, y)| = |x||y||x|+ |y| ≤

√x2 + y2

√x2 + y2

√x2 + y2

=√

x2 + y2�

ãîP, ¿àô^�§, Çÿ(x, y) → (0, 0)`, f(x, y) → 0�Æu�f(0, 0) = 0, Jf ×Õ�=��Ðó�

»»»2.5.�5½D¡ì�ÞÐó�=�P�(i) f(x, y) = (1− x2 − y2)−1/2;

(ii) g(x, y) = x arctan(y/x), x 6= 0, g(0, y) = 0��.(i) f��L½ x2 + y2 < 1, vfù3h ½=��

(ii) g��L½ R2�´�3x 6= 0�, g(x, y)Q=��êE∀x 6= 0, | arctan(y/x)| < π/2, Æ

lim(x,y)→(0,b)

g(x, y) = 0 = g(0, b), ∀b ∈ R�

.hg ×Õ�=�ÝÐó�

»»»2.6.�

f(x, y) =x2y + 2x− 2xy − 4x + y + 2

x2 + y2 − 2x + 4y + 5,

�O

lim(x,y)→(1,2)

f(x, y)�

��.�;¶f(x, y), -�ÿ(x, y) → (1, 2)`,

f(x, y) =(x− 1)2(y + 2)

(x− 1)2 + (y + 2)2→ 0 · 4

0 + 16= 0�

êêê ÞÞÞ 9.2

1. �¼�ì�&Ðó�=�FÝ/)�

522 ÏÜa 9�ÐóCÍ�5��5

(1) f(x, y) = log(x2 + y2)�(2) f(x, y) = cos x2/y�(3) f(x, y) = tan(x2/y)�(4) f(x, y) = arctan(y/x)�(5) f(x, y) = arcsin(x/

√x2 + y2)�

(6) f(x, y) = xy2

�(7) f(x, y) = arctan((x + y)/(1− xy))�(8) f(x, y) = arccos

√x/y�

(9) f(x, y) = (1− x2 − y2)−1�(10) f(x, y) =

√y cos x�

(11) f(x, y) = (x2 − y2)−1�(12) f(x, y) = tan πy/ cos πx�

2. �D¡ì�&Ðó�=�P�

(1) f(x, y) = x2y/(1 + x), ux 6= −1; vf(−1, y) = y�(2) f(x, y) = x2y/(x3+y3),u(x, y) 6= (0, 0);vf(0, 0) = 0�(3) f(x, y) = (x3+y3)/(x2+y2),u(x, y) 6= (0, 0);vf(0, 0) =

0�(4) f(x, y) = (x3 +y3)/(x2 +y),u(x, y) 6= (0, 0); vf(0, 0) =

0�

3. ��Lf(0, 0)�Â, ¸ì�&Ðó3(0, 0)=��(1) f(x, y) = sin(x2 + y)�(2) f(x, y) = sin xy/

√x2 + y2�

(3) f(x, y) = (x3 + y3)/(x2 + y2)�(4) f(x, y) = x2 log(x2 + y2)�(5) f(x, y) = sin(x2 + y2)/(x2 + y2)�(6) f(x, y) = sin(x4 + y4)/(x2 + y2)�(7) f(x, y) = e−(x2+y2)−1

/(x4 + y4)�4. �®Eì�&Ðóf , ÎÍ��Lf(0, 0)�Â, ¸f3(0, 0)=��

êÞ 523

(1) f(x, y) = sin x/y�(2) f(x, y) = x2y2/(x4 + y4)�(3) f(x, y) = (x5 + y5)/(x4 + y4)�(4) f(x, y) = xy3/(x4 + y4)�(5) f(x, y) = x2y/(x2 + y2)�(6) f(x, y) = sin(xy)/xy�(7) f(x, y) = sin(x2 − y2)/(x2 + y2)�

5. �JÐóf3(0, 0, 0)=�, g3(0, 0, 0)�=�, Í�

f(x, y, z) =

{xyz

x2+y2+z2 , (x, y, z) 6= (0, 0, 0),

0, (x, y, z) = (0, 0, 0);

g(x, y, z) =

{x2+y2−z2

x2+y2+z2 , (x, y, z) 6= (0, 0, 0),

0, (x, y, z) = (0, 0, 0)�

6. �Jì�ÞÐó

f(x, y) =x4y4

(x2 + y4)3, g(x, y) =

x2

x2 + y2 − x,

(x, y)º½�×àa��(0, 0), /��0, ¬hÞÐó/P°�L3(0, 0)�Â, ¸ÿ3(0, 0)=��

7. �D¡ì�&Ðóf , (x, y) → (0, 0)`, f(x, y)�Á§ÎÍD3, uD3JO��(1) f(x, y) = (x2 − y2)/(x2 + y2)�(2) f(x, y) = (x + y)2/(x2 + y2)�(3) f(x, y) = (x2 + 3xy + y2)/(x2 + 4xy + y2)�(4) f(x, y) = |x− y|/(x− y)2�(5) f(x, y) = exp{−|x− y|/(x− y)2}�(6) f(x, y) = |x|y�(7) f(x, y) = |x|1/|y|�(8) f(x, y) = |y||x|

√x2 + y2/(

√x2 + y2 + |y/x|)�

524 ÏÜa 9�ÐóCÍ�5��5

8. �JE�×b ∈ R, D3×a, ¸ÿ(x, y)ºha��(0, 0)`, f(x, y)��b, Í�

f(x, y) = 3(x− y)/(x + y), x + y 6= 0�

9. 'lim(x,y)→(a,b) f(x, y) = L, limx→a f(x, y)Climy→b f(x, y)/D3��J

limx→a

(limy→b

f(x, y)) = limy→b

(limx→a

f(x, y)) = L�

Û. h��î, uË�óÐó�Á§CÞ��ÐóÝÁ§/D3, JÞ@gÁ§8��ÍY�Ë, êÞÏ11Þ ×»�

10. �f(x, y) = (x− y)/(x + y), x + y 6= 0��J

limx→0

(limy→0

f(x, y)) = 1,¬ limy→0

(limx→0

f(x, y)) = −1�

¿àh��CîÞ0�(x, y) → (0, 0)`, f(x, y)�Á§�D3�

11. �

f(x, y) =x2y2

x2y2 + (x− y)2, x2y2 + (x− y)2 6= 0�

�Jlimx→0

(limy→0

f(x, y)) = limy→0

(limx→0

f(x, y)) = 0,

¬(x, y) → (0, 0)`, f(x, y)�Á§�D3�

12. �

f(x, y) =

{x sin(1/y), y 6= 0,

0, y = 0��J(x, y) → (0, 0)`, f(x, y) → 0, ¬

limy→0

(limx→0

f(x, y)) 6= limx→0

(limy→0

f(x, y))�

�Õh¬�ÀDÏ9Þ��

9.3 ]'0ó 525

13. �

f(x, y) =

{0, uy ≤ 0Ty ≥ x2,

1, u0 < y < x2�(i) �Ju(x, y)º½×àa��(0, 0), Jf(x, y) → 0;

(ii) �0�×;Ä(0, 0)�à, ¸ÿtÝ3(0, 0)², 3hàîf(x, y)�Â/ 1;

(iii) �®f3(0, 0)ÎÍ=�?

14. '

f(x, y) =

{e−1/x2

y

e−2/x2+y2

, x 6= 0,

0, x = 0��Ju(x, y)º½àxm = (y/c)n��0, Jf(x, y)��0,

Í�c 6= 0 ×ðó, m�n Þ!²ÑJó�¬f3(0, 0)�=��

9.3 ]]]'''000óóó

&Æ��ÊËÍ�óÝ�µ�'f ×Þ�óÐó, P ×3f�L½S�ÝF, L R2î×f;ÄP�àa�uÞR2Ú R3��x-y¿«, J;ÄLvkàx-y¿«�¿«p, øf�%�y×àC, CÌ f�%�3¿«p Ý× ^½½½(cross section)�3R3�,

/){(x, y, f(x, y))|(x, y) ∈ S}Ç f�%��ã¿¢îÝP²á,

L��î L = {P + tv|t ∈ R},

Í�v ×Þî�&ë'��L ×BÄPFv]' v�àa�êE×'�v, &Æ|v = (v1, v2)�ÍÞ5�5½ v1Cv2�uÞf(P+tv)Ú ×t�Ðó,JhÐó3t = 0�0ó|Dvf(P )

��Ç

(3.1) Dvf(P ) = limt→0

f(P + tv)− f(P )

t �

526 ÏÜa 9�ÐóCÍ�5��5

Dvf(P )-Ì f3PFv]' v�0ó�u�g(t) = f(P + tv),

J(3.1)P��Ç g′(0), Æ

(3.2) Dvf(P ) = g′(0)�

êuÞLã ×2ý�, Pã æF, ¨×�ã �z�¿�, JC Ç 3¿«pîg�%��

»»»3.1.�f(x, y) = x2 − xy + 5y�Of3P = (−1, 2) v]'v =

(3,−4)�0ó��.´�

P + tv = (−1, 2) + (3t,−4t) = (−1 + 3t, 2− 4t)�

Æu�g(t) = f(P + tv), J

g(t) = f(−1 + 3t, 2− 4t)

= (−1 + 3t)2 − (−1 + 3t)(2− 4t) + 5(2− 4t)

= 13− 36t + 21t2�

.hg′(t) = −36 + 42t, vDvf(P ) = g′(0) = −36�

)�g(t) = f(P + tv), v�h(t) = g(ct) = f(P + tcv), c ∈ R, Jã=Å!Jÿh′(t) = cg′(ct)��t = 0, ÇJ�

Dcvf(P ) = h′(0) = cg′(0) = cDvf(P )�

.h

(3.3) Dcvf(P ) = cDvf(P )�

uv ×��'�, Ç||v|| = 1, JDvf(P )Ì f3PFv]' v �]]]'''000óóó(directional derivative)�uP = (a, b), v = (r, s), v||v|| = 1, Çr2 + s2 = 1, JL�¢¢¢óóó

PPP x(t) = a + rt, y(t) = b + st�

9.3 ]'0ó 527

.hG�`aCî�×F�|λ(t) = (a + rt, b + st, g(t))��ã¿¢�Ý��á, `aC3Fλ(t)Ý6'�

λ′(t) = (r, s, g′(t))�êQ = λ(0) = (a, b, f(P )) ETPF�fÝ%�îÝF�Æ3Q F�6'� λ′(0) = (r, s, g′(0)), h'�ÇÌ f%�3QF, v]' v�666'''��(tangent vector)�h'��E£ g′(0)/

√r2 + s2 =

g′(0) = Dvf(P ), Í�t¡×�PàÕ(3.2)P�Æ]'0óÇ f�%�î, 3X��]'�6'�ÝE£�

»»»3.2.�f(x, y) = (x + y)/(x− y), Of3P = (1,−1) v]' v =

(1/2,√

3/2)�]'0ó��..f(P ) = f(1,−1) = 0, v

f(P + tv) = f(1 + t/2,−1 +√

3t/2),

Æ

Dvf(P ) = limt→0

f(P + tv)− f(P )

t= lim

t→0

1

t

1 + t/2− 1 +√

3t/2

1 + t/2 + 1−√3t/2

= limt→0

1 +√

3

4 + (1−√3)t=

1 +√

3

4 �

ëÍ|î�óÝ]'0ó, ô�v«2�L, &Æ|ì»¼1��

»»»3.3.�f(x, y, z) = x + xy − yz, P = (−1, 1, 2), v = (3, 1, 1),

ODvf(P )��.´�P + tv = (−1, 1, 2) + t(3, 1, 1) = (−1 + 3t, 1 + t, 2 + t)�Æ

Dvf(P ) = limt→0

f(P + tv)− f(P )

t= lim

t→0

2t + 2t2

t= 2�

'b×Ðóf , JÄPF, ©��×]'v, -�b×]'0ó�Í�3v1 = (1, 0)Cv2 = (0, 1) �]'0ó©½¥��uP =

528 ÏÜa 9�ÐóCÍ�5��5

(x, y), J

Dv1f(x, y) = limt→0

f(x + t, y)− f(x, y)

t,

Dv2f(x, y) = limt→0

f(x, y + t)− f(x, y)

t �

�:�Dv1f(x, y) ü�y,ÞfÚ ×x�ÐóÝ0ó,�Dv2f(x, y)

J ü�x,ÞfÚ ×y�ÐóÝ0ó�;ð&Æ|D1f(x, y)

�Dv1f(x,y), |D2f(x, y)�Dv2f(x, y)�D1fCD2f) (x, y)�Ðó, ¬Ì f�×××$$$��000óóó(first partial derivative)��ó x,

y`, D1fCD2f5½Ì fExCy�×$�0ó�E�0ó$b×°ðàÝBr:

D1f = f1 =∂f

∂x= fx, D2f = f2 =

∂f

∂y= fy�

Ðr“∂”)s¯“d”, ©Î39�Ðó�, Ý��Ðó�0óÐr ½, &Æ|∂f/∂xã�df/dx�

»»»3.4.�f(x, y) = x3 − 3x2y + y2 + x− 7, Of�×$�0ó��.�Q

∂f

∂x= 3x2 − 6xy + 1,

∂f

∂y= −3x2 + 2y�

EëÍ|îÝ�ó, ù�v«2�LÍ�0ó�

»»»3.5.�f(x, y, z) = log(x2 + y3 + z4), �Of�×$�0ó��.¿à=Å!Jÿ

fx =2x

x2 + y3 + z4, fy =

3y2

x2 + y3 + z4, fz =

4z3

x2 + y3 + z4�

f ×Þ�óÐó, D1fCD2f/) Þ�óÐó, .h&Æ��D¡D1f CD2f�×$�0ó, ÇD1(D1f), D2(D1f), D1(D2f)

9.3 ]'0ó 529

CD2(D2f)�h°ÐóÌ f�ÞÞÞ$$$��000óóó(second partial deriva-

tive)�EÞ$�0ó, ðàÝBr

f11 = D11f = D1(D1f) =∂2f

∂x2= fxx,

f12 = D12f = D2(D1f) =∂2f

∂y∂x= fxy,

f21 = D21f = D1(D2f) =∂2f

∂x∂y= fyx,

f22 = D22f = D2(D2f) =∂2f

∂y2= fyy�

¥�, f12�f21��L¬�8!, Gï �Ex�5�Ey �5, ¡ï �Ey�5�Ex�5��5 ×nyÁ§ÝºÕ, �&Æ�èÄ9g, ËÍbnÁ§ÝºÕ, uøðÍºÕg�, ��¬�×�8!�}¡&Æº�×»�1��!§, uf ×ë�óÐó, Jfb9ÍÞ$�0ó, Q9°Þ$

�0ó�, b°��º8��bÝÞ$�0ó, Q��Lë$�0ó�{$�0ó�»A, 'b×Ðóf(x, y), J

f122 = D122f = D2(D2(D1f)) =∂3f

∂y∂y∂x= fxyy,

f2122 = D2122f = D2(D2(D1(D2f))) =∂4f

∂y∂y∂x∂y= fyxyy,

Í�f2122ô�¶W∂4f/(∂y2∂x∂y), ¬Q�×��y∂4f/(∂y3∂x)�

»»»3.6.�f(x, y, z) = xeyz + yzex, �Of213C∂3f/(∂x2∂y)��.&Æbì�.0�

f2 = xzeyz + zex,

f21 =∂f2

∂x= zeyz + zex,

f213 =∂f21

∂z= eyz + yzeyz + ex,

∂3f

∂x2∂y=

∂f21

∂x= zex�

530 ÏÜa 9�ÐóCÍ�5��5

»»»3.7.�

f(x, y) =xy(x2 − y2)

x2 + y2, (x, y) 6= (0, 0), f(0, 0) = 0�

�OD12f(0, 0)CD21f(0, 0)��.´�.f(h, 0) = 0, ∀h 6= 0, vf(0, 0) = 0, Æ

D1f(0, 0) = limh→0

f(h, 0)− f(0, 0)

h

= limh→0

0− 0

h= 0�

êE(x, y) 6= (0, 0),

D1f(x, y) =y(x4 + 4x2y2 − y4)

(x2 + y2)2,

Æ

D1f(0, k) = −k5

k4= −k, k 6= 0�

.h

D12f(0, 0) = limk→0

D1f(0, k)−D1f(0, 0)

k

= limk→0

−k − 0

k= −1�

!§�ÿD21f(0, 0) = 1�

Í»�îD12f(x, y)�D21f(x, y) �×�8��\ïô��JD12f(x, y)CD21f(x, y), 3(0, 0)/�=��3�§5.2, &ÆÞ�¸D12f(x, y)�D21f(x, y)8��f��

êÞ 531

êêê ÞÞÞ 9.3

1. �Oì�&Ðó3X�ÝFC]'�]'0ó�(1) f(x, y) = ax2 + 2bxy + cy2, P = (1, 1), v = (4/5,−3/5)�(2) f(x, y, z) = x2+y2+xyz, P = (1, 1, 1), v = (1/3, 2/3, 2/3)�(3) f(x, y, z) = (x/y)3, P = (1, 1, 1),v = (2/

√6, 1/

√6,−1/

√6)

�

2. 'f(x, y) = x2 + y2, P = (a, b)��®¢`f3P�]'0ó 0?

3. 'f(x, y, z) = x2 + y2 + z2, P = (a, b, c)��®f3P�t�]'0ó¢`t�?

4. 'f(x, y) = 3x2 + y2, P = (x, y) ix2 + y2 = 1î�×F��OP�×]'v, ¸ÿf3Pv]'v�]'0ó t��

5. 'f(x, y, z) = axy2+byz+cz2x3, P = (1, 2,−1)��Oa, b, c�Â, ¸ÿf3P�t�]'0ósß3]' ¿�z�, vh`�]'0ó 64�

6. �Oì�&Ðó�×$�0ó, ¬�JD12f(x, y)�D21f(x, y)

ÎÍ8��(1) f(x, y) = x2 + y2 sin(xy)�(2) f(x, y) = x/

√x2 + y2, (x, y) 6= (0, 0)�

(3) f(x, y) = cos(x2/y), y 6= 0�(4) f(x, y) = tan(x2/y), y 6= 0�(5) f(x, y) = arctan(y/x), x 6= 0�(6) f(x, y) = arctan((x + y)/(1− xy)), xy 6= 1�(7) f(x, y) = xy2

, x > 0�(8) f(x, y) = arccos

√x/y, y 6= 0�

532 ÏÜa 9�ÐóCÍ�5��5

7. ×Ðóf(x, y, z), uE∀(x, y, z) ∈ Sb=�ÝÞ$�0ó, v��

∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2= 0

(hÌ Laplace]]]��PPP(Laplace’s equation)), JÌf3G ��õõõÝÝÝ(harmonic)��Jì�ÞÐó/ �õÝ�(i) f(x, y) = log(x2 + y2)1/2, S ×R2��â{(0, 0)} � ½;

(ii) f(x, y) = (x2 +y2 +z2)−1/2, S ×R3��â{(0, 0, 0)}� ½�

8. �f(x, y) = yne−x2/(4y)��Oðón, ¸ÿf��

∂f

∂y=

1

x2

∂

∂x(x2∂f

∂x)�

9. �z = u(x, y)eax+by, Í�u��∂2u/(∂x∂y) = 0��OðóaCb, ¸ÿ

∂2z

∂x∂y− ∂z

∂x− ∂z

∂y+ z = 0�

10. 'f(x, y) = e−(x2+y2)−1, (x, y) 6= (0, 0), vf(0, 0) = 0��

Jfx(0, 0) = fy(0, 0) = 0�

11. �5½O×Ðóf(x, y), ��

(i) D1f(0, 0) = D2f(0, 0) = 0;

(ii) 3æFC]'(1, 1)�]'0óD3, vÍÂ 3, ¬�Õ ¢h`Ðóf , 3(0, 0)Ä��

12. �

f(x, y) =y(x2 − y2)

x2 + y2, (x, y) 6= (0, 0), f(0, 0) = 0�

�OD1f(0, 0), D2f(0, 0), D21f(0, 0), D12f(0, 0)�

9.4 �0 533

13. �

f(x, y) =xy3

x3 + y6, (x, y) 6= (0, 0), f(0, 0) = 0�

4Q&Æ$Î�Þ�óÐó��Ý�L, ��?f3(0, 0)ÎÍ��, ¬��§ã�

9.4 ��000

'f ×Þ�óÐó,vD1f(P )CD2f(P )/D3,Í�P ∈ R2�Jf3P�VVV��(gradient), |5f(P )��(\ del-f of P ), Í�L

(4.1) 5f(P ) = (D1f(P ), D2f(P ))�

»A, uf(x, y) = x2 − 3xy + y3, J

D1f(x, y) = 2x− 3y, D2f(x, y) = −3x + 3y2,

v5f(x, y) = (2x− 3y,−3x + 3y2)�

3Ï°aE×��óÐóg, &Æ��LÍ�0dg = g′(x)dx�Ædg Þ�óxCdx�Ðó�EÞ�óÐóf , àÌî&Æ��LÍ�0df

df = fxdx + fydy�Ædf ×°�óx, y, dxCdy�Ðó�u&Æ¥�ÕAî�L�df ,

5f(x, y) = (fx, fy)�'�(dx, dy)�/�,Jì��LTÎ��QÝ�×Þ�óÐóf��0df��L

(4.2) df(P, v) = 5f(P ) · v,

Í�P ∈ R2, v ×Þî'��df��L½ Xb(P, v) �/),

©�5f(P )D3�

534 ÏÜa 9�ÐóCÍ�5��5

Íg&Æ¼:��P�'f ×Þ�óÐó, u

(4.3) lim||v||→0

f(P + v)− f(P )− df(P, v)

||v|| = 0,

JÌf3P��9ì¼:, ×Þ�óÐó��0, �×��Ðó��0� Ýv

«n;�'f3PF��, v�

g(v) = f(P ) + df(P, v)�

Jã(4.3)P,

lim||v||→0

f(P + v)− g(v)

||v|| = 0�

Æ∀ε > 0, D3×�ÝÞî¦B(P ; δ), ¸ÿ

|f(P + v)− g(v)| < ε||v||, ∀||v|| < δ�

Æ©�||v||È�, g(v) = f(P ) + df(P, v) f(P + v) �×�?Ý£��hvAE×��Ðóη,©�dxÈ�,Jη(x)+dη η(x+

dx)�×�?Ý£��

»»»4.1.�O√

(3.01)2 + (4.02)2�×�«Â��.�f(x, y) =

√x2 + y2, P = (3, 4), v = (0.01, 0.02)�J

fx =x√

x2 + y2, fy =

y√x2 + y2�

Æ

f(P ) = 5, fx(P ) =3

5, fy(P ) =

4

5,

df(P, v) = (3

5,4

5) · (0.01, 0.02) =

0.11

5= 0.022�

.h

√(3.01)2 + (4.02)2 = f(P + v)

.= f(P ) + df(P, v) = 5.022�

9.4 �0 535

Âÿ¥�ÝÎ, 5f(P )D3, Çf3P�×$�0ó/D3, ¬�×�1Jf3P��, �ì»�

»»»4.2.�f(x, y) =√|xy|�J

D1f(0, 0) = limh→0

f(0 + h, 0)− f(0, 0)

h= lim

h→0

0− 0

h= 0,

!§D2f(0, 0) = 0�

ÆuP = (0, 0), JE∀v = (r, s), Í�r, s ∈ R,

df(P, v) = 0�¬

lim||v||→0

f(P + v)− f(P )− df(P, v)

||v|| = lim||v||→0

f(v)

||v||

= lim(r,s)→(0,0)

√|rs|√

r2 + s2= lim

(r,s)→(0,0)

∣∣∣∣rs

r2 + s2

∣∣∣∣1/2

�

�ã»2.2á, î�Á§�D3�Æµ�Lf3(0, 0)��

9ì&Æ�×Þ�óÐó��Ý�5f��

��§§§4.1.'f ×Þ�óÐó, D1fCD2f3×�¦B(P ; r)D3, v3P=��Jf3P��JJJ��.�P = (a, b), v�LÞ�óÐóg

g(v) = f(P + v)− f(P )− df(P, v),

Í�||v|| < r��v = (x− a, y − b), J(x, y) = P + v, v

g(v) = f(x, y)− f(a, b)−D1f(a, b)(x− a)−D2f(a, b)(y − b)�Bã3îP��×4C3×4f(a, y), ÿ

|g(v)| ≤ |f(x, y)− f(a, y)− (x− a)D1f(a, b)|(4.4)

+|f(a, y)− f(a, b)− (y − b)D2f(a, b)|�

536 ÏÜa 9�ÐóCÍ�5��5

ãy=#(x, y)�(a, y)C(a, y)�(a, b)�Það/3B(P ; r)�,Æã�', f3hÞaðî��0ó/D3�.h, ¿àíÂ�§ÿ, D3×x1+yx�a� , vD3×y1+yy�b� , ¸ÿ

f(x, y)− f(a, y) = (x− a)D1f(x1, y),

f(a, y)− f(a, b) = (y − b)D2f(a, y1)�

ÞîÞP�á(4.4)P�, ÿ

|g(v)| ≤ |x− a||D1f(x1, y)−D1f(a, b)|(4.5)

+|y − b||D2f(a, y1)−D2f(a, b)|�¨.ã�'D1f�D2f/3P=�, Æ∀ε > 0, D3×B(P ; δ), ¸

ÿE∀Q ∈ B(P ; δ),

|D1f(Q)−D1f(P )| < ε

2, |D2f(Q)−D2f(P )| < ε

2�

ê��2, uQ = (x, y) ∈ B(P ; δ), J(x1, y)�(a, y1) ù/3B(P ; δ)

��.h, uQ ∈ B(P ; δ), J

|D1f(x1, y)−D1f(a, b)| < ε

2, |D2f(a, y1)−D2f(a, b)| < ε

2�

Æã(4.4)P, E∀||v|| < δ,

(4.6) |g(v)| ≤ ε

2(|x− a|+ |y − b|)�

.||v|| = ((x − a)2 + (y − b)2)1/2, Æ|x − a| ≤ ||v||v|y − b| ≤ ||v||,

.h(4.6)P0lE∀||v|| < δ,

|g(v)| ≤ ε||v||,

ãîP¿àÁ§Ý�L, Çÿ

lim||v||→0

f(P + v)− f(P )− df(P, v)

||v|| = 0�

ÿJf3P��

9.4 �0 537

î�§¼�, 3ÊÝf�ì, �0óD3º0l��ì�§J¼�, 3Ø°f�ì, Ðóf3ØFP��0, �f3PFÝ�×]'�0óº8!�

��§§§4.2.'f3PF��, JEN×&ëÝÞî'�v, Dvf(P )D3,

v

(4.7) Dvf(P ) = df(P, v)�

JJJ��.´�E∀||v|| 6= 0,∣∣∣∣f(P + tv)− f(P )

t− df(P, v)

∣∣∣∣

= ||v| |f(P + tv)− f(P )− df(P, tv)|||tv|| �

.f3P��, îP��t → 0`, .||tv|| → 0, Æù��0�.hîP¼�ù��0��

limt→0

f(P + tv)− f(P )

t= Dvf(P ),

Æ(4.7)PWñ�

ã�§4.2á, uf3P��, JE∀v = (a, b), a2 + b2 6= 0, .ã(4.2)P,

df(P, v) =5f(P ) · v = (D1f(P ), D2f(P )) · (a, b)

= aD1f(P ) + bD2f(P ),

Æ

(4.8) Dvf(P ) = aD1f(P ) + bD2f(P )�

îPÇèº×O3Ø×]'�0óÝ��]°�'Ðóf(x, y)3P��, &Æ�á¼¢`]'0ót�? 3î

×;ÝêÞ�, ô�¯��Ä×°h®ÞÝYê�u5f(P ) =

538 ÏÜa 9�ÐóCÍ�5��5

(0, 0), JEN×ÞîÝ&ë'�v, ã(4.8)Pá, Dvf(P ) = 0, h`0!` Dvf(P )�Á�ÂCÁ�Â�gu5f(P ) 6= (0, 0), J¿à(1.3)P, EN×��'�v,

Dvf(P ) = | 5 f(P ) · v| ≤ || 5 f(P )|| · ||v|| = || 5 f(P )||�uã'�u = c5 f(P ), Í�c = || 5 f(P )||−1, J||u|| = 1, v

Duf(P ) =5f(P ) · c5 f(P ) = c(5f(P ) · 5f(P ))

= c|| 5 f(P )||2 = || 5 f(P )||,

Çh`]'0ó¾ÕÁ�Â�&Æ-J�Ýì��

»»»4.3.'Ðóf3P��, J|| 5 f(P )|| ]'0óDvf(P )�ÁÂ�êu5f(P ) 6= (0, 0), JG�Á�Âsß3]'

(4.9) v = || 5 f(P )||−1 5 f(P )�

»»»4.4.'f(x, y) = x2 + xy, P = (1,−1), �O]'0óDvf(P )�Á�Â��..D1f(x, y) = 2x + y, D2f(x, y) = x, Æ

5f(P ) = (D1f(P ), D2f(P )) = (1, 1)�

.|| 5 f(P )|| = √2, ÆDvf(P )�Á�Â

√2, vsß3]'

v = (1√2,

1√2)�

3��Ðó�, ��º0l=�, E×Þ�óÝÐóôbv«Ý��

��§§§4.3.'f ×Þ�óÝÐó, vf3P��, Jf3P=��JJJ��.ã�L, f3P��0l

lim||v||→0

f(P + v)− f(P )− df(P, v)

||v|| = 0�

9.4 �0 539

u�g(P + v) = (f(P + v)− f(P )− df(P, v))/||v||, JîPÇ

lim||v||→0

g(P + v) = 0�

Æ∀ε > 0,D3×δ > 0,¸ÿu||v|| < δvP +v 6= P ,J|g(P +v)| <ε�©½2, uãε = 1, Jb

|f(P + v)− f(P )−5f(P ) · v| < ||v||, ∀||v|| < δ�

ãîPêÿ

|f(P + v)− f(P )| < | 5 f(P ) · v|+ ||v||, ∀||v|| < δ�

�¿àÞ��P, îPê0l

|f(P + v)− f(P )| < || 5 f(P )|| · ||v||+ ||v||= (1 + || 5 f(P )||)||v||, ∀||v|| < δ�

ãhêñÇÿlim

||v||→0f(P + v) = f(P )�

J±�

|îÝ×°��, EëÍ|îÝ�óôKÊà�»A, 'b×Ðóh(x, y, z), JE×P ∈ R3,

5h(P ) = (D1h(P ), D2h(P ), D3h(P ))�

�0Ý�Lô)v«, Ç

(4.10) dh(P, v) = 5h(P ) · v�

êu(4.10)PWñ, Jh3P��^¡uÂÕëÍ|îÝ�ó�Ðó, &Æ-ÞÍ;nyË�ó

Ý��, ��2Ú EëÍ|îÝ�óùWñ�

540 ÏÜa 9�ÐóCÍ�5��5

3»4.2�¼�, yØF�×$�0ó/D3, �×�1JÐó3�F��9ì��×»�

»»»4.5.'

f(x, y) =

{xy

x2+y2 , (x, y) 6= (0, 0),

0, (x, y) = (0, 0)�Jã»2.2á, f3(0, 0)�=�, Æã�§4.3á, f3(0,0)��¬D1f(0, 0) = D2f(0, 0) = 0�

E×��óÐóf , u3ØFa�0óD3, J3Í%�îÝF(a, f(a))�6aùD3, vf ′(a) �6a�E£�uf ×�L3S�Þ�óÐó, JÍ%� R3�×`«, Ç/

){(x, y, f(x, y))|(x, y) ∈ S}�uP = (a, b) ∈ S, vf3P ��, JEN×Þî'�v = (r, s), (r, s, Dvf(P )) f3P F, v]'v�%�Ý6'��©½2,

w1 = (1, 0, D1f(P )) C w2 = (0, 1, D2f(P ))

5½ 3Þ]'(1, 0)C(0, 1)�6'��uv = (r, s) ×&ë'�, Jã(4.8)Pá

Dvf(P ) = rD1f(P ) + sD2f(P )�

.h

(r, s, Dvf(P )) = r(1, 0, D1f(P )) + s(0, 1, D2f(P ))

= rw1 + sw2�

¬ãÏ9.3;�D¡á, (r, s, Dvf(P ))�3PFf�%�îÝFQ =

(a, b, f(P ))y]'v�6'��.h3Q�N×6'�/ w1Cw2

�aPà)�ð�1, ÄQ F�N×6'�/a3¿«p, Í�

p = {Q + (rw1 + sw2)|r, s ∈ R}�

9.4 �0 541

&ÆÌp 3PF�fÝ%�Ý666¿¿¿«««(tangent plane)�©�¶�p�°'�(ÇÄQ��6¿«kà�'�), -�ÿÕp

�]�P��w1�w2�²²²��(cross product)w1 × w2Ç ×°'��3h, 'bÞëî'�u1 = (l1,m1, n1), u2 = (l2,m2, n2), JÍ²�u1 × u2 ��L

(4.11) u1 × u2 = (m1n2 −m2n1, n1l2 − n2l1, l1m2 − l2m1),

) ×ëî'��.h

w1 ×w2 = (−D1f(P ),−D2f(P ), 1)�

ÆuP = (x0, y0), J3PF�fÝ%�î, Í6¿«]�P

(4.12) D1f(P )(x−x0)+D2f(P )(y−y0)−(z−f(x0, y0)) = 0�

»»»4.6.�z = x2 + 4y2, �OÄ(−2, 1, 8)�6¿«��.�f(x, y) = x2 + 4y2, P = (−2, 1)�.

D1f(x, y) = 2x,D2f(x, y) = 8y,

ÆD1f(P ) = −4, D2f(P ) = 8�.hã(4.12)P, 6¿«]�P

−4(x + 2) + 8(y − 1)− (z − 8) = 0,

T4x− 8y + z + 8 = 0�

×��F (x, y, z) = 0à��×`«, F3×FP = (x0, y0, z0)�V�

5F (P ) = (D1F (P ), D2F (P ), D3F (P )),

JÄP�6¿«]�P

5F (P ) · (x− P ) = 0,

542 ÏÜa 9�ÐóCÍ�5��5

Í�x = (x, y, z)�hÇ (4.13)

D1F (P )(x− x0) + D2F (P )(y − y0) + D3F (P )(z − z0) = 0�

»A, uz = f(x, y) ×Ðó%�, J

F (x, y, z) = 0,

Í�F (x, y, z) = f(x, y) − z�JD1F = fx, D2F = fy, D3F = −1,

h`(4.13)PÇW (4.12)P�Æ(4.13) ´(4.12)?×�Ý2P�

êêê ÞÞÞ 9.4

1. �Oì�&Ðó3X�ÝFC]'v = (a, b, c)��0�(i) f(x, y, z) = log

√x2 + y2 + z2, P = (1, 1, 1);

(ii) f(x, y, z) =√

ex2 + ey2 + sin(x + y + z), P = (0, 0, 0)�

2. �Bã�0, Oì�&ó��«Â�(i)√

1002 + 1992 + 2012, (ii) sin 44◦ · cos 31◦�

3. �Oì�&Ðó3ØF�V�, ©�3�F�V�D3�(i) f(x, y) = x2 + y2 sin(xy), (ii) f(x, y) = ex cos y,

(iii) f(x, y, z) = xyz, (iv) f(x, y, z) = x2y3z4,

(v) f(x, y, z) = x2 − y2 + 2z2,

(vi) f(x, y, z) = log(x2 + 2y2 − 3z2)�4. �Oì�&]�PXà��`«, 3X��F�6¿«�

(i) z = x2 − 4y2, Q = (2, 1, 0),

(ii) x = y2 + 9z2, Q = (13,−2, 1),

(iii) x2 + y2 − 4z2 = 4, Q = (2,−2, 1)�5. 'fCg Þ�L3Sî�Þ�óÐó, v5f(P )C5g(P )/D3, ∀P ∈ S��J|ì&B��

9.5 )WÐóC2Ðó��5 543

(1) uf3S� ×ðó, J5f(P ) = (0, 0), ∀P ∈ S�(2) 5(f + g) = 5f +5g�(3) 5(cf) = c5 f , Í�c ×ðó�(4) 5(fg) = f 5 g + g5 f�(5) 3¸g� 0ÝF, 5(f/g) = (g5 f − f 5 g)/g2�

6. 'f(x, y) = 2xy/√

x2 + y2, (x, y) 6= (0, 0), vf(0, 0) = 0��Jfx(0, 0) = fy(0, 0), ¬f�%�3(0, 0)P6¿«�

7. 'f(x, y) = xy(1 + y2)/(x2 + y2), (x, y) 6= (0, 0), f(0, 0) =

0��Jfx(0, 0)Cfy(0, 0)/D3, ¬f3(0, 0)�=��

9.5 )))WWWÐÐÐóóóCCC222ÐÐÐóóó��555E×9�ÐóùbvA��ÐóÝíííÂÂÂ��§§§�

��§§§5.1.'f ×Þ�ó��Ðó, �L3×S ⊂ R2�ê'P ∈S, v ×Þî'�,¸ÿ{P+tv|t ∈ [0, 1]} ⊂ S�JD3×s ∈ (0, 1),

¸ÿ

(5.1) f(P + v)− f(P ) = Dvf(P + sv)�JJJ��.�

g(t) = f(P + tv), t ∈ [0, 1]�ã(3.1)P�pÿÕ

g′(t) = Dvf(P + v)�êã��Ðó�íÂ�§ÿ, D3×s ∈ (0, 1), ¸ÿ

g(1)− g(0) = g′(s) = Dvf(P + sv)��g(1)− g(0) = f(P + v)− f(P )�Æ(5.1)PWñ�

544 ÏÜa 9�ÐóCÍ�5��5

3»3.7&Æ�¼�, b°Þ�óÐóf , Í��)))ÝÝÝÞÞÞ$$$��000óóó(mixed second partial derivative) D12f�D21f�×�8��¬b`Þïº8��»A, '

f(x, y) = sin(xy2)�

JD1f(x, y) = y2 cos(xy2), D2f(x, y) = 2xy cos(xy2),

.h

D12f(x, y) = D21f(x, y) = 2y cos(xy2)− 2xy3 sin(xy2)�

¯@îE£°�� ???(well-behaved)ÝÐó, h��KWñ�

��§§§5.2.'b×Ðóf(x, y), S R2��×�/)�u3S�, D12f

CD21f/D3v=�, J

(5.2) D12f(P ) = D21f(P ), ∀P ∈ S�

3»3.7, 4D12f(0, 0)CD21f(0, 0)

/D3, ¬D12fCD21f3(0, 0)/�=�, .h�§5.2Ýf�¬��3»3.7�, D12f(0, 0)�D21f(0, 0)ô¬�8��¬�§5.2�Y¬�Wñ, Çb��(5.2)PWñ, �D12fCD21f3PF¬&/=��ì»�

»»»5.1.'

f(x, y) =

{x2 sin(1/x)y2 sin(1/y), xy 6= 0,

0, xy = 0�

J|�

D1f(0, y) = 0, ∀y ∈ R,

D2f(x, 0) = 0, ∀x ∈ R�

9.5 )WÐóC2Ðó��5 545

ê

D12f(0, 0) = limk→0

D1f(0, k)−D1f(0, 0)

k= 0,

!§D21f(0, 0) = 0�

ÆD12f(0, 0) = D21f(0, 0)�¬

D12f(x, y) = (2x sin1

x− cos

1

x2)(2y sin

1

y− cos

1

y2), xy 6= 0,

Ælim(x,y)→(0,0) D12f(x, y)�D3�!ñ×è, EÍ», \ïô��J

D12f(0, y) = limk→0

D1f(0, y + k)−D1f(0, 0)

k= 0,

vD12f(x, 0) = 0�

�§5.2�J�&Æº�9.8;, 6¿à×°ny�5Ý��¿à�5ÝJ�, �¢�Apostol (1969) Theorem 8.12�¯@î,

Apostol (1969)Ý�§×å&Æ�§5.2�f��3°�ÇD12fCD21f3S�D3, v3P=�(�à3JÍS �), -�¸(5.2)PWñ�Apostol (1969)ÝTheorem 8.13,ô��¨×¸(5.2)PWñ��5f��ã�§5.2á, ©�?{$Ý�0ó =�, J?{$Ý�0ó-

�øð�55��A

f211 = f121 = f112, f2211 = f2121 = f1221

��×9�Ðófu =�, vÍÏ×$�0óù=�, JfÌ

¿¿¿âââÐÐÐóóó�×��, ufCÍXb�Ïn$��0ó/=�, -Ìf n-¿âÐó�3ÏÞaE��Ðó, &Æ|===ÅÅÅ!!!JJJ¼O)))WWWÐÐÐóóó��

5�E9�ÐóùbETÝ��

546 ÏÜa 9�ÐóCÍ�5��5

��§§§5.3.'F ×�L3�/)S �Þ�óÝ¿¿¿âââÐÐÐóóó, fCg Þ�L3 I��Ðó, ¸ÿP (t) = (f(t), g(t)) ∈ S, ∀t ∈I��

G(t) = F (P (t)), t ∈ I,

J

(5.3) G′(t) = F1(P (t))f ′(t) + F2(P (t))g′(t)�

JJJ��.ã�L

(5.4) G′(t) = limh→0

F (P (t + h))− F (P (t))

h �

êãíÂ�§, D3t1Ct2+yt�t + h , ¸ÿ

f(t + h) = f(t) + hf ′(t1),

g(t + h) = g(t) + hg′(t2)�

�v = (f ′(t1), g′(t2)), JP (t + h) = P (t) + hv, vã�§5.1, D3×s ∈ (0, 1), ¸ÿ

(5.5) F (P (t + h)− F (P (t)) = DhvF (P (t) + shv)�

¨3Í�§Ý�'ì, ã�§4.1á, F3S��.h

DhvF (P (t) + shv) = hDvF (P (t) + shv)(5.6)

= hdF (P (t) + shv, v) = h5 F (P (t) + shv) · v,

Í�Ï×Í�PàÕ(3.3)P, ÏÞÍ�PàÕ(4.7)P, ÏëÍ�PàÕ(4.2)P��)(5.4)�(5.5)C(5.6)P, Çÿ

G′(t) = limh→0

5F (P (t) + shv) · v = 5F (P (t)) · (f ′(t), g′(t)),

h�àÕD1F , D2F , f ′Cg′/ =��¿à(4.1)P-ñÇÿÕ(5.3)P�J±�

9.5 )WÐóC2Ðó��5 547

�§5.3ô�;¶ ì��P: 3ÊÝf�ì, uz = f(x, y),

Í�x = x(t), y = y(t), J

(5.7)dz

dt=

∂f

∂x

dx

dt+

∂f

∂y

dy

dt�

¨², �§5.3��G ×��Ðó, E9�Ý)WÐó, &Æô�b=Å!J�»A, 'b×ÐóF (x, y, z), vx = x(s, t), y =

y(s, t), z = z(s, t), ê�

G(s, t) = F (x(s, t), y(s, t), z(s, t)),

J

∂G

∂s=

∂F

∂x

∂x

∂s+

∂F

∂y

∂y

∂s+

∂F

∂z

∂z

∂s,

∂G

∂t=

∂F

∂x

∂x

∂t+

∂F

∂y

∂y

∂t+

∂F

∂z

∂z

∂t,

©�î�9°0ó3×ÊÝ/)�D3v=��îÞP�J�tÝ�âëÍ�ó², ÍõI5��§5.3��8!�?×�2, ub×n�ÐóF (x1, x2, · · · , xn), v

xi = xi(t1, · · · , tm), i = 1, · · · , n,

ê�G(t1, · · · , tm) = F (x1(t1, · · · , tm), · · · , xn(t1, · · · , tm)), J

(5.8)∂G

∂tj=

n∑i=1

∂G

∂xi

∂xi

∂tj, j = 1, · · · , m�

QXb×$�0ó, )��'D3v=��

»»»5.2.'F (x, y) = xy2 +x3 +y, x = f(t) = t2−1, y = g(t) = 2t− t3,

ê�G(t) = F (f(t), g(t)), J

G′(t) =∂F

∂x

dx

dt+

∂F

∂y

dy

dt= (y3 + 3x2)(2t) + (2xy + 1)(2− 3t2)

= ((2t− t3)2 + 3(t2 − 1)2)2t + (2(t2 − 1)(2t− t3) + 1)(2− 3t2)�

548 ÏÜa 9�ÐóCÍ�5��5

»»»5.3.'z = f(x, y) = x2+y2, x = g(t) = cos t, y = h(t) = sin t�J

dz

dt= 2x(− sin t) + 2y cos t = 0�

Qu¥�Õz = 1, ô�ñÇÿÕdz/dt = 0�

»»»5.4.'z = x2 + y2, x = r cos θ, y = r sin θ�J

∂z

∂r=

∂z

∂x

∂x

∂r+

∂z

∂y

∂y

∂r= 2x cos θ + 2y sin θ = 2r,

∂z

∂θ=

∂z

∂x

∂x

∂θ+

∂z

∂y

∂y

∂θ= 2x(−r sin θ) + 2y(r cos θ) = 0�

¯@îz = x2 + y2 = r2, Æ∂z/∂r = 2r, ∂z/∂θ = 0�

»»»5.5.'b×Ðóf(x, y), x = r cos θ, y = r sin θ��

φ(r, θ) = f(r cos θ, r sin θ)�

�'Xm�f�/Wñ, J.

∂x

∂r= cos θ,

∂y

∂r= sin θ,

∂x

∂θ= −r sin θ,

∂y

∂θ= r cos θ,

Æ

∂φ

∂r=

∂f

∂xcos θ +

∂f

∂ysin θ,

∂φ

∂θ= −r

∂f

∂xsin θ + r

∂f

∂ycos θ�

Íg¼:Þ$�0ó�

∂2φ

∂θ2=

∂

∂θ(∂φ

∂θ) =

∂

∂θ(−r

∂f

∂xsin θ + r

∂f

∂ycos θ)(5.9)

= −r cos θ∂f

∂x− r sin θ

∂

∂θ(∂f

∂x)− r sin θ

∂f

∂y+ r sin θ

∂

∂θ(∂f

∂y)�

9.5 )WÐóC2Ðó��5 549

ê

∂

∂θ(∂f

∂x) =

∂

∂x(∂f

∂x)∂x

∂θ+

∂

∂y(∂f

∂x)∂y

∂θ(5.10)

=∂2f

∂x2(−r sin θ) +

∂2f

∂y∂x(r cos θ),

v

∂

∂θ(∂f

∂y) =

∂

∂x(∂f

∂y)∂x

∂θ+

∂

∂y(∂f

∂y)∂y

∂θ(5.11)

=∂2f

∂x∂y(−r sin θ) +

∂2f

∂y2(r cos θ)�

Þ(5.10)C(5.11)P�á(5.9)Pÿ

∂2φ

∂θ2= −r cos θ

∂f

∂x+ r2 sin2 θ

∂2f

∂x2− r2 sin θ cos θ

∂2f

∂y∂x

−r sin θ∂f

∂y− r sin θ cos θ

∂2f

∂x∂y+ r2 cos2 θ

∂2f

∂y2�

!§�ÿ

∂2φ

∂r2= cos2 θ

∂2f

∂x2+ cos θ sin θ(

∂2f

∂x∂y+

∂2f

∂y∂x) + sin2 θ

∂2f

∂y2,

∂2φ

∂r∂θ= −r cos θ sin θ

∂2f

∂x2+ r cos2 θ

∂2f

∂x∂y− r sin2 θ

∂2f

∂y∂x

+r cos θ sin θ∂2f

∂y2− sin θ

∂f

∂x+ cos θ

∂f

∂y�

Íg&Æ¼:222ÐÐÐóóó��555�'F (x, y) ×Þ�ó�¿âÐó, vf ×��Ðó, ¸ÿEN

×òyf��L½�Ýx,

F (x, f(x)) = 0�

¿à=Å!Jÿ

0 =dF

dx=

∂F

∂x

dx

dx+

∂F

∂y

dy

dx�

550 ÏÜa 9�ÐóCÍ�5��5

ãîP�O�dy/dx

dy

dx= −∂F/∂x

∂F/∂y,

îPEN×3f��L½�ÝxvF2(x, f(x)) 6= 0Wñ�

»»»5.6.'F (x, y) = x3 + y3 − 6xy = 0, �Ody/dx��.uï|G�§��Ðó��5, �ÿ

3x2 + 3y2 dy

dx− 6y − 6x

dy

dx= 0,

ãhÿ

(3x2 − 6y) + (3y2 − 6x)dy

dx= 0,

.hdy

dx= −3x2 − 6y

3y2 − 6x= −x2 − 2y

y2 − 2x�u|êGÝ]P, .

∂F

∂x= 3x2 − 6y,

∂F

∂y= 3y2 − 6x,

Ædy

dx= −∂F/∂x

∂F/∂y=

3x2 − 6y

3y2 − 6x,

�n8!�

'F (x, y, z) = 0,

ãîP��L�×Ðó

z = f(x, y)�

'F�f/ ¿âÐó�.F (x, y, f(x, y)) = 0, ã(5.8)Pÿ

0 =∂

∂xF (x, y, f(x, y)) =

∂F

∂x

∂x

∂x+

∂F

∂y

∂y

∂x+

∂F

∂z

∂z

∂x�

êÞ 551

ê∂x/∂x = 1, ∂y/∂x = 0, .x, y Ú Þ}ñÝ�ó�.h

∂z

∂x= −∂F/∂x

∂F/∂z�

!§�ÿ∂z

∂y= −∂F/∂y

∂F/∂z�

»»»5.7.'�xy2 + yz2 + z3 +x3−4 = 0,�L�z ×x, y�Ðó��O∂z/∂xC∂z/∂y��.�F (x, y, z) = xy2 + yz2 + z3 + x3 − 4, J

∂F

∂x= y2 + 3x2,

∂F

∂y= 2xy + z2,

∂F

∂z= 2yz + 3z2�

.h∂z

∂x= − y2 + 3x2

2yz + 3z2,

∂z

∂y= − 2xy + z2

2yz + 3z2�

»»»5.8.ãÞ]�P

2x = v2 − u2, y = uv,

��L�u, v xCy�Ðó��O∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y��.�ü�y, �ÞX��ÞP¼��5½Ex�5, ÿ

2 = 2v∂v/∂x− 2u∂u/∂x, 0 = u∂v/∂x + v∂u/∂x�

ãîÞP, Ç��∂u/∂xC∂v/∂x:

∂u

∂x= − u

u2 + v2,

∂v

∂x= − v

u2 + v2�

¨×]«, u�ü�x, Jã!øÝM»�ÿ

∂u

∂y=

v

u2 + v2,

∂v

∂y= − u

u2 + v2�

552 ÏÜa 9�ÐóCÍ�5��5

êêê ÞÞÞ 9.5

1. 3ì�&Þ, �Odw/dt�(1) w = x2 + y2, x = (t− 1)/t, y = t/(t + 1)�(2) w = t sin(xy), x = log t, y = t3�

2. 3ì�&Þ, �O∂z/∂uC∂z/∂v�(1) z = x2 + y2, x = u cos v, y = u sin v�(2) z = arcsin(xy), x = u + v, y = u− v�(3) z = ex/y, x = 2u− v, y = u + 2v�

3. 3ì�&Þ, �O∂w/∂r, ∂w/∂sC∂w/∂t�(1) w = (x+y)/z, x = r−2s+t, y = 2r+s−3t, z = r2+s2+t2�(2) w = xy + yz + zx, x = r cos s, y = r sin t, z = st�

4. 3ì�&Þ, �O∂w/∂uC∂w/∂v�(1) w =

√x2 + y2 + z2, x = u sin v, y = u cos v, z = uv�

(2) w = (x2 + y2)/(y2 + z2), x = uev, y = veu, z = u−1�

5. 'z = F (x, y), x = f(u, v), y = g(u, v)�' ∂2z∂x∂y

= ∂2z∂y∂x��

J

∂2z

∂u2=

∂2z

∂x2(∂x

∂u)2 + 2

∂2z

∂x∂y

∂x

∂u

∂y

∂u+

∂2z

∂y2(∂y

∂u)2 +

∂z

∂x

∂2x

∂u2+

∂z

∂y

∂2y

∂u2�

6. 'z = F (x, y), x = f(u, v), y = g(u, v)��O∂2z/(∂v∂u)�

7. 'z = x + f(u), u = xy��J

x∂z

∂x− y

∂z

∂y= x�

êÞ 553

8. 'z = f(u/v)/v��J

v∂z

∂v+ u

∂z

∂u+ z = 0�

(èî: �x = u/v, y = 1/v, Jz = yf(x))

9. 'z = f(u2 + v2)��J

u∂z

∂v− v

∂z

∂u= 0�

(èî: �x = u2 + v2)

10. 'z = f(x, y), x = r cos θ, y = r sin θ�' ∂2z∂x∂y

= ∂2z∂y∂x��J

(i) (∂z∂r

)2 + 1r2 (

∂z∂θ

)2 = ( ∂z∂x

)2 + (∂z∂y

)2;

(ii) ∂2z∂r2 + 1

r2∂2z∂θ2 = ∂2z

∂x2 + ∂2z∂y2�

11. 'F (x, y) = f(y + ax) + g(y − ax), Í�a ×ðó��J

∂2F

∂x2= a2∂2F

∂y2�

12. 'F (x, y) =√

x2 − y2 arcsin(y/x)��OxF1(x, y)+yF2(x, y)�

13. 'F (x, y) = x log y/y��J

2x2F11(x, y) + 2xyF12(x, y) + y2F22(x, y) = 0�

14. 'Ðóf(x, y)��L½ S, v��E∀t ∈ R, ©�(x, y)C(tx, ty) ∈ S, -b

f(tx, ty) = tnf(x, y),

JÌf nggg��PPP(homogeneous of degree n)��JEN×2-¿âÝn g�PÐóf ,

(i) xf1(x, y) + yf2(x, y) = nf(x, y);

(ii) x2f11(x, y)+2xyf12(x, y)+ y2f22(x, y) = n(n−1)f(x, y)�

554 ÏÜa 9�ÐóCÍ�5��5

15. 'F (x, y) = arctan(y/x)��JF��Laplace]�P, Ç

F11 + F22 = 0�

16. 'F (x1, x2, · · · , xn) = (x21 + x2

2 + · · · + x2n)1−n/2��JF�

�Laplace]�P, Ç

F11 + F22 + · · ·+ Fnn = 0�

17. 'F (x, y)��Laplace]�P�E×ðóa, �

G(x, y) = F

(ax

x2 + y2,

ay

x2 + y2

)�

�JGù��Laplace]�P�

9.6 aaa��555

|G&Æà«�ÝÃF, ¼�Õ×Ðó3× ��5�!ø2, &Æô�¿àÎ§î��(work)ÝÃF, ¼�L×'�ÂÐó3×¿âàîÝ�5�9Ë�5Ì aaa��555(line integral)�&Æ;ð¢ÞîÝà¼1�, ¬Eëî#��n îÝ�µ, Í²î¬P-²�'F ×ðóÝæææ(force), ®à3×º'�vÉ��Ô�î�J

ãÎ§�ÝáIá, hæX�Ý� F · v = ||F || · ||v|| cos θ, Í�θ

�F�v �ô��¨'λ R2�×¿âà, Í¢óP

λ(t) = (x(t), y(t)), t ∈ [a, b]�

uæ��Hbn, Ç'3λî×F(x, y)�æ F (x, y)�v'F�Þ5� A(x, y)CB(x, y), Ç

F (x, y) = (A(x, y), B(x, y)),

9.6 a�5 555

×x, y�'�ÂÐó�'A(x, y)CB(x, y)/=�, &Æ�OF®à3`aλîÝ��'p = {t0, t1, · · · , tn} [a, b]�×5v�Fλ(ti) = (x(ti), y(ti)),

i = 0, 1, · · · , n, Þ`aλ5Wnð�=(subarc)�Ei = 1, 2, · · · , n,

�

∆ti = ti − ti−1, ∆xi = x(ti)− x(ti−1), ∆yi = y(ti)− y(ti−1),

v|∆ri�=#λ(ti−1)Cλ(ti)�'�, �%6.1�

λ(t0)

λ(t1)

λ(ti−1)

λ(ti)

λ(tn−1)

λ(tn)

∆yi∆ri

∆xi

%6.1.

E∀i = 1, 2, · · · , n, ãíÂ�§, 5½D3ciCdi ∈ (ti−1, ti), ¸ÿ

∆xi = x′(ci)∆ti, ∆yi = y′(di)∆ti��Pi = λ(ci), Qi = λ(di), i = 1, 2, · · · , n�JE∀i, ðó'�F i =

(A(Pi), B(Qi))�à¼� F (x, y)º½�=ãλ(ti−1)�λ(ti)��«Â(4Pi�Qi� � Þ 8 ² F, ' �F i) º # �F (x, y))�êãλ(ti−1) �λ(ti)��=�|∆ri¼¿��Jðó'�F i ®à3×º½∆riÉ��Ô�, XÿÝ�

F i ·∆ri = A(Pi)∆xi + B(Qi)∆yi

= A(Pi)x′(ci)∆ti + B(Qi)y

′(di)∆ti�Æ&Æ�àõ

(6.1)n∑

i=1

A(Pi)x′(ci)∆ti +

n∑i=1

B(Qi)y′(di)∆ti

556 ÏÜa 9�ÐóCÍ�5��5

� F (x, y)º½λX��«Â�ên∑

i=1

A(Pi)x′(ci)∆ti =

n∑i=1

A(x(ci), y(ci))x′(ci)∆ti

ÐóA(x(t), y(t))x′(t)3 [a, b]�×Riemannõ�!§(6.1)P�ÏÞÍõ ÐóB(x(t), y(t))y′(t), 3 [a, b]�×Riemannõ�Æ�n →∞, (6.1)P��Þ4õ��

∫ b

a

(A(x(t), y(t))x′(t) + B(x(t), y(t))y′(t))dt(6.2)

=

∫ b

a

F (x(t), y(t)) · λ′(t)dt�

b`&Æô�bì�¶°

(6.3)

∫ b

a

F (x(t), y(t)) · λ′(t)dt =

∫ b

a

F (x(t), y(t)) · dλ(t)�

ã|îD¡, 9ìÝ�L-®ßÝ�

��LLL6.1.'λ ×�L3[a, b]�¿â`a, 3N×λîÝF(x, y)b×æF (x, y), J×Ô�º½λ Xÿ�À�

∫ b

a

F (x(t), y(t)) · λ′(t)dt�

�î��5ôÌ F3λîÝa�5�

uF (x, y) = (A(x, y), B(x, y))v¿àdx = x′(t)dt, dy = y′(t)dt,

J&Æ�b×´��Ýa�5Ðr, Ç∫

λ(Adx + Bdy),

Í�ACBQÄ63[a, b]��uλ ëî�×¿â`a, vb×ëîÝ'�ÂÐóF (x, y, z),

�L3λîN×F(x, y, z)�u

F (x, y, z) = (A(x, y, z), B(x, y, z), C(x, y, z)),

9.6 a�5 557

JETÝa�5�¶W∫

λ(Adx + Bdy + Cdz)�

3�9@jTà�, λ©�' @@@ððð¿¿¿âââ(piecewise smooth),

Çλ(t)b×b&Ý0óλ′(t), tÝ��3×°b§ÝF², λ′(t)/=��

»»»6.1.�O∫λ(x2ydx + y3dy), Í�λ ã(0, 0)�(1, 1)ÝeÎay =

x2Ý=��.´�λ(t)�¶Wλ(t) = (t, t2), t ∈ [0, 1]�Jdx = dt, dy = 2tdt,

v∫

λ(x2ydx + y3dy) =

∫ 1

0

(t2 · t2 + t6(2t))dt

=

∫ 1

0

(t4 + 2t7)dt =9

20�

Qλ(t)ô�¶Wλ(t) = (√

t, t), t ∈ [0, 1]�Jdx = (2√

t)−1dt,

dy = dt, v∫

λ(x2ydx + y3dy) =

∫ 1

0

(t2

2√

t+ t3)dt =

9

20,

)ÿÕ8!Ý�n�×a�5, �.Í=λ�¢óP�î°�!,

�b�!Ý�n��ÄÇ¸Î!×=, u�Ý]'8D, Ja�5ÝÂª?-×�r�AÍ»�, uλ ã(1, 1)�(0, 0)Ý=, Ía�5 ∫ 0

1

(t4 + 2t7)dt = − 9

20�

»»»6.2.�O ∫

λ(ydx + zdy + xdz),

Í�λ(t) = (cos t, sin t, t), tã0�2π�

558 ÏÜa 9�ÐóCÍ�5��5

��.

æP =

∫ 2π

0

(− sin2 t + t cos t + cos t)dt = −π�

êêê ÞÞÞ 9.6

1. �Oì�&a�5�(1) F (x, y) = (x2−2xy, y2−2xy),º½y = x2ã(−1, 1)�(1, 1)�(2) F (x, y) = (2a−y, x),º½λ(t) = (a(t− sin t), a(1−cos t)),

t ∈ [0, 1]�(3) F (x, y, z) = (y2 − z2, 2yz,−x2), º½λ(t) = (t, t2, t3), t ∈[0, 1]�(4) F (x, y) = (x2 + y2, x2 − y2), º½y = 1− |1− x|ã(0, 0)�(2, 0)�(5) F (x, y) = (x + y, x− y), ��Yib2x2 + a2y2 = a2b2, Y`j]'�(6) F (x, y, z) = (2xy, x2 + z, y), ã(1, 0, 2)�(3, 4, 1)º½×àa�(7) F (x, y, z) = (x, y, xz − y), ã(0, 0, 0)�(1, 2, 4)º½×àa�(8) F (x, y, z) = (x, y, xz − y), º½λ(t) = (t2, 2t, 4t3), t ∈[0, 1]�

2. �Oì�&a�5�(1)

∫λ(x2−2xy)dx+(y2−2xy)dy,Í�λ º½y = x2ã(−2, 4)

�(1, 1)�`a�(2)

∫λ

(x+y)dx−(x−y)dyx2+y2 , Í�λ Y`j]'�iøx2 + y2 =

a2�(3)

∫λ

dx+dy|x|+|y| ,Í�λ Y`j]'�cF (1, 0), (0, 1), (−1, 0)

C(0,−1)�Ñ]�ø��

9.7 ÁÂ 559

9.7 ÁÁÁÂÂÂ

��Ðó�ÁÁÁÂÂÂÝÃF, ô�.ÂÕ9�Ðó�'b×Ðóf(x, y), S ×�âyf��L½Ý/)�E×P ∈

S, uf(P ) ≥ f(Q), ∀Q ∈ S,

JÌfyS�, 3Pb�EÁ�Âf(P )�uD3×�¦B(P ; r), ¸ÿ

f(P ) > f(Q), ∀Q ∈ B(P ; r), Q 6= P,

JÌf3Pb×8EÁ��!§��L�EÁ�Â, C8EÁ�Â�8EÁ�ÂC8EÁ�ÂÙÌ8EÁÂ�'f(x, y)3¿«î×�TÝ ½(AÎ�T×i8)=�, Jf3

h ½�b�EÁ�C�EÁ��h��J�v«��óÝ�µ(�Ï×a�§6.6), 3h¯Ä�9ì&Æ¼:A¢´0×Þ�óÐóÝ8EÁÂ� Ý�-, E8EÁÂ, &Æð6¯“8E”ÞC,

�©ÌÁÂ(8EÁ�ÂT8EÁ�Âô×ø�§)�´�uf(a, b) f�×Á�Â, Jf�%�3¿«îx = aCy = b

�^½, /|(a, b, f(a, b)) Á�F��uf(a, b) f �×Á�Â,

J3G�Þ^û, (a, b, f(a, b))- Á�F�Æuf 3(a, b)b×8EÁÂ, vf3(a, b)�×$�0ó/D3, JÄb

f1(a, b) = 0, f2(a, b) = 0�

.h, A&ÆXï]Ý, 3×ÁÂsß�Ý6¿«¿�x-y¿«(�(4.12)6¿«�]�P)�ÆkOÐóf�ÁÂ, ��!`��ìÞ]�P�x, y:

f1(x, y) = 0, f2(x, y) = 0,

�l�N×à�ÎÍ Á�ÂTÁ�Â�Q, A!��óÝ�µ, ÁÂôb��sß3\&F, T�0ó�D3��Ä�I5&ÆÂÕÝ»��, &Æ¬��Ê9Ë��

»»»7.1.'f(x, y) = 4− x2 − y2, �Of�ÁÂ�

560 ÏÜa 9�ÐóCÍ�5��5

��.´�f1(x, y) = −2x, f2(x, y) = −2y�

�f1(x, y) = f2(x, y) = 0©b°×�x = y = 0�Æ(0, 0) °×��ÁÂsß��.

f(x, y) = 4− (x2 + y2) < 4 = f(0, 0), ∀(x, y) 6= (0, 0),

Æf(0, 0) = 4 f�Á�Â�

»»»7.2.�f(x, y) = 4 + x2 − y2, �Of�ÁÂ��.´�

f1(x, y) = 2x, f2(x, y) = −2y��x = y = 0) °×��ÁÂsß��3y = 0�^½ z =

4 + x2, Æ3(0, 0, 4)bÁ�Â�3x = 0�^½ z = 4 − y2, Æ3(0, 0, 4) bÁ�Â�.h3(0, 0, 4)ÉPÁ�ùPÁ��3hu3Ø×FP , f1(P ) = f2(P ) = 0, v3P�N×�¦�, Ä�0ÕQ1, Q2ÞF, ¸ÿf(Q1) > f(P ), vf(Q2) < f(P ), -ÌP ×ëëëFFF(saddle point)�.f(x, 0) > f(0, 0), ∀x 6= 0, f(0, y) < f(0, 0),

∀y 6= 0, Æ(0, 0) f�×ëF��yuf(x, y) = x3 − 3xy2, (0, 0)ù ×ëF�J�, º�\ï

��Yê�E´�ÓÝ»�, ã�×$�0ó 0�]�P��ÝF, ¬�

|X�ÎÍ ÁÂ, h�µE��óôÎv«�.h&ÆôÞ�×|Þ$0ó¼l�ÁÂÝ]°�'f1(a, b) = f2(a, b) = 0, vÞ$�0óf11(a, b)Cf22(a, b)/D

3v� 0�uf3(a, b)bÁ�, Jf�%�3¿«x = a Cy = b�^û, Ä/ ì�, v

f11(a, b) < 0, f22(a, b) < 0�

!§, uf3(a, b)bÁ�, JÄb

f11(a, b) > 0, f22(a, b) > 0�

9.7 ÁÂ 561

�uf11(a, b)�f22(a, b)Ðr8D, Jf3(a, b)PÁÂ�&ÆB�h��Aì, hÇË�ó�Þ$0óÁÂ¾�°�

��§§§7.1.'f ×Þ�ó�¿âÐó, �L3R2�×�/)S�v�

(7.1) F (Q) = f11(Q)f22(Q)− f 212(Q),

×�L½ S�Ðó�'P ∈ Sv��

f1(P ) = f2(P ) = 0�

J(i) uF (P ) > 0vf11(P ) < 0, Jf3PbÁ�;

(ii) uF (P ) > 0vf11(P ) > 0, Jf3PbÁ�;

(iii) uF (P ) < 0, Jf3PPÁÂ, P ×ëF;

(iv) uF (P ) = 0, Jh°´[�

uF (P ) > 0, Jf11(P )f22(P )Ä Ñ, .hf11(P )�f22(P )6!r�Æ�§7.1�(i)C(ii)�Ýf11(P )/�|f22(P )ã��3J��§7.1�G, &Æ�:¿Í»��

»»»7.3.�f(x, y) = x3 − 12xy + 8y3,

�Of�ÁÂ��.´�

f1(x, y) = 3x2 − 12y, f2(x, y) = −12x + 24y2,

f11(x, y) = 6x, f12(x, y) = −12, f22(x, y) = 48y�

�f1(x, y) = f2(x, y) = 0,��(x, y) = (0, 0)T(2, 1)�ãhÿF (0, 0)

= −144, F (2, 1) = 432�.F (0, 0) < 0, Æ(0, 0) f�ëF�ê.f11(2, 1) > 0, f(2, 1) = −8, Æf3(2, 1)bÁ�Â−8�

562 ÏÜa 9�ÐóCÍ�5��5

»»»7.4.�f(x, y) = x2 − y2, �Of�ÁÂ��.´�

f1(x, y) = 2x, f2(x, y) = −2y,

f11(x, y) = 2, f12(x, y) = 0, f22(x, y) = −2�

�f1(x, y) = f2(x, y) = 0, ��(x, y) = (0, 0)�.F (0, 0) < 0,

Æ(0, 0) f�ëF�

»»»7.5.�f(x, y) = x2 − 2xy2 + y4 − y5, Of�ÁÂ��.´�

f1(x, y) = 2x− 2y2, f2(x, y) = −4xy + 4y3 − 5y4,

f11(x, y) = 2, f12(x, y) = −4y, f22(x, y) = −4x + 12y2 − 20y3�

�f1(x, y) = f2(x, y) = 0, ��(x, y) = (0, 0)�.f11(0, 0) = 2,

f12(0, 0) = f22(0, 0) = 0, ÆF (0, 0) = 0�.hã�§7.1, ¬P°X�f3(0, 0)ÎÍbÁÂ�¬.f(0, 0) = 0, vux = y2, Jf(x, y) = −y5 > 0y < 0,

f(x, y) < 0, y > 0, Æf3(0, 0)PÁÂ, v(0, 0) f�×ëF�

»»»7.6.�f(x, y) = x2 + y2 + y3, �Of�ÁÂ��.´�

f1(x, y) = 2x, f2(x, y) = 2y + 3y2,

f11(x, y) = 2, f12(x, y) = 0, f22(x, y) = 2 + 6y�

ãf1(x, y)= f2(x, y)= 0, ��(x, y) = (0, 0), T(x, y) = (0,−2/3)�¨.F (0, 0) = 4 > 0, vf11(0, 0) = 2 > 0, Æf3(0, 0)bÁ��.F (0,−2/3) = −4 < 0, Æ(0, 0) f�×ëF�

»»»7.7.'�]��ë\��õü�, �X�&\�¸Í��t��

9.7 ÁÂ 563

��.'ë\�5½ x, y, z�ã�'x + y + z = a, vx, y, z > 0, Í�a > 0 ×ðó�'�� V , J

V = xyz = xy(a− x− y)��

∂V

∂x= ay − 2xy − y2 = 0,

∂V

∂y= ax− x2 − 2xy = 0,

��x = y = 0Tx = y = a/3�ê∂2V

∂x2= −2y,

∂2V

∂y2= −2x,

∂2V

∂y∂x= a− 2x− 2y�

.h

∂2V

∂x2(a

3,a

3) =

∂2V

∂y2(a

3,a

3) = −2

3a < 0,

∂2V

∂y∂x(a

3,a

3) = −a

3,

v

F (a

3,a

3) =

4a2

9− a2

9=

a2

3> 0�

Æx = y = z = a/3`, ��bÁ��ê�Qx = y = 0&Á�Â�

��§§§7.1��JJJ��(i) ã�'f11(P ) < 0, F (P ) > 0, vf�Þ$�0ó3S�/=

��ÆD3×ε > 0, ¸ÿE∀Q ∈ B(P ; ε), f11(Q) < 0 vF (Q) >

0�&ÆÞJ�E∀Q ∈ B(P ; ε)C&ëÞî'�v,

D2vf(Q) = Dv(Dvf)(Q) < 0�

ãh�ÿDvf3B(P ; ε)�,º½N×'�v/ �}�3,.hf(P )

f3B(P ; ε)�º½N×'�v�Á�Â,.hf(P ) f3B(P ; ε)��Á�Â�

564 ÏÜa 9�ÐóCÍ�5��5

¨�v = (r, s), r2 + s2 6= 0�J

Dvf(Q) = 5f(Q) · v = rf1(Q) + sf2(Q),

v

D2vf(Q) = Dv(Dvf)(Q) = 5Dvf(Q) · v

= (rf11 + sf21, rf12 + sf22) · v�Í�N×Þ$�0ó/��áQF, |ì!�.3Í�§Ý�'ì,

f12 = f21, Æ

D2vf(Q) = r2f11 + 2rsf12 + s2f22

= f11(r +f12

f11

s)2 +s2

f11

F (Q)�

.f11(Q) < 0, F (Q) > 0,vr�s�! 0, Æ

D2vf(Q) < 0, ∀Q ∈ B(P ; ε)�

.h(i)Wñ�!§�J(ii)��y(iii)C(iv)�J�º3êÞ��

Íg&Æ¼:bbb§§§×××ÝÝÝÁÁÁÂÂÂ(extrema with constrains)Ý®Þ,

¬+ÛLagrange¶¶¶óóó°°°(Method of Langrange’s Multipliers)�&Æ�:9ìËÍ»��Ï×» , ��×�;ÄæF�`«S, 0�Sît#�æFï�ÏÞ» ,�f(x, y, z)�3F(x, y, z)�á�,

��×ëîè �ÝàC, O3hàîá�t{Ct±F�îÞ»Ý×��P : X�×9�Ðóf(x1, · · · , xn)�ÁÂ, Í

�(x1, · · · , xn) §×3f��L½Ý×�/�3Ï×»�, Ç�OÐóf(x, y, z)= (x2+y2 + z2)1/2�Á�Â, Í�(x, y, z)§×3Ø×��Ý`«Sî�3ÏÞ»�, X§×Ý/) ×à�9vb§×ÝÁÂ®Þ, ×�¼1¬��|�, h.¬P×;J

��Xb®Þ�å§×Ý/)bÁ��Ý�x, »A, GÞ»�Ý×`«T×à, Jb©�Ý]°Êà�9µÎXÛLagrange

9.7 ÁÂ 565

¶ó°�&Æ�à�h]°Ý×��P, Q¡�|¿¢ÝD¡, ¼1�h°�ÿ;�

Lagrange¶ó°: 'b×Ðóf(x1, · · · , xn), J3

(7.2) g1(x1, · · · , xn) = 0, · · · , gm(x1, · · · , xn) = 0,

Í�m < n, �§×ì, ¸fb8EÁÂÝF6��

(7.3) 5f = λ1 5 g1 + · · ·+ λm 5 gm,

Í�λ1, · · · , λm ðó�kX�ÁÂ, &Æ�Êã(7.2)P�ÝmÍ§×]�P, C(7.3)P

�ÝnÍ]�P, À�m + nÍ]�P�'°��m + n ÍÎáóx1, · · · , xnCλ1, · · · , λm�Í��Ýx1, · · · , xn-b��Îf�ÁÂ�3�ÝÄ��, &Æ¢Ãλ1, · · · , λm, hmÍóÂ�Ì Lagrange¶ó�N×Í§×Pím�×Í¶ó�Ðóf , g1, · · · , gm

í' ��h°©b§×PÝóêm�y�óÍón, v¬&Xb§×Ðóg1, · · · , gm, Ex1, · · · , xn�Ý��mÍ3ÁÂ�ÝJacobian(Jacobian��L�10.5;)/ 0 �b[�h°�b[PÝJ�, {��5�×¥�Ý��,�\ï¢�Apostol (1974)

Section 13.7�9ì&Æ|¿¢ÝD¡, ¼1�h° ¢Wñ�3G�Ï×»�, &ÆkX�3×��Ý`«S�, t#�æF

ï��ëîè �×F(x, y, z)ûæF�ûÒ r, uv°u¸a3�5 rÝ¦î, Ç

x2 + y2 + z2 = r2�3h, E×��Ýc,

L(c) = {(x1, · · · , xn)|f(x1, · · · , xn) = c}

Ì Ðóf�×��«««(level surface)�¦«x2 + y2 + z2 = r2, Ç Ðóf(x, y, z) = (x2 + y2 + z2)1/2�×��«�&Æ-Î�Of�Á�Â�uãr = 0�s, @�¯r¦�, àÕ��«Ï×g#ÇÕX�

Ý`«S, JN×#ÇF, / Sît#�æFï�kX�£°#

566 ÏÜa 9�ÐóCÍ�5��5

ÇFÝ2ý, &Æ�'S ã×°]�Pg(x, y, z) = 0Xà��u3N×#ÇFS/b6¿«, h6¿«ùÄ #ÇFÝ��«Ý6¿«�.h, 3×#ÇF, `«g(x, y, z) = 0�V�'�5g(x, y, z),

��«f(x, y, z) = r�V�'�Ä6¿��.hD3×ðóλ, ¸ÿ3N×#ÇF

5f(x, y, z) = λ5 g(x, y, z)�

hÇb×§×PìÝLagrange]°�Ý(7.3)P�#½¼:G�ÝÏÞ»�&Æ�3×��ÝàCî, 0á�

Ðóf(x, y, z)�ÁÂ�uÞàCÚ Þ`«

g1(x, y, z) = 0 C g2(x, y, z) = 0

�ø/, J&Æb×3Þ§×PìÝÁÂ®Þ�ÞV�'�5g1C5g25½ î�Þ`«�°'�, Æù ø/àC �°'��}¡&ÆÞJ�, á�Ðóf�V�'�5f , 3ÁÂ�ù C�°'��Æ5f�5g1C5g23!×¿«î�Æu5g1C5g2 Þ}ñ'�, J5f ��î 5g1C5g2�aPà), Ç

5f = λ1 5 g1 + λ2 5 g2�

hÇbÞ§×PìÝ(7.3)P�kJ5f3ÁÂ� C�°'�, 'C ×'�ÂÐóα(t) X

à�, Í�t ∈ [a, b]�3àCî, á� ×t�Ðó, Aφ(t) =

f(α(t))�uφ3[a, b]�Ø×/Ft1b×ÁÂ, Jφ′(t1) = 0�¨×]«, =Å!Jê��

φ′(t) = 5f(α(t)) ·α′(t)�

h/�3t = t1 0, .h5fkàα′(t1)�¬α′(t1)�C86, Æ5f(α(t1))a3kàC�¿«î�¨ÞV�'�5g1C5g2 }ñ, uv°uÍ²�

(∂(g1, g2)

∂(y, z),∂(g1, g2)

∂(z, x),∂(g1, g2)

∂(x, y)

)6= (0, 0, 0)�

9.7 ÁÂ 567

.h5g1�5g2}ñ¬&�XbîP¼�'��ëJacobian/ 0��G&Æ-�¼�, Lagrange]°©bhf�Wñ�Êà�u5g1�5g28µ, Jh°´[�»A, O

f(x, y, z) = x2 + y2

�ÁÂ, §×f�

g1(x, y, z) = z = 0, g2(x, y, z) = z2 − (y − 1)3 = 0,

hÞ`«�ø/ {(x, 1, 0)|x ∈ R} ×àa��Qx = 0`, �¸f(x, y, z)t�, ÇÁ�Âsß3(0, 1, 0)�¬3hF

5g1 = (0, 0, 1), 5g2 = (0, 0, 0),

v5f = (0, 2, 0),

¬P°0Õðóλ1Cλ2, ¸ÿ

5f = λg1 + λg2�

»»»7.8.'×=ßºãëI5àW, � I5 ×iÖ, G¡ Þ8!ÝÑi�, i�Ý{�iÖÝ{��E×��Ý�«�, �O¯h=ßºbt��M��.�r�iÖÝ�5, h�{�X��f�

(7.4) 2πrh + 2πr√

h2 + r2 = C,

Í�C ×ðó�3h&ÆàÕ×9�5 r, { h�Ñi�Ý�«� πr

√h2 + r2 (�Ï0a»3.6)�h=ßº��

(7.5) V = πr2h +2

3πr2h =

5

3πr2h�

&Æ-Î�3(7.4)P�§×ì, OV�Á�Â��

5f = λ5 g,

568 ÏÜa 9�ÐóCÍ�5��5

Í�

f(r, h) =5

3πr2h,

g(r, h) = 2πrh + 2πr√

h2 + r2 − C�ÿ

10

3πrh− λ(2πh + 2π

√h2 + r2 + 2πr

r√h2 + r2

) = 0,(7.6)

5

3πr2 − λ(2πr + 2πr

h√h2 + r2

) = 0�(7.7)

ã(7.4)�(7.6)C(7.7)P��Á�Âsß3

r = r0 =4

√C2

20π2, h = h0 =

2√5r0�

Qô��ã(7.4)C(7.5)P��óh, �ÿ

V =5r

12C(C2 − 4π2r4), 0 < r <

4

√C2

4π2�h ×r��Ðó, )��3r0bÁ�Â�

êêê ÞÞÞ 9.7

1. �Oì�&Ðó�ÁÂCëF�(1) f(x, y) = x2 + (y − 1)2� (2) f(x, y) = x2 − (y − 1)2�(3) f(x, y) = (x− y + 1)2� (4) f(x, y) = x3 − 3xy2 + y3�(5) f(x, y) = sin x cosh y� (6) f(x, y) = x3 + y3 − 3xy�(7) f(x, y) = x2y3(6− x− y)�(8) f(x, y) = 2x2 − xy − 3y2 − 3x + 7y�(9) f(x, y) = x2 − xy + y2 − 2x + y�(10) f(x, y) = e2x+3y(8x2 − 6xy + 3y2)�(11) f(x, y) = (5x + 7y − 25)e−(x2+xy+y2)�(12) f(x, y) = (x2 + y2)e−(x2+y2)�(13) f(x, y) = sin x sin y sin(x + y), 0 ≤ x, y ≤ π�(14) f(x, y) = x− 2y + log

√x2 + y2 + 3 arctan(y/x), x > 0�

êÞ 569

2. �J�§7.1�(iii)C(iv)�

3. �f(x, y) = (3 − x)(3 − y)(x + y − 3), �D¡f�8EÁÂC�EÁÂ�

4. �f(x, y) = xy(1 − x2 − y2), 0 ≤ x, y ≤ 1, �D¡f�8EÁÂ�ëFC�EÁÂ�

5. �5½Ef(x) = x2C(x2+1)−1,X�ðóa, b�Â,¸ÿ∫ 1

0(ax+

b− f(x))2dx�Â t��

6. ��nÍ�!Ýóx1, · · · , xnCnÍóy1, · · · , yn (�×�8�),

×��k0×;ÄXb(xi, yi), i = 1, · · · , n, �àaf(x) =

ax + bÛ��Ä&Æ�0×¸000---¿¿¿]]]õõõ(total square

error)

E(a, b) =n∑

i=1

(f(xi)− yi)2

t�Ýàa��X�h`�aCb�hÇttt��¿¿¿]]]°°°(Method

of least squares)�

7. 3x + y = 1�f�ì, �Oz = xy�ÁÂ�

8. �5½O`a5x2 +6xy +5y2 = 8îûæFt�CtG�F�

9. 'a, b Þü�Ñó�(i) 3x2 + y2 = 1�f�ì, �Oz = x/a + y/b�ÁÂ;

(ii) 3x/a + y/b = 1�f�ì, �Oz = x2 + y2�ÁÂ��|¿¢îÝ�L�ÕhÞ®Þ�

10. 3x− y = π/4�f�ì, �Oz = cos2 x + cos2 y�ÁÂ�

11. 3¦x2 + y2 + z2 = 1î, �Of(x, y, z) = x− 2y + 2z�ÁÂ�

12. �0�3`«z2 = xy + 1î, t#�æFï�

570 ÏÜa 9�ÐóCÍ�5��5

13. �3Þ`«

x2 − xy + y2 − z2 = 1 C x2 + y2 = 1

�ø/�, 0�ûæFt�ï�

14. 'a, b, c ëÑÝðó, 3x + y + z = 1�ì, �Of(x, y, z) =

xaybzc �Á�Â�

15. ��×r > 0, 3x2 + y2 + z2 = 5r2, Í�x, y, z > 0, �f�ì,

�Of(x, y, z) = log x + log y + 3 log z�Á�Â�¬¿àh��J�E��a, b, c > 0,

abc3 ≤ 27(a + b + c

5)5�

9.8 999��ÐÐÐóóó��555

3ì×a&Æ�ºD¡9�Ðó�¥¥¥��555(multiple integral),Í;&Æ��Ê9�Ðó�×î�5�'f(x, y) ×xCy�=�Ðó, x ∈ [α, β], y ∈ [a, b]��Þxü

�, ��Êf(x, y)3y ∈ [a, b]��5, Ç∫ b

af(x, y)dy�h�5�xb

n�.h∫ b

af(x, y)dy ×x �Ðó�35��, &ÆðºÂÕ9Ë

�5¡) ÐóÝ�µ�»A, u¢Ã�ó�ð, �xy = u, �ÿ

∫ 1

0

x√1− x2y2

dy = arcsin x, −1 < x < 1�

¨², ×¶�Ðó, uÞÍ¼óÚ ×¢ó, J�5¡) ×Ðó,

A ∫ 1

0

yxdy =1

x + 1, x > −1�

9ì ×ÃÍÝP²�

9.8 9�Ðó��5 571

��§§§8.1.'Ðóf(x, y)3Î�[α, β] × [a, b] =��Jì�ÐóG) ×=�Ðó:

G(x) =

∫ b

a

f(x, y)dy, x ∈ [α, β]�

JJJ��.´�3×T =�Ý��óÐó, Ä í8=�(�Ï×a�§6.8)�àv«ÝJ°, �ÿh��E×3�TÎ�î=�ÝÞ�óÐóùWñ�Ç∀ε > 0, D3×δ > 0, ¸ÿ

|f(P )− f(Q)| < ε,

∀P, Q ∈ [α, β]× [a, b], ©�||P −Q|| < δ�ua = b, JG(x) = 0, ∀x ∈ [α, β], h`GQ=��g'a 6=

b�.f í8=�, Æ∀ε > 0, D3×δ > 0, ¸ÿ

|f(P )− f(Q)| < ε

b− a,

∀P, Q ∈ [α, β]× [a, b], ©�||P −Q|| < δ�¬E�Þx, c ∈ [α, β], ©�|x− c| < δ, J||(x, y)− (c, y)|| = |x− c| < δ, ∀y ∈ [a, b]�Æ

|f(x, y)− f(c, y)| < ε

b− a�

.h©�|x− c| < δ,

|G(x)−G(c)| = |∫ b

a

(f(x, y)− f(c, y))dt|

≤∫ b

a

|(f(x, y)− f(c, y)|dt ≤∫ b

a

ε

b− adt = ε,

ÆG3c=��ÿJ�

!§, u�

H(y) =

∫ β

α

f(x, y)dx, y ∈ [a, b],

572 ÏÜa 9�ÐóCÍ�5��5

JHù ×=�Ðó�ãyGCH/ =�Ðó, ÆÍ�5D3�&Æ-|

∫ β

α

(

∫ b

a

f(x, y)dy)dx T∫ β

α

dx

∫ b

a

f(x, y)dy

¼�î∫ β

αG(x)dx, |

∫ b

a

(

∫ β

α

f(x, y)dx)dy T∫ b

a

dy

∫ β

α

f(x, y)dx

¼�î∫ b

aH(y)dy�î�Þ�,5½Ì f3Î�[α, β]× [a, b]�@@@ggg

��555(repeated integral, TÌiterated integral), }¡&ÆºJ�hÞ@g�58��

»»»8.1.�f(x, y) = x2 − 2xy, �O∫ 2

−1(∫ 4

1f(x, y)dy)dx�

��.´�ü�x, ÿ∫ 4

1

(x2 − 2xy)dy = (x2y − xy2)∣∣∣y=4

y=1= (4x2 − 16x)− (x2 − x)

= 3x2 − 15x�

.h∫ 2

−1

(

∫ 4

1

(x2 − 2xy)dy)dx =

∫ 2

−1

(3x2 − 15x)dx = (x3 − 15

2x2)

∣∣∣2

−1

=27

2�

Íg&Æ��§8.1�×.Â�

��§§§8.2.'f ×�L3Î�[α, β]× [a, b] �=�Ðó, v�

G(x, y) =

∫ y

a

f(x, t)dt, y ∈ [a, b],

H(x, y) =

∫ x

α

f(t, y)dt, x ∈ [α, β]�

9.8 9�Ðó��5 573

JGCH/ [α, β]× [a, b]î�=�Ðó�JJJ��.&Æ©JG =�, H =��J�v«�)'a 6= b��M

|f(x, y)|3[α, β] × [a, b]�Á�Â�.f í8=�, E∀ε > 0, D3×δ > 0, v0 < δ < ε/(2(M + 1)), ¸ÿ

|f(Q)− f(P )| < ε

2(b− a),

∀P, Q ∈ [α, β] × [a, b], ©�||P − Q|| < δ�uÞP �W×ü�ÝF(x0, y0), Q = (x, y) Ú [α, β] × [a, b] �×��ÝF, v��||P−Q|| < δ,J||(x, t)−(x0, t)|| < δ, ∀t ∈ [a, b),v|y−y0| < δ�.h

|G(Q)−G(P )| = |∫ y

a

f(x, t)dt−∫ y0

a

f(x0, t)dt|

≤ |∫ y0

a

(f(x, t)− f(x0, t))dt|+ |∫ y

y0

f(x, t)dt|

< (y0 − a) · ε

2(b− a)+ |y − y0| ·M

<ε

2+ δM <

ε

2+

ε

2= ε�

Æ|G(Q)−G(P )| < ε, ∀Q ∈ B(P ; δ), .hG3P=��ÿJ�

b`&ÆÄ6EÐóG(x) =∫ b

af(x, y)dy�5, ì��§Ý��,

«{�pï�Õ�3hfx(x, y) = ∂f(x, y)/∂x�

��§§§8.3.'Ðóf�L3S = [α, β]×[a, b],vfCfx/3S�=��J

(8.1)d

dx

∫ b

a

f(x, y)dy =

∫ b

a

fx(x, y)dy�

JJJ��.�G(x) =∫ b

af(x, y)dy�.dG

dx= lim

h→0

G(x + h)−G(x)

h,

Æu�J�∀ε > 0, ©�hÈ�, J

(8.2)

∣∣∣∣G(x + h)−G(x)

h−

∫ b

a

fx(x, y)dy

∣∣∣∣ < ε,

574 ÏÜa 9�ÐóCÍ�5��5

Í�§-ÿJÝ�¨J�(8.2)P�´�

G(x + h)−G(x) =

∫ b

a

(f(x + h, y)− f(x, y))dy�

ãíÂ�§, D3×z+yx�x + h , ¸ÿ

f(x + h, y)− f(x, y) = hfx(z, y)�.h

∣∣∣∣G(x + h)−G(x)

h−

∫ b

a

fx(x, y)dy

∣∣∣∣

= |∫ b

a

fx(z, y)dy −∫ b

a

fx(x, y)dy| = |∫ b

a

(fx(z, y)− fx(x, y))dy|�

.ã�'fx =�, Æfx3TÎ�Sî í8=��.h∀ε > 0, D3×δ > 0, ¸ÿ

|fx(z, y)− fx(x, y)| < ε

b− a,

©�|h| < δ(B�zÎ+yx�x + h )�.hêÿ∣∣∣∣G(x + h)−G(x)

h−

∫ b

a

fx(x, y)dy

∣∣∣∣ <ε

b− a· (b− a) = ε,

©�|h| < δ�Æ(8.2)PWñ�

!§, &Æ�b×ny

d

dy

∫ β

α

f(x, y)dx =

∫ β

α

fy(x, y)dx

�v«Ý��¬u�§8.3�f��, J(8.1)P-�×�Wñ, �ì»�

»»»8.2.'

f(x, y) =

{(x3/y2)e−x2/y, y > 0,

0, y = 0,

9.8 9�Ðó��5 575

×�L3x ∈ R, y ≥ 0�Ðó�h ×x�=�Ðó, ô ×y�=�Ðó�¬f(x, y)3(0, 0)¬�=�(�º½`ay = x2��(0, 0)-�:�)�¨�

g(x) =

∫ 1

0

f(x, y)dy =

{xe−x2

, x 6= 0,

0, x = 0�

J

(8.3) g′(x) = e−x2

(1− 2x2), ∀x ∈ R�

êux 6= 0, J(Í�Ý�5Ä��\ï��W, v¥� ¢�Ox 6= 0)

∫ 1

0

fx(x, y)dy =

∫ 1

0

e−x2/y(3x2

y2− 2x4

y3)dye−x2

(1− 2x2)�

¨², .fx(0, y) = 0, ∀y ≥ 0, Æ

∫ 1

0

fx(0, y)dy = 0�

�ã(8.3)ÿ

g′(0) = 1 6= 0 =

∫ 1

0

fx(0, y)dy�

Æ3x = 0,d

dx

∫ 1

0

f(x, y)dy 6=∫ 1

0

fx(x, y)dy�

¿à�§8.3, &Æ-�J�Þ@g�58�Ý�

��§§§8.4.'f ×3[α, β]× [a, b]=�ÝÐó, J

(8.4)

∫ b

a

(

∫ β

α

f(x, y)dx)dy =

∫ β

α

(

∫ b

a

f(x, y)dy)dx�

576 ÏÜa 9�ÐóCÍ�5��5

JJJ��.�

G(t, y) =

∫ t

α

f(x, y)dy, F (t) =

∫ b

a

G(t, y)dy�

JF (α) = 0, v

(8.5) F (β) =

∫ b

a

(

∫ β

α

f(x, y)dx)dy�

ã��5ÃÍ�§, ÿ

Gt(t, y) = f(t, y),

�ã�§8.3, ÿ

F ′(t) =

∫ b

a

Gt(t, y)dy =

∫ b

a

f(t, y)dy�

.h

F (β) = F (β)− F (α) =

∫ β

α

F ′(x)dx(8.6)

=

∫ β

α

(

∫ b

a

f(x, y)dy)dx�

ã(8.5)C(8.6)PÇÿ(8.4)PWñ�J±�

�3ì×a&Æ.Ý¥�5, ô�à£`ÿÕÝ��J�(8.4)

P�¨3&Æ�J��§5.2Ý�&Æ��B�×Å��'b×Ðóf(x, y)vfxy�fyx3R2�×�/)S �/=��&

Æ�J�

(8.7) fxy(x, y) = fyx(x, y), ∀(x, y) ∈ S�

�h(x, y) = fy(x, y), ∀(x, y) ∈ S�Jhx(x, y) = fyx(x, y)�¨u(x, a) ∈ S, Í�a ×ðó, J

f(x, y) = f(x, a) + f(x, y)− f(x, a) = f(x, a) +

∫ y

a

h(x, t)dt�

9.8 9�Ðó��5 577

�ã�§8.3,

fx(x, y) = fx(x, a) +

∫ y

a

hx(x, t)dt�

îPË��5½Ey�5, v¿à��5ÃÍ�§, ÿ

fxy(x, y) = hx(x, y) = fyx(x, y)�

Æ(8.7)PWñ�

»»»8.3.�O ∫ 1

0

xα − 1

log xdx, α ≥ 0�

��.ãy�5Õ��5¬&��Ðó, ÆG��5¬P°à#O��

F (α) =

∫ 1

0

xα − 1

log xdx, α ≥ 0�

�f(x, α) = (xα− 1)/ log x, 0 < x < 1, α ≥ 0, f(0, α) = 0, f(1, α) =

α��JfCfα/3Î�[0, 1]× [0,∞] =��Jã�§8.3,

F ′(α) =

∫ 1

0

xα log x

log xdx =

∫ 1

0

xαdx =1

α + 1�

ÆF (α) = log(α + 1) + C,

Í�C ×ðó�.F (0) = 0, ÆC = 0, v

F (α) = log(α + 1)�

\ïÎÍ�:�, 3î»�k¿à�§8.3, b£×Þ;�º�?

9I5µº�&�� î?Ý�

»»»8.4.�

F (x) =

∫ u(x)

a

f(x, t)dt,

578 ÏÜa 9�ÐóCÍ�5��5

Í�u ×��Ðó, vf�fx/=�, �OF ′(x)��.�

H(x, u) =

∫ u

a

f(x, t)dt,

JF (x) = H(x, u(x))�

¿à)WÐó��5��§8.3C��5ÃÍ�§, ÿ

F ′(x) =dH(x, u(x))

dx= Hx(x, u)

dx

dx+ Hu(x, u)

du

dx

=

∫ u(x)

a

fx(x, t)dt + f(x, u(x))du

dx�

!§, 3ÊÝf�ì(Çu, v��, f�fx/=�), u

G(x) =

∫ u(x)

v(x)

f(x, t)dt,

J

G′(x) =

∫ u(x)

v(x)

fx(x, t)dt + f(x, u(x))du

dx− f(x, v(x))

dv

dx�

»»»8.5.(i) �

F (x) =

∫ x

0

sin(xy)dy,

Jãî»á,

F ′(x) =

∫ x

0

y cos(xy)dy + sin(x2)�

(ii) �

(8.8) F (x) =

∫ 1

0

x√1− x2y2

dy = arcsin x, −1 < x < 1,

Jã�§8.3,

F ′(x) =

∫ 1

0

1

(1− x2y2)3/2dy =

1√1− x2�

êÞ 579

Í�Ý�5,�¿à�ó�ð,�y = sin u/x��p:�uÞ(8.8)P�F (x) = arcsin xà#Ex�5ùÿ8!��

êêê ÞÞÞ 9.8

1. �à#�Jì�&@g�58��(1)

∫ b

a(∫ 1

0(3x2 +xy + y2)dy)dx =

∫ 1

0(∫ b

a(3x2 +xy + y2)dx)dy�

(2)∫ 3

1(∫ 2

1log(x + 2y)dy)dx =

∫ 2

1(∫ 3

1log(x + 2y)dx)dy�

2. 'f(x, y) = (x− y)/(x + y)3��J∫ 1

0

(

∫ 1

0

f(x, y)dy)dx = −∫ 1

0

(

∫ 1

0

f(x, y)dx)dy =1

2,

¬�Õ ¢�§8.43h¬�Êà�

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1969). Calculus, Vol II, 2nd ed. John Wiley

& Sons, New York, New York.



580 ÏÜa 9�ÐóCÍ�5��5

ÏÏÏèèèaaa

¥¥¥��555

10.1 GGG��

3ÏÞa&Æ×��L�5∫ b

af(x)dx`, ÎjE×�L3×

b§ [a, b]Ýb&Ðóf�#½3ÏÚa, &Æ.ÂÕ�5Ý , �|ÎP§vÐóô�|� b&�3î×a, &ÆêÞ�5ÝÃF.ÂÕa�5�Ía&ÆÞ�¨×]'Ý.Â�&Æ�Ê×�L3nîè �×/)S�ö�Ðóf , f3Sî��5, Ì n¥¥¥��555(n-fold integral)¬|∫

· · ·∫

S

f T∫· · ·

∫

S

f(x1, · · · , xn)dx1 · · · dxn

��, Í�bnÍ�5Ðr, T��2©|∫

Sf(x)dx��, Í�x =

(x1, · · · , xn)�un = 2`, &Æ??|(x, y)ã�(x1, x2), ¬|∫ ∫

S

f T∫ ∫

S

f(x, y)dxdy

�îÞ¥�5(double integral), !§|∫ ∫ ∫

S

f T∫ ∫ ∫

S

f(x, y, z)dxdydz

�îëëë¥¥¥��555(triple integral)�3hSÌ ��555 ½½½(region of in-

tegration)�A!3×îÝ�µ, Ðrdx�dyCdz�, 3¥�5Ý�L�¬Î6��¢��, �Ä3�ÕC��5»ð`, Q�bà�

581

582 Ïèa ¥�5

9�ÐóÝ�5ºÕ��ÐóÝ�5, b&9v«ÝP²�¬bn9�óÝ�5, Q�Ó�K�»A, E×ë�óÝÐóf(x, y, z), tÝa�5�ë¥�5, &Æ��Ê3×`«Ý�5(Ç«««��555(surface integral))�Ç¸Ah, Xb&vÝ�5, í��ÐóÝ�5n;Û6� Ý��, &Æ;ð©�ÊË�óÝ�µ, ¬XbÝD¡, K�ñÇ.ÂÕnÍ�ó�

10.2 ÞÞÞ¥¥¥��555��LLL'S ¿«î×Î�, v [a, b]C[c, d]�Î�É¶�, Ç

S = [a, b]× [c, d] = {(x, y)|x ∈ [a, b], y ∈ [c, d]}�ê'P1CP25½ [a, b]C[c, d]�5v, Í�

P1 = {x0, x1, · · · , xn}, P2 = {y0, y1, · · · , ym}�Î�É¶�P1 × P2-Ì S�×5v, vÞS5vWmnÍ��ÎÎÎ��(subrectangle)�S�×5vP ′u��P ⊆ P ′, -Ì P�×Þ5�'f ×�L3Sî�Ðó, uD3S�×5vP , ¸ÿf 3N×�Ý�Î�î ðó, -Ìf ×$VÐó�h�×��â\&�Î�-Ì�ÝÎ��×$VÐó3ÍN×�Î��\&ÝÂ,

3�5�¬�¥��|�ufCg Sî�Þ$VÐó, JÍaPà)c1f + c2g) ×$VÐó�¨'P = P1 × P2 S�×5v, Í�P1 = {x0, x1, · · · , xn}, P2 =

{y0, y1, · · · , ym}, �f ×3Sî�$VÐó, Çf3G�5v��N×�Î� ðó�|Sij�[xi−1, xi] × [yj−1, yj], v'f3Sij�/F(Ç��â\&ÝF)ãÂcij�ucij > 0, J|Sij 9, cij {��]�� cij(xi − xi−1)(yj − yj−1)�¨ÞXbG�¶��R¼, -�L f3Sî�Þ¥�5�Ç

(2.1)

∫ ∫

S

f =m∑

j=1

n∑i=1

cij(xi − xi−1)(yj − yj−1)�

10.2 Þ¥�5��L 583

A!3��óÝ�µ, u|P�×Þ5P ′ã�P , J�5Â��, ùÇ�5Â�5vPn, ©�f3N×�Ý�Î� ðó��∆xi = xi − xi−1, ∆yj = yj − yj−1, i = 1, · · · , n, j = 1, · · · ,m,

J(2.1)PW

(2.2)

∫ ∫

S

f =m∑

j=1

n∑i=1

cij∆xi∆yj�

îP¼�b`ô¶W∫ ∫

S

f(x, y)dxdy,

9ø¶Ý?�, Î|y�(2.2)P��Eï, èø&Æhõ�ã¼�uf3S�/F/ ðó, ÉA1

f(x, y) = k, ∀x ∈ (a, b), y ∈ (c, d),

J

(2.3)

∫ ∫

S

f = k(b− a)(d− c)�

ê.b− a =∫ b

adx, d− c =

∫ d

cdy, Æ(2.3)P�;¶

(2.4)

∫ ∫

S

f =

∫ d

c

(

∫ b

a

f(x, y)dx)dy =

∫ b

a

(

∫ d

c

f(x, y)dy)dx�

ôµÎE×ðóÐó, &Æ�|Þg@g�5, ¼O×Þ¥�5�hÞïÍ²îÎ�!Ý, Þg@g�5, ÛÎ3�Ëg��ÐóÝ�5�}¡��º:Õ, 4b`�ÕôÎb°�|, ¬N×g©Î3�×��ÐóÝ�5, ÍÄ�/�î´¥�5��9Ý�&��|ÏÞaS��ÐóÝ�5Ý]P, ��LìõCîõ,

Q¡�Lì�5Cî�5, �Þï8�, -Ìf3S��, Í�!ÂÇ f3S�Þ¥�5ÝÂ�9ì&Æ};�×ì]P, 9ôÎ&9>IhS��Ðó�5Ý]P, �Äæ§Í@Î×øÝ�&Æ��L$VÐó��5, Q¡��L´×�ÝÐó��5�¬)

584 Ïèa ¥�5

A��óÝ�µ, h�L¬P°b[2O�5Â�5?&ÆÂÕÝ¥�5, �K�; @g�5�E×$VÐóf ,∫ ∫

Sij

f =

∫ yj

yj−1

(

∫ xi

xi−1

f(x, y)dx)dy =

∫ xi

xi−1

(

∫ yj

yj−1

f(x, y)dy)dx�(2.5)

9ì9°ny$VÐóÝP², K�ã(2.2)PT¿à(2.5)PJ��§��ÐógCh/ �L3Î�Sî�$VÐó, êS ×&&&[[[;;;(nondegenerate)�Î�, ÇS � ×FT×að�

��§§§2.1.(aaaPPP). E�Þðóc1Cc2,

∫ ∫

S

(c1g(x, y) + c2h(x, y))dxdy

= c1

∫ ∫

S

g(x, y)dxdy + c2

∫ ∫

S

h(x, y)dxdy�

��§§§2.2.(��PPP). 'S�5WÞÎ�S1CS2, J∫ ∫

S

g(x, y)dxdy =

∫ ∫

S1

g(x, y)dxdy +

∫ ∫

S2

g(x, y)dxdy�

��§§§2.3.(fff´��§§§). ug(x, y) ≤ h(x, y), ∀(x, y) ∈ Q, J∫ ∫

S

g(x, y)dxdy ≤∫ ∫

S

h(x, y)dxdy�

©½2, uh(x, y) ≥ 0, ∀(x, y) ∈ Q, J∫ ∫

S

h(x, y)dxdy ≥ 0�

Íg&Æ�L3×Î�î b&�ÐóÝÞ¥�5�'f ×3Î�Sî�b&Ðó, Ç'

|f(x, y)| ≤ M, ∀(x, y) ∈ S�

10.2 Þ¥�5��L 585

Jf+yÞ$VÐóg�h� , Í�g(x, y)= −M , h(x, y) = M ,

∀(x, y) ∈S�uD3°×�@óI, ¸ÿ

(2.6)

∫ ∫

S

g ≤ I ≤∫ ∫

S

h,

EXb��

(2.7) g(x, y) ≤ f(x, y) ≤ h(x, y), ∀(x, y) ∈ S,

�$VÐóg�hWñ, JÌf3Sî��, v�5Â I, Ç∫ ∫

S

f = I�

9ìèÕÝgCh)/ $VÐó��

G = sup{∫ ∫

S

g, g(x, y) < f(x, y), ∀(x, y) ∈ S},

H = inf{∫ ∫

S

h, f(x, y) ≤ h(x, y), ∀(x, y) ∈ S}�

.f ×b&Ðó, ÆGCH/&è/)�ê.g(x, y) ≤ h(x, y),

∀(x, y) ∈ S, Æ ∫ ∫

S

g ≤∫ ∫

S

h,

.hG�N×-ô/�yT�yH��×-ô�ÇáGbt�î&,

�Hbt�ì&, ¬��∫ ∫

S

g ≤ lubG ≤ glbH ≤∫ ∫

S

h,

©�g, h��(2.7)P�ÆálubGCglbH/��(2.6)P��¡Î,

f3Sî��uv°ulubG=glbH, vh`∫ ∫

S

f = lubG = glbH�

&Æ|I(f)�lubG, ¬Ì f3S��ì�5, |I¯(f)�glbH, ¬

Ì� f�S��î�5�ã|îÝD¡-ÿì��§�

586 Ïèa ¥�5

��§§§2.4.'f ×3Î�Sî�b&Ðó, JÍî�5Cì�5��∫ ∫

S

g ≤ I¯(f) ≤ I(f) ≤

∫ ∫

S

h,

Í�g�h Þ��(2.7)P�$VÐó��f3Sî��, uv°u

I¯(f) = I(f),

h`, ∫ ∫

S

f = I¯(f) = I(f)�

\ïô��ûî�M», ¼�L��ÐóÝ�5�3�5§¡�, $VÐó6�½¥�Ý��¨², |áEÞ¥�5ùbA$VÐó�aP��PCf´�§, J�K��|, Æº�\ï� �W�&Æ�:�u�µ¥�5Ý�L, ¼O�5Â, ÞÎ×�ÝÜÝ

�®�9ìÝ�§, -èº×BãÞg@g�5, ¼OÞ¥�5Â�]°�

��§§§2.5.'f ×�L3Î�S = [a, b] × [c, d] �b&Ðó�v'f3S��E∀y ∈ [c, d],'

∫ b

af(x, y)dxD3,v|A(y)�ÍÂ�

Ju∫ d

cA(y)dyD3, ÍÂ-�yf3Sî�Þ¥�5, Ç

(2.8)

∫ ∫

S

f(x, y)dxdy =

∫ d

c

(

∫ b

a

f(x, y)dx)dy�

JJJ��.ãÞ$VÐógCh, ��(2.7)P�JuEx3[a, b]�5, -ÿ

∫ b

a

g(x, y)dx ≤ A(y) ≤∫ b

a

h(x, y)dx�

.ã�'∫ d

cA(y)dyD3, Æ�Eî��PEy3[c, d]�5, �ÿ

∫ ∫

S

g ≤∫ d

c

A(y)dy ≤∫ ∫

S

h�

10.2 Þ¥�5��L 587

Æ∫ d

cA(y)dyEXb��(2.7)P�gCh, ��(2.6)P�êã�'f3

S��, °×��(2.6)P�@ó f3Sî��5Â, ÆÿJ∫ d

cA(y)dy f3Sî�Þ¥�5, Æ(2.8)PWñ�

3¿à(2.8)POÞ¥�5`, &Æ�Þyü�, �ÞfExãa�b

�5, Q¡�ÞXÿ�×yÝÐó, Eyãc�d�5�3v«Ýf�ì, Ç'

∫ d

cf(x, y)dyD3, ∀x ∈ [a, b], v'h�53[a, b]��, J�

Þf�Ey�Ex�5, v

(2.9)

∫ ∫

S

f(x, y)dxdy =

∫ b

a

(

∫ d

c

f(x, y)dy)dx�

�uÞf�/��, J∫ ∫

S

f(x, y)dxdy =

∫ d

c

(

∫ b

a

f(x, y)dx)dy =

∫ b

a

(

∫ d

c

f(x, y)dy)dx,

Ç�øð�55��\ïô��Þh��ÏÜa�§8.4f´,

h�øð�55�Xmf��Q´3�&Æô�E�§2.5�×¿¢Ý�Õ�uf &�, Jëîè

�ÝF(x, y, z)Ý/),Í�(x, y) ∈ S, 0 ≤ z ≤ f(x, y),Ì f 3Sî�Á/�h/)|W��, ¸�âXb3`«z = f(x, y) �ì, �3Î�S�îÝXbF�E∀y ∈ [c, d], �5A(y) =

∫ b

af(x, y)dx,

�×¿�x-z¿«�¿«, X^��^«Ý«��ê.^«�A(y), 3[c, d]�, Æã6.2;��á,

∫ b

aA(y)dyÇ�yW��

�V (W )�Æf &�`, �§2.5J�f3Sî�Á/��, Ç f3S �¥�5�¨², (2.9)PJèº¨×�ÕÁ/��Ý]P�hg&Æ�

O¿�x-z¿«�¿«X^��«��

9ì&Æ�¿Í�§2.5�Tà»��

»»»2.1.�f(x, y) = x sin y − yex, S = [−1, 1] × [0, π/2], �Of3S�Þ¥�5�

588 Ïèa ¥�5

��.�Q�¿à�§2.5�&Æ�OA(y)�

A(y) =

∫ 1

−1

(x sin y − yex)dx = (1

2x2 sin y − yex)

∣∣∣x=1

x=−1= (e−1 − e)y�

.h

∫ ∫

S

f =

∫ π/2

0

A(y)dy =

∫ π/2

0

(e−1 − e)ydy = (e−1 − e)π2/8�

u��yù�ÿÕ8!Ý�n�∫ ∫

S

=

∫ 1

−1

(

∫ π/2

0

(x sin y − yex)dy)dx

=

∫ 1

−1

(−x cos y − 1

2y2ex)

∣∣∣y=π/2

y=0dx

=

∫ 1

−1

(−π2ex/8 + x)dx = (e−1 − e)π2/8�

»»»2.2.�f(x, y) =√|y − x2|, S = [−1, 1]× [0, 2]��Of3S�Þ¥

�5��.&Æ�ÞfEy�5, ¬|B(x)�h�5�.�5Õ�b�EÂ|y − x2|, �h4�yy − x2Tx2 − yµy ≥ x2Ty < x2 ��Æ�ÞB(x)tWË4�Ç

B(x) =

∫ 2

0

√|y − x2|dy =

∫ x2

0

√x2 − ydy +

∫ 2

x2

√y − x2dy�

¥�Ey�5`, xÛÚ ×ðó, Æ�3îP��Ï×Í�5�,

�t = x2 − y, �3ÏÞÍ�5�, �t = y − x2, vÿ

B(x) = −∫ 0

x2

√tdt +

∫ 2−x2

0

√tdt =

2

3|x|3 +

2

3(2− x2)3/2�

êÞ 589

-

6

x

y

O

y > x2¾

y = x2¾

y < x2¾

−1 1

1

2

%2.1.

�¿à�§2.5ÿ∫ ∫

S

√|y − x2|dxdy

=

∫ 1

−1

(2

3|x|3 +

2

3(2− x2)3/2)dx =

1

3+

4

3

∫ 1

0

(2− x2)3/2dx

=1

3+

1

3(x(2− x2)3/2 + 3x

√2− x2 + 3x

√2− x2 + 6 arcsin(

x√2))

∣∣∣1

0

=5

3+

π

2�

\ïô�|��Ex�5, QºÿÕ8!Ý�n, �Ä�Õº}��Ó°�

»»»2.3.�O3f(x, y) = 4− 1100

(25x2 +16y2)�%�ì,v3Î�S =

[0, 2]× [0, 3]�îÝñ�W��.ÇO

V (W ) =

∫ ∫

S

f =

∫ 2

0

(

∫ 3

0

(4− 1

100(25x2 + 16y2))dy)dx

=

∫ 2

0

(4y − 1

4x2y − 4

75y3)

∣∣∣y=3

y=0dx

=

∫ 2

0

(264

25− 3

4x2)dx = 19.12�

590 Ïèa ¥�5

êêê ÞÞÞ 10.2

1. �BãO@g�5, Oì�&Þ¥�5�(1)

∫∫S

xy(x + y)dxdy, S = [0, 1]× [0, 1]�(2)

∫∫S(x3 + 3x2y + y3)dxdy, S = [0, 1]× [0, 1]�

(3)∫∫

S(√

y + x− 3xy2)dxdy, S = [0, 1]× [1, 3]�(4)

∫∫S

sin2 x sin2 ydxdy, S = [0, π]× [0, π]�(5)

∫∫S

sin(x + y)dxdy, S = [0, π/2]× [0, π/2]�(6)

∫∫S| cos(x + y)|dxdy, S = [0, π]× [0, π]�

(7)∫∫

S[x + y]dxdy, S = [0, 2]× [0, 2], [ · ]�t�JóÐó�

(8)∫∫

Sy−3etx/ydxdy, S = [0, t]× [1, t], t > 0 ×ðó�

2. 'S = [a, b]× [c, d], �J3ÊÝf�ì,

∫ ∫

S

f(x)g(y)dxdy =

∫ b

a

f(x)dx

∫ d

c

g(y)dy�

3. �

f(x, y) =

{1− x− y, x + y ≤ 1,

0, Í�,

S = [0, 1]×[0, 1]�i�f3Sî�Á/Ý%�,¬BãOf3S�Þ¥�5(��h�5D3), �OG�Á/��

4. �¥�Ï3Þ, ¬Þf;

f(x, y) =

{x + y, x2 ≤ y ≤ 2x2,

0, Í��

5. �¥�Ï3Þ, ¬ÞS; [−1, 1]× [−1, 1], f;

f(x, y) =

{x2 + y2, x2 + y2 ≤ 1,

0, Í��

10.3 ¥�5�×MD¡ 591

6. 'f�L3S = [1, 2]× [1, 4]î, Í�

f(x, y) =

{(x + y)−2, x ≤ y ≤ 2x,

0, Í��Of3Sî�Þ¥�5(�'h�5D3)�

7. 'f�L3S = [0, 1]× [0, 1], Í�

f(x, y) =

{1, x = y,

0, x 6= y��Jf3Sî�Þ¥�5D3vÍÂ 0�

10.3 ¥¥¥��555��×××MMMDDD¡¡¡

3ÏÞa�§4.7, E�×�ó, ¿àT îÝ=�ÐóÄ í8=�ÝP², &ÆJ�T îÝ=�Ðó ��EËÍ�óÝÐó, &Æùbv«Ý��, J�3h¯�, �¢�Apostol

(1969) Theorem 11.6�

��§§§3.1.'f3Î�S = [a, b]× [c, d]=�, Jf3Sî��, v

(3.1)

∫ ∫

S

f =

∫ d

c

(

∫ b

a

f(x, y)dx)dy =

∫ b

a

(

∫ d

c

f(x, y)dy)dx�

ÏÞa�§4.2¼�, ©3b§ÍF(¯@î�óÍFù�) �=�Ý×ÍT îÝb&Ðó, ) ��E×TÎ�îÝËÍ�óÝb&Ðó, ©�Í�=�ÝF“�H9”, hÐó)b��&Æ��ì��L�

��LLL3.1.'A ¿«î×b&Ý/), u∀ε > 0, D3b§ÍÎ�Ð/�âA, vÍÀ«��yε, JÌA�«««�� 0(content zero)�

592 Ïèa ¥�5

ãî�Lá, ì�&/)�«�í 0:

(i) ¿«îb§ÍF�/);

(ii) ¿«îb§Í«� 0�/)�Ð/;

(iii) ×«� 0�/)Ý�×�/;

(iv) �×að�

��§§§3.2.'f ×3Î�S = [a, b]× [c, d]î�b&Ðó, vÍ�=�FÝ«� 0, Jf3Sî�Þ¥�5D3�

î�§�J�,�¢�Apostol (1969) Theorem 11.7�hÞ�§,

4ÍJ�K¬&�p, �Ä&Æ- £°J�, ¬&¨$ðD¡¥�5Ý¥F, X|K¯��¬ÍÞ´ÝJ�, Eè{&�5�]«Ý�æ, )b��ÝQÃ, b·¶Ý\ï�÷��åApostol (1969)

×h��h c, &ÆD¡ÝÞ¥�5, KÎ3×Î�î�&Æ�Þ�

5P�.ÂÕ?×�Ý ½�'Q ×¿«îb&Ý ½, v'Q�ây×Î�S��'f

×�L3Q�b&Ðó��L×±Ðóf3SAì:

(3.2) f(x, y) =

{f(x, y), (x, y) ∈ Q,

0, (x, y) ∈ S \Q�ÇÞf��L½U"�S, �Þ3Q�²ÝÐóÂ� 0�¨3Ý®ÞÎ, 9ø�L�¼ÝÐóf , ÎÍ3S��? uÎÝ�, &Æ-Ìf3Q��, v�L

∫ ∫

Q

f =

∫ ∫

S

f�

´�&Æ�Êì�x-y¿«îÝ ½:

Q = {(x, y)|a ≤ x ≤ b, φ1(x) ≤ y ≤ φ2(x)},

Í�φ1Cφ2 T [a, b]î�=�Ðó, vφ1(x) ≤ φ2(x), ∀x ∈[a, b]� Ý|y 5, &ÆÌh Ï×l� ½, %3.1 ×»�.

10.3 ¥�5�×MD¡ 593

φ1Cφ2/3[a, b]=�, Æ b&, .h9Ë ½ b&�kàax =

t �Q�ø/ ×að, ãy = φ1(x)�y = φ2(x)�

xO

y

a b

y = φ1(x)

Q

y = φ2(x)

xO

y

c

d

x = ψ1(y)

Q′

I

x = ψ2(y)

%3.1. Ï×lÝ ½ %3.2. ÏÞlÝ ½

¨×v&Æa�ÊÝx-y¿«îÝ ½ :

Q′ = {(x, y)|c ≤ y ≤ d, ψ1(y) ≤ x ≤ ψ2(y)},

Í�ψ1Cψ2 [c, d] î�=�Ðó, vψ1(y) ≤ ψ2(y), ∀y ∈ [c, d]�&ÆÌh ÏÞl� ½, %3.2 ×»�h`×i¿a�Q′�ø/ ×að�|�9Ë ½ù b&�&ÆX�ÊÝ ½, ÞÎÏ×lTÏÞl, T�|5Wb§ÍÏ

×lTÏÞlÝ ½�¨'f �L3×Ï×lÝ ½Q�b&Ðó�|×Î�S¼

�âQ, ¬�LÐófA(3.2)P�f3S��=�F, �âf3Q��=�F�3Q�\&îvf�Â� 0�F��Q �\& %3.1Ýφ1�φ2�%��ÞkàaðXàW�hÞaðÝ«� 0,

�hÞ%��«�ù 0(�Apostol (1969) Theorem 11.8)�ì�§¼�,uf3Q�/F(|intQ��)=�,JÞ¥�5

∫∫Q

f

D3�3hintQ = {(x, y)|a < x < b, φ1(x) < y < φ2(x)}�

��§§§3.3.'Q ×Ï×lÝ ½, +yÐóφ1�φ2Ý%� , x ∈[a, b]�ê'f ×�L3Q�b&Ðó, v3intQ=��JÞ¥�

594 Ïèa ¥�5

5∫∫

QfD3, vÍÂ�ãÞg@g�5Oÿ, Ç

(3.3)

∫ ∫

Q

f(x, y)dxdy =

∫ b

a

(

∫ φ2(x)

φ1(x)

f(x, y)dy)dx�

JJJ��.�S = [a, b]× [c, d] ×�âQ�Î�, v�f �LA(3.2)P�.f©��3Q�\&F�=�, �G«�¼�Q�\&FÝ«� 0, Æã�§3.2á, f3S��êE∀x ∈ (a, b), ×î�5∫ d

cf(x, y)dyD3, h.�5Õ�f3[c, d]�t9©bÞ�=�F�

Æã�§2.5( �(2.9)P) á,

(3.4)

∫ ∫

S

f =

∫ b

a

(

∫ d

c

f(x, y)dy)dx�

t¡, .E∀x ∈ [a, b]

f(x, y) =

{f(x, y), φ1(x) ≤ y ≤ φ2(x),

0, Í�,

Æ ∫ d

c

f(x, y)dy =

∫ φ2(x)

φ1(x)

f(x, y)dy,

.h(3.3) Wñ�

Q, &Æùb×ETÏÞl ½Ý��Çuf ×�L3ÏÞl ½Q′�b&Ðó, v3intQ′=�, Jf 3Q′��, v

(3.5)

∫ ∫

Q′f(x, y)dxdy =

∫ d

c

(

∫ ψ2(y)

ψ1(y)

f(x, y)dx)dy�

êb° ½!` Ï×lCÏÞl, A×i8(ÇiCÍ/IÝ ½), h`�55�-�¥�, v

∫ b

a

(

∫ φ2(x)

φ1(x)

f(x, y)dy)dx =

∫ d

c

(

∫ ψ2(y)

ψ1(y)

f(x, y)dx)dy�

10.3 ¥�5�×MD¡ 595

3Ø°�µì, hÞË�5�×��ºf¨×�Õî�|&9, hã9ìÝ×°»��:��3�5Gt?¯�Ý£×ì¢ï´|�Õ�'Q = {(x, y)|a ≤ x ≤ b, φ1(x) ≤ y ≤ φ2(x)} ×Ï×lÝ

½�ãf(x, y) = 1, ∀(x, y) ∈ Q, Jã�§3.3á

∫ ∫

Q

dxdy =

∫ b

a

(φ2(x)− φ1(x))dx,

�ã×îÝ�5��á, îP��y ½Q�«�, Æá¿àÞ¥�5�O«��¨², uf ×�L3Q�&�v=�ÝÐó, J�5

∫ φ2(x)

φ1(x)

f(x, y)dy

�×¿�y-z¿«Ef3Qî�Á/X^��¿«Ý«�, �%3.3

��YÅI5�(3.3)PÇ¼�f3Q��Þ¥�5�yh^««�Ý�5�ÆÞ¥�5

∫∫Q

f �yf3Qî�Á/Ý��×��,

uf�g/3Q=�, vf ≤ g, JÞ¥�5∫∫

Q(g − f)�y+yÞÐ

óf�g�%� Ýñ�Ý��E×ÏÞlÝ ½ô�bv«Ý�Õ�

x

y

z

a

b

y = ψ1(x) y = ψ2(x)

f^««�

=∫ ψ2(x)

ψ1(x)f(x, y)dy

%3.3.

9ì&Æ�×°»��

596 Ïèa ¥�5

»»»3.1.�f(x, y) = (2y−1)/(x+1), S ãëàax = 0, y = 0C2x−y − 4 = 0X�� ½�Of3Sî�Þ¥�5��.A%3.4, S�Ú 3Þ`ay = 0�y = 2x− 4 , ãx = 0�x = 2

� ½�Æ∫ ∫

S

f(x, y)dxdy =

∫ 2

0

(

∫ 0

2x−4

2y − 1

x + 1dy)dx =

∫ 2

0

y2 − y

x + 1

∣∣∣y=0

y=2x−4dx

= −∫ 2

0

4x2 − 18x + 20

x + 1dx = −2

∫ 2

0

(2x− 11 +21

x + 1)dx

= −2(x2 − 11x + 21 log(x + 1))∣∣∣2

0= −6(7 log 3− 6)�

¨², ù�ÞSÚ +y`ax = 0�x = (y + 4)/2 , ãy =

−4�y = 0 � ½, J

∫ ∫

S

f(x, y)dxdy =

∫ 0

−4

(

∫ (y+4)/2

0

2y − 1

x + 1dx)dy

=

∫ 0

−4

(2y − 1) log(x + 1)∣∣∣x=(y+4)/2

x=0dy =

∫ 0

−4

(2y − 1) log(y + 6

2)dy�

yõÝM», º�&�\ï� �W, )�ÿÕ8!Ý�n, �ÄÏ×Ë]P�Q´�|°�AGX�, Þ¥�5b`ºb9Ë�µ,

��Ø×�ó��ºf�¨×�ó�Õ�|°�

-

6

x

y

O 2

−4

%3.4.

»»»3.2.�O+yÞeÎay = x2�y = 4 − x2 � ½QÝ«�, �%3.5�

10.3 ¥�5�×MD¡ 597

-

6

x

y

O

(−√2, 2)(√

2, 2)

%3.5.

��.Aì-�O�«��∫ ∫

Q

dxdy =

∫ √2

−√2

(

∫ 4−x2

x2

dy)dx =

∫ √2

−√2

(4− 2x2)dx =16√

2

3 �

»»»3.3.�O3`«z = 4− x2 − 4y2�ì, �3x-y ¿«× ½Q�îÝñ�WÝ��, Í�Q x = 0, y = 0Cx + 2y − 2 = 0X�� ½, �%3.6�

x

2

z

4

1y

%3.6.

��.V (W ) =∫ 1

0(∫ 2−2y

0(4 − x2 − 4y2)dx)dy = 8

3�Í�E×ü�Ýy,∫ 2−2y

0(4− x2 − 4y2)dx�¿�x-z �¿«ÞW^��«��

598 Ïèa ¥�5

»»»3.4.�O3¿«z = x + 2y�ì, �°5�×iÖx2 + y2 ≤ 4, x,

y ≥ 0, z ≥ 0, �ñ�Ý��.�� ∫ 2

0

(

∫ √4−x2

0

(x + 2y)dy)dx =

∫ 2

0

(x√

4− x2 + 4− x2)dx

= (−1

3(4− x2)3/2 + 4x− 1

3x3)

∣∣∣2

0= 8�

»»»3.5.�Oì�YYY¦¦¦(ellipsoid)X��:

x2

a2+

y2

b2+

z2

c2= 1�

��.¿àEÌPáY¦��

V = 8

∫ ∫

S

f(x, y)dxdy,

Í�

f(x, y) = c√

1− x2/a2 − y2/b2, S = {(x, y)|x2

a2+

y2

b2≤ 1, x, y ≥ 0}�

Æ

V = 8c

∫ a

0

(

∫ b√

1−x2/a2

0

√1− x2/a2 − y2/b2dy)dx�

�k =√

1− x2/a2, J/IÝ�5�y∫ bk

0

√k2 − y2/b2dy = k2b

∫ π/2

0

cos2 tdt =π

4k2b =

πb

4(1− x2

a2),

h�àÕ�ó�ð, �y = bk sin t�Æ

V = 8c

∫ a

0

πb

4(1− x2

a2)dx =

4

3πabc�

©½2, ua = b = c, JÿÕ�5 a�¦�� 43πa3�

»»»3.6.�Þ∫ 1

0

∫ x

x2 f(x, y)dydxøð�55�, �'høð )°�

10.3 ¥�5�×MD¡ 599

��.�5 ½

Q = {(x, y)|x ∈ [0, 1], x2 ≤ y ≤ x},

h ×Ï×l� ½�êQù�Ú ×ÏÞl� ½, Ç

Q = {(x, y)|y ∈ [0, 1], y ≤ x ≤ √y}�

Ææ�5�y ∫ 1

0

(

∫ √y

y

f(x, y)dx)dy�

»»»3.7.�Þ ∫ 3

0

(

∫ √25−y2

4y/3

f(x, y)dx)dy

øð�55�, �'høð )°��.�5 ½A%3.7�

-

6

x

y

O 1 2 3 4 5

x = 4y/3 x =√

25− y2

%3.7.

�5 ½æ ×ÏÞlÝ ½

Q = {(x, y)|0 ≤ y ≤ 3, 4y/3 ≤ x ≤√

25− y2}�u�øð�55�, QÄ65WËÍÏ×lÝ ½, �ÿ

∫ 4

0

(

∫ 3x/4

0

f(x, y)dy)dx +

∫ 5

4

(

∫ √25−x2

0

f(x, y)dy)dx�

600 Ïèa ¥�5

�§3.1Ýøð=�ÐóÝ�55�, b&9�!ÝTà�»A,

b×°��ÐóÝ��5, u�ÐóÝ��5P�@Ý�P(Ç&��Ðó), b`�¢ãh�§O��5Â�

»»»3.8.�O∫ 1

0(x2 − 1)/ log xdx�

��.�Q�5Õ��5P°¶�, ¬&Æ�BãÞ¥�5¼O�5Â�´�E∀α > 0,

∫ α

0

∫ 1

0

xtdxdt =

∫ α

0

1

t + 1dt = log(α + 1)�

¬Bãøð�55�,

∫ α

0

∫ 1

0

xtdxdt =

∫ 1

0

∫ α

0

xtdtdx =

∫ 1

0

xt

log x

∣∣∣∣t=α

t=0

dx =

∫ 1

0

xα − 1

log xdx�

Æÿ ∫ 1

0

xα − 1

log xdx = log(α + 1), ∀α > 0�

|α = 2�áîP, ÇÿXO��5Â log 3�Í»��ÏÜa»8.3�®°f´�

9ì ¨×»�

»»»3.9.�Oì��5Â

I =

∫ ∞

0

e−ax − e−bx

xdx,

Í�a, b > 0��.´�h��5[eÝJ�º�&�\ï�¥�

∫∞0

e−ax/xdx�∫∞0

e−bx/xdx/s÷�ÞI;¶WÞg@g�5, Ç

I =

∫ ∞

0

(

∫ b

a

e−xydy)dx�

10.3 ¥�5�×MD¡ 601

ãy�5X3ÝÎ�� b§, .h��ñÇSà�§3.1¼øð�55�, �Äu�;¶Wì��P

I = limt→∞

∫ t

0

(

∫ b

a

e−xydy)dx,

-�øð�55��ÿ

I = limt→∞

∫ b

a

1− e−ty

ydy = log

b

a− lim

t→∞

∫ b

a

e−ty

ydy�

ub ≥ a > 0, Jt →∞`,

0 ≤∫ b

a

e−ty

ydy <

1

a

∫ b

a

e−tydy =1

at(e−at − e−bt) → 0,

!§ua > b > 0, Jt →∞`,∫ b

ae−ty/ydyù��0�Æ

I = logb

a�

&Æ�.Âî»Ý��Aì�'f(t)3t≥0 @ð¿â, Ç3N×b§Ý , tÝ��3b

§ÍFb®�Ý�=�Ý0ó, 3Íõ2]/b=�Ý0ó�ê'

∫∞1

f(t)/tdtD3�JE��a, b > 0,

I =

∫ ∞

0

f(ax)− f(bx)

xdx = f(0) log

b

a�

îP�J�, )Î�ÞI;¶WÞg@g�5

I =

∫ ∞

0

(

∫ a

b

f ′(x, y)dy)dx,

�øð�55��ny¥�5, $b×°�×î�v«Ý��, &Æ©��×°,

J�ôKv«×î�ÝJ�, .h/¯��(i) 'f(x, y) ≥ 0, ∀(x, y) ∈ S, J

∫ ∫

S

f(x, y)dxdy ≥ 0�

602 Ïèa ¥�5

(ii) 'f(x, y) ≥ g(x, y), ∀(x, y) ∈ S, J∫ ∫

S

f(x, y)dxdy ≥∫ ∫

S

g(x, y)dxdy�

(iii) | ∫∫S

f(x, y)dxdy| ≤ ∫∫S|f(x, y)|dxdy�

Í�$b¥�5�íÂ�§�, �¢�Courant and John (1974)

pp.384-385�

êêê ÞÞÞ 10.3

1. �Oì�&Þ¥�5∫∫

Sf�

(1) f(x, y) = x cos(x + y), S cF (0, 0), (π, 0), (π, π)�ë��(2) f(x, y) = (1 + x) sin y, S cF (0, 0), (1, 0), (1, 2), (0,

1)�°\��(3) f(x, y) = ex+y, S = {(x, y)| |x|+ |y| ≤ 1}�(4) f(x, y) = x2y2, S Ï×é§�, +yÔ`axy = 1�xy =

2, CÞàay = x�y = 4x � ½�(5) f(x, y) = x2 − y2, S +y`ay = sin x� [0, π] � ½�(6) f(x, y) = x2 + y2, S = {(x, y)| |x| ≤ 1, |y| ≤ 1}�(7) f(x, y) = 3x + y, S = {(x, y)| 4x2 + 9y2 ≤ 36, x > 0, y >

0}�(8) f(x, y) = y + 2x + 20, S = {(x, y)| x2 + y2 ≤ 16}�(9) f(x, y) = x2y2, S = {(x, y)| x2 + y2 ≤ 1}�(10) f(x, y) = (x3 + y3 − 3xy(x2 + y2))(x2 + y2)−3/2, S =

{(x, y)| x2 + y2 ≤ 1}�

êÞ 603

2. øðì�&�55�, �'Íøð )°�(1)

∫ 1

0(∫ y

0f(x, y)dx)dy� (2)

∫ 2

0(∫ 2y

y2 f(x, y)dx)dy�(3)

∫ 4

1(∫ 2√

xf(x, y)dy)dx� (4)

∫ 2

1(∫ √2x−x2

2−xf(x, y)dy)dx�

(5)∫ 2

−6(∫ 2−x

(x2−4)/4f(x, y)dy)dx� (6)

∫ e

1(∫ log x

0f(x, y)dy)dx�

(7)∫ 1

−1(∫ 1−x2

−√1−x2 f(x, y)dy)dx� (8)∫ 1

0(∫ x2

x3 f(x, y)dy)dx�(9)

∫ π

0(∫ sin x

− sin(x/2)f(x, y)dy)dx� (10)

∫ 4

0(∫ (y−4)/2

−√4−yf(x, y)dx)dy�

3. �O+y`«z = x2 − y2, x-y¿«, x = 1Cx = 3 �ñ��

4. �O3¿«z = 3x + y�ì, �°5�×Yi4x2 + 9y2 ≤ 36,

x ≥ 0, y ≥ 0, �ñ�Ý��

5. �O3¿«z = 2y�ì, �y = x2, y = 0Cx = 2X�� ½ �ñ�Ý��

6. �OeeeÎÎÎ«««(paraboloid)z = 16− x2 − 4y23Ï×ßßß§§§(octant)

X��

7. �OiÖx2 + y2 = 9C¿«y = 0, z = 0, z = xX��ñ�3Ï×ß§��

8. �J√

x/a +√

y/b +√

z/c = 1, a, b, c > 0, �%��ë2ý¿«X��ñ�Ý�� V = abc/90�

9. �J(x2)1/3 + (y2)1/3 + (z2)1/3 = a2/3, a > 0, Í%�X��ñ�Ý�� V = 4πa3/35�

10. 'bëÍ�5/ r�ÑiÖ, Í�T�â?xWëÍ2ý�,

�¿àÞg@g�5, OÍø/I5��

11. �J ∫ x

0

(

∫ u

0

f(t)dt)du =

∫ x

0

f(u)(x− u)du�

604 Ïèa ¥�5

×��, �J∫ x

0

(

∫ x1

0

· · · (∫ xn−1

0

(

∫ xn

0

f(t)dt)dxn) · · · )dx1

=1

n!

∫ x

0

f(u)(x− u)ndu�

12. �'&ÆBãì��5∫ 1

0

(

∫ y

0

(x2 + y2)dx)dy +

∫ 2

1

(

∫ 2−y

0

(x2 + y2)dx)dy

OØñ��øð�55�¬OÍÂ�

13. �øðì��5�5�, ¬O��5Â�∫ 4

2

(

∫ (20−4x)/(8−x)

4/x

(y − 4)dy)dx�

10.4 Green��§§§

Þ¥�5Ca�5 , b×¥�Ýn;, Ç�Þ3¿«î× ½ÝÞ¥�5, |h ½Ý\&Ýa�5¼�î�h��;ðÌ Green��§§§(Green’s theorem), 9Î ÝSFz»ó.�Green

(1793-1841) t\39]«Ý"D�3B�h�§�G,&Æ�+Û×Ðr�'λ ¿«î×@ð¿

âÝ��Tà(simple closed curve,Çuλ(a) = λ(b),vλ(t1) 6=λ(t2), ∀t1 6= t2, t1, t2 ∈ [a, b), Jλ(t), t ∈ [a, b], Xà��Ýà-Ì ×��Tà�¿«î×iÇ ×»)�J'�ÂÐó(A(x, y), B(x, y))3λîYYY`jjj(counterclockwise)]'Ýa�5|

∮

λ(Adx + Bdy)

��

10.4 Green�§ 605

��§§§4.1.(Green��§§§). 'S ¿«î×�/), λ S�×@ð¿âÝ��T`a, vÍ/Fù3S��ê'A(x, y)CB(x, y) Þ@ÂÐó, v3S�b=�Ý×$�0ó�J

(4.1)

∮

λ(Adx + Bdy) =

∫ ∫

Q

(∂B

∂x− ∂A

∂y)dxdy,

Í�Q ¿«îλCÍ/FXxW�/)�

u�J�

(4.2)

∮

λAdx =

∫ ∫

Q

∂A

∂ydxdy,

C

(4.3)

∮

λBdy =

∫ ∫

Q

∂B

∂xdxdy,

JhÞP¼��5½8�, -ÿ(4.1)PWñ�&Æ�EQ ×Ï×lÝ ½, J�(4.2)PWñ�Ç'

Q = {(x, y)|a ≤ x ≤ b, f(x) ≤ y ≤ g(x)},

Í�f�g/3[a, b]î=�, vf(x) ≤ g(x), ∀x ∈ [a, b]�´�ã@g�5�O¥�5

∫∫Q(∂A/∂y)dxdyAì:

∫ ∫

Q

∂A

∂ydxdy =

∫ b

a

(

∫ g(x)

f(x)

∂A

∂ydy)dx =

∫ b

a

A(x, y)

∣∣∣∣y=g(x)

y=f(x)

dx

=

∫ b

a

(A(x, g(x))− A(x, f(x)))dx�

êλ�â°I5: ì]Ýf�%�, î]g�%�, CÞkàa, A%4.1�

606 Ïèa ¥�5

x

y

O a b

C2 : y = g(x)

C1 : y = f(x)

%4.1.

.3kàaðî�a�5 0, Æ∮

λAdx =

∫

λ1

Adx +

∫

λ2

Adx =

∫ b

a

A(x, f(x))dx +

∫ a

b

A(x, g(x))dx

=

∫ b

a

(A(x, f(x))− A(x, g(x))dx = −∫ ∫

Q

∂A

∂ydxdy,

.h(4.2)PWñ�!§uQ ×ÏÞlÝ ½, ù�J�(4.3)PWñ�Æu×

½!` Ï×lCÏÞl, JGreen�§Wñ��yE´×�Ý ½, Green�§)Wñ, ¬ÍJ�ø�9�ÝP�, &Æ�a9D¡�

»»»4.1.�¿àGreen �§OI =∮λ(ydx + x2ydy), Í�λ +y(0,

0)�(1, 1) , ãy2 = x�y = xX�W��T`a��..A(x, y) = y, B(x, y) = x2y, v

∂B

∂x= 2xy,

∂A

∂y= 1,

Æ

I =

∫ 1

0

(

∫ y

y2

(2xy − 1)dx)dy = − 1

12�

»»»4.2.�OI =∮

λ(y+3x)dx+(2y−x)dy),Í�λ Yi4x2 +y2 =

4�ø��

10.4 Green�§ 607

��..A(x, y) = y + 3x, B(x, y) = 2y − x, v

∂B

∂x= −1,

∂A

∂y= 1,

Æ

I =

∫ ∫

Q

−2dxdy = −2A(Q),

Í�A(Q)�Yi4x2 + y2 = 4X��«��.hYi�Þ�5½ 1C2, Í«��y2π, ÆI = −4π�

»»»4.3.�OI =∫λ((5 − xy − y2)dx − (2xy − x2)dy), Í�λ cF

(0,0), (1,0), (1,1)C(10, 1)�°\�ø�, ]' Y`j��..A(x, y) = 5− xy − y2, B(x, y) = x2 − 2xy, v

∂B

∂x= 2x− 2y,

∂A

∂y= −x− 2y,

Æ

I = 3

∫ 1

0

∫ 1

0

xdxdy =3

2�

»»»4.4.'A(x, y)CB(x, y)/ =�Ý@ÂÐó, Í×$�0ó/D3v=�, vEXb(x, y)òy¿«îØ�/)S,

∂A

∂y=

∂B

∂x,

JGreen �§¼�∮

λ(Adx + Bdy) = 0,

Í�λ S��×@ð¿âÝ��T`a, vλÝ/Iù3S��

Í�nyGreen�§Ý×°aªeÿC×MÝD¡, �¢�Apostol (1969) pp.378-392�

608 Ïèa ¥�5

êêê ÞÞÞ 10.4

1. �¿àGreen�§, Oa�5∮

λ(y2dx + xdy), Í�

(i) λ |(0, 0), (2, 0), (2, 2)C(0, 2) cF�Ñ]�;

(ii) λ |(−1,−1), (1,−1), (1, 1)C(−1, 1) cF�Ñ]�;

(iii) λ |x(t) = 2 cos3 t, y(t) = 2 sin3 t, t ∈ [0, 2π], ¢óP�`a;

(iv) λ |(0, 0) iT, �5 2�i�

2. �O∮

λ(x2ydx+y3dx),Í�λ y = x�y3 = x2ã(0, 0)�(1, 1)

XxW��T`a�

10.5 ��óóó��ððð

3��ÐóÝ�5�, �ó�ð ×¥�Ý�5*», &9�ÓÝ�5, ??�¢Ã�ó�ð, �»W´��Ý�P.��h°ÎÃyì�2P:

(5.1)

∫ b

a

f(x)dx =

∫ g−1(b)

g−1(a)

f(g(t))g′(t)dt�

3Ï3.3;, yg =��Cf(g(t)) =�Ýf�ì, &ÆJ�îPWñ�EyËÍ�ó, &Æôb×v«(5.1)PÝ2P, ôµÎÞ¥

�5Ý�ó�ð2P, h2P�Þ×3x-y¿«î× ½S îÝ�5

∫∫S

f(x, y)dxdy, » ×3u-v¿«î× ½TÝÞ¥�5∫∫T

g(u, v)dudv�9ì&Æ-Þ��S�TÝn;Cf(x, y)�g(u, v)

Ýn;�

10.5 �ó�ð 609

Þ¥�5Ý�ó�ð, ´×îÝ�µ�Ó&9�h.bÞ�ó��ð, ùÇ�u3(5.1)P�, ©b×Ðóg�¨, ¨3ºbÞÐó,

|X,Y ��, hÞÐó�)x, y�u, vAì:

(5.2) x = X(u, v), y = Y (u, v)�î�ÞÐóÞu-v¿«î×F(u, v)Ì�x-y¿«î×F(x, y)��u-

v¿«î×/)T-Ì�x-y¿«î×/)S�b`ã(5.2)P��ÞP, ��u, v|xCy��, Çÿ

u = U(x, y), v = V (x, y)�î�ÞP�L�×ãx-y¿«,�u-v¿«ÝÌ ,Ì (5.2)PX�L�Ì ÝDÌ �1−1Ì ×©», ÇT��!ÝFÌ�S ��!ÝF�&ÆÞ©�ÊXCY =�Ðó, v∂X/∂u, ∂X/∂v, ∂Y/∂u,

∂Y/∂v, ��0ó/ =��EUCVô�v«Ý�'��I5&Æ@jÂÕÝÐó, /º��9°f��Þ¥�5Ý�ó�ð2P

(5.3)

∫ ∫

S

f(x, y)dxdy =

∫ ∫

T

f(X(u, v), Y (u, v))|J(u, v)|dudv�

îP��5Õ��ÝJ(u, v), �1î2P�Ýg′(t)X6�Ý��8!�h4ÌJacobian determinant(��PPP), T©ÌJacobian, ¸�y

J(u, v) =

∣∣∣∣∣∂X∂u

∂Y∂u

∂X∂v

∂Y∂v

∣∣∣∣∣ =

∣∣∣∣∣∂X∂u

∂X∂v

∂Y∂u

∂Y∂v

∣∣∣∣∣�

b`|∂(X,Y )∂(u,v)

ã�J(u, v)�(5.3)PÝJ�, �¢�Apostol (1974) Theorem 15.11�tÝG

�nyX,Y, UCV��'², $6'ãT�S�Ì 1−1vT (u, v)

6= 0��Äu©Î3×«� 0Ý/)�,hÌ � 1−1TJacobian

0, J(5.3)P)Wñ�S ×Î�vf(x, y) = 1, ∀(x, y) ∈ S,

J(5.2)PW

(5.4)

∫ ∫

S

dxdy =

∫ ∫

T

|J(u, v)|dudv�

610 Ïèa ¥�5

Ç¸Eh©», J�ô��|�¢ÃGreen�§, Apostol (1969)

Section 11.29 �Ý×(5.4)P�J��Q¡3Section 11.30, ãh©»0�(5.3) P�

Jacobian�Ú Þî�ó�ðÄ��, ãu-v¿«�x-y¿«, ×\�5½∆uC∆vÝÎ�Ý«��;.��ny9]«Ý¿¢�Õ, �¢�Apostol (1969) pp.394-396�&Æ¼:×°»��

»»»5.1.ÁÁÁ222ýýý(polar coordinates). �

x = r cos θ, y = r sin θ, r > 0, 0 ≤ θ < 2π,

Jh 3r-θ¿«îÎ�[0, a]×[0, 2π)��×�/îÝ1−1�ð�h�ð�Jacobian

J(r, θ) =

∣∣∣∣∣∂x∂r

∂y∂r

∂x∂θ

∂y∂θ

∣∣∣∣∣ =

∣∣∣∣∣cos θ sin θ

−r sin θ r cos θ

∣∣∣∣∣ = r(cos2 θ + sin2 θ) = r�

Æ»ð2PW ∫ ∫

S

f(x, y)dxdy =

∫ ∫

T

f(r cos θ, r sin θ)rdrdθ�

r-θ¿«î×Î�ET�x-y¿«î×G�,A%5.1�r = 0`Jacobian 0,

¬.r = 0�FÝ/)�«� 0, Æ�Å(»ð2PÝWñ�

rO

θ

xO

y

θ = ðó

r = ðó

x = r cos θy = r sin θ

r-`a(θ =ðó)θ-`a

(r=ðó)

%5.1. Á2ý�»ð

10.5 �ó�ð 611

u�5 ½Ý\&Îº½rTθ ðó, JÁ2ýÝ»ð-º�Ê)�»A, 3OÏ×ß§��5 a�¦Ý��, ÇO

∫ ∫

S

√a2 − x2 − y2dxdy,

Í�S = {(x, y)|x2 + y2 ≤ a2, x ≥ 0, y ≥ 0}�u�ð Á2ý, �5W ∫ ∫

T

√a2 − r2rdrdθ,

Í�T Î�[0, a]× [0, π/2)�î��5��|2O� πa3/6�

»»»5.2.aaaPPP��ððð (linear transformations). �Êì�aP�ð:

(5.5) x = Au + Bv, y = Cu + Dv,

Í�A,B,C,D ðó�JJacobian

J(u, v) = AD −BC�

Ý¸Jacobian� 0, &Æ�'AD − BC 6= 0, Ah��ã(5.5)P��uCv�æ¼¿�ÝÞàa, BÄaP�ð, ) ¿��Æu-v¿«î×

Î�, Bh�ð¡, W x-y¿«î×¿�°\�, �«� æ¼Î�Ý«�¶î|J(u, v)| = |AD −BC|��ð2P ∫ ∫

S

f(x, y)dxdy = |AD −BC|∫ ∫

T

f(Au + Bv, Cu + Dv)dudv�

Ü×Í»�¼:, �Ê�5∫∫

Se−(y−x)/(y+x)dxdy, Í�S ãà

ax + y = 2 CÞ2ý�XxW�ë��, �%5.2�ãy�5Õ��by − xCy + x, X|�

u = y − x, v = y + x�

��x = (v − u)/2, y = (v + u)/2, J(u, v) = −1/2�kO3u-v¿«îS�Ìé, .àax = 0Cy = 05½Ì�àau = vCu = −v, v

612 Ïèa ¥�5

àax + y = 2Ì�v = 2�ÆT ×ë��, 0 ≤ v ≤ 2, −v ≤ u ≤v�.h

∫ ∫

S

e(y−x)/(y+x)dxdy =1

2

∫ ∫

T

eu/vdudv,

1

2

∫ 2

0

(

∫ v

−v

eu/vdu)dv =1

2

∫ 2

0

v(e− e−1)dv = e− e−1�

uO

v

xO

y

u = −v u = v

v = 2

T

S

x + y = 2x = 12(v − u)

y = 12(v + u)

%5.2. aP�ð�Ì

êêê ÞÞÞ 10.5

1. �ÞÞ¥�5∫∫

Sf(x, y)dxdyBãÁ2ýÝ»ð, �îWÞg

@g�5, Í�S5½

(i) S = {(x, y)|x2 + y2 ≤ a2}, a > 0;

(ii) S = {(x, y)|x2 + y2 ≤ 2x};(iii) S = {(x, y)|a2 ≤ x2 + y2 ≤ b2}, 0 < a < b;

(iv) S = {(x, y)|0 ≤ y ≤ 1− x, 0 ≤ x ≤ 1};(v) S = {(x, y)|x2 ≤ y ≤ 1,−1 ≤ x ≤ 1}�

2. �|Á2ýÝ»ð, 5½O�ì�&�5, Í�a > 0�(i)

∫ 2a

0(∫ √2ax−x2

0(x2 + y2)dy)dx;

êÞ 613

(ii)∫ 1

0(∫ x

x2(x2 + y2)dy)dx;

(iii)∫ a

0(∫ x

0(x2 + y2)1/2dy)dx;

(iv)∫ a

0(∫√a2−y2

0(x2 + y2)dx)dy;

(v)∫ 1

−1(∫ 1

−1(x2 + y2)−1/2dx)dy�

3. �|Á2ýÝ»ð, �î�ì�&�5�(i)

∫ 1

0(∫ 1

0f(x, y)dy)dx;

(ii)∫ 1

0(∫ (1−x2)1/2

1−xf(x, y)dy)dx;

(iii)∫ 2

0(∫ x

√3

xf(

√x2 + y2)dy)dx;

(iv)∫ 1

0(∫ x2

0f(x, y)dy)dx�

4. 'S |(π, 0), (2π, π), (π, 2π)C(0, π) cFXxW�°\��|ÊÝaP�ðO

∫∫S(x− y)2 sin2(x + y)dxdy�

5. �OÞ¥�5

I(p, r) =

∫ ∫

S

(p2 + x2 + y2)−pdxdy,

Í�S = {(x, y)|x2 + y2 ≤ r2}�¬X�p�Â¸limr→∞ I(p, r)

D3�

6. �|ÊÝ�ó�ð, J�ì��PWñ�(i)

∫∫S

f(x + y)dxdy =∫ 1

−1f(u)du, S = {(x, y)||x|+ |y| ≤ 1};

(ii)∫∫

Sf(ax+by+c)dxdy = 2

∫ 1

−1

√1− u2f(u

√a2 + b2 +c)du,

S = {(x, y)|x2 + y2 ≤ 1}, a2 + b2 6= 0;

(iii)∫∫

Sf(xy)dxdy = log 2

∫ 2

1f(u)du, S 3Ï×é§, ã°

`axy = 1, xy = 2, y = xCy = 4xX�� ½�

7. �O+yx-y¿«, �eÎ«z = 2− x2 − y2 ��

614 Ïèa ¥�5

8. �O∫∫

S(1 + x2 + y2)−2dxdy, Í�S5½

(i) (x2 + y2)2 − (x2 − y2) = 0�%�Ý×�X�� ½;

(ii) |(0, 0), (2, 0)C(1,√

3) cF�ë��

9. �J

∫ 1

0

(

∫ 1

0

x2 − y2

(x2 + y2)2dy)dx 6=

∫ 1

0

(

∫ 1

0

x2 − y2

(x2 + y2)2dx)dy�

10. (i) ã

d

dxG(α(x), β(x), λ(x)) = α′(x)G1 + β′(x)G2 + λ′(x)G3,

�J

d

dx

∫ β(x)

α(x)

F (x, t)dt = β′(x)F (x, β(x))− α′(x)F (x, α(x))

+

∫ β(x)

α(x)

F1(x, t)dt�

(ii)ã∫ π

0(a+b cos t)−1dt = π/

√a2 + b2, a > b > 0,¬¿à(i)0

� ∫ π

0

cos t

(a + b cos t)3dt = −3π

2

ab

(a2 − b2)5/2�

11. �J ∫ ∫

S

e−(x2+y2)dxdy = ae−a2

∫ ∞

0

e−u2

a2 + u2du,

Í�S �¿«x ≥ a > 0, v¿àx2 + y2 = u2 + a2, y = vx ��ð�

10.6 {î¥�5 615

10.6 {{{îîî¥¥¥��555

¥�5ÝÃF�ãÞîè .Â�nîè , Í�n ≥ 3�ãyÍÚx��v«ÞîÝ�µ, X|&Æ©à�x�Ý��A10.1;X�, �'b×�L3nîè �ö�Ðóf , Jf3S

��5, Ì n¥�5, v|∫· · ·

∫

S

f T∫· · ·

∫

S

f(x1, · · · , xn)dx1 · · · dxn

��n = 3, &Æ|(x, y, z)ã�(x1, x2, x3), v|∫ ∫ ∫

S

f T∫ ∫ ∫

S

f(x, y, z)dxdydz

�ë¥�5�û�LÞ¥�5Ý]P, &Æ)�¢Ã$VÐó, �LÐóf�n

¥�5��f ��Ýf�ôv«Þ¥�5XmÝf�, �¢��§3.2�3ÊÝf�ì, n¥�5ô�Bãng@g�5O��5Â�Þ¥�5�Ý�ó�ð2P, ô�ñÇ.Â�n¥�5�'

x1 = X1(u1, · · · , un), · · · , xn = Xn(u1, · · · , un)�

�x = (x1, · · · , xn), u = (u1, · · · , un), X = (X1, · · · , Xn), Jî�9°�P�L�×ãnîè ��×/)TÌ�¨×nîè �/)SÝ'�ÂÐó

X : T → S�&Æ�'ÐóX 1− 1v3T�=��JnîÝ�5�ð2P

(6.1)

∫

S

f(X)dx =

∫

T

f(X(u))|J(u)|du,

Í�

(6.2) J(u) = detDX(u),

616 Ïèa ¥�5

�

DX(u) =

D1X1(u) D2X1(u) · · · DnX1(u). . . . . .

.... . .

D1Xn(u) D2Xn(u) · · · DnXn(u)

�

A!3ÞîÝ�µ, �ð2P(6.1)X 3T��1 − 1�ð, vJacobian J(u)3T�/� 0Wñ�Qu3T �×“��”(3nîè �&Æ|��ÝÌñã�«�) 0Ý�5�, X� 1 − 1, T39Ë/)�J(u) = 0, J�ð2P)Wñ�n = 3, &Æ|(x, y, z)ã�(x1, x2, x3), |(u, v, w)ã�(u1, u2,

u3), |(X, Y, Z)ã�(X1, X2, X3), Jë¥�5��ð2PW

∫ ∫ ∫

S

f(x, y, z)dxdydz

=

∫ ∫ ∫

T

f(X(u, v, w), Y (u, v, w), Z(u, v, w))|J(u, v, w)|dudvdw,

Í�

J(u, v, w) =

∂X∂u

∂Y∂u

∂Z∂u

∂X∂v

∂Y∂v

∂Z∂v

∂X∂w

∂Y∂w

∂Z∂w

�

ÝOÎp��PÝÂ, &��ê×°ÄnaP�óÝáI�9ì&Æ¼:ËÍëî�¥�Ý�ð�

»»»6.1.iiiÖÖÖ222ýýý(cylindrical coordinates). |r, θ, zã�u, v, w, v�

x = r cos θ, y = r sin θ, z = z�

Ç|x, y�Á2ý�ðxCy,�z��k¸h�ð 1−1, r6 Ñ,

�θ ∈ [0, 2π)�%6.1�î×3r-θ-zè �Ý�]�, Ì�x-y-zè �Ý�Ï�

10.6 {î¥�5 617

EiÖ2ý, Jacobian

J(r, θ, z) =

∣∣∣∣∣∣∣

cos θ sin θ 0

−r sin θ r cos θ 0

0 0 1

∣∣∣∣∣∣∣= r(cos2 θ + sin2 θ) = r�

Æ�ð2P ∫ ∫ ∫

S

f(x, y, z)dxdydz =

∫ ∫ ∫

T

f(r cos θ, r sin θ, z)rdrdθdz�

r = 0`, Jacobian = 0, ¬.9ËFÝ/)3ë�è �� 0, Æ�Å(�ð2PÝb[P�

r

θ

z

O y

x

O

z

r =ðó

θ =ðó

z =ðóx = r cos θ

y = r sin θ

z = z

%6.1. iÖ2ý�»ð

»»»6.2.¦¦¦222ýýý(spherical coordinates). &Æ|ρ, θ, φã�u, v, w,v�

x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ�

%6.2�îh�ðÝ¿¢�L�k¸h�ð 1 − 1, ãρ > 0, 0 ≤ θ < 2π, 0 ≤ φ < π�E�

�ðóc1, c2, c3, `«ρ = c1, �|×æF ¦T, �5 c�¦«;

`«θ = c2, ×;Äz��¿«; �`«φ = c3, �×|z �T��Ñi��Æ×ρ-θ-φè ��]�, 3x-y-zè �Ý�ÏA%6.2Xî�

618 Ïèa ¥�5

ρ

θ

φ

O y

x

z

O

ψ =ðó

θ =ðóρ =ðóx = ρ sin φ cos θ

y = ρ sin φ sin θz = ρ cos φ (x, y, z)

φ

θ

ρ

ρ sin φ ρ cos φ

-

%6.2. ¦2ý�»ð

9Ë�ð�Jacobian

J(ρ, θ, φ) =

∣∣∣∣∣∣∣

cos θ sin φ sin θ sin φ cos φ

−ρ cos θ sin φ ρ cos θ sin φ 0

ρ cos θ cos φ ρ sin θ cos φ −ρ sin φ

∣∣∣∣∣∣∣= −ρ2 sin φ�

.φ ∈ [0, π), sin φ ≥ 0, Æ|J(ρ, θ, φ)| = ρ2 sin φ, v�ð2PW ∫ ∫ ∫

S

f(x, y, z)dxdydz

=

∫ ∫ ∫

T

f(ρ cos θ sin φ, ρ sin θ sin φ, ρ cos φ)ρ2 sin φdρdθdφ�

��2, 4φ = 0`J(ρ, θ, φ) = 0, ¬.3ëîè �, φ = 0�FÝ/)�� 0, Æ�ð2P)b[�

¿à¥�5ô�O×nîñ�Ý��'b×nîÝ/)S,

JS��

V (S) =

∫· · ·

∫

S

dx1 · · · dxn,

QkîPWñ, Sô6b°§×��

»»»6.3.�O3eÎ«z = 4 − x2 − y2�ì, �3¿«z = 4 − 2x�îÝñ�Ý��

10.6 {î¥�5 619

��.B0�hñ�Ý�%, �ÿ

4− 2x ≤ z ≤ 4− x2 − y2,

�B��z, ÿ4− 2x = 4− x2 − y2, Ty2 = 2x− x2 = x(2− x) ≥ 0,

Æ−√

2x− x2 ≤ y ≤√

2x− x2, 0 ≤ x ≤ 2�.h��

V =

∫ 2

0

∫ √2x−x2

−√2x−x2

∫ 4−x2−y2

4−2x

dzdydx=

∫ 2

0

∫ √2x−x2

−√2x−x2

(2x− x2 − y2)dydx

=

∫ 2

0

(2xy − x2y − 1

3y3)

∣∣∣∣y=√

2x−x2

y=−√2x−x2

dx =4

3

∫ 2

0

(2x− x2)3/2dx =π

2�

»»»6.4.�Sn(a)�×nî�5aÝ@T¦, Ç�

Sn(a) = {(x1, · · · , xn)|x21 + · · ·+ x2

n ≤ a2},

v�

Vn(a) =

∫· · ·

∫

Sn(a)

dx1 · · · dxn,

�Sn(a)��J

(6.3) Vn(a) =πn/2

Γ(n/2 + 1)an, ∀n ≥ 1�

JJJ��.un = 1, JV1(a) = 2a� [−a, a]��, �

Γ(1

2+ 1) =

1

2Γ(

1

2) =

√π

2,

Æ(6.3)PWñ�un = 2, JV2(a) = πa2��5 a�i«�, �

Γ(2

2+ 1) = Γ(2) = Γ(1) = 1,

Æ(6.3)P)Wñ�9ì&ÆÞJ�n ≥ 3`(6.3)PWñ�

620 Ïèa ¥�5

&Æ�J�E∀a > 0, Cn ≥ 3,

(6.4) Vn(a) = anVn(1)�

ùÇ�5 a�¦Ý�� 5Î1�¦Ý��Ýan¹�àÌî9QÎEÝ�¿à�ó�ð, �x1 = au1, x2 = au2, · · · , xn = aun,

�ÞSn(1)Ì�Sn(a), vJacobian an�.h

Vn(a) =

∫· · ·

∫

Sn(a)

dx1 · · · dxn

=

∫· · ·

∫

Sn(1)

andu1 · · · dun = anVn(1)�

ÇÿJ(6.4)P�ÆkJ(6.3)P, ©mJ�

(6.5) Vn(1) =πn/2

Γ(n/2 + 1)

Ç��¨.

x21 + · · ·+ x2

n ≤ 1,

uv°u

x21 + · · ·+ x2

n−2 ≤ 1− x2n−1 − x2

n, v x2n−1 + x2

n ≤ 1,

Æ�ÞVn(1)¶W×(n− 2)¥�5C×Þ¥�5, Ç

(6.6) Vn(1) =

∫ ∫

x2n−1 + x2

n ≤ 1

(

∫· · ·

∫

x21 + · · ·+ x2

n−2 ≤ 1− x2n−1 − x2

n

dx1 · · · dxn−2)dxn−1dxn,

Í�/IÝ(n − 2)¥�5, Ç ×�5b�(n − 2)î¦��, Í�b =

√1− x2

n−1 − x2n, Æ�y

Vn−2(b) = bn−2Vn−2(1) = (1− x2n−1 − x2

n)n/2−1Vn−2(1)�

u5½|xCyã�xn−1Cxn, J(6.6)PW

Vn(1) = Vn−2(1)

∫ ∫

x2 + y2 ≤ 1

(1− x2 − y2)n/2−1dxdy�

êÞ 621

�¿àÁ2ýÝ�ð, -ÿ

Vn(1) = Vn−2(1)

∫ 2π

0

∫ 1

0

r(1− r2)n/2−1drdθ = Vn−2(1)2π

n�

.hVn(1)��ì�L]2P:

(6.7) Vn(1) =2π

nVn−2(1), ∀n ≥ 3,

vV1(1) = 2, V2(1) = π�t¡¿àhû°, �pÿÕ(6.7)P��Ç (6.5)P�J±�

êêê ÞÞÞ 10.6

1. �Oì�&¥�5�(1)

∫∫∫S

xy2z3dxdydz, Í�S ã`«z = xy, C¿«y = x,

y = 0, x = 1Cz = 0X�� ½�(2)

∫∫∫S(1 + x + y + z)−3dxdydz, Í�S ãëÍ2ý¿«, C

¿«x + y + z = 1X�� ½�(3)

∫∫∫S

xyzdxdydz,Í�S = {(x, y, z)|x2+y2+z2 ≤ 1, x ≥ 0,

y ≥ 0, z ≥ 0}�(4)

∫∫∫S(x2/a2 + y2/b2 + z2/c2)dxdydz, Í�S Y¦x2/a2 +

y2/b2 + z2/c2 = 1X�� ½�(5)

∫∫∫S(x2 + y2)1/2dxdydz, Í�S +yi�z2 = x2 + y2�

î�I, �¿«z = 1 � ½�(6)

∫∫∫S|xyz|dxdydz,Í�S Y¦x2/a2+y2/b2+z2/c2 = 1X

�� ½�(7)

∫∫∫S(x2 + y2 + z2)dxdydz, Í�S x2 + y2 + z2 = r2X�

�� ½�(8)

∫∫∫S

zdxdydz, Í�S ãx2 + y2 ≤ z2, x2 + y2 + z2 ≤1Cz ≥ 0X�L�� ½�

622 Ïèa ¥�5

2. �5½Þì�&¥�5; �Ey�5�(1)

∫ 1

0

∫ 1−x

0

∫ x+y

0f(x, y, z)dzdydx�

(2)∫ 1

−1

∫ √1−x2

−√1−x2

∫ 1√x2−y2 f(x, y, z)dzdydx�

(3)∫ 1

0

∫ 1

0

∫ x2+y2

0f(x, y, z)dzdydx�

3. �|iÖ2ýÝ�ð5½Oì�&¥�5�(1)

∫∫∫S(x2 + y2)dxdydz, Í�S `«x2 + y2 = 2z, C¿

«z = 2X�� ½�(2)

∫∫∫S

dxdydz, Í�S ëÍ2ý¿«, `«z = x2 + y2, C¿«x + y = 1X�� ½�(3)

∫∫∫S(y2 + z2)dxdydz, Í�S 93x-y¿«, |z �T�,

9�5 a, { h�Ñë��CÍ/I�

4. �|¦2ýÝ�ð, 5½Oì�&¥�5�(1)

∫∫∫S

dxdydz, Í�S |æF ¦T, �5 a�@T¦�(2)

∫∫∫S

dxdydz, Í�S +yÞ|æF ¦T, �55½ aCb, 0 < a < b, �!T¦ ÝI5�(3)

∫∫∫S((x − a)2 + (y − b)2 + (z − c)2)1/2dxdydz, Í�S

|(a, b, c) ¦T, �5 k�@T¦�

5. �5½Oì�&ñ�S��, CX��Ðóf3S�¥�5�(1) S ã¿«x = −1, x = 2, y = 0, y = 3, z = 1Cz = 4X��, f(x, y, z) = x− 2y + z�(2) S ãiÖx2 + y2 = 16CÞ¿«z = 0Cz = 2X��,

f(x, y, z) = xz + yz�(3) S ãÖ�x2 = zCx2 = 4−z Þ¿«y = 0Cz+2y = 4X��, f(x, y, z) = 2x− z�

êÞ 623

(4) S ã`«z = y/(1 + x2)C¿«x = 0, y = 0, z = 0,

Cx + y = 1X��, f(x, y, z) = y + x2y�(5) S 3Ï×ß§, +yiÖx2 + y2 = a2, Cy2 + z2 = a2 ,

f(x, y, z) = xyz�(6) S +yë2ý¿«C

√x/a +

√y/b +

√z/c = 1 ,

f(x, y, z) = xyz�

6. �Oì�&¥�5�(1)

∫∫∫S(x+y+z)x2y2z2dxdydz, S +yx+y+z ≤ 1, x ≥ 0,

y ≥ 0, z ≥ 0, � ½�(2)

∫∫∫S(x2 + y2 + (z − 2)2)−1dxdydz, S x2 + y2 + z2 = 1X

�� ½�(3)

∫∫∫S(x2 + y2 + (z − 1

2)2)−1dxdydz, S x2 + y2 + z2 = 1X

�� ½�

7. �O|¦«x2 + y2 + z2 = 5 c, �3eÎ«x2 + y2 = 4z�îÝ ½Ý��

8. �O+yÞiÖx2 + z2 ≤ 1, Cy2 + z2 ≤ 1 � ½Ý��

9. �O+yx-y¿«, iÖx2 + y2 = 2x, Ci�z =√

x2 + y2 � ½Ý��

10. �O((x2 + y2)1/2 − 1)2/a2 + z2/b2 ≤ 1, Í�a < 1, X��ñ��

11. �O+yx2/a2 + y2/b2 = z�z = h, Í�h > 0, ��

12. �O+yY¦x2/a2 +y2/b2 +z2/c2 = 1,�¿«lx+my+nz =

p ��

13. �OI =∫∫∫

Scos(ax+ by + cz)dxdydz, Í�S x2 + y2 + z2 ≤

1�

624 Ïèa ¥�5

14. E∀a > 0, Cn ≥ 1, �

Sn(a) = {(x1, · · · , xn)||x1|+ · · ·+ |xn| ≤ a}�n = 2, h ×|(0, a), (a, 0), (−a, 0), (0,−a) cF�Ñ]��Vn(a)�Sn(a)��(i) �JVn(a) = anVn(1);

(ii) En ≥ 2, �ÞVn(1)�îW×Í×î�5, C×(n− 1)î�5�à), vJ�

Vn(1) = Vn−1(1)

∫ 1

−1

(1− |x|)n−1dx =2

nVn−1(1);

(iii) �JVn(a) = 2nan/n!�

15. E∀a > 0, Cn ≥ 2, �

Sn(a) = {(x1, · · · , xn)||xi|+ |xn| ≤ a, ∀i = 1, · · · , n− 1},

ê�Vn(a)�Sn(a)��(i) �0S2(1);

(ii) �JVn(a) = anVn(1);

(iii) �ÞVn(1)�îW×Í×î�5, C×(n − 1)î�5�à), ¬J�Vn(a) = 2nan/n�

16. (i) 3»6.4�, �ÞVn(1)�îW×Í×î�5, C×(n − 1)î�5�à), ¬J�

Vn(1) = 2Vn−1(1)

∫ 1

0

(1− x2)(n−1)/2dx;

(ii) �¿à(i)C(6.5)PJ�∫ π/2

0

cosn tdt =

√π

2

Γ((n + 1)/2)

Γ(n/2 + 1) �

10.7 �¥�5 625

10.7 ��¥¥¥��555

A!��óÝ�µ, &Æ6Þ¥�5U"ÕXÛ��5�&Æx�)bËË��5, Í× Ðó� b&, ÍÞ �5 ½� b&�&Æ�:Ï×Ë��5�3�9ó&ÆaOÝ�5, Í�5

Õ�f ��3�5 ½S�=�, T©3×°âÒÝFTº½Ø×`a�=�(�âP�LT� b&)�¬�¡£×Ë�µ, 9°»²ÝF�«�6 ë�&Æ�|×� ½s�â9°»²ÝF,

Of3S \ s ��5, Q¡s�«��0`, Of3S \ s ��5ÝÁ§�uhÁ§D3, JÞhÁ§Â�L f3SîÝ��5�ãy&Æ�ThÁ§, ��SÝ]PPn, .h&ÆÞ�×´úÝ§×�Ç�O�GÎf , v|f |��5D3(8y3ùó��O�E[e, �&f�[e)�&Æ�B�ì�§�

��§§§7.1.'�5 ½S b&vS�«�D3�ê'D3×ó��¦v�TÝS�� ½{Sn, n ≥ 1}, ÇS1 ⊂ S2 ⊂ · · · ⊂Sn ⊂ · · · ⊂ S, ¸ÿf(x, y)3N×Sn/b�Lv=��ê'Sn�«�A(Sn)��S�«�A(S), Ç

(7.1) limn→∞

A(Sn) = A(S),

v'D3×M > 0, ¸ÿ

(7.2)

∫ ∫

Sn

|f(x, y)|dxdy ≤ M, ∀n ≥ 1�

(Ç|f |3Sn, n ≥ 1, Ý�5 í8b&)�J

(7.3) I = limn→∞

∫ ∫

Sn

f(x, y)dxdy

D3, v�{Sn}�óãPn�

&Æ-|î�§��I, ¼�Lf3Sî��5�Ç

(7.4) I =

∫ ∫

S

f(x, y)dxdy�

626 Ïèa ¥�5

3J��§7.1�G, &Æ:×°»�, |1�h�§��°�

»»»7.1.�ÊÐó

f(x, y) = log√

x2 + y2�hÐó(x, y) → (0, 0)`, ÍÂ��−∞� ÆuÞf3×�â(0, 0)� ½S = {(x, y)|x2 + y2 ≤ 1} �5, &ÆµÄ6�Þ�5 ½t�×�âæFÝ ½sn, Í�n → ∞ `, sn�«��0�uf3S \ snÝ�5D3, vÁ§ùD3, J|hÁ§Â�f3Sî��5«{Î)§Ý�Q, hÁ§Ä6�snÝóãPn��»A, ãsn ×|(0, 0) iT, �5 1/n�i8, v�Sn =

S \ sn�J¿àÁ2ýÝ»ð�ÿ∫ ∫

Sn

|f(x, y)|dxdy =

∫ 1

1/n

(

∫ 2π

0

r| log r|dθ)dr = 2π

∫ 1

1/n

r| log r|dr

u�g(r) = r| log r|, r > 0, g(0) = 0, Jg ×3r ≥ 0 �=��Ðó�Æ

∫ ∫

Sn

|f(x, y)|dxdy ≤ 2π

∫ 1

0

r| log r|dr = M < ∞�

.hã�§7.1á∫ ∫

S

log√

x2 + y2dxdy = limn→∞

∫ ∫

Sn

f(x, y)dxdy

=

∫ 1

0

(

∫ 2π

0

r log rdθ)dr = 2π

∫ 1

0

r log rdr = 2π(1

2r2 log r − 1

4r2)

∣∣∣1

0

= −π

2�

»»»7.2.�Ê�5

(7.5)

∫ ∫

S

1

(x2 + y2)α/2dxdy,

10.7 �¥�5 627

Í�S = {(x, y)|x2 + y2 ≤ a2}, a > 0��f(x, y) = (x2 + y2)−α/2,

Sn = {(x, y)|1/n2 < x2 + y2 ≤ a2}�J∫ ∫

Sn

|f(x, y)|dxdy =

∫ a

1/n

(

∫ 2π

0

r · r−αdθ)dr = 2π

∫ a

1/n

r1−αdr�

ãÏÚa»5.9á,∫ a

0r1−αdrD3, uv°uα < 2�ãM =

2π∫ a

0r1−αdr, ã�§7.1á, (7.5)P��5[euv°uα < 2�

ãî��&Æ�ÿÕ×¾½Þ¥��5[e��5f�(¬�&Ä�f�)Aì:

'Ðóf(x, y)3b& ½S, tÝ��3Ø×F², hF Ý�-ã (0, 0), /=��ê'D3×M > 0, C×ðóα < 2, ¸ÿ

(7.6) |f(x, y)| < M

(x2 + y2)α/2, ∀(x, y) ∈ S \ {(0, 0)}�

J∫∫

Sf(x, y)dxdy[e�

»»»7.3.&Æ�Þî»Ý��, .ÂÕëîÝ�µ��Êë¥�5∫ ∫ ∫

S

1

(x2 + y2 + z2)α/2dxdydz,

Í�b& ½S�âæF, ¿à¦2ý�»ð, æ�5W ∫ ∫ ∫

T

ρ2−α sin φdρdθdφ,

Í�T ETS�(ρ, θ, φ)�P��ûî»�D¡�ÿuα < 3Jæ�5[e�!ø2, 'f(x, y, z)tÝ3(0,0,0)², 3b& ½Sî=��ê

'D3×M > 0C×ðóα < 3¸ÿ

|f(x, y, z)| ≤ M

(x2 + y2 + z2)α/2,

J∫∫∫

Sf(x, y, z)dxdydz[e�

628 Ïèa ¥�5

?×�2, 'S ×b& ½, g3Sî=�, Juα < 3,

∫ ∫ ∫

S

g(x, y, z)

(x2 + y2 + z2)α/2dxdydz

[e�

u�5Õ�3Ø×`aî/ P§�(T�P§�), J��5ôb��D3�t��Ý�µ , �5Õ�3×àa�Ý×I5, ÉA1y�, P§��h`u

|f(x, y)| ≤ M

|x|α , ∀(x, y) ∈ S, x 6= 0,

Í�M > 0, α < 1, J∫∫

Sf(x, y)dxdyD3�ukJ�, ©��

ÞS�*×�âS ∩ y��]�, v�h�]��«��0Ç��

��§§§7.1 ��JJJ��.S1 ⊂ S2 ⊂ · · · , v(7.2)PWñ, Æ

∫∫Sn|f(x, y)|dxdy ×��

�¦vb&�ó�,.hÞ�f�Wñ,Çn →∞`,Á§D3�ãÏÚa�§1.1, ∀ε > 0, D3×n0 ≥ 1, ¸ÿm > n ≥ n0`,

∫ ∫

Sm

|f(x, y)|dxdy −∫ ∫

Sn

|f(x, y)|dxdy(7.7)

=

∫ ∫

Sm\Sn

|f(x, y)|dxdy < ε�

�

In =

∫ ∫

Sn

f(x, y)dxdy,

J.∣∣∣∫ ∫

Sm

f(x, y)dxdy −∫ ∫

Sn

f(x, y)dxdy∣∣∣

=∣∣∣∫ ∫

Sm\Sn

f(x, y)dxdy∣∣∣≤

∫ ∫

Sm\Sn

|f(x, y)|dxdy < ε,

10.7 �¥�5 629

∀m > n ≥ n0, Æ{In, n ≥ 1}ù��Þ�f��.h

I = limn→∞

∫ ∫

Sn

f(x, y)dxdy

D3�yì�J�Ý-ÎI�Â�{Sn, n ≥ 1}�ó°Pn�'T S �

×�T�/, vf3T�=��M |f |3T��×î&�J∣∣∣∫ ∫

T


T∩Sn

f(x, y)dxdy∣∣∣≤

∫ ∫

T\Sn

|f(x, y)|dxdy

≤ MA(T \ Sn) ≤ MA(S \ Sn) = M(A(S)− A(Sn))�

êã�'limn→∞ A(Sn) = A(S), ÿ

(7.8)

∫ ∫

T

f(x, y)dxdy = limn→∞

∫ ∫

T∩Sn

f(x, y)dxdy�

EÐó|f |¥�î�.0, v¿à(7.2)Pÿ∫ ∫

T

|f(x, y)|dx = limn→∞

∫ ∫

T∩Sn

|f(x, y)|dxdy(7.9)

≤ limn→∞

∫ ∫

Sn

|f(x, y)|dxdy ≤ M�

hÇÞ(7.2)P.ÂÕ´×�ÝS��/�(7.7)Pô�.Â�¿à(7.8)P, ÿE∀n ≥ n0(hn0�{Sn, n ≥

1}Pn),

∣∣∣∫ ∫

T


T∩Sn

f(x, y)dxdy∣∣∣(7.10)

= limm→∞

∣∣∣∫ ∫

T∩Sm


T∩Sn

f(x, y)dxdy∣∣∣

= limm→∞

∣∣∣∫ ∫

T∩(Sm\Sn)

f(x, y)dxdy∣∣∣≤ lim

m→∞

∫ ∫

Sm\Sn

|f(x, y)|dxdy

= limm→∞

(

∫ ∫

Sm

|f(x, y)|dxdy −∫ ∫

Sn

|f(x, y)|dxdy) < ε�

630 Ïèa ¥�5

¨�{Tm, m ≥ 1} S�×ó��¦Ý�T�/, ¸ÿf3N×Tm=�, vlimm→∞ A(Tm) = A(S)�.ã(7.9)P

∫ ∫

Tm

|f(x, y)|dxdy ≤ M,

Æ

J = limm→∞

∫ ∫

Tm

f(x, y)dxdy

D3�.hE∀ε > 0, ©�mÈ�,

|J −∫ ∫

Tm

f(x, y)dxdy| < ε�

�ã(7.10)P, ©�mCnÈ�,

|J −∫ ∫

Tm∩Sn

f(x, y)dxdy| < 2ε�

ÞTm�Sn��!ð, �ÿ©�m�nÈ�,

|I −∫ ∫

Tm∩Sn

f(x, y)dxdy| < 2ε�

�)îÞ��PÇÿ|J − I| < 4ε, ∀ε > 0�ÆJ = I, ÿJ�

Íg&Æ¼:�5 ½� b&ÝÏÞË��5��2, &Æ¬��Êt×�Ý�µ, �©�E&Æ@jîtð$ÕÝ�µ,

è�×[ePÝ¾½°�'S ×P&Ý ½, vÐóf3Sî=��)|×ó�S ��

��¦Ý�/)S1 ⊂ S2 ⊂ · · · ⊂ S¼¿�S, Í�N×Sn/' �Tvb&�¬.S� b&,

limn→∞

A(Sn) = A(S)

3hµ�×�b�LÝ�X|&Æ��'îPWñ, ��OS�N×�Tvb&Ý�/, Ä6�K�â3Í×Sn��»A, uS JÍx-y¿«, JSn�ãW|(0, 0) iTv�5 n �i�u

limn→∞

∫ ∫

Sn

f(x, y)dxdy

10.7 �¥�5 631

D3, v�{Sn, n ≥ 1}�óãPn, JhÁ§Â-�L f3S îÝ�5,v|

∫∫S

f(x, y)dxdy��&Æb9ìÝ¾½�§,ÍJ�.v«�§7.1Æ¯��

��§§§7.2.'�5 ½S� b&�uD3×ó��¦v�TÝS�� ½{Sn, n ≥ 1}, ¸ÿS �N×�Tv b&Ý�/, Ä�â3Ø×Sn�, v'D3×M > 0, ¸ÿ∫ ∫

Sn

|f(x, y)|dxdy ≤ M, ∀n ≥ 1�

J

I = limn→∞

∫ ∫

Sn

f(x, y)dxdy

D3, v�{Sn}�óãPn�

&Æ-|î�§��I® f3Sî��5�3ÏÚa»5.7, &Æ��×��5�Ý¥��, Ç

(7.11)

∫ ∞

0

e−x2/2dx =√

π/2�

ã�ó�ðá, (7.11)P��y

(7.12)

∫ ∞

0

e−x2

dx =√

π/2�

9ì&Æ¼:A¢Bã¥�5ÿÕ(7.12)P�

»»»7.4.�J(7.12)PWñ�JJJ��.�Ê¥�5

∫∫Sn

e−(x2+y2)dxdy, Í�S JÍx-y¿«, �Sn |(0, 0) iT, �5 n�i8�J9Ë{Sn, n ≥ 1}�Q��&ÆXm�¨¿àÁ2ý��ðÿ∫ ∫

Sn

e−(x2+y2)dxdy =

∫ ∫

x2+y2≤n2

e−(x2+y2)dxdy

=

∫ n

0

(

∫ 2π

0

re−r2

dθ)dr = 2π

∫ n

0

re−r2

dr

= −πe−r2∣∣∣n

0= π(1− e−n2

) ≤ π�

632 Ïèa ¥�5

Æ�5Õ�f(x, y) = e−(x2+y2), 3SnîÝ�5 í8b&, .hã�§7.2á, f3SîÝ�5D3��n →∞, Çÿ

∫ ∫

S

e−(x2+y2)dxdy = limn→∞

π(1− e−n2

) = π�

¨×]«, uãTm = {(x, y)| − m ≤ x ≤ m,−m ≤ y ≤ m} ×Ñ]�, Jm → ∞`, ã�§7.2á, f3TmîÝ�5�Á§) π�¬∫ ∫

Tm

e−(x2+y2)dxdy =

∫ m

−m

e−x2

dx

∫ m

−m

e−y2

dy = (

∫ m

−m

e−x2

dx)2�

Æ

limm→∞

∫ ∫

Tm

e−(x2+y2)dxdy = (

∫ ∞

−∞e−x2

dx)2 = π,

.h

(7.13)

∫ ∞

−∞e−x2

dx =√

π,

ãhñÇêÿ∫ ∞

0

e−x2

dx =1

2

∫ ∞

−∞e−x2

dx =√

π/2�

J±�

.hBãÞ¥�5, &Æ�ÿÕh×5��¥�Ý��5ÝÂ�ãye−x2

��5¬&��Ðó, Æukà#Oh��5ÂÎ�æpÝ�!ñ×è, �§7.1C�§7.2�OÞ¥�5Ý��5, ÍM»K

Î�){àÆÝ�¬u�áhÞ�§, ÞA¢ÿÕ(7.13)P? &Æ¼::9ìÝ®°, h°Ê)�5 ½� b&��5�E∀b > 0, �

I(b) =

∫ b

−b

e−u2

du�

10.7 �¥�5 633

J

I2(b) =

∫ b

−b

e−x2

dx

∫ b

−b

e−y2

dy =

∫ ∫

Sb

e−(x2+y2)dxdy,

Í�Sb = [−b, b]× [−b, b]�&Æ�Olimb→∞ I(b)��

C1 = {(x, y)|x2 + y2 ≤ b2},C2 = {(x, y)|x2 + y2 ≤ 2b2},

JC1 ⊂ Sb ⊂ C2�êe−(x2+y2) > 0, Æ∫ ∫

C1

e−(x2+y2)dxdy < I2(b) <

∫ ∫

C2

e−(x2+y2)dxdy�

�x = r cos θ, y = r sin θ, J∫ ∫

C1

e−(x2+y2)dxdy =

∫ 2π

0

∫ b

0

re−r2

drdθ

=

∫ 2π

0

1

2(1− e−b2)dθ = π(1− e−b2),

!§ ∫ ∫

C2

e−(x2+y2)dxdy = π(1− e−2b2)�

.hπ(1− e−b2) < I2(b) < π(1− e−2b2)�

�b →∞, ãô^�§ÿlimb→∞ I2(b) = π, Ælimb→∞ I(b) =√

π�

Íg�Êì��5:∫ 1

0

xa−1(1− x)b−1dx�

a, b > 1, h ×Ñð�5, ¬a < 1Tb < 1, Jh ×��5�9ì&Æ¼:, ¯@î©�a, b > 0, Jh��5[e�E�×c ∈ (0, 1), 0 < x ≤ c`,

(1− x)b−1 ≤{

1, b ≥ 1,

(1− c)b−1, 0 < b ≤ 1,

634 Ïèa ¥�5

�c ≤ x ≤ 1`,

xa−1 ≤{

1, a ≥ 1,

ca−1, 0 < a < 1�Æ

0 ≤∫ 1

0

xa−1(1− x)b−1dx

=

∫ c

0

xa−1(1− x)b−1dx +

∫ 1

c

xa−1(1− x)b−1dx

≤max{1, (1− c)b−1}∫ c

0

xa−1dx + max{1, ca−1}∫ 1

c

(1− x)b−1dx

≤max{1, (1− c)b−1}ca

a+ max{1, ca−1}(1− c)b

b< ∞�

&Æ-�LÐóB(a, b)

B(a, b) =

∫ 1

0

xa−1(1− x)b−1dx, a, b > 0,

vÌ� betaÐó�A!∫∞

0e−x2

dx,ôm¢Ã¥�5��O�B(a, b),

�ì»�

»»»7.5.�J

(7.14) B(a, b) =Γ(a)Γ(b)

Γ(a + b)�

JJJ��.�

f(x, y) = x2a−1y2b−1e−(x2+y2), x ≥ 0, y ≥ 0,

v�

(7.15) I(t) =

∫ t

0

∫ t

0

f(x, y)dxdy�

J ∫ ∫

C1

f(x, y)dxdy ≤ I(t) ≤∫ ∫

C2

f(x, y)dxdy,

10.7 �¥�5 635

Í�

C1 = {(x, y)|x2 + y2 ≤ t2, x ≥ 0, y ≥ 0},C2 = {(x, y)|x2 + y2 ≤ 2t2, x ≥ 0, y ≥ 0}�

�x = r cos θ, y = r sin θ, J

∫ ∫

C1

f(x, y)dxdy =

∫ π/2

0

∫ t

0

(r cos θ)2a−1(r sin θ)2b−1e−r2

rdrdθ

=

∫ t

0

r2a+b−1e−r2

dr

∫ π/2

0

cos2a−1 θ sin2b−1 θdθ

=1

2

∫ t2

0

ua+b−1e−udu

∫ π/2

0

cos2a−1 θ sin2b−1 θdθ,

Í�t¡×�PWñÎ¿àÕ�r2 = u�.∫ ∞

0

ua+b−1e−udu = Γ(a + b),

Æ

limt→∞

∫ ∫

C1

f(x, y)dxdy =1

2Γ(a + b)

∫ π/2

0

cos2a−1 θ sin2b−1 θdθ�

!§�ÿ

limt→∞

∫ ∫

C2

f(x, y)dxdy =1

2Γ(a + b)

∫ π/2

0

cos2a−1 θ sin2b−1 θdθ�

�ãô^�§Çÿ

limt→∞

I(t) =1

2Γ(a + b)

∫ π/2

0

cos2a−1 θ sin2b−1 θdθ(7.16)

= Γ(a + b)

∫ 1

0

va−1(1− v)b−1dvΓ(a + b)B(a, b),

636 Ïèa ¥�5

Í�&ÆêàÕ�ó�ð�cos2 θ = v�¬ã(7.15)P, v�w = x2,

s = y2, ÿ

limt→∞

I(t) =

∫ ∞

0

∫ ∞

0

x2a−1y2b−1e−(x2+y2)dxdy(7.17)

=

∫ ∞

0

x2a−1e−x2

dx

∫ ∞

0

y2b−1e−y2

dy

= (1

2

∫ ∞

0

wa−1e−wdw)(1

2

∫ ∞

0

sb−1e−sds)

=1

4Γ(a)Γ(b)�

f´(7.16)�(7.17)PÇÿJ(7.14)P�

ãî�D¡ñÇÿÕ×¥�Ý^£Û�Ðó, Ç

(7.18) g(x, y) =1

B(a, b)xa−1(1− x)b−1, x ∈ (0, 1),

Í�a, b > 0 Þðó�&Æ¼:î»�×Tà�

»»»7.6.�OK =∫∞

0(z2 +a)−`zkdz,Í�k > −1, 2`−k > 1, a > 0�

��.�(z2 + a)−1a = y, Jz2 = ay−1(1− y), .h�ÿ

K =1

2a

k+12−`

∫ 1

0

y`− k+32 (1− y)

k−12 dy =

1

2a

k+12−`B(`− k + 1

2,k + 1

2)

= ak+12−` Γ(`− (k + 1)/2)Γ((k + 1)/2)

2Γ(`) �

©½2, uk = 0, JÿE∀` > 1/2,

∫ ∞

0

(z2 + a)−`dz = a1/2−` Γ(`− 1/2)π1/2

2Γ(`) �

3ÏÚa»5.11, &Æ�D¡×°Dirichlet�5�e÷P, £`ô¼�

∫∞0

sin x/xdx = π/2�9ì&Æ:¿à¥�5, Þ�D|2O�h�5Â�

êÞ 637

»»»7.7.�J(i)∫∞0

sin x/xdx = π/2; (ii)∫∞0

(1−cos x)/x2dx = π/2��.&Æ©��/�ÝM», Í�Ý×°ºÕ¬�p��î�´�∫ ∞

0

sin x

xdx =

∫ ∞

0

sin x(

∫ ∞

0

e−xudu)dx =

∫ ∞

0

(

∫ ∞

0

e−xu sin xdx)du

=

∫ ∞

0

1

1 + u2du =

π

2,

ÿJ(i)�Íg¿à(i)�ÿ∫ ∞

0

1− cos x

x2dx =

∫ ∞

0

1

x2(

∫ x

0

sin udu)dx =

∫ ∞

0

sin u(

∫ ∞

u

1

x2dx)du

=

∫ ∞

0

sin u

udu =

π

2�

t¡, A!»7.2, Bã�(x2 + y2)�¶�8f, &Æù�b×¾½�5 ½� b&��5, Íe÷PÝl�°�Çuf(x, y)3×P&Ý ½Sî=�, vD3×M > 0Cα > 2, ¸ÿ

|f(x, y)| ≤ M

(x2 + y2)α/2, ∀x, y ∈ S,

J∫∫

Sf(x, y)dxdy[e�

êêê ÞÞÞ 10.7

1. (i) ¿àÁ2ý�»ð, �JE∀a > 0,

K =

∫ a sin β

0

(

∫ √a2−y2

y cot β

log(x2 + y2)dx)dy = a2β(log a− 1

2),

Í�0 < β < π/2�(ii) øð(i)��¥�5Ý5�, ¬O�5�

2. (i) �O∫∫

S(x2 + y2 + 1)−2dxdy, Í�S JÍx-y¿«�

(ii) �O∫∫∫

T(x2 + y2 + z2 + 1)−2dxdydz, Í�T JÍx-y-zè

�3. �J

∫ 1

0

∫ 1

0(y − x)/(x + y)−3dxdy��55��øð�

638 Ïèa ¥�5

10.8 ��+++

ÍaE¥�5�×°��Ý+Û, D¡tÃÍÝ®Þ, ¿à¥�5ô�O`«Ý«�, h�×°Í�8nÝÞC-ÎXÛ«�5"DÝP��åÝS»Ý§×, &ÆP°ÇC9°ôÎËb�¤ÝÞC�b·¶Ý\ï, �¢�Apostol (1969) Chapter 12CCourant

and John (1974) Chapter 4�

¢¢¢��ZZZ¤¤¤

1. Apostol, T. M. (1969). Calculus, Vol II, 2nd ed. John Wiley

& Sons, New York, New York.

2. Apostol, T. M. (1974). Mathematical Analysis, 2nd ed. Addi-

son -Wesley, Reading, Massachusetts.


Analysis, Vol II. Springer-Verlag, New York, New York.

ÏÏÏèèè×××aaa

��555]]]��PPP

11.1 GGG��5]�P5 Ëv:æææ��555]]]��PPP(ordinary differential equa-

tions, TÌððð��555]]]��PPP, �ÌODE), C��555]]]��PPP(partial

differential equations, �ÌPDE)��2ý, ×ÎáÝ� ÐóÝ]�P�, uâb0ó, -Ì �5]�P��5]�P, -ÎÞ]�P�ÎáÝÐó��u×�5]�P�ÝÎáÐó ��ó, -Ì æ�5]�P; uÎáÐó 9�ó, -Ì ��5]�P�»A,

(1.1) f ′(x) = f(x),

×��Ýæ�5]�P��Qf(x) = ex ×��}¡&Æº:Õ, �×��(1.1)P��Äbf(x) = CexÝ�P, Í�C ×ðó�êAuf ′(x) = g(x), Jf(x) =

∫g(x)dx + C, Çf g �×D0

ó�ÆOD0óù�Ú �×�5]�P�¨², ìP ×��Ý��5]�P:

(1.2)∂2f(x, y)

∂x2+

∂2f(x, y)

∂y2= 0�

639

640 Ïè×a �5]�P

h ×ËLaplace]�P, ®ß�é.�Â.�ø�æ.C×°Í�Ýr½��(1.2)Pb&9�!�PÝ�, Af(x, y) = ax + by, Í�a, b Þðó, f(x, y) = ex cos yCf(x, y) = log(x2 + y2)��3��CI.�Ý&9TàPÝ®Þ, ??�|; ×�

�5]�PÝ®Þ��èÚtSR, pñ�¾¾¹+CJohann

Bernoulli, �×°®ß�¿¢.C^_�, ��Ý�5]�P, �@Ý�5]�PÝ"D�hV��-1690OÝ\�"D, �£`Ýó.�, | Xb®ß�¿¢CÎ§�Ý�5]�PÝ�, /�|��5�ð�Ý��Ðó¼�î�.h\�Ý�®, KÎ�%|��Ý]°(ÇE��5�ð�ÝÐó, BÄb§M»Ý°JºÕ�)WC�5), s"×°��5]�PÝ*»�×°©�Ý��5]�PÝ]°, A555ÒÒÒ��óóó(separation of

variables), C¿à��555...��(integrating factors), -�9�|1Î3èÚtS�@G, �Q ®ßÝ�3èâtS, x�h�y�Z�LagrangeCLaplace�ß, s"�?b�Ù��5]�PÝM»��"2ó.�s¨, ©bÁKóÝ�5]�P, �|��Ý]°��@�2, ó.�ôÝ�, �0��Xb�5]�PÝ]°, Ûò��ã��ÝÎ, �Æs¨×�5]�PÎÍb�, |Cub�Ý�, ãX�Ý�5]�P0�Í�ÝP², 9Í�b?9Âÿ"DÝÞC�3hÚxì, ó.��Þ�5]�P, Ú ®ß±ÐóÝ¼Ù�3×�5]�P�, &Æð|y�îÐóf(x), y′�Ðóf ′(x), ×

��, |y(n)�f (n)(x)�Qô��ày, �àu, v, z��uG ×(n + 2)�óÝÐó, J

(1.3) G(x, y, y′, y′′, · · · , y(n)) = 0,

-Ì ×n$$$(order)�ð�5]�P�Ç×�5]�P�, X�¨0ó�t{$ï, Ç ��5]�P�$�»A, y′ = y, Cy′ =

x3y + sin(xy′′), 5½ ×$CÞ$��5]�P��t{$0ó�t{¶�, Ì ��5]�P�gggóóó(degree)�»A, 1 + (y′)2 =

(y + xy′)2, C(1 + (y′)2)3 = 3(y′′)3, 5½ Þg×$, CëgÞ$�5]�P�µ�5]�P�Í²��, ;ðgó�¥�P´�, .

11.1 G� 641

h&Æ??©èÍ$ó�×Ðóf , u��EN×òyf�L½�Ýx,

G(x, f(x), f ′(x), f ′′(x), · · · , f (n)(x)) = 0,

JÌ (1.3)P�×��3�-1820O¼�, Þ�ÿÕÏ×Íny�5]�P��DDD333

PPP��§§§(existence theorem)��J�EN×bì��P�×$]�P:

y′ = f(x, y),

©�f(x, y)��Ø°×�Ýf�, -b×�D3�Í�×Í¥�Ý»�ÎRicatti]]]��PPP(Ricatti equation):

(1.4) y′ + P (x)y + Q(x)y2 = R(x),

Í�P�QCR ��Ðó�Þ�Ý��0�E∀r > 0, ©�P�QCR3(−r, r)�, b�ùó"P, J(1.4)P, 3(−r, r)�b��3�-1841O, LiouvilleJ�3Ø°�µì, (1.4)P��P°|��Ðó�î��Ía©ÎEð�5]�P�×�MÝ+Û, ��×°ÃÍÝ�

��×Mny�5]�PÝD¡, ��Ý�Ýh°�´�, E�5]�P

(1.5) y′ = 2,

|�

(1.6) y = 2x + C

Í�, Í�C ��¢ó��×(1.5)P��Äb(1.6)PÝ�P�(1.6)P-Ì (1.5)P�×××��(general solution)�(1.6)PX��Ý� ¿«îXbE£ 2�àa�9°àaÇ ×`aaaHHH(family of curves)�!§

xy′ + y = 0

642 Ïè×a �5]�P

�×�� xy = C,

h ¿«î, |Þ2ý� ��a�Ô aX�W�×àH, C

Í¢ó�D�, ;ð×Ìb×Í¢ó�àH, ??ôºÎ×$�5]�

PÝ×��»A,

(1.7) y = Cx2,

à��×|(0, 0) cF, y� �T�eÎaH�Þ(1.7)P¼��5½Ex�5, ÿ

(1.8) y′ = 2Cx�

ã(1.7)PC(1.8)P��C, Çÿ

(1.9) xy′ = 2y�

�J�(1.7)P (1.9)P�×��, .h(1.9)P-��à��eÎaH(1.7)P�

»»»1.1.�O×à�ÔàH

(1.10) xy = Cx− 1

��5]�P��.Þ(1.10)P¼��5½Ex�5, ÿ

(1.11) xy′ + y = C�

f´(1.10)PC(1.11)P, ��C-ÿ

xy = (xy′ + y)x− 1�

ÆXO x2y′ = 1�

11.2 ×$aP�5]�P 643

Íg&Æ+ÛRRR��fff��(initial conditions)�´��J��5]�P

x + yy′ = 0

�×�� !Ti

x2 + y2 = C�u��¿«î×F, Jªb×i;ÄÄhF�ix2 + y2 = 2 Ì x + yy′ = 0�×©©©½½½��(particular solution), h ;Äx = 1,

y = 1 �°×��9Ëx = 1`, y = 2Ì ×�5]�P�R�f�(êÌ\\\&&&fff��(boundary condition))�×Í×$��5]�P, Í×��©b×¢ó, Æ×R�f

�-�|X�×©½��ub×Þ$�5]�P, -mÞR�f�(Ax = a`, y = bvy′ = c)��X�×©½��{$�5]�P�v.�

11.2 ×××$$$aaaPPP��555]]]��PPP

3Ï"a�§4.1&Æ�J�y��

(2.1) y′ = αy,

uv°u

y = Ceαx,

Í�C ×ðó�.x = 0`y = C, ÆC�ã��x = 0`, y �ÂX��î��µÎ×D3v°°°×××PPP��§§§(existence-uniqueness theo-

rem)�»�3��R�f�ì, (2.1)PD3×�(D3P), vt9ô©b×�(°×P)��5]�P�Ý&9@~, -Î0�Øv]�PD3v°×Ý�§�9ì&Æ¼D¡×¥��PÝ�5]�PÝ�, h�P (2.1)P�×.Â�

644 Ïè×a �5]�P

'P�Q Þ��Ý]�P, J

(2.2) y′ + P (x)y = Q(x),

-Ì ×××$$$aaaPPP��555]]]��PPP(first-order linear differential equa-

tion)�(2.2)P��y ×��x ÝÐó, P�Q �'3Ø×� I� =�, &Æ-Î�O3I��XbÝ�y�ãy(2.2)P�y�0ó ×$, vy′�Ú ×y�aPÐó: y′ = −P (x)y +Q(x),

X|ºAhú(�9Î×v�P��¬�¥�(ÇºðÂÕ)Ý�5]�P�×��, ×�5]�P�Ý&0óu/ ×g, -Ì aPÝ�A

a0(x)y(n) + a1(x)y(n−1) + · · ·+ an−1(x)y′ + an(x)y = Q(x)�

�ua0(x), a1(x), · · · , an(x)/ ðó, -Ì ð;óaP�5]�P�&Æ�:×©», ÇQ(x) ≡ 0Ý�µ, h`

(2.3) y′ + P (x)y = 0,

Ì ET(2.2)P�×�P]�P�&Æ�0��P]�PÝ�, Q¡¢hÿÕ(2.2)P�&�P]�PÝ��u3I�y 6= 0, J(2.3)P��yìP:

(2.4)y′

y= −P (x)�

¨'y ×��(2.4)P�ÑÐó, J(2.4)PW

d

dx(log y) = −P (x),

ãhÇÿ

log y = −∫

P (x)dx + C�.h

(2.5) y = e−A(x), Í� A(x) =

∫P (x)dx− C�

11.2 ×$aP�5]�P 645

ùÇu(2.3)Pb×ÑÝ�, JÄ (2.5)PÝ�P�e−A(x)Ì (2.3)

P�×�5.��¨², ô�Aì2J�N×ã(2.5)PX�L��ÐóÄ (2.3)P��:

y′ = −A′(x)e−A(x) = −P (x)e−A(x) = −P (x)y�

.h&ÆÇO�Xb��(2.3)P�ÑÝy�9ì&Æ��(2.3)P�Xb�, vÞ��B� ×D3v°×Ý�§�

��§§§2.1.'ÐóP3� I=��I��ã×Fa, v�b �×@ó�JD3°×ÝÐóy = f(x), ��

(2.6) y′ + P (x)y = 0, v f(a) = b, ∀x ∈ I�

êhÐó

(2.7) f(x) = be−A(x), Í� A(x) =

∫ x

a

P (t)dt�

JJJ��.'f�LA(2.7)P�.A(a) = 0, Æf(a) = be0 = b�Bã�5�ÿf��(2.6)P�Æf@ (2.6)P�×��Íg&ÆJ�h °×Ý��'g ��(2.6)P��×��&ÆaJ�g(x) = be−A(x), h��

yg(x)eA(x) = b��

(2.8) h(x) = g(x)eA(x)�

J

h′(x) = g′(x)eA(x) + g(x)A′(x)eA(x)(2.9)

= eA(x)(g′(x) + P (x)g(x))�

ê.g��(2.6)P��5]�P, Æg′(x) + P (x)g(x) = 0, ∀x ∈I�.h, h′(x) = 0, ∀x ∈ I�¬hÇ�h3I � ×ðó�Æ

h(x) = h(a) = g(a)eA(a) = g(a) = b�

646 Ïè×a �5]�P

�ã(2.8)PÇÿg(x) = be−A(x)�

.hg(x) = f(x)�J±�

ãî�§J��Ýt¡×I5, Çèº×�&�P�5]�P(2.2)PÝ]°�'g ��(2.2)P��×�,v�h(x) = g(x)eA(x),

Í�A(x) =∫ x

aP (t)dt�J(2.9)P)Wñ�ê.g��(2.2)P, g′(x)

+P (x)g(x) = Q(x), Æh`

h′(x) = eA(x)Q(x)�

�ã��5ÃÍ�§ÝÏÞI5ÿ

h(x) = h(a) +

∫ x

a

eA(t)Q(t)dt�

ê.h(a) = g(a), ÆN×��(2.2)P��Äbì��P:

g(x) = e−A(x)h(x)(2.10)

= g(a)e−A(x) + e−A(x)

∫ x

a

Q(t)eA(t)dt�

D�,Bãà#E(2.10)P�5,|�(2.10)PX�L��Ðó (2.2)

P�×��&Æ-0�Ý��(2.2)P�Xb��&ÆÞ��B�Aì�

��§§§2.2.'P�Q Þ3� Iî=�ÝÐó��ã×a ∈ I, v�b �×@ó�Jªb×Ðóy = f(x), ��

(2.11) y′ + P (x)y = Q(x), v f(a) = b, ∀x ∈ I�

êhÐó

(2.12) f(x) = be−A(x) + e−A(x)

∫ x

a

Q(t)eA(t)dt,

Í�A(x) =∫ x

aP (t)dt�

11.2 ×$aP�5]�P 647

»»»2.1.�Oxy′ + (1− x)y = e2x, x ∈ (0,∞),

�Xb��.�Þk��5]�P;¶W(2.2)PÝ�P�Ç

y′ + (1

x− 1)y =

e2x

x �

ÆP (x) = 1/x− 1, Q(x) = e2x/x�.PCQ/3(0,∞)=�, ÆD3×°×Ý�y = f(x), ��×X�ÝR�f�f(a) = b�´�b

A(x) =

∫ x

a

P (t)dt =

∫ x

a

(1

t− 1)dt = log

x

a− (x− a)�

ãhêÿe−A(x) = ex−a−log(x/a) =

a

xex−a,

CeA(x) =

x

aea−x�

.hã�§2.2á, �

f(x) = ba

xex−a +

a

xex−a

∫ x

a

e2t

t

t

aea−tdt = ab

ex−a

x+

ex

x

∫ x

a

etdt

= abex−a

x+

ex

x(ex − ea) =

(abe−a − ea)ex + e2x

x �

&Æô�Þf¶W

f(x) =e2x + Cex

x, x > 0,

Í�C = abe−a − ea�t¡&Æ¼:x → 0+`, fºA¢? |�klimx→0+ f(x) D3,

C6�y−1, vh`Á§Â −1�

»»»2.2.�5]�P

(2.13) y′ + R(x)y = S(x)yk, k 6= 0, 1,

648 Ïè×a �5]�P

Ì Bernoulli]]]��PPP(Bernoulli equation)�h Jacob Bernoulli

3�-1695OXè�, ¬9ìX�Ý�, ¾¾¹+3�-1696OXs��ãyk 6= 0, 1, Æ(2.13)P&aP�5]�P�¬uBÊ2»

ð, �; ×aP�5]�P�'y 6= 0, Þ(2.13)P�¼��&¶|(1− k)y−k, ÿ

(1− k)y−ky + (1− k)R(x)y1−k = (1− k)S(x)�

�v = y1−k, Jv′ = (1− k)y−ky′, vîP�;¶

v′ + (1− k)R(x)v = (1− k)S(x),

-W ×aP�5]�P, ETÝP (x) = (1 − k)R(x), Q(x) =

(1− k)S(x), ��¿à�§2.2��Ý�

»»»2.3.�

y′ +1

xy = x5y4�

��.h ×k = 4�Bernoulli]�P�ÞîP¼��&¶|−3y−4,

ÿ

−3y−4y′ − 3

xy−3 = −3x5,

��v = y−3, ÿ

v′ − 3

xv = −3x5�

¨.

A(x) =

∫ x

a

−3

tdt = −3 log

x

a,

.h

e−A(x) = e3 log(x/a) = (x

a)3,

v

eA(x) = (a

x)3�

11.2 ×$aP�5]�P 649

Æ

v = b(a

x)3 + (

a

x)3

∫ x

a

−3t5(a

t)3dt =

bx3

a3− 3x3

∫ x

a

t2dt

=bx3

a3− x3(x3 − a3)�

�|v = y−3�áîP, ÿ

y3((b

a3+ a3)x3 − x6) = 1

kO��vÍ�x = a`, v = b, Çy = b−1/3�îPô��îW

y3(Cx3 − x6) = 1�

&Æ�:¿Í»��

»»»2.4.'C1, C2 6= 0, O×y = f(x), vy 6= 0, ��

(2.14) y′ + C1y2 + C2y = 0�

��.�Þ(2.14)PË�!t|y2(Çy−2 ×�5.�), ÿ

y′

y2+ C1 +

C2

y= 0,

�z = 1/y, îPW

−z′ + C2z + C1 = 0,

ùÇz′ − C2z = C1�

'z(a) = b, .∫ x

a−C2dt = −C2(x− a), Æ

z(x) = beC2(x−a) + eC2(x−a)

∫ x

a

C1e−C2(t−a)dt

= beC2(x−a) +C1

C2

(eC2(x−a) − 1) = (b +C1

C2

)eC2(x−a) − C1

C2�

650 Ïè×a �5]�P

.h

y = ((b +C1

C2

)eC2(x−a) − C1

C2

)−1�

b°]�P�BÊ2»ð, �ºW ×�5]�P�A3ì»�, æ¼�ÝÎ×�5]�P, ¬B¿g»ð¡, -ÿÕ×ð��P��5]�P�

»»»2.5.'Ðóφ(s)�L3s ≥ 0, v��

(2.15) φk(s) =

∫ 1

0

rur−1φm(su)du,

Í�m > k, r > 0��JD3×λ ≥ 0, ¸ÿ

(2.16) φ(s) = (1 + λsc)−1/(m−k),

Í�c = r(m− k)/k�JJJ��.´�3(2.15)P��su = v, �ÿ

(2.17) φk(s) =1

sr

∫ s

0

rvr−1φm(v)dv�

.îP�� ×s�=�Ðó, Æ¼�ù =�, .hφ ×=�Ðó�Æã��5ÃÍ�§á, (2.17)P��, E∀s > 0��, .h¼�Esù��, Æφ Es > 0��¨Þ(2.17)PË��!¶|sr, �Þ¼��5½Es�5, Çÿ

(2.18) rsr−1φk(s) + ksr−1φk−1(s)φ′(s) = rsr−1φm(s),

�ÞîP�Ë�!t|rsr−1φm(s), -ÿ

1

φm−k(s)+

ksφ′(s)rφm−k+1(s)

= 1�

�φm−k(s) = (1 + H(s))−1, JîPW

−ks

r(m− k)H ′(s) + H(s) = 0�

êÞ 651

ãîPÿ∂

∂slog H(s) =

r(m− k)

k

1

s=

c

s�Æÿ

log H(s) = c log s + K = log eKsc = log λsc,

Í�K ×ðó, �λ = eK�.hH(s) = λsc, ãhÇÿ(2.16)P�

»»»2.6.�J

x +2

3x3 +

2

3

4

5x5 +

2

3

4

5

6

7x7 + · · · = arcsin x√

1− x2�

JJJ��.�

f(x) = x +2

3x3 +

2

3

4

5x5 +

2

3

4

5

6

7x7 + · · · ,

h�ùó3|x| < 1[§�ãÏÜa�§5.2á, f3|x| < 1��, v

f ′(x) = 1 + 2x2 +2

3· 4x4 +

2

3

4

5· 6x6 + · · ·

= 1 + xd

dx(x2 +

2

3x4 +

2

3

4

5x6 + · · · )

= 1 + xd

dx(xf(x)) = 1 + xf(x) + x2f ′(x)�

Æy = f(x)��(1− x2)y′ − xy = 1,

vR�Âf(0) = 0�Bã»�PÝ�Õ, ��

f(x) =arcsin x√

1− x2, |x| < 1�

êêê ÞÞÞ 11.2

1. ��ì�&�5]�P�(1) y′ − 3y = e2x, x ∈ (−∞,∞), vx = 0`y = 0�

652 Ïè×a �5]�P

(2) xy′ − 2y = x5, x ∈ (0,∞), vx = 1`y = 1�(3) y′ + y tan x = sin 2x, x ∈ (−π/2, π/2), vx = 0`y = 2�(4) y′ + xy = x3, x ∈ (−∞,∞), vx = 0`y = 0�(5) dx/dt + x = e2t, t ∈ (−∞,∞), vt = 0`x = 1�

2. ��ì�&�5]�P�(1) y′ + xy = x�(2) y′ + y tan x = sec x�(3) y′ = e2x + 3y�(4) (x2 + 1)y = 2x(x2 + 1)2 + 2xy�(5) y′ + xy = xy2�(6) yy′ − 2y2 = ex�

3. �Oy′ sin x + y cos x = 1, x ∈ (0, π), �Xb��¬J�39°��, ªb×Íx → 0`Á§D3, vªb×Íx → π

`Á§D3�

4. �Ox(x + 1)y′ + y = x(x + 1)2e−x2, 3x ∈ (−1, 0)��X

b��¬J�Xb�x → −1`, /��0, ¬©b×�x → 0`, ÍÁ§ b§�

5. �Oy′ + y cot x = 2 cos x, x ∈ (0, π), �Xb�, ¬J�9°��, ªb×Íù 3(−∞,∞)��

6. �5½O(x−2)(x−3)y′+2y = (x−1)(x−2)3ì�& �Xb�: (i) (−∞, 2), (ii) (2,3), (iii) (3,∞)�¬J�x → 2`,

N×�/��×b§ÝÁ§, ¬x → 3 `, ^b×�º��×b§ÝÁ§�

7. �Jªb×3(0,∞)=��Ðóf , ��

f(x) = 1 +1

x

∫ x

1

f(t)dt, ∀x > 0�

êÞ 653

8. 'Ðóf , Í�

f(x) = xe(1−x2)/2 − xe−x2

∫ x

1

t−2et2/2dt, x > 0,

��(i)3x > 0=�, (ii)

f(x) = 1− x

∫ x

1

f(t)dt, ∀x > 0�

�0�Xb��î�ÞP²�Ðó�

9. ��ì�&�5]�P�(1) y′ − 4y = 2exy1/2, x ∈ (−∞,∞), vx = 0`y = 2�(2) y′−y = −y2(x2 +x+1), x ∈ (−∞,∞),vx = 0`y = 1�(3) xy′ − 2y = 4x3y1/2, x ∈ (−∞,∞), vx = 1`y = 0�(4) xy′ + y = y2x2 log x, x ∈ (0,∞), vx = 1`y = 1/2�(5) 2xyy′ + (1 + x)y2 = ex, x ∈ (0,∞), v5½ (i) x = 1`,

y =√

e, (ii) x = 1`, y = −√e, (iii) x → 0`yb×b§ÝÁ§�

10. 312.1;&Æ��LRicatti]�P, Ç

y′ + P (x)y + Q(x)y2 = R(x)�

E×�ÝRicatti]�P¬P°��Juu îP�×�,

Jy = u+1/v ù ×�,Í�v��××$aP�5]�P�

11. Ricatti]�Py′ + y + y2 = 2bÞðó��5½�Í�×��s, v¿àîÞAì20�?9�: (i) u−2 ≤ b < 1,

3(−∞,∞)�0×��x = 0`y = b�(ii) ub ≥ 1Tb <

−2, 3(−∞,∞)�0×��x = 0`y = b�

12. ��yy′′ − 2(y′)2 = 0, v'x = 1`y = 1�¬D¡y��L½�

654 Ïè×a �5]�P

13. �Jx1

+ x3

1·3 + x5

1·3·5 + · · ·1 + x2

2+ x4

2·4 + · · · =

∫ x

0

e−t2/2dt�

(èî: �¼�5� f(x), �J�f(x)E∀x ∈ R[e, ê¼�5Ò�yex2/2)

11.3 ÞÞÞ$$$aaaPPP��555]]]��PPP

3î×;E×$aP�5]�P

y′ + P (x)y = Q(x),

&Æ�¬J�Ý�ÝD3C°×P, ¬�Þ��@2��Í;&Æ¼D¡ÞÞÞ$$$aaaPPP��555]]]��PPP, Ç

(3.1) y′′ + P1(x)y′ + P2(x)y = R(x),

Í�P1CP2Ì h]�P�;ó�4QE(3.1)P, ùb×ETÝD3v°×PÝ�§, ¬tÝ×°©»², E×�ÝÞ$aP�5]�P, &Æ¬P°�@2��ÍXb��3h&Æ¬�aD¡×��PÝÞ$aP�5]�PÝ�, �©D¡t��Ýð;óÝ�5]�P, ÇP1�P2/ ðó�´�&Æ:��PPP(homogeneous)Ý�µ, ÇR(x) = 0�ð;óÝ�PaP�5]�P, Ç

(3.2) a0y(n) + a1y

(n−1) + · · ·+ an−1y′ + any = 0,

Ï×Ë��5]�P��Z3�-1743O´�è��°�9Ë�5]�P, ®ß�&9TàÝ®Þ, �Ä3h&Æ©:n = 2 Ý�µ�'b×ð;óÝÞ$aP�5]�P

(3.3) y′′ + ay′ + by = 0�

11.3 Þ$aP�5]�P 655

&Æ�0×3JÍ@óRîÝ��Qy ≡ 0 ×�, 9Ë�Ì PPPììì��(trivial solution)�&Æb·¶ÝQÎ0&&&PPPììì��(nontrivial solution)�´�:a = 0Ý�µ, h`]�PW y′′ +

by = 0�9ì&Æ¼:3h�µì, �¬��|20�Í�, ¬�¢hñÇÿÕ(3.3)PÝ��

»»»3.1.'a = b = 0,Çb×]�Py′′ = 0�&Æ�0�3(−∞,∞)î��ãyy′ ×ðó, 'y′ = c1(Ahy′�0ó�º 0), .hyÄbì��P:

y = C1x + C2,

Í�C1CC2 ðó�D�, E�Þ��ðóC1CC2, ×g94Py = C1x + C2Ä��y′′ = 0�Æ&Æ0�Ýh`�Xb��

Íg'b 6= 0, &Æ5 b < 0Cb > 0Þ�µ�

»»»3.2.�Êy′′ + by = 0, Í�b < 0�.b < 0, Þb¶Wb = −k2, Í�k > 0, J�5]�PW

y′′ = k2y��p:�y = ekxCy = e−kx/ ��ãh�ÿhÞÐó�aPà)

y = C1ekx + C2e

−kx

/ �, Í�C1CC2 ��Þðó�}¡&ÆºJ�, îP-�âXbÝ��

»»»3.3.�Êy′′ + by = 0, Í�b > 0�Þb¶Wb = k2, Í�k > 0�J�5]�PW

y′′ = −k2y��ãÌD°ÿáy = cos kxCy = sin kx/ ��ãhêÿÍaPà)

y = C1 cos kx + C2 sin kx,

656 Ïè×a �5]�P

Í�C1CC2 �Þðó, ù ��}¡&ÆôºJ�, îP-�âÝXbÝ��

E×ð;óÝÞ$aP�5]�P, &Æ¼:A¢; a = 0Ý�P, .��¢îÞ»��&ÆÝ�°Î9øÝ�'y = uv, Í�u, v ÞÐó�J

y′ = uv′ + u′v, y′′ = uv′′ + 2u′v′ + u′′v�

v

y′′ + ay′ + by(3.4)

= uv′′ + 2u′v′ + u′′v + a(uv′ + u′v) + buv

= (v′′ + av′ + bv)u + (2v′ + av)u′ + vu′′�

¨óãv, ¸ÿu′�;ó 0, Çv��

2v′ + av = 0,

Æ�ãv = e−ax/2�Ehv, v′′ = −av′/2 = a2v/4, v(3.4)P��uÝ;óW

v′′ + av′ + bv =a2v

4− a2v

2+ bv =

4b− a2

4v�

.h(3.4)PW

y′′ + ay′ + by = (u′′ +4b− a2

4u)v�

.v = e−ax/2Ä� 0, Æuy��

y′′ + ay′ + by = 0,

Ju��

u′′ +4b− a2

4u = 0�

11.3 Þ$aP�5]�P 657

ÇJ�ì��§�

��§§§3.1.'yCu ÞÐó, vy = ue−ax/2�J3(−∞,∞)î, y��

y′′ + ay′ + by = 0,

uv°uu��

u′′ +4b− a2

4u = 0�

î��§ÇÞ�y′′ + ay′ + by = 0�®Þ, » �y′′ + by = 0 Ý®Þ��»3.2C3.3, ��hË]�P�&Pì��¬tÝb = 0

Ý�µ²(�»3.1), &ÆÍ@$ÎJ��0�Xb��ì�°×PÝ�§�

��§§§3.2.'fCgÞÐó3(−∞,∞)î��y′′ + by = 0, v'f�g��ì�R�f�

f(0) = g(0), f ′(0) = g′(0)�

Jf(x) = g(x), ∀x ∈ R�JJJ��.�h(x) = f(x)− g(x), &Æ�J�h(x) = 0, ∀x ∈ R�´��:�h ù y′′ + by = 0�×�, v��R�f�h(0) = 0Ch′(0) =

0�êuy��y′′ + by = 0, J

y′′′ = (y′′)′ = (−by)′ = −by′,

y(4) = (y′′′)′ = (−by′)′ = −by′′ = b2y2,

õv.��:�y3RîP§g��, v

y(2n) = (−1)nbny, y(2n−1) = (−1)n−1bn−1y′, n ≥ 1�

¨.h(0) = h′(0) = 0, Æh(n)(0) = 0, ∀n ≥ 1�.hh30 �N×��94PÝ;ó/ 0�ãÏ°a(3.18)P��2Pÿ

h(x) = R2n−1(x), n ≥ 1,

658 Ïè×a �5]�P

Í�R2n−1(x)�õ4�u�J�©�nÈ�, JR2n−1(x) ��,

Í�§-ÿJÝ�&ÆÞ¿àÏ°a�§3.5¼£�õ4��.h&Æ6

�£�0óh(2n)��ã�×b§ÝT [−c, c], Í�c >

0�.h =�, Æ3[−c, c] b&�ÇD3×M > 0, ¸ÿ|h(x)| ≤M , ∀x ∈ [−c, c]�ê.

h(2n)(x) = (−1)nbnh(x),

Æ|h(2n)(x)| ≤ M |b|n, ∀x ∈ [−c, c]�

.hãÏ°a�§3.5á

|R2n−1(x)| ≤ M |b|nx2n

(2n)! �

ÆE∀x ∈ [−c, c]Cn ≥ 1,

(3.5) 0 ≤ |h(x)| ≤ M |b|nx2n

(2n)!≤ M |b|nc2n

(2n)!=

MA2n

(2n)!,

Í�A = |b|1/2c��ãÏÜa(4.1)Pá

limn→∞

A2n

(2n)!= 0,

.h(3.5)PÇ0�h(x) = 0, ∀x ∈ [−c, c]�ê.c ��Ñó, .hh(x) = 0, ∀x ∈ R�Í�§J±�

ÛÛÛ.î�§¼�, uy′′ + by = 0bÞ�, vhÞ�30 �ÂC0ó/8!, JhÞ�Ä��8��¯@î, �p:�óã“0”¬�©½¥��uÞ0ð �×@óξ, Ç'Þ�3ξ�ÂC0ó/8!, JhÞ�)Ä��8��J�)v«, ©�ÞE0 ��"P; Eξ��"PÇ��

¿àG�°×P�§, &Æ-�X�y′′ + by = 0�Xb��

11.3 Þ$aP�5]�P 659

��§§§3.3.E�×b ∈ R, y′′ + by = 0, 3(−∞,∞)��bì��P:

(3.6) y = C1u1(x) + C2u2(x),

Í�C1CC2 Þðó, �(i) ub = 0, Ju1(x) = 1, u2(x) = x;

(ii) ub = −k2 < 0, Ju1(x) = ekx, u2(x) = e−kx;

(iii) ub = k2 > 0, Ju1(x) = cos kx, u2(x) = sin kx�JJJ��.ã»3.1-3.3á,E��C1CC2, (3.6)P y′′+by = 0�×��9ì&ÆJ�Xb�/b(3.6)PÝ�P�êãyb = 0Ý�µ�3»3.1��XÝ, Æ'b 6= 0�¨�y = f(x)��y′′+ by = 0�'�J�ÄD3ðóC1CC2,¸

ÿ

(3.7) C1u1(0) + C2u2(0) = f(0), C1u′1(0) + C2u

′2(0) = f ′(0)�

J.f(x)Cu1(x) + u2(x)/ y′′ + by = 0��, v30�ÂC30�0ó/8�, Æã�§3.2°×PÝ��á,

f(x) = C1u1(x) + C2u2(x), ∀x ∈ R�

3�µ(ii)�, .u1(x) = ekx, u2(x) = e−kx, Æu1(0) = u2(0) = 1,

vu′1(0) = k, u′2(0) = −k�h`(3.7)PW

C1 + C2 = f(0), C1 − C2 = f ′(0)/k�

ãh��

C1 =1

2(f(0) + f ′(0)/k), C2 =

1

2(f(0)− f ′(0)/k)�

3�µ(iii)�, .u1(x) = cos kx, u2(x) = sin kx, Æu1(0) = 0,

u2(0) = 0, vu′1(0) = 0, u′2(0) = k�ãh��

C1 = f(0), C2 = f ′(0)/k�

.�¡�µ(ii)T(iii), /�0�C1CC2��(3.7)P, ÆÍ�§ÿJ�

660 Ïè×a �5]�P

á)|îÝ��, &Æ-�X�×�Ýy′′ + ay′ + by = 0 ��´��§3.1¼�y y′′ + ay′ + by = 0�×�, uv°uy

= ue−ax/2,Í�u u′′+ 14(4b−a2)u = 0�×��ã�§3.3á, u�

�ê�4b−a2�Ðrbn�&Æ-Þa2−4bÌ y′′+ay′+ by = 0�¾¾¾½½½PPP(discriminant), ¬|d�hÂ�&ÆW��Aì�

��§§§3.4.�d = a2 − 4b�J3(−∞,∞), y′′ + ay′ + by = 0 ��bì��P:

(3.8) y = e−ax/2(C1u1(x) + C2u2(x)),

Í�C1CC2 Þðó, �(i) ud = 0, Ju1(x) = 1, u2(x) = x;

(ii) ud > 0, Ju1(x) = ekx, u2(x) = e−kx, Í�e =√

d/2;

(iii)ud < 0,Ju1(x) = cos kx, u2(x) = sin kx,Í�k =√−d/2�

3î�§�, ud > 0, J(3.8)P�;¶

y = C1e(−a/2+k)x + C2e

(−a/2−k)x = C1er1x + C2e

r2x,

Í�

(3.9) r1 = −a

2+ k =

−a +√

d

2, r2 = −a

2− k =

−a−√

d

2,

ª ]�P

(3.10) r2 + ar + b = 0

�Þq�(3.10)P-Ì �5]�Py′′ + ay′ + by = 0�©©©ÇÇÇ]]]��PPP(characteristic function)�ud < 0, J(3.9)P�r1Cr2 (3.10)P�Þ�óq�ã¼óÝP

²(�Ïâa(4.6)P)á, (3.8)P��y)�¶WC1er1x + C2e

r2x, ©ÎC1CC2µ�×� @óÝ�

11.3 Þ$aP�5]�P 661

(3.8)PX�Ýy-Îy′′+ay′+by = 0�×��,��ÞðóC1CC2,

XÿÝy- ×©½��»A,

v1(x) = e−ax/2u1(x), v2(x) = e−ax/2u2(x),

/ ©½��v1�v2�aPà), -��XbÝ��Þ�uÌbhP², -Ì �/)�×ÃÃÃ999(basis)�×�5]�P�Ã9¬�°×�»A, 'y′′ = 9y, Jv1 = e3x�v2 = e−3x ×àÃ9,

�w1 = cosh 3x�w2 = sinh 3x ù ×àÃ9�¯@î, .

v1 = w1 + w2, v2 = w1 − w2,

ÆN×v1�v2�aPà)ù w1�w2�aPà)�Æw1�w2@ ×àÃ9�#��J�(º3êÞ), �×Ey′′ + ay′ + by = 0 ��v1�v2, ©�v2/v1� ðó, -�W×àÃ9�Íg, &Æ¼:&�P�Þ$aP�5]�P��'b×]

�P

(3.11) y′′ + ay′ + by = g(x),

Í�a, b ðó, g ×�L3(−∞,∞)�Ðó�uy1�y2/ (3.11)

P��, J.

y′′1 + a1y′1 + by1 = g(x),

y′′2 + a1y′2 + by2 = g(x),

Æ(y2 − y1)

′′ + a1(y2 − y1)′ + b(y2 − y1) = 0,

.hy2 − y1 ]�Py′′ + ay′ + by = 0�×��Æ

y2 − y1 = c1v1 + c2v2,

Í�c1v1 + c2v2 �P]�Py′′ + ay′ + by = 0(Ì (3.11)PBBBÃÃÃ]]]��PPP(complementary equation))�×��Æ�Þ(3.11)P��y1Cy2, ��

y2 = c1v1 + c2v2 + y1�

662 Ïè×a �5]�P

Æu�0Õ(3.11)P�×©½�yp, J

(3.12) y = c1v1 + c2v2 + yp

��î�Xb�,Í�c1Cc2 ðó��(3.12)P��yôµÎ(3.11)

P�×��&ÆÇJ�ì��§�

��§§§3.5.uyp (3.11)P�×©½�, yc ETÝ�P]�P�×��, Jyp + yc- (3.11)P�×��

»»»3.4.�Oì�]�P��

y′′ + y = 2x�

��.ãÌD°ÿyp = 2x, �BÃ]�P

y′′ + y = 0

�×�� yc = c1 cos x + c2 sin x�

Æy′′ + y = 2x�×�� y = c1 cos x + c2 sin x + 2x�

9ì&Æ�×ÿÕ©½�Ý]°, h°Ì ¢¢¢óóó��555°°°(Method

of variation of parameters), Johann Bernoulli3�-1679O´�à¼�×$aP�5]�P, �Lagrange3�-1774Oà¼�Þ$aP�5]�P�

��§§§3.6.'

v1(x) = e−ax/2u1(x), v2(x) = e−ax/2u2(x)

(3.8)PX�y′′ + ay′ + by = 0��

(3.13) W (x) = v1(x)v′2(x)− v2(x)v′1(x),

11.3 Þ$aP�5]�P 663

v'W (x)� 0�Jyp y′′ + ay′ + by = g(x)�×©½�, Í�

(3.14) yp(x) = t1(x)v1(x) + t2(x)v2(x),

�

(3.15) t1(x) = −∫

v2(x)g(x)

W (x)dx, t2(x) =

∫v1(x)

g(x)

W (x)dx�

JJJ��.&Æ�0�Ðót1�t2,¸ÿyp = t1v1+t2v2��y′′p+ay′p+byp =

g(x)�Ehyp,

y′p = t1v′1 + t2v

′2 + (t′1v1 + t′2v2),

y′′p = t1v′′1 + t2v

′′2 + (t′1v

′1 + t′2v

′2) + (t′1v1 + t′2v2)

′�

.v1�v2/ y′′ + ay′ + by = 0��, Æ

v′′1 + av′1 + bv1 = 0, v′′2 + av′2 + bv2 = 0�

.h

y′′p + ay′p + byp = (t′1v′1 + t′2v

′2) + (t′1v1 + t′2v2)

′ + a(t′1v1 + t′2v2)�

&Æ�ót1�t2, ¸ÿy′′p + ay′p + byp = g(x)��Qut1�t2��

t′1v1 + t′2v2 = 0, v t′1v′1 + t′2v

′2 = g(x)

Ç��.W (x)� 0, ÆîÞP��

t′1 =−v2g(x)

W, t′2 =

v1g(x)

W �

ãhÇÿ(3.15)P�(3.13)P�W (x)-Ì v1�v2�Wronskian�êî��§��t1

Ct2Î|��5¼�î, ��5�b�5ðó�¬�:�E�Þt1Ct2, (t1(x) + c1)v1(x) + (t2(x) + c2)v2(x), Í�c1Cc2 Þðó,

) ×©½��

664 Ïè×a �5]�P

ÛÛÛ.��ÞÐóu1Cu2, W (x) = u1(x)u′2(x)−u2(x)u′1(x)-Ì u1Cu2

�Wronskian, Wronski(1778-1853)X�L�

»»»3.5.�Oì�]�P��(i) y′′ + y = tan x; (ii) y′′ + y = sec x�

��.BÃ]�Py′′+y = 0�×�� yc = c1 cos x+c2 sin x�Çv1(x)

= cos x, v2(x) = sin x�J

W (x) = v1(x)v′2(x) = v2(x)v′1(x) = cos2 x + sin2 x = 1�

.h

t1(x) = −∫

sin x tan xdx = sin x− log | sec x + tan x|,

t2(x) =

∫cos x tan xdx =

∫sin xdx = − cos x�

Æ

yp = t1(x)v1(x) + t2(x)v2(x)

= sin x cos x− cos x log | sec x + tan x| − sin x cos x

= − cos x log | sec x + tan x|

×©½��.hy′′ + y = tan x�×��

y = c1 cos x + c2 sin x− cos x log | sec x + tan x|�

!§�ÿyp = x sin x + (log | cos x|) cos x, y′′ + y = sec x �×©½��Æ×��

y = c1 cos x + (c2 + x) sin x + (log | cos x|) cos x�

4Q�§3.6èº×0©½�Ý]°, ¬g(x) b×°©½Ý�P`, b`ºb×°©½¬Q´�|Ý]°�&Æ|9ì×°»�¼1��

11.3 Þ$aP�5]�P 665

»»»3.6.'g(x) ×ng94P, vb 6= 0, J��×ng94Pyp(x) =∑nk=0 akx

k ©½�, Þy�áy′′ + ay′ + by = 0, �f´!×x¶�Ý;ó, -��a0, a1, · · · , an�»A, �y′′ + y = x3�'yp(x) = a3x

3 + a2x2 + a1x + a0�J

y′′p(x) + ay′p(x) + byp(x) = (6a3x + 2a2) + (a3x3 + a2x

2 + a1x + a0)

= x3�Æÿa3 = 1, a2 = 0, a1 = −6, a0 = 0, .hyp(x) = x3 − 6x2 ×©½�, v

y = c1 cos x + c2 sin x + x3 − 6x

×��&Æô��¢ó°f´�ã(3.15)Pÿ

t1(x) = −∫

x3 sin xdx = −(3x2 − 6) sin x + (x3 − 6x) cos x,

t2(x) =

∫x3 cos xdx = (3x2 − 6) cos x + (x3 − 6x) sin x�

Í��5ðó/6¯, .�G�1��5ðó3h¬�¥�, &Æ�yÎÞ�5ðóã 0�.v1(x) = cos x, v2(x) = sin x, Æ

t1(x)v1(x) + t2(x)v2(x) = x3 − 6x,

Ç)ÿ8!Ý©½��f´Þ®°, �:�uà¢ó�5°J6O�5

∫x3 sin xdx

�∫

x3 cos xdx, �ÕÄ��}3�°�ub = 0,J×ng94P¬P°��y′′+ay′ = g(x),Í�g ×n

g94P�¬×(n + 1)g94P, -��º��y′′ + ay′ = g(x), ©�a 6= 0�ua = b = 0, h`]�PW y′′ = g(x), �Q×�� ×(n + 2) g94P�

»»»3.7.'g(x) = p(x)emx, Í�p ×ng94P, m ×ðó�h`u�y = u(x)emx, JÞy′′ + ay′ + by = g(x)»;

u′′ + (2m + a)u′ + (m2 + am + b)u = p(x),

666 Ïè×a �5]�P

W ×î»�Ý�P�.h�0Õ×94PÝ�u1�Æyp =

u1(x)emx æP�×©½��um2+am+b 6= 0, u1�gó�p(x)8!�um2 + am + b = 0v2m + a 6= 0, Ju1 �gó´p(x)91�um2 +

am + b = 2m + a = 0, Ju1�gó´p(x)92�»A, Oy′′ + y = xe3x��y = ue3x, Jÿ

u′′ + 6u′ + 10u = x�

u�u1(x) = a1x1 + a0, ÿu1(x) = (5x − 3)/50, ÆæP�×©½�

yp(x) = e3x(5x− 3)/50�

»»»3.8.'g(x) = p(x)emx cos αxTg(x) = p(x)emx sin αx, Í�p ×94P, m�α / ðó�hÞË�µ/��©½�

yp(x) = emx(q(x) cos αx + r(x) sin αx),

Í�q�r/ 94P�

êêê ÞÞÞ 11.3

1. ��ì�&�5]�P�(1) y′′ − 4y = 0� (2) y′′ + 4y = 0�(3) y′′ − 4y′ = 0� (4) y′′ + 4y′ = 0�(5) y′′ − 2y′ + 3y = 0� (6) y′′ + 2y′ − 3y = 0�(7) y′′ − 2y′ + 2y = 0� (8) y′′ − 2y′ + 5y = 0�(9) y′′ + 2y′ + y = 0� (8) y′′ − 2y′ + y = 0�

2. ��ì�&�5]�P�(1) 2y′′ + 3y′ = 0, vx = 0`, y = 1, y′ = 1�(2) y′′ + 25y = 0, vx = 3`, y = −1, y′ = 0�(3) y′′ − 4y′ − y = 0, vx = 1`, y = 2, y′ = −1�(4) y′′ + 4y′ + 5y = 0, vx = 0`, y = 2, y′ = y′′�

êÞ 667

3. ��ì�&�5]�P�(1) y′′ − y = x� (2) y′′ − y′ = x2�(3) y′′ + y′ = x2 + 2x� (4) y′′ − 2y′ + 3y = x3�(5) y′′ − 4y = e2x� (6) y′′ + 4y = e−2x�(7) y′′ + y′ − 2y = ex� (8) y′′ + y′ − 2y = e2x�(9) y′′ + y′ − 2y = ex + e2x� (10) y′′ − y = 2/(1 + ex)�(11) y′′ + 2y′ + y = e−x/x2� (12) y′′ + y = cot2 x�(13) y′′ + y′ − 2y = ex/(1 + ex)�(14) y′′ − 2y′ + y = x + 2xex�(15) y′′ − 5y′ + 4y = x2 − 2x + 1�(16) y′′ + y′ − 6y = 2x3 + 5x2 − 7x + 2�(17) y′′ + 6y′ + 9y = f(x), Í�1 ≤ x ≤ 2`, f(x) = 1, EÍõÝx, f(x) = 0�

4. ��ì�&�5]�P�(1) y′′ + y = sin x� (2) y′′ + y = cos x�(3) y′′ + 4y = 3x cos x� (4) y′′ + 4y = 3x sin x�(5) y′′ − 3y′ = 2e2x sin x� (5) y′′ + y = e2x cos 3x�

5. (i)'ÞÐóu1Cu2,xòyØ� I`,ÍWronskian W (x)

0��Ju2(x)/u1(x)3I� ×ðó�ùÇu3I�u2/u1

� ðó, J�KD3×c ∈ I, ¸ÿW (c) 6= 0�(ii) �JW ′(x) = u1u

′′2 − u2u

′′1�

6. 'u1Cu2 y′′+ay′+by = 0�Þ�, W u1�u2�Wronskian�(i) �JW��W ′ + aW = 0, .hW (x) = W (0)e−ax�hP�îuW (0) 6= 0, JW (x)� 0�(ii) 'u1� 0, �JW (0) = 0uv°uu2/u1 ×ðó�

7. �v1Cv2 y′′ + ay′ + by = 0�Þ�, vv2/v1� ðó�(i) �y = f(x) y′′ + ay′ + by = 0��¿àWronskian�P², �JD3ðóc1Cc2, ¸ÿ

c1v1(0) + c2v2(0) = f(0), c1v′1(0) + c2v

′2(0) = f ′(0)�

668 Ïè×a �5]�P

(ii) �Jy′′ + ay′ + by = 0�N×�Ý�P/ y = c1v1 + c2v2,

ùÇv1�v2�W�/)�×àÃ9�

8. 'k 6= 0 ×ðó��Jy1 y′′ − k2y = g(x)�×©½�, Í�

y1 =1

k

∫ x

0

g(t) sinh k(x− t)dt�

Oy′′ − 9y = e3x��

9. 'k 6= 0 ×ðó��Jy1 y′′ + k2y = g(x)�×©½�, Í�

y1 =1

k

∫ x

0

g(t) sin k(x− t)dt�

Oy′′ + 9y = sin 3x��

10. �®λ ¢Â∫ 1

0

min{x, y}f(y)dy = λf(x)

3(0,1)b×� ë��, ¬Oh`��

11.4 ��555ÒÒÒÝÝÝ��555]]]��PPP

�5]�Py′ = f(x, y),

uf(x, y) = Q(x)R(y), -Ìh ×$��555ÒÒÒ��555]]]��PPP(separa-

ble differentiable equation)��A, y′ = x3, y′ = sin y log x�, / �5Ò��5]�P�uR(y) 6= 0, Jy′ = Q(x)R(y)�;¶

(4.1) A(y)y′ = Q(x),

Í�A(y) = 1/R(y)�ì��§��5ÒÝ�5]�P��

11.4 �5ÒÝ�5]�P 669

��§§§4.1.'y = f(x) (4.1)P��×�, v'f ′ 3Ø×� I =��ê'QC)WÐóA ◦ fù/3I =��G A�×D0ó,

ÇG = A�JG��

(4.2) G(y) =

∫Q(x)dx + C,

Í�C ×ðó�D�, uy��(4.2)P, Jy (4.1)P�×��JJJ��..y = f(x) (4.1)P�×�, Æ

(4.3) A(f(x))f ′(x) = Q(x), ∀x ∈ I�

ê.G′ = A, ÆîPW

G′(f(x))f ′(x) = Q(x)�

êã=Å!J,îP¼� )WÐóG◦f�0ó�.hG◦f Q�×D0ó, ùÇ

(4.4) G(f(x)) =

∫Q(x)dx + C,

Í�C ×ðó�îPÇ (4.2)P�D�, uy = f(x)��(4.2)P, Þ(4.2)P¼��5½Ex�5, Ç

ÿ(4.3)P�Æf (4.1)P�×��J±�(4.2)Pù�|A¼�î�ã(4.3)Pÿ

∫A(f(x))f ′(x)dx =

∫Q(x)dx + C�

3îP�, u�y = f(x), Jdy = f ′(x)dx, Çÿ

(4.5)

∫A(y)dy =

∫Q(x)dx + C�

.∫

A(y)dy A�×D0ó, ÆîP (4.2)P�¨×¶°�¯@î, ã(4.1)Pà#ÿÕ(4.5)P ¾¾¹+Ðr�b[PÝ¨×

670 Ïè×a �5]�P

¤J�&Æ�3(4.1)P�, �Þy′¶Wdy/dx, Q¡Þdy/dx�' dyt|dx, �Ë�&¶|dxÿ

A(y)dy = Q(x)dx�t¡, ¼��&!î×�5Ðr, ��î×ðó-ÿ(4.2)P�9Ë]Pðð�ÿ;, �A�§4.1��J�/ 9õ�

»»»4.1.��(x + sec2 x) + (y − ey)y′ = 0�

��.îP�� ∫(x + sec2 x)dx +

∫(y − ey)dy = C,

T1

2x2 + tan x +

1

2y2 − ey = C�

»»»4.2.��1√

1− x2+

1

y

dy

dx= 0, y > 0�

��.îP�� ∫1√

1− x2dx +

∫1

ydy = C,

Tarcsin x + log y = C�

êêê ÞÞÞ 11.4

1. �Oì�&�5]�P��(1) y′ = x3/y2� (2) tan x cos y = −y′ tan y�(3) (x + 1)y′ + y2 = 0� (4) y′ = (y − 1)(y − 2)�(5) y

√1− x2y′ = x� (6) (1− x2)1/2y′ + 1 + y2 = 0�

(7) (x− 1)y′ = xy� (8) xy(1 + x2)y′ − (1 + y2) = 0�(9) (x2 − 4)y′ = y� (10) xyy′ = 1 + x2 + y2 + x2y2�(11) yy′ = ex+2y sin x� (12) xdx + ydy = xy(xdy − ydx)�

11.5 ª�5]�P 671

11.5 ªªª��555]]]��PPP

Í;&Æ¼:¨×v��Ý×$aP�5]�P�´�×�5]�P

(5.1) M(x, y) + N(x, y)y′ = 0,

u��∂M

∂y=

∂N

∂x,

-Ì ªªª��555]]]��PPP(exact differential equation)��:�(4.1)P��5Ò�5]�P,ù ×Ëª�5]�P��Ät&M(x, y)

G ×x�Ðó, vNG ×y�Ðó, ÍJ×ª�5]�P, ¬� ×�5ÒÝ�5]�P�'b×]�P(2x− y) + (y2 − x)y′ = 0�.

∂

∂y(2x− y) = −1 =

∂

∂x(y2 − x),

Æh ×ª�5]�P�9ì&Æ/'(5.1)P��M�N , Kb=�Ý×$�0ó�

��§§§5.1.'F (x, y)�Þ×$�0ó/=�, v

(5.2) 5F = (M,N)�

J

(5.3) F (x, y) = C,

Í�C ×ðó, ª�5]�P

(5.4) M(x, y) + N(x, y)y′ = 0

�×��

672 Ïè×a �5]�P

��.'y = f(x) (5.4)P�×��Jã=Å!J

DF (x, f(x)) = D1F (x, f(x)) + D2F (x, f(x))f ′(x)

= M(x, f(x)) + N(x, f(x))f ′(x) = 0�

ÆF (x, f(x)) = C, Í�C ×ðó�D�, uF (x, f(x)) = C, J

DF (x, f(x)) = D(C) = 0�

.h¿à=Å!J)ÿ

M(x, f(x)) + N(x, f(x))f ′(x) = 0�

Æy = f(x)@ ×��J±�

»»»5.1.��2x− y + (y2 − x)y′ = 0�

��.�M(x, y) = 2x− y, N(x, y) = y2 − x�G«�1Äh ×ª�5]�P�ÆF (x, y) = C Í�, Í�F��

D1F (x, y) = 2x− y,(5.5)

D2F (x, y) = y2 − x�(5.6)

ã(5.5)P, B�5ÿ

F (x, y) = x2 − xy + g(y),

Í�g(y) ×y�Ðó�ê.(5.6)P�Wñ, Æg′(y) = 2y�.h�ãg(y) = y3/3, v

F (x, y) = x2 − xy +1

3y3 = C

��

11.5 ª�5]�P 673

»»»5.2.��

sin y + (x cos y + y cos y + sin y)y′ = 0�

��.�M(x, y) = sin y, N(x, y) = x cos y + y cos y + sin y�.

∂M

∂y= cos y =

∂N

∂x,

Æh ×ª�5]�P�&Æ60×ÐóF (x, y), ¸ÿ

D1F (x, y) = sin y,(5.7)

D2F (x, y) = x cos y + y cos y + sin y�(5.8)

ã(5.7)PÿF (x, y) = x sin y + g(y)�

ÞhF�ÿ(5.8)P, ÿ

g′(y) = y cos y + sin y�

ãhÇÿg(y) = y sin y�Æ

F (x, y) = x sin y + y sin y = C

��

t¡, E×�5Ò�5]�P(4.1), Ç

Q(x)− A(y)y′ = 0�

ûî��°,�ÿ∫

Q(x)dx−∫A(y)dy = C �,�hÇ(4.2)P�h

�§4.1�¨×J��ê!ñ×è, (5.1)Pb`ô¶W

(5.9) M(x, y)dx + N(x, y)dy = 0�

674 Ïè×a �5]�P

êêê ÞÞÞ 11.5

1. ��ì�&�5]�P�(1) x + y + (x + 2y)y′ = 0�(2) yex − x + (ex + 1)y′ = 0�(3) (x sin y − y)y′ = cos y�(4) yexy + 2xy + (xexy + x2)y′ = 0�(5) y sec2 x + y′ tan x = 0�(6) (ex sin y + y)y′ = ex cos y�

2. ��ì�&�5]�P�(1) x− y + (2y3 − x)y′ = 0, vx = 2`y = 1�(2) x cos xy + (1 + x cos xy)y′ = 0, vx = π/4`y = −1�

3. uI(x, y)M(x, y) + I(x, y)N(x, y)y′ = 0 ×ª�5]�P,

JI(x, y)Ì M(x, y) + N(x, y)y′ = 0�×�5.��Jì�&Þ��I(x, y)/ �5.��(1) y − xy′ = 0, I(x, y) = y−2�(2) y − xy′ = 0, I(x, y) = x−2�(3) y − xy′ = 0, I(x, y) = (xy)−1�(4) x + y + y′ = 0, I(x, y) = ex�(5) xy′ = x− 3y, I(x, y) = x2�(6) y + x + (y − x)y′ = 0, I(x, y) = (x2 + y2)−1�

11.6 �P×$aP�5]�P 675

11.6 ��PPP×××$$$aaaPPP��555]]]��PPP

A9.5;�êÞÏ14ÞX�L, ×ÍÞ�óÐóF (x, y), u��

F (tx, ty) = tnF (x, y),

E∀tC(x, y), ©�(x, y)C(tx, ty)/3F��L½, -Ì ×ng�PÐó��Af(x, y) = ax + by, g(x, y) = ax2 + bxy + cy2Ch(x, y) =

ax3 + bx2y + cxy2 + dy3-5½ 1g�2gC3g�PÐó�¨²,

u�

F (x, y) = x2 +x3 + 2y3

y,

J.

F (tx, ty) = (tx)2 +(tx)3 + 2(ty)3

ty= t2F (x, y),

ÆF ×ÍÞg�PÐó�êu

G(x, y) =1

x + ysin

x− y

x + y,

J.

G(tx, ty) =1

tx + tysin

tx− ty

tx + ty= t−1G(x, y),

ÆG ×�×g�PÐó�¨uR(x, y)�S(x, y) !gÝ�PÐó, J

(6.1) R(x, y) + S(x, y)y′ = 0

-Ì ��PPP��555]]]��PPP(homogeneous differential equation)�9ì&Æ¼:9Ë]�PÝ�°�&ÆÞ�0×��ÝÐóg, ¸ÿy = xg(x), x 6= 0, (6.1)P�

��u�v = g(x), J

y = xv, y′ = v + xv′,

�á(6.1)Pÿ

R(x, xv) + S(x, xv)(v + xv′) = 0�

676 Ïè×a �5]�P

uR�S ng�PÐó, J.

R(x, xv) = xnR(1, v), S(x, xv) = xnS(1, v),

Æ(6.1)PW (N×4!t|xn)

R(1, v) + S(1, v)(x + xv′) = 0,

ãhêÿ

(6.2)1

x+

S(1, v)

R(1, v) + vS(1, v)v′ = 0�

Æuv = g(x) (6.2)P�×�, Jy = xg(x) (6.1)P��ã|îÝ.0á, ×�P�5]�P, Ä�»ð ×�5ÒÝ�

5]�P�u2à�0ÝBr, �y = xv, Jdy = xdv + vdx, -Þ

R(x, y)dx + S(x, y)dy = 0

»ð ×�5ÒÝ�5]�P

1

xdx +

S(1, v)

R(1, v) + vS(1, v)dv = 0�

»»»6.1.��(y − 4x)dx + (y + 2x)dy = 0�

��.�R(x, y) = y−4x, S(x, y) = y+2x,/ ×g�PÐó��y =

xv, dy = xdv + vdx, Jk��5]�PW

(xv − 4x)dx + (xv + 2x)(xdv + vdx) = 0,

Ë�&t|x, x 6= 0, ÿ

(v − 4)dx + (v + 2)(xdv + vdx) = 0�

.hÿÕì��5Ò��5]�P:

1

xdx +

v + 2

v2 + 3v − 4dv = 0,

êÞ 677

Í�

log |x|+ 2

5log |v + 4|+ 3

5log |v − 1| = C1�

B;�ÿlog |x5(v + 4)2(v − 1)3| = 5C1,

hÇx5(v + 4)2(v − 1)3 = C,

Í�C = e5C1�îPê0l

(xv + 4x)2(xv − x)3 = C,

�Þxv = y�á, ÿ

(y + 4x)2(y − x)3 = C�

»»»6.2.��(x− y tan

y

x)dx + x tan

y

xdy = 0�

��.h ×�P�5]�P�)|y = xv�áÿ

1

xdx + tan vdv = 0,

Ælog |x|+ log | sec v| = C1,

hê��yx sec v = C�Æ� x sec(y/x) = C�

êêê ÞÞÞ 11.6

1. ��ì�&�5]�P�(1) y′ = −x/y� (2) y′ = 1 + y/x�(2) xy′ = y −

√x2 − y2� (4) xy′ = y −

√x2 + y2�

(5) x2y′ + xy + 2y2 = 0� (6) y2 + (x2 − xy + y2)y′ = 0�(7) y′ = y/x + sin(y/x)� (8) y′ = (x2 + y2)/x2�(9) (x− 2y) + xy′ = 0�(10) x(y + 4x)y′ + y(x + 4y) = 0�

678 Ïè×a �5]�P

(11) (2yey/x − x)y + (2x + y) = 0�(12) (y2 − x2 + 2xy) + (y2 − x2 − 2xy)y′ = 0�(13) (x2 + y2) + 2xyy′ = 0�(14) (2x + y) + (x + 3y)y′ = 0�(15) y′ = y(x2 + xy + y2)/(x(x2 + 3xy + y2))�

2. E×Þ$��5]�PF (x, y′, y′′) = 0, uP�y¬Î�¨, JBz = y′, z′ = y′′��ð, -�» ×$��5]�P��ì�&�5]�P�(1) y′′ = y′� (2) y′′ = −2y′�(3) y′′ = sin x� (4) xy′′ = y′ + 1�(5) y′′ =

√(y′)2 + 1� (6) y′′ − exy′ = 0�

3. �JE×Þ$��5]�PF (x, y′, y′′) = 0, uP�x¬Î�¨, JBp = y′, p(dp/dy) = y′′��ð, �»ð ×$��5]�P�

4. �¿àîÞ�(1) 2y′′ = ey, (2) (y′′)2 = (1 + (y′)2)3�

11.7 ��555]]]��PPP��ùùùóóó��

ùó

f(x) =∞∑

n=0

an(x− x0)n

3Í[e �L�×Ðó�E×�5]�P, b`ô�¢ãùó¼�îÍ��ì»�

»»»7.1.�Oy′′ + xy′ + y = 0�×ùó��.'

y =∞∑

n=0

anxn

11.7 �5]�P�ùó� 679

×��JãÏâa�§5.2á

y′ =∞∑

n=1

nanxn−1,

y′′ =∞∑

n=2

n(n− 1)anxn−2 =

∞∑n=0

(n + 2)(n + 1)an+2xn�

.h

y′′ + xy′ + y =∞∑

n=0

(an + nan + (n + 2)(n + 1)an+2)xn�

uy@ y′′ + xy′ + y = 0�×�, JîP��y0, .hÍN×x �¶�Ý;óÄ6 0�Ç

(1 + n)an + (n + 2)(n + 1)an+2 = 0,

.h

an+2 = − 1

n + 2an, n ≥ 0�

ãhÇÿ

a2n =(−1)n

2 · 4 · · · (2n)a0, n ≥ 1,

a2n+1 =(−1)n

1 · 3 · · · (2n + 1)a1, n ≥ 1�

Æy��îWÞùóÝõ, Í��×�â�g4, ¨×�â�g4:

y = a0

∞∑n=0

(−1)n

2 · 4 · · · (2n)x2n + a1

∞∑n=0

(−1)n

1 · 3 · · · (2n + 1)x2n+1,

Í�un = 0, J2 · 4 · · · (2n)�L 1�t¡, ãfÂ�§l�°á,

î�Þùó/3Rî[e�

E×�5]�P

an(x)y(n) + a1(x)y(n−1) + · · ·+ a1(x)y = G(x),

680 Ïè×a �5]�P

uan(x0) = 0, Jx = x0Ì h�5]�P�×��²²²FFF(singular

point)�ùóÝ�©3��âx0 Ý D3�

»»»7.2.��(1− x2)y′′ − 6xy′ − 4y = 0�

��..x = 1C−1/ �²F, Æ�ÊÝùó�

y =∞∑

n=0

anxn, |x| < 1�

h`X��5]�PW ∞∑

n=0

n(n−1)anxn−2−∞∑

n=0

n(n−1)anxn−∞∑

n=0

6nanxn−∞∑

n=0

4a0xn = 0�

B;�ÿ∞∑

n=0

(n(n− 1)an − (n− 1)(n + 2)an−2)xn−2 = 0�

.hn(n− 1)an − (n− 1)(n + 2)an−2 = 0,

Ç

an =n + 2

nan−2, n ≥ 2�

ãhÇÿ

a2n = (n + 1)a0, a2n+1 =2n + 3

3a1, n ≥ 1�

Æ�

y = a0

∞∑n=0

(n + 1)x2n + a1

∞∑n=0

2n + 3

3x2n+1�

×Ðóu3Ø ��îW×�ùó, JÌhÐó3� � ��ÝÝÝ(analytic), î�®°-Î0×�5]�P��&Æ�ì��§�

11.7 �5]�P�ùó� 681

��§§§7.1.'y = f(x) y′′ = ky3[x1, x2]��, x1 < x2�Jf3G� ��Ý�JJJ��..f y′′ = ky�×�,Æf ′�f3[x1, x2]/D3,ãhêÿf ′′(x)

= kf(x), f ′′′(x) = kf ′(x), f (4)(x) = kf ′′(x) = k2f(x), · · ·�Æf3[x1, x2]�N×$0ó/D3�ãÏ°a(3.24)P, E∀x0 ∈ [x1, x2], D3×ξn+yx�x0 , ¸ÿ

f(x) = Pn(x) + Rn(x),

Í�

Pn(x) =n∑

i=0

f (i)(x0)

i!(x− x0)

i, Rn(x) =f (n+1)(ξn)

(n + 1)!(x− x0)

n+1,

u�J�lim

n→∞Rn(x) = 0, ∀x ∈ [x1, x2],

Jã8.5;��á

(7.1) f(x) =∞∑

n=0

f (n)(x0)

n!(x− x0)

n,

∀x ∈ [x1, x2], ÇÿJf3[x1, x2] ��Ý�.f�f ′/3[x1, x2]=�, ÆD3×M > 0, ¸ÿ

|f(x)| ≤ M, |f ′(x)| ≤ M, ∀x ∈ [x1, x2]�

¨êu|k| > 1, J�m = |k|, u|k| ≤ 1, -�m = 1�J

|f ′′(x)| = |kf(x)| ≤ mM ≤ m2M,

|f ′′′(x)| = |kf ′(x)| ≤ mM ≤ m3M,

|f (4)(x)| = |k2f(x)| ≤ m2M ≤ m4M,

×��,

|f (n)(x)| ≤ mnM, ∀x ∈ [x1, x2]�

682 Ïè×a �5]�P

.h

|Rn(x)| = |f (n+1)(ξn)

(n + 1)!|x− x0|n+1 ≤ Mmn+1

(n + 1)!|x− x0|n+1,

�hÇ

|Rn(x)| ≤ M

(n + 1)!|m(x− x0)|n+1�

.

limn→∞

M

(n + 1)!|m(x− x0)|n+1 = 0,

Æ

limn→∞

Rn(x) = 0�ÿJ�

¿àùó�ô�J�°×P�§(��§3.2)�'g1Cg2 y′′ =

ky�Þ�, v��

g1(x0) = g2(x0), g′1(x0) = g′2(x0),

Í�x0 ∈ [x1, x2]��

f(x) = g1(x)− g2(x),

Jf) y′′ = ky�×�, vf(x0) = f ′(x0) = 0�.

f ′′(x0) = kf(x0), f ′′′(x0) = kf ′(x0), f (4)(x0) = k2f(x0),

ÇE∀n ≥ 1,

f (n)(x0) = 0�ã�§7.1(Ç(7.1)P),

f(x) =∞∑

n=0

f (n)(x0)

n!(x− x0)

n, ∀x ∈ [x1, x2],

êÞ 683

�h�ùó�N×;ó/ 0, Æf(x) ≡ 0, ∀x ∈ [x1, x2]�.h

g1(x) = g2(x), ∀x ∈ [x1, x2]�

ð�1, ã��R�f�x = x0`, y = y0, y′ = y1, �°×X�y′′ = ky ��

êêê ÞÞÞ 11.7

1. �Oì�&�5]�P�ùó�, ¬X�&ùó�[e�5�(1) y′ = y� (2) y′ = xy�(3) y′′ = xy� (4) y′′ + x2y′ + xy = 0�(5) (1− x2)y′′ + 2xy′ + 4y = c�

2. �|ùó¼�y′′ = y, ¬J��îW

y = C1ex + C2e

−x�

3. �Jy′′′ = ky�� Ý�

4. �Jy′ = x + y2��

y = a + a2x + (1

2+ a3)x2 + (

a

3+ a4)x3 + · · ·�

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684 Ïè×a �5]�P

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õõõ SSS�� ZZZ õõõ SSS

×××iii×l[e, 469

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×�4, 389

×��, 641

×$�0ó, 528

×$aP�5]�P, 644

ÞÞÞiiiÜaÕ�, 2

Þ¥ó�, 467

Þ$�0ó, 529

Þ$aP�5]�P, 654

Þ$0ó, 132

Þ$0ó¼¾½ÁÂ, 202

Þ4P�§, 15, 17, 499

Þ4Pùó, 496

æ, 554

ëëëiii��Ý�$, 343

�Î�, 582

ë��P, 13

ë�Ðó, 50

ë�ÐóÝ�5, 105

ë�»�§, 14

ë¥�5, 581

ì�, 200

ìõ, 78

ì@&, 8

ì�5, 80

î�, 200

îõ, 78

î@&, 7

î�5, 80

°°°iii�Ñð�5, 438

��, 52, 231

��5, 112

� Â�§, 59

!Ê, 100

/�, 516

/F, 516

2�6, 2

5µÐó, 357

5I�5°, 161

5v, 75

5Ò�ó, 640

6¿«, 541

6'�, 527

686 õS

6a, 120

Dë�Ðó, 257

DÑ6, 320

DÑ<, 318

D`F, 203

D æF, 245

Dõ<, 319

D0ó, 153

M��;�, 90

]'0ó, 526

fÂl�°, 410

f´�§, 91

f´l�°, 403, 441

i¿��a, 38

pñ, 55, 71

pñ°, 249

"""iii�TÏ½, 26

�5Ò�5]�P, 668

�É�Ý�=�, 45

��, 119

��Ðó, 119

�óÍ, 93

��, 83

�=�, 45

�Á§, 27, 32

²�, 541

²F, 516

¼=�, 45

¼Á§, 27, 33

¿íÂ, 99

¿É��P, 90

¿â, 373

¿âÐó, 545, 546

Í², 46

Ñð�5, 438

Ñ!5v, 75

Ñ4ùó, 402

�ó]�P, 20

�óÐó, 257

�óó, 20

�Ðó, 115

�Ðó, 115

�P, 90

�JíÂ�§, 99

�, 554

�� , 100

000iiiøýùó, 419

�aÝ� æ§, 446

�IÁ�, 182

9�Ðó, 515

9�óÐó, 515

?Ý� , 473

D3P�§, 641

[e, 2, 5, 446

[e�5, 482

[e , 482

[ei, 485

`aH, 641

b&, 8

b§×ÝÁÂ, 564

�ZõS 687

b§Í, 93

b§Ë�, 65

g�P, 553

gó, 640

�QEó, 258

��ó, 515

�, 468

��P, 609

� �?, 544

�$, 225

'��óÐó, 515

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ÚÚÚiii¦9¯, 71

�, 468

í8[e, 469, 470

í8b&, 478

í8=�, 63

í8=�P, 63

íÂ�§, 97, 187, 543

ô^�§, 14

ô^æ§, 14

IÝP², 63

IÁ�, 182

�I, 5

¾½P, 660

�ý, 227

X�ÿP, 356

âââiiiÐóÝÁ§, 26

Ðóùó, 461

Ðóó�, 461

ß§, 603

å×y, 403

��5, 156

9, 262

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eÎ«, 603

eÎa°, 384

jF, 203

�@ÝÝ�P, 334

õ, 84

ø�P, 105

ü�F�§, 60

�²F, 680

°a, 125

½Ð]�P, 263

��Ðó, 222

�«8�, 404

�]9\�, 76

�Ã{Æ, 23

&[;, 584

&Pì�, 655

ÜÜÜiiikà��a, 36

ª�5]�P, 671

¼óÐó, 257

÷Ú2, 25

øó, 150

øó°, 150

8EÁ�Â, 182

688 õS

8EÁ�Â, 182

8EÁÂ, 182

8n>£, 247

Ö�, 365

Þ�, 43

Þ�xÂ, 440

Þ�[e¾½°, 392

Þ�íÂ�§, 188

Þ��P, 224

Þ�f�, 476

Þ�ó�, 391

¥�5, 570

¥Pùó, 400

ª6èâß, 25

«�, 75

«�5, 582

«� 0, 591

èèèiiiÅó, 143

æÐó, 153

æ�5]�P, 639

-5¤, 132

-5ºÕ, 132

qPl�°, 409

��2P, 219

��94P, 217

��ùó, 491

Îp, 467

Þ]�, 23

ö�, 515

ùó, 389

ùó�¥4, 429

©½�, 643

©Ç]�P, 660

Y`j, 604

{ú, 443

{úl�°, 417

{$0ó, 132

R�f�, 643

©�, 50

èèè×××iii��5]�P, 639

¢óP, 526

¢ó°, 376

¢ó�5°, 662

°×P�§, 490, 643

Ã9, 661

I»�, 370

V�°, 383

V�, 533

f�[e, 424

ðà�5�, 160

ð�5]�P, 639

¦2ý, 617

Î�É¶�, 517

Ï×l��5, 438

ÏÞl��5, 444

Þ5, 78

�)ÝÞ$�0ó, 544

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�ZõS 689

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=�Ðó�b&�§, 61

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=�ÐóÝÁÂ�§, 61

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@gÁ§, 520

@g�5, 572

@ð¿â, 557

@ð=�Ðó, 56

@ð��, 86

@ðaP, 117

@F[e, 463

@FÝ, 56

I55P, 329

I55P°, 161

I5�5, 438

Jì, 25

èèèÞÞÞiiimñf, 42

t�ì&, 8

t�î&, 7

t�î&2§, 7

t�¿]°, 569

t·�, 181

vi�, 2

va, 124

��'�, 526

��Á§, 27

��0ó, 119

��vb&, 7

��ó�, 7

¿¢¿í, 15

¿¢ùó, 395, 457

s÷, 5

��«, 565

�E[e, 424

�EÁ�, 60

�EÁ�, 60

¾¾¹+, 71

¾¾¹+!J, 419

Ìa�ó, 84

P§�Ý�, 72

P§g��, 491

P§Ý�=�, 45

P§Ý¶�, 503

P§ùó, 389

P§�5, 438

Pì�, 655

�é�, 71

�Ë�, 65

$, 640

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/)Ðó, 100

ðy, 71

ðyty` æ§, 245

ø÷Ðó, 257

ø÷ó, 20

èèèëëëiiiiø£, 23, 459

iÖ2ý, 616

690 õS

Á§, 1

Á§f´l�°, 403, 441

Á§Ðó, 463

ÁÂ, 60, 559

Á2ý, 610

�¥�5, 625

�5]�P, 292

�5Ý, 133

�5ÝíÂ�§, 187

�5¤, 133

�5ºÕ, 132

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151

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154

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EÌÝÁ§, 440

EÌÏ½, 26

EóÐó, 257

��a, 36, 204

�3, 7

�¦, 7

Õ�¿í, 15

Õ�¿¢¿íÂ��P, 16

Õ¿��P, 16

å[P, 65

^«, 365

^½, 365, 525

0-¿]õ, 569

0-§¡, 443

BÃ]�P, 661

L]2P, 173

�P, 654

�P�5]�P, 675

èèè"""iiiÂL�5, 438

ó�ÝÁ§, 1

�Z, 19

�Z2P, 506

�Zðó, 423

�Zó, 19

�S, 2

¦�, 213

Pó, 91

aP, 90

aP�ð, 611

a�5, 554

|Ï�, 150

�õÝ, 532

�õùó, 397

õ4, 219

Ë�, 65

Ï½, 26

ëF, 560

èèè000iii�ùó, 458

�ùó"P, 488

�ZõS 691

0ó, 119

Y¦, 598

Yi�5, 354

^£Û�Ðó, 443

�5, 438

�5�íÂ�§, 97

�52P, 328

�5.�, 640

�5 ½, 581

�5Õ�, 83

�5l�°, 406

�^Ý, 356

�^ÿP, 356

èèèÚÚÚiiiT�ó, 515

e÷P, 403

� >�, 119

Û&F, 184

2Ðó��5, 549

Á/, 100

èèèâââiii»ð, 357

Ô`Ðó, 317

èèèÜÜÜiii%�, 11

1Ä¾!J, 232

\&f�, 643

\&F, 516

ÞÞÞèèèÞÞÞiiiJ¥Ðó, 99

ÞÞÞèèèëëëiii�5°, 72

�;£, 72

�ó�ð°, 161

��, 616

692 õS

zZõS 693

zzz ZZZ õõõ SSS

A

αg�Lipschitzf�, 195

αg�í8Lipschitzf�, 195

AbelI5õ2P, 426

AbelÁ§�§, 511

Abell�°, 427

Abel partial summation formula,

426

Abel’s limit theorem, 511

Abel’s test, 427

absolute convergence, 424

absolute maximum, 60

absolute minimum, 60

algebraic function, 257

algebraic number, 20

alternating series, 419

analytic, 680

antiderivative, 153

Arbogast, 132

arc cosine, 319

arc length, 373

arc sine, 318

arc tangent, 320

Archimedes, 23

area, 75

arithmetic mean, 15

asymptote, 36

asymptotically equal, 404

B

Barrow, 71

base, 262

basis, 661

Bernoulli equation, 648

Bernoulli inequality, 15

Bernoulli��P, 15

Bernoulli]�P, 648

Bernstein, 495

binomial series, 496

Binomial theorem, 17

Bolzano, 57

Bolzano theorem, 57

Bolzano�§, 57

boundary condition, 643

boundary point, 516

Boundedness theorem for con-

tinuous function, 61

Brouncker, 390

C

calculus of variations, 72

Cartesian product, 517

694 õS

Cauchy, 43

Cauchy condition, 476

Cauchy principal value, 440

Cauchy��P, 104

Cauchy’s convergence criterion,

392

Cauchy’s inequality for integrals,

104

Cauchy’s mean-value formula,

188

chain rule, 138

change of variable, 161

characteristic function, 660

circle of convergence, 485

column, 468

compactness, 65

comparison test, 403

complementary equation, 661

concave downward, 200

concave function, 115

concave upward, 200

conditional convergence, 424

content zero, 591

continuity, 42

continuous, 42

continuous from the right, 45

continuously differentiable, 161

convergent, 5

converges pointwise, 463

converges to zero, 2

convex function, 115

counterclockwise, 604

covering, 65

critical point, 184

cross product, 541

cross section, 365, 525

cylinder, 365

cylindrical coordinates, 616

D

De Moivre, 506

De Moivre2P, 506

definite integral, 156

degree, 640

del-f , 533

deleted neighborhood, 26

dependent variable, 515

derivative, 119

difference operator, 132

difference quotient, 132

differentiable, 119

differentiable function, 119

differential equation, 292

differential quotient, 133

differentials, 133

differentiation operation, 132

directional derivative, 526

Dirichlet integral, 445

Dirichlet�5, 445

Dirichletl�°, 427

Dirichlet’s test, 427

discriminant, 660

distribution function, 357

zZõS 695

divergent, 5

double sequence, 467

dummy variable, 84

E

ellipsoid, 598

essential, 46

Euler, 19

Euler’s constant, 423

Euler’s formula, 506

Euler’s number, 19

exact differential equation, 671

existence theorem, 641

existence-uniqueness theorem,

643

exponential function, 257

exterior point, 516

extrema, 182

extrema with constrains, 564

extreme value, 60

Extreme-value theorem for con-


extremum, 182

F

family of curves, 641

Fermat’s principle of least time,

245

first partial derivative, 528

first-order linear differential equa-

tion, 644

Fixed-point theorem, 60

fluxions, 150

force, 554

Fourier, 42

Fresnel integral, 446

Fresnel�5, 446

G

gammaÐó, 446

Gauss’ test, 417

general solution, 641

general term, 389

geometric mean, 15

geometric series, 395

glb, 8

global maximum, 182

good behavior, 473

gradient, 533

greatest lower bound, 8

Green, 604

Green�§, 604

Green’s theorem, 604

Gregory, 459

H

half-open interval, 100

harmonic series, 397

Hermite, 20

higher derivatives, 132

homogeneous, 654

homogeneous differential equa-

tion, 675

homogeneous of degree n, 553

696 õS

horizontal asymptote, 38

Huygens, 150

hyperbolic cosine, 317

hyperbolic functions, 317

hyperbolic sine, 317

hyperbolic tangent, 317

I

improper integral, 438

improper integral of the first

kind, 438

improper integral of the second

kind, 444

increment, 213

indefinite integral, 112

independent variables, 515

indeterminate form, 52

infimum, 8

infinite discontinuity, 45

infinite integral, 438

infinite product, 503

infinite series, 389

infinitely differentiable, 491

infinitely small quantities, 72

infinitesimals, 72

initial conditions, 643

inner product, 516

instantaneous velocity, 119

integral test, 406

integrating factors, 640

integration by partial fractions,

161

integration by substitution, 161

integration formula, 328

interior point, 516

Intermediate-value theorem, 59

interval of convergence, 482

inverse cosine, 319

inverse sine, 318

inverse tangent, 320

inverses of the trigonometric func-

tions, 257

is dominated by, 403

iterated integral, 572

J

Jacob Bernoulli, 648

Jacobian, 565

Jacobian determinant, 609

jump discontinuity, 45

L

L’Hospital’s rule, 232

L’Hosptial, 232

Lagrange, 131

Lagrange¶ó°, 564

Lalpace»ð, 449

Landau, 225

Laplace transform, 449

Laplace]�P, 532

Laplace’s equation, 532

least upper bound, 7

left-hand limit, 27

Leibniz, 71

zZõS 697

Leibniz rule, 419

Leibniz-Gregory series, 459

level surface, 565

limit comparison test, 403

line integral, 554

linear transformations, 611

Liouville, 641

Lipschitz, 191

Lipschitz condition, 191

Lipschitz condition of order α,

195

Lipschitz continuous, 191

Lipschitzf�, 191

Lipschitz=�, 191

little-oh notation, 224

local maximum, 182

local property, 63

logarithmic function, 257

lower integral, 80

lower sum, 78

lub, 7

M

Machin, 460

Maclaurin, 224

Maclaurin’s formula, 224

Maclaurin2P, 224

matrix, 467

mean value, 99

Mean-value theorem for deriva-

tives, 187

Mercater, 390

Mercator, 459

Method of Langrange’s Multi-

pliers, 564

Method of least squares, 569

Method of variation of param-

eters, 662

mixed second partial derivative,

544

multiple integral, 570

mutually disjoint, 100

N

n-fold integral, 581

n¥�5, 581

Naiper, 267

Napierian logarithm, 267

natural logarithm, 258

neighborhood, 26

Newton, 71

Newton’s method, 249

nondegenerate, 584

nonnegative term series, 402

nontrivial solution, 655

norm, 91

normal line, 125

O

o-notation, 224

octant, 603

ODE, 639

one-sided derivative, 119

one-sided limit, 27

698 õS

open covering, 65

optical law of reflection, 245

optimum solution, 181

order, 225, 640

order of magnitude, 343

ordinary differential equations,

639

ordinate set, 100

P

p series, 407

pùó, 407

parabolic rule, 384

paraboloid, 603

parameter method, 376

partial differential equations, 639

partial fractions, 329

partial integral, 438

particular solution, 643

partition, 75

periodicity, 105

piecewise continuous function,

56

piecewise linear, 117

piecewise monotonic, 86

piecewise smooth, 557

point of inflection, 203

pointwise, 56

polar coordinates, 610

positive term series, 402

power series, 458

power-series expansion, 488

primitive function, 153

probability density function, 443

proper integral, 438

R

Raabel�°, 417

Raabe’s test, 417

radian measure, 50

radius of convergence, 482

rate of change, 72

ratio test, 410

rearrangements of series, 429

rectangular polygon, 76

recursive formula, 173

refinement, 78

region of integration, 581

regular partition, 75

related rates, 247

relative extreme value, 182

relative extremum, 182

relative maximum, 182

relative minimum, 182

remainder, 219

removable discontinuity, 45

repeated integral, 572

Ricatti equation, 641

Ricatti]�P, 641

Riemann, 93

Riemann sum, 93

Riemann zeta-function, 405

Riemann zetaÐó, 405

Riemannõ, 93, 366

zZõS 699

Riemann¥4�§, 432

Riemann’s Rearrangement The-

orem, 432

right-hand limit, 27

Rolle, 186

Rolle�§, 186

Rolle’s theorem, 186

root test, 409

row, 468

S

saddle point, 560

Sandwich rule, 14

scalar, 515

secant line, 124

second derivative, 132

second derivative test for ex-

trema, 202

second partial derivative, 529

Seidel, 469

separable differentiable equation,

668

separation of variables, 640

sequence of functions, 461

series, 389

series of functions, 461

set function, 100

sign-preserving property of con-


Simpson, 384

Simpson°, 384

Simpson’s rule, 384

singular point, 680

smooth, 373

Snell’s law of refraction, 246

spherical coordinates, 617

Squeezing principle, 14

stationary, 11

Stirling2P, 416

Stirling’s formula, 416

Stokes, 469

subrectangle, 582

subscript, 227

successive bisection, 61

sum, 84

supermum, 7

surface integral, 582

symmetric limit, 440

symmetric neighborhood, 26

T

tail, 5

tangent line, 120

tangent plane, 541

tangent vector, 527

Tauber�§, 513

TauberÏ×�§, 513

Tauber’s first theorem, 513

Tauber’s theorem, 513

Taylor, 217

Taylor polynomial, 217

Taylor’s formula, 219

telescoping series, 400

total square error, 569

700 õS

transcendental function, 257

transcendental number, 20

transform, 357

trapezoidal rule, 383

trial and error, 56

triple integral, 581

trivial solution, 655

U

uniform continuity, 63

uniform convergence, 469

uniform Lipschitz condition of

order α, 195

uniformly bounded, 478

uniformly continuous, 63

Uniqueness theorem, 490

upper integral, 80

upper sum, 78

V

vertical asymptote, 36

W

Wallis, 175

Weierstrass, 469

Weierstrass M -test for Uniform

Convergence, 475

Weierstrass M -l�°, 475

weight function, 99

well-behaved, 544

work, 554

Wronski, 664

Wronskian, 663

Z

zetaÐó, 405

璋編著國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

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Transcript of 璋編著國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

璋 編著 國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

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Transcript of 璋 編著 國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

璋編著國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...

Transcript of 璋編著國立高雄大學應用數學系stat.nuk.edu.tw/huangwj/book/calculus.pdfCalculus, Vol I...