Calculus One and Several Variables 10th Ed Textbook + Student Solutions Manual

726 Pages·2007·52.364 MB·English

by  Salas, Saturnino L., Etgen, Garret J., Hille, Einar

TENTH
EDITION
CALCULUS
ONE VARIABLE
JOHN WILEY & SONS, INC.
CHAPTER I PRECALCULUS REVIEW
1.1 WhatisCalculus? I
1.2 Review of Elementary Mathematics J
1.3 Review of Inequalities II
1.4 Coordinate I'lane; Analytic Geometry 17
1.5 Functions 24
1.6 The Elementary Functions 32
1.7 Combinations of Functions 41
1.8 A Note on Mnthcmutical Proof; Mathematical Induction 41
CHAPTER 2 LIMITS AND CONTINUITY 53
2.1 The Limit Process (An Intuitive Introduction) S3
2.2 Definition of Limit 64
2.3 Some Limit Theorems 7J
2.4 Continuity 82
2.5 The Pinching Theorem; Trigonometric Limits 91
2.6 T"'O Basic Theorems 97
Project 2.6 The Bisection Method for Finding the Roots of f(x) = 0 1(}2
xv
xvi • CONTENTS
CHAPTER 3 THE DERIVATIVE; THE PROCESS OF DIFFERENTIATION 105
3.1 The Derivative 105
3.2 Some Ditl'crentimion Formulas 115
3.3 The d/dr Notation; Derivatives of Higher Order 124
3.4 The Derivative As A RateorChange 130
3.5 The Chain I{u[e 133
3.6 Differentiating The Trigonomt:trie runClions 142
3.7 Implicit Difii:rcntiation; Rational Powers 147
CHAPTER 4 THE MEAN-VALUE THEOREM; APPLICATIONS
OF THE FIRST AND SECOND DERIVATIVES 154
4.1 The Mean-Value Theorem 154
4.2 Increasing and Decreasing Functions 160
4.3 Local Extreme Values 167
4.4 Endpoint Extrellll! Values; Absolute Extreme Values 174
4.5 Some Max-Min Problems 182
Projeet4.S Flight Paths ofBird~ 190
4.6 Concavily lmd Points of Inflection 190
4.7 Vertical and Horizontal Af.ymptotcs; Vertical Tangents alld Cusps 11)5
4.8 Some Curve Sketching 201
4.9 Velocity and Acceleration; Speed 209
Prujet.:t 4.9A Angular Vdoeity; Unifornl Cin:ular Motion 217
Project 4.98 Energy ofa railing Body (Ncar the Surface of tile Earth) 217
4.10 Related Rates of Change per Unit Time 218
4.11 Differentials 223
Project 4.11 Mar};inal Cost, Marginal ReV'cllue, Marginal Profit 228
4.12 Newton-Raph:>on Approximations 229
CHAPTER 5 INTEGRATION 234
5. I An Area Problem; a Speed·Distance Problem 234
5.2 The Definite Integral ora ContinuoL.!s Function 237
5.3 The Function j(x) = 1~ j(t)dt 246
5.4 The Fundamental Theorem of Integral Calculus 254
5.5 Some Area Problems 260
Project 5.5 Integrability; Integrating Discontinuous Functions 266
5.6 Indefinite Integrals 268
5.7 Working l3ack from the Chain R\lle; the II-Substitution 274
5.8 Additional Properties of the Definite Integral 281
5.9 Mean-Value Theorems for Integrals; Average Value ora Function 285
CONTENTS • xvii
CHAPTER 6 SOME APPLICATIONS OF THE INTEGRAL 292
6.1 MoreonArca 292
6.2 Volume by Parallel Cross Sections; Disks and Washers 296
6.3 Volume by the Shell Method 306
6.4 The Centroid ora Region; j'appus's Theorem on Volumes 312
Project 6.4 Cenlmid ofa Solid of Revolution 319
6.5 ThcNotionofWork 319
"6.6 Fluid Force 327
CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS 333
7.1 One-la-One runclil)ns; Jnver~es 333
7.2 The Lognrithm Function, Part r 342
7.3 The LAgarithm Function, Pm1 II 347
7.4 The Exp()ncntial Function 356
Project 7.4 Some RationalOounds for the Number e 364
7.5 Arbitmry Po ..... 'Cf'Sj Other Bases 364
7.6 Exponential Growth and Decay 370
7.7 The Inverse Trigonometric Functions 378
Projccl7.7Refrnction 387
7.8 The Hyperbolic Sine and Cosine 388
'7.9 The Other Hyperbolic Functions 392
CHAPTER 8 TECHNIQUES OF INTEGRATION 398
8.1 jnlegrdl Tables and Review 398
8.2 Intcgnltion by Parts 402
Project 8.2 Sine Wavesy = sinnx and Cosine Wavesy = cosnx 410
8.3 Powers 1l11d Products ofTrigonomelric Functions 411
8.4 Integrals Featuring.J'ii'f=X!, ..rar+:?, ~ 417
8.5 Rational Functions; Partial Fractions 422
'8.6 Somc Rationalizing Substitutions 430
8.7 NUJllericallmegration 433
CHAPTER 9 SOME DIFFERENTIAL EQUATIONS 443
9.1 First-Order Linear Equ31ions 444
9.2 Integral Curves; Scp3r.Ib1c Equations 451
Projecl9.2 Orthogonal Trajectories 458
9.3 The Equation y" + ar' + by = 0 459
'Denotcsoptional !.Celion.
r
xviii. CONTENTS
CHAPTER 10 THE CONIC SECTIONS; POLAR COORDINATES;
PARAMETRIC EQUATIONS 469
10.1 Geometry of Parubola. Ellipse, Hyperbola 469
10.2 Polar Coordinates 478
10.3 Sketching Curves in Polar Coordin&les 484
Projttt 10.3 Parabola, Ellip!>e, Hyperbola in Polar Coordinates 491
10.4 Area in PolarCoordinales 492
10.5 Curves Given Parametrically 496
Project 10.5 ParaboHc Trajectories 503
10.6 Tangents to Curves Given Panunctricnlly S03
10.7 Arc Lengtband Speed 509
10.8 The Area of A Surface of Rev oJu ti on; The Centroidofa Curve; Pappus's Theorem
on Surface Area 517
Project 10.8 The Cycloid 525
CHAPTER II SEQUENCES; INDETERMINATE FORMS;
IMPROPER INTEGRALS 528
11.1 The Least UppcrBound Axiom 521:1
11.2 Sequences or Real Numbers 532
11.3 Limit of a Sequence 538
Project 11.3 Sequences and the Ncwton-Raphson Method 541
11.4 Some Important limitS 550
11.5 The Indetenninate Fonn (0/0) SS4
11.6 The Indetenninate Form (00/00); Other Indeterminate Forms 560
11.7 Improper Integrals 565
CHAPTER 12 INFINITE SERIES 575
12.1 Sigma Norutioll 575
12.2 Infinite Series 577
12.3 The Integral Test; Basic Comparison, Limit CompOlrison 585
12.4 The Root Test; the: Ratio Test 593
12.5 Absolute Convergencc and Conditional Convergence; Alternating Series 597
12.6 Taylor Polynomials in .f; Taylor Series in x 602
12.7 Taylor Polynomials and Taylor Series in;r -a 613
12.8 Power Serics 616
12.9 Differclltiation and Integration of Power Series 623
Project 12.9A The Binomial Serics 633
Project 12.98 Estimating 1l 634
CONTENTS • xix
APPENDIX A SOME ADDITIONAL TOPICS A-I
A.l Rotation ofAxeR; Eliminating the xy-Term A-I
A.2 Determinants A-3
APPENDIX B SOME ADDITIONAL PROOFS A-8
8.1 The Intermediate-Value Theorem A-8
B.2 l3oundcdncss; Extn:me-Valuc Theorem A-9
B.3 Inverses A-\O
B.4 The Integrability of Continuous Functions A-ll
8.5 The Integral as the Limit of Riemann Sums A-14
ANSWERS TO ODD-NUMBERED EXERCISES A-15
Index 1-1
Tobie of Integrals Inside Covers
TABLES OF INTEGRALS
POWERS
1. J~ du=--nu++·C"1. n:;f':""l 2. J-d ;u; = In lui + C
EXPONENTIALS AND LOGARITHMS
J
3. e"du=e"+C 4. / P"du=i!....+C
Inp
f f
5. ue"du=ue"-e"+C 6. u2ell du = u2e" - 2ue" + 2e" + C
J J
7. u"e" du = u"e" - n Un-leu du 8. !lnudu=Ulnu- u+C
J
9. ! (lnU)2du=U(lnu)2-2ulnu+2u+C 10. ulnudu=4u21nu- *u2+C
,[ 1
11. Ju "lnudu=u"+ -11In+-" 1 -(n- +I- If +C 12. / -u dIn"u =inllnul+C
! SINES AND COSfIN ES
13. sinudu = -cosu + C 14. cosudu = sinu + C
J f
15. sin2udu= ~u - ~sin211 +C 16. cos2udu= 4u+*sin lu+C
f J
17. sinJudu = f cosJu - cosu + C 18. cosJudu = sinu - t sin3,t + C
.
19. Jsm"udu=-sin-,,--l,u,-c0-5+11 -nn--Ijf sin .. -2 IIdu
11-1/
20. / cos" II du = -cos"--I,,u-sin-u + -n- cos"-2 II du
J f
21. II sinuciu = -/I casu + sinu + C 22. ucosudu=usinu+cosu+C
f J
23. ju"sinudu=-ullcosu+n! u,,-lcosudu 24. u"cosudu=u"sinu- n un-Isinudu
25. s.ml.l/ usmllsinl [lemd -un=)u~] -sin [em~ + n)u] +C,m 2 #n 2
J/
26 . cosmucosnu d u=sin[~(m- n+)u] ~sin[(m+n)u] + C ,m2 ,# n 2
. cos[(m-n)u] cos[(m+n)u] 2 2
27. / smmucosnudu= 2(m- n) ~+c, 111 ":f:.n
f
28. (!"usinbudu= a2:Jil(aSinbu-bCOSbU)+C
29. / ellU cosbudu = -a 'e-+"-" b2 (acosbu + bsinbu) + C
I
J TANGENTS AND SfEC ANTS
30. lanudu = lnlsecul +C 31. seeudu=lnlsecu+tanul+C
f f
32. tan2udu = lanu - u + C 33. seel udll = tanu +C
f f
34. seeulanudu=seeu+C 35. tan3udu = 4t an2 u + Inleosul + C
J
36. sec3lldll= ~seclltanu+~lnlseeu+tanlll+C
n-2/ "
37. / lanl/udll = -lannn---I1I I - / tann-1udu 38. / sec" II dll = -sec"-n- 2-I-11ta n-ll + -n --1 sec -2 u du
J CONTANGENTS AND fCO SECANTS
39. cOll/du = In I sinlll + C 40. eseudu = In Icscu - cotul + C
! 42.! 2
4J. co~ u du = -col u - u + C csc udll= -cotu+C
43.! J
cscucotudu=-cscu+C 44. eOl31ldu = -~ COl2 II - In Isinul + C
f
45. ese3 udu = -4escli COlu + i In Icscll - cmul + C
-/eorr-
46. / COt"Udu=_cot"-lu 2udll
11-1 n-2/
47. / cscnutiu = -C-SCnn- --21-l1C o-tu + -n - -I esc"-2Ildu
J HYPERBOLIC FUNfCT IONS
48. sinhudu=eoshu+C 49. coshudu = sinhu + C
J J
50. tanh u du = In (cosh u) + C 51. cothudu = In I sinh III + C
J !
52. scchudu = arctan(sinhu)+C 53. cschudll = In I tanh 4111+ C
J J
54. scch2udu = tanh II + C 55. csch 2udu = -cath 1/ + C
J f
56. sech u tanh II du = -scch II + C 57. csehueoliludll=-cschll+C
f f
58. sinhl udu = ~sinh2u - 4" + C 59. cosh2udll= ~sinh2u + 4u +C
f f
60. lanh2II du = u - Ianhu + C 61. eoth2udu=u-cothu+C
f f
62. usinhudu =ucoshu - sinhu + C 63. u coshudu = usinhu - coshu + C
(rabiecofilinued<JIlhehadr;)
(continued/rom/herronl)
f INVERSE TRIGONOMETRICf FUNCTIONS
64. arcsinudll=lIarcsinu+~+C 65. arccosudu=lIarccosu -~+C
f f
66. arctanu du = u arctanu - 4 In(1 + u2) + C 67. arccotu du = u arccotu + 41n(1 + u2) + C
f f
68. arcsecudu = uarcsecu -Inlu +~I +C 69. arccscudu= uarccscu+lnlu+~1 +C
f f
70. u arcsinudu = ~[(2u2-I)arcsinu + u~ + C] 71. uarctanlldu = 4(u2 + l)arctanu - 4u + C
f
72. lIarccosudu = ~[(2U2 - 1) arccos u - u~ + c]
J
73. Ju narCSinlldu=- I - [ un+1arCS.l TIU- ~u"+d' u ] ,n#--i
n+1 v1- u2
J
74. Ju " arccosu du = - I - [ un+1 arccosu + "-u--"-+---'- --.,, du ] ,n #--I
n+l vl-u2
J 1
75. Ju narctanudun= +1- -1 [ u,,+larctanu- -1u +"+-u' 2du ,n#--l
~,a>O
76. J-a 2d+u- u2 = -a1 arctan -au + C 77. f .ja~:u2 =In(u+Ja2+u2)+C
f
78. ..,Ja2 +u2du = ~.Ja2 +u2 + ~In(u +.Ja2 +u2) +C
79.!
1I2Ja2+u2du=
80. J-.J-a2u+-ud2 u=.Ja2+u2- aln la+-..-/au-2+-u21+ C
81. J-~u-,- du=-,/-a2-u+-u' +1n ( u+v~a2+)u2 +C
82. J ~u=2d-u. .,Ju 02+u2-a-2 1n ( u+.Ja2+u2 ) +C
v02+u2 2 2
83 . J -u-du~- _ _ --a1 In l-a+-~u- -1 + C· 84. J -u2../-ad2u + - u2 ~ -.J-aa2- 2+u -u2 + C
85. J (-a2 +-d uu 2-)3/2~ a-2../-0 121 +- u2 +C
~,a>O
86. J- d-u - = arcsin -u + c 87. J /a2 -u2du = Zu .Ja2 _u2 + a22a rcsin~u + C
../a2 _ u2 a
88. Ju 2.Ja2 - u2 du = _u8 ( 2u2 - (2))02 - u2 + -a84 arcsin -au + C
89. j-~-,,- dl/= ..r;;0--2----u-2, -0In /-a+v-'a,, _'-_"'/ +C
90. j-~-I12 - dll= --IlIv r=a;-2- _--1-1:2; -arcst.n1-1a + C 91. j -~/l-2du- = --U2v ra=2;- _--1-1-:2; + -02a2 rcsinaI-I +C
du r;------,
92. j u./02_U2 =-;1; In /-a-+u~-/- +C 93. j --d-" =--1v a2-u2+C
u2~ a1u
95. j -(a2 -_d l1-l/ 2)3=/2 -a2.-jaI2I- _+112 C 96. j -02 d--u u=l -210 In la-a +--"II1 +C
Ju2_a2, u >0
97. j 11./11t2il l_ 02 = -;I:; arcsec -lu;;i + C 98. j-~-- dll = 'l/rU=2;- _---0-:2; - aarcsec-II + C
f " a
99. Ju2 -a1du = ~ju2 _ 02 - ~ln)u +/112 - a2) +C
j ~ -I,,' _a' I r;------,I
101. -,-,,-dU=---,,-+ln u+..;u2_02 +C
I r;------,I
102. j -JI1l1-22 d_l-l0 2 = -2/IV 1r=l2;- _---0-:2; + 0-22 In II + "u2 _02 +C
j
103. j -",-.;-,d,-u, - -a' = -./1a12- '-,, -0-2 + C 104. -u(-,- d-au' -)'-/' = -a-'-';"-u- '- -a-' + C
l05.j(~u2_a2)3/2 --~~+ 1 n II I +~II Ia +C
a + bu,...;a+bii
j udu I
106. a+bu =/J.(a+bu- aln1a+bul)+C
107. j ou+2bdull = W1[ (a+bu) 2 - 4a(a+bu)+2a21nla+bul]+C
108.ju~(a=+b.u)! .a lnla-+"b_u I+C 109. j -u2(-a d+-u b=ll) --a+1u -ab2 I n l-a+/Ib- "l +C
I1l1l0.. jj -((aau+u~-'ddbbUuuu- r) ~' == -ObIJ' ((aaQ+ +bbuu) +-pa-I- +alnb' -1ua- +2baull+nlCa+ bulI )l l+. Cj -u(a- d+u - bul=l a-(a -+1 b-ll) --a12 In l-a+II- bul +C
j
113. j ,,~0u= -d-u-b u- -;3(2tbr u-,2-a)-",,a,+b-u+C 114. uvra-+:-b;l-ldu=I~S2(b 3bu-2a)(a+buiI2+C