Table Of ContentI ’
NSTRUCTOR S
S M
OLUTIONS ANUAL
M
ULTIVARIABLE
W A
ILLIAM RDIS
Collin County Community College
T ’ C
HOMAS ALCULUS
T E
WELFTH DITION
B O W
ASED ON THE RIGINAL ORK BY
George B. Thomas, Jr.
Massachusetts Institute of Technology
R
AS EVISED BY
Maurice D. Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
The author and publisher of this book have used their best efforts in preparing this book. These efforts
include the development, research, and testing of the theories and programs to determine their
effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to
these programs or the documentation contained in this book. The author and publisher shall not be liable in
any event for incidental or consequential damages in connection with, or arising out of, the furnishing,
performance, or use of these programs.
Reproduced by Addison-Wesley from electronic files supplied by the author.
Copyright © 2010, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-60072-1
ISBN-10: 0-321-60072-X
1 2 3 4 5 6 BB 14 13 12 11 10
PREFACE TO THE INSTRUCTOR
This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS
by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises. The corresponding Student's
Solutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (because
the CAS command templates would give them all away).
In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised or
rewritten every solution which appeared in previous solutions manuals to ensure that each solution
ì conforms exactly to the methods, procedures and steps presented in the text
ì is mathematically correct
ì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra
ì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation
ì is formatted in an appropriate style to aid in its understanding
Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems. A template showing
an example of the CAS commands needed to execute the solution is provided for each exercise type. Similar exercises within
the text grouping require a change only in the input function or other numerical input parameters associated with the problem
(such as the interval endpoints or the number of iterations).
For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
10.4 Comparison Tests 590
10.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 602
10.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627
Practice Exercises 634
Additional and Advanced Exercises 642
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
13.6 Velocity and Acceleration in Polar Coordinates 784
Practice Exercises 785
Additional Exercises 791
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
14 Partial Derivatives 795
14.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
16.6 Surface Integrals 972
16.7 Stokes's Theorem 980
16.8 The Divergence Theorem and a Unified Theory 984
Practice Exercises 989
Additional Exercises 997
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 10 INFINITE SEQUENCES AND SERIES
10.1 SEQUENCES
1. a" œ 11(cid:1)#1 œ0, a# œ "#(cid:1)#2 œ(cid:1)4", a$ œ 13(cid:1)#3 œ(cid:1)92, a% œ 14(cid:1)#4 œ(cid:1)136
2. a" œ 11! œ1, a# œ #"! œ 2", a$ œ 31! œ 61, a% œ 41! œ 214
# $ % &
3. a" œ (#(cid:1)(cid:1)1)1 œ1, a# œ (4(cid:1)(cid:1)")1 œ(cid:1)"3, a$ œ (6(cid:1)(cid:1)1)1 œ "5, a% œ (8(cid:1)(cid:1)1)1 œ(cid:1)7"
4. a" œ2(cid:2)((cid:1)1)" œ1, a# œ2(cid:2)((cid:1)1)# œ3, a$ œ2(cid:2)((cid:1)1)$ œ1, a% œ2(cid:2)((cid:1)1)% œ3
# $ %
5. a" œ #2# œ #", a# œ 22$ œ #", a$ œ 2#% œ #", a% œ 22& œ #"
# $ %
6. a" œ 2(cid:1)#" œ "#, a# œ 22(cid:1)#1 œ 43, a$ œ 22(cid:1)$1 œ 87, a% œ 22(cid:1)%" œ 1165
7. a" œ1, a# œ1(cid:2) "# œ #3, a$ œ #3 (cid:2) #"# œ 74, a% œ 74 (cid:2) #"$ œ 185, a& œ 185 (cid:2) #"% œ 1361, a' œ 3623,
a( œ 127, a) œ 255, a* œ 511, a"! œ 1023
64 128 256 512
" " "
8. a" œ1, a# œ "#, a$ œ ˆ3#‰ œ "6, a% œ ˆ46‰ œ #"4, a& œ ˆ#54‰ œ 1#"0, a' œ 7#"0, a( œ 50"40, a) œ 40,"320,
a* œ " , a"! œ "
362,880 3,628,800
9. a" œ2, a# œ ((cid:1)1#)#(2) œ1, a$ œ ((cid:1)12)$(1) œ(cid:1)"#, a% œ ((cid:1)1)%#ˆ(cid:1)"#‰ œ(cid:1)4", a& œ ((cid:1)1)&#ˆ(cid:1)"4‰ œ 8",
a' œ 1"6, a( œ(cid:1)3"#, a) œ(cid:1)6"4, a* œ 1#"8, a"! œ 2"56
"
10. a" œ(cid:1)2, a# œ 1†(#(cid:1)2) œ(cid:1)1, a$ œ 2†(3(cid:1)1) œ(cid:1)32, a% œ 3†ˆ(cid:1)4 23‰ œ(cid:1)"#, a& œ 4†ˆ(cid:1)5 #‰ œ(cid:1)52, a' œ(cid:1)3",
a( œ(cid:1)2, a) œ(cid:1)", a* œ(cid:1)2, a"! œ(cid:1)"
7 4 9 5
11. a" œ1, a# œ1, a$ œ1(cid:2)1œ2, a% œ2(cid:2)1œ3, a& œ3(cid:2)2œ5, a' œ8, a( œ13, a) œ21, a* œ34, a"! œ55
" "
12. a" œ2, a# œ(cid:1)1, a$ œ(cid:1)"#, a% œ ˆ(cid:1)(cid:1)1#‰ œ "#, a& œ ˆ(cid:1)ˆ#‰"#‰ œ(cid:1)1, a' œ(cid:1)2, a( œ2, a) œ(cid:1)1, a* œ(cid:1)"#, a"! œ "#
13. a œ((cid:1)1)n(cid:2)1, nœ1, 2, á 14. a œ((cid:1)1)n, nœ1, 2, á
n n
(cid:0)
15. an œ((cid:1)1)n(cid:2)1n#, nœ1, 2, á 16. an œ ((cid:1)"n)#n 1 , nœ1, 2, á
(cid:1)
17. a œ 2n 1 , nœ1, 2, á 18. a œ 2n(cid:1)5 , nœ1, 2, á
n 3an(cid:2)2b n nan(cid:2)1b
19. a œn#(cid:1)1, nœ1, 2, á 20. a œn(cid:1)4, nœ1, 2, á
n n
21. a œ4n(cid:1)3, nœ1, 2, á 22. a œ4n(cid:1)2, nœ1, 2, á
n n
23. an œ 3nn(cid:2)!2, nœ1, 2, á 24. an œ 5nn(cid:0)31 , nœ1, 2, á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
570 Chapter 10 Infinite Sequences and Series
25. a œ 1(cid:2)((cid:1)1)n(cid:0)1, nœ1, 2, á 26. a œ n(cid:1)"#(cid:2)((cid:1)1)nˆ"#‰ œÚnÛ, nœ1, 2, á
n # n # #
27. lim 2(cid:2)(0.1)n œ2 Ê converges (Theorem 5, #4)
nÄ_
28. lim n(cid:2)((cid:1)")n œ lim 1(cid:2) ((cid:1)1)n œ1 Ê converges
nÄ_ n nÄ_ n
"
29. lim "(cid:1)2n œ lim ˆ"n‰(cid:1)2 œ lim (cid:1)2 œ(cid:1)1 Ê converges
nÄ_ 1(cid:2)#n nÄ_ ˆ ‰(cid:2)2 nÄ_ #
n
"
30. n lÄim_ 12(cid:1)n3(cid:2)È"n œn lÄim_ 2ÈŠnÈ "(cid:2)(cid:1) Š3È‹n‹ œ(cid:1)_ Ê diverges
n
"
31. lim "%(cid:1)5n%$ œ lim Šn%‹(cid:1)5 œ(cid:1)5 Ê converges
nÄ_ n (cid:2)8n nÄ_ 1(cid:2)ˆ8‰
n
32. lim #n(cid:2)3 œ lim n(cid:2)3 œ lim " œ0 Ê converges
nÄ_ n (cid:2)5n(cid:2)6 nÄ_ (n(cid:2)3)(n(cid:2)2) nÄ_ n(cid:2)#
#
33. lim n (cid:1)2n(cid:2)1 œ lim (n(cid:1)1)(n(cid:1)1) œ lim (n(cid:1)1)œ_ Ê diverges
nÄ_ n(cid:1)1 nÄ_ n(cid:1)1 nÄ_
"
34 lim "(cid:1)n$# œ lim Šn#‹(cid:1)n œ_ Ê diverges
nÄ_ 70(cid:1)4n nÄ_ Š7#0‹(cid:1)4
n
35. lim a1(cid:2)((cid:1)1)nb does not exist Ê diverges 36. lim ((cid:1)1)nˆ1(cid:1) "‰ does not exist Ê diverges
nÄ_ nÄ_ n
37. lim ˆn(cid:2)"‰ˆ1(cid:1) "‰œ lim ˆ" (cid:2) "‰ˆ1(cid:1) "‰œ " Ê converges
nÄ_ #n n nÄ_ # #n n #
(cid:0)
38. lim ˆ2(cid:1) "‰ˆ3(cid:2) "‰œ6 Ê converges 39. lim ((cid:1)")n 1 œ0 Ê converges
nÄ_ #n #n nÄ_ #n(cid:1)1
40. lim ˆ(cid:1)"‰n œ lim ((cid:1)")n œ0 Ê converges
nÄ_ # nÄ_ #n
41. lim É 2n œÉ lim 2n œÊ lim Š 2"‹œÈ2 Ê converges
nÄ_ n(cid:2)1 nÄ_ n(cid:2)1 nÄ_ 1(cid:1)
n
42. lim " œ lim ˆ"0‰n œ_ Ê diverges
nÄ_ (0.9)n nÄ_ 9
43. lim sinˆ1 (cid:2) "‰œsinŠ lim ˆ1 (cid:2) "‰‹œsin 1 œ1 Ê converges
nÄ_ # n nÄ_ # n #
44. lim n1 cos(n1)œ lim (n1)((cid:1)1)n does not exist Ê diverges
nÄ_ nÄ_
45. lim sin n œ0 because (cid:1)" Ÿ sin n Ÿ " Ê converges by the Sandwich Theorem for sequences
nÄ_ n n n n
# #
46. lim sin n œ0 because 0Ÿ sin n Ÿ " Ê converges by the Sandwich Theorem for sequences
nÄ_ #n #n #n
47. lim n œ lim " œ0 Ê converges (using l'Ho^pital's rule)
nÄ_ #n nÄ_ #n ln 2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.1 Sequences 571
# $
48. lim 3n$ œ lim 3n ln# 3 œ lim 3n(ln 3) œ lim 3n(ln 3) œ_ Ê diverges (using l'Ho^pital's rule)
nÄ_ n nÄ_ 3n nÄ_ 6n nÄ_ 6
"
49. n lÄim_ ln(Èn(cid:2)n") œn lÄim_ Šˆn#È(cid:0)"1‹‰ œn lÄim_ n2È(cid:2)n1 œn lÄim_ 1Š(cid:2)È2Šn"‹‹ œ0 Ê converges
n n
"
ˆ ‰
50. lim ln n œ lim n œ1 Ê converges
nÄ_ ln 2n nÄ_ ˆ2‰
2n
Î
51. lim 81n œ1 Ê converges (Theorem 5, #3)
nÄ_
Î
52. lim (0.03)1n œ1 Ê converges (Theorem 5, #3)
nÄ_
53. lim ˆ1(cid:2) 7‰n œe( Ê converges (Theorem 5, #5)
nÄ_ n
54. lim ˆ1(cid:1) "‰n œ lim ’1(cid:2) ((cid:1)")“n œe(cid:1)" Ê converges (Theorem 5, #5)
nÄ_ n nÄ_ n
55. lim Èn 10nœ lim 101În†n1În œ1†1œ1 Ê converges (Theorem 5, #3 and #2)
nÄ_ nÄ_
56. lim Èn n# œ lim ˆÈn n‰# œ1# œ1 Ê converges (Theorem 5, #2)
nÄ_ nÄ_
Î
57. lim ˆ3‰1În œ n lÄim_ 31În œ " œ1 Ê converges (Theorem 5, #3 and #2)
nÄ_ n lÄim_ n1n 1
n
58. lim (n(cid:2)4)1ÎÐn(cid:2)4Ñ œ lim x1Îx œ1 Ê converges; (let xœn(cid:2)4, then use Theorem 5, #2)
nÄ_ xÄ_
59. lim ln În œ n lÄim_ ln În œ _ œ_ Ê diverges (Theorem 5, #2)
nÄ_ n1n lÄim_ n1n 1
n
60. lim cln n(cid:1)ln(n(cid:2)1)dœ lim lnˆ n ‰œlnŠ lim n ‹œln 1œ0 Ê converges
nÄ_ nÄ_ n(cid:2)1 nÄ_ n(cid:2)1
61. lim Èn 4nnœ lim 4Èn nœ4†1œ4 Ê converges (Theorem 5, #2)
nÄ_ nÄ_
62. lim Èn 32n(cid:2)1 œ lim 32(cid:2)a1Înb œ lim 3#†31În œ9†1œ9 Ê converges (Theorem 5, #3)
nÄ_ nÄ_ nÄ_
63. lim n! œ lim "†2†3â(n(cid:1)1)(n) Ÿ lim ˆ"‰œ0 and n! 0 Ê lim n! œ0 Êconverges
nÄ_ nn nÄ_ n†n†nân†n nÄ_ n nn nÄ_ nn
64. lim ((cid:1)4)n œ0 Ê converges (Theorem 5, #6)
nÄ_ n!
65. lim n! œ lim "' œ_ Ê diverges (Theorem 5, #6)
nÄ_ 106n nÄ_ Š(10)n‹
n!
66. lim n! œ lim " œ_ Ê diverges (Theorem 5, #6)
nÄ_ 2n3n nÄ_ ˆ6n‰
n!
67. lim ˆ"‰1ÎÐlnnÑ œ lim expˆ " lnˆ"‰‰œ lim expˆln 1(cid:1)ln n‰œe(cid:1)" Ê converges
nÄ_ n nÄ_ ln n n nÄ_ ln n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
572 Chapter 10 Infinite Sequences and Series
68. lim lnˆ1(cid:2) "‰n œlnŠ lim ˆ1(cid:2) "‰n‹œln eœ1 Ê converges (Theorem 5, #5)
nÄ_ n nÄ_ n
69. lim ˆ3n(cid:2)"‰n œ lim expˆn lnˆ3n(cid:2)"‰‰œ lim expŠln(3n(cid:2)1)(cid:1)"ln(3n(cid:1)1)‹
nÄ_ 3n(cid:1)1 nÄ_ 3n(cid:1)1 nÄ_
n
œn lÄim_ exp(cid:1)3n3(cid:0)Š1(cid:2)(cid:2)"#3‹n3(cid:1)1(cid:2)œn lÄim_ expŠ(3n(cid:2)16)n(#3n(cid:1)1)‹œexpˆ69‰œe#Î$ Ê converges
n
" "
70. n lÄim_ ˆn(cid:2)n1‰n œn lÄim_ expˆn lnˆn(cid:2)n1‰‰œn lÄim_ expŠln n(cid:1)ˆln"‰(n(cid:2)1)‹œn lÄim_ exp(cid:1)nŠ(cid:2)(cid:2)n"#(cid:0)‹1(cid:2)
n
n
#
œ lim expŠ(cid:1) n ‹œe(cid:1)" Ê converges
nÄ_ n(n(cid:2)1)
71. lim ˆ xn ‰1În œ lim xˆ " ‰1În œx lim expˆ" lnˆ " ‰‰œx lim expŠ(cid:1)ln(2n(cid:2)1)‹
nÄ_ 2n(cid:2)1 nÄ_ #n(cid:2)1 nÄ_ n #n(cid:2)1 nÄ_ n
œx lim expˆ (cid:1)2 ‰œxe! œx, x(cid:4)0 Ê converges
nÄ_ 2n(cid:2)1
" "
lnŠ1(cid:2) #‹ Š2$‹‚Š1(cid:2) #‹
72. n lÄim_ ˆ1(cid:1) n"#‰n œn lÄim_ expˆn lnˆ1(cid:1) n"#‰‰œn lÄim_ exp(cid:1) ˆ"‰n (cid:2)œn lÄim_ exp– n Š(cid:2) "#‹n —
n
n
œ lim expˆ (cid:1)#2n ‰œe! œ1 Ê converges
nÄ_ n (cid:1)1
73. lim 3(cid:0)n†6n œ lim 36n œ0 Ê converges (Theorem 5, #6)
nÄ_ 2 n†n! nÄ_ n!
74. lim ˆ1101‰n œ lim ˆ1121‰nˆ1101‰n œ lim ˆ112201‰n œ0 Ê converges
nÄ_ ˆ9‰n(cid:2)ˆ11‰n nÄ_ ˆ12‰nˆ9‰n(cid:2)ˆ12‰nˆ11‰n nÄ_ ˆ108‰n(cid:2)1
10 12 11 10 11 12 110
(Theorem 5, #4)
(cid:0)
75. lim tanh nœ lim en(cid:1)e(cid:0)n œ lim e2n(cid:1)" œ lim 2e2n œ lim "œ1 Ê converges
nÄ_ nÄ_ en(cid:2)e n nÄ_ e2n(cid:2)1 nÄ_ 2e2n nÄ_
76. lim sinh(ln n)œ lim elnn(cid:1)e(cid:0)lnn œ lim n(cid:1)ˆn"‰ œ_ Ê diverges
nÄ_ nÄ_ 2 nÄ_ #
77. n lÄim_ n#2 snin(cid:1)ˆ1n"‰ œn lÄim_ Šs2in(cid:2)ˆ"n"#‰‹ œn lÄim_ (cid:2)ˆŠc(cid:2)os2ˆ#"n(cid:1)‰‰2Š$‹n"#‹ œn lÄim_ (cid:2)(cid:2)#co(cid:1)sˆˆ2n"‰‰ œ #" Ê converges
n n n n n
78. n lÄim_ nˆ1(cid:1)cos "n‰œn lÄim_ ˆ"(cid:2)ˆc"o‰s "n‰ œn lÄim_ (cid:1)sinˆŠ"n‰"#‘‹Šn"#‹ œn lÄim_ sinˆ"n‰œ0 Ê converges
n
n
79. lim ÈnsinŠ 1 ‹œ lim sinŠÈ1n‹ œ lim cosŠÈ1n‹Š(cid:1)2n31Î2‹ œ lim cosŠ 1 ‹œcos0œ1Êconverges
nÄ_ Èn nÄ_ È1 nÄ_ (cid:1) 1Î nÄ_ Èn
n 2n32
80. lim a3n(cid:2)5nb1În œ lim exp’lna3n(cid:2)5nb1În“œ lim exp’lna3n(cid:2)5nb“œ lim exp–3nln33n(cid:0)(cid:0)55nnln5—
nÄ_ nÄ_ nÄ_ n nÄ_ 1
œ lim exp’Š35nn‹ln3(cid:2)ln5“œ lim exp’ˆ35‰nln3(cid:2)ln5“œexpaln5bœ5
nÄ_ ˆ35nn‰(cid:2)1 nÄ_ ˆ35‰n(cid:2)1
81. lim tan(cid:1)"nœ 1 Ê converges 82. lim " tan(cid:1)"nœ0† 1 œ0 Ê converges
nÄ_ # nÄ_ Èn #
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