Solucionario Calculo Varias Variables, 12va Edició

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INSTRUCTOR’S SOLUTIONS MANUAL MULTIVARIABLE WILLIAM ARDIS Collin County Community College

THOMAS’ CALCULUS TWELFTH EDITION BASED ON THE ORIGINAL WORK BY

George B. Thomas, Jr. Massachusetts Institute of Technology

AS REVISED 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 11.2 11.3 11.4 11.5 11.6 11.7

Parametrizations of Plane Curves 647 Calculus with Parametric Curves 654 Polar Coordinates 662 Graphing in Polar Coordinates 667 Areas and Lengths in Polar Coordinates 674 Conic Sections 679 Conics in Polar Coordinates 689 Practice Exercises 699 Additional and Advanced Exercises 709

12 Vectors and the Geometry of Space 715 12.1 12.2 12.3 12.4 12.5 12.6

Three-Dimensional Coordinate Systems 715 Vectors 718 The Dot Product 723 The Cross Product 728 Lines and Planes in Space 734 Cylinders and Quadric Surfaces 741 Practice Exercises 746 Additional Exercises 754

13 Vector-Valued Functions and Motion in Space 759 13.1 13.2 13.3 13.4 13.5 13.6

Curves in Space and Their Tangents 759 Integrals of Vector Functions; Projectile Motion 764 Arc Length in Space 770 Curvature and Normal Vectors of a Curve 773 Tangential and Normal Components of Acceleration 778 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 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10

Functions of Several Variables 795 Limits and Continuity in Higher Dimensions 804 Partial Derivatives 810 The Chain Rule 816 Directional Derivatives and Gradient Vectors 824 Tangent Planes and Differentials 829 Extreme Values and Saddle Points 836 Lagrange Multipliers 849 Taylor's Formula for Two Variables 857 Partial Derivatives with Constrained Variables 859 Practice Exercises 862 Additional Exercises 876

15 Multiple Integrals 881 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

Double and Iterated Integrals over Rectangles 881 Double Integrals over General Regions 882 Area by Double Integration 896 Double Integrals in Polar Form 900 Triple Integrals in Rectangular Coordinates 904 Moments and Centers of Mass 909 Triple Integrals in Cylindrical and Spherical Coordinates 914 Substitutions in Multiple Integrals 922 Practice Exercises 927 Additional Exercises 933

16 Integration in Vector Fields 939 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8

Line Integrals 939 Vector Fields and Line Integrals; Work, Circulation, and Flux 944 Path Independence, Potential Functions, and Conservative Fields 952 Green's Theorem in the Plane 957 Surfaces and Area 963 Surface Integrals 972 Stokes's Theorem 980 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

1
1 1#

2. a" œ

1 1!

3.

a" œ

"
2 ##

œ 0, a# œ

œ 1, a# œ (
1)# #
1

" #!

œ

œ 1, a# œ

œ
4" , a$ œ

1
3 3#

" 2

1 6

, a$ œ

(
")$ 4
1

œ

1 3!

, a% œ

œ
"3 , a$ œ

1
4 4#

œ
92 , a% œ œ

1 4!

(
1)% 6
1

œ

" 5

3 œ
16

1 24 (
1)& 8
1

, a% œ

œ
7"

4. a" œ 2  (
1)" œ 1, a# œ 2  (
1)# œ 3, a$ œ 2  (
1)$ œ 1, a% œ 2  (
1)% œ 3 5. a" œ

2 ##

6. a" œ

2
" #

" #

œ

, a# œ

œ

" #

a( œ

, a) œ

8. a" œ 1, a# œ a* œ

" 362,880

" #

œ

" # 255 128

" #

œ

3 #

œ

511 256

, a$ œ 3 #



" #

œ

" ##

, a"! œ

ˆ #" ‰ " 3 œ 6 " 3,628,800

, a$ œ

, a"! œ

3 4

, a$ œ

, a* œ

2$ #%

, a$ œ

2#
1 2#

, a# œ

7. a" œ 1, a# œ 1  127 64

2# 2$

, a% œ

, a% œ

2$
1 2$

œ

7 4

œ

2% 2& 7 8

œ

" #

, a% œ

, a% œ

7 4



2%
" 2%

" #$

œ

a' œ

,

15 8

ˆ "6 ‰ 4

œ

" #4

, a& œ

ˆ #"4 ‰ 5

œ

$ (
1)% ˆ
"# ‰ (
1)# (2) œ 1, a$ œ (
1)2 (1) œ
"# , a% œ # # " " a( œ
3"# , a) œ
64 , a* œ 1#"8 , a"! œ 256

1†(
2) œ
1, a$ œ 2†(3
1) œ
32 , a% # a) œ
"4 , a* œ
29 , a"! œ
"5

10. a" œ
2, a# œ a( œ
27 ,

15 16

, a& œ



15 8

" #%

œ

œ

31 16 , a'

63 32

,

1023 512

9. a" œ 2, a# œ " 16

œ

œ

3†ˆ
23 ‰ 4

" 1 #0

, a' œ

" 7 #0

œ
4" , a& œ

œ
"# , a& œ

, a( œ

" 5040

(
1)& ˆ
"4 ‰ #

4†ˆ
"# ‰ 5

, a) œ

œ

" 8

" 40,320

,

,

œ
52 , a' œ
3" ,

11. a" œ 1, a# œ 1, a$ œ 1  1 œ 2, a% œ 2  1 œ 3, a& œ 3  2 œ 5, a' œ 8, a( œ 13, a) œ 21, a* œ 34, a"! œ 55 12. a" œ 2, a# œ
1, a$ œ
"# , a% œ

ˆ
"# ‰
1

œ

" #

, a& œ

ˆ "# ‰ ˆ
"# ‰

œ
1, a' œ
2, a( œ 2, a) œ
1, a* œ
"# , a"! œ

13. an œ (
1)n1 , n œ 1, 2, á

14. an œ (
1)n , n œ 1, 2, á

15. an œ (
1)n1 n# , n œ 1, 2, á

16. an œ

(
")n n#

1

, n œ 1, 2, á

18. an œ

2n
5 nan  1b

, n œ 1, 2, á

17. an œ

2n
1 3an  2b ,

n œ 1, 2, á

19. an œ n#
1, n œ 1, 2, á

20. an œ n
4 , n œ 1, 2, á

21. an œ 4n
3, n œ 1, 2, á

22. an œ 4n
2 , n œ 1, 2, á

23. an œ

3n  2 n! ,

n œ 1, 2, á

24. an œ

n3 5n 1

, n œ 1, 2, á

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" #

570

Chapter 10 Infinite Sequences and Series

25. an œ

1  (
1)n #

1

, n œ 1, 2, á

26. an œ

27. n lim 2  (0.1)n œ 2 Ê converges Ä_ n  (
")n n

29. n lim Ä_

"
2n 1  #n

30. n lim Ä_

2n  " 1
3È n

œ n lim Ä_

31. n lim Ä_

"
5n% n%  8n$

œ n lim Ä_

32. n lim Ä_

n3 n#  5n  6

œ n lim Ä_

n3 (n  3)(n  2)

œ n lim Ä_

33. n lim Ä_

n#
2n  1 n
1

œ n lim Ä_

(n
1)(n
1) n
1

œ n lim (n
1) œ _ Ê diverges Ä_

34 n lim Ä_

"
n$ 70
4n#

ˆ "n ‰
2 ˆ "n ‰  2

œ n lim Ä_

œ 1 Ê converges

2Èn  Š È"n ‹ Š È"n
3‹

1  ˆ 8n ‰

" ‹
n n# 70 Š #‹
4 n

Š

œ n lim Ä_


2 #

œ n lim Ä_

Š n"% ‹
5

œ
1 Ê converges

œ
_ Ê diverges

œ
5 Ê converges " n#

œ 0 Ê converges

œ _ Ê diverges 36. n lim (
1)n ˆ1
"n ‰ does not exist Ê diverges Ä_

35. n lim a1  (
1)n b does not exist Ê diverges Ä_ ˆ n #n " ‰ ˆ1
"n ‰ œ lim ˆ "#  37. n lim Ä_ nÄ_ ˆ2
38. n lim Ä_

" ‰ˆ 3 #n



"‰ #n

ˆ
"# ‰n œ lim 40. n lim Ä_ nÄ_

É n 2n 41. n lim  1 œ É n lim Ä_ Ä_ 42. n lim Ä_

" (0.9)n

" ‰ˆ 1 #n


n" ‰ œ

œ 6 Ê converges

(
")n #n

œ Ú n# Û, n œ 1, 2, á

(Theorem 5, #4)

28. n lim Ä_

œ n lim 1 Ä_

(
1)n n

n
"#  (
1)n ˆ "# ‰ #

" #

Ê converges 39. n lim Ä_

(
")nb1 #n
1

œ 0 Ê converges

œ 0 Ê converges

2n n1

œ Ên lim Š 2 ‹ œ È2 Ê converges Ä _ 1
" n

ˆ "0 ‰n œ _ Ê diverges œ n lim Ä_ 9

ˆ 1  n" ‰‹ œ sin 43. n lim sin ˆ 1#  n" ‰ œ sin Šn lim Ä_ Ä_ #

1 #

œ 1 Ê converges

44. n lim n1 cos (n1) œ n lim (n1)(
1)n does not exist Ê diverges Ä_ Ä_ 45. n lim Ä_

sin n n

46. n lim Ä_

sin# n #n

47. n lim Ä_

n #n

œ 0 because
n" Ÿ œ 0 because 0 Ÿ

œ n lim Ä_

" #n ln 2

sin n n

sin# n #n

Ÿ

Ÿ

" n

" #n

Ê converges by the Sandwich Theorem for sequences Ê converges by the Sandwich Theorem for sequences

^ œ 0 Ê converges (using l'Hopital's rule)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.1 Sequences 48. n lim Ä_

3n n$

49. n lim Ä_

ln (n  ") Èn

50. n lim Ä_

ln n ln 2n

œ n lim Ä_

3n ln 3 3n#

œ n lim Ä_

œ n lim Ä_

œ n lim Ä_ ˆn " 1‰

" ‹ Š #È n

ˆ "n ‰ 2 ‰ ˆ 2n

3n (ln 3)# 6n

œ n lim Ä_

œ n lim Ä_

2È n n1

3n (ln 3)$ 6

œ n lim Ä_

^ œ _ Ê diverges (using l'Hopital's rule)

Š È2n ‹

1  Š n" ‹

œ 0 Ê converges

œ 1 Ê converges

51. n lim 81În œ 1 Ê converges Ä_

(Theorem 5, #3)

52. n lim (0.03)1În œ 1 Ê converges Ä_

(Theorem 5, #3)

ˆ1  7n ‰n œ e( Ê converges 53. n lim Ä_ ˆ1
"n ‰n œ lim ’1  54. n lim Ä_ nÄ_

(
") n “

(Theorem 5, #5) n

œ e
" Ê converges

(Theorem 5, #5)

n È 55. n lim 10n œ n lim 101În † n1În œ 1 † 1 œ 1 Ê converges Ä_ Ä_

# n n È ˆÈ 56. n lim n# œ n lim n‰ œ 1# œ 1 Ê converges Ä_ Ä_

ˆ 3 ‰1În œ nÄ_ 1În œ 57. n lim lim n Ä_ n nÄ_ lim 31În

" 1

œ 1 Ê converges

(Theorem 5, #3 and #2)

(Theorem 5, #2)

(Theorem 5, #3 and #2)

58. n lim (n  4)1ÎÐn4Ñ œ x lim x1Îx œ 1 Ê converges; (let x œ n  4, then use Theorem 5, #2) Ä_ Ä_ 59. n lim Ä_

ln n n1În

lim Ä_ ln1Înn œ œ nlim n n

Ä_

_ 1

œ _ Ê diverges

(Theorem 5, #2)

60. n lim cln n
ln (n  1)d œ n lim ln ˆ n n 1 ‰ œ ln Šn lim Ä_ Ä_ Ä_ n n È 61. n lim 4n n œ n lim 4È n œ 4 † 1 œ 4 Ê converges Ä_ Ä_

n n1‹

œ ln 1 œ 0 Ê converges

(Theorem 5, #2)

n È 62. n lim 32n1 œ n lim 32 a1Înb œ n lim 3# † 31În œ 9 † 1 œ 9 Ê converges Ä_ Ä_ Ä_

œ n lim Ä_

"†2†3â(n
1)(n) n†n†nân†n

63. n lim Ä_

n! nn

64. n lim Ä_

(
4)n n!

65. n lim Ä_

n! 106n

œ n lim Ä_

" 'n Š (10n! ) ‹

66. n lim Ä_

n! 2n 3n

œ n lim Ä_

" ˆ 6n!n ‰

œ 0 Ê converges

ˆ " ‰ œ 0 and Ÿ n lim Ä_ n

n! nn

  0 Ê n lim Ä_

n! nn

(Theorem 5, #3)

œ 0 Ê converges

(Theorem 5, #6)

œ _ Ê diverges

œ _ Ê diverges

(Theorem 5, #6)

(Theorem 5, #6)

ˆ " ‰1ÎÐln nÑ œ lim exp ˆ ln"n ln ˆ n" ‰‰ œ lim exp ˆ ln 1ln
nln n ‰ œ e
" Ê converges 67. n lim Ä_ n nÄ_ nÄ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

571

572

Chapter 10 Infinite Sequences and Series

n ˆ1  n" ‰n ‹ œ ln e œ 1 Ê converges 68. n lim ln ˆ1  "n ‰ œ ln Šn lim Ä_ Ä_

(Theorem 5, #5)

 " ‰‰ ˆ 3n  " ‰n œ lim exp ˆn ln ˆ 3n 69. n lim œ n lim exp Š ln (3n  1)
" ln (3n
1) ‹ 3n
1 Ä _ 3n
1 nÄ_ Ä_ n 3



3

6n #Î$ ˆ6‰ œ n lim exp
3n 1 "3n
1  œ n lim exp Š (3n  1)(3n Ê converges
1) ‹ œ exp 9 œ e Ä_ Ä_ Š ‹ #

n#

"

"



ˆ n ‰n œ lim exp ˆn ln ˆ n n 1 ‰‰ œ lim exp Š ln n
ln" (n  1) ‹ œ lim exp
n n 1  70. n lim ˆn‰ Ä _ n1 nÄ_ nÄ_ nÄ_ Š "# ‹ n

œ n lim exp Š
Ä_

n# n(n  1) ‹


"

œe

Ê converges

 1) ˆ x ‰1În œ lim x ˆ #n " 1 ‰1În œ x lim exp ˆ "n ln ˆ #n " 1 ‰‰ œ x lim exp Š
ln (2n 71. n lim ‹ n Ä _ 2n  1 nÄ_ nÄ_ nÄ_
2 ! œ x n lim exp ˆ 2n1 ‰ œ xe œ x, x  0 Ê converges Ä_ n

ˆ1
72. n lim Ä_

" ‰n n#

œ n lim exp ˆn ln ˆ1
Ä_

" ‰‰ n#

œ n lim exp
Ä_

ln Š1  n"# ‹

exp –  œ n lim Ä_

ˆ n" ‰

Š n2$ ‹‚Š1  n"# ‹ Š n"# ‹

—

œ n lim exp ˆ n
#
2n1 ‰ œ e! œ 1 Ê converges Ä_ 73. n lim Ä_

3 n †6 n 2cn †n!

œ n lim Ä_

36n n!

œ 0 Ê converges

ˆ 10 ‰n

ˆ 12 ‰n ˆ 10 ‰n 11 11 12 ‰n ˆ 9 ‰n 12 ‰n ˆ 11 ‰n ˆ 11 ˆ  11 10 12

11 74. n lim lim n 11 ‰n œ Ä _ ˆ 109 ‰  ˆ 12 nÄ_ (Theorem 5, #4)

75. n lim tanh n œ n lim Ä_ Ä_

en
e en  e

76. n lim sinh (ln n) œ n lim Ä_ Ä_

77. n lim Ä_

n# sin ˆ n" ‰ 2n
1

œ n lim Ä_

(Theorem 5, #6)

n n

œ n lim Ä_

eln n
e 2

ln n

sin ˆ "n ‰

Èn sinŠ È1 ‹ œ lim 79. n lim n Ä_ nÄ_

ˆ"  cos "n ‰ ˆ "n ‰

sinŠ È1n ‹

Èn 1

œ n lim Ä_ n
ˆ n" ‰ #

œ n lim Ä_

œ n lim Ä_

Š 2n  n"# ‹

78. n lim n ˆ1
cos "n ‰ œ n lim Ä_ Ä_

e2n
" e2n  1

ˆ 120 ‰n 121 n ˆ 108 ‰ 1 110

œ n lim Ä_

2e2n 2e2n

Š n2#
n2$ ‹

œ n lim Ä_

œ n lim " œ 1 Ê converges Ä_

œ _ Ê diverges

 ˆcos ˆ "n ‰‰ Š n"# ‹

œ n lim Ä_

œ 0 Ê converges

œ n lim Ä_


sin ˆ "n ‰‘ Š "# ‹ n Š n"# ‹

cos Š È1n ‹Š



1 2n3Î2

1 ‹ 2n3Î2

 cos ˆ n" ‰ #
ˆ 2n ‰

œ

" #

Ê converges

œ n lim sin ˆ "n ‰ œ 0 Ê converges Ä_

œ n lim cos Š È1n ‹ œ cos 0 œ 1 Ê converges Ä_

80. n lim a3n  5n b1În œ n lim exp’lna3n  5n b1În “ œ n lim exp’ lna3 n 5 b “ œ n lim exp– Ä_ Ä_ Ä_ Ä_ n

n

œ n lim exp’ Ä_

Š 35n ‹ln 3  ln 5

81. n lim tan
" n œ Ä_

ˆ 35nn ‰  1 1 #

exp’ “ œ n lim Ä_

Ê converges

ˆ 35 ‰n ln 3  ln 5 ˆ 35 ‰n  1 “

n

3n ln 3 b 5n ln 5 3n b 5n

1

—

œ expaln 5b œ 5 82. n lim Ä_

" Èn

tan
" n œ 0 †

1 #

œ 0 Ê converges

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.1 Sequences ˆ " ‰n  83. n lim Ä_ 3

" È 2n

573

n

n œ n lim Šˆ 3" ‰  Š È"2 ‹ ‹ œ 0 Ê converges Ä_

(Theorem 5, #4)

#

n 1‰ ! È 84. n lim n#  n œ n lim exp ’ ln ann  nb “ œ n lim exp ˆ 2n n#  n œ e œ 1 Ê converges Ä_ Ä_ Ä_

85. n lim Ä_

(ln n)#!! n

86. n lim Ä_

(ln n)& Èn

œ n lim Ä_

200 (ln n)"** n

œ n lim Ä_

200†199 (ln n)"*) n

œ á œ n lim Ä_

200! n

œ 0 Ê converges

%

œ n lim Ä_ –

Š 5(lnnn) ‹ "

Š #Èn ‹

— œ n lim Ä_

10(ln n)% Èn

œ n lim Ä_ È

80(ln n)$ Èn

œ á œ n lim Ä_

#

87. n lim Šn
Èn#
n‹ œ n lim Šn
Èn#
n‹ Š n  Èn#
n ‹ œ n lim Ä_ Ä_ Ä_ n n
n œ

" #

88. n lim Ä_

œ 0 Ê converges

œ n lim Ä_

" 1  É1


" n

Ê converges " È n#
1
È n#  n

œ n lim Š Ä_ È

É1
n"#  É1  "n

œ n lim Ä_ 89. n lim Ä_

n n  È n#
n

3840 Èn

ˆ
n"
1‰

' 90. n lim Ä_ 1

n

" xp

œ n lim Ä_

È n#
1  È n#  n
1
n

œ
2 Ê converges

'1n x" dx œ n lim Ä_

" n

È # È # " ‹ Š Èn#
1  Èn#  n ‹ n#
1
È n#  n n
1 n n

ln n n

dx œ n lim ’ " Ä _ 1
p

œ n lim Ä_ n

" xpc1 “ 1

" n

œ 0 Ê converges

œ n lim Ä_

" 1
p

ˆ np"c1
1‰ œ

(Theorem 5, #1) " p
1

if p  1 Ê converges

72 91. Since an converges Ê n lim a œ L Ê n lim a œ n lim ÊLœ Ä_ n Ä _ n1 Ä _ 1  an Ê L œ
9 or L œ 8; since an  0 for n   1 Ê L œ 8

72 1L

Ê La1  Lb œ 72 Ê L2  L
72 œ 0

an  6 92. Since an converges Ê n lim a œ L Ê n lim a œ n lim ÊLœ Ä_ n Ä _ n1 Ä _ an  2 Ê L œ
3 or L œ 2; since an  0 for n   2 Ê L œ 2

L6 L2

Ê LaL  2b œ L  6 Ê L2  L
6 œ 0

È8  2an Ê L œ È8  2L Ê L2
2L
8 œ 0 Ê L œ
2 93. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ or L œ 4; since an  0 for n   3 Ê L œ 4 È8  2an Ê L œ È8  2L Ê L2
2L
8 œ 0 Ê L œ
2 94. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ or L œ 4; since an  0 for n   2 Ê L œ 4 È5an Ê L œ È5L Ê L2
5L œ 0 Ê L œ 0 or L œ 5; since 95. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ an  0 for n   1 Ê L œ 5 ˆ12
Èan ‰ Ê L œ Š12
ÈL‹ Ê L2
25L  144 œ 0 96. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ Ê L œ 9 or L œ 16; since 12
Èan  12 for n   1 Ê L œ 9 97. an  1 œ 2 

n   1, a1 œ 2. Since an converges Ê n lim a œ L Ê n lim a œ n lim Š2  Ä_ n Ä _ n1 Ä_ Ê L2
2L
1 œ 0 Ê L œ 1 „ È2; since an  0 for n   1 Ê L œ 1  È2 1 an ,

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 an ‹

ÊLœ2

1 L

574

Chapter 10 Infinite Sequences and Series

È1  an Ê L œ È1  L 98. an  1 œ È1  an , n   1, a1 œ È1. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ Ê L2
L
1 œ 0 Ê L œ

1 „ È5 2 ;

since an  0 for n   1 Ê L œ

1  È5 2

99. 1, 1, 2, 4, 8, 16, 32, á œ 1, 2! , 2" , 2# , 2$ , 2% , 2& , á Ê x" œ 1 and xn œ 2nc2 for n   2 100. (a) 1#
2(1)# œ
1, 3#
2(2)# œ 1; let f(aß b) œ (a  2b)#
2(a  b)# œ a#  4ab  4b#
2a#
4ab
2b# œ 2b#
a# ; a#
2b# œ
1 Ê f(aß b) œ 2b#
a# œ 1; a#
2b# œ 1 Ê f(aß b) œ 2b#
a# œ
1 #

‰
2œ (b) r#n
2 œ ˆ aa2b b

a#  4ab  4b#
2a#
4ab
2b# (a  b)#

In the first and second fractions, yn   n. Let

a b

œ


aa#
2b# b (a  b)#

œ

„" y#n

#

Ê rn œ Ê2 „ Š y"n ‹

represent the (n
1)th fraction where

for n a positive integer   3. Now the nth fraction is lim rn œ È2.

a  2b ab

a b

  1 and b   n
1

and a  b   2b   2n
2   n Ê yn   n. Thus,

nÄ_

101. (a) f(x) œ x#
2; the sequence converges to 1.414213562 ¸ È2 (b) f(x) œ tan (x)
1; the sequence converges to 0.7853981635 ¸

1 4

(c) f(x) œ ex ; the sequence 1, 0,
1,
2,
3,
4,
5, á diverges 102. (a) n lim nf ˆ n" ‰ œ lim b f(??xx) œ lim b f(0??x)x
f(0) œ f w (0), where ?x œ Ä_ ?x Ä ! ?x Ä ! "
" ˆ " ‰ w
" (b) n lim n tan œ f (0) œ x # œ 1, f(x) œ tan  n 1 0 Ä_

" n

(c) n lim n ae1În
1b œ f w (0) œ e! œ 1, f(x) œ ex
1 Ä_ (d) n lim n ln ˆ1  2n ‰ œ f w (0) œ 1 22(0) œ 2, f(x) œ ln (1  2x) Ä_ #

103. (a) If a œ 2n  1, then b œ Ú a# Û œ Ú 4n

#

 4n  1 Û # #

#

œ Ú2n#  2n  "# Û œ 2n#  2n, c œ Ü a# Ý œ Ü2n#  2n  "# Ý #

œ 2n#  2n  1 and a#  b# œ (2n  1)  a2n#  2nb œ 4n#  4n  1  4n%  8n$  4n# #

œ 4n%  8n$  8n#  4n  1 œ a2n#  2n  1b œ c# . (b) a lim Ä_

# Ú a# Û # Ü a# Ý

œ a lim Ä_

2n#  2n 2n#  2n  1

œ 1 or a lim Ä_

#

Ú a# Û #

Ü a# Ý

œ a lim sin ) œ Ä_

2n1 ‰ 104. (a) n lim (2n1)1Î a2nb œ n lim exp ˆ ln2n œ n lim exp
Ä_ Ä_ Ä_

21 Š 2n 1‹

#

(b)

n 40 50 60

15.76852702 19.48325423 23.19189561

sin ) œ 1

exp ˆ #"n ‰ œ e! œ 1;  œ n lim Ä_

n n n! ¸ ˆ ne ‰ È 2n1 , Stirlings approximation Ê È n! ¸ ˆ ne ‰ (2n1)1Î a2nb ¸ n È n!

lim

) Ä 1 Î2

n e

for large values of n

n e

14.71517765 18.39397206 22.07276647

ˆ"‰

ln n " n 105. (a) n lim œ n lim œ n lim œ0 Ä _ nc Ä _ cncc1 Ä _ cnc
Ðln %ÑÎc (b) For all %  0, there exists an N œ e such that n  e
Ðln %ÑÎc Ê ln n 
lnc % Ê ln nc  ln ˆ "% ‰ Ê nc  "% Ê n"c  % Ê ¸ n"c
0¸  % Ê lim n"c œ 0 nÄ_

106. Let {an } and {bn } be sequences both converging to L. Define {cn } by c2n œ bn and c2nc1 œ an , where n œ 1, 2, 3, á . For all %  0 there exists N" such that when n  N" then kan
Lk  % and there exists N# such that when n  N# then kbn
Lk  %. If n  1  2max{N" ß N# }, then kcn
Lk  %, so {cn } converges to L.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.1 Sequences

575

107. n lim n1În œ n lim exp ˆ "n ln n‰ œ n lim exp ˆ n" ‰ œ e! œ 1 Ä_ Ä_ Ä_ 108. n lim x1În œ n lim exp ˆ "n ln x‰ œ e! œ 1, because x remains fixed while n gets large Ä_ Ä_ 109. Assume the hypotheses of the theorem and let % be a positive number. For all % there exists a N" such that when n  N" then kan
Lk  % Ê
%  an
L  % Ê L
%  an , and there exists a N# such that when n  N# then kcn
Lk  % Ê
%  cn
L  % Ê cn  L  %. If n  max{N" ß N# }, then L
%  an Ÿ bn Ÿ cn  L  % Ê kbn
Lk  % Ê n lim b œ L. Ä_ n 110. Let %  !. We have f continuous at L Ê there exists $ so that kx
Lk  $ Ê kf(x)
f(L)k  %. Also, an Ä L Ê there exists N so that for n  N kan
Lk  $ . Thus for n  N, kf(an )
f(L)k  % Ê f(an ) Ä f(L). 111. an1   an Ê

3(n  1)  1 (n  1)  1



3n  1 n1

3n  4 n#

Ê



3n  1 n1

Ê 3n#  3n  4n  4  3n#  6n  n  2

Ê 4  2; the steps are reversible so the sequence is nondecreasing;

3n  " n1

 3 Ê 3n  1  3n  3

Ê 1  3; the steps are reversible so the sequence is bounded above by 3 112. an1   an Ê

(2(n  1)  3)! ((n  1)  1)!



(2n  3)! (n  1)!

Ê

(2n  5)! (n  2)!



(2n  3)! (n  1)!

Ê

(2n  5)! (2n  3)!



(n  2)! (n  1)!

Ê (2n  5)(2n  4)  n  2; the steps are reversible so the sequence is nondecreasing; the sequence is not bounded since 113. an1 Ÿ an Ê

(2n  3)! (n  1)!

œ (2n  3)(2n  2)â(n  2) can become as large as we please

2nb1 3nb1 (n  1)!

Ÿ

2n 3n n!

2nb1 3nb1 2n 3n

Ê

(n  1)! n!

Ÿ

Ê 2 † 3 Ÿ n  1 which is true for n   5; the steps are

reversible so the sequence is decreasing after a& , but it is not nondecreasing for all its terms; a" œ 6, a# œ 18, a$ œ 36, a% œ 54, a& œ 324 5 œ 64.8 Ê the sequence is bounded from above by 64.8 114. an1   an Ê 2


2 n 1




" #nb1

 2


2 n




" #n

Ê

reversible so the sequence is nondecreasing; 2
115. an œ 1


" n

converges because

116. an œ n


" n

diverges because n Ä _ and

117. an œ

2 n
1 2n

œ1


" #n

and 0 

" #n

" n

2 n 2 n





2 " n1   #nb1 " #n Ÿ 2 Ê




" #n

Ê

2 n(n  1)

 
#n"b1 ; the steps are

the sequence is bounded from above

Ä 0 by Example 1; also it is a nondecreasing sequence bounded above by 1



" n

; since

" n " n

Ä 0 by Example 1, so the sequence is unbounded Ä 0 (by Example 1) Ê

" #n

Ä 0, the sequence converges; also it is

a nondecreasing sequence bounded above by 1 118. an œ

2 n
1 3n

n

œ ˆ 23 ‰


" 3n

; the sequence converges to ! by Theorem 5, #4

119. an œ a(
1)n  1b ˆ nn 1 ‰ diverges because an œ 0 for n odd, while for n even an œ 2 ˆ1  n" ‰ converges to 2; it diverges by definition of divergence 120. xn œ max {cos 1ß cos 2ß cos 3ß á ß cos n} and xn1 œ max {cos 1ß cos 2ß cos 3ß á ß cos (n  1)}   xn with xn Ÿ 1 so the sequence is nondecreasing and bounded above by 1 Ê the sequence converges. 121. an   an1 Í and

1  È2n Èn

1  È2n Èn

 

"  È2(n  1) Èn  1

Í Èn  1  È2n#  2n   Èn  È2n#  2n Í Èn  1   Èn

  È2 ; thus the sequence is nonincreasing and bounded below by È2 Ê it converges

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

576

Chapter 10 Infinite Sequences and Series

122. an   an1 Í

n1 n

 

(n  1)  " n1

Í n#  2n  1   n#  2n Í 1   0 and

n1 n

  1; thus the sequence is

nonincreasing and bounded below by 1 Ê it converges 123.

4nb1  3n œ4 4n 3 ‰n ˆ 4  4   4;

ˆ 34 ‰n so an   an1 Í 4  ˆ 34 ‰n   4  ˆ 34 ‰n" Í ˆ 34 ‰n   ˆ 34 ‰n1 Í 1  

3 4

and

thus the sequence is nonincreasing and bounded below by 4 Ê it converges

124. a" œ 1, a# œ 2
3, a$ œ 2(2
3)
3 œ 2#
a22
"b † 3, a% œ 2 a2#
a22
"b † 3b
3 œ 2$
a2$
1b 3, a& œ 2 c2$
a2$
1b 3d
3 œ 2%
a2%
1b 3, á , an œ 2n
"
a2n
"
1b 3 œ 2n
"
3 † 2n
1  3 œ 2n
1 (1
3)  3 œ
2n  3; an   an1 Í
2n  3  
2n1  3 Í
2n  
2n1 Í 1 Ÿ 2 so the sequence is nonincreasing but not bounded below and therefore diverges 125. Let 0  M  1 and let N be an integer greater than Ê n  M  nM Ê n  M(n  1) Ê

n n1

M 1
M

. Then n  N Ê n 

 M.

M 1
M

Ê n
nM  M

126. Since M" is a least upper bound and M# is an upper bound, M" Ÿ M# . Since M# is a least upper bound and M" is an upper bound, M# Ÿ M" . We conclude that M" œ M# so the least upper bound is unique. 127. The sequence an œ 1 

(
")n #

is the sequence

" #

,

3 #

,

" #

,

3 #

, á . This sequence is bounded above by

3 #

,

but it clearly does not converge, by definition of convergence. 128. Let L be the limit of the convergent sequence {an }. Then by definition of convergence, for corresponds an N such that for all m and n, m  N Ê kam
Lk  kam
an k œ kam
L  L
an k Ÿ kam
Lk  kL
an k 

% #



% #

% #

% #

there

and n  N Ê kan
Lk  #% . Now

œ % whenever m  N and n  N.

129. Given an %  0, by definition of convergence there corresponds an N such that for all n  N, kL"
an k  % and kL#
an k  %. Now kL#
L" k œ kL#
an  an
L" k Ÿ kL#
an k  kan
L" k  %  % œ 2%. kL#
L" k  2% says that the difference between two fixed values is smaller than any positive number 2%. The only nonnegative number smaller than every positive number is 0, so kL"
L# k œ 0 or L" œ L# . 130. Let k(n) and i(n) be two order-preserving functions whose domains are the set of positive integers and whose ranges are a subset of the positive integers. Consider the two subsequences akÐnÑ and aiÐnÑ , where akÐnÑ Ä L" , aiÐnÑ Ä L# and L" Á L# . Thus ¸akÐnÑ
aiÐnÑ ¸ Ä kL"
L# k  0. So there does not exist N such that for all m, n  N Ê kam
an k  %. So by Exercise 128, the sequence Öan × is not convergent and hence diverges. 131. a2k Ä L Í given an %  0 there corresponds an N" such that c2k  N" Ê ka2k
Lk  %d . Similarly, a2k1 Ä L Í c2k  1  N# Ê ka2k1
Lk  %d . Let N œ max{N" ß N# }. Then n  N Ê kan
Lk  % whether n is even or odd, and hence an Ä L. 132. Assume an Ä 0. This implies that given an %  0 there corresponds an N such that n  N Ê kan
0k  % Ê kan k  % Ê kkan kk  % Ê kkan k
0k  % Ê kan k Ä 0. On the other hand, assume kan k Ä 0. This implies that given an %  0 there corresponds an N such that for n  N, kkan k
0k  % Ê kkan kk  % Ê kan k  % Ê kan
0k  % Ê an Ä 0. 133. (a) f(x) œ x#
a Ê f w (x) œ 2x Ê xn1 œ xn


x#n
a #xn

Ê x n 1 œ

2x#n
ax#n
ab 2xn

œ

x#n  a 2xn

œ

ˆxn  xa ‰ n #

(b) x" œ 2, x# œ 1.75, x$ œ 1.732142857, x% œ 1.73205081, x& œ 1.732050808; we are finding the positive number where x#
3 œ 0; that is, where x# œ 3, x  0, or where x œ È3 .

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.2 Infinite Series

577

134. x" œ 1, x# œ 1  cos (1) œ 1.540302306, x$ œ 1.540302306  cos (1  cos (1)) œ 1.570791601, x% œ 1.570791601  cos (1.570791601) œ 1.570796327 œ 1# to 9 decimal places. After a few steps, the arc axnc1 b and line segment cos axnc1 b are nearly the same as the quarter circle. 135-146. Example CAS Commands: Mathematica: (sequence functions may vary): Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] Mathematica: (sequence functions may vary): Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] The last command (Limit) will not always work in Mathematica. You could also explore the limit by enlarging your table to more than the first 25 values. If you know the limit (1 in the above example), to determine how far to go to have all further terms within 0.01 of the limit, do the following. Clear[minN, lim] lim= 1 Do[{diff=Abs[a[n]
lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN For sequences that are given recursively, the following code is suggested. The portion of the command a[n_]:=a[n] stores the elements of the sequence and helps to streamline computation. Clear[a, n] a[1]= 1; a[n_]; = a[n]= a[n
1]  (1/5)(n
1) first25= Table[N[a[n]], {n, 1, 25}] The limit command does not work in this case, but the limit can be observed as 1.25. Clear[minN, lim] lim= 1.25 Do[{diff=Abs[a[n]
lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN 10.2 INFINITE SERIES 1. sn œ

a a 1
rn b (1
r)

œ

n 2 ˆ1
ˆ "3 ‰ ‰ " 1
ˆ3‰

2. sn œ

a a 1
rn b (1
r)

œ

" ‰n ‰ 9 ‰ˆ ˆ 100 1
ˆ 100 " 1
ˆ 100 ‰

3. sn œ

a a 1
rn b (1
r)

œ

1
ˆ
"# ‰ 1
ˆ
"# ‰

4. sn œ

1
(
2)n 1
(
2)

, a geometric series where krk  1 Ê divergence

5.

" (n  1)(n  #)

œ

" n1

n




Ê n lim s œ Ä_ n

Ê n lim s œ Ä_ n

Ê n lim s œ Ä_ n

" n#

2 1
ˆ "3 ‰

" ˆ #3 ‰

œ3 9 ‰ ˆ 100

" ‰ 1
ˆ 100

œ

œ

" 11

2 3

Ê sn œ ˆ #"
3" ‰  ˆ 3"
4" ‰  á  ˆ n " 1


" ‰ n#

œ

" #




" n#

Ê n lim s œ Ä_ n

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" #

578 6.

Chapter 10 Infinite Sequences and Series œ

5 n(n  1)




5 n

5 n1

Ê sn œ ˆ5
25 ‰  ˆ 25
35 ‰  ˆ 35
45 ‰  á  ˆ n
5 1
n5 ‰  ˆ n5


5 ‰ n1

œ 5


5 n1

Ê n lim s œ5 Ä_ n 7. 1


8.

" 16

9.

7 4

" 4





10. 5


" 16

 " 64

7 16

5 4

" 256







" 64




7 64

5 16

 á , the sum of this geometric series is

 á , the sum of this geometric series is

 á , the sum of this geometric series is




5 64

5 1
ˆ "# ‰

" 1
ˆ "3 ‰



œ 10 

œ

3 #

" 1
ˆ "3 ‰




œ 10


œ

3 #

14. 2 

4 5

" 1  ˆ "5 ‰

 

8 25



œ2

16 125

5 6

" ‰ 25 œ 17 6

œ

4 5

" 1#

7 3

5 1
ˆ
"4 ‰

œ4

" ‰ #7

 á , is the sum of two geometric series; the sum is

" ‰ #7

 á , is the difference of two geometric series; the sum is

 ˆ 18


 á œ 2 ˆ1 

15. Series is geometric with r œ

œ

" 1  ˆ "4 ‰

17 #

13. (1  1)  ˆ 1#
"5 ‰  ˆ 14  1 1
ˆ "# ‰

ˆ 74 ‰

1
ˆ "4 ‰

œ

œ

23 #

12. (5
1)  ˆ 5#
"3 ‰  ˆ 54
9" ‰  ˆ 58
5 1
ˆ "# ‰

" ‰ ˆ 16 1
ˆ "4 ‰

 á , the sum of this geometric series is

11. (5  1)  ˆ 5#  "3 ‰  ˆ 54  9" ‰  ˆ 58 

" 1
ˆ
"4 ‰

2 5



" ‰ 1#5

4 25



 á , is the sum of two geometric series; the sum is

8 125

 á ‰ ; the sum of this geometric series is 2 Š 1
"ˆ 2 ‰ ‹ œ 5

Ê ¹ 25 ¹  1 Ê Converges to

2 5

1 1
25

œ

5 3

1 8

œ

1 7

16. Series is geometric with r œ
3 Ê ¹
3¹  1 Ê Diverges 17. Series is geometric with r œ

Ê ¹ 18 ¹  1 Ê Converges to

1 8

1
18

18. Series is geometric with r œ
23 Ê ¹
23 ¹  1 Ê Converges to _

19. 0.23 œ !

n œ0

_

21. 0.7 œ !

nœ0

23 100

7 10

ˆ 10" # ‰n œ

" ‰n ˆ 10 œ

23 Š 100 ‹

"

1
ˆ 100 ‰

7 Š 10 ‹

1


" Š 10 ‹

œ

œ

_

nœ0

n œ0

_

n œ0

414 1000

n œ0

22. 0.d œ !

" 1
Š 10 ‹

24. 1.414 œ 1  !

_

7 9

6 Š 100 ‹

ˆ 10" $ ‰n œ 1 

œ
25

20. 0.234 œ !

23 99

_

1 ‰ ˆ 6 ‰ ˆ " ‰n 23. 0.06 œ ! ˆ 10 œ 10 10


23 1
ˆ
23 ‰

œ

6 90

414 Š 1000 ‹

" 1
Š 1000 ‹

œ

d 10

234 1000

ˆ 10" $ ‰n œ

" ‰n ˆ 10 œ

234 Š 1000 ‹

" 1
Š 1000 ‹

d Š 10 ‹

" 1
Š 10 ‹

œ

d 9

" 15

œ1

414 999

œ

"413 999

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

œ

234 999

10 3

Section 10.2 Infinite Series 25. 1.24123 œ

124 100

_

!

123 10&

n œ0

_

26. 3.142857 œ 3  !

n œ0

œ lim

124 100

28.

lim nan  1b nÄ_ an  2ban  3b

29.

lim 1 nÄ_ n  4

œ 0 Ê test inconclusive

30.

lim 2 n nÄ_ n  3

œ lim

33. 34.

10

1
Š

" ‹ 10$

Š 142,857 ' ‹ 10

1
Š

" ‹ 10'

œ

124 100

œ3



123 10&
10#

142,857 10'
1

œ

œ

124 100

3,142,854 999,999



123 99,900

œ

œ

123,999 99,900

œ

41,333 33,300

116,402 37,037

œ 1 Á 0 Ê diverges

lim n nÄ_ n  10

32.

Š 123& ‹



ˆ 10" ' ‰n œ 3 

142,857 10'

27.

31.

1 nÄ_ 1

ˆ 10" $ ‰n œ

579

n2  n 2 nÄ_ n  5n  6

2n  1 nÄ_ 2n  5

œ lim

œ lim

œ lim

2 nÄ_ 2

œ 1 Á 0 Ê diverges

œ 0 Ê test inconclusive

1 nÄ_ 2n

lim cos 1n œ cos 0 œ 1 Á 0 Ê diverges

nÄ_

n lim ne nÄ_ e  n

œ

n lim n e nÄ_ e  1

en n nÄ_ e

œ lim

œ lim

1 nÄ_ 1

œ 1 Á 0 Ê diverges

lim ln 1n œ
_ Á 0 Ê diverges

nÄ_

lim cos n 1 œ does not exist Ê diverges

nÄ_

35. sk œ ˆ1
"2 ‰  ˆ "2
"3 ‰  ˆ "3
"4 ‰  á  ˆ k
" 1
k" ‰  ˆ k"
œ lim ˆ1
kÄ_

" ‰ k1

kÄ_

œ 1


" k1

Ê

œ 1, series converges to 1

36. sk œ ˆ 31
34 ‰  ˆ 34
39 ‰  ˆ 39
œ lim Š3


" ‰ k1

3 ‹ ak  1b2

3 ‰ 16

 á  Š ak
3 1b2


3 k2 ‹

 Š k32


3 ‹ ak  1b2

œ 3


lim sk

kÄ_

3 ak  1b2

Ê

lim sk

kÄ_

œ 3, series converges to 3

37. sk œ ŠlnÈ2
lnÈ1‹  ŠlnÈ3
lnÈ2‹  ŠlnÈ4
lnÈ3‹  á  ŠlnÈk
lnÈk
1‹  ŠlnÈk  1
lnÈk‹ œ lnÈk  1
lnÈ1 œ lnÈk  1 Ê

lim sk œ lim lnÈk  1 œ _; series diverges

kÄ_

kÄ_

38. sk œ atan 1
tan 0b  atan 2
tan 1b  atan 3
tan 2b  á  atan k
tan ak
1bb  atan ak  1b
tan kb œ tan ak  1b
tan 0 œ tan ak  1b Ê lim sk œ lim tan ak  1b œ does not exist; series diverges kÄ_

kÄ_

39. sk œ ˆcos
1 ˆ 12 ‰
cos
1 ˆ 13 ‰‰  ˆcos
1 ˆ 13 ‰
cos
1 ˆ 14 ‰‰  ˆcos
1 ˆ 14 ‰
cos
1 ˆ 15 ‰‰  á  ˆcos
1 ˆ 1k ‰
cos
1 ˆ k 1 1 ‰‰  ˆcos
1 ˆ k 1 1 ‰
cos
1 ˆ k 1 # ‰‰ œ 13
cos
1 ˆ k 1 # ‰ Ê

lim sk œ lim ’ 13
cos
1 ˆ k 1 # ‰“ œ

kÄ_

kÄ_

1 3




1 2

œ 16 , series converges to

1 6

40. sk œ ŠÈ5
È4‹  ŠÈ6
È5‹  ŠÈ7
È6‹  á  ŠÈk  3
Èk  2‹  ŠÈk  4
Èk  3‹ œ Èk  4
2 Ê

lim sk œ lim ’Èk  4
2“ œ _; series diverges

kÄ_

kÄ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

580 41.

42.

Chapter 10 Infinite Sequences and Series 4 " " "‰ " ‰ ˆ ˆ" "‰ ˆ" (4n
3)(4n  1) œ 4n
3
4n  1 Ê sk œ 1
5  5
9  9
13   ˆ 4k "
3
4k " 1 ‰ œ 1
4k " 1 Ê lim sk œ lim ˆ1
4k " 1 ‰ œ 1 kÄ_ kÄ_

œ

6 (2n
1)(2n  1)

A 2n
1



A(2n  1)  B(2n
1) (2n
1)(2n  1)

œ

B 2n  1

á  ˆ 4k "
7


" ‰ 4k
3

Ê A(2n  1)  B(2n
1) œ 6 Ê (2A  2B)n  (A
B) œ 6

k k 2A  2B œ 0 ABœ0 6 Ê œ Êœ Ê 2A œ 6 Ê A œ 3 and B œ
3. Hence, ! (2n
1)(2n œ 3 ! ˆ #n "
1
 1) A
Bœ6 A
Bœ6 n œ1 nœ1

œ 3 Š "1


" 3



lim 3 ˆ1


kÄ_

43.

40n (2n
1)# (2n1)#

" 3




" 5

" ‰ #k  1

œ

A (2n
1)

" 5






" 7

á


" #(k
1)  1



" 2k
1

" #k  1 ‹





œ

A(2n
1)(2n1)#  B(2n1)#  C(2n1)(2n
1)#  D(2n
1)# (2n
1)# (2n1)# # #




œ 3 ˆ1


" ‰ #k  1

" ‰ #n  1

Ê the sum is

œ3 

B (2n
1)#

C (2n1) #

D (2n1)#

Ê A(2n
1)(2n  1)#  B(2n  1)  C(2n  1)(2n
1)  D(2n
1) œ 40n Ê A a8n$  4n#
2n
1b  B a4n#  4n  1b  C a8n$
4n#
2n  1b œ D a4n#
4n  1b œ 40n Ê (8A  8C)n$  (4A  4B
4C  4D)n#  (
2A  4B
2C
4D)n  (
A  B  C  D) œ 40n Ú Ú 8A  8C œ 0 8A  8C œ 0 Ý Ý Ý Ý B Dœ 0 4A  4B
4C  4D œ 0 A B
C Dœ 0 Ê œ Ê 4B œ 20 Ê B œ 5 Ê Û Ê Û


œ


œ
2D œ 20 2A 4B 2C 4D 40 A 2 B C 2D 20 2B Ý Ý Ý Ý Ü
A  B  C  D œ 0 Ü
A  B  C  D œ 0 k ACœ0 Ê C œ 0 and A œ 0. Hence, ! ’ (#n
1)40n and D œ
5 Ê œ # (2n1)# “
A  5  C
5 œ 0 n œ1 k

œ 5 ! ’ (#n
" 1)#
n œ1

44.

" (#n1)# “

œ 5 Š1


" (2k1)# ‹

2n  1 n# (n  1)#

" n#

Ê

œ

45. sk œ Š1
Ê

 Š È"2
kÄ_

" ‰ #"Î#

"  ˆ #"Î#


lim sk œ

kÄ_

47. sk œ ˆ ln"3
œ
ln"# 

" ‰ ln #

" #




" 1

œ

" 9




" #5



" #5


á


" (2k1)# ‹

Ê



" (#k
1)#




" (#k1)# ‹

œ5 " ‰ 16

 á  ’ (k
" 1)#


" k# “

 ’ k"#


" (k  1)# “

" È4 ‹

 á  ŠÈ "

k
1



" Èk ‹

 Š È"
k

" Èk  1 ‹

œ1


" Èk  1

œ1

"  ˆ #"Î$


" ‰ ln 3

" (2(k
1)  1)#

œ1

 Š È"3


" Èk  1 ‹

" ‰ #"Î$
"#

 ˆ ln"4


" ln (k  2)

" (k  1)# “

" È3 ‹

lim sk œ lim Š1


kÄ_

46. sk œ ˆ "#
Ê

kÄ_

" È2 ‹



Ê sk œ ˆ1
4" ‰  ˆ 4"
9" ‰  ˆ 9"


lim sk œ lim ’1


kÄ_

" 9

Ê the sum is n lim 5 Š1
Ä_

" (n  1)#




œ 5 Š 1"


" ‰ #"Î%

 ˆ ln"5


 á  ˆ #1ÎÐ"k

" ‰ ln 4

1Ñ




" ‰ #1Îk

 á  Š ln (k" 1)


 ˆ #1"Îk


" ln k ‹

" ‰ #1ÎÐk
1Ñ

 Š ln (k" 2)


œ

" #




" #1ÎÐk
1Ñ

" ln (k  1) ‹

lim sk œ
ln"#

kÄ_

48. sk œ ctan
" (1)
tan
" (2)d  ctan
" (2)
tan
" (3)d  á  ctan
" (k
1)
tan
" (k)d  ctan
" (k)
tan
" (k  1)d œ tan
" (1)
tan
" (k  1) Ê lim sk œ tan
" (1)
kÄ_

49. convergent geometric series with sum

" 1
Š È" ‹ 2

50. divergent geometric series with krk œ È2  1

œ

È2 È 2
1

1 #

œ

1 4




1 #

œ
14

œ 2  È2

51. convergent geometric series with sum

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Š 3# ‹ 1
Š "# ‹

œ1

Section 10.2 Infinite Series 52. n lim (
1)n1 n Á 0 Ê diverges Ä_

53. n lim cos (n1) œ n lim (
1)n Á 0 Ê diverges Ä_ Ä_

54. cos (n1) œ (
1)n Ê convergent geometric series with sum " 1
Š

55. convergent geometric series with sum

56. n lim ln Ä_

" 3n

" ‹ e#

2 " 1
Š 10 ‹

58. convergent geometric series with sum

" 1
Š "x ‹

59. difference of two geometric series with sum ˆ1
"n ‰n œ lim ˆ1  60. n lim Ä_ nÄ_

_

63. ! n œ1

n! 1000n

2n  3n 4n

since r œ _

! n œ1

64.

2n  3n 4n

_

n œ1

5 6


" ‰n n


2œ œ

Ê

2n 4n

_

!

¹ 12 ¹

nœ1

3n 4n

_

20 9




œ

18 9

2 9

x x
1

" 1
Š 23 ‹




" 1
Š 3" ‹

œ3


œ

3 #

3 #

œ e
" Á 0 Ê diverges 62. n lim Ä_ _

n

_

n

nn n!

œ n lim Ä_ _

n

n †n â n 1†#ân

 n lim n œ _ Ê diverges Ä_

n

œ ! ˆ 21 ‰  ! ˆ 43 ‰ ; both œ ! ˆ 21 ‰ and ! ˆ 43 ‰ are geometric series, and both converge nœ1

 1 and r œ

nœ1

3 4

Ê

¹ 34 ¹

n œ1

 1, respectivley Ê

n œ1

_

! ˆ 1 ‰n 2

n œ1

œ

1 2

1
12

_

n

œ 1 and ! ˆ 34 ‰ œ nœ1

3 4

1
34

œ3Ê

œ 1  3 œ 4 by Theorem 8, part (1)

2n  4n n n nÄ_ 3  4

lim

œ

e# e #
1

œ _ Á 0 Ê diverges

œ! 1 2

œ

" 1
Š "5 ‹

œ
_ Á 0 Ê diverges

57. convergent geometric series with sum

61. n lim Ä_

581

œ

lim

nÄ_

_

_

n œ1

n œ1

2n 4n 3n 4n

" "

ˆ 12 ‰n  " 3 n nÄ_ ˆ 4 ‰  "

œ lim

œ

1 1

œ 1 Á 0 Ê diverges by nth term test for divergence

65. ! ln ˆ n n 1 ‰ œ ! cln (n)
ln (n  1)d Ê sk œ cln (1)
ln (2)d  cln (2)
ln (3)d  cln (3)
ln (4)d  á  cln (k
1)
ln (k)d  cln (k)
ln (k  1)d œ
ln (k  1) Ê

lim sk œ
_, Ê diverges

kÄ_

66. n lim a œ n lim ln ˆ 2n n 1 ‰ œ ln ˆ "# ‰ Á 0 Ê diverges Ä_ n Ä_ 67. convergent geometric series with sum 68. divergent geometric series with krk œ _

_

n œ0

n œ0

" 1
ˆ 1e ‰ e1 1e

¸

œ

23.141 22.459

1 1
e

1

69. ! (
1)n xn œ ! (
x)n ; a œ 1, r œ
x; converges to _

_

n œ0

n œ0

" 1
(
x)

n 70. ! (
1)n x2n œ ! a
x# b ; a œ 1, r œ
x# ; converges to

œ

" 1  x#

" 1x

for kxk  1

for kxk  1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

582

Chapter 10 Infinite Sequences and Series

71. a œ 3, r œ _

72. ! n œ0

œ

(
1)n #

x
1 #

; converges to _

ˆ 3  "sin x ‰n œ !

n œ0

3  sin x 2(4  sin x)

œ

3  sin x 8  2 sin x

3 1
Šx

ˆ 3 
"sin x ‰n ; a œ

" #

" 1
2x

74. a œ 1, r œ
x"# ; converges to

for k2xk  1 or kxk 

" #

; converges to

77. a œ 1, r œ sin x; converges to

_

79. (a) ! nœ
2 _

80. (a) ! nœ
1

" 1  (x  1)

" 1
Š3

x # ‹

" 1
sin x

œ

œ

" #x

for kx  1k  1 or
2  x  0

for kln xk  1 or e
"  x  e _

5 (n  2)(n  3)

(b) !

n œ0 _

n œ3



" 4

(b) one example is
3#
(c) one example is 1


" #

for all x‰

for x Á (2k  1) 1# , k an integer

(b) !

" #

" ‹ 1
Š 3
sin x

for ¸ 3
# x ¸  1 or 1  x  5

2 x
1

" (n  4)(n  5)

81. (a) one example is

ˆ "# ‰

#

" 1
ln x

78. a œ 1, r œ ln x; converges to

" 3  sin x

Ÿ

; converges to

x ¸1¸ " ‹ œ x#  1 for x#  1 or kxk  1. # x

75. a œ 1, r œ
(x  1)n ; converges to 3
x #


" 3  sin x

,rœ

" #

" 1
Š

" 4

" #

Ÿ

for all x ˆsince

73. a œ 1, r œ 2x; converges to

76. a œ 1, r œ

6 x
" " œ 3
x for
1  #  1 or
1  x  3 # ‹



" 8



" 16

á œ

3 4




3 8




3 16




" 4




" 8




Š "# ‹ 1
Š "# ‹


á œ " 16

_

" (n  2)(n  3)

(c) !

5 (n
2)(n
1)

(c) !

n œ5 _

nœ20

" (n
3)(n
#)

5 (n
19)(n
18)

œ1

Š
3# ‹ 1
Š "# ‹


á œ 1


œ
3 Š "# ‹ 1
Š "# ‹

œ 0.

_

Š k# ‹

n œ0

1
Š "# ‹

n 1 82. The series ! kˆ 12 ‰ is a geometric series whose sum is

œ k where k can be any positive or negative number.

_

_

_

_

_

nœ1

nœ1

nœ1

nœ1

nœ1

_

_

_

_

_

nœ1

nœ1

nœ1

nœ1

nœ1

n n 83. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! Š bann ‹ œ ! (1) diverges.

n n n 84. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! aan bn b œ ! ˆ 4" ‰ œ

n

n

_

85. Let an œ ˆ 4" ‰ and bn œ ˆ "# ‰ . Then A œ ! an œ n œ1

" 3

_

_

_

nœ1

nœ1

nœ1

" 3

Á AB.

n , B œ ! bn œ 1 and ! Š bann ‹ œ ! ˆ "# ‰ œ 1 Á

86. Yes: ! Š a"n ‹ diverges. The reasoning: ! an converges Ê an Ä 0 Ê

" an

A B

.

Ä _ Ê ! Š a"n ‹ diverges by the

nth-Term Test.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.3 The Integral Test 87. Since the sum of a finite number of terms is finite, adding or subtracting a finite number of terms from a series that diverges does not change the divergence of the series. 88. Let An œ a"  a#  á  an and n lim A œ A. Assume ! aan  bn b converges to S. Let Ä_ n Sn œ (a"  b" )  (a#  b# )  á  (an  bn ) Ê Sn œ (a"  a#  á  an )  (b"  b#  á  bn ) Ê b"  b#  á  bn œ Sn
An Ê n lim ab"  b#  á  bn b œ S
A Ê ! bn converges. This Ä_ contradicts the assumption that ! bn diverges; therefore, ! aan  bn b diverges. 89. (a) (b)

2 1
r

œ5 Ê

Š 13 2 ‹ 1
r

2 5

œ5 Ê

œ1
r Ê rœ

13 10

90. 1  eb  e2b  á œ

#

; 2  2 ˆ 35 ‰  2 ˆ 35 ‰  á

3 5

3 œ 1
r Ê r œ
10 ; " 1
e b

" 9

œ9 Ê

13 2




13 #

3 ‰ ˆ 10 

œ 1
eb Ê eb œ

13 #

3 ‰# ˆ 10


13 #

3 ‰$ ˆ 10 á

Ê b œ ln ˆ 89 ‰

8 9

91. sn œ 1  2r  r#  2r$  r%  2r&  á  r2n  2r2n1 , n œ 0, 1, á Ê sn œ a1  r#  r%  á  r2n b  a2r  2r$  2r&  á  2r2n1 b Ê n lim s œ Ä_ n 1  2r œ 1
r# , if kr# k  1 or krk  1 92. L
sn œ

a 1
r




a a1
rn b 1
r

œ

#

#

#

94. (a) L" œ 3, L# œ 3 ˆ 43 ‰ , L$ œ 3 ˆ 43 ‰ , á , Ln œ 3 ˆ 43 ‰

nc1

" #

á œ

4 1


" #

An œ lim

È3 4

È3 ˆ " ‰2 4 ‹ 3

nÄ_

 ! 3a4bk
2 Š

È3 8 ˆ5‰ 4

œ

kœ2

An œ

È3 4

œ

È3 ˆ " ‰2 4 ‹ 33

n

2r 1
r#



È3 1#

, A$ œ A#  3a4bŠ

, A5 œ A4  3a4b3 Š

È3 ˆ " ‰k
1 4 ‹ 32

È3 lim nÄ_ Œ 4

œ

n

 3 È 3 Œ! kœ2

È3 4

È3 ˆ " ‰2 4 ‹ 32

È3 ˆ " ‰2 4 ‹ 34 ,

œ

n

k
1

œ

œ_

È3 2 4 s , we see that È3 È3 È3 4  12  #7 ,

È3 4 ,

A" œ

n

4kc$ . 9k 1 

œ

È3 4

 3È3Œ!

 3È 3Œ 1
4  œ

È3 4

1 ‰  3È3ˆ 20 œ

kœ2

È3 4

nc1

...,

 ! 3È3a4bk
$ ˆ "9 ‰

4kc$ 9k 1 

œ 8 m#

Ê n lim L œ n lim 3 ˆ 43 ‰ Ä_ n Ä_

(b) Using the fact that the area of an equilateral triangle of side length s is

A% œ A$  3a4b2 Š



arn 1
r

93. area œ 2#  ŠÈ2‹  (1)#  Š È"2 ‹  á œ 4  2  1 

A# œ A"  3Š

" 1
r#

1 36

9

kœ2

È3 ˆ 4 1

 53 ‰

œ 85 A"

10.3 THE INTEGRAL TEST 1. faxb œ

1 x2

œ lim

bÄ_

2. faxb œ

1 x0.2

œ lim

bÄ_

_1

is positive, continuous, and decreasing for x   1; '1

_1

ˆ
1b  1‰ œ 1 Ê ' 1

_

x2

dx converges Ê !

n œ1

1 n2

x2

_

ˆ 54 b0.8
54 ‰ œ _ Ê ' 1

1 x0.2

_

bÄ_

'1b x1

2

dx œ lim

bÄ_

b

’
1x “

1

converges

is positive, continuous, and decreasing for x   1; '1

_

dx œ lim

1 x0.2

dx œ lim

bÄ_

'1b x1

0.2

dx œ lim

bÄ_

dx diverges Ê ! n10.2 diverges n œ1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

b

’ 54 x0.8 “

1

583

584

Chapter 10 Infinite Sequences and Series

3. faxb œ œ

1 x2  4

_

is positive, continuous, and decreasing for x   1; '1

lim ˆ 1 tan
1 b2 bÄ_ 2

4. faxb œ

1
1 1 ‰ 2 tan 2




œ

1 4

bÄ_

2

_

bÄ_

_

œ lim

ˆ
2e12b 

bÄ_

_

'3

x x2  4

x x2  4

1 2e2

_

_ ln x2

'3

œ

1 ‰ ln 2

1 ln 2

_

Ê '2

_

dx converges Ê !

1 xaln xb2

'3

b

bÄ_

x x2  4

_

n œ3

n œ1

bÄ_

dx œ lim

bÄ_

'3

ln x x

2

_

'7

2

x exÎ3

dx œ lim _

n œ3

n œ2

n enÎ3

n œ1

10. faxb œ

2

x exÎ3


18b Š 3a
6b ‹ ebÎ3 2

Ê !

'7

b

bÄ_

bÄ_

œ

1 e1Î3

x
4 x2
2x  1



œ

x
4 a x
1 b2

_

bÄ_ _

Ê !

nœ8

’lnlx
1l 

bÄ_

9 e1

ln 4 2

bÄ_



_

n
4 n2
2n  1 diverges

16 e4Î3



x
4 ax
1b2

36 e2

2
ln x2 x2

bÄ_

3

18x exÎ3

_

Ê !

nœ2

bÄ_

 !

n œ7

11. converges; a geometric series with r œ

b

b

54 “ exÎ3 7 327 e7Î3

2

n enÎ3

x
1 ax
1b2

ˆlnlb
1l 

n
4 n2
2n  1

" 10

bÄ_

’
ln1x “

2

converges

ˆ 12 lnab2  4b
12 lna13b‰ œ _ Ê ' 3

x x2  4

dx

 0 for x  e, thus f is decreasing for x   3;

_ ln x2

a2aln bb
2aln 3bb œ _ Ê '3


x a x
6 b 3exÎ3




œ

_

”'8 bÄ_

dx œ lim

œ lim

b

dx œ lim

2

n œ3

327 e7Î3

e



bÄ_

'2b xaln1xb

x

dx

n œ3

’
e3xxÎ3


25 e5Î3

dx œ lim

 0 for x  6, thus f is decreasing for x   7;

œ lim

bÄ_

_

Ê '7

x2 exÎ3

Š
3b

2


18b
54 ebÎ3

œ
2


1 4

b

dx converges Ê !

n œ7

13. diverges; by the nth-Term Test for Divergence, n lim Ä_

3 ax
1b2

œ

n2 converges enÎ3

1 16



2 25



3 36

7
x ax
1 b 3

dx• œ lim ”'8 bÄ_

b

_


ln 7
37 ‰ œ _ Ê '8

0

1

327 ‹ e7Î3

converges

dx
'8

3 b
1



_

is continuous for x   2, f is positive for x  4, and f w axb œ

b

1

2  ! lnnn diverges

ˆ
b54 ‰ Î3

œ lim

3 x
1 “8

b

’
12 e
2x “

lim

 ! n2 n 4 diverges

_

2

dx œ lim

327 e7Î3



4 e2Î3

decreasing for x   8; '8 œ lim

2 8

’2aln xb“ œ lim

bÄ_

_

dx œ lim

œ lim



1 5

is positive and continuous for x   1, f w axb œ

x2 exÎ3

_

1

_

bÄ_

3

b

2 2 diverges Ê ! lnnn diverges Ê ! lnnn œ

9. faxb œ

bÄ_

b

’lnlx  4l“

lim

 0 for x  2, thus f is decreasing for x   3;

’ 21 lnax2  4b“ œ lim

is positive and continuous for x   2, f w axb œ b

4
x2 ax2  4b2 b

dx œ lim

_

1 xaln xb2

1 naln nb2

n œ2

is positive and continuous for x   1, f w axb œ

dx œ lim

ln x2 x

x

1

n œ1

diverges Ê ! n2 n 4 diverges Ê ! n2 n 4 œ 8. faxb œ

'1b e
2x dx œ

Ê '1 e
2x dx converges Ê ! e
2n converges

_

ˆ
ln1b 

bÄ_

œ

1 ‰ 2e2

_

bÄ_

is positive, continuous, and decreasing for x   2; '2

1 xaln xb2

œ lim 7. faxb œ

’ 21 tan
1 x2 “

n œ1

5. faxb œ e
2x is positive, continuous, and decreasing for x   1; '1 e
2x dx œ lim

6. faxb œ

bÄ_

bÄ_

_

bÄ_

'1b x 1 4 dx œ

dx œ lim

1 x4

b

lim

2

n œ1

_ alnlb  4l
ln 5b œ _ Ê '1 x 1 4 dx diverges Ê ! n 1 4 diverges

œ lim

2

_

is positive, continuous, and decreasing for x   1; '1

1 x4

'1b x 1 4 dx œ

dx œ lim

_ Ê '1 x 1 4 dx converges Ê ! n 1 4 converges

1
1 1 2 tan 2




1 x2  4

_

 !

n œ8

x
4 ax
1b2

n
4 n2
2n  1

 0 for x  7, thus f is

1 x
1

dx
'8

3 ax
1b2

dx diverges

diverges

12. converges; a geometric series with r œ n n1

b

œ1Á0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" e

1

dx•

Section 10.3 The Integral Test 14. diverges by the Integral Test; '1

n

_

15. diverges; ! n œ1

3 Èn

_

16. converges; ! n œ1

_

" Èn

œ3!

nœ1


2 nÈ n

_

dx œ 5 ln (n  1)
5 ln 2 Ê '1

5 x1

_

n œ1


8 n

_

œ
2 !

n œ1

_

œ
8 !

nœ1

dx Ä _

, which is a divergent p-series (p œ #" ) " n$Î#

, which is a convergent p-series (p œ 3# )

17. converges; a geometric series with r œ 18. diverges; !

5 x1

" 8

1 _

and since !

1 n

nœ1

19. diverges by the Integral Test:

_

" n

diverges,
8 !

n œ1

'2n lnxx dx œ "# aln# n
ln 2b

Ê

t œ ln x × dt œ dx Ä x Õ dx œ et dt Ø œ lim 2ebÎ2 (b
2)
2eÐln 2ÑÎ2 (ln 2
2)‘ œ _

20. diverges by the Integral Test:

'2_ lnÈxx dx; Ô

1 n

diverges

'2_ lnxx dx

'ln_2 tetÎ2 dt œ

Ä _

b

lim 2tetÎ2
4etÎ2 ‘ ln 2

bÄ_

bÄ_

21. converges; a geometric series with r œ 22. diverges; n lim Ä_ _

23. diverges; ! n œ0


2 n 1

5n 4n  3

_

œ
2 !

n œ0

" n1

1

ˆ ln 5 ‰ ˆ 54 ‰n œ _ Á 0 œ n lim Ä _ ln 4

5n ln 5 4n ln 4

œ n lim Ä_

2 3

, which diverges by the Integral Test

24. diverges by the Integral Test:

'1n 2xdx
1 œ "# ln (2n
1)

25. diverges; n lim a œ n lim Ä_ n Ä_

2n n1

26. diverges by the Integral Test:

'1n Èx ˆÈdxx  1‰ ; – u œ

27. diverges; n lim Ä_

Èn ln n

œ n lim Ä_

œ n lim Ä_

2n ln 2 1

œ_Á0 Èx  "

du œ

" Š 2È ‹ n

Š "n ‹

œ n lim Ä_

Èn #

Ä _ as n Ä _

dx Èx

Ènb1 du

' —Ä 2

u

œ ln ˆÈn  1‰
ln 2 Ä _ as n Ä _

œ_Á0

ˆ1  n" ‰n œ e Á 0 28. diverges; n lim a œ n lim Ä_ n Ä_ 29. diverges; a geometric series with r œ

" ln #

30. converges; a geometric series with r œ

31. converges by the Integral Test:

¸ 1.44  1

" ln 3

¸ 0.91  1

'3_ (ln x) ÈŠ(ln‹x)
1 dx; ” " x

#

u œ ln x Ä du œ "x dx •

'ln_3

" uÈ u#
1

du

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

585

586

Chapter 10 Infinite Sequences and Series b œ lim csec
" kukd ln 3 œ lim csec
" b
sec
" (ln 3)d œ lim cos
" ˆ "b ‰
sec
" (ln 3)‘

bÄ_

bÄ_

œ cos
" (0)
sec
" (ln 3) œ

1 #

32. converges by the Integral Test:

'1_ x a1 "ln xb dx œ '1_ 1 Š(ln‹x) " x

#

œ lim ctan
" ud 0 œ lim atan
" b
tan
" 0b œ b

bÄ_

bÄ_


sec
" (ln 3) ¸ 1.1439

bÄ_

1 #


0œ

dx; ”

#

'0_ 1"u

u œ ln x Ä du œ "x dx •

#

du

1 #

33. diverges by the nth-Term Test for divergence; n lim n sin ˆ "n ‰ œ n lim Ä_ Ä_

sin ˆ "n ‰ ˆ "n ‰

œ lim

34. diverges by the nth-Term Test for divergence; n lim n tan ˆ "n ‰ œ n lim Ä_ Ä_

tan ˆ "n ‰ ˆ "n ‰

œ n lim Ä_

xÄ0

œ1Á0

sin x x

Š
n"# ‹ sec# ˆ n" ‰ Š
n"# ‹

œ n lim sec# ˆ "n ‰ œ sec# 0 œ 1 Á 0 Ä_ 35. converges by the Integral Test:

'1_ 1 e e x

1 #

œ lim atan
" b
tan
" eb œ bÄ_

36. converges by the Integral Test: œ lim 2 ln bÄ_

u ‘b u1 e

dx; ”

2x

'e_

u œ ex Ä du œ ex dx •

" 1  u#

ctan
" ud e du œ n lim Ä_

b


tan
" e ¸ 0.35

_

'1

u œ ex × _ _ dx; du œ ex dx Ä 'e u(1 2 u) du œ 'e ˆ 2u
Õ dx œ " du Ø u Ô

2 1  ex

2 ‰ u1

du

œ lim 2 ln ˆ b b 1 ‰
2 ln ˆ e e 1 ‰ œ 2 ln 1
2 ln ˆ e e 1 ‰ œ
2 ln ˆ e e 1 ‰ ¸ 0.63 bÄ_

37. converges by the Integral Test:

38. diverges by the Integral Test:

'1_ 81tancx x dx; ” u œ tan dx x •
"

"

#

du œ

1  x#

'1_ x x1 dx; ” u œ x

39. converges by the Integral Test:

#

#

1 Ä du œ 2x dx •

'1_ sech x dx œ 2

Ä

x

x #

bÄ_

#

'2_ du4 œ

" #

'1b 1 eae b

lim

'11ÎÎ42 8u du œ c4u# d 11ÎÎ24 œ 4 Š 14




b lim  #" ln u‘ 2 œ lim

1# 16 ‹

"

bÄ_ #

bÄ_

œ

31 # 4

(ln b
ln 2) œ _

dx œ 2 lim ctan
" ex d 1 b

bÄ_

œ 2 lim atan
" eb
tan
" eb œ 1
2 tan
" e ¸ 0.71 bÄ_

40. converges by the Integral Test:

'1_ sech# x dx œ

œ 1
tanh 1 ¸ 0.76 41.

'1_ ˆ x a 2
x " 4 ‰ dx œ a lim (bb2)4 bÄ_

lim

bÄ_

'1b sech# x dx œ

lim ca ln kx  2k
ln kx  4kd 1 œ lim ln b

bÄ_

œ a lim (b  2) bÄ_

bÄ_

a
1

lim ctanh xd b1 œ lim (tanh b
tanh 1)

bÄ_

(b  2)a b4

bÄ_


ln ˆ 35 ‰ ; a

_, a  1 œœ Ê the series converges to ln ˆ 53 ‰ if a œ 1 and diverges to _ if 1, a œ 1

a  1. If a  1, the terms of the series eventually become negative and the Integral Test does not apply. From that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges. 42.

'3_ ˆ x
" 1
x 2a 1 ‰ dx œ " 2ac1 b Ä _ #a(b  1)

œ lim

b

lim ’ln ¹ (xx
1)12a ¹“ œ lim ln

bÄ_

œ

3

bÄ_

b
1 (b  1)2a

b
"


ln ˆ 422a ‰ ; lim

2a b Ä _ (b  1)

1, a œ "# Ê the series converges to ln ˆ #4 ‰ œ ln 2 if a œ _, a  "#

" #

and diverges to _ if

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.3 The Integral Test if a 

" #

. If a 

" #

587

, the terms of the series eventually become negative and the Integral Test does not apply.

From that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges. 43. (a)

(b) There are (13)(365)(24)(60)(60) a10* b seconds in 13 billion years; by part (a) sn Ÿ 1  ln n where n œ (13)(365)(24)(60)(60) a10* b Ê sn Ÿ 1  ln a(13)(365)(24)(60)(60) a10* bb œ 1  ln (13)  ln (365)  ln (24)  2 ln (60)  9 ln (10) ¸ 41.55 _

44. No, because ! n œ1

" nx

œ

" x

_

! n œ1

" n

_

and ! n œ1

" n

diverges

_

_

_

nœ1

nœ1

nœ1

45. Yes. If ! an is a divergent series of positive numbers, then ˆ "# ‰ ! an œ ! ˆ a#n ‰ also diverges and

an #

 an .

_

There is no “smallest" divergent series of positive numbers: for any divergent series ! an of positive numbers n œ1

_

! ˆ an ‰ has smaller terms and still diverges. #

n œ1

_

_

_

nœ1

nœ1

nœ1

46. No, if ! an is a convergent series of positive numbers, then 2 ! an œ ! 2an also converges, and 2an   an . There is no “largest" convergent series of positive numbers. 47. (a) Both integrals can represent the area under the curve faxb œ

1 Èx  1 ,

and the sum s50 can be considered an 50

approximation of either integral using rectangles with ?x œ 1. The sum s50 œ !

n œ1

integral

1 Èn  1

is an overestimate of the

'151 Èx1 1 dx. The sum s50 represents a left-hand sum (that is, the we are choosing the left-hand endpoint of

each subinterval for ci ) and because f is a decreasing function, the value of f is a maximum at the left-hand endpoint of each sub interval. The area of each rectangle overestimates the true area, thus '1

51

manner, s50 underestimates the integral '0

50

1 Èx  1 dx.

1 Èx  1 dx

50

!

n œ1

1 Èn  1 .

In a similar

In this case, the sum s50 represents a right-hand sum and because

f is a decreasing function, the value of f is aminimum at the right-hand endpoint of each subinterval. The area of each 50

rectangle underestimates the true area, thus ! n œ1

1 Èn  1

œ ’2Èx  1“ œ 2È52
2È2 ¸ 11.6 and '0 51

50

1

50

11.6  !

n œ1

1 Èn  1

1 Èx  1 dx

50

1 Èx  1 dx.

Evaluating the integrals we find '1

51

1 Èx  1 dx

50

œ ’2Èx  1“ œ 2È51
2È1 ¸ 12.3. Thus, 0

 12.3.

nb1

(b) sn  1000 Ê '1

 '0

1 Èx  1 dx

nb1

œ ’2Èx  1“

1

2

œ 2Èn  1
2È2  1000 Ê n  Š500  2È2‹
¸ 251414.2

Ê n   251415.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

588

Chapter 10 Infinite Sequences and Series 30

48. (a) Since we are using s30 œ !

n œ1

1 n4

_

to estimate ! n œ1

of the area under the curve faxb œ

_

the error is given by !

1 n4 ,

nœ31

1 n4 .

We can consider this sum as an estimate

when x   30 using rectangles with ?x œ 1 and ci is the right-hand endpoint of

1 x4

each subinterval. Since f is a decreasing function, the value of f is a minimum at the right-hand endpoint of each _

subinterval, thus ! nœ31

 '30

_

1 n4

1 x4 dx

œ lim '30

b

bÄ_

1 x4 dx

b

œ lim ’
3x1 3 “ œ lim Š
3b1 3  bÄ_

bÄ_

30

1 ‹ 3a30b3

¸ 1.23 ‚ 10
5 .

Thus the error  1.23 ‚ 10
5 Þ (b) We want S
sn  0.000001 Ê 'n

_

œ

lim ˆ
3b1 3 bÄ_



1 ‰ 3n3

œ

1 3n3

49. We want S
sn  0.01 Ê 'n

_

œ

1 2n2

 0.000001 Ê n 

1 x3 dx

 0.01 Ê 'n

1 x2  4 dx

bÄ_

1 n2  4

_

10 n0.1

 0.1 Ê lim 'n

b

1 4




bÄ_ 1
1 ˆ n ‰ 2 tan 2

b

bÄ_

1 x4 dx

b

œ lim ’
3x1 3 “ bÄ_

n

œ lim 'n

b

1 x3 dx

bÄ_

bÄ_

bÄ_

n

1 ‰ 2n2

¸ 1.195

1 n3

1 x2  4 dx

b

œ lim ’
2x1 2 “ œ lim ˆ
2b1 2 

b

œ lim ’ 21 tan
1 ˆ 2x ‰“ bÄ_

n

 0.1 Ê n  2tanˆ 12
0.2‰ ¸ 9.867 Ê n   10 Ê S ¸ s10

1 x1.1 dx

 0.00001 Ê 'n

_

1 x1.1 dx

œ lim 'n

b

bÄ_

1 x1.1 dx

b

œ lim ’
x10 lim ˆ
b10 0.1 “ œ 0.1  bÄ_

bÄ_

n

10 ‰ n0.1

 0.00001 Ê n  100000010 Ê n  1060

52. S
sn  0.01 Ê 'n

_

œ

1 x3 dx

œ lim 'n

¸ 0.57

51. S
sn  0.00001 Ê 'n œ

_

1 x4 dx

¸ 69.336 Ê n   70.

É 1000000 3 3

n œ1

œ lim ˆ 12 tan
1 ˆ b2 ‰
12 tan
1 ˆ n2 ‰‰ œ n œ1

_

8

_

10

 0.000001 Ê 'n

 0.01 Ê n  È50 ¸ 7.071 Ê n   8 Ê S ¸ s8 œ !

50. We want S
sn  0.1 Ê 'n

œ!

1 x4 dx

lim Š
2aln1bb2 bÄ_



1 dx xaln xb3

1 ‹ 2aln nb2

n

n

k œ1

k œ1

 0.01 Ê 'n

_

œ

1 2aln nb2

œ lim 'n

b

1 dx xaln xb3

bÄ_

È50

 0.01 Ê n  e

1 dx xaln xb3

b

œ lim ’
2aln1xb2 “ bÄ_

n

¸ 1177.405 Ê n   1178

53. Let An œ ! ak and Bn œ ! 2k aa2k b , where {ak } is a nonincreasing sequence of positive terms converging to 0. Note that {An } and {Bn } are nondecreasing sequences of positive terms. Now, Bn œ 2a#  4a%  8a)  á  2n aa2n b œ 2a#  a2a%  2a% b  a2a)  2a)  2a)  2a) b  á ˆ2aa2n b  2aa2n b  á  2aa2n b ‰ Ÿ 2a"  2a#  a2a$  2a% b  a2a&  2a'  2a(  2a) b  á  ðóóóóóóóóóóóóóóñóóóóóóóóóóóóóóò 2n
1 terms _

 ˆ2aa2nc1 b  2aa2nc1 1b  á  2aa2n b ‰ œ 2Aa2n b Ÿ 2 ! ak . Therefore if ! ak converges, k œ1

then {Bn } is bounded above Ê ! 2k aa2k b converges. Conversely, _

An œ a"  aa#  a$ b  aa%  a&  a'  a( b  á  an  a"  2a#  4a%  á  2n aa2n b œ a"  Bn  a"  ! 2k aa2k b . k œ1

_

Therefore, if ! 2 aa2k b converges, then {An } is bounded above and hence converges. k

k œ1

54. (a) aa2n b œ _

Ê !

n œ2

" 2n ln a2n b " n ln n

œ

" 2n †n(ln 2)

_

_

n œ2

n œ2

Ê ! 2 n a a2 n b œ ! 2 n

" #n †n(ln 2)

œ

" ln #

_

! n œ2

" n

, which diverges

diverges.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.3 The Integral Test " #np

(b) aa2n b œ

55. (a)

_

_

n œ1

nœ1

Ê ! 2 n a a2 n b œ ! 2 n † "

#pc1

'2_ x(lndxx)

u œ ln x • Ä du œ dx x

œœ

;”

_

n œ ! ˆ #p"c1 ‰ , a geometric series that nœ1

'2_ x dxln x œ

p œ 1.

cpb1

lim ’
up  1 “

bÄ_

b ln 2

œ lim Š 1
" p ‹ cb
p1
(ln 2)
p1 d bÄ_

Ê the improper integral converges if p  1 and diverges if p  1.

_, p  "

For p œ 1:

" a2n bpc1

nœ1

'ln_2 ucp du œ

(ln 2)cpb1 , p  1

" p
1

_

œ!

 1 or p  1, but diverges if p Ÿ 1.

converges if

p

" #np

589

lim cln (ln x)d b2 œ lim cln (ln b)
ln (ln 2)d œ _, so the improper integral diverges if

bÄ_

bÄ_

_

" n(ln n)p

(b) Since the series and the integral converge or diverge together, ! n œ2

converges if and only if p  1.

56. (a) p œ 1 Ê the series diverges (b) p œ 1.01 Ê the series converges _

(c) ! n œ2

" n aln n$ b

" 3

œ

_

" n(ln n)

! n œ2

; p œ 1 Ê the series diverges

(d) p œ 3 Ê the series converges 57. (a) From Fig. 10.11(a) in the text with f(x) œ Ÿ 1  '1 f(x) dx Ê ln (n  1) Ÿ 1  n

Ÿ ˆ1 

" #

" 3



á 

"‰ n

" #

" x

and ak œ



" 3

á 

(b) From the graph in Fig. 10.11(b) with f(x) œ Ê 0


cln (n  1)
ln nd œ ˆ1 

If we define an œ 1 

" #

œ

nb1

, we have '1 " n

" 3



" n

" x

" n1 " " #  3 

,

" x

dx Ÿ 1 

" #



" 3

á 

" n

Ÿ 1  ln n Ê 0 Ÿ ln (n  1)
ln n


ln n Ÿ 1. Therefore the sequence ˜ˆ1 

1 and below by 0. " n1

" k

nb1

 'n

" x

á

" n 1

" #



" 3

 á  n" ‰
ln n™ is bounded above by

dx œ ln (n  1)
ln n
ln (n  1)‰
ˆ1 

" #



" 3

á 

" n


ln n‰ .


ln n, then 0  an1
an Ê an1  an Ê {an } is a decreasing sequence of

nonnegative terms.

_

_

# # b 58. e
x Ÿ e
x for x   1, and '1 ecx dx œ lim c
e
x d " œ lim ˆ
e
b  e
1 ‰ œ ec1 Ê '1 ecx dx converges by

bÄ_

bÄ_

_


n#

the Comparison Test for improper integrals Ê ! e n œ0

10

59. (a) s10 œ !

'10_ x1

n œ1

" n3

dx œ lim

3

bÄ_

Ê 1.97531986  _

(b) s œ !

n œ1

" n3

10

60. (a) s10 œ !

'10_ x1

nœ1

4

¸

" n4

(b) s œ !

n œ1

" n4

¸

c2 b

lim ’
x2 “

bÄ_

'11_ x1

'10b x
4 dx œ 1 3993

10

4

dx œ lim c3 b

bÄ_

c2 b

lim ’
x2 “

bÄ_

œ lim ˆ
2b1 2  bÄ_

bÄ_ 10

 s  1.082036583 

1.08229  1.08237 2

'11b x
3 dx œ

1 ‰ 200

œ

11

œ lim ˆ
2b1 2  bÄ_

1 ‰ 242

œ

1 242

and

1 200

Ê 1.20166  s  1.20253

1 200

lim ’
x3 “

#

nœ1

œ 1.202095; error Ÿ

œ 1.082036583;

Ê 1.082036583 

bÄ_

 s  1.97531986 

1.20166  1.20253 2

bÄ_

dx œ lim

3

'10b x
3 dx œ

1 242

dx œ lim

_

'11_ x1

œ 1.97531986;

_

œ 1  ! e
n converges by the Integral Test.

1.20253
1.20166 2

'11b x
4 dx œ

c3 b

lim ’
x3 “

bÄ_

œ lim ˆ
3b1 3  bÄ_

1 3000

œ 1.08233; error Ÿ

œ 0.000435

1 ‰ 3000

œ

11

œ lim ˆ
3b1 3  bÄ_

1 3000

Ê 1.08229  s  1.08237

1.08237
1.08229 2

œ 0.00004

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 ‰ 3993

œ

1 3993

and

590

Chapter 10 Infinite Sequences and Series

10.4 COMPARISON TESTS _

1. Compare with ! n œ1

" n2 ,

which is a convergent p-series, since p œ 2  1. Both series have nonnegative terms for n   1. For

n   1, we have n2 Ÿ n2  30 Ê _

2. Compare with ! n œ1

" n3 ,

1 n2

 

1 n2  30 .

_

Then by Comparison Test, !

1 n2  30

n œ1

converges.

which is a convergent p-series, since p œ 3  1. Both series have nonnegative terms for n   1. For

n   1, we have n4 Ÿ n4  2 Ê

 

1 n4

Ê

1 n 4 2

 

n n4

Ê

n n 4 2

1 n3

 

n n 4 2

 

_

n
1 n 4 2 .

Then by Comparison Test, ! n œ1

n
1 n 4 2

converges. _

3. Compare with ! n œ2

" Èn ,

which is a divergent p-series, since p œ

n   2, we have Èn
1 Ÿ Èn Ê _

4. Compare with ! n œ2

" n,

_

n œ1

" , n3Î2

n2

1
n

 

_

n œ1

" 3n ,

_

nœ1 _

œ È5 !

n œ1

1 n3Î2

Ê

1 n2

cos2 n n3Î2

n œ2

1 Èn
1

diverges.

n2

n
n

 

n n2

œ

1 n

Ê

Ÿ

1 . n3Î2

n2 n2
n

3 2

 

n2

n
n

_

  n1 . Thus !

n œ2

n2 n2
n

 1. Both series have nonnegative terms for n   1. _

Then by Comparison Test, ! n œ1

cos2 n n3Î2

converges.

È5 . n3Î2

_

The series ! nœ1

1 n †3 n

1 n3Î2

Ÿ

1 3n .

_

Then by Comparison Test, ! n œ1

is a convergent p-series, since p œ

3 2

1 n †3 n

converges. _

n 3 an  4 b n4  4

È5 n3Î2

 1, and the series !

nœ1

converges by Theorem 8 part 3. Both series have nonnegative terms for n   1. For n   1, we have

n3 Ÿ n4 Ê 4n3 Ÿ 4n4 Ê n4  4n3 Ÿ n4  4n4 œ 5n4 Ê n4  4n3 Ÿ 5n4  20 œ 5an4  4b Ê Ê

diverges.

which is a convergent geometric series, since lrl œ ¹ 13 ¹  1. Both series have nonnegative terms for

n   1. For n   1, we have n † 3n   3n Ê

7. Compare with !

_

Then by Comparison Test, !

1 Èn .

which is a convergent p-series, since p œ

For n   1, we have 0 Ÿ cos2 n Ÿ 1 Ê

6. Compare with !

 

Ÿ 1. Both series have nonnegative terms for n   2. For

which is a divergent p-series, since p œ 1 Ÿ 1. Both series have nonnegative terms for n   2. For

n   2, we have n2
n Ÿ n2 Ê

5. Compare with !

1 Èn
1

" #

Ÿ5Ê _

8. Compare with ! n œ1

n4 n4  4

" Èn ,

Ÿ

5 n3

Ê É nn444 Ÿ É n53 œ

È5 n3Î2

_

n4  4n3 n4  4

Ÿ 5.

Then by Comparison Test, ! É nn444 converges. n œ1

which is a divergent p-series, since p œ

" #

Ÿ 1. Both series have nonnegative terms for n   1. For

n   1, we have Èn   1 Ê 2Èn   2 Ê 2Èn  1   3 Ê nˆ2Èn  1‰   3n   3 Ê 2 nÈn  n   3 Ê n2  2 nÈn  n   n2  3 Ê Ê

Èn  1 È n2  3

 

1 Èn .

n ˆn  2 È n  1 ‰ n2  3

 1Ê

n  2È n  1 n2  3

 

1 n

Ê

ˆÈ n  1 ‰2 n2  3

 

1 n

ÊÊ

ˆÈ n  1 ‰ 2 n2  3

_

Èn  1 È n2  3 n œ1

Then by Comparison Test, !

diverges.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

  É 1n

Section 10.4 Comparison Tests _

" n2 ,

9. Compare with ! nc2 n3 c n2 b 3 1 În 2

œ lim _

! n œ1

n œ1

nÄ_

n
2 n3
n2  3

which is a convergent p-series, since p œ 2  1. Both series have positive terms for n   1. lim

an nÄ_ bn

n3
2n2 3 2 nÄ_ n
n  3

œ lim

3n2
4n 2 nÄ_ 3n
2n

œ lim

6n
4 nÄ_ 6n
2

œ lim

œ lim

6 nÄ_ 6

œ 1  0. Then by Limit Comparison Test,

converges. _

" Èn ,

10. Compare with ! n œ1

É nn2bb12

œ lim

591

which is a divergent p-series, since p œ

n œ lim É nn2   2 œ É lim

n2  n 2 nÄ_ n  2

2

nÄ_ 1ÎÈn

nÄ_

œ É lim

nÄ_

" #

2n  1 2n

Ÿ 1. Both series have positive terms for n   1. lim

an nÄ_ bn

œ É lim

2 nÄ_ 2

œ È1 œ 1  0. Then by Limit Comparison

_

Test, ! É nn212 diverges. n œ1

_

" n,

11. Compare with ! nan b 1b Šn2

œ lim

n œ2

b 1‹an c 1b

_

Test, ! n œ2

n3 + n2 3 2 nÄ_ n
n  n
1

n an  1 b an2  1ban
1b

_

n œ1

lim an nÄ_ bn

nÄ_

_

n œ1 5n

È n 4n

œ lim

†

nÄ_ 1ÎÈn

6n  2 nÄ_ 6n
2

œ lim

œ lim

6 nÄ_ 6

œ 1  0. Then by Limit Comparison

which is a convergent geometric series, since lrl œ ¹ 12 ¹  1. Both series have positive terms for

œ lim

13. Compare with !

3n2  2n 2 nÄ_ 3n
2n  1

œ lim

diverges.

" 2n ,

12. Compare with ! n   1.

an nÄ_ bn

œ lim

1 În

nÄ_

which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n   2. lim

" Èn ,

2n 3 b 4n 1Î2 n

4n 3  4n nÄ_

œ lim

4n ln 4 n nÄ_ 4 ln 4

œ lim

_

œ 1  0. Then by Limit Comparison Test, !

which is a divergent p-series, since p œ

n œ1

1 2

an nÄ_ bn

_

nÄ_

converges.

Ÿ 1. Both series have positive terms for n   1. lim

n œ lim ˆ 54 ‰ œ _. Then by Limit Comparison Test, !

5n n nÄ_ 4

œ lim

2n 3  4n

n œ1

5n Èn†4n

diverges.

_

n 14. Compare with ! ˆ 25 ‰ , which is a convergent geometric series, since lrl œ ¹ 25 ¹  1. Both series have positive terms for n œ1

n   1. œ exp

b 3 ‰n ˆ 2n 5n b 4

n  15 ‰n  15 ‰ lim ˆ 10n  15 ‰ œ exp lim lnˆ 10n œ exp lim n lnˆ 10n n œ 10n  8 10n  8 nÄ_ a2Î5b nÄ_ 10n  8 nÄ_ nÄ_ b 15 ‰ 10 lnˆ 10n
10b 8 70n2 70n2 10n b 8 lim œ exp lim 10n b
151În10n œ exp lim a10n  15 2 2 1 În ba10n  8b œ exp nlim nÄ_ nÄ_ nÄ_ Ä_ 100n  230n  120

lim an nÄ_ bn

œ lim

œ exp lim

œ exp lim

140n nÄ_ 200n  230 _

15. Compare with ! n œ2

œ lim

" ln n

nÄ_ 1În

n œ1

lnŠ1  n"2 ‹ 1În2

_

 3 ‰n œ e7Î10  0. Then by Limit Comparison Test, ! ˆ 2n converges. 5n  4 n œ1

which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n   2. lim

an nÄ_ bn

n nÄ_ ln n _

nÄ_

" n,

œ lim

16. Compare with ! œ lim

140 nÄ_ 200

" n2 ,

œ lim

1 nÄ_ 1În

_

œ lim n œ _. Then by Limit Comparison Test, ! nÄ_

n œ2

" ln n

diverges.

which is a convergent p-series, since p œ 2  1. Both series have positive terms for n   1. lim

an nÄ_ bn

1

œ lim

nÄ_

1

2 " Š
n3 ‹

n2

Š
n23 ‹

œ lim

1 " nÄ_ 1  n2

_

œ 1  0. Then by Limit Comparison Test, ! lnˆ1  n œ1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

"‰ n2

converges.

592

Chapter 10 Infinite Sequences and Series _

" Èn

17. diverges by the Limit Comparison Test (part 1) when compared with ! n œ1

, a divergent p-series:

"

lim

Œ #Èn È $ n
Š È"n ‹

nÄ_

Èn $ n 2È n  È

œ n lim Ä_

ˆ " œ n lim Ä _ #n

1Î6

‰œ

" #

18. diverges by the Direct Comparison Test since n  n  n  n  Èn  0 Ê

_

" n

term of the divergent series ! nœ1

3 n  Èn



" n

, which is the nth

" n

or use Limit Comparison Test with bn œ

19. converges by the Direct Comparison Test;

sin# n 2n

Ÿ

" #n

, which is the nth term of a convergent geometric series

20. converges by the Direct Comparison Test;

1  cos n n#

Ÿ

2 n#

2n 3n
1

21. diverges since n lim Ä_

œ

2 3

Š nn# È"n ‹

" n#

converges

Á0

22. converges by the Limit Comparison Test (part 1) with lim nÄ_

and the p-series !

" n$Î#

, the nth term of a convergent p-series:

ˆ n n " ‰ œ 1 œ n lim Ä_

" ‹ Š $Î# n

23. converges by the Limit Comparison Test (part 1) with lim

Š n(n 10n1)(n" 2) ‹ Š n"# ‹

nÄ_

10n#  n n#  3n  2

œ n lim Ä_

œ n lim Ä_

20n  1 2n  3

24. converges by the Limit Comparison Test (part 1) with lim

 n# (n

" n#

, the nth term of a convergent p-series:

œ n lim Ä_

" n#

œ 10

20 2

, the nth term of a convergent p-series:

5n$

3n 2) Šn#
5‹ 

Š n"# ‹

nÄ_

œ n lim Ä_

5n$
3n n$
2n#  5n
10

15n#
3 3n#
4n  5

œ n lim Ä_ n

œ n lim Ä_

30n 6n
4

œ5

n

n

n ‰ 25. converges by the Direct Comparison Test; ˆ 3n n 1 ‰  ˆ 3n œ ˆ "3 ‰ , the nth term of a convergent geometric series

26. converges by the Limit Comparison Test (part 1) with "

Š $Î# ‹ n

lim nÄ_ Š " È$ n

2

$

‹

É n n$ 2 œ lim É1  œ n lim Ä_ nÄ_

" n$Î#

, the nth term of a convergent p-series:

œ1

2 n$

27. diverges by the Direct Comparison Test; n  ln n Ê ln n  ln ln n Ê

" n

_

28. converges by the Limit Comparison Test (part 2) when compared with ! n œ1 #

lim nÄ_

’ (lnn$n) “ Š n"# ‹

œ n lim Ä_

(ln n)# n

œ n lim Ä_

2(ln n) Š n" ‹ 1

œ 2 n lim Ä_

29. diverges by the Limit Comparison Test (part 3) with lim

nÄ_

’È

1 “ n ln n ˆ n" ‰

œ n lim Ä_

Èn ln n

" n

Š 2È n ‹ ˆ n" ‰

" n#

" ln n



" ln (ln n)

_

and ! n œ3

" n

, a convergent p-series:

œ0

, the nth term of the divergent harmonic series:

"

œ n lim Ä_

ln n n



œ n lim Ä_

Èn 2

œ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

diverges

Section 10.4 Comparison Tests " n&Î%

30. converges by the Limit Comparison Test (part 2) with lim

n)# ’ (ln$Î# “ n

nÄ_ Š

" ‹ n&Î%

(ln n)# n"Î%

œ n lim Ä_

œ n lim Ä_

ˆ 2 lnn n ‰

" n

31. diverges by the Limit Comparison Test (part 3) with lim

nÄ_

ˆ 1 "ln n ‰ ˆ "n ‰

œ n lim Ä_

n 1  ln n

32. diverges by the Integral Test:

" Š "n ‹

œ n lim Ä_

, the nth term of a convergent p-series:

œ 8 n lim Ä_

" Š $Î% ‹ 4n

ln n n"Î%

œ 8 n lim Ä_

ˆ n" ‰ Š

" ‹ 4n$Î%

œ 32 n lÄ im_

" n"Î%

œ 32 † 0 œ 0

, the nth term of the divergent harmonic series:

œ n lim nœ_ Ä_

'2_ lnx(x11) dx œ 'ln_3 u du œ

 " u# ‘ b œ lim ln 3

lim bÄ_ 2

"

bÄ_ #

ab#
ln# 3b œ _

" 33. converges by the Direct Comparison Test with n$Î# , the nth term of a convergent p-series: n#
1  n for " " n   2 Ê n# an#
1b  n$ Ê nÈn#
1  n$Î# Ê $Î#  or use Limit Comparison Test with nÈ n#
1

n

" n$Î# Èn n#  1

34. converges by the Direct Comparison Test with n#  1 Èn

Ê n#  1  Ènn$Î# Ê _

35. converges because ! n œ1 _

! nœ1

" n2n

"
n n2n

 n$Î# Ê

_

œ!

n œ1

" n2n

_


" #n

!

n œ1

593

, the nth term of a convergent p-series: n#  1  n# " n$Î#



_

n œ1

or use Limit Comparison Test with

" . n$Î#

which is the sum of two convergent series:

converges by the Direct Comparison Test since

36. converges by the Direct Comparison Test: !

1 n# .

" n #n



" #n

_

n  2n n# 2n

_

, and !

œ ! ˆ n2" n  nœ1

nœ1

"‰ n#


" 2n

and

is a convergent geometric series

" n2n



" n#

Ÿ

" #n



" n#

, the sum of

the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: 38. diverges; n lim Š3 Ä_

nc1

" 3n ‹

ˆ"  œ n lim Ä_ 3

" 3nc1  1 "‰ 3n

" 3

œ

" 3nc1



, which is the nth term of a convergent geometric series

Á0 _

n 39. converges by Limit Comparison Test: compare with ! ˆ 15 ‰ , which is a convergent geometric series with lrl œ n œ1

lim nÄ_

1 1 Š n2n b b 3n † 5n ‹

a 1 Î5 b n

œ n lim Ä_

n1 n2  3n

œ n lim Ä_

1 2n  3

_

n œ1

3 Š 23n b b 4n ‹ n

n

a 3 Î4 b n

œ n lim Ä_

8n  12n 9n  12n

œ n lim Ä_

8 ‰n ˆ 12 1 9 ‰n ˆ 12 1

œ

1 1

_

n œ1

œ

œ

2 n lim 2 aln 2b n Ä _ 2n aln 2b2

1 5

 1,

œ 1  0.

41. diverges by Limit Comparison Test: compare with ! n lim 2 ln 2
1 n Ä _ 2n ln 2

 1,

œ 0.

n 40. converges by Limit Comparison Test: compare with ! ˆ 34 ‰ , which is a convergent geometric series with lrl œ

lim nÄ_

1 5

Š 2n 2cnn ‹ n

1 n,

which is a divergent p-series, n lim Ä_

†

1 În

2
n œ n lim Ä _ 2n n

œ 1  0. _

_

n œ1

n œ1

42. diverges by the definition of an infinite series: ! lnˆ n n 1 ‰ œ ! ln n
ln an  1b‘, sk œ aln 1
ln 2b  aln 2
ln 3b  Þ Þ Þ  alnak
1b
ln kb  aln k
ln ak  1bb œ
ln ak  1b Ê lim sk œ
_ kÄ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

594

Chapter 10 Infinite Sequences and Series _

43. converges by Comparison Test with ! n œ2

_

which converges since !

1 nan
1b

sk œ ˆ1
12 ‰  ˆ 12
13 ‰  Þ Þ Þ  ˆ k
1 2


n œ2

1 ‰ k
1

 ˆ k
1 1
k1 ‰ œ 1


Ê nan
1ban
2b!   nan
1b Ê n!   nan
1b Ê

1 n!

_

1 n3 ,

n œ1

œ

œ

n2 lim n Ä _ n2 3n  2

œ n lim Ä_

45. diverges by the Limit Comparison Test (part 1) with "‰ n

ˆsin lim n Ä _ ˆ "n ‰

œ lim

xÄ0

sin x x

nœ2

Ê lim sk œ 1; for n   2, an
2b!   1

1 k

kÄ_

1 nan
1b

an

which is a convergent p-series, n lim Ä_

2n 2n 3

œ n lim Ä_

c 1bx 2bx

1 În 3

œ10

2 2

" n

, the nth term of the divergent harmonic series:

" n

, the nth term of the divergent harmonic series:

œ1

46. diverges by the Limit Comparison Test (part 1) with ˆtan "n ‰ lim n Ä _ ˆ "n ‰

Ÿ

_

œ ! ’ n
1 1
n1 “, and

an

44. converges by Limit Comparison Test: compare with ! n 3 a n
1 bx lim n Ä _ an  2ban  1bnan
1bx

1 nan
1b

œ n lim Š " ‹ Ä _ cos " n

ˆsin n" ‰ ˆ "n ‰

œ lim ˆ cos" x ‰ ˆ sinx x ‰ œ 1 † 1 œ 1 xÄ0

tanc" n n1.1

47. converges by the Direct Comparison Test:



_

1 #

n1.1

1

and ! n œ1

1 #

œ

#

n1.1

_

! nœ1

" n1.1

is the product of a

convergent p-series and a nonzero constant 48. converges by the Direct Comparison Test: sec
" n 

1 #

Ê

secc" n n1 3 Þ



ˆ 1# ‰ n1 3 Þ

_

and ! n œ1

ˆ 1# ‰ n1 3 Þ

œ

1 #

_

! n œ1

" n1 3 Þ

is the

product of a convergent p-series and a nonzero constant

49. converges by the Limit Comparison Test (part 1) with œ n lim Ä_

"  ec2n 1
ec2n

"
ec2n 1  ec2n

: n lim Ä_

" n#

: n lim Ä_

52. converges by the Limit Comparison Test (part 1) with " 123án

lim nÄ_ 54.

œ

Š nan 2b 1b ‹ Š n"# ‹

" 1  2#  3#  á  n#

Š n"# ‹

œ n lim coth n œ n lim Ä_ Ä_

en  ecn en
ecn

n Š tanh ‹ n#

Š n"# ‹

œ n lim tanh n œ n lim Ä_ Ä_

en
e en  e

n n

œ1

51. diverges by the Limit Comparison Test (part 1) with 1n : n lim Ä_

53.

n Š coth ‹ n#

œ1

50. converges by the Limit Comparison Test (part 1) with œ n lim Ä_

" n#

" ˆ n(n #

1) ‰

œ

œ n lim Ä_ œ

"

2 n(n  1) .

2n# n#  n

n(n b 1)(2n b 1) 6

œ

1 Š nÈ n n‹

ˆ 1n ‰

" n# : n lim Ä_

Š

œ n lim Ä_

Èn n ‹ n#

Š n"# ‹

1 n n È

œ 1.

n È œ n lim nœ1 Ä_

The series converges by the Limit Comparison Test (part 1) with

œ n lim Ä_

4n 2n  1

6 n(n  1)(2n  1)

Ÿ

œ n lim Ä_ 6 n$

4 2

" n# :

œ 2.

Ê the series converges by the Direct Comparison Test

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.4 Comparison Tests

595

an 55. (a) If n lim œ 0, then there exists an integer N such that for all n  N, ¹ bann
0¹  1 Ê
1  bann  1 Ä _ bn Ê an  bn . Thus, if ! bn converges, then ! an converges by the Direct Comparison Test. an (b) If n lim œ _, then there exists an integer N such that for all n  N, bann  1 Ê an  bn . Thus, if Ä _ bn ! bn diverges, then ! an diverges by the Direct Comparison Test. _

56. Yes, ! n œ1

an n

converges by the Direct Comparison Test because

an n

 an

an 57. n lim œ _ Ê there exists an integer N such that for all n  N, Ä _ bn ! then bn converges by the Direct Comparison Test

an bn

 1 Ê an  bn . If ! an converges,

58. ! an converges Ê n lim a œ 0 Ê there exists an integer N such that for all n  N, 0 Ÿ an  1 Ê an#  an Ä_ n Ê ! a#n converges by the Direct Comparison Test 59. Since an  0 and n lim a œ _ Á 0, by nth term test for divergence, ! an diverges. Ä_ n 60. Since an  0 and n lim a n2 † an b œ 0, compare !an with ! n"# , which is a convergent p-series; n lim Ä_ Ä_

an 1 În 2

œ n lim a n2 † an b œ 0 Ê !an converges by Limit Comparison Test Ä_ _

61. Let
_  q  _ and p  1. If q œ 0, then !

n œ2

_

! n œ2

1 nr

where 1  r  p, then n lim Ä_

œ n lim Ä_

1 aln nbcq npcr

qc1 lim qaln nb n Ä _ ap
rbnpcr

aln nbq np 1 În r

œ 0. If q  0, n lim Ä_

œ n lim Ä_ qc2

q ap
rbnpcr aln nb1cq

aln nbq np

œ n lim Ä_ aln nbq npcr

_

œ!

nœ2

aln nbq npcr ,

œ n lim Ä_

1 np ,

which is a convergent p-series. If q Á 0, compare with

and p
r  0. If q  0 Ê
q  0 and n lim Ä_

qaln nbqc1 ˆ 1n ‰ ap
rbnpcrc1

œ n lim Ä_

qaln nbqc1 ap
rbnpcr .

aln nbq npcr

If q
1 Ÿ 0 Ê 1
q   0 and

œ 0, otherwise, we apply L'Hopital's Rule again. n lim Ä_ qc2

qaq
1baln nbqc2 ˆ 1n ‰ ap
rb2 npcrc1

qaq
1baln nb qaq
1baln nb q aq
1 b œ n lim . If q
2 Ÿ 0 Ê 2
q   0 and n lim œ n lim œ 0; otherwise, we Ä _ ap
rb2 npcr Ä _ ap
rb2 npcr Ä _ ap
rb2 npcr aln nb2cq apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that q
k Ÿ 0 Ê k
q   0. Thus, after k qaq
1bâaq
k  1baln nbqck q a q
1 bâa q
k  1 b œ n lim Ä _ ap
rbk npcr aln nbkcq ap
rbk npcr _ q series ! alnnnpb converges. n œ1

applications of L'Hopital's Rule we obtain n lim Ä_ 0 in every case, by Limit Comparison Test, the

_

62. Let
_  q  _ and p Ÿ 1. If q œ 0, then !

n œ2

_

! nœ2

1 np ,

aln nbq

np

which is a divergent p-series. Then n lim Ä_

where 0  p  r Ÿ 1. n lim Ä_ lim

aln nbq np

ar
pbn

rcpc1

n Ä _ a
qbaln nbcqc1 ˆ 1n ‰

œ n lim Ä_

aln nbq

np

1 În r

œ

q lim aln nb n Ä _ npcr rcp

ar
pbn . a
qbaln nbcqc1

1 În p

œ

_

œ!

nœ2

1 np ,

which is a divergent p-series. If q  0, compare with _

œ n lim aln nbq œ _. If q  0 Ê
q  0, compare with ! Ä_

nœ2

nrcp lim cq n Ä _ aln nb

otherwise, we apply L'Hopital's Rule again to obtain n lim Ä_

1 nr ,

since r
p  0. Apply L'Hopital's to obtain

If
q
1 Ÿ 0 Ê q  1   0 and n lim Ä_

a r
pb2 nrcp a
qba
q
1baln nbcqc2

œ 0. Since the limit is

2 rcpc1

a r
pb n a
qba
q
1baln nbcqc2 ˆ 1n ‰ 2 rcp

ar
pbnrcp aln nbqb1 a
qb

œ n lim Ä_

qb2

œ _,

a r
pb2 nrcp . a
qba
q
1baln nbcqc2

If

a r
pb n aln nb œ n lim œ _, otherwise, we a
qba
q
1b Ä_ apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that
q
k Ÿ 0 Ê q  k   0. Thus, after


q
2 Ÿ 0 Ê q  2   0 and n lim Ä_

k applications of L'Hopital's Rule we obtain n lim Ä_

a r
pbk nrcp a
qba
q
1bâa
q
k  1baln nbcqck

œ n lim Ä_

a r
pbk nrcp aln nbqbk a
qba
q
1bâa
q
k  1b

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

œ _.

596

Chapter 10 Infinite Sequences and Series _

q

Since the limit is _ if q  0 or if q  0 and p  1, by Limit comparison test, the series ! alnnpncbr diverges. Finally if q  0 _

q

and p œ 1 then !

aln nb np

Ê aln nbq   1 Ê

aln nbq n

nœ2

_

_

œ !

nœ2

aln nb n

q

_

. Compare with ! nœ2 _

  1n . Thus !

n œ2

aln nbq n

nœ1

1 n,

which is a divergent p-series. For n   3, ln n   1

diverges by Comparison Test. Thus, if
_  q  _ and p Ÿ 1,

q

the series ! alnnpncbr diverges. nœ1

63. Converges by Exercise 61 with q œ 3 and p œ 4. 64. Diverges by Exercise 62 with q œ

1 2

and p œ 12 .

65. Converges by Exercise 61 with q œ 1000 and p œ 1.001. 66. Diverges by Exercise 62 with q œ

1 5

and p œ 0.99.

67. Converges by Exercise 61 with q œ
3 and p œ 1.1. 68. Diverges by Exercise 62 with q œ
12 and p œ 12 . 69. Example CAS commands: Maple: a := n -> 1./n^3/sin(n)^2; s := k -> sum( a(n), n=1..k ); # (a)] limit( s(k), k=infinity ); pts := [seq( [k,s(k)], k=1..100 )]: # (b) plot( pts, style=point, title="#69(b) (Section 10.4)" ); pts := [seq( [k,s(k)], k=1..200 )]: # (c) plot( pts, style=point, title="#69(c) (Section 10.4)" ); pts := [seq( [k,s(k)], k=1..400 )]: # (d) plot( pts, style=point, title="#69(d) (Section 10.4)" ); evalf( 355/113 ); Mathematica: Clear[a, n, s, k, p] a[n_]:= 1 / ( n3 Sin[n]2 ) s[k_]= Sum[ a[n], {n, 1, k}] points[p_]:= Table[{k, N[s[k]]}, {k, 1, p}] points[100] ListPlot[points[100]] points[200] ListPlot[points[200] points[400] ListPlot[points[400], PlotRange Ä All] To investigate what is happening around k = 355, you could do the following. N[355/113] N[1
355/113] Sin[355]//N a[355]//N N[s[354]]

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.5 The Ratio and Root Tests

597

N[s[355]] N[s[356]] _

_

70. (a) Let S œ ! n12 , which is a convergent p-series. By Example 5 in Section 10.2, ! nan 1 1b converges to 1. By Theorem 8, n œ1

_

Sœ!

n œ1 _

1 n2

n œ1

_

œ!

_

!

1 nan  1b

n œ1

_


!

1 n2

n œ1

_

œ!

1 nan  1b

nœ1

nœ1

1 n an  1 b

_

!

nœ1

Š n12




1 nan  1b ‹

also converges.

_

(b) Since ! nan 1 1b converges to 1 (from Example 5 in Section 10.2), S œ 1  ! Š n12
n œ1

n œ1

_

(c) The new series is comparible to

! 13 , n

n œ1

1 nan  1b ‹

_

œ 1  ! n2 an1 1b n œ1

_

so it will converge faster because its terms Ä 0 faster than the terms of ! n12 . n œ1

1000

1000

(d) The series 1  ! n2 an1 1b gives a better approximation. Using Mathematica, 1  ! n2 an1 1b œ 1.644933568, while nœ1

1000000

!

n œ1

1 n2

nœ1

œ 1.644933067. Note that

1 6

œ 1.644934067. The error is 4.99 ‚ 10
7 compared with 1 ‚ 10
6 .

2

10.5 THE RATIO AND ROOT TESTS

1.

2.

3.

2n n!

 0 for all n   1; lim Œ nÄ_

n2 3n

2nb" an

"b !

2n n!

 0 for all n   1; lim Œ

an

nÄ_

an
1 b! an  1b2

2 †2 lim Š an" b†n! † n

œ

nÄ_

b1b b 2 3nb1 nb2  n 3

3 lim ˆ n3 n †3 †

œ

nÄ_

b1bc1b! b1bb1b2  anc1b! anb1b2

aan

 0 for all n   1; lim Œ nÄ_

aan

_

n! 2n ‹

œ

_

n

œ lim ˆ n 2 " ‰ œ 0  1 Ê ! 2n! converges nÄ_

3n ‰ n2

n3 ‰ ˆ1‰ œ lim ˆ 3n  6 œ lim 3 œ nÄ_

lim Š na†nan
21b2b! †

nÄ_

nœ1

nÄ_

an  "b2 an
1b! ‹

_

 1 Ê ! n 3n 2 converges

1 3

nœ1

n 3n  4n  1 œ lim Š nn22n 4n  4 ‹ œ lim Š 2n  4 ‹ 3

2

2

nÄ_

nÄ_


1b! œ lim ˆ 6n 2 4 ‰ œ _  1 Ê ! aann  diverges 1 b2 nÄ_

4.

2nb1 n†3n 1

n œ1

 0 for all n   1; lim
nÄ_

_

an

2an
1b
1
1b†3an
1b 2n
1 n†3n 1

1

nb1

lim Š an21b†3†n2 1 †3 †

œ

nÄ_

n †3 n 1 2n
1 ‹

œ lim ˆ 3n2n 3 ‰ œ lim ˆ 23 ‰ œ nÄ_

nÄ_

2 3

1

nb1

Ê ! n2†3nc1 converges nœ1

5.

n4 4n

 0 for all n   1; lim Œ nÄ_

œ lim ˆ 14  nÄ_

6.

3nb2 ln n

1 n



3 2n2



1 n3

b1b4 4nb1 n4 4n

an



1 ‰ 4n4

 0 for all n   2; lim Œ nÄ_ _

œ œ

3anb1bb2 ln anb1b 3nb2 ln n

1 4

4

lim Š an4n †14b †

4n n4 ‹

nÄ_

_

œ lim Š n

4

nÄ_

 4n3  6n2  4n  1 ‹ 4n4

4

 1 Ê ! n4n converges

œ

n œ1

nb2

lim Š ln3an †31b †

nÄ_

ln n 3nb2 ‹

œ lim Š ln 3anlnn1b ‹ œ lim Š nÄ_

nÄ_

3 n 1

nb1

‹ œ lim ˆ 3n n 3 ‰ nÄ_

nb2

œ lim ˆ 31 ‰ œ 3  1 Ê ! 3ln n diverges nÄ_

7.

n 2 a n  2 b! nx32n

n œ2

an

 0 for all n   1; lim
nÄ_

b 1b2aan b 1b b 2b! b 1bx32an 1b

an

n2 an 2b! nx32n

œ

7 ˆ 6n  15 ‰ ˆ6‰ œ lim Š 3n27n2 15n  18n ‹ œ lim 54n  18 œ lim 54 œ 2

nÄ_

nÄ_

nÄ_

2

3ban  2b! lim Š an an1ba1nb † †nx32n †32

nÄ_ 1 9

_

nx32n n 2 an  2 b! ‹

œ lim Š n nÄ_

 1 Ê ! n annx32n2b! converges 2

n œ1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

3

 5n2  7n  3 ‹ 9n3  9n2

598 8.

Chapter 10 Infinite Sequences and Series n †5 n a2n  3b lnan  1b

 0 for all n   1;

1b†a2n  3b lim Š 5an  † na2n  5b

œ

nÄ_

lim Œ

a2an

nÄ_

lnan  1b lnan  2b ‹

an b 1b†5n 1 1b 3b lnaan 1b n†5n a2n 3b lnan 1b

n  1b†5 †5 lim Š a2na 5b lnan  2b † n

œ

1b

nÄ_

a2n  3b lnan  1b ‹ n †5 n

lnan  1b  15  25 ‰ lim Š 10n 2n2 25n † lim Š ‹ † lim Š lnan  2b ‹ œ lim ˆ 20n  5n 4n  5 2

œ

nÄ_

nÄ_ _

nÄ_

nÄ_

1 nb1 1 nb2

‹

n †5 ! ‰ ˆ n2‰ ˆ 1‰ œ lim ˆ 20 4 † lim n  1 œ 5 † lim 1 œ 5 † 1 œ 5  1 Ê a2n  3b lnan  1b diverges nÄ_

9.

10.

7 a2n  5bn

4n a3n bn

nÄ_

nÄ_

n œ2

7 n   0 for all n   1; lim É œ a2n  5bn nÄ_

nÄ_

nÄ_

3‰ lim ˆ 4n 3n
5 œ

nÄ_

n 1

nÄ_

n 1

lim ˆ 43 ‰ œ

nÄ_

n   0 for all n   1; lim Ê’lnˆe2  1n ‰“

Ê ! ’lnˆe2  1n ‰“

n

n œ1

n  3 ‰n  3 ‰n ˆ 4n 11. ˆ 4n   0 for all n   2; lim É œ 3n
5 3n
5

_

n œ1

4 ‰ lim ˆ 3n œ 0  1 Ê ! a3n4 bn converges

n

n 1

_

n È

lim Š 2n 7 5 ‹ œ 0  1 Ê ! a2n 7 5bn converges

nÄ_

_

4 n   0 for all n   1; lim É œ a3n bn

12. ’lnˆe2  1n ‰“

n

œ

4 3

nÄ_

_

n

3‰  1 Ê ! ˆ 4n diverges 3n
5 n œ1

1 1 În

lim ’lnˆe2  1n ‰“

nÄ_

œ lnae2 b œ 2  1

diverges

n œ1

13.

8 ˆ3  1n ‰2n

8 n   0 for all n   1; lim É œ ˆ3  1 ‰2n nÄ_

lim Œ ˆ

nÄ_

n

n

n È 8

3  1n ‰

16.

"

n1bn

17. converges by the Ratio Test:

converges

nÄ_

n

n œ1

nÄ_

2

nœ1

n È

nÄ_

”

œ n lim Ä_

nÄ_

È

(n b 1) 2 2 n b1 •

”

_

n È

1 ! 1"bn converges lim Š n È n n‹ œ 0  1 Ê n

lim Š n1În 1 1 ‹ œ

lim anb1 n Ä _ an

8 1 ‰2n n œ1 3  n

n n lim ˆ1
n1 ‰ œ e
1  1 Ê ! ˆ1
n1 ‰ converges

nÄ_

nÄ_

_

 1Ê !ˆ

_

2

n n lim Ɉ1
n1 ‰ œ

" n   0 for all n   2; lim É n1bn œ

1 9

lim sinŠ È1n ‹ œ sina0b œ 0  1 Ê ! ’sinŠ È1n ‹“ converges

nÄ_

2

œ

_

n

n 14. ’sinŠ È1n ‹“   0 for all n   1; lim Ê ’sinŠ È1n ‹“ œ

n 15. ˆ1
n1 ‰   0 for all n   1;

2

È n 2 #n

•

Š (nenbb1)1 ‹

n œ2

È È n (n  1) 2 ˆ1  n" ‰ 2 ˆ "# ‰ œ œ n lim † 2È2 œ n lim Ä _ #nb1 Ä_ n

" #

1

2

18. converges by the Ratio Test:

lim anb1 n Ä _ an

œ n lim Ä_

19. diverges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

20. diverges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

# Š nen ‹

Š (nenbb1)! 1 ‹ ˆ en!n ‰

b 1)! ‹ Š (n 10nb1 ˆ 10n!n ‰

œ n lim Ä_

21. converges by the Ratio Test:

œ n lim Ä_

(n  ")! enb1

†

en n!

œ n lim Ä_

(n  ")! 10nb1

†

10n n!

Š (n10bn1)1 ‹

"!

Š n10n ‹

†

œ n lim Ä_

"!

lim anb1 n Ä _ an

(n  1)2 enb1

œ n lim Ä_

(n  ")"! 10n 1

†

en lim n2 œ n Ä _

œ n lim Ä_

œ n lim Ä_

10n n"!

ˆ1  n" ‰# ˆ "e ‰ œ

n" e

n 10

" e

1

œ_

œ_

ˆ1  n" ‰"! ˆ 1"0 ‰ œ œ n lim Ä_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" 10

1

Section 10.5 The Ratio and Root Tests ˆ n
n 2 ‰n œ lim ˆ1  22. diverges; n lim a œ n lim Ä_ n Ä_ nÄ_ 23. converges by the Direct Comparison Test:

2(
1)n (1.25)n


2 ‰ n n

599

œ e
# Á 0 n

n

œ ˆ 45 ‰ c2  (
1)n d Ÿ ˆ 45 ‰ (3) which is the nth term of a convergent

geometric series 24. converges; a geometric series with krk œ ¸
23 ¸  1 ˆ1
3n ‰n œ lim ˆ1  25. diverges; n lim a œ n lim Ä_ n Ä_ nÄ_ ˆ1
26. diverges; n lim a œ n lim Ä_ n Ä_

" ‰n 3n


3 ‰ n n

œ n lim 1 Ä_


27. converges by the Direct Comparison Test:

ln n n$



n n$

œ

œ e
$ ¸ 0.05 Á 0

Š
"3 ‹ n

" n#

n
"Î$ ¸ 0.72 Á 0  œe

for n   2, the nth term of a convergent p-series. n

n (ln n) n È É 28. converges by the nth-Root Test: n lim an œ n lim nn œ n lim Ä_ Ä_ Ä_

29. diverges by the Direct Comparison Test: with "n .

" n




" n#

œ

n
1 n#

ln n n



" n

" ‰n n#

ˆˆ n"
œ n lim Ä_

anb1 an

œ n lim Ä_

(n  1) ln (n  1) #nb1

†

33. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

(n  2)(n  3) (n  1)!

†

n! (n  1)(n  2)

34. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

(n  1)$ en 1

œ

35. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

(n  4)! 3! (n  1)! 3nb1

anb1 an

œ n lim Ä_

†

anb1 an

œ n lim Ä_

(n  1)! (2n  3)!

38. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

(n  1)! (n  1)nb1

"

ˆ1  "n ‰n

œ

" e

en n$

†

(n  1)2nb1 (n  2)! 3nb1 (n  1)!

37. converges by the Ratio Test: n lim Ä_

œ n lim Ä_

ln n n

œ n lim Ä_

Š "n ‹ 1

œ01

" ‰n ‰1În n#

ˆ"
œ n lim Ä_ n

"‰ n#

for n   3

32. converges by the Ratio Test: n lim Ä_

36. converges by the Ratio Test: n lim Ä_

œ n lim Ä_

 "# ˆ "n ‰ for n  2 or by the Limit Comparison Test (part 1)

n n ˆ n"
È É 30. converges by the nth-Root Test: n lim an œ n lim Ä_ Ä_

31. diverges by the Direct Comparison Test:

a(ln n)n b1În ann b1În

2n n ln (n)

" e

†

nn n!

1

œ01

œ n lim Ä_

3n n! n2n (n  1)!

(2n  1)! n!

†

" #

1

3! n! 3n (n  3)!

†

œ

n4 3(n  1)

" 3

œ

1

2‰ ˆ n n 1 ‰ ˆ 32 ‰ ˆ nn  œ n lim 1 œ Ä_

œ n lim Ä_

n" (2n  3)(2n  2)

œ01

ˆ n ‰n œ lim œ n lim Ä _ n1 nÄ_

"

ˆ n bn " ‰n

1

n n n È 39. converges by the Root Test: n lim an œ n lim œ n lim Ä_ Ä _ É (ln n)n Ä_

n n È ln n

œ n lim Ä_

" ln n

œ01

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

2 3

1

œ01

600

Chapter 10 Infinite Sequences and Series n n È Èln n

n n n È 40. converges by the Root Test: n lim an œ n lim œ n lim Ä_ Ä _ É (ln n)nÎ2 Ä_

Ä_ Ä_

n n lim È n Èln n lim n

œ

œ01

n È n œ 1‹ Šn lim Ä_

41. converges by the Direct Comparison Test:

œ

n! ln n n(n  2)!

ln n n(n  1)(n  2)



" (n  1)(n  #)

œ

n n(n  1)(n  2)



" n#

which is the nth-term of a convergent p-series an 1 an

42. diverges by the Ratio Test: n lim Ä_

œ n lim Ä_

3n 1 (n  1)$ 2n

1

†

n$ 2n 3n

†

a2nbx
nx‘2

43. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_


an1bx‘2
2(n  1)‘x

44. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

a2n  5bˆ2nb1  3‰ 3nb1  2

œ n lim ’ 2n  5 “ † n lim ’ 2 † 6  4 † 2  3† 3  6 “ œ 1 † Ä _ 2n  3 Ä _ 3 † 6 n  9 † 3 n  2† 2 n  6 n

n

n

45. converges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

2 3

œ

ˆ 1 b nsin n ‰ an an

"n

Š 1 b tan n

47. diverges by the Ratio Test: n lim Ä_

anb1 an

œ n lim Ä_

œ n lim Ä_ †

ˆ #3 ‰ œ

an1b2 (2n  2)(2n  1)

3n  2 a2n  3ba2n  3b

3 #

1

œ n lim Ä_

œ n lim ’ 2n  5 † Ä _ 2n  3

n2  2n  1 4n2  6n  2

œ

œ n lim Ä_

3n
1 2n  5

œ n lim Ä_

"  tan " n n

œ

3 #

œ 0 since the numerator

1


2 ‰ 48. diverges; an1 œ n n 1 an Ê an1 œ ˆ n n 1 ‰ ˆ n
n 1 an
1 ‰ Ê an1 œ ˆ n n 1 ‰ ˆ n
n 1 ‰ ˆ nn
1 an
2 a

" n n 1 n 2 3 " Ê an1 œ ˆ n  1 ‰ ˆ n ‰ ˆ n
1 ‰ â ˆ # ‰ a" Ê an1 œ n  1 Ê an1 œ n  1 , which is a constant times the

general term of the diverging harmonic series

49. converges by the Ratio Test: n lim Ä_

50. converges by the Ratio Test:

n  ln n n  10

 0 and a" œ

Ê an1 œ

n  ln n n  10

" #

œ n lim Ä_

lim anb1 n Ä _ an

œ n lim Ä_

anb1 an

œ n lim Ä_

51. converges by the Ratio Test: n lim Ä_ 52.

anb1 an

Š 2n ‹ an an

Œ

Èn n #

œ n lim Ä_

 an

an

œ n lim Ä_

Š 1 bnln n ‹ an an

2 n

œ01 n n È

œ n lim Ä_

n

œ

"ln n n

" #

1

œ n lim Ä_

Ê an  0; ln n  10 for n  e"! Ê n  ln n  n  10 Ê

an  an ; thus an1  an  

53. diverges by the nth-Term Test: a" œ

" 3

" #

" n

œ01

n  ln n n  10

1

Ê n lim a Á 0, so the series diverges by the nth-Term Test Ä_ n

3 3 6 " %! " 2 " 2 " 2 " É É É É , a# œ É 3 , a$ œ Ê 3 œ 3 , a% œ ËÊ 3 œ 3 ,á ,

%

n! " n! " n " É an œ É a œ 1 because šÉ 3 Ê n lim 3 › is a subsequence of š 3 › whose limit is 1 by Table 8.1 Ä_ n

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 4

1

2†6n  4†2n  3†3n  6 3 †6 n  9 † 3 n  2 † 2 n  6 “

œ01

‹ an

an

c1‰ ˆ 3n 2n b 5 an an

n3 (n  1)3

1

2 3

anb1 46. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ 1 approaches 1  # while the denominator tends to _

œ n lim Ä_

Section 10.5 The Ratio and Root Tests 54. converges by the Direct Comparison Test: a" œ n!

" #

# $

#

' %

'

#%

, a# œ ˆ "# ‰ , a$ œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , a% œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , á

n

Ê an œ ˆ "# ‰  ˆ "# ‰ which is the nth-term of a convergent geometric series anb1 an

55. converges by the Ratio Test: n lim Ä_ n" " œ n lim œ  1 # Ä _ 2n  1

2nb1 (n  1)! (n  1)! (2n  2)!

œ n lim Ä_

†

(2n)! 2n n! n!

2(n  1)(n  1) (2n  #)(2n  1)

œ n lim Ä_

(3n  3)! 1)! (n  2)! anb1 56. diverges by the Ratio Test: n lim œ n lim † n! (n (3n)! Ä _ an Ä _ (n  1)! (n  2)! (n  3)! (3n  3)(3  2)(3n  1)  2 ‰ ˆ 3n  1 ‰ œ n lim œ n lim 3 ˆ 3n n# n  3 œ 3 † 3 † 3 œ 27  1 Ä _ (n  1)(n  2)(n  3) Ä_ n

n (n!) n È 57. diverges by the Root Test: n lim an ´ n lim œ n lim Ä_ Ä _ É an n b# Ä_

n

œ_1

n! n#

n

n (n!) n (n!) É 58. converges by the Root Test: n lim œ n lim œ n lim É ann bn Ä_ Ä_ Ä_ nn# " Ÿ n lim œ01 Ä_ n

n! nn

ˆ " ‰ ˆ 2n ‰ ˆ 3n ‰ â ˆ n
n 1 ‰ ˆ nn ‰ œ n lim Ä_ n

" #n ln 2

n n n È 59. converges by the Root Test: n lim an œ n lim œ n lim Ä_ Ä _ É 2 n# Ä_

n #n

œ n lim Ä_

n n n È 60. diverges by the Root Test: n lim an œ n lim œ n lim Ä_ Ä _ É a#n b# Ä_

n 4

œ_1

n

n

anb1 an

61. converges by the Ratio Test: n lim Ä_

1†3 â (2n
1) (2†4 â #n) a3n  1b

62. converges by the Ratio Test: an œ Ê n lim Ä_

(2n  2)! c2nb1 (n  1)!d# a3nb1  1b #

œ n lim Š 4n  6n  2 ‹ Ä _ 4n#  8n  4 63. Ratio: n lim Ä_

anb1 an

†

a1  3cn b a3  3cn b

œ n lim Ä_

œ n lim Ä_

a2n n!b# a3n  1b (2n)!

œ1†

" (n  1)p

†

" 3

np 1

œ

" 3

anb1 an

œ n lim Ä_

" (ln (n  1))p

†

†

1†2†3†4 â (2n
1)(2n) (2†4 â 2n)# a3n  1b

œ

œ n lim Ä_

4n 2n n! 1†3† â †(2n
1)

œ

œ n lim Ä_

2n  " (4†#)(n  1)

(2n)! a2n n!b# a3n  1b

(2n  ")(2n  2) a3n  1b 2# (n  1)# a3n 1  1b

1

(ln n)p 1

" n n ‰p ˆÈ

" (1)p

œ

œ ’n lim Ä_

œ 1 Ê no conclusion

ln n ln (n  1) “

p

œ ”n lim Ä_

ˆ "n ‰

p

ˆ n b 1 ‰ • œ Šn lim Ä_ "

n" n ‹

œ (1)p œ 1 Ê no conclusion " n n È Root: n lim an œ n lim É (ln n)p œ Ä_ Ä_

"

p

lim (ln n)1În ‹ ŠnÄ_ ˆ

"

‰

; let f(n) œ (ln n)1În , then ln f(n) œ

ln (ln n) n ln n Ê n lim ln f(n) œ n lim œ n lim œ n lim n 1 Ä_ Ä_ Ä_ Ä_ " ln fÐnÑ ! n È an œ œ n lim e œ e œ 1; therefore lim Ä_ nÄ_

" n ln n p

lim (ln n)1În ‹ ŠnÄ_

65. an Ÿ

n 2n

_

_

for every n and the series ! nœ1

n #n

œ

ˆ n ‰p œ 1p œ 1 Ê no conclusion œ n lim Ä_ n1

n " n È É Root: n lim an œ n lim np œ n lim Ä_ Ä_ Ä_

64. Ratio: n lim Ä_

1†3† â †(2n
1)(2n  1) 4nb1 2nb1 (n  1)!

œ01

ln (ln n) n

œ 0 Ê n lim (ln n)1În Ä_ œ (1)" p œ 1 Ê no conclusion

converges by the Ratio Test since n lim Ä_

(n  ") 2nb1

†

2n n

œ

" #

Ê ! an converges by the Direct Comparison Test nœ1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1

p

" 4

1

601

602

Chapter 10 Infinite Sequences and Series 2

66.

2n n!

 0 for all n   1; lim
nÄ_ _

2 2anb1b anb1b! 2 2n n!

œ

n2 b2nb1

lim Š a2n1b†n! †

nÄ_

n! ‹ 2n2

2nb1

†4 ‰ ˆ 2†4 1ln 4 ‰ œ lim Š 2n1 ‹ œ lim ˆ n2 1 œ lim n

nÄ_

n

nÄ_

nÄ_

n2

œ _  1 Ê ! 2n! diverges n œ1

10.6 ALTERNATING SERIES, ABSOLUTE AND CONDITIONAL CONVERGENCE 1.

converges by the Alternating Convergence Test since: un œ Ê

1 Èn1

Ÿ

1 Èn

Ê un1 Ÿ un ;

lim un œ

nÄ_

lim 1 nÄ_ Èn

1 Èn

 0 for all n   1; n   1 Ê n  1   n Ê Èn  1   Èn

œ 0. _

_

n œ1

n œ1

2. converges absolutely Ê converges by the Alternating Convergence Test since ! kan k œ !

" n$Î#

which is a

convergent p-series 3. converges Ê converges by Alternating Series Test since: un œ Ê an  1b3n1   n 3n Ê

1 an1b3nb1

Ÿ

1 n 3n

Ê un1 Ÿ un ;

1 n3n

 0 for all n   1; n   1 Ê n  1   n Ê 3n1   3n

lim un œ

4. converges Ê converges by Alternating Series Test since: un œ

œ 0.

lim 1 n nÄ_ n 3

nÄ_

4 aln nb2

 0 for all n   2; n   2 Ê n  1   n

Ÿ

1 aln nb2

Ê

5. converges Ê converges by Alternating Series Test since: un œ

n n2  1

 0 for all n   1; n   1 Ê 2n2  2n   n2  n  1

Ê ln an  1b   ln n Ê aln an  1bb2   aln nb2 Ê lim un œ

nÄ_

lim 4 2 nÄ_ aln nb

1 aln an1bb2

4 aln an1bb2

Ÿ

4 aln nb2

Ê un  1 Ÿ u n ;

œ 0.

Ê n3  2n2  2n   n3  n2  n  1 Ê nan2  2n  2b   n3  n2  n  1 Ê nŠan  1b2  1‹   an2  1ban  1b Ê

n n 2 1

 

n 1 an1b2 1

Ê un1 Ÿ un ;

lim un œ

nÄ_

lim 2 n nÄ_ n  1

œ 0.

6. diverges Ê diverges by nth Term Test for Divergence since:

2 lim n2  5 nÄ_ n  4

7. diverges Ê diverges by nth Term Test for Divergence since:

2n 2 nÄ_ n

lim

œ1Ê

œ_Ê

5 lim a
1bn1 nn2   4 œ does not exist 2

nÄ_

lim a
1bn1 2n2 œ does not exist n

nÄ_

_

_

n œ1

n œ1

8. converges absolutely Ê converges by the Absolute Convergence Test since ! kan k œ ! anb1 nÄ_ an

Ratio Test, since lim

œ

lim 10 nÄ_ n  2

10n a n  1 bx ,

which converges by the

œ01

9. diverges by the nth-Term Test since for n  10 Ê

n 10

_

n ‰n ˆ n ‰n Á 0 Ê ! (
1)n1 ˆ 10  1 Ê n lim diverges Ä _ 10 n œ1

10. converges by the Alternating Series Test because f(x) œ ln x is an increasing function of x Ê Ê un   un1 for n   1; also un   0 for n   1 and

" lim n Ä _ ln n

11. converges by the Alternating Series Test since f(x) œ

ln x x

is decreasing

œ0

Ê f w (x) œ

Ê un   un1 ; also un   0 for n   1 and n lim u œ n lim Ä_ n Ä_

" ln x

ln n n

1
ln x x#

œ n lim Ä_

 0 when x  e Ê f(x) is decreasing Š "n ‹ 1

œ0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.6 Alternating Series, Absolute and Conditional Convergence 12. converges by the Alternating Series Test since f(x) œ ln a1  x
" b Ê f w (x) œ


" x(x  1)

 0 for x  0 Ê f(x) is decreasing

ˆ1  n" ‰‹ œ ln 1 œ 0 Ê un   un1 ; also un   0 for n   1 and n lim u œ n lim ln ˆ1  "n ‰ œ ln Šn lim Ä_ n Ä_ Ä_ 13. converges by the Alternating Series Test since f(x) œ Ê un   unb1 ; also un   0 for n   1 and n lim u œ Ä_ n 3È n  1 Èn  1

14. diverges by the nth-Term Test since n lim Ä_

_

_

nœ1

nœ1

Èx  " x1

1
x
2È x 2Èx (x  1)#

Ê f w (x) œ

Èn  " lim n Ä _ n1

 0 Ê f(x) is decreasing

œ0

3É 1 

œ n lim Ä_

" n

"

1  Š Èn ‹

œ3Á0

" ‰n 15. converges absolutely since ! kan k œ ! ˆ 10 a convergent geometric series

16. converges absolutely by the Direct Comparison Test since ¹ (
1)

nb1

(0.1)n

n

¹œ

" (10)n n

n

" ‰  ˆ 10 which is the nth term

of a convergent geometric series 17. converges conditionally since

" Èn



" Èn  1

" Èn

 0 and n lim Ä_

_

_

n œ1

n œ1

œ 0 Ê convergence; but ! kan k œ !

" n"Î#

is a divergent p-series 18. converges conditionally since _

_

! kan k œ !

nœ1

nœ1

" 1  Èn

" 1  Èn



" 1  Èn  1

is a divergent series since _

_

n œ1

nœ1

19. converges absolutely since ! kan k œ !

n n $ 1

n! #n

20. diverges by the nth-Term Test since n lim Ä_ 21. converges conditionally since _

œ!

n œ1

" n3

diverges because

" n3 " n3



" (n  1)  3

 

" 4n

" 1  Èn

 0 and n lim Ä_

" 1 È n

and

 

" #È n

n n $ 1



_

and !

" n#

nœ1

" n"Î#

œ 0 Ê convergence; but is a divergent p-series

which is the nth-term of a converging p-series

œ_  0 and n lim Ä_

_

and ! n œ1

" n

" n 3

_

œ 0 Ê convergence; but ! kan k n œ1

is a divergent series

_

22. converges absolutely because the series ! ¸ sinn# n ¸ converges by the Direct Comparison Test since ¸ sinn# n ¸ Ÿ n œ1

3n 5n

23. diverges by the nth-Term Test since n lim Ä_

œ1Á0 nb1

24. converges absolutely by the Direct Comparison Test since ¹ (
n2)5n ¹ œ

2nb1 n 5 n

n

 2 ˆ 25 ‰ which is the nth term

of a convergent geometric series 25. converges conditionally since f(x) œ

" x#



" x

Ê f w (x) œ
ˆ x2$ 

"‰ x#

 0 Ê f(x) is decreasing and hence _

_

n œ1

n œ1

ˆ "  "n ‰ œ 0 Ê convergence; but ! kan k œ ! un  unb1  0 for n   1 and n lim Ä _ n# _

œ!

n œ1

" n#

_

!

nœ1

" n

603

1 n n#

is the sum of a convergent and divergent series, and hence diverges

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" n#

604

Chapter 10 Infinite Sequences and Series

26. diverges by the nth-Term Test since n lim a œ n lim 101În œ 1 Á 0 Ä_ n Ä_ 27. converges absolutely by the Ratio Test: n lim Š uunbn 1 ‹ œ n lim Ä_ Ä_ ” 28. converges conditionally since f(x) œ

'2_ x dxln x œ

lim

Š "x ‹

bÄ_

_

_

n œ1

n œ1

Ê ! kan k œ !

'2b
ln x  dx œ " n ln n

n 1

•œ

2 3

1

 1d Ê f w (x) œ
cln(x(x) ln x)#  0 Ê f(x) is decreasing

" x ln x

" n ln n

Ê un  unb1  0 for n   2 and n lim Ä_

(n")# ˆ 23 ‰ n n# ˆ 23 ‰

œ 0 Ê convergence; but by the Integral Test,

lim cln (ln x)d b2 œ lim cln (ln b)
ln (ln 2)d œ _

bÄ_

bÄ_

diverges

_

" xb#

29. converges absolutely by the Integral Test since '1 atan
" xb ˆ 1 " x# ‰ dx œ lim ’ atan # bÄ_

œ lim ’atan bÄ_


"

#


"

bb
atan

#

1b “ œ

30. converges conditionally since f(x) œ œ

1
Š lnxx ‹
ln

x  Š lnxx ‹

(x
ln x)#

œ n lim Ä_

Š "n ‹ 1
Š n" ‹

_

_

n œ1

nœ1

! kan k œ !

œ

" #

1
ln x (x
ln x)#

# # ’ˆ 1# ‰
ˆ 14 ‰ “ œ

ln x x
ln x

Ê f w (x) œ

1

31 # 32

Š "x ‹ (x
ln x)
(ln x) Š1
x" ‹ (x
ln x)#

 0 Ê un   un1  0 when n  e and n lim Ä_

œ 0 Ê convergence; but n
ln n  n Ê

ln n n
ln n

b

“

" n
ln n

" n



Ê

ln n n
ln n

ln n n
ln n



" n

so that

diverges by the Direct Comparison Test

31. diverges by the nth-Term Test since n lim Ä_ _

_

n œ1

nœ1

n n1

œ1Á0

n 32. converges absolutely since ! kan k œ ! ˆ "5 ‰ is a convergent geometric series

33. converges absolutely by the Ratio Test: n lim Š uunbn 1 ‹ œ n lim Ä_ Ä_

("00)nb1 (n1)!

_

_

n œ1

n œ1

34. converges absolutely by the Direct Comparison Test since ! kan k œ !

†

n! (100)n

œ n lim Ä_

" n#  2n  1

and

"00 n1

œ01

" n#  2n  1



" n#

nth-term of a convergent p-series _

_

n œ1

n œ1

_

35. converges absolutely since ! kan k œ ! ¹ (n
È1)n ¹ œ ! _

36. converges conditionally since ! n œ1 _

_

n œ1

n œ1

! kan k œ !

" n

cos n1 n

n

n œ1

_

œ!

nœ1

(
1)n n

" n$Î#

is a convergent p-series

is the convergent alternating harmonic series, but

diverges

 1) n È kan k œ n lim 37. converges absolutely by the Root Test: n lim Š (n(2n) n ‹ Ä_ Ä_ n

1 În

œ n lim Ä_

n" #n

œ

" #

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1

which is the

Section 10.6 Alternating Series, Absolute and Conditional Convergence 38. converges absolutely by the Ratio Test: n lim ¹ anb1 ¹ œ n lim Ä _ an Ä_

a(n  1)!b# ((2n  2)!)

39. diverges by the nth-Term Test since n lim kan k œ n lim Ä_ Ä_

œ n lim Ä_

ˆ n # 1 ‰n
1 œ _ Á 0  n lim Ä_

(n  1)(n  2)â(n  (n
1)) #nc1

œ n lim Ä_

(2n)! 2n n! n

(n  1)# 3 (2n  2)(2n  3)

œ

3 4

Èn  1
Èn 1

†

Èn  1  Èn Èn  1  Èn

œ

" Èn  1  Èn _

decreasing sequence of positive terms which converges to 0 Ê !

n œ1

_

_

n œ1

nœ1

" Èn  1  Èn

Èn

lim nÄ_


" 1

Èn

"

Èn

(n  1)# (2n  2)(2n  1)

œ n lim Ä_

œ

" 4

1

(n  ")(n  2)â(2n) 2n n

†

(2n  1)! n! n! 3n

1

41. converges conditionally since

! kan k œ !

(2n)! (n!)#

(n  1)! (n  1)! 3nb1 (2n  3)!

40. converges absolutely by the Ratio Test: n lim ¹ anabn 1 ¹ œ n lim Ä_ Ä_ œ n lim Ä_

†

605

and š Èn  1"  Èn › is a (
")n Èn  1  Èn

diverges by the Limit Comparison Test (part 1) with

 œ n lim Ä_

Èn Èn  1  Èn

œ n lim Ä_

1 É1  1n 1

œ

converges; but " Èn ;

a divergent p-series:

" #

È

#

n n 42. diverges by the nth-Term Test since n lim ŠÈn#  n
n‹ œ n lim ŠÈn#  n
n‹ † Š Ènn#  ‹ Ä_ Ä_  n n

œ n lim Ä_

n Èn# nn

œ n lim Ä_

" É1 "n 1

" #

œ

Á0

É n  Èn  Èn

43. diverges by the nth-Term Test since n lim ŠÉn  Èn
Èn‹ œ n lim ŠÉ n  È n
È n ‹
Ä_ Ä_ – Én  Èn  Èn — Èn

œ n lim Ä_

É n  Èn  Èn

œ n lim Ä_

" É1 

"

Èn  1

" #

œ

Á0

44. converges conditionally since š Èn  "Èn  1 › is a decreasing sequence of positive terms converging to 0 _

(
")n Èn  Èn  1

Ê !

n œ1

_

so that ! nœ1

converges; but n lim Ä_

" Èn  Èn  1

"

Èn Š È"n ‹

Š Èn

1

‹

Èn È n È n 1

œ n lim Ä_

_

diverges by the Limit Comparison Test with ! nœ1

45. converges absolutely by the Direct Comparison Test since sech (n) œ

" Èn

œ n lim Ä_

" 1É1 "n

" #

œ

which is a divergent p-series

2 en  ecn

œ

2en e2n  1



2en e2n

œ

2 en

which is the

nth term of a convergent geometric series _

_

n œ1

nœ1

46. converges absolutely by the Limit Comparison Test (part 1): ! kan k œ ! Apply the Limit Comparison Test with lim

nÄ_

47.

1 4




1 6

Œ



2 en c ecn 1 en

1 8




1 10

 œ n lim Ä_ 

1 12




1 14

2en en
ecn

1 en ,

the n-th term of a convergent geometric series:

œ n lim Ä_ _

 ÞÞÞ œ !

n œ1

2 1
ec2n

(
")nb1 2 an  1 b ;

n  2   n  1 Ê 2an  2b   2an  1b Ê

2 en
ecn

œ2

converges by Alternating Series Test since: un œ

1 2aan  1b  1b

Ÿ

1 2 an 1 b

Ê u n  1 Ÿ un ;

lim un œ

nÄ_

1 2 an  1 b

lim 1 nÄ_ 2an1b

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

 0 for all n   1;

œ 0.

606

Chapter 10 Infinite Sequences and Series

48. 1 

1 4







1 9



1 16

1 25



1 36




1 49




1 64

_

_

_

n œ1

n œ1

n œ1

 Þ Þ Þ œ ! an ; converges by the Absolute Convergence Test since ! kan k œ !

" n#

which is a convergent p-series 49. kerrork  ¸(
1)' ˆ "5 ‰¸ œ 0.2 51. kerrork  ¹(
1)'

(0.01)& 5 ¹

50. kerrork  ¸(
1)' ˆ 10" & ‰¸ œ 0.00001

œ 2 ‚ 10
""

52. kerrork  k(
1)% t% k œ t%  1

53. kerrork  0.001 Ê un1  0.001 Ê

1 an  1b2  3

 0.001 Ê an  1b2  3  1000 Ê n 
1  È997 ¸ 30.5753 Ê n   31

54. kerrork  0.001 Ê un1  0.001 Ê

n1 an  1b2  1

 0.001 Ê an  1b2  1  1000an  1b Ê n 

998È9982  4a998b 2

¸ 998.9999 Ê n   999 55. kerrork  0.001 Ê un1  0.001 Ê

1 3 ˆan  1b  3Èn  1‰

3

 0.001 Ê Šan  1b  3Èn  1‹  1000

2

È Ê ŠÈn  1‹  3Èn  1
10  0 Ê Èn  1 œ
3  29  40 œ 2 Ê n œ 3 Ê n   4

56. kerrork  0.001 Ê un1  0.001 Ê

1 lnalnan  3bb

 0.001 Ê lnalnan  3bb  1000 Ê n 
3  ee

1000

¸ 5.297 ‚ 10323228467

which is the maximum arbitrary-precision number represented by Mathematica on the particular computer solving this problem.. 57.

" (2n)!

58.

" n!





Ê (2n)! 

5 10'

10' 5

Ê

5 10'

10' 5

 n! Ê n   9 Ê 1
1 

59. (a) an   an1 fails since _

_

nœ1

nœ1

" 3

" #!

œ 200,000 Ê n   5 Ê 1




" #

" #!




" 3!



" 4!






_

_

nœ1

nœ1

" 4!

" 5!

" 6!


" 6!








" 7!

" 8!



¸ 0.54030 " 8!

¸ 0.367881944

n n n n (b) Since ! kan k œ ! ˆ 3" ‰  ˆ "# ‰ ‘ œ ! ˆ "3 ‰  ! ˆ "# ‰ is the sum of two absolutely convergent

series, we can rearrange the terms of the original series to find its sum: ˆ "3 

" 9



" 27

60. s#! œ 1


" #



" 3

 á ‰
ˆ "# 


" 4

á 

" 19

" 4






" 20

" 8

 በœ

ˆ "3 ‰

1
ˆ 3" ‰




ˆ "# ‰

1
ˆ "# ‰

œ

" #


1 œ
#"

" #

†

" #1

¸ 0.6687714032 Ê s#! 

¸ 0.692580927

_

61. The unused terms are ! (
1)j 1 aj œ (
1)n 1 aan 1
an 2 b  (
1)n 3 aan 3
an 4 b  á jœn 1

œ (
1)n 1 caan 1
an 2 b  aan 3
an 4 b  á d . Each grouped term is positive, so the remainder has the same sign as (
1)n 1 , which is the sign of the first unused term. 62. sn œ

" 1 †2



" #†3



" 3 †4

á 

" n(n  1)

n

œ!

k œ1

" k(k  1)

n

œ ! ˆ k"
k œ1

œ ˆ1
"# ‰  ˆ "#
3" ‰  ˆ 3"
4" ‰  ˆ 4"
5" ‰  á  ˆ n"


" ‰ k1

" ‰ n1

which are the first 2n terms

of the first series, hence the two series are the same. Yes, for n

sn œ ! ˆ k"
k œ1

" ‰ k1

œ ˆ1
"# ‰  ˆ "#
3" ‰  ˆ 3"
4" ‰  ˆ 4"
5" ‰  á  ˆ n
" 1
n" ‰  ˆ n"


" ‰ n1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

œ 1


" n1

Section 10.6 Alternating Series, Absolute and Conditional Convergence

607

ˆ1
n " 1 ‰ œ 1 Ê both series converge to 1. The sum of the first 2n  1 terms of the first Ê n lim s œ n lim Ä_ n Ä_ ˆ1
n " 1 ‰ œ 1. series is ˆ1
n " 1 ‰  n " 1 œ 1. Their sum is n lim s œ n lim Ä_ n Ä_ _

_

_

_

n œ1

n œ1

n œ1

n œ1

63. Theorem 16 states that ! kan k converges Ê ! an converges. But this is equivalent to ! an diverges Ê ! kan k diverges _

_

n œ1

n œ1

64. ka"  a#  á  an k Ÿ ka" k  ka# k  á  kan k for all n; then ! kan k converges Ê ! an converges and these imply that _

_

n œ1

n œ1

º ! an º Ÿ ! kan k _

65. (a) ! kan  bn k converges by the Direct Comparison Test since kan  bn k Ÿ kan k  kbn k and hence n œ1 _

! aan  bn b converges absolutely

n œ1 _

_

_

(b) ! kbn k converges Ê !
bn converges absolutely; since ! an converges absolutely and nœ1 _

nœ1

nœ1 _

_

!
bn converges absolutely, we have ! can  (
bn )d œ ! aan
bn b converges absolutely by part (a)

nœ1 _

_

_

nœ1

n œ1

nœ1

nœ1

nœ1 _

(c) ! kan k converges Ê kkk ! kan k œ ! kkan k converges Ê ! kan converges absolutely

66. If an œ bn œ (
1)n

" Èn

_

, then ! (
1)n nœ1

67. s" œ
"# , s# œ
"#  1 œ " #

s$ œ
 1
s% œ s$  s& œ s%
s' œ s&  s( œ s'


" 4




" 6




" 8

" 3 ¸
0.1766, " " " #4
#6
#8
" 5 ¸
0.312, " " " 46
48
50


" #

" Èn

nœ1

_

_

nœ1

nœ1

converges, but ! an bn œ !

" n

diverges

,




" 10




" 1#




" 14




" 16




" 18




" #0




" 2#

¸
0.5099,

" 30




" 3#




" 34




" 36




" 38




" 40




" 42




" 44

¸
0.512,

" 52




" 54




" 56




" 58




" 60




" 62




" 64




" 66

¸
0.51106

N"
1

68. (a) Since ! kan k converges, say to M, for %  0 there is an integer N" such that º ! kan k
Mº  nœ1

N"
1

N"
1

_

nœ1

nœ1

nœN"

Í » ! kan k

! kan k  ! kan k » 

% #

_

Í »
! k an k »  nœN"

% #

_

Í ! kan k  nœN"

% #

% #

. Also, ! an

converges to L Í for %  0 there is an integer N# (which we can choose greater than or equal to N" ) such that ksN#
Lk 

% #

_

. Therefore, ! kan k  nœN"

% #

and ksN#
Lk 

% #

.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

608

Chapter 10 Infinite Sequences and Series _

k

nœ1

nœ1

(b) The series ! kan k converges absolutely, say to M. Thus, there exists N" such that º ! kan k
Mº  % whenever k  N" . Now all of the terms in the sequence ekbn kf appear in ekan kf. Sum together all of the N terms in ekbn kf, in order, until you include all of the terms ekan kf nœ" 1 , and let N# be the largest index in the N#

N#

_

n œ1

nœ1

n œ1

sum ! kbn k so obtained. Then º ! kbn k
Mº  % as well Ê ! kbn k converges to M. 10.7 POWER SERIES _

nb1 1. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ xxn ¹  1 Ê kxk  1 Ê
1  x  1; when x œ
1 we have ! (
1)n , a divergent Ä_ Ä_ n œ1

_

series; when x œ 1 we have ! 1, a divergent series n œ1

(a) the radius is 1; the interval of convergence is
1  x  1 (b) the interval of absolute convergence is
1  x  1 (c) there are no values for which the series converges conditionally nb1

2. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (x(x5)5)n ¹  1 Ê kx  5k  1 Ê
6  x 
4; when x œ
6 we have Ä_ Ä_ _

_

n œ1

nœ1

! (
1)n , a divergent series; when x œ
4 we have ! 1, a divergent series

(a) the radius is 1; the interval of convergence is
6  x 
4 (b) the interval of absolute convergence is
6  x 
4 (c) there are no values for which the series converges conditionally nb1

 1) " " 3. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (4x (4x  1)n ¹  1 Ê k4x  1k  1 Ê
1  4x  1  1 Ê
#  x  0; when x œ
# we Ä_ Ä_ _

_

_

_

_

n œ1

n œ1

n œ1

n œ1

n œ1

have ! (
1)n (
1)n œ ! (
1)2n œ ! 1n , a divergent series; when x œ 0 we have ! (
1)n (1)n œ ! (
1)n , a divergent series (a) the radius is "4 ; the interval of convergence is
"#  x  0 (b) the interval of absolute convergence is
"#  x  0

(c) there are no values for which the series converges conditionally nb1

4. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (3xn
2)1 Ä_ Ä_ Ê
1  3x
2  1 Ê

" 3

†

n (3x
2)n ¹

ˆ n ‰  1 Ê k3x
2k  1  1 Ê k3x
2k n lim Ä _ n1

 x  1; when x œ

" 3

_

n œ1

(b) the interval of absolute convergence is

" 3

(c) the series converges conditionally at x œ nb1


2) 5. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ (x 10 nb1 Ä_ Ä_

n œ1

" n

conditionally convergent; when x œ 1 we have ! (a) the radius is "3 ; the interval of convergence is

_

we have !

" 3

(
")n n

which is the alternating harmonic series and is

, the divergent harmonic series

Ÿx1

x1 " 3

10n (x
2)n ¹

1 Ê

kx
2 k 10

 1 Ê kx
2k  10 Ê
10  x
2  10

_

_

nœ1

nœ1

Ê
8  x  12; when x œ
8 we have ! (
")n , a divergent series; when x œ 12 we have ! 1, a divergent series (a) the radius is "0; the interval of convergence is
8  x  12 (b) the interval of absolute convergence is
8  x  12 (c) there are no values for which the series converges conditionally

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.7 Power Series nb1

6. n lim k2xk  1 Ê k2xk  1 Ê
"#  x  ¹ uunbn 1 ¹  1 Ê n lim ¹ (2x) (2x)n ¹  1 Ê n lim Ä_ Ä_ Ä_ _

! (
")n , a divergent series; when x œ

n œ1

" #

" #

; when x œ
"# we have

_

we have ! 1, a divergent series n œ1

(a) the radius is "# ; the interval of convergence is
"#  x  (b) the interval of absolute convergence is
"#  x 

" #

" #

(c) there are no values for which the series converges conditionally nb1

 1)x 7. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (n (n  3) † Ä_ Ä_

(n  2) nxn ¹

 1 Ê kxk n lim Ä_

_

Ê
1  x  1; when x œ
1 we have ! (
")n n œ1

_

have ! n œ1

n n#,

n n#

(n  1)(n  2) (n  3)(n)

 1 Ê kxk  1

, a divergent series by the nth-term Test; when x œ " we

a divergent series

(a) the radius is "; the interval of convergence is
"  x  " (b) the interval of absolute convergence is
"  x  " (c) there are no values for which the series converges conditionally nb1

8. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (x n2)1 Ä_ Ä_

†

n (x  2)n ¹

ˆ  1 Ê kx  2k n lim Ä_ _

Ê
1  x  2  1 Ê
3  x 
1; when x œ
3 we have !

n œ1

_

! n œ1

(
1)n n ,

" n,

n ‰ n1

 1 Ê kx  2k  1

a divergent series; when x œ
" we have

a convergent series

(a) the radius is "; the interval of convergence is
3  x Ÿ
" (b) the interval of absolute convergence is
3  x 
" (c) the series converges conditionally at x œ
1 nb1

x 9. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ (n  1)Èn  1 3nb1

Ê

kx k 3

nÈ n 3n xn ¹

1 Ê

kxk 3

n n ‹ n  1 ‹ ŠÉ n lim Ä _ n1

Šn lim Ä_ _

(1)(1)  1 Ê kxk  3 Ê
3  x  3; when x œ
3 we have !

n œ1

_

when x œ 3 we have !

n œ1

1 , n$Î#

(
")n , n$Î#

1

an absolutely convergent series;

a convergent p-series

(a) the radius is 3; the interval of convergence is
3 Ÿ x Ÿ 3 (b) the interval of absolute convergence is
3 Ÿ x Ÿ 3 (c) there are no values for which the series converges conditionally nb1

10. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (xÈ
n1) 1 † Ä_ Ä_

Èn (x
1)n ¹

 1 Ê kx
1k Én lim Ä_ _

Ê
1  x
1  1 Ê 0  x  2; when x œ 0 we have !

n œ1

_

we have ! n œ1

1 , n"Î#

(
")n , n"Î#

n n1

609

 1 Ê kx
1k  1

a conditionally convergent series; when x œ 2

a divergent series

(a) the radius is 1; the interval of convergence is 0 Ÿ x  2 (b) the interval of absolute convergence is 0  x  2 (c) the series converges conditionally at x œ 0 nb1

ˆ " ‰  1 for all x 11. n lim † n! ¹  1 Ê kxk n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ x Ä_ Ä _ (n  1)! xn Ä _ n1 (a) the radius is _; the series converges for all x

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

610

Chapter 10 Infinite Sequences and Series

(b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb1

nb1

ˆ " ‰  1 for all x 12. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ 3 x † 3nn!xn ¹  1 Ê 3 kxk n lim Ä_ Ä _ (n  1)! Ä _ n1 (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally 13. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹4 Ä_ Ä_

nb1 2nb2

x n1

†

n 4n x2n ¹

ˆ 4n ‰ œ 4x#  1 Ê x#   1 Ê x# n lim Ä _ n1

_

_

n œ1

nœ1

n 2n Ê
12  x  12 ; when x œ
12 we have ! 4n ˆ
12 ‰ œ !

_

! n œ1

4n ˆ 1 ‰2n n 2

_

œ!

n œ1

1 n,

1 n

1 4

, a divergent p-series; when x œ

1 2

we have

a divergent p-series

(a) the radius is 12 ; the interval of convergence is
12  x  (b) the interval of absolute convergence is
12  x 

1 2

1 2

(c) there are no values for which the series converges conditionally nb1

14. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ (x
1) Ä_ Ä _ an  1b2 3nb1

n2 3n (x
1)n ¹

_

2

 1 Ê lx
1l n lim Š n ‹ œ 13 lx
1l  1 Ä _ 3an  1b2 _

Ê
2  x  4; when x œ
2 we have ! (n
2 3)3n œ ! (
n1) , an absolutely convergent series; when x œ 4 we have 2 n

n œ1

_

n

nœ1

_

n

! (3) ! 12 , an absolutely convergent series. n2 3n œ n

n œ1

n œ1

(a) the radius is 3; the interval of convergence is
2 Ÿ x Ÿ 4 (b) the interval of absolute convergence is
2 Ÿ x Ÿ 4 (c) there are no values for which the series converges conditionally nb1

x 15. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ È(n  1)#  3

È n#  3 ¹ xn

_

Ê
1  x  1; when x œ
1 we have !

n œ1

_

! n œ1

" È n#  3

 1 Ê kxk Én lim Ä_

(
")n È n#  3

n#  3 n#  2n  4

 " Ê kxk  1

, a conditionally convergent series; when x œ 1 we have

, a divergent series

(a) the radius is 1; the interval of convergence is
1 Ÿ x  1 (b) the interval of absolute convergence is
1  x  1 (c) the series converges conditionally at x œ
1 n 1

x 16. n lim † ¹ uun n 1 ¹  1 Ê n lim ¹ Ä_ Ä _ È(n  1)#  3

È n#  3 ¹ xn

_

Ê
1  x  1; when x œ
1 we have !

nœ1

 1 Ê kxk Én lim Ä_

" È n#  3

n#  3 n#  2n  4

 " Ê kxk  1 _

, a divergent series; when x œ 1 we have !

nœ1

a conditionally convergent series (a) the radius is 1; the interval of convergence is
1  x Ÿ 1 (b) the interval of absolute convergence is
1  x  1 (c) the series converges conditionally at x œ 1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

(
")n È n#  3

,

Section 10.7 Power Series  3) 17. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (n  1)(x 5nb1 Ä_ Ä_

nb1

†

5n n(x  3)n ¹

1 Ê

kx  3 k lim 5 nÄ_

ˆ n n " ‰  1 Ê _

Ê kx  3k  5 Ê
5  x  3  5 Ê
8  x  2; when x œ
8 we have !

n œ1

_

series; when x œ 2 we have !

n œ1

n5n 5n

n(
5)n 5n

kx  3 k 5

611

1

_

œ ! (
1)n n, a divergent n œ1

_

œ ! n, a divergent series n œ1

(a) the radius is 5; the interval of convergence is
8  x  2 (b) the interval of absolute convergence is
8  x  2 (c) there are no values for which the series converges conditionally nb1

18. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ (n  1)x Ä_ Ä _ 4nb1 an#  2n  2b _

Ê
4  x  4; when x œ
4 we have !

n œ1

4 n an #  1 b ¹ nxn n(
1)n n#  1

1 Ê

kx k 4 n lim Ä_

#

(n 1) n 1 ¹ n an# a2n  2bb ¹  1 Ê kxk  4 _

, a conditionally convergent series; when x œ 4 we have !

n œ1

n n#  1

,

a divergent series (a) the radius is 4; the interval of convergence is
4 Ÿ x  4 (b) the interval of absolute convergence is
4  x  4 (c) the series converges conditionally at x œ
4 19. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä_

Èn  1 xnb1 3nb1

†

3n È n xn ¹

1 Ê

kx k 3

ˆ n n 1 ‰  1 Ê Én lim Ä_

kx k 3

 1 Ê kxk  3

_

_

n œ1

n œ1

Ê
3  x  3; when x œ
3 we have ! (
1)n Èn , a divergent series; when x œ 3 we have ! Èn, a divergent series (a) the radius is 3; the interval of convergence is
3  x  3 (b) the interval of absolute convergence is
3  x  3 (c) there are no values for which the series converges conditionally 20. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä_

nbÈ 1

n  1 (2x5)nb1 ¹ n n (2x5)n È

 1 Ê k2x  5k n lim Š Ä_

nbÈ 1

n1 ‹ n n È

1

t È

lim t Ä_ Ê k2x  5k Œ tlim n n   1 Ê k2x  5k  1 Ê
1  2x  5  1 Ê
3  x 
2; when x œ
3 we have È n

_

Ä_

_

n n n ! (
1) È È n, a divergent series since n lim n œ 1; when x œ
2 we have ! È n, a divergent series Ä_ n œ1 n œ1

(a) the radius is "# ; the interval of convergence is
3  x 
2

(b) the interval of absolute convergence is
3  x 
2 (c) there are no values for which the series converges conditionally _

_

_

n œ1

n œ1

21. First, rewrite the series as ! a2  (
1)n bax  1bn
1 œ ! 2ax  1bn
1  ! (
1)n ax  1bn
1 . For the series n œ1

_

n

! 2ax  1bn
1 : lim ¹ unb1 ¹  1 Ê lim ¹ 2ax1nbc1 ¹  1 Ê lx  1l lim 1 œ lx  1l  1 Ê
2  x  0; For the un nÄ_ n Ä _ 2 ax  1 b nÄ_ n œ1 _

nb1

n

(
1) ax1b series ! (
1)n ax  1bn
1 : n lim 1 œ lx  1l  1 ¹ uunbn 1 ¹  1 Ê n lim ¹ ¹  1 Ê lx  1ln lim Ä_ Ä _ (
1)n ax1bnc1 Ä_ n œ1 _

Ê
2  x  0; when x œ
2 we have ! a2  (
1)n ba
1bn
1 , a divergent series; when x œ 0 we have n œ1

_

! a2  (
1)n b, a divergent series

n œ1

(a) the radius is 1; the interval of convergence is
2  x  0 (b) the interval of absolute convergence is
2  x  0 (c) there are no values for which the series converges conditionally

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612

Chapter 10 Infinite Sequences and Series

( 1) 22. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹
Ä_ Ä_

Ê _

! n œ1

17 9

x

19 9 ;

(
1)n 32n ˆ 1 ‰n 3n 9

when x œ _

œ!

n œ1

(
1)n 3n ,

17 9

3 ax
2bnb1 3an  1b

nb1 2nb2

_

we have ! n œ1

†

(
1)n 32n ˆ 1 ‰n
9 3n

(b) the interval of absolute convergence is

17 9

(c) the series converges conditionally at x œ

23.

_

œ!

nœ1

1 3n ,

9n n1

œ 9lx
2l  1

a divergent series; when x œ

19 9

we have

a conditionally convergent series.

(a) the radius is 19 ; the interval of convergence is

lim ¹ uunbn 1 ¹ nÄ_

 1 Ê lx
2ln lim Ä_

3n ¹ (
1)n 32n ax
2bn

 1 Ê n lim Ä_ »

Š1 

n

"

nb1

1‹

xnb1

Š1  "n ‹ xn n

17 9

x

xŸ

19 9

19 9

19 9 "

t

lim Š1  t ‹ e Ä_ »  1 Ê kxk
lim Š1  " ‹n   1 Ê kxk ˆ e ‰  1 Ê kxk  1 n nÄ_ t

_

n Ê
1  x  1; when x œ
1 we have ! (
1)n ˆ1  "n ‰ , a divergent series by the nth-Term Test since n œ1

lim ˆ1  nÄ_

" ‰n n

_

n œ e Á 0; when x œ 1 we have ! ˆ1  n" ‰ , a divergent series n œ1

(a) the radius is "; the interval of convergence is
1  x  1 (b) the interval of absolute convergence is
1  x  1 (c) there are no values for which the series converges conditionally 24. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ ln (nxnln1)xn Ä_ Ä_

nb1

¹  1 Ê kxk n lim Ä_ º

ˆn " 1‰ ˆ n" ‰ º

ˆ n ‰  1 Ê kxk  1  1 Ê kxk n lim Ä _ n1

_

Ê
1  x  1; when x œ
1 we have ! (
1)n ln n, a divergent series by the nth-Term Test since n lim ln n Á 0; Ä_ n œ1

_

when x œ 1 we have ! ln n, a divergent series n œ1

(a) the radius is 1; the interval of convergence is
1  x  1 (b) the interval of absolute convergence is
1  x  1 (c) there are no values for which the series converges conditionally nb1 nb1

x ˆ1  n" ‰n ‹ Š lim (n  1)‹  1 25. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (n  1) ¹  1 Ê kxk Šn lim nn xn Ä_ Ä_ Ä_ nÄ_ Ê e kxk n lim (n  1)  1 Ê only x œ 0 satisfies this inequality Ä_

(a) the radius is 0; the series converges only for x œ 0 (b) the series converges absolutely only for x œ 0 (c) there are no values for which the series converges conditionally nb1

26. n lim (n  1)  1 Ê only x œ 4 satisfies this inequality ¹ uunbn 1 ¹  1 Ê n lim ¹ (n n!1)!(x(x

4)4)n ¹  1 Ê kx
4k n lim Ä_ Ä_ Ä_ (a) the radius is 0; the series converges only for x œ 4 (b) the series converges absolutely only for x œ 4 (c) there are no values for which the series converges conditionally nb1

27. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ (x  2) Ä_ Ä _ (n  1) 2nb1

n2n (x  2)n ¹

1 Ê

kx  2 k lim # nÄ_

ˆ n n 1 ‰  1 Ê

kx  2 k #

 1 Ê kx  2k  2

_

_

n œ1

n œ1

! (
1) Ê
2  x  2  2 Ê
4  x  0; when x œ
4 we have !
" n , a divergent series; when x œ 0 we have n the alternating harmonic series which converges conditionally (a) the radius is 2; the interval of convergence is
4  x Ÿ 0 (b) the interval of absolute convergence is
4  x  0 (c) the series converges conditionally at x œ 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

nb1

,

Section 10.7 Power Series nb1

613

nb1

(n  2)(x
1) ˆ n  2 ‰  1 Ê 2 kx
1k  1 28. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (
(2)
2)n (n  1)(x
1)n ¹  1 Ê 2 kx
1k n lim Ä_ Ä_ Ä _ n1

Ê kx
1k  _

" #

Ê
"#  x
1 

" #

" #

Ê

 x  3# ; when x œ

" #

_

we have ! (n  1) , a divergent series; when x œ n œ1

we have ! (
1) (n  1), a divergent series n

n œ1

(a) the radius is "# ; the interval of convergence is (b) the interval of absolute convergence is

" #

" #

x

x

3 #

3 #

(c) there are no values for which the series converges conditionally nb1

x 29. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ (n  1) aln (n  1)b#

Ê kxk (1) Œn lim Ä_ _

! nœ1

(
1)n n(ln n)#

ˆ "n ‰ ˆ nb" 1 ‰ 

#

n(ln n)# xn ¹

 1 Ê kxk Šn lim Ä_

n1 n ‹

 1 Ê kxk Šn lim Ä_

#

n ln n ‹ n  1 ‹ Šn lim Ä _ ln (n  1)

#

1

 1 Ê kxk  1 Ê
1  x  1; when x œ
1 we have _

which converges absolutely; when x œ 1 we have !

nœ1

" n(ln n)#

which converges

(a) the radius is "; the interval of convergence is
1 Ÿ x Ÿ 1 (b) the interval of absolute convergence is
1 Ÿ x Ÿ 1 (c) there are no values for which the series converges conditionally nb1

x 30. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ (n  1) ln (n  1)

n ln (n) xn ¹

 1 Ê kxk Šn lim Ä_

ln (n) n ‹ n  1 ‹ Šn lim Ä _ ln (n  1) _

(
1)n n ln n

Ê kxk (1)(1)  1 Ê kxk  1 Ê
1  x  1; when x œ
1 we have !

n œ2

_

when x œ 1 we have !

n œ2

" n ln n

1

, a convergent alternating series;

which diverges by Exercise 38, Section 9.3

(a) the radius is "; the interval of convergence is
1 Ÿ x  1 (b) the interval of absolute convergence is
1  x  1 (c) the series converges conditionally at x œ
1 2nb3


5) 31. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ (4x (n  1)$Î# Ä_ Ä_

n$Î# (4x
5)2n

1

¹  1 Ê (4x
5)# Šn lim Ä_

Ê k4x
5k  1 Ê
1  4x
5  1 Ê 1  x  absolutely convergent; when x œ

3 #

_

we have ! n œ1

(")2nb1 n$Î#

3 #

_

; when x œ 1 we have !

n œ1

$Î#

 1 Ê (4x
5)#  1

(
1)2nb1 n$Î#

_

œ!

n œ1


" n$Î#

which is

, a convergent p-series

(a) the radius is "4 ; the interval of convergence is 1 Ÿ x Ÿ (b) the interval of absolute convergence is 1 Ÿ x Ÿ

n n1‹

3 #

3 #

(c) there are no values for which the series converges conditionally nb2

32. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (3x2n1)4 Ä_ Ä_

†

2n  2 (3x  1)nb1 ¹

ˆ 2n  2 ‰  1 Ê k3x  1k  1  1 Ê k3x  1k n lim Ä _ 2n  4 _

Ê
1  3x  1  1 Ê
23  x  0; when x œ
23 we have !

n œ1

_

when x œ 0 we have !

n œ1

(")nb1 2n  1

_

œ!

nœ1

" #n  1

(
1)nb1 2n  1

, a conditionally convergent series;

, a divergent series

(a) the radius is "3 ; the interval of convergence is
32 Ÿ x  0 (b) the interval of absolute convergence is
23  x  0 (c) the series converges conditionally at x œ
23

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

3 #

614

Chapter 10 Infinite Sequences and Series nb1

x ˆ 1 ‰  1 for all x 33. n lim † 2†4†6xân a2nb ¹  1 Ê kxk n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ 2†4†6âa2nba2an  1bb Ä _ 2n  2 (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb2

3 5 7 2n 1 2 n 1 1x 2n 3 n 34. n lim † 3†5†7âan2n2 1bxnb1 ¹  1 Ê kxk n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ † † âa an ba1ba2 2nb1 b  b Š a2an 1bb2 ‹  1 Ê only Ä_ Ä_ Ä_ x œ 0 satisfies this inequality (a) the radius is 0; the series converges only for x œ 0 (b) the series converges absolutely only for x œ 0 (c) there are no values for which the series converges conditionally _

35. For the series ! n œ1

12ân n 12  22  â  n2 x ,

recall 1  2  â  n œ

nan b 1b

_

2

2 n

nan  1b 2

and 12  22  â  n2 œ

nan  1ba2n  1b 6

_

nb1 rewrite the series as ! Œ n n b 1 2 2n b 1 xn œ ! ˆ 2n 3 1 ‰xn ; then n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ a2an3x 1 b  1 b † Ä _ Ä _ 6 n œ1 nœ1 a

ba

b

so that we can

a2n  1b 3xn ¹

1

_

Ê kxk n lim ¹ a2n  1b ¹  1 Ê kxk  1 Ê
1  x  1; when x œ
1 we have ! ˆ 2n 3 1 ‰a
1bn , a conditionally Ä _ a2n  3b n œ1 _

convergent series; when x œ 1 we have ! ˆ 2n 3 1 ‰, a divergent series. n œ1

(a) the radius is 1; the interval of convergence is
1 Ÿ x  1 (b) the interval of absolute convergence is
1  x  1 (c) the series converges conditionally at x œ
1 _

36. For the series ! ŠÈn  1
Èn‹ax
3bn , note that Èn  1
Èn œ n œ1

_

can rewrite the series as ! n œ1

Ê lx
3ln lim Ä_

a x
3 bn Èn  1  Èn ;

Èn  1  Èn Èn  2  Èn  1

Èn  1
Èn 1

†

nb1

Èn  1  Èn Èn  1  Èn

then n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ ax
3 b Ä_ Ä _ Èn  2  Èn  1

ax
3bn

_

n œ1

1 Èn  1  Èn ,

so that we

¹1

a
1 b n Èn  1  Èn ,

n œ1

_

1 Èn  1  Èn

Èn  1  Èn

 1 Ê lx
3l  1 Ê 2  x  4; when x œ 2 we have !

convergent series; when x œ 4 we have !

œ

a conditionally

a divergent series;

(a) the radius is 1; the interval of convergence is 2 Ÿ x  4 (b) the interval of absolute convergence is 2  x  4 (c) the series converges conditionally at x œ 2 nb1

an  1bxx 37. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ Ä_ Ä _ 3†6†9âa3nba3an  1bb

3†6†9âa3nb ¹ nx xn 2 nb1

2 4 6 2n 2 n 1 x 38. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ a † † âa ba a  bbb Ä_ Ä _ a2†5†8âa3n
1ba3an  1b
1bb2 9 9 Ê lxl  4 Ê R œ 4 2 nb1

n 1 x 39. n lim † ¹ uunbn 1 ¹  1 Ê n lim ¹ aa  bxb Ä_ Ä _ 2nb1 a2an  1bbx

2n a2nbx ¹ an x b 2 x n

an  1 b  1 Ê lxln lim ¹ ¹1Ê Ä _ 3 an  1 b

a2†5†8âa3n
1bb2 ¹ a2†4†6âa2nbb2 xn

lx l 3

 1 Ê lxl  3 Ê R œ 3 2

 1 Ê lxln lim ¹ a2n  2b ¹  1 Ê Ä _ a3n  2b2

2

an  1 b  1 Ê lxln lim ¹ ¹1Ê Ä _ 2a2n  2ba2n  1b

lx l 8

4 lx l 9

1

 1 Ê lxl  8 Ê R œ 8

2

n n ‰n n n Ɉ ˆ n ‰n  1 Ê lxle
1  1 Ê lxl  e Ê R œ e È 40. n lim un  1 Ê n lim x  1 Ê lxl n lim n1 Ä_ Ä_ Ä _ n1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.7 Power Series nb1

nb1

41. n lim 3  1 Ê lxl  ¹ uunbn 1 ¹  1 Ê n lim ¹ 3 3n xxn ¹  1 Ê lxl n lim Ä_ Ä_ Ä_ _

_

! 3n ˆ
1 ‰n œ ! a
1bn , which diverges; at x œ 3

n œ0

n œ0

1 3

1 3

_

_

n œ0

nœ0

615

Ê
31  x  31 ; at x œ
31 we have _

n we have ! 3n ˆ 13 ‰ œ ! 1 , which diverges. The series ! 3n xn

_

œ ! a3xbn is a convergent geometric series when
13  x  n œ0

1 3

and the sum is

nœ0

1 1
3x .

nb1

e 4 42. n lim 1  1 Ê lex
4l  1 Ê 3  ex  5 Ê ln 3  x  ln 5; ¹ uunbn 1 ¹  1 Ê n lim ¹ a aex

4b bn ¹  1 Ê lex
4l n lim Ä_ Ä_ Ä_ x

_

_

_

_

nœ0 _

nœ0

nœ0

nœ0

n n at x œ ln 3 we have ! ˆeln 3
4‰ œ ! a
1bn , which diverges; at x œ ln 5 we have ! ˆeln 5
4‰ œ ! 1, which

diverges. The series ! aex
4bn is a convergent geometric series when ln 3  x  ln 5 and the sum is n œ0

2nb2

43. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (x
4n1)1 Ä_ Ä_

4n (x
1)2n ¹

†

1 Ê

(x
1)# lim 4 nÄ_ _

Ê
2  x
1  2 Ê
1  x  3; at x œ
1 we have !

n œ0

_

we have ! n œ0 _

! n œ0

(x
")2n 4n "

#

4 4n

nœ0

œ!

nœ0 4

(x 4

9n (x  1)2n ¹

†

1 Ê

(x  1)# lim 9 nÄ_

n œ0

!

n œ0

nœ0

n œ0

k1k  1 Ê (x  1)#  9 Ê kx  1k  3

(
3)2n 9n

_

œ ! 1 which diverges; at x œ 2 we have n œ0

_

œ ! " which also diverges; the interval of convergence is
4  x  2; the series

(x  1) 9n "

_

4 4 ")# “ œ 4
x#  2x
1 œ 3  2x
x#

_

!

_

n œ ! 44n œ ! 1, which diverges; at x œ 3

is a convergent geometric series when
1  x  3 and the sum is

Ê
3  x  1  3 Ê
4  x  2; when x œ
4 we have !

n œ0 _

k1k  1 Ê (x
1)#  4 Ê kx
1k  2

nœ0

2nb2

32n 9n

1 5
ex .

œ ! 1, a divergent series; the interval of convergence is
1  x  3; the series

44. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (x 9n1)1 Ä_ Ä_

_

œ

_

# n Šˆ x
# 1 ‰ ‹ "

’

n

œ!

2 4n

_

œ

" 1
Šxc # ‹

_

2n

(
2)2n 4n

1 1
ae x
4 b

nœ0

2n

_

n

# œ ! Šˆ x3 1 ‰ ‹ is a convergent geometric series when
4  x  2 and the sum is n œ0

1 1
Šxb 3 ‹

#

œ

"

’

9

(x
1)# “ 9

œ

9 9
x#
2x
1

45. n lim ¹ uunbn 1 ¹  1 Ê n lim Ä_ Ä_ º

œ

9 8
2x
x#

ˆÈx
2‰nb1 2nb1

†

2n ˆÈ x
2 ‰ n º

 1 Ê ¸Èx
2¸  2 Ê
2  Èx
2  2 Ê 0  Èx  4

_

_

Ê 0  x  16; when x œ 0 we have ! (
1)n , a divergent series; when x œ 16 we have ! (1)n , a divergent nœ0

nœ0

_

series; the interval of convergence is 0  x  16; the series !

n œ0

0  x  16 and its sum is

1
Œ

" Èx c 2 œ

# 

Œ

2c

"

Èx #

2

œ 

Èx
2 n Š # ‹

is a convergent geometric series when

2 4
Èx

nb1

46. n lim ¹ uunbn 1 ¹  1 Ê n lim ¹ (ln(lnx)x)n ¹  1 Ê kln xk  1 Ê
1  ln x  1 Ê e
"  x  e; when x œ e
" or e we Ä_ Ä_ _

_

_

nœ0

nœ0

nœ0

obtain the series ! 1n and ! (
1)n which both diverge; the interval of convergence is e
"  x  e; ! (ln x)n œ when e
"  x  e

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" 1
ln x

616

Chapter 10 Infinite Sequences and Series

47. n lim ¹ uunbn 1 ¹  1 Ê n lim Šx Ä_ Ä_ º

#

1 3 ‹

n1

n † ˆ x# 3 1 ‰ º  1 Ê

ax#  1b lim 3 nÄ_

x#  " 3

k1k  1 Ê

 1 Ê x#  2

_

Ê kxk  È2 Ê
È2  x  È2 ; at x œ „ È2 we have ! (1)n which diverges; the interval of convergence is n œ0

_


È2  x  È2 ; the series !

n œ0

" # 1
Š x 3b 1 ‹

œ

" # Š 3 c x3 c 1 ‹

œ

# Š x 3 1 ‹

n

is a convergent geometric series when
È2  x  È2 and its sum is

3 #
x#

ax 48. n lim ¹ uun n 1 ¹  1 Ê n lim ¹ Ä_ Ä_

#
1 bn 2n

1

2n ¹ a x #  1 bn

†

1

 1 Ê kx#
1k  2 Ê
È3  x  È3 ; when x œ „ È3 we

_

_

n œ0

n œ0

have ! 1n , a divergent series; the interval of convergence is
È3  x  È3 ; the series ! Š x convergent geometric series when
È3  x  È3 and its sum is

nb1

49. n lim ¹ (x
#n3) b1 Ä_

†

2n (x
3)n ¹

" # 1
Šx 2 1‹

"

œ

2

œ

Šx# 1 ‹

#



#

n
1 ‹ 2

is a

2 3
x#



_

 1 Ê kx
3k  2 Ê 1  x  5; when x œ 1 we have ! (1)n which diverges; nœ1

_

when x œ 5 we have ! (
1) which also diverges; the interval of convergence is 1  x  5; the sum of this n

n œ1

convergent geometric series is œ

2 x
1

" 3 1  Šxc # ‹

œ

2 x
1

n

. If f(x) œ 1
#" (x
3)  4" (x
3)#  á  ˆ
#" ‰ (x
3)n  á n

then f w (x) œ
#"  "# (x
3)  á  ˆ
#" ‰ n(x
3)n
1  á is convergent when 1  x  5, and diverges
2 (x
1)#

when x œ 1 or 5. The sum for f w (x) is

, the derivative of

2 x
1

.

n

50. If f(x) œ 1
"# (x
3)  4" (x
3)#  á  ˆ
"# ‰ (x
3)n  á œ œx


(x
3)# 4

(x
3)$ 12



_

n (x
3)n n 1

1

 á  ˆ
"# ‰

2 x
1

then ' f(x) dx _

 á . At x œ 1 the series ! n
21 diverges; at x œ 5 n œ1

2 the series ! (
n1) 1 converges. Therefore the interval of convergence is 1  x Ÿ 5 and the sum is n

n œ1

2 ln kx
1k  (3
ln 4), since '

dx œ 2 ln kx
1k  C, where C œ 3
ln 4 when x œ 3.

2 x
1

51. (a) Differentiate the series for sin x to get cos x œ 1
œ

x# #!

x% 4!

x' 6!

) x"! 1

 x8!
10! á . 2nb2 a b # n ! lim ¹ x † x#8 ¹ œ x2 n lim n Ä _ (2n  2)! Ä_

(b) sin 2x œ 2x


2$ x$ 3!

2& x& 5!

2( x( 7!

" 6!

œ 2x
52. (a) (b)

d x






5x% 5!




7x' 7!



9x) 9!




11x"! 11!

á

The series converges for all values of x since Š a2n  1ba" 2n  2b ‹ œ 0  1 for all x.

1†00†

" 4!

$ $

( (

2 x 3!

aex b œ 1 

& &



2 x 5!




2x 2!



3x# 3!

2 x 7!



0† 

4x$ 4!

' ex dx œ ex  C œ x  x#

#

#

(c) e
x œ 1
x  x#!
 ˆ1 † 3!"
1 † #"!   ˆ1 † 5!"
1 † 4!" 

x$ 3! " #! " #!

" 3!

* *

2 x 9!





2* x* 9!




2"" x"" 11!

" #

0†

"" ""

á

0†


5x% 5!

2 x 11!

 á œ 2x


8x$ 3!

&

(

*

""

128x 512x 2048x  32x 5!
7!  9!
11!  á " "‰ $ ‰ # ˆ (c) 2 sin x cos x œ 2 (0 † 1)  (0 † 0  1 † 1)x  ˆ0 †
" #  1 † 0  0 † 1 x  0 † 0
1 † #  0 † 0
1 † 3! x  ˆ0 † 4!"  1 † 0
0 † #"
0 † 3!"  0 † 1‰ x%  ˆ0 † 0  1 † 4!"  0 † 0  #" † 3!"  0 † 0  1 † 5!" ‰ x&

 ˆ0 †



3x# 3!

" 5!

 0 † 1‰ x'  á ‘ œ 2 ’x


á œ1x

x# #!



x$ 3!



x% 4!

4x$ 3!



16x& 5!


á“

 á œ ex ; thus the derivative of ex is ex itself

x$ x% x& x 3!  4!  5!  á  C, which is the general antiderivative of e % &  x4!
x5!  á ; ecx † ex œ 1 † 1  (1 † 1
1 † 1)x  ˆ1 † #"!
1 † 1  #"! † 1
3!" † 1‰ x$  ˆ1 † 4!"
1 † 3!"  #"! † #"!
3!" † 1  4!" † 1‰ x% † 3!"
3!" † #"!  4!" † 1
5!" † 1‰ x&  á œ 1  0  0  0  0  0  á



Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

† 1‰ x#

Section 10.8 Taylor and Maclaurin Series 53. (a) ln ksec xk  C œ ' tan x dx œ ' Šx  #

œ

x #



%



x 1#

'

)



x 45

17x 2520 1 # 

converges when
#

(b) sec x œ

d(tan x) dx

œ

when
1#  x 

d dx

 x

x$ 3

œ1x 

x$ 6

œx

$

x 6



x& 24



&

x 24

(b) sec x tan x œ when


1 #

17x' 45



61x( 5040



(



" 4 62x) 315

x# #



17x( 315





5x% 24

5 ‰ % 24 x



62x* 2835

61x' 720



62x* 2835

 á ‹ dx

 á ‹ œ 1  x# 

d(sec x) dx

*



œ

277x 72,576

d dx

 á ‹ Š1 

61  ˆ 720 

á ,


277x* 72,576



61x 5040

2x% 3



x# #

17x' 45

x% 12

 

62x) 315



x' 45

17x) 2520





31x"! 14,175

á ,

 á , converges

1 #

x x# 2



5 48 1 #



5 48

5x% 24





x# #



5x% 24

61 ‰ ' 720 x

61x' 720



61x' 720

 á‹

á

 á ‹ dx

 á  C; x œ 0 Ê C œ 0 Ê ln ksec x  tan xk  á , converges when
1#  x 

Š1 

x# #



5x% 24



61x' 720

5x$ 6

 á‹ œ x 

1 #



61x& 120



277x( 1008

 á , converges

1 #

(c) (sec x)(tan x) œ Š1 

x# #



2 œ x  ˆ "3  #" ‰ x$  ˆ 15 


1#  x 



1 #



x

17x( 315

2x& 15

54. (a) ln ksec x  tan xk  C œ ' sec x dx œ ' Š1  œx





5 œ 1  ˆ "#  "# ‰ x#  ˆ 24  2x% 3

2x& 15

 á  C; x œ 0 Ê C œ 0 Ê ln ksec xk œ

(c) sec# x œ (sec x)(sec x) œ Š1  #



"!

31x 14,175  1#

Šx 

x$ 3

617

1 #

5x% 24 " 6

_





61x' 720

 á ‹ Šx 

5 ‰ & 24 x

17  ˆ 315 

" 15

x$ 3





5 72

2x& 15





17x( 315

61 ‰ ( 720 x

á‹ á œ x

5x$ 6



61x& 120



277x( 1008

á ,

_

55. (a) If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n
1)(n
2)â(n
(k
1)) an xn
k and f ÐkÑ (0) œ k!ak n œ0

Ê ak œ

f ÐkÑ (0) k!

n œk _

; likewise if f(x) œ ! bn xn , then bk œ n œ0

f ÐkÑ (0) k!

Ê ak œ bk for every nonnegative integer k

_

(b) If f(x) œ ! an xn œ 0 for all x, then f ÐkÑ (x) œ 0 for all x Ê from part (a) that ak œ 0 for every nonnegative integer k n œ0

10.8 TAYLOR AND MACLAURIN SERIES 1. f(x) œ e2x , f w (x) œ 2e2x , f ww (x) œ 4e2x , f www (x) œ 8e2x ; f(0) œ e2a0b œ ", f w (0) œ 2, f ww (0) œ 4, f www (0) œ 8 Ê P! (x) œ 1, P" (x) œ 1  2x, P# (x) œ 1  x  2x# , P$ (x) œ 1  x  2x#  43 x3 2. f(x) œ sin x, f w (x) œ cos x , f ww (x) œ
sin x , f www (x) œ
cos x; f(0) œ sin 0 œ 0, f w (0) œ 1, f ww (0) œ 0, f www (0) œ
1 Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x, P$ (x) œ x
16 x3 3. f(x) œ ln x, f w (x) œ

" x

, f ww (x) œ
x"# , f www (x) œ

2 x$ ;

f(1) œ ln 1 œ 0, f w (1) œ 1, f ww (1) œ
1, f www (1) œ 2 Ê P! (x) œ 0,

P" (x) œ (x
1), P# (x) œ (x
1)
"# (x
1)# , P$ (x) œ (x
1)
"# (x
1)#  3" (x
1)$ 4. f(x) œ ln (1  x), f w (x) œ f w (0) œ 5. f(x) œ

œ 1, f ww (0) œ
(1)

1 1 " x

(1  x)
" , f ww (x) œ
(1  x)
# , f www (x) œ 2(1  x)
$ ; f(0) œ ln 1 œ 0,

œ
1, f www (0) œ 2(1)
$ œ 2 Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x


œ x
" , f w (x) œ
x
# , f ww (x) œ 2x
$ , f www (x) œ
6x
% ; f(2) œ

Ê P! (x) œ P$ (x) œ

" 1x œ
#

" #




" " " " # , P" (x) œ #
4 (x
2), P# (x) œ # " " " # $ 4 (x
2)  8 (x
2)
16 (x
2)

" #

x# #,

P$ (x) œ x


, f w (2) œ
4" , f ww (2) œ 4" , f www (x) œ
83


"4 (x
2)  "8 (x
2)# ,

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

x# #



x$ 3

618

Chapter 10 Infinite Sequences and Series

6. f(x) œ (x  2)
" , f w (x) œ
(x  2)
# , f ww (x) œ 2(x  2)
$ , f www (x) œ
6(x  2)
% ; f(0) œ (2)
" œ œ
4" , f ww (0) œ 2(2)
$ œ P$ (x) œ

" #




x 4



x# 8




" 4

, f www (0) œ
6(2)
% œ
38 Ê P! (x) œ

x$ 16

" #

, P" (x) œ

f ww ˆ 14 ‰ œ
sin P# (x) œ

È2 #



1 4 œ
È2 ˆx
#

È , f www ˆ 14 ‰ œ
cos 14 œ
#2 Ê È2 È 1‰ ˆx
14 ‰# , P$ (x) œ #2  4
4


x4 , P# (x) œ

" #

, f w (0) œ
(2)
#




x 4



È2 È2 1 w ˆ1‰ # ,f 4 œ cos 4 œ # È È È P! œ #2 , P" (x) œ #2  #2 ˆx
14 ‰ , È2 È È ˆx
14 ‰
42 ˆx
14 ‰#
1#2 ˆx
14 ‰$ #

7. f(x) œ sin x, f w (x) œ cos x, f ww (x) œ
sin x, f www (x) œ
cos x; f ˆ 14 ‰ œ sin È2 #

" #

" #

1 4

œ

x# 8

,

,

8. f(x) œ tan x, f w (x) œ sec2 x, f ww (x) œ 2sec2 x tan x, f www (x) œ 2sec4 x  4sec2 x tan2 x; f ˆ 14 ‰ œ tan 14 œ 1 , f w ˆ 14 ‰ œ sec2 ˆ 14 ‰ œ 2 , f ww ˆ 14 ‰ œ 2sec2 ˆ 14 ‰ tan ˆ 14 ‰ œ 4 , f www ˆ 14 ‰ œ 2sec4 ˆ 14 ‰  4sec2 ˆ 14 ‰ tan2 ˆ 14 ‰ œ 16 Ê P! (x) œ 1 , 2 2 3 P" (x) œ 1  2 ˆx
14 ‰ , P# (x) œ 1  2 ˆx
14 ‰  2 ˆx
14 ‰ , P$ (x) œ 1  2 ˆx
14 ‰  2 ˆx
14 ‰  83 ˆx
14 ‰

9. f(x) œ Èx œ x"Î# , f w (x) œ ˆ "# ‰ x
"Î# , f ww (x) œ ˆ
4" ‰ x
$Î# , f www (x) œ ˆ 83 ‰ x
&Î# ; f(4) œ È4 œ 2, " 3 f w (4) œ ˆ "# ‰ 4
"Î# œ "4 , f ww (4) œ ˆ
"4 ‰ 4
$Î# œ
32 ,f www (4) œ ˆ 38 ‰ 4
&Î# œ 256 Ê P! (x) œ 2, P" (x) œ 2  "4 (x
4), P# (x) œ 2  4" (x
4)


" 64

(x
4)# , P$ (x) œ 2  "4 (x
4)


" 64

(x
4)# 

" 51#

(x
4)$

10. f(x) œ (1
x)"Î# , f w (x) œ
"# (1
x)
"Î# , f ww (x) œ
"4 (1
x)
$Î# , f www (x) œ
38 (1
x)
&Î# ; f(0) œ (1)"Î# œ 1, f w (0) œ
"# (1)
"Î# œ
"# , f ww (0) œ
"4 (1)
$Î# œ
"4 , f www (0) œ
83 (1)
&Î# œ
83 Ê P! (x) œ 1, P" (x) œ 1
2" x, P# (x) œ 1
2" x
8" x# , P$ (x) œ 1
2" x
8" x#


1 16

x$

11. f(x) œ e
x , f w (x) œ
e
x , f ww (x) œ e
x , f www (x) œ
e
x Ê á f ÐkÑ (x) œ a
1bk e
x ; f(0) œ e
a0b œ ", f w (0) œ
1, _

f ww (0) œ 1, f www (0) œ
1, á ß f ÐkÑ (0) œ (
1)k Ê e
x œ 1
x  12 x#
16 x3  á œ !

n œ0

(
1)n n n! x

12. f(x) œ x ex , f w (x) œ x ex  ex , f ww (x) œ x ex  2ex , f www (x) œ x ex  3ex Ê á f ÐkÑ (x) œ x ex  k ex ; f(0) œ a0bea0b œ 0, _

f w (0) œ 1, f ww (0) œ 2, f www (0) œ 3, á ß f ÐkÑ (0) œ k Ê x  x#  12 x3  á œ !

n œ0

1 n a n
1 b! x

13. f(x) œ (1  x)
" Ê f w (x) œ
(1  x)
# , f ww (x) œ 2(1  x)
$ , f www (x) œ
3!(1  x)
% Ê á f ÐkÑ (x) œ (
1)k k!(1  x)
k
1 ; f(0) œ 1, f w (0) œ
1, f ww (0) œ 2, f www (0) œ
3!, á ß f ÐkÑ (0) œ (
1)k k! _

_

n œ0

nœ0

Ê 1
x  x#
x$  á œ ! (
x)n œ ! (
1)n xn 14. f(x) œ

2x 1
x

Ê f w (x) œ

œ 6(1
x)
$ , f www (x) œ 18(1
x)
% Ê á f ÐkÑ (x) œ 3ak!b(1
x)

3 ww (1
x)# , f (x)

_

f w (0) œ 3, f ww (0) œ 6, f www (0) œ 18, á ß f ÐkÑ (0) œ 3ak!b Ê 2  3x  3x#  3x$  á œ 2  ! 3xn n œ1

_

15. sin x œ !

n œ0 _

16. sin x œ !

nœ0

_

(
")n x2nb1 (#n1)!

Ê sin 3x œ !

(
")n x2nb1 (#n1)!

Ê sin

n œ0

_

17. 7 cos (
x) œ 7 cos x œ 7 !

n œ0

x #

_

œ!

nœ0

(
")n x2n (2n)!

(
")n (3x)2nb1 (#n1)!

2n 1

(
")n ˆ x# ‰ (#n1)!

œ7


7x# #!

_

(
")n 32nb1 x2nb1 (#n1)!

œ 3x


(
")n x2nb1 #2n 1 (2n1)!

x #

œ!

n œ0

_

œ!

nœ0



7x% 4!




7x' 6!

œ




3$ x$ 3!

x$ 2$ †3!





3& x& 5!

x& 2& †5!


á

á

 á , since the cosine is an even function

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

k 1

; f(0) œ 2,

Section 10.8 Taylor and Maclaurin Series _

18. cos x œ !

n œ0

19. cosh x œ _

œ!

n œ0

œ!

n œ0

ex  ecx #

Ê 5 cos 1x œ 5 !

nœ0

œ

" #

’Š1  x# 

œ

" #

’Š1  x 

x# #!



x$ 3!

(
1)n (1x)2n (#n)!



x% 4!

œ5


51 # x# 2!

51 % x% 4!



 á ‹  Š1
x 

x# #!

5 1 ' x' 6!







x$ 3!



á

x% 4!


á ‹“ œ 1 

x# #!



x% 4!



x' 6!

á

x2n (2n)!

20. sinh x œ _

_

(
1)n x2n (2n)!

619

ex
ecx #

x# #!



x$ 3!



x% 4!

 á ‹
Š1
x 

x# #!




x$ 3!



x% 4!


á ‹“ œ x 

x$ 3!



x& 5!



x' 6!

á

x2n 1 (2n  1)!

21. f(x) œ x%
2x$
5x  4 Ê f w (x) œ 4x$
6x#
5, f ww (x) œ 12x#
12x, f www (x) œ 24x
12, f Ð4Ñ (x) œ 24 Ê f ÐnÑ (x) œ 0 if n   5; f(0) œ 4, f w (0) œ
5, f ww (0) œ 0, f www (0) œ
12, f Ð4Ñ (0) œ 24, f ÐnÑ (0) œ 0 if n   5 24 % $ % $ Ê x%
2x$
5x  4 œ 4
5x
12 3! x  4! x œ x
2x
5x  4 22. f(x) œ

x# x1

Ê f w (x) œ

2x  x# ; f ww (x) ax  1b2

œ

2 ; ax  1 b 3

f www (x) œ


6 ax  1 b 4

Ê f ÐnÑ (x) œ

a
1bn nx ; ax  1bnb1

f(0) œ 0, f w (0) œ 0, f ww (0) œ 2,

_

f www (0) œ
6, f ÐnÑ (0) œ a
1bn nx if n   2 Ê x#
x3  x4
x5  Þ Þ Þ œ ! a
1bn xn n œ2

23. f(x) œ x$
2x  4 Ê f w (x) œ 3x#
2, f ww (x) œ 6x, f www (x) œ 6 Ê f ÐnÑ (x) œ 0 if n   4; f(2) œ 8, f w (2) œ 10, 6 # $ f ww (2) œ 12, f www (2) œ 6, f ÐnÑ (2) œ 0 if n   4 Ê x$
2x  4 œ 8  10(x
2)  12 2! (x
2)  3! (x
2) œ 8  10(x
2)  6(x
2)#  (x
2)$

24. f(x) œ 2x$  x#  3x
8 Ê f w (x) œ 6x#  2x  3, f ww (x) œ 12x  2, f www (x) œ 12 Ê f ÐnÑ (x) œ 0 if n   4; f(1) œ
2, f w (1) œ 11, f ww (1) œ 14, f www (1) œ 12, f ÐnÑ (1) œ 0 if n   4 Ê 2x$  x#  3x
8 12 # $ # $ œ
2  11(x
1)  14 2! (x
1)  3! (x
1) œ
2  11(x
1)  7(x
1)  2(x
1) 25. f(x) œ x%  x#  1 Ê f w (x) œ 4x$  2x, f ww (x) œ 12x#  2, f www (x) œ 24x, f Ð4Ñ (x) œ 24, f ÐnÑ (x) œ 0 if n   5; f(
2) œ 21, f w (
2) œ
36, f ww (
2) œ 50, f www (
2) œ
48, f Ð4Ñ (
2) œ 24, f ÐnÑ (
2) œ 0 if n   5 Ê x%  x#  1 48 24 # $ % # $ % œ 21
36(x  2)  50 2! (x  2)
3! (x  2)  4! (x  2) œ 21
36(x  2)  25(x  2)
8(x  2)  (x  2) 26. f(x) œ 3x&
x%  2x$  x#
2 Ê f w (x) œ 15x%
4x$  6x#  2x, f ww (x) œ 60x$
12x#  12x  2, f www (x) œ 180x#
24x  12, f Ð4Ñ (x) œ 360x
24, f Ð5Ñ (x) œ 360, f ÐnÑ (x) œ 0 if n   6; f(
1) œ
7, f w (
1) œ 23, f ww (
1) œ
82, f www (
1) œ 216, f Ð4Ñ (
1) œ
384, f Ð5Ñ (
1) œ 360, f ÐnÑ (
1) œ 0 if n   6 216 384 360 # $ % & Ê 3x&
x%  2x$  x#
2 œ
7  23(x  1)
82 2! (x  1)  3! (x  1)
4! (x  1)  5! (x  1) œ
7  23(x  1)
41(x  1)#  36(x  1)$
16(x  1)%  3(x  1)& 27. f(x) œ x
# Ê f w (x) œ
2x
$ , f ww (x) œ 3! x
% , f www (x) œ
4! x
& Ê f ÐnÑ (x) œ (
1)n (n  1)! x
n
2 ; f(1) œ 1, f w (1) œ
2, f ww (1) œ 3!, f www (1) œ
4!, f ÐnÑ (1) œ (
1)n (n  1)! Ê x"# _

œ 1
2(x
1)  3(x
1)#
4(x
1)$  á œ ! (
1)n (n  1)(x
1)n n œ0

28. f(x) œ

1 a1
xb3

Ê f w (x) œ 3(1
x)
4 , f ww (x) œ 12(1
x)
5 , f www (x) œ 60 (1
x)
6 Ê f ÐnÑ (x) œ

fa0b œ 1, f w a0b œ 3, f ww a0b œ 12, f www a0b œ 60, á , f ÐnÑ a0b œ _

œ!

n œ0

an  2b! 2

Ê

1 a1
xb3

an  2b! 2

(1
x)
n
3 ;

œ 1  3x  6x#  10x3  á

an  2ban  1b n x 2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

620

Chapter 10 Infinite Sequences and Series

29. f(x) œ ex Ê f w (x) œ ex , f ww (x) œ ex Ê f ÐnÑ (x) œ ex ; f(2) œ e# , f w (2) œ e# , á f ÐnÑ (2) œ e# Ê ex œ e#  e# (x
2) 

e# #

(x
2)# 

e$ 3!

_

(x
2)$  á œ !

n œ0

e# n!

(x
2)n

30. f(x) œ 2x Ê f w (x) œ 2x ln 2, f ww (x) œ 2x (ln 2)# , f www (x) œ 2x (ln 2)3 Ê f ÐnÑ (x) œ 2x (ln 2)n ; f(1) œ 2, f w (1) œ 2 ln 2, f ww (1) œ 2(ln 2)# , f www (1) œ 2(ln 2)$ , á , f ÐnÑ (1) œ 2(ln 2)n 2(ln 2)# #

Ê 2x œ 2  (2 ln 2)(x
1) 

(x
1)# 

2(ln 2)3 3!

_

(x
1)3  á œ !

n œ0

2(ln 2)n (x
1)n n!

31. f(x) œ cosˆ2x  12 ‰, f w (x) œ
2 sinˆ2x  12 ‰, f ww (x) œ
4 cosˆ2x  12 ‰, f www (x) œ 8 sinˆ2x  12 ‰, f a4b axb œ 24 cosˆ2x  12 ‰ß f a5b axb œ
25 sinˆ2x  12 ‰ß . . ; fˆ 14 ‰ œ
1, f w ˆ 14 ‰ œ 0, f ww ˆ 14 ‰ œ 4, f www ˆ 14 ‰ œ 0, f a4b ˆ 14 ‰ œ 24 , 2 4 f a5b ˆ 14 ‰ œ 0, . . ., f Ð2nÑ ˆ 14 ‰ œ a
1bn 22n Ê cosˆ2x  12 ‰ œ
1  2ˆx
14 ‰
32 ˆx
14 ‰  . . . _

œ!

n œ0

a
1bn 22n ˆ x a2nbx

2n
14 ‰


7 Î2 32. f(x) œ Èx  1, f w (x) œ 12 ax  1b
1Î2 , f ww (x) œ
14 ax  1b
3Î2 , f www (x) œ 38 ax  1b
5Î2 , f a4b (x) œ
15 , . . .; 16 ax  1b 1 1 3 15 1 1 1 5 f(0) œ 1, f w (0) œ , f ww (0) œ
, f www (0) œ , f a4b (0) œ
, . . . Ê Èx  1 œ 1  x
x2  x3
x4  Þ Þ Þ 2

4

8

16

_

a
1bn 2n a2nbx x

33. The Maclaurin series generated by cos x is ! n œ0

by _

! n œ0

2 1
x

2

8

16

which converges on a
_, _b and the Maclaurin series generated

_

is 2 ! xn which converges on a
1, 1b. Thus the Maclaurin series generated by faxb œ cos x
n œ0

a
1bn 2n a2nbx x

128

2 1
x

is given by

_


2 ! xn œ
1
2x
25 x2
Þ Þ Þ Þ which converges on the intersection of a
_, _b and a
1, 1b, so the nœ0

interval of convergence is a
1, 1b. _

34. The Maclaurin series generated by ex is ! n œ0

xn nx

which converges on a
_, _b. The Maclaurin series generated by _

faxb œ a1
x  x2 bex is given by a1
x  x2 b !

n œ0

_

35. The Maclaurin series generated by sin x is ! n œ0 _

generated by lna1  xb is !

n œ1

a
1bnc1 n x n

xn nx

œ 1  12 x2  23 x3 Þ Þ Þ Þ which converges on a
_, _bÞ

a
1bn 2n1 a2n  1bx x

which converges on a
_, _b and the Maclaurin series

which converges on a
1, 1b. Thus the Maclaurin series genereated by

_

faxb œ sin x † lna1  xb is given by Œ !

n œ0

_

a
1bn a
1bnc1 n 2n1 Œ ! n x  a2n  1bx x n œ1

œ x2
12 x3  61 x4
Þ Þ Þ Þ which converges on

the intersection of a
_, _b and a
1, 1b, so the interval of convergence is a
1, 1b. _

36. The Maclaurin series generated by sin x is ! n œ0

a
1bn 2n1 a2n  1bx x

_

genereated by faxb œ x sin2 x is given by xŒ !

n œ0

œ x3
13 x5  _

37. If ex œ !

n œ0

f ÐnÑ (a) n!

2 7 45 x

which converges on a
_, _b. The Maclaurin series

2 a
1 b n 2n1  a2n  1bx x

_

œ xŒ !

nœ0

_

a
1 b n a
1 b n 2n1 2n1 Œ ! a2n  1bx x  a2n  1bx x n œ0

 . . . which converges on a
_, _bÞ

(x
a)n and f(x) œ ex , we have f ÐnÑ (a) œ ea f or all n œ 0, 1, 2, 3, á !

Ê ex œ ea ’ (x
0!a) 

(x
a)" 1!



(x
a)# 2!

 á “ œ ea ’1  (x
a) 

(x
a)# 2!

 á “ at x œ a

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 10.9 Convergence of Taylor Series

621

38. f(x) œ ex Ê f ÐnÑ (x) œ ex for all n Ê f ÐnÑ (1) œ e for all n œ 0, 1, 2, á Ê ex œ e  e(x
1) 

e #!

(x
1)# 

e 3!

(x
1)$  á œ e ’1  (x
1) 

f ww (a) f www (a) # $ w # (x
a)  3! (x
a)  á Ê f (x) www œ f w (a)  f ww (a)(x
a)  f 3!(a) 3(x
a)#  á Ê f ww (x) œ f ww (a)  f www (a)(x Ðn 2Ñ Ê f ÐnÑ (x) œ f ÐnÑ (a)  f Ðn1Ñ (a)(x
a)  f # (a) (x
a)#  á w w Ðn Ñ Ðn Ñ

(x
1)# 2!

39. f(x) œ f(a)  f w (a)(x
a) 

Ê f(a) œ f(a)  0, f (a) œ f (a)  0, á , f

(a) œ f


a) 



(x
1)$ 3!

f Ð4Ñ (a) 4!

á“

4 † 3(x
a)#  á

(a)  0

40. E(x) œ f(x)
b!
b" (x
a)
b# (x
a)#
b$ (x
a)$
á
bn (x
a)n Ê 0 œ E(a) œ f(a)
b! Ê b! œ f(a); from condition (b), lim

xÄa

Ê Ê

f(x)
f(a)
b" (x
a)
b# (x
a)#
b$ (x
a)$
á
bn (x
a)n (x
a)n

œ0

w a)#
á
nbn (x
a)n 1 lim f (x)
b"
2b# (x
a)
n(x3b
$ (xa)
œ0 n 1 xÄa f ww (x)
2b#
3! b$ (x
a)
á
n(n
")bn (x
a)n w b" œ f (a) Ê xlim n(n
1)(x
a)n 2 Äa " #

f ww (a) Ê xlim Äa " www œ b$ œ 3! f (a) Ê xlim Äa Ê b# œ

g(x) œ f(a)  f w (a)(x
a) 

f www (x)
3! b$
á
n(n
1)(n
2)bn (x
a)n n(n
1)(n
#)(x
a)n

f

ÐnÑ

(x)
n! bn n!

f ww (a) 2!

œ 0 Ê bn œ

(x
a)#  á 

3

3

" n!

f ÐnÑ (a) n!

2

œ0

œ0

f ÐnÑ (a); therefore, (x
a)n œ Pn (x) #

41. f(x) œ ln (cos x) Ê f w (x) œ
tan x and f ww (x) œ
sec# x; f(0) œ 0, f w (0) œ 0, f ww (0) œ
1 Ê L(x) œ 0 and Q(x) œ
x2 42. f(x) œ esin x Ê f w (x) œ (cos x)esin x and f ww (x) œ (
sin x)esin x  (cos x)# esin x ; f(0) œ 1, f w (0) œ 1, f ww (0) œ 1 Ê L(x) œ 1  x and Q(x) œ 1  x 
"Î#

43. f(x) œ a1
x# b

x# #

Ê f w (x) œ x a1
x# b

f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1 


$Î#

and f ww (x) œ a1
x# b


$Î#

 3x# a1
x# b


&Î#

; f(0) œ 1, f w (0) œ 0,

x# #

44. f(x) œ cosh x Ê f w (x) œ sinh x and f ww (x) œ cosh x; f(0) œ 1, f w (0) œ 0, f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1  45. f(x) œ sin x Ê f w (x) œ cos x and f ww (x) œ
sin x; f(0) œ 0, f w (0) œ 1, f ww (0) œ 0 Ê L(x) œ x and Q(x) œ x 46. f(x) œ tan x Ê f w (x) œ sec# x and f ww (x) œ 2 sec# x tan x; f(0) œ 0, f w (0) œ 1, f ww œ 0 Ê L(x) œ x and Q(x) œ x 10.9 CONVERGENCE OF TAYLOR SERIES _

1. ex œ 1  x 

x# #!

á œ !

2. ex œ 1  x 

x# #!

á œ !

nœ0 _

nœ0

xn n!

Ê e
5x œ 1  (
5x) 

(
5x)# #!

 á œ 1
5x 

xn n!

Ê e
xÎ2 œ 1  ˆ
#x ‰ 

ˆ
x# ‰# #!

á œ1


_

3. sin x œ x


x$ 3!



x& 5!


á œ!

4. sin x œ x


x$ 3!



x& 5!


á œ!

n œ0 _

nœ0

(
1)n x2n 1 (#n1)!

Ê 5 sin (
x) œ 5 ’(
x)


(
1)n x2n 1 (#n1)!

Ê sin

1x #

œ

1x #




ˆ 1#x ‰$ 3!



(
x)$ 3!

ˆ 1#x ‰& 5!

x #








x# 2# #!

(
x)& 5!

ˆ 1#x ‰( 7!

5# x# #!







_

5$ x$ 3!

x$ 2$ 3!

á œ!

nœ0

_

á œ !

nœ0

_

(
1)n xn 2n n!

x
á “ œ ! 5(
(1) #n1)!

n 1 2n 1

n œ0

_

1 x  á œ ! (
21) 2n 1 (#n1)! nœ0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

(
1)n 5n xn n!

n 2n 1 2n 1

x# #

622

Chapter 10 Infinite Sequences and Series _

5. cos x œ !

n œ0

_

6. cos x œ !

n œ0

x$ 2†2!

œ1


Ê cos 5x2 œ !

a
1bn x2n (2n)!

$Î# cos Š xÈ ‹ 2

x' 2# †4!



_

7. lna1  xb œ !

n œ1

_

nœ0

10.

Ê

x* 2$ †6!




1 1
x

œ ! xn Ê _

n œ0

n œ0

_

nœ0 _

(
1)n x2n (2n)!

13. cos x œ !

n œ0

œ

x% 4!




x' 6!



_

(
1)n x2nb1 (2n1)!

14. sin x œ !

n œ0

œ Šx


x$ 3!

_

n œ0 _

16. cos x œ !

n œ0

œ1


" #




œ!

n œ1

nœ0

nœ0

_

xnb1 n!

œ!

n œ0

nœ0

x"! 10!

x# #


1  cos x œ

x( 7!

_

x# #

_

a
1bn 32nb1 x8nb4 n

œ!

nœ0

x4 2

x$ #!

œ x  x# 

_

(
1)n x2nb1 (#n1)! 

nœ0

_

(
1)n x2n (#n)!

n œ0

x& 4!



(
1)n x2nb3 (2n1)!

œ!


1!

 14 x  18 x2 

x% 3!



x# #

œ




x8 4

œ 3x4
9x12 

n

" #

x6 3






9 6 16 x

...

243 20 5 x

27 9 64 x

1 3 16 x

...

x( 5!

x* 7!




2187 28 7 x

...

á

œ x$


x& 3!


11




x# 2






x% 4!




á

x' 6!

x) 8!






x"! 10!

á

n œ2



x* 9!




x"" 11!

x$ 3!

_

œ Œ!

n œ0

 á‹
x _

(
1)n x2n (2n)!

Ê x# cos ax# b œ x# !

nœ0

n œ0

_

" #

(2x)% 2†4!




(2x)' 2†6!



" #

" #

_

œ!

nœ1






" #

! (
1) (2x) œ (2n)!

œ

n

2n

n œ0

(2x)) 2†8!

(
1)n x2nb1 (#n1)!  x$ 3!

(
1)n (1x)2n (#n)!

_

cos 2x #

(
1)nb1 (2x)2n #†(2n)!

nœ1

œ x2


n 2n

Ê x cos 1x œ x !



á

x  á œ ! (
(1) #n)!

Ê sin x
x 

2x ‰ 18. sin# x œ ˆ 1
cos œ # _

_

(
1)n x2n (2n)!



(2x)# 2†2!




x& 5!



15. cos x œ !

17. cos# x œ

x) 8!

_

Ê x# sin x œ x# Œ !

Ê

15625x12 6!




(
1)n x3n 2n (2n)!

nœ0

_

(
1)n x2nb1 (2n1)!

_

œ!

nœ0

a
1bnc1 x2n n

œ!

n n 1 œ #" ! ˆ #" x‰ œ ! ˆ #" ‰ xn œ

xn n! 

Ê xex œ x Œ !

12. sin x œ !

_

n

_

n

nœ0

_

xn n!

11. ex œ !

(#n)!

nœ0

2nb1 a
1bn ˆ3x4 ‰ 2n  1

_

" 1 # 1
"# x

œ

1 2
x



œ ! a
1bn ˆ 34 x3 ‰ œ ! a
1bn ˆ 34 ‰ x3n œ 1
34 x3 

1 1  34 x3

n œ0

n œ0

nœ1

nœ0

_

œ!

625x8 4!



2n

"Î#

$

a
1bn ŒŠ x# ‹

_

a
1bnc1 ˆx2 ‰ n

Ê lna1  x2 b œ ! _

œ ! a
1bn xn Ê

n œ0

25x4 #!

œ1


á

Ê tan
1 a3x4 b œ !

1 1x

_

œ

(
1)n 52n x4n (2n)!

œ!

"Î# $ cos ŒŠ x# ‹ 

_

a
1bnc1 xn n

_

2n

(
1)n
5x2 ‘ (2n)!

n œ0

a
1bn x2nb1 2n  1

8. tan
1 x œ !

9.

_

(
1)n x2n (2n)!

cos 2x œ

_

n œ1

" #

_

nœ0

2n


"# Š1


x( 7!

_

n œ0

(2x)# 2!

(
1)n (2x)2n 2†(2n)!

(2x)# #!



x* 9!

x$ 3!




(
")n 12n x2nb1 (#n)!

œ!

 "# ’1



á œ1!




œ!

(
1)n ax# b (#n)!

" #

x& 5!

œ


x



(
")n x4n (#n)!



(2x)% 4!

x"" 11!

œx


2

(2x)' 6!


_

n œ1




(2x)' 6!

n œ2

1 # x$ 2!

œ x#


œ1!

(2x)% 4!

_

á œ!



x' 2!





(2x)) 8!

(
1)n x2n 1 (2n1)!

1 % x& 4!




1 ' x( 6!

x"! 4!




x"% 6!

á

á


á“

(
1)n 22n 1 x2n (2n)!

 á‹ œ

(2x)# 2†2!




(2x)% 2†4!



(2x)' 2†6!

(
1)n 22n 1 x2n (2n)!

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.


á

...

Section 10.9 Convergence of Taylor Series 19.

x# 1
2x

_

n œ1

22.

" 1
x

_

n œ0

n œ0

(
1)nc1 (2x)n n

20. x ln (1  2x) œ x !

21.

_

œ x# ˆ 1
"2x ‰ œ x# ! (2x)n œ ! 2n xn2 œ x#  2x$  2# x%  2$ x&  á _

(
1)nc1 2n xn n

œ!

n œ1

_

œ ! xn œ 1  x  x#  x$  á Ê

2 a 1
x b$

d dx

nœ0

œ

d# dx#

ˆ 1
" x ‰ œ

d dx

Š (1
"x)# ‹ œ

d dx

1

œ 2x#


ˆ 1
" x ‰ œ

2# x$ #

" (1
x)#



2$ x% 4




2% x& 5

á _

_

nœ1

nœ0

œ 1  2x  3x#  á œ ! nxn
1 œ ! (n  1)xn _

a1  2x  3x#  á b œ 2  6x  12x#  á œ ! n(n
1)xn
2 n œ2

_

œ ! (n  2)(n  1)xn n œ0

3

5

7

23. tan
1 x œ x
13 x3  15 x5
17 x7  Þ Þ Þ Ê x tan
1 x2 œ xŠx2
13 ax2 b  15 ax2 b
17 ax2 b  Þ Þ Þ ‹ _

œ x3
13 x7  15 x11
17 x15  Þ Þ Þ œ !

n œ1

x3 3!

24. sin x œ x
œx


4 x3 3!

16 x5 5!



x2 2!

25. ex œ 1  x  œ Š1  x  26. sin x œ x
œ Š1
_

2

x 2!

x5 5!



x2 2!

x3 3!





4

1) x œ ! Š (
(2n)!
n œ0

x3 3!




x 4!

n 2n

x3 3!

x5 5!

x7 7!

64 x7 7!











a
1bn x4nc1 2n
1

 á Ê sin x † cos x œ "# sin 2x œ "# Š2x
á œx


 á and

1 1x

2 x3 3



2x5 15

4 x7 315




_

á œ!

n œ0

a2xb3 3!



x7 7!

œ 1
x  x2
x3  á Ê ex 

6

x 6!

 á and cos x œ 1


 á ‹
Šx


3

x 3!



x2 2!

x4 4!




x6 6!

a2xb7 7!

 á‹

1 1x 25 4 24 x

_

 á œ ! ˆ n!1  a
1bn ‰xn n œ0

 á Ê cos x
sin x

5




x 3

lna1  x2 b œ x3 Šx2
12 ax2 b  13 ax2 b
14 ax2 b  á ‹

x 5!

7






(
1)n 22n x2nb1 (#n1)!

 á ‹  a1
x  x2
x3  á b œ 2  32 x2
56 x3 




a2xb5 5!

x 7!

á‹ œ 1
x


x2 2!



x3 3!



x4 4!




x5 5!




x6 6!



x7 7!

á

(
1)n x2nb1 (#n1)! ‹

27. lna1  xb œ x
12 x2  13 x3
14 x4  á Ê œ 13 x3
16 x5  19 x7


1 9 12 x

_

2

3

4

nc1

 á œ ! a
13nb x2n1 n œ1

28. lna1  xb œ x
12 x2  13 x3
14 x4  á and lna1
xb œ
x
12 x2
13 x3
14 x4  á Ê lna1  xb
lna1
xb _

œ ˆx
12 x2  13 x3
14 x4  á ‰
ˆ
x
12 x2
13 x3
14 x4  á ‰ œ 2x  23 x3  25 x5  á œ ! 2n 2 1 x2n1 n œ0

29. ex œ 1  x  œ Š1  x 

x2 2! x2 2!

 

x3 3! x3 3!

 á and sin x œ x
 á ‹Šx


x3 3!



x5 5!




x3 3!



x5 5!

x7 7!

 á ‹ œ x  x2  13 x3





x7 7!

 á Ê ex † sin x 1 5 30 x


ÞÞÞÞ

30. lna1  xb œ x
12 x2  31 x3
41 x4  á and 1
" x œ 1  x  x#  x$  á Ê ln1a1
xxb œ lna1  xb † 7 4 œ ˆx
12 x2  13 x3
14 x4  á ‰a1  x  x#  x$  á b œ x  12 x2  56 x3  12 x  ÞÞÞÞ

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" 1
x

623

624

Chapter 10 Infinite Sequences and Series 2

31. tan
1 x œ x
13 x3  15 x5
17 x7  Þ Þ Þ Ê atan
1 xb œ atan
1 xbatan
1 xb 44 8 6 œ ˆx
13 x3  15 x5
17 x7  Þ Þ Þ‰ˆx
13 x3  15 x5
17 x7  Þ Þ Þ‰ œ x2
23 x4
23 45 x
105 x  Þ Þ Þ Þ 32. sin x œ x


x3 3!

x5 5!



x7 7!




 á and cos x œ 1


œ cos x † "# sin 2x œ "# Š1
33. sin x œ x


x3 3!

x5 5!



x7 7!




3

Ê esin x œ 1  Šx


x 3!

x2 2!



x4 4!




x6 6!

x2 2!






x 5!

7

x 7!

x2 2!

 á ‹  12 Šx


x6 6!


a2xb 3!

 á ‹Š2x


 á and ex œ 1  x  5

x4 4!



3

 á Ê cos2 x † sin x œ cos x † cos x † sin x



x3 3!

á

3



x5 5!

x 3!

a2xb5 5!






x7 7!




a2xb7 7!

 á ‹ œ x
76 x3 

2

 á ‹  16 Šx


x3 3!



x5 5!




x7 7!

61 5 120 x




1247 7 5040 x

 ÞÞÞ

3

á‹ á

œ 1  x  12 x2
18 x4  Þ Þ Þ Þ x3 x5 x7 1 3 1 5 1 7 1 3 1 5
1
1 ˆ 3!  5!
7!  á and tan x œ x
3 x  5 x
7 x  Þ Þ Þ Ê sinatan xb œ x
3 x  5 x 3 5 1 ˆ 1 ˆ
16 ˆx
31 x3  51 x5
71 x7  Þ Þ Þ‰  120 x
13 x3  15 x5
17 x7  Þ Þ Þ‰
5040 x
13 x3  15 x5
17 x7  5 7 x
12 x3  38 x5
16 x  ÞÞÞ

34. sin x œ x
œ


71 x7  Þ Þ Þ‰ 7

Þ Þ Þ‰  á

35. Since n œ 3, then f a4b axb œ sin x, lf a4b axbl Ÿ M on Ò0, 0.1Ó Ê lsin xl Ÿ 1 on Ò0, 0.1Ó Ê M œ 1. Then lR3 a0.1bl Ÿ 1 l0.14
x 0l

4

œ 4.2 ‚ 10
6 Ê error Ÿ 4.2 ‚ 10
6

36. Since n œ 4, then f a5b axb œ ex , lf a5b axbl Ÿ M on Ò0, 0.5Ó Ê lex l Ÿ Èe on Ò0, 0.5Ó Ê M œ 2.7. Then lR4 a0.5bl Ÿ 2.7 l0.55
x 0l œ 7.03 ‚ 10
4 Ê error Ÿ 7.03 ‚ 10
4 5

kxk& 5!

37. By the Alternating Series Estimation Theorem, the error is less than 5 Ê kxk  È 6 ‚ 10
# ¸ 0.56968 38. If cos x œ 1


Ê kxk&  a5!b a5 ‚ 10
% b Ê kxk&  600 ‚ 10
%

%

x# #

and kxk  0.5, then the error is less than ¹ (.5) 24 ¹ œ 0.0026, by Alternating Series Estimation Theorem;

since the next term in the series is positive, the approximation 1


x# #

is too small, by the Alternating Series Estimation

Theorem 39. If sin x œ x and kxk  10
$ , then the error is less than

a10c$ b 3!

$

¸ 1.67 ‚ 10
10 , by Alternating Series Estimation Theorem; $

The Alternating Series Estimation Theorem says R# (x) has the same sign as
x3! . Moreover, x  sin x Ê 0  sin x
x œ R# (x) Ê x  0 Ê
10
$  x  0.

40. È1  x œ 1 

x #




x# 8



x$ 16

#


á . By the Alternating Series Estimation Theorem the kerrork  ¹
8x ¹ 

œ 1.25 ‚ 10
& c $

3Ð0Þ1Ñ (0.1)$ 3!

c $

(0.1)$ 3!

41. kR# (x)k œ ¹ e3!x ¹  42. kR# (x)k œ ¹ e3!x ¹ 

2x ‰ 43. sin# x œ ˆ 1
cos œ #

Ê

d dx

asin# xb œ

œ 2x


(2x)$ 3!



d dx

(2x)& 5!

" #

 1.87 ‚ 10
4 , where c is between 0 and x

œ 1.67 ‚ 10
% , where c is between 0 and x
#

" #

Š 2x 2!



(2x)( 7!

cos 2x œ 2$ x% 4!



2& x' 6!

" #


"# Š1


(2x)# 2!


á ‹ œ 2x




(2x)% 4!

(2x)$ 3!






(2x)' 6!

(2x)& 5!




 á‹ œ (2x)( 7!

2x# #!




2$ x% 4!



2& x' 6!


á

 á Ê 2 sin x cos x

 á œ sin 2x, which checks

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

(0.01)# 8

Section 10.9 Convergence of Taylor Series 44. cos# x œ cos 2x  sin# x œ Š1
œ1


#

2x #!



$ %

2 x 4!




& '

2 x 6!

(2x)# #!



(2x)% 4!




(2x)' 6!

 á œ 1
x#  "3 x%


2 45



(2x)) 8!

x' 

#

 á ‹  Š 2x #!


" 315

2$ x% 4!



2& x' 6!




2( x) 8!

625

 á‹

x)
á

45. A special case of Taylor's Theorem is f(b) œ f(a)  f w (c)(b
a), where c is between a and b Ê f(b)
f(a) œ f w (c)(b
a), the Mean Value Theorem. 46. If f(x) is twice differentiable and at x œ a there is a point of inflection, then f ww (a) œ 0. Therefore, L(x) œ Q(x) œ f(a)  f w (a)(x
a). 47. (a) f ww Ÿ 0, f w (a) œ 0 and x œ a interior to the interval I Ê f(x)
f(a) œ Ê f(x) Ÿ f(a) throughout I Ê f has a local maximum at x œ a (b) similar reasoning gives f(x)
f(a) œ local minimum at x œ a

f ww (c# ) #

f ww (c# ) #

(x
a)# Ÿ 0 throughout I

(x
a)#   0 throughout I Ê f(x)   f(a) throughout I Ê f has a

48. f(x) œ (1
x)
" Ê f w (x) œ (1
x)
# Ê f ww (x) œ 2(1
x)
$ Ê f Ð3Ñ (x) œ 6(1
x)
% Ê f Ð4Ñ (x) œ 24(1
x)
& ; therefore

" 1
x

¸ 1  x  x#  x$ . kxk  0.1 Ê

&

%

Ð4Ñ

10 11



" 1
x



10 9

‰ Ê ¹ (1
"x)& ¹  ˆ 10 9

&

%

‰ Ê the error e$ Ÿ ¹ max f 4! (x) x ¹  (0.1)% ˆ 10 ‰ œ 0.00016935  0.00017, since ¹ f Ê ¹ (1
x x)& ¹  x% ˆ 10 9 9

Ð4Ñ

&

(x) 4! ¹

œ ¹ (1
"x)& ¹ .

49. (a) f(x) œ (1  x)k Ê f w (x) œ k(1  x)k
1 Ê f ww (x) œ k(k
1)(1  x)k
2 ; f(0) œ 1, f w (0) œ k, and f ww (0) œ k(k
1) Ê Q(x) œ 1  kx  k(k #
") x# " (b) kR# (x)k œ ¸ 3†3!2†" x$ ¸  100 Ê kx$ k 

" 100

Ê 0x

" 100"Î$

or 0  x  .21544

50. (a) Let P œ x  1 Ê kxk œ kP
1k  .5 ‚ 10
n since P approximates 1 accurate to n decimals. Then, P  sin P œ (1  x)  sin (1  x) œ (1  x)
sin x œ 1  (x
sin x) Ê k(P  sin P)
1k œ ksin x
xk Ÿ

kxk$ 3!



0.125 3!

‚ 10
3n  .5 ‚ 10
3n Ê P  sin P gives an approximation to 1 correct to 3n decimals.

_

_

n œ0

n œk

51. If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n
1)(n
2)â(n
k  1)an xn
k and f ÐkÑ (0) œ k! ak Ê ak œ

f ÐkÑ (0) k!

for k a nonnegative integer. Therefore, the coefficients of f(x) are identical with the corresponding

coefficients in the Maclaurin series of f(x) and the statement follows. 52. Note: f even Ê f(
x) œ f(x) Ê
f w (
x) œ f w (x) Ê f w (
x) œ
f w (x) Ê f w odd; f odd Ê f(
x) œ
f(x) Ê
f w (
x) œ
f w (x) Ê f w (
x) œ f w (x) Ê f w even; also, f odd Ê f(
0) œ f(0) Ê 2f(0) œ 0 Ê f(0) œ 0 (a) If f(x) is even, then any odd-order derivative is odd and equal to 0 at x œ 0. Therefore, a" œ a$ œ a& œ á œ 0; that is, the Maclaurin series for f contains only even powers. (b) If f(x) is odd, then any even-order derivative is odd and equal to 0 at x œ 0. Therefore, a! œ a# œ a% œ á œ 0; that is, the Maclaurin series for f contains only odd powers. 53-58. Example CAS commands: Maple: f := x -> 1/sqrt(1+x); x0 := -3/4; x1 := 3/4; # Step 1: plot( f(x), x=x0..x1, title="Step 1: #53 (Section 10.9)" );

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

626

Chapter 10 Infinite Sequences and Series

# Step 2: P1 := unapply( TaylorApproximation(f(x), x = 0, order=1), x ); P2 := unapply( TaylorApproximation(f(x), x = 0, order=2), x ); P3 := unapply( TaylorApproximation(f(x), x = 0, order=3), x ); # Step 3: D2f := D(D(f)); D3f := D(D(D(f))); D4f := D(D(D(D(f)))); plot( [D2f(x),D3f(x),D4f(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 3: #57 (Section 9.9)" ); c1 := x0; M1 := abs( D2f(c1) ); c2 := x0; M2 := abs( D3f(c2) ); c3 := x0; M3 := abs( D4f(c3) ); # Step 4: R1 := unapply( abs(M1/2!*(x-0)^2), x ); R2 := unapply( abs(M2/3!*(x-0)^3), x ); R3 := unapply( abs(M3/4!*(x-0)^4), x ); plot( [R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 4: #53 (Section 10.9)" ); # Step 5: E1 := unapply( abs(f(x)-P1(x)), x ); E2 := unapply( abs(f(x)-P2(x)), x ); E3 := unapply( abs(f(x)-P3(x)), x ); plot( [E1(x),E2(x),E3(x),R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], linestyle=[1,1,1,3,3,3], title="Step 5: #53 (Section 10.9)" ); # Step 6: TaylorApproximation( f(x), view=[x0..x1,DEFAULT], x=0, output=animation, order=1..3 ); L1 := fsolve( abs(f(x)-P1(x))=0.01, x=x0/2 ); # (a) R1 := fsolve( abs(f(x)-P1(x))=0.01, x=x1/2 ); L2 := fsolve( abs(f(x)-P2(x))=0.01, x=x0/2 ); R2 := fsolve( abs(f(x)-P2(x))=0.01, x=x1/2 ); L3 := fsolve( abs(f(x)-P3(x))=0.01, x=x0/2 ); R3 := fsolve( abs(f(x)-P3(x))=0.01, x=x1/2 ); plot( [E1(x),E2(x),E3(x),0.01], x=min(L1,L2,L3)..max(R1,R2,R3), thickness=[0,2,4,0], linestyle=[0,0,0,2], color=[red,blue,green,black], view=[DEFAULT,0..0.01], title="#53(a) (Section 10.9)" ); abs(`f(x)`-`P`[1](x) ) u*cos(v); y := (u,v) ->u*sin(v); z := (u,v) -> u; plot3d( [x(u,v),y(u,v),z(u,v)], u=0..2, v=0..2*Pi, axes=boxed, style=patchcontour, contours=[($0..4)/2], shading=zhue, title="#77 (Section 14.1)" ); 69-60. Example CAS commands: Mathematica: (assigned functions and bounds will vary) For 69 - 72, the command ContourPlot draws 2-dimensional contours that are z-level curves of surfaces z = f(x,y). Clear[x, y, f] f[x_, y_]:= x Sin[y/2]  y Sin[2x] xmin= 0; xmax= 51; ymin= 0; ymax= 51; {x0, y0}={31, 31}; cp= ContourPlot[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, ContourShading Ä False]; cp0= ContourPlot[[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, Contours Ä {f[x0,y0]}, ContourShading Ä False, PlotStyle Ä {RGBColor[1,0,0]}]; Show[cp, cp0] For 73 - 76, the command ContourPlot3D will be used. Write the function f[x, y, z] so that when it is equated to zero, it represents the level surface given. For 73, the problem associated with Log[0] can be avoided by rewriting the function as x2 + y2 +z2 - e1/4 Clear[x, y, z, f] f[x_, y_, z_]:= x2  y2  z2
Exp[1/4] ContourPlot3D[f[x, y, z], {x,
5, 5}, {y,
5, 5}, {z,
5, 5}, PlotPoints Ä {7, 7}]; For 77 - 80, the command ParametricPlot3D will be used. To get the z-level curves here, we solve x and y in terms of z and either u or v (v here), create a table of level curves, then plot that table. Clear[x, y, z, u, v] ParametricPlot3D[{u Cos[v], u Sin[v], u}, {u, 0, 2}, {v, 0, 2p}]; zlevel= Table[{z Cos[v], z sin[v]}, {z, 0, 2, .1}]; ParametricPlot[Evaluate[zlevel],{v, 0, 21}]; 14.2 LIMITS AND CONTINUITY IN HIGHER DIMENSIONS 3x#
y#  5

1.

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x  y  2

2.

lim Ðxß yÑ Ä Ð0ß 4Ñ Èy

x

œ

0 È4

œ

3(0)#
0#  5 0#  0#  2

œ

5 #

œ0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.2 Limits and Continuity in Higher Dimensions 3.

4.

5.

6.

7.

8.

9.

10.

lim

Ðxß yÑ Ä Ð3ß 4Ñ

Èx#  y#
1 œ È3#  4#
1 œ È24 œ 2È6 #

lim

sec x tan y œ (sec 0) ˆtan 14 ‰ œ (1)(1) œ 1

lim

cos Š xxy y 1 ‹ œ cos Š 000 0 1 ‹ œ cos 0 œ 1

Ðxß yÑ Ä Ð2ß
3Ñ

Ðxß yÑ Ä ˆ0ß 14 ‰

Ðxß yÑ Ä Ð0ß 0Ñ

lim

#

lim

Ðxß yÑ Ä Ð1ß 1Ñ

Ðxß yÑ Ä Ð1Î27ß 13 Ñ

œ

12.

lim y
sin x Ðxß yÑ Ä ˆ 12 ß 0‰

15.

16.

17.

1†sinˆ 16 ‰ 1#  1

x#
2xy  y# x
y

lim

x#
y# x
y

lim

xy
y
2x  2 x
1

œ

œ

œ

œ

lim

Ðxß yÑ Ä Ð1ß 1Ñ

œ

œ

(x  y)(x
y) x
y

œ

lim

Ðx ß y Ñ Ä Ð 1 ß 1Ñ xÁ1

y4

x
y  2È x
2È y Èx
Èy

œ

œ

œ
2

(x
y)# x
y

lim

1 2

1 4

11
1

Ðx ß y Ñ Ä Ð 1 ß 1Ñ

lim # # Ðxß yÑ Ä Ð2ß
4Ñ x y
xy  4x
4x # y Á
4, x Á x lim

xÄ0

1Î2 2

œ

acos 0b  " 0
sin ˆ 1# ‰

œ

lim

Ðxß yÑ Ä Ð0ß 0Ñ xÁy

" #

aey b ˆ sinx x ‰ œ e! † lim ˆ sinx x ‰ œ 1 † 1 œ 1

lim

Ðxß yÑ Ä Ð0ß 0Ñ

œ

cos y  1

Ðxß yÑ Ä Ð1ß 1Ñ xÁ1

$

3 1 ‰ 3 3 xy œ cos É ˆ 27 cos È 1 œ cos ˆ 13 ‰ œ

lim

Ðxß yÑ Ä Ð1ß 1Ñ xÁy

" 36

ln k1  x# y# k œ ln k1  (1)# (1)# k œ ln 2

ey sin x x Ðxß yÑ Ä Ð0ß 0Ñ

Ðxß yÑ Ä Ð1ß 1Ñ xÁy

#

ex
y œ e0
ln 2 œ eln ˆ 2 ‰ œ

lim

lim

$

1

Ðxß yÑ Ä Ð0ß ln 2Ñ

x sin y # Ðxß yÑ Ä Ð1ß 1Î6Ñ x  1

14.

#

Š x"  y" ‹ œ
#"  ˆ
"3 ‰‘ œ ˆ 6" ‰ œ

11.

13.

#

lim

lim

(x
y) œ ("
1) œ 0

lim

(x  y) œ (1  1) œ 2

Ðxß yÑ Ä Ð1ß 1Ñ

Ðxß yÑ Ä Ð1ß 1Ñ

(x
1)(y
2) x
1

œ

lim

Ðxß yÑ Ä Ð1ß 1Ñ

y4

lim Ðxß yÑ Ä Ð2ß
4Ñ x(x
1)(y  4) y Á
4, x Á x# lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ xÁy

œ

(y
2) œ (1
2) œ
1

1

lim Ðxß yÑ Ä Ð2ß
4Ñ x(x
1) x Á x#

ˆÈ x
È y ‰ ˆ È x  È y  2 ‰ Èx
Èy

œ

lim

Ðxß yÑ Ä Ð0ß 0Ñ

œ

" #(2
1)

Note: (xß y) must approach (0ß 0) through the first quadrant only with x Á y. xy
4

lim Ðxß yÑ Ä Ð2ß 2Ñ Èx  y
2 xyÁ4

œ

lim

Ðxß yÑ Ä Ð2ß 2Ñ xyÁ4

ˆÈx  y  2‰ ˆÈx  y
2‰ Èx  y
2

œ

lim

Ðxß yÑ Ä Ð2ß 2Ñ xyÁ4

" #

ˆÈ x  È y  2 ‰

œ ŠÈ0  È0  2‹ œ 2

18.

œ

ˆÈ x  y  2 ‰

œ ŠÈ2  2  2‹ œ 2  2 œ 4

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

805

806

Chapter 14 Partial Derivatives

19.

lim

œ 20.

22.

23.

24.

25.

26.

27.

28.

29.

30.

" È(2)(2)
0  #

" 22

œ

" È4  È3  1

œ

" 22

È2x
y
2

œ

lim Ðxß yÑ Ä Ð2ß 0Ñ ˆÈ2x
y  2‰ ˆÈ2x
y
2‰ 2x
y Á 4

œ

" 4

È x
È y 1 x
y
1

lim

Ðxß yÑ Ä Ð4ß 3Ñ x
yÁ1

œ 21.

È2x
y
2 2x
y
4

Ðxß yÑ Ä Ð2ß 0Ñ 2x
y Á 4

œ

œ

Èx
Èy  1

lim Ðxß yÑ Ä Ð4ß 3Ñ ˆÈx  Èy  1‰ ˆÈx
Èy  1‰ x
yÁ1

sinax#  y# b x#  y#

œ lim

sinar# b r#

lim

1
cosaxyb xy

œ lim

1
cos u u

Ðxß yÑ Ä Ð0ß 0Ñ

x3  y3 Ðxß yÑ Ä Ð1ß
"Ñ x  y

lim

x
y

lim 4 4 Ðx ß y Ñ Ä Ð 2 ß 2 Ñ x
y lim

T Ä Ð1ß 3ß 4Ñ

Š "x 

lim

T Ä Ð 1 ß
1 ß
1Ñ

lim

T Ä Ð3ß 3ß 0Ñ

lim

lim

lim

œ

uÄ0

œ lim

rÄ0

œ lim

2r†cosar# b 2r

uÄ0

sin u 1

rÄ0

œ0

ax  ybˆx2
xy  y2 ‰ xy Ðxß yÑ Ä Ð1ß
"Ñ x
y

lim 2 2 Ðxß yÑ Ä Ð2ß 2Ñ ax  ybax
ybax  y b

œ

"

lim Ðxß yÑ Ä Ð4ß 3Ñ Èx  Èy  1

œ lim cosar# b œ 1

œ

lim

 "z ‹ œ

2xy  yz x #  z#

œ

" 1



" 3



" 4

œ

2(1)(
1)  (
1)(
1) 1#  (
1)#

12  4  3 12

œ


2  " 11

œ

œ

lim

Ðxß yÑ Ä Ð1ß
"Ñ

ax2
xy  y2 b œ Š12
a1ba
1b  a
1b2 ‹ œ 3 1

lim 2 2 Ðxß yÑ Ä Ð2ß 2Ñ ax  ybax  y b

œ

1 a2  2ba22  22 b

œ

1 32

19 12

œ
#"

asin# x  cos# y  sec# zb œ asin# 3  cos# 3b  sec# 0 œ 1  1# œ 2

T Ä ˆ
14 ß 12 ß 2‰

T Ä Ð1ß 0ß 3Ñ

" y

rÄ0

œ

"

lim Ðxß yÑ Ä Ð2ß 0Ñ È2x
y  #

" 4

lim

Ðxß yÑ Ä Ð0ß 0Ñ

œ

ze

T Ä Ð2 ß
3 ß 6 Ñ

tan
" (xyz) œ tan
" ˆ
"4 † 2y

1 #

† 2‰ œ tan
" ˆ
14 ‰

cos 2x œ 3e 2Ð0Ñ cos 21 œ (3)(1)(1) œ 3

ln Èx#  y#  z# œ ln È2#  (
3)#  6# œ ln È49 œ ln 7

31. (a) All axß yb

(b) All axß yb except a0ß 0b

32. (a) All axß yb so that x Á y

(b) All axß yb

33. (a) All axß yb except where x œ 0 or y œ 0

(b) All axß yb

34. (a) All axß yb so that x#
3x  2 Á 0 Ê ax
2bax
1b Á 0 Ê x Á 2 and x Á 1 (b) All axß yb so that y Á x# 35. (a) All axß yß zb

(b) All axß yß zb except the interior of the cylinder x#  y# œ 1

36. (a) All axß yß zb so that xyz  0

(b) All axß yß zb

37. (a) All axß yß zb with z Á 0

(b) All axß yß zb with x#  z# Á 1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.2 Limits and Continuity in Higher Dimensions 38. (a) All axß yß zb except axß 0ß 0b

(b) All axß yß zb except a0ß yß 0b or axß 0ß 0b

39. (a) All axß yß zb such that z  x2  y2  1

(b) All axß yß zb such that z Á Èx2  y2

807

40. (a) All axß yß zb such that x2  y2  z2 Ÿ 4 (b) All axß yß zb such that x2  y2  z2   9 except when x2  y2  z2 œ 25 41.

lim




x È x#  y#

œ lim b
Èx#x x# œ lim b
È2x kxk œ lim b
Èx2 x œ lim b
È"2 œ
È"2 ; xÄ0 xÄ0 xÄ0 xÄ0

lim




x È x#  y#

œ lim c
È2x kxk œ lim c
È2(x
x) œ lim c

Ðxß yÑ Ä Ð0ß 0Ñ along y œ x x0 Ðxß yÑ Ä Ð0ß 0Ñ along y œ x x0

42.

43.

44.

45.

46.

47.

48.

49.

50.

lim

x% x%  y#

œ lim

lim

x%
y# x%  y#

œ lim

lim

xy kxyk

lim

x
y xy

lim

x2
y x
y

œ lim

lim

x#  y y

œ lim

lim

x# y x4  y2

œ lim

lim

xy2
1 y
1

œ lim

Ðxß yÑ Ä Ð0ß 0Ñ along y œ 0

Ðxß yÑ Ä Ð0ß 0Ñ along y œ kx#

Ðxß yÑ Ä Ð0ß 0Ñ along y œ kx kÁ0

Ðx ß y Ñ Ä Ð 0 ß 0 Ñ along y œ kx k Á
1

Ðxß yÑ Ä Ð0ß 0Ñ along y œ kx kÁ1

Ðxß yÑ Ä Ð0ß 0Ñ along y œ kx# kÁ0

Ðxß yÑ Ä Ð0ß 0Ñ along y œ kx#

Ðxß yÑ Ä Ð1ß 1Ñ along x œ 1

lim

Ðxß yÑ Ä Ð1ß
1Ñ along y œ
1

œ

xÄ0

xÄ0

x%

œ 1;

% # x Ä 0 x 0

x%
akx# b

#

œ lim

x(kx)

x Ä 0 kx(kx)k

œ lim

x
kx

x Ä 0 x  kx

1
k 1k

x#  kx# kx#

kx4

y2
1 y Ä 1 y
1

x
k

œ


k 1
k

# x Ä 0 x%  ax# b

" È2

œ lim

x%

% x Ä 0 2x

œ

" #

Ê different limits for different values of k

; if k  0, the limit is 1; but if k  0, the limit is
1

k 1  k2

Ê different limits for different values of k

yÄ1


1 x Ä 1 x1

œ lim

Ê different limits for different values of k, k Á 1

Ê different limits for different values of k, k Á 0

œ lim ay  1b œ 2;


x  1 2 x Ä 1 x
1

œ lim

1k k

œ

k

x Ä 0 kkk

x%

œ lim

1
k# 1  k#

œ

œ lim

x Ä 0 1
k

œ

x% x%  y#

œ

Ê different limits for different values of k, k Á
1

œ lim

4 2 4 x Ä 0 x k x

xy  1 x2
y2

kx#

# x Ä 0 kkx k

œ

x %
k# x%

% # % x Ä 0 x k x

œ lim

x2
kx x Ä 0 x
kx

xÄ0

lim

Ðxß yÑ Ä Ð0ß 0Ñ along y œ x#

œ lim

# x Ä 0 x%  akx# b

" È2

xÄ0

lim

Ðx ß y Ñ Ä Ð 1 ß 1Ñ along y œ x

œ
21 ;

lim

xy2
1 y
1

Ð x ß y Ñ Ä Ð 1 ß
1Ñ along y œ
x2

y3
1 y Ä 1 y
1

œ lim

xy  1 x2
y2

œ lim ay2  y  1b œ 3


x 3  1 2 4 x Ä 1 x
x

œ lim

3 2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

yÄ1

x2  x  1 a x  1bax2  1b xÄ1

œ lim

808

Chapter 14 Partial Derivatives

Ú 1 if y   x% 51. fax, yb œ Û 1 if y Ÿ 0 Ü 0 otherwise (a) (b) (c)

lim

fax, yb œ 1 since any path through a0, 1b that is close to a0, 1b satisfies y   x%

lim

fax, yb œ 0 since any path through a2, 3b that is close to a2, 3b does not satisfiy either y   x% or y Ÿ 0

lim

fax, yb œ 1 and

Ðxß yÑ Ä Ð0ß 1Ñ Ðxß yÑ Ä Ð2ß 3Ñ Ðx ß y Ñ Ä Ð 0 ß 0 Ñ along x œ 0

(b) (c)

fax, yb does not exist

lim

Ðxß yÑ Ä Ð0ß 0Ñ

if x   0 if x  0 fax, yb œ 32 œ 9 since any path through a3,
2b that is close to a3,
2b satisfies x   0

x2 x3 lim

52. fax, yb œ œ (a)

fax, yb œ 0 Ê

lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ along y œ x2

Ðxß yÑ Ä Ð3ß
2Ñ

fax, yb œ a
2b3 œ
8 since any path through a
2, 1b that is close to a
2, 1b satisfies x  0

lim

Ðxß yÑ Ä Ð
2ß 1Ñ

fax, yb œ 0 since the limit is 0 along any path through a0, 0b with x  0 and the limit is also zero along

lim

Ðxß yÑ Ä Ð0ß 0Ñ

any path through a0, 0b with x   0 53. First consider the vertical line x œ 0 Ê

2x2 y 4  y2 x Ðxß yÑ Ä Ð0ß 0Ñ

lim

2a0b2 y a b4  y2 0 yÄ0

œ lim

along x œ 0

œ lim 0 œ 0. Now consider any nonvertical yÄ0

through a0, 0b. The equation of any line through a0, 0b is of the form y œ mx Ê œ

2 lim 2x amxb 2 x Ä 0 x4  amxb

œ

3 lim 4 2mx 2 2 x Ä 0 x m x

54. If f is continuous at (x! ß y! ), then

3 lim 2 2mx 2 2 x Ä 0 x ax  m b

œ

lim

Ðxß yÑ Ä Ðx! ß y! Ñ

œ

lim

Ðxß yÑ Ä Ð0ß 0Ñ along y œ mx

œ 0. Thus

lim 22mx 2 x Ä 0 ax  m b

faxß yb œ

2x2 y 4  y2 x Ðxß yÑ Ä Ð0ß 0Ñ

lim

along y

lim

Ðxß yÑ Ä Ð0ß 0Ñ any line though a0, 0b

2x2 y x4  y2

œ mx

œ 0.

f(xß y) must equal f(x! ß y! ) œ 3. If f is not continuous at

(x! ß y! ), the limit could have any value different from 3, and need not even exist. 55.

lim

Ðxß yÑ Ä Ð0ß 0Ñ

Š1


x# y# 3 ‹

œ 1 and

lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ

1œ1 Ê

# #

56. If xy  0,

2 kxyk
Š x 6y ‹

lim

kxyk

Ðx ß y Ñ Ä Ð 0 ß 0Ñ

tan " xy xy

lim

Ðxß yÑ Ä Ð0ß 0Ñ

œ 1, by the Sandwich Theorem

# #

œ

2xy
Š x 6y ‹

lim

xy

Ðxß yÑ Ä Ð0ß 0Ñ

œ

lim

Ðxß yÑ Ä Ð0ß 0Ñ

ˆ2


xy ‰ 6

œ 2 and

# #

2 kxyk Ðxß yÑ Ä Ð0ß 0Ñ kxyk

lim

œ

lim

Ðx ß y Ñ Ä Ð 0 ß 0 Ñ

œ

lim

Ðxß yÑ Ä Ð0ß 0Ñ

ˆ2 

xy ‰ 6

2 œ 2; if xy  0,

œ 2 and

lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ

lim

2 kxyk
Š x 6y ‹ kxyk

Ðxß yÑ Ä Ð0ß 0Ñ

2 kxyk kxyk

œ2 Ê

lim

Ðxß yÑ Ä Ð0ß 0Ñ

# #

œ

lim


2xy
Š x 6y ‹

Ðxß yÑ Ä Ð0ß 0Ñ

4
4 cos Èkxyk kxyk


xy

œ 2, by the Sandwich Theorem

57. The limit is 0 since ¸sin ˆ "x ‰¸ Ÿ 1 Ê
1 Ÿ sin ˆ x" ‰ Ÿ 1 Ê
y Ÿ y sin ˆ x" ‰ Ÿ y for y   0, and
y   y sin ˆ "x ‰   y for y Ÿ 0. Thus as (xß y) Ä (!ß !), both
y and y approach 0 Ê y sin ˆ "x ‰ Ä 0, by the Sandwich Theorem. 58. The limit is 0 since ¹cos Š "y ‹¹ Ÿ 1 Ê
1 Ÿ cos Š y" ‹ Ÿ 1 Ê
x Ÿ x cos Š y" ‹ Ÿ x for x   0, and
x   x cos Š y" ‹   x for x Ÿ 0. Thus as (xß y) Ä (!ß !), both
x and x approach 0 Ê x cos Š "y ‹ Ä 0, by the Sandwich Theorem. 59. (a) f(xß y)k yœmx œ

2m 1  m#

œ

2 tan ) 1  tan# )

œ sin 2). The value of f(xß y) œ sin 2) varies with ), which is the line's

angle of inclination.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.2 Limits and Continuity in Higher Dimensions (b) Since f(xß y)k yœmx œ sin 2) and since
1 Ÿ sin 2) Ÿ 1 for every ),

lim

Ðxß yÑ Ä Ð0ß 0Ñ

809

f(xß y) varies from
1 to 1

along y œ mx. 60. kxy ax#
y# bk œ kxyk kx#
y# k Ÿ kxk kyk kx#  y# k œ Èx# Èy# kx#  y# k Ÿ Èx#  y# Èx#  y# kx#  y# k #

#

#

œ ax#  y# b Ê ¹ xyxa#x
y#y b ¹ Ÿ Ê

61.

62.

63.

lim

Ðxß yÑ Ä Ð0ß 0Ñ

x$
xy#

œ lim

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x  y lim

Ðxß yÑ Ä Ð0ß 0Ñ

lim

Ðxß yÑ Ä Ð0ß 0Ñ

x#
y# x#  y# ‹

Šxy

rÄ0

$

ax#  y# b x#  y#

#

œ x#  y# Ê
ax #  y # b Ÿ

œ 0 by the Sandwich Theorem, since r$ cos$ )
(r cos )) ar# sin# )b r# cos# )  r# sin# )

$

œ lim

rÄ0

$

$

$

y# x#  y#

r# sin# ) r# rÄ0

œ lim

lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ

r acos$ )
cos ) sin# )b 1

y r cos )
r sin ) cos Š xx#
 y# ‹ œ lim cos Š r# cos# )  r# sin# ) ‹ œ lim cos ’ $

rÄ0

xy ax#
y# b x#  y#

rÄ0

Ÿ a x#  y# b

„ ax#  y# b œ 0; thus, define fa0ß 0b œ 0

œ0

r acos$ )
sin$ )b “ 1

œ cos 0 œ 1

œ lim asin# )b œ sin# ); the limit does not exist since sin# ) is between rÄ0

0 and 1 depending on ) 64.

65.

lim

Ðxß yÑ Ä Ð0ß 0Ñ

2r cos )

œ lim

2x

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x  x  y

# r Ä 0 r  r cos )

œ lim

2 cos )

r Ä 0 r  cos )

œ

2 cos ) cos )

ky k krk akcos )k  ksin )kb
" kr cos )k  kr sin )k tan
" ’ kxx#k  ’ “ œ lim tan
" ’ “;  y# “ œ lim tan r# r#

rÄ0

if r Ä 0 , then lim b rÄ!

rÄ0

tan
" ’ krk akcos )rk# ksin )kb “

œ lim b tan
" ’ kcos )k r ksin )k “ œ rÄ!

lim tan
" ’ krk akcos )rk# ksin )kb “ œ lim c tan
" Š kcos )k
r ksin )k ‹ œ rÄ!

r Ä !c

66.

; the limit does not exist for cos ) œ 0

x#
y#

œ lim

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x  y

rÄ0

r# cos# )
r# sin# ) r#

1 #

1 #

Ê the limit is

; if r Ä 0
, then 1 #

œ lim acos# )
sin# )b œ lim (cos 2)) which ranges between rÄ0

rÄ0


1 and 1 depending on ) Ê the limit does not exist 67.

lim

Ðx ß y Ñ Ä Ð 0 ß 0Ñ

ln Š 3x

#


x# y#  3y# ‹ x#  y#

œ lim ln Š 3r rÄ0

#

cos# )
r% cos# ) sin# )  3r# sin# ) ‹ r#

œ lim ln a3
r# cos# ) sin# )b œ ln 3 Ê define f(0ß 0) œ ln 3 rÄ0

68.

lim

Ðxß yÑ Ä Ð0ß 0Ñ

3xy# x# y#

(3r cos )) ar# sin# )b r# rÄ0

œ lim

œ lim 3r cos ) sin# ) œ 0 Ê define f(0ß 0) œ 0 rÄ0

69. Let $ œ 0.1. Then Èx#  y#  $ Ê Èx#  y#  0.1 Ê x#  y#  0.01 Ê kx#  y#
0k  0.01 Ê kf(xß y)
f(!ß !)k  0.01 œ %. 70. Let $ œ 0.05. Then kxk  $ and kyk  $ Ê kfaxß yb
fa0ß 0bk œ ¸ x# y 1
0¸ œ ¸ x# y 1 ¸ Ÿ kyk  0.05 œ %. 71. Let $ œ 0.005. Then kxk  $ and kyk  $ Ê kfaxß yb
fa0ß 0bk œ ¸ xx#y1
0¸ œ ¸ xx#y1 ¸ Ÿ kx  yk  kxk  kyk  0.005  0.005 œ 0.01 œ %. kx  yk " 72. Let $ œ 0.01. Since
1 Ÿ cos x Ÿ 1 Ê 1 Ÿ 2  cos x Ÿ 3 Ê "3 Ÿ #cos Ÿ ¸ 2 x cosy x ¸ Ÿ kx  yk x Ÿ 1 Ê 3 Ÿ kxk  kyk . Then kxk  $ and kyk  $ Ê kfaxß yb
fa0ß 0bk œ ¸ 2 x cosy x
0¸ œ ¸ 2 x cosy x ¸ Ÿ kxk  kyk  0.01  0.01

œ 0.02 œ %.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

810

Chapter 14 Partial Derivatives y2 x2  y2

73. Let $ œ 0.04. Since y2 Ÿ x2  y2 Ê

Ÿ1Ê

lxly2 x2  y2

Ÿ lxl œ Èx2 Ÿ Èx2  y2  $ Ê kfaxß yb
fa0ß 0bk

2

œ ¹ x2xy y2
0¹  0.04 œ %. 74. Let $ œ 0.01. If lyl Ÿ 1, then y2 Ÿ lyl œ Èy2 Ÿ Èx2  y2 , so lxl œ Èx2 Ÿ Èx2  y2 Ê lxl  y2 Ÿ 2Èx2  y2 . Since x2 x2  y2

x2 Ÿ x 2  y 2 Ê

Ÿ 1 and y2 Ÿ x2  y2 Ê

y2 x2  y2

Ÿ 1. Then

lx3  y4 l x2  y2

Ÿ

x2 x2  y2 lxl



y2 2 x2  y2 y

Ÿ lxl  y2  2$

y Ê kfaxß yb
fa0ß 0bk œ ¹ xx2   y2
0¹  2a0.01b œ 0.002 œ % . 3

4

75. Let $ œ È0.015. Then Èx#  y#  z#  $ Ê kf(xß yß z)
f(!ß 0ß 0)k œ kx#  y#  z#
0k œ kx#  y#  z# k #

#

œ ŠÈx#  t#  x# ‹  ŠÈ0.015‹ œ 0.015 œ %. 76. Let $ œ 0.2. Then kxk  $ , kyk  $ , and kzk  $ Ê kf(xß yß z)
f(!ß 0ß 0)k œ kxyz
0k œ kxyzk œ kxk kyk kzk  (0.2)$ œ 0.008 œ %. 77. Let $ œ 0.005. Then kxk  $ , kyk  $ , and kzk  $ Ê kf(xß yß z)
f(!ß 0ß 0)k œ ¹ x# x y# yz#z 1
0¹ œ ¹ x# x y# yz#z 1 ¹ Ÿ kx  y  zk Ÿ kxk  kyk  kzk  0.005  0.005  0.005 œ 0.015 œ %. 78. Let $ œ tan
" (0.1). Then kxk  $ , kyk  $ , and kzk  $ Ê kf(xß yß z)
f(!ß 0ß 0)k œ ktan# x  tan# y  tan# zk Ÿ ktan# xk  ktan# yk  ktan# zk œ tan# x  tan# y  tan# z  tan# $  tan# $  tan# $ œ 0.01  0.01  0.01 œ 0.03 œ %. 79.

f(xß yß z) œ

lim

Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ

lim

(x  y
z) œ x!  y!
z! œ f(x! ß y! ß z! ) Ê f is continuous at

lim

ax#  y#  z# b œ x!#  y!#  z!# œ f(x! ß y! ß z! ) Ê f is continuous at

Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ

every (x! ß y! ß z! ) 80.

f(xß yß z) œ

lim

Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ

Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ

every point (x! ß y! ß z! ) 14.3 PARTIAL DERIVATIVES 1.

`f `x

œ 4x,

`f `y

3.

`f `x

œ 2x(y  2),

5.

`f `x

œ 2y(xy
1),

7.

`f `x

œ

9.

`f `x

œ
(x " y)# †

10.

`f `x

œ

11.

`f `x

œ œ

2.

`f `x

œ 2x
y,

4.

`f `x

œ 5y
14x  3,

œ 2x(xy
1)

6.

`f `x

œ 6(2x
3y)# ,

y È x#  y#

8.

`f `x

œ

œ
3 `f `y

x `f È x#  y# , ` y

œ x#
1

`f `y

œ

` `x

(x  y) œ
(x " y)# ,

ax#  y# b (1)
x(2x) ax#  y# b#

œ

y#
x# ax#  y# b#

(xy
1)(1)
(x  y)(y) (xy
1)#

œ

,

`f `y


y#
1 (xy
1)#

œ ,

`f `y

œ
(x " y)# †

ax#  y# b (0)
x(2y) ax#  y# b# `f `y

œ

` `y

2x# $ $ É x  ˆ #y ‰

`f `y

,

œ
x  2y

`f `y

`f `y

œ 5x
2y
6

`f `y

œ
9(2x
3y)#

œ

" $ $ 3É x  ˆ #y ‰

(x  y) œ
(x " y)#

œ
ax# 2xy  y # b#

(xy
")(1)
(x  y)(x) (xy
1)#

œ


x #
" (xy
1)#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.3 Partial Derivatives 12.

`f `x

œ

13.

`f `x

œ eÐxy1Ñ †

14.

`f `x

œ
e
x sin (x  y)  e
x cos (x  y),

15.

`f `x

œ

16.

`f `x

œ exy †

`f `x `f `y

œ 2 sin (x
3y) †

`f `x

œ 2 cos a3x
y# b †

17.

18.

" # 1  ˆ xy ‰

" xy

†

` `x

†

` `x

` `x

ˆ yx ‰ œ


` `x

œ
x# y y# ,

y #

x# ’1  ˆ xy ‰ “

(x  y  1) œ eÐxy1Ñ ,

(x  y) œ

" xy

,

`f `y

(xy) † ln y œ yexy ln y, ` `x ` `y

œ 2 sin (x
3y) †

" xy

œ

`f `y

`f `y

`f `y

†

`f `y

" # 1  ˆ xy ‰

œ

œ eÐxy1Ñ †

` `y

†

` `y

1 # x ’1  ˆ xy ‰ “

œ

x x#  y#

(x  y  1) œ eÐxy1Ñ

œ e
x cos (x  y)

` `y

" xy

(x  y) œ

œ exy †

` `y

(xy) † ln y  exy †

sin (x
3y) œ 2 sin (x
3y) cos (x
3y) † sin (x
3y) œ 2 sin (x
3y) cos (x
3y) †

` `x

ˆ yx ‰ œ

" y

œ xexy ln y 

` `x ` `y

(x
3y) œ 2 sin (x
3y) cos (x
3y),

exy y

(x
3y) œ
6 sin (x
3y) cos (x
3y)

cos a3x
y# b œ
2 cos a3x
y# b sin a3x
y# b †

œ
6 cos a3x
y# b sin a3x
y# b , `f ` # # # # ` y œ 2 cos a3x
y b † ` y cos a3x
y b œ
2 cos a3x
y b sin a3x
y b †

` `x

a3x
y# b

` `y

a3x
y# b

œ 4y cos a3x
y# b sin a3x
y# b 19.

`f `x

œ yxyc1 ,

21.

`f `x

œ
g(x),

`f `y

œ xy ln x

`f `y

20. f(xß y) œ

Ê

`f `x

" x ln y

œ

and

`f `y

œ


ln x y(ln y)#

œ g(y)

_

22. f(xß y) œ ! (xy)n , kxyk  1 Ê f(xß y) œ n œ0

`f `y

ln x ln y

œ
(1
"xy)# †

` `y

(1
xy) œ

" 1
xy

Ê

`f `x

œ
(1
"xy)# †

` `x

(1
xy) œ

y (1
xy)#

and

x (1
xy)#

23. fx œ y# , fy œ 2xy, fz œ
4z

24. fx œ y  z, fy œ x  z, fz œ y  x

25. fx œ 1, fy œ
Èy#y z# , fz œ
Èy#z z# 26. fx œ
x ax#  y#  z# b 27. fx œ

yz È 1
x # y# z#

28. fx œ

" kx  yzk È(x  yz)#
1

29. fx œ

" x  2y  3z

30. fx œ yz †

" xy

†

, fy œ

, fy œ ` `x

xz È 1
x # y# z#

, fy œ

(xy) œ

#

, fy œ
y ax#  y#  z# b

#

` `z

, fz œ

(yz)(y) xy

, fz œ œ

yz x


$Î#

, fz œ
z ax#  y#  z# b


$Î#

xy È 1
x# y# z#

z kx  yzk È(x  yz)#
1

2 x  2y  3z

fz œ y ln (xy)  yz † #


$Î#

, fz œ

y kx  yzk È(x  yz)#
1

3 x  2y  3z

, fy œ z ln (xy)  yz †

` `y

ln (xy) œ z ln (xy) 

yz xy

†

` `y

(xy) œ z ln (xy)  z,

ln (xy) œ y ln (xy) #

#

#

#

#

#

31. fx œ
2xe
ax y z b , fy œ
2ye
ax y z b , fz œ
2ze
ax y z b 32. fx œ
yze
xyz , fy œ
xze
xyz , fz œ
xye
xyz

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

811

812

Chapter 14 Partial Derivatives

33. fx œ sech# (x  2y  3z), fy œ 2 sech# (x  2y  3z), fz œ 3 sech# (x  2y  3z) 34. fx œ y cosh axy
z# b , fy œ x cosh axy
z# b , fz œ
2z cosh axy
z# b 35.

`f `t

œ
21 sin (21t
!),

36.

`g `u

œ v# eÐ2uÎvÑ †

37.

`h `3

œ sin 9 cos ),

38.

`g `r

œ 1
cos ),

` `u

`f `!

œ sin (21t
!) `g `v

Ð2uÎvÑ ˆ 2u ‰ , v œ 2ve

`h `9

`g `)

39. Wp œ V, Wv œ P 

`h `)

œ 3 cos 9 cos ), `g `z

œ r sin ), $ v# 2g ,

W$ œ

Vv# 2g

m `A q , `m

œ

2V$ v 2g

, Wv œ

#

V$ v g

, Wg œ
V#$gv#

`A `h

œ

q #

41.

`f `x

œ 1  y,

`f `y

œ 1  x,

42.

`f `x

œ y cos xy,

43.

`g `x

œ 2xy  y cos x,

44.

`h `x

œ ey ,

45.

`r `x

œ

46.

`s `x

œ”

` #s ` x#

œ

`w `x

œ 2x tanaxyb  x2 sec2 axyb † y œ 2x tanaxyb  x2 y sec2 axyb,

47.

` #w ` x#

`h `y

`f `y

œ x cos xy, `g `y

œ xey  1,

" `r x y , ` y

" #• 1  ˆ xy ‰

y(2x) ax#  y# b#

†

œ

` `x

` #f ` y#

` #f ` x#

œ
km q# 

` #f ` y` x

œ 0,

` #h ` x#

œ 0,

œ

` #h ` y#


" (xy)#

,

` #r ` y#

` #s ` y#

,

œ

œ

` #f ` x` y

` #f ` y#

œ

` #h ` x` y

` #w ` x#

`w `x ` #w ` x# ` #w ` y#

œ

` #w ` x` y

œ yex

2

,

` #f ` x` y

œ
cos y,

œ cos xy
xy sin xy ` #g ` y` x

œ

` #g ` x` y

œ 2x  cos x

` #r ` x` y

`s `y

œ
ax# 2xy ,  y # b#

œ


" (xy)#

œ”

" #• 1  ˆ xy ‰

` #s ` y` x

œ

` #s ` x` y `w `y

†

` `y

ˆ xy ‰ œ ˆ 1x ‰ ”

" #• 1  ˆ xy ‰

ax#  y# b (
1)  y(2y) ax #  y # b #

œ

œ

œ

x x #  y#

,

y#
x# ax #  y # b #

œ x2 sec2 axyb † x œ x3 sec2 axyb,

œ x3 a2secaxybsecaxyb tanaxyb † xb œ 2x4 sec2 axyb tanaxyb

† 2x œ 2xy ex


y

 2xyŠex ` #w ` y` x

2

2


y `w , `y


y

` #w ` x` y

2


y

2


y

œ a1bex

† 2x‹ œ 2yex

2


y

a1  2x2 b,

† a
1b œ ex ` #w ` y#

œ Šex
y

œ sinax2 yb  x cosax2 yb † 2xy œ sinax2 yb  2x2 ycosax2 yb,

`w `y

ay
2b,

œ

œ Šex

2


y

 yex

2

2


y

2

` #w ` y#

œ 3x2 sec2 axyb  x3 a2secaxybsecaxyb tanaxyb † yb œ 3x2 sec2 axyb  x3 y sec2 axyb tanaxyb


y

œ 2y ex

œ ex 49.

` #g ` y#

œ

œ 2tanaxyb  2x sec2 axyb † y  2xy sec2 axyb  x2 y a2secaxybsecaxyb tanaxyb † yb

` #w ` y` x `w `x

` #f ` y` x

œ ey

œ

œ 2tanaxyb  4xy sec2 axyb  2x2 y2 sec2 axyb tanaxyb,

48.

œ1

œ 2y
y sin x,


y x#  y#

œ

h #

œ
x# sin xy,


" ` #r (xy)# , ` y` x

" #• 1  ˆ xy ‰


x(2y) ax #  y # b#

` #g ` x#

` #h ` y` x

œ xey ,

ˆ xy ‰ œ ˆ
xy# ‰ ”

2xy ax #  y # b #

œ

œ
y# sin xy,

œ x#
sin y  sin x,

" ` #r x y , ` x #

œ

 c,

k q

œ 0,

`A `q

œ

œ m,

` #f ` x#

Ð2uÎvÑ ˆ 2u ‰
2ueÐ2uÎvÑ v œ 2ve

œ
1

`A `c

œ

` `v

œ
3 sin 9 sin )

40.

,

`A `k

œ 2veÐ2uÎvÑ  v# eÐ2uÎvÑ †

† 2x‹a1
yb œ 2x ex

2

2


y


y

a1
yb ,

† a
1b‹a1
yb  ex

2


y

a
1b

a1
yb œ x cosax2 yb † x2 œ x3 cosax2 yb,

œ cosax2 yb † 2xy  4xy cosax2 yb
2x2 y sinax2 yb † 2xy œ 6xy cosax2 yb
4x3 y2 sinax2 yb, œ
x3 sinax2 yb † x2 œ
x5 sinax2 yb,

` #w ` y` x

œ

` #w ` x` y

œ 3x2 cosax2 yb
x3 sinax2 yb † 2xy œ 3x2 cosax2 yb
2x4 y sinax2 yb

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.3 Partial Derivatives 50.

`w `x ` #w ` x# ` #w ` y#

œ œ œ

ˆx2  y‰
ax
yba2xb ax2  yb2

œ


x2  2xy  y ` w , `y ax 2  y b 2

œ

ˆx2  y‰a
1b
ax
yb ax2  yb2

ˆx2  y‰2 a
2x  2yb
ˆ
x2  2xy  y‰2ˆx2  y‰a2xb 2

2

’ax2  yb “ 2

ˆx  y‰ † 0
ˆ
x
x‰2ˆx2  y‰† 1 2

2

2

’ax2  yb “

2

œ

œ

2x2  2x ` # w , ax 2  y b 3 ` y ` x

œ

` #w ` x` y

2x3  3x2
2xy
y ax2  yb3

51.

`w `x

œ

52.

`w `x

œ ex  ln y  yx ,

53.

`w `x

œ y#  2xy$  3x# y% ,

`w `y

œ 2xy  3x# y#  4x$ y$ ,

54.

`w `x

œ sin y  y cos x  y,

`w `y

œ x cos y  sin x  x,

,

`w `y

œ

55. (a) x first

3 2x  3y `w `y

` #w ` y` x

,

œ

x y

57. fx a1ß 2b œ lim

hÄ0

hÄ0

` #w ` y` x

, and " y

œ œ

` #w ` x` y

œ

hÄ0

2 ’ax2  yb “

` #w ` x` y

` #w ` y` x

` #w ` y` x

2

" y

œ



" x #

œ 2y  6xy#  12x# y$ , and ``x`wy œ 2y  6xy#  12x# y$ ` #w ` x` y

œ cos y  cos x  1

(e) y first

(f) y first

œ cos y  cos x  1, and

(d) x first

(b) y first three times œ lim

ˆx2  y‰2 a2x  1b
ˆ
x2  2xy  y‰2ˆx2  y‰† 1


6 (2x  3y)#

œ

 x" , and

(c) x first

f(1  hß 2)
f(1ß 2) h


13h
6h# h


6 (2x  3y)#

 ln x,

(b) y first

56. (a) y first three times

œ lim

œ


x 2
x , ax2  yb2

2ˆx3
3x2 y
3 xy  y2 ‰ , ax 2  y b 3

œ

2 2x  3y

œ

(c) y first twice

c1
(1  h)  2
6(1  h)# d
(2
6) h

(d) x first twice
h
6 a1  2h  h# b  6 h

œ lim

hÄ0

œ lim (
13
6h) œ
13, hÄ0

f(1ß 2  h)
f(1ß 2) h hÄ0

fy (1ß 2) œ lim

œ lim (
2) œ
2

c1
1  (2  h)
3(2  h)d
(2
6) h

œ lim

hÄ0

(2
6
2h)
(2
6) h

œ lim

hÄ0

hÄ0

58. fx a
2ß 1b œ lim

hÄ0

œ lim

hÄ0

fa
2  hß 1b
fa
2ß 1b h

a2h
1
hb  1 h

œ lim

hÄ0

c4  2a
2  hb
3
a
2  hbd
a
3  2b h

œ lim 1 œ 1, hÄ0


4
4
3a1  hb  2a1  hb# ‘
a
3  2b fy a
2ß 1b œ lim fa
2ß 1  hhb
fa
2ß 1b œ lim h hÄ0 hÄ0 a
3
3h  2  4h  2h# b  1 h  2h# œ lim œ lim œ lim a1  2hb œ 1 h h hÄ0 hÄ0 hÄ0

59. fx a
2ß 3b œ lim

hÄ0

fa
2  hß 3b
fa
2ß 3b h

È2h  4
2 h hÄ0

œ lim

fy a
2ß 3b œ lim

hÄ0

œ lim

hÄ0

œ lim Š hÄ0

œ lim Š hÄ0

hÄ0

œ lim

œ lim

fa0ß 0  hb
fa0ß 0b h hÄ0

œ lim

fy a0ß 0b œ lim

hÄ0

hÄ0 hÄ0

œ lim

2

h Ä 0 È2h  4  2

œ 12 ,

È
4  3 a3  h b
1
È
4  9
1 h

È3h  4
2 È3h  4  2 È3h  4  2 ‹ h

fa0  hß 0b
fa0ß 0b h hÄ0

60. fx a0ß 0b œ lim

È 2 a
2  h b  9
1
È
4  9
1 h

È2h  4
2 È2h  4  2 È2h  4  2 ‹ h

fa
2ß 3  hb
fa
2ß 3b h

È3h  4
2 h

œ lim

sinŠh3 b 0‹ h2 b 0


0

sinŠ0 b h4 ‹ 0 b h2


0

h

h

œ lim

3

h Ä 0 È2h  4  2

œ

3 4

œ lim

sin h3 h3

œ1

œ lim

sin h4 h3

œ lim Šh †

hÄ0 hÄ0

hÄ0

sin h4 h4 ‹

œ0†1œ0

61. (a) In the plane x œ 2 Ê fy axß yb œ 3 Ê fy a2ß
1b œ 3 Ê m œ 3 (b) In the plane y œ
1 Ê fx axß yb œ 2 Ê fy a2ß
1b œ 2 Ê m œ 2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

813

814

Chapter 14 Partial Derivatives

62. (a) In the plane x œ
1 Ê fy axß yb œ 3y2 Ê fy a
1ß 1b œ 3a1b2 œ 3 Ê m œ 3 (b) In the plane y œ 1 Ê fx axß yb œ 2x Ê fy a
1ß 1b œ 2a
1b œ
2 Ê m œ
2 63. fz ax! ß y! ß z! b œ lim

hÄ0

fz a1ß 2ß 3b œ lim

hÄ0

fax! ß y! ß z!  hb
fax! , y! ß z! b h

fa1ß 2ß 3  hb
fa1, 2ß 3b h

64. fy ax! ß y! ß z! b œ lim

hÄ0

hÄ0

fax! ß y!  hß z! b
fax! , y! ß z! b h

`z `x

 z$
2y

œ lim

Ê 2c cos A
2b œ a2bc sin Ab ``Ab Ê Ê

"2h  2h# h

œ lim a12  2hb œ 12 hÄ0

œ lim a2h  9b œ 9 hÄ0

(sin A) ``Aa
a cos A sin# A

`A `b

œ

`A `a c cos A
b bc sin A

œ

a bc sin A

ba
csc B cot Bb Ê

œ
2 `x `z

œ

" 6

; also 0 œ 2b
2c cos A  a2bc sin Ab ``Ab

œ 0 Ê asin Ab `` xa
a cos A œ 0 Ê `a `B

`z `x


2x‰ `` xz œ
x Ê at (1ß
1ß
3) we have (
3
1
2) `` xz œ
1 or

y x

67. a# œ b#  c#
2bc cos A Ê 2a œ a2bc sin Ab ``Aa Ê

a b sin A œ sin B ˆ sin" A ‰ ``Ba œ

hÄ0

œ 0 Ê a3xz#
2yb `` xz œ
y
z$ Ê at (1ß 1ß 1) we have (3
2) `` xz œ
1
1 or

66. ˆ `` xz ‰ z  x  ˆ yx ‰ `` xz
2x `` xz œ 0 Ê ˆz 

68.

œ lim

;

a2h#  9hb
0 h hÄ0

fa
1ß hß 3b
fa
1, 0ß 3b h hÄ0

`z ‰ `x x

2a3  hb#
2a9b h

œ lim

fy a
1ß 0ß 3b œ lim 65. y  ˆ3z#

;

`a `A

œ

a cos A sin A

; also

œ
b csc B cot B sin A

69. Differentiating each equation implicitly gives 1 œ vx ln u  ˆ vu ‰ ux and 0 œ ux ln v  ˆ uv ‰ vx or " º0

aln ub vx  ˆ vu ‰ ux œ 1 Ê vx œ ˆ uv ‰ vx  aln vb ux œ 0 Ÿ

º

v u

ln v º

ln u u v

œ

v u

ln v º

ln v aln ubaln vb
1

70. Differentiating each equation implicitly gives 1 œ a2xbxu
a2ybyu and 0 œ a2xbxu
yu or a2xbxu
a2ybyu œ 1 Ê xu œ a2xbxu
yu œ 0  yu œ

" 0º
#x  4xy 2x º 2x

œ


2x
2x  4xy

œ

0 0


2y
1 º 2x
2y º 2x
1 º

2x 2x
4xy

œ 2x Š 2x
" 4xy ‹  2y Š 1
" 2y ‹ œ 71. fx axß yb œ œ

" º0

œ

" 1
#y

œ

1 1
2y




1
2x  4xy

œ

1 2x
4xy

and

; next s œ x#  y# Ê

2y 1
2y

œ

`s `u

œ 2x

`x `u

 2y

`y `u

1  2y 1
2y

if y   0 Ê fx axß yb œ 0 for all points ax, yb; at y œ 0, fy axß 0b œ lim fax, 0  hhb
fax, 0b œ lim fax, hhb
0 if y  0 h Ä0 h Ä0

œ lim fax,h hb œ 0 because h Ä0

lim

hÄ0c

fax, hb h

œ

3 lim h h Ä0 c h

œ 0 and

limb fax,h hb œ

h Ä0

2

limb hh œ 0 Ê fy axß yb œ œ

h Ä0

3y2
2y

if y   0 ; if y  0

fyx axß yb œ fxy axß yb œ 0 for all points ax, yb 72. At x œ 0, fx a0ß yb œ lim fa0  h, yhb
fa0, yb œ lim fah, yhb
0 œ lim fah,h yb which does not exist because h Ä0

œ

2

limc hh œ 0 and

hÄ0

fy axß yb œ œ

limb fah,h yb œ

h Ä0

h Ä0

Èh h Ä0 h

limb

hÄ0

1

œ

lim 1 œ  _ Ê fx axß yb œ  2Èx h Ä0 b È h 2x

if x  0 if x  0

lim

hÄ0c

fah, yb h

;

0 if x   0 Ê fy axß yb œ 0 for all points ax, yb; fyx axß yb œ 0 for all points ax, yb, while fxy axß yb œ 0 for all 0 if x  0

points ax, yb such that x Á 0.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.3 Partial Derivatives 73.

`f `x

œ 2x,

`f `y

74.

`f `x

œ
6xz,

œ 2y, `f `y

`f `z

œ
4z Ê `f `z

œ
6yz,

` #f ` x#

` #f ` y#

œ 2,

œ 2,

` #f ` z#

` #f ` x#

œ 6z#
3 ax#  y# b ,

` #f ` x#

œ
4 Ê ` #f ` y#

œ
6z,

` #f ` y#



œ
6z,



` #f ` z#

` #f ` z#

815

œ 2  2  (
4) œ 0

œ 12z Ê

` #f ` x#



` #f ` y#



` #f ` z#

œ
6z
6z  12z œ 0 `f `x

œ
2ec2y sin 2x,

76.

`f `x

œ

77.

`f `x

œ 3,

78.

`f `x

œ

75.

`f `y

œ
2ec2y cos 2x,

œ
4ec2y cos 2x  4ec2y cos 2x œ 0

`f `x

,

`f `y

`f `y

œ

# 1  Š xy ‹



œ

œ 2,

1 Îy

` #f ` x#

Ê 79.

x x#  y#

` #f ` y#

y x#  y#

` #f ` x#

œ 0,

y y#  x#

œ

` #f ` x#

,

,

`f `y


2xy ay#  x# b2

œ
"# ax#  y#  z# b

œ

y#
x# ax#  y# b#

œ
4ec2y cos 2x,

` #f ` x#

` #f ` y#

,

œ

` #f ` y#

œ0 Ê

` #f ` x#

œ


x Îy 2


x y #  x#



2xy ay#  x# b2


$Î#

œ

# 1  Š xy ‹



` #f ` y#

x#
y# ax#  y# b#

` #f ` x#

œ 4ec2y cos 2x Ê

` #f ` y#



œ

y#
x# ax#  y# b#



` #f ` x#

x#
y# ax#  y# b#

œ

ay#  x# b†0
y†2x ay#  x# b2

œ


2xy ay#  x# b2

,

` #f ` y#

œ

80.

`f `x


$Î# ` f , `y
$Î# #

a2xb œ
x ax#  y#  z# b

 3x# ax#  y#  z# b


$Î#

 3z# ax#  y#  z# b

`f `y

œ 3e3x4y cos 5z,

#


&Î#

œ 4e3x4y cos 5z,

` f ` z#

œ
25e3x4y cos 5z Ê

81.

`w `x

œ cos (x  ct),

82.

`w `x

œ
2 sin (2x  2ct),

Ê 83.

84. 85.

#

` w ` t#

`w `t

#

` f ` x#



#

` f ` y#



`f `z #

` f ` z#

` #w ` x#

œ c cos (x  ct); `w `t

œ
"# ax#  y#  z# b

œ


&Î#

“ œ
3 ax#  y#  z# b

œ
5e3x4y sin 5z;

,

`w `t

œ

c x  ct

;

` #w ` x#

œ

2xy ay #  x # b 2


$Î#

` #f ` x#


1 (x  ct)#

a2yb ax #  y #  z # b
&Î#

œ
sin (x  ct),

#

` #w ` t#

,

 3y# ax#  y#  z# b


$Î#

 a3x#  3y#  3z# b ax#  y#  z# b

œ 9e3x4y cos 5z,

` #f ` y#

“
&Î#

œ0

œ 16e3x4y cos 5z,

` #w ` x#

` #w ` t#

œ
c# sin (x  ct) Ê

œ
4 cos (2x  2ct),

` #w ` t#

` #w ` t#

œ c# [
sin (x  ct)] œ c#

œ
4c# cos (2x  2ct)

` w ` x#

,


&Î#

œ 9e3x4y cos 5z  16e3x4y cos 5z
25e3x4y cos 5z œ 0

œ
2c sin (2x  2ct);

œ c# [
4 cos (2x  2ct)] œ c#

" x  ct

œ


$Î#

“  ’
ax#  y#  z# b

`w `w ` x œ cos (x  ct)
2 sin (2x  2ct), ` t œ c cos (x  ct)
2c sin (2x ` #w ` #w # # ` x# œ
sin (x  ct)
4 cos (2x  2ct), ` t# œ
c sin (x  ct)
4c # # Ê `` tw# œ c# [
sin (x  ct)
4 cos (2x  2ct)] œ c# `` xw# `w `x

œ0

œ0


$Î#

 ’
ax#  y#  z# b

` #f ` y#

ay#  x# b†0
a
xb†2y ay#  x# b2


$Î# ` f
$Î# œ
y ax#  y#  z# b , ` z œ
"# ax#  y#  z b a2zb œ
z ax#  y#  z# b ; ` #f # # #
$Î# # # # #
&Î# ` # f # # #
$Î#  3x ax  y  z b , ` y # œ
ax  y  z b  3y# ` x # œ
ax  y  z b # # #
&Î# ` #f # # #
$Î#  3z# ax#  y#  z# b Ê `` xf#  `` yf#  `` zf# ` z # œ
ax  y  z b

œ ’
ax#  y#  z# b



œ00œ 0

` #f ` x#

,

Ê

` #f ` y#

œ


c# (x  ct)#

Ê

` #w ` t#

 2ct); cos (2x  2ct)


" # œ c# ’ (x  ct)# “ œ c

` #w ` x#

`w `w ` #w # # # ` x œ 2 sec (2x
2ct), ` t œ
2c sec (2x
2ct); ` x# œ 8 sec (2x
2ct) tan (2x
2ct), ` #w ` #w # # # # ` t# œ 8c sec (2x
2ct) tan (2x
2ct) Ê ux ` t# œ c [8 sec (2x
2ct) tan (2x
2ct)] œ

c#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

` #w ` x#

` #w ` x#

816 86.

87.

Chapter 14 Partial Derivatives # `w xbct ` w , ` t œ
15c sin (3x  3ct)  cexbct ; `` xw# œ
45 cos (3x  ` x œ
15 sin (3x  3ct)  e # # ` #w # # xbct Ê `` tw# œ c# c
45 cos (3x  3ct)  exbct d œ c# `` xw# ` t# œ
45c cos (3x  3ct)  c e

`w `t

œ

`f `u `u `t

œ

` #f ` u#

Ê

œ a#

`f `u

(ac) Ê

` #w ` t#

œ a# c#

` #w ` t#

œ (ac) Š `` uf# ‹ (ac) œ a# c#

#

` #f ` u#

œ c# Ša#

` #f ` u# ‹

œ c#

` #f ` u#

;

`w `x

œ

`f `u `u `x

œ

`f `u

†a Ê

3ct)  exbct ,

` #w ` x#

#

œ Ša `` uf# ‹ † a

` #w ` x#

88. If the first partial derivatives are continuous throughout an open region R, then by Theorem 3 in this section of the text, f(xß y) œ f(x! ß y! )  fx (x! ß y! ) ?x  fy (x! ß y! ) ?y  %" ?x  %# ?y, where %" , %# Ä 0 as ?x, ?y Ä 0. Then as (xß y) Ä (x! ß y! ), ?x Ä 0 and ?y Ä 0 Ê lim f(xß y) œ f(x! ß y! ) Ê f is continuous at every point (x! ß y! ) in R. Ðxß yÑ Ä Ðx! ß y! Ñ

89. Yes, since fxx , fyy , fxy , and fyx are all continuous on R, use the same reasoning as in Exercise 76 with fx (xß y) œ fx (x! ß y! )  fxx (x! ß y! ) ?x  fxy (x! ß y! ) ?y  %" ?x  %# ?y and fy (xß y) œ fy (x! ß y! )  fyx (x! ß y! ) ?x  fyy (x! ß y! ) ?y  s%" ?x  s%# ?y. Then lim fx (xß y) œ fx (x! ß y! ) Ðxß yÑ Ä Ðx! ß y! Ñ

and

lim

Ðxß yÑ Ä Ðx! ß y! Ñ

fy (xß y) œ fy (x! ß y! ).

90. To find ! and " so that ut œ uxx Ê ut œ
" sina! xbe
" t and ux œ ! cosa! xbe
" t Ê uxx œ
!2 sina! xbe
" t ; then ut œ uxx Ê
" sina! xbe
" t œ
!2 sina! xbe
" t , thus ut œ uxx only if " œ !2 h†02 04

91. fx a0, 0b œ lim fa0  hß 0hb
fa0ß 0b œ lim h2 h Ä0

lim

Ðxß yÑ Ä Ð0ß 0Ñ along x œ ky2

f ax, yb œ

values of k Ê


0

0†h2 h4

œ lim 0h œ 0; fy a0, 0b œ lim fa0ß 0  hhb
fa0ß 0b œ lim 02

h hÄ0 hÄ0 4 ˆky2 ‰y2 lim œ lim k2 yky 2 4  y4 y Ä 0 aky2 b  y4 yÄ0

lim

Ðxß yÑ Ä Ð0ß 0Ñ

hÄ0

œ

lim 2 k y Ä 0 k 1

œ

hÄ0

k k2  1

h


0

œ lim 0h œ 0; hÄ0

Ê different limits for different

f ax, yb does not exist Ê f ax, yb is not continuous at a0, 0b Ê by Theorem 4, f ax, yb is not

differentiable at a0, 0b. 92. fx a0, 0b œ lim fa0  hß 0hb
fa0ß 0b œ lim fahß 0hb
1 œ lim 1
h 1 œ 0; fy a0, 0b œ lim fa0ß 0  hhb
fa0ß 0b œ lim fa0ß hhb
1 œ lim 1
h 1 œ 0; h Ä0

lim

Ðxß yÑ Ä Ð0ß 0Ñ along y œ x2

h Ä0

h Ä0

f ax, yb œ lim 0 œ 0 but yÄ0

lim

Ðxß yÑ Ä Ð0ß 0Ñ along y œ 1.5x2

h Ä0

f ax, yb œ lim 1 œ 1 Ê yÄ0

h Ä0

lim

Ðxß yÑ Ä Ð0ß 0Ñ

h Ä0

f ax, yb does not exist

Ê f ax, yb is not continuous at a0, 0b Ê by Theorem 4, f ax, yb is not differentiable at a0, 0b. 14.4 THE CHAIN RULE 1. (a)

`w `x

œ 2x,

`w `y œ #

2y,

dx dt

#

œ
sin t, #

dy dt #

œ cos t Ê

œ 0; w œ x  y œ cos t  sin t œ 1 Ê (b)

dw dt

(1 ) œ 0

2. (a)

`w `x

œ 2x,

`w `y

œ 2y,

dx dt

œ
sin t  cos t,

dy dt

dw dt

dw dt

œ
2x sin t  2y cos t œ
2 cos t sin t  2 sin t cos t

œ0

œ
sin t
cos t Ê

dw dt

œ (2x)(
sin t  cos t)  (2y)(
sin t
cos t) œ 2(cos t  sin t)(cos t
sin t)
2(cos t
sin t)(sin t  cos t) œ a2 cos# t
2 sin# tb
a2 cos# t
2 sin# tb œ 0; w œ x#  y# œ (cos t  sin t)#  (cos t
sin t)# œ 2 cos# t  2 sin# t œ 2 Ê dw dt œ 0 (b)

dw dt

(0) œ 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.4 The Chain Rule 3. (a)

`w `x

œ

Ê

" z

dw dt

,

`w `y

œ

" z

,

`w `z

œ


(x  y) z#

œ
2z cos t sin t 

2 z

dx dt

,

œ
2 cos t sin t,

sin t cos t 

x y z# t#

dy dt

œ 2 sin t cos t,

cos# t  sin# t Š "# ‹ at# b

œ

œ 1; w œ

x z

dz dt

œ
t"#



y z

œ

t

(b)

dw dt

(3) œ 1

4. (a)

`w `x

œ

2x x #  y #  z#

`w `y

,

œ

2y x #  y#  z#

2y cos t dw
2x sin t dt œ x#  y#  z#  x#  y#  z# œ 11616t ; w œ ln ax#  y#  z# b dw 16 dt (3) œ 49

Ê

(b) 5. (a)

`w `x

œ 2yex ,

`w `y

œ 2ex ,

`w `z

,

`w `z



œ

2z x #  y#  z#

4zt "Î# x#  y#  z# #

œ

œ ln acos t 

œ
"z ,

dx dt

œ

2t t# 1

,

,

dx dt

œ
sin t,

dy dt

6. (a)

œ

dy dt

" t# 1

,

œ et Ê

dz dt

# t (4t) atan " tb at#  1b  2 at#t 11b
eet œ 4t tan
" t  1; w œ 2yex
ln t#  1
" ˆ 2 ‰ # Ê dw tb (2t)
1 œ 4t tan
" t  1 dt œ t# 1 at  1b  a2 tan dw ˆ1‰ dt (1) œ (4)(1) 4  1 œ 1  1 `w `x

œ
y cos xy,

`w `y

œ
x cos xy,

œ
(ln t)[cos (t ln t)]
tc1

œe


sin (t ln t) Ê

(b)

(1) œ 1
(1  0)(1) œ 0

7. (a)

`z `u

œ



`z `y `y `u

œ 1,

dx dt

œ 1,

dy dt

œ

" t

,

dz dt

dw dt

œ



sin# t Š "t ‹

œt Ê

x

v ‰ 4e œ a4ex ln yb ˆ ucos cos v  Š y ‹ (sin v) œ

dw dt

4ytex t#  1

œ



16 1  16t

2ex t#  1




et z

z œ a2 tan
" tb at#  1b
t

œ etc1 Ê

dw dt

œ
y cos xy


4ex ln y u



x cos xy t

xy

4ex sin v y

œ

4(u cos v) ln (u sin v) v)(sin v)  4(u cos œ (4 cos v) ln (u sin v)  4 cos v; u u sin v `z `z `x `z `y 4ex x x ˆ
u sin v ‰ ` v œ ` x ` v  ` y ` v œ a4e ln yb u cos v  Š y ‹ (u cos v) œ
a4e

ln yb (tan v) 

4ex u cos v y

4(u cos v)(u cos v) cos# v œ (
4u sin v) ln (u sin v)  4usin u sin v v ; `z sin x z œ 4e ln y œ 4(u cos v) ln (u sin v) Ê ` u œ (4 cos v) ln (u sin v)  4(u cos v) ˆ u sinvv ‰ v‰ œ (4 cos v) ln (u sin v)  4 cos v; also `` vz œ (
4u sin v) ln (u sin v)  4(u cos v) ˆ uu cos sin v # cos v œ (
4u sin v) ln (u sin v)  4usin v At ˆ2ß 14 ‰ : `` uz œ 4 cos 14 ln ˆ2 sin 14 ‰  4 cos 14 œ 2È2 ln È2  2È2 œ È2 (ln 2  2); (4)(2) ˆcos# 14 ‰ `z 1 1‰ ˆ œ
4È2 ln È2  4È2 œ
2È2 ln 2  4È2 ˆsin 1 ‰ ` v œ (
4)(2) sin 4 ln 2 sin 4 

œ [
4(u cos v) ln (u sin v)](tan v) 

(b)

4

8. (a)

`z `u `z `v

œ– œ–

Š "y ‹ #

Š xy ‹

Š

Š xy ‹

y cos v x#  y#




x sin v x # y #

œ

(u sin v)(cos v)
(u cos v)(sin v) u#

Š

x ‹ y#

— (
u sin v)  – Š x ‹#  1 — u cos v œ
1

yu sin v x#  y#




(b) At

xu cos v x#  y#

œ


(u sin v)(u sin v)
(u cos v)(u cos v) u#

y

œ
sin# v
cos# v œ
1; z œ tan
" Š xy ‹ œ tan
" (cot v) Ê œ

œ 0;

y

Š "y ‹ #

x ‹ y#

— cos v  – Š x ‹#  1 — sin v œ 1


" sin# v  cos# v œ
1 ˆ1.3ß 16 ‰ : `` uz œ 0

and

`z `v

`z `u

œ1

œ 2t
"Î#

t cos (t ln t)  etc1 œ
(ln t)[cos (t ln t)]
cos (t ln t)  etc1 ; w œ z
sin t dw tc1
[cos (t ln t)]
ln t  t ˆ "t ‰‘ œ etc1
(1  ln t) cos (t ln t) dt œ e

dw dt

`z `x `x `u

`w `z

dz dt


2 cos t sin t  2 sin t cos t  4 ˆ4t"Î# ‰ t "Î# cos# t  sin# t  16t # sin t  16tb œ ln (1  16t) Ê dw dt

œ

(b)

œ cos t,

cos# t Š "t ‹

œ 0 and

`z `v

" # ‰ œ ˆ 1  cot # v a
csc vb

œ
1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

 etc1

817

818 9. (a)

Chapter 14 Partial Derivatives `w `u

œ

`w `x `x `u



`w `y `y `u



`w `z `z `u

œ (y  z)(1)  (x  z)(1)  (y  x)(v) œ x  y  2z  v(y  x)

œ (u  v)  (u
v)  2uv  v(2u) œ 2u  4uv;

`w `v

œ

`w `x `x `v



`w `y `y `v



`w `z `z `v

œ (y  z)(1)  (x  z)(
1)  (y  x)(u) œ y
x  (y  x)u œ
2v  (2u)u œ
2v  2u# ; w œ xy  yz  xz œ au#
v# b  au# v
uv# b  au# v  uv# b œ u#
v#  2u# v Ê ``wu œ 2u  4uv and `w `v

œ
2v  2u#

(b) At ˆ "# ß 1‰ : 10. (a)

`w `u

`w `u

œ 2 ˆ "# ‰  4 ˆ "# ‰ (1) œ 3 and

`w `v

#

œ
2(1)  2 ˆ "# ‰ œ
#3

2y 2z v v v v v œ Š x#  2x y#  z# ‹ ae sin u  ue cos ub  Š x#  y#  z# ‹ ae cos u
ue sin ub  Š x#  y#  z# ‹ ae b v

u ‰ aev sin u  uev cos ub œ ˆ u# e2v sin# u 2ueu# esin 2v cos# u  u# e2v v cos u v ‰ v  ˆ u# e2v sin# u 2ue  u# e2v cos# u  u# e2v ae cos u
ue sin ub v

‰ aev b œ 2u ;  ˆ u# e2v sin# u  u2ue # e2v cos# u  u# e2v `w `v

2y 2z v v v œ Š x#  2x y#  z# ‹ aue sin ub  Š x#  y#  z# ‹ aue cos ub  Š x#  y#  z# ‹ aue b v

u ‰ auev sin ub œ ˆ u# e2v sin# u 2ueu# esin 2v cos# u  u# e2v v cos u ‰ v  ˆ u# e2v sin# u 2ue  u# e2v cos# u  u# e2v aue cos ub

‰ auev b œ 2; w œ ln au# e2v sin# u  u# e2v cos# u  u# e2v b œ ln a2u# e2v b  ˆ u# e2v sin# u  u2ue # e2v cos# u  u# e2v v

œ ln 2  2 ln u  2v Ê (b) At a
2ß 0b: 11. (a)

`w `u

œ

œ
1 and

2 `w u and ` v `w `v œ 2

œ2

r
p p
q q
rr
pp
q `u `p `u `q `u `r " œ 0; ` p ` x  ` q ` x  ` r ` x œ q
r  (q
r)#  (q
r)# œ (q
r)# r
p p
q q
r
rpp
q 2p
2r `u `u `p `u `q `u `r " œ (q ` y œ ` p ` y  ` q ` y  ` r ` y œ q
r
(q
r)#  (q
r)# œ (q
r)#
r)# (2x  2y  2z)
(2x  2y
2z) z `u `u `p `u `q `u `r œ œ (z
y)# ; ` z œ ` p ` z  ` q ` z  ` r ` z (2z
2y)# r
p p
q p
pq
2p
4y y " œ q
r  (q
r)#
(q
r)# œ q
r (qr
œ 2q
r)# (q
r)# œ (2z
2y)# œ
(z
y)# ; y (z
y)
y(
1)
y(1) `u `u u œ pq

qr œ 2z 2y œ (z
z y)# , and `` uz œ (z
(zy)(0)
2y œ z
y Ê ` x œ 0, ` y œ (z
y)#
y)# œ
(z
yy)# `u `x

œ

(b) At ŠÈ3ß 2ß 1‹ : 12. (a)

œ

2
#

`w `u

`u `x

œ

`u `y `u `z

`u `x

œ 0,

œ

" (1
2)#

œ 1, and

`u `z

œ


2 (1
2)#

œ
2 œ yz if
1#  x 

eqr È 1
p#

(cos x)  areqr sin
" pb (0)  aqeqr sin
" pb (0) œ

œ

eqr È 1
p#

(0)  areqr sin
" pb Š zy ‹  aqeqr sin
" pb (0) œ

œ

eqr È 1
p#

(0)  areqr sin
" pb (2z ln y)  aqeqr sin
" pb ˆ
z"# ‰ œ a2zreqr sin
" pb (ln y)


#

œ (2z) ˆ "z ‰ ayz x ln yb
`u `y

`u `y

œ xzyz
1 , and

(b) At ˆ 14 ß "# ß
"# ‰ :

`u `z `u `x

az# ln yb ayz b x z# z

eqr cos x È 1
p#

œ

z# reqr sin " p y

ez ln y cos x È1
sin# x

œ

z# ˆ " ‰ y z x z

y

1 #

;

œ xzyz
1 ;

œ xyz ln y; u œ ez ln y sin
" (sin x) œ xyz if
1# Ÿ x Ÿ

qeqr sin " p z# 1 #

Ê

œ ˆ 14 ‰ ˆ "# ‰


"Î#

`u `x

œ yz ,

œ œ xy ln y from direct calculations œ ˆ "# ‰


"Î#

œ È2,

`u `y

œ ˆ 14 ‰ ˆ
"# ‰ ˆ "# ‰

Ð
"Î#Ñ
"

È

œ
14 2 ,

`u `z

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

ln ˆ "# ‰ œ
1

È2 ln 2 4

Section 14.4 The Chain Rule œ

` z dx ` x dt



` z dy ` y dt

` z du ` u dt



` z dv ` v dt



` x dw ` w dt

dz dt

15.

`w `u

œ

`w `x `x `u



`w `y `y `u



`w `z `z `u

`w `v

œ

`w `x `x `v



`w `y `y `v



`w `z `z `v

16.

`w `x

œ

`w `r `r `x



`w `s `s `x



`w `t `t `x

`w `y

œ

`w `r `r `y



`w `s `s `y



`w `t `t `y

17.

`w `u

œ

`w `x `x `u



`w `y `y `u

`w `v

œ

`w `x `x `v



`w `y `y `v

14.

dz dt

œ

13.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

819

820

Chapter 14 Partial Derivatives

18.

`w `x

œ

`w `u `u `x

19.

`z `t

œ

`z `x `x `t

20.

`y `r

œ

dy ` u du ` r

22.

`w `p

œ

`w `x `x `p



`w `y `y `p



`w `z `z `p



23.

`w `r

œ

` w dx ` x dr



` w dy ` y dr

œ

` w dx ` x dr

since





`w `v `v `x

`z `y `y `t

21.

`w `y

œ

`w `u `u `y

`z `s

œ

`z `x `x `s

`w `s

œ

dw ` u du ` s

`w `s

œ

` w dx ` x ds





`w `v `v `y

`z `y `y `s

`w `t

œ

dw ` u du ` t

`w `v `v `p

dy dr

œ0



` w dy ` y ds

œ

` w dy ` y ds

since

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

dx ds

œ0

Section 14.4 The Chain Rule 24.

`w `s

œ

`w `x `x `s



25. Let F(xß y) œ x$
2y#  xy œ 0 Ê Jx (xß y) œ 3x#  y

`w `y `y `s

and Fy (xß y) œ
4y  x Ê Ê

dy dx

(1ß 1) œ

dy dx

dy dx

#

œ
FFxy œ
(
3x4yyx)

3 œ
FFxy œ
xy
2y

dy dx

(
1ß 1) œ 2

27. Let F(xß y) œ x#  xy  y#
7 œ 0 Ê Fx (xß y) œ 2x  y and Fy (xß y) œ x  2y Ê Ê

dy dx

4 3

26. Let F(xß y) œ xy  y#
3x
3 œ 0 Ê Fx (xß y) œ y
3 and Fy (xß y) œ x  2y Ê Ê

821

dy dx

y œ
FFxy œ
2x x  2y

(1ß 2) œ
45

28. Let F(xß y) œ xey  sin xy  y
ln 2 œ 0 Ê Fx (xß y) œ ey  y cos xy and Fy (xß y) œ xey  x sin xy  1 Ê

dy dx

œ
FFxy œ
xeye xysincosxyxy 1 Ê y

dy dx

(!ß ln 2) œ
(2  ln 2)

29. Let F(xß yß z) œ z$
xy  yz  y$
2 œ 0 Ê Fx (xß yß z) œ
y, Fy (xß yß z) œ
x  z  3y# , Fz (xß yß z) œ 3z#  y Ê Ê

Fx `z ` x œ
Fz `z ` y (1ß 1ß 1)

30. Let F(xß yß z) œ Ê

`z `x

œ
3z
# y y œ

y 3z#  y

Ê

`z `x

(1ß 1ß 1) œ

" 4

;

`z `y

#

œ
Fyz œ

x3z#zy3y œ F

x
z
3y# 3z#  y

œ
34 " x



" y

œ
FFxz œ




" z


1 œ 0 Ê Fx (xß yß z) œ
x"# , Fy (xß yß z) œ
y"# , Fz (xß yß z) œ
z"#

Š
x"# ‹ Š
z"# ‹

#

œ
xz# Ê

`z `x

(2ß 3ß 6) œ
9;

`z `y

F

œ
Fyz œ


Š
y"# ‹ Š
z"# ‹

#

œ
yz# Ê

`z `y

(2ß 3ß 6) œ
4

31. Let F(xß yß z) œ sin (x  y)  sin (y  z)  sin (x  z) œ 0 Ê Fx (xß yß z) œ cos (x  y)  cos (x  z), Fy (xß yß z) œ cos (x  y)  cos (y  z), Fz (xß yß z) œ cos (y  z)  cos (x  z) Ê `` xz œ
FFxz (x  y)  cos (x  z) œ
cos cos (y  z)  cos (x  z) Ê

`z `x

(1ß 1ß 1) œ
1;

`z `y

(x  y)  cos (y  z) œ
Fyz œ
cos cos (y  z)  cos (x  z) Ê F

`z `y

(1 ß 1 ß 1 ) œ
1

32. Let F(xß yß z) œ xey  yez  2 ln x
2
3 ln 2 œ 0 Ê Fx (xß yß z) œ ey  2x , Fy (xß yß z) œ xey  ez , Fz (xß yß z) œ yez Ê 33.

`w `r

`z `x

œ

œ
FFxz œ


`w `x `x `r



ˆey  2x ‰ yez

`w `y `y `r



Ê

`w `z `z `r

`z `x

(1ß ln 2ß ln 3) œ
3 ln4 2 ;

`z `y

œ
Fyz œ
xeyez e Ê F

y

z

`z `y

(1ß ln 2ß ln 3) œ
3 ln5 2

œ 2(x  y  z)(1)  2(x  y  z)[
sin (r  s)]  2(x  y  z)[cos (r  s)]

œ 2(x  y  z)[1
sin (r  s)  cos (r  s)] œ 2[r
s  cos (r  s)  sin (r  s)][1
sin (r  s)  cos (r  s)] Ê ``wr ¸ rœ1ßsœ
1 œ 2(3)(2) œ 12 34.

`w `v

œ

`w `x `x `v



`w `y `y `v



35.

`w `v

œ

`w `x `x `v



`w `y `y `v

œ ˆ2x


`w ¸ ` v uœ0ßvœ0

œ
7

Ê

`w `z `z `v

‰ ˆ"‰ ˆ 2v ‰ œ y ˆ 2v u  x(1)  z (0) œ (u  v) u  y‰ x# (
2)

 ˆ "x ‰ (1) œ ’2(u
2v  1)


v# u

Ê

`w ¸ ` v uœ
1ßvœ2

2u  v
2 (u
2v  1)# “ (
2)



œ (1) ˆ
41 ‰  ˆ
41 ‰ œ
8

" u
2v  1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

822 36.

Chapter 14 Partial Derivatives `z `u

œ

`z `x `x `u



`z `y `y `u

œ (y cos xy  sin y)(2u)  (x cos xy  x cos y)(v)

$

œ cuv cos au v  uv b  sin uvd (2u)  cau#  v# b cos au$ v  uv$ b  au#  v# b cos uvd (v) Ê `` uz ¸ uœ0ßvœ1 œ 0  (cos 0  cos 0)(1) œ 2 37.

38.

$

`z `u

œ

dz ` x dx ` u

œ ˆ 1 5 x# ‰ eu œ ’ 1  aeu 5 ln vb# “ eu Ê

`z `v

œ

dz ` x dx ` v

œ ˆ 1 5 x# ‰ ˆ "v ‰ œ ’ 1  aeu 5 ln vb# “ ˆ "v ‰ Ê

`z `u

œ

dz ` q dq ` u

œ Š q" ‹ Š

`z `v

œ

Èv  3 1  u# ‹

dz ` q dq ` v

œ Š "q ‹ Š 2Èv  3 ‹ œ

`V `I

41. V œ IR Ê

œ (600 ohms) 42. V œ abc Ê ¸ Ê dV dt

ˆts2 ‰2 2

†

1 t

œ 2s4 t 

s4 t 2

† ˆ
ts2 ‰ œ s5
`V `R

œ R and

œ

aœ1ßbœ2ßcœ3

œ

` V da ` a dt



œ ’ 1 5(2)# “ (1) œ 1 "

dw ` x dx ` s

œ I;

dV dt

œ

5s4 t ` w 2 ; `t

œ

s5 2

`z

` V db ` b dt



` V dI ` I dt

` V dc ` c dt

`w `s

œ

œ f w axb † 3s2 œ 3s2 es t , 3

`w `x `x `s

œ

`w `x `x `t





`w `y `y `s

`w `y `y `t

2

`w `t

œ

œ

" atan " 1b a1  1# b

dw ` x dx ` t

œ

2 1

;

œ f w axb † 2t œ 2t es t 3

œ fx ax, yb † 2t s  fy ax, yb †

œ fx ax, yb † s2  fy ax, yb †

1 t


s t2

s5 2

œ



 (0.04 amps)(0.5 ohms/sec)

dI dt

dV dt

`w `s

Ê w œ fˆt s2 ß st ‰ œ faxß yb Ê

ˆts2 ‰2 2

œ at s2 bˆ st ‰ † 2t s  œ at s2 bˆ st ‰ † s2 

s t

`z ¸ ` v uœln 2ßvœ1

¸ " u ‹ Š 1  u# ‹ œ atan " ub a1  u# b Ê ` u uœ1ßvœ
2 "u " `z ¸ " Š Èv  3"tan " u ‹ Š 2tan Èv  3 ‹ œ #(v  3) Ê ` v uœ1ßvœ
2 œ #

39. Let x œ s3  t2 Ê w œ fas3  t2 b œ faxb Ê 40. Let x œ t s2 and y œ

œ ’ 1 5(2)# “ (2) œ 2;

Èv  3

œ Š Èv  3"tan

tan " u

`z ¸ ` u uœln 2ßvœ1

` V dR dI dR ` R dt œ R dt  I dt Ê
0.01 Ê dI dt œ
0.00005 amps/sec

volts/sec

db dc œ (bc) da dt  (ac) dt  (ab) dt

œ (2 m)(3 m)(1 m/sec)  (1 m)(3 m)(1 m/sec)  (1 m)(2 m)(
3 m/sec) œ 3 m$ /sec

and the volume is increasing; S œ 2ab  2ac  2bc Ê db dc dS ¸ œ 2(b  c) da dt  2(a  c) dt  2(a  b) dt Ê dt

œ

dS dt

` S da ` a dt

` S db ` b dt





` S dc ` c dt

aœ1ßbœ2ßcœ3

œ 2(5 m)(1 m/sec)  2(4 m)(1 m/sec)  2(3 m)(
3 m/sec) œ 0 m# /sec and the surface area is not changing; " ˆa da  b db  c dc ‰ Ê dD ¸ D œ Èa#  b#  c# Ê dD œ ` D da  ` D db  ` D dc œ dt

œ

" Š È14 ‹ [(1 m

` a dt

` b dt

` c dt

È a#  b#  c#

dt

m)(1 m/sec)  (2 m)(1 m/sec)  (3 m)(
3 m/sec)] œ


dt

6 È14

dt

dt

aœ1ßbœ2ßcœ3

m/sec  0 Ê the diagonals are

decreasing in length 43.

`f `x `f `y `f `z

44. (a) (b)

œ œ œ

`f `u `f `u `f `u

`w `r `w `r

`u `x `u `y `u `z

  

`f `v `f `v `f `v

  

 fy

Ê fy œ (sin )) œ

`f `w `f `w `f `w

`w `x `w `y `w `z

œ œ œ

`f `u `f `u `f `u

`f ` w (
1) (
1)  `` vf (1)  ``wf (0) (0)  `` vf (
1)  ``wf (1)

(1) 

`f `v

(0) 

œ

`f `u

œ
œ


`f `w


`f `u `f `v

 

,

`f `v , `f `w

`y `r

and Ê

`f `x



`f `y

œ fx cos )  fy sin ) and ``w) œ fx (
r sin ))  fy (r cos )) Ê sin ) œ fx sin ) cos )  fy sin# ) and ˆ cosr ) ‰ ``w) œ
fx sin ) cos )  fy cos# ) œ fx

`x `r

`v `x `v `y `v `z

`w `r #


asin# )b

`w `r `w `r



`f `z

" `w r `)

œ0 œ
fx sin )  fy cos )

 ˆ cosr ) ‰ ``w) ; then ``wr œ fx cos ) 
(sin )) ``wr  ˆ cosr ) ‰ ``w) ‘ (sin )) Ê fx cos )
ˆ sin ) rcos ) ‰ ``w) œ a1
sin# )b ``wr
ˆ sin ) rcos ) ‰ ``w) Ê fx œ (cos )) ``wr
ˆ sinr ) ‰ #

`w ‰ `)

 Š sinr# ) ‹ ˆ ``w) ‰ and

#

`w ‰ `)

 Š cosr# ) ‹ ˆ ``w) ‰ Ê afx b#  afy b# œ ˆ ``wr ‰ 

(c) afx b œ acos# )b ˆ ``wr ‰
ˆ 2 sin )r cos ) ‰ ˆ ``wr afy b# œ asin# )b ˆ ``wr ‰  ˆ 2 sin )r cos ) ‰ ˆ ``wr

#

#

#

#

#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" r#

ˆ ``w) ‰#

`w `)

2

Section 14.4 The Chain Rule `w `x

45. wx œ

œ

`w `u `u `x #

œ

`w `u

 x Š `` uw#

œ

`w `u

 x#

` #w ` u#

`w `v `v `x



`u `x

` #w ` v ` v` u ` x ‹



 2xy

` #w ` v` u #

Ê wyy œ
``wu
y Š `` uw#

`w `u

œx

#

`w `v

#

 y Š ``u`wv ` #w ` v#

 y# `u `y

y

`u `x

; wy œ

` #w ` v ` v` u ` y ‹





` #w ` v ` v# ` x ‹

`w `y

œ #

`w `u `u `y

 x Š ``u`wv

#

`w `u

Ê wxx œ

#

`u `y

x

œ 



`w `u

` `x

ˆ ``wu ‰  y

wxx  wyy œ ax#  y# b 46.

`w `x `w `y

` w ` u#

 ax #  y # b

#

` w ` v#

ˆ ``wv ‰

#

#

 x Šx `` uw#  y ``v`wu ‹  y Šx

`w `v `v `y

œ
y

`w `u

x

` #w ` u` v

#

 y `` vw# ‹

`w `v

` #w ` v ` v# ` y ‹

#

œ
``wu
y Š
y `` uw#  x ``v`wu ‹  x Š
y ``u`wv  x `` vw# ‹ œ
``wu  y# #

` `x

823

` #w ` u#


2xy

` #w ` v` u

 x#

` #w ` v#

; thus

œ ax#  y# b (wuu  wvv ) œ 0, since wuu  wvv œ 0

œ f w (u)(1)  gw (v)(1) œ f w (u)  gw (v) Ê wxx œ f ww (u)(1)  gww (v)(1) œ f ww (u)  gww (v); œ f w (u)(i)  gw (v)(
i) Ê wyy œ f ww (u) ai# b  gww (v) ai# b œ
f ww (u)
gww (v) Ê wxx  wyy œ 0

47. fx (xß yß z) œ cos t, fy (xß yß z) œ sin t, and fz (xß yß z) œ t#  t
2 Ê œ (cos t)(
sin t)  (sin t)(cos t)  at#  t
2b(1) œ t#  t
2;

df dt

df dt

` f dx ` x dt #

œ



` f dy ` y dt



` f dz ` z dt

œ 0 Ê t  t
2 œ 0 Ê t œ
2

or t œ 1; t œ
2 Ê x œ cos (
2), y œ sin (
2), z œ
2 for the point (cos (
2)ß sin (
2)ß
2); t œ 1 Ê x œ cos 1, y œ sin 1, z œ 1 for the point (cos 1ß sin 1ß 1) 48.

dw dt

` w dx ` x dt

œ



` w dy ` y dt



` w dz ` z dt

" ‰ œ a2xe2y cos 3zb (
sin t)  a2x# e2y cos 3zb ˆ t#  a
3x# e2y sin 3zb (1)

2x# e2y cos 3z
3x# e2y t# 2(1)# (4)(1)
0œ4 #

œ
2xe2y cos 3z sin t  Ê 49. (a)

dw ¸ dt Ð1ßln 2ß0Ñ `T `x

œ0

œ 8x
4y and

`T `y

œ 8y
4x Ê

dT dt

sin 3z; at the point on the curve z œ 0 Ê t œ z œ 0

œ

` T dx ` x dt



` T dy ` y dt

œ (8x
4y)(
sin t)  (8y
4x)(cos t)

œ (8 cos t
4 sin t)(
sin t)  (8 sin t
4 cos t)(cos t) œ 4 sin# t
4 cos# t Ê dT dt

d# T dt#

œ 16 sin t cos t;

œ 0 Ê 4 sin t
4 cos t œ 0 Ê sin t œ cos t Ê sin t œ cos t or sin t œ
cos t Ê t œ 14 , #

#

#

#

51 31 71 4 , 4 , 4

on

the interval 0 Ÿ t Ÿ 21; d# T dt# ¹ tœ 1

1 4

œ 16 sin

1 4

cos

 0 Ê T has a minimum at (xß y) œ Š

4

È2 #

ß

È2 # ‹;

d# T dt# ¹ tœ 31

œ 16 sin

31 4

cos

31 4

 0 Ê T has a maximum at (xß y) œ Š


È2 #

ß

d# T dt# ¹ tœ 51

œ 16 sin

51 4

cos

51 4

 0 Ê T has a minimum at (xß y) œ Š


È2 #

ß


d# T dt# ¹ tœ 71

œ 16 sin

71 4

cos

71 4

 0 Ê T has a maximum at (xß y) œ Š

4

4

4

`T `T ` x œ 8x
4y, and ` y œ 8y È2 È2 È2 È2 # ß # ‹ œ TŠ # ß
# ‹ œ

(b) T œ 4x#
4xy  4y# Ê found in part (a): T Š
TŠ 50. (a)

`T `x

È2 #

ß

È2 # ‹

œ y and

œ T Š


`T `y

È2 #

œx Ê

ß


dT dt

È2 # ‹

œ

È2 #

ß


È2 # ‹; È2 # ‹;

È2 # ‹


4x so the extreme values occur at the four points 4 ˆ "# ‰
4 ˆ
"# ‰  4 ˆ "# ‰ œ 6, the maximum and

œ 4 ˆ #" ‰
4 ˆ #" ‰  4 ˆ #" ‰ œ 2, the minimum

` T dx ` x dt



` T dy ` y dt

œ y Š
2È2 sin t‹  x ŠÈ2 cos t‹

œ ŠÈ2 sin t‹ Š
2È2 sin t‹  Š2È2 cos t‹ ŠÈ2 cos t‹ œ
4 sin# t  4 cos# t œ
4 sin# t  4 a1
sin# tb œ 4
8 sin# t Ê 31 51 71 4 , 4 , 4 #

d T dt# ¹ tœ 1

d# T dt#

œ
16 sin t cost t;

dT dt

œ 0 Ê 4
8 sin# t œ 0 Ê sin# t œ

" #

Ê sin t œ „

on the interval 0 Ÿ t Ÿ 21;

œ
8 sin 2 ˆ 14 ‰ œ
8 Ê T has a maximum at (xß y) œ (2ß 1);

4

d# T dt# ¹ tœ 31

œ
8 sin 2 ˆ 341 ‰ œ 8 Ê T has a minimum at (xß y) œ (
2ß 1);

4

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" È2

Ê t œ 14 ,

824

Chapter 14 Partial Derivatives d# T dt# ¹ tœ 51

œ
8 sin 2 ˆ 541 ‰ œ
8 Ê T has a maximum at (xß y) œ (
2ß
1);

d# T dt# ¹ tœ 71

œ
8 sin 2 ˆ 741 ‰ œ 8 Ê T has a minimum at (xß y) œ (2ß
1)

4

4

(b) T œ xy
2 Ê

`T `x

œ y and

`T `y

œ x so the extreme values occur at the four points found in part (a):

T(2ß 1) œ T(
2ß
1) œ 0, the maximum and T(
2ß 1) œ T(2ß
1) œ
4, the minimum 51. G(uß x) œ 'a g(tß x) dt where u œ f(x) Ê u

dG dx

œ

` G du ` u dx



` G dx ` x dx

F(x) œ '0 Èt%  x$ dt Ê Fw (x) œ Éax# b%  x$ (2x)  '0 x#

x#

` `x

œ g(uß x)f w (x)  'a gx (tß x) dt; thus u

Èt%  x$ dt œ 2xÈx)  x$  '

52. Using the result in Exercise 51, F(x) œ 'x# Èt$  x# dt œ
'1 Èt$  x# dt Ê Fw (x) x#

1

œ ’
Éax# b$  x# x#
'

x#

` 1 `x

Èt$  x# dt “ œ
x# Èx'  x#  ' # È $x # dt x t x 1

14.5 DIRECTIONAL DERIVATIVES AND GRADIENT VECTORS 1.

`f `x

œ
1,

`f `y

œ 1 Ê ™ f œ
i  j ; f(2ß 1) œ
1

Ê
1 œ y
x is the level curve

2.

`f `x

œ

Ê

2y 2x `f `f x#  y# Ê ` x ("ß ") œ 1; ` y œ x#  y# `f ` y ("ß ") œ 1 Ê ™ f œ i  j ; f(1ß 1) # # # #

œ ln 2 Ê ln 2

œ ln ax  y b Ê 2 œ x  y is the level curve

3.

`g `x

`g ` x a2ß
1b

œ y2 Ê

œ 1;

`g `y

œ 2x y Ê

Ê ™ g œ i
4j ; ga2ß
1b œ 2 Ê x œ

`g ` x a2ß
1b œ
4; 2 y# is the level

curve

4.

`g `x

œx Ê

Ê

`g `y

`g `x

ŠÈ2ß "‹ œ È2;

`g `y

œ
y

ŠÈ2ß "‹ œ
1 Ê ™ g œ È2 i
j ;

g ŠÈ2ß "‹ œ

" #

Ê

" #

œ

x# #




y# #

x#

0

or 1 œ x#
y# is the level

curve

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

3x# 2Èt%  x$

dt

Section 14.5 Directional Derivatives and Gradient Vectors 5.

`f `x

œ

1 È2x  3y

`f `x

Ê

`f `x

Ê

`f `y

(
1ß 2) œ 21 ;

œ

3 2È2x  3y

(
1ß 2) œ 43 ; Ê ™ f œ 12 i  34 j ; f(
1ß 2) œ 2

Ê 4 œ 2x  3y is the level curve

6.

`f `x

œ

`f `y

œ
2y2  x Ê

`f `x

Ê

y 2y2 Èx  2x3Î2 Èx

`f `y

1 a4ß
2b œ
16 ;

1 a4ß
2b œ
14 Ê ™ f œ
16 i
41 j ;

f a4ß
2b œ
14 Ê y œ
Èx is the level curve

7.

`f `x

œ 2x 

z x

Ê

`f `x

(1ß 1ß 1) œ 3;

`f `y

`f `y

œ 2y Ê

("ß "ß ") œ 2;

`f `z

œ
4z  ln x Ê

`f `z

("ß "ß ") œ
4;

thus ™ f œ 3i  2j
4k 8.

`f `x

œ
6xz 

Ê 9.

10.

`f `x

œ


`f `z

œ


`f `x

œ exy cos z 

`f `z

x ax#  y#  z# b$Î# z ax#  y#  z# b$Î#

A kAk

œ

4i  3j È 4#  3#

`f `y

(1ß "ß ") œ
11 # ;

`f `y

œ
6yz Ê

("ß "ß ") œ
6;

`f `z

œ 6z#
3 ax#  y# b 

x x # z#  1

" thus ™ f œ
11 # i
6j  # k



" x

Ê

`f `x

(
1ß 2ß
2) œ
26 27 ;



" z

Ê

`f `z

(
1ß 2ß
2) œ
23 54 ; thus ™

y1 È 1
x# `f `z

œ
exy sin z Ê

11. u œ

`f `x

Ê

z x # z#  1 `f " ` z ("ß "ß ") œ # ;

œ

Ê

`f `x

ˆ!ß !ß 16 ‰ œ

È3 #

 1;

`f `y

`f `y

ˆ!ß !ß 16 ‰ œ
#" ; thus ™ f œ

œ


y  y" Ê `` yf ax#  y#  z# b$Î# 23 23 f œ
26 27 i  54 j
54 k

œ exy cos z  sin
" x Ê

È Š 3#2 ‹ i



È3 #

(
1ß 2ß
2) œ

`f `y

ˆ0ß 0ß 16 ‰ œ

23 54

È3 #

;

;

j
"# k

i  35 j ; fx (xß y) œ 2y Ê fx (5ß 5) œ 10; fy (xß y) œ 2x
6y Ê fy (5ß 5) œ
20

4 5

Ê ™ f œ 10i
20j Ê (Du f)P! œ ™ f † u œ 10 ˆ 45 ‰
20 ˆ 35 ‰ œ
4 12. u œ

A k Ak

œ

3i
4j È3#  (
4)#

œ

3 5

i
45 j ; fx (xß y) œ 4x Ê fx (
1ß 1) œ
4; fy (xß y) œ 2y Ê fy (
1ß 1) œ 2

Ê ™ f œ
4i  2j Ê (Du f)P! œ ™ f † u œ
12 5
13. u œ

A kAk

œ

12i  5j È12#  5#

œ

12 13

i

5 13

A kAk

œ

hy (xß y) œ

3i
2j È3# (
2)# ˆ "x ‰ y ˆ x ‰#  1



œ

3 È13

i


ˆ #x ‰ È3 x# y# Ê1
Š 4 ‹

œ
4

y2  2 Ê gx a1ß
1b axy  2b2 15 21 œ 36 13
13 œ 13

j ; gx axß yb œ

Ê ™ g œ 3i
3j Ê aDu gbP! œ ™ g † u 14. u œ

8 5

2 È13

j ; hx (xß y) œ

Ê hy (1ß 1) œ

3 #

Š x#y ‹ y ˆ x ‰#  1



x 2 œ 3; gy axß yb œ
axy Ê gy a1ß
1b œ
3  2b2

ˆ #y ‰ È3 Ê1
Š

Ê ™hœ

" #

x# y# 4 ‹

2

Ê hx (1ß 1) œ "# ;

i  #3 j Ê (Du h)P! œ ™ h † u œ

œ
2È313

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

3 2È13




6 2È13

825

826

Chapter 14 Partial Derivatives

15. u œ

A k Ak

œ

3 i  6 j
#k È3#  6#  (
2)#

œ

i  67 j
27 k ; fx (xß yß z) œ y  z Ê fx (1ß
1ß 2) œ 1; fy (xß yß z) œ x  z

3 7

Ê fy (1ß
1ß 2) œ 3; fz (xß yß z) œ y  x Ê fz (1ß
1ß 2) œ 0 Ê ™ f œ i  3j Ê (Du f)P! œ ™ f † u œ 16. u œ

A kAk

œ

ijk È 1 #  1#  1#

œ

1 È3

i

1 È3

j

1 È3

3 7



18 7

œ3

k ; fx (xß yß z) œ 2x Ê fx (1ß 1ß 1) œ 2; fy (xß yß z) œ 4y

Ê fy (1ß 1ß 1) œ 4; fz (xß yß z) œ
6z Ê fz (1ß 1ß 1) œ
6 Ê ™ f œ 2i  4j
6k Ê (Du f)P! œ ™ f † u œ 2 Š È"3 ‹  4 Š È"3 ‹
6 Š È"3 ‹ œ 0 17. u œ

A k Ak

œ

2i  j
2k È2#  1#  (
2)#

œ

i  13 j
23 k ; gx (xß yß z) œ 3ex cos yz Ê gx (0ß 0ß 0) œ 3; gy (xß yß z) œ
3zex sin yz

2 3

Ê gy (0ß 0ß 0) œ 0; gz (xß yß z) œ
3yex sin yz Ê gz (0ß 0ß 0) œ 0 Ê ™ g œ 3i Ê (Du g)P! œ ™ g † u œ 2 18. u œ

A k Ak

œ

i  2j  2k È 1#  2#  2#

œ

1 3

i  23 j  23 k ; hx (xß yß z) œ
y sin xy 

" x

Ê hx ˆ1ß 0ß "# ‰ œ 1;

hy (xß yß z) œ
x sin xy  zeyz Ê hy ˆ"ß !ß #" ‰ œ #" ; hz (xß yß z) œ yeyz  Ê (Du h)P! œ ™ h † u œ

" 3



" 3



4 3

Ê hz ˆ"ß !ß #" ‰ œ 2 Ê ™ h œ i  #" j  2k

œ2

19. ™ f œ (2x  y) i  (x  2y) j Ê ™ f(
1ß 1) œ
i  j Ê u œ most rapidly in the direction u œ


" z

" È2

i

" È2

™f k™f k

œ


i  j È(
1)#  1#

œ
È" i  2

" È2

j ; f increases

" È2

j and decreases most rapidly in the direction
u œ

i


" È2

j;

(Du f)P! œ ™ f † u œ k ™ f k œ È2 and (D
u f)P! œ
È2 ™f k™ f k

20. ™ f œ a2xy  yexy sin yb i  ax#  xexy sin y  exy cos yb j Ê ™ f(1ß 0) œ 2j Ê u œ

œ j ; f increases most

rapidly in the direction u œ j and decreases most rapidly in the direction
u œ
j ; (Du f)P! œ ™ f † u œ k ™ f k œ 2 and (D
u f)P! œ
2 21. ™ f œ

" y

i
Š yx#  z‹ j
yk Ê ™ f(4ß "ß ") œ i
5j
k Ê u œ " 3È 3

f increases most rapidly in the direction of u œ "
u œ
3È i 3

5 3È 3

j

" 3È 3

i


5 3È 3

j


" 3È 3

™f k™f k

œ

i
5j
k È1#  (
5)#  (
1)#

œ

" 3È 3

i


5 3È 3

j


" 3È 3

k and decreases most rapidly in the direction

k ; (Du f)P! œ ™ f † u œ k ™ f k œ 3È3 and (D
u f)P! œ
3È3

22. ™ g œ ey i  xey j  2zk Ê ™ g ˆ1ß ln 2ß "# ‰ œ 2i  2j  k Ê u œ g increases most rapidly in the direction u œ

2 3

™g k™gk

œ

2i  2j  k È 2#  2#  1#

œ

2 3

i  32 j  3" k ;

i  23 j  3" k and decreases most rapidly in the direction


u œ
23 i
23 j
3" k ; (Du g)P! œ ™ g † u œ k ™ gk œ 3 and (D
u g)P! œ
3 23. ™ f œ ˆ "x  x" ‰ i  Š y"  y" ‹ j  ˆ "z  "z ‰ k Ê ™ f("ß "ß ") œ 2i  2j  2k Ê u œ f increases most rapidly in the direction u œ
u œ
È"3 i


" È3

j


" È3

" È3

i

" È3

j

" È3

™f k™f k

2 7

" È3

2 7

™h k™hk

œ

j

" È3

k;

2i  3 j  6k È 2#  3#  6#

i  37 j  67 k and decreases most rapidly in the

direction
u œ
i
j
k ; (Du h)P! œ ™ h † u œ k ™ hk œ 7 and (D
u h)P! œ
7 3 7

" È3

k; (Du f)P! œ ™ f † u œ k ™ f k œ 2È3 and (D
u f)P! œ
2È3

i  37 j  67 k ; h increases most rapidly in the direction u œ 2 7

i

k and decreases most rapidly in the direction

2y 24. ™ h œ Š x# 2x y#
1 ‹ i  Š x#  y#
1  1‹ j  6k Ê ™ h("ß "ß 0) œ 2i  3j  6k Ê u œ

œ

œ

6 7

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

k;

Section 14.5 Directional Derivatives and Gradient Vectors

827

25. ™ f œ 2xi  2yj Ê ™ f ŠÈ2ß È2‹ œ 2È2 i  2È2 j Ê Tangent line: 2È2 Šx
È2‹  2È2 Šy
È2‹ œ 0 Ê È2x  È2y œ 4

26. ™ f œ 2xi
j Ê ™ f ŠÈ2ß 1‹ œ 2È2 i
j Ê Tangent line: 2È2 Šx
È2‹
(y
1) œ 0 Ê y œ 2È2x
3

27. ™ f œ yi  xj Ê ™ f(2ß
2) œ
2i  2j Ê Tangent line:
2(x
2)  2(y  2) œ 0 Ê yœx
4

28. ™ f œ (2x
y)i  (2y
x)j Ê ™ f(
1ß 2) œ
4i  5j Ê Tangent line:
4(x  1)  5(y
2) œ 0 Ê
4x  5y
14 œ 0

29. ™ f œ a2x
ybi  a
x  2y
1bj (a) ™ fa1,
1b œ 3i
4j Ê l ™ fa1,
1bl œ 5 Ê Du fa1,
1b œ 5 in the direction of u œ 35 i
45 j (b)
™ fa1,
1b œ
3i  4j Ê l ™ fa1,
1bl œ 5 Ê Du fa1,
1b œ
5 in the direction of u œ
35 i  45 j (c) Du fa1,
1b œ 0 in the direction of u œ 45 i  35 j or u œ
45 i
35 j (d) Let u œ u1 i  u2 j Ê lul œ Èu12  u22 œ 1 Ê u12  u22 œ 1; Du fa1,
1b œ ™ fa1,
1b † u œ a3i
4jb † au1 i  u2 jb 2

œ 3u1
4u2 œ 4 Ê u2 œ 43 u1
1 Ê u12  ˆ 43 u1
1‰ œ 1 Ê

25 2 3 16 u1
2 u1 7 œ 24 25 i
25 j

œ 0 Ê u1 œ 0 or u1 œ

24 25 ;

7 u1 œ 0 Ê u2 œ
1 Ê u œ
j, or u1 œ 24 25 Ê u2 œ
25 Ê u (e) Let u œ u1 i  u2 j Ê lul œ Èu12  u22 œ 1 Ê u12  u22 œ 1; Du fa1,
1b œ ™ fa1,
1b † u œ a3i
4jb † au1 i  u2 jb 2

œ 3u1
4u2 œ
3 Ê u1 œ 43 u2
1 Ê ˆ 43 u2
1‰  u22 œ 1 Ê u2 œ 0 Ê u1 œ
1 Ê u œ
i, or u2 œ

24 25

Ê u2 œ

7 25

Êuœ

25 2 8 9 u2
3 u2 7 24 25 i  25 j

œ 0 Ê u2 œ 0 or u2 œ

24 25 ;

. 30. ™ f œ

2y i ax  yb2




2x j a x  y b2

(a) ™ fˆ
21 , 23 ‰ œ 3i  j Ê l ™ fˆ
21 , 23 ‰l œ È10 Ê Du fˆ
21 , 23 ‰ œ È10 in the direction of u œ

3 È10 i



1 È10 j

(b)
™ fˆ
21 , 23 ‰ œ
3i
j Ê l ™ fˆ
21 , 23 ‰l œ È10 Ê Du fa1,
1b œ
È10 in the direction of u œ
È310 i


Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 È10 j

828

Chapter 14 Partial Derivatives

(c) Du fˆ
12 , 23 ‰ œ 0 in the direction of u œ

1 È10 i




3 È10 j

or u œ
È110 i 

3 È10 j

(d) Let u œ u1 i  u2 j Ê lul œ Èu12  u22 œ 1 Ê u12  u22 œ 1; Du fˆ
12 , 32 ‰ œ ™ fˆ
12 , 32 ‰ † u œ a3i  jb † au1 i  u2 jb œ 3u1  u2 œ
2 Ê u2 œ
3u1
2 Ê u12  a
3u1
2b2 œ 1 Ê 10u12  12u1  3 œ 0 Ê u1 œ u1 œ Êu

È
6  È 6 Ê u2 œ
2
103 6 10 È6 È œ
6
i 
2 103 6 j 10


6  È 6 i 10

Êuœ




2
3È 6 j, 10

or u1 œ


6
È 6 10

Ê u2 œ


6 „ È 6 10


2  3È 6 10

(e) Let u œ u1 i  u2 j Ê lul œ Èu12  u22 œ 1 Ê u12  u22 œ 1; Du fˆ
12 , 32 ‰ œ ™ fˆ
12 , 32 ‰ † u œ a3i  jb † au1 i  u2 jb œ 3u1  u2 œ 1 Ê u2 œ 1
3u1 Ê u12  a1
3u1 b2 œ 1 Ê 10u12
6u1 œ 0 Ê u1 œ 0 or u1 œ 35 ; u1 œ 0 Ê u2 œ 1 Ê u œ j, or u1 œ

3 5

Ê u2 œ
45 Ê u œ 35 i
45 j

31. ™ f œ yi  (x  2y)j Ê ™ f(3ß 2) œ 2i  7j ; a vector orthogonal to ™ f is v œ 7i
2j Ê u œ œ

7 È53

32. ™ f œ

i


4xy# a x #  y # b#

Ê uœ

j and
u œ
È753 i 

2 È53

v kv k

2 È53

v kvk

œ

7i
2j È7#  (
2)#

j are the directions where the derivative is zero

4x# y j Ê ™ f("ß ") œ i
j ; a vector orthogonal to ™ f is v œ i  j a x #  y # b# ij 1 1 1 1 È1#  1# œ È2 i  È2 j and
u œ
È2 i
È2 j are the directions where the

i


œ

derivative is zero

33. ™ f œ (2x
3y)i  (
3x  8y)j Ê ™ f(1ß 2) œ
4i  13j Ê k ™ f(1ß 2)k œ È(
4)#  (13)# œ È185 ; no, the maximum rate of change is È185  14 34. ™ T œ 2yi  (2x
z)j
yk Ê ™ T(1ß
1ß 1) œ
2i  j  k Ê k ™ T(1ß
1ß 1)k œ È(
2)#  1#  1# œ È6 ; no, the minimum rate of change is
È6 
3 35. ™ f œ fx ("ß #)i  fy ("ß #)j and u" œ

ij È 1#  1#

œ

" È2

i

" È2

j Ê (Du" f)(1ß 2) œ fx (1ß 2) Š È"2 ‹  fy (1ß 2) Š È"2 ‹

œ 2È2 Ê fx (1ß 2)  fy (1ß 2) œ 4; u# œ
j Ê (Du# f)(1ß 2) œ fx (1ß 2)(0)  fy (1ß 2)(
1) œ
3 Ê
fy (1ß 2) œ
3 Ê fy (1ß 2) œ 3; then fx (1ß 2)  3 œ 4 Ê fx (1ß 2) œ 1; thus ™ f(1ß 2) œ i  3j and u œ œ
È15 i


2 È5

j Ê (Du f)P! œ ™ f † u œ
È"5


36. (a) (Du f)P œ 2È3 Ê k ™ f k œ 2È3; u œ

v kvk

œ

v kv k

œ

ij È 1#  1#

œ

" È2

i

" È2

" È3


i
2j È(
1)#  (
2)#

œ

œ
È75

ij
k È1#  1#  (
1)#

Ê ™ f œ k ™ f k u Ê ™ f œ 2È3 Š È"3 i  (b) v œ i  j Ê u œ

6 È5

v kvk

j


" È3

œ

1 È3

i

1 È3

j


" È3

k; thus u œ

™f k ™f k

k‹ œ 2i  2j
2k

j Ê (Du f)P! œ ™ f † u œ 2 Š È"2 ‹  2 Š È"2 ‹
2(0) œ 2È2

37. The directional derivative is the scalar component. With ™ f evaluated at P! , the scalar component of ™ f in the direction of u is ™ f † u œ (Du f)P! . 38. Di f œ ™ f † i œ (fx i  fy j  fz k) † i œ fx ; similarly, Dj f œ ™ f † j œ fy and Dk f œ ™ f † k œ fz 39. If (xß y) is a point on the line, then T(xß y) œ (x
x! )i  (y
y! )j is a vector parallel to the line Ê T † N œ 0 Ê A(x
x! )  B(y
y! ) œ 0, as claimed. 40. (a) ™ (kf) œ

` (kf) `x

i

` (kf) `y

j

` (kf) `z

k œ k ˆ `` xf ‰ i  k Š `` yf ‹ j  k ˆ `` zf ‰ k œ k Š `` xf i 

`f `y

j

`f `z

k‹ œ k ™ f

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.6 Tangent Planes and Differentials ` (f  g) `x

(b) ™ (f  g) œ œ

`f `x

i

`g `x

i

`f `y

i

j

` (f  g) `y `g `y

j

j `f `z

` (f  g) `z

k

`g `z

k œ Š `` xf  k œ Š `` xf i 

`g `x ‹ i `f `y

 Š `` yf 

j

`f `z

`g `y ‹ j

 Š `` zf 

k‹  Š `` gx i 

`g `y

829

`g `z ‹ k

j

`g `z

k‹ œ ™ f  ™ g

f ‹ j  Š `` zf g 

`g `z

f‹ k

(c) ™ (f
g) œ ™ f
™ g (Substitute
g for g in part (b) above) ` (fg) `x

(d) ™ (fg) œ

i

` (fg) `y

j

` (fg) `z

`g `x

k œ Š `` xf g 

f ‹ i  Š `` yf g 

`g `y

œ ˆ `` xf g‰ i  Š `` xg f ‹ i  Š `` yf g‹ j  Š `` gy f ‹ j  ˆ `` zf g‰ k  Š `` gz f ‹ k œ f Š `` gx i  (e) ™ Š gf ‹ œ œŒ œ

`g `y

j

` Š gf ‹ `x

`g `z

i

k‹  g Š `` xf i 

` Š gf ‹ `y

g ``xf i  g ``yf j  g `` fz k  g#

g ™f g#




f™g g#

œ

j


Œ

` Š gf ‹ `z

`f `y

j

kœŠ

`f `z

k‹ œ f ™ g  g ™ f

g ``xf
f `` gx ‹i g#

f `` gx i  f `` gy j  f ``gz k  g#

œ

Œ

g ``yf
f `` gy j g#

g Š ``xf i  ``yf j  `` fz k‹ g#




Š

g `` zf
f ``gz ‹k g#

f Š `` gx i  `` gy j  ``gz k‹ g#

g™f
f™g g#

14.6 TANGENT PLANES AND DIFFERENTIALS 1. (a) ™ f œ 2xi  2yj  2zk Ê ™ f(1ß 1ß 1) œ 2i  2j  2k Ê Tangent plane: 2(x
1)  2(y
1)  2(z
1) œ 0 Ê x  y  z œ 3; (b) Normal line: x œ 1  2t, y œ 1  2t, z œ 1  2t 2. (a) ™ f œ 2xi  2yj
2zk Ê ™ f(3ß 5ß
4) œ 6i  10j  8k Ê Tangent plane: 6(x
3)  10(y
5)  8(z  4) œ 0 Ê 3x  5y  4z œ 18; (b) Normal line: x œ 3  6t, y œ 5  10t, z œ
4  8t 3. (a) ™ f œ
2xi  2k Ê ™ f(2ß 0ß 2) œ
4i  2k Ê Tangent plane:
4(x
2)  2(z
2) œ 0 Ê
4x  2z  4 œ 0 Ê
2x  z  2 œ 0; (b) Normal line: x œ 2
4t, y œ 0, z œ 2  2t 4. (a) ™ f œ (2x  2y)i  (2x
2y)j  2zk Ê ™ f(1ß
1ß 3) œ 4j  6k Ê Tangent plane: 4(y  1)  6(z
3) œ 0 Ê 2y  3z œ 7; (b) Normal line: x œ 1, y œ
1  4t, z œ 3  6t 5. (a) ™ f œ a
1 sin 1x
2xy  zexz b i  a
x#  zb j  axexz  yb k Ê ™ f(0ß 1ß 2) œ 2i  2j  k Ê Tangent plane: 2(x
0)  2(y
1)  1(z
2) œ 0 Ê 2x  2y  z
4 œ 0; (b) Normal line: x œ 2t, y œ 1  2t, z œ 2  t 6. (a) ™ f œ (2x
y)i
(x  2y)j
k Ê ™ f(1ß 1ß
1) œ i
3j
k Ê Tangent plane: 1(x
1)
3(y
1)
1(z  1) œ 0 Ê x
3y
z œ
1; (b) Normal line: x œ 1  t, y œ 1
3t, z œ
1
t 7. (a) ™ f œ i  j  k for all points Ê ™ f(0ß 1ß 0) œ i  j  k Ê Tangent plane: 1(x
0)  1(y
1)  1(z
0) œ 0 Ê x  y  z
1 œ 0; (b) Normal line: x œ t, y œ 1  t, z œ t 8. (a) ™ f œ (2x
2y
1)i  (2y
2x  3)j
k Ê ™ f(2ß
3ß 18) œ 9i
7j
k Ê Tangent plane: 9(x
2)
7(y  3)
1(z
18) œ 0 Ê 9x
7y
z œ 21; (b) Normal line: x œ 2  9t, y œ
3
7t, z œ 18
t

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

830

Chapter 14 Partial Derivatives

9. z œ f(xß y) œ ln ax#  y# b Ê fx (xß y) œ

and fy (xß y) œ

2x x#  y#

2y x#  y#

Ê fx (1ß 0) œ 2 and fy (1ß 0) œ 0 Ê from

Eq. (4) the tangent plane at (1ß 0ß 0) is 2(x
1)
z œ 0 or 2x
z
2 œ 0 #

#

#

#

#

#

10. z œ f(xß y) œ e
ax y b Ê fx (xß y) œ
2xe
ax y b and fy (xß y) œ
2ye
ax y b Ê fx (0ß 0) œ 0 and fy (!ß !) œ 0 Ê from Eq. (4) the tangent plane at (0ß 0ß 1) is z
1 œ 0 or z œ 1 11. z œ f(Bß y) œ Èy
x Ê fx (xß y) œ
"# (y
x)
"Î# and fy (xß y) œ

" #

(y
x)
"Î# Ê fx (1ß 2) œ
"# and fy ("ß #) œ

Ê from Eq. (4) the tangent plane at (1ß 2ß 1) is
"# (x
1)  "# (y
2)
(z
1) œ 0 Ê x
y  2z
1 œ 0

" #

12. z œ f(Bß y) œ 4x#  y# Ê fx (xß y) œ 8x and fy (xß y) œ #y Ê fx (1ß 1) œ 8 and fy ("ß 1) œ # Ê from Eq. (4) the tangent plane at (1ß 1ß 5) is 8(x
1)  2(y
1)
(z
5) œ 0 or 8x  2y
z
5 œ 0 13. ™ f œ i  2yj  2k Ê ™ f(1ß 1ß 1) œ i  2j  2k and ™ g œ i for all points; v œ ™ f ‚ ™ g â â â i j kâ â â Ê v œ â " 2 2 â œ 2j
2k Ê Tangent line: x œ 1, y œ 1  2t, z œ 1
2t â â â" 0 0â 14. ™ f œ yzi  xzj  xyk Ê ™ f(1ß 1ß 1) œ i  j  k; ™ g œ 2xi  4yj  6zk Ê ™ g(1ß 1ß 1) œ 2i  4j  6k ; â â â i j kâ â â Ê v œ ™ f ‚ ™ g Ê â " 1 1 â œ 2i
4j  2k Ê Tangent line: x œ 1  2t, y œ 1
4t, z œ 1  2t â â â2 4 6â 15. ™ f œ 2xi  2j  2k Ê ™ f ˆ1ß 1ß "# ‰ œ 2i  2j  2k and ™ g œ j for all points; v œ ™ f ‚ ™ g â â â i j kâ â â Ê v œ â 2 2 2 â œ
2i  2k Ê Tangent line: x œ 1
2t, y œ 1, z œ "#  2t â â â0 1 0â 16. ™ f œ i  2yj  k Ê ™ f ˆ "# ß 1ß "# ‰ œ i  2j  k and ™ g œ j for all points; v œ ™ f ‚ ™ g â â â i j kâ â â Ê v œ â 1 2 1 â œ
i  k Ê Tangent line: x œ "#
t, y œ 1, z œ "#  t â â â0 1 0â 17. ™ f œ a3x#  6xy#  4yb i  a6x# y  3y#  4xb j
2zk Ê ™ f(1ß 1ß 3) œ 13i  13j
6k ; ™ g œ 2xi  2yj  2zk â â j k â â i â â Ê ™ g("ß "ß $) œ 2i  2j  6k ; v œ ™ f ‚ ™ g Ê v œ â "3 13
6 ⠜ 90i
90j Ê Tangent line: â â 2 6 â â 2 x œ 1  90t, y œ 1
90t, z œ 3 18. ™ f œ 2xi  2yj Ê ™ f ŠÈ2ß È2ß 4‹ œ 2È2 i  2È2 j ; ™ g œ 2xi  2yj
k Ê ™ g ŠÈ2ß È2ß 4‹ â i j k ââ â â â œ 2È2i  2È2j
k ; v œ ™ f ‚ ™ g Ê v œ â 2È2 2È2 0 â œ
2È2 i  2È2 j Ê Tangent line: â â â 2È2 2È2
1 â x œ È2
2È2 t, y œ È2  2È2 t, z œ 4 19. ™ f œ Š x#  yx#  z# ‹ i  Š x#  yy#  z# ‹ j  Š x#  yz#  z# ‹ k Ê ™ f(3ß 4ß 12) œ uœ

v kvk

œ

3i  6j
2k È3#  6#  (
2)#

œ

3 7

i  76 j
27 k Ê ™ f † u œ

9 1183

3 169

i

4 169

j

12 169

k;

9 ‰ and df œ ( ™ f † u) ds œ ˆ 1183 (0.1) ¸ 0.0008

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.6 Tangent Planes and Differentials 20. ™ f œ aex cos yzb i
azex sin yzb j
ayex sin yzb k Ê ™ f(0ß 0ß 0) œ i ; u œ œ

1 È3

i

1 È3

j


k Ê ™f†uœ

1 È3

1 È3

and df œ ( ™ f † u) ds œ

v kvk

œ

831

2i  2j
2k È2#  2#  (
2)#

(0.1) ¸ 0.0577

1 È3

Ä 21. ™ g œ (1  cos z)i  (1
sin z)j  (
x sin z
y cos z)k Ê ™ g(2ß
1ß 0) œ 2i  j  k; A œ P! P" œ
2i  2j  2k Ê uœ

v kvk

œ


2 i  2 j  2 k È(
2)#  2#  2#

œ
È13 i 

1 È3

j

1 È3

k Ê ™ g † u œ 0 and dg œ ( ™ g † u) ds œ (0)(0.2) œ 0

22. ™ h œ c
1y sin (1xy)  z# d i
c1x sin (1xy)d j  2xzk Ê ™ h(
1ß
1ß
1) œ (1 sin 1  1)i  (1 sin 1)j  2k Ä k œ i  2k ; v œ P! P" œ i  j  k where P" œ (!ß !ß !) Ê u œ kvvk œ È1i#j1#  œ È13 i  È13 j  È13 k 1# Ê ™h†uœ

œ È3 and dh œ ( ™ h † u) ds œ È3(0.1) ¸ 0.1732

3 È3

23. (a) The unit tangent vector at Š "# ß

È3 # ‹

in the direction of motion is u œ

™ T œ (sin 2y)i  (2x cos 2y)j Ê ™ T Š "# ß È3 #

œ

sin È3


" #

È3 # ‹

È3 #

i
#" j ;

œ Šsin È3‹ i  Šcos È3‹ j Ê Du T Š "# ß

œ ™T†vœŠ™T† dT dt

œŠ

È3 #

œ ™T†u

cos È3 ¸ 0.935° C/ft ` T dx ` x dt



` T dy ` y dt

we have u œ

È3 #

i
#" j from part (a)

(b) r(t) œ (sin 2t)i  (cos 2t)j Ê v(t) œ (2 cos 2t)i
(2 sin 2t)j and kvk œ 2;

Ê

È3 # ‹

v kvk ‹

sin È3


" #

kvk œ (Du T) kvk , where u œ

v kv k

; at Š "# ß

È3 # ‹

dT dt

œ

cos È3‹ † 2 œ È3 sin È3
cos È3 ¸ 1.87° C/sec

24. (a) ™ T œ (4x
yz)i
xzj
xyk Ê ™ T(8ß 6ß
4) œ 56i  32j
48k ; r(t) œ 2t# i  3tj
t# k Ê the particle is at the point P()ß 6ß
4) when t œ 2; v(t) œ 4ti  3j
2tk Ê v(2) œ 8i  3j
4k Ê u œ kvvk

(b)

œ

8 È89

dT dt

œ

i

` T dx ` x dt

3 È89



j


` T dy ` y dt

4 È89

k Ê Du T(8ß 6ß
4) œ ™ T † u œ

" È89

œ ™ T † v œ ( ™ T † u) kvk Ê at t œ 2,

[56 † 8  32 † 3
48 † (
4)] œ

dT dt

736 È89

° C/m

736 œ Du T¸ tœ2 v(2) œ Š È ‹ È89 œ 736° C/sec 89

25. (a) f(!ß 0) œ 1, fx (xß y) œ 2x Ê fx (0ß 0) œ 0, fy (xß y) œ 2y Ê fy (0ß 0) œ 0 Ê L(xß y) œ 1  0(x
0)  0(y
0) œ 1 (b) f(1ß 1) œ 3, fx (1ß 1) œ 2, fy (1ß 1) œ 2 Ê L(xß y) œ 3  2(x
1)  2(y
1) œ 2x  2y
1 26. (a) f(!ß 0) œ 4, fx (xß y) œ 2(x  y  2) Ê fx (0ß 0) œ 4, fy (xß y) œ 2(x  y  2) Ê fy (0ß 0) œ 4 Ê L(xß y) œ 4  4(x
0)  4(y
0) œ 4x  4y  4 (b) f(1ß 2) œ 25, fx (1ß 2) œ 10, fy (1ß 2) œ 10 Ê L(xß y) œ 25  10(x
1)  10(y
2) œ 10x  10y
5 27. (a) f(0ß 0) œ 5, fx (xß y) œ 3 for all (xß y), fy (xß y) œ
4 for all (xß y) Ê L(xß y) œ 5  3(x
0)
4(y
0) œ 3x
4y  5 (b) f(1ß 1) œ 4, fx (1ß 1) œ 3, fy (1ß 1) œ
4 Ê L(xß y) œ 4  3(x
1)
4(y
1) œ 3x
4y  5 28. (a) f(1ß 1) œ 1, fx (xß y) œ 3x# y% Ê fx (1ß 1) œ 3, fy (xß y) œ 4x$ y$ Ê fy (1ß 1) œ 4 Ê L(xß y) œ 1  3(x
1)  4(y
1) œ 3x  4y
6 (b) f(0ß 0) œ 0, fx (!ß 0) œ 0, fy (0ß 0) œ 0 Ê L(xß y) œ 0 29. (a) f(0ß 0) œ 1, fx (xß y) œ ex cos y Ê fx (0ß 0) œ 1, fy (xß y) œ
ex sin y Ê fy (0ß 0) œ 0 Ê L(xß y) œ 1  1(x
0)  0(y
0) œ x  1 (b) f ˆ0ß 1# ‰ œ 0, fx ˆ0ß 1# ‰ œ 0, fy ˆ0ß 1# ‰ œ
1 Ê L(xß y) œ 0  0(x
0)
1 ˆy
1# ‰ œ
y 

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 #

832

Chapter 14 Partial Derivatives

30. (a) f(0ß 0) œ 1, fx (xß y) œ
e2y
x Ê fx (!ß !) œ
1, fy (xß y) œ 2e2y
x Ê fy (0ß 0) œ 2 Ê L(xß y) œ 1
1(x
0)  2(y
0) œ
x  2y  1 (b) f(1ß 2) œ e$ , fx (1ß 2) œ
e$ , fy (1ß 2) œ 2e$ Ê L(xß y) œ e$
e$ (x
1)  2e$ (y
2) œ
e$ x  2e$ y
2e$ 31. (a) Wa20, 25b œ 11‰ F; Wa30,
10b œ
39‰ F; Wa15, 15b œ 0‰ F (b) Wa10,
40b œ
65.5‰ F; Wa50,
40b œ
88‰ F; Wa60, 30b œ 10.2‰ F; 5.72 0.0684t `W (c) Wa25, 5b œ
17.4088‰ F; ``W V œ
v0.84  v0.84 Ê ` V a25, 5b œ
0.36; Ê

`W ` T a25,

`W `T

œ 0.6215  0.4275v0.16

5b œ 1.3370 Ê LaV, Tb œ
17.4088
0.36aV
25b  1.337aT
5b œ 1.337T
0.36V
15.0938

(d) i) Wa24, 6b ¸ La24, 6b œ
15.7118 ¸
15.7‰ F ii) Wa27, 2b ¸ La27, 2b œ
22.1398 ¸
22.1‰ F ii) Wa5,
10b ¸ La5,
10b œ
30.2638 ¸
30.2‰ F This value is very different because the point a5,
10b is not close to the point a25, 5b. 32. Wa50,
20b œ
59.5298‰ F; Ê

`W ` T a50,

`W `V

œ
v5.72 0.84 

0.0684t v0.84

Ê

`W ` V a50,


20b œ
0.2651;

`W `T

œ 0.6215  0.4275v0.16


20b œ 1.4209 Ê LaV, Tb œ
59.5298
0.2651aV
50b  1.4209aT  20b

œ 1.4209T
0.2651V
17.8568 (a) Wa49,
22b ¸ La49,
22b œ
62.1065 ¸
62.1‰ F (b) Wa53,
19b ¸ La53,
19b œ
58.9042 ¸
58.9‰ F (c) Wa60,
30b ¸ La60,
30b œ
76.3898 ¸
76.4‰ F 33. f(2ß 1) œ 3, fx (xß y) œ 2x
3y Ê fx (2ß 1) œ 1, fy (xß y) œ
3x Ê fy (2ß 1) œ
6 Ê L(xß y) œ 3  1(x
2)
6(y
1) œ 7  x
6y; fxx (xß y) œ 2, fyy (xß y) œ 0, fxy (xß y) œ
3 Ê M œ 3; thus kE(xß y)k Ÿ ˆ "# ‰ (3) akx
2k  ky
1kb# Ÿ ˆ 3# ‰ (0.1  0.1)# œ 0.06

34. f(2ß 2) œ 11, fx (xß y) œ x  y  3 Ê fx (2ß 2) œ 7, fy (xß y) œ x 

y #


3 Ê fy (2ß 2) œ 0

Ê L(xß y) œ 11  7(x
2)  0(y
2) œ 7x
3; fxx (xß y) œ 1, fyy (xß y) œ "# , fxy (xß y) œ 1 Ê M œ 1; thus kE(xß y)k Ÿ ˆ "# ‰ (1) akx
2k  ky
2kb# Ÿ ˆ #1 ‰ (0.1  0.1)# œ 0.02

35. f(0ß 0) œ 1, fx (xß y) œ cos y Ê fx (0ß 0) œ 1, fy (xß y) œ 1
x sin y Ê fy (0ß 0) œ 1 Ê L(xß y) œ 1  1(x
0)  1(y
0) œ x  y  1; fxx (xß y) œ 0, fyy (xß y) œ
x cos y, fxy (xß y) œ
sin y Ê Q œ 1; thus kE(xß y)k Ÿ ˆ "# ‰ (1) akxk  kykb# Ÿ ˆ #1 ‰ (0.2  0.2)# œ 0.08 36. f("ß #) œ 6, fx (xß y) œ y#
y sin (x
1) Ê fx (1ß 2) œ 4, fy (xß y) œ 2xy  cos (x
1) Ê fy (1ß 2) œ 5 Ê L(xß y) œ 6  4(x
1)  5(y
2) œ 4x  5y
8; fxx (xß y) œ
y cos (x
1), fyy (xß y) œ 2x, fxy (xß y) œ 2y
sin (x
1); kx
1k Ÿ 0.1 Ê 0.9 Ÿ x Ÿ 1.1 and ky
2k Ÿ 0.1 Ê 1.9 Ÿ y Ÿ 2.1; thus the max of kfxx (xß y)k on R is 2.1, the max of kfyy (xß y)k on R is 2.2, and the max of kfxy (xß y)k on R is 2(2.1)
sin (0.9
1) Ÿ 4.3 Ê M œ 4.3; thus kE(xß y)k Ÿ ˆ "# ‰ (4.3) akx
1k  ky
2kb# Ÿ (2.15)(0.1  0.1)# œ 0.086 37. f(0ß 0) œ 1, fx (xß y) œ ex cos y Ê fx (0ß 0) œ 1, fy (xß y) œ
ex sin y Ê fy (0ß 0) œ 0 Ê L(xß y) œ 1  1(x
0)  0(y
0) œ 1  x; fxx (xß y) œ ex cos y, fyy (xß y) œ
ex cos y, fxy (xß y) œ
ex sin y; kxk Ÿ 0.1 Ê
0.1 Ÿ x Ÿ 0.1 and kyk Ÿ 0.1 Ê
0.1 Ÿ y Ÿ 0.1; thus the max of kfxx (xß y)k on R is e0Þ1 cos (0.1) Ÿ 1.11, the max of kfyy (xß y)k on R is e0Þ1 cos (0.1) Ÿ 1.11, and the max of kfxy (xß y)k on R is e0Þ1 sin (0.1) Ÿ 0.12 Ê M œ 1.11; thus kE(xß y)k Ÿ ˆ "# ‰ (1.11) akxk  kykb# Ÿ (0.555)(0.1  0.1)# œ 0.0222

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.6 Tangent Planes and Differentials 38. f(1ß 1) œ 0, fx (xß y) œ

" x

Ê fx (1ß 1) œ 1, fy (xß y) œ

œ x  y
2; fxx (xß y) œ
" (0.98)#

kfxx (xß y)k on R is " (0.98)#

" x#

, fyy (xß y) œ


" y#

" y

833

Ê fy (1ß 1) œ 1 Ê L(xß y) œ 0  1(x
1)  1(y
1)

, fxy (xß y) œ 0; kx
1k Ÿ 0.2 Ê 0.98 Ÿ x Ÿ 1.2 so the max of

Ÿ 1.04; ky
1k Ÿ 0.2 Ê 0.98 Ÿ y Ÿ 1.2 so the max of kfyy (xß y)k on R is

Ÿ 1.04 Ê M œ 1.04; thus kE(xß y)k Ÿ ˆ #" ‰ (1.04) akx
1k  ky
1kb# Ÿ (0.52)(0.2  0.2)# œ 0.0832

39. (a) f("ß "ß ") œ 3, fx (1ß 1ß 1) œ y  zkÐ1ß1ß1Ñ œ 2, fy (1ß 1ß 1) œ x  zkÐ1ß1ß1Ñ œ 2, fz (1ß 1ß 1) œ y  xkÐ1ß1ß1Ñ œ 2 Ê L(xß yß z) œ 3  2(x
1)  2(y
1)  2(z
1) œ 2x  2y  2z
3 (b) f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ 0, fy (1ß 0ß 0) œ 1, fz (1ß 0ß 0) œ 1 Ê L(xß yß z) œ 0  0(x
1)  (y
0)  (z
0) œ y  z (c) f(0ß 0ß 0) œ 0, fx (0ß 0ß 0) œ 0, fy (0ß 0ß 0) œ 0, fz (0ß 0ß 0) œ 0 Ê L(xß yß z) œ 0 40. (a) f(1ß 1ß 1) œ 3, fx (1ß 1ß 1) œ 2xkÐ"ß"ß"Ñ œ 2, fy (1ß 1ß 1) œ 2ykÐ"ß"ß"Ñ œ 2, fz (1ß 1ß 1) œ 2zkÐ"ß"ß"Ñ œ 2 Ê L(xß yß z) œ 3  2(x
1)  2(y
1)  2(z
1) œ 2x  2y  2z
3 (b) f(0ß 1ß 0) œ 1, fx (0ß 1ß 0) œ 0, fy (!ß 1ß 0) œ 2, fz (0ß 1ß 0) œ 0 Ê L(xß yß z) œ 1  0(x
0)  2(y
1)  0(z
0) œ 2y
1 (c) f(1ß 0ß 0) œ 1, fx (1ß 0ß 0) œ 2, fy (1ß 0ß 0) œ 0, fz (1ß 0ß 0) œ 0 Ê L(xß yß z) œ 1  2(x
1)  0(y
0)  0(z
0) œ 2x
1 41. (a) f(1ß 0ß 0) œ 1, fx (1ß 0ß 0) œ fz (1ß 0ß 0) œ

z È x #  y#  z# ¹

x È x #  y #  z# ¹

Ð1ß0ß0Ñ

(b) f(1ß 1ß 0) œ È2, fx (1ß 1ß 0) œ Ê L(xß yß z) œ È2 

" È2

Ð1ß0ß0Ñ

œ 1, fy (1ß 0ß 0) œ

" 3

Ð1 ß0 ß0 Ñ

œ 0,

œ 0 Ê L(xß yß z) œ 1  1(x
1)  0(y
0)  0(z
0) œ x " È2

, fy (1ß 1ß 0) œ

(x
1) 

" È2

" È2

, fz (1ß 1ß 0) œ 0

(y
1)  0(z
0) œ

(c) f(1ß 2ß 2) œ 3, fx (1ß 2ß 2) œ "3 , fy (1ß 2ß 2) œ 23 , fz (1ß 2ß 2) œ œ

y È x #  y #  z# ¹

2 3

" È2

x

" È2

y

Ê L(xß yß z) œ 3  "3 (x
1)  23 (y
2)  23 (z
2)

x  32 y  32 z

42. (a) f ˆ 12 ß 1ß 1‰ œ 1, fx ˆ 1# ß 1ß 1‰ œ fz ˆ 1# ß 1ß 1‰ œ


sin xy z# ¹ ˆ 1 ß"ß"‰

y cos xy ¸ ˆ 1# ß"ß"‰ z

œ 0, fy ˆ 1# ß 1ß 1‰ œ

x cos xy ¸ ˆ 1# ß"ß"‰ z

œ 0,

œ
1 Ê L(xß yß z) œ 1  0 ˆx
1# ‰  0(y
1)
1(z
1) œ 2
z

#

(b) f(2ß 0ß 1) œ 0, fx (2ß 0ß 1) œ 0, fy (2ß 0ß 1) œ 2, fz (2ß 0ß 1) œ 0 Ê L(xß yß z) œ 0  0(x
2)  2(y
0)  0(z
1) œ 2y 43. (a) f(0ß 0ß 0) œ 2, fx (0ß 0ß 0) œ ex k Ð!ß!ß!Ñ œ 1, fy (0ß 0ß 0) œ
sin (y  z)k Ð!ß!ß!Ñ œ 0, fz (0ß 0ß 0) œ
sin (y  z)k Ð!ß!ß!Ñ œ 0 Ê L(xß yß z) œ 2  1(x
0)  0(y
0)  0(z
0) œ 2  x (b) f ˆ0ß 1# ß 0‰ œ 1, fx ˆ0ß 1# ß 0‰ œ 1, fy ˆ0ß 1# ß 0‰ œ
1, fz ˆ0ß 1# ß 0‰ œ
1 Ê L(xß yß z) œ 1  1(x
0)
1 ˆy
12 ‰
1(z
0) œ x
y
z  1#  1

(c) f ˆ0ß 14 ß 14 ‰ œ 1, fx ˆ0ß 14 ß 14 ‰ œ 1, fy ˆ0ß 14 ß 14 ‰ œ
1, fz ˆ0ß 14 ß 14 ‰ œ
1 Ê L(xß yß z) œ 1  1(x
0)
1 ˆy
14 ‰
1 ˆz
14 ‰ œ x
y
z  1#  1 44. (a) f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ fz (1ß 0ß 0) œ

xy (xyz)#  1 ¹ Ð"ß!ß!Ñ

yz (xyz)#  1 ¹ Ð"ß!ß!Ñ

œ 0, fy (1ß 0ß 0) œ

xz (xyz)#  1 ¹ Ð"ß!ß!Ñ

œ 0,

œ 0 Ê L(xß yß z) œ 0

(b) f(1ß 1ß 0) œ 0, fx (1ß 1ß 0) œ 0, fy (1ß 1ß 0) œ 0, fz (1ß 1ß 0) œ 1 Ê L(xß yß z) œ 0  0(x
1)  0(y
1)  1(z
0) œ z (c) f(1ß 1ß 1) œ 14 , fx (1ß 1ß 1) œ "# , fy (1ß 1ß 1) œ "# , fz (1ß 1ß 1) œ "# Ê L(xß yß z) œ 14  "# (x
1)  "# (y
1)  "# (z
1) œ

" #

x  "# y  "# z 

1 4




3 #

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

834

Chapter 14 Partial Derivatives

45. f(xß yß z) œ xz
3yz  2 at P! (1ß 1ß 2) Ê f(1ß 1ß 2) œ
2; fx œ z, fy œ
3z, fz œ x
3y Ê L(xß yß z) œ
2  2(x
1)
6(y
1)
2(z
2) œ 2x
6y
2z  6; fxx œ 0, fyy œ 0, fzz œ 0, fxy œ 0, fyz œ
3 Ê M œ 3; thus, kE(xß yß z)k Ÿ ˆ "# ‰ (3)(0.01  0.01  0.02)# œ 0.0024 46. f(xß yß z) œ x#  xy  yz  "4 z# at P! (1ß 1ß 2) Ê f(1ß 1ß 2) œ 5; fx œ 2x  y, fy œ x  z, fz œ y  "# z

Ê L(xß yß z) œ 5  3(x
1)  3(y
1)  2(z
2) œ 3x  3y  2z
5; fxx œ 2, fyy œ 0, fzz œ "# , fxy œ 1, fxz œ 0, fyz œ 1 Ê M œ 2; thus kE(xß yß z)k Ÿ ˆ "# ‰ (2)(0.01  0.01  0.08)# œ 0.01

47. f(xß yß z) œ xy  2yz
3xz at P! (1ß 1ß 0) Ê f(1ß 1ß 0) œ 1; fx œ y
3z, fy œ x  2z, fz œ 2y
3x Ê L(xß yß z) œ 1  (x
1)  (y
1)
(z
0) œ x  y
z
1; fxx œ 0, fyy œ 0, fzz œ 0, fxy œ 1, fxz œ
3, fyz œ 2 Ê M œ 3; thus kE(xß yß z)k Ÿ ˆ "# ‰ (3)(0.01  0.01  0.01)# œ 0.00135 48. f(xß yß z) œ È2 cos x sin (y  z) at P! ˆ0ß 0ß 14 ‰ Ê f ˆ0ß 0ß 14 ‰ œ 1; fx œ
È2 sin x sin (y  z), fy œ È2 cos x cos (y  z), fz œ È2 cos x cos (y  z) Ê L(xß yß z) œ 1
0(x
0)  (y
0)  ˆz
14 ‰ œ y  z
14  1; fxx œ
È2 cos x sin (y  z), fyy œ
È2 cos x sin (y  z), fzz œ
È2 cos x sin (y  z), fxy œ
È2 sin x cos (y  z), fxz œ
È2 sin x cos (y  z), fyz œ
È2 cos x sin (y  z). The absolute value of each of these second partial derivatives is bounded above by È2 Ê M œ È2; thus kE(xß yß z)k Ÿ ˆ " ‰ ŠÈ2‹ (0.01  0.01  0.01)# œ 0.000636. #

49. Tx (xß y) œ ey  e
y and Ty (xß y) œ x aey
e
y b Ê dT œ Tx (xß y) dx  Ty (xß y) dy œ aey  e
y b dx  x aey
e
y b dy Ê dTkÐ#ßln 2Ñ œ 2.5 dx  3.0 dy. If kdxk Ÿ 0.1 and kdyk Ÿ 0.02, then the maximum possible error in the computed value of T is (2.5)(0.1)  (3.0)(0.02) œ 0.31 in magnitude. #

21rh dr  1r dh 50. Vr œ 21rh and Vh œ 1r# Ê dV œ Vr dr  Vh dh Ê dV œ 2r dr  "h dh; now ¸ drr † 100¸ Ÿ 1 and 1 r# h V œ ¸ dh ¸ ¸ dV ¸ ¸ˆ2 drr ‰ (100)  ˆ dh ‰ ¸ ¸ dr ¸ ¸ dh ¸ h † 100 Ÿ 1 Ê V † 100 Ÿ h (100) Ÿ 2 r † 100  h † 100 Ÿ 2(1)  1 œ 3 Ê 3%

51.

dx x

Ÿ 0.02,

dy y

Ÿ 0.03

dy 2 (a) S œ 2x2  4xy Ê dS œ a4x  4ybdx  4x dy œ a4x2  4xyb dx x  4xy y Ÿ a4x  4xyba0.02b  a4xyba0.03b

œ 0.04a2x2 b  0.05a4xyb Ÿ 0.05a2x2 b  0.05a4xyb œ a0.05ba2x2  4xyb œ 0.05S 2 dy 2 2 2 (b) V œ x2 y Ê dV œ 2xy dx  x2 dy œ 2x2 y dx x  x y y Ÿ a2x yba0.02b  ax yba0.03b œ 0.07ax yb=0.07V

52. V œ

41 3 3 r

 1 r2 h Ê dV œ a41 r2  21 rhbdr  1 r2 dh; r œ 10, h œ 15, dr œ

1 2

and dh œ 0 Ê

dV œ Š41a10b2  21 a10ba15b‹ˆ 12 ‰  1 a10b2 a0b œ 3501 cm3 53. Vr œ 21rh and Vh œ 1r# Ê dV œ Vr dr  Vh dh Ê dV œ 21rh dr  1r# dh Ê dVkÐ5ß12Ñ œ 1201 dr  251 dh; kdrk Ÿ 0.1 cm and kdhk Ÿ 0.1 cm Ê dV Ÿ (1201)(0.1)  (251)(0.1) œ 14.51 cm$ ; V(5ß 12) œ 3001 cm$ 1 Ê maximum percentage error is „ 14.5 3001 ‚ 100 œ „ 4.83% 54. (a)

" R

œ

" R"



" R#

Ê
R"# dR œ
R"# dR"
"

" R##

#

" (b) dR œ R# ’Š R"# ‹ dR"  Š R"# ‹ dR# “ Ê dRk Ð100 400Ñ œ R# ’ (100) # dR"  "

ß

#

sensitive to a variation in R" since

" (100)#



#

dR# Ê dR œ Š RR" ‹ dR"  Š RR# ‹ dR# " (400)#

dR# “ Ê R will be more

" (400)#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.6 Tangent Planes and Differentials #

835

#

(c) From part (a), dR œ Š RR" ‹ dR"  Š RR# ‹ dR# so that R" changing from 20 to 20.1 ohms Ê dR" œ 0.1 ohm and R# changing from 25 to 24.9 ohms Ê dR# œ
0.1 ohms; Ê dRk Ð20 25Ñ œ ß

œ

0.011 ˆ 100 ‰ 9

ˆ 100 ‰# 9 (20)#

(0.1) 

ˆ 100 ‰# 9 (25)#

" R

œ

" R"



" R#

Ê Rœ

(
0.1) ¸ 0.011 ohms Ê percentage change is

100 9

ohms

dR ¸ R Ð20ß25Ñ

‚ 100

‚ 100 ¸ 0.1%

55. A œ xy Ê dA œ x dy  y dx; if x  y then a 1-unit change in y gives a greater change in dA than a 1-unit change in x. Thus, pay more attention to y which is the smaller of the two dimensions. 56. (a) fx (xß y) œ 2x(y  1) Ê fx (1ß 0) œ 2 and fy (xß y) œ x# Ê fy (1ß 0) œ 1 Ê df œ 2 dx  1 dy Ê df is more sensitive to changes in x dx " (b) df œ 0 Ê 2 dx  dy œ 0 Ê 2 dx dy  1 œ 0 Ê dy œ
# 57. (a) r# œ x#  y# Ê 2r dr œ 2x dx  2y dy Ê dr œ œ „ œ

0.07 5


y y#  x#

œ „ 0.014 Ê ¸ drr ‚ 100¸ œ ¸ „

dx 

x y#  x#

0.014 5

x r

dx 

y r

dy Ê dr|Ð$ß%Ñ œ ˆ 35 ‰ a „ 0.01b  ˆ 45 ‰ a „ 0.01b

‚ 100¸ œ 0.28%; d) œ

3 ‰ dy Ê d)|Ð$ß%Ñ œ ˆ
254 ‰ a „ 0.01b  ˆ 25 a „ 0.01b œ

y ‹ x# # y ˆ ‰ 1 x

Š


…0.04 25

dx 



Š x" ‹ y ˆ ‰#  1 x

dy

„0.03 #5

Ê maximum change in d) occurs when dx and dy have opposite signs (dx œ 0.01 and dy œ
0.01 or vice „0.0028
" ˆ 4 ‰ ¸ d)) ‚ 100¸ œ ¸ 0.927255218 versa) Ê d) œ „#0.07 ‚ 100¸ 5 ¸ „ 0.0028; ) œ tan 3 ¸ 0.927255218 Ê

¸ 0.30% (b) the radius r is more sensitive to changes in y, and the angle ) is more sensitive to changes in x

58. (a) V œ 1r# h Ê dV œ 21rh dr  1r# dh Ê at r œ 1 and h œ 5 we have dV œ 101 dr  1 dh Ê the volume is about 10 times more sensitive to a change in r " (b) dV œ 0 Ê 0 œ 21rh dr  1r# dh œ 2h dr  r dh œ 10 dr  dh Ê dr œ
10 dh; choose dh œ 1.5 Ê dr œ
0.15 Ê h œ 6.5 in. and r œ 0.85 in. is one solution for ?V ¸ dV œ 0 59. f(aß bß cß d) œ º

a b œ ad
bc Ê fa œ d, fb œ
c, fc œ
b, fd œ a Ê df œ d da
c db
b dc  a dd; since c dº

kak is much greater than kbk , kck , and kdk , the function f is most sensitive to a change in d. 60. ux œ ey , uy œ xey  sin z, uz œ y cos z Ê du œ ey dx  axey  sin zb dy  (y cos z) dz Ê duk ˆ2ßln 3ß 12 ‰ œ 3 dx  7 dy  0 dz œ 3 dx  7 dy Ê magnitude of the maximum possible error Ÿ 3(0.2)  7(0.6) œ 4.8 61. QK œ

" #

ˆ 2KM ‰
"Î# ˆ 2M ‰ , QM œ h h

" #

ˆ 2KM ‰
"Î# ˆ 2K ‰ h h , and Qh œ

" #

ˆ 2KM ‰
"Î# ˆ
2KM ‰ h h#

" ˆ 2KM ‰
"Î# ˆ 2M ‰ ‰
"Î# ˆ 2K ‰ dM  "# ˆ 2KM ‰
"Î# ˆ
2KM ‰ dh dK  "# ˆ 2KM # h h h h h h#
"Î# 2K 2KM ˆ 2KM ‰
2M ‘ h h dK  h dM
h# dh Ê dQk Ð2ß20ß0Þ0.05Ñ
"Î# (2)(2) (2)(2)(20) ’ (2)(2)(20) ’ (2)(20) 0.05 “ 0.05 dK  0.05 dM
(0.05)# dh“ œ (0.0125)(800 dK  80 dM

Ê dQ œ œ

" #

œ

" #


32,000 dh)

Ê Q is most sensitive to changes in h ab sin C Ê Aa œ "# b sin C, Ab œ "# a sin C, Ac œ "# ab cos C Ê dA œ ˆ "# b sin C‰ da  ˆ "# a sin C‰ db  ˆ "# ab cos C‰ dC; dC œ k2°k œ k0.0349k radians, da œ k0.5k ft,

62. A œ

" #

db œ k0.5k ft; at a œ 150 ft, b œ 200 ft, and C œ 60°, we see that the change is approximately dA œ "# (200)(sin 60°) k0.5k  "# (150)(sin 60°) k0.5k  "# (200)(150)(cos 60°) k0.0349k œ „ 338 ft#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

836

Chapter 14 Partial Derivatives

63. z œ f(xß y) Ê g(xß yß z) œ f(xß y)
z œ 0 Ê gx (xß yß z) œ fx (xß y), gy (xß yß z) œ fy (xß y) and gz (xß yß z) œ
1 Ê gx (x! ß y! ß f(x! ß y! )) œ fx (x! ß y! ), gy (x! ß y! ß f(x! ß y! )) œ fy (x! ß y! ) and gz (x! ß y! ß f(x! ß y! )) œ
1 Ê the tangent plane at the point P! is fx (x! ß y! )(x
x! )  fy (x! ß y! )(y
y! )
[z
f(x! ß y! )] œ 0 or z œ fx (x! ß y! )(x
x! )  fy (x! ß y! )(y
y! )  f(x! ß y! ) 64. ™ f œ 2xi  2yj œ 2(cos t  t sin t)i  2(sin t
t cos t)j and v œ (t cos t)i  (t sin t)j Ê u œ œ

(t cos t)i
(t sin t)j È(t cos t)#
(t sin t)#

v kvk

œ (cos t)i  (sin t)j since t  0 Ê (Du f)P! œ ™ f † u

œ 2(cos t  t sin t)(cos t)  2(sin t
t cos t)(sin t) œ 2 65. ™ f œ 2xi  2yj  2zk œ (2 cos t)i  (2 sin t)j  2tk and v œ (
sin t)i  (cos t)j  k Ê u œ œ

( sin t)i
(cos t)j
k È(sin t)#
(cos t)#
1#

t œ Š Èsin t ‹ i  Š cos È ‹j  2

2

" È2

k Ê (Du f)P! œ ™ f † u

t " œ (2 cos t) Š Èsin2 t ‹  (2 sin t) Š cos È2 ‹  (2t) Š È2 ‹ œ

(Du f) ˆ 14 ‰ œ

" #

" "Î# i  "# t"Î# j # t # #

" "Î# i # t # #

67. r œ Èti  Ètj  (2t
1)k Ê v œ v(1) œ

Ê (Du f) ˆ 41 ‰ œ

1 2È 2

, (Du f)(0) œ 0 and


4" k ; t œ 1 Ê x œ 1, y œ 1, z œ
1 Ê P! œ (1ß 1ß
1)

i  "# j
"4 k ; f(xß yß z) œ x  y
z
3 œ 0 Ê ™ f œ 2xi  2yj
k

Ê ™ f(1ß 1ß
1) œ 2i  2j
k ; therefore v œ

" #

2t È2

1 2È 2

66. r œ Èti  Ètj
4" (t  3)k Ê v œ and v(1) œ

v kvk

" #

" 4

( ™ f) Ê the curve is normal to the surface

 "# t"Î# j  2k ; t œ 1 Ê x œ 1, y œ 1, z œ 1 Ê P! œ (1ß 1ß 1) and

i  j  2k ; f(xß yß z) œ x  y
z
1 œ 0 Ê ™ f œ 2xi  2yj
k Ê ™ f(1ß 1ß 1) œ 2i  2j
k ;

now va1b † ™ fa1ß 1ß 1b œ 0, thus the curve is tangent to the surface when t œ 1 14.7 EXTREME VALUES AND SADDLE POINTS 1. fx (xß y) œ 2x  y  3 œ 0 and fy (xß y) œ x  2y
3 œ 0 Ê x œ
3 and y œ 3 Ê critical point is (
3ß 3); # œ 3  0 and fxx  0 Ê local minimum of fxx (
3ß 3) œ 2, fyy (
3ß 3) œ 2, fxy (
3ß 3) œ 1 Ê fxx fyy
fxy f(
3ß 3) œ
5 2. fx (xß y) œ 2y
10x  4 œ 0 and fy (xß y) œ 2x
4y  4 œ 0 Ê x œ 23 and y œ 43 Ê critical point is ˆ 23 ß 43 ‰ ; # œ 36  0 and fxx  0 Ê local maximum of fxx ˆ 23 ß 43 ‰ œ
10, fyy ˆ 23 ß 43 ‰ œ
4, fxy ˆ 23 ß 43 ‰ œ 2 Ê fxx fyy
fxy f ˆ 23 ß 43 ‰ œ 0 3. fx (xß y) œ 2x  y  3 œ 0 and fy (xß y) œ x  2 œ 0 Ê x œ
2 and y œ 1 Ê critical point is (
2ß 1); # œ
1  0 Ê saddle point fxx (
2ß 1) œ 2, fyy (
2ß 1) œ 0, fxy (
2ß 1) œ 1 Ê fxx fyy
fxy ˆ 6 69 ‰ 4. fx (xß y) œ 5y
14x  3 œ 0 and fy (xß y) œ 5x
6 œ 0 Ê x œ 65 and y œ 69 #5 Ê critical point is 5 ß 25 ; # ‰ ˆ 6 69 ‰ ˆ 6 69 ‰ fxx ˆ 65 ß 69 25 œ
14, fyy 5 ß 25 œ 0, fxy 5 ß 25 œ 5 Ê fxx fyy
fxy œ
25  0 Ê saddle point 5. fx (xß y) œ 2y
2x  3 œ 0 and fy (xß y) œ 2x
4y œ 0 Ê x œ 3 and y œ 3# Ê critical point is ˆ3ß 32 ‰ ; # œ 4  0 and fxx  0 Ê local maximum of fxx ˆ3ß 32 ‰ œ
2, fyy ˆ3ß 32 ‰ œ
4, fxy ˆ3ß 32 ‰ œ 2 Ê fxx fyy
fxy f ˆ3ß 3# ‰ œ

17 #

6. fx (xß y) œ 2x
4y œ 0 and fy (xß y) œ
4x  2y  6 œ 0 Ê x œ 2 and y œ 1 Ê critical point is (2ß 1); # œ
12  0 Ê saddle point fxx (2ß 1) œ 2, fyy (2ß 1) œ 2, fxy (2ß 1) œ
4 Ê fxx fyy
fxy

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.7 Extreme Values and Saddle Points

837

7. fx (xß y) œ 4x  3y
5 œ 0 and fy (xß y) œ 3x  8y  2 œ 0 Ê x œ 2 and y œ
1 Ê critical point is (2ß
1); # œ 23  0 and fxx  0 Ê local minimum of f(2ß
1) œ
6 fxx (2ß
1) œ 4, fyy (2ß
1) œ 8, fxy (2ß
1) œ 3 Ê fxx fyy
fxy 8. fx (xß y) œ 2x
2y
2 œ 0 and fy (xß y) œ
2x  4y  2 œ 0 Ê x œ 1 and y œ 0 Ê critical point is (1ß 0); # œ 4  0 and fxx  0 Ê local minimum of f(1ß 0) œ 0 fxx (1ß 0) œ 2, fyy (1ß 0) œ 4, fxy (1ß 0) œ
2 Ê fxx fyy
fxy 9. fx (xß y) œ 2x
2 œ 0 and fy (xß y) œ
2y  4 œ 0 Ê x œ 1 and y œ 2 Ê critical point is (1ß 2); fxx (1ß 2) œ 2, # œ
4  0 Ê saddle point fyy (1ß 2) œ
2, fxy (1ß 2) œ 0 Ê fxx fyy
fxy 10. fx (xß y) œ 2x  2y œ 0 and fy (xß y) œ 2x œ 0 Ê x œ 0 and y œ 0 Ê critical point is (0ß 0); fxx (0ß 0) œ 2, # œ
4  0 Ê saddle point fyy (0ß 0) œ 0, fxy (0ß 0) œ 2 Ê fxx fyy
fxy 11. fx axß yb œ

112x  8x È56x2  8y2  16x  31


8 œ 0 and fy axß yb œ

8y È56x2  8y2  16x  31

8 8 # ‰ ˆ 16 ‰ ˆ 16 ‰ fxx ˆ 16 7 ß 0 œ
15 , fyy 7 ß 0 œ
15 , fxy 7 ß 0 œ 0 Ê fxx fyy
fxy œ 16 ‰ fˆ 16 7 ß0 œ
7

12. fx axß yb œ

2x 3ax2
y2 b2Î3

œ 0 and fy axß yb œ

2y 3ax2
y2 b2Î3

‰ œ 0 Ê critical point is ˆ 16 7 ß0 ;

64 225

 0 and fxx  0 Ê local maximum of

œ 0 Ê there are no solutions to the system fx axß yb œ 0 and

fy axß yb œ 0, however, we must also consider where the partials are undefined, and this occurs when x œ 0 and y œ 0 Ê critical point is a0ß 0b. Note that the partial derivatives are defined at every other point other than a0ß 0b. We cannot use the second derivative test, but this is the only possible local maximum, local minimum, or saddle point. faxß yb has a local 3 maximum of fa0ß 0b œ 1 at a0ß 0b since faxß yb œ 1
È x2  y2 Ÿ 1 for all axß yb other than a0ß 0b. 13. fx (xß y) œ 3x#
2y œ 0 and fy (xß y) œ
3y#
2x œ 0 Ê x œ 0 and y œ 0, or x œ
23 and y œ 23 Ê critical points are (0ß 0) and ˆ
23 ß 23 ‰ ; for (0ß 0): fxx (0ß 0) œ 6xk Ð0ß0Ñ œ 0, fyy (0ß 0) œ
6yk Ð0ß0Ñ œ 0, fxy (0ß 0) œ
2 # Ê fxx fyy
fxy œ
4  0 Ê saddle point; for ˆ
32 ß 32 ‰ : fxx ˆ
32 ß 32 ‰ œ
4, fyy ˆ
32 ß 32 ‰ œ
4, fxy ˆ
32 ß 32 ‰ œ
2 # Ê fxx fyy
fxy œ 12  0 and fxx  0 Ê local maximum of f ˆ
23 ß 32 ‰ œ 170 27

14. fx (xß y) œ 3x#  3y œ 0 and fy (xß y) œ 3x  3y# œ 0 Ê x œ 0 and y œ 0, or x œ
1 and y œ
1 Ê critical points # are (0ß 0) and (
1ß
1); for (!ß !): fxx (0ß 0) œ 6xk Ð0ß0Ñ œ 0, fyy (0ß 0) œ 6yk Ð0ß0Ñ œ 0, fxy (0ß 0) œ 3 Ê fxx fyy
fxy # œ
9  0 Ê saddle point; for (
1ß
1): fxx (
1ß
1) œ
6, fyy (
1ß
1) œ
6, fxy (
1ß
1) œ 3 Ê fxx fyy
fxy

œ 27  0 and fxx  0 Ê local maximum of f(
1ß
1) œ 1 15. fx (xß y) œ 12x
6x#  6y œ 0 and fy (xß y) œ 6y  6x œ 0 Ê x œ 0 and y œ 0, or x œ 1 and y œ
1 Ê critical # points are (0ß 0) and (1ß
1); for (!ß !): fxx (0ß 0) œ 12
12xk Ð0ß0Ñ œ 12, fyy (0ß 0) œ 6, fxy (0ß 0) œ 6 Ê fxx fyy
fxy œ 36  0 and fxx  0 Ê local minimum of f(0ß 0) œ 0; for (1ß
1): fxx (1ß
1) œ 0, fyy (1ß
1) œ 6, # fxy (1ß
1) œ 6 Ê fxx fyy
fxy œ
36  0 Ê saddle point 16. fx (xß y) œ 3x#  6x œ 0 Ê x œ 0 or x œ
2; fy (xß y) œ 3y#
6y œ 0 Ê y œ 0 or y œ 2 Ê the critical points are (0ß 0), (0ß 2), (
2ß 0), and (
2ß 2); for (!ß !): fxx (0ß 0) œ 6x  6k Ð0ß0Ñ œ 6, fyy (0ß 0) œ 6y
6k Ð0ß0Ñ œ
6, # fxy (0ß 0) œ 0 Ê fxx fyy
fxy œ
36  0 Ê saddle point; for (0ß 2): fxx (0ß 2) œ 6, fyy (0ß 2) œ 6, fxy (0ß 2) œ 0 # Ê fxx fyy
fxy œ 36  0 and fxx  0 Ê local minimum of f(0ß 2) œ
12; for (
2ß 0): fxx (
2ß 0) œ
6, # fyy (
2ß 0) œ
6, fxy (
2ß 0) œ 0 Ê fxx fyy
fxy œ 36  0 and fxx  0 Ê local maximum of f(
2ß 0) œ
4; # for (
2ß 2): fxx (
2ß 2) œ
6, fyy (
2ß 2) œ 6, fxy (
2ß 2) œ 0 Ê fxx fyy
fxy œ
36  0 Ê saddle point

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

838

Chapter 14 Partial Derivatives

17. fx axß yb œ 3x2  3y2
15 œ 0 and fy axß yb œ 6x y  3y2
15 œ 0 Ê critical points are a2ß 1b, a
2ß
1b, Š0ß È5‹, and Š0ß
È5‹; for a2ß 1b: fxx a2ß 1b œ 6xk a2ß1b œ 12, fyy a2ß 1b œ a6x  6ybk a2ß1b œ 18, fxy a2ß 1b œ 6yk a2ß1b œ 6 # Ê fxx fyy
fxy œ 180  0 and fxx  0 Ê local minimum of fa2ß 1b œ
30; for a
2ß
1b: fxx a
2ß
1b œ 6xk a2ß1b # œ
12, fyy a
2ß
1b œ a6x  6ybk a2ß1b œ
18, fxy a
2ß
1b œ 6yk a2ß1b œ
6 Ê fxx fyy
fxy œ 180  0 and

fxx  0 Ê local maximum of fa
2ß
1b œ 30; for Š0ß È5‹: fxx Š0ß È5‹ œ 6x¹ œ a6x  6ybk Š0ßÈ5‹ œ 6È5, fxy Š0ß È5‹ œ 6y¹ for Š0ß
È5‹: fxx Š0ß
È5‹ œ 6x¹ fxy Š0ß
È5‹ œ 6y¹

Š0ßÈ5‹

Š0ßÈ5‹

Š0ßÈ5‹

Š0ßÈ5‹

œ 0, fyy Š0ß È5‹

# œ 6È5 Ê fxx fyy
fxy œ
180  0 Ê saddle pointà

œ 0, fyy Š0ß
È5‹ œ a6x  6ybk Š0ßÈ5‹ œ
6È5,

# œ
6È5 Ê fxx fyy
fxy œ
180  0 Ê saddle point.

18. fx (xß y) œ 6x#
18x œ 0 Ê 6x(x
3) œ 0 Ê x œ 0 or x œ 3; fy (xß y) œ 6y#  6y
12 œ 0 Ê 6(y  2)(y
1) œ 0 Ê y œ
2 or y œ 1 Ê the critical points are (0ß
2), (0ß 1), (3ß
2), and (3ß 1); fxx (xß y) œ 12x
18, fyy (xß y) œ 12y  6, and fxy (xß y) œ 0; for (!ß
2): fxx (0ß
2) œ
18, fyy (0ß
2) œ
18, fxy (0ß
2) œ 0 # Ê fxx fyy
fxy œ 324  0 and fxx  0 Ê local maximum of f(0ß
2) œ 20; for (0ß 1): fxx (0ß 1) œ
18, # fyy (0ß 1) œ 18, fxy (0ß 1) œ 0 Ê fxx fyy
fxy œ
324  0 Ê saddle point; for (3ß
2): fxx (3ß
2) œ 18, # fyy (3ß
2) œ
18, fxy (3ß
2) œ 0 Ê fxx fyy
fxy œ
324  0 Ê saddle point; for (3ß 1): fxx (3ß 1) œ 18, # fyy (3ß 1) œ 18, fxy (3ß 1) œ 0 Ê fxx fyy
fxy œ 324  0 and fxx  0 Ê local minimum of f(3ß 1) œ
34

19. fx (xß y) œ 4y
4x$ œ 0 and fy (xß y) œ 4x
4y$ œ 0 Ê x œ y Ê x a1
x# b œ 0 Ê x œ 0, 1,
1 Ê the critical points are (0ß 0), (1ß 1), and (
1ß
1); for (!ß !): fxx (0ß 0) œ
12x# k Ð0ß0Ñ œ 0, fyy (0ß 0) œ
12y# k Ð0ß0Ñ œ 0, # fxy (0ß 0) œ 4 Ê fxx fyy
fxy œ
16  0 Ê saddle point; for (1ß 1): fxx (1ß 1) œ
12, fyy (1ß 1) œ
12, fxy (1ß 1) œ 4 # Ê fxx fyy
fxy œ 128  0 and fxx  0 Ê local maximum of f(1ß 1) œ 2; for (
1ß
1): fxx (
1ß
1) œ
12, # fyy (
1ß
1) œ
12, fxy (
1ß
1) œ 4 Ê fxx fyy
fxy œ 128  0 and fxx  0 Ê local maximum of f(
1ß
1) œ 2

20. fx (xß y) œ 4x$  4y œ 0 and fy (xß y) œ 4y$  4x œ 0 Ê x œ
y Ê
x$  x œ 0 Ê x a1
x# b œ 0 Ê x œ 0, 1,
1 Ê the critical points are (0ß 0), (1ß
1), and (
1ß 1); fxx (xß y) œ 12x# , fyy (xß y) œ 12y# , and fxy (xß y) œ 4; # for (!ß 0): fxx (0ß 0) œ 0, fyy (0ß 0) œ 0, fxy (0ß 0) œ 4 Ê fxx fyy
fxy œ
16  0 Ê saddle point; for (1ß
1): # fxx (1ß
1) œ 12, fyy (1ß
1) œ 12, fxy (1ß
1) œ 4 Ê fxx fyy
fxy œ 128  0 and fxx  0 Ê local minimum of # f("ß
1) œ
2; for (
1ß 1): fxx (
1ß 1) œ 12, fyy (
1ß 1) œ 12, fxy (
1ß 1) œ 4 Ê fxx fyy
fxy œ 128  0 and

fxx  0 Ê local minimum of f(
1ß 1) œ
2 21. fx (xß y) œ

2x ax#
y#  1b#

œ 0 and fy (xß y) œ

2y ax#
y#  1b#

œ 0 Ê x œ 0 and y œ 0 Ê the critical point is (!ß 0);

# # 4x#  2y#
2 , fyy œ ax2x#

y#4y 1
b$2 , fxy œ ax#
8xy ; fxx (!ß !) œ
2, fyy (0ß 0) y#  1b$ ax#
y#  1b$ # fxx fyy
fxy œ 4  0 and fxx  0 Ê local maximum of f(0ß 0) œ
1

fxx œ Ê

22. fx (xß y) œ
x1#  y œ 0 and fy (xß y) œ x


1 y#

œ
2, fxy (0ß 0) œ 0

œ 0 Ê x œ 1 and y œ 1 Ê the critical point is (1ß 1); fxx œ

fxy œ 1; fxx (1ß 1) œ 2, fyy (1ß 1) œ 2, fxy (1ß 1) œ 1 Ê fxx fyy


# fxy

2 x$

, fyy œ

2 y$

œ 3  0 and fxx  2 Ê local minimum of f(1ß 1) œ 3

23. fx (xß y) œ y cos x œ 0 and fy (xß y) œ sin x œ 0 Ê x œ n1, n an integer, and y œ 0 Ê the critical points are (n1ß 0), n an integer (Note: cos x and sin x cannot both be 0 for the same x, so sin x must be 0 and y œ 0); fxx œ
y sin x, fyy œ 0, fxy œ cos x; fxx (n1ß 0) œ 0, fyy (n1ß 0) œ 0, fxy (n1ß 0) œ 1 if n is even and fxy (n1ß 0) œ
1 # if n is odd Ê fxx fyy
fxy œ
1  0 Ê saddle point.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

,

Section 14.7 Extreme Values and Saddle Points

839

24. fx (xß y) œ 2e2x cos y œ 0 and fy (xß y) œ
e2x sin y œ 0 Ê no solution since e2x Á 0 for any x and the functions cos y and sin y cannot equal 0 for the same y Ê no critical points Ê no extrema and no saddle points 25. fx axß yb œ a2x
4bex fyy a2ß 0b œ

2 e4

2


y2  4x

Ê fxx fyy


# fxy

œ 0 and fy axß yb œ 2yex œ

4 e8

2


y2  4x

œ 0 Ê critical point is a2ß 0b; fxx a2ß 0b œ

 0 and fxx  0 Ê local mimimum of fa2ß 0b œ

2 e4 , fxy a2ß 0b

œ 0,

1 e4

26. fx axß yb œ
yex œ 0 and fy axß yb œ ey
ex œ 0 Ê critical point is a0ß 0b; fxx a2ß 0b œ 0, fxy a2ß 0b œ
1, fyy a2ß 0b œ 1 # Ê fxx fyy
fxy œ
1  0 Ê saddle point 27. fx axß yb œ 2xey œ 0 and fy axß yb œ 2yey
ey ax2  y2 b œ 0 Ê critical points are a0ß 0b and a0ß 2b; for a0ß 0b: fxx a0ß 0b œ 2ey k a0ß0b œ 2, fyy a0ß 0b œ a2ey
4yey  ey ax2  y2 bbk a0ß0b œ 2, fxy a0ß 0b œ
2xey k a0ß0b œ 0 # Ê fxx fyy
fxy œ 4  0 and fxx  0 Ê local mimimum of fa0ß 0b œ 0; for a0ß 2b: fxx a0ß 2b œ 2ey k a0ß2b œ # fyy a0ß 2b œ a2ey
4yey  ey ax2  y2 bbk a0ß2b œ
e22 , fxy a0ß 2b œ
2xey k a0ß2b œ 0 Ê fxx fyy
fxy œ

2 e2 ,
e44 

0

Ê saddle point 28. fx axß yb œ ex ax2
2x  y2 b œ 0 and fy axß yb œ
2yex œ 0 Ê critical points are a0ß 0b and a
2ß 0b; for a0ß 0b: fxx a0ß 0b œ ex ax2  4x  2
y2 bk a0ß0b œ 2, fyy a0ß 0b œ
2ex k a0ß0b œ
2, fxy a0ß 0b œ
2yex k a0ß0b œ 0 # Ê fxx fyy
fxy œ
4  0 and fxx  0 Ê saddle point; for a
2ß 0b: fxx a
2ß 0b œ ex ax2  4x  2
y2 bk a2ß0b œ
e22 , # fyy a
2ß 0b œ
2ex k a2ß0b œ
e22 , fxy a
2ß 0b œ
2yex k a2ß0b œ 0 Ê fxx fyy
fxy œ

of fa
2ß 0b œ

4 e4

 0 and fxx  0 Ê local maximum

4 e2

29. fx axß yb œ
4 

2 x

œ 0 and fy axß yb œ
1 

1 y

œ 0 Ê critical point is ˆ 21 , 1‰ ; fxx ˆ 21 , 1‰ œ
8, fyy ˆ 12 , 1‰ œ
1,

# œ 8  0 and fxx  0 Ê local maximum of fˆ 12 , 1‰ œ
3
2ln 2 fxy ˆ 12 , 1‰ œ 0 Ê fxx fyy
fxy

30. fx axß yb œ 2x 

1 x
y

œ 0 and fy axß yb œ
1 

1 x
y

œ 0 Ê critical point is ˆ
21 , 23 ‰ ; fxx ˆ
21 , 23 ‰ œ 1, fyy ˆ
12 , 32 ‰ œ
1,

# œ
2  0 Ê saddle point fxy ˆ
12 , 32 ‰ œ
1 Ê fxx fyy
fxy

On OA, f(xß y) œ f(0ß y) œ y#
4y  1 on 0 Ÿ y Ÿ 2; f w (0ß y) œ 2y
4 œ 0 Ê y œ 2; f(0ß 0) œ 1 and f(!ß #) œ
3 (ii) On AB, f(xß y) œ f(xß 2) œ 2x#
4x
3 on 0 Ÿ x Ÿ 1; f w (xß 2) œ 4x
4 œ 0 Ê x œ 1; f(0ß 2) œ
3 and f(1ß #) œ
5 (iii) On OB, f(xß y) œ f(xß 2x) œ 6x#
12x  1 on 0 Ÿ x Ÿ 1; endpoint values have been found above; f w (xß 2x) œ 12x
12 œ 0 Ê x œ 1 and y œ 2, but ("ß #) is not an interior point of OB (iv) For interior points of the triangular region, fx (xß y) œ 4x
4 œ 0 and fy (xß y) œ 2y
4 œ 0 Ê x œ 1 and y œ 2, but (1ß 2) is not an interior point of the region. Therefore, the absolute maximum is 1 at (0ß 0) and the absolute minimum is
5 at ("ß #).

31. (i)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

840

Chapter 14 Partial Derivatives

On OA, D(xß y) œ D(0ß y) œ y#  1 on 0 Ÿ y Ÿ 4; Dw (0ß y) œ 2y œ 0 Ê y œ 0; D(!ß !) œ 1 and D(!ß %) œ 17 (ii) On AB, D(xß y) œ D(xß 4) œ x#
4x  17 on 0 Ÿ x Ÿ 4; Dw (xß 4) œ 2x
4 œ 0 Ê x œ 2 and (2ß 4) is an interior point of AB; D(#ß %) œ 13 and D(%ß %) œ D(!ß %) œ 17 (iii) On OB, D(xß y) œ D(xß x) œ x#  1 on 0 Ÿ x Ÿ 4; Dw (xß x) œ 2x œ 0 Ê x œ 0 and y œ 0, which is not an interior point of OB; endpoint values have been found above (iv) For interior points of the triangular region, fx (xß y) œ 2x
y œ 0 and fy (xß y) œ
x  2y œ 0 Ê x œ 0 and y œ 0, which is not an interior point of the region. Therefore, the absolute maximum is 17 at (!ß %) and (%ß %), and the absolute minimum is 1 at (0ß 0).

32. (i)

On OA, f(xß y) œ f(!ß y) œ y# on 0 Ÿ y Ÿ 2; f w (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0; f(0ß 0) œ 0 and f(0ß #) œ 4 (ii) On OB, f(xß y) œ f(xß 0) œ x# on 0 Ÿ x Ÿ 1; f w (xß 0) œ 2x œ 0 Ê x œ 0 and y œ 0; f(0ß 0) œ 0 and f(1ß 0) œ 1 (iii) On AB, f(xß y) œ f(xß
2x  2) œ 5x#
8x  4 on 0 Ÿ x Ÿ 1; f w (xß
2x  2) œ 10x
8 œ 0 Ê x œ 45 and y œ 25 ; f ˆ 45 ß 25 ‰ œ 45 ; endpoint values have been found above.

33. (i)

(iv) For interior points of the triangular region, fx (xß y) œ 2x œ 0 and fy (xß y) œ 2y œ 0 Ê x œ 0 and y œ 0, but (!ß 0) is not an interior point of the region. Therefore the absolute maximum is 4 at (0ß 2) and the absolute minimum is 0 at (0ß 0). 34. (i)

(ii)

On AB, T(xß y) œ T(!ß y) œ y# on
3 Ÿ y Ÿ 3; Tw (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0; T(0ß 0) œ 0, T(!ß
3) œ 9, and T(!ß 3) œ 9 On BC, T(xß y) œ T(xß 3) œ x#
3x  9 on 0 Ÿ x Ÿ 5; Tw (xß 3) œ 2x
3 œ 0 Ê x œ 3# and y œ 3; T ˆ 3# ß 3‰ œ 27 4 and T(5ß 3) œ 19

(iii) On CD, T(xß y) œ T(5ß y) œ y#  5y
5 on
3 Ÿ y Ÿ 3;Tw (5ß y) œ 2y  5 œ 0 Ê y œ
5# and x œ 5;T ˆ5ß
5# ‰ œ
45 4 , T(&ß
3) œ
11 and T(5ß 3) œ 19

(iv) On AD, T(xß y) œ T(xß
3) œ x#
9x  9 on 0 Ÿ x Ÿ 5; Tw (xß
3) œ 2x
9 œ 0 Ê x œ T ˆ 9# ß
3‰ œ
45 4 , T(!ß
3) œ 9 and T(&ß
3) œ
11

(v)

9 #

and y œ
3;

For interior points of the rectangular region, Tx (xß y) œ 2x  y
6 œ 0 and Ty (xß y) œ x  2y œ 0 Ê x œ 4 and y œ
2 Ê (4ß
2) is an interior critical point with T(4ß
2) œ
12. Therefore the absolute maximum is 19 at (5ß 3) and the absolute minimum is
12 at (4ß
2).

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.7 Extreme Values and Saddle Points 35. (i)

(ii)

841

On OC, T(xß y) œ T(xß 0) œ x#
6x  2 on 0 Ÿ x Ÿ 5; Tw (xß 0) œ 2x
6 œ 0 Ê x œ 3 and y œ 0; T(3ß 0) œ
7, T(0ß 0) œ 2, and T(5ß 0) œ
3 On CB, T(xß y) œ T(5ß y) œ y#  5y
3 on
3 Ÿ y Ÿ 0; Tw (5ß y) œ 2y  5 œ 0 Ê y œ
5# and x œ 5; T ˆ5ß
5# ‰ œ
37 4 and T(5ß
3) œ
9

(iii) On AB, T(xß y) œ T(xß
3) œ x#
9x  11 on 0 Ÿ x Ÿ 5; Tw (xß
3) œ 2x
9 œ 0 Ê x œ 9# and y œ
3; T ˆ 9# ß
3‰ œ
37 4 and T(!ß
3) œ 11

(iv) On AO, T(xß y) œ T(!ß y) œ y#  2 on
3 Ÿ y Ÿ 0; Tw (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0, but (0ß 0) is not an interior point of AO (v) For interior points of the rectangular region, Tx (xß y) œ 2x  y
6 œ 0 and Ty (xß y) œ x  2y œ 0 Ê x œ 4 and y œ
2, an interior critical point with T(%ß
2) œ
10. Therefore the absolute maximum is 11 at (!ß
3) and the absolute minimum is
10 at (4ß
2). 36. (i)

(ii)

On OA, f(xß y) œ f(!ß y) œ
24y# on 0 Ÿ y Ÿ 1; f w (0ß y) œ
48y œ 0 Ê y œ 0 and x œ 0, but (0ß 0) is not an interior point of OA; f(!ß 0) œ 0 and f(!ß 1) œ
24 On AB, f(xß y) œ f(xß 1) œ 48x
32x$
24 on 0 Ÿ x Ÿ 1; f w (xß 1) œ 48
96x# œ 0 Ê x œ È"2 and y œ 1, or x œ
È"2 and y œ 1, but Š
È"2 ß 1‹ is not in the interior of AB; f Š È"2 ß 1‹ œ 16È2
24 and f(1ß 1) œ
8

(iii) On BC, f(xß y) œ f("ß y) œ 48y
32
24y# on 0 Ÿ y Ÿ 1; f w ("ß y) œ 48
48y œ 0 Ê y œ 1 and x œ 1, but ("ß ") is not an interior point of BC; f("ß 0) œ
32 and f("ß ") œ
8 (iv) On OC, f(xß y) œ f(xß 0) œ
32x$ on 0 Ÿ x Ÿ 1; f w (xß 0) œ
96x# œ 0 Ê x œ 0 and y œ 0, but (0ß 0) is not an interior point of OC; f(!ß 0) œ 0 and f("ß 0) œ
32 (v) For interior points of the rectangular region, fx (xß y) œ 48y
96x# œ 0 and fy (xß y) œ 48x
48y œ 0 Ê x œ 0 and y œ 0, or x œ "# and y œ "# , but (0ß 0) is not an interior point of the region; f ˆ "# ß "# ‰ œ 2. Therefore the absolute maximum is 2 at ˆ "# ß "# ‰ and the absolute minimum is
32 at (1ß 0). 37. (i)

On AB, f(xß y) œ f(1ß y) œ 3 cos y on
14 Ÿ y Ÿ w

1 4

;

1 4

;

f (1ß y) œ
3 sin y œ 0 Ê y œ 0 and x œ 1; f("ß 0) œ 3, f ˆ1ß
14 ‰ œ

(ii)

3È 2 #

, and f ˆ1ß 14 ‰ œ

3È 2 #

On CD, f(xß y) œ f($ß y) œ 3 cos y on
14 Ÿ y Ÿ f w (3ß y) œ
3 sin y œ 0 Ê y œ 0 and x œ 3;

È 3È 2 ˆ 1‰ 3 2 # and f 3ß 4 œ # È2 1‰ # 4 œ # a4x
x b on

f(3ß 0) œ 3, f ˆ3ß
14 ‰ œ (iii) On BC, f(xß y) œ f ˆxß

1 Ÿ x Ÿ 3; f w ˆxß 14 ‰ œ È2(2
x) œ 0 Ê x œ 2 and y œ f ˆ3ß 14 ‰ œ

3È 2 #

; f ˆ2ß 14 ‰ œ 2È2, f ˆ1ß 14 ‰ œ

È2 # w # a4x
x b on 1 Ÿ x Ÿ 3; f È È œ 3 # 2 , and f ˆ3ß
14 ‰ œ 3 # 2

(iv) On AD, f(xß y) œ f ˆxß
14 ‰ œ f ˆ2ß
14 ‰ œ 2È2, f ˆ1ß
14 ‰

1 4

3È 2 #

, and

ˆxß
14 ‰ œ È2(2
x) œ 0 Ê x œ 2 and y œ
14 ;

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

842

Chapter 14 Partial Derivatives For interior points of the region, fx (xß y) œ (4
2x) cos y œ 0 and fy (xß y) œ
a4x
x# b sin y œ 0 Ê x œ 2 and y œ 0, which is an interior critical point with f(2ß 0) œ 4. Therefore the absolute maximum is 4 at

(v)

(2ß 0) and the absolute minimum is

3È 2 #

at ˆ3ß
14 ‰ , ˆ3ß 14 ‰ , ˆ1ß
14 ‰ , and ˆ1ß 14 ‰ .

On OA, f(xß y) œ f(!ß y) œ 2y  1 on 0 Ÿ y Ÿ 1; f w (0ß y) œ 2 Ê no interior critical points; f(0ß 0) œ 1 and f(0ß 1) œ 3 (ii) On OB, f(xß y) œ f(xß 0) œ 4x  1 on 0 Ÿ x Ÿ 1; f w (xß 0) œ 4 Ê no interior critical points; f(1ß 0) œ 5 (iii) On AB, f(xß y) œ f(xß
x  1) œ 8x#
6x  3 on 0 Ÿ x Ÿ 1; f w (xß
x  1) œ 16x
6 œ 0 Ê x œ 38 and y œ 58 ; f ˆ 38 ß 58 ‰ œ 15 8 , f(0ß 1) œ 3, and f("ß 0) œ 5

38. (i)

(iv) For interior points of the triangular region, fx (xß y) œ 4
8y œ 0 and fy (xß y) œ
8x  2 œ 0 Ê y œ "# and x œ 4" which is an interior critical point with f ˆ 4" ß #" ‰ œ 2. Therefore the absolute maximum is 5 at (1ß 0) and the absolute minimum is 1 at (0ß 0).

39. Let F(aß b) œ 'a a6
x
x# b dx where a Ÿ b. The boundary of the domain of F is the line a œ b in the ab-plane, and b

F(aß a) œ 0, so F is identically 0 on the boundary of its domain. For interior critical points we have: `F `F # # ` a œ
a6
a
a b œ 0 Ê a œ
3, 2 and ` b œ a6
b
b b œ 0 Ê b œ
3, 2. Since a Ÿ b, there is only one

interior critical point (
3ß 2) and F(
3ß 2) œ 'c3 a6
x
x# b dx gives the area under the parabola y œ 6
x
x# that is 2

above the x-axis. Therefore, a œ
3 and b œ 2. 40. Let F(aß b) œ 'a a24
2x
x# b b

"Î$

dx where a Ÿ b. The boundary of the domain of F is the line a œ b and on this line F is

identically 0. For interior critical points we have: `F `b

# "Î$

œ a24
2b
b b

`F `a

œ
a24
2a
a# b

"Î$

œ 0 Ê a œ 4,
6 and

œ 0 Ê b œ 4,
6. Since a Ÿ b, there is only one critical point (
6ß 4) and

F(
6ß 4) œ 'c6 a24
2x
x# b dx gives the area under the curve y œ a24
2x
x# b 4

"Î$

that is above the x-axis.

Therefore, a œ
6 and b œ 4.

41. Tx (xß y) œ 2x
1 œ 0 and Ty (xß y) œ 4y œ 0 Ê x œ

" #

and y œ 0 with T ˆ "# ß 0‰ œ
4" ; on the boundary

x#  y# œ 1: T(xß y) œ
x#
x  2 for
1 Ÿ x Ÿ 1 Ê Tw (xß y) œ
2x
1 œ 0 Ê x œ
"# and y œ „ T Š


" È3 #ß # ‹

Š
"# ß


œ

È3 # ‹;

9 4

, T Š


œ 2
ln

" #

œ

9 4

" 4

, T(
1ß 0) œ 2, and T("ß 0) œ 0 Ê the hottest is 2 ° at Š


" È3 #ß # ‹

2 x

" y# ¹ ˆ 1 ß2‰

œ 0 and fy (xß y) œ x
œ

2

" 4

" y

œ0 Ê xœ

" #

and y œ 2; fxx ˆ "# ß 2‰ œ

2¸ x# ˆ 12 ß2‰

œ 8,

# , fxy ˆ "# ß 2‰ œ 1 Ê fxx fyy
fxy œ 1  0 and fxx  0 Ê a local minimum of f ˆ "# ß 2‰

œ 2  ln 2

43. (a) fx (xß y) œ 2x
4y œ 0 and fy (xß y) œ 2y
4x œ 0 Ê x œ 0 and y œ 0; fxx (0ß 0) œ 2, fyy (0ß 0) œ 2, # œ
12  0 Ê saddle point at (0ß 0) fxy (0ß 0) œ
4 Ê fxx fyy
fxy (b) fx (xß y) œ 2x
2 œ 0 and fy (xß y) œ 2y
4 œ 0 Ê x œ 1 and y œ 2; fxx (1ß 2) œ 2, fyy (1ß 2) œ 2, # œ 4  0 and fxx  0 Ê local minimum at ("ß #) fxy (1ß 2) œ 0 Ê fxx fyy
fxy

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

;

and

the coldest is
"4 ° at ˆ "# ß 0‰ .

42. fx (xß y) œ y  2
fyy ˆ #" ß 2‰ œ

È3 " #ß
# ‹

È3 #

Section 14.7 Extreme Values and Saddle Points

843

(c) fx (xß y) œ 9x#
9 œ 0 and fy (xß y) œ 2y  4 œ 0 Ê x œ „ 1 and y œ
2; fxx (1ß
2) œ 18xk Ð1ß2Ñ œ 18, # œ 36  0 and fxx  0 Ê local minimum at ("ß
#); fyy (1ß
2) œ 2, fxy (1ß
2) œ 0 Ê fxx fyy
fxy # fxx (
1ß
2) œ
18, fyy (
"ß
2) œ 2, fxy (
"ß
2) œ 0 Ê fxx fyy
fxy œ
36  0 Ê saddle point at (
"ß
2)

44. (a) (b) (c) (d) (e) (f)

Minimum at (0ß 0) since f(xß y)  0 for all other (xß y) Maximum of 1 at (!ß !) since f(xß y)  1 for all other (xß y) Neither since f(xß y)  0 for x  0 and f(xß y)  0 for x  0 Neither since f(xß y)  0 for x  0 and f(xß y)  0 for x  0 Neither since f(xß y)  0 for x  0 and y  0, but f(xß y)  0 for x  0 and y  0 Minimum at (0ß 0) since f(xß y)  0 for all other (xß y)

45. If k œ 0, then f(xß y) œ x#  y# Ê fx (xß y) œ 2x œ 0 and fy (xß y) œ 2y œ 0 Ê x œ 0 and y œ 0 Ê (0ß 0) is the only critical point. If k Á 0, fx (xß y) œ 2x  ky œ 0 Ê y œ
2k x; fy (xß y) œ kx  2y œ 0 Ê kx  2 ˆ
2k x‰ œ 0 4‰ ˆ ˆ 2‰ Ê kx
4x k œ 0 Ê k
k x œ 0 Ê x œ 0 or k œ „ 2 Ê y œ
k (0) œ 0 or y œ „ x; in any case (0ß 0) is a critical point. # 46. (See Exercise 45 above): fxx (xß y) œ 2, fyy (xß y) œ 2, and fxy (xß y) œ k Ê fxx fyy
fxy œ 4
k# ; f will have a saddle point

at (0ß 0) if 4
k#  0 Ê k  2 or k 
2; f will have a local minimum at (0ß 0) if 4
k#  0 Ê
2  k  2; the test is inconclusive if 4
k# œ 0 Ê k œ „ 2. 47. No; for example f(xß y) œ xy has a saddle point at (aß b) œ (0ß 0) where fx œ fy œ 0. # 48. If fxx (aß b) and fyy (aß b) differ in sign, then fxx (aß b) fyy (aß b)  0 so fxx fyy
fxy  0. The surface must therefore have a

saddle point at (aß b) by the second derivative test. 49. We want the point on z œ 10
x#
y# where the tangent plane is parallel to the plane x  2y  3z œ 0. To find a normal vector to z œ 10
x#
y# let w œ z  x#  y#
10. Then ™ w œ 2xi  2yj  k is normal to z œ 10
x#
y# at (xß y). The vector ™ w is parallel to i  2j  3k which is normal to the plane x  2y  3z œ 0 if " ‰ 6xi  6yj  3k œ i  2j  3k or x œ "6 and y œ "3 . Thus the point is ˆ "6 ß "3 ß 10
36
9" ‰ or ˆ 6" ß 3" ß 355 36 . 50. We want the point on z œ x#  y#  10 where the tangent plane is parallel to the plane x  2y
z œ 0. Let w œ z
x#
y#
10, then ™ w œ
2xi
2yj  k is normal to z œ x#  y#  10 at (xß y). The vector ™ w is parallel ‰ to i  2j
k which is normal to the plane if x œ "# and y œ 1. Thus the point ˆ "# ß 1ß 4"  1  10‰ or ˆ #" ß 1ß 45 4 is the point on the surface z œ x#  y#  10 nearest the plane x  2y
z œ 0.

51. daxß yß zb œ Éax
0b2  ay
0b2  az
0b2 Ê we can minimize daxß yß zb by minimizing Daxß yß zb œ x2  y2  z2 ; 3x  2y  z œ 6 Ê z œ 6
3x
2y Ê Daxß yb œ x2  y2  a6
3x
2yb2 Ê Dx axß yb œ 2x
6a6
3x
2yb œ 0 and Dy axß yb œ 2y
4a6
3x
2yb œ 0 Ê critical point is ˆ 97 , 67 ‰ Ê z œ 37 ; Dxx ˆ 97 , 67 ‰ œ 20, Dyy ˆ 12 , 1‰ œ 10, Dxy ˆ 12 , 1‰ œ 12 Ê Dxx Dyy
D#xy œ 56  0 and Dxx  0 Ê local minimum of dˆ 97 , 67 , 37 ‰ œ

3È14 7

52. daxß yß zb œ Éax
2b2  ay  1b2  az
1b2 Ê we can minimize daxß yß zb by minimizing Daxß yß zb œ ax
2b2  ay  1b2  az
1b2 ; x  y
z œ 2 Ê z œ x  y
2 Ê Daxß yb œ ax
2b2  ay  1b2  ax  y
3b2 Ê Dx axß yb œ 2ax
2b  2ax  y
3b œ 0 and Dy axß yb œ 2ay  1b  2ax  y
3b œ 0 Ê critical point is ˆ 83 ,
13 ‰ Ê z œ 13 ; Dxx ˆ 83 ,
13 ‰ œ 4, Dyy ˆ 83 ,
13 ‰ œ 4, Dxy ˆ 83 ,
13 ‰ œ 2 Ê Dxx Dyy
D#xy œ 12  0 and Dxx  0 Ê local minimum of dˆ 83 ,
13 , 13 ‰ œ È2 3

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

844

Chapter 14 Partial Derivatives

53. saxß yß zb œ x2  y2  z2 ; x  y  z œ 9 Ê z œ 9
x
y Ê saxß yb œ x2  y2  a9
x
yb2 Ê sx axß yb œ 2x
2a9
x
yb œ 0 and sy axß yb œ 2y
2a9
x
yb œ 0 Ê critical point is a3, 3b Ê z œ 3; sxx a3, 3b œ 4, syy a3, 3b œ 4, sxy a3, 3b œ 2 Ê sxx syy
s#xy œ 12  0 and sxx  0 Ê local minimum of sa3, 3, 3b œ 27 54. paxß yß zb œ xyz; x  y  z œ 3 Ê z œ 3
x
y Ê paxß yb œ x ya3
x
yb œ 3x y
x2 y
x y2 Ê px axß yb œ 3y
2xy
y2 œ 0 and py axß yb œ 3x
x2
2xy œ 0 Ê critical points are a0, 0b, a0, 3b, a3, 0b, and a1, 1b; for a0, 0b Ê z œ 3; pxx a0, 0b œ 0, pyy a0, 0b œ 0, pxy a0, 0b œ 3 Ê pxx pyy
p#xy œ
9  0 Ê saddle point; for a0, 3b Ê z œ 0; pxx a0, 3b œ
6, pyy a0, 3b œ 0, pxy a0, 3b œ
3 Ê pxx pyy
p#xy œ
9  0 Ê saddle point; for a3, 0b Ê z œ 0; pxx a3, 0b œ 0, pyy a3, 0b œ
6, pxy a3, 0b œ
3 Ê pxx pyy
p#xy œ
9  0 Ê saddle point; for a1, 1b Ê z œ 1; pxx a1, 1b œ
2, pyy a1, 1b œ
2, pxy a1, 1b œ
1 Ê pxx pyy
p#xy œ 3  0 and pxx  0 Ê local maximum of pa1, 1, 1b œ 1 55. saxß yß zb œ xy  yz  xz; x  y  z œ 6 Ê z œ 6
x
y Ê saxß yb œ xy  ya6
x
yb  xa6
x
yb œ 6x  6y
xy
x2
y2 Ê sx axß yb œ 6
2x
y œ 0 and sy axß yb œ 6
x
2y œ 0 Ê critical point is a2, 2b Ê z œ 2; sxx a2, 2b œ
2, syy a2, 2b œ
2, sxy a2, 2b œ
1 Ê sxx syy
s#xy œ 3  0 and sxx  0 Ê local maximum of sa2, 2, 2b œ 12 56. daxß yß zb œ Éax  6b2  ay
4b2  az
0b2 Ê we can minimize daxß yß zb by minimizing Daxß yß zb œ ax  6b2  ay
4b2  z2 ; z œ Èx2  y2 Ê Daxß yb œ ax  6b2  ay
4b2  x2  y2 œ 2x2  2y2  12x
8y  52 Ê Dx axß yb œ 4x  12 œ 0 and Dy axß yb œ 4y
8 œ 0 Ê critical point is a
3, 2b Ê z œ È13; Dxx a
3, 2b œ 4, Dyy a
3, 2b œ 4, Dxy a
3, 2b œ 0 Ê Dxx Dyy
D# œ 16  0 and Dxx  0 Ê local xy

minimum of dŠ
3, 2, È13‹ œ È26 57. Vaxß yß zb œ a2xba2yba2zb œ 8xyz; x2  y2  z2 œ 4 Ê z œ È4
x2
y2 Ê Vaxß yb œ 8xyÈ4
x2
y2 , x   0 and y   0 Ê Vx axß yb œ a0, 0b, Š È#3 ,

# È3 ‹,

32y  16x2 y  8y3 È 4  x2  y2

Š È#3 ,
È#3 ‹, Š
È#3 ,

Va0ß 0b œ 0 and VŠ È#3 ,

# È3 ‹

œ

64 ; 3È 3

# È3 ‹,

œ 0 and Vy axß yb œ

32x  16x y2  8x3 È 4  x2  y2

œ 0 Ê critical points are

and Š
È#3 ,
È#3 ‹. Only a0, 0b and Š È#3 ,

# È3 ‹

satisfy x   0 and y   0

On x œ 0, 0 Ÿ y Ÿ 2 Ê Va0ß yb œ 8a0byÈ4
02
y2 œ 0, no critical points,

Va0ß 0b œ 0, Va0ß 2b œ 0; On y œ 0, 0 Ÿ x Ÿ 2 Ê Vaxß 0b œ 8xa0bÈ4
x2
02 œ 0, no critical points, Va0ß 0b œ 0, 2

Va0ß 2b œ 0; On y œ È4
x2 , 0 Ÿ x Ÿ 2 Ê VŠxß È4
x2 ‹ œ 8xÈ4
x2 Ê4
x2
ŠÈ4
x2 ‹ œ 0 no critical points, Va0ß 2b œ 0, Va2ß 0b œ 0. Thus, there is a maximum volume of 58. Saxß yß zb œ 2xy  2yz  2xz; xyz œ 27 Ê z œ y  0; Sx axß yb œ 2y


54 x2

27 xy

Syy a3, 3b œ 4, Dxy a3, 3b œ 2 Ê Dxx Dyy


if the box is

# È3

‚

27 27 Ê Saxß yß zb œ 2xy  2yŠ xy ‹  2xŠ xy ‹ œ 2xy 

œ 0 and Sy axß yb œ 2x
D#xy

64 3È 3

54 y2

# È3

54 x



‚

54 y ,

œ 0 Ê Critical point is a3, 3b Ê z œ 3; Sxx a3, 3b œ 4,

œ 12  0 and Dxx  0 Ê local minimum of Sa3ß 3ß 3b œ 54

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

# È3 .

x  0,

Section 14.7 Extreme Values and Saddle Points

845

59. Let x œ height of the box, y œ width, and z œ length, cut out squares of length x from corner of the material See diagram at right. Fold along the dashed lines to form the box. From the diagram we see that the length of the material is 2x  y and the width is 2x  z. Thus a2x  yba2x  zb œ 12 2ˆ6 2 x2
xy‰ . Since Vax, y, zb œ x y z 2x
y ˆ 2x y 6 2 x2
xy‰ Vax, yb œ , where x  0, y  2x
y 2 3 2 2 ˆ 4 3y  4x y  4x y  xy3 ‰

Êzœ Ê

Vx ax, yb œ

Vy ax, yb œ

œ 0 and

a2x
yb2 2ˆ12x 4 x  4x y  x y a2x
yb2 2

4

0.

2 2‰

3

œ 0 Ê critical points are ŠÈ3, 0‹, Š
È3, 0‹, Š È13 ,

and Š
È13 ,
È43 ‹. Only ŠÈ3, 0‹ and Š È13 ,

4 È3 ‹

4 È3 ‹,

satisfy x  0 and y  0. For ŠÈ3, 0‹: z œ 0; Vxx ŠÈ3, 0‹ œ 0,

# Vyy ŠÈ3, 0‹ œ
2È3, Vxy ŠÈ3, 0‹ œ
4È3 Ê Vxx Vyy
Vxy œ
48  0 Ê saddle point. For Š È13 ,

Vxx Š È13 ,

4 È3 ‹

1 œ
380 È3 , Vyy Š È3 ,

4 È3 ‹

Vxx  0 Ê local maximum of VŠ È13 ,

2 œ
3È , Vxy Š È13 , 3

4 4 È3 , È3 ‹

œ

4 È3 ‹

4 # œ
3È Ê Vxx Vyy
Vxy œ 3

16 3

4 È3 ‹:

zœ

 0 and

16 3È 3

60. (a) (i) On x œ 0, f(xß y) œ f(0ß y) œ y#
y  1 for 0 Ÿ y Ÿ 1; f w (0ß y) œ 2y
1 œ 0 Ê y œ f ˆ0ß "# ‰ œ 34 , f(0ß 0) œ 1, and f(0ß 1) œ 1

" #

and x œ 0;

On y œ 1, f(xß y) œ f(xß 1) œ x#  x  1 for 0 Ÿ x Ÿ 1; f w (xß 1) œ 2x  1 œ 0 Ê x œ
"# and y œ 1, but ˆ
"# ß 1‰ is outside the domain; f(0ß 1) œ 1 and f("ß ") œ 3

(ii)

(iii) On x œ 1, f(xß y) œ f("ß y) œ y#  y  1 for 0 Ÿ y Ÿ 1; f w (1ß y) œ 2y  1 œ 0 Ê y œ
"# and x œ 1, but ˆ1ß
"# ‰ is outside the domain; f(1ß 0) œ 1 and f("ß ") œ 3 (iv) On y œ 0, f(xß y) œ f(xß 0) œ x#
x  1 for 0 Ÿ x Ÿ 1; f w (xß 0) œ 2x
1 œ 0 Ê x œ f ˆ "# ß 0‰ œ 34 ; f(0ß 0) œ 1, and f("ß 0) œ 1

" #

and y œ 0;

On the interior of the square, fx (xß y) œ 2x  2y
1 œ 0 and fy (xß y) œ 2y  2x
1 œ 0 Ê 2x  2y œ 1 Ê (x  y) œ "# . Then f(xß y) œ x#  y#  2xy
x
y  1 œ (x  y)#
(x  y)  1 œ 34 is the absolute

(v)

minimum value when 2x  2y œ 1. (b) The absolute maximum is f("ß ") œ 3. 61. (a)

df dt

œ

` f dx ` x dt



` f dy ` y dt

œ

dx dt



dy dt

œ
2 sin t  2 cos t œ 0 Ê cos t œ sin t Ê x œ y

On the semicircle x#  y# œ 4, y   0, we have t œ

(i)

1 4

and x œ y œ È2 Ê f ŠÈ2ß È2‹ œ 2È2. At the

endpoints, f(
2ß 0) œ
2 and f(#ß !) œ 2. Therefore the absolute minimum is f(
2ß 0) œ
2 when t œ 1; the absolute maximum is f ŠÈ2ß È2‹ œ 2È2 when t œ 1 . 4

On the quartercircle x#  y# œ 4, x   0 and y   0, the endpoints give f(!ß 2) œ 2 and f(#ß 0) œ 2. Therefore the absolute minimum is f(2ß 0) œ 2 and f(!ß 2) œ 2 when t œ 0, 1# respectively; the absolute

(ii)

maximum is f ŠÈ2ß È2‹ œ 2È2 when t œ (b) (i)

dg dt

œ

` g dx ` x dt



` g dy ` y dt

œy

dx dt

x

dy dt

1 4

.

œ
4 sin# t  4 cos# t œ 0 Ê cos t œ „ sin t Ê x œ „ y.

On the semicircle x#  y# œ 4, y   0, we obtain x œ y œ È2 at t œ tœ

31 4

1 4

and x œ
È2, y œ È2 at

. Then g ŠÈ2ß È2‹ œ 2 and g Š
È2ß È2‹ œ
2. At the endpoints, g(
2ß 0) œ g(#ß 0) œ 0.

Therefore the absolute minimum is g Š
È2ß È2‹ œ
2 when t œ g ŠÈ2 ß È2‹ œ 2 when t œ

1 4

31 4

; the absolute maximum is

.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

4 È3 ;

846

Chapter 14 Partial Derivatives On the quartercircle x#  y# œ 4, x   0 and y   0, the endpoints give g(!ß 2) œ 0 and g(#ß 0) œ 0. Therefore the absolute minimum is g(2ß 0) œ 0 and g(!ß 2) œ 0 when t œ 0, 1# respectively; the absolute

(ii)

maximum is g ŠÈ2ß È2‹ œ 2 when t œ dh dt

(c)

œ

` h dx ` x dt



` h dy ` y dt

1 4

.

dy œ 4x dx dt  2y dt œ (8 cos t)(
2 sin t)  (4 sin t)(2 cos t) œ
8 cos t sin t œ 0

Ê t œ 0, 1# , 1 yielding the points (2ß 0), (0ß 2) for 0 Ÿ t Ÿ 1.

On the semicircle x#  y# œ 4, y   0 we have h(2ß 0) œ 8, h(0ß 2) œ 4, and h(
2ß 0) œ 8. Therefore, the absolute minimum is h(!ß 2) œ 4 when t œ 1# ; the absolute maximum is h(2ß 0) œ 8 and h(
2ß 0) œ 8

(i)

when t œ 0, 1 respectively. On the quartercircle x#  y# œ 4, x   0 and y   0 the absolute minimum is h(0ß 2) œ 4 when t œ

(ii)

absolute maximum is h(2ß 0) œ 8 when t œ 0. df dt

62. (a) (i)

œ

` f dx ` x dt



` f dy ` y dt

1 4

x# 9

y# 4



œ 1, y   0, f(xß y) œ 2x  3y œ 6 cos t  6 sin t œ È

1 4

.

On the quarter ellipse, at the endpoints f(0ß 2) œ 6 and f(3ß 0) œ 6. The absolute minimum is f(3ß 0) œ 6 È and f(0ß 2) œ 6 when t œ 0, 1 respectively; the absolute maximum is f Š 3 2 ß È2‹ œ 6È2 when t œ 1 .

(ii)

#

` g dy dx œ `` gx dx dt  ` y dt œ y dt Ê t œ 14 , 341 for 0 Ÿ t Ÿ

dg dt

x

dy dt

#

1. È

31 4

. At the endpoints, g(
3ß 0) œ g($ß 0) œ 0. The absolute minimum is

È

31 4

; the absolute maximum is g Š 3 # 2 ß È2‹ œ 3 when t œ

#

dh dt

œ

` h dx ` x dt

Ê t œ 0, (i) (ii)

œ

(ii)

, and

È

1 4

.

On the quarter ellipse, at the endpoints g(!ß 2) œ 0 and g($ß 0) œ 0. The absolute minimum is g(3ß 0) œ 0 È and g(0ß 2) œ 0 at t œ 0, 1 respectively; the absolute maximum is g Š 3 2 ß È2‹ œ 3 when t œ 1 .

(ii)

(i)

1 4

È

g Š
3 # 2 ß È2‹ œ
3 when t œ

df dt

#

œ (2 sin t)(
3 sin t)  (3 cos t)(2 cos t) œ 6 acos t
sin tb œ 6 cos 2t œ 0

g Š
3 # 2 ß È2‹ œ
3 when t œ

63.

4

#

On the semi-ellipse, g(xß y) œ xy œ 6 sin t cos t. Then g Š 3 # 2 ß È2‹ œ 3 when t œ

(i)

(c)

œ 6È 2

. At the endpoints, f(
3ß 0) œ
6 and f(3ß 0) œ 6. The absolute minimum is f(
3ß 0) œ
6 when

t œ 1; the absolute maximum is f Š 3 # 2 ß È2‹ œ 6È2 when t œ

(b)

1 4 for 0 Ÿ t Ÿ 1. È È 6 Š #2 ‹  6 Š #2 ‹

; the

dy œ 2 dx dt  3 dt œ
6 sin t  6 cos t œ 0 Ê sin t œ cos t Ê t œ

On the semi-ellipse, at t œ

1 #

 1 #

` h dy ` y dt

œ 2x

dx dt

 6y

#

dy dt

4

œ (6 cos t)(
3 sin t)  (12 sin t)(2 cos t) œ 6 sin t cos t œ 0

, 1 for 0 Ÿ t Ÿ 1, yielding the points (3ß 0), (0ß 2), and (
3ß 0).

On the semi-ellipse, y   0 so that h(3ß 0) œ 9, h(0ß 2) œ 12, and h(
3ß 0) œ 9. The absolute minimum is h(3ß 0) œ 9 and h(
3ß 0) œ 9 when t œ 0, 1 respectively; the absolute maximum is h(!ß 2) œ 12 when t œ On the quarter ellipse, the absolute minimum is h(3ß 0) œ 9 when t œ 0; the absolute maximum is h(!ß 2) œ 12 when t œ 1# . ` f dx ` x dt



` f dy ` y dt

1 #

dy œ y dx dt  x dt

" " x œ 2t and y œ t  1 Ê df dt œ (t  1)(2)  (2t)(1) œ 4t  2 œ 0 Ê t œ
# Ê x œ
1 and y œ # with f ˆ
1ß "# ‰ œ
"# . The absolute minimum is f ˆ
1ß "# ‰ œ
"# when t œ
"# ; there is no absolute maximum.

For the endpoints: t œ
1 Ê x œ
2 and y œ 0 with f(
2ß 0) œ 0; t œ 0 Ê x œ 0 and y œ 1 with f(!ß 1) œ 0. The absolute minimum is f ˆ
1ß "# ‰ œ
"# when t œ
"# ; the absolute maximum is f(0ß 1) œ 0

and f(
#ß 0) œ 0 when t œ
1, 0 respectively. (iii) There are no interior critical points. For the endpoints: t œ 0 Ê x œ 0 and y œ 1 with f(0ß 1) œ 0; t œ 1 Ê x œ 2 and y œ 2 with f(2ß 2) œ 4. The absolute minimum is f(0ß 1) œ 0 when t œ 0; the absolute maximum is f(2ß 2) œ 4 when t œ 1.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

.

Section 14.7 Extreme Values and Saddle Points df dt

64. (a)

` f dx ` x dt

œ



` f dy ` y dt

dy œ 2x dx dt  2y dt

4 4 x œ t and y œ 2
2t Ê df dt œ (2t)(1)  2(2
2t)(
2) œ 10t
8 œ 0 Ê t œ 5 Ê x œ 5 and y œ 4 f ˆ 45 ß 25 ‰ œ "#65  25 œ 45 . The absolute minimum is f ˆ 45 ß 25 ‰ œ 45 when t œ 45 ; there is no absolute

(i)

2 5

with

maximum along the line. For the endpoints: t œ 0 Ê x œ 0 and y œ 2 with f(0ß 2) œ 4; t œ 1 Ê x œ 1 and y œ 0 with f(1ß 0) œ 1. The absolute minimum is f ˆ 45 ß 25 ‰ œ 45 at the interior critical point when t œ 45 ; the absolute maximum is

(ii)

f(0ß 2) œ 4 at the endpoint when t œ 0. œ

dg dt

(b)

` g dx ` x dt



` g dy ` y dt

œ ’ ax#
2xy# b# “

 ’ ax#
2yy# b# “

dx dt

dy dt

x œ t and y œ 2
2t Ê x#  y# œ 5t#
8t  4 Ê

(i)

#

œ
a5t#
8t  4b (
10t  8) œ 0 Ê t œ maximum is g ˆ 45 ß 25 ‰ œ

5 4

when t œ

4 5

4 5

dg dt

#

œ
a5t#
8t  4b [(
2t)(1)  (
2)(2
2t)(
2)]

Ê xœ

4 5

and y œ

" 4

The absolute minimum is g(0ß 2) œ

with g ˆ 45 ß 25 ‰ œ

" ˆ 45 ‰

œ

5 4

. The absolute

; there is no absolute minimum along the line since x and y can be

as large as we please. For the endpoints: t œ 0 Ê x œ 0 and y œ 2 with g(0ß 2) œ

(ii)

2 5

" 4

; t œ 1 Ê x œ 1 and y œ 0 with g(1ß 0) œ 1. when t œ 0; the absolute maximum is g ˆ 45 ß 52 ‰ œ 45 when t œ 54 .

65. w œ am x1  b
y1 b2  am x2  b
y2 b2  â  am xn  b
yn b2 w Ê `` m œ 2am x1  b
y1 bax1 b  2am x2  b
y2 bax2 b  â  2am xn  b
yn baxn b Ê `w `m

`w `b

œ 2am x1  b
y1 ba1b  2am x2  b
y2 ba1b  â  2am xn  b
yn ba1b œ 0 Ê 2
am x1  b
y1 bax1 b  am x2  b
y2 bax2 b  â  am xn  b
yn baxn b‘ œ 0

Ê m x21  b x1
x1 y1  m x#2  b x2
x2 y2  â  m xn2  b xn
xn yn œ 0 Ê max21  x2#  â  xn2 b  bax1  x2  â  xn b
ax1 y1  x2 y2  â  xn yn b œ 0 n

n

n

k œ1

k œ1

k œ1

Ê m! ax2k b  b! xk
! axk yk b œ 0 `w `b

œ 0 Ê 2
am x1  b
y1 b  am x2  b
y2 b  â  am xn  b
yn b‘ œ 0

Ê m x1  b
y1  m x2  b
y2  â  m xn  b
yn œ 0 Ê max1  x2  â  xn b  ab  b  â  bb
ay1  y2  â  yn b œ 0 n

n

n

n

n

k œ1

k œ1

k œ1

k œ1

k œ1

n

n

kœ1 n

kœ 1 n

n

n

k œ1

k œ1

k œ1

Ê m ! xk  b ! 1
! yk œ 0 Ê m ! xk  bn
! yk œ 0 Ê b œ 1n Œ ! yk
m! xk . Substituting for b in the equation obtained for

`w `m

n

we get m ! ax2k b  1n Π! yk
m! xk ! xk
! axk yk b œ 0.

n

k œ1 n

n

k œ1

n

n

k œ1

kœ1

Multiply both sides by n to obtain m n ! ax2k b  Π! yk
m! xk ! xk
n ! axk yk b œ 0 k œ1

n

n

k œ1

k œ1

k œ1

n

2

n

k œ1

n

Ê m n ! ax2k b  Œ ! xk Œ ! yk 
mΠ! xk 
n ! axk yk b œ 0 n

k œ1 2

n

k œ1

kœ1

n

n

n

k œ1

k œ1

k œ1

n

n

n

Ê m n ! ax2k b
mŒ ! xk  œ n ! axk yk b
Œ ! xk Œ ! yk  kœ1

kœ1

n

2

n

Ê m–n! ax2k b
Œ ! xk  — œ n ! axk yk b
Œ ! xk Œ ! yk  k œ1

k œ1

n

Êmœ

k œ1

n

n

n ! axk yk bΠ! xk
Π! yk
kœ1

kœ1

n

n

kœ1

kœ1

kœ1 2

n! ax2k bΠ! xk


k œ1

n

œ

n

kœ1

n

Π! xk
Π! yk
 n ! axk yk b kœ1

kœ1

n

kœ1

2

n

2 Π! x k
 n ! ax k b kœ1

kœ1

To show that these values for m and b minimize the sum of the squares of the distances, use second derivative test. ` 2w ` m2

n

œ 2 x21  2 x#2  â  2 x2n œ 2 ! ax2k b; k œ1

` 2w `m `b

n

œ 2 x1  2 x2  â  2 xn œ 2! xk ; k œ1

` 2w ` b2

œ 2 2 â  2 œ 2n

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

847

848

Chapter 14 Partial Derivatives 2

n

n

kœ1

kœ1

#

n

n

kœ1

k œ1

2

The discriminant is: Š `` mw2 ‹Š `` bw2 ‹
Š ``m w` b ‹ œ ”2 ! ax2k b•a2 nb
”2 ! xk • œ 4–n ! ax2k b
Œ ! xk  —. 2

n

2

2

2

n

Now, n ! ax2k b
Œ ! xk  œ nax12  x#2  â  x#n b
ax1  x2  â  xn bax1  x2  â  xn b k œ1

kœ1 2 œ  n x#  â  n x#n
x21
x1 x2
â
x1 xn
x2 x1
x2#
â
x2 xn
xn x1
xn x2
â
x#n œ an
1b x21  an
1b x2#  â  an
1b x#n
2 x1 x2
2 x1 x3
â
2 x1 xn
2 x2 x3
â
2 x2 xn
â
2 xn1 xn œ a x21
2 x1 x2  x#2 b  a x12
2 x1 x3  x23 b  â  ax21
2 x1 xn  xn# b  ax2#
2 x2 x3  x23 b  â  a x#2
2 x2 xn  xn# b  â  ax2n1
2 xn1 xn  x#n b œ ax1
x2 b2  ax1
x3 b2  â  ax1
xn b2  ax2
x3 b2  â  ax2
xn b2  â  axn1
xn b2   0. 2 n n 2 2 2 2 Thus we have : Š `` mw2 ‹Š `` bw2 ‹
Š ``m w` b ‹ œ 4–n ! ax2k b
Œ ! xk  —   4a0b œ 0. If x1 œ x2 œ â œ xn then kœ1 k œ1

n x21

2

Š `` mw2 ‹Š `` bw2 ‹
Š ``m w` b ‹ œ 0. Also, 2

2

2

` 2w ` m2

n

` 2w ` m2

œ 2 ! ax2k b   0. If x1 œ x2 œ â œ xn œ 0, then k œ1

œ 0. 2

Provided that at least one xi is nonzero and different from the rest of xj , j Á i, then Š `` mw2 ‹Š `` bw2 ‹
Š ``m w` b ‹  0 and 2

` 2w ` m2

bœ Ê

67. m œ bœ Ê

68. m œ Ê

2

 0 Ê the values given above for m and b minimize w.

66. m œ

bœ

2

(0)(5)  3(6) 3 (0)#  3(8) œ 4 and "
3 ‘ 5 3 5
4 (0) œ 3 y œ 34 x  53 ; y¸ xœ4

œ

14 3

(2)(1)  3("4) œ
20 (2)#  3(10) 13 and "
20 9 ˆ ‰ ‘ 3
1

13 (2) œ 13 9 ¸ y œ
20 13 x  13 ; y xœ4 œ

(3)(5)  3(8) 3 (3)#  3(5) œ 2 and "
3 ‘ 1 3 5
2 (3) œ 6 y œ 32 x  16 ; y¸ xœ4

œ

37 6


71 13

k 1 2 3 D

xk
2 0 2 0

yk 0 2 3 5

x#k 4 0 4 8

xk yk 0 0 6 6

k 1 2 3 D

xk
1 0 3 2

yk 2 1
4
1

x#k 1 0 9 10

xk yk
2 0
12
14

k 1 2 3 D

xk 0 1 2 3

yk 0 2 3 5

x#k 0 1 4 5

xk yk 0 2 6 8

69-74. Example CAS commands: Maple: f := (x,y) -> x^2+y^3-3*x*y; x0,x1 := -5,5; y0,y1 := -5,5; plot3d( f(x,y), x=x0..x1, y=y0..y1, axes=boxed, shading=zhue, title="#69(a) (Section 14.7)" ); plot3d( f(x,y), x=x0..x1, y=y0..y1, grid=[40,40], axes=boxed, shading=zhue, style=patchcontour, title="#69(b) (Section 14.7)" ); fx := D[1](f); # (c) fy := D[2](f); crit_pts := solve( {fx(x,y)=0,fy(x,y)=0}, {x,y} ); fxx := D[1](fx); # (d) fxy := D[2](fx);

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.8 Lagrange Multipliers

849

fyy := D[2](fy); discr := unapply( fxx(x,y)*fyy(x,y)-fxy(x,y)^2, (x,y) ); for CP in {crit_pts} do # (e) eval( [x,y,fxx(x,y),discr(x,y)], CP ); end do; # (0,0) is a saddle point # ( 9/4, 3/2) is a local minimum Mathematica: (assigned functions and bounds will vary) Clear[x,y,f] f[x_,y_]:= x2  y3
3x y xmin=
5; xmax= 5; ymin=
5; ymax= 5; Plot3D[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, AxesLabel Ä {x, y, z}] ContourPlot[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, ContourShading Ä False, Contours Ä 40] fx= D[f[x,y], x]; fy= D[f[x,y], y]; critical=Solve[{fx==0, fy==0},{x, y}] fxx= D[fx, x]; fxy= D[fx, y]; fyy= D[fy, y]; discriminant= fxx fyy
fxy2 {{x, y}, f[x, y], discriminant, fxx} /.critical 14.8 LAGRANGE MULTIPLIERS 1.

™ f œ yi  xj and ™ g œ 2xi  4yj so that ™ f œ - ™ g Ê yi  xj œ -(2xi  4yj) Ê y œ 2x- and x œ 4yÊ x œ 8x-# Ê - œ „

È2 4

or x œ 0.

CASE 1: If x œ 0, then y œ 0. But (0ß 0) is not on the ellipse so x Á 0. CASE 2: x Á 0 Ê - œ „

È2 4

Therefore f takes on its extreme values at Š „ are „ 2.

È2 #

#

Ê x œ „ È2y Ê Š „ È2y‹  2y# œ 1 Ê y œ „ "# . È2 " 2 ß #‹

and Š „

È2 " 2 ß
#‹ .

The extreme values of f on the ellipse

.

™ f œ yi  xj and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê yi  xj œ -(2xi  2yj) Ê y œ 2x- and x œ 2yÊ x œ 4x-# Ê x œ 0 or - œ „ 12 .

CASE 1: If x œ 0, then y œ 0. But (0ß 0) is not on the circle x#  y#
10 œ 0 so x Á 0. CASE 2: x Á 0 Ê - œ „ 12 Ê y œ 2x ˆ „ "# ‰ œ „ x Ê x#  a „ xb#
10 œ 0 Ê x œ „ È5 Ê y œ „ È5. Therefore f takes on its extreme values at Š „ È5ß È5‹ and Š „ È5ß
È5‹ . The extreme values of f on the circle are 5 and
5. 3.

™ f œ
2xi
2yj and ™ g œ i  3j so that ™ f œ - ™ g Ê
2xi
2yj œ -(i  3j) Ê x œ
-# and y œ
3#Ê ˆ
-# ‰  3 ˆ
3#- ‰ œ 10 Ê - œ
2 Ê x œ 1 and y œ 3 Ê f takes on its extreme value at (1ß 3) on the line.

The extreme value is f("ß $) œ 49
1
9 œ 39. 4.

™ f œ 2xyi  x# j and ™ g œ i  j so that ™ f œ - ™ g Ê 2xyi  x# j œ -(i  j) Ê 2xy œ - and x# œ Ê 2xy œ x# Ê x œ 0 or 2y œ x. CASE 1: If x œ 0, then x  y œ 3 Ê y œ 3.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

850

Chapter 14 Partial Derivatives

CASE 2: If x Á 0, then 2y œ x so that x  y œ 3 Ê 2y  y œ 3 Ê y œ 1 Ê x œ 2. Therefore f takes on its extreme values at (!ß 3) and (2ß "). The extreme values of f are f(0ß 3) œ 0 and f(2ß 1) œ 4. 5. We optimize f(xß y) œ x#  y# , the square of the distance to the origin, subject to the constraint g(xß y) œ xy#
54 œ 0. Thus ™ f œ 2xi  2yj and ™ g œ y# i  2xyj so that ™ f œ - ™ g Ê 2xi  2yj œ - ay# i  2xyjb Ê 2x œ -y# and 2y œ 2-xy. CASE 1: If y œ 0, then x œ 0. But (0ß 0) does not satisfy the constraint xy# œ 54 so y Á 0. CASE 2: If y Á 0, then 2 œ 2-x Ê x œ -" Ê 2 ˆ -" ‰ œ -y# Ê y# œ -2# . Then xy# œ 54 Ê ˆ -" ‰ ˆ -2# ‰ œ 54 Ê -$ œ " Ê - œ " Ê x œ 3 and y# œ 18 Ê x œ 3 and y œ „ 3È2. 27

3

Therefore Š$ß „ 3È2‹ are the points on the curve xy# œ 54 nearest the origin (since xy# œ 54 has points increasingly far away as y gets close to 0, no points are farthest away). 6. We optimize f(xß y) œ x#  y# , the square of the distance to the origin subject to the constraint g(xß y) œ x# y
2 œ 0. Thus ™ f œ 2xi  2yj and ™ g œ 2xyi  x# j so that ™ f œ - ™ g Ê 2x œ 2xy- and 2y œ x# - Ê - œ 2y x# , since 2y x œ 0 Ê y œ 0 (but g(0ß 0) Á 0). Thus x Á 0 and 2x œ 2xy ˆ x# ‰ Ê x# œ 2y# Ê a2y# b y
2 œ 0 Ê y œ 1 (since y  0) Ê x œ „ È2 . Therefore Š „ È2ß 1‹ are the points on the curve x# y œ 2 nearest the origin (since x# y œ 2 has points increasingly far away as x gets close to 0, no points are farthest away). 7. (a) ™ f œ i  j and ™ g œ yi  xj so that ™ f œ - ™ g Ê i  j œ -(yi  xj) Ê 1 œ -y and 1 œ -x Ê y œ xœ

" -

Ê

" -#

œ 16 Ê - œ „

" 4.

Use - œ

" 4

" -

and

since x  0 and y  0. Then x œ 4 and y œ 4 Ê the minimum value is 8

at the point (4ß 4). Now, xy œ 16, x  0, y  0 is a branch of a hyperbola in the first quadrant with the x-and y-axes as asymptotes. The equations x  y œ c give a family of parallel lines with m œ
1. As these lines move away from the origin, the number c increases. Thus the minimum value of c occurs where x  y œ c is tangent to the hyperbola's branch. (b) ™ f œ yi  xj and ™ g œ i  j so that ™ f œ - ™ g Ê yi  xj œ -(i  j) Ê y œ - œ x y  y œ 16 Ê y œ 8 Ê x œ 8 Ê f()ß )) œ 64 is the maximum value. The equations xy œ c (x  0 and y  0 or x  0 and y  0 to get a maximum value) give a family of hyperbolas in the first and third quadrants with the x- and y-axes as asymptotes. The maximum value of c occurs where the hyperbola xy œ c is tangent to the line x  y œ 16. 8. Let f(xß y) œ x#  y# be the square of the distance from the origin. Then ™ f œ 2xi  2yj and ™ g œ (2x  y)i  (2y  x)j so that ™ f œ - ™ g Ê 2x œ -(2x  y) and 2y œ -(2y  x) Ê

2y 2y
x

œ-

Ê 2x œ Š 2y2y
x ‹ (2x  y) Ê x(2y  x) œ y(2x  y) Ê x# œ y# Ê y œ „ x. CASE 1: y œ x Ê x#  x(x)  x#
1 œ 0 Ê x œ „

" È3

and y œ x.

CASE 2: y œ
x Ê x#  x(
x)  (
x)#
1 œ 0 Ê x œ „ 1 and y œ
x. Thus f Š È"3 ß È"3 ‹ œ

2 3

œ f Š
È"3 ß
È"3 ‹ and f(1ß
1) œ 2 œ f(
1ß 1). Therefore the points (1ß
1) and (
1ß 1) are the farthest away; Š È"3 ß È"3 ‹ and Š
È"3 ß
È"3 ‹ are the closest points to the origin. 9. V œ 1r# h Ê 161 œ 1r# h Ê 16 œ r# h Ê g(rß h) œ r# h
16; S œ 21rh  21r# Ê ™ S œ (21h  41r)i  21rj and ™ g œ 2rhi  r# j so that ™ S œ - ™ g Ê (21rh  41r)i  21rj œ - a2rhi  r# jb Ê 21rh  41r œ 2rh- and 21r œ -r# Ê r œ 0 or - œ 2r1 . But r œ 0 gives no physical can, so r Á 0 Ê - œ 2r1 Ê 21h  41r œ 2rh ˆ 2r1 ‰ Ê 2r œ h Ê 16 œ r# (2r) Ê r œ 2 Ê h œ 4; thus r œ 2 cm and h œ 4 cm give the only extreme surface area of 241 cm# . Since r œ 4 cm and h œ 1 cm Ê V œ 161 cm$ and S œ 401 cm# , which is a larger surface area, then 241 cm# must be the minimum surface area.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.8 Lagrange Multipliers

851

10. For a cylinder of radius r and height h we want to maximize the surface area S œ 21rh subject to the constraint # g(rß h) œ r#  ˆ h# ‰
a# œ 0. Thus ™ S œ 21hi  21rj and ™ g œ 2ri  h# j so that ™ S œ - ™ g Ê 21h œ 2-r and 21r œ

-h #

Ê

1h r

4r# 4

œ - and 21r œ ˆ 1rh ‰ ˆ #h ‰ Ê 4r# œ h# Ê h œ 2r Ê r# 

œ a# Ê 2r# œ a# Ê r œ

a È2

Ê h œ aÈ2 Ê S œ 21 Š Èa2 ‹ ŠaÈ2‹ œ 21a# . #

#

x 11. A œ (2x)(2y) œ 4xy subject to g(xß y) œ 16  y9
1 œ 0; ™ A œ 4yi  4xj and ™ g œ x8 i  2y 9 j so that ™ A 2y 2y 32y x x ‰ ˆ 32y ‰ œ - ™ g Ê 4yi  4xj œ - ˆ 8 i  9 j‰ Ê 4y œ ˆ 8 ‰ - and 4x œ ˆ 9 ‰ - Ê - œ x and 4x œ ˆ 2y 9 x

Ê y œ „ 34 x Ê Then y œ

3 4

x# 16

Š2È2‹ œ

ˆ „43 x‰# œ 1 Ê x# 9 3È 2 # , so the length is



12. P œ 4x  4y subject to g(xß y) œ

x# a#



y# b#

and height œ 2y œ

2b# È a#
b#

2x œ 4È2 and the width is 2y œ 3È2.


1 œ 0; ™ P œ 4i  4j and ™ g œ

‰ ˆ 2y ‰ Ê 4 œ ˆ 2x a# - and 4 œ b# - Ê - œ œ 1 Ê aa#  b# b x# œ a% Ê x œ

œ 8 Ê x œ „ 2È2 . We use x œ 2È2 since x represents distance.

2a# x

a# È a#
b#

#

2x a#

i

#

b ‰ 2a and 4 œ ˆ 2y b# Š x ‹ Ê y œ Š a# ‹ x Ê #

, since x  0 Ê y œ Š ba# ‹ x œ

Ê perimeter is P œ 4x  4y œ

4a#
4b# È a#
b#

b# È a#
b#

2y b# x# a#

j so that ™ P œ - ™ g #

#



Š ba# ‹ x# b#

œ1 Ê

Ê width œ 2x œ

x# a#



b# x# a%

2a# È a#
b#

œ 4Èa#  b#

13. ™ f œ 2xi  2yj and ™ g œ (2x
2)i  (2y
4)j so that ™ f œ - ™ g œ 2xi  2yj œ -[(2x
2)i  (2y
4)j] 2# # Ê 2x œ -(2x
2) and 2y œ -(2y
4) Ê x œ - 1 and y œ -1 , - Á 1 Ê y œ 2x Ê x
2x  (2x)
4(2x) œ 0 Ê x œ 0 and y œ 0, or x œ 2 and y œ 4. Therefore f(0ß 0) œ 0 is the minimum value and f(2ß 4) œ 20 is the maximum value. (Note that - œ 1 gives 2x œ 2x
2 or ! œ
2, which is impossible.)

14. ™ f œ 3i
j and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê 3 œ 2-x and
1 œ 2-y Ê - œ #

Ê y œ
x3 Ê x#  ˆ
x3 ‰ œ 4 Ê 10x# œ 36 Ê x œ „ yœ

2 È10 .

Therefore f Š È610 ß
È210 ‹ œ

20 È10

6 È10

Ê xœ

6 È10

3 2x

3 ‰ and
1 œ 2 ˆ 2x y

and y œ
È210 , or x œ
È610 and

 6 œ 2È10  6 ¸ 12.325 is the maximum value, and f Š
È610 ß È210 ‹

œ
2È10  6 ¸
0.325 is the minimum value. 15. ™ T œ (8x
4y)i  (
4x  2y)j and g(xß y) œ x#  y#
25 œ 0 Ê ™ g œ 2xi  2yj so that ™ T œ - ™ g Ê (8x
4y)i  (
4x  2y)j œ -(2xi  2yj) Ê 8x
4y œ 2-x and
4x  2y œ 2-y Ê y œ -2x1 , - Á 1 Ê 8x
4 ˆ -2x1 ‰ œ 2-x Ê x œ 0, or - œ 0, or - œ 5. CASE 1: x œ 0 Ê y œ 0; but (0ß 0) is not on x#  y# œ 25 so x Á 0. CASE 2: - œ 0 Ê y œ 2x Ê x#  (2x)# œ 25 Ê x œ „ È5 and y œ 2x. CASE 3: - œ 5 Ê y œ and y œ È5 .

2x 4

#

œ
#x Ê x#  ˆ
#x ‰ œ 25 Ê x œ „ 2È5 Ê x œ 2È5 and y œ
È5, or x œ
2È5

Therefore T ŠÈ5ß 2È5‹ œ 0° œ T Š
È5ß
2È5‹ is the minimum value and T Š2È5ß
È5‹ œ 125° œ T Š
2È5ß È5‹ is the maximum value. (Note: - œ 1 Ê x œ 0 from the equation
4x  2y œ 2-y; but we found x Á 0 in CASE 1.) 16. The surface area is given by S œ 41r#  21rh subject to the constraint V(rß h) œ #

4 3

1r$  1r# h œ 8000. Thus

#

™ S œ (81r  21h)i  21rj and ™ V œ a41r  21rhb i  1r j so that ™ S œ - ™ V œ (81r  21h)i  21rj œ - ca41r#  21rhb i  1r# jd Ê 81r  21h œ - a41r#  21rhb and 21r œ -1r# Ê r œ 0 or 2 œ r-. But r Á 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

852

Chapter 14 Partial Derivatives

so 2 œ r- Ê - œ 4 3

2 r

Ê 4r  h œ

1r$ œ 8000 Ê r œ 10 ˆ 16 ‰

"Î$

2 r

a2r#  rhb Ê h œ 0 Ê the tank is a sphere (there is no cylindrical part) and

.

17. Let f(xß yß z) œ (x
1)#  (y
1)#  (z
1)# be the square of the distance from (1ß 1ß 1). Then ™ f œ 2(x
1)i  2(y
1)j  2(z
1)k and ™ g œ i  2j  3k so that ™ f œ - ™ g Ê 2(x
1)i  2(y
1)j  2(z
1)k œ -(i  2j  3k) Ê 2(x
1) œ -, 2(y
1) œ 2-, 2(z
1) œ 3Ê 2(y
1) œ 2[2(x
1)] and 2(z
1) œ 3[2(x
1)] Ê x œ y
# 1 Ê z  2 œ 3 ˆ y
# 1 ‰ or z œ 3y # 1 ; thus y
1 ˆ 3y # 1 ‰
13 œ 0 Ê y œ 2 Ê x œ 3# and z œ #5 . Therefore the point ˆ #3 ß 2ß 5# ‰ is closest (since no #  2y  3 point on the plane is farthest from the point (1ß 1ß 1)).

18. Let f(xß yß z) œ (x
1)#  (y  1)#  (z
1)# be the square of the distance from (1ß
1ß 1). Then ™ f œ 2(x
1)i  2(y  1)j  2(z
1)k and ™ g œ 2xi  2yj  2zk so that ™ f œ - ™ g Ê x
1 œ -x, y  1 œ -y # ‰#  ˆ 1 " - ‰# œ 4 and z
1 œ -z Ê x œ 1 " - , y œ
1 " - , and z œ 1" - for - Á 1 Ê ˆ 1 " - ‰  ˆ 1" Ê

" "-

œ „

2 È3

Ê xœ

2 È3

, y œ
È23 , z œ

2 È3

or x œ
È23 , y œ

2 È3

, z œ
È23 . The largest value of f

occurs where x  0, y  0, and z  0 or at the point Š
È23 ß È23 ß
È23 ‹ on the sphere. 19. Let f(xß yß z) œ x#  y#  z# be the square of the distance from the origin. Then ™ f œ 2xi  2yj  2zk and ™ g œ 2xi
2yj
2zk so that ™ f œ - ™ g Ê 2xi  2yj  2zk œ -(2xi
2yj
2zk) Ê 2x œ 2x-, 2y œ 2y-, and 2z œ
2z- Ê x œ 0 or - œ 1. CASE 1: - œ 1 Ê 2y œ
2y Ê y œ 0; 2z œ
2z Ê z œ 0 Ê x#
1 œ 0 Ê x#
1 œ 0 Ê x œ „ 1 and y œ z œ 0. CASE 2: x œ 0 Ê y#
z# œ 1, which has no solution. Therefore the points on the unit circle x#  y# œ 1, are the points on the surface x#  y#
z# œ 1 closest to the originÞ The minimum distance is 1. 20. Let f(xß yß z) œ x#  y#  z# be the square of the distance to the origin. Then ™ f œ 2xi  2yj  2zk and ™ g œ yi  xj
k so that ™ f œ - ™ g Ê 2xi  2yj  2zk œ -(yi  xj
k) Ê 2x œ -y, 2y œ -x, and 2z œ
Ê xœ

-y #

Ê 2y œ - Š -#y ‹ Ê y œ 0 or - œ „ 2.

CASE 1: y œ 0 Ê x œ 0 Ê
z  1 œ 0 Ê z œ 1. CASE 2: - œ 2 Ê x œ y and z œ
1 Ê x#
(
1)  1 œ 0 Ê x#  2 œ 0, so no solution. CASE 3: - œ
2 Ê x œ
y and z œ 1 Ê (
y)y
1  1 œ 0 Ê y œ 0, again. Therefore (0ß 0ß 1) is the point on the surface closest to the origin since this point gives the only extreme value and there is no maximum distance from the surface to the origin. 21. Let f(xß yß z) œ x#  y#  z# be the square of the distance to the origin. Then ™ f œ 2xi  2yj  2zk and ™ g œ
yi
xj  2zk so that ™ f œ - ™ g Ê 2xi  2yj  2zk œ -(
yi
xj  2zk) Ê 2x œ
y-, 2y œ
x-, and 2z œ 2z- Ê - œ 1 or z œ 0. CASE 1: - œ 1 Ê 2x œ
y and 2y œ
x Ê y œ 0 and x œ 0 Ê z#
4 œ 0 Ê z œ „ 2 and x œ y œ 0. CASE 2: z œ 0 Ê
xy
4 œ 0 Ê y œ
4x . Then 2x œ

4 x

- Ê -œ

x# #

#

, and
x8 œ
x- Ê
x8 œ
x Š x# ‹

Ê x% œ 16 Ê x œ „ 2. Thus, x œ 2 and y œ
2, or x =
2 and y œ 2. Therefore we get four points: (#ß
2ß 0), (
2ß 2ß 0), (0ß 0ß 2) and (!ß 0ß
2). But the points (!ß 0ß 2) and (!ß !ß
2) are closest to the origin since they are 2 units away and the others are 2È2 units away. 22. Let f(xß yß z) œ x#  y#  z# be the square of the distance to the origin. Then ™ f œ 2xi  2yj  2zk and ™ g œ yzi  xzj  xyk so that ™ f œ - ™ g Ê 2x œ -yz, 2y œ -xz, and 2z œ -xy Ê 2x# œ -xyz and 2y# œ -yxz Ê x# œ y# Ê y œ „ x Ê z œ „ x Ê x a „ xb a „ xb œ 1 Ê x œ „ 1 Ê the points are (1ß 1ß 1), ("ß
1ß
1), (
"ß
"ß "), and (
1ß 1,
1).

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.8 Lagrange Multipliers

853

23. ™ f œ i
2j  5k and ™ g œ 2xi  2yj  2zk so that ™ f œ - ™ g Ê i
2j  5k œ -(2xi  2yj  2zk) Ê 1 œ 2x-,
2 œ 2y-, and 5 œ 2z- Ê x œ #"- , y œ
-" œ
2x, and z œ #5- œ 5x Ê x#  (
2x)#  (5x)# œ 30 Ê x œ „ 1. Thus, x œ 1, y œ
2, z œ 5 or x œ
1, y œ 2, z œ
5. Therefore f(1ß
2ß 5) œ 30 is the maximum value and f(
1ß 2ß
5) œ
30 is the minimum value.

24. ™ f œ i  2j  3k and ™ g œ 2xi  2yj  2zk so that ™ f œ - ™ g Ê i  2j  3k œ -(2xi  2yj  2zk) Ê 1 œ 2x-, 2 œ 2y-, and 3 œ 2z- Ê x œ #"- , y œ -" œ 2x, and z œ #3- œ 3x Ê x#  (2x)#  (3x)# œ 25 Ê x œ „ È514 . Thus, x œ

5 È14

,yœ

10 È14

,zœ

15 È14

or x œ
È514 , y œ
È1014 , z œ
È1514 . Therefore f Š È514 ß È1014 ß È1514 ‹

œ 5È14 is the maximum value and f Š
È514 ß
È1014 ,
È1514 ‹ œ
5È14 is the minimum value. 25. f(xß yß z) œ x#  y#  z# and g(xß yß z) œ x  y  z
9 œ 0 Ê ™ f œ 2xi  2yj  2zk and ™ g œ i  j  k so that ™ f œ - ™ g Ê 2xi  2yj  2zk œ -(i  j  k) Ê 2x œ -, 2y œ -, and 2z œ - Ê x œ y œ z Ê x  x  x
9 œ 0 Ê x œ 3, y œ 3, and z œ 3. 26. f(xß yß z) œ xyz and g(xß yß z) œ x  y  z#
16 œ 0 Ê ™ f œ yzi  xzj  xyk and ™ g œ i  j  2zk so that ™ f œ - ™ g Ê yzi  xzj  xyk œ -(i  j  2zk) Ê yz œ -, xz œ -, and xy œ 2z- Ê yz œ xz Ê z œ 0 or y œ x. But z  0 so that y œ x Ê x# œ 2z- and xz œ -. Then x# œ 2z(xz) Ê x œ 0 or x œ 2z# . But x  0 so that 32 x œ 2z# Ê y œ 2z# Ê 2z#  2z#  z# œ 16 Ê z œ „ È45 . We use z œ È45 since z  0. Then x œ 32 5 and y œ 5 32 4 which yields f Š 32 5 ß 5 ß È5 ‹ œ

4096 25È5

.

27. V œ xyz and g(xß yß z) œ x#  y#  z#
1 œ 0 Ê ™ V œ yzi  xzj  xyk and ™ g œ 2xi  2yj  2zk so that ™ V œ - ™ g Ê yz œ -x, xz œ -y, and xy œ -z Ê xyz œ -x# and xyz œ -y# Ê y œ „ x Ê z œ „ x Ê x#  x#  x# œ 1 Ê x œ È"3 since x  0 Ê the dimensions of the box are È13 by È13 by È13 for maximum volume. (Note that there is no minimum volume since the box could be made arbitrarily thin.) 28. V œ xyz with xß yß z all positive and

x a



y b



z c

œ 1; thus V œ xyz and g(xß yß z) œ bcx  acy  abz
abc œ 0

Ê ™ V œ yzi  xzj  xyk and ™ g œ bci  acj  abk so that ™ V œ - ™ g Ê yz œ -bc, xz œ -ac, and xy œ -ab Ê xyz œ -bcx, xyz œ -acy, and xyz œ -abz Ê - Á 0. Also, -bcx œ -acy œ -abz Ê bx œ ay, cy œ bz, and a cx œ az Ê y œ ba x and z œ ac x. Then xa  by  zc œ 1 Ê xa  b" ˆ ba x‰  "c ˆ ca x‰ œ 1 Ê 3x a œ 1 Ê xœ 3 Ê y œ ˆ ba ‰ ˆ 3a ‰ œ b3 and z œ ˆ ca ‰ ˆ 3a ‰ œ 3c Ê V œ xyz œ ˆ 3a ‰ ˆ b3 ‰ ˆ 3c ‰ œ abc 27 is the maximum volume. (Note that there is no minimum volume since the box could be made arbitrarily thin.) 29. ™ T œ 16xi  4zj  (4y
16)k and ™ g œ 8xi  2yj  8zk so that ™ T œ - ™ g Ê 16xi  4zj  (4y
16)k œ -(8xi  2yj  8zk) Ê 16x œ 8x-, 4z œ 2y-, and 4y
16 œ 8z- Ê - œ 2 or x œ 0. CASE 1: - œ 2 Ê 4z œ 2y(2) Ê z œ y. Then 4z
16 œ 16z Ê z œ
43 Ê y œ
43 . Then #

#

4x#  ˆ
43 ‰  4 ˆ
43 ‰ œ 16 Ê x œ „ 43 . CASE 2: x œ 0 Ê - œ

2z y

# # # # # Ê 4y
16 œ 8z Š 2z y ‹ Ê y
4y œ 4z Ê 4(0)  y  ay
4yb
16 œ 0

Ê y#
2y
8 œ 0 Ê (y
4)(y  2) œ 0 Ê y œ 4 or y œ
2. Now y œ 4 Ê 4z# œ 4#
4(4) Ê z œ 0 and y œ
2 Ê 4z# œ (
2)#
4(
2) Ê z œ „ È3. °

°

The temperatures are T ˆ „ 43 ß
43 ß
43 ‰ œ 642 23 , T(0ß 4ß 0) œ 600°, T Š0ß
2ß È3‹ œ Š600
24È3‹ , and °

T Š0ß
2ß
È3‹ œ Š600  24È3‹ ¸ 641.6°. Therefore ˆ „ 43 ß
43 ß
43 ‰ are the hottest points on the space probe.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

854

Chapter 14 Partial Derivatives

30. ™ T œ 400yz# i  400xz# j  800xyzk and ™ g œ 2xi  2yj  2zk so that ™ T œ - ™ g Ê 400yz# i  400xz# j  800xyzk œ -(2xi  2yj  2zk) Ê 400yz# œ 2x-, 400xz# œ 2y-, and 800xyz œ 2z-. Solving this system yields the points a!ß „ 1ß 0b , a „ 1ß 0ß 0b , and Š „ "# ß „ "# ß „ temperatures are T a!ß „ 1ß 0b œ 0, T a „ 1ß 0ß 0b œ 0, and T Š „ "# ß „ "# ß „ È2 # ‹

maximum temperature at Š "# ß "# ß „ Š "# ß
"# ß „

È2 # ‹

and Š
#" ß #" ß „

and Š
"# ß
"# ß „

È2 # ‹;

È2 # ‹

È2 # ‹.

The corresponding

œ „ 50. Therefore 50 is the


50 is the minimum temperature at

È2 # ‹.

31. ™ U œ (y  2)i  xj and ™ g œ 2i  j so that ™ U œ - ™ g Ê (y  2)i  xj œ -(2i  j) Ê y  # œ 2- and x œ - Ê y  2 œ 2x Ê y œ 2x
2 Ê 2x  (2x
2) œ 30 Ê x œ 8 and y œ 14. Therefore U(8ß 14) œ $128 is the maximum value of U under the constraint. 32. ™ M œ (6  z)i
2yj  xk and ™ g œ 2xi  2yj  2zk so that ™ M œ - ™ g Ê (6  z)i
2yj  xk œ -(2xi  2yj  2zk) Ê 6  z œ 2x-,
2y œ 2y-, x œ 2z- Ê - œ
1 or y œ 0. CASE 1: - œ
1 Ê 6  z œ
2x and x œ
2z Ê 6  z œ
2(
2z) Ê z œ 2 and x œ
4. Then (
4)#  y#  2#
36 œ 0 Ê y œ „ 4. x x ‰ CASE 2: y œ 0, 6  z œ 2x-, and x œ 2z- Ê - œ 2z Ê 6  z œ 2x ˆ 2z Ê 6z  z# œ x#

Ê a6z  z# b  0#  z# œ 36 Ê z œ
6 or z œ 3. Now z œ
6 Ê x# œ 0 Ê x œ 0; z œ 3 Ê x# œ 27 Ê x œ „ 3È3.

Therefore we have the points Š „ 3È3ß 0ß 3‹ , (0ß 0ß
6), and a
4ß „ 4ß 2b . Then M Š3È3ß 0ß 3‹ œ 27È3  60 ¸ 106.8, M Š
3È3ß 0ß 3‹ œ 60
27È3 ¸ 13.2, M(0ß 0ß
6) œ 60, and M(
4ß 4ß 2) œ 12 œ M(
4ß
4ß 2). Therefore, the weakest field is at a
4ß „ 4ß 2b . 33. Let g" (xß yß z) œ 2x
y œ 0 and g# (xß yß z) œ y  z œ 0 Ê ™ g" œ 2i
j , ™ g# œ j  k , and ™ f œ 2xi  2j
2zk so that ™ f œ - ™ g"  . ™ g# Ê 2xi  2j
2zk œ -(2i
j)  .(j  k) Ê 2xi  2j
2zk œ 2-i  (.
-)j  .k Ê 2x œ 2-, 2 œ .
-, and
2z œ . Ê x œ -. Then 2 œ
2z
x Ê x œ
2z
2 so that 2x
y œ 0 Ê 2(
2z
2)
y œ 0 Ê
4z
4
y œ 0. This equation coupled with y  z œ 0 implies z œ
43 and y œ 43 . Then xœ

2 3

#

#

so that ˆ 23 ß 43 ß
43 ‰ is the point that gives the maximum value f ˆ 23 ß 43 ß
43 ‰ œ ˆ 23 ‰  2 ˆ 43 ‰
ˆ
43 ‰ œ

4 3

.

34. Let g" (xß yß z) œ x  2y  3z
6 œ 0 and g# (xß yß z) œ x  3y  9z
9 œ 0 Ê ™ g" œ i  2j  3k , ™ g# œ i  3j  9k , and ™ f œ 2xi  2yj  2zk so that ™ f œ - ™ g"  . ™ g# Ê 2xi  2yj  2zk œ -(i  2j  3k)  .(i  3j  9k) Ê 2x œ -  ., 2y œ 2-  3., and 2z œ 3-  9.. Then 0 œ x  2y  3z
6 ‰ œ "# (-  .)  (2-  3.)  ˆ 9# -  27 # .
6 Ê 7-  17. œ 6; 0 œ x  3y  9z
9 " 9 27 81 Ê # (-  .)  ˆ3-  # .‰  ˆ # -  # .‰
9 Ê 34-  91. œ 18. Solving these two equations for - and . gives -
. 2-
3. 3-
9. 78 81 9 - œ 240 œ 123 œ 59 . The minimum value is 59 and . œ
59 Ê x œ # œ 59 , y œ # 59 , and z œ # 21,771 81 123 9 369 f ˆ 59 ß 59 ß 59 ‰ œ 59# œ 59 . (Note that there is no maximum value of f subject to the constraints because

at least one of the variables x, y, or z can be made arbitrary and assume a value as large as we please.) 35. Let f(xß yß z) œ x#  y#  z# be the square of the distance from the origin. We want to minimize f(xß yß z) subject to the constraints g" (xß yß z) œ y  2z
12 œ 0 and g# (xß yß z) œ x  y
6 œ 0. Thus ™ f œ 2xi  2yj  2zk , ™ g" œ j  2k, and ™ g# œ i  j so that ™ f œ - ™ g"  . ™ g# Ê 2x œ ., 2y œ -  ., and 2z œ 2-. Then 0 œ y  2z
12 œ ˆ -#  .# ‰  2-
12 Ê #5 -  "# . œ 12 Ê 5-  . œ 24; 0 œ x  y
6 œ .#  ˆ -#  .# ‰
6 Ê "# -  . œ 6 Ê -  #. œ 12. Solving these two equations for - and . gives - œ 4 and . œ 4 Ê x œ

. #

œ 2, y œ

-
. #

œ 4, and

z œ - œ 4. The point (2ß 4ß 4) on the line of intersection is closest to the origin. (There is no maximum distance from the origin since points on the line can be arbitrarily far away.)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.8 Lagrange Multipliers 36. The maximum value is f ˆ 23 ß 43 ß
43 ‰ œ

4 3

855

from Exercise 33 above.

37. Let g" (xß yß z) œ z
1 œ 0 and g# (xß yß z) œ x#  y#  z#
10 œ 0 Ê ™ g" œ k , ™ g# œ 2xi  2yj  2zk , and ™ f œ 2xyzi  x# zj  x# yk so that ™ f œ - ™ g"  . ™ g# Ê 2xyzi  x# zj  x# yk œ -(k)  .(2xi  2yj  2zk) Ê 2xyz œ 2x., x# z œ 2y., and x# y œ 2z.  - Ê xyz œ x. Ê x œ 0 or yz œ . Ê . œ y since z œ 1. CASE 1: x œ 0 and z œ 1 Ê y#
9 œ 0 (from g# ) Ê y œ „ 3 yielding the points a0ß „ 3ß 1b. CASE 2: . œ y Ê x# z œ 2y# Ê x# œ 2y# (since z œ 1) Ê 2y#  y#  1
10 œ 0 (from g# ) Ê 3y#
9 œ 0 #

Ê y œ „ È3 Ê x# œ 2 Š „ È3‹ Ê x œ „ È6 yielding the points Š „ È6ß „ È3ß "‹ . Now f a!ß „ 3ß 1b œ 1 and f Š „ È6ß „ È3ß "‹ œ 6 Š „ È3‹  1 œ 1 „ 6È3. Therefore the maximum of f is 1  6È3 at Š „ È6ß È3ß 1‹, and the minimum of f is 1
6È3 at Š „ È6ß
È3ß "‹ . 38. (a) Let g" (xß yß z) œ x  y  z
40 œ 0 and g# (xß yß z) œ x  y
z œ 0 Ê ™ g" œ i  j  k , ™ g# œ i  j
k , and ™ w œ yzi  xzj  xyk so that ™ w œ - ™ g"  . ™ g# Ê yzi  xzj  xyk œ -(i  j  k)  .(i  j
k) Ê yz œ -  ., xz œ -  ., and xy œ -
. Ê yz œ xz Ê z œ 0 or y œ x. CASE 1: z œ 0 Ê x  y œ 40 and x  y œ 0 Ê no solution. CASE 2: x œ y Ê 2x  z
40 œ 0 and 2x
z œ 0 Ê z œ 20 Ê x œ 10 and y œ 10 Ê w œ (10)(10)(20) œ 2000 â â âi j k â â â " â œ
2i  2j is parallel to the line of intersection Ê the line is x œ
2t  10, (b) n œ â " " â â â " "
" â y œ 2t  10, z œ 20. Since z œ 20, we see that w œ xyz œ (
2t  10)(2t  10)(20) œ a
4t#  100b (20) which has its maximum when t œ 0 Ê x œ 10, y œ 10, and z œ 20. 39. Let g" (Bß yß z) œ y
x œ 0 and g# (xß yß z) œ x#  y#  z#
4 œ 0. Then ™ f œ yi  xj  2zk , ™ g" œ
i  j , and ™ g# œ 2xi  2yj  2zk so that ™ f œ - ™ g"  . ™ g# Ê yi  xj  2zk œ -(
i  j)  .(2xi  2yj  2zk) Ê y œ
-  2x., x œ -  2y., and 2z œ 2z. Ê z œ 0 or . œ 1. CASE 1: z œ 0 Ê x#  y#
4 œ 0 Ê 2x#
4 œ 0 (since x œ y) Ê x œ „ È2 and y œ „ È2 yielding the points Š „ È2ß „ È2ß !‹ . CASE 2: . œ 1 Ê y œ
-  2x and x œ -  2y Ê x  y œ 2(x  y) Ê 2x œ 2(2x) since x œ y Ê x œ 0 Ê y œ 0 Ê z#
4 œ 0 Ê z œ „ 2 yielding the points a!ß !ß „ 2b . Now, f a!ß !ß „ 2b œ 4 and f Š „ È2ß „ È2ß !‹ œ 2. Therefore the maximum value of f is 4 at a!ß !ß „ 2b and the minimum value of f is 2 at Š „ È2ß „ È2ß !‹ . 40. Let f(xß yß z) œ x#  y#  z# be the square of the distance from the origin. We want to minimize f(xß yß z) subject to the constraints g" (xß yß z) œ 2y  4z
5 œ 0 and g# (xß yß z) œ 4x#  4y#
z# œ 0. Thus ™ f œ 2xi  2yj  2zk , ™ g" œ 2j  4k , and ™ g# œ 8xi  8yj
2zk so that ™ f œ - ™ g"  . ™ g# Ê 2xi  2yj  2zk œ -(2j  4k)  .(8xi  8yj
2zk) Ê 2x œ 8x., 2y œ 2-  8y., and 2z œ 4-
2z. Ê x œ 0 or . œ "4 . CASE 1: x œ 0 Ê 4(0)#  4y#
z# œ 0 Ê z œ „ 2y Ê 2y  4(2y)
5 œ 0 Ê y œ Ê y œ
56 yielding the points ˆ!ß "# ß "‰ and ˆ!ß
56 ß 53 ‰ . CASE 2: . œ

" 4

" #

, or 2y  4(
2y)
5 œ 0

Ê y œ -  y Ê - œ 0 Ê 2z œ 4(0)
2z ˆ 4" ‰ Ê z œ 0 Ê 2y  4(0) œ 5 Ê y œ # 4 ˆ #5 ‰

(0)# œ 4x#  Ê no solution. " Then f ˆ!ß "# ß 1‰ œ 54 and f ˆ!ß
56 ß 35 ‰ œ 25 ˆ 36  "9 ‰ œ

125 36

Ê the point ˆ!ß "# ß 1‰ is closest to the origin.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

5 #

and

856

Chapter 14 Partial Derivatives

41. ™ f œ i  j and ™ g œ yi  xj so that ™ f œ - ™ g Ê i  j œ -(yi  xj) Ê 1 œ y- and 1 œ x- Ê y œ x Ê y# œ 16 Ê y œ „ 4 Ê (4ß 4) and (
%ß
4) are candidates for the location of extreme values. But as x Ä _, y Ä _ and f(xß y) Ä _; as x Ä
_, y Ä 0 and f(xß y) Ä
_. Therefore no maximum or minimum value exists subject to the constraint. 4

42. Let f(Aß Bß C) œ ! (Axk  Byk  C
zk )# œ C#  (B  C
1)#  (A  B  C
1)#  (A  C  1)# . We want k œ1

to minimize f. Then fA (Aß Bß C) œ 4A  2B  4C, fB (Aß Bß C) œ 2A  4B  4C
4, and fC (Aß Bß C) œ 4A  4B  8C
2. Set each partial derivative equal to 0 and solve the system to get A œ
"# , B œ 3# , and C œ
"4 or the critical point of f is ˆ
#" ß 3# ß
"4 ‰ . 43. (a) Maximize f(aß bß c) œ a# b# c# subject to a#  b#  c# œ r# . Thus ™ f œ 2ab# c# i  2a# bc# j  2a# b# ck and ™ g œ 2ai  2bj  2ck so that ™ f œ - ™ g Ê 2ab# c# œ 2a-, 2a# bc# œ 2b-, and 2a# b# c œ 2cÊ 2a# b# c# œ 2a# - œ 2b# - œ 2c# - Ê - œ 0 or a# œ b# œ c# . CASE 1: - œ 0 Ê a# b# c# œ 0. #

$

CASE 2: a# œ b# œ c# Ê f(aß bß c) œ a# a# a# and 3a# œ r# Ê f(aß bß c) œ Š r3 ‹ is the maximum value. (b) The point ŠÈaß Èbß Èc‹ is on the sphere if a  b  c œ r# . Moreover, by part (a), abc œ f ŠÈaß Èbß Èc‹ #

$

Ÿ Š r3 ‹ Ê (abc)"Î$ Ÿ

r# 3

œ

a
b
c 3

, as claimed.

n

44. Let f(x" ß x# ß á ß xn ) œ ! ai xi œ a" x"  a# x#  á  an xn and g(x" ß x# ß á ß xn ) œ x"#  x##  á  xn#
1. Then we i œ1

want ™ f œ - ™ g Ê a" œ -(2x" ), a# œ -(2x# ), á , an œ -(2xn ), - Á 0 Ê xi œ n

n

iœ1

i œ1

"Î#

Ê 4-# œ ! a#i Ê 2- œ Œ! a#i 

n

n

i œ1

i œ1

ai 2-

Ê f(x" ß x# ß á ß xn ) œ ! ai xi œ ! ai ˆ #a-i ‰ œ

Ê " #-

a#" 4- #



a## 4- #

an# 4- # "Î#

á 

n

n

i œ1

i œ1

! a#i œ Œ! a#i 

the maximum value. 45-50. Example CAS commands: Maple: f := (x,y,z) -> x*y+y*z; g1 := (x,y,z) -> x^2+y^2-2; g2 := (x,y,z) -> x^2+z^2-2; h := unapply( f(x,y,z)-lambda[1]*g1(x,y,z)-lambda[2]*g2(x,y,z), (x,y,z,lambda[1],lambda[2]) ); hx := diff( h(x,y,z,lambda[1],lambda[2]), x ); hy := diff( h(x,y,z,lambda[1],lambda[2]), y ); hz := diff( h(x,y,z,lambda[1],lambda[2]), z ); hl1 := diff( h(x,y,z,lambda[1],lambda[2]), lambda[1] ); hl2 := diff( h(x,y,z,lambda[1],lambda[2]), lambda[2] ); sys := { hx=0, hy=0, hz=0, hl1=0, hl2=0 }; q1 := solve( sys, {x,y,z,lambda[1],lambda[2]} ); q2 := map(allvalues,{q1}); for p in q2 do eval( [x,y,z,f(x,y,z)], p ); ``=evalf(eval( [x,y,z,f(x,y,z)], p )); end do;

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

# (a) #(b)

# (c) # (d)

is

œ1

Section 14.9 Taylor's Formula for Two Variables Mathematica: (assigned functions will vary) Clear[x, y, z, lambda1, lambda2] f[x_,y_,z_]:= x y  y z g1[x_,y_,z_]:= x2  y2
2 g2[x_,y_,z_]:= x2  z2
2 h = f[x, y, z]
lambda1 g1[x, y, z]
lambda2 g2[x, y, z]; hx= D[h, x]; hy= D[h, y]; hz= D[h,z]; hL1=D[h, lambda1]; hL2= D[h, lambda2]; critical=Solve[{hx==0, hy==0, hz==0, hL1==0, hL2==0, g1[x,y,z]==0, g2[x,y,z]==0}, {x, y, z, lambda1, lambda2}]//N {{x, y, z}, f[x, y, z]}/.critical 14.9 TAYLOR'S FORMULA FOR TWO VARIABLES 1. f(xß y) œ xey Ê fx œ ey , fy œ xey , fxx œ 0, fxy œ ey , fyy œ xey Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 0  x † 1  y † 0  "# ax# † 0  2xy † 1  y# † 0b œ x  xy quadratic approximation;

fxxx œ 0, fxxy œ 0, fxyy œ ey , fyyy œ xey Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (!ß !)  3x# yfxxy (0ß 0)  3xy# fxyy (!ß !)  y$ fyyy (0ß 0)d

œ x  xy  "6 ax$ † 0  3x# y † 0  3xy# † 1  y$ † 0b œ x  xy  "# xy# , cubic approximation

2. f(xß y) œ ex cos y Ê fx œ ex cos y, fy œ
ex sin y, fxx œ ex cos y, fxy œ
ex sin y, fyy œ
ex cos y Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (!ß 0)  "# cx# fxx (!ß !)  2xyfxy (!ß !)  y# fyy (0ß 0)d

œ 1  x † 1  y † 0  "# cx# † 1  2xy † 0  y# † (
1)d œ 1  x  "# ax#
y# b , quadratic approximation;

fxxx œ ex cos y, fxxy œ
ex sin y, fxyy œ
ex cos y, fyyy œ ex sin y Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d œ 1  x  "# ax#
y# b  6" cx$ † 1  3x# y † 0  3xy# † (
1)  y$ † 0d œ 1  x  "# ax#
y# b  6" ax$
3xy# b , cubic approximation

3. f(xß y) œ y sin x Ê fx œ y cos x, fy œ sin x, fxx œ
y sin x, fxy œ cos x, fyy œ 0 Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (!ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 0  x † 0  y † 0  "# ax# † 0  2xy † 1  y# † 0b œ xy, quadratic approximation;

fxxx œ
y cos x, fxxy œ
sin x, fxyy œ 0, fyyy œ 0 Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d œ xy  "6 ax$ † 0  3x# y † 0  3xy# † 0  y$ † 0b œ xy, cubic approximation

4. f(xß y) œ sin x cos y Ê fx œ cos x cos y, fy œ
sin x sin y, fxx œ
sin x cos y, fxy œ
cos x sin y, fyy œ
sin x cos y Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 0  x † 1  y † 0  "# ax# † 0  2xy † 0  y# † 0b œ x, quadratic approximation;

fxxx œ
cos x cos y, fxxy œ sin x sin y, fxyy œ
cos x cos y, fyyy œ sin x sin y Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d

œ x  "6 cx$ † (
1)  3x# y † 0  3xy# † (
1)  y$ † 0d œ x
6" ax$  3xy# b, cubic approximation

5. f(xß y) œ ex ln (1  y) Ê fx œ ex ln (1  y), fy œ

ex 1
y

, fxx œ ex ln (1  y), fxy œ

ex 1
y

Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d

œ 0  x † 0  y † 1  "# cx# † 0  2xy † 1  y# † (
1)d œ y  "# a2xy
y# b , quadratic approximation; fxxx œ ex ln (1  y), fxxy œ

ex 1
y

x

, fxyy œ
(1
e y)# , fyyy œ

x

, fyy œ
(1
e y)#

2ex (1
y)$

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

857

858

Chapter 14 Partial Derivatives Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d

œ y  "2 a2xy
y# b  6" cx$ † 0  3x# y † 1  3xy# † (
1)  y$ † 2d

œ y  "# a2xy
y# b  6" a3x# y
3xy#  2y$ b , cubic approximation 4 2 (2x
y
1)# , fxy œ (2x
y
1)# , " # # fyy œ (2x
" y
1)# Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  # cx fxx (0ß 0)  2xyfxy (0ß 0)  y fyy (0ß 0)d œ 0  x † 2  y † 1  "# cx# † (
4)  2xy † (
2)  y# † (
1)d œ 2x  y  "# a
4x#
4xy
y# b œ (2x  y)
"# (2x  y)# , quadratic approximation; fxxx œ (2x
16y
1)$ , fxxy œ (2x
8y
1)$ , fxyy œ (2x
4y
1)$ , fyyy œ (2x
2y
1)$ Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d œ (2x  y)
"# (2x  y)#  6" ax$ † 16  3x# y † 8  3xy# † 4  y$ † 2b œ (2x  y)
"# (2x  y)#  3" a8x$  12x# y  6xy#  y# b œ (2x  y)
"# (2x  y)#  3" (2x  y)$ , cubic approximation

6. f(xß y) œ ln (2x  y  1) Ê fx œ

2 2x
y
1

, fy œ

" #x
y
1

, fxx œ

7. f(xß y) œ sin ax#  y# b Ê fx œ 2x cos ax#  y# b , fy œ 2y cos ax#  y# b , fxx œ 2 cos ax#  y# b
4x# sin ax#  y# b , fxy œ
4xy sin ax#  y# b , fyy œ 2 cos ax#  y# b
4y# sin ax#  y# b Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 0  x † 0  y † 0  "# ax# † 2  2xy † 0  y# † 2b œ x#  y# , quadratic approximation;

fxxx œ
12x sin ax#  y# b
8x$ cos ax#  y# b , fxxy œ
4y sin ax#  y# b
8x# y cos ax#  y# b , fxyy œ
4x sin ax#  y# b
8xy# cos ax#  y# b , fyyy œ
12y sin ax#  y# b
8y$ cos ax#  y# b Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d œ x#  y#  "6 ax$ † 0  3x# y † 0  3xy# † 0  y$ † 0b œ x#  y# , cubic approximation

8. f(xß y) œ cos ax#  y# b Ê fx œ
2x sin ax#  y# b , fy œ
2y sin ax#  y# b , fxx œ
2 sin ax#  y# b
4x# cos ax#  y# b , fxy œ
4xy cos ax#  y# b , fyy œ
2 sin ax#  y# b
4y# cos ax#  y# b Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 1  x † 0  y † 0  "# cx# † 0  2xy † 0  y# † 0d œ 1, quadratic approximation;

fxxx œ
12x cos ax#  y# b  8x$ sin ax#  y# b , fxxy œ
4y cos ax#  y# b  8x# y sin ax#  y# b , fxyy œ
4x cos ax#  y# b  8xy# sin ax#  y# b , fyyy œ
12y cos ax#  y# b  8y$ sin ax#  y# b Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0)  y$ fyyy (0ß 0)d œ 1  "6 ax$ † 0  3x# y † 0  3xy# † 0  y$ † 0b œ 1, cubic approximation

9. f(xß y) œ

" 1xy

Ê fx œ

" (1  x  y)#

œ fy , fxx œ

Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0) 

2 (1  x  y)$ " # # cx fxx (0ß 0)

œ fxy œ fyy  2xyfxy (0ß 0)  y# fyy (0ß 0)d

œ 1  x † 1  y † 1  "# ax# † 2  2xy † 2  y# † 2b œ 1  (x  y)  ax#  2xy  y# b

œ 1  (x  y)  (x  y)# , quadratic approximation; fxxx œ Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0) 

6 œ fxxy œ fxyy œ fyyy (1  x  y)% # 3xy fxyy (0ß 0)  y$ fyyy (0ß 0)d $

œ 1  (x  y)  (x  y)#  "6 ax$ † 6  3x# y † 6  3xy# † 6  y † 6b

œ 1  (x  y)  (x  y)#  ax$  3x# y  3xy#  y$ b œ 1  (x  y)  (x  y)#  (x  y)$ , cubic approximation 10. f(xß y) œ fxy œ

" 1  x  y
xy

1 ("  x  y
xy)#

Ê fx œ

, fyy œ

1y (1  x  y
xy)#

, fy œ

1x ("  x  y
xy)#

, fxx œ

2(1  y)# (1  x  y
xy)$

,

2("  x)# (1  x  y
xy)$

Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d

œ 1  x † 1  y † 1  "# ax# † 2  2xy † 1  y# † 2b œ 1  x  y  x#  xy  y# , quadratic approximation;

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.10 Partial Derivatives with Constrained Variables fxxx œ

6(1  y)$ (1  x  y
xy)%

, fxxy œ

[4(1  x  y
xy)
6(1  y)(1  x)](1  y) (1  x  y
xy)%

,

$

[4(1  x  y
xy)
6(1  x)(1  y)](1  x)  x) , fyyy œ (1 6(1 (1  x  y
xy)% x  y
xy)% Ê f(xß y) ¸ quadratic  "6 cx$ fxxx (0ß 0)  3x# yfxxy (!ß 0)  3xy# fxyy (0ß 0) œ 1  x  y  x#  xy  y#  "6 ax$ † 6  3x# y † 2  3xy# † 2  y$ † 6b # # $ # # $

fxyy œ

 y$ fyyy (0ß 0)d

œ 1  x  y  x  xy  y  x  x y  xy  y , cubic approximation 11. f(xß y) œ cos x cos y Ê fx œ
sin x cos y, fy œ
cos x sin y, fxx œ
cos x cos y, fxy œ sin x sin y, fyy œ
cos x cos y Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d œ 1  x † 0  y † 0  "# cx# † (
1)  2xy † 0  y# † (
1)d œ 1


x# #




y# #

, quadratic approximation. Since all partial

derivatives of f are products of sines and cosines, the absolute value of these derivatives is less than or equal to 1 Ê E(xß y) Ÿ "6 c(0.1)$  3(0.1)$  3(0.1)$  0.1)$ d Ÿ 0.00134. 12. f(xß y) œ ex sin y Ê fx œ ex sin y, fy œ ex cos y, fxx œ ex sin y, fxy œ ex cos y, fyy œ
ex sin y Ê f(xß y) ¸ f(0ß 0)  xfx (0ß 0)  yfy (0ß 0)  "# cx# fxx (0ß 0)  2xyfxy (0ß 0)  y# fyy (0ß 0)d

œ 0  x † 0  y † 1  "# ax# † 0  2xy † 1  y# † 0b œ y  xy , quadratic approximation. Now, fxxx œ ex sin y,

fxxy œ ex cos y, fxyy œ
ex sin y, and fyyy œ
ex cos y. Since kxk Ÿ 0.1, kex sin yk Ÿ ke0Þ1 sin 0.1k ¸ 0.11 and kex cos yk Ÿ ke0Þ1 cos 0.1k ¸ 1.11. Therefore, E(xß y) Ÿ "6 c(0.11)(0.1)$  3(1.11)(0.1)$  3(0.11)(0.1)$  (1.11)(0.1)$ d Ÿ 0.000814. 14.10 PARTIAL DERIVATIVES WITH CONSTRAINED VARIABLES 1. w œ x#  y#  z# and z œ x#  y# : Î x œ x(yß z) Ñ y yœy Ä w Ê Š ``wy ‹ œ (a) Œ  Ä z z Ï zœz Ò œ 2x `` xy  2y Ê 0 œ 2x `` xy  2y Ê

`x `y

œ

" #y

`x `z

œ

1 2x

`w `x `x `z

`w `x `x `z

œ 2x `` yx  2y `` yy



`w `y `y `z



`w `z `x `z `z ; `z

œ 0 and

`z `z

œ 2x `` xz  2y `` yz



`w `y `y `z



`w `z `y `z `z ; `z

œ 0 and

`z `z

œ 2x `` xz  2y `` yz

Ê ˆ ``wz ‰y œ (2x) ˆ #"x ‰  (2y)(0)  (2z)(1) œ 1  2z

2. w œ x#  y
z  sin t and x  y œ t: Î xœx Ñ ÎxÑ Ð yœy Ó y Ä Ð Ä w Ê Š ``wy ‹ œ (a) Ó zœz xz ÏzÒ Ït œ x  yÒ ß

`t `y

`z `y

œ 0 and

" Ê ˆ ``wz ‰x œ (2x)(0)  (2y) Š 2y ‹  (2z)(1) œ 1  2z

Î x œ x(yß z) Ñ y yœy Ä w Ê ˆ ``wz ‰y œ (c) Œ  Ä z Ï zœz Ò Ê 1 œ 2x `` xz Ê

`w `z `z `z `y ; `y



z

xœx Ñ x y œ y(xß z) Ä w Ê ˆ ``wz ‰x œ (b) Œ  Ä z Ï zœz Ò `y `z

`w `y `y `y



œ
xy Ê Š ``wy ‹ œ (2x) ˆ
xy ‰  (2y)(1)  (2z)(0) œ
2y  2y œ 0

Î

Ê 1 œ 2y `` yz Ê

`w `x `x `y

`w `x `x `y



`w `y `y `y



`w `z `z `y



`w `t `x `t `y ; `y

œ 0,

`z `y

œ 0, and

œ 1 Ê Š ``wy ‹ œ (2x)(0)  (1)(1)  (
1)(0)  (cos t)(1) œ 1  cos t œ 1  cos (x  y) xßt

Îx œ t
yÑ ÎyÑ Ð yœy Ó z Ä Ð Ä w Ê Š ``wy ‹ œ (b) Ó z z œ zt ÏtÒ Ï tœt Ò ß

Ê

`x `y

œ

`t `y




`y `y

`w `x `x `y



`w `y `y `y



`w `z `z `y



`w `t `z `t `y ; `y

œ 0 and

`t `y

œ0

œ
1 Ê Š ``wy ‹ œ (2x)(
1)  (1)(1)  (
1)(0)  (cos t)(0) œ 1
2at
yb œ 1  2y
2t zßt

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859

860

Chapter 14 Partial Derivatives

Î xœx Ñ ÎxÑ Ð yœy Ó y Ä Ð Ä w Ê ˆ ``wz ‰x y œ (c) Ó œ z z ÏzÒ Ït œ x  yÒ ß

`w `x `x `z



`w `y `y `z



`w `z `z `z



`w `t `x `t `z ; `z

œ 0 and

`y `z

œ0



`w `z `z `z



`w `t `y `t `z ; `z

œ 0 and

`t `z

œ0



`w `z `z `t



`w `t `x `t `t ; `t

œ 0 and

`z `t

œ0

`w `z `z `t



`w `t `y `t `t ; `t

œ 0 and

`z `t

œ0

Ê ˆ ``wz ‰x y œ (2x)(0)  (1)(0)  (
1)(1)  (cos t)(0) œ
1 ß

Îx œ t
yÑ ÎyÑ Ð yœy Ó z Ä Ð Ä w Ê ˆ ``wz ‰y t œ (d) Ó zœz ÏtÒ Ï tœt Ò ß

`w `x `x `z



`w `y `y `z

Ê ˆ ``wz ‰y t œ (2x)(0)  (1)(0)  (
1)(1)  (cos t)(0) œ
1 ß

Î xœx Ñ ÎxÑ Ð y œ t
xÓ z Ä Ð Ä w Ê ˆ ``wt ‰x z œ (e) Ó zœz ÏtÒ Ï tœt Ò ß

`w `x `x `t



`w `y `y `t

Ê ˆ ``wt ‰x z œ (2x)(0)  (1)(1)  (
1)(0)  (cos t)(1) œ 1  cos t ß

Îx œ t
yÑ ÎyÑ Ð yœy Ó z Ä Ð Ä w Ê ˆ ``wt ‰y z œ (f) Ó zœz ÏtÒ Ï tœt Ò ß

`w `x `x `t



`w `y `y `t



Ê ˆ ``wt ‰y z œ (2x)(1)  (1)(0)  (
1)(0)  (cos t)(1) œ cos t  2x œ cos t  2(t
y) ß

3. U œ f(Pß Vß T) and PV œ nRT Î PœP Ñ P VœV Ä U Ê ˆ ``UP ‰V œ (a) Œ  Ä V PV ÏT œ Ò

`U `P `P `P

`U `V `V `P



`U `T `T `P



œ

`U `P

V ‰ ‰ ˆ ` U ‰ ˆ nR  ˆ `` U V (0)  ` T

nR

œ

`U `P

V ‰  ˆ ``UT ‰ ˆ nR

nRT ÎP œ V Ñ V Ä U Ê ˆ ``UT ‰V œ (b) Œ  Ä VœV T Ï TœT Ò ‰ `U œ ˆ ``UP ‰ ˆ nR V  `T

`U `P `P `T

4. w œ x#  y#  z# and y sin z  z sin x œ 0 Î xœx Ñ x yœy Ä w Ê ˆ ``wx ‰y œ (a) Œ  Ä y Ï z œ z(xß y) Ò (y cos z) Ê

`z `x

 (sin x)

ˆ ``wx ‰ yk (0ß1ß1)

`z `x

 z cos x œ 0 Ê

`z `x

`x `z

`w `x `x `x

œ

`U `V `V `T





`w `y `y `x

z cos x y cos z
sin x . #

`U `T `T `T

‰ ˆ `U ‰ œ ˆ ``UP ‰ ˆ nR V  ` V (0) 

`w `z `y `z `x ; `x



œ 0 and œ

1 1

œ1

œ (2x)

`x `z

 (2y)(0)  (2z)(1)

At (0ß 1ß 1),

`z `x

`U `T

œ (2x)(1)  (2y)(0)  (2z)(1)k Ð0ß1ß1Ñ œ 21

Î x œ x(yß z) Ñ y yœy Ä w Ê ˆ ``wz ‰y œ (b) Œ  Ä z Ï zœz Ò œ (2x)



`y `z  y x) `` xz œ

 2z. Now (sin z)

Ê y cos z  sin x  (z cos

`w `x `x `z



`w `y `y `z

cos z  sin x  (z cos x) 0 Ê

`x `z

œ

y cos z  sin x . z cos x

`w `z `z `z



`x `z

`y `z œ 0 (!ß "ß 1), `` xz œ (11)(1)0

œ 0 and

At

œ

" 1

Ê ˆ ``wz ‰Ck (!,"ß1Ñ œ 2(0) ˆ 1" ‰  21 œ 21 5. w œ x# y#  yz
z$ and x#  y#  z# œ 6 Î xœx Ñ x yœy (a) Œ  Ä Ä w Ê Š ``wy ‹ œ y x Ï z œ z(xß y) Ò

`w `x `x `y



`w `y `y `y



`w `z `z `y

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 14.10 Partial Derivatives with Constrained Variables œ a2xy# b (0)  a2x# y  zb (1)  ay
3z# b `x `y

`z `y

œ 0 Ê 2y  (2z)

`z `y

œ0 Ê

`z `y

`z `y

`x `y

`w `x `x `y

`x `y

 a2x# y  zb (1)  ay
3z# b (0) œ a2x# yb

œ 0 Ê (2x)

`x `y

`x `y

 2y œ 0 Ê

`w `y `y `y





`x `y

Now (2x) `z `y

œ
yz . At (wß xß yß z) œ (4ß 2ß 1ß
1),

œ c(2)(2)# (1)  (
1)d  c1
3(
1)# d (1) œ 5 Î x œ x(yß z) Ñ y yœy (b) Œ  Ä Ä w Ê Š ``wy ‹ œ z z Ï zœz Ò œ a2xy# b

`z `y .

œ 2x# y  z  ay
3z# b

 2y  (2z)

`z `y

œ 0 and

œ
"1 œ 1 Ê Š ``wy ‹ ¹

x (4ß2ß1ßc1)

`w `z `z `y

 2x# y  z. Now (2x)

œ
yx . At (wß xß yß z) œ (4ß 2ß 1ß
1),

`x `y

`x `y

 2y  (2z)

œ
"2 Ê Š ``wy ‹ ¹ z

œ (2)(2)(1) ˆ
"# ‰  (2)(2)# (1)  (
1) œ 5

`z `y

œ 0 and

(4ß2ß1ßc1)

#

6. y œ uv Ê 1 œ v œv

`u `y

 u Š
uv

`u `y

u

`u `y ‹

`v `y ;

œ Šv

#

x œ u#  v# and u v

#

‹

`u `y

Ê

`u `y

`x `y

œ

œ 0 Ê 0 œ 2u

`u `y

 2v

`v `y

At (uß v) œ ŠÈ2ß 1‹ ,

v v#  u# .

`v `y

Ê `u `y

œ ˆ
uv ‰ "

œ

#

1#  ŠÈ2‹

`u `y

Ê 1

œ
1

Ê Š `` uy ‹ œ
1 x

r x œ r cos ) 7. Œ  Ä Œ Ê ˆ ``xr ‰) œ cos ); x#  y# œ r# Ê 2x  2y ) y œ r sin )  Ê ``xr œ xr Ê ˆ ``xr ‰ œ È #x #

`y `x

œ 2r

`r `x

and

`y `x

8. If x, y, and z are independent, then ˆ ``wx ‰y z œ ß

`w `x `x `x



`w `y `y `x

`w `z `z `x





`w `t `t `x

œ (2x)(1)  (
2y)(0)  (4)(0)  (1) ˆ ``xt ‰ œ 2x  ``xt . Thus x  2z  t œ 25 Ê 1  0  Ê ˆ ``wx ‰ œ 2x
1. On the other hand, if x, y, and t are independent, then ˆ ``wx ‰ yßz

œ

Ê 1

`r `x

x
y

y

`w `x `x `x

œ 0 Ê 2x œ 2r

`t `x

œ0 Ê

`t `x

œ
1

yßt

 ``wy `` yx  ``wz `` xz  ``wt ``xt œ (2x)(1)  (
2y)(0)  4 `` xz  (1)(0) œ 2 `` xz  0 œ 0 Ê `` xz œ
#" Ê ˆ ``wx ‰yßt œ 2x  4 ˆ
#" ‰ œ 2x
2.

9. If x is a differentiable function of y and z, then f(xß yß z) œ 0 Ê

`f `x `x `x



2x  4

`f `y `y `x



`z `x .

`f `z `z `x

Thus, x  2z  t œ 25

œ0 Ê

`f `x



`f `y `y `x

œ0

Ê Š `` xy ‹ œ
`` f/f/`` yz . Similarly, if y is a differentiable function of x and z, Š `` yz ‹ œ
`` f/f/`` xz and if z is a z

x

differentiable function of x and y, ˆ `` xz ‰y œ
`` f/f/`` xy . Then Š `` xy ‹ Š `` yz ‹ ˆ `` xz ‰y z

œ Š


` f/` y ˆ ` f/` z ‰ ` f/` x ` f/` z ‹
` f/` x Š
` f/` y ‹

10. z œ z  f(u) and u œ xy Ê œ x ˆ1  y

df ‰ du


y ˆx

df ‰ du

`z `x

x

œ
1.

œ1

df ` u du ` x

œ1y

df du ;

also

`z `y

œ0

df ` u du ` y

œx

df du

so that x

`z `x


y

œ 0 and

`x `y

œ0 Ê

`g `y



`z `y

œx

11. If x and y are independent, then g(xß yß z) œ 0 Ê

`g `x `x `y



`g `y `y `y



`g `z `z `y

`y Ê Š `` yz ‹ œ
`` g/ g/` z , as claimed. x

12. Let x and y be independent. Then f(xß yß zß w) œ 0, g(xß yß zß w) œ 0 and Ê `` xf `` xx  `` yf `` yx  `` zf `` xz  ``wf ``wx œ `` xf  `` zf `` xz  ``wf ``wx `g `x `g `y `g `z `g `w `g `g `z `g `w `x `x  `y `x  `z `x  `w `x œ `x  `z `x  `w `x œ 0

`y `x

œ0

œ 0 and imply

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

`g `z `z `y

œ0

861

862

Chapter 14 Partial Derivatives



`f `z `g `z

`z `x `z `x

 

`f `w `g `w

`w `x `w `x

œ
œ


`f `x `g `x

Ê ˆ `` xz ‰y œ

c ``xf » c `g »

`x `f `z `g `z

`f `w `g » `w `f `w `g » `w

œ

`x `y œ 0 `g `g `z `y  `z `y

Likewise, f(xß yß zß w) œ 0, g(xß yß zß w) œ 0 and œ



`f `y `f `z `g `z

 `z `y `z `y

`f `z `z `y

 

`f `w `g `w

 `w `y `w `y

`f `w `w `y

œ 0 and (similarly)

œ
`` yf œ


`g `y

Ê Š ``wy ‹ œ x

`g `g `f `w
`x `w `g `f `f `g `z `w  `z `w

 ``xf

`f `z » `g `z `f `z » `g `z

c ``yf c `` gy » `f `w `g » `w

œ

œ


`f `x `f `z

`g `w `g `w

`g `x `f `g `w `z

 ``wf



`f `x `f `y `f `z `x `y  `y `y  `z `y `g `w ` w ` y œ 0 imply

Ê 

`g `g `f `y
`z `y `g `f `f `g `z `w  `z `w

 `` fz

œ


`f `z `f `z

`g `y `g `w

`g `z `f `g `w `z

 ``yf 

, as claimed. 

`f `w `w `y

, as claimed.

CHAPTER 14 PRACTICE EXERCISES 1. Domain: All points in the xy-plane Range: z   0 Level curves are ellipses with major axis along the y-axis and minor axis along the x-axis.

2. Domain: All points in the xy-plane Range: 0  z  _ Level curves are the straight lines x  y œ ln z with slope
1, and z  0.

3. Domain: All (xß y) such that x Á 0 and y Á 0 Range: z Á 0 Level curves are hyperbolas with the x- and y-axes as asymptotes.

4. Domain: All (xß y) so that x#
y   0 Range: z   0 Level curves are the parabolas y œ x#
c, c   0.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 14 Practice Exercises 5. Domain: All points (xß yß z) in space Range: All real numbers Level surfaces are paraboloids of revolution with the z-axis as axis.

6. Domain: All points (xß yß z) in space Range: Nonnegative real numbers Level surfaces are ellipsoids with center (0ß 0ß 0).

7. Domain: All (xß yß z) such that (xß yß z) Á (0ß !ß 0) Range: Positive real numbers Level surfaces are spheres with center (0ß 0ß 0) and radius r  0.

8. Domain: All points (xß yß z) in space Range: (0ß 1] Level surfaces are spheres with center (0ß 0ß 0) and radius r  0.

9.

lim

Ðxß yÑ Ä Ð1ß ln 2Ñ

ey cos x œ eln 2 cos 1 œ (2)(
1) œ
2 2
y

10.

lim Ðxß yÑ Ä Ð0ß 0Ñ x
cos y

11.

lim # # Ðxß yÑ Ä Ð1ß 1Ñ x  y xÁ „y

12.

13.

14.

xy

x$ y$  1 Ðxß yÑ Ä Ð1ß 1Ñ xy  1

lim

lim

P Ä Ð1ß
1ß eÑ

lim

œ

œ

2
0 0
cos 0

œ2 xy

lim Ðxß yÑ Ä Ð1ß 1Ñ (x  y)(x
y) xÁ „y

œ

œ

(xy  1) ax# y#
xy
1b xy  1 Ðx ß y Ñ Ä Ð 1 ß 1Ñ

lim

1

lim Ðxß yÑ Ä Ð1ß 1Ñ x
y

œ

lim

œ

Ðxß yÑ Ä Ð1ß 1Ñ

" 1
1

œ

" #

ax# y#  xy  1b œ 1# † 1#  1 † 1  1 œ 3

ln kx  y  zk œ ln k1  (
1)  ek œ ln e œ 1

P Ä Ð1 ß
1 ß
1 Ñ

tan" (x  y  z) œ tan" (1  (
1)  (
1)) œ tan" (
1) œ
14

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

863

864

Chapter 14 Partial Derivatives

15. Let y œ kx# , k Á 1. Then

kx#

œ

y

lim # Ðxß yÑ Ä Ð0ß 0Ñ x  y y Á x#

lim # # axß kx# b Ä Ð0ß 0Ñ x  kx

œ

k 1  k#

which gives different limits for

œ

1
k# k

which gives different limits for

different values of k Ê the limit does not exist. 16. Let y œ kx, k Á 0. Then

x#
y# xy Ðxß yÑ Ä Ð0ß 0Ñ

lim

x#
(kx)# x(kx) (xß kxÑ Ä Ð0ß 0Ñ

œ

lim

xy Á 0

different values of k Ê the limit does not exist. 17. Let y œ kx. Then

x#  y#

œ

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x
y

x#  k# x# x #
k# x#

1  k# 1
k#

œ

which gives different limits for different values

of k Ê the limit does not exist so f(0ß 0) cannot be defined in a way that makes f continuous at the origin. 18. Along the x-axis, y œ 0 and

sin (x  y) kxk
kyk

lim

Ðxß yÑ Ä Ð0ß 0Ñ

œ lim

sin x k xk

xÄ0

œœ

1, x  0 , so the limit fails to exist
", x  0

Ê f is not continuous at (0ß 0). 19.

`g `r

œ cos )  sin ),

20.

`f `x

œ

21.

`f ` R"

" #

Š x# 2x
y# ‹ 

œ
R"# , "

`f ` R#

`g `)

œ
r sin )  r cos )

y ‹ x# y # 1
ˆx‰

Š

œ

œ
R"# ,

x#

`f ` R$

#

x
y#




x#

y
y#

œ

xy x#
y#

`f `y

,

œ

" #

Š x# 2y
y# ‹ 

Š 1x ‹ y #

1
ˆx‰

œ

y x#
y#



x x#
y#

œ

x
y x#
y#

œ
R"# $

22. hx (xß yß z) œ 21 cos (21x  y
3z), hy (xß yß z) œ cos (21x  y
3z), hz (xß yß z) œ
3 cos (21x  y
3z) 23.

`P `n

œ

RT V

,

`P `R

œ

nT V

`P `T

,

œ

nR V

,

`P `V

œ
nRT V#

24. fr (rß jß Tß w) œ
2r"# j É 1Tw , fj (rß jß Tß w) œ
#r"j# É

25.

œ

" 4rj

É T1"w œ

`g `x

œ

" y

,

`g `y

" 4rjT

œ1


, fT (rß jß Tß w) œ ˆ #"rj ‰ Š È"1w ‹ Š 2È" T ‹

T 1w

É 1Tw , fw (rß jß Tß w) œ ˆ #"rj ‰ É T1 ˆ
"# w$Î# ‰ œ
4r"jw É 1Tw Ê

x y#

` #g ` x#

œ 0,

` #g ` y#

œ

2x y$

,

` #g ` y` x

œ

` #g ` x` y

œ
y"#

26. gx (xß y) œ ex  y cos x, gy (xß y) œ sin x Ê gxx (xß y) œ ex
y sin x, gyy (xß y) œ 0, gxy (xß y) œ gyx (xß y) œ cos x 27.

`f `x

œ 1  y
15x# 

2x x#
1

,

`f `y

œx Ê

` #f ` x#

œ
30x 

22x# ax#
1b#

,

` #f ` y#

œ 0,

` #f ` y` x

œ

` #f ` x` y

œ1

28. fx (xß y) œ
3y, fy (xß y) œ 2y
3x
sin y  7ey Ê fxx (xß y) œ 0, fyy (xß y) œ 2
cos y  7ey , fxy (xß y) œ fyx (xß y) œ
3 29.

`w `x

Ê Ê 30.

`w `x

Ê Ê

œ y cos (xy  1),

`w `y

œ x cos (xy  1),

dx dt

œ et ,

dy dt

dw t ˆ " ‰ dt œ [y cos (xy  1)]e  [x cos (xy  1)] t
1 ; dw ¸ ˆ " ‰ dt tœ0 œ 0 † 1  [1 † (
1)] 0
1 œ
1

œ ey ,

`w `y

œ xey  sin z,

dw y "Î#  axey  dt œ e t dw ¸ dt tœ1 œ 1 † 1  (2 † 1

`w `z

œ y cos z  sin z,

sin zb ˆ1 

"‰ t

dx dt

œ

" t
1

t œ 0 Ê x œ 1 and y œ 0

œ t"Î# ,

dy dt

œ 1  "t ,

dz dt

œ1

 (y cos z  sin z)1; t œ 1 Ê x œ 2, y œ 0, and z œ 1


0)(2)  (0  0)1 œ 5

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 14 Practice Exercises 31.

`w `x

Ê Ê

33.

`w `u `w `v

œ

`x `u `x `v

œ

ˆ 1
x x#

`f `x

œ y  z,

`f `y

œ x  z,

œ

Ê

`w `x

dw dx dw dx

œ ˆ 1
x x#


`x `r

`x `s

œ 1,

œ cos s,




" ‰ u x#
1 a2e cos vb ; u œ v œ 0 Ê " ‰ `w ¸ u x#
1 a
2e sin vb Ê ` v Ð0ß0Ñ œ `f `z

œ y  x,

dx dt

df dt œ
(y  z)(sin t)  (x  z)(cos df ¸ dt tœ1 œ
(sin 1  cos 2)(sin 1) 

Ê 34.

œ
cos (2x
y),

`y `r

`y `s

œ s,

œr

`w ` r œ [2 cos (2x
y)](1)  [
cos (2x
y)](s); r œ 1 and s œ 0 Ê x œ 1 and y œ 0 `w ¸ `w ` r Ð1ß0Ñ œ (2 cos 21)
(cos 21 )(0) œ 2; ` s œ [2 cos (2x
y)](cos s)  [
cos (2x
y)](r) `w ¸ ` s Ð1ß0Ñ œ (2 cos 21)(cos 0)
(cos 21)(1) œ 2
1

Ê

32.

`w `y

œ 2 cos (2x
y),

œ

dw ` s ds ` x

œ (5)

dw ds

and

`w `y

œ

dw ` s ds ` y

œ
sin t,

dy dt

`w ¸ ` u Ð0ß0Ñ

xœ2 Ê ˆ 52




œ cos t,

dz dt

"‰ 5 (0)

œ ˆ 52
5" ‰ (2) œ

1
y cos xy 2y  x cos xy

œ
2 sin 2t

(cos 1  cos 2)(cos 1)
2(sin 1  cos 1)(sin 2)

œ (1)

dw ds

œ

`w `x

Ê

dw ds


5

`w `y

œ5

œ

dy dx ¹ Ð0ß1Ñ

1
" 2

dw ds


5

dw ds

dy dx ¹ Ð0ßln 2Ñ

1  y cos xy œ
FFxy œ
2y  x cos xy

dy dx

xby


e œ
FFxy œ
2y 2x
exby

ß

1‰ 4

#

œ


i


j Ê f increases most rapidly in the direction u œ


È2 #

i


Ê f increases most rapidly in the direction u œ
u œ
È12 i 

1 È2

i


1 È2

" È2

œ

È2 #

and decreases most

38. ™ f œ 2xec2y i
2x# ec2y j Ê ™ f k Ð1ß0Ñ œ #i
#j Ê k ™ f k œ È2#  (
2)# œ 2È2; u œ 1 È2

#

œ
"# i
"# j Ê k ™ f k œ Ɉ
"# ‰  ˆ
"# ‰ œ

È2 # j È È È È rapidly in the direction
u œ #2 i  #2 j ; (Du f)P! œ k ™ f k œ #2 and (Dcu f)P! œ
#2 ; 7 u" œ kvvk œ È33i #

4j4# œ 53 i  54 j Ê (Du" f)P! œ ™ f † u" œ ˆ
"# ‰ ˆ 35 ‰  ˆ
"# ‰ ˆ 45 ‰ œ
10 ™f k™f k

È2 #

dy dx

2 œ
2 ln0
2
2 œ
(ln 2  1)

37. ™ f œ (
sin x cos y)i
(cos x sin y)j Ê ™ f k ˆ 14 È2 #

œ0

œ
1

36. F(xß y) œ 2xy  ex
y
2 Ê Fx œ 2y  ex
y and Fy œ 2x  ex
y Ê

uœ

;

t)
2(y  x)(sin 2t); t œ 1 Ê x œ cos 1, y œ sin 1, and z œ cos 2

Ê at (xß y) œ (!ß 1) we have

Ê at (xß y) œ (!ß ln 2) we have

2 5

œ0

35. F(xß y) œ 1
x
y#
sin xy Ê Fx œ
1
y cos xy and Fy œ
2y
x cos xy Ê œ

865

™f k™f k

œ

1 È2

i


1 È2

j

j and decreases most rapidly in the direction

j ; (Du f)P! œ k ™ f k œ 2È2 and (Dcu f)P! œ
2È2 ; u" œ

v kv k

œ

i
j È 1#
1#

œ

1 È2

i

1 È2

j

Ê (Du" f)P! œ ™ f † u" œ (2) Š È"2 ‹  (
2) Š È"2 ‹ œ 0 2 3 6 39. ™ f œ Š 2x
3y
6z ‹ i  Š 2x
3y
6z ‹ j  Š 2x
3y
6z ‹ k Ê ™ f k Ð 1ß 1ß1Ñ œ 2i  3j  6k ;

uœ

™f k™f k

œ

2i
3j
6k È 2#
3#
6#

œ

2 7

i  73 j  76 k Ê f increases most rapidly in the direction u œ

2 7

i  37 j  67 k and

decreases most rapidly in the direction
u œ
27 i
37 j
67 k ; (Du f)P! œ k ™ f k œ 7, (Du f)P! œ
7; u" œ

v kv k

œ

2 7

i  37 j  67 k Ê (Du" f)P! œ (Du f)P! œ 7

40. ™ f œ (2x  3y)i  (3x  2)j  (1
2z)k Ê ™ f k Ð0ß0ß0Ñ œ 2j  k ; u œ rapidly in the direction u œ

2 È5

j

" È5

™f k™f k

œ

2 È5

j

" È5

k Ê f increases most

k and decreases most rapidly in the direction
u œ
È25 j


(Du f)P! œ k ™ f k œ È5 and (Du f)P! œ
È5 ; u" œ

v kvk

œ

i
j
k È 1#
1#
1#

Ê (Du" f)P! œ ™ f † u" œ (0) Š È"3 ‹  (2) Š È"3 ‹  (1) Š È"3 ‹ œ

3 È3

œ

" È3

i

" È3

j

" È3

k

œ È3

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" È5

k;

;

866

Chapter 14 Partial Derivatives

41. r œ (cos 3t)i  (sin 3t)j  3tk Ê v(t) œ (
3 sin 3t)i  (3 cos 3t)j  3k Ê v ˆ 13 ‰ œ
3j  3k Ê u œ
È"2 j 

" È2

k ; f(xß yß z) œ xyz Ê ™ f œ yzi  xzj  xyk ; t œ

Ê ™ f k Ð 1ß0ß1Ñ œ
1j Ê ™ f † u œ (
1j) † Š
È"2 j 

" È2

k‹ œ

1 3

yields the point on the helix (
1ß 0ß 1)

1 È2

42. f(xß yß z) œ xyz Ê ™ f œ yzi  xzj  xyk ; at (1ß 1ß 1) we get ™ f œ i  j  k Ê the maximum value of Du f k œ k ™ f k œ È3 Ð1ß1ß1Ñ

43. (a) Let ™ f œ ai  bj at (1ß 2). The direction toward (2ß 2) is determined by v" œ (2
1)i  (2
2)j œ i œ u so that ™ f † u œ 2 Ê a œ 2. The direction toward (1ß 1) is determined by v# œ (1
1)i  (1
2)j œ
j œ u so that ™ f † u œ
2 Ê
b œ
2 Ê b œ 2. Therefore ™ f œ 2i  2j ; fx a1, 2b œ fy a1, 2b œ 2. (b) The direction toward (4ß 6) is determined by v$ œ (4
1)i  (6
2)j œ 3i  4j Ê u œ 35 i  45 j Ê ™f†uœ

14 5

.

44. (a) True

(b) False

(c) True

(d) True

45. ™ f œ 2xi  j  2zk Ê ™ f k Ð0ß 1ß 1Ñ œ j
2k , ™ f k Ð0ß0ß0Ñ œ j , ™ f k Ð0ß 1ß1Ñ œ j  2k

46. ™ f œ 2yj  2zk Ê ™ f k Ð2ß2ß0Ñ œ 4j , ™ f k Ð2ß 2ß0Ñ œ
4j , ™ f k Ð2ß0ß2Ñ œ 4k , ™ f k Ð2ß0ß 2Ñ œ
4k

47. ™ f œ 2xi
j
5k Ê ™ f k Ð2ß 1ß1Ñ œ 4i
j
5k Ê Tangent Plane: 4(x
2)
(y  1)
5(z
1) œ 0 Ê 4x
y
5z œ 4; Normal Line: x œ 2  4t, y œ
1
t, z œ 1
5t 48. ™ f œ 2xi  2yj  k Ê ™ f k Ð1ß1ß2Ñ œ 2i  2j  k Ê Tangent Plane: 2(x
1)  2(y
1)  (z
2) œ 0 Ê 2x  2y  z
6 œ 0; Normal Line: x œ 1  2t, y œ 1  2t, z œ 2  t 49.

`z `x

œ

2x x#
y#

Ê

`z ¸ ` x Ð0ß1ß0Ñ

œ 0 and

`z `y

œ

2y x#
y#

Ê

`z ` y ¹ Ð0ß1ß0Ñ

œ 2; thus the tangent plane is

2(y
1)
(z
0) œ 0 or 2y
z
2 œ 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 14 Practice Exercises 50.

`z `x

œ
2x ax#  y# b

#

`z ¸ ` x ˆ1ß1ß 12 ‰

Ê

œ
#" and

`z `y

œ
2y ax#  y# b

#

Ê

`z ` y ¹ ˆ1ß1ß 1 ‰ 2

867

œ
"# ; thus the tangent

plane is
"# (x
1)
"# (y
1)
ˆz
"# ‰ œ 0 or x  y  2z
3 œ 0 51. ™ f œ (
cos x)i  j Ê ™ f k Ð1ß1Ñ œ i  j Ê the tangent line is (x
1)  (y
1) œ 0 Ê x  y œ 1  1; the normal line is y
1 œ 1(x
1) Ê y œ x
1  1

52. ™ f œ
xi  yj Ê ™ f k Ð1ß2Ñ œ
i  2j Ê the tangent line is
(x
1)  2(y
2) œ 0 Ê y œ

" #

x  #3 ; the normal

line is y
2 œ
2(x
1) Ê y œ
2x  4

53. Let f(xß yß z) œ x#  2y  2z
4 and g(xß yß z) œ y
1. Then ™ f œ 2xi  2j  2kk a1 1 12 b œ 2i  2j  2k â â â i j kâ â â and ™ g œ j Ê ™ f ‚ ™ g œ â 2 2 2 â œ
2i  2k Ê the line is x œ 1
2t, y œ 1, z œ "#  2t â â â0 " 0â ß ß

54. Let f(xß yß z) œ x  y#  z
2 and g(xß yß z) œ y
1. Then ™ f œ i  2yj  kk a 12 1 12 b œ i  2j  k and â â â i j kâ â â ™ g œ j Ê ™ f ‚ ™ g œ â 1 2 1 â œ
i  k Ê the line is x œ "#
t, y œ 1, z œ "#  t â â â0 " 0â ß ß

55. f ˆ 14 ß 14 ‰ œ

" #

, fx ˆ 14 ß 14 ‰ œ cos x cos yk Ð1Î4ß1Î4Ñ œ

Ê L(xß y) œ

" #

 "# ˆx
14 ‰
"# ˆy
14 ‰ œ

" #

" # " #

, fy ˆ 14 ß 14 ‰ œ
sin x sin yk Ð1Î4ß1Î4Ñ œ
"#

 x
"# y; fxx (xß y) œ
sin x cos y, fyy (xß y) œ
sin x cos y, and

fxy (xß y) œ
cos x sin y. Thus an upper bound for E depends on the bound M used for kfxx k , kfxy k , and kfyy k . With M œ

È2 #

we have kE(xß y)k Ÿ

with M œ 1, kE(xß y)k Ÿ

" #

" #

Š

È2 ˆ¸ # ‹ x

#
14 ¸  ¸y
14 ¸‰ Ÿ

# (1) ˆ¸x
14 ¸  ¸y
14 ¸‰ œ

" #

È2 4

(0.2)# Ÿ 0.0142;

(0.2)# œ 0.02.

56. f(1ß 1) œ 0, fx (1ß 1) œ yk Ð1ß1Ñ œ 1, fy (1ß 1) œ x
6yk Ð1ß1Ñ œ
5 Ê L(xß y) œ (x
1)
5(y
1) œ x
5y  4; fxx (xß y) œ 0, fyy (xß y) œ
6, and fxy (xß y) œ 1 Ê maximum of kfxx k , kfyy k , and kfxy k is 6 Ê M œ 6 Ê kE(xß y)k Ÿ

" #

(6) akx
1k  ky
1kb# œ

" #

(6)(0.1  0.2)# œ 0.27

57. f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ y
3zk Ð1ß0ß0Ñ œ 0, fy (1ß 0ß 0) œ x  2zk Ð1ß0ß0Ñ œ 1, fz (1ß 0ß 0) œ 2y
3xk Ð1ß0ß0Ñ œ
3 Ê L(xß yß z) œ 0(x
1)  (y
0)
3(z
0) œ y
3z; f(1ß 1ß 0) œ 1, fx (1ß 1ß 0) œ 1, fy (1ß 1ß 0) œ 1, fz ("ß "ß !) œ
1 Ê L(xß yß z) œ 1  (x
1)  (y
1)
1(z
0) œ x  y
z
1 58. f ˆ0ß !ß 14 ‰ œ 1, fx ˆ!ß 0ß 14 ‰ œ
È2 sin x sin (y  z)¹

ˆ0ß0ß 1 ‰

œ 0, fy ˆ!ß 0ß 14 ‰ œ È2 cos x cos (y  z)¹

4

fz ˆ!ß 0ß 14 ‰ œ È2 cos x cos (y  z)¹

ˆ0ß0ß 1 ‰

È2 #

œ 1 Ê L(xß yß z) œ 1  1(y
0)  1 ˆz
14 ‰ œ 1  y  z


Ê L(xß yß z) œ

È2 È2 È2 ˆ1 1 ‰ ˆ1 1 ‰ # , fy 4 ß 4 ß 0 œ # , fz 4 ß 4 ß 0 œ # È È È È È  #2 ˆy
14 ‰  #2 (z
0) œ #2
#2 x  #2

, fx ˆ 14 ß 14 ß 0‰ œ
È2 #




È2 #

ˆx
14 ‰

œ 1,

4

4

f ˆ 14 ß 14 ß 0‰ œ

ˆ0ß0ß 1 ‰

y

È2 #

z

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

1 4

;

868

Chapter 14 Partial Derivatives

59. V œ 1r# h Ê dV œ 21rh dr  1r# dh Ê dVk Ð1Þ5ß5280Ñ œ 21(1.5)(5280) dr  1(1.5)# dh œ 15,8401 dr  2.251 dh. You should be more careful with the diameter since it has a greater effect on dV. 60. df œ (2x
y) dx  (
x  2y) dy Ê df k Ð1ß2Ñ œ 3 dy Ê f is more sensitive to changes in y; in fact, near the point (1ß 2) a change in x does not change f. 61. dI œ

" R

dV


V R#

" 100

dR Ê dI¸ Ð24ß100Ñ œ

dV


24 100#

dR Ê dI¸ dVœ
1ßdRœ
20 œ
0.01  (480)(.0001) œ 0.038,

" ‰ 20 ‰ or increases by 0.038 amps; % change in V œ (100) ˆ
24 ¸
4.17%; % change in R œ ˆ
100 (100) œ
20%;

Iœ

24 100

œ 0.24 Ê estimated % change in I œ

dI I

‚ 100 œ

0.038 0.24

‚ 100 ¸ 15.83% Ê more sensitive to voltage change.

62. A œ 1ab Ê dA œ 1b da  1a db Ê dAk Ð10ß16Ñ œ 161 da  101 db; da œ „ 0.1 and db œ „ 0.1 ¸ ¸ 2.61 ¸ Ê dA œ „ 261(0.1) œ „ 2.61 and A œ 1(10)(16) œ 1601 Ê ¸ dA A ‚ 100 œ 1601 ‚ 100 ¸ 1.625% 63. (a) y œ uv Ê dy œ v du  u dv; percentage change in u Ÿ 2% Ê kduk Ÿ 0.02, and percentage change in v Ÿ 3% Ê kdvk Ÿ 0.03;

dy y

Ÿ 2%  3% œ 5% (b) z œ u  v Ê dzz œ

œ

v du
u dv uv

du
dv u
v

œ

œ

du u
v

Ê ¸ dzz ‚ 100¸ Ÿ ¸ du u ‚ 100  64. C œ Ê

dv v



du u



dv u
v

Þ

Þ

Þ

Þ

Þ

Þ

Ÿ

¸ du Ê ¹ dy y ‚ 100¹ œ u ‚ 100  du u



dv v

dv v

¸ ¸ dv ¸ ‚ 100¸ Ÿ ¸ du u ‚ 100  v ‚ 100

(since u  0, v  0)

‚ 100¸ œ ¹ dy y ‚ 100¹

(0.425)(7) 7 71.84w0 425 h0 725 Ê Cw œ 71.84w1 425 h0 725 2.975 5.075 dC œ 71.84w 1 425 h0 725 dw  71.84w0 425 h1 725 Þ

dv v

Þ

and Ch œ

(0.725)(7) 71.84w0 425 h1 725 Þ

Þ

dh; thus when w œ 70 and h œ 180 we have

dCk Ð70ß180Ñ ¸
(0.00000225) dw
(0.00000149) dh Ê 1 kg error in weight has more effect 65. fx (xß y) œ 2x
y  2 œ 0 and fy (xß y) œ
x  2y  2 œ 0 Ê x œ
2 and y œ
2 Ê (
2ß
2) is the critical point; # œ 3  0 and fxx  0 Ê local minimum value fxx (
2ß
2) œ 2, fyy (
#ß
2) œ 2, fxy (
#ß
2) œ
1 Ê fxx fyy
fxy of f(
#ß
2) œ
8 66. fx (xß y) œ 10x  4y  4 œ 0 and fy (xß y) œ 4x
4y
4 œ 0 Ê x œ 0 and y œ
1 Ê (0ß
1) is the critical point; # œ
56  0 Ê saddle point with f(0ß
1) œ 2 fxx (0ß
1) œ 10, fyy (0ß
1) œ
4, fxy (0ß
1) œ 4 Ê fxx fyy
fxy 67. fx (xß y) œ 6x#  3y œ 0 and fy (xß y) œ 3x  6y# œ 0 Ê y œ
2x# and 3x  6 a4x% b œ 0 Ê x a1  8x$ b œ 0 Ê x œ 0 and y œ 0, or x œ
"# and y œ
"# Ê the critical points are (0ß 0) and ˆ
"# ß
"# ‰ . For (!ß !):

# fxx (!ß !) œ 12xk Ð0ß0Ñ œ 0, fyy (!ß !) œ 12yk Ð0ß0Ñ œ 0, fxy (!ß 0) œ 3 Ê fxx fyy
fxy œ
9  0 Ê saddle point with # f(0ß 0) œ 0. For ˆ
"# ß
"# ‰: fxx œ
6, fyy œ
6, fxy œ 3 Ê fxx fyy
fxy œ 27  0 and fxx  0 Ê local maximum " " " value of f ˆ
# ß
# ‰ œ 4

68. fx (xß y) œ 3x#
3y œ 0 and fy (xß y) œ 3y#
3x œ 0 Ê y œ x# and x%
x œ 0 Ê x ax$
1b œ 0 Ê the critical points are (0ß 0) and (1ß 1) . For (!ß !): fxx (!ß !) œ 6xk Ð0ß0Ñ œ 0, fyy (!ß !) œ 6yk Ð0ß0Ñ œ 0, fxy (!ß 0) œ
3 # Ê fxx fyy
fxy œ
9  0 Ê saddle point with f(0ß 0) œ 15. For (1ß 1): fxx (1ß 1) œ 6, fyy (1ß 1) œ 6, fxy (1ß 1) œ
3 # Ê fxx fyy
fxy œ 27  0 and fxx  0 Ê local minimum value of f(1ß 1) œ 14

69. fx (xß y) œ 3x#  6x œ 0 and fy (xß y) œ 3y#
6y œ 0 Ê x(x  2) œ 0 and y(y
2) œ 0 Ê x œ 0 or x œ
2 and y œ 0 or y œ 2 Ê the critical points are (0ß 0), (0ß 2), (
2ß 0), and (
2ß 2) . For (!ß !): fxx (!ß !) œ 6x  6k Ð0ß0Ñ # œ 6, fyy (!ß !) œ 6y
6k Ð0ß0Ñ œ
6, fxy (!ß 0) œ 0 Ê fxx fyy
fxy œ
36  0 Ê saddle point with f(0ß 0) œ 0. For # (0ß 2): fxx (!ß 2) œ 6, fyy (0ß #) œ 6, fxy (!ß 2) œ 0 Ê fxx fyy
fxy œ 36  0 and fxx  0 Ê local minimum value of

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 14 Practice Exercises

869

# f(!ß 2) œ
4. For (
#ß 0): fxx (
2ß 0) œ
6, fyy (
#ß 0) œ
6, fxy (
2ß 0) œ 0 Ê fxx fyy
fxy œ 36  0 and fxx  0

Ê local maximum value of f(
2ß 0) œ 4. For (
2ß 2): fxx (
2ß 2) œ
6, fyy (
2ß 2) œ 6, fxy (
2ß 2) œ 0 # Ê fxx fyy
fxy œ
36  0 Ê saddle point with f(
2ß 2) œ 0. 70. fx (xß y) œ 4x$
16x œ 0 Ê 4x ax#
4b œ 0 Ê x œ 0, 2,
2; fy (xß y) œ 6y
6 œ 0 Ê y œ 1. Therefore the critical points are (0ß 1), (2ß 1), and (
2ß 1). For (!ß 1): fxx (!ß 1) œ 12x#
16k Ð0ß1Ñ œ
16, fyy (!ß 1) œ 6, fxy (!ß 1) œ 0 # Ê fxx fyy
fxy œ
96  0 Ê saddle point with f(0ß 1) œ
3. For (2ß 1): fxx (2ß 1) œ 32, fyy (2ß 1) œ 6, # fxy (2ß 1) œ 0 Ê fxx fyy
fxy œ 192  0 and fxx  0 Ê local minimum value of f(2ß 1) œ
19. For (
#ß 1): # fxx (
2ß 1) œ 32, fyy (
#ß 1) œ 6, fxy (
2ß 1) œ 0 Ê fxx fyy
fxy œ 192  0 and fxx  0 Ê local minimum value of

f(
2ß 1) œ
19. 71. (i)

On OA, f(xß y) œ f(0ß y) œ y#  3y for 0 Ÿ y Ÿ 4 Ê f w (!ß y) œ 2y  3 œ 0 Ê y œ
3# . But ˆ!ß
3# ‰

is not in the region. Endpoints: f(0ß 0) œ 0 and f(0ß 4) œ 28. (ii) On AB, f(xß y) œ f(xß
x  4) œ x#
10x  28 for 0 Ÿ x Ÿ 4 Ê f w (xß
x  4) œ 2x
10 œ 0 Ê x œ 5, y œ
1. But (5ß
1) is not in the region. Endpoints: f(4ß 0) œ 4 and f(!ß 4) œ 28. (iii) On OB, f(xß y) œ f(xß 0) œ x#
3x for 0 Ÿ x Ÿ 4 Ê f w (xß 0) œ 2x
3 Ê x œ critical point with f ˆ 3# ß !‰ œ
94 .

3 #

and y œ 0 Ê ˆ 3# ß 0‰ is a

Endpoints: f(0ß 0) œ 0 and f(%ß 0) œ 4. (iv) For the interior of the triangular region, fx (xß y) œ 2x  y
3 œ 0 and fy (xß y) œ x  2y  3 œ 0 Ê x œ 3 and y œ
3. But (3ß
3) is not in the region. Therefore the absolute maximum is 28 at (0ß 4) and the absolute minimum is
94 at ˆ 3# ß !‰ .

On OA, f(xß y) œ f(0ß y) œ
y#  4y  1 for 0 Ÿ y Ÿ 2 Ê f w (!ß y) œ
2y  4 œ 0 Ê y œ 2 and x œ 0. But (0ß 2) is not in the interior of OA. Endpoints: f(0ß 0) œ 1 and f(0ß 2) œ 5. (ii) On AB, f(xß y) œ f(xß 2) œ x#
2x  5 for 0 Ÿ x Ÿ 4 Ê f w (xß 2) œ 2x
2 œ 0 Ê x œ 1 and y œ 2 Ê (1ß 2) is an interior critical point of AB with f(1ß 2) œ 4. Endpoints: f(4ß 2) œ 13 and f(!ß 2) œ 5. (iii) On BC, f(xß y) œ f(4ß y) œ
y#  4y  9 for 0 Ÿ y Ÿ 2 Ê f w (4ß y) œ
2y  4 œ 0 Ê y œ # and x œ 4. But (4ß 2) is not in the interior of BC. Endpoints: f(4ß 0) œ 9 and f(%ß 2) œ 13. (iv) On OC, f(xß y) œ f(xß 0) œ x#
2x  1 for 0 Ÿ x Ÿ 4 Ê f w (xß 0) œ 2x
2 œ 0 Ê x œ 1 and y œ 0 Ê (1ß 0) is an interior critical point of OC with f(1ß 0) œ 0. Endpoints: f(0ß 0) œ 1 and f(4ß 0) œ 9. (v) For the interior of the rectangular region, fx (xß y) œ 2x
2 œ 0 and fy (xß y) œ
2y  4 œ 0 Ê x œ 1 and y œ 2. But (1ß 2) is not in the interior of the region. Therefore the absolute maximum is 13 at (4ß 2) and the absolute minimum is 0 at (1ß 0).

72. (i)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

870 73. (i)

Chapter 14 Partial Derivatives On AB, f(xß y) œ f(
2ß y) œ y#
y
4 for
2 Ÿ y Ÿ 2 Ê f w (
2ß y) œ 2y
1 Ê y œ "# and x œ
2 Ê ˆ
2ß "# ‰ is an interior critical point in AB

with f ˆ
2ß "# ‰ œ
17 4 . Endpoints: f(
2ß
2) œ 2 and

f(2ß 2) œ
2. On BC, f(xß y) œ f(xß 2) œ
2 for
2 Ÿ x Ÿ 2 Ê f w (xß 2) œ 0 Ê no critical points in the interior of BC. Endpoints: f(
2ß 2) œ
2 and f(2ß 2) œ
2. (iii) On CD, f(xß y) œ f(2ß y) œ y#
5y  4 for
2 Ÿ y Ÿ 2 Ê f w (2ß y) œ 2y
5 œ 0 Ê y œ 5# and x œ 2. But ˆ#ß 5# ‰ is not in the region. (ii)

Endpoints: f(2ß
2) œ 18 and f(2ß 2) œ
2. (iv) On AD, f(xß y) œ f(xß
2) œ 4x  10 for
2 Ÿ x Ÿ 2 Ê f w (xß
2) œ 4 Ê no critical points in the interior of AD. Endpoints: f(
2ß
2) œ 2 and f(2ß
2) œ 18. (v) For the interior of the square, fx (xß y) œ
y  2 œ 0 and fy (xß y) œ 2y
x
3 œ 0 Ê y œ 2 and x œ 1 Ê (1ß 2) is an interior critical point of the square with f(1ß 2) œ
2. Therefore the absolute maximum "‰ ˆ is 18 at (2ß
2) and the absolute minimum is
17 4 at
#ß # . On OA, f(xß y) œ f(0ß y) œ 2y
y# for 0 Ÿ y Ÿ 2 Ê f w (!ß y) œ 2
2y œ 0 Ê y œ 1 and x œ 0 Ê (!ß 1) is an interior critical point of OA with f(0ß 1) œ 1. Endpoints: f(0ß 0) œ 0 and f(0ß 2) œ 0. (ii) On AB, f(xß y) œ f(xß 2) œ 2x
x# for 0 Ÿ x Ÿ 2 Ê f w (xß 2) œ 2
2x œ 0 Ê x œ 1 and y œ 2 Ê (1ß 2) is an interior critical point of AB with f(1ß 2) œ 1. Endpoints: f(0ß 2) œ 0 and f(2ß 2) œ 0. (iii) On BC, f(xß y) œ f(2ß y) œ 2y
y# for 0 Ÿ y Ÿ 2 Ê f w (2ß y) œ 2
2y œ 0 Ê y œ 1 and x œ 2 Ê (2ß 1) is an interior critical point of BC with f(2ß 1) œ 1. Endpoints: f(2ß 0) œ 0 and f(2ß 2) œ 0. (iv) On OC, f(xß y) œ f(xß 0) œ 2x
x# for 0 Ÿ x Ÿ 2 Ê f w (xß 0) œ 2
2x œ 0 Ê x œ 1 and y œ 0 Ê (1ß 0) is an interior critical point of OC with f(1ß 0) œ 1. Endpoints: f(0ß 0) œ 0 and f(0ß 2) œ 0. (v) For the interior of the rectangular region, fx (xß y) œ 2
2x œ 0 and fy (xß y) œ 2
2y œ 0 Ê x œ 1 and y œ 1 Ê (1ß 1) is an interior critical point of the square with f(1ß 1) œ 2. Therefore the absolute maximum is 2 at (1ß 1) and the absolute minimum is 0 at the four corners (0ß 0), (0ß 2), (2ß 2), and (2ß 0).

74. (i)

On AB, f(xß y) œ f(xß x  2) œ
2x  4 for
2 Ÿ x Ÿ 2 Ê f w (xß x  2) œ
2 œ 0 Ê no critical points in the interior of AB. Endpoints: f(
2ß 0) œ 8 and f(2ß 4) œ 0. (ii) On BC, f(xß y) œ f(2ß y) œ
y#  4y for 0 Ÿ y Ÿ 4 Ê f w (2ß y) œ
2y  4 œ 0 Ê y œ 2 and x œ 2 Ê (2ß 2) is an interior critical point of BC with f(2ß 2) œ 4. Endpoints: f(2ß 0) œ 0 and f(2ß 4) œ 0. (iii) On AC, f(xß y) œ f(xß 0) œ x#
2x for
2 Ÿ x Ÿ 2 Ê f w (xß 0) œ 2x
2 Ê x œ 1 and y œ 0 Ê (1ß 0) is an interior critical point of AC with f(1ß 0) œ
1. Endpoints: f(
2ß 0) œ 8 and f(2ß 0) œ 0. (iv) For the interior of the triangular region, fx (xß y) œ 2x
2 œ 0 and fy (xß y) œ
2y  4 œ 0 Ê x œ 1 and y œ 2 Ê (1ß 2) is an interior critical point of the region with f(1ß 2) œ 3. Therefore the absolute maximum is 8 at (
2ß 0) and the absolute minimum is
1 at (1ß 0).

75. (i)

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Chapter 14 Practice Exercises 76. (i)

(ii)

871

On AB, faxß yb œ faxß xb œ 4x#
2x%  16 for
2 Ÿ x Ÿ 2 Ê f w axß xb œ 8x
8x$ œ 0 Ê x œ 0 and y œ 0, or x œ 1 and y œ 1, or x œ
1 and y œ
1 Ê a0ß 0b, a1ß 1b, a
1ß
1b are all interior points of AB with fa0ß 0b œ 16, fa1ß 1b œ 18, and fa
1ß
1b œ 18. Endpoints: fa
2ß
2b œ 0 and fa2ß 2b œ 0. On BC, faxß yb œ fa2ß yb œ 8y
y% for
2 Ÿ y Ÿ 2 3 Ê f w a2ß yb œ 8
4y$ œ 0 Ê y œ È 2 and x œ 2 3 Ê Š2ß È 2‹ is an interior critical point of BC with 3 3 f Š2ß È 2‹ œ 6 È 2. Endpoints: fa2ß
2b œ
32 and fa2ß 2b œ 0.

3 (iii) On AC, faxß yb œ faxß
2b œ
8x
x% for
2 Ÿ x Ÿ 2 Ê f w axß
2b œ
8
4x$ œ 0 Ê x œ È
2 and y œ
2 3 3 3 Ê ŠÈ
2ß
2‹ is an interior critical point of AC with f ŠÈ
2ß
2‹ œ 6 È 2. Endpoints:

fa
2ß
2b œ 0 and fa2ß
2b œ
32. (iv) For the interior of the triangular region, fx axß yb œ 4y
4x$ œ 0 and fy axß yb œ 4x
4y$ œ 0 Ê x œ 0 and y œ 0, or x œ 1 and y œ 1 or x œ
1 and y œ
1. But neither of the points a0ß 0b and a1ß 1b, or a
1ß
1b are interior to the region. Therefore the absolute maximum is 18 at (1ß 1) and (
1ß
1), and the absolute minimum is
32 at a2ß
2b. On AB, f(xß y) œ f(
1ß y) œ y$
3y#  2 for
1 Ÿ y Ÿ 1 Ê f w (
1ß y) œ 3y#
6y œ 0 Ê y œ 0 and x œ
1, or y œ 2 and x œ
1 Ê (
1ß 0) is an interior critical point of AB with f(
1ß 0) œ 2; (
1ß 2) is outside the boundary. Endpoints: f(
1ß
1) œ
2 and f(
1ß 1) œ 0. (ii) On BC, f(xß y) œ f(xß 1) œ x$  3x#
2 for
1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x#  6x œ 0 Ê x œ 0 and y œ 1, or x œ
2 and y œ 1 Ê (0ß 1) is an interior critical point of BC with f(!ß 1) œ
2; (
2ß 1) is outside the boundary. Endpoints: f(
"ß 1) œ 0 and f("ß 1) œ 2. (iii) On CD, f(xß y) œ f("ß y) œ y$
3y#  4 for
1 Ÿ y Ÿ 1 Ê f w ("ß y) œ 3y#
6y œ 0 Ê y œ 0 and x œ 1, or y œ 2 and x œ 1 Ê ("ß 0) is an interior critical point of CD with f("ß 0) œ 4; (1ß 2) is outside the boundary. Endpoints: f(1ß 1) œ 2 and f("ß
1) œ 0. (iv) On AD, f(xß y) œ f(xß
1) œ x$  3x#
4 for
1 Ÿ x Ÿ 1 Ê f w (xß
1) œ 3x#  6x œ 0 Ê x œ 0 and y œ
1, or x œ
2 and y œ
1 Ê (0ß
1) is an interior point of AD with f(0ß
1) œ
4; (
#ß
1) is outside the boundary. Endpoints: f(
1ß
1) œ
2 and f("ß
1) œ 0. (v) For the interior of the square, fx (xß y) œ 3x#  6x œ 0 and fy (xß y) œ 3y#
6y œ 0 Ê x œ 0 or x œ
2, and y œ 0 or y œ 2 Ê (0ß 0) is an interior critical point of the square region with f(!ß 0) œ 0; the points (0ß 2), (
2ß 0), and (
2ß 2) are outside the region. Therefore the absolute maximum is 4 at (1ß 0) and the absolute minimum is
4 at (0ß
1).

77. (i)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

872

Chapter 14 Partial Derivatives

On AB, f(xß y) œ f(
1ß y) œ y$
3y for
1 Ÿ y Ÿ 1 Ê f w (
1ß y) œ 3y#
3 œ 0 Ê y œ „ 1 and x œ
1 yielding the corner points (
1ß
1) and (
1ß 1) with f(
1ß
1) œ 2 and f(
1ß 1) œ
2. (ii) On BC, f(xß y) œ f(xß 1) œ x$  3x  2 for
1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x#  3 œ 0 Ê no solution. Endpoints: f(
"ß 1) œ
2 and f("ß 1) œ 6. (iii) On CD, f(xß y) œ f("ß y) œ y$  3y  2 for
1 Ÿ y Ÿ 1 Ê f w ("ß y) œ 3y#  3 œ 0 Ê no solution. Endpoints: f(1ß 1) œ 6 and f("ß
1) œ
2. (iv) On AD, f(xß y) œ f(xß
1) œ x$
3x for
1 Ÿ x Ÿ 1 Ê f w (xß
1) œ 3x#
3 œ 0 Ê x œ „ 1 and y œ
1 yielding the corner points (
1ß
1) and (1ß
1) with f(
1ß
1) œ 2 and f(1ß
1) œ
2 (v) For the interior of the square, fx (xß y) œ 3x#  3y œ 0 and fy (xß y) œ 3y#  3x œ 0 Ê y œ
x# and x%  x œ 0 Ê x œ 0 or x œ
1 Ê y œ 0 or y œ
1 Ê (!ß 0) is an interior critical point of the square region with f(0ß 0) œ 1; (
1ß
1) is on the boundary. Therefore the absolute maximum is 6 at ("ß 1) and the absolute minimum is
2 at (1ß
1) and (
1ß 1).

78. (i)

79. ™ f œ 3x# i  2yj and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê 3x# i  2yj œ -(2xi  2yj) Ê 3x# œ 2x- and 2y œ 2y- Ê - œ 1 or y œ 0. CASE 1: - œ 1 Ê 3x# œ 2x Ê x œ 0 or x œ 23 ; x œ 0 Ê y œ „ 1 yielding the points (0ß 1) and (!ß
1); x œ Ê yœ „

È5 3

yielding the points Š 32 ß

È5 3 ‹

and Š 32 ß


2 3

È5 3 ‹.

CASE 2: y œ 0 Ê x#
1 œ 0 Ê x œ „ 1 yielding the points (1ß 0) and (
1ß 0). Evaluations give f a!ß „ 1b œ 1, f Š 23 ß „

È5 3 ‹

œ

23 27

, f("ß 0) œ 1, and f(
"ß 0) œ
1. Therefore the absolute

maximum is 1 at a!ß „ 1b and (1ß 0), and the absolute minimum is
1 at (
"ß !). 80. ™ f œ yi  xj and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê yi  xj œ -(2xi  2yj) Ê y œ 2-x and xy œ 2-y Ê x œ 2-(2-x) œ 4-# x Ê x œ 0 or 4-# œ 1. CASE 1: x œ 0 Ê y œ 0 but (0ß 0) does not lie on the circle, so no solution. CASE 2: 4-# œ 1 Ê - œ "# or - œ
"# . For - œ "# , y œ x Ê 1 œ x#  y# œ 2x# Ê x œ C œ „ È"2 yielding the points Š È"2 ß È"2 ‹ and Š
È"2 ,
È"2 ‹ . For - œ
#" , y œ
x Ê 1 œ x#  y# œ 2x# Ê x œ „

" È2

and

y œ
x yielding the points Š
È"2 ß È"2 ‹ and Š È"2 ,
È"2 ‹ . Evaluations give the absolute maximum value f Š È"2 ß È"2 ‹ œ f Š
È"2 ß
È"2 ‹ œ

" #

and the absolute minimum

value f Š
È"2 ß È"2 ‹ œ f Š È"2 ß
È"2 ‹ œ
#" . 81. (i) f(xß y) œ x#  3y#  2y on x#  y# œ 1 Ê ™ f œ 2xi  (6y  2)j and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê 2xi  (6y  2)j œ -(2xi  2yj) Ê 2x œ 2x- and 6y  2 œ 2y- Ê - œ 1 or x œ 0. CASE 1: - œ 1 Ê 6y  2 œ 2y Ê y œ
"# and x œ „

È3 #

yielding the points Š „

È3 #

ß
#" ‹ .

CASE 2: x œ 0 Ê y# œ 1 Ê y œ „ 1 yielding the points a!ß „ 1b . Evaluations give f Š „

È3 #

ß
"# ‹ œ

" #

, f(0ß 1) œ 5, and f(!ß
1) œ 1. Therefore

" #

and 5 are the extreme

values on the boundary of the disk. (ii) For the interior of the disk, fx (xß y) œ 2x œ 0 and fy (xß y) œ 6y  2 œ 0 Ê x œ 0 and y œ
"3 Ê ˆ!ß
13 ‰ is an interior critical point with f ˆ!ß
3" ‰ œ
3" . Therefore the absolute maximum of f on the disk is 5 at (0ß 1) and the absolute minimum of f on the disk is
"3 at ˆ!ß
3" ‰ .

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Chapter 14 Practice Exercises

873

82. (i) f(xß y) œ x#  y#
3x
xy on x#  y# œ 9 Ê ™ f œ (2x
3
y)i  (2y
x)j and ™ g œ 2xi  2yj so that ™ f œ - ™ g Ê (2x
3
y)i  (2y
x)j œ -(2xi  2yj) Ê 2x
3
y œ 2x- and 2y
x œ 2yÊ 2x("
-)
y œ 3 and
x  2y(1
-) œ 0 Ê 1
- œ

x 2y

x and (2x) Š 2y ‹
y œ 3 Ê x#
y# œ 3y

Ê x# œ y#  3y. Thus, 9 œ x#  y# œ y#  3y  y# Ê 2y#  3y
9 œ 0 Ê (2y
3)(y  3) œ 0 Ê y œ
3, 3# . For y œ
3, x#  y# œ 9 Ê x œ 0 yielding the point (0ß
3). For y œ 3# , x#  y# œ 9 Ê x# 

9 4

œ 9 Ê x# œ

Ê xœ „

27 4

È

¸ 20.691, and f Š 3 # 3 , 3# ‹ œ 9


27È3 4

3È 3 #

È

. Evaluations give f(0ß
3) œ 9, f Š
3 # 3 ß 3# ‹ œ 9 

27È3 4

¸
2.691.

(ii) For the interior of the disk, fx (xß y) œ 2x
3
y œ 0 and fy (xß y) œ 2y
x œ 0 Ê x œ 2 and y œ 1 Ê (2ß 1) is an interior critical point of the disk with f(2ß 1) œ
3. Therefore, the absolute maximum of f on the disk is 9 

27È3 4

È

at Š
3 # 3 ß 3# ‹ and the absolute minimum of f on the disk is
3 at (2ß 1).

83. ™ f œ i
j  k and ™ g œ 2xi  2yj  2zk so that ™ f œ - ™ g Ê i
j  k œ -(2xi  2yj  2zk) Ê 1 œ 2x-,
1 œ 2y-, 1 œ 2z- Ê x œ
y œ z œ -" . Thus x#  y#  z# œ 1 Ê 3x# œ 1 Ê x œ „ È"3 yielding the points Š È"3 ß
È"3 ,

" È3 ‹

and Š
È"3 ,

f Š È"3 ß
È"3 ß È"3 ‹ œ

3 È3

" È3

,
È"3 ‹ . Evaluations give the absolute maximum value of

œ È3 and the absolute minimum value of f Š
È"3 ß È"3 ß
È"3 ‹ œ
È3.

84. Let f(xß yß z) œ x#  y#  z# be the square of the distance to the origin and g(xß yß z) œ x#
zy
4. Then ™ f œ 2xi  2yj  2zk and ™ g œ 2xi
zj
yk so that ™ f œ - ™ g Ê 2x œ 2-x, 2y œ
-z, and 2z œ
-y Ê x œ 0 or - œ 1. CASE 1: x œ 0 Ê zy œ
4 Ê z œ
4y and y œ
4z Ê 2 Š
y4 ‹ œ
-y and 2 ˆ
4z ‰ œ
-z Ê 8 -

8 -

œ y# and

œ z# Ê y# œ z# Ê y œ „ z. But y œ x Ê z# œ
4 leads to no solution, so y œ
z Ê z# œ 4

Ê z œ „ 2 yielding the points (0ß
2ß 2) and (0ß 2ß
2). CASE 2: - œ 1 Ê 2z œ
y and 2y œ
z Ê 2y œ
ˆ
y# ‰ Ê 4y œ y Ê y œ 0 Ê z œ 0 Ê x#
4 œ 0 Ê

x œ „ 2 yielding the points (
2ß 0ß 0) and (2ß !ß 0). Evaluations give f(0ß
2ß 2) œ f(0ß 2ß
2) œ 8 and f(
2ß 0ß 0) œ f(2ß !ß 0) œ 4. Thus the points (
2ß 0ß 0) and (2ß !ß 0) on the surface are closest to the origin.

85. The cost is f(xß yß z) œ 2axy  2bxz  2cyz subject to the constraint xyz œ V. Then ™ f œ - ™ g Ê 2ay  2bz œ -yz, 2ax  2cz œ -xz, and 2bx  2cy œ -xy Ê 2axy  2bxz œ -xyz, 2axy  2cyz œ -xyz, and 2bxz  2cyz œ -xyz Ê 2axy  2bxz œ 2axy  2cyz Ê y œ ˆ bc ‰ x. Also 2axy  2bxz œ 2bxz  2cyz Ê z œ ˆ ca ‰ x. Then x ˆ bc x‰ ˆ ca x‰ œ V Ê x$ œ #

Height œ z œ ˆ ac ‰ Š cabV ‹

"Î$

#

c# V ab

œ Š abcV ‹

#

Ê width œ x œ Š cabV ‹

"Î$

"Î$

#

, Depth œ y œ ˆ bc ‰ Š cabV ‹

"Î$

#

œ Š bacV ‹

"Î$

, and

.

86. The volume of the pyramid in the first octant formed by the plane is V(aß bß c) œ

" 3

ˆ "# ab‰ c œ

" 6

abc. The point

(2ß 1ß 2) on the plane Ê  "b  2c œ 1. We want to minimize V subject to the constraint 2bc  ac  2ab œ abc. ac ab Thus, ™ V œ bc 6 i  6 j  6 k and ™ g œ (c  2b
bc)i  (2c  2a
ac)j  (2b  a
ab)k so that ™ V œ ac ab abc Ê bc 6 œ -(c  2b
bc), 6 œ -(2c  2a
ac), and 6 œ -(2b  a
ab) Ê 6 œ -(ac  2ab
abc), abc abc 6 œ -(2bc  2ab
abc), and 6 œ -(2bc  ac
abc) Ê -ac œ 2-bc and 2-ab œ 2-bc. Now - Á 0 since 2 a

a Á 0, b Á 0, and c Á 0 Ê ac œ 2bc and ab œ bc Ê a œ 2b œ c. Substituting into the constraint equation gives y 2 2 2 x z a  a  a œ 1 Ê a œ 6 Ê b œ 3 and c œ 6. Therefore the desired plane is 6  3  6 œ 1 or x  2y  z œ 6.

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™g

874

Chapter 14 Partial Derivatives

87. ™ f œ (y  z)i  xj  xk , ™ g œ 2xi  2yj , and ™ h œ zi  xk so that ™ f œ - ™ g  . ™ h Ê (y  z)i  xj  xk œ -(2xi  2yj)  .(zi  xk) Ê y  z œ 2-x  .z, x œ 2-y, x œ .x Ê x œ 0 or . œ 1. CASE 1: x œ 0 which is impossible since xz œ 1. CASE 2: . œ 1 Ê y  z œ 2-x  z Ê y œ 2-x and x œ 2-y Ê y œ (2-)(2-y) Ê y œ 0 or 4-# œ 1. If y œ 0, then x# œ 1 Ê x œ „ 1 so with xz œ 1 we obtain the points (1ß 0ß 1) and (
1ß 0ß
1). If 4-# œ 1, then - œ „ "# . For - œ
"# , y œ
x so x#  y# œ 1 Ê x# œ "# Ê xœ „

" È2

with xz œ 1 Ê z œ „ È2, and we obtain the points Š È"2 ß
È"2 ß È2‹ and

Š
È"2 ß È"2 ß
È2‹ . For - œ

" #

, y œ x Ê x# œ

" #

Ê xœ „

" È2

with xz œ 1 Ê z œ „ È2,

and we obtain the points Š È"2 ß È"2 , È2‹ and Š
È"2 ß
È"2 ß
È2‹ . Evaluations give f(1ß 0ß 1) œ 1, f(
1ß 0ß
1) œ 1, f Š È"2 ß
È"2 ß È2‹ œ f Š È"2 ß È"2 ß È2‹ œ

3 #

, and f Š
È"2 ß
È"2 ß
È2‹ œ

3 #

" #

, f Š
È"2 ß È"2 ,
È2‹ œ 3 #

. Therefore the absolute maximum is

Š È"2 ß È"2 ß È2‹ and Š
È"2 ß
È"2 ß
È2‹ , and the absolute minimum is

" #

" #

,

at

at Š
È"2 ß È"2 ß
È2‹ and

Š È"2 ß
È"2 ß È2‹ . 88. Let f(xß yß z) œ x#  y#  z# be the square of the distance to the origin. Then ™ f œ 2xi  2yj  2zk , ™ g œ i  j  k , and ™ h œ 4xi  4yj
2zk so that ™ f œ - ™ g  . ™ h Ê 2x œ -  4x., 2y œ -  4y., and 2z œ -
2z. Ê - œ 2x(1
2.) œ 2y(1
2.) œ 2z(1  2.) Ê x œ y or . œ "# . CASE 1: x œ y Ê z# œ 4x# Ê z œ „ 2x so that x  y  z œ 1 Ê x  x  2x œ 1 or x  x
2x œ 1 (impossible) Ê x œ "4 Ê y œ "4 and z œ "# yielding the point ˆ "4 ß "4 ß "# ‰ . CASE 2: . œ

" #

Ê - œ 0 Ê 0 œ 2z(1  1) Ê z œ 0 so that 2x#  2y# œ 0 Ê x œ y œ 0. But the origin

(!ß 0ß 0) fails to satisfy the first constraint x  y  z œ 1. Therefore, the point ˆ "4 ß 4" ß "# ‰ on the curve of intersection is closest to the origin. 89. (a) y, z are independent with w œ x# eyz and z œ x#
y# Ê œ a2xeyz b

`x `y

`w `y

`w `x `w `y `w `z `x `y  `y `y  `z `y œ 2x `` xy
2y Ê `` xy œ yx

œ

 azx# eyz b (1)  ayx# eyz b (0); z œ x#
y# Ê 0

; therefore,

Š ``wy ‹ œ a2xeyz b ˆ xy ‰  zx# eyz œ a2y  zx# b eyz z

(b) z, x are independent with w œ x# eyz and z œ x#
y# Ê œ a2xeyz b (0)  azx# eyz b

`y `z

`w `z

œ

`w `x `x `z



 ayx# eyz b (1); z œ x#
y# Ê 1 œ 0
2y

1 ˆ ``wz ‰ œ azx# eyz b Š
2y ‹  yx# eyz œ x# eyz Šy
x

`w `y `w `z `y `z  `z `z `y `y " ` z Ê ` z œ
#y

; therefore,

z 2y ‹

(c) z, y are independent with w œ x# eyz and z œ x#
y# Ê

`w `z

œ

œ a2xeyz b `` xz  azx# eyz b (0)  ayx# eyz b (1); z œ x#
y# Ê 1 1 ‰ ˆ ``wz ‰ œ a2xeyz b ˆ 2x  yx# eyz œ a1  x# yb eyz

`w `x `w `y `w `z `x `z  `y `z  `z `z œ 2x `` xz
0 Ê `` xz œ #"x

; therefore,

y

90. (a) T, P are independent with U œ f(Pß Vß T) and PV œ nRT Ê ``UT œ ``UP `` TP  ‰ ˆ ``VT ‰  ˆ ``UT ‰ (1); PV œ nRT Ê P ``VT œ nR Ê ``VT œ œ ˆ ``UP ‰ (0)  ˆ `` U V ˆ ``UT ‰ œ ˆ `` U ‰ ˆ nR ‰ V P  P

`U `T

`U `V `U `T `V `T  `T `T nR P ; therefore,

`U `P `U `V (b) V, T are independent with U œ f(Pß Vß T) and PV œ nRT Ê `` U V œ `P `V  `V `V  U‰ œ ˆ ``UP ‰ ˆ ``VP ‰  ˆ `` V (1)  ˆ ``UT ‰ (0); PV œ nRT Ê V ``VP  P œ (nR) ˆ ``VT ‰ œ 0 Ê

ˆ `` U ‰ V T

œ

ˆ ``UP ‰ ˆ
VP ‰



`U `V

`U `T `T `V `P P `V œ
V

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; therefore,

Chapter 14 Practice Exercises

875

91. Note that x œ r cos ) and y œ r sin ) Ê r œ Èx#  y# and ) œ tan" ˆ yx ‰ . Thus, `w `x

œ

`w `r `r `x



`w `) `) `x

œ ˆ ``wr ‰ Š Èx#x
y# ‹  ˆ ``w) ‰ Š x#
yy# ‹ œ (cos ))

`w `r


ˆ sinr ) ‰

`w `)

`w `y

œ

`w `r `r `y



`w `) ` ) )y

œ ˆ ``wr ‰ Š Èx#y
y# ‹  ˆ ``w) ‰ Š x#
x y# ‹ œ (sin ))

`w `r

 ˆ cosr ) ‰

`w `)

92. zx œ fu 93.

`u `y

`v `x

 fv

œ afu  afv , and zy œ fu

`u `x œ a " `w " `w a `x œ b `y

œ b and

Ê 94.

`u `x

`w `x

œ

and

`w `z

œ

2 r
s

œ

Ê

œ

2 x#
y#
2z

and

`w `s

œ

`w dw ` x œ du b ``wx œ a

" (r
s)#

`w `x `x `s

`w `y `y `s



Solving this system yields Ê ae cos vb

`u `y

`u `x


ae sin vb

u

`v `y

dw ` u du ` y

œ

2(r
s) 2 ar#
2rs
s# b

œ

" r
s

œ

`w `x `x `r

`g `)

œ

Ê

`f `x `x `) ` #g ` )#



`f `y `y `)

œ (
r sin )) Š `` x) 

`y `) ‹



" r
s






rs (r
s)#

`w `y

`w `z `z `r

dw du

œ œ

Ê

`f `x

`v `y

" r
s



œ

dw du

2(r  s) #(r
s)#

and

" `w b `y

œ

rs (r
s)#

œ

dw du

,

 ’ (r
" s)# “ (2s) œ

2r
2s (r
s)#

2 r
s


y œ 0 Ê aeu sin vb

`u `x

 aeu cos vb

`v `x

œ 0.

Similarly, e cos v
x œ 0 u

`u `y

œ eu cos v. Therefore Š `` ux i 

 (r cos ))

œ

rs (r
s)#

 ’ (r
" s)# “ (2r) œ

`v u ` x œ 1; e sin v `v u sin v. ` x œ
e

 aeu cos vb `u `y

`v `y

j‹ † Š `` vx i 

œ 1. Solving this

`v `y

j‹

u

cos vb jd œ 0 Ê the vectors are orthogonal Ê the angle

`f `x

 (r cos )) Š ``x`fy

sin vb i  ae

` #f ` y ` y` x ` ) ‹

" `w a `x

2y x#
y#
2z

œ 0 and e sin v
y œ 0 Ê aeu sin vb u

`x `)

`w `y `y `r



,

œb

u

u

#

œ

cos v and

œ eu sin v and

œ (
r sin )) Š `` xf#

and

aeu sin vb

`u `y

œ (
r sin ))

dw du

`w `z `z `s

œ cae cos vb i  ae sin vb jd † ca
e between the vectors is the constant 1# . 96.

œ bfu
bfv

œ

`u `x
u

œe

u

second system yields



`v `y

 fv

`w `y

œa

`w `r

Ê

95. eu cos v
x œ 0 Ê aeu cos vb u

`u `x `w `y

2(r
s) (r
s)#
(r  s)#
4rs -

œ

2x x#
y#
2z

Ê

`u `y

;

`f `y


(r cos ))


(r cos ))  (r cos )) Š `` x) 

`y `) ‹

#

`x `)



` #f ` y ` y# ` ) ‹


(r sin ))

`f `y


(r sin ))

œ (
r sin )  r cos ))(
r sin )  r cos ))
(r cos )  r sin )) œ (
2)(
2)
(0  2) œ 4
2 œ 2 at (rß )) œ ˆ2ß 1# ‰ . 97. (y  z)#  (z
x)# œ 16 Ê ™ f œ
2(z
x)i  2(y  z)j  2(y  2z
x)k ; if the normal line is parallel to the yz-plane, then x is constant Ê `` xf œ 0 Ê
2(z
x) œ 0 Ê z œ x Ê (y  z)#  (z
z)# œ 16 Ê y  z œ „ 4. Let x œ t Ê z œ t Ê y œ
t „ 4. Therefore the points are (tß
t „ 4ß t), t a real number.

98. Let f(xß yß z) œ xy  yz  zx
x
z# œ 0. If the tangent plane is to be parallel to the xy-plane, then ™ f is perpendicular to the xy-plane Ê ™ f † i œ 0 and ™ f † j œ 0. Now ™ f œ (y  z
1)i  (x  z)j  (y  x
2z)k so that ™ f † i œ y  z
1 œ 0 Ê y  z œ 1 Ê y œ 1
z, and ™ f † j œ x  z œ 0 Ê x œ
z. Then
z(1
z)  ("
z)z  z(
z)
(
z)
z# œ 0 Ê z
2z# œ 0 Ê z œ "# or z œ 0. Now z œ "# Ê x œ
"# and y œ Ê ˆ
"# ß "# ß "# ‰ is one desired point; z œ 0 Ê x œ 0 and y œ 1 Ê (0ß 1ß 0) is a second desired point. 99. ™ f œ -(xi  yj  zk) Ê

`f `x

œ -x Ê f(xß yß z) œ

" #

-x#  g(yß z) for some function g Ê -y œ

`f `y

œ

`g `y

-y#  h(z) for some function h Ê -z œ `` zf œ `` gz œ hw (z) Ê h(z) œ #" -z#  C for some arbitrary constant C Ê g(yß z) œ "# -y#  ˆ "# -z#  C‰ Ê f(xß yß z) œ "# -x#  "# -y#  "# -z#  C Ê f(0ß 0ß a) œ "# -a#  C Ê g(yß z) œ

" #

and f(0ß 0ß
a) œ

" #

-(
a)#  C Ê f(0ß 0ß a) œ f(0ß 0ß
a) for any constant a, as claimed.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

" #

876

Chapter 14 Partial Derivatives ß

f(0
su" ß 0
su# ß 0
su$ )f(0ß 0ß 0)

œ lim sÄ0

‰ 100. ˆ df ds u (0 0 0) ß ß

,s0

s

És# u#"
s# u##
s# u#$  0

œ lim sÄ0

s

,s0

sÉu#"
u##
u#$

œ lim œ lim kuk œ 1; s sÄ0 sÄ0 however, ™ f œ Èx#
xy#
z# i  Èx#
yy#
z# j  Èx#
zy#
z# k fails to exist at the origin (0ß 0ß 0) 101. Let f(xß yß z) œ xy  z
2 Ê ™ f œ yi  xj  k . At (1ß 1ß 1), we have ™ f œ i  j  k Ê the normal line is x œ 1  t, y œ 1  t, z œ 1  t, so at t œ
1 Ê x œ 0, y œ 0, z œ 0 and the normal line passes through the origin. 102. (b) f(xß yß z) œ x#
y#  z# œ 4 Ê ™ f œ 2xi
2yj  2zk Ê at (2ß
3ß 3) the gradient is ™ f œ 4i  6j  6k which is normal to the surface (c) Tangent plane: 4x  6y  6z œ 8 or 2x  3y  3z œ 4 Normal line: x œ 2  4t, y œ
3  6t, z œ 3  6t

CHAPTER 14 ADDITIONAL AND ADVANCED EXERCISES fx (0ß h)  fx (0ß 0) h

1. By definition, fxy (!ß 0) œ lim

hÄ0

so we need to calculate the first partial derivatives in the

numerator. For (xß y) Á (0ß 0) we calculate fx (xß y) by applying the differentiation rules to the formula for

fy (xß y) œ

00 h

œ lim

hÄ0

2.

`w `x

x$  xy# x#
y#

x# y  y$ x#
y#

ax#
y# b (2x)  ax#  y# b (2x) ax #
y # b #

4x# y$ Ê fx (0ß h) ax #
y # b# f(0ß0) For (xß y) œ (0ß 0) we apply the definition: fx (!ß 0) œ lim f(hß 0)  œ lim 0 h 0 œ h hÄ0 hÄ0 f (hß 0)  fy (!ß 0) fxy (0ß 0) œ lim hh 0 œ
1. Similarly, fyx (0ß 0) œ lim y , so for (xß y) Á h hÄ0 hÄ0

f(xß y): fx (xß y) œ




 (xy)

4x$ y# ax#
y# b#

Ê fy (hß 0) œ

h$ h#

x# y  y $ x#
y#

œ



$

œ
hh# œ
h. 0. Then by definition (0ß 0) we have

œ h; for (xß y) œ (0ß 0) we obtain fy (0ß 0) œ lim h0 h

œ 0. Then by definition fyx (0ß 0) œ lim

hÄ0

œ 1  ex cos y Ê w œ x  ex cos y  g(y);

`w `y

hÄ0

f(0ß h)  f(!ß 0) h

œ 1. Note that fxy (0ß 0) Á fyx (0ß 0) in this case.

œ
ex sin y  gw (y) œ 2y
ex sin y Ê gw (y) œ 2y

Ê g(y) œ y#  C; w œ ln 2 when x œ ln 2 and y œ 0 Ê ln 2 œ ln 2  eln 2 cos 0  0#  C Ê 0 œ 2  C Ê C œ
2. Thus, w œ x  ex cos y  g(y) œ x  ex cos y  y#
2. 3. Substitution of u  u(x) and v œ v(x) in g(uß v) gives g(u(x)ß v(x)) which is a function of the independent variable x. Then, g(uß v) œ 'u f(t) dt Ê v

œ Š
``u

#

#

fzz œ Š ddr#f ‹ ˆ ``zr ‰  ` #r ` y#

œ

` g du ` u dx



` g dv ` v dx

œ Š ``u

#

df ` # r dr ` x#

'uv f(t) dt‹ dxdu  Š ``v 'uv f(t) dt‹ dxdv

'vu f(t) dt‹ dudx  Š ``v 'uv f(t) dt‹ dvdx œ
f(u(x)) dudx  f(v(x)) dvdx œ f(v(x)) dvdx
f(u(x)) dudx

4. Applying the chain rules, fx œ

Ê

dg dx

œ

#

df ` r dr ` z#

x#
z# 3 ˆÈx#
y#
z# ‰

df ` r dr ` x

#

Ê fxx œ Š ddr#f ‹ ˆ ``xr ‰ 

. Moreover,

; and

`r `z

œ

`r `x

œ

x È x #
y #
z#

z È x #
y #
z#

Ê

` #r ` z#

Ê

œ

#

` r ` x#

œ

#

#

. Similarly, fyy œ Š ddr#f ‹ Š ``yr ‹  #

#

y
z 3 ˆÈx#
y#
z# ‰

x#
y# 3 ˆÈ x #
y #
z # ‰

;

`r `y

œ

y È x #
y #
z#

. Next, fxx  fyy  fzz œ 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

df ` # r dr ` y#

and

Chapter 14 Additional and Advanced Exercises #

#

df dr

d dr

x#

(f w ) œ ˆ
2r ‰ f w , where f w œ

œ Cr# Ê f(r) œ
Cr  b œ


y #
z# ‰

Ê

df dr

y#

d# f

x #
y #
z# ‰

x#
y#

#

‰  Š ddr#f ‹ Š x#
yz #
z# ‹  ˆ df dr Œ ˆÈ Ê

y #
z#

#

‰ Ê Š ddr#f ‹ Š x#
xy#
z# ‹  ˆ df dr Œ ˆÈ

3

df f

3

877

x #
z#

  Š dr# ‹ Š x#
y#
z# ‹  ˆ dr ‰ Œ ˆÈx#
y#
z# ‰3  d# f dr#

œ0 Ê

df

 Š Èx#
2y#
z# ‹

d# f dr#

œ0 Ê

df dr



2 df r dr

œ0

œ
2 rdr Ê ln f w œ
2 ln r  ln C Ê f w œ Cr# , or

w

w

 b for some constants a and b (setting a œ
C)

a r

5. (a) Let u œ tx, v œ ty, and w œ f(uß v) œ f(u(tß x)ß v(tß y)) œ f(txß ty) œ tn f(xß y), where t, x, and y are independent variables. Then ntnc1 f(xß y) œ ``wt œ ``wu ``ut  ``wv ``vt œ x ``wu  y ``wv . Now, `w `w `u `w `v `w `w ˆ `w ‰ ˆ `w ‰ ˆ " ‰ ˆ ``wx ‰ . Likewise, ` x œ ` u ` x  ` v ` x œ ` u (t)  ` v (0) œ t ` u Ê ` u œ t `w `y

œ

`w `u `u `y



ntnc1 f(xß y) œ x `w `x

Ê

œ

`f `x

`w `v `v `y `w `u

`w `v

y

`w `y

and

œ ˆ ``wu ‰ (0)  ˆ ``wv ‰ (t) Ê œ

`f `x

Ê nf(xß y) œ x

Also from part (a), œ

` `y

œ t#

ˆt

`w ‰ `v

` #w ` v` u

œt

` #w ` x#

` `x

œ

x

y

ˆ ``wx ‰

` `x

`w ‰ `u

t

œ

`f `x

`w `v . ` w `v ` v` u ` t

` w `u ` u# ` t

` #w ` u ` u` v ` y

Ê ˆ t"# ‰

`w `u

#

` #w ` x#

œ ˆ "t ‰ Š ``wy ‹ . Therefore,

œ ˆ xt ‰ ˆ ``wx ‰  ˆ yt ‰ Š ``wy ‹. When t œ 1, u œ x, v œ y, and w œ f(xß y)

(b) From part (a), ntnc1 f(xß y) œ x n(n
1)tnc2 f(xß y) œ x

`w `v

y

` #w ` v ` v# ` y

` #w ` u#

ˆt

` #w ` v#

œ t#

, ˆ t"# ‰

` #w ` y#

`f `y

, as claimed.

Differentiating with respect to t again we obtain

#

œ

y

œ

` #w ` u ` u` v ` t

œt

, and

` #w ` v#

` #w ` v ` v# ` t

y

` #w ` u ` u# ` x

t

` #w ` y` x

` `y

œ

, and ˆ t"# ‰

œ x#

` #w ` u#

 2xy

` #w ` v ` v` u ` x

œ

# t# `` uw#

ˆ ``wx ‰ œ

` `y

ˆt

` #w ` y` x

œ

`w ‰ `u

,

` #w ` u` v

` #w ` y#

œt

œ

 y# ` `y

` #w ` v#

.

Š ``wy ‹

` #w ` u ` u# ` y

t

` #w ` v ` v` u ` y

` #w ` v` u

‰ Š ``y`wx ‹  Š yt# ‹ Š `` yw# ‹ for t Á 0. When t œ 1, w œ f(xß y) and Ê n(n
1)tnc2 f(xß y) œ Š xt# ‹ Š `` xw# ‹  ˆ 2xy t# #

#

#

#

#

#

#

#

we have n(n
1)f(xß y) œ x# Š `` xf# ‹  2xy Š ``x`fy ‹  y# Š `` yf# ‹ as claimed. 6. (a) lim

rÄ0

sin 6r 6r

œ lim

tÄ0

sin t t

œ 1, where t œ 6r

f(0
hß 0)  f(0ß 0) h hÄ0 36 sin 6h lim œ0 12 hÄ0

(b) fr (0ß 0) œ lim œ

f(rß )
h)  f(rß )) h hÄ0

(c) f) (rß )) œ lim

ˆ sin6h6h ‰ 1 h hÄ0

œ lim

œ lim

hÄ0

œ lim

hÄ0

6 cos 6h  6 12h

(applying l'Hopital's rule twice) s ˆ sin6r6r ‰  ˆ sin6r6r ‰ h hÄ0

œ lim

œ lim

0

hÄ0 h

7. (a) r œ xi  yj  zk Ê r œ krk œ Èx#  y#  z# and ™ r œ (b) rn œ ˆÈx#  y#  z# ‰

sin 6h  6h 6h#

œ0

x È x #
y #
z#

i

y È x #
y #
z#

j

z È x #
y #
z#

kœ

r r

n

ÐnÎ2Ñ

1

(d) dr œ dxi  dyj  dzk Ê r † dr œ x dx  y dy  z dz, and dr œ rx dx  ry dy  rz dz œ

x r

Ê ™ arn b œ nx ax#  y#  z# b (c) Let n œ 2 in part (b). Then

" #

ÐnÎ2Ñ 1

ÐnÎ2Ñ

i  ny ax#  y#  z# b j  nz ax#  y#  z# b k œ nrn 2 r # ™ ar# b œ r Ê ™ ˆ "# r# ‰ œ r Ê r# œ #" ax#  y#  z# b is the function. 1

dx 

y r

dy 

z r

dz

Ê r dr œ x dx  y dy  z dz œ r † dr (e) A œ ai  bj  ck Ê A † r œ ax  by  cz Ê ™ (A † r) œ ai  bj  ck œ A 8. f(g(t)ß h(t)) œ c Ê 0 œ

df dt

œ

` f dx ` x dt



` f dy ` y dt

œ Š `` xf i 

`f `y

j‹ † Š dx dt i 

dy dt

j‹ , where

dx dt

i

dy dt

j is the tangent vector

Ê ™ f is orthogonal to the tangent vector 9. f(xß yß z) œ xz#
yz  cos xy
1 Ê ™ f œ az#
y sin xyb i  (
z
x sin xy)j  (2xz
y)k Ê ™ f(0ß 0ß 1) œ i
j Ê the tangent plane is x
y œ 0; r œ (ln t)i  (t ln t)j  tk Ê rw œ ˆ "t ‰ i  (ln t  1)j  k ; x œ y œ 0, z œ 1 Ê t œ 1 Ê rw (1) œ i  j  k . Since (i  j  k) † (i
j) œ rw (1) † ™ f œ 0, r is parallel to the plane, and r(1) œ 0i  0j  k Ê r is contained in the plane.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

878

Chapter 14 Partial Derivatives

10. Let f(xß yß z) œ x$  y$  z$
xyz Ê ™ f œ a3x#
yzb i  a3y#
xzb j  a3z#
xyb k Ê ™ f(0ß
1ß 1) œ i  3j  3k $

Ê the tangent plane is x  3y  3z œ 0; r œ Š t4
2‹ i  ˆ 4t
3‰ j  (cos (t
2)) k #

Ê rw œ Š 3t4 ‹ i
ˆ t4# ‰ j
(sin (t
2)) k ; x œ 0, y œ
1, z œ 1 Ê t œ 2 Ê rw (2) œ 3i
j . Since rw (2) † ™ f œ 0 Ê r is parallel to the plane, and r(2) œ
i  k Ê r is contained in the plane. 11.

`z `x

œ 3x#
9y œ 0 and

`z `y

œ 3y#
9x œ 0 Ê y œ

" 3

#

x# and 3 ˆ "3 x# ‰
9x œ 0 Ê

" 3

x%
9x œ 0

Ê x ax$
27b œ 0 Ê x œ 0 or x œ 3. Now x œ 0 Ê y œ 0 or (!ß 0) and x œ 3 Ê y œ 3 or (3ß 3). Next ` #z ` x#

œ 6x,

` #z ` y#

œ 6y, and

and for (3ß 3),

` #z ` #z ` x# ` y#

` #z ` x` y #

` #z ` #z ` x# ` y#

œ
9. For (!ß 0), #

` #z ` x#


Š ``x`zy ‹ œ 243  0 and

#

#


Š ``x`zy ‹ œ
81 Ê no extremum (a saddle point),

œ 18  0 Ê a local minimum.

12. f(xß y) œ 6xyeÐ2x
3yÑ Ê fx (xß y) œ 6y(1
2x)eÐ2x
3yÑ œ 0 and fy (xß y) œ 6x(1
3y)eÐ2x
3yÑ œ 0 Ê x œ 0 and y œ 0, or x œ "# and y œ 3" . The value f(0ß 0) œ 0 is on the boundary, and f ˆ "# ß "3 ‰ œ e"2 . On the positive y-axis,

f(0ß y) œ 0, and on the positive x-axis, f(xß 0) œ 0. As x Ä _ or y Ä _ we see that f(xß y) Ä 0. Thus the absolute maximum of f in the closed first quadrant is e"2 at the point ˆ #" ß 3" ‰ .

13. Let f(xß yß z) œ P! (x! ß y! ß y! ) is

y# x# a#  b# !‰ ˆ 2x a# x

 

z# c#
1 !‰ ˆ 2y b# y

Ê ™fœ  ˆ 2zc#! ‰ z

2y 2x a# i  b# j  # 2y#! ! œ 2x a#  b# 

#

2z c# k Ê an equation of the plane tangent 2z#! ˆ x! ‰ ˆ y! ‰ ˆ z! ‰ c# œ 2 or a# x  b# y  c# z œ 1.

#

at the point

#

The intercepts of the plane are Š xa! ß 0ß 0‹ , Š0ß by! ß 0‹ and Š!ß !ß zc! ‹ . The volume of the tetrahedron formed by the #

#

#

plane and the coordinate planes is V œ ˆ "3 ‰ ˆ #" ‰ Š xa! ‹ Š by! ‹ Š cz! ‹ Ê we need to maximize V(xß yß z) œ subject to the constraint f(xß yß z) œ #

" and ’
(abc) 6 “ Š xyz# ‹ œ

2z c#

x# a#



y# b#



#

z# c#

" œ 1. Thus, ’
(abc) 6 “ Š x# yz ‹ œ

2x a#

(abc)# 6

#

" -, ’
(abc) 6 “ Š xy# z ‹ œ

(xyz)"

2y b#

-,

-. Multiply the first equation by a# yz, the second by b# xz, and the third by c# xy. Then equate

the first and second Ê a# y# œ b# x# Ê y œ substitute into f(xß yß z) œ 0 Ê x œ

a È3

b a

x, x  0; equate the first and third Ê a# z# œ c# x# Ê z œ ca x, x  0;

Ê yœ

Ê zœ

b È3

c È3

Ê Vœ

È3 #

abc.

14. 2(x
u) œ
-, 2(y
v) œ -,
2(x
u) œ ., and
2(y
v) œ
2.v Ê x
u œ v
y, x
u œ
.# , and y
v œ .v Ê x
u œ
.v œ
.# Ê v œ

" #

or . œ 0.

CASE 1: . œ 0 Ê x œ u, y œ v, and - œ 0; then y œ x  1 Ê v œ u  1 and v# œ u Ê v œ v#  1 1 „ È1  4 Ê # " " " " # v œ # and u œ v Ê u œ 4 ; x
4 œ #
Ê y œ 78 . Then f ˆ
8" ß 87 ß "4 ß "# ‰ œ ˆ
8"

Ê v#
v  1 œ 0 Ê v œ

CASE 2:

no real solution. " 4 œ # 2 ˆ 38 ‰

y and y œ x  1 Ê x
# #
"4 ‰  ˆ 78
#" ‰ œ

Ê 2x œ
4" Ê x œ
8" Ê the minimum distance is 38 È2.


x


" #

(Notice that f has no maximum value.) 15. Let (x! ß y! ) be any point in R. We must show lim

Ðhß kÑ Ä Ð0ß 0Ñ

lim

Ðxß yÑ Ä Ðx! ß y! Ñ

f(xß y) œ f(x! ß y! ) or, equivalently that

kf(x!  hß y!  k)
f(x! ß y! )k œ 0. Consider f(x!  hß y!  k)
f(x! ß y! )

œ [f(x!  hß y!  k)
f(x! ß y!  k)]  [f(x! ß y!  k)
f(x! ß y! )]. Let F(x) œ f(xß y!  k) and apply the Mean Value Theorem: there exists 0 with x!  0  x!  h such that Fw (0 )h œ F(x!  h)
F(x! ) Ê hfx (0ß y!  k) œ f(x!  hß y!  k)
f(x! ß y!  k). Similarly, k fy (x! ß () œ f(x! ß y!  k)
f(x! ß y! ) for some ( with y!  (  y!  k. Then kf(x!  hß y!  k)
f(x! ß y! )k Ÿ khfx (0ß y!  k)k  kkfy (x! ß ()k . If M, N are positive real numbers such that kfx k Ÿ M and kfy k Ÿ N for all (xß y) in the xy-plane, then kf(x!  hß y!  k)
f(x! ß y! )k Ÿ M khk  N kkk . As (hß k) Ä 0, kf(x!  hß y!  k)
f(x! ß y! )k Ä 0 Ê lim kf(x!  hß y!  k)
f(x! ß y! )k Ðhß kÑ Ä Ð0ß 0Ñ

œ 0 Ê f is continuous at (x! ß y! ).

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 14 Additional and Advanced Exercises 16. At extreme values, ™ f and v œ

dr dt

df dt

are orthogonal because

œ ™f†

879

œ 0 by the First Derivative Theorem for

dr dt

Local Extreme Values. 17.

`f `x

œ 0 Ê f(xß y) œ h(y) is a function of y only. Also,

Moreover,

`f `y

œ

`g `x

`g `y

œ

`f `x

œ 0 Ê g(xß y) œ k(x) is a function of x only.

Ê hw (y) œ kw (x) for all x and y. This can happen only if hw (y) œ kw (x) œ c is a constant.

Integration gives h(y) œ cy  c" and k(x) œ cx  c# , where c" and c# are constants. Therefore f(xß y) œ cy  c" and g(xß y) œ cx  c# . Then f(1ß 2) œ g(1ß 2) œ 5 Ê 5 œ 2c  c" œ c  c# , and f(0ß 0) œ 4 Ê c" œ 4 Ê c œ Ê c# œ

9 #

. Thus, f(xß y) œ

" #

y  4 and g(xß y) œ

" #

" #

x  #9 .

18. Let g(xß y) œ Du f(xß y) œ fx (xß y)a  fy (xß y)b. Then Du g(xß y) œ gx (xß y)a  gy (xß y)b œ fxx (xß y)a#  fyx (xß y)ab  fxy (xß y)ba  fyy (xß y)b# œ fxx (xß y)a#  2fxy (xß y)ab  fyy (xß y)b# . 19. Since the particle is heat-seeking, at each point (xß y) it moves in the direction of maximal temperature increase, that is in the direction of ™ T(xß y) œ aec2y sin xb i  a2ec2y cos xb j . Since ™ T(xß y) is parallel to 2ec2y cos x ec2y sin x œ È œ
2 ln #2

the particle's velocity vector, it is tangent to the path y œ f(x) of the particle Ê f w (x) œ

2 cot x.

Integration gives f(x) œ 2 ln ksin xk  C and f ˆ 14 ‰ œ 0 Ê 0 œ 2 ln ¸sin 14 ¸  C Ê C

œ ln Š È22 ‹

#

œ ln 2. Therefore, the path of the particle is the graph of y œ 2 ln ksin xk  ln 2. 20. The line of travel is x œ t, y œ t, z œ 30
5t, and the bullet hits the surface z œ 2x#  3y# when 30
5t œ 2t#  3t# Ê t#  t
6 œ 0 Ê (t  3)(t
2) œ 0 Ê t œ 2 (since t  0). Thus the bullet hits the surface at the point (2ß 2ß 20). Now, the vector 4xi  6yj
k is normal to the surface at any (xß yß z), so that n œ 8i  12j
k is normal to the surface at (2ß 2ß 20). If v œ i  j
5k , then the velocity of the particle †25 ‰ after the ricochet is w œ v
2 projn v œ v
Š 2knvk†#n ‹ n œ v
ˆ 2209 n œ (i  j
5k)
ˆ 400 209 i 

œ
191 209 i


391 209

j


995 209

600 209

j


50 209

k‰

k.

21. (a) k is a vector normal to z œ 10
x#
y# at the point (!ß 0ß 10). So directions tangential to S at (!ß 0ß 10) will be unit vectors u œ ai  bj . Also, ™ T(xß yß z) œ (2xy  4) i  ax#  2yz  14b j  ay#  1b k Ê ™ T(!ß 0ß 10) œ 4i  14j  k . We seek the unit vector u œ ai  bj such that Du T(0ß 0ß 10) œ (4i  14j  k) † (ai  bj) œ (4i  14j) † (ai  bj) is a maximum. The maximum will occur when ai  bj has the same direction as 4i  14j , or u œ È"53 (2i  7j). (b) A vector normal to S at (1ß 1ß 8) is n œ 2i  2j  k . Now, ™ T(1ß 1ß 8) œ 6i  31j  2k and we seek the unit vector u such that Du T(1ß 1ß 8) œ ™ T † u has its largest value. Now write ™ T œ v  w , where v is parallel to ™ T and w is orthogonal to ™ T. Then Du T œ ™ T † u œ (v  w) † u œ v † u  w † u œ w † u. Thus Du T(1ß 1ß 8) is a maximum when u has the same direction as w . Now, w œ ™ T
Š ™knTk#†n ‹ n 62
2 ‰ œ (6i  31j  2k)
ˆ 124
(2i  2j  k) œ ˆ6

4
1

œ
98 9 i

127 9

j


58 9

k Ê uœ

w kwk

152 ‰ i 9

 ˆ31


152 ‰ j 9

 ˆ2


76 ‰ 9 k

" œ
È29,097 (98i
127j  58k).

22. Suppose the surface (boundary) of the mineral deposit is the graph of z œ f(xß y) (where the z-axis points up into the air). Then
`` xf i
`` yf j  k is an outer normal to the mineral deposit at (xß y) and `` xf i  `` yf j points in the direction of steepest ascent of the mineral deposit. This is in the direction of the vector

`f `x

i

`f `y

j at (0ß 0) (the location of the 1st borehole)

that the geologists should drill their fourth borehole. To approximate this vector we use the fact that (0ß 0ß
1000), (0ß 100ß
950), and (100ß !ß
1025) lie on the graph of z œ f(xß y). The plane containing these three points is a good â â j k â â i â â "00 50 â approximation to the tangent plane to z œ f(xß y) at the point (0ß 0ß 0). A normal to this plane is â 0 â â
25 â â "00 0 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

880

Chapter 14 Partial Derivatives œ
2500i  5000j
10,000k, or
i  2j
4k. So at (0ß 0) the vector

geologists should drill their fourth borehole in the direction of

" È5

`f `x

`f `y

i

j is approximately
i  2j . Thus the

(
i  2j) from the first borehole.

23. w œ ert sin 1x Ê wt œ rert sin 1x and wx œ 1ert cos 1x Ê wxx œ
1# ert sin 1x; wxx œ positive constant determined by the material of the rod Ê
1# ert sin 1x œ

" c#

" c#

wt , where c# is the

arert sin 1xb

# #

Ê ar  c# 1# b ert sin 1x œ 0 Ê r œ
c# 1# Ê w œ ec 1 t sin 1x 24. w œ ert sin kx Ê wt œ rert sin kx and wx œ kert cos kx Ê wxx œ
k# ert sin kx; wxx œ Ê
k# ert sin kx œ

" c#

" c#

wt # #

arert sin kxb Ê ar  c# k# b ert sin kx œ 0 Ê r œ
c# k# Ê w œ ec k t sin kx. # #

Now, w(Lß t) œ 0 Ê ec k t sin kL œ 0 Ê kL œ n1 for n an integer Ê k œ

n1 L

# # # # Ê w œ ec n 1 tÎL sin ˆ nL1 x‰ .

# # # # As t Ä _, w Ä 0 since ¸sin ˆ nL1 x‰¸ Ÿ 1 and ec n 1 tÎL Ä 0.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

CHAPTER 15 MULTIPLE INTEGRALS 15.1 DOUBLE AND ITERATED INTEGRALS OVER RECTANGLES 1.

'12 '04 2xy dy dx œ '12 cx y# d 40 dx œ '12 16x dx œ c8 x# d 21

2.

'02 'c11 ax
yb dy dx œ '02
xy
12 y# ‘ "
"

3.

'c01 'c11 (x  y  1) dx dy œ 'c01 ’ x2

4.

'01 '01 Š1
x
2 y ‹ dx dy œ '01 ’x
x6

5.

'03 '02 a4
y# b dy dx œ '03 ’4y
y3 “ # dx œ '03 163 dx œ
163 x‘30

6.

'03 'c02 ax# y
2xyb dy dx œ '03 ’ x 2y

7.

'01 '01 1
yx y dx dy œ '01 clnl1  x yld"0 dy œ '01 lnl1  yldy œ cy lnl1  yl
y  lnl1  yld 10 œ 2 ln 2
1

8.

'14 '04 ˆ 2x  Èy‰ dx dy œ '14
41 x2  xÈy‘ !4 dy œ '14 ˆ4  4 y1/2 ‰dy œ
4y  38 y3/2 ‘41

9.

'0ln 2 '1ln 5 e2x
y dy dx œ '0ln 2 ce2x
y dln" 5 dx œ '0ln 2 a5e2x
e2x
1 b dx œ
52 e2x
"# e2x
1 ‘0ln 2

10.

'01 '12 x y ex dy dx œ '01
"# x y2 ex ‘2" dx œ '01 32 x ex dx œ
32 x ex
32 ex ‘10

11.

'c21 '01Î2 y sin x dx dy œ 'c21 c
y cos xd10 Î2 dy œ 'c21 y dy œ
"# y2 ‘2 1 œ 32

12.

'121 '01 asin x  cos yb dx dy œ '121 c
cos x  x cos yd01 dy œ '121 a2  1 cos yb dy œ c2y  1 sin yd121

13.

' ' a6 y#
2 xbdA œ ' ' a6 y#
2 xb dy dx œ ' c2 y3
2 x yd20 dx œ ' a16
4 xb dx œ c16 x
2 x2 d10 œ 14 0 0 0 0

#

#

dx œ '0 2x dx œ c x# d 0 œ 4 2

"

 yx  x“

#

3




2

dy œ 'c1 (2y  2) dy œ cy#  2yd
" œ 1 0


" "

x y# 2 “0 dy

œ '0 Š 65
1

$

!

# #

1

œ 24

!


xy# “


#

!

y# 2 ‹dy

œ ’ 56 y


œ

2 3

œ 16

dx œ '0 a4x
2x# b dx œ ’2x#
3

2

1

y3 6 “0

œ

1

3

2x$ 3 “!

œ0

œ

92 3

œ 32 a5
eb

3 2

œ 21

1

R

14.

'' R

Èx y2 dA

œ '0

4

'12 Èy x dy dx œ '04 ’
Èyx “2 dx œ '04 "# x1Î2 dx œ
31 x3Î2 ‘40 2

1

8 3

' ' x y cos y dA œ ' ' x y cos y dy dx œ ' cx y sin y  x cos yd10 dx œ ' a
2xb dx œ c
x2 dc1 1 œ 0 c1 0 c1 c1 1

15.

œ

1

1

1

R

' ' y sinax  yb dA œ ' ' y sinax  yb dy dx œ ' c
y cosax  yb  sinax  ybd10 dx c1 0 c1 0

16.

1

0

R

œ 'c1 asinax  1b
1 cosax  1b
sin xbdx œ c
cosax  1b
1 sinax  1b  cos xdc0 1 œ 4 0

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

882

Chapter 15 Multiple Integrals

' ' ex  y dA œ ' ' ex  y dy dx œ ' c
ex  y dln0 2 dx œ ' a
ex  ln 2  ex b dx œ c
ex  ln 2  ex dln0 2 œ 0 0 0 0 ln 2

17.

ln 2

ln 2

ln 2

R

' ' x y ex y2 dA œ ' ' x y ex y2 dy dx œ ' ’ "# ex y2 “ dx œ ' ˆ "# ex
"# ‰ dx œ
"# ex
"# x‘20 œ "# ae2
3b 0 0 0 0 2

18.

1

1

2

'' R

20.

'' R

2

0

R

19.

" #

x y3 x2
1 dA

œ '0

y x2 y2
1 dA

1

'02 xx
y 1 dy dx œ '01 ’ 4axx y
1b “2 dx œ '01 x 4
x 1 dx œ c2 lnlx2  1ld10 3

4

2

2

2

0

œ '0

1

œ 2 ln 2

'01 ax yby
1 dx dy œ '01 ctan1 ax ybd10 dy œ '01 tan1 y dy œ
y tan1 y
"# lnl1  y2 l‘10 2

21.

'12 '12

22.

'01 '01 y cos xy dx dy œ '01 csin xyd 1! dy œ '01 sin 1y dy œ

1" cos 1y‘ "! œ
1" (
1
1) œ 12

1 xy

dy dx œ '1

2

" x

(ln 2
ln 1) dx œ (ln 2) '1

2

" x

œ

1 4


"# ln 2

dx œ (ln 2)#

" " 23. V œ ' ' fax, yb dA œ 'c1 'c1 ax2  y2 b dy dx œ 'c1
x2 y  31 y3 ‘ 1 dx œ 'c1 ˆ2 x2  32 ‰ dx œ
32 x3  32 x‘ 1 œ 1

1

1

1

R

24. V œ ' ' fax, yb dA œ '0

2

R

œ

8 3

'02 a16
x2
y2 b dy dx œ '02
16 y
x2 y
13 y3 ‘20 dx œ '02 ˆ 883
2 x2 ‰ dx œ
883 x
23 x3 ‘20

160 3

25Þ V œ ' ' fax, yb dA œ '0

'01 a2
x
yb dy dx œ '01
2 y
x y
"# y2 ‘ "! dx œ '01 ˆ 32
x‰ dx œ
32 x
"# x2 ‘ "! œ 1

26Þ V œ ' ' fax, yb dA œ '0

'02 y2 dy dx œ '04 ’ y4 “2 dx œ '04 1 dx œ cxd40 œ 4

1

R

4

R

27Þ V œ ' ' fax, yb dA œ '0

2

0

1Î2

R

'01Î4 2 sin x cos y dy dx œ '01Î2 c2 sin x sin yd01Î4 dx œ '01Î2 ŠÈ2 sin x‹ dx œ ’
È2 cos x“1Î2 0

œ È2 28. V œ ' ' fax, yb dA œ '0

1

R

'02 a4
y2 b dy dx œ '01
4 y
13 y3 ‘20 dx œ '01 ˆ 163 ‰ dx œ
163 x‘ "! œ 163

15.2 DOUBLE INTEGRALS OVER GENERAL REGIONS 1.

2.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 15.2 Double Integrals Over General Regions 3.

4.

5.

6.

7.

8.

9. (a)

'!# 'x8 dy dx 3

(b)

'!8 '0y

10. (a)

'!3 '02x dy dx

(b)

'!6 'y3Î2 dx dy

11. (a)

'!3 'x3x dy dx

(b)

'!9 'yÈÎ3y dx dy

12. (a)

'!# '1e dy dx

(b)

'1e 'ln2 y dx dy

13. (a)

'!9 '0

2

x

1Î3

dx dy

2

Èx

(b)

dy dx

'0 'y dx dy 3

9 2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

883

884

Chapter 15 Multiple Integrals

14. (a)

'!1Î4 'tan1 x dy dx

(b)

15. (a)

'01 '0tan

16. (a)

dx dy

'!ln 3 'e1c dy dx x

'1Î3 'ln y dx dy 1

(b)

c1 y

ln 3

'!1 '01 dy dx  '1e 'ln1 x dy dx '01 '0e dx dy y

(b)

17. (a) (b)

18. (a)

'!1 'x3  2x dy dx

'01 '0y dx dy  '13 '0a3  ybÎ2 dx dy

'21 'xx
2 dy dx 2

'0 'Èy dx dy  '13 'yÈy2 dx dy 1

(b)

19.

Èy

'01 '0x (x sin y) dy dx œ '01 c
x cos yd x! dx 1 1 œ '0 (x
x cos x) dx œ ’ x2
(cos x  x sin x)“ #

œ

1# #

!

2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 15.2 Double Integrals Over General Regions 20.

'01 '0sin x y dy dx œ '01 ’ y2 “ sin x dx œ '01 "# sin# x dx #

!

œ

21.

" 4

'01 (1
cos 2x) dx œ "4
x
"2 sin 2x‘ !1 œ 14

'1ln 8 '0ln yexby dx dy œ '1ln 8 cexbyd !ln y dy œ '1ln 8 ayey
eyb dy œ c(y
1)ey
ey d 1ln 8 œ 8(ln 8
1)
8  e œ 8 ln 8
16  e

'12 'yy

#

22.

dx dy œ '1 ay#
yb dy œ ’ y3
2

$

œ ˆ 83
2‰
ˆ "3
"# ‰ œ

7 3




œ

3 #

5 6

'01 '0y 3y$ exy dx dy œ '01 c3y# exy d 0y #

23.

# y# # “"

#

dy

œ '0 Š3y# ey
3y# ‹ dy œ ’ey
y$ “ œ e
2 1

$

"

$

!

24.

Èx

'14 '0 œ

3 #

3 #

eyÎÈx dy dx œ

'14
32 Èx eyÎÈx ‘ 0Èx dx

% (e
1) '1 Èx dx œ
23 (e
1) ˆ 32 ‰ x$Î# ‘ " œ 7(e
1) 4

25.

'12 'x2x

26.

'01 '01cx ax#  y# b dy dx œ '01 ’x# y  y3 “ "

x y

dy dx œ '1 cx ln yd x2x dx œ (ln 2) '1 x dx œ 2

2

$

x

0

$

œ ’ x3
27.

x% 4




" (1x)% 1# “ !

œ ˆ "3


" 4

#

œ '0 Š "#
u  1

u# #


vÈ u “

ln 2

dx œ '0 ’x# (1
x)  1


0‰
ˆ0
0


'01 '01cu ˆv
Èu‰ dv du œ '01 ’ v2

3 #

"

u

0


u"Î#  u$Î# ‹ du œ ’ u2


" ‰ 1#

œ

(1x)$ 3 “

dx œ '0 ’x#
x$  1

(1x)$ 3 “

dx

" 6

du œ '0 ’ 12u#
u
Èu(1
u)“ du 1

u# #



u$ 6

#

"


32 u$Î#  25 u&Î# “ œ !

" #




" #



" 6




2 3



Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

2 5

œ
#" 

2 5

" œ
10

885

886 28.

Chapter 15 Multiple Integrals

'12 '0ln t es ln t ds dt œ '12 ces ln td 0ln t dt œ '12 (t ln t
ln t) dt œ ’ t2

#

œ (2 ln 2
1
2 ln 2  2)
ˆ
"4  1‰ œ 29.

" 4

ln t


t# 4


t ln t  t“

# "

'c02 'vcv 2 dp dv œ 2'c02 cpd vv dv œ 2'c02
2v dv œ
2 cv# d c2 œ 8 0

30.

È1cs

'01 '0

#

È1cs

8t dt ds œ '0 c4t# d 0 1

œ '0 4 a1
s# b ds œ 4 ’s
1

31.

#

ds

" s$ 3 “!

œ

8 3

'c11ÎÎ33 '0sec t 3 cos t du dt œ ' 11ÎÎ33 c(3 cos t)ud 0sec t 1Î3

œ 'c1Î3 3 dt œ 21

32.

'03Î2 '14 2u 4 v 2u dv du œ '03Î2
2u v 4 ‘ 14 2u du 3Î2 $Î2 œ '0 a3
2ub du œ c3u
u# d ! œ 92 #

33.

'24 '0Ð4

y)Î2

34.

' 02 '0x
2 dy dx

dx dy

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 15.2 Double Integrals Over General Regions 35.

'01 'xx dy dx

36.

'01 '1cy1cydx dy

37.

'1e 'ln1ydx dy

38.

'12 '0ln x dy dx

39.

'09 '0

40.

'04 '0

#

È

1 2

È9cy

È4cx

16x dx dy

y dy dx

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

887

888

Chapter 15 Multiple Integrals È1cx

41.

'c11 '0

42.

'c22 '0

43.

'01 'ee x y dx dy

44.

'01Î2 '0sin

45.

'1e 'ln3 x ax  ybdy dx

46.

'01Î3 'tan 3x Èx y dy dx

È4cy

#

3y dy dx

#

6x dx dy

y

c1 y

x y2 dx dy

3

È

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 15.2 Double Integrals Over General Regions 47.

48.

'01 'x1 siny y dy dx œ '01 '0y siny y dx dy œ '01 sin y dy œ 2

'02 'x2 2y# sin xy dy dx œ '02 '0y2y# sin xy dx dy 2 2 œ '0 c
2y cos xyd 0y dy œ '0 a
2y cos y#  2yb dy #

œ c
sin y#  y# d ! œ 4
sin 4

49.

'01 'y1 x# exy dx dy œ '01 '0x x# exy dy dx œ '01 cxexyd 0x dx œ '0 axex
xb dx œ ’ "2 ex
1

È4cy

'02 '04cx 4xey dy dx œ '04 '0 #

50.

2y

œ '0 ’ #x(4ey) “ 4

51.

'02

# 2y

Èln 3 Èln 3

'y/2

Èln 3

œ '0

52.

" x# # “!

#

#

È4cy

0

dy œ '0

4 2y e

Èln 3

ex dx dy œ '0 #

#

#

Èln 3

$

dy dx œ '0

1

'03y

#

e2 #

dx dy 2y

%

dy œ ’ e4 “ œ

#

2xex dx œ cex d 0

'03 'È1xÎ3 ey

xe2y 4 y

œ

!

'02x ex

#

e)  " 4

dy dx

œ eln 3
1 œ 2

$

ey dx dy

œ '0 3y# ey dy œ cey d ! œ e
1 1

53.

$

$

"

'01Î16 'y1Î2 cos a161x& b dx dy œ '01Î2 '0x "Î%

%

cos a161x& b dy dx

161x b œ '0 x% cos a161x& b dx œ ’ sin a80 “ 1 1Î2

&

"Î# !

œ

" 801

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

889

890 54.

Chapter 15 Multiple Integrals

'08 'È2x $

œ '0

2

55.

dy dx œ '0

2

"

y %
1 y$ y %
1

dy œ

" 4

'0y y "
1 dx dy $

%

#

cln ay%  1bd ! œ

ln 17 4

' ' ay
2x# b dA R

xb1

œ 'c1 'cxc1 ay
2x# b dy dx  '0 0

1

'x1cc1x ay
2x# b dy dx

x " 1
x œ 'c1
"2 y#
2x# y‘
x
1 dx  '0
2" y#
2x# y‘ x
1 dx 0

1

œ 'c1
"# (x  1)#
2x# (x  1)
"# (
x
1)#  2x# (
x
1)‘dx 0

 '0
"# (1
x)#
2x# (1
x)
"# (x
1)#  2x# (x
1)‘ dx 1

œ
4 'c1 ax$  x# b dx  4 '0 ax$
x# b dx 0

1

%

œ
4 ’ x4 

56.

0

x$ 3 “ c1

" x$ 3 “!

%

 4 ’ x4


%

œ 4 ’ (41) 

(1)$ 3 “

3  4 ˆ 4"
3" ‰ œ 8 ˆ 12


4 ‰ 12

8 œ
12 œ
32

' ' xy dA œ ' ' xy dy dx  ' ' xy dy dx 0 x 2Î3 x 2Î3

R

2x

2Î3

1

2x 2 œ '0
"2 xy# ‘ x dx  '2Î3
2" xy# ‘ x 1

x

2 x

dx

œ '0 ˆ2x$
"# x$ ‰ dx  '2Î3
"# x(2
x)#
"# x$ ‘ dx 2Î3

1

œ '0

2Î3

3 #

x$ dx  '2Î3 a2x
x# b dx 1

2Î3 " 2‰ 8 ‰‘ ‰ ˆ
4 ˆ 2 ‰ ˆ 27 œ
38 x% ‘ 0 
x#
23 x$ ‘ #Î$ œ ˆ 38 ‰ ˆ 16 œ 81  1
3
9
3

57. V œ '0

1

'x2cx ax#  y# b dy dx œ '01 ’x# y  y3 “ 2cx dx œ '01 ’2x#
7x3 $

$

x

œ ˆ 23


7 12




2cx#

58. V œ 'c2 'x 1

œ ˆ 23


" 5

" ‰ 12


ˆ0
0


4cx#

1

œ

x# dy dx œ 'c2 cx# yd x 32 5




16 ‰ 4



(2x)$ 3 “

È 4 cx

40 œ ˆ 60


2

7x% 12




13 81 " (2x)% 12 “ !

12 60




15 ‰ 60


ˆ
320 60 

384 60




240 ‰ 60

œ

189 60

œ

63 20

4cx (x  4) dy dx œ 'c4 cxy  4yd 3x dx œ 'c4 cx a4
x# b  4 a4
x# b
3x#
12xd dx 1

1

#

#

(3
y) dy dx œ '0 ’3y
2

œ ’ 32 xÈ4
x#  6 sin" ˆ x# ‰
2x  61. V œ '0

$

œ

"

"

'0

16 ‰ 81

dx œ ’ 2x3


1

1

2


ˆ 36 81


dx œ 'c2 a2x#
x%
x$ b dx œ
23 x$
15 x&
14 x% ‘ #

œ 'c4 a
x$
7x#
8x  16b dx œ

41 x%
37 x$
4x#  16x‘ % œ ˆ
4"


60. V œ '0

27 81

4 3

2cx#

1


4" ‰
ˆ
16 3 

59. V œ 'c4 '3x

16 ‰ 12



6 81

È 4c x

y# 2 “0

# x$ 6 “!

#

7 3

‰  12‰
ˆ 64 3
64 œ

dx œ '0 ’3È4
x#
Š 4#x ‹“ dx 2

œ 6 ˆ 1# ‰
4 

#

8 6

œ 31


16 6

œ

918 3

'03 a4
y# b dx dy œ '02 c4x
y# xd $! dy œ '02 a12
3y# b dy œ c12y
y$ d !# œ 24
8 œ 16

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

157 3




" 4

œ

625 12

Section 15.2 Double Integrals Over General Regions 62. V œ '0

2

'04cx

#

2

œ
8x
43 x$  63. V œ '0

2

4cx#

a4
x#
yb dy dx œ '0 ’a4
x# b y
" 10

#

x& ‘ ! œ 16




32 3

32 10

œ

y# 2 “!

480320
96 30

œ

" #

a4
x# b dx œ '0 Š8
4x#  2

#

!

xb1

1

1Îx

2

66. V œ 4 '0

1Î3

'x1cc1x (3
3x) dy dx œ 6 'c01 a1
x# b dx  6 '01 (1
x)# dx œ 4  2 œ 6

2

2

" x


ˆ
1
x" ‰‘dx œ 2 '1 ˆ1  x" ‰ dx œ 2 cx  ln xd #" 2

'0sec x a1  y# b dy dx œ 4 '01Î3 ’y  y3 “ sec x dx œ 4 '01Î3 Šsec x  sec3 x ‹ dx $

$

0

1Î$

c7 ln ksec x  tan xk  sec x tan xd !

œ

’7 ln Š2  È3‹  2È3“

2 3

67.

68.

'1_ 'ec1 x"y dy dx œ '1_ ’ lnx y “ " x

$

$

ec x

_

dx œ '1
ˆ x$x ‰ dx œ
lim

bÄ_

1/ ˆ1cx ‰ È1cx 1 70. 'c1 'c1/È1cx (2y  1) dy dx œ 'c1 cy#  ydº 1

1/

# 1Î#

#

#

c1/ a1c

x# b1Î#

œ 4 lim c csin" b
0d œ 21 bÄ1 71.

dx

128 15

65. V œ '1 'c1Îx (x  1) dy dx œ '1 cxy  yd 1Î1xÎx dx œ '1
1  œ 2(1  ln 2)

69.

x% #‹

%

0

2 3

2

# 2 x '02cx a12
3y# b dy dx œ '02 c12y
y$ d # dx œ '0 c24
12x
(2
x)$ d dx œ ’24x
6x#  (24x) “ œ 20 !

64. V œ 'c1 'cxc1 (3
3x) dy dx  '0

œ

dx œ '0

_ _ 'c_ ' _ ax
1b"ay
1b -dx dy œ 2 '0_ Š y 2
1 ‹ Š #

#

#

lim

bÄ_

dx œ 'c1 È 2


x" ‘ b œ
lim 1

1

1 x #

bÄ_

ˆ "b
1‰ œ 1

dx œ 4 lim c csin" xd ! bÄ1 b

tan" b
tan" 0‹ dy œ 21 lim

bÄ_

'0b y "
1 dy #

œ 21 Š lim tan" b
tan" 0‹ œ (21) ˆ 1# ‰ œ 1# bÄ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

891

892 72.

Chapter 15 Multiple Integrals

'0_ '0_ xecÐx

2yÑ

_

_

c
xe
x
e
x d b0 dy œ '0 e
2y lim

bÄ_

œ '0 ec2y dy œ 73.

_

dx dy œ '0 e
2y lim " # b lim Ä_

a
be
b
e
b  1b dy

bÄ_

a
ec2b  1b œ

" #

' ' f(xß y) dA ¸ "4 f ˆ
"# ß 0‰  8" f(0ß 0)  8" f ˆ "4 ß 0‰ œ "4 ˆ
"# ‰  8" ˆ0  "4 ‰ œ
323 R

74.

' ' f(xß y) dA ¸ "4 ’f ˆ 47 ß 114 ‰  f ˆ 94 ß 114 ‰  f ˆ 74 ß 134 ‰  f ˆ 94 ß 134 ‰“ œ R

75. The ray ) œ

1 6

" 16

(29  31  33  35) œ

128 16

œ8

meets the circle x#  y# œ 4 at the point ŠÈ3ß 1‹ Ê the ray is represented by the line y œ

È

È

È

$Î# 3 4cx 3 x# b ' ' f(xß y) dA œ ' ' È È4
x# dy dx œ ' ’a4
x# b
Èx3 È4
x# “ dx œ ”4x
x3$  a4È 0 xÎ 3 0 3 3 • #

R

76.

'2_ '02 ax xb "(y1) #

bÄ_ bÄ_

77. V œ '0

1

0

cln (x
1)
ln xd 2b œ 6 lim

lim

bÄ_

_

dx œ 6 '2

bÄ_

dx x(x1)

[ln (b
1)
ln b
ln 1  ln 2]

$

x

7x 3

œ ˆ 23


" ‰ 1#

7 12




$



(2x)$ 3 “

$

dx œ ’ 2x3



ˆ0
0


16 ‰ 12

œ

œ '0

 '2

œ

2 tan

ˆ1 1" ‰ y 1
y# dy 1  1 ˆ 21 ‰ ln 5  "

21

ˆ2  1 1
y # "

"

21
2 tan

y‰

21


2


%

7x 12




" (2x)% 12 “ !

4 3

'02 atan" 1x
tan" xb dx œ '02 'x1x 1
"y

œ 2 tan

3 ‰ x # x

'x2cx ax#  y# b dy dx œ '01 ’x# y  y3 “ 2cx dx

œ '0 ’2x#


2

2

"Î$

0

ln ˆ1
"b ‰  ln 2“ œ 6 ln 2

1

78.

_

1) ' ˆ x#3x  dy dx œ '2 ’ 3(y ax# xb “ dx œ 2

'2b ˆ x" 1
"x ‰ dx œ 6

œ 6 lim

œ 6 ’ lim

_

#Î$

È3

dy dx œ '0

2

#

'yyÎ1

" 1
y #

dx dy  '2

21

# # " ‰
dy œ ˆ 12" y 1 cln a1  y bd !  2 tan " 21

" 21

ln a1  41# b
2 tan" 2  #

ln a1  41 b 

" #1

" #1

'y2Î1 1
"y

#

dx dy 21

ln a1  y# b‘ 2

ln 5

ln 5 #

79. To maximize the integral, we want the domain to include all points where the integrand is positive and to exclude all points where the integrand is negative. These criteria are met by the points (xß y) such that 4
x#
2y#   0 or x#  2y# Ÿ 4, which is the ellipse x#  2y# œ 4 together with its interior. 80. To minimize the integral, we want the domain to include all points where the integrand is negative and to exclude all points where the integrand is positive. These criteria are met by the points (xß y) such that x#  y#
9 Ÿ 0 or x#  y# Ÿ 9, which is the closed disk of radius 3 centered at the origin. 81. No, it is not possible. By Fubini's theorem, the two orders of integration must give the same result.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

x È3

œ

. Thus,

20È3 9

Section 15.2 Double Integrals Over General Regions 82. One way would be to partition R into two triangles with the line y œ 1. The integral of f over R could then be written as a sum of integrals that could be evaluated by integrating first with respect to x and then with respect to y:

' ' f(xß y) dA R

œ '0

1

'22ccÐ2yyÎ2Ñ f(xß y) dx dy  '12 'y2c1ÐyÎ2Ñ f(xß y) dx dy.

Partitioning R with the line x œ 1 would let us write the integral of f over R as a sum of iterated integrals with order dy dx. 83.

' bb ' bb e

x# y#

dx dy œ '

b ' e b b

b

#

y#

e

x#

dx dy œ ' b e b

#

y#

Œ' b e b

x#

dx dy œ Œ' b e b

x#

dx Œ' b e b

y#

dy

#

# # # œ Œ'cb ecx dx œ Œ2 '0 ecx dx œ 4 Œ'0 ecx dx ; taking limits as b Ä _ gives the stated result.

b

84.

'01 '03 (yx1)

dy dx œ '0

3

#

œ

b

#Î$

" 3 b lim Ä 1c

'0

b

dy (y1)#Î$



'01 (yx1)

dx dy œ '0

3

#

#Î$

" 3

b

'b

3

lim

b Ä 1b

dy (y1)#Î$

œ

" (y1)#Î$

lim

b Ä 1c

$

"

’ x3 “ dy œ !

" 3

'03 (ydy1)

#Î$


(y
1)"Î$ ‘ b  lim
(y
1)"Î$ ‘ 3 0 b b Ä 1b

3 3 œ ’ lim c (b
1)"Î$
(
1)"Î$ “
’ lim b (b
1)"Î$
(2)"Î$ “ œ (0  1)
Š0
È 2‹ œ 1  È 2 bÄ1 bÄ1

85-88. Example CAS commands: Maple: f := (x,y) -> 1/x/y; q1 := Int( Int( f(x,y), y=1..x ), x=1..3 ); evalf( q1 ); value( q1 ); evalf( value(q1) ); 89-94. Example CAS commands: Maple: f := (x,y) -> exp(x^2); c,d := 0,1; g1 := y ->2*y; g2 := y -> 4; q5 := Int( Int( f(x,y), x=g1(y)..g2(y) ), y=c..d ); value( q5 ); plot3d( 0, x=g1(y)..g2(y), y=c..d, color=pink, style=patchnogrid, axes=boxed, orientation=[-90,0], scaling=constrained, title="#89 (Section 15.2)" ); r5 := Int( Int( f(x,y), y=0..x/2 ), x=0..2 ) + Int( Int( f(x,y), y=0..1 ), x=2..4 ); value( r5); value( q5-r5 ); 85-94. Example CAS commands: Mathematica: (functions and bounds will vary) You can integrate using the built-in integral signs or with the command Integrate. In the Integrate command, the integration begins with the variable on the right. (In this case, y going from 1 to x).

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

893

894

Chapter 15 Multiple Integrals

Clear[x, y, f] f[x_, y_]:= 1 / (x y) Integrate[f[x, y], {x, 1, 3}, {y, 1, x}] To reverse the order of integration, it is best to first plot the region over which the integration extends. This can be done with ImplicitPlot and all bounds involving both x and y can be plotted. A graphics package must be loaded. Remember to use the double equal sign for the equations of the bounding curves. Clear[x, y, f] ˆ "# ‰ œ '0 t
"Î# e
t dt œ '0 ay# b 24. Q œ '0

" #

#

È1 # ‹

c)
cos )d #!1 œ

'cÈhh 'cÈhhccxx 'xhby

œ È1, where y œ Èt

21kR$ 3

#

#

#

#

dz dy dx

ah  r# b r dr d) œ '0 ’ hr2  r4 “ 21

#

%

Èh !

d) œ '0

21

h# 4

d) œ

h# 1 #

.

Since the top of the bowl has area 101, then we calibrate the bowl by comparing it to a right circular cylinder whose cross sectional area is 101 from z œ 0 to z œ 10. If such a cylinder contains to a depth w then we have 101w œ rain, w œ 3 and h œ È60.

h# 1 #

Ê wœ

h# 20

h# 1 #

cubic inches of water . So for 1 inch of rain, w œ 1 and h œ È20; for 3 inches of

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Chapter 15 Additional and Advanced Exercises 26. (a) An equation for the satellite dish in standard position is z œ "# x#
"# y# . Since the axis is tilted 30°, a unit vector v œ 0i
aj
bk normal to the plane of the È3 # È
#3

water level satisfies b œ v † k œ cos ˆ 16 ‰ œ Ê a œ È1  b# œ  "# Ê v œ  "# j Ê  "# (y  1)
Ê zœ

" È3

È3 #

y
Š "# 

k

ˆz  "# ‰ œ 0 " È3 ‹

is an equation of the plane of the water level. Therefore

' Èx byby c È dz dy dx, where R is the interior of the ellipse

the volume of water is V œ ' '

1

1 2

3

#

1 2

R

1 2

1

3

#

È 3  Ê 3
4 Š È 3
1‹ 2

x#
y# 

È

2

and " œ

y1


2 È3


Ê 43 3

œ 0. When x œ 0, then y œ ! or y œ " , where ! œ

2 È3

Ê Vœ

3

#

(b) x œ 0 Ê z œ

" #

y# and

!

œ y; y œ 1 Ê

dz dy

"Î#

Š yb1c È3 cy ‹

' 'c "


4 Š È
1‹ 2

2 3

#

2

3

"Î#

Š yb1c È3 cy# ‹ 2 3

2

yb

' Èb 1

1 2

x#

1 2

1 2

4

2

#

c È3 1

1 dz dx dy

y#

œ 1 Ê the tangent line has slope 1 or a 45° slant

dz dy

Ê at 45° and thereafter, the dish will not hold water. 27. The cylinder is given by x#
y# œ 1 from z œ 1 to _ Ê œ '0

21

_

'0 '1 1

z ar#  z# b&Î#

'0 '0 ' 21

dz r dr d) œ a lim Ä_

1

' ' ' z ar#
z# b
&Î# dV D

a

rz 1 ar#  z# b&Î#

dz dr d)

œ a lim Ä_

'021 '01 ’ˆ "3 ‰

œ a lim Ä_

'021 ’ 3" ar#
a# b
"Î#  3" ar#
1b
"Î# “ " d) œ a lim ' 21 ’ 3" a1
a# b
"Î#  3" ˆ2
"Î# ‰  3" aa# b
"Î#
3" “ d) Ä_ 0

œ a lim 21 ’ 3" a1
a# b Ä_

a

r “ ar#  z# b$Î# 1


"Î#

 3" Š

'021 '01 ’ˆ 3" ‰

dr d) œ a lim Ä_

È2 # ‹


ˆ "3 ‰

r ar#  a# b$Î#

!

r “ ar#  1b$Î#

dr d)

È2 # “.

 3" ˆ "a ‰
3" “ œ 21 ’ 3"  ˆ 3" ‰

28. Let's see? The length of the "unit" line segment is: L œ 2'0 dx œ 2. 1

The area of the unit circle is: A œ 4'0

1

È1 c x

'0

The volume of the unit sphere is: V œ 8'0

1

2

dy dx œ 1.

È1 c x

'0

2

È1 c x c y

'0

2

2

dz dy dx œ 43 1.

Therefore, the hypervolume of the unit 4-sphere should be: Vhyper œ 16'0

1

È1cx

'0

2

È1cx cy

'0

2

2

È1cx cy cz

'0

2

2

2

dw dz dy dx.

Mathematica is able to handle this integral, but we'll use the brute force approach. Vhyper œ 16'0

1

È1cx

œ 16'0

'0

œ 16'0

'0

œ 16'0

'0

1

1

1

È1cx È1cx

È1cx

'0

2

2

2

2

È1cx cy

'0

È1cx cy

'0

2

2

È1cx cy cz

'0

2

2

2

dw dz dy dx œ 16'0

1

2

z2 È 1
x2
y2 É 1
1 c x2 c y2

dz dy dx œ –

È1cx

'0

dz œ

a1  x2  y2 b'1/2 È1  cos2 ) sin ) d) dy dx œ 16'0 0

1 4 a1

1

2

È1cx cy

'0

2

2

È 1
x 2
y 2
z2

z È1
x2
y2 œ cos ) È1  x2  y2 sin

È1cx

'0

2

) d)

—

a1  x2  y2 b'1/2 sin2 ) d) dy dx 0

1

1
x3 $

‘ dx œ 83 1' a1  x2 b3/2 dx œ ” 0 1

0 x œ cos ) œ  83 1'1/2 sin4 ) d) dx œ sin ) d) •

2 ) ‰2 œ  83 1'1/2 ˆ 1
cos d) œ  23 1'1/2 a1  2 cos 2)
cos2 2)bd) œ  23 1'1/2 ˆ #3  2 cos 2)
2 0

dz dy dx

3/2  x2  y2 b dy dx œ 41'0 ŠÈ1  x2  x2 È1  x2  3" a1  x2 b ‹ dx

œ 41'0 È1  x2
a1  x2 b  1

2

0

0

cos 4) ‰ d) 2

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

œ

12 2

937

938

Chapter 15 Multiple Integrals

NOTES:

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

CHAPTER 16 INTEGRATION IN VECTOR FIELDS 16.1 LINE INTEGRALS 1. r œ ti
a"  tbj Ê x œ t and y œ 1  t Ê y œ 1  x Ê (c) 2. r œ i
j
tk Ê x œ 1, y œ 1, and z œ t Ê (e) 3. r œ a2 cos tbi
a2 sin tbj Ê x œ 2 cos t and y œ 2 sin t Ê x#
y# œ 4 Ê (g) 4. r œ ti Ê x œ t, y œ 0, and z œ 0 Ê (a) 5. r œ ti
tj
tk Ê x œ t, y œ t, and z œ t Ê (d) 6. r œ tj
a2  2tbk Ê y œ t and z œ 2  2t Ê z œ 2  2y Ê (b) 7. r œ at#  1b j
2tk Ê y œ t#  1 and z œ 2t Ê y œ

z# 4

 1 Ê (f)

8. r œ a2 cos tbi
a2 sin tbk Ê x œ 2 cos t and z œ 2 sin t Ê x#
z# œ 4 Ê (h) 9. ratb œ ti
a1  tbj , 0 Ÿ t Ÿ 1 Ê

œ i  j Ê ¸ ddtr ¸ œ È2 j ; x œ t and y œ 1  t Ê x
y œ t
("  t) œ 1

dr dt

Ê 'C faxß yß zb ds œ '0 fatß 1  tß 0b ¸ ddtr ¸ dt œ '0 (1) ŠÈ2‹ dt œ ’È2 t“ œ È2 1

"

1

!

10. r(t) œ ti
(1  t)j
k , 0 Ÿ t Ÿ 1 Ê œ t  (1  t)
1  2 œ 2t  2 Ê

dr dt

œ i  j Ê ¸ ddtr ¸ œ È2; x œ t, y œ 1  t, and z œ 1 Ê x  y
z  2

'C f(xß yß z) ds œ '01 (2t  2) È2 dt œ È2 ct#  2td "! œ È2

11. r(t) œ 2ti
tj
(2  2t)k , 0 Ÿ t Ÿ 1 Ê

dr dt

œ 2i
j  2k Ê ¸ ddtr ¸ œ È4
1
4 œ 3; xy
y
z

œ (2t)t
t
(2  2t) Ê 'C f(xß yß z) ds œ '0 a2t#  t
2b 3 dt œ 3
23 t$  "# t#
2t‘ ! œ 3 ˆ 23  1

12. r(t) œ (4 cos t)i
(4 sin t)j
3tk , 21 Ÿ t Ÿ 21 Ê Ê ¸ ddtr ¸ œ È16 sin# 1 œ c20td #
# 1 œ 801

"

dr dt

dr dt

Ê ¸ ddtr ¸ œ È1
9
4 œ È14 ; x
y
z œ (1  t)
(2  3t)
(3  2t) œ 6  6t Ê

œ  i  3 j  2k

'C f(xß yß z) ds

œ '0 (6  6t) È14 dt œ 6È14 ’t  t2 “ œ Š6È14‹ ˆ "# ‰ œ 3È14 " !

14. r(t) œ ti
tj
tk , 1 Ÿ t Ÿ _ Ê

_

dr dt

13 #

'C f(xß yß z) ds œ 'c2211 (4)(5) dt

13. r(t) œ (i
2j
3k)
t(i  3j  2k) œ (1  t)i
(2  3t)j
(3  2t)k , 0 Ÿ t Ÿ 1 Ê

#


2‰ œ

œ (4 sin t)i
(4 cos t)j
3k

t
16 cos# t
9 œ 5; Èx#
y# œ È16 cos# t
16 sin# t œ 4 Ê

1

" #

œ i
j
k Ê ¸ ddtr ¸ œ È3 ;

È3 x #  y#  z#

_ Ê 'C f(xß yß z) ds œ '1 Š 3t#3 ‹ È3 dt œ
 1t ‘ " œ lim ˆ b"
1‰ œ 1

œ

È3 t#  t#  t#

œ

È3 3t#

È

bÄ_

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

940

Chapter 16 Integration in Vector Fields

15. C" : r(t) œ ti
t# j , 0 Ÿ t Ÿ 1 Ê

œ i
2tj Ê ¸ ddtr ¸ œ È1
4t# ; x
Èy  z# œ t
Èt#  0 œ t
ktk œ 2t

dr dt

$Î# since t   0 Ê 'C f(xß yß z) ds œ '0 2tÈ1
4t# dt œ ’ "6 a"
4t# b “ œ "

1

!

"

C# : r(t) œ i
j
tk, 0 Ÿ t Ÿ 1 Ê

dr dt

1

"

#

"

5 6

#

16. C" : r(t) œ tk , 0 Ÿ t Ÿ 1 Ê

dr dt

(5)$Î# 

" 6

" 6

œ

Š5È5  1‹ ;

œ k Ê ¸ ddtr ¸ œ 1; x
Èy  z# œ 1
È1  t# œ 2  t#

Ê 'C f(xß yß z) ds œ '0 a2  t# b (1) dt œ
2t  "3 t$ ‘ ! œ 2 

œ 'C f(xß yß z) ds
'C f(xß yß z) ds œ

" 6

È5


" 3

œ

5 3

; therefore 'C f(xß yß z) ds

3 #

œ k Ê ¸ ddtr ¸ œ 1; x
Èy  z# œ 0
È0  t# œ t#

Ê 'C f(xß yß z) ds œ '0 at# b (1) dt œ ’ t3 “ œ  3" ; 1

"

$

!

"

C# : r(t) œ tj
k, 0 Ÿ t Ÿ 1 Ê

œ j Ê ¸ ddtr ¸ œ 1; x
Èy  z# œ 0
Èt  1 œ Èt  1

dr dt

" Ê 'C f(xß yß z) ds œ '0 ˆÈt  1‰ (1) dt œ
23 t$Î#  t‘ ! œ 1

#

C$ : r(t) œ ti
j
k , 0 Ÿ t Ÿ 1 Ê

dr dt #

œ  "6 17. r(t) œ ti
tj
tk , 0  a Ÿ t Ÿ b Ê Ê

" !

$

dr dt

 1 œ  3" ;

œ i Ê ¸ ddtr ¸ œ 1; x
Èy  z# œ t
È1  1 œ t

Ê 'C f(xß yß z) ds œ '0 (t)(1) dt œ ’ t2 “ œ 1

2 3

" #

Ê

'C f(xß yß z) ds œ 'C

"

œ i
j
k Ê ¸ ddtr ¸ œ È3 ;

f ds
'C f ds
'C f ds œ  3"
ˆ 3" ‰
#

xyz x #  y #  z#

œ

'C f(xß yß z) ds œ 'ab ˆ 1t ‰ È3 dt œ ’È3 ln ktk “ b œ È3 ln ˆ ba ‰ , since 0  a Ÿ b

$

ttt t#  t#  t#

œ

" #

1 t

a

18. r(t) œ aa cos tb j
aa sin tb k , 0 Ÿ t Ÿ 21 Ê

dr dt

œ (a sin t) j
(a cos t) k Ê ¸ ddtr ¸ œ Èa# sin# t
a# cos# t œ kak ;

21 1  kak sin t, 0 Ÿ t Ÿ 1 Èx#
z# œ È0
a# sin# t œ œ Ê 'C f(xß yß z) ds œ '0  kak# sin t dt
'1 kak# sin t dt kak sin t, 1 Ÿ t Ÿ 21

1

#1

œ ca# cos td !  ca# cos td 1 œ ca# (1)  a# d  ca#  a# (1)d œ 4a# Ê 'C x ds œ '0 t

È5 2

4

È5 2 dt

È5 2

'04 t dt œ ’ È45 t2 “ 4 œ 4È5

19. (a) ratb œ ti
"# tj , 0 Ÿ t Ÿ 4 Ê

dr dt

œ i
"# j Ê ¸ ddtr ¸ œ

(b) ratb œ ti
t j , 0 Ÿ t Ÿ 2 Ê

dr dt

œ i
2tj Ê ¸ ddtr ¸ œ È1
4t2 Ê 'C x ds œ '0 t È1
4t2 dt

2

3Î2 2

1 œ ’ 12 a1
4t2 b

“ œ !

œ

2

17È17
" 12

20. (a) ratb œ ti
4tj , 0 Ÿ t Ÿ 1 Ê

dr dt

œ i
4j Ê ¸ ddtr ¸ œ È17 Ê 'C Èx
2y ds œ '0 Èt
2a4tb È17 dt 1

œ È17'0 È9t dt œ 3È17'0 Èt dt œ ’2È17 t2Î3 “ œ 2È17 1

1

1

!

(b) C" : ratb œ ti , 0 Ÿ t Ÿ 1 Ê

'C Èx
2y ds œ 'C

1

œ i Ê ¸ ddtr ¸ œ 1; C2 : ratb œ i
tj, 0 Ÿ t Ÿ 1 Ê

dr dt

C2

œ '0 Èt dt
'0 È1
2t dt œ 1

2

21. ratb œ 4ti  3tj , 1 Ÿ t Ÿ 2 Ê

dr dt 2

16t œ 15'c1 t e16t dt œ ’ 15 “ 32 e 2

2

dr dt

œ j Ê ¸ ddtr ¸ œ 1

Èx
2y ds
' Èx
2y ds œ ' Èt
2a0b dt
' È1
2atb dt 1

2

c1


23 t2Î3 ‘ 1 !

2

0

0

2


’ 13 a1
2tb2Î3 “ œ !

2 3




È Š5 3 5

 31 ‹ œ

5È 5  1 3

œ 4i  3j Ê ¸ ddtr ¸ œ 5 Ê 'C y ex ds œ 'c1 a3tb ea4tb † 5dt 2

64 œ  15 32 e


15 16 32 e

œ

15 16 32 ae

2

2

 e64 b

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

!

Section 16.1 Line Integrals 22. ratb œ acos tbi
asin tbj , 0 Ÿ t Ÿ 21 Ê

941

œ asin tbi
acos tbj Ê ¸ ddtr ¸ œ Èsin2 t
cos2 t œ 1 Ê 'C ax  y
3b ds

dr dt

œ '0 acos t  sin t
3b † 1 dt œ csin t
cos t
3td 201 œ 61 21

23. ratb œ t2 i
t3 j , 1 Ÿ t Ÿ 2 Ê œ '1

2

œ '1Î2

œ 2ti
3t2 j Ê ¸ ddtr ¸ œ Éa2tb2
a3t2 b2 œ tÈ4
9t2 Ê 'C

3Î2 1 a4
9t2 b “ œ † tÈ4
9t2 dt œ '1 t È4
9t2 dt œ ’ 27

ˆt2 ‰2

2

2

at3 b4Î3

ŸtŸ1Ê

1 2

dr dt

1

25. C" : ratb œ ti
t2 j , 0 Ÿ t Ÿ 1 Ê Ê

dr dt

ds

œ 3t2 i
4t3 j Ê ¸ ddtr ¸ œ Éa3t2 b2
a4t3 b2 œ t2 È9
16t2 Ê 'C

1 a9
16t2 b † t2 È9
16t2 dt œ '1Î2 t È9
16t2 dt œ ’ 48

Èt4 t3

x2 y4Î3

80È10
13È13 27

1

24. ratb œ t3 i
t4 j , 1

dr dt

3 Î2

1

“

1Î2

œ

Èy x

ds

125
13È13 48

œ i
2tj Ê ¸ ddtr ¸ œ È1
4t2 ; C2 : ratb œ a1  tbi
a1  tbj, 0 Ÿ t Ÿ 1

dr dt

œ i  j Ê ¸ ddtr ¸ œ È2 Ê 'C ˆx
Èy‰ds œ 'C ˆx
Èy‰ds
'C ˆx
Èy‰ds 1

2

œ '0 Št
Èt2 ‹È1
4t2 dt
'0 Ša1  tb
È1  t‹ È2dt œ '0 2tÈ1
4t2 dt
'0 Š1  t
È1  t‹ È2dt 1

1

œ ’ 16 a1
4t2 b

1

3 Î2 1

1

0

0

Î “
È2’t  "# t2  23 a1  tb3 2 “ œ

26. C" : ratb œ ti , 0 Ÿ t Ÿ 1 Ê

5È 5
1 6




7È 2 6

1

œ

5È 5  7È 2
1 6

œ i Ê ¸ ddtr ¸ œ 1; C2 : ratb œ i
tj, 0 Ÿ t Ÿ 1 Ê ddtr œ j Ê ¸ ddtr ¸ œ 1; C3 : ratb œ a1  tbi
j, 0 Ÿ t Ÿ 1 Ê ddtr œ i Ê ¸ ddtr ¸ œ 1; C4 : ratb œ a1  tbj, 0 Ÿ t Ÿ 1 Ê ddtr œ j Ê ¸ ddtr ¸ œ 1; Ê 'C œ '0

1

1 x2  y2  1 ds dt t2  1


'0

œ ctan
1 td 0
1

1

œ 'C

1 x2  y2  1 ds

1

dt t2  2

dr dt


'0

1

x# #

2

dt a1
tb2  2

1 t
1 È2 ’tan Š È2 ‹“

27. r(x) œ xi
yj œ xi



'C

1 0





'0

1 x2  y2  1 ds

1


'C

œ '0 (2x)È1
x# dx œ ’ 23 a1
x# b 2

4

1 x2  y2  1 ds

1 0


ctan
1 a1  tbd 0 œ 1

1 2




2 1
1 È2 tan Š È2 ‹

#

“ œ !


'C

dr ¸ œ i
xj Ê ¸ dx œ È1
x# ; f(xß y) œ f Šxß x# ‹ œ

dr dx

$Î# #

1 x2  y2  1 ds

dt a1
tb2  1

1
1 t
1 È2 ’tan Š È2 ‹“

j, 0 Ÿ x Ÿ 2 Ê

3

2 3

ˆ5$Î#  1‰ œ

#

Š x# ‹

10È5
2 3

28. r(t) œ a1  tbi
#1 a1  tb2 j, 0 Ÿ t Ÿ 1 Ê ¸ ddtr ¸ œ É1
a1  tb# ; f(xß y) œ f Ša1  tbß #1 a1  tb2 ‹ œ Ê

'C f ds œ '01 a1
tb 

œ 0  ˆ "# 

4 1 4 a1
tb #

É1  a1
tb

" ‰ #0

œ

œ 2x Ê 'C f ds

x$

É1
a1  tb# dt œ ' Ša1  tb
14 a1  tb4 ‹ dt œ ’ "# a1  tb2  0 1

a1
tb  14 a1
tb4 É1  a1
tb#

1 20 a1

 tb5 “

" !

11 #0

29. r(t) œ (2 cos t) i
(2 sin t) j , 0 Ÿ t Ÿ

1 #

Ê

dr dt

œ (2 sin t) i
(2 cos t) j Ê ¸ ddtr ¸ œ 2; f(xß y) œ f(2 cos tß 2 sin t)

œ 2 cos t
2 sin t Ê 'C f ds œ '0 (2 cos t
2 sin t)(2) dt œ c4 sin t  4 cos td ! 1Î2

30. r(t) œ (2 sin t) i
(2 cos t) j , 0 Ÿ t Ÿ œ 4 sin# t  2 cos t Ê

1Î#

1 4

Ê

dr dt

œ 4  (4) œ 8

œ (2 cos t) i
(2 sin t) j Ê ¸ ddtr ¸ œ 2; f(xß y) œ f(2 sin tß 2 cos t)

'C f ds œ '01Î4 a4 sin# t  2 cos t b (2) dt œ c4t  2 sin 2t  4 sin td 01Î% œ 1  2Š1
È2‹

31. y œ x2 , 0 Ÿ x Ÿ 2 Ê ratb œ ti
t2 j , 0 Ÿ t Ÿ 2 Ê

dr dt

œ i
2tj Ê ¸ ddtr ¸ œ È1
4t2 Ê A œ 'C fax, yb ds

3 Î2 œ 'C ˆx
Èy‰ds œ '0 Št
Èt2 ‹È1
4t2 dt œ '0 2tÈ1
4t2 dt œ ’ 16 a1
4t2 b “ œ 2

2

2 0

17È17
1 6

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

942

Chapter 16 Integration in Vector Fields

32. 2x
3y œ 6, 0 Ÿ x Ÿ 6 Ê ratb œ ti
ˆ2  23 t‰j , 0 Ÿ t Ÿ 6 Ê œ 'C a4
3x
2ybds œ '0 ˆ4
3t
2ˆ2  23 t‰‰ 6

33. r(t) œ at#  1b j
2tk , 0 Ÿ t Ÿ 1 Ê

dr dt

È13 3

dt œ

È13 3

dr dt

È13 3

œ i  23 j Ê ¸ ddtr ¸ œ

Ê A œ 'C fax, yb ds

'06 ˆ8
35 t‰dt œ È313
8t
65 t2 ‘ 60 œ 26È13

œ 2tj
2k Ê ¸ ddtr ¸ œ 2Èt#
1; M œ 'C $ (xß yß z) ds œ '0 $ (t) Š2Èt#
1‹ dt 1

3/2 œ '0 ˆ 3# t‰ Š2Èt#
1‹ dt œ ’at#
1b “ œ 2$Î#  1 œ 2È2  1 "

1

!

34. r(t) œ at#  1b j
2tk , 1 Ÿ t Ÿ 1 Ê ddtr œ 2tj
2k Ê ¸ dr ¸ œ 2Èt#
1; M œ ' $ (xß yß z) ds dt

C

œ 'c1 1

ˆ15Èat#

 1b
2‰ Š2Èt#
1‹ dt

œ 'c1 30 at#
1b dt œ ’30 Š t3
t‹“ 1

$

"
"

œ 60 ˆ 3"
1‰ œ 80;

Mxz œ 'C y$ (xß yß z) ds œ 'c1 at#  1b c30 at#
1bd dt 1

œ 'c1 30 at%  1b dt œ ’30 Š t5  t‹“ 1

&

œ 48 Ê y œ

Mxz M

"


"

œ 60 ˆ 5"  1‰

48 œ  80 œ  53 ; Myz œ 'C x$ (xß yß z) ds œ 'C 0 $ ds œ 0 Ê x œ 0; z œ 0 by symmetry (since $ is

independent of z) Ê (xß yß z) œ ˆ!ß  35 ß 0‰ 35. r(t) œ È2t i
È2t j
a4  t# b k , 0 Ÿ t Ÿ 1 Ê

dr dt

œ È2i
È2j  2tk Ê ¸ ddtr ¸ œ È2
2
4t# œ 2È1
t# ;

(a) M œ 'C $ ds œ '0 (3t) Š2È1
t# ‹ dt œ ’2 a1
t# b 1

$Î# "

“ œ 2 ˆ2$Î#  1‰ œ 4È2  2 !

(b) M œ 'C $ ds œ '0 a1b Š2È1
t# ‹ dt œ ’tÈ1
t#
ln Št
È1
t# ‹“ œ ’È2
ln Š1
È2‹“  a0
ln 1b "

1

!

œ È2
ln Š1
È2‹ 36. r(t) œ ti
2tj
23 t$Î# k , 0 Ÿ t Ÿ 2 Ê

dr dt

œ i
2j
t"Î# k Ê ¸ ddtr ¸ œ È1
4
t œ È5
t;

# M œ 'C $ ds œ '0 ˆ3È5
t‰ ˆÈ5
t‰ dt œ '0 3(5
t) dt œ
32 (5
t)# ‘ ! œ 2

2

3 #

a7#  5# b œ

Myz œ 'C x$ ds œ '0 t[3(5
t)] dt œ '0 a15t
3t# b dt œ
"25 t#
t$ ‘ ! œ 30
8 œ 38; 2

2

2

2

# œ '0 ˆ10t$Î#
2t&Î# ‰ dt œ
4t&Î#
47 t(Î# ‘ ! œ 4(2)&Î#
47 (2)(Î# œ 16È2
2

œ

38 36

œ

19 18

,yœ

Mxz M

œ

76 36

œ

19 9

, and z œ

(24) œ 36;

#

Mxz œ 'C y$ ds œ '0 2t[3(5
t)] dt œ 2 '0 a15t
3t# b dt œ 76; Mxy œ 'C z$ ds œ '0 2

3 #

Mxy M

œ

144È2 7†36

37. Let x œ a cos t and y œ a sin t, 0 Ÿ t Ÿ 21. Then

dx dt

œ

4 7

32 7

È2 œ

dz dt

œ0

2 $Î# [3(5 3 t

144 7


t)] dt

È2 Ê x œ

Myz M

È2

œ a sin t,

dy dt

œ a cos t,

‰
Š dy ˆ dz ‰ dt œ a dt; Iz œ ' ax#
y# b $ ds œ ' aa# sin# t
a# cos# tb a$ dt Ê Êˆ dx dt dt ‹
dt C 0 #

#

21

#

œ '0 a$ $ dt œ 21$ a$ . 21

38. r(t) œ tj
(2  2t)k , 0 Ÿ t Ÿ 1 Ê

dr dt

œ j  2k Ê ¸ ddtr ¸ œ È5; M œ 'C $ ds œ '0 $ È5 dt œ $ È5; 1

" Ix œ 'C ay#
z# b $ ds œ '0 ct#
(2  2t)# d $ È5 dt œ '0 a5t#  8t
4b $ È5 dt œ $ È5
53 t$  4t#
4t‘ ! œ 1

1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

5 3

$ È5 ;

Section 16.1 Line Integrals " Iy œ 'C ax#
z# b $ ds œ '0 c0#
(2  2t)# d $ È5 dt œ '0 a4t#  8t
4b $ È5 dt œ $ È5
43 t$  4t#
4t‘ ! œ 1

1

Iz œ 'C ax#
y# b $ ds œ '0 a0#
t# b $ È5 dt œ $ È5 ’ t3 “ œ 1

"

$

!

39. r(t) œ (cos t)i
(sin t)j
tk , 0 Ÿ t Ÿ 21 Ê

" 3

4 3

$ È5 ;

$ È5

œ ( sin t)i
(cos t)j
k Ê ¸ ddtr ¸ œ Èsin# t
cos# t
1 œ È2;

dr dt

(a) Iz œ 'C ax#
y# b $ ds œ '0 acos# t
sin# tb $ È2 dt œ 21$ È2 21

(b) Iz œ 'C ax#
y# b $ ds œ '0 $ È2 dt œ 41$ È2 41

40. r(t) œ (t cos t)i
(t sin t)j


2È2 $Î# k, 3 t

0ŸtŸ1 Ê

dr dt

œ (cos t  t sin t)i
(sin t
t cos t)j
È2t k

" Ê ¸ ddtr ¸ œ È(t
1)# œ t
1 for 0 Ÿ t Ÿ 1; M œ 'C $ ds œ '0 (t
1) dt œ
"2 (t
1)# ‘ ! œ 1

Mxy œ 'C z$ ds œ '

È Š 2 3 2 t$Î# ‹ (t 0

œ

2È 2 3

ˆ 27
52 ‰ œ

1

2È 2 3

ˆ 24 ‰ 35 œ


1) dt œ

16È2 35

Ê zœ

2È 2 3

'0 ˆt&Î#
t$Î# ‰ dt œ

Mxy M

œ Š 1635 2 ‹ ˆ 23 ‰ œ

È

32È2 105

œ '0 at# cos# t
t# sin# tb (t
1) dt œ '0 at$
t# b dt œ ’ t4
t3 “ œ 1

2È 2 3

1

1

%

"

$

!

" 4

" #

a2#  1# b œ

3 #

;


27 t(Î#
25 t&Î# ‘ " !

; Iz œ 'C ax#
y# b $ ds




" 3

œ

7 12

41. $ (xß yß z) œ 2  z and r(t) œ (cos t)j
(sin t)k , 0 Ÿ t Ÿ 1 Ê M œ 21  2 as found in Example 3 of the text; also ¸ ddtr ¸ œ 1; Ix œ 'C ay#
z# b $ ds œ '0 acos# t
sin# tb (2  sin t) dt œ '0 (2  sin t) dt œ 21  2 1

42. r(t) œ ti


2È2 $Î# j 3 t




t# #

k, 0 Ÿ t Ÿ 2 Ê

1

dr dt

œ i
È2 t"Î# j
tk Ê ¸ ddtr ¸ œ È1
2t
t# œ È(1
t)# œ 1
t for

0 Ÿ t Ÿ 2; M œ 'C $ ds œ '0 ˆ t"1 ‰ (1
t) dt œ '0 dt œ 2; Myz œ 'C x$ ds œ '0 t ˆ t"1 ‰ (1
t) dt œ ’ t2 “ œ 2; 2

Mxz œ 'C y$ ds œ '

2È2 $Î# 3 t 0

yœ

Mxz M

œ

16 15

2

, and z œ

Mxy M

œ

2

dt œ # 3

# È ’ 4152 t&Î# “ !

œ

2

2

$

œ '0 ˆt#
89 t$ ‰ dt œ ’ t3
29 t% “ œ $

; Mxy œ 'C z$ ds œ '0

2 # t

#

dt œ

#

$ # ’ t6 “ !

; Ix œ 'C ay#
z# b $ ds œ '0 ˆ 98 t$
"4 t% ‰ dt œ ’ 92 t%


Iy œ 'C ax#
z# b $ ds œ '0 ˆt#
4" t% ‰ dt œ ’ t3
2

32 15

2

# !

8 3




32 9

œ

# t& 20 “ !

œ

8 3




32 20

œ

64 15

œ

# t& 20 “ !

œ

; Iz œ 'C ax#
y# b $ ds

56 9

43-46. Example CAS commands: Maple: f := (x,y,z) -> sqrt(1+30*x^2+10*y); g := t -> t; h := t -> t^2; k := t -> 3*t^2; a,b := 0,2; ds := ( D(g)^2 + D(h)^2 + D(k)^2 )^(1/2): 'ds' = ds(t)*'dt'; F := f(g,h,k): 'F(t)' = F(t); Int( f, s=C..NULL ) = Int( simplify(F(t)*ds(t)), t=a..b ); `` = value(rhs(%));

# !

% 3

# (a) # (b) # (c)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Ê xœ

Myz M




œ

32 9

32 20

œ 1, 232 45

;

943

944

Chapter 16 Integration in Vector Fields

Mathematica: (functions and domains may vary) Clear[x, y, z, r, t, f] f[x_,y_,z_]:= Sqrt[1
30x2
10y] {a,b}= {0, 2}; x[t_]:= t y[t_]:= t2 z[t_]:= 3t2 r[t_]:= {x[t], y[t], z[t]} v[t_]:= D[r[t], t] mag[vector_]:=Sqrt[vector.vector] Integrate[f[x[t],y[t],z[t]] mag[v[t]], {t, a, b}] N[%] 16.2 VECTOR FIELDS, WORK, CIRCULATION, AND FLUX 1. f(xß yß z) œ ax#
y#
z# b `f `y

#

#


"Î#

#
$Î#

œ  y ax
y
z b

and

`f `y

œ

y x #  y#  z#

and

`f `z

" #

2. f(xß yß z) œ ln Èx#
y#
z# œ similarly,

`f `x

Ê

`f `z

#


$Î#

#
$Î#

#

œ z ax
y
z b

ln ax#
y#
z# b Ê

œ

3. g(xß yß z) œ ez  ln ax#
y# b Ê

œ  #" ax#
y#
z# b

z x #  y #  z#

`g `x

Ê ™fœ

œ  x# 2x  y# ,

`g `y

`f `x

(2x) œ x ax#
y#
z# b

Ê ™fœ

œ

" #


$Î#

; similarly,


xi
yj
zk ax#  y#  z# b$Î#

Š x# y"# z# ‹ (2x) œ

x x#  y#  z#

;

xi  yj  zk x #  y#  z#

œ  x# 2y  y# and

`g `z

œ ez

z Ê ™ g œ Š x#
2xy# ‹ i  Š x# 2y  y# ‹ j
e k

`g `x

4. g(xß yß z) œ xy
yz
xz Ê

œ y
z,

`g `y

œ x
z, and

`g `z

œ y
x Ê ™ g œ (y
z)i
(B
z)j
(x
y)k

5. kFk inversely proportional to the square of the distance from (xß y) to the origin Ê È(M(xß y))#
(N(xß y))# œ

k x#  y#

y
x È x #  y# i  È x#  y# j Then M(xß y) œ Èx
#ax and N(xß y) œ Èx
#ay  y#  y# ky k
kx a œ x#  y# Ê F œ # # $Î# i  # # $Î# j , for any constant ax  y b ax  y b

, k  0; F points toward the origin Ê F is in the direction of n œ

Ê F œ an , for some constant a  0. Ê È(M(xß y))#
(N(xß y))# œ a Ê

k0

6. Given x#
y# œ a#
b# , let x œ Èa#
b# cos t and y œ Èa#
b# sin t. Then r œ ŠÈa#
b# cos t‹ i  ŠÈa#
b# sin t‹ j traces the circle in a clockwise direction as t goes from 0 to 21 Ê v œ ŠÈa#
b# sin t‹ i  ŠÈa#
b# cos t‹ j is tangent to the circle in a clockwise direction. Thus, let F œ v Ê F œ yi  xj and F(0ß 0) œ 0 . 7. Substitute the parametric representations for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F , and calculate 'C F †

dr dt

.

(a) F œ 3ti
2tj
4tk and

dr dt

œi
j
k Ê F†

(b) F œ 3t# i
2tj
4t% k and œ

7 3


2œ

dr dt

dr dt

œ 9t Ê

œ i
2tj
4t$ k Ê F †

dr dt

'01 9t dt œ 9#

œ 7t#
16t( Ê

'01 a7t#
16t( b dt œ
37 t$
2t) ‘ "!

13 3

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.2 Vector Fields, Work, Circulation, and Flux (c) r" œ ti
tj and r# œ i
j
tk ; F" œ 3ti
2tj and F# œ 3i
2j
4tk and

œ k Ê F# †

d r# dt

d r# dt

d r" dt

œ i
j Ê F" †

'01 4t dt œ 2

œ 4t Ê

Ê

d r" dt 5 #

'01 5t dt œ #5 ;

œ 5t Ê


2œ

9 #

8. Substitute the parametric representation for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F, and calculate 'C F †

dr dt

.

" ‰ (a) F œ ˆ t#  1 j and

dr dt

œi
j
kÊF†

" ‰ (b) F œ ˆ t#  1 j and

dr dt

œ i
2tj
4t$ k Ê F †

dr dt

" t#  1

œ

dr dt

" ‰ (c) r" œ ti
tj and r# œ i
j
tk ; F" œ ˆ t#  1 j

Ê F# †

d r# dt

œ 0 Ê '0

1

" t#  1

dt œ

Ê '0

1

œ

2t t#  1 and ddtr"

" t#  1

Ê '0

1

"

dt œ ctan
" td ! œ 2t t#  1

1 4 "

dt œ cln at#
1bd ! œ ln 2

œ i
j Ê F" †

d r" dt

œ

" t#  1

; F# œ

" #

j and

d r# dt

œk

1 4

9. Substitute the parametric representation for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F, and calculate 'C F †

dr dt

.

'01 ˆ2Èt  2t‰ dt œ
43 t$Î#  t# ‘ "! œ "3 1 " F œ t# i  2tj
tk and ddtr œ i
2tj
4t$ k Ê F † ddtr œ 4t%  3t# Ê '0 a4t%  3t# b dt œ
45 t&  t$ ‘ ! œ  "5 1 r" œ ti
tj and r# œ i
j
tk ; F" œ 2tj
Èt k and ddtr œ i
j Ê F" † ddtr œ 2t Ê '0 2t dt œ 1; 1 F# œ Èti  2j
k and ddtr œ k Ê F# † ddtr œ 1 Ê '0 dt œ 1 Ê 1
1 œ 0

(a) F œ Èti  2tj
Ètk and (b) (c)

dr dt

œi
j
k Ê F†

œ 2Èt  2t Ê

dr dt

"

#

"

#

10. Substitute the parametric representation for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F, and calculate 'C F †

dr dt

. œ 3t# Ê '0 3t# dt œ 1 1

(a) F œ t# i
t# j
t# k and

dr dt

œi
j
k Ê F†

(b) F œ t$ i  t' j
t& k and

dr dt

œ i
2tj
4t$ k Ê F †

%

œ ’ t4


t) 4

"


94 t* “ œ !

dr dt

œ t$
2t(
4t) Ê '0 at$
2t(
4t) b dt 1

17 18

(c) r" œ ti
tj and r# œ i
j
tk ; F" œ t# i and F# œ i
tj
tk and

dr dt

d r# dt

œ k Ê F# †

d r# dt

d r" dt

œ i
j Ê F" †

œ t Ê '0 t dt œ 1

" #

Ê

d r" dt " 3

œ t# Ê '0 t# dt œ




1

" #

œ

" 3

;

5 6

11. Substitute the parametric representation for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F, and calculate 'C F †

dr dt

.

(a) F œ a3t#  3tb i
3tj
k and

†

(b) F œ a3t#  3tb i
3t% j
k

Ê F†

Ê

dr dt œ i
j
k Ê F and ddtr œ i
2tj
4t$ k

dr dt

œ 3t#
1 Ê

œ 6t&
4t$
3t#  3t

'0 a6t&
4t$
3t#  3tb dt œ
t'
t%
t$  3# t# ‘ "! œ 3# 1

(c) r" œ ti
tj and r# œ i
j
tk ; F" œ a3t#  3tb i
k and Ê

dr dt

'01 a3t#
1b dt œ ct$
td "! œ 2

d r" dt

œ i
j Ê F" †

d r" dt

œ 3t#  3t

œ k Ê F# †

d r# dt

œ1 Ê

'0 a3t#  3tb dt œ
t$  32 t# ‘ "! œ  "# ; F# œ 3tj
k and ddtr 1

Ê  "#
1 œ

#

'01 dt œ 1

1 2

12. Substitute the parametric representation for r(t) œ x(t)i
y(t)j
z(t)k representing each path into the vector field F, and calculate 'C F †

dr dt

.

(a) F œ 2ti
2tj
2tk and

dr dt

œi
j
k Ê F†

dr dt

œ 6t Ê

'01 6t dt œ c3t# d "! œ 3

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

945

946

Chapter 16 Integration in Vector Fields

(b) F œ at#
t% b i
at%
tb j
at
t# b k and Ê '0 a6t&
5t%
3t# b dt œ ct'
t&
1

dr dt œ i " t$ d ! œ 3


2tj
4t$ k Ê F †

(c) r" œ ti
tj and r# œ i
j
tk ; F" œ ti
tj
2tk and F# œ (1
t)i
(t
1)j
2k and

d r# dt

œ k Ê F# †

d r# dt

dr dt

œ 6t&
5t%
3t#

œ i
j Ê F" †

dr" dt

œ 2t Ê '0 2t dt œ "; 1

d r" dt

œ 2 Ê '0 2 dt œ 2 Ê "
2 œ 3 1

13. x œ t, y œ 2t
1, 0 Ÿ t Ÿ 3 Ê dx œ dt Ê 'C ax  yb dx œ '0 at  a2t
1bb dt œ '0 at  1b dt œ
 "# t2  t‘ ! œ  15 2 3

14. x œ t, y œ t2 , 1 Ÿ t Ÿ 2 Ê dy œ 2t dt Ê 'C

x y

dy œ '1

2

t t2 a2tb dt

3

3

œ '1 2 dt œ c2td21 œ 2 2

15. C1 : x œ t, y œ 0, 0 Ÿ t Ÿ 3 Ê dy œ 0; C2 : x œ 3, y œ t, 0 Ÿ t Ÿ 3 Ê dy œ dt Ê 'C ax2
y2 b dy

œ 'C ax2
y2 b dx
'C ax2
y2 b dx œ '0 at2
02 b † 0
'0 a32
t2 b dt œ '0 a9
t2 bdt œ
9t
13 t3 ‘ ! œ 36 3

1

3

3

3

2

16. C1 : x œ t, y œ 3t, 0 Ÿ t Ÿ 1 Ê dx œ dt; C2 : x œ 1  t, y œ 3, 0 Ÿ t Ÿ 1 Ê dx œ dt; C3 : x œ 0, y œ 3  t, 0 Ÿ t Ÿ 3 Ê dx œ 0 Ê 'C Èx
y dx œ 'C Èx
y dx
'C Èx
y dx
'C Èx
y dx 1

2

3

œ '0 Èt
3t dt
'0 Èa1  tb
3 a1bdt
'0 È0
a3  tb † 0 œ '0 2Èt dt  '0 È4  t dt 1

1

3

1

œ
43 t2Î3 ‘ !
’ 23 a4  tb2Î3 “ œ 1

4 3

!


Š2È3 

16 3 ‹

1

1

œ 2È3  4

17. ratb œ ti  j
t2 k , 0 Ÿ t Ÿ 1 Ê dx œ dt, dy œ 0, dz œ 2t dt (a) (b) (c)

'C ax
y  zb dx œ '01 at  1  t2 b dt œ
12 t2  t  13 t3 ‘ 1! œ  56 'C ax
y  zb dy œ '01 at  1  t2 b † 0 œ 0

'C ax
y  zb dz œ '01 at  1  t2 b 2t dt œ '01 a2t2  2t  2t3 b dt œ

1

œ
23 t3  t2  12 t4 ‘ ! œ  56

18. ratb œ acos tbi
asin tbj  acos tbk , 0 Ÿ t Ÿ 1 Ê dx œ sin t dt, dy œ cos t dt, dz œ sin t dt (a)

'C x z dx œ '01 acos tb acos tbasin tbdt œ '01 cos2 t sin tdt œ ’ 13 acos tb3 “ 1 œ 23

(b)

'C x z dy œ '01 acos tb acos tbacos tbdt œ '01 cos3 t dt œ '01 a1  sin2 tb cos t dt œ ’ 13 asin tb3  sin t“ 1 œ 0

(c)

!

'C x y z dz œ '0 acos tbasin tb acos tbasin tbdt œ '0 1 1 1 œ  18 '0 a1  cos 4tb dt œ
 18 t
32 sin 4t‘ ! œ  18 1

1

cos t sin t dt œ 2

2

 14

'0

1

19. r œ ti
t# j
tk , 0 Ÿ t Ÿ 1, and F œ xyi
yj  yzk Ê F œ t$ i
t# j  t$ k and Ê F†

dr dt

œ 2t$ Ê work œ '0 2t$ dt œ 1

sin 2t dt œ

dr dt

2

 41

œ i
2tj
k

" #

20. r œ (cos t)i
(sin t)j
6t k , 0 Ÿ t Ÿ 21, and F œ 2yi
3xj
(x
y)k Ê F œ (2 sin t)i
(3 cos t)j
(cos t
sin t)k and œ 3 cos# t  2sin2 t
œ
32 t


3 4

" 6

sin 2t  t


cos t
sin 2t 2




" 6

" 6

dr dt

œ ( sin t)i
(cos t)j
6" k Ê F †

sin t Ê work œ '0 ˆ3 cos# t  2 sin2 t


sin t 

" 6

cos

#1 t‘ !

21

" 6

cos t


" 6

dr dt

sin t‰ dt

œ1

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

'

1

!

1
cos 4t 2 0

dt

Section 16.2 Vector Fields, Work, Circulation, and Flux 21. r œ (sin t)i
(cos t)j
tk , 0 Ÿ t Ÿ 21, and F œ zi
xj
yk Ê F œ ti
(sin t)j
(cos t)k and dr dt

œ (cos t)i  (sin t)j
k Ê F †

œ
cos t
t sin t 

t 2




sin 2t 4

dr dt

œ t cos t  sin# t
cos t Ê work œ '0 at cos t  sin# t
cos tb dt 21

#1


sin t‘ ! œ 1

22. r œ (sin t)i
(cos t)j
6t k , 0 Ÿ t Ÿ 21, and F œ 6zi
y# j
12xk Ê F œ ti
acos# tbj
(12 sin t)k and dr dt

œ (cos t)i  (sin t)j
6" k Ê F †

dr dt

œ t cos t  sin t cos# t
2 sin t

Ê work œ '0 at cos t  sin t cos# t
2 sin tb dt œ
cos t
t sin t
21

1 3

#1

cos$ t  2 cos t‘ ! œ 0

23. x œ t and y œ x# œ t# Ê r œ ti
t# j , 1 Ÿ t Ÿ 2, and F œ xyi
(x
y)j Ê F œ t$ i
at
t# b j and dr dt

œ i
2tj Ê F †

dr dt

œ t$
a2t#
2t$ b œ 3t$
2t# Ê 'C xy dx
(x
y) dy œ 'C F †

#

œ
34 t%
32 t$ ‘
" œ ˆ12


16 ‰ 3

 ˆ 34  23 ‰ œ

45 4




18 3

œ

dr dt

dt œ 'c" a3t$
2t# b dt #

69 4

24. Along (0ß 0) to (1ß 0): r œ ti , 0 Ÿ t Ÿ 1, and F œ (x  y)i
(x
y)j Ê F œ ti
tj and

dr dt

œi Ê F†

dr dt

œ t;

Along (1ß 0) to (0ß 1): r œ (1  t)i
tj , 0 Ÿ t Ÿ 1, and F œ (x  y)i
(x
y)j Ê F œ (1  2t)i
j and dr dr dt œ i
j Ê F † dt œ 2t; Along (0ß 1) to (0ß 0): r œ (1  t)j , 0 Ÿ t Ÿ 1, and F œ (x  y)i
(x
y)j Ê F œ (t  1)i
(1  t)j and dr dt

œ j Ê F †

dr dt

œ t  1 Ê 'C (x  y) dx
(x
y) dy œ '0 t dt
'0 2t dt
'0 (t  1) dt œ '0 (4t  1) dt 1

1

1

1

dr dy

œ 2yi
j and F †

"

œ c2t#  td ! œ 2  1 œ 1 25. r œ xi
yj œ y# i
yj , 2   y   1, and F œ x# i  yj œ y% i  yj Ê Ê

dr dy

œ 2y&  y

4‰ 3 63 39 'C F † T ds œ '2c1 F † dydr dy œ '2c1 a2y&  yb dy œ
3" y'  "# y# ‘
" œ ˆ 3"  #" ‰  ˆ 64 3  # œ #  3 œ # #

26. r œ (cos t)i
(sin t)j , 0 Ÿ t Ÿ ÊF†

dr dt

1 #

, and F œ yi  xj Ê F œ (sin t)i  (cos t)j and

œ  sin# t  cos# t œ 1 Ê

'C F † dr œ '0

1Î2

dr dt

œ ( sin t)i
(cos t)j

(1) dt œ  1#

27. r œ (i
j)
t(i
2j) œ (1
t)i
(1
2t)j , 0 Ÿ t Ÿ 1, and F œ xyi
(y  x)j Ê F œ a1
3t
2t# b i
tj and dr dt

œ i
2j Ê F †

dr dt

œ 1
5t
2t# Ê work œ 'C F †

dr dt

dt œ '0 a1
5t
2t# b dt œ
t
25 t#
23 t$ ‘ ! œ 1

"

28. r œ (2 cos t)i
(2 sin t)j , 0 Ÿ t Ÿ 21, and F œ ™ f œ 2(x
y)i
2(x
y)j Ê F œ 4(cos t
sin t)i
4(cos t
sin t)j and ddtr œ (2 sin t)i
(2 cos t)j Ê F †

25 6

dr dt

œ 8 asin t cos t
sin# tb
8 acos# t
cos t sin tb œ 8 acos# t  sin# tb œ 8 cos 2t Ê work œ 'C ™ f † dr œ 'C F †

dr dt

dt œ '0 8 cos 2t dt œ c4 sin 2td #!1 œ 0 21

29. (a) r œ (cos t)i
(sin t)j , 0 Ÿ t Ÿ 21, F" œ xi
yj , and F# œ yi
xj Ê F" œ (cos t)i
(sin t)j , and F# œ ( sin t)i
(cos t)j Ê F" †

dr dt

dr dt

œ ( sin t)i
(cos t)j ,

œ 0 and F# †

dr dt

œ sin# t
cos# t œ 1

Ê Circ" œ '0 0 dt œ 0 and Circ# œ '0 dt œ 21; n œ (cos t)i
(sin t)j Ê F" † n œ cos# t
sin# t œ 1 and 21

21

F# † n œ 0 Ê Flux" œ '0 dt œ 21 and Flux# œ '0 0 dt œ 0 21

21

(b) r œ (cos t)i
(4 sin t)j , 0 Ÿ t Ÿ 21 Ê F# œ (4 sin t)i
(cos t)j Ê F" †

dr dt

dr dt

œ ( sin t)i
(4 cos t)j , F" œ (cos t)i
(4 sin t)j , and

œ 15 sin t cos t and F# †

dr dt

œ 4 Ê Circ" œ '0 15 sin t cos t dt 21

œ
"25 sin# t‘ ! œ 0 and Circ# œ '0 4 dt œ 81; n œ Š È417 cos t‹ i
Š È"17 sin t‹ j Ê F" † n #1

21

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

947

948

Chapter 16 Integration in Vector Fields œ

4 È17

cos# t


sin# t and F# † n œ  È1517 sin t cos t Ê Flux" œ '0 (F" † n) kvk dt œ '0 Š È417 ‹ È17 dt 21

4 È17

21

# ‘ œ 81 and Flux# œ '0 (F# † n) kvk dt œ '0 Š È1517 sin t cos t‹ È17 dt œ
 15 2 sin t ! œ 0 21

21

#1

30. r œ (a cos t)i
(a sin t)j , 0 Ÿ t Ÿ 21, F" œ 2xi  3yj , and F# œ 2xi
(x  y)j Ê

œ (a sin t)i
(a cos t)j ,

dr dt

F" œ (2a cos t)i  (3a sin t)j , and F# œ (2a cos t)i
(a cos t  a sin t)j Ê n kvk œ (a cos t)i
(a sin t)j , F" † n kvk œ 2a# cos# t  3a# sin# t, and F# † n kvk œ 2a# cos# t
a# sin t cos t  a# sin# t Ê Flux" œ '0 a2a# cos# t  3a# sin# tb dt œ 2a#
2t
21

sin 2t ‘ #1 4 !

Flux# œ '0 a2a# cos# t  a# sin t cos t  a# sin# tb dt œ 2a#
2t
21

31. F" œ (a cos t)i
(a sin t)j ,

d r" dt

sin 2t ‘ #1 4 !

œ 1a# , and

a# #

#1

 3a#
2t 

œ (a sin t)i
(a cos t)j Ê F" †

sin 2t ‘ #1 4 ! d r" dt




csin# td !  a#
2t 

sin 2t ‘ #1 4 !

œ 1a#

œ 0 Ê Circ" œ 0; M" œ a cos t,

N" œ a sin t, dx œ a sin t dt, dy œ a cos t dt Ê Flux" œ 'C M" dy  N" dx œ '0 aa# cos# t
a# sin# tb dt œ '0 a# dt œ a# 1;

1

1

F # œ ti ,

d r# dt

œ i Ê F# †

d r# dt

œ t Ê Circ# œ 'ca t dt œ 0; M# œ t, N# œ 0, dx œ dt, dy œ 0 Ê Flux# a

œ 'C M# dy  N# dx œ 'ca 0 dt œ 0; therefore, Circ œ Circ"
Circ# œ 0 and Flux œ Flux"
Flux# œ a# 1 a

32. F" œ aa# cos# tb i
aa# sin# tb j ,

d r" dt

œ (a sin t)i
(a cos t)j Ê F" †

d r" dt

œ a$ sin t cos# t
a$ cos t sin# t

Ê Circ" œ '0 aa$ sin t cos# t
a$ cos t sin# tb dt œ  2a3 ; M" œ a# cos# t, N" œ a# sin# t, dy œ a cos t dt, 1

$

dx œ a sin t dt Ê Flux" œ 'C M" dy  N" dx œ '0 aa$ cos$ t
a$ sin$ tb dt œ 1

F # œ t# i ,

d r# dt

œ i Ê F# †

d r# dt

œ t# Ê Circ# œ 'ca t# dt œ a

2a$ 3

4 3

a$ ;

; M# œ t# , N# œ 0, dy œ 0, dx œ dt

Ê Flux# œ 'C M# dy  N# dx œ 0; therefore, Circ œ Circ"
Circ# œ 0 and Flux œ Flux"
Flux# œ 33. F" œ (a sin t)i
(a cos t)j ,

d r" dt

œ (a sin t)i
(a cos t)j Ê F" †

d r" dt

4 3

a$

œ a# sin# t
a# cos# t œ a#

Ê Circ" œ '0 a# dt œ a# 1 ; M" œ a sin t, N" œ a cos t, dx œ a sin t dt, dy œ a cos t dt 1

Ê Flux" œ 'C M" dy  N" dx œ '0 aa# sin t cos t
a# sin t cos tb dt œ 0; F# œ tj , 1

dr# dt

œ i Ê F# †

d r# dt

œ0

Ê Circ# œ 0; M# œ 0, N# œ t, dx œ dt, dy œ 0 Ê Flux# œ 'C M# dy  N# dx œ 'ca t dt œ 0; therefore, a

Circ œ Circ"
Circ# œ a# 1 and Flux œ Flux"
Flux# œ 0 34. F" œ aa# sin# tb i
aa# cos# tb j ,

d r" dt

œ (a sin t)i
(a cos t)j Ê F" †

Ê Circ" œ '0 aa$ sin$ t
a$ cos$ tb dt œ 1

4 3

d r" dt

œ a$ sin$ t
a$ cos$ t

a$ ; M" œ a# sin# t, N" œ a# cos# t, dy œ a cos t dt, dx œ a sin t dt

Ê Flux" œ 'C M" dy  N" dx œ '0 aa$ cos t sin# t
a$ sin t cos# tb dt œ 1

2 3

a$ ; F# œ t# j ,

d r# dt

œ i Ê F# †

d r# dt

œ0

Ê Circ# œ 0; M# œ 0, N# œ t# , dy œ 0, dx œ dt Ê Flux# œ 'C M# dy  N# dx œ 'ca t# dt œ  23 a$ ; therefore, a

Circ œ Circ"
Circ# œ

4 3

a$ and Flux œ Flux"
Flux# œ 0

35. (a) r œ (cos t)i
(sin t)j , 0 Ÿ t Ÿ 1, and F œ (x
y)i  ax#
y# b j Ê F œ (cos t
sin t)i  acos# t
sin# tb j Ê F †

dr dt

(b) r œ (1  2t)i , 0 Ÿ t Ÿ 1, and F œ (x
y)i F†

dr dt

œ 4t  2 Ê 'C F † T ds œ '0 (4t  1

œ (sin t)i
(cos t)j and

œ  sin t cos t  sin# t  cos t Ê 'C F † T ds

œ '0 a sin t cos t  sin# t  cos tb dt œ
 2" sin# t  1

dr dt

sin 2t 1 ‘1 4  sin t ! œ  #  ax#
y# b j Ê ddtr œ 2i and F œ (1 " 2) dt œ c2t#  2td ! œ 0 t #




 2t)i  (1  2t)# j Ê

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.2 Vector Fields, Work, Circulation, and Flux (c) r" œ (1  t)i  tj , 0 Ÿ t Ÿ 1, and F œ (x
y)i  ax#
y# b j Ê Ê F†

d r" dt

œ (2t  1)
a1  2t
2t# b œ 2t# Ê Flow" œ 'C F †

d r" dt

"

#

#

0 Ÿ t Ÿ 1, and F œ (x
y)i  ax
y b j Ê œ i  a2t#  2t
1b j Ê F † "

œ
t#  23 t$ ‘ ! œ

" 3

d r# dt

œ i  j and F œ (1  2t)i  a1  2t
2t# b j

d r" dt

œ '0 2t# dt œ 1

#

2 3

; r# œ ti
(t  1)j ,

#

œ i
j and F œ i  at
t  2t
1b j

œ 1  a2t#  2t
1b œ 2t  2t# Ê Flow# œ 'C F †

dr # dt

949

#

Ê Flow œ Flow"
Flow# œ

2 3

" 3




dr # dt

œ '0 a2t  2t# b dt 1

œ1

36. From (1ß 0) to (0ß 1): r" œ (1  t)i
tj , 0 Ÿ t Ÿ 1, and F œ (x
y)i  ax#
y# b j Ê

d r" dt

œ i
j ,

F œ i  a1  2t
2t# b j , and n" kv" k œ i
j Ê F † n" kv" k œ 2t  2t# Ê Flux" œ '0 a2t  2t# b dt 1

"

œ
t#  23 t$ ‘ ! œ

" 3

;

From (0ß 1) to (1ß 0): r# œ ti
(1  t)j , 0 Ÿ t Ÿ 1, and F œ (x
y)i  ax#
y# b j Ê

d r# dt

œ i  j ,

#

F œ (1  2t)i  a1  2t
2t b j , and n# kv# k œ i
j Ê F † n# kv# k œ (2t  1)
a1
2t  2t# b œ 2
4t  2t# Ê Flux# œ '0 a2
4t  2t# b dt œ
2t
2t#  23 t$ ‘ ! œ  23 ; 1

"

From (1ß 0) to (1ß 0): r$ œ (1
2t)i , 0 Ÿ t Ÿ 1, and F œ (x
y)i  ax#
y# b j Ê #

d r$ dt

œ 2i ,

#

F œ (1
2t)i  a1  4t
4t b j , and n$ kv$ k œ 2j Ê F † n$ kv$ k œ 2 a1  4t
4t b Ê Flux$ œ 2 '0 a1  4t
4t# b dt œ 2
t  2t#
43 t$ ‘ ! œ 1

"

37. (a) y œ 2x, 0 Ÿ x Ÿ 2 Ê ratb œ ti
2tj , 0 Ÿ t Ÿ 2 Ê œ 4t2
8t2 œ 12t2 Ê Flow œ 'C F †

dr dt

2 3




2 3

œ

œ Ša2tb2 i
2atba2tbj‹ † ai
2jb

2

dr dt

œ Šat2 b i
2atbat2 bj‹ † ai
2tjb

dr dt

œ i
2tj Ê F †

2

dt œ '0 5t4 dt œ ct5 d ! œ 32 2

dr dt

2

œ Šˆ "# t3 ‰ i
2atbˆ "# t3 ‰j‹ † ai
3t2 jb œ 14 t6
32 t6 œ 74 t6 Ê Flow œ 'C F † 2

dr dt

dr dt

œ i
3t2 j

dt œ '0 74 t6 dt œ
14 t7 ‘ ! 2

2

œ 32 38. (a) C1 : ratb œ a1  tbi
j , 0 Ÿ t Ÿ 2 Ê

C4 : ratb œ i
at  1bj , 0 Ÿ t Ÿ 2 Ê Ê Flow œ 'C F †

dr dt

dt œ 'C F † 1

dr dt

œ i Ê F †

dr dt

C2 : ratb œ i
a1  tbj , 0 Ÿ t Ÿ 2 Ê C3 : ratb œ at  1bi  j , 0 Ÿ t Ÿ 2 Ê

dr dt

dr dt dr dt

dr dt

œ j Ê F †

œiÊF† œjÊF†

dt
'C F †

dr dt

2

dr dt dr dt

œ aa1bi
aa1  tb
2a1bbjb † aib œ 1; dr dt

œ aa1  tbi
aa1b
2a1  tbbjb † ajb œ 2t  1;

œ aa1bi
aat  1b
2a1bbjb † aib œ 1; œ aat  1bi
aa1b
2at  1bbjb † ajb œ 2t  1;

dt
'C F † 3

dr dt

dt
'C F † 4

dr dt

dt

œ '0 a1b dt
'0 a2t  1b dt
'0 a1b dt
'0 a2t  1b dt œ ctd 2!
ct2  td !
ctd !2
ct2  td ! 2

2

2

2

2

œ 2
2  2
2 œ 0 (b) x2
y2 œ 4 Ê ratb œ a2cos tbi
a2sin tbj , 0 Ÿ t Ÿ 21 Ê ÊF†

dr dt

dr dt

2

œ a2sin tbi
a2cos tbj

œ aa2sin tbi
a2cos t
2a2sin tbbjb † aa2sin tbi
a2cos tbjb œ 4sin2 t
4cos2 t
8sin t cos t

œ 4cos 2t
4sin 2t Ê Flow œ 'C F †

dr dt

(c) answers will vary, one possible path is: C1 : ratb œ ti , 0 Ÿ t Ÿ 1 Ê ddtr œ i Ê F † C2 : ratb œ a1  tbi
tj , 0 Ÿ t Ÿ 1 Ê C3 : ratb œ a1  tbj , 0 Ÿ t Ÿ 1 Ê

dr dt

" 3

2

(c) answers will vary, one possible path is y œ 12 x3 , 0 Ÿ x Ÿ 2 Ê ratb œ ti
"# t3 j , 0 Ÿ t Ÿ 2 Ê ÊF†



dr dt

œ i
2tj Ê F †

dr dt

" 3

dt œ '0 12t2 dt œ c4t3 d ! œ 32

dr dt

(b) y œ x2 , 0 Ÿ x Ÿ 2 Ê ratb œ ti
t2 j , 0 Ÿ t Ÿ 2 Ê œ t4
4t4 œ 5t4 Ê Flow œ 'C F †

Ê Flux œ Flux"
Flux#
Flux$ œ

2 3

dr dt

dt œ '0 a4cos 2t
4sin 2tb dt œ c2sin 2t  2cos 2td 2!1 œ 0 21

dr dt

œ aa0bi
at
2a1bbjb † aib œ 0;

œ i
j Ê F †

œ j Ê F †

dr dt

dr dt

œ ati
aa1  tb
2tbjb † ai
jb œ 1;

œ aa1  tbi
a0
2a1  tbbjb † ajb œ 2t  1;

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

950

Chapter 16 Integration in Vector Fields Ê Flow œ 'C F †

dt œ 'C F †

dr dt

1

dr dt

dt
'C F † 2

dr dt

dt
'C F † 3

dr dt

dt œ '0 a0b dt
'0 a1b dt
'0 a2t  1b dt 1

1

1

1

œ 0
ctd 1!
ct2  td ! œ 1
a1b œ 0 39. F œ  Èx#y y# i


j on x#
y# œ 4;

x È x#  y#

at (2ß 0), F œ j ; at (0ß 2), F œ i ; at (2ß 0), È F œ j ; at (!ß 2), F œ i ; at ŠÈ2ß È2‹ , F œ  #3 i
#" j ; at ŠÈ2ß È2‹ , F œ Fœ

È3 #

È3 #

i
#" j ; at ŠÈ2ß È2‹ ,

i  #" j ; at ŠÈ2ß È2‹ , F œ

È3 #

i  #" j

40. F œ xi
yj on x#
y# œ 1; at (1ß 0), F œ i ; at (1ß 0), F œ i ; at (0ß 1), F œ j ; at (0ß 1), F œ j ; at Š "# ß at Š "# ß

È3 # ‹,

at Š "# ß 

È3 # ‹,

at Š "# ß 

È3 # ‹,

Fœ

" #

F œ  "# i
Fœ

È3 # ‹,

" #

i

i
È3 #

È3 #

È3 #

j;

j;

j;

F œ  "# i 

È3 #

j.

41. (a) G œ P(xß y)i
Q(xß y)j is to have a magnitude Èa#
b# and to be tangent to x#
y# œ a#
b# in a counterclockwise direction. Thus x#
y# œ a#
b# Ê 2x
2yyw œ 0 Ê yw œ  xy is the slope of the tangent line at any point on the circle Ê yw œ  ba at (aß b). Let v œ bi
aj Ê kvk œ Èa#
b# , with v in a counterclockwise direction and tangent to the circle. Then let P(xß y) œ y and Q(xß y) œ x Ê G œ yi
xj Ê for (aß b) on x#
y# œ a#
b# we have G œ bi
aj and kGk œ Èa#
b# . (b) G œ ˆÈx#
y# ‰ F œ ŠÈa#
b# ‹ F . 42. (a) From Exercise 41, part a, yi
xj is a vector tangent to the circle and pointing in a counterclockwise direction Ê yi  xj is a vector tangent to the circle pointing in a clockwise direction Ê G œ Èyxi #
xjy# is a unit vector tangent to the circle and pointing in a clockwise direction. (b) G œ F 43. The slope of the line through (xß y) and the origin is pointing away from the origin Ê F œ 

xi  yj È x#  y#

y x

Ê v œ xi
yj is a vector parallel to that line and

is the unit vector pointing toward the origin.

44. (a) From Exercise 43,  Èxxi #yjy# is a unit vector through (xß y) pointing toward the origin and we want kFk to have magnitude Èx#
y# Ê F œ Èx#
y# Š Èxxi #yjy# ‹ œ xi  yj . (b) We want kFk œ

C È x#  y#

where C Á 0 is a constant Ê F œ

C È x#  y#

yj Š Èxxi #yjy# ‹ œ C Š xx#i   y# ‹.

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.2 Vector Fields, Work, Circulation, and Flux

951

45. Yes. The work and area have the same numerical value because work œ 'C F † dr œ 'C yi † dr œ 'b [f(t)i] †
i
a

df dt

j‘ dt

[On the path, y equals f(t)]

œ 'a f(t) dt œ Area under the curve b

46. r œ xi
yj œ xi
f(x)j Ê from the origin Ê F †

'C

Ê

dr dx

dr dx

œ

F † T ds œ 'C F †

[because f(t)  0]

œ i
f w (x)j ; F œ k†y†f (x) È x#  y# w




kx È x#  y#

dx œ 'a k b

dr dx

d dx

k È x#  y#

œ

(xi
yj) has constant magnitude k and points away

kx  k†f(x)†f (x) Èx#  [f(x)]# w

œk

d dx

Èx#
[f(x)]# , by the chain rule

Èx#
[f(x)]# dx œ k
Èx#
[f(x)]# ‘ b a

œ k ˆÈb#
[f(b)]#  Èa#
[f(a)]# ‰ , as claimed. 47. F œ 4t$ i
8t# j
2k and 48. F œ 12t# j
9t# k and

dr dt

œ i
2tj Ê F †

dr dt

œ 3j
4k Ê F †

49. F œ (cos t  sin t)i
(cos t)k and

dr dt

dr dt

œ 12t$ Ê Flow œ '0 12t$ dt œ c3t% d ! œ 48 2

œ 72t# Ê Flow œ '0 72t# dt œ c24t$ d ! œ 24 1

œ ( sin t)i
(cos t)k Ê F †

dr dt

#

dr dt

"

œ  sin t cos t
1

Ê Flow œ '0 ( sin t cos t
1) dt œ
2" cos# t
t‘ ! œ ˆ #"
1‰  ˆ #"
0‰ œ 1 1

1

50. F œ (2 sin t)i  (2 cos t)j
2k and

dr dt

œ (2 sin t)i
(2 cos t)j
2k Ê F †

dr dt

œ 4 sin# t  4 cos# t
4 œ 0

Ê Flow œ 0 1 #

51. C" : r œ (cos t)i
(sin t)j
tk , 0 Ÿ t Ÿ Ê F†

dr dt

Ê F œ (2 cos t)i
2tj
(2 sin t)k and

1Î2

C# : r œ j


1 #

1Î#

( sin 2t
2t cos t
2 sin t) dt œ
2" cos 2t
2t sin t
2 cos t  2 cos t‘ !

(1  t)k , 0 Ÿ t Ÿ 1 Ê F œ 1(1  t)j
2k and

Ê Flow# œ '0 1 dt œ 1

c1td "!

Ê Flow$ œ '0 2t dt œ 1

œx

dx dt


y

dy dt


z

" ct# d ! dz dt

dr dt

œ  1# k Ê F †

dr dt

œ 1
1;

œ 1

œ 1 ;

C$ : r œ ti
(1  t)j , 0 Ÿ t Ÿ 1 Ê F œ 2ti
2(1  t)k and

dr dt

œ ( sin t)i
(cos t)j
k

œ 2 cos t sin t
2t cos t
2 sin t œ  sin 2t
2t cos t
2 sin t

Ê Flow" œ '0

52. F †

dr dt

dr dt

œij Ê F†

dr dt

œ 2t

œ 1 Ê Circulation œ (1
1)  1
1 œ 0

œ

` f dx ` x dt




` f dy ` y dt

by the chain rule Ê Circulation œ 'C F †

dr dt




` f dz ` z dt

dt œ 'a

, where f(xß yß z) œ b

d dt afaratbbb

" #

ax#
y#
x# b Ê F †

dr dt

œ

d dt afaratbbb

dt œ farabbb  faraabb. Since C is an entire ellipse,

rabb œ raab, thus the Circulation œ 0. 53. Let x œ t be the parameter Ê y œ x# œ t# and z œ x œ t Ê r œ ti
t# j
tk , 0 Ÿ t Ÿ 1 from (0ß 0ß 0) to (1ß 1ß 1) Ê œ

dr dt

œ i
2tj
k and F œ xyi
yj  yzk œ t$ i
t# j  t$ k Ê F †

œ t$
2t$  t$ œ 2t$ Ê Flow œ '0 2t$ dt 1

" #

54. (a) F œ ™ axy# z$ b Ê F † œ 'a

(b)

dr dt

b

d dt afaratbbb

dr dt

œ

` f dx ` x dt




` f dy ` y dt




` z dz ` z dt

œ

df dt

, where f(xß yß z) œ xy# z$ Ê )C F †

dr dt

dt œ farabbb  faraabb œ 0 since C is an entire ellipse.

Ð2ß1ß 1Ñ

'C F † ddtr œ 'Ð1ß1ß1Ñ

d dt

Ð#ß"ß
"Ñ

axy# z$ b dt œ cxy# z$ d Ð"ß"ß"Ñ œ (2)(1)# (1)$  (1)(1)# (1)$ œ 2  1 œ 3

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

dt

952

Chapter 16 Integration in Vector Fields

55-60. Example CAS commands: Maple: with( LinearAlgebra );#55 F := r -> < r[1]*r[2]^6 | 3*r[1]*(r[1]*r[2]^5+2) >; r := t -> < 2*cos(t) | sin(t) >; a,b := 0,2*Pi; dr := map(diff,r(t),t); # (a) F(r(t)); # (b) q1 := simplify( F(r(t)) . dr ) assuming t::real; # (c) q2 := Int( q1, t=a..b ); value( q2 ); Mathematica: (functions and bounds will vary): Exercises 55 and 56 use vectors in 2 dimensions Clear[x, y, t, f, r, v] f[x_, y_]:= {x y6 , 3x (x y5
2)} {a, b}={0, 21}; x[t_]:= 2 Cos[t] y[t_]:= Sin[t] r[t_]:={x[t], y[t]} v[t_]:= r'[t] integrand= f[x[t], y[t]] . v[t] //Simplify Integrate[integrand,{t, a, b}] N[%] If the integration takes too long or cannot be done, use NIntegrate to integrate numerically. This is suggested for exercises 57 - 60 that use vectors in 3 dimensions. Be certain to leave spaces between variables to be multiplied. Clear[x, y, z, t, f, r, v] f[x_, y_, z_]:= {y
y z Cos[x y z], x2
x z Cos[x y z], z
x y Cos[x y z]} {a, b}={0, 21}; x[t_]:= 2 Cos[t] y[t_]:= 3 Sin[t] z[t_]:= 1 r[t_]:={x[t], y[t], z[t]} v[t_]:= r'[t] integrand= f[x[t], y[t],z[t]] . v[t] //Simplify NIntegrate[integrand,{t, a, b}] 16.3 PATH INDEPENDENCE, POTENTIAL FUNCTIONS, AND CONSERVATIVE FIELDS 1.

`P `y

œxœ

`N `z

2.

`P `y

œ x cos z œ

3.

`P `y

œ 1 Á 1 œ

5.

`N `x

œ0Á1œ

6.

`P `y

œ0œ

`N `z

,

,

`M `z `N `z

œyœ ,

`N `z

`M `y `M `z

`M `z

`P `x

,

`N `x

`M `y

œzœ

œ y cos z œ

`P `x

,

`N `x

Ê Conservative œ sin z œ

Ê Not Conservative

`M `y

4.

Ê Conservative `N `x

œ 1 Á 1 œ

`M `y

Ê Not Conservative

Ê Not Conservative œ0œ

`P `x

,

`N `x

œ ex sin y œ

`M `y

Ê Conservative

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.3 Path Independence, Potential Functions, and Conservative Fields 7.

`f `x

`f `z

Ê 8.

`f `x

`f `y

œ 2x Ê f(xß yß z) œ x#
g(yß z) Ê

`f `z

œ xe `f `x


h(z) Ê

`f `z

œ 2xe

y2z

`f `y œ y2z

`f `y

œ y sin z Ê f(xß yß z) œ xy sin z
g(yß z) Ê `f `z

œ

Ê f(xß yß z) œ

z y #  z#

" #

œ

y 1  x# y# `g `y

œ

z È 1
y # z#

Ê

`f `z

œ

y È 1
y # z#

`g `y

œx
z Ê

œ z Ê g(yß z) œ zy
h(z)

w

`g `y

`g `y

œ xey2z Ê

œ 0 Ê f(xß yß z)

w

Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xey2z
C œ x sin z


`g `y

`g `y

œ x sin z Ê

œ 0 Ê g(yß z) œ h(z)

w

`f `x

ln ay#
z# b
g(xß y) Ê " #

`g `x œ #

œ

ln x
sec# (x
y) Ê g(xß y)

ln ay#
z b
(x ln x  x)
tan (x
y)
h(y)

y)


Ê f(xß yß z) œ tan
" (xy)
g(yß z) Ê

Ê


h(z)

œ xy cos z
h (z) œ xy cos z Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z)

Ê `` yf œ y# y z#
sec# (x
y)
hw (y) œ sec# (x
œ "# ln ay#
z# b
(x ln x  x)
tan (x
y)
C `f `x

3y# #


2z#
C

w

œ (x ln x  x)
tan (x
y)
h(y) Ê f(xß yß z) œ

12.

`g `y

xey2z



h (z) œ 2xe

œ xy sin z
C `f `z

œx



h(z) Ê f(xß yß z) œ x#


œ x
y
h (z) œ x
y Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z)

w

Ê f(xß yß z) œ xy sin z
h(z) Ê

11.

`f `y

3y #

#

3y# #

w

œ ey2z Ê f(xß yß z) œ xey2z
g(yß z) Ê y2z

10.

œ 3y Ê g(yß z) œ

œ y
z Ê f(xß yß z) œ (y
z)x
g(yß z) Ê

œ (y
z)x
zy
C `f `x

`g `y

œ hw (z) œ 4z Ê h(z) œ 2z#
C Ê f(xß yß z) œ x#


Ê f(xß yß z) œ (y
z)x
zy
h(z) Ê

9.

œ

953

`f `y

y y#  z#

œ

Ê hw (y) œ 0 Ê h(y) œ C Ê f(xß yß z)




x 1  x# y#

`g `y

œ

x 1  x# y#




z È1
y# z#

Ê g(yß z) œ sin
" (yz)
h(z) Ê f(xß yß z) œ tan
" (xy)
sin
" (yz)
h(z)
hw (z) œ

y È 1
y # z#

" z




Ê hw (z) œ

" z

Ê h(z) œ ln kzk
C

Ê f(xß yß z) œ tan
" (xy)
sin
" (yz)
ln kzk
C 13. Let F(xß yß z) œ 2xi
2yj
2zk Ê exact; Ê

`f `x

`f `z

`P `y

`N `z

`M `P `N `M `z œ 0 œ `x , `x œ 0 œ `y `g `f # ` y œ ` y œ 2y Ê g(yß z) œ y


œ0œ

#

œ 2x Ê f(xß yß z) œ x
g(yß z) Ê

œ f(2ß 3ß 6)  f(!ß !ß !) œ 2#
3#
(6)# œ 49

exact;

`f `x

`N `z

œxœ

œ yz Ê f(xß yß z) œ xyz
g(yß z) Ê

œ xyz
h(z) Ê Ê

`P `y

Ð3ß5ß0Ñ

'Ð1ß1ß2Ñ

`f `z

w

,

`f `y

`M `z

œyœ

œ xz


`g `y

`P `x

,

`N `x

œzœ

œ xz Ê

`g `y

h(z) Ê f(xß yß z) œ x#
y# œ h(z)

'Ð0Ð2ß0ß3ß0ßÑ 6Ñ 2x dx
2y dy
2z dz

œ hw (z) œ 2z Ê h(z) œ z#
C Ê f(xß yß z) œ x#
y#
z#
C Ê

14. Let F(xß yß z) œ yzi
xzj
xyk Ê

Ê M dx
N dy
P dz is

,

`M `y

Ê M dx
N dy
P dz is

œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z)

w

œ xy
h (z) œ xy Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xyz
C

yz dx
xz dy
xy dz œ f(3ß 5ß 0)  f(1ß 1ß 2) œ 0  2 œ 2

15. Let F(xß yß z) œ 2xyi
ax#  z# b j  2yzk Ê Ê M dx
N dy
P dz is exact;

`f `x

`P `y

œ 2z œ

`N `z

,

`M `z

œ0œ

`P `x

œ 2xy Ê f(xß yß z) œ x# y
g(yß z) Ê

Ê g(yß z) œ yz#
h(z) Ê f(xß yß z) œ x# y  yz#
h(z) Ê

`f `z

,

`N `x

`f `y w

œ 2x œ

œ x#


`g `y

`M `y

œ x#  z# Ê

`g `y

œ z#

œ 2yz
h (z) œ 2yz Ê hw (z) œ 0 Ê h(z) œ C

Ê f(xß yß z) œ x# y  yz#
C Ê 'Ð0ß0ß0Ñ 2xy dx
ax#  z# b dy  2yz dz œ f("ß #ß $)  f(!ß !ß !) œ 2  2(3)# œ 16 Ð1ß2ß3Ñ

16. Let F(xß yß z) œ 2xi  y# j  ˆ 1 4 z# ‰ k Ê Ê M dx
N dy
P dz is exact;

`f `x

`P `y

œ0œ

`N `z

,

`M `z

œ0œ

`P `x

,

`N `x

œ 2x Ê f(xß yß z) œ x#
g(yß z) Ê

œ0œ `f `y

œ

`M `y

`g `y

$

œ y# Ê g(yß z) œ  y3
h(z)

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

954

Chapter 16 Integration in Vector Fields Ê f(xß yß z) œ x# 

y$ 3

`f `z


h(z) Ê

œ hw (z) œ  1 4 z# Ê h(z) œ 4 tan
" z
C Ê f(xß yß z)

œ x# 

y$ 3

 4 tan
" z
C Ê 'Ð0ß0ß0Ñ 2x dx  y# dy 

œ ˆ9 

27 3

 4 † 14 ‰  (!  !  0) œ 1

Ð3ß3ß1Ñ

17. Let F(xß yß z) œ (sin y cos x)i
(cos y sin x)j
k Ê Ê M dx
N dy
P dz is exact; `g `y

œ cos y sin x Ê

`f `x

4 1  z#

`P `y

dz œ f(3ß 3ß 1)  f(!ß !ß !)

œ0œ

`N `z

`M `z

,

`P `x

œ0œ

,

`N `x

œ cos y cos x œ `f `y

œ sin y cos x Ê f(xß yß z) œ sin y sin x
g(yß z) Ê `f `z

œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z) œ sin y sin x
h(z) Ê

`M `y

œ cos y sin x


`g `y

œ hw (z) œ 1 Ê h(z) œ z
C

Ê f(xß yß z) œ sin y sin x
z
C Ê 'Ð1ß0ß0Ñ sin y cos x dx
cos y sin x dy
dz œ f(0ß 1ß 1)  f(1ß !ß !) Ð0ß1ß1Ñ

œ (0
1)  (0
0) œ 1 18. Let F(xß yß z) œ (2 cos y)i
Š "y  2x sin y‹ j
ˆ "z ‰ k Ê Ê M dx
N dy
P dz is exact; " y

œ

`g `y

 2x sin y Ê

" y

œ

`f `x

`P `y

`N `z

œ0œ

`M `z

,

œ0œ

`P `x

œ 2 cos y Ê f(xß yß z) œ 2x cos y
g(yß z) Ê

, `f `y

`N `x

œ 2 sin y œ

œ 2x sin y
`f `z

Ê g(yß z) œ ln kyk
h(z) Ê f(xß yß z) œ 2x cos y
ln kyk
h(z) Ê

`M `y

`g `y

œ hw (z) œ

" z

Ê h(z) œ ln kzk
C Ê f(xß yß z) œ 2x cos y
ln kyk
ln kzk
C

Ê 'Ð0ß2ß1Ñ

Ð1ß1Î2ß2Ñ

2 cos y dx
Š "y  2x sin y‹ dy


œ ˆ2 † 0
ln

1 #

" z

dz œ f ˆ1ß 1# ß 2‰  f(!ß #ß ")


ln 2‰  (0 † cos 2
ln 2
ln 1) œ ln #

`P `y

19. Let F(xß yß z) œ 3x# i
Š zy ‹ j
(2z ln y)k Ê Ê M dx
N dy
P dz is exact;

`f `x

œ

2z y

1 # `N `z

œ

`M `z

,

œ0œ

`P `x

`f `y

œ 3x# Ê f(xß yß z) œ x$
g(yß z) Ê

Ê f(xß yß z) œ x$
z# ln y
h(z) Ê œ x$
z# ln y
C Ê 'Ð1ß1ß1Ñ 3x# dx
Ð1ß2ß3Ñ

`N `x

,

œ0œ

œ

`g `y

œ

`M `y z# y

Ê g(yß z) œ z# ln y
h(z)

`f `z

œ 2z ln y
hw (z) œ 2z ln y Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z)

z# y

dy
2z ln y dz œ f(1ß 2ß 3)  f("ß "ß ")

œ (1
9 ln 2
C)  (1
0
C) œ 9 ln 2 #

`P `y

20. Let F(xß yß z) œ (2x ln y  yz)i
Š xy  xz‹ j  (xy)k Ê Ê M dx
N dy
P dz is exact; x# y

œ

 xz Ê

`g `y

`f `x

œ x œ

`N `z

,

`M `z

œ y œ

`P `x

,

`N `x

œ 2x ln y  yz Ê f(xß yß z) œ x# ln y  xyz
g(yß z) Ê `f `z

œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z) œ x# ln y  xyz
h(z) Ê

œ

2x y

`f `y

œ

zœ x# y

`M `y

 xz


`g `y

œ xy
hw (z) œ xy Ê hw (z) œ 0

Ê h(z) œ C Ê f(xß yß z) œ x# ln y  xyz
C Ê 'Ð1ß2ß1Ñ (2x ln y  yz) dx
Š xy  xz‹ dy  xy dz Ð2ß1ß1Ñ

#

œ f(2ß 1ß 1)  f("ß 2ß 1) œ (4 ln 1  2
C)  (ln 2  2
C) œ  ln 2 21. Let F(xß yß z) œ Š "y ‹ i
Š 1z 

x y# ‹ j

Ê M dx
N dy
P dz is exact; Ê

`g `y

œ

" z

Ê g(yß z) œ

Ê f(xß yß z) œ

x y




y z

y z

 ˆ zy# ‰ k Ê

`f `x

œ

" y

Ð2ß2ß2Ñ

" y

œ  z"# œ

Ê f(xß yß z) œ


h(z) Ê f(xß yß z) œ


C Ê 'Ð1ß1ß1Ñ

`P `y

x y

dx
Š 1z 




y z

x y# ‹

x y

`N `z

`M `z

,

œ0œ `f `y  zy#


g(yß z) Ê `f `z


h(z) Ê dy 

y z#

œ

`P `x

,

`N `x

œ  y1# œ

œ  yx#


`g `y

œ

" z



Ê `f `x

œ

`P `y

2xi  2yj  2zk x #  y #  z#

œ  4yz œ 3%

2x x #  y #  z#

`N `z

,

`M `z

Šand let 3# œ x#
y#
z# Ê œ  4xz œ 3% #

`P `x #

,

`N `x

œ  4xy œ 3% #

`3 `x

dz œ f(2ß 2ß 2)  f("ß 1ß 1) œ ˆ 2#


`M `y

Ê f(xß yß z) œ ln ax
y
z b
g(yß z) Ê

œ

x 3

,

`3 `y

œ

y 3

,

`3 `z

œ 3z ‹

Ê M dx
N dy
P dz is exact; `f `y

œ

2y x #  y #  z#

x y#


hw (z) œ  zy# Ê hw (z) œ 0 Ê h(z) œ C

œ0 22. Let F(xß yß z) œ

`M `y




`g `y

œ

2y x #  y #  z#

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

2 #


C‰  ˆ "1


" 1


C‰

Section 16.3 Path Independence, Potential Functions, and Conservative Fields Ê œ

`g `y œ 0 2z x #  y #  z#

`f `z

Ê g(yß z) œ h(z) Ê f(xß yß z) œ ln ax#
y#
z# b
h(z) Ê

Ð2ß2ß2Ñ

Ê 'Ð 1ß 1ß 1Ñ

œ

955


hw (z)

2z x #  y#  z#

Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ ln ax#
y#
z# b
C 2x dx  2y dy  2z dz x #  y #  z#

œ f(2ß 2ß 2)  f("ß 1ß 1) œ ln 12  ln 3 œ ln 4

23. r œ (i
j
k)
t(i
2j  2k) œ (1
t)i
(1
2t)j
(1  2t)k, 0 Ÿ t Ÿ 1 Ê dx œ dt, dy œ 2 dt, dz œ 2 dt Ð2ß3ß 1Ñ

Ê 'Ð1ß1ß1Ñ y dx
x dy
4 dz œ '0 (2t
1) dt
(t
1)(2 dt)
4(2) dt œ '0 (4t  5) dt œ c2t#  5td ! œ 3 1

1

24. r œ t(3j
4k), 0 Ÿ t Ÿ 1 Ê dx œ 0, dy œ 3 dt, dz œ 4 dt Ê

' 000304

Ð ß ß Ñ

Ð ß ß Ñ

"

#

x# dx
yz dy
Š y# ‹ dz

œ '0 a12t# b (3 dt)
Š 9t# ‹ (4 dt) œ '0 54t# dt œ c18t# d ! œ 18 1

25.

`P `y

1

#

œ0œ

`N `z

,

`M `z

œ 2z œ

`P `x

,

`N `x

,

`M `z

"

`M `y

œ0œ

Ê M dx
N dy
P dz is exact Ê F is conservative

Ê path independence 26.

`P `y

œ  ˆÈ

yz x #  y#  z# ‰

œ

$

`N `z

œ  ˆÈ

xz $ x #  y#  z# ‰

œ

`P `x

,

`N `x

œ  ˆÈ

xy x #  y#  z# ‰

$

œ

`M `y

Ê M dx
N dy
P dz is exact Ê F is conservative Ê path independence 27.

`P `y `f `x

œ0œ œ

2x y

`N `z

,

œ0œ

Ê f(xß y) œ

Ê f(xß y) œ 28.

`M `z

x# y



" y

`N `z

,

`M `z

#

x y

`P `x

`N `x

,

œ  2x y# œ

œ  xy#
gw (y) œ


C Ê F œ ™ Šx `P `x

`N `x

#

œ cos z œ

`f `x

œ ex ln y Ê f(xß yß z) œ ex ln y
g(yß z) Ê

,

œ

ex y

1
x# y#

" y#

Ê gw (y) œ

Ê g(y) œ  "y
C


1 y ‹

`P `y

œ0œ

Ê F is conservative Ê there exists an f so that F œ ™ f;

#

`f `y


g(y) Ê

`M `y

œ

`M `y

Ê F is conservative Ê there exists an f so that F œ ™ f; `f `y

œ

ex y




œ y sin z
h(z) Ê f(xß yß z) œ e ln y
y sin z
h(z) Ê x

`g ex `y œ y `f `z œ y x

`g `y


sin z Ê

œ sin z Ê g(yß z)

w

cos z
h (z) œ y cos z Ê hw (z) œ 0

Ê h(z) œ C Ê f(xß yß z) œ ex ln y
y sin z
C Ê F œ ™ ae ln y
y sin zb 29.

`P `y `f `x

œ0œ

`N `z

,

`M `z

#

`P `x

œ x
y Ê f(xß yß z) œ

Ê f(xß yß z) œ œ

œ0œ

" 3

x$
xy


" $ 3 x
xy " $ z 3 y
ze

(a) work œ 'A F † B

dr dt

, " 3 " 3

`N `x

œ1œ

`M `y

Ê F is conservative Ê there exists an f so that F œ ™ f; `f `y

$

x
xy
g(yß z) Ê


y$
h(z) Ê

œx


`g `y

œ y#
x Ê

`f `z

`g `y z

œ y# Ê g(yß z) œ

" 3

y$
h(z)

œ hw (z) œ zez Ê h(z) œ zez  e
C Ê f(xß yß z)  ez
C Ê F œ ™ ˆ "3 x$
xy
3" y$
zez  ez ‰

dt œ 'A F † dr œ
3" x$
xy
3" y$
zez  ez ‘ Ð"ß!ß!Ñ œ ˆ 3"
0
0
e  e‰  ˆ 3"
0
0  1‰ B

Ð"ß!ß"Ñ

œ1

(b) work œ 'A F † dr œ
"3 x$
xy
3" y$
zez  ez ‘ Ð"ß!ß!Ñ œ 1 B

Ð"ß!ß"Ñ

(c) work œ 'A F † dr œ
"3 x$
xy
3" y$
zez  ez ‘ Ð"ß!ß!Ñ œ 1 B

Ð"ß!ß"Ñ

Note: Since F is conservative, 'A F † dr is independent of the path from (1ß 0ß 0) to (1ß 0ß 1). B

30.

`P `y

œ xeyz
xyzeyz
cos y œ

that F œ ™ f;

`f `x

œe

yz

`N `z

,

`M `z

œ yeyz œ

`P `x

,

`N `x

œ zeyz œ

Ê f(xß yß z) œ xe
g(yß z) Ê yz

`f `y

`M `y

œ xze


Ê g(yß z) œ z sin y
h(z) Ê f(xß yß z) œ xe
z sin y
h(z) Ê yz

Ê F is conservative Ê there exists an f so

yz

`f `z

`g `y

œ xzeyz
z cos y Ê w

`g `y

œ z cos y

œ xye
sin y
h (z) œ xyeyz
sin y yz

Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xeyz
z sin y
C Ê F œ ™ axeyz
z sin yb

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

956

Chapter 16 Integration in Vector Fields

(a) work œ 'A F † dr œ cxeyz
z sin yd Ð"ß!ß"Ñ B

Ð"ß1Î#ß!Ñ

œ (1
0)  (1
0) œ 0

(b) work œ 'A F † dr œ cxeyz
z sin yd Ð"ß!ß"Ñ B

Ð"ß1Î#ß!Ñ

(c) work œ 'A F † dr œ cxeyz
z sin yd Ð"ß!ß"Ñ B

Ð"ß1Î#ß!Ñ

œ0 œ0

Note: Since F is conservative, 'A F † dr is independent of the path from (1ß 0ß 1) to ˆ1ß 1# ß 0‰ . B

31. (a) F œ ™ ax$ y# b Ê F œ 3x# y# i
2x$ yj ; let C" be the path from (1ß 1) to (0ß 0) Ê x œ t  1 and y œ t
1, 0 Ÿ t Ÿ 1 Ê F œ 3(t  1)# (t
1)# i
2(t  1)$ (t
1)j œ 3(t  1)% i  2(t  1)% j and r" œ (t  1)i
(t
1)j Ê dr" œ dt i  dt j Ê

'C

"

F † dr" œ '0 c3(t  1)%
2(t  1)% d dt 1

1 " œ '0 5(t  1)% dt œ c(t  1)& d ! œ 1; let C# be the path from (0ß 0) to (1ß 1) Ê x œ t and y œ t, 1 0 Ÿ t Ÿ 1 Ê F œ 3t% i
2t% j and r# œ ti
tj Ê dr# œ dt i
dt j Ê 'C F † dr# œ '0 a3t%
2t% b dt 1 œ '0 5t% dt œ 1

Ê 'C F † dr œ 'C F † dr"
'C "

#

#

F † dr# œ 2 Ð1ß1Ñ

(b) Since f(xß y) œ x$ y# is a potential function for F, 'Ð 1ß1Ñ F † dr œ f(1ß 1)  f(1ß 1) œ 2 32.

`P `y `f `x

œ0œ

`N `z

,

`M `z

œ0œ

`P `x

,

`N `x

œ 2x sin y œ

#

œ 2x cos y Ê f(xß yß z) œ x cos y
g(yß z) Ê #

Ê f(xß yß z) œ x cos y
h(z) Ê (a) (b) (c) (d)

`M `y

`f `z

Ê F is conservative Ê there exists an f so that F œ ™ f; `f `y

œ x# sin y


w

`g `y

œ x# sin y Ê

`g `y

œ 0 Ê g(yß z) œ h(z)

#

œ h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ x cos y
C Ê F œ ™ ax# cos yb

'C 2x cos y dx  x# sin y dy œ cx# cos yd Ð!ß"Ñ Ð"ß!Ñ œ 0  1 œ 1 'C 2x cos y dx  x# sin y dy œ cx# cos yd Ð"ß!Ñ Ð
"ß1Ñ œ 1  (1) œ 2 'C 2x cos y dx  x# sin y dy œ cx# cos yd Ð"ß!Ñ Ð
"ß!Ñ œ 1  1 œ 0 'C 2x cos y dx  x# sin y dy œ cx# cos yd Ð"ß!Ñ Ð"ß!Ñ œ 1  1 œ 0

33. (a) If the differential form is exact, then all x, and

`N `x

œ

`M `y

`P `y

œ

`N `z

Ê 2ay œ cy for all y Ê 2a œ c,

`M `z

œ

`P `x

Ê 2cx œ 2cx for

Ê by œ 2ay for all y Ê b œ 2a and c œ 2a

(b) F œ ™ f Ê the differential form with a œ 1 in part (a) is exact Ê b œ 2 and c œ 2 34. F œ ™ f Ê g(xß yß z) œ 'Ð0ß0ß0Ñ F † dr œ 'Ð0ß0ß0Ñ ™ f † dr œ f(xß yß z)  f(0ß 0ß 0) Ê ÐxßyßzÑ

`g `z

œ

`f `z

ÐxßyßzÑ

`g `x

œ

`f `x

 0,

`g `y

œ

`f `y

 0, and

 0 Ê ™ g œ ™ f œ F, as claimed

35. The path will not matter; the work along any path will be the same because the field is conservative. 36. The field is not conservative, for otherwise the work would be the same along C" and C# . 37. Let the coordinates of points A and B be axA , yA , zA b and axB , yB , zB b, respectively. The force F œ ai
bj
ck is conservative because all the partial derivatives of M, N, and P are zero. Therefore, the potential function is fax, y, zb œ ax
by
cz
C, and the work done by the force in moving a particle along any path from A to B is faBb  faAb œ f axB , yB , zB b  faxA , yA , zA b œ aaxB
byB
czB
Cb  aaxA
byA
czA
Cb Ä œ aaxB  xA b
bayB  yA b
cazB  zA b œ F † BA

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.4 Green's Theorem in the Plane 38. (a) Let GmM œ C Ê F œ C ’ `P `y

Ê

œ


3yzC ax#  y#  z# b&Î#

`f `x

œ

xC ax#  y#  z# b$Î# yC Ê `` gy œ ax#  y#  z# b$Î#

some f; œ

œ

`N `z

,

x ax#  y#  z# b$Î# `M `z

œ

i


y ax#  y#  z# b$Î#


3xzC ax#  y#  z# b&Î#

Ê f(xß yß z) œ 

œ

,

C ax#  y#  z# b"Î#

0 Ê g(yß z) œ h(z) Ê

Ê h(z) œ C" Ê f(xß yß z) œ 

`P `x

C ax#  y#  z# b"Î#

`f `z

œ

j


`N `x

œ

z ax#  y#  z# b$Î#
3xyC ax#  y#  z# b&Î#


g(yß z) Ê

`f `y

k“ `M `y

œ

œ


hw (z) œ

zC ax#  y#  z# b$Î#

Ê F œ ™ f for

yC ax#  y#  z# b$Î#


C" . Let C" œ 0 Ê f(xß yß z) œ




`g `y

zC ax#  y#  z# b$Î# GmM ax#  y#  z# b"Î#

is a potential

function for F. (b) If s is the distance of (xß yß z) from the origin, then s œ Èx#
y#
z# . The work done by the gravitational field F is work œ 'P F † dr œ ’ Èx#GmM “  y #  z# P#

T#

"

T"

œ

GmM

s#



GmM

s"

œ GmM Š s"# 

"

s" ‹ ,

as claimed.

16.4 GREEN'S THEOREM IN THE PLANE 1. M œ y œ a sin t, N œ x œ a cos t, dx œ a sin t dt, dy œ a cos t dt Ê `N `y

`M `x

œ 0,

`M `y

œ 1,

`N `x

œ 1, and

œ 0;

Equation (3):

)C M dy  N dx œ '021 [(a sin t)(a cos t)  (a cos t)(a sin t)] dt œ '021 0 dt œ 0;

' ' Š ``Mx
``Ny ‹ dx dy œ ' ' 0 dx dy œ 0, Flux R

R

Equation (4):

)C M dx
N dy œ '021 [(a sin t)(a sin t)  (a cos t)(a cos t)] dt œ '021 a# dt œ 21a# ; Èa c x

' ' Š ``Nx  ``My ‹ dx dy œ ' ' ca cc a

R

#

œ 2a

ˆ 1#




1‰ #

#

#

2 dy dx œ 'ca 4Èa#  x# dx œ 4 ’ x2 Èa#  x#
a

sin
" xa “

a

ca

#

œ 2a 1, Circulation

2. M œ y œ a sin t, N œ 0, dx œ a sin t dt, dy œ a cos t dt Ê Equation (3):

a# #

)C M dy  N dx œ '0

21

`M `x

œ 0,

`M `y

œ 1,

`N `x

œ 0, and

`N `y

œ 0;

#1 a# sin t cos t dt œ a#
2" sin# t‘ ! œ 0; ' ' 0 dx dy œ 0, Flux

R

21 #1 Equation (4): )C M dx
N dy œ '0 aa# sin# tb dt œ a#
2t  sin4 2t ‘ ! œ 1a# ; ' ' Š ``Nx  ``My ‹ dx dy

œ ' ' 1 dx dy œ '0

21

R

'0

a

r dr d) œ '0  21

R

a# #

d) œ 1a# , Circulation

3. M œ 2x œ 2a cos t, N œ 3y œ 3a sin t, dx œ a sin t dt, dy œ a cos t dt Ê `N `y

`M `x

œ 2,

`M `y

`N `x

œ 0,

œ 0, and

œ 3;

Equation (3):

)C M dy  N dx œ '021 [(2a cos t)(a cos t)
(3a sin t)(a sin t)] dt

œ '0 a2a# cos# t  3a# sin# tb dt œ 2a#
2t
21

sin 2t ‘ #1 4 !

 3a#
2t 

sin 2t ‘ #1 4 !

œ 21a#  31a# œ 1a# ;

' ' Š ``Mx
``Ny ‹ œ ' ' 1 dx dy œ ' ' r dr d) œ '  a## d) œ 1a# , Flux 0 0 0 21

R

a

21

R

Equation (4):

)C M dx
N dy œ '021 [(2a cos t)(a sin t)
(3a sin t)(a cos t)] dt

#1 œ '0 a2a# sin t cos t  3a# sin t cos tb dt œ 5a#
12 sin# t‘ ! œ 0; ' ' 0 dx dy œ 0, Circulation 21

R

4. M œ x# y œ a$ cos# t, N œ xy# œ a$ cos t sin# t, dx œ a sin t dt, dy œ a cos t dt Ê ``Mx œ 2xy, ``My œ x2 , ``Nx œ y# , and ``Ny œ 2xy; Equation (3):

)C M dy  N dx œ '021 aa% cos$ t sin t
a% cos t sin$ tb œ ’ a4

%

cos% t


a% 4

sin% t“

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

#1 !

œ 0;

957

958

Chapter 16 Integration in Vector Fields

' ' Š ``Mx
``Ny ‹ dx dy œ ' ' (2xy
2xy) dx dy œ 0, Flux R

R

21 21 Equation (4): )C M dx
N dy œ '0 aa% cos# t sin# t
a% cos# t sin# tb dt œ '0 a2a% cos# t sin# tb dt 21 41 %1 œ '0 "# a% sin# 2t dt œ a4 '0 sin# u du œ a4
u2  sin42u ‘ ! œ 1#a ; ' ' Š ``Nx  ``My ‹ dx dy œ ' ' ay#
x# b dx dy %

%

R

21 a 21 œ '0 '0 r# † r dr d) œ '0 a4

%

`M `x

5. M œ x  y, N œ y  x Ê Circ œ ' '

%

d) œ

1 a% #

, Circulation

œ 1,

`M `y

œ 1,

`N `x

`N `y

œ 1,

R

œ 1 Ê Flux œ ' ' 2 dx dy œ '0

1

R

'01 2 dx dy œ 2;

[1  (1)] dx dy œ 0

R `M `x

6. M œ x#
4y, N œ x
y# Ê

`M `y

œ 2x,

œ 4,

`N `x

œ 1,

`N `y

œ 2y Ê Flux œ ' ' (2x
2y) dx dy R

1 1 1 1 " " œ '0 '0 (2x
2y) dx dy œ '0 cx#
2xyd ! dy œ '0 (1
2y) dy œ cy
y# d ! œ 2; Circ œ ' '

œ '0

1

'01 3 dx dy œ 3 `M `x

7. M œ y#  x# , N œ x#
y# Ê œ '0

3

œ 2x,

'0 (2x
2y) dy dx œ '0 a2x x

3

#

`M `y

œ 2y,

`N `x

#


x b dx œ


" 3

œ 2x, $ x$ ‘ !

`N `y

œ 2y Ê Flux œ ' ' (2x
2y) dx dy R

œ 9; Circ œ ' ' (2x  2y) dx dy R

3 x 3 œ '0 '0 (2x  2y) dy dx œ '0 x# dx œ 9

8. M œ x
y, N œ  ax#
y# b Ê

`M `x

`M `y

œ 1,

œ 1,

`N `x

œ 2x,

1 x 1 œ '0 '0 (1  2y) dy dx œ '0 ax  x# b dx œ "6 ; Circ œ ' '

œ '0 a2x  xb dx œ  1

#

R

œ '0 œ '0

1

Èx

'x

2

`M `x

`M `y

œ y,

œ x
2y,

`N `x

œ 1,

`N `y

œ 1 Ê Flux œ ' ' ay
a1bb dy dx R

' ' a1  ax
2ybb dy dx ay  1b dy dx œ '0 ˆ "# x  Èx  "# x4
x# ‰ dx œ  11 60 ; Circ œ

2

7 a1  x  2yb dy dx œ '0 ˆÈx  x3Î2  x  x#
x3
x4 ‰ dx œ  60

Circ œ ' ' R

1

`M `x

œ 1,

`M `y

œ 3,

`N `x

œ 2,

`N `y

œ 1 Ê Flux œ ' ' a1
a1bb dy dx œ 0 R

È2 È2 2 x Î2 a2  3b dy dx œ 'cÈ2 '
È 2 c x Î2 a1b dy dx œ  È22 'cÈ2 È2  x2 dx œ  1È2 Èa

2b

a

11. M œ x3 y2 , N œ "# x4 y Ê 2

R

1 x (2x  1) dx dy œ '0 '0 (2x  1) dy dx

1

10. M œ x
3y, N œ 2x  y Ê

œ '0

œ 2y Ê Flux œ ' ' (1  2y) dx dy

R

Èx

'x

`N `y

7 6

9. M œ xy
y2 , N œ x  y Ê 1

(1  4) dx dy

R

`M `x

œ 3x2 y2 ,

2b

`M `y

œ 2x3 y,

`N `x

œ 2x3 y,

`N `y

œ "# x4 Ê Flux œ ' ' ˆ3x2 y2
"# x4 ‰ dy dx

'xx
x ˆ3x2 y2
"# x4 ‰ dy dx œ '02 ˆ3x5  72 x6
3x7  x8 ‰ dx œ 649 ; Circ œ ' ' 2

R

a2x3 y  2x3 yb dy dx œ 0

R

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

Section 16.4 Green's Theorem in the Plane 12. M œ

x 1  y2 ,

È1
y2

œ 'c1 '
È1
y2 1

`M `x

N œ tan
1 y Ê 2 1  y2

1 `M 1  y2 , ` y

œ


2x y , `N a 1  y 2 b2 ` x

œ

œ 0,

`N `y

œ

Ê Flux œ ' ' Š 1 1 y2


1 1  y2

R

1 1  y2 ‹

dx dy

dx dy œ 'c1 4 1 1
y2y dx œ 41È2  41 ; Circ œ ' ' Š0  Š a1
2xy2yb2 ‹‹ dy dx È

1

2

R

È1
y2

y œ 'c1 '
È1
y2 Š a1 2x ‹ dy dx œ 'c1 a0b dx œ 0  y 2 b2 1

1

`M `x

13. M œ x
ex sin y, N œ x
ex cos y Ê

Ècos 2)

1Î4

Ê Flux œ ' ' dx dy œ 'c1Î4 '0 R

œ 1
ex sin y, Î

`M `y

œ ex cos y,

1 4

1Î%

Ècos 2)

1Î4

R

R

y x

, N œ ln ax#
y# b Ê

Ê Flux œ ' ' Š x#
yy#
R

Circ œ ' ' Š x# 2x  y# 

x x#  y# ‹

R

`M `x

15. M œ xy, N œ y# Ê œ '0 Š 3x#  1

#

3x% # ‹

2y x#  y# ‹

dx œ

`M `x

dx dy œ '0

1

dx dy œ '0

1

`M `y

œ y,


y x#  y#

œ

Ê Flux œ ' ' (x sin y) dx dy œ '0 R

œ 0,

'0

1Î2

1Î2

R

`M `x

, N œ ex
tan
" y Ê

R

`N `y

" 1  y#




1 3cx œ 'c1 'x b 1 %

#

Ê

œ '0

'0

x$

2y x#  y#

'x

#

`N `x

,

`M `y

2xy dy dx œ ' #

1

2 0 3

x

œ 6xy# ,

`N `x

"!

2 33

20. M œ 4x  2y, N œ 2x  4y Ê

dx œ `M `y

œ 2,

x dy dx œ '0 ax#
x$ b dx œ  1"# `N `y

œ cos y,

œ x sin y #

#

œ 3y 

" 1  y#

,

`N `y

œ

" 1  y#

`N `x

œ

ex y

#1 !

'0aÐ1

cos )Ñ

(3r sin )) r dr d)

œ 4a$  a4a$ b œ 0

Ê Circ œ ' ' ’ ey  Š1
x

R

ex y ‹“

dx dy œ ' ' (1) dx dy R

œ 8xy# Ê work œ )C 2xy$ dx
4x# y# dy œ ' ' a8xy#  6xy# b dx dy R

`N `x

œ 2 Ê work œ )C (4x  2y) dx
(2x  4y) dy

œ ' ' [2  (2)] dx dy œ 4 ' ' dx dy œ 4(Area of the circle) œ 4(1 † 4) œ 161 R

#

'01Î2 2 cos y dx dy œ '01Î2 1 cos y dy œ c1 sin yd 1Î# œ1 !

R

ex y

'xx 3y dy dx

1

21

œ1


œ

1 1  dy dx œ  'c1 ca3  x# b  ax%
1bd dx œ 'c1 ax%
x#  2b dx œ  44 15

19. M œ 2xy$ , N œ 4x# y# Ê 1

`M `y

`N `y

(x sin y) dx dy œ '0 Š 18 sin y‹ dy œ  18 ;

$

ex y

,

R

x

1Î2

œ '0 a$ (1
cos ))$ (sin )) d) œ ’ a4 (1
cos ))% “ 18. M œ y
ex ln y, N œ

2x x#  y#

1

dx dy œ ' ' 3y dx dy œ '0

" 1  y# ‹

21

œ

œ 2y Ê Flux œ ' ' (y
2y) dy dx œ '0

œ  cos y,

Circ œ ' ' [cos y  ( cos y)] dx dy œ '0

Ê Flux œ ' ' Š3y 

`N `x

" #

#

`M `y

œ 0,

,

Î

1 4

#

R

`M `x

x x#  y#

;

r dr d) œ ' 1Î4 ˆ "# cos 2)‰ d) œ

'12 ˆ r sinr ) ‰ r dr d) œ '01 sin ) d) œ 2;

; Circ œ ' ' x dy dx œ '0

" 5

1Î2

x 1  y#

œ

1

16. M œ  sin y, N œ x cos y Ê

17. M œ 3xy 

`M `y

" #

œ ex sin y

'12 ˆ r cosr ) ‰ r dr d) œ '01 cos ) d) œ 0

`N `x

œ x,

,

`N `y

œ 1
ex cos y,

r dr d) œ ' 1Î4 ˆ "# cos 2)‰ d) œ
4" sin 2)‘
1Î% œ

Circ œ ' ' a1
ex cos y  ex cos yb dx dy œ ' ' dx dy œ 'c1Î4 '0 14. M œ tan
"

`N `x

R

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

959

960

Chapter 16 Integration in Vector Fields `M `y

21. M œ y# , N œ x# Ê œ '0

1

1cx

'0

œ 2y,

œ 2x Ê )C y# dx
x# dy œ ' ' (2x  2y) dy dx

`N `x

R

(2x  2y) dy dx œ '0 a3x
4x  1b dx œ cx
2x#  xd ! œ 1
2  1 œ 0 1

22. M œ 3y, N œ 2x Ê

`M `y

œ 3,

#

`N `x

œ 2 Ê )C 3y dx
2x dy œ ' ' a2  3b dx dy œ '0

`M `y

œ 6,

1

R

1 œ '0 sin x dx œ 2

23. M œ 6y
x, N œ y
2x Ê

"

$

'0sin x a1bdy dx

œ 2 Ê )C (6y
x) dx
(y
2x) dy œ ' ' (2  6) dy dx

`N `x

R

œ 4(Area of the circle) œ 161 24. M œ 2x
y# , N œ 2xy
3y Ê

`M `y

œ 2y,

`N `x

œ 2y Ê

)C a2x
y# b dx
(2xy
3y) dy œ ' ' (2y  2y) dx dy œ 0 R

25. M œ x œ a cos t, N œ y œ a sin t Ê dx œ a sin t dt, dy œ a cos t dt Ê Area œ œ

'0

21

" #

'0

21

" #

aa# cos# t
a# sin# tb dt œ

'021 aab cos# t
ab sin# tb dt œ "# '021 ab dt œ 1ab

" #

)C

" #

)C x dy  y dx

x dy  y dx

a# dt œ 1a#

26. M œ x œ a cos t, N œ y œ b sin t Ê dx œ a sin t dt, dy œ b cos t dt Ê Area œ œ

" #

)C x dy  y dx 41 3 ' œ "# '0 a3 sin# t cos# tb acos# t
sin# tb dt œ "# '0 a3 sin# t cos# tb dt œ 38 '0 sin# 2t dt œ 16 sin# u du 0

27. M œ x œ cos$ t, N œ y œ sin$ t Ê dx œ 3 cos# t sin t dt, dy œ 3 sin# t cos t dt Ê Area œ 21

œ

3 16


u2 

21

sin 2u ‘ %1 4 !

œ

3 8

21

" #

1

28. C1 : M œ x œ t, N œ y œ 0 Ê dx œ dt, dy œ 0; C2 : M œ x œ a21  tb  sina21  tb œ 21  t
sin t, N œ y œ 1  cosa21  tb œ 1  cos t Ê dx œ acos t  1b dt, dy œ sin t dt Ê Area œ œ

" #

" #

)C x dy  y dx œ "# )C

"

x dy  y dx


" #

)C

2

x dy  y dx

'021 a0bdt
"# '021 ca21  t
sin tbasin tb  a1  cos tb acos t  1bd dt œ  "# '021 a2 cos t
t sin t  2  21 sin tb dt

œ  12 c3 sin t  t cos t  2t  21 cos td20 1 œ 31 29. (a) M œ f(x), N œ g(y) Ê (b) M œ ky, N œ hx Ê

`M `y

`M `y

œ 0,

œ k,

`N `x

`N `x

œ 0 Ê )C f(x) dx
g(y) dy œ ' ' Š ``Nx  R

œh

`M `y ‹

dx dy œ ' ' 0 dx dy œ 0 R

Ê )C ky dx
hx dy œ ' ' Š ``Nx  ``My ‹ dx dy

œ ' ' (h  k) dx dy œ (h  k)(Area of the region)

R

R

30. M œ xy# , N œ x# y
2x Ê

`M `y

œ 2xy,

`N `x

œ 2xy
2 Ê )C xy# dx
ax# y
2xb dy œ ' ' Š ``Nx 

œ ' ' (2xy
2  2xy) dx dy œ 2 ' ' dx dy œ 2 times the area of the square R

R

R

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

`M `y ‹

dx dy

Section 16.4 Green's Theorem in the Plane

961

31. The integral is 0 for any simple closed plane curve C. The reasoning: By the tangential form of Green's Theorem, with M œ 4x$ y and N œ x% , )C 4x$ y dx
x% dy œ ' ' ’ ``x ax% b 

` `y

R

œ ' ' ðóóñóóò a4x$  4x$ b dx dy œ 0.

a4x$ yb“ dx dy

R

0 32. The integral is 0 for any simple closed curve C. The reasoning: By the normal form of Green's theorem, with ` ` $ $ M œ x$ and N œ y$ , )C y$ dy
x$ dx œ ' ' ”ðñò ` x ay b  ï ` y ax b • dx dy œ 0.

R

0 `M `x

33. Let M œ x and N œ 0 Ê

œ 1 and

`N `y

œ0 Ê

0

)C M dy  N dx œ ' ' Š ``Mx
``Ny ‹ dx dy

œ ' ' (1
0) dx dy Ê Area of R œ ' ' dx dy œ )C x dy; similarly, M œ y and N œ 0 Ê R

`N `x

R

œ 0 Ê )C M dx
N dy œ ' ' Š ``Nx
R

œ ' ' dx dy œ Area of R

Ê )C x dy

R

`M `y ‹

`M `y

œ 1 and

dy dx Ê )C y dx œ ' ' (0  1) dy dx Ê  )C y dx R

R

34.

'ab f(x) dx œ Area of R œ )C y dx, from Exercise 33

35. Let $ (xß y) œ 1 Ê x œ

My M

' ' x $ (xßy) dA

œ 'R '

$ (xßy) dA

' ' x dA

œ 'R '

R

dA

' ' x dA

œ

Ê Ax œ ' ' x dA œ ' ' (x
0) dx dy

R

A

R

R

R

œ )C x#

dy, Ax œ ' ' x dA œ ' ' (0
x) dx dy œ  ) xy dx, and Ax œ ' ' x dA œ ' ' ˆ 23 x
"3 x‰ dx dy

œ)

#

#

" C 3

R

R

" 3

" #

x dy  xy dx Ê

)C x

C

dy œ )C xy dx œ

#

" 3

)C x

dy  xy dx œ Ax

36. If $ (xß y) œ 1, then Iy œ ' ' x# $ (xß y) dA œ ' ' x# dA œ ' ' ax#
0b dy dx œ R

R

R

R

#

R

" 3

)C

x$ dy,

' ' x# dA œ ' ' a0
x# b dy dx œ  ) x# y dx, and ' ' x# dA œ ' ' ˆ 34 x#
4" x# ‰ dy dx C R

R

œ)

" C 4

37. M œ 38. M œ

`f `y

" 4

ellipse

$

" 4

#

x dy  x y dx œ , N œ  `` xf Ê

`M `y

" 4

œ

x# y
"3 y$ , N œ x Ê " 4

)C x ` #f ` y#

`M `y

,

R

$

#

dy  x y dx Ê

`N `x

œ

1 4

" 3

œ  `` xf# Ê )C #

x#
y# ,

`N `x

)C x

`f `y

R

$

dy œ  )C x# y dx œ

dx 

`f `x

œ 1 Ê Curl œ

" 4

)C

dy œ ' ' Š `` xf#  #

R

`N `x



`M `y

x$ dy  x# y dx œ Iy

` #f ` y# ‹

dx dy œ 0 for such curves C

œ 1  ˆ "4 x#
y# ‰  0 in the interior of the

x#
y# œ 1 Ê work œ 'C F † dr œ ' ' ˆ1  4" x#  y# ‰ dx dy will be maximized on the region R

R œ {(xß y) | curl F}   0 or over the region enclosed by 1 œ 2y 39. (a) ™ f œ Š x# 2x  y# ‹ i
Š x#  y# ‹ j Ê M œ

2x x#  y#

,Nœ

" 4

x#
y#

2y x#  y#

; since M, N are discontinuous at (0ß 0), we

compute 'C ™ f † n ds directly since Green's Theorem does not apply. Let x œ a cos t, y œ a sin t Ê dx œ a sin t dt, dy œ a cos t dt, M œ

2 a

cos t, N œ

2 a

sin t, 0 Ÿ t Ÿ 21, so 'C ™ f † n ds œ 'C M dy  N dx

œ '0
ˆ 2a cos t‰aa cos tb  ˆ 2a sin t‰aa sin tb ‘dt œ '0 2acos2 t
sin2 tbdt œ 41. Note that this holds for any 21

21

Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.

962

Chapter 16 Integration in Vector Fields a  0, so 'C ™ f † n ds œ 41 for any circle C centered at a0, 0b traversed counterclockwise and 'C ™ f † n ds œ 41 if C is traversed clockwise.

(b) If K does not enclose the point (0ß 0) we may apply Green's Theorem: 'C ™ f † n ds œ 'C M dy  N dx œ ' ' Š ``Mx
R

`N `y ‹

dx dy œ ' ' Š ax2  y2 b2
2 ˆy 2
x 2 ‰

R

2 ˆx 2
y 2 ‰ ‹ ax2  y2 b2

dx dy œ ' ' 0 dx dy œ 0. If K does enclose the point R

(0ß 0) we proceed as follows: Choose a small enough so that the circle C centered at (0ß 0) of radius a lies entirely within K. Green's Theorem applies to the region R that lies between K and C. Thus, as before, 0 œ ' ' Š ``Mx
R

`N `y ‹

dx dy

œ 'K M dy  N dx
'C M dy  N dx where K is traversed counterclockwise and C is traversed clockwise.

Hence by part (a) 0 œ ’ ' M dy  N dx “  41 Ê 41 œ K

'K ™ f † n ds œ œ 0

'K M dy  N dx

œ 'K ™ f † n ds. We have shown:

if (0ß 0) lies inside K if (0ß 0) lies outside K

41

40. Assume a particle has a closed trajectory in R and let C" be the path Ê C" encloses a simply connected region R" Ê C" is a simple closed curve. Then the flux over R" is )C F † n ds œ 0, since the velocity vectors F are "

tangent to C" . But 0 œ )C F † n ds œ )C M dy  N dx œ ' ' Š ``Mx
"

"

R"

`N `y ‹

dx dy Ê Mx
Ny œ 0, which is a

contradiction. Therefore, C" cannot be a closed trajectory. 41.

'gg yy

#Ð Ñ

"Ð Ñ

`N `x

dx dy œ N(g# (y)ß y)  N(g" (y)ß y) Ê

'cd 'gg yy ˆ ``Nx dx‰ dy œ 'cd [N(g# (y)ß y)  N(g" (y)ß y)] dy #Ð Ñ

"Ð Ñ

œ 'c N(g# (y)ß y) dy  'c N(g" (y)ß y) dy œ 'c N(g# (y)ß y) dy
'd N(g" (y)ß y) dy œ 'C N dy
'C N dy d

œ )C dy

d

Ê

)C N dy œ ' ' R

d

c

#

`N `x

"

dx dy

42. The curl of a conservative two-dimensional field is zero. The reasoning: A two-dimensional field F œ Mi
Nj can be considered to be the restriction to the xy-plane of a three-dimensional field whose k component is zero, and whose i and j components are independent of z. For such a field to be conservative, we must have `N `M `N `M ` x œ ` y by the component test in Section 16.3 Ê curl F œ ` x  ` y œ 0. 43-46. Example CAS commands: Maple: with( plots );#43 M := (x,y) -> 2*x-y; N := (x,y) -> x+3*y; C := x^2 + 4*y^2 = 4; implicitplot( C, x=-2..2, y=-2..2, scaling=constrained, title="#43(a) (Section 16.4)" ); curlF_k := D[1](N) - D[2](M): # (b) 'curlF_k' = curlF_k(x,y); top,bot := solve( C, y ); # (c) left,right := -2, 2; q1 := Int( Int( curlF_k(x,y), y=bot..top ), x=left..right ); value( q1 ); Mathematica: (functions and bounds will vary) The ImplicitPlot command will be useful for 43 and 44, but is not needed for 43 and 44. In 44, the equation of the line from (0, 4) to (2, 0) must be determined first.

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Section 16.5 Surfaces and Area

963

Clear[x, y, f]