Instructor's Solutions Manual for Thomas' Calculus, Multivariable, Twelfth Edition vol 2

INSTRUCTOR’S SOLUTIONS MANUAL MULTIVARIABLE WILLIAM ARDIS Collin County Community College THOMAS’ CALCULUS TWELFTH EDIT...

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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.

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

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

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

(")$ 41

œ

1 3!

, a% œ

œ 3" , a$ œ

1 4 4#

œ 92 , a% œ œ

1 4!

(1)% 61

œ

" 5

3 œ 16

1 24 (1)& 81

, 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 ##

œ

6. a" œ

2" #

" #

, a# œ

œ

" #

, a# œ " # 255 128

7. a" œ 1, a# œ 1 a( œ

127 64

, a) œ

8. a" œ 1, a# œ a* œ

" 362,880

" #

2# 2$

œ

œ

" #

2# 1 2# 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

, 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†(31) 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 n an 1 b

, n œ 1, 2, á

17. an œ

2n 1 3 an 2 b ,

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 n1

œ n lim Ä_

(n 1)(n 1) n1

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 ‹

1 ˆ 8n ‰

" ‹n n# 70 Š #‹4 n

Š

œ 1 Ê converges

œ _ Ê diverges

Š È"n 3‹

œ n lim Ä_

2 #

œ n lim Ä_

Š n"% ‹ 5

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)


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 1lnnln n ‰ œ e" Ê converges 67. n lim Ä_ n nÄ_ nÄ_


571

572


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

11 74. n lim œ n lim n ‰n Ä _ ˆ 109 ‰ ˆ 11 Ä_ 12 (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

sin ˆ "n ‰ œ 0 Ê converges œ n lim 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


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 ,


1 an ‹

ÊLœ2

1 L

574


È 1 an Ê L œ È 1 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 eÐln %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.


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


576


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 43 ‰n so an an1 Í 4 ˆ 43 ‰n 4 ˆ 43 ‰n" Í ˆ 43 ‰n ˆ 43 ‰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 † 2n1 3 œ 2n1 (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 1M

. Then n N Ê n

M.

M 1M

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 xn1 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 .


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)(n1) 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 a1 r n b (1 r)

œ

n 2 ˆ1 ˆ "3 ‰ ‰ " 1 ˆ3‰

2. sn œ

a a1 r n b (1 r)

œ

9 ‰ˆ " ‰n ‰ ˆ 100 1 ˆ 100 " 1 ˆ 100 ‰

3. sn œ

a a1 r n 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


" #

578 6.

Chapter 10 Infinite Sequences and Series œ

5 n(n 1)

5 n

5 n1

Ê sn œ ˆ5 52 ‰ ˆ 52 53 ‰ ˆ 53 54 ‰ 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



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 41 1 1 ˆ "# ‰

ˆ 74 ‰

1 ˆ "4 ‰

œ

œ

23 #

12. (5 1) ˆ 5# "3 ‰ ˆ 45 9" ‰ ˆ 85 5 1 ˆ "# ‰

" ‰ ˆ 16 1 ˆ 4" ‰


11. (5 1) ˆ 5# "3 ‰ ˆ 45 9" ‰ ˆ 85

" 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 ¹3¹ 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 !

nœ0

_

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

nœ0

_

nœ0

414 1000

nœ0

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


œ

234 999

10 3

Section 10.2 Infinite Series 25. 1.24123 œ

124 100

_

!

123 10&

nœ0

_

26. 3.142857 œ 3 !

nœ0

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

1Š

" ‹ 10$

Š 142,857 ' ‹ 10

1Š

" ‹ 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

1 nÄ_ 2n

œ lim

œ lim

2 nÄ_ 2


œ 0 Ê test inconclusive

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


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 1 b 2

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 ˆcos1 ˆ 12 ‰ cos1 ˆ 13 ‰‰ ˆcos1 ˆ 13 ‰ cos1 ˆ 14 ‰‰ ˆcos1 ˆ 14 ‰ cos1 ˆ 15 ‰‰ á ˆcos1 ˆ 1k ‰ cos1 ˆ k 1 1 ‰‰ ˆcos1 ˆ k 1 1 ‰ cos1 ˆ k 1 # ‰‰ œ 13 cos1 ˆ k 1 # ‰ Ê

lim sk œ lim ’ 13 cos1 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_


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 ABœ6 nœ1 nœ1

œ 3 Š "1

" 3

lim 3 ˆ1

kÄ_

43.

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

" 3

" 5

" ‰ #k 1

œ

A (2n1)

" 5

" 7

á

" #(k 1) 1

" 2k 1

" #k 1 ‹

œ

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

œ 3 ˆ1

" ‰ #k 1

" ‰ #n 1

Ê the sum is

œ3

B (2n1)#

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 4A 4B 4C 4D œ 0 A BC Dœ 0 B Dœ 0 Ê Û Ê Û Ê œ Ê 4B œ 20 Ê B œ 5 œ 2A 4B 2C 4D 40 A 2 œ 2D œ 20 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, ! ’ (#n1)40n and D œ 5 Ê œ # (2n1)# “ A 5 C 5 œ 0 nœ1 k

œ 5 ! ’ (#n" 1)# nœ1

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

" (#k1)#

" (#k1)# ‹

œ5 " ‰ 16

á ’ (k " 1)#

" k# “

’ k"#

" (k 1)# “

" È4 ‹

á ŠÈ "

k1

" Èk ‹

Š È" k

" Èk 1 ‹

œ1

" Èk 1

œ1

" ˆ #"Î$

" ‰ ln 3

" (2(k1) 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 ‹

" ‰ #1ÎÐk1Ñ

Š ln (k" 2)

œ

" #

" #1ÎÐk1Ñ

" 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



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 " 1Š


56. n lim ln Ä_

" 3n

" ‹ e#

2 " 1 Š 10 ‹


" 1 Š "x ‹

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

_

63. ! nœ1

n! 1000n

2n 3n 4n

since r œ _

! nœ1

64.

2n 3n 4n

_

nœ1

5 6

" ‰n n

2œ œ

Ê

2n 4n

_

!

¹ 12 ¹

nœ1

3n 4n

_

20 9

œ

18 9

2 9

x x1

" 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†nân 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

nœ1

1, respectivley Ê

nœ1

_

! ˆ 1 ‰n 2

nœ1

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


61. n lim Ä_

581

œ

lim

nÄ_

_

_

nœ1

nœ1

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 _

_

nœ0

nœ0

" 1 ˆ 1e ‰ e1 1e

¸

œ

23.141 22.459

1 1e

1

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

_

nœ0

nœ0

" 1 (x)

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

œ

" 1 x#

" 1x

for kxk 1

for kxk 1


582


71. a œ 3, r œ _

72. ! nœ0

œ

(1)n #

x1 #

; converges to _

ˆ 3 "sin x ‰n œ !

nœ0

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) ! nœ2 _

80. (a) ! nœ1

" 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) !

nœ0 _

nœ3

" 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 x1

" (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 3x #

" 3 sin x

,rœ

" #

" 1Š

" 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) !

nœ5 _

nœ20

" (n 3)(n #)

5 (n 19)(n 18)

œ1

Š 3# ‹ 1 Š "# ‹

á œ1

œ 3 Š "# ‹ 1 Š "# ‹

œ 0.

_

Š k# ‹

nœ0

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 nœ1

" 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.


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 1r

œ5 Ê

Š 13 2 ‹ 1r

2 5

œ5 Ê

œ1r Ê rœ

#

; 2 2 ˆ 53 ‰ 2 ˆ 53 ‰ á

3 5

3 œ 1 r Ê r œ 10 ;

13 10

90. 1 eb e2b á œ

" 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 98 ‰

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 1r

a a1 r n b 1r

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_

! 3a4bk2 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

3È3Œ! kœ2

È3 4

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" ‹ á œ 4 2 1 2

A# œ A" 3Š

" 1 r#

È3 ˆ " ‰2 4 ‹ 32

È3 ˆ " ‰2 4 ‹ 34 ,

œ

kœ2

œ

È3 4 ,

A" œ

...,

n

È3 4

œ_

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

k 1 ! 3È3a4bk$ ˆ 9" ‰ œ

4kc$ 9k 1

nc1

1 36

3È 3 Œ 1 4 œ 9

n

4kc$ . 9k 1

È3 4

3È3Œ!

È3 4

1 ‰ 3È3ˆ 20 œ

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


_

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


b

’ 45 x0.8 “

1

583

584


3. faxb œ œ

1 x2 4

_


lim ˆ 1 tan1 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 3a6b ‹ ebÎ3 2

Ê !

'7

b

bÄ_

bÄ_

œ

1 e1Î3

x4 x2 2x 1

œ

x4 ax 1 b 2

_

bÄ_ _

Ê !

n œ8

’lnlx 1l

bÄ_

9 e1

œ lim

bÄ_

_

ln 4 2

n4 n2 2n 1 diverges

16 e4Î3

x4 ax 1 b 2 b

1

2 ln x2 x2

bÄ_

3

36 e2

18x exÎ3

_

Ê !

n œ2

!

n œ7

11. converges; a geometric series with r œ

b

b

54 “ exÎ3 7 327 e7Î3

2

n enÎ3

x1 ax 1 b 2

ˆlnlb 1l

n4 n2 2n 1

" 10

2

converges

ˆ 21 lnab2 4b 21 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 ax 6 b 3exÎ3

œ

_

”'8 bÄ_

bÄ_

bÄ_

’ ln1x “

n œ3

327 e7Î3

e

dx œ lim

œ lim

b

dx œ lim

2

x

dx

n œ3

ˆ b54 ‰ Î3

bÄ_

'2b xaln1xb

2 ! lnnn diverges

’ 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 1 b 2

œ

n2 converges enÎ3

1 16

2 25

3 36

7x ax 1 b 3

dx• œ lim ”'8 bÄ_

b

_

ln 7 37 ‰ œ _ Ê '8

0

1

327 ‹ e7Î3

converges

dx '8

3 b1

_

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

3 x 1 “8

b

’ 12 e2x “

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 ax 2 4 b 2 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 e2x dx œ

Ê '1 e2x dx converges Ê ! e2n converges

_

ˆ ln1b

bÄ_

œ

1 ‰ 2e2

_

bÄ_


1 xaln xb2

œ lim 7. faxb œ

’ 21 tan1 2x “

n œ1

5. faxb œ e2x is positive, continuous, and decreasing for x 1; '1 e2x 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

_


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

x4 ax 1 b 2

n4 n2 2n 1

0 for x 7, thus f is

1 x1

dx '8

3 ax 1 b 2

dx diverges

diverges

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

b

œ1Á0


" e

1

dx•

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

n

_

15. diverges; ! nœ1

3 Èn

_

16. converges; ! nœ1

_

" Èn

œ3!

nœ1

2 nÈ n

_

dx œ 5 ln (n 1) 5 ln 2 Ê '1

5 x1

_

nœ1

8 n

_

œ 2 !

nœ1

_

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 !

nœ1

'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 _


'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; ! nœ0

2 n 1

5n 4n 3

_

œ 2 !

nœ0

" 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


'1n 2xdx 1 œ #" ln (2n 1)

25. diverges; n lim a œ n lim Ä_ n Ä_

2n n1


'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


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 #


'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 u2 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_



'1_ 81tancx x dx; ” u œ tan dx x • "

"

#

du œ

1 x#

'1_ x x1 dx; ” u œ x


#

#

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_


'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 (xx1)12a ¹“ œ lim ln

bÄ_

œ

3

bÄ_

b1 (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


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 ! nœ1

" nx

œ

" x

_

! nœ1

" n

_

and ! nœ1

" 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 nœ1

_

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

nœ1

_

_

_

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 !

nœ1

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

!

nœ1

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 ! nœ1

1 Èn 1

œ ’2Èx 1“ œ 2È52 2È2 ¸ 11.6 and '0 51

50

1

50

11.6 !

nœ1

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.


588

Chapter 10 Infinite Sequences and Series 30

48. (a) Since we are using s30 œ !

nœ1

1 n4

_

to estimate ! nœ1

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 ‚ 105 .

Thus the error 1.23 ‚ 105 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 tan1 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

nœ1

œ lim ˆ 12 tan1 ˆ b2 ‰ 12 tan1 ˆ n2 ‰‰ œ nœ1

_

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

kœ1

kœ1

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 2n1 terms _

ˆ2aa2nc1 b 2aa2nc1 1b á 2aa2n b ‰ œ 2Aa2n b Ÿ 2 ! ak . Therefore if ! ak converges, kœ1

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 . kœ1

_

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

kœ1

54. (a) aa2n b œ _

Ê !

nœ2

" 2n ln a2n b " n ln n

œ

" 2n †n(ln 2)

_

_

nœ2

nœ2

Ê ! 2 n a a2 n b œ ! 2 n

" #n †n(ln 2)

œ

" ln #

_

! nœ2

" n

, which diverges

diverges.


Section 10.3 The Integral Test " #np

(b) aa2n b œ

55. (a)

_

_

nœ1

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 ‹ cbp1 (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

" p1

_

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, ! nœ2

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) ! nœ2

" n aln n$ b

" 3

œ

_

" n(ln n)

! nœ2

; 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. ex Ÿ ex for x 1, and '1 ecx dx œ lim cex d " œ lim êb e1 ‰ œ ec1 Ê '1 ecx dx converges by

bÄ_

bÄ_

_

n #

the Comparison Test for improper integrals Ê ! e nœ0

10

59. (a) s10 œ !

'10_ x1

nœ1

" n3

dx œ lim

3

bÄ_

Ê 1.97531986 _

(b) s œ !

nœ1

" n3

10

60. (a) s10 œ !

'10_ x1

nœ1

4

¸

" n4

(b) s œ !

nœ1

" n4

¸

c2 b

lim ’ x2 “

bÄ_

'11_ x1

'10b x4 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 x3 dx œ

1 ‰ 200

œ

11


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 x3 dx œ

1 242

dx œ lim

_

'11_ x1

œ 1.97531986;

_

œ 1 ! en converges by the Integral Test.

1.20253 1.20166 2

'11b x4 dx œ

c3 b

lim ’ x3 “

bÄ_


1 3000

œ 1.08233; error Ÿ

œ 0.000435

1 ‰ 3000

œ

11


1 3000

Ê 1.08229 s 1.08237

1.08237 1.08229 2

œ 0.00004


1 ‰ 3993

œ

1 3993

and

590


10.4 COMPARISON TESTS _

1. Compare with ! nœ1

" 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 _


" n3 ,

1 n2

1 n2 30 .

_

Then by Comparison Test, !

1 n2 30

nœ1

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

_

n1 n 4 2 .

Then by Comparison Test, ! nœ1

n1 n 4 2

converges. _


" Èn ,

which is a divergent p-series, since p œ

n 2, we have Èn 1 Ÿ Èn Ê _


" n,

_

nœ1

" , n3Î2

n2

1 n

_

nœ1

" 3n ,

_

nœ1 _

œ È5 !

nœ1

1 n3Î2

Ê

1 n2

cos2 n n3Î2

nœ2

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 !

nœ2

n 3 an 4 b n4 4

diverges.

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

cos2 n n3Î2


converges.

È5 . n3Î2

_

The series ! nœ1

1 n †3 n

1 n3Î2

Ÿ

1 3n .

_


is a convergent p-series, since p œ

3 2

1 n †3 n

converges. _

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

n2 n2 n

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 !

_


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 _


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. nœ1


" #

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 nœ1


diverges.


É 1n

Section 10.4 Comparison Tests _

" n2 ,

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

œ lim _

! nœ1

nœ1

nÄ_

n2 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 ,


É nn2bb12

œ lim

591


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. nœ1

_

" n,

11. Compare with ! nan b 1b Šn2

œ lim

nœ2

b 1‹an c 1b

_

Test, ! nœ2

n3 + n2 3 2 nÄ_ n n n 1

n an 1 b an2 1ban 1b

_

nœ1

lim an nÄ_ bn

nÄ_

_

nœ1 5n

Èn 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 21 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 1 Î2 n

4n 3 4n nÄ_

œ lim

4n ln 4 n 4 nÄ_ ln 4

œ lim

_

œ 1 0. Then by Limit Comparison Test, !


nœ1

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

nœ1

5n È n †4 n

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 nœ1

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 b151În10n 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 _


œ lim

" ln n

nÄ_ 1În

nœ1

lnŠ1 n"2 ‹ 1 În 2

_

3 ‰n œ e7Î10 0. Then by Limit Comparison Test, ! ˆ 2n converges. 5n 4 nœ1

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_

nœ2

" 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 nœ1


"‰ n2

converges.

592

Chapter 10 Infinite Sequences and Series _

" Èn

17. diverges by the Limit Comparison Test (part 1) when compared with ! nœ1

, 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#

21. diverges since n lim Ä_

2n 3n 1

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


n# (n

" n#


œ n lim Ä_

" n#

œ 10

20 2


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#


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 ! nœ1 #

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 ! nœ3

" 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_


diverges

Section 10.4 Comparison Tests " n&Î%


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


" Š n" ‹

œ n lim Ä_


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


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 ! nœ1 _

! nœ1

" n2n

"n n2n

n$Î# Ê

_

œ!

nœ1

" n2n

_

" #n

!

nœ1

593

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

_

nœ1

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 nœ1

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

_

nœ1

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

_

nœ1

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_

†

1În

2 n œ n lim Ä _ 2n n

œ 1 0. _

_

nœ1

nœ1

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_


594


43. converges by Comparison Test with ! nœ2

_

which converges since !

1 n an 1 b

sk œ ˆ1 12 ‰ ˆ 12 13 ‰ Þ Þ Þ ˆ k 1 2

nœ2

1 ‰ k1

ˆ k 1 1 1k ‰ œ 1

Ê nan 1ban 2b! nan 1b Ê n! nan 1b Ê

1 n!

_

1 n3 ,

nœ1

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 n an 1 b

an

which is a convergent p-series, n lim Ä_

2n 2n 3

œ n lim Ä_

c 1bx 2bx

1În3

œ10

2 2

" n


" n


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 n an 1 b

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 ! nœ1

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 ! nœ1

ˆ 1# ‰ n1 3 Þ

œ

1 #

_

! nœ1

" 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


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, ! nœ1

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 1În2

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 !

nœ2

_

! nœ2

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 1Înr

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 qaq 1bâaq k 1b 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 !

nœ2

_

! 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 p b n

rcpc1

n Ä _ aqbaln nbcqc1 ˆ 1n ‰

œ n lim Ä_

aln nbq

np

1 În r

œ

q lim aln nb n Ä _ npcr rcp

ar p bn . aqbaln nbcqc1

1Înp

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 aqbaq 1baln nbcqc2

œ 0. Since the limit is

2 rcpc1

a r pb n aqbaq 1baln nbcqc2 ˆ 1n ‰ 2 rcp

ar pbnrcp aln nbqb1 a q b

œ n lim Ä_

qb2

œ _,

a r pb2 nrcp . aqbaq 1baln nbcqc2

If

a r pb n aln nb œ n lim œ _, otherwise, we aqbaq 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 aqbaq 1bâaq k 1baln nbcqck

œ n lim Ä_

a r pbk nrcp aln nbqbk aqbaq 1bâaq k 1b


œ _.

596


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 !

nœ2

aln nbq n

n œ1

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. n œ1

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]]


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, nœ1

_

Sœ!

nœ1 _

1 n2

nœ1

_

œ!

_

!

1 n an 1 b

nœ1

_

!

1 n2

nœ1

_

œ!

1 n an 1 b

nœ1

nœ1

1 n an 1 b

_

!

nœ1

Š n12

1 n an 1 b ‹

also converges.

_

(b) Since ! nan 1 1b converges to 1 (from Example 5 in Section 10.2), S œ 1 ! Š n12 nœ1

nœ1

_

(c) The new series is comparible to

! 13 , n

nœ1

1 n an 1 b ‹

_

œ 1 ! n2 an1 1b nœ1

_

so it will converge faster because its terms Ä 0 faster than the terms of ! n12 . nœ1

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

!

nœ1

1 n2

nœ1

œ 1.644933067. Note that

1 6

œ 1.644934067. The error is 4.99 ‚ 107 compared with 1 ‚ 106 .

2

10.5 THE RATIO AND ROOT TESTS

1.

2.

3.

2n n!

0 for all n 1; lim Œ nÄ_

n2 3n

2nb" "b! 2n n!

an

0 for all n 1; lim Œ

n

b1b b 2 3nb1 nb2 n 3

3 lim ˆ n3 n †3 †

œ

nÄ_

b1bc1b! b1bb1b2 c1b! anb1b2

aan


aan

an

_

n! 2n ‹

nÄ_

an

nÄ_

an 1 b ! an 1 b2

2 †2 lim Š an" b†n! †

œ

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†nan21b2b! †

nÄ_

n œ1

nÄ_

a n "b 2 an 1 b ! ‹

_

1 Ê ! n 3n 2 converges

1 3

n œ1

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

2

2

nÄ_

nÄ_

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

4.

2nb1 n †3 n 1

n œ1

0 for all n 1; lim nÄ_

_

2an1b1 1b†3an1b 2n1 n†3n 1

1

an

nb1

lim Š an21b†3†n2 1 †3 †

œ

nÄ_

n †3 n 1 2n1 ‹

œ lim ˆ 3n2n 3 ‰ œ lim ˆ 23 ‰ œ nÄ_

nÄ_

2 3

1

nb1

Ê ! n2†3nc1 converges n œ1

5.

n4 4n


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 an 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


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 1 b† 5 † 5 lim Š a2na 5b lnan 2b † n

œ

1b

nÄ_

a2n 3b lnan 1b ‹ n †5 n

a b 15 25 ‰ lim Š 10n 2n2 25n † lim Š ‹ † lim Š lnlnann 21b ‹ œ lim ˆ 20n 4n 5 5n 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’lnê2 1n ‰“

Ê ! ’lnê2 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. ’lnê2 1n ‰“

n

œ

4 3

nÄ_

_

n

3‰ 1 Ê ! ˆ 4n diverges 3n 5 n œ1

1 1 În

lim ’lnê2 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 ‰

n " 0 for all n 2; lim É n1bn œ

nÄ_

17. converges by the Ratio Test:

converges

nÄ_

n

n œ1

nÄ_

2

n œ1

n È

nÄ_

”

œ n lim Ä_

nÄ_

È

(n b 1) 2 2nb1 •

”

_

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 1n ‰ œ e1 1 Ê ! ˆ1 1n ‰ converges

nÄ_

"

_

1Ê !ˆ

_

2

n n n 15. ˆ1 1n ‰ 0 for all n 1; lim Éˆ1 1n ‰ œ

n1bn

1 9

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

nÄ_

16.

œ

_

n

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

2

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


lim anb1 n Ä _ an

œ n lim Ä_

19. diverges by the Ratio Test: n lim Ä_

anb1 an

œ 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_


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_


" 10

1

Section 10.5 The Ratio and Root Tests ˆ nn 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_


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

n1 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)


anb1 an

œ n lim Ä_

(n 1)$ en 1

œ


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)!


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)!


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



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


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_


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


œ n lim Ä_

3n 1 (n 1)$ 2n

1

†

n$ 2n 3n

†

a2nbx nx‘2


anb1 an

œ n lim Ä_

an1bx‘2 2(n 1)‘x


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


anb1 an

œ n lim Ä_

2 3

œ

ˆ 1 b nsin n ‰ an an

"n

Š 1 b tan n


anb1 an

œ n lim Ä_

œ n lim Ä_ †

ˆ #3 ‰ œ

an 1 b 2 (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 an1 ‰ Ê an1 œ ˆ n n 1 ‰ ˆ n n 1 ‰ ˆ nn 1 an2 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



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


1 4

1

2 †6 n 4 † 2 n 3 † 3 n 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 an n b n 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 2n# 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


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


1

p

" 4

1

601

602


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 È n 1

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. _

_

nœ1

nœ1

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 un1 Ÿ un ;

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 an 1 b 2 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 a1bn1 nn2 4 œ does not exist 2

nÄ_

lim a1bn1 2n2 œ does not exist n

nÄ_

_

_

nœ1

nœ1

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 nœ1

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


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_

_

_

nœ1

nœ1

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 _

_

nœ1

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!

nœ1

" 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 ! nœ1

" n

" n 3

_

œ 0 Ê convergence; but ! kan k nœ1

is a divergent series

_

22. converges absolutely because the series ! ¸ sinn# n ¸ converges by the Direct Comparison Test since ¸ sinn# n ¸ Ÿ nœ1

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 _

_

nœ1

nœ1

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

œ!

nœ1

" n#

_

!

nœ1

" n

603

1 n n#

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


" n#

604


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_

_

_

nœ1

nœ1

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

_

" x b#

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" ‹

_

_

nœ1

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

“

" nln n

" n

Ê

ln n n ln n

ln n nln n

" n

so that

diverges by the Direct Comparison Test

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

_

nœ1

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)!

_

_

nœ1

nœ1

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 _

_

nœ1

nœ1

_

35. converges absolutely since ! kan k œ ! ¹ (nÈ1)n ¹ œ ! _

36. converges conditionally since ! nœ1 _

_

nœ1

nœ1

! kan k œ !

" n

cos n1 n

n

nœ1

_

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

" #


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 ‰n1 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 !

nœ1

_

_

nœ1

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 È n # n n

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 ŠÉn Èn Èn‹ Ä_ Ä_ – É n È n È n — 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 !

nœ1

_

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 _

_

nœ1

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 !

nœ1

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 2 aa n 1 b 1 b

Ÿ

1 2 an 1 b

Ê un1 Ÿ un ;

lim un œ

nÄ_

1 2 an 1 b

lim 1 nÄ_ 2an1b


0 for all n 1;

œ 0.

606


48. 1

1 4

1 9

1 16

1 25

1 36

1 49

1 64

_

_

_

nœ1

nœ1

nœ1

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 1 b 2 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 1 b 2 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 ˆ an 1 b 3 È n 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

á‰ œ

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!

kœ1

" k(k 1)

n

œ ! ˆ k" kœ1

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" kœ1

" ‰ k 1

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

" ‰ n1


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_ _

_

_

_

nœ1

nœ1

nœ1

nœ1

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

_

nœ1

nœ1

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 _

_

nœ1

nœ1

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 nœ1 _

! aan bn b converges absolutely

nœ1 _

_

_

(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

nœ1

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 » ! kan 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

% #

.


608


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#

_

nœ1

nœ1

nœ1

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_ nœ1

_

series; when x œ 1 we have ! 1, a divergent series nœ1

(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_ _

_

nœ1

nœ1

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


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_ _

_

_

_

_

nœ1

nœ1

nœ1

nœ1

nœ1

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 (3xn2)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

_

nœ1

(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_

nœ1

" 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


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

nœ1

" #

" #

; when x œ "# we have

_

we have ! 1, a divergent series nœ1

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

" #

" #


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 nœ1

_

have ! nœ1

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 !

nœ1

_

! nœ1

(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

kx k 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 !

nœ1

_

when x œ 3 we have !

nœ1

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 (xÈn1) 1 † Ä_ Ä_

Èn (x 1)n ¹

1 Ê kx 1k Én lim Ä_ _


nœ1

_

we have ! nœ1

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


610


(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

_

_

nœ1

nœ1

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

_

! nœ1

4n ˆ 1 ‰2n n 2

_

œ!

nœ1

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


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 31 lx 1l 1 Ä _ 3 an 1 b 2 _

Ê 2 x 4; when x œ 2 we have ! (n2 3)3n œ ! (n1) , an absolutely convergent series; when x œ 4 we have 2 n

nœ1

_

n

nœ1

_

n

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

nœ1

nœ1

(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 !

nœ1

_

! nœ1

" 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

_


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


(")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 !

nœ1

_

series; when x œ 2 we have !

nœ1

n5n 5n

n(5)n 5n

kx 3 k 5

611

1

_

œ ! (1)n n, a divergent nœ1

_

œ ! n, a divergent series nœ1


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


nœ1

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 !

nœ1

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

_

_

nœ1

nœ1

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_ nœ1 nœ1

(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 _

_

_

nœ1

nœ1

21. First, rewrite the series as ! a2 (1)n bax 1bn1 œ ! 2ax 1bn1 ! (1)n ax 1bn1 . For the series nœ1

_

n

! 2ax 1bn1 : 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_ nœ1 _

nb1

n

series ! (1)n ax 1bn1 : n lim 1 œ lx 1l 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ ( 1) ax 1b ¹ 1 Ê lx 1ln lim Ä_ Ä _ (1)n ax1bnc1 Ä_ nœ1 _

Ê 2 x 0; when x œ 2 we have ! a2 (1)n ba1bn1 , a divergent series; when x œ 0 we have nœ1

_

! a2 (1)n b, a divergent series

nœ1

(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


612


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

Ê _

! nœ1

17 9

x

19 9 ;

(1)n 32n ˆ 1 ‰n 3n 9

when x œ _

œ!

nœ1

(1)n 3n ,

17 9

3 ax 2bnb1 3 an 1 b

nb1 2nb2

_

we have ! nœ1

†

(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 nœ1

lim ˆ1 nÄ_

" ‰n n

_

n œ e Á 0; when x œ 1 we have ! ˆ1 n" ‰ , a divergent series nœ1

(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_ nœ1

_

when x œ 1 we have ! ln n, a divergent series nœ1

(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(x4)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

_

_

nœ1

nœ1

! (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


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 nœ1

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 #


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 ‹


#

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


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 !

nœ2

_


nœ2

" 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 ! nœ1

(")2nb1 n$Î#

3 #

_

; when x œ 1 we have !

nœ1

$Î#

1 Ê (4x 5)# 1

(1)2nb1 n$Î#

_

œ!

nœ1

" 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 #


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 !

nœ1

_


nœ1

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


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

a2n 3bn 3 5 7 a2n 1ba2an 1b 1bx 34. n lim † 3†5†7âan2n2 1bxnb1 ¹ 1 Ê kxk n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ † † â an 1b2 2nb1 Š 2an 1b2 ‹ 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 ! nœ1

12ân n 12 22 â n2 x ,

recall 1 2 â n œ

nan b 1b

_

2

2 n

n an 1 b 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 nœ1 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 ‰a1bn , a conditionally Ä _ a2n 3b nœ1 _

convergent series; when x œ 1 we have ! ˆ 2n 3 1 ‰, a divergent series. nœ1

(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 nœ1

_

can rewrite the series as ! nœ1

Ê lx 3ln lim Ä_

ax 3 b n Èn 1 Èn ;

Èn 1 Èn Èn 2 Èn 1

Èn 1 Èn 1

†

nb1

Èn 1 Èn Èn 1 Èn

x3 then n lim † ¹ uunbn 1 ¹ 1 Ê n lim ¹ a b Ä_ Ä _ Èn 2 Èn 1

ax 3 b n

_

nœ1

1 Èn 1 Èn ,

so that we

¹1

a 1 b n Èn 1 Èn ,

nœ1

_

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

a n 1 bx x 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 a a bx b Ä_ Ä _ 2nb1 a2an 1bbx

2n a2nbx ¹ an xb 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 Éˆ ˆ n ‰n 1 Ê lxle1 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



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 ! a1bn , which diverges; at x œ 3

nœ0

nœ0

1 3

1 3

_

_

nœ0

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 nœ0

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 ! êln 3 4‰ œ ! a1bn , which diverges; at x œ ln 5 we have ! êln 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 nœ0

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 !

nœ0

_

we have ! nœ0 _

! nœ0

(x ")2n 4n "

#

œ!

2 4n

nœ0

_

œ!

nœ0

œ

" 1 Šxc # ‹

_

2n

4

4 4n

(x 4

n

9n (x 1)2n ¹

†

1 Ê

(x 1)# lim 9 nÄ_

nœ0

!

nœ0

nœ0

nœ0

k1k 1 Ê (x 1)# 9 Ê kx 1k 3

(3)2n 9n

_

œ ! 1 which diverges; at x œ 2 we have nœ0

_

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


nœ0 _

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

_

# Šˆ x # 1 ‰ ‹ "

’

n

(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 nœ0

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 ¸È x 2 ¸ 2 Ê 2 È x 2 2 Ê 0 È x 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 !

nœ0

0 x 16 and its sum is

1Œ

" 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


" 1 ln x

616


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

#

1 3 ‹

n 1

n † ˆ x# 3 1 ‰ º 1 Ê

ax # 1 b 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 nœ0

_

È2 x È2 ; the series !

nœ0

" # 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 ax # 1 b n

†

1

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

_

_

nœ0

nœ0

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 ‹

#

#

1 2 ‹

n

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

nœ1

convergent geometric series is œ

2 x1

" 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)n1 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 x1

.

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

_

á ˆ "# ‰

n (x 3)n n 1

1

2 x1

then ' f(x) dx _

á . At x œ 1 the series ! n21 diverges; at x œ 5 nœ1

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

nœ1

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 x1

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 n ! a b # 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) ex 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


† 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 xnk and f ÐkÑ (0) œ k!ak nœ0

Ê ak œ

f ÐkÑ (0) k!

nœk _

; likewise if f(x) œ ! bn xn , then bk œ nœ0

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 nœ0

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)# ,


x# #

x$ 3

618


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 83 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

4x , P# (x) œ

" #

, f w (0) œ (2)#

x 4

È2 È2 1 w ˆ1‰ 4 œ cos 4 œ # # ,f 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$Î# , f www (x) œ ˆ 38 ‰ x&Î# ; f(4) œ È4 œ 2, " 3 f w (4) œ ˆ "# ‰ 4"Î# œ 4" , f ww (4) œ ˆ 4" ‰ 4$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 ex , f w (x) œ ex , f ww (x) œ ex , f www (x) œ ex Ê á f ÐkÑ (x) œ a1bk ex ; f(0) œ ea0b 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 ex œ 1 x 12 x# 16 x3 á œ !

nœ0

(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 !

nœ0

1 n an 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)k1 ; 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! _

_

nœ0

nœ0

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

2x 1x

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 nœ1

_

15. sin x œ !

nœ0 _

16. sin x œ !

nœ0

_

(")n x2nb1 (#n1)!

Ê sin 3x œ !

(")n x2nb1 (#n1)!

Ê sin

nœ0

_

17. 7 cos (x) œ 7 cos x œ 7 !

nœ0

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!

nœ0

_

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


k 1

; f(0) œ 2,

Section 10.8 Taylor and Maclaurin Series _

18. cos x œ !

nœ0

19. cosh x œ _

œ!

nœ0

œ!

nœ0

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# #!

51 ' 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 1 b 2

œ

2 ; ax 1 b 3

f www (x) œ

6 ax 1 b 4

Ê f ÐnÑ (x) œ

a1 b n n x ; 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 a1bn nx if n 2 Ê x# x3 x4 x5 Þ Þ Þ œ ! a1bn xn nœ2

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)! xn2 ; 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 nœ0

28. f(x) œ

1 a1 x b 3

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!

nœ0

an 2 b ! 2

Ê

1 a1 x b 3

an 2 b ! 2

(1 x)n3 ;

œ 1 3x 6x# 10x3 á

an 2ban 1b n x 2


620


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 !

nœ0

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 !

nœ0

2(ln 2)n (x1)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 a1bn 22n Ê cosˆ2x 12 ‰ œ 1 2ˆx 14 ‰ 23 ˆx 14 ‰ . . . _

œ!

nœ0

a1bn 22n ˆ x a2nbx

2n 14 ‰

7 Î2 32. f(x) œ Èx 1, f w (x) œ 12 ax 1b1Î2 , f ww (x) œ 14 ax 1b3Î2 , f www (x) œ 38 ax 1b5Î2 , 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

_

a1bn 2n a2nbx x

33. The Maclaurin series generated by cos x is ! nœ0

by _

! nœ0

2 1x

2

8

16

which converges on a_, _b and the Maclaurin series generated

_

is 2 ! xn which converges on a1, 1b. Thus the Maclaurin series generated by faxb œ cos x nœ0

a1bn 2n a2nbx x

128

2 1x

is given by

_

2 ! xn œ 1 2x 25 x2 Þ Þ Þ Þ which converges on the intersection of a_, _b and a1, 1b, so the nœ0

interval of convergence is a1, 1b. _

34. The Maclaurin series generated by ex is ! nœ0

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 !

nœ0

_

35. The Maclaurin series generated by sin x is ! nœ0 _

generated by lna1 xb is !

nœ1

a1bnc1 n x n

xn nx

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

a 1 b n 2n1 a2n 1bx x

which converges on a_, _b and the Maclaurin series

which converges on a1, 1b. Thus the Maclaurin series genereated by

_

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

nœ0

_

a 1 b n a1bnc1 n 2n1 Œ ! n x a2n 1bx x nœ1

œ x2 21 x3 61 x4 Þ Þ Þ Þ which converges on

the intersection of a_, _b and a1, 1b, so the interval of convergence is a1, 1b. _

36. The Maclaurin series generated by sin x is ! nœ0

a 1 b n 2n1 a2n 1bx x

_

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

nœ0

œ x3 13 x5 _

37. If ex œ !

nœ0

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 nœ0

. . . 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


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 43. f(x) œ a1 x# b

"Î#

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 e5x œ 1 (5x)

(5x)# #!

á œ 1 5x

xn n!

Ê exÎ2 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!

nœ0 _

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

nœ0

_

1 x á œ ! (21) 2n 1 (#n1)! nœ0


(1)n 5n xn n!

n 2n 1 2n 1

x# #

622


5. cos x œ !

nœ0

_

6. cos x œ !

nœ0

x$ 2†2!

œ1

Ê cos 5x2 œ !

a1bn x2n (2n)!

$Î# cos Š xÈ ‹ 2

x' 2# †4!

_

7. lna1 xb œ !

nœ1

_

nœ0

10.

Ê

x* 2$ †6!

1 1x

œ ! xn Ê nœ0

_

nœ0

n œ0

_

nœ0 _

(1)n x2n (2n)!

13. cos x œ !

nœ0

œ

x% 4!

x' 6!

_

(1)n x2nb1 (2n1)!

14. sin x œ !

nœ0

œ Šx

x$ 3!

_

nœ0 _

16. cos x œ !

nœ0

œ1

" #

œ!

nœ1

nœ0

nœ0

_

xnb1 n!

œ!

nœ0

nœ0

x"! 10!

x# #

1 cos x œ

x( 7!

_

x# #

a1bn 32nb1 x8nb4 n

nœ0

x$ #!

œ x x#

_

(1)n x2nb1 (#n1)!

nœ0

_

(1)n x2n (#n)!

nœ0

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

nœ2

x* 9!

x"" 11!

x$ 3!

_

œ Œ!

nœ0

á‹ x _

(1)n x2n (2n)!

Ê x# cos ax# b œ x# !

nœ0

nœ0

_

" #

(2x)% 2†4!

(2x)' 2†6!

" #

" #

_

œ!

nœ1

" #

! (1) (2x) œ (2n)!

œ

n

2n

nœ0

(2x)) 2†8!

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

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

_

cos 2x #

(1)nb1 (2x)2n #†(2n)!

_

œ!

x4 2

n 2n

Ê x cos 1x œ x !

á

x á œ ! ((1) #n)!

Ê sin x x

2x ‰ 18. sin# x œ ˆ 1cos 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

nœ1

œ x2

nœ0

_

(1)n x2nb1 (2n1)!

15625x12 6!

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

n œ0

a1bnc1 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 !

_

" 1 # 1 "# x

œ

1 2x

_

œ!

œ ! a1bn ˆ 34 x3 ‰ œ ! a1bn ˆ 34 ‰ x3n œ 1 34 x3

1 1 34 x3

nœ0

(#n)!

nœ0

2nb1 a1bn ˆ3x4 ‰ 2n 1

nœ0

_

_

nœ1

_

œ ! a1bn xn Ê

œ!

625x8 4!

2n

"Î#

$

a1bn ŒŠ x# ‹

_

a1bnc1 ˆx2 ‰ n

Ê lna1 x2 b œ !

Ê tan1 a3x4 b œ !

1 1x

"Î#

25x4 #!

œ1

á _

a1bnc1 xn n

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

œ!

nœ0

$ cos ŒŠ x# ‹

œ

_

2n

(1)n 5x2 ‘ (2n)!

nœ0

a1bn x2nb1 2n 1

8. tan1 x œ !

9.

_

(1)n x2n (2n)!

cos 2x œ

_

nœ1

" #

_

nœ0

2n

"# Š1

x( 7!

_

nœ0

(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!

_

nœ1

(2x)' 6!

nœ2

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)!


á

...

Section 10.9 Convergence of Taylor Series 19.

x# 12x

_

nœ1

22.

" 1x

_

nœ0

nœ0

(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!

nœ1

_

œ ! xn œ 1 x x# x$ á Ê

2 a1 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$ #

" (1x)#

2$ x% 4

2% x& 5

á _

_

nœ1

n œ0

œ 1 2x 3x# á œ ! nxn1 œ ! (n 1)xn _

a1 2x 3x# á b œ 2 6x 12x# á œ ! n(n 1)xn2 nœ2

_

œ ! (n 2)(n 1)xn nœ0

3

5

7

23. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê x tan1 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 !

nœ1

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)! nœ0

x3 3!

x 4!

n 2n

x3 3!

x5 5!

x7 7!

64 x7 7!

a1bn 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!

nœ0

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 a1bn ‰xn nœ0

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 ! a13nb x2n1 nœ1

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 ‰ ˆx 12 x2 13 x3 14 x4 á ‰ œ 2x 23 x3 25 x5 á œ ! 2n 2 1 x2n1 nœ0

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 13 x3 14 x4 á and 1 " x œ 1 x x# x$ á Ê ln1a1xxb 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


" 1x

623

624


31. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê atan1 xb œ atan1 xbatan1 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!

Ê esin x œ 1 Šx

3

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 13 x3 15 x5 17 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

17 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 l0.14x 0l

4

œ 4.2 ‚ 106 Ê error Ÿ 4.2 ‚ 106

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 l0.55x 0l œ 7.03 ‚ 104 Ê error Ÿ 7.03 ‚ 104 5

kx k & 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 ‚ 1010 , 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 ‚ 104 , 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


(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 (1x 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)k1 Ê f ww (x) œ k(k 1)(1 x)k2 ; 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 ‚ 10n 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

kx k $ 3!

0.125 3!

‚ 103n .5 ‚ 103n Ê P sin P gives an approximation to 1 correct to 3n decimals.

_

_

nœ0

nœk

51. If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n 1)(n 2)â(n k 1)an xnk 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)" );


626


# 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) ) [3*cos(t),5*sin(t)]; lo := 0; hi := 2*Pi; t0 := Pi/4; P1 := plot( [r(t)[], t=lo..hi] ): display( P1, scaling=constrained, title="#27(a) (Section 13.4)" ); CURVATURE := (x,y,t) ->simplify(abs(diff(x,t)*diff(y,t,t)-diff(y,t)*diff(x,t,t))/(diff(x,t)^2+diff(y,t)^2)^(3/2)); kappa := eval(CURVATURE(r(t)[],t),t=t0); UnitNormal := (x,y,t) ->expand( [-diff(y,t),diff(x,t)]/sqrt(diff(x,t)^2+diff(y,t)^2) ); N := eval( UnitNormal(r(t)[],t), t=t0 ); C := expand( r(t0) + N/kappa ); OscCircle := (x-C[1])^2+(y-C[2])^2 = 1/kappa^2; evalf( OscCircle );

777

778

Chapter 13 Vector-Valued Functions and Motion in Space

P2 := implicitplot( (x-C[1])^2+(y-C[2])^2 = 1/kappa^2, x=-7..4, y=-4..6, color=blue ): display( [P1,P2], scaling=constrained, title="#27(e) (Section 13.4)" ); Mathematica: (assigned functions and parameters may vary) In Mathematica, the dot product can be applied either with a period "." or with the word, "Dot". Similarly, the cross product can be applied either with a very small "x" (in the palette next to the arrow) or with the word, "Cross". However, the Cross command assumes the vectors are in three dimensions For the purposes of applying the cross product command, we will define the position vector r as a three dimensional vector with zero for its z-component. For graphing, we will use only the first two components. Clear[r, t, x, y] r[t_]={3 Cos[t], 5 Sin[t] } t0= 1 /4; tmin= 0; tmax= 21; r2[t_]= {r[t][[1]], r[t][[2]]} pp=ParametricPlot[r2[t], {t, tmin, tmax}]; mag[v_]=Sqrt[v.v] vel[t_]= r'[t] speed[t_]=mag[vel[t]] acc[t_]= vel'[t] curv[t_]= mag[Cross[vel[t],acc[t]]]/speed[t]3 //Simplify unittan[t_]= vel[t]/speed[t]//Simplify unitnorm[t_]= unittan'[t] / mag[unittan'[t]] ctr= r[t0] + (1 / curv[t0]) unitnorm[t0] //Simplify {a,b}= {ctr[[1]], ctr[[2]]} To plot the osculating circle, load a graphics package and then plot it, and show it together with the original curve. x*sin(y/2) + y*sin(2*x); xdomain := x=0..5*Pi; ydomain := y=0..5*Pi; x0,y0 := 3*Pi,3*Pi; plot3d( f(x,y), xdomain, ydomain, axes=boxed, style=patch, shading=zhue, title="#69(a) (Section 14.1)" );

804

Chapter 14 Partial Derivatives plot3d( f(x,y), xdomain, ydomain, grid=[50,50], axes=boxed, shading=zhue, style=patchcontour, orientation=[-90,0], title="#69(b) (Section 14.1)" ); # (b) L := evalf( f(x0,y0) ); # (c) plot3d( f(x,y), xdomain, ydomain, grid=[50,50], axes=boxed, shading=zhue, style=patchcontour, contours=[L], orientation=[-90,0], title="#45(c) (Section 13.1)" );

73-76. Example CAS commands: Maple: eq := 4*ln(x^2+y^2+z^2)=1; implicitplot3d( eq, x=-2..2, y=-2..2, z=-2..2, grid=[30,30,30], axes=boxed, title="#73 (Section 14.1)" ); 77-80. Example CAS commands: Maple: x := (u,v) -> 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

œ0

œ

5 #

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 ˆ0ß 14 ‰


lim

#

lim


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# xy

lim

x# y# xy

lim

xy y 2x 2 x1

œ

œ

œ

œ

lim


œ

œ

(x y)(x y) xy

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)# xy

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


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


" 36

ln k1 x# y# k œ ln k1 (1)# (1)# k œ ln 2

ey sin x x Ðxß yÑ Ä Ð0ß 0Ñ


#

exy 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



(x 1)(y 2) x1

œ

lim


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


œ

" #(2 1)

Note: (xß y) must approach (0ß 0) through the first quadrant only with x Á y. xy4

lim Ðxß yÑ Ä Ð2ß 2Ñ Èx y 2 xyÁ4

œ

lim

Ðxß yÑ Ä Ð2ß 2Ñ xyÁ4

œ ŠÈ2 2 2‹ œ 2 2 œ 4

ˆÈx y 2‰ ˆÈx y 2‰ Èx y 2

œ

lim


" #

ˆÈ x È y 2 ‰

œ ŠÈ0 È0 2‹ œ 2

18.

œ

ˆÈ x y 2 ‰

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 xy1

lim


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‰ xyÁ1

sinax# y# b x# y#

œ lim

sinar# b r#

lim

1 cosaxyb xy

œ lim

1 cos u u


x3 y3 Ðxß yÑ Ä Ð1ß "Ñ x y

lim

xy

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 xy

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 a1ba1b a1b2 ‹ 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


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

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(xx) 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

xy xy

lim

x2 y xy

œ lim

lim

x# y y

œ lim

lim

x# y x4 y2

œ lim

lim

xy2 1 y1

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

3 2

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

1k 1k

x# kx# kx#

kx4

y2 1 y Ä 1 y1

xk

œ

k 1k

# 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


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 1k

œ

x% x% y#

œ


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 y1

Ð x ß y Ñ Ä Ð 1 ß 1Ñ along y œ x2

y3 1 y Ä 1 y1

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

yÄ1

x2 x 1 a x 1bax2 1b xÄ1

œ lim

808


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


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


fax, yb œ a2b3 œ 8 since any path through a2, 1b that is close to a2, 1b satisfies x 0

lim


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


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

lim

faxß yb œ

2x2 y 4 y2 x Ðxß yÑ Ä Ð0ß 0Ñ

lim

along y

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


Š1

x# y# 3 ‹

œ 1 and

lim


1œ1 Ê

# #

56. If xy 0,

2 kxyk Š x 6y ‹

lim

kxyk


tan " xy xy

lim


œ 1, by the Sandwich Theorem

# #

œ

2xy Š x 6y ‹

lim

xy


œ

lim


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


ˆ2

xy ‰ 6

2 œ 2; if xy 0,

œ 2 and

lim


lim

2 kxyk Š x 6y ‹ kxyk


2 kxyk kxyk

œ2 Ê

lim


# #

œ

lim

2xy Š x 6y ‹


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#

angle of inclination.

œ

2 tan ) 1 tan# )

œ sin 2). The value of f(xß y) œ sin 2) varies with ), which is the line's

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


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#xy#y b ¹ Ÿ Ê

61.

62.

63.

lim


Šxy

x$ xy#

œ lim

lim # # Ðxß yÑ Ä Ð0ß 0Ñ x y lim


lim


x# y# x# y# ‹

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


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


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 )kr 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


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


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 %.

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


every (x! ß y! ß z! ) 80.

f(xß yß z) œ

lim



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

(x y) œ (x " y)#

œ ax# 2xy y # b#

(xy ")(1) (x y)(x) (xy 1)#

œ

x # " (xy 1)#

`f `y

œ 5x 2y 6

`f `y

œ 9(2x 3y)#

œ

" $ $ 3É x ˆ #y ‰

Section 14.3 Partial Derivatives 12.

`f `x

œ

13.

`f `x

œ eÐxy1Ñ †

14.

`f `x

œ ex sin (x y) ex 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 ex 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)

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 yzexyz , fy œ xzexyz , fz œ xyexyz

yz xy

†

` `y

(xy) œ z ln (xy) z,

811

812


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 ù

œ v# eÐ2uÎvÑ †

37.

`h `3

œ sin 9 cos ),

38.

`g `r

œ 1 cos ),

` ù

`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 À q , `m

œ

2V$ v 2g

, Wv œ

#

V$ v g

, Wg œ V#$gv#

À `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,

† a1b 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,

À `q

œ

œ m,

` #f ` x#

Ð2uÎvÑ ˆ 2u ‰ 2ueÐ2uÎvÑ v œ 2ve

œ 1

À `c

œ

` `v

œ 3 sin 9 sin )

40.

,

À `k

œ 2veÐ2uÎvÑ v# eÐ2uÎvÑ †

† 2x‹a1 yb œ 2x ex

2

2

y

y

a1 yb ,

† a1b‹a1 yb ex

2

y

a1b

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


`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‰a1b ax yb ax2 yb2

ˆx2 y‰2 a2x 2yb ˆx2 2xy y‰2ˆx2 y‰a2xb 2

2

’ax2 yb “ 2

ˆx y‰ † 0 ˆx 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 ˆx2 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


hÄ0

œ lim

hÄ0

fa2 hß 1b fa2ß 1b h

a2h 1 hb 1 h

œ lim

hÄ0

c4 2a2 hb 3 a2 hbd a3 2b h

œ lim 1 œ 1, hÄ0

4 4 3a1 hb 2a1 hb# ‘ a3 2b fy a2ß 1b œ lim fa2ß 1 hhb fa2ß 1b œ lim h hÄ0 hÄ0 a3 3h 2 4h 2h# b 1 h 2h# œ lim œ lim œ lim a1 2hb œ 1 h h hÄ0 hÄ0 hÄ0


hÄ0

fa2 hß 3b fa2ß 3b h

È2h 4 2 h hÄ0

œ lim

fy a2ß 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


È 2 a 2 h b 9 1 È 4 9 1 h

È2h 4 2 È2h 4 2 È2h 4 2 ‹ h

fa2ß 3 hb fa2ß 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

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

hÄ0

sin h4 h4 ‹

œ0†1œ0

813

814


62. (a) In the plane x œ 1 Ê fy axß yb œ 3y2 Ê fy a1ß 1b œ 3a1b2 œ 3 Ê m œ 3 (b) In the plane y œ 1 Ê fx axß yb œ 2x Ê fy a1ß 1b œ 2a1b 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 `Àb Ê Ê

"2h 2h# h

œ lim a12 2hb œ 12 hÄ0

œ lim a2h 9b œ 9 hÄ0

(sin A) `Àa a cos A sin# A

À `b

œ

À à 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 `Àb

œ 0 Ê asin Ab `` xa a cos A œ 0 Ê à `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 `Àa 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

fa1ß hß 3b fa1, 0ß 3b h hÄ0

`z ‰ `x x

2a3 hb# 2a9b h

œ lim

fy a1ß 0ß 3b œ lim 65. y ˆ3z#

;

à À

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 ù

œ 2x

`x ù

2y

`y ù

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

0 0

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

;

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 if x 0

points ax, yb such that x Á 0.


`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

œ 3e3x4y cos 5z,

#

`f `y

&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 axb†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#

` #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# c45 cos (3x 3ct) exbct d œ c# `` xw# ` t# œ 45c cos (3x 3ct) c e

`w `t

œ

`f ù ù `t

œ

` #f ` u#

Ê

œ a#

`f ù

(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 ù ù `x

œ

`f ù

†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


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


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


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

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 ù

œ

`z `y `y ù

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 ù `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

" sin# v cos# v œ 1 ˆ1.3ß 16 ‰ : `` uz œ 0

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

and

`z `v

œ 1

`z ù

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 ù

`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

etc1

817

818 9. (a)

Chapter 14 Partial Derivatives `w ù

œ

`w `x `x ù

`w `y `y ù

`w `z `z ù

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 ù

`w ù

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 a2ß 0b: 11. (a)

`w ù

œ

œ 1 and

2 `w u and ` v `w `v œ 2

œ2

rp pq qrrppq ù `p ù `q ù `r " œ 0; ` p ` x ` q ` x ` r ` x œ q r (q r)# (q r)# œ (q r)# rp pq qrrppq 2p 2r ù ù `p ù `q ù `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 ù ù `p ù `q ù `r œ œ (z y)# ; ` z œ ` p ` z ` q ` z ` r ` z (2z 2y)# rp pq ppq 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) ù ù 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 (zyy)# ù `x

œ

(b) At ŠÈ3ß 2ß 1‹ : 12. (a)

œ

2 #

`w ù

ù `x

œ

ù `y ù `z

ù `x

œ 0,

œ

" (1 2)#

œ 1, and

ù `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 ù `y

ù `y

œ xzyz1 , and

(b) At ˆ 14 ß "# ß "# ‰ :

ù `z ù `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 xzyz1 ;

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#

ù `x

œ yz ,

œ œ xy ln y from direct calculations œ ˆ "# ‰

"Î#

œ È2,

ù `y

œ ˆ 14 ‰ ˆ "# ‰ ˆ "# ‰

Ð"Î#Ñ"

È

œ 14 2 ,

ù `z

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 ù

œ

`w `x `x ù

`w `y `y ù

`w `z `z ù

`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 ù

œ

`w `x `x ù

`w `y `y ù

`w `v

œ

`w `x `x `v

`w `y `y `v

14.

dz dt

œ

13.

819

820


18.

`w `x

œ

`w ù ù `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 ù ù `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

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 xy2y

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

êy 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 ¸ rœ1ßsœ1 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 uœ1ßvœ2

2u v 2 (u 2v 1)# “ (2)

œ (1) ˆ 41 ‰ ˆ 41 ‰ œ 8

" u 2v 1

822 36.

Chapter 14 Partial Derivatives `z ù

œ

`z `x `x ù

`z `y `y ù

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 ù

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 ù

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 Ì

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 uœ1ßvœ2 "u " `z ¸ " Š Èv 3"tan " u ‹ Š 2tan Èv 3 ‹ œ #(v 3) Ê ` v uœ1ßvœ2 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; " â 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 ù `f ù `f ù

`w `r `w `r

ù `x ù `y ù `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 ù `f ù `f ù

`f ` w (1) (1) `` vf (1) ``wf (0) (0) `` vf (1) ``wf (1)

(1)

`f `v

(0)

œ

`f ù

œ œ

`f `w

`f ù `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

#

#

#

#

#

" r#

ˆ ``w) ‰#

`w `)

2

Section 14.4 The Chain Rule `w `x

45. wx œ

œ

`w ù ù `x #

œ

`w ù

x Š `` uw#

œ

`w ù

x#

` #w ` u#

`w `v `v `x

ù `x

` #w ` v ` v` u ` x ‹

2xy

` #w ` v` u #

Ê wyy œ ``wu y Š `` uw#

`w ù

œx

#

`w `v

#

y Š `ù`wv ` #w ` v#

y# ù `y

y

ù `x

; wy œ

` #w ` v ` v` u ` y ‹

` #w ` v ` v# ` x ‹

`w `y

œ #

`w ù ù `y

x Š `ù`wv

#

`w ù

Ê wxx œ

#

ù `y

x

œ

`w ù

` `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 ù

x

` #w ` u` v

#

y `` vw# ‹

`w `v

` #w ` v ` v# ` y ‹

#

œ ``wu y Šy `` uw# x ``v`wu ‹ x Šy `ù`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# a3x# 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

œ 16 sin

1 4

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 Š2È2 sin t‹ x ŠÈ2 cos t‹

œ ŠÈ2 sin t‹ Š2È2 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

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 4

œ 8 sin 2 ˆ 341 ‰ œ 8 Ê T has a minimum at (xß y) œ (2ß 1);

" #

Ê sin t œ „

" 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

`g `y

œ y

ŠÈ2ß "‹ œ 1 Ê ™ g œ È2 i j ;

g ŠÈ2ß "‹ œ curve

ŠÈ2ß "‹ œ È2;

" #

Ê

" #

œ

x# #

y# #

or 1 œ x# y# is the level

x#

0

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 œ 2È313

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

3 2È13

6 2È13

825

826


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 (Du 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 (Du 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 (Du 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 (Du 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

2 7

" È3

j

" È3

k;

6 7

™h k™hk

œ

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 (Du h)P! œ 7 3 7

i

k; (Du f)P! œ ™ f † u œ k ™ f k œ 2È3 and (Du f)P! œ 2È3

i 37 j 67 k ; h increases most rapidly in the direction u œ 2 7

" È3

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

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 yœx4

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 ax 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

1 È10 j

828


(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 a3u1 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

œ

i 2j È(1)# (2)#

œ È75

ijk 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

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 a1 sin 1x 2xy zexz b i ax# 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

830


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 â œ 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 67 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

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

1 È3

k Ê ™f†uœ

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 c1y 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

1 #

832


30. (a) f(0ß 0) œ 1, fx (xß y) œ e2yx Ê fx (!ß !) œ 1, fy (xß y) œ 2e2yx 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

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 #

834


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 ey and Ty (xß y) œ x aey ey b Ê dT œ Tx (xß y) dx Ty (xß y) dy œ aey ey b dx x aey ey 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)#

" (400)#

dR# “ Ê R will be more

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#

836


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

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

838


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, a2ß 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 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 maximum of fa2ß 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 1b$2 , fxy œ ax# 8xy ; fxx (!ß !) œ 2, fyy (0ß 0) y# 1b$ 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.

,


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 a2ß 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 a2ß 0b: fxx a2ß 0b œ ex ax2 4x 2 y2 bk a2ß0b œ e22 , # fyy a2ß 0b œ 2ex k a2ß0b œ e22 , fxy a2ß 0b œ 2yex k a2ß0b œ 0 Ê fxx fyy fxy œ

of fa2ß 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 xy

œ 0 and fy axß yb œ 1

1 xy

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)

840


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).

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 ;

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 à

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

;

and

the coldest is "4 ° at ˆ "# ß 0‰ .

42. fx (xß y) œ y 2 fyy ˆ #" ß 2‰ œ

È3 " #ß # ‹

È3 #


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

844


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 a3, 2b Ê z œ È13; Dxx a3, 2b œ 4, Dyy a3, 2b œ 4, Dxy a3, 2b œ 0 Ê Dxx Dyy D# œ 16 0 and Dxx 0 Ê local xy

minimum of dŠ3, 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

# È3 .

x 0,


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‹, ŠÈ3, 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 ŠÈ2ß È2‹ œ 2. At the endpoints, g(2ß 0) œ g(#ß 0) œ 0.

Therefore the absolute minimum is g ŠÈ2ß È2‹ œ 2 when t œ g ŠÈ2 ß È2‹ œ 2 when t œ

1 4

.

31 4

; the absolute maximum is

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 ˆ1ß "# ‰ œ "# . The absolute minimum is f ˆ1ß "# ‰ 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 ˆ1ß "# ‰ 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.

.

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 2am 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 2am 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 Œ ! 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 Œ ! 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 mŒ ! 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 bŒ ! xk Œ ! yk kœ1

kœ1

n

n

kœ1

kœ1

kœ1 2

n! ax2k bŒ ! xk

k œ1

n

œ

n

kœ1

n

Œ ! xk Œ ! yk n ! axk yk b kœ1

kœ1

n

kœ1

2

n

2 Œ ! 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

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);

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.

850


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 2yx

œ-

Ê 2x œ Š 2y2yx ‹ (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.


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 ŠÈ5ß 2È5‹ is the minimum value and T Š2È5ß È5‹ œ 125° œ T Š2È5ß 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

852


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 y1 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).


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.

854


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 a4ß „ 4ß 2b . Then M Š3È3ß 0ß 3‹ œ 27È3 60 ¸ 106.8, M Š3È3ß 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 a4ß „ 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.)

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 a4t# 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.

5 #

and

856


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

abc 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;

# (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 1y

, fxx œ ex ln (1 y), fxy œ

ex 1y

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 1y

x

, fxyy œ (1 e y)# , fyyy œ

2ex (1 y)$

x

, fyy œ (1 e y)#

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 "# a4x# 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;

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 yx Ê Š ``wy ‹ œ (2x) ˆ yx ‰ (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

859

860


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 `ÙP ‰V œ (a) Œ Ä V PV ÏT œ Ò

Ù `P `P `P

Ù `V `V `P

Ù `T `T `P

œ

Ù `P

V ‰ ‰ ˆ ` U ‰ ˆ nR ˆ `` U V (0) ` T

nR

œ

Ù `P

V ‰ ˆ `ÙT ‰ ˆ nR

nRT ÎP œ V Ñ V Ä U Ê ˆ `ÙT ‰V œ (b) Œ Ä VœV T Ï TœT Ò ‰ Ù œ ˆ `ÙP ‰ ˆ nR V `T

Ù `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

Ù `V `V `T

`w `y `y `x

z cos x y cos z sin x . #

Ù `T `T `T

‰ ˆ Ù ‰ œ ˆ `ÙP ‰ 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

Ù `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

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

œ

" 1

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

ù `y

u Š uv

ù `y

u

ù `y ‹

`v `y ;

œ Šv

#

x œ u# v# and u v

#

‹

ù `y

Ê

ù `y

`x `y

œ

œ 0 Ê 0 œ 2u

ù `y

2v

`v `y

At (uß v) œ ŠÈ2ß 1‹ ,

v v# u# .

`v `y

Ê ù `y

œ ˆ uv ‰ "

œ

#

1# ŠÈ2‹

ù `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

œ 0 and imply

`y `x

œ0

`g `z `z `y

œ0

861

862


`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

Level curves are ellipses with major axis along the y-axis and minor axis along the x-axis.

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.

`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

`g `g `f `y `z `y `g `f `f `g `z `w `z `w

`` fz

1. Domain: All points in the xy-plane Range: z 0

Level curves are the straight lines x y œ ln z with slope 1, and z 0.

`f `x `f `z

Ê

CHAPTER 14 PRACTICE EXERCISES

2. Domain: All points in the xy-plane Range: 0 z _

œ

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 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 2y

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

20 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


" 11

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

863

864


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) kxkkyk

lim


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

xy 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

x y#

Ê

` #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)] t1 ; dw ¸ ˆ " ‰ dt tœ0 œ 0 † 1 [1 † (1)] 01 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

" t1

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

Chapter 14 Practice Exercises 31.

`w `x

œ 2 cos (2x y),

Ê Ê

33.

`w ù `w `v

œ

`x ù `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 a2e 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

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 Éˆ "# ‰ 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 35 i 45 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 exy 2 Ê Fx œ 2y exy and Fy œ 2x exy 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

ij 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 37 j 67 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

ijk 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

œ È3

i

" È3

j

" È3

k

" È5

k;

;

866


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#

2(y 1) (z 0) œ 0 or 2y z 2 œ 0

Ê

`z ` y ¹ Ð0ß1ß0Ñ

œ 2; thus the tangent plane is


`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

1 4

;

868


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¸ dVœ1ßdRœ20 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 uv

œ

œ

du uv

Ê ¸ dzz ‚ 100¸ Ÿ ¸ du u ‚ 100 64. C œ Ê

dv v

du u

dv uv

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

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)

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 ˆ2ß "# ‰ is an interior critical point in AB

with f ˆ2ß "# ‰ 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)

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, a1ß 1b are all interior points of AB with fa0ß 0b œ 16, fa1ß 1b œ 18, and fa1ß 1b œ 18. Endpoints: fa2ß 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:

fa2ß 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 a1ß 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)

872


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" ‰ .


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.

™g

874


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

. 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

" #

3 #

" #

,

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 `ÙT œ `ÙP `` TP ‰ ˆ ``VT ‰ ˆ `ÙT ‰ (1); PV œ nRT Ê P ``VT œ nR Ê ``VT œ œ ˆ `ÙP ‰ (0) ˆ `` U V ˆ `ÙT ‰ œ ˆ `` U ‰ ˆ nR ‰ V P P

Ù `T

Ù `V Ù `T `V `T `T `T nR P ; therefore,

Ù `P Ù `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 `ÙP ‰ ˆ ``VP ‰ ˆ `` V (1) ˆ `ÙT ‰ (0); PV œ nRT Ê V ``VP P œ (nR) ˆ ``VT ‰ œ 0 Ê

ˆ `` U ‰ V T

œ

ˆ `ÙP ‰ ˆ VP ‰

Ù `V

Ù `T `T `V `P P `V œ V

; therefore,


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.

ù `y

`v `x

fv

œ afu afv , and zy œ fu

ù `x œ a " `w " `w a `x œ b `y

œ b and

Ê 94.

ù `x

`w `x

œ

and

`w `z

œ

2 rs

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

ù `y

ù `x

ae sin vb

u

`v `y

dw ` u du ` y

œ

2(r s) 2 ar# 2rs s# b

œ

" rs

œ

`w `x `x `r

`g `)

œ

Ê

`f `x `x `) ` #g ` )#

`f `y `y `)

œ (r sin )) Š `` x)

`y `) ‹

" rs

rs (r s)#

`w `y

`w `z `z `r

dw du

œ œ

Ê

`f `x

`v `y

" rs

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 rs

y œ 0 Ê aeu sin vb

ù `x

aeu cos vb

`v `x

œ 0.

Similarly, e cos v x œ 0 u

ù `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 ù `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

ù `y

œ (r sin ))

dw du

`w `z `z `s

œ cae cos vb i ae sin vb jd † cae between the vectors is the constant 1# . 96.

œ bfu bfv

œ

ù `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

ù `x `w `y

2(r s) (r s)# (r s)# 4rs -

œ

2x x# y# 2z

Ê

ù `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.

" #

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 `ù

#

#

fzz œ Š ddr#f ‹ ˆ ``zr ‰ ` #r ` y#

œ

` g du ` u dx

` g dv ` v dx

œ Š `ù

#

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

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 `ùt ``wv ``vt œ x ``wu y ``wv . Now, `w `w ù `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 ù ù `y

ntnc1 f(xß y) œ x `w `x

Ê

œ

`f `x

`w `v `v `y `w ù

`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 ‰ ù

t

œ

`f `x

`w `v . ` w `v ` v` u ` t

` w ù ` u# ` t

` #w ` u ` u` v ` y

Ê ˆ t"# ‰

`w ù

#

` #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 ‰ ù

,

` #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.

878


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 6xyeÐ2x3yÑ Ê fx (xß y) œ 6y(1 2x)eÐ2x3yÑ œ 0 and fy (xß y) œ 6x(1 3y)eÐ2x3yÑ 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


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! ).

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 41

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

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 # # # # As t Ä _, w Ä 0 since ¸sin ˆ nL1 x‰¸ Ÿ 1 and ec n 1 tÎL Ä 0.

n1 L

# # # # Ê w œ ec n 1 tÎL sin ˆ nL1 x‰ .

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 cy 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 ccos 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 ' a2xb dx œ cx2 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 ' cy cosax yb sinax ybd10 dx c1 0 c1 0

16.

R

1

0

œ 'c1 asinax 1b 1 cosax 1b sin xbdx œ ccosax 1b 1 sinax 1b cos xdc0 1 œ 4 0

882

Chapter 15 Multiple Integrals

' ' ex y dA œ ' ' ex y dy dx œ ' cex y dln0 2 dx œ ' aex ln 2 ex b dx œ cex 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 4x 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


'02 y2 dy dx œ '04 ’ y4 “2 dx œ '04 1 dx œ cxd40 œ 4

1

R

4

R


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.

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

Èx

(b)

dy dx

'0 'y dx dy 3

9 2

2

1Î3

dx dy

883

884


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


'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

2 5

œ #"

2 5

" œ 10

885

886 28.


'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

'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 '0x2 dy dx

dx dy

ln t

t# 4

t ln t t“

# "


'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

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


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 c2y cos xyd 0y dy œ '0 a2y 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

889

890 54.


'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 " 1x œ 'c1 "2 y# 2x# y‘ x1 dx '0 2" y# 2x# y‘ x1 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 ax$ 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

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 “!

48032096 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

ˆ1 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 21 ‹ Š #

#

#

lim

bÄ_

œ 21 Š lim tan" b tan" 0‹ œ (21) ˆ 1# ‰ œ 1# 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 #

891

892 72.


'0_ '0_ xecÐx

2yÑ

_

_

cxex ex d b0 dy œ '0 e2y lim

bÄ_

œ '0 ec2y dy œ 73.

_

dx dy œ '0 e2y lim " # b lim Ä_

abeb eb 1b dy

bÄ_

aec2b 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

$Î# 3 4cx 3 x# b ' ' f(xß y) dA œ ' ' È È4x# dy dx œ ' ’a4 x# b Èx3 È4 x# “ dx œ ”4x x3$ 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 1y# 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

" 1y #

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.

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).

893

894


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] x^2*y^2*z; q1 := Int( Int( Int( F(x,y,z), y=-sqrt(1-x^2)..sqrt(1-x^2) ), x=-1..1 ), z=0..1 ); value( q1 ); Mathematica: (functions and bounds will vary) Clear[f, x, y, z]; f:= x2 y2 z Integrate[f, {x,1,1}, {y,Sqrt[1 x2 ], Sqrt[1 x2 ]}, {z, 0, 1}] N[%] topolar={x Ä r Cos[t], y Ä r Sin[t]}; fp= f/.topolar //Simplify Integrate[r fp, {t, 0, 21}, {r, 0, 1},{z, 0, 1}] N[%] 15.6 MOMENTS AND CENTERS OF MASS 1. M œ '0

1

'x2cx 3 dy dx œ 3'01 a2 x# xb dx œ 7# ; My œ '01 'x2cx #

œ 3'0 a2x x$ x# b dx œ 1

Ê xœ 2. M œ $ '0

3

Iy œ $ '0

3

3. M œ '0

2

œ

" #

and y œ

5 14

'03

'0

5 4

; Mx œ '0

'x2cx

#

3y dy dx œ

3x dy dx œ 3 '0 cxyd x2cx dx 1

'01 cy# d x2cx

3 #

#

dx œ

3 #

#

'01 a4 5x# x% b dx œ 195

38 35

dy dx œ $ '0 3 dx œ 9$ ; Ix œ $ '0 3

3

x dy dx œ $ '0 cx

3

1

#

3

#

#

$ yd !

'03 y# dy dx œ $ '03 ’ y3 “ 3 dx œ 27$ ; $

0

dx œ $ '0 3x dx œ 27$ 3

#

'y4Î2ydx dy œ '02 Š4 y y# ‹ dy œ 143 ; My œ '02 'y4Î2y #

#

#

4cy

x dx dy œ

" #

'02 cx# d y4 Îy2 dy #

'0 Š16 8y y# y4 ‹ dy œ 128 ' 'y Î2 y dx dy œ '0 Š4y y# y# ‹ dy œ 103 15 ; Mx œ 0 2

2

%

2

$

#

Ê xœ 4. M œ '0

3

64 35

and y œ

5 7

'03cx dy dx œ '03 (3 x) dx œ 9# ; My œ '03 '03cx x dy dx œ '03 cxyd 03cx dx œ '03 a3x x# b dx œ 9#

Ê x œ 1 and y œ 1, by symmetry

Èa cx

5. M œ '0 '0 a

#

Ê xœyœ

#

4a 31

dy dx œ

1a# 4

; My œ

, by symmetry

Èa cx

'0a '0

#

#

a a È# # x dy dx œ '0 cxyd 0 a cx dx œ '0 xÈa# x# dx œ

a$ 3

909

910


6. M œ '0

'0sin x dy dx œ '01 sin x dx œ 2; Mx œ '01 '0sin x y dy dx œ "# '01 cy# d 0sin x dx œ "# '01 sin# x dx

1

œ

" 4

'01 (1 cos 2x) dx œ 14 È4cx

1 #

Ê xœ

and y œ

È4cx

7. Ix œ 'c2 'cÈ4cx# y# dy dx œ 'c2 ’ y3 “ dx œ cÈ4cx# I o œ I x I y œ 81 2

8. Iy œ '1

21

#

2

$

#

#

ˆ

œ

'c22 a4 x# b$Î# dx œ 41; Iy œ 41, by symmetry;

#

‰

ex

0

b Ä _

'c_ e 0

2x

_

10. My œ '0

dx œ

'0e

'b0 ex dx œ 1

0

'b0 xex dx œ

lim

" #

2 3

'0 sin x Îx x# dy dx œ '121 asin# x 0b dx œ "# '121 (1 cos 2x) dx œ 1#

9. M œ 'c_ '0 dy dx œ ' _ ex dx œ lim b Ä _ œ

1 8

x# Î2

cxex ex d b0 œ 1

lim

b Ä _

" lim # bÄ _

'b0 e2x dx œ "4

x dy dx œ lim

bÄ_

ycy#

'0b xe

x# Î2

eb œ 1; My œ ' _ '0 x dy dx œ ' _ xex dx b Ä _ ex

0

abeb eb b œ 1; Mx œ 'c_ '0 y dy dx b Ä _

2

Ê x œ 1 and y œ

dx œ lim

bÄ_

ycy#

#

" ex# Î2

" 4

b

1‘ 0 œ 1

ycy#

2

%

ycy#

&

Ix œ '0 'cy y# (x y) dx dy œ '0 ’ x 2y xy$ “ dy œ '0 Š y2 2y& 2y% ‹ dy œ cy 2

È3Î2

2

È12

12. M œ 'cÈ3Î2 '4y#

4y#

# #

È3Î2

È12

5x dx dy œ 5 ' È3Î2 ’ x2 “ 4y# #

4y#

ex

0

lim

y 11. M œ '0 'cy (x y) dx dy œ '0 ’ x2 xy“ dy œ '0 Š y2 2y$ 2y# ‹ dy œ ’ 10 cy 2

0

lim

2

dy œ

5 #

'

y% #

64 105

# 2y$ 3 “!

œ

8 15

;

;

È

' È33ÎÎ22 a12 4y# 16y% b dy œ 23È3

'x2cx (6x 3y 3) dy dx œ '01 6xy 3# y# 3y‘ x2cx dx œ '01 a12 12x# b dx œ 8; 1 2cx 1 1 2cx My œ '0 'x x(6x 3y 3) dy dx œ '0 a12x 12x$ b dx œ 3; Mx œ '0 'x y(6x 3y 3) dy dx 1 3 17 œ '0 a14 6x 6x# 2x$ b dx œ 17 # Ê x œ 8 and y œ 16

13. M œ '0

1

14. M œ '0

1

'y2ycy (y 1) dx dy œ '01 a2y 2y$ b dy œ "# ; Mx œ '01 'y2ycy #

#

#

My œ '0

1

#

2ycy#

'y

#

x(y 1) dx dy œ '0 a2y# 2y% b dy œ 1

œ 2 '0 ay$ y& b dy œ 1

4 15

Ê xœ

8 15

y(y 1) dx dy œ '0 a2y# 2y% b dy œ 1

and y œ

8 15

; Ix œ '0

1

2ycy#

'y

#

4 15

;

y# (y 1) dx dy

" 6

15. M œ '0

'06 (x y 1) dx dy œ '01 (6y 24) dy œ 27; Mx œ '01 '06 y(x y 1) dx dy œ '01 y(6y 24) dy œ 14; 1 6 1 14 ' 1' 6 # My œ '0 '0 x(x y 1) dx dy œ '0 (18y 90) dy œ 99 Ê x œ 11 3 and y œ 27 ; Iy œ 0 0 x (x y 1) dx dy 1 ‰ œ 216 '0 ˆ y3 11 6 dy œ 432 1

16. M œ 'c1 'x# (y 1) dy dx œ 'c1 Š x# x# 3# ‹ dx œ 1

œ

48 35

1

1

%

; My œ 'c1 'x# x(y 1) dy dx œ 'c1 Š 3x # 1

œ 'c1 Š 3x2 1

#

x' 2

1

x% ‹ dx œ

1

16 35

x& #

32 15

; Mx œ 'c1 'x# y(y 1) dy dx œ 'c1 Š 56 1

1

x$ ‹ dx œ 0 Ê x œ 0 and y œ

1

9 14

x' 3

x% #‹

dx

; Iy œ 'c1 'x# x# (y 1) dy dx 1

1

Section 15.6 Moments and Centers of Mass 17. M œ 'c1 '0 (7y 1) dy dx œ 'c1 Š 7x# x# ‹ dx œ x#

1

1

%

31 15

; Mx œ 'c1 '0 y(7y 1) dy dx œ 'c1 Š 7x3 x#

1

1

My œ 'c1 '0 x(7y 1) dy dx œ 'c1 Š 7x# x$ ‹ dx œ 0 Ê x œ 0 and y œ 1

x

#

1

œ 'c1 Š 7x# x% ‹ dx œ 1

'

18. M œ '0

My œ '0

'c1 1

x ˆ1

x ‰ 20

dy dx œ '0 Š2x 20

y

x# 10 ‹

dx œ

Ê xœ

2000 3

1

1

y

; Iy œ '0 'cy x# (y 1) dx dy œ 1

1

y

y

" 3

1

3 #

x

and y œ 0; Ix œ '0

20

100 9

21. Ix œ '0 '0 a

b

œ 'c3 'c2 ’ 8y3 4

#

œ 'c3 'c2 ’ (4 812y) 3

4

$

Ð4

2yÑÎ3

Iz œ 'c3 'c2 'c4Î3 3

4

23. M œ 4 '0

1

œ 2 '0

8(2 y)$ 81

x# (4 2y) 3

64 81 “

4x# 3

11 30

aa# c# b and Iz œ

M 3

y

4

1

$

c$ b 3 ‹

Ð4

2yÑÎ3

6 5

dx œ

abc ab# c# b 3

ay# z# b dz dy dx 3

3

ax# y# b dz dy dx œ '

' 3

4

16 15

aa# b# b , by symmetry

dy dx œ 'c3 ˆ12x# 3

;

1

Ê Io œ Ix Iy œ

3

64 81 “

7 6

6 5

Ð4

2yÑÎ3

' ' ' dy dx œ 'c3 104 3 dx œ 208; Iy œ c3 c2 c4Î3

dy dx

ax# y# b ˆ 83 2

4

ax# z# b dz dy dx

32 ‰ 3

dx œ 280;

2y ‰ 3

dy dx œ 12 ' 3 ax# 2b dx œ 360 3

'01 '4y4 dz dy dx œ 4 '01 '01 a4 4y# b dy dx œ 16 '01 23 dx œ 323 ; Mxy œ 4 '01 '01 '4y4 #

#

z dz dy dx

12 '0 a16 16y% b dy dx œ 128 '0 dx œ 128 5 5 Ê z œ 5 , and x œ y œ 0, by symmetry; 1 1 4 1 1 64y 7904 % ' 1 1976 ‰ Ix œ 4 '0 '0 '4y ay# z# b dz dy dx œ 4 '0 '0 ’ˆ4y# 64 3 Š4y 3 ‹“ dy dx œ 4 0 105 dx œ 105 ; 1

1

1

'

#

Iy œ 4 '0

1

œ

4832 63

'01 '4y4 ax# z# b dz dy dx œ 4 '01 '01 ’ˆ4x# 643 ‰ Š4x# y# 64y3 ‹“ dy dx œ 4 '01 ˆ 83 x# 128 ‰ dx 7 '

#

; Iz œ 4 '0

1

œ 16 '0 Š 2x3 1

#

'01 '4y4 ax# y# b dz dy dx œ 16 '01 '01 ax# x# y# y# y% b dy dx

2 15 ‹

#

dx œ

ŠÈ4 x# ‹Î2

256 45

24. (a) M œ 'c2 'ŠcÈ4cx#‹Î2 '0 2

ŠÈ4 x# ‹Î2

2 x

Myz œ 'c2 'ŠcÈ4cx#‹Î2 '0 2

dz dy dx œ '

2 x

2 2

'ŠŠ È44 xx ‹‹ÎÎ22 (2 x) dy dx œ ' 22 (2 x) ŠÈ4 x# ‹ dx œ 41;

x dz dy dx œ '

È

#

#

2 2

ŠÈ4 x# ‹Î2

'Š

È4 x# ‹Î2

;

; Ix œ '0 'cy y# a3x# 1b dx dy œ '0 a2y& 2y$ b dy œ 56 ;

32 45

3

2y$ 3

x ‰ 20

1

y

is the top of the wedge Ê Ix œ 'c3 'c2 'c4Î3

4 2y 3

y

Ê Io œ Ix Iy œ

$

22. The plane z œ 3

1

'0c ay# z# b dz dy dx œ '0a '0b Šcy# c3 ‹ dy dx œ '0a Š cb3 M 3

y# ˆ1

dx œ 0;

; Ix œ '0 'cy y# (y 1) dx dy œ '0 a2y% 2y$ b dy

1

1

ab# c# b where M œ abc; Iy œ

"

1

1

Iy œ '0 'cy x# a3x# 1b dx dy œ 2 '0 ˆ 35 y& 3" y$ ‰ dy œ y

'c1 1

; Mx œ '0 'cy y a3x# 1b dx dy œ '0 a2y% 2y# b dy œ

y

1

7 10

'01 a2y% 2y$ b dy œ 103

My œ '0 'cy x a3x# 1b dx dy œ 0 Ê x œ 0 and y œ 1

M 3

1

y

1

20. M œ '0 'cy a3x# 1b dx dy œ '0 a2y$ 2yb dy œ

œ

; Iy œ 'c1 '0 x# (7y 1) dy dx

13 31

; Mx œ '0 'cy y(y 1) dx dy œ 2 ' ay$ y# b dy œ 0

5 3

My œ '0 'cy x(y 1) dx dy œ '0 0 dy œ 0 Ê x œ 0 and y œ 1

;

#

'020 ˆ1 20x ‰ dx œ 20

1

9 10

13 15

7 5

19. M œ '0 'cy (y 1) dx dy œ '0 a2y# 2yb dy œ

œ

dx œ

#

20

2 3

x% 2‹

'c11 ˆ1 20x ‰ dy dx œ '020 ˆ2 10x ‰ dx œ 60; Mx œ '020 'c11 y ˆ1 20x ‰ dy dx œ '020 ’ˆ1 #x0 ‰ Š y# ‹“ "

20

œ

&

'

911

x(2 x) dy dx œ '

2 2

x(2 x) ŠÈ4 x# ‹ dx œ 21;

912

Chapter 15 Multiple Integrals ŠÈ4 x# ‹Î2

Mxz œ 'c2 'ŠcÈ4cx#‹Î2 '0 2

œ

2 x

'c2 (2 x) ’ 44x 2

" #

#

4 x # 4 “

ŠÈ4 x# ‹Î2

(b) Mxy œ 'c2 'ŠcÈ4cx#‹Î2 '0 2

œ 51 Ê z œ 25. (a) M œ 4 '0

2

Mxy œ '0

2 x

È4cx

#

È

#

y(2 x) dy dx

#

dx œ 0 Ê x œ "# and y œ 0 " #

z dz dy dx œ

' 22 'ŠŠ È44 xx ‹‹ÎÎ22 È

#

#

' 22 (2 x)# ŠÈ4 x# ‹ dx

" #

(2 x)# dy dx œ

'x4 y dz dy dx œ 4 '01Î2 '02 'r 4 r dz dr d) œ 4 '01Î2 '02 a4r r$ b dr d) œ 4 '01Î2 4 d) œ 81; #

#

#

'0 'r zr dz dr d) œ '0 '0 2

Š 4 x ‹Î2 ' 2 Š È4 x ‹Î2

2

5 4

'0

21

y dz dy dx œ '

21

4

#

(b) M œ 81 Ê 41 œ '0

21

Èc

'0 'r

c #

2

r #

a16 r% b dr d) œ

r dz dr d) œ '0

21

Èc

'0

32 3

'021 d) œ 6431

acr r$ b dr d) œ '0

21

Ê zœ

8 3

c# 4

c# 1 #

d) œ

, and x œ y œ 0, by symmetry Ê c# œ 8 Ê c œ 2È2,

since c 0 26. M œ 8; Mxy œ 'c1 '3 'c1 z dz dy dx œ 'c1 '3 ’ z2 “ dy dx œ 0; Myz œ 'c1 '3 'c1 x dz dy dx " 1

5

1

1

5

"

#

1

5

1

œ 2 'c1 '3 x dy dx œ 4 'c1 x dx œ 0; Mxz œ 'c1 '3 'c1 y dz dy dx œ 2 'c1 '3 y dy dx œ 16 'c1 dx œ 32 1 5 1 1 5 1 Ê x œ 0, y œ 4, z œ 0; Ix œ ' ' ' ay# z# b dz dy dx œ ' ' ˆ2y# 23 ‰ dy dx œ 32 ' 100 dx œ 400 3 ; 1

5

1

1

c1

3

5

1

1

c1

c1

5

1

3

1 5 1 1 5 1 Iy œ 'c1 '3 'c1 ax# z# b dz dy dx œ 'c1 '3 ˆ2x# 23 ‰ dy dx œ 43 'c1 a3x# 1b dx œ 16 3 ; 1 5 1 1 5 1 400 ‰ Iz œ 'c1 '3 'c1 ax# y# b dz dy dx œ 2 'c1 '3 ax# y# b dy dx œ 2 'c1 ˆ2x# 98 3 dx œ 3

Ð2 yÑÎ2

27. The plane y 2z œ 2 is the top of the wedge Ê IL œ 'c2 'c2 'c1 2

œ 'c2 'c2 ’ (y 6)#(4 y) 2

Mœ

" #

4

#

(2 y)$ 24

4

4

$

49 3 ‹

dt œ 1386;

(3)(6)(4) œ 36 2

" #

c(y 6)# z# d dz dy dx

# "3 “ dy dx; let t œ 2 y Ê IL œ 4 'c2 Š 13t 24 5t 16t

Ð2 yÑÎ2

28. The plane y 2z œ 2 is the top of the wedge Ê IL œ 'c2 'c2 'c1 œ

c1

4

c(x 4)# y# d dz dy dx

'c22 'c42 ax# 8x 16 y# b (4 y) dy dx œ 'c22 a9x# 72x 162b dx œ 696; M œ "# (3)(6)(4) œ 36

'02cx '02cxcy 2x dz dy dx œ '02 '02cx a4x 2x# 2xyb dy dx œ '02 ax$ 4x# 4xb dx œ 43 2 2cx 2cxcy 2 2cx 2 8 8 Mxy œ '0 '0 '0 2xz dz dy dx œ '0 '0 x(2 x y)# dy dx œ '0 x(23 x) dx œ 15 ; Mxz œ 15 by 2 2cx 2cxcy 2 2cx 2 # symmetry; Myz œ '0 '0 '0 2x# dz dy dx œ '0 '0 2x# (2 x y) dy dx œ '0 a2x x# b dx œ 16 15

29. (a) M œ '0

2

(b)

$

Ê xœ 30. (a) M œ '0

2

4 5

, and y œ z œ

È

'0 x '04cx

(b) Myz œ '0

2

Ê xœ œ

256È2k 231

œ

k 4

#

È

kxy dz dy dx œ k'0

'0 x '04cx

5 4

2

#

2

2

È

'0 x '04cx

40È2 77

#

1

'01 '01

xy a4 x# b dy dx œ

Èx

'0

2

; Mxy œ '0

2

È

'0 x '04cx

#

Ê zœ

(x y z 1) dz dy dx œ '0

1

'02 a4x# x% b dx œ 32k 15

k #

x# y a4 x# b dy dx œ

kxy# dz dy dx œ k'0

'02 a16x# 8x% x' b dx œ 256k 105

31. (a) M œ '0

Èx

'0

kx# y dz dy dx œ k '0

; Mxz œ '0 Ê yœ

2 5

Èx

'0

k #

'02 a4x$ x& b dx œ 8k3

xy# a4 x# b dy dx œ

kxyz dz dy dx œ '0

2

Èx

'0

k 3

'02 ˆ4x&Î# x*Î# ‰ dx #

xy a4 x# b dy dx

8 7

'01

ˆx y 3# ‰ dy dx œ ' (x 2) dx œ 0 1

5 #

Section 15.6 Moments and Centers of Mass (b) Mxy œ '0

1

'01 '01 z(x y z 1) dz dy dx œ "# '01 '01 ˆx y 53 ‰ dy dx œ "# '01 ˆx 136 ‰ dx œ 43

Ê Mxy œ Myz œ Mxz œ (c) Iz œ '0

1

4 3

, by symmetry Ê x œ y œ z œ

8 15

'01 '01 ax# y# b (x y z 1) dz dy dx œ '01 '01 ax# y# b ˆx y 3# ‰ dy dx

œ '0 ˆx$ 2x# "3 x 43 ‰ dx œ 1

Ê I x œ Iy œ Iz œ

11 6

11 6

, by symmetry

32. The plane y 2z œ 2 is the top of the wedge. Ð2 yÑÎ2

(a) M œ 'c1 'c2 'c1 1

4

4

1

Ð2 yÑÎ2

(b) Myz œ 'c1 'c2 'c1 1

(x 1) dz dy dx œ 'c1 'c2 (x 1) ˆ2 y# ‰ dy dx œ 18

Ð2 yÑÎ2

Mxz œ 'c1 'c2 'c1 1

4

Ð2 yÑÎ2

Mxy œ 'c1 'c2 'c1 1

4

Ð2 yÑÎ2

(c) Ix œ 'c1 'c2 'c1 1

4

Ð2 yÑÎ2

Iy œ 'c1 'c2 'c1 1

4

Ð2 yÑÎ2

Iz œ 'c1 'c2 'c1 1

33. M œ '0

1

4

Èz

'zc1c1z '0

4

x(x 1) dz dy dx œ 'c1 'c2 x(x 1) ˆ2 y# ‰ dy dx œ 6; 1

4

y(x 1) dz dy dx œ 'c1 'c2 y(x 1) ˆ2 y# ‰ dy dx œ 0; 1

z(x 1) dz dy dx œ

" #

4

'c11 'c42 (x 1) Š y4

#

y‹ dy dx œ 0 Ê x œ

(x 1) ay# z# b dz dy dx œ 'c1 'c2 (x 1) ’2y# 1

4

(x 1) ax# z# b dz dy dx œ 'c1 'c2 (x 1) ’2x# 1

4

1 3

, and y œ z œ 0

$ 3" ˆ" 2y ‰ “ dy dx œ 45;

" 3

y$ #

" 3

x# y #

$ 3" ˆ" y2 ‰ “ dy dx œ 15;

(x 1) ax# y# b dz dy dx œ 'c1 'c2 (x 1) ˆ2 y# ‰ ax# y# b dy dx œ 42 1

(2y 5) dy dx dz œ '0

1

4

'zc1c1z ˆz 5Èz‰ dx dz œ '01 2 ˆz 5Èz‰ (1 z) dz

" $Î# œ 2 '0 ˆ5z"Î# z 5z$Î# z# ‰ dz œ 2 10 "# z# 2z&Î# 3" z$ ‘ ! œ 2 ˆ 93 3# ‰ œ 3 3 z 1

È4cx

16c2 ˆx# by# ‰

34. M œ 'c2 'cÈ4cx# '2 ax#by# b 2

œ 4 '0

21

35. (a) x œ

#

2

'02 r a4 r# b r dr d) œ 4 '021 ’ 4r3

$

Myz M

œ0 Ê

È4cx

Èx# y# dz dy dx œ ' ' È # Èx# y# c16 4 ax# y# bd dy dx c2 c 4cx #

r5 “ d) œ 4 '0 &

#

21

!

64 15

d) œ

5121 15

' ' ' x$ (xß yß z) dx dy dz œ 0 Ê Myz œ 0 R

(b) IL œ ' ' ' kv hik# dm œ ' ' ' k(x h) i yjk# dm œ ' ' ' ax# 2xh h# y# b dm D

D

D

œ ' ' ' ax# y# b dm 2h ' ' ' x dm h# ' ' ' dm œ Ix 0 h# m œ Ic m h# m Þ

D

D

36. IL œ Ic m mh# œ Þ

Þ

2 5

ma# ma# œ

7 5

ma# #

37. (a) (xß yß z) œ ˆ #a ß #b ß #c ‰ Ê Iz œ Ic m abc ŠÉ a4 Þ

œ

abc aa# b# b 3

abc aa# b# b 4

# (b) IL œ IcÞmÞ abc ŒÉ a4

œ

abc aa# 7b# b 3

Ð4

3

4

Þ

# b# 4‹

Ê IcÞmÞ œ Iz

# b # abc aa# b# b ; RcÞmÞ œ É IcMÞmÞ œ É a 12 1# # # # # # ˆ b# 2b‰# œ abc aa12 b b abc aa 4 9b b

abc aa# b# b 4

œ

IL ; RL œ É M œ Éa

2yÑÎ3

38. M œ 'c3 'c2 'c4Î3

Þ

D

dz dy dx œ '

#

œ

abc a4a# 28b# b 1#

7b# 3

% 4 3 y 2 2 ' ' (4 y) dy dx œ 4y ’ “ 3 3 2 3 2 3

3

#

#

#

dx œ 12 ' 3 dx œ 72; 3

x œ y œ z œ 0 from Exercise 22 Ê Ix œ IcÞmÞ 72 ŠÈ0# 0# ‹ œ IcÞmÞ Ê IL œ IcÞmÞ 72 ŠÉ16 ‰ œ 1488 œ 208 72 ˆ 160 9

16 9 ‹

#

913

914


15.7 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES 1.

'021 '01 'r

È2cr

#

dz r dr d) œ '0

21

œ '0 Š 2 3 23 ‹ d) œ 21

2.

È

'021 '03 'r Î318cr

4.

21

24r#

1

'03 ’r a18 r# b"Î# r3 “ dr d) œ '021 ’ "3 a18 r# b$Î# 12r “ $ d) $

%

!

$

’ 12)1#

21

21

)& 201% “ !

È

'01 '0 Î ' 3È44 rr ) 1

dz r dr d) œ '0

%

ˆ

#‰

'0 Î2 a3r 24r$ b dr d) œ '02 32 r# 6r% ‘ !Î2 )

"Î#

) Î1 !

1

1

)

1

d) œ

'02 Š 4)1 1

3 #

# #

'0 Î

) 1

œ 4 '0 Š 21)# 1

#

3 dz r dr d) œ 3 '0

21

" #

c9 a4 r# b a4 r# bd r dr d) œ 4 '0

)% 41 % ‹

1

d)

'0 Î a4r r$ b dr d) ) 1

371 15

d) œ

'01 ’r a2 r# b"Î# r# “ dr d) œ 3 '021 ’ a2 r# b"Î# r3 “ " d) $

!

21 œ 3 '0 ŠÈ2 43 ‹ d) œ 1 Š6È2 8‹

6.

'021 '01 'c11ÎÎ22 ar# sin# ) z# b dz r dr d) œ '021 '01 ˆr$ sin# ) 12r ‰ dr d) œ '021 Š sin4 ) 24" ‹ d) œ 13

7.

'021 '03 '0zÎ3

8.

'c11 '021 '01bcos

9.

'01 '0 z '021 ar# cos# ) z# b r d) dr dz œ '01 '0

#

r$ dr dz d) œ '0

21

)

21 z 3 '03 324 dz d) œ '0 20 d) œ 3101 %

4r dr d) dz œ 'c1 '0 2(1 cos ))# d) dz œ 'c1 61 d) œ 121 1

21

È 1

È

1

Èz

œ '0 ’ 14r 1r# z# “

10.

4) % 161% ‹

171 5

1

1

'021 '01 'r 2cr

œ

z dz r dr d) œ '0

#

#

œ 4 '0 ’2r# r4 “

5.

3

2 )

3 #

!

91 Š8È2 7‹

'021 '0 Î2 '03 œ

$

41 ŠÈ2 "‹

dz r dr d) œ '0

#

#

œ 3.

$Î#

'01 ’r a2 r# b"Î# r# “ dr d) œ '021 ’ "3 a2 r# b$Î# r3 “ " d)

%

Èz

!

1

#

$

È

#

$Î#

r# sin 2) 4

dz œ '0 Š 14z 1z$ ‹ dz œ ’ 112z

'02 'rc24cr '021 (r sin ) 1) r d) dz dr œ '02 'rc24cr œ 21 ’ "3 a4 r# b

#

’ r2)

r$ 3

#

#

z# )“

" 1 z% 4 “!

!

r dr dz œ '0

1

Èz

'0

a1r$ 21rz# b dr dz

1 3

œ

21r dz dr œ 21'0 ’r a4 r# b 2

r# “ œ 21 38 4 3" (4)$Î# ‘ œ 81 !

#1

"Î#

r# 2r“ dr

Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates È4cr

'021 '01 '0

11. (a)

È3

#

dz r dr d)

È4cz

'021 '0 '01 r dr dz d) '021 'È23 '0

(b)

È4cr

'01 '0

(c)

#

#

r dr dz d)

'021 r d) dz dr

'021 '01 'r 2cr dz r dr d) #

12. (a)

È2cz

(b)

'021 '01 '0z r dr dz d) '021 '12 '0

(c)

'0 'r '0

2cr#

1

21

r dr dz d)

r d) dz dr

13.

'c11ÎÎ22 '0cos '03r

14.

'c11ÎÎ22 '01 '0r cos

15.

'01 '02 sin '04cr sin

17.

'c1ÎÎ22 '11cos '04

19.

'0 Î4 '0sec '02

21.

'01 '01 '02 sin 9 3# sin 9 d3 d9 d) œ 83 '01 '01 sin% 9 d9 d) œ 83 '01 Š’ sin 94cos 9 “ 1 34 '01 sin# 9 d9‹ d)

)

)

#

)

1

)

Î

r$ dz dr d) œ ' 1Î2 '0 r% cos ) dr d) œ

)

1

f(rß )ß z) dz r dr d) 1 2

1

" 5

'

Î

1 2

1Î2

cos ) d) œ

2 5

f(rß )ß z) dz r dr d)

16.

' ÎÎ22 '03 cos '05


18.

' 1ÎÎ22 'cos2 cos '03

20.

' ÎÎ42 '0csc '02

)

r sin )


1

)

r cos )

1

1

)

r sin )

)

1

)

r sin )

1

23.

24.



$

!

1 1 1 1 1 œ 2 '0 '0 sin# 9 d9 d) œ '0 ) sin#2) ‘ ! d) œ '0 1 d) œ 1#

22.


'021 '01Î4 '02 (3 cos 9) 3# sin 9 d3 d9 d) œ '021 '01Î4 4 cos 9 sin 9 d9 d) œ '021 c2 sin# 9d 1! Î% d) œ '021 d) œ 21 '021 '01 '0Ð1 cos 9ÑÎ2 3# sin 9 d3 d9 d) œ 24" '021 '01 (1 cos 9)$ sin 9 d9 d) œ 96" '021 c(1 cos 9)% d 1! d) 21 " ' " 1 ' 21 œ 96 a2% 0b d) œ 16 96 0 d) œ 6 (21) œ 3 0 '031Î2 '01 '01 53$ sin$ 9 d3 d9 œ

5 6

d) œ

5 4

'031Î2 '01 sin$ 9 d9 d) œ 54 '031Î2 Š’ sin 93cos 9 “ 1 23 '01 sin 9 d9‹ d)

'031Î2 c cos 9d 1! d) œ 53 '031Î2 d) œ 5#1

#

!

915

916 25.


'021 '01Î3 'sec2 9 33# sin 9 d3 d9

d) œ '0

21

œ '0 (4 2) ˆ8 "# ‰‘ d) œ 21

5 #

'01Î3 a8 sec$ 9b sin 9 d9 d) œ '021 8 cos 9 "2 sec# 9‘ !1Î$ d)

'021 d) œ 51

26.

'021 '01Î4 '0sec 9 3$ sin 9 cos 9 d3 d9 d) œ "4 '021 '01Î4 tan 9 sec# 9 d9 d) œ "4 '021 "2 tan# 9‘ !1Î% d) œ "8 '021 d) œ 14

27.

2 0 '02 'c01 '11ÎÎ42 3$ sin 29 d9 d) d3 œ '02 ' 01 3$ cos229 ‘ 11Î# d) d3 œ '0 ' 1 3# Î%

28.

'11ÎÎ63 'csc2 csc9 9 '021 3# sin 9 d) d3 d9 œ 21 '11ÎÎ63 'csc2 csc9 9 3# sin 9 d3 d9 œ 231 '11ÎÎ63 c3$ sin 9d csc2 csc9 9 d9 œ 1431 '11ÎÎ63 csc# 9 d9 œ

29.

'01 '01 '01Î4 123 sin$ 9 d9 d) d3 œ '01 '01 Œ123 ’ sin 39 cos 9 “ 1Î% 83 '01Î4 sin 9 d9 d) d3

$

2 $ 3 1

#

%

#

d3 œ ’ 138 “ œ 21 !

#

!

œ '0

1

œ

30.

d) d3 œ '0

'0 Š È23

83 ccos 9d ! ‹ d) d3 œ '0

1

1

1Î%

2

'0 Š83 10È3 ‹ d) d3 œ 1'01 Š83 10È3 ‹ d3 œ 1 ’43# È53 1

#

2

“ 2

2

" !

Š4È2 5‹ 1 È2

'11ÎÎ62 ' 11ÎÎ22 'csc2 9 53% sin$ 9 d3 d) d9 œ '11ÎÎ62 ' 11ÎÎ22 a32 csc& 9b sin$ 9 d) d9 œ '11ÎÎ62 ' 11ÎÎ22 a32 sin$ 9 csc# 9b d) d9 œ 1 '1Î6 a32 sin$ 9 csc# 9b d9 œ 1 ’ 32 sin 39 cos 9 “ 1Î2

œ

#

È 1 Š 3224 3 ‹

641 3

ccos

1Î# 9d 1Î'

1 ŠÈ 3‹ œ

È3 3

1Î# 1Î'

1 ˆ 6431 ‰ Š

641 3

'11ÎÎ62 sin 9 d9 1 ccot 9d 11Î# Î'

È3 # ‹

œ

331È3 3

œ 111È3

31. (a) x# y# œ 1 Ê 3# sin# 9 œ 1, and 3 sin 9 œ 1 Ê 3 œ csc 9; thus

'021 '01Î6 '02 3# sin 9 d3 d9 d) '021 '11ÎÎ62 '0csc 9 3# sin 9 d3 d9 d) '021 '12 '1sinÎ6

(b)

3# sin 9 d9 d3 d) '0

21

'02 '01Î6 3# sin 9 d9 d3 d)

'021 '01Î4 '0sec 9 3# sin 9 d3 d9 d) '021 '01 '01Î4 3# sin 9 d9 d3 d)

32. (a) (b)

'0

21

33. V œ '0

21

œ

" Ð1Î3Ñ

" 3

È

'1 2 'cos1Î4" Ð"Î3Ñ 3# sin 9 d9 d3 d)

'01Î2 'cos2 9 3# sin 9 d3 d9 d) œ "3 '021 '01Î2 a8 cos$ 9b sin 9 d9 d)

'021 ’8 cos 9 cos4 9 “ 1Î# d) œ 3" '021 ˆ8 4" ‰ d) œ ˆ 3112 ‰ (21) œ 3161 %

!

'021 '01Î2 a3 cos 9 3 cos# 9 cos$ 9b sin 9 d9 d) 21 21 1Î# 111 ' 21 ˆ 11 ‰ œ 3" '0 3# cos# 9 cos$ 9 14 cos% 9‘ ! d) œ 3" '0 ˆ 32 1 "4 ‰ d) œ 11 12 0 d) œ 12 (21) œ 6

34. V œ '0

'01Î2 '11

35. V œ '0

1 9) '01 '01ccos 9 3# sin 9 d3 d9 d) œ "3 '021 '01 (1 cos 9)$ sin 9 d9 d) œ 3" '021 ’ (" cos “ d) 4

21

21

œ

" 12

(2)

cos 9

3# sin 9 d3 d9 d) œ

" 3

%

%

'021 d) œ 34 (21) œ 831

!

281 3È 3

Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 36. V œ '0

21

œ

" 12

21

'11ÎÎ42 '02 cos 9

38. V œ '0

21

1Î2

(c) 8 '0

2

%

!

3# sin 9 d3 d9 d) œ

8 3

'021 '11ÎÎ42 cos$ 9 sin 9 d9 d) œ 83 '021 ’ cos4 9 “ 1Î# d) %

1Î%

'01Î2 '02 3# sin 9 d3 d9 d) È4cx

'0

#

È4cx cy

'0

È

'01Î2 '03Î 2 'r

#

È9

r#

(b) 8'0

1Î2

È4cr

'02 '0

41. (a) V œ '0

21

È3cx

dz r dr d)

È4cx cy

(c) V œ 'cÈ3 'cÈ3cx# '1 (d) V œ '0

21

œ

È3

'0

(b)

'01Î2 '01Î4 '03 3# sin 9 d3 d9 d)

#

#

’r a4 r# b

(b) V œ '0

21

È

È

1‹ d) œ

2

È4cr

'0 3 '1

#

91 Š2 È2‹ 4

dz r dr d)

#

"Î#

dz dy dx r“ dr d) œ '0 ” a4 3r b 21

# $Î#

#

r# •

'021 d) œ 531

5 6

dz r dr d)

dz dy dx

'01Î3 'sec2 9 3# sin 9 d3 d9 d)

È3

#

#

'01Î2 '01Î4 '03 3# sin 9 d3 d9 d) œ 9 '01Î2 '01Î4 sin 9 d9 d) œ 9 '01Î2 Š È"

(c)

È$ !

d) œ '0 Š 3" 21

3 #

4$Î# 3 ‹

d)

'01 '0 1cr r# dz r dr d) 1Î2 21 1 Iz œ '0 '0 '0 a3# sin# 9b a3# sin 9b d3 d9 d), since r# œ x# y# œ 3# sin# 9 cos# ) 3# sin# 9 sin# ) œ 3# sin# 9

42. (a) Iz œ '0

21

(c) Iz œ '0 œ

2 15

#

!

(21) œ

41 15

43. V œ 4 '0

'01 'r 4 14r

44. V œ 4'0

'01 ' 1Èr1

1Î2

œ 4 '0

ˆ "#

45. V œ '31Î2 '0 21

9 4

#

%

1Î2

1Î2

#

'01Î2 "5 sin$ 9 d9 d) œ "5 '021 Œ’ sin 93cos 9 “ 1Î# 32 '01Î2 sin 9 d9 d) œ 152 '021 c cos 9d !1Î# d)

21

œ

1Î# 9) '021 '01Î2 (1 cos 9)$ sin 9 d9 d) œ 3" '021 ’ (" cos d) “ 4

21 4 ' 81 '11ÎÎ32 '02 3# sin 9 d3 d9 d) œ 83 '021 '11ÎÎ32 sin 9 d9 d) œ 83 '021 c cos 9d 11Î# Î$ d) œ 3 0 d) œ 3

39. (a) 8'0

(b)

" 3

3# sin 9 d3 d9 d) œ

'021 d) œ "6 (21) œ 13

" ‰ ˆ 83 ‰ ˆ 16

40. (a)

cos 9

'021 d) œ 12" (21) œ 16

37. V œ '0 œ

'01Î2 '01

917

" 3

3 cos )

0œ

'01 a5r 4r$ r& b dr d) œ 4 '01Î2 ˆ 5# 1 "6 ‰ d) œ 4 '01Î2 d) œ 831

1 Î2

r#

dz r dr d) œ 4 '0

"‰ 3

'0cr sin

)

d) œ 2'0 d) œ 1Î2

'01 Šr r# rÈ1r# ‹ dr d) œ 4 '01Î2 ’ r2

#

2 ˆ 1# ‰

dz r dr d) œ '31Î2 '0 21

3 cos )

c3 cos )

œ 18 Œ’ cos

#

'0r dz r dr d) œ 2 '1Î2 '0c3 cos

1 ) sin ) “ 3 1Î#

1

2 3

r$ 3

"3 a1 r# b

$Î# "

“ d) !

œ1 r# sin ) dr d) œ '31Î2 a9 cos$ )b (sin )) d) œ 94 cos% )‘ $1Î# 21

9 4

46. V œ 2 '1Î2 '0 1

1Î2

dz r dr d) œ 4 '0

)

r# dr d) œ

2 3

'1Î2 27 cos$ ) d) 1

'11Î2 cos ) d) œ 12 csin )d 11Î# œ 12

#1

918


47. V œ '0

1Î2

È1

'0sin '0 )

r#

1 2

Î

rÈ1r# dr d) œ '0 ’ "3 a1 r# b

)

1 2

$Î# sin )

“

!

d)

#

!

œ csin 2 9

48. V œ '0

1Î2

1Î# )d !

1 6

'0cos '03 )

È1

œ

1Î#

Î

1 2

1Î2

32 ccos )d !

4 3 1 18

dz r dr d) œ '0

r#

œ '0 ’ a1 cos# )b 1 #

'0sin

'01Î2 ’a1 sin# )b$Î# 1“ d) œ "3 '01Î2 acos$ ) 1b d) œ "3 Œ’ cos )3 sin ) “ 1Î# 32 '01Î2 cos ) d) 3) ‘ 1! Î#

œ "3

œ

Î

dz r dr d) œ '0

$Î#

1 #

œ

'0cos

)

Î

3rÈ1r# dr d) œ '0 ’ a1 r# b 1 2

1“ d) œ '0 a1 sin$ )b d) œ ’)

1Î2

2 3

œ

'12Î13Î3 '0a 3# sin 9 d3 d9 d) œ '021 '12Î13Î3

50. V œ '0

'01Î2 '0a 3# sin 9 d3 d9 d) œ a3 '01Î6 '01Î2

51. V œ '0

'01Î3 'sec2 9

1Î6

21

“

!

d)

'01Î2 sin ) d)

2 3

31 4 6

49. V œ '0

21

1Î# sin# ) cos ) “ 3 !

$Î# cos )

$

a$ 3

sin 9 d9 d) œ

'021 c cos 9d #11Î$Î$ d) œ a3 '021 ˆ "# "# ‰ d) œ 213a

a$ 3

sin 9 d9 d) œ

a$ 3

$

'01Î6 d) œ a181 $

3# sin 9 d3 d9 d)

'021 '01Î3 a8 sin 9 tan 9 sec# 9b d9 d) 21 1Î$ œ "3 '0 8 cos 9 "2 tan# 9‘ ! d) 21 21 œ "3 '0 4 #" (3) 8‘ d) œ 3" '0 #5 d) œ 65 (21) œ 531 œ

" 3

52. V œ 4 '0

1Î2

œ

28 3

'01Î4 'sec2 sec9 9 3# sin 9 d3 d9 d)

œ

4 3

'01Î2 '01Î4 a8 sec$ 9 sec$ 9b sin 9 d9 d)

'01Î2 '01Î4 sec$ 9 sin 9 d9 d) œ 283 '01Î2 '01Î4 tan 9 sec# 9 d9 d) œ 283 '01Î2 2" tan# 9‘ !1Î% d) œ 143 '01Î2 d) œ 731

53. V œ 4 '0

'01 '0r

#

54. V œ 4 '0

'01 'r r

#

55. V œ 8 '0

'1 2 '0r

56. V œ 8 '0

'1 2 '0

1Î2

1Î2

1Î2

1Î2

dz r dr d) œ 4 '0

1Î2

1

#

È

1Î2

dz r dr d) œ 4 '0

dz r dr d) œ 8 '0

1Î2

È2

È

r#

58. V œ '0

'02 '04cr cos cr sin

21

È2

'1

1Î2

'02 '04cr sin

)

'01 r dr d) œ 2 '01Î2 d) œ 1

dz r dr d) œ 8 '0

57. V œ '0

21

'01 r$ dr d) œ '01Î2 d) œ 1#

dz r dr d) œ '0

21

)

)

r# dr d) œ 8 Š 2

È2

'1

È2" ‹ 3

È

'01Î2 d) œ 41 Š2 3 2"‹ 1Î2

rÈ2 r# dr d) œ 8 '0 ’ "3 a2 r# b

$Î#

1Î2

'01 '4r5

#

r#

È# 1

d) œ

8 3

'01Î2 d) œ 431

'02 a4r r# sin )b dr d) œ 8 '02 ˆ1 sin3 ) ‰ d) œ 161 1

dz r dr d) œ '0

21

'02 c4r r# (cos ) sin ))d dr d) œ 83 '02

1

(3 cos ) sin )) d) œ 161

59. The paraboloids intersect when 4x# 4y# œ 5 x# y# Ê x# y# œ 1 and z œ 4 Ê V œ 4 '0

“

1Î2

dz r dr d) œ 4 '0

'01 a5r 5r$ b dr d) œ 20 '01Î2 ’ r2

#

1Î2

r4 “ d) œ 5'0 %

" !

d) œ

51 #

$

Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 60. The paraboloid intersects the xy-plane when 9 x# y# œ 0 Ê x# y# œ 9 Ê V œ 4 '0

1Î2

œ 4 '0

1Î2

'1

3

61. V œ 8 '0

21

'0

#

$

%

1Î2

17 ‰ 4

"

È4cr

'01 '0

21

œ 83

a9r r$ b dr d) œ 4 '0 ’ 9r2 r4 “ d) œ 4 '0 ˆ 81 4 1Î2

#

dz r dr d) œ 8 '0

21

ˆ3$Î# 8‰ d) œ

'13 '09cr

#

919

dz r dr d)

d) œ 64 '0 d) œ 321 1Î2

'01 r a4 r# b"Î# dr d) œ 8 '021 ’ "3 a4 r# b$Î# “ " d) !

41 Š8 3È3‹ 3

62. The sphere and paraboloid intersect when x# y# z# œ 2 and z œ x# y# Ê z# z 2 œ 0 Ê (z 2)(z 1) œ 0 Ê z œ 1 or z œ 2 Ê z œ 1 since z 0. Thus, x# y# œ 1 and the volume is given by the triple integral V œ 4 '0

1Î2

œ 4 '0 ’ "3 a2 r# b 1Î2

63. average œ

" #1

œ

È2

r#

#

1Î2

dz r dr d) œ 4 '0

r4 “ d) œ 4 '0 Š 2 3 2

$Î#

%

"

1Î2

È

!

'021 '01 'c11

" #1

r# dz dr d) œ

È

'021 '01 'cÈ11ccrr

" ˆ 431 ‰

64. average œ

'01 'r

# #

7 12 ‹

'01 ’r a2 r# b"Î# r$ “ dr d)

1 Š8 È 2 7 ‹

d) œ

6

'021 '01 2r# dr d) œ 3"1 '021 d) œ 23

r# dz dr d) œ

3 41

'021 '01 2r# È1 r# dr d)

'021 ’ "8 sin" r "8 rÈ1 r# a1 2r# b“ " d) œ 1631 '021 ˆ 1# 0‰ d) œ 323 '021 d) œ ˆ 323 ‰ (21) œ 3161

3 21

!

65. average œ

" ˆ 431 ‰

'021 '01 '01 3$ sin 9 d3 d9 d) œ 1631 '021 '01 sin 9 d9 d) œ 831 '021 d) œ 43

66. average œ

" ˆ 231 ‰

'021 '01Î2 '01 3$ cos 9 sin 9 d3 d9 d) œ 831 '021 '01Î2

œ

3 161

cos 9 sin 9 d9 d) œ

3 81

'021 ’ sin2 9 “ 1Î# d) #

!

'021 d) œ ˆ 1631 ‰ (21) œ 38

67. M œ 4 '0

'01 '0r dz r dr d) œ 4 '01Î2 '01 r# dr d) œ 43 '01Î2 d) œ 231 ; Mxy œ '021 '01 '0r z dz r dr d) 21 1 21 œ "# '0 '0 r$ dr d) œ 18 '0 d) œ 14 Ê z œ MM œ ˆ 14 ‰ ˆ 231 ‰ œ 38 , and x œ y œ 0, by symmetry 1Î2

xy

68. M œ '0

'02 '0r dz r dr d) œ '01Î2 '02 r# dr d) œ 83 '01Î2 d) œ 431 ; Myz œ '01Î2 '02 '0r x dz r dr d) 2 2 r 2 1Î2 1Î2 1Î2 1Î2 œ '0 '0 r$ cos ) dr d) œ 4 '0 cos ) d) œ 4; Mxz œ '0 '0 '0 y dz r dr d) œ '0 '0 r$ sin ) dr d) 2 r 2 1Î2 1Î2 1Î2 1Î2 M œ 4 '0 sin ) d) œ 4; Mxy œ '0 '0 '0 z dz r dr d) œ "# '0 '0 r$ dr d) œ 2 '0 d) œ 1 Ê x œ M 1Î2

yz

yœ 69. M œ

œ

Mxz M

, and z œ

3 1

; Mxy œ '0

21

81 3

œ 4 '0 ’ sin2 9 “ 21

70. M œ '0

#

1Î# 1Î$

Mxy M

œ

3 1

,

3 4

'11ÎÎ32 '02 z3# sin 9 d3 d9 d) œ '021 '11ÎÎ32 '02 3$ cos 9 sin 9 d3 d9 d) œ 4 '021 '11ÎÎ32

d) œ 4 '0 ˆ "# 38 ‰ d) œ 21

" #

'021 d) œ 1

Ê zœ

Mxy M

œ (1) ˆ 831 ‰ œ

3 8

$

$

$

%

%

cos 9 sin 9 d9 d)

, and x œ y œ 0, by symmetry È

'01Î4 '0a 3# sin 9 d3 d9 d) œ a3 '021 '01Î4 sin 9 d9 d) œ a3 '021 2 #È2 d) œ 1a Š23 2‹ ; 1Î4 1Î4 21 a 21 21 a ' Mxy œ '0 '0 '0 3$ sin 9 cos 9 d3 d9 d) œ a4 '0 '0 sin 9 cos 9 d9 d) œ 16 d) œ 18a 0 21

œ

%

920

Chapter 15 Multiple Integrals È2

‰ 2 œ Š 18a ‹ – $ 3 È — œ ˆ 3a 8 Š # 1a Š2 2‹ %

Mxy M

Ê zœ

‹œ

3 Š2È2‹ a 16

, and x œ y œ 0, by symmetry

È

71. M œ '0

È

'04 '0 r dz r dr d) œ '021 '04 r$Î# dr d) œ 645 '021 d) œ 1285 1 ; Mxy œ '021 '04 '0 r z dz r dr d) 21 4 '021 d) œ 6431 Ê z œ MM œ 65 , and x œ y œ 0, by symmetry œ "# '0 '0 r# dr d) œ 32 3 21

xy

1Î3

È

' È11 rr dz r dr d) œ ' 11ÎÎ33 '01 2rÈ1 r# dr d) œ ' 11ÎÎ33 ’ 23 a1 r# b$Î# “ " d) ! È1 r 1 1 1Î3 1Î3 1 Î3 2 ' 2 2 1 4 1 # œ 3 c1Î3 d) œ ˆ 3 ‰ ˆ 3 ‰ œ 9 ; Myz œ ' 1Î3 '0 ' È1 r r cos ) dz dr d) œ 2 ' 1Î3 '0 r# È1 r# cos ) dr d)

72. M œ 'c1Î3 '0

1

#

#

#

#

1 Î3

œ 2 'c1Î3 ’ 18 sin" r "8 rÈ1 r# a1 2r# b“ cos ) d) œ "

Myz M

Ê xœ

œ

!

9È 3 32

1 8

' 11ÎÎ33 cos ) d) œ 18 csin )d 1Î13Î3 œ ˆ 18 ‰ Š2 † È#3 ‹ œ 1È8 3

, and y œ z œ 0, by symmetry

73. We orient the cone with its vertex at the origin and axis along the z-axis Ê 9 œ which is through the vertex and parallel to the base of the cone Ê Ix œ '0

21

œ '0

%

74. Iz œ '0

21

'0a

Èa cr #

#

'

cÈa cr #

#

r$ dz dr d) œ '0

21

#

'0a 2r$ Èa# r# dr d) œ 2 '021 ’Š r5

#

2a# # 15 ‹ aa

“ d) œ 2 '0

21

$Î# a

r# b

!

2 15

a& d)

81 a& 15

75. Iz œ '0

21

'0a ' h r ax# y# b dz r dr d) œ '021 '0a ˆh‰ a

œ '0 h Š a4 21

%

a& 5a ‹

76. (a) M œ '0

'01 '0r

21

d) œ #

ha% 20

'021 d) œ 110ha

z dz r dr d) œ '0

21

21

'01 '0r

#

r# dz dr d) œ '0

21

" 2

77. (a) M œ '0

hr a

Šhr$

hr% a ‹

dr d) œ '0

21

#

Ê zœ

" #

%

h ’ r4

a

r& 5a “ !

d)

Ê zœ

5 14

z# dz r dr d)

, and x œ y œ 0, by symmetry; Iz œ '0

21

'01 r% dr d) œ 5" '021 d) œ 215 ; Mxy œ '021 '01 '0r

'021 '01 r' dr d) œ 14" '021 d) œ 17 21 1 21 œ '0 '0 r' dr d) œ 7" '0 d) œ 21 7 œ

hr a

' h r$ dz dr d) œ '021 '0a

'01 "# r& dr d) œ 12" '021 d) œ 16 ; Mxy œ '021 '01 '0r

" 3

(b) M œ '0

h

'

%

'021 '01 r( dr d) œ 24" '021 d) œ 121 21 1 21 " ' œ "# '0 '0 r( dr d) œ 16 d) œ 18 0

œ

#

'01 '0r

#

zr$ dz dr d)

zr# dz dr d)

, and x œ y œ 0, by symmetry; Iz œ '0

21

'01 '0r

#

r% dz dr d)

'01 'r 1 z dz r dr d) œ "# '021 '01 ar r$ b dr d) œ 8" '021 d) œ 14 ; Mxy œ '021 '01 'r 1 z# dz r dr d) 21 1 21 21 1 1 " ' œ 3" '0 '0 ar r% b dr d) œ 10 d) œ 15 Ê z œ 45 , and x œ y œ 0, by symmetry; Iz œ '0 '0 'r zr$ dz dr d) 0 21 1 21 1 " ' œ "# '0 '0 ar$ r& b dr d) œ 24 d) œ 12 0 21 1 1 21 1 1 21 1 M œ '0 '0 'r z# dz r dr d) œ 15 from part (a); Mxy œ '0 '0 'r z$ dz r dr d) œ 4" '0 '0 ar r& b dr d) 21 21 1 1 21 1 " ' œ 12 d) œ 16 Ê z œ 65 , and x œ y œ 0, by symmetry; Iz œ '0 '0 'r z# r$ dz dr d) œ "3 '0 '0 ar$ r' b dr d) 0 21 " ' 1 œ 28 d) œ 14 0 21

(b)

. We use the the x-axis

1

'01 Šr$ sin# ) r% sin# ) 3r r3 ‹ dr d) œ '021 Š sin20 ) 10" ‹ d) œ 40) sin802) 10) ‘ #!1 œ #10 15 œ 14

21

œ

1 4

'0 'r ar# sin# ) z# b dz r dr d) 1

Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 78. (a) M œ '0

'01 '0a 3% sin 9 d3 d9 d) œ a5 '021 '01 sin 9 d9 d) œ 2a5 '021 d) œ 415a ; Iz œ '021 '01 '0a '021 '01 a1 cos# 9b sin 9 d9 d) œ a7 '021 ’ cos 9 cos3 9 “ 1 d) œ 4a#1 '021 d) œ 8a211 21

a( 7

œ (b)

&

&

&

$

(

(

3' sin$ 9 d3 d9 d)

(

!

21 a 21 21 1 1 29 ) M œ '0 '0 '0 3$ sin# 9 d3 d9 d) œ a4 '0 '0 (1cos d9 d) œ 18a '0 d) œ 1 4a ; # 1 1 21 a 21 Iz œ '0 '0 '0 3& sin% 9 d3 d9 d) œ a6 '0 '0 sin% 9 d9 d) 1 21 21 21 1 1 œ a6 '0 Š’ sin 49 cos 9 “ 43 '0 sin# 9 d9‹ d) œ a8 '0 9# sin429 ‘ ! d) œ 116a '0 d) œ a 81 %

%

# %

'

$

'

'

'

' #

!

79. M œ '0

21

'0a '0

h a

Èa cr #

#

dz r dr d) œ '0

21

'0a

h a

rÈa# r# dr d) œ

Èa cr

'021 ’ 3" aa# r# b$Î# “ a d)

h a

!

h ' ' '0 3 d) œ 2ha3 1 ; Mxy œ '0 '0 '0 z dz r dr d) œ 2a aa# r r$ b dr d) 0 0 21 h ' œ 2a Š a# a4 ‹ d) œ a h4 1 Ê z œ Š 1a4h ‹ ˆ 2ha3 1 ‰ œ 83 h, and x œ y œ 0, by symmetry 0

œ

21 $ a

h a

21

#

h a

a

#

#

21

#

a

#

#

%

%

# #

# #

#

#

80. Let the base radius of the cone be a and the height h, and place the cone's axis of symmetry along the z-axis with the vertex at the origin. Then M œ œ

h# #

'021 ’ r2

#

a r% 4a# “ !

d) œ

h# #

'021 Š a#

#

1a# h 3

and Mxy œ '0

a# 4‹

d) œ

21

h # a# 8

'0a ' h r z dz r dr d) œ "# '021 '0a Šh# r ha

#

#

ˆh‰

r$ ‹ dr d)

a

'021 d) œ h a4 1 # #

Ê zœ

Mxy M

# #

œ Š h a4 1 ‹ ˆ 1a3# h ‰ œ

3 4

h, and

x œ y œ 0, by symmetry Ê the centroid is one fourth of the way from the base to the vertex 81. The density distribution function is linear so it has the form $ (3) œ k3 C, where 3 is the distance from the center of the planet. Now, $ (R) œ 0 Ê kR C œ 0, and $ (3) œ k3 kR. It remains to determine the constant k: M œ '0

21

œ' Ê

21

'

'01 '0R (k3 kR) 3# sin 9 d3 d9 d) œ '021 '01 ’k 34

%

'

1

$

R

kR 33 “ sin 9 d9 d)

21

% % k Š R4 R3 ‹ sin 9 d9 d) œ 0 1k# R% c cos 9d 1! d) œ 0 0 $ (3) œ 13M 3 13M R . At the center of the planet 3 œ 0 Ê R% R%

'0

21

!

%

6k R% d) œ k13R Ê k œ 13M R%

‰R œ $ (0) œ ˆ 13M R%

3M 1R$

.

82. The mass of the plant's atmosphere to an altitude h above the surface of the planet is the triple integral M(h) œ '0

'01 'Rh .! ecÐ3RÑ 3# sin 9 d3 d9 d) œ 'Rh '021 '01 .! ecÐ3RÑ 3# sin 9 d9 d) d3 h 21 h 21 h 1 œ 'R '0 .! ecÐ3RÑ 3# ( cos 9)‘ ! d) d3 œ 2 'R '0 .! ecR ec3 3# d) d3 œ 41.! ecR 'R ec3 3# d3 21

#

œ 41.! ecR ’ 3 ec #

c3

œ 41.! ecR Š h ec

ch

23e c3 c# 2he ch c#

h 2e c3 c$ “ R 2e ch c$

(by parts) R# e c

cR

2Re cR c#

2e cR c$ ‹ .

The mass of the planet's atmosphere is therefore M œ lim

hÄ_

#

M(h) œ 41.! Š Rc

2R c#

2 c$ ‹ .

83. (a) A plane perpendicular to the x-axis has the form x œ a in rectangular coordinates Ê r cos ) œ a Ê r œ Ê r œ a sec ), in cylindrical coordinates. (b) A plane perpendicular to the y-axis has the form y œ b in rectangular coordinates Ê r sin ) œ b Ê r œ Ê r œ b csc ), in cylindrical coordinates. 84. ax by œ c Ê aar cos )b bar sin )b œ c Ê raa cos ) b sin )b œ c Ê r œ

c a cos ) b sin ) .

921

a cos ) b sin )

922


85. The equation r œ fazb implies that the point ar, ), zb œ afazb, ), zb will lie on the surface for all ). In particular afazb, ) 1, zb lies on the surface whenever afazb, ), zb does Ê the surface is symmetric with respect to the z-axis.

86. The equation 3 œ fa9b implies that the point a3, 9, )b œ afa9b, 9, )b lies on the surface for all ). In particular, if afa9b, 9, )b lies on the surface, then afa9b, 9, ) 1b lies on the surface, so the surface is symmetric wiith respect to the z-axis. 15.8 SUBSTITUTIONS IN MULTIPLE INTEGRALS 1. (a) x y œ u and 2x y œ v Ê 3x œ u v and y œ x u Ê x œ ` (xßy) ` (ußv)

œ

" 3 » 2 3

" 3 " 3

»œ

" 9

2 9

œ

" 3

(u v) and y œ

" 3

(2u v);

" 3

(u v) and x œ

" 3

(u 2v);

" 3

(b) The line segment y œ x from (!ß 0) to (1ß 1) is x y œ 0 Ê u œ 0; the line segment y œ 2x from (0ß 0) to (1ß 2) is 2x y œ 0 Ê v œ 0; the line segment x œ 1 from (1ß 1) to ("ß 2) is (x y) (2x y) œ 3 Ê u v œ 3. The transformed region is sketched at the right. 2. (a) x 2y œ u and x y œ v Ê 3y œ u v and x œ v y Ê y œ ` (xßy) ` (ußv)

œ»

" 3 1 3

2 3 "3

" »œ9

2 9

œ 3"

(b) The triangular region in the xy-plane has vertices (0ß 0), (2ß 0), and ˆ 23 ß 23 ‰ . The line segment y œ x from (0ß 0) to ˆ 23 ß 23 ‰ is x y œ 0 Ê v œ 0; the line segment y œ 0 from (0ß 0) to (#ß 0) Ê u œ v; the line segment x 2y œ 2 from ˆ 23 ß 23 ‰ to (2ß 0) Ê u œ 2. The transformed region is sketched at the right. 3. (a) 3x 2y œ u and x 4y œ v Ê 5x œ 2u v and y œ ` (xßy) ` (ußv)

œ»

2 5 1 10

15 3 10

»œ

6 50

1 50

œ

" 10

(b) The x-axis y œ 0 Ê u œ 3v; the y-axis x œ 0 Ê v œ 2u; the line x y œ 1 " Ê "5 (2u v) 10 (3v u) œ 1 Ê 2(2u v) (3v u) œ 10 Ê 3u v œ 10. The transformed region is sketched at the right.

" #

(u 3x) Ê x œ

" 5

(2u v) and y œ

" 10

(3v u);

Section 15.8 Substitutions in Multiple Integrals 4. (a) 2x 3y œ u and x y œ v Ê x œ u 3v and y œ v x Ê x œ u 3v and y œ u 2v; " 3 ` (xßy) ` (ußv) œ º 1 2 º œ 2 3 œ 1 (b) The line x œ 3 Ê u 3v œ 3 or u 3v œ 3; x œ 0 Ê u 3v œ 0; y œ x Ê v œ 0; y œ x 1 Ê v œ 1. The transformed region is the parallelogram sketched at the right.

5.

'04 'yÐÎy2Î2Ñ " #

œ 6.

1

ˆx y# ‰ dx dy œ ' ’ x2 0 4

#

y

xy 2 # “y

"

dy œ

2

" #

'04 ’ˆ y# 1‰# ˆ y# ‰# ˆ y# 1‰ y ˆ y# ‰ y“ dy

'04 (y 1 y) dy œ "# '04 dy œ "# (4) œ 2

' ' a2x# xy y# b dx dy œ ' ' (x y)(2x y) dx dy R

R

ßy) " '' œ ' ' uv ¹ `` (x uv du dv; (ußv) ¹ du dv œ 3

G

G

We find the boundaries of G from the boundaries of R, shown in the accompanying figure: xy-equations for

Corresponding uv-equations

Simplified

for the boundary of G

uv-equations

the boundary of R y œ 2x 4

" 3

(2u v) œ (u v) 4

vœ4

y œ 2x 7

" 3

(2u v) œ 32 (u v) 7

vœ7

yœx2

" 3

yœx1

" 3

Ê

7.

" 3

2 3

(2u v) œ

1 3

(u v) 2

uœ2

(2u v) œ

1 3

(u v) 1

u œ 1

' ' uv du dv œ "3 ' ' uv dv du œ "3 ' u ’ v2# “ du œ c1 4 c1 2

7

2

( %

G

11 #

'c21 u du œ ˆ 11# ‰ ’ u2 “ #

' ' a3x# 14xy 8y# b dx dy R

œ ' ' (3x 2y)(x 4y) dx dy R

ßy) œ ' ' uv ¹ `` (x (ußv) ¹ du dv œ

G

" 10

' ' uv du dv; G

We find the boundaries of G from the boundaries of R, shown in the accompanying figure: xy-equations for the boundary of R


Simplified


uv-equations

3 #

yœ x1

" 10

(3v u) œ

(2u v) 1

uœ2

y œ 3# x 3

" 10

3 (3v u) œ 10 (2u v) 3

uœ6

y œ 4" x

" 10

1 (3v u) œ 20 (2u v)

vœ0

" 10

(3v u) œ

vœ4

" 4

yœ x1

3 10

1 20

(2u v) 1

#

"

‰ œ ˆ 11 4 (4 1) œ

33 4

923

924

Chapter 15 Multiple Integrals " 10

Ê

8.

' ' uv du dv œ

" 10

G

'26 '04 uv dv du œ 10" '26 u ’ v2 “ % du œ 45 '26 u du œ ˆ 54 ‰ ’ u2 “ ' œ ˆ 54 ‰ (18 2) œ 645 #

#

!

#

' ' 2(x y) dx dy œ ' ' 2v ¹ `` (x(ußßy)v) ¹ du dv œ ' ' 2v du dv; the region G is sketched in Exercise 4 R

G

G

3c3v

" ' ' 2v du dv œ ' ' 2v du dv œ '0 2v(3 3v 3v) dv œ '0 6v dv œ c3v# d ! œ 3 0 c3v 1

Ê

1

1

G

9. x œ

v" uv# œ v" u v" u œ 2u v ; v u º Ê v œ 1, and y œ 4x Ê v œ 2; xy œ 1 Ê u œ 1, and xy œ 9 Ê u œ 3; thus

and y œ uv Ê

u v

y œ x Ê uv œ

u v

œ v# and xy œ u# ;

y x

` (xßy) ` (ußv)

œ J(uß v) œ º

' ' ŠÉ yx Èxy‹ dx dy œ ' ' (v u) ˆ 2uv ‰ dv du œ ' ' Š2u 2uv # ‹ dv du œ ' c2uv 2u# ln vd #" du 1 1 1 1 1 3

2

3

2

3

R

œ '1 a2u 2u# ln 2b du œ u# 23 u# ln 2‘ " œ 8 23 (26)(ln 2) œ 8 3

$

` (xßy) ` (ußv)

10. (a)

œ J(uß v) œ º

52 3

(ln 2)

" 0 œ u, and v uº

the region G is sketched at the right

(b) x œ 1 Ê u œ 1, and x œ 2 Ê u œ 2; y œ 1 Ê uv œ 1 Ê v œ "u , and y œ 2 Ê uv œ 2 Ê v œ

'1 '1 2

œ

2

3 #

y x

dy dx œ '1

2

'1Îu ˆ uvu ‰ u dv du œ '1 '1Îu uv dv du œ '1 2Îu

R

` (xßy) ` (ußv)

u ’ v2 “

2Îu 1Îu

du œ '1 u ˆ u2# 2

" ‰ 2u#

; thus,

du

2

1

21

12.

#

#

I! œ ' ' ax# y# b dA œ '0 œ

2

'12 u ˆ u" ‰ du œ 3# cln ud #" œ 3# ln 2; '12 '12 yx dy dx œ '12 ’ x1 † y2 “ 2 dx œ 3# '12 dxx œ 3# cln xd #" œ 3# ln 2

11. x œ ar cos ) and y œ ar sin ) Ê

ab 4

2Îu

2

2 u

` (xßy) ` (rß))

œ J(rß )) œ º

#

#

sin 2) 4

b# ) 2

21

b# sin 2) “ 4 !

œ

ab1 aa# b# b 4

È1cu# 1 a 0 œ ab; A œ ' ' dy dx œ ' ' ab du dv œ 'c1 'cÈ1cu# ab dv du º 0 b R G

œ 2ab 'c1 È1u# du œ 2ab ’ u2 È1 u# 1

ar sin ) œ abr cos# ) abr sin# ) œ abr; br cos ) º

'01 r# aa# cos# ) b# sin# )b kJ(rß ))k dr d) œ '021 '01 abr$ aa# cos# ) b# sin# )b dr d)

'021 aa# cos# ) b# sin# )b d) œ ab4 ’ a2) a œ J(uß v) œ º

a cos ) b sin )

" #

sin" u“

"

"

œ ab csin" 1 sin" (1)d œ ab 1# ˆ 1# ‰‘ œ ab1

Section 15.8 Substitutions in Multiple Integrals 13. The region of integration R in the xy-plane is sketched in the figure at the right. The boundaries of the image G are obtained as follows, with G sketched at the right:

xy-equations for


Simplified


uv-equations

the boundary of R xœy

" 3

(u 2v) œ

x œ 2 2y

" 3

(u 2v) œ 2 32 (u v)

yœ0

0œ

Also, from Exercise 2,

` (xßy) ` (ußv)

1 3

(u v)

vœ0 uœ2

(u v)

1 3

vœu

œ J(uß v) œ "3 Ê

'02Î3 'y2

(x 2y) eÐy xÑ dx dy œ '0

2y

2

œ

" 3

'02 u cecv d !u du œ 3" '02 u a1 ecu b du œ 3" ’u au ecu b u#

œ

" 3

a3ec2 1b ¸ 0.4687

#

14. x œ u ` (xßy) ` (ußv)

v #

#

ecu “ œ !

" 3

'0u ue

v

¸ 3" ¸ dv du

c2 a2 ec2 b 2 ec2 1d

and y œ v Ê 2x y œ (2u v) v œ 2u and

" "# v º œ 1; next, u œ x # 0 " and v œ y, so the boundaries of the region of

œ J(uß v) œ º

œx

y #

integration R in the xy-plane are transformed to the boundaries of G: xy-equations for


Simplified


uv-equations

œ

uœ0

the boundary of R xœ xœ

u

y # y #

2

u

v # v #

œ

v # v #

2

uœ2

yœ0

vœ0

vœ0

yœ2

vœ2

vœ2

Ê '0

2

œ

" 4

15. x œ

'yÐÎy2Î2Ñ

ae 16

u v

2

y$ (2x y) eÐ2xyÑ dx dy œ '0 #

% # 1b ’ v4 “ !

u v

'02 v$ (2u) e4u

#

du dv œ '0 v$ ’ "4 e4u “ dv œ 2

#

# !

" 4

'02 v$ ae16 1b dv

œe 1 16

v" uv# œ v" u v" u œ 2u v ; v u º Ê v œ 1, and y œ 4x Ê v œ 2; xy œ 1 Ê u œ 1, and xy œ 4 Ê u œ 2; thus

and y œ uv Ê

y œ x Ê uv œ

2

y x

œ v# and xy œ u# ;

` (xßy) ` (ußv)

œ J(uß v) œ º

'12 '1yÎyax2 y2 b dx dy '24 'y4ÎÎ4yax2 y2 b dx dy œ '12 '12 Š uv

2 2

' ‰ u2 v2 ‹ ˆ 2u v du dv œ 1

2

'12 Š 2uv

3

3

2u3 v‹ du dv

925

926

Chapter 15 Multiple Integrals u 1 4 15 œ '1 ’ 2v dv œ '1 ˆ 2v 3 2 u v“ 3 2

#

4

2

"

16. x œ u2 v2 and y œ 2uv;

` (xßy) ` (ußv)

15v ‰ 2

15 dv œ ’ 4v 2

œ J(uß v) œ º

2 15v2 4 “"

œ

225 16

2v œ 4u2 4v2 œ 4au2 v2 b ; 2u º

2u 2v

y œ 2È1 x Ê y2 œ 4a1 xb Ê a2uvb2 œ 4a1 au2 v2 bb Ê u œ „ 1; y œ 0 Ê 2uv œ 0 Ê u œ 0 or v œ 0; x œ 0 Ê u2 v2 œ 0 Ê u œ v or u œ v; This gives us four triangular regions, but only the one in the quadrant where both u, v are positive maps into the region R in the xy-plane. È

'01 '02 1 x Èx2 y2 dx dy œ '01 '0u Éau2 v2 b2 a2uvb2 † 4au2 v2 b dv du œ 4'01 '0u au2 v2 b2 dv du 2 u 112 1 6 ‘ 2 56 '2 5 œ 4'1 u4 v 23 u2 v3 15 v5 ‘0 du œ 112 15 1 u du œ 15 6 u " œ 45 17. (a) x œ u cos v and y œ u sin v Ê

` (xßy) ` (ußv)

œº

cos v u sin v œ u cos# v u sin# v œ u sin v u cos v º

(b) x œ u sin v and y œ u cos v Ê

` (xßy) ` (ußv)

œº

sin v u cos v œ u sin# v u cos# v œ u cos v u sin v º

18. (a) x œ u cos v, y œ u sin v, z œ w Ê

(b) x œ 2u 1, y œ 3v 4, z œ

â â sin 9 cos ) â 19. â sin 9 sin ) â â cos 9 œ (cos 9) º

3 cos 9 cos ) 3 cos 9 sin ) 3 sin 9 3 cos 9 cos ) 3 cos 9 sin )

" #

` (xßyßz) ` (ußvßw)

(w 4) Ê

â â cos v â œ â sin v â â 0 ` (xßyßz) ` (ußvßw)

u sin v u cos v 0

â â2 â œ â0 â â0

0 3 0

â 0â â 0 â œ u cos# v u sin# v œ u â "â

â 0â 0 ââ œ (2)(3) ˆ #" ‰ œ 3 " â # â

â 3 sin 9 sin ) â â 3 sin 9 cos ) â â 0 â 3 sin 9 sin ) sin 9 cos ) (3 sin 9) º 3 sin 9 cos ) º sin 9 sin )

3 sin 9 sin ) 3 sin 9 cos ) º

œ a3# cos 9b asin 9 cos 9 cos# ) sin 9 cos 9 sin# )b a3# sin 9b asin# 9 cos# ) sin# 9 sin# )b œ 3# sin 9 cos# 9 3# sin$ 9 œ a3# sin 9b acos# 9 sin# 9b œ 3# sin 9 20. Let u œ gaxb Ê Jaxb œ

du dx

œ gw axb Ê 'a faub du œ 'gaab fagaxbbgw axb dx in accordance with Theorem 7 in gabb

b

Section 5.6. Note that gw axb represents the Jacobian of the transformation u œ gaxb or x œ g" aub. 21.

'03 '04 'y1Î2 ÐyÎ2Ñ ˆ 2x # y 3z ‰ dx dy dz œ '03 '04 ’ x2

#

œ '0 ’ (y 4 1) 3

#

y# 4

% yz 3 “!

dz œ '0 ˆ 49 3

4z 3

xy #

"ÐyÎ2Ñ xz 3 “ yÎ2

4" ‰ dz œ '0 ˆ2 3

dy dz œ '0

4z ‰ 3

3

'04 "# (y 1) y# 3z ‘ dy dz

dz œ ’2z

$ 2z# 3 “!

œ 12

â â âa 0 0â # # # â â 22. J(uß vß w) œ â 0 b 0 â œ abc; the transformation takes the ellipsoid region xa# by# cz# Ÿ 1 in xyz-space â â â0 0 câ into the spherical region u# v# w# Ÿ 1 in uvw-space ˆwhich has volume V œ 43 1‰ Ê V œ ' ' ' dx dy dz œ ' ' ' abc du dv dw œ R

G

41abc 3


927

â â âa 0 0â â â 23. J(uß vß w) œ â 0 b 0 â œ abc; for R and G as in Exercise 22, ' ' ' kxyzk dx dy dz â â R â0 0 câ œ ' ' ' a# b# c# uvw dw dv du œ 8a# b# c# G

œ

4a# b# c# 3

'01Î2 '01Î2

'01Î2 '01Î2 '01 (3 sin 9 cos ))(3 sin 9 sin ))(3 cos 9) a3# sin 9b d3 d9 d) a # b # c# 3

sin ) cos ) sin$ 9 cos 9 d9 d) œ

'01Î2 sin ) cos ) d) œ a b6 c

# # #

â 1 â â 24. u œ x, v œ xy, and w œ 3z Ê x œ u, y œ vu , and z œ "3 w Ê J(uß vß w) œ â uv# â â 0

0 " u

0

0 ââ 0 ââ œ " â â 3

" 3u

;

' ' ' ax# y 3xyzb dx dy dz œ ' ' ' u# ˆ vu ‰ 3u ˆ vu ‰ ˆ w3 ‰‘ kJ(uß vß w)k du dv dw œ "3 ' ' ' ˆv vw ‰ du dv dw u 0 0 1 3

D

2

2

G

œ

" 3

'0 '0 (v vw ln 2) dv dw œ 3" '03 (1 w ln 2) ’ v2 “ # dw œ 32 '03 (1 w ln 2) dw œ 32 ’w w2

œ

2 3

ˆ3

3

2

#

#

!

9 #

ln 2“

$ !

ln 2‰ œ 2 3 ln 2 œ 2 ln 8

25. The first moment about the xy-coordinate plane for the semi-ellipsoid,

x# a#

y# b#

z# c#

œ 1 using the

transformation in Exercise 23 is, Mxy œ ' ' ' z dz dy dx œ ' ' ' cw kJ(uß vß w)k du dv dw D

œ abc#

G

' ' ' w du dv dw œ aabc# b † aMxy of the hemisphere x# y# z# œ 1, z 0b œ G

the mass of the semi-ellipsoid is

#

2abc1 3

3 ‰ Ê z œ Š abc4 1 ‹ ˆ 2abc 1 œ

3 8

abc# 1 4

;

c

26. A solid of revolution is symmetric about the axis of revolution, therefore, the height of the solid is solely a function of r. That is, y œ faxb œ farb. Using cylindrical coordinates with x œ r cos ), y œ y and z œ r sin ), we have V œ ' ' ' r dy d) dr œ 'a G

b

'021 '0farb

r dy d) dr œ 'a

b

'021 c r y df0arb d) dr œ 'ab '021 r farb d) dr œ 'ab c r)farb d201 dr

'ab 21rfarbdr. In the last integral, r is a dummy or stand-in variable and as such it can be replaced by any variable name. b Choosing x instead of r we have V œ 'a 21xfaxbdx, which is the same result obtained using the shell method. CHAPTER 15 PRACTICE EXERCISES 1.

'110 '01Îyyexy dx dy œ '110 cexy d !"Îy dy 10 œ '1 (e 1) dy œ 9e 9

'01 '0x eyÎx dy dx œ '01 x eyÎx ‘ !x $

2.

$

œ '0 Šxex x‹ dx œ ’ "2 ex 1

#

#

dx " x# # “!

œ

e2 #

928 3.

Chapter 15 Multiple Integrals È

È9

'03Î2 ' È99 4t4t t ds dt œ '03Î2 ctsd È dt 9 4t 3Î2 $Î# $Î# œ '0 2tÈ9 4t# dt œ ’ "6 a9 4t# b “ #

#

œ 6" ˆ0$Î# 9$Î# ‰ œ

4.

!

œ

27 6

9 #

'01 'È2cy Èy xy dx dy œ '01 y ’ x2 “ 2cÈy #

Èy

œ

" #

dy

'01 y ˆ4 4Èy y y‰ dy

œ '0 ˆ2y 2y$Î# ‰ dy œ ’y# 1

'c02 '2x4 cb x4

" 4y&Î# 5 “!

" 5

œ

dy dx œ 'c2 ax# 2xb dx

#

5.

4t#

#

0

$

œ ’ x3 x# “

!

#

œ ˆ 38 4‰ œ

4 3

'04 'c(Èy c4 c4)/2y dx dy œ '04 ˆ y c2 4 È4 y‰ dy 4

2

œ ’ y2 2y 32 a4 yb3/2 “ œ 4 8

2 3

0

œ 4 6.

œ

16 3

† 43/2

4 3

'01 'yÈy Èx dx dy œ '01 23 x$Î# ‘ yÈy dy œ œ

2 3 2 3

'01 ˆy$Î% y$Î# ‰ dy œ 32 47 y(Î% 52 y&Î# ‘ "! ˆ 47 25 ‰ œ

4 35

'01 'xx Èx dy dx œ '01 x1/2 ax x2 b dx œ '01 ˆx3/2 x5/2 ‰ dx 2

1

œ 25 x5/2 27 x7/2 ‘0 œ 7.

È9

'c33 '0Ð1Î2Ñ

x#

2 5

œ

2 7

y dy dx œ 'c3 ’ y2 “ 3

#

œ 'c3 8" a9 x# b dx œ ’ 9x 8 3

œ ˆ 27 8

27 ‰ 24

È

'03Î2 'È99 4y4y #

#

4 35

ˆ 27 8

!

dx

$ x$ 24 “ $

œ

27 ‰ 24

È

Ð1Î2Ñ 9 x#

3Î2

27 6

œ

9 #

y dx dy œ '0 2yÈ94y# dy 3/2

œ "4 † 23 a94y# b3/2 º

œ

0

" 6

† 93/2 œ

27 6

œ

9 #

'02 '04 x 2x dy dx œ '02 c2xyd 04 x dx 2 2 œ '0 a2xa4 x2 bb dx œ '0 a8x 2x3 b dx 2

8.

2

œ ’4x2

È4 c y

'04 '0

2

x4 2 “!

œ 16

16 2

È4 c y

2x dx dy œ '0 cx2 d 0 4

œ '0 a4 yb dy œ ’ 4y 4

œ8

y2 2

4

dy

“ œ 16 0

16 2

œ8


'01 '2y2 4 cos ax# b dx dy œ '02 '0x/2 4 cos ax# b dy dx œ '02 2x cos ax# b dx œ csin ax# bd #! œ sin 4

10.

'02 'y1Î2 ex

11.

'08 'È2x

y%

12.

'01 'È1y

21 sin a1x# b x#

$

$

dx dy œ '0

1

#

" 1

'02x ex

dy dx œ '0

2

dy dx œ '0 2xex dx œ cex d ! œ e 1 1

#

'0y

$

" y% 1

dx dy œ '0

1

4 c x#

#

dx dy œ

'02 y 4y 1 dy œ ln417 $

" 4

%

'0x 21 sinx a1x b dy dx œ '01 21x sin a1x# b dx œ c cos a1x# bd "! œ (1) (1) œ 2 $

#

#

13. A œ 'c2 '2x b 4 dy dx œ 'c2 ax# 2xb dx œ 0

15. V œ '0

1

0

4

'2Ècyy

$

$

x

" 12

6 c x#

16. V œ 'c3 'x 2

7 ‰ 12

18. average value œ

È

'c11 'cÈ11ccxx

20.

'c11 'cÈ11ccyy

'01

"

ˆ1‰ 4

È

# #

2 y # b2

dx œ 'c3 a6x# x% x$ b dx œ

6 c x#

2

xy dy dx œ '0 ’ xy2 “ dx œ '0 1

"

#

1

!

È1 c x

a1 x #

dx dy œ '1 ˆÈy 2 y‰ dy œ 4

7x$ 3 “

$

dx œ ’ 2x3

#

xy dy dx œ

dy dx œ '0

21

'01

4 1

'01 ’ xy2 “ #

21

dx œ

È1 c x

#

!

dx œ

2 1

'01 ax x$ b dx œ #"1

21

'01 r ln ar# 1b dr d) œ '021 '12

" #

" #

'021 d) œ 1

ln u du d) œ

" #

'021 cu ln u ud #" d)

'021 (2 ln 2 1) d) œ [ln (4) 1] 1 1Î4

1Î4

œ 'c1Î4 ’ 2 a1 " r# b “

22. (a)

1Î4

Ècos 2)

!

d) œ "#

1Î4 " " ' " ' 11ÎÎ44 ˆ1 1 cos ‰ ˆ1 # cos ‰ ) d) 2) d) œ # 1Î4 #

'c1Î4 Š1 sec# ) ‹ d) œ "# ) tan2 ) ‘ 1Î14Î4 œ 14 2

''

#

" a1 x# y# b#

R

œ '0 ’ "# 1Î3

œ (b)

" 7x% 12 “ !

" 4

" dr d) œ '0 1 " r# ‘ ! d) œ

2r a1 r# b #

ln ax# y# 1b dx dy œ '0

x 2

#

" #

125 4

Ècos 2)

21. ax# y# b ax# y# b œ 0 Ê r% r# cos 2) œ 0 Ê r# œ cos 2) so the integral is 'c1Î4 '0

œ

(2x)% 12

37 6

4 3

'01 '0

# #

œ

2

1

19.

2% 12

x# dy dx œ 'c3 cx# yd x

17. average value œ '0

" #

14. A œ '1

4 3

'x2 c x ax# y# b dy dx œ '01 ’x# y y3 “ 2cx dx œ '01 ’2x# (23x)

œ ˆ 23

œ

"

#

'' R

" #

" a1 x # y # b #

œ '0

1Î3

" # a1 sec# )b “

’ È"2 tan"

1Î2

dx dy œ '0

u È2 “

È$ !

d) œ È2 4

œ

dx dy œ '0

lim ’ " bÄ_ #

1Î2

'0sec

)

dr d) œ '0

1Î3

r a1 r# b#

'01Î3 1 secsec) ) d); ” #

" #

#

" # a1 b# b “d)

r a1 r # b #

œ

" #

'0

dr d) œ '0

1Î2

Î

1 2

d) œ

1 4

sec )

!

d)

u œ tan ) Ä du œ sec# ) d) •

tan" É #3

'0_

’ 2 a1 " r# b “

" #

b

lim

bÄ_

È3

'0

’ 2 a1 " r# b “ d) 0

du 2 u #

r a1 r# b#

dr d)

929

930 23.


'01 '01 '01 cos (x y z) dx dy dz œ '01 '01 [sin (z y 1) sin (z y)] dy dz 1 œ '0 [ cos (z 21) cos (z 1) cos z cos (z 1)] dz œ 0

24.

'lnln67 '0ln 2 'lnln45 eÐxyzÑ dz dy dx œ 'lnln67 '0ln 2 eÐxyÑ dy dx œ 'lnln67 ex dx œ 1

25.

'01 '0x '0xby (2x y z) dz dy dx œ '01 '0x Š 3x#

26.

'1e '1x '0z 2yz dy dz dx œ '1e '1x "z dz dx œ '1e ln x dx œ cx ln x xd 1e œ 1

#

#

#

3y# # ‹

dy dx œ '0 Š 3x# 1

%

x' #‹

dx œ

8 35

$

27. V œ 2 '0

1Î2

28. V œ 4 '0

2

' 0cos y '0 2x dz dx dy œ 2 '01Î2 ' 0cos y È4cx

'0

œ ’x a4 x# b " 3

29. average œ

#

'04cx

$Î#

#

dz dy dx œ 4 '0

2

È4cx

'0

#

1 Î2

2x dx dy œ 2 '0 cos# y dy œ 2 ’ y2

a4 x# b dy dx œ 4 '0 a4 x# b 2

$Î#

1 Î2

sin 2y 4 “!

œ

1 #

dx

#

6xÈ4 x# 24 sin" x2 “ œ 24 sin" 1 œ 121 !

'01 '03 '01

30xzÈx# y dz dy dx œ

'01 '03 15xÈx# y dy dx œ 3" '03 '01 15xÈx# y dx dy

" 3

œ

" 3

'03 ’5 ax# yb$Î# “ " dy œ "3 '03 5(1 y)$Î# 5y$Î# ‘ dy œ "3 2(1 y)&Î# 2y&Î# ‘ $! œ "3 2(4)&Î# 2(3)&Î# 2‘

œ

" 3

2 ˆ31 3&Î# ‰‘

30. average œ

31. (a)

3 4 1 a$

È

È

!

'021 '01 '0a È

3$ sin 9 d3 d9 d) œ

3a 161

'021 '01 sin 9 d9 d) œ 83a1 '021 d) œ 3a4

'cÈ22 'cÈ22ccyy 'Èx4bcyx cy 3 dz dx dy #

#

#

#

#

#

(b)

'021 '01Î4 '02 33# sin 9 d3 d9 d)

(c)

'021 '0 2 'r

È

È4cr

#

3 dz r dr d) œ 3 '0

21

È2

'0

’r a4 r# b

"Î#

r# “ dr d) œ 3 '0 ’ "3 a4 r# b 21

$Î#

œ '0 ˆ2$Î# 2$Î# 4$Î# ‰ d) œ Š8 4È2‹'0 d) œ 21 Š8 4È2‹ 21

21

'c11ÎÎ22 '01 ' rr

1Î2

21(r cos ))(r sin ))# dz r dr d) œ ' 1Î2 '0 #

#

32. (a)

'c11ÎÎ22 '01 ' rr

#

(b)

33. (a) (b)

34. (a) (c) (d)

#

1Î2

21r$ cos ) sin# ) dz r dr d) œ 84 '0

1

' rr

# #

$

r3 “

È# !

d)

21r$ cos ) sin# ) dz r dr d)

'01 r' sin# ) cos ) dr d) œ 12'01Î2 sin# ) cos ) d) œ 4

'021 '01Î4 '0sec 9 3# sin 9 d3 d9 d) '021 '01Î4 '0sec 9 3# sin 9 d3 d9 d) œ 3" '021 '01Î4 (sec 9)(sec 9 tan 9) d9 d) œ 3" '021 2" tan# 9‘ !1Î4 d) œ 6" '021 d) œ 13 È

È

1 r 1Î2 '01 '0 1cx '0 x y (6 4y) dz dy dx (b) '0 '0 '0 (6 4r sin )) dz r dr d) '01Î2 '11ÎÎ42 '0csc 9 (6 43 sin 9 sin )) a3# sin 9b d3 d9 d) #

#

#

'01Î2 '01 '0r (6 4r sin )) dz r dr d) œ '01Î2 '01 a6r# 4r$ sin )b dr d) œ '01Î2 c2r$ r% sin )d "! d) 1Î2 1Î# œ '0 (2 sin )) d) œ c2) cos )d ! œ 1 1


È

È3 È3cx

È4cx cy

'01 'È13ccxx '1 #

#

z# yx dz dy dx '1

#

#

'0

#

È4cx cy

'1

#

#

z# yx dz dy dx

36. (a) Bounded on the top and bottom by the sphere x# y# z# œ 4, on the right by the right circular cylinder (x 1)# y# œ 1, on the left by the plane y œ 0

È

'01Î2 '02 cos 'cÈ44ccrr dz r dr d) )

(b)

#

#

37. (a) V œ '0

21

È8cr

'02 '2

dz r dr d) œ '0

21

#

'02 ŠrÈ8 r# 2r‹ dr d) œ '021 ’ "3 a8 r# b$Î# r# “ # d) !

œ '0 (4)$Î# 4 (8)$Î# ‘ d) œ '0 21

" 3

(b) V œ '0

21

œ œ

Š2 3 2È8‹ d) œ

'0 '2 sec 9 3# sin 9 d3 d9 d) œ 83 '0 '0 1Î4

21

1Î4

4 3

Š4È2 5‹ '0 d) œ 21

81 Š4È2 5‹ 3

Š2È2 sin 9 sec$ 9 sin 9‹ d9 d)

'021 '01Î4 Š2È2 sin 9 tan 9 sec# 9‹ d9 d) œ 83 '021 ’2È2 cos 9 "# tan# 9“ 1Î% d)

8 3

'0

21

32 5

È8

4 3

8 3

38. Iz œ '0 œ

21

" 3

!

21

Š2

" #

2È2‹ d) œ

8 3

'

21

È Š 5 #4 2 ‹ 0

d) œ

81 Š4È2 5‹ 3

'01Î3 '02 (3 sin 9)# a3# sin 9b d3 d9 d) œ '021 '01Î3 '02 3% sin$ 9 d3 d9 d)

'021 '01Î3 asin 9 cos# 9 sin 9b d9 d) œ 325 '021 ’ cos 9 cos3 9 “ 1Î$ d) œ 831 $

!

39. With the centers of the spheres at the origin, Iz œ '0

21

'01 'ab $(3 sin 9)# a3# sin 9b d3 d9 d)

'021 '01 sin$ 9 d9 d) œ $ ab 5 a b '021 '01 asin 9 cos# 9 sin 9b d9 d) 1 21 21 œ $ ab 5 a b '0 ’ cos 9 cos3 9 “ d) œ 4$ ab15 a b '0 d) œ 81$ ab15 a b

œ

$ ab& a& b 5 &

&

&

&

&

$

&

&

&

!

'01 '01ccos (3 sin 9)# a3# sin 9b d3 d9 d) œ '02 '0 '01ccos 3% sin$ 9 d3 d9 d) 21 21 1 1 œ "5 '0 '0 (1 cos 9)& sin$ 9 d9 d) œ '0 '0 (1 cos 9)' (1 cos 9) sin 9 d9 d);

40. Iz œ '0

21

)

1

u œ 1 cos 9 ” du œ sin 9 d9 • Ä œ

" 5

'021 2 56†2 $

41. M œ '1

2

&

d) œ

32 35

" 5

'021 '02 u' (2 u) du d) œ 5" '021 ’ 2u7

2

(

# u) 8 “!

d) œ

" 5

'021 ˆ 7" 8" ‰ 2) d)

'021 d) œ 64351

'22Îx

y dy dx œ '1 ˆ2

2y c y#

2

2‰ x#

dx œ 1 Ê x œ y œ

42. M œ '0 'c2y dx dy œ '0 a4y y# b dy œ 4

2y c y#

4

32 3

4

# #

" # ln 4 2y c y#

; Mx œ '0 'c2y y dx dy œ '0 a4y# y$ b dy œ ’ 4y3 4

y My œ '0 'c2y x dx dy œ '0 ’ a2y#y b 2y# “ dy œ ’ 10 4

2

)

'22Îx dy dx œ '12 ˆ2 2x ‰ dx œ 2 ln 4; My œ '12 '22Îx x dy dx œ '12 x ˆ2 2x ‰ dx œ 1;

Mx œ '1

43. Io œ '0

1

&

%

y% 2 “!

4

œ 128 5 Ê xœ

'2x4 ax# y# b (3) dy dx œ 3 '02 Š4x# 643 14x3 ‹ dx œ 104

44. (a) Io œ

$

'c22 'c11 ax# y# b dy dx œ 'c22 ˆ2x# 23 ‰ dx œ 403

$

My M

œ 12 5 and y œ

% y% 4 “! Mx M

œ

œ2

64 3

;

931

932


(b) Ix œ 'ca 'cb y# dy dx œ 'ca $ 4ab ab# a# b 4a$ b œ 4ab 3 3 œ 3 a

45. M œ $ '0

3

46. M œ '0

b

a

2b$ 3

dx œ

4ab$ 3

; Iy œ 'cb 'ca x# dx dy œ 'cb b

a

b

2a$ 3

dy œ

4a$ b 3

Ê Io œ Ix Iy

'02xÎ3 dy dx œ $ '03 2x3 dx œ 3$ ; Ix œ $ '03 '02xÎ3 y# dy dx œ 818$ '03 x$ dx œ ˆ 818$ ‰ Š 34 ‹ œ 2$ %

"3 'xx (x 1) dy dx œ '01 ax x$ b dx œ "4 ; Mx œ '01 'xx y(x 1) dy dx œ #" '01 ax$ x& x# x% b dx œ 120 ; 1 x 1 1 x 2 8 13 My œ '0 'x x(x 1) dy dx œ '0 ax# x% b dx œ 15 Ê x œ 15 and y œ 30 ; Ix œ '0 'x y# (x 1) dy dx 1 1 x 1 I 17 17 1 œ "3 '0 ax% x( x$ x' b dx œ 280 Ê Rx œ É M œ É 70 ; Iy œ '0 'x x# (x 1) dy dx œ '0 ax$ x& b dx œ 12 1

#

#

#

#

x

#

47. M œ 'c1 'c1 ˆx# y# 3" ‰ dy dx œ 'c1 ˆ2x# 34 ‰ dx œ 4; Mx œ 'c1 'c1 y ˆx# y# "3 ‰ dy dx œ 'c1 0 dx œ 0; 1 1 1 My œ ' ' x ˆx# y# "3 ‰ dy dx œ ' ˆ2x$ 34 x‰ dx œ 0 1

1

1

c1 c1

1

1

1

c1

48. Place the ?ABC with its vertices at A(0ß 0), B(bß 0) and C(aß h). The line through the points A and C is yœ

h a

x; the line through the points C and B is y œ

œ b$ '0 ˆ1 yh ‰ dy œ h

1Î3

$ bh #

49. M œ ' 1Î3 '0 r dr d) œ 9# and y œ 0 by symmetry 50. M œ '0

1Î2

yœ

13 31

3

; Ix œ '0

h

(x b). Thus, M œ '0

h

'ayÐaÎh bÑyÎh b $ dx dy

'ayÐaÎh bÑyÎh b y# $ dx dy œ b$ '0h Šy# yh ‹ dy œ $1bh# $

$

' 11ÎÎ33 d) œ 31; My œ ' 11ÎÎ33 '03 r# cos ) dr d) œ 9 ' 11ÎÎ33 cos ) d) œ 9È3

'13 r dr d) œ 4 '01Î2 d) œ 21; My œ '01Î2 '13

r# cos ) dr d) œ

26 3

'01Î2 cos ) d) œ 263

Ê xœ

Ê xœ

by symmetry

51. (a) M œ 2 '0

1Î2

'11bcos

)

1Î2

1Î2

My œ 'c1Î2 '1

1 cos )

1 cos 2) ‰ #

œ 'c1Î2 Šcos# ) cos$ ) Ê xœ

d) œ

81 4

;

(r cos )) r dr d)

1Î2

32 151 24

(b)

r dr d)

œ '0 ˆ2 cos )

œ

h ab

151 32 61 48

cos% ) 3 ‹

d)

, and

y œ 0 by symmetry 52. (a) M œ 'c! '0 r dr d) œ 'c!

d) œ a# !; My œ 'c! '0 (r cos )) r dr d) œ 'c! ! 2a sin ! Ê x œ 2a 3sin œ0 ! , and y œ 0 by symmetry; lim c x œ lim c 3! !

(b) x œ

2a 51

a

and y œ 0

!

!

a# #

!Ä1

!

a

!Ä1

a$ cos ) 3

d) œ

2a$ sin ! 3

13 31

3È 3 1

, and

,

Chapter 15 Additional and Advanced Exercises 53. x œ u y and y œ v Ê x œ u v and y œ v " " Ê J(uß v) œ º œ 1; the boundary of the 0 "º image G is obtained from the boundary of R as follows:

xy-equations for


Simplified

the boundary of R


uv-equations

yœx yœ0 Ê

vœuv

uœ0

vœ0

_

_ _

'0 '0 esx f(x yß y) dy dx œ '0 '0 x

vœ0 esÐuvÑ f(uß v) du dv $s "t !$ "#

54. If s œ !x " y and t œ # x $ y where (!$ "# )# œ ac b# , then x œ " (!$ "# )#

and J(sß t) œ œ

" Èac b#

$ º #

'021 '0_ rer

#

" œ ! º

dr d) œ

" !$ "#

" #Èac b#

_ _ 'c_ ' _ e as t b È #

Ê

'021 d) œ È

1 ac b#

#

" ac b#

,yœ

,

ds dt

1 Èac b#

. Therefore,

# s !t !$ "#

œ 1 Ê ac b# œ 1# .

CHAPTER 15 ADDITIONAL AND ADVANCED EXERCISES 6cx#

1. (a) V œ 'c3 'x 2

6cx#

(c) V œ 'c3 'x 2

6cx#

(b) V œ 'c3 'x 2

x# dy dx 6cx#

x# dy dx œ 'c3 'x 2

a6x# x% x$ b dx œ ’2x$

&

x 5

'0x %

x 4

#

“

dz dy dx # $

œ

125 4

2. Place the sphere's center at the origin with the surface of the water at z œ 3. Then 9 œ 25 x# y# Ê x# y# œ 16 is the projection of the volume of water onto the xy-plane Ê V œ '0

21

'04 'ccÈ325cr

dz r dr d) œ '0

21

#

'04 ŠrÈ25 r# 3r‹ dr d) œ '021 ’ "3 a25 r# b$Î# 3# r# “ % d)

21 21 œ '0 "3 (9)$Î# 24 3" (25)$Î# ‘ d) œ '0

3. Using cylindrical coordinates, V œ '0

21

œ '0 ˆ1 21

4. V œ 4 '0

1Î2

" 3

cos )

'01 'r

È2 #

œ 4 '0 Š 1Î2

" 3

" 4

r#

" 3

'01 '02crÐcos

sin )‰ d) œ ) 1Î2

dz r dr d) œ 4 '0 2È 2 3 ‹

d) œ

!

26 3

" 3

d) œ

)

sin )

sin )Ñ

" 3

521 3

dz r dr d) œ '0

21

'01 a2r r# cos ) r# sin )b dr d)

#1

cos )‘ ! œ 21

'01 ŠrÈ2 r# r$ ‹ dr d) œ 4'01Î2 ’ "3 a2 r# b$Î# r4 “ " d) %

!

'

1Î2

È Š 8 327 ‹ 0

d) œ

1 Š8È2 7‹ 6

933

934


5. The surfaces intersect when 3 x# y# œ 2x# 2y# Ê x# y# œ 1. Thus the volume is V œ 4 '0

1

È1 c x

'0

6. V œ 8 '0

1Î2

œ

#

'2x3cx2ycy #

#

#

#

1Î2

dz dy dx œ 4 '0

'01 '2r3

1Î2

dz r dr d) œ 4 '0

r#

#

'01 a3r 3r$ b dr d) œ 3'01Î2 d) œ 31#

'01Î2 '02 sin 9 3# sin 9 d3 d9 d) œ 643 '01Î2 '01Î2 sin% 9 d9 d)

'01Î2 ” sin 94cos 9 ¹1Î# 43 '01Î2 sin# 9 d9• d) œ 16 '01Î2 92 sin429 ‘ 1! Î# d) œ 41 '01Î2 d) œ 21# $

64 3

!

7. (a) The radius of the hole is 1, and the radius of the sphere is 2.

(b) V œ 2 '0

21

8. V œ '0

1

È4cz

È

'0 3 '1 È9cr

'03 sin '0 )

r dr dz d) œ '0

21

#

dz r dr d) œ '0

#

'03 sin

1

œ '0 ’ 3" a9 9 sin# )b 1

$Î#

)

È3

'0

a3 z# b dz d) œ 2È3 '0 d) œ 4È31 21

rÈ9 r# dr d) œ '0 ’ "3 a9 r# b 1

3" (9)$Î# “ d) œ 9'0 ’1 a1 sin# )b 1

œ '0 a1 cos ) sin# ) cos )b d) œ 9 ’) sin ) 1

9. The surfaces intersect when x# y# œ

'01 'r r

1‰Î2

'12 '0r sin

) cos )

V œ 4 '0

1Î2

10. V œ '0

1Î2

œ

11.

#

#

'0

1Î2

15 4

'0_ ec

ax

ˆ

#

_

dx œ '0

1 Î2

dz r dr d) œ 4 '0 dz r dr d) œ '0

1Î2

sin ) cos ) d) œ

ecbx x

x# y# 1 #

15 4

#

’ sin2 ) “

cxy

tÄ_

1

Ê x# y# œ 1. Thus the volume in cylindrical coordinates is $

#

%

"

r8 “ d) œ !

" #

1Î# !

1

%

"

œ

15 8

tÄ_

ecyt y ‹

dy œ 'a

b

" y

lim

tÄ_

'0t exy dx‹ dy

dy œ cln yd ab œ ln ˆ ba ‰

12. (a) The region of integration is sketched at the right Ê '0

a sin "

œ '0

"

È

'y cota "c y #

#

ln ax# y# b dx dy

'0a r ln ar# b dr d);

u œ r# ” du œ 2r dr • Ä

" #

'0" '0a ln u du d) #

'0" [u ln u u] !a d) " œ "# '0 ’2a# ln a a# lim œ

" #

#

t ln t“ d) œ

a# #

'0a cos " '0(tan ")x ln ax# y# b dy dx 'aacos " '0

#

tÄ0

(b)

d)

'01Î2 d) œ 14

'12 r$ sin ) cos ) dr d) œ '0 Î2 ’ r4 “ # sin ) cos ) d)

b

!

!

œ 91

'ab exy dy dx œ 'ab '0_ exy dx dy œ 'ab Š

t

“

“ d) œ 9'0 a1 cos$ )b d)

'01 Š #r r# ‹ dr d) œ 4'01Î2 ’ r4

œ 'a lim ’ e y “ dy œ 'a lim Š "y b

1 sin$ ) 3 “!

$Î#

$Î# 3 sin )

'0" (2 ln a 1) d) œ a# " ˆln a "# ‰

Èa cx

#

ln ax# y# b dy dx

Chapter 15 Additional and Advanced Exercises 13.

'0x '0u emÐxtÑ f(t) dt du œ '0x 't x emÐxtÑ f(t) du dt œ '0x (x t)emÐxtÑ f(t) dt; also '0x '0v '0u emÐxtÑ f(t) dt du dv œ '0x 't x 't v emÐxtÑ f(t) du dv dt œ '0x 't x (v t)emÐxtÑ f(t) dv dt x x x œ '0 "2 (v t)# emÐxtÑ f(t)‘ t dt œ '0 (x # t) emÐxtÑ f(t) dt #

14.

'01 f(x) Š'0x g(xy)f(y) dy‹ dx œ '01 '0x œ '0

1

g(xy)f(x)f(y) dy dx

'y1 g(xy)f(x)f(y) dx dy œ '01 f(y) Œ'y1 g(xy)f(x) dx dy;

'01 '01 g akxykb f(x)f(y) dx dy œ '01 '0x g(xy)f(x)f(y) dy dx '01 'x1 g(yx)f(x)f(y) dy dx 1 1 1 1 œ '0 'y g(xy)f(x)f(y) dx dy '0 'x g(yx)f(x)f(y) dy dx œ '0

1

'y1 g(xy)f(x)f(y) dx dy '01 'y1 g(xy)f(y)f(x) dx dy

ðóóóóóóóóóóóóñóóóóóóóóóóóóò simply interchange x and y variable names

œ 2'0

1

'y1 g(xy)f(x)f(y) dx dy, and the statement now follows.

15. Io (a) œ '0 '0

xÎa#

a

œ

a# 4

" 1#

ax# y# b dy dx œ '0 ’x# y a

a# ; Iow (a) œ

" #

dx œ '0 Š xa#

xÎa#

a

y$ 3 “!

a "6 a$ œ 0 Ê a% œ

$

x$ 3a' ‹

Ê a œ %É "3 œ

" 3

" % È 3

%

x dx œ ’ 4a #

a

x% 12a' “ !

. Since Iwwo (a) œ

" #

#" a% 0, the

value of a does provide a minimum for the polar moment of inertia Io (a). 16. Io œ '0

2

'2x4 ax# y# b (3) dy dx œ 3 '02 Š4x# 14x3

17. M œ 'c) 'b sec ) r dr d) œ )

a

'c Š a# )

#

)

b# #

$

64 3 ‹

dx œ 104

sec# )‹ d)

œ a# ) b# tan ) œ a# cos" ˆ ba ‰ b# Š

È a# b# ‹ b

œ a# cos" ˆ ba ‰ bÈa# b# ; Io œ 'c) 'b sec ) r$ dr d) )

a

'c aa% b% sec% )b d) œ "4 'c ca% b% a1 tan# )b asec# )bd d) œ

)

" 4

)

)

)

) b% tan$ ) “ 3 )

œ

" 4

œ

% $ a% ) b% tan ) ) b tan # # 6 " % " $È # " ˆ b ‰ a # a cos a # b

œ

%

’a ) b% tan )

2 ˆy# Î2‰

b# 6" b$ aa# b# b

18. M œ 'c2 '1cay#Î4b dx dy œ ' 2 Š1 2

œ 'c2 ’ x2 “ 2

œ 19.

3 16

#

2c ˆy Î2‰

1

ˆ32

64 3

y# 4‹

dy œ ’y

Î4b

3 dy œ 'c2 32 a4 y# b dy œ

32 ‰ 5

#

ay#

2

2

3 ‰ ˆ 32†8 ‰ œ ˆ 16 œ 15

48 15

'0a '0b emax ab x ßa y b dy dx œ '0a '0bxÎa eb x # #

# #

# #

3 32

$Î#

# y$ 12 “ #

œ

8 3

; My œ '

' 2 y Î2 2 1 y Î4 ˆ

2

a

#

#

b

‰

x dx dy

'c22 a16 8y# y% b dy œ 163 ’16y 8y3

Ê xœ

My M

dy dx '0

b

‰ ˆ 83 ‰ œ œ ˆ 48 15

'0ayÎb ea y

# #

dx dy

6 5

$

# y& 5 “!

, and y œ 0 by symmetry

935

936

Chapter 15 Multiple Integrals # # # # # # # # " " œ '0 ˆ ba x‰ eb x dx '0 ˆ ba y‰ ea y dy œ ’ 2ab eb x “ ’ 2ba ea y “ œ

a

" ab

œ

20.

b

b

!

!

" #ab

# #

Šeb a 1‹

" #ab

# #

Šea b 1‹

# #

Šea b 1‹

ßy) 'yy 'xx ``F(x ' y ` F(xßy) x x ` y dx dy œ y ’ ` y “ "

!

a

#

"

"

!

!

" ßy) dy œ 'y ’ ` F(x `y

y"

"

!

x!

` F(x! ßy) `y “

dx œ cF(x" ß y) F(x! ß y)d yy!"

œ F(x" ß y" ) F(x! ß y" ) F(x" ß y! ) F(x! ß y! ) 21. (a) (i) (ii) (iii) (iv)

Fubini's Theorem Treating G(y) as a constant Algebraic rearrangement The definite integral is a constant number

(b)

'0ln 2 '01Î2 ex cos y dy dx œ Œ'0ln 2 ex dx Œ'01Î2 cos y dy œ aeln 2 e0 b ˆsin 1# sin 0‰ œ (1)(1) œ 1

(c)

'12 'c11 yx

#

dx dy œ Œ'1

2

" y#

dy Œ'c1 x dx œ ’ y" “ ’ x2 “ #

1

#

"

" "

œ ˆ "# 1‰ ˆ "# "# ‰ œ 0

22. (a) ™ f œ xi yj Ê Du f œ u" x u# y; the area of the region of integration is Ê average œ 2'0

1

#

œ 2 ’u" Š x2 " area

(b) average œ

_ _

23. (a) I# œ '0 œ "#

'0

'01Î2

_

e

x$ 3‹

'01cx (u" x u# y) dy dx œ 2 '01 u" x(1 x) "# u# (1 x)# ‘ dx

ˆ "# u# ‰

" (1x)$ 3 “!

' ' (u" x u# y) dA œ R

ˆx# y# ‰

lim

bÄ_

1Î2

dx dy œ '0 #

_

21

œ 2 ˆ 6" u" 6" u# ‰ œ

" #

R

È

r#

1Î2

b r dr d) œ '0 ” lim bÄ_

'01Î2 d) œ 14

"Î# y#

e

Ê Iœ

_

$

Èh

'cÈhh 'cÈhhccxx ah x# y# b dy dx œ '021 '0 #

#

'0b re

È

$

È

r#

dr• d)

È1 #

(2y) dy œ 2 '0 ey dy œ 2 Š

'0R kr# (1 sin )) dr d) œ kR3 '021 (1 sin )) d) œ kR3 È

(u" u# )

R

25. For a height h in the bowl the volume of water is V œ œ

" 3

M u# ' ' ' ' x dA area y dA œ u" Š My ‹ u# ˆ MMx ‰ œ u" x u# y

u" area

'0_ ae

aecb 1b d) œ

(b) > ˆ "# ‰ œ '0 t"Î# et 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

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

œ

12 2

937

938


NOTES:

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 È

bÄ_

œ

È3 t# t# t#

œ

È3 3t#

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

e64 b

2

2

!

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 ctan1 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

ctan1 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

942


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

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

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(%));

# (a) # (b) # (c)

# !

œ

# t& 20 “ !

% 3

œ

; Iz œ 'C ax# y# b $ ds

Ê xœ

Myz M

œ

32 9

32 20

œ 1, 232 45

;

943

944


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#

;

x i y j 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

œijk Ê F†

(b) F œ 3t# i 2tj 4t% k and œ

7 3

2œ

13 3

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) ‘ "!

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

œijkÊF†

" ‰ (b) F œ ˆ t# 1 j and

dr dt


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


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

œijk Ê F†

œ 2Èt 2t Ê

dr dt

"

#

"

#


dr dt

. œ 3t# Ê '0 3t# dt œ 1 1

(a) F œ t# i t# j t# k and

dr dt

œijk Ê F†

(b) F œ t$ i t' j t& k and

dr dt


%

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


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


dr dt

.

(a) F œ 2ti 2tj 2tk and

dr dt

œijk Ê F†

dr dt

œ 6t Ê

'01 6t dt œ c3t# d "! œ 3

945

946


(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

œ1

" 6

cos t

" 6

dr dt

sin t‰ dt

'

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

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


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


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


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

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;

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# ‹.

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

dt

952


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

Ê Not Conservative œ0œ

`P `x

,

`N `x

œ ex sin y œ

`M `y

Ê Conservative

`M `y

Ê Not Conservative

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

œxz 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)

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#

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


$

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

956


(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


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

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“

#1 !

œ 0;

957

958


' ' 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 2u 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

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

Èa

2b

a

`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

R

a2x3 y 2x3 yb dy dx œ 0

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 tan1 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 a12xy2yb2 ‹‹ 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# R

15. M œ xy, N œ y# Ê œ '0 Š 3x# 1

#

3x% # ‹

2y x# y# ‹

x x# y# ‹

`M `x

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$

`M `y

'x

#

`N `x

ex y

,

2xy dy dx œ ' #

1

2 0 3

x

`N `x

"!

2 33

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

'xx 3y dy dx

'01Î2 2 cos y dx dy œ '01Î2 1 cos y dy œ c1 sin yd 1Î# œ1 !

R

œ 6xy# ,

20. M œ 4x 2y, N œ 2x 4y Ê R

2y x# y#

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

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

`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

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.

Section 16.5 Surfaces and Area

963

Clear[x, y, f]

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