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SOLUTIONS MANUAL
Thomas' Calculus: Multivariable
George B. Thomas Jr., Maurice D. Weir, and Joel R. Hass
ST
12th Edition
UV
IA
?_
AP
PR
OV
ED
? ?
, TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
ST
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
UV
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
IA
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
?_
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
AP
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
PR
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
OV
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
ED
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
??
13.6 Velocity and Acceleration in Polar Coordinates 784
Practice Exercises 785
Additional Exercises 791
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, 14 Partial Derivatives 795
14.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
ST
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
UV
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
IA
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
?_
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
AP
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
PR
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
16.6 Surface Integrals 972
16.7 Stokes's Theorem 980
16.8 The Divergence Theorem and a Unified Theory 984
OV
Practice Exercises 989
Additional Exercises 997
ED
??
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, CHAPTER 10 INFINITE SEQUENCES AND SERIES
10.1 SEQUENCES
1 1 "2
1. a" œ œ 0, a# œ œ 4" , a$ œ 13
œ 92 , a% œ 14
œ 16
3
ST
1# ## 3# 4#
" "
2. a" œ 1
1! œ 1, a# œ #! œ 2 , a$ œ 1
3! œ 1
6 , a% œ 1
4! œ 1
24
UV
(1)# (")$ (1)% (1)&
3. a" œ #1 œ 1, a# œ 41 œ "3 , a$ œ 61 œ "
5 , a% œ 81 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
" 2# " 2$ " 2% "
5. a" œ 2
## œ # , a# œ 2$ œ # , a$ œ #%
œ # , a% œ 2& œ #
IA
2" " 2# 1 2$ 1 2% "
6. a" œ # œ # , a# œ 2# œ 3
4 , a$ œ 2$ œ 7
8 , a% œ 2%
œ 15
16
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" " " "
7. a" œ 1, a# œ 1 # œ 3
# , a$ œ 3
# ## œ 7
4 , a% œ 7
4 #$ œ 15
8 , a& œ 15
8 #%
œ 31
16 , a' œ 63
32 ,
a( œ 127
64 , a) œ 255
128 , a* œ 511
256 , a"! œ 1023
512
" ˆ #" ‰ " ˆ "6 ‰ " ˆ #"4 ‰ " " " "
8. a" œ 1, a# œ , a$ œ 3 œ 6 , a% œ œ , a& œ œ , a' œ , a( œ , a) œ ,
AP
# 4 #4 5 1 #0 7 #0 5040 40,320
" "
a* œ 362,880 , a"! œ 3,628,800
(1)# (2) $ (1)% ˆ "# ‰ (1)& ˆ "4 ‰
9. a" œ 2, a# œ # œ 1, a$ œ (1)2 (1) œ "# , a% œ # œ 4" , a& œ # œ "
8 ,
"
a' œ , a( œ 3"# , a) œ 64 "
, a* œ 1#"8 , a"! œ 256
"
PR
16
1†(2) 3†ˆ 23 ‰ 4†ˆ "# ‰
10. a" œ 2, a# œ # œ 1, a$ œ 2†(31) œ 32 , a% œ 4 œ "# , a& œ 5 œ 52 , a' œ 3" ,
a( œ 27 , a) œ "4 , a* œ 29 , a"! œ "5
OV
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
ˆ "# ‰ ˆ "# ‰
12. a" œ 2, a# œ 1, a$ œ "# , a% œ 1 œ "
# , a& œ ˆ "# ‰ œ 1, a' œ 2, a( œ 2, a) œ 1, a* œ "# , a"! œ "
#
ED
13. an œ (1)n1 , n œ 1, 2, á 14. an œ (1)n , n œ 1, 2, á
(")n 1
15. an œ (1)n1 n# , n œ 1, 2, á 16. an œ n# , n œ 1, 2, á
2n 1 2n 5
??
17. an œ 3an 2b , n œ 1, 2, á 18. an œ nan 1b , 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
3n 2 n3
23. an œ n! , n œ 1, 2, á 24. an œ 5n 1 , n œ 1, 2, á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
SOLUTIONS MANUAL
Thomas' Calculus: Multivariable
George B. Thomas Jr., Maurice D. Weir, and Joel R. Hass
ST
12th Edition
UV
IA
?_
AP
PR
OV
ED
? ?
, TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
ST
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
UV
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
IA
11 Parametric Equations and Polar Coordinates 647
11.1 Parametrizations of Plane Curves 647
11.2 Calculus with Parametric Curves 654
?_
11.3 Polar Coordinates 662
11.4 Graphing in Polar Coordinates 667
11.5 Areas and Lengths in Polar Coordinates 674
11.6 Conic Sections 679
11.7 Conics in Polar Coordinates 689
AP
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
PR
12.1 Three-Dimensional Coordinate Systems 715
12.2 Vectors 718
12.3 The Dot Product 723
12.4 The Cross Product 728
12.5 Lines and Planes in Space 734
OV
12.6 Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
ED
13.1 Curves in Space and Their Tangents 759
13.2 Integrals of Vector Functions; Projectile Motion 764
13.3 Arc Length in Space 770
13.4 Curvature and Normal Vectors of a Curve 773
13.5 Tangential and Normal Components of Acceleration 778
??
13.6 Velocity and Acceleration in Polar Coordinates 784
Practice Exercises 785
Additional Exercises 791
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, 14 Partial Derivatives 795
14.1 Functions of Several Variables 795
14.2 Limits and Continuity in Higher Dimensions 804
14.3 Partial Derivatives 810
14.4 The Chain Rule 816
14.5 Directional Derivatives and Gradient Vectors 824
14.6 Tangent Planes and Differentials 829
14.7 Extreme Values and Saddle Points 836
ST
14.8 Lagrange Multipliers 849
14.9 Taylor's Formula for Two Variables 857
14.10 Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
UV
15 Multiple Integrals 881
15.1 Double and Iterated Integrals over Rectangles 881
15.2 Double Integrals over General Regions 882
IA
15.3 Area by Double Integration 896
15.4 Double Integrals in Polar Form 900
15.5 Triple Integrals in Rectangular Coordinates 904
15.6 Moments and Centers of Mass 909
?_
15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914
15.8 Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
AP
16 Integration in Vector Fields 939
16.1 Line Integrals 939
16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944
PR
16.3 Path Independence, Potential Functions, and Conservative Fields 952
16.4 Green's Theorem in the Plane 957
16.5 Surfaces and Area 963
16.6 Surface Integrals 972
16.7 Stokes's Theorem 980
16.8 The Divergence Theorem and a Unified Theory 984
OV
Practice Exercises 989
Additional Exercises 997
ED
??
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
, CHAPTER 10 INFINITE SEQUENCES AND SERIES
10.1 SEQUENCES
1 1 "2
1. a" œ œ 0, a# œ œ 4" , a$ œ 13
œ 92 , a% œ 14
œ 16
3
ST
1# ## 3# 4#
" "
2. a" œ 1
1! œ 1, a# œ #! œ 2 , a$ œ 1
3! œ 1
6 , a% œ 1
4! œ 1
24
UV
(1)# (")$ (1)% (1)&
3. a" œ #1 œ 1, a# œ 41 œ "3 , a$ œ 61 œ "
5 , a% œ 81 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
" 2# " 2$ " 2% "
5. a" œ 2
## œ # , a# œ 2$ œ # , a$ œ #%
œ # , a% œ 2& œ #
IA
2" " 2# 1 2$ 1 2% "
6. a" œ # œ # , a# œ 2# œ 3
4 , a$ œ 2$ œ 7
8 , a% œ 2%
œ 15
16
?_
" " " "
7. a" œ 1, a# œ 1 # œ 3
# , a$ œ 3
# ## œ 7
4 , a% œ 7
4 #$ œ 15
8 , a& œ 15
8 #%
œ 31
16 , a' œ 63
32 ,
a( œ 127
64 , a) œ 255
128 , a* œ 511
256 , a"! œ 1023
512
" ˆ #" ‰ " ˆ "6 ‰ " ˆ #"4 ‰ " " " "
8. a" œ 1, a# œ , a$ œ 3 œ 6 , a% œ œ , a& œ œ , a' œ , a( œ , a) œ ,
AP
# 4 #4 5 1 #0 7 #0 5040 40,320
" "
a* œ 362,880 , a"! œ 3,628,800
(1)# (2) $ (1)% ˆ "# ‰ (1)& ˆ "4 ‰
9. a" œ 2, a# œ # œ 1, a$ œ (1)2 (1) œ "# , a% œ # œ 4" , a& œ # œ "
8 ,
"
a' œ , a( œ 3"# , a) œ 64 "
, a* œ 1#"8 , a"! œ 256
"
PR
16
1†(2) 3†ˆ 23 ‰ 4†ˆ "# ‰
10. a" œ 2, a# œ # œ 1, a$ œ 2†(31) œ 32 , a% œ 4 œ "# , a& œ 5 œ 52 , a' œ 3" ,
a( œ 27 , a) œ "4 , a* œ 29 , a"! œ "5
OV
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
ˆ "# ‰ ˆ "# ‰
12. a" œ 2, a# œ 1, a$ œ "# , a% œ 1 œ "
# , a& œ ˆ "# ‰ œ 1, a' œ 2, a( œ 2, a) œ 1, a* œ "# , a"! œ "
#
ED
13. an œ (1)n1 , n œ 1, 2, á 14. an œ (1)n , n œ 1, 2, á
(")n 1
15. an œ (1)n1 n# , n œ 1, 2, á 16. an œ n# , n œ 1, 2, á
2n 1 2n 5
??
17. an œ 3an 2b , n œ 1, 2, á 18. an œ nan 1b , 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
3n 2 n3
23. an œ n! , n œ 1, 2, á 24. an œ 5n 1 , n œ 1, 2, á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
