Thomas Calculus 13th [Solutions]

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Chapter 1 Additional and Advanced Exercises 53 5. For (x, y) in the 1st quadrant, | x| | y| 1 x x y 1 x y 1. For (x, y) in the 2nd quadrant, | x| | y| x 1 x y x 1 y 2x 1. In the 3rd quadrant, | x| | y| x 1 x y x 1 y 2x 1. In the 4th quadrant, | x| | y | x 1 x ( y) x 1 y 1. The graph is given at the right. 6. We use reasoning similar to Exercise 5. (1) 1st quadrant: y | y| x | x| 2y 2 x y x. (2) 2nd quadrant: y | y | x | x| 2 y x ( x) 0 y 0. (3) 3rd quadrant: y | y| x | x| y ( y ) x ( x) 0 0 all points in the 3rd quadrant satisfy the equation. (4) 4th quadrant: y | y| x | x| y ( y) 2x 0 x . Combining these results we have the graph given at the right: 2 2 2 2 sin x 1 cos x 7. (a) sin x cos x 1 sin x 1 cos x (1 cos x) (1 cos x) 1 cos x 1 cos x sin x (b) Using the definition of the tangent function and the double angle formulas, we have 1 cos 2 x 2 x 2 2 2 2 x 1 cos 2 x 2 2 2 tan 2 sin x 1 cos x 2 cos 1 cos x . 2 sin x 1 cos x 8. The angles labeled in the accompanying figure are equal since both angles subtend arc CD. Similarly, the two angles labeled are equal since they both subtend arc AB. Thus, triangles AED and BEC are similar which a c 2a cos b implies b a c ( a c)( a c) b(2a cos b) 2 2 2 a c 2ab cos b 2 2 2 c a b 2ab cos . 9. As in the proof of the law of sines of Section 1.3, Exercise 61, ah bc sin A ab sin C ac sin B the area of ABC 1 (base)(height) 1 ah 1 bc sin A 1 ab sin C 1 ac sin B. 2 2 2 2 2 2 2 2 10. As in Section 1.3, Exercise 61, (Area of ABC) 1 4 (base) (height) 1 2 2 1 2 2 2 a h a b sin C 4 4 1 2 2 2 a b (1 cos C ) . By the law of cosines, 2 2 2 2 2 2 a b c c a b 2ab cos C cos C . Thus, 4 2ab 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 (area of ABC) a b (1 cos C) 1 a b c 2 2 a b 1 a b ( a b c ) 1 4 4 2ab 4 2 2 4a b 1 2 2 2 2 2 2 16 4 a b ( a b c ) 1 16 [(2ab 2 2 2 2 2 2 ( a b c )) (2 ab ( a b c ))] 1 2 2 2 2 [(( a b) c )( c ( a b) )] 1 [(( a b) c)(( a b) c)( c ( a b))( c ( a b))] 16 16 a b c a b c a b c a b c a b c s( s a)( s b)( s c), where s . 2 2 2 2 2 Therefore, the area of ABC equals s( s a)( s b)( s c ). Copyright 2014 Pearson Education, Inc.
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1 Functions 1 TABLE OF CONTENTS 1.1
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8.3 Trigonometric Integrals 569 8.4
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2 Chapter 1 Functions 15. The domai
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4 Chapter 1 Functions 3 0 0 ( 1) 1
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6 Chapter 1 Functions 43. Symmetric
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8 Chapter 1 Functions 68. (a) From
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10 Chapter 1 Functions The complete
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12 Chapter 1 Functions 35. 36. 37.
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14 Chapter 1 Functions (c) domain:
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16 Chapter 1 Functions 69. 3 y f (
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18 Chapter 1 Functions (c) (d) 80.
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20 Chapter 1 Functions 21. 22. peri
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22 Chapter 1 Functions 40. sin(2 x)
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24 Chapter 1 Functions 65. A 2, B 2
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26 Chapter 1 Functions 1.4 GRAPHING
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Page 73: 30 Chapter 1 Functions 35. 36. 37.
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Page 89: 38 Chapter 1 Functions y 1 36. y =
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Section 2.3 The Precise Definition
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Section 2.4 One-Sided Limits 81 2.4
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Section 2.4 One-Sided Limits 83 17.
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Section 2.4 One-Sided Limits 85 44.
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Section 2.5 Continuity 87 19. Disco
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Section 2.5 Continuity 89 49. The f
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Section 2.5 Continuity 91 65. Yes,
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13. (a) 2 3 2 lim x x lim 2 x x 5 7
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Section 2.6 Limits Involving Infini
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Chapter 2 Practice Exercises 103 2.
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Chapter 2 Practice Exercises 105 17
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Chapter 2 Additional and Advanced E
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CHAPTER 3 DERIVATIVES 3.1 TANGENTS
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Section 3.1 Tangents and the Deriva
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Section 3.1 Tangents and the Deriva
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Section 3.2 The Derivative as a Fun
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Right-hand derivative: When h 0,1 h
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63-68. Example CAS commands: Maple:
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Section 3.3 Differentiation Rules 1
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Section 3.4 The Derivative as a Rat
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Section 3.5 Derivatives of Trigonom
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Section 3.6 The Chain Rule 153 2 2
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Section 3.6 The Chain Rule 155 44.
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Section 3.6 The Chain Rule 159 11 2
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Section 3.6 The Chain Rule 161 sin
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Section 3.7 Implicit Differentiatio
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Section 3.8 Derivatives of Inverse
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109 110. Example CAS commands: Sect
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Section 3.9 Inverse Trigonometric F
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Section 3.10 Related Rates 187 5. y
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Section 3.10 Related Rates 189 27.
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Section 3.10 Related Rates 191 43.
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Section 3.11 Linearization and Diff
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plot(err(x), x 1..2, title #absolut
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Chapter 3 Practice Exercises 207 (b
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Chapter 3 Practice Exercises 213 (b
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CHAPTER 4 APPLICATIONS OF DERIVATIV
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Section 4.1 Extreme Values of Funct
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Section 4.2 The Mean Value Theorem
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27. r ( ) sec 1 5 ( ) (sec )(tan )
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Section 4.2 The Mean Value Theorem
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Section 4.3 Monotonic Functions and
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33. (a) 34. (a) 35. (a) 2 2 1/2 g(
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(b) The graph of f rises when f 0,
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86. f(x) is increasing since 1 5/3
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Section 4.4 Concavity and Curve Ske
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5 4 21. When y x 5 x , then 4 3 3 3
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31. When y x , then y 1 and x 2 2 3
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2 2 40. When y x , then x y 2 x 2 2
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The line x 2 is a vertical asymptot
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104. 105. Section 4.4 Concavity and
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3 2 2 Section 4.4 Concavity and Cur
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Section 4.5 Indeterminate Forms and
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Section 4.6 Applied Optimization 29
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16. (a) The base measures 10 2x in.
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2 3 Section 4.6 Applied Optimizatio
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Section 4.7 Newtons Method 305 4. 2
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Section 4.7 Newtons Method 307 20.
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Section 4.8 Antiderivatives 309 4.8
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Section 4.8 Antiderivatives 311 42.
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85. (a) Wrong: (b) (c) d dx 3 (2x 1
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Section 4.8 Antiderivatives 315 2 2
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Section 4.8 Antiderivatives 317 128
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45. (a) y 6 x( x 1)( x 2) (b) 3 2 6
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52. The graph of the first derivati
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92. (a) The distance between the pa
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Chapter 4 Practice Exercises 335 x
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CHAPTER 5 INTEGRATION 5.1 AREA AND
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Section 5.1 Area and Estimating wit
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Section 5.1 Area and Estimating wit
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Section 5.2 Sigma Notation and Limi
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Section 5.3 The Definite Integral 3
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65. Consider the partition P that s
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74. See Exercise 73 above. On [0, 0
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Section 5.4 The Fundamental Theorem
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93. Example CAS commands: Maple: f
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Section 5.5 Indefinite Integrals an
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80. Let 2 u t du dt du dt Section 5
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1 1 11. (a) Let u 4 5 t t ( u 4), d
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Section 5.6 Substitution and Area B
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61. AREA A1 A2 A3 A1: For the sketc
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Chapter 5 Practice Exercises 407 CH
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80. Let u 7 5r du 5 dr 1 du dr; r 0
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29. (a) (b) 1 g(1) f ( t) dt 0 1 3
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CHAPTER 6 APPLICATIONS OF DEFINITE
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Section 6.1 Volumes Using Cross-Sec
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Section 6.2 Volumes Using Cylindric
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Section 6.3 Arc Length 455 2. dy 3
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Section 6.3 Arc Length 457 9. dx 1
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Section 6.3 Arc Length 459 19. (a)
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Section 6.3 Arc Length 461 32. (a)
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Section 6.4 Areas of Surfaces of Re
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Section 6.5 Work and Fluid Forces 4
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(d) In a location where water weigh
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38. Using the coordinate system giv
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Section 6.6 Moments and Centers of
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mass: 2 dm dA 3 1 x dx . The moment
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Chapter 6 Practice Exercises 493 6.
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(b) shell method: Chapter 6 Practic
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Chapter 6 Practice Exercises 497 x
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(a) Chapter 6 Additional and Advanc
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10. Converting to pounds and feet,
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CHAPTER 7 INTEGRALS AND TRANSCENDEN
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Section 7.1 The Logarithm Defined a
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Section 7.1 The Logarithm Defined a
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61. y = ln kx y = ln x + ln k; thus
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Section 7.2 Exponential Change and
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Section 7.2 Exponential Change and
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37. 0 kt A A e and Section 7.2 Expo
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Section 7.3 Hyperbolic Functions 52
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Section 7.4 Relative Rates of Growt
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Chapter 7 Practice Exercises 535 CH
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Chapter 7 Practice Exercises 537 (d
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CHAPTER 8 TECHNIQUES OF INTEGRATION
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Section 8.1 Using Basic Integration
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Section 8.2 Integration by Parts 55
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Section 8.3 Trigonometric Integrals
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Section 8.4 Trigonometric Substitut
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Section 8.5 Integration of Rational
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Section 8.6 Integral Tables and Com
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Section 8.7 Numerical Integration 6
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Section 8.8 Improper Integrals 617
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1 sin x 56. 2 x 1 sin x dx; 0 2 2 2
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Section 8.9 Probability 629 In orde
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Section 8.9 Probability 631 20. 3/2
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Section 8.9 Probability 633 36. For
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Section 8.9 Probability 635 52 . 20
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Chapter 8 Practice Exercises 639 (3
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Chapter 8 Practice Exercises 643 d
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Chapter 8 Practice Exercises 647 3/
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Chapter 8 Practice Exercises 649 li
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t b t t Chapter 8 Additional and Ad
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CHAPTER 9 FIRST-ORDER DIFFERENTIAL
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Section 9.1 Solutions, Slope Fields
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Section 9.2 First-Order Linear Equa
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Section 9.2 First-Order Linear Equa
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Section 9.3 Applications 675 6. 2 2
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14. (a) dV (5 3) 2 V 100 2t dt The
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Section 9.4 Graphical Solutions of
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Section 9.4 Graphical Solution of A
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Implies coexistence is not possible
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Section 9.5 Systems of Equations an
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Section 9.5 Systems of Equations an
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CHAPTER 10 INFINITE SEQUENCES AND S
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Section 10.1 Sequences 703 40. 1 n
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Section 10.1 Sequences 705 70. 1 n
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95. Since a n converges Section 10.
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111. 3( n 1) 1 3 1 3 4 3 1 2 2 a n
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Section 10.1 Sequences 711 133. a2k
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Section 10.2 Infinite Series 713 12
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6. f ( x ) 1 is positive, continuou
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Section 10.3 The Integral Test 723
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Section 10.4 Comparison Tests 729 4
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Section 10.5 Absolute Convergence;
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Section 10.6 Alternating Series and
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Section 10.7 Power Series 753 6. n
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Section 10.7 Power Series 755 17. l
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Section 10.7 Power Series 757 25. n
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Section 10.7 Power Series 759 1 2 1
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Section 10.7 Power Series 761 inter
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Section 10.7 Power Series 763 3 5 1
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Section 10.8 Taylor and Maclaurin S
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Section 10.8 Taylor and Maclaurin S
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Section 10.9 Convergence of Taylor
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Section 10.10 The Binomial Series a
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Chapter 10 Additional and Advanced
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CHAPTER 11 PARAMETRIC EQUATIONS AND
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Section 11.1 Parametrizations of Pl
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Section 11.2 Calculus with Parametr
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Section 11.3 Polar Coordinates 819
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Section 11.4 Graphing Polar Coordin
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9. r 2cos and r 2sin 2cos 2sin cos
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17. r sec and A r 2 4cos 4cos sec c
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Section 11.5 Areas and Lengths in P
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Section 11.6 Conic Sections 839 9.
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Section 11.7 Conics in Polar Coordi
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75. (a) Perihelion a ae a(1 e ), Ap
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CHAPTER 11 ADDITIONAL AND ADVANCED
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CHAPTER 12 VECTORS AND THE GEOMETRY
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Section 12.1 Three-Dimensional Coor
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Section 12.2 Vectors 881 66. 2 2 2
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Section 12.2 Vectors 883 25. length
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Section 12.2 Vectors 885 F 1 sin 40
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Section 12.3 The Dot Product 887 11
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Section 12.4 The Cross Product 893
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23. (a) u v 6, u w 81, v w 18 none
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Section 12.4 The Cross Product 897
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Section 12.5 Lines and Planes in Sp
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L1 & L3: x 3 2t 3 2r 2t 2r 0 t r 0
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Section 12.6 Cylinders and Quadric
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i j k Chapter 12 Practice Exercises
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Chapter 12 Practice Exercises 919 (
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CHAPTER 13 VECTOR-VALUED FUNCTIONS
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Section 13.1 Curves in Space and Th
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Section 13.1 Curves in Space and Th
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Section 13.2 Integrals of Vector Fu
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dr dt Section 13.2 Integrals of Vec
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t Section 13.3 Arc Length in Space
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Section 13.3 Arc Length in Space 94
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Section 13.4 Curvature and Normal V
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8. (a) Section 13.4 Curvature and N
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Section 13.5 Tangential and Normal
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Section 13.6 Velocity and Accelerat
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13. Assuming Earth has a circular o
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Chapter 13 Practice Exercises 963 i
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i j k 9. (a) ur u cos sin 0 k a rig
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CHAPTER 14 PARTIAL DERIVATIVES 14.1
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17. (a) Domain: all points in the x
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Section 14.1 Functions of Several V
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44. (a) (b) Section 14.1 Functions
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Section 14.2 Limits and Continuity
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Section 14.3 Partial Derivatives 99
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Section 14.4 The Chain Rule 1001 w
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Section 14.4 The Chain Rule 1003 18
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Section 14.4 The Chain Rule 1005 y
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Section 14.4 The Chain Rule 1007 46
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Section 14.5 Directional Derivative
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Section 14.6 Tangent Planes and Dif
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Section 14.7 Extreme Values and Sad
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2 33. (i) On OA, f ( x, y) f (0, y)
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38. (i) On OA, f ( x, y) f (0, y) 2
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5. We optimize Section 14.8 Lagrang
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15. T (8x 4 y) i ( 4x 2 y) j and Se
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Section 14.8 Lagrange Multipliers 1
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34. Q( p, q, r) 2( pq pr qr ) and G
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Section 14.8 Lagrange Multipliers 1
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49-54. Example CAS commands: Maple:
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Section 14.9 Taylors Formula for Tw
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Section 14.10 Partial Derivatives w
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Section 14.10 Partial Derivatives w
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Chapter 14 Practice Exercises 1059
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74. (i) On OA, 2 f ( x, y) f (0, y)
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(iii) On CD, 3 f ( x, y) f (1, y) y
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96. Let Chapter 14 Practice Exercis
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21. (a) k is a vector normal to wil
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CHAPTER 15 MULTIPLE INTEGRALS 15.1
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Section 15.1 Double and Iterated In
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7. 8. Section 15.2 Double Integrals
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Section 15.2 Double Integrals Over
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Section 15.3 Area by Double Integra
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Section 15.3 Area by Double Integra
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Section 15.4 Double Integrals in Po
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6. The projection of D onto the xy
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Section 15.5 Triple Integrals in Re
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Section 15.5 Triple Integrals in Re
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Section 15.7 Triple Integrals in Cy
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66. average 1 2 3 Section 15.7 Trip
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Section 15.8 Substitutions in Multi
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53. x u y and y v x u v and y v 1 1
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CHAPTER 16 INTEGRALS AND VECTOR FIE
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Section 16.1 Line Integrals 1157 1
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Section 16.1 Line Integrals 1159 34
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Section 16.2 Vector Fields and Line
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(b) Section 16.2 Vector Fields and
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11. 2 2 Section 16.3 Path Independe
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Section 16.3 Path Independence, Pot
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Section 16.3 Path Independence, Pot
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Section 16.4 Greens Theorem in the
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Section 16.5 Surfaces and Area 1185
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(b) In a fashion similar to cylindr
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Section 16.6 Surface Integrals 1197
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Section 16.7 Stokes Theorem 1207 i
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Section 16.7 Stokes Theorem 1209 2
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Section 16.7 Stokes Theorem 1211 C
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Section 16.8 The Divergence Theorem
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30. By Exercise 29, 2 f g d f g f g
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CHAPTER 17 SECOND-ORDER DIFFERENTIA
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ww w w 2 2 Section 17.1 Second-Orde
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Section 17.1 Second-Order Linear Eq
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Section 17.2 Nonhomogeneous Linear
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Section 17.2 Nonhomogeneous Linear
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2 dy dx dy dx Section 17.2 Nonhomog
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dy w dx œc1sin x c2cos x cos xlnks
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Section 17.3 Applications 1017 ! !
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5 10 5È 2 2 5 dy 5È2 dy 1 16 $ È
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Section 17.4 Euler Equations 1021 "
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Section 17.5 Power-Series Soutions
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∞ ∞ ∞ # ww w # 5. x y 2xy 2
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# ww # 11. x 1y 6y 0 x 1 n2 nn 1c x
Page 2535:
∞ ∞ ∞ ww w 17. y xy 3y œ 0
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Thomas Calculus 13th [Solutions]

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