Instructor's Solutions Manual to Thomas' Calculus: Early Transcendentals, 13th Edition

1267 Pages·2013·74.994 MB·English

by  George B. Thomas , Maurice D. Weir , Joel R. Hass

TABLE OF CONTENTS
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 8
1.3 Trigonometric Functions 18
1.4 Graphing with Software 26
1.5 Exponential Functions 31
1.6 Inverse Functions and Logarithms 34
Practice Exercises 43
Additional and Advanced Exercises 52
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 62
2.3 The Precise Definition of a Limit 73
2.4 One-Sided Limits 81
2.5 Continuity 86
2.6 Limits Involving Infinity; Asymptotes of Graphs 92
Practice Exercises 102
Additional and Advanced Exercises 108
3 Differentiation 115
3.1 Tangents and the Derivative at a Point 115
3.2 The Derivative as a Function 121
3.3 Differentiation Rules 131
3.4 The Derivative as a Rate of Change 138
3.5 Derivatives of Trigonometric Functions 144
3.6 The Chain Rule 152
3.7 Implicit Differentiation 162
3.8 Derivatives of Inverse Functions and Logarithms 170
3.9 Inverse Trigonometric Functions 180
3.10 Related Rates 186
3.11 Linearization and Differentials 192
Practice Exercises 199
Additional and Advanced Exercises 214
4 Applications of Derivatives 219
4.1 Extreme Values of Functions 219
4.2 The Mean Value Theorem 233
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 253
4.5 Indeterminate Forms and L(cid:1)H(cid:244)pital(cid:1)s Rule 280
4.6 Applied Optimization 290
4.7 Newton’s Method 304
4.8 Antiderivatives 309
Practice Exercises 318
Additional and Advanced Exercises 336
5 Integration 343
5.1 Area and Estimating with Finite Sums 343
5.2 Sigma Notation and Limits of Finite Sums 348
5.3 The Definite Integral 354
5.4 The Fundamental Theorem of Calculus 369
5.5 Indefinite Integrals and the Substitution Method 379
5.6 Substitution and Area Between Curves 387
Practice Exercises 407
Additional and Advanced Exercises 422
6 Applications of Definite Integrals 431
6.1 Volumes Using Cross-Sections 431
6.2 Volumes Using Cylindrical Shells 443
6.3 Arc Length 454
6.4 Areas of Surfaces of Revolution 462
6.5 Work and Fluid Forces 468
6.6 Moments and Centers of Mass 479
Practice Exercises 492
Additional and Advanced Exercises 501
7 Integrals and Transcendental Functions 507
7.1 The Logarithm Defined as an Integral 507
7.2 Exponential Change and Separable Differential Equations 515
7.3 Hyperbolic Functions 521
7.4 Relative Rates of Growth 529
Practice Exercises 535
Additional and Advanced Exercises 540
8 Techniques of Integration 543
8.1 Using Basic Integration Formulas 543
8.2 Integration by Parts 555
8.3 Trigonometric Integrals 569
8.4 Trigonometric Substitutions 577
8.5 Integration of Rational Functions by Partial Fractions 585
8.6 Integral Tables and Computer Algebra Systems 594
8.7 Numerical Integration 607
8.8 Improper Integrals 617
8.9 Probability 629
Practice Exercises 637
Additional and Advanced Exercises 650
9 First-Order Differential Equations 661
9.1 Solutions, Slope Fields, and Euler’s Method 661
9.2 First-Order Linear Equations 670
9.3 Applications 674
9.4 Graphical Solutions of Autonomous Equations 678
9.5 Systems of Equations and Phase Planes 986
Practice Exercises 692
Additional and Advanced Exercises 698
10 Infinite Sequences and Series 701
10.1 Sequences 701
10.2 Infinite Series 712
10.3 The Integral Test 720
10.4 Comparison Tests 728
10.5 Absolute Convergence; The Ratio and Root Tests 738
10.6 Alternating Series and Conditional Convergence 744
10.7 Power Series 752
10.8 Taylor and Maclaurin Series 764
10.9 Convergence of Taylor Series 769
10.10 The Binomial Series and Applications of Taylor Series 777
Practice Exercises 786
Additional and Advanced Exercises 795
11 Parametric Equations and Polar Coordinates 801
11.1 Parametrizations of Plane Curves 801
11.2 Calculus with Parametric Curves 809
11.3 Polar Coordinates 819
11.4 Graphing Polar Coordinate Equations 825
11.5 Areas and Lengths in Polar Coordinates 832
11.6 Conic Sections 838
11.7 Conics in Polar Coordinates 849
Practice Exercises 860
Additional and Advanced Exercises 871

CHAPTER 1 FUNCTIONS
1.1 FUNCTIONS AND THEIR GRAPHS
1. domain ( , );range [1, ) 2. domain [0, );range ( ,1]
3. domain [ 2, );y in range and y 5x 10 0 y can be any positive real number range [0, ).
4. domain ( ,0] [3, );y in range and y x2 3x 0 y can be any positive real number
range [0, ).
5. domain ( ,3) (3, );y in range and y 4 , now if t 3 3 t 0 4 0, or if t 3
3 t 3 t
3 t 0 4 0 y can be any nonzero real number range ( ,0) (0, ).
3 t
6. domain ( , 4) ( 4,4) (4, );y in range and y 2 , now if t 4 t2 16 0 2 0, or if
t2 16 t2 16
4 t 4 16 t2 16 0 2 2 , or if t 4 t2 16 0 2 0 y can be any nonzero
16 t2 16 t2 16
real number range ( , 1] (0, ).
8
7. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Is the graph of a function of x since any vertical line intersects the graph at most once.
8. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Not the graph of a function of x since it fails the vertical line test.
2
9. base x;(height)2 x x2 height 3x; area is a(x) 1 (base)(height) 1(x) 3x 3x2;
2 2 2 2 2 4
perimeter is p(x) x x x 3x.
10. s sidelength s2 s2 d2 s d ; and area is a s2 a 1d2
2 2
2 2 2
11. Let D diagonal length of a face of the cube and the length of an edge. Then D d and
3/2
D2 2 2 3 2 d2 d . The surface area is 6 2 6d2 2d2 and the volume is 3 d2 d3 .
3 3 3 3 3
12. The coordinates of P are x, x so the slope of the line joining P to the origin is m x 1 (x 0).
x x
Thus, x, x 1 , 1 .
m2 m
13. 2x 4y 5 y 1x 5;L (x 0)2 (y 0)2 x2 ( 1x 5)2 x2 1x2 5x 25
2 4 2 4 4 4 16
5x2 5x 25 20x2 20x 25 20x2 20x 25
4 4 16 16 4
2 2 2 2 2 2 2 2 2
14. y x 3 y 3 x;L (x 4) (y 0) (y 3 4) y (y 1) y
y4 2y2 1 y2 y4 y2 1
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2 Chapter 1 Functions
15. The domain is ( , ). 16. The domain is ( , ).
17. The domain is ( , ). 18. The domain is ( ,0].
19. The domain is ( ,0) (0, ). 20. The domain is ( ,0) (0, ).
21. The domain is ( , 5) ( 5, 3] [3,5) (5, ) 22. The range is [2, 3).
23. Neither graph passes the vertical line test
(a) (b)
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Section 1.1 Functions and Their Graphs 3
24. Neither graph passes the vertical line test
(a) (b)
x y 1 y 1 x
x y 1 or or
x y 1 y 1 x
25. x 0 1 2 26. x 0 1 2
y 0 1 0 y 1 0 0
4 x2, x 1 1, x 0
27. F(x) 28. G(x) x
x2 2x, x 1 x, 0 x
29. (a) Line through (0, 0) and (1, 1): y x; Line through (1, 1) and (2, 0): y x 2
x, 0 x 1
f(x)
x 2, 1 x 2
2, 0 x 1
0, 1 x 2
(b) f(x)
2, 2 x 3
0, 3 x 4
30. (a) Line through (0, 2) and (2, 0): y x 2
Line through (2, 1) and (5, 0): m 0 1 1 1, so y 1(x 2) 1 1x 5
5 2 3 3 3 3 3
x 2, 0 x 2
f(x)
1x 5, 2 x 5
3 3
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4 Chapter 1 Functions
3 0
(b) Line through ( 1,0) and (0, 3):m 3, so y 3x 3
0 ( 1)
Line through (0, 3) and (2, 1): m 1 3 4 2, so y 2x 3
2 0 2
3x 3, 1 x 0
f(x)
2x 3, 0 x 2
31. (a) Line through ( 1,1) and (0, 0): y x
Line through (0, 1) and (1, 1): y 1
Line through (1, 1) and (3, 0): m 0 1 1 1, so y 1(x 1) 1 1x 3
3 1 2 2 2 2 2
x 1 x 0
f(x) 1 0 x 1
1x 3 1 x 3 1x 2 x 0
2 2 2
(b) Line through ( 2, 1) and (0, 0): y 1x f(x) 2x 2 0 x 1
2
Line through (0, 2) and (1, 0): y 2x 2 1 1 x 3
Line through (1, 1) and (3, 1): y 1
32. (a) Line through T,0 and (T, 1): m 1 0 2, so y 2 x T 0 2 x 1
2 T (T/2) T T 2 T
0, 0 x T
2
f(x)
2 x 1, T x T
T 2
A, 0 x T
2
A, T x T
2
(b) f(x)
A, T x 3T
2
A, 3T x 2T
2
33. (a) x 0for x [0,1) (b) x 0for x ( 1,0]
34. x x only when x is an integer.
35. For any real number x,n x n 1, where n is an integer. Now: n x n 1 (n 1) x n.
By definition: x nand x n x n. So x x for all real x.
36. To find f(x) you delete the decimal or
fractional portion of x, leaving only
the integer part.
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