Calculus & Its Applications, Brief Version, 15th Edition — Goldstein — Solutions

STUDY NOTES & REFERENCE GUIDE




Covers All Chapters




SOLUTIONS MANUAL




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CONTENTS
Chapter 0 Functions ......................................................................................... 1
Chapter 1 The Derivative ............................................................................... 26
Chapter 2 Applications of the Derivative.........................................................77
Chapter 3 Techniques of Differentiation........................................................ 115
Chapter 4 The Exponential and Natural Logarithmic Functions..................... 141
Chapter 5 Applications of the Exponential and Natural Logarithm
Functions ...................................................................................... 168
Chapter 6 The Definite Integral.....................................................................182
Chapter 7 Functions of Several Variables ...................................................... 215
Chapter 8 The Trigonometric Functions ........................................................ 245
Chapter 9 Techniques of Integration. ............................................................ 263
Chapter 10 Differential Equations ...................................................................302
Chapter 11 Taylor Polynomials and Infinite Series.......................................... 334
Chapter 12 Probability and Calculus ............................................................... 356




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Chapter 0 Functions
0.1 Functions and Their Graphs t
16. h(t)
1. (1 t)
1 1
1 1
2. h 2 2
3
2 1 1 3
2 2
3 3
3. 3
h 2 2
1
3
2 1 3
4. 2 2
a 1 a 1
h(a 1)
5. 1 (a 1) a 2

6. 17. ƒ (x) x 2 2x

3 ƒ (a 1) (a 1) 2 2(a 1)
7. [2, 3) 8. 1, (a 2 2a 1) 2a 2 a2 1
2
ƒ (a 2) (a 2) 2 2(a 2)
9. [–1, 0) 10. [–1, 8)
(a 2 4a 4) 2a 4 a 2 2a
11. ,3 12. 2,
18. ƒ (x) x 2 4x 3
13. ƒ (x) x 2 3x ƒ (a 1) (a 1) 2 4(a 1) 3
ƒ (0) 02 3(0) 0 (a 2 2a 1) (4a 4) 3
2 a 2 2a
ƒ (5) 5 3(5) 25 15 10
ƒ (a 2) (a 2) 2 4(a 2) 3
ƒ (3) 3 2
3(3) 9 9 0
(a 2 4a 4) (4a 8) 3
ƒ ( 7) ( 7) 2
3( 7) 49 21 70 a 2
1
19. a. ƒ (0) represents the number of laptops sold
14. ƒ (x) x3 x 2 x 1
in 2010.
ƒ (1) 1 1 3 2
1 1 0
b. ƒ (6) 150 2(6) 62
ƒ ( 1) ( 1)3 ( 1)2 ( 1) 1 0
150 12 36 198
3 2
1 1 1 1 9
ƒ 1 100x
2 2 2 2 8 20. R(x) , x 0
b x
ƒ (a) a3 a 2 a 1 a. b = 20, x 60
100(60)
15. g(t) t3 3t 2 t R(60) 75
20 60
g(2) 23 3(2)2 2 8 12 2 2 The solution produces a 75% response.
1 1 3
1 2
1 b. If R(50) = 60, then
g 3
2 2 2 2 100(50)
60
1 3 1 11 b 50
8 4 2 8 60b 3000 5000
100
2 2 3
2 2 2 b
g 3 3
3 3 3 3 This particular frog has a positive constant
8 12 2 10
.37037 of 33.3.
27 9 3 27
g(a) a 3a 2 a
3




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2


8x 39. 40.
21. ƒ (x)
(x 1)(x 2) 41. , 1 5, 9 42. 1, 5 9,
all real numbers such that x 1, 2 or
, 1 1, 2 2, 43. ƒ 1 .03; ƒ 5 .037

1 44. ƒ 6 .03
22. ƒ (t)
45. 0, .05 46. t 3
all real numbers such that t > 0 or 0,
1
1 47. ƒ (x) x x 2
23. g(x) ____ 2
3 x
1 25
all real numbers such that x < 3 or , 3 ƒ (3) 3 (3 2)
2 2
4 Thus, (3, 12) is not on the graph.
24. g(x)
x(x 2) 48. ƒ(x) = x(5 + x)(4 – x)
all real numbers such that x 0, –2 or ƒ(–2) = –2(5 + (–2))(4 – (–2)) = –36
, 2 2, 0 0, So (–2, 12) is not on the graph.

3x 5 49. g(x) 3x 1
25. ƒ x x2 1
x2 x 6
The denominator is zero for x x = 2, 1 3 21 1 1
2
g 2
so the domain consists of all real numbers 3 1 2 5 5
such that x 2
1 4

, 3 3, 2 2, . 1 2
So , is on the graph.
2 5
1
26. ƒ x
2
3x 1 (x 2 4)
The domain consists of all real numbers, or 50. g(x)
(x 2)
, . 2
40
2 4
g 2 3 9 5
27. ƒ x 2x 7 x 3 2
2 8 3
3 3
The domain consists of all real numbers 2 5
So , is on the graph.
greater than or equal to 0, or 0, .
3 3
2x 1 51. ƒ (x) x3
28. ƒ x
1 x
ƒ (a 1) (a 1)3
The denominator is greater than 0 for all x < 1,
1 5
and the numerator is defined for all x . 52. ƒ (x) x
2 x
1 1,1 .
Thus, the domain is x 1, or 5
2 2 ƒ (2 h) (2 h)
(2 h)
29. function 30. not a function 5 (2 h)2 1 4h h 2
31. not a function 32. not a function (2 h) 2 h

33. not a function 34. function x for 0 x 2
53. ƒ (x)
35. ƒ 0 1; ƒ 7 1 1 x for 2 x 5
ƒ (1) 1 1; ƒ (2) 1 2 3
36. ƒ 2 3; ƒ 1 0
ƒ (3) 1 3 4
37. positive 38. negative




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