Smith R., Minton R. Calculus
Подождите немного. Документ загружается.
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
Appendix B
ANSWERS TO ODD-NUMBERED EXERCISES
CHAPTER 0
Exercises 0.1, page 7
1. x < 3 3. x < −2 5. x > −
2
3
7. 3 ≤ x < 6 9. −
1
2
< x < 2
11. x < −4orx > 1 13. −2 < x < 3
15. all reals 17. −1 < x < 7
19. 2 < x < 4 21. x < −
3
2
or x >
1
2
23. x < −2orx > 2
25. x < −4or−4 < x < −1orx > 2
27. x < −1orx > 0 29.
√
13 31. 4
33.
√
20 35. yes 37. yes
39. increases by 550, 650, 750; predict 3200 + 850 = 4050
41. decreases by 10, 30, 50; predict 3910 − 70 = 3840
43. finite number of digits terminate
45.
1
2
7
−
2
3
12
1
2
7
= 0.013
47. P: 0.551, 0.587, 0.404, 0.538, 0.605
win%: 0.568, 0.593, 0.414, 0.556, 0.615
Exercises 0.2, page 18
1. yes 3. no 5. 2
7. −
5
2
9. −
5
7
11. (2, 5), y = 2(x − 1) +3
y
x
2
4
6
8
10
12
1056432
13. (0, 1), y = 1
y
x
0.5
1
1.5
2
1056432
15. (3.3, 2.3), y = 1.2(x − 2.3) +1.1
y
x
1
0
1
2
3
4
5
6
5
4321
17. parallel 19. perpendicular 21. perpendicular
23. (a) y = 2(x − 2) +1 (b) y =−
1
2
(x − 2) +1
25. (a) y = 2(x − 3) +1 (b) y =−
1
2
(x − 3) +1
27. y = 2(x − 1) +1; 7 29. yes
31. no 33. both 35. rational 37. neither
39. x ≥−2 41. x ≤
3
2
43. all reals 45. x =±1
47. −1, 1, 11, −
5
4
49. 1, 2, 0,
3
2
51. x > 0
53. 0 ≤ x ≤ number made, x an integer 55. no: many y’s for one x
57. no: many y’s for one x
59. constant, increasing, decreasing; graph going down;
graph going up
61. x-intercepts: −2, 4; y-intercept: −8
63. x-intercept: 2; y-intercept: −8
65. x-intercepts: ±2; y-intercept: −4
67. 1, 3 69. 2 +
√
2, 2 −
√
2 71. 0, 1, 2 73. 1, −
3
√
2
75. 63,000 feet 77. T =
1
4
R + 39 79. 51
A-13
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-14 APPENDIX B
..
Answers to Odd-Numbered Exercises
Exercises 0.3, page 26
1. (a)
5
5
5
5
y
x
(b)
10 50510
10
15
5
y
x
3. (a)
5
10
50
10
y
x
(b)
60
30
y
5
5
x
5. (a)
5
2
5
10
x
y
(b)
58
20
x
y
7. (a)
5
4
5
10
y
x
(b)
5
30
5
30
x
y
9. (a)
10
10
10
10
x
y
(b)
10
5
10
10
y
x
(c)
5
10
30
10
y
x
11. (a)
5
5
55
x
y
(b)
50
50
y
5
5
x
(c)
5
5
y
5
5
x
13. (a)
0.1
0.9
66
x
y
(b)
0.75
y
10
10
x
15. (a) (b)
4
4
88
x
y
10
10
y
10
10
x
17. x =−2, x = 2 19. x =−5, x = 2
21. x =−2, x =−1, x = 0
23.
1E4
1E4
0.1 0.1
x
y
25.
80
80
14
14
x
y
27.
5
5
y
5
5
x
29. 2; 1, 1.2 (approx.) 31. 1, −1
33. 2; 0, 9.53 (approx.) 35. 2; −1.18, 1.18 (approx.)
37. −1.88, 0.35, 1.53 39. 0.56, 3.07 41. −5.25, 10.01
43. possible answer: −9 ≤ x ≤ 11, −17 ≤ y ≤ 23
45. parabola y =
1
4
x
2
+ 1
Exercises 0.4, page 34
1. (a) 45
◦
(b) 60
◦
(c) 30
◦
(d) 240
◦
3. (a) π (b)
3π
2
(c)
2π
3
(d)
π
6
5. −
π
3
+ 2nπ;
π
3
+ 2nπ 7. −
π
4
+ 2nπ;
π
4
+ 2nπ 9.
π
2
+ 2nπ
11.
π
2
+ nπ;2nπ 13. π + 2nπ;
π
2
+ nπ
15.
2
1
2
1
x
y
17.
10
10
x
y
19.
2
3
2
3
y
x
21.
2
3
2
3
y
x
23.
1
1
2
2
y
x
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-15
25. A = 3, period = π, frequency =
1
π
27. A = 5, period =
2π
3
, f =
3
2π
29. A = 3, period = π, f =
1
π
31. A = 4, period = 2π, f =
1
2π
37. β ≈ 0.6435 39. no 41. yes, 2π 43.
2
√
2
3
45. −
√
3
2
47. 3; x = 0, x ≈ 1.109, x ≈ 3.698
49. 2; x ≈−1.455, x ≈ 1.455
51. 2tan 20
◦
≈ 0.73 mile
53. 100tan 50
◦
≈ 119 feet
55. f =
30
π
,
170
√
2
≈ 120.2 volts 57. $24,000 per year
Exercises 0.5, page 41
1. ( f ◦ g)(x) =
√
x − 3 +1, x ≥ 3
(g ◦ f )(x) =
√
x − 2, x ≥ 2
3. ( f ◦ g)(x) =
1
x
3
+ 4
, x =
3
√
−4
(g ◦ f )(x) =
1
x
3
+ 4, x = 0
5. ( f ◦ g)(x) = sin
2
x + 1, all reals
(g ◦ f )(x) = sin(x
2
+ 1), all reals
7. possible answer: f (x) =
√
x, g(x) = x
4
+ 1
9. possible answer: f (x) =
1
x
, g(x) = x
2
+ 1
11. possible answer: f (x) = x
2
+ 3, g(x) = 4x + 1
13. possible answer: f (x) = x
3
, g(x) = sin x
15. possible answer: f (x) =
3
x
, g(x) =
√
x, h(x) = sin x + 2
17. possible answer: f (x) = x
3
, g(x) = cos x, h(x) = 4x − 2
19. possible answer: f (x) = 4x − 5, g(x) = cosx, h(x) = x
2
21.
y
1
x
0
1
23.
y
1
x
0
1
25.
y
1
x
0
0.5
27.
20
5
y
4
4
x
29.
y
2
x
0
1
31.
y
2
x
0.5
33.
y
2
x
0
1
3
35.
y
2
x
0
1
3
37. y = (x + 1)
2
, shift left one
39. y = (x + 1)
2
+ 3, shift left one, up three
41. y = 2[(x + 1)
2
+ 1], shift left one, up one, double vertical scale
43. reflect across x-axis, double vertical scale
45. reflect across x-axis, triple vertical scale, shift up two
47. reflect across y-axis
49. shift right one
51. reflect across x-axis, vertical scale times |c|
53.
y
x
04 2
120
100
80
60
40
20
24
57. go to 0 59. 0.739085
Chapter 0 Review Exercises, page 43
1. −2 3. parallel 5. no
7. y =
1
2
(x − 1) +1, y =
5
2
9. y =−
1
3
(x + 1) −1
11. yes 13. −2 ≤ x ≤ 2
15.
10
10
10
10
x
y
17.
5
2
5
10
y
x
19.
12
8
8
12
y
x
21.
3
1
3
1
x
y
23.
2
3
2
3
y
x
25.
5
5
y
x
4
3
4
3
4
4
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-16 APPENDIX B
..
Answers to Odd-Numbered Exercises
27. x =−4, x = 2, y =−8 29. x =−2 31. −2, 5
33. 1, 1 +
√
3, 1 −
√
3 35. 3 37. 50 tan34
◦
≈ 33.7 feet
39. (a)
1
√
5
(b)
1
9
41. ( f ◦ g)(x) = x − 1, x ≥ 1
(g ◦ f )(x) =
√
x
2
− 1, x ≤−1orx ≥ 1
43. f (x) = cos x, g(x) = 3x
2
+ 2
45. (x − 2)
2
− 3, shift two right and three down
47.
π
4
+ nπ
CHAPTER 1
Exercises 1.1, page 51
1. (a) 2 (b) 4 3. (a) 0 (b) −1 5. (a) 1 (b) 3.43
7. (a) 1.90626 (b) 1.90913 (c) 1.91010
9. (a) 3.16732 (b) 3.16771 (c) 3.16784
11. (a) 9.15298 (b) 9.25345 (c) 9.29357
13. (a) 1.55 (b) 1.56; quarter circle
15. (a)
11
8
(b)
43
32
17. (a) 2.05 (b) 2.01
Exercises 1.2, page 57
1. 2 3.
1
4
5. 3
7. (a) −2 (b) 2 (c) does not exist (d) 2 (e) 2
(f) 2 (g) 0 (h) 1
9. (a) 4 (b) 4 (c) 4 (d) 2 (e) 9
11. 2.2247, 2.0488, 2.0050, 2.0005 → 2; 13. 1 15. −1
1.7071, 1.9487, 1.9950, 1.9995 → 2
17.
3
2
19. limit does not exist 21. does not exist
23.
x
y
1–1
–3
3
25.
x
y
0
1
3
27. does not exist
29. The first argument is correct.
31. One possibility: f (x) =
sin x
x
, g(x) =
2x, x = 0
x + 1, x > 0
33. w → 0, h(w) → 0 and | f (w) −h(w)|→0
35. 8; does not exist 37. m =
1
2
Exercises 1.3, page 66
1. 1 3. 0 5. 5 7.
3
4
9. 1 11. cos(1) 13.
1
4
15. 2 17.
1
2
19. 2 21. 4 23. does not exist
25. 4 27. 1 29. 0, f (x) =−x
2
, h(x) = x
2
31. f (x) = 0, h(x) =
√
x 33. 4 35. 0 37.
1
2
39. h(a) 41. (a) −1 (b) −2 43. 13 45. −
4
3
47. 0
51. f (x) =
1
x
, g(x) =−
1
x
53. yes
57. for 2 ≤ x < 3, [x] = 2 and for 3 ≤ x < 4, [x] = 3,
so lim
x→3
−
[x] = lim
x→3
+
[x].
59. 0, does not exist
Exercises 1.4, page 75
1. x =−2; g(x) = x −1
3. x =±1, g(x) =
1
x + 1
5. all real numbers 7. x =
nπ
2
for odd integers n
9. x > 1 11. x = 1 13. x = 1
15. f (1) is not defined and lim
x→1
f (x) does not exist
17. f (0) is not defined and lim
x→0
f (x) does not exist
19. lim
x→2
f (x) = f (2)
21. [−3, ∞) 23. (−1, ∞)
25. (−∞, ∞). 27. −700
29. b = $12,747.50, c = $23,801.30
31. (a)
2
20
32
, 2
21
32
(b)
−2
21
32
, −2
20
32
33.
23
32
,
24
32
35. (−7, −2), (−2, −1), (1, 4), (4, 7)
37. a = b = 2 39. a =
1
3
, b =
nπ
3
41. no 43. #41 is 49. No
55. x =−2, x =−1, x = 0
57.
x
y
0 204060 10080
20
x 100,
force that moves box
40
60
80
100
Friction
61. One answer: g(T) = 100 −25(T−30)
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-17
Exercises 1.5, page 84
1. (a) ∞ (b) −∞ (c) does not exist
3. (a) −∞ (b) −∞ (c) −∞
5. does not exist 7. does not exist 9.
1
3
11. 1 13.
1
2
15. −2 17.
3
4
19. −∞
21. (a) vertical asymptotes at x =±2; horizontal asymptote at y = 0
(b) vertical asymptotes at x =±2; horizontal asymptote at y =−1
23. vertical asymptotes at x =−1 and x = 3; horizontal asymptote at
y = 3
25. vertical asymptotes at x = 2πn
27. horizontal asymptotes at y = 0
29. vertical asymptotes at x =±2; slant asymptote at y =−x
31. vertical asymptotes at x =−
1
2
±
17
4
; slant asymptote at y = x − 1
33. with no light, 40 mm; with an infinite amount of light, 12 mm
35. f (x) =
80x
−0.3
+ 60
10x
−0.3
+ 30
37. 1 39. 1
41. −
1
2
43. no. 45. one larger
47. −2(x − 3)
2
49. x
2
+ 1
51. true 53. false 55. true
57. g(x) = sinx, h(x) = x
59. vertical asymptotes at x = 2; horizontal asymptotes at
y = 4 (as x →−∞) and y = 0 (as x →∞)
61. ∞, c 63. v
e
=
√
19.6R
Exercises 1.6, page 96
1.
ε
3
3.
ε
3
5.
ε
4
7. δ ≤ ε 9. min
1,
ε
3
11. min
1,
ε
5
13.
ε
|m|
, no
15. (a)
√
0.1 ≈ 0.32 (b)
√
0.05 ≈ 0.22
17. (a) 0.39 (b) 0.19 21. (a) 0.02 (b) 0.02
23. 12 25. −3.4
27. N =−
1
ε
− 2, for 0 <ε≤
1
2
29. δ =
4
−
2
N
31. M =
k
1
ε
33. 2 35. 1.9 41. min
1,
ε
10
Exercises 1.7, page 103
1.
1
4
;
x
√
4x
2
+ 1 +2x
3. 1;
2
√
x
√
x + 4 +
√
x + 2
5. 1;
2x
√
x
2
+ 4 +
√
x
2
+ 2
7.
1
6
;
sin
2
2x
12x
2
(1 +cos2x)
9.
1
2
;
sin
2
(x
3
)
x
6
(1 +cos(x
3
))
11.
2
3
;
2x
4/3
3
(x
2
+ 1)
2
+
3
(x
2
+ 1)(x
2
− 1) +
3
(x
2
− 1)
2
13. 3, does not exist
15. f (x) = 0, g(x) = 0.0016, −0.0159, −0.1586, −0.9998
17. 20, 0
Chapter 1 Review Exercises, page 104
1. 2 3. (a) 1.05799 (b) 1.05807 5. 0
7. does not exist 9. −
1
3
11. (a) 1 (b) −2 (c) does not exist (d) 0
13. x =−1, x = 1 15.
3
4
17. does not exist
19. does not exist 21. 5 23.
2
3
25. ∞
27.
1
3
29. −∞ 31. 0 33. does not exist
37. x =−3, x = 1
39. x = 2 41. (−∞, −2) ∪(−2, 3) ∪ (3, ∞)
43. (−∞, ∞) 45. x = 1, x = 2, y = 0
47. x =−1, x = 1, y = 1 49. y = x −1
51. x = 2πn 53.
1
4
;
sin
2
x
2x
2
(1 +cos x)
CHAPTER 2
Exercises 2.1, page 115
1. y = 2(x − 1) −1 3. y =−7(x + 2) +10
5. y =−
1
2
(x − 1) +1 7. y =
1
2
(x + 2) +1
9. (a) 6 (b) 18 (c) 8.25 (d) 14.25 (e) 10.41
(f) 11.61 (g) 11
11. (a) 0.33 (b) 0.17 (c) 0.27 (d) 0.19 (e) 0.23
(f) 0.22 (g) 0.22
13. C, B, A, D
15. (a) −9.8 m/s (b) −19.6 m/s
19. (a) 32 ft/s (b) 48 ft/s (c) 62.4 ft/s
(d) 63.84 ft/s (e) 64 ft/s
21. (a) 2.236 ft/s (b) 1.472 ft/s (c) 1.351 ft/s
(d) 1.343 ft/s (e) 1.342 ft/s
23. sharp corner
25. jump discontinuity
27.
y
0.5
1.0
x
0
1
29. No tangent line
31. (a) from 2002 to 2004, the balance increased at an average rate of
$21,034 per year
35. (a) (
2/3, 5
2/3 +1), (−
2/3, −5
2/3 +1)
37. (a) y = 6(x − 1) +5 (b) x =−2, x = 1
39. −10; −4.5
41. about 1.75 hours; 1.5 hours; 4 hours; rest
Exercises 2.2, page 124
1. 3 3.
3
4
5. 6x 7. 3x
2
+ 2
9.
−3
(x + 1)
2
11.
3
2
√
3x + 1
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-18 APPENDIX B
..
Answers to Odd-Numbered Exercises
13. (a) (b)
8
6
2
4
−2
0
−10
10
10
8
4
6
0
−4
2
−6
−8
−2−4−6−8−10
5.0
−5.0
54321
0
−1−2−3−4
2.5
−5
0.0
−2.5
15. (a) (b)
y
2
x
01
4
−1
y
4
23
x
0
0
5
3
1
−3
−1−3 1−5 −4 −2
−2
−4
−5
2
5
17. (a) (b)
y
8
8
4
4
−4
−8
0
x
10
10
6
2
6
0
−2
−6
2
−10
−2
−4
−6−8
−10
2
4
8
−4
64
−2
0−4
8
10
6
10
2
−8
0
−10
−6
−2−8 −6−10
y
x
19. D
+
f (0) = 3, D
−
f (0) = 2; no
21. D
+
f (0) = D
−
f (0) = 0; yes 23. 10
25. (a) x = 0, x = 2 (b) x = 0, x = 4
27. p ≥ 1 29. f (x) =−1 − x
2
31.
f (a) f
(a)
a
33. f
(1),
f (1.5) − f (1)
0.5
, f (2) − f (1), f (1)
37. 2x,3x
2
,4x
3
, nx
n−1
39. D
−
f (0) does not exist
43. 1.64 degrees per meter
45. (a) 0.4 ton per year (b) 0.2 ton per year
47. (a) meters per second (b) items per dollar
49. losing value; sell
53. f
(t) =
⎧
⎨
⎩
0, 0 < t < 20
10, 20 < t < 80
8, t > 80
Exercises 2.3, page 133
1. 3x
2
− 2 3. 9t
2
−
1
√
t
5. −
3
w
2
− 8
7.
−10
3
x
−4/3
− 2 9. 3s
1/2
+ s
−4/3
11.
3
2
−
1
2
x
−2
13. 9x
2
−
3
2
x
1/2
15. 12t
2
+ 6 17. 24x
2
−
9
4
x
−5/2
19. 24
21. v(t) =−32t + 40, a(t) =−32
23. v(t) =
1
2
t
−1/2
+ 4t, a(t) =−
1
4
t
−3/2
+ 4
25. (a) v(1) = 8 (going up); a(1) =−32
(b) v(2) =−24 (going down); a(2) =−32
27. y = 4(x − 2) +2 29. y =−x + 4
31. (a) (b)
y
4
4
2
2
−2
−4
0
x
5
5
3
1
3
0
−1
−3
1
−5
−1−2−3−4−5
−10
5
0
0
−5
10
10
5
−5
−10
33. x =−1 (peak); x = 1 (trough); x =
±
2
3
,
±
4
3
35. (a) x = 0; vertical tangent (b) x = 5; sharp corner
(c) x =−1, x = 4; sharp corners
37. (a) x =±
4
3
(b) x =±
√
1 +3
−3/2
39. (a)
3
2
x
2
+ 2x −2 (b)
1
2
x
2
+ 5x
41. 2 43. (a) f
(x) (b) 0 45. x
4
47.
2
3
x
3/2
49. b >
4
9c
2
51. f
(2000) ≈ 174.4, f
(2000) ≈−160
Exercises 2.4, page 140
1. 2x(x
3
− 3x +1) + (x
2
+ 3)(3x
2
− 3)
3.
1
2
x
−1/2
+ 3
5x
2
−
3
x
+ (
√
x + 3x)(10x + 3x
−2
)
5.
3(5t + 1) −(3t − 2)5
(5t + 1)
2
=
13
(5t + 1)
2
7.
(3 −3x
−1/2
)(5x
2
− 2) −(3x − 6
√
x)10x
(5x
2
− 2)
2
9.
(2u − 1)(u
2
− 5u + 1) −(u
2
− u − 2)(2u − 5)
(u
2
− 5u + 1)
2
11.
3
2
x
1/2
+
3
2
x
−1/2
+ x
−3/2
13.
4
3
t
1/3
+ 3
15. 2x
x
3
+ 3x
2
x
2
+ 2
+ (x
2
− 1)
(3x
2
+ 6x)(x
2
+ 2) −(x
3
+ 3x
2
)(2x)
(x
2
+ 2)
2
17. y = 2x 19. y =
1
4
x +
1
2
21. (a) y =−2x − 3 (b) y = 7(x − 1) −2
23. (a) y =−(x − 1) −2 (b) y = 0
25. P
(t) = 0.03P(t); 3 − 4 =−1
27. $65,000 per year.
29.
19.125
(m + 0.15)
2
; bigger bat gives greater speed
31.
−14.11
(m + 0.05)
2
; heavier club gives less speed
33. f
(x)g(x)h(x) + f (x)g
(x)h(x) + f (x)g(x)h
(x)
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-19
35.
2
3
x
−1/3
(x
2
− 2)(x
3
− x + 1) + x
2/3
(2x)(x
3
− x + 1) +
x
2/3
(x
2
− 2)(3x
2
− 1)
39. maximum slope at x = 0; minimum slope at x =±
√
3
45. F
(x) = f
(x)g(x) + 3 f
(x)g
(x) + 3 f
(x)g
(x) + f (x)g
(x)
49. 0; 1;
2.7x
1.7
(1+x
2.7
)
2
Exercises 2.5, page 145
1. 6x
2
(x
3
− 1) 3. 6x(x
2
+ 1)
2
5. (a) (9x
2
− 3)(x
3
− x)
2
(b)
x
√
x
2
+ 4
7. (a) 5t
4
t
3
+ 2 +
3t
7
2
√
t
3
+ 2
(b)
7
2
t
5/2
+ t
−1/2
9. (a)
u
2
+ 8u − 1
(u + 4)
2
(b)
12w
2
− w
4
(w
2
+ 4)
3
11. (a)
1
(x
2
+ 1)
3/2
(b)
1
2
(1 − x
2
)x
−1/2
(x
2
+ 1)
−3/2
13. (a) −6x(x
2
+ 4)
−3/2
(b)
x
6
√
x
2
+ 4
15. (a) −2(
√
x
3
+ 2 +2x)
−3
3x
2
2
√
x
3
+2
+ 2
(b)
3x
2
−4x
−3
2
√
x
3
+2+2x
−2
17. chain rule; product
19. product rule; chain, quotient
21. y =
3
5
(x − 3) +5 23.
1
√
3
25. −(2x + 1)
−3/2
;3(2x + 1)
−5/2
;−15(2x + 1)
−7/2
;
(−1)
n+1
3.5···(2n −3)(2x + 1)
−(2n+1)/2
27. −6 31. (a) 2 xf
x
2
(b) 2 f
(
x
)
f
(
x
)
33. (a) −
f
(1/x)
x
2
(b) −
f
(x)
[ f (x)]
2
35. (a) 3 (b) does not exist (c) 9
37. (a) 4(x
2
+ 4)
−3/2
(b)
8 −4t
2
(t
2
+ 4)
5/2
39. (a) x = 0, x = 1, x = 2; vertical tangents
(b) x = 0; vertical tangent
41. (a)
1
3
(x
2
+ 3)
3
43.
√
x
2
+ 1
Exercises 2.6, page 153
1. 12cos 3x − 1 3. 6 tan
2
2t sec
2
2t + 12csc
4
3t cot3t
5. cos(5x
2
) −10x
2
sin(5x
2
)
9. 3sec
2
3t 11. −4 csc(4w)cot(4w)
13. 4cos
2
2x − 4sin
2
2x 15.
x
√
x
2
+ 1
sec
2
x
2
+ 1
17. −3
3x
2
+4x
2
√
x
3
+2x
2
sin
2
cos
√
x
3
+ 2x
2
cos
cos
√
x
3
+ 2x
2
sin
√
x
3
+ 2x
2
19. 2x cos x
2
(b) 2sin x cosx (c) 2 cos2x
21. (a) 2x cos x
2
tan x + sin x
2
sec
2
x
(b) 2sin(tan x)cos(tanx)sec
2
x
(c) 2cos(tan
2
x)tan x sec
2
x
23. y = 1 25. y =−
π
2
4
x −
π
2
27. −2ft/s 29.
1
π
2
ft/s
31. (a) 12 cos3t (b) 12 (c) f = 0
33. (a) 3 (b)
1
4
(c) 0 (d) 1
35. −2
75
cos2x; −2
150
sin2x
45. (a) f
(x) =
(x cos x −sin x)/x
2
, x = 0
0, x = 0
Exercises 2.7, page 160
1. −
1
2
3. 0 5.
4 −2xy
2
3 +2x
2
y
7.
y
16y
√
xy − x
9.
y − 4y
2
x + 3 +2y
3
11.
−2xysin(x
2
y) −1
x
2
sin(x
2
y) +cos y
13.
16x
√
x + y − y
2
4y
(
x + y
)
+ y
2
− 2
√
x + y
15.
y
2
+ 2
2y sec
2
(y
2
+ 3) −2xy
17. y =
1
3
(x − 2) +1 19. y =−
4
5
(
x − 2
)
+ 1
21. y =
√
3
3
(
x − 1
)
+
√
3
2
23. y
=
27x
2
+ 48y
2
− 180x +240y − 144xy + 200
4(x
2
y − 2)
3
25. y
=
3[4x(y +2sin y)
2
− 3(x
2
− 2)
2
(1 +2cos y)]
4(y +2 sin y)
3
27.
6y(6y − 7 −9x +12sin 4y −8 cos4y)
(2y −2 −3x +4 sin4y)
3
29. horizontal tangent at (0, 0), (0, 3);
vertical tangent at (3/2, 3/2), (−3/2, 3/2)
31. (a) implicit (b) direct (c) direct (d) implicit
33. vertical asymptotes at x =±
√
2 horizontal asymptotes at y = 0
39. x
2
+ ny
2
= k
41. (a) y = x + 3;(4, 7) (b) y =−
1
2
(x + 1) +3;(9/4, 11/8)
Exercises 2.8, page 168
1. c = 0 3. c =
√
7 −1
3
5. c = cos
−1
2
π
7. 3x
2
+ 5 > 0 9. f
(x) = 4x
3
+ 6x has one zero
11. 3x
2
+ a > 0 13. 5x
4
+ 3ax
2
+ b > 0
15.
1
3
x
3
+ c 17. −
1
x
+ c 19. −cos x + c
21. sec x + c
23. f (x) > 0 in an interval (b, 0) for some b < 0
29. increasing 31. decreasing
33. decreasing for x > 0(x < 0)
41. f is not continuous at x = 0
Chapter 2 Review Exercises, page 169
1. 0.8 3. 2 5.
1
2
7. 3x
2
+ 1
9. y = 2x − 2 11. y = 6x 13. y =−3(x −1) + 1
15. v(t) =−32t + 40; a(t) =−32
17. v(t) = 40cos4t;a(t) =−160 sin4t
19. 8 ft/s going up, −24 ft/s coming down
21. (a) 0.3178 (b) 0.3339 (c) 0.3492 (d) 0.35
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-20 APPENDIX B
..
Answers to Odd-Numbered Exercises
23. 4x
3
− 9x
2
+ 2 25. −
3
2
x
−3/2
− 10x
−3
27. 2t(t + 2)
3
+ 3t
2
(t + 2)
2
29.
(3x
2
− 1) − x(6x)
(3x
2
− 1)
2
31. 2x sin x + x
2
cos x 33.
1
2
x
−1/2
sec
2
√
x
35. csct − t csct cott 37.
−2x
(x
2
+ 2)
3/2
39.
3x sin
√
4 − x
2
√
4 − x
2
41. 2cos 4x(sin 4x)
−1/2
43. 2
x + 1
x − 1
−2
(x − 1)
2
45. −
4
√
x
sin(4 −
√
x)cos(4 −
√
x)
49. 12x
2
− 18x +4
51. 8x sin 2x − 12cos 2x 53. 2sec
2
x tan x 55. −3
26
sin3x
57. (a) f (t) =±4 (b) f (t) = 0 (c) f (t) = 0
59.
2x − 2xy
x
2
− 9y
2
61.
sec
2
x +
y
(x + 1)
2
1
(x + 1)
− 3
63. y(0) =−
1
3
√
3
; slope of tangent line at x = 0is0;
y
(0) =
−2
3
√
3 −2
3
√
9
9
≈−0.78
65. (a) (0, 1) and (4, −31) (b) none
67. (a) (0, 0) (b) none 69. 3x
2
+ 7 > 0
75. c = 1 77. x
3
− sin x + c 79. m = 3
CHAPTER 3
Exercises 3.1, page 182
1.
1
2
x +
1
2
; 1.1 3.
1
3
x + 3; 2.967 5. 3x; 0.3
7. (a) 2.00125 (b) 2.0025 (c) 2.005
9. (a) 16.4 thousand (b) 12.8 thousand
11. (a) 133.6 (b) 138.4 13.
2
3
,
79
144
, 0.53209
15.
1
2
,
5
8
, 0.61803 17. −4.685780 19. 0.525261
21. −0.636733, 1.409624 23. f (x) = x
2
− 11;3.316625
25. f (x) = x
3
− 11;2.223980 27. f (x) = x
4.4
− 24;2.059133
29. f
(0) = 0;−0.3454, 0.4362, 1.6592
31. f
(0) = 0; no root 33. f
(−1) does not exist; 0.1340, 1.8660
35. (a) 1 (b) 2, slower 37. (a) −1 (b) 2, faster
39. 0.01 and 0.0100003; 0.1 and 0.1003; 1 and 1.557
41. 2.0025 and 2.002498; 2.025 and 2.0248; 2.25 and 2.2361
43. all three are the same; (a) y = 2x + 1
45. 0.00000117; 0.00000467; 0.0000186; 0.0007
47. x = 1; x =±1 49. too large
51. (d) F
2n+1
/F
2n
(e)
1 +
√
5
2
53. 0.133; 1
55. (a) 0.6407 (b) 0.6492, 3, 3.8508
57. P(1 −2x/R);104,500 ft
Exercises 3.2, page 193
1. (a) none (b) max of −1atx = 0 (c) none
(d) max at x = 0, min at x =±
1
2
.
3. (a) −
5
2
, local minimum (b) 2, local maximum
5. (a) none (b) 1, neither 7. 0, neither;
9
4
, local minimum
9. 0, neither;
16
9
, local minimum
11.
π
4
,
5π
4
, local maxima;
3π
4
,
7π
4
, local minima
13. −2 +
√
2, local minimum; −2 −
√
2, local maximum
15. −2, 1, local minima
17. −
2
3
, local minimum; −1, endpoint
19. 0, local maximum; ±1, local minima
21. −1, 2: local minima; 0, local maximum
23. (a) minimum, −1; maximum, 3
(b) minimum −17; maximum, 3
25. (a) maximum, 2
4/3
; minimum, 2
2/3
(b) maximum, 3
2/3
; minimum, 0
27. (a) maximum, 0; minimum, −12
(b) no maximum; no minimum
29. (a) maximum,
1
2
; minimum, 0
(b) maximum,
1
2
; minimum,
−1
2
31. (a) absolute min at (−1, −3);
absolute max at (0.3660, 1.3481)
(b) absolute min at (−1.3660, −3.8481);
absolute max at (−3, 49)
33. (a) absolute minimum at (0.6371 −1.1305);
absolute maximum at (1.2269, 2, 7463)
(b) absolute minimum at (−2.8051, −0.0748);
absolute maximum at (−5, 29.2549)
39. c ≥ 0, none; c < 0, one relative maximum, one relative minimum
41. 4b
2
− 12c > 0ifc < 0
43. c ≥ 0, one relative minimum; c < 0, two relative minima, one
relative maximum
47.
1
3
49.
m
n
53. t =
1
r
55. bottom; steepest at x =±2
57. max W
= ae
−1
at t = lnb
Exercises 3.3, page 201
1. increasing: x < −1, x > 1; decreasing: −1 < x < 1;
local maximum at x =−1; local minimum at x = 1
y
x
5
4
3
2
1
1
2
3
4
5
1234521
0
345
3. increasing: −2 < x < 0, x > 2; decreasing: x < −2, 0 < x < 2;
local maximum at x = 0; local minimum at x =±2
y
x
15
10
5
5
10
15
1234521
0
345
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-21
5. increasing: x > −1; decreasing: x < −1;
local minimum at x =−1
y
x
4
5
2
3
1
1
2
3
5
4
5
0
5
7. y is increasing on
−
3π
4
+ 2nπ,
π
4
+ 2nπ
;
y is decreasing on
π
4
+ 2nπ,
5π
4
+ 2nπ
;
local maximum at x =
π
4
+ 2nπ;
local minimum at x =
5π
4
+ 2nπ
y
x
4
5
2
3
1
1
2
3
5
4
5π
0
5π
11. x = 0 (local maximum) and x =±(3/2)
4/3
(local minima)
13. x =
1
3
√
2
(local maximum)
15. local max at x =−2,
local min at x = 0
17. vertical asymptotes at x =−1 and x = 1;
horizontal asymptote at y = 0
y
x
5
10
5
5
10
5
0
10
19. vertical asymptotes at x = 1 and x = 3; horizontal asymptote at
y = 1; local minimum at x = 0, local maximum at x = 3/2
y
x
10
20
10
10
20
20 10
0
20
21. horizontal asymptotes at y =−1 and y = 1
y
x
2
1
1
2
2
42
0
4
23. local max: x = 0.9374;
local min: x =−0.9474, x = 11.2599
−20 −16 −12 −8
−44
8121620
2200
−5500
−4400
−3300
−2200
−1100
1100
y
x
25. local max: x =−10.9079, x = 1.0084;
local min: x =−1.0084, x = 10.9079
−20−16−12−8
−44
8121620
−100000
−80000
−60000
−40000
−20000
20000
40000
60000
80000
100000
y
x
33. critical numbers: x ≈ 0.20 (local min) and x ≈ 40 (local max)
35. critical numbers: x ≈−120.008 (min) and x ≈ 0.008 (max)
37. f (x) = 3 + e
−x
has no zeros
39. yes 41. f
(0) = 1
45. If f (x) = 2
√
x and g(x) = 3 −
1
x
, f (1) = g(1) = 2
and for x > 1, f
(x) =
1
√
x
>
1
x
2
= g
(x)
49. c ≥
9
20
b
2
51. s
(t) =
1
2
√
t + 4
= rate of increase of sales function
53. (a) −0.000125 (b) 0.000125 (c) easier if c < −8
Exercises 3.4, page 210
1. concave up for x > 1, concave down for x < 1;
inflection point at x = 1
3. concave up for x > 0, concave down for x < 0;
no inflection points
5. concave up on
−
3π
4
+ 2nπ,
π
4
+ 2nπ
;
concave down on
π
4
+ 2nπ,
5π
4
+ 2nπ
;
inflection points at x =
π
4
+ nπ
7. concave up for x < 0, x > 2; concave down for 0 < x < 2;
inflection points at x = 0, x = 2
9. critical numbers:
x =−3 (min), x = 0 (inflection point)
11. local max at x =−2,
local min at x = 2
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-22 APPENDIX B
..
Answers to Odd-Numbered Exercises
13. min at x = 0, no inflection points
y
2
x
0
1
15. local max at x = 0; asymptotes: x =±3, y = 1
y
x
1020
20
10
10
10
0
20
20
17. local minimum at x =
16
9
; inflection point at x = 16
0
10 20 30
4
4
y
x
19. inflection point at x = 0
y
x
2
5
15
10
42
0
4
15
5
10
21. local minimum at x =−
1
6
; inflection points at x = 0,
2
3
y
x
2
2
42
0
4
2
4
23. local min at x =−0.1129 and x = 19.4993,
local max at x = 0.1135, inflection points at x = 0, x = 13
y
40,000
x
0
20
5
25. minimum at x = 0; inflection points at x =±
1
2
10 0 10
5
5
10
y
x
27. local min at x = 0.8952 and x = 9.9987,
local max at x = 1.106, inflection points at x = 1, x = 7
y
400
x
0
2
29. inflection points at x =−
√
6 and x =
√
6
y
x
2
20
10
10
20
42
0
4
35. cubic has one inflection point at x =−
b
3a
; quartic has two inflection
points if and only if 3b
2
− 8ac > 0
37. f (x) =−1 − x
2
39. increasing for −1 < x < 0, x > 1; decreasing for
x < −1, 0 < x < 1; local max at x = 0; local min at x =±1;
concave up for x < −
1
2
, x >
1
2
; concave down for −
1
2
< x <
1
2
;
inflection points at x =±
1
2
41. (a) (for 39) increasing for x < −1.5, x > 1.5; decreasing for
−1.5 < x < 1.5; local max at x =−1.5; local min at x = 1.5;
concave up for −1 < x < 0, x > 1; concave down for
x < −1, 0 < x < 1; inflection points at x = 0, x =±1
(b) (for 39) increasing, decreasing: no information; concave up for
x < −1.5, x > 1.5; concave down for −1.5 < x < 1.5; inflection
points at x =±1.5
45. need to know w
(0) 47. x = 30 49. x = 600
Exercises 3.5, page 221
1. inflection point at x = 1
−5
− 60
5
x
y
90
3. local max at x =−
6
5
, local min at x =
6
5
,
inflection points at x =−
3
5
, x = 0, x =
3
5
−2
−5
2
x
y
5