Smith R., Minton R. Calculus
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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-23
5. local max at x =−2, local min at x = 2;
asymptotes: x = 0, y = x
−20
−16
−12
4
8
12
16
20
−10−8 −6 −4 −2246810
y
x
−8
−4
7. No extrema, vertical asymptote x = 0,
horizontal asymptote y = 0.
100
60
2
20
0
−60
−100
80
3
40
0
1
−20
−40
−80
−1−2−3
9. No extrema, inflection point x = 0, vertical asymptote x =±1,
horizontal asymptote y = 0.
0
10
8
2
−2
−6
8
10
6
−4
4
0
624
−4
−8
−2
−8
−6
−10
−10
11. No extrema, inflection point x = nπ (n is an integer).
5
4
1
−1
2
−3
4
5
3
2
0
3
−2
−4
1
−5
0
−1−2−3−4−5
13. min at x = 0
−5
0
5
8
x
y
15. local max at x = 1 −
1
√
3
, local min at x = 1 +
1
√
3
, vertical tangent
lines at x = 0, 1, 2, asymptote is y = x −1, inflection points at
x = 0, 1, 2
y
x
10
40
30
20
10
20
10
40
30
20 10
0
20
17. Local max at x = 0, local min at x = 2.
y
8
8
4
4
−4
−8
0
x
10
10
6
2
6
0
−2
−6
2
−10
−2−4−6−8−10
19. vertical asymptote at x = 0, horizontal asymptote at y = 3
⫺2
8
y
⫺5
5
x
21. Local maximum at x = 3, local minimum at x =−1, vertical
asymptotes at x =−1.9304, 0.17074, and 4.8231; horizontal
asymptote y = 0.
4321
0
−6
2.5
0.0
−4
−2.5
6
−2−3
5
−1
−5.0
−5
5.0
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-24 APPENDIX B
..
Answers to Odd-Numbered Exercises
23. local max at x = 1 −
√
3
3
, x = 1 +
√
3
3
; local min at x = 0, 1, 2;
inflection points at x ≈−0.1883, 2.1883
y
x
10
5
5
15
121
0
3
25. local max at x = 0, vertical asymptotes at x =±
1
3
; horizontal
asymptote at y =
1
3
y
x
2
2
2
4
4204
4
27. local minima at x =±3, inflection points at x ≈±3.325
y
32
48
64
80
16
16
x
5
4
23
112345
29. horizontal asymptotes at y =−50 and y = 50
y
x
2
50
50
100
100
4204
31. local minima at x = 0.895 and x = 9.999, local maximum at
x = 1.106, inflection points at x = 1 and x = 7
−16−12 −8
x
y
−4481216
880
−2200
−1760
−1320
−880
−440
440
33. c < 0: 3 extrema, 2 inflection points;
c ≥ 0: 1 extremum, 0 inflection points;
as c →−∞, the graph widens and lowers;
as c →+∞, the graph narrows
35. min at x = 0; inflection points at x =±
c
√
3
; graph widens as
c
2
→∞; y = 1 for c = 0 (undefined at x = 0)
37. |c|=frequency of oscillation
39. y = 3x
4
3 2
1 1
2 3
4
4
6
8
10
10
8
6
4
2
2
x
y
41. y = x −2
2
2
6
8
4
22464
y
x
8 10
4
43. y = x
2
2
4
6
8
4
2248104
y
x
6
45. f (x) =
3x
2
(x − 1)(x −2)
47. f (x) =
2x
√
(x − 1)(x +1)
51. max at x =
√
b = most common gestation time; most common
lifespan
Exercises 3.6, page 230
1. 30
× 60
; the perimeter is 120
3. 20
× 30
7.
8
3
−
√
19
3
≈ 1.2137 9. (a) x = 3 (b) x = 0.9117
11.
1
2
,
1
2
or
−
1
2
,
1
2
13. (0, 1)
15.
−1
√
2
,
1
√
2
17. r = 1.1989
, h = 4.7959
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-25
19.
15
7
≈ 2.143 miles east of first development
21. 1.2529 miles east of bridge; $1.964 million 23. y = 0.568
27. r =
2
3
r
0
, contracts 29. x = R 31. 2
× 2
33. printed region:
√
46
× 2
√
46
; overall: (
√
46 +2)
× (2
√
46 +4)
35. (a) 12.7 ft (c) 1.1 ft (d) (L
2/3
− a
2/3
)
3/2
37. 4 39. (a) 50
◦
(b) 45
◦
(c) 40
◦
41. max area is 2ab 43. A =
P
2
12
√
3
Exercises 3.7, page 237
1. (a) 1.22 ft/min (b) 0.61 ft/min
3. (a) 58.9 gal/min (b) halved
7. (a) −2.25 ft/s (b)
−3
8
rad/s
9. (a) 24
√
101 ≈ 241 mph (b) 242.7 mph
13. s
(20) = 1.47152 thousand dollars per year
15. −2 dollars per year 17. (a) −65 rad/s (b) 6 ft/s
19. (a) 1 ft/s (b) −1.5 ft/s
21. 2.088 when x = 20, 2.332 when x = 10
23. 1760 Hz/s;
1
8
second 25. (a) 0 (b)
60
√
2
ft
29. (a) 8 − x = 0.568 at t = 1.16
Exercises 3.8, page 245
1. 3x
2
+ 40x +90; 9590 vs. 9421
3. 3x
2
+ 42x +110; 34,310 vs. 33,990
5. x = 10; costs rise more sharply 7. x =
20,000 ≈ 141
9. x =
3
√
18
11. (a) C
(100) = 42, C(100) = 77; C(101) = 76.65 < C(100)
(c) min at x = 600;C
(600) = C(600) = 52
13. (a)
p
p − 30
(b) 15 < p < 30
15. (a)
2p −20
p − 20
(b)
40
3
< p < 20
19. (a) 2 (b) 8 21. r = cK 23. 0; same
25. −
5
7
c
5/7
p
−12/7
27. 4 −cos x; less dense at ends 29. 4; homogeneous
31. 2.5
33. f
(x) =
−816x
−1.4
4x
−0.4
+ 15
2
< 0
37. the critical number is 2c, representing a minimum
39. x =
cr
2
r
1
, y =
cr
1
r
2
Chapter 3 Review Exercises, page 247
1. 1 3. 2 +
1
12
(7.96 −8) ≈ 1.99666
5. 0.198437 7. f
(1) = 0
9. (a) x =−3, 1
(b) increase: x < −3, x > 1; decrease: −3 < x < 1
(c) local max at x =−3, local min at x = 1
(d) up: x > −1, down: x < −1 (e) x =−1
11. (a) x = 0, 3 (b) increase: x > 3; decrease: x < 3(x = 0)
(c) min at x = 3 (d) up: x < 0, x > 2; down: 0 < x < 2
(e) x = 0, x = 2
13. (a) x = 180 (b) 0 < x < 180; x < 0 and x > 180
(c) local maximum at x = 180
(d) f is concave up for x > 270; f is concave
down for x < 0 and 0 < x < 270
(e) there is an inflection point at x = 270
15. (a) x =−2, 2
(b) increase: −2 < x < 2, decrease: x < −2, x > 2
(c) min at x =−2, max at x = 2
(d) up: −
√
12 < x < 0, x >
√
12;
down: x < −
√
12, 0 < x <
√
12
(e) x =−
√
12, x = 0, x =
√
12
17. min =−5atx = 1, max = 76 at x = 4
19. min = 0atx = 0, max = 3
4/5
at x = 3
21. local max at x =−
4
3
−
√
10
3
, local min at x =−
4
3
+
√
10
3
23. local max at x ≈ 0.2553, local min at x ≈ 0.8227
27. min at x =−3, inflection points at x =−2, x = 0
⫺
5
⫺
30
5
30
x
y
29. min at x =−1
⫺
55
⫺
40
60
x
y
31. min at x =−1, max at x = 1,
inflection points at x =−
√
3, x = 0, and x =
√
3,
horizontal asymptote at y = 0
⫺
5
⫺
2
5
2
x
y
33. min at x = 0, inflection points at x =−
1
3
, x =
1
3
horizontal
asymptote at y = 1
⫺
3
⫺
1
3
3
x
y
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-26 APPENDIX B
..
Answers to Odd-Numbered Exercises
35. local max at x =−
√
3, local min at x =
√
3, inflection point
at x = 0, vertical asymptotes at x =−1 and x = 1, slant asymptote
at y = x
y
x
⫺5
⫺10
⫺5
5
10
⫺10 5
0
10
37. (0.8237, 1.3570) 39. 1.136 miles east of point A
41. r = 1.663, h = 3.325
43. 2x, denser to the right
45. 0.04x + 20, 20.8 versus 20.78
CHAPTER 4
Exercises 4.1, page 307
1.
x
4
4
,
x
4
4
+ 3,
x
4
4
− 2
−3
2
10
−2 3
20
−1
5
1
15
3.
⫺5
10
y
⫺4
8
x
g2
g3
g1
5.
3
5
x
5
−
3
2
x
2
+ c 7. 2x
3/2
+
1
3
x
−3
+ c
9.
3
2
x
2/3
− 9x
1/3
+ c 11. −2 cos x + sin x + c
13. 2sec x + c 15. 5tan x + c
17. 3sin x − 2x + c 19.
5
2
x
2
+
3
x
+ c
21. x −
4
x
+ c 23.
2
5
x
5/2
−
16
5
x
5/4
+ c
25. (a) N/A (b)
2
5
x
5/2
+ 4x +c
27. (a) N/A (b) tan x + c
29.
1
3
x
3
+
1
2
x
2
+ 4 31. x
4
+ x
2
+ 2x +3
33.
1
3
t
3
+ t
2
− 6t + 2 35. −3 sin x +
1
3
x
4
+ c
1
x + c
2
37.
2
3
x
3
+
1
3
x
−1
+ c
1
x
2
+ c
2
x + c
3
39. 3t − 6t
2
+ 3 41. −3sin t +
1
2
t
2
+ 3t + 4
43. (a) (b)
y
x
1
0
1
4.8
x
31
7.2
20
8.8
−1−3
4.0
5.6
6.4
3.2
2.4
8.0
−2
45. f (x) =
x
3
6
−
x
2
2
+
7
2
x −
7
6
47. sin(x
2
) +c
49.
1
2
(x
2
sin2x) +c 51.
sin(x
2
) +c
57. a =
1
720
mi/s
2
;
2
45
mi ≈ 235 ft
59. distances fallen: 5.95, 12.925, 17.4, 19.3; accelerations:
−31.6, −24.2, −11.6, −3.6
61. speeds: 70, 69.55, 70.3, 70.35, 70.65; distances: 0, 34.89, 69.85,
105.01, 104.26
Exercises 4.2, page 315
1.
14
i=1
2i
2
3. (a)
50
i=1
i
2
= 42,925 (b)
50
i=1
i
2
= 1,625, 625
5. 3 +12 +27 +48 +75 +108 = 273
7. 26 +30 +34 +38 +42 = 170
9. 7385 11. −21,980 13. 323,400
15. 7308 17.
n(n + 1)(2n +1)
6
− 3n + 1
19. 2.84 21. 24.34
23.
(n + 1)(2n +1)
6n
2
+
n + 1
n
→
4
3
25.
8
3
(n + 1)(2n +1)
n
2
−
n + 1
n
→
13
3
29. 2870
31. 171,707,655,800 35. 1 −
1
4
n+1
;1
37. 375 miles 39. 217.75 ft
Exercises 4.3, page 321
1. (a) 0.125, 0.375, 0.625, 0.875; 1.328125
(b) 0.25, 0.75, 1.25, 1.75; 4.625
3. (a)
π
8
,
3π
8
,
5π
8
,
7π
8
;2.05234
(b)
π
16
,
3π
16
,
5π
16
, ···,
13π
16
,
15π
16
;2.0129
5. (a) 1.3027 (b) 1.3330 (c) 1.3652
7. (a) 6.2663 (b) 6.3340 (c) 6.4009
9. (a) 1.0156 (b) 1.00004 (c) 0.9842
11. (a)
4
3
(b)
14
3
(c)
32
3
13. (a)
5
3
(b)
10
3
(c)
140
3
15.
32
3
17. 18
19. (a) less than, (b) less than, (c) greater than
21. (a) greater than, (b) less than, (c) less than
23. For example, use x =
1
√
12
on [0, 0.5] and
7/12 on [0.5, 1]
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-27
25. (b) a −
1
2
x + ix for i = 1,...,n 27. A
2
29. 3.75, 1.75
31. (a)
4
3
1 +
1
n
2 +
1
n
(b)
4
3
1 −
1
n
2 −
1
n
; limit is
8
3
35. left: 1.81, right: 1.67 37. left: 1.18, right: 1.26
39. 0.092615
Exercises 4.4, page 331
1. 24.47 3. 0.8685
5. area bounded by y = x
2
, y = 0 and x = 1
7. area bounded by y = x
2
− 2, y = 0 and x = 2 minus the area
bounded by y = x
2
− 2, y = 0 and x = 0
9. 1 11.
8
3
13.
8
3
15.
2
−2
(4 − x
2
)dx
17. −
2
−2
(x
2
− 4)dx 19.
π
0
sin xdx
21. 2.095 23. 13 25. 5 27.
10
3
29. between −1.23 and 0.72 31. between 2 and 6
33.
2
√
3
35. (a)
3
0
f (x)dx (b)
2
0
f (x)dx
37. (a) 1 (b) 8
39. (a) (b)
0
1
1
2
2
x
y
12
8
4
3421
45. positive 47. negative 49. 6 51. π
53. (a)
π
0
sin xdx (b)
1
0
(
1 + x
)
dx (c)
1
0
f (x)dx
55. t < 40;t < 40;t > 40;t = 40 57. 6.93
59.
Q
2
61.
2c
0
D
c
c
(1 −r/ p)
63. 9000 lb; 360 ft/s
Exercises 4.5, page 341
1. −2 3. 0 5.
62
5
+ 3ln4 7. −
4
7
9. 3 11.
√
2 −1 13. −
4
3
15.
16
3
17.
1
3
t
3
+ t
2
+ t 19.
32
3
21.
8
3
23. 2
25. x
2
− 3x +2 27. 2x[cos(3x
2
) +1]
29. (sin x + x cos x)
x
2
sin
2
x + 1
31. 3x
2
sin
2x
3
− 2x sin
2x
2
33. 40t + cost +1 35. 2t
2
−
1
6
t
3
+ 8t
37. (a) Increasing 0 < t <π, decreasing π<t < 2π .
(b) 120 gallons
39. y = 0 41. y = x − 2
43. 2.96 45. 1.71 47.
√
2 −1
49.
3
0
f (x)dx,
2
0
f (x)dx,
1
0
f (x)dx;
increasing 0 < t < 1, t > 3, decreasing 1 < t < 3
51. (a)
1
x
2
> 0 (b)
1
x
2
is discontinuous at x = 0
53. b and c 55.
10
3
57.
2
π
59.
x
0
sin
t
2
+ 1 dt
61. local max at x = 1, local min at x = 2
63. F
(2) does not exist, but f (2) = 3
65. 5x − x
2
+ 10sin x = Katie’s lead 67.
Q
2
r
2
;
Q
3
Exercises 4.6, page 349
1.
2
9
(x
3
+ 2)
3/2
+ c 3.
1
2
(
√
x + 2)
4
+ c
5.
1
6
(x
4
+ 3)
3/2
+ c 7. −2
√
cos x + c
9.
1
3
sint
3
+ c 11. sin(tan x) + c
13. 2sin
√
x + c 15. −
1
6
(sin3x + 1)
−2
+ c
17. −2(
√
u + 1)
−1
+ c 19. (
2
x
+ 1)
−2
+ c
21. 27(3 −x)
−3
−
27
2
(3 − x)
−2
+ 9(3 − x)
−1
+ c
23.
1
6
(4x − 1)
3/2
+ c 25. −2(t + 7)
−1
+
11
2
(t + 7)
−2
+ c
27.
4
3
(1 +
√
x)
3/2
− 4(1 +
√
x)
1/2
+ c
29.
5
3
√
5 −
1
3
31. 0 33.
1
3
sin(1)
35.
8
3
37. (a) 0.77 (b)
1
2
−
1
2
cosπ
2
39. (a) 1.414 (b) 2ln 5 −
8
5
41.
1
2
4
0
f (u)du
43.
1
0
f (u)du 47. (a) 5 (b)
a
2
;
π
4
49. 1 51. x =±u
1/4
Exercises 4.7, page 361
1. midpoint
85
64
, trapezoidal
43
32
, Simpson
4
3
3. midpoint
3776
3465
, trapezoidal
67
60
, Simpson
11
10
5. (a) 3.146 >π (b) 3.131 <π (c) 3.14157 <π
7. (a) 0.8437 (b) 0.8371 (c) 0.8415
9.
n Midpoint Trapezoidal Simpson
10 0.5538 0.5889 0.5660
20 0.5629 0.5713 0.5657
50 0.5652 0.5666 0.5657
11.
n Midpoint Trapezoidal Simpson
10 51.707 51.831 51.755
20 51.738 51.769 51.749
50 51.747 51.752 51.749
13.
Midpoint Trapezoidal Simpson
n Error Error Error
10 0.00832 0.01665 0.00007
20 0.00208 0.00417 4.2 ×10
−6
40 0.00052 0.00104 2.6 ×10
−7
80 0.00013 0.00026 1.6 ×10
−8
15.
Midpoint Trapezoidal Simpson
n Error Error Error
10 5.5 ×10
−17
0 0
20 2.7 ×10
−17
1.6 × 10
−16
1.1 × 10
−16
40 2.9 ×10
−16
6.9 × 10
−17
1.3 × 10
−16
80 1.7 ×10
−16
3.1 × 10
−16
1.5 × 10
−16
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-28 APPENDIX B
..
Answers to Odd-Numbered Exercises
17. 4, 4, 16 19. (a) 9.1 (b) 9.033
21. (a) (i) y
24
(ii) y
48
, (b) (i) 817 (ii) 578
23. (a) 145 (b) 103 (c) 9
25. (a) 1081 (b) 764 (c) 37
27. answers for n = 80: midpoint bound 0.000391, midpoint error
0.00013; trapezoidal bound 0.000781, trapezoidal error 0.00026;
Simpson’s bound 1.63 × 10
−8
, Simpson’s error 1.63 × 10
−8
29. (a) 0.75 (b) 0.7 (c) 0.75 (d) 0.66
31. (a) under (b) over (c) can’t tell
33. (a) over (b) under (c) can’t tell
35. (a) under (b) over (c) exact
41. L = 1 43. 0.66 if T
n
− I ≈−2(M
n
− I) then
1
3
T
n
+
2
3
M
n
≈ I
47. 529 ft 49. 2.6 liters
Chapter 4 Review Exercises, page 374
1.
4
3
x
3
− 3x +c 3. 4ln|x|+c 5. −
1
2
cos4x + c
7.
1
2
x
2
−
1
4
e
4x
+ c 9.
1
2
x
2
+ 4ln|x|+c 11. e
x
− x + c
13.
1
3
(x
2
+ 4)
3/2
+ c 15. 2 sin x
3
+ c 17. −e
1/x
+ c
19. −ln |cos x|+c 21. x
3
+ x + 2 23. −16t
2
+ 10t + 2
25. 4 +10 +18 +28 +40 +54 = 154 27. 338,250
29.
n(n + 1)(2n +1)
6n
3
−
n(n + 1)
2n
3
→
1
3
31. 2.65625
33. 4.668 35. (a) 2.84 (b) 2.92 (c) 2.88 (d) 2.907
37. Simpson 39.
2
3
41.
3
0
(3x − x
2
)dx =
9
2
43. 25
45.
1
2
(e
2
− 1) ≈ 3.19 47. −
4
3
49. 1 51. 6 +4e
−5/2
53.
1
2
ln5 55.
1
3
(8
3/2
− 8) 57.
15
2
+
1
2
e
2
− 4e
59. sin x
2
− 2 61. (a) 2.079 (b) 2.083 (c) 2.080
63. (a) 2.08041, 2.08045 (b) 2.08055, 2.08048
(c) 2.08046, 2.08046
65. (a) −
2
1 +tanh(t/2)
+ c = sinh t − cosht + c
(b) t + ln(cosht + 1) − tanh(t/2) + c
CHAPTER 5
Exercises 5.1, page 321
1.
40
3
3.
52
5
5.
64
3
7.
27
4
9.
7
3
11.
125
6
13. 4.01449 15. 0.135698
17.
1
0
(2 −2y)dy = 1 19.
1
0
2xdx= 1
21.
1
0
(3 − x
2
− 2x)dx =
5
3
23. 35.08% 25. 93.02%
27. 3;
√
3
0
3 − x
2
dx =
3
√
3
x
2
− 3
dx = 2
√
3
33. L =
3
16
35. (a) A
2
(b) A
1
+ A
2
(c) A
3
(d) A
2
37. Critical numbers
1
2
kπ, for odd k; no extrema; inflection points at
1
2
nπ, for all integers n.
39. 16.2 million barrels 41. 2 million people
43.
time 1 2 3 4 5
amount 397 403 401 412 455
(answers will vary)
01 2 3 4 5
350
400
450
time
gallons
45. (q
∗
, p
∗
) = (80, 8); consumer surplus is 80
47. assuming C(0) = 0:
(a) loss from selling the first 2000 items
(b) profit from selling items 2001 through 5000
(c) The sum of parts (a) and (b)
(d) loss from selling items 5001 through 6000
Exercises 5.2, page 335
1. 12 3.
56π
3
5. (a)
500
0
750 −
3
2
x
2
dx = 93,750,000 ft
3
(b)
250
0
750 −
3
2
x
2
dx = 82,031,250 ft
3
; wider at the bottom
7.
30
0
3 −
1
12
x
2
dx =
215
2
ft
3
9.
60
0
π[60(60 − y)] dy = 108,000π ft
3
11.
2π
0
π
4 +sin
x
2
2
dx = 33π
2
+ 32π in
3
13. 0.2467 cm
3
15. 2.5 ft
3
17. (a)
8π
3
(b)
28π
3
19. (a)
32π
5
(b)
224π
15
21. (a) 28.99 (b)
2
3
π (24
√
6 −17)
23. (a) 0.637 (b) 7.472
25. (a)
16π
3
(b)
32π
3
(c)
64π
3
(d)
128π
3
(e)
32π
3
(f)
64π
3
27. (a)
π
2
(b)
π
5
(c)
π
6
(d)
7π
15
(e)
7π
6
(f)
13π
15
29.
πh
2
2a
=
1
2
πh
h
a
2
31.
1
−1
π(1)
2
dy = 2π
33.
1
−1
π
1 − y
2
2
dy =
2π
3
35.
r
−r
π(r
2
− y
2
)dy =
4
3
πr
3
37. same volume
39. (a)
16
3
(b)
2π
3
41. (a)
64
15
(b)
8π
15
(c)
16
√
3
15
43. using Simpson’s rule, 12.786
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-29
45. answers will vary; one possibility is shown:
0
1
2
3
4
5
height
time
47.
5π
12
Exercises 5.3, page 343
1. r = 2 − x, h = x
2
, V =
8π
3
3. r = x, h = 2x, V =
4π
3
5. r = x, h =
x
2
+ 1, V =
2π
3
17
3/2
− 1
7. r = 2 − y, h = 2
1 − y
2
, V = 4π
2
9.
32π
3
11.
128π
3
13.
27π
2
15. 288π
17. (a)
80π
3
(b) 16π (c) 16π (d)
16π
3
19. (a)
625π
6
(b)
625π
3
(c)
875π
6
(d)
500π
3
21. (a)
32π
15
(b)
5π
6
(c)
3π
2
(d)
38π
15
23. (a)
5π
6
(b)
64π
15
25. (a) 16.723 (b) 12.635 (c) 4.088 (d) 1.497
27. x =
√
y and x = y about x = 0
y
x
0
exercise 27
29. same as #27
33.
√
3π
2
35.
1 −
√
0.9 ≈ 0.2265
Exercises 5.4, page 350
1. 1.4604;1.4743 3. 3.7242;3.7901 5. 2
√
5
7.
73
√
73 −37
√
37
54
9.
10
3
11.
1
−1
1 +9x
4
dx ≈ 3.0957
13.
2
0
1 +(2 −2x)
2
dx ≈ 2.9579
15.
π
0
1 +sin
2
xdx ≈ 3.8202
17.
π
0
1 +(x sin x)
2
dx ≈ 4.6984 19. 29.58 ft 21. 5ft
23. 60 yards; 60 yards; 139.4 yards; 104.55 ft/s
25.
1
0
2π x
2
1 +4x
2
dx ≈ 3.8097
27.
2
0
2π(2x − x
2
)
1 +(2 −2x)
2
dx ≈ 10.9655
29.
π/2
0
2π cos x
1 +sin
2
xdx ≈ 7.2118
31. 1.672, 1.720, 1.754 → 2
33. (a) 0.9998 (b) 0.9749
35. (a) 6π (b) 4π (c) (
√
5 +1)π
37. t = 3 +
√
12
Exercises 5.5, page 358
1. y(0) = 80, y
(0) = 0 3. y(0) = 60, y
(0) = 10
5. −8
√
30 ft/s ≈−30 mph 7.
√
h 9. 256 ft
11. −4.9t
2
+ 19.6t;19.6 m; 4 sec; −19.6m/s
13. 8
5
3
ft/sec
17. 10
√
3 ≈ 17sec, 490
√
3 ≈ 849 m; the same
19. The serve is not in; 7.7
◦
, 9.0
◦
21. 2.59 ft
23. ball bounces (h ≈−0.62)
25. 40
5
√
3
≈ 68 ft/s; 20
√
10 ≈ 63 ft/s
27. (a)
25
16
sin4t −
25
4
t (b)
25
16
sin(4t + π/2) −
25
16
29.
20π
√
30
≈ 11.5 rad/s
33. (a) Goal! (x ≈−0.288 at y = 90)
(b) yes (x ≈ 1.7 when y = 30)
35. 25 ft; 30.25 ft
39. maximum allowed error is 0.01 rad ≈ 0.6
◦
41. 86.18 ft; 530.34 ft
45. (a) v =
2gH (b) 32 ft/s
(c) 22.63 ft/s (d) 3.94 ft/s; 22.28 ft/s
Exercises 5.6, page 369
1.
15
8
ft-lb 3.
1250
3
ft-lb
5. 270,000,000 ft-lb 7. 50,000 ft-lb
9. (a) 704,000 ft-lb (b) 16 hp
11. (a) 44,100π N-m
(b) 9800π N-m
13. 7.07 ft 15. J ≈ 2.133;113 ft/s
17. maximum thrust: f (3) = 11; impulse: ≈ 53.11
19. m = 15 kg,
x =
16
5
m; heavier to right of center
21. 0.0614 slug, 31.5 oz
23. 16.6 in.; same mass,
x differs by 3
25. 0.0614 slug, 31.4 oz;
x = 17.86 in.
27. 2, 4,
1
2
;
5
12
, 3,
16
3
29.
8
3
, 2
31. (0, 1.6) 33. 8,985,600 lb
35. 196,035 lb 37. 12,252 lb 39. 10,667 hp
41. 27.22, 20.54, 24.53% 43.
1
4
ρπa
3
b
45.
midsized
wooden
≈ 1.35;
oversized
wooden
≈ 1.78
Chapter 5 Review Exercises, page 372
1.
π
3
3
+ 2π −2 3.
1
12
5.
16
√
2
3
7.
5
6
9. 10,054 11.
98π
3
13. 4.373
15. (a)
256π
5
(b) 8π (c)
128π
3
(d)
1408π
15
17. (a)
2π
3
(b) 2π (c) 4π (d)
22π
3
19.
1
−1
1 +16x
6
dx ≈ 3.2
P1: JXR
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CONFIRMING PAGES
A-30 APPENDIX B
..
Answers to Odd-Numbered Exercises
21.
3
0
1
4(x + 1)
+ 1dx ≈ 3.1678
23.
1
0
2π
1 − x
2
1 +4x
2
dx ≈ 5.483 25. −64 ft/s
27. 1.026 sec, 46.3 ft 29. no, ball bounces
31. 64
√
2 ft/s, −64
√
2 ft/s 33.
40
3
ft-lb
35. m =
112
3
, x =
16
7
, heavier to right of x = 2
37. 22,630,400 lb 39. J ≈ 1.52; 32 ft/s
CHAPTER 6
Exercises 6.1, page 382
1. ln4 =
4
1
1
t
dt 3. ln8.2 =
8.2
1
1
t
dt 5. 1.3916
7.
2
x
9. −tan x 11. −3x
2
tan x
3
[cos(ln(cos x
3
))]
13.
x
x
2
+ 1
15.
4
x
−
5x
4
x
5
+ 1
17. ∞ 19. 4
21. ln(x
2
+ 1) +c 23.
−1
2
ln|cos 2x|+c 25. ln|ln x|+c
27.
1
3
(ln x + 1)
3
+ c 29.
2
3
(ln2)
3/2
31.
7
2
ln2 33.
1
2
ln3 37. x
sin x
cos x ln x +
sin x
x
39. (sin x)
x
[ln(sin x) + x cot x]
41.
y
x ⫽ 2
1
x
0
1
y ⫽ ln(x − 2)
increasing and concave down on (2, ∞)
43.
y ⫽ ln(x
2
+ 1)
y
1
x
0
1
increasing on (0, ∞),
decreasing on (−∞, 0);
concave up on (−1, 1),
concave down on (−∞, −1) ∪ (1, ∞)
45.
y ⫽ x lnx
y
x
0
1
1
decreasing on (0, 0.37), increasing on (0.37, ∞);
concave up on (0, ∞)
49. ∞
51. approximately 0.6
53. 2.744 s; x = 150; relay; straight throw is faster
55. x = 25; times equal with 0.0062-s delay
57. f =
1
2
;pH→∞
Exercises 6.2, page 389
5. f
−1
(x) =
3
√
x + 2 7. f
−1
(x) =
5
√
x + 1
9. not one-to-one 11. f
−1
(x) =
3
√
x
2
− 1, x ≥ 0
13. (a) 0;
1
4
; y =
1
4
(x + 1) (b) 1;
1
7
; y =
1
7
(x − 4) +1
15. (a) −1;
1
15
; y =
1
15
(x + 5) −1 (b) 1;
1
15
; y =
1
15
(x − 5) +1
17. (a) 2;
7
4
; y =
7
4
(x − 4) +2 (b) 0;2; y = 2(x − 2)
19.
y
x
24−2−4
−4
−2
2
4
21.
y
x
22⫺2⫺4
⫺4
⫺2
2
4
23. k ≥ 0
25.
y
x
⫺30 ⫺20 ⫺10 10
⫺1
⫺2
⫺3
3
2
1
20 30
27. not one-to-one
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-31
29.
y
x
⫺10 ⫺5
⫺5
5
10
⫺10
510
31.
y
x
⫺5
⫺5
5
10
⫺10
510
⫺15
33.
x
2
3
4
0.50
1
0
21.51
y
y ⫽ f (x)
y ⫽ f
−1
(x)
35. f
−1
(x) =−
√
x, x ≥ 0
y
x
0
⫺2⫺1.5⫺1
⫺0.5
0.5 1 1.5 2
⫺1
⫺2
⫺3
⫺4
3
4
2
1
y ⫽ f(x)
y ⫽ f
−1
(x)
37. f is one-to-one for x ≥ 2; f
−1
(x) =
√
x + 2, x ≥ 0
x
2
3
5
4
10
1
0
5432
y
y ⫽ f
−1
(x)
y ⫽ f(x)
39. f is one-to-one for x ≥ 2; f
−1
(x) =
√
x
2
+ 1 +1, x ≥ 0
y
x
1
0
2
3
4
5
7
6
1056432
y ⫽ f(x)
y ⫽ f
−1
(x)
41. f is one-to-one for −
π
2
≤ x ≤
π
2
y
x
0⫺1.5 ⫺1 ⫺0.5 0.5
⫺0.5
⫺1
⫺1.5
1.5
1
0.5
1 1.5
y ⫽ f(x)
y ⫽ f
−1
(x)
49. no; subtract 9.0909%
Exercises 6.3, page 396
1. 12e
3x
3.
(4x − 1)e
4x
x
2
5. 2(3x
2
− 3)e
x
3
−3x
7. 2xe
x
2
9.
1
3
e
3x
+ c 11.
1
2
e
x
2
+ c
13. −e
cos x
+ c 15. −e
1/x
+ c 17. x +2e
x
+
1
2
e
2x
+ c
19.
1
3
x
3
+ c 21.
1
3
e
3
−
1
3
23. 0
25.
y ⫽ 3e
2x
x
y
0
1
1
P1: JXR
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LT (Late Transcendental)
CONFIRMING PAGES
A-32 APPENDIX B
..
Answers to Odd-Numbered Exercises
27.
y ⫽ 3xe
−2x
x
y
0
1
1
29.
2.5
5
7.5
10
2.5 50
0
−2.5
−5
y
y ⫽ e
1/x
x
31. max at x =
1
2
, inflection point at x = 1
33. min at x = 0, local max at x = 1,
inflection points at x = 1 ±
√
2
2
37. 2(ln3)3
2x
39. 2x(ln 3)3
x
2
41.
2
x ln4
43.
1
ln2
2
x
+ c 45.
1
2ln2
2
x
2
+ c
47.
0
10
⫺44
49.
0
10
⫺4
4
51. increasing, concave up; decreasing, concave up 61. x =±1
65. t =
1
r
67. 23; 3.105 is percent per hour to hear rumor at t = 2; 70 is percent to
eventually hear rumor
69. 1; 0; x = 0;
1
1 +e
−(x−4)
⫺22
2
0
y
x
1
1⫹e
⫺x
f (x) ⫽
71. k =
1
3
ln2; 2
2/3
− 2
1/3
≈ 0.327
Exercises 6.4, page 403
1. (a) 0 (b) −
π
6
(c) −
π
2
3. (a)
π
4
(b) 0 (c) −
π
4
5. (a) 0 (b)
π
3
(c)
π
4
7.
√
1 − x
2
, −1 ≤ x ≤ 1
9.
√
x
2
− 1, x ≥ 1or−
√
x
2
− 1, x ≤−1 11.
√
3
2
13.
4
3
15.
π
8
17.
2
9
x
√
9 − x
2
, |x|≤3
19.
2
1
0
y
x
y
⫽ cos
−1
(2x)
21.
1
y
x
y
⫽ sin
⫺1
(3x)
2
2
⫺
3
1
⫺
3
1
23.
⫺2.5
1.25
2.5
⫺1.25
⫺2.5 2.5 5−5
0
0
y
x
2
y ⫽ tan
⫺1
(
)
x
2
− 1
1
2
⫺
25.
nπ
4
, odd n 27.
nπ
3
29.
π
20
+
nπ
5
31. ±
π
6
+ πn 35. tan
−1
5
3x
Exercises 6.5, page 409
1.
6x
√
1 −9x
4
3.
2
x
√
x
4
− 1
5. cos
−1
2x −
2x
√
1 −4x
2
7.
−cos x
1 −sin
2
x
=±1 9.
sec x tan x
1 +sec
2
x
11. 6 tan
−1
x + c
13. tan
−1
x
2
+ c 15. 2 sin
−1
x
2
+ c 17. sec
−1
x
2
+ c
19. tan
−1
x
2
+ c 21. sin
−1
(e
x
) +c 23.
3π
4
25.
π
4
31.
248
3
≈ 9.09 feet 35. cos
−1
x + sin
−1
x =
π
2
37. Integrating
1
1 + x
2
gives better accuracy for a given n.
39. x = 0 41. (a) decreases (b) decreases 43. no
45.
√
3 ≈ 1.73 feet