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 A-33
Exercises 6.6, page 417
1.
2
0
1
y
x
y
⫽ cosh(2x)
3.
0
1
1
y
x
y
⫽ tanh 4x
5. (a) 4 sinh4x (b) 4 cosh
3
x sinh x
7. (a) 2x sech
2
(x
2
) (b) 2tanh xsech
2
x
9. (a) 2x sinh5x + 5x
2
cosh5x
(b)
2x + 3(x
2
+ 1)coth x
csch
3
x
11. (a)
2
√
4x
2
− 1
(b)
2x
√
1 + x
4
13.
1
6
sinh 6x + c 15.
1
3
ln (cosh 3x) + c 17.
1
4
cosh 4 −
1
4
19. cosh(sin x) + c 21. e
sinh1
− 1 23. 27.62
31. f
(x) < 0 for x < 0, f
(x) > 0 for x > 0, f
(x) > 0 for all x
33. ln
x +
√
x
2
− 1
39. (g)
m
k
ln
cosh
kg
m
t
41. 9.8 m/s
2
Chapter 6 Review Exercises, page 418
1.
3x
2
x
3
+ 5
3.
4x
3
+ 1
2(x
4
+ x)
5. −2xe
−x
2
7. (3x
2
ln4)4
x
3
9.
2
√
1 −4x
2
11.
−2sin 2x
1 +cos
2
2x
13.
1
2
x
−1/2
sinh
√
x
15.
3
√
9x
2
+ 1
17.
1
3
ln|x
3
+ 4|+c 19.
1
2
ln2
21. −cos(ln x) + c 23. −
1
4
e
−4x
+ c 25. 2e
√
x
+ c
27.
1
3
e
6
−
1
3
29.
1
4ln3
3
4x
+ c 31.
3
2
tan
−1
x
2
+ c
33
1
3
sin
−1
(x
3
) +c 35.
9
2
sec
−1
x
2
+ c 37. 4 sinh
−1
x + c
39.
1
4
sinh 4x + c 41. yes,
3
√
x + 1 43. no
45.
1
f
(1)
=
1
11
47.
1
f
(2)
=
1
2
49.
π
2
51. −
π
4
53.
√
3
2
55.
π
4
57.
π
4
+ πn
59.
2
0
1
y
x
y
⫽ cosh(2x)
61.
1
y
x
y
⫽ sin
⫺1
(2x)
2
2
⫺
2
1
⫺
2
1
63.
y
y ⫽ e
⫺x
2
x
0
1
1
65. 17.77 67. e
x
69. x =
√
H
2
− P
2
2
CHAPTER 7
Exercises 7.1, page 425
1.
1
a
e
ax
+ c 3. sin
−1
x
a
+ c 5. −
1
6
cos6t + c
7.
1
5
x
5
+
8
3
x
3
+ 16x +c 9.
3
4
tan
−1
x
4
+ c
11. sin
−1
x + 1
2
+ c 13. 2tan
−1
x + 1
2
+ c
15. 2ln|t
2
+ 2t + 5|−2tan
−1
t + 1
2
+ c
17. −
1
2
e
3−2x
+ c 19. 6ln(1+ x
2/3
) +c
21. −2cos
√
x + c 23. 0 25. 1 −
√
2
27.
1
3
tan
−1
x
3
+ c 29. sin
−1
x
2
+ c 31.
1
2
sin
−1
x
2
+ c
33. tan
−1
x +
1
2
ln(1 + x
2
) +c 35. −
(
ln2
)
2
37.
12
5
39.
72
5
41.
5
3 + x
2
dx =
5
√
3
tan
−1
x
√
3
+ c
43.
ln x
2x
dx =
1
4
(ln x)
2
+ c 45.
xe
−x
2
dx =−
1
2
e
−x
2
+ c
47. 1 +
1
2
ln2 −tan
−1
(2) +
1
4
π
49. tan
−1
x + c;
1
2
ln(1 + x
2
) +c; x − tan
−1
x + c;
1
2
x
2
−
1
2
ln(1 + x
2
) +c
Exercises 7.2, page 431
1. x sin x +cos x + c 3.
1
2
xe
2x
−
1
4
e
2x
+ c
5.
1
3
x
3
ln x −
1
9
x
3
+ c 7. −
1
3
x
2
e
−3x
−
2
9
xe
−3x
−
2
27
e
−3x
+ c
9.
1
17
e
x
sin4x −
4
17
e
x
cos4x + c
11.
2
3
sin2x cos x −
1
3
cos2x sin x + c
13. x tan x +ln |cos x|+c 15.
1
2
x
2
e
x
2
−
1
2
e
x
2
+ c
17. sin x ln(sin x) −sin x + c 19.
1
4
sin2 −
1
2
cos2
21. −
2
π
2
23. 10 ln20 − ln2 − 9
25.
1
a
x
2
e
ax
−
2
a
2
xe
ax
+
2
a
3
e
ax
+ c
27.
1
n + 1
x
n+1
ln x −
1
(n + 1)
2
x
n+1
+ c
31. e
x
x
3
− 3x
2
+ 6x −6
+ c
33.
1
3
cos
2
x sin x +
2
3
sin x + c 35. 9e − 24 37.
8
15
39. m even:
(m − 1)(m − 3)···1
m(m −2) ···2
·
π
2
;
m odd:
(m − 1)(m − 3)···2
m(m −2) ···3
41. x cos
−1
x −
1 − x
2
+ c 43. −2
√
x cos
√
x + 2sin
√
x + c
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-34 APPENDIX B
..
Answers to Odd-Numbered Exercises
45.
x
2
[sin(ln x) − cos(ln x)] +c
47.
1 −
e
4x
2
cos(e
2x
) +e
2x
sin(e
2x
) +c
49. 6(e
2
− 1) 51. n times
53. (a) Substitution (b) Parts
(c) Parts (d) Substitution
55. first column: derivatives; second column: antiderivatives
57. x
4
sin x + 4x
3
cos x − 12x
2
sin x − 24x cos x + 24 sin x + c
59.
1
2
x
4
− x
3
+
3
2
x
2
−
3
2
x +
3
4
e
2x
+ c
61. −
1
3
x
3
e
−3x
−
1
3
x
2
e
−3x
−
2
9
xe
−3x
−
2
27
e
−3x
+ c
67. e
x
ln x + c
Exercises 7.3, page 440
1.
1
5
sin
5
x + c 3.
1
8
5.
1
3
7.
1
2
x +
1
4
sin2(x + 1) +c
9.
1
3
sec
3
x + c 11.
1
6
sec
3
(x
2
+ 1) −
1
2
sec(x
2
+ 1) +c
13. −
1
3
cot
3
x −
1
5
cot
5
x + c 15.
12
35
17.
1
8
x −
1
32
sin4x + c
19. −
8
21
+
25
√
2
168
21. −
1
9
√
9 − x
2
x
+ c
23. 8sin
−1
x
4
−
1
2
x
16 − x
2
+ c 25. π
27.
3x
2
x
3
2
− 1 +
9
2
ln
x
3
+
x
3
2
− 1
+ c
29. 2ln|
√
x
2
− 4 + x|+c
31.
4x
2
− 9 −3tan
−1
√
4x
2
− 9
3
+ c
33.
3x
2
x
3
2
+ 1 −
9
2
ln
x
3
+
x
3
2
+ 1
+ c
35.
1
2
x
16 + x
2
+ 8ln
1
4
16 + x
2
+
x
4
+ c
37. 9 −
16
√
2
3
39.
1
3
(1 + x
2
)
3/2
−
1 + x
2
+ c
41.
x
2
+ 4x −cosh
−1
x + 2
2
+ c
43.
x
2
+ 2x +10 −
1
3
sinh
−1
x + 1
3
+ c
45.
1
4
tan
4
x +
1
2
tan
2
x + c;
1
4
sec
4
x + c
47. (b)
1
2
sec x tan x +
1
2
|
sec x + tan x
|
+ c
(c)
1
3
sec
2
x tan x +
2
3
tan x + c
(d)
1
4
sec
3
x tan x +
3
8
sec x tan x +
3
8
ln
|
sec x + tan x
|
+ c
49. −
1
2
cot x csc x +
1
2
ln|csc x + cot x|+c
53.
1
2
RI
2
Exercises 7.4, page 449
1.
3
x + 1
−
2
x − 1
;3ln|x + 1|−2ln|x − 1|+c
3.
2
x + 1
+
4
x − 2
;2ln|x + 1|+4ln|x − 2|+c
5.
2
x + 1
+
1
2
x − 2
−
5
2
x
;2ln|x + 1|+
1
2
ln|x − 2|−
5
2
ln|x|+c
7.
3
2x + 1
−
2
3x − 7
;
3
2
ln|2x + 1|−
2
3
ln|3x − 7|+c
9.
1
4
x + 2
+
3
2
(x + 2)
2
−
1
4
x
;
1
4
ln|x + 2|−
3
2
(x + 2)
−1
−
1
4
ln|x|+c
11.
−2x + 1
x
2
+ 1
+
2
x
;−ln(x
2
+ 1) +tan
−1
x + 2ln|x|+c
13.
2
3
+
1
3
5
3x + 2
−
3
2x − 5
;
2
3
x +
5
9
ln|3x + 2|−
1
2
ln|2x − 5|+c
15.
2
x + 1
+
1
(x + 1)
2
;2ln|x + 1|−(x +1)
−1
+ c
17. 1 −
2
x
+
2
x
2
+ 2x +2
; x − 2ln|x|+2 tan
−1
(x + 1) +c
19. 3 +
2
x − 1
+
x − 2
x
2
+ 1
;
3x + 2ln|x − 1|+
1
2
ln
x
2
+ 1
−2 tan
−1
x + c
21. 11ln |x + 4|+2ln|x − 2|+
1
2
x
2
− 2x +c
23. ln|x + 2|−3ln|x + 1|+2ln|x|+c 25. ln|x
4
− x|+c
27.
1
2
ln|2x + 1|−
1
4
ln(4x
2
+ 1) +
1
2
tan
−1
(2x) + c
29.
1
6
x
2
−
2
9
x +
5
27
ln
x
2
+
3
2
x +
1
3
−
2
√
2
27
tan
−1
3x+1
√
2
+ c
31. 3ln|x|+
1
2
ln|x
2
+ x + 1|−
7
√
3
tan
−1
2x+1
√
3
+ c
33. −x
2
cos x + 2x sin x + 2 cos x + c
35.
1
2
ln(4 −sin
2
x) +c
37.
4
x
2
+ 1
−
2
x
2
+ 1
2
39.
4
x
2
+ x + 1
−
4x + 1
x
2
+ x + 1
2
41. ln|x
3
|−ln|x
3
+ 1|+c; −ln |1 +1/x
3
|+c
43. (a) partial fractions (b) substitution
(c) partial fractions (d) basic formula
45.
1
4(1−sin x)
−
1
4
ln(1 −sin x) −
1
4(1+sin x)
+
1
4
ln(1 +sin x) +c
Exercises 7.5, page 456
1.
1
8(2 +4x)
+
1
16
ln|2 +4x|+c
3.
2
15
(3e
x
− 2)(1 +e
x
)
3/2
+ c
5.
1
4
x
1/4 + x
2
−
1
16
ln
x +
1/4 + x
2
+ c
7. −
√
3
12
+
π
9
9. ln
2 +
√
8
− ln
1 +
√
5
11.
−1
x − 3
9 −(x −3)
2
− sin
−1
x − 3
3
+ c
13.
1
5
tan
5
u −
1
3
tan
3
u + tan u −u +c
15.
1
2
ln
√
4 +sin x − 2
√
4 +sin x + 2
+ c
17.
1
2
cos x
2
+
1
2
x
2
sin x
2
+ c
19. −
4
3
(cos x − 2)
√
1 +cos x + c
21.
1
2
sint
4 +sin
2
t − 2ln
sint +
4 +sin
2
t
+ c
23.
1
4
e
−2/x
2
+ c
25. −
4x − x
2
+ 2cos
−1
2 − x
2
+ c
27. e
x
tan
−1
(e
x
) −ln
1 +e
2x
+ c
Exercises 7.6, page 464
1. −
1
4
3. 3 5. 2 7. 1 9. −2
11. −
1
6
13.
1
2
15. 0 17. −
1
3
19. 2 −
2
sin2
21. 0 23. 0 25. does not exist 27. 1
29. 0 31. 0 33. e 35. ∞ 37. 1
39. e
−5
41. not indeterminate
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-35
43. 0 is the correct value, but the original expression is
not indeterminate
45. no 47. no
49. (a) sin3x ≈ 3x, sin2x ≈ 2x (b)
n
m
51. (a)
(x + 1)(2 +sin x)
x(2 +cos x)
(b)
x
e
x
(c)
3x + 1
x − 7
(d)
3 −8x
1 +2x
53. e
x
55. ∞ 57.
3
2
;
degree of p
degree of q
63.
3
4
65. (a)
√
40mg (b) 0 (c) gt
Exercises 7.7, page 476
1. (a) improper (b) not
3. (a) converges to
3
2
(b) diverges
5. (a) converges to 2 (b) converges to 8
7. (a) diverges (b) converges to
5
4
e
−2
9. (a) diverges (b) diverges
11. (a) converges to −1 (b) diverges
13. (a) diverges (b) diverges
15. (a) converges to π (b) diverges
17. (a) converges to 2 (b) diverges
19. (a) p < 1 (b) p > 1 21.
x
1 + x
3
<
1
x
2
, converges
23.
x
x
3/2
− 1
>
1
√
x
, diverges 25.
3
x + e
x
<
3
e
x
, converges
27.
sin
2
x
1 +e
x
<
1
e
x
, converges 29.
x
2
e
x
ln x
> e
x
, diverges
31.
1
2
ln4 −
1
4
35. true 37. false
39. (a)
π
k
(b)
√
π
2
43. (d) −
π
2
ln2
45. g(x) =
⎧
⎪
⎨
⎪
⎩
tan x
1 +tan x
if 0 ≤ x <
π
2
1ifx =
π
2
;
π
2
0
1
1 +tan x
dx =
π
4
49. (a) 2 (b) 4 (c) r 51.
1
r
53. e
−1/8
≈ 0.882
55. (a)
1
(r) =
1
(2r −1)e
2r
+ 1
(b)
2
(r) =
1 −
1
r
2
(c) μ =
1
2
(d) f
1
(x) > f
2
(x) for 0 < x < 0.34 and x > 1
(e) c =
1
2
; higher risk needed for higher gain
57. I(p
1
) = 0, I(p
2
) = ln2 −
1
2
Exercises 7.8, page 485
7. 4 9.
4
1 −e
−4
11.
4
π
13. 0.157
15. 7.77 ×10
−11
17. 1 − e
−3/2
≈ 0.77687
19. e
−6
− e
−12
≈ 0.00247 21. 0.594 23. 0.9999995
25. (a)
3
4
(b)
3
1
2
≈ 0.7937 27. (a)
2
π
ln2 (b) tan
π
8
29. (a)
π
2
≈ 1.57 (b)
π
2
≈ 1.57 31. c =
4
1 −e
−4b
→ 4
33. c =
6
1 −e
−6b
→ 6; mean =
c
36
[1 −(6b + 1)e
−6b
] →
1
6
35. (a) 0.0646 (b) 0.9354 (c) 0.0132 (d) 0.4147
39.
5
2
41. p =
m
n
43. m =
1
b
47. (b)
16
15
x
2
−
4
5
x + 1 (b)
1
2
Chapter 7 Review Exercises, page 487
1. 2e
√
x
+ c 3.
1
2
sin
−1
x −
1
2
x
√
1 − x
2
+ c
5. −
1
3
x
2
e
−3x
−
2
9
xe
−3x
−
2
27
e
−3x
+ c 7.
1
2
tan
−1
x
2
+ c
9.
1
4
ln(4 + x
4
) +c 11.
1
3
x
3
+ c
13.
1
9
sin3 −
1
3
cos3 15.
3π
16
17.
2
π
19. 4ln2−
15
16
21.
1
3
sin
3
x + c
23.
1
4
sin
4
x −
1
6
sin
6
x + c 25.
1
5
tan
5
x +
1
3
tan
3
x + c
27.
2
3
(sin x)
3/2
−
2
7
(sin x)
7/2
+ c 29. tan
−1
x + 2
2
+ c
31. −
4 − x
2
2x
+ c 33. −
x
2
3
9 − x
2
− 6
9 − x
2
+ c
35.
x
2
3
x
2
+ 9 −6
x
2
+ 9 +c
37. 3ln|x + 1|−2ln|x + 2|+c
39. 3ln|x|+2ln|x − 2|−ln|x + 2|+c
41.
1
5
e
x
cos2x +
2
5
e
x
sin2x + c 43.
1
3
(x
2
+ 1)
3/2
+ c
45.
4
5
x − 4
−
4
5
x + 1
47.
3
x
−
1
x + 2
−
2
x − 1
49.
1
x + 2
−
4
(x + 2)
2
51.
1
8
e
x
(4 +2e
2x
)
√
4 +e
2x
− 2ln(e
x
+
√
4 +e
2x
) +c
53.
1
3
tan x sec
2
x +
2
3
tan x + c
55.
4
3(3 − x)
+
4
9
ln
x
3 − x
+ c
57. −
9 +4x
2
x
+ 2ln
x +
9
4
+ x
2
+ c
59.
4 − x
2
− 2ln
2 +
4 − x
2
x
+ c
61. diverges 63. converges to 3 65. converges to π
67. diverges 69. RT;
4RT
3
3(2T
2
− 1)e
2T
2
+ 3
73. c 75. 24.75%; 135
CHAPTER 8
Exercises 8.1, page 498
1. 2e
4t
3. 5e
−3t
5. 2e
2(t−1)
7. 50 + 20e
t
9. (a) 3200 (b) 400 ·2
t
= 400e
(ln2)t
(c) 4525
11. (a) 8 hours (b) 100 ·2
t/4
= 100e
(ln2/4)t
(c) 23.6 hours
13. 20
ln10
ln2
≈ 66.4 minutes 17. (a) 12.5% (b) 8.4%
19. 0.4e
−(ln2/3)t
mg;15.97 hours; (a) 6 hours (b) 15.97 hours
21. 13,305 years 23.
ln
5
13
ln
11
13
≈ 5.72 minutes
25. (a) 70 −20e
(ln·7/2)t
(b) 66.6
◦
(c) 9.02 min
27. 9:46
P.M. 29. $1080, $1083, $1083.28, $1083.29
31. (a) A = $110,232; B = $66,402 (b) A = $22,255; B = $29,836
(c) 6.9%
33. (a) $14,715.18 (b) $5413.41
with linear depreciation: 10 years, $20,000; 20 years, $0
37. $7300;$7860 vs. $7665
39. p(x) ≈ e
1.468x+0.182
; answers will vary
41. p(t) ≈ e
−0.055t+2.015
, where t is years since 1960
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-36 APPENDIX B
..
Answers to Odd-Numbered Exercises
45. for 31-hour half-life: 87.71 mg; for 46-hour half-life: 104.48 mg
47. E(t) = e
−(ln2/4)t
0 6 12 18 24
0.5
1
t
E
49. $4493.29 51. A: $167,150.43; B: $179,373.42; C: $180,000
53. (a) lump sum ($1,271,249 to $1,267,853 after 3 years)
(b) 4 payments ($1,228,234 to $1,197,217 after 3 years)
(c) lump sum ($1,349,859 to $1,309,401 after 3 years)
Exercises 8.2, page 507
1. (a) yes (b) no 3. (a) yes (b) no
5. y = ce
x+x
3
/3
7. y =−
1
2
3
x
3
+ c
9. y =±
4ln
1 + x
3
+ c
11. (y +1)e
−y
= 2(x + 1)e
−x
+ c
13. cos y =−sin x + c 15. y = c
√
1 + x
2
17. y = ce
−x
2
/2
A=3
A=2
A=1
1.0
3.0
0.0
x
0.5
0−4 −2
2.0
1.5
2.5
24
19. y =±
√
2x + c
c=3
c=2
c=1
x
2.0
02
3.6
0.0
−4
0.4
4
1.6
0.8
−2
1.2
2.4
3.2
2.8
21. y = e
−1
e
(x+1)
3
23. y =
8
3
x
3
+ 4 25. y = (x + 3)
4
27. sin y = 2x
2
29. y =
2e
6t
1 +e
6t
31. y =
20e
10t
1 +4e
10t
33. y =
3e
t
1 +3e
t
35. (a) y =
8 ×10
7
e
0.71t
3 +e
0.71t
(b)
0–2
5E7
–4 42
1E8
(c) 4.642 years
37. (a) $277,901 (b) $25,002
39. (a) $1,131,949 (b) $998,258 (c) 10.5%
41. (a) y =
3
x
3
+
21
2
x
2
+ 9x (b) vertical tangent
⫺10 ⫺50510
5
10
15
y
x
(c) the cubic has three real solutions for
−217 −37
√
37
12
< c <
−217 +37
√
37
12
43. x(t) =
4
11 −3e
−(4/5)t
5
33 −e
−(4/5)t
; limiting concentration is
4
15
05
0.3
x
t
10
45. x(t) =
3
14 −5e
−(27/50)t
5
28 −e
−(27/50)t
0510
0.3
x
y
47. (a) the long-term population is about 682,800
(b) fish die off in 27 years
49. x(t) =
2 −6ce
0.2t
1 −ce
0.2t
→ 6; if p(0) < 2, then c > 1 and there is a
vertical asymptote
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-37
51. r
(t) = k(S −r);r(t) = 1 +13e
−k
(values in thousands) where
k =
1
4
ln
7
13
≈−0.155
53. k =−m = 0.0000847, M =
b
k
=
0.18
k
= 2125 55. 70 m/s
Exercises 8.3, page 518
1.
3.
5.
7. C 9. D 11. A
13. h = 0.1: y
1
= 1, y
2
= 1.02, y(1) ≈ 2.3346, y(2) ≈ 29.4986;
h = 0.05: y
1
= 1, y
2
= 1.005, y(1) ≈ 2.5107, y(2) ≈ 39.0930
15. h = 0.1: y
1
= 1.3, y
2
= 1.651, y(1) ≈ 3.8478, y(2) ≈ 3.9990;
h = 0.05: y
1
= 1.15, y
2
≈ 1.3139, y(1) ≈ 3.8188, y(2) ≈ 3.9978
17. h = 0.1: y
1
= 2.9, y
2
≈ 2.8005, y(1) ≈ 2.0943, y(2) ≈ 1.5276;
h = 0.05: y
1
= 2.95, y
2
≈ 2.9001, y(1) ≈ 2.0990, y(2) ≈ 1.5345
19. h = 0.1: y
1
= 1.1, y
2
≈ 1.2095, y(1) ≈ 2.3960, y(2) ≈ 4.5688;
h = 0.05: y
1
= 1.05, y
2
≈ 1.1024, y(1) ≈ 2.4210, y(2) ≈ 4.6203
21. y(1) = e ≈ 2.7183, y(2) = e
4
≈ 54.5982;
y(1) =
√
5 ≈ 2.236068, y(2) =
√
8 ≈ 2.828427
23.
25. y = 0 (unstable), y = 2 (stable)
27. y = 0 (unstable), y =−1 (unstable), y = 1 (stable)
29. y = 1 (stable)
33. estimates are 31.64, 218.12, overflow
35. x =
ln2
2
37. (a) y
n
alternates values above and below 8.
(b) y
n
increases to 8; solution increases to 8
39. y(0) = 3; y =
1
3
x
3
− 2x
2
+ 2x +3
41. y(0) =−5.5
43. a =
3 −
√
5
2
, b =
3 +
√
5
2
; g = 0 and g = b are stable; g(t) → 0if
3.98 < g(0) < 5.44 and 10.27 < g(0) < 11.72
Exercises 8.4, page 525
1. (0, 0): no prey or predators; (1, 0): prey but no predators;
(1/2, 1/4): twice as many prey as predators
3. (0, 0): no prey or predators; (3, 0): prey but no predators;
(2, 1/2): four times as many prey as predators
5. (0, 0): no prey or predators; (2, 0): prey but no predators
7. (0, 0) and (1, 0) are unstable, (0.5, 0.25) is stable
9. (0, 0) is unstable, (2, 0) is stable
11. unstable 13. stable
15. (0, 0): none of either species; (3/2, 0): some of first species,
none of second; (0, 2): some of second species, none of first;
(1, 1): equal amounts of each species
17. (0, 0): none of either species; (3/2, 0): some of first species,
none of second; (0, 2): some of second species, none of first;
(1/2, 1): second species is double the first species
19. (0, 0): none of either species; (1, 0): some of first species, none of
second; (0, 1): some of second species, none of first
21. (1, 1) is stable, the others are unstable
23. (a)
x
y
1.510.50⫺0.5
⫺0.5
0.5
1
1.5
(b)
x
y
1.510.50⫺0.5
⫺0.5
0.5
1
1.5
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-38 APPENDIX B
..
Answers to Odd-Numbered Exercises
(c)
x
y
1.510.50⫺0.5
⫺0.5
0.5
1
1.5
25. u
= v; v
=−4u − 2xv + 4x
2
27. u
= v; v
=−xu
2
+ (cosu)v + 2x
29. u
1
= u
2
;u
2
= u
3
;u
3
=−2xu
3
+ 4u
2
− 2u
1
+ x
2
31. u
1
= u
2
;u
2
= u
3
;u
3
= u
4
;u
4
= 2u
4
− xu
2
+ 2 −e
x
33. x(1) ≈ 0.253718, y(1) ≈ 0.167173
35. (0, 3), (0, −3), (2, 1), (−2, −1), (6, 3), (−6, −3)
37. (0, 0), (−2, 2), (4, 4)
39.
0.5
4
0.0
2
y(x)
0
−1.0 −0.5
x
1.0
−2
41. death rate < 0.4
Chapter 8 Review Exercises, page 528
1. 3e
2x
3.
√
2x
2
+ 4
5.
1
3
x
3/2
+
5
3
2
7. 10
4
e
(ln2/2)t
;
2ln 100
ln2
≈ 13.29 hours
9. 2e
−(ln2/2)t
;
−2ln0.05
ln2
≈ 8.64 hours
11.
ln2
0.08
≈ 8.66 years
13. 112e
[ln(108/112)]t
+ 68;21.10 minutes
15. y = ce
(1/2)x
4
17.
1
3
y
3
+
1
2
y
2
= 4 tan
−1
x + c
19. y = 0 (unstable), y = 2 (stable)
21. y = 0 (stable)
23.
25.
27. (a) x(t) = 0.01849
38.239e
−0.36056t−1.5700
− 12.239
e
−0.36056t−1.5700
− 1
(b) x(t) = 0.01849
38.239e
−0.36056t−0.56968
+ 12.239
e
−0.36056t−0.56968
+ 1
0 5 10 15 20
0.2
0.4
t
x
29. 0 ≤ x ≤ a; x(t) =
x(0) +t
ra
2
−ra · x(0)
1 +t
(
ra − r · x(0)
)
;
thelimiting concentrationwill bea
31. 10ln 4 ≈ 13.86 years
33. predator-prey model; equilibrium solutions are (0, 0) (no prey or
predators) and (1, 0) (prey but no predators)
35. competing species model; equilibrium solutions are (0, 0) (none of
either species); (0, 4) (none of first species, some of second); (5, 0)
(some of first species, none of second); (1, 2) (twice as many of
second species as first species)
37. (0, 0) is unstable, (1, 0) is stable
39. Let u = y and v = u
to get the system
u
= v; v
=−2u + 4x
2
v + 4xu − 1
CHAPTER 9
Exercises 9.1, page 542
1. 1,
3
4
,
5
9
,
7
16
,
9
25
,
11
36
3. 4, 2,
2
3
,
1
6
,
1
30
,
1
180
5. converges to 0 7. converges to 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-39
9.
1.0
0.5
4
0.0
108
0.75
6
0.25
2
Converges to 0
0.8
0.4
0.6
0.7
40
0.3
20 503010
0.9
0.5
Converges to 0
0.9
30
0.8
0.5
20
0.7
15 25105
0.6
Converges to 1
105
2.0
1.0
0.0
15
3.0
2.5
1.5
0.5
20
Converges to 2
11. converges to
3
2
13. diverges 15. diverges
17. converges to 0 19. converges to 0 21. converges to 0
23. converges to 0 25. 1 27. ln 2 29. 0
35. decreasing 37. increasing 39. |a
n
| < 3
41. |a
n
| <
1
2
43. −
(−2)
n
16
45.
2n−1
n
2
49. (a) lim
n→∞
1 +
2
n
n
= e
2
; lim
n→∞
1 −
2
n
n
= e
−2
51. 1.8312 53. converges to
1
2
;
1
0
xdx
55. (b) (c
3
− c
1
)
2
+ (r
3
−r
1
)
2
= (r
1
+r
3
)
2
; |c
3
− c
1
|=2
√
r
1
r
3
;
(c
3
− c
2
)
2
+ (r
3
−r
2
)
2
= (r
2
+r
3
)
2
; |c
3
− c
2
|=2
√
r
2
r
3
(d) r
n
= 1/F
2
n
57. S
48
=
2 −
2 +
2 +
√
3 ≈ 0.1308,
π
24
≈ 0.1309
59. a
n
= 144 − 36π ≈ 30.9
Exercises 9.2, page 551
1. converges to
15
4
3. converges to
3
8
5. diverges
7. converges to 3 9. diverges 11. diverges
13. converges to 1 15. converges to
2/e
e−1
17. diverges
19. converges to
5
6
21. diverges 23. diverges
25. −1 < c < 0 27. c = 0 29. converges
31. converges 33. (a) L −
m−1
k=1
a
k
37. 64; 256; 4
m−1
39. (a)
0.9
1 −0.1
= 1 41. a
k
=
1
k
and b
k
=−
1
k
43.
1
1−r
>
1
2
if −1 < r < 1 45. yes 47. no
49. 1.3589L;n = 4 51.
p
2
1 −2p(1 − p)
> p if p >
1
2
;0.692
53. 2
1 −e
−0.1
≈ 0.19 55. $400,000; save $150, 000
57. 1; you eventually win a game 59. 1.002004008 ...
Exercises 9.3, page 562
1. (a) diverges (b) diverges
3. (a) diverges (b) converges
5. (a) diverges (b) converges
7. (a) converges (b) converges
9. (a) diverges (b) diverges
11. (a) converges (b) converges
13. (a) diverges (b) diverges
15. (a) diverges (b) converges
17. (a) converges (b) converges
19. (a) diverges (b) diverges
(c) diverges (d) diverges
21. p > 1 23. p > 1 25.
1
3·100
3
27.
6
7·50
7
29. e
−1600
≈ 6.73 × 10
−696
31. 101 33. 4
35. (a) can’t tell (b) converges (c) converges (d) can’t tell
45.
π
2
6
,
π
4
90
,
π
6
945
,
π
8
9450
,
π
10
93,555
47. (a) y = x
n
is concave up for x > 0
49. 2 53. (a) 10
10
k=1
1
k
≈ 29.3 (b) 3
Exercises 9.4, page 570
1. convergent 3. convergent 5. convergent
7. convergent 9. divergent 11. diverges
13. convergent 15. convergent 17. diverges
19. convergent 21. divergent 23. convergent
25. 3.61 27. −0.22 29. 1.10 31. −0.21
33. 20,000 35. 34 terms (k = 0tok = 33)
37. converges 39. diverges 41. f
(k) < 0 for k ≥ 2
43. positives diverge, negatives converge
45.
2
1
1
x
dx 47.
3
4
1 +
3
4
=
3
7
Exercises 9.5, page 577
1. absolutely convergent 3. divergent
5. conditionally convergent 7. absolutely convergent
9. divergent 11. absolutely convergent
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-40 APPENDIX B
..
Answers to Odd-Numbered Exercises
13. absolutely convergent 15. divergent
17. conditionally convergent 19. absolutely convergent
21. absolutely convergent 23. absolutely convergent
25. conditionally convergent 27. conditionally convergent
29. absolutely convergent 31. absolutely convergent
33. divergent 35. absolutely convergent
37. absolutely convergent 39. absolutely convergent
41. ratio 43. alternating 45. p-series
47. ratio 49. integral 51. root
53. comparison 55. comparison
57. integral 59. ratio
63. (a) −1 ≤ p ≤ 1
Exercises 9.6, page 585
1. ∞, (−∞, ∞) 3. 4, (−4, 4)
5. 3, (−2, 4] 7. 0, {x =−1}
9. r =
1
2
, (1, 2) 11. r =
1
8
,
−
5
8
, −
3
8
13. r = 2, (−4, 0) 15. r =∞, all x
17.
0
5.0
−1−2−5 −3−4
2.5
−5.0
0.0
−2.5
x
365
−6 2
14
y
19. (−3, −1),
−1
1 + x
21. (0, 1),
1
2 −2x
23. (−2, 2),
2
2 + x
25.
∞
k=0
2x
k
, r = 1, (−1, 1)
27.
∞
k=0
(−1)
k
3x
2k
, r = 1, (−1, 1)
29.
∞
k=0
2x
3k+1
, r = 1, (−1, 1)
31.
∞
k=0
(−1)
k
1
2
2k+1
x
k
, r = 4, (−4, 4)
33.
∞
k=0
(−1)
k
3
2k + 1
x
2k+1
, r = 1
35.
∞
k=1
2kx
2k−1
, r = 1 37.
∞
k=0
(−1)
k
1
k + 1
x
2k+2
, r = 1
39. (−∞, ∞);
∞
k=1
−k sin(k
3
x), {x = nπ}
41. (−∞, 0);
∞
k=0
ke
kx
, (−∞, 0)
43. (a − b, a + b), r = b 45. r
47. r =1, 1.003005007...
49.
∞
k=0
2x
3k+2
3k + 2
;r = 1
4
0.75
0
x
3
1.0
2
1
0.50.250.0
51. (a)
∞
k=0
(−1)
k
1
4k + 1
+
1
4k + 3
= 1 +
1
3
−
1
5
−
1
7
+···
(b)
1
√
2
[tan
−1
(1 +
√
2) −tan
−1
(1 −
√
2)]
53.
2
(1 − p)
3
,
6
(1 − p)
4
,
1
(1 − p)
n+1
Exercises 9.7, page 597
1.
∞
k=0
(−1)
k
x
2k
(2k)!
, (−∞, ∞) 3.
∞
k=0
2
k
k!
x
k
, (−∞, ∞)
5.
∞
k=0
(−1)
k
x
k+1
k + 1
, (−1, 1] 7.
∞
k=0
(−1)
k
(k + 1)x
k
, (−1, 1)
9.
∞
k=0
(x − 1)
k
k!
, (−∞, ∞)
11. 1 +
∞
k=1
(−1)
k+1
e
−k
k
(x − e)
k
, (0, 2e]
13.
∞
k=0
(−1)
k
(x − 1)
k
, (0, 2)
15. 1 +
1
2
(x − 1) −
1
8
(x − 1)
2
+
1
16
(x − 1)
3
−
5
128
(x − 1)
4
+
7
256
(x − 1)
5
−
21
1024
(x − 1)
6
17. e
2
1 +(x −2) +
1
2
(x − 2)
2
+
1
6
(x − 2)
3
+
1
24
(x − 2)
4
+
1
120
(x − 2)
5
+
1
720
(x − 2)
6
19. x +
1
6
x
3
+
3
40
x
5
21. |R
n
(x)|≤
|x|
n+1
(n + 1)!
→ 0 23. |R
n
(x)|=
1
n + 1
x − 1
z
n+1
→ 0
25. (a) 0.04879 (b)
0.05
5
5
(c) 7
27. (a) 1.0488 (b)
7(0.1)
5
256
(c) 9
29.
∞
k=0
(−3)
k
k!
x
k
;r =∞ 31.
∞
k=0
(−1)
k
x
2k+1
k!
;r =∞
33.
∞
k=0
(−1)
k
2
2k+1
(2k + 1)!
x
2k+2
;r =∞
35. e
x
with x = 2 37. tan
−1
x with x = 1
39. (a) differentiation (b) Taylor series
41. (a) antidifferentiation (b) antidifferentiation
43. x; converges to f for all x ≥ 0
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-41
47. 1 +
∞
k=1
(−1)
k
(2k − 3)(2k − 1)···(−1)
2
k
k!
x
k
49. 1 +(x −1) = x =|x|, for x < 0
51. the first three terms of the Taylor series for f and g are identical
53. x + x
2
+
1
3
x
3
−
1
30
x
5
−
1
90
x
6
; equals the product
55. 1 −
x
2
3!
+
x
4
5!
−
x
6
7!
; equals to the Maclaurin series for sin x, divided
by x
57.
∞
k=0
x
2k
(2k)!
;
∞
k=0
x
2k+1
(2k + 1)!
59. 4; 6; Taylor is more efficient
Exercises 9.8, page 605
1. 0.99923163442 3. 0.94275466553
5. 0.81873075307 7. −
1
2
9. −
1
2
11. 1
13.
1703
900
15.
5651
3780
17.
2
5
19. ∞ 21. 12
23. 1 +
1
2
x +
3
8
x
2
+
5
16
x
3
+
35
128
x
4
+···
25. 6 −6x +12x
2
− 28x
3
+ 70x
4
−···
27. (a) 5.0990200 (b) 4.8989800
29. 64 +48x +12x
2
+ x
3
;1− 8x + 24x
2
− 32x
3
+ 16x
4
;n +1
31. x +
1
6
x
3
+
3
40
x
5
+
5
112
x
7
+
35
1152
x
9
+···
33. (b) m = 1.1m
0
if V =
√
0.2C ≈ 83,000 miles per second
(c) m
0
1 +
1
2c
2
V
2
+
9
24c
4
V
4
;m = 1.1m
0
if
V =
4
√
10−10
15
C ≈ 78,284 miles per second
35. (b) x =
R
20
≈ 200 miles
(c) mg
1 −
2
R
x +
3
R
2
x
2
; x = R
1−
√
0.7
3
≈ 217 miles
37. too large 41. λ ≈
0.0016
T
Exercises 9.9, page 618
1.
∞
k=1
(−1)
k+1
2
k
sinkx 3. π −
∞
k=1
8
(2k − 1)
2
π
cos[(2k −1)x]
5.
∞
k=1
−4
π(2k −1)
sin[(2k −1)x] 7. 3 sin2x
9.
∞
k=1
(−1)
k
2
kπ
sinkπ x 11.
1
3
+
∞
k=1
(−1)
k
4
k
2
π
2
coskπ x
13.
1
4
+
∞
k=1
−2
(2k − 1)
2
π
2
cos(2k −1)π x +
∞
k=1
(−1)
k+1
1
kπ
sinkπ x
15.
1
1
y
x
17.
0
y
x
1
1–3 –2 –1 32
19.
y
x
246–2
–1
–4–6
1
2
31. If f and g are even then fg is even.
35. sine 37. both
41.
0
y
x
21
43
–1
–2 –3–4
1
2
–1
–2
3 11 3
3
3
x
y
3 11 3
3
3
x
y
43. The amplitude varies slowly because the frequency of 2cos(0.2t)is
small compared to the frequency of sin(8.1t). The variation of the
amplitude explains why the volume varies, since the volume is
proportional to the amplitude.
47.
⫺8 ⫺40 84
0.2
0.4
0.6
0.8
1
y
n = 8
x
The modified Fourier series is
1
2
+
n
k=1
2n
[(2k − 1)π]
2
sin
(2k − 1)π
n
sin[(2k − 1)x].
Chapter 9 Review Exercises, page 621
1. converges to 0 3. converges to 0 5. converges to 0
7. diverges 9. diverges 11. can’t tell 13. diverges
15. converges 17. converges 19. 8 21.
4
3
23. −0.41 25. diverges 27. converges 29. diverges
31. converges 33. converges 35. converges
37. converges 39. diverges 41. converges 43. converges
45. converges conditionally 47. converges absolutely
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-42 APPENDIX B
..
Answers to Odd-Numbered Exercises
49. p > 1 51. 1732 53.
∞
k=0
(−1)
k
x
k
4
k+1
, r = 4
55.
∞
k=0
(−1)
k
x
2k
3
k
, r =
√
3 57.
∞
k=0
(−1)
k
x
k+1
(k + 1)4
k+1
+ ln4, r = 4
59. (−1, 1) 61. (−1, 1] 63. (−∞, ∞)
65.
5
3
,
7
3
67.
∞
k=0
(−1)
k
x
2k+1
(2k + 1)!
69. (x − 1) −
1
2
(x − 1)
2
+
1
3
(x − 1)
3
−
1
4
(x − 1)
4
71. 0.1822666, 10 73.
∞
k=0
(−1)
k
3
k
x
2k
k!
, r =∞ 75.
1117
2520
77.
∞
k=1
(−1)
k+1
4
kπ
sin
kπ
2
x
79.
y
x
3
15
5 1
2
4
3
4
2
81.
y
x
5
5
83.
2
3
89. contains powers of 2
CHAPTER 10
Exercises 10.1, page 630
1.
x
2
4
+
y
2
9
= 1
y
x
2 10
1
2
3
2
1
1
2
3
3. y =
3
2
x +
3
2
x
5
3
5
3
y
5. y = x
2
− 2x +3
x
5
0
5
6
y
7. x =
1
4
y
2
− 1
x
5
3
5
3
y
9. y = sin
(
sin x
)
,
−π
2
≤ x ≤
π
2
0.25
−0.25
1.0
−0.75
0.75
0.5
0.0
1.5
−0.5
0.5
0.0
−0.5−1.0−1.5
11. y = e
−x
2
, x > 0
1
0.75
0.25
−1
x
2
1.0
0.5
0
0.0
−2
13.
y
x
4
3
4
3
15.
x
2
2
2
2
y
17.
x
66
4
4
y