97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 65
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
Answers to Odd-Numbered Exercises
I
35. 共⫺⬁, 0兲 傼 共5, ⬁兲
CHAPTER 1
EXERCISES 1.1
N
37. 关0, 4兴
39. 共⫺⬁, ⬁兲
PAGE 19
41. 共⫺⬁, ⬁兲
y
1. Yes
3. (a) 3
y
2
(b) ⫺0.2
(c) 0, 3
(d) ⫺0.8
(e) 关⫺2, 4兴, 关⫺1, 3兴 (f) 关⫺2, 1兴
5. 关⫺85, 115兴
7. No
9. Yes, 关⫺3, 2兴, 关⫺3, ⫺2兲 傼 关⫺1, 3兴
11. Diet, exercise, or illness
13.
A65
5
0
x
1
0
_2
_1
t
T
43. 关5, ⬁兲
45. 共⫺⬁, 0兲 傼 共0, ⬁兲
y
y
4
t
0
15. (a) 500 MW; 730 MW
17. T
midnight
19.
2
(b) 4 AM; noon
0
0
47. 共⫺⬁, ⬁兲
t
noon
49. 共⫺⬁, ⬁兲
y
amount
y
(0, 2)
(0, 1)
_2
0
21.
23. (a)
0
Wed.
Wed.
⫺1
5
再
11
2
,1艋x艋5
0
53. f 共x兲 苷 1 ⫺ s⫺x
⫺x ⫹ 3 if 0 艋 x 艋 3
2 x ⫺ 6 if 3 ⬍ x 艋 5
57. A共L兲 苷 10L ⫺ L2, 0 ⬍ L ⬍ 10
59. A共x兲 苷 s3 x 2兾4, x ⬎ 0
61. S共x兲 苷 x 2 ⫹ 共8兾x兲, x ⬎ 0
3
2
63. V共x兲 苷 4x ⫺ 64x ⫹ 240x, 0 ⬍ x ⬍ 6
55. f 共x兲 苷
Wed.
1
x
1
51. f 共x兲 苷 2 x ⫺
price
height
of grass
Wed.
x
x
5
Wed. t
N
再
65. F共x兲 苷
200
15共40 ⫺ x兲 if 0 艋 x ⬍ 40
0
if 40 艋 x 艋 65
15共x ⫺ 65兲 if x ⬎ 65
F
600
150
(100, 525)
100
50
1996
2000
2004
t
(midyear)
(b) 126 million; 207 million
25. 12, 16, 3a 2 ⫺ a ⫹ 2, 3a 2 ⫹ a ⫹ 2, 3a 2 ⫹ 5a ⫹ 4,
6a 2 ⫺ 2a ⫹ 4, 12a 2 ⫺ 2a ⫹ 2, 3a 4 ⫺ a 2 ⫹ 2,
9a 4 ⫺ 6a 3 ⫹ 13a 2 ⫺ 4a ⫹ 4, 3a 2 ⫹ 6ah ⫹ 3h 2 ⫺ a ⫺ h ⫹ 2
27. ⫺3 ⫺ h
29. ⫺1兾共ax兲
31. 共⫺⬁, ⫺3兲 傼 共⫺3, 3兲 傼 共3, ⬁兲
33. 共⫺⬁, ⬁兲
0
67. (a)
40
65
100
x
(b) $400, $1900
R (%)
15
10
0
10,000
20,000
I (in dollars)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 66
A66
(c)
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
15. (a) T 苷 6 N ⫹
(b) 6 , change in ⬚F for every chirp per
minute change
(c) 76⬚F
17. (a) P 苷 0.434d ⫹ 15
(b) 196 ft
19. (a) Cosine
(b) Linear
21. (a) 15
Linear model is
appropriate.
1
T (in dollars)
2500
1000
0
10,000 20,000 30,000 I (in dollars)
307
6
1
69. f is odd, t is even
71. (a) 共⫺5, 3兲
(b) 共⫺5, ⫺3兲
73. Odd
75. Neither
77. Even
79. Even; odd; neither (unless f 苷 0 or t 苷 0)
61,000
0
EXERCISES 1.2
N
PAGE 33
1. (a) Logarithmic
(b) Root
(c) Rational
(d) Polynomial, degree 2
(e) Exponential
(f) Trigonometric
3. (a) h
(b) f (c) t
5. (a) y 苷 2x ⫹ b,
y b=3 b=0
b=_1
where b is the y-intercept.
(b) y 苷 ⫺0.000105x ⫹ 14.521
(c) y 苷 ⫺0.00009979x ⫹ 13.951
15
(b)
(c)
61,000
0
(d) About 11.5 per 100 population
y=2x+b
23. (a)
x
6.0
(e) About 6%
(f) No
Linear model is
appropriate.
height (m)
5.5
5.0
(b) y 苷 mx ⫹ 1 ⫺ 2m,
where m is the slope.
(c) y 苷 2x ⫺ 3
4.5
y
m=1
4.0
m=_1
3.5
m=0
(2, 1)
1896
x
1920
1940
1960
1980
2000
year
(b) y 苷 0.0265x ⫺ 46.8759
(c) 6.27 m; higher
25. Four times as bright
27. (a) N 苷 3.1046 A0.308
(b) 18
y-1=m(x-2)
7. Their graphs have slope ⫺1.
1900
y
(d) No
c=_1
EXERCISES 1.3
c=_2
0
x
c=2
c=1
c=0
9. f 共x兲 苷 ⫺3x共x ⫹ 1兲共x ⫺ 2兲
11. (a) 8.34, change in mg for every 1 year change
(b) 95 , change in ⬚F for every
1⬚C change; 32, Fahrenheit
temperature corresponding
to 0⬚C
(100, 212)
F= 95 C+32
32
(_40, _40)
(b)
y
0
(c)
y
1
1
(b) 8.34 mg
F
PAGE 42
1. (a) y 苷 f 共x兲 ⫹ 3
(b) y 苷 f 共x兲 ⫺ 3 (c) y 苷 f 共x ⫺ 3兲
(d) y 苷 f 共x ⫹ 3兲 (e) y 苷 ⫺f 共x兲 (f) y 苷 f 共⫺x兲
(g) y 苷 3f 共x兲 (h) y 苷 13 f 共x兲
3. (a) 3
(b) 1
(c) 4
(d) 5
(e) 2
5. (a)
13. (a)
N
0
x
1
(d)
y
1
0
2
x
y
1
1 x
0
1 x
C
7. y 苷 ⫺s⫺x 2 ⫺ 5x ⫺ 4 ⫺ 1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 67
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
9.
11.
y
x=_2
y
y=_ Œ„x
1
y=
x+2
0
0
x
x
33. (a) 共 f ⴰ t兲共x兲 苷 1 ⫺ 3 cos x, 共⫺⬁, ⬁兲
(b) 共t ⴰ f 兲共x兲 苷 cos 共1 ⫺ 3x兲, 共⫺⬁, ⬁兲
(c) 共 f ⴰ f 兲共x兲 苷 9x ⫺ 2, 共⫺⬁, ⬁兲
(d) 共t ⴰ t兲共x兲 苷 cos 共cos x兲, 共⫺⬁, ⬁兲
2x 2 ⫹ 6x ⫹ 5
, 兵 x x 苷 ⫺2, ⫺1其
共x ⫹ 2兲共x ⫹ 1兲
x2 ⫹ x ⫹ 1
(b) 共t ⴰ f 兲共x兲 苷
, {x x 苷 ⫺1, 0其
共x ⫹ 1兲2
x 4 ⫹ 3x 2 ⫹ 1
(c) 共 f ⴰ f 兲共x兲 苷
, {x x 苷 0其
x共x 2 ⫹ 1兲
2x ⫹ 3
, {x x 苷 ⫺2, ⫺ 53 }
(d) 共t ⴰ t兲共x兲 苷
3x ⫹ 5
ⱍ
35. (a) 共 f ⴰ t兲共x兲 苷
13.
15.
y
ⱍ
y
y=sin(x/2)
x-2-1
y=œ„„„„
0
1
ⱍ
2π
x
0
x
3
ⱍ
(2, _1)
17.
37. 共 f ⴰ t ⴰ h兲共x兲 苷 3 sin共x 兲 ⫺ 2
2
39. 共 f ⴰ t ⴰ h兲共x兲 苷 sx 6 ⫹ 4x 3 ⫹ 1
41. t共x兲 苷 2x ⫹ x 2, f 共x兲 苷 x 4
y
1
π
0
19.
y=21 (1-cos x)
21.
y
y=_(x+1)@+2
2
3
43. t共x兲 苷 s
x , f 共x兲 苷 x兾共1 ⫹ x兲
x
45. t共t兲 苷 t 2, f 共t兲 苷 sec t tan t
47. h共x兲 苷 sx , t共x兲 苷 x ⫺ 1, f 共x兲 苷 sx
y
2
49. h共x兲 苷 sx, t共x兲 苷 sec x, f 共x兲 苷 x 4
y=|x - 2|
51. (a) 4
(b) 3
in the domain of t.
1
x
_1 0
2
x
y
1
(c) 0
(e) 4
(d) Does not exist; f 共6兲 苷 6 is not
(f) ⫺2
(b) 共A ⴰ r兲共t兲 苷 3600␲ t 2; the area of the
circle as a function of time
55. (a) s 苷 sd 2 ⫹ 36
(b) d 苷 30t
(c) 共 f ⴰ t兲共t兲 苷 s900t 2 ⫹ 36; the distance between the lighthouse
and the ship as a function of the time elapsed since noon
53. (a) r共t兲 苷 60t
0
23.
57. (a)
y=| œ„
x-1 |
(b)
H
1
0
1
0
冋
V
120
x
t
0
V共t兲 苷 120H共t兲
册
2␲
共t ⫺ 80兲
25. L共t兲 苷 12 ⫹ 2 sin
365
27. (a) The portion of the graph of y 苷 f 共x兲 to the right of the
y-axis is reflected about the y-axis.
(b)
(c)
y
A67
(c)
V
240
V共t兲 苷 240H共t ⫺ 5兲
0
t
5
y
y= sin |x|
59. Yes; m1 m 2
61. (a) f (x兲 苷 x 2 ⫹ 6
63. Yes
y=œ„„
|x|
0
x
0
29. (a) 共 f ⫹ t兲共x兲 苷 x ⫹ 5x ⫺ 1, 共⫺⬁, ⬁兲
(b) 共 f ⫺ t兲共x兲 苷 x 3 ⫺ x 2 ⫹ 1, 共⫺⬁, ⬁兲
(c) 共 ft兲共x兲 苷 3x 5 ⫹ 6x 4 ⫺ x 3 ⫺ 2x 2, 共⫺⬁, ⬁兲
x 3 ⫹ 2x 2
(d) 共 f兾t兲共x兲 苷
, {x x 苷 ⫾1兾s3 }
3x 2 ⫺ 1
3
2
ⱍ
(b) t共x兲 苷 x 2 ⫹ x ⫺ 1
x
EXERCISES 1.4
N
PAGE 50
1. (c)
3.
100
_10
40
31. (a) 共 f ⴰ t兲共x兲 苷 4x ⫹ 4x, 共⫺⬁, ⬁兲
2
(b) 共t ⴰ f 兲共x兲 苷 2x 2 ⫺ 1, 共⫺⬁, ⬁兲
(c) 共 f ⴰ f 兲共x兲 苷 x 4 ⫺ 2x 2, 共⫺⬁, ⬁兲
(d) 共t ⴰ t兲共x兲 苷 4x ⫹ 3, 共⫺⬁, ⬁兲
_300
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
t
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 68
A68
5.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
7.
8
31.
3500
_2.5
_20
_50
260
11.
1.1
_4
1.5
0
_0.01
0.01
0
13.
2.5
20
_3500
0
9.
100
_1.5
11
If c ⬍ ⫺1.5, the graph has three humps: two minimum points and
a maximum point. These humps get flatter as c increases until at
c 苷 ⫺1.5 two of the humps disappear and there is only one minimum point. This single hump then moves to the right and
approaches the origin as c increases.
33. The hump gets larger and moves to the right.
35. If c ⬍ 0, the loop is to the right of the origin; if c ⬎ 0, the loop
is to the left. The closer c is to 0, the larger the loop.
2
EXERCISES 1.5
_2π
_
2π
π
25
π
25
_11
_2
15. (b) Yes; two are needed
17.
2 -1 -2 -3
1 _1.5
1
(c) 共0, ⬁兲
(d) See Figures 4(c), 4(b), and 4(a), respectively.
7.
All approach 0 as x l ⫺⬁,
5 y=20 ® y=5® y=´
all pass through 共0, 1兲, and
y=2®
all are increasing. The larger
the base, the faster the rate of
increase.
9.
1
PAGE 57
1. (a) 4
(b) x ⫺4兾3
12
3. (a) 16b
(b) 648y 7
5. (a) f 共x兲 苷 a x, a ⬎ 0
(b) ⺢
_1
_1
N
2
0
The functions with base
greater than 1 are increasing
and those with base less than
1 are decreasing. The latter
are reflections of the former
about the y-axis.
1 ®
y=” 13 ’® y=” 10
’ 5 y=10® y=3®
_1
19. No
21. ⫺0.72, 1.22
27. ⫺0.31 ⬍ x ⬍ 0.31
29. (a)
23. 0.65
25. t
_2
(b)
3
11.
2
x
x %œ„
_3
^œx„
y
y=10
3
0
_1
x+2
x
y=_2–®
4
_1
_1
1
_2
_2
(c)
13.
y
Œ„
x
$œx„
x
œ„
2
0
_1
0
x
2
x
Œ„
œ„
x
$œx„
15.
y
y=1
1
x
%œ„
”0, 2 ’
3
_1
0
1
y=1- 2 e–®
_1
(d) Graphs of even roots are similar to sx, graphs of odd roots are
3
n
similar to s
x. As n increases, the graph of y 苷 s
x becomes steeper
near 0 and flatter for x ⬎ 1.
x
17. (a) y 苷 e x ⫺ 2
⫺x
(d) y 苷 e
(b) y 苷 e x⫺2
(e) y 苷 ⫺e⫺x
(c) y 苷 ⫺e x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 69
APPENDIX I
19. (a) 共⫺⬁, ⫺1兲 傼 共⫺1, 1兲 傼 共1, ⬁兲
(b) 共⫺⬁, ⬁兲
21. f 共x兲 苷 3 ⴢ 2 x
27. At x ⬇ 35.8
29. (a) 3200
(b) 100 ⴢ 2 t兾3 (c) 10,159
(d)
ANSWERS TO ODD-NUMBERED EXERCISES
49. (a) 共0, ⬁兲; 共⫺⬁, ⬁兲
(c)
y
A69
(b) e ⫺2
x=0
t ⬇ 26.9 h
60,000
ƒ=ln x+2
0
40
0
31. P 苷 2614.086共1.01693兲t; 5381 million; 8466 million
EXERCISES 1.6
N
PAGE 69
1. (a) See Definition 1.
(b) It must pass the Horizontal Line Test.
3. No
5. No
7. Yes
9. No
11. Yes
13. No
15. (a) 6
(b) 3
17. 0
9
19. F 苷 5 C ⫹ 32; the Fahrenheit temperature as a function of the
Celsius temperature; 关⫺273.15, ⬁兲
1
2
21. y 苷 3 共x ⫺ 1兲 2 ⫺ 3, x 艌 1
1
23. y 苷 2 共1 ⫹ ln x兲
25. y 苷 e x ⫺ 3
⫺1
4
27. f 共x兲 苷 sx ⫺ 1
29.
6
y
f
f –!
f
x
0
f –!
6
0
51.
53.
55.
57.
59.
(a)
(a)
(a)
(a)
x
e–@
共7 ⫺ ln 6兲 (b) 13 共e 2 ⫹ 10兲
5 ⫹ log 2 3 or 5 ⫹ 共ln 3兲兾ln 2 (b) 12 (1 ⫹ s1 ⫹ 4e )
0⬍x⬍1
(b) x ⬎ ln 5
共ln 3, ⬁兲 (b) f ⫺1共x兲 苷 ln共e x ⫹ 3兲; ⺢
1
4
5
The graph passes the
Horizontal Line Test.
4
_2
_1
3
3
3
3
f ⫺1(x) 苷 ⫺ 16 s
4 (s
D ⫺ 27x 2 ⫹ 20 ⫺ s
D ⫹ 27x 2 ⫺ 20 ⫹ s
2 ),
where D 苷 3 s3 s27x 4 ⫺ 40x 2 ⫹ 16; two of the expressions are
complex.
61. (a) f ⫺1共n兲 苷 共3兾ln 2兲 ln共n兾100兲; the time elapsed when there
are n bacteria
(b) After about 26.9 hours
63. (a) ␲兾3
(b) ␲
65. (a) ␲兾4
(b) ␲兾4
67. (a) 10
(b) ␲兾3
71. x兾s1 ⫹ x 2
73.
π
2
y=sin– ! x
31. (a) f ⫺1 共x兲 苷 s1 ⫺ x 2 , 0 艋 x 艋 1; f ⫺1 and f are the same
function.
(b) Quarter-circle in the first quadrant
33. (a) It’s defined as the inverse of the exponential function with
base a, that is, log a x 苷 y &? a y 苷 x.
(b) 共0, ⬁兲 (c) ⺢
(d) See Figure 11.
35. (a) 3
(b) ⫺3
37. (a) 3
(b) ⫺2
39. ln 1215
41. ln
43.
sx
x⫹1
π
_2
π
2
The second graph is
the reflection of the
first graph about the
line y 苷 x.
π
_2
y=log 1.5 x
3
y=ln x
y=log 10 x
0
4
y=log 50 x
All graphs approach
⫺⬁ as x l 0⫹, all pass
through 共1, 0兲, and all
are increasing. The larger
the base, the slower the
rate of increase.
45. About 1,084,588 mi
47. (a)
(b)
y
2
0
y=-ln x
x
0
(b) 关⫺␲兾2, ␲兾2兴
(b) h⫺1共x兲 苷 共1兾c兲 f ⫺1共x兲
77. (a) t⫺1 共x兲 苷 f ⫺1共x兲 ⫺ c
CHAPTER 1 REVIEW
N
PAGE 72
True-False Quiz
3. False
13. False
5. True
7. False
9. True
Exercises
y
y=log 10 (x+5)
_4
75. (a) 关⫺3, 0兴
1. False
11. False
⫺5
_5
y=sin x
1
x
1. (a) 2.7
(b) 2.3, 5.6
(c) 关⫺6, 6兴 (d) 关⫺4, 4兴
(e) 关⫺4, 4兴 (f) No; it fails the Horizontal Line Test.
(g) Odd; its graph is symmetric about the origin.
1
1
3. 2a ⫹ h ⫺ 2
5. ( ⫺⬁, 3 ) 傼 ( 3 , ⬁), 共⫺⬁, 0兲 傼 共0, ⬁兲
7. 共⫺6, ⬁兲, ⺢
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/14/10 5:02 PM Page 70
A70
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
9. (a) Shift the graph 8 units upward.
(b) Shift the graph 8 units to the left.
(c) Stretch the graph vertically by a factor of 2, then shift it
1 unit upward.
(d) Shift the graph 2 units to the right and 2 units downward.
(e) Reflect the graph about the x-axis.
(f) Reflect the graph about the line y 苷 x (assuming f is
one-to-one).
11.
13.
y
9. (a)
_1
ƒ=max{x, 1/x}
y
(b)
y
1
x
π
x
1
y=_sin 2x
0
y
ƒ=max{sin x, cos x}
1
y= 2 (1+´)
_3π
4
1
1
y= 2
0
5π
4
0
_7π
4
x
π
4
x
_œ„2/2
15.
y
1
x=_2
y= x+2
1
2
(c)
0
y
x
4
(b) Odd
(c) Even
(d) Neither
17. (a) Neither
19. (a) 共 f ⴰ t兲共x兲 苷 ln共x 2 ⫺ 9兲, 共⫺⬁, ⫺3兲 傼 共3, ⬁兲
(b) 共t ⴰ f 兲共x兲 苷 共ln x兲 ⫺ 9, 共0, ⬁兲
(c) 共 f ⴰ f 兲共x兲 苷 ln ln x, 共1, ⬁兲
(d) 共t ⴰ t兲共x兲 苷 共x 2 ⫺ 9兲2 ⫺ 9, 共⫺⬁, ⬁兲
21. y 苷 0.2493x ⫺ 423.4818; about 77.6 years
23. 1
25. (a) 9
(b) 2
(c) 1兾s3
(d)
ƒ=max{≈, 2+x, 2-x}
2
27. (a)
_2
3
5
11. 5
⬇ 4.4 years
1000
0
2
x
13. x 僆 [⫺1, 1 ⫺ s3 ) 傼 (1 ⫹ s3, 3]
19. fn共x兲 苷 x 2
15. 40 mi兾h
n⫹1
CHAPTER 2
EXERCISES 2.1
10
0
冉
冊
1000 ⫺ P
; the time required for the population
9P
to reach a given number P.
(c) ln 81 ⬇ 4.4 years
(b) t 苷 ⫺ln
PRINCIPLES OF PROBLEM SOLVING
N
PAGE 80
a 苷 4 sh ⫺ 16兾h, where a is the length of the altitude and
h is the length of the hypotenuse
7
3. ⫺ 3 , 9
2
1.
5.
7.
y
y
1
x
x
PAGE 86
1. (a) ⫺44.4 , ⫺38.8 , ⫺27.8 , ⫺22.2 , ⫺16.6
(b) ⫺33.3
(c) ⫺33 13
3. (a)(i) 2
(ii) 1.111111 (iii) 1.010101 (iv) 1.001001
(v) 0.666667 (vi) 0.909091 (vii) 0.990099
(viii) 0.999001
(b) 1
(c) y 苷 x ⫺ 3
5. (a) (i) ⫺32 ft兾s (ii) ⫺25.6 ft兾s (iii) ⫺24.8 ft兾s
(iv) ⫺24.16 ft兾s
(b) ⫺24 ft兾s
7. (a) (i) 4.65 m兾s
(ii) 5.6 m兾s
(iii) 7.55 m兾s
(iv) 7 m兾s
(b) 6.3 m兾s
9. (a) 0, 1.7321, ⫺1.0847, ⫺2.7433, 4.3301, ⫺2.8173, 0,
⫺2.1651, ⫺2.6061, ⫺5, 3.4202; no
(c) ⫺31.4
EXERCISES 2.2
0
N
N
PAGE 96
1. Yes
3. (a) lim x l⫺3 f 共x兲 苷 ⬁ means that the values of f 共x兲 can be
made arbitrarily large (as large as we please) by taking x sufficiently close to ⫺3 (but not equal to ⫺3).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 71
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
(b) lim x l 4⫹ f 共x兲 苷 ⫺⬁ means that the values of f 共x兲 can be made
arbitrarily large negative by taking x sufficiently close to 4 through
values larger than 4.
5. (a) 2
(b) 1
(c) 4
(d) Does not exist
(e) 3
7. (a) ⫺1
(b) ⫺2
(c) Does not exist
(d) 2
(e) 0
(f) Does not exist
(g) 1
(h) 3
9. (a) ⫺⬁
(b) ⬁
(c) ⬁
(d) ⫺⬁
(e) ⬁
(f) x 苷 ⫺7, x 苷 ⫺3, x 苷 0, x 苷 6
11. lim f 共x兲 exists for all a except a 苷 ⫺1.
xla
13. (a) 1
15.
(b) 0
(c) Does not exist
17.
y
y
2
(ii) ⫺5
49. (a) (i) 5
(c)
0
0
1
x
x
_1
1
2
9. 3
21. 5
23. 4
25.
29. ⫺⬁
31. ⬁
33. ⫺⬁
39. ⫺⬁; ⬁
41. (a) 2.71828 (b)
3
5
27. (a) ⫺1.5
35. ⫺⬁
37. ⬁
6
(2, 5)
0
_3
_3
4
_2
43. (a) 0.998000, 0.638259, 0.358484, 0.158680, 0.038851,
0.008928, 0.001465; 0
(b) 0.000572, ⫺0.000614, ⫺0.000907, ⫺0.000978, ⫺0.000993,
⫺0.001000; ⫺0.001
45. No matter how many times we zoom in toward the origin, the
graph appears to consist of almost-vertical lines. This indicates
more and more frequent oscillations as x l 0 .
47. x ⬇ ⫾0.90 , ⫾2.24 ; x 苷 ⫾sin⫺1共␲兾4兲 , ⫾共␲ ⫺ sin⫺1共␲兾4兲兲
EXERCISES 2.3
N
x
(2, _5)
51. (a) (i) ⫺2 (ii) Does not exist (iii) ⫺3
(b) (i) n ⫺ 1 (ii) n (c) a is not an integer.
57. 8
63. 15; ⫺1
N
PAGE 116
1. 0.1 (or any smaller positive number)
3. 1.44 (or any smaller positive number)
5. 0.0906 (or any smaller positive number)
7. 0.011 (or any smaller positive number)
9. (a) 0.031
(b) 0.010
11. (a) s1000兾␲ cm
(b) Within approximately 0.0445 cm
(c) Radius; area; s1000兾␲ ; 1000; 5; ⬇0.0445
13. (a) 0.025
(b) 0.0025
35. (a) 0.093
(b) ␦ 苷 共B 2兾3 ⫺ 12兲兾共6B 1兾3 兲 ⫺ 1 , where
B 苷 216 ⫹ 108␧ ⫹ 12 s336 ⫹ 324␧ ⫹ 81␧ 2
41. Within 0.1
EXERCISES 2.5
_4
(b) Does not exist
y
EXERCISES 2.4
1
A71
N
PAGE 127
1. lim x l 4 f 共x兲 苷 f 共4兲
3. (a) f 共⫺4兲 is not defined and lim f 共x兲 [for a 苷 ⫺2, 2, and 4]
xla
does not exist
(b) ⫺4 , neither; ⫺2 , left; 2, right; 4, right
5.
y
7. y
0
0
2
x
3
5
x
PAGE 106
1. (a) ⫺6
(b) ⫺8
(c) 2
(d) ⫺6
(e) Does not exist
(f) 0
7
3
3. 105
5. 8
7. 390
9. 2
11. 4
6
1
13. Does not exist
15. 5
17. ⫺10
19. 12
1
1
1
1
27. 128
29. ⫺2
21. 6
23. ⫺16
25. 1
2
31. 3x 2
33. (a), (b) 3
37. 7
41. 6
43. ⫺4
45. Does not exist
y
47. (a)
(b) (i) 1
(ii) ⫺1
1
(iii) Does not exist
(iv) 1
0
x
9. (a)
T
7
5
0
11. 4
7 10
16 19
24
t
17. f 共⫺2兲 is undefined.
y
y=
0
x=_2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1
x+2
x
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 72
A72
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
19. lim f 共x兲 does not exist.
21. lim f 共x兲 苷 f 共0兲
39. (a), (b) ⫺2
43. y 苷 2; x 苷 ⫺2, x 苷 1
y
y
2⫺x
49. f 共x兲 苷 2
x 共x ⫺ 3兲
53. ⫺⬁, ⫺⬁
y=≈
1
y=´
0
_π
x
1
45. x 苷 5
51. (a)
5
4
23. Define f 共2兲 苷 3.
25. 共⫺⬁, ⬁兲
3
3
2 ) 傼 (s
2 , ⬁)
27. (⫺⬁, s
29. 关⫺1, 0兴
31. 共⫺⬁, ⫺1兴 傼 共0, ⬁兲
33. x 苷 0
47. y 苷 3
(b) 5
55. ⫺⬁, ⬁
y
x
0
41. y 苷 2; x 苷 2
1
xl0
xl0
y
0
x
1
3
3
0
57. (a) 0
x
1
(b) An infinite number of times
1
4
_4
_1
7
35. 3
37. 1
41. 0, left
-25
25
43. 0, right; 1, left
y
y
_0.5
(1, e)
(0, 2)
(0, 2)
59. (a) 0
(1, 1)
(0, 1)
(0, 1)
0
x
(2, 0)
63. (a) v *
0
(b) ⫾⬁
(b)
61. 5
⬇ 0.47 s
1.2
x
45. 3
47. (a) t 共x兲 苷 x 3 ⫹ x 2 ⫹ x ⫹ 1
(b) t 共x兲 苷 x 2 ⫹ x
55. (b) 共0.86, 0.87兲
57. (b) 70.347
63. None
65. Yes
2
EXERCISES 2.6
N
PAGE 140
65. N 艌 15
67. N 艋 ⫺6, N 艋 ⫺22
69. (a) x > 100
1. (a) As x becomes large, f 共x兲 approaches 5.
(b) As x becomes large negative, f 共x兲 approaches 3.
3. (a) ⫺2
(b) 2
(c) ⬁
(d) ⫺⬁
(e) x 苷 1, x 苷 3, y 苷 ⫺2, y 苷 2
5. y=5
7.
y
0
y
x
EXERCISES 2.7
N
PAGE 150
f 共x兲 ⫺ f 共3兲
f 共x兲 ⫺ f 共3兲
1. (a)
(b) lim
x
l
3
x⫺3
x⫺3
3. (a) 2
(b) y 苷 2x ⫹ 1
(c)
6
x=2
0
1
0
x
y=_5
9.
_1
y
5. y 苷 ⫺8x ⫹ 12
9. (a) 8a ⫺ 6a
y=3
0
x
(c)
2
7. y 苷 2 x ⫹
1
0
5
1
2
(b) y 苷 2x ⫹ 3, y 苷 ⫺8x ⫹ 19
10
x=4
3
2
1
6
3
2
1
2
11. 0
13.
23. 3
31. ⫺⬁
25.
27. 共a ⫺ b兲
1
33. ␲兾2
35. ⫺2
15.
17. 0
19. ⫺1
29. ⬁
37. 0
21. 4
_2
4
_3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 73
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
11. (a) Right: 0 ⬍ t ⬍ 1 and 4 ⬍ t ⬍ 6 ; left: 2 ⬍ t ⬍ 3 ;
standing still: 1 ⬍ t ⬍ 2 and 3 ⬍ t ⬍ 4
(b)
49. The rate at which the temperature is changing at 8:00 AM;
3.75⬚F兾h
51. (a) The rate at which the oxygen solubility changes with
respect to the water temperature; 共mg兾L兲兾⬚C
(b) S⬘共16兲 ⬇ ⫺0.25; as the temperature increases past 16⬚C ,
the oxygen solubility is decreasing at a rate of 0.25 共mg兾L兲兾⬚C .
53. Does not exist
v (m/s)
1
0
t
(seconds)
1
EXERCISES 2.8
13.
15.
17.
19.
21.
N
PAGE 162
1. (a) ⫺0.2
(b) 0
(c) 1
(e) 1 (f) 0
(g) ⫺0.2
⫺24 ft兾s
⫺2兾a 3 m兾s ; ⫺2 m兾s ; ⫺ 14 m兾s ; ⫺ 272 m兾s
t⬘共0兲, 0, t⬘共4兲, t⬘共2兲, t⬘共⫺2兲
f 共2兲 苷 3; f ⬘共2兲 苷 4
(d) 2
y
2
fª
1
3
_3
y
_2
1
0
5.
0
_1
3. (a) II
x
1
1
(b) IV
2
x
(c) I
(d) III
7.
y
fª
23. y 苷 3x ⫺ 1
3
3
25. (a) ⫺ 5 ; y 苷 ⫺ 5 x ⫹
16
5
y
fª
x
0
(b)
A73
4
0
_1
x
6
_2
27.
33.
35.
37.
39.
41.
5
29.
共a ⫹ 3兲2
1
6a ⫺ 4
31. ⫺
s1 ⫺ 2a
f 共x兲 苷 x 10, a 苷 1 or f 共x兲 苷 共1 ⫹ x兲10 , a 苷 0
f 共x兲 苷 2 x, a 苷 5
f 共x兲 苷 cos x, a 苷 ␲ or f 共x兲 苷 cos共␲ ⫹ x兲 , a 苷 0
1 m兾s ; 1 m兾s
Temperature
Greater (in magnitude)
72
(in °F)
38
9.
1
y
fª
fª
x
0
0
y=C ª(t)
Time
2
(in hours)
43. (a) (i) 23 million兾year
0
(ii) 20.5 million兾year
(iii) 16 million兾year
(b) 18.25 million兾year (c) 17 million兾year
45. (a) (i) $20.25兾unit (ii) $20.05兾unit
(b) $20兾unit
47. (a) The rate at which the cost is changing per ounce of gold
produced; dollars per ounce
(b) When the 800th ounce of gold is produced, the cost of
production is $17兾oz.
(c) Decrease in the short term; increase in the long term
x
13. (a) The instantaneous rate of change of percentage of full
capacity with respect to elapsed time in hours
(b) y
The rate of change of percentage of
full
capacity is decreasing and
40
approaching 0.
20
0
11.
y
15.
2
4
6
8
10
12 x
1963 to 1971
y
0.1
y=Mª(t)
0.05
_0.03
t
1950 1960 1970 1980 1990
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 74
A74
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
17.
f ⬘共x兲 苷 e x
y
53. f ⬘共x兲 苷
再
⫺1
1
f, fª
or f ⬘共x兲 苷
ⱍ
1
0
if x ⬍ 6
if x ⬎ 6
x⫺6
x⫺6
y
fª
1
0
ⱍ
x
6
_1
x
1
(b) ⫺1, ⫺2, ⫺4
19. (a) 0, 1, 2, 4
21. f ⬘共x兲 苷 2 , ⺢, ⺢
1
(c) f ⬘共x兲 苷 2x
55. (a)
(b) All x
(c) f ⬘共x兲 苷 2 x
y
ⱍ ⱍ
23. f ⬘共t兲 苷 5 ⫺ 18t, ⺢, ⺢
25. f ⬘共x兲 苷 2x ⫺ 6x 2 , ⺢, ⺢
0
⫺1
27. t⬘共x兲 苷
, 共⫺⬁, 9兴, 共⫺⬁, 9兲
2s9 ⫺ x
⫺7
, 共⫺⬁, ⫺3兲 傼 共⫺3, ⬁兲, 共⫺⬁, ⫺3兲 傼 共⫺3, ⬁兲
29. G⬘共t兲 苷
共3 ⫹ t兲2
31. f ⬘共x兲 苷 4x 3, ⺢, ⺢
33. (a) f ⬘共x兲 苷 4x 3 ⫹ 2
35. (a) The rate at which the unemployment rate is changing, in
percent unemployed per year
(b)
t
U⬘共t兲
t
U⬘共t兲
1999
2000
2001
2002
2003
⫺0.2
0.25
0.9
0.65
⫺0.15
2004
2005
2006
2007
2008
⫺0.45
⫺0.45
⫺0.25
0.6
1.2
59. 63⬚
CHAPTER 2 REVIEW
N
PAGE 166
True-False Quiz
1. False
11. True
21. False
3. True
13. False
23. True
5. False
15. True
7. True
9. True
17. True
19. False
Exercises
1. (a) (i) 3 (ii) 0 (iii) Does not exist
(v) ⬁ (vi) ⫺⬁ (vii) 4 (viii) ⫺1
(b) y 苷 4 , y 苷 ⫺1
(c) x 苷 0 , x 苷 2
37. ⫺4 共corner兲; 0 共discontinuity兲
39. ⫺1 共vertical tangent兲; 4 共corner兲
41.
2
3. 1
5.
15. ⫺⬁
Differentiable at ⫺1;
not differentiable at 0
_2
x
3
2
7. 3
17. 2
9. ⬁
11.
19. ␲兾2
4
7
(iv) 2
(d) ⫺3, 0, 2, 4
13.
1
2
2 1. x 苷 0, y 苷 0
29. (a) (i) 3 (ii) 0 (iii) Does not exist
(b) At 0 and 3
(c)
y
(iv) 0
(v) 0
23. 1
(vi) 0
3
1
_1
0
43. a 苷 f, b 苷 f ⬘, c 苷 f ⬙
45. a 苷 acceleration, b 苷 velocity, c 苷 position
47. 6x ⫹ 2; 6
7
f·
f
x
31. ⺢
(b) y 苷 ⫺8x ⫹ 17
35. (a) ⫺8
37. (a) (i) 3 m兾s (ii) 2.75 m兾s (iii) 2.625 m兾s
(iv) 2.525 m兾s
(b) 2.5 m兾s
39. (a) 10
(b) y 苷 10 x ⫺ 16
(c)
12
fª
_4
3
4
_1
–4
49.
3
f·
f
⫺4
6
fª
fªªª
⫺7
1
3
51. (a) a
⫺2兾3
f ⬘共x兲 苷 4x ⫺ 3x 2 ,
f ⬙共x兲 苷 4 ⫺ 6x ,
f ⵮共x兲 苷 ⫺6 ,
f 共4兲共x兲 苷 0
4
–12
41. (a) The rate at which the cost changes with respect to the
interest rate; dollars兾(percent per year)
(b) As the interest rate increases past 10%, the cost is increasing
at a rate of $1200兾(percent per year).
(c) Always positive
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA065-075.qk_97909_Ans_Ans_pA065-A075 10/7/10 2:30 PM Page 75
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
43.
47. (a) v共t兲 苷 3t 2 ⫺ 3, a共t兲 苷 6t
y
x
]
(b) (⫺⬁, 35 , (⫺⬁, 35 )
45. (a) f ⬘共x兲 苷 ⫺ 2 共3 ⫺ 5x兲⫺1兾2
5
(c)
(b) 12 m兾s2
(c) a共1兲 苷 6 m兾s2
49. (a) V 苷 5.3兾P
(b) ⫺0.00212; instantaneous rate of change of the volume with
respect to the pressure at 25⬚C; m3兾 kPa
51. 共⫺2, 21兲, 共1, ⫺6兲
1
1
55. y 苷 12x ⫺ 15, y 苷 12x ⫹ 17
57. y 苷 3 x ⫺ 3
3 3
9
2
59. 共⫾2, 4兲
63. P共x兲 苷 x ⫺ x ⫹ 3
65. y 苷 16 x ⫺ 4 x ⫹ 3
67. No
y
y
fª
0
6
2
f
(1, 2)
fª
0
_6
5 (vertical tangent)
49. The rate at which the total value of US currency in circulation
is changing in billions of dollars per year; $22.2 billion兾year
51. 0
f ⬘共x兲 苷
再
(b)
y
y
ƒ
(b) 1
(c) Yes; no
⫺3
0
PAGE 181
50
1
N
11. F⬘共 y兲 苷 5 ⫹
25.
27.
33.
7. t⬘共x兲 苷
14
9
⫹ 4
y2
y
x 2共3 ⫺ x 2兲
共1 ⫺ x 2兲 2
35. (a) y 苷 2 x ⫹ 1
1
⫺3
⫺10
3. f ⬘共x兲 苷 e x共x 3 ⫹ 3x 2 ⫹ 2x ⫹ 2兲
10
共3 ⫺ 4x兲2
15. y⬘ 苷
(b)
1.5
5
(_1, 0.5)
⫺40
43. f ⬘共x兲 苷 100x 9 ⫹ 25x 4 ⫺ 1; f ⬙共x兲 苷 900x 8 ⫹ 100x 3
15
4
x⫺1兾4, f ⬙共x兲 苷
15
16
x⫺5兾4
9. H⬘共u兲 苷 2u ⫺ 1
2t共⫺t 4 ⫺ 4t 2 ⫹ 7兲
共t 4 ⫺ 3t 2 ⫹ 1兲 2
3
p
y⬘ 苷 e (1 ⫹ 2 sp ⫹ p ⫹ psp )
19. y⬘ 苷 2v ⫺ 1兾sv
1兾2
4⫹t
⫺ACe x
f ⬘共t兲 苷
23. f ⬘共x兲 苷
2
(2 ⫹ st )
共B ⫹ Ce x 兲 2
2cx
f ⬘共x兲 苷 2
共x ⫹ c兲2
4
共x ⫹ 4x 3兲e x; 共x 4 ⫹ 8x 3 ⫹ 12x 2 兲e x
2
2x 2 ⫹ 2x
2
2
;
31. y 苷 3 x ⫺ 3
2
共1 ⫹ 2x兲 共1 ⫹ 2x兲3
y 苷 2x ; y 苷 ⫺ 12 x
13. y⬘ 苷
21.
75. m 苷 4, b 苷 ⫺4
PAGE 189
1⫺x
5. y⬘ 苷
ex
29.
100
73. a 苷 ⫺ 2 , b 苷 2
1. 1 ⫺ 2x ⫹ 6x 2 ⫺ 8x 3
17.
x
79. 3; 1
EXERCISES 3.2
(b) 0.99, 1.03; 2.7 ⬍ e ⬍ 2.8
2
3. f ⬘共x兲 苷 0
5. f ⬘共t兲 苷 ⫺3
7. f ⬘共x兲 苷 3x 2 ⫺ 4
2
9. t⬘共x兲 苷 2x ⫺ 6x
11. t⬘共t兲 苷 ⫺ 32 t ⫺7兾4
13. A⬘共s兲 苷 60兾s 6
1
15. R⬘共a兲 苷 18a ⫹ 6
17. S⬘共 p兲 苷 2 p⫺1兾2 ⫺ 1
19. y⬘ 苷 3e x ⫺ 43 x ⫺4兾3
21. h⬘共u兲 苷 3Au 2 ⫹ 2Bu ⫹ C
2
3
3
23. y⬘ 苷 2 sx ⫹
25. j⬘共x兲 苷 2.4 x 1.4
⫺
2xsx
sx
27. H⬘共x兲 苷 3x 2 ⫹ 3 ⫺ 3x ⫺2 ⫺ 3x ⫺4
1
29. u⬘ 苷 5 t ⫺4兾5 ⫹ 10t 3兾2
1
3
31. z⬘ 苷 ⫺10A兾y 11 ⫹ Be y
33. y 苷 4 x ⫹ 4
1
35. Tangent: y 苷 2x ⫹ 2; normal: y 苷 ⫺2 x ⫹ 2
37. y 苷 3x ⫺ 1
39. f ⬘共x兲 苷 4 x 3 ⫺ 6x 2 ⫹ 2x
41. (a)
(c) 4x 3 ⫺ 9x 2 ⫺ 12x ⫹ 7
3
x
3
71. y 苷 2x 2 ⫺ x
1. (a) See Definition of the Number e (page 180).
45. f ⬘共x兲 苷 2 ⫺
ƒ⬘
⫺3
CHAPTER 3
5
x
ⱍ ⱍ
ⱍ ⱍ
9
77. 1000
⫺3
1
0
2
N
0
if x ⬎ 3
if x ⬍ 3
2x
⫺2x
PAGE 170
1. 3
3. ⫺4
5. (a) Does not exist
1
1
3
7. a 苷 2 ⫾ 2 s5
9. 4
11. (b) Yes
13. (a) 0
(b) 1
(c) f ⬘共x兲 苷 x 2 ⫹ 1
EXERCISES 3.1
y=fª(x)
x
69. (a) Not differentiable at 3 or ⫺3
47. ⫺4 (discontinuity), ⫺1 (corner), 2 (discontinuity),
N
1
y=ƒ
1
_3
PROBLEMS PLUS
A75
⫺4
4
⫺0.5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A76
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
37. (a) e x 共x 3 3x 2 x 1兲
(b)
37. 5 ft兾rad
2
45.
fª
10
2
f
_2
(a) f 共x兲 苷
39.
4共1 3x 2 兲
4x
; f 共x兲 苷
2
共x 1兲
共x 2 1兲3
f·
fª
_6
41.
47. s2
(b) (c) 20
45. 7
2
xt共x兲 t共x兲
t共x兲 xt共x兲
(b) y 苷
(c) y 苷
2
关t共x兲兴
x2
1
53. Two, (2 s3, 2 (1 s3 ))
55. 1
57. $1.627 billion兾year
59. (c) 3e 3x
61. f 共x兲 苷 共x 2 2x兲e x, f 共x兲 苷 共x 2 4x 2兲e x,
f 共x兲 苷 共x 2 6x 6兲e x, f 共4兲共x兲 苷 共x 2 8x 12兲e x,
f 共5兲共x兲 苷 共x 2 10x 20兲e x; f (n)共x兲 苷 关x 2 2nx n共n 1兲兴e x
N
PAGE 205
1
s1 2x
11. f 共z兲 苷 PAGE 197
1
2 tan x x sec 2 x
sec tan
11. f 共 兲 苷
共2 tan x兲 2
共1 sec 兲2
共t 2 t兲 cos t sin t
y 苷
共1 t兲2
f 共x兲 苷 e x csc x 共x cot x x 1兲
y 苷 2s3x 23 s3 2
23. y 苷 x 1
(a) y 苷 2 x (b) 3π
9. y 苷
25.
15. y 苷 ekx 共k x 1兲
2
12 x 共x 2 1兲 2
共x 2 1兲 4
25. y 苷 51兾x共ln 5兲兾x 2
23. y 苷
3e 3x
s1 2e 3x
27. y 苷 共r 2 1兲3兾2
29. F共t兲 苷 e t sin 2t 共2t cos 2t sin 2t兲
33. y 苷 2sin x共
31. y 苷 2 cos共tan 2x兲 sec 2 共2x兲
35. y 苷
ln 2兲 cos x
4e
1e
sin
共1 e 2x 兲 2
1 e 2x
2x
2x
37. y 苷 2 cos
cot共sin 兲 csc 2 共sin 兲
39. f 共t兲 苷 sec 共e 兲e t e tan t sec 2t
2
t
41. f 共t兲 苷 4 sin 共e sin t 兲 cos 共e sin t 兲 e sin t sin t cos t
1. f 共x兲 苷 6x 2 sin x
3. f 共x兲 苷 cos x 2 csc2x
2
2
5. y 苷 sec 共sec tan 兲
7. y 苷 c sin t t共t cos t 2 sin t兲
21.
2z
共z 2 1兲2
19. h共t兲 苷 3 共t 1兲1兾3共2t 2 1兲 2 共20t 2 18t 1兲
2
15.
1
1
sin x
(b) sec x tan x 苷
2
cos x
cos 2x
cot x 1
(c) cos x sin x 苷
csc x
55. 1
21. y 苷
49. (a) 0
(b) 3
51. (a) y 苷 xt共x兲 t共x兲
13.
3
17. f 共x兲 苷 共2x 3兲 3 共x 2 x 1兲 4 共28x 2 12x 7兲
20
9
47. y 苷 2x 18
EXERCISES 3.3
51. A 苷 10, B 苷 10
49. cos x
13. y 苷 3x 2 sin共a 3 x 3 兲
43. (a) 16
N
3
53. (a) sec 2x 苷
9. F共x兲 苷 6
_2
1
4
43. 4
41. 3
4
e sx
2
1. 3
3.
sec
x
5.
3s共1 4x兲2
2sx
7. F共x兲 苷 10 x共x 4 3x 2 2兲 4 共2x 2 3兲
4
f
39. 3
EXERCISES 3.4
2
(b)
1
2
2
2
43. t共x兲 苷 2r 2 p共ln a兲 共2ra rx n兲 p1 a rx
x兲 sec 2 共 x兲 sinssin 共tan x兲
2ssin 共tan x兲
47. y 苷 2x sin共x 2 兲; y 苷 4x 2 cos共x 2 兲 2 sin共x 2 兲
45. y 苷
cos共tan
49. e x共 cos x e x 关共 2 2兲 sin x
51. y 苷 20x 1
1
55. (a) y 苷 2 x 1
sin x兲;
2 cos x兴
53. y 苷 x (b)
3
(0, 1)
2
_3
π
3
” 2 , π’
_1.5
0
π
27. (a) sec x tan x 1
29.
31.
33.
35.
(b)
cos sin ; 2 cos sin
(a) f 共x兲 苷 共1 tan x兲兾sec x
(b) f 共x兲 苷 cos x sin x
共2n 1兲 13 , n an integer
(a) v共t兲 苷 8 cos t, a共t兲 苷 8 sin t
4 s3, 4, 4s3; to the left
59.
61.
65.
2 2x 2
(a) f 共x兲 苷
s2 x 2
共共 兾2兲 2n , 3兲, 共共3 兾2兲 2n , 1兲, n an integer
24
63. (a) 30
(b) 36
(a) 34
(b) Does not exist
(c) 2
67.
69.
71.
77.
16 s2
(a) F共x兲 苷 e x f 共e x 兲 (b) G共x兲 苷 e f 共x兲 f 共x兲
120
73. 96
5
250 cos 2x
79. v共t兲 苷 2 cos共10 t兲 cm兾s
57.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 77
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
dB
7
2 t
(b) 0.16
苷
cos
dt
54
5.4
83. v 共t兲 苷 2e1.5t共2 cos 2 t 1.5 sin 2 t兲
81. (a)
2
53. G共x兲 苷 1 x arccos x
s1 x 2
57. y 苷 sin1 x
59. y 苷
15
61. 1 √
s
0
2
1
55. h共t兲 苷 0
sa 2 b 2
a b cos x
x arcsin x
s1 x 2
65.
0
67.
y
y
2
7
x
85. dv兾dt is the rate of change of velocity with respect to time;
dv兾ds is the rate of change of velocity with respect to displacement
87. (a) y 苷 ab t where a ⬇ 100.01244 and b ⬇ 0.000045146
(b) 670.63 A
89. (b) The factored form
93. (b) n cos n1x sin关共n 1兲x兴
EXERCISES 3.5
PAGE 215
N
1. (a) y 苷 9x兾y
(b) y 苷 s9x 2 1, y 苷 9x兾s9x 2 1
3. (a) y 苷 y 2兾x 2
(b) y 苷 x兾共x 1兲, y 苷 1兾共x 1兲2
x2
2x y
5. y 苷 2
7. y 苷
y
2y x
3y 2 5x 4 4x 3 y
2x y sin x
9. y 苷 4
11. y 苷
2
x 3y 6xy
cos x 2y
y共 y e x兾y 兲
13. y 苷 tan x tan y
15. y 苷 2
y xe x兾y
4 2
2
4 4
1 x y y x y 2xy
17. y 苷
x 2 2xy 2x 5y 3
y
e sin x y cos共x y兲
16
19. y 苷 y
21. 13
e cos x x cos共x y兲
2x 4y x 3 6xy 2
23. x 苷
25. y 苷 12 x
4x 3y 2 3x 2 y 2y 3
1
9
40
27. y 苷 x 2
29. y 苷 x 2
31. y 苷 13 x 13
9
5
(b)
33. (a) y 苷 2 x 2
5
x
V 3共nb V 兲
(b) 4.04 L兾atm
PV n2aV 2n3ab
73. (s3, 0)
75. 共1, 1兲, 共1, 1兲
77. (b)
79. (a) 0
(b) 12
71. (a)
3
EXERCISES 3.6
N
3
2
PAGE 223
1. The differentiation formula is simplest.
cos共ln x兲
1
5. f 共x兲 苷 x
x
3x 2
sin x
7. f 共x兲 苷 3
9. f 共x兲 苷
cos x ln共5x兲
共x 1兲 ln 10
x
2x 2 1
y
10
11. t共x兲 苷
13. G共 y兲 苷
2
x 共x 2 1兲
2y 1
y 1
1
a
15. F共s兲 苷
17. y 苷 sec 2共ln共ax b兲兲
s ln s
ax b
x
1
19. y 苷
21. y 苷
log10 x
ln 10
1x
3. f 共x兲 苷
23. y 苷 x 2x ln共2x兲; y 苷 3 2 ln共2x兲
1
x
25. y 苷
; y 苷
共1 x 2 兲 3兾2
s1 x 2
2x 1 共x 1兲 ln共x 1兲
;
共x 1兲关1 ln共x 1兲兴 2
共1, 1 e兲 傼 共1 e, 兲
2共x 1兲
; 共, 0兲 傼 共2, 兲
29. f 共x兲 苷
x 共 x 2兲
27. f 共x兲 苷
(1, 2)
_2
2
_2
35. 81兾y
41. (a)
3
37. 2x兾y
5
39. 1兾e
2
Eight; x ⬇ 0.42, 1.58
4
33. y 苷 3x 9
5
冑
x1
x4 1
1
1
(b) y 苷 x 1, y 苷 3 x 2
(c) 1 3 s3
5
5
43. ( 4 s3, 4 )
45. 共x0 x兾a 2 兲 共 y0y兾b 2 兲 苷 1
1
2 tan x
1
49. y 苷
51. y 苷
1 x2
sx 2 x
冉
冉
冉
16x 3
4x
4
x 2
x 4
2
51. y 苷
冉
ln tan x
sec2 x
x tan x
x2
2x
x y 2 2y
2
冊
冊
冊
sin x
cos x ln x
x
47. y 苷 共cos x兲 x共x tan x ln cos x兲
49. y 苷 共tan x兲1兾x
31. 1
37. 7
1
2x 3
4
2x 2
x 1
43. y 苷 x x共1 ln x兲
45. y 苷 x sin x
_3
35. cos x 1兾x
39. y 苷 共x 2 2兲 2 共x 4 4兲 4
41. y 苷
_2
A77
冊
53. f 共n兲共x兲 苷
共1兲n1共n 1兲!
共x 1兲n
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 78
A78
APPENDIX I
EXERCISES 3.7
ANSWERS TO ODD-NUMBERED EXERCISES
PAGE 233
N
1. (a) 3t 24t 36
(b) 9 ft兾s (c) t 苷 2, 6
(d) 0 t 2, t 6 (e) 96 ft
(g) 6t 24; 6 ft兾s2
(f)
t 8,
2
s 32
t 6,
s0
t 2,
s 32
t 0,
s0
0
(h)
s
20
(i) Speeding up when
2 t 4 or t 6;
slowing down when
0 t 2 or 4 t 6
40
s
√
0
8
a
25
冉 冊
t
1
sin
(b) 8 s2 ft兾s
4
4
(d) 4 t 8 (e) 4 ft
(f)
t =10,
3. (a) s=0
t 8, s 1
t 4,
s _1
t 0, s 1
s
0
(g) 161
(h) 1
(c) t 苷 0, 4, 8
2
cos共 t兾4兲; 321
s
2
s2 ft兾s 2
a
8
0
√
_1
(i) Speeding up when 0 t 2, 4 t 6, 8 t 10;
slowing down when 2 t 4, 6 t 8
5. (a) Speeding up when 0 t 1 or 2 t 3;
slowing down when 1 t 2
(b) Speeding up when 1 t 2 or 3 t 4;
slowing down when 0 t 1 or 2 t 3
5
7. (a) 4.9 m兾s; 14.7 m兾s
(b) After 2.5 s
(c) 32 16 m
(d) ⬇5.08 s (e) ⬇25.3 m兾s
9. (a) 7.56 m兾s
(b) 6.24 m兾s; 6.24 m兾s
11. (a) 30 mm2兾mm; the rate at which the area is increasing
with respect to side length as x reaches 15 mm
(b) A ⬇ 2x x
13. (a) (i) 5
(ii) 4.5
(iii) 4.1
(b) 4
(c) A ⬇ 2 r r
15. (a) 8 ft 2兾ft
(b) 16 ft 2兾ft (c) 24 ft 2兾ft
The rate increases as the radius increases.
17. (a) 6 kg兾m
(b) 12 kg兾m
(c) 18 kg兾m
At the right end; at the left end
2
19. (a) 4.75 A
(b) 5 A; t 苷 3 s
23. (a) dV兾dP 苷 C兾P 2
(b) At the beginning
25. 400共3t 兲 ln 3; ⬇ 6850 bacteria兾h
27. (a) 16 million兾year; 78.5 million兾year
(b) P共t兲 苷 at 3 bt 2 ct d, where a ⬇ 0.00129371,
b ⬇ 7.061422, c ⬇ 12,822.979, d ⬇ 7,743,770
(c) P共t兲 苷 3at 2 2bt c
(d) 14.48 million兾year; 75.29 million兾year (smaller)
(e) 81.62 million兾year
29. (a) 0.926 cm兾s; 0.694 cm兾s; 0
(b) 0; 92.6 共cm兾s兲兾cm; 185.2 共cm兾s兲兾cm
(c) At the center; at the edge
31. (a) C共 x 兲 苷 12 0.2 x 0.0015x 2
(b) $32兾yard ; the cost of producing the 201st yard
(c) $32.20
33. (a) 关xp共x兲 p共x兲兴兾x 2; the average productivity increases as
new workers are added.
35. 0.2436 K兾min
37. (a) 0 and 0
(b) C 苷 0
(c) 共0, 0兲, 共500, 50兲; it is possible for the species to coexist.
EXERCISES 3.8
N
PAGE 242
1. About 235
3. (a) 100共4.2兲 t
(b) ⬇7409
(c) ⬇10,632 bacteria兾h
(d) 共ln 100兲兾共ln 4.2兲 ⬇ 3.2 h
5. (a) 1508 million, 1871 million
(b) 2161 million
(c) 3972 million; wars in the first half of century, increased life
expectancy in second half
7. (a) Ce0.0005t
(b) 2000 ln 0.9 ⬇ 211 s
9. (a) 100 2t兾30 mg
(b) ⬇ 9.92 mg (c) ⬇199.3 years
11. ⬇2500 years
13. (a) ⬇137F
(b) ⬇116 min
15. (a) 13.3C
(b) ⬇67.74 min
17. (a) ⬇64.5 kPa
(b) ⬇39.9 kPa
19. (a) (i) $3828.84 (ii) $3840.25 (iii) $3850.08
(iv) $3851.61 (v) $3852.01 (vi) $3852.08
(b) dA兾dt 苷 0.05A, A共0兲 苷 3000
EXERCISES 3.9
N
PAGE 248
1. dV兾dt 苷 3x 2 dx兾dt
3. 48 cm 2兾s
5. 3兾共25 兲 m兾min
7. (a) 1
(b) 25
9. 18
11. (a) The plane’s altitude is 1 mi and its speed is 500 mi兾h.
(b) The rate at which the distance from the plane to the station is
increasing when the plane is 2 mi from the station
(c)
(d) y 2 苷 x 2 1
x
(e) 250 s3 mi兾h
1
y
13. (a) The height of the pole (15 ft), the height of the man (6 ft),
and the speed of the man (5 ft兾s)
(b) The rate at which the tip of the man’s shadow is moving when
he is 40 ft from the pole
15
xy
(c)
(d)
(e) 253 ft兾s
苷
6
y
15
6
x
y
15. 65 mi兾h
17. 837兾s8674 ⬇ 8.99 ft兾s
720
19. 1.6 cm兾min
21. 13 ⬇ 55.4 km兾h
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 79
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
23. 共10,000 800,000 兾9兲 ⬇ 2.89 10 5 cm3兾min
10
25. 3 cm兾min
27. 6兾共5 兲 ⬇ 0.38 ft兾min
29. 0.3 m 2兾s
107
33. 80 cm 3兾min
35. 810 ⬇ 0.132 兾s
31. 5 m
(b) 0.096 rad兾s
37. 0.396 m兾min
39. (a) 360 ft兾s
10
41. 9 km兾min
43. 1650兾s31 ⬇ 296 km兾h
45. 4 s15 ⬇ 6.78 m兾s
7
EXERCISES 3.10
N
PAGE 255
1. L共x兲 苷 10x 6
3. L共x兲 苷 4 x 1
1
5. s1 x ⬇ 1 x ;
1
2
3
y=1- 21 x
s0.9 ⬇ 0.95,
s0.99 ⬇ 0.995
y=œ„„„„
1-x
3
(1, 0)
_1
11. (a)
13. (a)
15. (a)
17. (a)
19. y
9. 0.368 x 0.677
t
dy 苷 2x共x cos 2x sin 2x兲 dx (b) dy 苷
dt
1 t2
2
4 v
sec st
dy 苷
dt (b) dy 苷
dv
共1 v 2兲2
2st
dy 苷 101 e x兾10 dx (b) 0.01
x
dx (b) 0.05
dy 苷
s3 x 2
苷 0.64, dy 苷 0.8
y
dy Îy
0
1
dx=Îx
21. y 苷 0.1, dy 苷 0.125
y
dx=Îx
Îy
2
y= x
2
1. True
11. True
dy
27. 1 25. 10.003
7.
8t 3
共t 1兲2
4
13. 3
9.
1 ln x
x ln x
e1兾x共1 2x兲
x4
(b) 36 cm , 0.006, 0.6%
⬇ 27 cm 2; 841 ⬇ 0.012 苷 1.2%
1
(b) 1764兾 2 ⬇ 179 cm 3; 56 ⬇ 0.018 苷 1.8%
37. (a) 2 rh r
(b) 共r兲2h
43. (a) 4.8, 5.2
(b) Too large
9. True
1
1
2sx
sx 3
cos sx sx sin sx
2sx
2xy cos y
15.
1 x sin y x 2
11.
冉 冊
1
1 t2
t
19.
sec2
2
共1 t 2兲2
1 t2
2sarctan x 共1 x 兲
21. 3 x ln x共ln 3兲共1 ln x兲
23. 共x 1兲2
2
2x y cos共xy兲
25.
27.
x cos共xy兲 1
共1 2x兲 ln 5
4x
29. cot x sin x cos x
31.
tan1共4x兲
1 16x 2
33. 5 sec 5x
35. 6x csc 2 共3x 2 5兲
3x 2
37. cos(tan s1 x 3 )(sec 2s1 x 3 )
2 s1 x 3
2
39. 2 cos tan共sin 兲 sec 共sin 兲
49.
2
33. (a) 270 cm , 0.01, 1%
35. (a) 84兾
兾90 ⬇ 0.965
7. False
3. 2 sx 5. x共 x cos x 2 sin x兲
45.
x
3
5. False
15. True
1. 4x 7共x 1兲 3共3x 2兲
1
23. 15.968
3. True
13. True
Exercises
41.
0 1
PAGE 264
17.
y=2x-≈
1
N
True-False Quiz
_3
7. 0.383 x 0.516
23. (a) 1
(b) 1
(c) (d) (e) 0
(f) 1
(g) (h) (i) 0
31. f 共x兲 苷 x cosh x
33. h共x兲 苷 tanh x
35. y 苷 3e cosh 3x sinh 3x
37. f 共 t 兲 苷 2e t sech 2 共e t 兲 tanh共e t 兲
1
2 sinh x
39. G共x兲 苷
41. y 苷
共1 cosh x兲2
2sx共x 1兲
43. y 苷 sinh1共x兾3兲
45. y 苷 csc x
51. (a) 0.3572
(b) 70.34°
53. (a) 164.50 m
(b) 120 m; 164.13 m
55. (b) y 苷 2 sinh 3x 4 cosh 3x
57. ( ln 共1 s2 ), s2 )
CHAPTER 3 REVIEW
(0, 1)
53.
59.
63.
共x 2兲4共3x 2 55x 52兲
43. 2x 2 cosh共x 2 兲 sinh共x 2 兲
2 sx 1 共x 3兲8
cosh x
47.
3 tanh 3x
ssinh 2x 1
3 sin (e stan 3x ) e stan 3x sec 2共3x兲
4
51. 27
2stan 3x
57. y 苷 2s3x 1 s3兾3
5x 4兾y 11
61. y 苷 x 2; y 苷 x 2
y 苷 2x 1
10 3x
7
1
(b) y 苷 4 x 4 , y 苷 x 8
(a)
2 s5 x
10
(c)
(4, 4)
EXERCISES 3.11
N
PAGE 262
_10
1. (a) 0
(b) 1
3. (a) 4
(b) 2 共e 2 e2 兲 ⬇ 3.62686
5. (a) 1
(b) 0
3
4
3
4
5
21. sech x 苷 5 , sinh x 苷 3 , csch x 苷 4 , tanh x 苷 5 , coth x 苷 4
3
A79
(1, 2)
10
1
ƒ
_10
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 80
A80
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
65. ( 兾4, s2 ), (5 兾4, s2 )
69. (a) 2
71. 2xt共x兲 x 2t共x兲
73. 2t共x兲t共x兲
11. (a)
(b) 44
0
1
0
x
3
1
2
3
x
_1
y
2
1
a
v
0
0
2
_1
(c)
y
1
1
f 共x兲关t共 x兲兴 2 t共x兲 关 f 共x兲兴 2
75. t共e x 兲e x
77. t共x兲兾t共x兲
79.
关 f 共x兲 t共x兲兴 2
81. f 共t共sin 4x兲兲t共sin 4x兲共cos 4x兲共4兲
2
14
83. 共3, 0兲
85. y 苷 3 x 2 3 x
ct
87. v共t兲 苷 Ae 关c cos共t 兲 sin共t 兲兴,
a共t兲 苷 Aect 关共c 2 2 兲 cos共t 兲 2c sin共t 兲兴
89. (a) v 共t兲 苷 3t 2 12; a共t兲 苷 6t
(b) t 2; 0 t 2
(c) 23
(d) 20
(e) t 2; 0 t 2
y
(b)
y
1
2
x
3
_1
t 3
position
15
13. (a)
91. 4 kg兾m
93. (a) 200共3.24兲 t
PROBLEMS PLUS
1. (s3兾2,
1
4
)
PAGE 269
N
5. 3s2
11. (0,
5
4
)
(b) 40 (cos s8 cos 2
480 sin (1 cos 兾s8 cos 2 ) cm兾s
xT 僆 共3, 兲, y T 僆 共2, 兲, xN 僆 (0, 53 ), yN 僆 (52 , 0)
(b) (i) 53 (or 127 ) (ii) 63 (or 117 )
R approaches the midpoint of the radius AO.
25. 2se
29. 共1, 2兲, 共1, 0兲
sin a
375
33. 2 128 ⬇ 11.204 cm3兾min
s29兾58
13. (a) 4 s3兾s11 rad兾s
(c)
17.
19.
21.
23.
31.
) cm
(b)
y
(b) ⬇22,040
(c) ⬇25,910 bacteria兾h (d) 共ln 50兲兾共ln 3.24兲 ⬇ 3.33 h
4
95. (a) C 0 ekt
(b) ⬇100 h
97. 3 cm2兾min
99. 13 ft兾s
101. 400 ft兾h
3
3
1 3x ⬇ 1 x; s
1.03 ⬇ 1.01
103. (a) L共x兲 苷 1 x; s
(b) 0.235 x 0.401
1
1
3
1
105. 12 2 ⬇ 16.7 cm 2
107. 32
109. 4
111. 8 x 2
y
_1
0
15.
19.
21.
23.
27.
35.
43.
49.
53.
x
2
0
x
Abs max f 共3兲 苷 4
17. Abs max f 共1兲 苷 1
Abs min f 共0兲 苷 0
Abs max f 共 兾2兲 苷 1; abs min f 共 兾2兲 苷 1
Abs max f 共2兲 苷 ln 2
25. Abs max f 共0兲 苷 1
1
Abs max f 共3兲 苷 2
29. 3
31. 2, 3
33. 0
4
8
0, 2
37. 0, 9
39. 0, 7, 4
41. n 共n an integer兲
0, 23
45. 10
47. f 共2兲 苷 16, f 共5兲 苷 7
f 共1兲 苷 8 , f 共2兲 苷 19
51. f 共2兲 苷 33, f 共2兲 苷 31
55. f (s2 ) 苷 2 , f 共1兲 苷 s3
f 共0.2兲 苷 5.2, f 共1兲 苷 2
57. f 共 兾6兲 苷 2s3, f 共 兾2兲 苷 0
3
8
59. f 共2兲 苷 2兾se , f 共1兲 苷 1兾s
e
61. f 共1兲 苷 ln 3, f ( 2 ) 苷 ln 4
1
CHAPTER 4
EXERCISES 4.1
N
PAGE 280
63. f
Abbreviations: abs, absolute; loc, local; max, maximum;
min, minimum
9.
y
y
3
3
2
2
1
1
0
1
2
3
4
5
x
0
1
2
3
4
5
a
ab
苷
a abb
共a b兲ab
6
6
(b) 25 s35 2, 25 s35 2
3
67. (a) 0.32, 0.00
(b) 16 s3, 0
69. ⬇3.9665C
71. Cheapest, t ⬇ 0.855 (June 1994);
most expensive, t ⬇ 4.618 (March 1998)
65. (a) 2.19, 1.81
1. Abs min: smallest function value on the entire domain of the
function; loc min at c: smallest function value when x is near c
3. Abs max at s, abs min at r, loc max at c, loc min at b and r,
neither a max nor a min at a and d
5. Abs max f 共4兲 苷 5 , loc max f 共4兲 苷 5 and f 共6兲 苷 4 ,
loc min f 共2兲 苷 2 and f 共1兲 苷 f 共5兲 苷 3
7.
冉 冊
3
x
73. (a) r 苷 3 r0
2
(c)
4
(b) v 苷 27 kr 30
√
4
27 kr#̧
0
2
3
r¸
r¸ r
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 81
APPENDIX I
EXERCISES 4.2
N
PAGE 288
29.
ANSWERS TO ODD-NUMBERED EXERCISES
A81
y
9
4
1. 2
3.
5. f is not differentiable on (1, 1)
7. 0.3, 3, 6.3
9. 1
11. 3兾 ln 4
13. 1
15. f is not continous at 3
23. 16
25. No
31. No
EXERCISES 4.3
N
PAGE 297
Abbreviations: inc, increasing; dec, decreasing; CD, concave
downward; CU, concave upward; HA, horizontal asymptote;
VA, vertical asymptote; IP, inflection point(s)
1. (a) 共1, 3兲, 共4, 6兲
(b) 共0, 1兲, 共3, 4兲
(d) 共2, 4兲, 共4, 6兲 (e) 共2, 3兲
(c) 共0, 2兲
3. (a) I / D Test
(b) Concavity Test
(c) Find points at which the concavity changes.
5. (a) Inc on 共1, 5兲; dec on 共0, 1兲 and 共5, 6兲
(b) Loc max at x 苷 5, loc min at x 苷 1
7. (a) 3, 5
(b) 2, 4, 6
(c) 1, 7
11. (a) Inc on 共1, 0兲, 共1, 兲; dec on 共, 1兲, 共0, 1兲
(b) Loc max f 共0兲 苷 3; loc min f 共1兲 苷 2
(c) CU on (, s3兾3), (s3兾3, );
CD on (s3兾3, s3兾3); IP (s3兾3, 229)
兾4兲, 共5 兾4, 2 兲; dec on 共 兾4, 5 兾4兲
(b) Loc max f 共 兾4兲 苷 s2 ; loc min f 共5 兾4兲 苷 s2
(c) CU on 共3 兾4, 7 兾4兲; CD on 共0, 3 兾4兲, 共7 兾4, 2 兲;
IP 共3 兾4, 0兲, 共7 兾4, 0兲
13. (a) Inc on 共0,
15. (a) Inc on ( ln 2, ); dec on (, ln 2)
(b) Loc min f ( ln 2) 苷 22兾3 21兾3 (c) CU on 共, 兲
1
3
17. (a) Inc on 共1, 兲; dec on 共0, 1兲
(b) Loc min f 共1兲 苷 0
(c) CU on 共0, 兲; No IP
19. Loc max f 共1兲 苷 2; loc min f 共0兲 苷 1
21. Loc min f (16 ) 苷 4
1
1
23. (a) f has a local maximum at 2.
(b) f has a horizontal tangent at 6.
27.
1
2
3
4
0
2
0
2
2
4
x
_2
0
2
8
x
1
23
’
9
6
x
(2, _14)
35. (a) Inc on 共, 1兲, 共0, 1兲;
dec on 共1, 0兲, 共1, 兲
(b) Loc max f 共1兲 苷 3, f 共1兲 苷 3;
loc min f 共0兲 苷 2
(c) CU on (1兾s3, 1兾s3 );
CD on (, 1兾s3 ), 共1兾s3, );
IP (1兾s3, 239)
(d) See graph at right.
39. (a) Inc on 共, 4兲; dec on 共4, 6兲
(b) Loc max f 共4兲 苷 4s2
(c) CD on 共, 6兲; No IP
(d) See graph at right.
y
x=2
0
1
”_ œ„3 ,
y
23
’
9
(_1, 3)
x
” œ„3 ,
(1, 3)
1
0
1
x
y
7
(_2, 7)
(_1, 3)
_1
x
y
_2
_2
37. (a) Inc on 共, 2兲, 共0, 兲;
dec on 共2, 0兲
(b) Loc max h 共2兲 苷 7;
loc min h 共0兲 苷 1
(c) CU on 共1, 兲;
CD on 共, 1兲; IP 共1, 3兲
(d) See graph at right.
y
0
y
(_2, 18)
(b) Loc max f 共3兲 苷 81; loc min f 共2兲 苷 44
(c) CU on 共12, 兲; CD on (, 12); IP ( 12, 372 )
1
3
1
3
31. (a) Inc on (0, 2), (4, 6), 共8, 兲;
dec on (2, 4), (6, 8)
(b) Loc max at x 苷 2, 6;
loc min at x 苷 4, 8
(c) CU on (3, 6), 共6, 兲;
CD on (0, 3)
(d) 3
(e) See graph at right.
33. (a) Inc on 共, 2兲, 共2, 兲; dec on 共2, 2兲
(b) Loc max f 共2兲 苷 18; loc min f 共2兲 苷 14
(c) CU on 共0, 兲, CD on 共, 0兲; IP 共0, 2兲
(d)
y
9. (a) Inc on 共, 3兲, 共2, 兲; dec on 共3, 2兲
25.
x
0
y
x
(0, _1)
{ 4, 4œ„2 }
0
4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
6 x
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 82
A82
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
41. (a) Inc on 共1, 兲;
dec on 共, 1兲
(b) Loc min C共1兲 苷 3
(c) CU on 共, 0兲, 共2, 兲;
CD on 共0, 2兲;
3
IP 共0, 0兲, (2, 6 s
2)
(d) See graph at right.
43. (a) Inc on 共 , 2 兲;
dec on 共0, 兲
(b) Loc min f 共 兲 苷 1
(c) CU on 共 兾3, 5 兾3兲;
CD on 共0, 兾3兲, 共5 兾3, 2 兲;
IP ( 兾3, 54 ), (5 兾3, 54 )
(d) See graph at right.
45. (a) VA x 苷 0; HA y 苷 1
(b) Inc on 共0, 2兲;
dec on 共, 0兲, 共2, 兲
(c) Loc max f 共2兲 苷 54
(d) CU on 共3, 兲;
CD on 共, 0兲, 共0, 3兲; IP (3, 119)
(e) See graph at right.
y
{ 2, 6 Œ„
2}
0
_4
x
(_1, _3)
y
61. (a) The rate of increase is initially very small, increases to a
maximum at t ⬇ 8 h, then decreases toward 0.
(b) When t 苷 8
(c) CU on 共0, 8兲; CD on 共8, 18兲 (d) 共8, 350兲
63. K共3兲 K共2兲; CD
65. 28.57 min, when the rate of increase of drug level in the bloodstream is greatest; 85.71 min, when rate of decrease is greatest
1
67. f 共x兲 苷 9 共2x 3 3x 2 12x 7兲
69. (a) a 苷 0, b 苷 1
(b) y 苷 x at 共0, 0兲
EXERCISES 4.4
5π 5
π 5
” 3, 4’
” 3, 4’
1
0
_1
2π ¨
π
(π, _1)
y
y=1
(2, 5/4)
0
47. (a) HA y 苷 0
2
3
x
x
0
y
(b) Inc on 共, 0兲, dec on 共0, 兲
(c) Loc max f 共0兲 苷 1
(d) CU on (, 1s2 ), (1兾s2 , );
CD on (1兾s2 , 1兾s2 );
IP (1兾s2 , e 1兾2)
(e) See graph at right.
_1
0
1
(1, 0)
53. 共3, 兲
55. (a) Loc and abs max f 共1兲 苷 s2, no min
(b)
1
4
(3 s17 )
57. (b) CU on 共0.94, 2.57兲, 共3.71, 5.35兲;
CD on 共0, 0.94兲, 共2.57, 3.71兲, 共5.35, 2 兲;
IP 共0.94, 0.44兲, 共2.57, 0.63兲, 共3.71, 0.63兲, 共5.35, 0.44兲
59. CU on 共, 0.6兲, 共0.0, 兲; CD on 共0.6, 0.0兲
(2, 32)
(4, 16)
0
3
3. A. ⺢
B. y-int 0; x-int 0, s
4
C. None
D. None
E. Inc on 共1, 兲; dec on 共, 1兲
F. Loc min f 共1兲 苷 3
G. CU on 共, 兲
H. See graph at right.
x=e
0
y
(6, 0)
x
x
y
x=0
1
PAGE 317
1. A. ⺢
B. y-int 0; x-int 0, 6
C. None
D. None
E. Inc on 共, 2兲, 共6, 兲;
dec on 共2, 6兲
F. Loc max f 共2兲 苷 32;
loc min f 共6兲 苷 0
G. CU on 共4, 兲; CD on 共, 4兲;
IP 共4, 16兲
H. See graph at right.
1
49. (a) HA y 苷 0
N
Abbreviation: int, intercept; SA, slant asymptote
y
Dec on 共, 兲
None
CU on 共, 兲
See graph at right.
51. (a) VA x 苷 0, x 苷 e
(b) Dec on 共0, e兲
(c) None
(d) CU on (0, 1); CD on 共1, e兲;
IP (1, 0)
(e) See graph at right.
PAGE 307
(b) 0
(c) 0
(d) , , or does not exist
(e) Indeterminate
3. (a) (b) Indeterminate
(c) 9
1
1
5. 4
7. 2
9. 3
11. 13. 2
15. 4
1
8
17. 0
19. 21. 5
23. 3
25. 2
27. 1
1
29. 1
31. 1兾 ln 3
33. 0
35. 1兾 2
37. 2 a共a 1兲
1
1
39. 24
41.
43. 3
45. 0
47. 2兾
49. 2
1
51. 2
53. 55. 1
57. e2
59. 1兾e
1
4
61. 1
63. e
65. 1兾se
67. e 2
69. 4
73. 1
75. f has an absolute minimum for c 0. As c increases, the minimum points get farther away from the origin.
1
81. 169 a
83. 2
85. 56
89. (a) 0
EXERCISES 4.5
(b)
(c)
(d)
(e)
N
1. (a) Indeterminate
y
0
x
(1, _3)
x
5. A. ⺢
B. y-int 0; x-int 0, 4
C. None
D. None
E. Inc on 共1, 兲; dec on 共, 1兲
F. Loc min f 共1兲 苷 27
G. CU on 共, 2兲, 共4, 兲;
CD on 共2, 4兲;
IP 共2, 16兲, 共4, 0兲
H. See graph at right.
y
0
(4, 0)
x
(2, _16)
(1, _27)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 83
APPENDIX I
7. A. ⺢
B. y-int 0; x-int 0
C. About 共0, 0兲 D. None
E. Inc on 共, 兲
F. None
G. CU on 共2, 0兲, 共2, 兲;
CD on 共, 2兲, 共0, 2兲;
256
IP (2, 256
15 ), 共0, 0兲, (2, 15 )
H. See graph at right.
19. A. ⺢
B. y-int 0; x-int 0
C. About y-axis D. HA y 苷 1
E. Inc on 共0, 兲; dec on 共, 0兲
F. Loc min f 共0兲 苷 0
G. CU on 共1, 1兲;
CD on 共, 1兲, 共1, 兲; IP (1, 14 )
H. See graph at right.
y
{2, 256
15 }
(0, 0)
{_2, _ 256
15 }
ⱍ
9. A. 兵x x 苷 1其
B. y-int 0; x-int 0
C. None
D. VA x 苷 1, HA y 苷 1
E. Dec on 共, 1兲, 共1, 兲
F. None
G. CU on 共1, 兲; CD on 共, 1兲
H. See graph at right.
11. A. 共, 1兲 傼 共1, 2兲 傼 共2, 兲
B. y-int 0; x-int 0
C. None
D. HA y 苷 1; VA x 苷 2
E. Inc on 共, 1兲, 共1, 2兲, 共2, 兲
F. None
G. CU on 共, 1兲, 共1, 2兲;
CD on 共2, 兲
H. See graph at right.
x
y
y1
x
0
x1
y
(1, 1)
0
x
13. A. 兵x x 苷 3其
15. A. ⺢
B. y-int 0; x-int 0
C. About (0, 0)
D. HA y 苷 0
E. Inc on 共3, 3兲;
dec on 共, 3兲, 共3, 兲
F. Loc min f 共3兲 苷 16 ;
loc max f 共3兲 苷 16 ;
G. CU on (3 s3, 0), (3 s3, );
CD on (, 3 s3 ), (0, 3 s3 );
IP (0, 0), (3 s3, s3兾12)
H. See graph at right.
17. A. 共, 0兲 傼 共0,兲
B. x-int 1
C. None
D. HA y 苷 0; VA x 苷 0
E. Inc on 共0, 2兲;
dec on 共, 0兲, 共2, 兲
F. Loc max f 共2兲 苷 14
G. CU on 共3, 兲;
CD on 共, 0兲, 共0, 3兲; IP (3, 29 )
H. See graph at right.
1
4
1
4
”1, ’
”_1, ’
(0, 0)
y
0
(1, _2)
y
y
1
”3, 6 ’
x
1
”_3, _ 6 ’
y
1
4
2
9
”2, ’ ”3, ’
0
1
x
0
_2
x
1
y
y=1
(0, 0)
x
y=_1
ⱍⱍ ⱍ
x3
x
3
27. A. { x x 1, x 苷 0} 苷 关1, 0兲 傼 共0, 1兴
B. x-int 1
C. About (0, 0)
D. VA x 苷 0
E. Dec on 共1, 0兲, 共0, 1兲
1
F. None
G. CU on (1, s2兾3 ), (0, s2兾3 );
CD on (s2兾3, 0), (s2兾3, 1);
IP (s2兾3, 1兾s2 )
H. See graph at right.
x
x
y
B. x-int 2, 1
C. None
D. None
E. Inc on (1, ); dec on (, 2)
F. None
G. CD on 共, 2兲, 共1, 兲
H. See graph at right.
25. A. ⺢
B. y-int 0; x-int 0
C. About the origin
D. HA y 苷 1
E. Inc on 共, 兲 F. None
G. CU on 共, 0兲;
CD on 共0, 兲; IP 共0, 0兲
H. See graph at right.
ⱍ
B. y-int 19
C. About y-axis
D. VA x 苷 3, HA y 苷 0
E. Inc on 共, 3兲, 共3, 0兲;
dec on (0, 3), 共3, 兲
F. Loc max f 共0兲 苷 19
G. CU on 共, 3兲, 共3, 兲;
CD on 共3, 3兲
x 3
H. See graph at right.
21. A. 关0, 兲
B. y-int 0; x-int 0, 3
C. None
D. None
E. Inc on 共1, 兲; dec on 共0, 1兲
F. Loc min f 共1兲 苷 2
G. CU on 共0, 兲
H. See graph at right.
A83
y=1
y
23. A. 共, 2兴 傼 关1, 兲
x=2
y=_1
ANSWERS TO ODD-NUMBERED EXERCISES
y
1 x
0
29. A. ⺢
B. y-int 0; x-int 0, 3s3 C. About the origin
D. None
E. Inc on 共, 1兲, 共1, 兲; dec on 共1, 1兲
y
F. Loc max f 共1兲 苷 2;
共_1, 2兲
loc min f 共1兲 苷 2
共3œ„3, 0兲
共0, 0兲
G. CU on 共0, 兲; CD on 共, 0兲;
0
x
IP 共0, 0兲
共_3œ„
3, 0兲
共1,
_2兲
H. See graph at right.
31. A. ⺢
B. y-int 1; x-int 1
C. About y-axis
D. None
E. Inc on 共0, 兲; dec on 共, 0兲
F. Loc min f 共0兲 苷 1
G. CU on 共1, 1兲;
CD on ;
IP
H. See graph at right.
y
(_1, 0)
0
(1, 0)
(0, _1)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 84
A84
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
33. A. ⺢
43. A. ⺢
B. y-int 0; x-int n (n an integer)
C. About 共0, 0兲, period 2
D. None
E–G answers for 0 x :
E. Inc on 共0, 兾2兲; dec on 共 兾2, 兲 F. Loc max f 共 兾2兲 苷 1
G. Let 苷 sin1s2兾3 ; CU on 共0, 兲, 共 , 兲;
CD on 共 , 兲; IP at x 苷 0, , , H.
y
B. y-int. 12
C. None
D. HA y 苷 0, y 苷 1
E. Inc on ⺢
F. None
G. CU on 共, 0兲; CD on 共0, 兲;
IP (0, 12 ) H. See graph at right.
45. A. 共0, 兲
B. None
C. None
D. VA x 苷 0
E. Inc on 共1, 兲; dec on 共0, 1兲
F. Loc min f 共1兲 苷 1
G. CU on 共0, 兲
H. See graph at right.
x
2π
y1
x
0
4π
0
_2π
y
y
(1, 1)
0
兾2兲 B. y-int 0; x-int 0
C. About y-axis
y
D. VA x 苷 兾2
E. Inc on 共0, 兾2兲;
π
π
dec on 共 兾2, 0兲
x 2
x 2
F. Loc min f 共0兲 苷 0
G. CU on 共 兾2, 兾2兲
H. See graph at right.
x
0
x
35. A. 共 兾2,
47. A. ⺢
B. y-int 14
C. None
D. HA y 苷 0, y 苷 1
E. Dec on ⺢
F. None
G. CU on ; CD on 共, ln 12);
IP (ln 12 , 49 )
H. See graph at right.
37. A. 共0, 3 兲
C. None
D. None
E. Inc on 共 兾3, 5 兾3兲, 共7 兾3, 3 兲;
dec on 共0, 兾3兲, 共5 兾3, 7 兾3兲
F. Loc min f 共 兾3兲 苷 共 兾6兲 12 s3, f 共7 兾3兲 苷 共7 兾6兲 12 s3;
y
loc max f 共5 兾3兲 苷 共5 兾6兲 12 s3
G. CU on 共0, 兲, 共2 , 3 兲;
CD on 共 , 2 兲;
IP 共 , 兾2兲, 共2 , 兲
π
5π
7π
3
3
3
H. See graph at right.
π
0
39. A. All reals except 共2n 1兲
2π
3π x
_4π _3π _2π _π
”ln 12 , 49 ’
0
π
2π
3π
x
4π
0
(n an integer)
51. A. 共, 0兲 傼 共0, 兲
B. None
C. None
D. VA x 苷 0
E. Inc on 共, 1兲, 共0, 兲;
dec on 共1, 0兲
F. Loc max f 共1兲 苷 e
G. CU on 共0, 兲; CD on 共, 0兲
H. See graph at right.
y
0
y=1
49. A. All x in 共2n , 共2n 1兲 兲 (n an integer)
B. x-int 兾2 2n
C. Period 2
D. VA x 苷 n
E. Inc on 共2n , 兾2 2n 兲; dec on 共 兾2 2n , 共2n 1兲 兲
F. Loc max f 共 兾2 2n 兲 苷 0 G. CD on 共2n , 共2n 1兲 兲
H.
y
B. y-int 0; x-int 2n
C. About the origin, period 2
D. VA x 苷 共2n 1兲
E. Inc on 共共2n 1兲 , 共2n 1兲 兲 F. None
G. CU on 共2n , 共2n 1兲 兲; CD on 共共2n 1兲 , 2n 兲;
IP 共2n , 0兲
H.
x=_3π
x=_π
x=π
x=3π
_2π
y
2π
x
y
x
(0 0)
(_1, _e)
x
53. A. ⺢
41. A. ⺢
B. y-int 兾4
C. None
D. HA y 苷 0, y 苷 兾2
E. Inc on 共, 兲 F. None
G. CU on 共, 0兲; CD on 共0, 兲;
IP 共0, 兾4兲
H. See graph at right.
y
B. y-int 2
C. None D. None
E. Inc on ( 15 ln 23, ); dec on (, 15 ln 23 )
y=π/2
F. Loc min f ( 15 ln 23) 苷
G. CU on 共, 兲
H. See graph at right.
(0, π/4)
0
y=0
y
( 23) 3兾5 ( 23) 2兾5
local
minimum
(0, 2)
x
0
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA076-085.qk_97909_Ans_Ans_pA076-085 10/7/10 2:32 PM Page 85
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
55.
71.
m
A85
y
π
yx2
(0, m¸)
√=c
0
√
x
0
yx
π
2
75. VA x 苷 0, asymptotic to y 苷 x 3
57. (a) When t 苷 共ln a兲兾k
(c)
(b) When t 苷 共ln a兲兾k
y
y
10
ƒ
1
y=˛
2
1/2
0
0
EXERCISES 4.6
59.
N
PAGE 324
1. Inc on 共0.92, 2.5兲, 共2.58, 兲; dec on , 共2.5, 2.58兲;
y
0
x
10
x
ln a
k
2
L/2
L
loc max f 共2.5兲 苷 4; loc min f 共0.92兲 ⬇ 5.12, f 共2.58兲 ⬇ 3.998;
CU on 共, 1.46兲, 共2.54, 兲;
CD on 共1.46, 2.54兲; IP 共1.46, 1.40兲, 共2.54, 3.999兲
x
10
61. y 苷 x 1
4.04
63. y 苷 2 x 2
ƒ
65. A. 共, 1兲 傼 共1, 兲
y
B. y-int 0; x-int 0
C. None
D. VA x 苷 1; SA y 苷 x 1
E. Inc on 共, 0兲, 共2, 兲;
dec on 共0, 1兲, 共1, 2兲
F. Loc max f 共0兲 苷 0;
loc min f 共2兲 苷 4
G. CU on 共1, 兲; CD on 共, 1兲
H. See graph at right.
67. A. 共, 0兲 傼 共0, 兲
3
B. x-int s
C. None
4
D. VA x 苷 0; SA y 苷 x
E. Inc. on 共, 0兲, 共2, 兲;
dec on 共0, 2兲
F. Loc min f 共2兲 苷 3
G. CU on 共, 0兲, 共0, 兲
H. See graph at right.
69. A. ⺢
B. y-int 2
C. None
D. SA y 苷 1 12 x
E. Inc. on 共ln 2, 兲;
dec on 共, ln 2兲
F. Loc min f 共ln 2兲 苷 32 12 ln 2
G. CU on 共, 兲
H. See graph at right.
0
(2, 4)
2.4
_6
x
(0, 0)
ƒ
4
2.7
3.96
3. Inc on 共15, 4.40兲, 共18.93, 兲;
dec on 共, 15兲, 共4.40, 18.93兲;
loc max f 共4.40兲 ⬇ 53,800; loc min f 共15兲 ⬇ 9,700,000,
f 共18.93兲 ⬇ 12,700,000; CU on 共, 11.34兲, 共0, 2.92兲,
共15.08, 兲; CD on 共11.34, 0兲, 共2.92, 15.08兲;
IP 共0, 0兲, ⬇ 共11.34, 6,250,000兲, 共2.92, 31,800兲,
共15.08, 8,150,000兲
y
10,000,000
y=x
(2, 3)
60,000
f
f
_25
25
_10
_2
0
10
x
_13,000,000
_30,000
5. Inc on 共, 1.47兲, 共1.47, 0.66兲; dec on 共0.66, 兲;
loc max f 共0.66兲 ⬇ 0.38; CU on 共, 1.47兲, 共0.49, 0兲,
共1.10, 兲; CD on 共1.47, 0.49兲, 共0, 1.10兲;
IP 共0.49, 0.44兲, 共1.10, 0.31兲
y
{ln 2, 32 + 21 ln 2}
2
3
f
1
2
1
y=1+ x
0
1
_4
x
3
xÅ_1.47
_3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA086-095.qk_97909_Ans_Ans_pA086-095 10/7/10 2:36 PM Page 86
A86
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
7. Inc on 共1.40, 0.44兲, 共0.44, 1.40兲; dec on 共, 1.40兲,
17. Inc on 共9.41, 1.29兲, 共0, 1.05兲; dec on 共, 9.41兲,
共0.44, 0兲, 共0, 0.44兲, 共1.40, 兲; loc max f 共0.44兲 ⬇ 4.68,
f 共1.40兲 ⬇ 6.09; loc min f 共1.40兲 ⬇ 6.09, f 共0.44兲 ⬇ 5.22;
CU on 共, 0.77兲, 共0, 0.77兲; CD on 共0.77, 0兲, 共0.77, 兲;
IP 共0.77, 5.22兲, 共0.77, 5.22兲
共1.29, 0兲, 共1.05, 兲; loc max f 共1.29兲 ⬇ 7.49, f 共1.05兲 ⬇ 2.35;
loc min f 共9.41兲 ⬇ 0.056, f 共0兲 苷 0.5; CU on 共13.81, 1.55兲,
共1.03, 0.60兲, 共1.48, 兲; CD on 共, 13.81兲, 共1.55, 1.03兲,
共0.60, 1.48兲; IP 共13.81, 0.05兲, 共1.55, 5.64兲, 共1.03, 5.39兲,
共0.60, 1.52兲, 共1.48, 1.93兲
8
f
8
_π
1
f
π
_8
9. Inc on (8 s61, 8 s61 ); dec on (, 8 s61 ),
f
_6
6
_15
0
_1
_0.1
共8 s61, 0兲, 共0, 兲; CU on (12 s138, 12 s138 ),
19. Inc on 共4.91, 4.51兲, 共0, 1.77兲, 共4.91, 8.06兲, 共10.79, 14.34兲,
共0, 兲; CD on (, 12 s138 ), (12 s138, 0)
75
1
f
f
_1
_100
1
_1
0.95
11. (a)
_10
1
共17.08, 20兲;
dec on 共4.51, 4.10兲, 共1.77, 4.10兲, 共8.06, 10.79兲, 共14.34, 17.08兲;
loc max f 共4.51兲 ⬇ 0.62, f 共1.77兲 ⬇ 2.58, , f 共14.34兲 ⬇ 4.39;
loc min f 共10.79兲 ⬇ 2.43, f 共17.08兲 ⬇ 3.49;
CU on 共9.60, 12.25兲, 共15.81, 18.65兲;
CD on 共4.91, 4.10兲, 共0, 4.10兲, 共4.91, 9.60兲, 共12.25, 15.81兲,
共18.65, 20兲;
IP at 共9.60, 2.95兲, 共12.25, 3.27兲, 共15.81, 3.91兲, 共18.65, 4.20兲
5
冢 œ„1e , 2e 冣
1
f
1.75
_0.25
_0.25
_5
(b) limx l 0 f 共x兲 苷 0
(c) Loc min f (1兾se ) 苷 1兾共2e兲; CD on 共0, e3兾2 兲;
CU on 共e3兾2, 兲
13. Loc max f 共5.6兲 ⬇ 0.018, f 共0.82兲 ⬇ 281.5,
f 共5.2兲 ⬇ 0.0145; loc min f 共3兲 苷 0
y
21. Inc on 共, 0兲, 共0, 兲;
CU on 共, 0.42兲, 共0, 0.42兲;
CD on 共0.42, 0兲, 共0.42, 兲;
IP 共 0.42, 0.83兲
0.02
8
1
20
0
1
ƒ
_3
3
ƒ
3.5
_1
x
500
0.01
0.6
23.
0.04
0.03
1
2
_2
2.5
1500
f 共x兲 苷 2
8
25. (a)
2
2
0
3
2
x 共x 1兲 共x 18x 44x 16兲
共x 2兲3共x 4兲5
2
15. f 共x兲 苷 0
0
3
2
共x 1兲共x 6 36x 5 6x 4 628x 3 684x 2 672x 64兲
共x 2兲4共x 4兲6
CU on 共35.3, 5.0兲, 共1, 0.5兲, 共0.1, 2兲, 共2, 4兲, 共4, 兲;
CD on 共, 35.3兲, 共5.0, 1兲, 共0.5, 0.1兲;
IP 共35.3, 0.015兲, 共5.0, 0.005兲, 共1, 0兲, 共0.5, 0.00001兲,
共0.1, 0.0000066兲
0
8
_1
(b) lim x l0 x 1兾x 苷 0, lim x l x 1兾x 苷 1
(c) Loc max f 共e兲 苷 e 1兾e (d) IP at x ⬇ 0.58, 4.37
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
27. Max f 共0.59兲 ⬇ 1, f 共0.68兲 ⬇ 1, f 共1.96兲 ⬇ 1;
min f 共0.64兲 ⬇ 0.99996, f 共1.46兲 ⬇ 0.49, f 共2.73兲 ⬇ 0.51;
IP 共0.61, 0.99998兲, 共0.66, 0.99998兲, 共1.17, 0.72兲,
共1.75, 0.77兲, 共2.28, 0.34兲
ⱍ ⱍ
35. For c
1, the graph has loc max and min values; for
1 it does not. The function increases for c 1 and decreases
for c 1. As c changes, the IPs move vertically but not
horizontally.
ⱍcⱍ
10
c=3
c=1
1.2
c=0.5
c=0
f
0
_15
π
1
5
c=_0.5
1.2
_10 c=_3 c=_1
37.
1.2
1
3
f
f
_2π
1
_2
0
2π
3
_3
0.55
0.9997
0.73
_1.2
29. For c
0, there is an absolute minimum at the origin. There
are no other maxima or minima. The more negative c becomes, the
farther the two IPs move from the origin. c 苷 0 is a transitional
value.
10
c=4
2
1
_3
For c 0, lim x l f 共x兲 苷 0 and lim x l f 共x兲 苷 .
For c 0, lim x l f 共x兲 苷 and lim x l f 共x兲 苷 0.
As c increases, the max and min points and the IPs get closer
to the origin.
39. (a) Positive
(b)
c=4
ⱍ ⱍ
c=1
c=0.5
c=0.2
c=0.1
12
c=_4
_4
4
c=0
0
31. For c
0, there is no extreme point and one IP, which
decreases along the x-axis. For c 0, there is no IP, and one
minimum point.
c=5
_6
6
c=0
10
_12
1
c=
5
_10
c=0
10
EXERCISES 4.7
c=_5
1
c=_
5
N
_10
3. 10, 10
33. For c
5.
9
4
12,500 ft@
50
0, the maximum and minimum values are always
1
2 , but the extreme points and IPs move closer to the y-axis as c
increases. c 苷 0 is a transitional value: when c is replaced by c,
the curve is reflected in the x-axis.
c=_1
c=_4
PAGE 331
1. (a) 11, 12
(b) 11.5, 11.5
7. 25 m by 25 m
9. N 苷 1
11. (a)
250
12,500 ft@
100
9000 ft@
120
0.6
0.2
0.5
1 2
5 4
5
1
0.6
A87
125
75
(b)
x
y
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A88
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
(c) A 苷 xy
(d) 5x 2y 苷 750
(e) A共x兲 苷 375x 52 x 2
2
(f) 14,062.5 ft
13. 1000 ft by 1500 ft
15. 4000 cm3
17. $191.28
6 3
1
4
19. (5 , 5)
21. (3 , 3 s2 )
23. Square, side s2 r
25. L兾2, s3 L兾4
27. Base s3 r, height 3r兾2
29. 4r 3兾(3 s3 )
31. r 2(1 s5 )
33. 24 cm, 36 cm
35. (a) Use all of the wire for the square
(b) 40 s3兾(9 4 s3 ) m for the square
3
37. Height 苷 radius 苷 s
V兾 cm
3
39. V 苷 2R 兾(9 s3 )
43. E 2兾共4r兲
3 2
45. (a) 2 s csc 共csc s3 cot 兲 (b) cos1(1兾s3 ) ⬇ 55
(c) 6s[h s兾(2 s2 )]
47. Row directly to B
49. ⬇ 4.85 km east of the refinery
3
3
51. 10 s
3兾(1 s
3 ) ft from the stronger source
53. 共a 2兾3 b 2兾3 兲3兾2
55. 2s6
57. (b) (i) $342,491; $342兾unit; $390兾unit
(ii) 400
(iii) $320兾unit
1
59. (a) p共x兲 苷 19 3000 x
(b) $9.50
1
61. (a) p共x兲 苷 550 10 x
(b) $175
(c) $100
65. 9.35 m
69. x 苷 6 in.
71. 兾6
1
73. At a distance 5 2 s5 from A
75. 2 共L W 兲2
77. (a) About 5.1 km from B
(b) C is close to B; C is close
to D; W兾L 苷 s25 x 2兾x, where x 苷 BC
(c) ⬇1.07 ; no such value
(d) s41兾4 ⬇ 1.6
ⱍ
EXERCISES 4.8
N
ⱍ
PAGE 342
1. (a) x2 ⬇ 2.3, x3 ⬇ 3
(b) No
9
3. 2
5. a, b, c
7. 1.1785
9. 1.25
11. 1.82056420
13. 1.217562
15. 1.964636
17. 3.637958, 1.862365, 0.889470
19. 1.412391, 3.057104
21. 0, 0.902025
23. 1.93822883, 1.21997997, 1.13929375, 2.98984102
25. 0.76682579
27. 0.21916368, 1.08422462
29. (b) 31.622777
35. (a) 1.293227, 0.441731, 0.507854
(b) 2.0212
37. 共1.520092, 2.306964兲
39. 共0.410245, 0.347810兲
41. 0.76286%
EXERCISES 4.9
N
PAGE 348
1. F 共x兲 苷 x 3x C
3. F 共x兲 苷 2 x 4 x 3 5 x 4 C
1 2
3
5. F 共x兲 苷 x 2 x x C
7. F 共x兲 苷 5x 7兾5 40x 1兾5 C
3
9. F 共x兲 苷 s2 x C
11. F 共x兲 苷 2x 3兾2 2 x 4兾3 C
1
2
2
3
13. F 共x兲 苷
1
2
再
1
5
1
5
ⱍ ⱍ
ⱍ ⱍ
x 2 ln x C1 if x
x 2 ln x C2 if x
1
1
37. 2 x 2兾3 2 if x
3
0
39. x 2x x 12x 4
41. sin cos 5 4
1
43. f 共x兲 苷 2x 2 x 3 2x 4 2x 3
45. x 2 cos x 2 x
47. ln x 共ln 2兲x ln 2
49. 10
51. b
2
53.
ⱍ ⱍ
23. F共x兲 苷 x 5 3 x 6 4
1
4
y
F
0
x
1
55.
57.
y
y
(2, 2)
2
1
(1, 1)
0
1
F
(3, 1)
2
_2π
0
2π x
x
3
_1
59.
61.
63.
65.
(c)
69.
75.
79.
(d)
s共t兲 苷 1 cos t sin t
s共t兲 苷 13 t 3 12 t 2 2t 3
s共t兲 苷 10 sin t 3 cos t 共6兾兲t 3
(a) s共t兲 苷 450 4.9t 2
(b) s450兾4.9 ⬇ 9.58 s
9.8s450兾4.9 ⬇ 93.9 m兾s (d) About 9.09 s
130
225 ft
71. $742.08
73. 11 ⬇ 11.8 s
88
2
77. 62,500 km兾h2 ⬇ 4.82 m兾s2
15 ⬇ 5.87 ft兾s
(a) 22.9125 mi
(b) 21.675 mi
(c) 30 min 33 s
55.425 mi
CHAPTER 4 REVIEW
N
PAGE 351
True-False Quiz
1. False
11. True
3. False
13. False
5. True
15. True
7. False
9. True
17. True
19. True
Exercises
1. Abs max f 共4兲 苷 5, abs and loc min f 共3兲 苷 1
2
1
9
3. Abs max f 共2兲 苷 5, abs and loc min f (3 ) 苷 2
5. Abs and loc max f 共兾6兲 苷 兾6 s3 ,
abs min f 共2兲 苷 2, loc min f 共5兾6兲 苷 5兾6 s3
1
7. 1
9. 8
11. 0
13. 2
15.
0
0
y
x6
2 0
1
2 5兾2
5
n an integer
19. F 共x兲 苷 5e x 3 sinh x C
1
21. F 共x兲 苷 2 x 2 ln x 1兾x 2 C
3
1
15. G共t兲 苷 2t t t C
17. H共 兲 苷 2 cos tan Cn on 共n 兾2, n 兾2兲,
2 3兾2
3
1兾2
0; 32 x 2兾3 52 if x
1
17.
9
x
12
y
y=2
25. f 共x兲 苷 x 5 x 4 x 3 Cx D
27. x 8兾3 Cx D
29. f 共t兲 苷 sin t Ct 2 Dt E
31. f 共x兲 苷 x 2x 3兾2 5
33. f 共t兲 苷 4 arctan t 3
20
x
y=_2
35. 2 sin t tan t 4 2s3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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APPENDIX I
19. A. ⺢
B. y-int 2
C. None
D. None
E. Dec on 共, 兲 F. None
G. CU on 共, 0兲;
CD on 共0, 兲; IP 共0, 2兲
H. See graph at right.
31. A.
2
x
y
2
1
0
1
ⱍ
23. A. 兵x x 苷 0, 3其
B. None
C. None
D. HA y 苷 0; VA x 苷 0, x 苷 3
E. Inc on 共1, 3兲; dec on 共, 0兲,
共0, 1兲, 共3, 兲
F. Loc min f 共1兲 苷 14
G. CU on 共0, 3兲, 共3, 兲; CD on 共, 0兲
H. See graph at right.
x
2
y
0
x
x3
ⱍ
25. A. 兵x x 苷 8其
y
B. y-int 0, x-int 0
C. None
D. VA x 苷 8; SA y 苷 x 8
E. Inc on 共, 16兲, 共0, 兲;
dec on 共16, 8兲, 共8, 0兲
F. Loc max f 共16兲 苷 32;
loc min f 共0兲 苷 0
G. CU on 共8, 兲; CD on 共, 8兲
H. See graph at right.
x 8
0
共16, 32兲
x
y x 8
x
1
_
π
2
y
{3, e_#}
1
0
x
2
_2
1.5
ƒ
_5
5
_1.5
2.5
f
2.1
_0.5
_20
x
4
”_ 3 , _
}
e# π/4
π
_1 0
y
_π/4
0
π
2
37. Inc on 共0.23, 0兲, 共1.62, 兲; dec on 共, 0.23兲, 共0, 1.62兲;
loc max f 共0兲 苷 2; loc min f 共0.23兲 ⬇ 1.96, f 共1.62兲 ⬇ 19.2;
CU on 共, 0.12兲, 共1.24, 兲;
CD on 共0.12, 1.24兲; IP 共0.12, 1.98兲, 共1.24, 12.1兲
_1
{_π4 , _ œ„22 e }
_π
35. Inc on (s3, 0), (0, s3 );
dec on (, s3 ), (s3, );
loc max f (s3 ) 苷 29 s3,
loc min f (s3 ) 苷 29 s3;
CU on (s6, 0), (s6, );
CD on (, s6 ), (0, s6 );
IP (s6, 365 s6 ), (s6, 365 s6 )
f
4œ„
6
’
9
29. A. 关, 兴
B. y-int 0; x-int , 0, C. None
D. None
E. Inc on (兾4, 3兾4 ); dec on (, 兾4 ), (3兾4, )
F. Loc max f 共3兾4兲 苷 12 s2 e 3兾4, loc min f 共兾4兲 苷 12 s2 e 3兾4
G. CU on 共兾2, 兾2兲; CD on 共, 兾2兲, 共兾2, 兲;
IP 共兾2, e兾2 兲, 共兾2, e 兾2 兲
H.
y
3π œ„2
2
33. A. ⺢
B. y-int 2; x-int 2
C. None
D. HA y 苷 0
E. Inc on 共, 3兲; dec on 共3, 兲
F. Loc max f 共3兲 苷 e3
G. CU on 共4, 兲; CD on 共, 4兲;
IP 共4, 2e4 兲
H. See graph at right.
A89
y
15
27. A. 关2, 兲
B. y-int 0; x-int 2, 0
C. None
D. None
E. Inc on ( 43 , ), dec on (2, 43 )
F. Loc min f ( 43 ) 苷 49 s6
G. CU on 共2, 兲
H. See graph at right.
{4,
{x ⱍ ⱍ x ⱍ
1}
B. None
C. About (0, 0)
D. HA y 苷 0
E. Dec on 共, 1兲, 共1, 兲
F. None
G. CU on 共1, 兲; CD on 共, 1兲
H. See graph at right.
y
21. A. ⺢
B. y-int 0; x-int 0, 1
C. None
D. None
E. Inc on ( 14 , ), dec on (, 14 )
27
F. Loc min f ( 14 ) 苷 256
1
G. CU on (, 2 ), 共1, 兲;
CD on ( 12 , 1); IP ( 12 , 161 ), 共1, 0兲
H. See graph at right.
ANSWERS TO ODD-NUMBERED EXERCISES
x
39.
0.4
共 0.82, 0.22兲; ( s2兾3, e3兾2 )
1
_5
1.5
5
0
41. 2.96, 0.18, 3.01; 1.57, 1.57; 2.16, 0.75, 0.46, 2.21
43. For C
1, f is periodic with period 2 and has local
maxima at 2n 兾2, n an integer. For C 1, f has no graph.
For 1 C 1, f has vertical asymptotes. For C 1, f is continuous on ⺢. As C increases, f moves upward and its oscillations
become less pronounced.
49. (a) 0
(b) CU on ⺢
53. 3 s3 r 2
57. L 苷 C
59. $11.50
55. 4兾s3 cm from D
61. 1.297383
63. 1.16718557
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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A90
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
65. f 共x兲 苷 sin x sin1x C
67. f 共x兲 苷 x
2
5
5兾2
x
3
5
5兾3
5. (a) 8, 6.875
C
(b) 5, 5.375
y
y
2
2
y
y
69. f 共t兲 苷 t 2 3 cos t 2
71. f (x兲 苷 2 x 2 x 3 4x 4 2x 1
1
73. s共t兲 苷 t 2 tan1 t 1
2
2
75. (b) 0.1e x cos x 0.9
(c)
5
0
1
x
0
0
x
1
0
x
1
1
x
F
(c) 5.75, 5.9375
_4
y
y
2
2
4
_1
77. No
79. (b) About 8.5 in. by 2 in.
83. (a) 20 s2 ⬇ 28 ft
(c) 20兾s3 in., 20 s2兾3 in.
0
dI
480k共h 4兲
(b)
, where k is the constant
苷
dt
关共h 4兲2 1600兴 5兾2
of proportionality
1
0
x
1
x
(d) M6
7. n 苷 2: upper 苷 3 ⬇ 9.42, lower 苷 2 ⬇ 6.28
y
3
PROBLEMS PLUS
N
PAGE 356
3. Abs max f 共5兲 苷 e 45, no abs min
7. 24
9. 共2, 4兲, 共2, 4兲
13. 共m兾2, m 2兾4兲
15. a
2
e
1兾e
19. (a) T1 苷 D兾c1, T2 苷 共2h sec 兲兾c1 共D 2h tan 兲兾c2,
T3 苷 s4h 2 D 2兾c1
(c) c1 ⬇ 3.85 km兾s, c2 ⬇ 7.66 km兾s, h ⬇ 0.42 km
3
23. 3兾(s
2 1) ⬇ 1112 h
1
0
π
x
3π
4
π
x
3π
4
π
x
π
2
n 苷 4: upper 苷 (10 s2 )共兾4兲 ⬇ 8.96,
lower 苷 (8 s2 )共兾4兲 ⬇ 7.39
y
3
CHAPTER 5
2
EXERCISES 5.1
N
PAGE 369
1
1. (a) L 4 苷 33, R 4 苷 41
y
6
y
6
4
4
2
2
0
π
4
π
2
n 苷 8: upper ⬇ 8.65, lower ⬇ 7.86
y
0
2
4
6
8x
0
2
4
6
3
8x
(b) L8 ⬇ 35.2, R8 ⬇ 39.2
3. (a) 0.7908, underestimate
2
(b) 1.1835, overestimate
y
y
1
1
1
0
0
ƒ=cos x
π
8
π
4
3π
8
π
2
x
0
ƒ=cos x
π
8
π
4
3π
8
π
2
x
π
4
π
2
9. 0.2533, 0.2170, 0.2101, 0.2050; 0.2
11. (a) Left: 0.8100, 0.7937, 0.7904;
right: 0.7600, 0.7770, 0.7804
13. 34.7 ft, 44.8 ft
15. 63.2 L, 70 L
17. 155 ft
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA086-095.qk_97909_Ans_Ans_pA086-095 10/7/10 2:36 PM Page 91
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
n
2共1 2i兾n兲
2
21. lim 兺 ssin共 i兾n兲 ⴢ
ⴢ
2
n l i苷1 共1 2i兾n兲 1
n l i苷1
n
n
23. The region under the graph of y 苷 tan x from 0 to 兾4
25. (a) L n
A Rn
n
19. lim
兺
27. (a) lim
nl
n 2共n 1兲2共2n 2 2n 1兲
12
n
64
n6
兺i
5
(b)
i苷1
(c)
32
3
EXERCISES 5.3
N
A91
PAGE 394
1. One process undoes what the other one does. See the
Fundamental Theorem of Calculus, page 393.
3. (a) 0, 2, 5, 7, 3
(d) y
(b) (0, 3)
(c) x 苷 3
g
29. sin b, 1
EXERCISES 5.2
1
PAGE 382
N
0
1. 6
y
3
ƒ=3- 1 x
2
The Riemann sum represents
the sum of the areas of the two 2
1
rectangles above the x-axis
minus the sum of the areas of
0
2
the three rectangles below the
x-axis; that is, the net area of the
rectangles with respect to the
x-axis.
3. 2.322986
The Riemann sum represents the sum
of the areas of the three rectangles
above the x-axis minus the area of the
rectangle below the x-axis.
5.
y=t@
x
6
0
6
11. F共x兲 苷 s1 sec x
3
15. y 苷 stan x stan x sec2 x
17. y 苷
1
2
5. (a) 6
(b) 4
(c) 2
7. Lower, L 5 苷 64; upper, R5 苷 16
17.
6
2
x
5
10
50
100
1.933766
1.983524
1.999342
1.999836
x ln共1 x 兲 dx
2
21. 9
23.
29. lim
兺
n l i苷1
n
31. lim
兺
n l i苷1
45. e e
3
47.
E
59. 3
x14 sx dx
63. 0
y
0
A
x
xe
41. e 2 1
43. 0
39.
5
1
x
D
dx
40
3
1
e1
e1
23.
31. 1
52
3
33.
49
3
39. 4兾3
关2, 1兴, so FTC2 cannot be applied.
47. The function f 共 兲 苷 sec tan is not continuous on the
interval 关兾3, 兴, so FTC2 cannot be applied.
243
49. 4
51. 2
53. 3.75
y=˛
1
0
(d) 2
5
2
41. 0
61.
12
71.
55. t共x兲 苷
43. 3
兾3
y
兾4
1
0
x
4
x dx
s3
12
tan x dx
73.
1
2
2
x
2共4x 2 1兲
3共9x 2 1兲
2
4x 1
9x 2 1
4
49. 122
53. 15
C
2兾e
29.
21. 63
共5x 4x兲 dx
f 共x兲 dx
6
3
4
3
冊
51. B
2
x
19.
9
5
37
37.
3
4
冉
27. 6
35. ln 2 7
25. 2
3
37. 3 4 3
2
25. 1 s3兾2
19.
y
7
2
5 i 2
sin
苷
n
n
5
(b) 10
(c) 3
33. (a) 4
35.
The values of Rn appear to be
approaching 2.
2 4i兾n
4
ⴢ
1 共2 4i兾n兲5 n
n
x
3共1 3x兲3
1 共1 3x兲2
13. h共x兲 苷 xe x
45. The function f 共x兲 苷 x 4 is not continuous on the interval
13. 0.9029, 0.9018
Rn
9. t共s兲 苷 共s s 2 兲8
4
0
_1
n
t
x
7. t共x兲 苷 1兾共x 3 1兲
ƒ=´-2
5
1
11. 0.9071
1
y
2
9. 6.1820
15.
(a), (b) x 2
y
8 10 12 14
4
x
1
2
57. F共x兲 苷 2xe x e x
59. y 苷 sin x ln共1 2 cos x兲 cos x ln共1 2 sin x兲
61. 共4, 0兲
63. 29
65. (a) 2 sn, s4n 2, n an integer
0
(b) 共0, 1兲, (s4n 1, s4n 3 ), and (s4n 1, s4n 1 ),
(c) 0.74
n an integer 0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA086-095.qk_97909_Ans_Ans_pA086-095 10/7/10 2:36 PM Page 92
A92
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
67. (a) Loc max at 1 and 5;
y
loc min at 3 and 7
(b) x 苷 9
1
2
0
(d) See graph at right.
_1
4
6
8
x
ⱍ
ⱍ
43. ln sin1 x C
1
40
47.
1
3
37.
39. ln共1 cos 2 x兲 C
2
( 12 , 2), 共4, 6兲, 共8, 9兲
(c)
35. 3 共cot x兲3兾2 C
sinh3x C
ⱍ
_2
1
共2x 5兲10 36 共2x 5兲 9 C
5
49. 共x 1兲 C
1
8
2
51. e cos x C
4
2
1
69.
77. f 共x兲 苷 x 3兾2, a 苷 9
79. (b) Average expenditure over 关0, t兴; minimize average
1
4
expenditure
ⱍ
41. ln sin x C
45. tan1x 2 ln共1 x 2 兲 C
f
F
0
_2
2π
2
F
f
EXERCISES 5.4
N
PAGE 403
5. x 共1兾x兲 C
1
3
2
3
7. x x x 2x C
1 4
8
1 3
11. 3
1
15. 2 2
5
9. u 3 2 u 2 4u C
1
8
2
x 4 sx C
9
10 5
53. 2兾
55.
63. 3
csc C
13. cos x cosh x C
17. tan C
1
19. sin x 4 x 2 C
20
_3
_1
1
5
3
65.
1
3
45
28
冉
1. True
11. False
_6
55
63
31.
39.
256
5
27. 5e 1
25. 2
21
5
29. 36
1
9
33. 2 ln 2
35.
11
ln 10
41. 兾3
43. 兾6
45. 3.5
37. 1 兾4
3
4
47. ⬇1.36
PAGE 416
3. True
13. True
5. False
15. False
1. (a) 8
7. True
9. True
17. False
(b) 5.7
y
y=ƒ
2
ages of 5 and 10
1000 to 5000 units
3
57. Newton-meters
59. (a) 2 m
(b) 416 m
1
61. (a) v共t兲 苷 2 t 2 4t 5 m兾s
(b) 416 23 m
2
63. 46 3 kg
65. 1.4 mi
67. $58,000
19.
2
1
1
2
1
3
sin 1
5. 4 cos 4 C
1
9. 20 共1 2x兲10 C
1
13. 76
21. 0
25. sx 4x C
ⱍ
ⱍ
29. 2e sx C
1
4
ln共1 x 4 兲 C
13. 3 ln 5 3x C
33.
15. (1兾兲 cos t C
1
C
1 eu
1
21. 3 共ln x兲3 C
39. 2 s1 sin x C
19. 3 s3ax bx 3 C
2
25. 3 共1 e x 兲3兾2 C
2
1
cos共5 t 兲 C
ln 5
1
17.
27.
1
15
23.
1
4
tan4 C
31. e tan x C
33. 45. t共x兲 苷 4x 3 cos共x 8 兲
49. 4
共x 3 3x兲 5 C
1
C
sin x
15.
21
4
17. Does not exist
ⱍ ⱍ
23. 共1兾x兲 2 ln x x C
1
sin 2 t C
27.
2
31. 2 关ln共cos x兲兴 2 C
11. 3 共2x x 2 兲 3兾2 C
1
x
2
7. f is c, f is b, x0x f 共t兲 dt is a
5. 3
9
10
2
3. 9 共x 3 1兲3兾2 C
7. 2 cos共x 2 兲 C
6
0
x
2
兾4
9. 37
11.
PAGE 413
y=ƒ
2
6
0
3.
71. 4.75 10 5 megawatt-hours
69. 5443 bacteria
29. 91. 2兾4
85. 5
Exercises
53. Number of gallons of oil leaked in the first 2 hours
55. Increase in revenue when production is increased from
1. e x C
N
L
y
51. The increase in the child’s weight (in pounds) between the
N
61. 0
69. 2
4
3
49.
EXERCISES 5.5
16
15
1
3
True-False Quiz
10
23.
冊
5
2 t
1 cos
4
5
83.
0
CHAPTER 5 REVIEW
21. 103
67.
71. ln共e 1兲
73. 16
75. s3 77. 6
79. All three areas are equal.
81. ⬇ 4512 L
_5
_10
59. e se
57. 4
(2 s2 1) a 3
x13 sx 2 3 dx
1
ⱍ
ⱍ
35. ln 1 sec C
41.
64
5
37.
23
3
43. F共x兲 苷 x 2兾共1 x 3 兲
47. y 苷 (2e x e sx )兾共2x兲
4 s3
55. 0.280981
57. Number of barrels of oil consumed from Jan. 1, 2000, through
Jan. 1, 2008
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
59. 72,400
61. 3
63. c ⬇ 1.62
2
65. e 2x共2x 1兲兾共1 ex 兲
71. 3
PROBLEMS PLUS
1. 兾2
7. 4 兾21
y
y
(1, 1)
PAGE 420
N
y=x
9. 关1, 2兴
7. e2
5. 1
3. 2k
A93
y=˛
11. (a) 2 共n 1兲n
1
0
0
x
x
(b) 12 冀b冁共2b 冀b冁 1兲 12 冀a冁共2a 冀a冁 1兲
17. 2(s2 1)
CHAPTER 6
EXERCISES 6.1
1.
N
3. e 共1兾e兲 32
3
9. ln 2 1
2
19. 2兾 9. 64兾15
PAGE 427
8
3
11.
5. e 共1兾e兲 10
3
23.
1
2
5
2
17.
25.
31. 2 s3 1
33. 0, 0.90; 0.04
35. 1.11, 1.25, 2.86; 8.38
37. 2.80123
29.
7.
15. e 2
13. 72
21. 2 2 ln 2
2
3
4
3
59
12
y
y
¥=x
9
2
32
3
(4, 2)
x=2y
27. ln 2
x
0
x
0
3
41. 12s6 9
1
39. 0.25142
11. 11兾30
45. 4232 cm2
43. 117 3 ft
y
y
y=≈
47. (a) Car A
(b) The distance by which A is ahead of B after
1 minute
(c) Car A
(d) t ⬇ 2.2 min
24
49. 5 s3
51. 4 2兾3
53. 6
55. 0
m 1; m ln m 1
2
y=1
y=1
0
0
x
1
x
1
x=y@
EXERCISES 6.2
1. 19兾12
N
PAGE 438
y
y
1
y=2- 2 x
13. 2 ( 3 s3 )
4
y
1
π
x=1
”_ 3 , 3’
x=2
1 y=0 2
0
y=3
y
π
” 3 , 3’
y=1+sec x
x
0
x
y=1
0
3. 8
x
0
x
y
y
x-1
y=œ„„„„
15. 3兾5
x=5
0
y=1
1
5
y=0
x
0
x
y
y=x#
1
5. 162
y
0
y
x
1
x=2
y=9
(6, 9)
y
x=0
1
x=2œ„
y
0
x
0
x
0
1
3
4
x
x=2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA086-095.qk_97909_Ans_Ans_pA086-095 10/7/10 2:36 PM Page 94
A94
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
17. 10s2 兾3
33. 0.13
y
43. 117兾5
x=3
{ 21 , œ„21 }
x=¥
9. (a)
y
x=3
0
N
25
24
29. 17兾45
EXERCISES 6.5
1 2x 2
0
2x 2
31. (a) 2 x e
dx ⬇ 3.75825
2
2ex ) dx ⬇ 13.14312
(b) 2 x01 (e
47. r h
11. W2 苷 3W1
(b) 10.8 cm
(b)
1875
4
ft-lb
冉 冊
23. 兾3
27. 兾3
41. 4兾3
2
PAGE 449
⬇ 1.04 J
1
1
a
b
29. (a) Gm1 m2
21. 兾3
39. 4s3 1
3
3
15. 650,000 ft-lb
17. 3857 J
19. 2450 J
21. ⬇1.06 10 6 J
5
23. ⬇1.04 10 ft-lb
25. 2.0 m
x
25. 13兾45
37. 8
45. r
13. (a) 625 ft-lb
1
19. 兾3
3
1. (a) 7200 ft-lb
(b) 7200 ft-lb
15
3. 4.5 ft-lb
5. 180 J
7. 4 ft-lb
x
1
1
32
4
3
EXERCISES 6.4
y=1-¥
0
35.
N
(b) ⬇8.50 10 9 J
PAGE 453
1.
3.
5. 共2兾兲共e 1兲
9. (a) 1
(b) 2, 4
(c)
8
3
7. 2兾共5兲
45
28
33. (a) 2 x02 8s1 x 2兾4 dx ⬇ 78.95684
(b) 2 x01 8s4 4y 2 dy ⬇ 78.95684
11
35. 1.288, 0.884; 23.780
37. 8 2
x ,
39. Solid obtained by rotating the region 0
0 y ssin x about the x-axis
41. Solid obtained by rotating the region above the x-axis bounded
by x 苷 y 2 and x 苷 y 4 about the y-axis
43. 1110 cm 3
45. (a) 196
(b) 838
1
1
2
47. 3 r 2h
49. h2 (r 3 h)
51. 3 b 2h
1
8
3
53. 10 cm
55. 24
57. 3
59. 15
61. (a) 8R x0r sr 2 y 2 dy
(b) 2 2r 2R
63. (b) r 2h
65.
EXERCISES 6.3
N
5
12
r 3
11. (a) 4兾
(b) ⬇1.24, 2.81
(c)
67. 8 x0r sR 2 y 2 sr 2 y 2 dy
PAGE 444
1. Circumference 苷 2 x, height 苷 x 共x 1兲2; 兾15
9
8
15.
17. 共50 28兾兲 F ⬇ 59 F
19. 6 kg兾m
21. About 4056 million (or 4 billion) people
23. 5兾共4兲 ⬇ 0.4 L
CHAPTER 6 REVIEW
3. 6兾7
5. 共1 1兾e兲
7. 8
15. 7兾15
17. 8兾3
13. 16兾3
21. (a) 2 x x e
2
0
2 x
dx
11. 768兾7
19. 5兾14
x0 2 共4 y兲 ssin y dy
PAGE 457
Exercises
1.
8
3
3.
7
12
4
5. 3
2 3兾2
11. 共2ah h 兲
4
3
(b) 4.06300
23. (a) 4 x兾2
共 x兲 cos4 x dx
兾2
25. (a)
9. 4
N
4兾
13.
7. 64 兾15
15. (a) 2兾15
(b) 兾6
(c) 8兾15
17. (a) 0.38
(b) 0.87
19. Solid obtained by rotating the region 0
(b) 46.50942
(b) 36.57476
9. 1656兾5
x兾3兾3 2 (兾2 x)(cos 2x 14 ) dx
x 兾2 about the y-axis
21. Solid obtained by rotating the region 0
0 y 2 sin x about the x-axis
125
23. 36
25. 3 s3 m 3
27. 3.2 J
29. (a) 8000兾3 ⬇ 8378 ft-lb
(b) 2.1 ft
y
cos x ,
x
,
0
27. 3.68
29. Solid obtained by rotating the region 0
y
x 4, 0
x
3
about the y-axis
31. Solid obtained by rotating the region bounded by
(i) x 苷 1 y 2, x 苷 0, and y 苷 0, or (ii) x 苷 y 2, x 苷 1, and y 苷 0
about the line y 苷 3
31. f 共x兲
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
PROBLEMS PLUS
N
PAGE 459
EXERCISES 7.2
1. (a) f 共t兲 苷 3t
(b) f 共x兲 苷 s2x兾
(c) 0.6736 m
5. (b) 0.2261
2
3.
32
27
(c) f 共 y兲 苷 sk A兾共 C兲 y 1兾4. Advantage: the markings on the
container are equally spaced.
13. b 苷 2a
15. B 苷 16A
N
1
3
ⱍ
ⱍ
37.
41.
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
F
_π
e 2x
2
23.
C
4共2x 1兲
2 2
81
1
3
27. 4 ln 3 5
29. 4 4 e2
1 1兾e
1
33. sin x 共ln sin x 1兲 C
6 ( 6 3 s3 )
32
64
62
2
5 共ln 2兲 25 ln 2 125
1
39. 2 兾4
2 sx sin sx 2 cos sx C
1
1 2
1
3
2
2 共x 1兲 ln共1 x兲 4 x 2 x 4 C
π
f
_π
53.
1
6
sin 3x 1
18
sin 9x C
1
ƒ
F
_2
43. 12 xe2x 14 e2x C
ⱍ
π
1
25.
31.
35.
ⱍ
ⱍ
17. 13 e 2共2 sin 3 3 cos 3 兲 C
19. z 3e z 3z 2e z 6ze z 6e z C
21.
1
5
3
ⱍ
PAGE 468
1
1
x 3 ln x 19 x 3 C
3. 5 x sin 5x 25 cos 5x C
1 3t
1 3t
5. 3 te 9 e C
7. 共x 2 2x兲 sin x 共2x 2兲 cos x 2 sin x C
1
3
9. x ln s
11. t arctan 4t 8 ln共1 16t 2 兲 C
x 13 x C
1
1
13. 2 t tan 2t 4 ln sec 2t C
15. x 共ln x兲2 2x ln x 2x C
1.
1
3
ⱍ
CHAPTER 7
EXERCISES 7.1
PAGE 476
N
1
sin x sin5 x C
3. 120
1
2
1
5.
sin 3 共 x兲 sin 5 共 x兲 sin 7 共 x兲 C
3
5
7
7. 兾4
9. 3兾8
11. 兾16
1
1
1
13. 4 t 2 4 t sin 2t 8 cos 2t C
2
15. 45 ssin 共45 18 sin 2 5 sin 4 兲 C
1
17. 2 cos 2x ln cos x C
19. ln sin x 2 sin x C
1
21. 3 sec3 x C
23. tan x x C
1
2
1
117
25. 9 tan9 x 7 tan7 x 5 tan5 x C
27. 8
1
3
29. 3 sec x sec x C
1
31. 4 sec 4x tan 2x ln sec x C
1
33. x sec x ln sec x tan x C
35. s3 3 22
8
37. 105 s2 105
39. ln csc x cot x C
1
1
1
1
41. 6 cos 3x 26 cos 13x C
43. 8 sin 4 12 sin 6 C
1
1
45. 2 s2
47. 2 sin 2x C
1
49. x tan x ln sec x 2 x 2 C
1 2
1
2
2
51. 4 x 4 sin共x 兲 cos共x 兲 C
1.
(d) (i) 1兾共105兲 ⬇ 0.003 in兾s (ii) 370兾3 s ⬇ 6.5 min
32
9. y 苷 9 x 2
11. (a) V 苷 x0h 关 f 共 y兲兴 2 dy
A95
2
1
f
_1
3
F
1
55. 0
57. 1
59. 0
65. s 苷 共1 cos 3 t兲兾共3兲
61. 2兾4
63. (2s2 5
2
)
_1
EXERCISES 7.3
45. 3 x 2 共1 x 2 兲3兾2 1
2
15
共1 x 2 兲5兾2 C
1. 4
_2
2
f
_4
47. (b) cos x sin x x 49.
55.
57.
61.
67.
sin 2x C
(b) 23 ,
x 关共ln x兲 3共ln x兲2 6 ln x 6兴 C
16
29
59. 1.75119, 1.17210; 3.99926
3 ln 2 9
4 8兾
63. 2 e
65. 1 共2兾 兲 ln 2
69. 2
2 et共t 2 2t 2兲 m
3
3
8
PAGE 483
s4 x 2
C
4x
3. sx 2 4 2 sec1
冉冊
x
2
C
1
1
s3
7.
24
8
4
s2 a 2
1
1
9. ln (sx 2 16 x) C
11. 4 sin1共2x兲 2 x s1 4x 2 C
1
1
2
2
13. 6 sec 共x兾3兲 sx 9兾共2x 兲 C
1
15. 16 a 4
17. sx 2 7 C
9
19. ln (s1 x 2 1)兾x s1 x 2 C
21. 500 9
1
23. 2 sin1共共x 2兲兾3兲 2 共 x 2兲s5 4x x 2 C
1
1
25. sx 2 x 1 2 ln (sx 2 x 1 x 2 ) C
1
1
2
2
27. 2 共x 1兲 sx 2x 2 ln x 1 sx 2x C
1
1
29. 4 sin1 共x 2 兲 4 x 2 s1 x 4 C
1
3
3
33. 6 (s48 sec1 7)
37. 8 2 4
41. 2 2Rr 2
43. r sR 2 r 2 r 2兾2 R 2 arcsin共r兾R兲
5.
F
1
4
8
15
3
N
3
16
ⱍ
ⱍ
ⱍ
ⱍ
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:38 PM Page 96
A96
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
EXERCISES 7.4
PAGE 492
N
(b)
A
A
B
B
C
1. (a)
(b)
2 4x 3
2x 5
x
x
5 2x
B
C
Dx E
A
3. (a)
2 3 2
x
x
x
x 4
B
D
C
A
(b)
x3
共x 3兲 2
x3
共x 3兲 2
B
A
5. (a) x 4 4x 2 16 x2
x2
Cx D
Ex F
Ax B
2
2
(b) 2
x x1
x 2
共x 2兲2
1
1
1
7. 4 x 4 3 x 3 2 x 2 x ln x 1 C
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
27
5
ⱍ
ⱍ
ⱍ
ⱍ
ⱍ
3
29.
1
2
ln共x 2 2x 5兲 32 tan1
31.
1
3
ln x 1 16 ln共x 2 x 1兲 ⱍ
ⱍ
33.
1
4
ln 83
35.
冉 冊
1
16
ⱍ ⱍ
ln x 321
冉 冊
37. s2 tan1
7
8
x2
s2
x1
2
ⱍ
ⱍ
53. ( x 1
2
ⱍ
ⱍ
1
57.
1
2
ln
冟
冟
x2
C
x
61.
1
5
ln
冟
21. 共x 1兲 arctan sx sx C
ⱍ
23
3
ⱍ
ⱍ
ⱍ
冉 冊
35.
1
8
x1
2
x1
s3 2x x 2 C
2
sin 4x 161 sin 8x C
ⱍ
ⱍ
ⱍ
ⱍ
37.
ⱍ
ⱍ
1
4
39. ln sec 1 ln sec C
1
3
ⱍ
ⱍ
43.
2
3
tan1共 x 3兾2 兲 C
47. ln x 1 3共x 1兲1 2 共x 1兲2 3 共x 1兲3 C
冟
3
1
冟
冟
冟
s4x 1 1
s4x 2 1 1
51. ln
C
C
2x
s4x 1 1
2
2
1 2
53.
x cosh共mx兲 2 x sinh共mx兲 3 cosh共mx兲 C
m
m
m
55. 2 ln sx 2 ln (1 sx ) C
3
3
57. 7 共x c兲7兾3 4 c共x c兲4兾3 C
59. sin共sin x兲 3 sin 3 共sin x兲 C
61. csc cot C or tan共兾2兲 C
1
冉
2x 1
s7
冊
C
冟
2 tan共x兾2兲 1
C
tan共x兾2兲 2
11
24,110
1
668
1
9438
1
4879 5x 2
323 2x 1
80,155 3x 7
1
22,098x 48,935
260,015
x2 x 5
69. (a)
25. 3x 33. 2 sin1
49. ln
63. 4 ln 3 2
65. 1 3 ln 2
1
67. t 苷 ln P 9 ln共0.9P 900兲 C, where C ⬇ 10.23
2
C
45. 共x 3 1兲ex C
ⱍ
) ln共x 2 x 2兲 2x s7 tan1
55. 2 ln 3 ⬇ 0.55
19. e
1
3
ⱍ
2
2 ln(sx 1) C
sx
3
3
43. 10 共x 2 1兲5兾3 4 共x 2 1兲2兾3 C
3
6
6
x 6 ln s
x1 C
45. 2 sx 3 sx 6 s
ⱍ
15. x兾s1 x 2 C
1
ex
41. tan 2 2 ln sec C
39. 2sx 1 ln(sx 1 1) ln sx 1 1 C
ⱍ
2
1
2
4
4097
45
31. sin1x s1 x 2 C
C
3x 8
C
4共x 2 4x 6兲
共e x 2兲2
C
47. ln
ex 1
49. ln tan t 1 ln tan t 2 C
51. x ln共e x 1兲 C
ⱍ
ln x 4 53 ln x 2 C
27. x ln 共1 e x 兲 C
29. x ln( x sx 2 1 ) sx 2 1 C
23.
1
2x 1
tan1
C
s3
s3
1
ln共x 2 4兲 C
8共x 2 4兲
41. 2 ln sx ⱍ
冉 冊
17. ⱍ
ln共x 2 1兲 (1兾s2 ) tan1(x兾s2 ) C
3
1
25. 2 ln x 1 ln共x 2 1兲 2 tan1 x C
1
2
ⱍ
PAGE 499
N
13. 5 cos5 t 7 cos7 t 9 cos9 t C
1
27.
ⱍ
1
t
tan1
C
7. e 兾4 e兾4
2s2
s2
243
242
1
9. 5 ln 3 25
11. 2 ln共x 2 4x 5兲 tan1共x 2兲 C
5.
23. ln x 1 2 ln共x 2 9兲 3 tan1共x兾3兲 C
ⱍ
ⱍ
2
ⱍ
1
ⱍ
1. sin x sin x C
3. sin x ln csc x cot x C
1
3
5
19. 10 ln x 3 9 ln x 2 C
x2
1 2
2
1
21. 2 x 2 ln共x 4兲 2 tan 共x兾2兲 C
ⱍ
ⱍ
EXERCISES 7.5
ln 2 95 ln 3 (or 95 ln 83)
ⱍ
ⱍ
11,049
75,772
2x 1
tan1
C
ln共x 2 x 5兲 260,015
260,015 s19
s19
The CAS omits the absolute value signs and the constant of
integration.
1
1
1
1
73. n
n n1 2 a 共x a兲
a x
a x
ax n
9. 12 ln 2x 1 2 ln x 1 C
11. 2 ln 2
7
2
13. a ln x b C
15. 6 ln 3
17.
334
3146
4822
ln 5x 2 ln 2x 1 ln 3x 7 4879
323
80,155
63. 2( x 2sx 2) e sx C
65. tan1 共cos 2 x兲 C
67. 3 关共x 1兲3兾2 x 3兾2 兴 C
2
69. s2 2兾s3 ln (2 s3 ) ln (1 s2 )
71. e x ln共1 e x 兲 C
73. s1 x 2 2 共arcsin x兲 2 C
1
75.
1
8
ⱍ
ⱍ
ln x 2 161 ln共x 2 4兲 18 tan1共x兾2兲 C
s1 e x 1
C
s1 e x 1
1
1
1
79. 3 x sin3 x 3 cos x 9 cos3x C
77. 2共x 2兲s1 e x 2 ln
81. 2s1 sin x C
2
83. xe x C
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:38 PM Page 97
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
EXERCISES 7.6
1. 5.
N
23. (a) 2.8
(b) 7.954926518
(c) 0.2894
(d) 7.954926521
(e) The actual error is much smaller.
(f ) 10.9
(g) 7.953789422
(h) 0.0593
(i) The actual error is smaller.
( j) n 50
PAGE 504
3. s13 4 ln(4 s13 ) 2 4 ln 3
5
21
3
1
1
1
4 ln(1 16 2)
8
7.
9. s4x 2 9兾共9x兲 C
1
6
ln
冟
3
冟
sin x 3
C
sin x 3
25.
11. e 2
ⱍ
ⱍ
13. 2 tan2共1兾z兲 ln cos共1兾z兲 C
1
15. 共e 2x 1兲 arctan共e x 兲 e x C
1
2
17.
1
2
冉
2y 1
2y 1
7
s6 4y 4y 2 sin1
8
8
s7
1
2 3兾2
12 共6 4y 4y 兲 C
冊
冟
1
9
sin 3x 关3 ln共sin x兲 1兴 C
23.
1
4
tan x sec 3x tan x sec x ln sec x tan x C
3
8
ⱍ
3
8
[
]
ⱍ
25. 2 共ln x兲s4 共ln x兲 2 2 ln ln x s4 共ln x兲 2 C
1
27. x
1
2
2
1
2
cos 共x 兲 s1 x
1
2
4
冟
e x s3
1
ln x
C
2s3
e s3
19.
21.
37.
1
5
1
3
1
4
1
4
1
4
ⱍ
ⱍ
n
Ln
5
10
20
0.742943
0.867782
0.932967
n
EL
ER
ET
EM
5
10
20
0.257057
0.132218
0.067033
0.286599
0.139610
0.068881
0.014771
0.003696
0.000924
0.007379
0.001848
0.000462
27.
43.
C
冟
冟
ⱍ
N
PAGE 516
1. (a) L 2 苷 6, R2 苷 12, M2 ⬇ 9.6
(b) L 2 is an underestimate, R2 and M2 are overestimates.
(c) T2 苷 9 I
(d) L n Tn I Mn Rn
3. (a) T4 ⬇ 0.895759 (underestimate)
(b) M4 ⬇ 0.908907 (overestimate)
T4 I M4
5. (a) M10 ⬇ 0.806598, E M ⬇ 0.001879
(b) S10 ⬇ 0.804779, E S ⬇ 0.000060
7. (a) 1.506361
(b) 1.518362
(c) 1.511519
9. (a) 2.660833
(b) 2.664377
(c) 2.663244
11. (a) 2.591334
(b) 2.681046
(c) 2.631976
13. (a) 4.513618
(b) 4.748256
(c) 4.675111
15. (a) 0.495333
(b) 0.543321
(c) 0.526123
17. (a) 8.363853
(b) 8.163298
(c) 8.235114
19. (a) T8 ⬇ 0.902333, M8 ⬇ 0.905620
(b) ET 0.0078, EM 0.0039
(c) n 苷 71 for Tn , n 苷 50 for Mn
21. (a) T10 ⬇ 1.983524, ET ⬇ 0.016476;
M10 ⬇ 2.008248, EM ⬇ 0.008248;
S10 ⬇ 2.000110, ES ⬇ 0.000110
(b) ET 0.025839, EM 0.012919, E S 0.000170
(c) n 苷 509 for Tn , n 苷 360 for Mn , n 苷 22 for Sn
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
0.992621
0.998152
0.999538
Mn
Sn
6.695473
6.474023
n
ET
EM
ES
6
12
0.295473
0.074023
0.147428
0.036992
0.003292
0.000206
6.252572
6.363008
6.403292
6.400206
Observations are the same as after Example 1.
(a) 19.8
(b) 20.6
(c) 20.53
(a) 14.4
(b) 12
64.4 F
35. 37.73 ft兾s
37. 10,177 megawatt-hours
(a) 190
(b) 828
6.0
43. 59.4
29.
31.
33.
39.
41.
45.
1 s1 x 2
C;
x
both have domain 共1, 0兲 傼 共0, 1兲
EXERCISES 7.7
Mn
3
cos3 x sin x 38 x 38 sin x cos x C
tan 4 x 12 tan 2 x ln cos x C
45. (a) ln
1.014771
1.003696
1.000924
6
12
tan x sec 2 x 23 tan x C
ⱍ
1.286599
1.139610
1.068881
Tn
39. x共x 2 2兲 sx 2 4 2 ln (sx 2 4 x) C
41.
Tn
n
33. 8 2
ln x 5 sx 10 2 C
Rn
Observations are the same as after Example 1.
29. se 2x 1 cos1共ex 兲 C
31.
A97
ⱍ ⱍ
y
1
0
0.5
EXERCISES 7.8
1
N
1.5
2
x
PAGE 527
Abbreviations: C, convergent; D, divergent
1. (a), (d) Infinite discontinuity
(b), (c) Infinite interval
3. 1兾共2t 兲; 0.495, 0.49995, 0.4999995; 0.5
1
5. 2
7. D
9. 5 e 10
11. D
13. 0
15. D
1
1
17. ln 2
19. 4
21. D
23. 兾9
25. 2
27. D
32
9
29. 3
31. D
33. 2
35. D
37. 2兾e
8
8
39. 3 ln 2 9
1
41. 1兾e
43. 2 ln 2
1
2
2
2
y
y=e_x
y=
1
˛+x
x=1
0
1
x
0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
3
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:38 PM Page 98
A98
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
45. Infinite area
ⱍ
冟
π
2
0
y
t
t
1
关共sin 2x兲兾x 2 兴 dx
2
5
10
100
1,000
10,000
33.
x
x
sin1
2
s4 x 2
1
4
41.
1
36
47. 1
2
sin 2x 18 cos 4x C
43. D
49. 兾4
4
3
57.
61.
63.
65.
67.
69.
71.
73.
sin x s4 sin 2 x 2 ln (sin x s4 sin 2 x ) C
No
(a) 1.925444
(b) 1.920915
(c) 1.922470
(a) 0.01348, n 368
(b) 0.00674, n 260
8.6 mi
(a) 3.8
(b) 1.7867, 0.000646
(c) n 30
(a) D
(b) C
3
2
75. 16 2
1
2
PROBLEMS PLUS
y=F(t)
t
(in hours)
ⱍ
(b) The rate at which the
fraction F共t兲 increases as t
increases
(c) 1; all bulbs burn out
eventually
N
PAGE 534
1. About 1.85 inches from the center
3. 0
7. f 共兲 苷 兾2
11. 共b baa 兲1兾共ba兲e1
13. 18 121
15. 2 sin1 (2兾s5 )
CHAPTER 8
69. 1000
71. (a) F共s兲 苷 1兾s, s
0 (b) F共s兲 苷 1兾共s 1兲, s
(c) F共s兲 苷 1兾s 2, s 0
77. C 苷 1; ln 2
79. No
1
EXERCISES 8.1
1. 4s5
7.
13.
PAGE 530
19.
True-False Quiz
N
PAGE 543
3. 3.8202
5. 3.6095
(82 s82 1) 9. 5924 11. 323
3
1
15. 4 2 ln 2
17.
ln(s2 1)
21. 10.0556
s2 ln (1 s2 )
2
243
23. 15.374568
27. (a), (b)
1. False
3. False
5. False
7. False
9. (a) True
(b) False
11. False
13. False
ln 3 12
25. 7.118819
L 1 苷 4,
L 2 ⬇ 6.43,
L 4 ⬇ 7.50
Exercises
1.
7.
7
2
2
15
ln 2
ⱍ
ln 2x 1 s4x 2 4x 3 C
55. 57. p 1, 1兾共1 p兲
65. s2GM兾R
700
37.
C
1
1
N
冉冊
3
55. 4 共2x 1兲s4 x 2 4x 3 10
CHAPTER 7 REVIEW
31. 6 2 29. 0
51. 共x 1兲 ln共x 2 2x 2兲 2 arctan共x 1兲 2x C
53. 0
0.1
0
2
5
45. 4 ln 4 8
sin@ x
≈
49. C
51. D
53. D
59. p 1, 1兾共 p 1兲2
y
67. (a)
27.
1
1
≈
1
3
2
39. 8 e 0.447453
0.577101
0.621306
0.668479
0.672957
0.673407
©=
ln共x 2 1兲 3 tan1x s2 tan1(x兾s2 ) C
25.
35. 4s1 sx C
It appears that the integral is convergent.
(c) 1
ƒ=
冟
sx 2 1 1
C
23. ln
x
y=sec@ x
47. (a)
ⱍ
21. ln x 2 sx 2 4x C
20
ⱍ
3. e 1
ⱍ
ⱍ
ⱍ
5. ln 2t 1 ln t 1 C
11. s3 3 9. cos共ln t兲 C
1
13. 3e sx 共x 2兾3 2x 1兾3 2兲 C
3
ⱍ ⱍ
ⱍ
ⱍ
15. 2 ln x 2 ln x 2 C
1
3
ⱍ
ⱍ
17. x sec x ln sec x tan x C
19.
1
18
ln共9x 2 6x 5兲 1
9
tan1
[ (3x 1)] C
1
2
(c)
x04 s1 关4共3 x兲兾共3共4 x兲2兾3 兲兴 2 dx
(d) 7.7988
29. s5 ln ( 2 (1 s5 )) s2 ln (1 s2 )
1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:38 PM Page 99
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
31. 6
EXERCISES 8.5
N
A99
PAGE 573
1. (a) The probability that a randomly chosen tire will have a
33. s共x兲 苷
2
27
[共1 9x兲3兾2 10 s10 ]
37. 209.1 m
39. 29.36 in.
EXERCISES 8.2
1. (a) (i)
(ii)
兾3
0
x
(ii)
5.
x
1
27
x
N
PAGE 550
2 tan x s1 sec 4 x dx
2 x s1 sec 4 x dx
3. (a) (i)
1
0
兾3
0
1
1
x
35. 2 s2 (s1 x 1)
41. 12.4
x 2
2 e
s1 (b) (i) 10.5017
(ii) 7.9353
2
4x 2e 2x dx
2 x s1 4x 2e 2x dx
2
(145 s145 1)
7.
(b) (i) 11.0753
98
3
(ii) 3.9603
11.
21
2
23.
[
[ln(s10 3) 3 s10 ]
27. (a) 3 a 2
1
(b)
冋
29. (a) 2 b 2 冋
(b) 2 a 2 31.
56
45
a 2b sin1(sa 2 b 2兾a)
sa 2 b 2
ab 2 sin1(sb 2 a 2兾b)
sb 2 a 2
册
xab 2 关c f 共x兲兴 s1 关 f 共x兲兴 2 dx
EXERCISES 8.3
N
CHAPTER 8 REVIEW
]
册
冉
冉(
冊
3. (a)
冊
7.
13. (2, 3 )
15. 2 2
21. (a) 1 e3兾8 ⬇ 0.31
(c) 8 ln 2 ⬇ 5.55 min
11.
2
3
25.
( 23, 23)
3. (a) 2r共r d 兲
(b) ⬇3.36
(d) ⬇7.84
2
15. 1.19
104 cm 3兾s
7
10 mi
10 6 mi 2
7. Height s2 b, volume ( 27 s6 2)b 3
11. 2兾, 1兾
13. 共0, 1兲
28
9. 0.14 m
CHAPTER 9
45. 3 r 2h
1
5. $407.25
13.
PAGE 577
r
e x兾H ⴢ 2 sr 2 x 2 dx
(b) 共P0 0 tH兲共r 2 兲 0 tHe L兾H xr
共1 k兲共b 2k a 2k 兲
共2 k兲共b 1k a 1k 兲
17. 6.60 L兾min
19. 5.77 L兾min
(16s2 8) ⬇ $9.75 million
N
(b) e5兾4 ⬇ 0.29
5. (a) P共z兲 苷 P0 t x0z 共x兲 d x
PAGE 566
3. $140,000; $60,000
9. 3727; $37,753
( 85 , 1)
1
2
EXERCISES 9.1
3. (a) , 1
1. $21,104
7. $12,000
11.
1. s3
2
3
1
2
EXERCISES 8.4
9. ⬇ 458 lb
124
5
5. 3.8202
PROBLEMS PLUS
10 5 lb
23. 10; 14; 共1.4, 1兲
41
10
(b)
17. $7166.67
0 for all x and x
f 共x兲 dx 苷 1
(b) ⬇ 0.3455 (c) 5, yes
33. 4 2r 2
1
e1
9
9
29. ( 20 , 20 )
,
e1
4
s2 4
1
8
1
31.
33. ( 5, 2 )
,
4 s2 1) 4 (s2 1)
8
1
12
1
35. 60; 160; ( 3, 1)
37. ( 5, 35 )
41. (0, 12 )
N
21
16
19. (a) f 共x兲
1. (a) 187.5 lb兾ft 2
(b) 1875 lb
(c) 562.5 lb
3. 6000 lb
5. 6.7
7. 9.8
10 4 N
10 3 N
2
4
2
9. 1.2
11. 3 ah
10 lb
13. 5.27
15. (a) 314 N
10 5 N
(b) 353 N
17. (a) 5.63
10 3 lb (b) 5.06 10 4 lb
27.
15
2
2
PAGE 560
(c) 4.88 10 4 lb (d) 3.03
19. 4148 lb
21. 330; 22
PAGE 575
N
Exercises
1.
s3 a 2
3
(e) 2 a0
21. 4 4 ln(s17 4) 4 ln(s2 1) s17 4 s2
1
6
4x10–10
0
(d) 1 41e8 ⬇ 0.986
9. 2s1 2 共2兾兲 ln ( s1 2 )
1
13. 27 (145 s145 10 s10 )
15. a 2
19. 24.144251
17. 1,230,507
1
lifetime between 30,000 and 40,000 miles
(b) The probability that a randomly chosen tire will have a
lifetime of at least 25,000 miles
3. (a) f 共x兲
0 for all x and x
f 共x兲 dx 苷 1 (b) 17
81
1
5. (a) 1兾
(b) 2
7. (a) f 共x兲
0 for all x and x
f 共x兲 dx 苷 1 (b) 5
4兾2.5
11. (a) e
⬇ 0.20 (b) 1 e2兾2.5 ⬇ 0.55 (c) If you
aren’t served within 10 minutes, you get a free hamburger.
13. ⬇44%
15. (a) 0.0668
(b) ⬇5.21%
17. ⬇0.9545
19. (b) 0; a0
(c) 1x10 10
N
PAGE 584
5. (d)
7. (a) It must be either 0 or decreasing
(c) y 苷 0
(d) y 苷 1兾共x 2兲
9. (a) 0 P 4200
(b) P 4200
(c) P 苷 0, P 苷 4200
13. (a) III
(b) I
(c) IV
(d) II
15. (a) At the beginning; stays positive, but decreases
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 100
A100
(c)
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
17.
P(t)
M
2 c 2; 2, 0, 2
c=3
y
2
c=1
P(0)
0
_1
t
0
t
1
c=_1
_2
EXERCISES 9.2
N
1. (a)
PAGE 592
c=_3
y
2.0
(iv)
19. (a) (i) 1.4
1.5
1.0
0.5
(b)
_2
_1
(iii) 1.4641
(iii)
(ii)
Underestimates
y=´
h=0.1
h=0.2
h=0.4
1.5
1.4
1.3
1.2
(i)
0
(ii) 1.44
y
2 x
1
1.1
1.0
(b) y 苷 0.5, y 苷 1.5
0
3. III
7.
0.1
0.2
0.3
0.4
x
5. IV
9.
y
(c) (i) 0.0918 (ii) 0.0518 (iii) 0.0277
It appears that the error is also halved (approximately).
y
(b)
(a)
3
21. 1, 3, 6.5, 12.25
23. 1.7616
(ii) 2.3928 (iii) 2.3701 (iv) 2.3681
(c) (i) 0.6321 (ii) 0.0249 (iii) 0.0022
(iv) 0.0002
It appears that the error is also divided by 10 (approximately).
27. (a), (d)
(b) 3
(c) Yes; Q 苷 3
Q
6
(e) 2.77 C
25. (a) (i) 3
3 x
_3
_3
(c)
_3
0
3 x
_3
4
11.
13.
y
3
y
3
2
0
3x
_3
3x
_3
EXERCISES 9.3
_3
4 t
2
_3
N
PAGE 600
2
1. y 苷
,y苷0
K x2
ⱍ ⱍ
3
3x 3 ln x K
3. y 苷 s
5. 2 y 2 cos y 苷 2 x 2 4 x 4 C
1
1
15.
1
1 t 2
2
3
9. p 苷 Ke t 兾3t 1
7. e 共 y 1兲 苷 C e
y
4
11. y 苷 sx 2 9
13. u 苷 st 2 tan t 25
15. 2 y 2 3 共3 y 2 兲 3兾2 苷 2 x 2 ln x 4 x 2 1
_3
3
17. y 苷
1
1
41
12
4a
sin x a
s3
2
_2
1
19. y 苷 e x 兾2
21. y 苷 Ke x x 1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 101
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
23. (a) sin1y 苷 x 2 C
43. (a) C共t兲 苷 共C0 r兾k兲ekt r兾k
(b) y 苷 sin共x 兲, s兾2 x s兾2
2
(b) r兾k; the concentration approaches r兾k regardless of the
value of C0
45. (a) 15et兾100 kg
(b) 15e0.2 ⬇ 12.3 kg
47. About 4.9%
49. t兾k
(b) B 苷 KV 0.0794
51. (a) L 1 苷 KL 2k
(c) No
1
y=sin (≈)
53. (a) dA兾dt 苷 k sA 共M A兲
_œ„„„
_œ
π/2
A101
Ϲ/2
„„„
0
25. cos y 苷 cos x 1
where C 苷
5
(b) A共t兲 苷 M
sM sA0
and A0 苷 A共0兲
sM sA0
EXERCISES 9.4
N
冉
CesM kt 1
Ce sM kt 1
冊
2
,
PAGE 613
1. (a) 100; 0.05
(b) Where P is close to 0 or 100;
on the line P 苷 50; 0 P0 100; P0 100
(c)
P
2.5
27. (a)
2.5
P¸=140
P¸=120
1
(b) y 苷
Kx
P¸=80
P¸=60
0
y
3
150
100
50
P¸=40
P¸=20
0
0
_3
3 x
_3
29. y 苷 Cx
2
4
_4
4
_4
31. x 2 y 2 苷 C
≈-¥=C
4
20
Solutions approach 100; some increase and some decrease, some
have an inflection point but others don’t; solutions with P0 苷 20
and P0 苷 40 have inflection points at P 苷 50
(d) P 苷 0, P 苷 100; other solutions move away from P 苷 0 and
toward P 苷 100
10 7 kg (b) ⬇1.55 years
3. (a) 3.23
5. 9000
1
7. (a) dP兾dt 苷 265 P共1 P兾100兲, P in billions
(b) 5.49 billion
(c) In billions: 7.81, 27.72
(d) In billions: 5.48, 7.61, 22.41
y0
9. (a) dy兾dt 苷 ky共1 y兲
(b) y 苷
y0 共1 y0 兲ekt
(c) 3:36 PM
13. PE 共t兲 苷 1578.3共1.0933兲t 94,000;
32,658.5
PL共t兲 苷
94,000
1 12.75e0.1706t
xy=k
_4
60 t
40
130,000
4
PL
P
(in thousands)
PE
_4
33. y 苷 1 e 2x 兾2
35. y 苷
2
4t
37. Q共t兲 苷 3 3e
;3
kt
39. P共t兲 苷 M Me
4
41. (a) x 苷 a (kt 2兾sa ) 2
冉 冑
2
tan1
(b) t 苷
ksa b
( 12 x 2 2) 2
;M
冑 冊
b
tan1
ab
0
90,000
1960
bx
ab
冉
45
1980
t (year)
2000
冊
m
m kt
P0 e
(b) m kP0
k
k
(c) m 苷 kP0 , m kP0 (d) Declining
15. (a) P共t兲 苷
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 102
A102
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
17. (a) Fish are caught at a rate of 15 per week.
(c) P 苷 250, P 苷 750
0 P0 250: P l 0;
P0 苷 250: P l 250;
P0 250: P l 750
(b) See part (d).
(d)
P
1200
800
27. (a) I共t兲 苷 4 4e5t
(b) 4 4e1兾2 ⬇ 1.57 A
29. Q共t兲 苷 3共1 e4t 兲, I共t兲 苷 12e4t
31. P共t兲 苷 M Cekt
P(t)
M
400
P(0)
0
0
40
33. y 苷 5 共100 2t兲 40,000共100 2t兲3兾2; 0.2275 kg兾L
2
250 750ke t兾25
1 ke t兾25
1
where k 苷 11 , 19
(e) P共t兲 苷
P 1200
(c) 共mt兾c兲关t 共m兾c兲ect兾m 兴 m 2t兾c 2
M
37. (b) P共t兲 苷
1 MCe kt
35. (b) mt兾c
120
t
0
19. (b)
0 P0 200: P l 0;
P0 苷 200: P l 200;
P0 200: P l 1000
P
1400
1200
1000
800
600
400
200
0
(c) P共t兲 苷
20
40
60
100 t
80
m共M P0兲 M共P0 m兲e 共Mm兲共k兾M 兲t
M P0 共P0 m兲e 共Mm兲共k兾M 兲t
21. (a) P共t兲 苷 P0 e 共k兾r兲关sin共rt 兲 sin 兴
EXERCISES 9.5
1. Yes
N
7. y 苷 x 1 Ce
11. y 苷
1. (a) x 苷 predators, y 苷 prey; growth is restricted only by
predators, which feed only on prey.
(b) x 苷 prey, y 苷 predators; growth is restricted by carrying
capacity and by predators, which feed only on prey.
3. (a) Competition
(b) (i) x 苷 0, y 苷 0: zero populations
(ii) x 苷 0, y 苷 400: In the absence of an x-population, the
y-population stabilizes at 400.
(iii) x 苷 125, y 苷 0: In the absence of a y-population, the
x-population stabilizes at 125.
(iv) x 苷 50, y 苷 300: Both populations are stable.
5. (a) The rabbit population starts at about 300, increases to 2400,
then decreases back to 300. The fox population starts at 100,
decreases to about 20, increases to about 315, decreases to 100,
and the cycle starts again.
(b)
R
F
R
2000
5. y 苷 1 Ce
x
PAGE 627
N
2500
x
9. y 苷 sx C兾x
x sin共x 2兲 dx C
200
1000
100
500
t 2 2t 2C
13. u 苷
2共t 1兲
0
17. u 苷 t 2 t 3
7.
C=3
C=5
C=7
Species 2
t=2
150
C=3
C=5
C=7
5
t
t¡ t™ t£
200
共x 1兲e x C
x2
300
F
1500
2
3
sin x
1
1
3
15. y 苷 ln x 2
x
x
x
19. y 苷 x cos x x
21. y 苷
(b) Does not exist
EXERCISES 9.6
PAGE 620
3. No
t
120 t
80
t=3
t=1
100
t=4
C=1
_3
50
t=0, 5
3
0
C=_5
C=_3
C=_1
冉
25. y 苷 Cx 4 2
5x
冊
1兾2
_5
C=_5
C=_3
C=_1
50
100
150
200
250 Species 1
11. (a) Population stabilizes at 5000.
(b) (i) W 苷 0, R 苷 0: Zero populations
(ii) W 苷 0, R 苷 5000: In the absence of wolves, the rabbit
population is always 5000.
(iii) W 苷 64, R 苷 1000: Both populations are stable.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 103
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
(d)
(c) The populations stabilize at 1000 rabbits and 64 wolves.
(d)
W
R
(birds) y
250
35,000
80
W
1500
x (insects)
45,000
birds
200
150
25,000
60
insects
100
15,000
1000
40
R
500
0
50
5,000
20
0
t
t
25. (a) y 苷 共1兾k兲 cosh kx a 1兾k or
y 苷 共1兾k兲 cosh kx 共1兾k兲 cosh kb h
CHAPTER 9 REVIEW
N
PAGE 629
True-False Quiz
1. True
3. False
PROBLEMS PLUS
5. True
1. f 共x兲 苷 10e
7. True
(b) 共2兾k兲 sinh kb
PAGE 633
5. y 苷 x 1兾n
冉冊
7. 20 C
x
x L
12 L ln
4L
L
9. (b) f 共x兲 苷
1. (a)
N
x
2
Exercises
A103
2
(c) No
11. (a) 9.8 h
(b) 31,900 ft 2; 2000 ft 2兾h
(c) 5.1 h
13. x 2 共 y 6兲2 苷 25
15. y 苷 K兾x, K 苷 0
y
6
(iv) 4
(iii)
2
CHAPTER 10
(ii)
(i) 0
t
1
EXERCISES 10.1
1.
y
y共0.3兲 ⬇ 0.8
y
3
2
2
0
PAGE 641
3.
t=_2
(2, 6)
(b) 0 c 4; y 苷 0, y 苷 2, y 苷 4
3. (a)
N
_2
_1
0
1
2
t=2
(6, 2)
t=
t=0
(0, 0)
t=
2
5. (a)
3x
y
y苷
(12 x 2 C) esin x
0
9. r共t兲 苷 5e tt
2
11. y 苷 2 x 共ln x兲2 2x
1
13. x 苷 C 2 y 2
1
2000
15. (a) P共t兲 苷
; ⬇560
1 19e0.1t
(b) t 苷 10 ln 572 ⬇ 33.5
17. (a) L共t兲 苷 L 关L L共0兲兴ekt
(b) L共t兲 苷 53 43e0.2t
19. 15 days
21. k ln h h 苷 共R兾V 兲t C
23. (a) Stabilizes at 200,000
(b) (i) x 苷 0, y 苷 0: Zero populations
(ii) x 苷 200,000, y 苷 0: In the absence of birds, the insect
population is always 200,000.
(iii) x 苷 25,000, y 苷 175: Both populations are stable.
(c) The populations stabilize at 25,000 insects and 175 birds.
π
3
1
x
(b) y 苷 34 x 14
(7, 5)
t=_1
(3, 2)
t=0
x
(_1, _1)
t=1
7. y 苷 sln共x 2 2x 3兾2 C兲
t=
π
2
π
6
(0, 0)
0
x
(b) 0.75676
(c) y 苷 x and y 苷 x; there is a loc max or loc min
5.
t=0
(1, 1)
1
1
_3
y
(_5, _4)
t=2
7. (a)
(b) x 苷 共 y 2兲2 1,
4 y 0
y
(_3, 0)
t=2
(0, _1)
t=1
x
(1, _2)
t=0
(_3, _4)
t=_2
(0, _3)
t=_1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 104
A104
9. (a)
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
(b) y 苷 1 x 2, x
y
π
29.
0
(0, 1) t=0
(1, 0) t=1
x
0
_4
4
(2, _3) t=4
_π
11. (a) x 2 y 2 苷 1, y
0
(b)
_1
13. (a) y 苷 1兾x, y
(b)
1
31. (b) x 苷 2 5t, y 苷 7 8t, 0 t 1
33. (a) x 苷 2 cos t, y 苷 1 2 sin t, 0 t 2
y
1
1 x
0
(b) x 苷 2 cos t, y 苷 1 2 sin t, 0 t 6
(c) x 苷 2 cos t, y 苷 1 2 sin t, 兾2 t 3兾2
37. The curve y 苷 x 2兾3 is generated in (a). In (b), only the portion
with x 0 is generated, and in (c) we get only the portion with
x 0.
41. x 苷 a cos , y 苷 b sin ; 共x 2兾a 2 兲 共 y 2兾b 2 兲 苷 1, ellipse
43.
y
y
2a
x
O
(1, 1)
0
x
45. (a) Two points of intersection
4
15. (a) y 苷 ln x 1
1
2
(b)
y
6
1
0
6
x
1
4
17. (a) y 2 x 2 苷 1, y
(b)
(b) One collision point at 共3, 0兲 when t 苷 3兾2
(c) There are still two intersection points, but no collision point.
0, there is a loop whose size
47. For c 苷 0, there is a cusp; for c
increases as c increases.
1
y
3
1
1
_1
1
2
0
x
0
1
0
1.5
0
1.5
19. Moves counterclockwise along the circle
共x 3兲2 共 y 1兲2 苷 4 from 共3, 3兲 to 共3, 1兲
21. Moves 3 times clockwise around the ellipse
共x 2兾25兲 共 y 2兾4兲 苷 1, starting and ending at 共0, 2兲
23. It is contained in the rectangle described by 1 x 4
and 2 y 3.
25.
27.
y
y
1
_3
_1
49. The curves roughly follow the line y 苷 x, and they start having
loops when a is between 1.4 and 1.6 (more precisely, when
a s2 ). The loops increase in size as a increases.
51. As n increases, the number of oscillations increases;
a and b determine the width and height.
1
t= 2
(0, 1) t=1
(_1, 0)
t=0
x
(0, _1) t=_1
t=0
EXERCISES 10.2
1
x
N
2t 1
t cos t sin t
7. y 苷 2x 1
1.
PAGE 651
3. y 苷 2 x 7
3
5. y 苷 x 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA096-105.qk_97909_Ans_Ans_pA096-105 10/7/10 2:39 PM Page 105
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
9. y 苷 6 x
1
57.
20
59.
61.
65.
_10
15.
17.
19.
2
1215
24
5
(247 s13 64)
(949 s26 1)
63. 5 a 2
6
71.
1
4
10
_2
11.
x0兾2 2 t cos t st 2 1 dt ⬇ 4.7394
x01 2 共t 2 1兲e tse 2t 共t 1兲 2共t 2 2t 2兲 dt ⬇ 103.5999
EXERCISES 10.3
2t 1
1
, 3, t 0
13. e 2t共1 t兲, e 3t共2t 3兲, t
2t
4t
32 tan t, 34 sec 3 t, 兾2 t 3兾2
Horizontal at 共0, 3兲, vertical at 共2, 2兲
Horizontal at ( 12 , 1) and ( 12 , 1), no vertical
21. 共0.6, 2兲; (5 ⴢ 6 6兾5, e 6
23.
7.5
1兾5
)
25. y 苷 x, y 苷 x
3
2
PAGE 662
N
1. (a)
π
”2, 3 ’
(b)
π
3
O
O
共2, 7兾3兲, 共2, 4兾3兲
(c)
y
_ 3π
”1, _ 3π ’
4
4
共1, 5兾4兲, 共1, 兾4兲
π
2
8.5
3
0
x
O
1
π
”_1, 2 ’
27. (a) d sin 兾共r d cos 兲
29. ( 27, 9 ), 共2, 4兲
31. ab
33. 3 e
35. 2r 2 d 2
37. x02 s2 2e 2t dt ⬇ 3.1416
共1, 3 兾2兲, 共1, 5兾2兲
16 29
39.
x04 s5 4 cos t
dt ⬇ 26.7298
43. s2 ln (1 s2 )
45. s2 共e 1兲
1
2
3. (a)
(b)
π
(1, π)
41. 4s2 2
O
O
_ 2π
1
2
3
”2, _ 2π ’
3
8
(1, s3 )
共1, 0兲
(c)
3π
4
25
47. 16.7102
0
2.5
O
1.4
3π
”_2, 4 ’
(s2, s2 )
_2.1
2.1
5. (a) (i) (2s2, 7兾4)
(b) (i) 共2, 2兾3兲
(ii) (2s2, 3兾4)
(ii) 共2, 5兾3兲
7.
_1.4
r=1
49. 612.3053
55. (a)
51. 6 s2, s2
t 僆 关0, 4兴
15
15
15
O
9.
¨=
15
(b) 294
3π
4
π
¨=
4
O
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A105
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 106
A106
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
11.
¨=
49
7π
3
51.
x=1
r=3
r=2
(2, 0)
(6, 0)
O
O
¨=
13.
17.
19.
21.
25.
29.
5π
3
53. (a) For c ⬍ ⫺1, the inner loop begins at ␪ sin⫺1 共⫺1兾c兲 and
ends at ␪ ␲ ⫺ sin⫺1 共⫺1/c兲; for c ⬎ 1, it begins at
␪ ␲ ⫹ sin⫺1 共1兾c兲 and ends at ␪ 2␲ ⫺ sin⫺1 共1兾c兲.
2s3
15. Circle, center O, radius s5
Circle, center 共1, 0兲, radius 1
Hyperbola, center O, foci on x-axis
r 2 csc ␪
23. r 1兾共sin ␪ ⫺ 3 cos ␪ 兲
27. (a) ␪ ␲兾6
r 2c cos ␪
(b) x 3
55. s3
57. ⫺␲
59. 1
61. Horizontal at (3兾s2, ␲兾4), (⫺3兾s2, 3␲兾4);
vertical at 共3, 0兲, 共0, ␲兾2兲
3
3
63. Horizontal at ( 2 , ␲兾3), 共0, ␲兲 [the pole], and ( 2 , 5␲兾3);
31.
vertical at (2, 0), ( 2 , 2␲兾3), ( 2 , 4␲兾3)
1
O
(4, 0)
1
65. Center 共b兾2, a兾2兲, radius sa 2 ⫹ b 2兾2
O
67.
69.
2.6
3.5
(2, 3π/2)
_3.4
33.
35.
1.8
_3
3
π
¨=
3
5
2
π
”4, 6 ’
(2π, 2π)
O
_2.5
_2.6
1
6
71.
3
37.
¨=
39.
π
8
¨=
4
5π
6
¨=
(2, 0)
π
6
73. By counterclockwise rotation through angle ␲兾6, ␲兾3,
or ␣ about the origin
O
75. For c 0, the curve is a circle. As c increases, the left side gets
flatter, then has a dimple for 0.5 ⬍ c ⬍ 1, a cusp for c 1, and a
loop for c ⬎ 1.
41.
43.
(3, π/6)
(3, π/4)
EXERCISES 10.4
⫺␲ 兾4
1. e
9. ␲
O
O
⫺e
N
PAGE 668
⫺␲ 兾2
3.
5. ␲ 2
7.
11. 11␲
9
2
(2, π/2)
41
4
␲
(3, π/2)
(1, π)
(5, 0)
O
45.
¨=
2π
3
π
¨= 3
47.
r=2 sin ¨
13. 2␲
9
1
(3, π)
(3, 3π/2)
O
2
3
(3, 0)
O
1
_4
4
_3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 107
APPENDIX I
15. 2␲
3
ANSWERS TO ODD-NUMBERED EXERCISES
11. 共0, ⫾2兲, (0, ⫾s2 )
1.4
13. 共⫾3, 0兲, (⫾2s2 , 0)
y
2
_2.1
2.1
y
1
2
œ„
2œ„2
_2œ„2
0
_3
_1.4
17. 3␲
4
19.
1
16
␲
33. 1 ⫺ 2 s2
1
37.
35.
1
4
29.
5
24
1
␲ ⫺ 14 s3
1
31. 2 ␲ ⫺ 1
1
(␲ ⫹ 3 s3 )
_1
15. 共1, ⫾3兲, (1, ⫾s5 )
( 32, ␲兾6), ( 32, 5␲兾6), and the pole
y
and 共⫺1, ␪ 兲 where ␪ 7␲兾12, 11␲兾12, 19␲兾12, 23␲兾12
1
1
41. ( 2 s3, ␲兾3), ( 2 s3, 2␲兾3), and the pole
43. Intersection at ␪ ⬇ 0.89, 2.25; area ⬇ 3.46
8
45. 2␲
47. 3 关共␲ 2 ⫹ 1兲3兾2 ⫺ 1兴
16
3
x
_2
39. 共1, ␪ 兲 where ␪ ␲兾12, 5␲兾12, 13␲兾12, 17␲兾12
49.
2
œ„
3 x
2
_œ„
23. 3 ␲ ⫹ 2 s3
3
27. ␲
25. 4 s3 ⫺ 43 ␲
0
2
_œ„
21. ␲ ⫺ 2 s3
A107
1
17.
x2
y2
⫹
1, foci (0, ⫾s5 )
4
9
(1, 3)
⫺1 0
3 x
(1,_3)
19. 共0, ⫾5兲; (0, ⫾s34 ); y ⫾ 3 x
5
y
y=53 x
{0, œ„„
34}
_0.75
1.25
(3, 5)
(0, 5)
x
_1
(0, _5)
51. 2.4221
53. 8.0091
55. (b) 2␲ (2 ⫺ s2 )
y=_ 53x
{0, _ œ„„
34}
EXERCISES 10.5
1. 共0, 0兲, (0,
3
2
N
PAGE 676
), y ⫺ 32
3. 共0, 0兲, (⫺2,, 0), x 1
y
2
y
6
1
2
21. 共⫾10, 0兲, (⫾10 s2 , 0), y ⫾x
y
y=x
(10, 10)
x=21
{_10œ„
2, 0}
(_10, 0)
(10, 0)
{10œ„
2, 0}
x
”0, 32’
(_1/2, 0)
6 x
x
_2
y=_x
y=-32
23. 共4, ⫺2兲, 共2, ⫺2兲;
y
(3⫾s5, ⫺2);
5. 共⫺2, 3兲, 共⫺2, 5兲, y 1
7. 共⫺2, ⫺1兲, 共⫺5, ⫺1兲, x 1
y
y ⫹ 2 ⫾2共x ⫺ 3兲
0
y
(2, _2)
(_2, 5)
(3-œ„5, _2)
0
(_5, _1)
y=1
x
(4, _2)
(3+œ„5, _2)
x
(_2, _1)
x
25. Parabola, 共0, ⫺1兲, (0, ⫺4 )
27. Ellipse, (⫾s2, 1), 共⫾1, 1兲
29. Hyperbola, 共0, 1兲, 共0, ⫺3兲; (0, ⫺1 ⫾ s5 )
3
x=1
9. x ⫺y 2, focus (⫺ 4 , 0), directrix x 1
1
4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 108
A108
31. y 2 4x
41.
45.
49.
51.
55.
59.
61.
63.
35. y ⫺ 3 2共x ⫺ 2兲2
33. y 2 ⫺12共x ⫹ 1兲
x
x
y
共 y ⫺ 4兲
⫹
1
⫹
1
39.
25
21
12
16
2
2
2
x
共x ⫹ 1兲
共 y ⫺ 4兲
y2
⫹
1
⫺
1
43.
12
16
9
16
2
2
2
2
x
共 y ⫺ 1兲
共x ⫹ 3兲
y
⫺
1
⫺
1
47.
25
39
9
36
2
2
x
y
⫹
1
3,763,600
3,753,196
121x 2
121y 2
(a)
⫺
1 (b) ⬇248 mi
1,500,625
3,339,375
(a) Ellipse (b) Hyperbola
(c) No curve
15.9
b 2c
a
⫹ ab ln
where c 2 a 2 ⫹ b 2
a
b⫹c
共0, 4兾␲ 兲
2
37.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
2
2
2
15. (a) 2
(d)
3
1
”- 4 , 0’
” 4 , π’
O
3
x=_ 8
17. (a) 2, y ⫺ 2
1
冉 冊
EXERCISES 10.6
N
(c) x ⫺ 38
(b) Hyperbola
1
_2
2
1
y=-2
_3
PAGE 684
6
4
3. r 2 ⫹ cos ␪
2 ⫹ 3 sin ␪
8
4
5. r 7. r 1 ⫺ sin ␪
2 ⫹ cos ␪
4
(b) Ellipse
(c) y ⫺1
9. (a) 5
(d)
y
1. r 1
(b) r 1 ⫺ 2 sin共␪ ⫺ 3␲兾4兲
2
_2
2
(4, π/2)
_2
” 45 , π’
19. The ellipse is nearly circular
when e is close to 0 and becomes
more elongated as e l 1⫺. At
e 1, the curve becomes a
parabola.
” 45 , 0’
O
x
4 3π
”9, 2 ’
11. (a) 1
y=_1
(c) y 23
(b) Parabola
(d)
y
e=0.4
e=1.0
e=0.6
e=0.8
2.26 ⫻ 10 8
1 ⫹ 0.093 cos ␪
27. 35.64 AU
29. 7.0 ⫻ 10 7 km
25. r 31. 3.6 ⫻ 10 8 km
y=2/3
”31 , π
2’
” 23, π’
CHAPTER 10 REVIEW
” 23, 0’
O
N
True-False Quiz
1. False
3. False
x
PAGE 685
5. True
7. False
9. True
Exercises
1. x y 2 ⫺ 8y ⫹ 12
13. (a)
1
3
(b) Ellipse
(d)
(c) x 9
2
3. y 1兾x
y
y
(0, 6), t=_4
9
x= 2
3 π
” 2, 2 ’
9
9
” 4 , π’
” 8 , 0’
(5, 1),
t=1
(1, 1), ¨=0
O
x
x
3 3π
” 2, 2 ’
5. x t, y st ; x t 4, y t 2;
x tan 2 t, y tan t, 0 艋 t ⬍ ␲兾2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 109
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
7. (a)
43. All curves have the vertical asymptote x 1. For c ⬍ ⫺1, the
curve bulges to the right. At c ⫺1, the curve is the line x 1.
For ⫺1 ⬍ c ⬍ 0, it bulges to the left. At c 0 there is a cusp at
(0, 0). For c ⬎ 0, there is a loop.
(b) (3s2, 3␲兾4),
(⫺3s2, 7␲兾4)
”4, 2π ’
3
A109
2π
3
O
47. (⫺ 24 , 3), 共⫺1, 3兲
45. 共⫾1, 0兲, 共⫾3, 0兲
25
y
y
2 œ„2
(1, 0)
(⫺2, 2s3 )
⫺3
9.
0
11.
(_1, 3)
x
3
⫺2œ„2
¨=
π
”1, 2 ’
0
x
π
6
49.
(1, 0)
(2, π)
O
y2
x2
⫹
1
25
9
y2
x2
⫺
1
72兾5
8兾5
51.
共8y ⫺ 399兲2
4
x2
55. r ⫹
1
25
160,801
3 ⫹ cos ␪
3 3
57. (a) At (0, 0) and ( 2 , 2 )
3
3
(b) Horizontal tangents at (0, 0) and (s
2, s
4 );
3
3
vertical tangents at (0, 0) and (s4, s2 )
(g) 32
(d)
y
53.
3π
”1, 2 ’
13.
15.
”_3, 3π
’
2
1
(2, π)
(2, 0)
3
y= 2
O
_1
”1, π
’
2
17. r 2
cos ␪ ⫹ sin ␪
19.
x
y ⫽ ⫺x ⫺ 1
O
0.75
r= sin ¨
¨
-0.3
1.2
PROBLEMS PLUS
N
1. ln共␲兾2兲
[
PAGE 688
]
3. ⫺4 s3, 4 s3 ⫻ 关⫺1, 2兴
3
3
CHAPTER 11
-0.75
21. 2
25.
EXERCISES 11.1
23. ⫺1
1 ⫹ sin t 1 ⫹ cos t ⫹ sin t
,
1 ⫹ cos t
共1 ⫹ cos t兲3
27.
1. (a) A sequence is an ordered list of numbers. It can also be
defined as a function whose domain is the set of positive integers.
(b) The terms an approach 8 as n becomes large.
(c) The terms an become large as n becomes large.
y
( 32 a, ⫾ 12 s3 a), 共⫺3a, 0兲;
horizontal tangent at
1
3
共a, 0兲, (⫺2 a, ⫾ 2 s3 a)
PAGE 700
Abbreviations: C, convergent; D, divergent
( 118 , 34 )
29. Vertical tangent at
N
4 3
8
5. 5, ⫺ 25, 125, ⫺ 625, 3125
5
1
3. 1, 5, 5, 17 , 13
(⫺3a, 0)
(a, 0)
x
0
9. 1, 2, 7, 32, 157
33. 共2, ⫾␲兾3兲
37. 2(5 s5 ⫺ 1)
39.
35. 2 共␲ ⫺ 1兲
1
冉
2 s␲ 2 ⫹ 1 ⫺ s4␲ 2 ⫹ 1
2␲ ⫹ s4␲ 2 ⫹ 1
⫹ ln
2␲
␲ ⫹ s␲ 2 ⫹ 1
41. 471,295␲兾1024
冊
1
1
2 2 2 2
3 5 7 9
11. 2, , , ,
1
1 1
1
1
1
7. 2, 6, 24, 120 , 720
13. a n 1兾共2n ⫺ 1兲
n2
n⫹1
19. 0.4286, 0.4615, 0.4737, 0.4800, 0.4839, 0.4865, 0.4884,
1
0.4898, 0.4909, 0.4918; yes; 2
21. 0.5000, 1.2500, 0.8750, 1.0625, 0.9688, 1.0156, 0.9922,
1.0039, 0.9980, 1.0010; yes; 1
23. 1
25. 5
27. 1
29. 1
31. D
33. 0
35. D
37. 0
39. 0
41. 0
43. 0
45. 1
47. e 2
49. ln 2
51. ␲兾2
53. D
55. D
1
57. 1
59. 2
61. D
63. 0
65. (a) 1060, 1123.60, 1191.02, 1262.48, 1338.23
(b) D
2 n⫺1
15. a n ⫺3(⫺ 3 )
31. 18
1
17. a n 共⫺1兲 n⫹1
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 110
A110
67.
69.
71.
73.
77.
79.
85.
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
(a) Pn 1.08Pn⫺1 ⫺ 300
(b) 5734
⫺1 ⬍ r ⬍ 1
Convergent by the Monotonic Sequence Theorem; 5 艋 L ⬍ 8
Decreasing; yes
75. Not monotonic; no
Decreasing; yes
1
1
2
81. 2 (3 ⫹ s5 )
83. (b) 2 (1 ⫹ s5 )
(a) 0
(b) 9, 11
EXERCISES 11.2
N
1. (a) A sequence is an ordered list of numbers whereas a series is
the sum of a list of numbers.
(b) A series is convergent if the sequence of partial sums is a convergent sequence. A series is divergent if it is not convergent.
3. 2
5. 1, 1.125, 1.1620, 1.1777, 1.1857, 1.1903, 1.1932, 1.1952; C
7. 0.5, 1.3284, 2.4265, 3.7598, 5.3049, 7.0443, 8.9644, 11.0540; D
(b) 5
(b) 157.895 mg
73.
1
2
(s3 ⫺ 1)
1
79. The series is divergent.
n共n ⫹ 1兲
85. 兵sn 其 is bounded and increasing.
77.
1
2
5
23
1
2
7
1
8
共n ⫹ 1兲! ⫺ 1
共n ⫹ 1兲!
119
89. (a) 2 , 6 , 24 , 120 ;
EXERCISES 11.3
1. C
N
(c) 1
PAGE 720
y
y=
a™
1
x 1.3
a£
10
0
ssn d
2
1
a¢
3
a∞
...
4
x
3. D
5. C
7. D
9. C
11. C
13. D
15. C
17. C
19. C
21. D
23. C
25. C
27. f is neither positive nor decreasing.
29. p ⬎ 1
31. p ⬍ ⫺1
33. 共1, ⬁兲
9
35. (a) 10␲ 4
(b) 901 ␲ 4 ⫺ 17
16
37. (a) 1.54977, error 艋 0.1
(b) 1.64522, error 艋 0.005
10
(c) 1.64522 compared to 1.64493
39. 0.00145
45. b ⬍ 1兾e
(d) n ⬎ 1000
ssn d
EXERCISES 11.4
san d
11
0
1
N
PAGE 726
1. (a) Nothing
(b) C
3. C
5. D
7. C
9. D
11. C
13. C
15. D
17. D
19. D
21. C
23. C
25. D
27. C
29. C
31. D
33. 1.249, error ⬍ 0.1
35. 0.0739, error ⬍ 6.4 ⫻ 10 ⫺8
45. Yes
ssn d
EXERCISES 11.5
san d
11
0
25
1
(a) C
(b) D
17. D
19. 3
21. 60
23. 7
5
D
27. D
29. D
31. 2
33. D
35. D
3
11
D
39. D
41. e兾共e ⫺ 1兲
43. 2
45. 6
47. e ⫺ 1
(b) 1
(c) 2
(d) All rational numbers with a terminating
decimal representation, except 0.
8
838
51. 9
53. 333
55. 5063兾3300
1
1 ⫺5x
3
57. ⫺ ⬍ x ⬍ ;
59. ⫺1 ⬍ x ⬍ 5;
5
5 1 ⫹ 5x
5⫺x
x
1
61. x ⬎ 2 or x ⬍ ⫺2;
63. x ⬍ 0;
x⫺2
1 ⫺ ex
2
65. 1
67. a1 0, an for n ⬎ 1, sum 1
n共n ⫹ 1兲
15.
25.
37.
49.
D共1 ⫺ c n 兲
1⫺c
71. (a) Sn san d
0
1.98637, 2.88080,
3.80927, 4.75796,
5.71948, 6.68962,
7.66581, 8.64639;
divergent
13. 0.29289, 0.42265,
0.50000, 0.55279,
0.59175, 0.62204,
0.64645, 0.66667,
0.68377, 0.69849;
convergent, sum 1
共1 ⫺ 0.05 n兲
1
_3
11. 0.44721, 1.15432,
3000
19
87. (a) 0, 9 , 9 , 3 , 3 , 9 , 9 , 1
PAGE 711
9. ⫺2.40000, ⫺1.92000,
⫺2.01600, ⫺1.99680,
⫺2.00064, ⫺1.99987,
⫺2.00003, ⫺1.99999,
⫺2.00000, ⫺2.00000;
convergent, sum ⫺2
69. (a) 157.875 mg;
N
PAGE 731
1. (a) A series whose terms are alternately positive and
negative
(b) 0 ⬍ bn⫹1 艋 bn and lim n l ⬁ bn 0,
where bn an
(c) Rn 艋 bn⫹1
3. C
5. C
7. D
9. C
11. C
13. D
15. C
17. C
19. D
21. ⫺0.5507
23. 5
25. 4
27. ⫺0.4597
29. 0.0676
31. An underestimate
33. p is not a negative integer
35. 兵bn 其 is not decreasing
ⱍ ⱍ
EXERCISES 11.6
N
ⱍ ⱍ
PAGE 737
Abbreviations: AC, absolutely convergent;
CC, conditionally convergent
1. (a) D
3. AC
15. AC
27. AC
(b) C
5. CC
17. CC
29. D
(c) May converge or diverge
7. AC
9. D
11. AC
13. AC
19. AC
21. AC
23. D
25. AC
31. D
33. AC
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 111
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
⬁
35. (a) and (d)
661
39. (a) 960 ⬇ 0.68854, error ⬍ 0.00521
19.
(b) n 艌 11, 0.693109
⬁
共⫺1兲 n ⬁ 共⫺1兲 n⫺1
45. (b) 兺
; 兺
n
n2 n ln n n1
21.
兺 共2n ⫹ 1兲x , R 1
n
n0
⬁
兺 共⫺1兲
n
n0
1
x 2n⫹1, R 4
16n⫹1
s£
0.25
EXERCISES 11.7
1. C
13. C
25. C
33. C
N
EXERCISES 11.8
s¡
PAGE 740
3. D
15. C
27. C
35. D
4
s™
s¢
f
s∞
s¡
1. A series of the form 冘⬁n0 cn共x ⫺ a兲n, where x is a variable
13. 4, 共⫺4, 4兴
25. ,
[ , 1]
31. k k
1
23.
]
13
兺
n0
2x 2n⫹1
,R1
2n ⫹ 1
s£
1
29. (a) Yes
⫺2
2
(b) No
33. No
⫺3
(b), (c)
s¸ s™ s¢
2
8
41. 2
EXERCISES 11.10
J¡
_8
37. 共⫺1, 1兲, f 共x兲 共1 ⫹ 2x兲兾共1 ⫺ x 2 兲
EXERCISES 11.9
⬁
1. 10
1.
PAGE 751
兺 共⫺1兲 x , 共⫺1, 1兲
n n
3.
⬁
5. 2
n0
⬁
兺 共⫺1兲
1
n
9 n⫹1
冋
n0
⬁
兺
共⫺1兲
n0
n⫹1
x
⫺
2 n⫹1
册
n0
x , 共⫺1, 1兲
n
⬁
n
n
n0
(b)
兺 共⫺1兲 共n ⫹ 2兲共n ⫹ 1兲x , R 1
n
n
n0
兺 共⫺1兲 n共n ⫺ 1兲x , R 1
n
n
n2
⬁
xn
,R5
n5 n
兺
n1
⬁
兺 共⫺1兲 4 共n ⫹ 1兲x
n
n0
n
b 8 f 共5兲兾8!
兺 x , 共⫺1, 1兲
n
, R 14
兺 共n ⫹ 1兲x , R 1
n
n0
n
7.
兺 共⫺1兲
␲ 2n⫹1
x 2n⫹1, R ⬁
共2n ⫹ 1兲!
n
n0
⬁
⬁
共ln 2兲 n n
x 2n⫹1
x ,R⬁
,R⬁
11. 兺
n!
n0
n0 共2n ⫹ 1兲!
2
3
13. ⫺1 ⫺ 2共x ⫺ 1兲 ⫹ 3共x ⫺ 1兲 ⫹ 4共x ⫺ 1兲 ⫹ 共x ⫺ 1兲4, R ⬁
⬁
1
15. ln 2 ⫹ 兺 共⫺1兲 n⫹1
共x ⫺ 2兲 n, R 2
n 2n
n1
9.
兺
⬁
17.
兺
n0
2 ne 6
共x ⫺ 3兲 n, R ⬁
n!
⬁
19.
兺 共⫺1兲
n⫹1
n0
25. 1 ⫺
n⫹1
⬁
3.
兺 共n ⫹ 1兲x , R 1
⬁
⬁
PAGE 765
n0
⬁
15. ln 5 ⫺
17.
5.
⬁
1
2
1
(c)
2
3
n⫹1
x n, 共⫺3, 3兲
N
共8兲
⬁
1
n1
兺 共⫺1兲 共n ⫹ 1兲x , R 1
13. (a)
兺
9. 1 ⫹ 2
, 共⫺3, 3兲
2n⫹1
1
兺
s¡ s£ s∞
_2
N
⬁
t 8n⫹2
,R1
n0 8n ⫹ 2
⬁
x n⫹3
,R1
27. C ⫹ 兺 共⫺1兲 n
n共n ⫹ 3兲
n1
31. 0.000983
33. 0.19740
29. 0.199989
35. (b) 0.920
39. 关⫺1, 1兴, 关⫺1, 1兲, 共⫺1, 1兲
25. C ⫹
11.
s¡
f
35. (a) 共⫺⬁, ⬁兲
7.
s™
3
11
23. 0, { 2 }
21. b, 共a ⫺ b, a ⫹ b兲
27. ⬁, 共⫺⬁, ⬁兲
3
5
[
1 1
17. 3 , ⫺ 3 , ⫺ 3 )
15. 1, 关1, 3兴
19. ⬁, 共⫺⬁, ⬁兲
1
5
1
_0.25
s£
⬁
[
11. 3, ⫺ 3, 3
s¢
s™
_4
PAGE 745
and a and the cn’s are constants
3. 1, 共⫺1, 1兲
5. 1, 关⫺1, 1兲
7. ⬁, 共⫺⬁, ⬁兲
9. 2, 共⫺2, 2兲
s∞
f
5. C
7. D
9. C
11. C
17. C
19. C
21. D
23. D
29. C
31. D
37. C
N
A111
⬁
3 ⴢ 7 ⴢ ⭈ ⭈ ⭈ ⴢ 共4n ⫺ 5兲 n
1
x⫺ 兺
x ,R1
4
4 n ⴢ n!
n2
⬁
27.
兺 共⫺1兲
n
n0
1
共x ⫺ ␲兲2n, R ⬁
共2n兲!
共n ⫹ 1兲共n ⫹ 2兲 n
x ,R2
2 n⫹4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 112
A112
⬁
29.
兺 共⫺1兲
n
n0
⬁
31.
兺
n0
␲ 2n⫹1
x 2n⫹1, R ⬁
共2n ⫹ 1兲!
EXERCISES 11.11
兺 共⫺1兲
n
n0
N
PAGE 774
1. (a) T0 共x兲 1 T1共x兲, T2共x兲 1 ⫺ 2 x 2 T3共x兲,
1
1
1
T4共x兲 1 ⫺ 2 x 2 ⫹ 24 x 4 T5共x兲,
2n ⫹ 1 n
x ,R⬁
n!
⬁
33.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
1
1
1
T6共x兲 1 ⫺ 2 x 2 ⫹ 24 x 4 ⫺ 720 x 6
T¢=T∞
1
x 4n⫹1 , R ⬁
2 共2n兲!
2
T¸=T¡
2n
35.
⬁
1
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲 2n⫹1
x ⫹ 兺 共⫺1兲n
x
,R2
2
n! 2 3n⫹1
n1
37.
兺 共⫺1兲
⬁
f
_2π
2π
2 2n⫺1 2n
x ,R⬁
共2n兲!
n⫹1
n1
_2
Tß
⬁
1
x 4n, R ⬁
39. 兺 共⫺1兲n
共2n兲!
n0
(b)
T¸=T¡=T™=T£
_1.5
1.5
Tˆ=T˜=T¡¸=T¡¡
T2 T3
T4 T5
0.7071
1
0.6916
0.7074
0.7071
0
1
⫺0.2337
0.0200
⫺0.0009
⫺1
1
⫺3.9348
0.1239
⫺1.2114
T6
(c) As n increases, Tn共x兲 is a good approximation to f 共x兲 on a
larger and larger interval.
f
3.
T¢=T∞=Tß=T¶
_1.5
T0 T1
x
␲
4
␲
2
␲
1.5
f
T™=T£
1
2
⫺ 14 共x ⫺ 2兲 ⫹ 18 共x ⫺ 2兲 2 ⫺ 161 共x ⫺ 2兲3
2
⬁
41.
共⫺1兲
x n, R ⬁
共n ⫺ 1兲!
n⫺1
兺
n1
f
T£
6
T£
T∞
T¡
冉 冊 冉 冊
4
T¡
T™
T™
5. ⫺ x ⫺
T¢ Tß
␲
2
⫹
f
T£
1
␲
x⫺
6
2
f
T£
45. (a) 1 ⫹
0
⬁
兺
n1
(b) x ⫹
⬁
兺
n1
T£
f
_1.1
7. 共x ⫺ 1兲 ⫺ 2 共x ⫺ 1兲 2 ⫹ 3 共x ⫺ 1兲 3
1
⬁
x 6n⫹2
47. C ⫹ 兺 共⫺1兲 n
,R⬁
共6n ⫹ 2兲共2n兲!
n0
1
2
T£
⬁
1
49. C ⫹ 兺 共⫺1兲 n
x 2n, R ⬁
2n 共2n兲!
n1
1
53. 0.40102
55. 2
51. 0.0059
3
8
65. ln 5
25
24
x4
67. 1兾s2
61. 1 ⫹ 6 x 2 ⫹
1
69. e 3 ⫺ 1
π
π
2
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲 2n
x
2 n n!
1 ⴢ 3 ⴢ 5 ⴢ ⭈ ⭈ ⭈ ⴢ 共2n ⫺ 1兲 2n⫹1
x
共2n ⫹ 1兲2 n n!
59. 1 ⫺ 2 x 2 ⫹
3
1.1
_6
T¢ T∞ Tß
43. 0.99619
4
0
f
_3
f
_1
57.
7
360
x4
3
1
120
63. e ⫺x
4
_4
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 113
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
A113
⬁
1
1 ⴢ 5 ⴢ 9 ⴢ ⭈ ⭈ ⭈ ⴢ 共4n ⫺ 3兲 n
x , R 16
⫹ 兺
2
n! 2 6n⫹1
n1
⬁
xn
55. C ⫹ ln x ⫹ 兺
n1 n ⴢ n!
9. x ⫺ 2x 2 ⫹ 2x 3
53.
3
ⱍ ⱍ
_1
1.5
57. (a) 1 ⫹ 2 共x ⫺ 1兲 ⫺ 8 共x ⫺ 1兲2 ⫹
1
(b)
T£ f
1
1
16
共x ⫺ 1兲3
(c) 0.000006
1.5
_4
冉 冊 冉 冊 冉 冊
冉 冊 冉 冊
␲
4
11. T5 共x兲 1 ⫺ 2 x ⫺
␲
4
⫹2 x⫺
8
␲
x⫺
3
4
2
⫺
10
␲
⫹
x⫺
3
4
4
T£
3
64
␲
⫺
x⫺
15
4
5
f
2
0
59. ⫺
1
6
T¢ T∞
5
T£
T™
PROBLEMS PLUS
f
0
T™
f
T∞
T£
1
11. ln 2
_2
13. (a) 2 ⫹ 4 共x ⫺ 4兲 ⫺
1
1
64
1
9
(b) 1.5625 ⫻ 10⫺5
共x ⫺ 4兲2
(a) 1 ⫹ 共x ⫺ 1兲 ⫺ 共x ⫺ 1兲 ⫹ 共x ⫺ 1兲 3
(b) 0.000097
(a) 1 ⫹ x 2 (b) 0.0014
1
(a) 1 ⫹ x 2
(b) 0.00006
21. (a) x 2 ⫺ 6 x 4
(b) 0.042
0.17365
25. Four
27. ⫺1.037 ⬍ x ⬍ 1.037
⫺0.86 ⬍ x ⬍ 0.86
31. 21 m, no
(c) They differ by about 8 ⫻ 10⫺9 km.
2
3
1
2
15.
17.
19.
23.
29.
37.
4
81
2
x 3 ⫹ 4x 2 ⫹ x
共1 ⫺ x兲4
9. 共⫺1, 1兲,
2
π
4
13. (a)
␲
19.
⫺1
2s3
冉
21. ⫺
N
True-False Quiz
1. False
3. True
11. True
13. True
19. True
21. True
5. False
15. False
7. False
9. False
17. True
250
101
␲ 共e⫺共n⫺1兲␲兾5 ⫺ e⫺n␲兾5 兲
1
27.
N
PAGE 790
3. C; A
5. A vertical plane that
intersects the xy-plane in
the line y 2 ⫺ x, z 0
z
y=2-x
31. e⫺e
37. 0.18976224, error ⬍ 6.4 ⫻ 10
41. 4, 关⫺6, 2兲
1
2
兺 共⫺1兲
n
n0
⬁
47.
43. 0.5, [2.5, 3.5)
⬁
兺 共⫺1兲 x
冋 冉 冊
1
␲
x⫺
共2n兲!
6
,R1
n n⫹2
2n
51.
兺 共⫺1兲
n
n0
冉 冊 册
␲
s3
x⫺
共2n ⫹ 1兲!
6
49. ln 4 ⫺
n0
⬁
⫹
⬁
兺
n1
8n⫹4
x
,R⬁
共2n ⫹ 1兲!
x
ⱍ
ⱍ
ⱍ
ⱍ
y
2
ⱍ
ⱍ
7. (a) PQ 6 , QR 2 s10 , RP 6 ; isosceles triangle
9. (a) No
(b) Yes
11. 共x ⫹ 3兲 2 ⫹ 共 y ⫺ 2兲 2 ⫹ 共z ⫺ 5兲 2 16;
35. 0.9721
⫺7
45.
y=2-x, z=0
2
29. ␲兾4
1
11
␲
where k is a positive integer
0
3. D
5. 0
7. e 12
9. 2
11. C
13. C
17. C
19. C
21. C
23. CC
25. AC
250
101
2
Exercises
1. 2
15. D
(b)
CHAPTER 12
1. 共4, 0, ⫺3兲
PAGE 778
冊
␲
⫺ ␲k
2
EXERCISES 12.1
CHAPTER 11 REVIEW
PAGE 781
1. 15!兾5! 10,897,286,400
3. (b) 0 if x 0, 共1兾x兲 ⫺ cot x if x k␲, k an integer
2
5. (a) sn 3 ⴢ 4 n, ln 1兾3 n, pn 4 n兾3 n⫺1
(c) 5 s3
T¢
π
2
N
2n⫹1
xn
,R4
n 4n
共 y ⫺ 2兲 2 ⫹ 共z ⫺ 5兲 2 7, x 0 (a circle)
13. 共x ⫺ 3兲2 ⫹ 共 y ⫺ 8兲2 ⫹ 共z ⫺ 1兲2 30
15. 共1, 2, ⫺4兲, 6
17. 共2, 0, ⫺6兲, 9兾s2
5 1
1
19. (b) 2 , 2 s94 , 2 s85
21. (a) 共x ⫺ 2兲2 ⫹ 共 y ⫹ 3兲2 ⫹ 共z ⫺ 6兲2 36
(b) 共x ⫺ 2兲2 ⫹ 共 y ⫹ 3兲2 ⫹ 共z ⫺ 6兲2 4
(c) 共x ⫺ 2兲2 ⫹ 共 y ⫹ 3兲2 ⫹ 共z ⫺ 6兲2 9
23. A plane parallel to the yz-plane and 5 units in front of it
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 114
A114
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
25. A half-space consisting of all points to the left of the
plane y 8
27. All points on or between the horizontal planes z 0 and z 6
29. All points on a circle with radius 2 with center on the z-axis
that is contained in the plane z ⫺1
31. All points on or inside a sphere with radius s3 and center O
33. All points on or inside a circular cylinder of radius 3 with axis
the y-axis
35. 0 ⬍ x ⬍ 5
37. r 2 ⬍ x 2 ⫹ y 2 ⫹ z 2 ⬍ R 2
39. (a) (2, 1, 4)
(b) L¡
z
C
0
k3, 8, 1l
x
3
7
8
1
4
i⫹
j
25. 9 i ⫺ 9 j ⫹ 9 k
27. 60⬚
s58
s58
具 2, 2s3 典 31. ⬇ 45.96 ft兾s, ⬇38.57 ft兾s
100 s7 ⬇ 264.6 N, ⬇139.1⬚
s493 ⬇ 22.2 mi兾h, N8⬚W
T1 ⫺196 i ⫹ 3.92 j, T2 196 i ⫹ 3.92 j
(a) At an angle of 43.4⬚ from the bank, toward upstream
20.2 min
⫾共i ⫹ 4 j兲兾s17
43. 0
y
(a), (b)
(d) s 97 , t 117
23. ⫺
35.
37.
39.
(b)
41.
45.
sa
a
PAGE 798
0
1. (a) Scalar
(b) Vector
(c) Vector
(d) Scalar
l
l l
l l
l l
l
3. AB DC, DA CB, DE EB, EA CE
5. (a)
(b)
u
u+v
v
w
(d)
(e)
_v
u-v
(f)
u
v
15. cos⫺1
_w
u
w
19. cos⫺1
u-w-v
7. c 2 a ⫹ 2 b, d 2 b ⫺ 2 a
1
1
1
9. a 具4, 1 典
⬇ 63⬚
冉 冊
17. cos⫺1
5
s1015
⬇ 81⬚
[
27. 共i ⫺ j ⫺ k兲兾s3 or 共⫺i ⫹ j ⫹ k兲兾s3
25. Yes
y
y
]
35. 1兾s14 , ⫺2兾s14 , ⫺3兾s14 ; 74⬚, 122⬚, 143⬚
A(_1, 3)
B(2, 2)
A(_1, 1)
0
x
9
7
41. ,
x
a
39. 4, 具 ⫺ 13, 13 典
20 48
37. 1兾s3, 1兾s3, 1兾s3; 55⬚, 55⬚, 55⬚
a
0
1
s5
7. 1
29. 45⬚
31. 0⬚ at 共0, 0兲, 8.1⬚ at 共1, 1兲
2 1 2
33. 3, 3, 3 ; 48⬚, 71⬚, 48⬚
11. a 具3, ⫺1 典
B(3, 2)
冉 冊
冉 冊
5. 19
7
21. 48⬚, 75⬚, 57⬚
⬇ 52⬚
s130
23. (a) Neither
(b) Orthogonal
(c) Orthogonal
(d) Parallel
_v
v+u+w
1
PAGE 806
N
1. (b), (c), (d) are meaningful
3. 14
1
1
9. ⫺15
11. u ⴢ v 2 , u ⴢ w ⫺ 2
u
v+w
x
b
47. A sphere with radius 1, centered at 共x0, y0, z0 兲
EXERCISES 12.3
v
c
tb
w
u+w
u
(c)
y
k0, 8, 0l
19. 具2, ⫺18典 , 具1, ⫺42典 , 13, 10
21. ⫺i ⫹ j ⫹ 2 k, ⫺4 i ⫹ j ⫹ 9 k, s14 , s82
y
41. 14x ⫺ 6y ⫺ 10z 9, a plane perpendicular to AB
43. 2 s3 ⫺ 3
N
k3, 0, 1l
L™
A
B
EXERCISES 12.2
z
29.
33.
P
x
17. 具3, 8, 1典
具
27
49
, ,⫺
54
49
18
49
典
43. 1兾s21,
2
21
i⫺
1
21
j⫹
4
21
k
47. 具0, 0, ⫺2 s10 典 or any vector of the form
具s, t, 3s ⫺ 2 s10 典, s, t 僆 ⺢
13. a 具 2, 0, ⫺2 典
49. 144 J
51. 2400 cos共40⬚兲 ⬇ 1839 ft-lb
13
53. 5
55. cos⫺1(1兾s3 ) ⬇ 55⬚
15. 具5, 2 典
y
z
A (0, 3, 1)
y
a
x
EXERCISES 12.4
k6, _2l
0
N
PAGE 814
1. 16 i ⫹ 48 k
3. 15 i ⫺ 3 j ⫹ 3 k
5. 2 i ⫺ j ⫹ 2 k
7. 共1 ⫺ t兲 i ⫹ 共t 3 ⫺ t 2 兲 k
9. 0
11. i ⫹ j ⫹ k
13. (a) Scalar
(b) Meaningless
(c) Vector
1
k_1, 4l
B (2, 3, _1)
k5, 2l
0
x
(d) Meaningless
(e) Meaningless
3
(f) Scalar
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA106-115.qk_97909_Ans_Ans_pA106-115 10/14/10 6:24 PM Page 115
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
17. 具⫺7, 10, 8 典, 具7, ⫺10, ⫺8典
15. 96s3; into the page
19.
冓
27. 16
31.
33.
41.
45.
冔冓
1
1
5
⫺
,⫺
,
,
3s3
3s3 3s3
1
1
5
,
,⫺
3s3 3s3
3s3
29. (a) 具0, 18, ⫺9 典
69. s61兾14
79. 13兾s69
冔
9
2
71.
EXERCISES 12.6
(b) s5
(a) 具13, ⫺14, 5 典
(b) s390
9
35. 16
39. 10.8 sin 80⬚ ⬇ 10.6 N ⴢ m
⬇417 N
43. 60⬚
(b) s97兾3
53. (a) No
(b) No
(c) Yes
1
2
N
73. 5兾 (2s14 )
18
7
77. 1兾s6
PAGE 832
1. (a) Parabola
(b) Parabolic cylinder with rulings parallel to the z-axis
(c) Parabolic cylinder with rulings parallel to the x-axis
3. Circular cylinder
5. Parabolic cylinder
z
z
EXERCISES 12.5
N
A115
PAGE 824
1. (a) True
(b) False
(c) True
(d) False
(e) False
(f ) True
(g) False
(h) True
(i) True
( j) False
(k) True
3. r 共2 i ⫹ 2.4 j ⫹ 3.5 k兲 ⫹ t共3 i ⫹ 2 j ⫺ k兲;
x 2 ⫹ 3t, y 2.4 ⫹ 2t, z 3.5 ⫺ t
5. r 共i ⫹ 6 k兲 ⫹ t共i ⫹ 3 j ⫹ k兲;
x 1 ⫹ t, y 3t, z 6 ⫹ t
7. x 2 ⫹ 2t, y 1 ⫹ 2 t, z ⫺3 ⫺ 4t;
y
x
y
x
1
7. Hyperbolic cylinder
共x ⫺ 2兲兾2 2y ⫺ 2 共z ⫹ 3兲兾共⫺4兲
y⫺1
x⫹8
,z4
11
⫺3
11. x 1 ⫹ t, y ⫺1 ⫹ 2t, z 1 ⫹ t ;
x ⫺ 1 共 y ⫹ 1兲兾2 z ⫺ 1
13. Yes
15. (a) 共x ⫺ 1兲兾共⫺1兲 共 y ⫹ 5兲兾2 共z ⫺ 6兲兾共⫺3兲
(b) 共⫺1, ⫺1, 0兲, (⫺ 32 , 0, ⫺ 32), 共0, ⫺3, 3兲
17. r共t兲 共2 i ⫺ j ⫹ 4 k兲 ⫹ t共2 i ⫹ 7 j ⫺ 3 k兲, 0 艋 t 艋 1
19. Skew
21. 共4, ⫺1, ⫺5兲
23. x ⫺ 2y ⫹ 5z 0
25. x ⫹ 4y ⫹ z 4
27. 5x ⫺ y ⫺ z 7
29. 6x ⫹ 6y ⫹ 6z 11
31. x ⫹ y ⫹ z 2
33. ⫺13x ⫹ 17y ⫹ 7z ⫺42
35. 33x ⫹ 10y ⫹ 4z 190
37. x ⫺ 2y ⫹ 4z ⫺1
39. 3x ⫺ 8y ⫺ z ⫺38
y
9. x ⫺8 ⫹ 11t, y 1 ⫺ 3t, z 4;
41.
43.
z
z
3
2
(0, 0, 10)
”0, 0, ’
y
x
9. (a) x k, y 2 ⫺ z 2 1 ⫺ k 2, hyperbola 共k ⫾1兲;
y k, x 2 ⫺ z 2 1 ⫺ k 2, hyperbola 共k ⫾1兲;
z k, x 2 ⫹ y 2 1 ⫹ k 2, circle
(b) The hyperboloid is rotated so that it has axis the y-axis
(c) The hyperboloid is shifted one unit in the negative y-direction
11. Elliptic paraboloid with axis the x-axis
z
(0, _2, 0)
0
(1, 0, 0)
0
(5, 0, 0)
(0, 2, 0)
y
x
y
x
y
x
45.
51.
55.
57.
59.
63.
65.
67.
47. 共2, 3, 1兲
49. 1, 0, ⫺1
共2, 3, 5兲
1
Perpendicular
53. Neither, cos⫺1 ( 3 ) ⬇ 70.5⬚
Parallel
5
(a) x 1, y ⫺t, z t (b) cos⫺1
⬇ 15.8⬚
3s3
61. x ⫹ 2y ⫹ z 5
x 1, y ⫺ 2 ⫺z
共x兾a兲 ⫹ 共 y兾b兲 ⫹ 共z兾c兲 1
x 3t, y 1 ⫺ t, z 2 ⫺ 2t
P2 and P3 are parallel, P1 and P4 are identical
13. Elliptic cone with axis the x-axis
z
冉 冊
y
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 116
A116
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
15. Hyperboloid of two sheets
35. 共 y 1兲2 苷 共x 2兲2 共z 1兲2
z
(2,-1,1) z
Circular cone with vertex 共2, 1, 1兲
and axis parallel to the y-axis
y
x
y
x
37.
17. Ellipsoid
39.
4
2
z 0
z 0
z
(0, 0, 1)
_4
_4
(0, 6, 0)
x
y
y
(1, 0, 0)
z
41.
19. Hyperbolic paraboloid
0
_2
_2
_4
0
44
x
y
0
2 2
0x
_2
z=2
z
z=œ„„„„„
≈+¥
0
y
y
x
x
21. VII
23. II
29. y 2 苷 x 2 25. VI
43. y 苷 x 2 z 2
27. VIII
45. 4x 苷 y 2 z 2, paraboloid
x2
y2
z2
苷1
共6378.137兲2
共6378.137兲2
共6356.523兲2
(b) Circle
(c) Ellipse
47. (a)
z
z2
9
51.
Elliptic cone with axis the y-axis
y
2
z1
0
1
x
31. y 苷 z 2 1
y
0
0
CHAPTER 12 REVIEW
x2
2
z
1
1
N
PAGE 834
True-False Quiz
1. False
3. False
11. True
13. True
19. False
21. True
Hyperbolic paraboloid
x
5. True
15. False
7. True
9. True
17. False
Exercises
y
x
共 y 2兲2
33. x 2 共z 3兲2 苷 1
4
(b) 共 y 2兲2 共z 1兲2 苷 68, x 苷 0
(c) Center 共4, 1, 3兲, radius 5
3. u ⴢ v 苷 3 s2; u v 苷 3 s2; out of the page
5. 2, 4
7. (a) 2
(b) 2
(c) 2
(d) 0
1
9. cos1( 3 ) ⬇ 71
11. (a) 具4, 3, 4典
(b) s41兾2
13. 166 N, 114 N
15. x 苷 4 3t, y 苷 1 2t, z 苷 2 3t
17. x 苷 2 2t, y 苷 2 t, z 苷 4 5t
19. 4x 3y z 苷 14
21. (1, 4, 4)
23. Skew
25. x y z 苷 4
27. 22兾s26
ⱍ
z
(0, 0, 3)
(0, 4, 3)
Ellipsoid with center 共0, 2, 3兲
0
x
1. (a) 共x 1兲2 共 y 2兲2 共z 1兲2 苷 69
y
ⱍ
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 117
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
29. Plane
31. Cone
z
y
15.
1
z
_2π
x
_1
0
z
y
2
y
x
2π
x
2π x
_2π
33. Hyperboloid of two sheets
35. Ellipsoid
z
_2
z
(0, 1, 2)
z
z
2
y
(1, 1, 0)
(0, 2, 0)
x
(0, 0, 2)
(0, 2, 0)
x
y
_1
(0, 1,-2)
y
y
1
x
37. 4x 2 y 2 z 2 苷 16
PROBLEMS PLUS
N
_2
17. r共t兲 苷 具2 4t, 2t, 2t典 , 0 t 1;
PAGE 837
1. (s3 2 ) m
3. (a) 共x 1兲兾共2c兲 苷 共 y c兲兾共c 2 1兲 苷 共z c兲兾共c 2 1兲
3
(b) x 2 y 2 苷 t 2 1, z 苷 t
5. 20
(c) 4兾3
x 苷 2 4t, y 苷 2t, z 苷 2t, 0 t 1
19. r共t兲 苷
具 12 t, 1 43 t, 1 34 t 典 , 0 t 1;
x 苷 12 t, y 苷 1 43 t, z 苷 1 34 t, 0 t 1
21. II
23. V
25. IV
27.
29. 共0, 0, 0兲, 共1, 0, 1兲
z
CHAPTER 13
0
EXERCISES 13.1
1. 共1, 2兴
7.
N
y
PAGE 845
3. i j k
5. 具1, 兾2, 0典
z
9.
y
x
31.
π
1
(0, 2, 0)
y
x
x
1
z 0
_1
0 x
_1
_1
0
y
1
1
33.
11.
13.
z
z
10
y
y=≈
z 0
x
1
x
y
_10
10
0
x
_10
10
0
_10
y
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A117
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 118
A118
35.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
z
7. (a), (c)
37.
1
2
rª(0)
(1, 1)
z
0
r(0)
0
2
_1
_1
y 0 1
1
0
y
0
0
0
x
2 2
_1
x
41. r共t兲 苷 t i 2 共t 2 1兲 j 2 共t 2 1兲 k
43. r共t兲 苷 cos t i sin t j cos 2t k, 0 t 2
45. x 苷 2 cos t, y 苷 2 sin t, z 苷 4 cos 2t
47. Yes
1
EXERCISES 13.2
1. (a)
(b) r共t兲 苷 2e 2t i e t j
y
N
1
PAGE 852
y
R
C
r(4.5)-r(4)
r(4.5)
1
Q
r(4.2)
2
x
9. r共t兲 苷 具t cos t sin t, 2t, cos 2t 2t sin 2t典
11. r共t兲 苷 i (1兾st ) k
2
13. r共t兲 苷 2te t i [3兾共1 3t兲兴 k
15. r共t兲 苷 b 2t c
1 2 2
3
4
17. 具 3 , 3 , 3 典
19. 5 j 5 k
具1, 2t, 3t 2 典 , 具1兾s14, 2兾s14, 3兾s14 典, 具0, 2, 6t典 , 具 6t 2, 6t, 2 典
x 苷 3 t, y 苷 2t, z 苷 2 4t
x 苷 1 t, y 苷 t, z 苷 1 t
r共t兲 苷 共3 4t兲 i 共4 3t兲 j 共2 6t兲 k
x 苷 t , y 苷 1 t , z 苷 2t
x 苷 t, y 苷 t, z 苷 t
35. 2 i 4 j 32 k
37. i j k
66°
tan t i 18 共t 2 1兲4 j (13 t 3 ln t 19 t 3) k C
2
2
41. t 2 i t 3 j ( 3 t 3兾2 3 ) k
47. 2t cos t 2 sin t 2 cos t sin t
49. 35
21.
23.
25.
27.
29.
31.
33.
39.
r(4.2)-r(4)
P
EXERCISES 13.3
r(4)
0
(b), (d)
y
R
r(4.5)
Q
T(4)
r(4.2)
P
(c) r共4兲 苷 lim
hl0
x
r共4 h兲 r共4兲
r共4兲
; T共4兲 苷
h
r共4兲
ⱍ
3. (a), (c)
冊
ⱍ
ⱍ
33. (a) P
(b) 1.3, 0.7
y=x –@
y=k(x)
ⱍ
(b) r共t兲 苷 具 1, 2t典
y
7. 18.6833
冊 冉
1
1
共13 3兾2 8兲
31. ( 2 ln 2, 1兾s2 ); approaches 0
35.
4
r(4)
0
1
27
冉
13. r共t共s兲兲 苷
r(4.5)-r(4)
0.5
1
5.
2
3
4
si 1
s j 5
s k
s29
s29
s29
15. 共3 sin 1, 4, 3 cos 1兲
17. (a) 具 1兾s10 , (3兾s10 )sin t, (3兾s10 ) cos t 典,
具0, cos t, sin t典 (b) 103
1
1
19. (a) 2t
具
具 1 e 2t, s2 e t, s2 e t 典
s2 e t, e 2t, 1 典 , 2t
e 1
e 1
(b) s2 e 2t兾共e 2t 1兲2
1 19
4
21. 6t 2兾共9t 4 4t 2兲3兾2
23. 25
25. 7 s14
2
6 3兾2
x
27. 12x 兾共1 16x 兲
29. e x 2 兾关1 共xe x e x 兲2 兴 3兾2
r(4.2)-r(4)
0.2
C
PAGE 860
3. e e1
11. 42
1. 10s10
9. 1.2780
x
1
N
4
_4
_1
(_3, 2)
37.
rª(_1)
␬(t)
r(_1)
0.6
5
0
x
z
5. (a), (c)
(b) r共t兲 苷 cos t i 2 sin t j
y
0
_5
0
2
” œ„
, œ„2 ’
2
50
y
100
0
250 x
500
_5
π
rª” 4 ’
0
π
r” 4 ’
x
0
5 t
39. a is y 苷 f 共x兲, b is y 苷 共x兲
41.
共t兲 苷
6s4 cos 2 t 12 cos t 13
共17 12 cos t兲 3兾2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/18/10 10:18 AM Page 119
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
11. s2 i e t j et k, e t j et k, e t et
13. e t 关共cos t sin t兲 i 共sin t cos t兲 j 共t 1兲 k兴,
k(t)
e t 关2 sin t i 2 cos t j 共t 2兲 k兴, e tst 2 2t 3
1
15. v共t兲 苷 t i 2t j k, r共t兲 苷 ( 2 t 2 1兲 i t 2 j t k
1 3
1
1
17. (a) r共t兲 苷 ( 3 t t ) i 共t sin t 1兲 j ( 4 4 cos 2t) k
(b)
integer multiples of 2
0
2π
4π
6π
t
43. 6t 2兾共4 t 2 9t 4 兲3兾2
z
45. 1兾(s2e t)
47. 具 3 , 3 , 3 典, 具 3 , 3 , 3 典, 具 3 , 3 , 3 典
49. y 苷 6x , x 6y 苷 6
5 2
81
5 2
16
51. ( x 2 ) y 2 苷 4 , x 2 ( y 3 ) 苷 9
2
2
1
1
2
2
2
1
2
7.5
0.6
0.4
0.2
0
_10
200
x
5
19.
21.
23.
25.
29.
31.
33.
2.5
0
0 y
_200
10
t苷4
r共t兲 苷 t i t j 52 t 2 k, v共t兲 苷 s25t 2 2
(a) ⬇3535 m
(b) ⬇1531 m
(c) 200 m兾s
30 m兾s
27. ⬇10.2, ⬇79.8
13.0 36.0, 55.4 85.5
共250, 50, 0兲; 10s93 ⬇ 96.4 ft兾s
(a) 16 m
(b) ⬇23.6 upstream
ⱍ
ⱍ
12
20
5
53. 共1, 3, 1兲
55. 2x y 4z 苷 7, 6x 8y z 苷 3
63. 2兾共t 4 4t 2 1兲
65. 2.07 1010 Å ⬇ 2 m
EXERCISES 13.4
N
PAGE 870
40
_12
_4
1. (a) 1.8 i 3.8 j 0.7 k, 2.0 i 2.4 j 0.6 k,
2.8 i 1.8 j 0.3 k, 2.8 i 0.8 j 0.4 k
(b) 2.4 i 0.8 j 0.5 k, 2.58
3. v共t兲 苷 具t, 1典
a共t兲 苷 具 1, 0典
v共t兲 苷 st 2 1
35. The path is contained in a circle that lies in a plane perpendicular to c with center on a line through the origin in the direction
of c.
37. 6t, 6
39. 0, 1
41. e t et, s2
43. 4.5 cm兾s2, 9.0 cm兾s2
45. t 苷 1
y
v(2)
(_2, 2)
ⱍ
40
0
0
ⱍ
A119
a(2)
CHAPTER 13 REVIEW
N
PAGE 873
0 x
True-False Quiz
1. True
3. False
9. True
11. False
5. v共t兲 苷 3 sin t i 2 cos t j
ⱍ
Exercises
1. (a)
y
a共t兲 苷 3 cos t i 2 sin t j
v共t兲 苷 s5 sin 2 t 4
v” π3 ’
(0, 2)
ⱍ
3
” 2 , œ„
3’
a” π3 ’
5. False
13. True
7. False
z
(3, 0)
x
0
(0, 1, 0)
y
x
7. v共t兲 苷 i 2t j
z
a共t兲 苷 2 j
v共t兲 苷 s1 4t 2
ⱍ
ⱍ
a(1)
(1, 1, 2)
v(1)
y
x
9. 具2t 1, 2t 1, 3t 2 典 , 具2, 2, 6t典 , s9t 4 8t 2 2
(2, 1, 0)
(b) r 共t兲 苷 i sin t j cos t k,
r 共t兲 苷 2 cos t j 2 sin t k
t 2
3. r共t兲 苷 4 cos t i 4 sin t j 共5 4 cos t兲 k, 0
1
5. 3 i 共2兾 2兲 j 共2兾 兲 k
7. 86.631
9. 兾2
11. (a) 具t 2, t, 1典兾st 4 t 2 1
(b) 具t 3 2t, 1 t 4, 2t 3 t典兾st 8 5t 6 6t 4 5t 2 1
(c) st 8 5t 6 6t 4 5t 2 1兾共t 4 t 2 1兲2
13. 12兾17 3兾2
15. x 2y 2 苷 0
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 120
A120
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
17. v共t兲 苷 共1 ln t兲 i j et k,
ⱍ
ⱍ
23. (a) v 苷 R共sin t i cos t j兲
PROBLEMS PLUS
N
ⱍ
17. 兵共x, y兲 1 x 1, 1 y 1其
v共t兲 苷 s2 2 ln t 共ln t兲2 e2t , a共t兲 苷 共1兾t兲 i et k
19. (a) About 3.8 ft above the ground, 60.8 ft from the athlete
(b) ⬇21.4 ft
(c) ⬇64.2 ft from the athlete
21. (c) 2et vd et R
y
1
_1
(c) a 苷 r
PAGE 876
ⱍ
19. 兵共x, y兲 y
x 2, x 苷 1其
y
(b) ⬇7.6
(c) ⬇2.13 ft to the right of the table’s edge
5. 56
7. r共u, v兲 苷 c u a v b where a 苷 具a1, a 2 , a 3 典,
b 苷 具b1, b 2 , b 3 典, c 苷 具 c1, c 2 , c 3 典
y=≈
_1
CHAPTER 14
EXERCISES 14.1
x
_1
1. (a) 90, v02兾共2t兲
3. (a) ⬇0.94 ft to the right of the table’s edge, ⬇15 ft兾s
ⱍ
21. 兵共x, y, z兲 x 2 y 2 z 2 1其
N
1
0
2
0
x
1
z
PAGE 888
1. (a) 27; a temperature of 15C with wind blowing at
0
40 km兾h feels equivalent to about 27C without wind.
(b) When the temperature is 20C, what wind speed gives a wind
chill of 30C ? 20 km兾h
(c) With a wind speed of 20 km兾h, what temperature gives a wind
chill of 49C ? 35C
(d) A function of wind speed that gives wind-chill values when the
temperature is 5C
(e) A function of temperature that gives wind-chill values when the
wind speed is 50 km兾h
3. ⬇94.2; the manufacturer’s yearly production is valued at $94.2
million when 120,000 labor hours are spent and $20 million in
capital is invested.
5. (a) ⬇20.5; the surface area of a person 70 inches tall who
weighs 160 pounds is approximately 20.5 square feet.
7. (a) 25; a 40-knot wind blowing in the open sea for 15 h will
create waves about 25 ft high.
(b) f 共30, t兲 is a function of t giving the wave heights produced by
30-knot winds blowing for t hours.
(c) f 共v, 30兲 is a function of v giving the wave heights produced by
winds of speed v blowing for 30 hours.
9. (a) 1
(b) ⺢ 2
(c) 关1, 1兴
11. (a) 3
(b) 兵共x, y, z兲 x 2 y 2 z 2 4, x 0, y 0, z 0其,
interior of a sphere of radius 2, center the origin, in the first octant
13. 兵共x, y兲 y 2x其
y
y
x
23. z 苷 1 y, plane parallel to x-axis
z
(0, 0, 1)
0
(0, _1, 0)
x
y
25. 4x 5y z 苷 10, plane
z
(0, 0, 10)
ⱍ
ⱍ
0
(0, 2, 0)
(2.5, 0, 0)
y
x
0
x
27. z 苷 y 2 1 , parabolic cylinder
y=2x
15. 兵共x, y兲
ⱍ
1
9
x2 y2
1其, 共, ln 9兴
z
y
1
9 ≈+¥=1
0
x
x
y
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 121
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
29. z 苷 9 x 2 9y 2,
51. x 2 9y 2 苷 k
z
elliptic paraboloid
A121
z
y
(0, 0, 9)
z=4
2
1
x
4
0
(0, 1, 0)
(3, 0, 0)
3
x
z=3
y
z=2
z=1
31. z 苷 s4 4x 2 y 2 ,
z
top half of ellipsoid
y
(0, 0, 2)
x
53.
(0, 2, 0)
(1, 0, 0)
y
x
33. ⬇56, ⬇35
39.
35. 11C, 19.5C
41.
z
y
0
x
37. Steep; nearly flat
55.
z
5
z 0
14
_2
y
x
0
2 2
0
_2
x
57.
y
x
y
1.0
z 0.5
43. 共 y 2x兲 苷 k
45. y 苷 sx k
2
0.0
y
y
_4
y 0
x
2
1
x
0
_1
43 2 1
0
_2
1 234
47. y 苷 kex
49. y 2 x 2 苷 k 2
y
1 2
20
2
1
0
z
_20
0
0
_2
_1
x
0
1
2
_3
(a) C
(b) II
61. (a) F
(b) I
(a) B
(b) VI
65. Family of parallel planes
Family of circular cylinders with axis the x-axis 共k 0兲
(a) Shift the graph of f upward 2 units
(b) Stretch the graph of f vertically by a factor of 2
(c) Reflect the graph of f about the xy-plane
(d) Reflect the graph of f about the xy-plane and then shift it
upward 2 units
59.
63.
67.
69.
3
0
3
_4
x
71.
y
3
0
4 4
x
_40
_5
y
0
5
5
0x
_5
f appears to have a maximum value of about 15. There are two
local maximum points but no local minimum point.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 122
A122
73.
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
5. (a) Positive
(b) Negative
7. (a) Positive
(b) Negative
9. c 苷 f, b 苷 fx, a 苷 fy
11. fx 共1, 2兲 苷 8 苷 slope of C1, fy共1, 2兲 苷 4 苷 slope of C2
10
5
z
0
z
_5
_10
2
z
16
16
x 0
0
_2 2
_2
The function values approach 0 as x, y become large; as 共x, y兲
approaches the origin, f approaches or 0, depending on the
direction of approach.
75. If c 苷 0, the graph is a cylindrical surface. For c 0, the level
curves are ellipses. The graph curves upward as we leave the origin, and the steepness increases as c increases. For c 0, the level
curves are hyperbolas. The graph curves upward in the y-direction
and downward, approaching the xy-plane, in the x-direction giving
a saddle-shaped appearance near (0, 0, 1).
77. c 苷 2, 0, 2
79. (b) y 苷 0.75x 0.01
C¡
0
y
x
(1, 2)
0
2
_2
ⱍ
ⱍ
ⱍ
2 _2
y
20
z
0
_20
2
_2
0
x
ⱍ
2
_2
0 y
fx 共x, y兲 苷 2xy 3
35. 兵共x, y, z兲 x 2 y 2 z 2 1其
39. 0
0
0
f 共x, y兲 苷 x 2 y 3
bers along different lines.
25. h共x, y兲 苷 共2 x 3y 6兲 2 s2x 3y 6 ;
兵共x, y兲 2x 3y 6其
27. Along the line y 苷 x
29. ⺢ 2
31. 兵共x, y兲 x 2 y 2 苷 1其
37. 兵共x, y兲 共x, y兲 苷 共0, 0兲其
(1, 2)
_20
1. Nothing; if f is continuous, f 共3, 1兲 苷 6
3. 2
7. 7
9. Does not exist
11. Does not exist
5. 1
13. 0
15. Does not exist
17. 2
19. s3
21. Does not exist
23. The graph shows that the function approaches different num-
33. 兵共x, y兲 x 2 y 2 4其
y
x
20
z
5
2
ⱍ
4
2
13.
PAGE 899
N
C™
0
4
2
x
EXERCISES 14.2
(1, 2, 8)
(1, 2, 8)
y
41. 1
43.
40
z
2
1
z
0
_1
y
0
2
2
0x
x
EXERCISES 14.3
PAGE 911
1. (a) The rate of change of temperature as longitude varies, with
latitude and time fixed; the rate of change as only latitude varies;
the rate of change as only time varies.
(b) Positive, negative, positive
3. (a) fT 共15, 30兲 ⬇ 1.3; for a temperature of 15C and wind
speed of 30 km兾h, the wind-chill index rises by 1.3C for each
degree the temperature increases. fv 共15, 30兲 ⬇ 0.15; for a
temperature of 15C and wind speed of 30 km兾h, the wind-chill
index decreases by 0.15C for each km兾h the wind speed
increases.
(b) Positive, negative
(c) 0
0
0
2 _2
y
fy 共x, y兲 苷 3x 2 y 2
f is continuous on ⺢ 2
N
2
0
_2
_2
_2
20
fx 共x, y兲 苷 3y, fy 共x, y兲 苷 5y 4 3x
fx 共x, t兲 苷 e t sin x, ft 共x, t兲 苷 et cos x
z兾x 苷 20共2x 3y兲 9, z兾y 苷 30共2x 3y兲 9
fx 共x, y兲 苷 1兾y, fy 共x, y兲 苷 x兾y 2
共ad bc兲y
共bc ad兲x
23. fx 共x, y兲 苷
, fy 共x, y兲 苷
共cx dy兲2
共cx dy兲2
2
3 4
25. tu共u, v兲 苷 10uv共u v v 兲 , tv共u, v兲 苷 5共u 2 3v 2兲共u 2v v 3 兲 4
q2
2pq
27. Rp共 p, q兲 苷
, R q 共 p, q兲 苷
1 p 2q 4
1 p 2q 4
29. Fx共x, y兲 苷 cos共e x 兲, Fy 共x, y兲 苷 cos共e y兲
31. fx 苷 z 10xy 3z 4, fy 苷 15x 2 y 2z 4, fz 苷 x 20x 2 y 3z 3
33. w兾x 苷 1兾共x 2y 3z兲, w兾y 苷 2兾共x 2y 3z兲,
w兾z 苷 3兾共x 2y 3z兲
15.
17.
19.
21.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 123
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
35. u兾x 苷 y sin1 共 yz兲, u兾y 苷 x sin1 共 yz兲 xyz兾s1 y 2 z 2,
u兾z 苷 xy 兾s1 y z
37. h x 苷 2xy cos共z兾t兲, h y 苷 x 2 cos共z兾t兲,
h z 苷 共x 2 y兾t兲 sin共z兾t兲, h t 苷 共x 2 yz兾t 2 兲 sin共z兾t兲
39. u兾xi 苷 xi兾sx 12 x 22 x n2
1
1
41. 5
43. 4
45. fx 共x, y兲 苷 y 2 3x 2 y , fy 共x, y兲 苷 2xy x 3
z
x z
2y
47.
苷 ,
苷
x
3z y
3z
z
z
yz
xz
49.
苷 z
,
苷 z
x
e xy y
e xy
51. (a) f 共x兲, t共 y兲
(b) f 共x y兲, f 共x y兲
53. fxx 苷 6xy 5 24x 2 y, fxy 苷 15x 2 y 4 8x 3 苷 fyx , fyy 苷 20x 3 y 3
55. wuu 苷 v 2兾共u 2 v 2 兲3兾2, wuv 苷 uv兾共u 2 v 2 兲3兾2 苷 wvu,
wvv 苷 u 2兾共u 2 v 2 兲3兾2
57. zxx 苷 2x兾共1 x 2 兲 2, zxy 苷 0 苷 zyx , zyy 苷 2y兾共1 y 2 兲 2
2
63. 24xy 2 6y, 24x 2 y 6x
65. 共2x 2 y 2z 5 6xyz 3 2z兲e xyz
67. e r 共2 sin cos r sin 兲
69. 4兾共 y 2z兲 3 , 0
71. 6yz 2
73. ⬇12.2, ⬇16.8, ⬇23.25
83. R 2兾R 21
T
V nb P
2n 2a
nRT
87.
苷
,
苷
P
nR
V
V3
共V nb兲2
93. No
95. x 苷 1 t, y 苷 2, z 苷 2 2t
99. 2
101. (a)
2
2 2
0.2
_0.2
_1
y 0
1
0
1
_1
x
x 4y 4x 2y 3 y 5
x 5 4x 3y 2 xy 4
, fy共x, y兲 苷
2
2 2
共x y 兲
共x 2 y 2 兲2
(e) No, since fxy and fyx are not continuous.
(b) fx 共x, y兲 苷
EXERCISES 14.4
N
PAGE 922
1. z 苷 7x 6y 5
5. x y z 苷 0
7.
3. x y 2z 苷 0
3. 关共x兾t兲 y sin t兴兾s1 x 2 y 2
5. e y兾z 关2t 共x兾z兲 共2xy兾z 2 兲兴
7. z兾s 苷 2xy 3 cos t 3x 2 y 2 sin t,
z兾t 苷 2sxy 3 sin t 3sx 2 y 2 cos t
9. z兾s 苷 t 2 cos cos 2st sin sin ,
z兾t 苷 2st cos cos s 2 sin sin 11.
z
t
13.
17.
u
t
冉
z
苷 e r t cos
s
冉
_1
0
11. 6x 4y 23
23.
25.
27.
29.
31.
37.
39.
0
0
y
_5
0
2
2
21.
3
7
冊
w
w r
w s
w t
苷
,
x
r x
s x
t x
w
w r
w s
w t
苷
y
r y
s y
t y
21. 1582, 3164, 700
23. 2, 2
19.
2x y sin x
cos x 2y
1 x 4 y 2 y 2 x 4 y 4 2xy
29.
x 2 2xy 2x 5y 3
x
2y
yz
xz
,
31. , 33. z
3z
3z
e xy e z xy
35. 2C兾s
37. ⬇ 0.33 m兾s per minute
39. (a) 6 m3兾s (b) 10 m 2兾s (c) 0 m兾s
41. ⬇ 0.27 L兾s
43. 1兾 (12 s3 ) rad兾s
45. (a) z兾r 苷 共z兾x兲 cos 共z兾y兲 sin ,
z兾 苷 共z兾x兲r sin 共z兾y兲r cos
51. 4rs 2z兾x 2 共4r 2 4s 2 兲2z兾x y 4rs 2z兾y 2 2 z兾y
5
144
5
, 965 , 144
27.
13. x y 1
9
6
7
2
9
2
3
N
PAGE 943
x
(b) 具2, 3典
(c) s3 32
9. (a) 具2xyz yz 3, x 2z xz 3, x 2 y 3xyz 2 典
(b) 具3, 2, 2典
(c) 25
y
15. 1 y
x 27 y z; 6.9914
4T H 329; 129F
dz 苷 2e 2x cos 2 t dx 2 e 2x sin 2 t dt
dm 苷 5p 4q 3 dp 3p 5q 2 dq
dR 苷 2 cos d 2 cos d 2 sin d z 苷 0.9225, dz 苷 0.9
33. 5.4 cm 2
35. 16 cm 3
⬇0.0165mg; decrease
1
41. 2.3%
43. 1 苷 x, 2 苷 y
17 ⬇ 0.059 19. 6.3
1. ⬇ 0.08 mb兾km
3. ⬇ 0.778
5. 2 s3兾2
7. (a) f 共x, y兲 苷 具 2 cos共2x 3y兲, 3 cos共2x 3y兲 典
z 0
z 200
5
冊
s
sin ,
ss 2 t 2
t
苷 e r s cos sin
ss 2 t 2
62
15. 7, 2
u
u x
u y u
u x
u y
苷
,
苷
,
r
x r
y r s
x s
y s
u x
u y
苷
x t
y t
EXERCISES 14.6
1
x 0 _10
PAGE 930
9.
400
10
N
1. 共2x y兲 cos t 共2y x兲e t
25.
z 0
(c) 0, 0
EXERCISES 14.5
A123
11.
17.
23.
27.
29.
31.
33.
4 3s3
10
13. 8兾s10
15. 4兾s30
19. 2兾5
21. s65 , 具1, 8 典
25. 1, 具 3, 6, 2 典
1, 具0, 1典
(b) 具12, 92典
All points on the line y 苷 x 1
(a) 40兾(3 s3 )
(a) 32兾s3
(b) 具38, 6, 12典
(c) 2 s406
23
42
327
774
35. 13
39. 25
41. (a) x y z 苷 11
(b) x 3 苷 y 3 苷 z 5
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 124
A124
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
x3
y2
z1
苷
苷
2
3
12
(b) x 苷 y 苷 z 1
49. 具2, 3典 , 2x 3y 苷 12
43. (a) 2x 3y 12z 苷 24
45. (a) x y z 苷 1
47.
41. (2, 1, s5 ), (2, 1, s5 )
100
, 100
3 , 3
45. 8r 兾 (3s3 )
47.
49. Cube, edge length c兾12
51. Square base of side 40 cm, height 20 cm
53. L 3兾 (3s3 )
39. 2兾s3
(b)
43.
100
3
4
3
3
y
EXERCISES 14.8
xy=6
2
Î
f (3, 2)
2x+3y=12
z 1
(3, 2)
0
0
x
N
PAGE 963
1. ⬇59, 30
3. No maximum, minimum f 共1, 1兲 苷 f 共1, 1兲 苷 2
5. Maximum f 共0, 1兲 苷 1, minimum f 共2, 0兲 苷 4
7. Maximum f 共2, 2, 1兲 苷 9, minimum f 共2, 2, 1兲 苷 9
9. Maximum 2兾s3, minimum 2兾s3
11. Maximum s3, minimum 1
13. Maximum f ( 2 , 2 , 2 , 2 ) 苷 2,
1
_1
1
x
2
1
y
5
5
25
8
)
63. x 苷 1 10t, y 苷 1 16t, z 苷 2 12t
67. If u 苷 具 a, b典 and v 苷 具c, d 典 , then afx bfy and c fx d fy are
known, so we solve linear equations for fx and fy .
EXERCISES 14.7
N
PAGE 953
1. (a) f has a local minimum at (1, 1).
(b) f has a saddle point at (1, 1).
3. Local minimum at (1, 1), saddle point at (0, 0)
1
2
1
5. Minimum f ( 3, 3 ) 苷 3
7. Saddle points at 共1, 1兲, 共1, 1兲
9. Maximum f 共0, 0兲 苷 2, minimum f 共0, 4兲 苷 30,
saddle points at 共2, 2兲, 共2, 2兲
11. Minimum f 共2, 1兲 苷 8, saddle point at 共0, 0兲
15. Minimum f 共0, 0兲 苷 0, saddle points at 共1, 0兲
13. None
17. Minima f 共0, 1兲 苷 f 共, 1兲 苷 f 共2, 1兲 苷 1,
saddle points at 共兾2, 0兲, 共3兾2, 0兲
21. Minima f 共1, 1兲 苷 3, f 共1, 1兲 苷 3
23. Maximum f 共兾3, 兾3兲 苷 3 s3兾2,
minimum f 共5兾3, 5兾3兲 苷 3 s3兾2, saddle point at 共, 兲
25. Minima f 共0, 0.794兲 ⬇ 1.191, f 共1.592, 1.267兲 ⬇ 1.310,
saddle points 共0.720, 0.259兲,
lowest points 共1.592, 1.267, 1.310兲
27. Maximum f 共0.170, 1.215兲 ⬇ 3.197,
minima f 共1.301, 0.549兲 ⬇ 3.145, f 共1.131, 0.549兲 ⬇ 0.701,
saddle points 共1.301, 1.215兲, 共0.170, 0.549兲, 共1.131, 1.215兲,
no highest or lowest point
29. Maximum f 共0, 2兲 苷 4, minimum f 共1, 0兲 苷 1
31. Maximum f 共1, 1兲 苷 7, minimum f 共0, 0兲 苷 4
33. Maximum f 共3, 0兲 苷 83, minimum f 共1, 1兲 苷 0
35. Maximum f 共1, 0兲 苷 2, minimum f 共1, 0兲 苷 2
37.
1
1
minimum f ( , , , 12 ) 苷 2
2
59. (4 , 4 ,
55. No
1
1
2
(_1, 0, 0)
1
2
15. Maximum f (1, s2, s2 ) 苷 1 2 s2,
minimum f (1, s2, s2 ) 苷 1 2 s2
3
1
17. Maximum 2 , minimum 2
19. Maximum f (3兾s2 , 3兾s2 ) 苷 9 12s2 ,
minimum f 共2, 2兲 苷 8
21. Maximum f (1兾s2, 1兾(2 s2 )) 苷 e 1兾4,
minimum f (1兾s2, 1兾(2 s2 )) 苷 e1兾4
29– 41. See Exercises 39–53 in Section 14.7.
1 1 1
43. Nearest ( 2 , 2 , 2 ), farthest 共1, 1, 2兲
45. Maximum ⬇9.7938, minimum ⬇5.3506
47. (a) c兾n
(b) When x1 苷 x2 苷 苷 xn
CHAPTER 14 REVIEW
N
PAGE 967
True-False Quiz
1. True
3. False
11. True
5. False
Exercises
1. 兵共x, y兲 y x 1其
ⱍ
7. True
3.
9. False
z
1
y
_1
x
1
x
y
_1
y=_x-1
5.
y
7.
y
2
(1, 2, 0)
1
0
0
1
2
2
34
5
1
x
_1
z
_2
_3
_1
0
x
1
4
2y
_2
0
1
2
x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA116-125.qk_97909_Ans_Ans_pA116-125 10/14/10 5:00 PM Page 125
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
2
9. 3
11. (a) ⬇3.5C兾m , 3.0C兾m
(b) ⬇ 0.35C兾m by
Equation 14.6.9 (Definition 14.6.2 gives ⬇1.1C兾m .)
(c) 0.25
13. fx 苷 32xy共5y 3 2x 2y兲7, fy 苷 共16x 2 120y 2 兲共5y 3 2x 2 y兲7
15. F 苷
17. 9 ln 2
23.
1
2
19.
(s3 1) 121 1
5
2
z
4
2 3
2 2
2 ln共 2 2 兲, F 苷 2
2
2
2
usw
uv
, Sw 苷
1 v 2w
2sw 共1 v 2w兲
19. f xx 苷 24x, f xy 苷 2y 苷 f yx, f yy 苷 2x
21. f xx 苷 k共k 1兲x k2 y lz m, f xy 苷 klx k1 y l1z m 苷 f yx,
f xz 苷 kmx k1 y lz m1 苷 f zx, f yy 苷 l共l 1兲x k y l2z m,
f yz 苷 lmx k y l1z m1 苷 f zy, f zz 苷 m共m 1兲x k y lz m2
y2
z1
x1
25. (a) z 苷 8x 4y 1
(b)
苷
苷
8
4
1
x2
y1
z1
27. (a) 2x 2y 3z 苷 3
(b)
苷
苷
4
4
6
29. (a) x 2y 5z 苷 0
(b) x 苷 2 t, y 苷 1 2t, z 苷 5t
1
1
31. (2, 2 , 1), (2, 2 , 1)
24
32
33. 60x 5 y 5 z 120; 38.656
35. 2xy 3共1 6p兲 3x 2 y 2共 pe p e p兲 4z 3共 p cos p sin p兲
37. 47, 108
4
43. 具 2xe yz , x 2 z 2e yz , 2x 2 yze yz 典
45. 5
9
5
47. s145兾2, 具4, 2 典
49. ⬇ 8 knot兾mi
51. Minimum f 共4, 1兲 苷 11
53. Maximum f 共1, 1兲 苷 1; saddle points (0, 0), (0, 3), (3, 0)
55. Maximum f 共1, 2兲 苷 4, minimum f 共2, 4兲 苷 64
57. Maximum f 共1, 0兲 苷 2, minima f 共1, 1兲 苷 3,
saddle points 共1, 1兲, 共1, 0兲
59. Maximum f (s2兾3, 1兾s3 ) 苷 2兾(3 s3 ),
minimum f (s2兾3, 1兾s3 ) 苷 2兾(3 s3 )
61. Maximum 1, minimum 1
63. (31兾4, 31兾4s2, 31兾4 ), (31兾4, 31兾4s2, 31兾4 )
65. P(2 s3 ), P(3 s3 )兾6, P(2 s3 3)兾3
0
17. Su 苷 arctan (v sw ), Sv 苷
2
2
PROBLEMS PLUS
N
y
x
25. 51
27.
33. 21e 57
166
27
31.
z
0
0
y
5
discontinuity at the origin.
EXERCISES 15.3
1. 32
11. (a)
3.
N
3
10
PAGE 995
1
3
5.
sin 1
7.
y
y
D
D
0
x
0
x
ⱍ
(b) Yes
ⱍ
15.
2
sx
x01 xsxsx y dy dx x14 xx2
y dy dx苷 x1
xyy2 y dx dy 苷 94
2
17. 共1 cos 1兲
27. 6
37.
N
9. 4
3
(b)
29.
128
15
19.
11
3
31.
1
3
21. 0
23.
17
60
33. 0, 1.213; 0.713
25.
31
8
35.
z
(0, 0, 1)
PAGE 981
0
(0, 1, 0)
(1, 0, 0)
y
x
PAGE 987
1. 500y , 3x
3. 222
5. 32共e 1兲
21
31
9. 2 ln 2
11. 30
13. 15. 0
2
0
x
1 1
35. 6
37. 0
39. Fubini’s Theorem does not apply. The integrand has an infinite
1. (a) 288
(b) 144
3. (a) 0.990
(b) 1.151
5. (a) 4
(b) 8
7. U
V L
9. (a) ⬇248
(b) ⬇15.5
11. 60
13. 3
15. 1.141606, 1.143191, 1.143535, 1.143617, 1.143637, 1.143642
EXERCISES 15.2
64
3
2
1
2
N
29. 2
13. Type I: D 苷 兵共x, y兲 0 x 1, 0 y x其,
type II: D 苷 兵共x, y兲 0 y 1, y x 1其; 13
PAGE 971
CHAPTER 15
EXERCISES 15.1
1
1
2
3. (a) x 苷 w兾3, base 苷 w兾3
1
1. L2W 2, 4 L2W 2
7. s3兾2, 3兾s2
3
21. 2 e 6 A125
4
7. 18
39. 13,984,735,616兾14,549,535
41. 兾2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
64
3
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 126
A126
43.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
x01 xx1 f 共x, y兲 dy dx
23. a 4兾16, a 4兾16; a兾2, a兾 2
y
(0, 1)
y=x
0
45.
x01 x0cos
1
y
f 共x, y兲 dx dy
x
y
1
y=cos x
or
x=cos_1y
0
47.
x0ln 2 xe2 f 共x, y兲 dx dy
y
y
33. (a)
x
π
2
y=ln x or x=e†
0
1
1.
5.
3兾2 4
0
0
x
x
N
1
f 共r cos , r sin 兲r dr d
3π
¨=
4
π
¨=
4
1
1
共x1兲兾2
0
x x
3.
EXERCISES 15.7
27
1. 4
65
13. 28
_1
0
]
(b) ⬇1.8616
f 共x, y兲 dy dx
3兾4
23. (a)
s1y 2
0
x x x
25. 0.985
27.
1
9.
17. 16兾3
1
60
x
0
7. 3
5
3
5.
15.
1
0
PAGE 1025
N
16
15
3.
R
_2
[
ln (11s5 3s70 )兾(3s5 s70 )
23. 共兾6兲(101 s101 1)
15
16
2 2
PAGE 1002
y
17. 8 s14 19. 3.3213
45
x
2
49. 6 共e 9 1兲
51. 3 ln 9
53. 3 (2 s2 1)
55. 1
3
57. 共兾16兲e1兾16 xxQ e共x y 兲 dA 兾16
59. 4
63. 9
3
65. a 2b 2 ab 2
67. a 2b
EXERCISES 15.4
PAGE 1016
N
2
x=2
2
disk with radius 10 mi centered at the center of the city
8
(b) 200k兾3 ⬇ 209k, 200(兾2 9 )k ⬇ 136k, on the edge
1. 15 s26
3. 3 s14
5. 12 sin1 ( 3 )
7. 共兾6兲(17 s17 5 s5 )
9. 共2兾3兲(2 s2 1)
11. a 2共 2兲
13. 13.9783
15. (a) ⬇1.83
y=0
1
冊
xxD k [1 201 s共x x0 兲2 共 y y0 兲2 ] dA, where D is the
EXERCISES 15.6
ln 2
1
冉
16384 s2
,0 ,
10395
5
5
4
4
5
, Iy 苷
, I0 苷
Ix 苷
384
105
384
105
192
1
27. (a) 2
(b) 0.375
(c) 485 ⬇ 0.1042
0.2
29. (b) (i) e
⬇ 0.8187
(ii) 1 e1.8 e0.8 e1 ⬇ 0.3481 (c) 2, 5
31. (a) ⬇0.500
(b) ⬇0.632
25. m 苷 3兾64, 共x, y兲 苷
(1, 1)
dz dy dx
19.
27
2
16
3
1
3
11. 9兾8
21.
8
15
(b) 1
4
z
1
1
2
x
2
0
1
y
x
7. 3
9. 共兾4兲共cos 1 cos 9兲
3
11. 共兾2兲共1 e4 兲
13. 64 2
1250
2
s4x y兾2
x2
x04x xs4x
f 共x, y, z兲 dz dy dx
y兾2
4 s4y
s4x y兾2
苷 x0 xs4y xs4x y兾2 f 共x, y, z兲 dz dx dy
1
s4y4z
x044z xs4y4z
f 共x, y, z兲 dx dy dz
苷 x1
s4y兾2
s4y4z
xs4y4z
f 共x, y, z兲 dx dz dy
苷 x04 xs4y兾2
2
4x 4z
s4x 兾2
f 共x, y, z兲 dy dz dx
苷 x2 xs4x 兾2 x0
1
4x 4z
s44z
f 共x, y, z兲 dy dx dz
苷 x1 xs44z x0
s3
4
16
4
19. 3 21. 3 23. 3 a 3
3
2
25. 共2兾3兲[1 (1兾s2 )]
27. 共8兾3兲(64 24 s3 )
1
29. 2 共1 cos 9兲
31. 2s2兾3
33. 4.5951
37. 2兾共a b兲
41. (a) s 兾4
(b) s 兾2
EXERCISES 15.5
N
39.
2
2
2
2
2
2
2
15
16
2
2
2
2
2
2
2
x2
xx4 x02y兾2 f 共x, y, z兲 dz dy dx
sy
x02y兾2 f 共x, y, z兲 dz dx dy
苷 x04 xsy
sy
f 共x, y, z兲 dx dy dz
苷 x02 x042z xsy
4 2y兾2 sy
xsy f 共x, y, z兲 dx dz dy
苷 x0 x0
2
x02x 兾2 xx42z f 共x, y, z兲 dy dz dx
苷 x2
s42z
xx42z f 共x, y, z兲 dy dx dz
苷 x02 xs42z
31.
1. 285 C
3. 42k, (2, )
5. 6, ( , )
7. k, (0, )
3
9. L兾4, 共L兾2, 16兾共9兲兲
11. ( 8 , 3兾16)
13. 共0, 45兾共14兲兲
15. 共2a兾5, 2a兾5兲 if vertex is (0, 0) and sides are along positive axes
64
8
88
17. 315 k, 105 k, 315 k
3
4
3
2
8
15
4
7
19. 7 ka 6兾180, 7 ka 6兾180, 7 ka 6兾90 if vertex is 共0, 0兲 and sides are
along positive axes
21. bh 3兾3, b 3h兾3; b兾s3, h兾s3
2
2
PAGE 1012
85
28
2
2
17.
35. 1800 ft 3
2
29.
15. 兾12
2
2
2
2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 127
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
x01 xsx1 x01y f 共x, y, z兲 dz dy dx 苷 x01 x0y x01y f 共x, y, z兲 dz dx dy
苷 x01 x01z x0y f 共x, y, z兲 dx dy dz 苷 x01 x01y x0y f 共x, y, z兲 dx dz dy
1z
1z
f 共x, y, z兲 dy dz dx 苷 x01 x0共1z兲 xsx
f 共x, y, z兲 dy dx dz
苷 x01 x01sx xsx
2
33.
2
15.
A127
4
z
2
2
x01 xy1 x0y f 共x, y, z兲 dz dx dy 苷 x01 x0x x0y f 共x, y, z兲 dz dy dx
苷 x01 xz1 xy1 f 共x, y, z兲 dx dy dz 苷 x01 x0y xy1 f 共x, y, z兲 dx dz dy
苷 x01 x0x xzx f 共x, y, z兲 dy dz dx 苷 x01 xz1 xzx f 共x, y, z兲 dy dx dz
35.
37. 64
33 571
, ( 358
553 , 79 , 553 )
79
30
39.
x
y
41. a , 共7a兾12, 7a兾12, 7a兾12兲
5
2
47. (a) m 苷
8
128
4
384
19. 3 15
21. 2兾5
23. 3 (s2 1)
(a) 162
(b) 共0, 0, 15兲
Ka 2兾8, 共0, 0, 2a兾3兲
29. 0
(a) xxxC h共P兲t共P兲 dV, where C is the cone
(b) ⬇3.1 1019 ft-lb
45. 2 kha 4
43. Ix 苷 Iy 苷 Iz 苷 3 kL5
17.
25.
27.
31.
1
1
x1
xx1 x01y sx 2 y 2 dz dy dx
2
(b) 共x, y, z兲, where
1
x 苷 共1兾m兲 x1
xx1 x01y x sx 2 y 2 dz dy dx
2
1
xx1
y 苷 共1兾m兲 x1
x y sx 2 y 2 dz dy dx
z 苷 共1兾m兲 x x x zsx 2 y 2 dz dy dx
1
(c) x1
xx1 x 共x y 2 兲3兾2 dz dy dx
2
49. (a)
(b)
(c)
冉
3
32
1y
2
0
1
1 1y
1 x 2 0
1y
2
0
11
24
28
30 128 45 208
,
,
9 44 45 220 135 660
1
240
EXERCISES 15.9
冊
(b)
z
z
π π
”6, 3 , 6 ’
0
π
6 6
(b)
(c)
53. L3兾8
55. (a) The region bounded by the ellipsoid x 2 2y 2 3z 2 苷 1
(b) 4 s6 兾45
51. (a)
1
64
x
1. (a)
π
3
z
π
”2, _ 2 , 1’
π
_2
y
冉
冊
z
2
4
π
3
”3, 2 , 4 ’
冊
11.
1
0
π 3π
y
3 3s3
3s2
3s2
,
, 3s3
0,
,
2
2
2
2
3. (a) 共2, 3兾2, 兾2兲
(b) 共2, 3兾4, 3兾4兲
1
1
5. Half-cone
7. Sphere, radius 2 , center (0, 2, 0)
2
2
2
2
2
9. (a) cos 苷 sin
(b) 共sin cos cos2 兲 苷 9
(b)
z
y
3
0
冉
PAGE 1031
N
3π
4
π
2
1
5760
x
EXERCISES 15.8
PAGE 1037
1. (a)
共68 15兲
1
8
N
∏=4
0
y
_2
x
x
π
π
”4, 3 , _2’
˙= 3
∏=2
(2, 2s3 , 2)
3. (a) (s2 , 3兾4, 1)
(b) 共4, 2兾3, 3兲
5. Vertical half-plane through the z-axis
7. Circular paraboloid
9. (a) z 2 苷 1 r cos r 2
(b) z 苷 r 2 cos 2
11.
y
共0, 2, 1兲
x
13.
z
z
1
z=1
x
y
˙=
2
x
3π
4
y
∏=1
2
13. Cylindrical coordinates: 6 r 7, 0 2, 0 z 20
15. 0 兾4, 0 cos
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 128
A128
17.
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
共9兾4兲 (2 s3 )
z
3
45. (a)
47.
1
15
1. 30
x
3.
13. abc
y
(c)
1
45
sy
x01 x01z xsy
f 共x, y, z兲 dx dy dz
PROBLEMS PLUS
π
6
19.
1
3
(b)
冉
1
2
N
49. ln 2
51. 0
PAGE 1053
sin 1
2
8
3
9s3
冊
7. (b) 0.90
x0兾2 x03 x02 f 共r cos , r sin , z兲 r dz dr d
21. 312,500 兾7
23. 1688兾15
25. 兾8
27. (s3 1) a 3兾3
29. (a) 10
7
31. (a) (0, 0, 12)
(b) 11K兾960
33. (a) (0, 0, 8 a)
(b) (0, 0, 2.1)
(b) 4Ka 5兾15
1
35. 3 (2 s2 ), (0, 0, 3兾 8(2 s2 ) )
37. 5兾6
39. (4 s2 5 )兾15
41. 4096兾21
43.
45. 136兾99
3
[
]
CHAPTER 16
EXERCISES 16.1
N
1.
PAGE 1061
y
2
1
_2
0
_1
x
1
_1
EXERCISES 15.10
PAGE 1047
N
3.
1. 16
3. sin cos 5. 0
7. The parallelogram with vertices (0, 0), (6, 3), (12, 1), (6, 2)
9. The region bounded by the line y 苷 1 , the y-axis, and y 苷 sx
1
1
11. x 苷 3 共v u兲 , y 苷 3 共u 2v兲 is one possible transformation,
where S 苷 兵共u, v兲 1 u 1, 1 v 3其
13. x 苷 u cos v, y 苷 u sin v is one possible transformation,
where S 苷 兵共u, v兲 1 u s2, 0 v 兾2 其
15. 3
17. 6
19. 2 ln 3
4
21. (a) 3 abc
(b) 1.083 10 12 km 3
(c) 154 共a 2 b 2 兲abck
3
8
1
23. 5 ln 8
25. 2 sin 1
27. e e
2
y
2
2
ⱍ
_2
ⱍ
CHAPTER 15 REVIEW
True-False Quiz
1. True
3. True
N
2
x
_2
5.
y
PAGE 1049
0
5. True
7. True
x
9. False
Exercises
1
2
1. ⬇64.0
3. 4e 2 4e 3
5. 2 sin 1
7. 3
4
9. x0 x2 f 共r cos , r sin 兲 r dr d
11. The region inside the loop of the four-leaved rose r 苷 sin 2 in
the first quadrant
1
1
7
1
13. 2 sin 1
15. 2 e 6 2
17. 4 ln 2
19. 8
81
64
21. 81兾5
23. 2
25. 兾96
27. 15
2
29. 176
31. 3
33. 2ma 3兾9
1
1 8
35. (a) 4
(b) ( 3 , 15 )
(c) Ix 苷 121 , Iy 苷 241 ; y 苷 1兾s3, x 苷 1兾s6
37. (a) 共0, 0, h兾4兲
(b) a 4h兾10
486
39. ln(s2 s3 ) s2兾3
41. 5
43. 0.0512
7.
13. I
15. IV
z
x
y
x
11. IV
9.
z
y
17. III
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 129
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
19.
The line y 苷 2x
4.5
29. (a)
11
8
1兾e
(b)
A129
2.1
F{r(1)}
4.5
4.5
1
F ”r” œ„2 ’’
f 共x, y兲 苷 共xy 1兲e i x e j
x
i
23. f 共x, y, z兲 苷
sx 2 y 2 z 2
y
z
j
k
sx 2 y 2 z 2
sx 2 y 2 z 2
25. f 共x, y兲 苷 2x i j
xy
21.
F{r(0)}
0
4.5
2.1
_0.2
2 xy
31.
172,704
5,632,705
s2 共1 e14 兲
33. 2k, 共4兾, 0兲
35. (a) x 苷 共1兾m兲 xC x 共x, y, z兲 ds,
y 苷 共1兾m兲 xC y 共x, y, z兲 ds,
z 苷 共1兾m兲 xC z 共x, y, z兲 ds, where m 苷 xC 共x, y, z兲 ds
y
2
(b) 共0, 0, 3兲
_6
_4
_2
0
4
_2
27.
6
x
37. Ix 苷 k ( 2 1
), Iy 苷 k ( 12 23 )
39. 2 2
41.
43. (a) 2ma i 6mbt j, 0 t 1
(b) 2ma mb
45. ⬇1.67 10 4 ft-lb
47. (b) Yes
51. ⬇22 J
2
9
2
7
3
2
4
EXERCISES 16.3
_4
_4
33. 共2.04, 1.03兲
31. II
(b) y 苷 1兾x, x
y
0
EXERCISES 16.4
1. 8
13. 4
y 苷 C兾x
N
5
共145 3兾2 1兲
3. 1638.4
5. 8
7. 2
2
1
6
9. s5 11. 12 s14 共e 1兲
13. 5 共e 1兲
15.
17. (a) Positive
(b) Negative
19. 45
6
21. 5 cos 1 sin 1
23. 1.9633
25. 15.0074
1
54
27. 3 3.
N
2
3
PAGE 1089
5. 12
7.
1
15. 8e 48e
1
3
9. 24
17. 1
12
243
19. 3
EXERCISES 16.5
N
2
2.5
.5
21. (c)
PAGE 1097
(b) y共e z e x 兲
5. (a) 0
(b) 2兾sx 2 y 2 z 2
7. (a) 具e y cos z, e z cos x, e x cos y典
2.5
16
35
3
1. (a) 0
(b) 3
3. (a) ze x i 共xye z yze x 兲 j xe z k
2.5
11. 3
23. 共4a兾3 , 4a兾3兲 if the region is the portion of the disk
x 2 y 2 苷 a 2 in the first quadrant
27. 0
PAGE 1072
2
3
PAGE 1082
x
0
EXERCISES 16.2
N
1. 40
3. f 共x, y兲 苷 x 2 3xy 2y 2 8y K
7. f 共x, y兲 苷 ye x x sin y K
5. Not conservative
2 3
9. f 共x, y兲 苷 x ln y x y K
1
13. (a) f 共x, y兲 苷 2 x 2 y 2
(b) 2
11. (b) 16
2
15. (a) f 共x, y, z兲 苷 xyz z
(b) 77
17. (a) f 共x, y, z兲 苷 ye xz
(b) 4
19. 4兾e
21. It doesn’t matter which curve is chosen.
23. 30
25. No
27. Conservative
31. (a) Yes
(b) Yes
(c) Yes
33. (a) No
(b) Yes
(c) Yes
4
29. III
35. (a)
1.
4
3
(b) e x sin y e y sin z e z sin x
9. (a) Negative
(b) curl F 苷 0
11. (a) Zero
(b) curl F points in the negative z-direction
13. f 共x, y, z兲 苷 xy 2z 3 K
15. Not conservative
17. f 共x, y, z兲 苷 xe yz K
19. No
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
9
2
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 130
A130
APPENDIX I
EXERCISES 16.6
N
ANSWERS TO ODD-NUMBERED EXERCISES
PAGE 1108
1. P: no; Q: yes
3. Plane through 共0, 3, 1兲 containing vectors 具 1, 0, 4典 , 具 1, 1, 5典
5. Hyperbolic paraboloid
7.
2
1
s3
x yz苷
2
2
3
37. x 2z 苷 1
39. 3 s14
41. s14 4
43. 15 共3 5兾2 2 7兾2 1兲
45. 共2兾3兲(2 s2 1)
3x y 3z 苷 3
33.
47. 2 s21 1
35.
[ln(2 s21 ) ln s17 ]
17
4
49. 4
51. A共S兲 s3 R 2
53. 13.9783
55. (a) 24.2055
(b) 24.2476
√ constant
57.
z 0
45
8
[
]
15
s14 16 ln (11s5 3s70 )兾(3s5 s70 )
59. (b)
0
2
_2
x
0
u constant
y
1
1
9.
z 0
2
u constant
2
y
1
0
2
1
1 0x
(c) x02 x0 s36 sin 4u cos 2v 9 sin 4u sin 2v 4 cos 2u sin 2u du dv
√ constant
61. 4
z 0
63. 2a 2共 2兲
EXERCISES 16.7
1. 49.09
_1
_1
_1
y
0
0
1 1
PAGE 1120
3. 900
9. 171 s14
x
N
5. 11s14
15. 共兾60兲(391s17 1)
713
1
2
3
(2s2 1)
13. 364 s2 兾3
11. s21兾3
17. 16
23. 180
25. 3 27. 0
33. 4.5822
35. 3.4895
11.
7.
4
19. 12
21. 4
31. 2 29. 48
8
3
xxS F ⴢ dS 苷 xxD 关P共 h兾 x兲 Q R共 h兾 z兲兴 dA,
where D 苷 projection of S on xz-plane
39. 共0, 0, a兾2兲
41. (a) Iz 苷 xxS 共x 2 y 2 兲 共x, y, z兲 dS (b) 4329 s2 兾5
8
43. 0 kg兾s
45. 3 a 3 0
47. 1248
37.
z 0
√ constant
_1
_1
y
_1
0
0
1 1
x
EXERCISES 16.8
u constant
IV
15. II
17. III
x 苷 u, y 苷 v u, z 苷 v
y 苷 y, z 苷 z, x 苷 s1 y 2 14 z 2
x 苷 2 sin cos , y 苷 2 sin sin ,
z 苷 2 cos , 0 兾4, 0 2
or x 苷 x, y 苷 y, z 苷 s4 x 2 y 2, x 2 y 2 2
25. x 苷 x, y 苷 4 cos , z 苷 4 sin , 0 x 5, 0 2
13.
19.
21.
23.
[
3. 0
5. 0
11. (a) 81兾2
z 苷 ex sin , 0 x 3,
0 2
PAGE 1127
9. 80
7. 1
(b)
5
z 0
5
]
29. x 苷 x, y 苷 ex cos ,
N
2
0
(c) x 苷 3 cos t, y 苷 3 sin t,
z 苷 1 3共cos t sin t兲,
0 t 2
1
z 0
2
2
y
0
2
x
4
z
2
0
_2
1
1
y
31. (a) Direction reverses
0
1 0
x
2
(b) Number of coils doubles
_2
y
0
2
2
0
_2
x
17. 3
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 131
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
EXERCISES 16.9
5. y 苷 e 2x 共c1 cos x c2 sin x兲 PAGE 1133
N
5.
7. 9兾2
9. 0
11. 32兾3
13. 2
81
15. 341 s2兾60 20 arcsin(s3兾3)
17. 13兾20
19. Negative at P1 , positive at P2
9
2
7. y 苷 cos x 3
2
CHAPTER 16 REVIEW
N
x
11.
3
_3
8
yp
Exercises
1. (a) Negative
7.
110
3
11
12
9.
17. 8
5. False
11. True
(b) Positive
4兾e
1
6
25.
29. 64兾3
5.
19. y 苷 c1 cos ( 2 x) c 2 sin ( 2 x) 3 cos x
1
13. 0
27. 共兾60兲(391 s17 1)
37. 4
1
13. yp 苷 共Ax B兲e x cos x 共Cx D兲e x sin x
15. yp 苷 Axe x B cos x C sin x
17. yp 苷 xex 关共Ax 2 Bx C 兲 cos 3x 共Dx 2 Ex F兲 sin 3x兴
4
15
11. f 共x, y兲 苷 e y xe xy
33. 2
The solutions are all asymptotic
to yp 苷 101 cos x 103 sin x as
x l . Except for yp , all
solutions approach either or as x l .
_3
3. 6 s10
(27 5 s5 )
e
1
PAGE 1136
True-False Quiz
1. False
3. True
7. False
9. True
1 x
10
3
sin x e x 6x
1
2
9. y 苷 e x ( 2 x 2 x 2)
0 in quadrants I, II; div F 0 in quadrants III, IV
21. div F
11
2
A131
39. 21
1
1
21. y 苷 c1e c2 xe e
23. y 苷 c1 sin x c 2 cos x sin x ln共sec x tan x兲 1
25. y 苷 关c1 ln共1 ex 兲兴e x 关c2 ex ln共1 ex 兲兴e 2x
x
x
2x
[
]
27. y 苷 e x c1 c 2 x 2 ln共1 x 2 兲 x tan1 x
1
CHAPTER 17
EXERCISES 17.1
N
EXERCISES 17.3
PAGE 1148
1. y 苷 c1 e 3x c 2 e2x
3. y 苷 c1 cos 4x c 2 sin 4x
5. y 苷 c1 e 2x兾3 c 2 xe 2x兾3
7. y 苷 c1 c 2 e x兾2
2x
9. y 苷 e 共c1 cos 3x c 2 sin 3x兲
11. y 苷 c1 e (s31) t兾2 c 2 e (s31) t兾2
[
1
0.02
1.4
I共t兲 苷 e
sin 20t
3
3
15. Q共t兲 苷 e10t [ 250 cos 20t 500 sin 20t]
3
3
250 cos 10t 125 sin 10t
17. y 苷 3e e 4x
19. y 苷 e 2x兾3 3 xe 2x兾3
3x
21. y 苷 e 共2 cos x 3 sin x兲
3
125
,
2
4x4
2x
e
1
7
33x
2xe
2x
25. y 苷 5 cos 2x 3 sin 2x
29. y 苷
e2
ex
e1
e1
31. No solution
33. (b) 苷 n 2 2兾L2, n a positive integer; y 苷 C sin共n x兾L兲
35. (a) b a 苷 n, n any integer
(b) b a 苷 n and
c
cos a
苷 e ab
unless cos b 苷 0, then
d
cos b
c
sin a
苷 e ab
d
sin b
(c) b a 苷 n and
c
cos a
苷 e ab
unless cos b 苷 0, then
d
cos b
c
sin a
苷 e ab
d
sin b
EXERCISES 17.2
1. y 苷 c1e
49
12
c=20
c=25
c=30
3 10t
5
_10
3x
5.
c=15
13. Q共t兲 苷 共e10t兾250兲共6 cos 20t 3 sin 20t兲 2x
27. y 苷 2e
6
_0.11
3
23. y 苷 e
1
0
]
f
1
7
3. x 苷 5 e6t 5 et
0 or as x l .
g
_3
PAGE 1163
1. x 苷 0.35 cos (2 s5 t)
7.
c=10
13. P 苷 et c1 cos (10 t) c 2 sin (10 t)
15.
All solutions approach either
10
1
N
EXERCISES 17.4
N
PAGE 1168
xn
x 3n
3
1. c0 兺
苷 c0 e x
3. c0 兺 n 苷 c0 e x 兾3
n!
3
n!
n苷0
n苷0
共1兲n 2n
共2兲n n! 2n1
x c1 兺
x
5. c0 兺 n
n苷0 2 n!
n苷0 共2n 1兲!
n
x
苷 c0 c1 ln共1 x兲 for x 1
7. c0 c1 兺
n苷1 n
x 2n
2
9. 兺 n 苷 e x 兾2
n苷0 2 n!
共1兲n225 2 ⴢ ⴢ 共3n 1兲2 3n1
x
11. x 兺
共3n 1兲!
n苷1
ⱍ ⱍ
CHAPTER 17 REVIEW
N
PAGE 1169
True-False Quiz
1. True
3. True
N
PAGE 1155
c2 e x 657 cos 2x 654 sin 2x
3. y 苷 c1 cos 3x c2 sin 3x 1 2x
13
e
Exercises
1. y 苷 c1e x兾2 c2 e x兾2
3. y 苷 c1 cos(s3 x) c2 sin(s3 x)
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
kg
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 132
A132
APPENDIX I
ANSWERS TO ODD-NUMBERED EXERCISES
21. y 苷 6x 15
23. 2x 3y 19 苷 0
25. 5x y 苷 11
27. y 苷 3x 2
29. y 苷 3x 3
31. y 苷 5
33. x 2y 11 苷 0
35. 5x 2y 1 苷 0
5. y 苷 e 2x共c1 cos x c2 sin x 1兲
1
1
7. y 苷 c1e x c2 xe x 2 cos x 2 共x 1兲 sin x
1
1
3x
2x
2x
6 5 xe
9. y 苷 c1e c2 e
11. y 苷 5 2e6共x1兲
13. y 苷 共e 4x e x 兲兾3
15. No solution
37. m 苷 3 ,
39. m 苷 0,
1
b苷0
共2兲nn! 2n1
17. 兺
x
n苷0 共2n 1兲!
19. Q共t兲 苷 0.02e10t共cos 10t sin 10t兲 0.03
21. (c) 2兾k ⬇ 85 min
(d) ⬇17,600 mi兾h
41. m 苷 4 ,
3
b 苷 2
y
b 苷 3
y
y
0
0
x
0
x
y=_2
x
_2
_3
APPENDIXES
EXERCISES A
3. 1. 18
ⱍ
ⱍ
9. x 1 苷
43.
PAGE A9
N
5. 5 s5
再
x1
x 1
11. x 2 1
0
_1
17. 共3, 兲
x
0
0
19. 共2, 6兲
47.
2
3
49.
y
y
6
y4
23. [1, 2 )
21. 共0, 1兴
1
0
1
_1
1
_2
1
2
0
0
_1
_œ„
3
33. 共, 1兴
51.
y
y=1+x
0
œ„
3
共0, 1兲
35. 共1, 0兲 傼 共1, 兲
_1
1
0
x
0
1
y=1-2x
( 14 , )
0 1
4
39. 10 C 35
41. (a) T 苷 20 10h, 0 h 12
3
4
(b) 30C T 20C
43. 2
45. 2, 3
47. 共3, 3兲
49. 共3, 5兲
51. 共, 7兴 傼 关3, 兲
53. 关1.3, 1.7兴
55. 关4, 1兴 傼 关1, 4兴
共c b兲兾a
57. x 共a b兲c兾共ab兲
59. x
(b) 共3.5, 3兲
53. 共0, 4兲
55. (a) 共4, 9兲
59. y 苷 x 3
61. (b) 4x 3y 24 苷 0
EXERCISES C
N
PAGE A15
5. 2 s37
3. s74
y
7. 2
19.
9
7. ( 2 , 0), 2
1
1
3. x 2 y 2 苷 65
9.
( 14 , 14 ), s10兾4
13. Ellipse
y
y
y
0
0
2
x
xy=0
x
57. 共1, 2兲
PAGE A23
1. 共x 3兲 共 y 1兲2 苷 25
11. Parabola
9. 2
x=3
3
N
2
5. 共2, 5兲, 4
EXERCISES B
x
1
2
31. (s3, s3 )
29. 共, 兲
37. 共, 0兲 傼
x2
x
1
2
0
2
27. [1, 2 ]
25. 共, 1兲 傼 共2, 兲
0
x
0
15. 关1, 兲
_2
1. 5
17.
y
7. 2 x
for x 1
for x 1
13. 共2, 兲
45.
y
x
_4
0
4 x
_2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 133
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
15. Hyperbola
y
_5
17. Ellipse
y
4
y= 5 x
1
5
0
EXERCISES D
x
0
1
_2
4
PAGE A32
N
1. 7兾6
3. 兾20
5. 5
11. 67.5
13. 3 cm
15.
2
3
17.
19.
y
7. 720°
9. 75°
rad 苷 共120兾兲
y
x
1
2
0
_1
y=_ 5 x
0
x
x
315°
19. Parabola
21. Hyperbola
y
x
y=_ 3
x
y
x
y= 3
1
0
_1
_1
3π
_ 4
21.
y
1
A133
2 rad
0
x
0
x
_1
23. Hyperbola
23. sin共3兾4兲 苷 1兾s2, cos共3兾4兲 苷 1兾s2, tan共3兾4兲 苷 1,
csc共3兾4兲 苷 s2, sec共3兾4兲 苷 s2, cot共3兾4兲 苷 1
25. sin共9兾2兲 苷 1, cos共9兾2兲 苷 0, csc共9兾2兲 苷 1, cot共9兾2兲 苷 0,
tan共9兾2兲 and sec共9兾2兲 undefined
25. Ellipse
y
y
27. sin共5兾6兲 苷 2 , cos共5兾6兲 苷 s3兾2, tan共5兾6兲 苷 1兾s3,
1
csc共5兾6兲 苷 2, sec共5兾6兲 苷 2兾s3, cot共5兾6兲 苷 s3
(1, 2)
0
29. cos 苷 5 , tan 苷 4 , csc 苷 3 , sec 苷 4 , cot 苷
4
x
x
29. Parabola
y
y
0
(3, 4)
1
15
(3 8 s2 )
4 x
4
3
24
25
71. 0, , 2
65. 兾3, 5兾3
69. 兾6, 兾2, 5兾6, 3兾2
73. 0 x 兾6 and 5兾6 x 2
75. 0 x 兾4, 3兾4 x 5兾4, 7兾4 x 2
2
77.
x
y
1
1
2
33.
y
63.
67. 兾4, 3兾4, 5兾4, 7兾4
2
31. Ellipse
5
苷 s5兾3, cos 苷 23 , tan 苷 s5兾2, csc 苷 3兾s5,
cot 苷 2兾s5
1
33. sin 苷 1兾s10, cos 苷 3兾s10, tan 苷 3 ,
csc 苷 s10, sec 苷 s10兾3
1
35. 5.73576 cm
37. 24.62147 cm
59. 15 (4 6 s2 )
61.
0
5
31. sin
0
27. Parabola
3
0
π
3
y
x
5π
6
(3, 9)
79.
0
1
3
y
x
5
0
x
0
35. y 苷 x 2 2x
37.
3π 2π
2
π
x
5π 3π
2
39.
y
y
81.
y
1
0
π
2
1
1
x
0
1
1
x
_π
0
π
2π
x
89. 14.34457 cm 2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
97909_Ans_Ans_pA126-134.qk_97909_Ans_Ans_pA126-134 10/14/10 5:11 PM Page 134
A134
ANSWERS TO ODD-NUMBERED EXERCISES
APPENDIX I
EXERCISES E
N
PAGE A38
1. s1 s2 s3 s4 s5
3. 3 4 3 5 3 6
1
3
5
7
10
5. 1 3 5 7 9
7. 1 210 310 n10
10
9. 1 1 1 1 共1兲n1
11.
兺i
兺
13.
i苷1
n
i
i1
21. 80
15.
5
兺 2i
17.
i苷1
23. 3276
25. 0
31. n共n 2 6n 17兲兾3
27. 61
43.
45. 14
EXERCISES G
N
19.
兺x
i
49. 2
n1
25. 3 s2 关cos共3兾4兲 i sin共3兾4兲兴
[
27. 5{cos tan
( )] i sin[tan1( 43)]}
1 4
3
关cos共兾6兲 i sin共兾6兲兴
31. 4 s2 关cos共7兾12兲 i sin共7兾12兲兴,
i苷1
29. n共n 1兲
33. n共n 2 6n 11兲兾3
35. n共n 3 2n 2 n 10兲兾4
41. (a) n 4
(b) 5 100 1
(c)
1
3
兺2
i苷0
23. (s7兾2)i
1
2
1
2
n
i
21. 1 2i
3
29. 4关cos共兾2兲 i sin共兾2兲兴, cos共兾6兲 i sin共兾6兲,
i苷1
19
19. 2 i
17. 4i, 4
(2 s2 )关cos共13兾12兲 i sin共13兾12兲兴, 14 关cos共兾6兲 i sin共兾6兲兴
35. 512 s3 512i
33. 1024
37. 1, i, (1兾s2 )共1 i 兲
(d) an a0
n n2
97
300
39. (s3兾2) 2 i, i
Im
i
2
Im
0
PAGE A54
1
Re
1. 8 4i
9.
1
2
i
1
2
0
Re
_i
1. (b) 0.405
EXERCISES H
1
N
3. 13 18i
11. i
13. 5i
41. i
43. 2 (s3兾2) i
45. e 2
3
47. cos 3 苷 cos 3 cos sin2,
1
PAGE A64
5. 12 7i
7.
11
13
15. 12 5i, 13
10
13
i
sin 3 苷 3 cos2 sin sin3
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