Stewart J. Calculus
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A118
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APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
13. Cylindrical coordinates: , ,
15.
17. 19. 21.
23.
(a) (b)
25. 27. 0
29. (a) , where is the cone
(b) ft-lb
EXERCISES 16.8
N
PAGE 1046
1. (a)
(0,0,1)
(b)
3. (a) (b)
5. Half-cone
7. Sphere, radius , center
9. (a) (b)
11.
13.
15.
0
兾4, 0
cos
x
z
y
˙=
3π
4
∏=1
∏=2
z
y
x
2
2
2
2
共sin
2
cos
2
cos
2
兲 苷 9cos
2
苷 sin
2
(
0,
1
2
, 0
)
1
2
(
s
2, 3
兾2, 3
兾4
)
共4,
兾3,
兾6兲
(
1
2
s
2,
1
2
s
6,
s
2
)
0
z
y
x
”2, , ’
2
π
4
π
3
π
3
π
4
0
z
y
x
(1,0,0)
1
⬇3.1 10
19
Cxxx
C
h共P兲t共P兲 dV
Ka
2
兾8, 共0, 0, 2a兾3兲
共0, 0, 15兲162
2
兾5
共e
6
e 5兲384
64
兾3
x
4
y
4
z
4
0 z 200
2
6 r 7
17.
19.
21. 23. 25.
27. 29.
(a) (b) (0, 0, 2.1)
31.
33.
(a) (b)
35.
37. 39.
41. 43.
EXERCISES 16.9
N
PAGE 1056
1. 16 3. 5. 0
7. The parallelogram with vertices (0, 0), (6, 3), (12, 1), (6, 2)
9. The region bounded by the line , the y-axis, and
11. 13. 15. 2 ln 3
17. (a) (b)
19. 21. 23.
CHAPTER 16 REVIEW
N
PAGE 1057
True-False Quiz
1.
True 3. True 5. True 7. False
Exercises
1. 3. 5. 7.
9.
11.
The region inside the loop of the four-leaved rose in
the first quadrant
13. 15. 17. 19.
21. 23.
40.5 25. 27. 29. 176
31. 33.
35.
(a) (b)
(c)
37.
39.
97.2 41. 0.0512
43. (a) (b) (c)
45. 47. 49. 0
PROBLEMS PLUS
N
PAGE 1060
1. 30 3. 7. (b) 0.90
1
2
sin 1
ln 2
x
1
0
x
1z
0
x
s
y
s
y
f 共x, y, z兲 dx dy dz
1
45
1
3
1
15
共0, 0, h兾4兲
I
x
苷
1
12
, I
y
苷
1
24
; y 苷 1兾
s
3, x 苷 1兾
s
6
(
1
3
,
8
15
)
1
4
2ma
3
兾9
2
3
64
15
兾9681
兾5
8
1
4
ln 2
1
2
e
6
7
2
1
2
sin 1
r 苷 sin 2
x
0
x
4
2
f 共r cos
, r sin
兲
r dr d
2
3
1
2
sin 14e
2
4e 3⬇64.0
e e
1
3
2
sin 1
8
5
ln 8
1.083 10
12
km
3
4
3
abc
6
3
y 苷
s
x
y 苷 1
sin
2
cos
2
136
兾99
(
4
s
2 5
)
兾155
兾6
共2
兾3兲
[
1
(
1兾
s
2
)]
,
(
0, 0, 3兾
[
8
(
2
s
2
)])
4K
a
5
兾15
(
0, 0,
3
8
a
)
(
0,
525
296
, 0
)
10
(
s
3 1
)
a
3
兾3
1562
兾1515
兾16312,500
兾7
x
兾2
0
x
3
0
x
2
0
f 共r cos
, r sin
, z兲 r dz dr d
共9
兾4兲
(
2
s
3
)
x
y
z
π
6
3
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
||||
A119
CHAPTER 17
EXERCISES 17.1
N
PAGE 1068
1.
3. 5.
7. 9.
11.
II 13. I 15. IV 17. III
19. The line
21.
23.
25.
0
_2_4_6 4 6
x
y
_2
2
f 共x, y兲 苷 2x i j
y
s
x
2
y
2
z
2
j
z
s
x
2
y
2
z
2
k
f 共x, y, z兲 苷
x
s
x
2
y
2
z
2
i
f 共x, y兲 苷 共xy 1兲e
xy
i x
2
e
xy
j
y 苷 2x
4.5
4.5
4.5
4.5
z
y
x
z
y
x
y
x
0
y
x
0
_2
2
2
y
1
0
_1
1
x
_1
_2
_2
27.
29.
III 31. II 33.
35.
(a) (b)
EXERCISES 17.2
N
PAGE 1079
1. 3. 1638.4 5. 7. 9.
11. 13. 15.
17.
(a) Positive (b) Negative
19. 45 21. 23. 1.9633 25. 15.0074
27.
29.
(a) (b)
31. 33.
35.
(a) ,
,
, where
(b)
37. ,
39. 41. 26 43. 45. (b) Yes
47.
⬇22 J
1.67 10
4
ft-lb2
2
I
y
苷 k
(
1
2
2
3
)
I
x
苷 k
(
1
2
4
3
)
共0, 0, 3
兲
m 苷
x
C
共x, y, z兲 dsz 苷 共1兾m兲 x
C
z
共x, y, z兲 ds
y 苷 共1兾m兲 x
C
y
共x, y, z兲 ds
x 苷 共1兾m兲 x
C
x
共x, y, z兲 ds
2
k,
共4兾
, 0兲
172,704
5,632,705
s
2共1 e
14
兲
1.6
0.2
01
F(r(1))
F(r(0))
F
”
r
”
’’
1
œ
„
2
1.6
11
8
1兾e
2.5
2.5
2.5 2 .5
3
2
3
6
5
cos 1 sin 1
97
3
1
5
1
12
s
14
共e
6
1兲
s
5
17
3
243
8
1
54
共145
3兾2
1兲
y 苷 C兾x
y 苷 1兾x, x 0
y
x
0
共2.04, 1.03兲
6
66
6
A120
||||
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
EXERCISES 17.3
N
PAGE 1089
1. 40 3.
5. 7.
9.
11.
(b) 16 13. (a) (b) 2
15. (a) (b) 77
17. (a) (b) 0
19. 2 21. 23. No 25. Conservative
29. (a) Yes (b) Yes (c) Yes
31. (a) Yes (b) Yes (c) No
EXERCISES 17.4
N
PAGE 1096
1. 3. 5. 12 7. 9. 11.
13. 15. 17. 19. 21.
(c)
23. if the region is the portion of the disk
in the first quadrant
EXERCISES 17.5
N
PAGE 1104
1. (a) (b)
3. (a) (b)
5. (a) (b)
7. (a) (b)
9. (a) Negative (b)
11. (a) Zero (b) curl F points in the negative -direction
13. 15.
17.
Not conservative 19. No
EXERCISES 17.6
N
PAGE 1114
1. : no; : yes
3. Plane through containing vectors ,
5. Hyperbolic paraboloid
7.
9.
_1
0
x
1
_1
0
y
z
u constant
√ constant
_1
0
1
1
2
0
_2
2
1
0
2
1.5
1
x
y
z
√ constant
u constant
具1, 1, 5典具1, 0, 4典共0, 3, 1兲
QP
f 共x, y, z兲 苷 x
2
y y
2
z Kf 共x, y, z兲 苷 xy
2
z
3
K
z
curl F 苷 0
1兾x 1兾y 1兾z具1兾y, 1兾x, 1兾x典
2兾
s
x
2
y
2
z
2
0
z 1兾
(
2
s
z
)
共x y兲
i y j k
yzx
2
i 3xy j xz k
x
2
y
2
苷 a
2
共4a兾3
, 4a兾3
兲
9
2
3
1
12
8e 48e
1
625
2
4
3
2
24
1
3
2
3
8
30
f 共x, y, z兲 苷 xy
2
cos z
f 共x, y, z兲 苷 xyz z
2
f 共x, y兲 苷
1
2
x
2
y
2
f 共x, y兲 苷 x ln y x
2
y
3
K
f 共x, y兲 苷 ye
x
x sin y Kf 共x, y兲 苷 e
x
sin y K
f 共x, y兲 苷 x
2
3xy 2y
2
8y K
11.
13.
IV 15. II 17. III
19.
21.
23.
,,
,,
25. ,,,
29. ,
,,
31. (a) Direction reverses (b) Number of coils doubles
33. 35. 37.
39. 41.
43.
45. 47.
4
49.
51.
(a) 24.2055 (b) 24.2476
53.
55.
(b)
(c)
57. 59.
EXERCISES 17.7
N
PAGE 1127
1. 49.09 3. 5. 7.
9. 11.
13. 15. 17.
1216
共
兾60兲
(
391
s
17 1
)
364
s
2
兾35
s
5兾48 1兾240
s
3
兾24171
s
14900
2a
2
共
2兲4
x
2
0
x
0
s
36 sin
4
u cos
2
v 9 sin
4
u sin
2
v 4 cos
2
u sin
2
u du dv
2
0
2
2
1
0
21
0
z
y
x
45
8
s
14
15
16
ln
[
(
11
s
5 3
s
70
)
兾
(
3
s
5
s
70
)
]
13.9783
1
2
s
21
17
4
[
ln
(
2
s
21
)
ln
s
17
]
共
兾6兲
(
17
s
17 5
s
5
)
共2
兾3兲
(
2
s
2 1
)
4
15
共3
5兾2
2
7兾2
1兲
3
s
14
x 2z 苷 13x y 3z 苷 3
0
2
0 x 3z 苷 e
x
sin
2
0
1
0
1
1
0
1
y
z
x
x 苷 x, y 苷 e
x
cos
0 x 5, 0
2
z 苷 4 sin
y 苷 4 cos
x 苷 x
[
or x 苷 x, y 苷 y, z 苷
s
4 x
2
y
2
, x
2
y
2
2
]
0
2
0
兾4z 苷 2 cos
y 苷 2 sin
sin
x 苷 2 sin
cos
x 苷 x, z 苷 z, y 苷
s
1 x
2
z
2
x 苷 1 u v, y 苷 2 u v, z 苷 3 u v
_1
0
x
1
_1
0
y
1
_1
0z
1
√ constant
u constant
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
||||
A121
19. 21. 23. 25. 0 27. 48
29. 31. 33. 3.4895
35. ,
where projection of on -plane
37.
39.
(a) (b)
41. 43. 45.
EXERCISES 17.8
N
PAGE 1133
3. 0 5. 0 7. 1 9.
11.
(a) (b)
(c) ,
,
17. 3
EXERCISES 17.9
N
PAGE 1139
5. 2 7.
9.
0 11. 13.
15. 17.
19.
Negative at , positive at
21. in quadrants I, II; in quadrants III, IV
CHAPTER 17 REVIEW
N
PAGE 1142
True-False Quiz
1.
False 3. True 5. False 7. True
Exercises
1.
(a) Negative (b) Positive 3. 5.
7. 9. 11. 13.
0
17. 25.
27. 29.
33. 37. 39.
214
1
2
64
兾3共
兾60兲
(
391
s
17 1
)
1
6
(
27 5
s
5
)
8
f 共x, y兲 苷 e
y
xe
xy
11
12
4兾e
110
3
4
15
6
s
10
div F
0div F 0
P
2
P
1
13
兾20341
s
2兾60
81
20
arcsin
(
s
3兾3
)
032
兾3
9
兾2
0 t 2
z 苷 1 3共cos t sin t兲
_2
0
2
4
_2
0
2
2
0
_2
z
y
x
x 苷 3 cos t, y 苷 3 sin t
2
5
0
5
z
0
y
2
2
2
0
x
81
兾2
80
1248
8
3
a
3
0
0 kg兾s
4329
s
2
兾5I
z
苷 xx
S
共x
2
y
2
兲
共x, y, z兲 dS
共0, 0, a兾2兲
xzSD 苷
xx
S
F ⴢ dS 苷 xx
D
关P共h兾x兲 Q R共h兾z兲兴 dA
0.16422
8
3
4
3
1
6
713
180
CHAPTER 18
EXERCISES 18.1
N
PAGE 1153
1. 3.
5. 7.
9.
11.
13.
15.
All solutions approach either
or as .
17. 19.
21. 23.
25. 27.
29.
No solution
31.
33.
(b) , n a positive integer;
EXERCISES 18.2
N
PAGE 1160
1.
3.
5.
7.
9.
11.
The solutions are all asymptotic
to as
. Except for , all
solutions approach either
or as .
13.
15.
17.
19.
21.
23.
25.
27.
EXERCISES 18.3
N
PAGE 1168
1. 3. 5.
49
12
kgx 苷
1
5
e
6t
6
5
e
t
x 苷 0.35 cos
(
2
s
5 t
)
y 苷 e
x
[
c
1
c
2
x
1
2
ln共1 x
2
兲 x tan
1
x
]
y 苷 关c
1
ln共1 e
x
兲兴e
x
关c
2
e
x
ln共1 e
x
兲兴e
2x
y 苷 c
1
sin x c
2
cos x sin x ln共sec x tan x兲 1
y 苷 c
1
e
x
c
2
xe
x
e
2x
y 苷 c
1
cos
(
1
2
x
)
c
2
sin
(
1
2
x
)
1
3
cos x
y
p
苷 xe
x
关共Ax
2
Bx C 兲 cos 3x 共Dx
2
Ex F兲 sin 3x兴
y
p
苷 Ax 共Bx C 兲e
9x
y
p
苷 Ae
2x
共Bx
2
Cx D兲 cos x 共Ex
2
Fx G兲 sin x
x l
y
p
x l
y
p
苷
1
10
cos x
3
10
sin x
3
_3
_3 8
y
p
y 苷 e
x
(
1
2
x
2
x 2
)
y 苷
3
2
cos x
11
2
sin x
1
2
e
x
x
3
6x
y 苷 e
2x
共c
1
cos x c
2
sin x兲
1
10
e
x
y 苷 c
1
c
2
e
2x
1
40
cos 4x
1
20
sin 4x
y 苷 c
1
e
2x
c
2
e
x
1
2
x
2
3
2
x
7
4
y 苷 C sin共n
x兾L兲
苷 n
2
2
兾L
2
y 苷 e
2x
共2 cos 3x e
sin 3x兲
y 苷
e
x3
e
3
1
e
2x
1 e
3
y 苷 3 cos
(
1
2
x
)
4 sin
(
1
2
x
)
y 苷 e
x
共2 cos x 3 sin x兲 y 苷 3 cos 4x sin 4x
y 苷 e
x/2
2xe
x兾2
y 苷 2e
3x兾2
e
x
x l 0
10
_10
_3 3
g
f
P 苷 e
t
[
c
1
cos
(
1
10
t
)
c
2
sin
(
1
10
t
)
]
y 苷 c
1
e
(
s
3
1
)
t兾2
c
2
e
(
s
3
1
)
t兾2
y 苷 e
2x
共c
1
cos 3x c
2
sin 3x兲
y 苷 c
1
c
2
e
x兾2
y 苷 c
1
e
2x兾3
c
2
xe
2x兾3
y 苷 c
1
cos 4x c
2
sin 4xy 苷 c
1
e
3x
c
2
e
2x
A122
||||
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
7.
13.
,
15.
EXERCISES 18.4
N
PAGE 1173
1. 3.
5.
7.
for
9.
11.
CHAPTER 18 REVIEW
N
PAGE 1174
True-False Quiz
1.
True 3. True
Exercises
1. 3.
5.
7.
9.
11. 13.
15.
17.
19.
(c) (d)
APPENDIXES
EXERCISES A
N
PAGE A9
1. 18 3. 5. 7.
9. 11.
13. 15.
0
_1
0
_2
关1, 兲共2, 兲
x
2
1
ⱍ
x 1
ⱍ
苷
再
x 1
x 1
for x 1
for x
1
2 x5
s
5
⬇17,600 mi兾h2
兾k ⬇ 85 min
Q共t兲 苷 0.02e
10t
共cos 10t sin 10t兲 0.03
兺
n苷0
共2兲
n
n!
共2n 1兲!
x
2n1
y 苷 共e
4x
e
x
兲兾3y 苷 5 2e
6共x1兲
y 苷 c
1
e
3x
c
2
e
2x
1
6
1
5
xe
2x
y 苷 c
1
e
x
c
2
xe
x
1
2
cos x
1
2
共x 1兲 sin x
y 苷 e
2x
共c
1
cos x c
2
sin x 1兲
y 苷 c
1
cos
(
s
3x
)
c
2
sin
(
s
3x
)
y 苷 c
1
e
5x
c
2
e
3x
x
兺
n苷1
共1兲
n
2
2
5
2
ⴢ ⴢ 共3n 1兲
2
共3n 1兲!
x
3n1
兺
n苷0
x
2n
2
n
n!
苷 e
x
2
兾2
ⱍ
x
ⱍ
1c
0
c
1
兺
n苷1
x
n
n
苷 c
0
c
1
ln共1 x兲
c
0
兺
n苷0
共1兲
n
2
n
n!
x
2n
c
1
兺
n苷0
共2兲
n
n!
共2n 1兲!
x
2n1
c
0
兺
n苷0
x
3n
3
n
n!
苷 c
0
e
x
3
兾3
c
0
兺
n苷0
x
n
n!
苷 c
0
e
x
3
250
cos 10t
3
125
sin 10t
Q共t兲 苷 e
10t
[
3
250
cos 20t
3
500
sin 20t
]
I共t兲 苷
3
5
e
10t
sin 20t
Q共t兲 苷 共e
10t
兾250兲共6 cos 20t 3 sin 20t兲
3
125
c=30
c=25
c=20
c=15
c=10
0.02
_0.11
0
1.4
17. 19.
21. 23.
25. 27.
29. 31.
33. 35.
37.
39. 41.
(a)
(b)
43. 45.
47. 49. 51.
53. 55.
57. 59.
EXERCISES B
N
PAGE A15
1. 5 3. 5. 7. 2 9.
17. 19.
21. 23.
25. 27. 29.
31. 33. 35.
37.
, 39. , 41. ,
43. 45.
0
y
x
0
y
x
0
x
y
_3
0
_2
x
y
y=_2
0
x
y
b 苷 3b 苷 2b 苷 0
m 苷
3
4
m 苷 0m 苷
1
3
5x 2y 1 苷 0x 2y 11 苷 0y 苷 5
y 苷 3x 3y 苷 3x 25x y 苷 11
2x 3y 19 苷 0y 苷 6x 15
0
x
y
xy=0
0
3
x
y
x=3
9
2
2
s
37
s
74
x 共c b兲兾ax 共a b兲c兾共ab兲
关4, 1兴 傼 关1, 4兴关1.3, 1.7兴
共, 7兴 傼 关3, 兲共3, 5兲共3, 3兲
2,
4
3
3
2
30C T 20C
T 苷 20 10h, 0 h 1210 C 35
0 1
4
共, 0兲 傼
(
1
4
,
)
_1 10
0 1
共1, 0兲 傼 共1, 兲共, 1兴
_œ„3 0œ„3
(
s
3,
s
3
)
共, 兲
_1
1
2
12
[
1,
1
2
]
共, 1兲 傼 共2, 兲
_1
1
2
01
[
1,
1
2
)
共0, 1兴
26
3
共2, 6兲共3, 兲
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
||||
A123
47. 49.
51.
53. 55.
(a) (b) 57.
59. 61.
(b)
EXERCISES C
N
PAGE A23
1. 3.
5. 7. 9.
11.
Parabola 13. Ellipse
15. Hyperbola 17. Ellipse
19. Parabola 21. Hyperbola
23. Hyperbola 25. Ellipse
y
0
x
(1,2)
y
0
x
y
0
x
y=
x
3
y=_
x
3
1
_1
0
x
y
_1
1
_1
0
x
y
1
_1
1
2
1
2
_
0
x
y
_5 5
y=_ x
4
5
y= x
4
5
0
x
y
_4 4
2
_2
0
x
y
(
1
4
,
1
4
)
,
s
10兾4
(
1
2
, 0
)
,
1
2
共2, 5兲, 4
x
2
y
2
苷 65共x 3兲
2
共y 1兲
2
苷 25
4x 3y 24 苷 0y 苷 x 3
共1, 2兲共3.5, 3兲共4, 9兲共0, 4兲
0
y
x
y=1-2
x
y=1+x
共0, 1兲
0
y
x
x 2
y 4
0
y
x
_2
2
27. Parabola 29. Parabola
31. Ellipse 33.
35.
37. 39.
EXERCISES D
N
PAGE A32
1. 3. 5. 7. 720° 9. 75°
11. 13. 15.
17. 19.
21.
23.
,,,
,,
25. ,,,,
and undefined
27.
29.
,,,,
31. ,, , ,
cot
苷 2兾
s
5
csc
苷 3兾
s
5tan
苷
s
5兾2cos
苷
2
3
sin
苷
s
5兾3
cot
苷
4
3
sec
苷
5
4
csc
苷
5
3
tan
苷
3
4
cos
苷
4
5
cot共5
兾6兲 苷
s
3sec共5
兾6兲 苷 2兾
s
3,csc共5
兾6兲 苷 2,
tan共5
兾6兲 苷 1兾
s
3,cos共5
兾6兲 苷
s
3兾2,sin共5
兾6兲 苷
1
2
,
sec共9
兾2兲tan共9
兾2兲
cot共9
兾2兲 苷 0csc共9
兾2兲 苷 1cos共9
兾2兲 苷 0sin共9
兾2兲 苷 1
cot共3
兾4兲 苷 1sec共3
兾4兲 苷
s
2csc共3
兾4兲 苷
s
2
tan共3
兾4兲 苷 1cos共3
兾4兲 苷 1兾
s
2sin共3
兾4兲 苷 1兾
s
2
0
x
y
2 rad
0
x
y
_
3π
4
0
x
y
315°
2
3
rad 苷 共120兾
兲3
cm67.5
5
兾207
兾6
y
0
x
1
1
y
0
x
1
1
y 苷 x
2
2x
0
x
y
(3, 9)
0
x
y
1 35
y
0
x
2
2
4
(3,4)
y
0
x
A124
||||
APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
33. ,,,
,
35. 5.73576 cm 37. 24.62147 cm 59.
61. 63. 65.
67. 69.
71. 73.
and
75.
77.
79.
81.
89.
EXERCISES E
N
PAGE A38
1. 3.
5. 7.
9. 11.
兺
10
i苷1
i1 1 1 1 共1兲
n1
1
10
2
10
3
10
n
10
1
1
3
3
5
5
7
7
9
3
4
3
5
3
6
s
1
s
2
s
3
s
4
s
5
14.34457 cm
2
y
0
x
1
π 2π
_π
y
0
x
3π
2
2ππ
π
2
5π
2
3π
y
0
x
1
1
2
π
3
5π
6
0 x
兾4, 3
兾4
x
5
兾4, 7
兾4
x 2
5
兾6 x 2
0 x
兾60,
, 2
兾6,
兾2, 5
兾6, 3
兾2
兾4, 3
兾4, 5
兾4, 7
兾4
兾3, 5
兾3
24
25
1
15
(
3 8
s
2
)
1
15
(
4 6
s
2
)
sec
苷
s
10兾3csc
苷
s
10
tan
苷
1
3
cos
苷 3兾
s
10sin
苷 1兾
s
10
13. 15. 17. 19.
21.
80 23. 3276 25. 0 27. 61 29.
31. 33.
35.
41.
(a) (b) (c) (d)
43. 45. 14 49.
EXERCISES G
N
PAGE A55
1. 3. 5. 7.
9. 11. 13. 15.
17. 19. 21.
23. 25.
27.
29.
,
31. ,
,
33. 35.
37. 39.
41. 43. 45.
47.
,
sin 3
苷 3
cos
2
sin
sin
3
cos 3
苷 cos
3
3 cos
sin
2
e
2
1
2
(
s
3兾2
)
i
i
0
Im
Re
_i
0
Im
Re
i
1
(
s
3兾2
)
1
2
i, i
1, i,
(
1兾
s
2
)
共1 i 兲
512
s
3
512i
1024
1
4
关cos共
兾6兲 i sin共
兾6兲兴
(
2
s
2
)
关cos共13
兾12兲 i sin共13
兾12兲兴
4
s
2
关cos共7
兾12兲 i sin共7
兾12兲兴
1
2
关cos共
兾6兲 i sin共
兾6兲兴
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
5
{
cos
[
tan
1
(
4
3
)
]
i sin
[
tan
1
(
4
3
)
]
}
3
s
2 关cos共3
兾4兲 i sin共3
兾4兲兴
1
2
(
s
7兾2
)
i
1 2i
3
2
i
4i, 4
12 5i, 135ii
1
2
1
2
i
11
13
10
13
i
12 7i13 18i8 4i
2
n1
n
2
n 2
1
3
a
n
a
0
97
300
5
100
1n
4
n共n
3
2n
2
n 10兲兾4
n共n
2
6n 11兲兾3n共n
2
6n 17兲兾3
n共n 1兲
兺
n
i苷1
x
i
兺
5
i苷0
2
i
兺
n
i苷1
2i
兺
19
i苷1
i
i 1
INDEX
A125
Abel, Niels, 167
absolute maximum and minimum, 205,
959, 964
absolute maximum and minimum
values, 205, 959, 964
absolute value, 17, A6, A47
absolute value function, 17
absolutely convergent series, 750
acceleration, 130, 171
acceleration of a particle, 875
components of, 878
as a vector, 875
Achilles and the tortoise, 6
adaptive numerical integration, 540
addition formulas for sine and
cosine, A28, A29
addition of vectors, 806, 809
Airy, Sir George, 764
Airy function, 764
algebraic function, 31
alternating harmonic series, 747, 750
alternating series, 746
Alternating Series Estimation Theorem, 748
Alternating Series Test, 746
analytic geometry, A10
angle, A24
between curves, 202
of deviation, 213
negative, A25
positive, A25
standard position, A25
between vectors, 815
angular momentum, 884
angular speed, 876
antiderivative, 274
antidifferentiation formulas, 276
aphelion, 703
apolune, 697
approach path of an aircraft, 164
approximate integration, 531
approximating cylinder, 356
approximating surface, 568
approximation
by differentials, 191
to e, 397
by the Midpoint Rule, 532
by Newton’s method, 269
by an nth-degree Taylor
polynomial, 195
quadratic, 195
by Riemann sums, 301
by Simpson’s Rule, 536, 537
tangent line, 189
by Taylor polynomials, 785
by Taylor’s Inequality, 773
by the Trapezoidal Rule, 533
approximation, linear, 189, 930, 934
to a tangent plane, 930
Archimedes’ Principle, 381, 1140
arc length, 561, 670
of a parametric curve, 669
of a polar curve, 688
of a space curve, 866, 867
arc length contest, 568
arc length formula, 562
arc length function, 564
area, 3, 289
of a circle, 505
under a curve, 289, 294, 300
between curves, 347, 350
of an ellipse, 505
by exhaustion, 3, 80
by Green’s Theorem, 1094
enclosed by a parametric curve, 668
in polar coordinates, 686
of a sector of a circle, 686
surface, 570, 671, 1111, 1113
of a surface of a revolution, 568, 574
area function, 312
Area Problem, 3, 289
argument of a complex number, A48
arithmetic-geometric mean, 722
arrow diagram, 12
astroid, 169, 665
asymptote, 244
horizontal, 232, 244
of a hyperbola, 694, A20
slant, 247
vertical, 73, 244
asymptotic curve, 249
autonomous differential
equation, 611
auxiliary equation, 1148
complex roots of, 1150
real roots of, 1149
average cost function, 265
average rate of change, 117, 171
average speed of molecules, 552
average value of a function, 374, 375,
593, 992, 1036
average velocity, 5, 63, 115, 171, 874
axes, coordinate, 801, A11
axes of an ellipse, A19
axis of a parabola, 691
bacterial growth, 627, 634
Barrow, Isaac, 4, 123, 313, 332
baseball and calculus, 637
RP denotes Reference Page numbers.
base of a cylinder, 354
base of a logarithm, 405, 442*
change of, 408, 443*
basis vectors, 810, 811
Bernoulli, James, 616, 643
Bernoulli, John, 481, 616, 661, 772
Bernoulli differential equation, 643
Bessel, Friedrich, 760
Bessel function, 710, 760, 764
Bézier, Pierre, 675
Bézier curves, 660, 675
binomial coefficients, 778
binomial series, 778, 784
discovery by Newton, 784
Binomial Theorem, 778,
RP1
binormal vector, 870
blackbody radiation, 793
blood flow, 176, 267, 587
boundary curve, 1129
boundary-value problem, 1152
bounded sequence, 718
bounded set, 964
Boyle’s Law, 181, 188, 421
Brache, Tycho, 880
brachistochrone problem, 661
branches of hyperbola, 665, 694, A20
Buffon’s needle problem, 601
bullet-nose curve, 51, 162
transformation, 1049
cable (hanging), 464
calculator, graphing, 46, 250,
660, 682
calculus, 9
invention of, 332
cancellation equations
for inverse functions, 387
for inverse trigonometric functions, 455
for logarithms, 405, 430*, 442*
cans, minimizing manufacturing cost
of, 268
Cantor, Georg, 732
Cantor set, 732
capital formation, 590
cardiac output, 588
cardioid, 169, 679
carrying capacity, 182, 604
Cartesian coordinate system, A11
Cartesian plane, A11
CAS. See computer algebra system
Cassini, Giovanni, 685
catenary, 464
Cauchy, Augustin-Louis, 91, 477, 997
Cauchy’s Mean Value Theorem, 477
Cauchy-Schwarz Inequality, 822
Cavalieri’s Principle, 364
center of gravity, 578
center of mass, 578
of a lamina, 1017
of a solid, 1032
of a surface, 1119
of a wire, 1072
centripetal force, 888
centroid of a plane region, 579
centroid of a solid, 1032
Chain Rule, 155, 156
for several variables, 937, 939, 940
change of base, formula for, 408, 443*
change of variables
in a double integral, 1011,
1049, 1052
in integration, 334
in a triple integral, 1038, 1043, 1055
characteristic equation, 1148
charge, electric, 173, 1016, 1032
in a circuit, 1165
charge density, 1016, 1032
chemical reaction, 174
circle, area of, 505
circle, equation of, A16
circle of curvature, 871
circular cylinder, 354
circular paraboloid, 846
circulation of a vector field, 1132
cissoid of Diocles, 664, 684
Clairaut, Alexis, 921
Clairaut’s Theorem, 921, A3
clipping planes, 840
closed curve, 1084
closed interval, A3
Closed Interval Method, 209
for a function of two variables, 965
closed set, 964
closed surface, 1122
Cobb, Charles, 892
Cobb-Douglas production function, 893,
923, 976
cochleoid, 706
coefficient(s)
binomial, 778
of friction, 155, 212
of inequality, 331
of a polynomial, 28
of a power series, 759
static friction, 851
combinations of functions, 41
comets, orbits of, 704
common ratio, 725
comparison properties of the
integral, 309
Comparison Test, 550, 741
comparison tests for series, 741
Comparison Theorem for integrals, 550
complementary equation, 1153
Completeness Axiom, 718
complex conjugate, A47
complex exponentials, A52
complex number(s), A46
argument of, A48
division of, A47, A49
equality of, A46
imaginary part of, A46
modulus of, A47
multiplication of, A47, A49
polar form, A48
powers of, A50
principal square root of, A47
real part of, A46
roots of, A51
component function, 853, 1064
components of acceleration, 878
components of a vector, 808, 818
composition of functions, 41, 156
continuity of, 103, 911
derivative of, 157
compound interest, 451, 480, 654
compressibility, 175
computer algebra system, 69, 660
computer algebra system, graphing
with, 46, 250, 682
for creating a three-dimensional
scene, 840
function of two variables, 895
level curves, 900
parametric equations, 660
parametric surface, 1108
partial derivatives, 921
polar curve, 682
sequence, 716
space curve, 856
vector field, 1065
computer algebra system, integrating
with, 527, 768
concavity, 224
Concavity Test, 224, A43
concentration, 174
conchoid, 662, 684
conditionally convergent series, 751
conductivity (of a substance), 1126
cone, 363, 844
parametrization of, 1109
conic section, 690, 698
directrix, 698
eccentricity, 698
focus (foci), 698
polar equations for, 700
shifted, 695, A21
vertex (vertices), 691
C
1
A126
||||
INDEX
conjugates, properties of, A47
connected region, 1084
conservation of energy, 1088
conservative vector field, 1068, 1089
constant function, 135
constraint, 970, 974
Constant Multiple Law of limits, 78
Constant Multiple Rule, 137
consumer surplus, 586, 587
continued fraction expansion, 722
continuity
of a function, 97
of a function of three variables, 912
of a function of two variables, 910
on an interval, 99
from the left, 99
from the right, 99
of a vector function, 854
continuous compounding of
interest, 451, 480, 654
continuous random variable, 591
contour curves, 896
contour map, 896
convergence
absolute, 750
conditional, 751
of an improper integral, 545, 548
interval of, 761
radius of, 761
of a sequence, 713
of a series, 724
convergent improper integral, 545, 548
convergent sequence, 713
convergent series, 724
properties of, 729
cooling tower, hyperbolic, 846
coordinate(s), A2
Cartesian, A11
polar, 675
rectangular, A11
coordinate axes, 801, A11
coordinate planes, 801
coordinate system
cylindrical, 1037
polar, 675
spherical, 1041
three-dimensional rectangular, 802
coplanar vectors, 827
Coriolis acceleration, 887
Cornu’s spiral, 673
cosine function, A26
derivative of, 151
graph of, 32, A30
power series for, 776
cost function, 177, 261
critically damped vibration, 1163
critical point(s), 959, 969
critical number, 208
cross product, 822
direction of, 824
geometric characterization of, 825
magnitude of, 825
properties of, 826
cross-section, 354
cross-section of a surface, 840
cubic function, 28
curl of a vector field, 1098
current, 173
curvature, 674, 868
curve(s)
asymptotic, 249
Bézier, 660, 675
boundary, 1129
bullet-nose, 51, 162
cissoid of Diocles, 664, 684
closed, 1084
Devil’s, 169
dog saddle, 904
epicycloid, 666
equipotential, 904
grid, 1107
helix, 854
length of, 561, 866
level, 890, 896
monkey saddle, 904
orientation of, 1075, 1091
orthogonal, 170
ovals of Cassini, 685
parametric, 657, 854
piecewise-smooth, 1071
polar, 677
simple, 1085
smooth, 561, 867
space, 854
strophoid, 689, 707
swallowtail catastrophe, 665
toroidal spiral, 856
trefoil knot, 856
trochoid, 664
twisted cubic, 856
witch of Maria Agnesi, 664
curve fitting, 25
curve-sketching procedure, 243, 244
cusp, 662
cycloid, 660
cylinder, 354
parabolic, 841
parametrization of, 1109
cylindrical coordinate system, 1037
conversion equations for, 1037
triple integrals in, 1038
cylindrical shell, 365
damped vibration, 1162
damping constant, 1162
decay, law of natural, 447
decay, radioactive, 449
decreasing function, 20
decreasing sequence, 717
definite integral, 300, 987
properties of, 307, 309
Substitution Rule for, 336
of a vector function, 863
definite integration
by parts, 489, 492
by substitution, 336
degree of a polynomial, 28
del (
ⵜ
), 949
delta (⌬) notation, 117
demand curve, 261, 586
demand function, 261, 586
De Moivre, Abraham, A50
De Moivre’s Theorem, A50
density
of a lamina, 1016
linear, 173
liquid, 576
mass vs. weight, 576
of a solid, 1032
dependent variable, 11, 891, 939
derivative(s), 113, 116, 123
of a composite function, 157
of a constant function, 135
of a difference, 138
directional, 946, 947, 950
domain of, 123
of exponential functions, 396, 398, 433*
as a function, 123
higher, 129
higher partial, 920
of hyperbolic functions, 463, 465
of an integral, 315
of an inverse function, 456
of inverse trigonometric functions,
454, 456
left-hand, 134
of logarithmic functions, 412, 422*, 443*
normal, 1105
notation, 126
notation for partial, 916
partial, 915, 916
of a power function, 135, 142
of a power series, 765
of a product, 139
of a quotient, 140
as a rate of change, 117
right-hand, 134
second, 129
second partial, 862
INDEX
||||
A127