Smith R., Minton R. Calculus
Подождите немного. Документ загружается.
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-63
33. (a) 16, 4, 24, −16, −4, −24,
√
848, −
√
848
(b)
0, 16, 0
,
0, −16, 0
, 16, −16
35. (a) 1.5 (b) −1 37. parallel
39.
2a, −a
, a = 0
−3
5.0
−2
x
2.5
−1
−1.0
−0.5
0.00
0.0
1
0.5
y
−2.5
1.0
2
−5.0
3
−3
−3
−2
y
−2
x
−1.0
−1
−1
−0.5
0.0
0
0
0.5
1
1.0
1
2
2
3
3
41. 2(x − 1) +3(y +1) −z = 0,
The equation of the normal line is
⎧
⎨
⎩
x = 1 + 2t
y =−1 +3t
z =−t
43. −2(x + 1) +4(y −2) +2(z −1) = 0,
The equation of the normal line is
⎧
⎨
⎩
x =−1 −2t
y = 2 + 4t
z = 1 + 2t
45. (0, 0, 0), (1, 1, −1), (−1, −1, −1)
47. a = 2, b = 1
49.
y
x
51. 2, −2 53. (a) true (b) true 55. 0
57. −tan10
◦
, tan 6
◦
≈−0.176, 0.105, 11.6
◦
59. (a) 8, 4 (b) Ellipse 61. 0.8, 0.3, −0.004
63. If the shark moves toward higher charge, it moves in the direction
12, −20, 5.
Exercises 13.7, page 884
1. (0, 0) saddle 3. (0, 0) saddle, (1, 1) local min
5. (0, 1) local min, (±2, −1) saddle 7. (0, 0) max
9.
(
1, 1
)
local min 11.
1
√
2
, 0
local max,
−
1
√
2
, 0
local min
13. (0, 0) saddle, (1, 1) and (−1, −1) local min
15. (0, 0) saddle, ±
1
2
,
1
2
local max, ±
1
2
, −
1
2
local min
17. (2.82, 0.18) local min, (−2.84, −0.18) saddle, (0.51, 0.99) saddle
19. (±1, 0) local max,
0, −
3
2
local max,
0,
3
2
local min, (0, 0) saddle,
±
√
19
3
√
3
, −
2
3
saddle
23. (a) 247, 104 (b) 276, 95
25. (−0.3210, −0.5185), (−0.1835, −0.4269)
27. (0.9044, 0.8087), (3.2924, −0.3853)
29. (0, 0) is a saddle point 31. f (2, 0) = 4, f (2, 2) =−2
33. f (3, 0) = 9, f (0, 0) = 0
35. max : e
−1
at (1, 1);
min:0at(0, y) and (x, 0)
37. a = b = c =
2
3
s
39. (a) f (x, 0) = f (0, y) = 0; minima
(b) f (x, 0) = f (0, y) = 0; minima
43. (a) Local max for k = 0, inflection at x = 0 for k = 0
(b) All traces have a local min at origin
45. false 47. false
49. extrema at
±
π
2
, ±
π
2
, saddles at (±nπ, ±mπ)
51. (1, 0), f (1, 0) < f (−10, 0)
53. (a) d(x, y) =
(x − 3)
2
+ (y +2)
2
+ (3 − x
2
− y
2
)
2
,
(1.55, −1.03, 0.54)
(b)
1, −
3
2
,
√
13
2
55. (a) 4ftby4ftby4ft (b)
4
3
√
6ftby
4
3
√
6ftby2
√
6ft
57. x = y ≈ 1.42 m; 24.55 > 17.85
Exercises 13.8, page 894
1. x =
6
5
, y =−
2
5
3. x = 2, y =−1
5. x = 1, y = 1 7. x = 1, y = 1
9. max : f (2, 2) = f (−2, −2) = 16,
min : f (2, −2) = f (−2, 2) =−16
11. max : f
4
3
,
4
3
=
256
27
,
min : f
(
x, 0
)
= f (0, y) = 0
13. max : f
(17 −1)/8, (
√
17 −1)/2
=
√
17 −1
8
e
(
√
17−1)/2
,
min : f
−
(17 −1)/8, (
√
17 −1)/2
=−
√
17 −1
8
e
(
√
17−1)/2
15. max : f (±
√
2, 1) = 2e, min = f (0, ±
√
3) = 0
17. max : f
±
2
(
18
)
1/4
, ±
1
(
18
)
1/4
, ±
1
(
18
)
1/4
=
√
18,
min : f
(
0, 0, ±1
)
= f
(
0, ±1, 0
)
= f
(
±1, 0, 0
)
= 1
19. no max or min
21. max : f
a
2
√
1 +a
2
,
1
√
1 +a
2
=
√
1 +a
2
,
min : f
−
a
2
√
1 +a
2
, −
1
√
1 +a
2
=−
1 +a
2
23. If a and b both are even then for all points
f
±
a
a + b
, ±
b
a + b
=
a
a + b
a
b
a + b
b
, maxima
f (0, ±1) = f (±1, 0) = 0, minima
If a and b both are odd then
f
±
a
a + b
, ±
b
a + b
=
a
a + b
a
b
a + b
b
,
maxima
f
±
a
a + b
, ∓
b
a + b
=−
a
a + b
a
b
a + b
b
,
minima
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-64 APPENDIX B
..
Answers to Odd-Numbered Exercises
25. max = f (±
√
2, 1) = 8,
min = f (±
√
2, −1) =−8
27. max = f (2
−1/4
, 2
−1/4
) = 2
1/4
,
min = f (−2
−1/4
, −2
−1/4
) =−2
29. max = f
(
1, 0
)
= f
(
3, 2
)
= 2,
min = f
(
5, 0
)
= f
(
1, 4
)
=−2
31. max = f
±
1
3
1/4
, ±
1
3
1/4
, ±
1
3
1/4
=
√
3,
min = f
(
0, 0, 0
)
= 0
33. u =
128
3
, z = 195 feet
37. P
(
20, 80, 20
)
= 660
39. P
8801
22
, 4
8801
22
,
8801
22
≈ 660.0374989; 660 +λ = 660.0375
41. x = y 43. f
−
√
2
2
,
√
2
2
=
√
2
47. f (4, −2, 2) = 24 49. f (1, 1, 2) = 2
51. (a) (±1, 0, 0) (b) (0, ±1, ±1) 55. (0.5898, 0, 0.3478)
57. α = β = θ =
π
6
; f
π
6
,
π
6
,
π
6
=
1
8
59. C(L, K) = C(64, 8) = 2400 61. c = d = 12
63. L = 50, K = 10
Chapter 13 Review Exercises, page 896
1.
–2
–1
0
1
2
–2
–1
0
1
2
–5
–2.5
0
2.5
5
x
y
z
3.
–2
–1
0
1
2
–2
–1
0
1
2
–4
–2
0
2
x
y
z
–4
–2
0
2
5.
−5.0
10
−2.5
5
x
0
0.0
0.0
y
−5
2.5
−10
2.5
5.0
z
5.0
7.5
10.0
7.
–2
–1
0
1
2
–2
–1
0
1
2
–2
–1
0
1
2
x
y
z
9.
–2
–1
0
1
2
–3
–2
–1
0
1
–5
–2.5
0
2.5
5
x
y
z
11. (a) D (b) B (c) C (d) A (e) F (f) E
13. (a) C (b) A (c) D (d) B
15. Along y = 0, L
1
= 0; along y = x
2
, L
2
=
3
2
; therefore L does not
exist.
17. Along x = 0, L
1
= 1; along y = x, L
2
=
2
3
; therefore L does not
exist.
19. 0 21. x = 0
23. f
x
=
4
y
+ (1 + xy)e
xy
, f
y
=−
4x
y
2
+ x
2
e
xy
25. f
x
= 6xycos y −
1
2
√
x
, f
y
= 3x
2
cos y − 3x
2
y sin y
29. −0.04, 0.06 31. 45 −10(x + 2) +9(y − 5)
33. (x − π) +2
y −
π
2
35. f
xx
= 24x
2
y + 6y
2
, f
yy
= 6x
2
, f
xy
= 8x
3
+ 12xy
37. 3(y +1) − z = 0 39. 4x + 4(y − 2) +2(z −1) = 0
41. 8e
8t
sint + (e
8t
+ 2sin t)cos t
43. g
(t) =
∂ f
∂x
x
(t) +
∂ f
∂y
y
(t) +
∂ f
∂z
z
(t) +
∂ f
∂w
w
(t)
45.
∂z
∂x
=−
x + y
z
,
∂z
∂y
=−
x + y
z
47.
'
−
1
2
, 12π −
1
2
(
49. −4 51. −
7
√
5
53. ±
1
√
145
9, −8, ±4
√
145
55. ±1, 0, ±4 57. 16, 2
59. (0, 0) relative min, (2, ±8) saddles
61.
4
3
,
4
3
relative max, (0, 0) saddle 63. 212, 112
65. f (4, 0) = 512, f (0, 0) = 0 67. f (1, 2) = 5, f (−1, −2) =−5
69. f
1
2
,
1
2
= f
−
1
2
, −
1
2
=
1
2
, f
1
2
, −
1
2
=
f
−
1
2
,
1
2
=−
1
2
71. (1, 1)
CHAPTER 14
Exercises 14.1, page 914
1. (a) 6 (b) 6.5 3. (a) 10 (b) 9 5. 90
7. 39 +3e
2
+ 4sin 4 +4 cos1 −16 cos4 − 4sin 1
9.
19
2
−
1
2
e
6
11. 2 13.
62
21
15. e
4
− 1 17. 2(ln 2)(sin1) 19.
13e
6
− 1
27
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-65
21.
1
6
+
1
3
ln2 23. 2 −
e
2
−
3
2e
25.
1
−1
1
x
2
(x
2
+ y
2
)dy dx =
88
105
27.
2
−2
4−y
2
0
(6 − x − y)dx dy =
704
15
29.
2
0
x
0
y
2
dydx =
4
3
31.
1
−1
1
−1
(−x
2
− y
2
+ 4)dxdy =
40
3
33. −1.5945 35. 1.6697 37.
2
0
1
y/2
f (x, y)dx dy
39.
4
0
x/2
0
f (x, y)dydx 41.
4
1
ln y
0
f (x, y)dx dy
43.
2
0
y
0
2e
y
2
dx dy = e
4
− 1 45.
1
0
x
0
3xe
x
3
dydx = e −1
47. (a)
1
0
2x
0
x
2
dydx =
1
2
;
2
0
y/2
0
x
2
dx dy =
1
6
(b)
2
1.5
1
0.5
y
0
1
0.8
0.6
x
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1
0.8
0.6
y
0.4
0.2
0
2
1.5
x
1
0.5
0
0
1
2
3
4
49. (a) (b)
0
2
4
0
2
6
8
0
2
4
4
6
2
2
2
6
5
4
3
1
2
0
0
0.0
0.25
0.5
2
0.75
1.0
4
6
51. (a) (b)
0
2
4
0
2
6
0
2
4
4
4
4
2
2
2
4
3
y
2
1
0
0
0
1
2
1
3
4
2
x
5
3
4
53.
2π
3
55. (b) 0 57. (a) 2.375 (b) 1.81875
59. 0.93125 61. (a) 4 (b) 2 63. no
65. (a) ln5 −
1
2
ln17 +4(tan
−1
4 −tan
−1
2)
(b)
π
3
+ 1 −
√
3
Exercises 14.2, page 923
1.
2
−2
8−x
2
x
2
1dy dx =
64
3
3.
2
0
3−y
y/2
1dx dy = 3
5.
1
0
√
x
x
2
1dy dx =
1
3
7. 6 9.
40
3
11. 3 −2sin 1 13.
5
12
15.
10,816
105
17.
279
20
19.
256
15
21.
7
3
23.
2
0
4−x
2
0
x
2
+ y
2
dydx = 10.275
25.
4
0
2−x/2
0
e
xy
dydx = 9.003
27. m =
1
3
, x =
3
5
, y =
12
35
29. m =
12
5
, x =
41
63
, y = 0
31. m = 4,
x =
16
15
, y =
8
3
33. (a) In exercise 30, ρ(x, y)isnotx-axis symmetric.
(b) ρ
(
x, −y
)
= ρ
(
x, y
)
35. 1164 37. (a)
y =
1
3
(b) x =
c
3
39. (a)
a
0
b−(b/a)x
0
c −
c
a
x −
c
b
y
dydx =
abc
6
41. (a)
12
5
(b) same
43. (a) 3.792
(b) The average distance of points in the region from the origin
45.
∞
0
∞
0
e
−x
e
−y
dxdy = 1 47. c =
1
32
49.
2
0
min{y,x
2
}
0
f (x, y)dydx =
1
0
x
2
0
f (x, y)dydx +
2
1
x
0
f (x, y)dydx
51. c =
3
16
;
17
√
17 −25
64
≈ 0.705
53. (a) 1200(1 −e
−2/3
) ≈ 583.899
(b) 984(1 −e
−2/3
) ≈ 478.80
55. I
y
=
68
3
,
5
3
; second spin rate 13.6 times faster
57. (a) total rainfall in region
(b) average rainfall per unit area in region
Exercises 14.3, page 931
1. 11π 3.
π
12
5.
π
9
+
√
3
6
7. 18π 9. π − πe
−4
11. 0 13.
81π
2
15.
16
3
17.
81π
2
19.
16π
3
21.
π
12
[61 −15
√
15] 23. 36 25.
π
2
27.
8π
3
29.
4π
3
8 −3
√
3
31.
2π
0
2
0
r
2
drdθ =
16π
3
33.
π/2
−π/2
2
0
re
−r
2
dr dθ =
π
2
(1 −e
−4
)
35. 4−π 37.
π
2
39.
x = 0, y =
2
3
41. I
y
=
π R
4
4
, if radius is doubled then moment of inertia is multiplied
by 16.
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-66 APPENDIX B
..
Answers to Odd-Numbered Exercises
43.
4
3
πa
3
45. V
cone
=
2π
0
k
0
(
k − r
)
rdrdθ =
1
3
πk
3
, V
paraboloid
=
2π
0
√
k
0
k − r
2
rdrdθ =
1
2
πk
2
47. 2
π/2
−π/2
2cosθ
0
9 −r
2
rdrdθ ≈ 17.164
49.
π
0
4sinθ
1+sin
2
θ
0
4 −r
2
− (2 −r sinθ)
rdrdθ
51. 1.2859 53. 20,000π(1 − e
−1
) ≈ 39,717
55. 1 −e
−1/16
≈ 0.06
57.
1
20
(e
−225/16
− e
−16
) ≈ 0.000000033 59.
31π
320
Exercises 14.4, page 936
1.
1
12
(69
3/2
− 5
3/2
) ≈ 46.831 3.
π
6
(17
3/2
− 1) ≈ 36.177
5. 4
√
2π 7. 6
√
11 9. 4
√
6 11. 8π 13. 583.7692
15. 31.3823 17. 37.174 19. 12.045
21. (a)
√
2A (b) 6
√
2 23. 4L
25. k =
3
4
+
1
16
(4 +68
√
17)
2/3
≈ 3.453
29. π[
√
2 −ln(
√
2 −1)] ≈ 7.212 33. 4acπ
2
Exercises 14.5, page 946
1. 16 3. −
2
3
5.
4
15
7.
171
5
9. 0
11. 0 13. 64 15.
5248
35
17.
1
3
(e
3
− 1) 19.
1
4e
21. symmetry; yes; no
–1
–0.5
0
0.2
0
–1
0.4
0.6
–0.5
0.8
0.5
0
1
1.2
0.5
1.4
1
1
23.
2
0
1
−1
1
x
2
dzdx dy =
8
3
25.
1
−1
1−y
2
0
4
2−z/2
dx dzdy =
44
15
27.
√
10
−
√
10
4−x
2
−6
y+6
0
dzdydx =
160
√
10
3
29.
1
−1
1
x
2
3−x
0
dydzdx = 4 31.
4
3
33. 8π
35. m = 32π,
x = y = 0, z =
8
3
37. m = 138, x =
186
115
, y =
56
115
, z =
168
115
39. right side is heavier in #36
41.
1
0
2−2y
0
2−x−2y
0
4yz dz dx dy
=
1
0
2−2y
0
2−2y−z
0
4yz dx dz dy
=
2
0
2−x
0
1−x/2−z/2
0
4yz dy dz dx
43.
2
0
4−2y
0
4−2y−x
0
dzdx dy
45.
1
0
√
1−x
2
0
√
1−x
2
−z
2
0
dydzdx
47.
2
0
4
x
√
y−x
2
0
dzdydx
49. The volume of the parallelepiped is abc; the volume of the
tetrahedron is abc/6.
51.
Q
f
(
x, y, z
)
dV =
max
{
x, y, z
}
dV
=
40881
3680
≈ 11.10897
53. total amount is 0.01563 gram; average density is 1.356 ×10
−5
gram
per cubic foot
55. c =
3
2
57. k = 2 − 2
2/3
≈ 0.413
Exercises 14.6, page 954
1. r = 4 3. r = 4cosθ 5. z = r
2
7. θ = tan
−1
(2)
9.
2π
0
2
0
√
8−r
2
r
rf(r cosθ,r sin θ,z) dzdr dθ
11.
2π
0
3
0
9−r
2
0
rf(r cosθ,r sin θ,z) dzdr dθ
13.
2π
0
√
8
√
3
8
r
2
−1
rf(r cosθ,r sin θ,z) dzdr dθ
15.
2π
0
2
0
4−r
2
0
rf(r cosθ, y, r sinθ) dydr dθ
17.
2π
0
1
0
2−r
2
r
2
rf(x, r cosθ,r sin θ) dx dr dθ
19.
3
2
2π
0
z
0
rf(r cosθ,r sin θ,z) dr dθ dz
21.
2π
0
√
15
0
4
√
1+r
2
rf
(
x, r cos θ,r sinθ
)
dx dr dθ
23.
2π
0
ln2
0
2
e
r
rf
(
x, r cos θ,r sinθ
)
dx dr dθ
25.
2π
0
2
0
2
1
re
r
2
dzdr dθ = π(e
4
− 1)
27.
2
0
3−3z/2
0
6−2y−3z
0
(x + z) dx dydz = 12
29.
2π
0
√
2
0
√
4−r
2
r
zr dz dr dθ = 2π
31.
2
0
4−2y
0
4−x−2y
0
(x + y)dzdx dy = 8
33.
2π
0
2
√
3
0
−
√
4−r
2
e
2
rdzdrdθ = π
4
e
− 1
35.
π
0
2sinθ
0
r
0
2r
2
cosθ dzdr dθ = 0
37.
2π
0
4
0
4
r
r sinrdxdrdθ =−4π
(
2sin 4 +cos4 − 1
)
39.
0
−1
1+x
0
1
e
xy
1
z
dzdydx =
1
24
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-67
41.
2π
0
1
0
r
0
3z
2
rdzdrdθ =
2π
5
43.
π
0
2
0
√
8−r
2
r
2rdzdrdθ =
32π
3
√
2 −1
45.
2π
π
3
0
r
2
0
r
3
dydr dθ =
243π
2
47.
0
2
0
2
0
2
2
2
2
49.
0
2
4
0
2
4
0
2
4
4
4
4
2
2
2
51.
0
2
4
0
2
4
0
2
4
4
4
4
4
2
2
2
53.
0
2
4
0
2
4
0
2
4
4
2
2
2
4
4
55. cosθ,sinθ,0
57.
d
ˆ
r
dt
=
ˆ
θ
dθ
dt
,
d
ˆ
θ
dt
=−
ˆ
r
dθ
dt
59. c = 2
√
2
61. c =
√
2 63. v =−2
√
2
ˆ
r =
√
2
π/4
−3π/4
ˆ
θ dθ
65. m =
128π
3
,
x = y = 0, z =
16
5
67. m = 10π,
x = 0, y =
4
5
,
z =
38
15
Exercises 14.7, page 961
1. (0, 0, 4) 3. (
√
2, 0,
√
2) 5.
√
2
4
,
√
6
4
,
√
6
2
7. ρ = 3 9. θ =
π
4
or
5π
4
11. ρ cosφ = 2 13. φ =
π
6
15. x
2
+ y
2
+ z
2
= 2
0
2
0
2
44
0
2
4
4
4
4
2
2
2
17. z =
x
2
+ y
2
0
2
0
1
2
3
0
2
2
1
2
2
19. y = 0;x ≥ 0
0
2
4
0
2
4
0
2
4
4
4
4
2
2
2
21.
0
2
0
2
0
2
6
4
4
2
2
2
4
4
23.
4.0
3.2
2.4
−3.2
−3
1.6
−2.4
−2
−1.6
0.8
−1
−0.8
0.0
0
0.0
1
−0.8
0.8
2
−1.6
1.6
−2.4
2.4
3.2
−3.2
4
25.
−4
−3
−2
−2
−1
−1
0
0
0
1
1
2
2
3
4
27.
2π
0
π/2
0
2
0
e
ρ
3
ρ
2
sinφ dρ dφ dθ =
2
3
π(e
8
− 1)
29.
2π
0
3π/4
π/4
√
2
cscφ
ρ
4
cos
2
φ sinφdρd φθ
=
2π
0
√
2
1
√
2−r
2
−
√
2−r
2
z
2
rdzdrdθ =
4π
15
31.
1
0
2
1
4
3
(x
2
+ y
2
+ z
2
)dz dydx = 15
33.
2π
0
2
0
4−r
2
0
r
3
dzdr dθ =
32π
3
35.
2π
0
π/4
0
√
2
0
ρ
3
sinφ dρ dφ dθ = (2 −
√
2)π
37.
2π
0
π/4
0
4cosφ
0
ρ
2
sinφ dρ dφ dθ = 8π
39.
2π
0
4
2
z/
√
2
0
rdrdzdθ =
28π
3
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-68 APPENDIX B
..
Answers to Odd-Numbered Exercises
41.
1
−1
1
−1
√
x
2
+y
2
0
dzdydx ≈ 3.061
43.
π/2
0
π/4
0
2
0
ρ
2
sinφ dρ dφ dθ =
4 −2
√
2
3
π
45.
2π
0
2
0
r
0
rdzdrdθ =
16π
3
47.
2π
0
π/4
0
2cosφ
0
ρ
2
sinφ dρ dφ dθ = π
49.
π/2
−π/2
π
0
1
0
ρ
3
sinφ dρ dφ dθ =
π
2
51.
π
0
π/4
0
√
8
0
ρ
5
sinφ dρ dφ dθ =
256 −128
√
2
3
π
53.
2
−2
√
4−x
2
0
0
−
√
4−x
2
−y
2
e
√
x
2
+y
2
+z
2
dzdydx = 2π
e
2
− 1
55.
π
0
π/2
0
2
0
ρ
2
dρ dφ dθ =
4
3
π
2
57. x = y = 0, z =
3
4
+
3
√
2
8
59. cosθ sinφ,sinθ sinφ,cosφ 61. c = 2 63. c = 2
65.
1
2
2
2
1
1
2
0
0
1
67.
1
0.5
0.5
1
0.5
0.5
1
0
0
1
Exercises 14.8, page 971
1. x =
1
6
(v − u), y =
1
3
(u + 2v), 2 ≤ u ≤ 5, 1 ≤ v ≤ 3
3. x =
1
4
(u + v), y =
1
4
(u − 3v), 1 ≤ u ≤ 3, −1 ≤ v ≤−3
5. x = r cosθ, y = r sinθ,1 ≤ r ≤ 2, 0 ≤ θ ≤
π
2
7. x = r cosθ, y = r sinθ,2 ≤ r ≤ 3,
π
4
≤ θ ≤
3π
4
9. x =
1
2
(v − u), y =
1
2
(u + v), 0 ≤ u ≤ 2, 2 ≤ v ≤ 4
11. x = ln
1
2
(v − u)
, y =
1
2
(u + v), 0 ≤ u ≤ 1, 3 ≤ v ≤ 5
13.
7
2
15.
13
3
17.
7
3
19.
ln3
6
e
5
− e
2
22.
9
4
23. −2u 25. 2
27. x = u −w, y =
1
2
(−u + v +w), z =
1
2
(u − v +w),
1 ≤ u ≤ 2, 0 ≤ v ≤ 1, 2 ≤ w ≤ 4
29. 1 31. e
2
− 1 33.
e
3
− e
6
37.
π
2
8
Chapter 14 Review Exercises, page 972
1. 18 3. 207 5. π (e
−1
− e
−4
) 7.
2
3
9. 0
11. −19.92 13.
4
3
15. 16π 17.
128
3
19.
64π
3
21.
π
3
(16 −8
√
2) 23.
11π
2
25.
4
0
2
√
y
f (x, y)dx dy
27.
π/2
−π/2
2
0
2r
2
cosθ dr dθ =
32
3
29. m =
16
3
,
x =
3
2
,
y =
9
4
31. m =
64
15
,
x = 0, y =
23
28
,
z =
5
14
33.
1
0
2−y
√
y
dx dy =
5
6
35.
1
2
37. 2
√
21 39.
13π
3
41. 16π
√
2
43.
2
0
1
−1
1
−1
z(x + y)dzdydx = 0
45.
2π
0
π/4
0
2
0
ρ
3
sinφ dρ dφ dθ = π(8 − 4
√
2)
47.
2
0
2
x
6−x−y
0
f (x, y, z)dzdydx
49.
2π
0
π/2
0
2
0
f (ρ sinφ cos θ,ρ sinφ sinθ,ρ cosφ)ρ
2
sinφ dρ dφ dθ
51.
π/2
π/4
√
2
0
r
0
e
z
rdzdrdθ =
πe
√
2
(
√
2 −1)
4
53.
π
0
π/4
0
√
2
0
ρ
3
sinφ dρ dφ dθ = π
1 −
√
2
2
55. (a) r sinθ = 3 (b) ρ sinφ sin θ = 3
57. (a) r
2
+ z
2
= 4 (b) ρ = 2
59. (a) z = r (b) φ =
π
4
61.
4
4
4 2
2
2
0
2
4
0
2
4
0
2
4
63.
0
2
4
0
2
4
0
2
4
4
4
4
2
2
2
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-69
65.
0
2
0
2
0
2
2
2
2
67. x =
1
4
(v − u), y =
1
2
(u + v), −1 ≤ u ≤ 1, 2 ≤ v ≤ 4
69.
1
2
(e −e
−1
) 71. 4uv
2
− 4u
2
CHAPTER 15
Exercises 15.1, page 987
1.
–2 2
–2
2
3.
–2 2
–2
2
5.
x
y
−1
−2
−2 0
−1
2
12
0
1
7.
9.
11. F
1
= D, F
2
= B, F
3
= A, F
4
= C
13. 2y = sin x +c 15. x
2
− y
2
= c
17.
4−1
x
−4 501
−5.0
−3−5
−2.5
2
y
0.0
2.5
3−2
5.0
Flows point in opposite direction to the right and left of (0,1)
19. 2x, 2y 21.
x, y
x
2
+ y
2
23. ∇ f =e
−xy
− xye
−xy
, −x
2
e
−xy
25.
x, y, z
x
2
+ y
2
+ z
2
27. ∇ f =
%
2xyz
2
, x
2
x
2
+ z
2
, −2x
2
yz
&
x
2
+ z
2
2
29. f (x, y) = xy + c 31. not
33. f (x, y) =
1
2
x
2
− x
2
y +
1
3
y
3
+ c
35. f (x, y) =−cos xy + c
37. f (x, y, z) = 2x
2
− xz +
3
2
y
2
+ yz +c 39. not
41. y
2
= x
3
+ c 43. (y + 1)e
−y
=−
1
2
x
2
+ c
45. y
2
+ 1 = ce
2x
49. 3rr
53.
–2
wire
2
2
55. f (x, y, z) =
x
0
f (u)du +
y
0
g(u)du +
z
0
h(u) du + c
57.
3x, y
x
2
+ y
2
(undefined at the origin)
59.
x
2
+ y
2
+ z
2
−x, −y, −z
61.
qx + 1, y
[(x + 1)
2
+ y
2
]
3/2
+
qx − 1, y
[(x − 1)
2
+ y
2
]
3/2
−
qx, y
[(x
2
+ y
2
)]
3/2
63. from hot to cold (assuming k > 0)
Exercises 15.2, page 1001
1. 4
√
13 3. 6
√
6 5. 12 7.
3
80
17
5/2
−
1
16
17
3/2
+
1
40
9.
9
2
11.
1
6
17
3/2
− 5
3/2
13. 0 15. 2(e
2
+ 1)
17. −2π 19. 0 21. −2
√
5 23. e
−1
− e
−2
25. cos3 −cos 6 27. 2(e
2
− 1) +
e
10
− 1
5
29. 31 31. −
64
3
33.
11
6
+
e
2
35. 0
37. positive 39. zero 41. negative
43. 4π 45.
32
3
53. 18.67
55.
x = 2.227, y = 5.324 57. 99.41 59. 359.9
61.
π
3
3
√
5 63. Temperature is constant along horizontal lines.
Exercises 15.3, page 1011
1. f (x, y) = x
2
y − x + c
3. f (x, y) =
x
y
− x
2
+
1
2
y
2
+ c
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-70 APPENDIX B
..
Answers to Odd-Numbered Exercises
5. not 7. f (x, y) = e
xy
+ sin y + c
9. f (x, y, z) = xz
2
+ x
2
y + y − 3z +c
11. f (x, y, z) = xy
2
z
2
−
1
2
x +
1
4
e
−2x
+
1
3
(y
2
+ 1)
3/2
13. f (x, y) = x
2
y − y;8
15. f (x, y) = e
xy
− y
2
; −16
17. f (x, y, z) = xz
2
+ x
2
y;−38
19. f (x, y) =
x
3
3
+ x +
y
5
5
−
2
3
y
3
+ y;
152
3
21.
√
30 −
√
14 23. −2 25. 10 −e
18
27.
π
4
+ 1 −e
4
+
4
3
(4 −
√
2) 29. 32π
31. yes 33. no 35. no
37. C
1
: x = t, y = 0, −2 ≤ t ≤ 2;
C
2
: x = 2cost, y = 2sint,π ≤ t ≤ 2π
39. C
1
: x =−2 +2t, y = 2 − 2t, 0 ≤ t ≤ 1;
C
2a
: x = t, y = 2, −2 ≤ t ≤ 0,
C
2b
: x = 0, y = 2 − t, 0 ≤ t ≤ 2
41. false 43. true 45. tan
−1
y
x
+ c, x = 0; 0
47. (a) simply-connected (b) not simply-connected
49. a potential for F is
−kq
x
2
+ y
2
+ z
2
51. (a) RT
2
ln
P
2
P
1
− R(T
2
− T
1
) (b) RT
1
ln
P
2
P
1
− R(T
2
− T
1
)
53. 0; 0
Exercises 15.4, page 1021
1. π 3. 16 5. −54 7.
32
3
9. 6π 11.
1
3
13.
4
3
+
1
2
e
2
+
3
2
e
−2
15. 8 17. 4e
2
+ 12e
−2
19. 2π e
21. 8π 23.
32
3
25.
3
8
π
29.
x = 0, y =
4
7
33. 0 35. 0
37. π 41. 2; −ln(cos 1);2 + ln(cos1)
Exercises 15.5, page 1029
1. 0, 0, −3y, −x 3. −3, 2x, 0, 2z
5. −y, −2x, −x, y + z
7. x
2
e
xy
+ 1, −
(
1 + xy
)
e
xy
, 0;2x + 1
9.
'
−x sin y −
√
x
2
√
z
, −
3y
z
2
− cos y,
√
z
2
√
x
−
3
z
(
;0
11.
+
−
2ze
y
2
/z
y
3
− 2z, 2x, 0
,
;2z + 1 +
e
y
2
/z
y
2
13. conservative 15. incompressible 17. conservative
19. neither 21. incompressible
23. conservative 25. conservative
27. (a) scalar (b) undefined (c) undefined (d) vector (e) vector
37. positive 39. negative 41. negative
43. ∇×F =
'
0, 0,
−2x
(1 + x
2
)
2
(
; rotational, axis of rotation perpendicular
to the xy-plane
45. 0, 0, 0, wheel moves clockwise, but does not spin
1.0
y
0.8
0.2
0.0
–0.8
x
0.6
0.5
0.4
−0.2
−0.5
−0.4
−1.0
1.00.0−1.0
−0.6
47. 0, incompressible
55. (a)
2
√
x
2
+ y
2
+z
2
(b) 0
57. f
(
x, y, z
)
exists for x, y > 0
61. xy
2
−
x
2
2
+ c 73.
'
0, 0, ±
9
8
1
3
(
Exercises 15.6, page 1041
1. x = x, y = y, z = 3x + 4y
3. x = cos u coshv, y = sin u coshv, z = sinhv, 0 ≤ u ≤ 2π,
−∞ <v<∞
5. x = 2 cosθ, y = 2 sinθ,z = z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 2
7. x = r cosθ, y = r sinθ,z = 4 −r
2
, 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 2
9.
–2
0
2
–2
0
2
0
2
4
6
x
y
z
0
2
4
6
11.
4
2
0
2
4
4
2
0
2
4
0
2.5
5
7.5
10
x
y
z
13.
2
0
2
2
0
2
x
y
z
2
0
2
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-71
15.
5
0
5
2
1
0
1
2
2
1
0
1
2
x
y
z
19. (a) 1 (b) 3 (c) 2 21. 16π
√
2
23. 2π
√
14 25.
√
2
2
27. 4π 29. 4acπ
2
31. π
√
3 +
π
√
2
sinh
−1
(
√
2) ≈ 7.988
33.
3
1
2
1
(2x
2
+ 3xy)
√
14dxdy =
82
√
14
3
35. 486π (the integrand is the constant 27 on a hemisphere of radius 3)
37.
2π
0
√
3
√
2
(2r
3
− 4r)
4r
2
+ 1dr dθ =
π
10
[81 −13
√
13]
39. 2
2π
0
4
0
√
2r
3
dr dθ = 256
√
2π
41. 0, by symmetry 43. 24π 45. −18π 47.
5
2
49.
9π
2
51. 96π
2
53. 0 55.
3π
2
57. 198.8π 59. 0.47π 61. 0
63. (a) flow lines don’t cross boundary
(b)
S
F · ndsis equal to the surface area of the surface
65. (a) (“Show that” type question)
(b)
π
√
c
2
+ 1
c
2
67. (a)
2π
3c
2
(b) As c tends to zero, the radius of cone approaches infinity and the
cone becomes a much larger surface
71. m = 8
√
14π, x = y = 0, z = 6
73. m = 2π,
x =
1
3
,
y = 0, z =
1
2
Exercises 15.7, page 1051
1.
3
2
3. π 5. 0 7. 12 9. 32π
11. 56π 13. (a) 8 (b) 0 15. (a) 4π (b) 2π
17. 0 19.
π
2
21. π e −
2
3
π 23. 16π
25.
27
5
27.
512
3
+ 8e
2
+ 24e
−2
29. (a) 2π
2
a
2
c (b) 2π
2
a
2
c 37. (b) c =
ρ
2ε
0
Exercises 15.8, page 1060
1. 0 3. 4π 5. −
4
3
7. −π 9. 0 11. 0
13. 4π 15. 4π 17. 0 19. 3π
21. (a) 0 (b) 0 23. (a)
88
3
π (b) 24π
31. Both boundary curves have the same orientation.
33. Use ∇×( f ∇g) = (∇ f ) ×(∇g).
Exercises 15.9, page 1067
1. 1 3. 0 5. 4πε
0
R
3
7. ∞ 9. −
16
3
11. 64π
13. B and
dB
dt
must be continuous as well E must have continuous
derivatives in all variables.
15. Equality of mixed partial derivatives, E and B have continuous
second derivative.
19. For the given field, ∇·E = 0 in any region not containing
the origin.
21. E ·n =
q
4πε
0
R
2
on the sphere and the area of the sphere is
4π R
2
25.
S
∇ u ·n dS =
Q
∇·(∇u) dV =
Q
∇
2
udV where Q is
the region enclosed by S
29. Use exercise 24 to show that ∇( f − g) = 0 everywhere inside S;
since f − g = 0onS, f = g inside S as well.
Chapter 15 Review Exercises, page 1068
1.
2
2
2
2
3. F
1
is D, F
2
is C, F
3
is B, F
4
is A
5. f (x, y) = xy = x
2
y
2
+ y + c 7. not conservative
9. y
3
= 3x
2
+ c 13. 18 15. 18π 17. 0 19. 0
21. 3π −4 23. zero 25. 40π 27. 66 29. 3
31. 10 33. conservative 35.
1
3
37. −2 39.
32
3
41. 6π 43. 0, 0, 0, 3x
2
− 3y
2
45. 0, 0, 0, 2 +2z
2
+ 2y
2
47. neither 49. both 51. positive
53.
0
2.5
5
7.5
10
0
2.5
5
7.5
10
10
5
0
5
10
x
y
z
55. (a) B (b) C (c) A 57.
π
6
(17
3/2
− 5
3/2
)
59. −8
√
14 61. 4π
√
26 63. 0
65. m =
π
3
(17
√
17 −1), x = y = 0, z =
1 +391
√
17
10(17
√
17 −1)
67.
16
3
69.
304
5
71.
8π
3
73. 0
75. 0 77. 0
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-72 APPENDIX B
..
Answers to Odd-Numbered Exercises
CHAPTER 16
Exercises 16.1, page 1080
1. y(t) = c
1
e
4t
+ c
2
e
−2t
3. y(t) = c
1
e
2t
+ c
2
te
2t
5. y(t) = e
t
(c
1
cos2t + c
2
sin2t)
7. y(t) = c
1
+ c
2
e
2t
9. y(t) = c
1
e
(1+
√
7)t
+ c
2
e
(1−
√
7)t
11. y(t) = c
1
e
√
5+1
2
t
+ c
2
e
√
5−1
2
t
13. y(t) =−
3
2
sin2t + 2 cos2t 15. y(t) =−e
t
+ e
2t
17. y(t) = e
t
(2cos 2t − sin2t) 19. y(t) = (3t − 1)e
t
21. A =
√
13,δ ≈−0.983 23. A =
√
105
5
,δ ≈−1.351
25. u(t) =
2
3
cos8t
01 4
⫺0.5
0.5
t
u
23
27. u(t) =
1
5
cos(7
√
2t) −
2
√
2
7
sin(7
√
2t); A =
√
249
35
,δ ≈−0.460
01 2 6543
⫺0.5
0.5
t
u
29. u(t) = e
−12t
−
3
2
e
−8t
0 0.5 1
⫺1
1
t
u
31. u(t) = e
−(1/4)t
−
1
2
cos
√
15655
20
t
−
√
15655
6262
sin
√
15655
20
t
0
5
10 15
⫺0.5
0.5
t
u
37. c = 8
√
2
41. The solution to y
+ a
2
y = 0 has ordinary sine and cosine functions
instead of hyperbolic sine and cosine.
Exercises 16.2, page 1088
1. u(t) = e
−t
(c
1
cos2t + c
2
sin2t) +3e
−2t
3. u(t) = e
−2t
(c
1
+ c
2
t) +t
2
− 2t +
3
2
5. u(t) = e
−t
(c
1
cos3t + c
2
sin3t) +2e
−3t
7. u(t) = e
−t
(c
1
+ c
2
t) −
25
2
cost
9. u(t) = c
1
e
2t
+ c
2
e
−2t
−
1
2
t
3
−
3
4
t
11. u(t) = Ae
−t
+ Bte
−t
cos3t + Cte
−t
sin3t + D cos3t + E sin3t
13. u(t) = t (C
3
t
3
+C
2
t
2
+C
1
t + C
0
) + Ae
2t
15. u(t) = Ae
t
cos3t + Be
t
sin3t + (Ct
2
+ Dt)cos3t
+ (Et
2
+ Ft)sin3t
17. u(t) = t
2
e
−2t
(At
2
+ Bt + C) + e
−2t
(Dt + E)cost
+ e
−2t
(Ft + G)sin t
19. u(t) = e
−t
−
1221
5963380
cos(
√
4899t) −
1229
√
4899
29214598620
sin(
√
4899t)
+
1221
5963380
cos4t +
1
2981690
sin4t
21. u(t) =
−1024 −543
√
2
14368
e
(−16+8
√
2)t
+
−1024 +543
√
2
14368
e
(−16−8
√
2)t
+
64
449
e
−t/2
23. u(t) =−cos 3t + 2sin 3t; amplitude =
√
5; phase shift ≈−0.464
25. u(t) = cost +2 sint; amplitude =
√
5; phase shift ≈ 0.464
27. u(t) =−
640
5881
cos2t +
3000
5881
sin2t; amplitude ≈ 0.522,
phase shift ≈−0.210
29. natural frequency and value of ω that produces resonance,
√
3; beats,
for example, at ω = 1.9
31. natural frequency and value of ω that produces resonance, 8
√
2;
beats, for example, at ω = 11