Smith R., Minton R. Calculus
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P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-53
21. 23.
–5
7.5
5.0
2.5
0.0
–2.5
00
–5.0
–7.5
25
50
5
75
100
125
150
–2
0
2
–2
0
2
–2
0
2
x
y
z
0
Not Periodic
25. (a) (b)
–2 –1 0 1 2
–3
–2
–1
0
1
–3
–2
2
3
–3 –223
Periodic, 2π Periodic, 2π
25. (c) Determines the number and smoothness of endpoints
(i) Depicts a circle when either of a or b = 0. If a = 0, then the
center of the circle lies on the x-axis and if b = 0, the center of
the circle lies on the y-axis.
(ii) Depicts an ellipse, when a = ba = b.
(iii) Depicts either an n-point star of an n-leaves structure where
n =
a + b
gcd(a, b)
for a, b > 0.
27. (a) 6 (b) 3 (c) 5 (d) 1 (e) 2 (f) 4
29. 2π(π + 1) 31. 8 +4ln3
33. 10.54 35. 21.56 37. 9.57
39. (a)
4π
0
9sin
2
t + 9cos
2
t +
100
16π
2
dt ≈ 39.00 ft
(b)
(
12π
)
2
+ 10
2
≈ 39 ft
41. x = 2 cost, y = 2sint, z = 2, 0 ≤ t ≤ 2π, arc length = 4π
–4–4
–2–2
–4
–2
0
00
22
xy
44
z
2
4
–2
–2
–1 –1
0
00
1
2
3
1
4
1
2
2
43. x = 3 cost, y = 3sint, z = 2 −3 sint, 0 ≤ t ≤ 2π
arc length ≈ 22.9
x
–4
–2–4 –2
–4
–2
0
z
0
0
2
4
6
2
y
4
2
4
−3
−2
−1
−3
−2
−1
−1
00
0
1
1
2
2
3
3
4
1
5
2
3
47. same except for domains: −∞ < x < ∞, −1 ≤ x ≤ 1, 0 ≤ x
49. cos2t = cos
2
t − sin
2
t
Exercises 12.2, page 767
1. −1, 1, 0 3. 1, 1, −1 5. does not exist
7. t ≤−1, 1 ≤ t < 2, t > 2 9. t =
nπ
2
, n odd
11. t < −3, −3 < t ≤−1, t > 0
13. 0 ≤ t <
π
2
,
π
2
< t ≤ 4 15.
'
4t
3
,
1
2
√
t + 1
, −
6
t
3
(
17. cost, 2t cost
2
, −sint
19.
)
2te
t
2
, 2te
2t
(
t + 1
)
, 2 sec2t tan 2t
*
21. Smooth except t = 1 23. Smooth except t =
nπ
2
, n odd
25. Smooth for all t > 0
27. 29.
0
y
x
2
2
–2
–2
–4
0
2
4
–4
–2
0
2
4
–4
–2
0
2
4
x
y
z
0
31.
'
3
2
t
2
− t,
2
3
t
3/2
(
+ c
33.
'
1
3
t sin3t +
1
9
cos9t, −
1
2
cost
2
,
1
2
e
2t
(
+ c
35.
'
4ln
t − 1
t
, ln(t
2
+ 1), 4 tan
−1
t
(
+ c 37.
'
−
2
3
,
3
2
(
39. (4ln 3, 1 −e
−2
, e
2
+ 1 41. all t 43. t = 0
47. (a) t = 0 (b) No such t exists (c) For all real t
49. (a) t =
nπ
4
, n odd (b) t = nπ, n is an integer
(c) t =
3π
4
+ nπ, n an integer
51.
3
2
1
0
−1.0
−3
−1
−2
−0.5
−1
0.0
0
0.5
−2
1
1.0
2
3
−3
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-54 APPENDIX B
..
Answers to Odd-Numbered Exercises
53.
−2
−1
0
−2
−1.0
−0.5
−1
1
0.0
0
0.5
1
1.0
2
2
55. Period is 2π for all rational numbers a and b except 0.
57. false 59. false
61.
x
y
P
Q
Q
′
a = 3, b = 2, t =
π
6
63. f
(t) ·[g(t) ×h(t)] + f(t) ·[g
(t) ×h(t) +g(t) × h
(t)]
71. No [f(t) = g(t), for all t]
100
75
50
40
50
30
20
25
10
0
00.0
2.5
5.0
7.5
10.0
Exercises 12.3, page 777
1. −10sin 2t, 10cos2t, −20cos 2t, −20 sin2t
3. 25, −32t + 15, 0, −32
5.
+
4
(
1 −2t
)
e
−2t
,
t
√
t
2
+ 1
,
1 −t
2
1 +t
2
2
,
;
+
16
(
t − 1
)
e
−2t
,
1
1 +t
2
3/2
,
2t
t
2
− 3
1 +t
2
3/2
,
7. 10t + 3, −16t
2
+ 4t + 8 9. 5t, −16t
2
+ 16
11.
'
8
1 +t
3/2
,
1
2
ln
1 +t
2
− 2, −e
−t
(
t + 1
)
+ 2
(
13.
'
1
6
t
3
+ 12t + 5, −4t, −8t
2
+ 2
(
15. −160cos 2t, sin 2t
17. −960cos4t, sin4t 19. −120 cos2t, −200sin 2t
21.
120, 0
23. Max altitude: 367.5 m; Horizontal range: 490
√
3 m; speed
at impact: 98 m/s
25. Max altitude: 61.25 m; Horizontal range: 245 m; speed at
impact: 49 m/s
27. Max altitude: 147.751 m; Horizontal range: 323.8 m; speed
at impact: 61.6119 m/s
29. Quadruples the horizontal range
31. r
(
t
)
= v
0
cosθ i +
v
0
sinθ t −
1
2
gt
2
+ h
j
33. 5rad/s 35.
225
2π
rad/s
2
37. the additional rotation increases speed by a factor of
√
3
43. 271, 117, 0 45. a = 100, b =−1, c = 10
47. r
(
t
)
=
'
49
√
2
t,
49
√
2
t, 49
√
3t − 4.9t
2
(
;
point of impact:
+
490
3
2
, 490
3
2
, 0
,
49. 33.64 m/s
51. (a)
)
60
√
3t, 3 +60t − 16t
2
*
(b) No-hits wall 5.7
up
(c)
%
120cos
(
31
◦
)
t, 3 +120 sin
(
31
◦
)
t − 16t
2
&
; clear the wall
53. (a)
%
120t, 8 −16t
2
&
(b) Out
(c)
%
80t, 8 −16t
2
&
; in (d)
%
65t, 8 −16t
2
&
; hits the net
55. 56.57ft/s 57. 1275.5m
59. w =
π
43082
, b ≈ 42, 168 km 61. 0.2473 mi/s
Exercises 12.4, page 784
1. x = 2 cos
s
2
, y = 2 sin
s
2
, 0 ≤ s ≤ 4π
2cos t, 2 sint
3. x =
3
5
s, y =
4
5
s, 0 ≤ s ≤ 5
3t, 4t
5.
1
27
(4 +9t
2
)
3/2
−
8
27
;
+
(27s + 8)
2/3
− 4
9
,
(27s + 8)
2/3
− 4
9
3/2
,
7. 1, 0,
1
√
13
3, −2,
1
√
13
3, 2
9. 0, 1, 1, 0, −1, 0
11.
1
√
13
3, 0, 2,
1
√
13
3, 0, 2,
1
√
13
3, 0, 2
13.
1
√
5
2, 1
;
1
√
5
2cos 1 −sin1, 2sin 1 +cos 1
;
1
√
5
2cos k − sink, 2 sink +cos k
15. 17.
0
y
x
–10
0
10
–10
–5
0
5
10
–10
–5
0
5
10
x
y
z
0
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
APPENDIX B
..
Answers to Odd-Numbered Exercises A-55
19. 2
−3/2
≈ 0.3536 21. 0 23. (6)(37
−3/2
) ≈ 0.0267
25. 1 27. smaller
29.
8
25
,
8
25
−2
−2
−1
−1
−5
0
0
0
5
z
10
y
1
x
15
1
2
2
31. 1,
1
27
−2
−1
−2
−1
0
0
0
1
5
2
y
10
3
1
4
15
z
5
20
2
3
x
4
5
33. maxat 0, ±3, min at ±2, 0
3
3
2
2
1
0
1
−1
−2
0
−3
−1−2−3
35. max at x =
1
45
1/4
and min at x =−
1
45
1/4
−1
0
−4
8
x
10
−8
−2
y
4
10
2
−6
2
6
−10
37. 0 39. 0
x
0.2
1
2
0.4
0.6
0
−1
0−2
−1
−10
x
−4
−2
0
10
−2
2
−6
4
y
6
2
−8
8
10
41. curve straightens 43. false
45. true 47. κ
C
<κ
B
<κ
A
51.
2
3
, 10 53.
1
√
45
,
1
√
45
e
−2
55.
25
52
>
1
10
Exercises 12.5, page 797
1. 1, 0 and 0, 1,
1
√
5
1, 2 and
1
√
5
−2, 1
3. 0, 1 and −1, 0, −1, 0 and 0, −1
5.
1
√
5
0, 1, 2, and −1, 0, 0,
1
√
5
0, 1, −2 and 1, 0, 0
7.
1
√
2
1, 0, 1 and 0, 1, 0,
1
√
6
1, 2, 1 and
1
√
3
−1, 1, −1
9. x
2
+
y −
1
2
2
=
1
4
11.
x +
5
2
2
+ y
2
=
81
4
13. a
T
=−
64
√
5
and a
N
=
32
√
5
, a
T
=
64
√
5
and a
N
=
32
√
5
15. a
T
= 0 and a
N
=
√
20, a
T
=
2π
√
16 +π
2
and a
N
= 4
20 +π
2
16 +π
2
17. neither; increasing 19. a
T
= 0 and a
N
= a
21.
1
√
5
2, −1, 0,
1
√
5
2, −1, 0
−4
−2
−4
−4
−2
−2
0
0
0
2
2
4
4
2
4
23.
1
√
1 +16π
2
0, −1, 4π,
1
√
1 +16π
2
0, 1, 4π
−4
−4 −2
−4
−2
−2
0
0
0
2
2
4
2
4
4
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 21, 2011 15:28
LT (Late Transcendental)
CONFIRMING PAGES
A-56 APPENDIX B
..
Answers to Odd-Numbered Exercises
25. true 27. true
29. (a) 10,000π
2
−cosπ t, −sin πt
(b) 20,000π
2
−cos π t, −sinπ t
(c) 40,000π
2
−cos2π t, −sin 2πt
31. 1 <
3
√
3
2
, |−1| >
−
1
√
2
33.
10
x
y
0
5−5
10
5
0−10
0
10
10
0
4
1−1
Exercises 12.6, page 802
1.
–2
0
2
–2
0
2
0
2
4
6
x
y
z
0
2
4
6
3.
⫺2
⫺1
⫺2
⫺1
0
0
0
1
y
1
2
x
z
1
2
3
4
2
5.
–2
0
2
–2
0
2
x
y
z
–2
0
2
7.
⫺
6
⫺
6
⫺
4
⫺
4
⫺
2
⫺
2
2
00
3
z
4
22
5
6
x y
7
4 4
6 6
9.
–5
0
5
–2
–1
0
1
2
–2
–1
0
1
2
x
y
z
11.
⫺3
⫺3
⫺2
⫺3
⫺2
⫺2
⫺1 ⫺1
⫺1
00
0
1
z
1
1
x
y
2
2
2
3
3
3
13.
5
⫺5
0
⫺2
⫺5
0
0
2
5
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-57
15.
−5.0
−2.5
2
2
−1
1
0
1
2
0
0.0
−1
2
2.5
5.0
17. x = x, y = y, z = 3x + 4y
19. x = cos u coshv, y = sin u coshv, z = sinhv, 0 ≤ u ≤ 2π,
−∞ <v<∞
21. x = 2 cosθ, y = 2 sinθ,z = z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 2
23. x = r cosθ, y = r sinθ,z = 4 −r
2
, 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 2
25. x =±cosh u, y = sinh u cosv, z = sinhu sinv, −∞ < u < ∞,
0 ≤ v ≤ 2π
27. (a) A (b) C (c) B
31. u = v = 0 gives (2, −1, 3);
u = 1 and v = 0 gives (3, 1, 0);
u = 0 and v = 1 gives (4, −2, 5)
⫺4
⫺4
⫺4
⫺2
⫺2
00
0
2
2
4
4
4
6
8
33. a normal is v
1
× v
2
35. r =3, 1, 1+−1, −2, 2u +1, 1, 0v
37.
10,000
10,000
8,000
10,000
5,000
6,000
4,000
5,000
2,000
0
0
0
2,000
5,000
4,000
10,000
5,000
6,000
8,000
10,000
10,000
39.
−10
−10
−5
−10
−5
0
0
0
5
10
5
10
15
20
10
41.
⫺5.0
⫺4
⫺3
⫺2.5
⫺2
−2
⫺1
0
0.0
0
1
2
2
2.5
3
4
5.0
43.
1
1
0.5
0.5
0
0.5
0
1
0.51
1
1
0.5
0.5 0
0
0
0.2
0.5
0.5
0.4
0.6
1
1
0.8
1
45. x = r cosθ, y = r sinθ,z = z with 0 ≤ r ≤ 2,
0 ≤ θ ≤
π
4
, 0 ≤ z ≤ 1
47.
1
0.8
0.6
0.4
0.2
0
t
21.510.50
⫺2
⫺2
⫺1
⫺1
0
0
0
0.2
y
1
0.4
x
0.6
1
2
0.8
1
2
49. x = r cosθ, y = r sinθ,z = sinr with r ≥ 0, 0 ≤ θ ≤ 2π
51. (a) a sphere of radius 3 (b) a sphere of radius k
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-58 APPENDIX B
..
Answers to Odd-Numbered Exercises
53. (a) half-plane y = x, x ≥ 0
(b) half-plane y =−x, x ≤ 0
(c) half-plane y = (tan k)x
55. (a) x = ρ cos θ sin
π
4
, y = ρ sin θ sin
π
4
,
z = ρ cos
π
4
with 0 ≤ θ ≤ 2π, 0 ≤ ρ
(b) x = ρ cos θ sin
3π
4
, y = ρ sin θ sin
3π
4
,
z = ρ sin
3π
4
,0≤ θ ≤ 2π,0≤ ρ
(c) x =
ρ
√
3
cosθ , y =
ρ
√
3
sinθ , z =
2
3
ρ,0≤ θ ≤ 2π ,
0 ≤ φ ≤ π ,0≤ ρ
Chapter 12 Review Exercises, page 804
1. 3.
–4
–2
0
2
4
–4
–2
0
2
4
–4
–2
0
2
4
x
y
z
(0, 2, 1)
(4, –2, 1)
(1, 1, 1)
0
y
x
5. 7.
0
y
x
–4
–2
0
2
4
–4
–2
0
2
4
–4
–2
0
2
4
x
y
z
9. 11.
–10
0
10
–10
0
10
–10
0
10
x
y
z
0
–4
–2
0
2
4
–4
–2
0
2
4
–4
–2
0
2
4
x
y
z
13. (a) B (b) C (c) A (d) F (e) D (f) E
15. 2π
√
37
1
1
0.5
0.5
0
0
0
5
0.5
0.5
10
15
1
1
20
25
30
35
17. 0, e
2
, −1 19. t = 0 21.
'
t
√
t
2
+ 1
, 4 cos4t,
1
t
(
23.
'
−
1
4
e
−4t
, −t
−2
, 2t
2
− t
(
+ c 25. 0, 2, 2
27. −8sin 2t, 8cos2t, 4, −16cos 2t, −16 sin2t, 0
29. t
2
+ 4t + 2, −16t
2
+ 1 31. 4t + 2, −16t
2
+ 3t + 6
33. 0, −128 35. 25(2 −
√
3) ≈ 6.70 feet, 100 feet, 80 ft/s
37.
1
√
2
−1, 1, 0,
1
√
e
−4
+ 1
−e
−2
, 1, 0 39.
1
2
,
4
3
√
3
41. 0, 0 43.
1
√
2
0, 1, 1, −1, 0, 0
45. a
T
= 0, a
N
= 2; a
T
=
√
2, a
N
=
√
2
47. 345,600−cos6t, −sin6t
49. (assuming 0 ≤ u ≤ 2π and 0 ≤ r)
x
y
z
51.
0
2.5
5
7.5
10
0
2.5
5
7.5
10
–10
–5
0
5
10
x
y
z
53. (a) B (b) C (c) A
CHAPTER 13
Exercises 13.1, page 818
1. y =−x 3. x
2
+ y
2
> 1 5. x
2
+ y
2
+ z
2
< 4
7. (a) f ≥ 0 (b) 0 ≤ f ≤ 2
9. (a) f ≥−1 (b) −
π
4
≤ f <
π
2
11. (a) 312 (b) 333 (c) 350 (d) about 19 feet
13. 77.4, 80.4, 82.8, 2.7
◦
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-59
15.
z=3
z=2
z=1
y
1.5
1.5
–0.5
x
–0.5–1.5 1.00.0–1.0
0.5
–1.0
0.5
1.0
0.0
–1.5
0.5−1.0 −0.5
0.5
1.0
2.0
0.0
z
0.0
1.5
y
1.0
–2
y
–1
2 1
0
0
0
2
4
x
–1 –2
6
8
1
2
17.
z=3
z=2
z=1
1
−3
2
3
−1
y
−2
x
3
0
2
−1
03−
−2
1
z
0
−2.5
5
5.0
1
0.0−5.0
2
y
2.5
3
4
4
−4
2
x
−2
0
0
0
1
2
3
4
2
5
y
−2
4
−4
19.
5
3
1
−3
−5
z
4
54
2
0
3
−1
−2
−4
10−1−2−3−4−5 2
y
z
−4
2.5
x
543210−1−2−3
5.0
0.0
−5
−2.5
−5.0
−1.6
y
1.6
x
1−1
2.0
0.8
1.2
−1.2
−0.4
−0.8
−2.0
20
0.4
0.0
−2
21. (a)
−3
−2
3
y
2
−1
1
2.0
1.6
1.2
0.8
0.4
0
0
0.0
−0.4
−0.8
−1.2
−1.6
−1
−2.0
x
1
−2
−3
2
3
−3
−2
x
3
−2.0
−1
2
−1.6
−1.2
1
−0.8
−0.4
0
0
0.0
0.4
−1
y
0.8
−2
1.2
1
1.6
−3
2.0
2
3
(b)
4
3
y
2
3
1
2
x
1
−1
0
0
0
1
2
−1
−2
−1
−2
−3
4
2
3
y
2
−1.0
−0.5
1
x
0
0.5
0
1.0
1.5
−1
2.0
−2
−2
0.0
23. (a)
3
2
1
−2
x
−0.8
−1
−0.6
−0.4
−0.2
0.0
0
0
0.2
0.4
0.6
1
0.8
2
−1
y
−2
−3
−3
−2
y
3
−1
2
−1.0
−0.8
1
−0.6
−0.4
0
0
−0.2
0.0
0.2
−1
x
0.4
0.6
−2
1
0.8
1.0
−3
2
3
(b)
3
−3
2
−0.75
−2
x
−0.5
1
−1
−0.25
0
0
0.0
0.25
1
−1
0.5
y
2
0.75
−2
3
−3
3
2
−3
−2
1
y
−1
−0.5
−0.25
0.0
0
0
0.25
0.5
0.75
1
−1
2
x
3
−2
−3
25. (a)
−3
−3 −3
−2
y
zx−2 −2
−1
−1 −1
00
0
11
1
22
2
33
3
3
2
1
0
3
2
1
0
−1
−2
−3
x
−1
z
−2
−3
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-60 APPENDIX B
..
Answers to Odd-Numbered Exercises
(b)
3.0
2.5
2.0
y
1.5
1.0
0.5
−5.0
z
x−3
−2.5
−2
−1
0
0.0
0.0
1
2
3
2.5
5.0
5.0
2.5
3
2
1
0.0
0.0
0
−1
−2
−3
x
−2.5
z
0.5
−5.0
1.0
1.5
y
2.0
2.5
3.0
27. (a) 1 (b) 4 (c) 2 (d) 3
29.
–2 2
–2
2
x
y
31.
–2 2
–2
2
x
y
33.
–2 2
–2
2
x
y
35.
–2 –1 0 1 2
–2
–1
0
1
2
37.
–6 –4 –2 0 2 4 6
–6
–4
–2
0
2
4
6
39. (a)
–3
–3
–2
–2
–1
–1
0
0
0
2
4
1
1
y
6
x
8
22
3
3
–3
–3
–2
–2
–1
-1
0
0
0
2
1
4
1
6
2
8
2
3
3
41. x = r cost, y = r sin t, z = cos(r
2
)
43. (a) 1 (b) 4 (c) 3 (d) 2
45. f
(
x, y, z
)
= 0 f
(
x, y, z
)
= 2
3
2
1
x
0
–1
–2
–3
-3
–2
–1
y
0
1
2
3
–3
–2
–1
0
z
1
2
3
3
2
1
x
0
–1
–2
–3
-3
–2
–1
0
1
y
2
3
–3
–2
-1
0
z
1
2
3
f
(
x, y, z
)
=−2
3
2
1
x
0
–1
–2
–3
–3
–2
–1
0
1
y
2
3
–3
-2
-1
0
z
1
2
3
47. f =−2, 2
–3
–2
–1
0
3
y
2
0
1
1
2
x
0
4
–1
2
6
–2
–3
3
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-61
49. (a) B (b) A
51. one z for each (x, y)
53. A: 480, B: 470, C: about 475
57. max = 3.942, min =−0.57, HS
59. 60 mph, impossible
Exercises 13.2, page 831
5. 3 7. −
1
2
9. Along x = 0, L
1
= 0; along y = 0, L
2
= 3, therefore
L does not exist.
11. Along x = 0, L
1
= 0; along y = x, L
2
= 2, therefore
L does not exist.
13. Along x = 0, L
1
= 0; along y = x
2
, L
2
= 1, therefore
L does not exist.
15. Along x = 0, L
1
= 0; along y
3
= x, L
2
=
1
2
, therefore
L does not exist.
17. Along x = 0, L
1
= 0; along y = x, L
2
=
1
2
, therefore
L does not exist.
19. Along x = 1, L
1
= 0; along y = x +1, L
2
=
1
2
, therefore
L does not exist.
21. Along x = 0, L
1
= 0; along x
2
= y
2
+ z
2
, L
2
=
3
2
, therefore
L does not exist.
23. Along x = 0, L
1
= 0; along x = y = z, L
3
=
1
3
, therefore
L does not exist.
25. 0 27. 0 29. 2 31. 0 33. 0
35. x
2
+ y
2
≤ 9 37. x
2
− y < 3 39. x
2
+ y
2
+ z
2
≥ 4
41. (0, 2) and all other points for which x = 0 43. all (x, y)
45. Limit does not exist 47. Limit does not exist
49.
1
2
51. true 53. false
61. no; no 63. 1 65. 0
Exercises 13.3, page 841
1. f
x
= 3x
2
− 4y
2
, f
y
=−8xy + 4y
3
3. f
x
= 2x sin xy + x
2
y cosxy, f
y
= x
3
cos xy − 9y
2
5. f
x
=
4e
x
y
y
−
y
x
2
+ y
2
, f
y
=−
4xe
x
y
y
2
+
x
x
2
+ y
2
7. f
x
=−sin x
2
, f
y
= sin y
2
9. f
x
= 3ln
x
2
yz
+ 6 +
y
z
x
y
z
−1
, f
y
=
3x
y
+
ln x
z
x
y
z
,
f
z
=
3x
z
−
y lnx
z
2
x
y
z
11.
∂
2
f
∂x
2
= 6x,
∂
2
f
∂y
2
=−8x,
∂
2
f
∂y∂ x
=−8y
13. f
xx
=−
4
x
2
− 6y
3
, f
xy
=−18xy
2
+
5
1 + y
2
,
f
xyy
=−36xy −
10y
1 + y
2
2
15. f
xx
=
xy
3
1 − x
2
y
2
3/2
, f
yz
= yz sin
(
yz
)
− cos
(
yz
)
, f
xyz
= 0
17. f
ww
= 2 tan
−1
(
xy
)
− z
2
e
wz
, f
wxy
=
2w
1 − x
2
y
2
1 + x
2
y
2
2
, f
wwxyz
= 0
19.
nRV
3
PV
3
− n
2
aV +2n
3
ab
Hint: Hold pressure constant.
21. about
12
V
, assuming V is much greater than 0.004
25. h 27. 1.4, −2.4 29. 2.2, 0.0195
31. (a) (b)
-4
-2
0
2
4
-4
-2
0
2
4
-20
-10
0
10
x
z
y
3
2
1
0
y
−3
−1
−2
−12.5
x
−1
−10.0
0
−2
1
−7.5
2
3
−5.0
−3
−2.5
0.0
2.5
33. (0, 0, 0) = min
35.
π
2
+ mπ,
π
2
+ nπ, 1
= max for m, n odd or m, n even;
π
2
+ mπ,
π
2
+ nπ, −1
= min for m odd and n even or
m even and n odd; (mπ, nπ, 0) neither max nor min
37. f
xy
= f
yx
=−
c
(y −b)
2
+ sin(x + y)
39. (a) 2.5, 2 (b) 1.5, 1.3 (c) 1.1, 0.8 45. −2y
3
sin(xy
2
)
47. concavity of intersection of z = f (x, y) with y = y
0
at x = x
0
49. (a) 2 (b) 0 (c) 0 (d) −1
51. x
2
sin y + x
3
y
2
+
2
3
y
3/2
+ c
53. ln
x
2
+ y
2
+ ln
x − 1
x + 1
+ 3tan
−1
y + c
55. (a)
∂
2
f
∂x
2
=−n
2
π
2
sinnπ x cosnπ ct,
∂
2
f
∂t
2
=−c
2
n
2
π
2
sinnπ x cosnπ ct
57. cos x cost, −sin x sint
61. 400,
1
4
, decrease by 27
63.
∂ P
∂ L
= 0.75L
−0.25
K
0.25
,
∂ P
∂ K
= 0.75L
0.75
K
−0.75
Exercises 13.4, page 852
1. (a) 4(x − 2) +2(y − 1) = z − 4; x = 2 +4t, y = 1 + 2t, z = 4 − t
(b) 4(y − 2) = z − 3; x = 0, y = 2 + 4t, z = 3 − t
3. (a) −(x − 0) = z −0orx + z = 0; x =−t, y = π, z =−t
(b) z + 1 = 0; x =
π
2
, y = π, z =−1 − t
5. (a) −
3
5
(x + 3) +
4
5
(y −4) = z − 5; x =−3 −
3
5
t, y = 4 +
4
5
t,
z = 5 − t
(b)
4
5
(x − 8) −
3
5
(y +6) = z − 10; x = 8 +
4
5
t, y =−6 −
3
5
t,
z = 10 − t
7. (a) L(x, y) = x (b) L(x, y) =−y
9. (a) 1 + x +
1
2
(y −π ) + 2π
z −
1
4
(b)
π
4
+
√
2
x −
1
√
2
+ 2z
11. (a) L(w, x, y, z) =−12w + 4x + 12y +2z − 37
(b) L(w, x, y, z) = 2w − 1
13. values of L are 3, 3.1, 3.1; values of f are 3.002, 3.1, 3.102
15. Values of L are 0.93, 1.1, 0.69; values of f are 0.932, 1.1, 0.73
17. 1.5552 ±0.6307 19. 3.85 21. 4.03
23. 2y
x + (2x +2y)y + (2 y)x +(y) y,
differentiable for all
(
a, b
)
.
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-62 APPENDIX B
..
Answers to Odd-Numbered Exercises
25. 2x
x + 2y y + (x)x + (y)y,
differentiable for all
(
a, b
)
.
27. e
x+2y
(x) + 2e
x+2y
(y)
+
e
x+2y
(
x) + (2y)
2!
+
(
x)
2
+ 3(x)(2y)
3!
+···
(
x)
+
e
x+2y
(
x) + (2y)
2!
+
3(
x)(2 y) +(2 y)
2
3!
+···
(2
y);
differentiable for all
(
a, b
)
.
29.
2x
y
x −
x
2
y
2
y +
yx −2xy
y(y +y)
x +
x
2
y
y
2
(y +y)
y;
differentiable for all
(
a, b
)
.
31. (ye
x
+ cos x) dx + e
x
dy
33. dw =
1
x
−
1
1 +(x − y − z)
2
dx
+
1
y
+
1
1 +(x − y − z)
2
dy
+
1
z
+
1
1 +(x − y − z)
2
dz
35. f
x
(0, 0) = f
y
(0, 0) = 0
37. (a) 6 +4x +2y (b) 8 +
8
7
(
x − 1
)
+
8
9
y
(c) 9 + x + (y − 2)
39. −9 +1.4(t − 10) −2.4(s − 10);−13.4
45. 4x + 4y − z = 8
47. x = 1
Exercises 13.5, page 861
1.
4t +
t
2
− 1
cost
t
2
− 1
e
sint
3. (2t + t
2
+ 1 −cose
t
)e
t
5.
∂g
∂u
= 512u
6
(3u
2
− v cos u)(u
3
− v sin u) +1536u
5
(u
3
− v sin u)
2
;
∂g
∂v
= 512u
6
sinu(v sin u − u
3
)
7. g
(t) =
∂ f
∂x
x
(t) +
∂ f
∂y
y
(t) +
∂ f
∂z
z
(t)
9.
∂g
∂u
=
∂ f
∂x
∂x
∂u
+
∂ f
∂y
∂y
∂u
,
∂g
∂ν
=
∂ f
∂x
∂x
∂ν
+
∂ f
∂y
∂y
∂ν
,
∂g
∂w
=
∂ f
∂x
∂x
∂w
+
∂ f
∂y
∂y
∂w
11.
∂g
∂u
=
∂ f
∂x
+
∂ f
∂y
+ 2u
∂ f
∂z
,
∂ f
∂v
=
∂ f
∂x
−
∂ f
∂y
+ 2v
∂ f
∂z
13.
∂g
∂u
= v
∂ f
∂x
+
1
v
∂ f
∂y
,
∂ f
∂v
= u
∂ f
∂x
−
u
v
2
∂ f
∂y
,
∂ f
∂w
= 2w
∂ f
∂z
15. −0.6271 17. 0.0587
21.
∂z
∂x
=
−6xz
3x
2
+ 6z
2
− 3y
,
∂z
∂y
=
3z
3x
2
+ 6z
2
− 3y
23.
∂z
∂x
=
3yze
xyz
− 4z
2
+ cos y
8xz − 3xye
xyz
,
∂z
∂y
=
3xze
xyz
− x sin y
8xz − 3xye
xyz
25.
∂z
∂x
=−
yz + sin
(
x + y + z
)
xy + sin
(
x + y + z
)
,
∂z
∂y
=−
xz + sin
(
x + y + z
)
xy + sin
(
x + y + z
)
33. k =
1
cv
0
35. g
(
t
)
= f
xx
x
(
t
)
2
+ 2 f
xy
x
(
t
)
y
(
t
)
+ f
yy
y
(
t
)
2
+ f
x
x
(
t
)
+ f
y
y
(
t
)
37.
∂
2
g
∂u
2
=
∂
2
f
∂x
2
∂x
∂u
2
+ 2
∂
2
f
∂x∂ y
∂x
∂u
∂y
∂u
+
∂
2
f
∂y
2
∂y
∂u
2
+
∂ f
∂x
∂
2
x
∂u
2
+
∂ f
∂y
∂
2
y
∂u
2
39.
∂
2
f
∂x
2
−
∂
2
f
∂y
2
+ 4uv
∂
2
f
∂z
2
+ 2v
∂
2
f
∂x∂ z
+
∂
2
f
∂y∂ z
+ 2u
∂
2
f
∂x∂ z
−
∂
2
f
∂y∂ z
41.
dg
dt
= vu
v−1
du
dt
+ (lnu)u
v
dv
dt
;
6t
2
2t + 1
+ 6t ln(2t + 1)
(2t + 1)
3t
2
43. g
(0) = f
xxx
u
3
1
+ 3 f
xxy
u
2
1
u
2
+ 3 f
xyy
u
1
u
2
2
+ f
yyy
u
3
2
g
(4)
(0) = f
xxxx
u
4
1
+ 4 f
xxxy
u
3
1
u
2
+ 6 f
xxyy
u
2
1
u
2
2
+4 f
xyyy
u
1
u
3
2
+ f
yyyy
u
4
2
45. (a) f (x, y) = x −
1
6
x
3
−
1
2
x y
2
(b) x −
1
6
x
3
−
1
2
xy
2
; x −
1
6
x
3
−
1
2
xy
2
+
1
120
x
5
+
1
12
x
3
y
2
+
1
24
xy
4
47. f (x, y) = 1 +2x + y +2x
2
+ 2xy +
1
2
y
2
+
4
3
x
3
+ 2x
2
y +xy
2
+
1
6
y
3
49.
t
2
+8t
(t +4)
2
+ 2t
3
+ t
2
+ 8t + 1
51. 0.2πr
(
2h − r
)
; h >
r
2
53. (a) R(1 + c, 1 +h) = 1 + 0.55c +0.45h
(b)
30
30
20
10
20
0
100
c
50
R
40
40
55. (a) 2 points; (b) halved; (c) about one point
Exercises 13.6, page 871
1. 2x + 4y
2
, 8xy − 5y
4
3. e
xy
2
+ xy
2
e
xy
2
, 2x
2
ye
xy
2
− 2y sin y
2
5. −8e
−8
− 2, −16e
−8
7. 0, 0, −1 9. −4, 0, 3e
4π
, 0
11. 2 +6
√
3 13.
17
5
√
13
15. 0 17. −
19
√
5
19. −
3
√
29
21.
−2
√
30
23.
37
10
25.
2
√
10
27. (a)
4, −3
,
−4, 3
, 5, −5 (b)
−2, −12
,
2, 12
,
√
148,−
√
148
29. (a)
4, 0
,
−4, 0
, 4, −4 (b)
−4, 0
,
4, 0
, 4, −4
31. (a)
'
3
5
, −
4
5
(
,
'
−
3
5
,
4
5
(
, 1, −1
(b)
'
−
4
√
41
,
5
√
41
(
,
'
4
√
41
, −
5
√
41
(
, 1, −1