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-43
19.
−0.4
−2
2
1.2
3
1.6
0.0
−1.6
−3
2.0
1
−0.8
−1.2
−2.0
−1
0.8
0.4
0
21. x = 3t, y = 1 + 3t, 0 ≤ t ≤ 1
23. x =−2 +8t, y = 4 −3t, 0 ≤ t ≤ 1
25. x = t, y = t
2
+ 1, 1 ≤ t ≤ 2
27. x = 2 +3 cost, y = 1 + 3sin t, 0 ≤ t ≤ 2π
29. (a) x = 12t, y = 16 − 16t
2
(b) x = (12cos6
◦
)t, y = 16 +(12 cos6
◦
)t − 16t
2
31. (a) x = 2t, y = 10 − 4.9t
2
(b) x = (2cos8
◦
)t, y = 10 −(2 cos8
◦
)t − 4.9t
2
33. yes, at (250, 100)
35. y = 0att = 2;d = 0ord = 5 (impractical)
37. (2, 3) and (−3, 8) 39. (2, 1) and (3, 0)
41. Integer values for k lead to closed curves,
but irrational values for k do not.
43. k = 2
x
⫺1.5 0.5⫺0.5⫺1
y
⫺1
⫺0.5
1
0.5
k = 3
y
x
1 0.5
1.5
0.5
1
10.5
1.5
1
0.5
k = 4
y
x
11.5 0.5
0.5
1
10.5
1
0.5
k = 5
y
x
11.5 0.5
0.5
1
10.5
1
0.5
45. x = 1 −2y
2
, −1 ≤ y ≤ 1; y =±2x
√
1 − x
2
47. C 49. B 51. A
53. (a) a circle with radius t at time t
(b) x = a + (t − c)cosθ, y = b +(t − c)sin θ
(c)
y
x
5
5
5
5
(d)
4
–1
5
–1–5
–2
5
–4
–4 2–3
1
2
–3
1
3
–5
43–2
(e)
x
5
5
5
5
y
(f) If sinθ =
1
1.4
, then tanθ =
1
√
0.96
.
(g) a cone
55. (a) x = (v sinθ )t, y = D −(v cosθ)t (e) vγ
Exercises 10.2, page 639
1. (a) −1 (b) 1 (c) undefined 3. (a) −
3
2
(b) 0 (c) 0
5. (a) 0 (b) −
2
π
(c) −π 7. 1att = 1; −1att =−1
9. (a)
√
2
2
, 1
,
√
2
2
, −1
,
−
√
2
2
, 1
−
√
2
2
, −1
(b) (1, 0), (−1, 0)
11. (a) (0, −3) (b) (−1, 0) 13. (a) (0, 1) (b) (0, −3)
15. (a) x
= 0, y
= 3; speed is 3; up
(b) x
=−2; y
= 0; speed is 2; left
17. (a) x
(0) = 20, y
(0) =−2, speed = 2
√
101, right/down
(b) x
(2) = 20, y
(2) =−66, speed =
√
4756, right/down
19. (a) x
(0) = 5, y
(0) = 4, speed =
√
41, right/up
(b) x
π
2
= 0, y
π
2
=−9, speed = 9, down
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-44 APPENDIX B
..
Answers to Odd-Numbered Exercises
21. 6π 23.
3π
8
25.
4
3
27.
256
15
29. At (3, 0), speed is 0 and acceleration is x
=−6, y
= 0; at (−1, 0),
speed is 4 and acceleration is x
=−2, y
= 0
35. x(t) = vt + r cos
v
r
t
, y(t) = r − r sin
v
r
t
;
min speed = 0 at bottom (y = 0), max speed = 2v at top (y = 2r)
37. x(t) = (a −b)cos t + b cos(
a
b
t), y(t) = (a −b)sint − b sin(
a
b
t)
39. speed = 4, (tan4t)(−cot4t) =−1
41. 5: 3
x
4
3
4
3
y
43. x = 2 cost + sin 3t, y = 2 sint + cos 3t;
x
44
3
3
y
Min/max speeds: 1, 5
Exercises 10.3, page 647
1. (a) 19.38 (b) 4π 3. (a) 2π (b) 55.09
5. (a)
π
2
(b) 4.29 7. (a) e
8
− e
−8
(b) 2980.2
9. (a) 4.4859k (b) 4.4859k (c) ∞; 3.89
11. (a) 4.4569k (b) 4.4569k (c) ∞; 4.07
13. (a) 85.8 (b) 83.92 15. (a) 85.8 (b) 162.60
17. (a) 40.30 (b) 43.16 19. x = 4u, y = 4
√
1 −u
2
;2π
21. (a) 4π (b) b − a
Exercises 10.4, page 658
1. (2, 0) 3. (2, 0) 5. (−3, 0)
7.
2
√
2, −
π
4
+ 2πn
,
−2
√
2,
3π
4
+ 2πn
9.
3,
π
2
+ 2πn
,
−3,
3π
2
+ 2πn
11.
5, tan
−1
4
3
+ 2nπ
,
−5, tan
−1
4
3
+ 2(n + 1)π
13. (1, −
√
3) 15. (0, 0) 17. (3.80, 1.24)
19.
x
8
6
8
6
y
x
2
+ y
2
= 16
21.
x
44
3
3
y
y =
1
√
3
x
23.
x
1
22
1
y
x
2
+ y
2
= x
25.
x
44
2
4
y
x
2
+ y
2
= 3y
27.
x
1
22
1
y
r = 0atθ =
kπ
4
(k odd), 0 ≤ θ ≤ 2π
29.
x
1
22
1
y
r = 0atθ =
nπ
3
, 0 ≤ θ ≤ π
31.
x
6
2
6
6
y
r > 0, 0 ≤ θ ≤ 2π
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-45
33.
y
x
66
7
1
r = 0atθ =
π
6
+ 2πn,
5π
6
+ 2πn;0≤ θ ≤ 2π
35.
y
x
5
1
5
5
r = 0atθ =
3π
2
+ 2πn, 0 ≤ θ ≤ 2π
37.
y
x
⫺5
5
3
⫺3
r = 0atθ = 0, −∞ <θ<∞
39.
y
x
2
1
2
2
r = 0atθ =
3π
4
+ πn, 0 ≤ θ ≤ π
41.
3
2
1
0
–1
–2
–3
321
0
–1–2–3
θ =
3π
4
+ 2πn,
7π
4
+ 2πn
43.
21
0
–1–2
2
1
0
–1
–2
θ = 0
45.
4
0
2
–2
–4
642
0–2
θ =
2π
9
+
2π
3
n,
4π
9
+
2π
3
n
47.
0
–100000
–50000
–150000
–200000
8000 400–400–800
θ =−
π
2
+ 2πn
49.
800
0
400
–400
–800
0
–200000 –50000–150000 –100000
θ = (2n +1)π
51. r =±2
√
−sec 2θ
53. r = 4
55. r = 3cscθ
57. circles of radius
1
2
|a| and center
1
2
a, 0
59. For integer a, rose with 2n leaves (n even) or n leaves (n odd)
61. For |a| > 1, larger |a| gives larger inner loop.
63.
y
x
0.5
1
10.80.60.40.2
1
0.5
65. n wide overlapping petals on a flower when 0 ≤ θ ≤ nπ,
up to n = 24; graph repeats for larger domains
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-46 APPENDIX B
..
Answers to Odd-Numbered Exercises
67.
0
4.8
1
3.2 4.00.8
2
1.6
−2
6.40.0 5.6 7.22.4
−1
69. (a)
x
B
(d, o)
h
h
C
y
sin B =
h
d
⇒ B = sin
−1
h
d
;
sinC =−
h
d
⇒ C = sin
−1
−
h
d
=−sin
−1
h
d
(c)
r
1
(A) = d cos A −
d
2
cos
2
A −(d
2
− h
2
);
r
2
(A) = d + b
⎛
⎝
1 −
A
sin
−1
h
d
2
⎞
⎠
;
A
1
=−sin
−1
h
d
, A
2
= sin
−1
h
d
Exercises 10.5, page 667
1. (a)
√
3 2. (a) undefined 3. (a) 0
5. (a)
1
2
(b)
2sin 1 +cos1
2cos 1 −sin1
7. (a)
√
3
2
,
1
2
,
−
√
3
2
,
1
2
, (0, −1)
(b) concave up at (0, 0) and (0, −1); concave down at (±0.73, 0.56)
9. (a) (3
√
2, −3
√
2), (−3
√
2, 3
√
2)
(b) concave up at (−0.98, −0.13), (−1.22, −1.49) and (2.97, −4.74);
concave down at (1.22, 1.49), (0.98, 0.13) and (−2.97, 4.74)
11.
π
12
13. π 15.
π
2
−
3
√
3
4
≈ 0.2718 17. 0.3806
19. 0.1470 21.
11
√
3
2
+
14π
3
≈ 24.187
23.
5π
3
+
√
3 ≈ 6.9680 25.
5π
4
− 2 ≈ 1.9270
27.
5π
12
−
√
3
2
29.
√
3
4
−
π
12
31. (0, 0), (0.3386, −0.75), (1.6614, −0.75)
33. (0, 0), (1.2071, 1.2071), (−0.2071, −0.2071)
35. 16 37. 6.6824 39. 20.0158
41. (a) 31.2% (b) 37.35%
43. (a) 0;
√
3;−
√
3
(b)
0.16
0.12
0.08
0.04
0
0.60.40.20–0.2–0.6 –0.4
0.2
–0.2
–0.6
0.50.40.30.20.1–0.1
0.4
0
–0.4
0
0.2
–0.2
–0.6
0.1
0
–0.1–0.2–0.3–0.5
0.4
0
–0.4
–0.4
Exercises 10.6, page 675
1. y =−
1
4
x
2
3. x =
1
4
y
2
+ 2 5.
x
2
12
+
(y −3)
2
16
= 1
7.
(x − 4)
2
16
+
(y −1)
2
12
= 1 9.
(x − 2)
2
1
−
y
2
3
= 1
11.
(y −4)
2
1
−
(x − 2)
2
3
= 1
13. parabola, (−1, −1), (−1, −
7
8
), y =−
9
8
15. ellipse, (1, −1) and (1, 5), (1, 2 −
√
5) and (1, 2 +
√
5)
17. hyperbola, (−2, 0) and (4, 0), (1 −
√
13, 0) and (1 +
√
13, 0)
19. hyperbola, (−2, −5) and (−2, 3), (−2, −1 −
√
20) and
(−2, −1 +
√
20)
21. ellipse, (−1, 0) and (5, 0), (2 −
√
8, 0) and (2 +
√
8, 0)
23. parabola, (−1, −2), (−1, −1), y =−3
25. y =
1
8
(x − 2)
2
− 1
0
y
x
27.
(x − 2)
2
16
+
(y −2)
2
12
= 1
0
y
x
–3 7
7
–3
29.
(y −1)
2
4
−
x
2
5
= 1
0
y
x
–5 5
5
–5
31.
1
16
, 0
33.
0,
1
8
35. 20 inches 37. (0, −8)
39. (−2, 0) 41. (
√
300, 0) and (−
√
300, 0)
43. −t
2
+ 16t, boomerang
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-47
Exercises 10.7, page 681
1. r =
1.2
0.6cos θ +1
x
3
44
3
y
3. r =
2
cosθ + 1
x
44
3
y
3
5. r =
1.2
0.6sin θ +1
y
x
4
4
2
4
7. r =
2
sinθ + 1
x
44
3
y
3
9. r =
−0.8
0.4cos θ −1
y
x
2
1
2
1
11. r =
−4
2cos θ −1
y
6
6
x
12 6
13. r =
−0.8
0.4sin θ −1
y
x
2
2
2
2
15. r =
−2
sinθ − 1
x
44
3
y
3
17. rotated hyperbola
y
2
6
x
48
19. rotated ellipse
y
x
8
6
8
6
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-48 APPENDIX B
..
Answers to Odd-Numbered Exercises
21. rotated parabola
10
2
6
2
x
y
23. x =−1 +3cos t, y = 1 +2 sint, 0 ≤ t ≤ 2π
25. x =−1 +4cosh t, y = 3sinht for the right half;
x =−1 −4 cosht, y = 3 sinht for the left half
27. x = t, y =−
1
4
t
2
+ 1 29. 2.6 times as fast
Chapter 10 Review Exercises, page 682
1. (x + 1)
2
+ (y −2)
2
= 9
y
x
5
2
5
5
3. y = x
2
− 2x +1
y
x
2 6
5
1
5.
y
x
22
1
1
7.
y
x
2
1
2
1
9. C 11. B 13. x = 2 + 2t, y = 1 + 6t, 0 ≤ t ≤ 1
15. (a)
1
3
(b) undefined (c) undefined at t =−1;
1
3
at t = 2
17. x
(0) =−3, y
(0) = 2, speed =
√
13, left/up 19. 6π
21. 1.9467 23. 5.2495 25. 27.18 27. 128.075
29. x
2
+ y
2
= 3x
x
1
4
y
2
2
31.
y
x
⫺4
4
3
⫺3
r = 0atθ = nπ;0≤ θ ≤ π
33.
x
55
1
y
5
r = 0atθ = sin
−1
2
3
+ 2πn,π − sin
−1
2
3
+ 2πn;0≤ θ ≤ 2π
35.
2
2
x
33
y
r = 0atθ =
π
2
n;0≤ θ ≤
π
2
37.
2
4
x
4
4
y
r = 0;0 ≤ θ ≤ 2π
θ =
7π
6
,
11π
6
39. r = 3 41.
1
√
3
43. 0.157 45. 0.543
47. 2.828 49. 28.814 51. y =
1
4
(x − 1)
2
+ 1
53.
(y −2)
2
1
−
(x − 2)
2
3
= 1
55. ellipse, (−1, −2) and (−1, 8), (−1, −1) and (−1, 7)
57. parabola, (1, 4), (1,
15
4
), y =
17
4
59. (0,
1
2
)
61. r =
2.4
0.8cos θ +1
63. r =
2.8
1.4sin θ +1
65. x =−1 +3cos t, y = 3 +5 sint, 0 ≤ t ≤ 2π
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-49
CHAPTER 11
Exercises 11.1, page 695
1.
a
b
a + b
3. 5, 3, −4, 6, 6, 12,
√
290
5. −2i +3j, 7i, 3i +6j,
√
290
7.
0
y
x
a
b
a + b
a – b
b
a
3
2
1.5
2.5
1
0.5
2
0
–0.5
1.5
–1
10.50
9. parallel 11. not 13. parallel 15. 3, 1
17. 2, −3 19. (a)
%
4
5
, −
3
5
&
(b) 5
%
4
5
, −
3
5
&
21. (a)
1
√
5
i −
2
√
5
j (b) 2
√
5
'
1
√
5
, −
2
√
5
(
23. (a)
'
3
√
10
,
1
√
10
(
(b)
√
10
'
3
√
10
,
1
√
10
(
25.
9
5
i +
12
5
j 27.
%
2
√
29, 5
√
29
&
29. 4, 0
31.
)
2
√
2, 2
√
2
*
33.
)
−
√
2
2
,
√
6
2
*
35.
c
a
b
5634
0.0
210
10.0
7.5
5.0
2.5
7
c
3b
2a
76
0.0
543210
10.0
7.5
5.0
2.5
39. 7, 1, 5 43. (a) a +b=
√
58 <
√
13 +
√
17 =a+b
43. (c) a, b in same direction
45. 10 pounds down, 20 pounds to the right
47. 30 pounds to the left and 190 pounds up
49. 13, 17; right and up
51. −80
√
14, 20 or 3.8
◦
north of west
53. 20, −20
√
399 or 2.86
◦
east of due south 55. 20 feet
57. (a)
√
17ft/s ≈ 4.123 ft/s; angle, tan
−1
4 ≈ 76.0
◦
(b) tan
−1
√
15 ≈ 75.5
◦
Exercises 11.2, page 702
1. (a)
y
x
z
(2, 1, 5)
(b)
y
x
z
(3, 1, 2)
(c)
y
x
z
(1, 2, 4)
3. 5 5. 3 7. 3, 4, −2, −1, −8, −2, 2
√
66
9. 8i +4k, −12i − 4j +4k, 2
√
186
11. (a) ±
1
√
14
3, 1, 2 (b)
√
14
'
3
√
14
,
1
√
14
,
2
√
14
(
13. (a) ±
1
3
(2i −j +2k) (b) 3
2
3
i −
1
3
j +
2
3
k
15. (a) ±
1
√
2
1, 0, −1 (b) 2
√
2
'
1
√
2
, 0, −
1
√
2
(
17. 4, 4, −2 19.
4
√
14
(2i −j +3k)
21. (x − 3)
2
+ (y −1)
2
+ (z −4)
2
= 4
23. (x − π)
2
+ (y −1)
2
+ (z +3)
2
= 5
25. sphere, center (1, 0, −2), radius 2
27. sphere, center (1, 0, 2), radius
√
5
29. point (−1, 2, 0)
31.
x
z
y
33. plane parallel to xz-plane 35. plane parallel to xy-plane
37. y = 0 39. x = 0 43. 2, −1, 1, 4, −2, 2, yes
45. not equilateral 47. they form a right triangle
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-50 APPENDIX B
..
Answers to Odd-Numbered Exercises
49. not a square 51. 4, −1, 7, 7
53. 0, −8, 10, 1, −9, −1 55.
√
31 57. 7
59. (a)
√
2 −1 59. (b)
√
3 −1 (c)
√
n − 1 > 2ifn ≥ 10
61. net force is 149 pounds in the direction −1, 5, −14; force required
to balance is 10, −50, 140 pounds
63. direction is 41, 38, 20 and speed is 593.72 mph
65. endpoints trace line segment from A to B
Exercises 11.3, page 711
1. 10 3. −14 5. 1 7. cos
−1
1
√
26
≈ 1.37
9. cos
−1
−8
√
234
≈ 2.12 11. yes 13. yes
15. (a) one possible answer: 1, 2, 3 (b) 1, 2, −3
17. (a) one possible answer: i −3j (b) −
7
6
i +2j −3k
19. 2,
%
6
5
,
8
5
&
21. 2,
2
3
1, 2, 2 23. −
8
5
, −
8
25
0, −3, 4
25. 105,600 foot-pounds 29. 920 foot-pounds
31. (a) false (b) true (c) true (d) false (e) false
33. a ·c, a ·b, b ·c
35. (a) 0, x or
%
x, −
3
4
x
&
, for any x > 0
(b) 0, x or
%
x, −
3
4
x
&
, for any x < 0
37. cos
−1
1
3
√
2
≈ 76.4
◦
, cos
−1
1
3
√
2
≈ 76.4
◦
,
cos
−1
8
9
≈ 27.3
◦
41. (a)
π
4
(b) cos
−1
1
√
3
≈ 54.7
◦
(c) cos
−1
1
√
n
43. a = cb 47. 15 49.
n
k=1
1
k
3
≤
π
3
6
√
15
51. (c) P
k
=
1
n
for each k
57. (a) a =
%
1
2
, 1,
3
2
&
, b =
%
5
2
, −2,
1
2
&
(b) a =1, 2, 3, b =−1, 2, −1
59. cos
−1
1
3
≈ 109.5
◦
61. (a) v ·n = 0, comp
v
w =−w sin θ,comp
n
w =−w cos θ
63. (a) −2000sin 10
◦
≈−347 pounds, ratio = tan10
◦
≈ 0.18
(b) 2500sin 15
◦
≈ 647 pounds, ratio = tan 15
◦
≈ 0.27
65. $190,000; monthly revenue
Exercises 11.4, page 723
1. 1 3. 4 5. 4, −3, −2 7. −9, −4, 1
9. 4, −2, 8 11. ±
1
√
69
8, 1, −2 13. ±
1
√
46
−3, −6, 1
15. ±
1
√
154
−1, −3, 12 17.
7
2
≈ 1.87 19.
61
5
≈ 3.49
21. 5 23.
11
√
3
2
25. 10
27.
20
√
2
3
≈ 9.4 foot-pounds
31. (a) spin right, force up (b) spin down right, force up right
33. (a) spin up left, force down left (b) spin up right, force up left
35. false 37. false 39. true
41. sin
−1
7
√
85
≈ 0.86 43. sin
−1
13
√
170
≈ 1.49
47. 0 51. coplanar 53. not coplanar
55. −i 57. −3j 59. Figure A; 12 61. (b) and (c)
63. ball rises 65. ball drops 67. no effect 69. ball rises
Exercises 11.5, page 732
1. (a) x = 1 +2t, y = 2 − t, z =−3 +4t
(b)
x − 1
2
=
y − 2
−1
=
z + 3
4
3. (a) x = 2 + 2t, y = 1 − t, z = 3 + t
(b)
x − 2
2
=
y − 1
−1
=
z − 3
1
5. (a) x = 1 −3t, y = 2, z = 1 + t (b)
x − 1
−3
= z − 1, y = 2
7. (a) x = 2 − 4t, y =−t, z = 1 +2t (b)
x − 2
−4
=
y
−1
=
z − 1
2
9. (a) x = 1 +2t, y = 2 − t, z =−1 +3t
(b)
x − 1
2
=
y − 2
−1
=
z + 1
3
11. intersect, (4, 2, 3) 13. parallel
15. 2(x − 1) −(y −3) +5(z −2) = 0
17. 2(x − 2) −7y −3(z −3) = 0
19.
x
a
+
y
b
+
z
c
= 1
21. −2x + 4(y +2) = 0
23. (x − 1) −(y −2) +(z −1) = 0
25. x = 4, y = 4, z = 4 27. x = 2, y = 1, z =−6
4
2
0
2
4
4
2
0
2
4
5
2.5
0
2.5
5
x
y
z
4
2
4
4
2
0
2
4
10
5
0
5
0
2
y
z
x
29. x = 4 31. z = 2
4
2
0
2
4
4
4
2
2
0
2
4
0
2
4
x
y
z
4
2
0
2
4
4
2
0
2
4
4
2
0
2
4
x
y
z
33. x = 1, z =−2
4
2
0
2
4
4
2
0
2
4
2
1
0
1
2
y
z
x
35. x = t, y =
5
3
t −
4
3
, z =
1
3
t −
8
3
37. x = 4t + 11, y =−3t − 8, z = t
39.
2
3
41.
2
√
3
43.
3
√
6
45. cos
−1
−13
√
234
≈ 2.59 47. perpendicular
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-51
49. parallel 53.
8
3
,
7
3
,
4
3
55. true (if the planes coincide, they intersect and are parallel)
57. false
59. false (true if we take all lines perpendicular to a given line through a
given point on the line)
61. true 63. yes 65. no
67. (a) intercepts x = 2, y = 2, z = 2/c (b) parallel to x + y + 2z = 0
(c) parallel to 2x + y − z = 0 (d) parallel to x + 2y −3z = 0
Exercises 11.6, page 743
1.
2
0
2
2
0
2
x
y
z
0
2.5
5
7.5
10
cylinder
3.
2
0
2
2
0
2
0
2
x
y
z
2
ellipsoid
5. 7.
2
1
1
2
2
1
0
1
2
0
5
10
15
20
x
y
z
0
4
2
0
2
4
4
2
0
2
4
4
2
0
2
4
x
y
z
circular paraboloid elliptic cone
9. 11.
2
1
0
1
2
2
1
0
1
2
5
0
5
x
y
z
2
1
0
1
2
2
1
0
1
2
2
1
0
1
2
x
y
z
hyperbolic paraboloid
hyperboloid of 1 sheet
13. 15.
4
0
4
10
5
0
5
10
5
2.5
0
2.5
5
x
y
z
5
0
5
5
0
5
2
1
0
1
2
x
y
z
hyperboloid of 2 sheets
cylinder
17. 19.
2
0
2
2
0
2
5
2.5
0
2.5
5
x
y
z
−2
x
−1
2
1
−2
0
0
−1
0
1
z
2
−1
y
−2
1
2
circular paraboloid
cylinder
21. 23.
4
0
4
4
2
0
2
4
2
0
2
4
6
8
x
y
z
4
0
4
2
0
2
4
6
8
4
2
0
2
4
x
y
z
circular cone, z ≥ 0
cylinder
25. 27.
2
0
2
0
2.5
5
7.5
10
2
0
2
x
y
z
4
0
4
4
2
0
2
4
4
2
0
2
4
x
y
z
circular paraboloid ellipsoid
29. 31.
−3
−2
x
−3
−1.0
y
−2
−1
z
−0.5
−1
0
0.0
0
0.5
1
1
1.0
2
32
3
–4
0
4
–4
–2
0
2
4
–5
–2.5
0
2.5
5
x
y
z
hyperbolic paraboloid hyperboloid of 1 sheet
33. 35.
–10
0
10
–10
–5
0
5
10
–10
–5
0
5
10
x
y
z
–2
0
2
–2
–1
0
1
2
–5
0
5
x
y
z
hyperboloid of 2 sheets
cylinder (plane)
37. 39.
–2
0
2
–2
–1
0
1
2
–2
0
2
x
y
z
−10
y
−5
z
7.5
5.0
2.5
0.0
−2.5
−5.0
−7.5
0
5
10
cylinder circular paraboloid
41. ellipsoid (c > 0) or hyperboloid of one sheet (c < 0)
43. elliptical paraboloid (c > 0) or hyperbolic paraboloid (c < 0)
45. (a) −x +2y
2
+ z
2
= 0 (b) −x
2
+ 2y
2
+ z
2
= 1
(c) −x
2
+ 2y
2
− z
2
= 1
53. exercise 3: x = sins cost, y = 3sins sint, z = 2coss;
exercise 5: x =
1
2
s cost, y =
1
2
s sint, z = s
2
;
exercise 7: x =
1
2
s cost, y = s sin t, z = s
55. possible answer: x = s cost, y = s sint, z = 4 −s
2
P1: JXR
MHDQ256-APP-B MHDQ256-Smith-v1.cls January 12, 2011 7:22
LT (Late Transcendental)
CONFIRMING PAGES
A-52 APPENDIX B
..
Answers to Odd-Numbered Exercises
Chapter 11 Review Exercises, page 745
1. −1, 3, 4, 0, 5 3. 6i +5j, −16i +12j +8k, 2
√
94
5. neither 7. parallel 9. −1, −2, 3 11.
)
1
√
5
,
2
√
5
*
13.
1
3
√
3
(5i +j −k) 15.
%
−
3
5
, 0,
4
5
&
17.
√
46
19.
2
√
3
(i −j + k) 21. 20
√
609, 80 or 9.2
◦
north of east
23. x
2
+ (y +2)
2
+ z
2
= 36 25. 0 27. −8
29. cos
−1
1
√
84
≈ 1.46 31.
1
√
6
,
1
6
(i +2j +k)
33. −2, 1, 4 35. −4i + 4j − 8k 37. ±
1
√
21
−2, 1, 4
39. 1700 foot-pounds 41. 3 43.
√
41
45.
25
2
foot-pounds
47. (a) x = 2 − 2t, y =−1 +3t, z =−3
(b)
x − 2
−2
=
y + 1
3
, z =−3
49. (a) x = 2 + 2t, y =−1 +
1
2
t, z = 1 − 3t
(b)
x − 2
2
= 2(y + 1) =
z − 1
−3
51. cos
−1
5
√
30
≈ 0.42 53. skew
55. 4(x + 5) + y − 2(z −1) = 0
57. 4(x − 2) −(y −1) +2(z −3) = 0
59. 61.
–4
–2
–4
–2
0
2
4
–10
–5
0
5
10
x
y
z
0
2
4
–2
–1
–2
–1
0
1
2
–2
–1
0
1
2
x
y
z
0
1
2
elliptic paraboloid
cylinder
63. 65.
–1
0
1
2
3
–
2
–1
0
1
2
–2
–1
0
1
2
x
y
z
–2
0
2
–2
0
2
–2
0
2
x
y
z
sphere
plane
67. 69.
–2
–1
0
1
2
–2
–1
0
1
2
–10
–5
0
5
10
x
y
z
–4
–2
0
2
4
–4 –2 0 2 4
–2
0
2
4
x
y
z
–4
4
plane
hyperboloid of 1 sheet
71.
–4
–2
0
2
4
–4
–2
0
2
4
–
4
–
2
0
2
4
y
z
x
hyperboloid of 2 sheets
CHAPTER 12
Exercises 12.1, page 756
1. 3.
30
10
5.02.50.0–2.5–5.0
40
–7.5
20
0
1
0
–1
–2
–3
210–1–2
5. 7.
–4
–2
0
2
4
–4
–2
0
2
4
–4
–2
2
4
x
y
z
0
–4
–2
0
2
4
–4
–2
0
2
4
–4
–2
0
2
4
x
y
z
0
9. 11.
–4
–2
0
0
0
10
20
30
0.5
2
x
40
z
50
60
1
70
4
y
1.5
2
–5
0
5
–5
–2.5
0
2.5
5
–5
–2.5
0
2.5
5
x
y
z
0
13. 15.
–5
0
5
–5
0
5
–5
0
5
x
y
z
0
–3
–2
–2
–5
–1
–1
0
0
0
5
1
1
10
2
2
3
–10
17. 19.
–2
–1.5
–3
–1
–2
–0.5
–6
–1
–4
–2
y
0
0
0
2
0.5
1
4
x
z
6
2
3
–1.0
–1.0
–0.5
–1.0
–0.5
–0.5
0.0
0.0
0.0
0.5
0.5
1.0
0.5
1.0
1.0
Periodic, 2π