Answers A-59
14. lim
nS∞
a
1+2
n
b
n
=e2≈7.389 06 (10.1) 15. 7[1 ln z26+1]
z (10.4)
23. x′=
4tx
3x2–2t2; –4 (10.5) 24. y′=
1
ey+2y; 1 (10.5)
2
1
x2+x
2
3ln
1
x2+x
2
xe
39. The rate of increase of area is proportional to the radius R, so the rate
1
69.
R
1
p
2
=40p
1
p–15
22
Inelastic
70.
R
1
p
2
=10p
1
p–9
22
Inelastic
71.
R
1
p
2
=30p–10p1p
Inelastic
72.
R
1
p
2
=30p–5p3>2
Inelastic
Chapter 11
Exercises 11.1
1. Decreasing 2. Increasing 3. Increasing 4. Decreasing
9. 1a, b2;
1d, f2;
1g, h2 10.
1
b,
d
2
;
1
f,
g
2
11.
1
b,
c
2
;
1
c,
d
2
;
1
f,
g
2
1
2
2
20.
f1
3
2
=1 is a local minimum; b 21. No local extremum; d 22. No
local extremum; h 23.
f1
3
2
=5 is a local maximum; f 24. No local
f1
2
27. (A)
f
′
1
x
2
=3x
2
–12 (B) –2,
2 (C) –2,
2
f
1
2
2
29. (A)
f′1x2=–6
1x+222 (B) –2 (C) 0
1
1
1
1
1
1
1
2
2
2
1
f1
2
1
f1
2
1
f1
2
1
1
1
1
1
f1
2
2
1
1
1
1
2
1
2
1
1
1
1
1
f1
f1
1
1
1
f1
f1
1
1
f1
f1
1
1
f1
5
ANSWERS
51. Increasing on 1–∞,
–12,
11,
∞2
Decreasing on
1
–1,
1
2
Horizontal tangents at x=–1,
1
f(x)
x
5
5
53. Decreasing for all x
f(x)
x
5
54. Increasing for all x
f(x)
A-62 Answers
17. 12x–8 18. –
6
x+4 19. 4x–
3
–18x–
4
55. Domain: All real numbers
f(x)
10
A-64 Answers
76. x
f
′1x2
f
1x2
–36x6-2Positive and
Increasing and
f
f
f
f
f1
f1
79. Domain: All real numbers
x int.: –1.18, 0.61, 1.87, 3.71;
Local maximum:
f1
1.24
2
=2.36;
A-66 Answers
26. Domain: All real numbers, except x=3
y int.: 2
f(x)
10
y int.: 10
20
35. Domain: All real numbers, except {2;
y int.: 0; x int.: 0;
f(x)
5
36. Domain: All real numbers, except x=–2,
x=2
1
Answers A-69
67. y=90 +0.02x 68. y=65 +0.01x 69. y=210 +0.1x
70. y=340 +0.4x 71. Domain: All real numbers except –1, 2
79. R(x)
60,000
1
1
2
1
1
2
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
24. (A) (B)
1
28. $95; $21,675 29. 40 trees; 1,600 lb 30. 2.5 weeks; $18.75
31. 110 –2172>3
=1.57 in. squares 32. (A) 18 in. by 18 in. by 36 in.;
1
1
1
1
1
1
y int.: 0; x int.: 0, 4
Increasing on
1
–∞,
3
2
1
1
2
x
50
5
y int.: –3; x int.: –3,
1
No vertical or horizontal asymptotes
25
A-72 Answers
31. 3 (11.3) 32. –1
41. x
f
′1x2
f
1x2
decreasing
concave downward
2
3
3
1
1
2
1
1
Inflection points:
1
2.92,
–46.41
2
,
1
7.08,
–88.04
2
(11.4)
57. (A) For the first 15 months, the graph of the price is increasing and
readers are manufactured and sold for $425
61. 12 orders/yr (11.6)
1
1
Chapter 12
Exercises 12.1
1.
f
(x)=5x–
4
2.
f
(x)=–6x–
9
3.
f
(x)=3x–4–2x–
5
f
f
1
2
3
6.
f
(x)=x1>3–4x–1>3 7.
f
(x)=4x1>3+x4>3–3x7>3
f
2
2
3
3
26. Yes 27. Ye s 28. Ye s 29. No 30. No 31. No 32. Ye s 33. True
34. False 35. False 36. True 37. True 38. False 39. No, since one
41. Yes, since one graph can be obtained from another by a vertical translation.
42. Yes, since one graph can be obtained from another by a vertical translation.
2
3
1
0
C
2
2
f
C
7. f′(x)=
4x3
x4–10 8. f′(x)=
2x+5
x2+5x+4 9.
1
3 (3x+5)3+
C
10. 1
4 (6x–1)4+C 11.
1
6 (x2–1)6+C 12.
1
5 (x6+1)5+C
13. –1
2 (5x3+1) –2+
C
14. –1
5 (4x2–3) –5+
C
15. e
C
3
3 (x–3)–3+C 23. –1
6 (6t–7) –1+
20 (5t+1)4+
25. 1
12 (t2+1)6+
C
26. –1
3 (t3+4) –1+
C
27.
1
2 ex2+
C
31. –e
1
–t+
C
32. –3
ln
0
2–t
0
+
C
33. –1
18 (3t2+1) –3+
C
2
ANSWERS
A-74 Answers
f
015
0
010
0
67. 100
68. 1,000
(C) 6.7 years 85. W(t)=2e
0
.
1
t+2; 6.45
g
Exercises 12.3
1. The derivative of f(x) is 5 times
f
(x); y′=5y 2. The derivative of f(x) is –2
f
4. The derivative of f(x) is 0.04 times f(x); y
′
=0.04y 5. The derivative of
f
f
Answers A-75
85. I=I
0
e–
0
.
00942
x; x≈117 ft 86. P(t)=P
0
e–at 87. (A) Q=5e–
0
.
04
t
5
5
5
5
3
(B) 12 minutes (C) After 15 minutes
Exercises 12.4
1. 80 in.2 2. 180 cm2 3. 36 m2 4. 90 ft2 5. No, p–271 m
2
16.
u(x)
v(x)
74. L4=5,520 …L100
N(t)
dt …6,180 =R4
Exercises 12.5
1. 500 2. 200 3. 28 4. 8 5. 48 6. 62,500 7. 4.
5
p≈14.14
8. 12.5p≈39.27 9. (A) F(15) –F(10) =375 (B) F9(x)
F9(x) 5 9
3
3
3
3
20. 2
1
3– ln
2≈0.4055 23. 2e3–2≈38.171
3
3
37. 56=15,625 38. 510 39.
l
n 4 ≈1.38
6
40.
l
n
3≈1.099
41. 20
1
e0.2
5
–e–0.
52
≈13.550 42. 32.637 43.
1
7
3
3
5
2 11–e–12≈0.316 46.
2 1e–12≈0.859 47. 0 48. 0
3
6
6
(B)
500
0
Ave f(x) 5 250
50. (A) Average g(x)=12
5
5
the reverse is true.
is decreasing on [1, 4], L
3
overestimates the area and R
3
underestimates the
area. 21. In both figures, the error bound for L3 and R3 is 7.
24. S4=–1,
6
2
6
25. S4=–1,194 26. S
5
=10 27. S
3
=–33.01
32. 10.667 33. 8.533 34. 2.132 35. 4.949 36. 7.083 37. –10.
66
7
43. 15 44. 63 45. 58.5 46. 10.5 47. –54 48. 16.5 49. 248
6
3 51. 0 52. 0 53. –183 54. 13.5 55. False 56. True
57. False 58. True 59. False 60. False 61. L1
0
=286,100 ft
2
; error
6
2.63. Geometrically, the definite integral over the interval [2, 5] is the sum of
67. Decreasing; L4≈–9.308 68. Decreasing; L4≈–5.245
73. L
3
=2,580, R
3
=3,900; error bound for L3 and R3 is 1,320