Answers6-4

Answers A-53
41. 3500
42. 20,000
43. 100
44.
1,000
0 15
0
0 10
0
0 30
0
0 40
0
45. 100
46. 1,000
47. 1000
48.
400
0 100
0
0 70
0
0 40
0
0 30
0
49. Apply the second-derivative test to f1y2 = ky1M - y2. 50. t = ln c 51. 2009 53. A = 1,000e 0.03t 54. A(t) = 5,250e 0.02t
kM
55. A = 8,000e 0.016t 56. A(t) = 5,000e 0.022t
57. (A) p1x2 = 100e -0.05x
(B) $60.65 per unit
(C)
p
100
58. (A) p(x) = 10e 0.005x
(B) $16.49 per unit
(C) p
35
59. (A) N = L11 - e -0.051t 2
(B) 22.5%
(C) 32 days
(D) N
L
60. (B) N(t) = L(1 - e -0.011t )
(C) 63 days
(D) N
L
25
61. I = I 0 e -0.00942x ; x L 74 ft
62. P(t) = P 0 e -at
x
63. (A)
(B)
250 x
Q = 3e -0.04t
Q1102 = 2.01 mL
(C) 27.47 hr
(D) 3
300
t
64. (A) 52 people; 749 people
(B) 17 days
(C) 1,000
(D) N
1000
300
t
0 90
0
65. 0.023 117 66. 0.057 536 67. Approx. 24,200 yr
68. 1.64 words per minute/hour of practice; 0.9 words per minute/hour of practice 69. 104 times; 67 times 70.
71. (A) 7 people; 353 people (B) 400 (C) 400
72. 15 minutes
30
t
S = k ln R R 0
0 30
0
Exercises 6-4
1. C, E 2. A, D 3. B 4. None 5. H, I 6. G, H 7. H 8. F, J
9. 10.
f(x)
10
g(x)
y f(x) 10
6 x
u(x)
y g(x)
10
6 x
v(x)
y u(x)
10
1 2 5 x
y v(x)
5 x
11. Figure A: L 3 = 13, R 3 = 20; Figure B:
L 3 = 14, R 3 = 7
12. Figure C: L 3 = 7, R 3 = 14
Figure D: L 3 = 20, R 3 = 13
4
13. L 3 … 1 f1x2 dx … R 4
3; R 3 … 1 g1x2 dx … L 3;
since f(x) is increasing, L 3 underestimates the
area and R 3 overestimates the area; since g(x)
is decreasing, the reverse is true.
4
4
14. L 3 … u(x) dx … R 3 , R 3 … v(x) dx … L 3 ; since u(x) is increasing on [1, 4], L 3 underestimates the area and R 3 overestimates
L L
1
1
the area; since v(x) is decreasing on [1, 4], L 3 overestimates the area and R 3 underestimates the area.
15. In both figures, the error bound for L 3 and is 7. 16. In both figures, the error bound for and is 7.
R 3
L 3
R 3

A-54 Answers
17. S 5 = -260 18. S 4 = -1,626 19. S 4 = -1,194 20. S 5 = 10 21. S 3 = -33.01 22. S 3 = -30.37 23. S 6 = -38
24. S 6 = -44 25. -2.475 26. 5.333 27. 4.266 28. 1.066 29. 2.474 30. 3.541 31. -5.333 32. -2.474 33. 1.067
34. -4.266 35. -1.066 36. -2.858 37. 15 38. 63 39. 58.5 40. 10.5 41. -54 42. 16.5 43. 248 44. - 496
3
45. 0 46. 0
47. -183 48. 13.5 49. False 50. True 51. False 52. True 53. False 54. False
55. L error bound is 50,000 ft 2 56. R 10 = 336,100 ft 2 ; error bound is 50,000 ft 2 10 = 286,100 ft 2 ;
; n Ú 200
; n Ú 500
57. L 6 = -3.53, R 6 = -0.91; error bound for L 6 and R 6 is 2.63. Geometrically, the definite integral over the interval [2, 5] is the sum
of the areas between the curve and the x axis from x = 2 to x = 5, with the areas below the x axis counted negatively and those
above the x axis counted positively.
58. L 5 = -6.25; R 5 = 2.5; error bound for L 5 and R 5 is 8.75. Geometrically, the definite integral over the interval [1, 6] is the sum of
the areas between the curve and the x axis from x = 1 to x = 6, with the areas below the x axis counted negatively and those
above the x axis counted positively.
59. Increasing on 1- q, 04; decreasing on 30, q2 60. Increasing on (-q, q)
61. Increasing on 3-1, 04 and 31, q2; decreasing on 1- q, -14 and [0, 1] 62. Decreasing on (-q, 0]; increasing on [0, q)
63. n Ú 22 64. n Ú 93 65. n Ú 104 66. n Ú 1,530 67. L 3 = 2,580, R 3 = 3,900; error bound for L 3 and R 3 is 1,320
68. L 4 = 5,520, R 4 = 6,180; error bound for L 4 and R 4 is 660
5
69. (A) L 5 = 3.72; R 5 = 3.37 (B) R 5 = 3.37 … 1 0 A¿1t2 dt … 3.72 = L 5
70. L 5 = 2.25, R 5 = 2.03; error bound for L 5 and R 5 is 0.22 71. L 3 = 114, R 3 = 102; error bound for L 3 and R 3 is 12
6
72. R 3 = 140 … N œ (x) dx … 156 = L 3
L
0
Exercises 6-5
1. (A)
(B)
F1152 - F1102 = 375
F(x)
120
F(x) 6x
2. (A)
(B)
F(15) - F(10) = 45
F(x)
20
3. (A)
(B)
F1152 - F1102 = 85
F(x)
50
F(x) 2x 42
4. (A)
(B)
F(15) - F(10) = 275
F(x)
80
F(x) 2x 30
Area 375
F(x) 9
Area 45
Area 85
Area 275
20
x
20
x
20
x
20
x
5. 40 6. 288 7. 72 8. 64 9. 46.5 10. 18 11. e - 1 L 1.718 12. 4e 2 - 4 L 25.556 13. ln 2 L 0.693 14. 2 ln 5 L 3.219
15. 0 16. 0 17. 48 18. 30 19. -48 20. -30 21. -10.25 22. -156 23. 0 24. 0 25. -2 26. -1 27. 14 28. 12
1 7
29. 5 6 = 15,625 30. 510 31. ln 4 L 1.386 32. ln 3 L 1.099 33. 201e 0.25 - e -0.5 2 L 13.550 34. 32.637 35. 36.
1
1
37.
2 11 - 2 3 L 2.333
e-1 2 L 0.316 38.
2
(e - 1) L 0.859 39. 0 40. 0
41. (A) Average f1x2 = 250
(B)
500
Ave f(x) 250
42. (A) Average g(x) = 12
(B)
20
g(x) 2x 7
43. (A) Average
(B)
f1t2 = 2
10
f(t) 3t 2 2t
Ave g(x) 12
0 10
0
f(x) 500 50x
0 5
0
1 2
2
Ave f(t) 2
44. (A) Average g(t) = -4
(B)
2
2 2
g(t) 4t 3t 2
45. (A) Average
(B)
3
f1x2 = 45
28 L 1.61
3
f(x) x
46. (A) Average g(x) L 2.53
(B)
4
g(x) x 1
Ave g(t) 4
Ave g(x) 2.53
20
47. (A) Average f1x2 = 211 - e -2 2 L 1.73
(B)
5
Ave f(x) 1.73
0 10
0
f(x) 4e 0.2x
1 8
0
Ave f(x) 1.61
48. (A) Average f(x) L 98.04
(B)
150
0 10
0
f(x) 64e 0.08x
Ave f(x) 98.04
3 8
0
1
49.
6 1153>2 - 5 3>2 2 L 7.819
1
50.
9 (53/2 - 2 3/2 )
1
51.
21ln 2 - ln 32 L -0.203
1
52. 1 4 ln 2 53. 0 54.
2
(ln 2)2
55. 4.566 56. 2.925 57. 2.214
58. 7.069