Type:
Solution Manual
Resource:
Calculus and Its Applications
Edition:
2nd Edition
Author(s):
Marvin L. Bittinger David Ellenbogen Scott Surgent Gene Kramer
A-14
INSTRUC TOR ANSWERS
INSTRUCTOR ANSWERS: CHAPTER 1
69.
Exercise Set 1.1, p. 112 1. 0.3, x S 0.3-
2. 1.7, x S 1.7+
4. - 4.9, x S - 4.9 + 7. 0.3, x S 0.3-
5.
2 3, x S
12 2 3
-
8. 1.2, x S 1.2-
10. 0, x S 0 -
11. -2
14. limx S 3-
15. limx S 5
3. - 3, x S -36.
4 3, x S
12 4 3
-
9. 1, x S 1+
-2 -1-1
61.
13. limx S 2+ 1 lim 16. 17. “the limit, xS
-5-4-3 -2 -1-1
f(x) = ƒ x ƒ 1 2 3 4 5 x
-2 -3 -4 -5
-5-4-3 -2 -1-1
-2 -3 -4 -5
1 2 3 4 5 x
y 6 5 4 3 2 1
66.
y 5 4 3 2 1
G
1, 3, does not exist
1 2 3 4 5 6 7 x
-2
74.
y 7 6 5 4 3 2 1
y 5 4 3 2 1
g
-5-4-3 -2 -1-1
1 2 3 4 5 x
-2 -3 -4 -5
1 2 3 4 5 6 7 8 x
f
-1, -1, - 1
1, 0, does not exist 75.
76.
y
y 4
3
1 2 3 4 5 x
3
-3
68.
1 2 3 4 5 x f(x) =
-2, does not exist
1 - 2x x
G
-2
-1
1
2 x
-2
-1 g(x) = ƒ x ƒ + 1 1 2 3 4 5 x
F(x) =
2 x-3
1
2
x
-1
1
-1 77.
-1
y 5 4 3 2 1
y
78.
8 6
H
-2 -3 -4 -5
does not exist, 2
G
4
1 2 3 4 5 x
2 -4
-2
2
4
x
1, 9
1 2 3 4 5 6 7 8 x
does not exist, 2
y 3 2 1
2 1
-2 -3 -4 -5
4, does not exist
-2 -3 -4 -5 -6 -7
1 2 3 4 5 x
3, 1, does not exist
-5-4-3 -2 -1-1
y 5 4 3 2 1 -2 -1-1
1 2 3 x
-2 -3 -4 -5
-5-4-3 -2 -1-1
F
-2 -3
-2 -3 -4
g(x) = x2 - 5
-7 -6 -5-4-3 -2 -1-1
67.
y 8 7 6 5 4 3 2 1 -3 -2 -1-1
-2 -1-1
4, 1
4 x+2
4, does not exist
-2 -3 -4 -5
73.
f(x) = x2
1 2 3 x
1
-5-4-3 -2 -1-1
-5, - 4 65.
-7 -6 -5-4-3 -2 -1-1
F
64.
y 5 4 3 2 1 -5 -4 -3 -2 -1-1
2x - 5 x-3
72.
-5-4-3 -2 -1-1
1, 0
63.
4x + 9 x+2
2
0, 2
G(x) =
y 9 8 7 6 5 4 3 2 1
g(x) =
y 5 4 3 2 1
71.
y 9 8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8 x
2, does not exist
2
62.
y 5 4 3 2 1
g(x) =
-2 -3
12. 7
as x approaches 4, of f1x2” or “the limit of f1x2 as x approaches 4” 18. “the limit, as x approaches 1, of g1x2” or “the limit of g1x2 as x approaches 1” 19. “the limit, as x approaches 5 from the left, of F1x2” or “the limit of F1x2 as x approaches 5 from the left” 20. “the limit, as x approaches 4 from the right, of G1x2” or “the limit of G1x2 as x approaches 4 from the right” 21. (a) -3; (b) - 3; (c) - 3 22. (a) 1; (b) 2; (c) does not exist 23. (a) - 1; (b) -1; (c) - 1 24. (a) 4; (b) 2; (c) does not exist 25. 5 26. 4 27. Does not exist 28. 0 29. 2 30. 0 31. 2 32. 4 33. 1 34. 3 35. 4 36. -1 37. 0 38. Does not exist 39. 0 40. 0 41. 0 42. 1 43. 1 44. 1 45. 4 46. 2 47. 1 48. Does not exist 49. 1 50. 1 51. - 1 52. 1 53. 2 54. Does not exist 55. Does not exist 56. 0 57. 0 58. 3 59. 2 60. 1
70.
y 7 6 5 4 3 2 1
y 8 7 6 5 4 3 2 1 -5-4-3 -2 -1-1
f(x) =
1 + 3x x
1 2 3 4 5 x
-2
3, does not exist
79. $3.50, $3.50, $3.50 80. $3.00, $3.50, does not exist 81. $4.00, $4.50, does not exist 82. $1.00, $1.21, does not exist 83. $1.21, $1.42, does not exist 84. $1.42, $1.42, $1.42 85. Does not exist 86. $1.63 87. 10%, 12%, does not exist 88. 12%, 12%, 12% 89. 22%, does not exist 90. 10%, 10%, 10% 91. 12%, 22%, does not exist 92. 22%, does not exist 93. 3 94. -3 95. - 1 96. (a) 0; (b) 2; (c) answers may vary. 97. (a) 4; (b) 4, (c) 4; (d) 4; (e) 4; (f) no; (g) yes 98. Does not exist, 2 99. 0, 0 100. Does not exist, 16
INSTRUC TOR ANSWERS
A-15
Exercise Set 1.2, p. 124
Exercise Set 1.3, p. 133
1. True 2. False 3. True 4. True 5. True 6. False 7. False 8. True 9. 5 10. 3 11. - 3 12. 7 13. 4 14. 4 15. 15 16. 10 17. 1 18. - 4 19. 10 20. 6 21. 72 22. 32 23. - 13 24. 13 25. 3 26. - 12 4 6 1 27. - 6 28. 10 29. Does not exist 30. Does not exist 31. 54 32. 73 33. 16 34. 3 35. 13 36. 12 37. 0 38. 0 39. Does not exist 40. Does not exist 41. 3 42. 27 43. Does not exist 44. Does not exist 45. 0 46. 0 47. Not continuous 48. Not continuous 49. Continuous 50. Not continuous 51. Not continuous 52. (a) - 2, -2, - 2; (b) - 2; (c) yes, the limit exists and equals the function value; (d) does not exist; (e) -3; (f) no, the limit does not exist. 53. (a) - 1, 2, does not exist; (b) -1; (c) no, the limit does not exist; (d) 3; (e) 3; (f) yes, the limit exists and equals the function value. 54. (a) 2; (b) 2; (c) yes, the limit exists and equals the function value; (d) 0; (e) 0; (f) yes, the limit exists and equals the function value. 55. (a) 2; (b) does not exist; (c) no, the function value does not exist; (d) -2; (e) - 2; (f) yes, the limit exists and equals the function value. 56. (a) 0.25; (b) 0.25; (c) yes, the limit exists and equals the function value; (d) does not exist; (e) does not exist; (f) no, neither the limit nor t1- 22 exists. 57. (a) 3; (b) 1; (c) does not exist; (d) 1; (e) no, the limit does not exist; (f) yes, the limit exists and equals the function value; (g) yes, the limit exists and equals the function value. 58. (a) 1; (b) - 1; (c) does not exist; (d) 1; (e) no, the limit does not exist; (f) yes, the limit exists and equals the function value. 59. Yes, the limit exists and equals the function value at 4. 60. Yes, the limit exists and equals the function value at 5. 61. No, the limit does not exist and G is not defined at 0. 62. No, F is not defined at - 1. 63. Yes, the limit exists and equals the function value at 4. 64. Yes, the limit exists and equals the function value at 3. 65. No, the limit does not exist at 3. 66. No, the limit does not exist and G is not defined at 4. 67. No, g is not defined at 4. 68. Yes, the limit exists and equals the function value at 3. 69. No, the limit value does not equal the function value at 2. 70. No, the limit value does not equal the function value at 1. 71. Yes, the limit exists and equals the function value at 4. 72. Yes, the limit exists and equals the function value at 5. 73. No, the limit does not exist and the function is not defined at 5. 74. Yes, the limit exists and equals the function value at 3. 75. No, the limit does not exist and the function is not defined at 2. 76. Yes, the limit exists and equals the function value at 4. 77. Yes, because limx S a g1x2 = g1a2 for all a such that -4 6 a 6 4 78. Yes, because limx S a F1x2 = F1a2 for all a such that - 5 6 a 6 5 79. Yes, because limx S a g1x2 = g1a2 for all a such that 1 6 a 6 ` , and limx S 1+ g1x2 = g112 80. Yes, because limx S a h1x2 = h1a2 for all a such that - 3 6 a 6 ` , and limx S -3+ h1x2 = h1- 32 81. Yes, because limx S a F1x2 = F1a2 for all a such that - 5 6 a 6 5, and limx S -5+ F1x2 = F1-52 and limx S 5- F1x2 = F152 82. Yes, because limx S a G1x2 = G1a2 for all a such that - 3 6 a 6 3, and limx S -3+ G1x2 = G1-32 and limx S 3- G1x2 = G132 83. 30, 25, does not exist 84. 8, 6, does not exist 85. 120, 120, 120 86. Yes 87. Yes 88. Yes 89. Yes 90. No 91. No 92. Yes 93. Yes 19 94. 5 95. 2 96. (a) Does not exist; (b) -3 97. a = , 4 1 1 100. 6 101. b = - 3 98. c = 9 99. 2 2 1 1 3 1 102. 103. 104. 105. 0 106. 4 4 223 27
1. The temperature rose 3 degrees/hr. 2. Jennifer hiked 3 mi/hr. 3. Marcus delivered 7 packages/hr. 4. The population of Felton grew by 100 people/yr. 5. Tanya scored 25 points/game. 6. Chris grew 7.5 cm/yr. 7. Burnham Industries had 5,000,000 dollars/month in revenue. 8. Juan spent 2.25 dollars/gallon on gasoline. 9. Unemployment changed by -0.333 percentage point/month. 10. Shannon spent 3.1 dollars/ day on electricity for April. 11. 3 12. 5 13. 2 1 14. 6 15. - 32 16. - 35 17. 8 18. 8 19. 4.25 20. 1.6 21. (a) 10x + 5h; (b) 60, 55, 50.5, 50.05 22. (a) 8x + 4h; (b) 48, 44, 40.4, 40.04 23. (a) - 10x - 5h; (b) - 60, -55, - 50.5, - 50.05 24. (a) -8x - 4h; (b) - 48, - 44, -40.4, -40.04 25. (a) 2x + h - 1; (b) 11, 10, 9.1, 9.01 26. (a) 2x + h + 1; (b) 13, 12, 11.1, 11.01 9 9 3 6 60 27. (a) ; (b) - , - , - , x1x + h2 35 10 17 167 2 2 1 4 40 28. (a) ; (b) - , - , - , x1x + h2 35 15 51 501 29. (a) 2; (b) 2, 2, 2, 2 30. (a) -2; (b) -2, - 2, -2, - 2 31. (a) 36x 2 + 36xh + 12h2; (b) 1308, 1092, 918.12, 901.8012 32. (a) - 3x 2 - 3xh - h2; (b) -109, -91, - 76.51, -75.1501 33. (a) 2x + h - 4; (b) 8, 7, 6.1, 6.01 34. (a) 2x + h - 3; (b) 9, 8, 7.1, 7.01 35. (a) 2x + h - 3; (b) 9, 8, 7.1, 7.01 36. (a) 2x + h + 4; (b) 16, 15, 14.1, 14.01 37. 0.36 percentage point/yr, -0.3 percentage point>yr, 0.09 percentage point/yr 38. 1.7 percentage point/yr, -0.2 percentage point>yr, 0.66 percentage point/yr 39. 0.66 percentage point/yr, - 0.15 percentage point>yr, 0.21 percentage point/yr 40. - 0.31 percentage point>yr, 0.1 percentage point/yr, - 0.08 percentage point>yr 41. 0.1 percentage point/yr, - 0.04 percentage point>yr, 0.03 percentage point/yr 42. -0.7 percentage point>yr, 0.46 percentage point/yr, -0.05 percentage point>yr 43. 2.1 percentage points/yr, -4.3 percentage points>yr, -1.5 percentage points>yr 44. 1 percentage point/yr, -0.43 percentage point>yr, 0.2 percentage point/yr 45. - +450>yr, $1266.67/yr, $580/yr 46. $1.467 billion/yr, $0.4 billion/yr, $0.93 billion/yr 47. (a) 70, 39, 29, 23; (b) answers may vary. 48. (a) 300, 180, 120, 100; (b) answers may vary. 49. (a) $26.62; (b) $31.16; (c) $4.54; (d) 0.7567, which means that prices increased by an average of about $0.76 per year 50. (a) $2391.24; (b) $2693.71; (c) $302.47; (d) $151.24, which means the amount grew on average $151.24 per year for 2 yr 51. The average cost of production of between 300 and 305 holders is $19.75 per unit. 52. The average revenue from sales of between 300 and 305 holders is $149.40 per unit. 53. 17.62; the average rate of change in Amazon’s revenue between 2014 and 2017 was $17.62 billion per year. 54. 41.35; the average rate of change in Panera Bread Co.’s income between 2014 and 2017 was $41.35 million per year. 55. (a) 1.49 hectares/g; (b) home range increases by 1.0902 hectares per gram as the animal’s weight grows from 200 g to 300 g. 56. (a) 50.33; the population grew by about 50 condors per year between 2010 and 2017. (b) 42.064; the population increased by about 42 condors per year between 2007 and 2015. 57. (a) 1.25, 1.25, 0.625, 0, 0; (b) answers may vary. 58. (a) 29.4 mi/gal; (b) 0.034 gal/mi 59. (a) 256 ft; (b) 128 ft/sec 60. (a) 184.05 mi, or the distance traveled from t = 2 hr to t = 5 hr; (b) 61.35 mi/hr 61. (a) 125 thousand people/yr; (b) answers may vary; (c) A: 290 thousand people/year, - 40 thousand people/yr, -50 thousand people/yr, 300 thousand people/yr;
A-16
INSTRUC TOR ANSWERS
B: 125 thousand people/yr, 125 thousand people/yr, 125 thousand people/yr, 125 thousand people/yr (d) answers may vary. 62. - 116; Payton County lost an average of 116 people per year between the 5th and 8th years. 63. 825.46; Harbor University’s undergraduate population was increasing at the rate of 825.46 students per year between the 2nd and 6th years. 64. m 65. 2ax + b + ah 66. 3ax 2 + 3axh + ah2 + 2bx + bh 67. 4x 3 + 6x 2h + 4xh2 + h3 68. 5x 4 + 10x 3h + 10x 2h2 + 5xh3 + h4 69. 5ax 4 + 10ax 3h + 10ax 2h2 + 5axh3 + ah4 + 4bx 3 - 2x - h + 6bx 2h + 4bxh2 + bh3 70. 1x + h2 2x 2 1 71. 72. (a) Multiplying by 1; 11 - x - h211 - x2 (b) performing multiplication in the numerator; (c) combining like terms and simplifying; (d) when h is not 0, then h>h = 1. 2 73. 221x + h2 + 1 + 22x + 1 -1 74. 2x 2x + h12x + 2x + h2
5. (a) and (b)
y 10 8 6 4 2
6. (a) and (b) (-2, 8)
-2
-1 -1
7. (a) and (b)
2
(c) f91x2 = x; (d) - 2, 0, 1
9. (a) and (b)
(1, 12) 1
-2 -1-1
2
(c) f91x2 = - 6x; (d) 12, 0, - 6
x-axis is tangent to curve at (0, 0). y 2 -1 -2
1
-4 -6 -8 -10 -12 -14 -16 -18
-1 -1
-2 -3 -4 -5 -6 -7 -8 (-2, -8)-9
1 2 3 4 5 6 7 8 x
-5-4-3 -2 -1-1
-2 -3 -4 -5
11. (a) and (b) (c) f91x2 = - 4x; (d) 8, 0, -4 1
2
x
(1, -2)
f
f
(c) f91x2 = 34;
y 5 4 3 2 1
f
y 1 -2
(d) 12, 12, 12
10. (a) and (b)
2 x (1, -3)
is tangent 4. (a) and (b) x-axis to curve at (0, 0).
1 2 3 4 5 x
(c) f91x2 = 12;
-2 -3 -4 -5 -6
x
x-axis is tangent to curve at (0, 0).
(-2, -12)
f
y 4 3 2 1
-1
-2
(c) f91x2 = 2; (d) 2, 2, 2
-2
f
1
3. (a) and (b)
1 2 3 4 5 x
y 8 7 6 5 4 3 2 1
2
-1
f
-5-4-3 -2 -1-1
3
-2
(c) f91x2 = -2; (d) -2, - 2, -2
-2
x
y
(-2, 2)
2 3 4 5 x (1, -1)
-5-4-3 -2 -1-1
x-axis is tangent to curve at (0, 0).
2. (a) and (b)
x-axis is tangent to curve at (0, 0).
y 8 7 6 5 4 3 2 1
(1, 1.5) 1
(c) f91x2 = -3x 2; (d) -12, 0, - 3
y 10 8 6 4 2 -4 -6 -8 -10
8. (a) and (b)
f
x-axis is tangent to curve at (0, 0).
-5-4-3 -2 -1-2
(c) f91x2 = 3x; (d) - 6, 0, 3
y 9 8 (-2, 6) 7 6 5 4 3 2 1
1 2 3 4 5 x
-4 -6 -8 -10
f
Exercise Set 1.4, p. 144 1. (a) and (b)
(1, 1)
-5-4-3 -2 -1-2 (-2, -8)
(c) f91x2 = 3x 2; (d) 12, 0, 3
f
(-2, 6)
y 8 7 6 5 4 3 2 1
-3 -2 -1 -1 -2
(d) 34, 34, 34 1 2 3 4 5 x f
f
(0, 0) 1
2
3 x
(1, 0)
(c) f91x2 = 2x - 1; (d) -5, - 1, 1
INSTRUC TOR ANSWERS y 8 7 6 5 4 3 2 1
12. (a) and (b)
(-2, 2)
(1, 2)
-3 -2 -1 -1 -2
13. (a) and (b)
(22, 0)
y 6 5 4 3 2 1
(c) f91x2 = 2x + 1; (d) - 3, 1, 3
f
1 2 (0, 0)
3 x
(c) f91x2 = 4x + 3; (d) - 5, 3, 7 (1, 3) y 5 2x 2 1 3x 2 2
24 232221 1 2 3 4 5 21 22 (0, 22) 23 24
14. (a) and (b)
y 32
(22, 31)
x
28
(c) f91x2 = 10x - 2; (d) -22, - 2, 8
24 20 16 (1, 10)
(0, 7) 4 23 22 21
1 y 5 4 3 2 1
15. (a) and (b)
-5 -4 -3 -2 -1-1 (-2, -1) -2
2
3
f (1, 2)
x
2 (c) f91x2 = - 2 ; x (d) - 12, not defined, -2
1 2 3 4 5 x
-3 -4 -5
16. (a) and (b)
y 5 4 3 2 1 -5-4-3 1 -2, - 2
(
-1-1 -2 ) -3 -4 -5
1 (c) f91x2 = - 2 ; x (d) - 14, not defined, -1
f
(b) answers may vary; (c) 1 54. (a) x = 0; (b) answers may vary; (c) 12 (description of method left to the student) 55. (a) x = 3; (b) answers may vary; (c) - 1, - 1, 1, 1 56. (a) x = -5; (b) answers may vary; (c) -2, -2, 2, 2 57. Answers may vary. 58. (a) Limit is 5 and F122 = 5, so F is continuous at x = 2. (b) No, there is a corner at x = 2. 59. (a) Limit is 1 and G112 = 1, so G is continuous at x = 1. (b) Yes, the graph is a smooth nonvertical curve at x = 1. 5 60. m = 11, b = -18 61. 32 62. 63. The trucker’s 23 average speed was 290/4 = 72.5, and his distance function is differentiable for the entire 4-hr drive, so he must have driven 72.5 miles per hour at least once in that 4-hr period. 64. Because g is not differentiable everywhere in the interval
Exercise Set 1.5, p. 154
y 5 5x 2 2 2x 1 7
12
A-17
(1, 1) 1 2 3 4 5 x
17. (a) y = 12x + 16; (b) y = 0; (c) y = 48x - 128 18. (a) y = 6x - 9; (b) y = - 2x - 1; (c) y = 20x - 100 19. (a) y = - 2x + 4; (b) y = - 2x - 4; (c) y = - 0.0002x + 0.04 1 1 2 20. (a) y = x + 2; (b) y = x - 1; (c) y = x + 4 25 5 21. (a) y = - 6x - 4; (b) y = - 1; (c) y = 6x - 16 22. (a) y = 2x + 5; (b) y = 4; (c) y = - 10x + 29 23. f91x2 = m 24. f91x2 = 2ax + b 25. x1, x3, x4, x5, x7 26. x0, x2, x4, x5, x6, x7 27. x0, x3, x4, x6, x12 28. x2, x4, x5, x7, x8 29. x1, x2, x3, x4 30. x1, x2, x3, x4, x5, x6, x7, x9, x10 31–38. Left to the student 39. False 40. True 41. False 42. False 43. Answers may vary. 44. Answers may vary. 2 1 45. f91x2 = 46. f91x2 = 5x 4 47. f91x2 = - 3 11 - x2 2 x 1 1 1 48. f91x2 = 49. f91x2 = 50. f91x2 = 22x 22x + 1 2x 2x 51. f91x2 = 2ax + b 52. Answers may vary. 53. (a) x = - 3;
1. The derivative of u with respect to v is f91v2; the derivative of du u with respect to v is u9; the derivative of u with respect to v is , dv d the derivative of u with respect to v is f1v2. 2. The derivadv tive of s with respect to t is g91t2; the derivative of s with respect ds to t is s9; the derivative of s with respect to t is ; the derivative dt d 3. The derivative of p with of s with respect to t is g1t2. dt respect to q is R91q2; the derivative of p with respect to q is p9; dp the derivative of p with respect to q is ; the derivative of p with dq d respect to q is R1q2. 4. The derivative of m with respect to dq n is G91n2; the derivative of m with respect to n is m9; the derivadm ; the derivative of m with respect tive of m with respect to n is dn d to n is G1n2. 5. The derivative of h with respect to k is dn m91k2; the derivative of h with respect to k is h9; the derivative of dh h with respect to k is ; the derivative of h with respect to k is dk d m1k2. 6. The derivative of C with respect to z is T91z2; the dk derivative of C with respect to z is C9; the derivative of C with dC d respect to z is ; the derivative of C with respect to z is T1z2. dz dz 7. 7x 6 8. 8x 7 9. - 3 10. - 0.5 11. 0 12. 0 13. 30x 14 14. 30x 9 15. -6x -7 16. - 8x -9 17. - 8x -3 4 18. - 15x -6 19. 3x 2 + 6x 20. 4x 3 - 7 21. 2x 2 2 22. 23. 0.9x -0.1 24. 1.7x 0.7 25. x -1>5 5 2x 21 24 1 3 28. - 5 29. + 2 26. -1.6x -2>3 27. - 4 4 3 x x x 42x 1 2 10 3 2 3 5 30. 31. 2 x 32. 33. 3 2 4 3 11 3 2x x 2x 42x 2 12 12 4 3 34. 36. - 4 37. - 2 - x -2>5 35. - 7 3 5 5x 7x x 5 2 -1>3 1 38. - 2 - x 39. 7 40. 4 41. 2x 3 2 x 1 3 42. 2x 43. -0.02x + 0.4 44. - 0.02x - 0.5 3 3 5 8 45. - x -7/4 - 2x -1/3 + x 1/4 - 5 4 4 x 7 3 -1/4 6 1/5 24 1 -5/3 46. - 2x + x + x - 4 47. - 2 4 5 7 x x
A-18
INSTRUC TOR ANSWERS
2 1 48. - 2 2 x 54. 14
49.
55. -
3 640
1 4
50. 24
51. - 5
52. 1
4 3
57. (a) y =
3 3 x - ; 2 2
56.
53.
1 12
103. f91x2 = 3x 2 + a; when a Ú 0, then the derivative is always nonnegative for all x, but when a 6 0, the derivative changes sign at least twice. 104. 2x + 1 105. 2x 106. 3x 2 - 1 1x ? 02 1
107. 3x 2 -
1x ? 02
108.
31 107 165 x - 17; (c) y = x 4 6 2 58. (a) y = 10x - 15; (b) y = x + 3; (c) y = - 2x + 1 1 2 1 1 4 2 59. (a) y = - x + ; (b) y = x + ; (c) y = x + 3 3 3 3 3 3 2 1 1 3 60. (a) y = - 2x + 3; (b) y = - x + ; (c) y = x + 27 3 4 4 61. 10, 42 62. 10, - 32 63. 10, - 22 64. 10, 12
113.
69. None 70. None 71. All points (the graph is a horizontal line) 72. All points (the graph is a horizontal line) 1 1 5 13 73. a1, -55 b, a11, 111 b 74. a , 5 b, 1- 1, - 42 3 3 3 27
x = -1.225, x = 0, x = 1.225
(b) y =
66. 1 56, 23 12 2
65. 10.3, 7.552
67. 120, 542
87. (a) w91t2 = 1.82 - 0.1192t + 0.002274t 2; (b) 21.148 lb; (c) 0.855 pound per month 88. (a) T91t2 = - 0.2t + 1.2; (b) 100.2 degrees (c) 0.9 degree per day 6000 89. (a) R91v2 = - 2 ; (b) 75 beats per minute; v 4 (c) -0.9375 beat per minute 90. (a) S91r2 = - 5 ; r (b) 0.4823 unit of resistance; (c) -12.2 units of resistance per millimeter dP 91. (a) = 4000t; (b) 300,000 people (c) 40,000 people per year dt (d) answers will vary. 92. (a) A91t2 = 0.08; (b) answers will dv 0.61 vary. 93. (a) = ; (b) 244 mi; (c) 0.00305 mile per dh 2h ds 1.55 foot gained; (d) answers will vary. 94. (a) = ; dd 2d (b) 9.8 meters per second; (c) 0.49 meter per second per meter dp 95. (a) About $1490; (b) = 1.716t - 18.864; dt (c) about $72.08 per year 96. 12, ` 2 97. 1- ` , - 12 h 13, ` 2 2 40 2 40 , - b, a,- b A3 9 A3 9
99. 10, -22, a
1
,-
55 1 55 ,- b b, a27 23 27
23 100. f91x2 = 5x 4 + 3x 2 is always nonnegative for all x. 101. g91x2 = - 2 - 3x 2 is always negative for all x. 2 102. k91x2 = - 3 is negative for all x 7 0. x
1
27 22x
111. 3x 2 + 6x + 3
x2
109.
32x 2
112. Left to the student
114.
y = x 4 - 3x 2 + 1
2 3
y=
2
-2
68. 1- 25, 76.252
75. 1- 22, 1 + 4 222, 122, 1 - 4 222 76. 1- 23, 2 + 2 232, 123, 2 - 2 232 77. 13, 02 78. 10, - 22, 1 - 1, - 11 79. 12.5, 8.752 80. 19.5, 99.752 6 2 81. 150, 752 82. 160, 1502 11 11 83. a1 + 26, - 3 26 b, a1 - 26, + 3 26 b 3 3 4 4 84. a- 2 + 23, - 23 b, a- 2 - 23, + 23 b 3 3 1 11 1 11 3 85. a- , - b, a , b 86. 1 1, 2 2 8 24 8 24
98. 10, - 42, a
110. 1 -
x2
2
-4
4
-2
-3
5
115.
5x 2 + 8x - 3 3x 2 + 2 3
x = -0.346, x = 1.929 1.5
116.
f f
-1
2 f’
0
5 -1.5
0
x < 0.432
f9112 = 1 5
117.
6
118.
f’
f’
f
f -5
5
-4
4
-3
-3
f9112 = 1.2
f9112 = 0
Technology Connection, p. 160 1. (c)
2–4. Left to the student
Exercise Set 1.6, p. 162 1. 11x 10
2. 13x 12
5. 20x 4 + 60x 2 8. 14x 5>2 + 9x 2
3. 12x + 7
4. 24x - 5 15 3>2 7. x + 4x 2 9. 24x 2 + 12x + 11 10. 18x 2 + 14x - 18
6. 18x 5 - 60x 4
1 9 2t + 2 2 2 2t
+ 11 13. 4x 3 1x ? 02 2 2t 14. 2x 1x ? 02 15. 18x 5 - 2x 1x ? 02 16. 8x 3 + 1 1x ? 02 19. 1 1t ? - 42 17. 2x - 3 1x ? - 32 18. 8x + 2 1 x ? 12 2 20. 1 1t ? 52 21. 48x 3 + 3x 2 + 22x + 17 - 10x 4 + 6x 2 + 30x 22. 40x 3 - 21x 2 - 26x + 13 23. 12x 3 + 32 2 6 3 3x - 16x - 2 25 3>2 3 24. 25. 120x 2 + x + x -1>2 + 24 3 2 2 2 1x - 12 105 3>2 3 26. x - 6x + 42x 1>2 + 4 27. + 15t 2 2 13 - t2 2 5 - 8t 3 29. 2x + 6 30. 50x - 40 28. 15 + 2t2 2 31. 2x1x 2 - 4213x 2 - 42 32. 413x - 2213x 2 - 4x + 52 11.
12. 32t +
17
INSTRUC TOR ANSWERS 33. 5 - 100x -3 + 30x -4 34. -36x -2 - 120x -3 + 144x -4 - 72x -5 6 35. 3t 2 - 1 + 2 36. 18t 5 - 60t 3 - 3t 2 + 5 1t ? 02 t x 4 + 3x 2 + 2x x 4 + 3x 2 + 2x 37. - 10x 38. + 12x 2 3 2 1x - 12 1x 2 + 12 2 39. 41.
- 2x + 21x 1>6 + 6
40.
6x 2>312x + 32 2
x1x + 22
42. -
1x + 12 2
44. -
1
45.
1t + 22 2
2x - 15x 1>6 - 8
2x
43. -
- 2x 2 + 6x + 2 1x 2 + 12 2
t 2 + 18t - 22
67. 72x 7 + 108x 5 - 30x 4 + 84x 3 + 84x - 14 68.
1x 2 - 12 2
(c) answers may vary. (c) answers may vary.
69. (a)
1
; (b) ; 1x + 12 2 1x + 12 2 2x 2x 70. (a) - 2 ; (b) - 2 ; 1x - 12 2 1x - 12 2 71. Answers may vary. 72. Answers
dR = - Q2 + kQ; (b) answers may vary. dQ 74. (a) Definition of derivative; (b) adding and subtracting same quantity; (c) the limit of the sum is the sum of the limits; (d) factoring; (e) the limit of a product is the product of the limits and evaluating lim f1x + h2 and lim g1x + h2;
may vary.
f’
(0, 0), 1- 1.414, - 42, 11.414, -42,
5 f’ -3
f
3
-4
Exercise Set 1.7, p. 171 1. 8x - 12 2. 8x + 4 3. - 5517 - x2 54 4. -10018 - x2 99 3x 1 4x 4 5. 6. 7. 8. 23x 2 - 4 24x 2 + 1 2 21 - x 21 + 8x 9. - 400x14x 2 + 12 -51
10. - 640x18x 2 - 62 -41
11. 41x - 42 712x + 32 517x - 62
12. 1x + 52 614x - 12 9168x + 1932
14. -
6
13x + 82 3
15.
13. -
4x15x + 142
16.
17 - 5x2 4
17. -512x - 32 3110x - 32
20. 13x - 12 612x + 12 4172x + 112 22. 25. 28.
2 25x + 2 4413x - 12 3 15x + 22 5
6815x - 12 3 12x + 32 5
30. 32. -
23.
2x - 5
29.
14 - 9x2 6
24x 21x 2 - 12 1x 2 + 12 4
2x15x - 12 24x - 1 5x 4 + 6 24. 2/3 4 3x 1x + 62 2/3 27. -
517x - 22 4
15x - 42 61120x + 1072
12x + 32 316x + 612 13x - 22 6 7
21.
41x 2 - 5x + 22 3/4
26. -
42x 213x + 22
18. - 315x + 42 5135x + 42
19. 415x + 22 312x - 32 7130x - 72 x 2135x + 122
8
14x + 52 3
2 21x + 32 314 - x2 1 x 34. , 2x, 2 2u 2x 2 - 1
13x - 12 6
16x + 12 4
31.
13
2 212x + 3215 - x2 3 45 90 33. - 4 , 2, u 12x + 12 4
35. 50u49, 12x 2 - 4x, 200x14x 3 - 2x 22 4913x - 12 3
-5
81. y3
f
-3
hS0
(f) definition of the derivative; (g) using Leibniz notation 76. $13.24/belt; 75. $0.79/bottle; $0.0024/bottle at x = 328 $0.078/belt at x = 169
1-1, - 22, 11, 22
5
80.
73. (a)
hS0
77.
22
1x 2 - 12 2
(c) - 0.48 degree/hr; (d) 98.77 degrees; (e) - 0.0069 degree/hr 3 4 63. (a) 0; (b) ; (c) - 2; (d) 64. (a) 0; (b) does not exist; 4 3 15x 2 + 112x - 12 3 65. 66. 30t 2 + 10t - 15 (c) -4; (d) 2 215x 2 + 42 2 1
4
5x 2 - 6x + 5
; (b) 100.2 degrees;
3x 6 - 4x 4 - 3x 2 + 32x
1-0.596, - 0.8942, 10, - 12
2
f
1000t 2 - 4500
1t 2 + 12 2
-20
23
; (b) 133,824 residents; 12t 2 + 92 2 (c) 1730 residents/yr; (d) 95,202 residents; (e) -1581 residents/yr 62. (a) T91t2 = -
4 f
1t - 42 2
14t 2 + 4t + 6
4t 2 - 4
f
f9
1 48. 49. (a) y = ; 47. - 2 2 2 2 2 1t - 2t + 42 1- t + 4t + 12 12 7 1 (b) y = x + 50. (a) y = 2; (b) y = x + 2 25 25 2 51. (a) y = 4x; (b) y = - 2 52. (a) y = x + 5; 21 21 (b) y = x 53. - +0.0925/belt 54. - +0.0579/bottle 4 4 56. - +0.0074>bottle 57. $0.0772/belt 55. - +0.0153/belt 58. $0.0505/bottle 59. $1.64/vase 60. $1.95/skateboard 61. (a) P91t2 = -
none
f’
-4
1
46.
f’
79.
3 6x 2>312 x - 52 2
1x 2 + 12 2
20
78.
A-19
A-20
INSTRUC TOR ANSWERS
36. -
2
1u - 12 2
,
38. 2x1x 2 + 3213x 2 - 52 40. -
210t17t 2 + 12 4
1317t 2 + 12 5 - 72 2
43. y = 4x - 3
37. 3x 2110x 3 + 132
1 , - 3/2 22x x 1
39.
- 6t - 11
31t + 22 213t + 52 2
41. y = 0
44. y = -735x + 1813
(b) same as in part (a); (c) they are the same. 4 3
2169
, or about 0.72
50. -
77.
5 3 42. y = x + 4 4 6416x + 12 45. (a) ; 12x - 52 3
(b) same as in part (a); (c) answers may vary.
49.
75.
2x - 3x 2
; 11 + x2 6 47. - 216 48. - 12 46. (a)
70(13)4 , or about - 42.85 66
51. (a) f1x2 = x 5, g1x2 = 3x 2 + 2x; (b) 4,587,520
61. $510,429/airplane
31x 2 - x - 12 21x 2 + 4x - 12 1x 2 + 12 4
x2 + x - 2
80.
212x + 12 31x 2 - 4x2
6 2t + 1 4 2t 23t + 1t
81. -7216x13 - x2 5 + 22 313 - x2 412x - 12
82. (a) Rewrit1 -1 ing = 3g1x24 ; (b) using Leibniz notation, differentiating g1x2 both sides; (c) using the Product Rule for differentiation; (d) using the Extended Power Rule; (e) simplifying by moving negative powers to the denominator; (f) multiplying the second fraction by g1x2 = 1; (g) writing over a common denominator and simplifying. g1x2 1-2.14476, - 7.7282, (2.14476, 7.728)
8 f’ -3
3 f
1 22x + 1
-8
b
3
62. $44,700/month
dA r 19 = 5000 a1 + b ; units are dollars per interest rate. dr 4 (b) It is the rate of change in the amount as the interest rate r changes.
63. (a)
r 35 dA = 3000 a1 + b ; units are dollars per interest rate. dr 12 (b) It is the rate of change in the amount as the interest rate r changes. 64. (a)
65. (a) P1t2 = 2t 2 + 400.8t + 80.08; (b) $592.80/month 80,000 ; (b) - 4.482 units/day 1.6t + 9 95w = 2.199w; 67. (a) D1c2 = 4.25c + 106.25, c1w2 = 43.2 (b) 4.25 mg/unit of creatine clearance; (c) 2.199 units of creatine clearance/kg; (d) 9.35 mg/kg; (e) it is the rate of change in dosage as the patient’s weight, in kilograms, varies. 68. (a) About 15.75, which means that an object with a weight of 100 kg at sea level has an equivalent weight of 15.75 stone; (b) 0.157, which means that an object’s weight increases by 0.157 stone per kilogram. 69. (a) 1F + C2130002 = F1C1300022 = F186.672 = 188, which means that water at 3000 m boils at 188°F; (b) - 0.008, which means that at 3000 m in altitude, the boiling point of water drops by 0.008°F/m as altitude increases. 70. (a) 36.35, which means that a woman with a 9-in. foot wears a size 36.35 shoe in Europe; (b) 4.05, which means that for every inch a woman’s foot grows, her European shoe size increases by 4.05 units per inch. 1 1 1 71. 4x1x 2 + 12 72. 1 + + a1 + b 2 2x 2 2x + 1x 2 2x 1 -26>27 73. 8x11x 2 + 12 2 + 121x 2 + 12 74. x 27
66. (a) D1t2 =
11 - x221 - x 2
78.
84.
a2x b 22x 2 + 11 - 3x 2 21 - 3x 1 58. 12x + 1x 2 + x2 42 -2>312 + 41x 2 + x2 312x + 122 3 59. $1,000,000/airplane 60. $489,574/airplane 57.
1
76. 13x 2 + 6x 4221 + x 2
1x 3 + 6x + 12 2>3 1
83. Left to the student
52. (a) f1x2 = 2x, g1x2 = 1 + 5x 2; (b) 20 9 x + 1 9 3 53. (a) f1x2 = ; g1x2 = x ; (b) 25 x + 4 1 54. (a) f1x2 = ; g1x2 = x 2 + 2x; (b) 49 x 55. 612x 3 + 14x - 52 22 516x 2 + 814x - 522
56. 3 1 -x 5 + 4x + 22x + 1 2 2 a-5x 4 + 4 +
79.
6x 7 + 32x 5 + 5x 4
1-1.47481, 9.48782
10
85. -5
f
5 f’ -10
Exercise Set 1.8, p. 179 1. 20x 3
2. 12x 2 3. 30x 4. 24x 2 5. 8 6. 8 7. 0 6 12 3 1 8. 0 9. 4 10. 5 11. 12. x x 16x 7>4 4x 3>2 2 4 6 10 2 13. 12x + 3 14. 6x - 3 15. 16. 5>3 x x 9x 25x 9>5 48 12 3 4 4 2 17. 5 18. 4 19. 61x + 2x2 151x + 72x + 202 x x 31x 2 + 22 20. 141x 2 + 3x2 5113x 2 + 39x + 272 21. 41x 2 + 12 5>4 2 41x - 32 3 2 22. 23. 24. 2 4>3 91x - 12 9x 4>3 42x 25.
45x 4 - 54x 2 - 3
26.
40x 6 + 56x 3 - 2
161x 3 - x2 5>4 91x 4 + x2 4>3 44 170 27. 28. 29. 120x 30. 24 12x - 32 3 15x - 12 3 2520 1760 31. 720x 32. 5040x 33. - 8 + x 243x 14>3 120 15 34. 6 + 35. (a) v1t2 = 3t 2 + 1; (b) a1t2 = 6t; x 16x 7>2 (c) 49 ft/sec, 24 ft/sec2 36. (a) v1t2 = -20t + 2; (b) a1t2 = -20; (c) -18 m/sec, -20 m/sec2 1 37. (a) v1t2 = 2t - ; (b) a1t2 = 2; (c) 1.5 m/sec, 2 m/sec2 2
INSTRUC TOR ANSWERS 38. (a) v1t2 = 3; (b) a1t2 = 0; (c) 3 mi/hr, 0 mi>hr2; (d) answers may vary. 39. (a) 144 ft; (b) 96 ft/sec; (c) 32 ft/sec2 40. (a) 64 ft; (b) 64 ft/sec; (c) 32 ft/sec2 41. 19.6 m/sec, 9.8 m/sec2
3
70.
s
3
43. (a) At a time t near 0, since the slope is steepest there; (b) negative, since the slopes of the tangent lines are decreasing 44. (a) At t = 20 sec, since the slope of the tangent line is steeper; (b) positive, since the slopes of the tangent lines are increasing 45. (a) 17, 112; (b) 12, 42, 17, 112, and 113, 152; (c) 10, 22 and 111, 132; (d) 14, 72; (e) answers may vary. 46. (a) 10, 22 and 18, 92; (b) 14, 52 and 111, 132; (c) 12, 42, 15, 82, 19, 112, and 113, 182 47. (a) $146,000/ month, $84,000/month, - +4000/month; (b) - +68,000/month2, - +56,000/month2, - +32,000/month2; (c) answers may vary. 48. (a) 2 pendants/day, 14 pendants/day, 74 pendants/day; (b) 6 pendants/day2, 18 pendants/day2, 42 pendants/day2; (c) answers may vary. 49. (a) 11.34, 1.98, 0.665; (b) - 0.789, - 0.0577, - 0.0112; (c) answers may vary. 50. (a) 1.2%/min, 0%/min, -0.0888%/min, - 0.0028%/min; (b) - 3.52%/min2, - 1.25%/min2, 0.031%/min2, 0.00018%/min2; (c) answers may vary. 51. (a) Height: 74.7 m, horizontal distance: 54.5 m; (b) vertical: 2.55 m/sec, horizontal: 27.25 m/sec; (c) vertical: - 9.8 m/sec2, horizontal: 0; (d) it is moving upward, since v122 7 0; (e) and (f) answers may vary. 52. (a) Height: 5.1 m, horizontal distance: 2 m; (b) vertical: -9.8 m/sec, horizontal: 2 m/sec; (c) vertical: - 9.8 m/sec2, horizontal: 0; (d) answers may vary. 53. Graph I 57. False
54. Graph IV 58. True
55. Graph II
59. False
56. Graph III 1 60. f91x2 = - 2 , x
6 24 2 , f901x2 = - 4 , f 142 = 5 . Signs alternate, and x x3 x odd-order derivatives are negative. The denominator’s power is always one higher than the order of the derivative. The numerator is - 1 for the 1st derivative, 1- 121- 22 = 2 for the 2nd derivative, 1- 121- 221- 32 = - 6 for the 3rd derivative, and so on. Thus, the 12th derivative will have 1- 121- 221- 32 g 1- 1121-122 = 479,001,600 479,001,600 as the numerator. f 11221x2 = . x 13 3 6 18 , f991x2 = , f9991x2 = , 61. f91x2 = 2 3 1x + 22 1x + 22 1x + 22 4 72 5 f99991x2 = 62. f91x2 = , 1x + 22 5 1x - 22 2 10 30 120 f991x2 = , f9991x2 = , f99991x2 = 1x - 22 5 1x - 22 3 1x - 22 4 63. (a) 3.24 m; (b) 3.24 m/sec; (c) 1.62 m/sec2; (d) it is the gravitational constant on the moon. 64. Answers may vary. Maximum is probably around 1.1 sec, but this would require a near-record vertical leap. 65. 42.33 ft/sec 66. Answers may vary. 67. (a) 6 6 x 6 8; (b) 3 6 x 6 6; (c) 10 6 x 6 12; (d) 8 6 x 6 10; (e) 8 6 x 6 12; (f) answers may vary. 68. (a) 0 6 x 6 9; (b) 0 6 x 6 4 or 10.5 6 x 6 12; (c) 4 6 x 6 6; (d) 9 6 x 6 12; (e) answers may vary.
a -3
At t < - 1.29, 1.29
3 a
v
-5
s 5
-3
At t < -1.104, 0.604
9
71.
a
v
s
-3
3
-9
72.
At t = 1
3 a
v
s
-2
4
-3
Chapter 1 Review Exercises, p. 188 1. False 2. False 3. True 4. False 5. True 6. True 7. False 8. True 9. (e) 10. (c) 11. (a) 12. (f) 13. (b) 14. (d) 15. (a) -10.1, - 10.01, - 10.001 and - 9.9, - 9.99, -9.999; (b) - 10, -10, -10 y 16. 10 8 6 4 2
f991x2 =
69.
At t = 0
v
-3
42. 29.4 m/sec, 9.8 m/sec2
A-21
-10 -8 -6-4-2-2 f
2 4 6 8 10 x
lim f1x2 = - 10
x S -7
-4 -6 -8 -10
1x + 721x - 32 x 2 + 4x - 21 = = x - 3, for x ? - 7. x + 7 x + 7 Thus, lim 1x - 32 = - 10. 18. -4 19. 10 20. - 12 17.
xS - 7
25. - 4 21. 3 22. 4 23. Does not exist 24. -4 26. Yes, since lim g1x2 = g112 27. Does not exist 28. - 2
xS1
29. No, since lim g1x2 does not exist xS - 2
30. No
32. 0 33. Not defined at x = - 2, since g is discon31. - 23 tinuous there 34. Not defined at x = -2, since there is a corner there 35. - 2 36. -3 37. Yes, since lim f1x2 = f1- 22 xS - 2
41. y = x - 1 40. -6x - 3h, h ? 0 8 -2>3 24 4 42. 14, 52 43. 45x 44. x 45. 9 46. 6x -3/5 3 x 5 47. 0.7x 6 - 12x 3 - 3x 2 48. x 5 + 32x 3 - 2 2 1 3 + 4 b + 12x + 4x213x 22 49. 1x + 52a 22x - x 2 + 16x + 8 50. 51. 215 - x212x - 12 4126 - 7x2 18 - x2 2 11x 2 + 4x 52. 35x 41x 5 - 32 6 53. 54. -48x -5 14x + 22 1/4 56. (a) P91t2 = 100t; (b) 30,000 55. 3x 5 - 60x + 26 people; (c) 2000 people/yr 57. 13, 52, 17, 82, 19, 132, 115, 182 58. 17, 82, 115, 182 59. 10, 32, 18, 92 60. 15, 72, 113, 152 38. Yes
39. 2
A-22
INSTRUC TOR ANSWERS
61. (a) 656 ft; (b) -96 ft/sec; (c) - 32 ft/sec2; (d) -226.27 ft/sec; (e) answers may vary. 5 2x + 100 5 2x + 100 , R = 40, P = 40 ; x x d (b) - +1.33/lamp 63. 1 f + g21x2 = - 411 - 2x2; dx - 9x 4 - 4x 3 + 9x + 2 d 1g + f21x2 = -4x 64. dx 2 21 + 3x11 + x 32 2 1 -242/243 65. x 66. - 0.25 67. 16 243
62. (a) C =
1- 1.7137, 37.4452, 10, 02, 11.7137, - 37.4452
50
68.
f’ -3
f
11.0835, 25.10292, 12.9502, 8.62472
50
45.
f
f’
0
5
-30
46. 0.5
Extended Technology Application, p. 192 160
160
1.
3 0
500 0
-50
0
500 0
Xscl = 50, Yscl = 20
Xscl = 50, Yscl = 20
2. (a) y = -0.0000045x 3 + 0.000204x 2 + 0.7806x + 4.6048; (c) acceptable fit; (b) 160
Chapter 1 Test, p. 190 1. (a) 11.9, 11.99, 11.999 and 12.1, 12.01, 12.001; (b) 12, 12, 12 y 18 16 14 12 10 8 6 4 2
2.
-8 -6-4-2-2
lim f1x2 = 12
0
xS6 f(x) =
500 0
x 2 - 36 x-6
Xscl = 50, Yscl = 20
(d) about 441 ft; (e) y9 = -0.0000135x 2 + 0.000408x + 0.7806; (f) at about (256, 142), meaning that the ball reached a maximum height of 142 ft at 256 ft from home plate. 3. (a) y = -0.0000000024x 4 - 0.0000026x 3 - 0.00026x 2 + 0.815x + 4.3026; (b) 160
2 4 6 8 10 12 x
1x + 621x - 62 x 2 - 36 = = x + 6, x ? 6. x - 6 x - 6 4. Does not exist 5. 0 Thus, lim 1x + 6) = 12
3.
xS6
6. Does not exist 7. 2 8. 4 9. 1 10. 1 11. 2 12. 0 13. 0 14. - 5, -3, - 2, 1, 4 15. - 5, -3, - 2, - 1, 1, 3, 4 16. Continuous 17. Not continuous, since lim f1x2
0
xS3
does not exist 18. (a) Does not exist; (b) 1; (c) no 19. 3 22. Does not exist, since the left-hand limit 20. 6 21. 18 does not equal the right-hand limit 23. 4x + 3 + 2h 3 24. y = x + 2 25. 10, 02, 12, - 42 26. 23x 22 4 4 5 10 5 27. x -2/3 + x -1/2 28. 2 29. x 1/4 30. - x + 0.61 3 2 4 x 3 2 31. x - 2x + 2 32. 1 3 2x + 1 2 12x - 12 + 1x 2 - x2a b 2 2x 5 33. 34. 1x + 32 317 - x2 4113 - 9x2 15 - x2 2 2x 2 + 5 35. -51x 5 - 4x 3 + x2 -615x 4 - 12x 2 + 12 36. 2x 2 + 5 2 37. 24x 38. (a) M91t2 = - 0.003t + 0.2t; (b) 9 words; (c) 1.7 words/min P = 50 -
39. (a) R = 50, C =
x 2>3 + 750 , x
x 2>3 + 750 ; (b) - +11.74/speaker x
40. 24x 5 - 6x 2 41. 61x 2 - x2 212x - 12 1 + 9x 43. 44. 27 211 - 3x2 2/311 + 3x2 5/6
42. Graph A
500 0
Xscl = 50, Yscl = 20
(c) acceptable fit; (d) about 440 feet; (e) y9 = -0.0000000096x 3 - 0.0000078x 2 0.00053x + 0.815; (f) at about 1257, 1422, meaning that the ball reached a maximum height of 142 ft at 257 ft from home plate. 4. (a) 160 (b) 450 ft;
0
500 0
Xscl = 50, Yscl = 20
303.75 - 0.003x 2
; (d) at about 1318, 1522, meaning 2202,500 - x 2 that the ball reached a maximum height of 152 ft at 318 ft from home plate. 5. The two models are similar; the main difference is that the maximum height is reached farther from home plate using the model in Exercise 4. 6. 466 ft, 442 ft, 430 ft 7. Left to the student 8. (a) 523 ft; (b) 490 ft, 450 ft; (c) 551 ft (c) y9 =
A-23
INSTRUC TOR ANSWERS
INSTRUCTOR ANSWERS: CHAPTER 2 Exercise Set 2.1, p. 203 1.
2.
y 10 9 8 7 6 5 4 3 2 1
g(x) = e-2x
-5-4-3-2-1
y 10 9 8 7 6 5 4 3 2 1
f(x) = e2x
-5-4-3-2-1
1 2 3 4 5 x
1 2 3 4 5 x
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 12; decreasing
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 12; increasing
3.
4.
y 10 9 8 7 6 5 4 3 2 1 -3-2-1
f(x) = e(1/3)x
1 2 3 4 5 6 7 x
y 10 9 8 7 6 5 4 3 2 1
y x = - 43 (1/2)x
-5-4-3-2-1
1 2 3 4 5 x
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 12; increasing 6. y
10 9 8 7 6 5 4 3 2 1
10 9 8 7 6 5 4 3 2 1
-5-4-3-2-1
1 2 3 4 5 x
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 122; decreasing y 7. 1
-4-3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9
1 2 3 4 5 6 x
F(x) = -e
(1/3)x
Domain: 1- ` , ` 2; range: 1- ` , 02; y-intercept: 10, - 12; decreasing
-5-4-3-2-1
0.5
-x
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 132; decreasing y 8. (1/2)x G(x) = -e
-4
-10
-8
54. 1- ` , 22
2
3
4
x=2
x
0.5 -5
f(x) = ln(4 - x)
-10
-6
-15
55. (-3, 5)
56. 1-1, 22
x = -3
y 4
x=5
x = -1
2
-2 -3 -4 -5 -6 -7 -8 -9
y 2
-2
5 x
-1
2
x
x=2
1
-2
-1
-4
-2
57. 10, ` 2
58. 10, ` 2
y 1
y 1
x=0 (vertical asymptote) 0.4
-1
2
0.6 x
x=0 (vertical asymptote) 0.05
0.10
0.15
0.20 x
-1 g(x) = ln(x + 2) + ln x
x
y = ln(2 - x) + ln(x + 1)
y = ln(x + 3) + ln(5 - x)
-2
1.5
1
-5
0.2
1
f(x) = ln(8 - 4x)
-4
1 2 3 4 5 x
9. (a) $1698.12; (b) $1701.14; (c) $1701.41; (d) $1701.42; (e) 22 yr 10. (a) $2941.92; (b) $2947.93; (c) $2948.46; (d) $2948.48; (e) 25.2 yr 11. (a) $85,587.46; (b) $85,818.59; (c) $85,839.54; (d) $85,840.26; (e) 15.4 yr 12. (a) $117,174.27; (b) $117,368.37; (c) $117,385.71; (d) $117,386.30; (e) 30.3 yr 13. (a) $304.45; (b) $305.00; (c) $305.04; (d) $305.05; (e) 34.8 yr 14. (a) $1920.34;
y 5
x=4
-2
y = ln(5x - 2)
-6
53. 1- ` , 42
1
Domain: 1- ` , ` 2; range: 1- ` , 02; y-intercept: 10, -12; decreasing
-2
x
y = ln(3x + 4) -5
1 g(x) = 13 e
x = 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x
-1.5 -1 -0.5
y 2
1 2 3 4 5 x
-5-4-3 -2 -1-1
y 2
5
g(x) = e
Domain: 1- ` , ` 2; range: 10, ` 2; y-intercept: 10, 12; increasing y 5.
f(x) = 12 e-x
(b) $1924.85; (c) $1925.25; (d) $1925.27; (e) 33.3 yr 15. (a) $10,621.26; (b) $10,633.19; (c) $10,634.27; (d) $10,634.30; (e) 16.9 yr 16. (a) $12,331.48; (b) $12,337.14; (c) $12,337.65; (d) $12,337.66; (e) 18.7 yr 17. 0.693; e 0.693 = 1.9997c < 2 2.197 18. 2.197; e = 8.9979c < 9 19. 1.504; e 1.504 = 4.4996c < 4.5 20. 1.203; e 1.203 = 3.33009c < 3.33 21. - 2.957; e -2.957 = 0.05197c < 0.052 22. -2.273; e -2.273 = 0.1030027c < 0.103 23. - 0.288; e -0.288 = 0.74976c < 0.75 24. 0.182; e 0.182 = 1.1996c < 1.2 25. 5; e 5 = e 5 26. 8; e 8 = e 8 27. 4.382 28. 2.9957 29. - 1.6094 30. 0.2231 31. 2.3863 32. 2.6094 33. 4 34. 3 35. -0.2231 36. - 1.3863 37. 0.3863 38. -0.6094 39. 2.079 40. 2.303 41. 2.267 42. 3.454 43. 4.605 44. 2.303 45. 140.671 46. -9.902 47. 0.549 48. 0.380 49. 24.414 50. 20.996 51. 1- 43, ` 2 52. 125, ` 2
-2
-3
-3
-4
-4
g(x) = ln(x + 6) + ln x
A-24
INSTRUC TOR ANSWERS
59. 1- ` , ` 2
60. 1- ` , ` 2 y 6
y 6
4
4
-5
5
2
y = ln(x + 2x + 4)
-15 -10 -5
10 x
5
10 15 x
65. 12 1e 1.5 - 1 2 < 1.741 66. 13 1e -1.4 + 2 2 < 0.749 67. 4 68. 4 69. (a) 2018: $222.7 billion, 2020: $268.7 billion;
(b) in 2.8 yr; (c) in 7.37 yr 70. (a) $48.9 billion; (b) in 3.2 yr, or 2017; (c) in 6.7 yr, or 2020 71. (a) A1t2 = 10,000e 0.0288t; (b) $11,548.84; (c) in 14.1 yr; (d) in 24.1 yr 72. (a) A1t2 = 25,000e 0.0403t; (b) $37,407.67; (c) in 11.7 yr; (d) in 17.2 yr 73. (a) $52.53; (b) $25; (c) after 30.1 weeks 74. (a) $103.60; (b) $60; (c) after 74.4 weeks 75. (a) $594.43; (b) about 2136 thousand units 76. (a) $529.04; (b) about 629 thousand units 77. (a) A1t2 = 45,000e 0.03t; (b) about 60,744; (c) in 6.7 yr; (d) in 23.1 yr 78. (a) A1t2 = 2500e 0.025t; (b) about 2905; (c) in 18.8 yr; (d) in 27.7 yr 79. (a) A1t2 = 2e 0.045t; (b) 3.14 mm2; (c) after 71.5 hr; (d) after 15.4 hr 80. (a) Q1t2 = 50e 0.55t; (b) about 191,400; (c) in 18 weeks; (d) in 1.26 weeks 81. (a) 75°C; (b) 58.4°C; (c) in 95.5 min; (d) 30, meaning that the coffee will decrease in temperature toward 30°C 82. (a) 100°C; (b) 64.2°C; (c) in 35.8 min; (d) 22, meaning that the water will cool toward 22°C 83. (a) 3.87 mg; (b) 6.26 days; (c) 2.7 days; (d) 0, which means that the sample will eventually decay toward zero. 84. (a) 12,054 people; (b) after 4.1 yr, or in 2021; (c) after 13.5 yr, in 2030; (d) 0, which means that the population will eventually approach 0 as a limit. 85. x = 0, x = ln 4 86. x = ln 3, x = ln 7 87. x = ln 4, x = ln 8 88. x = ln 3, x = ln 4 89. x = ln 4 90. x = ln 5 -1 92. x = e 1e - 22>3 < 0.58
94. x = e 5, x = e -3
96. 1520.18%; 0.05 yr (b) 2.857%
97. 3.45 hr
99. Answers may vary.
22.
2t - 3t 2
95. 6.077%; 11.4 yr
98. (a) A1t2 = 7500e t>35; 100. 10 yr
101. 20 yr
102. Left to the student 103. Answers may vary. 105. x = 0.933 104. x = 0.619, x = 1.512 106. x = 1.314 107. x = 0.27
33.
4x 1x
6. 5x 4 - 12e 6x 9.
2e 2x1x - 22 x5
3. 15e 5x
4. 8e 4x
7. 5x 4e 2x + 2x 5e 2x 10.
3e 3x1x - 22
13. 18 - 2x2e -x + 8x 2
x7
8. 7x 6e 4x + 4x 7e 4x
11. x 2e x 12. e x1x 2 + 5x - 62 2
15.
2
32. 16e 3 - 4x
e 1 t12t - 22 3
3
9t 2t 2
35. e x1x + 22
3
36. e x1x 2 + 4x + 22
37. 3e 3t16t + 132
38. 4e -2t15t - 92
39. 8e 2x12e 2x + 12
40. 144e 4x124e 4x + 27e 8x + 42
41. e 5t125t 2 + 70t + 972
42. 4e 11 - 2t214t 3 - 12t 2 + 5t + 12
43.
44.
45.
3e 3x13x 2 - 4x + 22 x4
9e 3t1e 3t - 22
46.
41e 3t - 12 3>2
4e 8x116x 2 - 16x + 52 3x 6
20e -4t12e 4t - 52
21 - 5e -4t1e 4t - 52
47. (a) dC>dx = 50e -x; (b) $50 million/yr; (c) $916,000/yr; (d) 100 and 0 48. (a) dC>dx = 40e -x; (b) $14.715 million/yr; (c) $270,000/yr; (d) 200 and 0 49. (a) $29,289.59; (b) $1596.28/yr 50. (a) $11,532.87; (b) - +961.84>yr 51. (a) - +2355.35>yr; (b) $20,019 52. (a) $2.66/week; (b) $75.35 53. (a) 0, 3.7, 5.4, 4.5, 0.05 (all in ppm); (b)
C 6 5 4 3 2 1
C(t) = 10t2 e-t
1 2 3 4 5 6 7 8 9 10 t
(c) C152 = 1.68 ppm, C9152 = -1.01 ppm/hr. After 5 hr, the concentration of the medication is about 1.68 ppm, and it is decreasing by about 1.01 ppm/hr. 54. (a) 242.6, 215.2, 174.4 (all in ppm);
200
5. 3x 2 - 10e 2x
14. 17 - 2x2e -x + 7x
29. 2e x 12x 2 + 12
31. 4e 2x + 1 34.
26. 9e -3x
25. 4e 2x
28. 49 e 12>32x
e 1x11x - 12
100
2. 3e 3x
2
3
3
e 2t
et
e 4t
30. 3xe x 13x 3 + 22
1 - 2t
2t 3 + 4t 2 - 2t - 2
12t 5 - 60t 3 - 3t 2 + 5
27. 14 e 1-1>22x
21.
(b) C(t)
Exercise Set 2.2, p. 210 1. 2e 2x
23. -
e 3t
24. -
62. e 6.5 < 665.142 61. e 10 < 22,026.466 4.5 63. e < 90.017 64. e 5>3 < 5.294
93. x = e, x = e 3
19. 18x 3 - 22x 2 - 13x + 32e x - 7x 20. 110x 3 - 36x 2 + 42x - 82e x - 4x
-2
2
18. -2xe -2x + e -2x - e -x + 3x 2
17. 3x 2 - xe x
22e x + 1
2
2
-2
91. x = e e - 1 < 595.29
ex
2
y = ln(x2 + 1)
2 -10
16.
ex 2 2e x - 1
C(t) = 250e-0.03t
10
20
t
(c) C182 = 196.7 ppm, C9182 = -5.9 ppm/min. After 8 min, the concentration of medication is about 196.7 ppm, and it is decreasing by about 5.9 ppm/min. 55. (a) 443.5, 370.4, 274.4 1all in mm22;
A-25
INSTRUC TOR ANSWERS 71.
(b) A(t)
f(x) = f ”(x) = e-x
f ’(x) = -e-x
4
1
400
10
20
30
-3
t
40
(c) A1122 = 243.4 mm2, A91122 = - 14.6 mm2 >hr. After 12 hr, the area of the colony is about 243.4 mm2, and it is decreasing by about 14.6 mm2 >hr. 56. (a) 1025 fish, 747 fish, 367 fish; (b)
3
-3
A(t) = 500e-0.06t
200
3 -1
72.
-4
f(x) = 2e 0.3x
f ’(x) = 0.6e 0.3 x
10
10
P(t)
1000 P(t) = 1200e-0.079t
500
10
20
30
-10
P(t) 100 90 80 70 60 50 40 30
-2
f ”(x) = 0.18e 0.3x
(c) P1202 = 247 fish, P91202 = -19.5 fish>week. After 20 weeks, the population of the invasive species is about 247 fish, and it is decreasing by about 19.5 fish/week. 57. (a) 100%, 69.8%, 54.8%, 40.9%, 40.1%; (b) 40%; (c)
10
-2
t
40
-10
10
10
10
-10 P(t) = 40 + 60e-0.7t
-2
73.
f ’(x) = -80e-0.08 x
f(x) = 1000e-0.08 x 2000
0
-10
40
1 2 3 4 5 6 7 8 9 10 t
(d) P91t2 = - 42e -0.7t; (e) answers may vary. 58. (a) 97 words, 73 words, 52 words; (b) 35; -10
(c) W(t)
40 0
100
-200 f ”(x) = 6.4e-0.08 x
20
50 W(t) = 35 + 85e-0.08t 5
t
10 15 20 25 30
(d) W172 = 84 words, W9172 = -3.88 words/day. After 7 days, the student retained 84 words but was forgetting them at a rate of about 3.88 words per day. 59. 61.
xe 11>22x
2 2x - 1 4
60.
- e -x1x 3 + x 2 + x - 12 11 + x 22 2
x
62. e e + x
1e x + e -x2 2
40
-10 0
74.
(
)
1 x f (x) = 1 + −− x 3
63. ex e - 1
64. (a) f91x2 = - 3e -3x, f01x2 = 9e -3x, f-1x2 = - 27e -3x;
(b) f 11021x2 = 1- 32 10e -3x = 59,049e -3x 65. g1721x2 = 128e 2x 66. y = 3x + 1 67. y = - 8x + 2 68. At x = 0, we have (0, a) and m = ab, so y - a = ab1x - 02, which simplifies to y = abx + a. 69. (1, e)
70.
0
40
0
75. (a) 1.0517, 1.005, 1.0005, 0.95163, 0.999502, 0.99995; (b) 1
Exercise Set 2.3, p. 216
f (x) = f ’(x) = f ”(x) = e x 4
1. -
9 x
2. -
8 x
7. x 511 + 6 ln x2 -3
3 -1
10.
1 - 4 ln x x5
3.
1 x
4.
1 x
5.
8. x 311 + 4 ln x2 2 11. x
4 12. x
1 x
13.
1 x 1 - 5 ln x
6. 9.
x6 6x + 2
3x 2 + 2x - 1
A-26 14. 17. 20.
INSTRUC TOR ANSWERS 14x + 5
7x 2 + 5x + 2 41ln x2 3 x
18.
15.
2x x2 + 5
31 ln x2 2 x
4 10x + 5 + 3 + 2 x x + x + 1
22. y = 1.778x - 1.358
-
19.
1 x
16.
2x x2 - 7
-
1 x
3x 2
2 - 2 + 3 x x + 1
21. y = 0.732x - 0.990 23. y = 4.329x - 6.75
24. y = 5.659x - 3.579 25. (a) 2000; 500 (b) N91a2 = , N91102 = 50; (c) $7390; (d) answers may a 200 vary. 26. (a) 1000; (b) N91a2 = ; N91102 = 20; a (c) $12,180; (d) answers may vary. 27. (a) R1x2 = 53.5x - 8x ln x; (b) R91x2 = 45.5 - 8 ln x; (c) 5400 28. (a) P91x2 = 1.7 - 0.3 ln x; (b) 0.197, which means that, when 150,000 candles are sold, profit is increasing by about 0.197 thousand dollars ($197) per unit; (c) about 23,700 units 29. (a) 78%; (b) 53.9%; (c) 29.7%; 15 (d) S91t2 = ; (e) S9142 = - 3, which means t + 1 that, after 4 months, the average score is decreasing by 3 percentage points per month; S91242 = -0.6, which means that, after 24 months, the average score is decreasing by 0.6 percentage point per month. 30. (a) 2.45 ft/s; (b) 3.40 ft/s; (c) v916602 = 0.000561 and v9185502 = 0.0000433, which indicate that as population increases, walking speed increases; (d) about 450,000 31. (a) t1152 = 20.27 min and t91152 = 5 min/°C, which mean that it takes 20.27 min to warm the water to 15°C, at which time the time needed for the water to warm by 1°C is changing by 5 min/degree; (b) 20.48°C 32. (a) t1602 = 216.25 min and t91602 = 15.625 min>8F, which mean that it takes 216.25 min to warm a frozen package of smoked salmon to 60°F, and at that moment, the time needed for the package to warm 1°F is changing by 15.625 min/°F; (b) 70.43°F 33. (a) t150002 = 22.34 months, t91p2 = 0.0152 month/bird. It takes about 22.34 months for the population to reach 5000 birds, and the time needed for the population to increase by one bird is about 0.0152 month/bird (about one new bird born every 10.9 hr); (b) 6570 birds 34. (a) t112,0002 = 30.65 weeks, t9112,0002 = 0.01 week/firefly. It takes about 30.65 weeks for the population of fireflies to reach 12,000, and the time needed for the population to increase by one firefly is about 0.01 week/firefly (about one new firefly every 1.68 hr). (b) 8800 fireflies t13t + 42 1 1 1 35. + 2 ln 1t + 12 37. - 2 36. 2 2 x 1t + 12 1w - 12 w 2 9 38. - 2 39. (a) T1t2 = 25 - 15e -0.02t; x 13x - 12 2 (b) 25, meaning that the water’s temperature approaches 25°C as a limit. 40. (a) T1t2 = 76 - 38e -0.004t; (b) 76, meaning that the package’s temperature approaches 76°F as a limit. 41. (a) p1t2 = 7500 - 4500e -0.0263t; (b) 7500, meaning that the population of birds approaches 7500 as a limit. 42. (a) p1t2 = 18,000 - 10,000e -0.0167t; (b) 18,000, meaning that the population of fireflies approaches 18,000 as a limit. 43. (1.35, 0.3) 44. (0.01832, - 4.0) and (2.187, 0.782) 45. a = e -1 46. a = - 1
Technology Connection, p. 221 1. 1.85 trillion (85.2 sextillion)
2. 2.1 quadrillion 3. 8.52 * 1022 4. Left to the student
Exercise Set 2.4, p. 226 1. f1x2 = ce 4x
3. A1t2 = ce -9t
2. g1x2 = ce 6x
4. P1t2 = ce -3t 5. Q1t2 = ce kt 6. R1t2 = ce kt 7. (a) N1t2 = 453,000e 0.039t; (b) 782,000; (c) about 30,499 applications/yr 8. (a) N1t2 = 50e 0.10t; (b) 369; (c) 36.9 franchises/year 9. (a) P1t2 = P0e 0.059t; (b) $1060.78, $1125.24; (c) $62.59/yr, $66.39/yr 10. (a) P1t2 = P0e 0.043t; (b) $20,878.76, $21,796.13; (c) $897.79/yr, $937.23/yr 11. (a) G1t2 = 11.8e 0.085t; (b) 27.6 billion gallons; (c) 2.347 billion gallons/yr 12. (a) A1t2 = 75e 0.292t; (b) 775.48 billion; (c) 226.4 billion/yr 13. (a) k = 0.151 (or 15.1%), V1t2 = 30,000e 0.151t; (b) $2,486,000,000; (c) $375,000,000/yr; (d) 78 yr 14. (a) I1t2 = 52,840e 0.0219t; (b) $70,244; (c) $1538/yr 15. (a) F1t2 = 2.77e 0.055314t; (b) $4.08 trillion; (c) after 23.2 yr, or in 2036 16. (a) c1t2 = 100e 0.026114t; (b) $284.21; (c) $7.42/yr 17. (a) y = 7.38473615411.4726301042 x, y = 7.384736154e 0.3870499885x; (b) 38.7%; (c) 163.3 EB; (d) after 8.5 yr; (e) about 77 EB/yr 18. (a) y = 7e 0.44365x; (b) 100.3 EB, 379.5 EB; (c) after 7.6 yr; (d) 1.56 yr; (e) answers may vary. 19. About $11 billion 20. R1t2 = 1.265e 0.1283t; about $188 billion 21. (a) S1t2 = 4e 0.0451t; (b) 4.5%/yr; (c) 60 cents, 66 cents; (d) $10,000; (e) answers may vary. 22. 10.89%; $11,698,000 23. (a) 2%; (b) 3.8%, 7%, 21.6%, 50.2%, 93.1%, 98%; (c) P91x2 = (d)
P(x) 100 90 80 70 60 50 40 30 20 10
637e -0.13x
11 + 49e -0.13x2 2
;
100 P(x) = ——–———– 1 + 49e-0.13x
10 20 30 40 50 60 x
24. (a) c1t2 = 0.05e 0.064t; (b) $2.82; (c) $0.18/yr 25. (a) V1t2 = 0.10e 0.227t; (b) about $12,132,700; (c) after 86 yr, or in 2024 26. After 77.8 yr, in 2016 27. After 80.8 yr, in 2019 28. (a) 2.23%; (b) answers may vary. 29. (a) 40, 185, 199; (b) P91t2 =
61,100e -0.0982t
117 + 183e -0.0982t2 2 (c) 3.13 people/yr, 1.34 people/yr, 0.13 people/yr; (d) 200 83,460e -0.214t 30. (a) 20, 46, 119, 146; (b) P91t2 = ; 120 + 130e -0.0214t2 2 (c) 3.71 tortoises/yr, 6.86 tortoises/yr, 5.28 tortoises/yr, 0.93 tortoise/yr; (d) 150 31. (a) 400, 520, 1214, 2059, 2396, 2478; (b) P91t2 =
4200e -0.32t
11 + 5.25e -0.32t2 2
;
A-27
INSTRUC TOR ANSWERS (c)
P(t) 3000 2500 2000 1500 1000 500
ka
1 + Ce -Lkt
2500 P(t) = ——–———–– 1 + 5.25e-0.32t
32. N1t2 = 48,869e 0.03644t, where t0 = 1930; exponential growth rate = 3.64% 33. (a) 0.244, 0.429, 0.753, 0.954, 0.989, 0.996; (b) p91t2 = 0.28e -0.28t; (c) p(t) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
p(t) = 1 - e-0.28t
2 4 6 8 10 12 14 16 18 20 t
34. (a) 11.8, 221.4, 547.2; (b) 4.4 new cases per week; (c) answers may vary. 35. (a) 0%, 33%, 55%, 70%, 86%, 99.2%, 99.8%; (b) 2.43, which means that after 7 months, the percentage of physicians prescribing the new drug is increasing by 2.43 percentage points per month; (c) P(t) 100 90 80 70 60 50 40 30 20 10
P(t) = 100(1 - e-0.4t)
2 4 6 8 10 12 14 16 t
29.47232081
36. (a) N1t2 =
1 + 79.56767122e -0.809743969t
;
(b) 29 students; 29.47232081 (c) N(t) =
1 + 79.56767122e-0.809743969t
30
0
15
1898.885181e -0.809743969t
11 + 79.56767122e -0.809743969t2 2
(e) answers may vary. P91t2 =
;
37–49. Answers may vary.
L2kCe -Lkt
. Also, 11 + Ce -Lkt2 2 L L kP1L - P2 = ka b aL b -Lkt 1 + Ce 1 + Ce -Lkt = ka
L
1 + Ce -Lkt Simplifying, we get
L + LCe -Lkt 1 + Ce -Lkt
ba
L11 + Ce -Lkt2 1 + Ce -Lkt
-
b.
1 + Ce -Lkt
1 + Ce -Lkt
ba
b LCe -Lkt
1 + Ce -Lkt
b = P91t2.
= k1- Ce -kt2 = P91t2
Exercise Set 2.5, p. 237 1. (a) N1t2 = N0e -0.096t; (b) 341 g; (c) - 32.7 g>day; (d) 7.2 days 2. (a) N1t2 = N0e -0.00012097t; (b) 182 g; (c) -0.022 g>yr; (d) 5730 yr 3. (a) N1t2 = N0e -0.000081547t; (b) 53.1 mg; (c) -0.00433 mg>yr; (d) 8500 yr 4. (a) N1t2 = N0e -0.0262t; (b) 51.9 mg; (c) -1.36 mg>sec; (d) 26.5 sec; (e) 176 sec 5. (a) N1t2 = N0e -0.2572t; (b) 1.65 mg; (c) - 0.425 mg>day; (d) 2.69 days; (e) 7.38 days 6. (a) N1t2 = N0e -0.025t; (b) 9.45 mg; (c) -0.236 mg>day; (d) 27.7 days; (e) 119.83 days 7. (a) P1t2 = 116,646e -0.0187t; (b) 60,621; (c) - 1134 people>yr; (d) 37 yr 8. (a) P1t2 = 92,124e -0.0272t; (b) 23,645; (c) - 643 people>yr; (d) 25.5 yr 9. (a) A1t2 = A0e -kt; (b) 11 hr 10. (a) A1t2 = A0e -kt; (b) 9 hr 11. (a) A1t2 = 35e -0.034t; (b) 47.34 min 12. (a) A1t2 = 4e -0.0666t; (b) 34.6 hr 13. (a) P1t2 = 5000e -0.116t; (b) 10.38 yr 14. (a) P1t2 = 10,000e -0.099t; (b) 9.26 yr 15. 5.78 yr 16. 12.05 months 17. 86.64 months 18. 17.77 yr 19. 23.1 yr 20. 39.61 days 21. 36.48 weeks 22. 138.63 weeks 23. 23.1%/min 24. 0.0433%/yr 25. 22 yr 26. 25 yr 27. 42.9 g 28. 9.9 g 29. 4223 yr 30. 19,034 yr 31. 25 days 32. 7575 yr 33. 3965 yr 34. $9035.83 35. $13,858.23 36. $12,098.12 37. $6,788,463 38. $10,579,378 39. $42,863.76 40. $35,677.60 41. (a) $40,000; (b) $5413.41; (c) answers may vary. 42. (a) $18,000; (b) $15,034.86; (c) answers may vary. 43. (a) $23,500; (b) $14,541; (c) answers may vary. 44. (a) $30,000; (b) $5936.96; (c) left to the student 45. (a) $2600; (b) $2171.70; (c) left to the student 46. (a) y = 34,001.7869710.67029777192 x, V1t2 = 34,001.78679e -0.4000332297t; (b) $2067.17, $622.56; (c) 8.8 yr; (d) 1.7 yr; (e) answers may vary. 47. (a) 0.022, 0.031, 0.069; (b) Q(t) Q(t) = 0.01345e 0.163t + 0.00055
48. (a) 0.037, 0.051, 0.115; (b) Q(t) 1 + Ce -Lkt
L L
1 2 3 4 5 6 7 8 9 10 t
L
-
51. P91t2 = - kCe -kt, and k1L - P2 = k 1L - 1L + Ce -kt2 2
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
(d) N91t2 =
ba
= ka
5 10 15 20 25 30 t
50.
L
0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
Q(t) = 0.02245e 0.163 t + 0.00055 1 2 3 4 5 6 7 8 9 10 t
A-28
INSTRUC TOR ANSWERS
(c) 53 49. (a) N1t2 = 5,650,000e -0.0153t; (b) 1,790,000; (c) in about 2063 50. (a) B1t2 = 64.6e -0.0068t; (b) 58.3 lb; (c) 2172 51. (a) P1t2 = 3.81e -0.00623t; (b) 3.36 million; (c) in 2028 52. (a) P1t2 = 2.37e -0.012t; (b) 1.86 million; (c) in 2025 53. (a) 60; (b) 0.01740; (c) 90°F; (d) 235 min; (e) answers may vary. 54. (a) 27; (b) 0.05878; (c) 83°F; (d) 28.7 min; (e) answers may vary. 55. At about 8 p.m. on the previous evening 56. At about 7 p.m. 57. (a) 112 lb; (b) -1 lb>day 58. (a) 145 lb; (b) - 1.2 lb>day 59. (a) 14.0 lb>in2; (b) 5.4 lb>in2; (c) 0 ft; (d) answers may vary. 60. (a) 11.2 W; (b) 173 days; (c) 402 days; (d) 50 W; (e) answers may vary. 61. (a) N1t2 = 69,895e -0.0336t; (b) 8138, 7115; (c) after 78 yr, or in 2034 62. (a) or (c) 63. (a) 64. (e) 65. (c) 66. (f) 67. (b) or (d) 68. (d) 69. (f) 70. (a) 71. (b) 72. (a) 12.05 weeks; (b) 24.1 weeks 73. (a) 4.27 yr; (b) 5.64 yr 74. Answers may vary. 75. Answers may vary. 76. Answers may vary. 77. Answers may vary. 78. (a) Answers may vary; (b) 25%; (c) 6.25%
Exercise Set 2.6, p. 248
1. P1t2 = 450 # 2t>5; P1t2 = 450e 0.1386t 2. P1t2 = 1300 # 2t>25; P1t2 = 1300e 0.02773t 3. P1t2 = 5000 # 3t>8; P1t2 = 5000e 0.1373t 4. P1t2 = 6 # 3t>9; P1t2 = 6e 0.1221t 5. P1t2 = 100 # 7t>12; P1t2 = 100e 0.1622t 6. P1t2 = 35,000 # 5t>20; P1t2 = 35,000e 0.08047t 7. P1t2 = 120010.92 t>2; P1t2 = 1200e -0.05268t
8. P1t2 = 100,00010.752 t>36; P1t2 = 100,000e -0.00799t
9. P1t2 = 60,00010.62 t>8; P1t2 = 60,000e -0.0639t
10. P1t2 = 25,00010.9252 t>9; P1t2 = 25,000e -0.008662t
11. P1t2 = 6500 # 1.5t>6; P1t2 = 6500e 0.06758t 12. P1t2 = 75 # 2.25t>2; P1t2 = 75e 0.4055t 13. ln 6 # 6x x t 14. ln 7 # 7 15. ln 15 # 15 16. ln 20 # 20t
17. ln 12.5 # 12.5x
18. ln 134 2 # 134 2 x
19. 3 ln 5 # 5x
20. 24 ln 9 # 9x 21. 4x 3 # ln 7 # 7x + 2 22. 2x # ln 4 # 4x + 5 23. 100 ln 0.52 # 0.52t 24. 3500 ln 0.038 # 0.038t 1 1 3 7 25. 26. 27. 28. x ln 6 x ln 13 x ln 4 x ln 11 3 4 10x + 5 29. 30. 31. 13x - 12ln 2 15 - 4x2ln 3 1x 2 + x2ln 6 4
16 - 24x 2
33.
4x
32.
12x - x 32ln 3
34.
7x + ln 7 # 7x # log 12 x x ln 12
x ln 5
+ ln 4 # 4x # log 5 x 35. 3xx1x ln 3 + 22
36. 2 # 5xx 31x ln 5 + 42
37. x 2 a
38. x 5 a
39.
1 + 6 log 4 x b ln 4
3 log 6 x 40.
3x + 2 x ln 6
1log 6 x2 2
2
1 + 3 log 7 x b ln 7
9x112x + 12ln 9 - 22 12x + 12 2
41. (a) P1t2 = 10,000 # 2t>9;
(b) P1t2 = 10,000e 0.07702t; (c) 7.702%/yr; (d) 80,000 people; (e) by 3081 people/yr
42. (a) A1t2 = 5000 # 2t>8;
(b) A1t2 = 5000e 0.08664t; (c) 8.664%/yr; (d) $20,000;
(e) by $3465.74/yr 43. (a) F1t2 = 50 # 3t>7; (b) F1t2 = 50e 0.1569t; (c) 15.69%/month; (d) 450 followers; (e) by 212 people/month 44. (a) M1t2 = 0.005 # 5t>36; 0.04471t (b) M1t2 = 0.005e ; (c) 4.471%/hr; (d) 0.125 g; (e) by 0.0279 mg/hr 45. (a) V1t2 = 10010.52 t>5; (b) V1t2 = 100e -0.1386t; (c) -13.86%>day; (d) $12.50; (e) by - +3.47>day 46. (a) M1t2 = 5010.82 t>2; -0.1116t (b) M1t2 = 50e ; (c) - 11.16%>week; (d) 32 g; (e) by - 2.856 g>week 47. (a) A1t2 = 50,000 # 1.2t>4; (b) A182 = 72,000, which means that after 8 yr, the account is worth $72,000; (c) A9182 = 3281.79, which means that after 8 years, the account is growing by $3281.79/yr 48. (a) V1t2 = 2500 # 1.35t>10; (b) V1202 = 4556.25, which means that after 20 yr, the autograph is worth about $4556.25; (c) V91202 = 136.74, which means that after 20 yr, the value of the autograph is increasing by $136.74/yr 49. (a) After 5 yr, the value of the machine is $1703.94; (b) after 5 yr, the value is changing by - +380.22>yr; (c) after 3.11 yr 50. (a) After 3 yr, there are 30,138.45 lb still is use; (b) after 3 yr, the amount in use is changing by - 21,254 lb>yr; (c) after 3.27 yr 51. (a) After 4 yr, there are 5408.51 lb of glass still in use; (b) after 4 yr, the amount of glass in use is changing by -5818.87 lb>yr; (c) after 2.78 yr 52. (a) After 10 yr, 0.817, or 81.7%, of the crop is nonheirloom; (b) in 10 yr, the percentage of the crop that is nonheirloom is changing by -1.65 percentage points/yr; (c) 34.31 yr 53. (a) I172 = I0107; (b) I182 = I0108; (c) a magnitude 8 quake is 10 times more powerful than a magnitude 7 quake; (d) I91R2 = I010R1ln 102; (e) answers may vary. 54. (a) I11002 = I01010; (b) I1102 = I010; (c) the sound in part (a) is 109 times as powerful as the sound in part (b); (d) I91L2 = I0100.1L1 ln 10210.12; (e) answers may vary. 1 55. (a) R91I2 = ; (b) answers may vary. I ln 10 10 56. (a) L91I2 = ; (b) answers may vary. I ln 10 57. (a) P1t2 = 35,000 # 2t>16.233; (b) P1t2 = 35,000 # 4t>32.466; (c) with base 4, the value of T is double that with base 2; (d) 48.699
58. (a) M1t2 = 25 112 2 t>6.359;
(b) M1t2 = 25 114 2 t>12.718; (c) with base 14, the value of T is double that with base 12; (d) 19.077 59. (a) P1t2 = 100,000 113 2 t>33.291;
(b) P1t2 = 100,000 119 2 t>66.582; (c) with base 19, the value of T is double that with base 13; (d) 99.873
60. (a) A1t2 = 2500 # 3t>43.0828; (b) A1t2 = 2500 # 9t>86.1657; (c) with base 9, the value of T is double that with base 3; (d) 129.2484 61. (a) 7.925 yr; (b) 800% 62. (a) 4.42 months; (b) 300% 63. (a) 37.6 hr; (b) 75% 64. (a) 2 yr; (b) 64% 65. ln a 66. ln 3 67. ln 7 68. 1
INSTRUC TOR ANSWERS
Chapter 2 Review Exercises, p. 255 1. (b) 2. (e) 3. (f) 4. (c) 5. (a) 6. (d) 7. False 8. False 9. False 10. False 11. True 12. True 13. False 14. True 15. False 16. False 17.
1 x
18. e x
19.
4x 3 x4 + 5
22. 3x 4e 3x + 4x 3e 3x
23.
20.
1 - 3 ln x
e 22x 2x 24.
x4 1 1 x 25. 4e 4x 27. 1ln 929x 26. x ex 2 # 3x 29. + 3x1ln 321log 4 12x + 122 12x + 12ln 4
21.
1 2x
2
2 ex + 2xe x 1ln 4x2 x 1 28. x ln 2
30. No x-intercept, y-intercept at 10, e -22 < 10, 0.1352, domain: 5 x u - ` 6 x 6 ` 6 , range: 5 y u 0 6 y 6 ` 6 y 10
f(x) = e
x-2
5 -5-4-3-2-1
1 2 3 4 5 x
31. x-intercept at 132, 0 2, y-intercept at (0, ln 4), domain: 5 x u - ` 6 x 6 2 6 , range: 5 y u - ` 6 y 6 ` 6 y 5 -2
-1
g(x) = ln(4 - 2x) 1
2 x
51. $186,373.98 52. (a) P1t2 = 2500 # 2t>6.5; 0.1066t (b) P1t2 = 2500e ; (c) 10.66%; (d) $10,000 53. (a) P1t2 = 15,000 # 1.25t>5; (b) 23,438 people; (c) 1045.99, meaning that Oak Fork was growing by about 1046 people/yr 3 in 2000 54. ln 4 # 4 x + 2x + 136x + ln 419x4 + 12x2 + 424 55. 19.02 yr 56. 18.68 yr 57. 0 58. (a) y = 989.966196511.0332237772 x, y = 989.9661965e 0.0326837949t, 3.27%; (b) $1616.4 billion; (c) after about 21.5 yr; (d) 21.2 yr
Chapter 2 Test, p. 257 1. 6e 3x
(d)
p 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
2
3. - 2xe -x
x
(c) P91t2 = (e)
672e -0.28t
11 + 24e -0.28t2 2
P(t) 100 90 80 70 60 50 40 30 20 10
1 x
5. e x - 15x 2
; (d) answers may vary;
100 P(t) = ——–———– 1 + 24e-0.28t
10
20
30
40
t
21. (a) A1t2 = 10,000 # 2t>8.25; (b) A1t2 = 10,000e 0.084t; (c) 8.4%; (d) 2301.70, which means that the value of Andres’s account is growing by about $2301.70 per year after 12 yr 22. (a) P1t2 = 7500 # 10.82 t>3; (b) 9.32 yr; (c) - 307.68, which means that after 8 yr, the population is decreasing by about 308 people/yr
23. 1ln x2 2
24.
24 - 18x 2
13x 2 + 42 2 ln 3
25. 2 26. (a) y = 740336.290811.0736572972 x, y = 740336.2908e 0.071070855t; (b) $7.2 million; (c) 101 yr; (d) about 9.8 yr
Extended Technology Application, p. 259 1. Linear: G = - 0.455t + 5.035 G = -0.455t + 5.035 10
p(t) = 1 - e-0.7t -10
10
-10 2 4 6 8 10 12 14 16 t
4.
1 3e x 7. 1ln 727x + 1ln 323x 8. + 3e x ln x x x ln 14 9. (a) 9.5427; (b) 0.2746 10. 2.302 11. 3.218 12. - 0.916 13. M1t2 = 2e 6t 14. 23.1%/hr 15. (a) A1t2 = 10,000e 0.06931t; (b) $13,194.83; (c) $914.53/yr 16. (a) C1t2 = 3.22e 0.0028t; (b) $3.39; (c) $0.009/yr 17. (a) A1t2 = 3e -0.1t; (b) 1.1 cc; (c) -0.11 cc>hr; (d) after 6.9 hr 18. About 16.47 centuries, or 1647 yr 19. (a) A1t2 = 14e -0.04077t; (b) 1.21 mg; (c) - 0.049 mg>s 20. (a) 4%; (b) 5.2%, 14.5%, 40.7%, 91.8%, 99.5%;
-10
45. (a) N1t2 = 60e 0.12t; (b) 157; (c) about 19 franchises/yr 46. (a) N1t2 = 24e 0.07t; (b) about 74; (c) about 5.15 franchises/yr 47. (a) N1t2 = 10e -0.13t; (b) 6.77 mg; (c) -0.88 mg>yr 48. (a) -18.2%>day; (b) N1t2 = 50e -0.182t; (c) -3.05 mg>day 49. (a) A1t2 = 800e -0.07t; (b) 197 g; (c) - 13.81 g>day 50. (a) 0.5, 0.75, 0.97, 0.999; (b) p91t2 = 0.7e -0.7t; (c) answers may vary;
41ln x2 3
2.
6.
-5
32. 2.639 33. - 1.253 34. 3.332 35. 1.253 36. 0.973 37. - 1.386 38. Q1t2 = 25e 7t 39. (a) r = 0.0433, or 4.33%; (b) P1t2 = 4000e 0.0433t; (c) 5416; (d) 235 people/yr 40. $411.51/yr 41. (a) C1t2 = 15.81e 0.0206t; (b) $34.59; (c) $0.71/yr 42. (a) C1t2 = 2.69e 0.0039t; (b) $3.14; (c) $0.012/yr 43. (a) P1t2 = 120 - 40e -0.0406t; (b) 0.721, meaning that the stock is increasing in value by about $0.72/week after 20 weeks; (c) after about 34 weeks 1000 44. (a) N1t2 = ; (b) after about 64.4 hr 1 + 49e -0.0604t
A-29
A-30
INSTRUC TOR ANSWERS
Quadratic: G = -0.2546428571t 2 + 1.836785714t + 1.215357143 2
G = -0.2546428571t + 1.836785714t + 1.215357143
Quadratic: G = 0.307797619t 2 - 6.459583333t + 40.21803571 G = 0.307797619t2 - 6.459583333t + 40.21803571 20
10
-8
12 0
20 0
-10
Cubic: G = 0.122979798t 3 - 1.91487013t 2 + 8.17024531t - 4.872142857 3
2
G = 0.122979798t - 1.91487013t + 8.17024531t - 4.872142857
Cubic: G = 0.0810858586t 3 - 0.7868614719t 2 - 2.283661616t + 36.20428571 G = 0.0810858586t3 - 0.7868614719t2 - 2.283661616t + 36.20428571 80
15
-20
20
15
-5
-40 -10
Exponential: G = 5.5893802410.82252452672 t
Exponential: G = 42.4747407610.81596457342 t G = 42.47474076(0.8159645734)t 50
t
G = 5.58938024(0.8225245267) 12
10
-10
-10
40 -10
-8
The cubic function fits well, but rises at higher values of t, which is not appropriate since weekly revenues are expected to fall as t increases. The exponential function also fits well and is better suited for forecasting future revenue. 2. Week 9: $0.96 million; week 10: $0.79 million; week 11: $0.65 million; week 12: $0.54 million; week 13: $0.44 million 3. Week 9: $24.86 million; week 10: $25.65 million; week 11: $26.3 million; week 12: $26.84 million; week 13: 23.38497984 $27.28 million 4. R1t2 = . 1 + 23.62209015e -1.055833743t The limiting value is about $23.4 million, which seems too low. 24.31102164 . The limiting value 5. R1t2 = 1 + 8.994071065e -0.7582056951t is about $24.3 million, which agrees better with the actual data. 6. Left to the student 7. Linear: G = - 3.689404762t + 35.60107143 G = -3.689404762t + 35.60107143 12
-6
14
-8
The exponential function is an excellent fit. 8. Week 9: $6.81 million; week 10: $5.56 million; week 11: $4.53 million; week 12: $3.7 million; week 13: $3.02 million; week 14: $2.46 million 153.2872607 9. R 1t 2 = . The limiting value is 1 + 5.563261043e -0.6327045804t about $153.29 million, which seems too low. 169.4506888 10. R 1t 2 = . The limiting value 1 + 3.215356463e -0.4138102809t is about $169.45 million. This seems to be a better estimation. 11. In the 9th week 12. Answers may vary.
INSTRUC TOR ANSWERS
INSTRUCTOR ANSWERS: CHAPTER 3
35. Relative minimum 1- 2, - 132, decreasing on 1- ` , - 22, increasing on 1- 2, ` 2
Technology Connection, p. 274 1.
2.
f(x) = 2 − (x − 1)2/3
f´(x) = − 2 (x − 1)−1/3 3 4
4
−4
6
−4
−2
−10 −8 −6 −4 −2 −2
The derivative is not defined at 11, 22.
Technology Connection, p. 275
175 5 2, - 4
2
Exercise Set 3.1, p. 276 1. (a) x = 1, x = 3; (b) relative minimum 1; (c) relative maximum 4; (d) relative minimum point 11, 12; (e) relative maximum point 13, 42 2. (a) x = -1, x = 3; (b) relative minimum - 1; (c) relative maximum 2; (d) relative minimum point 13, -12; (e) relative maximum point 1- 1, 22 3. (a) x = -2, x = 2; (b) relative minimum - 3; (c) relative maximum -1; (d) relative minimum point 12, -32; (e) relative maximum point 1- 2, - 12 4. (a) x = - 2, x = 1, x = 3; (b) relative minimum - 1; (c) relative maxima 2, 1; (d) relative minimum point 11, - 12; (e) relative maximum points 1-2, 22 and 13, 12 5–16. Answers may vary. 17. (a) x = -1; (b) relative minimum 1- 1, 12 18. (a) x = 3; (b) relative minimum 13, -82 19. (a) x = - 2, x = 2; (b) relative maximum 1- 2, 212, relative minimum 12, - 112 20. (a) x = -1, x = 1; (b) relative maximum 1- 1, -42, relative minimum 11, - 82 21. (a) x = - 1, x = 5; (b) relative minimum 15, -992, relative maximum 1-1, 92 22. (a) x = - 3, x = - 23; (b) relative 11 minimum 1 - 23, - 23 23. (a) x = 2; 27 2 , relative maximum 1 - 3, 2 2 (b) relative minimum 12, 02 24. (a) x = 13; (b) relative minimum ( 13, 2) 25. (a) x = 2; (b) relative minimum 12, 52 26. (a) x = 3; (b) relative minimum 13, -12 27. (a) No critical values; (b) no relative extrema 28. (a) No critical values; (b) no relative extrema 29. (a) x = - 14; (b) relative minimum 1 - 14, - 0.092 2 30. (a) x = 16; (b) relative maximum 1 1 6, 0.061 2 31. (a) x = - 1; (b) relative minimum 1- 1, 1.6092 32. (a) x = - 2; (b) relative minimum 1-2, 1.7922 33. Relative minimum 34. Relative maximum 1 - 12, 21 4 2, 1-3, -122, decreasing on 1 increasing on 1 - ` , - 2 2 , 1- ` , - 3), increasing on decreasing on 1 - 12, ` 2 1-3, ` 2 f(x) = x 2 + 6x − 3
−4 −6 −8 −10 −12 −14
−2
1. Relative maximum at 1- 1, 422; relative minimum at 1
y 10 8 6 4 2
−10 −8 −6 −4 −2 −2 −4 −6 −8 −10 −12 −14
2
x
y 7 6 f(x) = 5 − x − x 2 5 4 3 2 1 −5 −4 −3−2 −1−1 −2 −3 −4 −5
1 2 3 4 5 x
36. Relative minimum 1- 1, -22, decreasing on 1- ` , -12, increasing on 1-1, ` 2
y 6 4 2 6
−5 −4 −3 −2 −1−1
g(x) = x 3 +
1 2 3 4 5 x
−2 −3 −4 −5
F(x) = 0.5x 2 + 2x − 11
y 9 8 7 6 5 4 3 2 1 −5 −4 −3 −2 −1−1 −2 −3
y 5 4 3 g(x) = 1 + 6x + 3x 2 2 1
2 4 6 8 10 x
37. Relative minimum 1 23, 113 27 2 , relative maximum 1 - 1, 13 2 , 2 decreasing on 1 - 1, 23 2 , increasing on 1- ` , - 12 and on 1 23, ` 2
A-31
38. Relative minimum 11, 12,
relative maximum 1 - 13, 59 27 2 , decreasing on 1 - 13, 1 2 ,
increasing on 1 - ` , - 13 2 and
on 11, ` 2
y 5 4 3 2 1 1 2 3 4 5 x
1 2 x − 2x + 5 2
−5 −4 −3 −2 −1−1
1 2 3 4 5 x
−2 −3 −4 −5
G(x) = x 3 − x 2 − x + 2
39. Relative minimum 12, - 42, relative maximum 10, 02, decreasing on 10, 22, increasing on 1- ` , 02 and on 12, ` 2
y 5 4 3 2 f(x) = 3x2 + 2x3 1
y 5 4 3 2 1 −5 −4 −3 −2 −1−1 −2 −3 −4 −5
1 2 3 4 5 x
f(x) = x 3 − 3x2
41. No relative extrema, decreasing on (- ` , ` )
−2 −3 −4 −5
−5 −4 −3 −2 −1−1
1 2 3 4 5 x
−2 −3 −4 −5
42. No relative extrema, increasing on 1- ` , ` 2 y 15 10 5
y 5 4 3 2 1 −5 −4 −3 −2 −1−1
40. Relative minimum 10, 02, relative maximum 1-1, 12, decreasing on 1- 1, 02, increasing on 1- ` , - 12 and on 10, ` 2
−5 −4 −3 −2 −1−5 1 2 3 4 5 x F(x) = 1 − x3
−10 −15 −20 −25 −30 −35
1 2 3 4 5 x g(x) = 2x 3 − 16