I N S T R U C T OR A N S WE R S A- 83
APPENDIX A 136. No solution 137. - 23 138. - 145 139. 5
Exercise Set A, p. 843 140. - 27, 27 141. {8 142. {12 143. { 230
2 25
1. 5 # 5 # 5, or 125 2. 7 # 7, or 49
220
3. - 7 - 7 , or 49 144. { 297 145. { 7 146. { 13 147. { , or {
2 10
4. - 5 - 5 - 5 , or - 125 5. 1.0201 6. 1.030301 3 3
148. 23 149. 2 2 25
7. 1 8. 1 9. 1 10. 6x 11. - t 12. - 1 { { , or {
16 64 1 1 1 5 25 5
13. 1 14. 3 15. 9 16. 16 17. 8 1
18. 4 19. 9 20. 4 21. 0.1 22. 0.0001 23. 150. { 12 , or {4 23 151. 1, 1 152. 5, - 25
3 2 6
49
eb 23
1 1 1 5 7 -6 1
24. 25. 26. 27. x 28. t 29. x , or 2 + 210 2 - 210 1 + 22 1 - 22
153. , 154. ,
tk b h x6 5 5 4 4
30. x6 31. 35x 5 32. 8t 7 33. x4 34. x 35. 1 155. x Ú - 45 156. x Ú 3 157. x 7 - 12
1
4 -3 1 1 158. x 7 - 5 159. x 7 - 4 160. x … - 6
-4
36. 1 37. x6 38. x 39. x , or 3 40. x , or 4 13 7 2 5 5
x x 161. x … - 3 162. x Ú - 2 163. x 7 3 164. x … 3
41. 1 42. 1 43. et - 4 44. ek - 3 45. t14 46. t12 165. x 6 - 25 166. x … 54 167. 2 6 x 6 4
1 1
47. t2 48. t-4, or 49. a6b5 50. x7y7 51. t-6, or 168. 2 6 x … 4 169. 32 … x … 11
2 170. 65 … x 6 16
5
1 t4 t6 171. - 1 … x … 14 172. - 5 … x 6 -2 173. $650
-12 4x 5x 6 12 5
52. t , or
53. e 54. e 55. 8x y 174. $800 175. More than 7000 units 176. More than
t12 4200 units 177. 480 lb 178. 340 lb
1 x8y20
56. 32x10y20 57. x8y20z-16, or 179. 810,000 180. 5200 181. 60% … x 6 100%
81 81z16
1 -9 21 15 182. 50% … x 6 90% 183. 18 people 184. 75 tiles
y21z15 -16 14 4 9y14z4 185. Gina is 16 and Dave is 6. 186. $75,000
58. x y z , or 59. 9x y z , or
125 125x9 x16 187. 58, 59, 60 188. 8 189. 5 190. 7 and 24
13
625x 16 4 12 6y3
60. 625x16y-20z-12, or 20 12 61. c d
8 62. 64x 9 9
y z 16q a b APPENDIX B
63. 5x - 35 64. x + xt 65. x2 - 7x + 10 Exercise Set B, p. 850
66. x22- 7x + 12 67. a32- b3 68. x3 + y23 1. 5 2. - 4 3. 5 5. 0 6. 0 7. 19
69. 2x + 3x - 5 70. 3x + x - 4 71. a - 4 8. - 1 9. 2 4. 13
21
3 2 12. 1 13. - 5 2
2 10. 2 11. 3 4
72. 9x2 - 1 73. 25x2 - 4 74. t2 - 1 14. - 92 15. 13 16. 2 17. 1 18. 1 19. e2
75. a2 - 2ah + h2 76. a2 + 2ah + h2 20. e-3 21. e-4 22. e5 23. 1 24. 1 25. - 1
2
77. 25x2 + 10xt + t2 78. 49a2 - 14ac + c2 26. 52 27. 1 28. 1 29. Left to the student
79. 5x5 + 30x3 + 45x 80. - 3x6 + 48x2 30. Left to the3 student12 31. 1 32. - 1 33. - 2
3 2 2 3 3 2 2 3 34. - 7 2 3 5
81. a + 3a b + 3ab + b 82. a - 3a b + 3ab - b 35. (a) 0, 0, 0, 0; answers
3 (b) answers may vary (c)
83. x3 - 15x2 + 75x - 125 84. 8x3 + 36x2 + 54x + 27 2x 3x x
may vary 36. (a) e >e = 1>e , 0; (b) answers may vary
85. x 1 - t 86. x 1 + h 87. 1x + 3y22 37. (a) Subtract 0 from numerator and denominator;
88. 1x - 5y22 89. x - 5 x + 3 90. x + 5 x + 3 (b) since x ? a, multiply numerator and denominator by >1 1 x - a 2;
91. x - 5 x + 4 92. x - 10 x + 1 (c) property of limits, limit of quotient is quotient of limits;
93. 7x - t 7x + t 94. 3x - b 3x + b (d) definition of derivative at x = a; (e) differentiability of f and
g at x = a 38. l’Hôpital’s Rule does not apply since the first
95. 4 3t - 2m 3t + 2m 96. 5y - 3z 5y + 3z
expression is not indeterminate at x = 3. The correct limit is 0.
97. ab a + 4b a - 4b 98. 21x2 + 421x + 221x - 22
99. 1a4 + b421a2 + b221a + b21a - b2
100. 6y - 5 6y + 7 101. 10x a + 2b a - 2b APPENDIX C
102. xy x + 5y x - 5y 103. 211 + 4x2211 + 2x211 - 2x2 Exercise Set C, p. 855
104. 2x y + 5 y - 5 105. 9x - 1 x + 2 1. y = 3.45x + 0.6 2. y = 2.5283x + 13.34
106. 3x - 4 2x - 5 107. 1x + 221x2 - 2x + 42
3. y = 0.0732x2 + 2.3x + 11.813
108. 1a - 321a + 3a + 92 109. 1y - 4t21y2 + 4yt + 16t22
2
110. 1m + 10p21m2 - 10mp + 100p22
111. 13x2 - 121x - 22 112. 1y2 - 2215y + 22
APPENDIX E
113. 7x - 3 x 79+ 3 x - 5 114. t - 5 t + 5 t + 3 Exercise Set E, p. 863
115. 116. 117. - 8 118. 4 119. 120 1 -3x 2 3x 6x
4 12 5 4 1. - e 13x + 12 + C 2. e 13x - 12 + C 3. + C
120. 140 3121. 200 122. 200 123. 0, -3, 5 9 9 ln 6
124. 0, 2, - 125. 0, 2 126. 3, -3 127. 0, 3 2 1 5+x
2 1 1 4. ln u x + 2x + 9u + C 5. ln ` ` + C
128. 0, 5 129. 0, 7 130. 0, 3 131. 0, 3, - 3 10 5 - x
132. 0, 1, - 1 133. 1 134. 2 135. No solution 1 2 + 24 + x2
` ` 0 0
4 4 6. - ln +C 7. - x - 3 ln 3 - x + C
2 x
,A- 84 I N S T R U C T OR A N S WE R S
1 1 x 20
1 + ln 0 1 - x 0 + C 9. + ln ` ` + C 31. S1x2 = 100 J + ln120 - x2 R
8.
1 - x 818 - x2 64 8 - x 20 - x
10. 12 3x 2x2 + 9 + 9 ln 0 x + 2x2 + 9 0 4 + C 1 1
32. p1t2 = ln a t b + 0.8750
+
11. x ln 3x - x + C 12. x ln145 x2 - x + C x + t2
212 4 2 + -t 3 1
5 5 33. - 4 ln ` ` + C 34. + ln 0 2x - 3 0 + C
13. x ln x - x +
1 2 C
5 25 3x - 2 412x - 32 4
14. - 12x3e-2x - 34x2e-2x - 34 xe-2x - 38e-2x + C -1 1 x
35. + ln ` ` + C
15. x41ln x2 - x4 + C 16. x51ln x2 - x5 + C 21xx - 222x 4
1 x - x2 2x
4 16 5 36. 2 3e 2e + 1 + ln 0 e + 2e + 104 + C
1 + 21 - x2 -3
17. ln 0 x + 2x2 + 7 0 + C 37. + -x
18. - 3 ln ` ` + C e-x - 3 ln 0 e - 3 0 + C
x
2 2 1
+ ln ` x x 38. 14 3ln x 21ln x22 + 49 + 49 ln1ln x +121ln xx2+2 5+ 4924 + C
19. 5 - 7x 5 5 - 7x ` + C 20. 5 ln ` 7x + 2 ` + C 39. When a = -5, the antiderivative is - ln ` ` + C.
21. - 5 ln x - 1>2 + C 10 x - 5
` `
4 x + 1>2 Using a property of logarithms, we have
1 x+5 1
22. 3 3t 2t2 - 1 - 1 ln 0 t + 2t2 + 1 04 + C - ln ` ` = - 1ln 0 x + 5 0 - ln 0 x - 5 0 2
2 9 9 9
2 2 10 x - 5 10
1 1
23. m 2m + 4 + 4 ln 0 m + 2m + 4 0 + C =- ln 0 x + 5 0 + ln 0 x - 5 0
3 5 5
24. 1-ln x - 12 + C 25. 1ln x2 + + C 1 10 10
x 2x2 4x2 = 1ln 0 x - 0 - 0 + 02
5 ln x 5
26. x1ln x24 - 4x1ln x23 + 12x1ln x22 - 24x 1ln x2 + 24 + C 10
1 x - 5
27. x3ex - 3x2ex + 6xex - 6ex + C = ln ` `.
28. 3 ln 0 x + 2x2 + 25 0 + C 29. 1 13x - 1211 + 2x23>2 + C 10 x + 5
2 15
30. 13519x
2 - 4212 + 3x23>2 + C 40. Answers may vary. 41. Answers may vary.
,A-14 I N S T R U C T OR A N S WE R S
INSTRUCTOR ANSWERS: CHAPTER 1 69. y 70. y
Exercise Set 1.1, p. 112
1. 0.3, x S 0.3- 2. 1.7, x S 1.7+ 3. - 3, x S - 3-
g(x) = 4x + 9
x+2
4. -4.9, x S -4.9+ 5. 2, x S 12 2- 6. 4, x S 14 2-
3 3 3 3
7. 0.3, x S 0.3- 8. 1.2, x S 1.2- 9. 1, x S 1+ 1 1 2 3 4 5 6 7 8 x
2
10. 0, x S 0 - 11. - 2 12. 7 13. limx S2+ 3 x 3
14. limxS3- 15. limxS5 16. limxS1 17. “the limit, 2, does not exist 4, does not exist
2
as x approaches 4, of f1x2” or “the limit of f1x2 as x approaches 4” 71. y 72. y
18. “the limit, as x approaches 1, of g1x2” or “the limit of g1x2 as x 5
4
approaches 1” 19. “the limit, as x approaches 5 from the left, of 3 F
F1x 2” or “the limit of F 1x 2as x approaches 5 from the left” 2
1
20. “the limit, as x approaches 4 from the right, of G1 x2 ” or “the
limit of G1 x2 as x approaches 4 from the right” 21. (a) - 3; 2 G
(b) - 3; (c) - 3 22. (a) 1; (b) 2; (c) does not exist
1 1 2 3 4 5 6 7 x
23. (a) - 1; (b) - 1; (c) 1 24. (a) 4; (b) 2; (c) does not 2
exist 25. 5 26. 4 27. Does not exist 28. 0 29. 2
3, 1, does not exist 1, 3, does not exist
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 73. y 74. y
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
g
61. y 62. y
2 f
1 1 2 3 4 5 6 7 8 x 3
2 4
3 5
2
f(x) = x2 1, 0, does not exist - 1, - 1, -1
2 75. y 76. y
3
4
5
0, 2 1, 0 F
G
63. y 64. y
3 2 1
2 1
1
1
1
-1
y y
2
2 77. 78.
3
4 3
5 4
H
- 5, - 4 4, 1 G
65. y 66. y
2
3
4 4
G(x) = x 3 4 2
x+2 5
does not exist, 2 1, 9
1 1 2 3 4 5 6 7 8 x
2 2 79. $3.50, $3.50, $3.50 80. $3.00, $3.50, does not exist
3
4
3
4
81. $4.00, $4.50, does not exist 82. $1.00, $1.21, does not exist
5 5 83. $1.21, $1.42, does not exist 84. $1.42, $1.42, $1.42
4, does not exist does not exist, 2 85. Does not exist 86. $1.63 87. 10%, 12%, does not exist
88. 12%, 12%, 12% 89. 22%, does not exist
67. y 68. y
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.
2
97. (a) 4; (b) 4, (c) 4; (d) 4; (e) 4; (f) no; (g) yes
x
2 f(x) =
x 98. Does not exist, 2 99. 0, 0 100. Does not exist, 16
4
5
6
7 2
- 2, does not exist 3, does not exist
, I N S T R U C T OR A N S WE R S 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 1. The temperature rose 3 degrees/hr. 2. Jennifer hiked 3 mi/hr.
7. False 8. True 9. 5 10. 3 11. - 3 12. 7 13. 4 3. Marcus delivered 7 packages/hr. 4. The population of Felton
14. 4 15. 715 16.3 10 17.131 18. 13-4 19. 10 grew by 100 people/yr. 5. Tanya scored 25 points/game.
20. 6 21. 22. 23. - 24. 25. 3 26. - 12 6. Chris grew 7.5 cm/yr. 7. Burnham Industries had
27. - 6 2 1 2 6 4 Does not exist 5,000,000 dollars/month in revenue. 8. Juan spent
28. 10 29. Does not exist 30.
31. 54 32. 73 33. 16 34. 3 35. 13 36. 12 37. 0 2.25 dollars/gallon on gasoline. 9. Unemployment changed by
38. 0 39. Does not exist 40. Does not exist 41. 3 - 0.333 percentage point/month. 10. Shannon spent 3.1 dollars/
42. 27 43. Does not exist 44. Does not exist 45. 0 day on electricity for April. 11. 3 12. 5 13. 2
46. 0 47. Not continuous 48. Not continuous 14. 6 15. - 32 1 16. - 35 17. 8 18. 8 19. 4.25
49. Continuous 50. Not continuous 51. Not continuous 20. 1.6 21. (a) 10x + 5h; (b) 60, 55, 50.5, 50.05
52. (a) - 2, - 2, - 2; (b) - 2; (c) yes, the limit exists and equals 22. (a) 8x + 4h; (b) 48, 44, 40.4, 40.04 23. (a) - 10x - 5h;
the function value; (d) does not exist; (e) -3; (f) no, the limit (b) - 60, - 55, - 50.5, - 50.05 24. (a) - 8x - 4h;
does not exist. 53. (a) - 1, 2, does not exist; (b) - 1; (c) no, (b) - 48, - 44, - 40.4, - 40.04 25. (a) 2x + h - 1;
the limit does not exist; (d) 3; (e) 3; (f) yes, the limit exists and 27. 11, 10, 9.1, 9.01
(b) 9 26. (a)92x +3h + 1;6 (b) 13,
60 12, 11.1, 11.01
equals the function value. 54. (a) 2; (b) 2; (c) yes, the limit (a) - ; (b) - , - , - , -
exists and equals the function value; (d) 0; (e) 0; (f) yes, the x1x + h2 35 10 17 167
limit exists and equals the function value. 55. (a) 2; (b) does 2 2 1 4 40
not exist; (c) no, the function value does not exist; (d) - 2; (e) - 2; (a) - ; (b) - ,- ,- ,-
28. x1x + h2 35 15 51 501
(f) yes, the limit exists and equals the function value. 56. (a) 0.25; 29. (a) 2; (b) 2, 2, 2, 2 30. (a) -2; (b) - 2, - 2, - 2, -2
(b) 0.25; (c) yes, the limit exists and equals the function value; 31. (a) 36x2 + 36xh + 12h2; (b) 1308, 1092, 918.12, 901.8012
(d) does not exist; (e) does not exist; (f) no, neither the limit 32. (a) - 3x2 - 3xh - h2; (b) - 109, - 91, - 76.51, - 75.1501
nor t - 2 exists. 57. (a) 3; (b) 1; (c) does not exist; (d) 1; 33. (a) 2x + h - 4; (b) 8, 7, 6.1, 6.01 34. (a) 2x + h - 3;
(e) no, the limit does not exist; (f) yes, the limit exists and equals (b) 9, 8, 7.1, 7.01 35. (a) 2x + h - 3; (b) 9, 8, 7.1, 7.01
the function value; (g) yes, the limit exists and equals the function 36. (a) 2x + h + 4; (b) 16, 15, 14.1, 14.01 37. 0.36 percentage
value. 58. (a) 1; (b) - 1; (c) does not exist; (d) 1; (e) no, point/yr, - 0.3 percentage point >yr, 0.09 percentage point/yr
the limit does not exist; (f) yes, the limit exists and equals the 38. 1.7 percentage point/yr, - 0.2 percentage point >yr,
function value. 59. Yes, the limit exists and equals the function 0.66 percentage point/yr 39. 0.66 percentage point/yr,
value at 4. 60. Yes, the limit exists and equals the function value - 0.15 percentage point >yr, 0.21 percentage point/yr
at 5. 61. No, the limit does not exist and G is not defined at 0. 40. - 0.31 percentage point >yr, 0.1 percentage point/yr,
62. No, F is not defined at - 1. 63. Yes, the limit exists and - 0.08 percentage point >yr 41. 0.1 percentage point/yr,
equals the function value at 4. 64. Yes, the limit exists and equals - 0.04 percentage point >yr, 0.03 percentage point/yr
the function value at 3. 65. No, the limit does not exist at 3. 42. - 0.7 percentage point >yr, 0.46 percentage point/yr,
66. No, the limit does not exist and G is not defined at 4. - 0.05 percentage point >yr 43. 2.1 percentage points/yr,
67. No, g is not defined at 4. 68. Yes, the limit exists and equals - 4.3 percentage points >yr, - 1.5 percentage points >yr
the function value at 3. 69. No, the limit value does not equal 44. 1 percentage point/yr, - 0.43 percentage point >yr,
the function value at 2. 70. No, the limit value does not equal 0.2 percentage point/yr > $1266.67/yr, $580/yr
45. - +450 yr,
the function value at 1. 71. Yes, the limit exists and equals the 46. $1.467 billion/yr, $0.4 billion/yr, $0.93 billion/yr
function value at 4. 72. Yes, the limit exists and equals the function 47. (a) 70, 39, 29, 23; (b) answers may vary. 48. (a) 300,
value at 5. 73. No, the limit does not exist and the function is 180, 120, 100; (b) answers may vary. 49. (a) $26.62;
not defined at 5. 74. Yes, the limit exists and equals the function (b) $31.16; (c) $4.54; (d) 0.7567, which means that prices
value at 3. 75. No, the limit does not exist and the function is increased by an average of about $0.76 per year 50. (a) $2391.24;
not defined at 2. 76. Yes, the limit exists and equals the function (b) $2693.71; (c) $302.47; (d) $151.24, which means the amount
value at 4. 77. Yes, because limxSa 1g x2 = g1a 2for all a such grew on average $151.24 per year for 2 yr 51. The average cost
that - 4 6 a 6 4 78. Yes, because limxSa F x = F a for all a of production of between 300 and 305 holders is $19.75 per unit.
such that -5 6 a 6 5 79. Yes, because limxSa g 1 x2 = g1a 2for
52. The average revenue from sales of between 300 and 305 holders
all a such that 1 6 a 6 `, and limxS1+1 g2 x =1 g2 1 80. Yes, is $149.40 per unit. 53. 17.62; the average rate of change in
because limxSa h x = h a for all a such that - 3 6 a 6 `, and Amazon’s revenue between 2014 and 2017 was $17.62 billion per year.
limxS -3+ h x = h - 3 81. Yes, because limxSa F x = F a 54. 41.35; the average rate of change in Panera Bread Co.’s income
for all a such that - 5 6 a 6 5, and limxS -5+ F x = F - 5 and between 2014 and 2017 was $41.35 million per year.
limxS5- F x = F 5 82. Yes, because limxSa G x = G a 55. (a) 1.49 hectares/g; (b) home range increases by 1.0902 hect-
for all a such that - 3 6 a 6 3, and limxS -3+ G x = G - 3 and
ares per gram as the animal’s weight grows from 200 g to 300 g.
limxS3- G x = G 3 83. 30, 25, does not exist 84. 8, 6,
56. (a) 50.33; the population grew by about 50 condors per year
does not exist 85. 120, 120, 120 86. Yes 87. Yes
between 2010 and 2017. (b) 42.064; the population increased by
88. Yes 89. Yes 90. No 91. No 92. Yes 93. Yes
19 about 42 condors per year between 2007 and 2015. 57. (a) 1.25,
94. 5 95. 2 96. (a) Does not exist; (b) - 3 97. a = , 1.25, 0.625, 0, 0; (b) answers may vary. 58. (a) 29.4 mi/gal;
4 (b) 0.034 gal/mi 59. (a) 256 ft; (b) 128 ft/sec
1 1
b = - 3 98. c = 9 99. 100. 6 101. 60. (a) 184.05 mi, or the distance traveled from t = 2 hr to t = 5 hr;
2 2 (b) 61.35 mi/hr 61. (a) 125 thousand people/yr; (b) answers
102. 1 103. 1 104. 3 105. 1
- 0 106. may vary; (c) A: 290 thousand people/year, -40 thousand people/yr,
2 23 27 4 4 - 50 thousand people/yr, 300 thousand people/yr;
APPENDIX A 136. No solution 137. - 23 138. - 145 139. 5
Exercise Set A, p. 843 140. - 27, 27 141. {8 142. {12 143. { 230
2 25
1. 5 # 5 # 5, or 125 2. 7 # 7, or 49
220
3. - 7 - 7 , or 49 144. { 297 145. { 7 146. { 13 147. { , or {
2 10
4. - 5 - 5 - 5 , or - 125 5. 1.0201 6. 1.030301 3 3
148. 23 149. 2 2 25
7. 1 8. 1 9. 1 10. 6x 11. - t 12. - 1 { { , or {
16 64 1 1 1 5 25 5
13. 1 14. 3 15. 9 16. 16 17. 8 1
18. 4 19. 9 20. 4 21. 0.1 22. 0.0001 23. 150. { 12 , or {4 23 151. 1, 1 152. 5, - 25
3 2 6
49
eb 23
1 1 1 5 7 -6 1
24. 25. 26. 27. x 28. t 29. x , or 2 + 210 2 - 210 1 + 22 1 - 22
153. , 154. ,
tk b h x6 5 5 4 4
30. x6 31. 35x 5 32. 8t 7 33. x4 34. x 35. 1 155. x Ú - 45 156. x Ú 3 157. x 7 - 12
1
4 -3 1 1 158. x 7 - 5 159. x 7 - 4 160. x … - 6
-4
36. 1 37. x6 38. x 39. x , or 3 40. x , or 4 13 7 2 5 5
x x 161. x … - 3 162. x Ú - 2 163. x 7 3 164. x … 3
41. 1 42. 1 43. et - 4 44. ek - 3 45. t14 46. t12 165. x 6 - 25 166. x … 54 167. 2 6 x 6 4
1 1
47. t2 48. t-4, or 49. a6b5 50. x7y7 51. t-6, or 168. 2 6 x … 4 169. 32 … x … 11
2 170. 65 … x 6 16
5
1 t4 t6 171. - 1 … x … 14 172. - 5 … x 6 -2 173. $650
-12 4x 5x 6 12 5
52. t , or
53. e 54. e 55. 8x y 174. $800 175. More than 7000 units 176. More than
t12 4200 units 177. 480 lb 178. 340 lb
1 x8y20
56. 32x10y20 57. x8y20z-16, or 179. 810,000 180. 5200 181. 60% … x 6 100%
81 81z16
1 -9 21 15 182. 50% … x 6 90% 183. 18 people 184. 75 tiles
y21z15 -16 14 4 9y14z4 185. Gina is 16 and Dave is 6. 186. $75,000
58. x y z , or 59. 9x y z , or
125 125x9 x16 187. 58, 59, 60 188. 8 189. 5 190. 7 and 24
13
625x 16 4 12 6y3
60. 625x16y-20z-12, or 20 12 61. c d
8 62. 64x 9 9
y z 16q a b APPENDIX B
63. 5x - 35 64. x + xt 65. x2 - 7x + 10 Exercise Set B, p. 850
66. x22- 7x + 12 67. a32- b3 68. x3 + y23 1. 5 2. - 4 3. 5 5. 0 6. 0 7. 19
69. 2x + 3x - 5 70. 3x + x - 4 71. a - 4 8. - 1 9. 2 4. 13
21
3 2 12. 1 13. - 5 2
2 10. 2 11. 3 4
72. 9x2 - 1 73. 25x2 - 4 74. t2 - 1 14. - 92 15. 13 16. 2 17. 1 18. 1 19. e2
75. a2 - 2ah + h2 76. a2 + 2ah + h2 20. e-3 21. e-4 22. e5 23. 1 24. 1 25. - 1
2
77. 25x2 + 10xt + t2 78. 49a2 - 14ac + c2 26. 52 27. 1 28. 1 29. Left to the student
79. 5x5 + 30x3 + 45x 80. - 3x6 + 48x2 30. Left to the3 student12 31. 1 32. - 1 33. - 2
3 2 2 3 3 2 2 3 34. - 7 2 3 5
81. a + 3a b + 3ab + b 82. a - 3a b + 3ab - b 35. (a) 0, 0, 0, 0; answers
3 (b) answers may vary (c)
83. x3 - 15x2 + 75x - 125 84. 8x3 + 36x2 + 54x + 27 2x 3x x
may vary 36. (a) e >e = 1>e , 0; (b) answers may vary
85. x 1 - t 86. x 1 + h 87. 1x + 3y22 37. (a) Subtract 0 from numerator and denominator;
88. 1x - 5y22 89. x - 5 x + 3 90. x + 5 x + 3 (b) since x ? a, multiply numerator and denominator by >1 1 x - a 2;
91. x - 5 x + 4 92. x - 10 x + 1 (c) property of limits, limit of quotient is quotient of limits;
93. 7x - t 7x + t 94. 3x - b 3x + b (d) definition of derivative at x = a; (e) differentiability of f and
g at x = a 38. l’Hôpital’s Rule does not apply since the first
95. 4 3t - 2m 3t + 2m 96. 5y - 3z 5y + 3z
expression is not indeterminate at x = 3. The correct limit is 0.
97. ab a + 4b a - 4b 98. 21x2 + 421x + 221x - 22
99. 1a4 + b421a2 + b221a + b21a - b2
100. 6y - 5 6y + 7 101. 10x a + 2b a - 2b APPENDIX C
102. xy x + 5y x - 5y 103. 211 + 4x2211 + 2x211 - 2x2 Exercise Set C, p. 855
104. 2x y + 5 y - 5 105. 9x - 1 x + 2 1. y = 3.45x + 0.6 2. y = 2.5283x + 13.34
106. 3x - 4 2x - 5 107. 1x + 221x2 - 2x + 42
3. y = 0.0732x2 + 2.3x + 11.813
108. 1a - 321a + 3a + 92 109. 1y - 4t21y2 + 4yt + 16t22
2
110. 1m + 10p21m2 - 10mp + 100p22
111. 13x2 - 121x - 22 112. 1y2 - 2215y + 22
APPENDIX E
113. 7x - 3 x 79+ 3 x - 5 114. t - 5 t + 5 t + 3 Exercise Set E, p. 863
115. 116. 117. - 8 118. 4 119. 120 1 -3x 2 3x 6x
4 12 5 4 1. - e 13x + 12 + C 2. e 13x - 12 + C 3. + C
120. 140 3121. 200 122. 200 123. 0, -3, 5 9 9 ln 6
124. 0, 2, - 125. 0, 2 126. 3, -3 127. 0, 3 2 1 5+x
2 1 1 4. ln u x + 2x + 9u + C 5. ln ` ` + C
128. 0, 5 129. 0, 7 130. 0, 3 131. 0, 3, - 3 10 5 - x
132. 0, 1, - 1 133. 1 134. 2 135. No solution 1 2 + 24 + x2
` ` 0 0
4 4 6. - ln +C 7. - x - 3 ln 3 - x + C
2 x
,A- 84 I N S T R U C T OR A N S WE R S
1 1 x 20
1 + ln 0 1 - x 0 + C 9. + ln ` ` + C 31. S1x2 = 100 J + ln120 - x2 R
8.
1 - x 818 - x2 64 8 - x 20 - x
10. 12 3x 2x2 + 9 + 9 ln 0 x + 2x2 + 9 0 4 + C 1 1
32. p1t2 = ln a t b + 0.8750
+
11. x ln 3x - x + C 12. x ln145 x2 - x + C x + t2
212 4 2 + -t 3 1
5 5 33. - 4 ln ` ` + C 34. + ln 0 2x - 3 0 + C
13. x ln x - x +
1 2 C
5 25 3x - 2 412x - 32 4
14. - 12x3e-2x - 34x2e-2x - 34 xe-2x - 38e-2x + C -1 1 x
35. + ln ` ` + C
15. x41ln x2 - x4 + C 16. x51ln x2 - x5 + C 21xx - 222x 4
1 x - x2 2x
4 16 5 36. 2 3e 2e + 1 + ln 0 e + 2e + 104 + C
1 + 21 - x2 -3
17. ln 0 x + 2x2 + 7 0 + C 37. + -x
18. - 3 ln ` ` + C e-x - 3 ln 0 e - 3 0 + C
x
2 2 1
+ ln ` x x 38. 14 3ln x 21ln x22 + 49 + 49 ln1ln x +121ln xx2+2 5+ 4924 + C
19. 5 - 7x 5 5 - 7x ` + C 20. 5 ln ` 7x + 2 ` + C 39. When a = -5, the antiderivative is - ln ` ` + C.
21. - 5 ln x - 1>2 + C 10 x - 5
` `
4 x + 1>2 Using a property of logarithms, we have
1 x+5 1
22. 3 3t 2t2 - 1 - 1 ln 0 t + 2t2 + 1 04 + C - ln ` ` = - 1ln 0 x + 5 0 - ln 0 x - 5 0 2
2 9 9 9
2 2 10 x - 5 10
1 1
23. m 2m + 4 + 4 ln 0 m + 2m + 4 0 + C =- ln 0 x + 5 0 + ln 0 x - 5 0
3 5 5
24. 1-ln x - 12 + C 25. 1ln x2 + + C 1 10 10
x 2x2 4x2 = 1ln 0 x - 0 - 0 + 02
5 ln x 5
26. x1ln x24 - 4x1ln x23 + 12x1ln x22 - 24x 1ln x2 + 24 + C 10
1 x - 5
27. x3ex - 3x2ex + 6xex - 6ex + C = ln ` `.
28. 3 ln 0 x + 2x2 + 25 0 + C 29. 1 13x - 1211 + 2x23>2 + C 10 x + 5
2 15
30. 13519x
2 - 4212 + 3x23>2 + C 40. Answers may vary. 41. Answers may vary.
,A-14 I N S T R U C T OR A N S WE R S
INSTRUCTOR ANSWERS: CHAPTER 1 69. y 70. y
Exercise Set 1.1, p. 112
1. 0.3, x S 0.3- 2. 1.7, x S 1.7+ 3. - 3, x S - 3-
g(x) = 4x + 9
x+2
4. -4.9, x S -4.9+ 5. 2, x S 12 2- 6. 4, x S 14 2-
3 3 3 3
7. 0.3, x S 0.3- 8. 1.2, x S 1.2- 9. 1, x S 1+ 1 1 2 3 4 5 6 7 8 x
2
10. 0, x S 0 - 11. - 2 12. 7 13. limx S2+ 3 x 3
14. limxS3- 15. limxS5 16. limxS1 17. “the limit, 2, does not exist 4, does not exist
2
as x approaches 4, of f1x2” or “the limit of f1x2 as x approaches 4” 71. y 72. y
18. “the limit, as x approaches 1, of g1x2” or “the limit of g1x2 as x 5
4
approaches 1” 19. “the limit, as x approaches 5 from the left, of 3 F
F1x 2” or “the limit of F 1x 2as x approaches 5 from the left” 2
1
20. “the limit, as x approaches 4 from the right, of G1 x2 ” or “the
limit of G1 x2 as x approaches 4 from the right” 21. (a) - 3; 2 G
(b) - 3; (c) - 3 22. (a) 1; (b) 2; (c) does not exist
1 1 2 3 4 5 6 7 x
23. (a) - 1; (b) - 1; (c) 1 24. (a) 4; (b) 2; (c) does not 2
exist 25. 5 26. 4 27. Does not exist 28. 0 29. 2
3, 1, does not exist 1, 3, does not exist
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 73. y 74. y
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
g
61. y 62. y
2 f
1 1 2 3 4 5 6 7 8 x 3
2 4
3 5
2
f(x) = x2 1, 0, does not exist - 1, - 1, -1
2 75. y 76. y
3
4
5
0, 2 1, 0 F
G
63. y 64. y
3 2 1
2 1
1
1
1
-1
y y
2
2 77. 78.
3
4 3
5 4
H
- 5, - 4 4, 1 G
65. y 66. y
2
3
4 4
G(x) = x 3 4 2
x+2 5
does not exist, 2 1, 9
1 1 2 3 4 5 6 7 8 x
2 2 79. $3.50, $3.50, $3.50 80. $3.00, $3.50, does not exist
3
4
3
4
81. $4.00, $4.50, does not exist 82. $1.00, $1.21, does not exist
5 5 83. $1.21, $1.42, does not exist 84. $1.42, $1.42, $1.42
4, does not exist does not exist, 2 85. Does not exist 86. $1.63 87. 10%, 12%, does not exist
88. 12%, 12%, 12% 89. 22%, does not exist
67. y 68. y
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.
2
97. (a) 4; (b) 4, (c) 4; (d) 4; (e) 4; (f) no; (g) yes
x
2 f(x) =
x 98. Does not exist, 2 99. 0, 0 100. Does not exist, 16
4
5
6
7 2
- 2, does not exist 3, does not exist
, I N S T R U C T OR A N S WE R S 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 1. The temperature rose 3 degrees/hr. 2. Jennifer hiked 3 mi/hr.
7. False 8. True 9. 5 10. 3 11. - 3 12. 7 13. 4 3. Marcus delivered 7 packages/hr. 4. The population of Felton
14. 4 15. 715 16.3 10 17.131 18. 13-4 19. 10 grew by 100 people/yr. 5. Tanya scored 25 points/game.
20. 6 21. 22. 23. - 24. 25. 3 26. - 12 6. Chris grew 7.5 cm/yr. 7. Burnham Industries had
27. - 6 2 1 2 6 4 Does not exist 5,000,000 dollars/month in revenue. 8. Juan spent
28. 10 29. Does not exist 30.
31. 54 32. 73 33. 16 34. 3 35. 13 36. 12 37. 0 2.25 dollars/gallon on gasoline. 9. Unemployment changed by
38. 0 39. Does not exist 40. Does not exist 41. 3 - 0.333 percentage point/month. 10. Shannon spent 3.1 dollars/
42. 27 43. Does not exist 44. Does not exist 45. 0 day on electricity for April. 11. 3 12. 5 13. 2
46. 0 47. Not continuous 48. Not continuous 14. 6 15. - 32 1 16. - 35 17. 8 18. 8 19. 4.25
49. Continuous 50. Not continuous 51. Not continuous 20. 1.6 21. (a) 10x + 5h; (b) 60, 55, 50.5, 50.05
52. (a) - 2, - 2, - 2; (b) - 2; (c) yes, the limit exists and equals 22. (a) 8x + 4h; (b) 48, 44, 40.4, 40.04 23. (a) - 10x - 5h;
the function value; (d) does not exist; (e) -3; (f) no, the limit (b) - 60, - 55, - 50.5, - 50.05 24. (a) - 8x - 4h;
does not exist. 53. (a) - 1, 2, does not exist; (b) - 1; (c) no, (b) - 48, - 44, - 40.4, - 40.04 25. (a) 2x + h - 1;
the limit does not exist; (d) 3; (e) 3; (f) yes, the limit exists and 27. 11, 10, 9.1, 9.01
(b) 9 26. (a)92x +3h + 1;6 (b) 13,
60 12, 11.1, 11.01
equals the function value. 54. (a) 2; (b) 2; (c) yes, the limit (a) - ; (b) - , - , - , -
exists and equals the function value; (d) 0; (e) 0; (f) yes, the x1x + h2 35 10 17 167
limit exists and equals the function value. 55. (a) 2; (b) does 2 2 1 4 40
not exist; (c) no, the function value does not exist; (d) - 2; (e) - 2; (a) - ; (b) - ,- ,- ,-
28. x1x + h2 35 15 51 501
(f) yes, the limit exists and equals the function value. 56. (a) 0.25; 29. (a) 2; (b) 2, 2, 2, 2 30. (a) -2; (b) - 2, - 2, - 2, -2
(b) 0.25; (c) yes, the limit exists and equals the function value; 31. (a) 36x2 + 36xh + 12h2; (b) 1308, 1092, 918.12, 901.8012
(d) does not exist; (e) does not exist; (f) no, neither the limit 32. (a) - 3x2 - 3xh - h2; (b) - 109, - 91, - 76.51, - 75.1501
nor t - 2 exists. 57. (a) 3; (b) 1; (c) does not exist; (d) 1; 33. (a) 2x + h - 4; (b) 8, 7, 6.1, 6.01 34. (a) 2x + h - 3;
(e) no, the limit does not exist; (f) yes, the limit exists and equals (b) 9, 8, 7.1, 7.01 35. (a) 2x + h - 3; (b) 9, 8, 7.1, 7.01
the function value; (g) yes, the limit exists and equals the function 36. (a) 2x + h + 4; (b) 16, 15, 14.1, 14.01 37. 0.36 percentage
value. 58. (a) 1; (b) - 1; (c) does not exist; (d) 1; (e) no, point/yr, - 0.3 percentage point >yr, 0.09 percentage point/yr
the limit does not exist; (f) yes, the limit exists and equals the 38. 1.7 percentage point/yr, - 0.2 percentage point >yr,
function value. 59. Yes, the limit exists and equals the function 0.66 percentage point/yr 39. 0.66 percentage point/yr,
value at 4. 60. Yes, the limit exists and equals the function value - 0.15 percentage point >yr, 0.21 percentage point/yr
at 5. 61. No, the limit does not exist and G is not defined at 0. 40. - 0.31 percentage point >yr, 0.1 percentage point/yr,
62. No, F is not defined at - 1. 63. Yes, the limit exists and - 0.08 percentage point >yr 41. 0.1 percentage point/yr,
equals the function value at 4. 64. Yes, the limit exists and equals - 0.04 percentage point >yr, 0.03 percentage point/yr
the function value at 3. 65. No, the limit does not exist at 3. 42. - 0.7 percentage point >yr, 0.46 percentage point/yr,
66. No, the limit does not exist and G is not defined at 4. - 0.05 percentage point >yr 43. 2.1 percentage points/yr,
67. No, g is not defined at 4. 68. Yes, the limit exists and equals - 4.3 percentage points >yr, - 1.5 percentage points >yr
the function value at 3. 69. No, the limit value does not equal 44. 1 percentage point/yr, - 0.43 percentage point >yr,
the function value at 2. 70. No, the limit value does not equal 0.2 percentage point/yr > $1266.67/yr, $580/yr
45. - +450 yr,
the function value at 1. 71. Yes, the limit exists and equals the 46. $1.467 billion/yr, $0.4 billion/yr, $0.93 billion/yr
function value at 4. 72. Yes, the limit exists and equals the function 47. (a) 70, 39, 29, 23; (b) answers may vary. 48. (a) 300,
value at 5. 73. No, the limit does not exist and the function is 180, 120, 100; (b) answers may vary. 49. (a) $26.62;
not defined at 5. 74. Yes, the limit exists and equals the function (b) $31.16; (c) $4.54; (d) 0.7567, which means that prices
value at 3. 75. No, the limit does not exist and the function is increased by an average of about $0.76 per year 50. (a) $2391.24;
not defined at 2. 76. Yes, the limit exists and equals the function (b) $2693.71; (c) $302.47; (d) $151.24, which means the amount
value at 4. 77. Yes, because limxSa 1g x2 = g1a 2for all a such grew on average $151.24 per year for 2 yr 51. The average cost
that - 4 6 a 6 4 78. Yes, because limxSa F x = F a for all a of production of between 300 and 305 holders is $19.75 per unit.
such that -5 6 a 6 5 79. Yes, because limxSa g 1 x2 = g1a 2for
52. The average revenue from sales of between 300 and 305 holders
all a such that 1 6 a 6 `, and limxS1+1 g2 x =1 g2 1 80. Yes, is $149.40 per unit. 53. 17.62; the average rate of change in
because limxSa h x = h a for all a such that - 3 6 a 6 `, and Amazon’s revenue between 2014 and 2017 was $17.62 billion per year.
limxS -3+ h x = h - 3 81. Yes, because limxSa F x = F a 54. 41.35; the average rate of change in Panera Bread Co.’s income
for all a such that - 5 6 a 6 5, and limxS -5+ F x = F - 5 and between 2014 and 2017 was $41.35 million per year.
limxS5- F x = F 5 82. Yes, because limxSa G x = G a 55. (a) 1.49 hectares/g; (b) home range increases by 1.0902 hect-
for all a such that - 3 6 a 6 3, and limxS -3+ G x = G - 3 and
ares per gram as the animal’s weight grows from 200 g to 300 g.
limxS3- G x = G 3 83. 30, 25, does not exist 84. 8, 6,
56. (a) 50.33; the population grew by about 50 condors per year
does not exist 85. 120, 120, 120 86. Yes 87. Yes
between 2010 and 2017. (b) 42.064; the population increased by
88. Yes 89. Yes 90. No 91. No 92. Yes 93. Yes
19 about 42 condors per year between 2007 and 2015. 57. (a) 1.25,
94. 5 95. 2 96. (a) Does not exist; (b) - 3 97. a = , 1.25, 0.625, 0, 0; (b) answers may vary. 58. (a) 29.4 mi/gal;
4 (b) 0.034 gal/mi 59. (a) 256 ft; (b) 128 ft/sec
1 1
b = - 3 98. c = 9 99. 100. 6 101. 60. (a) 184.05 mi, or the distance traveled from t = 2 hr to t = 5 hr;
2 2 (b) 61.35 mi/hr 61. (a) 125 thousand people/yr; (b) answers
102. 1 103. 1 104. 3 105. 1
- 0 106. may vary; (c) A: 290 thousand people/year, -40 thousand people/yr,
2 23 27 4 4 - 50 thousand people/yr, 300 thousand people/yr;
