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EXERCISE 0.2 (page 3)
1. True. 3. False; the natural numbers are 1, 2, 3, and so on.5. True. 7. False; =5, a positive integer.9. True. 11. True.
EXERCISE 0.3 (page 7)
1. False. 3. False. 5. False. 7. True. 9. False.11. Distributive. 13. Associative. 15. Commutative.17. Definition of subtraction. 19. Distributive.
EXERCISE 0.4 (page 10)
1. –6. 3. 2. 5. 11. 7. –2. 9. –63.11. –6. 13. 6-x. 15. –12x+12y (or 12y-12x).
17. . 19. –2. 21. 18. 23. 64. 25. 3x-12.
27. –x+2. 29. . 31. . 33. . 35. 3.
37. . 39. . 41. . 43. . 45. .
47. . 49. Not defined. 51. Not defined.
EXERCISE 0.5 (page 16)
1. 25 (=32). 3. w12. 5. . 7. .
9. 8x6y9. 11. x6. 13. x14. 15. 5. 17. –2.
19. . 21. 7. 23. 8. 25. . 27. .
29. . 31. . 33. 4x2. 35. –2 +4 .
37. 3z2. 39. . 41. . 43. . 45. .
47. 71/3s2/3. 49. x1/2-y1/2. 51. .
53. . 55. . 57. .
59. . 61. . 63. . 65. 4.
67. 69. . 71. t2/3. 73. .
75. xyz. 77. . 79. . 81. x2y5/2. 83. .
85. x8. 87. . 89. .
EXERCISE 0.6 (page 22)
1. 11x-2y-3. 3. 6t2-2s2+6.5. .7. .9. . 11. –15x+15y-27.13. x2+9y2+xy. 15. 6x2+96.17. –6x2-18x-18. 19. x2+9x+20.21. w¤-3w-10. 23. 10x2+19x+6.25. x2+6x+9. 27. x2-10x+25.29. . 31. 4s¤-1.33. x3+4x2-3x-12.35. 3x4+2x3-13x2-8x+4. 37. 5x3+5x2+6x.39. 3x2+2y2+5xy+2x-8.41. x3+15x2+75x+125.43. 8x3-36x2+54x-27. 45. z-18.
47. . 49. .
51. . 53. .
55. .
EXERCISE 0.7 (page 25)
1. 2(3x+2). 3. 5x(2y+z).5. 4bc(2a3-3ab2d+b3cd2). 7. (z+7)(z-7).9. (p+3)(p+1). 11. (4x+3)(4x-3).13. (z+4)(z+2). 15. (x+3)2.17. 5(x+3)(x+2). 19. 3(x-1)(x+1).21. (6y+1)(y+2). 23. 2s(3s+4)(2s-1).25. x2/3y(1+2xy)(1-2xy). 27. 2x(x+3)(x-2).29. 4(2x+1)2. 31. x(xy-7)2.33. (x-2)2(x+2). 35. (y+4)2(y+1)(y-1).37. (x+2)(x2-2x+4).39. (x+1)(x2-x+1)(x-1)(x2+x+1).41. 2(x+3)2(x+1)(x-1). 43. P(1+r)2.45. (x2+4)(x+2)(x-2).47. (y4+1)(y2+1)(y+1)(y-1).49. (x2+2)(x+1)(x-1). 51. y(x+1)2(x-1)2.
EXERCISE 0.8 (page 31)
1. 3. . 5. .
7. . 9. .
11. . 13. . 15. . 17. .23
n
3x
22 1x + 4 2
1x - 4 2 1x + 2 2
3 - 2x
3 + 2x-
y2
1y - 3 2 1y + 2 2
3x + 2x + 2
x - 5x + 5
x + 2x
.
x - 2 +7
3x + 2
t + 8 +64
t - 83x2 - 8x + 17 +
-37x + 2
x +-1
x + 33x3 + 2x -
12x2
2y + 612y + 9
12y - 13z6x2 - 9xy - 2z + 12 - 421x + 12y + 13z
4x4z4
9y4-
4s5
y10
z2
4y4
x2
13
64y6x1>2x2
2x6
y3
2
0216a10b15
ab.
329x2
3x
212x
x
3177
352w3 -
15227w3
152x4
52 18x - y 2 4x9>4z3>4
y1>2
19t2
5m9
x3
y2z2
9t2
4
31213x 312412
116
14
12
a21
b20
x8
x17
k
9n
140
x - y
9-
16
56
7xy
23x
-
5x
7y
811
-
15
125
A N S W E R S T O O D D - N U M B E R E D P R O B L E M S
AN1
19. –27x2. 21. 1. 23. . 25. 1.
27. . 29. x+2. 31. .
33. . 35. .
37. . 39. .
41. . 43. . 45. .
47. . 49. .
51. 2- . 53. . 55. –4- .
57. . 59. 4 -5 +14.
MATHEMATICAL SNAPSHOT—CHAPTER 0 (page 33)
1. The results agree. 3. The results agree.
PRINCIPLES IN PRACTICE 1.1
1. P=2(w+2)+2w=2w+4+2w=4w+4.2. 200 specialty coffees. 3. 46 weeks; $1715.
4. . 5. .
EXERCISE 1.1 (page 41)
1. 0. 3. . 5. –2.
7. Adding 5 to both sides; equivalence guaranteed.9. Raising both sides to the fourth power; equivalence notguaranteed.11. Dividing both sides by x; equivalence not guaranteed.13. Multiplying both sides by x-1; equivalence notguaranteed.15. Multiplying both sides by (x-5)/x; equivalence notguaranteed.
17. . 19. 0. 21. 1. 23. . 25. –1.
27. 2. 29. . 31. 126. 33. 8. 35. .
37. . 39. . 41. . 43. 3. 45. .
47. . 49. . 51. .
53. . 55. 120 m.
57. c=x+0.0825x=1.0825x. 59. 3 years.
61. 31 hours. 63. 0.00001. 65. . 67. .
PRINCIPLES IN PRACTICE 1.2
1. 8 mi/h. 2. .
3. ramp is 5 feet long.
EXERCISE 1.2 (page 46)
1. . 3. �. 5. . 7. 2. 9. 0. 11. .
13. . 15. 3. 17. . 19. �. 21. 11.
23. . 25. . 27. 2. 29. 7. 31. .
33. . 35. . 37. .
39. 20. 41. .
43. Antenna B: 4 m; Antenna A: 12.25 m.
PRINCIPLES IN PRACTICE 1.3
1. The number is –5 or 6. 2. 50 feet by 60 feet.3. 1*1*5. 4. 15 items at $15 per item.5. 2.5 seconds and 7.5 seconds. 6. $100 7. Never.
EXERCISE 1.3 (page 53)
1. 2. 3. 4, 3. 5. 3, –1. 7. 4, 9. 9. —2.
11. 0, 8. 13. . 15. 1, . 17. 5, –2. 19. 0, .
21. 0, 1, –4. 23. 0, —8. 25. 0, . 27. –3, –1, 2.
29. 3, 4. 31. 4, –6. 33. . 35. .
37. No real roots. 39. . 41. 40, –25.
43. . 45. . 47. 2, .
49. . 51. –4, 1. 53. . 55. .
57. 6, –2. 61. 5, –2. 63. . 65. –2. 67. 6.
69. 4, 8. 71. 2. 73. 0, 4. 75. 4. 77. 64.15, 3.35.79. 6 inches by 8 inches. 83. 1 year and 10 years.85. 86.8 cm or 33.2 cm. 87. a. 9 s; b. 3 s or 6 s.89. 1.5, 0.75. 91. No real root. 93. 1.999, 0.963.
REVIEW PROBLEMS—CHAPTER 1 (page 56)
1. . 3. . 5. . 7. �. 9. . 11. .
13. . 15. . 17. 0, . 19. 5. 21. .
23. –3. 25. . 27. —2, —3. 29. .
31. . 33. 9. 35. 5. 37. No solution.
39. 10. 41. 4, 8. 43. –8, 1. 45. Q= .
47. C¿=l2(n-1-C). 49. T=;2� .
51. v=; . 55. .
57. –0.757, 0.384.
6, 54B2mgh - mv2
I
AL
g
EA
4�k
4 ; 1133
12
5 ; 1136
58
,
;
2163
75
-53
, 1-97
13
52
-12
-215
14
32
32
, -1157
, 115
;
155
, ;
12
-12
;13, ;12-2 ; 114
2
12
, -53
7 ; 1372
32
12
, -43
32
-52
12
t =d
r - c; r =
d
t+ c
n =2mI
rB- 1t =
r - d
rd-
94
4936
-
109
2625
513
18
53
83
15
2x2 + 16 - x = 2; x = 3;
t =d
r + w; w =
d
t- r
10r + 2
=6
r - 2;
1461
18
, -1
14
a1 =2S - nan
n
r =S - P
Ptq =
p + 18
P =I
rt
78
143
6017
-
3718
-
269
103
125
52
103
AS
πr =
d
t
1312x - 15x2 - 5
216-16 + 213
313
21x - 21x + h1x1x + h
1x + 2 2 16x - 1 22x2 1x + 3 2
4x + 13x
x
1 - xy
x2 + 2x + 1x2
35 - 8x
1x - 1 2 1x + 5 22x - 3
1x - 2 2 1x + 1 2 1x - 1 2
2x2 + 3x + 1212x - 1 2 1x + 3 2
11 - p2
73t
-12x + 3 2 11 + x 2
x + 4
2x2
x - 1
AN2 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN3
MATHEMATICAL SNAPSHOT—CHAPTER 1 (page 58)
1. a. $107.15; b. $10.26; c. 10 lb; d. 10.44 lb; e. 4.4%3. –1.9%.
EXERCISE 2.1 (page 66)
1. 120. 3. 48 of A, 80 of B. 5. . 7. 1 m.
9. 13,000. 11. $4000 at 6%, $16,000 at .
13. $4.25. 15. 4%. 17. 80. 19. $8000.21. 1138. 23. $116.25. 25. 40. 27. 46,000.29. Either $440 or $460. 31. $100. 33. 77.35. 80 ft by 140 ft. 37. 9 cm long, 4 cm wide.39. $112,000. 41. 60. 43. Either 125 units of A and100 units of B, or 150 units of A and 125 units of B.
PRINCIPLES IN PRACTICE 2.2
1. 5375.2. 150-x4 � 0; 3x4-210 � 0; x4+60 � 0; x4 � 0.
EXERCISE 2.2 (page 74)
1. (4, q). 3. (–q, 5]. 5. .
7. . 9. (0, q). 11. .
13. . 15. �. 17. .
19. (–q, 48). 21. (–q, –5]. 23. (–q, q).
25. . 27. . 29. (0, q).
31. (–q, 0). 33. (–q, –2].
35. 444,000<S<636,000. 37. x<70 degrees.
EXERCISE 2.3 (page 78)
1. 120,001. 3. 17,000. 5. 60,000. 7. $25,714.29.9. 1000. 11. t>36.5. 13. At least $67,400.
PRINCIPLES IN PRACTICE 2.4
1. |w-22 oz| � 0.3 oz.
EXERCISE 2.4 (page 82)
1. 13. 3. 6. 5. 5. 7. –4<x<4.9. . 11. a. |x-7|<3; b. |x-2|<3;c. |x-7| � 5; d. |x-7|=4; e. |x+4|<2;f. |x|<3; g. |x|>6; h. |x-6|>4; i. |x-105|<3;j. |x-850|<100. 13. |p1-p2| � 8. 15. —7.
17. —6. 19. 13, –3. 21. . 23. .
25. (–4, 4). 27. (–q ,–8) ´ (8, q). 29. (–9, –5).
31. (–q, 0) ´ (1, q). 33. .
35. (–q, 0] ´ . 37. |d-17.2| � 0.03 m
39. (–q, Â-hÍ) ´ (Â+hÍ, q).
REVIEW PROBLEMS—CHAPTER 2 (page 84)
1. (–q, 0]. 3. . 5. �. 7. .
9. (–q, q). 11. –2, 5. 13. .
15. ´ . 17. 542. 19. 6000.
21. c<$212,814.
MATHEMATICAL SNAPSHOT—CHAPTER 2 (page 85)
1. 1 hour. 3. 1 hour. 5. 600; 310.
PRINCIPLES IN PRACTICE 3.1
1. a. a(r)=�r2; b. all real numbers; c. r � 0.
2. a. t(r)= ; b. all real numbers except 0; c. r>0;
d. ;
e. The time is scaled by a factor c; .
3. a. 300 pizzas; b. $21.00 per pizza; c. $16.00 per pizza.
EXERCISE 3.1 (page 93)
1. All real numbers except 0. 3. All real numbers � 3.
5. All real numbers. 7. All real numbers except .
9. All real numbers except 0 and 1.
11. All real numbers except 4 and .
13. 1, 7, –7. 15. –62, 2-u2, 2-u4.17. 2, (2v)2+2v=4v2+2v, (–x2)2+(–x2)=x4-x2.19. 4, 0, (x+h)2+2(x+h)+1
=x2+2xh+h2+2x+2h+1.
21.
.=x + h - 4
x2 + 2xh + h2 + 51x + h 2 - 41x + h 2 2 + 5
130
, 3x - 413x 2 2 + 5
=3x - 49x2 + 5
-
12
-
72
t a x
cb =
300c
x
t 1x 2 =300x
; t a x
2b =
600x
; t a x
4b =
1200x
300r
c72
, qba-q, -
12d
a0, 12ba-q,
52da2
3, qb
c163
, qbc 12
, 34d
12
, 325
15 - 2
–2)0
0)
34–3
)179
c-
343
, qba179
, qb–5
)48
)– 22
3
)2–7
a - q, 13 - 2
2ba-
27
, qb7–5
0))
27
c-
75
, qba-q, 27b
1–2
54)
a-q, -
12d
712
%
513
23. 0, 256, . 25. a. 4x+4h-5; b. 4.
27. a. x2+2hx+h2+2x+2h; b. 2x+h+2.29. a. 2-4x-4h-3x2-6hx-3h2;
b. –4-6x-3h. 31. a. ; b. .
33. 9. 35. y is a function of x; x is a function of y.37. y is a function of x; x is not a function of y.39. Yes. 41. V=f(t)=20,000+800t.43. Yes; P; q. 45. 400 pounds per week; 1000 poundsper week; amount supplied increases as the price increases.47. a. 4; b. 8 ; c. f(2I0)=2 f(I0); doubling the intensity increases the response by a factor of 2 .49. a. 3000, 2900, 2300, 2000; 12, 10;b. 10, 12, 17, 20; 3000, 2300. 51. a. –5.13; b. 2.64;c. –17.43. 53. a. 11.33; b. 50.62; c. 2.29.
PRINCIPLES IN PRACTICE 3.2
1. a. p(n)=$125; b. The premiums do not change;c. constant function.2. a. quadratic function; b. 2; c. 3.
3. 4. 7!=5040.
EXERCISE 3.2 (page 98)
1. Yes. 3. No. 5. Yes. 7. No.9. All real numbers. 11. All real numbers.13. a. 3; b. 7. 15. a. 4; b. –3. 17. 8, 8, 8.19. 1, –1, 0, –1. 21. 8, 3, 1, 1. 23. 720. 25. 2.27. 5. 29. c(i)=$4.50; constant function.31. a. C=850+3q; b. 250.
33. 35. .
37. a. All T such that 30 � T � 39; b. 4, .
39. a. 237,077.34; b. –434.97; c. 52.19.41. a. 2.21; b. 9.98; c. –14.52.
PRINCIPLES IN PRACTICE 3.3
1. c(s(x))=c(x+3)=2(x+3)=2x+6.2. Let the length of a side be represented by the function l(x)=x+3 and the area of a square with sides of length x be represented by a(x)=x2. Then g(x)=(x+3)2=[l(x)]2=a(l(x)).
EXERCISE 3.3 (page 103)
1. a. 2x+8; b. 8; c. –2; d. x2+8x+15; e. 3;
f. ; g. x+8; h. 11; i. x+8. 3. a. 2x2+x;
b. –x; c. ; d. x4+x3; e. (for x ≠ 0);
f. –1; g. (x2+x)2=x4+2x3+x2; h. x4+x2; i. 90.
5. 6; –32. 7. .
9. . 11. f(x)=x5, g(x)=4x-3.
13. f(x)= , g(x)=x2-2.
15. f(x)= , g(x)= .
17. a. r(x)=9.75x; b. e(x)=4.25x+4500;c. (r-e)(x)=5.5x-4500.19. 400m-10m2; the total revenue received when the total output of m employees is sold.21. a. 14.05; b. 1169.64. 23. a. 345.03; b. –1.94.
PRINCIPLES IN PRACTICE 3.4
1. y=–600x+7250; x-intercept ;
y-intercept (0, 7250).2. y=24.95; horizontal line; no x-intercept;y-intercept (0, 24.95).3.
4.
EXERCISE 3.4 (page 112)
1.
3. a. 1, 2, 3, 0; b. all real numbers; c. all real numbers;d. –2. 5. a. 0, –1, –1; b. all real numbers;c. all nonpositive real numbers; d. 0.
x
y
(– , –2)12
(0, 0)
Q.I
Q.III Q.IV
(2, 7)
(8, –3)
–1–3
8
7
xtherms
y
80604020 100
20
40
60
Cos
t (do
llars
)
(0, 0)
(70, 37.1)
(100, 59.3)
xhours
y
4321 5
12
24
36M
iles
(0, 0)
(5, 0)
(2.5, 30)
a121
12, 0b
x + 13
51x
1x
1v + 3
; B2w2 + 3w2 + 1
41 t - 1 2 2 +
14t - 1
+ 1; 2
t2 + 7t
x2
x2 + x=
x
x + 112
x + 3x + 5
174
, 334
964
c 1n 2 = e8.50n
8.00n
ifif
n 6 10,n � 10.
c 1n 2 = •3.50n
3.00n
2.75n
ififif
n � 5,5 6 n � 10,n 7 10.
312
312312
-
1x 1x + h 2
1x + h
116
AN4 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN5
7. (0, 0); function; all real numbers; all real numbers.
9. (0, –5), ; function; all real numbers;
all real numbers.
11. (0, 0); function; all real numbers;all nonnegative real numbers.
13. Every point on y-axis; not a function of x.
15. (0, 0); function; all real numbers; all real numbers.
17. (0, 0); not a function of x.
19. (0, 2), (1, 0); function; all real numbers; all real numbers.
21. All real numbers; all real numbers � 4;(0, 4), (2, 0), (–2, 0).
23. All real numbers; 2; (0, 2).
25. All real numbers; all real numbers � –3; (0, 1), (2_ , 0).
x
y
2 +(2, –3)
3
2 – 3
1
13
x
y
2
t
s
2–2
4
x
y
2
1
x
y
x
y
x
y
x
y
x
y
5
–5
3
a53
, 0b
x
y
27. All real numbers; all real numbers; (0, 0).
29. All real numbers � 5; all nonnegative real numbers;(5, 0).
31. All real numbers; all nonnegative real numbers;
(0, 1), .
33. All nonzero real numbers; all positive real numbers;no intercepts.
35. All nonnegative real numbers; all real numbers cwhere 0 � c<6.
37. All real numbers; all nonnegative real numbers.
39. (a), (b), (d).41.
43. As price decreases, quantity increases; p is a function of q.
45.
47. –1, –0.35. 49. 0.62, 1.73, 4.65. 51. –0.84, 2.61.53. –0.49, 0.52, 1.25. 55. a. 3.94; b. –1.94.57. a. (–q, q); b. (–1.73, 0), (0, 4.00).59. a. 2.07; b. [2.07, q); c. (0, 2.39); d. no.
EXERCISE 3.5 (page 119)
1. (0, 0); sym. about origin.3. (—2, 0), (0, 8); sym. about y-axis.5. (—2, 0); sym. about x-axis, y-axis, origin.7. (–2, 0); sym. about x-axis. 9. Sym. about x-axis.11. (–21, 0), (0, –7), (0, 3).
13. (0, 0); sym. about origin. 15. .a0, 38b
x
y
4 5 12
4
q
p
5 25
20
5
x
y
20
16
12
8
4
10P.M.
86421210A.M.
Cos
t (do
llars
)
x
g (x )
3
9
p
c
6
65
t
F(t )
x
f (x )
1
1
2
a12
, 0b
r
s
5
t
f (t )
AN6 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN7
17. (2, 0), (0, —2); sym. about x-axis.
19. (—2, 0), (0, 0); sym. about origin.
21. (0, 0); sym. about x-axis, y-axis, origin.
23. (—2, 0), (0, —4); sym. about x-axis, y-axis, origin.
25. a. (—1.18, 0), (0, 2); b. 2; c. (–q, 2].
EXERCISE 3.6 (page 122)
1.
3.
5.
7.
9.
11.
13. Translate 4 units to the right and 3 units upward.15. Reflect about the y-axis and translate 5 units downward.
REVIEW PROBLEMS—CHAPTER 3 (page 123)
1. All real numbers except 1 and 2. 3. All real numbers.5. All nonnegative real numbers except 1.7. 7, 46, 62, 3t2-4t+7. 9. 0, 2, .
11. . 13. –8, 4, 4, –92.35
, 0, 1x + 4
x, 1u
u - 4
1t, 2x3 - 1
x
y
f (x) = xy = –x
x
y
1
1
y = 1 – (x – 1)2
f (x ) = x 2
x
y
–1
–2
f (x ) = x
y = x + 1 –2
x
y
1–1
1
2
–1
–2 y = 23x
f (x ) = 1x
x
y
f (x ) =
2
y = 1x – 2
1x
x
y
–2
y = x 2 – 2
f (x ) = x 2
x
y
–2
–4
4
2
x
y
x
y
2–2
x
y
2
2
–2
15. a. 3-7x-7h; b. –7.17. a. 4x2+8hx+4h2+2x+2h-5;b. 8x+4h+2. 19. a. 5x+2; b. 22; c. x-4;
d. 6x2+7x-3; e. 10; f. ;
g. 3(2x+3)-1=6x+8; h. 38;i. 2(3x-1)+3=6x+1.
21. . 23. , (x+2)3/¤.
25. (0, 0), (— , 0); sym. about origin.27. (0, 9), (—3, 0); sym. about y-axis.
29. (0, 2), (–4, 0); all u � –4; all real numbers � 0.
31. ; all t Z 4; all positive real numbers.
33. All real numbers; all real numbers � 1.
35.
37. a, c. 39. –0.67, 0.34, 1.73.41. –1.50, –0.88, –0.11, 1.09, 1.40.43. a. (–q, q); b. (1.92, 0), (0, 7)45. a. None; b. 1, 3.
MATHEMATICAL SNAPSHOT—CHAPTER 3 (page 125)
1. $28,321. 3. $87,507.90. 5. Answers may vary.
PRINCIPLES IN PRACTICE 4.1
1. –2000; the car depreciated $2000 per year.
2. S=14T+8. 3. .
4. slope= ; y-intercept= .
5. 9C-5F+160=0.6.
7. The slope of is 0; the slope of is 7; the slope ofis 1. None of the slopes are negative reciprocals of each
other, so the triangle does not have a right angle. The pointsdo not define a right triangle.
EXERCISE 4.1 (page 134)
1. 3. 3. . 5. Undefined. 7. 0.
9. 6x-y-4=0. 11. x+4y-18=0.13. 3x-7y+25=0. 15. 8x-5y-29=0.17. 2x-y+4=0. 19. x+2y+6=0.21. y+2=0. 23. x-2=0. 25. 4; –6.
27. . 29. Slope undefined; no y-intercept.
31. 3; 0. 33. 0; 1.
35. 2x+3y-5=0; y= .
37. 4x+9y-5=0; y= .
39. 3x-2y+24=0; y= .
41. Parallel. 43. Parallel. 45. Neither.47. Perpendicular. 49. Perpendicular.
32
x + 12
-49
x +59
-23
x +53
-
12
; 32
-
12
CABCAB
C
F
100–100
–100
100
1253
1253
F =95
C + 32
x
y
2
y = –
f (x ) = x 2
x 2 + 212
x
y
1
t
g(t)
4
12
a0, 12b
u
G (u )
–4
2
x
y
9
–3 3
12>32x3 + 2
1x - 1
, 1x
- 1 =1 - x
x
3x - 12x + 3
AN8 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN9
51. y=4x+14. 53. y=1. 55. y= .
57. x=7. 59. y= . 61. (5, –4).
63. –2; the stock price dropped an average of $2 per year.65. y=3x+5. 67. slope≠0.65; y-intercept ≠4.38
69. a. y= ; b. y=3x- .
71. y=–x+3300; without modification, the approach angle will cause the plane to crash 700 feet short of the airport. 73. R=50,000T+80,000.75. The lines are parallel. This is expected because theyeach have a slope of 1.5.
PRINCIPLES IN PRACTICE 4.2
1. x=number of skis produced; y=number of bootsproduced; 8x+14y=1000.
2. p= .
3. Answers may vary, but two possible points are (0, 60) and (2, 140).
4. f(t)=2.3t+32.2. 5. f(x)=70x+150.
EXERCISE 4.2 (page 141)
1. –4; 0. 3. 2; –4.
5. . 7. f(x)=4x.
9. f(x)=–2x+4.
11. f(x)= .
13. f(x)=x+1.
15. p= +28; $16.
17. p= q+190.
19. c=3q+10; $115. 21. f(x)=0.125x+4.15.
23. v=–800t+8000; slope=–800.
25. f(x)=45,000x+735,000. 27. f(x)=65x+85.
29. x+10y=100. 31. a. y= ; b. 12.
33. a. p=0.059t+0.025; b. 0.556.
35. a. t= b. add 37 to the number of chirps in
15 seconds. 37. P= 39. a. Yes; b. 1.8704.
PRINCIPLES IN PRACTICE 4.3
1. Vertex: (1, 400); y-intercept: (0, 399);x-intercepts: (–19, 0), (21, 0).
2. Vertex: (1, 24); y-intercept: (0, 8);
x-intercepts:.
3. 1000 units; $3000 maximum revenue.
EXERCISE 4.3 (page 149)
1. Quadratic. 3. Not Quadratic. 5. Quadratic.7. Quadratic. 9. a. (1, 11); b. highest.11. a. –8; b. –4, 2; c. (–1, –9).
x
y
5–5
30
a1 +162
, 0 b , a1 -162
, 0 b
x
y
25–25
100
400
T
4+ 80.
14
c + 37;
511
, x =60011
v
t10
8000
14
-25
q
-12
x +154
q
h (q )
27
-17
; 27
t
g (t )
–4
x
y
x
f (x)
2010
1000
500
-38
q + 1025
32
-13
x +16
-23
x -293
-13
x + 5
13. Vertex: (3, –4); intercepts: (1, 0), (5, 0), (0, 5);range: all y � –4.
15. Vertex: ; intercepts: (0, 0), (–3, 0);
range: all y � .
17. Vertex: (–1, 0); intercepts: (–1, 0), (0, 1); range: all s � 0.
19. Vertex: (2, –1); intercept: (0, –9); range: all y � –1.
21. Vertex: (4, –2); intercepts: (4+ ), (4- ), (0, 13); range: all t � –3.
23. Minimum; 24. 25. Maximum; –10.27. q=200; r=$120,000.29. 200 units; $240,000 maximum revenue.31. Vertex: (9, 225); y-intercept: (0, 144);
x-intercepts: (–6, 0), (24, 0).
33. 70 grams. 35. 132 ft; 2.5 sec.
37. Vertex: ; y-intercept: (0, 16),
x-intercepts:.
39. a. 2.5; b. 8.7 m. 41. a. ; b. ; c. 0 and l.
43. 50 ft*100 ft. 45. (1.11, 2.88).47. a. 0; b. 1; c. 2. 49. 4.89.
PRINCIPLES IN PRACTICE 4.4
1. $120,000 at 9% and $80,000 at 8%.2. 500 of species A and 1000 of species B.
3. Infinitely many solutions of the form A= ,
B=r where 0 � r � 5000.
4. lb of A; lb of B; lb of C.12
13
16
20,0003
-43
r
wl2
8l
2
x
h(t)
10–10
160
a 5 + 1292
, 0 b , a 5 - 1292
, 0 ba5
2, 116b
P(x)
400
x30–20
s
t
(4, –2)
14
4 – 2 4 + 2
12, 012, 0
x
y
–1
–9
2
t
s
–11
x
y
92
32
– 3
–
92
a -32
, 92b
x
y
1
(3, – 4)
5
5
AN10 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN11
EXERCISE 4.4 (page 161)
1. x=–1, y=1. 3. x=3, y=–1.5. v=0, w=18. 7. x=–3, y=2.9. No solution. 11. x=12, y=–12.
13. p= -3r, q=r; r is any real number.
15. . 17. x=1, y=1, z=1.
19. x=1+2r, y=3-r, z=r; r is any real number.
21. ; r is any real number.
23. r and s are any real
numbers.25. 420 gal of 20% solution, 280 gal of 30% solution.27. 0.5 lb of cotton; 0.25 lb of polyester; 0.25 lb of nylon.29. 275 mi/h (speed of airplane in still air),25 mi/h (speed of wind).31. 240 units (Early American), 200 units (Contemporary).33. 800 calculators from Exton plant, 700 from Whyton plant.35. 4% on first $100,000, 6% on remainder.37. 60 units of Argon I, 40 units of Argon II.39. 100 chairs, 100 rockers, 200 chaise lounges.41. 40 semiskilled workers, 20 skilled workers, 10 shippingclerks. 45. x=3, y=2. 47. x=8.3, y=14.0.
EXERCISE 4.5 (page 165)
1. x=4, y=–12; x=–1, y=3.3. p=–3, q=–4; p=2, q=1.5. x=0, y=0; x=1, y=1.7. x=4, y=8; x=–1, y=3.9. p=0, q=0; p=1, q=1.11. x= , y=2; x=– , y=2; x= ,y=–1; x= , y=–1. 13. x=21, y=15.15. At (10, 8.1) and (–10, 7.9). 17. Three.19. x=–1.3, y=5.1. 21. x=1.76. 23. x=–1.46.
EXERCISE 4.6 (page 174)
1.
3. (5, 212.50). 5. (9, 38). 7. (15, 5).9.
11. Cannot break even at any level of production.13. 15 units or 45 units. 15. a. $12; b. $12.18.17. 5840 units; 840 units; 1840 units. 19. $4.21. Total cost always exceeds total revenue—no break-evenpoint. 23. Decreases by $0.70.25. pA=5; pB=10. 27. 2.4 and 11.3.
REVIEW PROBLEMS—CHAPTER 4 (page 176)
1. 9. 3. y=–x+1; x+y-1=0.
5. y= -1; x-2y-2=0. 7. y=4; y-4=0.
9. y= +2; x-3y+6=0.
11. Perpendicular. 13. Neither. 15. Parallel.
17. y= . 19. y= .
21. –2; (0, 4).
23. (3, 0), (–3, 0), (0, 9); (0, 9).
25. (5, 0), (–1, 0), (0, –5); (2, –9).
27. 3; (0, 0).
t
p
t
y
2 5–1
–9
–5
x
y
3–3
9
x
y
2
4
43
; 032
x - 2; 32
13
x
12
x
q
yTR
TC
2000 6000
15,000(4500, 13,500)
5000
q
p
100 200
10
(100, 5)5
-114114117117
x =32
- r +12
s, y = r, z = s;
x = -13
r, y =53
r, z = r
x =12
, y =12
, z =14
32
29. (0, –3); (–1, –2).
31. . 33. x=2, y=–1.
35. x=8, y=4. 37. x=0, y=1, z=0.39. x=–3, y=–4; x=2, y=1.41. x=–2-2r, y=7+r, z=r; r is any real number.43. x=r, y=r, z=0; r is any real number.
45. a+b-3=0; 0. 47. f(x)= .
49. 50 units; $5000. 51. 6. 53. 1250 units; $20,000.55. 2.36 tons per square km. 57. x=230, y=–130.59. x=0.75, y=1.43.
MATHEMATICAL SNAPSHOT—CHAPTER 4 (page 170)
1. Advantage I is the best plan for airtimes from 85 to
153 minutes. Advantage II is the best plan for airtimes
from 153 to 233 minutes.
3. If the initial guess is on the horizontal portion of both graphs, the calculator may not be able to find the intersection point.
PRINCIPLES IN PRACTICE 5.1
1. The shape of the graphs are the same. The value of Ascales the ordinate of any point by A.2.
1.1; The investment increases by 10% every year(1+1(0.1)=1+0.1=1.1).
Between 7 and 8 years.
3.
0.85; The car depreciates by 15% every year (1-1(0.15)=1-0.15=0.85).
Between 4 and 5 years.4. y=1.08 ; Shift the graph 3 units to the right.5. $3684.87; $1684.87. 6. $2753.79; $753.79.7. 117 employees.8.
EXERCISE 5.1 (page 192)
1. 3.
5. 7.
x
y
–21
9
x
y
1–1
8
2
x
y
1–1
3
1
x
y
1
4
1
tyears
P
10 20
1
t - 3
xyears
y
4321 5
1
2
Year Multiplicative ExpressionDecrease
0 1 0.850
1 0.85 0.851
2 0.72 0.852
3 0.61 0.853
xyears
y
4321 5
1
2
Year Multiplicative ExpressionIncrease
0 1 1.10
1 1.1 1.11
2 1.21 1.12
3 1.33 1.13
4 1.46 1.14
13
13
13
-43
x +193
x =177
, y = -87
x
y
–1 – 2– 3
AN12 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN13
9. 11.
13. B. 15. 138,750. 17. .
19. a. $6014.52; b. $2014.52.21. a. $1964.76; b. $1264.76.23. a. $14,124.86; b. $10,124.86.25. a. $6256.36; b. $1256.36.27. a. $9649.69; b. $1649.69.29. $10,446.15.31. a. N=400(1.05)t; b. 420; c. 486.33.
1.3; The recycling increases by 30% every year(1+1(0.3)=1+0.3=1.3).
Between 4 and 5 years.35. 97,030. 37. 4.4817. 39. 0.4966.41. 43. 0.2240.
45. (ek)t, where b=ek.47. a. 10; b. 7.6;
c. 2.5; d. 25 hours.49. 32 years.51. 0.1465.55. 3.17.57. 4.2 min.59. 16.
PRINCIPLES IN PRACTICE 5.2
1. t=log2 16; t=the number of times the bacteria have
doubled. 2.
3.
4. 5. Approximately 13.9%.6. Approximately 9.2%.
EXERCISE 5.2 (page 201)
1. log 10,000=4. 3. 26=64. 5. ln 7.3891=2.
7. e1.09861=3.9. 11.
13.
15. 17. 2. 19. 3.21. 1. 23. –2.25. 0. 27. –3.29. 9. 31. 125.
33. . 35. e .
37. 2. 39. 6.
41. . 43. 2.
45. . 47. 4. 49. . 51. .
53. 1.60944. 55. 2.00013. 57. .
59. 41.50. 61. E=2.5*10 .63. a. 305.2 mm of mercury; b. 5.13 km.65. e . 67. 21.7 years.
69. . 71. (1, 0). 73. 7.39.y =13
ln 10 - x
2
3u0 - 1x22>224 >A
11 + 1.5M
1h
= 105.5
5 + ln 32
ln 23
53
181
-3110x
y
e1
1
–1
–2
x
y
4 6
1
x
y
4
1
1
–1x
y
31
1
x
y
1
8
4
multiplicativedecrease
y = log0.8
x
x
y
5 10
6
3
multiplicativeincrease
y = log1.5
x
I
I0= 108.3
x
y
1–1
xyears
y
4321 5
1
2
3
Year Multiplicative ExpressionIncrease
0 1 1.30
1 1.3 1.31
2 1.69 1.32
3 2.20 1.33
12
x
y
–2 –1 1
4
2
1x
y
1
3
1
PRINCIPLES IN PRACTICE 5.3
1.
2. log(10,000)=log(104)=4.
EXERCISE 5.3 (page 208)
1. b+c. 3. a-b. 5. 3a-b. 7. 2(a+b).
9. . 11. 48. 13. –4. 15. 5.01. 17. –2.
19. 2. 21. ln x+2 ln(x+1).23. 2 ln x-3 ln(x+1). 25. 3[ln x-ln(x+1)].27. ln x-ln(x+1)-ln(x+2).
29. .
31. .
33. log 24. 35. log2 . 37. log[79(23)5].
39. log[100(1.05)10]. 41. . 43. 1. 45. .
47. —2. 49. . 51. .
53. y=ln . 57. a. 3; b. 2+M1. 59. 3.5229.
61. 12.4771.63. 65. ln 4.
PRINCIPLES IN PRACTICE 5.4
1. 18. 2. Day 20. 3. The other earthquake is67.5 times as intense as a zero-level earthquake.
EXERCISE 5.4 (page 214)
1. 5.000. 3. 2.750. 5. –3.000. 7. 2.000.9. 0.083. 11. 1.099. 13. 0.203. 15. 5.140.17. –0.073. 19. 2.322. 21. 3.183. 23. 0.483.25. 2.496. 27. 1003.000. 29. 2.222. 31. 3.082.33. 3.000. 35. 0.500. 37. S=12.4A0.26.39. a. 100; b. 46. 41. 20.5.
43. . 45. 7.
47. a. 91; b. 432; c. 8. 49. 1.20.
51.
REVIEW PROBLEMS—CHAPTER 5 (page 216)
1. log3 243=5. 3. 161/4=2. 5. ln 54.598=4.
7. 3. 9. –4. 11. –2. 13. 4. 15. .
17. –1. 19. 3(a+1). 21. log . 23. ln .
25. log2 . 27. 2 ln x+ln y-3 ln z.
29. (ln x+ln y+ln z). 31. (ln y-ln z)-ln x.
33. . 35. 1.8295. 37. .
39. 2x. 41. .
43. 45. .
47. 1.49. 10.51. 2e.53. 0.880.55. –3.222.57. –1.596.59. a. $3829.04;
b. $1229.04.61. 14%.
63. a. P=8000(1.02)t; b. 8323.65. a. 10 mg; b. 4.4; c. 0.2; d. 1.7; e. 5.6.67. a. 6; b. 28. 71. (–q, 0.37]. 73. 2.93.75.
MATHEMATICAL SNAPSHOT—CHAPTER 5 (page 221)
1. a. ; b. .
3. a. 156; b. 65.
d =1
kI ln c P
P - T 1ekI - 1 2 dP =T 1ekI - 1 2
- e-dkI
10–10
7
–2
13
x
y
–3
1
8
y = ex2 + 2
2x +12
xln 1x + 5 2
ln 3
12
13
x9>21x + 1 2 3 1x + 2 2 4
x2y
z3
2527
1100
1–10
3
–3
p =log 180 - q 2
log 2; 4.32
10–3
4
–4
z
7
ln 1x2 + 1 2ln 3
ln 1x + 6 2ln 10
52
8164
2x
x + 1
25
ln x -15
ln 1x + 1 2 - ln 1x + 2 2
12
ln x - 2 ln 1x + 1 2 - 3 ln 1x + 2 2
b
a
= log 1100 2 = 2.
log 1900,000 2 - log 19000 2 = log a 900,0009000
b
AN14 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN15
PRINCIPLES IN PRACTICE 6.1
1. 3*2 or 2*3. 2. .
EXERCISE 6.1 (page 229)
1. a. 2*3, 3*3, 3*2, 2*2, 4*4, 1*2, 3*1, 3*3, 1*1; b. B, D, E, H, J; c. H, J upper triangular;D, J lower triangular; d. F, J; e. G, J.3. 2. 5. 4. 7. 0. 9. 7, 2, 1, 0.
11. . 13. 120 entries, 1, 0, 1, 0.
15. a. ; b. .
17. . 19. .
21. a. A and C; b. all of them.
25. x=6, y= . 27. x=0, y=0.
29. a. 7; b. 3; c. February; d. deluxe blue; e. February;
f. February; g. 38. 31. –2001. 33. .
PRINCIPLES IN PRACTICE 6.2
1. . 2. x1=670, x2=835, x3=1405.
EXERCISE 6.2 (page 237)
1. . 3. . 5. .
7. Not defined. 9. .
11. . 13. . 15. O.
17. . 19. Not defined. 21. .
23. . 29. . 31. .
33. Impossible. 35. x= .
37. x=6, y= . 39. x=–6, y=–14, z=1.
41. . 43. 1.1. 45. .
47. .
PRINCIPLES IN PRACTICE 6.3
1. $5780. 2. $22,843.75. 3. = .
EXERCISE 6.3 (page 249)
1. –12. 3. 19. 5. 7. 7. 2*2; 4.9. 3*5; 15. 11. 2*1; 2. 13. 3*3; 9.
15. 3*1; 3. 17. . 19. .
21. . 23. . 25. .
27. . 29. .
31. . 33. . 35. .
37. . 39. . 41.
43. . 45. Impossible. 47. .
49. . 51. .
53. . 55. . 57. .
59.
61. . 63. $2075.. 65. $1,133,850.
67. a. $180,000, $520,000, $400,000, $270,000, $380,000,$640,000; b. $390,000, $100,000, $800,000; c. $2,390,000;
d. . 71. .
73. .
PRINCIPLES IN PRACTICE 6.4
1. 5 blocks of A, 2 blocks of B, and 1 block of C.2. 3 of X; 4 of Y; 2 of Z. 3. A=3D; B=1000-2D; C=500-D; D=any amount (� 500).
c 15.606-739.428
64.08373.056
dc72.8251.32
-9.8-36.32
d110239
, 129239
£430
-103
3-1
2§ £
r
s
t
§ = £97
15§
c37
1-2d cx
yd = c6
5d .
c 6-7
-79dc1
0-1
101d£
200
020
002§
c 0-1
3-1
02dc 3
-2-1
2d
£020
0-1
0
-4-2
8§£
-121
51731§
£32
00
032
0
0032
§c -1-2
-2023d£
001
0-1
2
010§
c2x1 + x2 + 3x3
4x1 + 9x2 + 7x3d£
z
y
x
§c -5-5
-8-20d
c 78-21
84-12d≥
46
-82
69
-123
-4-6
8-2
69
-123
¥
3-6 16 10 -6 4£12
-3
-42
-2
243§c23
50d
c1210
-126d≥
1000
0100
0010
0001
¥
c 8553dcx
ydc1
1
8513d
c-1024
2236
12-44d
c154
-47
2630d£
357525
655515§
43
14613
, y = -2813
c-16
5-8dc4
72
-3202dc21
192
292
- 152d
c-22-11
-159dc 28
-2226d
c 6-2
53d£
50
-3
-473
1-213§
c-12-42
36-6
-42-36
-612d
3-9 -7 114£-5-9
5
559§£
4-210
-3105
153§
c230190
220255d
≥3142
1736
1412
¥
23
, z =72
≥1373
32
-20
-4501
¥c6-3
24d
F000000
000000
000000
000000
000000
000000
V≥0000
0000
0000
0000
¥
£61014
81216
101418
121620§
£111
222
444
888
161616§
EXERCISE 6.4 (page 261)
1. Not reduced. 3. Reduced. 5. Not reduced.
7. . 9. . 11. .
13. x=2, y=1. 15. No solution.
17. x= where r is any
real number. 19. No solution.21. x=–3, y=2, z=0. 23. x=2, y=–5, z=–1.25. x1=0, x2=–r, x3=–r, x4=–r, x5=r, where r is any real number. 27. Federal, $72,000; state, $24,000.29. A, 2000; B, 4000; C, 5000. 31. a. 3 of X, 4 of Z;2 of X, 1 of Y, 5 of Z; 1 of X; 2 of Y, 6 of Z; 3 of Y, 7 of Z;b. 3 of X, 4 of Z; c. 3 of X, 4 of Z; 3 of Y, 7 of Z.33. a. Let s, d, g represent the numbers of units S, D, G respectively. The six combinations are given by:
b. The combination s=0, d=3, g=5.
PRINCIPLES IN PRACTICE 6.5
1. Infinitely many solutions:
in parametric form: where r is
any real number.
EXERCISE 6.5 (page 267)
1. w=–r-3s+2, x=–2r+s-3, y=r, z=s(where r and s are any real numbers).3. w=–s, x=–3r-4s+2, y=r, z=s(where r and s are any real numbers).5. w=–2r+s-2, x=–r+4, y=r, z=s(where r and s are any real numbers).7. x1=–2r+s-2t+1, x2=–r-2s+t+4, x3=r, x4=s, x5=t (where r, s, and t are any real numbers).9. Infinitely many. 11. Trivial solution.13. Infinitely many. 15. x=0, y=0.
17. . 19. x=0, y=0.
21. x=r, y=–2r, z=r.23. w=–2r, x=–3r, y=r, z=r.
PRINCIPLES IN PRACTICE 6.6
1. Yes. 2. MEET AT NOON FRIDAY.
3. E–1= ; F is not invertible.
4. A: 5000 shares; B: 2500 shares; C:2500 shares
EXERCISE 6.6 (page 275)
1. 3. Not invertible. 5. .
7. Not invertible. 9. Not invertible (not a square matrix).
11. . 13. .
15. . 17. .
19. x1=10, x2=20. 21. x=17, y=–20.23. x=1, y=3. 25. x=–3r+1, y=r.
27. x=0, y=1, z=2. 29. x=1, .
31. No solution. 33. w=1, x=3, y=–2, z=7.
35. . 37. a. 40 of model A, 60 of model B;
b. 45 of model A, 50 of model B. 39. b. .
41. Yes. 43. D: 5000 shares; E:1000 shares; F:4000 shares.
45. a. b. .
47. .
49. w=14.44, x=0.03, y=–0.80, z=10.33.
PRINCIPLES IN PRACTICE 6.7
1. 6
EXERCISE 6.7 (page 285)
1. 1. 3. –16. 5. y. 7. 9. 12.
11. –12. 13. 6. 15. .
17. . 19. –16. 21. 98. 23. –89.
25. –1. 27. 2. 29. –90. 31. 1. 33. 24.
35. 0. 37. 0. 39. 3, 4. 41. 192. 43. b.
45. c=–1 or c=4. 47. –1630. 49. –3864.
EXERCISE 6.8 (page 290)
1. 3.
5. 7.
9. x=4, y=2, z=0. 11.
13. x=3-r, y=0, z=r. 15. x=1, y=3, z=5.
x =23
, y = -2815
, z = -2615
.
x =65
, z =165
.x = -13
, y = -1.
x =7
16, y =
138
.x =32
, y = -1.
13
.
3a21
a31
a41
a22
a32
a42
a24
a34
a44
33a11
a21
a41
a13
a23
a43
a14
a24
a44
3-
27
.
£1.800.350.44
1.101.310.42
-0.46-0.17
0.59§
c130894589
5089
12089dc1.46
0.510.561.35d;
c47
610d
c- 2313
- 13
- 13d
y =12
, z =12
£113
- 7323
-33
-1
13
- 2313
§£1
-1-1
- 2343
1
53
- 103
-2§
£103
010
207§£
100
-110
0-1
1§
£100
0-
13
0
0014
§c-17
1-6d.
£23
- 13
- 13
- 1656
- 16
- 13
- 1323
§
x = -65
r, y =8
15r, z = r
x = -12
r, y = -12
r, z = r,
x +12
z = 0, y +12
z = 0;
s
d g
3 5 8 0
471
362
253
144
035
-23
r +53
, y = -16
r +76
, z = r,
≥1000
0100
0010
0001
¥£100
200
300§c1
001d
AN16 Answers to Odd-Numbered Problems �
� Answers to Odd-Numbered Problems AN17
17. y=6, w=1. 19. Since ∆= =0,
Cramer’s rule does not apply. But the equations in
represent distinct parallel lines and hence
no solution exists. 21. Four games.23. x=17.85, y=–0.42, z=–24.09.
EXERCISE 6.9 (page 294)
1. 3. a. b.
5. . 7.
REVIEW PROBLEMS—CHAPTER 6 (page 296)
1. . 3. 5.
7. 9. 11.
13. x=3, y=21. 15. . 17. .
19. x=0, y=0. 21. No solution. 23. .
25. No inverse exists. 27. x=0, y=1, z=0.29. 18. 31. 3. 33. rich. 35. x=1, y=2.37. –2. 39. A2=I£, A–1=A, A¤‚‚‚=I£.
41.
43. a. Let x, y, z represent the weekly doses of capsules ofbrands I, II, III, respectively. The combinations are given by:
b. Combination 4:x=1, y=0, z=3.
45. 47.
MATHEMATICAL SNAPSHOT—CHAPTER 6 (page 298)
1. $151.40. 3. It is not possible, because guests 3 and 4 each cost the lodge the same amount per day.
PRINCIPLES IN PRACTICE 7.1
1. 2x+1.5y>0.9x+0.7y+50, y>–1.375x+62.5;sketch the dashed line y=–1.375x+62.5 and shade thehalf plane above the line. In order to produce a profit, thenumber of magnets of types A and B produced and soldmust be an ordered pair in the region.
2. x � 0, y � 0, x+y � 50, x � 2y; The region consists ofpoints on or above the x-axis and on or to the right of the y-axis. In addition, the points must be on or above the linex+y=50 and on or below the line x=2y.
EXERCISE 7.1 (page 306)
1. 3.
5. 7.
9. 11.
13. 15.
17. 19.
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
2
4
x
y
7
72
x
y
2
3
c40.840.56
d.c21589
87141d.
combination 1combination 2combination 3combination 4
x
4321
y
9630
z
0123
x = 2 -c
a, y =
c
a- 1, z = 1 -
a
c.
c- 3212
56
- 16d
£100
200
010§c1
001d
c20
017d.c-1
2-2
1d.c 6
32d.
c-15
-222d.£
121
42-18
0
5-7-2§.c 3
-168
-10d
£10731016952§.£
130112151188
§
£102.17125.28175.27
§.£297.80349.54443.12
§;c12901425
d; 1405.
ex + y = 2,x + y = -3,
2 11
112
21. 23.
25. 27.
29. x � 0, y � 0, 3x+2y � 240, 0.5x+y � 80.
EXERCISE 7.2 (page 315)
1. P=640 when x=40, y=20.3. Z=–10 when x=2, y=3.5. No optimum solution (empty feasible region).7. Z=3 when x=0, y=1.
9. C=2.4 when
11. No optimum solution (unbounded).13. 15 widgets, 25 wadgits; $210.15. 4 units of food A, 4 units of food B; $8.17. 10 tons of ore I, 10 tons of ore II; $1100.19. 6 chambers of type A and 10 chambers of type B.21. c. x=y=75.23. Z=15.54 when x=2.56, y=6.74.25. Z=–75.98 when x=9.48, y=16.67.
PRINCIPLES IN PRACTICE 7.3
1. Ship 10t+15 TV sets from C to A, –10t+30 TV setsfrom C to B, –10t+10 TV sets from D to A, and 10t TVsets from D to B, for 0 � t � 1; minimum cost $780.
EXERCISE 7.3 (page 318)
1. Z=33 when x=(1-t)(2)+5t=2+3t,y=(1-t)(3)+2t=3-t, and 0 � t � 1.3. Z=72 when x=(1-t)(3)+4t=3+t,y=(1-t)(2)+0t=2-2t, and 0 � t � 1.
PRINCIPLES IN PRACTICE 7.4
1. 0 gadgets of Type 1, 72 gadgets of Type 2, 12 gadgets ofType 3; maximum profit of $20,400.
EXERCISE 7.4 (page 330)
1. Z=8 when x1=0, x2=4.3. Z=14 when x1=1, x2=5.5. Z=28 when x1=3, x2=2.7. Z=20 when x1=0, x2=5, x3=0.
9. Z=2 when x1=1, x2=0, x3=0.
11. when x1= , x2=
13. W=13 when x1=1, x2=0, x3=3.15. Z=600 when x1=4, x2=1, x3=4, x4=0.17. 0 from A, 2400 from B; $1200.19. 0 chairs, 300 rockers, 100 chaise lounges; $10,800.
PRINCIPLES IN PRACTICE 7.5
1. 35-7t of device 1, 6t of device 2, 0 of device 3, for 0 � t � 1.
EXERCISE 7.5 (page 337)
1. Yes; for the tableau, x2 is the entering variable and the
quotients and tie for being the smallest.
3. No optimum solution (unbounded).5. Z=12 when x1=4+t, x2=t, and 0 � t � 1.7. No optimum solution (unbounded).
9. Z=13 when x1= x2=6t, x3=4-3t, and
0 � t � 1.11. $15,200. If x1, x2, x3 denote the number of chairs,rockers, and chaise lounges produced, respectively, then x1=100-100t, x2=100+150t, x3=200-50t, and 0 � t � 1.
PRINCIPLES IN PRACTICE 7.6
1. Plant I: 500 standard, 700 deluxe; plant II: 500 standard,100 deluxe; $89,500 maximum profit.
EXERCISE 7.6 (page 348)
1. Z=7 when x1=1, x2=5.3. Z=4 when x1=1, x2=2, x3=0.
5. Z= when x1= , x2= , x3=0.
7. Z=–17 when x1=3, x2=2.9. No optimum solution (empty feasible region).11. Z=2 when x1=6, x2=10.13. 255 Standard bookcases, 0 Executive bookcases.15. 30% in A, 0% in AA, 70% in AAA; 6.6%.
EXERCISE 7.7 (page 352)
1. Z=54 when x1=2, x2=8.3. Z=216 when x1=18, x2=0, x3=0.5. Z=4 when x1=0, x2=0, x3=4.7. Z=0 when x1=3, x2=0, x3=1.9. Z=28 when x1=3, x2=0, x3=5.11. Install device A on kilns producing 700,000 barrels annually, and device B on kilns producing 2,600,000 barrelsannually. 13. To Exton, 5 from A and 10 from B; toWhyton, 15 from A; $380. 15. a. Column 3: 1, 3, 3;column 4: 0, 4, 8; b. x1=10, x2=0, x3=20, x4=0;c. 90 in.
23
143
583
32
-32
t,
31
62
143
.23
Z =163
x =35
, y =65
.
x + y � 0
x
y
100
100
x + y � 100
x : number of lb from Ay : number of lb from B
x + x � 0
x
y
5
3
x
y
x
y
AN18 Answers to Odd-Numbered Problems �
PRINCIPLES IN PRACTICE 7.8
1. Minimize W=60,000y1+2000y2+120y3 subject to 300y1+20y2+3y3 � 300, 220y1+40y2+y3 � 200, 180y1+20y2+2y3 � 200, and y1, y2, y3 � 0.2. Maximize W=98y1+80y2 subject to20y1+8y2 � 6,6y1+16y2 � 2,and y1, y2 � 0.3. 5 device 1, 0 device 2, 15 device 3.
EXERCISE 7.8 (page 361)
1. Minimize W=6y1+4y2 subject toy1-y2 � 2,y1+y2 � 3,y1, y2 � 0.3. Maximize W=8y1+2y2 subject toy1-y2 � 1,y1+2y2 � 8,y1+y2 � 5,y1, y2 � 0.5. Minimize W=13y1-3y2-11y3 subject to–y1+y2-y3 � 1,2y1-y2-y3 � –1,y1, y2, y3 � 0.7. Maximize W=–3y1+3y2 subject to–y1+y2 � 4,y1-y2 � 4,y1+y2 � 6,y1, y2 � 0.
9. Z=11 when x1=0, x2= , x3= .
11. Z=26 when x1=6, x2=1.13. Z=14 when x1=1, x2=2.15. $250 on newspaper advertising, $1400 on radio advertising; $1650.17. 20 shipping clerk apprentices, 40 shipping clerks,90 semiskilled workers, 0 skilled workers; $1200.
REVIEW PROBLEMS—CHAPTER 7 (page 362)
1. 3.
5. 7.
9. 11. Z=3 when x=3, y=0.13. Z=–2 when x=0, y=2.15. No optimum solution
(empty feasible region).17. Z=36 when x=2+2t,
y=3-3t, and 0 � t � 1.19. Z=32 when x1=8, x2=0.21. Z=2 when x1=0, x2=0,
x3=2.
23. Z=24 when x1=0, x2=12.
25. Z= when x1= , x2=0, x3= .
27. No optimum solution (unbounded).29. Z=70 when x1=35, x2=0, x3=0.31. 0 units of X, 6 units of Y, 14 units of Z; $398.33. 500,000 gal from A to D, 100,000 gal from A to C,400,000 gal from B to C; $19,000.35. 10 kg of food A only.37. Z=117.88 when x=7.23, y=3.40.
MATHEMATICAL SNAPSHOT—CHAPTER 7 (page 365)
1. 2 minutes of radiation. 3. Answers may vary.
PRINCIPLES IN PRACTICE 8.1
1. 4.9%. 2. 7 years, 16 days. 3. 7.7208%.4. The $10,000 investment is slightly better over 20 years.
EXERCISE 8.1 (page 372)
1. a. $11,105.58; b. $5105.58. 3. 4.060%. 5. 4.081%.7. a. 10%; b. 10.25%; c. 10.381%; d. 10.471%; e. 10.516%.9. 8.08%. 11. 9.0 years. 13. $10,282.95.15. $38,503.23. 17. a. 18%; b. $19.56%.19. $3198.54. 21. 8% compounded annually.23. a. 5.47%; b. 5.39%. 25. 11.61%. 27. 6.29%.
EXERCISE 8.2 (page 377)
1. $2261.34. 3. $1751.83. 5. $5118.10.7. $4862.31. 9. $6838.95. 11. $9419.05.13. $14,091.10. 15. $1238.58. 17. $3244.6319. a. $515.62; b. profitable. 21. Savings account.23. $226.25. 25. 9.55%.
PRINCIPLES IN PRACTICE 8.3
1. 48 ft, 36 ft, 27 ft, 20 ft, 15 ft.
2. 750, 1125, 1688, 2531, 3797, 5695. 3. 35.72 m.
316
14
94
54
72
x
y
x
y
x
y
x
y
–3/2x
y
2
– 3
32
12
� Answers to Odd-Numbered Problems AN19
4. $176,994.65. 5. 6.20%. 6. $101,925; $121,925.7. $723.03. 8. $13,962.01. 9. $45,502.06.10. $48,095.67.
EXERCISE 8.3 (page 386)
1. 64, 32, 16, 8, 4. 3. 100, 102, 104.04. 5. .
7. 1.11111. 9. 18.664613. 11. 8.213180.13. $2050.10. 15. $29,984.06. 17. $8001.24.19. $90,231.01. 21. $204,977.46. 23. $24,594.36.25. $1937.14. 27. $458.40.29. a. $3048.85; b. $648.85. 31. $3474.12.33. $1725. 35. 102.91305. 37. 55,360.30.39. $131.34. 41. $1,872,984.02.43. $205,073; $142,146.
EXERCISE 8.4 (page 391)
1. $69.33. 3. $502.84.5. a. $221.43; b. $25; c. $196.43.
7.
9.
11. 11. 13. $1273.15. a. $2089.69; b. $1878.33; c. $211.36; d. $381,907.17. 23. 19. $113,302.45. 21. $38.64.
REVIEW PROBLEMS—CHAPTER 8 (page 394)
1. . 3. 8.5% compounded annually.
5. $586.60. 7. a. $1997.13; b. $3325.37.9. $936.85. 11. $886.98. 13. $314.00.
15.
17. $1279.36.
MATHEMATICAL SNAPSHOT—CHAPTER 8 (page 396)
1. $15,597.85. 3. When investors expect a drop in inter-est rates, long-term investments become more attractive relative to short-term ones.
EXERCISE 9.1 (page 403)
1.
3.
5. 20. 7. 96. 9. 1024. 11. 20. 13. 720.15. 720. 17. 1000; error message is displayed.19. 6. 21. 336. 23. 216. 25. 1320. 27. 336.29. 720. 31. 2520; 5040. 33. 624. 35. 24.37. a. 11,880; b. 19,008. 39. 48. 41. 2880.
Start
1
2
3
4
5
6
H
T
H
T
H
T
H
T
H
T
H
T
1, H
1, T
2, H
2, T
3, H
3, T
4, H
4, T
5, H
5, T
6, H
6, T
12 possible results
Die Coin Result
Start
AD
E
6 possible production routes
Assemblyline
Finishingline
Productionroute
BD
E
CD
E
AD
AEBD
BECD
CE
Prin. Outs. Interest Pmt. Prin.at for at Repaid
Period Beginning Period End at End
1 15,000.00 112.50 3067.84 2955.342 12,044.66 90.33 3067.84 2977.513 9067.15 68.00 3067.84 2999.844 6067.31 45.50 3067.84 3022.345 3044.97 22.84 3067.81 3044.97
Total 339.17 15,339.17 15,000.00
6316
Prin. Outs. Interest Pmt. Prin.at for at Repaid
Period Beginning Period End at End
1 900.00 22.50 193.72 171.222 728.78 18.22 193.72 175.503 553.28 13.83 193.72 179.894 373.39 9.33 193.72 184.395 189.00 4.73 193.73 189.00
Total 68.61 968.61 900.00
Prin. Outs. Interest Pmt. Prin.at for at Repaid
Period Beginning Period End at End
1 5000.00 350.00 1476.14 1126.142 3873.86 271.17 1476.14 1204.973 2668.89 186.82 1476.14 1289.324 1379.57 96.57 1476.14 1379.57
Total 904.56 5904.56 5000.00
422243
AN20 Answers to Odd-Numbered Problems �
EXERCISE 9.2 (page 412)
1. 15. 3. 1. 5. 18. 9. 2380. 11. 66.
13. 15. 56. 17. 1680. 19. 35.
21. 720. 23. 1680. 25. 252. 27. 756,756.29. a. 90; b. 330. 31. 17,325. 33. a. 1; b. 1; c. 18.35. 3744. 37. 5,250,960.
PRINCIPLES IN PRACTICE 9.3
1. 10,586,800.
EXERCISE 9.3 (page 421)
1. {9D, 9H, 9C, 9S}.3. {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}.5. {mo, mu, ms, me, om, ou, os, oe, um, uo, us, ue, sm, so, sn,se, em, eo, eu, es}.7. a. {RR, RW, RB, WR, WW, WB, BR, BW, BB};b. {RW, RB, WR, WB, BR, BW}.9. Sample space consists of ordered sets of six elements andeach element is H or T; 64.11. Sample space consists of ordered pairs where first ele-ment indicates card drawn and second element indicatesnumber on die; 312.13. Sample space consists of combinations of 52 cardstaken 13 at a time; 52C13.15. {1, 3, 5, 7, 9}. 17. {7, 9}. 19. {1, 2, 4, 6, 8, 10}.21. S. 23. E1 and E4, E2 and E3, E3 and E4.25. E and H, G and H, H and I.27. a. {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT};b. {HHH, HHT, HTH, HTT, THH, THT, TTH};c. {HHT, HTH, HTT, THH, THT, TTH, TTT}; d. S;e. {HHT, HTH, HTT, THH, THT, TTH}; f. �;g. {HHH, TTT}.29. a. {ABC, ACB, BAC, BCA, CAB, CBA};b. {ABC, ACB}; c. {BAC, BCA, CAB, CBA}.
EXERCISE 9.4 (page 433)
1. 600. 3. a. 0.8; b. 0.4. 5. No.
7. a. b. c. d. e. f. g.
9. a. b. c. d. e. f. g. h. i. 0.
11. a. b. c. d.
13. a. b.
15. a. b. c. d. 17. a. b.
19. a. 0.1; b. 0.35; c. 0.7; d. 0.95; e. 0.1, 0.35, 0.7, 0.95.
21. 23. a. b.
25.
27. a. ≠0.040; b. ≠0.026.
29. 31. a. 0.51; b. 0.44; c. 0.03. 33. 4:1.
35. 3:7. 37. 39. 41. 3:1.
EXERCISE 9.5 (page 447)
1. a. b. c. d. e. 3. 1. 5. 0.43.
7. a. b. 9. a. b. c. d.
11. a. b. c. d. e. f.
13. a. b. 15. 17. a. b. 19.
21. 23. 25. 27. 29.
31. 33. 35. 37. a. b.
39. a. b. 41. 43. 45.
47. 0.049. 49. a. 0.06; b. 0.155. 51.
EXERCISE 9.6 (page 458)
1. a. b. c. d. e. f. g. 3.
5. Independent. 7. Independent. 9. Dependent.11. Dependent. 13. a. Dependent; b. dependent;
c. dependent; d. no. 15. Dependent. 17.
19. 21. 23. a. b. c.
25. a. b. c. d. e. 27. a. b.
29. 31. 33. a. b.
35. a. b. c. 37. 0.0106.
EXERCISE 9.7 (page 468)
1. P(E | D)= P(F | D¿)= . 3. ≠0.387.
5. a. ≠0.275; b. ≠0.005. 7. 9.
11. ≠0.910. 13. ≠55.1%. 15. .
17. ≠0.828. 19. 21. ≠0.933.
23. a. =0.205; b. ≠0.585; c. =0.115.
25. a. 0.18; b. 0.23; c. 0.59; d. high quality.
27. ≠0.78.79
23200
2441
41200
1415
45
.2429
34
8189
631
.58
.14
3021258937
1231
47
14
,
53512
.164
;15
1024;
38
.1
1728;
3200
.139361
.
415
.2
15;
215
.1315
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;25
;
110
.140
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10;
3676
.125
.
118
.
56
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;112
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;13
;56
;14
;
431
.
125
.14
.920
.35
.34
;
2747
.47
100;
217
.11850
.8
16,575.
4051
.113
.12
.16
.111
.
23
.14
.12
;23
.49
.12
;
2586
.1047
;8
25;
1139
;3558
;58
;
29
.12
;25
;35
;23
.12
;
13
.23
;13
;35
;25
;
27
.59
.
19
.
4140161,700
6545161,700
13 · 4C3 · 12 · 4C2
52C5.
111024
.1
210 =1
1024;
110
.
15
.45
;78
.18
;38
;18
;
3382652
=13
102.
122652
=1
221;
39624
=116
.8
624=
178
;4
624=
1156
;1
624;
126
;413
;1
52;
12
;12
;113
;14
;1
52;
56
.12
;12
;1
36;
14
;112
;5
36;
74!10! # 64!
.
� Answers to Odd-Numbered Problems AN21
REVIEW PROBLEMS—CHAPTER 9 (page 473)
1. 336. 3. 36. 5. 608,400. 7. 32. 9. 210.11. 126. 13. a. 2024; b. 253. 15. 34,650. 17. 560.19. a. {1, 2, 3, 4, 5, 6, 7}; b. {4, 5, 6}; c. {4, 5, 6, 7, 8}; d. �;e. {4, 5, 6, 7, 8}; f. no.21. a. {R1R2R3, R1R2G3, R1G2R3, R1G2G3, G1R2R3,G1R2G3, G1G2R3, G1G2G3}; b. {R1R2G3, R1G2R3, G1R2R3};c. {R1R2R3, G1G2G3}.
23. 0.2. 25. 27. a. b.
29. a. b. 31. 3:5. 33. 35.
37. 0.42. 39. a. b. 41.
43. a. b. independent. 45. Dependent.
47. a. 0.0081; b. 0.2646; c. 0.3483.
49. 51. 53. a. 0.014; b. ≠0.57.
MATHEMATICAL SNAPSHOT—CHAPTER 9 (page 477)
1. ≠0.645.
EXERCISE 10.1 (page 486)
1. Â=1.7; Var(X)=1.01; Í≠1.00.
3. Â= =2.25; Var(X)= =0.6875; Í≠0.83.
5. a. 0.1; b. 5; c. 3.
7. E(X)= =1.5; Í2= =0.75; Í≠0.87.
9. E(X)= =1.2; Í2= =0.36; Í= =0.6.
11. f(0)= , f(1)= , f(2)= .
13. a. –$0.15 (a loss); b. –$0.30 (a loss). 15. $101.43.17. $3.00. 19. $410. 21. Loss of $0.25; $1.
PRINCIPLES IN PRACTICE 10.2
1.
EXERCISE 10.2 (page 492)
1. Â= ; Í= .
3. Â=2;
Í= . 5. 0.001536. 7. 9. .
11. ≠0.081. 13. 15. 0.002.
17. a. b. 19. ≠0.593. 21. 0.7599.
23. . 25. ≠0.267.
EXERCISE 10.3 (page 502)
1. No. 3. No. 5. Yes. 7.
9. a=0.3, b=0.6, c=0.1. 11. Yes. 13. No.
15. X1= , X2= , X3= .
17. X1= , X2= , X3= .19. X1= , X2= ,X3= .
21. a. T2= , T3= ; b. ; c. .
23. a. T2= ,
T3= ; b. 0.40; c. 0.369.
25. . 27. . 29. .30.5 0.25 0.254c37
47dc3
5
2
5d
£0.2300.3690.327
0.6900.5300.543
0.0800.1010.130
§
£0.500.230.27
0.400.690.54
0.100.080.19§
916
38
≥716916
916716
¥≥5838
3858
¥
30.1766 0.3138 0.50964 30.164 0.302 0.534430.26 0.28 0.46430.4168 0.58324 30.416 0.584430.42 0.584c 83108
25108dc25
361136dc11
12112d
a =13
, b =34
.
21878192
1316
1627
532
.9
64;
96625
= 0.1536.1652048
316
96625
= 0.1536.163
f 10 2 =127
, f 11 2 =29
, f 12 2 =49
, f 13 2 =8
27;
164
12
f 10 2 =916
, f 11 2 =38
, f 12 2 =1
16;
310
35
110
35
925
65
34
32
1116
94
x
f (x )
0 1 2 3
0.4
0.3
0.2
0.1
47
14
.2245
.
13
;
14
.118
.2
11;
313
.67
.14
.14
;
215
.4
25;
45512
.
AN22 Answers to Odd-Numbered Problems �
x P(x)
0
1
2
3
481
10,000
75610,000
264610,000
411610,000
240110,000
31. a. b. 37, 36.
33. a. b. 0.781.
35. a. b. 0.19; c. 40%.
37. a. ; b. 65%; c. 60%.
39. a.
b. 59.18% in compartment 1, 40.82% in compartment 2;c. 60% in compartment 1, 40% in compartment 2.
41. a. ; b. .
REVIEW PROBLEMS—CHAPTER 10 (page 506)
1. Â=1.5, Var(X)=0.65, Í=0.81.
3. a.
= b. 4. 5. $0.10 (a loss).
7. a. $176; b. $704,000.9.
Â=0.3; Í= ≠0.52. 11. . 13. .
15. . 17. a=0.3, b=0.2, c=0.5.
19. X1= , X2= .
21. a. T2= T3= ; b. ;
c. . 23. .
25. a. 76%; b. 74.4% Japanese, 25.6% non-Japanese;c. 75% Japanese, 25% non-Japanese.
MATHEMATICAL SNAPSHOT—CHAPTER 10 (page 508)
1. 7.
3. Against Always Defect: ;
Against Always Cooperate: ;
Against regular Tit-for-tat: .
PRINCIPLES IN PRACTICE 11.1
1. The limit as x S a does not exist if a is an integer, but itexists if a is any other value.
2. 36∏ cc. 3. 3616. 4. 20. 5. 2.
EXERCISE 11.1 (page 521)
1. a. 1; b. 0; c. 1. 3. a. 1; b. does not exist; c. 3.5. f(0.9)=2.8, f(0.99)=2.98, f(0.999)=2.998,f(1.001)=3.002, f(1.01)=3.02, f(1.1)=3.2; 3.7. f(–0.1)≠0.9516, f(–0.01)≠0.9950,f(–0.001)≠0.9995, f(0.001)≠1.0005, f(0.01)≠1.0050.f(0.1)≠1.0517; 1.
9. 16. 11. 20. 13. –1. 15. . 17. 0.
19. 5. 21. –2. 23. 3. 25. 0. 27. .
29. . 31. . 33. 4. 35. 2x. 37. –1.
39. 2x. 41. 2x-3. 43. . 45. a. 1; b. 0.
47. 11.00. 49. –7.00. 51. Does not exist.
PRINCIPLES IN PRACTICE 11.2
1. p(x)=0. The graph starts out high and quickly goes
down toward zero. Accordingly, consumers are willing to purchase large quantities of the product at prices closeto 0.
2. y(x)=500. The greatest yearly sales they can
expect with unlimited advertising is $500,000.3. C(x)= . This means that the cost continues to
increase without bound as more units are made.4. The limit does not exist; $250.
EXERCISE 11.2 (page 531)
1. a. 2; b. 3; c. does not exist; d. –q; e. q; f. q; g. q;h. 0; i. 1; j. 1; k. 1. 3. 1. 5. –q. 7. –q.9. q. 11. 0. 13. Does not exist. 15. 0.17. q. 19. 0. 21. 1. 23. 0. 25. q.
27. 0. 29. . 31. –q. 33. . 35. –q.25
-
25
qlimxSq
limxSq
limxSq
14
119
-
15
16
-
52
≥1
0.100
0 1 1 0.1
0 0.9 0 0
0 0 0
0.9
¥
≥1
0.11
0.1
0 0 0 0
0 0.9 0 0.9
0 0 0 0
¥
≥0000
1 0.1 1
0.1
0 0 0 0
0 0.9 0
0.9
¥
c12
12d117
343
3049
≥109343117343
234343226343
¥≥19491549
30493449
¥,
30.130 0.155 0.715430.10 0.15 0.7541127
881
164
10.27
f 10 2 = 0.729, f 11 2 = 0.243, f 12 2 = 0.027,
f 16 2 =16
, f 17 2 =1
12;
f 11 2 =1
12, f 12 2 = f 13 2 = f 14 2 = f 15 2
x
f (x )
1 2 30.10.2
0.7
3313
%c23
13d
1
2 ≥5
73
7
2
74
7
¥;1 2
c0.80.3
0.2 0.7
dACompet.
A Compet.
£0.80.10.3
0.10.80.2
0.10.10.5§;
DRO
D R O
AB
c0.90.2
0.10.8d;
A B
c0.10.2
0.90.8d;
Flu No Flu
� Answers to Odd-Numbered Problems AN23
37. . 39. . 41. q. 43. q. 45. q.
47. Does not exist. 49. –q. 51. 0. 53. 1.55. a. 1; b. 2; c. does not exist; d. 1; e. 2.57. a. 0; b. 0; c. 0; d. –q; e. –q.59. 61. 20,000. 63. 20.
65. 1, 0.5, 0.525, 0.631, 0.912, 0.986, 0.998; conclude limit is 1.67. 0. 69. a. 11; b. 9; c. does not exist.
EXERCISE 11.3 (page 535)
1. $5563.87; $1563.87. 3. $1456.87. 5. 4.08%.7. 3.05%. 9. $109.42. 11. $778,800.78.13. a. $21,911; b. $6599. 15. $4.88%.17. $1264. 19. 16 years.21. Option (a): $1072.51; Option (b): $1093.30;Option (c): $1072.18.23. a. $9458.51; b. This strategy is better by $26.90.
EXERCISE 11.4 (page 543)
7. Continuous at –2 and 0. 9. Discontinuous at —3.11. Continuous at 2 and 0. 13. f is a polynomial function.15. f is a rational function and the denominator is never zero.17. None. 19. x=–4. 21. None.23. x=–5, 3. 25. x=0, —1. 27. None.29. x=0. 31. None. 33. x=2.35. Discontinuities at t=1, 2, 3, 4.
37. Yes, no, no.
PRINCIPLES IN PRACTICE 11.5
1. 0<x<4.
EXERCISE 11.5 (page 547)
1. (–q, –1), (4, q). 3. [2, 3]. 5. .
7. No solution. 9. (–q, –6], [–2, 3].11. (–q, –4), (0, 5). 13. [0, q). 15. (–3, 0), (1, q).17. (–q, –3), (0, 3). 19. (1, q).21. (–q, –5), [–2, 1), [3, q). 23. (–5, –1).25. (–q, –1- ], [–1+ , q).27. Between 50 and 150 inclusive. 29. 17 in. by 17 in.31. (–q, –7.72]. 33. (–q, –0.5), (0.667, q).
REVIEW PROBLEMS—CHAPTER 11 (page 550)
1. –5. 3. 2. 5. x. 7. . 9. 0. 11. .
13. Does not exist. 15. –1. 17. . 19. –q.
21. q. 23. –q. 25. 1. 27. –q. 29. 8.31. 23. 33. a. $5034.38; b. $1241.46. 35. 6.18%.37. 20 ln 2.41. Continuous everywhere; f is a polynomial function.43. x=–3. 45. None. 47. x=–4, 1.49. x=–2. 51. (–q, –6), (2, q).53. [2, q), x=0. 55. (–q, –5), (–1, 1).57. (–q, –4), [–3, 0], (2, q). 59. 1.00.61. 0. 63. [2.00, q).
MATHEMATICAL SNAPSHOT—CHAPTER 11 (page 553)
1. 17%.3. An exponential model assumes a fixed repayment rate.
PRINCIPLES IN PRACTICE 12.1
1. =40-32t.
EXERCISE 12.1 (page 564)
1. a.
b. We estimate that mtan=12.3. 1. 5. 4. 7. –4. 9. 0. 11. 2x+4.
13. 4q+5. 15. . 17. . 19. –4.
21. 0. 23. y=x+4. 25. y=–3x-7.
27. y= . 29. .
31. –3.000, 13.445. 33. –5.120, 0.038.35. For the x-values of the points where the tangent to thegraph of f is horizontal, the corresponding values of f¿(x)are 0. This is expected because the slope of a horizontal lineis zero and the derivative gives the slope of the tangent line.
PRINCIPLES IN PRACTICE 12.2
1. 50-0.6q.
r
rL - r -dC
dD
-3x + 9
121x + 2
-
6x2
dH
dt
19
37
-83
1313
a-
72
, -2b
x
y
5 10 15100
600
x
y
1 2 3 4 4
0.340.280.220.160.10
12
q
c
5000
6
lim c = 6q → �
-
12
115
AN24 Answers to Odd-Numbered Problems �
x-value of Q 3 2.5 2.2 2.1 2.01 2.001
mPQ 19 15.25 13.24 12.61 12.0601 12.0060
EXERCISE 12.2 (page 571)
1. 0. 3. 6x5. 5. 80x79. 7. 18x. 9. 20w4.
11. . 13. . 15. 1. 17. 8x-2.
19. 4p3-9p2. 21. –8x7+5x4.
23. –39x2+28x-2. 25. –8x3. 27. .
29. 16x3+3x2-9x+8. 31. x3+7x2. 33. .
35. . 37. or . 39. 2r–2/3.
41. –4x–5. 43. –3x–4-5x–6+12x–7.
45. –x–2 or . 47. –40x–6. 49. –4x-4.
51. . 53. . 55. –3x–2/3-2x–7/5.
57. . 59. –x–3/2. 61. .
63. 9x2-20x+7. 65. 45x4.
67. . 69. .
71. 2(x+2). 73. 1. 75. 4, 16, –14. 77. 0, 0, 0.79. y=13x+2. 81. y=–4x+6.
83. y=x+3. 85. (0, 0), . 87. (3, –3).
89. 0. 91. The tangent line is y=9x-16.
PRINCIPLES IN PRACTICE 12.3
1. 2.5 units. 2. . =0 feet/s.
When t=0.5 the object reaches its maximum height.3. 1.2 and 120%.
EXERCISE 12.3 (page 582)
1.
We estimate the velocity t=1 to be 5.0000 m/s.With differentiation the velocity is 5 m/s.3. a. 4 m; b. 5.5 m/s; c. 5 m/s.5. a. 8 m; b. 6.1208 m/s; c. 6 m/s.7. a. 2 m; b. 10.261 m/s; c. 9 m/s. 9. 0.65.
11. . 13. 0.27.
15. dc/dq=10; 10. 17. dc/dq=0.6q+2; 3.8.19. dc/dq=2q+50; 80, 82, 84.21. dc/dq=0.02q+5; 6, 7.23. dc/dq=0.00006q2-0.02q+6; 4.6, 11.25. dr/dq=0.7; 0.7, 0.7, 0.7.27. dr/dq=250+90q-3q2; 625, 850, 625.29. dc/dq=6.750-0.000656q; 3.47.31. dP/dR=–4,650,000R–1.93. 33. a. –7.5; b. 4.5.
35. a. 1; b. c. 1; d. ≠0.111; e. 11.1%.
37. a. 6x; b. c. 12; d. ≠0.632; e. 63.2%.
39. a. –3x2; b. c. –3; d. ≠–0.429;
e. –42.9%. 41. 3.2; 21.3%.
43. a. dr/dq=30-0.6q; b. ≠0.089; c. 9%.
45. . 47. $3125. 49. $5.07/unit.
PRINCIPLES IN PRACTICE 12.5
1. 6.25-6x. 2. T¿(x)=2x-x2; T¿(1)=1.
EXERCISE 12.5 (page 594)
1. (4x+1)(6)+(6x+3)(4)=48x+18=6(8x+3).3. (8-7t)(2t)+(t2-2)(–7)=14+16t-21t2.5. (3r2-4)(2r-5)+(r2-5r+1)(6r)
=12r3-45r2-2r+20.7. 8x3-10x.9. (x2+3x-2)(4x-1)+(2x2-x-3)(2x+3)
=8x3+15x2-20x-7.11. (8w2+2w-3)(15w2)+(5w3+2)(16w+2)
=200w4+40w3-45w2+32w+4.13. (x2-1)(9x2-6)+(3x3-6x+5)(2x)-4(8x+2)
=15x4-27x2-22x-2.
15.
= .
17. 0. 19. 18x2+94x+31.
21. .
23. . 25. .
27. .
29. .
= .
31.
.
33. . 35. .
37. . 39. .
41.
.=- 1x2 - 10x + 18 23 1x + 2 2 1x - 4 2 4 2
3 1x + 2 2 1x - 4 2 4 11 2 - 1x - 5 2 12x - 2 23 1x + 2 2 1x - 4 2 4 2
41x - 8 2 2 +
213x + 1 2 2
15x2 - 2x + 13x4>3
4 1v5 + 2 2v2-
100x99
1x100 + 7 2 2=
5x2 - 8x + 112x2 - 3x + 2 2 2
12x2 - 3x + 2 2 12x - 4 2 - 1x2 - 4x + 3 2 14x - 3 212x2 - 3x + 2 2 2
-38x2 - 2x + 51x2 - 5x 2 2
1x2 - 5x 2 116x - 2 2 - 18x2 - 2x + 1 2 12x - 5 21x2 - 5x 2 2
1z2 - 4 2 1-2 2 - 16 - 2z 2 12z 21z2 - 4 2 2 =
2 1z2 - 6z + 4 21z2 - 4 2 2
1x - 1 2 11 2 - 1x + 2 2 11 21x - 1 2 2 = -
31x - 1 2 2-
9x7
1x - 1 2 15 2 - 15x 2 11 21x - 1 2 2 = -
51x - 1 2 2
34112p1>2 - 5p-1>2 - 32 2
32c 1p1>2 - 4 2 14 2 + 14p - 5 2 a 1
2p-1>2 b d
dR
dx=
0.432t
445
-
37
-3x2
8 - x3;
1219
6x
3x2 + 7;
19
1x + 4
;
dy
dx=
252
x3>2; 337.50
dy
dt`t = 0.5
dy
dt= 16 - 32t
a2, -
43b
8q +4q2
13
x-2>3 -103
x-5>3 =13
x-5>3 1x - 10 2
52
x3>2-
15
x-6>5
17
- 7x-2-
12
t-2
-
1x2
1121x
112
x-1>234
x-1>4 +103
x2>3
72
x5>265
-
43
x3
12
t883
x3
� Answers to Odd-Numbered Problems AN25
≤t 1 0.5 0.2 0.1 0.01 0.001
≤s/≤t 9 6.75 5.64 5.31 5.0301 5.003001
43.
= .
45. . 47. .
49. –6. 51. . 53. y=16x+24.
55. 1.5. 57. 1 m, –1.5 m/s. 59. .
61. . 63. .
65. . 67. 0.615; 0.385. 69. a. 0.32; b. 0.026.
71. . 73. . 75. .
77. . 79. .
PRINCIPLES IN PRACTICE 12.6
1. 288t.
EXERCISE 12.6 (page 604)
1. (2u-2)(2x-1)=4x3-6x2-2x+2.
3. . 5. –2. 7. 0.
9. 18(3x+2)5. 11. –6x(5-x2)2.13. 200(3x2-16x+1)(x3-8x2+x)99.15. –6x(x2-x)–4.17. .
19. . 21. .
23. . 25. –6(4x-1)(2x2-x+1)–2.
27. –2(2x-3)(x2-3x)–3. 29. –8(8x-1)–3/2.
31. .
33. (x2)[5(x-4)4(1)]+(x-4)5(2x)=x(x-4)4(7x-8).
35.
.37. (x2+2x-1)3(5)+(5x)[3(x2+2x-1)2(2x+2)]
=5(x2+2x-1)2(7x2+8x-1).39. (8x-1)3[4(2x+1)3(2)]+(2x+1)4[3(8x-1)2(8)]
=16(8x-1)2(2x+1)3(7x+1).
41.
.
43.
.
45.
.
47.
.
49. 6{(5x2+2)[2x3(x4+5)–1/2]+(x4+5)1/2(10x)}=12x(x4+5)–1/2(10x4+2x2+25).
51. 8+ .
53. .
55. 0. 57. 0. 59. y=4x-11.
61. . 63. 96%. 65. 20. 67. 13.99.
69. a. ; b. ;
c. .
71. –325. 73. . 75. 48�(10)–19.
77. a. –0.001424x‹+0.01338x¤+1.692x-34.8; –22.986;b. –0.001. 79. –4. 81. 40. 83. 86,111.22.
REVIEW PROBLEMS—CHAPTER 12 (page 608)
1. –2x. 3. . 5. 0.
7. 28x3-18x2+10x=2x(14x2-9x+5).
9. 4s3+4s=4s(s2+1). 11. .
13. (x2+6x)(3x2-12x)+(x3-6x2+4)(2x+6)=5x4-108x2+8x+24.
15. 100(2x2+4x)99(4x+4)=400(x+1)[(2x)(x+2)]99.
17. .
19. (8+2x)(4)(x2+1)3(2x)+(x2+1)4(2)=2(x2+1)3(9x2+32x+1).
21. .
23. .
25. .
27. (x-6)4[3(x+5)2]+(x+5)3[4(x-6)3]=(x-6)3(x+5)2(7x+2).
29. .
31.
= .-
3411 + 2-11>8 2x-11>8
2 a -
38bx-11>8 + a -
38b 12x 2-11>8 12 2
1x + 6 2 15 2 - 15x - 4 2 11 21x + 6 2 2 =
341x + 6 2 2
-
1211 - x 2-3>2 1-1 2 =
1211 - x 2-3>2
4314x - 1 2-2>3
1z2 + 4 2 12z 2 - 1z2 - 1 2 12z 21z2 + 4 2 2 =
10z
1z2 + 4 2 2
-
612x + 1 2 2
2x
5
1321x
dc
dq=
5q 1q2 + 6 21q2 + 3 2 3>2
100 -q22q2 + 20
- 2q2 + 20
-
q
1002q2 + 20 - q2 - 20-
q2q2 + 20
y = -
16
x +53
1x2 - 7 2 4 3 12x + 1 2 12 2 13x - 5 2 13 2 + 13x - 5 2 2 12 2 4 - 12x + 1 2 13x - 5 2 2 34 1x2 - 7 2 3 12x 2 4
1x2 - 7 2 8
51 t + 4 2 2 - 18t - 7 2 = 15 - 8t +
51 t + 4 2 2
=18x - 1 2 4 148x - 31 2
13x - 1 2 4
13x - 1 2 3 340 18x - 1 2 4 4 - 18x - 1 2 5 39 13x - 1 2 2 413x - 1 2 6
=-2 15x2 - 15x - 4 2
1x2 + 4 2 4
1x2 + 4 2 3 12 2 - 12x - 5 2 33 1x2 + 4 2 2 12x 2 41x2 + 4 2 6
=5
2 1x + 3 2 2 ax - 2x + 3
b -1>2
12a x - 2
x + 3b -1>2 c 1x + 3 2 11 2 - 1x - 2 2 11 2
1x + 3 2 2 d=
110 1x - 7 2 91x + 4 2 11
10 a x - 7x + 4
b 9 c 1x + 4 2 11 2 - 1x - 7 2 11 21x + 4 2 2 d
= 6x 16x - 1 2-1>2 + 216x - 1
12x 2 c 1216x - 1 2-1>2 16 2 d + 116x - 1 2 12 2
7317x 2-2>3 + 317
125
x2 1x3 + 1 2-3>5
1212x - 1 2-3>41
2110x - 1 2 15x2 - x 2-1>2
-10 14x - 3 2 12x2 - 3x - 1 2-13>3
a -
2w3 b 1-1 2 =
212 - x 2 3
6x2 + 2x - 13-
1120
0.735511 + 0.02744x 2 2
910
dc
dq=
5q 1q + 6 21q + 3 2 2
14
; 34
dC
dI= 0.672
dr
dq=
2161q + 2 2 2 - 3
dr
dq= 25 - 0.04q
y = -32
x +152
-2a
1a + x 2 23 -2x3 + 3x2 - 12x + 43x 1x - 1 2 1x - 2 2 4 2
-3t6 - 12t5 + t4 + 6t3 - 21t2 - 14t - 213 1 t2 - 1 2 1 t3 + 7 2 4 2
3 1 t2 - 1 2 1 t3 + 7 2 4 12t + 3 2 - 1 t2 + 3t 2 15t4 - 3t2 + 14t 23 1 t2 - 1 2 1 t3 + 7 2 4 2
AN26 Answers to Odd-Numbered Problems �
33. .
.
35. .
.
37. 7(1-2z). 39. y=–4x+3.
41. . 43. ≠0.714; 71.4%.
45. dr/dq=20-0.2q. 47. 0.569, 0.431.49. dr/dq=450-q.51. dc/dq=0.125+0.00878q; 0.7396.
53. 84 eggs/mm. 55. a. ; b. . 57. 8∏ ft3/ft.
59. 4q- . 61. a. 240; b. ;
c. no, since dr/dm<300 when m=80. 63. 0.305.65. –0.32.
MATHEMATICAL SNAPSHOT—CHAPTER 12 (page 612)
1. The slope is greater—above 0.9. More is spent; less is saved.3. Spend $705, save $295. 5. Answers may vary.
PRINCIPLES IN PRACTICE 13.1
1. . 2. .
EXERCISE 13.1 (page 618)
1. . 3. . 5. . 7. .
9.
11. (ln t)=1+ln t.
13. +2x ln(4x+3). 15. .
17. .
19. .
21. .
23. . 25. .
27. . 29. . 31. .
33. . 35. .
37. . 39. .
41. . 43. .
45. y=4x-12. 47. . 49. .
51. . 53. .
57. 1.36.
PRINCIPLES IN PRACTICE 13.2
1. .
EXERCISE 13.2 (page 623)
1. 7ex. 3. . 5. –5e9-5x.7. (6r+4) =2(3r+2) .9. x(ex)+ex(1)=ex(x+1). 11. (1-x2).
13. . 15. (6x) ln 4. 17. .
19. . 21. 5x4-5x ln 5. 23. .
25. 1. 27. (1+ln x)ex ln x. 29. –e.31. y-e–2=e–2(x+2) or y=e–2x+3e–2.33. dp/dq=–0.015e–0.001q, –0.015e–0.5.35. dc/dq=10eq/700; 10e0.5; 10e. 37. –5.39. e. 41. 100e–2. 47. –b(10A-bM) ln 10.51. 0.0036. 53. 0.68.
PRINCIPLES IN PRACTICE 13.3
1. .
2. =4∏r¤ and =2880∏ inches/minute
3. The top of the ladder is sliding down at a rate of
feet/second.
EXERCISE 13.3 (page 630)
1. . 3. . 5. . 7. . 9. .
11. . 13. . 15. .
17. . 19. . 21. .
23. 6e3x(1+e3x)(x+y)-1. 25. .
27. 0; . 29. . 31. .
33. . 35. –ÒI. 37. 1.5E ln 10.
39. . 41. .
EXERCISE 13.4 (page 634)
1. (x+1)2(x-2)(x2+3) .c 2x + 1
+1
x - 2+
2x
x2 + 3d
38
-
V
0.4T= -2.5
V
T
dq
dp= -
1q + 5 2 340
dq
dp= -
12q
y = -
34
x +54
-
4x0
9y0
-
35
-
ey
xey + 1xey - y
x 1 ln x - xey 21 - 6xy3
1 + 9x2y2
6y2>33y1>6 + 2
4y - 2x2
y2 - 4x
11 - y
x - 1
-
y
x-
y1>4x1>4-
1y1x
712y3-
x
4y
94
dV
dt`
r = 12
dr
dt
dV
dt
dP
dt= 0.5 1P - P2 2
2ex
1ex + 1 2 2e1 +1x
21x
2e2w 1w - 1 2w343x2ex - e-x
3
2xe-x2e3r2 + 4r + 4e3r2 + 4r + 4
2xex2 + 4
dT
dt= Ckekt
6a
1T - a2 + aT 2 1a - T 2dq
dp=
202p + 1
257
ln 13 2 - 1ln2 3
3
2x 14 + 3 ln x
x
2 1x - 1 2 + ln 1x - 1
4 ln3 1ax 2x
3 11 + ln2 x 2x
2 1x2 + 1 22x + 1
+ 2x ln 12x + 1 25x
+5
2x + 1
4x
x2 + 2+
3x2 + 1x3 + x - 1
x
1 - x4
21 - t2
9x
1 + x2
3 12x + 4 2x2 + 4x + 5
=6 1x + 2 2
x2 + 4x + 5
1 ln x 2 12x 2 - 1x2 - 1 2 a 1xb
1 ln x 2 2 =2x2 ln 1x 2 - x2 + 1
x ln2 x
z a 1zb - 1 ln z 2 11 2
z2 =1 - ln z
z2
2x c1 +1
1 ln 2 2 1x2 + 4 2 d
81 ln 3 2 18x - 1 2
4x2
4x + 3
t a 1tb +
6p2 + 32p3 + 3p
=3 12p2 + 1 2p 12p2 + 3 2 .
-2x
1 - x2
2x
33x - 7
4x
dR
dI=
1I ln 10
dq
dp=
12p
3p2 + 4
1100
10,000q2
124
43
57
y =112
x +43
=95
x 1x + 4 2 1x3 + 6x2 + 9 2-2>5
a 35b 1x3 + 6x2 + 9 2-2>5 13x2 + 12x 2
=x 1x2 + 4 21x2 + 5 2 3>2
2x2 + 5 12x 2 - 1x2 + 6 2 11>2 2 1x2 + 5 2-1>2 12x 2x2 + 5
� Answers to Odd-Numbered Problems AN27
3. .
5.
.
7. .
9. .
11. .
13. . 15. .
17. .
19. 4exx3x(4+3 ln x). 21. 12. 23. y=96x+36.
25. y=6ex-3e. 27. .
PRINCIPLES IN PRACTICE 13.5
1. feet/sec2 (Note: Negative values indicate
the downward direction.).2. c¿¿ (3)=14 dollars/unit2.
EXERCISE 13.5 (page 638)
1. 24. 3. 0. 5. ex. 7. 3+2 ln x. 9. .
11. . 13. . 15. .
17. . 19. ez(z2+4z+2).
21. 32. 23. . 25. . 27. .
29. . 31. . 33. .
35. 300(5x-3)2. 37. 0.6. 39. —1.41. –4.99 and 1.94.
REVIEW PROBLEMS—CHAPTER 13 (page 640)
1. 2ex+ (2x)=2(ex+x ).
3. .
5.(2x+4)=2(x+2) .7. ex(2x)+(x2+2)ex=ex(x2+2x+2).
9. .
11. = .
13. . 15. –7(ln 10)2-7x.
17. . 19. .
21. . 23. (x+1)x+1[1+ln(x+1)].
25. .
27.
,
where y is as given in the problem.29. (xx)x(x+2x ln x). 31. 4. 33. –2.35. y=6x+6(1-ln 2) or y=6x+6-ln 64.
37. (0, 4 ln 2). 39. 18. 41. 2. 43. .
45. . 47. .
49. .
51. f¿(t)=0.008e–0.01t+0.00004e–0.0002t. 53. 0.90.
MATHEMATICAL SNAPSHOT—CHAPTER 13 (page 643)
1. Figure 13.5 shows that the population reaches its finalsize in about 45 days.3. The tangent line will not coincide exactly with the curvein the first place. Smaller time steps could reduce the error.
PRINCIPLES IN PRACTICE 14.1
1. There is a relative maximum when x=1, and a relativeminimum when x=3.2. The drug is at its greatest concentration 2 hours after injection.
EXERCISE 14.1 (page 655)
1. Dec. on (–q, –1) and (3, q); inc. on (–1, 3); rel. min. (–1, –1); rel. max. (3, 4).3. Dec. on (–q, –2) and (0, 2); inc. on (–2, 0) and (2, q);rel. min. (–2, 1) and (2, 1); no rel. max.5. Inc. on (–q, –1) and (3, q); dec. on (–1, 3); rel max. when x=–1; rel. min. when x=3.7. Dec. on (–q, –1); inc. on (–1, 3) and (3, q); rel. min. when x=–1.9. Inc. on (–q, 0) and (0, q); no rel. min. or max.
11. Inc. on ; dec. on
rel. max. when x=
13. Dec. on (–q, –5) and (1, q); inc. on (–5, 1);rel. min. when x=–5; rel. max. when x=1.15. Dec. on (–q, –1) and (0, 1); inc. on (–1, 0) and (1, q);rel. max. when x=0; rel. min. when x=—1.
12
.
a 12
, q b ;a - q, 12b
dy
dx=
y + 1y
; d2y
dx2 = -y + 1
y3
49
xy2 - y
2x - x2y
-
y
x + y
= y c 3x
x2 + 2+
8x
9 1x2 + 9 2 -12 1x2 + 2 2
11 1x3 + 6x 2 d-
411a 1
x3 + 6xb 13x2 + 6 2 d
y c 32a 1
x2 + 2b 12x 2 +
49a 1
x2 + 9b 12x 2
2 a 1tb +
12a 1
2 - tb 1-1 2 =
5t - 82t 1 t - 2 2
1 + 2l + 3l2
1 + l + l2 + l3
1618x + 5 2 ln 2
4e2x + 1 12x - 1 2x2
2q + 1
+3
q + 2
1 - x ln xxex
ex a 1xb - 1 ln x 2 1ex 2
e2x
1 1x - 6 2 1x + 5 2 19 - x 22
c 1x - 6
+1
x + 5+
1x - 9
d
ex2 + 4x + 5ex2 + 4x + 5
1r2 + 5r
12r + 5 2 =2r + 5
r 1r + 5 2ex2
ex2
-
16125
y
11 - y 2 32 1y - 1 211 + x 2 2
18x3>2-
4y3-
1y3
- c 1x2 +
11x + 6 2 2 d
41x - 1 2 3
5015x - 6 2 3-
14 19 - r 2 3>2
-
10p6
d2h
dt2 = -32
13e13
2 13x + 1 2 2x c 3x
3x + 1+ ln 13x + 1 2 d
x1>x 11 - ln x 2x2x2x + 1 a 2x + 1
x+ 2 ln x b
12A 1x - 1 2 1x + 1 2
3x - 4c 1x - 1
+1
x + 1-
33x - 4
d
12x2 + 2 2 21x + 1 2 2 13x + 2 2 c
4x
x2 + 1-
2x + 1
-3
3x + 2d
21 - x2
1 - 2xc x
x2 - 1+
21 - 2x
dc 1x + 1
+2x
x2 - 2+
1x + 4
d
2x + 1 2x2 - 2 2x + 42
#
13x2 - 1 2 2 12x + 5 2 3 c 18x2
3x3 - 1+
62x + 5
dAN28 Answers to Odd-Numbered Problems �
17. Inc. on (–q, 1) and (3, q); dec. on (1, 3);rel. max. when x=1; rel. min. when x=3.
19. Inc. on and ; dec. on ;
rel. max. when x= ; rel. min. when x= .
21. Inc. on and ;
dec. on ; rel. max. when
x= ; rel. min. when x= .
23. Inc. on (–q, –1) and (1, q); dec. on (–1, 0) and (0, 1);rel. max. when x=–1; rel. min. when x=1.25. Dec. on (–q, –4) and (0, q); inc. on (–4, 0); rel. min. when x=–4; rel. max. when x=0.27. Inc. on (–q, ) and (0, ); dec. on ( , 0) and ( , q); rel. max. when x=— ; rel. min. when x=0.29. Inc. on (–q, –1), (–1, 0), and (0, q); never dec.;no rel. extremum.31. Dec. on (–q, 1) and (1, q); no rel. extremum.33. Dec. on (0, q); no rel. extremum.35. Dec. on (–q, 0) and (4, q); inc. on (0, 2) and (2, 4);rel. min. when x=0; rel. max. when x=4.37. Inc. on (–q, –3) and (–1, q); dec. on (–3, –2) and(–2, –1); rel. max. when x=–3; rel. min. when x=–1.
39. Dec. on and ;
inc. on ; rel. min. when
x= ; rel. max. when x=
41. Inc. on (–q, –2), , and (5, q); dec. on
; rel. max. when x= ; rel. min. when x=5.
43. Inc. on (–q, 0), , and (6, q); dec. on ;
rel. max. when x= ; rel. min. when x=6.
45. Dec. on (–q, q); no rel. extremum.
47. Dec. on ; inc. on ;
rel. min. when x= .
49. Dec. on (–q, 0); inc. on (0, q); rel. min. when x=0.51. Dec. on (0, 1); inc. on (1, q); rel. min. when x=1;no rel. max.
53. Dec. on (–q, 3); inc. on (3, q); rel. min. when x=3;intercepts: (7, 0), (–1, 0), (0, –7).
55. Dec. on (–q, –1) and (1, q); inc. on (–1, 1);rel. min. when x=–1; rel. max. when x=1;sym. about origin; intercepts: (— , 0), (0, 0).
57. Inc. on (–q, 1) and (2, q); dec. on (1, 2);rel. max. when x=1; rel. min. when x=2; intercept: (0, 0).
59. Inc. on (–2, –1) and (0, q); dec. on (–q, –2) and (–1, 0); rel. max. when x=–1; rel. min. when x=–2, 0;intercepts: (0, 0), (–2, 0).
x
y
–1– 2
1
x
y
1 2
54
x
y
1–1
2
–2
13
x
y
3–1 7
–16
–7
3122
a 3122
, q ba0, 312
2b
187
a187
, 6ba0, 187b
115
a 115
, 5 ba -2,
115b
-2 + 1295
.-2 - 129
5
a -2 - 1295
, -2 + 129
5ba -2 + 129
5, q ba - q,
-2 - 1295
b
1212-1212-12
-2 + 173
-2 - 173
a -2 - 173
, -2 + 17
3ba -2 + 17
3, q ba - q,
-2 - 173
b
52
-23
a -23
, 52ba 5
2, q ba - q, -
23b
� Answers to Odd-Numbered Problems AN29
61. Dec. on (–q, –2) and ; inc. on
and (1, q); rel. min. when x=–2, 1;
rel. max. when x= ; intercepts: (1, 0), (–2, 0), (0, 4).
63. Dec. on (1, q); inc. on (0, 1); rel. max. when x=1;intercepts: (0, 0), (4, 0).
65. 69. Never.
71. 40. 75. a. 25,300; b. 4; c. 17,200.77. Rel. min.: (–4.10, –2.21).79. Rel. max.: (2.74, 3.74); rel. min.: (–2.74, –3.74).81. Rel. min.: 0, 1.50, 2.00; rel. max.: 0.57, 1.77.83. a. f¿(x)=4-6x-3x2
c. Dec.: (–q, –2.53), (0.53, q); inc.: (–2.53, 0.53).
EXERCISE 14.2 (page 660)
1. Maximum: f(3)=6; minimum: f(1)=2.
3. Maximum: f(0)=1; minimum: f(2)= .
5. Maximum: f(3)=84; minimum: f(1)=–8.7. Maximum: f(–2)=56; minimum: f(–1)=–2.9. Maximum: f( )=4; minimum f(2)=–16.11. Maximum: f(0)=f(3)=2;
minimum: .
13. Maximum: f(3)≠2.08; minimum: f(0)=0.15. a. –3.22, –0.78; b. 2.75; c. 9; d. 14,283.
EXERCISE 14.3 (page 666)
1. Conc. up (–q, 0), ; conc. down ;
inf. pt. when x= .
3. Conc. up (– ; conc down (7, q);inf. pt. when x=7.5. Conc. up (–q, – ), ( , q); conc down (– , );no inf. pt.7. Conc. down. (–q, q).9. Conc down. (–q, –1); conc. up (–1, q); inf. pt. when x=–1.
11. Conc. down. ; conc. up ;
inf. pt. when x= .
13. Conc. up (–q, –1), (1, q); conc. down (–1, 1); inf. pt. when x=—1.15. Conc. up (–q, 0); conc. down (0, q);inf. pt. when x=0.
17. Conc. up , ; conc. down ;
inf. pt. when x= .
19. Conc. down ;
conc. up ;
inf. pt. when x=0, .
21. Conc. up (–q, – ), ; conc. down (– , – ), ;inf. pt. when x=— , — .23. Conc. down (–q, 1); conc. up (1, q).25. Conc. down. (–q, – ), ( , q);conc. up (– , ); inf. pt. when x=— .
27. Conc. down. (–q, –3), ; conc. up ;
inf. pt. when x= .
29. Conc. up. (–q, q).31. Conc. down (–q, –2); conc. up (–2, q);inf. pt. when x=–2.33. Conc. down (0, e3/2); conc. up (e3/2, q); inf. pt. when x=e3/2.35. Int. (–3, 0), (–1, 0), (0, 3); dec. (–q, –2);inc. (–2, q); rel. min. when x=–2; conc. up (–q, q).
x
y
27
a 27
, q ba-3, 27b
1>131>131>131>131>13
1215112, 15 212151-12, 12 2 , 115, q 215
3 ; 152
a0, 3 - 15
2b , a 3 + 15
2, q b
1- q, 0 2 , a 3 - 152
, 3 + 15
2b
-72
, 13
a -72
, 13ba 1
3, q ba - q, -
72b
74
a 74
, q ba - q, 74b
12121212
q, 1 2 , 11, 7 20,
32
a0, 32ba 3
2, q b
f a 3122b = -
734
12
-193
x
y
1 3
2
1
x
y
1 4
1
x
y
1– 2
4
-12
a -2, -12ba -
12
, 1 bAN30 Answers to Odd-Numbered Problems �
37. Int. (0, 0), (4, 0); inc. (–q, 2); dec. (2, q); rel. max. when x=2; conc. down (–q, q).
39. Int. (0, –19); inc. (–q, 2), (4, q); dec. (2, 4);rel. max. when x=2; rel. min. when x=4;conc. down (–q, 3); conc. up (3, q); inf. pt. when x=3.
41. Int. (0, 0), (— , 0); inc. (–q, –2), (2, q); dec. (–2, 2); rel. max. when x=–2; rel. min. when x=2; conc. down (–q, 0); conc. up (0, q); inf. pt. when x=0;sym. about origin.
43. Int. (0, –3); inc. (–q, 1), (1, q); no rel. max. or min.;conc. down (–q, 1); conc. up (1, q); inf. pt. when x=1.
45. Int. (0, 0), ; inc. (–q, 0), (0, 1); dec. (1, q); rel. max. when x=1; conc. up ; conc. down (–q, 0),
; inf. pt. when x=0, x=2/3.
47. Int. (0, –2); dec. (–q, –2), (2, q); inc. (–2, 2);rel. min. when x=–2; rel. max. when x=2;conc. up (–q, 0); conc. down (0, q); inf. pt. when x=0.
49. Int. (0, –6); inc. (–q, 2), (2, q); conc. down (–q, 2);conc. up (2, q); inf. pt. when x=2.
51. Int. (0, 0), ; dec. (–q, –1), (1, q); inc. (–1, 1); rel. min. when x=–1; rel. max. when x=1;conc. up (–q, 0); conc. down (0, q); inf. pt. when x=0;sym. about origin.
x
y
1; 415, 0 2x
y
x
y
x
y
12>3, q 2 10, 2>3 214>3, 0 2
x
y
x
y
213
x
y
x
y
� Answers to Odd-Numbered Problems AN31
53. Int. (0, 1), (1, 0); dec. (–q, 0), (0, 1); inc. (1, q);
rel. min. when x=1; conc. up (–q, 0), (2/3, q);
conc. down ; inf. pt. when x=0, x=2/3.
55. Int. (0, 0), (—2, 0); inc. (–q, – ), (0, ); dec. (– , 0), ( , q); rel. max. when x=— ;rel. min. when x=0; conc. down (–q, – ), ( , q); conc. up (– , ); inf. pt. when x=— ; sym. about y-axis.//
57. Int. (0, 0), (8, 0); dec. (–q, 0), (0, 2); inc. (2, q);rel. min. when x=2; conc. up (–q, –4), (0, q); conc. down (–4, 0); inf. pt. when x=–4, x=0.
59. Int. (0, 0), (–4, 0); dec. (–q, –1); inc. (–1, 0), (0, q);rel. min. when x=–1; conc. up (–q, 0), (2, q); conc. down (0, 2); inf. pt. when x=0, x=2.
61. Int. (0, 0), ; inc. (–q, –1), (0, q);
dec. (–1, 0); rel. min. when x=0; rel. max. when x=–1;conc. down (–q, 0), (0, q).
63. 65.
69.
73. b. c. 0.26.
75. Two. 77. Above tangent line; concave up.79. –2.61, –0.26.
EXERCISE 14.4 (page 670)
1. Rel. min. when x= ; abs. min.
3. Rel. max. when x= ; abs. max.
5. Rel. max. when x=–3; rel. min. when x=3.7. Rel. min. when x=0; rel. max. when x=2.9. Test fails, when x=0 there is a rel. min. by first-deriv. test.
11. Rel. max. when x= ; rel. min. when x= .
13. Rel. min. when x=–5, –2; rel. max. when x= .-72
13
-13
14
52
6.2
r
f (r )
1 10
60
A
S
625
x
y
1
1
x
y
2
4
1
x
y
–1– 278
a -278
, 0 b
x
y
2–1
–4–3
6 3 2
x
y
– 4 2 8
12 3 4
– 6 3 2
x
y
12>3 12>312>312>3 12>31212121212
x
y
10, 2>3 2
AN32 Answers to Odd-Numbered Problems �
EXERCISE 14.5 (page 678)
1. y=1, x=1. 3. y= , x= .
5. y=0, x=0. 7. y=0, x=1, x=–1.9. None. 11. y=2, x=2, x=–3.13. y=–7, x=–2 , x=2 . 15. y=7, x=6.
17. x=0, x=–1. 19. y= , x= .
21. y= , x= . 23. y=4.
25. Dec. (–q, 0), (0, q); conc. down (–q, 0); conc. up (0, q); sym. about origin; asymptotes x=0, y=0.
27. Int. (0, 0); inc. (–q, –1), (–1, q); conc. up (–q, –1);conc. down (–1, q); asymptotes x=–1, y=1.
29. Dec. (–q, –1), (0, 1); inc. (–1, 0), (1, q); rel. min. when x=—1; conc. up (–q, 0), (0, q); sym. about y-axis; asymptote x=0.
31. Int. (0, –1); inc. (–q, –1), (–1, 0); dec. (0, 1), (1, q);rel. max. when x=0; conc. up (–q, –1), (1, q); conc. down (–1, 1); asymptotes x=1, x=–1, y=0; sym. about y-axis.
33. Int. (–1, 0), (0, 1); inc. (–q, 1), (1, q); conc. up (–q, 1); conc. down (1, q); asymptotes x=1, y=–1.
35. Int. (0, 0); inc. , (0, q); dec. ,
; rel. max. when x= ; rel. min. when x=0;
conc. down ; conc. up ;
asymptote x= .
37. Int. ; inc. dec.
; rel. max. when x= ; conc. up
; conc. down ;
asymptotes y=0, x= .
x
y
23
43–
, –113( )
-23
, x =43
a -23
, 43ba - q, -
23b , a 4
3, q b
13
a 13
, 43b , a 4
3, q b
a - q, -23b , a -
23
, 13b ; a0, -
98b
–16/49x
y
87
–
–x = 47
-47
a -47
, q ba - q, -47b
-87
a -47
, 0 ba -
87
, -47ba - q, -
87b
x
y
1
–1
x
y
1–1 –1
x
y
–1 1
2
x
y
1
–1
x
y
-43
12
-12
14
1212
-32
12
� Answers to Odd-Numbered Problems AN33
39. Int. ; dec.
inc. rel. min. when x= ;
conc. down ; conc. up ;
inf. pt. when x= ; asymptotes x= , y=0.
41. Int. (–1, 0), (1, 0); inc. (– , 0), (0, ); dec. (–q, – ), ( , q); rel. max. when x= ; rel. min. when x=– ; conc. down (–q, – ), (0, ); conc. up (– , 0), ( , q); inf. pt. when x=— ; asymptotes x=0, y=0; sym. about origin.
43. Int. (0, 1); inc. (–q, –2), (0, q); dec. (–2, –1), (–1, 0); rel. max. when x=–2; rel. min when x=0;conc. down (–q, –1); conc. up (–1, q); asymptote x=–1.
45. Int. (0, 5); dec. ; inc. ,
(1, q); rel. min. when x= ; conc. down ,
(1, q); conc. up ; asymptotes x= , x=1,
y=–1.
47.
49.
55. x≠—2.45, x≠0.67, y=2. 57. y≠0.48.
REVIEW PROBLEMS—CHAPTER 14 (page 681)
1. y=3, x=4, x=–4. 3. y= , x= .
5. x=0, 4. 7. x= , –1.
9. Inc. (1, 3); dec. on (–q, 1) and (3, q).11. Dec. on (–q, – ), (0, ), ( , );inc. on (– , – ), (– , 0), ( , q).
13. Conc. up on (–q, 0) and ;
conc. down on .a0, 12b
a 12
, q b16131316
16131316
-158
-23
59
x
y
–1 2
x
y
1
2
x
y
1
–1
13
–
, ( )13
72
-13
a -13
, 1 ba - q, -
13b1
3
a 13
, 1 ba - q, -13b , a -
13
, 13b
x
y
–1
–3
x
y
3
3–
161616161613
1313131313
x
y
, ( )92
127
92
, (— )32
124—
——
92
-92
a -92
, 92b , a 9
2, q ba - q, -
92b
-32
a -32
, 92b ;
a - q, -32b , a 9
2, q b ;a 3
2, 0 b , a0,-
127b
AN34 Answers to Odd-Numbered Problems �
15. Conc. down on ; conc. up on .
17. Conc. up on ;
conc. down on .
19. Rel. max. at x=1; rel. min. at x=2.21. Rel. min. at x=–1.
23. Rel. max. at x= ; rel. min. at x=0.
25. At x=3. 27. At x=1. 29. At x=2_ .31. Maximum: f(2)=16; minimum: f(1)=–1.
33. Maximum: f(0)=0; minimum: .
35. a. f has no relative extrema;b. f is conc. down on (1, 3); inf. pts.: (1, 2e–1), (3, 10e–3).37. Int. (–4, 0), (6, 0), (0, –24); inc. (1, q); dec. (–q, 1);rel. min. when x=1; conc. up (–q, q).
39. Int. (0, 20); inc. (–q, –2), (2, q); dec. (–2, 2);rel. max. when x=–2; rel. min. when x=2;conc. up (0, q); conc. down (–q, 0); inf. pt. when x=0.
41. Int. (0, 0); inc. (–q, q); conc. down (–q, 0); conc. up (0, q); inf. pt. when x=0; sym. about origin.
43. Int. (–5, 0); inc. (–10, 0); dec. (–q, –10), (0, q); rel. min. when x=–10; conc. up (–15, 0), (0, q);conc. down (–q, –15); inf. pt. when x=–15;horiz. asym. y=0; vert. asym. x=0.
45. Int. (0, 0); inc. ; dec. , ;
rel. max. when x= ; conc. up ;
conc. down ; inf. pt. when x= ;
horiz. asym. y=0; vert. asym. x= .
47. Int. (0, 1); inc. (0, q); dec. (–q, 0); rel. min. when x=0; conc. up (–q, q); sym. about y-axis.
49. a. False; b. false; c. true; d. false; e. false.51. q>2.57. Rel. max. (–1.32, 12.28); rel. min. (0.44, 1.29).59. x=–0.60.
MATHEMATICAL SNAPSHOT—CHAPTER 14 (page 685)
1. The data for 1998–2000 fall into the same pattern as the1959–1969 data.
EXERCISE 15.1 (page 696)
1. 13 and 13. 3. 300 ft by 250 ft. 5. 100 units.
7. $15. 9. a. 110 grams; b. 51 grams.
11. 525 units; price=$51; profit=$10,525. 13. $22.15. 120 units; $86,000. 17. 625 units; $4.
911
x
f (x )
1
x
y
12
– , ( )14
227
– , ( )12
116
12
-12
a -12
, 12b
a - q, -12b , a 1
2, q b-
14
a 12
, q ba -14
, 12ba - q, -
14b
x
f (x)
x
y
x
y
(2, 4)
(–2, 36)
x
y
(1, – 25)
f a -65b = -
1120
12
-25
a -54
, -14b
a - q, -54b , a -
14
, q ba1
2, qba - q,
12b
� Answers to Odd-Numbered Problems AN35
19. $17; $86,700. 21. 4 ft by 4 ft by 2 ft.23. 2 in.; 128 in3.27. 130 units, p=$340, P=$36,980; 125 units, p=$350,P=$34,175. 29. 250 per lot (4 lots). 31. 35.33. 60 mi/h. 35. 7; $1000.37. 5- tons; 5- tons. 41. 10 cases; $50.55.
EXERCISE 15.2 (page 705)
1. 3 dx. 3. dx. 5.
7. 9. 3 +3(12x2+4x+3) dx.
11. ≤y=–0.14, dy=–0.14.13. ≤y=–2.5, dy=–2.75.
15. ≤y≠0.073, dy= =0.075. 17. a. –1; b. 2.9.
19. 9.95. 21. . 23. –0.03. 25. 1.01.
27. . 29. . 31. –p2. 33. .
35. . 37. 44; 41.80. 39. 2.04. 41. 0.7.
43. (1.69*10–11)p cm3. 45. c. 42 units.
EXERCISE 15.3 (page 711)
1. –3, elastic. 3. –1, unit elasticity.
5. , elastic. 7. , elastic.
9. –1, unit elasticity. 11. , inelastic.
13. , inelastic.
15. |Ó|= when p=10, |Ó|= when p=3, |Ó|=1
when p=6.50. 17. –1.2, 0.6% decrease.23. b. Ó=–2.5, elastic; c. 1 unit;d. increase, since demand is elastic.
25. a. Ó= ≠–13.8, elastic; b. 27.6%; c. Since
demand is elastic, lowering the price results in an increase
in revenue. 27. Ó=–1.6; .
29. Maximum at q=5; minimum at q=95.
PRINCIPLES IN PRACTICE 15.4
1. 43 and 1958.
EXERCISE 15.4 (page 716)
1. 0.25410. 3. 1.32472. 5. –2.38769. 7. 0.33767.9. 1.90785. 11. 4.141. 13. –4.99 and 1.94.15. 13.33. 17. 2.880. 19. 3.45.
REVIEW PROBLEMS—CHAPTER 15 (page 718)
1. 20. 3. 300. 5. $2800. 7. 200 ft by 100 ft.9. a. 200, $120; b. 300.
11. dx. 13.�.
15. 0.99. 17. . 19. elastic. 21. a. –1.
23. a. <p<100;
b. Ó= ; demand decreases by approximately 1.67%.
25. 0.619 and 1.512.
MATHEMATICAL SNAPSHOT—CHAPTER 15 (page 721)
1. F=$40, V=$20; yes. 3. No difference.
PRINCIPLES IN PRACTICE 16.1
1.
2.
3.
4.
5. S(t)=0.7t3-32.7t2+491.6t+C.
EXERCISE 16.1 (page 730)
1. 5x+C. 3. . 5. .
7. . 9. . 11. .
13. . 15. .
17. (7+e)x+C. 19. .
21. 6ex+C. 23.
25. . 27. .
29. . 31. .
33. . 35. .
37. .
39. .
41. . 43. .
45. . 47. .
49. . 51. x+ex+C.
53. No, F(x)-G(x) must be a constant.
55. .12x2 + 1
+ C
z3
6+
5z2
2+ C
2v3
3+ 3v +
12v4 + C
4u3
3+ 2u2 + u + C
2x5>25
+ 2x3>2 + Cx4
4- x3 +
5x2
2- 15x + C
-3x5>325
- 7x1>2 + 3x2 + C
4x3>23
-12x5>4
5+ C
xe + 1
e + 1+ 10ex + C
171z2 - 5z 2 + C
w3
2+
23w
+ Cx4
12+
32x2 + C
2 81x + C-4x3>2
9+ C
x9.3
9.3-
9x7
7-
1x3 -
12x2 + C.
x2
14-
3x5
20+ C
t3 - 2t2 + 5t + Cy6
6-
5y2
2+ C
8u +u2
2+ C-
56y6>5 + C-
29x9 + C
-5
6x6 + Cx9
9+ C
1500 + 3001t 2dt = 500t + 200t3>2 + C.3
-480t3 dt =
240t2 + C.3
0.12t2 dt = 0.04t3 + C.3
28.3 dq = 28.3q + C.3
-13
2003
18y + 7
a 910bc x2
x + 5+ 2x ln 1x + 5 2 d
dr
dq= 30
-20715
310
103
-12
-932
- a 150e
- 1 b-
5352
-45
133
16p 1p2 + 5 2 2
12
4132
340
e2x22x
x2 + 7 dx.
-2x3 dx.
2x32x4 + 6
1313
AN36 Answers to Odd-Numbered Problems �
PRINCIPLES IN PRACTICE 16.2
1. N(t)=800t+200et+6317.37.2. y(t)=14t3+12t2+11t+3.
EXERCISE 16.2 (page 735)
1. . 3. 18.
5. .
7. . 9. p=0.7.
11. p=275-0.5q-0.1q2. 13. c=1.35q+200.
15. 7715. 17. .
21. $80 (dc/dq=27.50 when q=50 is not relevant toproblem).
PRINCIPLES IN PRACTICE 16.3
1. T(t)=10e–0.5t+C. 2. 35 lnœt+1œ+C.
EXERCISE 16.3 (page 743)
1. . 3. .
5. . 7. .
9. . 11. .
13. . 15. .
17. e3x+C. 19. +C. 21. +C.
23. +C. 25. ln |x+5|+C.
27. ln |x3+x4| +C. 29. .
31. 4 ln |x|+C. 33. ln |s3+5|+C.
35. ln |5-3x|+C.
37. .
39. . 41. +1+C.
43. +1+C. 45. .
47. . 49. ln |x3+6x|+C.
51. 2 ln |3-2s+4s2|+C. 53. ln (2x2+1)+C.
55. (x3-x6)–9+C. 57. (x4+x2)2+C.
59. (4-9x-3x2)–4+C. 61. +C.
63. (8-5x2)5/2+C.
65. .
67. .
69. ln(x2+1)- +C.
71. ln |3x-5|+ (x3-x6)–9+C.
73. (3x+1)3/2-ln . 75. .
77. e–x+ ex+C. 79. ln2 (x2+2x)+C.
81. .
83. y=–ln |x|=ln |1/x|. 85. 160e0.05t+190.
87. .
EXERCISE 16.4 (page 749)
1. x2+3x-ln |x|+C. 3. (2x3+4x+1)3/2+C.
5. . 7. .
9. 7x2-4 +C.
11. |3x-1|+C.
13. ln(e2x+1)+C. 15. .
17. x2+4 ln |x2-4|+C. 19. ( +2)3+C.
21. 3(x1/3+2)5+C. 23. (ln2 x)+C.
25. ln3 (r+1)+C. 27. .
29. +C. 31. 8 ln |ln(x+3)|+C.
33. +x+ln |x2-3|+C.
35. ln3/2 [(x2+1)2]+C.
37. -(ln 4)x+C.
39. x2-8x-6 ln |x|- +C.
41. x+ln |x-1|+C. 43. .
45. . 47. (x2+e)5/2+C.
49. .
51. +C. 53. .
55. . 57. p= .
59. c=20 ln |(q+5)/5|+2000.
61. C=2( +1). 63. .C =34
I -131I +
7112
1I
100q + 2
ln2 x2
+ x + C
x2
2+ 2x + Ce-2s3-
23
13612
3 18x 2 3>2 + 3 4 3>2 + C
15
-1e-x + 6 2 3
3+ C
3ex2 + 2 + C
2x2
2x4 - 112
x2
2
e1x2 + 32>2
3ln x
ln 3+ C
13
12
1x29
-17
e7>x + C32
x2 - 3x +23
ln
e11>42x2
47x
7 ln 4+ C-614 - 5x + C
13
Rr2
4K+ B1 ln 0 r 0 + B2
y = -1613 - 2x 2 3 +
112
14
14
-
14
2e1x + C2x2 + 3 + C29
127
13
16 1x6 + 1 2
12
x5
5+
2x3
3+ x + C
12x 2 3>23
- 12x + C =212
3x3>2 - 12x1>2 + C
-125
e4x3 + 3x2 - 416
12
14
127
14
13
-1
2413 - 3x2 - 6x 2 4 + C
-15
e-5x + 2ex + Ce-2v3-16
ey412
2x2 - 4 + C
21515x 2 3>2 + C =
2153
x3>2 + C
-83
13
-341z2 - 6 2-4 + C
-3e-2x
e7x2114
et2 + t
35127 + x5 2 4>3 + C
1x2 + 3 2 13
26+ C
17x - 6 2 535
+ C1312x - 1 2 3>2 + C
-5 13x - 1 2-2
6+ C
351y3 + 3y2 + 1 2 5>3 + C
1x2 + 3 2 66
+ C1x + 5 2 8
8+ C
G = -P2
50+ 2P + 20
y =x4
12+ x2 - 5x + 13
y = -x4
12-
x3
3+
4x
3+
112
y =3x2
2- 4x + 1
� Answers to Odd-Numbered Problems AN37
65. a. $150 per unit; b. $15,000; c. $15,300.67. 2500-800 ≠$711 per acre. 69. I=3.
EXERCISE 16.5 (page 754)
1. 35. 3. 0. 5. 25. 7. . 9. .
11. . 13. . 15. .
17. 101,475. 19. 84. 21. 273. 23. 8; $850.
PRINCIPLES IN PRACTICE 16.6
1. $5975.
EXERCISE 16.6 (page 762)
1. square unit. 3. square unit.
5. .
7. a. ; b. . 9. square unit.
11. square unit. 13. square unit. 15. 6.
17. –18. 19. . 21. 0. 23. .
25. 4.3 square units. 27. 2.4. 29. –25.5.
PRINCIPLES IN PRACTICE 16.7
1. $32,830. 2. $28,750.
EXERCISE 16.7 (page 771)
1. 14. 3. . 5. –20. 7. . 9. .
11. . 13. 0. 15. . 17. . 19. .
21. 4 ln 8. 23. e5. 25. (e8-1). 27. .
29. . 31. . 33. ln 3. 35. .
37. . 39. . 41. 6+ln 19.
43. . 45. 6-3e. 47. 7. 49. 0. 51. a5/2T.
53. . 55. $8639. 57. 1,973,333.
59. $220. 61. $2000. 63. 696; 492. 65. 2Ri.69. 0.05. 71. 3.52. 73. 55.39.
EXERCISE 16.8 (page 777)
In Problems 1–33, answers are assumed to be expressed insquare units.
1. 8. 3. . 5. 8. 7. . 9. 9. 11. .
13. 36. 15. 8. 17. . 19. 1. 21. 18.
23. . 25. . 27. e2-1.
29. 31. 68. 33. 2.
35. 19 square units. 37. a. ; b. ; c. .
39. a. b. ln (4)-1; c. 2-ln 3.
41. 1.89 square units. 43. 11.41 square units.
EXERCISE 16.9 (page 784)
1. Area= .
3. Area=
.
5. Area= .
7. Area= .
In Problems 9–33, answers are assumed to be expressed insquare units.
9. . 11. . 13. . 15. 40. 17. .
19. . 21. . 23. . 25. .
27. . 29. . 31. .
33. 12. 35. . 37. square units. 39. 24/3.
41. 4.76 square units. 43. 7.26 square units.
EXERCISE 16.10 (page 788)
1. CS=25.6, PS=38.4.3. CS=50 ln (2)-25, PS=1.25.5. CS=800, PS=1000. 7. $426.67. 9. $254,000.11. CS≠1197, PS≠477.
REVIEW PROBLEMS—CHAPTER 16 (page 791)
1. . 3. .
5. 7. 2 ln |x3-6x+1|+C.
9. . 11. .
13. . 15. ln .
17. (3x3+2)3/2+C. 19. (e2y+e–2y)+C.
21. ln |x|- +C. 23. 111. 25. .
27. 4- . 29. . 31. .
33. . 35. 1. 37. .11 + e3x 2 3
9+ C4 1x3>2 + 1 2 3>2 + C
32
- 5 ln 23t
-21t
+ C3 312
73
2x
12
227
103
13
4z3>43
-6z5>6
5+ C
y4
4+
2y3
3+
y2
2+ C
11 31114
- 4
-3 1x + 5 2-2 + C.
1172
x4
4+ x2 - 7x + C
83m3
2063
25532
- 4 ln 212
431515 - 212 2
443
3281
12512
92
1256
816163
43
3 111 - 2x2 2 - 1x2 - 4 2 4dx32
-15
3 1y + 1 2 - 11 - y 4dy31
0
+ 34
33 1x2 - x 2 - 2x 4dx3
3
032x - 1x2 - x 2 4dx
33
-23 1x + 6 2 - x2 4dx
ln 53
;
716
34
116
32
+ 2 ln 2 =32
+ ln 4.
32
312263
323
503
193
192
3a
b-Ax-Bdx
4712
e3
21e12 - 1 23 -
2e
+1e2
e +1
2e2 -32
12
1528
389
34
13
-16
323
53
-76
152
73
152
114
56
163
13
12
32
Sn =n + 1
2n+ 1
Sn =1nc4 a 1
nb + 4 a 2
nb + … + 4 a n
nb d =
2 1n + 1 2n
1427
23
a10
k = 1k2a
4
k = 112k - 1 2a
19
k = 1k
-76
-316
15
AN38 Answers to Odd-Numbered Problems �
39. . 41. e2x+3x-1.
In Problems 43–57, answers are assumed to be expressed insquare units.
43. . 45. . 47. . 49. 6+ln 3. 51. .
53. 36. 55. . 57. e-1.
59. p=100- . 61. $1900. 63. 0.5507.
65. 15 square units. 67. CS=166 , PS=53 .
73. 24.71 square units. 75. CS≠1148, PS≠251.
MATHEMATICAL SNAPSHOT—CHAPTER 16 (page 795)
1. a. 225; b. 125.3. a. $2,002,500; b. 18,000; c. $111.25.
PRINCIPLES IN PRACTICE 17.1
1. S(t)=–40te0.1t+400e0.1t+4600.2. P(t)=0.025t2-0.05t2 ln t+0.05t2(ln t)2+C.
EXERCISE 17.1 (page 802)
1. .
3. . 5. .
7. x[ln(4x)-1]+C.9.
.
11. |2x+1|+C.
13. . 15. e2(3e2-1).
17. (1-e–1), parts not needed.
19. 2 .21. 2 .23. .
25. .
27. .
29. .
31. 2e3+1 square units. 33. square units.
PRINCIPLES IN PRACTICE 17.2
1.
2. V(t)=150t2-900 ln (t2+6)+C.
EXERCISE 17.2 (page 809)
1. . 3. .
5. . 7. .
9. 2 ln |x|+3 ln |x-1|+C=ln |x2(x-1)3|+C.11. –3 ln |x+1|+4 ln |x-2|+C
=ln +C.
13.
= .
15. ln |x|+2 ln |x-4|-3 ln |x+3|+C
=ln +C.
17. ln |x6+2x4-x2-2|+C, partial fractions not required.
19. -5 ln |x-1|+7 ln |x-2|+C
= +ln +C.
21. 4 ln |x|-ln (x2+4)+C= .
23. .
25. 5 ln(x2+1)+2 ln(x2+2)+C=ln [(x2+1)5(x2+2)2]+C.
27. ln(x2+1)+ .
29. 18 ln (4)-10 ln (5)-8 ln (3).
31. 11+24 ln square units.
EXERCISE 17.3 (page 815)
1. . 3. .
5. . 7. ln .
9. .
11. .
13. . 15. .
17. -3 ln œx+ œ)+C.
19. . 21. ex(x2-2x+2)+C.
23. .
25. .
27. .115a 1
217 ln ` 17 + 15x17 - 15x
` b + C
19a ln 01 + 3x 0 +
11 + 3x
b + C
2 a -24x2 + 1
2x+ ln 02x + 24x2 + 1 0 b + C
1144
2x2 - 3121x2x2 - 3
1 + ln 49
2 c 11 + x
+ ln ` x
1 + x` d + C
1812x - ln 34 + 3e2x 4 2 + C
12c 45
ln 04 + 5x 0 -23
ln 02 + 3x 0 d + C
` 2x2 + 9 - 3x
` + C13
16
ln ` x
6 + 7x` + C
-216x2 + 3
3x+ C
x
929 - x2+ C
23
1x2 + 1
+ C32
-12
ln 1x2 + 1 2 -2
x - 3+ C
ln c x4
x2 + 4d + C
` 1x - 2 2 71x - 1 2 5 `
4x - 2
4x - 2
` x 1x - 4 2 21x + 3 2 3 `
14a 3x2
2+ ln c x - 1
x + 1d 2 b + C
14c 3x2
2+ 2 ln 0x - 1 0 - 2 ln 0x + 1 0 d + C
` 1x - 2 2 41x + 1 2 3 `
3x
-2x
x2 + 14
x + 1-
91x + 1 2 2
1 +2
x + 2-
8x + 4
12x + 6
-2
x + 1
r 1q 2 =52
ln ` 3 1q + 1 2 3q + 3
` .
29815
22x - 1
ln 2+
2x + 1x
ln 2-
2x + 1
ln2 2+
x3
3+ C
ex2
21x2 - 1 2 + C
x3
3+ 2e-x 1x + 1 2 -
e-2x
2+ C
ex 1x2 - 2x + 2 2 + C
x 1x - 1 2 ln 1x - 1 2 - x2 + C
1913 - 1012 212
-1x11 + ln x 2 + C
-x
2 12x + 1 2 +14
ln
= 2 1x + 1 2 3>2 13x - 2 2 + C10x 1x + 1 2 3>2 - 4 1x + 1 2 5>2 + C
y4
4c ln 1y 2 -
14d + C-e-x 1x + 1 2 + C
23
x 1x + 5 2 3>2 -4
151x + 5 2 5>2 + C
13
23
12q
1253
23
1256
163
43
y =12
22103x
ln 10+ C
� Answers to Odd-Numbered Problems AN39
29.
.31. .
33. .
35. .
37. -ln œ∏+7e œ)+C.
39. . 41.
43. . 45. .
47. .49. x(ln x)2-2x ln(x)+2x+C.
51. . 53. .
55. . 57.
59. a. $37,599; b. $4924. 61. a. $5481; b. $535.
EXERCISE 17.4 (page 818)
1. . 3. –1. 5. 0. 7. 13. 9. $12,400.
11. $3155.
PRINCIPLES IN PRACTICE 17.5
1. 76.90 feet. 2. 5.77 grams.
EXERCISE 17.5 (page 823)
1. 413. 3. 5. 1.388; ln 4≠1.386.
7. 0.883. 9. 2,361,375. 11. 3.0 square units.
13. . 15. 0.771. 17. km2.
19. a. $29,750; b. $36,600; c. $5350.
PRINCIPLES IN PRACTICE 17.6
1. I=I0e–0.0085x.
EXERCISE 17.6 (page 830)
1. y= . 3. y= .
5. y=Cex, C>0. 7. y=Cx, C>0.
9. . 11. y= .
13. . 15. .
17. y=ln . 19. c=(q+1)e1/(q+1).
21. 46 weeks.23. N=40,000e0.018t; N=40,000(1.2)t/10; 57,600.25. 2e billion. 27. 0.01204; 57.57 sec.29. 2900 years. 31. N=N0 , t � t0.
33. 12.6 units. 35. A=400(1-e–t/2), 157 grams.37. a. V=21,000e(2 ln 0.9)t; b. June 2002.
EXERCISE 17.7 (page 838)
1. 58,800. 3. 860,000. 5. 1990. 7. b. 375.9. 1:06 A.M. 11. $62,500.13. N=M-(M-N0)e–kt.
PRINCIPLES IN PRACTICE 17.8
1. 20 ml.
EXERCISE 17.8 (page 843)
1. . 3. Div. 5. . 7. Div. 9. . 11. 0.
13. a. 800; b. . 15. 4,000,000. 17. square unit.
19. 20,000 increase.
REVIEW PROBLEMS—CHAPTER 17 (page 846)
1. [2 ln(x)-1]+C. 3. 5+ ln 3.
5. 9 ln |3+x|-2 ln |2+3x|+C.
7. .
9. . 11. .
13. (7x-1)+C. 15. ln |ln 2x|+C.
17. x- ln |3+2x|+C.
19. 2 ln |x|+ ln(x2+1)+C.
21. 2 [ln(x+1)-2]+C. 23. 34.25. a. 1.405; b. 1.388. 27. y=C , C>0.
29. . 31. Div. 33. 144,000. 35. 0.0005; 90%.
37. N= . 39. 4:16 P.M. 41. 1.
43. a. 207, 208; b. 157, 165; c. 41, 41.
MATHEMATICAL SNAPSHOT—CHAPTER 17 (page 848)
1. 114; 69. 5. Answers may vary.
PRINCIPLES IN PRACTICE 18.1
1. . 2. 0.607.
3. Mean 5 years, standard deviation 5 years.
EXERCISE 18.1 (page 859)
1. a. b. c. d. .-1 + 1101316
= 0.8125;1116
= 0.6875;5
12;
13
4501 + 224e-1.02t
118
ex3 + x21x + 1
32
32
12
e7x
32
ln ` x - 3x + 3
` + C-29 - 16x2
9x+ C
12 1x + 2 2 +
14
ln ` x
x + 2` + C
94
x2
4
12
23
-12
1e
13
ek1t - t021.08124
a 122x2 + 3 b
y = B a 3x2
2+
32b 2
- 1y =4x2 + 3
2 1x2 + 1 2
ln x3 + 3
3y = 12x
1x2 + 1 2 3>2 + C-1
x2 + C
356
83
0.340; 13
L 0.333.
163
ln ` qn 11 - q0 2q0 11 - qn 2 ` .
72
ln 12 2 -34
2 1212 - 17 2231913 - 1012 2
e2x 12x - 1 2 + C
x4
4c ln 1x 2 -
14d + Cln ` x - 3
x - 2` + C
12x2 + 1 2 3>2 + C.12
ln 1x2 + 1 2 + C
41x12p141x
-29 - 4x2
9x+ C
12
ln 02x + 24x2 - 13 0 + C
4 19x - 2 2 11 + 3x 2 3>2 + C= x6 36 ln 13x 2 - 1 4 + C
481c 13x 2 6 ln 13x 2
6-13x 2 6
36d + C
AN40 Answers to Odd-Numbered Problems �
3. a.
b. c. 0; d. e. f. 0; g. 1; h. i.
j.
5. a.
b. c. .
7. a. e–2-e–6≠0.133; b. 1-e–4≠0.982;c. e–9≠0.0001; d. 1-e–2≠0.865.
9. a. b. c. d. 1; e. f.
g. ; h. . 11. 5 min. 13. e–3≠0.050.
EXERCISE 18.2 (page 865)
1. a. 0.4641; b. 0.3239; c. 0.8980; d. 0.9983; e. 0.9147;f. 0.4721. 3. 0.13. 5. –1.08. 7. 0.34.9. a. 0.9970; b. 0.0668; c. 0.0873. 11. 0.3085.13. 0.8185. 15. 8. 17. 9.68%. 19. 90.82%.21. a. 1.7%; b. 85.6.
PRINCIPLES IN PRACTICE 18.3
1. 0.0396.
EXERCISE 18.3 (page 870)
1. 0.1056; 0.0122. 3. 0.0430; 0.9232. 5. 0.7507.7. 0.4129. 9. 0.2514; 0.0287. 11. 0.0336.
REVIEW PROBLEMS—CHAPTER 18 (page 871)
1. a. 2; b. c.
d.
3. a. ; b. 5. 0.3085. 7. 0.2417.
9. 0.1587. 11. 0.9817. 13. 0.0228.
MATHEMATICAL SNAPSHOT—CHAPTER 18 (page 874)
1. The result should correspond to the known distributionfunction. 3. Answers may vary.
PRINCIPLES IN PRACTICE 19.1
1. a. $3260; b. $4410.
EXERCISE 19.1 (page 881)
1. 3. 3. –2. 5. –1. 7. 88. 9. 3.11. 2x0+2h-5y0+4. 13. 2000. 15. y=–4.17. z=6.19. 21.
23.
25.
y
x
z
2
4
y
x
z
2
1
y
x
z
2
6
4
y
x
z
1
1
1
2123
L 0.94.83
F 1x 2 = µ0,x
3+
2x3
3,
1,
if x 6 0,
if 0 � x � 1,
if x 7 1.
34
;932
;
710
;716
212
2123
;83
;3964
L 0.609;516
;18
;
s2 =1b - a 2 2
12, s =
b - a112a + b
2;
f 1x 2 = •1
b - a,
0,
if a � x � b,
otherwise;
x
F (x )
1
1
4
P 1X 6 2 2 =13
, P 11 6 X 6 3 2 =23
.
F 1x 2 = µ0,x - 1
3,
1,
if x 6 1,
if 1 � x � 4,
if x 7 4.
132
;52
;13
;56
;13
;
x
f (x )
1 4
13
f 1x 2 = •130
,
if 1 � x � 4,
, otherwise;
� Answers to Odd-Numbered Problems AN41
27.
EXERCISE 19.2 (page 887)
1. fx(x, y)=8x; fy(x, y)=6y.3. fx(x, y)=0; fy(x, y)=2.5. gx(x, y)=3x2y2+4xy-4y;
gy(x, y)=2x3y+2x2-4x+3.
7. gp(p, q)= ; gq(p, q)= .
9. hs(s, t)= ; ht(s, t)= .
11. (q1, q2)= ; (q1, q2)= .
13. hx(x, y)=(x3+xy2+3y3)(x2+y2)–3/2;hy(x, y)=(3x3+x2y+y3)(x2+y2)–3/2.
15.
17. .
19. fr(r, s)= ;
fs(r, s)= .
21. fr(r, s)=–e3-r ln(7-s); fs(r, s)= .
23. gx(x, y, z)=6xy+2y2z; gy(x, y, z)=3x2+4xyz;gz(x, y, z)=2xy2+9z2.
25. gr(r, s, t)=2res+t;gs(r, s, t)=(7s3+21s2+r2)es+t;gt(r, s, t)=es+t(r2+7s3).
27. 50. 29. . 31. 0. 33. 26.
39.
EXERCISE 19.3 (page 893)
1. 20. 3. 1374.5.
5. .
7.
competitive.
9.
complementary.
11.
.
13. 4480; if a staff manager with an M.B.A. degree had anextra year of work experience before the degree, the manager would receive $4480 per year in extra compensation.15. a. –1.015; –0.846;b. One for which w=w0 and s=s0.
17. for VF>0. Thus if x increases and VF
and Vs are fixed, then g increases.
19. a. When pA=8 and pB=64, and
b. Demand for A decreases by approximately
units.
21. a. No; b. 70%. 23. .
25. .
EXERCISE 19.4 (page 898)
1. . 3. . 5. . 7. .
9. . 11. . 13. . 15. .
17. 4. 19. .
21. a. 36; b. With respect to qA, ; with respect to qB, .
EXERCISE 19.5 (page 901)
1. 8xy; 8x. 3. 3; 0; 0.5. 18xe2xy; 18e2xy(2xy+1); 72x(1+xy)e2xy.7. 3x2y+4xy2+y3; 3xy2+4x2y+x3; 6xy+4y2; 6xy+4x2. 9. x(x2+y2)–1/2; y2(x2+y2)–3/2.11. 0. 13. 28,758. 15. 2e.
17. . 23. .
EXERCISE 19.6 (page 905)
1. 3. .
5. 5(2xz2+yz)+2(xz+z2)-(2x2z+xy+2yz).7. 3(x2+xy2)2(2x+y2+16xy).9. –2s(2x+yz)+r(xz+3y2z2)-5(xy+2y3z).11. 19s(2x-7). 13. 324. 15. –1.
c2t +31t
2d ex + y∂z
∂r= 13;
∂z
∂s= 9.
-y2 + z2
z3 = -3x2
z3-18
28865
6013
52
-4e2-
910
-3x
z
yz
9 + z
-ey - zx 1yz2 + 1 2z 11 - x2y 2
4y
3z2-x
z
hpA= -1, hpB
= -12
hpA= -
546
, hpB=
146
158
∂qA
∂pB=
1532
;
∂qA
∂pA= -5
∂g
∂x=
1VF
7 0
∂P
∂C= 0.01A0.27B0.01C-0.99D0.23E0.09F0.27
∂P
∂B= 0.01A0.27B-0.99C0.01D0.23E0.09F0.27;
∂qB
∂pA= -
5003pBp
4>3A
; ∂qB
∂pB= -
500p
2Bp
1>3A
;
∂qA
∂pA= -
100p
2Ap
1>2B
; ∂qA
∂pB= -
50pAp
3>2B
;
∂qA
∂pA= -50;
∂qA
∂pB= 2;
∂qB
∂pA= 4;
∂qB
∂pB= -20;
∂P
∂k= 1.208648l0.192k-0.236;
∂P
∂l= 0.303744l-0.808k0.764
-ra
2 c1 + an - 1
2d 2
.
1114
e3 - r
s - 7
2 1s - r 21r + 2s +r3 - 2rs + s21r + 2s
1r + 2s 13r2 - 2s 2 +r3 - 2rs + s2
21r + 2s
∂z
∂x= 5 c 2x2
x2 + y+ ln 1x2 + y 2 d ; ∂z
∂y=
5x
x2 + y
∂z
∂x= 5ye5xy;
∂z
∂y= 5xe5xy.
14q2
uq2
34q1
uq1
-s2 + 41 t - 3 2 2
2s
t - 3
p
21pq
q
21pq
y
x
z
1
1
1
AN42 Answers to Odd-Numbered Problems �
17. When pA=25 and pB=4, .
19. a. b. –15.
EXERCISE 19.7 (page 913)
1. . 3. (2, 5), (2, –6), (–1, 5), (–1, –6).
5. (50, 150, 350). 7. , rel. min.
9. rel. max.
11. (1, 1), rel. min; , neither.
13. (0, 0), rel. max.; rel. min.; , (4, 0), neither.
15. (122, 127), rel. max. 17. (–1, –1), rel. min.19. (0, –2), (0, 2), neither. 21. l=24, k=14.23. pA=80, pB=85.25. qA=48, qB=40, pA=52, pB=44, profit=3304.27. qA=3, qB=2. 29. 1 ft by 2 ft by 3 ft.
31. , rel. min. 33. a=–8, b=–12, d=33.
35. a. 2 units of A and 3 units B;b. Selling price for A is 30 and selling price for B is 19.Relative maximum profit is 25.37. a. P=5T(1-e–x)-20x-0.1T2;c. Relative maximum at (20, ln 5); no relative extremum at
.
EXERCISE 19.8 (page 922)
1. (2, –2). 3. . 5. .
7. . 9. . 11. (3, 3, 6).
13. Plant 1, 40 units; plant 2, 60 units.15. 74 units (when l=8, k=7).17. $15,000 on newspaper advertising and $45,000 on TV advertising.19. x=5, y=15, z=5.21. x=12, y=8. 23. x=10, y=20, z=5.
EXERCISE 19.9 (page 929)
1. =0.98+0.61x; 3.12. 3. =0.057+1.67x; 5.90.5. =82.6-0.641p. 7. =100+0.13x; 105.2.9. =8.5+2.5x.11. a. =35.9-2.5x; b. =28.4-2.5x.
EXERCISE 19.11 (page 936)
1. 18. 3. . 5. . 7. 3. 9. 324. 11. .
13. . 15. –1. 17. . 19. .
21. . 23. e–4-e–2-e–3+e–1. 25. .
REVIEW PROBLEMS—CHAPTER 19 (page 939)
1.
3.
5. 8x+6y; 6x+2y. 7. .
9. . 11. . 13. 2(x+y).
15. xzeyz ln z; +yeyz ln z=eyz .
17. . 19. 2(x+y)er+2 .
21. . 23. .
25. Competitive. 27. (2, 2), rel. min.29. 4 ft by 4 ft by 2 ft.31. A, 89 cents per pound; B, 94 cents per pound.33. (3, 2, 1). 35. =12.67+3.29x
37. 8. 39. .
MATHEMATICAL SNAPSHOT—CHAPTER 19 (page 942)
1. y=9.50e–0.22399x+5. 3. T=79e–0.01113t+45.
EXERCISE A.1 (page 948)
1.
x
y
105
(0, 0)
(3, 5)
(4, 7)
(8, 10)
(10, 10)10
5
130
y
∂P
∂l= 14l-0.3k0.3;
∂P
∂k= 6l0.7k-0.72x + 2y + z
4z - x
a x + 3y
r + sb ; 2 a x + 3y
r + sb1
64
a 1z
+ y ln z beyz
z
2xzex2yz 11 + x2yz 2y
x2 + y2
y
1x + y 2 2; -x
1x + y 2 2
y
x
z
y
x
z
3
9
92
38
124
-274
e2
2- e +
12
83
-585
23
14
yyy
yqyy
a 23
, 43
, -43b16, 3, 2 2
a 43
, -43
, -83ba3,
32
, -32b
a5, ln 54b
a 10537
, 2837b
a0, 12ba4,
12b ,
a 12
, 14b
a -14
, 12b ,
a -2, 32b
a 143
, -133b
∂w
∂s=
∂w
∂x ∂x
∂s+
∂w
∂y ∂y
∂s;
∂c
∂pA= -
14
and ∂c
∂pB=
54
� Answers to Odd-Numbered Problems AN43
3.
5.
7. 75. 9. Between 1990 and 1993, 1995 and 1998, 1999 and 2000; positive. 11. Between 1994 and 1995; zero.13. Between 1993 and 1994. 15. 75 students; 1990.17. a. Possible graph:
; b. Wednesday.
19. a. ;
b. approximately 85 mi; c. between 5 and 6 h; 0;d. between 3 and 5 h; The slope of the line graph duringthis time interval is greater than the slope of the line graphduring the remaining intervals.
21. a. ;
b. between 6:00 A.M. and 8:00 A.M.;c. between 12:00 P.M. and 2:00 P.M.; 0;d. the number of fish caught per hour remained constant.
EXERCISE A.2 (page 954)
1. . 3. .
5. y=15,525(0.91)x. 7. P(E ´ F)=P(E)+P(F).9. ; linear; y=2x+5.
11. ; exponential; y=3x.
13. ; quadratic;f(x)=2(x-1)2+3 orf(x)=2x¤-4x+5.
x
y
–3 5
25
(2, 5)
(–2, 21) (4, 21)
(0, 5)(1, 3)
x
y
–2 5
85
(–1, )
(0, 1)
(1, 3)
(3, 27)
(4, 81)
13
x
y
–5 5
–5(–3, –1)
(–1, 3)(0, 5)
(2, 9)
(4, 13)15
y = -12
x +52
A =12
h 1b1 + b2 2
t
f
30
20
10
6 12
Num
ber
of fi
sh
Number of hoursafter 6 A.M.
(0, 0)
(2, 8)
(4, 14)
(6, 20)
(8, 20)(10, 22)
(12, 26)
t
d
400
300
200
100
5 10
Dis
tanc
e (m
iles)
Time (hours)
(0, 0)(1, 55)
(3, 115)
(5, 265) (6, 265)
(8, 325)
t
P
1 2 3 4
Pric
e pe
r sh
are
Number of daysafter Monday
t
P
6000
5500
5000
1 2 3 4
Pop
ulat
ion
Number of yearsafter 1996
(0, 5120)
(1, 5342)
(2, 5510)
(3, 5750)
(4, 6002)
x
y
105
(0, 4)
(2, 1)
(3, 9)
(7, 5)
(10, 3)
10
5
AN44 Answers to Odd-Numbered Problems �
15. ;
logarithmic; y=log™ x.17. ; linear.
19. ; quadratic.
21. ; exponential.
23. a. C=100+40r; b. $500; c. 22 reels.25. a. h=–16t¤+80t; b. 2.5 sec, 100 ft.27. a.
;
b. N=2000(3) ; c. 118,098,000 bacteria.29. a. (2, 1.3) and (6.5, 34.1); b. Answers may vary,but should be close to y=3.1x+1.5;c. Answers may vary, but should be close to y=44.9.
EXERCISE A.3 (page 960)
1. –3, –3, –3, –3; linear. 3. 1, 7, 19, 37; nonlinear.5. ; a. 1.5;
b. –1.5;c. 0;d. 2.005.
7. ; a. 16;b. 7;c. 4;d. 3.01.
9. 4.5 in. per yr. 11. $42 per h.13. a. 3.5 degrees per day; b. –1.25 degrees per day;c. 1 degree per day; d. ≠0.59 degree per day.15. a. 0; b. 0; c. 0; d. 0. 17. a. 7; b. 13; c. h+8;d. 2xº+h+2. 19. a. –2; b. –2; c. –2; d. –2;e. Since g(x) is linear, the average rate of change between any two points is constant. 21. x(t)=2t+3.23. Possible graph:
25. Average cost per unit over the interval.
EXERCISE A.4 (page 967)
1. Average. 3. Instantaneous. 5. –8. 7. .
9. 1. 11. y=x-1. 13. y= .
15. 0. 17. 5. 19. 20x. 21. .
23. . 25. .
27. . 29. =0.1q+28;
$35.50 per rug. 31. =–32t+32; a. 32 ft/sec;
b. –32 ft/sec; c. –64 ft/sec.
dh
dt
dc
dq-
52112 - 5x
+ 20x
5215x - 11
15112 - 5x 2 2
-5
15x + 11 2 2
12
x +12
-19
x
y
y
40
x5–5
y
10
x5–5
t
t
A
105
10,000
5000
Number of daysof decay
Am
ount
of s
ubst
ance
(mill
igra
ms)
x
y
–10 10–5
70
x
y
10
35
x
y
–4 32–1
5 (32, 5)
(8, 3)
(2, 1)
(1, 0)
( , –1)12
� Answers to Odd-Numbered Problems AN45
t 0 1 2 3 4 5
N 2000 6000 18,000 54,000 162,000 486,000
EXERCISE A.5 (page 972)
In Problems 1–13, answers are assumed to be expressed insquare units.
1. . 3. 34. 5. 28. 7. a. 41; b. 44;
c. 42; d. 42; e. parts (c) and (d).9. a. �12.57; b. �9.98; c. �11.36; d. �11.98; e. part (d).11. a. 54; b. 42; c. trapezoid.13. a. 104; b. 86; c. trapezoid.15. ; The area under f(x) can be
divided into 2 sections (seegraph). The top section isequivalent to the areaunder g(x), so they havethat area in common. Thebottom section is a rectangle that the areaunder g(x) does not include.
EXERCISE A.6 (page 977)
1. 12, 17, t. 3. 168. 5. 532. 7. . 9. .
11. . 13. 520. 15. 5. 17. 37,750.
19. 14,980. 21. 295,425. 23. .
25. 8- . 27. 4.500625 square units.
EXERCISE A.7 (page 985)
1. ; 20. 3. .
5. . 7. .
9. ; positive.
11. ; positive.
13. ; positive.
15. a. 7b; b. 70. 17. a. ; b. 170.
19. a. 14; b. G(x)=2x+b, where b can be any real number; c. 14.
21. 30. 23. –14. 25. 25 . 27. e‹-1≠19.09.
EXERCISE A.8 (page 993)
1. Integral. 3. Function itself. 5. Derivative.7. Function itself. 9. a. 50; The cost of the rental is $50.00 when you drive the truck 50 miles; b. 0.60; When you have driven the truck 50 miles, the cost is increasing at the rate of $0.60 per mile. 11. a. b(t)=300t;b. b�(t)=300; The employee’s bonus increases at the rate of $300 per year; c. The integral dt approximates the sum of an employee’s annual bonuses during the first ten years with the company. 13. a. 23; b. 2; In 1995,the number of books that Xul reads annually was increasing at the rate of about 2 books per year; c. 140.26;Between 1991 and 2000, Xul read about 140 books.15. a. 8; 11.95; In 2006, the program’s budget would be $8 billion with model bl and $11.95 billion with be; b. 1.6;1.11; In 2006, the program’s budget is increasing at the rate of $1.6 billion per year with model bl and about $1.11 billion per year with be; c. 20; 38.36; In the first five years of the program, the cumulative budget would be about $20 billion with model bl and about $38.36 billion with be; d. be.; e. bl;f. be. 17. a. 0.012 mi/sec; b. 0.002 mi/sec2;c. 0.035 mile.
REVIEW PROBLEMS (page 995)
1.
3. between October and November.5. C=550+22.50x.
t
A
4321
180
140
90
(0, 125)
(1, 98)
(3, 150)
(2, 175)
(4, 150)
Am
ount
(do
llars
)
Number of monthsafter October
110
0 b1t2
12
32
b2 + 2b
x
y
3
x
y
4–2
x
y
6
35
-513x + 5 2 dx3
5
-21x2 + x + 2 2 dx
39
48 dx8 a n + 1
nb + 12
41n+1212n+123n2
4 2325
a8
i = 12i
a8
j = 35ja
60
i = 36i
x
y
f(x)
g(x)
5
8
412
+p
4
AN46 Answers to Odd-Numbered Problems �
7. a. , quadratic;
b. h=–16t¤+100t; c. 24 ft; d. 6.25 s.9. a. –3, –3, –3, –3, linear; b. 1, 3, 5, 7, nonlinear.11. a. 300 kilobytes;b. ;
c. ≠–1.26 kilobytes per second,≠–1.18 kilobytes persecond,≠–0.76 kilobytes per second;d. The negative sign indicates that as the amount of time left decreases, the amount of the document which has been downloaded increases; e. 302 seconds to 204 seconds.
13. x(t)=4t-1. 15. y= .
17. . 19. 29 square units.
21. a. �7.07 square units; b. �5.07 square units;c. �6.145 square units; d. �6.57 square units; e. part (d).23. a. 45 square units; b. 42.5 square units;
c. neither is better. 25. 513. 27. .
29. 2.999975 square units.
31. 33. log x dx.
35.
37.
39. a. ; b. 80. 41. 100,000. 43. 36.
45. a. 2103.64; The energy costs for a 1900 square-foot home were about $2103.64 in 2001; b. 63.11; In 2001, the energy costs for a 1900 square-foot home were increasing at a rate of about $63.11 per year; c. 42,455.27;The cumulative energy costs for a 1900 square-foot home between 1970 and 2001 were about $42,455.27.47. a. 0.015 mi/sec; b. 0.005 mi/sec¤; c. 0.03 mi.
12
b2 + 3b
x
y
–5 5
–5
5
x
y
–5 5–2
12
3100
1 1-x
2 - x + 2 2 dx31
-1
n
4
dy
dx=
113 - x 2 2
14
x +32
t
h
54321
160
80 (5, 100)(1, 84)
(3, 156)
(2, 136)(4, 144)
Hei
ght (
feet
)
Time (Seconds)
� Answers to Odd-Numbered Problems AN47
time 302 sec 204 sec 130 sec 20 sec
size 3 K 126 K 213 K 297 K