HaeusslerAnswers

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.


1. False. 3. False. 5. False. 7. True. 9. False.11. Distributive. 13. Associative. 15. Commutative.17. Definition of subtraction. 19. Distributive.


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.


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. .


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. .


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.


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. .


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. .


1. 8 mi/h. 2. .

3. ramp is 5 feet long.


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.


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.


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


1. a. $107.15; b. $10.26; c. 10 lb; d. 10.44 lb; e. 4.4%3. –1.9%.


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.


1. 5375.2. 150-x4 � 0; 3x4-210 � 0; x4+60 � 0; x4 � 0.


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.


1. 120,001. 3. 17,000. 5. 60,000. 7. $25,714.29.9. 1000. 11. t>36.5. 13. At least $67,400.


1. |w-22 oz| � 0.3 oz.


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).


1. (–q, 0]. 3. . 5. �. 7. .

9. (–q, q). 11. –2, 5. 13. .

15. ´ . 17. 542. 19. 6000.

21. c<$212,814.


1. 1 hour. 3. 1 hour. 5. 600; 310.


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.


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.


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.


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.


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)).


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.


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.


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



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.


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 )



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].


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.


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.


1. $28,321. 3. $87,507.90. 5. Answers may vary.


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.


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



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.


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.


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.


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.


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.


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




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.


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.


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.


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.


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.


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.


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



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.


1. t=log2 16; t=the number of times the bacteria have

doubled. 2.

3.

4. 5. Approximately 13.9%.6. Approximately 9.2%.


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


1.

2. log(10,000)=log(104)=4.


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.


1. 18. 2. Day 20. 3. The other earthquake is67.5 times as intense as a zero-level earthquake.


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.


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.


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




1. 3*2 or 2*3. 2. .


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. .


1. . 2. x1=670, x2=835, x3=1405.


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. .


1. $5780. 2. $22,843.75. 3. = .


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. .


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§


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.


1. Infinitely many solutions:

in parametric form: where r is

any real number.


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.


1. Yes. 2. MEET AT NOON FRIDAY.

3. E–1= ; F is not invertible.

4. A: 5000 shares; B: 2500 shares; C:2500 shares


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.


1. 6


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.


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



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.


1. 3. a. b.

5. . 7.


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.


1. $151.40. 3. It is not possible, because guests 3 and 4 each cost the lodge the same amount per day.


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.


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.


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.


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.


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.


1. 0 gadgets of Type 1, 72 gadgets of Type 2, 12 gadgets ofType 3; maximum profit of $20,400.


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.


1. 35-7t of device 1, 6t of device 2, 0 of device 3, for 0 � t � 1.


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.


1. Plant I: 500 standard, 700 deluxe; plant II: 500 standard,100 deluxe; $89,500 maximum profit.


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%.


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



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.


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.


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.


1. 2 minutes of radiation. 3. Answers may vary.


1. 4.9%. 2. 7 years, 16 days. 3. 7.7208%.4. The $10,000 investment is slightly better over 20 years.


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%.


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%.


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


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.


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.


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.


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.


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.


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



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



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



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.


1. 10,586,800.


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}.


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.


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.


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.


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

;715

;15

;25

;

110

.140

;3

10;

3676

.125

.

118

.

56

.13

.12

;112

;23

;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!

.



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.


1. ≠0.645.


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.


1.


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.


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

.


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. .


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.


1. 7.

3. Against Always Defect: ;

Against Always Cooperate: ;

Against regular Tit-for-tat: .


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.


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.


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.


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


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.


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.


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.


1. 0<x<4.


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).


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).


1. 17%.3. An exponential model assumes a fixed repayment rate.


1. =40-32t.


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.


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


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


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.


1. 2.5 units. 2. . =0 feet/s.

When t=0.5 the object reaches its maximum height.3. 1.2 and 120%.


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.


1. 6.25-6x. 2. T¿(x)=2x-x2; T¿(1)=1.


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


≤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. .


1. 288t.


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.


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


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.


1. The slope is greater—above 0.9. More is spent; less is saved.3. Spend $705, save $295. 5. Answers may vary.


1. . 2. .


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.


1. .


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.


1. .

2. =4∏r¤ and =2880∏ inches/minute

3. The top of the ladder is sliding down at a rate of

feet/second.


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. .


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


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. .


1. feet/sec2 (Note: Negative values indicate

the downward direction.).2. c¿¿ (3)=14 dollars/unit2.


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.


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.


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.


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.


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


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).


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.


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


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.


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



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


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.


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


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.


1. The data for 1998–2000 fall into the same pattern as the1959–1969 data.


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


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.


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.


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.


1. 43 and 1958.


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.


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.


1. F=$40, V=$20; yes. 3. No difference.


1.

2.

3.

4.

5. S(t)=0.7t3-32.7t2+491.6t+C.


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



1. N(t)=800t+200et+6317.37.2. y(t)=14t3+12t2+11t+3.


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).


1. T(t)=10e–0.5t+C. 2. 35 lnœt+1œ+C.


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. .


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


65. a. $150 per unit; b. $15,000; c. $15,300.67. 2500-800 ≠$711 per acre. 69. I=3.


1. 35. 3. 0. 5. 25. 7. . 9. .

11. . 13. . 15. .

17. 101,475. 19. 84. 21. 273. 23. 8; $850.


1. $5975.


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.


1. $32,830. 2. $28,750.


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.


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.


1. Area= .

3. Area=

.

5. Area= .

7. Area= .


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.


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.


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


39. . 41. e2x+3x-1.


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.


1. a. 225; b. 125.3. a. $2,002,500; b. 18,000; c. $111.25.


1. S(t)=–40te0.1t+400e0.1t+4600.2. P(t)=0.025t2-0.05t2 ln t+0.05t2(ln t)2+C.


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.


1.

2. V(t)=150t2-900 ln (t2+6)+C.


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.


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


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.


1. . 3. –1. 5. 0. 7. 13. 9. $12,400.

11. $3155.


1. 76.90 feet. 2. 5.77 grams.


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.


1. I=I0e–0.0085x.


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.


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.


1. 20 ml.


1. . 3. Div. 5. . 7. Div. 9. . 11. 0.

13. a. 800; b. . 15. 4,000,000. 17. square unit.

19. 20,000 increase.


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.


1. 114; 69. 5. Answers may vary.


1. . 2. 0.607.

3. Mean 5 years, standard deviation 5 years.


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


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.


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.


1. 0.0396.


1. 0.1056; 0.0122. 3. 0.0430; 0.9232. 5. 0.7507.7. 0.4129. 9. 0.2514; 0.0287. 11. 0.0336.


1. a. 2; b. c.

d.

3. a. ; b. 5. 0.3085. 7. 0.2417.

9. 0.1587. 11. 0.9817. 13. 0.0228.


1. The result should correspond to the known distributionfunction. 3. Answers may vary.


1. a. $3260; b. $4410.


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;


27.


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.


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. .


1. . 3. . 5. . 7. .

9. . 11. . 13. . 15. .

17. 4. 19. .

21. a. 36; b. With respect to qA, ; with respect to qB, .


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. .


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


17. When pA=25 and pB=4, .

19. a. b. –15.


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

.


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.


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.


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. .


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. .


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


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.


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


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.


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.


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


t 0 1 2 3 4 5

N 2000 6000 18,000 54,000 162,000 486,000



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.


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.


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.


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


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)


time 302 sec 204 sec 130 sec 20 sec

size 3 K 126 K 213 K 297 K

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