ANSWERS TO ODD-NUMBERED PROBLEMSwps.prenhall.com/.../1246/1276309/HaeusslerAnswers.pdf ·...

EXERCISE 0.1 (page 3)

1. True 3. False; the natural numbers are 1, 2, 3, and so on.5. True 7. False; =5, a positive integer9. True 11. True


1. False 3. False 5. False 7. True 9. False11. Distributive 13. Associative 15. Commutative17. Definition of subtraction 19. Distributive29. –4 31. 5 33. 8 35. –18 37. 24

39. 9 41. –7x 43. 6+y 45. 47. –8

49. –8 51. 0 53. 20+4x 55. 0 57. 5

59. 61. 63. 65. 67.

69. 2 71. 73. 75. 77. 0

79. 0


1. 25(� 32) 3. w12 5. 7. 9. 8x6y9

11. x4 13. x14 15. 5 17. –2 19.

21. 7 23. 27 25. 27. 29.

31. 33. 4x2 35.

37. 3z2 39. 41. 43. 45.

47. 51/5x2/5 49. x1/2-y1/2 51.

53. 55. 57.

59. 61. 63. 65. 4

67. 69. 71. 9 73.

75. xyz 77. 79. 81. x2y5/2 83.

85. x8 87. 89.


1. 11x-2y-3 3. 6t2-2s2+65.

7.9. 11. –15x+15y-2713. x2+9y2+xy 15. 6x2+9617. –40x3+10x2-20x-50 19. x2+9x+2021. w¤-3w-10 23. 10x2+19x+625. x2+6x+9 27. x2-10x+2529. 3x+10 +25 31. 4s¤-133. x3+4x2-3x-1235. 3x4+2x3-13x2-8x+4 37. 5x3+5x2+6x39. 3x2+2y2+5xy+2x-841. 8a3+36a2+54a+2743. 8x3-36x2+54x-27 45. z-18

47. 49.

51.

53. x2-2x+4- 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. (a+3)(a+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. u2/3v2(u+2v1/2)(u-2v1/2) 27. 2x(x+3)(x-2)29. 4(2x+1)2 31. x(xy-7)2

33. (x+2)(x-2)2 35. (y+4)2(y+1)(y-1)37. (a+3)(a2-3a+9)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+3)(X+1)(X-1) 51. y(x+1)2(x-1)2


1. 3. 5.

7. 9.

11. 13. 15. 17.

19. –27x2 21. 1 23. 25. 12x2

x - 1

23

n

3X

22 1x + 4 2

1x - 4 2 1x + 2 2

3 - 2x

2x + 3-

y2

1y - 3 2 1y + 2 2

3x + 2x + 2

x - 5x + 5

a + 3a

x - 2 +7

3x + 28

x + 2

3x2 - 8x + 17 +-37

x + 2

x +-1

x + 33x3 + 2x -

12x2

13x

12y - 13z6x2 - 9xy - 2z + 12 - 4

1a + 513b - 1c

4x4z4

9y4-

4s5

a5c14

b24

4y4

x2

13

64y6x1>2x2

2x6

y3

2

0216a10b15

ab

23 9x2

3x

212x

x

6155

325 w3-

125 27w3

125 x425 18x - y 2 4

x9>4z3>4y1>2

19t2

5m9

x3

y2z2

9t2

4

412 - 1513 + 413 2x13 2

412116

14

12

a21

b20

x8

y17

-2514

6y

x

1712

76

10xy

6y

x

3ab

c-

32x

13

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

HP-IMA11e_Ans_00_06 3/4/04 4:28 AM Varekai:Users:jeanmarieperchalski:Desktop:CTA-HPIMA11e:HP 11e files:

27. 29. x+2 31.

33. 35.

37. 39.

41. 43. 45.

47. 49.

51. 2- 53. 55. –4-

57. 59. 4 -5 +14


1. 0 3. 5. –2

7. Adding 5 to both sides; equivalence guaranteed9. Raising both sides to the third power; equivalence notguaranteed11. Dividing both sides by x; equivalence not guaranteed13. Multiplying both sides by x-1; equivalence notguaranteed15. Multiplying both sides by (x-5)/x; equivalence notguaranteed

17. 19. 0 21. 1 23. 25. –1

27. 2 29. 31. 126 33. 15 35.

37. 39. 41. 43. 3 45.

47. 49. � 51. 53. 2 55. 0 57.

59. 61. 3 63. 65. � 67. 11

69. 71. 73. 2 75. 7 77.

79. 81. 83.

85. 87.

89. 91. 93. 170 m

95. c=x+0.0825x=1.0825x 97. 3 years

99. 31 hours 101. 20 103.

105. � 84 ft 107. 109. 0


1. 2 3. 3, 5 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, —2

29. 3, 4 31. 4, –6 33. 35.

37. No real roots 39. 41. 40, –25

43. 45. 47. 2,

49. 51. 3, 0 53. 55.

57. 6, –2 59. 61. 5, –2 63.

65. –2 67. 6 69. 4, 8 71. 2 73. 0, 475. 1 77. � 64.15, 3.35 79. 6 inches by 8 inches83. 1 year and 10 years 85. (a) 8 s; (b) 5.4 s or 2.6 s87. 1.5, 0.75 89. No real root 91. 1.999, 0.963

MATHEMATICAL SNAPSHOT—CHAPTER 0 (page 48)

1. The results agree. 3. The results agree.


1. 120 3. 48 of A, 80 of B 5. 7. 1 m

9. � 13,077 tons 11. $4000 at 6%, $16,000 at

13. $4.25 15. 4% 17. 80 19. $800021. 1209 cartridges must be sold to approximately breakeven. 23. $116.25 25. 40 27. 46,00029. Either $440 or $460 31. $100 33. 4235. 80 ft by 140 ft 37. 9 cm long, 4 cm wide39. $112,000 41. 60 43. Either 125 units of A and100 units of B, or 150 units of A and 125 units of B.

APPLICATIONS IN PRACTICE 1.2

1. 53752. 150-x4 � 0; 3x4-210 � 0; x4+60 � 0; x4 � 0


1. (4, q) 3. (–q, 4] 5.

7. 9. (0, q) 11.

13. 15. � 17.

)– 22

3

)2–7

a - q, 13 - 2

2ba-

27

, qb

7–5

0))

27

c-

75

, qba-q, 27b

1–2

44)

a-q, -

12d

712

%

513

-9 ; 1414

12

32

, -1157

, 115

;

155

, ;

12

-12

;13, ;12-2 ; 114

2

-5 ; 1578

7 ; 1372

32

12

, -43

32

23

12

-12

t =d

r - c; c = r -

d

t

n =2mI

rB- 1t =

r - d

rd

a1 =2S - nan

nr =

S - P

Pt

q =p + 1

8r =

I

Pt-

94

4936

-109

683

513

18

72

83

15

2552

143

6017

-

3718

-

269

103

125

52

103

1312x - 15x2 - 5

216-16 + 213

313

3 113 x - 13 x + h 213 x + h 13 x

1x + 2 2 16x - 1 22x2 1x + 3 2

4x + 13x

x

1 - xy

x2 + 2x + 1x2

35 - 8x

1x - 1 2 1x + 5 22 1x + 2 2

1x - 3 2 1x + 1 2 1x + 3 2

2x2 + 3x + 1212x - 1 2 1x + 3 2

11 - p2

73t

-12x + 3 2 11 + x 2

x + 4

AN2 Answers to Odd-Numbered Problems ■


■ Answers to Odd-Numbered Problems AN3

19. (–q, 48) 21. (–q, –5] 23. (–q, q)

25. 27. (–12, q) 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. $25,714.299. 1000 11. t>36.7 13. At least $67,400


1. |w-22 oz| � 0.3 oz


1. 13 3. 6 5. 7 7. –4<x<49. 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|<10013. |p1-p2| � 9 15. —7 17. —35 19. 13, –3

21. 23. 25. (–4, 4)

27. (–q ,–8) ´ (8, q) 29. (–14, –4)

31. (–q, 0) ´ (1, q) 33.

35. (–q, 0] ´ 37. |d-17.2| � 0.3 m

39. (–q, Â-hÍ) ´ (Â+hÍ, q)

REVIEW PROBLEMS—CHAPTER 1 (page 73)

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. f � g 3. h � k 5. All real numbers except 07. All real numbers � 3 9. All real numbers

11. All real numbers except

13. All real numbers except 2

15. All real numbers except 4 and

17. 1, 7, –7 19. –62, 2-u2, 2-u4

21. 2, 4v2+2v, x4-x2

23. 4, 0, x2+2xh+h2+2x+2h+1

25.

27. 0, 256, 29. (a) 4x+4h-5; (b) 4

31. (a) x2+2hx+h2+2x+2h; (b) 2x+h+233. (a) 2-4x-4h-3x2-6hx-3h2;

(b) –4-6x-3h 35. (a) ; (b)

37. 5 39. y is a function of x; x is a function of y.41. y is a function of x; x is not a function of y.43. Yes 45. V=f(t)=20,000+800t47. Yes; P; q 49. 402.72 pounds per week;935.52 pounds per week; amount supplied increases as theprice increases 51. (a) 4; (b) 8 ;(c) f(2I0)=2 f(I0); doubling the intensity increases the response by a factor of 253. (a) 3000, 2900, 2300, 2000; 12, 10;(b) 10, 12, 17, 20; 3000, 2300 55. (a) –5.13; (b) 2.64;(c) –17.43 57. (a) 11.33; (b) 50.62; (c) 2.29


1. (a) p(n)=$125; (b) The premiums do not change;(c) constant function2. (a) quadratic function; (b) 2; (c) 3

3. 4. 7!=5040


1. Yes 3. No 5. Yes 7. No9. All real numbers 11. All real numbers13. (a) 3; (b) 7 15. (a) 4; (b) –3 17. 8, 8, 819. 1, –1, 0, –1 21. 7, 2, 2, 2 23. 720 25. 227. 5 29. c(i)=$4.50; constant function31. (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

174

, 334

964

c 1n 2 = e9.50n

8.75n

ifif

n 6 12,n � 12

c 1n 2 = •3.50n

3.00n

2.75n

ififif

n � 55 6 n � 10n 7 1

312

312

312

-

1x 1x + h 2

1x + h

116

-2

27,

3x - 79x2 + 2

, x + h - 7

x2 + 2xh + h2 + 2

-

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

a -15

, 1 ba-q,

52da2

3, qb

c163

, qbc 12

, 34d

12

, 325

15 - 2

–2)0

0))

–12)

179

a179

, qb–5

)48


41. (a) 2.21; (b) 9.98; (c) –14.52


1. c(s(x))=c(x+3)=2(x+3)=2x+62. 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; (j) 11

3. (a) 2x2-x+1; (b) x+1; (c) ;

(d) x4-x3+x2-x; (e) ; (f) ;

(g) x4-2x3+x2+1; (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-450019. 400m-10m2; the total revenue received when the total output of m employees is sold21. (a) 14.05; (b) 1169.64 23. (a) 345.03; (b) –1.94


1. 3. F–1(x)=2x+14

5. 7. f(x)=5x+12 is one-to-one

9. h(x)=(5x+12)2, for , is one-to-one

11.


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

7. (0, 0); function; one-to-one; all real numbers;all real numbers

9. (0, –5), ; function; one-to-one; all real numbers;

all real numbers

x

y

5

–5

3

a53

, 0b

x

y

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

36

Mile

s

(0, 0)

(5, 0)

(2.5, 30)

a121

12, 0b

x =117

3+

23

x � -5

12

r 1A 2 = AA

p

f- 1 1x 2 =x

3-

73

x2 - 1x + 3

14 x

1x

1v + 3

; B2w2 + 3w2 + 1

41 t - 1 2 2 +

14t - 1

+ 1; 2

t2 + 7t

53

x2 + 1x2 - x

12

x + 3x + 5




11. (0, 0); function; not one-to-one; all real numbers;all nonnegative real numbers

13. Every point on y-axis; not a function of x

15. (0, 0); function; one-to-one; all real numbers; all realnumbers

17. (0, 0); not a function of x

19. (0, 2), (1, 0); function; one-to-one; all real numbers; allreal numbers

21. All real numbers; all real numbers � 4;(0, 4), (2, 0), (–2, 0)

23. All real numbers; 3; (0, 3)

25. All real numbers; all real numbers � –3; (0, 1), (2_ , 0)

27. All real numbers; all real numbers; (0, 0)

29. All real numbers � 5; all nonnegative real numbers;(5, 0)

r

s

5

t

f (t )

x

y

2 +(2, –3)

3

2 – 3

1

13

x

y

3

t

s

2–2

4

x

y

2

1

x

y

x

y

x

y

x

y


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 1 � c<8

37. All real numbers; all nonnegative real numbers

39. (a), (b), (d)

41. y=1800-175x, , (0, 1800)

43. As price increases, quantity increases; p is a function of q.

45.

47. 0.39 49. –0.61, –0.04 51. –1.1253. –0.57, 0, 0.76, 2.31 55. (a) 19.60; (b) –10.8657. (a) 5; (b) 4 59. (a) 28; (b) [–q, 28];(c) –4.02, 0.60 61. (a) 34.21; (b) 18.68;(c) [18.68, 34.21]; (d) no intercept


1. (0, 0); sym. about origin3. (—2, 0), (0, 8); sym. about y-axis

5. ; sym. about x-axis, y-axis, origin

7. (–2, 0); sym. about x-axis 9. Sym. about x-axis11. (–21, 0), (0, –7), (0, 3)

13. (0, 0); sym. about origin 15.

17. (3, 0), (0, —3); sym. about x-axis

x

y

3

3

–3

a0, 38b

a ;54

, 0 b

x

y

7 14 21

300

1000

q

p

21030 90 150

50

10

30

a1027

, 0 b

x

g(x)

3

9

7

8

5

p

c

t

F(t )

x

f(x)

1

1

2

a12

, 0b




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]27.


1.

3.

5.

7.

9.

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

21

1 2x

y

23

23

x

y

–2

–4

4

2

x

y

x

y

2–2


11.

13. Translate 2 units to the right and 1 unit upward15. Reflect about the y-axis and translate 5 units downward


1. All real numbers except 1 and 5 3. All real numbers5. All nonnegative real numbers except 17. 7, 46, 62, 3t2-4t+7 9. 0, 2,

11. 13. 20, –3, –3, undefined

15. (a) 3-7x-7h; (b) –717. (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) 6x+8;

(h) 38; (i) 6x+1 21.

23. , (x+2)3/¤

25. (0, 0), (— , 0); sym. about origin27. (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, 1.7341. –1.50, –0.88, –0.11, 1.09, 1.4043. (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=06.

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

13

2x3 + 2

1x - 1

, 1x

- 1 =1 - x

x

3x - 12x + 3

35

, 0, 1x + 4

x, 1u

u - 4

1t, 2x3 - 1

x

y

f(x) = xy = –x




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. 7x+y-2=0 11. x+4y-18=013. 3x-7y+25=0 15. 8x-5y-29=017. 2x-y+4=0 19. x+2y+6=021. y+3=0 23. x-2=0 25. 4; –6

27. 29. Slope undefined; no y-intercept

31. 3; 0 33. 0; 3

35. 2x+3y-5=0; y=

37. 4x+9y-5=0; y=

39. 3x-2y+24=0; y=

41. Parallel 43. Parallel 45. Neither47. Perpendicular 49. Perpendicular

51. y=4x+14 53. y=1 55. y=

57. x=5 59. y= 61. (5, –4)

63. –2; the stock price dropped an average of $2 per year65. y=28,000x-100,000 67. –t+d-1874=071. C=59.82T+769.58 73. R=50,000T+80,00075. The slope is 3.4.


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. 5; –7

5.

7. f(x)=4x 9. f(x)=–2x+4

11. f(x)= 13. f(x)=x+1

15. p= +28.75; $13.95 17. p= q+190

19. c=3q+10; $115 21. f(x)=0.125x+4.1523. v=–800t+8000; slope=–800

25. f(x)=45,000x+735,000 27. f(x)=64x+95

29. x+10y=100 31. (a) ; (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


1. Vertex: (1, 400); y-intercept: (0, 399);x-intercepts: (–19, 0), (21, 0)

x

y

25–25

100

400

14

c + 37;

y =5

11x +

60011

v

t10

8000

14

-25

q

-12

x +154

q

h(q)

27

-17

; 27

t

h(t )

—7

x

y

x

f (x)

2010

1000

500

-38

q + 1025

-23

x -293

-13

x + 5

32

x + 12

-49

x +59

-23

x +53

-

12

; 32

-

12

CABCAB


2. Vertex: (1, 24); y-intercept: (0, 8);

x-intercepts:

3. 1000 units; $3000 maximum revenue


1. Quadratic 3. Not Quadratic 5. Quadratic7. Quadratic 9. (a) (1, 11); (b) highest

11. (a) –6; (b) –3, 2; (c)

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, 14); range: all t � –2

23. Minimum; 25. Maximum; –10

27. 200 units; $120,000 maximum revenue29. 200 units; $240,000 maximum revenue31. Vertex: (9, 225); y-intercept: (0, 144);

x-intercepts: (–6, 0), (24, 0)

33. 70 grams 35.≠134.86 ft;≠2.7 sec

P (x)

400

x30–20

80849

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

a -12

, -254b

x

y

5–5

30

a1 +162

, 0 b , a1 -162

, 0 b




37. Vertex: ; y-intercept: (0, 16),

x-intercepts:

39. 50 ft � 100 ft 41. (1.11, 2.88)43. (a) 0; (b) 1; (c) 2 45. 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


1. x=–1, y=1 3. x=3, y=–15. u=6, v=–1 7. x=–3, y=29. No solution 11. x=12, y=–12

13. p= -3r, q=r; r is any real number

15. 17. x=2, y=–1, z=4

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

numbers25. 420 gal of 20% solution, 280 gal of 30% solution27. 0.5 lb of cotton; 0.25 lb of polyester; 0.25 lb of nylon29.≠285 mi/h (speed of airplane in still air),

≠23.2 mi/h (speed of wind)31. 240 units of early American, 200 units of Contemporary33. 800 calculators at Exton plant, 700 at Whyton plant35. 4% on first $100,000, 6% on remainder37. 60 units of Argon I, 40 units of Argon II39. 100 chairs, 100 rockers, 200 chaise lounges41. 10 semiskilled workers, 5 skilled workers, 55 shippingclerks 45. x=3, y=2 47. x=8.3, y=14.0


1. x=4, y=–12; x=–1, y=33. p=–3, q=–4; p=2, q=15. x=0, y=0; x=1, y=1.

7. x=4, y=8; x=–1, y=39. p=0, q=0; p=1, q=111. x= , y=2; x=– , y=2; x= ,y=–1; x= , y=–1 13. x≠13.53, y≠19.0615. At (10, 8.1) and (–10, 7.9) 17. Three19. x=–1.3, y=5.1 21. x=1.76 23. x=–1.46


1. The equilibrium point is (100, 7).

3. (5, 212.50) 5. (9, 38) 7. (15, 5)9.

11. Cannot break even at any level of production13. Cannot break even at any level of production15. (a) $12; (b) $12.1817. 5840 units; 840 units; 1840 units 19. $421. Total cost always exceeds total revenue—no break-evenpoint. 23. Decreases by $0.7025. pA=8; 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= -3; 2x-5y-15=0 11. Perpendicular

13. Neither 15. Parallel 17. y=

19. y=

21. –5; (0, 17)

x

y

175

17

43

; 0

32

x - 2; 32

25

x

12

x

q

yTR

TC

2000 6000

15,000(4500, 13,500)

5000

q

p

100 200

10

(100, 7)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

12

13

16

20,0003

-43

r

x

h(t)

10–10

160

a 5 + 1292

, 0 b , a 5 - 1292

, 0 ba5

2, 116b


23. (3, 0), (–3, 0), (0, 9); (0, 9)

25. (5, 0), (–1, 0), (0, –5); (2, –9)

27. 3; (0, 0)

29. (0, –3); (–1, –2)

31. 33. x=2, y=

35. x=8, y=4 37. x=0, y=1, z=039. x=–3, y=–4; x=2, y=141. x=–2-2r, y=7+r, z=r; r is any real number43. x=r, y=r, z=0; r is any real number

45. a-4b=–7; 13 47. f(x)=

49. 50 units; $5000 51. 6 53. 1250 units; $20,00055. 2.36 tons per square km 57. x=7.29, y=–0.7859. 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. 117 employees

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

-95

x =177

, y = -87

x

y

–1 – 2– 3

t

p

t

y

2 5–1

–9

–5

x

y

3–3

9




7.


1. 3.

5. 7.

9. 11.

13. B 15. 138,750

17.

19. (a) $6014.52;(b) $2014.52

21. (a) $1964.76;(b) $1264.76

23. (a) $11,983.37;(b) $8983.37

25. (a) $6256.36;(b) $1256.36

27. (a) $9649.69; (b) $1649.69 29. $10,446.1531. (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. 334, 485 37. 4.4817 39. 0.496641. 43. 0.2240

45. (ek)t, where b=ek

47. (a) 12; (b) 8.8;(c) 3.1; (d) 22 hours

49. 32 years51. 0.146555. 3.1757. 4.2 min59. 17


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 20.0855=3

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

1

–1

–2

2

3

4

5

6

7

8

9

–1 1 2 3–2x

y

x

y

1

3

1

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


7. e1.09861=39. 11.

13.

15. 17. 2 19. 321. 1 23. –225. 0 27. –329. 81 31. 125

33. 35. e

37. 2 39. 6

41. 43. 2

45. 47. 4 49. 51.

53. 1.60944 55. 2.00013 57. y=log1.10 x

59. 3 61. (a) 2N0;(b) k is the time it takes for the population to double.63. 36.1 minutes 65. z=y3/2

67. (a) (0, 1); (b) [–0.37, q) 69. 1.1071. 1.41, 3.06


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 59. log x=

61. ln ∏


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


1. 3 3. 2.75 5. –3 7. 2 9. 0.08311. 1.099 13. 0.028 15. 5.140 17. –0.07319. 2.322 21. 3.183 23. 0.483 25. 2.49627. 1003 29. 2.222 31. 3.082 33. 3 35. 0.537. S=12.4A0.26 41. 20.5

43. 49. 3.33


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. 5

47. 149. 1051. 2e

53. 0.88055. –3.22257. –1.59659. (a) $3829.04;

(b) $1229.0461. 14%

63. (a) P=8000(1.02)t; (b) 832365. (a) 10 mg; (b) 4.4; (c) 0.2; (d) 1.7; (e) 5.667. (a) 6; (b) 28 71. (–q, 0.37] 73. 2.53

x

y

–3

1

8

y = ex2 + 2

2y +12

xln 1x + 5 2

ln 3

12

13

x9>21x + 1 2 3 1x + 2 2 4

x2y

z3

73

52

1100

p =log 180 - q 2

log 2; 4.32

ln x

ln 10z

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

5 + ln 32

ln 23

53

127

-3110

x

y

e1

1

–1

–2

x

y

4 6

1

x

y

4

1

1

–1x

y

31

1




75.


1. (a) ; (b)

3. (a) 156; (b) 65


1. 4.9% 2. 7 years, 16 days 3. 7.7208%4. 11.25% compounded quarterly has the better effectiverate of interest. The $10,000 investment is slightly betterover 20 years.


1. (a) $11,105.58; (b) $5105.58 3. 3.023%5. 4.081% 7. (a) 10%; (b) 10.25%; (c) 10.381%;(d) 10.471%; (e) 10.516% 9. 8.08% 11. 9.0 years13. $10,282.95 15. $38,503.2317. (a) 18%; (b) $19.56% 19. $3198.5421. 8% compounded annually 23. (a) 5.47%;(b) 5.39% 25. 11.61% 27. 6.29%


1. $2261.34 3. $1751.83 5. $5118.107. $4862.31 9. $6838.95 11. $11,381.8913. $14,091.10 15. $1238.58 17. $3244.6319. (a) $515.62; (b) profitable 21. Savings account23. $226.25 25. 9.55%


1. 48 ft, 36 ft, 27 ft, 20 ft, 15 ft

2. 500(1.5), 500(1.5)2, 500(1.5)3, 500(1.5)4, 500(1.5)5,500(1.5)6 or 750, 1125, 1688, 2531, 3797, 5695 3. 35.72 m4. $176,994.65 5. 6.20% 6. $101,925; $121,9257. $723.03 8. $13,962.01 9. $45,502.0610. $48,095.67


1. 64, 32, 16, 8, 4 3. 100, 102, 104.04 5.

7. 1.11111 9. 18.664613 11. 8.21318013. $2050.10 15. $29,984.06 17. $8001.2419. $90,231.01 21. $204,977.46 23. $24,594.3625. $1937.14 27. $458.4029. (a) $3048.85; (b) $648.85 31. $3474.1233. $1725 35. 102.91305 37. 55,360.3039. $131.34 41. $1,872,984.0243. $205,073; $142,146


1. $69.33 3. $502.845. (a) $221.43; (b) $25; (c) $196.43

7.

9.

11. 11 13. $1415.5615. (a) $2089.69; (b) $1878.33; (c) $211.36; (d) $381,90717. 23 19. $113,302.45 21. $38.64


1. 3. 8.5% compounded annually

5. $586.60 7. (a) $1997.13; (b) $3325.379. $936.85 11. $886.98 13. $314.0015.

17. $1279.36


1. $15,597.85 3. When investors expect a drop in interest rates, long-term investments become more attractive relative to short-term ones.

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

66581



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

21,04416,807

316

14

d =1

kI ln c P

P - T 1ekI - 1 2 dP =T 1ekI - 1 21 - e-dkI

10–10

7

–2



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, J3. 6 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. 1 7. 2*2; 49. 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. $207565. $828,950

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. 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§

c32

1-9d 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§

£73

00

073

0

0073

§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 8553dcy

xdc1

1

8513d

c-1024

2236

12-44d

c154

-47

2630d£

357525

655515§

43

9029

, y = -2429

c-16

5-8dc4

72

-3202dc21

192

292

- 152d

c-22-11

-159dc 28

-2226d

c 6-2

53d£

3-2

5

-58

10

60

15§

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

¥c 6-3

24d

F000000

000000

000000

000000

000000

000000

V≥0000

0000

0000

0000

¥

£61014

81216

101418

121620§

£111

222

444

888

161616§





1. 5 blocks of A, 2 blocks of B, and 1 block of C2. 3 of X; 4 of Y; 2 of Z 3. A=3D; B=1000-2D; C=500-D; D=any amount between 0 and 500


1. Not reduced 3. Reduced 5. Not reduced

7. 9. 11.

13. x=5, y=2 15. No solution

17. x= where r is any

real number 19. No solution21. x=–3, y=2, z=0 23. x=2, y=–5, z=–125. x1=0, x2=–r, x3=–r, x4=–r, x5=r, where r is any real number 27. Federal, $72,000; state, $24,00029. 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 Z33. (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=–1+7r, x=2-5r, y=4-7r, z=r, (where r is any real number)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 solution13. Infinitely many 15. x=0, y=0

17. 19. x=0, y=0

21. x=r, y=–2r, z=r23. 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=–2023. 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. 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=029. A2=I£, A–1=A, A¤‚‚‚=I£

c- 3212

56

- 16d

£100

200

010§c1

001d

c20

09dc -1

2-2

1dc 6

32d

c -15

-222d£

121

42-18

0

5-7-2§c 3

-168

-10d

£10731016952§£

130112151188

§

£68.5984.50

108.69§£

134.29162.25234.35

§c12901425

d ; 1405

£1.800.350.44

1.101.310.42

-0.46-0.17

0.59§

c 84413041

105827041dc2.05

0.731.281.71d ;

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


31. (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

33. 35.


1. $151.40 3. It is not possible; different combinationsof lengths of stays can cost the same.


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 shaded 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.

21. 23.

25. 27.

29. x � 0, y � 0, 3x+2y � 240, 0.5x+y � 80


1. P=75 when x=15, y=03. Z=–10 when x=2, y=35. 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. 10 trucks, 20 spinning tops; $11015. 4 units of food A, 4 units of food B; $817. 10 tons of ore I, 10 tons of ore II; $110019. 6 chambers of type A and 10 chambers of type B21. (c) x=y=7523. Z=15.54 when x=2.56, y=6.74

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

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

c168171.6

dc21589

87141d

combination 1combination 2combination 3combination 4

x

4321

y

9630

z

0123



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 � 13. 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=43. Z=2 when x1=0, x2=15. Z=28 when x1=3, x2=27. Z=20 when x1=0, x2=5, x3=09. Z=2 when x1=1, x2=0, x3=0

11. when x1= , x2=

13. W=13 when x1=1, x2=0, x3=315. Z=600 when x1=4, x2=1, x3=4, x4=017. 0 from A, 2400 from B; $120019. 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 � 17. No optimum solution (unbounded)

9. Z=13 when x1= x2=6t, x3=4-3t, and

0 � t � 111. $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=05. Z=28 when x1=8, x2=2, x3=07. Z=–17 when x1=3, x2=29. No optimum solution (empty feasible region)11. Z=2 when x1=6, x2=1013. 255 Standard bookcases, 0 Executive bookcases15. 30% in A, 0% in AA, 70% in AAA; 6.6%


1. Z=54 when x1=2, x2=83. Z=216 when x1=18, x2=0, x3=05. Z=4 when x1=0, x2=0, x3=47. Z=0 when x1=3, x2=0, x3=19. Z=28 when x1=3, x2=0, x3=511. 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.


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 � 02. Maximize W=98y1+80y2 subject to20y1+8y2 � 66y1+16y2 � 2and y1, y2 � 03. 5 device 1, 0 device 2, 15 device 3


1. Minimize W=5y1+3y2 subject toy1-y2 � 1y1+y2 � 2y1, y2 � 03. Maximize W=8y1+2y2 subject toy1-y2 � 1y1+2y2 � 8y1+y2 � 5y1, y2 � 05. Minimize W=13y1-3y2-11y3 subject to–y1+y2-y3 � 12y1-y2-y3 � –1y1, y2, y3 � 07. Maximize W=–3y1+3y2 subject to–y1+y2 � 4y1-y2 � 4y1+y2 � 6y1, y2 � 0

9. Z=11 when x1=0, x2= , x3=

11. Z=26 when x1=6, x2=113. Z=14 when x1=1, x2=2

32

12

32

-32

t,

31

62

143

23

Z =163



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=013. Z=–2 when x=0, y=215. No optimum solution (empty feasible region)17. Z=36 when x=2+2t, y=3-3t, and 0 � t � 119. Z=32 when x1=8, x2=0

21. Z= 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=031. 0 units of X, 6 units of Y, 14 units of Z; $39833. 500,000 gal from A to D, 100,000 gal from A to C,400,000 gal from B to C; $19,00035. 10 kg of food A only37. Z=129.83 when x=9.38, y=1.63


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


1.

Start

AD

E

6 possible production routes

Assemblyline

Finishingline

Productionroute

BD

E

CD

E

AD

AEBD

BECD

CE

94

54

72

53

x

y

x

y

x

y

x

y

–3/2x

y

2

– 3



3.

5. 20 7. 96 9. 1024 11. 20 13. 72015. 720 17. 1000; error message is displayed19. 6 21. 336 23. 216 25. 1320 27. 36029. 720 31. 2520; 5040 33. 624 35. 2437. (a) 11,880; (b) 19,008 39. 48 41. 2880


1. 15 3. 1 5. 18 9. 2380 11. 715

13. 15. 56 17. 1680 19. 35

21. 720 23. 1680 25. 252 27. 756,75629. (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. {64, 69, 60, 61, 46, 49, 40, 41, 96, 94, 90, 91, 06, 04, 09, 01, 16,14, 19, 10}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 I27. (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) ≠0.0264140

161,7006545

161,700

13 · 4C3 · 12 · 4C2

52C5

111024

1210 =

11024

;

110

15

45

;78

18

;38

;18

;

6084132,600

=39

8504 · 3 · 2132,600

=1

5525

39624

=116

8624

=178

;

4624

=1

156;

1624

;126

;

413

;152

;12

;12

;113

;14

;152

;

56

12

;12

;136

;14

;112

;536

;

74!10! # 64!

1

RedDie

1 1, 1

36 possible results

GreenDie Result

2 1, 2

3 1, 3

4 1, 4

5 1, 5

6 1, 6

2

1 2, 1

2 2, 2

3 2, 3

4 2, 4

5 2, 5

6 2, 6

3

1 3, 1

2 3, 2

3 3, 3

4 3, 4

5 3, 5

6 3, 6

4

1 4, 1

2 4, 2

3 4, 3

4 4, 4

5 4, 5

6 4, 6

5

1 5, 1

2 5, 2

3 5, 3

4 5, 4

5 5, 5

6 5, 6

6

1 6, 1

2 6, 2

3 6, 3

4 6, 4

5 6, 5

6 6, 6

Start



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) 0 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. Dependent13. (a) Independent; (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. ≠0.453.

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


1. 336 3. 36 5. 608,400 7. 32 9. 21011. 462 13. (a) 2024; (b) 253 15. 34,65017. 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) no21. (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) 5.8; (c) 1.56

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. $101.4317. $3.00 19. $410 21. Loss of $0.25; $1

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

211

;

1013

67

14

14

;

215

425

;45512

79

23200

2441

41200

1415

45

2429

34

2749

8189

631

58

143021

258937

175386

47

14

,

53512

164

;15

1024;

38

11728

;

3200

139361

415

215

;

215

1315

;715

;15

;25

;

110

140

;310

;3

676

125

118

56

13

12

;112

;23

;13

;56

;14

;

431

125

14

920

35

34

;

2747

47100

;217

11850

816,575

4051

113

23

16

111

23

14

12

;23

49

12

;

2586

1047

;825

;1139

;3558

;58

;

29

12

;25

;35

;23

12

;

12

;14

;45

;15

;

27

712

19




1.


1. Â= ; Í=

3. Â=2;

Í= 5. 0.001536 7. 9.

11. ≠0.081 13. ≠0.3545 15. 0.002

17. (a) (b) 19. ≠0.655 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.

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) $704,0009.

Â=0.6; Í≠0.71

11. 13. 15.

17. a=0.3, b=0.2, c=0.519. X1= , X2= ,X3=

21. (a) T2= T3= ; (b) ;

(c) 23.

25. (a) 76%; (b) 74.4% Japanese, 25.6% non-Japanese;

c12

12d117

343

3049

≥109343117343

234343226343

¥≥19491549

30493449

¥,

30.1310 0.1595 0.7095 4 30.130 0.155 0.715430.10 0.15 0.754

1127

881

164

f 13 2 = 0.011, f 14 2 = 0.0005,f 10 2 = 0.522, f 11 2 = 0.368, f 12 2 = 0.098,

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

FluNo Flu

c0.10.2

0.90.8d ;

Flu No Flu

30.5 0.25 0.254c 37

47dc 4

737d

£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

≥7

16916

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

20483125

532

964

;

12253456

1652048

316

96625

= 0.1536163

f 10 2 =1

27, f 11 2 =

29

, f 12 2 =49

, f 13 2 =8

27;

164

12

f 10 2 =9

16, f 11 2 =

38

, f 12 2 =1

16;


x P(x)

0

1

2

3

481

10,000

75610,000

264610,000

411610,000

240110,000


(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. ∏ 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)=–3.7, f(–0.99)=–3.97,f(–0.999)=–3.997, f(–1.001)=–4.003,f(–1.01)=–4.03, f(–1.1)=–4.3; –47. 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. 3x2-8x

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)=q. 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. 15. –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 37. 39. 41. q

43. q 45. q 47. Does not exist 49. –q51. 0 53. 155. (a) 1; (b) 2; (c) does not exist; (d) 1; (e) 257. (a) 0; (b) 0; (c) 0; (d) –q; (e) –q59. 61. 20,000 63. 20

65. 1, 0.5, 0.525, 0.631, 0.912, 0.986, 0.998; conclude limit is 167. 0 69. (a) 11; (b) 9; (c) does not exist


1. $5819.97; $1819.97 3. $1456.87 5. 4.08%7. 3.05% 9. $109.42 11. $778,800.7813. (a) $39,066; (b) $13,671 15. $4.88%17. $1264 19. 16 years21. (a) $1072.51; (b) $1093.30; (c) $1072.1823. (a) $9458.51; (b) This strategy is better by $26.90.


7. Continuous at –2 and 0 9. Discontinuous at —411. Continuous at 2 and 0 13. f is a polynomial function15. f is a rational function and the denominator is never zero.17. None 19. x=–4 21. None 23. x=–5, 325. x=0, —1 27. None 29. x=0 31. None33. x=235. Discontinuities at t=1, 2, 3, 4

37. Yes, no, no

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

25

-

25

limxSq

limxSq

limxSq

14

119

-

15

16

-

52

43

≥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

¥




1. 0<x<4


1. (–q, –1), (4, q) 3. [3, 5] 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 65 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. 831. 23 33. (a) $5034.38; (b) $1241.46 35. 6.18%

37.

41. Continuous everywhere; f is a polynomial function.43. x=–3 45. None 47. x=–4, 149. 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. 1.0061. 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. 3 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


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. 2x+4 73. 1 75. 4, 16, –14 77. 0, 0, 079. 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 when t=1 to be 7.0000 m/s.With differentiation the velocity is 7 m/s.3. (a) 4 m; (b) 5.5 m/s; (c) 5 m/s5. (a) 8 m; (b) 6.1208 m/s; (c) 6 m/s7. (a) 2 m; (b) 10.261 m/s; (c) 9 m/s

9. 11. 0.27

13. dc/dq=10; 10 15. dc/dq=0.6q+2; 3.817. dc/dq=2q+50; 80, 82, 8419. dc/dq=0.02q+5; 6, 7

dy

dx=

252

x3>2; 337.50

dy

dt`t = 0.5

dy

dt= 16 - 32t

a 53

, 12554b

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

725

t683

x3

r

rL - r -dC

dD

-3x + 9

121x + 2

-

6x2

dH

dt

ln 20.065

19

59

-83

1313

a-

72

, -2b


≤t 1 0.5 0.2 0.1 0.01 0.001

≤s/≤t 9 8 7.4 7.2 7.02 7.002

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


21. dc/dq=0.00006q2-0.02q+6; 4.6, 1123. dr/dq=0.8; 0.8, 0.8, 0.825. dr/dq=250+90q-3q2; 625, 850, 62527. dc/dq=6.750-0.000656q; 3.4729. P=5,000,000R–0.93; dP/dR=–4,650,000R–1.93

31. (a) –7.5; (b) 4.5

33. (a) 1; (b) (c) 1; (d) ≠0.111; (e) 11.1%

35. (a) 6x; (b) (c) 12; (d) ≠0.632;

(e) 63.2% 37. (a) –3x2; (b) (c) –3;

(d) ≠–0.429; (e) –42.9% 39. 3.2; 21.3%

41. (a) dr/dq=30-0.6q; (b) ≠0.089; (c) 9%

43. 45. $3125 47. $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. (5-3t)(3t2-4t)+(t3-2t2)(–3)

=–12t3+33t2-20t5. (3r2-4)(2r-5)+(r2-5r+1)(6r)

=12r3-45r2-2r+207. 8x3-10x9. (x2+3x-2)(4x-1)+(2x2-x-3)(2x+3)

=8x3+15x2-20x-711. (8w2+2w-3)(15w2)+(5w3+2)(16w+2)

=200w4+40w3-45w2+32w+413. (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. 4v3+

37. 39.

41.

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.


1. 288t


1. (2u-2)(2x-1)=4x3-6x2-2x+2

3. 5. –2 7. 0

9. 18(3x+2)5 11. 30x2(3+2x3)4

13. 200(3x2-16x+1)(x3-8x2+x)99

15. –6x(x2-2)–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)

= 10x2 15x + 1 2-12 + 8x15x + 1

4x2 c 1215x + 1 2-1

2 15 2 d + 115x + 1 2 18x 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

-

1120

0.735511 + 0.02744x 2 2

910

dc

dq=

5q 1q + 6 21q + 3 2 2

76

; 16

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

=- 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

8v2-

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

34a45p

12 - 12 - 5p

-12 b

32c 15p

12 - 2 2 13 2 + 13p - 1 2 a5 �

12

p- 1

2 b d

dR

dx=

0.432t

445

-

37

-3x2

8 - x3;

1219

6x

3x2 + 7;

19

1x + 4

;



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. –481.5 73. 75. 48∏(10)–19

77. (a) –0.001416x‹+0.01356x¤+1.696x-34.9,–136.188; (b) –0.008 79. –4 81. 4083. 86,111.37


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)(2x2+4x)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.

=

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, 0.43149. dr/dq=450-q51. 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. 0.30565. –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. +3x2 ln(2x+5) 15.

17.

19.z a 1

zb - 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

2x3

2x + 5

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

-

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



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. 5ex 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. –e31. 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. –539. e 41. 100e–2 47. –b(10A-bM) ln 1051. 0.0036 53. 0.68


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% decrease23. (b) Ó=–2.5, elastic; (c) 1 unit;(d) increase, since demand is elastic

25. (a) Ó= =–13.8, elastic; (b) 27.6%;

(c) increase, since demand is elastic

27. Ó=–1.6;

29. Maximum at q=5; minimum at q=95


1.

2. =4∏r¤ and =2880∏ in3/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. 39.


1. (x+1)2(x-2)(x2+3)

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. 43 and 1958

13e1.3

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 + 3 2 1x - 2 2

2x - 1c 1x + 3

+1

x - 2-

22x - 1

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

13x3 - 1 2 2 12x + 5 2 3 c 18x2

3x3 - 1+

62x + 5

dc 2x + 1

+1

x - 2+

2x

x2 + 3d

38

-f

l

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

4y3>42y1>4 + 1

4y - 2x2

y2 - 4x

11 - y

x - 1

-

y

x-

y1>4x1>4-

1y1x

73y2-

x

4y

94

dV

dt`

r = 12

dr

dt

dV

dt

dP

dt= 0.5 1P - P2 2

dr

dq= 30

-20715

310

103

-12

-9

32

- a 150e

- 1 b-

5352

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 - l2

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




1. 0.25410 3. 1.32472 5. –0.68233 7. 0.337679. 1.90785 11. 4.141 13. –4.99 and 1.9415. 13.33 17. 2.880 19. 3.45


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. —141. –4.99 and 1.94


1. 3ex+0+ (2x)+(e2) =3ex+2x +e2

3.

5. (2x+4)=2(x+2)7. ex(2x)+(x2+2)ex=ex(x2+2x+2)

9.

11. =

13. 15. –7(ln 10)102-7x

17. 19.

21. 23. (x+1)x+1[1+ln(x+1)]

25.

27.

,

where y is as given in the problem29. (xx)x(x+2x ln x) 31. 4 33. –235. 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. 0.9055. Ó=–1, unit elasticity57. Ó=–0.5, demand is inelastic

59. ,≠ % increase 61. 1.7693


1. Figure 12.11 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 q=2, and a relativeminimum when q=5.2. 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. max5. Inc. on (–q, –2) and (1, q); dec. on (–2, 1); rel max. when x=–2; rel. min. when x=17. Dec. on (–q, –1); inc. on (–1, 3) and (3, q); rel. min. when x=–19. 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=115. Dec. on (–q, –1) and (0, 1); inc. on (–1, 0) and (1, q);rel. max. when x=0; rel. min. when x=—117. 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 (–q, 5- ) and (5+ , q);dec. on (5- , 5+ ); rel. max. when x=5- ;rel. min. when x=5+23. Inc. on (–q, –1) and (1, q); dec. on (–1, 0) and (0, 1);rel. max. when x=–1; rel. min. when x=125. Dec. on (–q, –4) and (0, q); inc. on (–4, 0); rel. min. when x=–4; rel. max. when x=027. Inc. on (–q, ) and (0, ); dec. on ( , 0) and ( , q); rel. max. when x=— ; rel. min. when x=01212

-1212-12

13131313

1313

52

-23

a -23

, 52ba 5

2, q ba - q, -

23b

12

a 12

, q b ;a - q, 12b

38

-916

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

1t

+12

� 1

4 - t2 � 1-2t 2 =1t

-t

4 - t2

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 2xe2 - 1ex2

xe2 - 1ex2

-

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

812x + 3 2 3-

14 19 - r 2 3>2

-

10p6

d2h

dt2 = -32


HP-IMA11e_Ans_13_AppA 3/4/04 4:33 AM Varekai:Users:jeanmarieperchalski:Desktop:CTA-HPIMA11e:HP 11e files:

29. Inc. on (–q, –1), (–1, 0), and (0, q); never dec.;no rel. extremum31. Dec. on (–q, 1) and (1, q); no rel. extremum33. Dec. on (0, q); no rel. extremum35. Dec. on (–q, 0) and (4, q); inc. on (0, 2) and (2, 4);rel. min. when x=0; rel. max. when x=437. 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=051. Dec. on (0, 1); inc. on (1, q); rel. min. when x=153. 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)

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)

x

y

1 4

1

x

y

1– 2

4

-12

a -2, -12ba -

12

, 1 b

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 - 1295

a -2 - 1295

, -2 + 129

5ba -2 + 129

5, q ba - q,

-2 - 1295

b



65. 69. Never

71. 40 75. (a) 25,300; (b) 4; (c) 17,20077. 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, 1.7783. (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(–1)= ; minimum: f(0)=1

5. Maximum: f(3)=84; minimum: f(1)=–87. Maximum: f(–2)=56; minimum: f(–1)=–29. Maximum: f( )=4; minimum f(2)=–1611. Maximum: f(0)=f(3)=2;

minimum:

13. Maximum: f(3)≠2.08; minimum: f(0)=015. (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=75. 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=—115. 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=–233. Conc. down (0, e3/2); conc. up (e3/2, q); inf. pt. when x=e3/2

35. Int. (–2, 0), (3, 0), (0, –6); dec. ;

inc. ; rel. min. when x= ; conc. up (–q, q)

37. Int. (0, 0), (4, 0); inc. (–q, 2); dec. (2, q); rel. max. when x=2; conc. down (–q, q)

x

y

x

y

12

a 12

, q ba - q,

12b

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

56

a 56

, q ba - q, 56b

12121212

q, 1 2 , 11, 7 20,

32

a0, 32ba 3

2, q b

f a 3122b = -

734

12

196

x

y

1 3

2

1



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.

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

x

y

10, 2>3 2

x

y

1; 415, 0 2

x

y

x

y

x

y

12>3, q 2 10, 2>3 214>3, 0 2

x

y

x

y

213

x

y



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 up79. –2.61, –0.26


1. Rel. min. when x= ; abs. min.

3. Rel. max. when x= ; abs. max.

5. Rel. max. when x=–5; rel. min. when x=17. Rel. min. when x=0; rel. max. when x=29. Test fails, when x=0 there is a rel. min. by first-deriv. test

11. Rel. max. when x= ; rel. min. when x=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



13. Rel. min. when x=–5, –2; rel. max. when x=


1. y=1, x=1 3. y= , x=

5. y=0, x=0 7. y=0, x=1, x=–19. None 11. y=2, x=2, x=–313. y=2, x=– , x= 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=

–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

1515

-32

12

-72



37. Int. ; inc. dec.

; rel. max. when x= ; conc. up

; conc. down ;

asymptotes y=0, x=

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

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

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



47.

49.

55. x≠—2.45, x≠0.67, y=2 57. y≠0.48


1. 41 and 41 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. $2215. 120 units; $86,000 17. 625 units; $419. $17; $86,700 21. 4 ft by 4 ft by 2 ft23. 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. 3533. 60 mi/h 35. 7; $100037. 5- tons; 5- tons 41. 10 cases; $50.55


1. y=3, x=4, x=–4 3. y= , x=

5. x=0 7. x= , –1

9. Inc. (–1, 7); dec. on (–q, –1) and (7, q)11. Dec. on (–q, – ), (0, ), ( , );inc. on (– , – ), (– , 0), ( , q)

13. Conc. up on (–q, 0) and ;

conc. down on

15. Conc. down on ; conc. up on

17. Conc. up on ;

conc. down on

19. Rel. max. at x=1; rel. min. at x=221. 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

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

a0, 12b

a 12

, q b16131316

16131316

-158

-23

59

1313

911

x

y

–1 2

x

y

1

2



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) false51. q>257. Rel. max. (–1.32, 12.28); rel. min. (0.44, 1.29)59. x≠–0.60 61. 20 63. 200 65. $280067. 100 ft by 200 ft 69. (a) 200 stands at $120 per stand;(b) 300 stands


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


1. 3 dx 3. dx 5.

7. 9. 3 +3(12x2+4x+3) dx

11. ≤y=–0.14, dy=–0.1413. ≤y=–2.5, dy=–2.7515. ≤y≠0.049, dy=0.050 17. (a) –1; (b) 2.9

19. 9.95 21. 23. –0.03 25. 1.01 27.

29. 31. –p2 33. 35.

37. 44; 41.8 39. 2.04 41. 0.743. (1.69*10–11)p cm3 45. (c) 42 units


1.

2.

3.

4.

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


1. 7x+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) might be a nonzero constant

55.


1. N(t)=800t+200et+6317.372. 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

y =x4

12+ x2 - 5x + 13

y = -x4

4+

2x3

3+ x +

1912

y =3x2

2- 4x + 1

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

ue + 1

e + 1+ eu + C

171z2 - 5z 2 + C

w3

2+

23w

+ Cx4

12+

32x2 + C

x3>43

+ 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

4t +t2

2+ C-

56y6>5 + C-

29x9 + C

-5

6x6 + Cx9

9+ C

1500 + 3001t 2dt = 500t + 200t3>2 + C3

-480t3 dt =

240t2 + C3

0.12t2 dt = 0.04t3 + C3

28.3 dq = 28.3q + C3

-45

136

16p 1p2 + 5 2 2

12

4132

e2x22x

x2 + 7 dx

-2x3 dx

2x32x4 - 9

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



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(x2+3) 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. -2x+C

3. (2x3+4x+1)3/2+C

5. 7.

9. 7x2-4 +C

11. |3x-1|+C

13. ln(7e2x+4)+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 7)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.

65. (a) $150 per unit; (b) $15,000; (c) $15,30067. 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

a10

k = 1k2a

4

k = 112k - 1 2a

19

k = 1k

56

-3

16

15

C =34

I -131I +

7112

1I

-200

q 1q + 2 2ln2 x

2+ 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 - 4x

2

x2

2

e1x2 + 32>2

3ln x

ln 3+ C

13

12

1x29

-17

e7>x + C5

14

x2 - 3x +23

ln

e11>42x2

47x

7 ln 4+ C-614 - 5x + C

13

x5

5+

43

x3

Rr2

4K+ B1 ln 0 r 0 + B2

y = -1613 - 2x 2 3 +

112

14

14

-

14

2e1x + C12

29

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

15u2 - 9 2 15

150+ 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




1. $5975


1. square unit 3. square unit

5.

7. (a) ; (b) 9. square unit

11. square unit 13. square units 15. 20

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. 2Ri69. 0.05 71. 3.52 73. 55.39


1. 76.90 feet 2. 5.77 grams


1. 413 3. 0.340; ≠0.333 5. ≠0.767; 0.750

7. 0.883 9. 2,430,733 11. 3.0 square units 13.

15. 0.771 17. km2


In Problems 1–33, answers are assumed to be expressed insquare units.

1. 8 3. 5. 8 7. 9. 9 11.

13. 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= [(7-2x2)-(x2-5)]dx


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. 6.17 square units


1. CS=25.6, PS=38.43. CS=50 ln 2-25, PS=1.255. CS=800, PS=1000 7. $426.67 9. $254,00011. 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.

39. 41. e2x+3x-1y =12

22103x

ln 10+ C

11 + e3x 2 39

+ C4 1x3>2 + 1 2 3>2 + C

32

- 5 ln 23t

-21t

+ C3 312

73

2x

12

227

575

13

4z3>43

-6z5>6

5+ C

y4

4+

2y3

3+

y2

2+ C

11 31114

- 4

-3 1x + 5 2-2 + C

2563

x4

4+ x2 - 7x + C

32m3

2063

25532

- 4 ln 212

431515 - 212 2

443

3281

12512

92

1256

816163

43

32

1

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

1256

503

193

192

356

83

13

3b

a-Ax-Bdx

4712

e3

21e12 - 1 23 -

2e

+1e2

12a e +

1e

- 2 b12

1528

389

34

13

-16

323

53

43

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

1532

23




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) 1253. (a) $2,002,500; (b) 18,000; (c) $111.25


1. S(t)=–40te0.1t+400e0.1t+46002. P(t)=0.025t2-0.05t2 ln t+0.05t2(ln t)2+C


1.

3. 5.

7. x[ln(4x)-1]+C9.

11.

13. 15. e2(3e2-1)

17. (1-e–1), parts not needed

19. 221. 223.

25.

27.

29.

31. 2e3+1 square units 33. square units

37.


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|+C11. –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

7 c 15 15 + 2x 2 +

125

ln ` x

5 + 2x` 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 + 11

x + 2+

21x + 2 2 2

1 +2

x + 2-

8x + 4

12x + 6

-2

x + 1

r 1q 2 =52

ln ` 3 1q + 1 2 3q + 3

`

3f-1 1x 2dx = xf-1 1x 2 - F 1f-1 1x 2 2 + C

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

10 15x + 2 2 2 -1

50 15x + 2 2 + C

= 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



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. $11,050

11. $3155


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. 120 weeks23. N=40,000e0.018t; N=40,000(1.2)t/10; 57,60025. 2e billion 27. 0.01204; 57.57 sec29. 2900 years 31. N=N0 , t � t0

33. 12.6 units 35. A=400(1-e–t/2), 157 g/m2

37. (a) V=21,000e(2 ln 0.9)t; (b) June 2002


1. 58,800 3. 500 5. 1990 7. (b) 3759. 3:21 A.M 11. $62,50013. 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. ln |3x+1|+4 ln |x-2|+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. y=C , C>0 27. 29. Div

31. 144,000 33. 0.0005; 90%

35. N= 37. 4:16 P.M 39. 1

41. (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)

3. (a)

(b) (c) 0; (d) (e) (f) 0; (g) 1; (h) 4; (i) ;213

34

;38

;14

;

x

f(x)

2 6

14

f 1x 2 = •140

,

if 2 � x � 6

, otherwise

-1 + 110

1316

= 0.8125;1116

= 0.6875;512

;

13

4501 + 224e-1.02t

118

ex3 + x2

1x + 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

13

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 = 13 3x - 2

1x2 + 1 2 3>2 + C-1

x2 + C

73

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 + C12

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



(j)

5. (a)

(b) (c)

7. (a) –e–15+e–6≠0.0248; (b) –e–9+1≠1;(c) e–15≠0.000; (d) –e–2+1≠0.865;

(f)

9. (a) (b) (c) (d) 1; (e) (f)

(g) ; (h) 11. 5 min 13. e–3≠0.050


1. (a) 0.4554; (b) 0.3317; (c) 0.8907; (d) 0.9982;(e) 0.8972; (f) 0.4880 3. 0.13 5. –1.08 7. 0.349. (a) 0.9970; (b) 0.0668; (c) 0.0873 11. 0.308513. 0.9078 15. 8 17. 95% 19. 90.82%21. (a) 1.7%; (b) 85.6


1. 0.0396


1. 0.1056; 0.0122 3. 0.0430; 0.9232 5. 0.75077. 0.4129 9. 0.5; 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. 311. 13. 2000 15. y=–417. z=619. 21.

23.

25.

27.

y

x

z

1

1

1

y

x

z

2

4

y

x

z

2

1

y

x

z

2

6

4

y

x

z

1

1

1

ex0 + h + y0

B2518

L 1.18103

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

;

F 1x 2 = e0,1 - e-3x,

if x 6 0if x � 0

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)

2

1

6

P 1X 6 3 2 =14

, P 12 6 X 6 5 2 =34

F 1x 2 = µ0,x - 2

4,

1,

if x 6 2

if 2 � x � 6

if x 7 6




1. fx(x, y)=8x; fy(x, y)=6y3. fx(x, y)=0; fy(x, y)=25. gx(x, y)=12x3y+2y2-5y+8;

gy(x, y)=3x4+4xy-5x-9

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. So cannot be

determined for x=y=z=0 15. 17. 4

19. 21. (a) 36;

(b) With respect to qA, ; with respect to qB,


1. 8xy; 8x 3. 3; 0; 05. 18xe2xy; 18e2xy(2xy+1); 72x(1+xy)e2xy

7. 3x2y+4xy2+y3; 3xy2+4x2y+x3; 6xy+4y2; 6xy+4x2

9. ; 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.

17. When pA=25 and pB=4,

19. (a) (b) -20

312 + 15e

∂w

∂t=

∂w

∂x ∂x

∂t+

∂w

∂y ∂y

∂t;

∂c

∂pA= -

14

and ∂c

∂pB=

54

40e9

c2t +31t

2d ex + y∂z

∂r= 13;

∂z

∂s= 9

-y2 + z2

z3 = -3x2

z3-18

z

1x2 + y2 2 32 3x2 + y22x2 + y2 4zy

2x2 + y2

28865

6013

52

-4e2

∂z

∂x

∂z

∂x12xz + y 2 = 28x3 - z2

-3x

z

yz

1 + 9z

-ey - zx 1yz2 + 1 2z 11 - x2y 2

4y

3z2-2x

5z

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




1. 3. (2, 5), (2, –6), (–1, 5), (–1, –6)

5. (50, 150, 350) 7. , rel. min.

9. rel. max.

11. ; D=–5<0 no relative extremum

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=1423. pA=80, pB=8525. qA=48, qB=40, pA=52, pB=44, profit=330427. 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 units15. 74 units (when l=8, k=7)17. $15,000 on newspaper advertising and $45,000 on TV advertising19. x=5, y=15, z=521. x=12, y=8 23. x=10, y=20, z=5


1. =0.98+0.61x; 3.12 3. =0.057+1.67x; 5.905. =82.6-0.641p 7. =100+0.13x; 105.29. =8.5+2.5x11. (a) =35.9-2.5x; (b) =28.4-2.5x


1. 18 3. 5. 7. 3 9. 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. ;

17. 19. 2(x+y)er+2

21. 23.

25. Competitive 27. (2, 2), rel. min.29. 4 ft by 4 ft by 2 ft31. A, 89 cents per pound; B, 94 cents per pound33. (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 905)

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

ex + y + z c ln xyz +1x

+1zdex + y + z c ln xyz +

1yd

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

5252

23

14

yyy

yqyy

a 23

, 43

, -43b16, 3, 2 2

a0, 14

, 58ba3,

32

, -32b

a5, ln 54b

a 10537

, 2837b

a0, 12ba4,

12b ,

a 25

, -35b

a -14

, 12b ,

a -2, 32b

a 143

, -133b



3.

5.

7. 75 9. Between 1990 and 1993, 1995 and 1998, 1999 and 2000; positive 11. Between 1994 and 1995; zero13. Between 1993 and 1994 15. 75 students; 199017. (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

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



13. ; quadratic;f(x)=2(x-1)2+3 orf(x)=2x¤-4x+5

15. ;

logarithmic; y=log™ x17. ; linear

19. ; quadratic

21. ; exponential

23. (a) C=100+40r; (b) $500; (c) 22 reels25. (a) h=–16t¤+80t; (b) 2.5 sec, 100 ft27. (a)

;

(b) N=2000(3) ; (c) 118,098,000 bacteria29. (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; nonlinear5. ; (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 h13. (a) 3.5 degrees per day; (b) –1.25 degrees per day;(c) 1 degree per day; (d) ≠0.59 degree per day15. (a) 0; (b) 0; (c) 0; (d) 017. (a) 7; (b) 13; (c) h+8; (d) 2xº+h+219. (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

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

x

y

–3 5

25

(2, 5)

(–2, 21) (4, 21)

(0, 5)(1, 3)


t 0 1 2 3 4 5

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


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



1. 3. 34 5. 61 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) trapezoid13. (a) 104; (b) 86; (c) trapezoid15. ; 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. Derivative7. 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;

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

dh

dt

dc

dq-

52112 - 5x

+ 20x

3213x + 7

15112 - 5x 2 2

-5

15x + 11 2 2

12

x +12

-19

x

y



(c) 140.26; Between 1991 and 2000, Xul read about 140books 15. (a) 8; 11.95; In 2006, the program’s budgetwould 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 atthe 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 fiveyears of the program, the cumulative budget would beabout $20 billion with model bl and about $38.36 billionwith 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 949)

1.

3. between October and November5. C=550+22.50x7. (a) , quadratic;

(b) h=–16t¤+100t; (c) 24 ft; (d) 6.25 s9. (a) –3, –3, –3, –3, linear; (b) 1, 3, 5, 7, nonlinear11. (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) ≠5.06 square units;(c) ≠6.14 square units; (d) ≠6.56 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. 4 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 $42,455.2747. (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)

t

A

4321

180

140

90

(0, 125)

(1, 98)

(3, 150)

(2, 175)

(4, 150)

Am

ount

(do

llars

)

Number of monthsafter October


time 302 sec 204 sec 130 sec 20 sec

size 3 K 126 K 213 K 297 K


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