A242 Answers to Odd-Numbered Exercises and Testshorneckermath.weebly.com/.../chapter_9-11_odd_answers.pdf · 2018. 8. 15. · A246 Answers to Odd-Numbered Exercises and Tests 29.

Review Exercises (page 651)1. 3. 5.

7. 9. 11.

13. 15. 30 17. 19. 51,005,000

21. 43,078 23.

25. 27. (a) (b)

29. (a) (b) 2

31. (a) 2512.50, 2525.06, 2537.69, 2550.38, 2563.13,2575.94, 2588.82, 2601.77

(b) $3051.99

33. Arithmetic sequence,

35. Arithmetic sequence,

37. 3, 7, 11, 15, 19 39. 1, 4, 7, 10, 13

41. 35, 32, 29, 26, 23;

43. 9, 16, 23, 30, 37;

45. 47. 80 49. 88

51. 25,250 53. (a) $43,000 (b) $192,500

55. Geometric sequence,

57. Geometric sequence,

59. 61. or

63.

65.

67.

69. 71. 127 73. 3277

75. 1301.01 77. 24.85 79. 32 81. 12

83. (a) (b) $20,168.40

85 and 87. Answers will vary. 89. 465 91. 4648

93. 5, 10, 15, 20, 25

First differences: 5, 5, 5, 5

Second differences: 0, 0, 0

Linear model

95. 16, 15, 14, 13, 12

First differences:

Second differences: 0, 0, 0

Linear model

97. 45 99. 126 101. 20 103. 70

105.

107.

109. 111. 10

113. (a) 216 (b) 108 (c) 36 115. 45

117. 239,500,800 119. 4950 121. 999,000

123. 5040 125. 3,628,800 127. 15,504

129. 131.

133. (a) 0.416 (b) 0.8 (c) 0.074 135. 0.0475

137. True.

139. (a) Each term is obtained by adding the same constant(common difference) to the preceding term.

(b) Each term is obtained by multiplying the sameconstant (common ratio) by the preceding term.

141. (a) Arithmetic. There is a constant difference betweenconsecutive terms.

(b) Geometric. Each term is a constant multiple of thepreceding term. In this case the common ratio isgreater than 1.

143. Each term of the sequence is defined using a previousterm or terms.

145. If is even, the expressions are the same. If is odd, theexpressions are negatives of each other.

Chapter Test (page 655)1. 2.

3.

4.

5. 7920 6. 7. 8.

9. 10.

11. 12. 13. 189

14. 28.80 15. 16. Answers will vary.

17.

18. 84 19. 1140 20. 72 21. 328,440

22. 23. 26,000 24. 12,650 25.

26. 27. (a) (b) (c) 28. 0.251601213600

14

1462

326n ! 3

16a 4 " 160a3b # 600a2b2 " 1000ab3 # 625b 4

257

!$

n!12"14#

n"1

!12

n!1

23n # 1

an ! 4$12%n"1an ! 5100 " 100nan ! n2 # 12n

1n # 1

"x3

6, "

x5

120, "

x7

5040, "

x9

362,880, "

x11

39,916,800

"x, x2

2, "

x3

3,

x 4

4, "

x5

5

12, 16, 20, 24, 281, "23, 49, "

827,

1681

nn

$n # 2%!n!

!$n # 2%$n # 1%n!

n!! $n # 2%$n # 1%

19n ! 3

1241 # 2520i

a5 " 20a4b # 160a3b2 " 640a2b3 # 1280ab4 " 1024b5x 4 # 20x3 # 150x2 # 500x # 625

"1, "1, "1, "1

at ! 120,000$0.7%t

an ! 100$1.05%n"1; 3306.60an ! 16$"12%n"1; 10.67an ! 25$"35%n"125, "15, 9, "275 ,

8125; r ! "

35;

an ! 120$13%n"1120, 40, 403 ,

409 ,

4027; r !

13;

9, "6, 4, "83, 1699, 6, 4,

83,

1694, "1,

14, "

116,

164

r ! "13

r ! 2

an ! 103 " 3n; 1430

an ! 2 # 7n

d ! 7;

an ! 38 " 3n

d ! "3;

d ! 12

d ! "2

158

59

11112000!

9

k!1

kk # 1

! 7.071

!20

k!1 12k

! 1.799

20524$n # 1%$n%

1380

9, 5, 1, "3, "7an !2

2n " 1

an ! 5n"1, 12, "

16,

124, "

1120

23,

45,

89,

1617,

3233

A242 Answers to Odd-Numbered Exercises and Tests

333350_08_ans_odd.qxp 1/10/07 12:06 PM Page A242

Chapter 9Section 9.1 (page 667)

Vocabulary Check (page 667)1. conic section 2. locus 3. circle, center

4. parabola, directrix, focus 5. vertex

6. axis 7. tangent

1. 3.

5.

7. Center: 9. Center:

Radius: 7 Radius: 4

11. Center: 13.

Radius: Center:

Radius: 2

15.

Center:

Radius:

17. 19.

Center: Center:

Radius: 1 Radius: 1


Radius: 4 Radius: 3


Radius: 5 Radius: 6

29. intercept: 31. intercepts:

intercepts:

intercepts:

33. intercept: 35. (a)

(b) Yes

intercept: none (c) 6 miles

37. e 38. b 39. d 40. f 41. a 42. c

43. 45. 47.

49. 51. 53.

55. Vertex: 57. Vertex:

Focus: Focus:

Directrix: Directrix:


Focus: Focus:



Focus: Focus:


1 32−1−3 −2−4−5

−2

3

4

5

6

x

y

–10 –8 –6 –4

–8

–6

–4

–2

2

x

y

y ! 1x ! 0

!"32, 3"!"4, "3"!"32, 2"!"2, "3"

x

y

−2

−4

−6

−8

−10

−12

4

2

2x

y

−4−6−8 4 6 8−2

−4

−6

−8

−10

4

6

2

y ! "1y ! 2

!"1, "5"!0, "2"!"1, "3"!0, 0"

–6 –5 –4 –3 –2 –1 1 2

–4

–3

3

4

y

x

–1

1

2

3

4

5

–3 –2 2 3

y

x

x ! 32y ! "12

!"32, 0"!0, 12"!0, 0"!0, 0"

y2 ! 9xy2 ! "8xx2 ! 4y

y2 ! "8xx2 ! "6yx2 ! 32 y

y-

!6 ± #7, 0"x2 # y2 ! 6561x-

!0, 9", !0, "3"y-!0, "3 ± #5"

!1 ± 2#2, 0"y-x-!2, 0"x-

−2−4−8−10 2 4 6 8 10

−4

−8−10

24

810

x

y

−2 4 6 8 10

−4−6−8

−10−12−14

246

x

y

14 16 18

!"1, 0"!7, "4"

−1−2−3−5−6−7 2 3

−2−3−4

−6−7

23

x

y

−1−2−3−5 1 2 3 5

−2−3

−5

123

5

x

y

!"2, "2"!0, 0"

!"32, 3"!1, "3"!x # 32"2 # !y " 3"2 ! 1!x " 1"2 # !y # 3"2 ! 1

#32

!0, 0"

x2 # y 2 !34

!0, 0"#15x2 # y2 ! 4!1, 0"

!"2, 7"!0, 0"!x # 3"2 # !y # 1"2 ! 7

!x " 3" # !y " 7"2 ! 53x2 # y2 ! 18

Answers to Odd-Numbered Exercises and Tests A243

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333350_09A_ans_odd.qxp 1/19/07 10:34 AM Page A243

x 0 200 400 500 600

y 0 14.844 59.375 92.773 133.59



Focus: Focus:


71. Vertex:

Focus:

Directrix:

73. 75.

77. 79.

81.

83.

85. 87.

89. televisions

91. (a) (b) 2.67 inches

93. (a) (b)

(c)

95.

97. (a) (b) Highest point: 7.125 feet

Distance: 15.687 feet

99. 101.

103. False. represents a circle with itscenter at and a radius of 5.

105. False. A circle is a conic section. 107. True

109. The resulting surface has the property that all incomingrays parallel to the axis are reflected through the focus ofthe parabola. Graphical representations will vary.

111.

113. Minimum: maximum:

115. Minimum:

Section 9.2 (page 677)

Vocabulary Check (page 677)1. ellipse 2. major axis, center

3. minor axis 4. eccentricity

1. b 2. c 3. d 4. f 5. a 6. e

7. Center:

Vertices:

Foci:

Eccentricity:

9. Center:

Vertices:

Foci:

Eccentricity: 35

!4, 2", !4, "4"!4, "6"!4, 4",

x

y

−2−4 2 6 10−2

−4

−6

−8

2

4

6

!4, "1"

#558

!±#55, 0"!±8, 0"

−2−4−10 2 4 10

−4−6−8

−10

2468

10

x

y!0, 0"

!"0.79, 0.81"!"0.67, 3.78"!0.67, 0.22";

y ! #6!x # 1" # 3

!0, "5"x2 # ! y # 5"2 ! 25

y !#22

x " 3#2y !34

x "254

00 16

10

y2 ! 640x

x2 !51,200

19yy

x

(−640, 152) (640, 152)

y 2 ! 6x

x ! 125

00 250

25,000

4x " y # 2 ! 0; !"12, 0"4x " y " 8 ! 0; !2, 0"

!2, 4"

−3

−6 6

5

!y " 2"2 ! 8xx2 ! 8!y " 4"!y " 2"2 ! "8!x " 5"y 2 ! 2!x # 2"!x " 3"2 ! "! y " 1"

−1 1−2

−2

1

2

−3x

y

x ! 12

!0, "12"! 14, "12"

2−1−3−4−6

−2

−3

3

2

1

4

5

x

y

–2 2 4

2

4

6

x

y

y ! 52y ! 0

!"2, "12"!1, 2"!"2, 1"!1, 1"



CH

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11. Center:

Vertices:

Foci:

Eccentricity:

13. (a) (c)

(b) Center:

Vertices:

Foci:

Eccentricity:

15. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

17. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

19. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

21. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

23. 25. 27.x2

16#

y2

7! 1

x2

9#

y2

5! 1

x2

4#

y2

16! 1

x

y

−1−2−3 1 2 3

1

2

−2

−3

−4

#105

$12 ± #2, "1%$12 ± #5, "1%$12, "1%

!x " 12"25

#!y # 1"2

3! 1

–2 –1 1 3

–3

–2

1

2

x

y

35

$74, "1%, $14

, "1%$94, "1%, $"

14

, "1%!1, "1"

!x " 1"22516

# !y # 1"2 ! 1

2

2

4

−4

−2

−6x

y

#63

$"32, 52 ± 2#2%$"32, 5 ± 4#32 %$"32, 52%

!x # 32"24

#!y " 52"2

12! 1

1 2−1−3 −2−4−5−6

−2

4

6

2

3

x

y

#53

!"2, 3 ± #5 "!"2, 6", !"2, 0"

!"2, 3"

!x # 2"24

#!y " 3"2

9! 1

2#23

!±4#2, 0"!±6, 0"

!0, 0"

−8−10 6 8 10

−4−6−8

−10

468

10

x

yx2

36#

y2

4! 1

#52

$"5 ± #52 , 1%$"72, 1%, $"132 , 1%

1−1−3 −2−4−5−6−7

−2

−3

−4

4

2

3

1

x

y!"5, 1"



29. 31.

33. 35.

37. 39.

41. 43. 45.

47. (a)

(b) (c) 17.4 feet

49. 6 feet 51. 40 units

53. 55. Answers will vary.

57. 59.

61. True

63. (a)

(b) The sum of the distances from the two fixed points is constant.

65. 67. Arithmetic

69. Geometric 71. 1093 73. 15.0990


Vocabulary Check (page 687)1. hyperbola 2. branches

3. transverse axis, center 4. asymptotes

5.

1. b 2. c 3. a 4. d

5. Center:

Vertices:

Foci:

Asymptotes:

7. Center:

Vertices:

Foci:

Asymptotes:

9. Center:

Vertices:

Foci:

Asymptotes:

11. Center:

Vertices:

Foci:

Asymptotes:

13. Center:

Vertices:

Foci:

Asymptotes:

y ! "5 ±23

!x " 1"

$1, "5 ± #136 %$1, "5 ± 13%

x

y

−1−2 1 2 3 4−1

−2

−3

−5

!1, "5"

y ! "2 ± 12!x " 1"

!1 ± #5, "2"!3, "2", !"1, "2"

1 2 3

–5

–4

1

2

3

x

y!1, "2"

y ! ±59x

!0, ±#106"!0, ±5"

x

y

−6−9 6 9 12 15−3

−9−12−15

3

91215

!0, 0"

y ! ±12x

!0, ±#5 "!0, ±1"

–3 –2 2 3

–3

–2

2

3

y

x

!0, 0"

y ! ±x

!±#2, 0"!±1, 0"

–2 2

–2

–1

1

2

x

y!0, 0"

Ax2 # Cy2 # Dx # Ey # F ! 0

!x " 6"2324

#! y " 2"2

308! 1

2a

−4 −2 2 4

−4

4

x

(

((

( , 2

, 2−, 2−

, 2 3 5

3 53 5

3 55

55

5 )

))

)

−

−

y

x

−

−

9

9 9

94

4 4

4,

, ,

7

7 7− −

, 7(

( (

()

) )

)

y

−2−4 2 4

−2

2

x2

4.88#

y2

1.39! 1

!±#5, 0";

x2

2500#

y2

1600! 1

−20−40 20 40

−20

20

60

80

(−50, 0) (50, 0)

(0, 40)

x

y

x2

2#

y2

9! 1

2#23

#53

x2

16#

!y " 4"212

! 1!x " 3"2

9#

!y " 5"216

! 1

x2

308#

!y " 4"2324

! 1!x " 4"2

16#

!y " 2"21

! 1

!x " 2"21

#!y " 3"2

9! 1

x2

400&21 #y2

25! 1


15. (a) (c)

(b) Center:

Vertices:

Foci:

Asymptotes:

17. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

19. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

21. (a)

(b) It is a degenerate conic. The graph of this equation istwo lines intersecting at

(c)

23. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

25. 27.

29. 31.

33. 35.

37. 39.

41.

43.

45. (a) (b)

47. 49. Ellipse

51. Hyperbola 53. Parabola 55. Circle

57. Parabola

59. True. For a hyperbola, The larger the ratioof to the larger the eccentricity of the hyperbola,

61. False. If or the graph is two intersectinglines. For example, the graph of istwo intersecting lines.

x2 " y 2 " 2x # 2y ! 0D ! "E,D ! E

e ! c&a.a,b

c2 ! a2 # b2.

!12#5 " 12, 0" ' !14.83, 0"1.89 feet ! 22.68 inchesx2 "

y2

27! 1

x2

98,010,000"

y2

13,503,600! 1

!x " 3"29

"!y " 2"2

4! 1

!x " 2"21

"!y " 2"2

1! 1

!y " 2"24

"x2

4! 1

y2

9"

4!x " 2"29

! 1!y " 5"2

16"

!x " 4"29

! 1

!x " 4"24

"y 2

12! 1

17y 2

1024"

17x2

64! 1

x2

1"

y 2

25! 1

y2

4"

x2

12! 1

x

y

2

−6

−8

−10

2

4

y ! "3 ±13

!x " 1"

!1, "3 ± 2#5 "!1, "3 ± #2 "

!1, "3"

!y # 3"22

"!x " 1"2

18! 1

–4 –2 2

–6

–4

–2

2

4

x

y

!"1, "3".

!x # 1"2 " 9!y # 3"2 ! 0

–6 –4 –2 2 4 6 8

–8

–6

–4

2

x

y

y ! "3 ± 3!x " 2"!2 ± #10, "3"

!3, "3", !1, "3"!2, "3"

!x " 2"2 " !y # 3"2

9! 1

−3−4 3 4

−3

−2

−4

1

2

3

4

x

y

y ! ±#63

x

!±#5, 0"!±#3, 0"

!0, 0"

x2

3"

y2

2! 1

y ! ±23

x

!±#13, 0"!±3, 0"

!0, 0"

−2−4−5 2 4 5

−2−3−4−5

12345

x

yx2

9"

y2

4! 1


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63. Proof 65. Answers will vary. 67. Proof

69. 71.

73. 75.

77.


Vocabulary Check (page 697)1. rotation, axes 2. invariant under rotation

3. discriminant

1.

3. 5.

7.

9. 11.

13. 15.

17. 19.

21. e 22. b 23. f 24. a 25. d 26. c

27. (a) Parabola

(b)

(c)

29. (a) Ellipse or circle

(b)

(c)

31. (a) Hyperbola

(b)

(c)

−6

−10 8

6

y !6x ± !56x2 " 80x # 440

#10

−4

−6 6

4

y !8x ± !#356x2 " 1260

14

−2

−4 8

6

y !24x " 40 ± !3000x " 1600

18

$ ! 31.72%$ ! 26.57%

−6

−9 9

6

−6

−9 9

6

$ ! 45%

−8

−12 12

8

x

x ′y′

−4 42

−2

2

4

6

y

y& ! 16"x2

# 13 x&

x

x′y′

2−4−6

2

−2

−4

y

x′y′

2

2

3

−3

−3−4

−4

4

3 4x

y

x& ! #"y"x& #2

6"

"y& #2

3$2! 1

x

x′

y ′4

6

8

−4

−4 2 4 6 8

y

"x& # 3!2 #216

#" y& # !2 #2

16! 1

−4−6−8 4 6 8

−6

−8

4

6

8

x

y

y ′ x ′

−4 −3 −2 4

−4−3−2

4y ′ x ′

x

y

"x& #2 # "y& #2

1$3 ! 1"y& #2

2#

"x& #2

2! 1

"3, 0#

2"2x " 3#"4x2 # 6x " 9#2x"x # 6#2x"x " 4#"x # 4#

x2 # 2x " 1 "2

x " 2x3 " x2 " 2x # 6

333350_09B_ans_odds.qxp 1/19/07 10:36 AM Page A248

33. (a) Parabola

(b)

(c)

35. 37.

39. 41.

43. 45.

47. 49.

51.

53. True. The discriminant will be greater than zero.

55. 57.

Intercept: Intercept:

Asymptotes: Asymptotes:

59. (a) (b) (c)

61. (a) (b) (c) Not possible

63. 65.

67. 69.

71. 45.11 73. 48.60


Vocabulary Check (page 704)1. plane curve, parametric equations, parameter

2. orientation 3. eliminating, parameter

1. c 2. d 3. b 4. a 5. f 6. e

7. (a)

(b)

(c)

−3

3

5−4

−1

−2

−1

−2 1 2

1

2

x

y

−1−2−3 1 2 5 6 7

123

−2−3−4

−7

x

y

−2−4−6−8−10 2 6 8 10

2468

10121416

−4

x

y

−1−2−3 1 2 3

1

3

−2

−1

−3

x

y

x2

4

6

8

10

−8 −6 −4 −2−2

4

y

% 12#1620#6

8#10

15#20

25&'45(%#512

#1819&%

123

30#20&%

#1525

97&

t ! 2, y ! #t # 2x ! 2, y ! 0

"0, 0#"0, 1#

−10 −5 5 10 15

−15

−10

−5

5

10

(0, 0)t

y

−2 −1 1 3

−4−3−2

2

3

4

x

(0, 1)

y

"#3, 0#, "0, 32#"8, 0#"!3, #2!3#, "#!3, 2!3#

"0, 4#"0, 8#, "12, 8#"#8, 12#"1, !3#, "1, #!3#

x

y

−1−2−3 1 2 3

1

2

3

−2

−3

−1 1 2 3

3

−2−3x

y

−5

−4 8

3

y !#4x " 1 ± !72x " 49

8


CH

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t 0 1 2 3 4

x 0 1 !2 !3 2

y 2 1 0 #1 #2


(d)

The graph is an entire parabola ratherthan just the right half.

9. b

11. 13.

15. 17.

19. 21.

23. 25.

27. 29.

31.

33. Each curve represents a portion of the line

Domain Orientation

(a) Left to right

(b) Depends on

(c) Right to left

(d) Left to right

35.

37.

39.

41. 43.

45.

47.

49.

−4

4

6−6

x ! 2t, y ! 24t 2 # 5

x ! t, y ! 6t 2 # 5

y !1t3

x ! t3,

y !1t

x ! t,

x ! 15 t, y ! t # 3y ! 3 sin $

x ! t, y ! 5t # 3x ! 5 cos $

x ! 1 " 5t, y ! 4 # 7t

"x # h#2a2

""y # k# 2

b 2! 1

y # y1 !y2 # y1x2 # x1

"x # x1#

"0, '#"0, '#

$'#1, 1("#', '#

y ! 2x " 1.

−8

8

12−12

12−12

−8

8

8−1

−5

1

y ! ln xy ! x#3, x > 0

−1

−3−2

−4−5

−2 32 4 5 6 7 8

12

345

x

y

−1−2 321 4 5 6 7 8

1

23456789

10

x

y

x2

4"

y2

9! 1y !

12)x # 4)

−1−3−4 1

1

2

4

3 4

−2

−4

−1

x

y

2 4

6

4

8

10

6 8 10−2−2

x

y

y ! "x # 2#2y ! 16x2

−1−2 1 2 3 4 5 6−1

1

2

4

3

x

y

1 2

5

3−3 −2 −1−1

x

y

y ! 23 x " 3y ! #4x

−1−2−3−4−6 1 2−1

−2

−3

1

2

4

5

x

y

−1

−2

−1

−2 1 2

2

x

y

−1

−2

−1

−2 1 2

1

x

y

y ! 2 # x2



51. b 52. c 53. d 54. a

55. (a)

(b) (c)

No Yes

(d) About

57. (a)

(b) 54.09 feet per second

(c)

22.04 feet

(d) 2.03 seconds

59. True. Both sets of parametric equations correspond to

61. False. does not correspond to as a functionof

63. Answers will vary. Sample answer:

65. Even 67. Neither


Vocabulary Check (page 711)1. pole 2. directed distance, directed angle

3. polar

1. 3.

5.

7.

9.

11.

*32, (2+, *#32, 3(2 +, *#32, #(2+

32

0

, −

1 2 3

)) 32π2π

*#!3, 11(6 +, *!3, #7(6 +, *#!3, #(6+

0

3, )) 56π

1 2 3

2π

*#1, 5(3 +, *1, 2(3 +, *1, #4(3 +

01 2 3

−1, −3π( (

π2

*3, #7(6 +, *#3, 11(6 +, *#3, #(6+

01 2 3

π2

3, 6π5( (

*!22 , !22 +"0, 4#

y ! #2 sin $

x ! cos $

x.yy ! tx ! t 2,

y ! x2 " 1.

0

24

900

y ! 7 " "v0 sin 35%#t # 16t 2x ! "v0 cos 35%#t

19.38%

0

60

50000

30

4500

y ! 3 " "146.67 sin $#t # 16t 2x ! "146.67 cos $#t


CH

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R 9


13. 15.

17. 19.

21. 23.

25. 27.

29. 31.

33. 35.

37. 39. 41.

43. 45. 47.

49. 51.

53. 55. 57.

59. 61.

63. 65. 67.

69. 71.

73. 75.

77. 79.

81. The graph is a circle centered at the origin with a radius of 7;

83. The graph consists of all points on the line that makes anangle of with the positive axis;

85. The graph is a vertical line through

87. True. Because is a directed distance, can berepresented by so

89. (a) Answers will vary.

(b) The points lie on a line passing through the pole.

(c) (Pythagorean Theorem)

Answers will vary.

(d) Answers will vary. The Distance Formula should givethe same result in both cases.

d ! !r1 2 " r2 2d ! !r12 " r22 # 2r1r2 ! )r1 # r2)

)r) ! )#r)."#r, $ ± "2n " 1#(#,"r, $#r

2

1

2 41

3

−3

−2

−2−1

−1x

y

x # 3 ! 0."3, 0#;

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

x # y ! 0.x-($4

x

y

−2−4−8 2 4 6 8−2

−4

−6

−8

2

4

6

8

x2 " y2 ! 49.

4x2 # 5y 2 ! 36y " 36y2 ! 2x " 1

"x2 " y 2#2 ! 6x2y # 2y 3"x2 " y2#3 ! x2y ! #3x2 " y2 ! 16

x ! 0y ! #!33

xy ! !3x

x2 " y2 ! 6yr ! tan2 $ sec $

r ! 2a cos $r ! 6 cos $r2 ! 9 cos 2$

r2 ! 8 csc 2$r ! #2

3 cos $ # 6 sin $

r ! 8 sec $r ! 4 csc $r ! 3

"2.83, 0.49#"2.65, 0.86#"3.61, #0.59#

"10.82, 0.98#, "#10.82, 4.12#*!6, 5(4 + , *#!6, (

4+

−3 3 6 9 12−3

3

6

9

12

x

y

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

*!2, (4+, *#!2, 5(4 +"7, (#, "#7, 0#

1

2

1

2

3

3−3

−3

−2

−2−1

−1x

y

−1−2−4−6−8 −3−5−7−9

1

1

−2−3−4−5

2

3

45

x

y

"#3.60, 1.97#"#0.02, 2.50#"#1.20, #4.34#"1.53, 1.29#

"#1.004, 0.996#"0, 0#

1 2 30

2π

1 2 30

2π

*!22 , !22 +"2, #2!3 #

1 2 30

2π

2 4 60

2π




CH

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91. 93. 95.

97. 99. 101.

103. Not collinear 105. Collinear


Vocabulary Check (page 720)

1. 2. polar axis 3. convex limaçon

4. circle 5. lemniscate 6. cardioid

1. Rose curve 3. Lemniscate 5. Rose curve

7. a 9. c 11. Polar axis 13.

15. 17. Pole

19. Maximum: 21. Maximum:

Zero of r: Zeros of r:

23. 25.

27. 29.

31. 33.

35. 37.

Answers will vary.

39. 41.

Answers will vary.

Answers will vary.

43. 45.

Answers will vary. Answers will vary.

47. 49.

Answers will vary.

Answers will vary.

51. 53.

Answers will vary.

55. 57.

0 ≤ $ <(2

0 ≤ $ < 4(

−1

1

1−1

−2

2

3−3

0 ≤ $ < 2(

−4

4

6−6

−2000

−1400

400

200

0 ≤ $ < 2(

18−18

−10

14

−2

2

3−3

−2

2

3−3

−4

4

6−6

0 ≤ $ < 2(

−6

6

4−1418−18

−14

10

0 ≤ $ < 2(

18−18

−12

12

04 5 6

2π

01 65

2π

4 6 8

π2

0

01 2

2π

2 40

2π

21 30

2π

2 4 6 8

π2

0

$ !(

6,

(

2,

5(6

$ !(

2

)r) ! 4)r) ! 20$ !

(2

$ !(

2

$ !(2

"2, #3, 3#"0, 0, 0#"2, 3#c , 5.25B , 86%C , 101.09%B , 25.91%b , 19.44B , 48.23%A , 119.09%a , 16.16A , 30.68%


59. 61.

63. True

65.

Negative values of produce the heart-shaped curves;positive values of produce the bell-shaped curves.

67. (a), (b), and (c) Answers will vary.

69. (a)

(b)

(c)

(d)

71.

circle convex limaçon

cardioid limaçon with inner loop


Vocabulary Check (page 726)1. conic 2. eccentricity,

3. (a) i (b) iii (c) ii

1. 3.

(a) Parabola (a) Parabola

(b) Ellipse (b) Ellipse

(c) Hyperbola (c) Hyperbola

5. b 6. c 7. f 8. e 9. d 10. a

11. Parabola 13. Ellipse 15. Ellipse

17. Ellipse 19. Hyperbola

21. 23.

Parabola Hyperbola

−3

9

9−9

−4

4

6−6

−8

4

9−9

a

c

b

−4

4

8−4

a

bc

e

k ! 3;k ! 2;

−4

4

8−4

−4

4

8−4

k ! 1;k ! 0;

−4

4

6−6

−4

4

6−6

r ! 4 sin $ cos $

r ! 4 sin*$ # 2(3 + cos*$ #2(3 +

r ! #4 sin $ cos $

r ! 4 sin*$ # (6+ cos*$ #(

6+

nn

−4

4

6−6

n ! 5

−4

4

6−6

−4

4

6−6

n ! 4n ! 3

−2

2

3−3

−2

2

3−3

n ! 2n ! 1

−2

2

3−3

−2

2

3−3

n ! 0n ! #1

−2

2

3−3

−4

4

6−6

n ! #2n ! #3

−4

4

6−6

−4

4

6−6

n ! #4n ! #5

−1

3

3−3

−4

4

6−6



25. 27.

Ellipse

29. 31.

33. 35.

37. 39.

41. 43.

45. 47.

49. Answers will vary.

51.

Perihelion: miles

Aphelion: miles

53.

Perihelion: kilometers

Aphelion: kilometers

55. (a)

(b) Neptune: Perihelion: kilometers


Pluto: Perihelion: kilometers


(c)

(d) Yes; because on average, Pluto is farther from the sunthan Neptune.

(e) Using a graphing utility, it would appear that the orbitsintersect. No, Pluto and Neptune will never collidebecause the orbits do not intersect in three-dimensionalspace.

57. False. The equation can be rewritten as

Because is negative, must be negative and since represents the distance between the pole and the directrix,the directrix has to be below the pole.

59. Answers will vary. 61.

63. 65.

67. (a) Ellipse

(b) is reflected about the line

is rotated counterclockwise.

69. Answers will vary. 71.

73. 75.

77. 79. 81. 220 83. 720

Review Exercises (page 730)1. 3.

5. 7.

Center: Center:

Radius: 6 Radius: 1

9. 11.

Center:

Radius: 4

!!2, !3"

!3 ± #6, 0"

−1−2−3−4−6−7 2 3

−2−3−4−5−6

−8

2

x

y

!12, !34"!0, 0"!x ! 12"2 " ! y " 34"2 # 1x2 " y2 # 36

!x ! 2"2 " !y ! 4"2 # 13x2 " y2 # 25

#210

#210

$2

" n$$3

" n$, 2$3

" n$

$6

" n$

90%r #4

1 ! 0.4 sin &

& #$2

.r #4

1 " 0.4 cos &

r 2 #144

25 sin2 & ! 16r 2 #

40025 ! 9 cos2 &

r 2 #24,336

169 ! 25 cos2 &

ppep

r #!4$3

1 " sin &.

−5 × 109 8 × 109

−7 × 109

7 × 109

7.3754 ' 1094.4366 ' 109

4.5367 ' 1094.4593 ' 109

rPluto #5.5404 ' 109

1 ! 0.2488 cos &

rNeptune #4.4977 ' 109

1 ! 0.0086 cos &

8.1609 ' 1087.4073 ' 108

r #7.7659 ' 108

1 ! 0.0484 cos &

9.4508 ' 10 79.1404 ' 10 7

r #9.2930 ' 107

1 ! 0.0167 cos &

r #8

3 " 5 sin &r #

203 ! 2 cos &

r #10

3 " 2 cos &r #

101 ! cos &

r #2

1 ! sin &r #

21 " 2 cos &

r #1

2 " sin &r #

11 ! cos &

−8

4

9−93−3

−2

2

20−10

−5

152

2−4

−2


CH

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R 9

333350_09B_ans_odds_pg A255.qxp 5/28/08 9:42 AM Page A255


Focus: Focus:


17. 19.

21. 23. meters


Vertices: Vertices:

Foci: Foci:

Eccentricity: Eccentricity:

29. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

31. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

33. 35.

37. The foci should be placed 3 feet on either side of the center at the same height as the pillars.

39.

41. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

43. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:54

"6, #1#, "#4, #1#"5, #1#, "#3, #1#

"1, #1#

"x # 1#216

#"y " 1#2

9! 1

−1−2−3−4−5 1 2 3 4 5

−3−4−5

1

345

x

y

32

"0, ±3#"0, ±2#

"0, 0#

y2

4#

x2

5! 1

e , 0.0543

"x # 2#225

"y 2

21! 1

x2

25"

y2

9! 1

−1−2−3−4

2

4

6

8

x

y

!304

*#2 ± !3012 , 7+*#2 ± !33 , 7+

"#2, 7#

"x " 2#21$3 "

"y # 7#21$8 ! 1

−1−2−3 1 2 3 4 5

−2

−3

−4

−5

−6

−8

x

y

!74

"1, #4 ± !7#"1, 0#, "1, #8#

"1, #4#

"x # 1#29

""y " 4#2

16! 1

−1−2 1 2 3 5 6 7 8

−2−3−4−5−6−7−8−9

1x

y

1 −1−3−4 3 4

–3

–2

1

2

3

x

y

!33

!32

"4, #4 ± !3#"0, ±2!3#"4, #1#, "4, #7#"0, ±4#

"4, #4#"0, 0#8!62x " y # 2 ! 0; "1, 0#

"y # 2#2 ! 12xy2 ! #24x

x

y

−4−8−12−16−20 4−4

4

12

–2 2 4 6 8 10

–6

–4

–2

2

4

6

x

y

x ! 9x ! #1

"#9, 0#"1, 0#"0, 0#"0, 0#



t #2 #1 0 1 2 3

x #8 #5 #2 1 4 7

y 15 11 7 3 #1 #5

(c)

45. (a)

(b) Center:

Vertices:

Foci:

Eccentricity:

(c)

47. 49.

51. miles 53. Ellipse 55. Hyperbola

57. 59.

61. (a) Parabola

(b)

(c)

63. (a) Parabola

(b)

(c)

65.

67.

69. 71.

73. 75.

77. 79.

−4

−6 6

4

−4

−6 6

4

y ! 12 x2$3

−4

−6 6

4

−1−2−3−4 1

1

2

3

4

5

2 3 4

−2

−3

−1

x

y

y ! 4x # 11, x ≥ 2y ! 25 x "275

4−4 −2 6 8 10 12

12

10

8

6

4

2

−4

x

y

−20

10 20

20

−1030

30

10

−10

40

x

y

−4−8−12 8 12−4

−8

4

12

16

x

y

"#10, 12#

0−15 0

10

y !#"2x # 2!2# ± !"2x # 2!2#2 # 4"x2 " 2!2x " 2#

2

−10

−5 2

2

y !8x # 5 ± !"5 # 8x#2 # 4"16x2 # 10x#

2

y′ x′

−1 1 2 3−2−3−1

−2

−3

1

2

3

x

y

y′ x′

−2

−2

−3

2

2 3 4 5

345

x

y

"x3

""y& #2

2! 1

"x& #28

#"y& #2

8! 1

,72

"x # 4#216$5 #

y2

64$5 ! 1x2

16#

y2

20! 1

−20−30 10 20 30−10

−20

−30

10

20

30

x

y

!5

*#6 ± !10102 , 1+*#6 ± !2022 , 1+

"#6, 1#

"x " 6#2101

2

#"y # 1#2

202! 1

−6 −4 4 62

2

4

6

8

−4

−2

−6

−8

x

y


CH

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R 9



81. 83.

85.

87. 89.

91. 93.

95. 54.22 feet per second

97. 21.91 feet

99.

101.

103.

105. 107.

109. 111.

113.

115. 117. 119.

121. 123.

125. 127. "x2 " y2#2 # x2 " y2 ! 0x2 " y2 ! 3x

x2 " y2 ! 25r2 !1

1 " 3 cos2 $

r2 ! 5 sec $ csc $r ! 4 cos $r ! 3

*#5!2, 3(4 +, *5!2, 7(4 +

x

y

−1−2 1 32 4 5 6−1

−3

−2

−5

−4

1

3

2

(5, −5)

*#9, (2+, *9, 3(2 +*#3!22

, 3!2

2 +

x

y

−3−6−9 3 6 9−3

−6

−9

−12

3

6

(0, −9)

01 2 3 4

π2

)(3, 43π

"1, !3#*#5!32 , 52+

01 2 3 4

π2

π ))2, 35−

01 2 3

π2

5, −6

7π( (

*!5, #2(3 +, *#!5, (3+, *#!5, #5(3 +

0

π)( ,

1 2 3

5 43

π2

−

*#2, (6+, *2, 7(6 +, *2, #5(6 +

01 2 3

π2

π ))−2, 611−

*1, #7(4 +, *#1, 5(4 +, *#1, #3(4 +

01 2 3

1,4π( (

π2

00 100

25

y ! 6 # 6t

x ! #1 " 11tx ! t, y ! 5

x ! 12 t, y !14 t 2 " 2x ! 2t, y ! 12t " 2

x ! t, y ! t 2 " 2x ! t, y ! 6t " 2

12−12

−8

8

−4

−3 9

4

−2

−4 8

6


129.

131. 133.

135.

137. Dimpled limaçon

Symmetry: Polar axis

Maximum:

Zeros of None

139. Limaçon with inner loop

Symmetry: The line

Maximum:

Zeros of

141. Rose curve

Symmetry: Pole, polar axis, and the line

Maximum:

Zeros of

143. Lemniscate

Symmetry: Pole

Maximum:

Zeros of

145. Parabola 147. Ellipse

149. Ellipse

151. 153.

155.

Perihelion: 1.383 astronomical units

Aphelion: 1.667 astronomical units

157. False. The equation of a hyperbola is a second-degreeequation.

r !1.512

1 # 0.093 cos $

r !5

3 # 2 cos $r !

41 # cos $

−2

−3 3

2

−2

−2 4

2

−2

−6 6

6

$ ! 0, (2

r:)r) ! !5

01 2 3

π2

$ !(4

, 3(4

, 5(4

, 7(4

r:

)r) ! 3$ !

(2

04

π2

$ , 0.64, 2.50r:)r) ! 8

$ !(2

01 2 3 5

π2

r:)r) ! 9

0

π2

2 4 6 8 1210

064321

π2

2 310

π2

4 620

π2

y ! #!33

x


CH

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R 9



159. (a) vertical translation (b) horizontal translation

(c) reflection in the y-axis (d) vertical shrink

161. 5; The ellipse gets closer and closer to circular andapproaches a circle of radius 5.

163. (a) The time it takes to make one revolution is halved.

(b) The length of the major axis is increased by two units.


Vertex: Vertex:

Focus: Focus:

3. 4.

Vertices:

Foci:

5. 6.

7. Answers will vary.

8. (a) 9. No real solution

(b)

10. 11.

12. 13.

14. 15.

16.

17.

18. 19.

20. Limaçon with inner loop 21. Parabola

22. Hyperbola

23. 24.

25. Maximum:

Zeros of r: $ !(6

, (2

, 5(6

)r) ! 8r !

104 " 5 sin $

r !4

4 " sin $

12−12

−6

10

20−4

−8

8

6−6

−2

6

x2 " "y # 1#2 ! 1r ! 12 sin $

*2!2, 7(4 +; *2!2, #(4+, *#2!2, 3(4 +"#7, 7!3#

x ! ±!16 # t 2, y ! 12 tx ! 4 # 4t 2, y ! 2t

x ! ±2!4 # t 2, y ! tx ! 4 # t 2, y ! t

"x # 2#29

"y2

4! 1

x ! 2t, y ! 4t 2 " 10

x ! t, y ! t 2 " 10

−2 1 2 3

2

3

4

4 6

−2

−3

−4

x

y

x2

2#

y2

1$8 ! 1, x ≥!2"y " 1#2 ! 1

4"x " 6#

−2−4 2

2

4

6

8

8 10 12

−4

−6

−8

−2

x

y

−2−4−8 2

2

4

6

4 6 8

−4

−6

−8

−10

x

y

y′ x′

−2

23

−3−4−5

−3−4−5 2 3

45°

y

x

45%

−6

9−9

6

y2

9#

x2

4! 1

"x " 6#216

""y # 3#2

49! 1

"2 ± !5, 0#"0, 0#, "4, 0#

84

4

12

8

−4

−4

−8

x

y

"y " 2#2 ! 8"x # 6#

2

6

−4

−6

862−2

4

−2−4x

y

"2, 0#"2, 0#"1, 0#"0, 0#

x3

3

6

9

−3−3

−6

−9

6 9 12 15

y

42

2

6

4

x

y

−2

−2

−4


Cumulative Test for Chapters 7–9(page 735)1. 2.

3. 4.

5. 6.

7. 8.

9. (a) (b) 1 10. 22

11. (a) (b) 3, 6, 12, 24, 48 12. 135

13. 14. 34.48 15. 66.67 16.

17. 18. 19. Answers will vary.

20.

21.

22.

23.

24. 30 25. 120 26. 453,600 27. 151,200

28. Hyperbola 29. Ellipse

30. Hyperbola 31. Circle

32.

33. 34.

35.

36. (a) and (b)

(c)

37. (a) and (b)

(c)

38. (a) and (b)

(c)

39. 40.

x ! ±4!1 " t 2, y ! 4tx ! 2t, y ! 6t # 2

x ! ±!16 " t 2, y ! tx ! t, y ! 3t # 2

y ! 0.5e0.5x, x ≥ 0

6

8

10

4

2

42−2

−2−4 6 8x

y

y ! 2 # 2x2, #1 ≤ x ≤ 1

x

y

−1−2−3 1 2 3−1

−2

1

3

4

y !x2 # 2x " 1

4

3

4

5

6

1

2

21 3 4 5

−2

−1−1−2−3

x

y

$ " 37.98%

−6

−9 9

6

#y " 4$24

#x2

16%3 ! 1#x # 1$2

25"

#y # 4$24

! 1

#x # 2$2 ! #43#y # 3$

−1 1 2−1

2

−2

4

5

3 4x

y

6

8

2

42 6 8

−6

−8

−2−4−6−8x

y

2

1

21 3 5

−3

−4

−2

−1−1

x

y

10

15

10−10

−15

−5−5

15 20x

y

# 393,216ab7 " 65,536b8" 1,451,520a4b4 # 1,548,288a3b5 " 1,032,192a2b6

6561a8 # 69,984a7b " 326,592a6b2 # 870,912a5b3# 192xy5 " 64y6

x6 # 12x5y " 60x 4y 2 # 160x3y3 " 240x2y432x 5 " 80x 4y 2 " 80x3y 4 " 40x 2y6 " 10xy 8 " y10x 4 " 12x3 " 54x2 " 108x " 81

83#

551

158

4752

15, #

17,

19, #

111,

113

37

#20#3

#1371'& 5#3616

36120

31#36

18'&3

2252

#3118

#40

26

14'ܤ#20

151152

#1434

#1'&#7#6

#12

#101816

#1697'

#1, #4, #4$#35, #4, #15$#8, 4$, #2, #2$#4, #3$


CH

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R 9

333350_09B_ans_odds.qxp 2/9/07 12:43 PM Page A261


41. 42.

43.

44.

45.

46.

47. 48.

49.

50. Circle 51. Dimpled limaçon

52. Limaçon with inner loop 53. $701,303.32

54. 55. meters


Vocabulary Check (page 747)1. three-dimensional

2. -plane, -plane, -plane 3. octants

4. Distance Formula 5.

6. sphere 7. surface, space 8. trace

1. A: B: C:

3. A: B: C:

5. 7.

y

x

12

4

1

2

3

−2−3−4−5

1 2 3−2

−3−4

−3−4−5(3, −1, 0)

(−4, 2, 2)z

y

x

5432−2

−2−3

1

3

5

2

4

1

2

3

4

5

(2, 1, 3)( 1, 2, 1)−

z

!!2, 2, !3"!3, !2, 0",!!2, !1, 4"!!3, 0, !2"!1, 3, !2",!!1, 4, 3"

#x1 " x22 , y1 " y2

2,

z1 " z22 $

yzxzxy

24%214

−4

4

8−4

−6

2

6−6

−2

2

3−3

!x " 109 "26481

!y249

# 1

!x ! 1"2 " y2 # 1r # ! 14 sin $ " 4 cos $

#!3, %6$, #3, !5%6 $, #3, 7%6 $

01 2 3

6π11( )

π2

−3, −

#!2, !3%4 $, #2, !7%4 $, #2, %4$

01 2

−2, 4π5( )

π2

#5, 5%4 $, #!5, !7%4 $, #!5, %4$

02 4 6 8

5, −( )4π3

π2

#8, !7%6 $, #!8, !%6$, #!8, 11%6 $

02 4 6 8

6π5( )

π2

8,

x # 2t, y #e4t

e4t " 1x #

1t, y # 2t

x # t, y #e2t

e2t " 1x # t, y #

2t

333350_10_ans_odds.qxp 1/17/07 9:29 AM Page A262

9.

11. 13. 15. Octant IV

17. Octants I, II, III, and IV 19. Octants II, IV, VI, and VIII

21. units 23. units 25. units

27. 29.

31. isosceles triangle

33. isosceles triangle

35. 37. 39.

41.

43.

45.

47.

49.

51. Center: radius:

53. Center: radius:

55. Center: radius: 2



61. Center: radius:


65. 67.

69. 71.

73.

75. 77.

79. False. z is the directed distance from the plane to

81. 0; 0; 0 83. A point or a circle

85.

87. 89.

91. 93.

95. 97.

99. 1, 2, 6, 15, 31First differences: 1, 4, 9, 16Second differences: 3, 5, 7Neither

101. 2, 5, 8, 11First differences: 3, 3, 3, 3Second differences: 0, 0, 0Linear

103.

105.

107. 109.


Vocabulary Check (page 755)1. zero 2.

3. component form 4. orthogonal 5. parallel

v # v1i " v2 j " v3k

!x ! 6"24

!y2

32# 1

!x ! 3"29

"!y ! 3"2

4# 1

!y ! 1"2 # !12!x ! 4"!x " 5"2 " !y ! 1"2 # 49

!1,

!7%41, 51.34&

3%2, 315&y # !1 ± %10

2

x #5 ± %5

2v # !

3 ± %172

!x2, y2, z2" # !2xm ! x1, 2ym ! y1, 2zm ! z1"

P.xy-

x2 " y2 " z2 #1652

4!3, 3, 3"

7

43

−3

6 57

z

x y

yx

2 233

4455

5

6

zz

xy

23

4

−1−2

32

−2

3

2

5y2 + (z − 2)2 = 3

y

x

2

2 2

( 3) + = 5y z− 2 2

( 2, 3, 0)−

z

y

x

2

−2

2

2

(1, 0, 0)

( 1) + = 36x z− 2 2z

!13, !1, 0";

%212

!1, !2, 0";

!1, 13, 4";!!2, 0, 4";!2, !1, 3";

%5!2, !1, 0";

52!52, 0, 0";

!x ! 32"2 " y2 " !z ! 3"2 # 454!x " 3"2 " !y ! 7"2 " !z ! 5"2 # 25x2 " !y ! 4"2 " !z ! 3"2 # 9!x " 1"2 " !y ! 2"2 " z2 # 3!x ! 3"2 " !y ! 2"2 " !z ! 4"2 # 16

!52, 2, 6"!1, 0, 112 "!0, !1, 7"6, 6, 2%10;

6, 6, 2%10;

3, 6, 3%52%5, 3, %29

%110%114%189

!10, 0, 0"!!3, 3, 4"

y

x

12

45

1

2

3

4

5

2

3

−−

−3

(3, −2, 5)

z

32

, 4 ,−2( (

1 2 3 5


CH

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R 10



1. (a) 3. (a)

(b) (b)

5. (a) (b) (c)

7. (a) (b) (c)

9. (a) (b)

(c) (d)

11. (a) (b)

(c) (d)

13. 15.

17. 19.

21. 23. 25. 27.

29. 31. (a) (b)

33. (a) (b)

35. 37. 39. 41. 0

43. 45. 47. Parallel

49. Orthogonal 51. Neither 53. Orthogonal

55. Not collinear 57. Collinear

59. Right triangle. Answers will vary.

61. Acute triangle. Answers will vary. 63.

65. 67.

69. or

71. 10.91 pounds 73. True

75. (a)

(b) Answers will vary.

(c)

(d) Answers will vary.

77. The angle between u and v is an obtuse angle.


Vocabulary Check (page 762)1. cross product 2. 3.

4. triple scalar product

1. 3.

y

x

1

2

−2

2

−1

−21

−2

−1

2

(0, 1, 0)−

z

y

x

1

−1−2

2

−1

−21

−2

−1

2

(0, 0, 1)−

z!j!k

&u & &v & sin $0

a # b # 1

y

x

3

2

1

−1−2

−3

−2−3

23

23

z

uv

'0, 2%2, !2%2('0, 2%2, 2%2(±

3%1414#6, 52, !74$

!3, 1, 7"

)109.92&)124.45&!4) 8.73'!26, 0, 48(

!%7474

!8i " 3j ! k"%7474

!8i " 3j ! k"

! 113!5i ! 12k"113 !5i ! 12k"%34

%74%21%219%2

z # ' 112 , !54, !6(z # '! 52, 12, 152 (z # ' 12, 6, 32(z # '!3, 7, 6(

y

x

12

34

2

3

4

321 4−2

−3−4

−3−2

−4

−3−4

z

〈0, 0, 0〉y

x

12

34

56

1

2

3

321 4 6−2

−3−2

−4−5

z

5, 5, − 52

y

x

12

34

2

3

4

321 4−2

−3−4

−3−2

−4

−3−4

z

〈−2, −2, 1〉y

x

12

34

2

1

321−2

−3−4

−3−4

−6−5

−3−4

z

〈4, 4, −2〉

y

x

12

34

2

3

4

321 4−2

−3−4

−3−2

−4

−3−4

z

〈0, 0, 0〉

y

x

12

34

2

1

3

4

5

32 4−2

−3−2

−3−4

z

32

, 32

, 92

y

x

12

34

2

3

4

321 4−2

−3−4

−3−4

−3−4

z

〈−1, −1, −3〉y

x

12

34

1

2

3

4

5

6

−2

31 4−2−3−4

z

〈2, 2, 6〉

%22

'1, 1, 0(2%2'2, 2, 0(

%1133

'7, !5, 5(3%11'7, !5, 5(

y

x

3

12

34

−2−3

−2−1

−4

−3−4

23

4

2

−4

1

−2

−3

z

(0, 0, −4)y

x

3

2

1

1

−1

−3

−2−3

12

3

23

z

(−2, 3, 1)

'0, 0, !4('!2, 3, 1(


p 15 20 25 30 35 40 45

T 5.75 7.66 9.58 11.49 13.41 15.32 17.24

5. 7. 9.

11. 13.

15. 17. 19.

21. 23.

25. 27.

29. 31.

33. 35. 1 37. 39. 14

41. (a) Answers will vary.

(b)

(c) The parallelogram is not a rectangle.

43. 45. 47. 49. 2

51. 2 53. 12 55. 84

57. (a)

(b)

59. True 61 and 63. Proofs

65. 67.



1. direction, 2. parametric equations

3. symmetric equations 4. normal

5.

1. (a)

(b)

3. (a)

(b)

5. (a)

(b)

7. (a)

(b)

9. (a)

(b)

11. (a)

(b) Not possible.

13. (a)

(b)

15. 17.

19. 21.

23. 25.

27. 29.

31. 33.

35.

37.

39.

41. Orthogonal 43. Orthogonal

45. (a) (b)

47. (a) (b)

49. 51.

53.

x y6

−1−2 −2−1

−6−7

54

3

65

4

z

(0, 0, −6)

(0, 3, 0)(2, 0, 0)

y

x

32

4

4

6

3

−1−2

564

32

56

(0, 2, 0)(4, 0, 0)

z

−1−2

y

x

34

4

56

2

−2

56

23

(0, 0, 2)

(0, 3, 0)

(6, 0, 0)

z

x # 1 " 6t, y # t, z # 1 " 7t77.83&x # 2 ! t, y # 8t, z # 7t60.67&

x # 2 ! t, y # 1 " t, z # 2 " t

x # 5 " 2t, y # !3 ! t, z # !4 " 3t

x # 2 " 3t, y # 3 " 2t, z # 4 ! t

x # 2, y # 3, z # 4 " t7x " y ! 11z ! 5 # 0

y ! z " 2 # 0y ! 5 # 0

6x ! 2y ! z ! 8 # 0!3x ! 9y " 7z # 0

!x ! 2y " z " 2 # 0!2x " y ! 2z " 10 # 0

x ! 2 # 0

y

x

(0, 2, 1)

z

y

x

3

2

1

1

−1−2

−1−2−3

12

3

23

z

x " 123

#y ! 2

!5#

z ! 12!1

x # !12 " 3t, y # 2 ! 5t, z #12 ! t

x # 3 ! 4t, y # 1, z # 2 " 3t

x " 34

#y ! 8!10

# z ! 15

x # !3 " 4t, y # 8 ! 10t, z # 15 " t

x ! 2!1

#y4

#z ! 2!5

x # 2 ! t, y # 4t, z # 2 ! 5t

x ! 22

#y " 3!3

# z ! 5

x # 2 " 2t, y # !3 ! 3t, z # 5 " t

x " 43

#y ! 1

8#

z!6

x # !4 " 3t, y # 1 " 8t, z # !6t

x #y2

#z3

x # t, y # 2t, z # 3t

a!x ! x1" " b!y ! y1" " c!z ! z1" # 0

PQ\

t

!12

!12

T!p" # p2

cos 40&

!1612%4290

3%132

6%10

%806%2

2!i ! j"

%76027602

!!71i ! 44j " 25k"%1919

!i ! 3j " 3k"

%166166

'9, 6, !7(!14

i !7

10j ! 2k

!6i ! 15j ! 6k'10, !2, !4(!i ! 2j ! k!18i ! 6j!76 i !

78 j

!17i " j " 2k!7i " 13j " 16k

'0, 42, 0('3, !3, !3('1, 1, 1(


CH

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R 10



55. 57. 59.

61. False. Lines that do not intersect and are not in the sameplane may not be parallel.

63. Parallel. is a scalar multiple of

65. 67.

69. 71.

Review Exercises (page 774)1. 3. 5.

7. 9. 11.

13.

15.


19. (a) (b)

21. (a) (b) (c)

23. (a) (b) (c)

25. 27. 1 29. 31.

33. Orthogonal 35. Parallel 37. Not collinear

39. Collinear

41. A: 159.10 pounds of tension

B: 115.58 pounds of tension

C: 115.58 pounds of tension

43.

45.

47. Answers will vary; 49. 75

51. (a)

(b)

53. (a)

(b)

55. (a) (b)

57. 59.

61. 63.

65. 67. 69. False.

71 and 73. Answers will vary.

75.

77. The magnitude of the cross product will increase by a factor of 4.

Chapter Test (page 776)1.

2. No. Answers will vary. 3.

4.

4

4

6

8

6

−2−4−4

−10−8

2

8

12

y

x

xz-trace

sphere

z

!x ! 7"2 " !y ! 1"2 " !z ! 2"2 # 19!7, 1, 2"

y

x

24

2

−44

−4

( 2, 2, 3)− −

( 1, 2, 1)−(5, 2, 3)−

z

" !u1v2 ! u2v1"ku ' v # !u2v3 ! u3v2"i ! !u1v3 ! u3v1"j

u ' v # !!v ' u"%11055

%110110

x

y

(3, 0, 0)

(0, 0, 2)−1 21

−2 −1

3

3

4

1

−1

−2

2

z

x

y

(0, 0, 2)

(2, 0, 0)

(0, 3, 0)−

11

23

1

−2

z

z ! 2 # 0!2x ! 12y " 5z # 0

x!2

#y

5*2 # zx # !2t, y #52

t, z # t

x " 14

#y ! 3

3#

z ! 5!6

x # !1 " 4t, y # 3 " 3t, z # 5 ! 6t

x ! 36

#y

11#

z ! 24

z # 2 " 4ty # 11t,x # 3 " 6t,

2%43

%76027602

!!71i ! 44j " 25k"

'!10, 0, !10(

90&90&!9

%185185

'!10, 6, 7(%185'!10, 6, 7(

%3333

'1, 4, !4(%33'1, 4, !4(

2

−2 −22

4

4 42

6x

y

( 3) + = 16y z− 2 2

(0, 3, 0)

z

2

−2− 42

4

4 46x

y

(0, 3, 0)

x z2 2+ = 7

z

!2, 3, 0";!x ! 1"2 " !y ! 5"2 " !z ! 2"2 # 36!x ! 2"2 " !y ! 3"2 " !z ! 5"2 # 1

!1, 2, !9"!0, !1, 0"%29, %38, %67

%41!!5, 4, 0"

y

x

12

34

1

2

3

−2−3−4−5

1 2 3−2

−3

−4−5(5, −1, 2)

(−3, 3, 0)

z

r # 5 csc $r # 7

x2 " y2 ! 3x # 0x2 " y2 # 100

'!15, 27, !30(.'10, !18, 20(

88.45&2%6

389


x 0.9 0.99 0.999 1

f !x" 0.2564 0.2506 0.2501 Error

x 1.001 1.01 1.1

f !x" 0.2499 0.2494 0.2439

x !0.1 !0.01 !0.001 0

f !x" 1.8127 1.9801 1.9980 Error

x 0.001 0.01 0.1

f !x" 2.0020 2.0201 2.2140

x !0.1 !0.01 !0.001 0

f !x" 1.9867 1.99987 1.9999987 Error

x 0.001 0.01 0.1

f !x" 1.9999987 1.99987 1.9867

x 2.9 2.99 2.999 3

f !x" 0.1695 0.1669 0.1667 Error

x 3.001 3.01 3.1

f !x" 0.1666 0.1664 0.1639

x 1.9 1.99 1.999 2

f !x" 13.5 13.95 13.995 14

x 2.001 2.01 2.1

f !x" 14.005 14.05 14.5

x 3 3.5 3.9 4

V 972 1011.5 1023.5 1024

x 4.1 4.5 5

V 1023.5 1012.5 980

5. 6.

7.

8. (a) (b) 84 (c) 9.

10. Answers will vary. Sample answer:

(a)

(b)

11. Neither 12. Orthogonal

13. Answers will vary;

14. 15. 200

16. 17.

18.


Vocabulary Check (page 788)1. limit 2. oscillates 3. direct substitution

1. (a) (b) Answers will vary.

(c)

(d)

3.

14; Yes

5.

0.1667; No7.

2; No

9.

2; No

11.

0.25

00

12

1200

limx→4

V " 1024

2(12 )− x

2(12 )− x

x

4#147

y

x

4

810

2

4

6

−6−8

−10

2 4 6

−6

−10

(0, −10, 0)

(0, 0, −5)

(2, 0, 0)

z

y

x

1

65

1

3

4

21 4−2

−3

−3−4

z

(0, 0, 3)

(6, 0, 0)

(0, 4, 0)

27x # 4y # 32z # 33 " 0

2#230

x ! 8!2

"y # 2

6"

z ! 5!6

x " 8 ! 2t, y " !2 # 6t, z " 5 ! 6t

46.23$$0, 62, 62%#194v " $!12, 5, !5%u " $!2, 6, !6%,

7#2$!3, !5, 8%;#129$2, 5, !10%;


CH

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333350_11_ans_odds.qxp 1/19/07 12:38 PM Page A267

x 0.9 0.99 0.999 1

f !x" 2.2314 2.0203 2.0020 Error

x 1.001 1.01 1.1

f !x" 1.9980 1.9803 1.8232

x !0.1 !0.01 !0.001 0

f !x" 0.9063 0.9901 0.9990 Error

x 0.001 0.01 0.1

f !x" 1.0010 1.0101 1.1070

x !0.1 !0.01 !0.001 0

f !x" !0.0997 !0.0100 !0.0010 Error

x 0.001 0.01 0.1

f !x" 0.0010 0.0100 0.0997

x !0.1 !0.01 !0.001 0

f !x" 0.9983 0.99998 0.9999998 Error

x 0.001 0.01 0.1

f !x" 0.9999998 0.99998 0.9983

x !0.1 !0.01 !0.001 0

f !x" 0.2247 0.2237 0.2236 Error

x 0.001 0.01 0.1

f !x" 0.2236 0.2235 0.2225

x !4.1 !4.01 !4.001 !4

f !x" 0.4762 0.4975 0.4998 Error

x !3.999 !3.99 !3.9

f !x" 0.5003 0.5025 0.5263


13.

0.2236

15.

0.5

17.

1

19.

0

21.

1

23.

2

25. 27.

Limit does not exist.

29. 13

31. Limit does not exist; one-sided limits do not agree.

33. Limit does not exist; function oscillates between and 2.

35. Limit does not exist; one-sided limits do not exist.

37. 39.

No No

41. 43.

No Yes

45.

Yes

47. (a) (b) 9 (c) (d)

49. (a) 8 (b) (c) 3 (d)

51. 53. 7 55. 57. 59.

61. 1 63. 65. 67. 0 69.

71. True 73. (a) and (b) Answers will vary.

75. (a) No. The function may approach different values fromthe right and left of 4.

(b) No. The function may approach 4 as approaches 2,but the function could be undefined at

77. 79.5x # 45x # 2

, x %13

!13

, x % 5

x " 2.x

&6

e3353

713!

910!3!15

!61838

#312!12

−6

−4

6

4

−1

−3

8

3

−1

−3

8

3

−3

−2

3

2

−3

−1

3

3

!2

limx→2

f !x" " 5

−2−4−6−8 2

2

4

6

8

10

4 6 8

−6

−4

x

y

−2 2 4 6 8

2

−2

4

6

8

x

y


81.


Vocabulary Check (page 798)1. dividing out technique 2. indeterminate form

3. one-sided limit 4. difference quotient

1. (a) 1 (b) 3 (c) 5

3. (a) 2 (b) 0 (c) 0

5. 7. 4 9. 12 11. 0 13. 15. 3

17. 19. 21. 23. Limit does not exist.

25. 0 27. 0

29. 31.

0.29 1.00

33. 35.

80.00

37. 39.

2.000 0.000

41. 43.

2.000 1.000

45. 47.

0.333 0.135

49. (a) and (b) 0.50 (c)

51. (a) and (b) (c)

53. 55.


57. 59.


61. 63.

65.

67. (a) Direct substitution; 0 (b) 1

limx→0

f !x" " 0

−3

−2

3

2

y = x y = −x

f(x) = x sin 1x

limx→0

f !x" " 0limx→0

f !x" " 0

−9

−6

9

6

y = x y = −x

f(x) = ⏐x⏐sin x

−9

−6

9

6

y = x y = −x

f(x) = x cos x

limx→2

f !x" " 1

x

y

−1−3 1 2 3 4 5−1

−2

−3

1

2

3

5

x

y

−1−2−3 1 2 3 4 5−1

−3

−4

1

2

3

4

limx→1

f !x" " 12

−1−1 1 2 3−2−3

−2

−3

2

3

x

y

8 10−2

−4

−6

2

4

6

x

y

!18!0.13

12

−1.5

−1

1.5

1

−3

−2

3

2

−3

−2

3

2

−6

−4

6

4

−2

−1

4

3

−6

−1

3

5

!0.06

−8

−3

1

3

−10

−20

10

200

−2

−1

4

3

−4

−2

2

2

!114

#510

13

112

g2!x" " x!x # 1"

g2!x" " !2x # 1

x2 ! 3x # 9x ! 2

, x % !3


CH

AP

TE

R 11


t 2 2.5 2.9 3 3.1 3.5 4

C 1.25 1.50 1.50 1.50 1.75 1.75 1.75

t 3 3.3 3.4 3.5 3.6 3.7 4

C 1.50 1.75 1.75 1.75 1.75 1.75 1.75


69. 3 71. 73. 75.

77. feet per second

79. (a)

(b)

(c)

Limit does not exist; the one-sided limits do not agree.

81. Answers will vary. 83. True.

85. (a) and (b) Answers will vary.

87.

89. Parabola 91. Hyperbola

93. Parabola 95. Orthogonal

97. Parallel


Vocabulary Check (page 808)1. Calculus 2. tangent line 3. secant line

4. difference quotient 5. derivative

1. 0 3. 5. 2 7. 9. 11.

13. (a) 0 (b) 2

15. (a) (b)

17. (a) (b)

19. 21.

2

23. 25. 0

27. 29. 31. 33.

35. 37.

39. (a) 4 41. (a) 1(b) (b)

(c) (c)

43. (a) 45. (a)

(b) (b)

(c) (c)

−4−8−10−12 2

2

4

6

8

4

−6

−8

−4

x

y

(−4, 1)

x

y

−1−2 1 2 3 4−1

−2

2

3

4

(3, 2)

y " !x ! 3y " 14x #54

!114

−2−3

2

3

2 3 4 5

−4

−5

−2

−1

x

y

(1, −1)

x

y

−3−4 3 4

−2

1

2

3

4

5

6

2−2

(2, 3)

y " x ! 2y " 4x ! 5

!1

2!x ! 9"3&2!1

!x # 2"2

12#x ! 4

!2x3

!6x!13

!1

−6

−4

6

4

(1, 2)

!12

−6

−4

6

4

(1, 1)

−9

−6

9

6

(1, −1)

16

14

m "1

2#x ! 1;

!14

!1

16m " !

1!x # 4"2;

m " !2x;

16!1!2

12

−9

−6

9

6

−8

−9

16

9

−12

−9

12

9

x ! 2y ! 26 " 0

limt→3.5

C!t" " 1.75

−1−0.5

1 2 3 4 5

0.5

1.0

1.5

2.0

2.5

x

y

!32

!1

!x # 2"22x ! 31

2#x


x !2 !1.5 !1 !0.5 0

f !x" 2 1.125 0.5 0.125 0

f' !x" !2 !1.5 !1 !0.5 0

x 0.5 1 1.5 2

f !x" 0.125 0.5 1.125 2

f' !x" 0.5 1 1.5 2

x !2 !1.5 !1 !0.5 0

f !x" 1 1.225 1.414 1.581 1.732

f' !x" 0.5 0.408 0.354 0.316 0.289

x 0.5 1 1.5 2

f !x" 1.871 2 2.121 2.236

f' !x" 0.267 0.25 0.236 0.224

47.

They appear to be the same.

49.

They appear to be the same.

51.

53.

55.

57.

59. 61.

63. (a)

(b)

38; The population is increasing at approximately38,000 people per year in 2020.

(c)


65. (a) (b)

(c) Cubic inches per inch; The derivative is a formula forrate of change.

67. (a) (b) 16 feet per second

(c) seconds; Answers will vary.

(d) feet per second

(e)

69. True. The graph of the derivative is a line, which is a one-to-one function.

71. b 72. a 73. d 74. c

75. Answers will vary. Sample answer: A sketch of any linearfunction with positive slope

77. 79.

81. 83.


Vocabulary Check (page 817)1. limit, infinity 2. converge 3. diverge

$0, 0, !36%$!2, 3, !1%

1

−1

−2

−3

−4

−3−4

2

3

4

321 4x

y

1

−2−3−4

2

3

4

3 4x

y

00

6

200

!96

t " 2

s'!t" " !32t # 64

'201.06V'!r" " 4&r2

f' !20" " 38.1f' !x" " !1.26x # 63.3;

00 200

13,000

P!t" " !0.63t 2 # 63.3t # 8448

!e!1, !e!1"!0, 0", !!2, 4e!2"

(&6, #3 # &6), (5&6 , 5&6 ! #3)!!1, !1", !0, 0", !1, !1"

!1, !6"!!1, 6",f'!x" " 9x2 ! 9;!2, !1"f'!x" " 2x ! 4;

−2

−1

2

3

−2

−2

2

2


CH

AP

TE

R 11


n 100 101 102 103

an 2 1.55 1.505 1.5005

n 104 105 106

an 1.5001 1.5000 1.5000

n 100 101 102 103

an 16 6.16 5.4136 5.3413

n 104 105 106

an 5.3341 5.3334 5.3333

x 100 101 102 103

f !x" !0.7321 !0.0995 !0.0100 !0.0010

x 104 105 106

f !x" !1.0 ( 10!4 !1.0 ( 10!5 !1.0 ( 10!6

x 100 101 102 103

f !x" !0.7082 !0.7454 !0.7495 !0.74995

x 104 105 106

f !x" !0.749995 !0.7499995 !0.7500


1. c 2. a 3. d 4. b 5. f 6. g 7. h

8. e 9. 0 11. 13. 15.

17. Limit does not exist. 19. 21. 2 23.

25. 27.

29. 31.

33.

35. (a)

0

(b)

0

37. (a)

(b)

39. 41.

Limit: 0 Limit:

43. 45. 2, 3, 4, 5, 6

Limit does not exist. Limit does not exist.

47.

Limit: 0

49.

1.5

51.

5.33

53. (a) (b) $471; $59.25

(c) $13.50; As the number of PDAs produced gets verylarge, the average cost approaches $13.50.

55. (a)

Answers will vary.

(b) 2004: 72,000,000; 2008: 73,800,000

(c) 78 million; As time passes, school enrollment in theUnited States approaches 78 million.


57. False. does not have a horizontal asymptote.

59. True

61. Answers will vary. Sample answer: Let

and Now and

limx→0

* f !x" ! g!x"+ " 0.

limx→0

1x2

" )c " 0.g!x" "1x2

,

f !x" " 1x2

,

y "x2

x # 1

0 1460

76

C "13.50x # 45,750

x

limn→)

an "163

limn→)

an "32

!1, 12, !13,

14, !

15

15,

12,

911,

87,

2517

12

13,

25,

37,

49,

5111,

35,

25,

517,

313

!0.75

−3

−5

6

1

−6

−10

12

2

−6

−5

6

3

−6

−4

6

4

−4

−6

8

2

!5!4

!143

!456!1


n 100 101 102 103 104

S!n" 0 0.615 0.6617 0.6662 0.6666

n 100 101 102 103 104

S!n" 1 0.3025 0.2550 0.2505 0.2501

n 100 101 102 103 104

S!n" 6 1.185 1.0154 1.0015 1.0002

n 100 101 102 103 104

S!n" 3 0.2385 0.0234 0.0023 0.0002

n 4 8 20 50

Approximate area 18 21 22.8 23.52

n 4 8 20 50

Approximate area

3.5156 2.8477 2.4806 2.3409

n 4 8 20 50 100 )

Area 40 38 36.8 36.32 36.16 36

n 4 8 20 50 100 )

Area 36 38 39.2 39.68 39.84 40

n 4 8 20 50 100 )

Area 14.25 14.81 15.13 15.25 15.29 463

n 4 8 20 50 100 )

Area 19 18.5 18.2 18.08 18.04 18

63. 65.

Converges to 0 Diverges

67.

69. 71.

73. 5, 0, 0 75. 3

77. 60 79. 150



1. 2. 3. area

1. 420 3. 44,140 5. 44,140 7. 5850

9. (a)

(b)

(c)

11. (a)

(b)

(c)

13. (a)

(b)

(c)

15. (a)

(b)

(c)

17. 14.25 19. 1.27

21.

23.

25.

27.

29.

31.

33. 3 35. 2 37. 39. 41. 43.

45.

square feet'105,208.33

−100

−100

600

500

514

34

174

103

limn→)

S!n" " 23

S!n" " 4n2 ! 3n ! 1

6n2

limn→)

S!n" " 0

S!n" " 14n2 # 3n # 1

6n3

limn→)

S!n" " 1

S!n" " 2n2 # 3n # 7

2n2

limn→)

S!n" " 14

S!n" " n2 # 2n # 1

4n2

n2!n # 1"24

n!n # 1"2

−9

−9

9

3

−10

−150

10

25

!4,

x3 # 5x2 ! 3 !2

3x # 2x2 # 2x # 1

x

y

−2−4 2 4 6

−4

2

4

6(a) (d)

(c)(b)

00

11

360

00

11

5


CH

AP

TE

R 11


n 100 101 102 103 104

S!n" 3 0.99 0.8484 0.8348 0.8335

n 4 8 20 50

Approximate area 7.5 6.375 5.74 5.4944

x 2.9 2.99 2.999 3

f !x" 16.4 16.94 16.994 17

x 3.001 3.01 3.1

f !x" 17.006 17.06 17.6

x !0.1 !0.01 !0.001 0

f !x" 1.0517 1.0050 1.0005 Error

x 0.001 0.01 0.1

f !x" 0.9995 0.9950 0.9516


47. True 49. Answers will vary. 51.

53. 55.

Review Exercises (page 829)1.

17; Yes

3.

1; No

5. 2 7. 2 9. (a) 64 (b) 7 (c) 20 (d)

11. 5 13. 77 15. 17. 19. 0

21. 23. 25. 27.

29. 31. 33. 35.

37. (a) and (b) 0.17 39. Limit does not exist.

41. (a) and (b) 2 43. (a) and (b) 0.577

45. 47.


49. 51.


53. 55. 2

57. 59.

2

61.

63. (a) (b) 6

65. (a) (b)

67. 69. 71.

73. 75.

77. 79. 2 81. 0

83. Limit does not exist. 85. 3

87. 89.

Limit: Limit: 0

91.

Limit:

93. (a)

(b)

(c)

95. 675

97.

99. 50 101. 15 103. 6 105. 34

107. (a)

! 0.168x # 132

y " !!3.376 ( 10!7"x3 # !3.753 ( 10!4"x2

56

S!n" " 5n2 # 9n # 4

6n2

!1

!12, !98, !

76, !

3732, !

5750

25

!1, 18, !1

27, 164, !

1125!

19,

114,

319,

524,

729

g'!x" " ! 12!x # 4"3&2

g'!s" " ! 4!s # 5"2f'!t" "1

2#t # 5

g' !x" " 4xh'!x" " !12f'!x" " 0

!1!4m " !4

!x ! 6"2;

!4m " 2x ! 4;

!32

20−10

−10

10

(2, −3)

14

−6

−2

6

6

(2, 2)−5

−4

7

4

(2, 0)

3 ! 2x

x

y

−1−2 1 2 3 4 5 6−1

1

2

3

4

5

6

7

x

y

6 8 10−2

−4

−6

2

4

6

x

y

−3−4 3 4

−2

−3

−4

1

2

3

4

2−2x

y

−1−2 1 2 4 5 6

−2

−3

−4

1

2

3

4

14

14!1!

13

115

!14

!&6

2e

!2103

45

#5 ! 2$24, !30%n&


(b)

(c) 88,700 square feet

109. False. The limit of the rational function as approachesdoes not exist.



3. 4.

Limit does not exist. 3.0000

5.

2.0000

6. (a) (b)

7. 8.

9. 10. 0 11.

12. Limit does not exist.

13. 14.

Limit: Limit: 0

15. 12.5 16. 8 17.

18. (a) (b) 81.7 feet per second

Cumulative Test for Chapters 10 and 11(page 833)1. 2. 3.

4. 5.

6.

7.

8.

9. Neither 10. Orthogonal 11. Parallel 12. 12

13. (a)

(b)

14.

15.

16. 17.

18. 19. 20. 21. 22.

23. 24. Limit does not exist. 25. 26.

27. 28. 29.

30. 31.

32. Limit does not exist. 33. 34. 3

35. 0 36. 0 37. Limit does not exist.

38. 39. 8190 40. 672,880

41. 10.5 42. 8.13 43. 2.69 44. 1.57 45.

46. 64 47. 28 48. 49. 50. 34163

763

52

!42,875

!7

m " 2x ! 1; 1m " !!x # 3"!2; ! 116

m " 12!x # 3"!1&2; 12m " !2x; 0

14

12!

14!1

14

114!

13484.26$

#302

y

x

4

8

4 62

−6

−4

2

4

6

−6

z

(0, −4, 0)

(8, 0, 0)

(0, 0, −2)

75x # 50y ! 31z " 0

z " ty " 2 ! 4t,x " !1 # 2t,

x # 27

"y ! 3

5"

z25

x " !2 # 7t, y " 3 # 5t, z " 25t

u ( v " $!18, !6, !14%u * v " !38

y

x

24

−2

2

−22

4 yz-trace(0, 1, 0)−

xy-trace( 2) + ( + 1) = 4x y− 2 2

z

!x ! 2"2 # !y ! 2"2 # !z ! 4"2 " 24!!1, 2, 12"3, 4, 5

#149!0, !4, 0"!!6, 1, 3"

y " 8.79x2 ! 6.2x ! 0.4

163

12

0, 1, 0, 12, 00, 34,

1419,

1217,

3653

!3f'!x" " ! 1!x # 3"2

f'!x" " 4x # 4f'!x" " !25

m " 6x2 # 6; 12m " 6x ! 5; 7

−6

−1

6

7

−2

−1

2

4

&&

−2

−4

10

4

limx→!2

f !x" " !34

!0.75

8−4

−6

2

−6

−4

6

4

)x

00

1000

150


CH

AP

TE

R 11



AppendicesAppendix B.1 (page A32)

Vocabulary Check (page A32)1. (a) iii (b) vi (c) i (d) iv (e) v (f) ii

2. Cartesian 3. Distance Formula

4. Midpoint Formula

5. center, radius

1.

3. 5.

7. 9. 11. Quadrant IV

13. Quadrant II 15. Quadrant III or IV

17. Quadrant III 19. Quadrant I or III

21.

23. 8 25. 5 27. 13 29. 31.

33. (a) 4, 3, 5 (b)

35. (a) (b)

37.

39. Two equal sides of length

41. Opposite sides have equal lengths of and

43. The diagonals are of equal length The slope of theline between and is The slope of the linebetween and is The slopes are negativereciprocals, making them perpendicular lines, which forma right angle.

45. (a) (b) 10

(c)

47. (a) (b) 17

(c)

49. (a) (b)

(c)

51. (a) (b)

(c)

53. (a) (b)

(c)

55. $3,093.5 million

57. (a) (b) !9, !3"!7, 0"!2xm ! x1, 2ym ! y1";

!1.25, 3.6"#110.97

2

4

6

8

2−2−2

−4 4 6

(6.2, 5.4)

( 3.7, 1.8)−

y

x

$!1, 76%

#823

( )( ), 1

−1 −−− −2

25

134

222

3

12

52

12

12

32

52

,−

x

y

!2, 3"2#10

(5, 4)

( 1, 2)−

−1 1−1

2 3 4 5

3

4

5

x

y

!0, 52"( 4, 10)−

(4, 5)−

−8 −6 −4 −2 4 6 8

−6

−4

2

6

8

10

y

x

!5, 4"

(1, 1)

(9, 7)

12

10

8

6

4

2

2−2 4 6 8 10

y

x

!52.!!3, 1"!!5, 6"

25.!0, 8"!!5, 6"

!#58 ".#85.2#5

#29

!#5 "2 " !#45 "2 # !#50 "2102 " 32 # !#109"210, 3, #109

42 " 32 # 52#71.78

#2776

Sale

s (i

n m

illio

nsof

dol

lars

)

1997

1998

1999

2000

2001

2002

2004

2003

Year

2,0004,0006,0008,000

10,00012,00014,00016,00018,000

2005

2006

!!6, !6"!!5, 4"

x

y

(−2, −2.5) (0.5, −1)(5, −6)

(3, 8)

−2−4−6−8 2 4 6 8

−4

−6

−8

2

4

6

8

−2−4 2 4 6−2

−4

−6

2

4

6

x

y

(−4, 2)

(−3, −6)

(1, −4)

(0, 5)

A: !2, 6"; B: !!6, !2"; C: !4, !4"; D: !!3, 2"

!x ! h"2 " !y ! k"2 # r2,

333350_APP_ans.qxp 1/22/07 9:27 AM Page A276

x !2 !1 0 1 2

y !72 !134 !3 !

114 !

52

x !2 0 23 1 2

y !4 !1 0 12 2

Solutionpoint !!2, !4" !0, !1" !

23, 0" !1, 12" !2, 2"

59. 61.

63.

65.

67.

69.

71.


Radius Radius

77. Center:

Radius

79. 81.

83. 65

85. (a) Answers will vary. Sample answer: The number of artistselected each year seems to be nearly steady except forthe first few years. Estimate: From 5 to 7 new membersin 2007

(b) Answers will vary. Sample answer: The Rock and RollHall of Fame was opened in 1986.

87. yards

89. (a)

(b) 2 P.M.: 40 miles; 4 P.M.: 80 miles; The yachts are twiceas far from each other at 4 P.M. as they were at 2 P.M.

91. The distance between and is The distance between and is

The distance between and is Because the distance between each set of points is thesides connecting those points are all the same length, mak-ing the coordinates the vertices of an equilateral triangle.

93. False. You would have to use the Midpoint Formula 15times.

95. False. It could be a rhombus.

97. No. The scales depend on the magnitudes of the quantitiesmeasured.

Appendix B.2 (page A43)

Vocabulary Check (page A43)1. solution point 2. graph 3. intercepts

1. (a) Yes (b) Yes 3. (a) No (b) Yes

5. (a) No (b) Yes

7.

9. (a)

−1−2−3−4−5 1 2 3 4 5

−3−4−5

12345

x

y

4#3,4#3.!2 ! 2#3, 0"!2, 6"4#3.

!2 ! 2#3, 0"!2 " 2#3, 0"4#3.!2 " 2#3, 0"!2, 6"

(0, 64)4 P.M.

(−48, 0)4 P.M.

(−24, 0)2 P.M.

(0, 32)2 P.M.

1 unit : 8 mi

Fisherman

Beach L

over

−72

−64

−56

−48

−40

−32

−24

−16 −8

8

2416

324048566472

N

S

EW

5#74 & 43

!!1, 5", !2, 8", !4, 5", !1, 2"!0, 1", !4, 2", !1, 4"

–1 1 2 3

1

3

y

x

# 32

! 12, 12"

–2 –1 1 2 3 4

–5

–3

–1

1

x

y

−1−2−3−4 1 2 3 4 6−2−3−4

−6

1234

6

y

x

# 2# 5

!1, !3"!0, 0"!x ! 2"2 " !y " 1"2 # 16!x ! 3"2 " !y " 6"2 # 16!x " 2"2 " !y ! 1"2 # 1!x ! 3" 2 " !y ! 4" 2 # 25!x " 1" 2 " !y ! 2" 2 # 5

!x ! 2"2 " !y " 1"2 # 16x 2 " y 2 # 9


AP

PE

ND

ICE

S

333350_APP_ans.qxp 1/22/07 9:27 AM Page A277

A242 Answers to Odd-Numbered Exercises and Testshorneckermath.weebly.com/.../chapter_9-11_odd_answers.pdf · 2018. 8. 15. · A246 Answers to Odd-Numbered Exercises and Tests 29.

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