Chapter 10...Precalculus with Limits, Answers to Section 10.2 3 Section 10.2 (page 740) Vocabulary...

Precalculus with Limits, Answers to Section 10.1 1

Chapter 10Section 10.1 (page 732)

Vocabulary Check (page 732)1. inclination 2.

3. 4.

1. 2. 1 3. 4.

5. 6. 7. 3.2236 8.

9. 10. 2.0344 radians,

11. 12. 1.1071 radians,

13. 0.6435 radian, 14. 1.9513 radians,

15. 1.0517 radians, 16. 0.6023 radian,

17. 2.1112 radians, 18. 2.0344 radians,

19. 1.2490 radians, 20. 2.4669 radians,

21. 2.1112 radians, 22.

23. 1.1071 radians, 24.

25. 0.1974 radian, 26. 1.1071 radians,

27. 1.4289 radians, 28. 1.4109 radians,

29. 0.9273 radian, 30. 1.0808 radians,

31. 0.8187 radian, 32. 1.0240 radians,

33.

34.

35.

36.

37. 0 38. 39.

40. 41. 7 42. 4

43. 44.

45. (a) (b) 4 (c) 8

46. (a) (b) (c)

47. (a) (b) (c)

48. (a) (b) (c)

49. 50. 51. 0.1003, 1054 feet

52. 0.2027, 1049 feet 53.

54. (a)

(b)(c) 15.8 meters

18.4�

3 m

1 mθ

31.0�

952�2

312

31�389389

y

x

A

CB

−4−6 2 4 6 8 10

−4

−6

4

6

8

10

12

358

35�3774

y

x−1−2 1 2 3 4 5

−1

−2

1

2

3

4

5

A

B

C

332

33�2929

y

x−1−2 3 4 6

−1

−2

−3

1

2

3

4

5

A

C

B

1−1 2 3 4 5 6−1

1

2

3

4

5

6

A

B

C

y

x

9�2 � 12.72798�37

37� 1.3152

5�22

� 3.5355

75

4�55

� 1.7889

��3, 4�: 32.5�; �2, 1�: 16.9�; ��2, 2�: 130.6�

�2, 1� ↔ ��2, 2�: slope � �14

��3, 4� ↔ �2, 1�: slope � �35

��2, 2� ↔ ��3, 4�: slope � �2

��4, �1�: 11.9�; �3, 2�: 21.8�; �1, 0�: 146.3�

��4, �1� ↔ �3, 2�: slope �37

��3, 2�: 35.8�; �1, 3�: 94.4�; �2, 0�: 49.8�

�2, 0� ↔ ��3, 2�: slope � �25

�1, 3� ↔ �2, 0�: slope � �3

��3, 2� ↔ �1, 3�: slope �14

�2, 1�: 42.3�; �4, 4�: 78.7�; �6, 2�: 59.0�

�6, 2� ↔ �2, 1�: slope �14

�4, 4� ↔ �6, 2�: slope � �1

�2, 1� ↔ �4, 4�: slope �32

58.7�46.9�

61.9�53.1�

80.8�81.9�

63.4�11.3�

�

4 radian, 45�63.4�

�

4 radian, 45�121.0�

141.3�71.6�

116.6�121.0�

34.5�60.3�

111.8�36.9�

63.4��

4 radian, 45�

116.6�3�

4 radians, 135�

�0.2677��33

�3

��3�1�3

3

�Ax1 � By1 � C��A2 � B2� m2 � m1

1 � m1m2�tan �

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333202CB10a_AN.qxd 4/13/06 6:34 PM

(Continued)

55.

56. (a) 0.6167 radian,

(b) 518.5 feet

(c)

(d)

57. True. The inclination of a line is related to its slope byIf the angle is greater than but less than

then the angle is in the second quadrant, where the tangentfunction is negative.

58. False. Substitute and for and in the formula for the angle between two lines.

59. (a)

(b)

(c)(d) The graph has a horizontal asymptote of As the

slope becomes larger, the distance between the originand the line, becomes smaller andapproaches 0.

60. (a)

(b)

(c) (d) Yes.

(e) As the line approaches the vertical, the distanceapproaches 3.

61. intercept: 62. intercept:

intercept: intercept:

63. intercepts: 64. No intercepts

intercept: intercept:

65. intercepts:

intercept:

66. intercepts:

intercept:

67. 68.

Vertex: Vertex:

69. 70.

Vertex: Vertex:

71. 72.

Vertex: Vertex:

73. 74.

75. 76. y

x−2 6 8

−2

−4

−6

−8

2

y

x−1−2−3−4 1 2 3 4

−1

−2

1

2

5

6

−2−4−6 2 4

−2

−4

2

4

y

x

−3 3 6 9 12

−3

3

6

9

12

y

x

��178 , 121

8 �� 112, �289

24 �f �x� � �8�x �

178 �2

�1218f �x� � 6�x �

112�2

�28924

��4, 1��175 , �324

5 �f �x� � ��x � 4�2 � 1f �x� � 5�x �

175 �2

�324

5

�14, �169

8 ��13, �49

3 �f �x� � 2�x �

14�2

�169

8f �x� � 3�x �13�2

�493

�0, �22�y-

��11, 0�, �2, 0�x-

�0, �1�y-

�7 ± �532

, 0�x-

�0, 133�y-�0, 20�y-

x-�5 ± �5, 0�x-

�0, 81�y-�0, 49�y-

��9, 0�x-�7, 0�x-

d � 3.

m � �1m � 1

m−2−4−6

8

−2

−4

2 4 6

6

4

d

d �3�m � 1��m2 � 1

y � mx � 4,

d � 0.m � 0

−4 −3 −2 −1 1 2 3 4

−2

1

2

5

6

m

d

d �4

�m2 � 1

m2m1tan �2tan �1

�,�2m � tan �.

100 200 300 400 500 600

100

200

300

400

500

600

y

x

y � 0.709x

35.3�

� � 33.69�; � 56.31�


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Section 10.2 (page 740)

Vocabulary Check (page 740)1. conic 2. locus 3. parabola; directrix; focus4. axis 5. vertex 6. focal chord 7. tangent

1. A circle is formed when a plane intersects the top or bottom half of a double-napped cone and is perpendicularto the axis of the cone.

2. An ellipse is formed when a plane intersects only the topor bottom half of a double-napped cone but is not parallelor perpendicular to the axis of the cone, is not parallel tothe side of the cone, and does not intersect the vertex.

3. A parabola is formed when a plane intersects the top orbottom half of a double-napped cone, is parallel to the sideof the cone, and does not intersect the vertex.

4. A hyperbola is formed when a plane intersects both halvesof a double-napped cone, is parallel to the axis of the cone,and does not intersect the vertex.

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

11. Vertex: 12. Vertex:

Focus: Focus:

Directrix: Directrix:


Focus: Focus:



Focus: Focus:



Focus: Focus:



Focus: Focus:



Focus: Focus:


2 4 6 10

−6

−4

−2

2

4

x

y

−2 2 4

2

4

6

x

y

x � 7y � 0

�9, �1��1, 2��8, �1��1, 1�

x4

6

−2

−2−4−6

4

8

2

y

x

8

7

6

5

4

3

−2

321−1−2−3−4−5−6−7

1

y

y � 0y � 1

��12, 2��3

2, 3��1

2, 1��32, 2�

x

6

−2

−4

−2−4−6 2

4

2

y

−3 −2 −1 1 2 3 4 5

−4

−3

1

2

3

4

x

y

x � �194y � 0

��214 , 1��1, �4�

��5, 1��1, �2�

y

x−1−2−3−4−5 1

−1

−2

−3

1

2

3

x

2

431−3−4

1

−1

−2

−3

−4

−5

−6

y

x �14y �

32

��14, 0��0, �3

2��0, 0��0, 0�

y

x−1 1 2 3 4 5

1

2

3

−6 −5 −4 −3 −2 −1 1 2

−4

−3

3

4

x

y

x � �34x �

32

�34, 0��3

2, 0��0, 0��0, 0�

x21

1

−3

−4

−5

−6

−7

43−1−2−3−4

y

1

2

3

4

5

x1 2 3

y

−1

y �18y � �

12

� 0, �18��0, 12�

�0, 0��0, 0�

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(Continued)


Focus: Focus:



Focus: Focus:


27. Vertex: 28. Vertex: Focus: Focus: Directrix: Directrix:

29. 30. 31.

32. 33. 34.

35. 36. 37.

38. 39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

units units

61.

62. (a) (b)

(c)

63. (a) (b) 8 feet

64.

65. (a) miles per hour 24,750 miles per hour

(b)

66. (a)

(b) Highest point: Range: 15.69 feet

67. (a) (b) 69.3 feet

68. 34,294.6 feet

69. False. If the graph crossed the directrix, there would existpoints closer to the directrix than the focus.

70. True. If the axis (line connecting the vertex and focus) ishorizontal, then the direction must be vertical.

71. (a)

As increases, the graph becomes wider.(b) �0, 1�, �0, 2�, �0, 3�, �0, 4�

p

18

21

−3

−18

p = 2

p = 1

p = 3

p = 4

x2 � �64� y � 75�

�6.25, 7.125�

00

18

12

x2 � �16,400�y � 4100��17,500�2

y 2 � 640x

y � �1

640 x2

y �19x2

51,200y

x

(−640, 152) (640, 152)

y �118 x2

x � 135x � 106

2750

30,000

00 225

15,000

0

8x � y � 8 � 0; �1, 0�4x � y � 2 � 0; ��12, 0�

6x � 2y � 9 � 0; ��32, 0�4x � y � 8 � 0; �2, 0�

�6, �3��2, 4�

−12 12

−10

6

25

10

−10

−5

y � ��2�x � 4� � 1y � �6�x � 1� � 3

x2 � �16�y � 4��y � 2�2 � 8x

�y � 1�2 � �12�x � 2�x2 � 8�y � 4�

�x � 1� 2 � �8�y � 2��y � 2�2 � �8�x � 5�

�x � 3�2 � 3�y � 3�y2 � 4�x � 4�

�y � 3�2 � �2�x � 5��x � 3�2 � ��y � 1�

x2 � �3yy2 � 9xy 2 � 12x

y2 � �8xx 2 � �12yx2 � 4y

x2 � �8yy2 � �8xy 2 � 10x

x2 � �6yy2 � �18xx2 �32 y

−4

−8

20

8

2

4

−4

−10

x � �2x �12

�0, 0��0, �12�

��1, 0��14, �1

2�

−8 10

−9

3

10

4

−12

−14

y � 1y �52

�1, �3��2, �12�

�1, �1��2, 1�

−4 2 4

−2

4

6

x

y

−10 −8 −6 −4

−8

−6

−4

−2

2

x

y

x � �2x � 0

�0, 2��4, �3��1, 2��2, �3�


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Distance, x 0 250 400 500 1000

Height, y 0 23.19 59.38 92.77 371.09

333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)

(c) 4, 8, 12, 16;(d) Easy way to determine two additional points on the

graph

72. (a)

(b) As approaches zero, the parabola becomes narrowerand narrower, thus the area becomes smaller andsmaller.

73. 74. Answers will vary. 75.

76. 77.

78.

79. 80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90. A � 41.85�, C � 67.15�, b � 29.76

B � 24.62�, C � 90.38�, a � 10.88

A � 43.53�, B � 19.42�, C � 117.05�

A � 16.39�, B � 23.77�, C � 139.84�

A � 50�, b � 10.87, c � 24.07

C � 89�, a � 1.93, b � 2.33

A � 96.37�, C � 29.63�, a � 22.11

B � 23.67�, C � 121.33�, c � 14.89

12, �1, ±3

5

−50

−5

20

12, �5

3, ±2

32, ±5if �x� � x3 � 7x2 � 17x � 15

±13, ±2

3, ±1, ±2, ±113 , ±22

3 , ±11, ±22

±12, ±1, ±2, ±4, ±8, ±16±1

2, ±1, ±2, ±52, ±5, ±10

±1, ±2, ±4m �x1

2p

p

64�23

� 30.17

4�p�

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Vocabulary Check (page 750)1. ellipse; foci 2. major axis; center3. minor axis 4. eccentricity

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

7. Ellipse 8. EllipseCenter: Center:Vertices: Vertices:Foci: Foci:

Eccentricity: Eccentricity:

9. Circle 10. CircleCenter: Center:Radius: 5 Radius: 3

11. Ellipse 12. EllipseCenter: Center:Vertices: Vertices:Foci: Foci:Eccentricity: Eccentricity:

13. EllipseCenter:Vertices:Foci:Eccentricity:

14. Ellipse 15. CircleCenter: Center:Vertices: Radius:Foci:Eccentricity:

16. EllipseCenter:

Vertices:

Foci:

Eccentricity:

17. EllipseCenter:Vertices:

Foci:

Eccentricity:

18. CircleCenter:Radius: 5

2

�3, 1�y

x−1 2 3 4 6

−1

−2

1

2

3

4

5

�3

2

��4 ± �3

2, �4�

��3, �4�, ��1, �4��2, �4�

−3 −2 −1 1

−5

−4

−3

−2

−1

x

y

�53

��5 ±�52

, 1��

72

, 1�, ��132

, 1�

x1−2−3−4−5−6−7

4

3

2

1

−2

−3

−4

−1

y��5, 1�

x2−1−2 3 4 5 6 8

−2

−8

1

2

−3

−4

−5

−6

−7

y

12

y

x−1−2 1 2

−1

−2

−3

1

�4, �1�, �4, �5�

23�4, 1�, �4, �7��0, �1��4, �3�

35

��3, 8�, ��3, 2��3, 10�, ��3, 0�

��3, 5�

x642

12

8

6

4

2

y

−4

−4

−2−6−8

x−2−4−6 42

8

−4

−10

4

2

−8

10

6

6 10

y

x431

4

2

1

y

−4

−4

−2

−3 −1

34

23

�±6, 0��0, ±2��±8, 0��0, ±3�

�0, 0��0, 0�

x−1−2−4−5 21

4

−2

−4

4

21

5

−5

5

yy

x−2−6 2 4 6

−2

−4

−6

2

4

6

�0, 0��0, 0�

x−3−6−12 63

9

−6−9

12

63

15

−15

15

y

−6 −2 2 4 6

−6

2

6

x

y

�74

35

�0, ±3�7��±3, 0��0, ±12��±5, 0�

�0, 0��0, 0�


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(Continued)

19. EllipseCenter:Vertices:Foci:

Eccentricity:

20. Ellipse 21. CircleCenter: Center:Vertices: Radius: 6Foci:

Eccentricity:


Eccentricity:


Eccentricity:

24. Ellipse

Center:

Vertices:

Foci:

Eccentricity:

25. Ellipse

Center:

Vertices:

Foci:

Eccentricity:

26. Circle 27. CircleCenter: Center:Radius: 4 Radius:

28. EllipseCenter: Vertices:

Foci:

Eccentricity:

29. Ellipse 30. CircleCenter: Center:

Vertices: Radius:

Foci:

Eccentricity:

31. Center:

Vertices:Foci: �0, ±�2 �

�0, ±�5 ��0, 0�

6

4

−4

−6

x21

3

3

1

2

y

45

y

x−1−2 2 4

−1

−2

−3

1

2

3

�3415, 1�, �26

15, 1�54�7

3, 1�, �53, 1�

�2, �1��2, 1�

35

� 74, �1�, � 1

4, �1�� 9

4, �1�, �� 14, �1�

�1, �1�

−2 −1 1 3

−3

−2

1

2

x

y

y

x−1−2−3 1

−1

2

3

y

x−2−4 2 4 6 8

−2

−4

−8

2

4

23

��1, 1��2, �3�

�32

�3 ± 3�3, �52�

�9, �52�, ��3, �

52�

�3, �52�

x

2

6

4

y

2 4 6 10

−8

−6

−2−4

�63

��32

, 52

± 2�2��

32

, 52

± 2�3��

32

, 52�

x

7

5

4

3

2

1

−2

−3

4321−3−4−5−6

y

�63

��3, 1 ± 2�6 ��3, 7�, ��3, �5�

��3, 1�

x

2

4

8

y

2 4−2−4−8

−6

−10

2�55

�4 ± 4�5, 3��14, 3�, ��6, 3�

�4, 3�

x1410−4−6

1412108

42

−4−6−8

−10

y

x2 4

−12

2

8−2−4−6

−10

y

�53

y

x−2−6−8 2 4 6 8

−2

−4

−6

−10

2

6

�3, �5 ± 2�5 ��3, 1�, �3, �11�

�1, �2��3, �5�

�5

3

��2, 3 ± �5 ��2, 6�, ��2, 0�

��2, 3�

−6 −4 −2 2

−2

2

4

6

x

y

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(Continued)

32. Center:

Vertices:

Foci:

33. Center:

Vertices:

Foci:

34. Center:

Vertices:

Foci:

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57. (a) (b)

(c) Yes

58. Positions: Length of string: 6 feet

59. (a)

(b) (c) Aphelion:35.29 astronomical unitsPerihelion:0.59 astronomical unit

60.

61. (a)

(b) (c) The bottom half

62. Answers will vary.

63. 64.

−1

−2

2

1x

(

(

(

(

, −

,

−1

1

1

1

2

2

2

2

)

)

)

)

− 3

− 3

3,

3,

y

x

− 9

9

94

4

4, 7

7

, 7(

(

()

)

)

y

−2−4 2 4

−2

2

, −94

7( ), −−

y

x−0.8 −0.4 0.4 0.8

−2

2

x2

0.04�

y2

2.56� 1

e � 0.052

−21 21

14

−14

x2

321.84�

y2

20.89� 1

�±�5, 0�;

x2

625�

y2

100� 1

x

y

(−25, 0) (25, 0)

(0, 10)

x2

48�

y2

64� 1

x2

25�

y2

16� 1

�x � 5�2

16�

�y � 6�2

36� 1

�x � 2�2

4�

� y � 2�2

1� 1

�x � 3�2

36�

�y � 2�2

32� 1

x2

16�

� y � 4�2

12� 1

�x � 2�2

1�

4�y � 1�2

9� 1

x2

48�

�y � 4�2

64� 1

�x � 2�2

16�

y 2

12� 1

�x � 2�2

4�

�y � 4�2

1� 1

�x � 2�2

4�

�y � 1�2

1� 1

�x � 2�2

16�

�y � 3�2

9� 1

�x � 4�2

9�

y2

16� 1

�x � 2�2

1�

�y � 3�2

9� 1

x2

4�

y 2

16� 1

21x2

400�

y 2

25� 1

x2

16�

y 2

12� 1

x2

36�

y 2

11� 1

x2

48�

y 2

64� 1

x2

36�

y 2

32� 1

x2

4�

4y 2

9� 1

x2

4�

y 2

16� 1

��23

, 2 ±�93

3 �

��23

, 2 ±2�31

3 ��

23

, 2�

−8 7

−3

7

� 12 ± �2, �1�

� 12 ± �5, �1�

� 12, �1�

5

2

−4

−4

�±1, 0�

�±2, 0�

�0, 0�

−3

−5 4

3


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333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)

65. 66.

67. False. The graph of is not an ellipse. The degree of is 4, not 2.

68. True. If is close to 1, the ellipse is elongated and the fociare close to the vertices.

69. (a) (b)

(c)

(d)

The shape of an ellipse with a maximum area is acircle. The maximum area is found when (ver-ified in part c) and therefore so the equationproduces a circle.

70. (a)

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

71. Geometric 72. Arithmetic

73. Arithmetic 74. Geometric

75. 547 76. 1093

77. 340.15 78. 15.10

2a

b � 10,a � 10

00

20

350

a � 10, circle

x2

196�

y 2

36� 1A � �a�20 � a�

e

yx24 � y4 � 1

−3 −1

−2

2

1 3x

( (

((

, ,− 5 − 5

5, 5 ,

4

4

4

4

3

3

3

3

) )

))

−

−

y

−2−4 42

−4

4

x

(

((

( , 2

, 2−, 2−

, 2 3 5

3 53 5

3 55

55

5 )

))

)

−

−

y

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a 8 9 10 11 12 13

A 301.6 311.0 314.2 311.0 301.6 285.9

333202CB10a_AN.qxd 4/13/06 6:34 PM


Vocabulary Check (page 760)1. hyperbola; foci 2. branches3. transverse axis; center 4. asymptotes5.

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

5. Center: 6. Center:Vertices: Vertices:Foci: Foci:Asymptotes: Asymptotes:

7. Center: 8. Center:Vertices: Vertices:Foci: Foci:Asymptotes: Asymptotes:

9. Center:Vertices:Foci:Asymptotes:


11. Center:Vertices:

Foci:

Asymptotes:

12. Center:

Vertices:

Foci:

Asymptotes:


14. Center:Vertices:Foci:Asymptotes: y � 2 ± 1

3 x�±2�10, 2�

�±6, 2�

−8 −4 4 8

−12

−8

−4

4

8

12

x

y�0, 2�

y � �3 ± 3�x � 2�

�2 ± �10, �3��3, �3�, �1, �3�

−6 −4 −2 2 4 6 8

−8

−6

−4

2

x

y�2, �3�

y � 1 ± 2�x � 3�

��3, 1 ±�54 �

��3, 32�, ��3,

12�

x1

−1

1

−1−3−5−6

2

3

−2

4

y��3, 1�

y � �6 ±23

�x � 2�

�2, �6 ±�13

6 ��2, �

173 �, �2, �

193 � x

2

642

y

−2

−6

−10

−12

−14

�2, �6�

y � 2 ± 512�x � 3�

�10, 2�, ��16, 2��9, 2�, ��15, 2�

x3

−6

6

−3

9

12

18

−9

−12

−6

15

y��3, 2�

1 2 3

−5

−4

1

2

3

x

y

y � �2 ± 12�x � 1�

�1 ± �5, �2��3, �2�, ��1, �2�

�1, �2�

−12 12

−12

−8

−4

4

8

12

x

y

x1086

10

8

6

4

2

y

−2

−4

−6

−6−8

−10

y � ±13 xy � ±5

9 x�±2�10, 0��0, ±�106 �

�±6, 0��0, ±5��0, 0��0, 0�

x21 4 5

−3

2

3

4

5

−4

−5

−1−2−4−5

y

−2 2

−2

−1

1

2

x

y

y � ±53 xy � ±x

�±�34, 0��±�2, 0��±3, 0��±1, 0�

�0, 0��0, 0�

Ax2 � Cy 2 � Dx � Ey � F � 0


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333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)

15. The graph of this 16. The graph of this equation is two lines equation is two linesintersecting at . intersecting at

17. Center:Vertices:Foci:

Asymptotes:

18. Center:Vertices:Foci:

Asymptotes:


20. Center:

Vertices:

Foci:

Asymptotes:

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33.

34.

35.

36.

37.

38.

39. (a) (b) feet

40. 41.

42. (a) miles (b) 57.0 miles

(c) 0.00129 second

(d) The ship is at the position

43.

44. (a) Circle

(b)

(c) Approximately 180.28 meters

45. Circle 46. Ellipse 47. Hyperbola

48. Parabola 49. Hyperbola 50. Circle

51. Parabola 52. Ellipse 53. Ellipse

54. Hyperbola 55. Parabola 56. Circle

57. Ellipse 58. Parabola 59. Circle 60. Hyperbola

61. True. For a hyperbola, The larger the ratio ofto the larger the eccentricity of the hyperbola,

62. False. For the trivial solution of two intersecting lines tooccur, the standard form of the equation of the hyperbolawould be equal to zero,

or


64. The extended diagonals of the central rectangle are asymp-totes of the hyperbola.

�y � k�2

a2 ��x � h�2

b2 � 0.�x � h�2

a2 ��y � k�2

b2 � 0

e � ca.a,bc2 � a2 � b2.

y

x100 200 300

−100

100

�x � 100�2

62,500�

y2

62,500� 1

�12��5 � 1�, 0� � �14.83, 0��144.2, 60�.

x � 110.3

�3300, �2750�x 2

98,010,000�

y 2

13,503,600� 1

� 2.403x 2

1�

y 2

1693� 1

�y � 2�2

4�

�x � 3�2

9� 1

�x � 3�2

9�

�y � 2�2

4� 1

�y � 3�2

9�

�x � 3�2

9� 1

�x � 2�2

1�

�y � 2�2

1� 1

y2

4�

�x � 1�2

4� 1

�y � 2�2

4�

x2

4� 1

x2

4�

7�y � 1�2

12� 1

y 2

9�

4�x � 2�2

9� 1

x2

4�

�y � 1�2

5� 1

�y � 5�2

16�

�x � 4�2

9� 1

y2

9�

�x � 2�2

27� 1

�x � 4�2

4�

y 2

12� 1

x2

64�

y2

36� 1

17y 2

1024�

17x2

64� 1

y2

9�

x2

1� 1

x2

1�

y 2

25� 1

x2

16�

y2

20� 1

y 2

4�

x2

12� 1

y � 5 ± 3�x � 3��3 ±

�103

, 5��

103

, 5�, ��83

, 5�

−2

−12 6

10��3, 5�

y � �3 ± 13�x � 1�

�1, �3 ± 2�5 ��1, �3 ± �2 �

2

−10

−8 10

�1, �3�

y � ±�22

x

�0, ±3��0, ±�3 �

6−6

−4

4�0, 0�

y � ±�63

x

�±�5, 0��±�3, 0�

12−12

8

−8

�0, 0�

−1 1 2 3

−4

−3

−2

−1

x

y

−4 −2 2

−6

−4

−2

2

4

x

y

�1, �2�.��1, �3�

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333202CB10a_AN.qxd 4/13/06 6:34 PM

(Continued)

65.

66.

The equation is a parabola that could intersectthe circle in zero, one, two, three, or four places dependingon its location on the axis.

(a) and (b)

(c) (d)

(e)

67. 68. 69.

70. 71.

72.

73. 74.

75. 76.

1

3

2

4

π2

−π π2

π π2

3−

y

x

1

2

3

4

−

y

xπ π4

π2

π4 4

3π4

3−

−2

−3

−4

1

2

3

4

ππ π2

− π4 4

3−

y

x

−2

−1

−3

−4

1

3

4

y

xππ

2 23π

23−

��x2 � 1��x � 4�2�2x � 3��4x2 � 6x � 9�x�3x � 2��2x � 5�

2x�x � 6�2�x � 7�2x�x � 4��x � 4�

�174 < C < �2

C � �2�2 < C < 2, C � �174

C � 2C < �174C > 2

y-

y � x 2 � C

x

y

−1−3 1 3−1

−3

1

3

y � 1 � 3��x � 3�2

4� 1


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333202CB10a_AN.qxd 4/13/06 6:34 PM



Vocabulary Check (page 769)1. rotation of axes2.3. invariant under rotation 4. discriminant

1. 2. 3.

4. 5.

6.

7. 8.

9.

10.

11.

12. 13.

14. 15.

16. 17.

x

x ′

y′

y

−2

−4 2 4

2

4

6

x1 3 4 62 5

1

x′

y′

y

�x � 1�2 � 6� y �16��y �2� 4�x � 1�

x

x′y′

2

2

y

−4

−4

−2

−6

2−2−3 3

−3

−2

3

x

y

x′

y′

� y�2 � �x�x �

1

2

��y �2

4� 1

y' x'

x32

3

2

y

−3

−3

x

x′

y′ 2

4

−4

4−2−4

y

�x �2

6�

�y �2

32

� 1�x �2

4�

�y �2

4� 1

x

x′

y ′4

6

8

y

−4

−4 4 6 82

�x � 3�2 �2

16�

� y � �2 �2

16� 1

x

x′y′

−4−6−8 4 6

4

6

8

−4

−6

−8

y

y � �3�22��2

10�

x � ��22��2

10� 1

x'

x

y'

21

2

1

y

−2 −1

−1

−2

y � ±�22

x

x′y′

4

2

3

4

32−2

y

4y ′ x ′

x

y

−4 −3 −2

−2

−3

−4

�x �2

4�

�y �2

4� 1

�y �2

2�

�x �2

2� 1

��3 � 2, 2�3 � 1�

�3�22

, ��22 ��3 � �3

2,

1 � 3�32 �

�3 � �32

, 3�3 � 1

2 ��3�2, 0 ��3, 0�

A�x �2 � C� y �2 � Dx � Ey � F � 0

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333202CB10a_AN.qxd 4/13/06 6:34 PM

(Continued)

18. 19.

20. 21.

22. 23.

24. 25.

26.

27. e 28. f 29. b 30. a 31. d 32. c

33. (a) Parabola

(b)

(c)

34. (a) Hyperbola

(b)

(c)

35. (a) Ellipse

(b)

(c)

36. (a) Ellipse

(b)

(c)

37. (a) Hyperbola

(b)

(c)

38. (a) Parabola

(b)

(c)−8

−5

y �60x � 9 ± ��60x � 9�2 � 3600x2

50

6

9−9

−6

y �6x ± �36x2 � 20�x2 � 4x � 22�

�10

4−8

−3

5

y ��4x � 4� ± ��4x � 4�2 � 20�2x2 � 3x � 20�

10

3

−3

−4 5

y �6x ± �36x2 � 28�12x2 � 45�

14

−12 12

−8

8

y �4x ± �16x2 � 8�x2 � 6�

�4

2

−3

−4

y ��8x � 5� ± ��8x � 5�2 � 4�16x2 � 10x�

2

� � 31.72�

−4

8−7

6

� � 33.69�� 28.16�

27

18

−6

−9

−3

5−4

3

� � 31.72�� 33.69�

−4

−6 6

4

−2

3−3

2

� � 26.57�� 37.98�

9

6

−6

−9

−6

9−9

6

� � 45�x

−1

1

2

3

x ′

y ′

−3 −2 −1 1

y

15

10

−10

−15

y �14�x �2


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333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)

39. (a) Parabola

(b)

(c)

40. (a) Ellipse

(b)

(c)

41. 42.

43. 44.

45. 46. 47.

48. 49. 50.

51. 52.

53. 54. 55. No solution

56. 57.

58.

59. True. The graph of the equation can be classified by find-ing the discriminant. For a graph to be a hyperbola, the discriminant must be greater than zero. If then thediscriminant would be less than or equal to zero.

60. False. The coefficients of the new equation after it has been rotated are obtained by making the substitutions

and


62. Major axis: 4; Minor axis: 2

63. 64.

65. 66.

67. 68.

69. 70.

71.72.73.74. Area � 310.39 square units

Area � 48.60 square unitsArea � 187.94 square unitsArea � 45.11 square units

y

t−1−2−3−4 1 2 3 4

−1

−2

−3

−4

1

2

3

4

t

y

−1 1 2 5 6 7−1

−2

−3

−4

1

2

y

t−1−2−3−4−6−7 1

−1

−2

−3

−4

1

2

3

4

y

t−1−2−3 1 2 3 4 5

−1

1

2

3

4

5

6

7

y

x−1−2 1 2 3 4 5 6

−1

−2

1

2

3

4

5

6

x

y

−1−2−3−4 1 2 3 4

−2

−3

−4

1

3

4

y

x−1−2 1 2 3 4 5 6

−1

−2

1

2

3

4

5

6

y

x−1−2−3−4−5−6 1 2

−1

−2

1

2

3

4

5

6

y � x sin � � y cos �.x � x cos � � y sin �

k ≥ 14,

�16�3 � �30�, 16�3 ��30��, �1

6�3 ��30�, 16�3 ��30��0, 32�, ��3, 0��1, 1�, �3, 1�

�±3, 2��1, �3 �, �1, ��3 ��2, 8�, �5, 8 ± 4�21 ��0, 4�

�1, 0��0, 8�, �12, 8��14, �8�, �6, �8�

��8, 12��7, 0�, ��1, 0��2, 2�, �2, 4�

−2 −1 1 2 3 4

−2

1

2

3

4

x

y

− 4 −3 −1−2 1 3 4

− 4

−3

−2

1

3

4

x

y

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

2

1

3

4

x

(1, −3)

y

x642−2−4−6

−6

6

y

−5 4

−3

3

y ��x � 1� ± ��x � 1�2 � 16�x2 � x � 4�

8

7

2

−4

−2

y ��4x � 1� ± ��4x � 1�2 � 16�x2 � 5x � 3�

8

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333202CB10a_AN.qxd 4/13/06 6:34 PM


Vocabulary Check (page 776)1. plane curve; parametric; parameter2. orientation 3. eliminating the parameter

1. (a)

(b)

(c)

The graph of the rectangular equation shows the entireparabola rather than just the right half.

2. (a)

(b)

(c)

The graph of the rectangular equation continues thegraph into the second and third quadrants.

3. (a) 4. (a)

(b) (b)

5. (a) 6. (a)

(b) (b)

7. (a) 8. (a)

(b) (b) y � 1 � x2, x ≥ 0y � x2 � 4x � 4

x

y

−1

1

−2

−3

2−1 3

x

4

2

1

3

y

1−1−2 2 3 4 5 6

−2

y � x3y � 16x2

x

y

−2

1

2

3

4

−3

−4

1−2−3−4 2 3 4

−2 −1

−1

1 2x

y

y � �32x �

132y �

23 x � 3

x1 2 3 4 6−1−2

−2

1

2

3

4

5

6

y

x

6

4

2

1

5

y

1 2 3−1−2

−2

−3

−4

−3−4−7

−4 −3 −2 −1 1 2

−4

−3

1

3

4

x

y

x � �y2 � 4

−4 −3 −2 −1 1 2

−4

−3

−2

1

2

3

4

x

y

x431

4

2

1

y

−4 −3

−3

−4

−2

−1

y � 3 � x2

x431

4

3

2

1

y

−2

−1−1−2


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d.

t 0 1 2 3 4

x 0 1 2

y 3 2 1 0 �1

�3�2

0

x 0 2 4 2 0

y 0 2�2��2�2

�

2�

4�

�

4�

�

2�

333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)9. (a) 10. (a)

(b) (b)

11. (a) 12. (a)

(b) (b)

and

13. (a) 14. (a)

(b) (b)

15. (a) 16. (a)

(b) (b)

17. (a)

(b)

18. (a)

(b)

19. (a) 20. (a)

(b) (b)

21. (a) 22. (a)

(b) (b) y �e2x

2y � ln x

1

2

4

x

3

5

7

1 2 3 4

6

−1−2−3−4

y

x

y

1

−1−1 1 2 3 4 5 6−2

−2

−3

−4

2

3

4

y � �xy �1x3, x > 0

x

y

−1

1

2

3

1 2 3 4x

y

4

3

2

1

−1

−1 1 2 3 4

�x � 4�2

4�

�y � 2�2

9� 1

1

2

4

x

3

5

6

1 2 4 6 7 8

y

�x � 4�2

4� � y � 1�2 � 1

x

y

−1 1 3 4 5 7

1

2

3

−2

−3

−4

−5

y � ±4x�1 � x2x2

16�

y2

4� 1

x

y

−2

−2

2

2x

y

4

3

1

−3

−3 −2 −1 1 2 3

−4

x2

4�

y2

9� 1

y2

9�

x2

9� 1

x−1−3−4 31 4

−4

−2

1

2

4

y

x

y

−4

−4

−2

1

2

4

−2 −1 1 2 4

y � x � 3, x ≥ 0

y � �x � 3, x > 0y � �x2 � 3�

x

y

8

6

4

2

−4

2−2 6 8 10

x10

10

2

6

8

4 6 122 8

12

14

14

y

−2

y �x � 1

xy �

�x � 1�x

x

y

1−2−3

1

2

3

2 3x

2

1

y

1−1−1

−2

−2−3 2 3

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333202CB10a_AN.qxd 4/13/06 6:34 PM

(Continued)

23. 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

24. Each curve represents a portion of the parabola

Domain Orientation

(a) Left to right

(b) Depends on

(c) Depends on

(d) Left to right

25. 26.

27.

28.

29. 30.

31. 32.

33. 34.

35. 36.

37. (a) (b)

38. (a) (b)

39. (a) (b)

40. (a) (b)

41. (a) (b)

42. (a) (b)

43. (a) (b)

44. (a) (b)

45. 46.

47. 48.

49. 50.

51. 52.

53. b 54. c

Domain: Domain:

Range: Range:

55. d 56. a

Domain: Domain:

Range: Range:

57. (a) Maximum height: 90.7 feet

Range: 209.6 feet

(b) Maximum height: 204.2 feet

Range: 471.6 feet

(c) Maximum height: 60.5 feet

Range: 242.0 feet

(d) Maximum height: 136.1 feet

Range: 544.5 feet

00

600

200

00

300

100

0 5000

220

0 2500

100

�2, 2��, ��, ��, ��

�6, 6��1, 1��4, 4��2, 2�

−6 6

−4

4

6

4

−4

−6

−6

14

0 100

6

4

−4

−6

−6

0 30

14

0

−6

18

6

00

12

8

0 510

34

x � �t � 2, y �1

�2t � 4x � t, y �

12t

x � �t � 2, y � �1

t � 2x � t, y �

1t

x � �t � 2, y � tx � t, y � 2 � t

x � �t � 2, y � t 2 � 4t � 5x � t, y � t 2 � 1

x � �t � 2, y � ��t � 2�3 x � t, y � t3

x � �t � 2, y � t 2 � 4t � 4x � t, y � t 2

x � �t � 2, y � �13�t � 4�x � t, y �

13�t � 2�

x � �t � 2, y � �3t � 4x � t, y � 3t � 2

y � 2�3 tan �y � 3 tan �

x � 2 sec �x � 4 sec �

y � 2 � 4 sin �y � �7 sin �

x � 4 � 5 cos �x � 4 cos �

y � 2 � 5 sin �y � 2 � 4 sin �

x � �3 � 5 cos �x � 3 � 4 cos �

y � 3 � 6ty � �3t

x � 2 � 4tx � 6t

�x � h�2

a2�

�y � k�2

b 2� 1

�x � h�2

a2�

�y � k�2

b 2� 1

�x � h�2 � �y � k�2 � r 2y � y1 � m�x � x1��0, ��

t�1, 1�t0, ��

��, ��

y � x2 � 1.

�0, ��0, ��

��1, 1��, ��

y � 2x � 1.


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333202CB10a_AN.qxd 4/13/06 6:34 PM


(Continued)

58. (a) Maximum height: 3.8 feet

Range: 56.3 feet

(b) Maximum height: 10.5 feet

Range: 156.3 feet


Range: 97.4 feet

(d) Maximum height: 39.1 feet

Range: 270.6 feet

59. (a)

(b) No

(c) Yes

(d)

60. (a)

(b) 643 feet(c) 32.1 feet

(d) 2.72 seconds


62. (a)

(b)


Range: 56.2 feet

63. 64.

65. True

66. False.

67. Parametric equations are useful when graphing two func-tions simultaneously on the same coordinate system. Forexample, they are useful when tracking the path of anobject so that the position and the time associated with thatposition can be determined.

68. Sketching a plane curve starts by choosing a numeric valuefor the parameter. Then, the coordinates can be determinedfrom the value chosen for the parameter. Finally, plottingthe resulting points in the order of increasing parametervalues shows the direction, or orientation, of the curve.

y � x for x ≥ 0

y � 9t 2 � 1 ⇒ y � x2 � 1x � 3ty � t 2 � 1 ⇒ y � x2 � 1x � t

y � 3 sin � � sin 3�y � a � b cos �

x � 3 cos � � cos 3�x � a� � b sin �

600

25

0

y � 7 � �40 sin 45��t � 16t 2

x � �40 cos 45��t

h � 7, v0 � 40, � � 45�

0 7000

40

y � 5 � �240 sin 10��t � 16t 2

x � �240 cos 10��t

19.3�

00

500

60

00

450

50

y � 3 � �146.67 sin ��t � 16t 2

x � �146.67 cos ��t

00

300

40

00

100

15

00

160

15

00 60

6

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333202CB10a_AN.qxd 4/13/06 6:34 PM

(Continued)

69. 70.

71. 72.

73.

74.

75.

76.

y

x

5π6

θ ′

� ��

6

y

x

2π3

−θ ′

� ��

3

y

x

105°θ ′

� � 50�

y

x

230°

θ ′

� � 75�

�3, 1, �2��1, �2, 1��2, 3��5, 2�


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333202CB10a_AN.qxd 4/13/06 6:34 PM

Precalculus with Limits, Answers to Section 10.7 21C

opyr

ight

©H

ough

ton

Mif

flin

Com

pany

. All

righ

ts r

eser

ved.


Vocabulary Check (page 783)1. pole 2. directed distance; directed angle

3. polar 4.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10. 11.

12. 13. 14.

15. 16.

17. 18. 19.

20. 21. 22.

23. 24. 25.

26. 27.

28. 29. 30.

31. 32. 33.

34. 35. 36.

37. 38.

39. 40.

41.

42.

43. or 44.

45. 46. 47.

48. 49.

50. 51.

52. 53.

54. 55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. The graph of the polar equation consists of all points that are six units from the pole.x2 � y2 � 36

x

2

4

2 4 8

8

y

−2−4

−4

−8

−8

2x � 3y � 64x2 � 5y 2 � 36y � 36 � 0

y2 � 2x � 1x2 � 4y � 4 � 0

�x2 � y2�3 � 9�x2 � y2�2�x2 � y 2�2 � 6x2y � 2y 3

�x2 � y2�2 � 2xyx2 � y2 � x2�3 � 0

x � �3y � 4x2 � y2 � 100

x2 � y2 � 16�3x � y � 0

�3x � y � 0x2 � y2 � 2x � 0

x2 � y 2 � 4y � 0r � 2a sin �

r � 2a cos �r � 3ar � a

r2 � 9 cos 2��4

1 � cos �r �

41 � cos �

r2 �12 sec � csc � � csc 2�

r2 � 16 sec � csc � � 32 csc 2�

r �2

3 cos � � 5 sin �r �

�2

3 cos � � sin �

r � 4a sec �r � 10 sec �

� ��

4r � 4 csc �r � 4

r � 3�2.3049, 0.7086��176 , 0.4900�

�6, �

4��7, 0.8571��29, 2.7611�

��13, 5.6952��13, 1.1760�

�3�13, 0.9828��2, 11�

6 ��6, 5�

4 �

��10, 5.9614��5, 2.2143��5, 3�

2 �

�6, ��3�2, 5�

4 ��2, �

4��7.7258, �2.8940��1.1340, �2.2280�

��3, 1��2, �2 ��0, 0�

��22

, �22 ��0, �3��0, 3�

��5, 3.9232�, �5, 0.7816��2�2, 10.99�, ��2�2, 7.85�

02 4 6 8

π

3π2

2π

01 2 3 4

3π2

π

2π

��3, 4.7132�, �3, 1.5716��2, 8.64�, ��2, �0.78�

01 2 3 4

π

3π2

2π

01 2 3 4

3π2

π

2π

�16, �

2�, ��16, 3�

2 ��0, 5�

6 �, �0, �13�

6 �

04 8 12 16

3π2

π

2π

01 2 3 4

3π2

π

2π

��1, 5�

4 �, �1, �

4��4, 5�

3 �, ��4, �4�

3 �

0

3π

π

2

1 2 3 4

2π

0π

2

3π2

π

1 2 3 4

y � r sin � r2 � x2 � y2

x � r cos � tan � �yx

333202CB10b_AN.qxd 4/13/06 5:21 PM

(Continued)

66. The graph of the polar equation consists of all points that are eight units from the pole.

67. The graph of the polar equation consists of all points on the line that make an angle of with the positive polar axis.

68. The graph of the polar equation consists of all points on the line that make an angle of with the positive polar axis.

69. The graph of the polar equation is not evident by simple inspection, so convertto rectangular form.

70. The graph of the polar equation is not evident by simple inspection, so convertto rectangular form.

71. True. Because is a directed distance, the point canbe represented as

72. False. If then and are differentpoints.

73.

Radius:

Center:

74. circle

75. (a) Answers will vary.(b) and the pole are collinear.

This represents the distance between two points on theline

(c)This is the result of the Pythagorean Theorem.

(d) Answers will vary. For example:Points: Distance: 2.053Points: Distance: 2.053

76. (a) Horizontal: coordinate changesVertical: coordinate changes

(b) Horizontal: and both changeVertical: and both change

(c) Unlike and , and measure horizontal and verticalchanges, respectively.

77.

78. 79.

80. 81.

82. 83. 84.

85. 86. 87.

88. 89.

90. Cramer’s Rule does not apply. 91. Not collinear

92. Collinear 93. Collinear 94. Not collinear

�2, �3, 3��29589 , 844

89 , �67289 �

� 87, 88

35, 85 ��4514, �55

14��2, 3�

ln 6y

x � 3ln �x�x � 2�log5 a�x � 1�8

log7 x3y

ln 5 � 2 ln x � ln�x2 � 1�

ln x � 2 ln�x � 4�14 �

12 log4 x � log4 y

2 log6 x � log6 z � log6 3 � log6 y

yx�r�r

�ry-

x-

��3, 7��6�, ��4, 4��3�

�3, ��6�, �4, ��3�

d � �r12 � r2

2

� � �1 � �2.

d � �r12 � r2

2 � 2r1r2 � �r1 � r2��r1, �1�, �r2, �2�

�x �12�2

� � y �32�2

�52;

�h, k�

�h2 � k2

�x � h�2 � �y � k�2 � h2 � k2

�r2, ��r1, ��r1 � �r2,

�r, � ± 2�n�.�r, ��r

y � 2 � 01−1−2−3−4 2 3 4

−4

−3

−2

1

3

4

x

y

x � 3 � 0−4 −3 −2 −1 1 2 4

−4

−3

−2

1

2

3

4

x

y

x � y � 0

3��4

x−4 −3 −2 −1

−2

1

2

4

3

2 3 41

−3

−4

y

��3x � 3y � 0

��6

x�

−4

3

1

−4 1 2 4−2−1

3−3

2

4

−2

−3

y�

x2 � y2 � 64x

−10

2

4

42−2−4−6

−4

10

6 10

6

−6

y


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Vocabulary Check (page 791)

1. 2. polar axis 3. convex limaçon

4. circle 5. lemniscate 6. cardioid

1. Rose curve with 4 petals 2. Cardioid

3. Limaçon with inner loop 4. Lemniscate

5. Rose curve with 4 petals 6. Circle

7. Polar axis 8. Polar axis

9. 10. Polar axis

11. polar axis, pole 12. Pole

13. Maximum: when

Zero: when

14. Maximum: when

Zeros: when

15. Maximum: when

Zero: when

16. Maximum: when

Zeros: when

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

01 2

π

2π

3π2

02 31

3π2

π

2π

02 4 6

π

2π

3π2

02 31

3π2

π

2π

01 52 3

π

2π

3π2

04 6

3π2

π

2π

02 4 6

π

2π

3π2

02

3π2

π

2π

01 52 3

π

2π

3π2

01 2 3

3π2

π

2π

01 43

π

2π

3π2

01 2

3π2

π

2π

01 3

π

2π

3π2

02 4 6

3π2

π

2π

� � 0, �

2, �,

3�

2r � 0

� ��

4,

3�

4,

5�

4,

7�

4�r� � 3

� ��

6,

�

2,

5�

6r � 0

� � 0, �

3,

2�

3�r� � 4

� �2�

3,

4�

3r � 0

� � 0�r� � 18

� ��

2r � 0

� �3�

2�r� � 20

� ��

2,

� ��

2

� ��

2

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333202CB10b_AN.qxd 4/13/06 5:21 PM

(Continued)

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

−6 6

−4

4

6

−4

−6

4

0 ≤ � < �0 ≤ � < �

−1 0

1.5−1.5

1.0

5

3

−3

−4

0 ≤ � < 4�0 ≤ � < 4�

−6 6

−4

4

3

−2

−3

2

0 ≤ � < 2�0 ≤ � < 2�

−7

7

14−75

−5

−10

5

−9 9

−4

8

5

3

−3

−4

−3 3

−2

2

10−11

−10

4

−3 3

−2

2

14

6

−6

−4

032

π

2π

3π2

04

3π2

π

2π

0321

π

2π

3π2

0321

3π2

π

2π

021 43

π

2π

3π2

01 3

3π2

π

2π

04

π

2π

3π2

04

3π2

π

2π

02 4 6 8

π

2π

3π2

02 4

3π2

π

2π


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333202CB10b_AN.qxd 4/13/06 5:21 PM


(Continued)

55. 56.

57. True. For a graph to have polar axis symmetry, replaceby or

58. False. For a graph symmetric with respect to the pole, oneportion of the graph coincides with the other portion whenrotated radians about the pole.

59. (a) (b)

Upper half of circle Lower half of circle(c) (d)

Full circle Left half of circle

60. (a) (b)

(c)

The angle controls rotation of the axis of symmetry.

61–62. Answers will vary.

63. (a) (b)

(c) (d)

64. (a)

(b)

(c)

(d)

65. (a) (b)

66. (a) (b)

(c) (d)

67. circleconvex limaçoncardioidlimaçon with inner loopk � 3,

k � 2,k � 1,k � 0,

8

7

k = 3 k = 2k = 1

k = 0−3

−7

02 4

π

2π

3π2

02

π

2π

3π2

02 4

π

2π

3π2

02 4

π

2π

3π2

01 2

3π2

π

2π

01 2

3π2

π

2π

r � 4 sin � cos �

r � 4 sin�� 2�

3 � cos�� 2�

3 �r � �4 sin � cos �

r � 4 sin��

6� cos��

6�

r � 2 � cos �r � 2 � sin �

r � 2 � cos �r � 2 ��22

�sin � � cos ��

r � 6�1 � sin ��

−12

−4

15

14

−9

−6

18

12

−8

−8 16

8

01 2 3 4 5 6 7

3π2

π

2π

041 2 3 5 7

3π2

π

2π

01 2 3 4 5 7

3π2

π

2π

01 2 3 4 5 6 7

3π2

π

2π

�

��r, � � ��.�r, ��r, ��

−6 6

−4

4

5

−2

−3

4

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333202CB10b_AN.qxd 4/13/06 5:21 PM

(Continued)

68. (a) (b)

(c) Yes. Explanations will vary.

69. 70. No zeros

71. 72. 3

73.

74.

1 2 3 4 5 6 7−1

−2

−3

−4

−5

1

2

3

y

x

−1−2−3−4−5 1 2 3−1

−2

−3

1

2

3

5

y

x

�x � 3�2

7�

� y � 1�2

16� 1

�x � 1�2

9�

� y � 2�2

4� 1

135

±3

0 ≤ � < 4�0 ≤ � < 4�

−6 6

−4

4

−6 6

−4

4


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333202CB10b_AN.qxd 4/13/06 5:21 PM



Vocabulary Check (page 797)1. conic 2. eccentricity; 3. vertical; right4. (a) iii (b) i (c) ii

1. parabola

ellipse

hyperbola

2. parabola

ellipse

hyperbola

3. parabola

ellipse

hyperbola

4. parabola

ellipse

hyperbola

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

11. Parabola 12. Parabola

13. Parabola 14. Parabola

15. Ellipse 16. Ellipse

17. Ellipse 18. Ellipse

02 104 86

π

2π

3π2

04 62

3π2

π

2π

02

π

2π

3π2

01 3

3π2

π

2π

04 8

π

2π

3π2

02 4

3π2

π

2π

01 2 3

π

2π

3π2

01 2 3 4

3π2

π

2π

−15 15

−6

e = 1.5

e = 1

e = 0.5

14

e � 1.5: r �6

1 � 1.5 sin �,

e � 0.5: r �2

1 � 0.5 sin �,

e � 1: r �4

1 � sin �,

17

6

e = 1

e = 1.5

−16

−16

e = 0.5

e � 1.5: r �6

1 � 1.5 sin �,

e � 0.5: r �2

1 � 0.5 sin �,

e � 1: r �4

1 � sin �,

−14 7

−7

e = 1.5

e = 1

e = 0.5

7

e � 1.5: r �6

1 � 1.5 cos �,

e � 0.5: r �2

1 � 0.5 cos �,

e � 1: r �4

1 � cos �,

5

3

−3

−4

e � 1.5: r �6

1 � 1.5 cos �,

e � 0.5: r �2

1 � 0.5 cos �,

e � 1: r �4

1 � cos �,

e

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333202CB10b_AN.qxd 4/13/06 5:21 PM

(Continued)

19. Hyperbola 20. Hyperbola

21. Hyperbola 22. Hyperbola

23. Ellipse 24. Hyperbola

25. 26.

Parabola Hyperbola27. 28.

Ellipse Hyperbola29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.


51.

Perihelion: miles

Aphelion: miles

52.

Perihelion: kilometers

Aphelion: kilometers

53.

Perihelion: kilometers

Aphelion: kilometers

54.

Perihelion: miles

Aphelion: miles

55.

Perihelion: miles

Aphelion: miles

56.

Perihelion: kilometersAphelion: kilometers8.1609 108

7.4073 108

r �7.7659 108

1 � 0.0484 cos �

1.5486 108

1.2840 108

r �1.4039 108

1 � 0.0934 cos �

4.3377 107

2.8583 107

r �3.4459 107

1 � 0.2056 cos �

1.0894 108

1.0747 108

r �1.0820 108

1 � 0.0068 cos �

1.5043 109

1.3497 109

r �1.4228 109

1 � 0.0542 cos �

9.7558 107

9.4354 107

r �9.5929 107

1 � 0.0167 cos �

r �8

3 � 5 sin�r �

9

4 � 5 sin �

r �16

3 � 5 cos �r �

20

3 � 2 cos �

r �8

3 � sin�r �

10

3 � 2 cos �

r �20

1 � sin�r �

10

1 � cos �

r �12

1 � cos�r �

2

1 � sin �

r �3

2 � 3 cos�r �

2

1 � 2 cos �

r �9

4 � 3 sin�r �

1

2 � sin �

r �2

1 � sin�r �

1

1 � cos �

−9 9

−9

3

6

3

−7

−9

−2

3−3

2

15

−3

−3

9

−5

−2

2

2

2

−2

−4

−4

0−3 3

3−3

−3

01

π

2π

3π2

03 521

3π2

π

2π

01

π

2π

3π2

01

3π2

π

2π

01 2 3 4 6

π

2π

3π2

01

3π2

π

2π


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(Continued)

57. astronomical unit

58. (a)

(b)

(c) 1467 miles (d) 394 miles

59. True. The graphs represent the same hyperbola.

60. False. The graph has a horizontal directrix below the pole.

61. True. The conic is an ellipse because the eccentricity is lessthan 1.


65. 66.

67. 68.

69. 70.

71. (a) Ellipse

(b) The given polar equation, has a vertical directrix to the left of the pole. The equation, has a verticaldirectrix to the right of the pole, and the equation, has a horizontal directrix below the pole.

(c)

72. If remains fixed and changes, then the lengths of both themajor axis and the minor axis change. For example, graph

with and and graph

with and , on the same set of

coordinate axes.

73. 74.

75. 76.

77. 78.

79. 80.

81. 82.

83. 84.

85. 86. 87.

88. 89. 220 90. 153

91. 720 92. 812

an � 1.5n � 3.5

an � 9nan � 10 � 3nan � �14 n �

14

tan 2u � �3tan 2u �247

cos 2u � �12

cos 2u � �725

sin 2u � ��32

sin 2u � �2425

�210

7�210

�7�210

�210

�

3� 2n�,

5�

3� 2n�

�

2� n�

�

3� n�,

2�

3� n�

�

3� n�,

2�

3� n�

�

3� 2n�,

5�

3� 2n�

�

6� n�

p � 9e �23r �

6

1 �23 sin �

,

p �152 ,e �

23r �

5

1 �23 sin �

,

pe

12−12

10

−6

1 − 0.4 cos 4

θr =r1 =

1 − 0.4 sin 4

θr2 =

1 + 0.4 cos 4

θ

r2,r1,

r,

r 2 �225

25 � 16 cos2 �r2 �

144

25 sin2 � � 16

r2 �36

10 cos2 � � 9r2 �

144

25 cos2 � � 9

r2 �400

25 � 9 cos2 �r2 �

24,336

169 � 25 cos2 �

−1,000

−10,000 10,000

5,000

r �8200

1 � sin �

r � 0.338r �0.624

1 � 0.847 sin ��2;

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333202CB10b_AN.qxd 4/13/06 5:21 PM

Review Exercises (page 801)

1. radian, 2. 2.6012 radians,

3. 1.1071 radians, 4. 0.7086 radian,

5. 0.4424 radian, 6. 0.4424 radian,

7. 0.6588 radian, 8. 1.4309 radians,

9. 10. 11. Hyperbola 12. Parabola

13. 14.

15. 16.

17. 18.

19. meters 20.

21. 22.

23. 24.

25. The foci occur 3 feet from the center of the arch on a lineconnecting the tops of the pillars.

26. Longest distance: 36 feet

Shortest distance: 28 feet

Distance between foci: feet

27. Center:

Vertices:

Foci:

Eccentricity:

28. Center: 29. Center:

Vertices: Vertices:

Foci: Foci:

Eccentricity: Eccentricity:

30. Center:

Vertices:

Foci:

Eccentricity:

31. 32.

33. 34.

35. Center:

Vertices:

Foci:

Asymptotes:y � �5 ± 1

2�x � 3�

�3 ± 2�5, �5��7, �5�, ��1, �5�

x4 6 8

2

2

y

−2

−8

−10

�3, �5�

5y2

16�

5�x � 3�2

4� 1

5�x � 4�2

16�

5y2

64� 1

x2

4�

� y � 2�2

12� 1y2 �

x2

8� 1

�215

��2 ± �21, 3��3, 3�, ��7, 3�

��2, 3�

�74

�356

�1, �4 ± �7 ��5, �3 ± �35 ��1, 0�, �1, �8��5, 3�, �5, �9�

�1, �4��5, �3�

�1910

��2, 1 ± �19 ��2, 11�, ��2, �9�

��2, 1�

16�2

y

x−2−6−8 2 4

−2

2

4

6

8

10

12

x

y

−1−2 1 2 3 4 5−1

−2

−3

1

2

3

4

�x � 4�2

4�

� y � 5�2

36� 1

�x � 2�2

4� � y � 1�2 � 1

y

x−1−2 1 2 3 4

−1

−2

1

2

3

4

x

y

−4−6−8 2 4 6 8 10

−6

−8

−10

2

6

8

10

�x � 2�2

3�

�y � 2�2

4� 1

�x � 2�2

25�

y2

21� 1

y2 � 6x8�6

y � 4x � 8; ��2, 0�y � �2x � 2; �1, 0�

y

x1−1−2 2 3 4 5 6

−1

−2

1

2

4

5

6

x

y

−1−2−3−4 1 2 43 5

−2

−3

1

2

3

4

5

6

7

�x � 2�2 � 8�y � 2��y � 2�2 � 12x

y

x−1−2−3−4 1 3 4

−1

−2

−3

−4

1

2

3

4

x

y

−1−2−3−4 1 2 3 4 5

−2

−3

−4

−5

1

2

3

4

5

y2 � �8�x � 2�y 2 � 16x

6�55

2�2

81.9837.75

25.3525.35

40.6063.43

149.0445�

4

Precalculus with Limits, Answers to Review Exercises 30

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(Continued)

36. Center:

Vertices:

Foci:

Asymptotes:

37. Center:

Vertices:

Foci:

Asymptotes:

38. Center:

Vertices:

Foci:

Asymptotes:

39. 72 miles 40.

41. Hyperbola 42. Parabola

43. Ellipse 44. Circle

45. 46.

47. 48.

49. (a) Parabola

(b)

(c)

50. (a) Ellipse

(b)

(c)

51. (a) Parabola

(b)

(c)

52. (a) Hyperbola

(b)

(c)

−2

3−3

2

y �10x ± �100x2 � 4�x2 � 1�

2

7

−1

−11

y ��2x � 2�2� ± ��2x � 2�2�2

� 4�x2 � 2�2x � 2�2

−4

−6 6

4

y �8x ± �64x2 � 28�13x2 � 45�

14

9

7

−1

−3

y �24x � 40 ± ��24x � 40�2 � 36�16x2 � 30x�

18

x ′y′

−3−4 2 3 4

−3

−4

2

3

4

y

xx

1

1 2

2

y

−2

−2

−1

−1

x′y′

y� � �4�x� �2 � 8x��x� �2

3�

�y� �2

2� 1

x

x ′y′

1

1

21

21

y

x

x′y′

2

3

4

3 42

y

−2

−2

−3

−3−4

�x� �2

1�4�

�y� �2

1�6� 1

�x� �2

8�

�y� �2

8� 1

64�x � 1�2

25�

64y2

39� 1

576x2

25�

576y2

227� 1

y � �3 ± 25�x � 1�

��1, �3 ± �29 ��1, �1�, ��1, �5� x

−2 4

1

−2

−4

−6

−7

y��1, �3�

y � �1 ± 34�x � 1�

�6, �1�, ��4, �1�

�5, �1�, ��3, �1�

x

6

4 6 8

2

4

y

−4

−4−6

−6

−8

�1, �1�

y � 1 ± 2x

�0, 1 ± �5 ��0, 3�, �0, �1�

x2−2−3−4 3

4

5

−3

4

y�0, 1�

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333202CB10b_AN.qxd 4/13/06 5:21 PM

(Continued)

53.

54.

55. (a) 56. (a)

(b) (b)

57. (a) 58. (a)

(b) (b)

59. (a)

(b)

60. (a)

(b)

61. 62.

63. 64. Answers will vary.

65. 66.

67. 68.

69. 70.

71. 72. 73. �2, �

2��0, 0��3�2

2,

3�22 �

��2, ��2��12

, ��32 �

��3, 8.90�, ��3, 5.76��7, 1.05�, ��7, 10.47�

01 2 3 4

π

2π

3π2

0π2 4 6 8

2π

3π2

��5, 5�

3 �, �5, 2�

3 ��2, 9�

4 �, ��2, 5�

4 �

02 4 6 8

π

2π

3π2

0π

π

1 2 3 4

23

2π

y � 4 sec �

x � 3 tan �

y � 4 � 3 sin �y � 4 � 6 sin �

x � �3 � 4 cos �x � 5 � 6 cos �

�x � 3�2

9�

�y � 2�2

25� 1

2−2 4 6 8 10

−4

−2

2

4

6

8

x

y

x2 � y 2 � 36

−8 − 4 −2 2 4 8

4

2

−4

−8

8

x

y

y � �x � 4�2y � 4�x

2−2 4 6 8

−2

2

4

6

8

x

y

x

1

2

4

3

y

1 2 3 4

y � �34 x �

114y � 2x

x1−1−2−3 2 3 5

−3

1

4

−2

2

5

y

−4 −3 −2 −1

−4

−3

1

2

3

4

x

y

1 2 3 4

y

x1 2 3 4 5 6

−3

−4

1

2

3

4

y

x−4−8−12 8 12

−4

−8

4

12

16

20


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0 1 2 3

1 4 7

19 15 11 7 3 �5�1y

�2�5�8�11x

�1�2�3t

0 2 3 4 5

0 1

4 2 143�4�2y

45

35

25�

15x

�1t

333202CB10b_AN.qxd 4/13/06 5:21 PM


(Continued)

74. 75. 76.

77. 78. 79.

80. 81.

82. 83.

84. 85.

86. 87.

88.

89. Symmetry: polar axis, pole

Maximum value of for all values of

No zeros of


Maximum value of for all values of

No zeros of


Maximum value of when

Zeros of when

92. Symmetry: polar axis

Maximum value of

when

Zeros of when



Zeros of when



Zeros of when

95. Symmetry:

Maximum value of when � ��

2�r� � 8�r�:

� ��

2

02 4

π

3π2

2π

2� � arccos 34� � arccos 34,r � 0r:

� � ��r� � 7�r�:

02

π

2π

3π2

� � �r � 0r:

� � 0�r� � 4�r�:

02

π

3π2

2π

9�

107�

10,

5�

10,

3�

10,� �

�

10,r � 0r:

8�

56�

5,

4�

5,

2�

5,� � 0,�r� � 1

�r�:

0π4

2π

3π2

3�

2�,

�

2,� � 0,r � 0r:

7�

45�

4,

3�

4,� �

�

4,�r� � 4�r�:

� ��

2,

40

3π2

π

2π

r

��r� � 11�r�:� �

�

2,

0π2

2π

3π2

r

��r� � 4�r�:� �

�

2,

�x2 � y2�2 � x2 � y2

x2 � y2 � y2�3x2 � y2 � 8y

x2 � y2 � 3xx2 � y2 � 144

x2 � y 2 � 25r2 � �4 csc 2�

r2 � 10 csc 2�r � 4 cos �

r � 6 sin �r � 2�5r � 7

�5, 5.3559��2�13, 0.9828��10, 3�

4 �

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(Continued)

Zeros of when



Zero of when



Zeros of when



Zeros of when

99. Limaçon 100. Limaçon

101. Rose curve 102. Lemniscate

103. Hyperbola 104. Parabola

105. Ellipse 106. Hyperbola

107. 108.

109. 110.

111.

112. 89,600,000 miles

113. False. When classifying an equation of the formits graph can be

determined by its discriminant. For a graph to be aparabola, its discriminant, must equal zero.So, if then or equals 0.

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

CAB � 0,B2 � 4AC,

Ax2 � Bxy � Cy2 � Dx � Ey � F � 0,

r �12,000,0001 � sin �

;

r �7978.81

1 � 0.937 cos �; 11,011.87 miles

r �7

3 � 4 cos �r �

53 � 2 cos �

r �4

1 � sin �r �

41 � cos �

063 129 15 21

π

3π2

2π

01 3 4

π

2π

3π2

0π

3π2

2π

01

π

2π

3π2

−6 6

4

−4

−6 6

−4

4

−12 6

6

−6

8−16

−8

8

20π

3π2

2π

7�

45�

4,

3�

4,� �

�

4,r � 0r:

3�

2�,

�

2,� � 0,�r� � 1�r�:

� ��

2,

04

π

2π

3π2

� ��

4,

3�

4,

5�

4,

7�

4r � 0r:

� � 0, �

2, �,

3�

2�r� � 3�r�:

� ��

2,

02 4

π

3π2

2π

� � 0r � 0r:

� � ��r� � 10�r�:

042 6

π

2π

3π2

� � 3.4814, 5.9433r � 0r:


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(Continued)

115. False. The following are two sets of parametric equationsfor the line.

116. False. and all representthe same point.

117. 5. The ellipse becomes more circular and approaches a circle of radius 5.

118. The orientation would be reversed.

119. (a) The speed would double.

(b) The elliptical orbit would be flatter; the length of themajor axis would be greater.

120. (a) Symmetric to the pole

(b) Symmetric to the polar axis

(c) Symmetric to

121. (a) The graphs are the same.

(b) The graphs are the same.

122. 40

� � ��2�2, 9��4��2, ��4�, ��2, 5��4�,

x � 3t, y � 3 � 6t

x � t, y � 3 � 2t

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333202CB10b_AN.qxd 4/13/06 5:21 PM

Chapter Test (page 805)

1. 0.2783 radian, 2. 0.8330 radian,

3.

4. Parabola:

Vertex:

Focus:

5. Hyperbola:

Center:

Vertices:

Foci:

Asymptotes:

6. Ellipse:

Center:

Vertices:

Foci:

7. Circle:

Center:

8. 9.

10. (a)

(b)

x

x′y′

4

4

6

6

y

−4

−4

−6

−6

45

5�y � 2�2

4�

5x2

16� 1�x � 3�2 �

32

�y � 2�

x

3

2

2

1

1 3

y

−1

−1

�2, 1�

�x � 2�2 � �y � 1�2 �12

��3 ± �7, 1��1, 1�, ��7, 1�

x

6

2

2

4

y

−2

−2−4−8

−4

��3, 1�

�x � 3�2

16�

�y � 1�2

9� 1

−4 2 6 8

−4

−6

2

4

6

x(2, 0)

y

y � ±12�x � 2�

�2 ± �5, 0��0, 0�, �4, 0�

�2, 0�

�x � 2�2

4� y2 � 1

−2 −1 2 3 4 5 6

−4

−3

−2

1

2

3

4

x

y

�2, 0�

�1, 0�

y2 � 4�x � 1�

7�22

47.715.9

Precalculus with Limits, Answers to Chapter Test 36

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Precalculus with Limits, Answers to Chapter Test 37

(Continued)

11.

12.

13.

14.

15.

16.

Parabola

17.

Ellipse18.

Limaçon with inner loop19.

Rose curve

20. Answers will vary. For example:

21. Slope: 0.1511; Change in elevation: 789 feet

22. No; Yes

r �1

1 � 0.25 sin �

043

π

2π

3π2

02 4

π

2π

3π2

02 3

π

2π

3π2

01 3 4

π

2π

3π2

r � 4 sin �

�2�2, 7�

4 �, ��2�2, 3�

4 �, �2�2, ��

4�

��3, �1�

y � 4 � 7tx � 6 � 4t

�x � 2�2

9�

y 2

4� 1

−2

−4

−2

2

4

x

y

2 4 6

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Problem Solving (page 809)

1. (a) 1.2016 radians (b) 2420 feet, 5971 feet

2. (a) (b) feet

(c) square feet square feet

3. 4.

5. (a) Since by definition, the outer bound thatthe boat can travel is an ellipse. The islands are the foci.

(b) Island 1:Island 2:

(c) 20 miles; Vertex:

(d)

6.


8. (a) The first set of parametric equations models projectilemotion along a straight line. The second set ofparametric equations models projectile motion of anobject launched at a height of units above the groundthat will eventually fall back to the ground.

(b)

(c) In the first case, the path of the moving object is notaffected by changing the velocity because eliminatingthe parameter removes

9. Answers will vary. For example:

10. (a) (b)

The graph is a line The graph is a three-sidedbetween and 2 on figure with counterclockwisethe -axis. orientation.

(c) (d)

The graph is a four-sided The graph is a 10-sidedfigure with counterclockwise figure with counterclockwiseorientation. orientation.(e) (f)

The graph is a three-sided The graph is a four-sidedfigure with clockwise figure with clockwiseorientation. orientation.

11. (a) (b)

(c)

12.

The graphs are rose curves where the petals never retrace,so there are infinitely many petals.

13. Circle

14. (a) No. Because of the exponential, the graph will continueto trace the butterfly curve at larger values of

(b) This value will increase if is increased.

15.

For a bell is produced.

For a heart is produced.

For a rose curve is produced.n � 0,

n ≤ �1,

n ≥ 1,

−4

−6 6

4

−4

−6 6

4

�r 4.1.r.

r � a sin�2.63��r � a cos�3.5��

−a

−a a

a

−a

−a a

a

3−3

2

−2

r � cos 2� sec �y2 � x2�1 � x1 � x�

−6

6−6

6

−6

6−6

6

−10

−10 10

10

−6

6−6

6

x�2

−6

6−6

6

−6

6−6

6

y � 2 sin��t�x � cos��t�

v0.

y � h � x tan � �16x2 sec2 �

v02y � �tan ��x;

h

�x � 6�2

9�

� y � 2�2

7� 1

x2

100�

y2

64� 1

�10, 0��6, 0��6, 0�;

d1 � dz ≤ 20,

A �4a2 b2

a2 � b2y2 � 4p�x � p�

3504.451115.5�

85.4x2

2352.25�

y2

529� 1

Precalculus with Limits, Answers to Problem Solving 38

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Precalculus with Limits, Answers to Problem Solving 39

(Continued)

16. (a)

(b) Neptune:

Pluto:

(c)

(d) Yes, the orbits intersect. No, the orbits will not collidedue to the way energy is transferred between the planets, gravitationally.

(e) Yes, at times Pluto can be closer to the sun thanNeptune. Pluto is called the ninth planet because it hasthe longest orbit around the sun and therefore alsoreaches the furthest distance away from the sun.

−1.8 × 1010 1.8 × 1010

−1.2 × 1010

1.2 × 1010

Neptune

Pluto

Perihelion � 8.123 109 km

Aphelion � 1.350 1010 km

Perihelion � 8.923 109 km

Aphelion � 9.077 109 km

rPluto �1.014 1010

1 � 0.2488 cos �

rNeptune �8.999 109

1 � 0.0086 cos �

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Chapter 10...Precalculus with Limits, Answers to Section 10.2 3 Section 10.2 (page 740) Vocabulary...

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Transcript of Chapter 10...Precalculus with Limits, Answers to Section 10.2 3 Section 10.2 (page 740) Vocabulary...