James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

668 Chapter 10 Parametric Equations and Polar Coordinates
;
65. Show that the polar equation r − a sin 1 b cos , where
ab ± 0, represents a circle, and find its center and radius.
66. Show that the curves r − a sin and r − a cos intersect
at right angles.
67–72 Use a graphing device to graph the polar curve. Choose
the parameter interval to make sure that you produce the entire
curve.
67. r − 1 1 2 sinsy2d (nephroid of Freeth)
68. r − s1 2 0.8 sin 2 (hippopede)
69. r − e sin 2 2 coss4d (butterfly curve)
where n is a positive integer. How does the shape change as
n increases? What happens as n becomes large? Explain the
shape for large n by considering the graph of r as a function
of in Cartesian coordinates.
77. Let P be any point (except the origin) on the curve r − f sd.
If is the angle between the tangent line at P and the radial
line OP, show that
tan −
r
dryd
[Hint: Observe that − 2 in the figure.]
70. r − | tan | | cot | (valentine curve)
71. r − 1 1 cos 999 (Pac-Man curve)
r=f(¨ )
ÿ
72. r − 2 1 coss9y4d
P
;
;
;
73. How are the graphs of r − 1 1 sins 2 y6d
and r − 1 1 sins 2 y3d related to the graph of
r − 1 1 sin ? In general, how is the graph of r − f s 2 d
related to the graph of r − f sd?
74. Use a graph to estimate the y-coordinate of the highest
points on the curve r − sin 2. Then use calculus to find
the exact value.
75. Investigate the family of curves with polar equations
r − 1 1 c cos , where c is a real number. How does the
shape change as c changes?
; 76. Investigate the family of polar curves
r − 1 1 cos n
;
O
¨
78. (a) Use Exercise 77 to show that the angle between the tangent
line and the radial line is − y4 at every point
on the curve r − e .
(b) Illustrate part (a) by graphing the curve and the tangent
lines at the points where − 0 and y2.
(c) Prove that any polar curve r − f sd with the property
that the angle between the radial line and the tangent
line is a constant must be of the form r − Ce k , where C
and k are constants.
˙
laboratory Project
; families of polar curves
In this project you will discover the interesting and beautiful shapes that members of families of
polar curves can take. You will also see how the shape of the curve changes when you vary the
constants.
1. (a) Investigate the family of curves defined by the polar equations r − sin n, where n is a
positive integer. How is the number of loops related to n?
(b) What happens if the equation in part (a) is replaced by r − | sin n | ?
2. A family of curves is given by the equations r − 1 1 c sin n, where c is a real number and
n is a positive integer. How does the graph change as n increases? How does it change as c
changes? Illustrate by graphing enough members of the family to support your conclusions.
3. A family of curves has polar equations
r − 1 2 a cos
1 1 a cos
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