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

666 Chapter 10 Parametric Equations and Polar Coordinates
In Exercise 53 you are asked to prove
analytically what we have discovered
from the graphs in Figure 19.
1
de creases from 2 to 0, the limaçon is shaped like an oval. This oval becomes more
circular as c l 0, and when c − 0 the curve is just the circle r − 1.
c=1.7 c=1 c=0.7 c=0.5 c=0.2
c=2.5
c=_2
c=0 c=_0.2 c=_0.5 c=_0.8 c=_1
FIGURE 19
Members of the family of
limaçons r − 1 1 c sin
The remaining parts of Figure 19 show that as c becomes negative, the shapes
change in reverse order. In fact, these curves are reflections about the horizontal axis of
the corresponding curves with positive c.
n
Limaçons arise in the study of planetary motion. In particular, the trajectory of Mars,
as viewed from the planet Earth, has been modeled by a limaçon with a loop, as in the
parts of Figure 19 with | c | . 1.
1–2 Plot the point whose polar coordinates are given. Then find
two other pairs of polar coordinates of this point, one with r . 0
and one with r , 0.
1. (a) s1, y4d (b) s22, 3y2d (c) s3, 2y3d
2. (a) s2, 5y6d (b) s1, 22y3d (c) s21, 5y4d
3–4 Plot the point whose polar coordinates are given. Then find
the Cartesian coordinates of the point.
3. (a) s2, 3y2d (b) (s2 , y4) (c) s21, 2y6d
4. (a) (4, 4y3) (b) s22, 3y4d (c) s23, 2y3d
5–6 The Cartesian coordinates of a point are given.
(i) Find polar coordinates sr, d of the point, where r . 0
and 0 < , 2.
(ii) Find polar coordinates sr, d of the point, where r , 0
and 0 < , 2.
5. (a) s24, 4d (b) (3, 3s3 )
6. (a) (s3 , 21) (b) s26, 0d
7–12 Sketch the region in the plane consisting of points whose
polar coordinates satisfy the given conditions.
7. r > 1
8. 0 < r , 2, < < 3y2
9. r > 0, y4 < < 3y4
10. 1 < r < 3, y6 , , 5y6
11. 2 , r , 3, 5y3 < < 7y3
12. r > 1, < < 2
13. Find the distance between the points with polar coordinates
s4, 4y3d and s6, 5y3d.
14. Find a formula for the distance between the points with polar
coordinates sr 1, 1d and sr 2, 2d.
15–20 Identify the curve by finding a Cartesian equation for the
curve.
15. r 2 − 5 16. r − 4 sec
17. r − 5 cos 18. − y3
19. r 2 cos 2 − 1 20. r 2 sin 2 − 1
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