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

690 Chapter 10 Parametric Equations and Polar Coordinates
exercises
;
;
1–4 Sketch the parametric curve and eliminate the parameter to
find the Cartesian equation of the curve.
1. x − t 2 1 4t, y − 2 2 t, 24 < t < 1
2. x − 1 1 e 2t , y − e t
3. x − cos , y − sec , 0 < , y2
4. x − 2 cos , y − 1 1 sin
5. Write three different sets of parametric equations for the
curve y − sx .
6. Use the graphs of x − f std and y − tstd to sketch the parametric
curve x − f std, y − tstd. Indicate with arrows the
direction in which the curve is traced as t increases.
x
_1
1
x=f(t)
t
y
1
1
y=g(t)
7. (a) Plot the point with polar coordinates s4, 2y3d. Then
find its Cartesian coordinates.
(b) The Cartesian coordinates of a point are s23, 3d. Find
two sets of polar coordinates for the point.
8. Sketch the region consisting of points whose polar coordinates
satisfy 1 < r , 2 and y6 < < 5y6.
9–16 Sketch the polar curve.
9. r − 1 1 sin 10. r − sin 4
11. r − cos 3 12. r − 3 1 cos 3
13. r − 1 1 cos 2 14. r − 2 cossy2d
15. r −
3
1 1 2 sin
16. r −
3
2 2 2 cos
17–18 Find a polar equation for the curve represented by the
given Cartesian equation.
17. x 1 y − 2 18. x 2 1 y 2 − 2
19. The curve with polar equation r − ssin dy is called a
cochleoid. Use a graph of r as a function of in Cartesian
coordinates to sketch the cochleoid by hand. Then graph it
with a machine to check your sketch.
20. Graph the ellipse r − 2ys4 2 3 cos d and its directrix.
Also graph the ellipse obtained by rotation about the origin
through an angle 2y3.
t
;
21–24 Find the slope of the tangent line to the given curve at
the point corresponding to the specified value of the parameter.
21. x − ln t, y − 1 1 t 2 ; t − 1
22. x − t 3 1 6t 1 1, y − 2t 2 t 2 ; t − 21
23. r − e 2 ; −
24. r − 3 1 cos 3; − y2
25–26 Find dyydx and d 2 yydx 2 .
25. x − t 1 sin t, y − t 2 cos t
26. x − 1 1 t 2 , y − t 2 t 3
27. Use a graph to estimate the coordinates of the lowest point
on the curve x − t 3 2 3t, y − t 2 1 t 1 1. Then use calculus
to find the exact coordinates.
28. Find the area enclosed by the loop of the curve in
Exercise 27.
29. At what points does the curve
x − 2a cos t 2 a cos 2t
y − 2a sin t 2 a sin 2t
have vertical or horizontal tangents? Use this information to
help sketch the curve.
30. Find the area enclosed by the curve in Exercise 29.
31. Find the area enclosed by the curve r 2 − 9 cos 5.
32. Find the area enclosed by the inner loop of the curve
r − 1 2 3 sin .
33. Find the points of intersection of the curves r − 2 and
r − 4 cos .
34. Find the points of intersection of the curves r − cot and
r − 2 cos .
35. Find the area of the region that lies inside both of the circles
r − 2 sin and r − sin 1 cos .
36. Find the area of the region that lies inside the curve
r − 2 1 cos 2 but outside the curve r − 2 1 sin .
37–40 Find the length of the curve.
37. x − 3t 2 , y − 2t 3 , 0 < t < 2
38. x − 2 1 3t, y − cosh 3t, 0 < t < 1
39. r − 1y, < < 2
40. r − sin 3 sy3d, 0 < <
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