16. Parametric Equations & Polar Coordinates

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).

r =3 − 6 cos θ

Find the slope of the tangent line of the polar curve $r=2\cos \theta$ at $\theta =\frac{\pi }{3}$.

Find the area enclosed by the cardioid $r=2-2\sin \theta$ between $\theta =\frac{\pi }{6}$ and $\theta =\frac{2\pi }{3}$.

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 1 - sin θ; (1/2, π/6)

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 + sin θ; (4, 0) and (3, 3π/2)

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 4 cos 2θ; at the tips of the leaves

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.

r = 2θ; (π/2, π/4)

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

r = 4 cos θ

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the curve r = √(cos θ)

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the circle r = 8 sin θ

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the limaçon r = 2 + cos θ

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside one leaf of r = cos 3θ

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves

r = 3 sin θ and r = 3 cos θ

45–60. Areas of regions Find the area of the following regions.

The region outside the circle r = 1/2 and inside the circle r = cos θ