16. Parametric Equations & Polar Coordinates

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

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.

44–49. Areas of regions Find the area of the following regions.

The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ

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)

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)

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)

A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.

Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

63–74. Arc length of polar curves Find the length of the following polar curves.

{Use of Tech} The complete limaçon r=4−2cosθ

44–49. Areas of regions Find the area of the following regions.

The region inside the limaçon r=2+cosθ and outside the circle r=2

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

The region common to the circles r = 2 sin θ and r = 1

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

The region inside the lemniscate r² = 6 sin 2θ

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 = 1 −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

Area of roses Assume m is a positive integer.

a. Even number of leaves: What is the relationship between the total area enclosed by the 4m-leaf rose r=cos(2mθ) and m?

Circles in general Show that the polar equation

r² - 2r r₀ cos(θ - θ₀) = R² - r₀²

describes a circle of radius R whose center has polar coordinates (r₀, θ₀)