Polar Coordinates and Polar Graphs Section 9.4 AP Calculus

Find the coordinate in rectangular and polar form.

Thm 9.10 Coordinate Conversion The polar coordinates (r,θ) of a point are related to the rectangular coordinates (x,y) of the point as follows: 1) 2)

A) If (r,θ) = , find (x,y). B) If (x,y) = (-2,-2), find (r,θ).

Graph: A) r=4 B) C) r=cscθ D) r=4cos5θ

Thm 9.11 Slope in Polar Form If f is a differentiable function of θ, then the slope of the tangent line to the graph of r=f(θ) at point (r,θ) is

Horizontal Tangent: Vertical Tangent:

Find and the slope of the tangent lines at (2,0) and

Find the Horizontal and Vertical Tangents of r=cosθ when 0≤θ≤π.

Thm 9.12 Tangent Lines at a Pole If f(α)=0 and f’(α)≠0, then the line θ=α is tangent at the pole to the graph r=f(θ).

Look at when -π≤θ≤π. Find where the graph has lines tangent to the pole.

Review of Polar Functions Limaçon Circle Rose Curve Lemniscate