10.3 Polar Functions Quick Review 5.Find dy / dx. 6.Find the slope of the curve at t = 2. 7.Find the points on the curve where the slope is zero. 8.Find.

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Presentation transcript:

10.3 Polar Functions

Quick Review

5.Find dy / dx. 6.Find the slope of the curve at t = 2. 7.Find the points on the curve where the slope is zero. 8.Find the points on the curve where the slope is undefined. 9.Find the length of the curve from t = 0 to t = .

What you’ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed by Polar Curves A Small Polar Gallery Essential Questions How can we use polar equations to define some interesting and important curves that would be difficult or impossible to define in the form y=f(x)?

Rectangular and Polar Coordinates 1. Find rectangular coordinates for the following polar coordinates a. (4,  /2) b. (8, 30 o ) c. (8, 240 o ) d. (6, 5  /6)

Example Rectangular and Polar Coordinates 1.Find two different sets of polar coordinates for the point with the rectangular coordinate (  3, 3).

Circles

Rose Curves

Limaçon Curves

Lemniscate Curves

Spiral of Archimedes

Polar-Rectangular Conversion Formulas Parametric Equations of Polar Curves

Converting Polar to Rectangular 3. Replace the polar equation by an equivalent rectangular equation. Then identify the graph. Multiply both sides by r. A circle with center: and radius:

Example Finding Slope of a Polar Curve 4. Find the slope of the rose curve r = 2sin3  at the point where  =  /6. Define parametrically.

Area in Polar Coordinates The area of the region between the origin and the curve r = f (  ) for  ≤  ≤  is

Example Finding Area 5. Find the area of the region in the plane enclosed by the cardioid

Area Between Polar Curves The area of the region between r 1 (  ) and r 2 (  ) for  ≤  ≤  is

Example Finding Area Between Curves 6. Find the area of the region that lies inside the circle r = cos  and outside the cardioid r = 1 – cos . Find points of intersection. The outer curve is r = cos  The inner curve is r = 1 – cos 

Example Finding Area Between Curves 6. Find the area of the region that lies inside the circle r = cos  and outside the cardioid r = 1 – cos .

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