Ch. 11 – Parametric, Vector, and Polar Functions 11.3 – Polar Functions.

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Presentation transcript:

Ch. 11 – Parametric, Vector, and Polar Functions 11.3 – Polar Functions

Polar Slope Ex: Find the slope of the circle r=2sinθ at θ=π/6. –Formula for finding slope in polar: –You must multiply the function for r by sinθ and cosθ BEFORE taking the derivative! –Let’s look at the example above:

Polar Slope Ex: Find the slope of the cardioid r=1-cosθ at θ=π/2. –Don’t forget about product rule when taking derivatives!

Polar Slope Ex: Find the equation of the tangent line to the rose r=2sin3θ at θ=π/4. –Always find tangent line equations using rectangular coordinates! –Find slope, then plug into x=rcosθ and y=rsinθ to find points for point- slope tangent line! –Answer:

Polar Area Area of a sector: When finding area of a sector in polar form, we need to find the sum of infinitely many tiny sectors: –It might be useful to recall the following: α β

Polar Area Ex: Find the area of the region enclosed by the cardioid r=2+2cosθ. –To form the entire cardioid, θ should go from 0 to 2π… –To evaluate sans calculator, use the following identity:

Polar Area Ex: Find the area of the region enclosed by one petal of the rose r=3sin2θ. –To form one petal, determine where one petal starts and stops… –So the first petal goes from 0 to π/2.

Polar Area between curves: r o = radius of outer function;r i = radius of inner function Ex: Find the area of the region inside the circle r=2sinθ and outside the cardioid r=2+2cosθ. –First, graph both functions to find where the circles intersect to get the limits of integration. –Now use the formula above:

Sometimes it may be better to think outside the box to find the area of a polar region… Ex: Find the area of the region inside the limacon r=2+sinθ and outside the circle r=2sinθ. –First, graph both functions. The defined region goes from 0 to 2π for the limacon but only from 0 to π for the circle… –Can’t we just do limacon area minus circle area?