# Type 9: Polar Equation Questions

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem, but students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graph may be given or easy to find.

What students should know how to do:

• Calculate the coordinates of a point on the graph,
• Find the intersection of two graphs (to use as limits of integration).
• Find the area enclosed by a graph or graphs: Area = $\displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}$θ $\displaystyle ){{)}^{2}}d$θ
• Use the formulas $x\left( \theta \right)\text{ }=~r\left( \theta \right)\text{cos}\left( \theta \right)~~\text{and}~y\left( \theta \right)\text{ }=~r(\theta )\text{sin}\left( \theta \right)~$  to convert from polar to parametric form,
• Calculate $\displaystyle \frac{dy}{d\theta }$ and $\displaystyle \frac{dx}{d\theta }$ (Hint: use the product rule on the equations in the previous bullet).
• Discuss the motion of a particle moving on the graph by discussing the meaning of $\displaystyle \frac{dr}{d\theta }$ (motion towards or away from the pole), $\displaystyle \frac{dy}{d\theta }$ (motion in the vertical direction), and/or $\displaystyle \frac{dx}{d\theta }$ (motion in the horizontal direction).
• Find the slope at a point on the graph, $\displaystyle \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }$.

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

Free-response questions:

• 2013 BC 2
• 2014 BC 2
• 2017 BC 2

Multiple-choice questions from non-secure exams:

• 2008 BC 26
• 2012 BC 26, 91

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