Consider the polar curve for

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the polar curve r=2+2sin⁡θr = 2 + 2\sin\theta for 0≤θ≤2π0 \le \theta \le 2\pi. Two specific directions, θ=π6\theta = \dfrac{\pi}{6} and θ=π2\theta = \dfrac{\pi}{2}, are of interest. (a) Compute dydx\displaystyle \frac{dy}{dx} at θ=π6\theta = \dfrac{\pi}{6}. Express your answer as an exact value. (b) Determine whether the tangent line to the curve at θ=π6\theta = \dfrac{\pi}{6} is horizontal, vertical, or neither. Justify your conclusion by referencing the relevant derivatives. (c) Repeat parts (a) and (b) for θ=π2\theta = \dfrac{\pi}{2}. Clearly state your final slope and classification. (d) Interpret your findings from parts (b) and (c) in terms of the local behavior of the curve near each of the two angles. Your discussion should connect the calculated slopes to the qualitative shape of the graph in a neighborhood of each point.

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