Mathematics

Problem

Consider the polar curve

r = 2 + 2\sin\theta

for

0 \le \theta \le 2\pi

. Two specific directions,

\theta = \dfrac{\pi}{6}

and

\theta = \dfrac{\pi}{2}

, are of interest. (a) Compute

\displaystyle \frac{dy}{dx}

\theta = \dfrac{\pi}{6}

. Express your answer as an exact value. (b) Determine whether the tangent line to the curve at

\theta = \dfrac{\pi}{6}

is horizontal, vertical, or neither. Justify your conclusion by referencing the relevant derivatives. (c) Repeat parts (a) and (b) for

\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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