A smooth plane curve is represented in polar coordinates by

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

Problem

r = f(\theta)

, where

f

is differentiable on all real numbers. The curve is traced exactly once as

\theta

increases from

0

2\pi

. (a) Starting from the relations

x = r\cos\theta

and

y = r\sin\theta

, derive formulas for

\frac{dx}{d\theta}

and

\frac{dy}{d\theta}

in terms of

r

\frac{dr}{d\theta}

\sin\theta

, and

\cos\theta

. Clearly state each step of your reasoning. (b) Using your results from part (a) and appropriate differentiation rules, obtain an explicit expression for

\displaystyle\frac{dy}{dx}

in terms of

r

\frac{dr}{d\theta}

\sin\theta

, and

\cos\theta

. Simplify completely. (c) Describe the conditions on

\frac{dy}{d\theta}

and

\frac{dx}{d\theta}

that must be satisfied for the tangent line to be (i) horizontal and (ii) vertical. Explain why these conditions are sufficient in each case. (d) Suppose both

\frac{dy}{d\theta}=0

and

\frac{dx}{d\theta}=0

at a particular angle

\theta=c

. Explain why the usual test for horizontal or vertical tangency is inconclusive at

\theta=c

. Outline a systematic strategy that could be employed to classify the tangent at

\theta=c

without explicitly computing any limits.

Related theory

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