A closed curve is generated by for

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

Problem

A closed curve is generated by

r=4\cos\!\left(\dfrac{\theta}{2}\right)

for

0\le\theta\le4\pi

. Portions of the curve overlap as

\theta

increases. (a) Determine all values of

\theta

[0,4\pi]

at which the curve crosses the

x

–axis. Provide justification. (b) Compute

\dfrac{dy}{dx}

and

\dfrac{d^2y}{dx^2}

for the curve. (c) Create a sign chart for

\dfrac{d^2y}{dx^2}

over

[0,4\pi]

and list every interval of

\theta

on which the curve is concave upward and every interval on which it is concave downward. (d) Using your work from part (c), identify all points of inflection that lie on distinct loops of the curve, and explain how overlapping portions affected—or did not affect—your conclusions.

Related theory

Polar Functions

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