A spiral is described by for .

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

Problem

A spiral is described by

r=\theta

for

0\le\theta\le2\pi

. (a) Find

\dfrac{dy}{dx}

in terms of

\theta

. (b) Derive an expression for

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

in terms of

\theta

. (c) Determine all intervals of

\theta

[0,2\pi]

on which the spiral is concave upward and all intervals on which it is concave downward. (d) Identify every value of

\theta

[0,2\pi]

for which the spiral has a point of inflection, and verify your conclusion with appropriate reasoning or sign analysis.