A smooth plane curve is given for by differentiable functions and

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

Problem

A smooth plane curve is given for

-3\le t\le4

by differentiable functions

x(t)

and

y(t)

. The table shows the signs of the first derivative of

x

and the

t

-derivative of

\dfrac{dy}{dx}

at several key

t

-values.

No sign changes occur within each open interval listed. (a) For each open interval in

t

shown in the table, determine the sign of

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

. Show your reasoning. (b) Translate the results from part (a) into statements about the concavity of the curve in terms of

t

. (c) Restate your conclusions from part (b) in terms of

x

-intervals, providing clear justification for how the intervals in

t

correspond to intervals in

x

. Assume

x(t)

is strictly monotonic on each open interval listed.