A curve is given for all real numbers by x=t^ 2

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

Problem

A curve is given for all real numbers

t

x=t^{2},\qquad y=t^{3}-3t.

Answer the following questions about this curve. (a) Determine

\dfrac{dy}{dx}

in terms of

t

. (b) Derive an expression for

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

in terms of

t

. Simplify completely. (c) Identify every value of

t

at which

\dfrac{d^{2}y}{dx^{2}}=0

or is undefined. Be sure to include points where

\dfrac{dx}{dt}=0

. List these

t

–values and the corresponding

(x,y)

coordinates. (d) For each candidate obtained in part (c), determine whether a sign change of

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

actually occurs. Clearly justify which points, if any, are inflection points. (e) State all open intervals of

t

over which the curve is concave upward and all intervals over which it is concave downward.