For the curve defined on by x=t^ 3 -3t

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

Problem

For the curve defined on

-3<t<3

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

answer the following. (a) Compute

\dfrac{dy}{dx}

in terms of

t

. Show your work. (b) Find an explicit expression for

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

in terms of

t

. (c) Solve for all values of

t

(-3,3)

at which

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

or is undefined. (d) For each

t

found in part (c), determine whether the sign of

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

changes. State the coordinates of every inflection point of the curve on the given interval.