A plane curve is given parametrically by and for all real .

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

Problem

A plane curve is given parametrically by

x=t^{2}

and

y=t^{3}-3t

for all real

t

. (a) Express

\dfrac{dy}{dx}

in terms of

t

. (b) Derive an explicit expression for

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

in terms of

t

. Simplify as much as algebraically convenient. (c) Determine all real values of

t

at which the concavity of the curve changes. Show all analytical steps used to reach your conclusion. (d) Give the exact coordinates of every point of inflection on the curve.

Related theory

Curve Analysis
Parametric Functions

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