Consider the parametric curve defined for all real by x(t)=t^ 3 -3t

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

Problem

Consider the parametric curve defined for all real

t

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

The curve resembles a flattened "S" shape that changes concavity as

t

varies. (a) Derive an explicit expression for

\displaystyle \frac{dy}{dx}

in terms of

t

. Do not simplify numerators or denominators prematurely. (b) Derive an explicit expression for

\displaystyle \frac{d^{2}y}{dx^{2}}

in terms of

t

. Clearly indicate each differentiation step. (c) Determine all real values of

t

at which the curve changes concavity. Justify your answer algebraically. (d) For each

t

value found in part (c), classify the corresponding point on the curve as an inflection point or explain why no inflection occurs even though the concavity changes sign in

y

with respect to

x

. Provide complete reasoning.

Related theory

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