Let the parametric equations x(t)=t^ 2 -4t

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let the parametric equations

x(t)=t^{2}-4t, \qquad y(t)=t^{3}-6t^{2}+9t

describe a curve for

0\le t\le 4

. Focus on the point corresponding to

t=1

. (a) Verify the sign of

\dfrac{dx}{dt}

t=1

and explain why this sign is relevant when analyzing concavity. (b) Determine

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

t=1

. (c) Based on the value in part (b), classify the concavity of the curve at the point where

t=1

and justify your conclusion.