The equations and describe a curve for all real

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

Problem

The equations

x = t^{2}

and

y = t^{3}

describe a curve for all real

t

. Study its concavity as follows: (i) Without calculating derivatives, express

y

directly in terms of

x

and sketch the general shape you expect for the curve. (ii) Compute

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

as a function of

t

and determine its sign for

t<0

t=0

, and

t>0

. State clearly whether the curve is concave upward or downward on each of these intervals of

t

. (iii) Explain why the point corresponding to

t=0

is an inflection point, relating your explanation to both the algebraic sign change found in part (ii) and the sketch made in part (i).