Consider the curve defined parametrically by x = t^ 2 +2t

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

Problem

Consider the curve defined parametrically by

x = t^{2}+2t,\qquad y = t^{3}-3t,\qquad t\in\mathbb R.

(i) Find

\dfrac{dy}{dx}

in terms of

t

. (ii) Find

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

in terms of

t

. (iii) Hence evaluate

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

t = 1

and state, with reason, whether the curve is concave upward or concave downward at this point.