Let be a constant and let the curve be given by x = a t

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

Problem

Let

a>0

be a constant and let the curve be given by

x = a\cos t,\qquad y = a(\sin t + t),\qquad -\pi< t < \pi.

(a) Determine

\dfrac{dy}{dx}

in terms of

t

and

a

. (b) Obtain

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

in terms of

t

and

a

. (c) For

0< t < \dfrac{\pi}{2}

, show whether the curve is concave upward or concave downward, giving clear justification from your answer in part (b).