Mathematics

Problem

For the curve defined parametrically by

x=t^{2}+e^{t},\qquad y=\sin t,\qquad t\in\mathbb R,

answer the following questions. a. (i) Find

\dfrac{dy}{dx}

t=\dfrac{\pi}{4}.

b. (ii) Calculate

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

at the same value of

t.

c. (iii) State whether the curve is concave upward or concave downward at the point corresponding to

t=\dfrac{\pi}{4}.

Justify your answer using the result obtained in part (ii).