Mathematics

Problem

The curve is defined piecewise by

r(\theta)=\begin{cases}\theta^{2}, & 0\le\theta\le\tfrac{\pi}{2}\\ \pi-\theta, & \tfrac{\pi}{2}<\theta\le\pi\end{cases}.

Compute

\frac{dy}{dx}

\theta=\tfrac{\pi}{4}

and at

\theta=\tfrac{3\pi}{4}

. State whether the tangent is vertical at either of those angles.