The parametric equations $x=t-4sin(\frac{t}{4})$ and $y=4-4cos(\frac{t}{4})$ describe a curve called a cycloid.

A$.$ Compute the separate derivatives, $\frac{dx}{dt}$ and $\frac{dy}{dt}.$

B$.$ Find and simplify a formula for $\frac{dy}{dx}$ as a function of $t.$

C$.$ Determine all values of $t$ where this cycloid has a horizontal tangent line.

D$.$ Determine all values of $t$ where this curve has a vertical tangent line.

E$.$ Find and simplify a formula for the speed of this parametric motion.

F$.$ Determine all values of $t$ where the speed is zero.

G$.$ Determine all values of $t$ where the speed is maximized. You could $"$do the optimization" using the critical points, etc., OR you could reason through by considering how the cosine function behaves.