Exercise 13

Consider the functions $r(t) = t^t$ and $s(t) = \arccos(t)$, for which you are given the facts that $r'(t) = t^t(\ln(t) + 1)$ and $s'(t) = -\frac{1}{\sqrt{1-t^2}}$. Do not be concerned with where these derivative formulas come from. We restrict our interest in both functions to the domain $0 \lt t \lt 1$.

1. Let $w(t) = t^t \arccos(t)$. Determine $w'(t)$.

2. Find an equation for the tangent line to $y = w(t)$ at the point $(\frac{1}{2}, w(\frac{1}{2}))$.

3. Let $v(t) = \frac{t^t}{\arccos(t)}$. Is $v$ increasing or decreasing at the instant $t = \frac{1}{2}$? Why?