Answer
a.)$\frac{-4}{\pi}$
b.)$\frac{-3\sqrt 3}{\pi}$
Work Step by Step
The definition of average rate of change is
$\frac{f(x2) - f(x1)}{x2-x1}$.
a.) h(t) = cot t
interval = [$\frac{\pi}{4}$, $\frac{3\pi}{4}$]
$\frac{h(\frac{3\pi}{4}) - h(\frac{\pi}{4})}{\frac{3\pi}{4}-\frac{\pi}{4}}$.
h($\frac{3\pi}{4}$) =cot($\frac{3\pi}{4}$) = -1
h($\frac{\pi}{4}$) =cot($\frac{\pi}{4}$) = 1
$\frac{-1 - 1}{\frac{3\pi}{4}-\frac{\pi}{4}}$. = $\frac{-2}{\frac{\pi}{2}}$
=$\frac{-4}{\pi}$
b.) h(t) = cot t
interval = [$\frac{\pi}{6}$, $\frac{\pi}{2}$]
$\frac{h(\frac{\pi}{2}) - h(\frac{\pi}{6})}{\frac{\pi}{2}-\frac{\pi}{6}}$.
h($\frac{\pi}{6}$) =cot($\frac{\pi}{6}$) = $\sqrt 3$
h($\frac{\pi}{2}$) =cot($\frac{\pi}{2}$) = 0
$\frac{0 - \sqrt 3}{\frac{\pi}{2}-\frac{\pi}{6}}$. = $\frac{-\sqrt 3}{\frac{\pi}{3}}$
=$\frac{-3\sqrt 3}{\pi}$
