Answer
(a) $\frac{\Delta y}{\Delta t}=-\frac{4}{\pi}$
(b) $\frac{\Delta y}{\Delta t}=-\frac{3\sqrt3}{\pi}$
Work Step by Step
*Average rates of change:
The average rate of change of $y=f(x)$ with respect to $x$ over the interval $[x_1,x_2]$ is:
$$\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2-x_1}$$
$$h(t)=\cot t$$
(a) $[\pi/4,3\pi/4]$
The average rate of change of $y=h(t)$: $$\frac{\Delta y}{\Delta t}=\frac{\cot(3\pi/4)-\cot(\pi/4)}{\frac{3\pi}{4}-\frac{\pi}{4}}$$ $$\frac{\Delta y}{\Delta t}=\frac{-1-1}{\frac{2\pi}{4}}=\frac{-2}{\frac{\pi}{2}}=-\frac{4}{\pi}$$
(b) $[\pi/6,\pi/2]$
The average rate of change of $y=h(t)$: $$\frac{\Delta y}{\Delta t}=\frac{\cot(\pi/2)-\cot(\pi/6)}{\frac{\pi}{2}-\frac{\pi}{6}}$$ $$\frac{\Delta y}{\Delta t}=\frac{0-\sqrt3}{\frac{2\pi}{6}}=\frac{-\sqrt3}{\frac{\pi}{3}}=-\frac{3\sqrt3}{\pi}$$
