Answer
(a) $\frac{\Delta y}{\Delta t}=-\frac{2}{\pi}$
(b) $\frac{\Delta y}{\Delta t}=0$
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:
$$\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2-x_1}$$
$$g(t)=2+\cos t$$
(a) $[0,\pi]$
The average rate of change of $y=h(t)$: $$\frac{\Delta y}{\Delta t}=\frac{(2+\cos\pi)-(2+\cos0)}{\pi-0}$$
$$\frac{\Delta y}{\Delta t}=\frac{\cos\pi-\cos0}{\pi}=\frac{-1-1}{\pi}=-\frac{2}{\pi}$$
(b) $[-\pi,\pi]$
The average rate of change of $y=h(t)$: $$\frac{\Delta y}{\Delta t}=\frac{(2+\cos\pi)-[2+\cos(-\pi)]}{\pi-(-\pi)}$$
$$\frac{\Delta y}{\Delta t}=\frac{\cos\pi-\cos(-\pi)}{2\pi}=\frac{-1-(-1)}{2\pi}=\frac{-1+1}{2\pi}=\frac{0}{2\pi}=0$$
