[tex] \text{Consider the function}\\ \\ h(t)=\cot(t) \text{ on the interval }\left [ \frac{\pi}{4}, \ \frac{3\pi}{4} \right ]\\ \\ \text{we know that the average rate of change of a funtion f(x) over the }\\ \text{interval [a, b] is given by}\\ \\ f_{avg}=\frac{f(b)-f(a)}{b-a}\\ \\ \text{so using this, the average rate of change of the given function is} [/tex]
[tex] h_{avg}=\frac{h\left ( \frac{3\pi}{4} \right )-h\left ( \frac{\pi}{4} \right )}{\frac{3\pi}{4}-\frac{\pi}{4}}\\ \\ =\frac{\cot \left ( \frac{3\pi}{4} \right )-\cot \left ( \frac{\pi}{4} \right )}{\frac{3\pi-\pi}{4}}\\ \\ =\frac{(-1)-(1)}{\frac{2\pi}{4}}\\ \\ =\frac{-2}{\frac{\pi}{2}}\\ \\ \text{Average rate of change of function}=\frac{-4}{\pi} [/tex]
