Question

1 pt) find the average rate of change of the function over the given intervals. h(t)=\cot (t);

a. \left[\displaystyle \frac{\pi}{4},\displaystyle \frac{3\pi}{4}\right]

Answer 1

`\text{Consider the function}\\ \\ h(t)=\cot(t) \text{ on the interval }\left [ (\pi)/(4), \ (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)=(f(b)-f(a))/(b-a)\\ \\ \text{so using this, the average rate of change of the given function is}`

`h_(avg)=(h\left ( (3\pi)/(4) \right )-h\left ( (\pi)/(4) \right ))/((3\pi)/(4)-(\pi)/(4))\\ \\ =(\cot \left ( (3\pi)/(4) \right )-\cot \left ( (\pi)/(4) \right ))/((3\pi-\pi)/(4))\\ \\ =((-1)-(1))/((2\pi)/(4))\\ \\ =(-2)/((\pi)/(2))\\ \\ \text{Average rate of change of function}=(-4)/(\pi)`