Q. 78 - Precalculus Enhanced with Graphing Utilities

Question

Find the average rate of change off from $0$to$\frac{\mathrm{\pi }}{2}$.

$f\left(x\right)=\cos x$

Step-by-Step Solution

Answer

The average rate of change of $f\left(x\right)=\cos x$from$0$ to $\frac{\mathrm{\pi }}{2}$is$\frac{-2}{\mathrm{\pi }}$

Step 1. Given data

The given function is

$f\left(x\right)=\cos x$

The given interval is$\left(0,\frac{\mathrm{\pi }}{2}\right)$

Step 2. The average rate of change

differentiate function 

$$\begin{gathered} f\left(x\right)=\cos x \\ f\text{'}\left(x\right)=-\sin x \end{gathered}$$

Consider  $(a,b)=\left(0,\frac{\mathrm{\pi }}{2}\right)$

The average rate of change

$$\begin{gathered} \frac{f\left(b\right)-f\left(a\right)}{b-a}=\left|\frac{f\text{'}\left({\displaystyle \frac{\mathrm{\pi }}{2}}\right)-f\text{'}\left(0\right)}{\frac{\mathrm{\pi }}{2}-0}\right| \\ =\left|\frac{-\sin \left({\displaystyle \frac{\mathrm{\pi }}{2}}\right)--\sin \left(0\right)}{\frac{\mathrm{\pi }}{2}-0}\right| \\ =\left|\frac{-1-0}{\frac{\mathrm{\pi }}{2}}\right| \\ =\left|\frac{-2}{\mathrm{\pi }}\right| \\ =\frac{-2}{\mathrm{\pi }} \end{gathered}$$