Question

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

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

Short answer

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

Step-by-step solution

Step 1. Given data

The given function is

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

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

Step 2. The average rate of change

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

The average rate of change of f

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