Question

Find the average rate of change offfrom $0$torole="math" localid="1646561530009" $\frac{\mathrm{\pi }}{2}$.

role="math" localid="1646561569314" $f\left(x\right)=\sin \left(\frac{x}{2}\right)$

Short answer

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

Step-by-step solution

Step 1. Given data

The given function is

$f\left(x\right)=\sin \left(\frac{x}{2}\right)$

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

role="math" localid="1646562090430" $$\begin{gathered} \frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left({\displaystyle \frac{\mathrm{\pi }}{2}}\right)-f\left(0\right)}{\frac{\mathrm{\pi }}{2}-0} \\ =\frac{\sin \left(\frac{\frac{\mathrm{\pi }}{2}}{2}\right)-\sin \left(\frac{0}{2}\right)}{\frac{\mathrm{\pi }}{2}-0} \\ =\frac{\sin \left(\frac{\mathrm{\pi }}{4}\right)-\sin \left(0\right)}{\frac{\mathrm{\pi }}{2}} \\ =\frac{{\displaystyle \frac{1}{\sqrt{2}}}-0}{\frac{\mathrm{\pi }}{2}} \\ =\frac{\sqrt{2}}{\mathrm{\pi }} \end{gathered}$$