Question

find the average rate of change of f from$0$ to$\frac{\mathrm{\pi }}{6}$

role="math" localid="1646338076356" $f\left(x\right)=\tan \left(2x\right)$

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Short answer

the average rate of change of f$f\left(x\right)=\tan \left(2x\right)$from$0$to$\frac{\mathrm{\pi }}{6}$is$\frac{6\sqrt{3}}{\mathrm{\pi }}$

Step-by-step solution

Step 1. Given data

The given function is

$f\left(x\right)=\tan \left(2x\right)$

The given range is$\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{6}\right]$

Step 2. average rate of change 

The average rate of change f

$$\begin{gathered} \frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{\tan \left(2b\right)-\tan \left(2a\right)}{b-a} \\ =\frac{\tan (2\times {\displaystyle \frac{\mathrm{\pi }}{6}})-\tan (2\times 0)}{\frac{\mathrm{\pi }}{6}-0} \\ =\frac{\tan {\displaystyle \frac{\mathrm{\pi }}{3}}-\tan 0}{\frac{\mathrm{\pi }}{6}} \\ =\frac{\sqrt{3}-0}{\frac{\mathrm{\pi }}{6}} \\ =\frac{6\sqrt{3}}{\mathrm{\pi }} \end{gathered}$$

So average rate of change is$\frac{6\sqrt{3}}{\mathrm{\pi }}$