Question

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

role="math" localid="1646337271984" $f\left(x\right)=\tan x$

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Short answer

the average rate of change of $f\left(x\right)=\tan x$from$0$ torole="math" localid="1646337601973" $\frac{\mathrm{\pi }}{6}$ is$\frac{2\sqrt{3}}{\mathrm{\pi }}$

Step-by-step solution

Step 1. Given data

The given function is

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

The given range isrole="math" localid="1646337698106" $\left[a,b\right]=\left[0,\frac{\mathrm{\pi }}{6}\right]$

Step 2. average rate of change

the average rate of change f

$$\begin{gathered} \frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{\tan \left(b\right)-\tan \left(a\right)}{b-a} \\ =\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{6}}\right)-\tan \left(0\right)}{\frac{\mathrm{\pi }}{6}-0} \\ =\frac{{\displaystyle \frac{1}{\sqrt{3}}}-0}{\frac{\mathrm{\pi }}{6}} \\ =\frac{2\sqrt{3}}{\mathrm{\pi }} \end{gathered}$$

So average rate of change is$\frac{2\sqrt{3}}{\mathrm{\pi }}$