# How do you find the average value of f(x)=2^x as x varies between [1,4]?

Mar 30, 2017

$6.7$

#### Explanation:

The formula for average value of a continuous function $F \left(x\right)$ on $\left[a , b\right]$ is $\frac{1}{b - a} {\int}_{a}^{b} F \left(x\right) \mathrm{dx}$

Our formula here will be

$A = \frac{1}{4 - 1} {\int}_{1}^{4} {2}^{x} \mathrm{dx}$

$A = \frac{1}{3} {\int}_{1}^{4} {2}^{x} \mathrm{dx}$

$A = \frac{1}{3} {\left[{2}^{x} / \ln 2\right]}_{1}^{4}$

$A = \frac{1}{3} \left[{2}^{4} / \ln 2 - {2}^{1} / \ln 2\right]$

$A = \frac{1}{3} \left[\frac{14}{\ln} 2\right]$

$A = \frac{14}{3 \ln 2}$

This can be approximated as $6.7$. The average value of $y = {2}^{x}$ on $\left[1 , 4\right]$ is $6.7$.

Hopefully this helps!