Maliyah Robles

2022-07-03

How do you find the average value of the function for $f\left(x\right)={e}^{x},-1\le x\le 1$?

Miguidi4y

Expert

Explanation:
By definition, the average value of a function f(x) over a domain [a,b] is given by:
$f\left(x\right)=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx$
So, for the given function, $f\left(x\right)={e}^{x}$ with $-1\le x\le 1$, we have:
$f\left(x\right)=\frac{1}{1-\left(-1\right)}{\int }_{-1}^{1}{e}^{x}dx$
$=\frac{1}{2}\left[{e}^{x}{\right]}_{-1}^{1}$
$=\frac{1}{2}\left(e-{e}^{-1}\right)$
$=\frac{1}{2}\left(e-\frac{1}{e}\right)$
$\approx 1.1752$

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