Question

How do you find the average value of the function for `f(x)=e^x, -1<=x<=1`?

Answer 1

`bar (f)(x) = 1/2 \ (e-1/e) ~~ 1.1752`

Explanation:

By definition, the average value of a function `f(x)` over a domain `[a,b]` is given by:

`bar (f)(x) = 1/(b-a) \ int_a^b \ f(x) \ dx`

So, for the given function, `f(x)=e^x` with `-1 le x le 1`, we have:

`bar (f)(x) = 1/(1-(-1)) \ int_(-1)^(1) \ e^x \ dx`

`\ \ \ \ \ \ \ = 1/2 \ [e^x]_(-1)^(1)`

`\ \ \ \ \ \ \ = 1/2 \ (e-e^(-1))`

`\ \ \ \ \ \ \ = 1/2 \ (e-1/e)`

`\ \ \ \ \ \ \ ~~ 1.1752`