Question

How do you find the average value of the function for `f(x)=2-1/2x, 0<=x<=4`?

Answer 1

The average value of the function on the given interval is `1`.

Explanation:

Average value of a function is given by

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

`A = 1/4 int_0^4 2 - 1/2x dx`

`A = 1/4[2x - 1/4x^2]_0^4`

`A = 1/4(2(4) - 1/4(4)^2)`

`A = 1/4(8 - 4)`

`A = 1/4(4)`

`A = 1`

Hopefully this helps!

Answer 2

The average value of `f` over `[0,4]` is 1.

Explanation:

The average value of a function over an interval is its (definite) integral over that interval divided by the length of the interval.

`int_0^4 (2 - x/2) dx = [2x-x^2/4]_0^4`

`= 4 - 0`

`= 4`

`(int_0^4 (2 - x/2) dx)/(4-0) = 4/4 = 1`

The average value of `f` over `[0,4]` is 1.