Question

Given a function, how do you find the average?

Answer 1

If you have two numbers `a` and `b`, then their average is `(a+b)/2`.

If you have three numbers `a`, `b` and `c`, then their average is
`(a+b+c)/3`

If you have a finite sequence of numbers: `a_1, a_2,...,a_n`, then their average is `(sum_(i=1)^(i=n) a_i)/n`

If you have a finite set `F` and a function `f:F->RR` then
the average value of `f` over `F` is

`(sum_(x in F)(f(x)))/|F|`

If instead of a finite set you have a measurable set `S` of finite measure such as a finite interval or the surface of a sphere or suchlike and `f:S->RR` then the average of `f` over `S` is

`(int_(x in S)f(x))/|S| = (int_(x in S)f(x))/(int_(x in S)1)`

Answer 2

Given the topic "Graphs of Linear Equations and Functions > Slope" you may be wondering how to calculate the "average slope" of a function between two points.

If you are given points `(x_1, f(x_1))` and `(x_2, f(x_2))` through which a curve described by a function `f` passes, and `f` is suitably continuous and differentiable between those points, then the average slope of `f(x)` between those points is:

`(f(x_2) - f(x_1))/(x_2 - x_1)`

It doesn't matter how much the curve wiggles in between - essentially we are dealing with the integral of the derivative of `f(x)`.

The average slope is:

`(int_(x=x_1)^(x=x_2) d/(dx)f(x)) / (x_2-x_1)`

which simplifies to `(f(x_2) - f(x_1))/(x_2 - x_1)`