Given a function, how do you find the average?

Jun 7, 2015

If you have two numbers $a$ and $b$, then their average is $\frac{a + b}{2}$.

If you have three numbers $a$, $b$ and $c$, then their average is
$\frac{a + b + c}{3}$

If you have a finite sequence of numbers: ${a}_{1} , {a}_{2} , \ldots , {a}_{n}$, then their average is $\frac{{\sum}_{i = 1}^{i = n} {a}_{i}}{n}$

If you have a finite set $F$ and a function $f : F \to \mathbb{R}$ then
the average value of $f$ over $F$ is

$\frac{{\sum}_{x \in F} \left(f \left(x\right)\right)}{|} F |$

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 $f : S \to \mathbb{R}$ then the average of $f$ over $S$ is

$\frac{{\int}_{x \in S} f \left(x\right)}{|} S | = \frac{{\int}_{x \in S} f \left(x\right)}{{\int}_{x \in S} 1}$

Jun 7, 2015

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 $\left({x}_{1} , f \left({x}_{1}\right)\right)$ and $\left({x}_{2} , f \left({x}_{2}\right)\right)$ 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 $f \left(x\right)$ between those points is:

$\frac{f \left({x}_{2}\right) - f \left({x}_{1}\right)}{{x}_{2} - {x}_{1}}$

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

The average slope is:

$\frac{{\int}_{x = {x}_{1}}^{x = {x}_{2}} \frac{d}{\mathrm{dx}} f \left(x\right)}{{x}_{2} - {x}_{1}}$

which simplifies to $\frac{f \left({x}_{2}\right) - f \left({x}_{1}\right)}{{x}_{2} - {x}_{1}}$