The average change

Let the function be $y=x^2$

This function covers the whole real straight line, since with every value of $x$ there is a different value of $y$ . Any given interval can be defined over the function.

We can choose, for example, the closed interval $[1,4]$. In this interval, $x$ is growing from an initial value, $1$, up to a final value, $4$. It is increasing, and we will call this increase $\mathrm{\Delta }x$, so we will have $\mathrm{\Delta }x=4-1=3$.

Given this interval it might be interesting to study how the value of $y$ evolves. Firstly, $y=x^2=1^2=1$, while at the end of the interval $y=x^2=4^2=16$. In this case, then, the entire increase is not $3$, but $\mathrm{\Delta }y=4^2-1^2=15$.

The average change is defined as:$${\displaystyle AC=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}}$$

In the previous example, the $AC=5$.

Obviously the concept can be generalized to any function $y=f(x)$, and any interval $[a,a+\mathrm{\Delta }x]$. The definition of average change is then $${\displaystyle AC=\frac{y}{x}=\frac{f(a+x)-f(a)}{x}}$$ In many textbooks it is usually called $h$ to the value of $\mathrm{\Delta }x$.