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 $\Delta x$, so we will have $\Delta x=4-1=3$.

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 $\Delta y=4^2-1^2=15$.

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

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

Obviously the concept can be generalized to any function $y = f (x)$, and any interval $[a,a+\Delta x]$. 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 $\Delta x$.