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 $$\Delta x$$.