Average rate of change

If you drive 120 km in 2 hours, you know at once: that was 60 km/h on average. Whether you raced or crawled in between does not matter for this average. This is exactly the idea behind the average rate of change: how much does one quantity change on average while another grows by one unit?

The difference quotient

You pick two positions $a$ and $b$ on the x-axis and look at how the function values differ there. You divide the change in height by the change in x. The result tells you how much y changes on average per unit of x. The two positions have to be different, otherwise the bottom of the fraction would be zero.

This is the slope of a secant

Draw the two points $(a, f(a))$ and $(b, f(b))$ on the graph and connect them with a straight line. This connecting line is called a secant. The difference quotient is nothing other than its slope, the familiar rise divided by the horizontal run.

An example to work through

Take $f(x) = x^2$ and the positions $a = 1$ and $b = 3$. We have $f(1) = 1$ and $f(3) = 9$. So the average rate of change is $\frac{9 - 1}{3 - 1} = \frac{8}{2} = 4$. Between $x = 1$ and $x = 3$ the function therefore rises by 4 units on average while x grows by 1.