Average Rate of Change (College Board AP® Calculus AB): Revision Note

Average rate of change

What is the average rate of change?

• The average rate of change between two points on a graph, is the slope of the line segment joining the two points

• The slope is equal to the change in $y$-values, divided by the change in $x$-values

• The average rate of change between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is therefore

  • $\frac{y_2-y_1}{x_2-x_1}$

• This means that the average rate of change is undefined at a point where the change in $x$ is zero for a given change in $y$

How can I write the average rate of change using function notation?

• For a function $f$,

  • a point with $x$-coordinate $x$ will have $y$-coordinate $f\left(x\right)$

  • a point with $x$-coordinate $a$ will have $y$-coordinate $f\left(a\right)$

• The average rate of change can therefore be written as

  • $\frac{f\left(x\right)-f\left(a\right)}{x-a}$

• If an $x$-coordinate of $a$ is used for the first point, and the second point lies $h$ units to the right,

  • the second point will have an $x$-coordinate of $a+h$

• The average rate of change can therefore also be written as

  • $\frac{f\left(a+h\right)-f\left(a\right)}{h}$

    • This is known as the difference quotient