Concept preview

What you will be able to do

1. 01Compute the average rate of change of a function over an interval $[a,\,a+h]$ and recognise it as the slope of a secant line.
2. 02Explain how letting the interval width $h\to 0$ turns a list of average rates into a single instantaneous rate of change at the point $x=a$.
3. 03Interpret an instantaneous rate in context (for example velocity) with correct units, and estimate it numerically from a shrinking table of average rates.

Pick a function $f$ and two input values, $a$ and $a+h$, where $h$ is the width of the interval between them. The average rate of change of $f$ over $[a,\,a+h]$ is the change in output divided by the change in input: $\dfrac{f(a+h)-f(a)}{h}$. This is a familiar object wearing new clothing — it is nothing more than "rise over run" between two points on the graph, the same slope formula you have used since algebra. The only novelty is the labels: the run is $h$ and the rise is $f(a+h)-f(a)$.