In this figure, a (poorly drawn) person looks from a cliff as a hot air balloon rises and falls in the distance.[br][list][*]Notice that there is a play/pause button in the lower left corner of the figure.[/*][*]When you select "Visualize average rate of change," you'll see two times marked on the horizontal axis. You can drag these two points. If you are using a keyboard, you can fine-tune these points by clicking on one and using the left- and right-arrow keys.[/*][*]Observe that the average rate of change is the same calculation as the slope of a line through two points.[/*][*]Observer that the [i]instantaneous [/i]rate of change at time [math]t_1[/math] can be approximated by making the value of [math]t_2[/math] very close to [math]t_1[/math]. [/*][/list]

For motion along a line (linear motion), suppose there is a function [math]s[/math] that describes the object's position at time [math]t[/math]. [br][list][*]The average rate of change on the interval from [math]t_1[/math] to [math]t_2[/math] is [math]\frac{\Delta s}{\Delta t}=\frac{s\left(t_2\right)-s\left(t_1\right)}{t_2-t_1}[/math]. [/*][*]The instantaneous rate of change at time [math]t[/math] is [math]lim_{\Delta t\to0}\frac{s\left(t+\Delta t\right)-s\left(t\right)}{\Delta t}=s'\left(t\right)[/math].[/*][/list]