The graph below shows the motion of a point along a vertical line over time. On the left is the actual motion of the point; one with an arrow that represents an instantaneous velocity and one with an arrow representing the average velocity. On the right is the position-time graph. [br][list][*]Use the input box to adjust the value of c or click and drag the point P or c on the graph to adjust to a particular time. [/*][*]Use the input box for h to adjust the value of h, which is the horizontal distance (difference in x-values) for the points P and Q on the graph. [/*][*]The check boxes for "Secant" and "Tangent" show the secant line through P and Q and the tangent line at P, respectively. [/*][*]Click on the "Let h approach 0" button to shorten the time interval between P and Q. [/*][/list]

"Instantaneous" rate of change is a bit of a paradox. For something to change, it must be different at two different times, but an "instant" means one single moment in time. You can't tell how fast something is moving from a single picture; you have to piece together two data points to talk about how fast something is changing. [br][br]However, we also have an intuitive understanding of instantaneous rate of change. The speedometer on your car is giving you an estimate of your current speed at any given moment. If you watched swimming events at the Olympics you could see the instantaneous speeds of the top 3 swimmers at any given time. [br][br]An informal definition: The [b]instantaneous rate of change[/b] of a function [math]f[/math] at input [math]a[/math] is defined to be the [b]limit [/b]of the [b]average rates[/b] of change of f over [i]progressively smaller intervals [/i]containing [math]a[/math]. This means that [i]we can [b]estimate [/b]instantaneous rates of change with average rates of change[/i], as long as we choose a small enough interval for the average rate.