On the Left Side: [br][list][*]Observe the point P with two different velocities represented as an arrow. The direction of the arrow tells you if P is moving up or down. The length of the arrow tells you how fast it is moving (longer means faster). [/*][*]The green arrow represents the instantaneous velocity (rate of change). This value tells you how fast a point at P is moving at that particular instant in time. [/*][*]The blue arrow represents the average velocity (rate of change). This value tells you how fast an object moves along the graph from P to Q on average. [/*][/list][br]On the Right Side: [list][*]Use the input box for f(x) to define the function. [/*][*]Use the input box for c to adjust the location of P. Or, use the play animation button (bottom left corner) to start/pause movement for P along the graph of f(x). [/*][*]Use the input box for h to set the horizontal distance between P and Q. Click the "Let h approach 0" button to bring Q closer to P. [/*][*]Use the checkboxes for Secant and Tangent to show/hide the secant and tangent lines. [/*][/list]

Instantaneous rate of change is a bit paradoxical; how can something change in an instant (i.e., without time passing)? [br][br]Intuitively, we can estimate how fast something moves in an instant by calculating its average rate of change over a very small interval. However, "small" is a relative term. A distance of 0.1 inches may be small if we're talking about distances between planets, but it would be very big if we were talking about distances between atoms. [br][br]Fortunately, in many cases, if we estimate average rate of change over a small interval and then repeat the calculation over a smaller interval, we get a new estimate that isn't too far off from the first. If we continue this process the values eventually "settle down" around a particular value, which we define to be the instantaneous rate of change.