3A-3. Instantaneous Rate of Change
Instructions:
On the Left Side:
- 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.
- 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.
Instantaneous Rate of Change
Instantaneous rate of change is a bit paradoxical; how can something change in an instant (i.e., without time passing)?
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.
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.