Average and Instantaneous Rate of Change
In this figure, a (poorly drawn) person looks from a cliff as a hot air balloon rises and falls in the distance.
- 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 instantaneous rate of change at time can be approximated by making the value of very close to .
For motion along a line (linear motion), suppose there is a function that describes the object's position at time .
- The average rate of change on the interval from to is .
- The instantaneous rate of change at time is .