Understanding the Calculation of Derivatives: The Relationship Between Average and Instantaneous Rates of Change

Before we learned how to compute derivatives, which of the rates of change (average or instantaneous) were we able to compute? What was our strategy for “computing” the other?

Before learning how to compute derivatives, we were able to compute average rates of change

Before learning how to compute derivatives, we were able to compute average rates of change.

The average rate of change measures the average rate at which a quantity is changing over a specific interval. To calculate the average rate of change, we divide the change in the quantity by the change in the independent variable (usually time).

For example, if we wanted to find the average velocity of an object moving from point A to point B, we would divide the change in position by the time it took to travel that distance.

On the other hand, we were not able to compute instantaneous rates of change directly. The instantaneous rate of change measures the rate at which a quantity is changing at a specific point or instant. It provides us with the rate of change at a particular moment rather than an average over a given interval.

To approximate the instantaneous rate of change, our strategy was to compute the average rate of change over smaller and smaller intervals. By taking smaller and smaller intervals, we could approach the instantaneous rate of change at a specific point. This approach is known as the “limiting process” and it forms the basis of calculus and the calculation of derivatives.

In summary, our strategy for “computing” the instantaneous rate of change was to approximate it by computing the average rate of change over smaller and smaller intervals. This approximation process becomes more precise as the intervals approach zero, leading to the calculation of derivatives.

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