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 represents the overall change in a function over a given interval.
Our strategy for “computing” instantaneous rates of change, which are represented by derivatives, was to approximate them using average rates of change. We would take smaller and smaller intervals and calculate the average rate of change over these intervals. As we decreased the interval size, the average rate of change approached the instantaneous rate of change at a specific point.
To illustrate this, let’s consider a function f(x) representing the position of an object at time x. To find the instantaneous velocity at a specific time t, we would choose a small interval around t, calculate the average rate of change of position over this interval, and then decrease the interval size. As the interval approaches zero, the average rate of change approaches the instantaneous rate of change, which is the velocity at time t. This process is the foundation of the concept of derivatives.
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