Derivative

A fairly simple, and not altogether impractical example is that of the falling apple. Observation tells us that the apple's initial speed (the instant before letting go from the tree) is zero, and that it accelerates rapidly. Scientists have found, from repeated measurements with Figure 3. Illustration by Hans & Cassidy. Courtesy of Gale Group. various falling objects (neglecting wind resistance), that the distance an object falls on the earth (call it S) in a specified time period (call it T) is given by the following equation (see Figure 2):

Suppose you are interested in the apple's speed after it has dropped 4 ft (1.2 m). As a first approximation, connect the points where Sl₁=0 and Sl₂=8 (see Figure 3 and line 1 of Table 1).

Using equation (1), find the corresponding times, and calculate the slope of the approximating line segment (use the formula in Figure 1). Repeat this process numerous times, each time letting the two points get closer together. If a calculator or computer spreadsheet is available this is rather simple. Table 1 shows the result for several approximating line segments.

The line segments corresponding to the first two entries in the table are drawn in Figure 3. Looking at Figure 3, it is clear that as the approximating line gets shorter, its slope approximates the rate of rise of the curve more accurately.