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Average Slope

If (x0, y0) and (x0 + Δx, y0 + Δy) are two points on the curve, then the "average slope" of the curve between these two points is defined as the ratio of the change in y to the change in x,


This is exactly the same as the slope of the straight line through the points (x0, y0) and (x0 + Δx, y0 + Δy), as shown in Figure 1.4.1.


Figure 1.4.1

Let us compute the average slope. The two points (x0, y0) and (x0 + Δx, y0 + Δy) are on the curve, so

y0 = x02

y0 + Δy = (x0 + Δx)2.


Δy = (x0 + Δx)2 - x02.

Dividing by Δx,


This can be simplified,


Thus the average slope is


Notice that this computation can only be carried out when Δx ≠ 0, because at Δx = 0 the quotient Δy/Δx is undefined.

Last Update: 2010-11-25