Average Slope Between Two Points

Our method of sketching curves in Section 3.7 depends on a consequence of Rolle's Theorem called the Mean Value Theorem. It deals with the average slope of a curve between two points.

We can see in Figure 3.8.17 that the average slope of f between a and b is equal to the slope of the line passing through the points (a, f(a)) and (b, f(b)). This is shown by the two-point equation for a line (Section 1.3). In particular, if f is already a linear function f(x) = mx + c, then the average slope of f between a and b is equal to the slope m of the line y =f(x).

Figure 3.8.17 Average Slope

This is shown by the two-point equation for a straight line (Section 1.2). In particular, if f is already a linear function f(x) = mx + c, then the average slope of f between a and b is equal to the slope m of the straight line y = f(x).