The applet below lets you compare the slope of the secant line between two points (A & B) to the slope of tangent line to point C. The x-coordinate of Point C is the midpoint between the x-coordinates of Points A and B. The slope of the secant line and tangent line are displayed at right. To change point A & B, type a value in the appropriate box at right then press enter. Zoom in or out by right clicking. Use the check boxes to try three different functions.

Change A and B and notice what happens to the slopes of the secant line AB and the tangent line to the midpoint C. Try values for A and B that are far away from each other and close together.

When are the slope of the secant line AB and the tangent line to the midpoint C closest in value? Are they ever the same? If so, under what conditions?

In what situations do the slope of the secant line AB and tangent line to midpoint C differ significantly?

Read up on 'Mean Value Theorem' or explore it in this applet and think about how it is related to the situation illustrated here.