Mean Value Theorem
The Mean Value Theorem says: If a function, f, is continuous on an interval a ≤ x ≤ b and differentiable on a < x < b, then there exists a number, c, with a < c < b, such that 
. This means that, somewhere in the interval, there is a place on the curve where the slope is the same as the average slope over the interval (or, equivalently, the slope of the secant line connecting the points (a, f (a)) and (b, f (b))).
Try the following:
- The applet initially shows the graph of a parabola. The red dot is (a, f (a)), the blue dot is (b, f (b)), and the magenta dot is (c, f (c)). The secant line segment connectiong a and b is shown, as is its slope. The Mean Value Theorem says that there exists a value of c between a and b such that the slope of f at c is the same as the slope of the secant. Move the c slider until you find a point where this is true (you will also see that the tangent line looks parallel to the secant). You can move the a and b sliders around, which changes the secant slope, and hence requires you to find a new c.
- Select the second example from the pull down menu, showing a sine curve. Move the c slider to find a place where the slopes are the same. Can you find more than one? The Mean Value Theorem only says that there is at least one value for c; there may be more than one.
- Select the third example, showing the absolute value function, which is not differentiable at a point in between a and b. Move the c slider; can you match up the slopes? This is why the Mean Value Theorem only holds if the function is differentiable on the interval.
- Select the fourth example, showing a function with a discontinuity. Move the c slider; can you make the slopes match? This is why the Mean Value Theorem only holds if the function is continuous on the interval.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.