

The Mean Value Theorem states:
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.
Put anothr way, somewhere in the interval, there is a place on the curve where the slope is the same as the slope of the secant line connecting the points ( a, f (a) ) and ( b, f (b) ).
1 Parabola
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.
2 Sine curve
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.
Where it doesn't work

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?
No. 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?
No. This is why the Mean Value Theorem only holds if the function is continuous on the interval.
Other differentiation topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.

