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) ).
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.