# The Mean Value Theorem

- Author:
- gbattaly

The blue curve below is a cubic function. Consider the interval [a, b] to be [0.5, 6.74]
The cubic function is 1. continuous on the closed interval [0.5, 6.74] and
2. differentiable on the open interval (0.5, 6.74)
The objects in green relate to the secant line connecting the endpoints of the interval.
msec is the slope of the secant line.
The objects in red relate to the tangent line at the point C on the cubic function along the interval.
mtan is the slope of the tangent line
xcoordinate is the x coordinate of the point C
Click and drag the slider to change the location of point C. Move the slider until the value of mtan equals msec.

At how many locations on the interval does mtan = msec? _______
How many locations on the interval are guaranteed by the Mean Value Theorum? ______
Does the Mean Value Theorum apply for all polynomials on any interval? ______
Does the Mean Value Theorum apply for all continuous functions on any interval? ______
Explain. __________________________________________________