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Mean Value Theorem


Main Concept

The Mean Value Theorem (MVT) states that if a function f is continuous on the closed interval a&comma;b and differentiable on the open interval a&comma;b where a<b, then there exists a point c in a&comma;b such that f  &apos;c&equals;fbfaba.

In other words, for a function which changes smoothly over an interval, there must be at least one point in the interval where the instantaneous rate of change of the function equals its average rate of change over the whole interval. In terms of the graph of the function, if the graph is smooth, then there must be some point between the two endpoints where the tangent at that point is parallel to the secant line connecting the two endpoints.


As can be seen in the proof below, the Mean Value Theorem follows from a specific statement of Rolle's Theorem: if a real-valued function f  is continuous on the closed interval a&comma;b, differentiable on the open interval a&comma;b, and fa &equals; fb, then there exists a c in a&comma;b such that f  &apos;c&equals;0.


The MVT is one of the most important theorems in differential calculus, as it is essential in proving the Fundamental Theorem of Calculus and the general statement of Taylor's Theorem (with the Lagrange form of the remainder term).

Proof of the Mean Value Theorem

Consider the function hx &equals; fxba  xfb  fa, which is continuous on a&comma;b because fx is continuous on this interval.

Taking the derivative, we obtain: h &apos;x &equals; f  &apos;xba  fb  fa which is differentiable on a&comma; b because fx is differentiable on this interval.

Note that: ha &equals; fab  a  afb  fa

                               &equals; bfa  afa  afb &plus; afa

                        &equals; bfa  afb

                        &equals; bfb  afb  bfb &plus; bfa

                        &equals; fbba bfb  fa

                        &equals; hb

Therefore by Rolle's Theorem, there exists a point c in a&comma; b such that: h &apos;c &equals; 0 &equals; f  &apos;cba  fb  fa.

And so, fb  fa &equals; f  &apos;cba which can be rearranged to give us f  &apos;c &equals; fb  fab  a.


Select a function by choosing a function from the drop-down menu below or entering your own function in the text box. View the plot of the function with a secant line (in blue) and a tangent line (in green). You can specify the interval on which you would like to investigate the Mean Value Theorem, and then click and drag on the plot to move the tangent line. The tangent line will turn orange when it is very close to a point at which the tangent is parallel to the secant, and it will turn red at any point which satisfies the MVT by having a tangent which is exactly parallel to the secant.

If you are having trouble finding a point at which the instantaneous rate of change is equal to the average rate of change on your interval, click the "Find Nearest Value of c" button to automatically move to one of these special points.



fx =







Current value of x :

Slope of the secant:

Slope of the tangent:





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