Integral Mean Value Theorem & Mean Value Theorem

Today we'll explore the integral mean value theorem along with the mean value theorem for differential calculus. They're very similar in concept and will be fundamental in proving the Fundamental Theorems of the Calculus.

Mean Value Theorem -

If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) - there's at least 1 point "c" between a and b where

the derivative of c is equal to the slope of the secant line between a and b. Now, said that way you might be thinking to yourself, "what's the big deal." Fair enough. Let's take a look at a visual representation made in Mathematica:

What this means is that if the condition of f being continuous over [a,b] and differentiable over (a,b) there must be some C such that a where f '(c) = m of the secant line between AB.

The secant line is the average derivative of the curve over the interval. At least one time the derivative at a point will = the average derivative.

So - you're speeding along in your Ferrari for an hour. You end up driving 145 miles from point A to point B in an all out sprint (not shown). Can you guarantee that for at least 1 instant (perhaps more but at least for an infinitesimally small quantity of time you did go 145mph? You sure can. The mean value theorem guarantees that over a curve (the drive), your instantaneous velocity (f prime of c) must be equal to your average speed (secant line) at some point.

Integral Mean Value Theorem -

Given a function f(x) that's continuous over [a,b] there exists some c such that a <> $\displaystyle f(c)=\frac{1}{b-a}\int\limits_a^bf(x)dx$

Said differently - f(c)*(b - a) equals the area beneath f(x) from a to b. Stated even more plainly, if you choose the right C, the area of the rectangle that you create whose height is f(c) and width if [a,c] will = the area under the curve. Here's an illustration done with Mathematica.

In this case, the gold rectangle with base 2.88-0 and height f(c) has area equal to the curve f(x) over the interval a to b.

Let's examine another image and see what this graph would look like had we chosen a value for C greater than ~2.88.

Clearly the area of the rectangle when using c = 4 results in an area that nearly doubles the area under the curve.

So the C you choose - is a function of the function and the mean value theorem for integration simply guarantees that there will be some C, where the area of the rectangle will equal the integral (area under f(x)) - just as in the above example for the mean value theorem used in differential calculus, there was some derivative of f(x) at c between [a,b] where the instantaneous slope was = average slope of the function.

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Resources:
Wolfram Integral Mean Value Theorem + Mean Value Theorem

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