How Does the Integral Mean Value Theorem Link to Fundamental Calculus Concepts?

So basically, the First Mean Value Theorem for Integration states that the definite integral of a function is equal to the difference between the values of the function at the endpoints of the interval, multiplied by the derivative of the function evaluated at some point within the interval. This is supported by both the ordinary mean value theorem and the fundamental theorem of calculus. In summary, the First Mean Value Theorem for Integration provides a generalization of the fundamental theorem of calculus by using the mean value theorem and showing that it holds for any function, not just the constant function.
  • #1
daudaudaudau
302
0
http://en.wikipedia.org/wiki/Mean_value_theorem#First_mean_value_theorem_for_integration"

Take a look at the Wikipedia proof. Now, wouldn't it be easier to prove it like this:
The ordinary mean value theorem says that
[tex]
G(b)-G(a)=(b-a)G'(\xi)
[/tex]

And the fundamental theorem of calculus says that
[tex]
G(b)-G(a)=\int_a^b G'(x)dx
[/tex]

So the conclusion is
[tex]
\int_a^b G'(x)dx=(b-a)G'(\xi)
[/tex]
 
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  • #2
You proved the case where [tex] \phi = 1[/tex] identically. Notice the wikipedia proof covers a far broader case than you did.

Also, you probably used the mean value theorem to prove the fundamental theorem of calculus, so this is circular
 
  • #3
I see.
 

FAQ: How Does the Integral Mean Value Theorem Link to Fundamental Calculus Concepts?

1. What is the integral mean value theorem?

The integral mean value theorem, also known as the first mean value theorem for integrals, is a mathematical theorem that establishes a relationship between the average value of a function over an interval and its derivative at a particular point within that interval.

2. How is the integral mean value theorem used?

This theorem is used to prove important results in calculus, such as the fundamental theorem of calculus. It is also used in applications such as physics and engineering to analyze the behavior of continuous functions.

3. What is the difference between the integral mean value theorem and the mean value theorem for derivatives?

The integral mean value theorem deals with the average value of a function over an interval, while the mean value theorem for derivatives deals with the instantaneous rate of change of a function at a specific point. The integral mean value theorem is a consequence of the mean value theorem for derivatives.

4. What are the conditions for the integral mean value theorem to hold?

The integral mean value theorem holds if the function is continuous on a closed interval and differentiable on the open interval. Additionally, the derivative of the function must be bounded on the interval.

5. Can the integral mean value theorem be applied to non-continuous functions?

No, the integral mean value theorem only applies to continuous functions. If a function is not continuous, then it does not have an average value over an interval, and the theorem cannot be applied.

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