How Can the Mean Value Theorem Prove the Derivative of an Indefinite Integral?

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Homework Help Overview

The discussion revolves around the application of the Mean Value Theorem in relation to the Fundamental Theorem of Calculus, specifically addressing the derivative of an indefinite integral.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore whether the Mean Value Theorem can be used to prove that the derivative of the indefinite integral is the original function. There are attempts to clarify the definitions of indefinite and definite integrals, and questions about the correct application of the theorem.

Discussion Status

The conversation includes various interpretations of the problem, with some participants suggesting that the Mean Value Theorem may not be the appropriate tool for this proof. Others are guiding the original poster towards using definitions and first principles to approach the derivative of the integral.

Contextual Notes

There is a reminder of the forum's rules against providing direct answers, emphasizing the importance of hints and guidance instead. The distinction between indefinite and definite integrals is also under discussion.

courtrigrad
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Hello all

Using the Mean Value Theorem, prove that the derivative of the indefinite integral [itex]\int f(x) \ dx[/itex] is [itex]f(x)[/itex]

So do I just use the fact that [itex]\int^b_a f(x) \ dx = f(\xi)(b-a)[/itex]?

Thanks
 
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is this right?
 
You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.
 
not prove but maybe show
 
hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer
 
vincentchan said:
hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer

Yes,you're right sorry about that.
 
I believe your question should to prove that the derivative of this function:

[tex]F(x)=\int_a^{x} f(t)dt[/tex]

is f(x). Am I right? The above is a definite integral. The derivative of the indefinite integral is just f(x) by definition. Indefinite integral means anti-derivative.

What is F'(x) from first principles i.e: using the definition of derivative?
 
[tex]F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x}[/tex]

forgot to put limit
 
Last edited:
courtrigrad said:
[tex]F'(x) = \frac{f(x+\Delta x) - f(x)}{\Delta x}[/tex]

forgot to put limit

Careful... we're looking for F'(x) not f'(x).
 
  • #10
is this right?
 
  • #11
courtrigrad said:
is this right?

Yes. So now use the definition of F(x) in post #7, to plug in the approriate F(x) and F(x+deltax) into your derivative equation...
 

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