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Proof Using Mean Value Theorem

  • #1
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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
 

Answers and Replies

  • #2
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is this right?
 
  • #3
HallsofIvy
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You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.
 
  • #4
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not prove but maybe show
 
  • #5
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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
 
  • #6
learningphysics
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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.
 
  • #7
learningphysics
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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?
 
  • #8
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[tex] F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x} [/tex]

forgot to put limit
 
Last edited:
  • #9
learningphysics
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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
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is this right?
 
  • #11
learningphysics
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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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