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Proof Using Mean Value Theorem

  1. Jan 26, 2005 #1
    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
     
  2. jcsd
  3. Jan 27, 2005 #2
    is this right?
     
  4. Jan 27, 2005 #3

    HallsofIvy

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    You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.
     
  5. Jan 27, 2005 #4
    not prove but maybe show
     
  6. Jan 27, 2005 #5
    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
     
  7. Jan 27, 2005 #6

    learningphysics

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    Yes,you're right sorry about that.
     
  8. Jan 27, 2005 #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?
     
  9. Jan 27, 2005 #8
    [tex] F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x} [/tex]

    forgot to put limit
     
    Last edited: Jan 27, 2005
  10. Jan 27, 2005 #9

    learningphysics

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    Careful... we're looking for F'(x) not f'(x).
     
  11. Jan 27, 2005 #10
    is this right?
     
  12. Jan 27, 2005 #11

    learningphysics

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    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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