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Rieman Integral: The Fundamental Theorem of Calculus

  1. Apr 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Let I := [a,b] and let f: I→ℝ be continuous on I. Also let J := [c,d] and let u: J→ℝ be differentiable on J and satisfy u(J) contained in I. Show that if G: J→ℝ is defined by

    G(x) :=∫u(x)af for x in J, then G'(x) = (f o u)(x)u'(x) for all x in J.


    2. The attempt at a solution

    I am not sure what theorem to use in this situation. I am thinking that the product and composition of differentiable functions means that G'(x) exists....but I don't believe that will help me in this situation. Any ideas as to how i should start this one?
     
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  3. Apr 21, 2012 #2

    Dick

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  4. Apr 21, 2012 #3

    HallsofIvy

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    The Leibniz formula that Dick links to is basically using the chain rule.

    If [itex]F(x)=\int_a^{g(x)} f(t)dt[/itex] let u= g(x) so that the integral is
    [itex]\int_a^u f(t)dt[/itex] and, by the Fundamental theorem of Calculus,
    [tex]\frac{d}{du}\int_a^u f(t)dt= f(u)[/tex]

    Now, use the chain rule to find the derivative with respect to x.
     
  5. Apr 22, 2012 #4
    I don't understand. Isn't the derivative of G(x) the f in the integral. Or am i supposed to find the derivative of F(X)?
     
  6. Apr 22, 2012 #5

    HallsofIvy

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    No, the derivative of G(x) is NOT "the f in the integral". If the upper limit were just "x" it would be.
     
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