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Homework Help: A very quick question about definite integrals

  1. Mar 10, 2010 #1
    1. The problem statement, all variables and given/known data

    F(x) = [tex]\int^{x}_{0}f(t)dt[/tex]

    Then F'(x) = f(x)

    what is f'(x)? is this equivalent to f(t)?

    Thanks for your help
    M
     
  2. jcsd
  3. Mar 10, 2010 #2

    Dick

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    F(x) is your integral of f from 0 to x. F'(x) is the derivative of F(x), which is f(x), the value of your integrand f(t) evaluated at t=x. This is just the fundamental theorem of calculus, that the integral is the antiderivative of the integrand.
     
  4. Mar 10, 2010 #3
    Thanks for the reply.

    so f '(x) IS indeed f(t)? the very same f(t) in F(x)?
     
  5. Mar 10, 2010 #4

    Dick

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    No. F'(x) is f(x). But, yes, the derivative of the integral is the function you are integrating, isn't that what the fundamental theorem of calculus is all about?
     
  6. Mar 10, 2010 #5
    I'm sorry, I don't think my notations are clear. I understand that big F'(x) = f(x), what I'm concerned with is whether small f '(x) is f(t).

    so can we write

    F(x) = [tex]\int^{x}_{0}f'(x)dt[/tex] = [tex]\int^{x}_{0}f(t)dt[/tex]
     
  7. Mar 10, 2010 #6

    Mark44

    Staff: Mentor

    You don't have enough information to determine f'(x). The only information you have is that F is an antiderivative of f. A nearly equivalent way to say this is that f is the derivative of F. IOW, F'(x) = f(x).

    For example, if F(x) = x3, F'(x) = f(x) = 3x2. To go a step further and find f'(x), you need to know what the function f(x) is.
     
  8. Mar 10, 2010 #7

    Dick

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    How can f'(x) be the same as f(t)? They don't even involve the same variable.
     
  9. Mar 10, 2010 #8
    Thanks guys, it now makes sense. I keep getting all the notations mixed up.

    I appreciate all of your help

    M
     
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