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Math Help - Approximate Fundamental Theorem

  1. #1
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    Approximate Fundamental Theorem

    I am looking for an approximate form of the Fundamental Theorem that looks like: let f be C^1 on [a,b]; for every \epsilon > 0, whenever P is a partition of [a,b] s.t. ||P|| < \cdots (some term depending on \epsilon), |\int_a^bf'(x)dx - (f(b)-f(a))| < \epsilon.

    -------------

    Let P be a partition of [a,b], P : a=x_0 \leq x_1^* \leq x_1 \leq x_2^* \leq \cdots \leq x_n = b, and write ||P|| := \max\ (x_i-x_{i-1})_{i=1}^n for its mesh size. \sum_P f := \sum_{i=1}^n f(x_i^*) (x_i - x_{i-1}) denotes a Riemann sum of f on this partition.

    Now:

    \left| \int_a^b f'(x) dx - \left( f(b) - f(a) \right) \right| \leq \left| \int_a^b f'(x) dx - \sum_P f' \right| + \left| \sum_P f' - \left( f(b)-f(a) \right) \right|

    The first term can be made \leq \epsilon by chusing P fine enough (this is the definition of integrability), so I ignore it and look at the second term. I can make it vanish by chusing the "right" x_i^* but suppose I cannot chuse these and can control only the mesh size of P.

    Using the formula f(x_i)-f(x_{i-1})=f'(x_i^*)(x_i - x_{i-1}) + \mathrm{error}(x_i^*;x_i-x_{i-1}) where the error term is o(x_i-x_{i-1}) as x_i - x_{i-1} \to 0...

    \sum_P f' - \left( f(b)-f(a) \right)
    =\sum_{i=1}^n f'(x_i^*) (x_i - x_{i-1}) - \sum_{i=1}^n \left( f(x_i) - f(x_{i-1}) \right)
    =\sum_{i=1}^n \mathrm{error}(x_i^*;x_i-x_{i-1})

    I want to show this last sum can be made arbitrarily small by shrinking ||P||. I do not know how to control the error terms. Can anyone help?
    Last edited by maddas; February 22nd 2010 at 11:21 AM.
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