Results 1 to 1 of 1

Thread: Approximate Fundamental Theorem

  1. #1
    Senior Member
    Joined
    Feb 2010
    Posts
    422

    Approximate Fundamental Theorem

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

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

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

    Now:

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

    The first term can be made $\displaystyle \leq \epsilon$ by chusing $\displaystyle P$ 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" $\displaystyle x_i^*$ but suppose I cannot chuse these and can control only the mesh size of $\displaystyle P$.

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

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

    I want to show this last sum can be made arbitrarily small by shrinking $\displaystyle ||P||$. I do not know how to control the error terms. Can anyone help?
    Last edited by maddas; Feb 22nd 2010 at 10:21 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fundamental Theorem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: Nov 10th 2009, 06:08 PM
  2. fundamental theorem of cal
    Posted in the Calculus Forum
    Replies: 8
    Last Post: Jan 3rd 2009, 01:40 PM
  3. fundamental theorem!!!
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 16th 2008, 12:22 PM
  4. Replies: 2
    Last Post: Jun 14th 2007, 06:35 AM
  5. fundamental theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 24th 2006, 05:44 AM

Search Tags


/mathhelpforum @mathhelpforum