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Math Help - Fundamental Theorem of Calc

  1. #1
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    Fundamental Theorem of Calc

    I'm attempting to re-familiarize myself with analysis by trying to work through this pdf. I know how to prove the Fundamental Theorem, and it's corollary, but not with the information provided to me to use to solve it in this booklet. Can anyone figure this out using only the information provided previous to it.

    It's presented as Theorem 49 on page 22 (labeled as 17 on the top of the page) of this pdf and only theorems from this are supposed to be used. That does include MVT, which I used for a different proof of this, but one that doesn't coincide with this booklet.

    http://www.jiblm.org/downloads/jiblm...S/V090212S.pdf
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  2. #2
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    Quote Originally Posted by bweiland View Post
    I'm attempting to re-familiarize myself with analysis by trying to work through this pdf. I know how to prove the Fundamental Theorem, and it's corollary, but not with the information provided to me to use to solve it in this booklet. Can anyone figure this out using only the information provided previous to it.

    It's presented as Theorem 49 on page 22 (labeled as 17 on the top of the page) of this pdf and only theorems from this are supposed to be used. That does include MVT, which I used for a different proof of this, but one that doesn't coincide with this booklet.

    http://www.jiblm.org/downloads/jiblm...S/V090212S.pdf
    Theorem 49 follows directly from Theorem 48 by a slight change of notation (write: f' instead of f, and f instead of F), and because

    \int_a^bf' =\int_x^b f'+\int_a^x f'=\int_x^b f'-\int_x^a f'=f(b)-f(a)
    Where we take x to be from [a;b]. Then the first equals sign holds, because we can split [a;b] into two intervalls of integration, [a;x] and [x;b]. Then we exchange the lower and the upper limit of the second integral, which changes its sign, finally we apply Theorem 48.
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  3. #3
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    Thank you for that. That was really easy. I had done a proof I remembered from my undergrad using MVT and a couple convergences and then realized I didn't have those to work with.
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