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Math Help - second fundamental calculus theorem

  1. #1
    Boo
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    second fundamental calculus theorem

    Dear All!
    Proof ofthe second fundamental theorem

    Please, how do we get in the sixth line that f(x) is right the one f(t), also, the sam f function as we have in the lines before?

    In other words, how do we KNOW that (F(x+\Delta x)-F(x))=f(x) The SAME f(x) we have in the right side before?
    I understand that it is an DERIVATIVE, SURE! "An derivative! " But how do we know it is the same FUNCTION as we have f(t) on the right?
    Many thanks!
    Last edited by Boo; February 14th 2013 at 07:04 AM.
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    Re: second fundamental calculus theorem

    I get the error:

    Forbidden

    You don't have permission to access /~djk/18_01/chapter14/proof01.html on this server.

    - Hollywood
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    Re: second fundamental calculus theorem

    It is the "mean value theorem". As long as F is differentiable, there exist c between x and x+ h such that (F(x+h)- F(x))/h= F'(c)= f(c) because F is defined as the anti-derivative of f. As h goes to 0, since c is always between x and x+h, c goes to x and f(c) becomes f(x).
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    Re: second fundamental calculus theorem

    Quote Originally Posted by HallsofIvy View Post
    It is the "mean value theorem". As long as F is differentiable, there exist c between x and x+ h such that (F(x+h)- F(x))/h= F'(c)= f(c) because F is defined as the anti-derivative of f.
    In the link, the fact that F is the anti-derivative of f is what has to be proved. F is defined as the integral of f.
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    Boo
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    Re: second fundamental calculus theorem

    Hello!
    Thank U all very much, but I think we did not understand each other.
    Sure it is clear that F'=f
    My question is:

    how do we know that this f from the left side of the eqation is the SAME f we have on the right side.
    Proof ofthe second fundamental theorem

    The functin on the right side we marked as "f".
    Then, the derivative function on the left side we get we name AGAIN WITH THE SAME NAME! (f)
    hOW can we know that f=f?
    p.s. we could name the derivative on the left g8x) couln't we?
    So, what woudl tell us then that g(x)=f(x)?

    many thanks if someone has nerves for this-and sory if I am too intrusive!
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    Re: second fundamental calculus theorem

    Quote Originally Posted by emakarov View Post
    In the link, the fact that F is the anti-derivative of f is what has to be proved. F is defined as the integral of f.
    Ah! "F" is defined as the area under the curve. In that case, we don't know that " F(x+ \Delta x)- F(x)= f(x)", it is NOT, in general, true. What is true is that \frac{F(x+ \Delta x)- F(x)}{\Delta x}= f(x_0) where x_0 is some number between x and x+ \Delta x. That is true because of the "intermediate value property".

    If we were to draw a horizontal line at (x_M, f(x_M)) where f(x_M) is the maximum of f on the interval from x to x+\Delta x, we get a rectangle, of area f(x_M)\Delta x which completely contains the area under the graph: f(x_M)\Delta x> F(x+\Delta x)- F(x).

    If we were to draw a horizontal line at (x_m f(x_m)) where f(x_m) is the minimum of f on the interval from x to x+ \Delta x, we get a rectangle, of area f(x_m)\Delta x which is completely contained in the area under the graph: F(x+ \Delta x)- F(x)> f(x_m)\Delta x.

    (Using the intermediate value theorem, as well as guarenteeing that those "maximum" and "minimum" values exist requires that f be continuous. More generally, if f is not continuous at some points (but stays finite) we can work between points of discontinuity. The anti-derivative is the same as the 'area under the curve' (Riemann integral) as long as f has only a finite number of points of discontinuity and is bounded on the interval.)

    Putting those together f(x_m)\Delta x< F(x+ \Delta x)- F(x)< f(x_M)\Delta x which, since \Delta x> 0, is the same as f(x_m)< \frac{F(x+ \Delta x)- F(x)}{\Delta x}< f(x_M.

    By the "intermediate value property then, there exist a value of x, x_0, between x_m and x_M, such that f(x_0)= \frac{F(x+ \Delta x)- F(x)}{\Delta x}.

    It is taking the limit as \Delta x goes to 0 so that x+ \Delta x goes to x, and x_0 is "trapped" between them, that gives F'(x)= f(x).
    Last edited by HallsofIvy; February 15th 2013 at 07:43 AM.
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