Re: Gabriel's Theorem - what I have learned thus far.

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/19/05 

Date: Sat, 19 Feb 2005 07:46:48 -0600

On 18 Feb 2005 08:06:32 -0800, "Jason" <logamath@yahoo.com> wrote:

>His theorems are found at:
>
> http://www.geocities.com/john_gabriel
>
>for those who are looking at this thread the first time.
>
>I have to agree that calling it the Average 'Tangent' theorem is a bit
>of a curveball, so perhaps we can understand it better as the Average
>Derivative Theorem.
>
>Some facts I have established:
>
>- Not a single contributor has been able to show me one reference of
>Gabriel's theorem being stated before Gabriel in any shape or form.
>About the closest resemblance is the Average Value Theorem and these
>are not the same.
>
>- Prof. Ullrich claims that for the definition of the derivative,
>differentiability is only required at a point and not the entire
>interval. However, Gabriel requires the entire interval to be
>differentiable except perhaps for the upper limit, i.e. x+w

Indeed he does - he requires this in the _definition_ of the
derivative, which makes the definition incoherent. Circular.
Not even wrong.

>- I claim that neither differentiability nor continuity are
>well-defined.

Giggle. They're not well-defined by anything you've told us
about Gabriel's math, no. There's no problem with the
standard definitions, however.

>- If the mvt requires differentiability over the interval (except at
>x+w), then so does the ftoc and Gabriel's ATT.
>
>MathWorld (Wolfram) states the following for the mvt:
>
>"Mean-Value Theorem
>Let f(x) be differentiable on the open interval (a, b) and continuous
>on the closed interval [a, b]. Then there is at least one point c in
>(a, b) such that ..."
>
>This appears to be incorrect

Only to you. It is in fact _correct_.

>because it must also be differentiable at
>'a'. Gabriel's theorem confirms this.

If so then that's proof that Gabriel's theorem is wrong.

>- So we have Gabriel's Average Derivative Theorem (ATT):
>
> 1 n-1 ws
> Average Derivative = Lim - Sigma f'(a + -- )
> n->Infinity n s=0 n
>
>and
>
> f(a+w) - f(a)
> Average Derivative = -------------
> w
>
>and
>
> 1 a+w
> Average Derivative = - Integral f'(a) dx
> w a
>
>b = a + w
>
>So,
>
> f(b) - f(a)
> Average Derivative = -----------
> b - a

************************

David C. Ullrich