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

From: Jason (logamath_at_yahoo.com)
Date: 02/18/05 

Date: 18 Feb 2005 08:06:32 -0800

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

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

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

- 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

Relevant Pages

  • Gabriels Theorem
    ... The ATT if true can be used to ... prove the mvt and ftoc and several others. ... Gabriel's theorem being stated before Gabriel in any shape or form. ... differentiability is only required at a point and not the entire ...
    (sci.math)
  • Re: Mean Value Theorem for Integrals
    ... see that he's talking about proving the MVT for integration _using_ the ... MVT for derivatives, so it is certainly not obvious that he is _not_ ... weaker condition than differentiability, you can't use anything ... calc student it is sufficient for them to simply master their ...
    (sci.math)
  • Re: Mean Value Theorem for Integrals
    ... see that he's talking about proving the MVT for integration _using_ the ... MVT for derivatives, so it is certainly not obvious that he is _not_ ... weaker condition than differentiability, you can't use anything ... calc student it is sufficient for them to simply master their ...
    (sci.math)
  • Re: Attempt epsilon-delta
    ... >Here is my attempt at an elementary epsilon-delta proof. ... >I used MVT which requires differentiability. ... How can I do the proof with just the continuity of ...
    (sci.math)