Re: John Gabriel's Average Tangent Theorem

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/16/05 

Date: Wed, 16 Feb 2005 08:10:31 -0600

On 16 Feb 2005 02:43:15 -0800, "Jason" <logamath@yahoo.com> wrote:

>> No, please state the definition as he has it - the one that
>> starts "The DERIVATIVE of f(x) (sic) denoted by f'(x) is the
>> gradient ..." Please state what he has exactly and thoroughly.
>
>Okay. He states:
>
>The DERIVATIVE of f(x) denoted by f'(x) is the gradient of the tangent
>to the curve f(x) at the point [x; f(x)] provided f(x) is
>differentiable over [x;x+w].
>
>So what's your problem with this?

Is this his _definition_ of the derivative, as oppposed to just
a statement about derivatives? If it's a statement about derivatives
then it's wrong because a few words are used incorrectly, and it's
also curiously weak. But if it's supposed to be the _definition_
of the derivative, as I gather it is, then it's simply incoherent,
exactly as Wade said/

>He is saying the derivative of the
>function f is the gradient of the tangent to the curve f at the point
>[x; f(x)] provided f is differentiable over [x;x+w].
>
>I see no problem here whatsoever. Am I missing something?

The smallest problem is the use of the word "gradient". In
fact the derivative of a function is the _slope_ of the
tangent - the tangent is a line, and lines have slopes,
lines do not have gradients.

A big problem is defining the derivative as the slope
("gradient") of the tangent. The usual definition of
tangent line is the line passing through the point
(x,f(x)) with slope equal to f'(x). Unless he has
previously given a coherent definition of "tangent"
that does not mention derivatives the definition
is circular.

The "provided f is differentiable" is also circular,
since "f is differentiable" means "f has a derivative".
This makes the definition incoherent: It says that
_if_ f has a derivative then the derivative is so
and so. If we don't know what the word "derivative"
means then this "definition" does not tell us what
it means, because we cannot understand the definition
until _after_ we know what the word "derivative" means.

(Another less important point is the "provided f is
differentiable over [x;x+w]". If this were a statement
about derivatives, as opposed to being the _definition_,
then it would be sort of correct, but requiring that
f be differentiable over [x;x+w] is very strange,
since all that is needed is that f be differentiable
at x.)

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

David C. Ullrich

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