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Topic: Matheology � 258
Replies: 53   Last Post: May 11, 2013 10:07 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology 258
Posted: May 9, 2013 4:38 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 9 Mai, 21:36, Virgil <vir...@ligriv.com> wrote:
> > In article
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > For all n: f(n) = 1 , lim_n-->oo f(n) = 1
> > > > This is required for correctly calculating differential quotients in
> > > > analysis. (Just this morning I explained that in class.)

> >
> > How is
> >    "For all n: f(n) = 1 , lim_n-->oo f(n) = 1"
> > needed  to calculate  the differential quotient of f(x) = e^x at x = pi?

>
> It is necessary to calculate the differential quotient of functions
> like f(x) = ax + b.

It is not at all necessary, as many calculus texts manage quite nicely
to find the differential quotients of such linear functions without it.
> >
> > It can ONLY be of any use in correctly calculating differential
> > quotients in the rare cases in which the difference quotients at a point
> > are all equal regardless of the differences in x.

>
> So it is. But even these "rare cases" belong to mathematics and have
> to be solved correctly.

I do not know of any such rare cases that cannot be solved much more
simply by ordinary difference quotiens,
> >
> > I.e., when the delta-y over delta-x ratio is constant, as in linear
> > functions.
> >
> > So apparently WM never gets anywhere beyond the derivatives of linear
> > functions.

>
> That is not an admissible conclussion.

Why not? Your sequential arguments are like using sledgehammers to crack
eggs.

Give y = f(x) = a*x + b for all real x, and son point x = x_0
The difference quotient from x = x_0 to x = x_0 + h, for any h =\= 0 is
[f(x_0 + h) - f(x_0)]/[(x_0 + h) - (x_0)] =
[a*(x_0 + h) + b - a*x_0 - b]/h =
[a*h]/h =
a.
Since this is true for all non-zero real h,
it is also the limit as h -> 0.
No sequences needed.

Is that too difficult for your students, or just too difficult for you?
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