Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Ralf Bader Posts: 488 Registered: 7/4/05
Re: Matheology 258
Posted: May 9, 2013 5:53 PM

Virgil wrote:

> 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.

Not really. They are the way you go if you unwind the definitions.

> 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?

Well, in your above computation, you still need to take lim_(x->x_0), and
the final a is to be seen as a function with constant value a, for
lim_(x->x_0) a to make sense. And the limit of that function at x_0 is taken
through sequences, and that boils down to something remotely resembling
Mückenheim's crap. One time at least you (resp. students) should see that
and how the machinery works before starting to be sloppy. Or imagine
programming a CAS system doing this stuff. You will need to distinguish the
number a from the constant function with value a, and probably you will
need to think a little bit to get things straight at this point.
Mückenheim-style crap like
For all n: f(n) = 1 , lim_n-->oo f(n) = 1
is necessary to calculate the differential quotient of functions
like f(x) = ax + b.
will make a mess out of your CAS. It does make some sense that the f(.)
notation is used for functions, and the index notation (x_i) is used for
sequences, but Mückenheim proudly screws everything up.

But this is just the first level of current Mückenheim crap. This strawman
about differentiating linear functions was brought in to start a new round
of Mückenheim's idiotic digit shuffling procedure allegedly proving that in
what Mückenheim believes to be set theory, "For all n: f(n) = 1 ,
lim_n-->oo f(n) = 0".