Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: paper on teaching calculus



Hi all-
The "modern approach" that I refer to is Robinson's non-standard
analysis. His approach provides a realization of the "infinitesimal"
that is smaller than any number. Such a realization is provided by an
extended number system in which numbers are represented by infinite
arrays. These are not very useful for numerical computations.
Regards,
Jack

Adam was by constitution and proclivity a scientist; I was the same, and
we loved to call ourselves by that great name...Our first memorable
scientific discovery was the law that water and like fluids run downhill,
not up.
Mark Twain, <Extract from Eve's Autobiography>

On Mon, 9 Oct 2000, brian whatcott wrote:

I'm not quite up to finding the eprint URL to
use with Jack's pointer, but on the
topic of an infinitesimal approach to sin and
cos functions, I recall the following code:

1 S = 0
2 C = 1
3 eT = 0
4 Input F
5 Display "Frequency is ", F
6 T = 1/(2*pi*F)
7 dT = T/1000
8 S = S + C*dT
9 C = C - S*dT
10 eT = eT + dT
11 DISPLAY "Sine, Cosine, Elapsed Time ", S,C,eT
12 goto 8

This from memory - input and display are generalised commands;
alter to suit. I was surprised that rounding errors did not
quickly drift the results.
From a concrete viewpoint, this could be a sampled illustration
of two integrators in a feedback loop.

Brian

At 11:03 10/9/00 -0500, Jack Uretsky wrote:
Hi all-
I have just posted a short paper on the Los Alamos e-print
archives that may be of interest to some of you:

math.GM/0010065 [abs, src, ps, other] :

Title: Introductory Calculus from the Viewpoint of Non-Standard
Analysis - Derivative of Sine and
Cosine
Authors: Jack L. Uretsky
Comments: LaTeX; 19 pp, 6 figs
Subj-class: General Mathematics
MSC-class: 00-01; 97-01

This article exemplifies a novel approach to the teaching of
introductory differential calculus using
the modern notion of ``infinitesimal'' as opposed to the traditional
approach using the notion of
``limit''. I illustrate the power of the new approach with a
discussion of the derivatives of the sine
and cosine functions. (45kb)
Regards,
Jack

-------------------------------------------------------




brian whatcott <inet@intellisys.net> Altus OK
Eureka!