Re: Isn't it the limit?

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

On Sun, 30 Mar 1997, al clark wrote:

> Would it be adequate to point out that in one case, the error introduced 
> is an infinitesmal fraction of the (infinitesmal) area of the strip, but 
> that in the "triangle" that isn't the case. In choosing patches, you need 
> to be able to show, or at least convince yourself, that the error becomes 
> an infinitesmal fraction of the area of each patch.

That's a very good point to make to students Al. In the problem at hand
the error is a constant percent as the limit is taken, hence this is an
invalid method. However, it doesn't answer the student's concern
(motivated by naive common sense) that "That skinny figure has *got to be*
a triangle bent into a curve along a longitude line from equator to pole.
After all, weren't the longitude lines of its two long sides straight
north-south at all times? And doesn't the base become indistinguishable
from a chord in the limit?" At least some of my students would surely
complain that "I just don't 'see' it." Others would become impatient with
having to sweat so many details, and a bit uneasy that their common sense
had been shown to be unreliable when applied to infinitesimals.

More and more students coast through math and physics courses with poorly
formed or incomplete concepts which happen to work most of the time. Even
when textbooks and instructors make efforts to combat this, so long as a
strategy works in homework and exam questions, most students will not
critically examine it. We don't want to bog students down with esoteric
bits of math, but we don't want them handicapped by a partial or
inadequate understanding which may cause them grief in later courses, or
in their chosen life work. 

	-- Donald

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Dr. Donald E. Simanek                            Office: 717-893-2079
Prof. of Physics                    Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745          CIS: 73147,2166 
Home page: http://www.lhup.edu/~dsimanek            FAX: 717-893-2047
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