Questions tagged [infinitesimals]

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37 votes
7 answers
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A calculus book that uses differentials?

All introductory calculus books that I have seen spend most of their chapters on differential calculus talking about derivatives, with at most a short section defining differentials as $dy = f'(x) \, ...
Mike Shulman's user avatar
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26 votes
5 answers
2k views

Should we "program" calculus students, like the physicists seem to want us to?

If it is true that we first learn formalism...how to do things that we don't understand, should we regard teaching students mathematics as programming dumb machines with formal rules (to the greatest ...
Jon Bannon's user avatar
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25 votes
6 answers
2k views

Practical experience with teaching differentials in freshman calc?

There is a well known essay by Dray and Manogue which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of ...
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22 votes
8 answers
951 views

Non-Rigorous Use of Differentials

Consider the following example of working "directly" with differentials. One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc ...
Austin Mohr's user avatar
22 votes
3 answers
1k views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
Mikhail Katz's user avatar
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21 votes
4 answers
1k views

Evaluating the reception of (epsilon, delta) definitions

Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians ...
Mikhail Katz's user avatar
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16 votes
7 answers
5k views

Why don’t we teach a topological view of continuity instead of epsilon-delta?

I would like a critique of this approach to teaching continuity to calculus 1 students. Show them that for an increasing function on (a,b) we have that (a,b) is contained in the set of solutions to $...
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14 votes
4 answers
728 views

Is there research for or against such an approach in teaching calculus?

Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914): CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS The preliminary terror, which chokes off most fifth-form boys from ...
Alecos Papadopoulos's user avatar
13 votes
7 answers
2k views

When should we get into limits in introductory calculus courses?

All of the calculus textbooks I've used (teaching at community colleges) start with the first chapter covering limits. (Perhaps after a review chapter.) I think this order is wrong. Historically, ...
Sue VanHattum's user avatar
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13 votes
3 answers
525 views

Teaching Infinitesmals and Non-Standard Analysis

This question is asked from a self-teacher standpoint(I am currently trying to learn more about non-standard analysis on my own), but I'd think it could be applicable to educators also. What are good ...
ruler501's user avatar
  • 530
9 votes
2 answers
683 views

The 'epsilon-delta' method for teaching limits

Weierstrass' method for handling limits with the epsilon and delta symbols is very useful for rigorous analysis of math but it is terrible in terms of any intuitive approach to limits. There are are ...
201044's user avatar
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8 votes
2 answers
488 views

Learning math through fun rather than rote learning

Is it easier to remember something if it is expressed in a funny and/or fascinating way rather than by learning through repetitious exercises that hopefully instill the necessary understanding ? The ...
user128932's user avatar
5 votes
4 answers
1k views

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

Nonstandard calculus is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, ...
Christopher King's user avatar
5 votes
2 answers
307 views

Is it to the students' advantage to learn the language of infinitesimals?

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
Mikhail Katz's user avatar
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