Questions tagged [infinitesimals]
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16 questions
2 votes
3 answers
571 views
Engineering applications of epsilon-delta?
Epsilon-delta definitions are obviously better than infinitesimal-based definitions because of tradition dating from the heroic era of Weierstrass's disciples. On the other hand, in an old comment, ...
- 2,500
17 votes
8 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 ...
user22312
7 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, ...
- 336
8 votes
3 answers
1k views
Teaching Calculus Less Formally
I'm wondering if anyone knows of calculus books or other work towards teaching calculus in a less mathematically rigorous way. I'm thinking mostly of American-style college level calculus courses ...
- 11.7k
6 votes
2 answers
361 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 - ...
- 2,500
29 votes
5 answers
3k 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 ...
- 6,213
14 votes
4 answers
851 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 ...
- 1,572
10 votes
2 answers
751 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 ...
- 275
22 votes
8 answers
1k 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 ...
- 596
23 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 ...
- 2,500
14 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, ...
- 22k
8 votes
2 answers
526 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 ...
- 479
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 ...
user507
22 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 ...
- 2,500
13 votes
3 answers
572 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 ...
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