Making epsilon-delta proofs not just precalculus

Question

In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it reduces the problems to "do you have the pre-calculus algebra to solve the question?" and all the questions are basically the same template-like format but with different numbers.

I'm not sure if viewers would know why the definition is the way it is or why the proof is structured that way.

When teaching $\epsilon$-$\delta$ proofs, what are you strategies to make sure students truly understand the concepts (beyond mere calculations)? Of course you can't completely escape computation and manipulation, but I don't want my students to be computers who can only do problems they've seen before; I want them to be free-thinkers who can expand beyond what I've shown them.

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My answer/suggestion:

To give a concrete example of what I'm talking about, in my own video at 25:25, I made a part that shows some creativity in picking the $\epsilon$-$\delta$ relationship for any line of $mx+b$.

Before that section, I proved that you could have $\frac{\epsilon}{\left|m\right|}=\delta$ but then you'd need to do another case for $m=0$. But if $\delta=\frac{\epsilon}{\left|m\right|+1}$ then you wouldn't have to do 2 cases.

To me, something like that shows a little bit of creative thinking can cut down on computation and show that one recognizes that the $\delta$ found in the stratch work is just the maximum value, and that smaller (positive) ones can work.

What are some of the creative ways you make $\epsilon$-$\delta$ proofs more than just computation?

Answer 1

I think you should consider that for most calculus students, they will have a limited exposure to e-d and then move on to massive amounts of other calculations. And that will serve the vast amount of physicists, chemists, engineers, geologists, etc. just fine. Even from a TIME usage perspective, going deeper and more intuitive into e-d may be a waste of time.

For what it's worth, I'm an experienced person in STEM (not math) and have never regretted not having a deeper feeling for e-d. It didn't hold me back. As far as I remember, my intuitive concept of it was that one gets smaller and smaller and another gets smaller and smaller and then there's an inequality or something. And you can sort of grok how that would pin the beast in and not let it escape the corral. But still a butt-load of algebra to slog through. (Good enough for concept. And then rock the algebra.)

But really for the purpose of the rest of the course, it wouldn't even matter if you had blown off the e-d proof stuff. I mean when you are off finding turning points of polynomials or volumes of rotation, who cares about e-d?

And I would not "dis" the value of practicing algebra. There will be other areas where tedious algebra is needed (series solution to diffyQs for instance). Some facility with tracking a multi-step calculation down the pages is a good thing to develop. Heck, solidifying algebra was like a side benefit of calculus itself (Richard Feynman makes the same point). And the ability to do detailed calculations is a frequent need in chemistry (equilibrium, stoichiomentry) or engineering (hydraulic circuits, power cycles, electrical circuits) or physics (all over the place).

Furthermore, it's sometimes useful to learn something mechanically and later develop the intuitions. (Or maybe at least DEEPEN the initial intuitions, later.) E.g. quantum mechanics. Although this point does not completely apply since I don't think it is a useful allocation time for students to go deeper into e-d AT THIS STAGE in their courses.