Timeline for Interesting but very easy epsilon-delta problems?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| May 8 at 8:44 | history | edited | Mikhail Katz | edited tags | |
| Feb 23, 2018 at 18:55 | answer | added | Acccumulation | timeline score: -2 | |
| Feb 23, 2018 at 17:10 | answer | added | user52817 | timeline score: 0 | |
| Feb 23, 2018 at 13:48 | history | edited | benblumsmith | CC BY-SA 3.0 | addendum with more ideas |
| Feb 23, 2018 at 9:40 | comment | added | Benoît Kloeckner | Well, for quadratics you could start with a limit at 0. Then, I would rather use the difficulty in finding the $\delta$ in your example as a motivation to prove the general result on limits of products. This is a great opportunity to convince students that general result are great, and that they actually prefer to prove them than to deal with all particular cases by hand. Honestly, I would not want to do your example by hand. | |
| Feb 23, 2018 at 1:24 | vote | accept | benblumsmith | ||
| Feb 22, 2018 at 19:49 | comment | added | James S. Cook | Sometimes making a graph of something you're trying to bound is helpful. Not that it constitutes a "proof", but it is often a helpful guide through the maze of otherwise opaque inequalities.... | |
| Feb 22, 2018 at 19:45 | answer | added | Brendan W. Sullivan | timeline score: 7 | |
| Feb 22, 2018 at 17:07 | comment | added | Santiago Canez | (continued) This way of phrasing such a problem then makes polynomial expressions with higher powers, for instance, clearer to handle. | |
| Feb 22, 2018 at 17:05 | comment | added | Santiago Canez | I think the work you demonstrated misses the point in such problems: using information about $|x-4|$ to bound $|3x^2-48|$. I think it's crucial to phrase it in this way, so that students directly see the relation between the two expressions. Get students to think about manipulating $3x^2-48$ in order to explicitly get $x-4$ appearing. Once they get down to $3|x-4||x+4|$, the entire goal should then be to bound $|x+4|$ by a constant, which they can do nicely by making additional assumptions on how small $|x-4|$ might be. The reason why we take a minimum in the end is simpler to see this way. | |
| Feb 22, 2018 at 12:27 | comment | added | pjs36 | I think any function that’s not linear requires a bit of ingenuity when dealing with the inequality (I think the use of “min” is a tricky thing, personally, and I can’t think of a nonlinear example that doesn’t require it. But, $\epsilon$-$\delta$ is certainly not my speciality). This is probably the first place most students ever manipulate inequalities, instead of just solving them. I think it’s inevitible that the algebra and subtlety of inequalities makes the logic hard to focus on, at first. | |
| Feb 22, 2018 at 1:38 | comment | added | guest | Maybe the quadratics are exactly what they need. Show them the trick but then make them do 5 or 10 or 20 of them. For you just the trick is the interesting part. But if you want to "build their muscles" of algebra manipulation and inequalities, they need to do some drill. I actually improved my algebra a lot when I took calculus and did every HW problem in the book. Same thing can work for Real Analysis. | |
| Feb 22, 2018 at 1:32 | comment | added | benblumsmith | @guest - If nobody answers the question and I don't otherwise find some other good examples, I will certainly use linear examples for them to consolidate the skill with. But these problems are too mechanically similar to each other; the mechanics call attention away from the true underlying logical structure of the problems. That's why I asked the question. | |
| Feb 22, 2018 at 1:10 | comment | added | guest | Also maybe an intermediate step is linear but more complicated like a(bx-c) or the like. | |
| Feb 22, 2018 at 1:08 | comment | added | guest | Just a comment since I don't know main answer. Make sure to give them some drill and homework with things at the same level of what they learned. Not just harder stuff. IOW, for what you mentioned here, some linear problems, not just show them linear in class and give quadratic in HW. After all, even the linear is new to them! So they need some significant drill even in that. Of course you need to get to the more complicated stuff eventually, but build the basics very well first. | |
| Feb 22, 2018 at 0:06 | history | asked | benblumsmith | CC BY-SA 3.0 |