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$\begingroup$ Generalizing differentiation and integration lead us to see that they differ as left- of right- sided inverses. One side generalizes to Lebesgue differentiation theorem, on the other side generalizes to bounded variation and absolute continuity. $\endgroup$

user2529
– user2529

2010-04-25 14:41:30 +00:00
Commented Apr 25, 2010 at 14:41
26

$\begingroup$ I disagree with this: I think it is a fantastic heuristic, indeed the single most important heuristic of first year calculus. To argue against it is mostly to say "I don't like heuristics", it seems to me. $\endgroup$

Pete L. Clark
– Pete L. Clark

2012-08-02 08:21:52 +00:00
Commented Aug 2, 2012 at 8:21
4

$\begingroup$ Well, I didn't really have first year calculus in mind when I wrote this answer. Sure, it's a great heuristic at that level, but it's not so great later on. I guess the lesson here is that you can't really talk about a heuristic without talking about the context as well. My answer was less about the heuristic being bad, and more about it being bad to cling onto a heuristic as you transition into territory where it ceases to be so fantastically useful. $\endgroup$

Zach Conn
– Zach Conn

2012-11-18 05:21:43 +00:00
Commented Nov 18, 2012 at 5:21
$\begingroup$ It sounds like Zach is saying that some unlearning has to take place if they go on in math. That's true, but at the same time there are so many viewpoints on what differentiation "is" (see for example Thurston's list in the beginning of his Proofs and Progress paper) that it's hard to get more than just a few across in a semester or even year-long course, so I suppose some unlearning will have to take place anyway. The inversion heuristic has an advantage of being memorable. $\endgroup$

Todd Trimble
– Todd Trimble

2018-10-27 11:29:48 +00:00
Commented Oct 27, 2018 at 11:29

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