3 ms·
"That change c never truly goes away, but it is so small that it doesn't really matter: if you make c small enough, the derivative is basically 8. So we're jus
by Patient0 8y ago
"That change c never truly goes away, but it is so small that it doesn't really matter: if you make
c small enough, the derivative is basically 8. So we're just going to pretend that all the
cs are so small they have become zero."
My first calculus teacher taught us derivatives in a similar way - but I have to say that, for me, this imprecise language confused me. When and why is it OK to pretend that c is zero!?
Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". But that's not what it means. Again, this is imprecise language that confuses more than it explains. What does "as small as possible" even mean? If we want to make it as small as possible, why not set it to 0!? How small is small enough?
I think somewhere in this article there should be a precise definition of what a limit is, using epsilon and delta.
It should be saying something using the words "arbitrarily close" and "sufficiently small".
Something like : "we can make the approximation on the left arbitrarily close to the expression on the right by choosing any value of c sufficiently close to 0, even though the expression might be undefined when c=0. Epsilon: you tell me how close you want the approximation to be. Delta: I tell you how small c needs to be".
There used to be a great website that was precise but also informal: karlscalculus.org - but sadly it appears to be down now.
- foxes 8y agoI think it was Weierstrass who first gave a proper definition in terms of epsilon / delta (e,d). To say it more precisely: We say that the limit lim x -> c f(x) = L, if for all e > 0, there exists a d, such that for all x, if | x - c | < d, then | f(x) - L | < e.
- Patient0 8y agoThe history seems to be summarized well in this wikipedia entry: https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...
- marsrover 8y ago> Further on, the official definition using a limit claims that the "c->0" means "make c as small as possible". Yeah.. I’m pretty sure the official definition of a limit says nothing like that. Any definition of a limit that I’ve ever seen (official or not) has been careful to note that you cannot set c to 0, but only approach it (whether in mathematical terms or English).
- Patient0 8y agoYes, that's my point. The article is giving an incorrect definition of what a limit is. The target audience does not know what a limit is or how it is defined. The use of imprecise and incorrect language will confuse these people more than it will help.
- noelwelsh 8y agoThe rigorous definition of infinitesimals is developed in non-standard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis IIRC you define e such that e^2 = 0 and then normal algebra just works. There is background work needed to precisely concepts but that's the gist of it (again, IIRC).
- Patient0 8y agoFound a link on the Web archive: https://web.archive.org/web/20080117115912/http://www.karlscalculus.org:80/calc1.html#s1_4 https://web.archive.org/web/20080117115912/http://www.karlsc...