Talk:Difference quotient

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[edit] Delta

Hi Kaimbridge. Wow, that's a lot of work! A couple of remarks.

1. The Delta needs to be empty delta, like this: Δ and not ▲. Same for ∇.

   Ouch! Yeah, okay: I'm blind in one eye and half blind in the other (thus the "one eyed smilies"—"P=)"), so I have my browser settings to white (or at least light colored) text against black/dark background, so I thought "▲" was showing up as white, like in the PNG images (which I see now it doesn't—GROSS!!), plus I couldn't find the upside down "Δ" (not considering "&nabla"). I'll fix them.

2. I don't understand that iota business. There is no iota in calculus. I think it should be zero instead. In usual calculus, there are no infinitesimals.

   Because if you just say "zero", the purists will jump down your throat saying "division by zero is undefined"—I do note when I first introduce it that it is usually expressed as a limit (and note infinitesimal does acknowledge its place in calculus theory), I just think, in the context of this article, it is best to define it as a concrete quantity (be it "iota", "jot" or whatever).

3. Some of those formulas are really big. You could as well say that they follow by analogy to what is above instead of writing them down explicitely. Or maybe some recursion is possible.

   If you notice, it's not as repetitive as it appears: The first order is thoroughly dissected for all three forms (general, derivative and finite), but for the second and third orders (and you need to do it to the third order for the pattern to be apparent), there are different and distinct focuses: General—Leibniz regression/breakdown (factoring?); derivative—derivative/function (F->G->H->I) relationships; finite/divided difference—average derivative usage and regression. You might have a better understanding, now, why I was making the fuss about f vs. G for F' in Fundamental theorem of calculus—or maybe not! P=)

4. Why do you put accents on things, like Ń? Maybe you just need to use a new variable?

   I purposely chose "ń", not as an "accent", but as a "prime"/derivative (i.e., the "nth" derivative); likewise, "ã" in P_ã denotes a "smoothing" average (and I use "P_(tn)" in the summation to distinguish "P₍₁₎" from "P₁", two different points).

5. Wonder what you think. I will help with some work on this article later. Cheers, Oleg Alexandrov (talk) 16:23, 7 December 2005 (UTC)

   Like all other articles, I expect people will pop in and tweak it here and there....I just hope the presentation retains a concrete, numerical focus (the separate derivative, divided differences and MVT articles delve into the more abstract, analytical aspects).

   (BTW, sorry about messing up your "#" order/list, but I figured it would be better to answer point-by-point P=)

~Kaimbridge~ 19:50, 7 December 2005 (UTC)

I see. Kaimbridge, one thing I have to say is that you have to use standard math notation, rather than the notation you feel comfortable with. Otherwise people will not understand what you are trying to say. For example, nobody uses Ń to denote the N-th derivative, people use f^(N) for that.

I still don't agree with the iota business; there is no division by zero, that divided formula is only used when the quotent goes to zero, but never actually reaches it. Oleg Alexandrov (talk) 01:07, 8 December 2005 (UTC)