Talk:Difference quotient

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Delta

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

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

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)[reply]

Citation and sources

Regarding the requests for sources, there are several given, including for Newton's attribution, which is given in the derivative and (now added) quotient rule articles. ~Kaimbridge~15:39, 25 July 2007 (UTC)[reply]

It was a citation request, not a request for links to webpages or citations that may or may not be on other wikipedia articles. (and I looked at the web page sources - absolutely none of them said "this method is attributed to Newton" or the ilk) If it is attributed to Newton somewhere, cite it. Here. I will restore the citation requests you deleted. 198.163.150.8 (talk) 23:53, 18 January 2008 (UTC)[reply]

Do either of these [1][2]count? ~Kaimbridge~01:27, 19 January 2008 (UTC)[reply]

The second link you provided is broken (i think you might have cut and paste erroneously). The first link, which is a fine link with a nice summary, may not fit the wikipedia standards for reference, since it is only a webpage of some undergraduate at Bath. (i.e. "some random webpage") Do you have a more authoritative source? I could make a webpage that says it came from Leibniz, and that should carry no more or less authority than a random undergraduate's. It probably came from Newton, but we should be formally correct about it. 198.163.150.8 (talk) 23:41, 24 January 2008 (UTC)[reply]

Similar quotients were used by Fermat and Barrow before Newton, and functional notation was missing in Newton's time anyway, so Newton would not have written quotients this way. (See https://www.brainm.com/software/pubs/math/471fermatMT.pdf, https://wiki.riteme.site/wiki/Isaac_Barrow#Calculating_tangents, for example.) The question is not whether Newton "invented" these quotients—he did not—but whether the term "Newton quotient" is standard terminology, where it appears or first appeared, whether (commonly used) competing terminology exists or not, and whether there is a trace of this debate "in the literature". 2A02:1210:5A6B:9600:9839:D2D:41D1:5B1C (talk) 10:15, 1 October 2022 (UTC)[reply]

The second link I think was on a professor's page, so that one was iffy to begin with (both the cached and original PDF links did work when I posted them). As for finding a more authoritative source (or accepting it as de facto), you might want to post that citation request to the derivative and quotient rule articles, as they are more heavily edited by (much) more qualified people who may know! P=) ~Kaimbridge~15:04, 25 January 2008 (UTC)[reply]

I seem to remember that the standard Calculus textbooks call it "Newton's Quotient", e.g., George B. Thomas. But I do not have my copy handy right now. Certainly it is in Serge Lang, Calculus, and one finds it on the web on google books on page 313 of The Concise Oxford Dictionary of Mathematics

By Christopher Clapham, James Nicholson.   This does not say Newton invented it, just that it is called the NEwton quotient.98.109.232.157 (talk) 14:02, 4 September 2014 (UTC)[reply]

Terminology usage: Point

I was confused by the usage of the word "point". It took some rereading to understand that whatever is meant by the word in the article, it is not the geometrical sense that I learned in school. The thing that the article calls a point is what I might call a value of a single variable (which would correspond to a geometrical point in a 1-dimensional geometry), commonly the x-coordinate as eventually mentioned in the article. 207.189.230.42 (talk) 21:40, 17 July 2008 (UTC)[reply]

Yes many places seem to use the word "point" when the correct term is just "real number". 2A02:1210:5A6B:9600:9839:D2D:41D1:5B1C (talk) 09:48, 1 October 2022 (UTC)[reply]

tried to typeset

Tried to typeset the forward, central, and backward differences, but for some odd reason the forward difference is duplicating and rendering in small font, for no apparent reason that I can discern, I've checked and rechecked the code, seems good but sometimes I ain't no good at being a TeX'an.:

Forward difference:

$\Delta F(P)=F(P+\Delta P)-F(P)$ $\Delta F(P)=F(P+\Delta P)-F(P)$

Central difference:

$\delta F(P)=F(P+{\frac {1}{2}}\Delta P)-F(P-{\frac {1}{2}}\Delta P);$

Backward difference:

$\nabla F(P)=F(P)-F(P-\Delta P).$

POV check

Infinitesimals

I see this page was written from a non-standard analysis point of view. I've added the more standard viewpoint at the beginning based on typical college textbooks, but I think the page needs more work throughout. Some1Redirects4You (talk) 09:44, 26 April 2015 (UTC)[reply]

I agree that this article is a mess. I had earlier proposed deleting it and replacing it with a redirect to finite difference (which should be expanded if necessary). cffk (talk) 14:39, 26 April 2015 (UTC)[reply]

The difference quotient(s) is/are probably notable by itself/themselves, but it seems some of the content of this page (gratuitously?) goes into other things. I'm not sure why the Central difference or the Backward difference are introduced in the lead, for example. They don't appear to play a role in the sequel. Also, I'm not sure from where to source the section of Higher-order difference quotients, which would be germane to this page. It appears the page was largely written (circa 2005) as a personal essay/expose rather than trying to follow some easily identifiable sources. Some1Redirects4You (talk) 11:41, 27 April 2015 (UTC)[reply]

Now that the article contains standard analysis, and non-standard analysis viewpoints, I find it very confusing. Since the difference quotient is a definition of the derivative that uses limits, I think the standard analysis view is more appropriate. Russphelan (talk) 00:26, 21 May 2015 (UTC)[reply]

Or, the two approaches should be clearly separated in the article. Otherwise, the meaning of symbols like dx, and dy are ambiguous. In modern real analysis, they can be of finite size. (see differential of a function for information supporting this) In non-standard analysis, they are infinitesimal. Russphelan (talk) 01:51, 21 May 2015 (UTC)[reply]

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