Talk:Calculus/Archive 4

power rule for derivatives?

What are everyone's thoughts on including the power rule in the derivatives section?Dannery4 20:24, 16 July 2007 (UTC)

Not so hot on the idea. Really not necessary for a general treatment of calculus.--Cronholm¹⁴⁴ 20:50, 16 July 2007 (UTC)

Perhaps. But, it's a very handy little tool to use and it's exponentially simpler than the difference quotient. Further, there are a lot of technical articles out there, and while I can understand that they are hard for some readers to understand, they are a valuable tool for just as many others, if not more.Dannery4 21:07, 16 July 2007 (UTC)

We do have an article on derivatives and we also have wikibooks. I think we also have a list somewhere that has all of the rules for derivatives. --Cronholm¹⁴⁴ 21:13, 16 July 2007 (UTC)

Calculus is about averages, not limits.

The section on infinitesimals and nonstandard analysis should be removed. It has no sound foundation and does not belong in this article. It should be explained that calculus is "not about limits" but about "averages". the "average" is the reason calculus works, not the limit. The limit itself is a last resort effort to add rigor to the definition of a derivative. It is based on the method of approximation. Netwon was trying to find a method to calculate an average at a point (instantaneous average). Long before Newton, mathematicians were able to find derivatives at a point using the same method as Newton, i.e. numerical differentiation. Newton separated himself by inventing a general method. Although the limit process helps in finding "general derivatives", it adds more confusion. The whole truth of the matter is this - calculus can be used rigorously "without limits". A simple example is the quadratic equation and a slightly more complex example is a polynomial with integral powers. See [[1]] for the more complex example. As for the quadratic equation there are three simple steps: 1) First form a new polynomial equation by taking the intersection of the actual equation to be differentiated and the straight line y=mx+k. 2) Secondly, solve the new polynomial equation and consider the case when the roots are equal. 3) Finally, rearrange the equation making m the subject. The final form is the derivative.

Limits are a concept that make the calculation of derivatives and integrals easier. The classic definition of a derivative:

f'(x) = Lim (h->0) [f(x+h)-f(x)]/h

rests on the fact that the gradient at a point can be approximated by finding averages using other points very close to it. In fact, an instantaneous average is not possible using any two points on a differentiable function f. In the classic definition, the average quotient becomes 0/0 (undefined) when h is zero. Frankly it does not matter whether h is zero or not, because the instantaneous average is equal to the slope of the line that intersects a function f in exactly one point and crosses it nowhere. One is really interested in the change of x/y on the tangent line, NOT the function f as in popular opinion. It just so happens that the instantaneous change of f is equal to the slope of a tangent line at the same point.

The classic definition of a derivative is based on approximations through a difference quotient. The difference quotient is used to approximate the slope of the tangent line. It just so happens that the slope we are interested in occurs when the difference quotient is 0/0, but because we cannot evaluate it, we us the fact that the limit of the difference quotient is equal to the slope of the tangent at the point where the difference quotient is in fact 0/0.76.31.201.0 14:45, 21 July 2007 (UTC)

I can also find the derivative of any polynomial using a simple linear transformation, should that be included too? Seriously though, do you have any references? Secondly, there are many different ways of looking at differentiation and integration. Limits and infinitesimals are just the most common...hence their inclusion. --Cronholm¹⁴⁴ 17:03, 21 July 2007 (UTC)

What are you going on about? As for references, did you notice that I provided a link? There are not many ways of looking at differentiation and integration. These are both average sums and very similar processes rather than inverses. The transformation you mentioned is equivalent to expanding polynomials in powers. This is what Prof. Doverman uses in the link provided. As for the simple example concerning quadratics, it is easy to verify using the 3-step process I provided. I don't know what you want? 'Seriously', you need to read more carefully before you respond. 198.22.21.50 21:01, 21 July 2007 (UTC)

Already the first textbook of calculus, Marquis de l'Hôpital's "Analyse des infiniment petits pour l'intelligence des lignes courbes" (based on the lectures of Johann Bernoulli), contained the famous l'Hôpital's rule for the limits. Thus it's accurate to say that although the modern definition of a limit is a product of nineteenth century mathematics, the notion of a limit has been part of calculus almost from its inception in the late seventeenth century. Arcfrk 22:52, 21 July 2007 (UTC)

Perhaps it might be more constructive, if 76.31.201.0 or 198.22.21.50 think that averages are how calculus is really defined, to give concrete proposals as to how the article might be changed. Wikipedia requires that all material is verifiable: if we have a book that outlines this approach (that is reasonably accessible to the average editor), then it might be possible to include this perspective. Without that, we just have your outline of this conceptional framework to go on, and that is, unfortunately, not grounds to include this material at this stage. Xantharius 02:50, 22 July 2007 (UTC)

I am sorry if I came off as dismissive, but editors come to this article frequently and announce...calculus is really all about X and Y and the article Must include it. These complaints rarely result in improvements to the article. I have yet to be convinced that this is an exception to that rule.--Cronholm¹⁴⁴ 03:05, 22 July 2007 (UTC)

The idea of calculus without limits is an old one -- older than the idea of calculus with limits. Modern treatments of calculus, however, use the definition of the limit to prove the mean value theorems and the various methods for taking derivatives and integrals. This is the standard way of doing calculus, and the many other ways of doing calculus are non-standard. See, for example, any calculus textbook at a major university, with very few if any exceptions, as well as most real and complex analysis books. (They did non-standard analysis at Tulane a few years back, but I don't believe that is the case today.) There is often an almost messianic fervor on the part of those who want to do away with the limit, but the use of the limit is still standard. Rick Norwood 14:10, 22 July 2007 (UTC)

Arcfrk - The notion of a limit has always been associated with calculus because calculus is about infinite averages. In some cases it happens that such averages are reducible so that no limit is required. As an example consider

the derivative of the quadratic equation I mentioned earlier. It is in fact also an infinite average but we can calculate it exactly because it is completely reducible. See this link [[2]] that will show you why the mean value theorem is about an infinite average and how it is possible to calculate this average because it is reducible (to boundary values of a primitive function). L'Hopitals rule uses preestablished theorems (e.g. mean value theorem) and as such does not command any extra respect. As for the fundamental theorem of calculus, it is really nothing except a different form of the mean value theorem. See [[3]]. Xantharius - The concept of average is at the heart of calculus. My first suggestion would be to include this. For references, you could use the following link [[4]]. It is an online book (first 5 chapters only) that is easily accessible. Norwood - The 'limit' is our last resort invention based on approximation because we could not find any other way till recently that helps us calculate a general derivative. Limits are important but nowhere nearly as important as the key concept which is the 'average'. In calculus we are most interested in finding instantaneous averages that work whether or not the idea of limit was ever conceived. 76.31.201.0 14:26, 22 July 2007 (UTC)

I followed the link [[5]] suggested. In it the author uses limits extensively, but in fact all of the main ideas of calculus are simply re-framed in terms of averages, which it is certainly possible to do, but, as far as I am concerned, far less illuminating than the current presentation. Indeed, the ideas of average, I feel, actually disguise what the main points of calculus are: instantaneous rate of change and area (not necessarily averages).

Quite apart from this, there is material in that reference which is just plain wrong, such as a discussion of how Stewart's version of the fundamental theorem of calculus (the standard one) is incorrect. The author of this PDF belives that it is not the case that g is the antiderivative of f if we define

$g(x)=\int _{a}^{x}f(t)\,dt$

since, apparently, the right hand side is equal to g(x) − g(a). He cites f(x) = x² as a counterexample. The author also thinks that the mean value theorem for integrals is useless since it's just a reformulation of other ideas. Indeed, 76.31.201.0 views the mean value theorem as "about an infinite average because it is reducible (to boundary values of a primitive function)" but that is precisely because the second fundamental theorem of calculus applies in this case. Editors may want to have a look for yourselves. Given that several of the remarks made by 76.31.201.0 appear to be based in part on an erroneous document which appears to be pushing a very limited (no pun intended), narrow agenda for a reframing of calculus (not to mention the glaring mistakes), I do not believe that there is anything to be gained from changing the generally accepted (certainly by editors here) approach to calculus.

This article is supposed to be an introductory article about calculus, not about how calculus can be completely reformulated if you think about it in terms of averages. By references I was assuming that the alternative approach would be presented in a published textbook, preferably more than one, not in a single PDF. I do not believe that pursuit of this idea any further is going to improve the article. Xantharius 16:31, 22 July 2007 (UTC)

So you feel the main points of calculus are instantaneous rate of change (IA) and area? Well, if you had read the link correctly and all the material related to it, then you would have realized that both the IA and area are indeed

averages. The IA is an infinite average and area is the product of two averages. Perhaps you should start reading from chapter 1? As for the references to Stewart - I think you are not correct in stating the counter-example is wrong. Given g(x) = x^2, g(x) = X^2 - a^2. This would only be true if a=0. Stewart does not say anything about this. The fundamental theorem of calculus does not contain two parts. Stewart adds nothing by breaking it up. As for your claim that the author is trying to reframe calculus, I don't get the same impression as you. You have a master of science in mathematics? This is very hard to believe given what you reported. I suggest that editors read this for themselves and decide. You have misrepresented what is written in this book and evidently understood much less. 76.31.201.0 18:51, 22 July 2007 (UTC)

76.31.201.0, please do not engage in personal attacks against editors on Wikipedia, on either myself or other editors who contribute. Questioning the credentials of, making personal accusations about his intentions, and generally denegrating an editor all fall into this category, and are neither purposeful nor permitted on Wikipedia. May I respectfully suggest that you stick to discussion of the article rather than the people who edit it.

As for the mathematical content of the remarks (most of which I still find troubling, in particular the erroneous remarks about the fundamental theorems of calculus—yes, there are definitely two of them), I leave further discussion of that for other editors as I do not wish to become embroiled in uncivil behaviour. I would point out, however, that editors who have strong feelings about making changes to the article page should just make them, and then we'll have something to work on. Xantharius 21:03, 22 July 2007 (UTC)

Look, I am sorry. Rereading my last edit, it could have been taken up wrongly and it was. As for making edits to the content of this article, I do not really care to do this. I was merely trying to point out that this article (like quite a few other articles on Wikipedia) either miss the point entirely or contain gross errors. It is too bad because when certain information is run by the same editors, sysops, etc, this is the final undersirable outcome that is slanted and limited to their interpretation. As for the fundamental theorem of calculus, let me assure you that there is only one. Stewart (who may or may not have been the first) is one of the mathematics educators who chose to present the theorem in two parts (evaluation and total change). This is incorrect and misleading. If you look at older textbooks, the fundamental theorem of calculus is stated similarly to Stewart's total change theorem. In fact, the fundamental theorem of calculus is really the mean value theorem in a different form. I am not surprised you find this troubling. I think that many educators do not know this. Finally, the 'average' concept is at the heart of calculus. Newton started off with finite differences (in fact these are in reality infinite averages), developed function approximation polynomials and finally ended up realizing the connection between the two processes of differentiation and integration - both are defined in terms of average sums. 76.31.201.0 22:00, 22 July 2007 (UTC)

Wikipedia is not the proper forum for an attempt to change the views of the mathematics community. Wikipedia reports the views of the mathematics community. Because calculus is so fundamental to all advanced mathematics, the ways in which it is formulated are and should be resistant to change. The most recent major attempt to change the way calculus is taught was Nonstandard Analysis. This was formulated, rigorously, by a group of mathematicians who had already made their reputation in other areas. Because it did not lead to any new theorems, it has become an area of relatively minor interest today.

If you want to promote your vision that calculus is really all about averages, you need first to publish research in other areas, make a reputation for yourself, gather other mathematicians to your cause, and then create a create a group of well-known mathematicians to work out the details. When that is done, and only then, will it be appropriate to report your work in Wikipedia. Rick Norwood 13:04, 23 July 2007 (UTC)

This is not my work as you presumptiously asserted. As for the views of the mathematics community (whoever you are referring to - I don't know, but suspect it is the Wiki panel of mathematics judges and not the community at large), these do not and should not matter when information is published. Since when did any community have to 'bless' information before it is published? Calculus was not formulated the way you think (i.e. using limits or non-standard analysis) so your argument about resisting change is as old-fashioned as the hills. Besides, no one here (including myself) is advocating any reformulation. That calculus works at all is because it is based on averages, not limits or ill-defined infinitesimals. Wikipedia freely publishes edits by its team of editors who are not the final authority in any community. You publish what you claim are the views of the community at large. Information as important as this is to fundamental understanding is not 'research'. It is only research if you can find something fundamentally wrong with it. If not, then why not publish it? Readers are not forced to believe or disbelieve anything they read on the internet. An astute reader always verifies his/her sources and investigates multiple views of the same subject. You publish so much worthless junk. Why not publish something that could make a great difference in the understanding of why calculus works? No one is forced to accept it and you have a disclaimer. The links I refered to certainly coaxed me to rethink many of the ideas I had accepted for granted. Let me end with this: quite a few of your editors claim the limit is the core of calculus. Nothing could be more misconstrued and plain wrong. In one sentence: The limit is the 'name' given to the definition of a process that is based on approximations. The founders of real analysis did not 'rigorize' anything except to add a formal definition to a process that approximates averages. Did they add any real value? No. Did they make calculus easier to grasp? No. Did any of them explain why it works? No. Did they know that the fundamental theorem and the mean value theorem are different forms of the same theorem? No. I do not advocate reformulation, because although it is possible to use Calculus without limits rigorously in polynomials with integral powers or simple quadratic (perhaps even cubic or quintic) equations, it is still necessary to retain the classic definition when trying to find the form of a general derivative. Making the connection between all the theorems cleared up some of my original confusion. The links I provided contain invaluable information that is worthy of publication and criticism. 76.31.201.0 14:30, 23 July 2007 (UTC)

To address the points you have made above, I would refer to you what Wikipedia is not and remind you that original research is not permitted on Wikipedia (the link defines what original research is). The editors who edit mathematics pages on Wikipedia judge material on the basis of whether it has been published, and whether it is encyclopedic (meaning, does it contribute to a better mathematics article for an encyclopedia). Many of us have views on whether a subject should be formulated in a certain way, but these views are immaterial; all that matters is whether the information in question has seen prior publication (again, usually, printed work, not merely one online source) and whether it contributes in a meaningful manner to an article. The editors are not the final authority on any given matter, but they do make sure that the content is verifiable, accurate, and sound.

Personal views about what a subject should be about, rather than what it appears to be about from the current literature, are therefore not permitted under the current Wikipedia guideliness.

In discussing an article editors are required to be civil with one another, and for the second time, from your side, this is in danger of not being the case. I have asked that the discussion can be kept civil: saying that editors are "presumptious", saying that they "could not be more wrong", accusing them of "publishing so much worthless junk", and generally attacking them in the process of editing is, as I have stated before, not productive, and not permitted on Wikipedia. The suggestions of Rick Norwood about how you should proceed would appear to be your best option, because not one editor, including yourself, has been able to find any mainstream text which supports your view. The material you quote is so different from the mainstream view (quite apart from the mistakes, which there surely are) that I find no merit in including it at this point.

I would therefore firmly ask, again, that you maintain civility. Xantharius 17:37, 23 July 2007 (UTC)

I am going to ignore your latest rantings about civility. Nothing I stated in my last edit was in any way uncivil. Let me remind you that I included two

sources: one from Professor Doverman and another by the author of the Calculus Of Averages. So, the information has been published. As far as most are concerned, an online document is as good as a printed one. Does paper make material sacrosanct?

Does this information contribute to a better understanding of the subject? Of course, yes - it is the reason why calculus works (averages). Regarding personal views - all editors have personal views. And why not express these

views? Just read the thousands of wiki edits and try to tell me the editors did not express personal views. Regarding mistakes in the sources I provided - how arrogant of you to mention these. Does not Wikipedia have thousands of mistakes in its articles? Are you infallible? So far no editor from Wikipedia has been able to find real errors in my sources. My advice to you - Quit writing long dissertations as responses. Address the subject at hand - how and why would this information enhance this article. 76.31.201.0 20:41, 23 July 2007 (UTC)

You sound quite angry at other editors' rejections of your views. Of course you're mad: You feel like nobody else here is treating you and your ideas fairly. You feel like you understand calculus and nobody else here does. And now you feel like people are insulting you for no good reason.

I don't think anyone here but you is qualified to explain this viewpoint on calculus. (Myself, I've never heard about these ideas before.) If you'd like this viewpoint represented in the calculus article, then you should write a clear and careful explanation and post it here so that we can read it. I know that you've already given links to an exposition of these ideas, but you're the only one here who is an expert in them.

Please try to make your explanation good enough that it could be added to the article. That way it will be easier for the rest of us (the non-experts) to understand. 141.211.120.81 21:56, 23 July 2007 (UTC)

I could not agree more. Changing the article itself will allow other editors to understand exactly the changes desired, and will allow them also to make their own edits to these if necessary. Xantharius 22:37, 23 July 2007 (UTC)

Appeal to close the discussion as leading nowhere

There are – literally - hundreds of conventionally published calculus textbooks. A recent search at amazon.com turned up 3309 items, although this no doubt includes many ancillaries and re-editions. An article in wikipedia cannot, and indeed, should not, report on all of them (indeed, some are known to be wrong, in whole or in parts). However, there does exist a conventional framework of caculus as it is being presently taught and understood, and wikipedia editors' job is to present it lucidly and accurately. Rick Norwood has made a very astute remark concerning the "messianic fervor" of those who want to do away with the limit, and this discussion is an excellent illustration. From what I have read so far, it appears that 76.31.201.0 feels that his understanding of calculus has increased after reading through the linked documents – good for him! At the same time, he has trouble transforming his admiration into a form suitable for an encyclopaedic article. Moreover, nothing on this page indicates that he is qualified to make pronouncements concerning what Calculus "is about", as he seems to struggle with fairly basic notions of it. No improvement to the present article seems likely to result from vague pronouncements on what calculus should be, or how a majority of people are "misled" about its true nature. I propose closing this discussion as non-constructive, and engage in productive editing! Arcfrk 23:07, 23 July 2007 (UTC)

Actually I don't have the time to transfer this knowledge into an encyclopedic article. For one thing, I don't know how to use your formula editor and am not an expert at editing using your 'sandbox'. I was hoping Wikipedia might include a short paragraph with a link to the sources. For example, the link to Dovermann (Calculus without limits) shows how to differentiate polynomials in integral powers without any use of limits either directly or indirectly. The other link (calculus of averages) has an example in chapter 3 that shows how the mean value in the mvt is in fact an infinite average. It goes on to show how the fundamental theorem is just a different form of the mean value theorem. Chapter 5 consolidates all the fundamental theorems of calculus (mvt, ftoc and definition of derivative). Including this information can only enhance this article. 76.31.201.0 02:45, 24 July 2007 (UTC)

All you have to do is click edit and type, like you do here. You have maintained a bad attitude throughout your interactions with us here. If you can't be bothered with editing the article, that is fine, but don't expect us to insert anything for you. Especially since no one has voiced support for your proposed changes.--Cronholm¹⁴⁴ 02:56, 24 July 2007 (UTC)

As I said above, nobody here but you is qualified to explain this point of view. If you'd like to work on your version privately, try the Wikipedia:Sandbox. If you want more information about the sandbox, see Wikipedia:About the Sandbox. If you want more information about writing mathematics on Wikipedia, see Wikipedia:Manual of Style (mathematics), and for more information on formulas specifically, see meta:Help:Formula.

Writing an article is tough but rewarding. I'm sure you have lots of good ideas—but you're the only one who can turn them into a finished article! 141.211.120.77 14:46, 24 July 2007 (UTC)

It would appear that there is indeed no need for further discussion: if edits are made, then we'll address those at the time. Otherwise, there's nothing more to say. Xantharius 18:24, 24 July 2007 (UTC)

I am not the author of either of these works and think it would be unfair to the orginal authors if I tried to transcribe the ideas therein into any article, let alone an 'encyclopedic' article. I'll leave it to you the expert edit-warriors. You have much experience in editing and seem to do a lot of it. Ergo, who better than you to make these edits? 198.22.21.50 22:25, 24 July 2007 (UTC)

I have once again removed the sections promoting the use of "averages". Nobody denies that non-standard analysis is possible. But this material is a minority view, and not important enough to appear in this article. If it were well-written (and it is not the obligation of Wikipedians to rewrite poorly written material!) then there might be a place for it in the article on non-standard analysis.

Calculus as averages?

198.22.21.50 has been generous enough to provide the article with his viewpoint that calculus is all about averages. I've moved his explanation to what I think is a more appropriate place in the article: We can't talk about derivatives and integrals until we've introduced them. However, I find his text somewhat confusing. I'm inclined to remove it, but I thought it would be nice to provide him with a chance to respond to my questions first:

1. You seem to make a distinction between "mean value" and "average". For example, at one point you say that "the mean value is an actual average". What is the difference?
2. You define the derivative of a function as the limit of the mean values of the function on small intervals as those small intervals shrink to the size of the point. But doesn't this give the value of the function? Indeed,

$\lim _{h\to 0}{\frac {1}{h}}\int _{x}^{x+h}f(x)\,dx=\lim _{h\to 0}{\frac {F(x+h)-F(x)}{h}}=F'(x)=f(x).$

1. You mention an "average sum theorem" but do not state it. I am not quite sure I understand its statement on web page you linked to, either.
   1. On the left hand side of the equation, the quantity f(x) is independent of n, so $\lim _{n\to \infty }f(x)$ is f(x). Why not write f(x)?
   2. The number p is said to satisfy $p\succ 1$ and $p\in Z$. What is $\succ$? Is it ≥? (It cannot be > because immediately after the statement the author gives as an example application the case p = 1.)
   3. Is Z meant to be Z, the set of integers?
   4. The hypotheses needed for the application to the mean value theorem seem too weak. Don't you need to assume that f is absolutely continuous to ensure that the antiderivative of f' is f? (See Cantor function for an example of a non-absolutely continuous function.)

Since I have indulged you by reading your links, may I ask if you read something I like? I think Walter Rudin's Principles of mathematical analysis is an excellent book. If you'd like to understand my point of view so that we can communicate better, then that book is a very good place to start. 141.211.62.20 15:38, 1 August 2007 (UTC)

1. In my opinion the author does not make a distinction between "mean value" and "average". I think that when he states the mean value is an actual average, he means that it is an average of an infinite sum of values of f over an interval w that is partitioned into infinitely many portions(indivisibles?). So, there is no difference between mean value and average. 2. Your formula appears to be incorrect. Adjust it as follows: lim (h->0) 1/h Sum (x to x+h) f(x)dx = lim (h->0) [f(x+h)-f(x)]/h = f'(x) <> f(x) 3. I think that when p = 1, the average sum theorem produces the statement of the mean value theorem. When p > 1, the average sum theorem produces the definition/value of the function (derivative or otherwise). Z denotes integers as far back as I can remember. 4. Of course f is absolutely continuous. Cantor functions are not differentiable so I do not see how this is relevant. Rudin's book is used in Real Analysis courses the world over - I am aware of its contents. If you have more questions, why don't you contact the author? 76.31.201.0 20:34, 1 August 2007 (UTC)

In point 2, could you please explain why lim (h->0) 1/h Sum (x to x+h) f(x)dx = lim (h->0) [f(x+h)-f(x)]/h ? I assume that you use that Sum (x to x+h) f(x)dx = [f(x+h)-f(x)], but why is this? If the Sum on the left is supposed to be an integral, then you need the antiderivative F on the right-hand side, as 141.211.62.20 claimed.

Furthermore, I could not find "a rigorous method for finding the derivatives of polynomials in integer powers" on page 24 of the manuscript by Dobermann. That page is about logarithms. Could you please check the page number? -- Jitse Niesen (talk) 05:20, 2 August 2007 (UTC)

It seems 141.211.62.20 is correct. The author shows the same formula [B5] on Page 4 [6]. I am now unsure what 141.211 has a problem with? [f(x+h)-f(x)/h] is a 'mean value'(average). Its limit is f'(x) as h->0. As for Dovermann's book, are you looking at the same PDF? Page 24-25 does not discuss logarithms. It shows an example (1.10) of expanding the polynomial f(x)=x^2+5x-2 in powers of (x-2). The result is f(x)=(x-2)^2+9(x-2)+12. Dovermann shows how y(x)=ax^2+bx+c can be expanded to A(x-x_0)^2+B(x-x_0)+C where A=a, B=2ax_0+b and C=ax_0^2+bx_0+c=y(x_0). B is the derivative of this polynomial. If P(x_0) is a polynomial, then L(x_0) is that part of P(x_0) which is a straight line. Dovermann expands a polynomial in powers of (x-x_0) and then uses the fact that P(x_0) = L(x_0) where P is the polynomial to be differentiated and L(x_0) is that part of P which is a straight line. It is clear that the gradient of P at a point x_0 is given by the gradient of L. 76.31.201.0 09:33, 2 August 2007 (UTC)

I understand now that you are talking about the 24th page of the PDF file, not about the page in the book that bears the number 24. That page is indeed about expanding polynomials around a point x_0, which can be used to find the derivative using the definition used in the book (I suppose that the book explains that later). -- Jitse Niesen (talk) 15:16, 3 August 2007 (UTC)

I still disagree with you regarding the Cantor function. It is continuous but not absolutely continuous. The only hypotheses assumed in the reference you gave are continuity, not differentiability.

However, I think I disagree with you more regarding the derivative as an average. Here is an example of what I mean. According to your definition, the following should be the derivative of f(x) = x at x = 0:

$\lim _{h\to 0}{\frac {1}{h}}\int _{0}^{h}x\,dx.$

However, when we evaluate this integral, we get:

$\lim _{h\to 0}{\frac {1}{2h}}x^{2}{\Big \vert }_{0}^{h}=\lim _{h\to 0}{\frac {h^{2}}{2h}}=\lim _{h\to 0}{\frac {h}{2}}=0.$

But the derivative of f(x) = x at x = 0 is 1.

If you're already familiar with Rudin's book, may I suggest another? Royden's Real Analysis is a thorough treatment of some very interesting topics. 141.211.63.35 19:20, 2 August 2007 (UTC)

I think you have misunderstood the source entirely. The derivative is not defined as you assumed. The last example you used is definitely incorrect and out of context. If you look at formula [B5], you will notice that you should be equating f(x) = Lim (h->0) 1/h Sum (0 to h) xdx. Now f(0)=0 and thus [B5] is true. As for the definition of the derivative, I believe the average sum formula refers to the case where p>1. Again, this makes sense.

The derivative is an average - Consider f'(x) = lim (h->0) [f(x+h)-f(x)/h]. What do you think f(x+h)-f(x)/h evaluates to? An average. You hastily removed the sections I edited. Therefore, if you want me to continue to 'enlighten' you, I expect that you place both the sections back into the article. Finally, I wrote nothing about the Cantor function being "continuous or absolutely continuous.". I stated that it is not differentiable, thus it is not (necessarily) continuous. I am not discussing the Cantor function with you because it has no relevance. Why were both sections removed? Are all sources without error? 76.31.201.0 23:03, 2 August 2007 (UTC)

Why do you think f(x+h)-f(x)/h is an average? That does not make sense to me. -- Jitse Niesen (talk) 15:16, 3 August 2007 (UTC)

Ah, I see. I agree that formula [B5] is correct, but it is not the same as what you stated in your comments above. (That's okay, we all make typos.)

I didn't remove the two sections you wrote. Someone else did; that's just part of how Wikipedia works. I agree that it can be frustrating sometimes, and I've gotten into my share of arguments while I've been here. But (and this is really more of a response to your comments below), WP articles aren't controlled by anyone. There are even warnings about that (for example, at WP:OWN). I don't think your viewpoint has been rejected out of prejudice or because we refuse to listen, but because your viewpoint is so distinctive and unique that it leaves the rest of us scratching our heads. It's certainly not like anything I've ever encountered before. (I'm with Jitse Nielsen when he asks why the difference quotient is an average. That doesn't make sense to me, either.)

I think I have a suggestion for another possible way forward: How about you write a thorough explanation of this approach and submit it to a journal? I have in mind a journal like American Mathematical Monthly or Mathematics Magazine (they're always on the lookout for clever stuff.) Once your paper is accepted, we'd get a reference we could cite in this article. How does that sound? 141.211.120.76 19:11, 3 August 2007 (UTC)

Look 141.211, you can do what you like. I cannot submit any papers because the work is not mine. Did you notice that I was careful in providing links to the sources? Why should Wiki articles require the blessing of AMM journal? As for writing a thorough explanation, I cannot do this for reasons I have already explained to you. I do not fully understand all the work. Dovermann's claims are easy to verify - they contain only algebra whereas Gabriel uses limits extensively. As for why the difference quotient is an average, you can look at gabriel's chapter 3. He explains this is much detail. This was mentioned in the link I placed in my original edit. 76.31.201.0 21:00, 4 August 2007 (UTC)

Oh, but this is one of the beauties of those two journals: They accept expository papers, so as long as you properly credit Dovermann and Gabriel, you're okay. I do think you should make an attempt. As I keep saying, your point of view is special and unusual, and you're the only one here who comes close to understanding it. I did look again at Gabriel's chapter 3, and now I understand why you say that the difference quotient is an average: It is the average value of the derivative function f'. What I now fail to understand is why this viewpoint is easier than the viewpoint explained by, say, Rudin.

You also say that you "do not fully understand all the work". Maybe this is why we are having such trouble communicating? If you were go to and study those things, maybe we would be able to communicate better? (May I again recommend Rudin's book? Or if you don't like that, Pugh, Real Mathematical Analysis or Bartle and Sherbert, Introduction to Real Analysis.) 141.211.120.44 20:32, 5 August 2007 (UTC)

So now you 'see' why the derivative is an average? Um. Perhaps you could help Jitse (another Calculus teacher) to understand? None of the sources you mentioned come close to explaining these concepts as well as the sources provided. I don't believe Dovermann or Gabriel have all the answers either, but they do have different approaches and both are worth mentioning. I have tried to contact the authors for more information, but have not received any responses yet. I do not think that submitting expository papers with credits is legal. Furthermore 141.211, it would be better if you quit saying "..you're the only one here who comes close to understanding it..", when what you really mean is: "..the material is such nonsense that it is only comprehensible by a fool..". Neither Dovermann nor Gabriel is a fool. As for Rudin's viewpoint on the derivative, I fail to understand how it is better? Please state Rudin's viewpoint and show me how it is easier to understand. Also show me why you think calculus without limits is not good enough to be part of an encyclopedic article? 76.31.201.0 03:27, 6 August 2007 (UTC)

Yes, I do understand how the derivative can be interpreted as an average: It is the average of f' on a sufficiently small neighborhood. However, I don't see how to take it as a definition: As far as I can tell, this interpretation requires an independent definition of differentiation and integration and the use of the Mean Value Theorem and the Fundamental Theorem of Calculus. This is why I don't understand why it's easier than the usual definition usually credited to Newton and Leibnitz and explained in the textbooks by Rudin, Pugh, Bartle and Sherbert, etc.: A real-valued function f of a real variable is differentiable at x if $\textstyle \lim _{h\to 0}(f(x+h)-f(x))/h$ exists; assuming that the limit exists, then its value is the derivative of f at x.

I do believe that calculus without limits is good enough to be part of an encyclopedic article, and have argued elsewhere on WP that the use of limits in calculus is a historical accident. While that may be overstating the case against limits, I certainly try to avoid them in my daily life.

If you prefer that I not compliment you, then I will cease doing it. 141.211.63.61 17:23, 6 August 2007 (UTC)

141.211.63.61

Well, from your first paragraph I can tell that you probably understand as much as I do. I guessed how you might respond, but I wanted to be sure. I agree that the use of limits in calculus is a historical accident. In fact, the very definition of a limit is really nothing rigorous or robust; it is simply the mathematical equivalent of saying: For every x close to c, a differentiable function f approaches some value f(c). Differentiability requires 'continuity' but the reverse is not true. The limit of the difference quotient is quite different to Weierstrass's so-called rigorous epsilon-delta arguments because it requires that f be defined at c; for otherwise how can you talk at all about f being differentiable at the point c. However, for a plain limit argument, f(c) need not necessarily be defined. If it is, then it gaurantees continuity at that point. Then there are different limit definitions for sequences/series, etc, the which I will not deal with here. Thus, unlike the standard limit definition, i.e. |f(x)-L| < epsilon <=> 0<|x-c|< delta, the limit of the difference quotient requires that f(c) be defined. This notion of differentiability is very sketchy and confusing. Now regarding gabriel's calculus of averages, it does not require that differentiation or integration be defined first. In fact, it starts off in Chapter 1 with the concept of area and develops into the integral. Later, in chapter 2, it discusses how the derivative was discovered. Chapter 3 makes the connection between a finite difference and a derivative based on the limit of a difference quotient. Chapter 4 demonstrates two very interesting finite difference formulas that directly link the finite difference quotient with its analog derivate. The one formula states this in terms of derivatives and the other in terms of integrals that are easier to work with. I have 'never' seen these formulas in any publication other than gabriel's. For many years I wondered how the derivative was connected to its ancestor - the finite difference/average. Gabriel's work was the first (I read) that showed this connection. In his fifth chapter, he demonstrates how all the fundemental theorems are directly related via the average sum theorem. I am certain there are errors but am also certain that understanding averages is key to understanding all calculus and how its parts relate to each other. Just as areas are a product of two averages, volumes are also a product of 3 averages. Tell me, how did the term "instantaneous average" arrive on the scene? Was it LaGrange who realized this when he published the mean value theorem? well, whoever it was, no one showed or proved why it should be called an instantaneous average. Truth is that it is better to call it the "average at a point" rather than an instantaneous average. In fact, the difference quotient makes absolutely ZERO sense when h = 0 in Rudin's definition. It is an approximation that was standardized. In other words, it is a 'fake' or misleading method (of appeasement rather than ingenuity) that was developed because no one knew any other way of finding a general derivative. Now Dovermann and Gabriel mention calculus without limits; Dovermann uses polynomials and gabriel mentions only the case of a quadratic. Dovermann's method will not work on polynomials with non-integral powers BUT it is a start. Gabriel's method will not work on general equations that are of higher order than a quintic but it is a start. Calculus does not work because of limits but because of averages. Perhaps if my edits were not deleted, someday a bright individual might arrive at a brand new method for computing general derivatives; one that is easier to understand and requires far less learning. Finally, there are many things I do not understand - although I know a lot of differential calculus and have used it often in applied mathematics such as boundary value problems and optimization, I still am afraid to say that I understand the recipe more than I do the process itself. 76.31.201.0 19:08, 6 August 2007 (UTC)

This is really an interesting discussion we're having! I learned analysis from Rudin and never had any serious objections. But I also had a few other references and people to talk to that helped me straighten things out when I didn't understand. Here's some stuff which I think you'll find interesting:

• Continuity makes more sense when you define it terms of open sets. This is the definition you need in a topological space. Open sets capture the idea of "nearness": Two points are "near each other" if there are "a lot" of open sets that contain both of them. That's vague, but you can translate it into a definition of continuity: f is continuous at x if and only if for every open set V containing f(x), there is an open set U such that f⁻¹(V) ⊆ U. In other words, if y is a point near f(x) (so there are lots of Vs containing both of them), then all the points near x are sent by f to a point near y.
• Notice that limits didn't show up in that definition. Limits are a way of going from a collection of approximations to a single object. The most basic example is a limit point in a topological space. A point x is a limit point of a set S if, for every open set U containing x, there is a point of S in U. In other words, there are points of S arbitrarily near x, where nearness is measured, as before, by being in a common open set.
  • To derive the definition of a limit of a sequence, let N^* denote the union of the natural numbers and a special point which we will denote by ω. Define an open set to be any union of intervals (a, b) and half-rays (a, ω]. Here (a, b) denotes all c such that a < c < b and (a, ω] denotes all c such that either a < c or c is ω. (This space is the ordinal space [0, ω], i.e., the one-point compactification of N with the order topology.) A sequence (of real numbers) is a function a from N to R. (The nth term of the sequence is usually denoted a_n instead of a(n).) The limit of the sequence, if it exists, is defined to be the number L such that if we define a^*(n) = a(n) and a^*(ω) = L, then a^* is a continuous function.
  • To derive the definition of a limit of a function, you do a similar trick: Let f be a function and let x₀ be the point that you're interested in taking the limit at. The limit of f at x, if it exists, is the number L such that the function f^* defined by f^*(x₀) = L and f^*(x) = f(x) for x not equal to x₀ is continuous at x₀.
• Differentiability is defined in terms of the existence of the limit of the difference quotient. The difference quotient should be thought of as a function of h. The definition of a limit of a function at a point pays no attention to whether or not the function is defined at that point, so we can take the limit of the difference quotient as h → 0 even though the difference quotient is not defined when h = 0.
• There are already foundations for the derivative that do not depend on calculus, such as non-standard analysis and (my favorite) the universal derivation d : O_X → Ω_X.

If you're curious about limits and the foundations of calculus, you might find Munkres's Topology or Kelley's General Topology interesting. Happy reading! 141.211.62.20 22:10, 6 August 2007 (UTC)

It's all just real nice 141.211. Do you honestly think all this is a lot simpler than the approach by Dovermann and Gabriel? It is possible to reject every step of your argument as follows: a point is an undefined concept. Open sets contain points; you use open sets to build a model for continuity and limits. My 'point'? The derivative and integral were conceived long before limits or set theory came into existence. Both limits and set theory are filled with vagueness, ambiguity and contradictions. I'll leave you with a simple example to play with because I can see that this discussion is going nowhere. Using the principles of real analysis, you can show that a circle is a union of rectangles. This is true. You can also show that a rectangle is a union of circles. However, the previous statement is untrue, both in theory and practice. It is in plain language a contradiction and a failure of the methods of real analysis. You try figure it out. 76.31.201.0 00:06, 7 August 2007 (UTC)

Since this discussion is ending, I'd just like to say that I enjoyed talking with you, even though we still disagree. Have a nice day. 141.211.62.20 17:19, 7 August 2007 (UTC)

Yes, it was good chatting with you even though we disagree. I leave you with one more reference that backs up my claim about calculus being about averages - [7]. See page 5 of 41. Have a nice day! 76.31.201.0 19:13, 7 August 2007 (UTC)

"Removal" notices added to article

If a section of an article is removed by an editor it is never appropriate to note this fact on the page itself. By all means rewrite the section, or address the concerns of the editor who removed it on the talk page (and these are usually necessary if the editor considers the portion too badly-written to revise). But please do not reference this on the article page itself: that is not an appropriate forum to discuss the revisions made (indeed, this is the place to do that). I have reverted the attempt to do this. Xantharius 23:42, 2 August 2007 (UTC)

The editor removed these sections out of ignorance and prejudice. It is evident he does not understand the contents (see previous section where he has demonstrated this). The way I see it, if your editors are not bright or capable enough to understand the material, then it is summarily removed. It is not my job to convince you (after the all, the work is not mine and I too do not fully understand it all, but am able to see it has merit. No source is 100% error-free. We all look at knowledge and decide for ourselves what has value and what does not). The removed sections were written as good or better than existing sections. Your team removed these out of prejudice. Well, I am not surprised. After all, Wikipedia is the world according to what the Wiki sysops and administrators think is 'encyclopedic'. Both of the sources I quoted are educated academics in their respective fields. 141.211.62.20 is the Wiki-puppet judge and his/her word is final. 141.211 became immediately prejudiced because one theorem was stated 'weakly', i.e. rather than being differentiable, it was only stated that it is continuous. I could see his point of view but if he had read all the material (chapters), he might have noticed that it mentions problems with the notion of differentiability. For example in Chapter 2, it shows how f(x)=abs(x) can be differentiable everywhere depending on how the derivative is defined. Newton's definition of the derivative leaves much to be desired. Newton himself could not defend it. You only have to look at how confused you all are and the discussions on these pages by so many 'educated' contributors and if you are smart enough (doubtful in my opinion), you will realize that they all know very little. Unlike you, I am uncertain about the foundations of the derivative. Its definition is problematic in many ways. Not only is it not defined rigorously but it has become more obscure. The removed section on Calculus without limits showed how one can differentiate without using the limit concept in any way. 76.31.201.0 12:03, 3 August 2007 (UTC)

You claim insight into human motives (prejudice) that is not justified by the evidence. Your attempts to insert this material into the article have met with no support. The reasons why the material is inappropriate have been explained to you repeatedly. Rick Norwood 14:45, 3 August 2007 (UTC)

In response to 76.31.201.0: 141.211.62.20 behaved exemplary. S/he did not remove the material you added even though it contained mistakes, took the time to look through some of the material you presented, and tried to engage you all the way. But this talk page is not to discuss the persons, it's to discuss the article (see "no personal attacks"). I've seen no evidence that John Gabriel's book, your main reference, is a reliable source (where is the evidence that he's an educated academic?). Thus, we should not use it as a reference for this article. Dovermann is a professor in mathematics so his lecture notes can be used, but he says himself that he's following a very unusual approach which does not give the results in full generality. Given the many books on calculus, I think his approach should not be given much coverage in the article, though I'm very interested in it and I'll consider it next time I teach calculus to biologists. -- Jitse Niesen (talk) 15:16, 3 August 2007 (UTC)

Let's deal with your statements one at a time. Regarding Gabriel's work: 141.211 was correct in only one thing - he stated that the theorem should

have been stronger by requiring differentiability, not only continuity. However, if s/he or you took time to read Gabriel's reasons, you would have seen that he questions the definition of differentiability. In chapter two he illustrates how using a different definition for the derivative (the central difference) produces a different result for differentiability (page 5 of chapter 2). 141.211 subsequently made a mistake in the follow up response. So who has been making the mistakes? Now you question whether Gabriel is qualified or not. Since when did Wiki required that the author of a source be suitably qualified? Heck, most of you are not suitably qualified. 141.211 claims that the derivative is not the limit of an average. If not, then tell me what f(x+h)-f(x)/h means? It is an average.

Now let's deal with Dovermann. It does not matter that his results do not work in generality. This is irrelevant. The point of my edits was to show that one can use calculus rigorously without limits. You have several articles that can't even be substantiated with any solid evidence and yet they pass by because of your blessing. Some examples are your article on infinitesimals and non-standard analysis just to mention a few. Sorry, it is about YOU and not the subject because YOU decide what gets to stay or go. Look, I know who sits on top of this heap - it's Michael Hardy and cohorts. And this is not a personal attack, it is a fact. Whoever removed the edits is prejudiced and unqualified. 76.31.201.0 20:52, 4 August 2007 (UTC)

76.31.201.0, I would appreciate if you examined a message on your talk page before this gets much further out of hand. Xantharius 22:35, 4 August 2007 (UTC)

Wikipedia has always required that the articles are based on reliable sources. In the sciences, this usually means that the sources have undergone peer review. This is one of Wikipedia's core policies (see Wikipedia:Verifiability for more information). -- Jitse Niesen (talk) 05:28, 5 August 2007 (UTC)

Okay, so you found some weak arguments in Gabriel's work but did you find any weak arguments in Dovermann's work? Why was the section 'calculus without limits' removed? I (and Dovermann) believe it is very important and presents a new perspective. Perhaps Newton (if he were alive today) would have been overjoyed to read Dovermann's work. Perhaps Archimedes would have been fascinated by "averages" because this is in effect why he was able to calculate the magnitude of areas and volumes. Perhaps we should think of 'indivisibles' and not 'infinitesimals' - these are two completely different concepts. The latter is non-sense and the former is part of the reason calculus is about averages - without collapsible/telescopic averages (composed of indivisibles), neither integral or differential calculus would have been possible. 76.31.201.0 03:36, 6 August 2007 (UTC)

Would you care to explain your comment about Michael? Do you have some sort of personal vendetta against WP:WPM or the math editors? If you do, then it would be best to take this up elsewhere or not at all. Your condescending attitude is not appreciated by anyone here. Everyone here(except me) has made an attempt to engage with you in civil discourse. You have rebuffed their every attempt to placate you. The frequent ad hominum ("prejudiced and unqualified" for one) attacks and insinuations do not strengthen your arguments and try the patience of everyone here. The reasons for the removal for the material you wanted inserted has been explained to you(WP:UNDUE is the most cogent IMO). If you cannot accept this policy then I am afraid that wikipedia is not the place for you to be. I realize that I have been rather terse, and I would prefer to continue this in a more amicable way. Here is the olive branch... take it or leave it.--Cronholm¹⁴⁴ 10:42, 5 August 2007 (UTC)