Talk:Calculus/Archive 6

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Terminology

About half-way through the section on the Development of calculus is the following sentence. "Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus." I think the integral link should actually link to Integer but I am not certain of this as it looks like both uses of the term integral are being used here. Also I think "powers" right after this usage of integral should be singular, I think an argument can be made either way. -AndrewBuck (talk) 04:10, 10 February 2008 (UTC)

Yes, it should link to integer (thanks). No, I think it is supposed to be plural. Would someone care to comment on what the "sum of integral powers" actually means? The most obvious meaning is the Riemann zeta ζ(s) for s an integer, but that's not exactly clear from the text. 04:17, 10 February 2008 (UTC)

I looked it up. I must be tired indeed, for the ibn al-Haytham formula is just that the sum of the first n fourth powers

$\sum _{k=1}^{n}k^{4}=\left({\frac {n}{5}}+{\frac {1}{5}}\right)n\left(n+{\frac {1}{2}}\right)\left((n+1)n-{\frac {1}{3}}\right).$

I'm not sure what it means by the sum of any integral powers. al-Haytham almost assuredly did not possess a general formula. Silly rabbit (talk) 04:46, 10 February 2008 (UTC)

Learning Calculus Faster

I really do appreciate all the information about this article, but it seems like their would be an faster way to learn this. Maybe just a list of formulas explaining step by step solutions. I have ADD so it makes it hard to read all of this information. Does anyone have any non ignorant ideas? —Preceding unsigned comment added by 76.126.198.46 (talk) 22:28, 25 February 2008 (UTC)

Like King Ptolemy I before you, you are asking for the Royal Road. If you don't understand the concepts behind the formulas well, you will have a hard time applying them appropriately. In the end, the investment in getting the basis right will pay off.  --Lambiam 11:08, 26 February 2008 (UTC)

Haha, thanks. —Preceding unsigned comment added by 76.126.198.46 (talk) 04:08, 1 March 2008 (UTC)

Problems solved in detail

If you have the smarts, can you type out what you do while solving 'any' of the problems in this article? The pictures are great, but i am left wondering what steps you took to get from A to B. Pretty sure most of it's just algebra; but i think if you do this it will help new calculus students.

Best Regards, The Nate DIZZLE

 —Preceding unsigned comment added by 76.126.198.46 (talk) 03:02, 12 March 2008 (UTC) 

This is not a calculus textbook and isn't intended to teach beginning students how to do problems in detail. I suggest you find a calculus textbook and work through the problems there to learn the process if you are having difficulty. PhySusie (talk) 10:53, 12 March 2008 (UTC)

Smooth infinitesimal analysis

I don't like the sentences concerning smooth infinitesimal analysis that have been added recently, like:

Another alternative is smooth infinitesimal analysis in which calculus is based on the concepts of microstraightness (of continuous functions) and of nilsquare infinitesimals. The logic of smooth infinitesimal analysis is intuitionistic logic which differs from classical logic in that it does not include the Axiom of Choice.

I feel that smooth infinitesimal calculus is used even less often than nonstandard analysis, and thus it should also get less space in this article. In other words, we only mention that it exists and let the reader go to the article for any further details. I amended this article accordingly.

Incidentally, I would replace the Axiom of Choice with the law of excluded middle in the above fragment. -- Jitse Niesen (talk) 14:16, 29 March 2008 (UTC)

I agree, WP:NPOV's undue weight section says that we can only give ideas space in articles in their relation to their importance to the topic. Unless reliable sources can be found that show that smooth infitesimal analysis is used extensively and is has widely impacted the use and teaching of calculus it should not get much or perhaps even any space in this article depending on how much impact can be shown to exist for it. This is the main calculus article and only the main topics can be given space. - Taxman ^Talk 16:48, 29 March 2008 (UTC)

It is not clear what the meaning is of "being used" with relation to smooth infinitesimal analysis. Like constructive mathematics, it is in some ways less powerful (and in other ways more powerful) than classical analysis. It is clearly more elegant than Bishop's Foundations of Constructive Analysis. Inasmuch as it may be viewed as an alternative, it is not in the calculus aspect (in the sense of a body of form-based rules), but in the foundational aspect. Can you measure how much a foundational approach is "used"? The people who have published on this are not the least among category theorists. With due respect to the undue-weight clause, I think it can at least be mentioned. The logic you can reconstruct from the approach goes beyond the propositional logic described in our article on intuitionistic logic and rejects both LEM and AC, but gives you back that all functions defined on R are continuous, a theorem of Brouwer.  --Lambiam 09:07, 30 March 2008 (UTC)

If you can argue for a different metric than "use" I'm fine with that, but there has to be some demonstrable bar of importance that it meets in order to justify mention at all. I probably just should have said generic importance in the first place. So basically you are saying you think there is more than enough published on this topic by people of importance to justify inclusion? Can you pull up anything that would show evidence to justify inclusion in the main calculus article? - Taxman ^Talk 17:11, 30 March 2008 (UTC)

I agree with Taxman that some specific reference linking smooth infinitesimal analysis and calculus should be found. At the very least, this will help to ascertain the extent to which it should be mentioned here. I have reverted 82...'s most recent edit [1] as it clearly places undue weight on smooth infinitesimal analysis (limits are no longer the standard approach to calculus? huh?) In this case, I recommend that we stick to reliable sources as much as possible. silly rabbit (talk) 15:44, 31 March 2008 (UTC)

Some minor edits

I agree with a previous poster that the history section does indeed go into sometimes unnecessary detail, and the overuse of names, places, dates and texts make the section hard to navigate and read. The Limits section only has a few sentences on the current use of limits (the first paragraph is mostly about the use of infinitesimals), and since limits are an important part of calculus, I think an example would greatly benefit the section. Though there is a 'limits' article on wikipedia that goes into much more detail, I think it's impossible to get a real feel for what limits are by just reading that short paragraph - one would be forced to go into the limits article to understand what they are. Notation should be added as well as one example, and this wouldn't add too much to the length of the article. In the derivatives section, there is an example shown where F(x)=x^2 and F'(x)=2x. Wouldn't it make more sense to explain that the derivative of x^n is nx^(n-1)? It gives a clearer understanding of how a derivative is obtained, and we could leave the previous example for explanation if need be. I would also suggest an equation for the antiderivative section as well. Finally, a very minor suggestion - when providing the equation "distance = speed x time", there is no need to have the equation written out like that, and formatted similarly to the other equations in the article - it makes it seem like that equation is a calculus term, and is unnecessarily formatted. Nessalora (talk) 03:41, 27 May 2008 (UTC)

I disagree with adding more about limits for two reasons: the main article on limits is very prominently linked, and there is an explicit calculation involving limits which gives a flavor of what elementary calculations in calculus are like. It is one of the beauties of an electronic encyclopedia that a quick check to find out what limits are and then returning to the main article to read more are both very easy. I don't know exactly what you mean about adding notation, but notation for limits is certainly covered in this article in the examples given.

Consensus against replacing the computation of the derivative of x² with that of xⁿ was reached on this talk page at Polynomial calculations above, and for good reasons: the current example is more concrete and therefore easier (at least in this case) to understand.Xantharius (talk) 18:02, 27 May 2008 (UTC)

Fundamental Theorem notes

As a somebody who took calculus in high school 20+ years ago, I think the article is useful and understandable quick summary, with the exception only of this part of the Fundamental theorem section:

"Furthermore, for every x in the interval (a, b),

${\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).$

I don't think a reader who doesn't already understand calculus will be able to meaningfully parse this equation.

Earlier in the article it is suggested the "dx" means "with respect to x. So what does "d/dx" mean? Further who or what is "t" and what does it have to do with the equation?

Really quite curious about this, since apparently the equation is "key to massive proliferation of analytic results"

Joshk6 (talk) 05:32, 20 June 2008 (UTC)

That notation is indeed not explained in this article, and therefore it should perhaps not be used. For the notation, see Leibniz's notation#Leibniz's notation for differentiation. A possible replacement text:

Conversely, the function F defined as a definite integral of ƒ, as follows,

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

has ƒ as its derivative for every x in the interval (a, b):

$F'(x)=f(x)\,.$

 --Lambiam 22:20, 26 June 2008 (UTC)

IMO the Liebniz notation is so important, and so widely used, that this article absolutely needs to explain it. I've added an explanation of it, and I've made a section about it below on the talk page.--76.167.77.165 (talk) 18:53, 7 March 2009 (UTC)

I'm happy to see the addition of an explanation of Liebniz notation...but it could be a little more. Nevertheless, I applaud the addition thus far Gingermint (talk) 03:04, 16 June 2009 (UTC)

Wouldn't it also be proper to include a link to the main article Leibniz's notation? I would think so, but would like other opinions. ---JamesWill (talk) 08:20, 7 July 2009 (UTC)

I'm in the same position as Joshk6 and would really appreciate someone who understands calculus well expanding this section, continuing with the previous example of x squared and its derivative 2x. It appears to me at this point you're finding the area under 2x, which is linear. How do you calculate the antiderivative of x squared? And what is "t"? That just comes out of nowhere. I found this article absolutely excellent for my needs right up to this point, where I'm left in a cloud of confusion ---- TomW

Topics in Calulus

I suggest that the following articles be merged

• Shell method

• Disc method

• solid of revolution --this one is a good one I think it has potential to be an A-class Article but it is not even mentioned in this article

I also suggest

that Arc length be added to topics in Calculus

I just want to hear what you all think --GlasGhost (talk) 02:32, 1 August 2008 (UTC)

Are you suggesting that these articles be merged here? That doesn't seem like a very good idea. The subject of calculus is quite large, and we cannot hope to accommodate all of it in the article. There is a List of calculus topics for other topics, to which I will add your suggestions. siℓℓy rabbit (talk) 03:15, 1 August 2008 (UTC)

thx I was looking for that article but couldn't find it--GlasGhost (talk) 22:12, 16 August 2008 (UTC)

Where are the Applications?

The Applications section of the Calculus page of Wikipedia states-

"Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields."

Wikipedia then gives between 7-10 examples, some of them pretty weak especially in the light of the claim that it is used in seven different major fields, one of which is "every branch of the physical sciences". (call them astronomy, physics, chemistry meteorology) which is a total of 10 major disciplines. 7 examples. We (I) do not seem to be proving the point that calculus can be applied to all of these disciplines.

Is calculus the BASIS for a lot of formulas but not really USED for calculating answers to problems on a day to day basis? Is that the problem. If so, I think that is an important distinction that should be emphasized.

Please allow me to digress for a paragraph. I am a government engineer who passed the calculus sequence 21 years ago and I don't feel I really know the answer to the question "How is calculus applied". I did apply one example to Wikipedia (The planimeter example) but otherwise I can't answer the question concerning how calculus is applied. I will throw in the caveat that I work for the government and they hire a bunch of engineers and then have us do mostly non engineering work.

Ultimately I want others to show me examples even though I can not do so myself. And I realize the hypocrasy (sp?) in that statement.

This is my first Wikipedia edit. So be gentle and guiding in your statements. —Preceding unsigned comment added by Tfeeney65 (talk • contribs) 21:07, 17 October 2008 (UTC)

I believe this deserves further discussion. As it would be unwise to list applications for every branch of science and statistics listed in the article, perhaps creating a page entitled Applications of Calculus would be the best course? Otherwise it seems this section is lacking and only adding to the not-up-to-par standard of the article. ---JamesWill (talk) 08:28, 7 July 2009 (UTC)

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