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| {{About|the branch of mathematics}}
| | If you're thinking how to be anorexic, we are not alone. Anorexia nervosa is the world's most normal eating disorder (after binge-eating). Unfortunately, it really is moreover pretty risky to a person's mental plus physical wellness.<br><br>"What is being shown is that there is an income gradient that relates to weight in the same means which alternative factors do. People are more worried about overweight plus obesity in certain regions it can well relate to the fact that they are becoming more aware of the health consequences," he added.<br><br>Some folks think which it is wise to test and shed a few pounds before computing for their body mass index. You need to remember which doing this type of thing is similar to cheating during an exam at school. We require to measure the real fat inside purchase to receive a obvious picture of how harmful we certainly are. If you try to get rid of weight before the computation, you would get a lower body mass index which may effectively mask the real condition. This usually create you think which you don't have to go to the doctor or which we don't have to lose fat at all.<br><br>So being the girl I am and having a should learn type of notice, I found an online [http://safedietplans.com/bmi-calculator bmi calculator females] and place inside my numbers. After my friend Sally's experience I was certain that mine would at the least say I was at the high-end of regular. Oh however no, my online BMI outcome was that I was underweight. WTF? Again I know which everyone has their own perception of what exactly is thick plus was is not however, when I were to really judge where I think my weight could fall I would have said average, incredibly with all the knowledge that Sally's came back on the high-end of overweight.<br><br>In purchase to recognize where you fall on the body fat charts, you'll need to calculate the body fat. There are several techniques to do this, from a do it yourself skin fold test, to online calculators. The skin fold test is not (omit) because correct because other techniques. Basically, you're pinching several skin at different body places, that is measured. This may show the changes inside body composition over time, even if it's not a totally correct system of calculating body fat.<br><br>When we recognize the BMI, merely compare it to the chart below. Please note a excellent BMI does not indicate you are at risk. Consult a doctor when you're not sure.<br><br>Note: If you don't know a BMI score you will find several free online calculators to aid you figure it. One such calculator can be found at: wdxcyber.com/bmi. Remember an ideal BMI score is one between 19 and 24.9. An obese BMI score is 1 that falls between 25.9 plus 29.9 plus anything over 30 is considered fat. |
| {{insufficient inline citations|date=December 2013}}
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| {{Calculus}}
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| '''Calculus''' is the [[mathematics|mathematical]] study of change,<ref>{{citation
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| |title=Calculus Concepts: An Applied Approach to the Mathematics of Change
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| |first1=Donald R.
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| |last1=Latorre
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| |first2=John W.
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| |last2=Kenelly
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| |first3=Iris B.
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| |last3=Reed
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| |first4=Sherry
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| |last4=Biggers
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| |publisher=Cengage Learning
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| |year=2007
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| |isbn=0-618-78981-2
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| |page=2
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| |url=http://books.google.com/books?id=bQhX-3k0LS8C}}, [http://books.google.com/books?id=bQhX-3k0LS8C&pg=PA2 Chapter 1, p 2]
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| </ref> in the same way that [[geometry]] is the study of shape and [[algebra]] is the study of operations and their application to solving equations. It has two major branches, [[differential calculus]] (concerning rates of change and slopes of curves), and [[integral calculus]] (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the [[fundamental theorem of calculus]]. Both branches make use of the fundamental notions of [[convergence (mathematics)|convergence]] of [[infinite sequence]]s and [[Series (mathematics)|infinite series]] to a well-defined [[limit (mathematics)|limit]]. Generally considered to have been founded in the 17th century by [[Isaac Newton]] and [[Gottfried Leibniz]], today calculus has widespread uses in [[science]], [[engineering]] and [[economics]] and can solve many problems that [[Elementary algebra|algebra]] alone cannot. | |
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| Calculus is a part of modern [[mathematics education]]. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of [[Function (mathematics)|function]]s and limits, broadly called [[mathematical analysis]]. Calculus has historically been called "the calculus of [[infinitesimal]]s", or "[[infinitesimal calculus]]". The word "calculus" comes from [[Latin]] (''[[wikt:en:calculus#Latin|calculus]]'') and refers to a small stone used for counting. More generally, ''calculus'' (plural ''calculi'') refers to any method or system of calculation guided by the symbolic manipulation of [[expression (mathematics)|expression]]s. Some examples of other well-known calculi are [[propositional calculus]], [[calculus of variations]], [[lambda calculus]], and [[process calculus]].
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| == History ==
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| <!--
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| Attention; leave dates as they are. We're not really that bothered, as the majority of Wikipedia dates state "BC". Just think of it as "Before Cronholm" :-)
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| -->
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| {{Main|History of calculus}}
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| === Ancient ===
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| The ancient period introduced some of the ideas that led to [[integral]] calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the [[Egyptian mathematics|Egyptian]] [[Moscow Mathematical Papyrus|Moscow papyrus]] (c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components.<ref>Morris Kline, ''Mathematical thought from ancient to modern times'', Vol. I</ref> From the age of [[Greek mathematics]], [[Eudoxus of Cnidus|Eudoxus]] (c. 408−355 BC) used the [[method of exhaustion]], which foreshadows the concept of the limit, to calculate areas and volumes, while [[Archimedes]] (c. 287−212 BC) [[Archimedes' use of infinitesimals|developed this idea further]], inventing [[heuristics]] which resemble the methods of integral calculus.<ref>Archimedes, ''Method'', in ''The Works of Archimedes'' ISBN 978-0-521-66160-7</ref> The [[method of exhaustion]] was later reinvented in [[Chinese mathematics|China]] by [[Liu Hui]] in the 3rd century AD in order to find the area of a circle.<ref>{{cite journal
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| |series=Chinese studies in the history and philosophy of science and technology
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| |volume=130
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| |title=A comparison of Archimdes' and Liu Hui's studies of circles
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| |first1=Liu
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| |last1=Dun
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| |first2=Dainian
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| |last2=Fan
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| |first3=Robert Sonné
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| |last3=Cohen
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| |publisher=Springer
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| |year=1966
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| |isbn=0-7923-3463-9
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| |page=279
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| |url=http://books.google.com/books?id=jaQH6_8Ju-MC}}, [http://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter , p. 279]
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| </ref> In the 5th century AD, [[Zu Chongzhi]] established a method that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].<ref>{{cite book
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| |title=Calculus: Early Transcendentals
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| |edition=3
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| |first1=Dennis G.
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| |last1=Zill
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| |first2=Scott
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| |last2=Wright
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| |first3=Warren S.
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| |last3=Wright
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| |publisher=Jones & Bartlett Learning
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| |year=2009
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| |isbn=0-7637-5995-3
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| |page=xxvii
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| |url=http://books.google.com/books?id=R3Hk4Uhb1Z0C}}, [http://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27]
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| </ref>
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| === Medieval ===
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| [[Alexander the Great]]'s invasion of northern India brought Greek trigonometry, using the [[Chord (geometry)|chord]], to India where the sine, cosine, and tangent were conceived. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, [[Alhazen]] derived a formula for the sum of [[fourth power]]s. He used the results to carry out what would now be called an [[Integral|integration]], where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref name=katz>Katz, V. J. 1995. "Ideas of Calculus in Islam and India." ''Mathematics Magazine'' (Mathematical Association of America), 68(3):163-174.</ref> In the 14th century, Indian mathematician [[Madhava of Sangamagrama]] and the [[Kerala school of astronomy and mathematics]] stated components of calculus such as the [[Taylor series]] and [[infinite series]] approximations.<ref>[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html Indian mathematics<!-- Bot generated title -->]</ref> Madhava developed some components of [[calculus]] such as [[derivative|differentiation]], term-by-term [[Integral|integration]], [[iterative method]]s for solutions of [[Nonlinearity|non-linear]] equations, and the theory that the area under a curve is its integral.
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| If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by [[James Gregory (mathematician)|James Gregory]] in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term).<ref name=rajag86>Madhava extended Archimedes' work on the geometric Method of Exhaustion to measure areas and numbers such as π, with arbitrary accuracy and error ''limits'', to an algebraic infinite series with a completely separate error ''term''.
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| {{cite journal
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| | title = On medieval Keralese mathematics,
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| | author = C T Rajagopal and M S Rangachari
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| | journal = Archive for History of Exact Sciences
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| | url = http://www.springerlink.com/content/t1343xktl7g52003/
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| | volume = 35
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| | year = 1986
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| | pages = 91–99
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| | doi = 10.1007/BF00357622
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| }}</ref> This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlying [[infinite series]] expansions of functions, [[power series]], [[trigonometric series]], and rational approximations of infinite series.<ref name="MAT 314">{{cite web
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| | publisher=Canisius College |
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| work=MAT 314
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| |url=http://www.canisius.edu/topos/rajeev.asp
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| | title=Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
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| | accessdate=2006-07-09
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| }}
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| </ref>
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| However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.
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| Among his many contributions, he discovered the infinite series for the [[trigonometric function]]s of [[sine]], [[cosine]], [[tangent (trigonometric function)|tangent]] and [[arctangent]], and many methods for calculating the [[circumference]] of a [[circle]]. One of Madhava's series is known from the text ''[[Yuktibhāṣā]]'', which contains the derivation and proof of the [[power series]] for [[Inverse trigonometric function|inverse tangent]], discovered by Madhava.<ref name="infinity">{{cite web
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| | publisher=D.P. Agrawal—Infinity Foundation |
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| work=Indian Mathemematics
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| |url=http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm
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| | title=The Kerala School, European Mathematics and Navigation
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| | accessdate=2006-07-09
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| }}
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| </ref> In the text, [[Jyeṣṭhadeva]] describes the series in the following manner:
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| {{cquote|The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.<ref name=Gupta>
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| {{cite journal
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| | author = R C Gupta
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| | title = The Madhava-Gregory series
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| | journal = Math. Education
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| | volume = 7
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| | year = 1973
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| | pages = B67–B70
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| }}</ref>}}
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| This yields:
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| :<math> r\theta={\frac {r\sin \theta }{\cos \theta
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| }}-(1/3)\,r\,{\frac { \left(\sin \theta \right) ^
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| {3}}{ \left(\cos \theta \right) ^{3}}}+(1/5)\,r\,{\frac {
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| \left(\sin \theta \right) ^{5}}{ \left(\cos
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| \theta \right) ^{5}}}-(1/7)\,r\,{\frac { \left(\sin \theta
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| \right) ^{7}}{ \left(\cos \theta \right) ^{
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| 7}}} + \cdots</math>
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| or equivalently:
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| :<math>\theta = \tan \theta - \frac{\tan^3 \theta}{3} + \frac{\tan^5 \theta}{5} - \frac{\tan^7 \theta}{7} + \cdots</math>
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| This series was traditionally known as the Gregory series (after [[James Gregory (astronomer and mathematician)|James Gregory]], who discovered it three centuries after Madhava). Even if we consider this particular series as the work of [[Jyeṣṭhadeva]], it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava. Today, it is referred to as the Madhava-Gregory-Leibniz series.<ref name=Gupta/><ref name=nair>{{cite web
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| | publisher=Prof. C.G.Ramachandran Nair |
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| work=Government of Kerala—Kerala Call, September 2004
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| |url=http://www.kerala.gov.in/keralcallsep04/p22-24.pdf
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| | title=Science and technology in free India
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| | accessdate=2006-07-09
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| |format=PDF}}
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| </ref>
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| The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the [[Malabar Coast]]. At the time, the port of [[Muziris]], near [[Sangamagrama]], was a major center for maritime trade, and a number of [[Jesuit]] missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including George Joseph of the [[University of Manchester]] have suggested<ref name=predated>
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| {{cite news
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| | title = Indians predated Newton 'discovery' by 250 years
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| | publisher = press release, University of Manchester
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| | url = http://www.humanities.manchester.ac.uk/aboutus/news/display/?id=121685
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| | date = 13 August 2007
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| | accessdate = 2007-09-05
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| }}</ref> that the writings of the Kerala school may have also been transmitted to Europe around this time, which was still about a century before Newton.
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| === Modern ===
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| {|class="toccolours" style="float: right; margin-left: 0.5em; margin-right: 0.5em; font-size: 84%; background:#white; color:black; width:30em; max-width: 30%;" cellspacing="5"
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| |style="text-align: left;"| "The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." —[[John von Neumann]]<ref>von Neumann, J., "The Mathematician", in Heywood, R. B., ed., ''The Works of the Mind'', University of Chicago Press, 1947, pp. 180–196. Reprinted in Bródy, F., Vámos, T., eds., ''The Neumann Compedium'', World Scientific Publishing Co. Pte. Ltd., 1995, ISBN 981-02-2201-7, pp. 618–626.</ref>
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| |}
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| In Europe, the foundational work was a treatise due to [[Bonaventura Cavalieri]], who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in [[Archimedes' use of infinitesimals|The Method]], but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
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| The formal study of calculus brought together Cavalieri's infinitesimals with the [[calculus of finite differences]] developed in Europe at around the same time. [[Pierre de Fermat]], claiming that he borrowed from [[Diophantus]], introduced the concept of [[adequality]], which represented equality up to an infinitesimal error term.<ref>[[André Weil]]: Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc., Boston, MA, 1984, ISBN 0-8176-4565-9, p. 28.</ref> The combination was achieved by [[John Wallis]], [[Isaac Barrow]], and [[James Gregory (astronomer and mathematician)|James Gregory]], the latter two proving the [[Fundamental theorem of calculus|second fundamental theorem of calculus]] around 1670.
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| [[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|200px|right|[[Isaac Newton]] developed the use of calculus in his [[Newton's laws of motion|laws of motion]] and [[gravitation]].]]
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| The [[product rule]] and [[chain rule]], the notion of [[higher derivative]]s, [[Taylor series]], and [[analytical function]]s were introduced by [[Isaac Newton]] in an idiosyncratic notation which he used to solve problems of [[mathematical physics]].<ref>Donald Allen: Calculus, http://www.math.tamu.edu/~dallen/history/calc1/calc1.html</ref> In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a [[cycloid]], and many other problems discussed in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the [[Taylor series]]. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
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| [[File:Gottfried Wilhelm von Leibniz.jpg|thumb|200px|left|[[Gottfried Wilhelm Leibniz]] was the first to publish his results on the development of calculus.]]
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| These ideas were arranged into a true calculus of infinitesimals by [[Gottfried Wilhelm Leibniz]], who was originally accused of [[plagiarism]] by Newton.<ref>Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. Page 228. [http://books.google.com/books?hl=en&lr=&id=7d8_4WPc9SMC&oi=fnd&pg=PA3&dq=Gottfried+Wilhelm+Leibniz+accused+of+plagiarism+by+Newton&ots=09h9BdTlbE&sig=hu5tNKpBJxHcpj8U3kR_T2bZqrY#v=onepage&q=plagairism&f=false|Online Copy]</ref> He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the [[product rule]] and [[chain rule]], in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts.
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| [[Gottfried Leibniz|Leibniz]] and [[Isaac Newton|Newton]] are usually both credited with the invention of calculus. Newton was the first to apply calculus to general [[physics]] and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
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| When Newton and Leibniz first published their results, there was [[Newton v. Leibniz calculus controversy|great controversy]] over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his ''[[Method of Fluxions]]''), but Leibniz published his ''[[Nova Methodus pro Maximis et Minimis]]'' first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the [[Royal Society]]. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "[[Method of fluxions|the science of fluxions]]".
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| Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on finite and infinitesimal analysis was written in 1748 by [[Maria Gaetana Agnesi]].<ref>{{cite web| url=http://www.agnesscott.edu/lriddle/women/agnesi.htm| title=Maria Gaetana Agnesi| first=Elif| last=Unlu|date=April 1995| publisher =Agnes Scott College}}</ref>
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| [[File:Maria Gaetana Agnesi.jpg|thumb|150px|right|[[Maria Gaetana Agnesi]]]]
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| === Foundations ===
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| In calculus, ''foundations'' refers to the [[Rigorous#Mathematical rigour|rigorous]] development of a subject from precise axioms and definitions. In early calculus the use of [[infinitesimal]] quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably [[Michel Rolle]] and [[George Berkeley|Bishop Berkeley]]. Berkeley famously described infinitesimals as the [[ghosts of departed quantities]] in his book ''[[The Analyst]]'' in 1734. A recent study argues that Leibnizian calculus was more solidly grounded than Berkeley's [[empiricism|empiricist]] critique thereof.<ref>{{citation
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| | last1 = Katz | first1 = Mikhail
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| | author1-link = Mikhail Katz
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| | last2 = Sherry | first2 = David
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| | author2-link =
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| | arxiv = 1205.0174
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| | doi = 10.1007/s10670-012-9370-y
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| | issue =
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| | journal = [[Erkenntnis]]
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| | pages =
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| | title = Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond
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| | volume =
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| | year = 2012}}.</ref> Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.
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| Several mathematicians, including [[Colin Maclaurin|Maclaurin]], tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of [[Augustin Louis Cauchy|Cauchy]] and [[Karl Weierstrass|Weierstrass]], a way was finally found to avoid mere "notions" of infinitely small quantities.<ref>{{Cite book |last= Russell |first= Bertrand |authorlink= Bertrand Russell |year= 1946 |title= [[A History of Western Philosophy|History of Western Philosophy]] |location= London |publisher= [[George Allen & Unwin Ltd]] |page= [http://archive.org/stream/westernphilosoph035502mbp#page/n857/mode/2up 857] |quote= The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. }}</ref> The foundations of differential and integral calculus had been laid. In Cauchy's writing (see [[Cours d'Analyse]]), we find a broad range of foundational approaches, including a definition of [[continuous function|continuity]] in terms of infinitesimals, and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit]] in the definition of differentiation. In his work Weierstrass formalized the concept of [[Limit of a function|limit]]<!--correct link?--> and eliminated infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities. [[Bernhard Riemann]] used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to [[Euclidean space]] and the [[complex plane]].
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| In modern mathematics, the foundations of calculus are included in the field of [[real analysis]], which contains full definitions and [[mathematical proof|proof]]s of the theorems of calculus. The reach of calculus has also been greatly extended. [[Henri Lebesgue]] invented [[measure theory]] and used it to define integrals of all but the most pathological functions. [[Laurent Schwartz]] introduced [[Distribution (mathematics)|distribution]]s, which can be used to take the derivative of any function whatsoever.
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| Limits are not the only rigorous approach to the foundation of calculus. Another way is to use [[Abraham Robinson]]'s [[non-standard analysis]]. Robinson's approach, developed in the 1960s, uses technical machinery from [[mathematical logic]] to augment the real number system with [[infinitesimal]] and [[Infinity|infinite]] numbers, as in the original Newton-Leibniz conception. The resulting numbers are called [[hyperreal number]]s, and they can be used to give a Leibniz-like development of the usual rules of calculus.
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| === Significance ===
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| While many of the ideas of calculus had been developed earlier in [[Egyptian mathematics|Egypt]], [[Greek mathematics|Greece]], [[Chinese mathematics|China]], [[Indian mathematics|India]], [[Islamic mathematics|Iraq, Persia]], and [[Japanese mathematics|Japan]], the use of calculus began in [[Europe]], during the 17th century, when [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
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| Applications of differential calculus include computations involving [[velocity]] and [[acceleration]], the [[slope]] of a curve, and [[Mathematical optimization|optimization]]. Applications of integral calculus include computations involving [[area]], [[volume]], [[arc length]], [[center of mass]], [[work (physics)|work]], and [[pressure]]. More advanced applications include [[power series]] and [[Fourier series]].
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| Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving [[division by zero]] or sums of infinitely many numbers. These questions arise in the study of [[Motion (physics)|motion]] and [[area]]. The [[ancient Greek]] [[philosopher]] [[Zeno of Elea]] gave several famous examples of such [[Zeno's paradoxes|paradoxes]]. Calculus provides tools, especially the [[Limit (mathematics)|limit]] and the [[infinite series]], which resolve the paradoxes.
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| == Principles ==
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| === Limits and infinitesimals ===
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| {{Main|Limit of a function|Infinitesimal|Infinitesimal Calculus}}
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| Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by [[infinitesimal]]s. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number ''dx'' could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and less than any positive [[real number]]. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the [[Archimedean property]]. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of [[non-standard analysis]] and [[smooth infinitesimal analysis]], which provided solid foundations for the manipulation of infinitesimals.
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| In the 19th century, infinitesimals were replaced by the [[epsilon, delta]] approach to [[Limit of a function|limit]]s. Limits describe the value of a [[function (mathematics)|function]] at a certain input in terms of its values at nearby input. They capture small-scale behavior in the context of the [[real number|real number system]]. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
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| === Differential calculus ===
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| {{Main|Differential calculus}}
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| [[File:Tangent derivative calculusdia.svg|thumb|300px|Tangent line at (''x'', ''f''(''x'')). The derivative ''f′''(''x'') of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.]]
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| Differential calculus is the study of the definition, properties, and applications of the [[derivative]] of a function. The process of finding the derivative is called ''differentiation''. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the ''derivative function'' or just the ''derivative'' of the original function. In mathematical jargon, the derivative is a [[linear operator]] which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.)
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| The most common symbol for a derivative is an apostrophe-like mark called [[prime (symbol)|prime]]. Thus, the derivative of the function of ''f'' is ''f′'', pronounced "f prime." For instance, if ''f''(''x'') = ''x''<sup>2</sup> is the squaring function, then ''f′''(''x'') = 2''x'' is its derivative, the doubling function.
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| If the input of the function represents time, then the derivative represents change with respect to time. For example, if ''f'' is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of ''f'' is how the position is changing in time, that is, it is the [[velocity]] of the ball.
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| If a function is [[linear function|linear]] (that is, if the [[Graph of a function|graph]] of the function is a straight line), then the function can be written as {{nowrap|''y'' {{=}} ''mx'' + ''b''}}, where ''x'' is the independent variable, ''y'' is the dependent variable, ''b'' is the ''y''-intercept, and:
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| :<math>m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.</math>
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| This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in ''y'' divided by the change in ''x'' varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let ''f'' be a function, and fix a point ''a'' in the domain of ''f''. (''a'', ''f''(''a'')) is a point on the graph of the function. If ''h'' is a number close to zero, then ''a'' + ''h'' is a number close to ''a''. Therefore (''a'' + ''h'', ''f''(''a'' + ''h'')) is close to (''a'', ''f''(''a'')). The slope between these two points is
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| :<math>m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.</math>
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| This expression is called a ''difference quotient''. A line through two points on a curve is called a ''secant line'', so ''m'' is the slope of the secant line between (''a'', ''f''(''a'')) and (''a'' + ''h'', ''f''(''a'' + ''h'')). The secant line is only an approximation to the behavior of the function at the point ''a'' because it does not account for what happens between ''a'' and ''a'' + ''h''. It is not possible to discover the behavior at ''a'' by setting ''h'' to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the [[limit (mathematics)|limit]] as ''h'' tends to zero, meaning that it considers the behavior of ''f'' for all small values of ''h'' and extracts a consistent value for the case when ''h'' equals zero:
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| :<math>\lim_{h \to 0}{f(a+h) - f(a)\over{h}}.</math>
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| Geometrically, the derivative is the slope of the [[tangent line]] to the graph of ''f'' at ''a''. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function ''f''.
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| Here is a particular example, the derivative of the squaring function at the input 3. Let ''f''(''x'') = ''x''<sup>2</sup> be the squaring function.
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| [[File:Sec2tan.gif|thumb|300px|The derivative ''f′''(''x'') of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is ''f''(''x'') = ''x''<sup>3</sup> − ''x''. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.]]
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| :<math>\begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\
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| &=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\
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| &=\lim_{h \to 0}{6h + h^2\over{h}} \\
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| &=\lim_{h \to 0} (6 + h) \\
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| &= 6.
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| \end{align}
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| </math>
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| The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the ''derivative function'' of the squaring function, or just the ''derivative'' of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.
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| === Leibniz notation ===
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| {{Main|Leibniz's notation}}
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| A common notation, introduced by Leibniz, for the derivative in the example above is
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| :<math>
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| \begin{align}
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| y&=x^2 \\
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| \frac{dy}{dx}&=2x.
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| \end{align}
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| </math>
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| In an approach based on limits, the symbol ''dy/dx'' is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, ''dy'' being the infinitesimally small change in ''y'' caused by an infinitesimally small change ''dx'' applied to ''x''. We can also think of ''d/dx'' as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:
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| :<math>
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| \frac{d}{dx}(x^2)=2x.
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| </math>
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| In this usage, the ''dx'' in the denominator is read as "with respect to x". Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like ''dx'' and ''dy'' as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the [[total derivative]].
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| === Integral calculus ===
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| {{Main|Integral}}
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| ''Integral calculus'' is the study of the definitions, properties, and applications of two related concepts, the ''indefinite integral'' and the ''definite integral''. The process of finding the value of an integral is called ''integration''. In technical language, integral calculus studies two related [[linear operator]]s.
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| The ''indefinite integral'' is the ''[[antiderivative]]'', the inverse operation to the derivative. ''F'' is an indefinite integral of ''f'' when ''f'' is a derivative of ''F''. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.)
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| The ''definite integral'' inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the [[x-axis]]. The technical definition of the definite integral is the [[limit (mathematics)|limit]] of a sum of areas of rectangles, called a [[Riemann sum]].
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| A motivating example is the distances traveled in a given time.
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| :<math>\mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time}</math>
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| If the speed is constant, only multiplication is needed, but if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a [[Riemann sum]]) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.
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| [[File:Constant velocity.png|left|thumb|Constant Velocity]]
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| [[File:Integral as region under curve.svg|right|thumb|280px|Integration can be thought of as measuring the area under a curve, defined by ''f''(''x''), between two points (here ''a'' and ''b'').]]
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| When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles. In the diagram on the left, when constant velocity and time are graphed, these two values form a rectangle with height equal to the velocity and width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve. This connection between the area under a curve and distance traveled can be extended to ''any'' irregularly shaped region exhibiting a fluctuating velocity over a given time period. If ''f(x)'' in the diagram on the right represents speed as it varies over time, the distance traveled (between the times represented by ''a'' and ''b'') is the area of the shaded region ''s''.
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| To approximate that area, an intuitive method would be to divide up the distance between ''a'' and ''b'' into a number of equal segments, the length of each segment represented by the symbol ''Δx''. For each small segment, we can choose one value of the function ''f''(''x''). Call that value ''h''. Then the area of the rectangle with base ''Δx'' and height ''h'' gives the distance (time ''Δx'' multiplied by speed ''h'') traveled in that segment. Associated with each segment is the average value of the function above it, ''f(x)''=h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for ''Δx'' will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as ''Δx'' approaches zero.
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| The symbol of integration is <math>\int \,</math>, an elongated ''S'' (the S stands for "sum"). The definite integral is written as:
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| :<math>\int_a^b f(x)\, dx.</math>
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| and is read "the integral from ''a'' to ''b'' of ''f''-of-''x'' with respect to ''x''." The Leibniz notation ''dx'' is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width ''Δx'' becomes the infinitesimally small ''dx''. In a formulation of the calculus based on limits, the notation
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| :<math>\int_a^b \ldots\, dx</math>
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| is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, ''dx'', is not a number, and is not being multiplied by ''f(x)'', although, serving as a reminder of the ''Δx'' limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.
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| The indefinite integral, or antiderivative, is written:
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| :<math>\int f(x)\, dx.</math>
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| Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function ''y'' = ''x''² + ''C'', where ''C'' is any constant, is ''y′'' = 2''x'', the antiderivative of the latter given by:
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| :<math>\int 2x\, dx = x^2 + C.</math>
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| The unspecified constant ''C'' present in the indefinite integral or antiderivative is known as the [[constant of integration]].
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| === Fundamental theorem ===
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| {{Main|Fundamental theorem of calculus}}
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| The [[fundamental theorem of calculus]] states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
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| The fundamental theorem of calculus states: If a function ''f'' is [[continuous function|continuous]] on the interval [''a'', ''b''] and if ''F'' is a function whose derivative is ''f'' on the interval (''a'', ''b''), then
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| :<math>\int_{a}^{b} f(x)\,dx = F(b) - F(a).</math>
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| Furthermore, for every ''x'' in the interval (''a'', ''b''),
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| :<math>\frac{d}{dx}\int_a^x f(t)\, dt = f(x).</math>
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| This realization, made by both [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]], who based their results on earlier work by [[Isaac Barrow]], was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for [[antiderivative]]s. It is also a prototype solution of a [[differential equation]]. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.
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| == Applications ==
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| [[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|right|The [[logarithmic spiral]] of the [[Nautilus|Nautilus shell]] is a classical image used to depict the growth and change related to calculus]]
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| Calculus is used in every branch of the physical sciences, [[actuarial science]], computer science, statistics, engineering, economics, business, medicine, [[demography]], and in other fields wherever a problem can be [[mathematical model|mathematically modeled]] and an [[optimization (mathematics)|optimal]] solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.
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| [[Physics]] makes particular use of calculus; all concepts in [[classical mechanics]] and [[electromagnetism]] are related through calculus. The [[mass]] of an object of known [[density]], the [[moment of inertia]] of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is [[Newton's laws of motion|Newton's second law of motion]]: historically stated it expressly uses the term "rate of change" which refers to the derivative saying ''The'' '''rate of change''' ''of momentum of a body is equal to the resultant force acting on the body and is in the same direction.'' Commonly expressed today as Force = Mass × acceleration, it involves differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.
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| Maxwell's theory of [[electromagnetism]] and [[Albert Einstein|Einstein]]'s theory of [[general relativity]] are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay. In biology, population dynamics starts with reproduction and death rates to model population changes.
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| Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with [[linear algebra]] to find the "best fit" linear approximation for a set of points in a domain. Or it can be used in [[probability theory]] to determine the probability of a continuous random variable from an assumed density function. In [[analytic geometry]], the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, [[Concave function|concavity]] and [[inflection points]].
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| [[Green's Theorem]], which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a [[planimeter]], which is used to calculate the area of a flat surface on a drawing. For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.
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| [[Discrete Green's Theorem]], which gives the relationship between a double integral of a function around a simple closed rectangular curve ''C'' and a linear combination of the antiderivative's values at corner points along the edge of the curve, allows fast calculation of sums of values in rectangular domains. For example, it can be used to efficiently calculate sums of rectangular domains in images, in order to rapidly extract features and detect object - see also the [[summed area table]] algorithm.
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| In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. From the decay laws for a particular drug's elimination from the body, it is used to derive dosing laws. In nuclear medicine, it is used to build models of radiation transport in targeted tumor therapies.
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| In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both [[marginal cost]] and [[marginal revenue]].
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| Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as [[Newton's method]], [[fixed point iteration]], and [[linear approximation]]. For instance, spacecraft use a variation of the [[Euler method]] to approximate curved courses within zero gravity environments.
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| == See also ==
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| {{Portal|Mathematics|Analysis}}
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| {{Main|Outline of calculus}}
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| === Lists ===
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| * [[List of calculus topics]]
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| * [[List of derivatives and integrals in alternative calculi]]
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| * [[List of differentiation identities]]
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| * [[List of publications in mathematics#Calculus|Publications in calculus]]
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| * [[Table of integrals]]
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| === See also ===
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| * [[Calculus of finite differences]]
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| * [[Calculus with polynomials]]
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| * [[Complex analysis]]
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| * [[Differential equation]]
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| * [[Differential geometry and topology|Differential geometry]]
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| * ''[[Elementary Calculus: An Infinitesimal Approach]]''
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| * [[Fourier series]]
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| * [[Integral equation]]
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| * [[Mathematical analysis]]
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| * [[Multivariable calculus]]
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| * [[Non-classical analysis]]
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| * [[Non-standard analysis]]
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| * [[Non-standard calculus]]
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| * [[Precalculus]] ([[Mathematics education|mathematical education]])
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| * [[Product integral]]
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| * [[Stochastic calculus]]
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| * [[Taylor series]]
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| == References ==
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| === Notes ===
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| {{Reflist|2}}
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| === Books ===
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| {{Refbegin|2}}
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| *[[Ron Larson (mathematician)|Larson, Ron]], Bruce H. Edwards (2010). ''Calculus'', 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2
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| *McQuarrie, Donald A. (2003). ''Mathematical Methods for Scientists and Engineers'', University Science Books. ISBN 978-1-891389-24-5
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| *{{Cite book |last1= Salas |first1= Saturnino L. |last2= Hille |first2= Einar |author2-link= Einar Hille |last3= Etgen |first3= Garret J. |year= 2006 |title= Calculus: One and Several Variables |edition= 10th |location= [[John Wiley & Sons|Wiley]] |isbn= 978-0-471-69804-3 }}
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| *[[James Stewart (mathematician)|Stewart, James]] (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8
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| *[[George B. Thomas|Thomas, George B.]], Maurice D. Weir, [[Joel Hass]], Frank R. Giordano (2008), ''Calculus'', 11th ed., Addison-Wesley. ISBN 0-321-48987-X
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| {{Refend}}
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| == Other resources ==
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| === Further reading ===
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| <!-- Please use APA style; it's a lot easier for it to stay universally readable. -->
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| {{Refbegin|2}}
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| * [[Carl Benjamin Boyer|Boyer, Carl Benjamin]] (1949). [http://books.google.com/books?id=KLQSHUW8FnUC&printsec=frontcover ''The History of the Calculus and its Conceptual Development'']. Hafner. Dover edition 1959, ISBN 0-486-60509-4
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| * [[Richard Courant|Courant, Richard]] ISBN 978-3-540-65058-4 ''Introduction to calculus and analysis 1.''
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| * [[Edmund Landau]]. ISBN 0-8218-2830-4 ''Differential and Integral Calculus'', [[American Mathematical Society]].
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| * Robert A. Adams. (1999). ISBN 978-0-201-39607-2 ''Calculus: A complete course''.
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| * Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) ''Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey'', Mathematical Association of America No. 7.
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| * [[John Lane Bell]]: ''A Primer of Infinitesimal Analysis'', Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses [[synthetic differential geometry]] and nilpotent infinitesimals.
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| * [[Florian Cajori]], "The History of Notations of the Calculus." ''Annals of Mathematics'', 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1–46.
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| * Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004.
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| * [[Cliff Pickover]]. (2003). ISBN 978-0-471-26987-8 ''Calculus and Pizza: A Math Cookbook for the Hungry Mind''.
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| * [[Michael Spivak]]. (September 1994). ISBN 978-0-914098-89-8'' Calculus''. Publish or Perish publishing.
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| * [[Tom M. Apostol]]. (1967). ISBN 978-0-471-00005-1 ''Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra''. Wiley.
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| * [[Tom M. Apostol]]. (1969). ISBN 978-0-471-00007-5 ''Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications''. Wiley.
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| * [[Silvanus P. Thompson]] and [[Martin Gardner]]. (1998). ISBN 978-0-312-18548-0 ''Calculus Made Easy''.
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| * [[Mathematical Association of America]]. (1988). ''Calculus for a New Century; A Pump, Not a Filter'', The Association, Stony Brook, NY. ED 300 252.
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| * Thomas/Finney. (1996). ISBN 978-0-201-53174-9 ''Calculus and Analytic geometry 9th'', Addison Wesley.
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| * Weisstein, Eric W. [http://mathworld.wolfram.com/SecondFundamentalTheoremofCalculus.html "Second Fundamental Theorem of Calculus."] From MathWorld—A Wolfram Web Resource.
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| *Howard Anton,Irl Bivens,Stephen Davis:"Calculus",John Willey and Sons Pte. Ltd.,2002.ISBN 978-81-265-1259-1
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| {{Refend}}
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| === Online books ===
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| {{Refbegin|2}}
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| * Boelkins, M. (2012). "''Active Calculus: a free, open text''". Retrieved 1 Feb 2013 from [http://gvsu.edu/s/km http://gvsu.edu/s/km]
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| * Crowell, B. (2003). "''Calculus''" Light and Matter, Fullerton. Retrieved 6 May 2007 from [http://www.lightandmatter.com/calc/calc.pdf http://www.lightandmatter.com/calc/calc.pdf]
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| * Garrett, P. (2006). "''Notes on first year calculus''" University of Minnesota. Retrieved 6 May 2007 from [http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf <nowiki>http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf</nowiki>]
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| * Faraz, H. (2006). "''Understanding Calculus''" Retrieved 6 May 2007 from Understanding Calculus, URL [http://www.understandingcalculus.com/ http://www.understandingcalculus.com/] (HTML only)
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| * Keisler, H. J. (2000). "''Elementary Calculus: An Approach Using Infinitesimals''" Retrieved 29 August 2010 from [http://www.math.wisc.edu/~keisler/calc.html <nowiki>http://www.math.wisc.edu/~keisler/calc.html</nowiki>]
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| * Mauch, S. (2004). "''Sean's Applied Math Book''" California Institute of Technology. Retrieved 6 May 2007 from [http://www.cacr.caltech.edu/~sean/applied_math.pdf <nowiki>http://www.cacr.caltech.edu/~sean/applied_math.pdf</nowiki>]
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| * Sloughter, Dan (2000). "''Difference Equations to Differential Equations: An introduction to calculus''". Retrieved 17 March 2009 from [http://synechism.org/drupal/de2de/ http://synechism.org/drupal/de2de/]
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| * Stroyan, K.D. (2004). "''A brief introduction to infinitesimal calculus''" University of Iowa. Retrieved 6 May 2007 from [http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm] (HTML only)
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| * Strang, G. (1991). "''Calculus''" Massachusetts Institute of Technology. Retrieved 6 May 2007 from [http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm]
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| * Smith, William V. (2001). "''The Calculus''" Retrieved 4 July 2008 [http://www.math.byu.edu/~smithw/Calculus/] (HTML only).
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| {{Refend}}
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| == External links ==
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| {{Sister project links|Calculus}}
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| * {{MathWorld | urlname=Calculus | title=Calculus}}
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| * {{PlanetMath | urlname=TopicsOnCalculus | title=Topics on Calculus | id=7592}}
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| * [http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf Calculus Made Easy (1914) by Silvanus P. Thompson] Full text in PDF
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| * {{In Our Time|Calculus|b00mrfwq|Calculus}}
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| * [http://www.calculus.org Calculus.org: The Calculus page] at University of California, Davis – contains resources and links to other sites
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| *[http://cow.math.temple.edu/ COW: Calculus on the Web] at Temple University – contains resources ranging from pre-calculus and associated algebra
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| *[http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
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| *[http://integrals.wolfram.com/ Online Integrator (WebMathematica)] from Wolfram Research
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| *[http://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematics] from ERICDigests.org
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| *[http://ocw.mit.edu/OcwWeb/Mathematics/index.htm OpenCourseWare Calculus] from the [[Massachusetts Institute of Technology]]
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| * [http://www.encyclopediaofmath.org/index.php?title=Infinitesimal_calculus&oldid=18648 Infinitesimal Calculus] – an article on its historical development, in Encyclopedia of Mathematics, [[Michiel Hazewinkel]] ed. .
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| *[http://math.mit.edu/~djk/calculus_beginners/ Calculus for Beginners and Artists] by Daniel Kleitman, MIT
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| *[http://www.math.ucdavis.edu/~kouba/ProblemsList.html Calculus Problems and Solutions] by D. A. Kouba
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| *[http://www.math.tamu.edu/~dallen/history/calc1/calc1.html Donald Allen's notes on calculus]
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| *[http://www.imomath.com/index.php?options=277 Calculus training materials at imomath.com]
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| *{{en icon}} {{ar icon}} [http://www.wdl.org/en/item/4327/ The Excursion of Calculus], 1772
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| {{Mathematics-footer}}
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| {{Use dmy dates|date=June 2011}}
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| [[Category:Calculus| ]]
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| {{Link FA|id}}
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