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• • • Calculus (from calculus, literally 'small pebble', used for counting and calculations, like on an ) is the study of continuous change, in the same way that is the study of shape and is the study of generalizations of. It has two major branches, (concerning rates of change and slopes of curves), and (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the. Both branches make use of the fundamental notions of of and to a well-defined. Generally, modern calculus is considered to have been developed in the 17th century by and. Today, calculus has widespread uses in,, and.

Calculus is a part of modern. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of and limits, broadly called. Calculus has historically been called 'the calculus of ', or 'infinitesimal calculus'. The term calculus (plural calculi) is also used for naming specific methods of calculation or notation as well as some theories, such as,,,, and.

Archimedes used the to calculate the area under a parabola. The ancient period introduced some of the ideas that led to calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of and, one goal of integral calculus, can be found in the (, c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components. From the age of, ( c.

408–355 BC) used the, which foreshadows the concept of the limit, to calculate areas and volumes, while ( c. 287–212 BC), inventing which resemble the methods of integral calculus. The was later discovered independently in by in the 3rd century AD in order to find the area of a circle. In the 5th century AD,, son of, established a method that would later be called to find the volume of a. Medieval [ ] In the Middle East, (c. 1040 CE) derived a formula for the sum of.

He used the results to carry out what would now be called an of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. And the thereby stated components of calculus. A complete theory encompassing these components is now well-known in the Western world as the or approximations.

However, they were not able to 'combine many differing ideas under the two unifying themes of the and the, show the connection between the two, and turn calculus into the great problem-solving tool we have today'. '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.' — In Europe, the foundational work was a treatise due to, 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, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Definition Of Data Files there.

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. The formal study of calculus brought together Cavalieri's infinitesimals with the developed in Europe at around the same time., claiming that he borrowed from, introduced the concept of, which represented equality up to an infinitesimal error term. The combination was achieved by,, and, the latter two proving the around 1670. Developed the use of calculus in his and. The and, the notions of and, and of [ ] were introduced by in an idiosyncratic notation which he used to solve problems of. 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, and many other problems discussed in his (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.

He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. Was the first to publish his results on the development of calculus. These ideas were arranged into a true calculus of infinitesimals by, who was originally accused of by Newton. He is now regarded as an 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 and, 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. Today, and are usually both given credit for independently inventing and developing calculus.

Newton was the first to apply calculus to general 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. When Newton and Leibniz first published their results, there was over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his ), but Leibniz published his ' first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the. This controversy divided English-speaking mathematicians from continental European 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. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus '. 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 both infinitesimal and was written in 1748. Foundations [ ] In calculus, foundations refers to the development of the subject from and definitions. In early calculus the use of quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably and.

Berkeley famously described infinitesimals as the in his book in 1734. 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. Several mathematicians, including, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of and, a way was finally found to avoid mere 'notions' of infinitely small quantities. The foundations of differential and integral calculus had been laid. In Cauchy's, we find a broad range of foundational approaches, including a definition of in terms of infinitesimals, and a (somewhat imprecise) prototype of an in the definition of differentiation. In his work Weierstrass formalized the concept of and eliminated infinitesimals (although his definition can actually validate infinitesimals).

Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called 'infinitesimal calculus'. 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 and the. In modern mathematics, the foundations of calculus are included in the field of, which contains full definitions and of the theorems of calculus. The reach of calculus has also been greatly extended. Invented and used it to define integrals of all but the most pathological functions. Introduced, which can be used to take the derivative of any function whatsoever.

Limits are not the only rigorous approach to the foundation of calculus. Another way is to use 's.

Robinson's approach, developed in the 1960s, uses technical machinery from to augment the real number system with and numbers, as in the original Newton-Leibniz conception. The resulting numbers are called, and they can be used to give a Leibniz-like development of the usual rules of calculus. There is also, which differs from non-standard analysis in that it mandates neglecting higher power infinitesimals during derivations.

Significance [ ] While many of the ideas of calculus had been developed earlier in,,,, and, the use of calculus began in Europe, during the 17th century, when and 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. Applications of differential calculus include computations involving and, the of a curve, and. Applications of integral calculus include computations involving area,,,,, and. More advanced applications include and.

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 or sums of infinitely many numbers. These questions arise in the study of and area.

The philosopher gave several famous examples of such. Calculus provides tools, especially the and the, that resolve the paradoxes. Principles [ ] Limits and infinitesimals [ ].

The of the is a classical image used to depict the growth and change related to calculus. Calculus is used in every branch of the physical sciences,, computer science, statistics, engineering, economics, business, medicine,, and in other fields wherever a problem can be and an 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. Makes particular use of calculus; all concepts in and are related through calculus.

The of an object of known, the 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: historically stated it expressly uses the term 'change of motion' which implies the derivative saying The 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 implies 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.

Maxwell's theory of and 's theory of 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. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with to find the 'best fit' linear approximation for a set of points in a domain. Or it can be used in to determine the probability of a continuous random variable from an assumed density function. In, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, and., 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, 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., 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; another algorithm that could be used is the. 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. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both and. 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,, and. For instance, spacecraft use a variation of the to approximate curved courses within zero gravity environments. Varieties [ ] Over the years, many reformulations of calculus have been investigated for different purposes.

Non-standard calculus [ ]. Main article: Imprecise calculations with infinitesimals were widely replaced with the rigorous starting in the 1870s. Meanwhile, calculations with infinitesimals persisted and often led to correct results. This led to investigate if it were possible to develop a number system with infinitesimal quantities over which the theorems of calculus were still valid. In 1960, building upon the work of and, he succeeded in developing. The theory of non-standard analysis is rich enough to be applied in many branches of mathematics. As such, books and articles dedicated solely to the traditional theorems of calculus often go by the title.

Smooth infinitesimal analysis [ ]. Dover edition 1959, • Introduction to calculus and analysis 1. Differential and Integral Calculus,. Calculus: A complete course. • 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. •: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998.. Uses and nilpotent infinitesimals. •, 'The History of Notations of the Calculus.' Annals of Mathematics, 2nd Ser., Vol. 1923), pp. 1–46.

Lebedev and Michael J. Cloud: 'Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus', Princeton Univ. Calculus and Pizza: A Math Cookbook for the Hungry Mind.

(September 1994). Publish or Perish publishing. Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra. Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications. Calculus Made Easy. Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY.

• Thomas/Finney. Calculus and Analytic geometry 9th, Addison Wesley. • Weisstein, Eric W. From MathWorld—A Wolfram Web Resource. • Howard Anton, Irl Bivens, Stephen Davis:'Calculus', John Willey and Sons Pte. Edwards (2010). Calculus, 9th ed., Brooks Cole Cengage Learning.

• McQuarrie, Donald A. Mathematical Methods for Scientists and Engineers, University Science Books. • Salas, Saturnino L.;; Etgen, Garret J. Calculus: One and Several Variables (10th ed.)... Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning. •, Maurice D. Weir,, Frank R.

Giordano (2008), Calculus, 11th ed., Addison-Wesley. Online books [ ].

• Boelkins, M. Archived from on 30 May 2013. Retrieved 1 February 2013. • Crowell, B.

Light and Matter, Fullerton. Retrieved 6 May 2007 from • Garrett, P. ' Notes on first year calculus'. University of Minnesota. Retrieved 6 May 2007 from • Faraz, H. Download Naruto Rise Of A Ninja Pc Tpbank.

' Understanding Calculus'. Retrieved 6 May 2007 from UnderstandingCalculus.com, URL (HTML only) • Keisler, H. ' Elementary Calculus: An Approach Using Infinitesimals'.

Retrieved 29 August 2010 from • Mauch, S. ' Sean's Applied Math Book' (pdf).

California Institute of Technology. Retrieved 6 May 2007 from • Sloughter, Dan (2000). ' Difference Equations to Differential Equations: An introduction to calculus'. Retrieved 17 March 2009 from • Stroyan, K.D. ' A brief introduction to infinitesimal calculus'. University of Iowa.

Retrieved 6 May 2007 from (HTML only) • Strang, G. ' Calculus' Massachusetts Institute of Technology. Retrieved 6 May 2007 from • Smith, William V. ' The Calculus'. Retrieved 4 July 2008 (HTML only). External links [ ].

Find more about Calculusat Wikipedia's • from Wiktionary • from Wikimedia Commons • from Wikinews • from Wikiquote • from Wikisource • from Wikibooks • from Wikiversity •, ed. (2001) [1994],,, Springer Science+Business Media B.V. / Kluwer Academic Publishers, •.. • Full text in PDF • • on at the. • at University of California, Davis – contains resources and links to other sites • at Temple University – contains resources ranging from pre-calculus and associated algebra • • from Wolfram Research • from ERICDigests.org • from the • – an article on its historical development, in Encyclopedia of Mathematics, ed.. • Daniel Kleitman, MIT..

Kouba • • • (in English) (in Arabic), 1772.