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Calculus is a central branch of mathematics, developed from algebra and geometry. The word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus."
Calculus is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behaviour of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus.
Examples of typical differential calculus problems include:
- finding the optimal number of units a company should produce to maximize its profit.
Examples of integral calculus problems include:
- finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure
- finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.
Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.
- Main article: Derivative
The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:
for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled, as measured by the odometer, as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.
Differential calculus can be used to determine the instantaneous speed at any given instant, while the formula "speed = distance divided by time" only gives the average speed, and cannot be applied to an instant in time because it then gives an undefined quotient zero divided by zero. Calculus avoids division by zero by using the concept of the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity. More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.
The derivative of a function, if it exists, gives information about its graph. It is useful for finding optimum solutions to problems, called maxima and minima of a function. It is proved mathematically that these optimum solutions exist either where the tangent of the graph is flat, so that the slope is zero; or where the graph has a sharp turn (cusp) where the derivative does not exist; or at the endpoints of the graph. Another application of differential calculus is Newton's method, a powerful equation solving algorithm. Differential calculus has been applied to many questions that were first formulated in other areas, such as business or medicine.
The derivative lies at the heart of the physical sciences. Newton's law of motion, Force = Mass × Acceleration, involves calculus because acceleration is the derivative of the velocity. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas.
The derivative of a function y = f(x) with respect to x is usually expressed as either y ′ (read "y-prime"), f ' (x) (read "f-prime of x") or as
which is commonly shortened to:
- Main article: Integral
There are two types of integral in calculus, the indefinite and the definite. The indefinite integral is simply the antiderivative. That is, F is an antiderivative of f when f is a derivative of F. (This use of capital letters and lower case letters is common in calculus. The lower case letter represents the derivative of the capital letter.)
The definite integral evaluates the cumulative effect of many small changes in a quantity. The simplest instance is the formula
for calculating the distance a car moves during a period of time when it is traveling at constant speed. The distance moved is the cumulative effect of the small distances moved in each instant. Calculus is also able to deal with the natural situation in which the car moves with changing speed.
Integral calculus determines the exact distance traveled during an interval of time by creating a series of better and better approximations, called Riemann sums, that approach the exact distance as a limit. More formally, we say that the definite integral of a function on an interval is a limit of Riemann sum approximations.
Applications of integral calculus arise whenever the problem is to compute a number that is in principle (approximately) equal to the sum of a large number of small quantities. The classic geometric application is to area computations. In principle, the area of a region can be approximated by chopping it up into many pieces (typically rectangles, or, in polar coordinates, circular sectors), and then adding the areas of those pieces. The length of an arc, the area of a surface, and the volume of a solid can also be expressed as definite integrals. Probability, the basis for statistics, provides another important application of integral calculus.
The symbol of integration is ∫, a stretched s (which stands for "sum"). The precise meanings of expressions involving integrals can be found in the main article Integral. The definite integral, written as:
is read "the integral from a to b of f(x) dx".
There is more than one rigorous approach to the foundation of calculus. The usual one is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. Calculus also can be rigorously developed by using explicit uniform estimates. This approach avoids both limits and infinitesimals, it is more in the spirit of constructive mathematics that fits much better with the computational aspect of the subject than the other two approches. It is also much simpler conceptually and technically. The tools of calculus include techniques associated with elementary algebra, and mathematical induction. The foundations of calculus are included in the field of real analysis, which contains all full definitions and proofs of the theorems of calculus as well as generalisations such as measure theory and distribution theory.
Fundamental theorem of calculus
- Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another continuous function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.
Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then
- Also, for every x in the interval [a, b],
This realization, made by both Newton and Leibniz, 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 antiderivatives. 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.
- Main article: Use of calculus in psychological research
The development and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins. The success of calculus has been extended over time to differential equations, vector calculus, calculus of variations, complex analysis, and differential topology.
- Main article: History of calculus
The origins of integral calculus are generally regarded as going back no further than to the time of the ancient Greeks, circa 200 BC. The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this method further, inventing heuristic methods which resemble integral calculus. After him, the development of calculus did not advance appreciably for over 500 years.1
In India, the mathematician-astronomer Aryabhata in 499 used infinitesimals and expressed an astronomical problem in the form of a basic differential equation. Manjula in the 10th century elaborated on this differential equation in a commentary. This equation eventually led Bhaskara in the 12th century to develop a number of ideas that are foundational to the development of modern calculus, including the statement of the theorem now known as "Rolle's theorem". He was also the first to define the notion of the derivative as a limit. In the 14th century, Madhava, along with other mathematician-astronomers of the Kerala School, studied infinite series, power series, Taylor series, differentiation, integration, and the mean value theorem. Yuktibhasa, which some consider to be the first text on calculus, summarizes these results. These developments would not be duplicated in Europe until much later.
Calculus started making great strides in Europe towards the end of the early modern period and into the first years of the eighteenth century. This was a time of major innovation in Europe, making accessible answers to old questions. Calculus provided a new method in mathematical physics. Several mathematicians contributed to this breakthrough, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668. In Japan at around this time, Seki Kowa expanded further upon Eudoxus's method of exhaustion.
Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the probably independent and nearly simultaneous "invention" of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The fundamental insight that both Newton and Leibniz had was the fundamental theorem of calculus. Virtually all modern methods of symbolic integration follow from this theorem, and it has proven indispensible in the development of modern mathematics and physics. For example, see Integration by parts and Integration by substitution.
When Newton and Leibniz first published their results, there was some controversy over whether Leibniz's work was independent of Newton's. While Newton derived his results years before Leibniz, it was only some time after Leibniz published in 1684 that Newton published. Later, Newton would claim that Leibniz got the idea from Newton's notes on the subject; however examination of the papers of Leibniz and Newton show they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Today, both Newton and Leibniz are given credit for independently developing calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluxions". Some others who contributed important ideas are Descartes, Barrow, Fermat, Huygens, and Wallis.
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 18th century, calculus was put on a much more rigorous footing by Cauchy, Riemann, Weierstrass, and others. It was also during this time period that the ideas of calculus were generalized to Euclidean space and the complex plane. Calculus continues to be further generalized, such as the development of the Lebesgue integral in 1900.
Archimedes, Method, in The Works of Archimedes ISBN 0-521-66160-9
- Basic calculus equations and formulas
- Calculus with polynomials
- Differential geometry
- List of calculus topics
- Publications in calculus
- Multivariable calculus
- Non-standard analysis
- Precalculus (mathematical education)
- Tom M Apostol (1967). Calculus, 2nd Ed. Wiley. ISBN 0-471-00005-1 and ISBN 0-471-00007-8.
- Archimedes. Method, in The Works of Archimedes. ISBN 0-521-66160-9.
- Carl B. Boyer (1949). The History of the Calculus and its Conceptual Development.
- James M. Henle and Eugene M. Kleinberg (2003). Infinitesimal Calculus, Dover Publications. ISBN 0-486-42886-9. Uses nonstandard analysis and hyperreal infinitesimals.
- George Gheverghese Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Ed., Penguin Books. ISBN 0-14-021118-1.
- Kalakriti (2002). 'History of Mathematics in India', South Asian History.
- John J. O'Connor and Edmund F. Robertson (1997). 'Takakazu Seki Kowa', MacTutor History of Mathematics archive, St Andrews University.
- John J. O'Connor and Edmund F. Robertson (2000-2004). 'Index of Ancient Indian mathematics', MacTutor History of Mathematics archive, St Andrews University.
- Ian G. Pearce (2002). 'Indian Mathematics: Redressing the balance', MacTutor History of Mathematics archive, St Andrews University.
- Robert A. Adams. (1999) ISBN 0-201-39607-6 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. 7,
- John L. Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN 0-521-62401-0. Uses synthetic differential geometry and nilpotent infinitesimals
- 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
- Cliff Pickover. (2003) ISBN 0-471-26987-5 Calculus and Pizza: A Math Cookbook for the Hungry Mind.
- Michael Spivak. (Sept 1994) ISBN 0-914098-89-6 Calculus. Publish or Perish publishing.
- Silvanus P. Thompson and Martin Gardner. (1998) ISBN 0-312-18548-0 Calculus Made Easy.
- Mathematical Association of America. (1988) Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300 252.
- Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin
- Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa
- Mauch, Sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus
- Crowell, Benjamin, Calculus, Fullerton College, an online textbook
- Garrett, Paul, Notes on First-Year Calculus
- Hussain, Faraz, Understanding Calculus, a complete online book with a conceptual focus
- Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus
- Online Integrator by Mathematica
- step-by-step derivatives and integrals
- The Role of Calculus in College Mathematics
- Calculus on MIT OpenCourseWare
- Example problems from all areas of Calculus
- Madhava of Sangamagramma
- Work of Bhaskaracharya II
- Online Calculus Graphing Calculator - WebGraphing.com
Major fields of mathematics
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