Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Historically, it was sometimes referred to as "the calculus of infinitesimals", but that usage is seldom seen today. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Most basically, calculus is the study of change, in the same way that geometry is the study of space. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Elementary algebra is a fundamental and relatively basic form of Algebra taught to students who are presumed to have little or no formal knowledge of Mathematics beyond Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions. Analysis has its beginnings in the rigorous formulation of Calculus. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

More generally, calculus (plural calculi) can refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, predicate calculus, relational calculus, and lambda calculus. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Mathematical logic, predicate logic is the generic term for symbolic Formal systems like First-order logic, Second-order logic, Many-sorted Relational calculus consist of two calculi the Tuple relational calculus and the Domain relational calculus, that are part of the Relational model In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function

History

Sir Isaac Newton is one of the most famous contributors to the development of calculus, with, among other things, the use of calculus in his laws of motion and gravitation. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements

Development

Main article: History of calculus

The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. This is a sub-article to Calculus and History of mathematics. "Ancient" redirects here For other uses see Ancient_(disambiguation. The term modern period or modern era (sometimes also modern times) is the period of history that followed the Middle Ages between c The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically A pyramid is a Building where the upper surfaces are triangular and converge on one point Elements special cases and related concepts Each plane section is a base of the frustum ^[1][2] From the school of Greek mathematics, Eudoxus (c. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space ^[3] The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Mathematics in China emerged independently by the 11th century BC Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere. Zu Chongzhi ( 429–500 Courtesy name Wenyuan (文遠 was a prominent Chinese mathematician and astronomer during the Liu "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe ^[2]

In the 12th century, the Indian mathematician Bhāskara II developed an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem". Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, a branch of Mathematics, Rolle's theorem essentially states that a differentiable function, which attains equal values at two points must ^[4] Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhazen) was the first to derive the formula for the sum of the fourth powers, and using mathematical induction, he developed a method that is readily generalizable to finding the formula for the sum of any integral powers, which was fundamental to the development of integral calculus. TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized In Mathematics and elsewhere the adjective quartic means fourth order, such as the function x^4 Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French ^[5] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus. layout and formatting it should ensure no clashes with the top of the infobox (1135 - 1213 was a Persian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages) In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. ^[6] In the 14th century, Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,^[7] which are treated in the text Yuktibhasa. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives Yuktibhasa ( Malayalam:യുക്തിഭാഷ; meaning &mdash rationale language) also known as Ganita Yuktibhasa (compendium of astronomical ^[8][9][10]

In the modern period, independent discoveries in calculus were being made in early 17th century Japan, by mathematicians such as Seki Kowa, who expanded upon the method of exhaustion. In the History of mathematics, Japanese mathematics or wasan (和算 denotes a genuinely distinct kind of mathematics developed in Japan during the or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the In Europe, the second half of the 17th century was a time of major innovation. Calculus provided a new opportunity in mathematical physics to solve long-standing problems. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the Isaac Barrow (October 1630 &ndash May 4, 1677) was an English scholar and Mathematician who is generally given credit for his early role James Gregory proved a special case of the second fundamental theorem of calculus in AD 1668. James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration.

Gottfried Wilhelm Leibniz was originally accused of plagiarism of Sir Isaac Newton's unpublished works, but is now regarded as an independent inventor and contributor towards calculus. Plagiarism is the unauthorized use or close imitation of the language and thoughts of another author and the representation of them as one's own original work

Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the independent and nearly simultaneous invention of calculus. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements 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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The basic insight that both Newton and Leibniz had was the fundamental theorem of calculus. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration.

When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. The Calculus controversy was an argument between Seventeenth-century Mathematicians Isaac Newton and Gottfried Leibniz over who had Newton derived his results first, but Leibniz published 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 "the science of fluxions". Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Lebesgue further generalized the notion of the integral. Henri Léon Lebesgue leɔ̃ ləˈbɛg ( June 28, 1875, Beauvais &ndash July 26, 1941, Paris) was a French

Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development. ^[11]

Significance

While some of the ideas of calculus were developed earlier, in Greece, China, India, Iraq, Persia, and Japan, the modern 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 the basic principles of calculus. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Mathematics in China emerged independently by the 11th century BC Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. In the History of mathematics, Japanese mathematics or wasan (和算 denotes a genuinely distinct kind of mathematics developed in Japan during the Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements This work had a strong impact on the development of physics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. In Physics, velocity is defined as the rate of change of Position. Slope is used to describe the steepness incline gradient or grade of a straight line. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Physics, mechanical work is the amount of Energy transferred by a Force. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface More advanced applications include power series and Fourier series. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.

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. In These questions arise in the study of motion and area. In Physics, motion means a constant change in the location of a body Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Zeno of Elea (ˈziːnoʊ əv ˈɛliə Greek: Ζήνων ὁ Ἐλεάτης (ca Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

Foundations

In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse 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.

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. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below In Mathematics, the real numbers may be described informally in several different ways An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory. Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions

Principles

Limits and infinitesimals

Main article: Limit (mathematics)

Calculus is usually developed by manipulating very small quantities. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Historically, the first method of doing so was by infinitesimals. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have These are objects which can be treated like numbers but which are, in some sense, "infinitely small". On a number line, these would be locations which are not zero, but which have zero distance from zero. No non-zero number is an infinitesimal, because its distance from zero is positive. Any multiple of an infinitesimal is still infinitely small, in other words, infinitesimals do not satisfy the Archimedean property. In Abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups From this viewpoint, calculus is a collection of techniques for manipulating infinitesimals. This viewpoint 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. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of Infinitesimals Based on the ideas of F

In the 19th century, infinitesimals were replaced by limits. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Limits describe the value of a function at a certain input in terms of its values at nearby input. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function They capture small-scale behavior, just like infinitesimals, but use ordinary numbers. From this viewpoint, 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 easy to put on rigorous foundations, and for this reason they are the standard approach to calculus.

Derivatives

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.

Main article: Derivative

Differential calculus is the study of the definition, properties, and applications of the derivative or slope of a graph. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Slope is used to describe the steepness incline gradient or grade of a straight line. The process of finding the derivative is called differentiation. In technical language, the derivative is a linear operator, which inputs a function and outputs a second function, so that at every point the value of the output is the slope of the input. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

The concept of the derivative is fundamentally more advanced than the concepts encountered in algebra. In algebra, students learn about functions which input a number and output another number. For example, if the doubling function inputs 3, then it outputs 6, while if the squaring function inputs 3, it outputs 9. But the derivative inputs a function and outputs another function. For example, if the derivative inputs the squaring function, then it outputs the doubling function, because the doubling function gives the slope of the squaring function at any given point.

To understand the derivative, students must learn mathematical notation. In mathematical notation, one common symbol for the derivative of a function is an apostrophe-like mark called prime. The prime symbol ( ′  double prime symbol ( &Prime  triple prime symbol ( ‴  etc Thus the derivative of f is f′ (spoken "f prime"). The last sentence of the preceding paragraph, in mathematical notation, would be written

If the input of a function is time, then the derivative of that function is the rate at which the function changes.

If a function is linear (that is, if the graph of the function is a straight line), then the function can be written y = mx + b, where:

. In Mathematics, the term linear function can refer to either of two different but related concepts In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x)

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, and we can use calculus to find an exact value at a given point. (Note that y and f(x) represent the same thing: the output of the function. This is known as function notation. ) A line through two points on a curve is called a secant line. The slope, or rise over run, of a secant line can be expressed as

where the coordinates of the first point are (x, f(x)) and h is the horizontal distance between the two points. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point

To determine the slope of the curve, we use the limit:

.

Working out one particular case, we find the slope of the squaring function at the point where the input is 3 and the output is 9 (i. e. , f(x) = x², so f(3) = 9).

The slope of 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 generalized to any point on the graph of any function. The procedure can be visualized as in the following figure.

Tangent line as a limit of secant lines. 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) = x³ − 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.

Integrals

Main article: Integral

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

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 upper- and lower-case letters for a function and its indefinite integral is common in calculus. )

The definite integral inputs a function and outputs a number, which gives the area between the graph of the input and the x-axis. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral.

A motivating example is the distances traveled in a given time.

If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. 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. In Mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph otherwise known as an Integral. 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.

Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).

If f(x) in the diagram on the left 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.

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.

The symbol of integration is , an elongated S (which stands for "sum"). The definite integral is written as:

and is read "the integral from a to b of f-of-x with respect to x. "

The indefinite integral, or antiderivative, is written:

.

Functions differing by only a constant have the same derivative, and therefore the antiderivative of a given function is actually a family of functions differing only by a constant. Since the derivative of the function y = x² + C, where C is any constant, is y′ = 2x, the antiderivative of the latter is given by:

.

An undetermined constant like C in the antiderivative is known as a constant of integration. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant

Fundamental theorem

Main article: Fundamental theorem of calculus

The fundamental theorem of calculus states that differentiation and integration are inverse operations. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. 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.

The Fundamental Theorem of Calculus states: 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

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

This realization, made by both Newton and 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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Isaac Barrow (October 1630 &ndash May 4, 1677) was an English scholar and Mathematician who is generally given credit for his early role The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative It is also a prototype solution of a differential equation. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Applications

The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus

Calculus is used in every branch of the physical sciences, in computer science, statistics, engineering, economics, business, medicine, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta Nautilus (from Greek ναυτίλος, 'sailor' is the common name of any marine creatures of the Cephalopod family Nautilidae, the sole Physical science is an encompassing term for the branches of Natural science and Science that study non-living systems in contrast to the biological sciences Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Economics is the social science that studies the production distribution, and consumption of goods and services. A business (also called firm or an enterprise) is a legally recognized organizational entity designed to provide goods and/or services to Medicine is the art and science of healing It encompasses a range of Health care practices evolved to maintain and restore Human Health by the Note The term model has a different meaning in Model theory, a branch of Mathematical logic. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function

Physics makes particular use of calculus; all concepts in classical mechanics are interrelated through calculus. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different This article is about the moment of inertia of a rotating object. In the subfields of electricity and magnetism calculus can be used to find the total flux of electromagnetic fields. In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks A more historical example of the use of calculus in physics is Newton's second law of motion, it expressly uses the term "rate of change" which refers to the derivative: 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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Even the common expression of Newton's second law as Force = Mass × Acceleration involves differential calculus because acceleration can be expressed as the derivative of velocity. Maxwell's theory of electromagnetism and Einstein's theory of general relativity are also expressed in the language of differential calculus. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Chemistry also uses calculus in determining reaction rates and radioactive decay.

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. Linear algebra is the branch of Mathematics concerned with

In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.

In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maximums and minimums), slope, concavity and inflection points. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, a concave function is the negative of a Convex function. In Differential calculus, an inflection point, or point of inflection (or inflexion) is a point on a Curve at which the Curvature

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue. In Economics and Finance, marginal cost is the change in Total cost that arises when the quantity produced changes by one unit In Microeconomics, Marginal Revenue ( MR) is the extra revenue that an additional unit of product will bring

Calculus can be used to find approximate solutions to equations, in methods such as Newton's method, fixed point iteration, and linear approximation. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the In Numerical analysis, fixed point iteration is a method of computing fixed points of Iterated functions More specifically given a function f In Mathematics, a linear approximation is an approximation of a general function using a Linear function (more precisely an Affine function For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments. In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving

See also

Lists

• List of basic calculus topics
• List of basic calculus equations and formulas
• List of calculus topics
• Publications in calculus
• Table of integrals

Related topics

• Calculus with polynomials
• Differential geometry
• Mathematics
• Multivariable calculus
• Non-standard analysis
• Precalculus (mathematical education)
• Product Integrals
• Stochastic calculus

References

Notes

Books

Other resources

Further reading

Online books

Web pages

• Eric W. Weisstein, Calculus at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Topics on Calculus at PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.
• Calculus Made Easy (1914) by Silvanus P. ThompsonFull text in PDF
• The Online Calculus course for transfer, notes, video lectures, active forum at San Francisco State University by Professor Arek Goetz
• Calculus.org: The Calculus page at University of California, Davis — contains resources and links to other sites
• COW: Calculus on the Web at Temple University - contains resources ranging from pre-calculus and associated algebra
• Online Integrator (WebMathematica) from Wolfram Research
• The Role of Calculus in College Mathematics from ERICDigests. org
• OpenCourseWare Calculus from the Massachusetts Institute of Technology
• Infinitesimal Calculus — an article on its historical development, in Encyclopaedia of Mathematics, Michiel Hazewinkel ed. .