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Encyclopedia > Calculus
Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem Look up Calculus in Wiktionary, the free dictionary. ... 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 analysis. ... 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 multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...

Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives This article is about derivatives and differentiation in mathematical calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... 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 derivatives exist. ... 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 may not be given explicitly in terms of the independent variable. ... The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial... 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 are known. ... The primary operation in differential calculus is finding a derivative. ...

Integration

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells
, substitution,
trigonometric substitution,
partial fractions This article is about the concept of integrals in calculus. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... 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. ...

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. Historically, it was sometimes referred to as "the calculus of infinitesimals", but that usage is seldom seen today. Calculus has widespread applications in science and engineering and is used to solve problems for which algebra alone is insufficient. 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. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions. For other uses, see Latins and Latin (disambiguation). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... This article is about derivatives and differentiation in mathematical calculus. ... This article is about the concept of integrals in calculus. ... In mathematics, a series is a sum of a sequence of terms. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... 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 arithmetic. ... This article is about the branch of mathematics. ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... This article deals with the concept of an integral in calculus. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... This article is about functions in mathematics. ...


More generally, calculus can refer to any method or system of calculation.

Contents

History

Aryabhatta along with other Indian mathematicians over the centuries made important contribution to the field of calculus.
Aryabhatta along with other Indian mathematicians over the centuries made important contribution to the field of calculus.
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 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.

Image File history File links Download high-resolution version (951x1361, 193 KB) [edit] Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... Image File history File links Download high-resolution version (951x1361, 193 KB) [edit] Summary This image of a public statue in IUCAA Pune was photographed in May 2006 by myself, and I release all rights. ... Aryabhata (आर्यभट) (Āryabhaṭa) is the first of the great astronomers of the classical age of India. ... Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Sir Isaac Newton in Knellers portrait of 1689. ...

Development

Main article: History of calculus

The history of calculus falls into several distinct time periods, most notably the ancient, medieval, and modern periods. 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. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum.[1][2] From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus.[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. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.[2] // The method of integration can be traced back to the Egyptians, in the Moscow Mathematical Papyrus circa 1800 BC, which gives the formula for finding the volume of a pyramidal frustrum. ... “Ancient” redirects here. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ... For other uses, see Volume (disambiguation). ... For other meanings, see pyramid (disambiguation). ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... 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 AD around the Eastern shores of the Mediterranean. ... Another article concerns Eudoxus of Cyzicus. ... 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 area of the containing shape. ... For other uses, see Archimedes (disambiguation). ... For heuristics in computer science, see heuristic (computer science) Heuristic is the art and science of discovery and invention. ... This article is about the concept of integrals in calculus. ... 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 area of the containing shape. ... A possible likeness of Liu Hui on japanpostage stamp This is a Chinese name; the family name is 劉 (Liu) Liu Hui 劉徽 was a Chinese mathematician who lived in the 200s in the Wei Kingdom. ... Zu Chongzhi (Traditional Chinese: ; Simplified Chinese: ; Hanyu Pinyin: ZÇ” ChōngzhÄ«; Wade-Giles: Tsu Chung-chih, 429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. ... For other uses, see Sphere (disambiguation). ...


In AD 499 the Indian mathematician Aryabhata used the notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[4] This equation eventually led Bhāskara II in the 12th century to develop an early derivative representing infinitesimal change, and he described an early form of "Rolle's theorem".[5] 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.[6] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.[7] 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,[8] which are treated in the text Yuktibhasa.[9][10][11] For other uses, see Aryabhata (disambiguation). ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician and astronomer. ... This article is about derivatives and differentiation in mathematical calculus. ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... In mathematics and elsewhere, the adjective quartic means fourth order, such as the function . ... “Exponent” redirects here. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... The integers are commonly denoted by the above symbol. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... Sharafeddin Muzzafar-i Tusi (1135 - 1213) was a Persian mathematician of the Middle Ages. ... This article is about derivatives and differentiation in mathematical calculus. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Series expansion redirects here. ... Yuktibhasa (Malayalam:യുക്തിഭാഷ ; meaning — rationale language ) also known as Ganita Yuktibhasa (compendium of astronomical rationale) is a major treatise on Mathematics and Astronomy, written by Indian astronomer Jyesthadeva of the Kerala School of Mathematics in AD 1530. ...


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 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. Several mathematicians contributed to these breakthroughs, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in AD 1668. Kowa Seki (Seki Takakazu, 関 孝和) (1642? – October 24, 1708) was a Japanese mathematician. ... 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 area of the containing shape. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the... James Gregory For other people with the same name, see James Gregory. ... 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.
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.

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. 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 basic insight that both Newton and Leibniz had was the fundamental theorem of calculus. Image File history File links Gottfried_Wilhelm_von_Leibniz. ... Image File history File links Gottfried_Wilhelm_von_Leibniz. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... For other uses, see Plagiarism (disambiguation). ... Leibniz redirects here. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... 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. 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". Isaac Newton began working on a form of the calculus in 1666. ... Method of Fluxions was a book by Isaac Newton. ...


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. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue further generalized the notion of the integral. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Bernhard Riemann. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. ...


Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development.[12]


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. This work had a strong impact on the development of physics. In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. ... For other uses, see Europe (disambiguation). ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...


Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway. This article is about velocity in physics. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... This article is about the mathematical term. ... 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 by systematically choosing the values of real or integer variables from within an allowed set. ... This article is about the physical quantity. ... For other uses, see Volume (disambiguation). ... Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... Work (abbreviated W) is the energy transferred in applying force over a distance. ... This article is about pressure in the physical sciences. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...


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 and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes. For the album by Hux Flux, see Division by Zero (album). ... This article or section is in need of attention from an expert on the subject. ... This article is about the physical quantity. ... Beginning of Homers Odyssey The Ancient Greek language is the historical stage of the Greek language[1] as it existed during the Archaic (9th–6th centuries BC) and Classical (5th–4th centuries BC) periods in Ancient Greece. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... Zeno of Elea (IPA:zɛnoʊ, ɛlɛɑː)(circa 490 BC? – circa 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. ... “Arrow paradox” redirects here. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a series is a sum of a sequence of terms. ...


Foundations

In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. 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. For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ...


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. 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. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, the word continuum sometimes denotes the real line. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... In mathematics, a measure is a function that assigns a number, e. ... 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. Historically, the first method of doing so was by infinitesimals. 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. 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. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ...


In the 19th century, infinitesimals were replaced by limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but using 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. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... This article is about functions in mathematics. ...


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. 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... This article is about derivatives and differentiation in mathematical calculus. ... This article is about derivatives and differentiation in mathematical calculus. ... This article is about the mathematical term. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...


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. Thus the derivative of f is f′ (spoken "f prime"). The last sentence of the preceding paragraph, in mathematical notation, would be written This article is not about the symbol for the set of prime numbers, â„™. The prime (′, Unicode U+2032, ′) is a symbol with many mathematical uses: A complement in set theory: A′ is the complement of the set A A point related to another (e. ...

 begin{align} f(x) &= x^2  f ' (x) &= 2x. end{align}

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: A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ...

m= frac{mbox{rise}}{mbox{run}}= {mbox{change in } y over mbox{change in } x} = {Delta y over{Delta 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

m = {f(x+h) - f(x)over{(x+h) - x}} = {f(x+h) - f(x)over{h}},

where the coordinates of the first point are (x, f(x)) and h is the horizontal distance between the two points. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...


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

lim_{h to 0}{f(x+h) - f(x)over{h}}.

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).

 begin{align} f'(3)&=lim_{h to 0}{(3+h)^2 - 9over{h}}  &=lim_{h to 0}{9 + 6h + h^2 - 9over{h}}  &=lim_{h to 0}{6h + h^2over{h}}  &=lim_{h to 0} (6 + h)  &= 6 end{align}

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) = x3x. 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 process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators. This article is about the concept of integrals in calculus. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...


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. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a Riemann sum is a method for approximating the values of integrals. ...


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

mathrm{Distance} = mathrm{Speed} cdot mathrm{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. 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. In mathematics, a Riemann sum is a method for approximating the values of integrals. ...

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


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

int_a^b f(x), dx

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


The indefinite integral, or antiderivative, is written:

int f(x), dx.

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:

int 2x, dx = x^2 + C.

An undetermined constant like C in the antiderivative is known as a constant of integration. In calculus, the indefinite integral of a given function (i. ...


Fundamental theorem

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. 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. ...


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 In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

int_{a}^{b} f(x),dx = F(b) - F(a).

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

frac{d}{dx}int_a^x f(t), dt = f(x).

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. 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. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Leibniz redirects here. ... Isaac Barrow (October 1630 - May 4, 1677) was an English divine, scholar and mathematician who is generally given minor credit for his role in the development of modern calculus; in particular, for his work regarding the tangent; for example, Barrow is given credit for being the first to calculate the... In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...


Applications

The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
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. Download high resolution version (1095x862, 107 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1095x862, 107 KB) Wikipedia does not have an article with this exact name. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Genera Allonautilus Nautilus Nautilus (from Greek ναυτίλος, sailor) is the common name of any marine creatures of the cephalopod family Nautilidae, the sole family of the suborder Nautilina. ... == Headline text ==cant there be some kind of picture somewhere so i can see by picture???? Physical science is a 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 of the theoretical foundations of information and computation and their implementation and application in computer systems. ... This article is about the field of statistics. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... In economics, a business is a legally-recognized organizational entity existing within an economically free country designed to sell goods and/or services to consumers, usually in an effort to generate profit. ... For the chemical substances known as medicines, see medication. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... 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 by systematically choosing the values of real or integer variables from within an allowed set. ...


Physics makes particular use of calculus; all concepts in classical mechanics are interrelated 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. In the subfields of electricity and magnetism calculus can be used to find the total flux of electromagnetic fields. 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. 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. Chemistry also uses calculus in determining reaction rates and radioactive decay. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... For other uses, see Mass (disambiguation). ... For other uses, see Density (disambiguation). ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... For other senses of this word, see magnetism (disambiguation). ... flux in science and mathematics. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... “Einstein” redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...


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 the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...


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 using the principles of algebra. ... In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. ... Plot of y = x3 with inflection point of (0,0). ...


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 a firm. ...


Calculus can be used to find approximate solutions to equations, in 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. In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... In numerical analysis, fixed point iteration is a method of computing fixed points of functions. ... Linear approximation is a method of approximating otherwise difficult to find values of a mathematical function by taking the value on a nearby tangent line instead of the function itself. ... In mathematics and computational science, Euler integration is the most basic kind of numerical integration for calculating trajectories from forces at discrete timesteps. ...


See also

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Lists

For a more comprehensive list, see the List of calculus topics. ... The primary operation in differential calculus is finding a derivative. ... This is a list of calculus topics. ... This is a list of important publications in mathematics, organized by field. ... It has been suggested that this article or section be merged with List of integrals. ...

Related topics

In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ... It has been suggested that this article be split into multiple articles. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... In mathematics education, precalculus, an advanced form of secondary school algebra, is a foundational mathematical discipline. ... Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ... Product integrals are a multiplicative version of standard integrals of infinitesimal calculus. ... Stochastic calculus is a branch of mathematics that operates on stochastic processes. ...

References

Notes

  1. ^ There is no exact evidence on how it was done; some, including Morris Kline (Mathematical thought from ancient to modern times Vol. I) suggest trial and error.
  2. ^ a b Helmer Aslaksen. Why Calculus? National University of Singapore. See
  3. ^ Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7
  4. ^ Aryabhata the Elder
  5. ^ Ian G. Pearce. Bhaskaracharya II.
  6. ^ Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), pp. 163-174.
  7. ^ J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), pp. 304-309.
  8. ^ Madhava. Biography of Madhava. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-09-13.
  9. ^ An overview of Indian mathematics. Indian Maths. School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved on 2006-07-07.
  10. ^ Science and technology in free India. Government of Kerala — Kerala Call, September 2004. Prof.C.G.Ramachandran Nair. Retrieved on 2006-07-09.
  11. ^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland. 
  12. ^ UNESCO-World Data on Education [1]

Morris Kline (1 May 1908 – 10 June 1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects. ... Malay name Malay: Universiti Nasional Singapura Tamil name Tamil: சிங்கப்பூர் தேசிய பல்கலைக்கழகம் University Cultural Centre The National University of Singapore (Abbreviation: NUS) is Singapores oldest university. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 256th day of the year (257th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 188th day of the year (189th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 190th day of the year (191st in leap years) in the Gregorian calendar. ... UNESCO (United Nations Educational, Scientific and Cultural Organization) is a specialized agency of the United Nations established in 1945. ...

Books

  • Donald A. McQuarrie (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 9781891389245
  • James Stewart (2002). Calculus: Early Transcendentals, 5th ed., Brooks Cole. ISBN 9780534393212

Other resources

Further reading

  • Courant, Richard ISBN 978-3540650584 Introduction to calculus and analysis 1.
  • Robert A. Adams. (1999). ISBN 978-0-201-39607-2 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 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals.
  • Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1-46.
  • 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 978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the Hungry Mind.
  • Michael Spivak. (September 1994). ISBN 978-0-914098-89-8 Calculus. Publish or Perish publishing.
  • Silvanus P. Thompson and Martin Gardner. (1998). ISBN 978-0-312-18548-0 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.
  • Thomas/Finney. (1996). ISBN 978-0-201-53174-9 Calculus and Analytic geometry 9th, Addison Wesley.
  • Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld--A Wolfram Web Resource.

Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ... Clifford A. Pickover is a writer in the fields of science, mathematics, and science fiction. ... Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ... Silvanus Phillips Thompson (June 19, 1851 – June 12, 1916). ... Martin Gardner (b. ... The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. ...

Online books

is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 126th day of the year (127th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ...

Web pages


  Results from FactBites:
 
calculus.org - THE CALCULUS PAGE . (1238 words)
Calculus demonstrations that you can use in the classroom to spice up your lectures and prevent those nodding heads.
Single Variable Calculus Mika Seppälä of Florida State University and the University of Helsinki presents classroom type notes on calculus, in pdf and powerpoint format.
Math Applets for Calculus at SLU: Mike May at St. Louis University has a selection of applets to illustrate important concepts of single and multivariable calculus.
Calculus graphics -- Douglas N. Arnold (1444 words)
The diagram illustrates the local accuracy of the tangent line approximation to a smooth curve, or--otherwise stated--the closeness of the differential of a function to the difference of function values due to a small increment of the independent variable.
The proof is based on a diagram depicting a circular sector in the unit circle together with an inscribed and a circumscribed triangle.
A brief graphical exploration of a continuous, nowhere differentiable function fits very well in the first semester of calculus, for example, to provide a strong counterexample to the converse of the theorem that differentiability implies continuity; or to show that it is only differentiable functions which look like straight lines under the microscope.
  More results at FactBites »

 
 

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