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[::MainTopicLength::#1500] [::ListTopicLength::#1000] [::ListLength::#15] [::ListAdRepeat::#3] picture info Continuous Function In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, consider the function h(t), which describes the height of a growing flower at time t [...More...] "Continuous Function" on: Wikipedia Google Yahoo Partial Fractions In Integration In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function. In symbols, one can use partial fraction expansion to change a rational fraction in the form f ( x ) g ( x ) displaystyle frac f(x) g(x) where f and g are polynomials, into an expression of the form ∑ j f j ( x ) [...More...] "Partial Fractions In Integration" on: Wikipedia Google Yahoo Faà Di Bruno's Formula Bruno's Supermarkets, LLC was an American chain of grocery stores with its headquarters in Birmingham, Alabama. It founded in 1932 by Joe Bruno in Birmingham. During the company's pinnacle, it operated over 300 stores under the names Bruno’s, Food World, Foodmax, Food Fair, Fresh Value, Vincent's Markets, Piggly Wiggly, Consumer Foods, and American Fare in Alabama, Florida, Georgia, Mississippi, Tennessee, and South Carolina. The chain was acquired by Birmingham-based Belle Foods Belle Foods which discontinued the brand in 2012.Contents1 History 2 Bankruptcy 3 Bruno's as a defunct brand 4 Sports Sponsorship 5 References 6 External linksHistory The company began as a market opened by Joseph Bruno in Birmingham, Alabama Alabama during the Great Depression [...More...] "Faà Di Bruno's Formula" on: Wikipedia Google Yahoo Differential (mathematics) In mathematics, differential refers to infinitesimal differences or to the derivatives of functions. The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.Contents1 Basic notions 2 Differential geometry 3 Algebraic geometry 4 Other meanings 5 References 6 External linksBasic notionsIn calculus, the differential represents a change in the linearization of a function.The total differential is its generalization for functions of multiple variables.In traditional approaches to calculus, the differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals [...More...] "Differential (mathematics)" on: Wikipedia Google Yahoo picture info Shell Integration Shell integration Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis parallel to the axis of revolution. This is in contrast to disk integration which integrates along the axis perpendicular to the axis of revolution.Contents1 Definition 2 Example 3 See also 4 ReferencesDefinition The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b] [...More...] "Shell Integration" on: Wikipedia Google Yahoo picture info Disc Integration Disc integration, also known in integral calculus as the disc method, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions [...More...] "Disc Integration" on: Wikipedia Google Yahoo picture info Integration By Parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found [...More...] "Integration By Parts" on: Wikipedia Google Yahoo picture info Integration By Reduction Formulae Integration by reduction formula in integral calculus is a technique of integration, in the form of a recurrence relation. It is used when an expression containing an integer parameter, usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree, can't be integrated directly. But using other methods of integration a reduction formula can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated [...More...] "Integration By Reduction Formulae" on: Wikipedia Google Yahoo picture info Series (mathematics) In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical by mathematicians and philosophers. This paradox was resolved using the concept of a limit during the 19th century [...More...] "Series (mathematics)" on: Wikipedia Google Yahoo Differentiation Rules This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.Contents1 Elementary rules of differentiation1.1 Differentiation is linear 1.2 The product rule 1.3 The chain rule 1.4 The inverse function rule2 Power laws, polynomials, quotients, and reciprocals2.1 The polynomial or elementary power rule 2.2 The reciprocal rule 2.3 The quotient rule 2.4 Generalized power rule3 Derivatives of exponential and logarithmic functions3.1 Logarithmic derivatives4 Derivatives of trigonometric functions 5 Derivatives of hyperbolic functions 6 Derivatives of special functions 7 Derivatives of integrals 8 Derivatives to nth order8.1 Faà di Bruno's formula 8.2 General Leibniz rule9 See also 10 References 11 Sources and further reading 12 External linksElementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that ret [...More...] "Differentiation Rules" on: Wikipedia Google Yahoo Lists Of Integrals Integration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful [...More...] "Lists Of Integrals" on: Wikipedia Google Yahoo picture info Order Of Integration (calculus) In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.Contents1 Problem statement 2 Relation to integration by parts 3 Principal-value integrals 4 Basic theorems 5 See also 6 References and notes 7 External linksProblem statement The problem for examination is evaluation of an integral of the form ∬ D   f ( x , y )   d x d y , displaystyle iint _ D f(x,y) dx,dy, where D is some two-dimensional area in the xy–plane [...More...] "Order Of Integration (calculus)" on: Wikipedia Google Yahoo General Leibniz Rule In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule") [...More...] "General Leibniz Rule" on: Wikipedia Google Yahoo picture info Geometric Series In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 1 2 + 1 4 + 1 8 + 1 16 + ⋯ displaystyle frac 1 2 ,+, frac 1 4 ,+, frac 1 8 ,+, frac 1 16 ,+,cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. Geometric series Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series [...More...] "Geometric Series" on: Wikipedia Google Yahoo Quotient Rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f ( x ) = g ( x ) / h ( x ) , displaystyle f(x)=g(x)/h(x), where both g displaystyle g and h displaystyle h are differentiable and h ( x ) ≠ 0. displaystyle h(x)neq 0 [...More...] "Quotient Rule" on: Wikipedia Google Yahoo Power Rule In calculus, the power rule is used to differentiate functions of the form f ( x ) = x r displaystyle f(x)=x^ r , whenever r displaystyle r is a real number [...More...] "Power Rule" on: Wikipedia Google Yahoo
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