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occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is <span>indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) <span><body><html>

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (November 2015) (Learn how and when to remove this template message) <span>In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization". Dynamic programming algorithms are often used for optimization. A dyna

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The Black–Scholes / ˌ b l æ k ˈ ʃ oʊ l z / [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of E

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen

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bution attributes probability zero to each of the intervals removed, and the lengths of these intervals add up to one. So all of the probability is concentrated on the Cantor set C ∞ , which is what the measure-theoretic jargon calls a set of <span>Lebesgue measure zero, Lebesgue measure being the measure-theoretic analog of ordinary length. <span><body><html>

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The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations

ackward algorithm - Wikipedia Forward–backward algorithm From Wikipedia, the free encyclopedia (Redirected from Forward-backward algorithm) Jump to: navigation, search <span>The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations/emissions o 1 : t := o 1

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at any point on the constraint surface the gradient ∇g(x) of the constraint function will be orthogonal to the surface

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In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor, when that linear transformation is applied to it.

m Wikipedia, the free encyclopedia (Redirected from Eigenvalue) Jump to: navigation, search "Characteristic root" redirects here. For other uses, see Characteristic root (disambiguation). <span>In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor, when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v)

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If a continuous function on an open interval satisfies the equality for all compactly supported smooth functions on , then is identically zero.

[hide] 1 Basic version 2 Version for two given functions 3 Versions for discontinuous functions 4 Higher derivatives 5 Vector-valued functions 6 Multivariable functions 7 Applications 8 Notes 9 References Basic version[edit source] <span>If a continuous function f {\displaystyle f} on an open interval ( a , b ) {\displaystyle (a,b)} satisfies the equality ∫ a b f ( x ) h ( x ) d x = 0 {\displaystyle \int _{a}^{b}f(x)h(x)\,\operatorname {d} x=0} for all compactly supported smooth functions h {\displaystyle h} on ( a , b ) {\displaystyle (a,b)} , then f {\displaystyle f} is identically zero. [1] [2] Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "co

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This inner product in the space ℂ 2 is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate:

2 . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w}}_{1}+z_{2}{\overline {w}}_{2}\,.} The real part of ⟨z,w⟩ is then the four-dimensional Euclidean dot product. <span>This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w

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separation of variables solves ordinary and partial differential equations by allowing one to rewrite an equation so that each of two variables occurs on a different side of the equation. </spa

People [show] Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta v t e <span>In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Contents [hide] 1 Ordinary differential equations (ODE) 1.1 Alternative notation 1.2 Example 2 Partial differential equations 2.1 Example: homogeneous case 2.2 Example:

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a Jump to: navigation, search This article is about the nabla symbol, its name, their usage, and their historical origin. See Del for an article focused on the mathematics. ∇ The nabla symbol <span>The nabla is a triangular symbol like an inverted Greek delta: [1] ∇ {\displaystyle \nabla } or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, [2] and was suggested by the encyclopedist William Robertson Smith to Pete

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The nabla is a triangular symbol like an inverted Greek delta: or ∇.

a Jump to: navigation, search This article is about the nabla symbol, its name, their usage, and their historical origin. See Del for an article focused on the mathematics. ∇ The nabla symbol <span>The nabla is a triangular symbol like an inverted Greek delta: [1] ∇ {\displaystyle \nabla } or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, [2] and was suggested by the encyclopedist William Robertson Smith to Pete

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( X ) . {\displaystyle \operatorname {Var} (Y)\geq {\frac {\operatorname {Cov} (Y,X)\operatorname {Cov} (Y,X)}{\operatorname {Var} (X)}}.} <span>After defining an inner product on the set of random variables using the expectation of their product, ⟨ X , Y ⟩ := E ( X Y ) , {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY),} then the Cauchy–Schwarz inequality becomes | E ( X Y ) | 2 ≤ E ( X 2 ) E ( Y 2 ) . {\displaystyle |\operatorname {E} (XY)|^{2}\leq \operatorname {E} (X^{2})\operatorname {E} (Y^{2}).} To prove the covariance inequality using the Cauchy–Schwarz inequality, let μ = E ( X ) {

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Hilbert curve From Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] First eight iterations of the Hilbert curve View first 10 iterations <span>A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, [1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. [2] Because it is space-fi

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A Hilbert curve is a continuous fractal space-filling curve

Hilbert curve From Wikipedia, the free encyclopedia Jump to: navigation, search [imagelink] First eight iterations of the Hilbert curve View first 10 iterations <span>A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, [1] as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. [2] Because it is space-fi

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/head>Analysis of variance - Wikipedia Analysis of variance From Wikipedia, the free encyclopedia (Redirected from Anova) Jump to: navigation, search Analysis of variance (ANOVA) is a collection of statistical models and their associated procedures (such as "variation" among and between groups) used to analyze the differences among group means. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVAs are useful for comparing (testing) three or more means (groups or variables) for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative (results in less type I error) [1] and is therefore suited to a wide range of practical problems. Contents [hide] 1 History 2 Motivating example 3 Background and terminology 3.1 Design-of-experiments terms 4 Classes of models 4.1 Fixed-effects models 4.2 Random-effe

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uing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. [17] [18] [imagelink] Maria Gaetana Agnesi Foundations[edit source] <span>In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley

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the complex plane. In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been greatly extended. <span>Henri Lebesgue invented measure theory and used it to define integrals of all but the most pathological functions. Laurent Schwartz introduced distributions, which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus

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otype of an (ε, δ)-definition of limit in the definition of differentiation. [20] In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can actually validate nilsquare infinitesimals). <span>Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during t

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ion of limit - Wikipedia (ε, δ)-definition of limit From Wikipedia, the free encyclopedia (Redirected from Epsilon, delta) Jump to: navigation, search [imagelink] <span>Whenever a point x is within δ units of c, f(x) is within ε units of L In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy,

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The (ε, δ)-definition of limit defines the limit of a function f at point c as: whenever a point x is within δ units of c, f(x) is within ε units of L

ion of limit - Wikipedia (ε, δ)-definition of limit From Wikipedia, the free encyclopedia (Redirected from Epsilon, delta) Jump to: navigation, search [imagelink] <span>Whenever a point x is within δ units of c, f(x) is within ε units of L In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy,

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infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. <span>The infinitesimal approach 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. In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a near

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ion 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. <span>In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, 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 were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus[edit source] Main article: Differential calculus [imagelink] Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise ov

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ll-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. <span>In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function i

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2 ", as an input, that is all the information —such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on— and uses this information to output another function, the function g(x) = 2x, as will turn out. <span>The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of a function called f is denoted by f′, pronounced "f prime". For instance, if f(x) = x 2 is the squaring function, then f′(x) = 2x is its derivative (t

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e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. <span>The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2

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= 2 x . {\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}} <span>In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimal

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2 ) = 2 x . {\displaystyle {\frac {d}{dx}}(x^{2})=2x.} In this usage, the dx in the denominator is read as "with respect to x". <span>Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative. Integral calculus[edit source]

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effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). <span>Calculation of the total derivative of f {\displaystyle f} with respect to t {\displaystyle t} does not assume that the other arguments are constant while t {\displaystyle t} varies; instead, it assumes that the other arguments too depend on t {\displaystyle t} . The total derivative includes these indirect dependencies to find the overall dependency of f {\displaystyle f} on t

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integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's <span>direct effect as well as indirect effects via the other arguments of the function. <span><body><html>

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous ar

integral Line integral Surface integral Volume integral Jacobian Hessian Specialized[show] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>In the mathematical field of differential calculus, a total derivative or full derivative of a function f {\displaystyle f} of several variables, e.g., t {\displaystyle t} , x {\displaystyle x} , y {\displaystyle y} , etc., with respect to an exogenous argument, e.g., t {\displaystyle t} , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). Calculation of the

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Calculation of the total derivative of with respect to assumes that the other arguments too depend on t.

effect as well as indirect effects via the other arguments of the function. The total derivative of a function is different from its corresponding partial derivative ( ∂ {\displaystyle \partial } ). <span>Calculation of the total derivative of f {\displaystyle f} with respect to t {\displaystyle t} does not assume that the other arguments are constant while t {\displaystyle t} varies; instead, it assumes that the other arguments too depend on t {\displaystyle t} . The total derivative includes these indirect dependencies to find the overall dependency of f {\displaystyle f} on t

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In calculus, foundations refers to the rigorous development of the subject from axioms and definitions.

uing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. [17] [18] [imagelink] Maria Gaetana Agnesi Foundations[edit source] <span>In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley

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In calculus, foundations refers to the rigorous development of the subject from axioms and definitions.

uing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi. [17] [18] [imagelink] Maria Gaetana Agnesi Foundations[edit source] <span>In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley

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Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities

otype of an (ε, δ)-definition of limit in the definition of differentiation. [20] In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can actually validate nilsquare infinitesimals). <span>Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during t

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Henri Lebesgue invented measure theory and used it to define integrals of all but the most pathological functions.

the complex plane. In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been greatly extended. <span>Henri Lebesgue invented measure theory and used it to define integrals of all but the most pathological functions. Laurent Schwartz introduced distributions, which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus

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The infinitesimal approach 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 manipulat

infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. <span>The infinitesimal approach 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. In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a near

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The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise.

infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. <span>The infinitesimal approach 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. In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a near

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The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise.

infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. <span>The infinitesimal approach 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. In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a near

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In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this

ion 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. <span>In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, 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 were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus[edit source] Main article: Differential calculus [imagelink] Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise ov

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In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals ge

ion 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. <span>In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, 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 were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus[edit source] Main article: Differential calculus [imagelink] Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise ov

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In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output.

ll-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. <span>In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function i

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The most common symbol for a derivative is an apostrophe-like mark called prime.

2 ", as an input, that is all the information —such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on— and uses this information to output another function, the function g(x) = 2x, as will turn out. <span>The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of a function called f is denoted by f′, pronounced "f prime". For instance, if f(x) = x 2 is the squaring function, then f′(x) = 2x is its derivative (t

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The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

e behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is undefined. <span>The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference

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The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients.

h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} Geometrically, the derivative is the slope of the tangent line to the graph of f at a. <span>The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2

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In an approach based on limits, the symbol dy / dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit

= 2 x . {\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}} <span>In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimal

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Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers

2 ) = 2 x . {\displaystyle {\frac {d}{dx}}(x^{2})=2x.} In this usage, the dx in the denominator is read as "with respect to x". <span>Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative. Integral calculus[edit source]

#spanish

The absolute superlative indicates the highest or lowest quality grade, i.e. it can express superiority or inferiority. The absolute superlative of superiority can be expressed in the following ways: Muy (very) + adjective: Este chico es muy guapo (This guy is very handsome). Esa obra de teatro parece muy interesante (That theatre play looks very interesting). Suffixes -ísimo, -ísima may also be added: Este chico es guapísimo. Esa obra de teatro parece interesantísima. The absolute superlative of inferiority can be expressed in the following ways: (Muy) poco [(very) little] + positive adjective. It is used to express politely that you dislike something. Es muy poco agradable (He/She is not very kind). El documental es poco interesante (The documentary is not very interesting). Suffixes -ísimo, -ísima with negative adjectives. Esa obra de teatro es aburridísima (That theatre play is very boring). Ese cuadro es feísimo (That painting

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3; The absolute superlative of inferiority can be expressed in the following ways: (Muy) poco [(very) little] + positive adjective. It is used to express politely that you dislike something. <span>Es muy poco agradable (He/She is not very kind). El documental es poco interesante (The documentary is not very interesting). Suffixes -ísimo, -ísima with negative adjectives. Esa obra de teatro es aburri

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Es muy poco agradable (He/She is not very kind).

3; The absolute superlative of inferiority can be expressed in the following ways: (Muy) poco [(very) little] + positive adjective. It is used to express politely that you dislike something. <span>Es muy poco agradable (He/She is not very kind). El documental es poco interesante (The documentary is not very interesting). Suffixes -ísimo, -ísima with negative adjectives. Esa obra de teatro es aburri

#spanish

A verb periphrasis is the combination of two verb forms, one conjugated and the other one a non-finite form (infinitive, gerund, past participle). Sometimes there is a conjunction or preposition that links those verbs. The most important verb phrases with infinitive are the following: Periphrasis Meaning Example haber que(impersonal use, 3rd person: hayque) + infinitive obligation, advice Hay que ir a la compra: no tenemos nada en lanevera (We have to do the shopping. There is nothing in our fridge). deber + infinitive obligation, advice Estás demasiado delgado: debes comer más(You are too thin. You should eat more). tener que + infinitive obligation, advice Tienes que estudiar mucho hoy porquemañana tienes el examen final (You must study a lot today because your final exam is taking place tomorrow). poder + infinitive possibility, permission No puedes comer aquí (You cannot eat in here). volver a + infinitive reiteration Vuelve a decirle que me llame (Tell her to call me again). empezar a

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hoy porquemañana tienes el examen final (You must study a lot today because your final exam is taking place tomorrow). poder + infinitive possibility, permission No puedes comer aquí (You cannot eat in here). volver a + infinitive reiteration <span>Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis a

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xam is taking place tomorrow). poder + infinitive possibility, permission No puedes comer aquí (You cannot eat in here). volver a + infinitive reiteration Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start <span>Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit Suelo comer en una tasc

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lver a + infinitive reiteration Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy <span>Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work). ir a + infinitive near future Voy a comer a casa de mis tíos (I am goi

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again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit <span>Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work). ir a + infinitive near future Voy a comer a casa de mis tíos (I am going to eat at my uncle and aunt’s). <span><body><html>

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Vuelve a decirle que me llame (Tell her to call me again).

hoy porquemañana tienes el examen final (You must study a lot today because your final exam is taking place tomorrow). poder + infinitive possibility, permission No puedes comer aquí (You cannot eat in here). volver a + infinitive reiteration <span>Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis a

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Empezamos a correr cuando vimos que llovía

xam is taking place tomorrow). poder + infinitive possibility, permission No puedes comer aquí (You cannot eat in here). volver a + infinitive reiteration Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start <span>Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit Suelo comer en una tasc

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Luis y Ana acaban de irse (Luis and Ana have just left).

lver a + infinitive reiteration Vuelve a decirle que me llame (Tell her to call me again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy <span>Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work). ir a + infinitive near future Voy a comer a casa de mis tíos (I am goi

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Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work).

again). empezar a + infinitive start Empezamos a correr cuando vimos que llovía(We started to run when we saw it was raining). acabar de + infinitive immediacy Luis y Ana acaban de irse (Luis and Ana have just left). soler + infinitive habit <span>Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work). ir a + infinitive near future Voy a comer a casa de mis tíos (I am going to eat at my uncle and aunt’s). <span><body><html>

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started reading on | finished reading on |

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started reading on | finished reading on |

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started reading on | finished reading on |