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Sandwich Theorem
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions. The squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute , and was formulated in modern terms by Carl Friedrich Gauss. Statement The squeeze theorem is formally stated as follows. * The functions and are said to be lower and upper bounds (respectively) of . * Here, is ''not'' required to lie in the interior of . Indeed, if is an endpoint of , then the above limits are left- or right-hand limits. * A similar statement holds for infinite intervals: for example, if , then the conclusion holds, taking the limits as . This theorem is also valid for sequences. Let be two seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior (topology)
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the closure of the complement of . In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem. Definitions Interior point If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is completely contained in S. (This is illustrated in the introductory section to this article.) This definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functions And Mappings
In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but '' transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limits (mathematics)
Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 song by Calvin Harris from ''Ready for the Weekend'' (album) * "Limits", a 2019 song by Paenda; see Austria in the Eurovision Song Contest 2019 * ''Limits'' (collection), a collection of short stories and essays by Larry Niven * The Limit, a Dutch band * "The Limit", an episode from ''Adventure Time'' * "The Limit", an episode from ''The Amazing World of Gumball'' * " The Limit is Just Me", a documentary film about world's longest triathlon. Mathematics * Limit (mathematics), the value that a function or sequence "approaches" as the input or index approaches some value ** Limit of a function ***(ε,_δ)-definition of limit, formal definition of the mathematical notion of limit ** Limit of a sequence ** One-sided limit, either of the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Parents' Choice Award in 2008. Wolfram Research's staff organizes and edits the Demonstrations, which may be created by any user of Wolfram Mathematica, Mathematica, then freely published and freely downloaded. Technology The Demonstrations run in Wolfram Mathematica, Mathematica 6 or above and in Wolfram Computable Document Format, CDF Player, which is a free modified version of Wolfram Mathematica and available for Microsoft Windows, Windows, Linux, and macOS and can operate as a web Browser extension, browser plugin. Demonstrations can also be embedded into a website. Each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar. The Demonstrations typically consist of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khan Academy
Khan Academy is an American non-profit educational organization created in 2006 by Sal Khan. Its goal is to create a set of online tools that help educate students. The organization produces short video lessons. Its website also includes supplementary practice exercises and materials for educators. It has produced over 10,000 video lessons teaching a wide spectrum of academic subjects, including mathematics, sciences, literature, history, and computer science. All resources are available free to users of the website and application. History Starting in 2004, Salman "Sal" Khan began tutoring one of his cousins in mathematics on the Internet using a service called Yahoo! Doodle Images. After a while, Khan's other cousins began to use his tutoring service. Due to the demand, Khan decided to make his videos watchable on the Internet, so he published his content on YouTube. Later, he used a drawing application called SmoothDraw, and now uses a Wacom tablet to draw using ArtRa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sal Khan
Salman Amin Khan (born October 11, 1976) is an American educator and the founder of Khan Academy, a free online non-profit educational platform with which he has produced over 6,500 video lessons teaching a wide spectrum of academic subjects, originally focusing on mathematics and science. He is also the founder of Khan Lab School, a private in-person school in Mountain View, California. , the Khan Academy channel on YouTube has 8.74 million subscribers, and its videos have been viewed more than two billion times. In 2012, Khan was named in the annual publication of ''Time'' 100. In the same year, he was featured on the cover of ''Forbes'', with the tagline "The $1 Trillion Opportunity." Early life and family Khan was born in Metairie, Louisiana, on October 11, 1976, into a Bengali Muslim family. His father Fakhrul Amin Khan (d. 1990) was a physician, originally hailing from the village of Rahmatpur in Babuganj Upazila, Bangladesh while his mother Masuda Khan i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Sin X X
Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 song by Calvin Harris from ''Ready for the Weekend'' (album) * "Limits", a 2019 song by Paenda; see Austria in the Eurovision Song Contest 2019 * ''Limits'' (collection), a collection of short stories and essays by Larry Niven * The Limit, a Dutch band * "The Limit", an episode from ''Adventure Time'' * "The Limit", an episode from ''The Amazing World of Gumball'' * " The Limit is Just Me", a documentary film about world's longest triathlon. Mathematics * Limit (mathematics), the value that a function or sequence "approaches" as the input or index approaches some value ** Limit of a function ***(ε,_δ)-definition of limit, formal definition of the mathematical notion of limit ** Limit of a sequence ** One-sided limit, either of the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Function
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inst Satsen , with reference to music
{{disambiguation ...
Inst may refer to: * As "inst.", abbreviation for instant, with reference to time * Alternative shortened Instagram name * As "inst.", abbreviation for '' instante mense'', meaning a date of the current month, such as "the 5th inst." * The Royal Belfast Academical Institution, a grammar school in Belfast, Northern Ireland * Tuskegee University, Alabama, United States * Bannari Amman Institute of Technology, Tamil Nadu, India * Instrumental An instrumental or instrumental song is music without any vocals, although it might include some inarticulate vocals, such as shouted backup vocals in a big band setting. Through Semantic change, semantic widening, a broader sense of the word s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
