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            "2155": {
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                "title": "Reader's Digest National Word Power Challenge",
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                        "*": "The '''Reader's Digest National Word Power Challenge''' is the first nationwide vocabulary competition for middle schoolers. Over 3 million students compete each year, and the top 3 are awarded a total of $50,000 in scholarships by Reader's Digest. \n\n==Format==\nAll middle-school students in grades 6-8 residing in the US or dependencies thereof, who have not previously won a scholarship from Word Power, can participate. \n\nThere are 5 levels of competition at Word Power, each of which is 25 multiple-choice questions. At the Classroom level, the highest score on a 25-question test from each classroom goes to the School competition. The School level narrows the field to the top student in each grade from the school, each of whom get to take the State Qualifier test. The top 100 scores on the 40-question State Qualifier are invited to the State Competition, where the top student on 2 25-question tests is awarded an all-expense-paid trip to the National Competition.\n\nAt Nationals, a 25-question preliminary round reduces the 55 state winners to the top 10 finalists, who compete in the televised National Finals. The National Finals, hosted by Al Roker, determine the top 3 verbophages in the nation, who get <nowiki>$25,000, $15,000, and $</nowiki>10,000 respectively in scholarship money.\n\n==Past Winners==\n====2003====\n*  William Brannon, VA, 8th grade\n*  Gordon Bourjaily, IA, 8th grade\n*  Richard Lyford, NY, 8th grade\n====2004====\n*  Spencer Gill, MI, 7th grade\n*  Jeffrey Seigal, MD, 8th grade\n*  Ajay Ravichandran, NC, 8th grade, 2-time finalist\n====2005====\n*  Ming-Ming Tran, NJ, 6th grade, sister of AoPSer\n*  [[User:solafidefarms | Billy Dorminy]], GA, 8th grade, 2-time finalist, AoPSer\n*  Levi Foster, AR, 8th grade, 3-time finalist\n\n====2006====\n*  Joe Shepherd, GA, 8th grade, AoPSer\n*  William Johnson, VA, 8th grade\n*  Christopher Molini, OK, 8th grade, 2-time state champion\n\nThere will be no 2008 Word Power Challenge.\n==See Also==\n[http://nwpc.rd.com Official Reader's Digest National Word Power Challenge site]"
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            "19688": {
                "pageid": 19688,
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                "title": "Real analysis",
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                        "*": "Broadly speaking, '''real analysis''' is the study of the real numbers and its topological properties, sequences and series of real numbers, and properties of real-valued functions. Some properties that are studied in the real numbers are the construction of the real numbers, convergence of sequences, subsets of the plane as metric spaces, limits, notions of [[continuity]], [[Derivative|differentiation]], and [[integration]].\n\nA common description of real analysis courses is that real analysis is the formal rigorous study of single-variable [[calculus]] with proofs. This view does have merit to it because most (if not all) of the theorems typically presented to students in courses in single-variable calculus are proven rigorously; however, one should note that courses in real analysis also spend considerable amount of time on pathological examples with little concern for applications, and one also aims to generalize and prove results rather than apply results to calculate numerical answers to exercises as one typically does in a calculus course.\n\n== Construction of the real numbers ==\n\nThe entirety of real analysis is built upon the real numbers, particularly with the notion of completeness in mind. Intuitively, this is described as the fact that the real numbers <math>\\mathbb R</math> lack the existence of any \"holes\" unlike the rational numbers <math>\\mathbb Q</math> (for instance, the set <math>\\{x \\in \\mathbb{Q} \\mid x^2 < 2\\}</math> has no largest element in the rational numbers). This property of the real numbers is known as the ''least upper bound property''.\n\nTwo particularly known constructions of the real numbers are via [[Cauchy sequence|Cauchy sequences]] and [[Dedekind cut|Dedekind cuts]], both of which take <math>\\mathbb{Q}</math> and construct <math>\\mathbb{R}</math> as a completion of <math>\\mathbb{Q}</math>.\n\n== Sequences of real numbers ==\n\nA [[sequence]] is a function <math>f:\\mathbb{N}\\to\\mathbb{R}</math>. Conventionally, sequences are typically denoted by the notation <math>(s_n)_{n = k}^{\\infty} = (s_k,s_{k + 1},\\ldots)</math> where <math>f(n)</math> is denoted by <math>s_n</math>. In the case where <math>k = 1</math>, we can denote <math>(s_n)_{n = k}^{\\infty}</math> by <math>(s_n)_{n \\in \\mathbb{N}}</math>.\n\nIn real analysis, particular attention is paid attention to the convergence and divergence of sequences. Intuitively, the idea of convergence is captured by the notion that the sequence \"approaches\" some value as <math>n</math> becomes arbitrarily large. Also important is the notion of Cauchy sequences which intuitively describe sequences whose terms become arbitrarily close to each other as <math>n</math> becomes arbitrarily large.\n\n== Limits ==\n\nA large problem with the intuitive notion of a sequence converging to some value is that \"approaching\" is not only vague, but is also handwavy and lacks mathematical precision. Limits solve this problem by precisely defining the notion of convergence of a sequence.\n\n'''Definition''': Let <math>(s_n)</math> be a sequence of real numbers. The sequence <math>(s_n)</math> converges to the limit <math>s\\in\\mathbb{R}</math> provided that for every <math>\\varepsilon > 0</math>, there exists <math>N\\in\\mathbb{N}</math> such that for every <math>n \\ge N</math>, we have <math>|s_n - s| < \\varepsilon.</math> If <math>(s_n)</math> converges to <math>s</math>, then we say that <math>\\lim_{n\\to\\infty}s_n = s</math> or <math>s_n \\rightarrow s</math> as <math>n \\to \\infty</math>.\n\nThis definition can be shown to be equivalent to the likely more familiar <math>\\varepsilon-\\delta</math> definition of a limit of a function.\n\n'''Definition''': Let <math>S\\subseteq\\mathbb{R}</math>. The limit of the function <math>f:S\\to\\mathbb{R}</math> as <math>x</math> approaches <math>x_0</math> is <math>L</math> provided that for every <math>\\varepsilon > 0</math>, there exists a <math>\\delta > 0</math> such that that for every <math>x \\in S</math> and <math>0<|x-x_0|<\\delta</math>, we have <math>|f(x)-L|<\\varepsilon</math>. Notationally, we say that <math>\\lim_{x\\to x_0}f(x)=L</math> or <math>f(x) \\rightarrow L</math> as <math>x \\rightarrow x_0</math>.\n\nLimits are a key tool in the definition of continuity, derivatives, and any result in real analysis that relies upon sequences.\n\n== Continuity ==\n\nA common analogy used in calculus classes for continuity is a function whose graph can be drawn without lifting up one's pencil--that is, the graph has no breaks or jumps. While intuitive, it turns out that this notion of continuity is actually very misleading, in fact, a continuous function may have discontinuities at points not in its domain (for example, <math>f(x) = 1/x</math> is continuous at all points in its domain yet is \"visually discontinuous\" at <math>x = 0</math>). This calls for a more precise notion of continuity.\n\n'''Definition''': Let <math>S\\subseteq\\mathbb{R}</math>. The function <math>f:S\\to\\mathbb R</math> is continuous at <math>x_0\\in S</math> provided that for every sequence <math>(x_n)</math> in <math>S</math> converging to <math>x_0</math>, we have <math>f(x_n)\\rightarrow f(x_0)</math> as <math>x_n\\rightarrow x_0</math>. In other words, <math>f</math> preserves convergence.\n\nThis definition of continuity is equivalent to the more familiar definition of continuity from calculus below.\n\n'''Definition'''. Let <math>S\\subseteq\\mathbb{R}</math>. The function <math>f:S\\to\\mathbb R</math> is continuous at <math>x_0\\in S</math> provided that for every <math>\\varepsilon > 0</math>, there exists a <math>\\delta > 0</math> such that for every <math>x \\in S</math> and <math>|x - x_0|<\\varepsilon</math>, we have <math>|f(x)-f(x_0)|<\\varepsilon</math>.\n\nContinuity can also be generalized to topological spaces and described in terms of preimages of open sets. Furthermore, various other notions of continuity such as Lipschitz continuity, uniform continuity, and absolute continuity are studied in real analysis.\n\n== Differentiation ==\n\nDerivatives are central notion in calculus and various fields such as the sciences and economics where derivatives can be interpreted as the instantaneous rate of change at a point. \n\n'''Definition''': The function <math>f:(a,b) \\rightarrow \\mathbb{R}</math> is differentiable at <math>x_0</math> if the limit <cmath>f'(x_0) = \\frac{f(x)-f(x_0)}{x-x_0}</cmath> exists.\n\nIn the context of real analysis, differentiability is a condition which guarantees continuity, but not the other way around. In particular, there exist functions that are continuous on <math>\\mathbb{R}</math> but differentiable nowhere (for instance, [[Weierstrass's function]]). One can also define higher-order derivatives by inductively differentiating the derivative of a function.\n\n== Integration ==\n\nIn calculus, integrals are introduced as the approximation of the area under the curve of a continuous functions by rectangles as the number of rectangles becomes arbitrarily large. This construction is widely used in many fields of knowledge including math itself and is known as a Riemann integral. In real analysis, Riemann integrals of non-continuous functions are also considered and the notion of Lebesgue integration is also developed.\n\nIn some texts, the notion of ''Darboux integrals'' are introduced first which are then noted as special cases of Riemann sums. \n\nLet <math>f:[a,b]\\rightarrow\\mathbb{R}</math> be bounded.\n\n'''Definition''':  Let <math>P</math> be a [[Partition of an interval|tagged partition]] of <math>[a,b]</math>. Then the Riemann sum corresponding to <math>f</math> and <math>P</math> is <cmath>R(f,P) = \\sum_{i=1}^{n} f(t_i)(x_{i} - x_{i-1}).</cmath> Furthermore, we define the mesh <math>|P|</math> of the partition <math>P</math> to be the length of the largest subinterval <math>[x_{i-1},x_i]</math>.\n\n'''Definition''': The real number <math>I</math> is the Riemann integral of <math>f</math> over <math>[a,b]</math> provided that for every <math>\\varepsilon > 0</math>, there exists some <math>\\delta>0</math> such that if <math>P</math> is any partition of <math>[a,b]</math>, then <math>|R - I| < \\varepsilon</math> whenever <math>|P| < \\delta</math>. If such an <math>I</math> exists, then it is denoted as <math>\\int_a^b f(x)\\,dx = I</math>.\n\nVarious shortcomings of Riemann integration exist. A famous example is the [[Dirchlet function]] which is not Riemann-integrable on any interval of <math>\\mathbb{R}</math>. Furthermore, the [[monotone convergence theorem]] is false in the context of Riemann-integrals which motivates the [[Lebesgue integral]].\n\n== See also ==\n\n* [[Complex analysis]]\n* [[Functional analysis]]\n* [[Metric space]]\n* [[Topology]]\n* [[Measure theory]]\n* [[Calculus]]\n\n{{stub}}[[Category:Calculus]]"
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