Critical Values of T Critical Values of T Critical value is used in many ways in maths. It is used in calculus, statistics, correlation coefficeint etc. In english language it means " a value having importance to that problem" By using differentiation, a function get differentiated say f ´(x). Now the solutions of f´(x) = 0 are called critical values. These critical values are very important to know the minimum and maximum value of the given function. In this article we will be discussing only to find the critical value. Before you reading this make sure to know the differentiation formula like = and so on. Critical Value in Calculus In calculus, critical values are the points at which a function has the maxima or minima . For finding the critical values of a function f(x) , we first find f '( x) then solve f ' (x) = 0, suppose we get x = c1, c2, c3, ....... Then c1, c2, c3, .....are the critical ponts, at which f( x ) can have maxima or minima. Example for critical value in calculus : Let f(x) = x2 - 6x + 5 Know More About Rate of Change Problems Math.Tutorvista.com
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To find the critical value, we will first differentiate it f ' (x) = 2x -6 and now equate it to zero. 2x -6 = 0 this implies x = 3, it is the x coordinate of the turning point, to find the y coordinate, we put this value in the equation. f(x) = 32 - 6 x 3 +5 = -4 so , the critical value is -4. Critical Value in Statistics In statistics, the critical value corresponds to a given significance level. This cutoff value determines the boundary between the two samples, on the basis of which , it is determined, whether to reject the null hypothesis or whether to not reject the null hypothesis. If the calculated value from the statistical data is greater than the critical value, then the null hypothesis is rejected in favour of the alternative hypothesis and vice versa. Example for critical value in statistics: Question 1: what is the critical value for a two tailed test and 95% accuracy. Answer : the critical value for a two tailed test is Za /2 or 5 /2 = 2.5 % Question 2: What is the critical value for a one tailed test and 95% accuracy. Answer : its critical value is 5%.
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Definite Integral Definite Integral Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics.
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Some definite integrals, the first two of which are due to Bailey and Plouffe (1997) and the third of which is due to GuĂŠnard and Lemberg (2001), which were identified by Borwein and Bailey (2003, p. 61) and Bailey et al. (2007, p. 62) to be "technically correct" but "not useful" as computed by Mathematica are reproduced below. Happily, Mathematica Version 5 returns them in the same simple form given by Borwein and Bailey without even the need for additional simplification: There are a wide range of methods available for numerical integration. Good sources for such techniques include Press et al. (1992) and Hildebrand (1956). The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called the trapezoidal rule and Simpson's rule, respectively. The 5-point formula is called Boole's rule. A generalization of the trapezoidal rule is romberg integration, which can yield accurate results for many fewer function evaluations. If the analytic form of a function is known (instead of its values merely being tabulated at a fixed number of points), the best numerical method of integration is called Gaussian quadrature. By picking the optimal abscissas at which to compute the function, Gaussian quadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additional complication of the Gaussian quadrature formalism often makes it less desirable than the brute-force method of simply repeatedly calculating twice as many points on a regular grid until convergence is obtained. The June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007) featured the following definite integral as a "hard" exam problem intended for a remedial math class but accidentally handed out to the normal class: The integral corresponds to integration over a spherical cone with opening angle and radius 4. However, it's not clear what the integrand physically represents (it resembles computation of a moment of inertia, but that would give a factor rather than the given ). Read More About Increasing and Decreasing Functions
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