In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality in many mathematical fields, such as linear algebra, analysis, probability theory, vector algebra and other areas. Suppose that a curve \(\gamma\) is described by the parametric equations \(x = f\left( t \right),\) \(y = g\left( t \right),\) where the parameter \(t\) ranges in the interval \(\left[ {a,b} \right].\) When changing the parameter \(t,\) the point of the curve in Figure \(2\) runs from \(A\left( {f\left( a \right), g\left( a \right)} \right)\) to \(B\left( {f\left( b \right),g\left( b \right)} \right).\) According to the theorem, there is a point \(\left( {f\left( {c} \right), g\left( {c} \right)} \right)\) on the curve \(\gamma\) where the tangent is parallel to the chord joining the ends \(A\) and \(B\) of the curve. Complex integration: Cauchy integral theorem and Cauchy integral formulas Deﬁnite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable t: f(t) = u(t) + iv(t), which is assumed to be a piecewise continuous function deﬁned in the closed interval a ≤ t … The mean value theorem says that there exists a time point in between and when the speed of the body is actually . The mathematician Baron Augustin-Louis Cauchy developed an extension of the Mean Value Theorem. x ∈ ( a, b). \end{array} \right.,\;\;}\Rightarrow on the closed interval , if , and Mean-value theorems (other than Cauchy's, Lagrange's or Rolle's) 1. If the function represented speed, we would have average spe… It is evident that this number lies in the interval \(\left( {1,2} \right),\) i.e. Walk through homework problems step-by-step from beginning to end. Join the initiative for modernizing math education. To see the proof see the Proofs From Derivative Applications section of the Extras chapter. The contour integral is taken along the contour C. It states that if f(x) and g(x) are continuous on the closed interval [a,b], if g(a)!=g(b), and if both functions are differentiable on the open interval (a,b), then there exists at least one c with a

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