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These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name. Additionally, notice that this is precisely the mean value theorem when k = 0 {\textstyle k=0} .
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. [ 6 ] [ 7 ] It states: if the functions f {\displaystyle f} and g {\displaystyle g} are both continuous on the closed interval [ a , b ] {\displaystyle [a,b]} and differentiable on the open interval ( a , b ) {\displaystyle ...
Extreme value theorem; Differential equation; Differential operator; Newton's method; Taylor's theorem; L'Hôpital's rule; General Leibniz rule; Mean value theorem; Logarithmic derivative; Differential (calculus) Related rates; Regiomontanus' angle maximization problem; Rolle's theorem
Demonstration of this result is fairly straightforward under the assumption that () is differentiable near the neighborhood of and ′ is continuous at with ′ ().To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
Multi-binomial theorem ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,x^{\nu }y^{\alpha -\nu }.} Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for ( x 1 + y 1 ) α 1 ⋯( x n + y n ) α n .
Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc Taylor's theorem – gives an approximation of a k {\displaystyle k} times differentiable function around a given point by a k {\displaystyle k} -th ...