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A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as T ( x ) = f ( a ) + ( x − a ) T D f ( a ) + 1 2 !
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.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, [2] although an earlier version of the result was already mentioned in 1671 by James Gregory. [ 3 ] Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis .
The Taylor expansion would be: + where / denotes the partial derivative of f k with respect to the i-th variable, evaluated at the mean value of all components of vector x. Or in matrix notation , f ≈ f 0 + J x {\displaystyle \mathrm {f} \approx \mathrm {f} ^{0}+\mathrm {J} \mathrm {x} \,} where J is the Jacobian matrix .
Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as: V a r [ f ( X ) ] ≈ ∑ n = 1 n m a x ( σ n n ! ( d n f d X n ) X = μ ) 2 V a r [ Z n ] + ∑ n = 1 n m a x ∑ m ≠ n σ n + m n ! m !
It can be heuristically derived by forming the Taylor series expansion of the ... The same factor of σ 2 / 2 appears in the d 1 and d 2 auxiliary variables ...
Taylor's theorem; Rules and identities ... but a function with two variables is a surface in 3D, ... is a function of one variable), we can write the Taylor expansion ...
In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the Taylor polynomial (truncated Taylor series) of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.