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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
The above is obtained using a second order approximation, following the method used in estimating the first moment. It will be a poor approximation in cases where () is highly non-linear. This is a special case of the delta method.
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 ...
The special case of calculus in three dimensional space is often called vector calculus. ... for example, for a 1D function ... using Taylor's theorem to construct ...
Taylor's theorem; Rules and identities; ... is a special case of the Euler–Lagrange equation in the ... An example of an application of the Beltrami identity is ...
Fermat's little theorem, which states "if p is a prime number, then for any integer a, then ()" is a special case of Euler's theorem, which states "if n and a are coprime positive integers, and () is Euler's totient function, then () ()", in the case that n is a prime number.
The judge overseeing former President Donald Trump's federal election interference case on Tuesday granted special counsel Jack Smith's request to file a 180-page brief on presidential immunity ...
Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...