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  2. Taylor's theorem - Wikipedia

    en.wikipedia.org/wiki/Taylor's_theorem

    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.

  3. Approximation theory - Wikipedia

    en.wikipedia.org/wiki/Approximation_theory

    The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing the ...

  4. Taylor series - Wikipedia

    en.wikipedia.org/wiki/Taylor_series

    The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations.

  5. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f. The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic ...

  6. Linear approximation - Wikipedia

    en.wikipedia.org/wiki/Linear_approximation

    In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and ...

  7. Finite difference method - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_method

    Where n! denotes the factorial of n, and R n (x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. Following is the process to derive an approximation for the first derivative of the function f by first truncating the Taylor polynomial plus remainder: f ( x 0 + h ) = f ( x 0 ) + f ...

  8. Function approximation - Wikipedia

    en.wikipedia.org/wiki/Function_approximation

    Several progressively more accurate approximations of the step function. An asymmetrical Gaussian function fit to a noisy curve using regression.. In general, a function approximation problem asks us to select a function among a well-defined class [citation needed] [clarification needed] that closely matches ("approximates") a target function [citation needed] in a task-specific way.

  9. Order of approximation - Wikipedia

    en.wikipedia.org/wiki/Order_of_approximation

    For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. In the case of a smooth function , the n th-order approximation is a polynomial of degree n , which is obtained by truncating the Taylor series to this degree.

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