First-order approximation

Fit approximation
Concepts
Orders of approximation Scale analysis · Big O notation Curve fitting · False precision Significant figures
Other fundamentals
Approximation · Generalization error Taylor polynomial Scientific modelling

In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth.

Formally, an nth-order approximation is one where the order of magnitude of the error is at most ${\displaystyle x^{n+1}}$, or in terms of big O notation, the error is ${\displaystyle O(x^{n+1}).}$ In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree.

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