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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.
Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor ...
The notes that Marx took have been collected into four independent treatises: On the Concept of the Derived Function, On the Differential, On the History of Differential Calculus, and Taylor's Theorem, MacLaurin's Theorem, and Lagrange's Theory of Derived Functions, along with several notes, additional drafts, and supplements to these four ...
Download as PDF; Printable version; ... move to sidebar hide. From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Taylor's theorem; Retrieved from ...
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
Taylor's theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals (terms of the signature) form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough ...
Toggle An example theorem subsection. 3.1 ... Download as PDF; ... where the last term (the remainder) depends on the exact version of Taylor's formula. For ...
For example, the p i may be the factors of the square-free factorization of g. When K is the field of rational numbers , as it is typically the case in computer algebra , this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition.