Approximating Functions with Taylor Polynomials in Calculus II

Dec 07, 2025

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This content explores the concept of approximating functions using Taylor polynomials, specifically focusing on the n-th order Taylor polynomial, which comprises the first n terms of the Taylor series. It discusses the remainder in the polynomial approximation as exemplified by Taylor’s Theorem and includes prerequisites such as continuous derivatives on an open interval. Understanding the properties of the Taylor polynomial, centered at a specific value, and estimating the remainder's impact on function approximation are crucial aspects emphasized in this study.

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Approximating Functions with Taylor Polynomials in Calculus II

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11.1 Approximating Functions with Polynomials Math 6B Calculus II
Taylor Polynomial • The n th-order Taylor Polynomial is the first n terms of the Taylor series.
Remainder in a Taylor Polynomial • Let be a Taylor polynomial of order n for f . The remainder in using to approximate f at the point x is
Taylor’s Theorem • Let f have continuous derivatives up to on an open interval I containing a. For all x in I. where is the n-th order Taylor polynomial for f centered at a, and the remainder is for some point c between x and a.
Taylor’s Inequality