Taylor's Theorem
In this lesson on Taylor's Theorem, we learn about the convergence of Taylor series to f(x) and the importance of Taylor polynomials in providing local approximations of f(x) near x0. We are also introduced to the remainder term and Lagrange's formula for the remainder, which allow us to estimate the error in the approximation of f(x) by a Taylor polynomial. The integral form of the remainder and the proof of its generalization are also discussed. Overall, this lesson provides a comprehensive understanding of Taylor's Theorem and its applications in mathematics.
Taylor polynomials and the remainder term. Convergence of Taylor series to f(x).
Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at http://online.math.uh.edu/HoustonACT/videocalculus/index.html.
- What is Taylor's Theorem?
- When does a Taylor series of f converge to f(x)?
- What are Taylor polynomials?
- What is the nth Taylor polynomial centered at x0 for a function f?
- What are the Taylor polynomials centered at 0 for sin x?
- What is the remainder term for Taylor polynomials?
- What is Lagrange's formula for the remainder?
- When does f(x) = its Taylor series?
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Staff Review
This video takes the idea of convergence of Taylor Series, and defines, explains and outlines the proof of Taylorâs Theorem. Again, many examples and great visuals are used. Taylor polynomials and remainder terms are explained and shown visually as well. Lagrangeâs formula for the remainder is also defined and used in several examples. This is a really complicated and high-level lesson using the ideas from the last few videos. -
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