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1. Find the Taylor series presentation and 4th, 6th order Taylor polynomials for the following functions. a) y = cos x b) y = x - x² cos x 2. Make table of error of three-point centered-difference formula for f'(0), where f(x) = sinx - cos x, with h 10-¹, 10-², 10-³.

1. Find the Taylor series presentation and 4th, 6th order Taylor polynomials for the following functions. a) y = cos x b) y = x - x² cos x 2. Make table of error of three-point centered-difference formula for f'(0), where f(x) = sinx - cos x, with h 10-¹, 10-², 10-³.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Sure! Here's a transcription of the text from the image, suitable for an educational website: --- ### Exercises on Taylor Series and Numerical Differentiation #### 1. Taylor Series Representation Find the Taylor series representation and the 4th, 6th order Taylor polynomials for the following functions: a) \( y = \cos x \) b) \( y = x - x^2 \cos x \) #### 2. Error Analysis in Numerical Differentiation Create a table showing the error of the three-point centered-difference formula for \( f'(0) \), where \( f(x) = \sin x - \cos x \), using \( h = 10^{-1}, 10^{-2}, 10^{-3} \). --- For further clarification: - **Taylor Series**: This is a mathematical series used to approximate functions. The exercises involve expanding the given functions into their Taylor series up to certain orders of approximation (4th and 6th). - **Three-Point Centered-Difference Formula**: This is a numerical method used to approximate the derivative of a function at a certain point. The error table involves comparing the approximated derivative using various small increments (\( h \)) to determine accuracy. This section is designed to enhance understanding of Taylor series expansions and the accuracy of numerical differentiation methods.
Transcribed Image Text:Sure! Here's a transcription of the text from the image, suitable for an educational website: --- ### Exercises on Taylor Series and Numerical Differentiation #### 1. Taylor Series Representation Find the Taylor series representation and the 4th, 6th order Taylor polynomials for the following functions: a) \( y = \cos x \) b) \( y = x - x^2 \cos x \) #### 2. Error Analysis in Numerical Differentiation Create a table showing the error of the three-point centered-difference formula for \( f'(0) \), where \( f(x) = \sin x - \cos x \), using \( h = 10^{-1}, 10^{-2}, 10^{-3} \). --- For further clarification: - **Taylor Series**: This is a mathematical series used to approximate functions. The exercises involve expanding the given functions into their Taylor series up to certain orders of approximation (4th and 6th). - **Three-Point Centered-Difference Formula**: This is a numerical method used to approximate the derivative of a function at a certain point. The error table involves comparing the approximated derivative using various small increments (\( h \)) to determine accuracy. This section is designed to enhance understanding of Taylor series expansions and the accuracy of numerical differentiation methods.
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