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an error of magnitude less than 1-cos(2x) 4 Use the identity sin² a to obtain the Maclaurin series for sin² x. Then differentiate this series to 2 obtain the Maclaurin series for 2 sin x cosx. 5 Use the Taylor series definition (determine the pattern for all derivatives, etc.) to obtain the Maclaurin series for sin(2x) and verify that this is the same as the series for 2 sin a cos z from the previous exercise.

an error of magnitude less than 1-cos(2x) 4 Use the identity sin² a to obtain the Maclaurin series for sin² x. Then differentiate this series to 2 obtain the Maclaurin series for 2 sin x cosx. 5 Use the Taylor series definition (determine the pattern for all derivatives, etc.) to obtain the Maclaurin series for sin(2x) and verify that this is the same as the series for 2 sin a cos z from the previous exercise.

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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### Mathematical Exercises on Series Approximations #### Problem 1 **Estimate the error if \( P_3(x) = x - x^3/6 \) is used to approximate the value of \( \sin x \) at \( x = 0.1 \).** #### Problem 2 **If \( \cos x \) is replaced by \( 1 - x^2/2 \) and \( |x| < 0.5 \), what estimate can be made of the error? Does \( 1 - x^2/2 \) tend to be too large, or too small?** #### Problem 3 **How many terms of the Maclaurin series for \( \ln(1+x) \) should you add up to be sure of calculating \( \ln(1.1) \) with an error of magnitude less than \( 10^{-8} \)?** #### Problem 4 **Use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) to obtain the Maclaurin series for \( \sin^2 x \). Then differentiate this series to obtain the Maclaurin series for \( 2 \sin x \cos x \).** #### Problem 5 **Use the Taylor series definition (determine the pattern for all derivatives, etc.) to obtain the Maclaurin series for \( \sin(2x) \) and verify that this is the same as the series for \( 2 \sin x \cos x \) from the previous exercise.** ### Graphs and Diagrams There are no graphs or diagrams included in this transcription. The problems focus on the application and understanding of series approximations in different mathematical contexts.
Transcribed Image Text:### Mathematical Exercises on Series Approximations #### Problem 1 **Estimate the error if \( P_3(x) = x - x^3/6 \) is used to approximate the value of \( \sin x \) at \( x = 0.1 \).** #### Problem 2 **If \( \cos x \) is replaced by \( 1 - x^2/2 \) and \( |x| < 0.5 \), what estimate can be made of the error? Does \( 1 - x^2/2 \) tend to be too large, or too small?** #### Problem 3 **How many terms of the Maclaurin series for \( \ln(1+x) \) should you add up to be sure of calculating \( \ln(1.1) \) with an error of magnitude less than \( 10^{-8} \)?** #### Problem 4 **Use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) to obtain the Maclaurin series for \( \sin^2 x \). Then differentiate this series to obtain the Maclaurin series for \( 2 \sin x \cos x \).** #### Problem 5 **Use the Taylor series definition (determine the pattern for all derivatives, etc.) to obtain the Maclaurin series for \( \sin(2x) \) and verify that this is the same as the series for \( 2 \sin x \cos x \) from the previous exercise.** ### Graphs and Diagrams There are no graphs or diagrams included in this transcription. The problems focus on the application and understanding of series approximations in different mathematical contexts.
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