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If cos x is replaced by 1-²/2 and [x] < 0.5, what estimate can be made of the error? Does 1-²/2 tend to be too large, or too small?

If cos x is replaced by 1-²/2 and [x] < 0.5, what estimate can be made of the error? Does 1-²/2 tend to be too large, or too small?

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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1 Estimate the error if P3(x) = x - x³/6 is used to approximate the value of sin x at x = 0.1. 2 If cos z is replaced by 1-²/2 and 2x < 0.5, what estimate can be made of the error? Does 1-2²/2 tend to be too large, or too small? 3 How many terms of the Maclaurin series for In(1+x) should you add up to be sure of calculating In(1.1) with an error of magnitude less than 10-8? 4 Use the identity sin²x = 1-cos(2x) to obtain the Maclaurin series for sin²x. Then differentiate this series to 2 obtain the Maclaurin series for 2 sin x cos x. 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 cos x from the previous exercise.
Transcribed Image Text:1 Estimate the error if P3(x) = x - x³/6 is used to approximate the value of sin x at x = 0.1. 2 If cos z is replaced by 1-²/2 and 2x < 0.5, what estimate can be made of the error? Does 1-2²/2 tend to be too large, or too small? 3 How many terms of the Maclaurin series for In(1+x) should you add up to be sure of calculating In(1.1) with an error of magnitude less than 10-8? 4 Use the identity sin²x = 1-cos(2x) to obtain the Maclaurin series for sin²x. Then differentiate this series to 2 obtain the Maclaurin series for 2 sin x cos x. 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 cos x from the previous exercise.
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