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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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### Calculus Problem Set: Taylor and Maclaurin Series 1. **Error Estimation Using Taylor Series** - Estimate the error if \( P_3(x) = x - x^3/6 \) is used to approximate the value of \(\sin x\) at \( x = 0.1 \). 2. **Error Analysis for Cosine Function Approximation** - 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? 3. **Determining Terms for Accurate approximation** - 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 \(10^{-8}\)? 4. **Using Identities to Derive Maclaurin Series** - 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 \(2 \sin x \cos x\). 5. **Taylor Series Definition and Verification** - 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. #### Notes and Guidance: - For problem 1, recall the general form of the Taylor series remainder term to provide an error estimate for the third-degree polynomial approximation. - For problem 2, consider the higher-order terms of the Taylor series expansion and how they impact the accuracy of the \(1 - x^2/2\) approximation. - Problem 3 involves determining the number of terms required to meet a specific precision. Use the remainder estimation for the logarithmic series. - In problem 4, the identity for \(\sin^2 x\) simplifies the process of deriving the Maclaurin series. - Problem 5 requires the use of differentiation to establish the series for \(\sin(2
Transcribed Image Text:### Calculus Problem Set: Taylor and Maclaurin Series 1. **Error Estimation Using Taylor Series** - Estimate the error if \( P_3(x) = x - x^3/6 \) is used to approximate the value of \(\sin x\) at \( x = 0.1 \). 2. **Error Analysis for Cosine Function Approximation** - 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? 3. **Determining Terms for Accurate approximation** - 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 \(10^{-8}\)? 4. **Using Identities to Derive Maclaurin Series** - 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 \(2 \sin x \cos x\). 5. **Taylor Series Definition and Verification** - 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. #### Notes and Guidance: - For problem 1, recall the general form of the Taylor series remainder term to provide an error estimate for the third-degree polynomial approximation. - For problem 2, consider the higher-order terms of the Taylor series expansion and how they impact the accuracy of the \(1 - x^2/2\) approximation. - Problem 3 involves determining the number of terms required to meet a specific precision. Use the remainder estimation for the logarithmic series. - In problem 4, the identity for \(\sin^2 x\) simplifies the process of deriving the Maclaurin series. - Problem 5 requires the use of differentiation to establish the series for \(\sin(2
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