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2. For small x, the approximation sin x x is often used. For what range of x is this good to a relative accuracy of × 10-¹4? The Taylor expansion of sin æ about 0 is sin x = x2k+1 Σ(-1)*, (2k + 1)! k=0

2. For small x, the approximation sin x x is often used. For what range of x is this good to a relative accuracy of × 10-¹4? The Taylor expansion of sin æ about 0 is sin x = x2k+1 Σ(-1)*, (2k + 1)! k=0

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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2. For small \( x \), the approximation \(\sin x \approx x\) is often used. For what range of \( x \) is this good to a relative accuracy of \( \frac{1}{2} \times 10^{-14} \)? The Taylor expansion of \(\sin x\) about 0 is \[ \sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \]
Transcribed Image Text:2. For small \( x \), the approximation \(\sin x \approx x\) is often used. For what range of \( x \) is this good to a relative accuracy of \( \frac{1}{2} \times 10^{-14} \)? The Taylor expansion of \(\sin x\) about 0 is \[ \sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \]
Expert Solution
Step 1: Description

For small x, the approximation sin open parentheses x close parentheses almost equal to x is often used.

Consider Taylor's series for sin(x) up to the x cubed term.

sin open parentheses x close parentheses equals 0 plus x minus 0 minus fraction numerator x cubed over denominator 3 factorial end fraction.

The error bound is given as open vertical bar R subscript n open parentheses x close parentheses close vertical bar less or equal than 10 to the power of negative 14 end exponent over 2.

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