a. Calculate the first four derivatives of sin(x²) and hence find the fourth order Tay- lor polynomial for sin(x²) centered at a = 0. b. Part (a) demonstrates the brute force approach to computing Taylor polynomials and series. Now we find an easier method that utilizes a known Taylor series. Recall that the Taylor series centered at 0 for f(x) = sin(x) is Σ(-1)* k=0 x2k+1 (2k + 1)! (8.5.7) i. Substitute x² for x in the Taylor series (8.5.7). Write out the first several terms and compare to your work in part (a). Explain why the substitution in this problem should give the Taylor series for sin(x²) centered at 0. ii. What should we expect the interval of convergence of the series for sin(x²) to be? Explain in detail.
a. Calculate the first four derivatives of sin(x²) and hence find the fourth order Tay- lor polynomial for sin(x²) centered at a = 0. b. Part (a) demonstrates the brute force approach to computing Taylor polynomials and series. Now we find an easier method that utilizes a known Taylor series. Recall that the Taylor series centered at 0 for f(x) = sin(x) is Σ(-1)* k=0 x2k+1 (2k + 1)! (8.5.7) i. Substitute x² for x in the Taylor series (8.5.7). Write out the first several terms and compare to your work in part (a). Explain why the substitution in this problem should give the Taylor series for sin(x²) centered at 0. ii. What should we expect the interval of convergence of the series for sin(x²) to be? Explain in detail.
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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show full and complete procedure. Do not ommit any steps
Transcribed Image Text:a. Calculate the first four derivatives of sin(x²) and hence find the fourth order Tay-
lor polynomial for sin(x²) centered at a = 0.
b. Part (a) demonstrates the brute force approach to computing Taylor polynomials
and series. Now we find an easier method that utilizes a known Taylor series.
Recall that the Taylor series centered at 0 for f(x) = sin(x) is
x2k+1
(2k + 1)!*
Σ(-1)*,
k=0
(8.5.7)
i. Substitute x² for x in the Taylor series (8.5.7). Write out the first several terms
and compare to your work in part (a). Explain why the substitution in this
problem should give the Taylor series for sin(x²) centered at 0.
ii. What should we expect the interval of convergence of the series for sin(x²)
to be? Explain in detail.
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