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Consider the Maclaurin expansion +10 6 120 x6 sin(x²) = x² + x14 5040 + x18 362880 + valid for all real numbers x. a) Use the expansion to give the first three terms of a power series expansion for sin(x²) dx. S b) Use your expansion in part a) to approximate 0.⁹ sin(x²) dx. Again, use three terms. Round your answer to four decimal places. c) Use the Alternating Series Estimation Theorem to give an upper bound on the error in your estimate in part b). a

Consider the Maclaurin expansion +10 6 120 x6 sin(x²) = x² + x14 5040 + x18 362880 + valid for all real numbers x. a) Use the expansion to give the first three terms of a power series expansion for sin(x²) dx. S b) Use your expansion in part a) to approximate 0.⁹ sin(x²) dx. Again, use three terms. Round your answer to four decimal places. c) Use the Alternating Series Estimation Theorem to give an upper bound on the error in your estimate in part b). a

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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Problem (3) Consider the Maclaurin expansion sin(x²) = x² to x10 + 6 120 x¹4 5040 + x18 362880 + valid for all real numbers x. a) Use the expansion to give the first three terms of a power series expansion for f sin(x²) dx. -0.9 b) Use your expansion in part a) to approximate ſº.⁹ sin(x²) dx. Again, use three terms. Round your answer to four decimal places. c) Use the Alternating Series Estimation Theorem to give an upper bound on the error in your estimate in part b). a
Transcribed Image Text:Problem (3) Consider the Maclaurin expansion sin(x²) = x² to x10 + 6 120 x¹4 5040 + x18 362880 + valid for all real numbers x. a) Use the expansion to give the first three terms of a power series expansion for f sin(x²) dx. -0.9 b) Use your expansion in part a) to approximate ſº.⁹ sin(x²) dx. Again, use three terms. Round your answer to four decimal places. c) Use the Alternating Series Estimation Theorem to give an upper bound on the error in your estimate in part b). a
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