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2. Is it possible to write sin (37x) as a linear combination of sin(2rx) and sin(Tx) for all values of xe 0,1? Prove your answer.

2. Is it possible to write sin (37x) as a linear combination of sin(2rx) and sin(Tx) for all values of xe 0,1? Prove your answer.

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. Consider the function f(x) = 4x, for 0<x<1/2, and f(x) = 2(1-x), for 1/2<x<1. Express f(x) in terms of the Fourier sine series, i.e., f' (s) - Σ. (sin πx). n=1 For N = 1,2 and 3, find f" (x). Show f(x). f' (x), f² (x) and f (x) on the same plot. For N = 1, 2 and 3, find ( S, (f(x) – f" (x))2 dx)/ ,f (x))2 dx). What, if anything, can we conclude from the approximations f (x). f* (x) and f (x) of f(x)? 2. Is it possible to write sin (37x) as a linear combination of sin(2nx) and sin(Tx) for all values of xe0,1? Prove your answer. 3. For xe0,1. express sin 7x a+ br+ cx + ex Find values of a, b, c and e by ensuring that the error (or the residual, r) defined by r(x)=-sin 7x+ a+bx+cx +ex' equals zero at x= 0, 1/4,1/2,3/4.
Transcribed Image Text:1. Consider the function f(x) = 4x, for 0<x<1/2, and f(x) = 2(1-x), for 1/2<x<1. Express f(x) in terms of the Fourier sine series, i.e., f' (s) - Σ. (sin πx). n=1 For N = 1,2 and 3, find f" (x). Show f(x). f' (x), f² (x) and f (x) on the same plot. For N = 1, 2 and 3, find ( S, (f(x) – f" (x))2 dx)/ ,f (x))2 dx). What, if anything, can we conclude from the approximations f (x). f* (x) and f (x) of f(x)? 2. Is it possible to write sin (37x) as a linear combination of sin(2nx) and sin(Tx) for all values of xe0,1? Prove your answer. 3. For xe0,1. express sin 7x a+ br+ cx + ex Find values of a, b, c and e by ensuring that the error (or the residual, r) defined by r(x)=-sin 7x+ a+bx+cx +ex' equals zero at x= 0, 1/4,1/2,3/4.
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