Solve c part only in 30 min and take a thumb up Consider the space of differentiable functions f: [0, 1] - R with the L 2 -inner product(f. g) = 5n 0 f(x)g(x) dx.(a)Show that the collection consisting of the featuressin(x),sin(2x), ...,sin( (k - 1)x), (8.1)defined for x € (0, n), are orthogonal for this inner product.(b)Explain that function f(x) = 1 is not perpendicular to sin(x) or to sin(3x), but is perpendicularto sin(2x) and sin(4x).(c)Determine the projection f(x) = 1 along sin(x), sin(3x)and find from there a function of theform 1 + c1 sin(x) + c3 sin(3x), which is perpendicular to both sin(x) and sin(3x). Explain whythis combination is perpendicular to all functions in (8.1) for k = 5.(d)For k = 5, determine in python the linear combination of functions in (8.1), which is closest tothe functionf(x) = 1 - e -x for 0< x

The aim is to find projection of function f(x)=1 on sin x and sin 2x. Then we are trying to provide…

Answered: Determine the projection f(x) = 1 along…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Determine the projection f(x) = 1 along sin(x), sin(3x)and find from there a function of the form 1 + c1 sin(x) + c3 sin(3x), which is perpendicular to both sin(x) and sin(3x). Explain why this combination is perpendicular to all functions in (8.1) for k = 5.