Question: Let B = {sin 4x, cos 4x, x sin 4x, x cos 4x} be a basis of the vector space V = span(B). 1)

Let B = {sin 4x, cos 4x, x sin 4x, x cos

4x} be a basis of the vector space V = span(B). 1)

Let B = {sin 4x, cos 4x, x sin 4x, x cos 4x} be a basis of the vector space V = span(B). 1) Find the matrix [D]+B of the derivative map D:V V D(f) = f'. [D]g+ 4 2)What is the fourth power of [D]g g? ([D]g-2)* = 3) Use the matrix in the previous part to find the fourth derivative of f(x) = 4 sin 4x + 5 cos 4x + 3x sin 4x + 8x cos 4x in the form f(4) (2) = a sin 4x + B cos 4x + y sin 4x + 8x cos 4x. a = B = Y = 8 = -1 4) What is the inverse of [D g? ([Da- 5) Use the matrix in the previous part to find the coefficients p, r,s, and t in the integral | (10 sin 4x + 2 cos 4x + 9x sin 4x + 4x cos 4x)dx = = psin 4x + r cos 4x + sx sin 4x + tx cos 4x + C. p = S = t =

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