Let B = {sin 5x, cos 5x, x sin 5x, x cos 5x} be a basis of the vector space V = span(B). 1) Find the matrix [D]a 2 of the derivative map D:V V D(f) = f'. [D]a. 2)What is the fourth power of [D]g? ([D]-R)* = 3) Use the matrix in the previous part to find the fourth derivative of f(x) f(4) (x) = a sin 5x + Bcos 5 + y sin 5x + dx cos 5x. = 10 sin 5x + 8 cos 5x + 5x sin 5x + 3x cos 5x in the form a = B = -1 4) What is the inverse of [D]a? ([D]g+B) 5) Use the matrix in the previous part to find the coefficients p, r,s, and t in the integral (6 sin 5x + 9 cos 5x + 3x sin 5x + 8x cos 5a)dx = = p sin 5x + r cos 5x + sx sin 5x + tx cos 5x + C. p = r = s = t =