Name:Math 4564 Homework 71. Show that the following three polynomials are orthogonal in L2 [−1, 1]φ0 (x) = 1,φ1 (x) = x,1φ2 (x) = (3×2 − 1)22. Compute the norms ||φi (x)|| for the polynomials in the previous problem.3. Compute the best approximation p2 (x) for the function f (x) = ex .p2 (x) = c0 φ0 (x) + c1 φ1 (x) + c2 φ2 (x),ck =f, φkφk , φkYou can use Integrate[…] command in Mathematica to compute c2 .Plot both f (x) and p2 (x) in Mathematica on the same set of axes.4. Use Mathematica to show that the functionsφk (x) = sin2kπxb−aand ψk (x) = cos2kπx ,b−ak = 1, 2, 3, …form an orthogonal set in L2 [a, b].Hint:Use Integrate[…,{x,a,b}] command to compute the inner products.
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