3) Consider the weighted inner product {p,q) = fo° p(x)q(x)e¯* dx on C[0, ∞) the space of functions continuous on the interval [0, ∞). Use Mathemat- ica to answer the following problems. a) Construct an orthonormal set from the set {1, x, x², x³, x², x³}. b) Use the orthonormal set to find an approximation to sin(x). Give the coefficients numerically. c) Plot sin(x) and the approximation from b.) on the same set of axes on the interval 0 ≤ x ≤ 10. d) Compute the metric distance between sin(x) and the approximation from b.). e) Why are the graphs so different despite the distance between the functions being so small?

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