(a) Using an orthogonal polynomial basis, find the best least squares polynomial approxi- mations, 92(t) of degree at most 2 and 93(t) of degree at most 3, to f (t) = e-³t over the interval [0,3]. Exercises 381 [Hint: For a polynomial p(x) of degree n and a scalar a > 0 we have fe¯ax p(x)dx = a ax 'n (Σ"; =0 (2)(x))), where p()(x) is the jth derivative of p(x). Alternatively, just use numerical quadrature, e.g., the MATLAB function quad.] (b) Plot the error functions f(t) - 92(t) and f(t) - 93(t) on the same graph on the interval [0,3]. Compare the errors of the two approximating polynomials. In the least squares sense, which polynomial provides the better approximation? [Hint: In each case you may compute the norm of the error, (for (f(t) - qn(t))² dt) 1/2, using the MATLAB function quad.] (c) Without any computation, prove that 93(t) generally provides a least squares fit, which is never worse than with 92(t).