# 11. In an introductory calculus course, you may have seen of the form approximation formulas for...

###### Question:

11. In an introductory calculus course, you may have seen of the form approximation formulas for integrals f(t)dt wf(a,), 2 where the a, are equally spaced points on the interval (a,b) and the u, are certain "weights (giving Riemann sums, trapezoidal sums, or showed that, with the same computational effort (same type of formula) we can get better approximations if we don't require the a, to be equally spaced. Simpson's rule depending on their values). Gauss Consider the space P, with inner product S.9) ()g(t)dt. Let fo f fn be an orthonormal basis of this space with degree of f k. We could make such a basis by doing Gram-Schmidt to 1,t, t2,..,. It is a fact that fn has n different roots a1, ... , an on (-1, 1). We covered in the additional notes how to find "weights" , w2 Wn such that for all polynomials of degree less than n. equal"). In fact this works for all polynomials of degree less than 2n (you don't need to prove this) (this is an honest equality, not "approximately For the case n 2, find a,a2, and w1, W2, and show that f()dt= wf(a1)+ wef (02) for all cubic polynomials f

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