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Gaussian Quadrature

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1. Consider approximating integrals of the form

I ( f ) = ∫ √x f(x)dx

in which f(x) has several continuous derivatives on [0, 1]
a. Find a formula

∫ √x f(x)dx ≈ w1 f(x1) ≡ I1( f )

which is exact if f(x) is any linear polynomial.

b. To find a formula

∫ √x f(x)dx ≈ w1 f(x1) + w2 f(x2) ≡ I2( f )

which is exact for all polynomial of degree ≤ 3, set up a system of four equations with unknown w1, w2, x1, x2 and find the values

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