2. (a) Briey describe how orthogonal polynomials can be used to fi nd the nodes of Gaussian quadra-ture rules for a weighted integral $${\int }_a^{b}f(x)w(x)\mathrm{d}x.$$(b) Using your described approach to find the nodes, and then the method of undetermined coefficients to find the weights, derive the Gauss quadrature rule of the form,$${\int }_{-1}^{1}f(x)(1-x^2{)}^{1/2}\mathrm{d}x\approx w_{0}f(x_0)+w_{1}f(x_1).$$You may use the following facts without proof: ${\varphi }_0(x)=1,{\varphi }_1(x)=x,{\varphi }_2(x)=xt,{\varphi }_2(x)=x^2-1/4.$ are orthogonal polynomials w.r.t. the weight function $w(x)=(1-x^2{)}^{1/2}$ on $(-1,1)$, and ${\int }_{-1}^1(1-x^2{)}^{1/2}\mathrm{d}x=\pi /2.$

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4. Let $f\in [-1,1]$ and let $p_n$ be the best weighted $L_2$ approximation to $f$ from polynomials of degree at most $n$ with respect to the weight function $w(x)=(1-x^2{)}^{-1/2}$ on $(-1;1).$