Construct a quadrature rule of the form* (` S ( ` ) da ~ AOf ( - 1 ) + AI. $ ( 0 ) + A28 ( ! ) ( 2) which is exact for polynomials of degree lt; 2....

Construct a quadrature rule of the form which is exactly for polynomials <= 2. (a) derive the 3-point (legendre) Gaussian quadrature to approximate f(x) (b) versify its degree of precision (c) compare the accuracy of this 3-point gaussian quadrature with that of the simple simpson rule for approximating e^x (d) show that the 3-point gaussian quadrature can be used for approximating f(x) by doing a simple change of variables and apply this to approximate sin(x)/x

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Construct a quadrature rule of the form*(` S ( ` ) da ~ AOf ( - 1 ) + AI. $ ( 0 ) + A28 ( ! )( 2)which is exact for polynomials of degree < 2.( a ) Derive the 3 - point ( Legendre ) Gaussian quadrature to appproximate [_ folder( i.e. you need to obtain the nodes In , II , I, and the corresponding weightsAN, AI , A2 ).( b ) Verify it's degree of precision .