Question

7. (15 pts) Numerical Integration. Given a continuous function f (x) on the interval [a, b], write the Lagrange form of the quadratic polynomial interpolating f(a), (a b)), f(b). Instead of calculating the integral I(f) Jaf(x)dx we could approximate it via Q(f) = | q(x)dx. Find an expression for this quadrature rule, the so-called Simpson's rule.

Answer 1

Given that a continuous function f(x) on the interval [a, b] f: [a, b] rightarrow R is continuous x_0 = a, x_1 = a + b/3 x_2 = b y_0 = f(a) y_1 = f(a + b/2) y_2 = f(b) Lagrange's from of the quadratic interpolating polynomial is q(x) = p_0(x)/p_0(x_0).y + p_1(x)/p_1(x_1) y_1 + p_2(x)/p_2(x_2) y_2 p_k(x) = (x - x_0)(x - x_1)(x - x_2) .... (x - x_k - 1)(x - x_k + 1) so, p_0(x) = (x - x_1)(x - x_2) = (x - a + b/2)(x - b) p_1(x) = (x - x_1)(x - x_2) = (x - a)(x - b) p_2(x) = (x - x_0)(x - a + b/2) \r\n p_0(x_0) = (a - a + b/2)(a - b) = (a - b)^2/2 p_1(x_1) = (a + b/2 - a)(a + b/2 - b) = (b - a)/2.(a - b)/2 = -(a - b)^2/4 p_2(x_2) = (b - a)(b - a + b/2) = (b - a)^2/2 so, the polynomial q(x) is q(x) = 2 + (a)/(a - b)^2(x - a + b/2)(x - b)