MATH 573 ASSIGNMENT 2 1a. Let I = f??1;??1=2; 0; 1=2; 1

MATH 573 ASSIGNMENT 2 1a. Let I = f??1;??1=2; 0; 1=2; 1g and f(x) = 1=(1 + x2). Find the value of each of the following at x = 3=4. i) The polynomial P(x) of degree 4 interpolating f(x) on the set I. ii) The piecewise linear function L(x) dened on a mesh of width 1=2 interpolating f(x) on I. iii) The piecewise quadratic function Q(x) dened on a mesh of width 1 interpolating f(x) on I. 1b. Using the error formula nd the smallest bound on the quantity maxx2[??1;1] jf(x)??L(x)j where L(x) is the approximation given in (ii). As a computational check f000(x) = ??24x(x2 ?? 1) (1 + x2)4 : 2. Using the Newton formula determine the cubic polynomial P(x) satisfying: P(a) = f(a); P(b) = f(b); P0(a) = 0; P0(b) = 0: 3. Let a = x0

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