MAT128A: Numerical Analysis, Homework 6

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1. Go over the midterm problems and the provided solutions!
2. Show that for all nonnegative integers n, Tn(1) = 1 and Tn(−1) = (−1)n
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3. Show that for all integers n ≥ 2 and all −1 < t ≤ 1, Z t −1 Tn(x) dx = 1 2 ˆ Tn+1(t) n + 1 − Tn−1(t) n − 1 ˙ − (−1)n n2 − 1 . 4. Let x0, x1, . . . , xN , w0, w1, . . . , wN denote the nodes and weights of the (N + 1)-point GaussLegendre quadrature rule. Suppose that f : [−1, 1] → R is continuously differentiable, and that c0, c1, . . . , cN are defined by the formula cn = X N j=0 f(xj )Pn(xj )wj . Show that the polynomial pN (x) = X N n=0 cnPn(x) interpolates f at the points x0, x1, . . . , xN . 1