MAT128A: Numerical Analysis, Section 2
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
.
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 .
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