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 .

1