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1. Find a polynomial p of degree 3 such that

p(0) = 0, p(1) = 1, p(2) = 1, and p

0

(0) = 1.

2. (a) Show that the roots of

pN (x) = TN+1(x) − TN−1(x)

are

xj = cos ´ π

N

j

¯

, j = 0, 1, . . . , N.

(b) Use (a) to prove that

(x − x0)· · ·(x − xN ) = 2−N pN (x).

(c) Show that

|(x − x0)· · ·(x − xN )| ≤ 2

−N+1

for all x ∈ [−1, 1].

3. Suppose that f : [a, b] → R is (N + 1)-times continuously differentiable, and that x0, . . . , xN are

the (N + 1) nodes of the Chebyshev extrema grid on the interval [a, b] so that

xj =

b − a

2

cos ˆ

j

N

π

˙

+

b + a

2

for all j = 0, 1 . . . , N.

Also, let pN be the polynomial of degree N which interpolates f at the nodes x0, x1, . . . , xN . Show

that there exists ξ ∈ (a, b) such that

|f(x) − pk(x)| ≤ 2

−N+1 ˆ

b − a

2

˙N+1 ˇ

ˇ

ˇ

ˇ

ˇ

f

(N+1)(ξ)

(N + 1)!

ˇ

ˇ

ˇ

ˇ

ˇ

Hint: Let g(x) = f

`

b−a

2

x +

b+a

2

˘

and use 2(c) to develop an error bound for g.

4. Suppose that f(x) = cos(x), that N is a positive integer, and that x0, . . . , xN are the nodes of

the Chebyshev extrema grid on the interval [0, 1]. Also, let pN denote the polynomial of degree N

which interpolates f at the nodes x0, . . . , xN . Show that

|f(x) − pN (x)| ≤

2

−2N

(N + 1)!

for all −1 ≤ x ≤ 1.

1

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