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Problem 1. Indicate which of the following statements are true and which are false. You do not need

to justify your answers.

1. If f : r´1, 1s Ñ R is a C

k

functions and tanu are the Chebyshev coefficients of f, then |an| “

O

`

1

nk

˘

.

2. The condition number of evaluation of the function fpxq “ 1

x

goes to 8 as x Ñ 0

`.

3. The condition number of evaluation of the function fpxq “ cospxq goes to 8 as x Ñ π{2.

4. The quadrature rule

ż π

´π

fptq dt «

2π

n ` 1

ÿn

j“0

f

ˆ

´π `

2π

n ` 1

j

˙

,

is exact for the collection of functions

expp´iktq, k “ ´n, ´n ` 1 ´ n ` 2, . . . , ´1, 0, 1, 2, . . . , n ´ 1, n.

5. If p is a monic polynomial of degree n, then

max

´1ďxď1

|ppxq| ě 2

´n`1

.

Problem 2. Find the unique polynomial p of degree less than or equal to 3 such that

ppx0q “ fpx0q

p

1

px0q “ f

1

px0q

ppx1q “ fpx1q

p

1

px1q “ f

1

px1q,

where

fpxq “ sinpxq

and

x0 “ 0, and x1 “

π

2

.

Problem 3. Let fpxq “ cospxq and, for each positive integer N, let pN be the polynomial of degree

less than or equal to N which interpolates f at the nodes

xj “ ´1 `

2j

N

, j “ 0, . . . , N.

Show that

max

´1ďxď1

|fpxq ´ pN pxq| Ñ 0 as N Ñ 8.

Problem 4. Find a quadrature rule of the form

ż 1

´1

fpxq |x| dx « fp´1qw0 ` fp0qw1 ` fp1qw2

which is exact whenever f is a polynomial of degree less than or equal to 2.

Problem 5. Compute the Chebyshev coefficients of the function

fpxq “ a

1 ´ x

2.

Problem 6. Suppose that f : R Ñ R is smooth, and that h ą 0. Find coefficients a, b and c such that

afp´hq ` bfphq ` cfp2hq “ f

1

p0q ` O

`

h

2

˘

.

Problem 7. Let fpxq “ cospxq and, for each positive integer N, let pN be the polynomial of degree

less than or equal to N which interpolates f at the nodes

xj “ cos ˜

j `

1

2

N ` 1

¸

, j “ 0, . . . , N.

Show that

max

´1ďxď1

|fpxq ´ pN pxq| ď 2

´N

pN ` 1q!

.

Problem 8. Show that the Legendre Polynomial Pn of degree n satisfies the differential equation

p1 ´ x

2

qf

2

pxq ´ 2xf1

pxq ` npn ` 1qfpxq “ 0.

Bonus problem 1: Find a quadrature rule of the form

ż 1

´1

fpxq |x| dx « fpx0qw0 ` fpx1qw1 ` fpx2qw2

which is exact whenever f is a polynomial of degree less than or equal to 5.

Bonus problem 2: Find the nodes t0, t1, t2, t3 and weights w0, w1, w2, w3 of a quadrature rule

ż π

´π

fptq dt «

ÿ

3

j“0

fptj qwj

which is exact for the functions

texppintq : n “ ´3, ´2, ´1, 0, 1, 2, 3u .

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