## Description

Problem 1. Derive the following formulas for approximating derivatives and show that they

are both O(h

4

) by establishing their error terms

f

0

(x) ≈

1

12h

(−f(x + 2h) + 8f(x + h) − 8f(x − h) + f(x − 2h))

f

00(x) ≈

1

12h

2

(−f(x + 2h) + 16f(x + h) − 30f(x) + 16f(x − h) − f(x − 2h)).

Problem 2. Derive the Newton-Cotes formula for R 1

0

f(x)dx based on the nodes 0, 1/3, 2/3, 1.

Problem 3. Verify that the following formula is exact for polynomials of degrees ≤ 4

Z 1

0

f(x)dx ≈ 1/90(7f(0) + 32f(1/4) + 12f(1/2) + 32f(3/4) + 7f(1)).

Problem 4. Derive a formula for approximating

Z 3

1

f(x)dx

in terms of f(0), f(2) and f(4). It should be exact for all f ∈ Π2.

Problem 5. Determine values for A, B, and C that make the formula

Z 2

0

xf(x)dx ≈ Af(0) + Bf(1) + Cf(2)

exact for all polynomials of degree as high as possible. What is the maximum degree?

1