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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
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