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MAT128A: Numerical Analysis, Section 2

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MAT128A: Numerical Analysis, Section 2

1. Let x0 = 0, x1 = 1/2 and x2 = 1. Find weights w0, w1, and w2 such that the formula
Z 1
0
f(x) dx = w1f(x0) + w2f(x1) + w3f(x2) (1)
holds whenever f is a polynomial of degree less than or equal to 2.
2. Let x0 = −
a
3/5, x1 = 0 and x2 =
a
3/5. Find weights w0, w1 and w2 such that
Z 1
−1
f(x) dx = w1f(x0) + w2f(x1) + w3f(x2) (2)
holds whenever f is a polynomial of degree less than or equal to 2. Show that the formula in fact
holds when f is a polynomial of degree less than or equal to 5.
3. Let x0 = 0, x1 = 1/2 and x2 = 1 Find weights w0, w1 and w2 such that
Z 1
0
f(x)
?
x dx = w0f(x0) + w1f(x1) + w2f(x2) (3)
when f is a polynomial of degree less than or equal to 2. Use this quadrature rule to approximate
Z 1
0
cos(x)
?
x dx.
How accurate is your approximation?
4. Suppose that
f(x) = 2T0(x) + 4T1(x) − 6T2(x) + 12T3(x) − 14T4(x).
Find
Z 1
−1
f(x) dx.
1

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