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