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1. Let f(x) be a function defined on the interval [−1, 1], and f ∈ C

4

[−1, 1] .

(a) Let h(x) be the Lagrange interpolation polynomial of f(x) at the nodes x = −1, 0, 1.

Write down the expression of h(x).

(b) Write down the error term E(x) := f(x) − h(x) in terms of the derivatives of f(x).

(Recall the theorem about the error between the interpolation formula h and the

exact function f.)

(c) Compute the integral

Z

1

−1

h(x)dx

exactly in terms of the values of f(x) at points x = −1, 0, 1.

(d) If we approximate the integral R

1

−1

f(x)dx by R

1

−1

h(x)dx, is it true that the above

approximation is exact if f is a polynomial of degree less than or equal to 2 ? Why ?

(e) Write down an error bound of this approximation rule suggested in (d) directly based

on the result in (b).

2. A function f has the values shown as below:

x 0 1 2 3 4

f(x) 1 2 1 2 1

(a) Use Simpson’s Rule and only the function values at x = 0, 2, 4 to approximate the

integral R

4

0

f(x)dx.

(b) Use composite Simpson’s Rule and the functions values at x = 0, 1, 2, 3, 4 to approximate the same integral R

4

0

f(x)dx.

3. (Programming problem) Consider the integral:

Zπ

0

cos xdx

1

(a) Write a program to use the composite trapezoidal to approximate the above integral

by dividing [0, π] to N equal spaces.

(b) Write a program to use the composite Simpson’s approximate the above integral by

dividing [0, π] to N equal spaces.

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