4401, Name
MAD Numerical Analysis
Keesling
Test 1
2/10/12
Do all problems. Show your work and explain your answers. Each problem is worth 20
points.
1.
Estimate the
3
2
sin( x 2 ) dx using Romberg Integration. Use up to 16
subdivisions in the Composite Trapezoidal Rule. Give seven digits beyond
the decimal for each number. Circle the best estimate of the integral.
n=1
n=2
n=4
n=8
n = 16
2.
Give the normalized Newton-Cotes coefficients for n = 1, 4, 8 . Give these as
fractions.
n =1
a0 =
a1 =
n=4
a0 =
a1 =
a2 =
a3 =
a4 =
n=8
a0 =
a1 =
a2 =
a3 =
a4 =
a5 =
a6 =
a8 =
a7 =
3.
Determine the coefficients to estimate the derivative of f ( x ) at x0 for the
following choices of points. Explain the calculations.
{ x0 , x0 + h, x0 + 2h}
f ( x0 ) A0 f ( x0 ) + A1 f ( x0 + h ) + A2 f ( x0 + 2 h )
A0 =
A1 =
A2 =
4.
What are the coefficients to estimate the integral
b
a
f ( x ) dx using the
following points in [ a, b ] ?
x0 = a +
(b a )
10
x1 = a +
7 (b a )
10
x2 = a +
9 (b a )
10
5.
Suppose that p( x ) is a polynomial of degree at most n. Show that if
p( xi ) = 0 for distinct { x0 , x1 ,, xn } , then p( x ) 0 .