INSTRUCTOR 
Dr. Andrew Knyazev
Office: CU (Dravo) 620G. Phone: 556-8102.
Office hours: Tue 3pm - 6pm
                                            Monday 5:30, SI203
NA I     FINAL  Exam     (Chapter 5)        December 11, 1995

Closed testbook, no calculators. You can use your notes!
Good luck!
_______________________________________________________________________                      


Part I. Trapezoid rule. Section 5.2. 
Use interval [-1, 3] and function  f(x) = x





1. Use nodes {-1,0,1,2,3}. Then T(f), an approximation to the integral
by Trapezoid rule, is

[A]   10    [B]    11   [C]   22   [D]    33   [E]   None





2. Using the Theorem of accuracy of the Trapezoid rule
find the estimate of absolute error in problem #1.

[A]   0    [B]    2   [C]   4   [D]    6   [E]   None



3. Use nodes {-1,3}. Then T(f), an approximation to the integral
by Trapezoid rule, is


[A] 32    [B]  42    [C]  52   [D]  62   [E]   None



4. In problem #3, find the actual absolute error


[A] 13    [B]  14    [C]  15   [D]  16   [E]   None





Part II. Simpson's rule. Section 5.4. 
Use interval [-1, 3] and function   f(x) = x


5. Use nodes {-1,0,1,2,3}. Then S(f), an approximation to the integral
by Simpson's rule, is

[A]   20    [B]    41   [C]   52   [D]    63   [E]   None









6. In the previous problem, the actual absolute error is

[A]   0    [B]    1   [C]   2   [D]    4/3   [E]   None

       
7. Using the Theorem of accuracy of the Simpson's rule
find the best estimate of absolute error in the problem #5.


[A]   0    [B]    1   [C]   2   [D]    4/3   [E]   None



Part III. Gaussian Quadrature Formulas. Section 5.5. 
Use interval [-1,1].


8. Determine the quadrature formula of the form
A  f(0) + A  f(1) on the interval [-1,1] with nodes {0,1}.
Then A  = 

[A] 0    [B] 1   [C] 2   [D]  3   [E]   None




9. Determine the quadrature formula of the form
A  f(-1) + A  f(1) on the interval [-1,1] with nodes {-1,1}.
Then A  = 

[A] 0    [B] 1   [C] 2   [D]  3   [E]   None





10. Find the polynomial q(x) of the first degree with q(1)=-1
referred to in the Theorem on Gaussian quadrature, i.e.
the polynomial q(x) that is "orthogonal" to a zero degree
polynomial, on the interval [-1,1]. Then q(1/2)=
 

[A]   0    [B]    1/2   [C]   -1/2   [D]    1   [E]   None