CS/Math 166 -- Spring 2008 -- Final Exam Postmortem

CS/Math 166 Final Exam Postmortem -- Spring 2008

CS/Math 166 -- D. C. Smolarski, S.J.
Santa Clara University, Department of Mathematics and Computer Science

NOTES:

General:

As a matter of policy, I keep finals for one regular quarter. You may examine your final at any time in my office. If you wish to collect your final, you may do you in December.

For your information, many of the multiple-choice questions were "borrowed" from on-line exams for Numerical Analysis courses or printed tests I obtained from other universities. These show you the types of questions that may be found elsewhere.

Prob 1: (cf. Final Review bullet 4; Notes 4-23)

Prob 2: (cf. Mid I Review bullet 5, Final Review Bullet 1; Notes 2-4)

Prob 3: (cf. Mid I Review bullet 6; Notes 3-10, 3-11)

Prob 4: (cf. Mid I Review bullet 12; Notes 4-26A)

Prob 5: (cf. Mid I Review bullet 12)

Prob 6: (cf. Notes 6-27)

Prob 7:

Prob 8: (cf. Final Review bullet 3; Notes 4-26A)

Prob 9: (cf. Mid II Review bullet 11, Final Review bullet 5; Notes 4-17)
I remember mentioning in class that "normal" equations are used in several ways in Numerical Analysis.

Prob 10: (cf. Final Review bullet 10; Notes 6-14)
Some had problems doing the simple evaluation of the function (i.e., the polynomial within the integral) correctly at 2 or 3 points to get the Gauss Quad values.

Prob 11: (cf. Final Review bullet 11)
This problem specified that we wanted an exact formula for a polynomial of degree 0.
A polynomial of degree 0 is a constant. Thus it can be taken out of the integral (on the left of the given equation) and "cancelled" with f(3/2) (which equals the same constant value) on the right. Then the left side of the initial equation can be evaluated exactly via the (integration by parts) formula given.
This was not intented so much as a "trick" question but as a question meant to see whether the reader took into account all the given information, which is very important in numerical problems.

Prob 12: (cf. Final Review bullet 12; Notes 6-18)
One introduces new variables which eliminate some of the derivatives of y and then one creates several equations with a derivative on the left side and an expression with no derivatives on the right. Since you are faced with a 3rd order DiffEq (i.e., one with a 3rd derivative in it), you should end up with 3 first order DiffEqs and new initial conditions related to the new variables.

Prob 13: (cf. Final Review bullet 9; Notes 6-8)
One needs to use the error formula to determinine the width needed and thus also the number of panels. In the error formula the argument of f⁽⁴⁾(x) is chosen to maximize the value over the interval of integration.
One should use the error formula for the "composite" version given in the notes (and on page 199 of the book).
Since Simpson's rule demands that n be an even number, the minimum answer was 32 (the next higher even number greater than 30.006).

Prob 14: (cf. Final Review bullet 13; Notes 6-38A, 6-39)
Some had the correct discretization form for y'' but, unfortunately, made various algebraic errors resulting in an incorrect final linear system.

Prob 15: (cf. Final Review bullet 7; Notes 6-4)
Some used all 5 points for each approximation or varied the interval final point depending on the panel widths. In each case, one is supposed to compute the approximation to the same integral over the same interval (0,1) but using various widths for the panels. When the width h is 1, then there is only one panel (since the distance between the two limits of the integral is 1) and only the two endpoints are used. When the width is 1/2, there are two panels and the two endpoints and the middle point is used. When the width is 1/4, all 5 points are used.

Prob 16: (cf. Final Review bullet 8; Notes 6-11)
From information given in class, the Trapezoidal rule is of order 2 so the base value in the Richardson formula should be 2. Since the increase in panels form the first value (5) to the second value (10) is 2-fold, the exponent is also 2. Thus the denominator of the fraction is 2²-1 or 3.

Statistics

Scores, raw and normalized

  
      final   nfinal
	148    79
	109    58
	 95    50
	 81    43
	 73    38
	 61    32

MAXIMUM 200      100

High score: 148/200
Low score: 61
Range: 88 pts
Median (middle score): < 88 >
Mean (average score): 94.500
Standard Deviation: 31.11
Number who took exam: 6

Distribution

   x    x
   x    x    x         x
   60-  80- 100- 120- 140-
   79   99  119  139  159

Number of Perfect Scores per Problem