Sample Exam Questions taken from Examples in Class Notes

The following are based on the topics listed at midsfinalstudy.html and the examples found in the class notes.

NOTE that questions on other topics may appear on exams, so the following should NOT be seen as a complete list of possible topics or types of questions. In particular, True/False questions, multiple choice questions, and "discuss"-type questions are also possible.

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

[Return to CS/Math 166 Homepage]

NOTE: the numbers in square brackets, e.g., [2-5], following each question below refers to the page number in the notes where the answer (or a reference to the definition or to the solution method) may be found.

  1. Convert 10011012 from binary to decimal. [2-1]
  2. Convert 0.110 from decimal to binary. [2-2]
  3. Using 3 digit rounding arithmetic, compute the value of 2/3 × 1/7. Do the same with 3 digit chopping arithmetic. [2-4], [2-4A], BBF sec 1.2 HW.
  4. Given x = 1/3 and the approximation fl(x) = 0.33, find (a) the error, (b) the relative error, and (c) the percentage error. [2-5]
  5. Using standard rules for "interval arithmetic," find [0,1] - [2,3] and [0,1]×[2,3]. Between [2-6] and [2-7]
  6. Using Horner's rule, convert x3 - 10 x2 + 3x -5 into a nested form. [2-9]
  7. Using Taylor's series expansion, convert x - sin(x) into a polynomial expression to avoid "catastrophic cancellation." [2-11]
  8. Does the following sequence converge linearly or quadratically? 2, 2.1, 2.11, 2.111.
    What about this sequence? 2, 2.1, 2.11, 2.1101, 2.11010001. [2-14B]
  9. Use 3 iterations of the Bisection Method to find the root of x2 -3 in the interval [1,2]. [3-5]
  10. Use 3 iterations of the Newton-Raphson Method to find the root of x2 -3 in the interval [1,2]. Use x = 1 as the initial value. [3-8]
  11. Use 3 iterations of the Secant Method to find the root of x2 -3 in the interval [1,2]. Use as the initial points 1 and 1.500. [3-9]
  12. Given y = (x-2)2(x-3), what is the rate of convergence via Newton Raphson to x=2? What is the rate of convergence to x = 3? [3-10]
  13. Find the characteristic polynomial of the following matrix [4-14]:
    ( 1 3   2 )
    ( 1 2 -1 )
    ( 2 -1 1 )
  14. Is the following matrix orthogonal? [4-15]
    ( 1 3   2 )
    ( 1 2 -1 )
    ( 2 -1 1 )
  15. Given x= (1 -3 2)T, find (a) the 1-norm, (b) the 2-norm, and (c) the infinity-norm of x. [4-16R]
  16. Given the following matrix, find (a) the 1-norm, and (b) the infinity-norm. [4-16R]
    ( 1 0 1 )
    ( 2 3 2 )
    ( 0 2 3 )
  17. Given the following matrix A, find cond*(A). [4-16R]
    ( 10 0 0 )
    ( 0 20 0 )
    ( 0 0   2 )
  18. Find the "normal equations" for the following linear system. [4-17]
    3x + 2y = 5
    5x - y = 10
  19. Find the LU decomposition of the following linear system. [4-21], [4-23], [4-24]
    ( 3 2 -1 ) (x)   ( 7 )
    ( 5 3  2 ) (y) = ( 4 )
    (-1 1 -3 ) (z)   (-1 )
  20. Suppose it takes 5 seconds to solve a linear system of 10 variables via Gaussian elimination on a given computer. How long would it take to solve a 100 × 100 system? [4-26A]
  21. Find the eigenvalues of the following matrix. [4-41]
    ( 2 1 )
    ( 1 2 )
  22. Use Gerschgorin's Disk Theorem to determine the interval in which the eigenvalues of the following matrix lie. [4-42]
    ( 2 1 )
    ( 1 2 )
  23. Use the Power Method for two iterations to determine an approximation to the dominant eigenvalue of the following matrix. Assume that x0 = (1,1,1)T. [4-44]
    ( 3 7 9 )
    ( 7 4 3 )
    ( 9 3 8 )
  24. Given the following six data points, find the linear least squares line which best fits the points. [5-5]
    (0, 1), (2, 5.1), (4, 9), (6, 13), (8, 17), (10, 21).
  25. Given the following three data points, find the Lagrange interpolating polynomial through the points. [5-13]
    (1, 1), (4, 64), (5, 125).
  26. Given the following six data points, create the forward difference table to the third difference. [5-18]
    (0, -7), (1, -3), (2, 6), (3, 25), (4, 62), (5, 129).
  27. Given the following five data points, create the divided difference table to the fourth difference. [5-28]
    (-2, -8), (0, 0), (1, 1), (4, 64), (5, 125).
  28. Given the following three data points for (x, f(x)), find an interpolated value for f(4) using the Newton Forward Difference Formula. [5-31]
    (3, 1), (5, 11), (7, 29).
  29. Why might one use a cubic spline through a set of data point in contrast to the Lagrange interpolating polynomial? [5-33B]-[5-35]
  30. Given the following data, set up the linear system to determine the sigma values (i.e., curvature) assuming a "natural spline" at the endpoints. Do not solve the system. [5-40A], [5-41]
    ( 0, -8 ), ( 1, -7 ), ( 2, 0 ), ( 3, 19 ), ( 4, 56 ).
  31. Use the Trapezoidal rule to approximate the integral of x3 from 0 to 2 using 4 panels. [6-4]
  32. How many panels are needed to get accuracy of less than 1/8 via the Trapezoidal rule to integrate x3? [6-5]
  33. Use Simpson's 1/3-rule to approximate the integral of x3 from 0 to 2 using 4 panels. [6-7]
  34. Use Richardson Extrapolation to obtain a "better" value, given that N1(h/2) = 2 and N1(h) = 3. [6-10]
  35. Use Romberg Integration to the third extrapolation to get better integration values for the following discrete function. [6-12]
    ( 0, 1 ), ( 2, -17 ), ( 4, -87 ), ( 6, -41 ), ( 8, 529 ).
  36. Use Gaussian quadrature to evaluate the integral of x2 from -1 to 1 using 2 nodes and then 3 nodes. [6-14]
  37. Change the following second order ODE into a coupled system of first order ODEs. Make sure to include the revised initial values. [6-18]
    A y'' + By' + Cy = g(t) with y(0) = 2, y'(0) = 1.
  38. Set up a linear system to solve this ODE-BVP by discretization techniques (but don't actually solve the system). [6-38A]-[6-40]
    y'' - ( 1 - t/5 ) y = t

This page is maintained by Dennis C. Smolarski, S.J. Email: dsmolarski "at" scu.edu
© Copyright 2016, 2017 Dennis C. Smolarski, SJ, All rights reserved.
Last changed: 25 March 2017.