Midterm Examination         CSCI 3321        Summer, 2015                      ___________

Please answer each question by entering the most nearly correct answer (a, b, c, d) in the blank on the left.

____1. When approximating ex by a truncated Taylor series (expanded about x=0), how many terms would be required to keep the absolute value of the error below 10-3 over the

interval [-1,1] ?

a. 1                b. 2                c. 3                d. 4

____2. The polynomial that interpolates the data shown in the table below can be written as:

a.     1 – ½ x2                                              b.    1 + ½ x – 7/2 x2 + 1/6   x3

c.     1 – ½ x2 + 1/6   x3                                           d.     1 – ½ x2 – 1/6   x3

 Xx x Y y 0 21   1 1 2/3 2 3    1/3 3 3 3     1

____3. Use Newton’s method to solve the equation x – sin(x-1) -1 = 0, with the initial guess

of x0 = 0.5. After 2 iterations, the approximation, x2 , is about

a. 0.68068                              c. 0.77912

b. 0.66068                              d. None of the previous answers

____4. The Newton form of the interpolating polynomial is:

a. usually faster to compute than the Lagrange form           b. less useful for theorem proving

c. harder to use when the f values are subject to change       d. all of the previous answers

____5. The error in the approximation f’(x) @ (f(x+h) – f(x-h))/ (2h) is:

a. O(h)                                              b. O(h2)

c. O(h3)                                             d. O(h4)

{Hint: You could use the Taylor series for f(x+h) and for f(x-h) about the point x }

6. Consider the following system of linear algebraic equations:

x + 2y + 3z = 14

2x + y + 2z = 10

2x + 4y + 2z = 16

Is it possible to use naïve Gauss elimination to solve this system? _____

If it is not possible, state why. If it is possible, what are x, y, and z?

____7. Circle each of the following numbers that is computer-representable:

0.75               0.1              1/3            10               -0.125          π

____8. Determine a 2-point quadrature formula (of highest degree possible) with

x1= – 1/4 and x2 = 1/4.   What are the weights? w1 = ____   and w2 = ____.

What is the degree of your formula?

a. one                                     b. two

c. three                                  d. four

______________________________________________________________________

___10. Rounding errors are::

a.partly due to problems representing numbers accurately in the computer memory

b.partly caused by a loss of significance when trying to calculate the result of an expression

c.occasionally able (perhaps surprisingly) actually to improve the accuracy of an answer

d.all of the above

____11.When generating an interpolating polynomial of degree 8, if you have control over the spacing of the selected points, how should these points be spaced?

a.                 c.

b.              d.

____12. When using the bisection method to find the zero of the function shown below when the initial interval (a,b) is (-1,1), the final answer is the root near:

a. -0.97685        b. -0.26595         c. 0.08855      d, 0.44221    e. 0.79477

____13.   The secant method has advantages over Newton’s method when:

a.             f’(x) is not known (and not easily determined)

b.            f’(x) is very expensive to evaluate (when compared to f(x) itself)

c.            two initial guesses for the root are known

d.            all of the above

____14.   The Gauss elimination with scaled partial pivoting::

a. can always find a solution to a linear system of equations

b. is always faster than naïve Gauss elimination

c. is less likely to find an accurate solution than naïve Gauss elimination

d. none of the above

____15. When trying to solve numerically the equation, x – 0.9 sin(x) – 0.2 = 0, using

x -1 = 0.5 and x0 = 0.75 as initial guesses, the Secant Method gives for the approximation x2:

a. 1.04618725                                          c. 0.90160425

b. 1.70168725                                          d. 0.80160425

Notes

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