1. Which of the following is true of the Fourier Transform?

2. Consider the following linear system.

14.1x + 2.5y + 2.78z = 12

8.8x + 49.2y - 12.7z = 8

16.4x + 32.7y - 81.4z = 7

Rewrite the system in matrix form as' A v = 'w, where 'v is the unknown vector (x, y, z) and w is (12, 8, 7).

The matrix A is then (in Scilab notation) [14.1, 2.5, 2.7; 8.8, 49.2, -12.7; 16.4, 32.7, -81.4].

Use forward substitution to convert this matrix to upper triangular form. Which of the following matrices best matches your result?

3. Find an approximation for the integral of the water flow rate over the time interval 0-300 minutes by using the trapezoidal rule on the following data.

Q(t) (gal/min)	t (min)
5.3	0
5.8	90
6.1	100
6.5	159
4.2	240
4.7	280
5.0	300

4. The binary representation of the decimal number 0.1 (or 1/10) is best described by which of the following?

5. For a continuous operator, what is the analog for the transpose of a matrix?

6. Calculate the second derivative of the function f(x) = 2sin(x^1.3)exp(-2x^1.7) about x = 1.1 with a step size h = 0.2. Which of the following most nearly equals your answer?

7. You have performed numerical integration using the trapezoidal rule for a tabulated function f(x) over the interval x = [0, 1]. The results of the integration for several step sizes are listed in the table below.

Step Size h	Integral
0.2	689
0.1	611
0.05	581
0.01	567

Which of the following is the best estimate of the true value of the integral?

8. Use the bisection method from a starting interval of [0.1,2] to find a zero of the function f(x) = 2x - 4 log (4x) (natural logarithm) in that interval. What is the search interval after applying the bisection method twice (i.e. two iterations)?

9. Consider the following linear system.

14.1x + 2.5y + 2.78z = 12

8.8x + 49.2y - 12.7z = 8

16.4x + 32.7y - 81.4z = 7

Rewrite the system in matrix form as' A v = 'w, where 'v is the unknown vector (x, y, z) and w is (12, 8, 7).

The matrix A is then (in Scilab notation) [14.1, 2.5, 2.7; 8.8, 49.2, -12.7; 16.4, 32.7, -81.4].

Implement the Gauss-Siedel iterative algorithm to find the vector 'v.
Which of the following vectors most closely resembles your estimate after two iterations of the Gauss-Siedel method from an initial guess of [1; 1; 1]?

10. Which of the following is the correct binary representation of the decimal integer 111?

11. The force exerted by a mechanical spring system on a weight sliding on a table is found to be well represented by the function F = 412 exp(-2.4 x^1.7)[Newtons], where x is measured in meters over the domain [0, 5].

Use Simpson's rule with a step size (h) of 0.5 m to find an approximate value for the work performed by the spring system on the weight for the displacement from [0 to 5] meters.

12. What is the value of all Fourier coefficients (except for the first) for the real series [1, 1, 1, 1, 1, 1, 1, 1]?

13. For the tabulated data below, calculate an approximation for the second derivative f''(t) at t = 0.53 seconds, using a centered difference approximation with a step size h of 0.03 seconds.

time (seconds)	f(t)
0.45	0.71
0.50	0.693
0.53	0.685
0.56	0.681
0.59	0.649

Which of the following most closely matches the result of your calculation?

14. Of the following, to whom are the fundamental theoretical origins of finite element methods most often attributed?

15. The data in the table below indicate one spatial coordinate of an incoming missile as a function of time. We wish to estimate the value of the spatial coordinate at intermediate times by quadratic interpolation.

Time (ms)	Position (m)
0	5085
200	5245
400	5301
600	5490
800	5605

You need to estimate the position at t = 500 ms.

First, fit a quadratic to the positions at 0, 200, and 400 ms. Then, fit a second quadratic to the data at 400 and 600 ms with the additional constraint that the slopes of the first and second quadratics are the same at t = 400 ms.

The first quadratic, that fits the first three positions, is x(t) = 5085 + 1.06t - 0.0013 t² (where x is in meters and t is in milliseconds).

Find the second quadratic and estimate the position at t = 500 ms.

Which of the following positions best matches the position predicted by your second quadratic for t = 500 ms?

16. Use the bisection method from a starting interval of [0.1,2] to find a zero of the function f(x) = 2x - 4 log (4x) (natural logarithm) in that interval. What is your estimate of the root after applying the bisection method 50 times (i.e. 50 iterations)?

17. Calculate the slope and intercept of the least-squares linear fit to the following data.

x-position (cm)	y-height (cm)
1.3	2.3
3.6	7.1
5.8	11.0
17.4	38.0

Which of the following most closely matches the results of your calculations?

18. Consider the following linear system.

14.1x + 2.5y + 2.78z = 12

8.8x + 49.2y - 12.7z = 8

16.4x + 32.7y - 81.4z = 7

Rewrite the system in matrix form as' A v = 'w, where 'v is the unknown vector (x, y, z) and w is (12, 8, 7).

The matrix A is then (in Scilab notation) [14.1, 2.5, 2.7; 8.8, 49.2, -12.7; 16.4, 32.7, -81.4].

Perform LU decomposition to find the lower triangular matrix (with ones on the diagonal) which corresponds to the upper triangular matrix found by forward elimination.

Which of the following matrices corresponds most closely to your result?

19. Calculate a₀ and a₁ for the least-squares fit of y = a₀ + a₁ x^1.5 to the following data.

x-position (cm)	y-height (cm)
1.3	2.3
3.6	7.1
5.8	11.0
17.4	38.0

Following the guide in subunit 6.3 of the course, minimization of square deviation for the model y = a₀ + a₁ x^1.5 may be written:

a₀ + Sxm a₁ = Sy

and

Sxm a₀ + Sx2m a₁ = Sxmy

where n is the number of observations, m is the exponent 1.5 in the model, Sxm = sum over n of x^1.5, Sy = sum over n of y, Sx2m = sum over n of x³, and Sxmy = sum over n of of x^1.5y.

Tabulate the sums and solve the above equations for a₀ and a₁.

Which of the following most closely matches the results of your calculations for a₀ and a₁, respectively?

20. For the following nonlinear equation, implement Newton's method for iterative solution using analytical derivatives for the following equation.

0 = 4t²+ 2t³cos(8t) (Hint: use radians.)

For a starting point of t = 1.0 ( at number of iterations = 0), what is the value of t after 2 iterations of the Newton method?

21. For the function, f(t) = B exp( — (t/T)), calculate an approximation for the first derivative f'(t) using the centered difference method about t = 2.0, for T = 4.0, and B = 50. You should use a step size h of 0.4. What is the percentage deviation of the numerically calculated derivative from the true value?

22. Consider the following linear system.

14.1x + 2.5y + 2.78z = 12

8.8x + 49.2y - 12.7z = 8

16.4x + 32.7y - 81.4z = 7

Rewrite the system in matrix form as' A v = 'w, where 'v is the unknown vector (x, y, z) and w is (12, 8, 7).

The matrix A is then (in Scilab notation) [14.1, 2.5, 2.7; 8.8, 49.2, -12.7; 16.4, 32.7, -81.4].Which of the following vectors most closely resembles the solution for 'v?

23. Use 3-point Gaussian quadrature to integrate the function f(x) = 2x^1.25/(1+ x^0.5sin(x^0.75)) from x = 0 to x = 3.

Which of the following most closely matches your result?

24. It is predicted that the cooling of a steel part under the influence of forced convection will follow the differential equation dT/dt = -0.003 (T-22)^1.8, where t is in minutes and T is in degrees C.

At t = 0, T = 482 degrees C.

Use Euler's method with a time step (h) of 0.2 minute to predict the temperature of the part in degrees C at t = 1 minute.

25. For the following nonlinear equation, implement Newton's method for iterative solution using numerical derivatives (forward difference) with a step size in t of 0.00001.

0 = 4t²+ 2t³cos(8t) (Hint: use radians.)

For a starting point of t = 1.0 ( at number of iterations = 0), what is the value of t after 50 iterations of the Newton method?