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

Choose one answer.

	A. 0.1
	B. An infinetly repeating binary number
	C. 0.10011
	D. 0.0100110011001
	E. 1/2

Question 2

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

Choose one answer.

	A. 01111111
	B. 111
	C. 010101
	D. 001111
	E. 01101111

Question 3

Which of the following is an adequate representation of 0.5 (decimal) in binary notation?

Choose one answer.

	A. 0.01010101
	B. 0.1
	C. 0.2
	D. 0.0001
	E. 0.11111...

Question 4

For the function, f(t) = B exp( - (t/T)^1.7), calculate an approximation for the first derivative f'(t) using the centered difference method about t = 0.5, for
T = 1.0, and B = 95. You should use a step size h of 0.01. What is the true relative error of the approximate derivative calculation?

Choose one answer.

	A. 0.008%
	B. 0.01 %
	C. 0.00008%
	D. 0.1%
	E. 0.0008%

Question 5

For the function, f(t) = B exp( - (t/T)^1.7), calculate an approximation for the first derivative f'(t) using the centered difference method about t = 0.5, for T = 1.0, and B = 95. You should use a step size h of 0.01. Which of the following best matches your calculation?

Choose one answer.

	A. 73.0770
	B. -73.0770
	C. -73.0712
	D. -7.30712
	E. -69.4662

Question 6

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?

Choose one answer.

	A. 4%
	B. 1%
	C. 40%
	D. 0.04%
	E. 0.004%

Question 7

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?

Choose one answer.

	A. 1.33
	B. 4.44
	C. -0.13333
	D. -0.26666
	E. 5.55

Question 8

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?

Choose one answer.

	A. 0.701
	B. 0.843
	C. 0.780
	D. 0.521
	E. 0.743

Question 9

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)?

Choose one answer.

	A. 0.1, 0.832
	B. 0.575, 1
	C. 0.1, 0.575
	D. 0.1, 0.3375
	E. 3.86, -2.18

Question 10

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)?

Choose one answer.

	A. 0.144
	B. 0.235
	C. 0.802
	D. -0.107
	E. 0.289

Question 11

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?

Choose one answer.

	A. 1.87
	B. 1.25
	C. 2.85
	D. 1.25
	E. 2.43

Question 12

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?

Choose one answer.

	A. 1.43
	B. 1.25
	C. 1.87
	D. 2.35
	E. 2.85

Question 13

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]?

Choose one answer.

	A. [0.432; 0.241; 0.527]
	B. [0.482; 0.241; 0.597]
	C. [0.582; 0.241; 0.597]
	D. [0.482; 0.241; 0.517]
	E. [0.430; 0.211; 0.527]

Question 14

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?

Choose one answer.

	A. [14.1, 2.5, 2.7; 0, 47.639716, -14.385106; 0, 0, -75.544487]
	B. [14.1, 2.5, 2.7; 0, 47.539716, -14.375106; 0, 0, -75.544487]
	C. [14.1, 2.5, 2.7; 0, -47.639716, -14.385106; 0, 0, -75.544487]
	D. [14.1, 2.5, 2.7; 0, 47.639716, -14.385106; 1, 0, -75.544487]
	E. [14.1, 2.5, 2.7; 0, 47.639716, -14.385106; 0, 0, 75.544487]

Question 15

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?

Choose one answer.

	A. [0.825; 0.0399; 0.1962]
	B. [0.835; 0.0391; 0.0922]
	C. [0.342; 0.198; 0.0578]
	D. [0.825; 0.0399; 0.0962]
	E. [0.342; 0.118; 0.0578]

Question 16

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?

Choose one answer.

	A. [1,0,0; 0.654,1,0; 1.163,0.695,1]
	B. [1,0,0; 0.624,1,0; 1.163,0.625,1]
	C. [1,0,0; 0.624,1,0; 1.143,0.625,0]
	D. [0,0,0; 0.614,0,0; 0.163,0.625,0]
	E. [1,0,0; 0.524,1,0; 1.263,0.625,1]

Question 17

The data given are available for temperature as a function of time.

t (minutes)	T (degrees C)
10	41.2
40	46.1
60	49.1
90	39.5

Interpolate to approximate the temperature at t = 70 minutes using a second order polynomial (quadratic function).

Which of the following temperatures most closely matches your result?

Choose one answer.

	A. 48.8 degrees C
	B. 45.9 degrees C
	C. 48.7 degrees C
	D. 45 degrees C
	E. 47.8 degrees C

Question 18

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?

Choose one answer.

	A. 5395 m
	B. 5396 m
	C. 5349 m
	D. 5355 m
	E. 5400 m

Question 19

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?

Choose one answer.

	A. 2.28, -1.09
	B. 2.23, -1.1
	C. 2.18, -1.11
	D. 2.25, 1.1
	E. 2, 0

Question 20

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?

Choose one answer.

	A. -3.1, 0.5
	B. 2.12, 0.5
	C. 1.09, 2.2
	D. 3.12, 0.484
	E. -1.1, 2.2

Question 21

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?

Choose one answer.

	A. 4.962948
	B. 4.696419
	C. 4.7053758
	D. 3.1369172
	E. 3.1368487

Question 22

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?

Choose one answer.

	A. 689
	B. 559
	C. 573
	D. 502
	E. 599

Question 23

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.

Choose one answer.

	A. 107 J
	B. 212 J
	C. 221 J
	D. 109 J
	E. 215 J

Question 24

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

Choose one answer.

	A. 1629 gal
	B. 1815 gal
	C. 1640 gal
	D. 1840 gal
	E. 1722 gal

Question 25

Step Size h	Integral
0.2	489
0.1	415
0.05	391
0.01	375

Which of the following is the best estimate for the absolute value of the percentage error in the integral estimate from the trapezoidal rule with h = 0.075?

Choose one answer.

	A. 1%
	B. 10%
	C. 15%
	D. 20%
	E. 5%

Question 26

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.

Choose one answer.

	A. 312
	B. 333
	C. 320
	D. 342
	E. 422

Question 27

It is predicted that the cooling of a steel part under the influence of forced convection will follow the differential equation dT/dt = -0.03 (T-22)^1.8, where t is in minutes and T is in degrees C.

At t = 0, T = 500 degrees C.

Use the Runge-Kutta method (2nd order with direct analytical evaluation of the second derivative) with a time step (h) of 0.22 minutes to predict the temperature of the part in degrees C at t = 5 minutes.

Choose one answer.

	A. 32
	B. 31
	C. 22
	D. 43
	E. 37

Question 28

Consider a projectile fired directly upward in a vacuum. At t = 0 (seconds), the projectile has x-coordinate (in meters) of 0.0 (it is on the ground). It is fired directly upward with an unknown initial velocity v0. The gravitational acceleration is a strange function of altitude and can be described by g(x) = -9.8/(1+0.001x). However, at exactly t = 20, the projectile returns to x = 0.

Use the shooting method to determine the initial upward velocity of the projectile. Which of the following numbers best represents that initial velocity (m/s)?

Choose one answer.

	A. 46
	B. 98
	C. 81
	D. 92
	E. 23

Question 29

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

Choose one answer.

	A. Conjunct
	B. Adjoint
	C. Complex conjugate
	D. Inverse
	E. Diagonalized form

Question 30

Which of the following is one of the important reasons for using finite element methods for multidimensional problems?

Choose one answer.

	A. Finite elements handle curved boundaries more efficiently than finite difference methods.
	B. Finite elements do not work at all for 1-d problems.
	C. Finite element methods are always faster than finite difference methods for higher dimensional problems.
	D. Finite element methods require less user intervention for multidimensional methods.
	E. Finite element methods are inherently more accurate for multidimensional problems.

Question 31

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

Choose one answer.

	A. 1
	B. pi
	C. e^pi
	D. pi/2
	E. Zero

Question 32

Which of the following is true of the Fourier Transform?

Choose one answer.

	A. It describes heat conduction.
	B. It interconverts between time and frequency domains.
	C. It may only be applied to real signals.
	D. It may only be applied to complex signals.
	E. None of the above

Question 33

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

Choose one answer.

	A. Strang
	B. Fourier
	C. Russel
	D. Galerkin
	E. Helmholtz