1
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?
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.
Which of the following most closely matches the result of your calculation?
| 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?
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?
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]?
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?
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?
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?
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.
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?
| 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.
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?
| 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.
Which of the following most closely matches the results of your calculations?
| 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 x1.5 to the following data.
Following the guide in subunit 6.3 of the course, minimization of square deviation for the model y = a0 + a1 x1.5 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 x1.5, Sy = sum over n of y, Sx2m = sum over n of x3, and Sxmy = sum over n of of x1.5y.
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?
| 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 x1.5 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 x1.5, Sy = sum over n of y, Sx2m = sum over n of x3, and Sxmy = sum over n of of x1.5y.
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) = 2x1.25/(1+ x0.5sin(x0.75)) from x = 0 to x = 3.
Which of the following most closely matches your result?
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.
Which of the following is the best estimate of the true value of the integral?
| 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 x1.7)[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.
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
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.
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?
| 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.
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.
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)?
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
|