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2009 Spring EXAM 2 MEEN 357 March 26, 2009 Dr. Bowen Name: ____________________________________________ "Aggies do not lie, cheat, or steal, nor do they tolerate those who do." Aggie Code of Honor By my signature below I pledge that my conduct on this exam is consistent in every way with the Aggie Code of Honor: Signature: ____________________________________________
1. This exam consists of six (6) problems that are weighted as shown below. The last question is take home and requires the use of MATLAB. You will receive this question and appropriate instructions by Email not later than midnight March 26, 2009. a) If you do not receive the take home question in your Email, notify rbowen@tamu.edu. b) The take home portion of the exam will be due by Email with a time stamp not later than 8am, Saturday, March 28, 2009. 2. This is a open textbook (Applied Numerical Methods with MATLAB, Second Edition, by Steven C. Chapra) open lecture note exam. a) The lecture notes are those posted on Vista Blackboard. 1. It is understood that you add incidental comments to your copies of these notes. b) You may also use graded homework solutions and the solutions posted on Vista Blackboard c) You may not use individual notes that you have created other than what you have added to the pages of the posted and printed class notes. 3. You may use handheld calculators unless the problem dictates otherwise. a) You may not use other computing devices for the in-class portion of the exam. 4. Be sure to read the question carefully. 5. Be sure to show all work and calculations, and organize your solution procedure as clearly and systematically as possible. a) Example: Do not simply enter a string of numbers in your calculators and write down the answer. Write out the numbers in a mathematical formula and then show the answers obtained from doing the mathematical operations on your calculators. 6. For the in-class portion of the exam, work problems in the space provided on the exam sheets. Clearly indicate continuation of the problem if you need extra pages. GRADE Problem 1: ____12.96 / 15% (In Class) Problem 2: ____13.35 / 15% (In Class) Problem 3: ____12.73 / 15% (In Class) Problem 4: ____13.50 / 15% (In Class) Problem 5: _____4.54 / 15% (In Class) Problem 6: ____18.79 / 25% (Take Home) TOTAL: ______75.88
Section: 502
MEEN 357 Exam 2-Solution 2009
RMB
1
Spring 2009 1. (15%) You are given the following system of equations. 15 x1 3 x2 x3 = 3800 3 x1 + 18 x2 6 x3 = 1200 4 x1 x2 + 12 x3 = 2350 a. (5%) If you were to attempt to solve this system by the Gauss-Seidel iteration method, are you assured in advance that it will converge? Why? Explain your
(10%) Give the MATLAB script sufficient to generate the solution (1.1) by the Gauss-Seidel method such that the final percentage relative error is less than 5%. You must clearly print your script. Otherwise, your results will be considered incorrect. You may assume you have full access to all of the MATLAB tools we have discussed. Solution: Part a): The convergence criteria is, from Lecture 10, aii > aij
j =1 j i n
(1.1)
(1.2)
for a system of n equations. Because 15 > 3 + 1 18 > 3 + 6 12 > 1 + 4 the convergence criteria is obeyed. Part b): The script clc, clear all A=[15,-3,-1;-3,18,-6;-4,-1,12] b=[3800;1200;2350] x0=[0;0;0] x=GaussSeidel(A,b,x0,5) will generate the solution. (1.3)
2. (15%) It is desired to determine the solution of the simultaneous nonlinear equations
MEEN 357 Exam 2-Solution 2009
RMB
2
Spring 2009
x + y 5= 0
2 2
y + 1 x2 = 0
(2.1)
Give the MATLAB script sufficient to generate this solution if you are given an initial guess of ( x, y ) = (1.4,1.5 ) . You must clearly print your script. Otherwise, your results will be considered incorrect. You may assume you have full access to all of the MATLAB tools we have discussed. Solution: This problem is best worked by the Newton Raphson method discussed in class. The script clc, clear all f=@(x)([x(2)^2+x(1)^2-5,... x(2)+1-x(1)^2]) x=fsolve(f,[1.4,1.5]) is the easiest way to find the roots. You can also use the function file newtmult.m. If you look at the solution to the homework for Lecture 10, you will see an almost identical example where this function file is used to find the solution. 3. (15%)You are given the data set x y 1 1 3 2 4 5 45 50 70 82 40 27
a. (10%)Use your calculators and perform a least squared linear regression for this data. Show sufficient work to allow an understanding of your solution method. (See instruction 5a. above) b. (5%)Give the MATLAB script sufficient to solve this problem. You must clearly print your script. Otherwise, your results will be considered incorrect. You may assume you have full access to all of the MATLAB tools we have discussed. Solution: Part a): Many of the solutions simply used the plug (11.41) in your notes. It is equivalent to form the matrices A and y in equation (11.27) of your notes. Given the above table, these equations are
MEEN 357 Exam 2-Solution 2009
RMB
3
Spring 2009 1 2 5 y= 50 82 27 The unknown coefficients
a c = 0 a1
and
1 1 1 3 1 4 A= 1 45 1 70 1 40
(3.1)
(3.2)
are, from (11.33) the solution of AT Ac = AT y Therefore, given (3.1), it follows that (3.3) is 1 1 1 1 3 2 1 1 1 1 1 1 1 4 1 1 1 1 1 1 5 1 2 4 45 70 40 1 45 c = 1 2 4 45 70 40 50 1 70 82 1 40 27 If you perform these multiplications, you get 163 6 167 c= 163 8551 9097 Therefore,
2647 a0 8551 163 167 1195 1 c= = 2 = 6 9097 1074 a1 6 (8551) (163) 163 971 2.2151 = 1.1061
(3.3)
(3.4)
(3.5)
(3.6)
and the resulting equation is
y = 2.2151 + 1.1061x
Part b): MEEN 357 Exam 2-Solution 2009 RMB
(3.7)
4
Spring 2009 clc, clear all x=[1,3,4,45,70,40]'; y=[1,2,5,50,82,27]'; A=[x.^0,x.^1] c=A\y %Or polyfit(x,y,1)
4. (15%) You are given the data table x y 2 -2 3 4 4 24 5 64
Use the Lagrange Interpolation procedure and find a cubic equation that interpolates this data.
Solution:
From the Lagrange interpolation formula in your notes, a cubic (third order polynomial) is given by
y ( x) =
( x x1 )( x x2 )( x x3 ) ( x x0 ) ( x x2 ) ( x x3 ) f ( x0 ) + f (x ) ( x0 x1 )( x0 x2 )( x0 x3 ) ( x1 x0 )( x1 x2 )( x1 x3 ) 1 ( x x0 )( x x1 )( x x2 ) ( x x0 )( x x1 )( x x3 ) + f ( x2 ) + f (x ) ( x3 x0 )( x3 x1 )( x3 x2 ) 3 ( x2 x0 )( x2 x1 )( x2 x3 )
(4.1)
You can identify the various entries by restating the above table as
x y x0=2 y0=-2 x1=3 y1=4 x2=4 y2=24 x3=5 y3=64
Therefore,
y ( x) =
( x 3)( x 4 ) ( x 5 ) 2 + ( x 2 ) ( x 4 ) x ( 5) 4 () () ( 2 3)( 2 4 )( 2 5) ( 3 2 )( 3 4 )( 3 5) ( x 2 )( x 3)( x 5) 24 + ( x 2 )( x 3)( x 4 ) 64 + () () ( 5 2 )( 5 3)( 5 4 ) ( 4 2 )( 4 3)( 4 5)
(4.2)
This expression simplifies to
MEEN 357 Exam 2-Solution 2009
RMB
5
y ( x) = y ( x) =
( x 3)( x 4 )( x 5) ( x 2 ) ( x 4 ) ( x 5) ( 2 ) + (4) ( 2 3)( 2 4 )( 2 5) ( 3 2 )( 3 4 )( 3 5) ( x 2 )( x 3)( x 5) ( x 2 )( x 3)( x 4 ) + ( 24 ) + ( 64 ) ( 4 2 )( 4 3)( 4 5) ( 5 2 )( 5 3)( 5 4 )
Spring 2009
1 ( x 3)( x 4 )( x 5) + 2 ( x 2 )( x 4 )( x 5) 3 32 12 ( x 2 )( x 3)( x 5) + ( x 2 )( x 3)( x 4 ) (4.3) 3 32 1 = ( x 4 )( x 5) ( x 3) + 2 ( x 2 ) + ( x 2 )( x 3) 12 ( x 5) + ( x 4 ) 3 3 1 1 = ( x 4 )( x 5)( 7 x 15) ( x 2 )( x 3)( 4 x 52 ) 3 3 1 1 = ( 7 x 3 78 x 2 + 275 x 300 ) ( 4 x 3 72 x 2 + 284 x 312 ) 3 3 3 2 = x 2 x 3x + 4 = Note: Full credit was given if the correct version of (4.2) was given. The algebra leading to the simplified form (4.3) is useful but not essential for the purposes of the exam. 5. (15%) You are given the values of a certain f ( x ) in the form of the following data table:
x f ( x)
0 20
1 12
2 7
3 4
4 3
5 1
The problem is to find an approximation to the integral
I = f ( x ) dx
5 0
(5.1)
a. (5%) Use the trapezoidal method to find an approximation to the integral (5.1). Solution: For equally spaced data, the trapezoidal rule is
I = f ( x )dx =
b a
h
f ( x1 ) + f ( x0 ) f ( x2 ) + f ( x1 ) f ( xN ) + f ( xN 1 ) +h + + h 2 2 2 N 1 h = f ( x0 ) + 2 f ( xi ) + f ( xN ) 2 i =1
(5.2)
where h = 1 and N = 5 . The N + 1 = 6 values of the function are given in the above table. Therefore,
MEEN 357 Exam 2-Solution 2009
RMB
6
Spring 2009
I=
1 ( 20 + 2 (12 + 7 + 4 + 3) + 1) = 1 ( 20 + 52 + 1) = 32.5 2 2
(5.3)
b. (10%) Use the Simpson Rules to find an approximation to the integral (5.1). Solution: Because N = 5 , neither of the two Simpson Rules can be applied alone. The simplest solution is to partition the region into one two section portion followed by one three section portion. It is perhaps instructive to display this choice as follows:
[0,5] = [0,2] [ 2,5]
1 Rule 3 3 Rule 8
(5.4)
The Simpson 1 rule works for the first interval because its number of elements is 2 3 and the Simpson 2 rule works for the second interval because its number of elements is 3 3. The area is then the sum
I=
h 3h ( f ( x0 ) + 4 f ( x1 ) + f ( x2 ) ) + 8 ( f ( x2 ) + 3 f ( x3 ) + 3 f ( x4 ) + f ( x5 ) ) 3
(15.38)of the notes (16.35)of the notes
1 3 (5.5) ( 20 + 4 (12 ) + 7 ) + 8 ( 7 + 3( 4 ) + 3 ( 3) + 1) 3 1 3 287 = ( 75) + ( 29 ) = = 35.875 3 8 8 A second, equally valid, approach is to place the three element portion first as follows: =
[0,5] = [0,3] [3,5]
3 Rule 8 1 Rule 3
(5.6)
The area is then the sum I= h 3h ( f ( x0 ) + 3 f ( x1 ) + 3 f ( x2 ) + f ( x3 ) ) + 3 ( f ( x3 ) + 4 f ( x4 ) + f ( x5 ) ) 8
(16.35)of the notes (15.38)of the notes
3 1 ( 20 + 3(12 ) + 3 ( 7 ) + 4 ) + 3 ( 4 + 4 ( 3) + 1) 8 3 1 865 = (81) + (17 ) = = 36.0417 8 3 24 = Note: Very few of you received full credit on this problem. There was rather poor understanding of the restrictions on when one can apply the Simpson 1 and the 3 3 methods. If you will examine page 16 of Lecture 16, you will find a Simpson 8 problem that is similar to the one above.
(5.7)
MEEN 357 Exam 2-Solution 2009
RMB
7
Spring 2009
6. (25%) Take Home Question: You are given the data table
x w
50 99
80 177
130 202
200 248
250 229
350 219
450 173
550 142
700 72
a. (10%) Fit this data to the equation
1 w = 0 xe 1 1
x
(6.1)
Solution: Given (6.1)
x ln w = ln 0 + ln x + 1 1 1
(6.2)
or
x ln w ln x = ln 0 + 1 1 1
(6.3)
Let y = ln w ln x a0 = ln 0 + 1 1 1 a1 = (6.4)
1
and the problem is transformed into a linear regression y = a0 + a1 x The MATLAB Script
clc clear all %From the data table in the problem statement x=[50,80,130,200,250,350,450,550,700]'; w=[99,177,202,248,229,219,173,142,72]'; y=log(w)-log(x); A=[x.^0,x.^1]; %The direct solution of A'*A*c=A'*y is c=inv(A'*A)*A'*y;
(6.5)
MEEN 357 Exam 2-Solution 2009
RMB
8
Spring 2009 %Or capitalizing on the special properties of the left division c=A\y; %Calculate the alphas alpha1=-1/c(2) alpha0=alpha1*exp(c(1)-1) Yields
alpha1 = 217.9415
alpha0 = 234.4190
Therefore,
1 1 234.419 1 217.9414 w = 0 xe 1 = xe = 1.0756 xe 217.9414 217.9414 1
x
x
x
(6.6)
Note: A few of you correctly used the approach given in Section 14.5 of the textbook. We did not discuss this section in class, but it contains another good way to solve this problem. You should be able to study the Example 14.5 and see how it is easily modified for the problem given on the exam. b. (10%) Plot the given data and the calculated function (6.1) on the same axis. Solution: If the MATLAB script
%Part b. %Plot of above results by the script plot(x,w,'o','MarkerFaceColor','k','MarkerSize',10) xlabel('x'),ylabel('w') grid on hold xvalues=[50:50:700]; wvalues=alpha0*(xvalues/alpha1).*(exp(1-xvalues/alpha1)) plot(xvalues,wvalues,'r','linewidth',2) title('Problem 4') legend('Data Points','Curve Fit')
is appended to the above, the result is the plot
MEEN 357 Exam 2-Solution 2009
RMB
9
Spring 2009
c. Use the trapazoidal method to calculate the area under each of the curves displayed on the plot for part b.
Solution: First we shall use the given data above to calculate the area under the curve one would get if the data represented points on some curve. There are eight unequal intervals represented in the above table. The area under this curve, by the trapezoidal rule is given by the script
% Part c. %Area by use of information in the data table I1=trapuneq(x,w)
appended to the script from parts a) and b). Recall that trapuneq.m is one of the function files in the book and it was discussed in class. The output from this script is
I1 = 115090
We can calculate the area under the curve (6.6) with the script
%Area by use of calculated function wvalues=alpha0*(x/alpha1).*(exp(1-x/alpha1)) I2=trapuneq(x,wvalues)
appended to the script above. The output from this calculation is
I2 =
MEEN 357 Exam 2-Solution 2009
RMB
10
Spring 2009
1.1173e+005
Many of you calculated the first area by simply adding up the elements. That works fine. The second area can be calculated with equal elements because you are free to evaluate w ( x ) at such points on the interval. Many of you made this choice. The number you will obtain will be slightly different than the one above.
MEEN 357 Exam 2-Solution 2009
RMB
11

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UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei Zhang Problem Set 2 Answer KeyGains from Trade and the Production Possibilities Frontier 1. Wheat Australia Wheata)600New Zealand200600Cotton200Cottonb) New Zealand has a comparative advantage

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei Zhang Problem Set 3 Due by September 26, 2005 (in class) Consumer Choice 1. The Budget Constraint. Suppose Lewis lives in a world with only two goods, beer and cheese. Lewis income each month is $500. Be

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei Zhang Problem Set 3: Answer Key 1. Recall that Lewis income each month is $500. Beer costs $2 a bottle, and cheese costs $5 per pound.a) If Lewis spent all of his income each month on beer, he could con

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei Zhang NOT TO BE TURNED IN Problem Set 4 Finish before the October 3rd Midterm. This problem set has been provided to help you prepare for Midterm 1. Answers are already posted. This problem set will be n

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 4 Answer Key1. Recall that annual demand for widgets is given by the equation QD=40-4P while the supply curve is given by QS=2P-2. a) Quantity demanded will be 0 when Price = $10. To s

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 5 Due By October 10, 2005 (in class) Elasticity of Demand and Marginal Revenue 1) a. Consider the demand curve for widgets, QD=100 5P. Find the price and quantity at which the price ela

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 5 - Answer Key1)Recall the demand curve for widgets, QD=100 5P.a. There are two ways to find the price and quantity at which the price elasticity of demand is equal to -1 (unit elast

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 6 Due By October 24, 2005 (in class) The Impact of a Price Ceiling 1) Demand for leaf blowers in Madison is given by the equation QD=90-P, while supply is given by QS=2P. a. b. c. What

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 6 ANSWER KEY The Impact of a Price Ceiling 1) Recall that the demand for leaf blowers in Madison is given by the equation Q =90-P, while supply is given by Q =2P.D Sa. In equilibrium

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 7 Due By October 31, 2005 (in class) Public Goods and Externalities: Multiple Choice. 1) An example of a good that is both rival and excludable is a) b) c) d) 2) a) b) c) d) the defense

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 7 ANSWER KEY Public Goods and Externalities: 1) An example of a good that is both rival and excludable is (b), a pair of pants. Excludable means that access can be denied (i.e. a theme

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 8 Due By November 14, 2005 (in class) Perfect Competition (Chapter 14) 1) A local microbrewery has total costs of production given by the equation TC=50+10Q+5Q2. This implies that the f

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 8 ANSWER KEY Perfect Competition (Chapter 14) 1) A local microbrewery has total costs of production given by the equation TC=50+10Q+5Q2. This implies that the firm's marginal cost is gi

Wisconsin - ECON - 101

UNIVERSITY OF WISCONSIN Economics 101 Fall 2005 Wei ZhangProblem Set 9 Due By Nov 21, 2005 Perfect Competition (Chapter 14): Short Run and Long Run 1) It may help to use Chapter 14 from the text to aid in answering the following questions.a. Show a pair