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Unformatted text preview: — 1'
g ‘5 Exam 2
ME 218Engineering Computational Methods
The University of Texas at Austin
Spring 2008 Directions: This is a closed book/closed note exam. You may only use a reference sheet and a
nongraphing calculator. Show all work and write neatly. Budget your time. Good luck! Problem 1 (30 points): Consider the following set of nonlinear equations we wish to solve. x2 =y2 —7
x2 =—cos(y)+4 ,a’) Find the J acobian of this equation set. (6 pts) ’5) Identify at least two unique points which would not be valid initial starting points for Newton—
Raphson. (5 pts)
Consider starting values of [xo, yo]=[1 2 ]T. After plugging this value into the Jacobian, you call
the MATLAB functions cond(J) and det(J) where J is the 2x2 J acobian matrix. MATLAB
returns a values 3.7497 and 6.1814 respectively. Describe the meaning of the values returned by each of these functions and your interpretation of these values returned? (4 pts)
[W \ﬂ Complete one iteration of the Newton’sRaphson Method. (15 pts)
\ i i “5 ‘ 3R ~ 3: ﬁg“, :2 H ~ £17. I} "can ‘v, f ———————fr Problem 2 (20 points): Consider the following data set measuring the daily population of a mold (in grams) in a cheese sample you found in your refrigerator. a) Find the best least—squares regression exponential curve fit to this data. (you may ﬁnd the linear curve ﬁt for partial credit if necessary) (16 pts) b) Using the curve found in part (a), estimate the population of mold in grams after one week and after 4 weeks. (4 pts)
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\ '“ mama — Problem 3 (30 points): The normal distribution ‘bell’ curve is symmetric and one of the most common functions found in statistics f (x) = Ce'("2). Here is a graph to help jog your memory.
(note: graph may not match C value given below) Despite the importance, there is no analytical solution for the integral to the function. a) Use Simpson’s 1/3 rule to determine the integral of the function if C = 3.1 from 3 to 3 using
2 sets of3 points (i.e. 3, —1.5, 0, 1.5, 3). (15 pts) b) If you use the MATLAB function QUAD to ﬁnd the integral over the region of interest,
which of the following are a correctly worded NLATLAB statements that will return the
correct result? Write ‘correct’ next to those statements you think will work and circle at least
one error in each that you suspect is incorrect. (10 pts) > QUAD('31*exp<(1) _xA2)'3,3> «97/
> QUAD(‘3.1.*exp((—1)*(x."2))',3,3) z
> QUAD( '31.*:§((1)*(x.A2))', [ 3 3 1) r: i a ~’— / > QUAD('3.1.*exp(1*(x.*x))', 3,3) 1' ( ,7, >QQUAD('(3.1.*exp(1*(x.*x))i)o,3 )*2 .. 1 r5";
,r‘ ‘ i ’t ‘ l H  'y \L “ I \ \l "f., (4 — Problem 4: (25 points) Compare the Euler’s Method (10 pts) to the Improved Euler’s Method (10
pts) to ﬁnd the value y(0.5) using h = 0.25 for Euler’s method and h=0.5 for the Improved Euler’s
Method. The initial condition is y(0) = 1.0. What is the absolute error between the two results at t=0.5 seconds? (5 pts) : U \‘l MG
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 matlab, pts, 218Engineering Computational Methods, regression exponential curve, closed book/closed note

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