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Unformatted text preview: CHBE 2120 Exam 1, Fall 2010
7:15 PM ~— 9200 PM, September 20, 2010 By signing below, I am agreeing to abide by the rules of this exam and by the Georgia Tech
Honor Code. In particular, I agree to use no printed materials other than a single 8.5x11” sheet of
paper with notes on one side, and I will only use standard scientiﬁc functions on my calculator (I
will not use any programmable or graphing functions on my calculator). Name: Signature:
Write your answers in the spaces pro ' . ank paper is available for scratch work. If you need
additional room for your answers, staple extra sheets to the exam and clearly indicate the
problem numbers for your solutions. ' In problems that ask for a speciﬁc calculated value, highlight your solution with a box. Note that credit will not be given for calculated results that are correct if incorrect reasoning for
that result is used. You must turn of all cell phones and other wireless devices for the duration of the exam. (for instructor use)
Part 1

Part 2
Part 3
Part 4
Part 5
Part 6
(max. 26) TOTAL  (MAX. 100) Part 1. Choose the best answer for each problem. (2 points each, 12 points total) 1) You need to perform Gaussian elimination on the following system: 0 4 8 x1 1
3 6 4 x 3 7 How do you proceed? 64?8 2 0 7 x1 1 K 6’ VQC
A) Pivot the system so that it looks like this: [0 4 8] [x2] = [4] 3 6 4 x3 7 3 6 4 x1 7
B) Pivo the system so that it looks like this: 2 0 7 [x2] = [4] 0 4 8 x3 1 3 6 4 xgp%%(yég
C) Pivot the system so that it looks like this: 2 0 7 352] = 4] 0 4o 8 x1 1 D) No pNng is necessary; calculate the ﬁrst coefﬁcient for row subtraction
E) Mum answers above are correct/valid 2) “Partial pivoting” is deﬁned as which of the following?
A. Only perform a row pivot if the diagonal element is equal to zero
B. Only perform a row pivot if the diagonal element is approximately equal to zero n1 exchange matrix rows, not the b vector elements
D. Pérfo a row pivot at every step 3) What is the order of complexity of the number of operations in the following MATLAB code?
foo = 2,
for 1 = 1 n §$;*’—‘—“ /1
for j = l:m {=;*_‘—“/h
for k = l:n. ﬁﬁ:‘—~——' /)
foo = foo + i*j*k;
end ‘Z
end (:j) /5 /ﬁ
foo = foo + i*foo,
end
A) 0013)
) 0(n m
Hm )
D) on?) 4) Consider the following MATLAB code; what is the value of bar after executing this code?
bar 2; {‘I z 3
for 1f:r132~ 1'2 .72 affz; 5’
_ ‘ J
bar = bar + i*j; 5'2}. end bar = bar + i*bar; 7‘3 ' 3 Z/I/Z.
end 0 j;— A 36 .
@339 w _ .
C) 54 [6% 746 " 43/
D) 66
E) None of the above 5) Given: §=y5+cost; y(0) = 1. You are instructed to use the Multipoint Heun method to integrate this equation with
h = 0.01. Which of the following is true? A) You can’t immediately use any Heun method because the ODE is nonlinear B) You can’t immediately use the Multipoint Heun method because the step size is too
II a ma ’t immediately use the Multipoint Heun method because you don’t have
 ough initial conditions/information/data points
D) You can immediately proceed with a Multipoint Heun integration 6) An nth order ODE needs how many boundary conditions or ﬁxed initial values in order to
be solved? A) 1
B) n—2 Cg n—l E)n+l Pan 2. (12 points total)
Suppose you have 3 matrices: A=[1 2 3].B=H,C=[§ i];0=[§ 5
6
Perform each of the following operations or state that they cannot be performed/are undeﬁned.
(2 points each) H
§
1) AB 4 _ 4) Give an example of an upper triangular matrix. (1 point) I23
0%?
004 5) Give an example of an identity matrix. (I point) [00"
010
[mil 6) You are integrating an ODE from t = 0 using the Euler method and a step size of h = 0.5.
You ﬁnd that y(0.5) = 1. Your friend is integrating the same ODE using the Heun method and a step size of h = 0.5. She
ﬁnds that y(0.5) = 11. Is it reasonable for you to continue integrating to t = 10 using the Euler method and h = 0.5? / If,
If so, justify why, and why the Euler method is a good choice here. //’ / If not, justify why not, and say what you would do if you were forced to continue integrating V5,
with the Euler method. 3 points) W ‘ a7 2
‘" gag/M dé g d 7
6 W p W0f f/l .WZQMM (‘5 ﬁg} 50 jwyﬂm/ae veg/W
gimm //’/»//=/O < , :— 3 % 070/;
9Wf@{’/a ' 245/6/
:DP w 7) You are given the following differential equation: You are told that y(3) = 10.
You are asked to integrate from t = 3 to t = 10. r! ./ _+l 0 Elk/54 QM 114$: 5‘7? $226 aw _ 2_
dt—3t+4y 2 Is this an initial value problem? Why or why not? (1 point) %5. [Maﬁa Mg M M WW 742 55
H: M 530/ p 27C 993/ M1 W a 5
M9 3);, M 7 5‘86 x/ (M M Part 3. (24 points total) The following system of equations describes the operation of a new reactor setup that your
coworkers have developed. ' 3’ —4x = — 8 '3: I
63‘3 3— x1 = 22"“ 8172 ’K ‘*%L+67{? — l 5x1‘3x3+10=° 57, ~37t5 ‘= —/(J t/ J , :21) Can you translate these problems into matrixequation form that can be solved using
r~ ‘ ﬂ 6”“ { . . s . n . . 
c) '{ Gaussum el1mmat10n’? If so, translate it into matrix form. If not, explaln why you can’t. WW 6’ (4 points) 0 20 All 7‘} a/ Q L 1" Z
_ 5’ 0 "3, 7‘3 '70 , ,l
WAR WM r; 5L2) Now, use Gaussian elimination to solve the following system of equations. Pivot only 2 2 2 x1 8
when necessary. (10 points) 6 4 1 x2] = 22]
4 0 9 x3 —3 2%) + 62H) :— 13 / U "middle; Wj'fwg') ULing your previous results, perform LU decomposition on the system above. Clearly
MU {119mg 21indicate what the L and U matrices are. Do not resolve the system. (4 pomts) ,. {00 . 222
[53/0 Wow26'
2 Z/ O 0/5“ _H':/l ' F‘ _f (21" {695/ [iota‘7 [whiff gm r {5.5%} l“
LVJI_,.‘H'ljg.t;f nir'ﬂ' 4; Dem nstrazte that yOstjLU ecorrigcdiéitiorgilsaébrrect s’igp che 3 points) era—'5’
0 0/“;
2 [@O 7’2 {00 22 H 3/0 04623/0 éb’zw’éd/
.22) 00 5 21/ 4,1 ‘M‘fﬂaf Lfooz LUZ 1’ 5) If you had 100 other b vectors to solve for, would it be easiest to use Gaussian if 1L, / Elimination for each of those problems, would it be easiest to use your LU decomposition H to solve for each vector, or would both approaches be equivalently easy? Justify your
[1‘ LU J35 y/w’ 75} mag /¢ M U _, Part 4. (ll points total) M g” for”, .' " / fl) Make a Taylor expansion of a function y(x) around x using a distance of Ax. Keep three L} m {x ,‘nst’ exact terms in your Taylor expansion, and include an error term for the fourth term that
p 77’ * ,r' M 6') approximates the order of the error. (3 points)
a J 2 9 J
6 ci(.4;[£:A jasj (1)1" ditjlw + m 2) Let’s say you know the value of y’(x) and of y’(x+Ax). Use the deﬁnition of a slope
between two points to approximate the value of y’ ’(x). (2 points) a [2 , M «j [+4 J’Z'C‘c)
L f“ f (.y , a : 2: 3r
5/0195 ﬂm @WQK “A”; 30 AK I'n
, I 9/ MﬁWéw/IyJ) Elug thllS aggrommanon Into your Taylor expan81pn, and reduce the equatlon untll you
'/ 6r M fen” ave on y ee exact terms (one in yet? 1n y (x), and oneln y (x+Ax)), plus one ﬁfth) {72331343333249 (A: 17%“ ’5" + 0&ng
mi—Ax a, +4; + ' wro/mj If 3 #1111 instead, you would likely use an approximation or guess of what that value is. Keeping
iML {It (‘7’ 1" i this in mind, look carefully at the equation you have derived, and indicate which I»— q r , { 7 + I integration method that you have learned is identical to this approximation. What is the / (“If "‘9" _ . local error of that mtegration method? Is that co srstent W1th your V2.1?ko 655% ‘" 0 Part 5. (15 points total) Your roommate, a biology major, has created a strain of yeast that turns essentially all of the glucose it is given into acetic acid. For every mole of glucose that it is supplied, the yeast cells
create 3 moles of acetic acid. The inﬂux of glucose into a yeast cell is driven by the concentration differenc of glucose
between the interior and exterior of the cell times a constant, K... (with units [5]). Since these cells grow in a huge vat of wellstirred liquid, the extracellular concentration is essentially
constant with a value of Ge (in [mol/LD. This makes the equation for the rate of glucose inﬂux: rateGlucose influx = m(Ge _ G)
Where G is the concentration of glucose in [Incl/L]. When exposed to high concentrations of glucose, this strain of yeast turns on genes that export acetic acid into the liquid reactor solvent. As these genes are being turned on, the kinetics of this
acetic acid transport are as follows: at
TateAcettc Acid export = b + t * A I
Where a is a constant with units Eds], b is a constant with units [3], t is the time (in [3]) since the
reaction was started by adding glucose, and A is the concentration of acetic acid in [moi/L]. Glucose never leaves the cell, and acetic acid cannot reenter the cell. The set of reactions from glucose to acetic acid can be approximated reasonably as one pseudo~
reaction that is ﬁrst order and has a rate constant of k with units Ms]. This can be represented by the following diagram: M restore, :5
(the éwc ~ v 1) Write two transient balancd‘on an individual cell in terms of the concentrations of  f ‘ glucose and acetic acid in the cell and all relevant parameters. One balance should be on
1 13 1 5M? tgﬂucose in he cell, the other should be on acetic acid in the cell. (8 points)
_ J . I ,' " __ Clam .2: Ill/~007’f’ 6’54/~ 604/5 Mose: Va 49Alen (Kgé)  0+0 are 2) Is your glucose balance a linear or non inear 1 erential equation? (1 point) Ltd/[W 3) Is your acetic acid balance a linear or nonlinear differential equation? (1 point) (“MW Now assume the following values of constants: Km = 10, Ge = 10, k = 100, a = 0.2, b = 2. 4) Is your glucose balance clearly a stiff equation? Explain why or why not (2 points) _ ([00406‘1000 5) Is your acetic acid balance clearly a stiff equation? Explain why or why not. (Hint: to decide whether it is stiff, consider what the equation looks like wherﬁ is extremely small! _Y
and when t is extremely large.) . (2 points) 4'} C’ 77 a 2A? & th ‘,Zé 7/7 «1.x,
.. '— f—  +d ' ' /,%“‘(W€
2; WZ/ git/(6066’ ‘Qkﬁﬁ Part 6. (26 points total) Given the following differential equation:
d4y d2 y WWW” y(0) = 1; HO) =4; y"(0) = 2; y”’(0) =3 L
1 f'  41) Circle all of the following terms that describe the above differential equatlon: (2 pomts) _ /’ .iX o
rt‘ V '
. layﬁwmnw 18‘ order it (w? PDE Q 2) Transform this equation into a system of coupled ﬁrst—order ODEs that DOES NOT
contain the original y variable in any form. Make sure to clearly deﬁne all variables that
you use. Also include the initial condition for our 5 stem of ﬁrstorder OD R I a ‘ m
"a 3) Use the Euler method with a step size of h =. 0.1 to inte . (8 points) ' \ : .. A  gt ‘
80” L If  y‘/ c) " Sillgof/l JCGOIéOJ .7 ’13 ,' ,rﬂr'rzl M75 39% y l : 3d 1'22: {3%} (*5) 5/ :"y°
w, (.r 2t" 111" yr ’5’  I ' A
[2/7” A 711% / yd?! : 9‘; f Aligygttz) fat/L)
l"? ‘M We Use the midpoint method with a step size of h = 0.2 to integrate this system to t = 0.2. “' _ ,bi (10 points) '
N“ W” “"0 ' " 1 yo T % féox'gd)
J fwd" L,  “NZ 6
{a L ck FA F/éirz) #1 JW
. /
.’//@/wth : 0 1' 5) If you had decreased the time step for each of parts (3) and (4) by a factor of 1—10 how much more accurate would your results have been for each part? (Hint: Don’t redo the
integrations, but describe what the impact on the error would be.) (2 points) 2551/, 50 error WM I? 3446/ “My; HH/ M1] errant, «J; illwin 72? Mr . asea/jééw/wﬂn} ’/ ...
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