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Unformatted text preview: Math 170 (Spring 2007)
Kouba Exam 2 Please PRINT your name here : ___________________________________________________________
Your Exam ID Number ____________ 1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY
WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM
ANOTHER STUDENT’S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR
CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE
EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK
YOU FOR YOUR COOPERATION. 2. No notes, books, or classmates may be used as resources for this exam. YOU MAY
USE A CALCULATOR ON THIS EXAM. 3. Read directions to each problem carefully. Show all work for full credit. In most
cases, a correct answer with no supporting work will NOT receive full credit. What you
write down and how you write it are the most important means of your getting a good
score on this exam. Neatness and organization are also important. 4. Make sure that you have 7 pages, including the cover page.
5. You will be graded on proper use of limit, derivative, and integral notation. 6. You have until 9:50 a.m. sharp to ﬁnish the exam. 1.) (10 pts.) Find the Jacobi Matrix Df(:1;,y)f0r ﬂay) ___ (81.22%) _ 2.) (7 pts) How many distinct 9~letter permutations are there of the letters in the word
POPPYSEED ? 3.) (7 pts) A group of 13 people includes 7 men and 6 women. In how many ways can a
committee of 5 people be chosen if the committee must have exactly 3 women and 2 men? 4.) (7 pts) You are going to buy 16 freshly—made doughnuts at Krispy Kreme. If you pick
from among vanilla, chocolate, maple, raspberry, and glazed doughnuts, in how many ways
can you make your selection ? 5.) (10 pts.) (Drug dissipation) A single dose of zolpidem (a drug for shortterm treatment
of insomnia) is administered to a patient. A laboratory measurement shows that the
patient’s body tissue retains 2.5 mg. of the drug after 6 hours. Let $1 and 1:2 be the mg. of zolpidem in the person’s body tissue and urinary tract, resp., at time t hours and assume dim that ~22? = ~0.231:c1. Use the mathematical model from class to determine formulas for $1 and x2. Determine the initial dosage (in mg.) of zolpidem. 6.) (Lotka—Voltera PredatorPrey Model) (10 pts.) Consider the system of differential
equations given below, where N (t) is the prey population at time t and P(t) is the predator
population at time t. Separate variables and integrate to get an implicit solution (in N
and P) for this system. dN gz4N—PN
——=PN5P
dt 7.) (12 pts.) Consider the two tanks containing water and salt mixtures and connected as
shown in the diagram. Tank 1 holds 200 gallons of mixture. Tank 2 holds 250 gallons of
mixture. Let :171 and x2 represent the pounds of salt in Tank 1 and Tank 2, resp., at time
t. Initially, Tank 1 contains 45 pounds of salt and Tank 2 contains 30 pounds of salt. The
mixture in each tank is kept uniform by stirring, and the mixtures are pumped from each
tank to the other at the rates indicated in the ﬁgure. In addition, a mixture containing
2/3 pound of salt per gallon is pumped into Tank 1 at 5 gal. /min.; a mixture containing
1/4 pound of salt per gallon is pumped into Tank 2 at 3 gal. / min; the mixture leaves
Tank 2 at 9 gal. / min. SET UP, BUT DO NOT SOLVE, a sytem of differential equations
with initial conditions, which represents the amount of salt in each tank. PAY CLOSE
ATTENTION TO FLOW RATES IN AND OUT OF EACH TANK i!!! a gilt/gal, a 741”{74' ‘8.) (7 pts.) A bag holds 7 red, 10 blue, and 6 white basketballs. In how many ways can
you select 8 balls, if you must choose at least 1 white ball ? 9.) (12 pts.) (Graphical Method) .Consider the given graph of zero isoclines with equi
librium (a,b). Use the given ' sign chart for f1 and f2
to create a signed Jacobi Matrix and determine if the the equilibrium is ~l1‘>0 stable or unstable. *1 10.) Consider the following nonlinear system of differential equations. 15;— = m?  $3 a.) ( 4 pts.) Determine all equilibria for this system.
dag?  cc 9: — 2:1:
dt  1 2 1 b.) (14 pts.) For each equilibrium use the analytical approach (Jacobi Matrix and
eigenvalues) to determine stability and classify it, or conclude that this method is incon’ elusive. The following EXTRA CREDIT PROBLEM is worth 10 points. This problem is OP
TIONAL. 1.) Determine the number of distinct 4letter permutations that can be made from the
letters in the word ZIGZAGS. Count the total number of permutations WITHOUT listing all the possibilities. ...
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 Spring '09
 Kouba
 Calculus

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