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Unformatted text preview: “7,14% $25M!” Last Name Mata 5 First NameEC MATH 211, First Midterm
October 5, 2006
Instructor: Adela Comanici idterm Total (100) 0 You have 90 minutes to complete the exam. Budget your time so that you will be
able to attempt all sections. Instructions: 0 Print your name on this paper before starting the exam and return this
paper with your exami 0 Show all your work. Answers Without proper work will not receive full credit.
0 Graphics calculators are allowed ONLY for problems 2(b) and 6.
0 Put a box around your ﬁnal answer. Upon finishing PLEASE write and sign your pledge on front page of your exam paper: On my honor I have neither given nor received any aid on this
eram. (Zn ”Z AOnpr 1 64V! MIW4/(«m 40f rééﬂ’ WA 44},
4M on 74/; MW , . ,r ,MWX 1.(10 points) Find the solution of the following initial value problem and indicate the the interval of existence t + 1
I : 7 —1 : 0.
y t(t + 4) M ) 2.(20 points)
(a) Find the general solution of the following differential
my, —— y = 25523;.
If possible, ﬁnd an explicit solution. (b) Tritium, 3H, is an isotope of hydrogen that is sometimes used as a biochemical tracer.
Suppose that 100 mg of 3H decays to 80 mg in 4 hours. Determine the half-life of 3H. Hint: The mathematical model is the exponential equation NI 2 ~AN, where N
represents the mass of remaining nuclei. The half-time T% of 3H is the amount of time required for 50% of 3H to decay and T; = 19:2
3.(15 points) Find the solution to the following initial value problem and indicate the
interval of existence I 1
cc +$cost = —2—sin2t, 93(0) 2 1. 4.(15 points) A tank contains 100 gal of pure water. At time zero, a salt—water solution
containing 0.21b of salt per gal enters the tank at a rate of 3 gal per minute. Simultaneously,
a drain is opened at the bottom of the tank allowing the salt solution to leave the tank at 3
gal per minute. Assume that the solution in the tank is kept perfectly mixed at all times.
Find the salt content in the tank at any time t. ‘ Hint: Write down a mathematical model by expressing the rate of change of the amount
of salt in the tank both as a derivative and as a difference between rate in and rate out. Find the solution for this mathematical model.
5.(20 points) (a) Verify if the following initial value problem I t
2 0 =0
3: :r 1, 117() is guaranteed a unique solution by the hypotheses of the uniqueness theorem of solu-
tions for ODEs. (b) Suppose that :c is a solution to the initial value problem , cps—x 1 Show that O < $(t) < l for all t for which :0 is deﬁned. 6.(20 points) Consider the following autonomous equation I y = (y+1)(y2-9). (1) Perform a qualitative analysis of the solutions to (1) using the graph of f , the phase line
information and the sketch of the equilibrium and non—equilibrium solutions in ty-plane. ...
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