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Unformatted text preview: Name: E E I .
ID # : .
Signature: Section: 4pm 5pm 6pm 7pm 8pm
TA's name: IﬁnalExanl
MAT 17C Spring 2009 Instructor: Tim Lewis
Thursday, June 11, 2009 When you receive your exam, ﬁll in the top of this page but do not begin the
exam until you are told to do so. Read each question carefully and do all your work on the exam; you must show
all your work to get credit. No books or notes are permitted during the exam. Cellphones must be turned off
and put away. /50
/50
/40
/60
/35
/30
/35 somewwr Total: / 300 Problem 1. [50 points] Suppose that the growth rate R of a population of bacteria is a function of the concentration of the bacteria ((2) and the concentration of nutrients (n)
R=f(n,c)=cn—c. : C (n w I) (a) [lSpts] On a single graph with n 2 0, c 2 0 , sketch and label the contours of R = f (n,c) for
R21, R=0 and R=1.
R=o é o= (Vi13 0A gal 3:0 om 5:)
I" IQ==> Hecwo
ﬁv/ I a: .L
, I “Mir!
1/ i Rs“ : “\1CZI)
. C: ‘
/ 1 ‘__‘_ L_.______~:’_Q+=O w—m (b) [10pts] Find the equation for the tangent plane of f (11,0) at (2,1), and use it to approximate
the growth rate R for (n,c) : (2.1,1.1) . é—‘Ec §£,mi 'p(9")='
an ac, & 7' \"C‘r ambit}; i 3Ci9.1) 'i ‘l
3
+
a" I
p (c) [15 pts] Find the derivative of the growth rate R = f (n,c) at the point (2,1) in the direction of
V = (4,3) , i.e. ﬁnd Dv f (2, 1). Is R = f (n,c) increasing or decreasing in this direction? We : (355%); (cu mi) V1C(a\i) : (17 I) CL: 52.: “be, [9i 4fl+32:'7: 5
WI 5“ Dvwc(2,i)= qunm) 1. LL \l
('\
\/ U1 1—
(\
i—P
()0
\,/ U
01“] \l 0 $ 1Q: 4(mC) is Mcreasinﬂ cc!“ (1.131” “R dwah‘on Laws) (d) [10 pts]
(i) What is the direction in which the function f(n,c) increases most rapidly at the point (2,1). What is the rate of change of f (n,c) in this direction?
At (1,1), ‘PCWQ Manages mos+ my;me m "ii/:6. darea+~iovm V—FCZI:)=C111>@ The, calf, 0? coma/me 0? {(WC) "W +1412, diraCi‘on 1g iV‘Ci : E
U
{.e. DWC (cm = Wm.) » WENQ —. \vwzim : Fiﬁ" : «4:2.
mhzmjl
(ii) Is there a direction in which the rate of change of f (n,c) at the point (2,1) is equal to 5?
Justify your answer. No] +Mefe Hg NET 0v d'i‘f‘cC“i‘m” 1“ Whit/94 We. rode, o—C (“Range 04‘ 41%;) r— 4)» (1‘)) ',g ccbwll +29 b 5 began/«Se, +171: mma‘immm Fate, 04 change )3 4? (9e: wova)‘ (iii) What is the direction in which the rate of change of f (n,c) at the point (2,1) is exactly equal
to 1? Justify your answer. Dunc (2,1) = V‘HQH) ‘ WWW) : (‘0 13‘ CM',.?._R%) : i
(“L vac/fur é IL: (,\‘o) m Cw (on). 94 Mole, +i/\o..7l’ we Corrcspondmﬂ' Amid—(mu dermal—1M owg, Problem 2. [50 points] 1
Consider the function f (x, y) = 3x2 4xy + y2 +14x, (a) [20 pts] Find—61, ﬂ, 9:, (Bl—f, 62f , and—:1.
ax 5y 6362 ayz 6x5y dyax
3;? — in 5:. _
 70  4 l @i: v — 4
37C ‘6‘ + 4 3,5" i cﬂéa
5i— '1
5‘3'Atmt2xa “4:1:32’ 31L :_<¥
a a?) WV (b) [30 pts] Find the critical point of f (x, y), and determine whether it is a local maximum, local
minimum, or a saddle. v£(ryl%)=(0,oy i7 3&4wa __> (wage)
$94 ONLY poSS'IéLE QRVTiQ/«n. PT‘. —4\r’y +2? '0 _ “L SE” 7. : own—mazuwo
$(V’ﬂv 'm A SADDLE,
“0‘34 Problem 3. [40 points]
Consider the system of differential equations Elx—X+3
dt y 05’ —=3x+
dz y (a) [5 pts] Write the system of differential equations in matrix form. aw ; I (2:5 (b) [5 pts] Assume that the eigenvalues k1, kg and corresponding eigenvectors V1, V2 of the matrix of coefﬁcients are
1 1
k1=2,v1:[ I] and k2 =4,v2 =[J Write down the general solution for the system of differential equations. VHV 1m): a. éwﬁ‘l) + £16412 (1” (6H) (0) [5 pts] Classify the steady state x = 0, y = 0 (according to stable node, unstable node, saddle,
stable spiral, unstable spiral, or center). Justify your answer. Typically, what does a solution of
the system (x(t), y(t)) do in the long run, i.e. what happens to x and y as t —) oo ? ’kfQ, 40 Mad, Q1I4>0 3 (0'0) is A Swoce‘ P7. TYWCALLY A8 +, —> 00 \ yL+)———?+I>Q OIL “(90 MD m6L+3v>+l9<) (ML—too (d) [5 pts] Find the solution of the system for the initial condition x(0)=1, y(0)=l. Describe what
happens to x and y as t —> so. What is it about the initial condition that makes this a special solution? X10) 1 \ W v1 \ _V l\
%(0) 1 l ( =§ cm: 9 TH‘S QJLU'T‘ION LS SPECJA'L‘, 6' Fr swws 0N we 'smste Wlﬁouﬂ" OF we SADDLE.) Mi. Awards “I'Hé smbbé aerexlvecj'oa 0F (0 o) Problem 4. [60 points] Chemostats are bioreactors that continually cultivate microorganisms (e. g. bacteria). They are
used extensively in many biological research labs and industrial settings. In the chemostat,
nutrients are continuously supplied to a culture vessel and the cells in the vessel consume the
nutrients to grow and proliferate. Residual nutrients and cells are removed from the vessel. A mathematical model for a chemostat is dn — — — +2 : me)
dt on n dc ’ E=CH—( :1 ,g‘fyhc) where n and c are the concentration of the nutrients and concentration of the microorganism
inside the culture vessel, respectively. (a) [25 pts] There is a steady state at (n=2, 0:0), which is a saddle point. Show that (n=1, c=l) is
also steady state. Use the J acobian matrix to analytically show that this steady state is a stable
node. Brieﬂy explain why the J acobian matrix can be used to determine stability of steady states. O Q—l '14
Mm); 9“ 6C ;
{n «(C (I V‘x'i
@0337 7cm); —o'l« ~
l O
eigeN/otlwes 0‘? 70,1) 1 (1401) 3 ‘r i = 124ml) + i t O
)= 'éL i44~4 : —1
2
13 ~r 9 AL“!
:3? (m=\,c:() is A smeté N086.
THE STACObUMJ Mrme EVALMATED AT A STEADY SF'A'TE is We 3 MilI'le 0F CO~E¥FlCl€N7€ 0F TH’Z WTS‘TyM o; DlFFéﬂémlAL gem/vﬂorxlé? (i L’NEA‘Q‘EQD AgoKT 'm‘” 57“»?!le ‘Zﬂ‘meté THC—I EIMENVAUAGQ or: THE “ViWM 0F, COEFF\C\€AT5 M: A; L\NéA,Q’ syy—IEH 0F DIFFéfLéNT/AL EQL‘AT‘MJS 1357’E7tu’lli‘15 THE S’MBHM'TY 0F \TS may gmrg , (b) [10 pts] The nullclines are drawn below. Properly label all curves (the nullclines and the axes) in the ﬁgure. For c 2 0, n 2 0, indicate the regions in the n,cplane where E and in are
dt dt positive or negative. hm“ C/\‘. keg ‘1  .2
agrarimwro T: c— 7b:
at ‘
airL/niqcﬂ), CL“:5;>0 0“:
ad (n=37c:o), 03E:—I<o
0%
dCan,QJ :O 1% Q10 0Q ﬁsl (c) [15 pts] Draw a phase—plane of the system for c 2 0, n 2 0. Include nullclines and arrows indicating direction of ﬂow. Label all curves. Sketch the solution trajectory starting at (n=1,
c=0.01). Make sure your trajectory is consistent with the information in (a) and (b). L 4}”:0 01L
& (c) [15 pts] Draw a phaseplane of the system for c 2 0, n 2 0. Include nullclines and arrows indicating direction of ﬂow. Label all curves. Sketch the solution trajectory starting at (n=l,
c=0.01). Make sure your trajectory is consistent with the information in (a) and (b). H r121 lNlTIA’L ’41“ C: 0'6,
CoNblTWON (d) [10 pts] Sketch n(t) and c(t) vs. time tfor the initial conditions n(0)=l, c(0)=0.01. Problem 5. [35 points] Proteins are the chief actors within the cell. They carry out the duties speciﬁed by the information encoded in genes. Proteins are large organic compounds made of a linear sequence
of amino acids. There are 20 different types of amino acids. (a) [10 pts] How many different 10 amino acid proteins are (mathematically) possible? 620.80.610, , 520 : 80H) _/_._,«r,___._w,_ v . .. _ ,1 I 0 Mug/MA) (gr/zigcila Ade/€241.24, ; WW Wu 60»ny V494, H M,£7’l:4bt£,al fl oﬂom MAT'rémé. (b) [10 pts] How many different 10 amino acid proteins that consia/of all different amino acids
are possible (i.e. each amino acid appears only once in the seguence that makes up the protein)? uoo/l :: ,,.s,,_._...WM_,_,., i w/ l O 'l ’0 Wmma cue/£004.] Mac/'5 gel; (cub. Wyfﬂjw, ,1 g) aggyx, $9.4“. (if “,4. 679,641. a 61. ' (c) [15 pts] The smallest protein, which is called Chignolin, consists of a sequence of 10 amino
acids. Chignolin has the sequence of amino acid residues @YDPEZQIWG}, where each letter
symbolizes a speciﬁc amino acid. How many distinguishable amino acid proteins can be formed
using the same set of 10 amino acids in Chignolin? #0,: WM: «m AMANG‘E Io chiMaA.c,»/JS = lo!
#01; WW) TD ﬂammme 3; @g : 3!
SL TS 1' 01 /
4W
(ca/ﬁnd, W AM) it OF DIS’Me—wSHABLE Ammo ACID $6QUBMCJCS H
O 7.. Problem 6. [30 pts] \ z balis
“A (a) An urn contains four green balls, blue You take three balls out of
the urn without replacement. Answer the following questions and justify your answers. i. [10 pts] What is the probability that all balls are the same color? l
4 4. 
ﬂoe WMSTDG‘ET ALL @593: (EB‘ET’TT*4
49cm: wms’mtreTALL E>LUE= <e>_ Ca'. :20
3 ataé
{i or/ wm/s "ID GET mi, R60 = o
TDTYH, it op w/wg 17; CHOOSE 3 2,ng = z 2.20
P<SAMG (,0ka = 4 + ‘90 _ 614' : é
33w) 3‘ 0 10¢!
Li H .
ii. [10 pts] What is the probability that all three balls are different colors? Gee?“ ecu E— 3‘ 12760
9 J4
it 0? wms TO CHOOSE? Au, DlF‘Pe—KGNT cm,on — 4i ’ E ' 2’ Le. om; gags»), 0N6 we. we ﬂab. Act, ~\ 4 . Q,  ,2 4 Mai (b) [10 pts] Assume that the probability that an insect lives more than ﬁve days is 0.1. Find the
probability that in a sample size of 10 of these insects at least one will still be alive after ﬁve days?
L ; Msle a‘ul‘we, ¢£ieve 5 clatfo
LC = Knead dead. «New 5 aha"1’s
PQALL to INSECTS; Dam \ a: (0.9)“)
AFTFKL ES DAMS )
P ( Almgéig‘r 1 0;: 10 thmiSB : \ —— P (
AUVE Apnea 590mg P(L>=OI
P053: lO.l *MW : 0.? (Lu, ﬂwmlﬂhmr~gb  I!
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l
A
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O Problem 7. [35 points] An initial screening test for the hepatitis C virus shows a positive (+) result in 95% of all cases
when the virus is actually present in an individual and in 8% of all cases when the Virus is not
present in the individual. The probability that a random person will test positive is 20%. (Where
applicable, assume that tests are independent for any individual). (a) [15 pts] What is the prevalence of hepatitis C, Le. what is the percentage of the population that
carry the hepatitis C virus? D = HAS Hep/Wine P ; res—rs past/'me DC: DOES I061” HAVE Heprmixg pf— : TESTS Mewrive
PCP\D\)= 0,94: P(PDCB:O,O§? PUD): 0‘; p(pc>10'8
PLPC\«>§=o.oe PiPClDUVWD'qL P(D)=VP/ tome): [102 W?) = P(PlD‘>P(D\) 4 Ptplb‘) Ptb‘)
0,10 : 0.515 via, + 0,08 (l’4nz)
(LEV—149.: o.IQ, m...”— %= 0.159 (b) [15 pts] Assume that the prevalence is 14%. If a person tests positive for the hepatitis C virus,
what is the probability that the person has the hepatitis C virus? Mum = NemlPtD) = (M‘simgotéo
90>} 0,10 = o. e ‘65" (c) [5 pts] If a person takes the test 2 times and tests positive for the hepatitis C Virus in both tests,
what is the probability that the person is a carrier of the Virus? P u. ‘ “‘ > 3 i p ' \
“if. P( D 1 Pl n P1) = LL13? \ D.) P (Dl n 923 mm?» P (m MWDB 4 P Cent’s \ch WW
13/40/5 mepeweme PCP,n131ib\)= PCPJM Whig): (NPWWL \ P magma} = p mime) P0139?) = (9mm); — M”’"‘“""“’” g; M I \ _ (WWW PEEL. ( 0 r45 31 f D“ " \l (0,45); (0,14) + (0.08)L(1— 0.14) H ...
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This note was uploaded on 10/27/2009 for the course MATH 17C taught by Professor Lewis during the Spring '09 term at UC Davis.
 Spring '09
 Lewis

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