Math
NAME:
Exam L
155
Fall2OO7
l ./
11ert
I
TIME:
SECTION:
INSTRUCTOR:
The exarn is closed book and closed notes. You may use an
fnstructions:
approved calculator, but be sure to show your work on each problem for full
credit. Work that is crossed out or erased will not be graded. Turn in any
scratch paper that you use during the exam. You will have one hour and 45
minutes to work on the exarn. Please turn off your cell phones.
Problem Points Score
I
T2
2
T2
12
15
12
12
13
3
4
5
6
7
8
TotaJ
12
100
1. (12 pts) a) An ice cube tray contains 12 cubical spaces for ice, each of
which is 1.25inchesper side. Ice has a density of 0.95 g lcms , and 2.54cm : I
inch. What is the mass in grams of a fully filled*traY? ,
.r cutb e
gI
l. ? 5 i, ' t
is
Volu.',*
l?
t l.?;,,r']'
i
\r \
i-:f -! *,0 *
= 36'{' 81 ?
j
i (., *0,;f
b) write the equation of the line passingthrough the points (1,3) and (3,7)
in slope-intercept form.
11)=1* :
J
7
3"-t
="'L
a .- 3 - 2 ( x - t)
-i'L'-2
t
B= 2x+ \
c) Where is the function tr(t) continuous?Justify your answer.
( t'
L (t): 1 z t - 1
I r*3
p rl t" :' tL \e
O^ \.V
y::r'l": * t
if t < l
if 1 < t < b
if 5< t
\ i: t ofi 4 , ,i ul.i{ t;
s:.
fl.-..'r-e-L*l ; L*'5
\ i,'.,-..,t-' = I
t-+t-
li.n {.?l-l )'i-l
L+r*
,, I
L
lio
;;t"-[
ll,n Lttl'\irv\
t4 5 L-es-
x
'L"u ;
1 ,,o'1 Lt fu,I * llr'v\
2
L.4 5 +
fl egu
?" d
So
L
L
rs
c l ;sli) " 4 ' r \ r t r ' ) u S
lil:*l
&i
r's
{ -& , ; {
*dl
t=5'
iE cCInL , c)c\ { ^ "'*., 'i } L' { tir"*l'; '
{XJ I
2. (72 pts) a) Graph the following discrete-time dynamical system and cobweb for three steps, starting from the initiat condition. Label the coordinates
of each point.
b t+r:2 .0 b t- 2' 0
far ting
fr om bo:2' 2
I'it{.1 r'-ril;':"1'
b) For what positive initial values of b; is the solution of the discrete-time
dynarnical system increasing? \4trhatis the long-term behavior of the system
for these values?
Lu'=*
ilrr."
i*er
- ' ilt *
"li: - ; ! ""' u' Fi, . ; :': '-
r r " r i : i i 'i
r.l L t *
'.r.1 p * 1",
t,o 61''''t *" r'il'{
'
c) Solve for the equilibrium point algebraically. Show all of yoru work to
receive full credit.
bx =? b*
r h*
0
l-,
*
,r'1
"/-
* "'
''''j"
€ c1 ,q 1 'l ' t3
L,j
*'"]
"{,
t' - 3, r \ "3
ri'
I
3. (L2pts) Let f (r) : 3r2+ 2r - 5
a) Find the equation of the seeant line to / that passesthrough (4, /(a)) and
(4+An,f(4+4")).
Jtq l . 5 |
rr1: t_ !1 :3 .)* 31
t{ tA x
3( . 1+ Ax ) "* ?( "1 + ax ) * S- F l
=
- 4
L/,
= 1!g 1 'tax +axt ) t..8r?Ax.:-:S
/}X
=
k.t"Lf"g"d : 2("t3,sx
AX
b) Find the slope of the secantline when
=21
1.Ar:1
26t3
2. Ar: o.L
2 G + O.3 a 4 6 .3
2 U * c} " 0 3 ' ]G . OT
3. Ar:0.01
c)Fin df(4). \rm s(q-t.:).:JLJ)
Ar+o *-tl;;-:T*-
E lit
?&+Bn x *2 ( ,
ax-"$
d) What is the relationship betweenthe answersto (b) and (c) ?
T h e a n s k ) €r lo ( c)
$he ftn$q"a1*r,ii
{* (b ) .
4
is
*h e
li,tr
Ax+ g
ES
4. (15 pts) Supposef (t) : 760e-t' representsthe amount of a radioactive
substa,ncepresent in a lake in parts per million (ppm), and the lake is considered sa,feto swim in when the amount of the substancein the lake is less
than 10-3 ppm, Here t ) 0 representstime in weeks.
a) Find the half-life of this radioactivtrsubstance.
DCI = \LoOe- "
L ,-VO = j"n, G O
= - L*
-.Gl? l
t -{fE[f*
*tt
=*
"&3?s i^-*e,shs
b) After how marry weekswill the lake be sa,feto swim in?
lo -3 z lt o * - L^
U - to - t - t * lt O = -t *
t-
3 . tl G I L, r,..re,lsl
c) Plot the function /(t) on a semilogplot and label your axes.
frri , lcae'L
j.nt ( {ttt)
5
,l
2
J
z
I
tn, tti-t * \"'f e$ - t"
u J* 5 ' c ?' 57* & '
S. (L2pts) The lung model, with absorption not proportional to the chemical
concentration in the lung, is given by
ct+r: (l - q )(q - s ) * u l
where 7 is the chemical concenteration of ambient ur, q is the fraction of
air exchanged and s is the fraction by which the chemical concentration is
reduced at each breatha) Consider a lung with volume 2.0L that replaces 0.75.L at each breath with
lilhat is the discrete
ambient air, which has oxygen concenteration of 2L%.
time dynamical system for this model if the oxygen concenteration of the
lnng is reduced by 3% at each breath?
-: dl".;:;-?
1. = W = il:"]-5
^5
-*. {.)
U V
16 :
2\ ''-lo -- "2-"1
S a 3"-l*-'"eT
c1*r: ( \-.3TS)( cr -"i]1i
+ o'clG
Cgrr'
' GiSce
*i'1"15){":;if J
b) what is the equilibrium concentrationfor the systemin part (a)?
C# ' ^U15 cn + "0G
I . oG
" 3 1 5 L*
Cx
2
"lL
a{
16%
c) Are there valuesof q, wherethe systemdoesnot makesense?If so, what
a re th e y?
t.' ' .!l +lr u.r i '- 'L'i
r
c. i, .' l) :
tlr *,' e
Y e t" T S
OX r1 1 d 'rr t,'\
V crIu e S r
*]f'e '
-F 'fr e' l' ' r ,' *6 $"
{ lt' [!' ;tr ' ' i3, *]tJ J' u"* - t} "r n''" 'S-
tE- /r :' *:'
f;' - '1:'fr ei"' r ii) lr :itr l"t' ' *wl ' *"n- ' o*k{
6. (12pts) LetVlal represent the voltage of the AV node in the Heart Model'
if e -o " V t S V "
if e-o"Vt > V"
f e -o " V + u
\ "-""V
1. Let e-a" : .6, L1,:10 a n d V : 2 0 .
,r
vt+r:
a) ff Vs - 27, will the hea,rt beat? Why or why not? CalculateV1C^ -'?tl
:
Y,t
V\
,). c :l {l
lG .
.Q (, r-] )'
\G"?
-
tl( )
*
qr'
/"\e.
h er^*-l h.t"l-\
so
r'l
c"
b) If y0 : 36,will the heartbeat?why or why not? calculatevr.
: g t " iL i
e.* * 'tl V *
So
f, go. {
-l \. e
s^,r l\
Yle'"1 i:r'c""[
v r - 2r ' C
2. Let a-ar : .25,u:
8 and V:
14'
a) If this system has an equilibrium, find it'
€ l-rr. r, l, rJ $ r\ -lh e
|hc
ft
=
v*
**1"*"''J
^7':"'/o* Y'
;l'iv"r:-E
it " ' . ' + ilh t '
Si' - *- o 'lO,G4V ()
Vx= IO.L
b) Graph this updating function below and cobweb to determine if
the heart is healthy or sick. Label yolrr axes'
efi'vu_= .!
{,14)
\/vtrl
f l"r, :
*l-re
tg
[.t5vt
(, ; 5v,
J L'
fs
ed,ur l'
)L
V 356
+ > ) to
-t
;-L
7. (13pts) Supposethe firnction b(t) : t*t2 represents the size of a bacterial
population in millions at time f hours.
a) Find the average rate of change of the function b(l) as a function of A/.
L + tL + (L n a L )' -& n t3 )
t ! |1: " P *r lJ i) =
At
=
( t .n L -r t - ^+ 2t r : L- ra L' - t - f )
*
:
f3
: A' L^ ;
At+? tle
l + ?L + a t
f;'h
b) Show that the instarrtaneousrate of changeof b(t) at time f : 0 is 1
million bacteriaper hour.
\i-
a f+o
b**
tt+,tt ) . - bli) :
1, , n1 l+ ) - t - + At
t-"2'-*
et
= \*? "b
At
t='J
" i i ' :e .j,,' ,,{ r ;- |
c) How small must Af be to get the average rate of change between t : 0
and At to be within 7% of the instantaneous rate of change?
i}
i* l' *
{'! " rc 4 ' -* } t , s -' " -L r ' ' lJ t ! ' 1 6 r: a { ' € '
t=C),
AU
**! .i d
-:tl
{
\ +aL
/- \*
$
* l.ol
.i' ' I
8. (12 pts) Evaluate the following limits, if they exist. Show all of your work
and justify your arlswers.
t ] + lJ r - ?
12+5r+'1
\ r.
a)nm---'
t-B
r-I
-:
-
'
-
b),,t*/(r)where
f (,): {tl:,\
li,-r
l?
' \J
ii; : 3
=Z
3 x''+.Z
X ..?c
:
c)Iy:GJi*cosr)
3j1"
ll
tt
d) rml
ll
r+O- fr
t\
fr"
1 '-**--{ i -'
t
I
I
-\"
\l
1l
'\l
t-
il
I
!l
,J
9
+ (o:if-f*
*'3$rr
-I