Math 155 Exam L Fall2OO7 NAME: l./ 11e rt I SECTION: TIME: INSTRUCTOR: fnstructions: The exarn is closed book and closed notes. You may use an 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 I T2 2 T2 3 12 4 15 5 12 6 12 7 13 8 12 TotaJ 100 Score
1. (12 pts) a) An ice cube tray contains 12 cubical spaces for ice, each of which is 1.25 inches per 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?, Volu.',* gi.r cutbe is l.?5- i,'t l? t l.?;,,r']' i \r\ i-: f -! *,0 i * (., *0,;f j = 36'{ ' 81? b) write the equation of the line passing through the points (1,3) and (3,7) in slope-intercept form. 11)=1*: J ="'L 3"-t 7 a.-3-2(x-t) t -i'l'-2 B= 2x+ \ c) Where is the function tr(t) continuous? Justify your answer. O^\.V prlt":'tl\e L(t): \ i,'.,-.., t-' = I t-+tli.n {.?l-l )'i-l L+r* ( t' if t<l 1zt-1 if 1<t<b I r*3 if 5< t y::r'l ": * t s:. \ i: t ofi 4,,i ul. i{ t; fl.-..'r-e- -L*l ; L*'5,, I lio L r's {- &,;{ {XJ I ll,n Lttl'\irv\ ;;t"-[ x lil:*l *dl L-es- t4 5-1,,o'1 Lt fu, I * llr'v\ 'L" u ; flegu L.45+ 2 So L rs cl;sli)"4'r\rtr')us L?" d &i t=5' ie ccin L, c)c\ { ^ "'*., 'i } L' { tir"*l'; '
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. bt+r:2.0bt - 2'0 farting from bo:2'2 I'it{.1 r'-ril;':"1' ri' I b) For what positive initial values of b; is the solution of the discrete-time dynarnical system increasing? \4trhat is the long-term behavior of the system for these values? i*er ilrr." Lu'=* -'ilt* "li:-;!""'u'f i,.;:':'- rr" r i:ii'i '.r.1 p * 1", r.l L t * 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 0 h* *'"] "{,,r'1 * "/- l-, * "' ''''j " c1,q1'l' t3 L,j t'-3, r\ "3
3. (L2 pts) Let f (r) : 3r2 + 2r - 5 a) Find the equation of the seeant line to / that passes through (4, /(a)) and (4+An,f(4+4")). Jtql. 5 rr1 : t_!1 :3.)*31 = 3(.1 +Ax)"*?("1 +ax)*s-fl t{ tax - 4 L/, = 1!g 1 'tax +axt ) t..8 r?ax.:-:s /}X = k.t"lf"g"d : 2("t 3,sx AX b) Find the slope of the secant line when 1.Ar:1 26t3 =21 2. Ar: o.l 2G + O.3 a 46.3 3. Ar:0.01 2U * c} "03 ' ]G. OT c)findf(4). \rm s(q-t.:).:jlj) E lit?&+bnx *2(, Ar+o *-tl;;-:t*- ax-"$ d) What is the relationship between the answers to (b) and (c)? The ansk) r lo (c) is *he li,tr Ax+g ES $he ftn$q"a1*r,ii {* (b ). 4
4. (15 pts) Suppose f (t) : 760e-t' represents the amount of a radioactive substa,nce present in a lake in parts per million (ppm), and the lake is considered sa,fe to swim in when the amount of the substance in the lake is less than 10-3 ppm, Here t ) 0 represents time in weeks. a) Find the half-life of this radioactivtrsubstance. DCI = \LoO e- " L,-VO = j"n,go *tt -.Gl?l = -L* t -{fe[f* = * "&3?s i^-*e,shs b) After how marry weeks will the lake be sa,fe to swim in? lo-3 z lto *-L^ U- to-t - t* lto = -t* t- 3. tl G I L, r,..r e,ls l c) Plot the function /(t) on a semilog plot and label your axes. j.nt ( {ttt) 5,l 2 J z I frri, lca e'l tn, tti-t * \"'f e$ - t" uj * 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) * ul 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 ambient air, which has oxygen concenteration of 2L%. lilhat is the discrete time dynamical system for this model if the oxygen concenteration of the lnng is reduced by 3% at each breath? 1. = W = il:"]-5 -: dl".;:;-? ^5 U V -*. {.) 16 : 2\ ''-lo-- "2-"1 S a 3"-l* -'"et c1*r: ( \-.3TS)( cr -"i]1i *i'1"15){":;if J Cgrr' ' GiSce + o'clg b) what is the equilibrium concentration for the system in part (a)? C# ' ^U15 cn + "0G " 31 5 L* I. og Cx 2 "ll a{ 16% c) Are there values of q, where the system does not make sense? If so, what arethey? Yet" TS c. i,.'l): tlr*,'e r t.''.!l +lru.ri'-'l'i OXr11d'rr t,'\ -F'fre' l''r,'*6 $" {lt'[!';tr''i3, *]tj J'u"* -t}"rn''"'s- VcrIueSr *]f'e' f;'-1 ':' frei"'r ii)lr:itrl" t''*wl '*"n-'o*k{ te-/r:'*: '
6. (12pts) LetVlal represent the voltage of the AV node in the Heart Model',r f e-o"v+u if e-o"vtsv" vt+r: \ "-""V if e-o"vt > V" 1. Let e-a" :.6, L1,:10 and V:20. a) ff Vs - 27, will the hea,rt beat? ^-'?tl :.Q(,r-])' C Y,t V\ - \G"? tl() Why or why not? Calculate V1- lg.,). c :l{l so h er^*-l h.t"l-\ * qr' r'l /"\e. c" b) If y0 : 36, will the heart beat? why or why not? calculate vr. e.**'tlv* : gt"ili So -l\. e f, go. { s^, r l\ Yle'"1 i:r'c""[ vr - 2r'C 2. Let a-ar :.25, u: 8 and V: 14' a) If this system has an equilibrium, find it' ft hc l-rr.r,l, rj $r\ -lhe v* = ^7':"'/o* Y' **1"*"''J it"'.'+ilht' ;l'iv"r:-e Vx=IO.L Si'-*-o' lo,g4v() b) Graph this updating function below and cobweb to determine if f l"r, : f s the heart is healthy or sick. Label yolrr axes' *l-re ed,ur l' efi'vu_ =.! {,14) JL' \/ vtrl [.t5vt (,;5v, tg V 356 )L -t + >)to ;-L
7. (13pts) Suppose the 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/. t! 1:"P*rlJi)= At L+tL + (LnaL)' -&nt3) = * ( t.n L-r t-^+2tr:l -r a L' - t- f ) : At+?tle f3 : A'L^ ; l +?L +a t f;'h b) Show that the instarrtaneous rate of change of b(t) at time f : 0 is 1 million bacteria per hour. \i- btt+,tt).-bli): ** 1,,n1 l+)-t-+at af+o et t-"2'-* = \*?"b At t='j "ii':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? AU t=c), {'!"rc 4'-*}t, s-'"-lr ''ljt!'16r:a{' ' i} i*l'* **!.i d \ +al * l.ol -:tl { /-\* $.i''i
8. (12 pts) Evaluate the following limits, if they exist. Show all of your work and justify your arlswers. \ r. 12+5r+'1 a)nm---- t] +ljr-? ' t-b r-i -: - - ' ' \J l? b),,t*/(r) where f (,) : {tl:,\ ii; : 3 li,-r 3x''+.Z X..?c =Z c)iy:gji*cosr) : 3j1" + (o:if-f* *'3$rr -I d) rml r+o- fr t- '\l!l,j ll tt ll t\ fr" 1 '-**--- t { i -' I -\" I \l 1l il I 9