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' )  ` Inthelastfewyears,thingshavestartedtochangeforavarietyofreasons. z(r!+ Traditionaldisciplinelinesarebecomingincreasinglymeaninglessascurrent )"- researchanddevelopmentsoccurinfieldsthatoverlaptwoormoredisciplines.This >+6$/ isreflectedstronglyinthescientificandengineeringworkplace,whichisincreasingly  interdisciplinaryinnature.Thereformmovementinmathematicshas,amongother jb things,placedarenewedemphasisonrealisticapplicationstomotivatethe   mathematicaldevelopments.Finally,moderntechnologynowallowsus,trivially, . & todothingsinthemathematicsclassroomthatpreviouslywouldhaverequireda   roomfulofwhatnotsolongagowerestateoftheartcomputers.     ` Thekeytointroducingsuchapproachesintothemathematicscurriculumis TL  tofocusonmathematicalmodelingasaunifyingtheme.Thus,insteadofhanding   thestudentsequationstosolveorfunctionstodifferentiateorintegrate,wewould   beginbypresentinginterestingsituationsthatrequirethestudentsto: zr   .#'()*+,-(Dx3"#3"  #32D3  0 `   identifythemathematicalcomponent,#3݌ ` `  Ќ  "#3"  #32D3  0 `   createanappropriatemathematicalmodel(oftenafunctionorequation),#3݌>6` `  Ќ  "#3"  #32D3  0 `   raisepertinentquestionsinthecontextofthesituation,#3݌` `  Ќ  "#3"  #32D3  0 `   usethemathematicalmodeltoanswerthosequestions,#3݌` `  Ќ  "#3"  #32D3  0 `   interpretthesolutionstoseeiftheymakesense.#3݌d\` `  Ќ  Themathematicaltechniquesneededmayariseandbedevelopedinthecontextof  theproblem.Alternatively,thesemethodsareoftenonesthatthestudentslikelyhave (   seenpreviously,thoughoftenhavenotmastered(probablybecausetheysawlittle " valueinpracticingskillswithoutcontextormeaningfulapplications). $  ` Inthepresentarticle,weillustratehowonesuchtopic,modelingthelevelof N!F& adrugormedicationinthebloodstream,canbeintroducedandmodeledatavariety "( ofdifferentlevelsthroughoutthecurriculumfromdevelopmentalmathematics $ * throughcollegealgebraandprecalculus,andthenagainatthecalculuslevel.Atall t%l, levelsofthecurriculum,thisapplicationdemonstratesdramaticallytheusefulness &. ofthemathematicsthestudentsarelearning.Inaddition,itprovidesanexcellent 8(0!0 vehicleforintroducingsomeexceptionallynicemathematicalideasandmethods. )"2 ToparaphraseMary_Poppins_,aspoonfulofmedicinemakesthemathematicsgo *#4 down,inamostdelightfulway.   @v\XX,0ModelingDrugResponses#X,0Xv\#    XXXX,0L_dopa_ԀisadministeredtopeoplesufferingfromParkinsonsdiseasetorelieve j b symptomssuchasextremetremorsandrigidity.#X,0XXX5#ԀConsiderthefollowingsetofdata   onthelevelsofL_dopa_XXXX,0Ԁintheblood,in_nanograms_Ԁpermilliliter,asafunctionof RJ  time,inminutesafterthedrugisadministered.XX   *Laeddd Xdd Xdd X  ,sdd ,dd ,dd ,5dd ,5dd ,5dd ,5dd ,5dd ,dd ,dd ,dd ,dd ,dd ,dd ,dd , dd +  ):2 :2  )t ;1  # ;0 ZP%  #   4@204@Z20 f\1   4@20 4@  D@40D@f40 f\1   D@40 D@  N@60N@f60 f\1   N@60 N@  T@80T@f80 g]1   T@80 T@ Y@100Y@g100 h^2  Y@100 Y@ ^@120^@h120 h^2  ^@120 ^@ a@140a@h140 h^2  a@140 a@ d@160d@h160 h^2  d@160 d@ f@180f@h180 h^2  f@180 f@ i@200i@h200 h^2  i@200 i@ k@220k@h220 h^2  k@220 k@ n@240n@h240 h^2  n@240 n@ r@300r@h300 h^2  r@300 r@ v@360v@h360 C97  v@360  v@ CL 2(  # 20 RH%  #  r@300r@R300 `V2 ! r@300 r@ @2700@`2700 aW3 " @2700 @  @2950 @a2950 aW3 #  @2950  @ P@2600P@a2600 aW3 $ P@2600 P@ 8@15508@a1550 aW3 % 8@1550 8@ 0@11000@a1100 `V3 & 0@1100 0@  @900 @`900 _U2 '  @900  @ @725@_725 _U2 ( @725 @ @600@_600 _U2 ) @600 @ @510@_510 _U2 * @510 @ {@440{@_440 _U2 + {@440 {@ r@300r@_300 _U2 , r@300 r@ @o@250@o@_250 _U2 - @o@250 @o@  l@225 l@_225B86 .  l@225   l@ B `     h   #T #% T  #XbX%]#XXbTable1#XbX+#%XXb#T %+# T  p    VN . #XX,#AplotofthesepointsisshowninFigure1.#X,0XXXa,# X,0XXX,0(Wenotethatcomparablegraphsand 2 consequentlythecorrespondingdatavaluesformanyotherdrugscanbefoundatthe 4 _websites_Ԁforthemajorpharmaceuticalcompanies.) rj6  ` Thepatternshowninthisdrugresponsecurveistypicalofwhathappenswith 8 anymedication.Theunderlyingassumptionisthatthereisnomedicationinthe 6.: bloodstreamwhenthedrugisfirsttaken,thatthelevelrisesrapidlyasthedrugis < absorbedintotheblood,andthelevelthendecreasesasthedrugiswashedoutofthe > bloodbythekidneys. \ T@ uJK?/+p|0W"3 `..E"3 3  &K" L u    ` Wefirstdevelopmodels,inseveral !B differentways,forthedecayinthedrug  #D levelasamedicationiswashedoutofthe $zF blood.Wewillcomebacktodevelop %H modelsfortheoverallpatterninthedrug F'> J responsecurvelater.Wenotethatdifferent (!L drugsare"washedout"ofthebloodat  *#N differentrates.Infact,everydrugisratedbyl+d$P howlongittakesfor50%tobeeliminatedfromthebodyandthisisknownasits  biologicalhalflifeorsimplythehalflife.#X,0XX X,0$-# X,0XXX,0ԀForexample,aspirinisremovedquite jb rapidly;itslevelisreducedbyabout50%every29minutes.#X,0XX X,0v3# X,0XXX,0    ` Wefirstconstructamathematicalmodelforthelevelofaspirininthe . & bloodstreamonceithasbeenfullyabsorbedintotheblood.Basedonthehalflifeof   29minutes,inanyonehourperiod,approximately76.2%oftheaspirinintheblood    iseliminated,leavingabout23.8%.Iftheinitialdosageofaspirinis500mg,wecan TL  modelthelevelofaspirininthebloodwiththeexponentialdecayfunction   @ A(t)=500(0.238)t,   wheretmeasuresthenumberofhoursfromwhentheaspirinwasfullyabsorbedinto zr  theblood.Alternatively,if#X,0XX X,014# X,0XXX,0Ԁtmeasuresthenumberofminutesfromwhentheaspirin   wasfullyabsorbed#X,0XX X,0u7# X,0XXX,0,wecanmodeltheaspirinlevelbytheexponentialfunction >6 @A(t)=500(0.9764)t.  Thebaseordecayfactor,0.9764,indicatesthatalmost2.5%oftheaspirinis  eliminatedeveryminute. d\  ` Oncesuchanexponentialmodelhasbeencreated,therearetwokindsof  questionsthatarisenaturally: (   (a)Howmuchaspirinisleftafteranygivenlengthoftime? " (b)Howlongdoesittakeuntilthelevelofaspirinisdowntoagivenamount? $  ` Bothtypesofquestionsmakesensetostudentsandtheyseethevaluein N!F& developingthemodelasameansofansweringsuchquestions.And,becausethe "( questionsmakesense,theyaremuchmorelikelytoexpendthetimeandeffort $ * solvingthecorrespondingalgebraicequationsthantheyaretosolvealonglistof t%l, contextfreeequationsjustforthesakeofpractice. &.  ` Alternatively,supposewestartwithacollectionofdataonactualdruglevels 8(0!0 insteadofbeinggivenahalflife.Consideragainthetableofdatavaluesonthelevel )"2 ofL_dopa_Ԁintheblood.WenowfocusonthedatapointswhilethelevelofL_dopa_ *#4 intheblooddecays.Wethenhavethefollowingtablewherewerelabelthestarting  timetobet=0whenthemedicationisapparentlyatitspeaklevel.#X,0XX X,0!8# jb XXXX,0XX*3L6aededsdd sdd dd 5dd 55dd 55dd 55dd 55dd 5dd dd dd dd dd dd dd  dd L  ,,dd ,(dd ,5dd ,5dd ,5dd ,dd ,dd ,dd ,dd ,dd ,dd ,dd ,Sdd +  )   )#XX;?# XXt ;10 ( # ;0 ZP%0 ( #   4@204@Z20 f\10 (  4@20 4@  D@40D@f40 f\10 (  D@40 D@  N@60N@f60 f\10 (   N@60 N@  T@80T@f80 g]10 (   T@80 T@ Y@100Y@g100 h^20 (  Y@100 Y@ ^@120^@h120 h^20 (  ^@120 ^@ a@140a@h140 h^20 (  a@140 a@ d@160d@h160 h^20 ( d@160 d@ f@180f@h180 h^20 ( f@180 f@ n@240n@h240 h^20 ( n@240 n@ r@300r@h300 C970 ( r@300  r@ CL @6x p  @2950 @@2950 aW3x p  @2950  @ P@2600P@a2600 aW3x p P@2600 P@ 8@15508@a1550 aW3x p 8@1550 8@ 0@11000@a1100 `V3x p 0@1100 0@  @900 @`900 _U2x p  @900  @ @725@_725 _U2x p @725 @ @600@_600 _U2x p @600 @ @510@_510 _U2x p @510 @ {@440{@_440 _U2x p {@440 {@ r@300r@_300 _U2x p r@300 r@ @o@250@o@_250 _U2x p @o@250 @o@  l@225 l@_225#XX B#XXB86x p  l@225   l@ B `     h   #T ?#% T#XbX%jL#XXb Table2 #XbXlM#%XXb#T %*M# T   #XXM##X,0XXXM#sJVd;+'p|0E  `EW    T Y s     WeplotthedatainFigure2,whichsuggests    thatadecayingexponentialfunctionisa &" reasonablemodel.(Adecayingpower  $ functionwouldnotbereasonablebecause  & thereclearlyisnoverticalasymptoteatthe LD ( origin.)  * sJ\{;+'p|1* p `E 8  o k s ` Thecorrespondingexponential ,    regressionequationis rj. @$ $  L(t)=2147.9(0.9909)t, 0 wheretmeasurestimeinminutesfrom 6.2 absorption.Thismodeltellsusthatthelevel 4 ofL_dopa_Ԁintheblooddecreasesbyalmost 6 1%everyminute.Weshowthisfunction \T8 superimposedoverthedatainFigure3and : seethatitisareasonablygood,thoughnot  !< anoutstanding,fit. X,0XXX,0ԀInaddition,wenotethat#X,0XX X,0[S# "z>    thecorrelationcoefficientr=0.9545 X,0XXX,0,whichalsosuggeststhatthefitisgood, #@ especiallyastincreases. F%>B Ѐ @  v\X X,0ModelingtheDrugResponseCurve# X,0X v\T#   (!F Nowletsconsidertheproblemofmodelingtheentiredrugresponsecurverather )"H thanjusttheportionofitwherethelevelofmedicationisdecreasing.Conventional  +$J wisdom[3]suggeststhatsuchcurvesaremodeledbysurgefunctionsoftheform   #X,0XX X,0T#mXX,0u7j?/+p|0 7  `(# E 7 { 7 T  u@ W!m#XX!WV#S(t)=Atpekt󀀀#X,0XXXBW#Ԁor@@*XXXX,0S(t)=Atp_bt_ #X,0XXXW#, jb  TT  whereA,p,andkareconstants.Thegraphof    TUTUTT    TTTUTUsuchafunctionisshowninFigure4;wesee R J  TUTUTT R J  TTTUTUthatitclearlyexhibitsthebehaviorpattern    TUTUTT    TTTUTUdisplayedinthedataonL_dopa_ԀinFigure1.    TUTUTT    TTTUTUXXXX,0Wethusseektheequationofasurgefunction xp   TUTUTT xp   TTTUTUthatmodelsthisdata.    TTTT ` FromtheplotofalltheL_dopa_Ԁdatain H@   TUTUTT H@   TTTUTUFigure1,weseethatthesurgefunctionshould    TUTUTT      TUTUreachitsmaximumataboutt=60,wherethemaximumvalueisapproximately3000.  Wealsoseethatthesurgefunctionhastwopointsofinflection,oneoneithersideof x thepeak.FromthedatainTable1,noticethatthegreatestincreaseinSoccurs  betweent=20andt=40,soweestimatethatoneinflectionpointoccursataboutt PH =30.ThegreatestdecreaseinLoccursbetweent=80andt=100,soweestimate  thattheotherinflectionpointoccursataboutt=90.    ` Thegeneralequationofasurgefunction,S(t)=Atpekt,involvesthree   parameters,A,p,andk,whosevalueswehavetodetermine.Wedifferentiatethis " functionwithrespecttottoobtain X P$  `     h S(t)=A[ptp1ekt󀄀ktpekt] !&  `     h   =A[p󀄀kt]tp1ekt.0  0   0x   0@x  0@xh h (1)(# (x x  Wesetthisexpressionto0andsolve.Sincetheexponentialfunctionisnever0and $* tp1iszeroonlywhent=0,themaximumoccurswhen %, @p󀄀kt=0ort=p/k.   `'X . Fromourpreviousexaminationofthedataandthecorrespondingplot,wedecided (!0 thatthisoccursatabout 0*(#2  `     h t=p/k=60orp=60k     x  +$4  ` 0 (2)   ` Wenextfindthepointsofinflection.WhenwedifferentiateEquation(1)for ph S(t)above,weget    `  򀀀S(t)=A +XX{#XX+ gf#[p(p1)tp2󀄀kptp1]ekt +XXԀ#XX+ g#Ԅk[ptp1󀄀ktp]ekt +XX}#XX+ g# @ 8  `    =A +XX{#XX+ h#[p(p1)tp2󀄀kptp1]k[ptp1󀄀ktp] +XX}#XX+ h#ekt +XXԀ#XX+ 4i#.   Aftersomealgebraicsimplification,thisreducesto    `  S(t)=A +XX{#XX+  j#p(p1)2kpt+k2t2 +XX}#XX+ j#tp2 +XXԀ#XX+ j#ekt +XXԀ#XX+ Fk#. h`  Theinflectionpointsoccurwhentheterminsidethebracketsiszero: H@  @p(p1)2kpt+k2t2=0.   Thisisaquadraticequationintwhoserootsare  eSY/` p  `E Z dAXe                    Thistellsusthatthetwoinflectionpointsarelocatedatequaldistancesof(Np)/kon  eithersideofthemaximumpointatt=p/k.Sincewedecidedthatthepointsof   inflectionoccuratapproximatelyt=30andt=90,wemusthave   `     h Np/k=30orNp=30k.    0 x (3) x x  UsingEquation(2),wefindthat X P"  `     h   p=900k2= p 60k !$ sothat (# & @kkk=1/15andp=60k=4. $(  ` Finally,toestimatethevalueforA,weusetheexpressionforthesurge %* function `'X , @S(t)=Atpekt󀀀=At4et/15, (!. andthefactthatS(60)=3000toget 0*(#0 @S(60)=A604e60/15=3000, +$2 sothat  @A=3000e60/15/604 0.0126. ph s[;+'p|1K Z `E# " Y r 9 s 99  Therefore,ourmodelforthesurgefunctionis    9999@  S(t)=0.0126t4et/15. @ 8  9999#X,0XXX[#XXXX,0Thegraphofthisfunctionsuperimposedover    9:9:99    999:9:theoriginaldatasetforthelevelofL_dopa_Ԁin    9:9:99    999:9:thebloodisshowninFigure5.Wenotethat xp   9:9:99 xp   999:9:itisareasonablygoodfitfortbetween0and    9:9:99    999:9:about120minutes,althoughthereafterthe H@   9:9:99 H@   999:9:functiondecreasesmuchmorerapidlythanthe    9:9:99    999:9:levelofthedrugdoes.#X,0XXXt#   9999 ` Wecanimproveonthefitsomewhat x  9:9:99 x    9:9:byapplyingtheleastsquarescriteriondirectlytominimizethesumofthesquaresof  thedeviationsbetweenthedatavaluesandthesurgefunctionratherthandepending D< oninspectiontoestimatethelocationofthemaximumandtheinflectionpoints.In  particular,thesumofthesquarescorrespondingtothefunctionshowninFigure5  is1,716,480;thesumofthesquaresisreducedbyabout10%to1,555,393usingthe jb  similarfunction " @ooXXXX,0S(t)=0.012764t3.962e0.0637t#X,0XXX`{#XXXX,0.#X,0XXX{#Ԉ . &$ Nevertheless,thisisstillaratherpoorfittothedata,especiallybeyondaboutt=140 !& minutes. "(  @v\XX,0UsingaRationalFunction#X,0Xv\|#  %,  X,0XXX,0Analternativeapproachtocreatingafunctiontofitthistypeofdataisdiscussedby Z'R . Brownand_Timchek_Ԁ[1],whosuggestusingarationalfunction.Atfirstthought,this (!0 likelyseemsunreasonable,sincetherearenoverticalasymptotes.However,further *#2 thoughtmightsuggestthatarationalfunctionhavingnolinearfactorsinthe +x$4 denominatorcouldhavethisproperty.Thefactthatthecurvepassesthroughthe  originmightthensuggestarationalfunctionoftheform jb @dd*k 7'#`^ `.lE  k߈   whichdiesoutasx.However,suchafunctiondoesnothavetheappropriate !  behaviorneartheorigin;weneedacurvewhoseslopeat0is0andwhichisconcave  { upforx>0,butthesecondderivativeofthisrationalfunctionisnegativefor0<x   <b.Afterabitofexperimentation,itturnsoutthatarationalfunctionhavingthe G?  desiredbehaviorpatternis   @dd*k$7'#`^H `.E  k#X,0XX X,0r}# X,0XXX,0Ԉ    yJC 3/p~1r  `..E8t % ) k yBrownand_Timchek_Ԁindicatethat,usingthecurvefittingroutinesinMaple,theleast     squarescriterionappliedtosuchafunctionleads  tothespecificfunction `X @ zk07 '#`^Hd] `. Ed]Qdd] k#X,0XX X,0,# X,0XXX,0Ԉ  Weshowthegraphofthisfunctionsuper LD imposedovertheL_dopa_ԀdatainFigure6,where  weobservethatitisactuallyquiteaccurate.The   correspondingvalueforthesumofthesquaresis r!j     #X,0XX X,0#620517.5,whichisasignificantimprovementoverthevaluesobtainedusingthe "" surgefunction. X,0XXX,0 6$.$  #X,0XX X,0#@!v\XX,0RepeatedDrugDoses#X,0Xv\#  &( Intheabovedevelopments,weconsideredthesituationwhereasingledoseofa (!* medicationistaken.Inmanycases,peopletakeadoseofamedicationonarepeated, )", oftendaily,basis.Itmightbeinsulinfordiabetes,anyofavarietyofmedications \+T$. tolowerhighbloodpressure,ordrugstoreducecholesterol.Ineachinstance,once  themedicationhasbeenabsorbedintotheblood,itisfilteredoutbythekidneysand jb excretedfromthebody.However,beforeitistotallyeliminated,thenextdosecomes   along.Theproblemthenis:Howdowemodelthelevelofthemedicationinthe . & bloodasafunctionoftime? X,0XXX,0   #X,0XX X,0# ` Tostudythistypeofsituation,supposeweconsiderProzac,whichis    administeredtoindividualstocountertheeffectsofextremedepression.Atypical TL  doseofProzacis40mgandapproximately25%oftheProzacinthebloodstreamis   eliminatedevery24hours.IfjustasingledosageofProzacweretaken,thelevelin   thebloodstreamcouldbemodeledbytheexponentialdecayfunction zr  @NN! X,0XXX,0@@NN;D(t)=40(0.75)t,   wheretmeasuresthenumberof24hourperiodssincetheProzacwasfirsttaken.#X,0XX X,0# >6  X,0XXX,0 ` Morerealistically,#X,0XX X,0# X,0XXX,0Ԁsupposethatapersontakes40mgofProzaceachday.  Afterthefirst24hourtimeperiod,25%oftheProzac,or10mg,iseliminated,  leaving30mgandthenextdaysdoseadds40mgtothat.Therefore,afterthefirst d\ 24hours,theamountofProzacinthebloodis  @kk D1=30+40=70mg. (   #X,0XX X,0D# X,0XXX,0 ` Duringthesecond24hourperiod,thekidneyseliminate25%oftheProzac " presentandthe#X,0XX X,02# X,0XXX,0Ԁpat#X,0XX X,0# X,0XXX,0ienttakesthenextdaysdoseof40mg.Thus,aftertwo24hour $ periods,theamountofProzacinthebloodis N!F& @D2=0.75(70)+40=92.5mg. "( Afterthreedays,thelevelofProzacis $ * @NND3=0.75(92.5)+40=109.375mg, t%l, andsoon,indefinitely.ThecorrespondingsequenceofProzaclevelsistherefore &. @ {40,70,92.5,109.375,122.031,131.523,...}. 8(0!0 Noticehowthelevelsofthedrugkeeprising,butinaconcavedownmanner. )"2  ` Letsnowlookataslightlymoresophisticatedwaytodescribethisprocess. *#4 TheinitialdosageisD0=40mg.Duringthefirst24hourtimeperiod,25%ofthis  amountisremovedfromthebloodandthepatientthentakesthenextdoseof40mg. jb Thus    ` @D1=0.75D0+40. . & Similarly,duringthesecondday,25%oftheProzaciseliminatedandtheperson   takesanother40mgdose,sothat    @ D2=0.75D1+40 TL  and,again,afterthreedays,   @ D3=0.75D2+40.   Ingeneral,attheendofn+1days,foranyvalueofn, zr  @||Dn+1=0.75Dn+40.   Thisequation,whichshowstherelationshipbetweenthelevelofProzaconanytwo >6 successivedays,#X,0XX X,03# X,0XXX,0Ԁiscalledadifferenceequationorrecurrencerelation.   ` SincetheinitialdoseisD0=40mg,ifwesetn=0,1,2,...inthisdifference  equation,weobtain,respectively, d\ Ifn=0:@ D1=0.75D0+40@@D=0.75(40)+40=70mg.  Ifn=1:@e e D2=0.75D1+40@@D#X,0XX X,0# X,0XXX,0=0.75(70)+40#X,0XX X,0ٛ# X,0XXX,0Ԁ=92.5mg. (   Ifn=2:@ D3=0.75D2+40@@D#X,0XX X,0+# X,0XXX,0=0.75(92.5)+40#X,0XX X,0# X,0XXX,0Ԁ=109.375mg " andsoforth,sothatwegeneratethesamesequence $ @? ? {40,70,92.5,109.375,122.031,131.523,...#X,0XX X,0B# X,0XXX,0Ԁ} N!F& uJw? /+p|0!$ 1 `..E$  1I# u   asabove.Thissequenceisthesolutiontothe "( differenceequation. $ *  ` Noticethateachsuccessivetermhas t%l, grownbysomewhatlessthanthetermbefore. &. ThecurvedrawnthroughthepointsinFigure7 8(0!0 isconcavedownandthesuccessivevalues )"2 seemtobelevelingoff.Ifyoucontinuethe*#4 aboveprocessnumerically,youwillseethatthelimitingamountLappearstobevery  closeto160mg. jb  ` Thisrecursivepatterncanbegeneratedinatrivialmanneronmostgraphing   calculators(orwithaspreadsheet).Withacalculator,startbykeyingintheinitial . & value,say40,andpressingENTER.Thenentertheexpression   @!.75*2ndANS+40    and,eachtimeyoupressENTERthereafter,thefollowingterminthesequenceis TL  displayed.    ` Intermsoftheoriginalsituation,thislimitLrepresentsthemaximumlevel   ofProzacthatwilleverbereachedintheblood.Itisknownasthemaintenancelevel zr  forthedrug.Furthermore,oncethatlevelofProzachasbeenreached,theamount   inthebloodwillremainconstantatthatlevelevery24hours,solongasthesame >6 dosageistakenrepeatedly.   ` Inpractice,medicalresearchersdeterminethataspecificlevelLofa  medicationismosteffective,consideringfactorssuchassafetyandeffectiveness. d\ Aninitialdoseof40mgoftheProzacmeansthatforsomeperiodoftime,the  amountinthebloodstreamisbelowtheoptimallevel.Becauseofthis,doctorsoften (   prescribeaninitialdosethatisabovethenormalrepeateddosesothatthedruglevel " approachesthemaintenancelevelLmorerapidly.Forexample,aninitialdoseof $ 120mgofProzacfollowedbydailydosesof40mgwillachievethemaintenance N!F& levelmuchmorerapidly.However,thisstrategymustbebalancedwiththesafety "( issueinvolvedintakingsuchalargedosage,especiallyasthefirstdoseofthedrug. $ * #X,0XX X,0=# X,0XXX,0 ` Whathappensifapatienttakesanoverdose,sothatthelevelofdruginthe t%l, systemexceedsthemaintenancelevel?Forinstance,supposethatapersontakes400 &. mgofProzacinitiallyandthereaftertakestheusual40mgdailydose.Weusethe 8(0!0  samedifferenceequation )"2 yJyC 3/p~0ZFg  `..EF Fg  | y@||Dn+1=0.75Dn+40,   butnowwithaninitialdoseD0=400,sothat jb ifn=0:D1=0.75D0+40=340mg   ifn=1:D2=0.75D1+40=295mg . & ifn=2:D3=0.75D2+40=261.25mg   andsoon.     ` Thelevelsofthedruginthebloodstreamarenowgivenbythesolution TL  sequence   @UU{400,340,295,261.25,235,937,216.953,202.715,192.036,184.027,   ...}. zr  WeplotthesepointsinFigure8andobservethattheyfallintoadecreasing,concave   uppatternthatapparentlyconvergestothesamelimitingvalueL=160mg,butthey >6 approachitfromaboveratherthanfrombelow.Consequently,weseethatifthedrug     levelrisestoohigh,therearesome  counteractingeffectsthatreducethe d\ level,assumingtheoverdosedoesnot  causeanyotherproblem.#X,0XX X,0# X,0XXX,0 (    ` Noticethatbychangingthe " initialconditionD0inthedifference $ equation,weobtainadifferent N!F& solutionsequence.Infact,forevery "( possibleinitialvalueD0,therewillbe $ *    adifferentsolution. t%l,  @  v\X X,0DeterminingtheMaintenanceLevelL# X,0X v\E#  8(0!0 WecandeterminethelimitingvalueLforProzacpreciselyusingthefollowing )"2 argument.Supposethatforsomevalueofn,DnreachesthelimitL,sothatall 8+0$4 _successivelevelsofProzacarethesame.Thusfornlargeenough,weassumethat  Dn+1=LandDn=L.Substitutingthesevaluesintothedifferenceequation jb @||Dn+1=0.75Dn+40,   #X,0XX X,0{# X,0XXX,0fortheProzacdrugmodel,#X,0XX X,0,# X,0XXX,0weobtain . & @ L󀀀=0.75L+40   sothat    @$0.25L=40 TL  andhencethelimitingvalueLis   @L=40/0.25=160mg.    ` Moregenerally,thismodelcanbeusedforanymedication.Supposethat,for zr  acertaindrug,afractionaiseliminatedfromthebloodstreamduringagiventime   period,sothatthefractionremaininginthebloodafterthatlengthoftimeisb=1 >6 a.IftherepeateddosageisC,thenthecorrespondingdifferenceequationis  @ee"Dn+1=bDn+C#X,0XX X,0# X,0XXX,0Ԉ  andtheassociatedmaintenancelevelforthemedicationis d\ @ L=C/(1b)=C/a.  #X,0XX X,0#   v\XX,05v\v\FindingaFormulafortheSolution#v\5v\@##X,0Xv\!#  X,0XXX,0 $ Wenowattempttofindaclosedformexpressionforthesolutionsequenceofthe !& differenceequation "( @ Dn+1=0.75Dn+40 N$F* forProzacandthemoregeneraldifferenceequationmodel %, @!Dn+1=bDn+C#X,0XX X,0ʺ# X,0XXX,0. ' . Althoughtherearesomepowerfultechniquesforfindingformulasforthesolution t(l!0 sequencestosuchdifferenceequations#X,0XX X,0# X,0XXX,0Ԁ(theymirrorthemethodsusedtofind )"2 solutionsofdifferentialequations),wewillnotgointothemhere.Theinterested 8+0$4 readercanfindthesemethodsineither[1]or[2].Instead,wewillconstructsucha  solutionusingsomefundamentalideasaboutfunctions. jb  ` Letslookagainatthetermswecalculatedaboveforthesolutionsequence   totheProzacdifferenceequationmodelbasedonaninitialvalueofD0=40mg: . & @& & @@ {40,70,92.5,109.375,122.03125,131.523,...#X,0XX X,0# X,0XXX,0Ԁ}#X,0XX X,0n# X,0XXX,0.   #X,0XX X,0# X,0XXX,0Wecanthinkofthesevaluesasasetofdatavalues:    *ILrd ed,dd ,(dd (5dd 55dd 55dd 5dd dd dd dd dd dd dd Sdd S3L6  I,dd ,dd", dd",Xdd",dd",dd",dd"+  $TL TL $n 8.  # 80 G= #   ?1?G1 RH'  ?1 ?  @2@R2 RH'  @2 @  @3@R3 RH'  @3 @  @4@R4 RH'  @4 @  @5@R5 <)'  @5 @ <Dn D:80   D@40D@D40 TJ(80   D@40 D@  Q@70Q@T70 VL(80   Q@70 Q@  W@92.5 W@V92.5 [Q*80   W@92.5  W@ X[@109.375X[@[109.375 ^T-80  X[@109.375 X[@ Dl^@122.031Dl^@^#X,0XX X,0 # X,0XXX,0122.031#X,0XX X,0# X,0XXX,0 ^T-80  Dl^@122.031 Dl^@ ~jp`@131.523~jp`@^#X,0XX X,0# X,0XXX,0131.523#X,0XX X,0# X,0XXX,0=3180  ~jp`@131.523  ~jp`@ =(Wecouldusemorevalues,butthesewillsuffice.)SincethevaluesforDnapproach   alimitingvalueofL=160mgasahorizontalasymptote,wecanshifteach#X,0XX X,0# X,0XXX,0Dn#X,0XX X,0# X,0XXX,0value ,$  toobtainthecorrespondingvaluesof#X,0XX X,0k# X,0XXX,0160Dn#X,0XX X,0# X,0XXX,0,asshowninthefollowingtableand  inFigure9.   #X,0XX X,0C# X,0XXX,0*/L~drddd dd" dd" Xdd"Xdd"dd"dd"L  ,9dd , dd", dd",Xdd",dd",dd",dd"+  (RJ"RJ @( &h8%X X,0n 8.# # 80 G=$ #   ?1?G1 RH'%  ?1 ?  @2@R2 RH'&  @2 @  @3@R3 RH''  @3 @  @4@R4 RH'(  @4 @  @5@R5 F3-)  @5 @0 @ @FDn D:*  D@40D@D40 TJ(+  D@40 D@  Q@70Q@T70 VL(,  Q@70 Q@  W@92.5 W@V92.5 [Q*-  W@92.5  W@ X[@109.375X[@[109.375 ^T-. X[@109.375 X[@ Dl^@122.031Dl^@^#&h8%8% &h# &h8%8%&h122.031#&h8%8% &h# &h8%8%&h ^T-/ Dl^@122.031 Dl^@ ~jp`@131.523~jp`@^#&h8%8% &h.# &h8%8%&h131.523#&h8%8% &h# &h8%8%&h L930 ~jp`@131.523 @0 ~jp`@ @L# X,0X8% &h# &h8%X X,0160Dn#&h8%8% &h# &h8%8%&h E;"1 ^@120^@E120 UK)"2 ^@120 ^@  V@90V@U90 VL("3  V@90 V@ P@67.5P@V67.5 ZP*"4 P@67.5 P@ PI@50.625PI@Z50.625 \R,"5 PI@50.625 PI@ y&1B@37.969y&1B@\37.969 \R,"6 y&1B@37.969 y&1B@ '1z<@28.477'1z<@\28.477B86"7 '1z<@28.477 @0  '1z<@ B# X,0X8% &h##X,0XX X,0# X,0XXX,0yJzC 3/p~0*!3 (  `..E!3 3 ( : &|! ySinceDnapproaches160asnincreases,t#X,0XX X,0# X,0XXX,0hevaluesof#X,0XX X,0# X,0XXX,0160Dn#X,0XX X,0# X,0XXX,0Ԁapproach0asngets 9 larger.Itthereforemakessensetofitadecayingexponentialfunctionto160Dnas ZR;  ||  afunctionofn.(Wewouldnotusea  = decayingpowerfunctionbecausethedata "? startswithn=0.)Theexponentialfunction #xA thatbestfitsthisdatausinganyofthe $C standardcalculatorsorExcelis D&<E @ 160Dn=119.99961(0.7500021)n, ' G withacorrelationcoefficientofr=1, )"I whichsuggestsavirtuallyperfectfit.Ifwej*b#K|| solveforthelevelofProzac,Dn,weobtain  @ Dn=160119.99961(0.7500021)n. jb Thenumbersinthisexpressionsuggestthatthe correctformulaforthesolution   mightbe . & @MMDn=160120(0.75)n.0  0   0x   0@xx(#x(#(4) x x  Wewillcomebacktothisproblemlatertoshowthattheformulatrulyholdsfor    everypossiblevalueofn,notjustforthefewparticularvaluesofnweusedin TL  constructingthebestfitexponentialfunction.#X,0XX X,0Z# X,0XXX,0   #X,0XX X,0# X,0XXX,0 ` Wenowextendtheaboveformulatosolvethecomparabledifference   equationforanymedicationwithanyfixedperiodicdose.#X,0XX X,0# X,0XXX,0ԀThecorresponding zr  differenceequationis   @LL"Dn+1=bDn+C. >6 Foranyvalueofbbetween0and1andanypositivevalueforC,thesuccessiveterms  inthesolutionsequenceforDnwillhavethecomparablebehaviortothatshownin  Figure7:thesolutionwillbeanincreasing,concavedownfunctionthatapproaches d\ ahorizontalasymptote.#X,0XX X,0# X,0XXX,0Ifyourcalculatordisplaysgraphsofsolutionsofdifference  equations,s#X,0XX X,0# X,0XXX,0electsometypicalvaluesandcheckoutthebehaviorofthesolutions. (    ` Wefoundabovethatthemaintenancelevelisgivenby " @ L=C/a#X,0XX X,0# X,0XXX,0Ԁ=C/(1b). $ Ifyouexamine#X,0XX X,0# X,0XXX,0Ԁthesolution(4)wecreatedforthelevelofProzacintheblood,itis N!F& apparentthatamoregeneralformulaforthesolutionwouldlikelybe "( @BB@@--Dn=L(L󀀄D0)bn $ * T%t%lforanyvalueofn,withanyparametersa,C,L=C/(1򀄀b),andD0.#X,0XX X,0S# X,0XXX,0Toverifythat %, thisexpressionfor#X,0XX X,0U# X,0XXX,0Dn#X,0XX X,0# X,0XXX,0Ԁisactuallyaformulaforthesolution,wemustshowthatit *'" . satisfiesthedifferenceequation (!0 @ee"Dn+1=bDn+C0 p 0p p 0   0x   0@xx(#x(#(5))"2x x  #X,0XX X,0# X,0XXX,0foreveryvalueofn.(Thisisessentiallyanapplicationofmathematicalinduction.) P+H$4 Wesubstitute#X,0XX X,0# X,0XXX,0boththeaboveassumedexpressionforDn#X,0XX X,0u# X,0XXX,0=L(L󀀄D0)bn#X,0XX X,0# X,0XXX,0Ԁandthe  correspondingexpressionforDn+1whennisreplacedbyn+1#X,0XX X,0n# X,0XXX,0intothedifference jb equation#X,0XX X,0## X,0XXX,0.Thelefthandsideofthedifferenceequation(5)canbeexpressedas#X,0XX X,0# X,0XXX,0   @Dn+1=L(L󀀄D0)bn+1. . & Therighthandsideis   @bDn+C=_bL_Ԁb(LD0)bn+C     `     h =_bL_Ԁ(LD0)bn+1+C. TL  However,   @99L=C/(1b)sothatC=L(1b).   Therefore,therighthandsidebecomes zr  @PPbDn+C=_bL_Ԁ(LD0)bn+1+C    `     h =_bL_Ԁ(LD0)bn+1+L(1b) >6  `     h =L(L󀀄D0)bn+1,  whichisidenticaltotheassumedexpressionforDn+1#X,0XX X,0+# X,0XXX,0Ԁonthelefthandside.Thus,the  expression d\ @))Dn=L(L󀀄D0)bn  istrulyaformulaforthesolutionsequenceDnanditholdsforallpossiblevaluesof (   n.#X,0XX X,0e# X,0XXX,0Ԁ#X,0XX X,0# X,0XXX,0Ԁ#X,0XX X,08#Ԁ "  ` Thissolutionappliestoanydifferenceequationoftheform X,0XXX,0 $ @kk!xn+1=bxn+C#X,0XX X,0# X,0XXX,0, N!F& notjus#X,0XX X,0z# X,0XXX,0tthedrugmodeldiscussedhere.Inparticular,anysituationthatcanbe "( representedinadecayreplenishorgrowthdiminishmodelcanbetreatedinthe $ * identicalfashion.Asseveralexamples,considerremovingafixedamountofmoney t%l, fromaretirementfundthatgrowsatagivenrateorharvestingafixednumberof &. animalsfromaherdthatisgrowingatagivenrateorcontributingafixedamount, 8(0!0 say$2000,annuallytoanIRAaccountgrowingatagivenrate. )"2 #X,0XX X,0# *#4  @%v\XX,0References#X,0Xv\#   1.GordonSheldonP.,FlorenceS.Gordon,etal,FunctioningintheRealWorld:A  _PreCalculus_ԀExperience,AddisonWesley,1997,Reading,MA.   2._Mickins_,Ronald,DifferenceEquations:TheoryandApplications,Van_Nostrand_ j b Reinhold,1990,NewYork.   3.Hughes_Hallett_,Deborah,Andrew_Gleason_,etal,AppliedCalculusforBusiness, .&  SocialScience,andLifeScienceStudents,JohnWiley&Sons,NewYork,1998.   4.Brown,DouglasandThomas_Timchek_,_Levodopa_ԀtotheBrain:ARateChange   Study(unpublishedmanuscript). TL    X,0XXX,0 v\X X,0  Acknowledgment# X,0X v\#  zr TheworkdescribedinthisarticlewassupportedbytheDivisionofUndergraduate  EducationoftheNationalScienceFoundationundergrant#DUE9555401forthe zr LongIslandConsortiumforInterconnectedLearning.However,theviewsexpressed  arenotnecessarilythoseofeithertheFoundationortheproject.#X,0XX X,0#