1.Considertheexponentialfunctiony=2x.Fillin the table ofvalues forthefunction, andfindtherateofchangebetweenconsecutivevaluesinthefunction(∆y). Whatpattern doyouseefor∆ychange?

x / y / ∆y
0
1
2
3
4
5

2.InStudent ActivitySheet3,youlearnedaboutthesometimesfatal antibiotic-resistantstaphbacteriamethicillin-resistantStaphylococcusaureus(MRSA)growinginanagardish.Theinitialareaoccupiedbythebacteriaintheagardishis2square millimeters,andtheyincreaseinarea by20%eachweek. Thetablebelowgives theareaofthe bacteriaoverseveralweeks.Usethetabletodescribetherateofgrowthofthearea ofthebacteria(∆a).

x
(no.ofweeks) / a
(areainmm2) / ∆a
0 / 2
1 / 2.4
2 / 2.88
3 / 3.456
4 / 4.147
5 / 4.977

3.Supposeaquantityincreasesatarateproportionaltothequantity,andtheconstantofproportionalityis0.2. Theinitial quantityis2. Writeadifferenceequationthat describesthe statementabove, and find severalvalues of thisquantity.

x / ∆y / y
0 / 2
1
2
3
4

4.Usespreadsheetsoftwaretogenerateabout 75values of thetable you started inQuestion3.The spreadsheetallows youtousearecursiveruletogeneratethedata.

5.TheagardishthattheMRSAbacteriaaregrowinginhasanareaof1,000square millimeters.Thegrowthofthebacteriaislimitedin the labbythesizeoftheagardish.Thegrowthofthebacteriacanstill bemodeledbya proportionaldifferenceequation,butnow the rateofincrease ofthebacteria’s area isdirectly proportional tothebacteria’s area and thedifferencebetween theagardish’s area andthe fungus’s area.Thisconstantofproportionalityistheratiooftheoriginalconstantofproportionality(0.2)and themaximum areathebacteriacan reach (1,000 squaremillimeters). Thedifferenceequationcanbe writtenas follows:

A

t

0.2

1,000

A(1,000 A)

Let ∆t=1 to simplify yourwork, and then use the difference equation to find the newvalues ofA. Use a spreadsheet tocalculateabout 75 values.

6.Comparethedatageneratedintheunrestrictedandrestrictedgrowthmodels.Record yourobservations.

7.Usespreadsheetsoftwareoryourgraphingcalculatortomakeagraphoftherestrictedgrowthmodel.Sketchthegraphbelow.Whatobservationscanyoumakeaboutthe graph?

8.REFLECTION:Theunrestrictedbacteriagrowthmodelsexponentialgrowthandhasacommonratioof1.2.Use the spreadsheetto find the ratiobetweensuccessive valuesintherestrictedgrowthmodel.Whatdoyounotice?Howdoesthissupportthegraph inQuestion7?

9.A rancher has decided to dedicate a 400-square-mile portion ofhis ranchas ablack bearhabitat.Workingwithhisstate,heplanstobring10youngblackbearstothehabitatin anefforttogrowthepopulation.Hisresearchshowsthattheannualgrowthrateofblack bears is about0.8. Blackbearsthrivewhen the populationdensityis no morethan about

1.5 blackbears per squaremile.

a.Whatisthemaximumsustainablenumberofblackbearsforthehabitat?

b.Writearecursiveruleshowingtherestrictedgrowthinpopulationfortheblackbears. (Hint:Theconstantofproportionalityisthe ratiooftheunrestrictedgrowthrateandthemaximumsustainablepopulation.)

c.Makeatableandgraphshowingtheyearlypopulationoftheblackbearsinthe habitat.(Includeenoughyearstoshowthepopulationreachingthemaximum sustainablepopulation.)

d.Whenwill thepopulation of bears in the habitat reach 500?

e.Therancherwantstorepopulatethestatewithblackbears.Therancher’soriginalplan was toreleasethe bears from his ranchwhenthe population reaches500. Do youthinkthisisagooddecisionbasedonthegrowthratewithinthehabitatovertime?Ifyouagreewiththerancher,supportthedecisionwithyourdataandgraph.Ifyou disagree,proposeadifferenttargetpopulationvaluetotherancher;again,support yourproposalwiththedataandgraph.

10.EXTENSION:Researchpopulationdata,eitherofhumansinvariouspartsoftheworldoranimalspecies.Youneedtofinddataoverasignificanttime,notjustafewyears. Citeyoursource.Makeascatterplotofthedata.Dothedatashow exponentialgrowthordotheyshowsignsthatthepopulation’sgrowthisslowing?Whatlimitationsdoesthe populationyouareanalyzinghave?Couldyoupredictamaximumpopulation?Support yourprediction.