AP真题及答案（Calculus BC）\Free-Response Questions\ap05_frq_calculus_bc_b[1]

AP®CalculusBC2005Free-ResponseQuestionsFormBTheCollegeBoard:ConnectingStudentstoCollegeSuccessTheCollegeBoardisanot-for-profitmembershipassociationwhosemissionistoconnectstudentstocollegesuccessandopportunity.Foundedin1900,theassociationiscomposedofmorethan4,700schools,colleges,universities,andothereducationalorganizations.Eachyear,theCollegeBoardservesoverthreeandahalfmillionstudentsandtheirparents,23,000highschools,and3,500collegesthroughmajorprogramsandservicesincollegeadmissions,guidance,assessment,financialaid,enrollment,andteachingandlearning.Amongitsbest-knownprogramsaretheSAT®,thePSAT/NMSQT®,andtheAdvancedPlacementProgram®(AP®).TheCollegeBoardiscommittedtotheprinciplesofexcellenceandequity,andthatcommitmentisembodiedinallofitsprograms,services,activities,andconcerns.Copyright©2005byCollegeBoard.Allrightsreserved.CollegeBoard,APCentral,APCD,AdvancedPlacementProgram,AP,APVerticalTeams,Pre-AP,SAT,andtheacornlogoareregisteredtrademarksoftheCollegeEntranceExaminationBoard.AdmittedClassEvaluationService,CollegeEd,Connecttocollegesuccess,MyRoad,SATProfessionalDevelopment,SATReadinessProgram,andSettingtheCornerstonesaretrademarksownedbytheCollegeEntranceExaminationBoard.PSAT/NMSQTisaregisteredtrademarkoftheCollegeEntranceExaminationBoardandNationalMeritScholarshipCorporation.Otherproductsandservicesmaybetrademarksoftheirrespectiveowners.PermissiontousecopyrightedCollegeBoardmaterialsmayberequestedonlineat:http://www.collegeboard.com/inquiry/cbpermit.html.VisittheCollegeBoardontheWeb:www.collegeboard.com.APCentralistheofficialonlinehomefortheAPProgramandPre-AP:apcentral.collegeboard.com.2005AP®CALCULUSBCFREE-RESPONSEQUESTIONS(FormB)CALCULUSBCSECTIONII,PartATime—45minutesNumberofproblems—3Agraphingcalculatorisrequiredforsomeproblemsorpartsofproblems.1.Anobjectmovingalongacurveinthexy-planehasposition(xt(),yt())attimet≥0withdxdy4=-12tt32and=+-ln(1()t4).dtdtAttimet=0,theobjectisatposition()-13,5.Attimet=2,theobjectisatpointPwithx-coordinate3.(a)Findtheaccelerationvectorattimet=2andthespeedattimet=2.(b)Findthey-coordinateofP.(c)WriteanequationforthelinetangenttothecurveatP.(d)Forwhatvalueoft,ifany,istheobjectatrest?Explainyourreasoning.WRITEALLWORKINTHETESTBOOKLET.Copyright©2005byCollegeEntranceExaminationBoard.Allrightsreserved.Visitapcentral.collegeboard.com(forAPprofessionals)andwww.collegeboard.com/apstudents(forAPstudentsandparents).GOONTOTHENEXTPAGE.22005AP®CALCULUSBCFREE-RESPONSEQUESTIONS(FormB)2.AwatertankatCampNewtonholds1200gallonsofwaterattimet=0.Duringthetimeinterval018££thours,waterispumpedintothetankattheratetWt()=95tsin2()gallonsperhour.6Duringthesametimeinterval,waterisremovedfromthetankattheratetRt()=275sin2()gallonsperhour.3(a)Istheamountofwaterinthetankincreasingattimet=15?Whyorwhynot?(b)Tothenearestwholenumber,howmanygallonsofwaterareinthetankattimet=18?(c)Atwhattimet,for018,££tistheamountofwaterinthetankatanabsoluteminimum?Showtheworkthatleadstoyourconclusion.(d)Fort>18,nowaterispumpedintothetank,butwatercontinuestoberemovedattherateR()tuntilthetankbecomesempty.Letkbethetimeatwhichthetankbecomesempty.Write,butdonotsolve,anequationinvolvinganintegralexpressionthatcanbeusedtofindthevalueofk.WRITEALLWORKINTHETESTBOOKLET.Copyright©2005byCollegeEntranceExaminationBoard.Allrightsreserved.Visitapcentral.collegeboard.com(forAPprofessionals)andwww.collegeboard.com/apstudents(forAPstudentsandparents).GOONTOTHENEXTPAGE.32005AP®CALCULUSBCFREE-RESPONSEQUESTIONS(FormB)3.TheTaylorseriesaboutx=0foracertainfunctionfconvergestofx()forallxintheintervalofconvergence.Thenthderivativeoffatx=0isgivenbyn+1()()-+!11(n)fn()0=forn≥2.51n()n-2Thegraphoffhasahorizontaltangentlineatx=0,andf()06.=(a)Determinewhetherfhasarelativemaximum,arelativeminimum,orneitheratx=0.Justifyyouranswer.(b)Writethethird-degreeTaylorpolynomialforfaboutx=0.(c)FindtheradiusofconvergenceoftheTaylorseriesforfaboutx=0.Showtheworkthatleadstoyouranswer.WRITEALLWORKINTHETESTBOOKLET.ENDOFPARTAOFSECTIONIICopyright©2005byCollegeEntranceExaminationBoard.Allrightsreserved.Visitapcentral.collegeboard.com(forAPprofessionals)andwww.collegeboard.com/apstudents(forAPstudentsandparents).42005AP®CALCULUSBCFREE-RESPONSEQUESTIONS(FormB)CALCULUSBCSECTIONII,PartBTime—45minutesNumberofproblems—3Nocalculatorisallowedfortheseproblems.4.Thegraphofthefunctionfaboveconsistsofthreelinesegments.x(a)Letgbethefunctiongivenbygx()=Úftdt().Foreachofg()-1,g¢()-1,andg¢¢()-1,findthevalue-4orstatethatitdoesnotexist.(b)Forthefunctiongdefinedinpart(a),findthex-coordinateofeachpointofinflectionofthegraphofgontheopeninterval-<43.x