欧美国际数学IB 15 May--Mathematics_paper_3_Calculus_HL_markscheme

M15/5/MATHL/HP3/ENG/TZ0/SE/MMarkschemeMay2015CalculusHigherlevelPaper312pages–2–M15/5/MATHL/HP3/ENG/TZ0/SE/MThismarkschemeisthepropertyoftheInternationalBaccalaureateandmustnotbereproducedordistributedtoanyotherpersonwithouttheauthorizationoftheIBAssessmentCentre.–3–M15/5/MATHL/HP3/ENG/TZ0/SE/MInstructionstoExaminersAbbreviationsMMarksawardedforattemptingtouseavalidMethod;workingmustbeseen.(M)MarksawardedforMethod;maybeimpliedbycorrectsubsequentworking.AMarksawardedforanAnswerorforAccuracy;oftendependentonprecedingMmarks.(A)MarksawardedforanAnswerorforAccuracy;maybeimpliedbycorrectsubsequentworking.RMarksawardedforclearReasoning.NMarksawardedforcorrectanswersifnoworkingshown.AGAnswergiveninthequestionandsonomarksareawarded.Usingthemarkscheme1GeneralMarkaccordingtoRM™Assessorinstructionsandthedocument“MathematicsHL:Guidancefore-markingMay2015”.Itisessentialthatyoureadthisdocumentbeforeyoustartmarking.Inparticular,pleasenotethefollowing:Marksmustberecordedusingtheannotationstamps.Pleasecheckthatyouareenteringmarksfortherightquestion.Ifapartiscompletelycorrect,(andgainsallthe“mustbeseen”marks),usethetickswithnumberstostampfullmarks.Ifapartiscompletelywrong,stampA0bythefinalanswer.Ifapartgainsanythingelse,itmustberecordedusingalltheannotations.AllthemarkswillbeaddedandrecordedbyRM™Assessor.2MethodandAnswer/AccuracymarksDonotautomaticallyawardfullmarksforacorrectanswer;allworkingmustbechecked,andmarksawardedaccordingtothemarkscheme.ItisnotpossibletoawardM0followedbyA1,asAmark(s)dependontheprecedingMmark(s),ifany.WhereMandAmarksarenotedonthesameline,egM1A1,thisusuallymeansM1foranattempttouseanappropriatemethod(egsubstitutionintoaformula)andA1forusingthecorrectvalues.Wherethemarkschemespecifies(M2),N3,etc.,donotsplitthemarks.Onceacorrectanswertoaquestionorpart-questionisseen,ignorefurthercorrectworking.However,iffurtherworkingindicatesalackofmathematicalunderstandingdonotawardthefinalA1.Anexceptiontothismaybeinnumericalanswers,whereacorrectexactvalueisfollowedbyanincorrectdecimal.However,iftheincorrectdecimaliscarriedthroughtoasubsequentpart,andcorrectFTworkingshown,awardFTmarksasappropriatebutdonotawardthefinalA1inthatpart.–4–M15/5/MATHL/HP3/ENG/TZ0/SE/MExamplesCorrectanswerseenFurtherworkingseenAction1.5.65685...AwardthefinalA182(incorrectdecimalvalue)(ignorethefurtherworking)2.1sin4xsinxDonotawardthefinalA143.logabloglog(ab)DonotawardthefinalA13NmarksAwardNmarksforcorrectanswerswherethereisnoworking.DonotawardamixtureofNandothermarks.TheremaybefewerNmarksavailablethanthetotalofM,AandRmarks;thisisdeliberateasitpenalizescandidatesfornotfollowingtheinstructiontoshowtheirworking.4ImpliedmarksImpliedmarksappearinbracketseg(M1),andcanonlybeawardedifcorrectworkisseenorifimpliedinsubsequentworking.Normallythecorrectworkisseenorimpliedinthenextline.Markswithoutbracketscanonlybeawardedforworkthatisseen.5FollowthroughmarksFollowthrough(FT)marksareawardedwhereanincorrectanswerfromonepartofaquestionisusedcorrectlyinsubsequentpart(s).ToawardFTmarks,theremustbeworkingpresentandnotjustafinalanswerbasedonanincorrectanswertoapreviouspart.IfthequestionbecomesmuchsimplerbecauseofanerrorthenusediscretiontoawardfewerFTmarks.Iftheerrorleadstoaninappropriatevalue(egsin1.5),donotawardthemark(s)forthefinalanswer(s).Withinaquestionpart,onceanerrorismade,nofurtherdependentAmarkscanbeawarded,butMmarksmaybeawardedifappropriate.Exceptionstothisrulewillbeexplicitlynotedonthemarkscheme.6Mis-readIfacandidateincorrectlycopiesinformationfromthequestion,thisisamis-read(MR).Acandidateshouldbepenalizedonlyonceforaparticularmis-read.UsetheMRstamptoindicatethatthishasbeenamisread.Thendeductthefirstofthemarkstobeawarded,evenifthisisanMmark,butawardallotherssothatthecandidateonlylosesonemark.IfthequestionbecomesmuchsimplerbecauseoftheMR,thenusediscretiontoawardfewermarks.IftheMRleadstoaninappropriatevalue(egsin1.5),donotawardthemark(s)forthefinalanswer(s).–5–M15/5/MATHL/HP3/ENG/TZ0/SE/M7Discretionarymarks(d)Anexaminerusesdiscretiontoawardamarkontherareoccasionswhenthemarkschemedoesnotcovertheworkseen.InsuchcasestheannotationDMshouldbeusedandabriefnotewrittennexttothemarkexplainingthisdecision.8AlternativemethodsCandidateswillsometimesusemethodsotherthanthoseinthemarkscheme.Unlessthequestionspecifiesamethod,othercorrectmethodsshouldbemarkedinlinewiththemarkscheme.Ifindoubt,contactyourteamleaderforadvice.AlternativemethodsforcompletequestionsareindicatedbyMETHOD1,METHOD2,etc.Alternativesolutionsforpart-questionsareindicatedbyEITHER...OR.Wherepossible,alignmentwillalsobeusedtoassistexaminersinidentifyingwherethesealternativesstartandfinish.9AlternativeformsUnlessthequestionspecifiesotherwise,acceptequivalentforms.Asthisisaninternationalexamination,acceptallalternativeformsofnotation.Inthemarkscheme,equivalentnumericalandalgebraicformswillgenerallybewritteninbracketsimmediatelyfollowingtheanswer.Inthemarkscheme,simplifiedanswers,(whichcandidatesoftendonotwriteinexaminations),willgenerallyappearinbrackets.Marksshouldbeawardedforeithertheformprecedingthebracketortheforminbrackets(ifitisseen).Example:fordifferentiatingfx()2sin(5x3),themarkschemegives:fx()2cos(5x3)510cos(5x3)A1AwardA1for2cos(5x3)5,evenif10cos(5x3)isnotseen.10AccuracyofAnswersCandidatesshouldNOLONGERbepenalizedforanaccuracyerror(AP).Ifthelevelofaccuracyisspecifiedinthequestion,amarkwillbeallocatedforgivingtheanswertotherequiredaccuracy.Whenthisisnotspecifiedinthequestion,allnumericalanswersshouldbegivenexactlyorcorrecttothreesignificantfigures.PleasecheckworkcarefullyforFT.11CrossedoutworkIfacandidatehasdrawnalinethroughworkontheirexaminationscript,orinsomeotherwaycrossedouttheirwork,donotawardanymarksforthatwork.–6–M15/5/MATHL/HP3/ENG/TZ0/SE/M12CalculatorsAGDCisrequiredforpaper3,butcalculatorswithsymbolicmanipulationfeatures(forexample,TI-89)arenotallowed.CalculatornotationTheMathematicsHLguidesays:Studentsmustalwaysusecorrectmathematicalnotation,notcalculatornotation.Donotacceptfinalanswerswrittenusingcalculatornotation.However,donotpenalizetheuseofcalculatornotationintheworking.13MorethanonesolutionWhereacandidateofferstwoormoredifferentanswerstothesamequestion,anexaminershouldonlymarkthefirstresponseunlessthecandidateindicatesotherwise.–7–M15/5/MATHL/HP3/ENG/TZ0/SE/M1.f(0)0A1f()xxxexxcosesin1M1A1f(0)0(M1)f()xx2esinxA1f(0)0f(3)(xxx)2exxsin2ecosA1f(3)(0)22x3x3thefirstnon-zerotermisA13!3Note:Awardnomarksforusingknownseries.[7marks]2.(a)METHOD1d1y1f()dxxfx()M1M1A1dxx2xdyxyfxx(),0AGdxNote:M1foruseofproductrule,M1foruseofthefundamentaltheoremofcalculus,A1forallcorrect.METHOD2dyxyfx()dxd(xy)f()x(M1)dxxyfxx()dM1A11yfxx()dAGx[3marks]continued…–8–M15/5/MATHL/HP3/ENG/TZ0/SE/MQuestion2continued11(b)yxc22A1A1xNote:A1forcorrectexpressionapartfromtheconstant,A1forincludingtheconstantinthecorrectposition.attempttousetheboundaryconditionM1c4A111yx242A1x[5marks]Note:Condoneuseofintegratingfactor.Total[8marks]3.(a)METHOD111(0),(forn3)A1nnn22ln()1convergesA12n2n11bythecomparisontest(convergesimplies)convergesR122n2nn2nn(ln)Note:Mentionofusingthecomparisontestmayhavecomeearlier.OnlyawardR1ifprevious2A1shavebeenawarded.METHOD21nn2ln1limlim0A1nn1lnnn21convergesA12n2nbythelimitcomparisontest(ifthelimitis0andtheseriesrepresentedbythedenominatorconverges,thensodoestheseriesrepresentedbythecontinued…–9–M15/5/MATHL/HP3/ENG/TZ0/SE/MQuestion3continued1numerator,hence)convergesR12n2nn(ln)Note:Mentionofusingthelimitcomparisontestmaycomeearlier.DonotawardtheR1ifincorrectjustificationsaregiven,forexampletheseries“convergeordivergetogether”.OnlyawardR1ifprevious2A1shavebeenawarded.[3marks](b)(i)EITHER11ln(nn)ln1ln1A1nnOR11nln(nn)ln1ln()lnnnln(nn)ln(1)ln(n)A1THENln(n1)AGnnln()(ii)attempttousetheratiotestM1(1)ln(1)nnn1asn(A1)n1ln()nnln()M1ln(n1)1ln(n)ln1n1(asn)(A1)nnln()1(asn)henceratiotestisinconclusiveR1(1)ln(1)nnNote:Alinkwiththelimitequalling1andtheresultbeinginconclusiveneedstobegivenforR1.[6marks]1(c)(i)considerfx()(forx1)A1xxlnf()xiscontinuousandpositiveA1andis(monotonically)decreasingA1Note:Ifacandidateusesnratherthanx,awardasfollows1ispositiveanddecreasingA1A1nnln1iscontinuousforn,n>1A1(onlyawardthismarkifthennlndomainhasbeenexplicitlychanged).continued…–10–M15/5/MATHL/HP3/ENG/TZ0/SE/MQuestion3continuedR1(ii)considerdxM12xlnxRln(lnx)(M1)A12asRR1henceseriesdivergesA1Note:CondonetheuseofinplaceofR.Note:Ifthelowerlimitisnotequalto2,buttheexpressionisintegratedcorrectlyawardM0M1A1R0A0.[8marks]Total[17marks]x22x4.(a)limlimM1A1xxeexx2lim0M1A1xexNote:AwardM1foranattemptatdifferentiatingforasecondtime.[4marks](b)attempttointegratebypartsM1x22edxxxxxe2edxx(A1)x2e2e2edxxxxx(A1)x2e2e2e()xxxxcA1Rx2exdxR2eR2ReR2eR2M1A10Rlimxx2edx2M1A10RNote:AwardM1forconsiderationofthelimitandA1forcorrectlimitingvalue.hencetheimproperintegralconvergesAGNote:DonotawardthefinalfourmarkstocandidateswhodonotconsiderR.[8marks]Total[12marks]–11–M15/5/MATHL/HP3/ENG/TZ0/SE/M5.(a)(i)f()xxx326A1gradientofchord1A1361cc2323c(2.15,0.155)A1A13Note:Acceptanyanswersthatroundtothecorrect2sfanswers(0.15)–2.2,.(ii)awardA1forcorrectshapeandclearindicationofcorrectdomain,A1forchord(fromx=3tox=1)andA1fortwotangentsdrawnattheirvaluesofcA1A1A1[7marks](b)(i)METHOD1(ifatheoremistruefortheinterval[a,b],itisalsotrueforanyinterval[,x12x]whichbelongsto[a,b])supposex12,[,]xabM1fx()fx()bytheMVT,thereexistscsuchthatfc()210M1A1xx21hencef()xfx12()R1asx12,xarearbitrarilychosen,fxisconstantonab,Note:IftheaboveisexpressedintermsofaandbawardM0M1A0R0.METHOD2(ifatheoremistruefortheinterval[a,b],itisalsotrueforanyinterval[,x12x]whichbelongsto[a,b])supposex[,ab]M1continued…–12–M15/5/MATHL/HP3/ENG/TZ0/SE/MQuestion5continuedfx()fa()bytheMVT,thereexistscsuchthatfc()0M1A1xahencef(x)f(a)constantR1(ii)attempttodifferentiatef(xx)2arccosarccos12x2M114x2A1A1221x112x214x20A1144xxx224Note:OnlyawardA1for0ifacorrectattempttosimplifythedenominatorisalsoseen.fx()f(0)20A1AG2Note:ThisA1isnotdependentonpreviousmarks.Note:Allowanyvalueofx[0,1].[9marks]Total[16marks]