JamesBoswellExam
VWOMathematicsB
Date: 2014
Time:
13:00–16:00hours
Numberofexercises:
Numberofsubexercises: 23
No.ofattachments:
1(doublesided)
Maximumscore:
85points
5
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Writeyournameoneverysheetofpaperthatyouhandin.
Useaseparatesheetofpaperforeachexercise.Usetheattachedworkingsheet
forexercise3ofthisexam.
Foreachexercise,showhowyouobtainedyouranswereitherbymeansofa
calculationor,ifyouusedagraphingcalculator,anexplanation.Nopointswillbe
awardedtoananswerwithoutanexplanation.
Makesurethatyourhandwritingislegibleandwriteinink.Nocorrectionfluidof
anykindispermitted.Useapencilonlytodrawgraphsandgeometricfigures.
Youmayusethefollowing:
o Graphingcalculator(withoutCAS);
o Drawingutensils;
o Protractorandcompass;
o Dictionary,subjecttotheapprovaloftheinvigilator.
Exercise1.Powerfunctions.
Letthefamilyoffunctions:
Figure1
!! ! = ! ! − !! ! begiven.Figure1showsthegraphsof!! and!! .
Thegraphof!! istangenttothe!-axisatthe
origin! 0,0 anditintersectsthe!-axisatpoint
A.
4p a. Inanexactmanner,deriveanequation
forthetangentlinetothegraphof!! atpointA.
Thegraphof!! hasthreepointsincommonwiththeline! = 3!.
3p b. Calculateanalyticallythecoordinatesofthesethreepoints.
!isthesurfaceareaenclosedbythegraphof!! andthe!-axis.
4p c.
!isrevolvedaroundthe!-axis.
Analyticallycalculatethevolumeoftheresultingsolidofrevolution.
Forevery! ≠ 0,thegraphof!! hastwo
Figure2
extremevalues.
InFigure2,thegraphsofseveralfunctions!! havebeenplotted.Inaddition,thedashedline
!
showsthegraphof! ! = − ! ! .
!
Thefiguresuggeststhatallextremepointsof!! lieonthegraphofg.
5p d. Provethatforallvaluesof! ≠ 0,the
extremepointsof!! lieonthegraphofthe
!
function! ! = − ! ! .
!
→
Exercise2.Twoexponentialfunctions.
Letthefollowingfunctionsbegiven:
! ! =
Figure3
! !! − 6! !
and ! ! = 2 − 2! ! .
!! + 2
Thegraphsof!and!havebeenplottedin
Figure3.Let!betheintersectionpointofthe
graphsof!and!.
3p a. Showanalyticallythat!! = ln 2 .
(ln (!)denotesthenaturallogarithmof!.)
3p b. Provethat
! ! ! !! + 4! ! − 12
!! ! =
.
!! + 2 !
Figure3showsoneextremevalueofthegraphof!.
Thequestionarises,whether!hasanymoreextremesthatarenotshowninthefigure.
4p c. Calculateanalyticallythenumberofextremesofthegraphof!.
LetVbetheareaenclosedbythegraphof!,thegraphof!andthe!-axis.
4p d. Provethat! ! = ! ! − 8 ln(! ! + 2)isanantiderivative(i.e.primitivefunction)of! ! .
4p e. Analyticallycalculatethesurfaceareaof!.
→
4p
4p
4p
3p
Exercise3.Fromakiteonacircletoacyclicsquare.
Usetheattachedworkingsheettoanswerexercise3.
Figure4showsacyclic(orinscribed)quadrilateral
Figure4
!"#$withitscircumscribedcircle.
Thecyclicquadrilateralisakite,meaningthatitisa
quadrilateralforwhich!" = !"and!" = !".
Answerthefollowingquestionsontheattached
workingsheet.
a. ProvethatACisthebisectoroftheangle∠!"#,
andofangle∠!"#.
b. Provethat∠!"# = ∠!"# = 90°.
Nowthebisectorsof∠!"#and∠!"#are
constructed.
Thebisectorof∠!"#intersectsthecircleatpoint!,
andatanadditionalpoint!.Thebisectorof∠!"#
Figure5
intersectsthecircleatpoint!,inadditiontopoint!.
SeeFigure5.
!isthecenterofthecircle.
c. Showthat∠!"# = ∠!"# = 90°.
d. ProvethatthecyclicquadrilateralPAQCisa
square.
→
Exercise4.Alogarithmicfunction.
Figure6
Let! ! = 2! ! ln !,whereln !denotesthe
naturallogarithmof!.
Thegraphof!isplottedinFigure6.
Point!isattheintersectionofthegraphof!
withthe!-axis.
4p a. Analyticallycalculatethecoordinatesof
theminimumoff.
4p b. Inanexactmanner,deriveanequation
forthetangentlinetothegraphof!at
point!.
4p c. Calculatethecoordinatesofthe
inflectionpointof!.Giveanexact
calculation.
!
!
!
!
Thefunction! ! = ! ! ln ! −
isanantiderivative(i.e.primitivefunction)ofthe
function!(!).
4p d. Showthisanalytically.
Visthepartoftheplaneenclosedbythegraphof!,theverticalline! = !and
the!-axis.
2p e. Calculateanalyticallythesurfaceareaof!.
→
Exercise5.Pincurve
LettheparametriccurveKbegivenby:
! ! = cos(2!) − 3 sin ! − 1
!:
! ! = sin 2! − 2 cos !
WeconsidercurveKontheinterval[0, 2!].See
alsoFigure7.
CurveKsharestwopoints!and!withthe
!
!-axis.Attime! = !,curveKisatpoint!.
Figure7
!
4p a. Calculateanalyticallythecoordinatesofpoints!and!.
4p b. Calculateanalyticallythecoordinatesofthetwopointsatwhichcurve!intercepts
the!-axis.
4p c. Usinganexactcomputation,showthattheparametriccurve!hasvelocity! = 0atpoint!.
!
Let!(!! , !! )bethepointonthecurveKattime! = ! + !.Point!(!! , !! )isthepointon
!
!
curveKattime! = ! − !.
!
Thefollowingrelationsholdbetweenthecoordinatesofthepoints!and!:
!! = !! = − cos 2! − 3 cos ! − 1and!! = −!! = − sin 2! + 2 sin ! .
4p d. Usetheanglesumanddifferenceidentitiestoshowanalyticallythat
!! = −!! = − sin 2! + 2 sin ! .
Together,theidentities!! = !! = − cos 2! − 3 cos ! − 1and!! = −!! = − sin 2! +
2 sin ! expressageometricpropertyofthecurve!.
2p e. Whichgeometricpropertyofthecurveisexpressedbytheserelations?
Explainyouranswer.
END
Workingsheetforexercise3–Name:________________________________________
Exercise3a. Proof:
Exercise3b. Proof:
Exercise3c. Proof:
Exercise3d. Proof: