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MATHEMATICAL METHODSWritten examination 2
Monday 31 May 2021 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
A 20 20 20B 5 5 60
Total 80
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof29pages• Formulasheet• Answersheetformultiple-choicequestions
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice
questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• Youmaykeeptheformulasheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2021
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2021
STUDENT NUMBER
Letter
GO
LD
S
TR
IPE
2021MATHMETHEXAM2(NHT) 2
SECTION A – continued
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Question 1Thegraphbelowshowsonecycleofacircularfunction.
x
y
4
2
O
–2
1
TheruleforthefunctioncouldbeA. y=3sin(x)+1
B. y x= −
+3
21sin
π
C. y=−3cos(2π x)+1
D. y=3sin(2π x)−1
E. y=−3sin(2π x)+1
Question 2If3f (x)=f(3x)forx>0,thentheruleforfcouldbe
A. f (x)=3x
B. f x x( ) = 3
C. 3
( ) =3xf x
D. ( ) log3exf x =
E. f (x)=x−3
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
3 2021MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 3
Thefunctionf : D → R, f x x x x x( ) = − − +4 3 2
4 39
29 willhaveaninversefunctionfor
A. D = RB. D=(–3,1)C. D=(1,∞)D. D=(–∞,0)E. D=(0,∞)
Question 4Thegraphoff :R → R,f (x)=x3+ax2+bx+chasaturningpointatx=3anday-interceptaty=9.Thevaluesofa,bandccouldbe,respectivelyA. −5,3and9
B. 7,−15and−9
C. − −2 32
9, and
D. 5,−3and−9
E. −1,−3and9
Question 5
Theexpression log3 25
q p
isequivalentto
A. log3(5)–log3(q)–log3(p)
B. 12
5 2 23 3 3log ( ) log ( ) log ( )− −q p
C. 12
5 23 3 3log ( ) log ( ) log ( )− −q p
D. 2 log3(5)–2 log3(q)–log3(p)
E. 2 log3(5)–2 log3(q)–2 log3(p)
Question 6Thesumofthefirstfourpositivesolutionstotheequationtan(2x)−1=0is
A. 32π
B. 52π
C. 2π
D. 72π
E. 4π
2021MATHMETHEXAM2(NHT) 4
SECTION A – continued
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Question 7Partofthegraphofy = f′(x)isshowninthediagrambelow.
3
2
1
–1O 3
2
1–1 4
y
x
Giventhatf(0)=1,thecorrespondingpartofthegraphofy = f(x)couldbe
2
–2
1O 2 3 4
y
x
2
–2
1O 2 3 4
y
x
2
–2
1O 2 3 4
y
x
2
–2
1O 2 3 4
y
x
2
–2
1O 2
3
4
y
x
A. B.
C. D.
E.
1
1
1
1
1
–1–1
–1–1
–1–1
–1–1
–1–1
5 2021MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 8
Partofthegraphofapolynomialfunctionf isshownbelow.Thisgraphhasturningpointsat − −( )2 2 1, and 2 2 1, −( ).
y
x–4 4O
f(x)isstrictlydecreasingfor
A. x ∈(–∞,–4]∪[4,∞)
B. x ∈[–4,4]
C. x∈ −2 2 2 2,
D. x∈ −∞ −( ∪
, ,2 2 0 2 2
E. x∈ − ∪ ∞
)2 2 0 2 2, ,
Question 9Thecontinuousanddifferentiablefunctionf :R → R hasrootsatx=1andx=6andarepeatedrootatx=4.
Giventhat4
1( )f x dx a=∫ and
6
4( )f x dx b=∫ ,wherea,b ∈ R, f x dx( ) +( )∫ 1
1
6isequalto
A. a+b+1B. a–b+1C. a+b+5D. a–b–5E. a–b
Question 10Considerthegraphoff:R → R,f (x)=– x2–4x+5.Thetangenttothegraphoff isparalleltothelineconnectingthenegativex-interceptandthey-interceptoffwhenxisequaltoA. −3
B. −52
C. −32
D. −1
E. −12
2021MATHMETHEXAM2(NHT) 6
SECTION A – continued
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Question 11Asurveyofalargerandomsampleofpeoplefoundthatanapproximate95%confidenceintervalfortheproportionofpeoplewhoownedayellowrubberduckwas(0.6299,0.6699).ThenumberofpeopleintherandomsampleisclosesttoA. 569B. 1793C. 2108D. 2179E. 2185
Question 12ThetransformationT :R2 → R2mapsthegraphofy = x3−xontothegraphofy=2(x −1)3−2(x −1)+4.ThetransformationTcouldbegivenby
A. Txy
xy
=
+
1 00 2
14
B. Txy
xy
=
+
1 0
0 12
14
C. Txy
xy
=
+
2 00 1
12
D. Txy
xy
=
+
12
0
0 1
12
E. Txy
xy
=
+
1 00 2
12
Question 13
Forthefunctionp (x)=ke−k x,wherex≥0andk>0,thevalueof aforwhich p a p( ) ( )=12
0 is
A. 1 1
2k elog
B. 1 2k elog ( )
C. kloge (2)
D. k elog 12
E. kkelog 1
2
7 2021MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 14Acontinuousrandomvariable,X,hastheprobabilitydensityfunction
f xe x
x
x
( ). .
=≥<
−0 2 00 0
0 2
ThevarianceofXisA. 25B. 12.5C. 6.25D. 3.125E. 0
Question 15Thegraphsoffunctionsfandgareshownbelow.Bothfunctionshavethesamedomainof[0,b],where b>0,andthesameaveragevalue.
O
2b
f
y
xb O
g
y
xb
8
ThevalueofbisA. 1B. 2C. 4D. 8E. 16
Question 16Inaparticularcity,itisknownthat70%ofalladultsgettheirhaircuteverymonth.Arandomsampleof 720adultsfromthiscityisselected.Fromthissample,theprobabilitythattheproportionofadultswhogettheirhaircuteverymonthisgreaterthan0.72isA. 0.2104B. 0.1359C. 0.1187D. 0.0847E. 0.0392
2021MATHMETHEXAM2(NHT) 8
SECTION A – continued
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Question 17Partofthegraphofthefunctionf isshownbelow.Thesmallestpositivex-interceptofthegraphoccursatx = a.Thehorizontallineisatangenttofatthelocalminimum b f b, ( )( ).Theshadedareaistheareaboundedbythegraphoff,thex-axis,they-axisandthegraphofy = f (b).
( ), ( )b f b
(a, 0)
y
Ox
Theareaoftheshadedregionis
A. a f b f x dxa
b( ) ( )+ ∫
B. a f b f x dxa
b( ) ( )− ∫
C. f x dx b f ba
b( ) ( )+∫
D. b f b f x dxa
b( ) ( )− ∫
E. f x dx b f ba
b( ) ( )−∫
Question 18
Giventhatd x x
dxx x x
cos( )cos( ) sin( )( )
= − , sin( )x x dx∫ isequalto
A. cos(x)−xcos(x)
B. cos cos( ) ( )x x x dx+ ∫C. x x x dxcos( ) cos( )− ∫D. cos cos( ) ( )x dx x x−∫E.
−x xx
cos( )cos( )
9 2021MATHMETHEXAM2(NHT)
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END OF SECTION ATURN OVER
Question 19Acubicpolynomialfunctionf :R → Rhasrootsatx=1andx=3onlyanditsgraphhasay-interceptat y =3.Whichoneofthefollowingstatementsmustbetrueaboutthefunctiong,whereg x f x( ) ( )= ?A. Thefunctionghasalocalmaximumatx = 2B. g(2)=1C. Thedomainofgdoesnotincludetheinterval(1,3)D. Thedomainofgincludestheinterval(1,3)E. Thedomainofgdoesnotincludetheinterval(3,∞)
Question 20TheprobabilitydistributionforthediscreterandomvariableX,whereb ∈ R,isshowninthetablebelow.
x 0 1 2 3
Pr(X = x) 45
110
3− b 15 250
2b − 9 550b +
ThevalueofbisA. –0.4B. –0.3C. –0.2D. 0.2E. 0.5
2021MATHMETHEXAM2(NHT) 10
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SECTION B – Question 1–continued
Question 1 (11marks)Let f R R f x x x: , ( ) cos( ) cos( )→ = − +( )2 4 andg :R → R,g (x)=2cos(x).
a. Statetheperiodandtheamplitudeofg. 1mark
b. Findthevalueofcforwhichf(c)=0,wherec∈
0
2, π . 1mark
c. Findtheminimumvalueoff. 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
11 2021MATHMETHEXAM2(NHT)
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SECTION B – Question 1–continuedTURN OVER
Partsofthegraphsofthefunctionsoffandgareshownbelow.
Thegraphsoffandgtouch,butdonotcross,atthepoints(p, q)and π3
, v
.
y
xO
g
(p, q)
f
,3
vπ
d. Findthevaluesofpandq. 2marks
e. i. Findthevalueofthederivativeoffandthevalueofthederivativeofg at x = π3. 2marks
ii. Findtheequationofthetangenttothegraphsoffandg at x = π3. 1mark
iii. Findtheequationofthelineperpendiculartothegraphsoffandg at x = π3. 1mark
2021MATHMETHEXAM2(NHT) 12
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SECTION B – continued
f. Theareaboundedbythegraphsoffandgisshadedinthediagrambelow.
y
xO
g
f
Findtheareaoftheshadedregion. 2marks
13 2021MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
2021MATHMETHEXAM2(NHT) 14
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SECTION B – Question 2–continued
Question 2 (10marks)
Thefunctionh xx
x xe( ) log , [ , ]=−( )
−
∈
32005
54
10 502 where ,modelstherateatwhichheatislost
fromthewaterinahot-waterpipewithinsulation,whereh(x) istherateatwhichunitsofheatarelostfromthewaterandx istheradiusofthehot-waterpipewithitsinsulation,inmillimetres.Thediagrambelowshowsacross-sectionofthepipewithitsinsulation.
radius of pipewith insulation
radius of pipe
O
x
Theradiusofthepipewithoutitsinsulationis10mm.Thegraphoftherateofheatlostfromthewateroverthegivendomainisshownbelow.
O
40
30
20
10
10 20 30 40 50x
h
a. Findtherateatwhichheatislostfromthewaterinapipewithnoinsulation,correcttothreedecimalplaces. 1mark
15 2021MATHMETHEXAM2(NHT)
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SECTION B – Question 2–continuedTURN OVER
b. i. Statethederivativeofh (x). 1mark
ii. Findthemaximumrateatwhichheatislostfromthewater,correcttothreedecimalplaces. 1mark
c. Aparticularinsulatedpipehasthesamerateofheatlostfromthewaterasapipewithnoinsulation.
Findthethicknessofinsulationforthispipe,inmillimetres,correcttothreedecimalplaces. 1mark
d. i. Ifboththeradiusofthepipewithoutinsulationandtheradiusofthepipewithinsulation,asshowninthediagramonpage14,aredoubled,showthattherateofheatlostfromthewater,h1,isnowgivenby
h x xx e1 212 800 10
810( ) log=
−
−( )
andstatethedomainofh1. 2marks
ii. Describethetransformationthatmapsthegraphofhtothegraphofh1. 1mark
2021MATHMETHEXAM2(NHT) 16
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SECTION B – continued
e. i. Findtheareabetweenthegraphofh1andthehorizontalaxisoveritsdomain.Giveyouranswercorrecttothreedecimalplaces. 2marks
ii. Lettheareafoundinpart e.i.beA.
Determinetheareabetweenthegraphofh andthehorizontalaxisoveritsdomain,intermsofA. 1mark
17 2021MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
2021MATHMETHEXAM2(NHT) 18
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SECTION B – Question 3–continued
Question 3 (10marks)Theparabolicarchofatunnelismodelledbythefunctionf:[−c,c] → R,f(x)=ax2+b,where a < 0,b ∈ Randc>0.Letxbethehorizontaldistance,inmetres,fromtheoriginandletybetheverticaldistance, inmetres,abovethebaseofthearch.Thegraphoff isshownbelow,wherethecoordinatesofthey-interceptare(0,k)andthecoordinatesofthex-interceptsare(−c,0)and(c,0).
y
c–c O
k
x
f
a. Expressa andb intermsofc andk. 2marks
19 2021MATHMETHEXAM2(NHT)
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SECTION B – Question 3–continuedTURN OVER
Aparticulartunnelhasanarchmodelledbyf.Ithasaheightof6matthecentreandawidthof 8matthebase.
b. i. Findtheruleforthisarch. 1mark
ii. Atruckthathasaheightof3.7mandawidthof2.7mwillfitthroughthearchwiththefunctionf foundinpart b.i.
y
x
k
–c cO
d
truck
f
Assumingthatthetruckdrivesdirectlythroughthemiddleofthearch,letdbetheminimumdistancebetweenthearchandthetopcornerofthetruck.
Finddandthevalueofxforwhichthisoccurs,correcttothreedecimalplaces. 3marks
2021MATHMETHEXAM2(NHT) 20
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SECTION B – Question 3–continued
Adifferenttunnelhasasemicirculararch.Thisarchcanbemodelledbythefunctiong R g x r x: [ , ] , ( )− → = −6 6 2 2 ,wherer > 0.Thegraphofgisshownbelow.
6
x6O–6
y
g
c. Statethevalueofr. 1mark
21 2021MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
d. Twolightshavebeenplacedonthearchtolighttheentranceofthetunnel.Thepositionsofthelightsare −( ) ( )11 5 11 5, ,and .Theareathatislitbytheselightsisshadedinthe
diagrambelow.
6
x6O–6
y
( )11, 5− ( )11, 5
Determinetheproportionofthecross-sectionofthetunnelentrancethatislitbythelights.Giveyouranswerasapercentage,correcttothenearestinteger. 3marks
2021MATHMETHEXAM2(NHT) 22
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SECTION B – Question 4–continued
Question 4 (17marks)Aparticularpetrolstationhastwoairpumps,AandB,toinflatetyres.Eachinflationofatyreisindependentofanyotherinflationofatyre.WhenpumpAissetto320kilopascals(kPa),thepressurethatthetyreswillbeinflatedtofollows anormaldistributionwithameanof320kPaandastandarddeviationof10kPa.
a. Findtheprobabilitythatatyrewillbeinflatedtoapressuregreaterthan330kPawheninflatedbypumpA,correcttofourdecimalplaces. 1mark
b. TheprobabilitythatatyreisinflatedbypumpA toapressuregreaterthanais0.9
Findthevalueofa,correcttothenearestkilopascal. 2marks
WhenpumpBissetto320kPa,thepressurethatthetyreswillbeinflatedtoismodelledbythefollowingprobabilitydensityfunction.
b xx x x
( )( ) ( )
=− − ≤ ≤
340000
310 330 310 330
0
2
elsewhere
c. DeterminethemeantyrepressurefortyresinflatedbypumpB. 2marks
d. ArandomlyselectedtyreisinflatedbypumpB.
FindtheprobabilitythatthistyrewillbeinflatedtoapressuregreaterthanthemeantyrepressureoftyresinflatedbypumpB,correcttofourdecimalplaces. 2marks
23 2021MATHMETHEXAM2(NHT)
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SECTION B – Question 4–continuedTURN OVER
e. TheprobabilitythatatyreisinflatedbypumpB toapressurelessthankis0.95
Findthevalueofk,correcttothenearestkilopascal. 2marks
f. AmotoristisequallylikelytouseeitherpumpAorpumpBtoinflateoneoftheircar’styres.
FindtheprobabilitythatthemotoristhasusedpumpAgiventhatthetyreisinflatedtoapressuregreaterthan325kPa.Giveyouranswercorrecttofourdecimalplaces. 2marks
Thecompanythatmanufacturesthepumpstestsallofitspumpsandremovesthosethataredefective.Theprobabilitythatarandomlyselectedpumpisdefective,fromallofthepumpstested,is0.08
g. Findtheprobabilitythatfourpumpsaredefectivefromasampleof25randomlyselectedpumps,correcttofourdecimalplaces. 2marks
h. Forrandomsamplesof25pumps,P̂ istherandomvariablethatrepresentstheproportionof pumpsthataredefective.
FindtheprobabilitythatP̂ isgreaterthan15%,correcttofourdecimalplaces. 2marks
2021MATHMETHEXAM2(NHT) 24
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SECTION B – continued
i. Forrandomsamplesofnpumps,P̂ nistherandomvariablethatrepresentstheproportionofpumpsthataredefective.
Findtheleastvalueofnsuchthat 1ˆPr 0.15nPn
< <
2marks
25 2021MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
2021MATHMETHEXAM2(NHT) 26
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SECTION B – Question 5–continued
Question 5 (12marks)
Let f R R f xx
g R R g x x: , ( ) : , ( )+ → = − → = −1
565
452 and .
Partsofthegraphsoffandgareshownbelow.
A
B
y
O
–1
–2
1 2x
f
g
a. Findthecoordinatesofthepointsofintersectionofthegraphsoff andg,labelledAandB inthediagramabove. 2marks
b. DeterminetheareaboundedbythegraphsoffandgbetweenAandB.Giveyouranswerin
theform r s tu+ ,wherer,s,tanduareintegers. 2marks
27 2021MATHMETHEXAM2(NHT)
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SECTION B – Question 5–continuedTURN OVER
Let 21 1 (1 ): , ( ) and : , ( ) for 1a a xp R R p x q R R q x a
ax a a+ + −→ = − → = > .
c. Findthevalueofaforwhichp (x)=f(x)andq (x)=g (x)forallx. 1mark
d. Findthepositivex-interceptofpintermsofa. 1mark
PointMliesonthegraphofy = p (x).Thetangenttop at Misparalleltoq.
e. Findthex-coordinateofMintermsofa. 2marks
2021MATHMETHEXAM2(NHT) 28
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SECTION B – Question 5–continued
f. i. Findthey-interceptofthetangenttop at Mintermsofa. 1mark
ii. Giventhat 23
2 113
23x x≥ −( ) forx>1,showthatthetangenttop paralleltoqwillhave
anegativey-interceptforalla>1. 1mark
iii. Thetangenttopparalleltoqhasanegativey-intercept.
Explainwhythisimpliespandqwillalwaysenclosearegionboundedbybothgraphsforalla>1. 1mark
29 2021MATHMETHEXAM2(NHT)
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END OF QUESTION AND ANSWER BOOK
g. Partsofthegraphsofpandqareshownbelowforwhena=100.
y
O
–1
1x
p
q
Theshadedareaisboundedbythegraphsofpandq.
Findthesmallestvalue,b,suchthattheshadedareaislessthanbforalla≥100. 1mark
MATHEMATICAL METHODS
Written examination 2
FORMULA SHEET
Instructions
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Victorian Certificate of Education 2021
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2021
MATHMETH EXAM 2
Mathematical Methods formulas
Mensuration
area of a trapezium 12a b h+( ) volume of a pyramid 1
3Ah
curved surface area of a cylinder 2π rh volume of a sphere
43
3π r
volume of a cylinder π r 2h area of a triangle12bc Asin ( )
volume of a cone13
2π r h
Calculus
ddx
x nxn n( ) = −1 x dxn
x c nn n=+
+ ≠ −+∫ 11
11 ,
ddx
ax b an ax bn n( )+( ) = +( ) −1 ( )( )
( ) ,ax b dxa n
ax b c nn n+ =+
+ + ≠ −+∫ 11
11
ddxe aeax ax( ) = e dx a e cax ax= +∫ 1
ddx
x xelog ( )( ) = 1 1 0x dx x c xe= + >∫ log ( ) ,
ddx
ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dx a ax c= − +∫ 1
ddx
ax a axcos( )( ) −= sin ( ) cos( ) sin ( )ax dx a ax c= +∫ 1
ddx
ax aax
a axtan ( )( )
( ) ==cos
sec ( )22
product ruleddxuv u dv
dxv dudx
( ) = + quotient ruleddx
uv
v dudx
u dvdx
v
=
−
2
chain ruledydx
dydududx
=
3 MATHMETH EXAM
END OF FORMULA SHEET
Probability
Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)
Pr(A|B) = Pr
PrA BB∩( )( )
mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2
Probability distribution Mean Variance
discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)
continuous Pr( ) ( )a X b f x dxa
b< < = ∫ µ =
−∞
∞
∫ x f x dx( ) σ µ2 2= −−∞
∞
∫ ( ) ( )x f x dx
Sample proportions
P Xn
=̂ mean E(P̂ ) = p
standard deviation
sd P p pn
(ˆ ) ( )=
−1 approximate confidence interval
,p zp p
np z
p pn
−−( )
+−( )
1 1ˆ ˆ ˆˆˆ ˆ