syllabus math hl
TRANSCRIPT
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8/9/2019 Syllabus Math HL
1/31
Mathematics HL guide 17
Syllabu
s
Syllabuscontent
Top
ic1Core:Algebra
30
hours
Theaimofthistopicistointroducestudentstosomebasicalgebraicconce
ptsandapplications.
Content
Furtherguidance
L
inks
1.1
Arithmeticsequencesandseries;sumoffinite
arithmeticseries;geometricseq
uencesand
series;sumoffiniteandinfinite
geometric
series.
Sigmanotation.
Sequencescanb
egeneratedanddisplayedin
severalways,includingrecursivefunctions.
Linkinfinitegeo
metricserieswithlimitsof
convergencein6.1.
I
nt:Thechesslegend(SissaibnDahir).
I
nt:Aryabhattaissometimesconsideredthe
fatherofalgebra.Comparewith
a
l-Khawarizmi.
I
nt:Theuseofseveralalphabetsin
m
athematicalnotation(egfirstterm
and
c
ommondifferenceofanarithmeti
csequence).
T
OK:Mathematicsandtheknowe
r.Towhat
e
xtentshouldmathematicalknowledgebe
c
onsistentwithourintuition?
T
OK:Mathematicsandtheworld.
Some
m
athematicalconstants(
,e,,
Fibonacci
n
umbers)appearconsistentlyinna
ture.
What
d
oesthistellusaboutmathematica
l
k
nowledge?
T
OK:Mathematicsandtheknowe
r.Howis
m
athematicalintuitionusedasaba
sisfor
f
ormalproof?(Gaussmethodforaddingup
integersfrom1to100.)
(continued)
Applications.
Examplesinclud
ecompoundinterestand
populationgrow
th.
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Mathematics HL guide18
Syllabus content
Content
Furtherguidance
Links
(see
notesabove)
Aim8:Short-termloansathighinterestrates.
H
owcanknowledgeofmathematicsresultin
individualsbeingexploitedorprotectedfrom
e
xtortion?
Appl:Physics7.2,1
3.2(radioactiv
edecayand
n
uclearphysics).
1.2
Exponentsandlogarithms.
Lawsofexponents;lawsoflogarithms.
Changeofbase.
Exponentsandlogarithmsarefurther
developedin2.4
.
Appl:Chemistry18.1,1
8.2(calculationofpH
a
ndbuffersolutions).
TOK:Thenatureofmathematicsandscience.
W
erelogarithmsaninventionordiscovery?(This
topicisanopportunityforteachersan
dstudentsto
reflectonthenatureofmathematics.)
1.3
Countingprinciples,includingp
ermutations
andcombinations.
Theabilitytofin
d
n r
andn
rPu
singboththe
formulaandtechnologyisexpected.L
inkto
5.4.
TOK:Thenatureofmathematics.
The
u
nforeseenlinksbetweenPascalstriangle,
c
ountingmethodsandthecoefficie
ntsof
p
olynomials.Isthereanunderlying
truththat
c
anbefoundlinkingthese?
Int:ThepropertiesofPascalstrianglewere
k
nowninanumberofdifferentcultureslong
b
eforePascal(egtheChinesemath
ematician
Y
angHui).
Aim8:Howmanydifferentticketsare
p
ossibleinalottery?Whatdoesthistellus
a
bouttheethicsofsellinglotteryticketsto
thosewhodonotunderstandtheim
plications
o
ftheselargenumbers?
Thebinomialtheorem:
expansionof
(
)n
a
b
+
,n
.
Notrequired:
Permutationswheresomeobjec
tsareidentical.
Circulararrangements.
Proofofbinomialtheorem.
Linkto5.6,b
ino
mialdistribution.
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Mathematics HL guide 19
Syllabus content
Content
Furtherguidance
L
inks
1.4
Proofbymathematicalinduction.
Linkstoawidevarietyoftopics,
forexample,
complexnumbers,d
ifferentiation,sumsof
seriesanddivisibility.
T
OK:Natureofmathematicsandscience.
W
hatarethedifferentmeaningsof
inductionin
m
athematicsandscience?
T
OK:Knowledgeclaimsinmathematics.Do
p
roofsprovideuswithcompletelycertain
k
nowledge?
T
OK:Knowledgecommunities.W
hojudges
thevalidityofaproof?
1.5
Complexnumbers:thenumber
i
1
=
;the
termsrealpart,imaginarypart,conjugate,
modulusandargument.
Cartesianform
i
z
a
b
=
+
.
Sums,productsandquotientsofcomplex
numbers.
Whensolvingproblems,studentsmayneedto
usetechnology.
A
ppl:Conceptsinelectricalengineering.
Impedanceasacombinationofresistanceand
reactance;alsoapparentpowerasa
c
ombinationofrealandreactivepo
wers.These
c
ombinationstaketheform
i
z
a
b
=
+
.
T
OK:Mathematicsandtheknowe
r.Dothe
w
ordsimaginaryandcomplexmak
ethe
c
onceptsmoredifficultthanifthey
had
d
ifferentnames?
T
OK:Thenatureofmathematics.
Hasi
b
eeninventedorwasitdiscovered?
T
OK:Mathematicsandtheworld.
Whydoes
iappearinsomanyfundamental
lawsof
p
hysics?
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Mathematics HL guide20
Syllabus content
Content
Furtherguidance
L
inks
1.6
Modulusargument(polar)form
i
(cos
isin
)
cis
e
z
r
r
r
=
+
=
=
.
i
er
isalsoknownasEulersform.
Theabilitytoconvertbetweenformsis
expected.
A
ppl:Conceptsinelectricalengineering.
P
haseangle/shift,powerfactorand
apparent
p
owerasacomplexquantityinpolarform.
T
OK:Thenatureofmathematics.Wasthe
c
omplexplanealreadytherebefore
itwasused
torepresentcomplexnumbersgeom
etrically?
T
OK:Mathematicsandtheknower.Why
m
ightitbesaidthat
ie
1
0
+
=
isbeautiful?
Thecomplexplane.
Thecomplexpla
neisalsoknownasthe
Arganddiagram.
1.7
Powersofcomplexnumbers:de
Moivres
theorem.
nthr
ootsofacomplexnumber.
Proofbymathem
aticalinductionforn
+
.
T
OK:Reasonandmathematics.W
hatis
m
athematicalreasoningandwhatroledoes
p
roofplayinthisformofreasoning
?Arethere
e
xamplesofproofthatarenotmath
ematical?
1.8
Conjugaterootsofpolynomiale
quationswith
realcoefficients.
Linkto2.5and2.7.
1.9
Solutionsofsystemsoflinearequations(a
maximumofthreeequationsinthree
unknowns),includingcaseswherethereisa
uniquesolution,aninfinityofsolutionsorno
solution.
Thesesystemsshouldbesolvedusingboth
algebraicandtec
hnologicalmethods,egrow
reduction.
Systemsthathav
esolution(s)maybereferred
toasconsistent.
Whenasystemhasaninfinityofsolutions,a
generalsolution
mayberequired.
Linktovectorsin4.7.
T
OK:Mathematics,sense,percept
ionand
reason.I
fwecanfindsolutionsinhigher
d
imensions,canwereasonthatthesespaces
e
xistbeyondoursenseperception?
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Mathematics HL guide 21
Syllabus content
Top
ic2Core:Functionsandequations
22
hours
Theaimsofthistopicaretoexplore
thenotionoffunctionasaunif
yingthemeinmathematics,and
toapplyfunctionalmethodsto
avarietyof
mathem
aticalsituations.Itisexpectedth
atextensiveusewillbemadeofte
chnologyinboththedevelopmentandtheapplicationofthistopic.
Content
Furtherguidance
Links
2.1
Conceptoffunction
:
(
)
f
x
f
x
:domain,
range;image(value).
Oddandevenfunctions.
Int:Thenotationforfunctionswasdeveloped
byanumberofdifferentmathematiciansinthe
17th
and18th
centuries.
Howdidthenotation
weusetodaybecomeinternational
lyaccepted?
TOK:Thenatureofmathematics.
Is
mathematicssimplythemanipulationof
symbolsunderasetofformalrules?
Compositefunctions
f
g
.
Identityfunction.
(
)()
(
())
f
g
x
fg
x
=
.Linkwith6.2.
One-to-oneandmany-to-onefu
nctions.
Linkwith3.4.
Inversefunction
1
f
,including
domain
restriction.
Self-inversefunctions.
Linkwith6.2.
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Mathematics HL guide22
Syllabus content
Content
Furtherguidance
L
inks
2.2
Thegraphofafunction;itsequ
ation
()
y
f
x
=
.
T
OK:Mathematicsandknowledgeclaims.
D
oesstudyingthegraphofafunctioncontain
t
hesamelevelofmathematicalrigouras
s
tudyingthefunctionalgebraically
(
analytically)?
A
ppl:Sketchingandinterpretingg
raphs;
G
eographySL/HL(geographicskills);
C
hemistry11.3.1.
I
nt:Bourbakigroupanalyticalapp
roachversus
M
andlebrotvisualapproach.
Investigationofkeyfeaturesofgr
aphs,suchas
maximumandminimumvalues,intercepts,
horizontalandverticalasymptotesandsymmetry,
andconsiderationofdomainandrange.
Thegraphsofthefunctions
(
)
y
f
x
=
and
(
)
y
f
x
=
.
Thegraphof
(
)
1
y
f
x
=
giventhegraphof
()
y
f
x
=
.
Useoftechnologytographavarietyof
functions.
2.3
Transformationsofgraphs:tran
slations;
stretches;reflectionsintheaxes.
Thegraphoftheinversefunctio
nasa
reflectioniny
x
=
.
Linkto3.4.
Studentsareexpectedtobeaware
oftheeffectoft
ransformationsonboththe
algebraicexpres
sionandthegraphofa
function.
A
ppl:EconomicsSL/HL1.1
(shiftindemand
a
ndsupplycurves).
2.4
Therationalfunction
,
ax
b
x
cx
d
+ +
andits
graph.
Thereciprocalfunctionisaparticularcase.
Graphsshouldincludebothasymptotesand
anyinterceptsw
ithaxes.
Thefunction
x
x
a
,
0
a
>
,anditsgraph.
Thefunction
log
a
x
x
,
0
x
>
,anditsgraph.
Exponentialand
logarithmicfunctionsas
inversesofeach
other.
Linkto6.2andthesignificanceofe.
Applicationofc
onceptsin2.1,
2.2and2.3.
A
ppl:GeographySL/HL(geograp
hicskills);
P
hysicsSL/HL7.2
(radioactivedecay);
C
hemistrySL/HL16.3
(activation
energy);
E
conomicsSL/HL3.2
(exchangerates).
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Mathematics HL guide 23
Syllabus content
Content
Furtherguidance
Links
2.5
Polynomialfunctionsandtheirgraphs.
Thefactorandremaindertheore
ms.
Thefundamentaltheoremofalg
ebra.
Thegraphicalsignificanceofrepeatedfactors.
Therelationship
betweenthedegreeofa
polynomialfunc
tionandthepossiblenumbers
ofx-intercepts.
2.6
Solvingquadraticequationsusingthequadratic
formula.
Useofthediscriminant
2
4
b
ac
=
to
determinethenatureoftheroots.
Maybereferred
toasrootsofequationsor
zerosoffunctions.
Appl:Chemistry17.2
(equilibrium
law).
Appl:Physics2.1
(kinematics).
Appl:Physics4.2
(energychanges
insimple
h
armonicmotion).
Appl:Physics(HLonly)9.1
(projectile
m
otion).
Aim8
:Thephraseexponentialgr
owthis
u
sedpopularlytodescribeanumberof
p
henomena.Isthisamisleadingus
eofa
m
athematicalterm?
Solvingpolynomialequationsb
othgraphically
andalgebraically.
Sumandproductoftherootsof
polynomial
equations.
Linkthesolutionofpolynomialequationsto
conjugaterootsin1.8.
Forthepolynom
ialequation
0
0
n
r
r
r
a
x
=
=
,
thesumis
1
n n
a a
,
theproductis
0
(
1)n n
a
a
.
Solutionof
x
a
b
=
usinglogarithms.
Useoftechnologytosolveavarietyof
equations,includingthosewher
ethereisno
appropriateanalyticapproach.
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Mathematics HL guide24
Syllabus content
Content
Furtherguidance
Links
2.
7
Solutionsof
(
)
(
)
g
x
f
x
.
Graphicaloralgebraicmethods,
forsimple
polynomialsuptodegree3.
Useoftechnologyfortheseandotherfunctions.
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Mathematics HL guide 25
Syllabus content
Top
ic3Core:Circularfunctionsandtr
igonometry
22
hours
Theaimsofthistopicaretoexplorethe
circularfunctions,tointroducesomeimportanttrigonometricidentitiesandtosolvetrianglesusingtrigonometry.
Onexa
minationpapers,radianmeasureshouldbeassumedunlessotherwiseindicated,
forexample,
by
sin
x
x
.
Content
Furtherguidance
Links
3.1
Thecircle:radianmeasureofangles.
Lengthofanarc;areaofasecto
r.
Radianmeasure
maybeexpressedasmultiples
of,ordecimals.
Linkwith6.2.
Int:Theoriginofdegreesinthem
athematics
ofMesopotamiaandwhyweusem
inutesand
secondsfortime.
TOK:Mathematicsandtheknowe
r.Whydo
weuseradians?(Thearbitrarynatureofdegree
measureversusradiansasrealnum
bersandthe
implicationsofusingthesetwome
asureson
theshapeofsinusoidalgraphs.)
TOK:Mathematicsandknowledg
eclaims.If
trigonometryisbasedonrighttriangles,how
canwesensiblyconsidertrigonom
etricratios
ofanglesgreaterthanarightangle
?
Int:Theoriginofthewordsine.
Appl:PhysicsSL/HL2.2
(forcesand
dynamics).
Appl:TriangulationusedintheGlobal
PositioningSystem(GPS).
Int:WhydidPythagoraslinkthes
tudyof
musicandmathematics?
Appl:Conceptsinelectricalengineering.
Generationofsinusoidalvoltage.
(continued)
3.2
Definitionofcos
,sin
andtan
interms
oftheunitcircle.
Exactvaluesofsin,cosandtan
of
0,
,
,
,
6
4
3
2
andtheirmultip
les.
Definitionofthereciprocaltrigonometric
ratiossec
,csc
andcot.
Pythagoreanidentities:
2
2
cos
sin
1
+
=
;
2
2
1
tan
sec
+
=
;
2
2
1
cot
csc
+
=
.
3.3
Compoundangleidentities.
Doubleangleidentities.
Notrequired:
Proofofcompoundangleidentities.
Derivationofdo
ubleangleidentitiesfrom
compoundangle
identities.
Findingpossiblevaluesoftrigonometricratios
withoutfinding,
forexample,
findingsin
2
givensin.
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Mathematics HL guide26
Syllabus content
Content
Furtherguidance
Links
3.4
Compositefunctionsoftheform
()
sin((
))
f
x
a
b
x
c
d
=
+
+
.
Applications.
(see
notesabove)
TOK:Mathematicsandtheworld.
Musiccan
b
eexpressedusingmathematics.D
oesthis
m
eanthatmusicismathematical,t
hat
m
athematicsismusicalorthatbothare
r
eflectionsofacommontruth?
Appl:PhysicsSL/HL4.1
(kinematicsof
s
impleharmonicmotion).
3.5
Theinversefunctions
arcs
in
x
x
,
arccos
x
x
,
arctan
x
x
;theirdomainsand
ranges;theirgraphs.
3.6
Algebraicandgraphicalmethod
sofsolving
trigonometricequationsinafiniteinterval,
includingtheuseoftrigonometricidentities
andfactorization.
Notrequired:
Thegeneralsolutionoftrigonometric
equations.
TOK:Mathematicsandknowledg
eclaims.
H
owcantherebeaninfinitenumb
erof
d
iscretesolutionstoanequation?
3.7
Thecosinerule
Thesineruleincludingtheamb
iguouscase.
Areaofatriangleas
1
sin
2
ab
C
.
TOK:Natureofmathematics.Ifth
eanglesof
a
trianglecanadduptolessthan180,180or
m
orethan180,whatdoesthistellusaboutthe
factoftheanglesumofatriangleandabout
t
henatureofmathematicalknowle
dge?
Applications.
Examplesinclud
enavigation,problemsintwo
andthreedimensions,includinganglesof
elevationandde
pression.
Appl:PhysicsSL/HL1.3
(vectors
andscalars);
P
hysicsSL/HL2.2
(forcesanddyn
amics).
Int:Theuseoftriangulationtofindthe
c
urvatureoftheEarthinordertosettlea
d
isputebetweenEnglandandFranceover
N
ewtonsgravity.
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Mathematics HL guide 27
Syllabus content
Top
ic4Core:Vecto
rs
24
hours
Theaimofthistopicistointroducetheu
seofvectorsintwoandthreedim
ensions,andtofacilitatesolvingp
roblemsinvolvingpoints,
linesan
dplanes.
Content
Furtherguidance
L
inks
4.1
Conceptofavector.
Representationofvectorsusing
directedline
segments.
Unitvectors;basevectorsi,j,k.
A
im8:Vectorsareusedtosolvem
any
p
roblemsinpositionlocation.
This
canbeused
t
osavealostsailorordestroyabuildingwitha
l
aser-guidedbomb.
Componentsofavector:
1 2
1
2
3
3
.
v v
v
v
v
v
=
=
+
+
v
i
j
k
A
ppl:PhysicsSL/HL1.3
(vectors
andscalars);
P
hysicsSL/HL2.2
(forcesanddyn
amics).
T
OK:Mathematicsandknowledgeclaims.
Y
oucanperformsomeproofsusin
gdifferent
m
athematicalconcepts.
Whatdoes
thistellus
a
boutmathematicalknowledge?
Algebraicandgeometricapproachestothe
following:
thesumanddifferenceoftwovectors;
thezerovector0
,thevectorv;
multiplicationbyascalar,
kv
;
magnitudeofavector,v
;
positionvectors
OA
=a.
Proofsofgeome
tricalpropertiesusingvectors.
AB
=
b
a
Distancebetwee
npointsAandBisthe
magnitudeofAB
.
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Mathematics HL guide28
Syllabus content
Content
Furtherguidance
L
inks
4.2
Thedefinitionofthescalarprod
uctoftwo
vectors.
Propertiesofthescalarproduct:
=
v
w
w
v;
(
)
+
=
+
u
v
w
u
v
u
w
;
(
)
(
)
k
k
=
v
w
v
w
;
2
=
v
v
v
.
Theanglebetweentwovectors.
Perpendicularvectors;parallelvectors.
cos
=
v
w
v
w
,where
istheangle
betweenvandw
.
Linkto3.6.
Fornon-zerovectors,
0
=
v
w
isequivalentto
thevectorsbeingperpendicular.
Forparallelvectors,
=
v
w
v
w
.
A
ppl:PhysicsSL/HL2.2
(forcesa
nd
d
ynamics).
T
OK:Thenatureofmathematics.
Whythis
d
efinitionofscalarproduct?
4.3
Vectorequationofalineintwo
andthree
dimensions:
=r
a+
b.
Simpleapplicationstokinemati
cs.
Theanglebetweentwolines.
Knowledgeofth
efollowingformsfor
equationsofline
s.
Parametricform
:
0
x
x
l
=
+
,
0
y
y
m
=
+
,
0
z
z
n
=
+
.
Cartesianform:
0
0
0
x
x
y
y
z
z
l
m
n
=
=
.
A
ppl:Modellinglinearmotioninthree
d
imensions.
A
ppl:Navigationaldevices,egGP
S.
T
OK:Thenatureofmathematics.
Whymight
itbearguedthatvectorrepresentationoflines
issuperiortoCartesian?
4.4
Coincident,parallel,intersectingandskew
lines;distinguishingbetweenth
esecases.
Pointsofintersection.
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Mathematics HL guide 29
Syllabus content
Content
Furtherguidance
Links
4.5
Thedefinitionofthevectorproductoftwo
vectors.
Propertiesofthevectorproduct:
=
v
w
w
v;
(
)
+
=
+
u
v
w
u
v
u
w
;
(
)
(
)
k
k
=
v
w
v
w
;
=
0
v
v
.
sin
=
v
w
v
w
n,w
herei
stheangle
betweenv
andw
andni
stheunitnormal
vectorwhosedirectionisgivenbytheright-
handscrewrule.
Appl:PhysicsSL/HL6.3(magneticforceand
field).
Geometricinterpretationof
v
w
.
Areasoftriangle
sandparallelograms.
4.6
Vectorequationofaplane
=
+
+
r
a
b
c.
Useofnormalvectortoobtaintheform
=
r
n
a
n.
Cartesianequationofaplaneax
by
cz
d
+
+
=
.
4.7
Intersectionsof:alinewithaplane;two
planes;threeplanes.
Anglebetween:alineandaplane;twoplanes.
Linkto1.9.
Geometricalinte
rpretationofsolutions.
TOK:Mathematicsandtheknowe
r.Whyare
s
ymbolicrepresentationsofthree-d
imensional
o
bjectseasiertodealwiththanvisual
representations?Whatdoesthistellusabout
o
urknowledgeofmathematicsino
ther
d
imensions?
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Mathematics HL guide30
Syllabus content
Top
ic5Core:Statisticsandprobability
36
hours
Theaimofthistopicistointroducebasicconcepts.
Itmaybeconsidered
asthreeparts:manipulationandp
resentationofstatisticaldata(5.1),thelawsof
probab
ility(5.25.4),andrandomvariablesandtheirprobabilitydistributio
ns(5.55.7).Itisexpectedthatm
ostofthecalculationsrequiredwillbedoneon
aGDC
.Theemphasisisonunderstandin
gandinterpretingtheresultsobtained.S
tatisticaltableswillnolong
erbeallowedinexaminations.
Content
Furtherguidance
Links
5.1
Conceptsofpopulation,sample,random
sampleandfrequencydistributionofdiscrete
andcontinuousdata.
Groupeddata:mid-intervalvalu
es,
interval
width,upperandlowerinterval
boundaries.
Mean,variance,standarddeviation.
Notrequired:
Estimationofmeanandvarianceofa
populationfromasample.
Forexamination
purposes,inpapers1and2
datawillbetreatedasthepopulation.
Inexaminations
thefollowingformulaeshould
beused:
1k
i
i
i
fx
n
=
=
,
2
2
2
2
1
1
(
)
k
k
i
i
i
i
i
i
f
x
fx
n
n
=
=
=
=
.
TOK:Thenatureofmathematics.
Whyhave
m
athematicsandstatisticssometim
esbeen
treatedasseparatesubjects?
TOK:Thenatureofknowing.Isth
erea
d
ifferencebetweeninformationanddata?
Aim8:Doestheuseofstatisticsle
adtoan
o
veremphasisonattributesthatcan
easilybe
m
easuredoverthosethatcannot?
Appl:PsychologySL/HL(descriptive
s
tatistics);GeographySL/HL(geographic
s
kills);BiologySL/HL1.1.2
(statistical
a
nalysis).
Appl:Methodsofcollectingdatainreallife
(
censusversussampling).
Appl:Misleadingstatisticsinmediareports.
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15/31
Mathematics HL guide 31
Syllabus content
Content
Furtherguidance
L
inks
5.2
Conceptsoftrial,outcome,equallylikely
outcomes,samplespace(U)and
event.
TheprobabilityofaneventAas
(
)
P(
)
(
)
nA
A
nU
=
.
ThecomplementaryeventsAan
dA(notA).
UseofVenndiagrams,treediag
rams,counting
principlesandtablesofoutcomestosolve
problems.
A
im8
:Whyhasitbeenarguedtha
ttheories
b
asedonthecalculableprobabilitiesfoundin
c
asinosareperniciouswhenapplie
dto
e
verydaylife(egeconomics)?
I
nt:Thedevelopmentofthemathe
matical
t
heoryofprobabilityin17thc
enturyFrance.
5.3
Combinedevents;theformulaf
orP(
)
A
B
.
Mutuallyexclusiveevents.
5.4
Conditionalprobability;thedefinition
(
)
P(
)
P
|
P(
)
A
B
A
B
B
=
.
A
ppl:Useofprobabilitymethodsinmedical
s
tudiestoassessriskfactorsforcertain
d
iseases.
T
OK:Mathematicsandknowledgeclaims.Is
i
ndependenceasdefinedinprobabilisticterms
t
hesameasthatfoundinnormalexperience?
Independentevents;thedefinition
(
)
(
)
(
)
P
|
P
P
|
A
B
A
A
B
=
=
.
UseofBayestheoremforama
ximumofthree
events.
UseofP(
)
P()P()
A
B
A
B
=
toshow
independence.
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Mathematics HL guide32
Syllabus content
Content
Furtherguidance
L
inks
5.5
Conceptofdiscreteandcontinu
ousrandom
variablesandtheirprobabilityd
istributions.
Definitionanduseofprobabilityd
ensityfunctions.
T
OK:Mathematicsandtheknowe
r.Towhat
e
xtentcanwetrustsamplesofdata
?
Expectedvalue(mean),mode,m
edian,
varianceandstandarddeviation
.
Foracontinuousrandomvariable,avalueat
whichtheprobabilitydensityfunctionhasa
maximumvalue
iscalledamode.
Applications.
Examplesinclud
egamesofchance.
A
ppl:Expectedgaintoinsurancecompanies.
5.6
Binomialdistribution,
itsmean
andvariance.
Poissondistribution,
itsmeanandvariance.
Linktobinomialtheoremin1.3.
Conditionsunde
rwhichrandomvariableshave
thesedistributions.
T
OK:Mathematicsandtherealworld.
Isthe
b
inomialdistributioneverauseful
modelfor
a
nactualreal-worldsituation?
Notrequired:
Formalproofofmeansandvari
ances.
5.7
Normaldistribution.
Probabilitiesandvaluesofthevariablemustbe
foundusingtech
nology.
Thestandardizedvalue(z)givesthenumberof
standarddeviationsfromthemean.
A
ppl:ChemistrySL/HL6.2
(collisiontheory);
P
sychologyHL(descriptivestatistics);Biology
S
L/HL1.1.3
(statisticalanalysis).
A
im8:Whymightthemisuseofthenormal
d
istributionleadtodangerousinferencesand
c
onclusions?
T
OK:Mathematicsandknowledgeclaims.To
w
hatextentcanwetrustmathemat
icalmodels
s
uchasthenormaldistribution?
I
nt:DeMoivresderivationofthe
normal
d
istributionandQueteletsuseofittodescribe
lhommemoyen.
Propertiesofthenormaldistribution.
Standardizationofnormalvaria
bles.
Linkto2.3.
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Mathematics HL guide 33
Syllabus content
Top
ic6Core:Calculus
48
hours
Theaimofthistopicistointroducestudentstothebasicconceptsandtech
niquesofdifferentialandintegral
calculusandtheirapplication.
Content
Furtherguidance
L
inks
6.1
Informalideasoflimit,continuityand
convergence.
Definitionofderivativefromfirstprinciples
0
(
)
()
()
lim
h
f
x
h
f
x
f
x
h
+
=
.
Thederivativeinterpretedasag
radient
functionandasarateofchange
.
Findingequationsoftangentsandnormals.
Identifyingincreasinganddecreasing
functions.
Includeresult
0sin
lim
1
=
.
Linkto1.1.
Useofthisdefin
itionforpolynomialsonly.
Linktobinomialtheoremin1.3.
Bothformsofnotation,
d dy x
and
(
)
f
x
,forthe
firstderivative.
T
OK:Thenatureofmathematics.
Doesthe
f
actthatLeibnizandNewtoncame
acrossthe
c
alculusatsimilartimessupportth
eargument
t
hatmathematicsexistspriortoits
discovery?
I
nt:HowtheGreeksdistrustofzeromeant
t
hatArchimedesworkdidnotleadtocalculus.
I
nt:InvestigateattemptsbyIndian
m
athematicians(5001000CE)to
explain
d
ivisionbyzero.
T
OK:Mathematicsandtheknowe
r.What
d
oesthedisputebetweenNewtonandLeibniz
t
ellusabouthumanemotionandm
athematical
d
iscovery?
A
ppl:EconomicsHL1.5
(theoryo
fthefirm);
C
hemistrySL/HL11.3.4
(graphica
l
t
echniques);PhysicsSL/HL2.1
(kinematics).
Thesecondderivative.
Higherderivatives.
Useofbothalge
braandtechnology.
Bothformsofnotation,
22
d dy
x
and
(
)
f
x
,for
thesecondderiv
ative.
Familiaritywith
thenotation
d dn
nyx
and
(
)(
)
n
f
x
.Linkw
ithinductionin1.4.
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Mathematics HL guide34
Syllabus content
Content
Furtherguidance
L
inks
6.2
Derivativesof
n
x
,sin
x,cosx,
tan
x,e
xa
nd
ln
x.
Differentiationofsumsandmultiplesof
functions.
Theproductandquotientrules.
Thechainruleforcompositefunctions.
Relatedratesofchange.
Implicitdifferentiation.
Derivativesofsecx,cscx,cot
x,
x
a
,log
a
x,
arcsin
x,arccosxa
ndarctan
x.
A
ppl:PhysicsHL2.4
(uniformcircularmotion);
P
hysics12.1
(inducedelectromotiveforce(emf)).
T
OK:Mathematicsandknowledgeclaims.
E
ulerwasabletomakeimportanta
dvancesin
m
athematicalanalysisbeforecalcu
lushadbeen
p
utonasolidtheoreticalfoundationbyCauchy
a
ndothers.However,someworkw
asnot
p
ossibleuntilafterCauchyswork.
Whatdoes
thistellusabouttheimportanceof
proofand
thenatureofmathematics?
T
OK:Mathematicsandtherealworld.T
he
s
eeminglyabstractconceptofcalculusallowsus
tocreatemathematicalmodelsthatp
ermithuman
feats,suchasgettingamanontheM
oon.
What
d
oesthistellusaboutthelinksbetween
m
athematicalmodelsandphysicalre
ality?
6.3
Localmaximumandminimumvalues.
Optimizationproblems.
Pointsofinflexionwithzeroandnon-zero
gradients.
Graphicalbehaviouroffunction
s,includingthe
relationshipbetweenthegraphs
of
f,
fandf.
Notrequired:
Pointsofinflexion,where
(
)
f
x
isnot
defined,
forexample,
1
3
y
x
=
at(0,
0).
Testingforthem
aximumorminimumusing
thechangeofsig
nofthefirstderivativeand
usingthesignof
thesecondderivative.
Useoftheterms
concaveupfor
(
)
0
f
x
>
,
concavedown
for
(
)
0
f
x