Maths IB SL Syllabus
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Transcript of Maths IB SL Syllabus
-
8/10/2019 Maths IB SL Syllabus
1/20
Mathematics SL guide 17
Syllabu
s
Syllabuscontent
Top
ic1
Algebra
9
hours
Theaimofthistopicistointroducestudentstosomebasicalgebraicconce
ptsandapplications.
Content
Furtherguidance
L
inks
1.1
Arithmeticsequencesandseries
;sumoffinite
arithmeticseries;geometricsequencesandseries;
sumoffiniteandinfinitegeometricseries.
Sigmanotation.
Technologymay
beusedtogenerateand
displaysequencesinseveralways.
Linkto2.6,exponentialfunctions.
I
nt:Thechesslegend(SissaibnDahir).
I
nt:Aryabhattaissometimesconsideredthe
fatherofalgebra.Comparewith
al-Khawarizmi.
T
OK:HowdidGaussaddupinteg
ersfrom
1
to100?Discusstheideaofmathe
matical
intuitionasthebasisforformalproof.
T
OK:Debateoverthevalidityofthenotionof
infinity:finitistssuchasL.
Kronecker
considerthatamathematicalobjectdoesnot
existunlessitcanbeconstructedfromnatural
n
umbersinafinitenumberofsteps
.
T
OK:WhatisZenosdichotomyp
aradox?
H
owfarcanmathematicalfactsbe
from
intuition?
Applications.
Examplesinclud
ecompoundinterestand
populationgrowth.
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2/20
Mathematics SL guide18
Syllabus content
Content
Furtherguidance
L
inks
1.2
Elementarytreatmentofexpone
ntsand
logarithms.
Examples:
3 4
16
8
;
16
3
log
8
4
;
log
32
5l
og
2
;
4
3
12
(2
)
2
.
A
ppl:Chemistry18.1
(Calculation
ofpH).
T
OK:Arelogarithmsaninvention
or
d
iscovery?(Thistopicisanopport
unityfor
t
eacherstogeneratereflectionon
thenatureof
m
athematics.)
Lawsofexponents;lawsoflogarithms.
Changeofbase.
Examples:
4
ln
7
log
ln
4
7
,
25
5 5
log
log
log
1
25
3
125
25
2
.
Linkto2.6,
loga
rithmicfunctions.
1.3
Thebinomialtheorem:
expansionof(
),
n
a
b
n
.
Countingprinciplesmaybeusedinthe
developmentof
thetheorem.
A
im8:Pascalstriangle.
Attributingtheorigin
o
famathematicaldiscoverytothe
wrong
m
athematician.
I
nt:Theso-calledPascalstrianglewas
k
nowninChinamuchearlierthanPascal.
Calculationofbinomialcoefficientsusing
Pascalstriangleand
n r
.
n r
s
houldbef
oundusingboththeformula
andtechnology.
Example:finding
6 r
f
rom
inputting
6n
r
y
C
X
and
thenreadingcoefficientsfrom
thetable.
Linkto5.8,
bino
mialdistribution.
Notrequired:
formaltreatmentofpermutation
sandformula
forn
rP
.
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Mathematics SL guide 19
Syllabus content
Top
ic2
Functions
andequations
24
hours
Theaimsofthistopicaretoexplore
thenotionofafunctionasaun
ifyingthemeinmathematics,an
dtoapplyfunctionalmethodstoavarietyof
mathematicalsituations.Itisexpectedthatextensiveusewillbemadeof
technologyinboththedevelopm
entandtheapplicationofthistop
ic,ratherthan
elaborateanalyticaltechniques.
Onexa
minationpapers,questionsmay
besetrequiringthegraphingof
functionsthatdonotexplicitly
appearonthe
syllabus,andstudentsmayneedtoch
oosetheappropriateviewingwindow.
Forthosefunctionsexplicitlymentioned,questionsmayalsobeseton
compo
sitionofthesefunctionswiththelinearfunctiony
ax
b
.
Content
Furtherguidance
L
inks
2.1
Conceptoffunction
:
(
)
f
x
fx
.
Domain,range;image(value).
Example:for
2
x
x
, domainis
2
x
,
rangeis
0
y
.
Agraphishelpf
ulinvisualizingtherange.
I
nt:Thedevelopmentoffunctions,
Rene
D
escartes(France),GottfriedWilh
elmLeibniz
(
Germany)andLeonhardEuler(Switzerland).
Compositefunctions.
(
)
(
(
))
f
g
x
f
g
x
.
T
OK:Iszerothesameasnothing
?
T
OK:Ismathematicsaformallanguage?
Identityfunction.
Inversefunction
1
f
.
1
1
(
)(
)
(
)(
)
f
f
x
f
f
x
x
.
Onexaminationpapers,studentswillonlybe
askedtofindthe
inverseofaone-to-onefunction.
Notrequired:
domainrestriction.
2.2
Thegraphofafunction;itsequation
()
y
fx
.
A
ppl:Chemistry11.3.1
(sketching
and
i
nterpretinggraphs);geographicsk
ills.
T
OK:Howaccurateisavisualrep
resentation
o
famathematicalconcept?(Limitsofgraphs
i
ndeliveringinformationaboutfun
ctionsand
p
henomenaingeneral,relevanceofmodesof
r
epresentation.)
Functiongraphingskills.
Investigationofkeyfeaturesof
graphs,suchas
maximumandminimumvalues
,intercepts,
horizontalandverticalasympto
tes,symmetry,
andconsiderationofdomainandrange.
Notethedifferenceinthecommandterms
drawandske
tch.
Useoftechnologytographava
rietyof
functions,includingonesnotsp
ecifically
mentioned.
Ananalyticapproachisalsoexpectedfor
simplefunctions,includingallthoselisted
undertopic2.
Thegraphof
1
(
)
y
f
x
asthereflectionin
theliney
x
ofthegraphof
()
y
fx
.
Linkto6.3,
loca
lmaximumandminimum
points.
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4/20
Mathematics SL guide20
Syllabus content
Content
Furtherguidance
L
inks
2.3
Transformationsofgraphs.
Technologyshouldbeusedtoinvestigatethese
transformations.
A
ppl:Economics1.1
(shiftingofsupplyand
d
emandcurves).
Translations:
(
)
y
f
x
b
;
(
)
y
f
x
a
.
Reflections(inbothaxes):
()
y
fx
;
(
)
y
f
x
.
Verticalstretchwithscalefactorp:
(
)
y
pf
x
.
Stretchinthex-directionwithscalefactor
1 q:
y
f
qx
.
Translationbythevector
3 2
d
enotes
horizontalshiftof3unitstotheright,and
verticalshiftof2down.
Compositetransformations.
Example:
2
y
x
usedtoobtain
2
3
2
y
x
by
astretchofscale
factor3inthey-direction
followedbyatranslationof
0 2
.
2.4
Thequadraticfunction
2
x
ax
bx
c
:its
graph,y-intercept(0,
)c.
Axiso
fsymmetry.
Theform
(
)(
)
x
ax
p
x
q
,
x-intercepts(
,0)
p
and(,
0)
q
.
Theform
2
(
)
x
a
x
h
k
,ve
rtex(,
)
h
k
.
Candidatesaree
xpectedtobeabletochange
fromoneformtoanother.
Linksto2.3,
transformations;2.7,quadratic
equations.
A
ppl:Chemistry17.2
(equilibrium
law).
A
ppl:Physics2.1
(kinematics).
A
ppl:Physics4.2
(simpleharmonicmotion).
A
ppl:Physics9.1
(HLonly)(proje
ctile
m
otion).
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Mathematics SL guide 21
Syllabus content
Content
Furtherguidance
L
inks
2.5
Thereciprocalfunction
1
x
x
,
0
x
:its
graphandself-inversenature.
Therationalfunction
ax
b
x
cx
d
andits
graph.
Examples:
4
2
()
,
3
2
3
hx
x
x
;
7
5
,
2
5
2
x
y
x
x
.
Verticalandhorizontalasympto
tes.
Diagramsshouldincludeallasymptotesand
intercepts.
2.6
Exponentialfunctionsandtheir
graphs:
x
x
a
,
0
a
,
ex
x
.
I
nt:TheBabylonianmethodofmu
ltiplication:
2
2
2
(
)2
a
b
a
b
ab
.SulbaSutr
asinancient
I
ndiaandtheBakhshaliManuscrip
tcontained
a
nalgebraicformulaforsolvingqu
adratic
e
quations.
Logarithmicfunctionsandtheir
graphs:
log
a
x
x
,
0
x
,
ln
x
x
,
0
x
.
Relationshipsbetweenthesefun
ctions:
ln
e
x
x
a
a
;log
x
a
a
x
;
loga
x
a
x
,
0
x
.
Linksto1.1,geo
metricsequences;1.2,
lawsof
exponentsandlogarithms;2.1,
inverse
functions;2.2,g
raphsofinverses;and6.1,
limits.
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Mathematics SL guide22
Syllabus content
Content
Furtherguidance
L
inks
2.7
Solvingequations,bothgraphicallyand
analytically.
Useoftechnologytosolveavarietyof
equations,includingthosewherethereisno
appropriateanalyticapproach.
Solutionsmaybereferredtoasrootsof
equationsorzerosoffunctions.
Linksto2.2,
fun
ctiongraphingskills;and2.3
2.6,equationsin
volvingspecificfunctions.
Examples:
4
5
6
0
e
sin
,
x
x
x
x
.
Solving
2
0
ax
bx
c
,
0
a
.
Thequadraticformula.
Thediscriminant
2
4
b
ac
andthenature
oftheroots,
thatis,
twodistinct
realroots,
two
equalrealroots,norealroots.
Example:Findk
giventhattheequation
2
3
2
0
kx
x
k
hastwoequalrealroots.
Solvingexponentialequations.
Examples:
1
2
10
x
,
1
1
9
3
x
x
.
Linkto1.2,exponentsandlogarithms.
2.8
Applicationsofgraphingskillsandsolving
equationsthatrelatetoreal-lifesituations.
Linkto1.1,geometricseries.
A
ppl:Compoundinterest,growthanddecay;
p
rojectilemotion;brakingdistance;electrical
c
ircuits.
A
ppl:Physics7.2.77.2.9,
13.2.5,
13.2.6,
1
3.2.8
(radioactivedecayandhalf-life)
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Mathematics SL guide 23
Syllabus content
Top
ic3
Circularfu
nctionsandtrig
onometry
16
hours
Theaimsofthistopicaretoexploreth
ecircularfunctionsandtosolve
problemsusingtrigonometry.
On
examinationpapers,radianmeasureshouldbe
assumedunlessotherwiseindicated.
Content
Furtherguidance
L
inks
3.1
Thecircle:radianmeasureofangles;lengthof
anarc;areaofasector.
Radianmeasure
maybeexpressedasexact
multiplesof,
ordecimals.
I
nt:SekiTakakazucalculatingtoten
d
ecimalplaces.
I
nt:Hipparchus,MenelausandPto
lemy.
I
nt:Whyarethere360degreesinacomplete
turn?LinkstoBabylonianmathematics.
T
OK:Whichisabettermeasureo
fangle:
r
adianordegree?Whatarethebestcriteria
b
ywhichtodecide?
T
OK:Euclidsaxiomsasthebuild
ingblocks
o
fEuclideangeometry.
Linktonon-Euclidean
g
eometry.
3.2
Definitionofcos
andsinintermsofthe
unitcircle.
A
im8:WhoreallyinventedPythagoras
theorem?
I
nt:Thefirstworktoreferexplicitlytothe
s
ineasafunctionofanangleisthe
A
ryabhatiyaofAryabhata(ca.510).
T
OK:Trigonometrywasdevelope
dby
s
uccessivecivilizationsandculture
s.Howis
m
athematicalknowledgeconsidere
dfroma
s
ocioculturalperspective?
Definitionoftana
ssin
cos
.
Theequationof
astraightlinethroughthe
originis
ta
n
y
x
.
Exactvaluesoftrigonometricratiosof
0,
,
,
,
6
4
3
2
andtheirmultipl
es.
Examples:
sin
,
cos
,
tan
210
3
2
4
3
2
.
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8/20
Mathematics SL guide24
Syllabus content
Content
Furtherguidance
L
inks
3.3
ThePythagoreanidentity
2
2
cos
sin
1
.
Doubleangleidentitiesforsine
andcosine.
Simplegeometricaldiagramsand/or
technologymay
beusedtoillustratethedouble
angleformulae(andothertrigonometric
identities).
Relationshipbetweentrigonometricratios.
Examples:
Givensin,
fin
dingpossiblevaluesoftan
withoutfinding
.
Given
3
cos
4
x
,andx
isacute,
findsin2x
withoutfinding
x.
3.4
Thecircularfunctionssin
x,co
sxa
ndtan
x:
theirdomainsandranges;amplitude,
their
periodicnature;andtheirgraphs.
A
ppl:Physics4.2
(simpleharmonicmotion).
Compositefunctionsoftheform
(
)
sin
(
)
f
x
a
b
x
c
d
.
Examples:
(
)
tan
4
f
x
x
,
(
)
2cos
3(
4)
1
f
x
x
.
Transformations.
Example:
sin
y
x
usedtoobtain
3sin
2
y
x
byastretchofscalefactor3inthey-direction
andastretchofscalefactor
1 2
inthe
x-direction.
Linkto2.3,
tran
sformationofgraphs.
Applications.
Examplesinclud
eheightoftide,motionofa
Ferriswheel.
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9/20
Mathematics SL guide 25
Syllabus content
Content
Furtherguidance
L
inks
3.5
Solvingtrigonometricequationsinafinite
interval,bothgraphicallyandanalytically.
Examples:2sin
1
x
,0
2
x
,
2sin
2
3cos
x
x
,
o
o
0
180
x
,
2tan
3(
4)
1
x
,
x
.
Equationsleadingtoquadraticequationsin
sin
,
cos
or
tan
x
x
x.
Notrequired:
thegeneralsolutionoftrigonom
etricequations.
Examples:
2
2sin
5cos
1
0
x
x
for0
4
x
,
2sin
cos2
x
x
,
x
.
3.6
Solutionoftriangles.
Pythagorastheorem
isaspecialcaseofthe
cosinerule.
A
im8:Attributingtheoriginofa
m
athematicaldiscoverytothewrong
m
athematician.
I
nt:Cosinerule:Al-KashiandPyt
hagoras.
Thecosinerule.
Thesinerule,
includingtheambiguouscase.
Areaofatriangle,
1
sin
2
ab
C.
Linkwith4.2,scalarproduct,notingthat:
2
2
2
2
c
a
b
c
a
b
a
b
.
Applications.
Examplesinclud
enavigation,problemsintwo
andthreedimen
sions,includinganglesof
elevationandde
pression.
T
OK:Non-Euclideangeometry:anglesum
on
a
globegreaterthan180.
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Mathematics SL guide26
Syllabus content
Top
ic4
Vectors
16
hours
Theai
mofthistopicistoprovideanelementaryintroductiontovectors,includingbothalgebraicandgeom
etricapproaches.
Theuseofdynamicgeometry
softwa
reisextremelyhelpfultovisualizesituationsinthreedimensions.
Content
Furtherguidance
Links
4.1
Vectorsasdisplacementsinthe
planeandin
threedimensions.
Linktothree-dimensionalgeometry,x,y
andz-
axes.
Appl:Physics1.3.2
(vectorsumsa
nd
differences)Physics2.2.2,
2.2.3
(v
ector
resultants).
TOK:Howdowerelateatheoryt
othe
author?Whodevelopedvectorana
lysis:
JWG
ibbsorOHeaviside?
Componentsofavector;colum
n
representation;
1 2
1
2
3
3v v
v
v
v
v
v
i
j
k.
Componentsare
withrespecttotheunit
vectorsi,jandk
(standardbasis).
Algebraicandgeometricapproachestothe
following:
Applicationstosimplegeometricfiguresare
essential.
thesumanddifferenceoftw
ovectors;the
zerovector,thevectorv;
Thedifferenceo
fva
ndwi
s
(
)
v
w
v
w
.Vectorsumsanddifferences
canberepresentedbythediagonalsofa
parallelogram.
multiplicationbyascalar,
k
v;parallel
vectors;
Multiplicationb
yascalarcanbeillustratedby
enlargement.
magnitudeofavector,v;
unitvectors;basevectors;i,
jandk;
positionvectorsOA
a
;
AB
OB
OA
b
a.
Distancebetwee
npointsAandBisthe
magnitudeofA
B
.
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Mathematics SL guide 27
Syllabus content
Content
Furtherguidance
L
inks
4.2
Thescalarproductoftwovectors.
Thescalarprodu
ctisalsoknownasthedot
product.
Linkto3.6,cosi
nerule.
Perpendicularvectors;parallelv
ectors.
Fornon-zerovectors,
0
v
w
isequivalentto
thevectorsbeingperpendicular.
Forparallelvect
ors,
k
w
v,
v
w
v
w
.
Theanglebetweentwovectors.
4.3
Vectorequationofalineintwo
andthree
dimensions:
t
r
a
b.
Relevanceofa
(position)andb
(direction).
Interpretationof
t
astimeandb
asvelocity,
withb
representingspeed.
A
im8:Vectortheoryisusedfortr
acking
d
isplacementofobjects,
including
forpeaceful
a
ndharmfulpurposes.
T
OK:Arealgebraandgeometrytwoseparate
d
omainsofknowledge?(Vectoralgebraisa
g
oodopportunitytodiscusshowgeometrical
p
ropertiesaredescribedandgenera
lizedby
a
lgebraicmethods.)
Theanglebetweentwolines.
4.4
Distinguishingbetweencoincidentandparallel
lines.
Findingthepointofintersection
oftwolines.
Determiningwhethertwolinesintersect.
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Mathematics SL guide28
Syllabus content
Top
ic5
Statisticsa
ndprobability
35
hours
Theaimofthistopicistointroducebas
icconcepts.
Itisexpectedthatmo
stofthecalculationsrequiredwillbedoneusingtechnology,
bute
xplanationsof
calculationsbyhandmayenhanceunderstanding.
Theemphasisisonunde
rstandingandinterpretingtheresu
ltsobtained,
incontext.Statisticaltableswillno
longer
beallowedinexaminations.Whilemanyofthecalculationsrequire
dinexaminationsareestimates,i
tislikelythatthecommandterms
writedown,
find
andcalculatewillbeused.
Content
Furtherguidance
L
inks
5.1
Conceptsofpopulation,sample,random
sample,
discreteandcontinuous
data.
Presentationofdata:frequencydistributions
(tables);frequencyhistogramswithequalclass
intervals;
Continuousand
discretedata.
A
ppl:Psychology:descriptivestatistics,
r
andomsample(variousplacesintheguide).
A
im8
:Misleadingstatistics.
I
nt:TheStPetersburgparadox,Ch
ebychev,
P
avlovsky.
box-and-whiskerplots;outliers.
Outlierisdefine
dasmorethan1.5
IQR
from
thenearestquartile.
Technologymaybeusedtoproduce
histogramsandbox-and-whiskerplots.
Groupeddata:useofmid-interv
alvaluesfor
calculations;intervalwidth;upp
erandlower
intervalboundaries;modalclass.
Notrequired:
frequencydensityhistograms.
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Mathematics SL guide 29
Syllabus content
Content
Furtherguidance
L
inks
5.2
Statisticalmeasuresandtheirin
terpretations.
Centraltendency:mean,median,mode.
Quartiles,percentiles.
Onexamination
papers,datawillbetreatedas
thepopulation.
Calculationofm
eanusingformulaand
technology.
Studentsshouldusemid-interval
valuestoestimatethemeanofgroupeddata.
A
ppl:Psychology:descriptivestatistics
(
variousplacesintheguide).
A
ppl:Statisticalcalculationstoshowpatterns
a
ndchanges;geographicskills;sta
tistical
g
raphs.
A
ppl:Biology1.1.2
(calculatingm
eanand
s
tandarddeviation);Biology1.1.4
(comparing
m
eansandspreadsbetweentwoor
more
s
amples).
I
nt:Discussionofthedifferentfor
mulaefor
v
ariance.
T
OK:Dodifferentmeasuresofce
ntral
t
endencyexpressdifferentpropertiesofthe
d
ata?Arethesemeasuresinvented
or
d
iscovered?Couldmathematicsmake
a
lternative,equallytrue,formulae?
Whatdoes
t
histellusaboutmathematicaltruths?
T
OK:Howeasyisittoliewithstatistics?
Dispersion:range,interquartile
range,
variance,standarddeviation.
Effectofconstantchangestoth
eoriginaldata.
Calculationofstandarddeviation/variance
usingonlytechn
ology.
Linkto2.3,
tran
sformations.
Examples:
If5issubtracted
fromallthedataitems,then
themeanisdecr
easedby5,
butthestandard
deviationisunchanged.
Ifallthedataite
msaredoubled,
themedianis
doubled,
butthe
varianceisincreasedbya
factorof4.
Applications.
5.3
Cumulativefrequency;cumulat
ivefrequency
graphs;usetofindmedian,quartiles,
percentiles.
Valuesofthemedianandquartilesproduced
bytechnologym
aybedifferentfromthose
obtainedfroma
cumulativefrequencygraph.
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Mathematics SL guide30
Syllabus content
Content
Furtherguidance
L
inks
5.4
Linearcorrelationofbivariated
ata.
Independentvar
iablex,
dependentvariabley.
A
ppl:Chemistry11.3.3
(curvesof
bestfit).
A
ppl:Geography(geographicskills).
M
easuresofcorrelation;geographicskills.
A
ppl:Biology1.1.6
(correlationdoesnot
i
mplycausation).
T
OK:Canwepredictthevalueof
xfromy,
u
singthisequation?
T
OK:Canalldatabemodelledby
a(known)
m
athematicalfunction?Considerthereliability
a
ndvalidityofmathematicalmode
lsin
d
escribingreal-lifephenomena.
Pearsonsproductmomentcorrelation
coefficientr.
Technologyshouldbeusedtocalculater.
However,handcalculationsofrmayenhance
understanding.
Positive,zero,negative;strong,weak,no
correlation.
Scatterdiagrams;linesofbestfit.
Thelineofbest
fitpassesthroughthemean
point.
Equationoftheregressionlineofyonx.
Useoftheequationforpredictionpurposes.
Mathematicalandcontextualinterpretation.
Notrequired:
thecoefficientofdetermination
R2
.
Technologyshouldbeusedfindtheequation.
Interpolation,ex
trapolation.
5.5
Conceptsoftrial,outcome,equallylikely
outcomes,samplespace(U)andevent.
Thesamplespac
ecanberepresented
diagrammaticall
yinmanyways.
T
OK:Towhatextentdoesmathem
aticsoffer
m
odelsofreallife?Istherealways
afunction
t
omodeldatabehaviour?
TheprobabilityofaneventAis
(
)
P(
)
(
)
nA
A
nU
.
ThecomplementaryeventsAan
dA
(notA).
UseofVenndiagrams,treediagramsand
tablesofoutcomes.
Experimentsusi
ngcoins,dice,cardsandsoon,
canenhanceund
erstandingofthedistinction
between(experimental)relativefrequencyand
(theoretical)pro
bability.
Simulationsmay
beusedtoenhancethistopic.
Linksto5.1,
frequency;5.3,cumulative
frequency.
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Syllabus content
Content
Furtherguidance
L
inks
5.6
Combinedevents,P(
)
A
B
.
Mutuallyexclusiveevents:P(
)
0
A
B
.
Conditionalprobability;thedef
inition
P(
)
P
|
P(
)
A
B
A
B
B
.
Independentevents;thedefiniti
on
P
|
P()
P
|
A
B
A
A
B
.
Probabilitieswithandwithoutr
eplacement.
Thenon-exclusi
vityofor.
Problemsareoft
enbestsolvedwiththeaidofa
Venndiagramortreediagram,withoutexplicit
useofformulae.
A
im8:Thegamblingissue:useof
probability
i
ncasinos.Couldorshouldmathem
aticshelp
i
ncreaseincomesingambling?
T
OK:Ismathematicsusefultome
asurerisks?
T
OK:Cangamblingbeconsideredasan
a
pplicationofmathematics?(Thisisagood
o
pportunitytogenerateadebateon
thenature,
r
oleandethicsofmathematicsregardingits
a
pplications.)
5.7
Conceptofdiscreterandomvariablesandtheir
probabilitydistributions.
Simpleexample
sonly,suchas:
1
P(
)
(4
)
18
X
x
x
for
1,
2,
3
x
;
5
6
7
P(
)
,
,
18
18
18
X
x
.
Expectedvalue(mean),E()Xf
ordiscretedata.
Applications.
E(
)
0
X
indicatesafairgamewhereX
representsthegainofoneoftheplayers.
Examplesinclud
egamesofchance.
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Syllabus content
Content
Furtherguidance
L
inks
5.8
Binomialdistribution.
Meanandvarianceofthebinom
ial
distribution.
Notrequired:
formalproofofmeanandvariance.
Linkto1.3,
bino
mialtheorem.
Conditionsunde
rwhichrandomvariableshave
thisdistribution.
Technologyisusuallythebestwayof
calculatingbinomialprobabilities.
5.9
Normaldistributionsandcurves
.
Standardizationofnormalvariables(z-values,
z-scores).
Propertiesofthenormaldistribution.
Probabilitiesand
valuesofthevariablemustbe
foundusingtech
nology.
Linkto2.3,
transformations.
Thestandardizedvalue(z
)givesthenumber
ofstandarddeviationsfromthemean.
A
ppl:Biology1.1.3
(linkstonorm
al
d
istribution).
A
ppl:Psychology:descriptivestatistics
(
variousplacesintheguide).
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Mathematics SL guide 33
Syllabus content
Top
ic6
Calculus
40
hours
Theaimofthistopicistointroducestudentstothebasicconceptsandtech
niquesofdifferentialandintegral
calculusandtheirapplications.
Content
Furtherguidance
L
inks
6.1
Informalideasoflimitandconv
ergence.
Example:0.3,0.3
3,
0.3
33,...convergesto
1 3
.
Technologyshouldbeusedtoexploreideasof
limits,numericallyandgraphically.
A
ppl:Economics1.5
(marginalco
st,marginal
r
evenue,marginalprofit).
A
ppl:Chemistry11.3.4
(interpretingthe
g
radientofacurve).
A
im8:ThedebateoverwhetherN
ewtonor
L
eibnitzdiscoveredcertaincalculu
sconcepts.
T
OK:Whatvaluedoestheknowledgeof
limitshave?Isinfinitesimalbehaviour
a
pplicabletoreallife?
T
OK:Opportunitiesfordiscussing
hypothesis
f
ormationandtesting,andthenthe
formal
p
roofcanbetackledbycomparing
certain
c
ases,
throughaninvestigativeapp
roach.
Limitnotation.
Example:
2
3
lim
1
x
x x
Linksto1.1,infi
nitegeometricseries;2.52.7,
rationalandexponentialfunctions,and
asymptotes.
Definitionofderivativefromfirstprinciplesas
0
(
)
()
()
lim
h
f
x
h
f
x
f
x
h
.
Useofthisdefin
itionforderivativesofsimple
polynomialfunc
tionsonly.
Technologycouldbeusedtoillustrateother
derivatives.
Linkto1.3,
bino
mialtheorem.
Useofbothform
sofnotation,
d dy x
and
f
x
,
forthefirstderivative.
Derivativeinterpretedasgradientfunctionand
asrateofchange.
Identifyinginter
valsonwhichfunctionsare
increasingordecreasing.
Tangentsandnormals,andtheirequations.
Notrequired:
analyticmethodsofcalculating
limits.
Useofbothanalyticapproachesand
technology.
Technologycan
beusedtoexploregraphsand
theirderivatives.
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Syllabus content
Content
Furtherguidance
Links
6.
2
Derivativeof
(
)
n
x
n
,sinx,cosx,
tanx
,
ex
andlnx
.
Differentiationofasum
andarealmultipleof
thesefunctions.
Thechainruleforcompositefu
nctions.
Theproductandquotientrules.
Linkto2.1,com
positionoffunctions.
Technologymay
beusedtoinvestigatethechain
rule.
Thesecondderivative.
Useofbothform
sofnotation,
2
2
d d
yx
and
()
f
x
.
Extensiontohigherderivatives.
d dn
nyx
and
(
)
n
f
x
.
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Syllabus content
Content
Furtherguidance
L
inks
6.3
Localmaximumandminimumpoints.
Testingformaximumorminimum.
Usingchangeof
signofthefirstderivativeand
usingsignofthe
secondderivative.
Useoftheterms
concave-upfor
(
)
0
f
x
,
andconcave-do
wnfor
(
)
0
f
x
.
A
ppl:profit,area,volume.
Pointsofinflexionwithzeroandnon-zero
gradients.
Atapointofinflexion,
(
)
0
f
x
andchanges
sign(concavitychange).
(
)
0
f
x
isnot
asufficientconditionfora
pointofinflexion:forexample,
4
y
x
at(0,
0).
Graphicalbehaviouroffunction
s,
includingtherelationshipbetwe
enthe
graphsoff,
f
and
f.
Optimization.
Bothglobal(forlarge
x
)andlocal
behaviour.
Technologycan
displaythegraphofa
derivativewitho
utexplicitlyfindingan
expressionforth
ederivative.
Useofthefirsto
rsecondderivativetestto
justifymaximum
and/orminimumvalues.
Applications.
Notrequired:
pointsofinflexionwhere
(
)
f
x
isnotdefined:
forexample,
1
3
y
x
at(0,0).
Examplesinclud
eprofit,area,volume.
Linkto2.2,grap
hingfunctions.
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Content
Furtherguidance
L
inks
6.4
Indefiniteintegrationasanti-dif
ferentiation.
Indefiniteintegralof
(
)
n
x
n
,sin
x
,cosx,
1 x
andex.
1
d
ln
x
x
C
x
,
0
x
.
Thecompositesofanyofthese
withthelinear
functionax
b
.
Example:
1
()
cos(2
3
)
()
sin(2
3)
2
f
x
x
fx
x
C
.
Integrationbyinspection,orsubstitutionofthe
form
(())'()d
fgx
g
x
x
.
Examples:
4
2
2
2
1
d
,
sin
d
,
d
sin
cos
x
x
x
x
x
x
x
x x
.
6.5
Anti-differentiationwithaboun
darycondition
todeterminetheconstantterm.
Example:
if
2
d
3
dy
x
x
x
and
10
y
when
0
x
,then
3
2
1
1
0
2
y
x
x
.
I
nt:Successfulcalculationofthev
olumeof
thepyramidalfrustumbyancientE
gyptians
(
EgyptianMoscowpapyrus).
U
seofinfinitesimalsbyGreekgeo
meters.
Definiteintegrals,
bothanalyticallyandusing
technology.
(
)d
()
(
)
b a
g
x
x
gb
g
a
.
Thevalueofsom
edefiniteintegralscanonly
befoundusingtechnology.
A
ccuratecalculationofthevolume
ofa
c
ylinderbyChinesemathematician
LiuHui
Areasundercurves(betweenthecurveandthe
x-axis).
Areasbetweencurves.
Volumesofrevolutionaboutthex-axis.
Studentsareexp
ectedtofirstwriteacorrect
expressionbeforecalculatingthearea.
Technologymay
beusedtoenhance
understandingofareaandvolume.
I
nt:IbnAlHaytham:firstmathematicianto
c
alculatetheintegralofafunction,
inorderto
f
indthevolumeofaparaboloid.
6.6
Kinematicproblemsinvolvingd
isplacements,
velocityvandaccelerationa.
d ds
v
t
;
22
d
d
d
d
v
s
a
t
t
.
A
ppl:Physics2.1
(kinematics).
Totaldistancetravelled.
Totaldistancetravelled
21
d
t t
v
t
.