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10 Mathematical studies SL guide
Syllabus outline
Syllabus
Syllabus component
Teaching
hours
SL
All topics are compulsory. Students must study all the sub-topics in each of the topics in the
syllabus as listed in this guide. Students are also required to be familiar with the topics listed
as prior learning.
Topic 1
Number and algebra
20
Topic 2
Descriptive statistics
12
Topic 3
Logic, sets and probability
20
Topic 4
Statistical applications
17
Topic 5
Geometry and trigonometry
18
Topic 6
Mathematical models
20
Topic 7
Introduction to differential calculus
18
Project
The project is an individual piece of work involving the collection of information or
the generation of measurements, and the analysis and evaluation of the information or
measurements.
25
Total teaching hours 150
It is essential that teachers are allowed the prescribed minimum number of teaching hours necessary to meet
the requirements of the mathematical studies SL course. At SL the minimum prescribed number of hours is
150 hours.
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14 Mathematical studies SL guide
Syllabus
Prior learning topics
As noted in the previous section on prior learning, it is expected that all students have extensive
previous mathematical experiences, but these will vary. It is expected that mathematical studies SL
students will be familiar with the following topics before they take the examinations, because
questions assume knowledge of them. Teachers must therefore ensure that any topics listed here that
are unknown to their students at the start of the course are included at an early stage. They should also
take into account the existing mathematical knowledge of their students to design an appropriate
course of study for mathematical studies SL.
Students must be familiar with SI (Systme International) units of length, mass and time, and their
derived units.
The reference given in the left-hand column is to the topic in the syllabus content; for example, 1.0
refers to the prior learning for Topic 1Number and algebra.
Learning how to use the graphic display calculator (GDC) effectively will be an integral part of the
course, not a separate topic. Time has been allowed in each topic of the syllabus to do this.
Content Further guidance
1.0 Basic use of the four operations of
arithmetic, using integers, decimals
and fractions, including order of
operations.
Prime numbers, factors and
multiples.
Simple applications of ratio,
percentage and proportion.
Examples: 2(3 4 7) 62 ; 2 3 4 7 34 .
Basic manipulation of simple
algebraic expressions, including
factorization and expansion.
Examples: ( );ab ac a b c 2( 1)( 2) 3 2x x x x .
Rearranging formulae.Example:
1 2
2
AA bh h
b .
Evaluating expressions by
substitution.Example: If 3x then
2 22 3 ( 3) 2( 3) 3 18x x .
Solving linear equations in one
variable.Examples: 3( 6) 4( 1) 0x x ;
64 7
5
x .
Solving systems of linear equations
in two variables.Example: 3 4 13x y ,
12 1
3x y .
Evaluating exponential expressionswith integer values.
Examples: ,ba b ; 4 1216
;
4( 2) 16 .
Use ofinequalities , , , .
Intervals on the real number line.
Example: 2 5,x x .
Solving linear inequalities. Example: 2 5 7x x .
Familiarity withcommonly
accepted world currencies.
Examples: Swiss franc (CHF); United States dollar (USD);
British pound sterling (GBP); euro (EUR); Japanese yen
(JPY); Australian dollar (AUD).
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Mathematical studies SL guide 15
Prior learning topics
Content Further guidance
2.0 The collection of data and its
representation in bar charts, pie
charts and pictograms.
5.0 Basic geometric concepts: point,
line, plane, angle.
Simple two-dimensional shapes and
their properties, including perimeters
and areas of circles, triangles,
quadrilaterals and compound
shapes.
SI units for length and area.
Pythagoras theorem.
Coordinates in two dimensions.
Midpoints, distance between points.
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16 Mathematical studies SL guide
Syllabuscontent
Syllabus
Topic1
Numberandalgebra
20hours
Theaimsofthistopicareto
introducesomebasicelementsandc
onceptsofmathematics,andtolinkthesetofinancialandotherapplicatio
ns.
Content
Furtherguidance
Links
1.1
Naturalnumbers,;integers,;rational
numbers,
;andrealnumbers,.
Notrequired:
proofofirrationali
ty,
forexample,of
2
.
Linkwithdomainandrange6.1.
Int:Historicaldevelopmentofnumbersystem.
Awarenessthatourmodernnu
meralsare
developedfromtheArabicnotation.
TOK:Domathematicalsymb
olshavesensein
thesamewaythatwordshave
sense?Iszero
different?Arethesenumberscreatedor
discovered?Dothesenumbersexist?
1.2
Approximation:de
cimalplaces,significant
figures.
Percentageerrors.
Studentsshouldbeawareoftheerrorsthat
can
result
fromprematurerounding.
Appl:Currencyapproximationstonearest
wholenumber,egpeso,yen.C
urrency
approximationstonearestcent/penny,egeuro,
dollar,pound.
Appl:Physics1.1
(rangeofm
agnitudes).
Appl:Meteorology,alternativ
erounding
methods.
Appl:Biology2.1.5
(microscopic
measurement).
TOK:Appreciationofthedifferencesofscale
innumber,andofthewaynum
bersareused
thatarewellbeyondourevery
dayexperience.
Estimation.
Studentsshouldbeabletorecognizewheth
er
theresultsofcalculationsarereasonable,
includ
ingreasonablevaluesof,forexample,
length
s,anglesandareas.
Forex
ample,
lengthscannotbenegative.
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Mathematical studies SL guide 17
Syllabus content
Content
Furtherguidance
Links
1.3
Expressingnumbersintheform
10k
a
,where
1
10
a
andkis
aninteger.
Studentsshouldbeabletousescientificmode
onthe
GDC.
Appl:Verylargeandverysm
allnumbers,eg
astronomicaldistances,sub-atomicparticles;
Physics1.1;globalfinancialfigures.
Appl:Chemistry1.1(Avogad
rosnumber).
Appl:Physics1.2(scientificn
otation).
Appl:Chemistryandbiology
(scientific
notation).
Appl:Earthscience(earthqua
kemeasurement
scale).
Operationswithnu
mbersinthisform.
Calculatornotationisnotacceptable.
Forex
ample,5.2E3isnotacceptable.
1.4
SI(SystmeInterna
tional)andotherbasicunits
ofmeasurement:forexample,kilogram(kg),
metre(m),second(s),litre(l),metrepersecond
(ms
1),Celsiussca
le.
Studentsshouldbeabletoconvertbetween
differentunits.
Linkwiththeformofthenotationin1.3,for
example,
6
5km
5
10
mm
.
Appl:Speed,acceleration,force;Physics2.1,
Physics2.2;concentrationofasolution;
Chemistry1.5.
Int:SInotation.
TOK:DoestheuseofSInota
tionhelpusto
thinkofmathematicsasauniversal
language?
TOK:Whatismeasurable?H
owcanone
measuremathematicalability?
1.5
Currencyconversions.
Studentsshouldbeabletoperformcurrenc
y
transa
ctionsinvolvingcommission.
Appl:Economics3.2(exchangerates).
Aim8:Theethicalimplicationsoftradingin
currencyanditseffectondifferentnational
communities.
Int:Theeffectoffluctuations
incurrencyrates
oninternationaltrade.
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Syllabus content
Content
Furtherguidance
Links
1.6
UseofaGDCtosolve
pairsoflinearequationsintwovariables
Inexa
minations,nospecificmethodof
solutionwillberequired.
TOK:Equationswithnosolu
tions.Awareness
thatwhenmathematicianstalk
about
imaginaryorrealsolution
stheyareusing
precisetechnicaltermsthatdo
nothavethe
samemeaningastheeveryday
terms.
quadraticequ
ations.
Standardterminology,suchaszerosorroo
ts,
shouldbetaught.
Linkwithquadraticmodelsin6.3.
1.7
Arithmeticsequencesandseries,andtheir
applications.
TOK:Informalandformalreasoningin
mathematics.Howdoesmathematicalproof
differfrom
goodreasoningin
everydaylife?Is
mathematicalreasoningdiffer
entfrom
scientificreasoning?
TOK:Beautyandelegancein
mathematics.
Fibonaccinumbersandconnectionswiththe
Goldenratio.
Useoftheformula
eforthenthterm
andthe
sum
ofthefirstntermsofthesequence.
StudentsmayuseaGDCforcalculations,but
theyw
illbeexpectedtoidentifythefirstte
rm
andth
ecommondifference.
1.8
Geometricsequencesandseries.
Useoftheformula
eforthenthterm
andthe
sum
ofthefirstntermsofthesequence.
Notrequired:
formalproofsoffo
rmulae.
StudentsmayuseaGDCforcalculations,but
theyw
illbeexpectedtoidentifythefirstte
rm
andth
ecommonratio.
Notrequired:
useoflogarithmstofindn,giventhesum
of
thefirstnterms;sumstoinfinity.
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Mathematical studies SL guide 19
Syllabus content
Content
Furtherguidance
Links
1.9
Financialapplicati
onsofgeometricsequences
andseries:
compoundinterest
annualdepreciation.
Notrequired:
useoflogarithms.
Useo
ftheGDCisexpected,
including
built-
infinancialpackages.
Theconceptofsimpleinterestmaybeused
as
anintroductiontocompoundinterestbutw
ill
notbe
examined.
Inexa
minations,questionsthataskstudentsto
derive
theformulawillnotbeset.
Comp
oundinterestcanbecalculatedyearly,
half-y
early,quarterlyormonthly.
Linkwithexponentialmodels6.4.
Appl:Economics3.2
(exchan
gerates).
Aim8
:Ethicalperceptionsof
borrowingand
lendingmoney.
Int:Doallsocietiesviewinve
stmentand
interestinthesameway?
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Syllabus content
Topic2
Descriptivestatistics
12hours
Theaimofthistopicistod
eveloptechniquestodescribeandinterpretsetsofdata,inpreparationfor
furtherstatisticalapplications.
Content
Furthe
rguidance
Links
2.1
Classificationofdataasdiscreteorcontinuous.
Studentsshouldunderstandtheconceptof
populationandofrepresentativeandrandom
sampling.Samplingwillnotbeexaminedb
ut
canbe
usedininternalassessment.
Appl:Psychology3(research
methodology).
Appl:Biology1(statisticalan
alysis).
TOK:Validityofdataandintroductionof
bias.
2.2
Simplediscretedata:frequencytables.
2.3
Groupeddiscreteo
rcontinuousdata:frequency
tables;mid-interva
lvalues;upperandlower
boundaries.
Frequencyhistogra
ms.
Inexa
minations,frequencyhistogramswill
havee
qualclassintervals.
Appl:Geography(geographic
alanalyses).
2.4
Cumulativefrequencytablesforgrouped
discretedataandforgroupedcontinuousdata;
cumulativefrequen
cycurves,medianand
quartiles.
Box-and-whiskerd
iagram.
Notrequired:
treatmentofoutliers.
UseofGDCtoproducehistogramsandbox-
and-w
hiskerdiagrams.
2.5
Measuresofcentra
ltendency.
Forsimplediscrete
data:mean;median;mode.
Forgroupeddiscre
teandcontinuousdata:
estimateofamean
;modalclass.
Studentsshouldusemid-intervalvaluesto
estima
tethemeanofgroupeddata.
Inexa
minations,questionsusing
notation
willnotbeset.
Aim8:Theethicalimplicationsofusing
statisticstomislead.
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Syllabus content
Content
Furthe
rguidance
Links
2.6
Measuresofdisper
sion:range,
interquartile
range,standarddeviation.
Studentsshouldusemid-intervalvaluesto
estima
tethestandarddeviationofgrouped
data.
Inexa
minations:
s
tudentsareexpectedtouseaGDCto
calculatestandarddeviations
thedatasetwillbetreatedasthe
population.
StudentsshouldbeawarethattheIBnotation
mayd
ifferfromthenotationontheirGDC.
Useofcomputerspreadsheetsoftwareis
encouragedinthetreatmentofthistopic.
Int:Thebenefitsofsharingan
danalysingdata
fromdifferentcountries.
TOK:Isstandarddeviationamathematical
discoveryoracreationofthehumanmind?
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Syllabus content
Topic3
Logic
,setsandprobability
20hours
Theaimsofthistopicareto
introducetheprinciplesoflogic,to
usesettheorytointroduceprobability,andtodeterminethelikelihoodof
randomeventsusinga
varietyoftechniques.
Content
Furtherguidance
Links
3.1
Basicconceptsofsymboliclogic:definitionof
aproposition;symbolicnotationof
propositions.
3.2
Compoundstatements:implication,;
equivalence,;n
egation,;conjunction,
;disjunction,
;exclusivedisjunction,.
Translationbetwee
nverbalstatementsand
symbolicform.
3.3
Truthtables:conce
ptsoflogicalcontradiction
andtautology.
Amaximumofthreepropositionswillbeu
sed
intruthtables.
Truth
tablescanbeusedtoillustratethe
associativeanddistributivepropertiesof
conne
ctives,andforvariationsofimplication
andequivalencestatements,forexample,
q
p
.
3.4
Converse,inverse,
contrapositive.
Logicalequivalenc
e.
Appl:Useofargumentsinde
velopinga
logicalessaystructure.
Appl:Computerprogramming;digitalcircuits;
PhysicsHL14.1;PhysicsSLC1.
TOK:Inductiveanddeductiv
elogic,fallacies.
Testingthevalidityofsimplearguments
throughtheuseof
truthtables.
Theto
picmaybeextendedtoinclude
syllog
isms.Inexaminationsthesewillnot
be
tested
.
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Syllabus content
Content
Furthe
rguidance
Links
3.5
Basicconceptsofs
ettheory:elementsx
A
,
subsetsA
B
;int
ersectionA
B
;union
A
B
;compleme
ntA
.
Venndiagramsand
simpleapplications.
Notrequired:
knowledgeofdeM
organslaws.
Inexaminations,theuniversalsetU
will
includenomorethanthreesubsets.
Theem
ptysetisdenotedby.
3.6
Samplespace;eventA;complementaryevent,
A
.Probabilityofanev
ent.
Probabilityofacomplementaryevent.
Expectedvalue.
Probabilitymaybeintroducedandtaughtina
practicalwayusingcoins,dice,playingcards
andotherexamplestodemonstraterandom
behaviour.
Inexaminations,questionsinvolvingplaying
cardswillnotbeset.
Appl:Actuarialstudies,proba
bilityoflife
spansandtheireffectoninsurance.
Appl:Governmentplanningb
asedon
projectedfigures.
TOK:Theoreticalandexperim
ental
probability.
3.7
Probabilityofcombinedevents,mutually
exclusiveevents,independentevents.
Studen
tsshouldbeencouragedtousethem
ost
approp
riatemethodinsolvingindividual
questions.
Appl:Biology4.3
(theoreticalgenetics);
Biology4.3.2
(Punnettsquares).
Appl:PhysicsHL13.1
(determ
iningthe
positionofanelectron);PhysicsSLB1.
Aim8:Theethicsofgambling.
TOK:Theperceptionofrisk,inbusiness,in
medicineandsafetyintravel.
Useoftreediagram
s,Venndiagrams,sample
spacediagramsand
tablesofoutcomes.
Probabilityusingwithreplacementand
withoutreplacement.
Conditionalprobab
ility.
Probab
ilityquestionswillbeplacedincontext
andwillmakeuseofdiagrammatic
representations.
Inexaminations,questionsrequiringthe
exclusiveuseoftheformulainsection3.7of
thefor
mulabookletwillnotbeset.
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Syllabus content
Topic4
Statis
ticalapplications
17hours
Theaimsofthistopicareto
developtechniquesininferentialsta
tisticsinordertoanalysesetsofdata
,drawconclusionsandinterpretthes
e.
Content
Furthe
rguidance
Links
4.1
Thenormaldistribution.
Theconceptofara
ndomvariable;ofthe
parameters
and
;ofthebellshape;the
symmetryaboutx
.
Studentsshouldbeawarethatapproximate
ly
68%o
fthedataliesbetween
,95%lies
between
2
and99%l
iesbetween
3
.
Appl:Examplesofmeasurements,ranging
frompsychologicaltophysicalphenomena,
thatcanbeapproximated,tovaryingdegrees,
bythenormaldistribution.
Appl:Biology1(statisticalan
alysis).
Appl:Physics3.2(kineticmoleculartheory).
Diagrammaticrepr
esentation.
Useofsketchesofnormalcurvesandshading
whenusingtheGDCisexpected.
Normalprobability
calculations.
StudentswillbeexpectedtousetheGDC
whencalculatingprobabilitiesandinverse
norma
l.
Expectedvalue.
Inversenormalcalculations.
Inexa
minations,inversenormalquestionswill
notinvolvefindingthemeanorstandard
deviat
ion.
Notrequired:
Transformationof
anynormalvariabletothe
standardizednormal.
Transformationofanynormalvariabletothe
standa
rdizednormalvariable,z,maybe
appropriateininternalassessment.
Inexa
minations,questionsrequiringtheus
eof
zscoreswillnotbeset.
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Syllabus content
Content
Furthe
rguidance
Links
4.2
Bivariatedata:the
conceptofcorrelation.
Studentsshouldbeabletomakethedistinc
tion
betweencorrelationandcausation.
Appl:Biology;Physics;Chem
istry;Social
sciences.
TOK:Doescorrelationimply
causation?
Scatterdiagrams;l
ineofbestfit,byeye,
passingthroughthemeanpoint.
Pearsonsproductmomentcorrelation
coefficient,r.
Hand
calculationsofr
mayenhance
understanding.
Inexa
minations,studentswillbeexpected
to
useaGDCtocalculater.
Interpretationofpo
sitive,zeroandnegative,
strongorweakcorrelations.
4.3
Theregressionline
fory
onx.
Hand
calculationsoftheregressionlinema
y
enhanceunderstanding.
Inexa
minations,studentswillbeexpected
to
useaGDCtofindtheregressionline.
Appl:Chemistry11.3(graphicaltechniques).
TOK:Canwereliablyusethe
equationofthe
regressionlinetomakepredictions?
Useoftheregressionlineforprediction
purposes.
Studentsshouldbeawareofthedangersof
extrap
olation.
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Syllabus content
Content
Furtherguidance
Links
4.4
The
2
testforindependence:formulationof
nullandalternativehypotheses;significance
levels;contingencytables;expected
frequencies;degreesoffreedom;p-values.
Inexa
minations:
t
hemaximum
numberofrowsorcolu
mns
inacontingencytablewillbe4
t
hedegreesoffreedomwillalwaysbe
greate
rthanone
t
he
2
criticalvaluewillalwaysbeg
iven
o
nlyquestionsonuppertailtestswith
comm
onlyusedsignificancelevels(1%,5%,
10%)
willbeset.
Calcu
lationofexpectedfrequenciesbyhan
dis
requir
ed.
Hand
calculationsof
2
mayenhance
understanding.
Inexa
minationsstudentswillbeexpectedto
useth
eGDCtocalculatethe
2
statistic.
Ifusin
g
2
testsininternalassessment,
studentsshouldbeawareofthelimitations
of
thetestforsmallexpectedfrequencies;
expec
tedfrequenciesmustbegreaterthan
5.
Ifthe
degreeoffreedom
is1,thenYatess
contin
uitycorrectionshouldbeapplied.
Appl:Biology(internalassessment);
Psychology;Geography.
TOK:Scientificmethod.
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Syllabus content
Topic5
Geom
etryandtrigonometry
18hours
Theaimsofthistopicaretodeveloptheabilitytodrawcleard
iagramsintwodimensions,andtoa
pplyappropriategeometricandtrigo
nometrictechniquesto
problem-solvingintwoand
threedimensions.
Content
Furtherguidance
Links
5.1
Equationofaline
intwodimensions:theforms
y
mx
c
and
0
ax
by
d
.
Linkwithlinearfunctionsin6.2.
Appl:Gradientsofmountainroads,eg
CanadianHighway.Gradients
ofaccessramps.
Appl:Economics1.2(elastici
ty).
TOK:Descartesshowedthatgeometric
problemscanbesolvedalgebraicallyandvice
versa.Whatdoesthistellusabout
mathematicalrepresentationandmathematical
knowledge?
Gradient;intercepts.
Pointsofintersectionoflines.
Linkwithsolutionsofpairsoflinearequations
in1.6.
Lineswithgradien
ts,
1
m
and
2
m
.
Parallellines
1
2
m
m
.
Perpendicularlines,
1
2
1
m
m
.
5.2
Useofsine,cosine
andtangentratiostofind
thesidesandangle
sofright-angledtriangles.
Anglesofelevationanddepression.
ProblemsmayincorporatePythagoras
theore
m.
Inexa
minations,questionswillonlybeset
in
degrees.
Appl:Triangulation,map-making,finding
practicalmeasurementsusing
trigonometry.
Int:DiagramsofPythagoras
theoremoccurin
earlyChineseandIndianmanuscripts.The
earliestreferencestotrigonom
etryareinIndian
mathematics.
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Syllabus content
Content
Furthe
rguidance
Links
5.3
Useofthesinerule:
sin
sin
sin
a
b
c
A
B
C
.
Inallareasofthistopic,studentsshouldbe
encouragedtosketchwell-labelleddiagram
sto
supporttheirsolutions.
Theambiguouscasecouldbetaught,butw
ill
notbe
examined.
Inexa
minations,questionswillonlybeset
in
degrees.
Appl:Vectors;Physics1.3;bearings.
Useofthecosiner
ule
2
2
2
2
cos
a
b
c
bc
A
;
2
2
2
cos
2
b
c
a
A
bc
.
Useofareaofatriangle=
1
sin
2
ab
C.
Constructionoflab
elleddiagramsfrom
verbal
statements.
TOK:Usethefactthatthecosineruleisone
possiblegeneralizationofPyth
agorastheorem
toexploretheconceptofgenerality.
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Syllabus content
Content
Furthe
rguidance
Links
5.4
Geometryofthree-dimensionalsolids:cuboid;
rightprism;rightp
yramid;rightcone;cylinder;
sphere;hemisphere;andcombinationsofthese
solids.
Thedistancebetwe
entwopoints;egbetween
twoverticesorverticeswithmidpointsor
midpointswithmidpoints.
Thesizeofananglebetweentwolinesor
betweenalineand
aplane.
Notrequired:
anglebetweentwo
planes.
Inexa
minations,onlyright-angled
trigonometryquestionswillbesetinreference
tothre
e-dimensionalshapes.
TOK:Whatisanaxiomaticsystem?
Dotheanglesinatrianglealw
aysaddto180?
Non-Euclideangeometry,suchasRiemanns.
Flightmapsofairlines.
Appl:Architectureanddesign
.
5.5
Volumeandsurfac
eareasofthethree-
dimensionalsolids
definedin5.4.
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Syllabus content
Topic6Math
ematicalmodels
20hours
Theaimofthistopicistodevelopunderstandingofsomemathematicalfunctionsthatcanbeusedtomodelpracticalsituations.Extensive
useofaGDCistobe
encouragedinthistopic.
Content
Furthe
rguidance
Links
6.1
Conceptofafunction,domain,rangeand
graph.
Functionnotation,
eg
(),(),()
fx
vt
C
n
.
Conceptofafunctionasamathematical
model.
Inexa
minations:
thedomainisthesetofallrealnumbe
rs
unless
otherwisestated
m
appingnotation
:f
x
y
willnotbe
used.
TOK:Whycanweusemathematicsto
describetheworldandmakep
redictions?Isit
becausewediscoverthemathematicalbasisof
theworldorbecauseweimposeourown
mathematicalstructuresontotheworld?
Therelationshipbetweenreal-worldproblems
andmathematicalmodels.
6.2
Linearmodels.
Linearfunctionsandtheirgraphs,
(
)
fx
mx
c
.
Linkw
ithequationofalinein5.1.
Appl:Conversiongraphs,egtemperatureor
currencyconversion;Physics3.1;Economics
3.2.
6.3
Quadraticmodels.
Quadraticfunction
sandtheirgraphs
(parabolas):
2
()fx
ax
bx
c
;
0
a
Linkw
iththequadraticequationsin1.6.
Functionswithzero,oneortworealrootsa
re
includ
ed.
Appl:Costfunctions;projectilemotion;
Physics9.1;areafunctions.
Propertiesofaparabola:symmetry;vertex;
interceptsonthex-axisandy-axis.
Equationoftheaxisofsymmetry,
2b
x
a
.
Thefo
rmoftheequationoftheaxisof
symmetrymayinitiallybefoundby
investigation.
PropertiesshouldbeillustratedwithaGDC
or
graphicalsoftware.
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Syllabus content
Content
Furthe
rguidance
Links
6.4
Exponentialmodel
s.
Exponentialfunctionsandtheirgraphs:
(
)
;
,
1,
0
x
f
x
ka
c
a
a
k
.
(
)
;
,
1,
0
x
f
x
ka
c
a
a
k
.
Conceptandequationofahorizontal
asymptote.
Inexa
minations,studentswillbeexpected
to
usegraphicalmethods,includingGDCs,to
solveproblems.
Appl:Biology5.3
(population
s).
Appl:Biology5.3.2
(populationgrowth);
Physics13.2
(radioactivedeca
y);PhysicsI2
(X-rayattenuation);coolingofaliquid;spread
ofavirus;depreciation.
6.5
Modelsusingfunctionsoftheform
(
)
...
;
,
m
n
fx
ax
bx
m
n
.
Inexa
minations,studentswillbeexpected
to
usegraphicalmethods,includingGDCs,to
solveproblems.
Functionsofthisty
peandtheirgraphs.
They-axisasaverticalasymptote.
Examples:
4
(
)
3
5
3
f
x
x
x
,
2
4
(
)
3
gx
x
x
.
6.6
Drawingaccurategraphs.
Creatingasketchfrominformationgiven.
Transferringagrap
hfromGDCtopaper.
Reading,
interpretingandmakingpredictions
usinggraphs.
Studentsshouldbeawareofthedifference
betweenthecommandtermsdrawand
sketc
h.
Allgraphsshouldbelabelledandhavesom
e
indica
tionofscale.
TOK:Doesagraphwithoutlabelsor
indicationofscalehavemeaning?
Includedallthefun
ctionsaboveandadditions
andsubtractions.
Examples:
3
2
(
)
5
fx
x
x
,
(
)
3
x
gx
x
.
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Mathematical studies SL guide32
Syllabus content
Content
Furthe
rguidance
Links
6.
7
UseofaGDCtosolveequationsinvolving
combinationsofth
efunctionsabove.
Examples:
3
2
2
3
1
x
x
x
,5
3x
x
.
Other
functionscanbeusedformodelling
in
internalassessmentbutwillnotbeseton
examinationpapers.
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Syllabus content
Topic7
Introductiontodifferentialcalculus
18hours
Theaim
ofthistopicistoin
troducetheconceptofthederivative
ofafunctionandtoapplyittooptim
izationandotherproblems.
Content
Furthe
rguidance
Links
7.1
Conceptofthederivativeasarateofchange.
Tangenttoacurve
.
Notrequired:
formaltreatmento
flimits.
Teach
ersareencouragedtointroduce
differentiationthroughagraphicalapproac
h,
rather
thanaformaltreatment.
Emph
asisisplacedoninterpretationofthe
conceptindifferentcontexts.
Inexa
minations,questionsondifferentiatio
n
from
firstprincipleswillnotbeset.
Appl:Ratesofchangeinecon
omics,
kinematicsandmedicine.
Aim8:Plagiarism
andacknow
ledgmentof
sources,egtheconflictbetwee
nNewtonand
Leibnitz,whoapproachedthe
developmentof
calculusfrom
differentdirections
TOK:Isintuitionavalidway
ofknowingin
maths?
Howisitpossibletoreachthe
sameconclusion
from
differentresearchpaths?
7.2
Theprinciplethat
1
(
)
(
)
n
n
f
x
ax
f
x
anx
.
Thederivativeoff
unctionsoftheform
1
(
)
...,
n
n
f
x
ax
bx
whereallexponentsare
integers.
Studentsshouldbefamiliarwiththealternative
notationforderivatives
d dy x
or
d dV r
.
Inexa
minations,knowledgeofthesecond
deriva
tivewillnotbeassumed.
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22/22
Syllabus content
Content
Furthe
rguidance
Links
7.3
Gradientsofcurve
sforgivenvaluesofx.
Valuesofxwhere
(
)
f
x
isgiven.
Theuseoftechnologytofindthegradientata
pointisalsoencouraged.
Equationofthetan
gentatagivenpoint.
Theuseoftechnologytodrawtangentand
norma
llinesisalsoencouraged.
Equationofthelin
eperpendiculartothe
tangentatagivenpoint(normal).
Links
withperpendicularlinesin5.1.
7.4
Increasinganddec
reasingfunctions.
Graphicalinterpretationof
(
)
0
f
x
,
(
)
0
f
x
and
(
)
0
f
x
.
7.5
Valuesofxwhere
thegradientofacurveis
zero.
Solutionof
(
)
0
f
x
.
Stationarypoints.
Theuseoftechnologytodisplay
(
)
f
x
and
(
)
f
x
,andfindthesolutionsof
(
)
0
f
x
is
alsoencouraged.
Localmaximumandminimumpoints.
Awarenessthatalocalmaximum/minimum
willnotnecessarilybethegreatest/leastvalue
ofthe
functioninthegivendomain.
Awarenessofpointsofinflexionwithzero
gradie
ntistobeencouraged,
butwillnotb
e
examined.
7.6
Optimizationproblems.
Examples:Maximizingprofit,minimizingcost,
maxim
izingvolumeforgivensurfacearea.
Inexa
minations,questionsonkinematicsw
ill
notbe
set.
Appl:Efficientuseofmateria
linpackaging.
Appl:Physics2.1
(kinematics
).