ib math studies syllabus 2014

8/11/2019 IB Math Studies Syllabus 2014

1/22

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.


2/22


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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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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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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Mathematical studies SL guide18

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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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


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.


12/22


Syllabus content

Topic4

Statis

ticalapplications

17hours


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


to

useaGDCtofindtheregressionline.

Appl:Chemistry11.3(graphicaltechniques).

TOK:Canwereliablyusethe

equationofthe

regressionlinetomakepredictions?

Useoftheregressionlineforprediction

purposes.

Studentsshouldbeawareofthedangersof

extrap

olation.


14/22


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.


16/22


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.


17/22


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.


18/22


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.


19/22


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


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


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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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.


21/22


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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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

).

ib math studies syllabus 2014

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