syllabus math hl

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


r.Howis

m

athematicalintuitionusedasaba

sisfor

f

ormalproof?(Gaussmethodforaddingup

integersfrom1to100.)

(continued)

Applications.

Examplesinclud

ecompoundinterestand

populationgrow

th.


2/31

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.


3/31


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


r.Dothe

w

ordsimaginaryandcomplexmak

ethe

c

onceptsmoredifficultthanifthey

had

d

ifferentnames?

T

OK:Thenatureofmathematics.

Hasi

b

eeninventedorwasitdiscovered?

T

OK:Mathematicsandtheworld.

Whydoes

iappearinsomanyfundamental

lawsof

p

hysics?


4/31


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?


5/31


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?


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.


6/31


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).


7/31


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.


8/31


Syllabus content

Content

Furtherguidance

Links

2.

7

Solutionsof

(

)

(

)

g

x

f

x

.

Graphicaloralgebraicmethods,

forsimple

polynomialsuptodegree3.

Useoftechnologyfortheseandotherfunctions.


9/31


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.


10/31


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?


(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.


(vectors

andscalars);

P

hysicsSL/HL2.2

(forcesanddyn

amics).

Int:Theuseoftriangulationtofindthe

c

urvatureoftheEarthinordertosettlea

d

isputebetweenEnglandandFranceover

N

ewtonsgravity.


11/31


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


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

.


12/31


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


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


Whymight

itbearguedthatvectorrepresentationoflines

issuperiortoCartesian?

4.4

Coincident,parallel,intersectingandskew

lines;distinguishingbetweenth

esecases.

Pointsofintersection.


13/31


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?


14/31


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

=

=

=

=

.


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.


15/31


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.


16/31


Syllabus content

Content

Furtherguidance

L

inks

5.5

Conceptofdiscreteandcontinu

ousrandom

variablesandtheirprobabilityd

istributions.

Definitionanduseofprobabilityd

ensityfunctions.

T


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.


17/31


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


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


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.


18/31


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


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

syllabus math hl

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