specialist maths 3/4 bound notes

8/9/2019 Specialist Maths 3/4 Bound Notes

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Writtenby MaoYuanLiu

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Lastupdate Version1.43,29th September2008


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TableofContents

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0.2 TableofContents0.2.1

TableofContents

1 Algebra........................................................................................................................................................ 5

1.1 LinearAlgebra....................................................................................................................................... 6

1.1.1

Systems

and

Matrices

...................................................................................................................

6

1.1.2 Vectors....................................................................................................................................... 18

1.2 AlgebraofFunctions........................................................................................................................... 28

1.2.1 CircularFunctions....................................................................................................................... 28

1.2.2 HyperbolicFunctions.................................................................................................................. 34

1.2.3 FunctionsandtheirGraphs........................................................................................................ 36

1.3 ComplexNumbers............................................................................................................................... 39

1.3.1 ArgandDiagram.......................................................................................................................... 40

1.3.2 Operations.................................................................................................................................. 40

1.3.3 PolarForm.................................................................................................................................. 41

1.3.4 ComplexRoots............................................................................................................................ 45

1.3.5 RelationshipsintheComplexPlane........................................................................................... 46

2 Calculus..................................................................................................................................................... 49

2.1 SingleVariableCalculus....................................................................................................................... 50

2.1.1 Limits.......................................................................................................................................... 50

2.1.2 MethodsofDifferentiation........................................................................................................ 54

2.1.3 ApplicationsofDifferentialCalculus.......................................................................................... 57

2.1.4 MethodsofAntidifferentiation.................................................................................................. 63

2.1.5 ApplicationsofIntegralCalculus................................................................................................ 73

2.1.6 DifferentialEquations................................................................................................................. 81

2.1.7 PhysicalApplications.................................................................................................................. 93

2.1.8 SequencesandSeries............................................................................................................... 107

2.2 VectorCalculus.................................................................................................................................. 115

2.2.1 SpaceCurveandContinuity..................................................................................................... 115

2.2.2 Derivative................................................................................................................................. 115

2.2.3 VectorTangent......................................................................................................................... 116

2.2.4 Curvature.................................................................................................................................. 1162.2.5 NormalandBiNormal.............................................................................................................. 116

2.3 MultivariableCalculus....................................................................................................................... 117

2.3.1 Continuity................................................................................................................................. 117

2.3.2 PartialDerivatives.....................................................................................................................117

2.3.3 Tangentplanes......................................................................................................................... 119

2.3.4 Chainrule................................................................................................................................. 119

2.3.5 DirectionalDerivatives............................................................................................................. 120

2.3.6 CriticalPointsofaSurface........................................................................................................ 121


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0.3

0.3.1


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Algebra

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1

1.0 Algebra1.0.1

AlgebraLinear Algebra (Systems and Matrices, Vectors), Algebra of Functions,Complex Numbers


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LinearAlgebra

SystemsandMatrices

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1.1LinearAlgebra

1.1.1 SystemsandMatricesA matrixisarectangulararrayofmrowsandncolumns,denotedby.Itsentriesatirowandjcolumnisdenotedas .Thematrixcanalsobeexpressedas Addition/subtractionareonlylegalwheretheorder(dimension)ofthematricesarethesame.

Twomatricesareconsideredequaliftheirorderandalltheirentriesarethesame.

Propertiesofaddition:

Commutative,(A+B)=(B+A)

Associative,A+(B+C)=(A+B)+C

Additionofzeromatrixhasnoeffect,A+0=A

ThereexistanegativematrixDofA,whereeachandallitsentriesarenegativethatofA,

suchthat,D+A=A+D=0

Scalarmultipleofamatrixisobtainedbymultiplyingeachentrybythescalar.

Propertiesofscalarmultiplication:

Scalar1hasnoeffect,1A=A Collective,kA+nA=(k+n)A

Distributive,k(A+B)=kA+kB

Associativeandcommutative,k(nA)=n(kA)=(nk)A

Zeroscalarnullsthematrix,0A=0


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1.1.1.1 MatrixMultiplicationMatrixmultiplicationbetweenAandBisonlylegalifthenumberofcolumnsisthesameasthe

numberofrows.TheproductinheritthenumberofrowsofA,andthenumberofcolumnsofB

Forexample

Wherethematriceshavetherightdimensionstomultiplyeachother,theyaresaidtobeconformable

Matrixdivisioninvolvesthematrixinverse,whichdoesntalwaysexist.Thiswillbeexploredlateron

inthenotes.

Propertiesofmultiplication:

Associative,A(BC)=(AB)C

Distributive,A(B+C)=AB+AC

NOTcommutative

1.1.1.2 TransposeAtransposematrixisamatrixwithrowsandcolumnsswitched,orinvertedaboutitsprimaryaxis[a11,

a22,a33]

Forexample

Propertiesoftranspose:

Thetransposeofatransposeisitself,(AT)T=A

Transposeofasumisthesumoftransposes,(A+B)T=A

T+B

T

Transposeofascalarmultipleisthescalarmultipleofthetranspose,(kA)T=k(A

T)

TransposeofamatrixproductistheproductofthetransposesintheREVERSEORDER,

(AB)T=B

TA

T


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1.1.1.3 SpecialTypesofMatrices

1.1.1.3.1 Zero

Zeromatricesarematriceswhereallentriesare0:

0 0 0 0 01.1.1.3.2 Square

Squarematricesarematriceswhichhasthesamehorizontalandverticaldimension,suchasOnlyasquarematrixcanhavedeterminants,inversesandpowers.

1.1.1.3.3 Symmetrical

Thetransposeofasymmetricalmatrixisequaltoitself.I.e.thematrixisequivalentoneithersideof

itsprimaryaxis.

issquare.1.1.1.3.4 Diagonal

Adiagonalmatrixisasymmetricalmatrixwhereallentriesexceptthoseontheprimaryaxisarezero.

0 00 00 0

isadiagonalmatrix.

Thenonzeroentriesoftheinverseofthediagonalmatrixarethereciprocalofthenonzeroentries

ofthediagonalmatrix. / 0 00 / 00 0 / 1.1.1.3.5 Identity

Anidentitymatrixisadiagonalmatrixwhereallthenonzeroentriesare1.

1 0 0 1,

1 0 00 1 00 0 1

Amatrixmultipliedbyaconformableidentitymatrixisitself.IA=AI=A.

Theinverseoftheidentitymatrixisitself.

1.1.1.3.6 Orthogonal

Orthogonalmatricesaresquarematriceswheretheirtransposeisequaltotheirinverse.

One

of

such

case

is

the

rotational

matrix,

cos sin sin cos.


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1.1.1.4 LinearTransformationForanndimensionalspace,thepositionvectorcanbetransformedlinearlybyan transformationmatrixT,suchthat:

1.1.1.4.1 HomogenousLinearTransformation

HomogenouslineartransformationaretransformationintheR2spacebya2x2matrix.

Generally,wherex2=mx1+c(astraightline),giventhat 0 :

, 1.1.1.4.1.1 SpecialCases

Where 0 , , 0

Or

wherea22isnot0.Or

wherebotha12anda22are0.

1.1.1.4.2 Dilation 00 ,wherehisthedilationfactorfromtheyaxis(paralleltox),andkisthedilationfactorfromxaxis(paralleltoy)

1.1.1.4.3 Reflection1 00 1reflectsabouttheyaxis.

1 00 1reflectsaboutthexaxis.

0 11 0reflectsaboutthey=xline,ortakestheinverseoftherelationship.1.1.1.4.4 Rotationcos sin sin cosrotatesbyintheanticlockwisedirection.1.1.1.4.5 Shearing

1 1

,wherehistheamountofsheerinthexdirection,andkistheamountofsheerinthey

direction.


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1.1.1.5 RowOperationsTherearethreeelementaryrowoperations:

Rowswapswapanyrowwithanotherrow

Multiplybyascalar(nonzero)multiplyanyrowbyanumber(notzero) Addamultipleofanotherrow

Byusingrowoperations,amatrixcanbemadeintoreforrref.

1.1.1.5.1 RowEchelonForm

Entriesbelowanyleadingentries(thefirstnonzeroentryinarow)arezero.Thisalsoimpliesthatall

entriestotheleftofanyleadingentriesarezero.Leadingentriesarepreferredtobe,butnot

necessarily,1.

1 0 0 1 0 0 0 1.1.1.5.2 ReducedRowEchelonForm

Entriesbelowandaboveanyleadingentries(thefirstnonzeroentryinarow)arezero.Thisalso

impliesthatallentriestotheleftofanyleadingentriesarezero.Allleadingentriesare1.

1 0 0 0 0 1 0 0 0 0 1 1.1.1.6 DeterminantsThedeterminantisanumericalvaluedfunctionofasquarematrixthatdetermineswhetheritis

invertible.Thisallowsthecalculationofthematrixinverse,andhenceallowsdivision.

Determinantsaredenotedbydet .Itisgrantedthatdet .Fora2by2matrix,wecanexpressalinearsystemas

Bytheprocessofelimination, Forx1tohaveauniquesolution,itscoefficientmustnotbe0.Thiscoefficientisthedeterminant.

det


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1.1.1.6.1 Minor

Theminor(ij)ofmatrixAisthematrixwiththeith

rowandjth

columnstruckout:

,

1.1.1.6.2 Cofactor

Thecofactoristhedeterminantoftheminormultipliedby1tothepoweroftherowpluscolumn:

1 Fortheabovecase, 1 The1termwillmeanthatthecofactorsofdifferententrieswillhavedifferingsigns.Theyfollowthis

generalpattern:

1.1.1.6.3 CofactorExpansion

Cofactorexpansioncantakeanyrowandanycolumn.Thedeterminantisthesumoftheproductof

eachentryanditscofactorinaparticularroworcolumn.FormatrixBycolumn:

det 1 | | , where is a constant Byrow:

det 1 , where is a constant

Forexample:

1 0 11 2 33 2 1,tofinddetA,expandingbyfirstrow:det 1 1 2 32 1 1 0 1 33 1 1 1 1 23 2 2 6 0 2 6 8


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1.1.1.6.4 PropertiesofDeterminants

Somepropertiesofdeterminantsare:

Det(A)=Det(AT)

Ifthereisanyroworcolumnthatisentirelyconsistedofzeros,thedeterminantiszero

Ifanyroworcolumnsareidentical,thedeterminantiszero

Ifanyroworcolumnaremultiplesofanotherrow/column,thenthedeterminantiszero

Scalarmultipleofasingleroworcolumngivesthescalarmultipleofthedeterminant,

Decompositionofrows/columns,

Multiplesofanotherroworcolumn,

Rowswap,

Det(AB)=Det(A)Det(B)

1.1.1.7

Inverse

AmatrixinverseisonesuchthatA(A1

)=(A1

)A=I

1.1.1.7.1 ByCofactor

Thematrixinverseisthetransposeofthecofactormatrixdividedbyitsdeterminant.Henceamatrix

witha0determinanthasnoinverse.

1det Thetransposeofthecofactormatrixiscalledtheadjointmatrix,denotedbyadj(A).

adj detForexample, 1 0 11 2 33 2 1,

4 8 42 2 22 2 2 , det 8

18

4 0 48 2 22 2 2

18 4 2 28 2 24 2 2

12 14 14

1 14 1412 14 14


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1.1.1.7.2 ByRowOperations

Byaugmentingthesquarematrixwiththeidentityofequaldimensionontheright,usingelementary

rowoperationstomakethelefthandmatrixintoidentity,therighthandmatrixwillbetheinverse.

1 0 1 1 0 01 2 3 0 1 03 2 1 0 0 1 1 0 0 12 14 140 1 0 1 14 140 0 1 12 14 14

Whereanentireroworcolumnofthelefthandsidebecomesentirely0,theidentitymatrixcan

neverbeobtained,i.e.nomatrixinverse.

1.1.1.7.3 PropertiesofInverse

Amatrixhasaninverseifandonlyifithasanonzerodeterminant.

o

Ifamatrixhasnoinverse,itissingular

o Ifamatrixhasaninverse,itisnonsingularorinvertible

det(A)det(A1

)=1

(AT)

1=(A

1)T

(AB)1

=B1

A1

1.1.1.8 ElementaryMatricesElementarymatricesaresquaretransformationmatricesthatperformasingleelementaryrow

operation.Thesematricesareinvertible.

An matrixisconsideredanelementarymatrixifitdiffersfromtheidentitymatrixbyasinglerowoperation.1.1.1.8.1 TypesandProperties

TypeIInterchangetworows

0 1 01 0 00 0 1TypeIIMultiplyarowbyanonzeronumber

1 0 00 2 00 0 1TypeIIIaddamultipleofanotherrow

1 0 20 1 00 0 1 IfweletEbean

elementarymatrix,andAbean

.ThematrixproductEAwouldbethe

sameasapplyingthatrowoperationtoA.


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1.1.1.8.2 InverseElementaryMatrices

TypeIInterchangetworows

0 1 01 0 00 0 1

0 1 01 0 00 0 1

(Thistypeofelementarymatrixisitsselfinverse.)

TypeIIMultiplybythereciprocal

1 0 00 2 00 0 1 1 0 00 1/2 00 0 1

TypeIIISubtractmultipleofanotherrow

1 0 20 1 00 0 1 1 0 20 1 00 0 1Elementarymatricesfundamentallycharacterisesmatrixinverses:

IfAistheproductofelementarymatricesEkE3E2E1,then:

Amatrixhasaninverseifandonlyifitistheproductofelementarymatrices.

1.1.1.9 SystemsofLinearEquationsAsystemoflinearequationcanbegeneralisedtobe:

Fornvariablenequationssystem,itcanbeexpressedasthematrixequation:


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1.1.1.9.1 Solving

1.1.1.9.1.1 UsingInverses

InthecasethatAisanonsingularmatrix,

1.1.1.9.1.2 CramersRule

Cramersruleisconsideredtobeeasierthanusingmatrixinverses.

Since

,thesolutionofthesystemcanbeexpressedas:

1det

1det

Itisevidentthattherightmostmatrixentriesisacofactorexpansionofacolumn,wheretheentry forthejthcolumn,andtherestoftheentriesareidenticaltothatofA.A

(j)isusedtodenotethematrixobtainedfromAbyreplacingthej

thcolumnofAbythecolumn

vectorB.

Forexample, Ingeneral,

1

detdetdet

det

Inparticular,


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1.1.1.9.1.3 GaussianElimination

Gaussianeliminationuseselementaryrowoperationsoftheaugmentedmatrix[A|B],carryingitto

itsrowechelonorreducedrowechelonform.

Forexample:

1 0 11 2 33 2 1 111

Augmentedmatrix

1 0 1 11 2 3 13 2 1 1Usingrowechelonforms

1 0 1 11 2 3 13 2 1 1 1 2/3 1/3 1/30 1 2 1 / 20 0 1 1 / 2 ; 2 ; Usingreducedrowechelonforms

1 0 1 11 2 3 13 2 1 1

1 0 0 1/20 1 0 1/20 0 1 1/2

12 , 12 , 12Gaussianeliminationcanbeusedtosolvenotonlynbynsystems,butanysystemoflinearequations.


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1.1.1.9.2 Consistency

Inasystemoflinearequations,exactlyoneofthefollowingoccurs.

1.1.1.9.2.1 NoSolutions

Asystemoflinearequationsissaidtobeinconsistentiftherearenosolutions.

Whentherearenosolutions,thedeterminantofthecoefficientmatrixAiszero,andthematrix

inversedoesnotexist(Theconverseisnotnecessarilytrue).UsingmatrixinverseorCramersruleto

solveasystemwouldyieldanindeterminateresult(dividebyzero).

UsingGaussianeliminations,therearenosolutionswhentherearerowofthetype0 0 0 | ,where*isanonzeronumber.1.1.1.9.2.2 UniqueSolution

Forasystemwithasetofuniquesolutions,

Thenumberofequationsmustbeequalormorethanthenumberofvariables.

Thedeterminantofthecoefficientmatrixmustnotbe0

Thecoefficientmatrixmustbeinvertible

Thereducedrowechelonformofthecoefficientmatrixmustresembletheidentitymatrix.

Thereducedrowechelonformofthecoefficientmatrixmustnothaveacolumnentirely

consistedofzeros.

unique solution

exists

det 0

1.1.1.9.2.3 InfiniteSolutions

Asystemmayhaveinfinitesolutions,suchastwoplanesintersectingonalineortwoequations

coincide.Generally,iftherearemorevariablesthanequations,thereareinfinitesolutionstothe

system.

Ifthecoefficientmatrixofasystemwithinfinitesolutionsisasquarematrix,itsdeterminantwillbe

zero(theconverseisnotnecessarilytrue).UsingmatrixinverseandCramersrulewillyieldan

indeterminateresult(dividebyzero).

UsingGaussianelimination,inreforrref,thecolumnsofthecoefficientwhichdonotcontainany

leadingentriesareunbound,anditscorrespondingcoefficientbecomesaparameter.

Forexample,1 3 12 6 31 3 0

141Theaugmentedmatrixwillhencebe1 3 1 12 6 3 41 3 0 1,anditsrrefis

1 3 0 10 0 1 20 0 0 0 Ignoringtherowofzeroentries,wecanseethat

, ,andyisunbound.

Let , , .ThisdescribesalineinR3space.


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1.1.2 Vectors

1.1.2.1 DefinitionAvectorisaquantitywithbothmagnitudeanddirection.Itisalsopossibletodefinethesenseofa

vector,i.e.oneofthetwowaysthevectorcanbepointingtowards.

Avectorisrepresentedbyastraightlinesegmentwithanarrow.

1.1.2.1.1 EqualityofVectors

Twovectorsareequalifandonlyiftheyhavethesamemagnitude,samedirectionandsamesense.

Vectorsarefree,wherethestartingpointofavectorisirrelevant.Thesamelinesegments

pointinginthesamedirectionalwaysrepresentthesamevector,regardlessoftheirstartingpoint.

Displacementvectorsarefreevectorswithoutaboundstartingpoint.Positionvectorsarefree

vectorswiththestartingpointboundatorigin.

1.1.2.1.2 SpecialVectors

1.1.2.1.2.1 UnitVectors

Unitvectorshaveamagnitudeof1.

1.1.2.1.2.2 SpatialDimensions

Thedefineddimensionsareawaytocoordinatendimensionalspace.

Inparticular,thesedimensionalvectorsareunitvectors,andareperpendiculartoeachother.iisthe

firstdimension,jistheseconddimensionperpendiculartoi,andkisthethirddimension

perpendiculartobothiandj.

Avectorisoftenresolvedintocomponentsinthedirectionofspatialdimensions.

1.1.2.1.2.3 ZeroVectors

Zerovectorisa0dimensionalvector,withzeromagnitude,unspecifieddirectionandsense.Itisa

singlepoint.

1.1.2.1.3 Magnitude

ThemagnitudeofavectorcanbecalculatedbyPythagorastheoremwhenitisexpressedasperpendicularcomponents.

||


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1.1.2.1.4 AnglewithAxis

Thecosineoftheangleavectormakeswithanaxisisitscomponentinthatdirectiondividedbyits

magnitude(ratioofcosine).

If

istheangleavectormakeswiththexaxis(idirection),then

cos | |Foravectoru,whereistheanglebetweenitandthexaxis,totheyaxisandtothezaxis,cos || , cos || , cos ||

And,since|| wouldbeaunitvector ||,itfollowsthat

cos cos cos 1For

example,

40 60 49 ,find

the

acute

angle

this

vector

makes

with

the

horizontal.

Letbetheanglemakeswiththezaxis.cos 4940 60 49 0.562 2.168 124.2

Hencetheacuteangleitmakeswiththehorizontalis

124.2 90 34.2


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1.1.2.1.5 Sums,DifferencesandScalarMultiples(ParallelVectors)

Sumofvectorsarecalculatedbyjoiningeachvectorheadtotail.Theresultantvectoristhevector

whichjoinsthetailofthefirstvectortotheheadofthelastvector.

Whenvectorsareexpressedintheircomponents,thesumofthevectorsinaparticulardirectionis

thesumofthecomponentsinthatdirection.Thecomponentsoftheresultantvectorarethesumof

thecomponents.

Vectorsubtractionareadditionofnegativevectors(i.e.reversedvector).

Scalarmultipleofavectorchangesthemagnitudebythefactorofthescalar,directionandsenseare

notchanged.

Vectorsadditionandscalarmultiplicationare

Commutative:u+v=v+u

Associative:u+(v+w)=(u+v)+w

Additionofzerovectorhasnoeffect:u+0=0+u=u

Scalarmultiplicationisassociative:n(ku)=(nk)u

Collective:nu+ku=(n+k)u

Distributiveoveraddition:n(u+v)=nu+nv

Multiplicationby1:1u=u

Multiplicationby0:0u=0

Twovectorsareconsideredparalleliftheyarescalarmultiplesofeachother.


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1.1.2.1.6 LinearDependence

Vectorsarelinearlydependantifthesumofmultiplesofvectorsisthezerovector.

Ifthisrelationshipistrueforasetofcoefficientsthatarenotallzero.

Thisimpliesthatanysetofvectorsareautomaticallylinearlydependantifoneofthemisazero

vector.

Linearindependencycanbedescribedaswhenaquantitycannotbedescribedintermsofmultiples

ofotherquantitiesinaset.Thisappliesnotonlytovectors,butalgebraicexpressionsalso.

1.1.2.2 ScalarDotProduct

1.1.2.2.1 DefinitionandInterpretation

Thescalardotproductisdefinedasfollowed:

| | | | cos Whereistheanglebetweenthevectors.Theimplicationofthisisthatunitvectorswhichpointinthesamedirectionhaveadotproductof1,

andperpendicularvectorshaveadotproductof0.

Inthreedimensionalspace(intermsofi,j,andk),thedotproductoftwovectorsis

cos || | | Anglebetweenvectorsisnevermorethan180

o().

Propertiesofthedotproduct:

Commutative,a.b=b.a

Distributiveoveraddition,a.(b+c)=a.b+a.c

Distributiveoverscalarmultiplication,a.(kb)=k(a.b)=(ka).b Thedotproductofanyvectorwithazerovectoris0.

Thedotproductofanyvectorwithitselfisitsmagnitudesquared,a.a=|a|2

Twononzerovectorsareorthogonalifandonlyiftheirdotproductiszero.

1.1.2.2.2 OrthogonalVectors

Orthogonalvectorsarevectorswhichpointinperpendiculardirections.

Fornonzeroorthogonalvectors,theirdotproductisalways0.


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1.1.2.2.3 Resolute

Thevectorresoluteofavectorinthedirectionofanothervector:

Thescalarresoluteofavectorinthedirectionofanothervectoris ,simplythemagnitudeofthevectorresolute.

Theperpendicularresoluteis .1.1.2.3 VectorCrossProduct

1.1.2.3.1 DefinitionandInterpretation

Thevectorcrossproductisdefinedas

| | | | sin

Whereuisaunitvectorperpendiculartobothaandb.

Thecrossproductoftwovectorsinthethreedimensionalspacecanbecomputedbyadeterminant:

Themagnitudeofthevectorcrossproductcanbeinterpretedastheareaofparallelogramformed

bythetwovectors.

Propertiesofvectorproduct:

Distributiveoverscalarmultiples,ax(kb)=k(axb)=(ka)xb

NOTcommutative.Bythepropertyofdeterminants,reversingtheorderswapstworows,

makingthedeterminantnegativeofwhatitwas.bxa=(axb)

Distributiveoveraddition,ax(b+c)=axb+axc

Vectorproductwithitselfisthezerovector

Vectorproductwithazerovectoristhezerovector

1.1.2.3.2 ScalarTripleProductandCo-Planarity

Thescalar

triple

product,

or

box

product

[a,b,c],

is

defined

as

Bythepropertiesofdeterminants,

Arowswapmakesthedeterminantnegative,hence, [a,b,c]=[b,a,c]=[a,c,b]=[c,b,a]

Swappingtworowsmakesthedeterminantpositivehence,[a,b,c]=[c,a,b]=[b,c,a]

Aninterpretationofthevalueoftheboxproduct(scalartripleproduct)isthevolumeofthe

parallelepipedofthethreevectors.

Ifthescalartripleproductiszero,thevectorsarecoplanar(theyexistonthesameplane).


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1.1.2.4 VectorGeometry

1.1.2.4.1 Line

1.1.2.4.2 Plane

Foraplanein3D,anormalvectornisperpendiculartotheplaneatallpoints.

1.1.2.4.2.1

EquationforaPlane 0 Wheren=

, , , , 1.1.2.4.2.2 PerpendicularDistanceFromOrigin

(x0,y0,z0)denotesthepointwheretheplaneisclosesttotheorigin,i.e.itsperpendiculardistance

fromorigin.

Atthatpoint,thepositionvectorisamultipleofthenormalvector, ,,Alsothat

||

|| | |||

1.1.2.4.2.3 AnglesBetweenPlanes

Theanglebetweenplanesaresimplytheanglebetweenthenormalvectors.

cos || | |


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1.1.2.4.3 ParameterisationandCartesianEquivalence

1.1.2.4.3.1Ellipses

cos , sin

or sin , cos 1 Ellipses with going to +/ a in the x direction and +/ b in the y direction, centred at (h,k)

1.1.2.4.3.2Hyperbola

Hyperbolaontheleftandright.

sec , tan 1 Hyperbolaonthetopandbottom.

tan , sec

1


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Forexample

Provethecosineruleforanyangle.

Let be a triangle

2

BC 2 2 QED

Provethemidpointofthehypotenuseofarightangledtriangleisequidistantfromallvertices.

Let be a right angled triangle

Let be the midpoint of 12

12||

||

12 Since 12|| | | Point is equidistant from , and

QED


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1.2AlgebraofFunctions

1.2.1 CircularFunctions

1.2.1.1 SymmetricalIdentities

2 2 2 2 sin cos cos sin sin sin cos cos sin sin

cos sin sin cos cos cos sin sin cos cos

tan cot cot tan tan tan cot cot tan tan

sec cosec cosec sec sec Sec cosec cosec sec sec

cosec sec sec cosec cosec cosec sec sec cosec cosec

cot tan tan cot cot cot tan tan cot cot

1.2.1.2

CartesianIdentitiessin cos 1tan 1 sec 1 cot csc 1.2.1.3 CompoundAngleFormulaesin sin cos sin cos cos cos cos sin sin

tan tan tan1 tan tan

sin2 2 sin cos cos2 cos sin 2cos 1 1 2 sin tan2 2 tan 1 tan Forexample,

1 cos sin tan 2

LHS 1 2cos 2 12sin 2 cos 22 1 cos 22sin 2 cos 2

sin 2cos 2 tan 2 RHS

tan8 1 cos 4sin 4

2 22 2 1

sec8 1 tan

8 4 2 2


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sec 8 4 221.2.1.3.1 MultipleAngleFormulae

sin cos 12 sin sin sin sin 12 cos cos cos cos 12 cos cos 1.2.1.4

SineandCosineRuleInatrianglewithsidesa,bandcandtheangleoppositethemA,BandC,

sin

sin

sin

2 cos


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1.2.1.6 RestrictedFunctionsandInversesSin is deined for

Sin

is deined for 1 1

Range: 2 2

Cos is deined for 0

Cos is deined for 1 1Range: 0

Tan is deined for 2 2 Tan is deined for

Range: 2 2


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1.2.1.6.1FurtherIdentitiessec cos 1

csc sin 1

cot tan 1sin cos 2

sincos 1

sintan

1 cossin 1

costan 11 tansin 1

tancos

1

1.2.1.7 CircularArcsandChords

a) Ifwebisecttheangle,thebisectorraywouldperpendicularlybisectthestraightline,cutting

itinhalfto135m.

sin 135 135sinAlso,thearclengthwouldbehalvedto150m

1502

2

150

Equating,

135150 sin910 sinConvertingtodegrees

910

180 sin

sin 200 , as required.


33/121

AlgebraofFunctions

CircularFunctions

Page33

a)

Thenewlengthcanbebrokenintotwosections,arcAPandthelinesegmentPB.

|| 2 2 2 10 2 || tan 10 tan || || 20

2 tan 2, as required


34/121

AlgebraofFunctions

HyperbolicFunctions

Page

34

1.2.2

Hyperbolic

Functions

1.2.2.1 DefinitionsandInterpretationsThehyperbolicfunctionsareoddandevenpartsofthenaturalexponential.

2

2

cosh sinh

sinh

2

Domain: Range:

cosh 2

Domain:

Range: 1

tanh sinhcosh

Domain:

Range: 1 1


35/121

AlgebraofFunctions

HyperbolicFunctions

Page

35

csch 1sinh

Domain: \0

Range: \0

sech 1cosh

Domain:

Range:0 1

coth coshsinh 1tanh

Domain: \0

Range: 1 1

Thehyperbolicfunctionsareverysimilartothe

circularfunctions.Wherethecircularfunctions

arefunctionsoftheareaofthesector,hyperbolic

functionsarefunctionsoftheareaenclosedby

theunithyperbolax2y

2=1,astraightlinefromthe

originto

the

hyperbola

and

its

vertical

reflection.

1.2.2.2 Identitiessinh sinh cosh cosh

cosh

sinh

1

1 tanh sech Compoundangleidentities:

sinh sinh cosh sinh coshsinh2 2 sinh cosh

cosh cosh cosh sinh sinhcosh2 cosh sinh 1 2sinh 2cosh 1

tanh tanh tanh

1 tanh tanh

tanh2 2tanh 1 tanh


36/121

AlgebraofFunctions

Page36

1.2.2.3 InverseHyperbolicFunctionssinh ln 1,

cosh ln 1 , 1

tanh 12 ln 1 1 ,1 1sech cosh 1csch sinh 1

coth tanh 1

sinhcosh 1

sinhtanh 1 coshsinh 1coshtanh 11 tanhsinh 1

tanhcosh

1

1.2.3 RelationshipsandtheirGraphs

1.2.3.1 EllipsesFollowsthegeneralequation

1

Whichisanellipsecentredat(h,k)spanningaunitstotheleftandright,andbunitstothetopandbottom.

Ellipsehavethedomain , andtherange , .Anellipsecanalsobedescribedbytheequation


37/121

AlgebraofFunctions

RelationshipsandtheirGraphs

Page37

1.2.3.2 HyperbolasAhyperbolacantaketwoforms:

1 Isaleftrighthyperbola,centredat(h,k).Thetwobranchesare

centredat(ha,k)and(h+a,k).

Thedomainis\ , ,andtherangeis.Thistypeofhyperbolacanalsobedescribedbytheequation

1Isanupdownhyperbola,centredat(h,k).Thetwobranches

arecentredat(h,kb)and(h,k+b).

Thedomainis

,andtherangeis

\ , .

Thistypeofhyperbolacanalsobedescribedbytheequation

Inbothcases,theequationsofthetangentscanbeobtainedasfollowed:

1,asxandygetlarge,the1canbeignored.


38/121

AlgebraofFunctions

RelationshipsandtheirGraphs

Page38

Forexample,considertherelationship 9 8 18 41 4 9 8 18 41 4 2 1 4 9 2 1 9 41 4 1 9 1 36

19 14 1 Thisisanupdownhyperbola,hencetherangeis,4 2,

Theequationoftheasymptotes

1 32 1 32 52 , 32 12


39/121

AlgebraofFunctions

Transformation

Page39

1.2.4 TransformationTransformation Rule

Dilation

Byafactorofafromtheyaxis(paralleltoxaxis)

1 Byafactorof1/afromtheyaxis(paralleltoxaxis) Byafactorofaaboutx=h Ingeneral,dilationinthehorizontaldirectionfromcanberepresentedbysubstitutingwith .Byafactorofafromthexaxis(paralleltoyaxis) 1 Byafactorof1/afromthexaxis(paralleltoyaxis) 2Byafactorofaabouty=k

Ingeneral,dilationintheverticaldirectionfromcanberepresentedbysubstitutingwith Reflection

Reflectionineitheraxiscanberepresentedbydilatingbyafactorof1fromtheaxis.

Forexample,considertherelationship 9 8 18 41 ,whataretheequationsoftheasymptotesafteradilationbyafactoroffromtheyaxisthenatranslationof 1unitsparallelto

thexaxis?

Beforetransformation,

32 52 , 32 12Dilationbyfactoroffromy, 2

3 52 , 3 12Translationof 1unitsparalleltox, 1

3 12 , 3 52


40/121

ComplexNumbers

ArgandDiagram

Page40

1.3ComplexNumbersAcomplexnumberhastwoparts:

Arealpart,consistingofanyrealnumber,

Andanimaginarypart,consistingofanyrealmultiplesoftheimaginarynumberi,where 1.Somepropertiesoftheimaginarynumber:

1, , 1, , 1 1.3.1 ArgandDiagramThearganddiagramcanbeusedtographicallyrepresentcomplexnumbers.

Itsxaxisistherealpart,Re(z).Itsyaxisistheimaginarypart,Im(z).

TheCartesianform(rectangularcoordinates)ofacomplexnumberis

1.3.2 Operations

1.3.2.1 Addition

1.3.2.2 Multiplication 1.3.2.3 ConjugatesTheconjugateofacomplexnumberisdenotedbyabar,

If , then Characteristically,

1.3.2.4

DivisionEvaluationofacomplexfractionisachievedwhenthefractionismultipliedbytheconjugateofthe

denominatorontopandbottom.

Morespecifically,thedenominatorofthefinalexpressionisthesquareofthemagnitudeofthe

complexnumber,orthesquareofitsmodulus.Also,thereciprocalofanimaginarynumber:

1 ||


41/121

ComplexNumbers

PolarForm

Page41

1.3.3 PolarFormAcomplexnumbercanalsobeexpressedinpolarform,i.e.directionandmagnitude.

1.3.3.1 ModulusThemodulusisthemagnitudeofacomplexnumber,i.e.itsdistancefromtheorigin.

UsingPythagorastheoremwiththerectangularcoordinates,

|| 1.3.3.2 ArgumentTheArgumentistheanglethecomplexnumbermakeswiththepositiverealaxis.Thisgivesusthree

relationshipsusingtherectangularcoordinates,

sin ||cos ||tan

TheArgumentisrestrictedto,.ItispossibletofindtheArgumentwithanyoftheabovethreerelationshipsinconjunctionwiththeknowledgeofquadrants.

Acomplexnumbercanhencebeexpressedas: | | || ||

| |cos sinThecos sincomponentisabbreviatedtocis,butitismoreformallyknownas

Exp

| | cis | | Exp | | Notethatforcis,sinceitisthesumofcosandi*sin,cis cis ,cis cis Theonlysymmetricalpropertiesithasarecis cis , and cos 2 cos , where .Moreconventionally,cisisusuallyexpressedas:

cis

Whereistheargumentofz.


42/121

ComplexNumbers

PolarForm

Page42

Somespecialnumbers:

0+0icannotbeexpressedinthepolarform,itsargumentisundefined.

Allpositiverealnumbers,a+0i,haveanArgumentof0.

Allnegativerealnumbers, a+0i,haveanArgumentof

.

Allpositiveimaginarynumbers,0+ai,haveanArgumentof/2.

Allnegativeimaginarynumbers,0ai,haveanArgumentof /2.Acomplexnumbercanhaveitsargumentexpressedintermsofthearctangent:

1stquadrant,a+bi

o

tan 2

ndquadrant, a+bi

o

tan

3rdquadrant, abi

o tan

4th

quadrant,abi

o

tan Acomplexnumbersconjugatecanbederivedasfollowed:

| |cossin

| |cossin

| | cis


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ComplexNumbers

PolarForm

Page43

Forexample,solveforzsuchthat| 3| 3 and arg 3 3 9, 1

3 tan 34 1, 2 3

2 3 9

322 3 3 22

|| 92 9 92 92 32 2

Sincezistheintersectionof[1]and[2],itisthepointofintersectionof

therayandthecircle.Itiseasytoconstructatriangletofindcosineof

cos 8322 23 2 22

8

2 5

8

Hence,thepolarformofzis

6cos 8 cis 58


44/121

ComplexNumbers

PolarForm

Page44

1.3.3.3 MultiplicationandDivisionMultiplicationanddivisionusingpolarformsisaloteasierthanusingtheCartesianform.

Fortwocomplexnumbers,

| | cisand

| | cis

| ||| cis | ||| cis Thereciprocalcanalsobeworkedoutfairlyeasily:

1 | 1||| cis0 1|| cis ||Withpolarcoordinates,theideaofrotationisintroduced.Whenacomplexnumberz1ismultiplied

byanothercomplexnumberz2suchthat|z2|=1,thenz1isrotatedanticlockwisebytheanglez2

makeswiththepositiverealaxis:

| ||| cis | | cis Inparticular,multiplyingbythecomplexnumberirotatesthecomplexnumber90

oanticlockwise.

1.3.3.4 DeMoivresTheorem | | cisForexample, 3 , 1 ,find 14 2 2

tan 17 2 tan13 tan 1

tan17 2 tan

13 4


45/121


46/121

ComplexNumbers

RelationshipsintheComplexPlane

Page46

1.3.5 RelationshipsintheComplexPlaneAlocusinthecomplexplaneisasetofpointsontheArganddiagram.

Forexample,

: || 1,

1.3.5.1 Line : | | | |, ThisdescribesastraightlineontheArganddiagram.Itistheperpendicularbisectorofthelinez1z2.

Particularcasesinclude:

: , : 2,where is a constant Thisdescribesaverticalline,x=c

: , : 2,where is a constant Thisdescribesahorizontalline,y=c

: , Thisdescribesthesetofpointsthatmakesanangleof90

owiththeirreflectioninthehorizontalaxis,

i.e.theliney=x

: |1 | | 1|

Thisdescribesalinealso:|1 | | | | | | | Inequalitiesinvolvingalinecanbefoundusingasimplepointsubstitution,usuallytheorigin.1.3.5.2 Ray : , ,where is a constantThisdescribesarayfrom(butnotincluding)z0atanangleoffromthehorizontal.Inequalitiescanbefoundusingasimplepointsubstitution.Itshouldalsobenotedthat

For: , or : , ,thehorizontaltotheleft: , istheendoftheregion,andisincluded. For: , or : , ,thehorizontaltotheleft: , istheendoftheregion,andisnotincluded. z0isnotincluded.


47/121

ComplexNumbers


Page47

1.3.5.3 HyperbolaBythegeometricdefinitionofhyperbola,

: | | | | ,

Thisdescribesahyperbolaonthesideofz1.Thecentreisat .Thecentreofthebranchislocated

unitsawayfromthecentreinthedirectionofz1. 2 2 | |1.3.5.4 Circles : | | , , where is a constant Thisdescribesacirclecentredatz0witharadiusofc.

Ifwelet , then ,| | : , , where is a constantThisalsodescribesacirclecentredatz0witharadiusofc.

1.3.5.4.1 Ellipses

Bythegeometricdefinitionofanellipses,

: | | | | , , where is a constant

Thisdescribestheellipseswithitstwofociatz0andz1.

Axisalongthetwofoci: Axisperpendiculartothefoci:

2 | |2 12 | |


48/121

ComplexNumbers


Page48

1.3.5.4.2 Arcs

Bythegeometrictheorem thatanglessubtendedbyachord/arcatallpointsonthecircumference

areequal:

arg arg

Thisdescribesanarcsuchthattheanglemadebetweentwosegments, and isalways,i.e.thisexistsasanarc.Notethatthisisonlyonesideofthearc,theothersideofthearcisdescribedbyarg arg .ItiseasiertoconvertthistoCartesianformtryingtodrawit.

Forexample,arg arg

tan 1 tan

1 4 , 0 1 11 1 1 1 1 1 1 1 2, 0

Axisintercepts: 1, 2+1

tan 1 tan 1 4 , 0, 1

1 2, 0, 1, FALSE tan 1 tan 1 4 , 0, 1 1 2, 0, 1, FALSE

For 0,1 1, isinthesecondquadrant,anditsargumentliesin , . isinthethirdquadrant,anditsargumentliesin

,

.Hence,

arg arg hasthe

maximum2,andtheminimum.Therefore,arg arg nottruefor 0, 1 1.


49/121

Calculus

Page49

2

2.0 Calculus2.0.1

CalculusSingle Variable Calculus, Vector Calculus, Multivariable Calculus


50/121

SingleVariableCalculus

Limits

Page50

2.1SingleVariableCalculus

2.1.1 LimitsAlimitofafunctionataisthefunctionsvalueasthevariableapproachesa.

2.1.1.1 DefinitionandInterpretationThelimithastwoparts,thelefthandlimitlim ,whichapproachesthevaluefromthelefthandnegativeside,andtherighthandlimitlim ,whichapproachesthevaluefromtherighthandpositiveside.

Alimitexistsifandonlyif:

Thelefthandlimit

lim

exists

Therighthandlimitlim exists lim lim Thefunctionmustbedefinedonsomeopenintervalthatcontains(withthepossibleexceptionof)a.

Underformaldefinition,

lim ifforeverynumber 0thereisanumber 0suchthat| | whenever 0 | |

Thetwosidelimitscanbesimilarlydefinedwiththeboundsonx:

0forlefthand

limits,and0 forrighthandlimits.Somelimitsmayevaluatetoinfinite.Thatis,lim foreverypositivenumberM,thereisapositivesuchthat

whenever 0 | |

Similarly,forlimitsthatevaluatetothenegativeinfinity:lim foreverynegativenumberNnumber,thereisapositivesuchthat

whenever 0 | |

Forlimitsattheinfinity,alimitcanbeeitherdivergent,divergenttoinfinity,orconvergent.

Alimitthatconvergestoanumberatthepositiveinfinity,underformaldefinition,

lim ifforeverynumber 0thereisapositivenumbersuchthat| | whenever

Asimilardefinitionmaybeformedforconvergencyatthenegativeinfinity.Convergencyateither

infinityindicatesahorizontalasymptote.


51/121


Limits

Page51

Alimitthatdivergestoinfinitygetsinfinitelylarge,(orsmall),underformaldefinition,

lim ifforeverypositivenumberthereisapositivenumbersuchthat

whenever

Asimilardefinitionmaybeformedfortheotherthreepossibilities(negativeinfinitytotheright,and

positive/negativeinfinitytotheleft).

Alimitcanalsodivergewithoutgettingtoinfinity.Thistypeoffunctionsusuallyoscillates.An

exampleisthesineratio.

2.1.1.2 LimitLawsEvaluationofalimitcanbeassimpleassubstitutingthenumber.However,theseareoftennot

enough.

Somelimitlawsinclude:

lim lim lim lim limlim lim lim

lim

limlim limlim


52/121


53/121


Limits

Page53

2.1.1.5 ContinuityAfunctioniscontinuousataif

lim

Ifapointiscontinuous,then

lim exists lim exists lim lim

Theleftorrighthandlimitcanbeusedtodefinecontinuityononeside,whereaistheendpointof

anopeninterval.

2.1.1.6 DifferentiabilityAfunctionisdifferentiableataif

exists iscontinuousata Thereisnoabruptchangeofdirectionata(i.e.thederivativeiscontinuousata)


54/121


MethodsofDifferentiation

Page54

2.1.2 MethodsofDifferentiationFirstprinciple

lim

2.1.2.1 DifferentiationRulesAdditionrule:

Chainrule:

Productrule:

Andhence,theconstantrule:

,where is a real constantQuotientrule:

Someparticularcases:

2.1.2.2 ImplicitDifferentiation

Bythechainrule,ifyisafunctionofx: Or,

,


55/121


56/121

2.1.2.6 ExponentialandLogarithms ln

ln 1 log 1ln ln 1

2.1.2.7 LogarithmicDifferentiationIf ln ln

ln ln

1 ln ln 2.1.2.8 SecondDerivatives 2.1.2.8.1

Concavity

Forat ,ifis Positive,theconcavityisupwards

o

Ifitisalsoastationarypoint,itisalocalminimum

Negative,theconcavityisdownwards

o Ifitisalsoastationarypoint,itisalocalmaximum

Zero,

o

Ifthethirdderivativeisalsozero,concavitytestisinconclusive

o

Otherwiseitisapointofinflection

Ifthethirdderivativeispositive(signchangefromnegativetopositive),itis

theminimumgradient

Ifthethirdderivativeisnegative(signchangefrompositivetonegative),itis

themaximumgradient

Ifitisalsoastationarypoint,itisastationarypointofinflection


57/121


ApplicationsofDifferentialCalculus

Page57

2.1.3 ApplicationsofDifferentialCalculus

2.1.3.1 GraphingADomain

BFindxandyintercepts

CLookforsymmetry:iff(x)=f(x),itisevenlysymmetrical;iff(x)=f(x),itisoddlysymmetrical

DAsymptotes,vertical,horizontal,slant

EIntervalswhichthegraphisincreasing/decreasing

FStationarypoints

G

Points

of

inflection

2.1.3.2 AdditionofOrdinatesWhengraphingahybridfunction,themethodofadditionofordinatesmaybeused.

Thenfor at ,


58/121



Page

58

2.1.3.2.1 SomeRationalFunctions

Vertical asymptote at x=0 Slant asymptote y=ax


59/121



Page

59

Vertical asymptote at x=0

Slant asymptote y=ax


60/121



Page

60


Parabolic asymptote y=ax2


61/121



Page61


Parabolic asymptote y=ax2


62/121



Page62

2.1.3.3 ReciprocalFunctionsAreciprocalfunctioncanbegraphedbytakingthereciprocalofthefunctionsvalue.

Whenthefunction

Crossesthexaxisfrompositivetonegative

o Thereciprocalfunctiongoestopositiveinfinityontheleft,andtonegativeinfinity

ontheright(asymmetricverticalasymptote)

Crossesthexaxisfromnegativetopositive

o

Thereciprocalfunctiongoestonegativeinfinityontheleft,andtopositiveinfinity

ontheright(asymmetricverticalasymptote)

Touchesthexaxis

o Dependingonwhetherthefunctiontouchestheaxisonthepositivesideorthe

negativeside,thereciprocalfunctiongoestopositive/negativeinfinityonbothsides

(symmetricverticalasymptote) Hasalocalminimum

o

Thereciprocalfunctionhasalocalmaximum

Hasalocalmaximum

o

Thereciprocalfunctionhasalocalminimum

Hasastationarypointofinflection(notwheny=0)

o

Thereciprocalfunctionhasastationarypointofinflection

Hasapointofinflection(notwheny=0)

o

Thereciprocalfunctionhasapointofinflection

1 1 1 1 1


63/121


MethodsofAntidifferentiation

Page63

2.1.4 MethodsofAntidifferentiationIndefiniteintegrals:

Somerulesoftheintegral:

, where is a real constant ,where C is a real constant

Andfordefiniteintegrals:

Fundamentaltheoremofcalculus

Fundamentaltheoremofcalculus(II)

If is the antiderivative of

Somerules:

0

Themostobviousmethodofantidifferentiationisviarecognition.

ln || 1


64/121


65/121



Page65

2.1.4.3 IntegrationbyRecognitionByfindingthedifferentiatingafunction,whichderivativecontainsallorpartoftheintegrand,can

allowintegrationbyrecognition.

Generally,ifg(x)

has

an

antiderivative

G(x),

and

Then

Somealgebraandidentitiesmaybeusedinthisprocess.

Forexample,differentiate sin ,henceantidifferentiate cos sin sin 1 cos sin 2 sin 2 cos 1

2 cos 1 sin

2 cos 1 2 cos cos cos 1


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Page66

2.1.4.4 IntegrationbyPartsReverseproductrule.

Or

Generally,f(x)ispreferredinthefollowingorder:

Logarithm

Inversetrig

Algebraic

Trigonometric

Exponential

sometimesmaynotbeapparent,butcanbeassimpleas1.

Forexample,

log log log 1 log 2

2 2 2 2 2 2


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Page67

2.1.4.5 TrigonometricIntegrationsin cos

cos sin sec tan

csc cot

sec tan sec csc cot csc Forothers,theconnectionmaynotbesoapparent.Theuseofthedoubleangleidentitiesandthe

Pythagoreanidentityandthesubstitutionmethodisextensive.

Ingeneral,

For

sincos

o

Ifnisodd,usethePythagoreanidentitytofactoroutallcosinebutone.Thenmakethesubstitutionu=sinxsin cos sin cos cos

sin 1 sin cos 1

o

Ifmisodd,usethePythagoreanidentitytofactoroutallsinebutone,thenmake

thesubstitutionu=cosx

sin cos sin cos sin cos 1 cos sin 1

o Ifbothmandnareeven

sin 1 cos 2 cos 1 cos2

sin cos sin2


68/121


69/121



Page69

sec sec sec tansec tan , let sec tan

ln| | Oralternatively,

sec cos cos cos1 sin , let sin

1

tanh , since 1 sin 112 ln 1 sin1 sin

sec Usingintegrationbyparts, sec , sec tan , sec , tan

sec sec tan tan sec sec tan sec 1 sec sec tan sec sec

2 sec sec tan ln|sec tan | sec 12 sec tan ln|sec tan |

Forintegralsinvolvingmultipleangles:

sincosor sinsinor coscos,sin cos 12 sin sin sin sin 12 cos cos

cos cos 12 cos cos


70/121



Page70

2.1.4.6 TrigonometricSubstitutionAtypeofinversesubstitution,theindependentvariableissubstitutedwithaonetoonefunction.

Ingeneral,welet

,andsubstitute

Fortheradical

,let sin , ,let tan , ,let sec , 0 or

Forexample,

9

Let 3 sin , 3cos 9 3 cos 3sin 3 cos cot

csc 1

cot

1tan 3 sin 3 9 sin 3 2.1.4.7 HyperbolicSubstitutionHyperbolicsubstitutionisalmostidenticaltotrigonometricsubstitution,andissometimespreferred

overusingthesubstitution tan or sec .Fortheradical

,let sinh , ,let cosh , 0 or 0


71/121



Page71

2.1.4.8 PartialFractionsPartialfractionscanbeusedtosimplifyarationalfunctionsothatitcanbeintegrated.

Arationalfunction

,whereallofthesefunctionsarepolynomials,andthe

degreeofRislessthanthedegreeofQ.

PartI

Thedenominatoristheproductofdistinctlinearfactors.

PartII

Thedenominator

isaproduct

of

linear

factors,

some

of

which

are

repeated

PartIII

Thedenominatorcontainsdistinctirreduciblequadraticfactors

PartIV

Thedenominatorcontainsirreduciblequadraticfactors,someofwhicharerepeated

2.1.4.8.1 Quartics

Forquarticsorhigherdegreepolynomialswithoutrealroots,itispossibletofactorisetheseusing

completethesquaretoirreduciblequadratics.

1 2 1 2 1 2 2 1 2 1 1 1 1 1 2 1 3 1 3 1 3 1


72/121


73/121


ApplicationsofIntegralCalculus

Page73

2.1.5 ApplicationsofIntegralCalculus

2.1.5.1 Area

2.1.5.1.1

Approximation

Areaapproximationworksusesvariousshapeswithdefinedareaformulaetoapproximatethearea

underagraph.

2.1.5.1.1.1Methods

Leftandrightendpointrectangles

Left:

Right:

MidpointRectangles

2

Trapeziums

2 2 2

2.1.5.1.1.2Bounds

Foranincreasingfunction

o Theleftrectanglesmethodgivesanunderestimation

o

Therightrectanglesmethodgivesanoverestimation Foradecreasingfunction

o

Theleftrectanglesmethodgivesanoverestimation

o

Therightrectanglesmethodgivesanunderestimation

Foraconcaveupfunction

o Themidpointrectanglesmethodgivesanunderestimation

o Thetrapeziummethodgivesanoverestimation

Foraconcavedownfunction

o Themidpointrectanglesmethodgivesanoverestimation

o

Thetrapeziummethodgivesanunderestimation


74/121



Page74

2.1.5.1.1.3Error

Forafunctionf(x)onaclosedinterval[a,b]:thereisanumberKsuchthat|| ,thentheerrorboundforthetrapezoidalandmidpointrulesare

|| 12 || 24 Hence,themidpointruleisabouttwiceasaccurateasthetrapezoidalmethod.

2.1.5.1.1.4TheIntegralasASum

Takingthemidpointmethod,asthenumberofrectanglesisincreased,theapproximationgets

moreandmoreaccurate.

lim Asngoestoinfinity, ,

lim

2.1.5.1.2 SignedArea

Yvaluesunderthexaxisarenegative,hencetheareacalculatedbyanintegralwouldalsobe

negative.

Iffisnegativebetweenintheinterval(a,b)[usuallyaandbwouldbetwoxintercepts],then

| | Whenfindingareaofafunctionthatcrossesthexaxisseveraltimes,thesignedareamustbetaken

intoaccount.

2.1.5.1.3 BetweenCurves

Iff(x)>g(x)foraninterval(a,b)[usuallyaandbarepointsofintersection],then

Whenfindingtheareabetweentwocurvesthatcrosseachotherseveraltimes,thesigned

differencemustbetakenintoaccount.

Ifg(x)>f(x)foranotherinterval(b,c),then

| |


75/121



Page

75

2.1.5.1.4 AlongtheYAxis

Whenintegratinginversefunctions(whichareratherdifficult),itisofteneasiertofindthearea

alongtheyaxis,andthensubtractthatfromarectangle.

Forexample

sin

Whenx=0,y=0.Whenx=1,y=/2. sin sin

sin

1 2 sin

2 cos

/ 2 1


76/121



Page76

2.1.5.2 SolidsofRevolutionWhenanareaisspunaroundanaxis,asolidofrevolutionisformed.

2.1.5.2.1 SlabMethod

Theslab

method

takes

the

solid

of

revolution

as

infinitely

thin

circular

disks

(cylinders).

The

slab

methodisapplicableonlywhentheareabeingspunisboundbytheaxiswhichitisbeingspun

around.

Ingeneral,

Aboutthexaxis:

Abouttheyaxis:

2.1.5.2.1.1 WasherMethod

Thewashermethod,akathedonutmethod,iswhenanareanotboundbytheaxisofrotationis

spun.Thecrosssectionofsuchavolumeresemblesawasher.

Thistypeofareaistheareabetweencurves,i.e. The

volume

is

calculated

by

the

summation

of

infinitely

thin

washers.

Ingeneral,asolidofrevolutionaboutthexaxis:

Abouttheyaxis:


77/121


78/121



Page78

2.1.5.4 Work Whenaforceisappliedonanobjectfromatob,andtheforceontheobjectatpointxisf(x),then

Examples

Aforceof40Nisrequiredtoholdaspringthathasbeenstretchedfromitsnaturallengthof10cmto

alengthof15cm.Howmuchworkisdoneinstretchingthespringfrom15cmto18cm?

ByHookeslaw, , 40 0.05 800 kg s 800 ... 1.56J

A20kgcableof10mlongishangeddownfromthetopofabuilding.Howmuchworkisrequiredto

liftthecabletothetopofthebuilding?

Letthetopofthebuildingbe0.Thedensityofthecableis2kgm1.

Eachsmallsectionsofcable,dx,musttraveluptothetopofthebuilding(xmetresabove)

Therefore,theworkoneachsmallsectionofcable:2*dx*g*x=2gx*dx

2 100J Aninvertedconicaltankwithheightof10mandbaseradiusof4isfilledwithwatertoaheightof

8m.Findtheworkrequiredtoemptythetankbypumpingallofthewatertothetopofthetank.

Letthetopofthetankbe0.Atsomexmetresbelowthetop,thereisalayerofwaterwithradiusof

randthicknessofdx. 10 Themassofthatlayerofwaterishence 10 Theworkonthatlayerofwaterishence 10

425 10 400025 10 3.4 10 J


79/121



Page79

2.1.5.5 MomentsandCentreofMassMoment ,wherexnisdistancefromtheorigin.Thecentreofmassiswherethesumofmomentsfromitequaltozero:

0

IfMisthetotalmassofthesystem,thenthecentreofmasswouldbe:

1

Inatwodimensionalplane,thecentreofmassofaboutthexandtheyaxis(i.e.distancefromthe

axis)arethen:

1 1 Forashapewithuniformdensityandthickness,representedbyanarea,themomentoftheshape

canbeconsideredtobethesumofmomentsofeachinfinitesimalrectangleinthexdirection:

Foreachoftheseinfinitesimalrectangle,theyhaveathicknessofdx,aheightof

,theircentre

ofmassisthecentreoftherectangle,, ,andtheirmassis ,wherepisthedensity.Hence,themomentabouttheyaxis: ,thesumofmomentishence Themomentaboutthexaxis: ,thesumofmomentishence ThemassMofthesystemis

Hence,thecentreofmassinthexandydirection(thecentroid)are,

, 12

Forashapedefinedby

,

12


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DifferentialEquations

Page81

2.1.6 DifferentialEquations

2.1.6.1 SeparableEquations

Ingeneral

Forsecondcase,itshouldbenotedthattheantiderivativewouldmostprobablybealogarithmic

functionofanabsolutefunction,whichcanbepositiveornegative.

Problemsgivinganinitialstatearecalledinitialvalueproblems,andtheinitialvaluedetermines

thepositionatwhichtherelationshipwouldmapto,aswellaswhichbranchthefunctionwilltake.

Forexample,

2 1, 0 1

12 ln|2 1| 12 ln|1| 0 | 2 1|2 1

Since

0 1,thelefthandsideisevaluatestoanegativenumber,implyingtherighthandside

mustbenegative. 2 1 12 1


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Page83

2.1.6.2.2 NewtonsLawofCooling

Abodycools/heatsatarateproportionaltothedifferencebetweenitstemperatureandits

surroundings.

Thistranslatestothedifferentialequation

,where is the temperature of the surroundings 1 ln| |

Ifatt=0,T=T0, ln| |

Sincethetemperaturefunctionismonotonic(itdoesnotovershoot),thenumeratorexpressionis

alwaysthesamesignwiththedenominatorexpression,andthemodulusisnotrequired.

Sincetherearethreeconstantsinthisexpression,thequestionwouldneedtogiveatleastthree

conditionstoworkouttheseconstants.

Inthisform,solvingforconstantswillbeverydifficult.Thefollowing[nonexplicit]formwouldbe

moreappropriate:

1 ln Forexample,athermometeristakenfromahouseat21degreestotheoutside.Oneminutelaterit

reads27degrees,anotherminutelateritreads30degrees.Findtemperatureoutsidehouse.

1 ln

21

1 1 ln 21 27 12 1 ln 21 30 22 1 2 21 27 21 30 30 21 2751 620 54 729

3 99

33


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Page84

2.1.6.2.3 DifferenceofRates

Foravolumeofsolution,withaninflow,andwhilstitiskeptevenlymixed,anoutflow,adifferential

equationforthissituationcanbemodelledby:

WhereQistheamountofsolute,

istherateofinflow,istheconcentrationofinflow,and istherateofoutflow.Thedifferentialequationcanbethoughtofas:

inlow outlow inlow concentration of inlow outlow

AmountVolume

Forsystemswheretherateofinflowisequaltotherateofoutflow:

Hence,

ln

Solvingthedifferentialequationwheretherateofinflowdoesnotequaltotherateofoutflow

involvesthemethodofintegratingfactor,whichisnotintheSpecialistMathematicscourse.An

exampleofthiswillbecoveredintherelevantsection.


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Page85

2.1.6.2.4 FiniteIntegral

Differentialequationscantranslateintoafiniteintegral,whichcanbesolvednumericallyusing

principlesofapproximation(ormachineapproximations).Thiscanbeusefulforfunctionswithoutan

antiderivative,orarathercomplexantiderivative.

LettherebeafunctionfsuchthatitsantiderivativeisF

Giventheinitialconditionwhenx=x0,y=y0,

BytheFundamentalTheoremofCalculus(II)

Hence,giventheinitialvalue,thedifferentialequationcanbesolvedforanyxwithinacontinuous

closedsetinitsdomain.

For


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Page87

2.1.6.2.6 SlopeField

Aslopefieldofadifferentialequationassignsthevalueofthegradienttoeachpointonaplane

P(x,y).

Thegradientisusuallygivenbyashortstraightlineinthedirectionoftheslopeatregularintervals

inthexandydirections.

Withaslopefield,anydifferentialequationcanbesolvedfornumericallygivenaninitialconditionat

anypointonP(x,y)

4 1 InitialCondition:x=0,y=02.1.6.2.7 OrthogonalTrajectories

Orthogonaltrajectoriesarecurvesthatarealwaysperpendiculartoeachotheratthepointof

intersection.Thesesatisfythedifferentialequation

1 Ingeneral,apairofrelationshipsthatareorthogonal:

, , \0 , , \0 Inparticular,whenn=1, , ,


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Page88

2.1.6.3 FirstOrderLinearDifferentialEquations(Integrating

Factors)Theordinarydifferentialequation

Isanonseparablefirstorderlineardifferentialequation,andcanbesolvedbymultiplyingbothsidesbyanintegratingfactor.

Usingtheintegratingfactorrecognisesthat:

Hence,multiplyingbothsidesoftheDEbytheintegratingfactorI(x)

Itcanbeseenthat

Forexample:

A20Ltankofsaltsolutioninitiallyhas2kgofdissolvedsalt.Saltispouredintothesolutionat

0.1kg/min,andthesolutionisflowingoutataconstantrateof1L/minwhilstthesolutioniskept

evenlymixed.

0.1 20 120 0.1 || | 20|

Theimplieddomainist


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Page89

2.1.6.4 SecondorderDifferentialEquationsAnordinarysecondorderlinearequationisintheformof:

2.1.6.4.1 Homogenous

AhomogenoussecondorderDEisintheformof

0Then,ify1andy2aresolutionstothisdifferentialequation,

Wherey(x)isthegeneralsolutiontothedifferentialequation,andc1andc2arearbitraryconstants.Thisimpliesthatiftwosolutionsareknown,thenallsolutionsareknown.Thisalsoimpliesthaty1

andy2arelinearlyindependent.

Notalldifferentialequationsaresolvable,butitisalwayspossibletosolveitwhenP,QandRare

constantfunctions.

0 Tosolvethislineardifferentialequation,lety=e

rx.

0 0 Theaboveequationiscalledtheauxiliaryequation(orcharacteristicequation).

Solvingforrcanhavethreedifferentoutcomes:

Twosolutions

o Onesolution

o Norealsolutiono , o

cos sino

Wherec1andc2canbecomplexnumbers.Thisgivessolutionintherealand

complexplane.


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Page90

2.1.6.4.2 Non-Homogenous

Forsecondordernonhomogenouslineardifferentialequationswithconstantcoefficients:

Thenthegeneralsolutiontakestheform Whereypisaparticularsolution,andycisthegeneralsolutiontothecomplementaryequation:

0 Whilstyccanbefoundwithreasonableease,findingypismoreinvolved,andtwoofthemethodsare

explainedbelow.

2.1.6.4.2.1

Method

of

Undetermined

Coefficients

2.1.6.4.2.1.1 PolynomialFunction

WhereG(x)isapolynomial.

TheparticularsolutionwillbeofthesamedegreeofG(x),andwilltaketheform:

Substitutingthisanditsderivativesintothedifferentialequationwillgiveasystemoflinear

equationsbyequatingthecoefficients.

2.1.6.4.2.1.2 ExponentialFunction

If then,

Substitutingthisanditsderivativesintothedifferentialequationwillgiveasystemoflinear

equationsbyequatingthecoefficientsoftheexponentialterms.

2.1.6.4.2.1.3 TrigonometricFunctions

If

cos or sinthen,

cos sin Substitutingthisanditsderivativesintothedifferentialequationwillgiveasystemoflinear

equationsbyequatingthecoefficientsofthetrigonometricterms.


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Page91

2.1.6.4.2.1.4 ProductofFunctions

Ifisaproductoftheprevioustypesoffunctions,thenatrialsolutionwouldbeaproductoftheparticularsolutions.

If

,then

If sin or cos ,then cos sin If sin or cos ,then cos sin 2.1.6.4.2.1.5 SumofFunctions

If ,thentheparticularsolutionswillbethesumofparticularsolutionsto and .2.1.6.4.2.2

Methodof

Variation

of

Parameters

Ifthecomplementaryequationhasalreadybeensolvedandisexpressedwitharbitraryconstants,

themethodofvariationofparametersthenletthearbitraryconstantsbearbitraryfunctionsand

triestofindaparticularsolution.

Hence,differentiatinggives:

Sinceu1andu2arearbitraryfunctions,conditionsmaybeimposedonthem,suchthat 0 . Hence,

Sincey1andy2areparticularsolutionstothecomplementaryequation,thissimplifiesto

Alsosince 0 ,solvingthesesimultaneouslycangiveexpressionsfor and ,whichcanbeantidifferentiatedandhencetheparticularsolutionisfound.


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Page92

Forexample,solve tan , 0 /2 Thecomplementaryequationis 0 ,whichgivesanauxiliaryequation

1 0

cos sin cos sin cos sinHence, 0 cos sin 0 Also, sin cos tan sin

cos cos sin

cos

sin cos sin cos sin cos

sin cos sin ln sec tan

sin lnsec tan cos cos sin cos ln sec tan

2.1.6.4.3 InitialValueProblemsandBoundaryValueProblems

Initialvalueproblemsforsecondorderdifferentialequationswillprovideinitialyvalueaswellasthe

initialgradient.Solvethesejustasinitialvalueproblemsinfirstorderdifferentialequations.

Aboundaryvalueproblemgivestwoyvaluesfortwoxvalues,andmaynotalwayshaveasolution.

Substitutethexandyvaluesintothegeneralsolutionandsolvesimultaneouslyforthearbitrary

constants.


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PhysicalApplications

Page93

2.1.7 PhysicalApplications

2.1.7.1 Kinematics

2.1.7.1.1

SUVATand

V

-tGraphs

Formotionwithconstantacceleration,thefollowingformulaecanbeused,whereuistheinitial

velocity,visthefinalvelocity,aistheacceleration,tisthetime,andsisthedistancetravelled.

2

12 2For a v-tgraph, the gradient is the acceleration, and the area under the graph is thedisplacement.For particular problems where an object has a maximum rate of acceleration a, a maximum rateof deceleration ntimes a, a maximum velocity vat which it reaches and travels at during thejourney, and a set distance to travel

s

, the timet

can be solved by the following:2 2 32

2.1.7.1.2 Acceleration 12 Forexample,findxintermsoftif 4 8andwhen 0, , 33 4 8 4 8 2 2 8 , 33

272 24 4


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Page94

10 4 4 5 2 5

4 4 4 since is negative

123 2 12 cos 23 0,

12 cos 12 3

cos 2 3 23 3 cos 2 3 2


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Page95

2.1.7.2 StaticsandDynamics

2.1.7.2.1 Force

Forceisavectorquantity,withadirectionandmagnitude.Itcanberesolvedintocomponents(in2D

or3D).

2.1.7.2.1.1 Equilibrium

Inequilibrium,theresultantoftheforcesisazero.Thesumofcomponentsofforcesinanydirection

isalsozero.

Forthreeforcesactingonaparticleatequilibrium:

Ifthemagnitudesofthethreeforcesareknown,thenthecosinerulecanbeapplied.

LetthevectoroppositeAbea,oppositeBbeb,andoppositeCbec.

|| | | || 2 |||| cos2.1.7.2.1.1.1 LamisTheorem

Whenanangleisknown,thesinerulecanbeappliedtofindthemagnitude/angleoftheother

forces.

||sin | |sin | |sin


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Page96

2.1.7.2.1.1.2 HangingMass

Sincethehangingmassisinequilibrium,wecanmake

thefollowinggeneralisations:

sin sin

cos cos Alternatively,wherethecomplementoftheanglesandareknown:

cos cos sin sin Solvingthesesimultaneouslywhentheangleisgivenwouldgivethemagnitudeofthetensionforces

alongthestrings.Alternatively,LamisTheoremcanbeapplied.

Generally,thelengthsofthestringsaregiven,andtheanglecanbeworkedouthenceforth.

Wheretheangleisrequiredandtheforcesareknown,thecosinerule(picturedasbefore)canbe

applied.

2.1.7.2.1.2 NewtonsLawsofMotion

Newtonsfirstlawofmotion

Aparticleatrestorinconstantmotionwillremainatrestorconstantmotionunlessactedonbyan

unbalancedforce.

Foraparticle/systeminequilibrium,theforcesactingonitmustbalance.

Newtonssecondlawofmotion

Theforceisproportionatetotherateofchangeoftheobjectsmomentum.

ThevalueofoneNewtonischosensuchthatkis1whenaisinms

2andmisinkg.

Newtonsthird

law

of

motion

Everyforcehasanequalandoppositeforce

Weight

Theweightisspecifictoparticulargravitationalfields.Onekilograminaparticularfieldweighs1kg

wt.Inotherwords,1kgwt=1gN.


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Page97

2.1.7.2.1.3 Friction

Thefrictionforcealwaysopposesthedirectionofmotion.

Themaximumfrictionforcebetweentwoparticularsurfacesisproportionaltothenormalforce

(opposingtheweightforce).

Whereisthecoefficientoffriction.Thecoefficientsoffrictionfortwostaticsurfacesandtwosurfacesslidingrelativelytoeachotheraredifferent.

2.1.7.2.1.3.1 Static Ifanexternalforceactsonastaticobjectonasurface,frictionopposesthisforceparalleltothe

planeofthesurface.Thefrictionforceopposestheexternalforceasmuchaspossibleuptoits

maximumlimit.Untiltheexternalforceisgreaterthanthemagnitudeofmaximumforce,thereisno

motion,andtheobject/systemissaidtobeinequilibrium.

Foranobject/systeminequilibrium,theminimumcoefficientoffrictionpossibleiswhenthe

object/systemisonthepointofsliding.I.e.thefrictionalforceisatmaximum.(Ifthefrictionalforce

isnotatmaximum,thenthecoefficientoffrictionwouldneedtobegreater.)

2.1.7.2.1.3.2 Sliding

Slidingfrictionopposesthedirectionofmotion,andhasthemagnitude


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Page98

2.1.7.2.2 SingleObject

2.1.7.2.2.1InclinedPlane

Forexample

AnobjectisprojectedwithspeedUuparoughplanewithcoefficientoffriction

andinclinationof

degreestothehorizontal.Thedistanceittravelsuptheplane(downtheplaneisnegative):

sin cos sin cos sin cos sin cos 1sin cos 2 2sin cos

Thespeedwhichitreturnstoground:

sin cos 2sin cos 2sin cos

2sin cos

2sin cos

sin cos sin cos


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Page99

Anotherexample

Anobjectofmass24kgisonthepointofslidingdownarough

inclinedplanewhenpulledbyaforceof10kgwtatanangleof

30ototheinclinedplane.Whenthesizeoftheforceis

increasedto12kgwt,theobjectisonthepointofslidingup.

Downtheplaneisnegative.Atpointofslidingdown,the

frictionalforceopposesgravityandpointsuptheplane.At

pointofslidingup,thefrictionalforceopposesthepullingforce

andpointsdowntheplane.Inbothinstances,thefrictional

forceismaximum,i.e. Pointofslidingdown:

Perpendiculartotheplane:10sin30 24cos 24 cos 5 Paralleltotheplane:10cos30 24 sin 5 3 24cos 5 24 sin Pointofslidingup:

Perpendiculartotheplane:12sin30 24cos 24cos 6 Paralleltotheplane:12cos30 24 sin 6 3 24 sin 24 cos 6

113 24 cos 5 48sin 24 cos 6

48sin 11353 48 sin 11324cos 5 24 sin 53 48 sin 11324cos 5 24sin , |0 2 2327or 7641

0.05243 27.6552


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Page101

MoveableWedge

Foramoveablewedgeonasurfaceandablock

sittingontop,theblockexertsaforce

perpendiculartotheslantface,causingthe

wedgetomoveaway.Theblockhasanet

accelerationtowardstheslantfaceaswellas

paralleltoit.

Inthediagramontheright,blueforcesactson

theblock,andredforcesactsonthewedge,and

thegreenforcesarecomponentsofR1,which

areparticularlyimportant.

Theblockisacceleratingdownwardsperpendiculartotheslantface(downwardsdirectionis

negative). cos Theblockexertsaforceonthewedge,R1,whichisequalandoppositetoR1.

cos Hence,theaccelerationofthewedgetotheleftwouldbe(leftisnegative)

sin

sin Also,astheblockacceleratesperpendicularlytotheslantface,itdoesso

thatitkeepsupwiththewedgewhichismovingaway.i.e.the

accelerationoftheblockisthecomponentofthenetaccelerationofthe

wedgeperpendiculartotheslantface.

sin Thenormalreactionforceexertedbythegroundonthewedge,R2,is

cos


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Page102

Forexample,a2kgsmoothwedgeisplacedonasmoothtable,andasmooth1kgblockisplacedon

theslantface.

cos cos cos 2 sin sin cos 2 sin sin 2 cossin2 2 sin 2 sin2

52 sin 12 2 sin2

52 12 cos2 2 sin2 sin 25 cos 2Nowthefrictionlesstableisreplacedbyaroughsurface.Theminimumcoefficientoffrictionwould

beifthesystemisnowonthepointofsliding.

2 cosSincethewedgeisnotmoving,theblockhasnonetaccelerationperpendiculartotheslantface.

cos sin 2 cos cos sin 12 sin21

2cos 2 5

2

sin2cos 2 5


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Page

103

2.1.7.2.3.2 ConnectedParticles

Inconnectedparticles,therope(inextensible)exertsanequaltensionforceonbothobjects

connectedtoit.

Horizontalplane

Pulley

Thepulleysystemmovestowardstheheavierside.

Ontheheavierside:

Onthelighterside:

Ifanobjecthastwostringsattached:

or


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Page104

2.1.7.2.4 VectorForce

Ifaforceisgivenasavector,theforceactsinthedirectionofthevector,andthemagnitudeofthe

forceisthemagnitudeofthevector.

Ifseveralforcesareinvolved,theresultantvectorcanbefound.

2.1.7.2.5 VariableForce

Ifforceisvariable,thenaccelerationisnotconstant.

If , 1 Forexample,anobjectmass3kgisprojectedverticallyupwardswithinitialspeedUm/s,andreturns

toitsstartingpointwithspeedVm/s.Assumethatairresistanceis,wherevisthespeedofthe

object.

Upwardsmotionispositive.

Maximumheight:

20 60 20 60 60

60 60 6060 6060 30 260 30 ln|60 | 30ln 60 60


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Page105

Thetimetakenfortheobjecttoreturntostartingpointfrommaximumheight(frictionisupwards

[positive]thistime)

20

60 20

60 60 60 60 60 60

15 2 60 60 60

15 1 60 1 60 15ln 60 60

Givenwhent=0(maximumheight),v=0

15 ln1 0 15ln 60 60Whenv=V(whenitreachestheground)

15ln 60 60 15ln 60 60


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Page106

Also,

60 60 60 60 60 60 30 2 60 30 ln| 60 | 30ln 6060

30ln 60 60 Notethatthemagnitudeofthedistancetravelledis

ln ,asthedistancetravelledisinthedownwardsdirection,i.e.negative.

Sincethedistancetravelledontheupwardsjourneyisthesameasthedistancetravelledonthe

downwardsjourney,

30ln 60 60 30ln 60 60 60 60 60 60


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SequencesandSeries

Page107

2.1.8 SequencesandSeries

2.1.8.1 SequenceAsequenceisalistofnumberswritteninadefiniteorder,usuallyobeyingaparticularrule.

, , , Foralternatingsequences,1isusuallyincorporatedinitsrule.2.1.8.1.1LimitsofSequences

Asequencemaybedefinedasafunctionofnaturalnumbers,andthisfunctionisasubsetofthe

functionoverR.

Thelimitlawsappliesforlimitsofsequences,whichmaybeevaluatedsimply.

Severalkeynotesare:

Forasequencethatdoesnotconvergeatinfinity,itiscalleddivergent(usuallyanoscillating

function).Forasequencethatgetsinfinitelylargeorsmall,itiscalleddivergenttoinfinity.

Aparticularusefulidentity,

lim || 0, then lim 0 Whereanindeterminateformisencounteredwhenevaluatingalimit,LHopitalsrulecannotbe

applieddirectly,butachangeofvariablesmay:

If where is an integer, and lim for ,then lim also.LHopitalsrulecanbeappliedforthelimitoverreals,butnotoverintegers.

Also,fortheseries ,lim 0, 1 11, 1divergent, elsewhere

Asequencecanbesaidtobeincreasingordecreasing.Ifasequenceisalwaysincreasingor

decreasing,itiscalledmonotonic.

Forasequencethathaveanupperboundorlowerbound,itissaidtobeboundedaboveor

boundedbelow,andinthecaseofboth,bounded.

Everyboundedmonotonicsequenceisconvergent.


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SequencesandSeries

Page109

2.1.8.2.1TestsofConvergence

2.1.8.2.1.1IntegralTest

Iffisacontinuous,positiveanddecreasingfunctionon ,

,and

,then

If isdivergent,then isalsodivergent.

If isconvergent,then isalsoconvergent.Bothofthesecanbeshowngraphicallybyconstructingleft/rightrectangles(overestimationtoshow

itisdivergent,underestimationtoshowitisconvergent).

Notethatthelowerboundisnotnecessarily1,iftheseriesisdefinedfromn=k,thentheintegral

wouldbecomputedfromk.

Whenthefirstntermsareusedtoestimatetheseries(i.e.usingapartialsum),theerrormade

(calledtheremainder, )isboundedsuchthat Also, .Thisgivesabetterestimationthanpartialsumsdo.Inthiscase,theerrorishalfwaybetweentheupperandlowerbounds.

2.1.8.2.1.1.1 p-Series

Thepseries

isconvergentif

1andisdivergentotherwise.

2.1.8.2.1.2ComparisonTest

Thecomparisontestcomparesagivenserieswithaseriesthatisknowntobeconvergentor

divergent.

FortwoseriesandIf foralln,andisdivergent,thenisalsodivergent.If foralln,andisconvergent,thenisalsoconvergent.If

both

sequences

are

positive

terms

and

/ ,theneitherbothseriesconvergesorbothdiverges.ThisrestonthefactthatthereexiststwonumbersmandMsuchthat ,hence .Ifwasconvergent,theupperboundisfinitehenceisalsoconvergent.Ifwasdivergent,thelowerboundisinfiniteandhenceisalsodivergent.Ifwasfoundtobedivergentbycomparisonwith,thencanbeestimatedbypartialseriesandtheerrorcouldbefoundbycomparingremainders.Notethatthisonlyworksif ,andtheremainderof

islessthantheremainderof

.


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SequencesandSeries

Page110

2.1.8.2.1.3AlternatingSeries

Foranalternatingseries or ,wherebnisamonotonicdecreasingsequencethatconvergestozero,converges.Orinotherwords,if

foralln,and

lim 0,then

converges.

Also,wheretheaboveconditionsaremet,theerror(remainder)isboundedsuchthat|| .2.1.8.2.1.4AbsoluteConvergenceandRatioTest

Foraseries,itsabsoluteseriesis||.If||isconvergent,thenisabsolutelyconvergent.

Allabsolutelyconvergentseriesareconvergent.Aconvergentserieswhichisnotabsolutely

convergentiscalledconditionallyconvergent.

The

ratio

test

uses

the

limit

IfL1,thentheseriesisdivergent IfL=1,thentheratiotestisinconclusive

However,itshouldbenotedthatthislimitevaluatesto1forallpseries,andhenceall

rational/algebraicfunctionsofn.

Similarly,thereisalsoaroottestforexponentialseries,using

|

|

IfL1,thentheseriesisdivergent IfL=1,thentheratiotestisinconclusive

2.1.8.2.1.5Mixed

Evaluatethelimitofthesequencefirst,itisnotzero,itisdivergent.

pseriesandgeometricseriesareeasilyidentifiable.

Forseriessimilartothepseriesorgeometricseries,usethecomparisontest.

Foralternatingseries,usethealternatingseriestest.

Iftheseriesinvolvesfactorialsorotherproducts(orexponentials),trytheratiotestorthe

roottest.

Iftheintegralcanbeeasilycomputed,thentheintegraltestiseffective.


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SequencesandSeries

Page111

2.1.8.2.2PowerSeries

Apowerseriesisdefinedas:

Wherecnisacoefficient.Thepowerseriesissimilartoapolynomial,exceptthatithasinfinite

numberofterms.Wherea=0andcnisaconstant,thepowerseriesbecomesageometricserieswith

x=r.

Thepowerseriescanbeconvergent/divergentdependingonthevalue(s)ofx.Ingeneral,thethree

possibilitiesare:

Theseriesconvergeswhenx=a

Theseriesconvergesforallx

Theseriesconvergesforarangeofvaluesuchthat| | Forthelastcase,Riscalledtheradiusofconvergence,andisequaltozero/infinityfortheother

twocases.Theintervalofconvergenceistheintervalonxatwhichthepowerseriesconverges.

Normally,theratiotestisusedtofind| | ,andothermethodsareusedto

specialist maths 3/4 bound notes

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