week beginning videos pagejune/july c3: integration as the inverse of differentiation 16 date lesson...
TRANSCRIPT
1
M
2
Weekbeginning Videos Page
June/July C3AlgebraicFractions 3
June/July C3AlgebraicDivision 4
June/July C3ReciprocalTrigFunctions 5
June/July C3PythagoreanIdentities 6
June/July C3TrigConsolidation 7
June/July C3ChainRule 8
June/July C3TheProductRule 9
June/July C3TheQuotientRule 10
June/July C3Differentiatingtrigfunctions 11
June/July C3TheCompoundAngleFormulae 12
June/July C3TheDoubleAngleFormulae 13
June/July C3TheFactorFormulae 14
June/July C3Naturallogsincludingderivativesofe^xandlnx 15
June/July C3:Integrationastheinverseofdifferentiation 16
Date Lesson Video Page
11thSeptember 1 C3:RevisionforCWC1:AlgebraicFractions 17
18thSeptember 1 C3:RevisionforCWC2:Differentiatingln 18
2 C3:RevisionforCWC3:ProductandQuotientRule 19
3 C3:RevisionforCWC4:Trigproofs 20
25thSeptember 1 C3:Functions–Domain,Range,Composite 21
2ndOctober 1 C3Functions–Modulus,Inverse,Transformations 22
9thOctober 1 C3NumericalMethods 23
16thOctober 2 C3Rcos(x+a) 24
2 C3Inversetrigfunctionsincludinggraphs 25
30thOctober 1 C3:dy/dx=1/(dx/dy) 26
2 C4ImplicitDifferentiation 27
6thNovember 1 C4BinomialExpansion 28
2 C4Partialfractions 29
13thNovember Readingweek–nolessons
20thNovember 1 M2Projectiles 31
27thNovember 1 C4IntegrationusingTrig,PartialFractions,lnx 32
2 C4TrapeziumRuleandpercentageerror 34
4thDecember 1 M2Motioninastraightline 35
11thDecember 1 C4IntegrationbyParts 36
2 C4IntegrationbySubstitution 37
1stJanuary 1 M2CentresofMass 38
2 M2Frameworks 39
8thJanuary 1 M2Tilting 40
15thJanuary 1 C4ConnectedRatesofChange 41
22ndJanuary 1 C4Vectors–Introduction 42
2 C4vectors–Scalar(dot)product 43
3 C4Vectors–Vectorequationofaline 44
29thJanuary 1 M2Work,Energy&Power 45
5thFebruary 1 C4FormingDifferentialEquations 46
2 C4DifferentialEquations 47
19thFebruary 1 M2Statics 48
26thFebruary 1 C4Parametrics 49
5thMarch 1 VolumeofSolidofRevolution 50
12thMarch M2Collisions 51
3
C3:AlgebraicFractions
https://youtu.be/MC90CB-s8QM
Copythetwoexamples
1) Simplify!!!!!!!
!!!!
!!
2) Simplify!!!!!!!!
!" ÷!!!!!"
4
C3AlgebraicDivision
https://youtu.be/tfjYIrkvalI
Copytheexample !!!!!!!!!!!!"!!!!!!!
5
C3:ReciprocalTrigFunctions
https://youtu.be/5mpYSXxktHU
Completethetable
secx=
1cos !
cosecx=
1sin !
cotx=
1tan !
Drawthegraphsofy=secx, y=cosecxyy=cotx
Whenprovingatrigidentity,howshouldyoustarttheproofandhowshouldyoufinishtheproof?
StartwithLHS≡ andendwith≡ RHS(orviceversa).Thelastthingyoushouldwriteis“ProofComplete”(orQ.E.D.ifyoulikeyourLatin)Whatisthedifferencebetween=and≡=means“equals”andisusedtosolveequations≡ means“isidenticalto”andisusedwhenalgebraicexpressionsareidenticale.g.!! = !isonlytruefortwovaluesofxe.g.!! − 4 �(! − 2)(! + 2)istrueforeveryvalueofx
6
C3PythagoreanIdentities
https://youtu.be/VcDtYgSDvRs
TherearethreePythagoreanidentities(oneofwhichyouhadtoknowforC2).
Whatarethey? sin! ! + cos! ! = 1
1 + tan! ! = sec! !
1 + cot! ! = !"#$!!!
Provethat!"#$!!�-cot! � ≡ !!!"#! �
!!!"#! �
7
C3TrigConsolidation
https://youtu.be/zk8Bled7bFc
Solvetheequation4!"#$!!! − 9=cotθ.Showfullworking
8
C3ChainRule
https://youtu.be/YLSm56VIa6U
Ify= 5!! − 2! !,whatis!"!".
!"!" = 6 5!! − 2! !(15!! − 2)
9
C3:TheProductRule
https://youtu.be/Jpg3QX5slg4
Ify=f(x)g(x),whatis!"!"
f’(x)g(x)+f(x)g’(x)
Ify=!! 2!! + 3! !,whatis!"!".Showthefullworking.
10
C3TheQuotientRule
https://youtu.be/aYzFUB5q7BI
Ify=!!whereuandvarefunctionsofx,whatis
!"!" ?
!"!" =
!′! − !"′!!
Ify=!(!)!(!)whatis
!"!" ?
!"!" =
!! ! ! ! − ! ! !! !! ! !
Ify=!!!! !
!! ,whatis!"!"?Showthefullworking.
11
C3:Differentiatingtrigfunctions
https://youtu.be/kQ0lvJtXIgY
Completethistable
y !"!"
sinx
cosx
cosx
-sinx
tanx
!"#! !cotx
− !"#$!!!secx
secxtanx
cosecx
-cosecxcotx
Whendifferentiatingatrigshouldyouuseradiansordegrees?RADIANS
12
C3:TheCompoundAngleFormulae
https://youtu.be/DyqQG7MzOPU
sin(A±B)=sinAcosB±cosAsinB
cos(A±B)=cosAcosB∓sinAsinB
tan(A±B)=!"#!±!"#! ! ∓!"#$ !"#$
13
C3:TheDoubleAngleFormulae
https://youtu.be/upkil94kk_g
Completethesedoubleangleformulae
sin2A=2sinAcosA
tan2A=! !"#!!!!"#! !
cos2A=cos! ! − sin! !
Therearetwootherformulaeforcos2A.
cos2A=2 cos! ! − 1
cos2A=1 − 2 sin! !
14
C3:TheFactorFormulae
https://youtu.be/sHPEY10RSOE
Thefactorformulae
2cos
2sin2sinsin
BABABA −+=+
2sin
2cos2sinsin
BABABA −+=−
2cos
2cos2coscos
BABABA −+=+
2sin
2sin2coscos
BABABA −+−=−
15
C3:Naturallogarithmsincludingderivativesof!!andlnxhttps://youtu.be/cxtkmtQmhUY
16
C4:Integrationastheinverseofdifferentiation
https://youtu.be/NRZJw-FtuSA
17
C3:RevisionforContinuingWithConfidenceTest1SimplifyingAlgebraicFractions
https://youtu.be/PY1lwgBLnqo
Express)2)(32(
32 2
−+
+
xxxx
–2
62 −− xx
asasinglefractioninitssimplestform.
18
C3:RevisionforContinuingWithConfidenceTest2Differentiatingln
https://youtu.be/8uUqYRl9T0Q
ThepointPliesonthecurvewithequationy=ln ⎟⎠
⎞⎜⎝
⎛ x31
.Thex-coordinateofPis3.
FindanequationofthenormaltothecurveatthepointP intheformy=ax+b,whereaandbareconstants.
19
C3:RevisionforContinuingWithConfidenceTest3ProductandQuotientRule
https://youtu.be/X8xxaHJtsJE
Differentiatewithrespecttox
(i) x2e3x+2,
(ii)xx
3)2(cos 3
.
20
C3:RevisionforContinuingWithConfidenceTest4Trigproofs
https://youtu.be/Csj_NSc1uOk
Showthat
(i)xx
xsincos2cos+
≡cosx–sinx,x≠(n– 41 )π,n∈ℤ,
(ii)!!(cos2x–sin2x)≡cos
2x–cosxsinx–!!.
21
C3:Functions–Domain,Range,Composite
https://youtu.be/obyZwc7EX3Y
Definetheseterms
Amapping……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Domain……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Range………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………Function……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………CompositeFunction…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………
Iff(x)=3x+2andg(x)=!! + 1,a) whatisfg(x)
b) whatisgf(x)
22
C3Functions–Modulus,Inverse,Transformations
https://youtu.be/dxzM3Yl0X5s
Iff(x)=3x+2,whatisf-1(x)?Showthefullworking.
Sketchthefollowingfunctions f(x)=|3x+2| f(x)=|x|-2
Thegraphshowsthefunctionf(x)Sketchthegraphsofy=|f(x)| y=f(|x|)
23
C3:NumericalMethods
https://youtu.be/4JYb-lPtspU
Showthatthereisarootoftheequation!" !=!!betweenx=1.7andx=1.8.Underlinethekeysentencethatyoumustwriteattheendofthesequestions.
Useiterationtofindasolutiontox2–4x+1=0correctto2d.p.
24
C3:Rcos(x+α )
Usehttps://youtu.be/dmrFYuNfkFg
Solvetheequation3cosx+5sinx=2
(0°≤x≤360°)
Showthefullworking.
25
C3:Inversetrigfunctionsincludinggraphs
https://youtu.be/hklOnHJx1t4
Whatisthedifferencebetweeny=sin-1xandy=(sinx)-1?…………………………………………………………………………………………………………………………Sketchthegraphsofy=arcsinx(y=sin-1x) y=arccosx(y=cos-1x)y=arctanx(y=tan-1x)
26
C3:dy/dx=1/(dx/dy)
https://youtu.be/KEoip8FNAp4
Findthevalueof!"!"atthepoint(2,1)onthecurvewith
equation!! + ! = !
27
C4:ImplicitDifferentiation
https://youtu.be/am9WPDZL76M
Theequationofacurveis3!! + 2!!!! + 4!! = 12
Findanexpressionfor!"!"
28
C4:BinomialExpansion
https://youtu.be/eC2I1eismWQ
Whatisthebinomialexpansionwhichisvalidforallvaluesofn?
Ifnisfractionalornegative,theexpansionisonlyvalidforcertainvaluesofx.Whatarethose
values?
29
C4PartialFractions
https://youtu.be/OeUCqui7bu0
Write!!!!
!!! !!! inpartialfractions.Showthefullmethod
Write!!!!!!"! !!!!! ! !!!! inpartialfractions.Showthefullmethod
….continuedonnextpage
30
C4PartialFractionsPage2
Write!!!!!!!!!!! !!! inpartialfractions.Showthefullmethod
31
S2:Projectiles
https://youtu.be/ZLbfw3iYIBc
AprojectilePisprojectedfromapointOonahorizontalplanewithspeed28 !!!!andwithangleofelevation30°.Afterprojection,theprojectilemovesfreelyundergravityuntilitstrikestheplaneat
pointA.Find
a) thegreatestheightabovetheplanereachedbyP,
b) thetimeofflightofP
c) thedistanceOA
32
C4:Integration(Trig,PartialFractions,lnx)
https://youtu.be/wHYbo3igKqs
Showhowtointegratethefollowing
∫tan! ! !"
∫!"#! ! !"
∫cos! ! !"**….continuedonthenextpage**Thisisn’tonthevideo.Weshalldothisinthelesson
33
C4:Integration(Trig,PartialFractions,lnx)page2
∫!! !! ! dx=
∫!!!
!!! !!! dx
34
C4:TrapeziumRuleandPercentageError
https://youtu.be/IBMPn_4eKqY
Usethetrapeziumrulewith4stripstofindanapproximatevaluefor sec ! !"!!!
Whencalculatingthepercentageerror,usethisformula
Percentage error =
35
S2:Motioninastraightline
https://youtu.be/xvVkAG7o1T8
36
C4:IntegrationbyParts
https://youtu.be/3ElTLQirl4E
Theformulaforintegrationbypartsis
� ! !"!" !" =
Useintegrationbypartstoworkout∫xcosxdx
37
C4:IntegrationbySubstitution
https://youtu.be/WeAKe8uGQ1M
Useintegrationbysubstitutiontoworkout
∫x√(2x+5)dx
Useintegrationbysubstitutiontoevaluate
cos ! √(1 + sin !) !"!!!
38
M2CentreofMass
https://youtu.be/6dVVxvKBYdY
Findthecentreofmassofa2kgmassatthepoint(3,0),a5kgmassatthepoint(4,0)anda3kgmass
atthepoint(6,0)
Findthecentreofmassofa2kgmassatthepoint(1,2),a5kgmassatthepoint(4,3)anda3kg
massatthepoint(3,1)
Whereisthecentreofmassofthefollowinglaminas
a) Auniformcirculardisc……………………………………………………………
b) Auniformrectangularlamina………………………………………………
c) Auniformsectorofacircle…………………………………………………
Findthecentreofmassofauniformtriangularlaminawithverticesat(1,1),(5,0)and(2,4)
39
M2Frameworks
https://youtu.be/yZy7eC1hJM8
Findthecentreofmassofthisframework
40
M2Tilting
https://youtu.be/TogyRaizaQw
FindtheanglethatthelineABmakeswiththeverticalifthis
laminaisfreelysuspendedfromA
41
C4:ConnectedratesofChange
https://youtu.be/OyeiYysYXZI
Therateofchangeoftheradiusofacircleis5cms-1.Findtherateofchangeoftheareaof
thecirclewhentheradiusis3cm.
42
C4:Vectors-Introduction
https://youtu.be/2OWPNzC7JBI
ThefirstpartofthisvideoisrevisionofGCSEvectors.Makeyourownnotesofstuffyoumayhave
forgotten.
Findaunitvectorwhichisinthedirection3a+4b
Whatarethevectorsi,jandk?
Findthedistancebetween(4,3,6)and(3,4,-2)
43
C4:Vectors–scalar(dot)product
https://youtu.be/zkAMAhqeXio
Whatisthedefinitionofa.b?
Ifa=a1i+a2j+a3kandb=b1i+b2j+b3kwhatisa.b?
Whatistheanglebetween!!!!
and!!!!
?
44
C4:Vectors–vectorequationofaline
https://youtu.be/ltVa0nqX7o8
Astraightlinepassesthrough(-1,1)and(0,1).Whatisthevectorequationofthisline?
Thisnextbitisn’tinthevideo.Trytowritedowntheanswer.Wewilldiscussinthelesson.
Thevectorequationofastraightlineisr=a+λb.Whatinformationisgivenbya?Whatinformationisgivenbyb?
45
M2:Work,EnergyandPower
https://youtu.be/gu9NCEa1GnQ
46
C4:FormingDifferentialEquations
https://youtu.be/dnWa5_3eNb8
47
C4:DifferentialEquations
https://youtu.be/q9a52OxY3Ww
Solvethedifferentialequation!"!" = ! cos !.Usetheboundaryconditionthaty=0whenx=1
48
M2:Statics
https://youtu.be/4XB-7EjmqtA
49
C4:Parametrics
https://youtu.be/nFClKAxutOc
Acurveisdefinedbytheparametricequationsx=2tandy=t2.WhatistheCartesianequationofthiscurve?
Acurveisdefinedbytheparametricequationsx=t2andy=2t(3-t).Findtheareabetweenthiscurveandthexaxiswhichisboundedbythey-axisandthelinex=3
50
C4:VolumesofSolidsofRevolution
https://youtu.be/djcBiBMwI7Q
Whatistheformulaforthevolumeofasolidofrevolution?
Whatistheformulafortheareaunderacurve?
51
M2:Collisions
https://youtu.be/fLtDbmdnWl0
WhatistheformulaforConservationofLinearMomentumWhatistheformulafore(thecoefficientofrestitution)Findv1andv2
! = !!
m1=200kg m2=400kg u1=5ms-1 u2=-4ms-1 v1=? v2=?
52
UnitC3:CoreMathematics3Thissectionlistsformulaethatcandidatesareexpectedtorememberandthatmaynotbeincludedinformulaebooklets.Trigonometry
Differentiationfunction Derivative sinkx kcoskxcoskx –ksinkxekx kekxlnx x
1
f(x)+g(x) )(g +)( f xx ʹʹ
f ( g (x x) ) )(g )( f)g( )( f xxxx ʹ+ʹ
f(g(x)) )(g ))( g( f xx ʹʹ
C3Syllabus1 AlgebraandfunctionsSimplificationofrationalexpressionsincludingfactorisingandcancelling,andalgebraicdivision.
Denominatorsofrationalexpressionswillbelinear
orquadratic,egb + ax
1,
r + qx + pxb + ax
2,
11
2
3
−
+
xx
.
Definitionofafunction.Domainandrangeoffunctions.Compositionoffunctions.Inversefunctionsandtheirgraphs.
Theconceptofafunctionasaone-oneormany-onemappingfromℝ(orasubsetofℝ)toℝ.Thenotationf:x! andf(x)willbeused.Candidatesshouldknowthatfgwillmean‘dogfirst,thenf’.
Candidatesshouldknowthatiff−1exists,thenf−1f(x)=ff−1(x)=x.
Themodulusfunction.
Candidatesshouldbeabletosketchthegraphsofy=⏐ax+b⏐andthegraphsofy=⏐f(x)⏐andy=f(⏐x⏐),giventhegraphofy=f(x).
Combinationsofthetransformationsy=f(x)asrepresentedbyy=af(x),y=f(x)+a,y=f(x+a),y=f(ax).
Candidatesshouldbeabletosketchthegraphof,
forexample,y=2f(3x),y=f(−x)+1,giventhegraphofy=f(x)orthegraphof,forexample,
y=3+sin2x,y=−cos ⎟⎠
⎞⎜⎝
⎛+4π
x .
Thegraphofy=f(ax+b)willnotberequired.
AAAA
AA
22
22
22
cot1cosectan1sec
1sincos
+≡
+≡
≡+
AAA
AAAAAA
2
22
tan1 tan 2
2tan
sincos2 cos cos sin 2 2sin
−≡
−≡
≡
53
2 TrigonometryKnowledgeofsecant,cosecantandcotangentandofarcsin,arccosandarctan.Theirrelationshipstosine,cosineandtangent.Understandingoftheirgraphsandappropriaterestricteddomains.
Anglesmeasuredinbothdegreesandradians.
Knowledgeanduseofsec2θ=1+tan2θandcosec2θ=1+cot2θ.
Knowledgeanduseofdoubleangleformulae;use
offormulaeforsin(A±B),cos(A±B)andtan(A±B)andofexpressionsforacosθ+bsinθintheequivalentformsofrcos(θ±a)orrsin(θ±a).
Toincludeapplicationtohalfangles.Knowledgeof
thet(tan θ21 )formulaewillnotberequired.
Candidatesshouldbeabletosolveequationssuchas
acosθ+bsinθ=cinagiveninterval,andtoprovesimpleidentitiessuchas
cosxcos2x+sinxsin2x≡cosx.
3 ExponentialsandlogarithmsThefunctionexanditsgraph.
Toincludethegraphofy=eax+b+c.
Thefunctionlnxanditsgraph;lnxastheinversefunctionofex.
Solutionofequationsoftheformeax+b=pandln(ax+b)=qisexpected.
4 DifferentiationDifferentiationofex,lnx,sinx,cosx,tanxandtheirsumsanddifferences.
Differentiationusingtheproductrule,thequotientruleandthechainrule.
Differentiationofcosecx,cotxandsecxarerequired.Skillwillbeexpectedinthedifferentiationoffunctionsgeneratedfromstandardformsusingproducts,quotientsandcomposition,suchas2x4sin
x,x
x3e,cosx2andtan22x.
Theuseof
⎟⎟⎠
⎞⎜⎜⎝
⎛=
yxx
y
dd1
dd
. E.g.findingxydd
forx=sin3y.
5 NumericalmethodsLocationofrootsoff(x)=0byconsideringchangesofsignoff(x)inanintervalofxinwhichf(x)iscontinuous.
Approximatesolutionofequationsusingsimpleiterativemethods,includingrecurrencerelationsoftheformxn+1=f(xn).
Solutionofequationsbyuseofiterativeproceduresforwhichleadswillbegiven.
54
UnitC4:CoreMathematics4Thissectionlistsformulaethatcandidatesareexpectedtorememberandthatmaynotbeincludedinformulaebooklets.Integrationfunction integral
coskx kx
k sin1
+c
sinkx kx
k cos
1− +c
ekx
kx
ke1 +c
x1
ln x +c, 0≠x
)( g)( f xx ʹ+ʹ )( g+)( f xx +c
)(g ))( (g f xx ʹʹ ))( g( f x +cVectors
zcybxacba
zyx
++=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
.
C4Syllabus1 AlgebraandfunctionsRationalfunctions.Partialfractions(denominatorsnotmorecomplicatedthanrepeatedlinearterms).
Partialfractionstoincludedenominatorssuchas(ax+b)(cx+d)(ex+f)and(ax+b)(cx+d)2.Thedegreeofthenumeratormayequalorexceedthedegreeofthedenominator.Applicationstointegration,differentiationandseriesexpansions.Quadraticfactorsinthedenominatorsuchas(x2+a),a>0,arenotrequired.
2 Coordinategeometryinthe(x,y)planeParametricequationsofcurvesandconversionbetweenCartesianandparametricforms.
Candidatesshouldbeabletofindtheareaunderacurvegivenitsparametricequations.Candidateswillnotbeexpectedtosketchacurvefromitsparametricequations.
55
3 SequencesandseriesBinomialseriesforanyrationaln.
Forab
x < ,candidatesshouldbeabletoobtainthe
expansionof(ax+b)n,andtheexpansionofrationalfunctionsbydecompositionintopartialfractions.
4 DifferentiationDifferentiationofsimplefunctionsdefinedimplicitlyorparametrically.
Thefindingofequationsoftangentsandnormalstocurvesgivenparametricallyorimplicitlyisrequired.
Exponentialgrowthanddecay.
Knowledgeanduseoftheresult
xdd(ax)=axlnaisexpected.
Formationofsimpledifferentialequations.
Questionsinvolvingconnectedratesofchangemaybeset.
5 Integration
Integrationofex,x1,sinx,cosx.
Toincludeintegrationofstandardfunctionssuchas
sin3x,sec22x,tanx,e5x,x21
.
Candidatesshouldrecogniseintegralsoftheform
xxx d)f()(f
⎮⌡⌠ ʹ
=lnf(x)+c.
Candidatesareexpectedtobeabletousetrigonometricidentitiestointegrate,forexample,sin2x,tan2x,cos23x.
Evaluationofvolumeofrevolution. xy d2
⎮⌡
⌠π isrequired,butnot yx d2⎮⌡
⌠π .
Candidatesshouldbeabletofindavolumeofrevolution,givenparametricequations.
Simplecasesofintegrationbysubstitutionandintegrationbyparts.Thesemethodsasthereverseprocessesofthechainandproductrulesrespectively.
Exceptinthesimplestofcasesthesubstitutionwillbegiven.
Theintegral ∫ xln dxisrequired.
Morethanoneapplicationofintegrationbyparts
mayberequired,forexample ∫ xx x de2 .
Simplecasesofintegrationusingpartialfractions.
Integrationofrationalexpressionssuchasthose
arisingfrompartialfractions,e.g.53
2+x
,2)1(
3−x
.
Notethattheintegrationofotherrational
56
expressions,suchas52 +x
xand
4)12(2−x
isalso
required(seeaboveparagraphs).
Analyticalsolutionofsimplefirstorderdifferentialequationswithseparablevariables.
Generalandparticularsolutionswillberequired.
Numericalintegrationoffunctions.
ApplicationofthetrapeziumruletofunctionscoveredinC3andC4.Useofincreasingnumberoftrapeziatoimproveaccuracyandestimateerrorwillberequired.Questionswillnotrequiremorethanthreeiterations.
Simpson’sRuleisnotrequired.
6 VectorsVectorsintwoandthreedimensions.
Magnitudeofavector.
Candidatesshouldbeabletofindaunitvectorinthe
directionofa,andbefamiliarwith⏐a⏐.
Algebraicoperationsofvectoradditionandmultiplicationbyscalars,andtheirgeometricalinterpretations.
Positionvectors.Thedistancebetweentwopoints.
ab−==−→→→
ABOAOB .
Thedistancedbetweentwopoints(x1,y1,z1)and(x2,y2,z2)isgivenbyd2=(x1–x2)2+(y1–y2)2+(z1–z2)2.
Vectorequationsoflines. Toincludetheformsr=a+tbandr=c+t(d–c).Intersection,orotherwise,oftwolines.
Thescalarproduct.Itsuseforcalculatingtheanglebetweentwolines.
Candidatesshouldknowthatfor→OA=a=a1i+a2j+a3kand→OB=b=b1i+b2j+b3kthena.b=a1b1+a2b2+a3b3and
cos∠AOB=baba ..
Candidatesshouldknowthatifa.b=0,andthataandbarenon-zerovectors,thenaandbareperpendicular.
57
UnitM2:Mechanics2
58
59
FormulaeandTablesCoreMathematicsC3Logarithmsandexponentials
xax a=lne Trigonometricidentities
BABABA sincoscossin)(sin ±=±
BABABA sinsincoscos)(cos ∓=±
))(( tantan1tantan)(tan 2
1 π+≠±±
=± kBABABABA
∓
2cos
2sin2sinsin
BABABA −+=+
2sin
2cos2sinsin
BABABA −+=−
2cos
2cos2coscos
BABABA −+=+
2sin
2sin2coscos
BABABA −+−=−
Differentiation
f(x) fʹ(x)tankx ksec2kxsecx secxtanxcotx –cosec2xcosecx –cosecxcotx
)g()f(xx
))(g(
)(g)f( )g()(f2x
xxxx ʹ−ʹ
CoreMathematicsC4Integration(+constant)f(x)
⎮⌡
⌠ xx d)f(
sec2kx k1tankx
xtan xsecln
xcot
xsinln
xcosec )tan(lncotcosecln 21 xxx =+−
xsec )tan(lntansecln 41
21 π+=+ xxx
⎮⌡⌠
⎮⌡⌠−= x
xu
vuvxxv
u dddd
dd
60
FormulaeandTables