最美夕阳红项目创业计划书

GCEEdexcelGCEinMathematicsMathematicalFormulaeandStatisticalTablesBTECFirstDiploma/sLevel2SubjectTitleAugust2004XXXXxxDraft8ForuseinEdexcelAdvancedSubsidiaryGCEandAdvancedGCEexaminationsCoreMathematicsC1–C4FurtherPureMathematicsFP1–FP3MechanicsM1–M5StatisticsS1–S4ForusefromJanuary2008UA018598UA018598–EdexcelAS/AlevelMathematicsFormulaeList:C1–C4,FP1–FP3–ContentsPage–Issue1–September20071TABLEOFCONTENTSPage4CoreMathematicsC14Mensuration4Arithmeticseries5CoreMathematicsC25Cosinerule5Binomialseries5Logarithmsandexponentials5Geometricseries5Numericalintegration6CoreMathematicsC36Logarithmsandexponentials6Trigonometricidentities6Differentiation7CoreMathematicsC47Integration8FurtherPureMathematicsFP18Summations8Numericalsolutionofequations8Coordinategeometry8Conics8Matrixtransformations9FurtherPureMathematicsFP29Areaofsector9Maclaurin’sandTaylor’sSeries10Taylorpolynomials11FurtherPureMathematicsFP311Vectors13Hyperbolics14Integration14Arclength15Surfaceareaofrevolution2UA018598–EdexcelAS/AlevelMathematicsFormulaeList:M1–M5,S1–S4ContentsPage–Issue1–September200716MechanicsM116TherearenoformulaegivenforM1inadditiontothosecandidatesareexpectedtoknow.16MechanicsM216Centresofmass16MechanicsM316Motioninacircle16Centresofmass16Universallawofgravitation17MechanicsM417TherearenoformulaegivenforM4inadditiontothosecandidatesareexpectedtoknow.17MechanicsM517Momentsofinertia17Momentsasvectors18StatisticsS118Probability18Discretedistributions18Continuousdistributions19Correlationandregression20TheNormaldistributionfunction21PercentagepointsoftheNormaldistribution22StatisticsS222Discretedistributions22Continuousdistributions23Binomialcumulativedistributionfunction28Poissoncumulativedistributionfunction29StatisticsS329Expectationalgebra29Samplingdistributions29Correlationandregression29Non-parametrictests30Percentagepointsofthe2distribution31Criticalvaluesforcorrelationcoefficients32Randomnumbers33StatisticsS433Samplingdistributions34PercentagepointsofStudent’stdistribution35PercentagepointsoftheFdistributionTherearenoformulaeprovidedforDecisionMathematicsunitsD1andD2.UA018598–EdexcelAS/AlevelMathematicsFormulaeList–Issue1–September20073Theformulaeinthisbooklethavebeenarrangedaccordingtotheunitinwhichtheyarefirstintroduced.Thusacandidatesittingaunitmayberequiredtousetheformulaethatwereintroducedinaprecedingunit(e.g.candidatessittingC3mightbeexpectedtouseformulaefirstintroducedinC1orC2).ItmayalsobethecasethatcandidatessittingMechanicsandStatisticsunitsneedtouseformulaeintroducedinappropriateCoreMathematicsunits,asoutlinedinthespecification.4UA018598–EdexcelAS/AlevelMathematicsFormulaeList:CoreMathematicsC1–Issue1–September2007CoreMathematicsC1MensurationSurfaceareaofsphere=4r2Areaofcurvedsurfaceofcone=rslantheightArithmeticseriesun=a+(n–1)dSn=21n(a+l)=21n[2a+(n1)d]UA018598–EdexcelAS/AlevelMathematicsFormulaeList:CoreMathematicsC2–Issue1–September20075CoreMathematicsC2CandidatessittingC2mayalsorequirethoseformulaelistedunderCoreMathematicsC1.Cosinerulea2=b2+c2–2bccosABinomialseries21)(221nrrnnnnnbbarnbanbanaba(nℕ)where)!(!!Crnrnrnrnnxxrrnnnxnnnxxrn,1(21)1()1(21)1(1)1(2ℝ)LogarithmsandexponentialsaxxbbalogloglogGeometricseriesun=arn1Sn=rran1)1(S=ra1forr<1NumericalintegrationThetrapeziumrule:baxyd21h{(y0+yn)+2(y1+y2+...+yn–1)},wherenabh6UA018598–EdexcelAS/AlevelMathematicsFormulaeList:CoreMathematicsC3–Issue1–September2007CoreMathematicsC3CandidatessittingC3mayalsorequirethoseformulaelistedunderCoreMathematicsC1andC2.LogarithmsandexponentialsxaxalneTrigonometricidentitiesBABABAsincoscossin)(sinBABABAsinsincoscos)(cos))((tantan1tantan)(tan21kBABABABA2cos2sin2sinsinBABABA2sin2cos2sinsinBABABA2cos2cos2coscosBABABA2sin2sin2coscosBABABADifferentiationf(x)f(x)tankxksec2kxsecxsecxtanxcotx–cosec2xcosecx–cosecxcotx)g()f(xx))(g()(g)f()g()(f2xxxxxUA018598–EdexcelAS/AlevelMathematicsFormulaeList:CoreMathematicsC4–Issue1–September20077CoreMathematicsC4CandidatessittingC4mayalsorequirethoseformulaelistedunderCoreMathematicsC1,C2andC3.Integration(+constant)f(x)xxd)f(sec2kxk1tankxxtanxseclnxcotxsinlnxcosec)tan(lncotcosecln21xxxxsec)tan(lntansecln4121xxxxxuvuvxxvudddddd8UA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP1–Issue1–September2007FurtherPureMathematicsFP1CandidatessittingFP1mayalsorequirethoseformulaelistedunderCoreMathematicsC1andC2.Summations)12)(1(6112nnnrnr224113)1(nnrnrNumericalsolutionofequationsTheNewton-Raphsoniterationforsolving0)f(x:)(f)f(1nnnnxxxxCoordinategeometryTheperpendiculardistancefrom(h,k)to0cbyaxis22bacbkahTheacuteanglebetweenlineswithgradientsm1andm2is21211arctanmmmmConicsParabolaRectangularHyperbolaStandardFormaxy42xy=c2ParametricForm(at2,2at)tcct,Foci)0,(aNotrequiredDirectricesaxNotrequiredUA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP1–Issue1–September20079MatrixtransformationsAnticlockwiserotationthroughaboutO:cossinsincosReflectioninthelinexy)(tan:2cos2sin2sin2cos10UA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP2–Issue1–September2007FurtherPureMathematicsFP2CandidatessittingFP2mayalsorequirethoseformulaelistedunderFurtherPureMathematicsFP1andCoreMathematicsC1–C4.AreaofasectorA=d212r(polarcoordinates)Complexnumberssinicosei)sini(cos)}sini(cos{nnrrnnTherootsof1nzaregivenbynkzi2e,for1,,2,1,0nkMaclaurin’sandTaylor’sSeries)0(f!)0(f!2)0(f)0f()f()(2rrrxxxx)(f!)()(f!2)()(f)()f()f()(2araxaaxaaxaxrr)(f!)(f!2)(f)f()f()(2arxaxaxaxarrxrxxxxrxallfor!!21)exp(e2)11()1(32)1(ln132xrxxxxxrrxrxxxxxrrallfor)!12()1(!5!3sin1253xrxxxxrrallfor)!2()1(!4!21cos242)11(12)1(53arctan1253xrxxxxxrrTaylorpolynomialserror)(f!2)(f)f()f(2ahahaha)0()(f!2)(f)f()f(2hahahahaerror)(f!2)()(f)()f()f(2aaxaaxax)()(f!2)()(f)()f()f(2xaaxaaxaxUA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP3–Issue1–September200711FurtherPureMathematicsFP3CandidatessittingFP3mayalsorequirethoseformulaelistedunderFurtherPureMathematicsFP1,andCoreMathematicsC1–C4.VectorsTheresolvedpartofainthedirectionofbisba.bThepointdividingABintheratio:isbaVectorproduct:122131132332321321ˆsinbababababababbbaaakjinbaba)()()(321321321bac.acb.cba.cccbbbaaaca.bba.ccba)()()(IfAisthepointwithpositionvectorkjia321aaaandthedirectionvectorbisgivenbykjib321bbb,thenthestraightlinethroughAwithdirectionvectorbhascartesianequation)(332211bazbaybaxTheplanethroughAwithnormalvectorkjin321nnnhascartesianequationa.nddznynxnwhere0321Theplanethroughnon-collinearpointsA,BandChasvectorequationcbaacabar)1()()(TheplanethroughthepointwithpositionvectoraandparalleltobandchasequationcbartsTheperpendiculardistanceof),,(from0321dznynxnis232221321nnndnnn.12UA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP3–Issue1–September2007Hyperbolicfunctions1sinhcosh22xxxxxcoshsinh22sinhxxx22sinhcosh2cosh)1(1lnarcosh}{2xxxx}{1lnarsinh2xxx)1(11lnartanh21xxxxConicsEllipseParabolaHyperbolaRectangularHyperbolaStandardForm12222byaxaxy4212222byax2cxyParametricForm)sin,cos(ba)2,(2atat(asec,btan)(acosh,bsinh)tcct,Eccentricity1e)1(222eab1e1e)1(222eabe=2Foci)0,(ae)0,(a)0,(ae(2c,2c)Directriceseaxaxeaxx+y=2cAsymptotesnonenonebyax0,0yxUA018598–EdexcelAS/AlevelMathematicsFormulaeList–Issue1–September200713Differentiationf(x)f(x)xarcsin211xxarccos211xxarctan211xxsinhxcoshxcoshxsinhxtanhx2sechxarsinh211xxarcosh112xartanhx211xIntegration(+constant;0awhererelevant)f(x)xxd)f(xsinhxcoshxcoshxsinhxtanhxcoshln221xa)(arcsinaxax221xaaxaarctan1221ax)(lnarcosh}{22axaxxax221xa}{22lnarsinhaxxax221xa)(artanh1ln21axaxaxaxaa221axaxaxaln2114UA018598–EdexcelAS/AlevelMathematicsFormulaeList:FurtherPureMathematicsFP3–Issue1–September2007Arclengthxxysddd12(cartesiancoordinates)ttytxsddddd22(parametricform)Surfaceareaofrevolution22dd2d2dddxxySysytttUA018598–EdexcelAS/AlevelMathematicsFormulaeList–Issue1–September200715BLANKPAGETURNOVERFORMECHANICS&STATISTICSFORMULAE16UA018598–EdexcelAS/AlevelMathematicsFormulaeList:MechanicsM1–M3–Issue1–September2007MechanicsM1TherearenoformulaegivenforM1inadditiontothosecandidatesareexpectedtoknow.CandidatessittingM1mayalsorequirethoseformulaelistedunderCoreMathematicsC1.MechanicsM2CandidatessittingM2mayalsorequirethoseformulaelistedunderCoreMathematicsC1,C2andC3.CentresofmassForuniformbodies:Triangularlamina:32alongmedianfromvertexCirculararc,radiusr,angleatcentre2:sinrfromcentreSectorofcircle,radiusr,angleatcentre2:3sin2rfromcentreMechanicsM3CandidatessittingM3mayalsorequirethoseformulaelistedunderMechanicsM2,andalsothoseformulaelistedunderCoreMathematicsC1–C4.MotioninacircleTransversevelocity:rvTransverseacceleration:rvRadialacceleration:rvr22CentresofmassForuniformbodies:Solidhemisphere,radiusr:r83fromcentreHemisphericalshell,radiusr:r21fromcentreSolidconeorpyramidofheighth:h41abovethebaseonthelinefromcentreofbasetovertexConicalshellofheighth:h31abovethebaseonthelinefromcentreofbasetovertexUniversallawofgravitation221ForcedmGmUA018598–EdexcelAS/AlevelMathematicsFormulaeList:MechanicsM4–M5–Issue1–September200717MechanicsM4TherearenoformulaegivenforM4inadditiontothosecandidatesareexpectedtoknow.CandidatessittingM4mayalsorequirethoseformulaelistedunderMechanicsM2andM3,andalsothoseformulaelistedunderCoreMathematicsC1–C4andFurtherPureMathematicsFP1.MechanicsM5CandidatessittingM5mayalsorequirethoseformulaelistedunderMechanicsM2andM3,andalsothoseformulaelistedunderCoreMathematicsC1–C4andFurtherPureMathematicsFP1.MomentsofinertiaForuniformbodiesofmassm:Thinrod,length2l,aboutperpendicularaxisthroughcentre:231mlRectangularlaminaaboutaxisinplanebisectingedgesoflength2l:231mlThinrod,length2l,aboutperpendicularaxisthroughend:234mlRectangularlaminaaboutedgeperpendiculartoedgesoflength2l:234mlRectangularlamina,sides2aand2b,aboutperpendicularaxisthroughcentre:)(2231bamHooporcylindricalshellofradiusraboutaxisthroughcentre:2mrHoopofradiusraboutadiameter:221mrDiscorsolidcylinderofradiusraboutaxisthroughcentre:221mrDiscofradiusraboutadiameter:241mrSolidsphere,radiusr,aboutdiameter:252mrSphericalshellofradiusraboutadiameter:232mrParallelaxestheorem:2)(AGmIIGAPerpendicularaxestheorem:yxzIII(foralaminainthex-yplane)MomentsasvectorsThemomentaboutOofFactingatrisFr18UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS1–Issue1–September2007StatisticsS1Probability)P()P()P()P(BABABA)|P()P()P(ABABA)P()|P()P()|P()P()|P()|P(AABAABAABBADiscretedistributionsForadiscreterandomvariableXtakingvaluesixwithprobabilitiesP(X=xi)Expectation(mean):E(X)==xiP(X=xi)Variance:Var(X)=2=(xi–)2P(X=xi)=2ixP(X=xi)–2Forafunction)g(X:E(g(X))=g(xi)P(X=xi)ContinuousdistributionsStandardcontinuousdistribution:DistributionofXP.D.F.MeanVarianceNormal),N(2221e21x2UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS1–Issue1–September200719CorrelationandregressionForasetofnpairsofvalues),(iiyxnxxxxSiiixx222)()(nyyyySiiiyy222)()(nyxyxyyxxSiiiiiixy))(())((TheproductmomentcorrelationcoefficientisnyynxxnyxyxyyxxyyxxSSSriiiiiiiiiiiiyyxxxy222222)()())(()()())((}}{{Theregressioncoefficientofyonxis2)())((xxyyxxSSbiiixxxyLeastsquaresregressionlineofyonxisbxaywherexbya20UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS1–Issue1–September2007THENORMALDISTRIBUTIONFUNCTIONThefunctiontabulatedbelowis(z),definedas(z)=teztd21221.z(z)z(z)z(z)z(z)z(z)0.000.50000.500.69151.000.84131.500.93322.000.97720.010.50400.510.69501.010.84381.510.93452.020.97830.020.50800.520.69851.020.84611.520.93572.040.97930.030.51200.530.70191.030.84851.530.93702.060.98030.040.51600.540.70541.040.85081.540.93822.080.98120.050.51990.550.70881.050.85311.550.93942.100.98210.060.52390.560.71231.060.85541.560.94062.120.98300.070.52790.570.71571.070.85771.570.94182.140.98380.080.53190.580.71901.080.85991.580.94292.160.98460.090.53590.590.72241.090.86211.590.94412.180.98540.100.53980.600.72571.100.86431.600.94522.200.98610.110.54380.610.72911.110.86651.610.94632.220.98680.120.54780.620.73241.120.86861.620.94742.240.98750.130.55170.630.73571.130.87081.630.94842.260.98810.140.55570.640.73891.140.87291.640.94952.280.98870.150.55960.650.74221.150.87491.650.95052.300.98930.160.56360.660.74541.160.87701.660.95152.320.98980.170.56750.670.74861.170.87901.670.95252.340.99040.180.57140.680.75171.180.88101.680.95352.360.99090.190.57530.690.75491.190.88301.690.95452.380.99130.200.57930.700.75801.200.88491.700.95542.400.99180.210.58320.710.76111.210.88691.710.95642.420.99220.220.58710.720.76421.220.88881.720.95732.440.99270.230.59100.730.76731.230.89071.730.95822.460.99310.240.59480.740.77041.240.89251.740.95912.480.99340.250.59870.750.77341.250.89441.750.95992.500.99380.260.60260.760.77641.260.89621.760.96082.550.99460.270.60640.770.77941.270.89801.770.96162.600.99530.280.61030.780.78231.280.89971.780.96252.650.99600.290.61410.790.78521.290.90151.790.96332.700.99650.300.61790.800.78811.300.90321.800.96412.750.99700.310.62170.810.79101.310.90491.810.96492.800.99740.320.62550.820.79391.320.90661.820.96562.850.99780.330.62930.830.79671.330.90821.830.96642.900.99810.340.63310.840.79951.340.90991.840.96712.950.99840.350.63680.850.80231.350.91151.850.96783.000.99870.360.64060.860.80511.360.91311.860.96863.050.99890.370.64430.870.80781.370.91471.870.96933.100.99900.380.64800.880.81061.380.91621.880.96993.150.99920.390.65170.890.81331.390.91771.890.97063.200.99930.400.65540.900.81591.400.91921.900.97133.250.99940.410.65910.910.81861.410.92071.910.97193.300.99950.420.66280.920.82121.420.92221.920.97263.350.99960.430.66640.930.82381.430.92361.930.97323.400.99970.440.67000.940.82641.440.92511.940.97383.500.99980.450.67360.950.82891.450.92651.950.97443.600.99980.460.67720.960.83151.460.92791.960.97503.700.99990.470.68080.970.83401.470.92921.970.97563.800.99990.480.68440.980.83651.480.93061.980.97613.901.00000.490.68790.990.83891.490.93191.990.97674.001.00000.500.69151.000.84131.500.93322.000.9772UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS1–Issue1–September200721PERCENTAGEPOINTSOFTHENORMALDISTRIBUTIONThevalueszinthetablearethosewhicharandomvariableZN(0,1)exceedswithprobabilityp;thatis,P(Z>z)=1(z)=p.pzpz0.50000.00000.05001.64490.40000.25330.02501.96000.30000.52440.01002.32630.20000.84160.00502.57580.15001.03640.00103.09020.10001.28160.00053.290522UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS2–Issue1–September2007StatisticsS2CandidatessittingS2mayalsorequirethoseformulaelistedunderStatisticsS1,andalsothoselistedunderCoreMathematicsC1andC2.DiscretedistributionsStandarddiscretedistributions:DistributionofX)P(xXMeanVarianceBinomial),B(pnxnxppxn)1(np)1(pnpPoisson)Po(!exxContinuousdistributionsForacontinuousrandomvariableXhavingprobabilitydensityfunctionfExpectation(mean):xxxXd)f()E(Variance:2222d)f(d)f()()Var(xxxxxxXForafunction)g(X:xxxXd)f()g())E(g(Cumulativedistributionfunction:000d)(f)P()F(xttxXxStandardcontinuousdistribution:DistributionofXP.D.F.MeanVarianceUniform(Rectangular)on[a,b]ab1)(21ba2121)(abUA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS2–Issue1–September200723BINOMIALCUMULATIVEDISTRIBUTIONFUNCTIONThetabulatedvalueisP(Xx),whereXhasabinomialdistributionwithindexnandparameterp.p=0.050.100.150.200.250.300.350.400.450.50n=5,x=00.77380.59050.44370.32770.23730.16810.11600.07780.05030.031210.97740.91850.83520.73730.63280.52820.42840.33700.25620.187520.99880.99140.97340.94210.89650.83690.76480.68260.59310.500031.00000.99950.99780.99330.98440.96920.94600.91300.86880.812541.00001.00000.99990.99970.99900.99760.99470.98980.98150.9688n=6,x=00.73510.53140.37710.26210.17800.11760.07540.04670.02770.015610.96720.88570.77650.65540.53390.42020.31910.23330.16360.109420.99780.98420.95270.90110.83060.74430.64710.54430.44150.343830.99990.99870.99410.98300.96240.92950.88260.82080.74470.656341.00000.99990.99960.99840.99540.98910.97770.95900.93080.890651.00001.00001.00000.99990.99980.99930.99820.99590.99170.9844n=7,x=00.69830.47830.32060.20970.13350.08240.04900.02800.01520.007810.95560.85030.71660.57670.44490.32940.23380.15860.10240.062520.99620.97430.92620.85200.75640.64710.53230.41990.31640.226630.99980.99730.98790.96670.92940.87400.80020.71020.60830.500041.00000.99980.99880.99530.98710.97120.94440.90370.84710.773451.00001.00000.99990.99960.99870.99620.99100.98120.96430.937561.00001.00001.00001.00000.99990.99980.99940.99840.99630.9922n=8,x=00.66340.43050.27250.16780.10010.05760.03190.01680.00840.003910.94280.81310.65720.50330.36710.25530.16910.10640.06320.035220.99420.96190.89480.79690.67850.55180.42780.31540.22010.144530.99960.99500.97860.94370.88620.80590.70640.59410.47700.363341.00000.99960.99710.98960.97270.94200.89390.82630.73960.636751.00001.00000.99980.99880.99580.98870.97470.95020.91150.855561.00001.00001.00000.99990.99960.99870.99640.99150.98190.964871.00001.00001.00001.00001.00000.99990.99980.99930.99830.9961n=9,x=00.63020.38740.23160.13420.07510.04040.02070.01010.00460.002010.92880.77480.59950.43620.30030.19600.12110.07050.03850.019520.99160.94700.85910.73820.60070.46280.33730.23180.14950.089830.99940.99170.96610.91440.83430.72970.60890.48260.36140.253941.00000.99910.99440.98040.95110.90120.82830.73340.62140.500051.00000.99990.99940.99690.99000.97470.94640.90060.83420.746161.00001.00001.00000.99970.99870.99570.98880.97500.95020.910271.00001.00001.00001.00000.99990.99960.99860.99620.99090.980581.00001.00001.00001.00001.00001.00000.99990.99970.99920.9980n=10,x=00.59870.34870.19690.10740.05630.02820.01350.00600.00250.001010.91390.73610.54430.37580.24400.14930.08600.04640.02330.010720.98850.92980.82020.67780.52560.38280.26160.16730.09960.054730.99900.98720.95000.87910.77590.64960.51380.38230.26600.171940.99990.99840.99010.96720.92190.84970.75150.63310.50440.377051.00000.99990.99860.99360.98030.95270.90510.83380.73840.623061.00001.00000.99990.99910.99650.98940.97400.94520.89800.828171.00001.00001.00000.99990.99960.99840.99520.98770.97260.945381.00001.00001.00001.00001.00000.99990.99950.99830.99550.989391.00001.00001.00001.00001.00001.00001.00000.99990.99970.999024UA018598–EdexcelAS/AlevelMathematicsFormulaeList:StatisticsS2–Issue1–September2007p=0.050.100.150.200.250.300.350.400.450.50n=12,x=00.54040.28240.14220.06870.03170.01380.00570.00220.00080.000210.88160.65900.44350.27490.15840.08500.04240.01960.00830.003220.98040.88910.73580.55830.39070.25280.15130.08340.04210.019330.99780.97440.90780.79460.64880.49250.34670.22530.13450.073040.99980.99570.97610.92740.84240.72370.58330.43820.30440.193851.00000.99950.99540.98060.94560.88220.78730.66520.52690.387261.00000.99990.99930.99610.98570.96140.91540.84180.73930.612871.00001.00000.99990.99940.99720.99050.97450.94270.88830.806281.00001.00001.00000.99990.99960.99830.99440.98470.96440.927091.00001.00001.00001.00001.00000.99980.99920.99720.99210.9807101.00001.00001.00001.00001.00001.00000.99990.99970.99890.9968111.00001.00001.00001.00001.00001