A Level数学公式书

GCEEdexcel GCE in MathematicsMathematical Formulae and Statistical TablesFor use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations Core Mathematics C1 – C4Further Pure Mathematics FP1 – FP3Mechanics M1 – M5Statistics S1 – S4For use from January 2008UA018598TABLE OF CONTENTSPage4 Core Mathematics C14 Mensuration4 Arithmetic series5 Core Mathematics C25 Cosine rule5 Binomial series5 Logarithms and exponentials5 Geometric series5 Numerical integration6 Core Mathematics C36 Logarithms and exponentials6 Trigonometric identities6 Differentiation7 Core Mathematics C47 Integration8 Further Pure Mathematics FP1 8 Summations8 Numerical solution of equations 8 Coordinate geometry8 Conics8 Matrix transformations9 Further Pure Mathematics FP2 9 Area of sector9 Maclaurin’s and Taylor’s Seri es10 Taylor polynomials11 Further Pure Mathematics FP3 11 Vectors13 Hyperbolics14 Integration14 Arc length15 Surface area of revolution16 Mechanics M116 There are no formulae given for M1 in addition to those candidates are expected to know.16 Mechanics M216 Centres of mass16 Mechanics M316 Motion in a circle16 Centres of mass16 Universal law of gravitation17 Mechanics M417 There are no formulae given for M4 in addition to those candidates are expected to know.17 Mechanics M517 Moments of inertia17 Moments as vectors18 Statistics S118 Probability18 Discrete distributions18 Continuous distributions19 Correlation and regression20 The Normal distribution function21 Percentage points of the Normal distribution22 Statistics S222 Discrete distributions22 Continuous distributions23 Binomial cumulative distribution function28 Poisson cumulative distribution function29 Statistics S329 Expectation algebra29 Sampling distributions29 Correlation and regression29 Non-parametric tests30 Percentage points of the 2 distribution31 Critical values for correlation coefficients32 Random numbers33 Statistics S433 Sampling distributions34 Percentage points of Student’s t distribution35 Percentage points of the F distributionThere are no formulae provided for Decision Mathematics units D1 and D2.The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2).It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification.MensurationSurface area of sphere = 4π r 2Area of curved surface of cone = π r ⨯ slant heightArithmetic series u n = a + (n – 1)dS n = 21n (a + l ) = 21n [2a + (n - 1)d ]Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rulea 2 =b 2 +c 2 – 2bc cos ABinomial series2 1)(221nr r n n n n n b b a r n b a n b a n a b a ++⎪⎪⎭⎫ ⎝⎛++⎪⎪⎭⎫ ⎝⎛+⎪⎪⎭⎫ ⎝⎛+=+--- (n ∈ ℕ) where )!(!!C r n r n r n r n -==⎪⎪⎭⎫ ⎝⎛ ∈<+⨯⨯⨯+--++⨯-++=+n x x r r n n n x n n nx x rn ,1( 21)1()1( 21)1(1)1(2 ℝ)Logarithms and exponentialsax x b b a log log log =Geometric series u n = ar n - 1S n = rr a n --1)1(S ∞ = ra-1 for ∣r ∣ < 1Numerical integrationThe trapezium rule: ⎜⎠⎛bax y d ≈ 21h {(y 0 + y n ) + 2(y 1 + y 2 + ... + y n – 1)}, where n a b h -=Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.Logarithms and exponentialsx a x a =ln eTrigonometric identitiesB A B A B A sin cos cos sin )(sin ±=±B A B A B A sin sin cos cos )(cos =±))(( tan tan 1tan tan )(tan 21π+≠±±=±k B A BA BA B A 2cos2sin 2sin sin BA B A B A -+=+ 2sin2cos 2sin sin BA B A B A -+=- 2cos2cos 2cos cos BA B A B A -+=+ 2sin2sin 2cos cos BA B A B A -+-=-Differentiationf(x ) f'(x )tan kx k sec 2 kxsec x sec x tan x cot x –cosec 2 x cosec x–cosec x cot x)g()f(x x))(g()(g )f( )g()(f 2x x x x x '-'Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.Integration (+ constant )f(x ) ⎜⎠⎛x x d )f(sec 2 kxk1tan kxx tan x sec lnx cotx sin lnx cosec )tan(ln cot cosec ln 21x x x =+- x sec)tan(ln tan sec ln 4121π+=+x x x ⎜⎠⎛⎜⎠⎛-=x x u v uv x xv u d d d d d dFurther Pure Mathematics FP1Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.Summations)12)(1(6112++=∑=n n n rnr 224113)1(+=∑=n n rnrNumerical solution of equationsThe Newton-Raphson iteration for solving 0)f(=x : )(f )f(1n n n n x x x x '-=+Coordinate geometryThe perpendicular distance from (h , k ) to 0=++c by ax is22ba c bk ah +++The acute angle between lines with gradients m 1 and m 2 is 21211arctan m m m m +-ConicsUA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 20079Matrix transformationsAnticlockwise rotation through θ about O : ⎪⎪⎭⎫⎝⎛-θθθθcos sin sin cosReflection in the line x y )(tan θ=: ⎪⎪⎭⎫ ⎝⎛-θθθθ2cos 2sin 2sin 2cos10UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 – Issue 1 – September 2007Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1–C4.Area of a sectorA = ⎜⎠⎛θd 212r (polar coordinates)Complex numbersθθθsin i cos e i +=)sin i (cos )}sin i (cos {θθθθn n r r n n +=+The roots of 1=nz are given by nk z i 2e π=, for 1 , ,2 ,1 ,0-=n kMaclaurin’s and Taylor’s Series)0(f !)0(f !2)0(f )0f()f()(2+++''+'+=r r r x x x x)(f !)( )(f !2)()(f )()f()f()(2+-++''-+'-+=a r a x a a x a a x a x r r)(f ! )(f !2)(f )f()f()(2+++''+'+=+a r x a x a x a x a r rx r x x x x rxall for !!21)ex p(e 2 +++++==)11( )1( 32)1(ln 132≤<-+-+-+-=++x rx x x x x rr x r x x x x x r rall for )!12()1( !5!3sin 1253 ++-+-+-=+x r x x x x r r all for )!2()1( !4!21cos 242 +-+-+-= )11( 12)1( 53arctan 1253≤≤-++-+-+-=+x r x x x x x r r Taylor polynomialserror )(f !2)(f )f()f(2+''+'+=+a h a h a h a)0( )(f !2)(f )f()f(2h a h a h a h a <<+''+'+=+ξξerror )(f !2)()(f )()f()f(2+''-+'-+=a a x a a x a x)( )(f !2)()(f )()f()f(2x a a x a a x a x <<''-+'-+=ξξUA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 200711Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1–C4. VectorsThe resolved part of a in the direction of b is ba.bThe point dividing AB in the ratio μλ: is μλλμ++baVector product: ⎪⎪⎪⎭⎫ ⎝⎛---===⨯122131132332321321ˆ sin b a b a b a b a b a b a b b b a a a k j inb a b a θ)()()(321321321b a c.ac b.c b a.⨯=⨯==⨯c c c b b b a a ac a.b b a.c c b a )()()(-=⨯⨯If A is the point with position vector k j i a 321a a a ++= and the direction vector b is given by k j i b 321b b b ++=, then the straight line through A with direction vector b has cartesian equation)( 332211λ=-=-=-b a z b a y b a xThe plane through A with normal vector k j i n 321n n n ++= has cartesian equationa.n -==+++d d z n y n x n where 0321The plane through non-collinear points A , B and C has vector equationc b a a c a b a r μλμλμλ++--=-+-+=)1()()(The plane through the point with position vector a and parallel to b and c has equationc b a r t s ++=The perpendicular distance of ) , ,(γβα from 0321=+++d z n y n x n is232221321nn n dn n n +++++γβα.12 UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007Hyperbolic functions1sinh cosh 22=-x x x x x cosh sinh 22sinh = x x x 22sinh cosh 2cosh +=)1( 1ln arcosh }{2≥-+=x x x x}{1ln arsinh 2++=x x x)1( 11ln artanh 21<⎪⎭⎫ ⎝⎛-+=x x x x ConicsUA018598 – Edexcel AS/A level Mathematics Formulae List – Issue 1 – September 200713Differentiationf(x )f'(x )x arcsin211x - x arccos211x--x arctan 211x + x sinh x cosh x coshx sinhx tanhx 2sech x arsinh 211x+ x arcosh112-xartanh x211x - Integration (+ constant ; 0>a where relevant )f(x )⎜⎠⎛x x d )f( x sinh x cosh x cosh x sinh x tanhx cosh ln221xa -)( arcsin a x a x <⎪⎭⎫⎝⎛221x a + ⎪⎭⎫ ⎝⎛a x a arctan 1 221a x - )( ln arcosh }{22a x a x x a x >-+=⎪⎭⎫⎝⎛221x a +}{22ln arsinh a x x a x ++=⎪⎭⎫⎝⎛221x a - )( artanh 1ln 21a x a x a x a x a a <⎪⎭⎫⎝⎛=-+ 221a x -ax a x a +-ln 2114 UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007Arc lengthx x y s d d d 12⎜⎠⎛⎪⎭⎫⎝⎛+= (cartesian coordinates)t t y t x s d d d d d 22⎜⎠⎛⎪⎭⎫ ⎝⎛+⎪⎭⎫ ⎝⎛= (parametric form)Surface area of revolution2d 2x S y s t ππ⎛⎜⎜⎜⎠==⎰BLANK PAGETURN OVER FOR MECHANICS & STATISTICS FORMULAEUA018598 – Edexcel AS/A level Mathematics Formulae List – Issue 1 – September 2007 1516 UA018598 – Edexcel AS/A level Mathematics Formulae List: Mechanics M1–M3 – Issue 1 – September 2007There are no formulae given for M1 in addition to those candidates are expected to know.Candidates sitting M1 may also require those formulae listed under Core Mathematics C1.Mechanics M2Candidates sitting M2 may also require those formulae listed under Core Mathematics C1, C2 and C3.Centres of mass For uniform bodies:Triangular lamina: 32 along median from vertex Circular arc, radius r , angle at centre 2α :ααsin r from centreSector of circle, radius r , angle at centre 2α : αα3sin 2r from centreMechanics M3Candidates sitting M3 may also require those formulae listed under Mechanics M2, and also those formulae listed under Core Mathematics C1–C4.Motion in a circleTransverse velocity: θr v = Transverse acceleration: θ r v= Radial acceleration: rvr 22-=-θCentres of mass For uniform bodies:Solid hemisphere, radius r : r 83from centre Hemispherical shell, radius r : r 21 from centre Solid cone or pyramid of height h : h 41 above the base on the line from centre of base to vertex Conical shell of height h : h 31 above the base on the line from centre of base to vertex Universal law of gravitation221Force d m Gm =UA018598 – Edexcel AS/A level Mathematics Formulae List: Mechanics M4–M5 – Issue 1 – September 200717There are no formulae given for M4 in addition to those candidates are expected to know.Candidates sitting M4 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP1.Mechanics M5Candidates sitting M5 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP1.Moments of inertiaFor uniform bodies of mass m :Thin rod, length 2l , about perpendicular axis through centre: 231ml Rectangular lamina about axis in plane bisecting edges of length 2l : 231ml Thin rod, length 2l , about perpendicular axis through end: 234ml Rectangular lamina about edge perpendicular to edges of length 2l : 234ml Rectangular lamina, sides 2a and 2b , about perpendicular axis through centre: )(2231b a m + Hoop or cylindrical shell of radius r about axis through centre: 2mrHoop of radius r about a diameter: 221mr Disc or solid cylinder of radius r about axis through centre: 221mr Disc of radius r about a diameter: 241mr Solid sphere, radius r , about diameter: 252mr Spherical shell of radius r about a diameter: 232mrParallel axes theorem: 2)(AG m I I G A +=Perpendicular axes theorem: y x z I I I += (for a lamina in the x -y plane) Moments as vectorsThe moment about O of F acting at r is F r ⨯18 UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007Statistics S1Probability)P()P()P()P(B A B A B A ⋂-+=⋃ )|P()P()P(A B A B A =⋂ )P()|P()P()|P()P()|P()|P(A A B A A B A A B B A ''+=Discrete distributionsFor a discrete random variable X taking values i x with probabilities P(X = x i )Expectation (mean): E(X ) = μ = ∑x i P(X = x i )Variance: Var(X ) = σ 2 = ∑(x i – μ )2 P(X = x i ) = ∑2i x P(X = x i ) – μ 2 For a function )g(X : E(g(X )) = ∑g(x i ) P(X = x i )Continuous distributionsStandard continuous distribution:Correlation and regressionFor a set of n pairs of values ) ,(i i y xn x x x x S i i i xx 222)()(∑-∑=-∑= ny y y y S i ii yy 222)()(∑-∑=-∑=ny x y x y y x x S i i i i i i xy ))(())((∑∑-∑=--∑=The product moment correlation coefficient is⎪⎪⎭⎫⎝⎛∑-∑⎪⎪⎭⎫ ⎝⎛∑-∑∑∑-∑=-∑-∑--∑==n y y n x x n y x y x y y x x y y x x S S S r i i i i i i i i i i i i yyxx xy 222222)( )())(()()())((}}{{The regression coefficient of y on x is 2)())((x x y y x x S S b i i i xxxy -∑--∑==Least squares regression line of y on x is bx a y += where x b y a -=THE NORMAL DISTRIBUTION FUNCTIONThe function tabulated below is Φ(z ), defined as Φ(z ) = t e zt d 21221⎜⎠⎛∞--π.PERCENTAGE POINTS OF THE NORMAL DISTRIBUTIONThe values z in the table are those which a random variable Z ~N(0, 1) exceeds with probability p; that is, P(Z > z) = 1 -Φ(z) = p.Statistics S2Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Core Mathematics C1 and C2.Discrete distributionsStandard discrete distributions:Continuous distributionsFor a continuous random variable X having probability density function fExpectation (mean): ⎰==x x x X d )f()E(μVariance: ⎰⎰-=-==2222d )f(d )f()()Var(μμσx x x x x x X For a function )g(X : ⎰=x x x X d )f()g())E(g(Cumulative distribution function: ⎜⎠⎛=≤=∞-000d )(f )P()F(xt t x X xStandard continuous distribution:BINOMIAL CUMULATIVE DISTRIBUTION FUNCTIONThe tabulated value is P(X x), where X has a binomial distribution with index n and parameter p.POISSON CUMULATIVE DISTRIBUTION FUNCTION The tabulated value is P(X ≤x), where X has a Poisson distribution with parameter λ.UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 200729Statistics S3Candidates sitting S3 may also require those formulae listed under Statistics S1 and S2.Expectation algebraFor independent random variables X and Y)E()E()E(Y X XY =, )V ar()V ar()V ar(22Y b X a bY aX +=±Sampling distributionsFor a random sample n X X X , , ,21 of n independent observations from a distribution having mean μ and variance 2σX is an unbiased estimator of μ , with nX 2)V ar(σ=2S is an unbiased estimator of 2σ, where 1)(22--∑=n X X S iFor a random sample of n observations from ) ,N(2σμ)1 ,0N(~/n X σμ-For a random sample of x n observations from ) ,N(2x x σμ and, independently, a random sample of y n observations from ) ,N(2y y σμ)1 ,0N(~)()(22yyxxy x n n Y X σσμμ+---Correlation and regressionSpearman’s rank correlation coefficient is )1(6122-∑-=n n d r sNon-parametric testsGoodness-of-fit test and contingency tables:22~)(νχ∑-ii i E E OPERCENTAGE POINTS OF THE χ2 DISTRIBUTIONThe values in the table are those which a random variable with the χ2 distribution on νdegrees of freedom exceeds with the probability shown.30 UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007CRITICAL VALUES FOR CORRELATION COEFFICIENTSThese tables concern tests of the hypothesis that a population correlation coefficient is 0. The values in the tables are the minimum values which need to be reached by a sample correlation coefficient in order to be significant at the level shown, on a one-tailed test.UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007 31RANDOM NUMBERS86 13 84 10 07 30 39 05 97 96 88 07 37 26 04 89 13 48 19 2060 78 48 12 99 47 09 46 91 33 17 21 03 94 79 00 08 50 40 1678 48 06 37 82 26 01 06 64 65 94 41 17 26 74 66 61 93 24 9780 56 90 79 66 94 18 40 97 79 93 20 41 51 25 04 20 71 76 0499 09 39 25 66 31 70 56 30 15 52 17 87 55 31 11 10 68 98 2356 32 32 72 91 65 97 36 56 61 12 79 95 17 57 16 53 58 96 3666 02 49 93 97 44 99 15 56 86 80 57 11 78 40 23 58 40 86 1431 77 53 94 05 93 56 14 71 23 60 46 05 33 23 72 93 10 81 2398 79 72 43 14 76 54 77 66 29 84 09 88 56 75 86 41 67 04 4250 97 92 15 10 01 57 01 87 33 73 17 70 18 40 21 24 20 66 6290 51 94 50 12 48 88 95 09 34 09 30 22 27 25 56 40 76 01 5931 99 52 24 13 43 27 88 11 39 41 65 00 84 13 06 31 79 74 9722 96 23 34 46 12 67 11 48 06 99 24 14 83 78 37 65 73 39 4706 84 55 41 27 06 74 59 14 29 20 14 45 75 31 16 05 41 22 9608 64 89 30 25 25 71 35 33 31 04 56 12 67 03 74 07 16 49 3286 87 62 43 15 11 76 49 79 13 78 80 93 89 09 57 07 14 40 7494 44 97 13 77 04 35 02 12 76 60 91 93 40 81 06 85 85 72 8463 25 55 14 66 47 99 90 02 90 83 43 16 01 19 69 11 78 87 1611 22 83 98 15 21 18 57 53 42 91 91 26 52 89 13 86 00 47 6101 70 10 83 94 71 13 67 11 12 36 54 53 32 90 43 79 01 95 15 32 UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 200733Statistics S4Candidates sitting S4 may also require those formulae listed under Statistics S1, S2 and S3.Sampling distributionsFor a random sample of n observations from ) ,N(2σμ2122~)1(--n S n χσ 1~/--n t nS X μ(also valid in matched-pairs situations)For a random sample of x n observations from ) ,N(2x x σμ and, independently, a random sample of y n observations from ) ,N(2y y σμ1,12222~//--y n x nyy xx F S S σσIf 222σσσ==y x (unknown) then22~11)()(-+⎪⎪⎭⎫ ⎝⎛+---yn x ny x py x t n n S Y X μμ where 2)1()1(222-+-+-=y x yy x x p n n S n S n SPERC ENTAGE POINTS OF STUDENT’S t DISTRIBUTIONThe values in the table are those which a random variable with Student’s t distribution on ν degrees of freedom exceeds with the probability shown.34 UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007PERCENTAGE POINTS OF THE F DISTRIBUTIONThe values in the table are those which a random variable with the F distribution on ν1and ν2 degrees of freedom exceeds with probability 0.05 or 0.01.If an upper percentage point of the F distribution on ν1 and ν2 degrees of freedom is f , then the corresponding lower percentage point of the F distribution on ν2 and ν1 degrees of freedom is 1/ f . UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007 35BLANK PAGEFurther copies of this publication are available fromEdexcel Publications, Adamsway, Mansfield, Notts, NG18 4FNTelephone 01623 467467Fax 01623 450481E-mail:*****************************Publication Code UA018598For more information on Edexcel qualifications please contactCustomer Response Centre on 0870 240 9800or or visit our website: London Qualifications Limited, trading as Edexcel. Registered in England and Wales No. 4496750 Registered Office: 190 High Holborn, London WC1V 7BH。