Tài liệu MEI STRUCTURED MATHEMATICS EXAMINATION FORMULAE AND TABLES ppt

MEI STRUCTURED MATHEMATICS EXAMINATION FORMULAE AND TABLES Arithmetic series General (kth) term, last (nth) term, l = Sum to n terms, Geometric series General (kth) term, Sum to n terms, Sum to infinity Infinite series f(x) uk = a + (k – 1)d un = a + (n – l)d – – Sn = n(a + l) = n[2a + (n – 1)d] 2 x2 xr = f(0) + xf'(0) + –– f"(0) + + –– f (r)(0) + 2! r! f(x) f(a + x) uk = a r k–1 a(1 – r n) a(r n – 1) Sn = –––––––– = –––––––– 1–r r–1 (x – a)2 (x – a)rf(r)(a) = f(a) + (x – a)f'(a) + –––––– f"(a) + + –––––––––– + r! 2! x2 xr = f(a) + xf'(a) + –– f"(a) + + –– f(r)(a) + 2! r! x2 xr ex = exp(x) = + x + –– + + –– + , all x 2! r! a S∞ = ––––– , – < r < 1–r x2 x3 xr = x – –– + –– – + (–1)r+1 –– + , – < x р r sin x x3 x5 x 2r+1 = x – –– + –– – + (–1)r –––––––– + , all x 3! 5! (2r + 1)! cos x x2 x4 x 2r = – –– + –– – + (–1)r –––– + , all x 2! 4! (2r)! arctan x x3 x5 x 2r+1 = x – –– + –– – + (–1)r –––––– + , – р x р 2r + General case sinh x n(n – 1) n(n – 1) (n – r + 1) (1 + x)n = + nx + ––––––– x2 + + ––––––––––––––––– xr + , |x| < 1, 2! 1.2 r n∈‫ޒ‬ x3 x5 x 2r+1 = x + –– + –– + + –––––––– + , all x 3! 5! (2r + 1)! cosh x x2 x4 x 2r = + –– + –– + + –––– + , all x 2! 4! (2r)! artanh x x3 x5 x 2r+1 = x + –– + –– + + –––––––– + , – < x < (2r + 1) Binomial expansions When n is a positive integer n n n (a + b)n = an + an –1 b + an–2 b2 + + r an–r br + bn , n ∈ ‫ގ‬ where n n n n+1 n! n r = Cr = –––––––– r + r+1 = r+1 r!(n – r)! () () () () () ( ) ( ) Logarithms and exponentials exln a = ax logbx loga x = ––––– logba Numerical solution of equations f(xn) Newton-Raphson iterative formula for solving f(x) = 0, xn+1 = xn – –––– f'(xn) Complex Numbers {r(cos θ + j sin θ)}n = r n(cos nθ + j sin nθ) ejθ = cos θ + j sin θ 2πk The roots of zn = are given by z = exp( –––– j) for k = 0, 1, 2, , n – n Finite series n n 1 ∑ r2 = – n(n + 1)(2n + 1) ∑ r3 = – n2(n + 1)2 r=1 r=1 ALGEBRA ln(1 + x) Hyperbolic functions cosh2x – sinh2x = 1, sinh2x = 2sinhx coshx, cosh2x = cosh2x + sinh2x arcosh x = ln(x + x + ), 1+x artanh x = – ln ––––– , |x| < 1–x arsinh x = ln(x + ( x – ), x у ) Matrices Anticlockwise rotation through angle θ, centre O: Reflection in the line y = x tan θ : ( cosθθ sin ( cos22θθ sin –sin θ cos θ ) sin 2θ –cos 2θ ) Cosine rule A b2 + c2 – a2 cos A = –––––––––– (etc.) 2bc c a2 = b2 + c2 –2bc cos A (etc.) B Trigonometry Perpendicular distance of a point from a line and a plane b a Line: (x1,y1) from ax + by + c = : a2 + b2 n1α + n2β + n3γ + d Plane: (α,β,γ) from n1x + n2y + n3z + d = : –––––––––––––––––– √(n12 + n22 + n32) C sin (θ ± φ) = sin θ cos φ ± cos θ sin φ cos (θ ± φ) = cos θ cos φ ϯ sin θ sin φ Vector product i a1 b1 a2b3 – a3b2 ^ = j a b = a b –a b a × b = |a| |b| sinθ n 2 1 k a3 b3 a1b2 – a2b1 tan θ ± tan φ tan (θ ± φ) = –––––––––––– , [(θ ± φ) ≠ (k + W)π] ϯ tan θ tan φ | a × (b × c) = (c a) b – (a b) c – – sin θ – sin φ = cos (θ + φ) sin (θ – φ) 2 Conics Ellipse Hyperbola Rectangular hyperbola y2 x2 –– + –– = b2 a y2 = 4ax y2 x2 –– – –– = b2 a x y = c2 Parametric form (acosθ, bsinθ) (at2, 2at) (asecθ, btanθ) c (ct, – –) t Parabola Standard form – (θ – φ) – (θ – φ) Vectors and 3-D coordinate geometry (The position vectors of points A, B, C are a, b, c.) The position vector of the point dividing AB in the ratio λ:µ µa + λb is ––––––– (λ + µ) Line: ) a1 b1 c1 a (b × c) = a2 b2 c2 = b (c × a) = c (a × b) a3 b3 c3 (1 + t ) – – sin θ + sin φ = sin (θ + φ) cos (θ – φ) 2 – cos θ + cos φ = cos (θ + φ) cos – cos θ – cos φ = –2 sin (θ + φ) sin | |( | Cartesian equation of line through A in direction u is x – a1 y – a2 z – a3 –––––– = –––––– = –––––– = t u1 u2 u3 ( ) e1 = a2 (e2 – 1) e = √2 l – = + e cos θ r TRIGONOMETRY, VECTORS AND GEOMETRY (1 – t2) 2t – For t = tan θ : sin θ = –––––– , cos θ = –––––– 2 (1 + t ) ax1 + by1 + c Differentiation f(x) tan kx sec x cot x cosec x arcsin x f'(x) ksec2 kx sec x tan x –cosec2 x –cosec x cot x ––––––– √(1 – x2) –1 ––––––– √(1 – x2) ––––– + x2 cosh x sinh x sech2 x ––––––– √(1 + x2) arccos x arctan x sinh x cosh x x arsinh x artanh x du dv v ––– – u ––– dx dx u dy Quotient rule y = – , ––– = v v dx Trapezium rule b–a – ∫a ydx ≈ h{(y0 + yn) + 2(y1 + y2 + + yn–1)}, where h = ––––– n b Integration by parts Area of a sector Arc length dv du ∫ u ––– dx = uv – ∫ v ––– dx dx dx – A = ∫ r dθ (polar coordinates) – y ˙ A = ∫ (x˙ – yx) dt (parametric form) ˙ ˙ s = ∫ √ (x + y ) dt (parametric form) dy s = ∫ √ (1 + [ –––] ) dx (cartesian coordinates) dx dr s = ∫ √ (r + [ –––] ) dθ (polar coordinates) dθ 2 2 2 sec x –––––– – a2 x ––––––– √(a2 – x2) –––––– a2 + x2 –––––– a2 – x2 sinh x cosh x x ––––––– √(a2 + x2) ––––––– √(x2 – a2) Surface area of revolution ∫f(x) dx (+ a constant) (l/k) tan kx ln |sec x| ln |sin x| x –ln |cosec x + cot x| = ln |tan – | x π ln |sec x + tan x| = ln tan – + – x– –– ln ––– a x +–– a 2a | ( | | x arcsin ( – ) , |x| < a a arctan – x – ( a) a x a +–– x –– ln | ––– x | = artanh ( – ) , |x| < a – a a a– 2a cosh x sinh x ln cosh x x arsinh – or ln (x + x + a ), a () x arcosh ( – ) or ln (x + x a S = 2π∫y ds = 2π∫y√(x ˙ S = 2π∫x ds = 2π∫x√(x ˙ – a ), x > a , a > + y 2) dt ˙ + y 2) dt ˙ x y Curvature )| d2y ––– dψ x x y ă dx2 = = –––––––– = ––––––––––––– + y2)3/2 ds dy 3/2 (˙ ˙ x + –– dx ( [ ]) Radius of curvature ρ = –– , κ ^ Centre of curvature c = r + ρ n L'Hôpital’s rule If f(a) = g(a) = and g'(a) ≠ then f(x) f'(a) Lim –––– = –––– x ➝a g(x) g'(a) Multi-variable calculus ∂g/∂x ∂w ∂w ∂w For w = g(x, y, z), δw = ––– δx + ––– δy + ––– δz grad g = ∂g/∂y ∂x ∂y ∂z ∂g/∂z ( ) CALCULUS ––––––– √(x2 – 1) ––––––– (1 – x2) arcosh x Integration f(x) sec2 kx tan x cot x cosec x Centre of mass (uniform bodies) Triangular lamina: Solid hemisphere of radius r: Hemispherical shell of radius r: Solid cone or pyramid of height h: Moments of inertia (uniform bodies, mass M) along median from vertex – 3 r from centre – r from centre – h above the base on the – line from centre of base to vertex Sector of circle, radius r, angle 2θ: 2r sin θ ––––––– from centre 3θ r sin θ Arc of circle, radius r, angle 2θ at centre: ––––––– from centre θ h above the base on the – Conical shell, height h: line from the centre of Thin rod, length 2l, about perpendicular axis through centre: Rectangular lamina about axis in plane bisecting edges of length 2l: Thin rod, length 2l, about perpendicular axis through end: Rectangular lamina about edge perpendicular to edges of length 2l: Rectangular lamina, sides 2a and 2b, about perpendicular – M(a2 + b2) axis through centre: Hoop or cylindrical shell of radius r about perpendicular axis through centre: Hoop of radius r about a diameter: Disc or solid cylinder of radius r about axis: base to the vertex Solid sphere of radius r about a diameter: Motion in polar coordinates Spherical shell of radius r about a diameter: ˙ Transverse velocity: v = rθ v2 ˙ Radial acceleration: r = r ă Transverse acceleration: v = rθ General motion Radial velocity: Transverse velocity: Radial acceleration: Transverse acceleration: r ˙ ˙ rθ ˙ r r ă d ă + 2r = –– (r2θ ) ˙ – ˙˙ rθ r dt Moments as vectors The moment about O of F acting at r is r × F Mr2 – Mr2 – Mr2 – Mr2 – Mr2 – Mr2 Parallel axes theorem: IA = IG + M(AG)2 Perpendicular axes theorem: Iz = Ix + Iy (for a lamina in the (x, y) plane) MECHANICS Disc of radius r about a diameter: Motion in a circle – Ml2 – Ml2 – Ml2 – Ml2 Probability Product-moment correlation: Pearson’s coefficient ∑ xi yi –xy Sxy Σ( xi – x )( yi – y ) n r= = = 2 Sxx Syy  ∑ xi   ∑ yi  Σ( xi – x ) Σ( yi – y ) − x2 − y2   n   n P(A∪B) = P(A) + P(B) – P(A∩B) P(A∩B) = P(A) P(B|A) P(B|A)P(A) P(A|B) = –––––––––––––––––––––– P(B|A)P(A) + P(B|A')P(A') [ P(Aj)P(B|Aj) Bayes’ Theorem: P(A j |B) = –––––––––––– ∑P(Ai)P(B|Ai) Rank correlation: Spearman’s coefficient Populations 6∑di2 rs = – –––––––– n(n2 – 1) Discrete distributions X is a random variable taking values xi in a discrete distribution with P(X = xi) = pi Expectation: µ = E(X) = ∑xi pi Variance: σ2 = Var(X) = ∑(xi – µ)2 pi = ∑xi2pi – µ2 For a function g(X): E[g(X)] = ∑g(xi)pi Regression X is a continuous variable with probability density function (p.d.f.) f(x) Expectation: µ = E(X) = ∫ x f(x)dx Variance: σ2 = Var (X) = ∫(x – µ)2 f(x)dx = ∫x2 f(x)dx – µ2 For a function g(X): E[g(X)] = ∫g(x)f(x)dx Cumulative x distribution function F(x) = P(X р x) = ∫–∞f(t)dt Sxx = ∑(xi – x )2 = ∑xi2 – n , Syy = ∑(yi – y )2 = ∑yi2 – (∑xi)(∑yi) Sxy = ∑(xi – x )(yi – y ) = ∑xi yi – ––––––––– n Covariance Sxy –––– = ∑( xi – x )( yi – y ) = ∑ xi yi – x y n n n S2 for population variance σ where S2 = –––– ∑(xi – x )2fi n–1 Probability generating functions For a discrete distribution For a sample of n pairs of observations (xi, yi) (∑xi)2 ––––– Least squares regression line of y on x: y – y = b(x – x ) ∑ xi yi –xy Sxy ∑(xi – x) (yi – y ) n b = ––– = ––––––––––––––– = Sxx ∑ xi ∑(xi – x )2 – x2 n Estimates Unbiased estimates from a single sample σ2 X for population mean µ; Var X = –– n (∑yi)2 ––––– n , G(t) = E(tX) E(X) = G'(1); Var(X) = G"(1) + µ – µ2 GX + Y (t) = GX (t) GY (t) for independent X, Y Moment generating functions: MX(θ) = E(eθX) E(X) = M'(0) = µ; E(Xn) = M(n)(0) Var(X) = M"(0) – {M'(0)}2 MX + Y (θ) = MX (θ) MY (θ) for independent X, Y STATISTICS Continuous distributions Correlation and regression ] Markov Chains pn + = pnP Long run proportion p = pP Bivariate distributions Covariance Cov(X, Y) = E[(X – µX)(Y – µY)] = E(XY) – µXµY Cov(X, Y) Product-moment correlation coefficient ρ = –––––––– σX σY Sum and difference Var(aX ± bY) = a2Var(X) + b2Var(Y) ± 2ab Cov (X,Y) If X, Y are independent: Var(aX ± bY) = a2Var(X) + b2Var(Y) E(XY) = E(X) E(Y) Coding X = aX' + b ⇒ Cov(X, Y) = ac Cov(X', Y') Y = cY' + d } One-factor model: xij = µ + αi + εij, where εij ~ N(0,σ2) Ti2 –– SSB = ∑ni ( x i – x )2 = ∑ ––– – T ni n i i –– SST = ∑ ∑ (xij – x )2 = ∑ ∑ xij2 – T n i j i j Yi α + βxi + εi RSS ∑(yi – a – bxi)2 α + βf(xi) + εi α + βxi + γzi + εi ∑(yi – a – εi ~ N(0, σ2) No of parameters, p bf(xi))2 ∑(yi – a – bxi – czi)2 a, b, c are estimates for α, β, γ For the model Yi = α + βxi + εi, Sxy σ2 b = ––– , b ~ N β, ––– , Sxx Sxx ( b−β ~ tn –2 ˆ σ / Sxx ) σ 2∑xi2 ––––––– a = y – b x , a ~ N α, n S xx ( RSS ^ –––– σ2 = n – p ) (x0 – x )2 a + bx0 ~ N(α + βx0, σ2 + ––––––– – n Sxx (Sxy) RSS = Syy – ––––– = Syy (1 – r2) Sxx { } Randomised response technique y – – (1 – θ) n ^ E(p) = –––––––––– (2θ – 1) [(2θ – 1) p + (1 – θ)][θ – (2θ – 1)p] ^ Var(p) = –––––––––––––––––––––––––––––– n(2θ – 1) Factorial design Interaction between 1st and 2nd of treatments (–) { (Abc – abc) + (AbC – abC) (ABc – aBc) + (ABC – aBC) ––––––––––––––––––––– – –––––––––––––––––––––– 2 } Exponential smoothing ^ yn+1 = α yn + α(1 – α)yn–1 + α(1 – α)2 yn–2 + + α(1 – α)n–1 y1 + (1 – α)ny0 ^ = y + α(y – y ) ^ ^ yn+1 n n n ^ ^ yn+1 = α yn + (1 – α) yn STATISTICS Analysis of variance Regression Description Pearson’s product moment correlation test r= Distribution ∑ xi yi –xy n  ∑ xi   ∑ yi  – x2 – y2   n  n   t-test for the difference in the means of samples 6∑di2 rs = – ––––––– n(n2 – 1) Normal test for a mean x–µ σ/ n N(0, 1) t-test for a mean x–µ s/ n tn – χ2 test t-test for paired sample Normal test for the difference in the means of samples with different variances Description ∑ (f o – fe ) ( x1 – x2 ) – µ s/ n ( x – y ) – ( µ1 – µ2 ) 1 + s n1 n2 See tables Wilcoxon Rank-sum (or Mann-Whitney) 2-Sample test Samples size m, n: m р n Wilcoxon W = sum of ranks of sample size m Mann-Whitney – T = W – m(m + 1) See tables p –θ Normal test on binomial proportion N(0, 1) + n2 – (n1 – 1)s12 + (n2 – 1)s22 where s2 = ––––––––––––––––––––––– n1 + n2 –  θ (1 – θ )   n  χ2 test for variance (n – 1)s σ2 ( x – y ) – ( µ1 – µ2 ) σ 12 σ 2 + n1 n2 tn A statistic T is calculated from the ranked data χ 2v t with (n – 1) degrees of freedom Distribution Wilcoxon single sample test fe Test statistic F-test on ratio of two variances s12 /σ12 ––––––– s22 /σ22 , s12 > s22 N(0, 1) χ2n – Fn –1, n2 –1 STATISTICS: HYPOTHESIS TESTS Spearman rank correlation test Test statistic Name Mean Function Variance p.g.f G(t) (discrete) m.g.f M(θ) (continuous) P(X = r) = nCr qn–rpr , for r = 0, 1, ,n , < p < 1, q = – p np npq G(t) = (q + pt)n Poisson (λ) Discrete λr P(X = r) = e–λ ––– , r! for r = 0, 1, , λ > λ λ G(t) = eλ(t – 1) µ σ2 M(θ) = exp(µθ + Wσ 2θ 2) x–µ exp – W ––––– σ ( ( Normal N(µ, σ2) Continuous f(x) = Uniform (Rectangular) on [a, b] Continuous f(x) = ––––– b–a Exponential Continuous f(x) = λe–λx Geometric Discrete P(X = r) = q r – 1p , σ 2π ) ), –∞ < x < ∞ Negative binomial Discrete , aрxрb , 0