MEI EXAMINATION FORMULAE AND TABLES (MF2)

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1 MEI EXAMINATION FORMULAE AND TABLES (MF) For use with: Advaced Geeral Certificate of Educatio Advaced Subsidiary Geeral Certificate of Educatio MEI STRUCTURED MATHEMATICS ad Advaced Subsidiary GCE QUANTITATIVE METHODS (MEI) MF CST5

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3 MEI STRUCTURED MATHEMATICS ad QUANTITATIVE METHODS (MEI) EXAMINATION FORMULAE AND TABLES

4 ALGEBRA Arithmetic series Geeral (kth) term, u k = a + (k )d last (th) term, l = u = a + ( l)d Sum to terms, S = (a + l) = [a + ( )d] Geometric series Geeral (kth) term, u k = a r k a( r Sum to terms, S = ) = r Sum to ifiity S = a, < r < r Biomial expasios Whe is a positive iteger (a + b) = a + ( ) a b + ( ) a b ( ) a r b r +... b, where ( ) =! C r = ( ) ( ) ( ) + = + r r r + r + r!( r)! r a(r ) r Geeral case ( )! ( )... ( r +).... r ( + x) = + x + x x r +..., x <, Logarithms ad expoetials e xl a = a x log a x = Numerical solutio of equatios Newto-Raphso iterative formula for solvig f(x) = 0, x + = x Complex Numbers {r(cos θ + j si θ)} = r (cos θ + j si θ) e jθ = cos θ + j si θ The roots of z = are give by z = exp( j) for k = 0,,,..., Fiite series r = ( + )( + ) 6 r= r= πk log b x log b a r 3 = 4 ( + ) f(x ) f'(x ) Ifiite series f(x) = f(0) + xf'(0) + f"(0) f (r) (0) +... f(x) = f(a)+(x a)f'(a) + f"(a) f(a + x) = f(a) + xf'(a) + f"(a) f (r) (a) +... e x = exp(x) = + x , all x x l( + x) = x ( ) r+ +..., < x x 3 3! si x = x ( ) r +..., all x x! cos x = ( ) r +..., all x x 3 3 arcta x = x ( ) r +..., x x 3 3! sih x = x , all x x! cosh x = , all x x 3 3 x! x 3 3 x 5 5! x 4 4! x 5 5 x 5 5! x 4 4! x 5 5 x! x! (x a)! x r+ (r + )! x r (r)! x r (r)! x r+ r + x r+ (r +) x r+ (r + )! artah x = x , < x < x r r! x r r x r r! x r r! (x a) r f (r) (a) r! Hyperbolic fuctios cosh x sih x =, sihx = sihx coshx, coshx = cosh x + sih x Matrices arsih x = l(x + ), arcosh x = l(x + x + x ), x + x x artah x = l ( ), x < cos θ si θ Aticlockwise rotatio through agle θ, cetre O: ( si θ cos θ ) ( ) Reflectio i the lie y = x ta θ : cos θ si θ si θ cos θ

5 TRIGONOMETRY, VECTORS AND GEOMETRY Cosie rule cos A = b + c a bc (etc.) a = b + c bc cos A (etc.) Trigoometry si (θ ± φ) = si θ cos φ ± cos θ si φ cos (θ ± φ) = cos θ cos φ si θ si φ ta θ ± ta φ ta (θ ± φ) =, [(θ ± φ) (k + )π] ta θ ta φ B c a A b C For t = ta θ : si θ = t, cos θ = ( t ) ( + t ) ( + t ) si θ + si φ = si (θ + φ) cos (θ φ) si θ si φ = cos (θ + φ) si (θ φ) cos θ + cos φ = cos (θ + φ) cos (θ φ) cos θ cos φ = si (θ + φ) si (θ φ) Vectors ad 3-D coordiate geometry (The positio vectors of poits A, B, C are a, b, c.) The positio vector of the poit dividig AB i the ratio λ:μ is μa + λb (λ + μ) Lie: Cartesia equatio of lie through A i directio u is y a z a 3 x a u = = ( = t ) u u 3 The resolved part of a i the directio u is a. u u Plae: Cartesia equatio of plae through A with ormal is x + y + 3 z + d = 0 where d = a. The plae through o-colliear poits A, B ad C has vector equatio r = a + s(b a) + t(c a) = ( s t) a + sb + tc The plae through A parallel to u ad v has equatio r = a + su + tv Perpedicular distace of a poit from a lie ad a plae Lie: (x,y ) from ax + by + c = 0 : ax + by + c a + b Plae: (α,β,γ) from x + y + 3 z + d = 0 : α + β + 3 γ + d ( ) Vector product a b = a b siθ ^ i a b ( a b 3 a 3 b ) = j a b = a 3 b a b 3 k a 3 b 3 a b a b a b c a. (b c) = a b c a 3 b 3 c 3 = b. (c a) = c. (a b) a (b c) = (c. a) b (a. b) c Coics Rectagular Ellipse Parabola Hyperbola hyperbola Stadard x y + = = form y x = 4ax y xy = c a b a b Parametric form (acosθ, bsiθ) (at, at) (asecθ, btaθ) (ct, ) c Eccetricity e < b = a ( e ) e > e = b = a (e e = ) Foci (± ae, 0) (a, 0) (± ae, 0) (±c, ±c ) Directrices x = ± a e x = a x = ± a e x + y = ±c Asymptotes oe oe x y a = ± b x = 0, y = 0 t Ay of these coics ca be expressed i polar l coordiates (with the focus as the origi) as: r = + e cos θ where l is the legth of the semi-latus rectum. Mesuratio Sphere : Surface area = 4πr Coe : Curved surface area = πr slat height 3

6 CALCULUS Differetiatio f(x) f'(x) ta kx ksec kx sec x sec x ta x cot x cosec x cosec x cosec x cot x arcsi x ( x ) arccos x arcta x ( x ) + x sih x cosh x cosh x sih x tah x sech x arsih x arcosh x artah x ( + x ) (x ) ( x ) du v u u Quotiet rule y =, dy = dx v dx v dv dx Trapezium rule b ydx h{(y a 0 + y ) + (y + y y Itegratio by parts u dx = uv v dx dv dx Area of a sector A = r dθ (polar coordiates) du dx )}, where h = A = (xẏ yẋ) dt (parametric form) Arc legth s = (ẋ + ẏ ) dt (parametric form) dy dx s = ( + [ ] ) dx (cartesia coordiates) s = dr (r + [ ] ) dθ (polar coordiates) dθ b a Itegratio f(x) f(x) dx (+ a costat) sec kx (l/k) ta kx ta x l sec x cot x l si x x cosec x l cosec x + cot x = l ta x x a a l x a x + a sec x l sec x + ta x = l ta ( + 4 π ) (a x ) a + x a x arcsi ( ), x < a arcta( a x ) sih x cosh x cosh x sih x tah x l cosh x (a + x ) (x a ) a a x a l a + x a x x a = artah ( ), x < a arsih ( ) or l (x + ), x a a arcosh ( ) or l (x + ), x > a, a > 0 x a x + a x a Surface area of revolutio S x = π y ds = π y (ẋ + ẏ ) dt Curvature dψ ds ẋ ÿ ẍ ẏ (ẋ + ẏ ) 3/ κ = = = S y = π x ds = π x (ẋ + ẏ ) dt d y dx dy ( + [ ] ) 3 / dx Radius of curvature ρ = κ, Cetre of curvature c = r + ρ ^ L'Hôpital s rule Lim x a f(x) g(x) If f(a) = g(a) = 0 ad g'(a) 0 the = f'(a) g'(a) Multi-variable calculus g/ x g/ y g/ z grad g = ( ) w x w y w z For w = g(x, y, z), δw = δx+ δy + δz 4

7 MECHANICS Cetre of mass (uiform bodies) Momets of iertia (uiform bodies, mass M) Triagular lamia: 3 alog media from vertex Thi rod, legth l, about perpedicular axis through cetre: 3 Ml Solid hemisphere of radius r: 3 8 r from cetre Rectagular lamia about axis i plae bisectig edges of legth l: 3 Ml Hemispherical shell of radius r: r from cetre 4 Thi rod, legth l, about perpedicular axis through ed: 3 Ml Solid coe or pyramid of height h: 4 h above the base o the lie from cetre of base to vertex Sector of circle, radius r, agle θ: r si θ from cetre 3θ Arc of circle, radius r, agle θ at cetre: 3 r si θ θ from cetre Coical shell, height h: h above the base o the lie from the cetre of base to the vertex 4 Rectagular lamia about edge perpedicular to edges of legth l: 3 Ml Rectagular lamia, sides a ad b, about perpedicular axis through cetre: 3 M(a + b ) Hoop or cylidrical shell of radius r about perpedicular axis through cetre: Mr Hoop of radius r about a diameter: Mr Disc or solid cylider of radius r about axis: Mr Disc of radius r about a diameter: 4 Mr Motio i polar coordiates Motio i a circle Trasverse velocity: v = rθ Radial acceleratio: rθ = Trasverse acceleratio: v = rθ v r Solid sphere of radius r about a diameter: 5 Mr Spherical shell of radius r about a diameter: 3 Mr Parallel axes theorem: I A = I G + M(AG) Perpedicular axes theorem: I z = I x + I y (for a lamia i the (x, y) plae) Geeral motio Radial velocity: ṙ Trasverse velocity: rθ Radial acceleratio: r rθ Trasverse acceleratio: rθ + ṙθ = r (r θ ) d dt Momets as vectors The momet about O of F actig at r is r F 5

8 STATISTICS Probability P(A B) = P(A) + P(B) P(A B) P(A B) = P(A). P(B A) P(A B) = P(B A)P(A) P(B A)P(A) + P(B A')P(A') Populatios Bayes Theorem: P(A j B) = P(A j )P(B A j ) P(A i )P(B A i ) Discrete distributios X is a radom variable takig values x i i a discrete distributio with P(X = x i ) = p i Expectatio: μ = E(X) = x i p i Variace: σ = Var(X) = (x i μ) p i = x i pi μ For a fuctio g(x): E[g(X)] = g(x i )p i Cotiuous distributios X is a cotiuous variable with probability desity fuctio (p.d.f.) f(x) Expectatio: μ = E(X) = x f(x)dx Variace: σ = Var(X) = (x μ) f(x)dx = x f(x)dx μ For a fuctio g(x): E[g(X)] = g(x)f(x)dx Cumulative distributio fuctio F(x) = P(X x) = x f(t)dt Correlatio ad regressio For a sample of pairs of observatios (x i, y i ) ( x i ) S xx = (x i x ) = x i, S yy = (y i y ) = y i, S xy = (x i x )(y i y ) = x i y i ( x i )( y i ) ( y i ) S xy Covariace = ( )( ) = i i i i x x y y xy xy Product-momet correlatio: Pearso s coefficiet S xy Σ ( xi x)( yi y) r = = = Sxx Syy [ Σ( xi x) Σ( yi y) ] x xy i i x xy y y i i Rak correlatio: Spearma s coefficiet r s = 6 d i ( ) Regressio Least squares regressio lie of y o x: y y = b(x x ) xy i i S b = xy (x = i x ) (y i y ) xy = S (x i ) xx x x x Estimates Ubiased estimates from a sigle sample X for populatio mea μ; Var X = σ S for populatio variace σ where S = (x i x ) f i Probability geeratig fuctios For a discrete distributio G(t) = E(t X ) E(X) = G'(); Var(X) = G"() + μ μ G X + Y (t) = G X (t) G Y (t) for idepedet X, Y Momet geeratig fuctios: M X (θ) = E(e θx ) E(X) = M'(0) = μ; E(X ) = M () (0) Var(X) = M"(0) {M'(0)} M X + Y (θ) = M X (θ) M Y (θ) for idepedet X, Y i 6

9 STATISTICS Markov Chais p + = p P Log ru proportio p = pp Bivariate distributios Covariace Cov(X, Y) = E[(X μ X )(Y μ Y )] = E(XY) μ X μ Y Cov(X, Product-momet correlatio coefficiet ρ = Y) σ X σ Y Sum ad differece Var(aX ± by) = a Var(X) + b Var(Y) ± ab Cov (X,Y) If X, Y are idepedet: Var(aX ± by) = a Var(X) + b Var(Y) E(XY) = E(X) E(Y) Codig X = ax' + b } Cov(X, Y) = ac Cov(X', Y') Y = cy' + d Aalysis of variace Oe-factor model: x ij = μ + α i + ε ij, where ε ij ~ N(0,σ ) SS B = i ( i ) x x = T i i i i T SS T = (x ij x ) = x ij T i j i j Regressio Y i α + βx i + ε i α + βf(x i ) + ε i α + βx i + γz i + ε i ε i ~ N(0, σ ) a, b, c are estimates for α, β, γ. For the model Y i = α + βx i + ε i, S xy S xx RSS (y i a bx i ) (y i a bf(x i )) (y i a bx i cz i ) σ S xx b =, b ~ N( β, ), ~ t a = b, a ~ N( α, σ x y x i S xx ) a + bx 0 ~ N(α + βx 0, σ { + (x 0 ) } (S xy ) RSS = S yy = S yy ( r ) S xx b β σˆ / S xx x S xx No. of parameters, p 3 ^σ = RSS p Radomised respose techique y E(^p) = ( θ) Var(^p) = (θ ) [(θ ) p + ( θ)][θ (θ )p] (θ ) Factorial desig Iteractio betwee st ad d of 3 treatmets (Abc abc) + (AbC abc) (ABc abc) + (ABC abc) ( ) { } Expoetial smoothig ^y+ = αy + α( α)y + α( α) y α( α) y + ( α) y 0 ^y+ = ^y + α(y ^y ) ^y+ = αy + ( α) ^y 7

10 STATISTICS: HYPOTHESIS TESTS Descriptio Pearso s product momet correlatio test Spearma rak correlatio test Normal test for a mea t-test for a mea χ test t-test for paired sample Normal test for the differece i the meas of samples with differet variaces r = Test statistic Distributio x xy i i xy y x i i y r s = 6 d i ( ) x μ N(0, ) σ / x μ s/ t ( f f ) o e f e χ v ( ) μ x x s / t with ( ) degrees of freedom ( x y) ( μ μ ) σ σ + N(0, ) Descriptio Test statistic Distributio t-test for the differece i the meas of samples where s = ( x y) ( μ μ) s + ( )s + ( )s + t + Wilcoxo sigle sample test A statistic T is calculated from the raked data. See tables Wilcoxo Rak-sum (or Ma-Whitey) -Sample test Samples size m, : m Wilcoxo W = sum of raks of sample size m Ma-Whitey T = W m(m + ) See tables Normal test o biomial proportio p θ θ( θ) N(0, ) χ test for variace ( ) σ s χ F-test o ratio of two variaces s /σ s /σ, s > s F, 8

11 STATISTICS: DISTRIBUTIONS Name Biomial B(, p) Discrete Poisso (λ) Discrete Normal N(μ, σ ) Cotiuous Uiform (Rectagular) o [a, b] Cotiuous Expoetial Cotiuous Geometric Discrete Negative biomial Discrete Fuctio P(X = r) = C r q r p r, for r = 0,,...,, 0 < p <, q = p P(X = r) = e λ λ r r!, for r = 0,,..., λ > 0 f(x) = exp ( ( ) ), σ π x μ σ < x < f(x) =, a x b b a f(x) = λe λx, x 0, λ >0 P(X = r) = q r p, r =,,..., 0 < p <, q = p P(X = r) = r C q r p, r =, +,..., 0 < p <, q = p Mea Variace p pq λ λ μ σ a + b (b a) λ λ q p p p q p p.g.f. G(t) (discrete) m.g.f. M(θ) (cotiuous) G(t) = (q + pt) G(t) = e λ(t ) M(θ) = exp(μθ + σ θ ) M(θ) = e bθ e aθ (b a)θ M(θ) = λ λ θ G(t) = pt qt pt qt G(t) = ( ) 9

12 NUMERICAL ANALYSIS DECISION & DISCRETE MATHEMATICS Numerical Solutio of Equatios The Newto-Raphso iteratio for solvig f(x) = 0 : x + = x Numerical itegratio The trapezium rule b ydx h{(y a 0 + y ) + (y + y y )}, where h = f(x ) f'(x ) b a The mid-ordiate rule b ydx h(y + y a y + y ), where h = b a Simpso s rule for eve b ydx 3 h{(y a 0 + y ) + 4(y + y y ) + (y + y y )}, where h = b a The Gaussia -poit itegratio rule h f ( ) h h x dx h + h f f 3 3 Iterpolatio/fiite differeces Lagrage s polyomial : P (x) = L r (x)f(x r ) where L r (x) = Newto s forward differece iterpolatio formula (x x 0 ) h (x x 0 )(x x )!h f(x) = f(x 0 ) + Δf(x 0 ) + Δ f(x 0 ) +... i=0 i r x x i x r x i Newto s divided differece iterpolatio formula f(x) = f[x 0 ] + (x x 0 )f[x 0, x ] + (x x 0 ) (x x )f[x 0, x, x ] +... Numerical differetiatio f"(x) f(x + h) f(x) + f(x h) h Taylor polyomials f(a + h) = f(a) + hf'(a) + h! f"(a) + error f(a + h) = f(a) + hf'(a) + h f"(a + ξ), 0 < ξ < h! (x f(x) = f(a) + (x a)f '(a) + a) f"(a) + error! (x f(x) = f(a) + (x a)f '(a) + a) f"(η), a < η < x! Numerical solutio of differetial equatios dy For = f(x, y): dx Euler s method : y r + = y r + hf(x r, y r ); x r+ = x r + h Ruge-Kutta method (order ) (modified Euler method) y r + = y r + (k + k ) where k = h f(x r, y r ), k = h f(x r + h, y r + k ) Ruge-Kutta method, order 4: y r+ = y r + 6 (k + k + k 3 + k 4 ), where k = hf(x r, y r ) k = hf(x r + h, y r + k ) k 3 = hf(x r + h, y r + k ) k 4 = hf(x r + h, y r + k 3 ). Logic gates NOT AND OR NAND 0

13 Statistical Tables 7 Cumulative biomial probability 8 0 Cumulative Poisso probability Critical values for correlatio coefficiets The Normal distributio ad its iverse 3 Percetage poits of the χ distributio 3 Percetage poits of the t-distributio 4 5 Critical values for the F-test 6 7 Critical values for the Ma-Whitey test 8 9 Critical values for the Wilcoxo Rak Sum -sample test 30 Critical values for the Wilcoxo Sigle sample ad Paired sample tests 30 Shewhart Chart: Actio ad Warig lies 3 Estimatio of stadard deviatio from rage 3 3 Radom permutatios

14 CUMULATIVE BINOMIAL PROBABILITY The Biomial distributio: cumulative probabilites P(X x) = x r=0 Cr ( p) r p r x p / / / /

15 CUMULATIVE BINOMIAL PROBABILITY x p / / / /

20 CUMULATIVE POISSON PROBABILITY The Poisso distributio: cumulative probabilities P(X x) = r=0 e x λ λr r! x λ x λ x λ x λ x λ x λ

21 CUMULATIVE POISSON PROBABILITY x λ x λ x λ x λ

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