THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE

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1 THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE UNIVERSITY PRESS, MSIDA, MALTA 208

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3 THE UNIVERSITY OF MALTA DEPARTMENT OF MATHEMATICS MATHEMATICAL FORMULAE UNIVERSITY PRESS, MSIDA, MALTA 208

4 THIS BOOKLET IS NOT AN ORIGINAL COPY, BUT A RE-TRANSCRIPTION OF THE BOOKLET PROVIDED TO STUDENTS DURING MATSEC EXAMINATION SESSIONS FOR PURE MATHEMATICS (AM27/IM27). EFFORT WAS MADE TO ENSURE THAT THIS RE-TRANSCRIPTION IS AS ACCURATE AS POSSIBLE. TRANSCRIBED BY LUKE COLLINS CREATED WITH L A TEX 2ε AVAILABLE TO DOWNLOAD FOR FREE AT MMXVIII

5 CONTENTS Pge MENSURATION ALGEBRA 2 HYPERBOLIC FUNCTIONS 3 CIRCULAR FUNCTIONS 4 COORDINATE GEOMETRY 5 CALCULUS I INFINITE SERIES 6 II DERIVATIVES 7 III INTEGRALS 8 IV APPLICATIONS 9 V APROXIMATIONS 2 VECTORS 2 MECHANICS 3 PROBABILITY 4 STATISTICS 6 I FORMULAE 6 II TABLES 8

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7 MENSURATION Circle Are of circle, rdius r is πr 2 Circumference of circle is 2πr Sphere Volume of sphere, rdius r, is 4 3 πr3 Surfce re of sphere is 4πr 2 Right circulr cylinder Volume of cylinder, rdius r nd height h is πr 2 h Curved surfce re is 2πrh Right circulr cone Volume of cone, rdius r, nd height h is 3 πr2 h Curved surfce re is πrl where l is the slnt height of the cone.

8 ALGEBRA Fctors 3 + b 3 = ( + b)( 2 b + b 2 ) 3 b 3 = ( b)( 2 + b + b 2 ) Permuttions nd Combintions ( ) n n n! C r = = r r!(n r)! n P r = n! (n r)! Finite Series n q=0 ( + qd) = n 2 [2 + (n )d]; n q=0 r q = ( rn ) r n = r=r n 2 n(n + ); r=r 2 = n 6 n(n + )(2n + ); r 3 = r= 4 n2 (n + ) 2 ( + x) n = + nx + n(n ).2 x ( ) n x r + + x n r (n +ve int.) de Moivre s Theorem If n is n integer, (cosθ + isinθ) n = cosnθ + isinnθ. If n is rtionl number, cosnθ + isinnθ is one of the vlues of (cosθ + isinθ) n. 2

9 HYPERBOLIC FUNCTIONS sinhx = ex e x 2 coshx = ex + e x 2 sinh x = ln[x + (x 2 + )] Principl vlue of cosh x = ln[x + (x 2 )] (x ) tnh x = / 2 ln + x x ( x < ) 3

10 CIRCULAR FUNCTIONS sin 2 A + cos 2 A = sec 2 A = + tn 2 A cosec 2 A = + cot 2 A If sinθ = sinα, If cosθ = cosα, If tnθ = tnα then θ = nπ + ( ) n α then θ = 2nπ ± α then θ = nπ + α where n = 0,±,±2,... sin(a ± B) = sinacosb ± cosasinb cos(a ± B) = cosacosb sinasinb tna ± tnb tn(a ± B) = tnatnb sina + sinb = 2sin / 2 (A + B)cos / 2 (A B) sina sinb = 2cos / 2 (A + B)sin / 2 (A B) cosa + cosb = 2cos / 2 (A + B)cos / 2 (A B) cosa cosb = 2sin / 2 (A + B)sin / 2 (A B) 2sinAcosB = sin(a + B) + sin(a B) 2cosAsinB = sin(a + B) sin(a B) 2cosAcosB = cos(a + B) + cos(a B) 2sinAsinB = cos(a B) cos(a + B) sin2a = 2sinAcosA cos2a = cos 2 A sin 2 A = 2sin 2 A = 2cos 2 A tn2a = 2tnA tn 2 A If tn A 2 = t, then sina = 2t t2 ; cosa = + t2 + t 2 4

11 COORDINATE GEOMETRY Perpendiculr distnce from (h,k) to x+by+c = 0 is h + bk + c ( 2 + b 2 ) The cute ngle between two lines with grdients m, m 2 is tn m m 2 + m m 2 Are of Tringle is x x 2 x 3 / 2 [x (y 2 y 3 ) + x 2 (y 3 y ) + x 3 (y y 2 )] = / 2 y y 2 y 3 Circle The eqution x 2 + y 2 + 2gx + 2 f y + c = 0 represents circle with centre t ( g, f ) nd rdius (g 2 + f 2 c). The prmetric equtions of circle with centre t (,b) nd rdius r re x = + r cost, y = b + r sint. Point dividing P P 2 in the rtio k : hs coordintes ( x + kx 2 + k, y + ky 2 + k, z ) + kz 2 + k Angle φ between two lines with direction cosines l, m, n : l, m, n ±(ll + mm + nn ) is given by cosφ = (l 2 + m 2 + n 2 ) (l 2 + m 2 + n 2 ) Distnce from P (x,y,z ) to plne Ax + By +Cz + D = 0 is Ax + By +Cz + D (A 2 + B 2 +C 2 ) Plne distnce p from origin, direction cosines of norml l, m, n, lx + my + nz = p. Line through (x,y,z ), direction cosines l, m, n. x x l = y y m = z z = t. n 5

12 CALCULUS I. INFINITE SERIES Tylor s Theorem f (+x) = f ()+x f ()+ x2 2! f ()+ + xr (r )! f (r ) ()+, with reminder term xr r! f (r) ( + θx), where 0 < θ <. Mclurin s Theorem f (x) = f (0) + x f (0) + x2 2! f (0) + + xr (r )! f (r ) (0) +, with reminder term xr r! f (r) (θx), where 0 < θ <. expx e x = + x + x2 xr + + 2! r! + * log e ( + x) ln( + x) = x x2 2 + x3 3 vlid for < x. + ( )r xr r + sinx = x x3 3! + x5 5! + ( )r x 2r+ (2r + )! + * cosx = x2 2! + x4 4! + ( )r x 2r (2r)! + * sinhx = / 2 (e x e x ) = x + x3 3! + x5 x2r ! (2r + )! + * coshx = / 2 (e x + e x ) = + x2 2! + x4 x2r + + 4! (2r)! + * * These series re vlid for ll finite x. 6

13 II DERIVATIVES f (x) x n sinx cosx tnx cotx secx cosecx e x x ( > o) log e x lnx sinhx coshx uv u v f (x) nx n cosx sinx sec 2 x cosec 2 x secxtnx cosecxcotx e x x ln x coshx sinhx uv + u v (vu uv )/v 2 7

14 III INTEGRALS (Constnts of integrtion re omitted; ln log e ) f (x) ( 2 x 2 ) ( 2 + x 2 ) ( 2 + x 2 ) x ( 2 + x 2 ) ( 2 x 2 ) sinx cosx f (x)dx ( sin x ) ( x ) tn ln{x + (x ( )} or sinh x ) ( 2 + x 2 ) ln{x + (x 2 2 ( )} or cosh x ) cosx sinx tn x ln(sec x) cot x ln(sin x) secx ln(secx + tnx) { ( x or ln tn 2 + π )} 4 cosecx lntn x 2 coshx sinhx u dv dx sinhx coshx uv v du dx dx 8

15 IV APPLICATIONS For curve y = f (x), x b. Are between curve nd x-xis = y dx Men vlue = y dx b Volume of revolution bout x-xis = π y 2 dx Centroid of re between curve nd x-xis hs coordintes x = xy dx y dx ; y = / 2 y 2 dx y dx Centroid of solid of revolution bout x-xis: x = xy 2 dx y 2 dx 9

16 For the re shown in Figure y First moment bout x-xis = 2 y 2 dx y = f (x) First moment bout y-xis = xy dx Second moment bout x-xis = 3 y3 dx 0 Fig. b x Second moment bout y-xis = x 2 y dx For the solid of revolution shown in Figure 2 First moment bout xy-plne = 0 y y = f (x) First moment bout yz-plne = π xy 2 dx 0 b x Second moment bout x-xis = π 2 y4 dx Second moment bout y-xis = π y 2 (x 2 + y2 4 ) dx Fig. 2 0

17 Length of rc = (dy) t 2 2} { + dx = (ẋ 2 + ẏ 2 )dt dx Are of surfce of revolution = 2π y { + ( dy) 2} dx dx t t 2 = 2π y (ẋ 2 + ẏ 2 )dt t { + ( dy) 2} 3/2 Rdius of curvture ρ = dx d 2 y dx 2 = (ẋ2 + ẏ 2 ) 3/2 ẋÿ ẍẏ Polr Coordintes θ 2 Are enclosed by curve = / 2 r 2 dθ θ Length of rc = θ 2 θ {r 2 + ( dr dθ ) r 2 2} dθ = r { + r 2 ( dθ ) 2} dr dr / d p Rdius of curvture ρ = r dr

18 V APPROXIMATIONS Trpezoidl Rule: y dx / 2 h{(y o + y n ) + 2(y + y y n )} Simpson s rule (n even) y dx / 3 h{(y o + y n ) + 4(y + y y n ) + 2(y 2 + y y n 2 ) Newton s pproximtion to root of f (x) = 0: x n+ = x n f (x n) f (x n ) VECTORS Line through point, position vector, prllel to b r = +tb Position vector of point dividing the line joining P, Q with position λp + µq vectors p, q in the rtio λ : µ is λ + µ Plne through point, position vector, perpendiculr to n (r ). n = 0 Sclr product =. 2 = 2 cosθ = x x 2 + y y 2 + z z 2 i j k Vector product = 2 = 2 = 2 sinθ ˆn = x y z x 2 y 2 z 2 2

19 MECHANICS Centre of mss Arc, rdius r, ngle 2θ r sinθ/θ from centre Sector of circle, rdius r, ngle 2θ 2 3 r sinθ/θ from centre Hemisphere, rdius r 3 8 r from centre Hemisphericl shell, rdius r 2 r from centre Solid cone, height h 2 from vertex Conicl shell, height h 2 3 h from vertex Moments of inerti Rod, length 2l, bout perpendiculr xis through centre 3 ml 2 Disc, rdius r, bout perpendiculr xis through centre 2 mr 2 Hoop, rdius r, bout dimeter 2 mr 2 Solid sphere, rdius r, bout dimeter 2 5 mr 2 Sphericl shell, rdius r, bout dimeter 2 3 mr 2 Prllel xes theorem I A = I G + M(GA) 2 Perpendiculr xes theorem for lmin I oz = I ox + I oy Simple hrmonic motion d 2 y dt 2 = ω2 x, ( dx) 2 = ω 2 ( 2 x 2 ), x = sin(ωt + ε) dt Compound pendulum Period = 2π (k 2 + h 2 )/gh Components of ccelertion r r θ 2 long rdius vector 2ṙ θ + r θ perpendiculr to rdius vector 3

20 PROBABILITY Probbility lws P(A B) = P(A) + P(B) P(A B) P(A B) = P(A) P(B A) Discrete vrible X with probbility function P(X = x) Continuous vrible X with probbility density function f (X) Distribution function F(X) F(x o ) = P(X x o ) = x x o P(x) Expecttion of X E(X) = ΣxP(X = x) Expecttion of g(x) E[g(x)] = Σg(x)P(X = x) F(x) = P(X < x) x o = f (x) dx E(X) = x f (x) dx E[g(X)] = g(x) f (x)dx Vrince σ 2 Vr(X) = E[{X E(X) 2 }] Covrince Cov(X,X 2 ) = E[{X E(X )}{X 2 E(X 2 )}] Correltion coefficient ρ 2 (X,X 2 ) ρ 2 = Cov(X,X 2 ) {Vr(X )Vr(X 2 )} Liner regression coefficient, β 2, for X on X 2 β 2 = Cov(X,X 2 ) Vr(X 2 ) 4

21 Probbility generting function G(z) G(z) = P(0) + P()z + P(2)z P(r)z r +, where P(r) = P(X = r) Binomil distribution (X, p,n) P(X = k) = ( N k E(X) = N p ) p k ( p) N k Vr(X) = N p( p) G(z) = [pz + ( p)] N Poisson distribution (X,m) P(X = k) = e m m k k! E(X) = m Vr(X) = m G(z) = e m e mz Norml distribution If X is distributed N(µ,σ 2 ) then X µ σ is distributed N(0,) where σ is the stndrd devition nd σ 2 is the vrince. 5

22 STATISTICS µ, σ 2 popultion men nd vrince X i ith rndom selection in smple size n Smple men X = n ΣX i Smple vrince S 2 = n Σ(X i X) 2 E(S 2 ) = σ 2 E(X) = µ Vr(X) = Vr(X) n = σ 2 n One smple t-test t n = X µ o S/ n Two smple t-test t n +n 2 2 = X X 2 S ( + ) n n 2 where S 2 = Σ(x x ) 2 + Σ(x 2 x 2 ) 2 n + n 2 2 nd n, n 2 re the sizes of the two smples 6

23 Pired smple t-test t n = s 2 = Vr(y) Y S ( ) where Y j = X j X 2 j ( j =,2,3,...,n) nd n Spermn s rnk correltion coefficient ρ ρ = 6Σ d2 n(n 2 ) Kendll s rnk correltion coefficient r ( ) ( ) Number of greed Number of different pir rnkings pir rnkings r = = Number of pirs S 2n(n ) Pired smple Wilcoxon (n > 8) T = (sum of the rnks with the less frequent sign) Z = T T s distributed N(0,); T = n(n + ) ; s 2 n(n + )(2n + ) =

24 Tble The stndrdised norml distribution Entry represents re under the stndrdized norml distribution from the men to Z Z Z 8

25 Tble 2 Percentge points of Student s t-distribution α v t α,v α t 9

26 Tble 3 Percentge points of the χ 2 distribution α v α χ 2 α,v 20

27 Tble 4 Upper percentge points of the F-distribution () α = 0.0 v v α F α,v,v 2 2

28 Tble 4 (continued) () α = v v

29 Tble 4 (continued) () α = 0.05 v v

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