GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS

Transcription

1 GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS FORMULA BOOKLET Fom Septembe 07 Issued 07

2 Mesuto Pue Mthemtcs Sufce e of sphee = 4 Ae of cuved sufce of coe = slt heght Athmetc Sees S l d ( ) Geometc Sees S S fo Summtos Boml Sees b b b b b whee! C!( )!,.. Logthms d epoetls l e Comple Numbes cos s cos s The oots of z e gve by e k z, fo k 0,,,,

3 Mclu s d Tylo s Sees ( ) f ( ) f (0) f (0) f (0) f (0)!! ( ) ( ) ( ) f ( ) f ( ) ( ) f ( ) f ( ) f ( )!! ( ) f ( ) f ( ) f ( ) f ( ) f ( )!! e ep( ) fo ll!! 3 l( ) s 3! 5!! 4 fo ll cos fo ll! 4!! 3 5 t sh 3! 5!! 4 fo ll cosh fo ll! 4!! th Hypebolc Fuctos cosh sh sh sh cosh cosh cosh sh cosh l sh l th l 3

4 Tgoometc Idettes s A B s Acos B cos As B cos AB cos Acos B s As B t A t A t B t At B B Fo t t t t A: s A, cos A t t AB AB s As B s cos AB AB s As B cos s AB AB cos Acos B cos cos AB AB cos A cos B s s A B k Vectos The esolute of the decto of b s b. b. The pot dvdg AB the to : s b. The equto of ple Ctes fom s y 3z k. ( ). The pepedcul dstce betwee two skew les s D b, whee d b e posto vectos of pots o ech le d s mutul pepedcul to both les. by c The pepedcul dstce betwee pot d le s D, whee the b coodtes of the pot e (, y) d the equto of the le s gve by by c. 3 k The pepedcul dstce betwee pot d ple s D, whee 3,, e the coodtes of the pot d y 3z k s the equto of the ple. 4

5 Mt tsfomtos -D cos s Atclockwse otto though bout O: s cos Reflecto the le y t cos s : s cos Mt tsfomtos 3-D Atclockwse otto though bout: -s y-s z-s 0 0 cos 0 s cos s 0 0 cos s 0 0 s cos 0 0 s cos s 0 cos 0 0 whee tclockwse (o postve) otto bout O (o -s) s the sese j k, tclockwse (o postve) otto bout Oy (o y-s) s the sese k, tclockwse (o postve) otto bout Oz (o z-s) s the sese j. 5

6 Dffeetto Fucto Devtve f( ) ( ) f ( ) g( ) f ( ) g( ) g g ( ) t sec sec sec t cot cosec cosec s cos t sh cosh th sh cosh th cosec cot cosh sh sech 6

7 Itegto (+ costt; 0 whee elevt) Fucto t Itegl l sec cot l s cosec l cosec cot l t sec l sec t l t sec sh cosh th t cosh sh l cosh s t 4 cosh l sh l l th l dv du u d uv v d d d Ae of secto A d (pol coodtes) 7

8 Numecl Mthemtcs Numecl tegto b The tpezum ule: y d h y0 y y y y, whee b h Numecl Soluto of Equtos f( ) The Newto-Rphso teto fo solvg f( ) 0 : f( ) Mechcs Moto ccle Tsvese velocty: v Rdl cceleto: Cetes of Mss of Ufom Bodes v Tgul lm: 3 log med fom vete Sem ccle: 4 fom stght edge log s of symmety 3 Qute ccle: y fom vete 3 8

9 Pobblty P( A B) P( A) P( B) P( A B) P( AB) P( A)P( B A) P( A)P( B A) P( A B) P( A)P( B A) P( A)P( B A) Byes Theoem: Dscete dstbutos Pobblty & Sttstcs P( Aj)P( B Aj) P( Aj B) P( A ) P( B A ) Fo dscete dom vble X tkg vlues wth pobbltes Epectto (me): E( X ) p Vce: Fo fucto ( ) p V( X ) ( ) p p gx : g X E ( ) g( ) p Stdd dscete dstbutos: Dstbuto of X P( X ) Me Vce Boml B( p, ) p ( p ) Posso Po( ) p p( p) e! Cotuous dstbutos Fo cotuous dom vble X hvg pobblty desty fucto f Epectto (me): E( X ) f ( )d Vce: Fo fucto ( ) V( X ) ( ) f ( )d f ( )d E g( X ) g( ) f ( )d gx : F( ) P X f ( t)dt Cumultve dstbuto fucto: Stdd cotuous dstbutos: Dstbuto of X P.D.F. Me Vce Ufom (Rectgul) o [ b, ] U[ b, ] b ( b) ( b ) Noml N, e Epoetl Ep( ) e 9

10 Epectto lgeb Fo depedet dom vbles X d Y E( XY ) E( X )E( Y), Smplg dstbutos V( X by) V( X ) b V( Y) Fo dom smple X, X,, X of depedet obsevtos fom dstbuto hvg me d vce X s ubsed estmto of, wth V X S s ubsed estmto of, whee Fo dom smple of obsevtos fom N, X X S N 0, t ( ) S X X If X s the obseved umbe of successes depedet Beoull tls ech of whch the X pobblty of success s p, d Y, the p( p) E( Y) p d V( Y) obsevtos fom N, y obsevtos fom N y, y X Y y Fo dom smple of of y y N 0, d, depedetly, dom smple 0

11 Coelto d Regesso Fo smple of ps of obsevtos, y S y Syy y y y Sy y y y y A mesue of le ssocto betwee two vbles X d Y s gve by the Peso poduct - momet coelto coeffcet. Sy Fo the smple, y,, y,,, y, t s gve by. S S Gve dt, the pmetes d of the le egesso model my be estmted usg the pcple of lest sques. The lest sques estmte ˆ of the pmete s gve by ˆ Sy. S The lest sques estmte ˆ of the pmete s gve by ˆ y ˆ. The lest sques egesso le s gve by y ˆ ˆ. yy Spem's k coelto coeffcet s gve by s 6 d. SS/LG/W7(8)E