AS and A Level Further Mathematics B (MEI)

Transcription

1 fod Cmbdge d RSA * * AS d A evel Futhe Mthemtcs B (MEI) The fomto ths booklet s fo the use of cddtes followg the Advced Subsd Futhe Mthemtcs B (MEI)(H635) o the Advced GCE Futhe Mthemtcs B (MEI) (H645) couse. The fomule booklet wll be pted fo dstbuto wth the emto ppes. Copes of ths booklet m be used fo techg. Ths documet cossts of 6 pges. CR 08 [603364X] DC (T) Advced Subsd GCE (H635) Advced GCE (H645) Futhe Mthemtcs B (MEI) Fomule Booklet Istuctos to Ems ffceivglto Do ot sed ths Dt Sheet fo mkg; t should be eted the cete o destoed. Futhe Mthemtcs B (MEI) CST33

2 Cotets A level Mthemtcs Coe ue Mechcs Futhe ue wth Techolog Et ue umecl Methods Sttstcs Sttstcl tbles A level Mthemtcs Athmetc sees S = ( + l) = { + ( - ) d} Geometc sees ( - ) S = - S fo 3 = - Boml sees ( + b) = + C b+ C b + f+ C b + f + b (! ),! whee C = C = =!( - )! ( - ) ( -) f( - + ) ( + ) = f + + f ^,! Rh!! Dffeetto f() f l() t k sec cot cosec du dv u d v - u Quotet Rule =, = d d v d v Dffeetto fom fst pcples ( ) () f () lm f + h - f l = h " 0 h ksec k sect -cosec -cosec cot CR 08 Futhe Mthemtcs B (MEI)

3 3 Itegto f l() d = lf() + c f() + fl() ^f() h d = ^f() h + c + Itegto b pts dv d d d u d uv v u = - d Smll Agle Appomtos s., cos. -, t. whee s mesued ds Tgoometc dettes s( A! B) = s AcosB! cos As B cos( A! B) = cos AcosB" s As B ta! t B t( A! B) = ( A! B! ( k + ) ) " tat B umecl methods b b Tpezum ule: d. h{( + ) + ( + + f + )} 0 -, whee h = - f( ) The ewto-rphso teto fo solvg f() = 0 : = - + f l( ) obblt ( A, B) = ( A) + ( B) - ( A+ B) ( A+ B) ( A+ B) = ( A) ( B A) = ( B) ( A B) o ( A B) = ( B) Smple Vce s ^ h = S whee S = ( - ) = - = - - Stdd devto, s = vce CR 08 Futhe Mthemtcs B (MEI) Tu ove

4 4 The Boml Dstbuto If X + B(, p) the X ( = ) = C p q - whee q = -p Me of X s p Hpothess test fo the me of oml dstbuto If X + (, v ) the X +, v X - d + ( 0,) v ecetge pots of the oml dstbuto p 0 5 z p% z p% emtcs Moto stght le Moto two d thee dmesos v = u+ t v = u+ t s = ut+ t s = ut+ t s = ( u+ vt ) s = ( u+ v) t v = u + s - s = vt- t s = vt- t CR 08 Futhe Mthemtcs B (MEI)

5 5 Coe ue Comple umbes De Move s theoem: {( cos s )} + = ( cos + s ) Roots of ut: The oots of z k = e gve b z = ep fo k = 0,,, f, - Vectos d 3-D geomet Ctes equto of ple s + + z+ d = 0 3 Ctes equto of le 3-D s - - z- = = 3 d d d b Vecto poduct # b = b b 3 - b - b - b b-b b 3 3 # b = b-b j b bs 3 3 = = t b-b k b 3 3 whee bt,,, tht ode, fom ght-hded tple. d # d Dstce betwee skew les s.( - ) whee s the posto vecto of pot o the fst le d d # d d s pllel to the fst le, smll fo the secod le. + b + c Dstce betwee pot (, ) d le + b + c = 0 s + b + + z+ d 3 Dstce betwee pot (,, z ) d ple + + z+ d = 0 s Hpebolc fuctos cosh - sh = sh = l[ + ( + )] cosh = l[ + ( - ) ], $ + th = l, CR 08 Futhe Mthemtcs B (MEI) Tu ove

6 6 Clculus f() f l() cs ccos ct f() f() d cs ^ h ct sh o l( + + ) cosh o l( + - ) ( ) The me vlue of f() o the tevl [,b] s b f() d b- Ae of secto eclosed b pol cuve s d Sees 3 = ( + )( + ) = ( + ) 6 4 = = () fm() 0 f () 0 f() = f() 0 + f l() f+ + f!! e = ep() = f+ + ffo ll!! 3 + l( + ) = f+ (- ) + f( - # ) s = f+ (-) + ffo ll 3! 5! ( + )! 4 cos = f+ (- ) + f fo ll! 4! ( )! ( - ) ( -) f( - + ) ( + ) = f + + f ^,! Rh!! CR 08 Futhe Mthemtcs B (MEI)

7 7 Mechcs Moto ccle Fo moto ccle, tgetl veloct s v = o v dl cceleto s o o towds the cete tgetl cceleto s p Futhe ue wth Techolog umecl soluto of dffeetl equtos d Fo = f(, ): d Eule s method: = + h = + hf (, ) + + Modfed Eule method (A Ruge-utt method of ode ): k hf(, ) = k = hf( + h, + k ) = + h, = + ( k + k ) + + Ruge-utt method of ode 4: k hf(, ) k k = 3 h k hf(, = + + ) h k hf(, = + + ) k = hf( + h, + k ) 4 3 ( k k k k ) + = Gdet of tget to pol cuve d Fo cuve = f( ), = d d s + cos d d cos - s d CR 08 Futhe Mthemtcs B (MEI) Tu ove

8 8 Futhe Mthemtcs B (MEI) CR 08 Et ue Multvble clculus g g g g g gd z d = =. If g(,, z) c be wtte s, ) f( z = the g f f gd = -

9 9 umecl methods Soluto of equtos f( ) The ewto-rphso teto fo solvg f() = 0 : = - + f l( ) Fo the teto g( ) = the eled teto s = ( - m) + mg( ) + +. umecl tegto b To estmte f() d: The mdpot ule: M = h( f ) - - whee h b = - The tpezum ule: b T = h{( + ) + ( + + f + )} 0 - whee h = - Smpso s ule S = h{( + ) + 4( + + f+ ) + ( + + f+ )} These e elted s follows: T ( M T ) = + S = ( M + T ) = ( 4 T -T ) 3 3 b whee h = - Itepolto ewto s fowd dffeece tepolto fomul: ( ) ( )( ) 0 f() = f( ) + - Tf( ) f( ) 0 h 0 T +! h gge s poloml: () = ()( f ) whee () = % = 0! f CR 08 Futhe Mthemtcs B (MEI) Tu ove

10 0 Sttstcs Dscete dstbutos X s dom vble tkg vlues dscete dstbuto wth ( X = ) = p Epectto: E( X) = = p Vce: v = V( X) = ( - ) p = p - obblt E(X) V(X) Ufom dstbuto ove,,, X ( = ) = Geometc dstbuto X ( = ) = q - p q = -p + p ( ) - - p p osso dstbuto m X ( = ) = e -m! Coelto d egesso Fo smple of ps of obsevtos (, ) S k k = -, S = -, S = - S poduct momet coelto coeffcet: = = S S lest sques egesso le of o s - = b ( - ) whee b R S S T - k - - S = = S S lest sques egesso le of o s - = bl ( - ) whee bl = = S Spem s coeffcet of k coelto: 6 d = - s ( - ) V k W W X - k k CR 08 Futhe Mthemtcs B (MEI)

11 Cofdece tevls To clculte cofdece tevl fo me o dffeece me dffeet ccumstces, use the gve dstbuto to clculte the ctcl vlue, k. To estmte Cofdece tevl Dstbuto me me dffeece me of ped popultos! k v (0, ) s! k t tet dffeeces s sgle dstbuto Hpothess tests Descpto Test sttstc Dstbuto eso s poduct momet coelto test Spem s k coelto test test S = S S - = R k S - - T 6 d = - s ( - ) ^f - f o eh f V k W W X v e oml test fo me t-test fo me Wlcoo sgle smple test - v - s A sttstc T s clculted fom the ked dt (0, ) t Cotuous dstbutos X s cotuous dom vble wth pobblt dest fucto (pdf) f() Epectto: E( X) f() d = = Vce: v = V( X) = ( - ) f() d = f() d - Cumultve dstbuto fucto F() = ( X # ) = f() t dt -3 E(X) V(X) Cotuous ufom dstbuto ove [,b] + b ( b- ) CR 08 Futhe Mthemtcs B (MEI) Tu ove

12 CR 08 Futhe Mthemtcs B (MEI) Ctcl vlues fo the poduct momet coelto coeffcet, 5% ½% % ½% 0% 5% % % -Tl Test -Tl Test 5% ½% % ½% 5% ½% % ½% 0% 5% % % 0% 5% % % Ctcl vlues fo Spem s k coelto coeffcet, s -Tl Test -Tl Test 5% ½% % ½% 0% 5% % %

13 ecetge pots of the 3 (ch-squed) dstbuto p% p% p% v = CR 08 Futhe Mthemtcs B (MEI) Tu ove

14 4 ecetge pots of the t dstbuto p% p% t v p% = pecetge pots of the oml dstbuto (0, ) ecetge pots of the oml dstbuto p% z p% p 0 5 z CR 08 Futhe Mthemtcs B (MEI)

15 5 Ctcl vlues fo the Wlcoo Sgle Smple test -tl 5% ½% % ½% -tl 5% ½% % ½% -tl 0% 5% % % -tl 0% 5% % % CR 08 Futhe Mthemtcs B (MEI)

16 6 BA AGE fod Cmbdge d RSA Copght Ifomto CR s commtted to seekg pemsso to epoduce ll thd-pt cotet tht t uses ts ssessmet mtels. CR hs ttempted to detf d cotct ll copght holdes whose wok s used ths ppe. To vod the ssue of dsclosue of swe-elted fomto to cddtes, ll copght ckowledgemets e epoduced the CR Copght Ackowledgemets Booklet. Ths s poduced fo ech sees of emtos d s feel vlble to dowlod fom ou publc webste ( fte the lve emto sees. If CR hs uwttgl fled to coectl ckowledge o cle thd-pt cotet ths ssessmet mtel, CR wll be hpp to coect ts mstke t the elest possble oppotut. Fo quees o futhe fomto plese cotct the Copght Tem, Fst Floo, 9 Hlls Rod, Cmbdge CB GE. CR s pt of the Cmbdge Assessmet Goup; Cmbdge Assessmet s the bd me of Uvest of Cmbdge ocl Emtos Sdcte (UCES), whch s tself deptmet of the Uvest of Cmbdge. CR 08 Futhe Mthemtcs B (MEI)