Mathematics HL and further mathematics HL formula booklet

Transcription

1 Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Publshed Jue 0 Itertol Bcclurete Orgzto

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3 Cotets Pror lerg Core Topc : Algebr Topc : Fuctos d equtos 4 Topc : Crculr fuctos d trgoometry 4 Topc 4: Vectors 5 Topc 5: Sttstcs d probblty 6 Topc 6: Clculus 8 Optos 0 Topc 7: Sttstcs d probblty 0 Further mthemtcs HL topc Topc 8: Sets, reltos d groups Further mthemtcs HL topc 4 Topc 9: Clculus Further mthemtcs HL topc 5 Topc 0: Dscrete mthemtcs Further mthemtcs HL topc 6 Formule for dstrbutos Topcs 5.6, 5.7, 7., further mthemtcs HL topc. Dscrete dstrbutos Cotuous dstrbutos Further mthemtcs 4 Topc : Ler lgebr 4

4 Formule Pror lerg Are of prllelogrm A b h, where b s the bse, h s the heght Are of trgle A ( b h ), where b s the bse, h s the heght Are of trpezum A ( b ) h, where d b re the prllel sdes, h s the heght Are of crcle A r, where r s the rdus Crcumferece of crcle C r, where r s the rdus Volume of pyrmd V (re of bse ertcl heght) Volume of cubod V l w h, where l s the legth, w s the wdth, h s the heght Volume of cylder V r h, where r s the rdus, h s the heght Are of the cured surfce of cylder A rh, where r s the rdus, h s the heght Volume of sphere 4 V r, where r s the rdus Volume of coe V r h, where r s the rdus, h s the heght Dstce betwee two pots (, y) d (, y ) Coordtes of the mdpot of le segmet wth edpots (, y) d (, y ) d ( ) ( y y ), y y Solutos of qudrtc equto The solutos of b c 0 re b b 4c

5 Core Topc : Algebr. The th term of rthmetc sequece u u ( ) d The sum of terms of rthmetc sequece S ( u ( ) d) ( u u ) The th term of geometrc sequece u u r The sum of terms of fte geometrc sequece The sum of fte geometrc sequece S S u( r ) u( r ) r r u r, r, r. Epoets d logrthms b log b, where 0, b 0, e l log log b logc log b c log. Combtos! r r!( r)! Permuttos! Pr ( r)! Boml theorem ( b) b b b r r r.5 Comple umbers z b r(cos s ) re r cs.7 De More s theorem r(cos s ) r (cos s ) r e r cs

6 Topc : Fuctos d equtos.5 As of symmetry of the grph of qudrtc fucto f ( ) b c s of symmetry b.6 Dscrmt b 4c Topc : Crculr fuctos d trgoometry. Legth of rc l r, where s the gle mesured rds, r s the rdus Are of sector A r, where s the gle mesured rds, r s the rdus. Idettes s t cos sec cosec cos s Pythgore dettes cos s t sec cot csc. Compoud gle dettes s( A B) s Acos B cos As B cos( A B) cos Acos B s As B t( A B) t A t B t At B Double gle dettes s s cos cos cos s cos s t t t

7 .7 Cose rule c b b cos C ; cosc b c b Se rule b c s A s B s C Are of trgle A bs C Topc 4: Vectors 4. Mgtude of ector, where Dstce betwee two pots (, y, z ) d (, y, z ) Coordtes of the mdpot of le segmet wth edpots (, y, z ), (, y, z ) d ( ) ( y y ) ( z z ), y y, z z 4. Sclr product w w cos, where s the gle betwee d w w w w w, where, w w w w Agle betwee two ectors cos w w w w 4. Vector equto of le r = +λb Prmetrc form of the equto of le Crtes equtos of le 0 l, y y0 m, z z y y z z l m

8 4.5 Vector product w w w w w where w w, w w w w w w s, where s the gle betwee d w Are of trgle A w where d w form two sdes of trgle 4.6 Vector equto of ple r = +λb+ c Equto of ple (usg the orml ector) r Crtes equto of ple by cz d Topc 5: Sttstcs d probblty 5. Populto prmeters Let f Me f Vrce f f Stdrd deto f 5. Probblty of eet A A ( ) P( A) U ( ) Complemetry eets P( A) P( A ) 5. Combed eets P( A B) P( A) P( B) P( A B ) Mutully ecluse eets P( A B) P( A) P( B )

9 5.4 Codtol probblty P( A B) P AB P( B) Idepedet eets P( A B) P( A) P( B ) Byes theorem P( B)P A B P B A P( B)P A B P( B )P A B P( B A) P( B ) P( A B ) P( B ) P( A B ) P( B ) P( A B ) P( B ) P( A B ) 5.5 Epected lue of dscrete rdom rble X Epected lue of cotuous rdom rble X E( X ) P( X ) E( X ) f ( )d Vrce Vr( X) E( X ) E( X ) E( X ) Vrce of dscrete rdom rble X Vrce of cotuous rdom rble X 5.6 Boml dstrbuto Me Vrce Posso dstrbuto Me Vrce Vr( X ) ( ) P( X ) P( X ) Vr( X ) ( ) f ( )d f ( )d X ~ B(, p) P( X ) p ( p), 0,,, E( X) p Vr( X) p( p ) m m e X ~ Po( m) P( X ), 0,,,! E( X) Vr( X) m m 5.7 Stdrdzed orml rble z

10 Topc 6: Clculus 6. Derte of f() 6. Derte of d y f ( h) f ( ) y f ( ) f ( ) lm d h 0 h f ( ) f ( ) Derte of s f ( ) s f ( ) cos Derte of cos f ( ) cos f ( ) s Derte of t f ( ) t f ( ) sec Derte of e f ( ) e f ( ) e Derte of l f ( ) l f ( ) Derte of sec f ( ) sec f ( ) sec t Derte of csc f ( ) csc f ( ) csc cot Derte of cot f ( ) cot f ( ) csc Derte of ( ) f f ( ) (l ) Derte of log f ( ) log f ( ) l Derte of rcs f ( ) rcs f ( ) Derte of rccos f ( ) rccos f ( ) Derte of rct f ( ) rct f ( ) Ch rule y g() u, where u f () dy dy du d du d Product rule dy d du y u u d d d Quotet rule y u dy du d u d d d

11 6.4 Stdrd tegrls d C, d l C s d cos C cos d s C e d e C d C l d rct C d rcs C, 6.5 Are uder cure Volume of reoluto (rotto) b A y d or A d y b b π d or π d V y V y b 6.7 Itegrto by prts d du u d u d d d or ud u du

12 Optos Topc 7: Sttstcs d probblty Further mthemtcs HL topc 7. (.) 7. (.) Probblty geertg fucto for dscrete rdom rble X Ler combtos of two depedet rdom rbles X, X G( t) E( t X ) P( X ) t E X X E X E X Vr X X Vr X Vr X 7. (.) Smple sttstcs Me f Vrce s f ( ) f s Stdrd deto s s f ( ) Ubsed estmte of populto rce s f ( ) f s s 7.5 (.5) Cofdece terls Me, wth ow rce z Me, wth uow rce t s 7.6 (.6) Test sttstcs Me, wth ow rce z / Me, wth uow rce t s /

13 7.7 (.7) Smple product momet correlto coeffcet r y y y y Test sttstc for H 0 : ρ = 0 t r r Equto of regresso le of o y Equto of regresso le of y o y y ( y y) y y y y y y ( ) Topc 8: Sets, reltos d groups Further mthemtcs HL topc 4 8. (4.) De Morg s lws ( A B) A B ( A B) A B Topc 9: Clculus Further mthemtcs HL topc (5.5) Euler s method y y h f (, y ) ; h, where h s costt (step legth) Itegrtg fctor for y P( ) y Q( ) ( )d e P

14 9.6 (5.6) Mclur seres f ( ) f (0) f (0) f (0)! Tylor seres ( ) f ( ) f ( ) ( ) f ( ) f ( )...! Tylor ppromtos (wth error term R () ) ( ) f f f f R! ( ) ( ) ( ) ( ) ( )... ( ) ( ) Lgrge form ( ) f () c R ( ) ( ) ( )!, where c les betwee d Mclur seres for specl fuctos e...! l( )... 5 s...! 5! 4 cos...! 4! 5 rct... 5 Topc 0: Dscrete mthemtcs Further mthemtcs HL topc (6.7) Euler s formul for coected plr grphs Plr, smple, coected grphs e f, where s the umber of ertces, e s the umber of edges, f s the umber of fces e 6 for e 4 f the grph hs o trgles

15 Formule for dstrbutos Topcs 5.6, 5.7, 7., further mthemtcs HL topc. Dscrete dstrbutos Dstrbuto Notto Probblty mss fucto Me Vrce Geometrc X ~ Geo p pq for,,... p q p Negte boml X ~ NB r, p r r r pq r p rq p for r, r,... Cotuous dstrbutos Dstrbuto Notto Probblty desty fucto Me Vrce Norml X ~ N, e π

16 Further mthemtcs Topc : Ler lgebr. Determt of mtr A b det c d A A d bc Ierse of mtr A b d b A c d det A c, d bc Determt of mtr b c e f d f d e A d e f det A b c h g g h g h