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 3 Topc : Algebr 3 Topc : Fuctos d equtos 4 Topc 3: 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 3 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 3 Topcs 5.6, 5.7, 7., further mthemtcs HL topc 3. Dscrete dstrbutos 3 Cotuous dstrbutos 3 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 vertcl heght) 3 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 curved surfce of cylder A rh, where r s the rdus, h s the heght Volume of sphere 4 3 V r, where r s the rdus 3 Volume of coe V r h, where r s the rdus, h s the heght 3 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 The sum of terms of fte geometrc sequece The sum of fte geometrc sequece u S S u r u( r ) u( r ) r r u r, r, r. Epoets d logrthms b log b, where 0, b 0, e l log logc logb log b c log.3 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 Movre 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 3: Crculr fuctos d trgoometry 3. 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 3. Idettes s t cos sec cos cosec s Pythgore dettes cos s t sec cot csc 3.3 Compoud gle dettes s( A B) s Acos B cos As B cos( AB) cos Acos B s As B t A t B t( AB) t At B Double gle dettes s s cos cos cos s cos s t t t

7 3.7 Cose rule c b bcos C ; b c cosc b Se rule b c s A s B s C Are of trgle A bsc Topc 4: Vectors 4. Mgtude of vector v v v v 3, where v v v v 3 Dstce betwee two pots (, y, z ) d (, y, z ) d ( ) ( y y ) ( z z ) Coordtes of the mdpot of le segmet wth edpots (, y, z ), (, y, z ), y y, z z 4. Sclr product v w v w cos, where s the gle betwee v d w v w vw vw v3w3, where v v v, v 3 w w w w 3 Agle betwee two vectors v w v w v w cos vw Vector equto of le r = +λb Prmetrc form of the equto of le Crtes equtos of le l, y y m, z z y y z z l m 0 0 0

8 4.5 Vector product vw3 v3w v w v3w vw 3 where vw vw v v v, v 3 w w w w 3 v w v w s, where s the gle betwee v d w Are of trgle A vw where v d w form two sdes of trgle 4.6 Vector equto of ple r = +λb + c Equto of ple (usg the orml vector) Crtes equto of ple r by cz d Topc 5: Sttstcs d probblty 5. Populto prmeters Let f Me Vrce f f f Stdrd devto f 5. Probblty of evet A A ( ) P( A) U ( ) Complemetry evets P( A) P( A) 5.3 Combed evets P( A B) P( A) P( B) P( A B) Mutully eclusve evets P( A B) P( A) P( B)

9 5.4 Codtol probblty P( A B) P ( AB) P( B) Idepedet evets P( AB) 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) P( A B) P( B A) P( B ) P( A B ) P( B ) P( A B ) P( B ) P( A B ) Epected vlue of dscrete rdom vrble X Epected vlue of cotuous rdom vrble X E( X ) P( X ) E( X ) f ( )d Vrce Vr( X ) E( X ) E( X ) E( X ) Vrce of dscrete rdom vrble X Vrce of cotuous rdom vrble X 5.6 Boml dstrbuto Me Vrce Posso dstrbuto Me Vrce 5.7 Stdrdzed orml vrble 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 z m

10 Topc 6: Clculus 6. Dervtve of f( ) 6. Dervtve of d y f ( h) f ( ) y f ( ) f ( ) lm d h0 h f ( ) f ( ) Dervtve of s f ( ) s f ( ) cos Dervtve of cos f ( ) cos f ( ) s Dervtve of t f ( ) t f ( ) sec Dervtve of e f ( ) e f ( ) e Dervtve of l f ( ) l f ( ) Dervtve of sec f ( ) sec f ( ) sec t Dervtve of csc f ( ) csc f ( ) csc cot Dervtve of cot f ( ) cot f ( ) csc Dervtve of ( ) f f ( ) (l ) Dervtve of log f ( ) log f ( ) l Dervtve of rcs f ( ) rcs f ( ) Dervtve of rccos f ( ) rccos f ( ) Dervtve of rct f ( ) rct f ( ) Ch rule y g( u), where dy dy du u f ( ) d du d Product rule dy dv du y uv u v d d d Quotet rule du dv v u u dy y d d v d v

11 6.4 Stdrd tegrls d C, d l C s d cos C cos d s C e de C d C l d rct C d rcs C, 6.5 Are uder curve Volume of revoluto (rotto) b A yd or A dy b b π d or π d V y V y b 6.7 Itegrto by prts dv du u d uv v d d d or d d u v uv v u

12 Optos Topc 7: Sttstcs d probblty Further mthemtcs HL topc 3 7. (3.) Probblty geertg fucto for dscrete rdom vrble X G( t) E( t ) P( X ) t E( X) G() Vr( X ) G() G() G() 7. (3.) 7.3 (3.3) Ler combtos of two depedet rdom vrbles X, X Smple sttstcs Me E( X X ) E ( X ) E ( X ) Vr( X X ) Vr ( X ) Vr ( X ) f Vrce s f ( ) f s Stdrd devto s s f ( ) Ubsed estmte of populto vrce s f ( ) f s s 7.5 (3.5) Cofdece tervls Me, wth ow vrce z 7.6 (3.6) Me, wth uow vrce Test sttstcs Me, wth ow vrce t s z /

13 Me, wth uow vrce t s / 7.7 (3.7) 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...! 3 l( ) s... 3! 5! 4 cos...! 4! 3 5 rct Topc 0: Dscrete mthemtcs Further mthemtcs HL topc (6.7) Euler s formul for coected plr grphs Plr, smple, coected grphs v e f, where v s the umber of vertces, e s the umber of edges, f s the umber of fces e3v 6 for v 3 ev 4 f the grph hs o trgles

15 Formule for dstrbutos Topcs 5.6, 5.7, 7., further mthemtcs HL topc 3. Dscrete dstrbutos Dstrbuto Notto Probblty mss fucto Me Vrce Geometrc X ~ Geo ( p ) pq for,,... Negtve boml X ~ NB ( r, p ) r pq r r for r, r,... p r p q p rq p Cotuous dstrbutos Dstrbuto Notto Probblty desty fucto Me Vrce Norml X ~ N (, ) e π

16 Further mthemtcs Topc : Ler lgebr. Determt of mtr Iverse of mtr Determt of 3 3 mtr b A det A A d bc c d b d b A A, d bc c d det A c b c e f d f d e A d e f det A b c h g g h g h