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 0 5048 Mthemtcs HL d further mthemtcs formul boolet
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 Mthemtcs HL d further mthemtcs formul boolet
Formule Pror lerg Are of prllelogrm Ab 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 lw 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 Mthemtcs HL d further mthemtcs formul boolet
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, b0, 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 br(cos s ) re rcs.7 De Movre s theorem r(cos s ) r (cos s ) r e r cs Mthemtcs HL d further mthemtcs formul boolet 3
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( A B) cos Acos B s As B t A t B t( AB) tatb Double gle dettes s scos cos cos s cos s t t t Mthemtcs HL d further mthemtcs formul boolet 4
3.7 Cose rule c b bcosc; 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 vw v w cos, where s the gle betwee v d w v w vw vw vw 3 3, where v v v, v 3 w w w w 3 Agle betwee two vectors vw vw vw cos v w 3 3 4.3 Vector equto of le r = +λb Prmetrc form of the equto of le Crtes equtos of le l, y y m, z z 0 0 0 y y zz l m 0 0 0 Mthemtcs HL d further mthemtcs formul boolet 5
4.5 Vector product vw 3 vw 3 vw vw 3 vw 3 where vw vw v v v, v 3 w w w w 3 vw v w s, where s the gle betwee v d w Are of trgle A v w 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( AB) P( A) P( B) P( A B) Mutully eclusve evets P( AB) P( A) P( B) Mthemtcs HL d further mthemtcs formul boolet 6
5.4 Codtol probblty P( A B) P ( AB) P( B) Idepedet evets 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) PB ( ) PAB ( ) PB ( A) PB ( ) PAB ( ) PB ( ) PA ( B) PB ( ) PA ( B) 3 3 5.5 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 ) m Vr( X ) z m Mthemtcs HL d further mthemtcs formul boolet 7
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( ) sect 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 Mthemtcs HL d further mthemtcs formul boolet 8
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 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 duv v d d d or d d u v uv v u Mthemtcs HL d further mthemtcs formul boolet 9
Optos Topc 7: Sttstcs d probblty Further mthemtcs HL topc 3 7. (3.) Probblty geertg fucto for dscrete rdom vrble X X Gt () Et ( ) PX ( t ) 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 Me, wth uow vrce t s z / t s / Mthemtcs HL d further mthemtcs formul boolet 0
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 ( AB) AB ( A B) AB Topc 9: Clculus Further mthemtcs HL topc 5 9.5 (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 Mthemtcs HL d further mthemtcs formul boolet
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( )... 3 3 5 s... 3! 5! 4 cos...! 4! 3 5 rct... 3 5 Topc 0: Dscrete mthemtcs Further mthemtcs HL topc 6 0.7 (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 e 3v 6 for v 3 e v 4 f the grph hs o trgles Mthemtcs HL d further mthemtcs formul boolet
Formule for dstrbutos Topcs 5.6, 5.7, 7., further mthemtcs HL topc 3. Dscrete dstrbutos Dstrbuto Notto Probblty mss fucto Geometrc X ~Geo ( p ) pq for,,... Negtve boml X ~NB ( r, p ) r r p q r for r, r,... Me p r p Vrce q p rq p Cotuous dstrbutos Dstrbuto Notto Probblty desty fucto Me Vrce Norml X ~N (, ) e π Mthemtcs HL d further mthemtcs formul boolet 3
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 Mthemtcs HL d further mthemtcs formul boolet 4