Mathematics HL and further mathematics HL formula booklet
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1 Dplom Progrmme Mthemtcs HL d further mthemtcs HL formul boolet For use durg the course d the emtos Frst emtos 04 Edted 05 (verso ) Itertol Bcclurete Orgzto
2 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
3 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 ( + bh ), where d b re the prllel sdes, h s the heght Are of crcle A π r, where r s the rdus Crcumferece of crcle C π 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 π rh, where r s the rdus, h s the heght Are of the curved surfce of cylder A π rh, where r s the rdus, h s the heght Volume of sphere Volume of coe V V π r, where r s the rdus 3 π rh, where r s the rdus, h s the heght Dstce betwee two pots (, y) d (, y ) d ( ) + ( y y ) Coordtes of the mdpot of le segmet wth edpots (, y) d (, y ) +, y + y Solutos of qudrtc equto The solutos of b c re ± b b c 4 Mthemtcs HL d further mthemtcs formul boolet
4 Core Topc : Algebr. The th term of rthmetc sequece u u + ( ) d The sum of terms of S u + ( ) d ( u + u ) rthmetc sequece ( ) The th term of geometrc sequece The sum of terms of fte geometrc sequece The sum of fte geometrc sequece u S S ur 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 Permuttos! r r!( r)!! Pr ( r)! Boml theorem ( + b) + b+ + b + + b r r r.5 Comple umbers z + b r(cosθ + s θ) re θ rcsθ.7 r(cosθ + s θ) r (cos θ + s θ) r e θ r cs θ De Movre s theorem [ ] Mthemtcs HL d further mthemtcs formul boolet 3
5 Topc : Fuctos d equtos.5 As of symmetry of the grph of qudrtc fucto b + + f ( ) b c s of symmetry.6 Dscrmt b 4c Topc 3: Crculr fuctos d trgoometry 3. Legth of rc l θr, where θ s the gle mesured rds, r s the rdus Are of sector A rdus θ, where θ s the gle mesured rds, r s the r 3. Idettes sθ tθ cosθ secθ cosθ cosecθ sθ Pythgore dettes cos θ + s θ + t θ sec θ + cot θ csc θ 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 ( A± B) t At B Double gle dettes s θ sθ cosθ cos cos s cos s θ θ θ θ θ tθ t θ t θ Mthemtcs HL d further mthemtcs formul boolet 4
6 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 bs C 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 cosθ, where θ 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 cosθ v w Vector equto of le r +λb Prmetrc form of the equto of le Crtes equtos of le + λl, y y + λm, z z + λ y y z z l m Mthemtcs HL d further mthemtcs formul boolet 5
7 4.5 Vector product vw 3 vw 3 v w vw 3 vw 3 where vw vw v v v, v 3 w w w w 3 v w v w sθ, where θ 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 +λ b+ µ 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 µ µ f Vrce σ ( ) µ f f σ µ Stdrd devto σ σ f ( µ ) 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) Mthemtcs HL d further mthemtcs formul boolet 6
8 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 ) 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) µ P( X ) E( X) µ f( )d Vr( X) E( X µ ) E( X ) E( X) Vrce [ ] Vrce of dscrete rdom vrble X Vr( X) ( µ ) P( X ) P( X ) µ 5.6 Vrce of cotuous rdom vrble X Boml dstrbuto Me Vrce Vr( X) ( µ ) f( )d f( )d µ X ~ B (, p) P ( X ) p ( p), 0,,, E( X ) p Vr ( X ) p( p) Posso dstrbuto Me Vrce m m e X ~ Po( m) P( X ), 0,,,! E( X) Vr ( X) m m 5.7 Stdrdzed orml vrble µ z σ Mthemtcs HL d further mthemtcs formul boolet 7
9 Topc 6: Clculus 6. Dervtve of f( ) d y f( + h) f( ) y f( ) f ( ) lm d h 0 h 6. Dervtve of 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 gu ( ), where dy dy du u f( ) d du d Product rule Quotet rule dy dv du y uv u + v d d d du dv v u u dy y d d v d v Mthemtcs HL d further mthemtcs formul boolet 8
10 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 V πy d or V π dy b b 6.7 Itegrto by prts dv du u d uv v d d d or d d u v uv v u Mthemtcs HL d further mthemtcs formul boolet 9
11 Optos Topc 7: Sttstcs d probblty Further mthemtcs HL topc 3 7. (3.) Probblty geertg fucto for dscrete rdom vrble X Gt ( ) E( t ) P( X t ) E ( X) G () ( ) Vr ( X) G () + G () G () 7. (3.) Ler combtos of two depedet rdom vrbles X, X ( X ± X ) ( X) ± ( X) ( X ± X ) ( X ) + ( X ) E E E Vr Vr Vr 7.3 (3.3) Smple sttstcs Me f Vrce s s f( ) f 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 σ 7.6 (3.6) Me, wth uow vrce Test sttstcs Me, wth ow vrce ± t s µ z σ / Mthemtcs HL d further mthemtcs formul boolet 0
12 Me, wth uow vrce t s µ / 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 y y of o y ( y y) y y Equto of regresso le y y of y o 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 + Py ( ) Q ( ) ( )d e P Mthemtcs HL d further mthemtcs formul boolet
13 9.6 (5.6) Mclur seres Tylor seres Tylor ppromtos (wth error term R ( )) Lgrge form f( ) f(0) + f (0) + f (0) +! ( ) f( ) f( ) + ( ) f ( ) + f ( ) +...! ( ) f f f f R! ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) ( + ) f () c R ( ) ( ) ( + )! +, where c les betwee d Mclur seres for specl fuctos e ! 3 l ( + ) s ! 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 e 3v 6 for v 3 e v 4 f the grph hs o trgles Mthemtcs HL d further mthemtcs formul boolet
14 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 pq r 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( µ, σ ) e σ π µ σ µ σ Mthemtcs HL d further mthemtcs formul boolet 3
15 Further mthemtcs Topc : Ler lgebr. Determt of mtr b A det A A d bc c d Iverse of mtr Determt of 3 3 mtr 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
Mathematics HL and further mathematics HL formula booklet
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
More informationMathematics HL and further mathematics HL formula booklet
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