DIPLOMA PROGRAMME MATHEMATICS HL FURTHER MATHEMATICS SL INFORMATION BOOKLET

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1 b DIPLOMA PROGRAMME MATHEMATICS HL FURTHER MATHEMATICS SL INFORMATION BOOKLET For use by techers d studets, durig the course d i the emitios First emitios 006 Itertiol Bcclurete Orgiztio Bueos Aires Crdiff Geev New York Sigpore

2 Diplom Progrmme Mthemtics HL Further Mthemtics SL Iformtio Booklet Itertiol Bcclurete Orgiztio, Geev, CH-8, Switzerld First published i November 004 by the Itertiol Bcclurete Orgiztio Peterso House, Mlthouse Aveue, Crdiff Gte Crdiff, Wles GB CF3 8GL UNITED KINGDOM Tel: F: Web site: Itertiol Bcclurete Orgiztio 004 The IBO is grteful for permissio to reproduce d/or trslte y copyright mteril used i this publictio. Ackowledgmets re icluded, where pproprite, d, if otified, the IBO will be plesed to rectify y errors or omissios t the erliest opportuity. IBO merchdise d publictios i its officil d workig lguges c be purchsed through the olie ctlogue t foud by selectig Publictios from the shortcuts bo. Geerl orderig queries should be directed to the sles deprtmet i Crdiff. Tel: F: E-mil: sles@ibo.org Prited i the Uited Kigdom by Atoy Rowe Ltd, Chippehm, Wiltshire. 56b

3 CONTENTS Formule Presumed kowledge Topic Core: Algebr Topic Core: Fuctios d equtios Topic 3 Core: Circulr fuctios d trigoometry 3 Topic 4 Core: Mtrices 3 Topic 5 Core: Vectors 4 Topic 6 Core: Sttistics d probbility 5 Topic 7 Core: Clculus 7 Topic 8 Optio: Sttistics d probbility (further mthemtics SL topic ) 9 Topic 9 Optio: Sets, reltios d groups (further mthemtics SL topic 3) 9 Topic 0 Optio: Series d differetil equtios (further mthemtics SL topic 4) 0 Topic Optio: Discrete mthemtics (further mthemtics SL topic 5) 0 Formule for distributios (topic 8., further mthemtics SL topic.) Discrete distributios Cotiuous distributios Sttisticl tbles 3 Are uder the stdrd orml curve (topic 6.) 3 Iverse orml probbilities (topic 6.) 4 Criticl vlues of the studet s t-distributio (topic 8.4, further mthemtics SL topic.4) 6 Criticl vlues of the χ distributio (topic 8.6, further mthemtics SL topic.6) 7

5 Formule Presumed kowledge Are of prllelogrm A= ( b h ), where b is the bse, h is the height Are of trigle Are of trpezium Are of circle A= ( b h ), where b is the bse, h is the height A= ( + b ) h, where d b re the prllel sides, h is the height A=π r, where r is the rdius Circumferece of circle C = πr, where r is the rdius Volume of pyrmid V = (re of bse verticl height) 3 Volume of cuboid V = l w h, where l is the legth, w is the width, h is the height Volume of cylider Are of the curved surfce of cylider Volume of sphere Volume of coe Distce betwee two poits (, y) d (, y ) Coordites of the midpoit of lie segmet with edpoits (, y) d (, y ) V =π r h, where r is the rdius, h is the height A= πrh, where r is the rdius, h is the height V V 4 = π 3 = π 3 3 r, where r is the rdius r h, where r is the rdius, h is the height d = ( ) + ( y y ) + y, + y Itertiol Bcclurete Orgiztio 004

6 Topic Core: Algebr. The th term of rithmetic sequece The sum of terms of rithmetic sequece The th term of geometric sequece u = u + ( ) d S = ( u + ( ) d) = ( u + u ) u = u r The sum of terms of fiite geometric sequece The sum of ifiite geometric sequece. Epoets d logrithms.3 Combitios S u( r ) u( r ) = = r r u S =, r < r = b =log b = e l log log = = log b log log b = c! = r r!( r)! c, r Biomil theorem ( b) b b b r + = r r + + i.5 Comple umbers z = + i b= r (cosθ + isi θ) = re θ = r cisθ.7 De Moivre s theorem [ ] i r(cosθ + isi θ) = r (cos θ + isi θ) = r e θ = r cis θ Topic Core: Fuctios d equtios.5 Ais of symmetry of the grph of qudrtic fuctio.6 Solutio of qudrtic equtio f( ) b c is of symmetry = + + = b b± b 4c + b + c = 0 =, 0 Discrimit = b 4c Itertiol Bcclurete Orgiztio 004

7 Topic 3 Core: Circulr fuctios d trigoometry 3. Legth of rc l = θ r, where θ is the gle mesured i rdis, r is the rdius Are of sector 3. Idetities Pythgore idetities 3.3 Compoud gle idetities θ rdius A = r, where θ is the gle mesured i rdis, r is the siθ tθ = cosθ cos θ + si θ = + t θ = sec θ + cot θ = csc θ si( A± B) = si Acos B± cos Asi B cos( A± B) = cos Acos B si Asi B t A± t B t( A± B) = tatb Double gle idetities si θ = siθcosθ cos = cos si = cos = si θ θ θ θ θ tθ t θ = t θ 3.6 Cosie rule Sie rule Are of trigle + = + cos ; cos = b c b c b b C C b c = = si A si B si C A= bsi C, where d b re djcet sides, C is the icluded gle Topic 4 Core: Mtrices 4.3 Determit of mtri Iverse of mtri Determit of 3 3 mtri A b = det = d bc c d A b d b A= A =, d bc c d d bc c b c e f d f d e A= d e f det A = b + c h k g k g h g h k Itertiol Bcclurete Orgiztio 004 3

8 Topic 5 Core: Vectors 5. Mgitude of vector v 3 = v + v + v, where v v = v v3 Distce betwee two poits (, y, z ) d (, y, z ) d = ( ) + ( y y ) + ( z z ) Coordites of the midpoit of lie segmet with edpoits (, y, z ), (, y, z ) + y, + y z, + z 5. Sclr product v w = v w cosθ, where θ is the gle betwee v d w v w = vw + vw + vw, where 3 3 v v = v, v3 w w = w w3 Agle betwee two vectors vw + vw + vw cosθ = v w Vector equtio of lie r = +λb Prmetric form of equtios of lie Crtesi equtios of lie 5.5 Vector product (Determit represettio) Are of trigle 5.6 Vector equtio of ple Equtio of ple (usig the orml vector) Crtesi equtio of ple = + λl, y = y + λm, z = z + λ y y z z = = l m i j k v w = v v v3 w w w 3 v w = v w siθ, whereθ is the gle betwee v d w A = v w r = + λb+ µ c r = + by + cz + d =0 4 Itertiol Bcclurete Orgiztio 004

9 Topic 6 Core: Sttistics d probbility 6.3 Let = k f i i= Popultio prmeters Me µ Vrice σ k f i i i= µ = k ( µ ) f f σ = = µ i i i i i = i = k Stdrd devitio σ Smple sttistics Me Vrice s k fi i i= σ = = s k i= f i i ( µ ) k k fi( i ) fii i= i= = = Stdrd devitio s s = k i= f ( ) i i 6.5 Ubised estimte of popultio vrice s Probbility of evet A k k fi( i ) fii i= i= = = = s s ( ) P( A) = A U ( ) Complemetry evets P( A) + P( A ) = 6.6 Combied evets P( A B) = P( A) + P( B) P( A B ) Mutully eclusive evets P( A B) = P( A) + P( B ) Itertiol Bcclurete Orgiztio 004 5

10 Topic 6 Core: Sttistics d probbility (cotiued) 6.7 Coditiol probbility P( AB) P( A B) = P( B) Idepedet evets P( A B) = P( A) P( B ) Byes Theorem P ( B A) P( B)P ( A B) ( ) + ( ) = P( B)P A B P( B)P A B 6.9 Epected vlue of discrete rdom vrible X Epected vlue of cotiuous rdom vrible X E( X ) = µ = P( X = ) E( X ) µ f( )d = = Vrice [ ] Vr( X ) = E( X µ ) = E( X ) E( X ) Vrice of discrete rdom vrible X Vrice of cotiuous rdom vrible X 6.0 Biomil distributio Me Vrice Poisso distributio Me Vrice 6. Stdrdized orml vrible Vr( X ) = ( µ ) P( X = ) = P( X = ) µ Vr( X ) = ( µ ) f ( )d = f ( )d µ X ~B(, p) P( X = ) = p ( p), = 0,,, E( X ) = p Vr( X ) = p( p ) m m e X ~ P o ( m) P( X = ) =, = 0,,,! E( X ) = m Vr( X ) = m µ z = σ 6 Itertiol Bcclurete Orgiztio 004

11 Topic 7 Core: Clculus 7. Derivtive of f( ) Derivtive of d y f( + h) f( ) y = f( ) = f ( ) = lim d h 0 h f ( ) = f ( ) = Derivtive of si f ( ) = si f ( ) = cos Derivtive of cos f ( ) = cos f ( ) = si Derivtive of t = = f( ) t f ( ) sec Derivtive of e f( ) = e f ( ) = e Derivtive of l f( ) = l f ( ) = Derivtive of sec f ( ) = sec f ( ) = sect Derivtive of csc f ( ) = csc f ( ) = csc cot Derivtive of cot = = f ( ) cot f ( ) csc Derivtive of = = f( ) f ( ) (l ) Derivtive of log f( ) = log f ( ) = l Derivtive of rcsi f ( ) = rcsi f ( ) = Derivtive of rccos f ( ) = rccos f ( ) = Derivtive of rct f ( ) = rct f ( ) = + 7. Chi rule y = gu ( ), where dy dy du u = f( ) = d du d Product rule Quotiet rule dy dv du y = uv = u + v d d d du dv v u u dy y = = d d v d v Itertiol Bcclurete Orgiztio 004 7

12 Topic 7 Core: Clculus (cotiued) 7.4 Stdrd itegrls + d = + C, + d = l + C sid= cos + C cosd= si + C e d = e + C d= + C l d rct = + C + d = rcsi + C, < 7.5 Are uder curve Volume of revolutio (rottio) A= b ydor A= d b y b b π d or π d V = y V = y 7.9 Itegrtio by prts dv du u d= uv d d v d or uv d = uv vu d 8 Itertiol Bcclurete Orgiztio 004

13 Topic 8 Optio: Sttistics d probbility (further mthemtics SL topic ) 8. (.) Lier combitios of two idepedet rdom vribles X, X ( X ± X ) = ( X) ± ( X) ( X ± X ) = ( X ) + ( X ) E E E Vr Vr Vr 8.4 (.4) Cofidece itervls Me, with kow vrice ± z σ 8.5 (.5) 8.6 (.6) Me, with ukow vrice Popultio Test sttistics Me, with kow vrice Me, with ukow vrice The χ test sttistic s ± t ˆ( ˆ ˆ P P) P± z, where ˆP is the proportio of successes i the smple µ z = σ / µ t = s / o e o ( f f ) f χclc = = f f e frequecies, f e re the epected frequecies, e, where f o re the observed = f o Topic 9 Optio: Sets, reltios d groups (further mthemtics SL topic 3) 9. (3.) De Morg s lws ( A B) = A B ( A B) = A B Itertiol Bcclurete Orgiztio 004 9

14 Topic 0 Optio: Series d differetil equtios (further mthemtics SL topic 4) 0.5 (4.5) Mcluri series Tylor series Tylor pproimtios (with error term R ( )) Lgrge form Itegrl form Other series f( ) = f(0) + f (0) + f (0) +! ( ) f( ) = f( ) + ( ) f ( ) + f ( ) +...! ( ) f f f f R! ( ) = ( ) + ( ) ( ) ( ) ( ) + ( ) ( + ) f () c R ( ) = ( ) ( + )! + ( + f ) () t R ( ) = ( t) dt! e = ! 3 l( + ) = si = ! 5! 4 cos = +...! 4! 3 5 rct = , where c lies betwee d 0.6 (4.6) Euler s method y+ = y + h f(, y ) ; = + + h, where h is costt Itegrtig fctor for y + Py ( ) = Q ( ) ( )d e P Topic Optio: Discrete mthemtics (further mthemtics SL topic 5).6 (5.6) Euler s reltio v e+ f =, where v is the umber of vertices, e is the umber of edges, f is the umber of fces Plr grphs e 3v 6 e v 4 0 Itertiol Bcclurete Orgiztio 004

15 Discrete distributios Formule for distributios (topic 8., further mthemtics SL topic.) Distributio Nottio Probbility mss fuctio Me Vrice Beroulli X ~B(, p ) p ( p ) for = 0, p p( p ) Biomil X ~B( p, ) p ( p ) for = 0,,..., p p( p ) Hypergeometric X ~Hyp( MN,, ) M N M N Poisso ( ) ~P o for = 0,,..., X m m e! m for = 0,... p where p = M N m N p( p) N where p = M N m Geometric X ~Geo( p ) pq for =,,... p q p Negtive biomil ~NB (, ) X r p r p q r r for = r, r +,... Discrete uiform X ~DU( ) for =,..., r p + rq p Itertiol Bcclurete Orgiztio 004

16 Cotiuous distributios Distributio Nottio Probbility desity fuctio Me Vrice Uiform X ~U ( b, ) +, b b ( ) b ( ) b Epoetil ( ) ~Ep λ Norml X ~N ( µ, σ ) X λ λ e, 0 λ e σ π µ σ µ λ σ Itertiol Bcclurete Orgiztio 004

17 Itertiol Bcclurete Orgiztio P( ) = p Z z Sttisticl tbles Are uder the stdrd orml curve (topic 6.) 0 z p z

18 Iverse orml probbilities (topic 6.) p = P( Z z) p 0 z p Itertiol Bcclurete Orgiztio 004

19 Iverse orml probbilities (topic 6., cotiued) p Itertiol Bcclurete Orgiztio 004 5

20 Criticl vlues of the studet s t-distributio (topic 8.4, further mthemtics SL topic.4) p = P( X t) p t p v = *** ν = umber of degrees of freedom 6 Itertiol Bcclurete Orgiztio 004

21 Criticl vlues of the χ -distributio (topic 8.6, further mthemtics SL topic.6) p p = P( X c ) 0 c p ν = ν = umber of degrees of freedom Itertiol Bcclurete Orgiztio 004 7

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DIPLOMA PROGRAMME MATHEMATICS SL INFORMATION BOOKLET

b DIPLOMA PROGRAMME MATHEMATICS SL INFORMATION BOOKLET For use by teachers ad studets, durig the course ad i the examiatios First examiatios 006 Iteratioal Baccalaureate Orgaizatio Bueos Aires Cardiff