Mathematics HL and Further mathematics HL Formula booklet

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1 Dploma Programme Mathematcs HL ad Further mathematcs HL Formula booklet For use durg the course ad the eamatos Frst eamatos 04 Mathematcal Iteratoal Baccalaureate studes SL: Formula Orgazato booklet 0

2 CONTENTS Formulae Pror learg Topc Core: Algebra Topc Core: Fuctos ad equatos 3 Topc 3 Core: Crcular fuctos ad trgoometry 4 Topc 4 Core: Vectors 5 Topc 5 Core: Statstcs ad probablty 7 Topc 6 Core: Calculus 9 Topc 7 Opto: Statstcs ad probablty (further mathematcs HL topc 3) Topc 8 Opto: Sets, relatos ad groups (further mathematcs HL topc 4) 3 Topc 9 Opto: Calculus (further mathematcs HL topc 5) 4 Topc 0 Opto: Dscrete mathematcs (further mathematcs HL topc 6) 5 Formulae for dstrbutos (topc 5.6, 5.7, 7., further mathematcs HL topc 3.) 6 Dscrete dstrbutos 6 Cotuous dstrbutos 7 Further Mathematcs Topc Lear algebra 8 Iteratoal Baccalaureate Orgazato 0

3 Mathematcal studes SL: Formula booklet 3

5 Formulae Pror learg Area of a parallelogram A b h, where b s the base, h s the heght Area of a tragle Area of a trapezum Area of a crcle A ( bh ), where b s the base, h s the heght A ( a b ) h, where a ad b are the parallel sdes, h s the heght A r, where r s the radus Crcumferece of a crcle C r, where r s the radus Volume of a pyramd V (area of base vertcal heght) 3 Volume of a cubod V l w h, where l s the legth, w s the wdth, h s the heght Volume of a cylder V r h, where r s the radus, h s the heght Area of the curved surface of a cylder Volume of a sphere A rh, where r s the radus, h s the heght 4 V 3 3 r, where r s the radus Volume of a coe V 3 r h, where r s the radus, h s the heght Dstace betwee two pots (, y) ad (, y ) Coordates of the mdpot of a le segmet wth edpots (, y ) ad (, y ) d ( ) ( y y ) y y, Solutos of a quadratc equato The solutos of a b c 0 are b b 4ac a Iteratoal Baccalaureate Orgazato 004

6 Topc Core: Algebra. The th term of a arthmetc sequece The sum of terms of a arthmetc sequece The th term of a geometrc sequece u u ( ) d S ( u ( ) d) ( u u ) u u r The sum of terms of a fte geometrc sequece The sum of a fte geometrc sequece S u( r ) u( r ) rr u S, r r, r. Epoets ad logarthms.3 Combatos a b log b, where a0, b 0 a e l a loga log a a a logc a logb a log b! r r!( r)! c a Permutatos Pr! ( r)! Bomal theorem ( a b) a a b a b b r r r.5 Comple umbers z a b r (cos s ) re r cs.7 De Movre s theorem r(cos s ) r (cos s ) r e r cs Iteratoal Baccalaureate Orgazato 004

7 Topc Core: Fuctos ad equatos.5 As of symmetry of the graph of a quadratc fucto y a b c as of symmetry b a.6 Dscrmat b 4ac Iteratoal Baccalaureate Orgazato 004 3

8 Topc 3 Core: Crcular fuctos ad trgoometry 3. Legth of a arc l r, where s the agle measured radas, r s the radus Area of a sector 3. Idettes Pythagorea dettes 3.3 Compoud agle dettes radus A r, where s the agle measured radas, r s the s ta cos sec cos cosec = s cos s ta sec cot csc s( A B) s Acos B cos As B cos( A B) cos Acos B s As B ta A ta B ta( AB) ta Ata B Double agle dettes s s cos cos cos s cos s ta ta ta 3.7 Cose rule Se rule Area of a tragle a b c ab c a b abcos C; cosc a b c s A s B sc A absc 4 Iteratoal Baccalaureate Orgazato 004

9 Topc 4 Core: Vectors 4. Magtude of a vector v 3 v v v, where v v v v3 Dstace betwee two pots (, y, z ) ad (, y, z ) d ( ) ( y y ) ( z z ) Coordates of the mdpot of a le segmet wth edpots (, y, z ), (, y, z ) y, y z, z 4. Scalar product v w v w cos, where s the agle betwee v ad w v w v w v w v w, where 3 3 v v v, v3 w w w w3 Agle betwee two vectors cos v w v w v w vw Vector equato of a le r = a +λb Parametrc form of the equato of a le Cartesa equatos of a le l, y y m, z z y y z z l m Vector product vw3 v3w v w v3w vw3 where vw vw v v v, v3 w w w w3 v w v w s, where s the agle betwee v ad w Area of a tragle 4.6 Vector equato of a plae Equato of a plae (usg the ormal vector) A vw where v ad w form two sdes of a tragle r = a + λb + c r a Cartesa equato of a a by cz d Iteratoal Baccalaureate Orgazato 004 5

10 plae 6 Iteratoal Baccalaureate Orgazato 004

11 Topc 5 Core: Statstcs ad probablty 5. Let k f Populato parameters Mea Varace k f k f f k Stadard devato k f 5. Probablty of a evet A ( ) P( A) A U ( ) Complemetary evets P( A) P( A ) 5.3 Combed evets P( A B) P( A) P( B) P( A B ) Mutually eclusve evets P( A B) P( A) P( B ) Iteratoal Baccalaureate Orgazato 004 7

12 Topc 5 Core: Statstcs ad probablty (cotued) 5.4 Codtoal probablty P AB P( A B) P( B) Idepedet evets P( A B) P( A) P( B ) Bayes Theorem P B A P( B)P A B P( B)P A B P( B )P A B P( A B) P( B) P( B A) P( A B ) P( B ) P( A B ) P( B ) P( A B ) P( B ) 5.5 Epected value of a dscrete radom varable X Epected value of a cotuous radom varable X E( X ) P( X ) E( X ) f ( )d Varace Var( X ) E( X ) E( X ) E( X ) Varace of a dscrete radom varable X Varace of a cotuous radom varable X 5.6 Bomal dstrbuto Mea Varace Posso dstrbuto Mea Varace 5.7 Stadardzed ormal varable Var( X ) ( ) P( X ) P( X ) Var( X ) ( ) f ( )d f ( )d ~ B(, ) P( ) ( ) X p X p p, 0,,, E( X ) p Var( X ) p( p ) m m e X ~ P o ( m) P( X ), 0,,,! E( X) m Var( X) m z 8 Iteratoal Baccalaureate Orgazato 004

13 Topc 6 Core: Calculus 6. Dervatve of f( ) Dervatve of d y f ( h) f ( ) y f ( ) f ( ) lm d h0 h f ( ) f ( ) Dervatve of s f ( ) s f ( ) cos Dervatve of cos f ( ) cos f ( ) s Dervatve of ta f ( ) ta f ( ) sec Dervatve of e f ( ) e f ( ) e Dervatve of l f ( ) l f ( ) Dervatve of sec f ( ) sec f ( ) sec ta Dervatve of csc f ( ) csc f ( ) csc cot Dervatve of cot f ( ) cot f ( ) csc Dervatve of a ( ) f a f ( ) a (l a ) Dervatve of log a f ( ) log a f ( ) l a Dervatve of arcs f ( ) arcs f ( ) Dervatve of arccos f ( ) arccos f ( ) Dervatve of arcta f ( ) arcta f ( ) Cha rule y g( u ), 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 Iteratoal Baccalaureate Orgazato 004 9

14 Topc 6 Core: Calculus (cotued) 6.4 Stadard tegrals d C, d l C sd cos C cos d s C e de C a d a C l a d arcta C a a a a d arcs C, a a 6.5 Area uder a curve Volume of revoluto (rotato) A b yd or A d a b y a b b π d or π d a a V y V y 6.7 Itegrato by parts dv du u d uv d d v d or udv uv vdu 0 Iteratoal Baccalaureate Orgazato 004

15 Topc 7 Opto: Statstcs ad probablty (further mathematcs HL topc 3) 7. (3.) Probablty geeratg fucto for a dscrete radom varable X G( t) E( t X ) P( X ) t 7. (3.) Lear combatos of two depedet radom varables X, X a X a X a X a X a X a X a X a X E E E Var Var Var 7.3 (3.3) Sample statstcs Mea Varace s s k f k k f ( ) f Stadard devato s s k f ( ) Ubased estmate of populato varace s k k f ( ) f s s 7.5 (3.5) Cofdece tervals Mea, wth kow varace z Mea, wth ukow varace s t 7.6 (3.6) Test statstcs Mea, wth kow varace Mea, wth ukow z / Iteratoal Baccalaureate Orgazato 004

16 Iteratoal Baccalaureate Orgazato 004 varace / t s 7.7 Sample product momet correlato coeffcet Test statstc for H 0 : ρ = 0 Equato of regresso le of o y Equato of regresso le of y o y y y y r r r t ) ( y y y y y y ) ( y y y y

17 Topc 8 Opto: Sets, relatos ad groups (further mathematcs HL topc 4) 8. (4.) De Morga s laws ( A B) A B ( A B) A B Iteratoal Baccalaureate Orgazato 004 3

18 Topc 9 Opto: Calculus (further mathematcs HL topc 5) 9.6 (5.6) Maclaur seres Taylor seres Taylor appromatos (wth error term R () ) Lagrage form f ( ) f (0) f (0) f (0)! ( a) f ( ) f ( a) ( a) f ( a) f ( a )...! ( a) f f a a f a f a R! ( ) ( ) ( ) ( ) ( )... ( ) ( ) ( ) f () c R ( ) ( a) ( )!, where c les betwee a ad Maclaur seres for specal fuctos e...! 3 l( ) s... 3! 5! 4 cos...! 4! 3 5 arcta (5.5) Euler s method y y h f (, y ) ; h, where h s a costat (steplegth) Itegratg factor for yp( ) y Q( ) ( )d e P 4 Iteratoal Baccalaureate Orgazato 004

19 Topc 0 Opto: Dscrete mathematcs (further mathematcs HL topc 6) 0.7 (6.7) Euler s formula for coected plaar graphs Plaar, smple, coected graphs v e f, where v s the umber of vertces, e s the umber of edges, f s the umber of faces e3v6for v 3 ev4f the graph has o tragles Iteratoal Baccalaureate Orgazato 004 5

20 Formulae for dstrbutos (topc 5.6, 5.7, 7., further mathematcs HL topc 3.) Dscrete dstrbutos Dstrbuto Notato Probablty mass fucto Mea Varace Geometrc X ~ Geo p pq for,,... p q p Negatve bomal ~ NB, X r p r pq r r for r, r,... r p rq p 6 Iteratoal Baccalaureate Orgazato 004

21 Cotuous dstrbutos Dstrbuto Notato Probablty desty fucto Mea Varace Normal X ~ N, e π Iteratoal Baccalaureate Orgazato 004 7

22 Further Mathematcs Topc Lear algebra. Determat of a matr Iverse of a matr Determat of a 3 3 matr a b A det A A ad bc c d a b d b A A, ad bc c d det A c a a b c e f d f d e A d e f det A a b c h k g k g h g h k 8 Iteratoal Baccalaureate Orgazato 004

23 Iteratoal Baccalaureate Orgazato 004 9

INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS FORMULAE AND STATISTICAL TABLES

INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS ORMULAE AND STATISTICAL TABLES To be used the teachg ad eamato of: Mathematcs HL Mathematcal Methods SL Mathematcal Studes SL urther Mathematcs