INTERNATIONAL BACCALAUREATE ORGANIZATION GROUP 5 MATHEMATICS FORMULAE AND STATISTICAL TABLES

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1 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 SL Thrd Edto: ebruary 00 Vald for Eamato Sessos from May 00

2 Group 5 Mathematcs ormulae ad Statstcal Tables rst publshed: August 998 Secod edto: Aprl 999 Thrd edto: ebruary 00 Iteratoal Baccalaureate Orgaato 999 Iteratoal Baccalaureate Orgaato Route des Morllos 5 8 Grad-Sacoe Geeva, SWITZERLAND

3 CONTENTS ormulae for: Pages Mathematcal Studes SL 4 Mathematcal Methods SL 7 Mathematcs HL urther Mathematcs SL Table : Area Uder the Stadard Normal Curve Table : Crtcal Values of the χ Dstrbuto 4 Table : Crtcal Values of the Studet s t-dstrbuto 5 IB Group 5 Mathematcs ormulae ad Statstcal Tables ebruary 00

4 MATHEMATICAL STUDIES SL MATHEMATICAL METHODS SL MATHEMATICS HL Plae ad Sold gures Area of a parallelogram: Area of a tragle: Area of a trapeum: Area of a crcle: Crcumferece of a crcle: Volume of a pyramd: Volume of a cubod: Volume of a cylder: Area of the curved surface of a cylder: A ( b h) A b h, 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 ( ) Aπr, where r s the radus C πr, where b s the base, h s the heght, where r s the radus V (area of base vertcal heght) V l w h, where l s the legth, w s the wdth, h s the heght V πr h, where r s the radus, h s the heght A πrh, where r s the radus, h s the heght Volume of a sphere: V 4 πr, where r s the radus Volume of a coe: V πr h, where r s the radus, h s the heght te Sequeces 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 ( u + d u + u ( ) ) ( ) u r The sum of terms of a geometrc sequece: S u( r ) u( r ) r r, r Trgoometry Se rule: Cose rule: Area of a tragle: a b c s A s B s C a b c bc A A b + c + cos ; cos a bc A absc, where a ad b are adjacet sdes, C s the cluded agle IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00

5 MATHEMATICAL STUDIES SL MATHEMATICAL METHODS SL MATHEMATICS HL 4 Geometry Dstace betwee two pots (, y ) ad (, y ): Coordates of the mdpot of a le segmet wth edpots (, y ) ad (, y ): Magtude of a vector: d ( ) + ( y y ) HG + y y, + + H G v I K J v v v, where v I K J v 5 acal Mathematcs Smple terest: Cr I, where Cs the captal, r % s the terest rate, s the umber of 00 Compoud terest: 6 Matrces ( ) Determat: Traspose: 7 Probablty HG tme perods, I s the terest I K J r I C + C, where Cs the captal, r % s the terest rate, 00 A A H G a bi K J det ad bc c d A A H G a bi K J H G T a ci K J c d b d umber of tme perods, I s the terest s the Probablty of a evet A: Complemetary evets: Combed evets: Mutually eclusve evets: Idepedet evets: ( A) P( A) U ( ) P( A ) P( A) P( A B) P( A) + P( B) P( A B) P( A B) P( A) + P( B) P( A B) P( A) P( B) Codtoal probablty: P( A B) PcABh P( B) IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00

6 MATHEMATICAL STUDIES SL MATHEMATICAL METHODS SL MATHEMATICS HL 8 Statstcs Populato mea: f µ,where f Populato stadard devato: σ fb µ g,where f Sample mea: f,where f Stadard devato of the sample: s f ( ), where f Stadarded ormal varable: Covarace: s µ σ y ( )( y y ) Product momet correlato coeffcet: s y r, where s ss y b g, s y by yg Regresso le for y o : sy y y s b g The χ test statstc: f f ( ), where fe are the epected frequeces, f e o χ calc e f o are the observed frequeces IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00

7 MATHEMATICAL STUDIES SL MATHEMATICAL METHODS SL MATHEMATICS HL 9 Dfferetal Calculus Dervatve of f ( ) : Dervatve of a : Dervatve of a polyomal: y f + h f y f( ) d f ( ) lm ( ) ( ) d h 0 h f ( ) a f ( ) a HG f ( ) a + b + f ( ) a + ( ) b + I K J At-dfferetato: dy d + y + C, + IB Group 5 Mathematcs Page 4 ormulae ad Statstcal Tables ebruary 00

8 MATHEMATICAL METHODS SL MATHEMATICS HL 0 Ifte Sequeces The sum of a fte geometrc sequece: u S, r < r Algebra Soluto of a quadratc equato: Epoets ad logarthms: Bomal theorem: b b 4ac a + b + c 0 ±, a 0 a a b log a b a e l a log log a b ( a b) a log a a a (log c a) a (log b) c a b r a b b r r + + H G I K J + + H G I K J + + Trgoometry Legth of a arc: Area of a sector: Idettes: Vectors Scalar product: l qr, where q s the agle measured radas, r s the radus A qr, where qs the agle measured radas, r s the radus s θ + cos θ sθ taθ cosθ s θ sθcosθ cosθ cos θ s θ cos θ s θ v v w v w cos θ vw vw, where v, w v + H G I K J H G wi K J w vw + vw cosθ v cosθ vw H G w I v K J w Vector equato of a le: r p+td IB Group 5 Mathematcs Page 5 ormulae ad Statstcal Tables ebruary 00

9 MATHEMATICAL METHODS SL MATHEMATICS HL 4 Matrces ( ) Iverse: Trasformato matr represetg a rotato through θ about the org: Trasformato matr represetg a reflecto y ta θ : 5 Statstcs P P H G a bi K J c d ad bc R M HG HG cosθ sθ sθ cosθ I KJ cosθ s θ s θ cosθ I KJ HG d c b a I KJ Stadard error of the mea: SE σ Test statstc for the mea of a ormal populato: µ σ / 6 Dfferetato Dervatve of s : Dervatve of cos : Dervatve of e : Dervatve of l : Dervatve of a : Dervatve of log a : Dervatve of ta : Cha rule: Product rule: Quotet rule: f( ) s f ( ) cos f( ) cos f ( ) s f( ) e f ( ) e f( ) l f ( ) f( ) a f ( ) a (l a) f( ) log a f ( ) l a f( ) ta f ( ) cos dy dy du y g( u), where u f( ) d du d y y uv d u v d +v d u d d d d v u d u v u dy y d d v d v IB Group 5 Mathematcs Page 6 ormulae ad Statstcal Tables ebruary 00

10 MATHEMATICAL METHODS SL MATHEMATICS HL 7 Itegrato Stadard tegrals: d l + C sd cos + C cosd s + C 8 Iterato e d e + C + ( a + b) ( a + b) d a ( + ) + C, Newto Raphso method: f + ( ) f ( ) 9 Appromate Itegrato b h Trapeum rule: fbg d y 0 + y + y + + y + y a b a where h ; y fba+ hg, 0,,,, IB Group 5 Mathematcs Page 7 ormulae ad Statstcal Tables ebruary 00

11 MATHEMATICS HL 0 Combatos HG I rk J! r!( r)! Seres The sum of the frst tegers: The sum of the squares of the frst tegers: The sum of the cubes of the frst tegers: ( + ) ( + )( + ) 6 ( +) 4 Comple Numbers a+ b r(cosθ + s θ ) De Movre s theorem: r(cosθ + s θ) r (cos θ + s θ) Trgoometry Idettes: s( A± B) s Acos B± cos As B cos( A± B) cos Acos B s As B ta A± ta B ta( A± B) ta Ata B taθ ta θ ta θ + ta θ sec θ + cot θ csc θ θ s ± θ cos ± θ ta ± cosθ + cosθ cosθ + cosθ Compoud formula: b acos ± bs Rcos( α), where R a + b, taα a IB Group 5 Mathematcs Page 8 ormulae ad Statstcal Tables ebruary 00

12 MATHEMATICS HL 4 Vector Geometry Magtude of a vector: Scalar product: Vector product: v v + v + v, where v v v w vw + vw + vw v w G, where v, J v cosθ vw + vw + vw vw j v w v v v w w w H G v v v I K J H I K H G w w w I K J v w v w sθ Area of a tragle: Vector equato of a le: Vector equato of a plae: Equato of a plae (usg the ormal vector): Cartesa equato of a le: Cartesa equato of a plae: A v w r a+λb r a+ λ b+ µ c r a l y y m a + by + c + d 0 5 Matrces ( ) Determat: A H G I a b c d e f A J det a e f b d f h + c d g g g h K e h IB Group 5 Mathematcs Page 9 ormulae ad Statstcal Tables ebruary 00

13 MATHEMATICS HL 6 Dfferetato Dervatve of sec : Dervatve of csc : Dervatve of cot : f ( ) sec f ( ) secta f ( ) csc f ( ) csc cot f ( ) cot f ( ) csc Dervatve of arcs : f ( ) arcs f ( ) Dervatve of arccos : f ( ) arccos f ( ) Dervatve of arcta : f ( ) arcta f ( ) + 7 Itegrato Itegrato by parts: Stadard tegrals: dv du u d uv v d d d d arcta a + a a H G I K J + C a d arcs C, a a H G I K J + < 8 Appromate Itegrato Trapeum rule (cludg error term): d l + C b a L M O P h ( b a) h f( ) d y + y + y f ( c ) NM QP 0 b a where h ; y f( a+ h), 0,,,, ; c a, b Smpso s rule, for eve (cludg error term): b 4 h ( b a) h f y y y y y y f ( 4) bgd ( c ) a 80 b a where h ; y fba+ hg, 0,,,, ; c a, b IB Group 5 Mathematcs Page 0 ormulae ad Statstcal Tables ebruary 00

14 MATHEMATICS HL 9 Probablty Epected value of a dscrete radom varable X: Epected value of a cotuous radom varable X: Varace: E( X ) µ P( X ) E( X) µ f ( ) d Var( X) E( X ) E( X ) E( X) µ Posso dstrbuto: Bomal dstrbuto: Bayes Theorem: µ e X ~ P ( µ ) P( X r), r 0,,, r! r r X ~ B (, p) P( X r) p ( p), r,,, H G I rk J 0 c P AB h c h P B A P( A) P( B) r µ 0 Statstcs Lear combatos of two radom varables X, X : Lear combatos of two depedet radom varables X, X : ( ax ± ax ) a ( X ) ± a ( X ) E E E ( ax ± ax ) a ( X ) + a ( X ) Var Var Var Ubased estmate of the populato varace: Pooled estmate of the populato mea for two samples of se ad m: s s where + m + m f( ) m f, Pooled estmate of the populato varace for two samples of se ad m: s + m m m s + ms + m ( ) s + ( m ) s + m Test statstc for the mea of a ormal populato of uow varace: t s µ µ / s / IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00

15 MATHEMATICS HL Seres ad Appromato Maclaur seres: Taylor seres: Taylor appromatos (cludg error term): f ( ) f ( 0 ) + f ( 0 ) + f ( 0) +! f ( a+ ) f ( a ) + f ( a ) + f ( a ) +! f a f a f a f ( ) a ( + ) ( + ) ( ) + ( ) + + ( ) + f ( c)! ( + )! where c s betwee a ad a +, (ecludg edpots). + Set Theory De Morga s Laws: ( A B) A B ( A B) A B Graph Theory Euler s relato: v e+ f, where v s the umber of vertces, e s the umber of edges, f s the umber of faces IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00

16 p Z P( ) IB Group 5 Mathematcs Page ormulae ad Statstcal Tables ebruary 00 0 p TABLE : AREA UNDER THE STANDARD NORMAL CURVE

17 p X c P( ) ν p ν umber of degrees of freedom IB Group 5 Mathematcs Page 4 ormulae ad Statstcal Tables ebruary 00 0 c p TABLE : CRITICAL VALUES O THE χ DISTRIBUTION

18 TABLE : CRITICAL VALUES O THE STUDENT'S t-distribution p P( X t) p t p ν *** ν umber of degrees of freedom IB Group 5 Mathematcs Page 5 ormulae ad Statstcal Tables ebruary 00

Mathematics HL and Further mathematics HL Formula booklet

Mathematics HL and Further mathematics HL Formula booklet 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