Mathematics HL and further mathematics HL …

1 Published June 2012 International Baccalaureate Organization 2012 5048 Mathematics HL and further Mathematics HL formula booklet For use during the course and in the examinations First examinations 2014 Diploma Programme Contents Prior learning 2 Core 3 Topic 1: Algebra 3 Topic 2: Functions and equations 4 Topic 3: Circular functions and trigonometry 4 Topic 4: Vectors 5 Topic 5: Statistics and probability 6 Topic 6: Calculus 8 Options 10 Topic 7: Statistics and probability 10 further Mathematics HL topic 3 Topic 8: Sets, relations and groups 11 further Mathematics HL topic 4 Topic 9: Calculus 11 further Mathematics HL topic 5 Topic 10: Discrete Mathematics 12 further Mathematics HL topic 6 Formulae for distributions 13 Topics , , , further Mathematics HL topic Discrete distributions 13 Continuous distributions 13 further Mathematics 14 Topic 1: Linear algebra 14 Formulae Prior learning Area of a parallelogram A b h , where b is the base, h is the height Area of a triangle 1()2Ab h , where b is the base, h is the height Area of a trapezium 1()2Aa b h , where a and b are the parallel sides, h is the height Area of a circle 2Ar , where r is the radius Circumference of a circle 2Cr , where r is the radius Volume of a pyramid 1(area of base vertical height)3V Volume of a cuboid Vl w h , where l is the length, w is the width, h is the height Volume of a cylinder 2Vr h , where r is the radius, h is the height Area of the curved surface of a cylinder 2 Arh , where r is the radius, h is the height Volume of a sphere 343Vr , where r is the radius Volume of a cone 213Vr h , where r is the radius, h is the height Distance between two points11( , )xyand22( ,)xy 221212()()

2 Dxxyy Coordinates of the midpoint of a line segment with endpoints 11( , )xyand22( ,)xy 1212, 22xxyy Solutions of a quadratic equation The solutions of 20axbx c are 242bbacxa Core Topic 1: Algebra The nth term of an arithmetic sequence 1(1)nuund The sum of n terms of an arithmetic sequence 11(2(1) )()22nnnnSunduu The nth term of a geometric sequence 11nnuu r The sum of n terms of a finite geometric sequence 11(1)(1)11nnnu rurSrr , 1r The sum of an infinite geometric sequence 11uSr , 1r Exponents and logarithms logxaabxb , where 0,0,1aba lnexx aa loglogaxxaax a logloglogcbcaab Combinations !!()!nnrr n r Permutations !()!nnPrnr Binomial theorem 1()1nnnn rrnnna baa ba bbr Complex numbers i(cosi sin )ecisiz ab rrr De Moivre s theorem (cosi sin )(cosi sin)ecisnnn innrrnnrrn Topic 2: Functions and equations Axis of symmetry of the graph of a quadratic function 2( )axis of symmetry 2bf xaxbx cxa Discriminant 24bac Topic 3.

3 Circular functions and trigonometry Length of an arc lr , where is the angle measured in radians, r is the radius Area of a sector 212Ar , where is the angle measured in radians, r is the radius Identities sintancos 1seccos 1cosecsin Pythagorean identities 222222cossin11 tansec1 cotcsc Compound angle identities sin() sin coscos sinA BABAB cos() cos cossin sinA BABAB tantantan()1 tan tanABABAB Double angle identities sin 22sin cos 2222cos 2cossin2cos1 1 2sin 22 tantan 21 tan Cosine rule 2222coscababC ; 222cos2abcCab Sine rule sinsinsinabcABC Area of a triangle 1sin2 AabC Topic 4: Vectors Magnitude of a vector 222123vvv v, where 123vvv v Distance between two points 111( , , )x y z and 222( , , )x y z 222121212()()()dxxyyzz Coordinates of the midpoint of a line segment with endpoints 111( , , )x y z, 222( , , )

4 X y z 121212, , 222xxyyzz Scalar product cos vwvw, where is the angle between v and w 1 12233v wv wv w vw, where 123vvv v, 123www w Angle between two vectors 1 12233cosv wv wv w vw Vector equation of a line =+ rab Parametric form of the equation of a line 000, , x xl yym z zn Cartesian equations of a line 000x xyyz zlmn Vector product 233231131221v wv wv wv wv wv w vw where 123vvv v, 123www w sin vwvw, where is the angle between v and w Area of a triangle 12A vw where v and w form two sides of a triangle Vector equation of a plane =+ rab+c Equation of a plane (using the normal vector) rnan Cartesian equation of a plane ax by cz d Topic 5: Statistics and probability Population parameters Let 1kiinf Mean 1kiiifxn Variance 2 222211kkiii iiif xf xnn Standard deviation 21kiiifxn Probability of an event A ()P( )()nAAnU Complementary events P( ) P( ) 1AA Combined events P() P( ) P( ) P()ABABAB Mutually exclusive events P() P( ) P( )ABAB Conditional probability P()P ()P( )ABABB Independent events P() P( ) P( )ABAB Bayes theorem P( )P ( | )P ( | )P( )P ( | ) P( )P ( |)BA BBABA BBA B 112233( ) ()(| )( ) ( |)() ( |)( ) ( |)iiiP B P A BP B AP B P A BP B P A BP B P A B Expected value of a discrete random variable X E( )P()XxXx Expected value of a continuous random variable X E( )( ) dXx f x x Variance 222 Var( ) E()E()E( )

5 XXXX Variance of a discrete random variable X 222 Var( )() P()P()XxXxxXx Variance of a continuous random variable X 222 Var( )()( ) d( ) dXxf x xx f x x Binomial distribution Mean Variance ~ B( , )P()(1),0, 1,,xn xnXn pXxppxnx E( )Xnp Var( )(1)Xnpp Poisson distribution Mean Variance e~ Po( )P(),0, 1, 2,!xmmXmXxxx E( )Xm Var( )Xm Standardized normal variable xz Topic 6: Calculus Derivative of ()fx 0d()( )( )( ) limdhyf x hf xyf xf xxh Derivative of nx 1( )( )nnf xxf xnx Derivative of sinx ( ) sin( ) cosf xxf xx Derivative of cosx ( ) cos( )sinf xxf xx Derivative of tanx 2( ) tan( ) secf xxf xx Derivative of ex ( ) e( ) exxf xf x Derivative of lnx 1( ) ln( )f xxf xx Derivative of secx ( ) sec( ) sec tanf xxf xxx Derivative of cscx ( ) csc( )csc cotf xxf xxx Derivative of cotx 2( ) cot( )cscf xxf xx Derivative of xa ( )( )(ln )xxf xaf xaa Derivative of logax 1( ) log( )lnaf xxf xxa Derivative of arcsinx 21( ) arcsin( )1f xxf xx Derivative of arccosx 21( ) arccos( )1f xxf xx Derivative of arctanx 21( ) arctan( )1f xxf xx Chain rule ()yg u , where ddd()

6 Dddyyuuf xxux Product rule ddddddyvuy uvuvxxx Quotient rule 2dddddduvvuuyxxyvxv Standard integrals 1d,11nnxxxCnn 1dlnxxCx sin dcosx xx C cos dsinx xx C e dexxxC 1dlnxxaxaCa 2211darctanxxCaxaa 221darcsin,xxCxaaax Area under a curve Volume of revolution (rotation) dbaAy x or dbaAx y 22 d or dbbaaVy xVx y Integration by parts ddddddvuux uvvxxx or ddu v uvv u Options Topic 7: Statistics and probability further Mathematics HL topic 3 ( ) Probability generating function for a discrete random variable X 2( ) E( )P()E( )(1)Var( )(1)(1)(1)xxxG ttXx tXGXGGG ( ) Linear combinations of two independent random variables 12,XX 112211222211221122E ()E ( )E ()Var ()Var ( )Var ()a Xa XaXaXa Xa XaXaX ( ) Sample statistics Mean x 1kiiifxxn Variance 2ns 222211()kkiii iiinf xxf xsxnn Standard deviation ns 21()kiiinf xxsn Unbiased estimate of population variance 21ns 22222111()1111kkiii iiinnf xxf xnnssxnnnn ( ) Confidence intervals Mean, with known variance xzn Mean, with unknown variance 1nsxtn ( ) Test statistics Mean, with known variance /xzn Mean, with unknown variance 1/nxtsn ( ) Sample product moment correlation coefficient 1222211niiinniiiix yn x yrxn xyn y Test statistic for H0.

7 = 0 221ntrr Equation of regression line of x on y 1221()niiiniix yn x yx xyyyn y Equation of regression line of y on x 1221()niiiniix yn x yyyx xxn x Topic 8: Sets, relations and groups further Mathematics HL topic 4 ( ) De Morgan s laws ()()ABABABAB Topic 9: Calculus further Mathematics HL topic 5 ( ) Euler s method 1( ,)nnnnyyh f x y ; 1nnxxh , where h is a constant (step length) Integrating factor for ( )( )yP x y Q x ( ) deP x x ( ) Maclaurin series 2( )(0)(0)(0)2!xf xfx ff Taylor series 2()( )( ) () ( )( ) ..2!xaf xf ax a f af a Taylor approximations (with error term ()nRx) ()()( )( ) () ( ) ..( )( )!nnnxaf xf ax a f afaR xn Lagrange form (1)1()( )()(1)! nnnfcR xx an, where c lies between a and x Maclaurin series for special functions !xxx 23ln(1).

8 23xxxx !5!xxxx !4!xxx Topic 10: Discrete Mathematics further Mathematics HL topic 6 ( ) Euler s formula for connected planar graphs 2v ef , where v is the number of vertices, e is the number of edges, f is the number of faces Planar, simple, connected graphs 36 ev for 3v 24 ev if the graph has no triangles Formulae for distributions Topics , , , further Mathematics HL topic Discrete distributions Distribution Notation Probability mass function Mean Variance Geometric ~ Geo ( )Xp 1xpq for 1, 2,..x 1p 2qp Negative binomial ~ NB ( , )Xr p 11rx rxpqr for ,1,..x r r rp 2rqp Continuous distributions Distribution Notation Probability density function Mean Variance Normal 2~ N ( ,)X 2121e2 x 2 further Mathematics Topic 1: Linear algebra Determinant of a 22 matrix detabad bccd AAA Inverse of a 22 matrix 11,deta bdbadbcc dca AAA Determinant of a 33 matrix deta bcefdfdedefabchkgkg hg hk AA