Mathematics HL and further mathematics HL formula booklet

1 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 Edited in 2015 (version 2) 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.

2 Linear algebra 14 Mathematics HL and further Mathematics formula booklet 1 Formulae Prior learning Area of a parallelogram A bh= , where b is the base, h is the height Area of a triangle 1()2 Abh= , where b is the base, h is the height Area of a trapezium 1()2Aa bh= +, 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 13area of basevertical height( )= V Volume of a cuboid V l wh= , where l is the length, w is the width, h is the height Volume of a cylinder 2 Vrh= , 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 213 Vrh= , where r is the radius, h is the height Distance between two points11(, )xyand22(, )xy 22121 2()()d xxyy= + Coordinates of the midpoint of a line segment with endpoints 11(, )xyand22(, )xy 1 21 2, 22xxyy++ Solutions of a quadratic equation The solutions of 20axbx c+ += are 242bbacxa = Mathematics HL and further Mathematics formula booklet 2 Core Topic 1.

3 Algebra The nth term of an arithmetic sequence 1(1)=+ nuu n d The sum of n terms of an arithmetic sequence ()112(1)()22= + = +nnnnSu n duu The nth term of a geometric sequence 11nnuur = The sum of n terms of a finite geometric sequence 11(1)(1)11nnnururSrr == , 1r The sum of an infinite geometric sequence 11uSr = , 1<r Exponents and logarithms logxaab xb= =, where 0,0,1ab a>> lnexxaa= loglogaxxaa xa= = logloglogcbcaab= Combinations !!()!nnrrn r = Permutations !()!nnPrnr= Binomial theorem 1()1nnnnr rnnnab aaba bbr + = +++++ Complex numbers i(cosi sin )ecisiza brrr =+= + = = De Moivre s theorem []i(cosi sin )(cosi sin)ecisnnnnnrr nnr rn + = +== Mathematics HL and further Mathematics formula booklet 3 Topic 2: Functions and equations Axis of symmetry of the graph of a quadratic function 2()2axis of symmetry bf xaxbx cxa= + + = Discriminant 24bac = Topic 3.

4 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 sinABA BA B = cos () cos cossin sinABA BA B = tantantan ()1 tan tanABABAB = Double angle identities sin 22 sin cos = 2222cos 2cossin2 cos1 1 2 sin = = = 22 tantan 21 tan = Mathematics HL and further Mathematics formula booklet 4 Cosine rule 2 222coscababC=+ ; 2 22cos2abcCab+ = Sine rule sinsinsinabcABC== Area of a triangle 1sin2 AabC= Topic 4: Vectors Magnitude of a vector 222123vv v= ++v, where 123vvv = v Distance between two points 1 11(, ,)xyz and 2 22(, ,)xyz 2 22121 212()()()d xxyyzz= + + Coordinates of the midpoint of a line segment with endpoints 1 11(, ,)xyz, 2 22(, ,)

5 Xyz 1 2 1 21 2, , 222xxyyzz+++ Scalar product cos =vw v w, where is the angle between v and w 112 23 3vw v wvw = + +vw, where 123vvv = v, 123www = w Angle between two vectors 112 23 3cos ++=vw v wvwvw Vector equation of a line =+ ra b Parametric form of the equation of a line 00 0, , x xly ymz zn =+=+ =+ Cartesian equations of a line 0 00xx yy zzl mn == Mathematics HL and further Mathematics formula booklet 5 Vector product 233231131221vw vwvw vwvwv w = vw where 123vvv = v, 123www = w sin =vw vw, where is the angle between v and w Area of a triangle 12= Avw where v and w form two sides of a triangle Vector equation of a plane =+ r ab+ c Equation of a plane (using the normal vector) = rn an Cartesian equation of a plane ax by czd++= Topic 5: Statistics and probability Population parameters Let 1kiinf== Mean 1kiiifxn == Variance 2 ()222211kki iiiiif xfxnn == == 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 = + Mathematics HL and further Mathematics formula booklet 6 Conditional probability P()P()P( )ABABB = Independent events P() P( ) P( )ABA B = Bayes theorem P( ) P( | )P( | )P( ) P( | ) P( ) P( | )BABBABABBAB= + 1 12 23 3P( ) P( |)P( | )P( ) P( |) P() P( |) P( ) P( |)iiiBABBABABBABBAB=++ Expected value of a discrete random variable X E( )P()Xx Xx = == Expected value of a continuous random variable X E( )( )

6 DXxf x x = = Variance []222 Var( ) E()E()E( )XXX X = = Variance of a discrete random variable X 222 Var( )() P()P()XxXxx Xx = == = Variance of a continuous random variable X 222 Var( )()( )d( )dXxfx xxfx x = = Binomial distribution Mean Variance ~ B ( , )P ()(1),0 , 1 ,,xnxnXnpX xppxnx == = E( )Xnp= V a r ( )(1)Xnpp= Poisson distribution Mean Variance e~ Po( )P(),0,1, 2,!xmmXmXxxx == = E( )Xm= Var ( )Xm= Standardized normal variable xz = Mathematics HL and further Mathematics formula booklet 7 Topic 6: Calculus Derivative of ()fx 0d()()()() limdhyfx h fxy fxf xxh + = = = Derivative of nx 1()()nnf xxf xnx = = Derivative of sinx ( ) sin( ) cosfxxf xx = = Derivative of cosx ( ) cos( )sinfxxf xx = = Derivative of tanx 2( )tan( ) secfxxf xx = = Derivative of ex () e() exxfxf x = = Derivative of lnx 1() ln()fxxf xx = = Derivative of secx ( ) sec( ) sec tanfxxfxxx = = Derivative of cscx ( ) csc( )csc cotfxxf xx x = = Derivative of cotx 2( ) cot( )cscfxxf xx = = Derivative of xa ()()(ln )xxfxafxa a = = Derivative of logax 1() log()lnafxxf xxa = = Derivative of arcsinx 21( ) arcsin( )1fxxf xx = = Derivative of arccosx 21( ) arccos( )1fxxf xx = = Derivative of arctanx 21( ) arctan( )1fxxf xx = =+ Chain rule ()y gu=, where d dd()

7 Dddy yuu fxx ux= = Product rule d ddd ddy vuy uvuvx xx= = + Quotient rule 2dddddduvvuuyxxyvx v = = Mathematics HL and further Mathematics formula booklet 8 Standard integrals 1d,11nnxxxC nn+= + + 1dlnxxCx=+ sin dcosxxx C= + cos dsinxxx C=+ ed exxxC= + 1dlnxxaxa Ca=+ 2211darctanxxCaxaa =+ + 221darcsin,xxC xaaax = +< Area under a curve Volume of revolution (rotation) dbaAyx= or dbaAxy= 2 dbaVyx= or 2 dbaVxy= Integration by parts ddddddvuux uvvxxx= or ddu v uvv u= Mathematics HL and further Mathematics formula booklet 9 Options Topic 7: Statistics and probability further Mathematics HL topic 3 ( ) Probability generating function for a discrete random variable X ( ) E( )P()xxxG ttX xt=== E ( )(1)XG = ()2V a r ( )(1)(1)(1)XG GG =+ ( ) Linear combinations of two independent random variables 12,XX ()( )( )()( )( )112 2112222112 21122 EEEVarVarVaraXa XaXaXaXa XaXaX = =+ ( ) Sample statistics Mean x 1kiiifxxn== Variance 2ns 222211()kki iiiiinf x xfxsxnn== == Standard deviation ns 21()kiiinfx xsn= = Unbiased estimate of population variance 21ns 22222111()1111kki iiiiinnf x xfxnnssxnnnn== === ( ) Confidence intervals Mean, with known variance xzn Mean, with unknown variance 1nsxtn ( )

8 Test statistics Mean, with known variance /xzn = Mathematics HL and further Mathematics formula booklet 10 Mean, with unknown variance 1/nxtsn = ( ) Sample product moment correlation coefficient 1222211niiinniiiix ynx yrxnxyny= = = Test statistic for H0: = 0 221ntrr = Equation of regression line of x on y 1221()niiiniix ynx yxxyyyny== = Equation of regression line of y on x 1221()niiiniix ynx yyyxxxnx== = Topic 8: Sets, relations and groups further Mathematics HL topic 4 ( ) De Morgan s laws ()()AB A BAB A B = = Topic 9: Calculus further Mathematics HL topic 5 ( ) Euler s method 1(, )nnnnyy h fx y+ = +; 1nnx xh+= +, where h is a constant (step length) Integrating factor for ()()y Pxy Qx += ( )dePx x Mathematics HL and further Mathematics formula booklet 11 ( ) Maclaurin series 2( )(0)(0)(0)2!xf xfxff =++ + Taylor series 2()()() () ()().

9 2!xafx fax af af a = + ++ Taylor approximations (with error term ()nRx) ()()()() () () ..()()!nnnxafx fax af af a R xn = + +++ Lagrange form (1)1()()()(1) !++= +nnnfcRxx an, where c lies between a and x Maclaurin series for special functions !xxx=++ + 23ln (1)..23xxxx+= + !5!xxxx= + !4!xxx= + + Topic 10: Discrete Mathematics further Mathematics HL topic 6 ( ) Euler s formula for connected planar graphs 2ve f + =, 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 Mathematics HL and further Mathematics formula booklet 12 Formulae for distributions Topics , , , further Mathematics HL topic Discrete distributions Distribution Notation Probability mass function Mean Variance Geometric ~ Geo ( )Xp 1xpq for 1, 2.

10 X= 1p 2qp Negative binomial ~ NB ( , )Xrp 11r xrxpqr for ,1,..x rr= + rp 2rqp Continuous distributions Distribution Notation Probability density function Mean Variance Normal 2~ N( ,)X 2121e2 x 2 Mathematics HL and further Mathematics formula booklet 13 further Mathematics Topic 1: Linear algebra Determinant of a 22 matrix detabad bccd = == AAA Inverse of a 22 matrix 11,detabd badbccdc a = = AAA Determinant of a 33 matrix detab ce fd fdede fabchkg kghghk = = + AA Mathematics HL and further Mathematics formula booklet 14