Mathematics: analysis and approaches formula booklet

1 International Baccalaureate Organization 2019 Diploma Programme Mathematics: analysis and approaches formula booklet For use during the course and in the examinations First examinations 2021 Version Contents Prior learning SL and HL 2 Topic 1: Number and algebra SL and HL 3 HL only 4 Topic 2: Functions SL and HL 5 HL only 5 Topic 3: Geometry and trigonometry SL and HL 6 HL only 7 Topic 4: Statistics and probability SL and HL 9 HL only 10 Topic 5: Calculus SL and HL 11 HL only 12 Mathematics: analysis and approaches formula booklet 2 Prior learning SL and HL 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 trapezoid 1()

2 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 cuboid V lwh=, 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 Volume of a prism =VAh, where A is the area of cross-section, h is the height Area of the curved surface of a cylinder 2 Arh= , 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++ Mathematics: analysis and approaches formula booklet 3 Topic 1: Number and algebra SL and HL SL The nth term of an arithmetic sequence 1(1)=+ nuu n d The sum of n terms of an arithmetic sequence ()112(1).

3 ()22nnnnnSu n d Suu= + = + SL The nth term of a geometric sequence 11nnuur = The sum of n terms of a finite geometric sequence 11(1)(1)11nnnururSrr == , 1r SL Compound interest 1100knrFVPVk = + , where FV is the future value, PV is the present value, n is the number of years, k is the number of compounding periods per year, r% is the nominal annual rate of interest SL Exponents and logarithms logxaab xb= =, where 0,0,1ab a>> SL Exponents and logarithms loglogloga aaxyxy=+ loglogloga aaxxyy= loglogmaaxm x= logloglogbabxxa= SL The sum of an infinite geometric sequence 1,11uSrr =< SL Binomial theorem 1()CC1nnnnr rnnnab aaba bbr + = +++++.

4 !C!()!nnrrn r= Mathematics: analysis and approaches formula booklet 4 Topic 1: Number and algebra HL only AHL Combinations !C!()!nnrrn r= Permutations !P()!nnrnr= AHL Complex numbers iz ab= + AHL Modulus-argument (polar) and exponential (Euler) form i(cosi sin )eciszrr r = +== AHL De Moivre s theorem []i(cosi sin )(cosi sin)ecisnnnnnrr nnr rn + = +== Mathematics: analysis and approaches formula booklet 5 Topic 2: Functions SL and HL SL Equations of a straight line y mxc= +; 0axbyd+ +=; ()11y y mx x = Gradient formula 2121 = yymxx SL Axis of symmetry of the graph of a quadratic function 2()2bf xaxbx cxa= + + = axis of symmetry is SL Solutions of a quadratic equation Discriminant 2240,02bbacaxbx cxaa + += = 24bac = SL Exponential and logarithmic functions lnexxaa=; loglogaxxaa xa= = where ,0,1axa> Topic 2: Functions HL only AHL Sum and product of the roots of polynomial equations of the form 00nrrrax== Sum is 1nnaa.

5 Product is ( )01nnaa Mathematics: analysis and approaches formula booklet 6 Topic 3: Geometry and trigonometry SL and HL SL 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 and 2 22(, ,)xyz 1 2 1 21 2, , 222+++ x xy yz z Volume of a right-pyramid 13 VAh=, where A is the area of the base, h is the height Volume of a right cone 213 Vrh= , where r is the radius, h is the height Area of the curved surface of a cone = Arl, where r is the radius, l is the slant height Volume of a sphere 343Vr= , where r is the radius Surface area of a sphere 24 =Ar, where r is the radius SL Sine rule sinsinsinabcABC== Cosine rule 2 222coscababC=+.

6 2 22cos2abcCab+ = Area of a triangle 1sin2 AabC= SL Length of an arc lr =, where r is the radius, is the angle measured in radians Area of a sector 212Ar =, where r is the radius, is the angle measured in radians Mathematics: analysis and approaches formula booklet 7 SL Identity for tan sintancos = SL Pythagorean identity 22cossin1 += Double angle identities sin 22 sin cos = 2222cos 2cossin2 cos11 2 sin = = = Topic 3: Geometry and trigonometry HL only AHL Reciprocal trigonometric identities 1seccos = 1cosecsin = Pythagorean identities 22221 tansec1 cotcosec +=+= AHL Compound angle identities sin () sin coscos sinABA BA B = cos () cos cossin sinABA BA B = tantantan ()1 tan tanABABAB = Double angle identity for tan 22 tantan 21 tan = AHL Magnitude of a vector 222123vv v= ++v, where 123vvv = v Mathematics.

7 analysis and approaches formula booklet 8 AHL Scalar product 112 23 3vw v w vw = + +vw, where 123vvv = v, 123www = w cos =vwvw, where is the angle between v and w Angle between two vectors 112 23 3cos ++=vw v w vwvw AHL Vector equation of a line =+ rab Parametric form of the equation of a line 00 0, , x xly ymz zn =+=+ =+ Cartesian equations of a line 0 00xx yy zzl mn == AHL Vector product 233231131221vw vwvw vwvw v w = vw, where 123vvv = v, 123www = w sin =vwvw, where is the angle between v and w Area of a parallelogram A= vw where v and w form two adjacent sides of a parallelogram AHL 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++= Mathematics: analysis and approaches formula booklet 9 Topic 4.

8 Statistics and probability SL and HL SL Interquartile range 31 IQRQQ= SL Mean, x, of a set of data 1kiiifxxn== , where 1kiinf== SL Probability of an event A ()P( )()nAAnU= Complementary events P( ) P( ) 1AA += SL 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( )ABA B = SL Expected value of a discrete random variable X E( )P()Xx Xx== SL Binomial distribution ~ B( , )Xnp Mean E( )Xnp= Variance V a r ( )(1)Xnpp= SL Standardized normal variable xz = Mathematics: analysis and approaches formula booklet 10 Topic 4.

9 Statistics and probability HL only AHL Bayes theorem P( ) P( | )P( | )P( ) P( | ) P( ) P( | )BABBABABBAB= + 1 12 23 3P( ) P( |)P( | )P( ) P( |) P() P( |) P( ) P( |)iiiBABBABABBABBAB=++ AHL Variance 2 ()222211kki iiiiif xfxnn == == Standard deviation ()21kiiifxn = = Linear transformation of a single random variable ()()2EE( )VarVar ( )aX baXbaX baX+= ++= Expected value of a continuous random variable X E( )( )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 = = Mathematics: analysis and approaches formula booklet 11 Topic 5.

10 Calculus SL and HL SL Derivative of nx 1()()nnf xxf xnx = = SL Integral of nx 1d,11nnxx xCnn+= + + Area between a curve ()y fx= and the x-axis, where () 0fx> dbaAyx= SL Derivative of sinx ( ) sin( ) cosfxxf xx = = Derivative of cosx ( ) cos( )sinfxxf xx = = Derivative of ex () e() exxfxf x = = Derivative of lnx 1() ln()fxxf xx = = Chain rule ()y gu=, where d dd()dddy yuu fxx ux= = Product rule d ddd ddy vuy uvuvx xx= = + Quotient rule 2dddddduvvuuyxxyvx v = = SL Acceleration 22ddddvsatt= = Distance travelled from 1t to 2t distance 21() dttvt t= Displacement from 1t to 2t displacement 21( )dttvt t= Mathematics: analysis and approaches formula booklet 12 SL Standard integrals 1dlnx xCx=+ sin dcosxxx C= + cos dsinxxx C=+ ed exxxC= + SL Area of region enclosed by a curve and x-axis dbaAyx= Topic 5.