Mathematics: applications and interpretation formula booklet

1 International Baccalaureate Organization 2019 Mathematics: applications and interpretation formula booklet For use during the course and in the examinations First examinations 2021 Version Diploma Programme Contents Prior learning SL and HL 2 HL only 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 11 Mathematics: applications and interpretation 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 Vlwh=, 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 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++ Prior learning HL only Solutions of a quadratic equation The solutions of 20axbx c+ += are 24,02bbacxaa = Mathematics: applications and interpretation 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 Percentage error AEE100%vvv = , where Ev is the exact value and Av is the approximate value of v Mathematics: applications and interpretation formula booklet 4 Topic 1: Number and algebra HL only AHL Laws of logarithms loglogloga aaxyxy=+ loglogloga aaxxyy= loglogmaaxm x= for ,,0axy> AHL The sum of an infinite geometric sequence 11uSr = , 1r< AHL Complex numbers iz ab= + Discriminant 24bac = AHL Modulus-argument (polar) and exponential (Euler) form i(cosi sin )eciszrrr = +== AHL Determinant of a 22 matrix detabad bccd = == AAA Inverse of a 22 matrix 11,detabd badbccdc a = = AAA AHL Power formula for a matrix 1nn =MPDP, where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues Mathematics.

4 applications and interpretation formula booklet 5 Topic 2: Functions SL and HL SL Equations of a straight line ymxc= +; 0axbyd+ +=; 11()y y mx x = Gradient formula 2121yymxx = SL Axis of symmetry of the graph of a quadratic function 2()f xaxbx c= + + axis of symmetry is 2bxa= Topic 2: Functions HL only AHL Logistic function ()1ekxLfxC =+, ,, 0L kC> Mathematics: applications and interpretation 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(, ,)

5 Xyz 1 2 1 21 2, , 222xxyyzz+++ 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=+ ; 2 22cos2abcCab+ = Area of a triangle 1sin2 AabC= SL Length of an arc 2360lr = , where is the angle measured in degrees, r is the radius Area of a sector 2360Ar = , where is the angle measured in degrees, r is the radius Mathematics: applications and interpretation formula booklet 7 Topic 3.

6 Geometry and trigonometry HL only AHL 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 AHL Identities 22cossin1 += sintancos = AHL Transformation matrices cos 2sin 2sin 2cos 2 , reflection in the line (tan )yx = 001k , horizontal stretch / stretch parallel to x-axis with a scale factor of k 100k , vertical stretch / stretch parallel to y-axis with a scale factor of k 00kk , enlargement, with a scale factor of k, centre (0 , 0) cossinsincos , anti clockwise/counter-clockwise rotation of angle about the origin (0 >) cossinsincos , clockwise rotation of angle about the origin (0 >) Mathematics.

7 applications and interpretation formula booklet 8 AHL Magnitude of a vector 222123vv v= ++v, where 123vvv = v AHL Vector equation of a line =+ rab Parametric form of the equation of a line 00 0, , x xly ymz zn =+=+ =+ AHL Scalar product 112 23 3vw v wvw = + +vw, where 123vvv = v, 123www = w cos =vwvw, where is the angle between v and w Angle between two vectors 112 23 3cosvw v wvw ++=vw Vector product 233231131221vw vwvw vwvwv 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 Mathematics: applications and interpretation 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= Mathematics: applications and interpretation formula booklet 10 Topic 4: Statistics and probability HL only AHL Linear transformation of a single random variable ()()2EE( )VarVar ( )aXbaXbaXbaX+= ++= Linear combinations of n independent random variables, 12.

9 ,nXXX ()( )( )( )()( )( )( )112 21122112 Xa XaXaXaXaXa Xa XaXaXaX = =+++ Sample statistics Unbiased estimate of population variance 21ns 2211nnnssn = AHL Poisson distribution ~ Po ( )Xm Mean E( )Xm= Variance Var ( )Xm= AHL Transition matrices 0nn=Tss, where 0s is the initial state Mathematics: applications and interpretation formula booklet 11 Topic 5: Calculus SL and HL SL Derivative of nx 1()()nnf xxf xnx = = SL Integral of nx Area of region enclosed by a curve ()y fx= and the x-axis, where () 0fx> 1d,11nnxxxC nn+= + + dbaAyx= SL The trapezoidal rule ()01211d() 2 (.)

10 2bnnayxh y yy yy + + + ++ , where bahn = Topic 5: Calculus HL only AHL Derivative of sinx ( ) sin( ) cosfxxf xx = = Derivative of cosx ( ) cos( )sinfxxf xx = = Derivative of tanx 21() tan()cosfxxf 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 = = Mathematics: applications and interpretation formula booklet 12 AHL Standard integrals 1dlnxxCx=+ sin dcosxxx C= + cos dsinxxx C=+ 21tancosxCx=+ ed exxxC= + AHL Area of region enclosed by a curve and x or y-axes dbaAyx= or dbaAxy= Volume of revolution about x or y-axes 2 dbaVyx= or 2 dbaVxy= AHL Acceleration 22dd ddd dvs vavtt s= = = Distance travelled from 1t to 2t distance 21() dttvt t= Displacement from 1t to 2t displacement 21( )dttvt t= AHL Euler s method 1(, )nnnnyy h fx y+ = +.