Mathematics: Applications And Interpretation Formula Booklet

Diploma ProgrammeMathematics: applications and interpretationformula bookletFor use during the course and in the examinationsFirst examinations 2021Version 1.1 International Baccalaureate Organization 2019

ContentsPrior learningSL and HL2HL only2Topic 1: Number and algebraSL and HL3HL only4Topic 2: FunctionsSL and HL5HL only5Topic 3: Geometry and trigonometrySL and HL6HL only7Topic 4: Statistics and probabilitySL and HLHL only910Topic 5: CalculusSL and HL11HL only11

Prior learning – SL and HLArea of a parallelogramA bh , where b is the base, h is the heightArea of a triangle1A (bh) , where b is the base, h is the height2Area of a trapezoid A1(a b) h , where a and b are the parallel sides, h is the height2Area of a circleA πr 2 , where r is the radiusCircumference of a circleC 2πr , where r is the radiusVolume of a cuboidV lwh , where l is the length, w is the width, h is the heightVolume of a cylinderV πr 2 h , where r is the radius, h is the heightVolume of prismV Ah , where A is the area of cross-section, h is the heightArea of the curved surface ofa cylinderA 2πrh , where r is the radius, h is the heightDistance between twopoints ( x1 , y1 ) and ( x2 , y2 )d Coordinates of the midpoint ofa line segment with endpoints( x1 , y1 ) and ( x2 , y2 ) x1 x2 y1 y2 , 2 2( x1 x2 ) 2 ( y1 y2 ) 2Prior learning – HL onlySolutions of a quadraticequationThe solutions of ax 2 bx c 0 are xMathematics: applications and interpretation formula booklet b b 2 4ac,a 02a2

Topic 1: Number and algebra – SL and HLSL1.2SL1.3The nth term of anarithmetic sequenceun u1 (n 1) dThe sum of n terms of anarithmetic sequenceS n The nth term of ageometric sequenceun u1r n 1nn( 2u1 (n 1) d ) ; Sn (u1 un )22u1 (r n 1) u1 (1 r n )The sum of n terms of a, r 1 Sn finite geometric sequencer 11 rSL1.4Compound interestr FV PV 1 , where FV is the future value, 100k 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 interestSL1.5Exponents and logarithmsa x b x log a b , where a 0, b 0, a 1Percentage errorε SL1.6knvA vE 100% , where vE is the exact value and vA isvEthe approximate value of vMathematics: applications and interpretation formula booklet3

Topic 1: Number and algebra – HL onlyAHL1.9Laws of logarithmslog log a x log a ya xyxlog log a x log a yaylog a x m m log a xfor a, x, y 0AHL1.11The sum of an infinitegeometric sequenceS AHL1.12Complex numbersz a biDiscriminant b 2 4acModulus-argument (polar)and exponential (Euler)formz r (cos θ isin θ ) re iθ r cis θAHL1.13AHL1.14AHL1.15Determinant of a 2 2matrixu1, r 11 r a b A A ad bc det A c d Inverse of a 2 2 matrix a b 1 d 1 A A det A c c d b , ad bca Power formula for a matrixM n PD n P 1 , where P is the matrix of eigenvectors and D isthe diagonal matrix of eigenvaluesMathematics: applications and interpretation formula booklet4

Topic 2: Functions – SL and HLSL2.1SL2.5Equations of a straight line 0 ; y y1 m ( x x1 )y mx c ; ax by d y2 y1x2 x1Gradient formulam Axis of symmetry of thegraph of a quadraticfunctionf ( x) ax 2 bx c axis of symmetry is x b2aTopic 2: Functions – HL onlyAHL2.9Logistic functionf ( x) L, L , k,C 01 Ce kxMathematics: applications and interpretation formula booklet5

Topic 3: Geometry and trigonometry – SL and HLSL3.1Distance between twopoints ( x1 , y1 , z1 ) andd Coordinates of themidpoint of a line segmentwith endpoints ( x1 , y1 , z1 ) x1 x2 y1 y2 z1 z2 ,, 22 2Volume of a right-pyramidV 1Ah , where A is the area of the base, h is the height3Volume of a right coneV 1 2πr h , where r is the radius, h is the height3Area of the curved surfaceof a coneA πrl , where r is the radius, l is the slant heightVolume of a sphereV Surface area of a sphereA 4πr 2 , where r is the radiusSine ruleabc sin A sin B sin CCosine rulec 2 a 2 b 2 2ab cos C ; cos C Area of a triangle1A ab sin C2( x1 x2 ) 2 ( y1 y2 ) 2 ( z1 z2 ) 2( x2 , y2 , z2 )and ( x2 , y2 , z2 )SL3.2SL3.4Length of an arc l4 3πr , where r is the radius3θ360a 2 b2 c22ab 2πr , where θ is the angle measured in degrees, r isthe radiusArea of a sector Aθ360 πr 2 , where θ is the angle measured in degrees, r isthe radiusMathematics: applications and interpretation formula booklet6

Topic 3: Geometry and trigonometry – HL onlyAHL3.7Length of an arcl rθ , where r is the radius, θ is the angle measured in radiansArea of a sector1A r 2θ , where r is the radius, θ is the angle measured in2radiansAHL3.8Identitiescos 2 θ sin 2 θ 1tan θ AHL3.9Transformation matrices cos 2θ sin 2θsin θcos θsin 2θ , reflection in the line y (tan θ ) x cos 2θ k 0 , horizontal stretch / stretch parallel to x-axis with a scale 0 1 factor of k 1 0 , vertical stretch / stretch parallel to y-axis with a scale 0 k factor of k k 0 , enlargement, with a scale factor of k, centre (0, 0) 0 k cos θ sin θ sin θ , anticlockwise/counter-clockwise rotation ofcos θ angle θ about the origin ( θ 0 ) cos θ sin θ(θ 0 )sin θ , clockwise rotation of angle θ about the origincos θ Mathematics: applications and interpretation formula booklet7

AHL3.10AHL3.11AHL3.13Magnitude of a vector v1 v v2 v3 , where v v2 v 3 21v 22Vector equation of a liner a λbParametric form of theequation of a linex x0 λ l , y y0 λ m, z z0 λ nScalar product v1 w1 v w v1w1 v2 w2 v3 w3 , where v v2 , w w2 v w 3 3 v w v w cos θ , where θ is the angle between v and wv1w1 v2 w2 v3 w3v wAngle between twovectorscos θ Vector product w1 v2 w3 v3 w2 v1 v w v3 w1 v1w3 , where v v2 , w w2 w v 3 v1w2 v2 w1 3 v w v w sin θ , where θ is the angle between v and wArea of a parallelogramA v w where v and w form two adjacent sides of aparallelogramMathematics: applications and interpretation formula booklet8

Topic 4: Statistics and probability – SL and HLSL4.2SL4.3SL4.5SL4.6Interquartile rangeIQR Q3 Q1kMean, x , of a set of datax fxi 1i i, where n nk fi 1in ( A)n (U )Probability of an event AP ( A) Complementary eventsP ( A) P ( A′) 1Combined eventsP ( A B ) P ( A) P ( B) P ( A B)Mutually exclusive eventsP ( A B ) P ( A) P ( B)Conditional probabilityP ( A B) Independent eventsP ( A B) P ( A) P ( B)SL4.7Expected value of aE(X ) discrete random variable XSL4.8Binomial distributionP ( A B)P ( B)x P(X x)X B (n , p)MeanE ( X ) npVarianceVar ( X ) np (1 p )Mathematics: applications and interpretation formula booklet9

Topic 4: Statistics and probability – HL onlyAHL4.14Linear transformation of asingle random variableE ( aX b ) aE ( X ) bLinear combinations of nindependent randomvariables, X 1 , X 2 , ., X na1E ( X 1 ) a2 E ( X 2 ) . an E ( X n )E ( a1 X 1 a2 X 2 . a n Xn )Var ( aX b ) a 2 Var ( X )Var ( a1 X 1 a2 X 2 . an X n ) a12 Var ( X 1 ) a2 2 Var ( X 2 ) . an 2 Var ( X n )Sample statisticsUnbiased estimate ofpopulation variance sn2 1AHL4.17AHL4.19sn2 1 n 2snn 1Poisson distributionX Po (m)MeanE(X ) mVarianceVar ( X ) mTransition matricesT n s0 sn , where s0 is the initial stateMathematics: applications and interpretation formula booklet10

Topic 5: Calculus – SL and HLSL5.3SL5.5Derivative of x nf ( x) x n f ′( x) nx n 1Integral of x ndx x Area of region enclosed bya curve y f ( x) and thex-axis, where f ( x) 0SL5.8The trapezoidal rulenx n 1 C , n 1n 1bA y dxa1y dx h ( ( y0 yn ) 2( y1 y2 . yn 1 ) ) ,2b awhere h n baTopic 5: Calculus – HL onlyAHL5.9Derivative of sin xf ( x) sin x f ′( x) cos xDerivative of cos xf ( x) cos xf ′( x) sin xDerivative of tan xf ( x) tan x f ′( x) Derivative of e xf ( x) e x f ′( x) exDerivative of ln x1f ( x) ln x f ′( x) xChain ruley g (u ) , where u f ( x) Product ruley uv Quotient ruledudvv uudydxdxy 2vdxvMathematics: applications and interpretation formula booklet1cos 2 xdy dy du dx du dxdydvdu u vdxdxdx11

AHL5.11Standard integrals1dx x ln x C cos x C sin x dx dx cos x 1 cos x2 eAHL5.12AHL5.13AHL5.16xsin x Ctan x Cx ex Cd bbaaArea of region enclosedby a curve and x or y-axesA y dx or A x dyVolume of revolutionabout x or y-axesV πy 2 dx or V πx 2 dyAcceleration aDistance travelled fromt1 to t2distance Displacement fromt1 to t2displacement Euler’s methodbbaadv d 2 sdv v2dt dtds t2t1v(t ) dt t2t1v(t ) dtxn h , where h is a constantyn yn h f ( xn , yn ) ; xn 11(step length)Euler’s method forcoupled systemsxn 1 xn h f1 ( xn , yn , tn )yn 1 yn h f 2 ( xn , yn , tn )tn 1 tn hwhere h is a constant (step length)AHL5.17Exact solution for coupled x Aeλ1t p1 Beλ2t p2linear differential equationsMathematics: applications and interpretation formula booklet12

Mathematics: applications and interpretation formula booklet 2 Prior learning – SL and HL Area of a parallelogram is the heightA bh , where b is the base, h Area of a triangle , whe. 1 2 A bh b is the base, re h is the height Area of a trapezoid 1 2 A a bh , where a and b are the parallel sides, h is the height