Diploma Programme 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) © International Baccalaureate Organization 2012 5048 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 5.6, 5.7, 7.1, further mathematics HL topic 3.1 Discrete distributions 13 Continuous distributions 13 Further mathematics 14 Topic 1: Linear algebra 14 Mathematics HL and further mathematics formula booklet 1 Formulae Prior learning Area of a parallelogram Abh= × , where b is the base, h is the height 1 Area of a triangle A=() bh × , where b is the base, h is the height 2 1 Area of a trapezium A=() a + bh, where a and b are the parallel sides, h is the height 2 Area of a circle Ar= π 2 , where r is the radius Circumference of a circle Cr=2 π , where r is the radius 1 Volume of a pyramid V = (area of base × vertical height) 3 Volume of a cuboid V=×× lwh, where l is the length, w is the width, h is the height Volume of a cylinder V= π rh2 , where r is the radius, h is the height Area of the curved surface of A=2 π rh , where r is the radius, h is the height a cylinder 4 Volume of a sphere Vr= π 3 , where r is the radius 3 1 Volume of a cone V= π rh2 , where r is the radius, h is the height 3 Distance between two 22 d=( xx12 − )( +− yy 12 ) points (,xy11 )and (,xy22 ) Coordinates of the midpoint of xxyy121++ 2 a line segment with endpoints , 22 (,xy11 )and (,xy22 ) Solutions of a quadratic −±b b2 −4 ac equation The solutions of ax2 + bx += c 0 are x = 2a Mathematics HL and further mathematics formula booklet 2 Core Topic 1: Algebra 1.1 The nth term of an uun =+−1 ( n 1) d arithmetic sequence The sum of n terms of an nn S=(2 u +− ( n 1) d) = ( uu + ) arithmetic sequence nn2211 The nth term of a n−1 un = ur1 geometric sequence The sum of n terms of a ur(nn−− 1) u (1 r ) S =11 = , r ≠1 finite geometric sequence n rr−−11 The sum of an infinite u S = 1 , r <1 geometric sequence ∞ 1− r 1.2 Exponents and logarithms x ab=⇔= xloga b, where aba>>≠0, 0, 1 a x= e xaln x loga x loga a= xa = logc a logb a = logc b 1.3 n n! Combinations = r rn!(− r )! n n! Permutations P = r (nr− )! n nnn n−−1 nr r n Binomial theorem ()ab+ = a + ab ++ ab ++ b 1 r 1.5 Complex numbers za=+=i br (cosθθ + isin ) =r eiθ = r cis θ n θ 1.7 De Moivre’s theorem [r(cosθθ+ isin )] =rn (cos n θ +== isin nr θ )nn ei rn n cis θ Mathematics HL and further mathematics formula booklet 3 Topic 2: Functions and equations 2.5 Axis of symmetry of the b f() x= ax2 + bx +⇒ c axis of symmetry x =− graph of a quadratic 2a function 2.6 Discriminant ∆=b2 −4 ac Topic 3: Circular functions and trigonometry 3.1 Length of an arc lr=θ , where θ is the angle measured in radians, r is the radius 1 Area of a sector Ar= θ 2 , where θ is the angle measured in radians, r is the 2 radius 3.2 Identities sinθ tanθ = cosθ 1 secθ = cosθ 1 cosecθ = sinθ Pythagorean identities cos22θθ+= sin 1 1+= tan22θθ sec 1+= cot22θθ csc 3.3 Compound angle identities sin (AB±= ) sin A cos B ± cos A sin B cos(AB±= ) cos A cos B sin A sin B tanAB± tan tan (AB±= ) 1 tanAB tan Double angle identities sin 2θ= 2sin θθ cos cos2θ= cos22 θθ − sin = 2cos 2 θ −=− 1 1 2sin2 θ 2tanθ tan 2θ = 1− tan 2 θ Mathematics HL and further mathematics formula booklet 4 3.7 abc222+− Cosine rule c2=+− a 22 b2 ab cos C ; cosC = 2ab abc Sine rule = = sinABC sin sin 1 Area of a triangle A= absin C 2 Topic 4: Vectors 4.1 v1 Magnitude of a vector =222 ++ = v vvv123, where v v2 v3 Distance between two 2 22 d=( xx12 − )( +− yy 1 2 )( +− zz 12 ) points (,xyz1 11 ,) and (,xyz2 22 ,) Coordinates of the xxyyzz1+++ 2 1 21 2 midpoint of a line segment , , 222 with endpoints (,xyz1 11 ,), (,xyz2 22 ,) 4.2 Scalar product vw⋅= v wcosθ , where θ is the angle between v and w v1 w1 ⋅= + + = = vw vw11 vw 2 2 vw 33, where v v2 , w w2 v3 w3 Angle between two vw11++ vw 2 2 vw 33 vectors cosθ = vw 4.3 Vector equation of a line ra=+λ b Parametric form of the x=+=+=+ x00λλ ly, y mz, z 0 λ n equation of a line Cartesian equations of a xx−−− yy zz 0= 00 = line l mn Mathematics HL and further mathematics formula booklet 5 4.5 vw23− vw 32 v1 w1 Vector product ×= − = = vwvw31 vw 13 where v v2 , w w2 vw12− vw 21 v3 w3 vw×= vwsinθ , where θ is the angle between v and w 1 Area of a triangle A =vw × where v and w form two sides of a triangle 2 4.6 Vector equation of a plane r=+ aλ b+µ c Equation of a plane rn⋅=⋅ an (using the normal vector) Cartesian equation of a ax++= by cz d plane Topic 5: Statistics and probability 5.1 k Population parameters Let nf= ∑ i i=1 k ∑ fxii Mean µ µ = i=1 n kk 2 2 ∑∑fi( x i− µ ) fxii Variance σ 2 σµ22= ii=11== − nn k 2 ∑ fxii( − µ ) Standard deviation σ σ = i=1 n 5.2 nA() Probability of an event A P(A )= nU() Complementary events P(AA )+= P(′ ) 1 5.3 Combined events P(AB∪= ) P( A ) + P( B ) − P( AB ∩ ) Mutually exclusive events P(AB∪= ) P( A ) + P( B ) Mathematics HL and further mathematics formula booklet 6 5.4 P(AB∩ ) Conditional probability P(AB )= P(B ) Independent events P(AB∩= ) P( A )P( B ) P(B ) P( AB | ) Bayes’ theorem P(BA | )= P(B ) P( AB | )+′ P( B ) P( AB |′ ) P(Bii )P( AB | ) P(BAi | )= P(B1 )P( AB | 12 )++ P( B )P( AB | 23 ) P( B )P( AB | 3 ) 5.5 Expected value of a E(X )=µ =∑ x P( Xx = ) discrete random variable X Expected value of a ∞ E(X )=µ = xf( x )d x continuous random ∫−∞ variable X 2 Variance Var(XX )= E( −=µ )22 E( X ) −[ E( X )] Variance of a discrete Var(X )=−===−∑∑ ( xµµ )2 P( Xx ) x 22 P( Xx ) random variable X Variance of a continuous ∞∞ Var(X )=−=− ( xµµ )2 fx ( )d x xfx 22 ( )d x random variable X ∫∫−∞ −∞ 5.6 n x nx− Binomial distribution X~ B ( np , )⇒==− P ( X x ) p (1 p ) , x = 0 , 1, , n x Mean E(X )= np Variance Var (X )= np (1 − p ) mxme− Poisson distribution X~ Po( m )⇒== P( Xx ) , x = 0,1,2, x! Mean E(Xm )= Variance Var (Xm ) = 5.7 Standardized normal x − µ z = variable σ Mathematics HL and further mathematics formula booklet 7 Topic 6: Calculus 6.1 dy fx(+− h ) fx () Derivative of fx() y= fx() ⇒= f′ () x = lim dxhh→0 6.2 Derivative of xn f() x=⇒= xnn f′ () x nx −1 Derivative of sin x fx( )=⇒= sin x f′ ( x ) cos x Derivative of cos x fx( )=⇒=− cos x f′ ( x ) sin x Derivative of tan x fx( )=⇒= tan x f′ ( x ) sec2 x Derivative of ex fx()=⇒= exx f′ () x e 1 Derivative of ln x fx()=⇒= ln x f′ () x x Derivative of sec x fxxfxxx()=⇒= sec′ () sectan Derivative of csc x fx()=⇒=− csc x f′ () x csccot x x Derivative of cot x fx( )=⇒=− cot x f′ ( x ) csc2 x Derivative of a x fxa()=⇒=xx fxa′ () (ln) a 1 Derivative of log x fx()= log x ⇒= f′ () x a a xaln 1 Derivative of arcsin x fx( )= arcsin x ⇒= f′ ( x ) 1− x2 1 Derivative of arccos x fx( )= arccos x ⇒=− f′ ( x ) 1− x2 1 Derivative of arctan x fx( )= arctan x ⇒= f′ ( x ) 1+ x2 dy dd yu Chain rule y= gu(), where u= fx() ⇒=× dddxux dy dd vu Product rule y=⇒= uv u + v dddxxx dduv vu− uyd Quotient rule y =⇒=ddxx vxd v2 Mathematics HL and further mathematics formula booklet 8 6.4 Standard integrals xn+1 xxn d= + C ,1 n ≠− ∫ n +1 1 dx= ln xC + ∫ x ∫sinxx d=−+ cos x C ∫cosxx d= sin x + C ∫edxxxC= e + 1 axxxd = a + C ∫ ln a 11x dxC= arctan + ∫ ax22+ a a 1 x ∫ dx= arcsin +< C , xa ax22− a 6.5 b b Area under a curve A= yxd or A= xyd ∫a ∫a Volume of revolution b b V= πd yx2 or V= πd xy2 (rotation) ∫a ∫a 6.7 ddvu Integration by parts udd x= uv − v x or udd v= uv − v u ∫∫ddxx∫∫ Mathematics HL and further mathematics formula booklet 9 Options Topic 7: Statistics and probability Further mathematics HL topic 3 7.1 Probability generating Gt( )= E( txx ) =∑ P( X = xt ) (3.1) function for a discrete x random variable X E (XG )= ′ (1) 2 Var (XG )=′′ (1) +− G ′ (1)( G ′ (1) ) 7.2 Linear combinations of two E(aX11±= aX 2 2) a 1 EE( X 1) ± a 2( X 2) (3.2) independent random 22 Var(aX11±= aX 2 2) a 1Var( X 1) + a 2 Var( X 2) variables XX12, 7.3 Sample statistics (3.3) k ∑ fxii Mean x x = i=1 n kk 22 ∑∑fi() x i− x fxii Variance s2 sx22= ii=11== − n n nn k 2 ∑ fxii()− x Standard deviation s s = i=1 n n n kk 22 Unbiased estimate of ∑∑fi() x i− x fxii 2 22nnii=11= 2 population variance s ssnn−1 = = = − x n−1 n−1 n − 1 nn −− 11 7.5 Confidence intervals (3.5) Mean, with known σ variance xz±× n Mean, with unknown sn−1 variance xt±× n 7.6 Test statistics (3.6) Mean, with known x − µ variance z = σ / n Mathematics HL and further mathematics formula booklet 10 Mean, with unknown x − µ variance t = snn−1 / 7.7 n x y− nx y (3.7) Sample product moment ∑ ii r = i=1 correlation coefficient nn 22 22 ∑∑xii−− nx y ny ii−=11 Test statistic for H0: n − 2 = ρ = 0 tr 2 1− r n Equation of regression line ∑ xii y− nx y i=1 of x on y xx−= n ()yy− 22 ∑ yi − ny i=1 n Equation of regression line ∑ xii y− nx y i=1 of y on x yy−= n ()xx− 22 ∑ xi − nx i=1 Topic 8: Sets, relations and groups Further mathematics HL topic 4 8.1 De Morgan’s laws ()AB∪=∩′′′ A B (4.1) ()AB∩=∪′′′ A B Topic 9: Calculus Further mathematics HL topic 5 Euler’s method 9.5 yn+1 = y n + h× fx(, nn y ); xnn+1 = xh + , where h is a constant (5.5) (step length) Integrating factor for Px( )d x e∫ y′ += Pxy() Qx () Mathematics HL and further mathematics formula booklet 11 9.6 x2 (5.6) Maclaurin series f( x )=++ f (0) xf′ (0) f ′′ (0) + 2! ()xa− 2 Taylor series fx()= fa () +− ( x af )′ () a + f′′ () a + ..