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Published June 2012 International Baccalaureate Organization2012 5048

Mathematics HL andfurther mathematics HL

formula bookletFor use during the course and in theexaminations

First examinations 2014

Diploma Programme

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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

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Formulae

Prior learning

Area of a parallelogram A b h , where b is the base, h is theheight

Area of a triangle 1( )

2A b h , where b is the base, h is the height

Area of a trapezium 1( )

2A a b h , where a and b are the parallel sides, h is theheight

Area of a circle 2A r , where r is the radius

Circumference of a circle 2C r , where r is the radius

Volume of a pyramid 1(area of base vertical height)

3V

Volume of a cuboid V l w h , where l is the length, w is thewidth, h is the height

Volume of a cylinder 2V r h , where r is the radius, h is theheight

Area of the curved surface of

a cylinder2A rh , where r is the radius, h is the height

Volume of a sphere34

3V r , where r is the radius

Volume of a cone21

3V r h , where r is the radius, h is the height

Distance between two

points1 1( , )x y and 2 2( , )x y

2 2

1 2 1 2( ) ( )d x x y y

Coordinates of the midpoint ofa line segment with endpoints

1 1( , )x y and 2 2( , )x y

1 2 1 2,2 2

x x y y

Solutions of a quadraticequation The solutions of

2 0ax bx c are2

4

2

b b acx

a

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Core

Topic 1: Algebra

1.1 The nt term of an

arithmetic sequence1 ( 1)nu u n d

The sum ofn terms of anarithmetic sequence 1 1(2 ( 1) ) ( )22

n n

n nS u n d u u

The nt term of a

geometric sequence

1

1

n

nu u r

The sum ofn terms of a

finite geometric sequence

1 1( 1) (1 )

1 1

n n

n

u r u r

S r r , 1r

The sum of an infinitegeometric sequence

1

1

uS

r, 1r

1.2 Exponents and logarithms logxa

a b x b , where 0, 0, 1a b a

lnex x aa

loglog a

xx

aa x a

loglog

logc

b

c

aa

b

1.3 Combinations !

!( )!

n n

r r n r

Permutations !

( )!

nnP

r n r

Binomial theorem1

( ) 1

n n n n r r nn n

a b a a b a b br

1.5 Complex numbers i (cos isin ) e cisiz a b r r r

1.7 De Moivres theorem (cos isin ) (cos isin ) e cisn n n in nrr n n r r n

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Topic 2: Functions and equations

2.5 Axis of symmetry of thegraph of a quadratic

function

2( ) axis of symmetry

2

bf x ax bx c x

a

2.6 Discriminant 2 4b ac

Topic 3: Circular functions and trigonometry

3.1 Length of an arc l r , where is the angle measured inradians, r is the radius

Area of a sector21

2

A r , where is the angle measured in radians, r is the

radius

3.2 Identities sintan

cos

1sec

cos

1cosec

sin

Pythagorean identities 2 2

2 2

2 2

cos sin 1

1 tan sec

1 cot csc

3.3 Compound angle identities sin( ) sin cos cos sinA B A B AB

cos( ) cos cos sin sinA B A B A B

tan tantan( )

1 tan tan

A BA B

A B

Double angle identities sin2 2sin cos

2 2 2 2cos2 cos sin 2cos 1 1 2sin

2

2tantan2

1 tan

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3.7 Cosine rule2 2 2

2 cosc a b ab C ;2 2 2

cos2

a b cC

ab

Sine rule

sin sin sin

a b c

A B C

Area of a triangle 1sin

2A ab C

Topic 4: Vectors

4.1 Magnitude of a vector

2 2 21 2 3v v vv , where

1

2

3

v

v

v

v

Distance between two

points1 1 1( , , )x y z and

2 2 2( , , )x y z

2 2 2

1 2 1 2 1 2( ) ( ) ( )d x x y y z z

Coordinates of themidpoint of a line segment

with endpoints1 1 1( , , )x y z ,

2 2 2( , , )x y z

1 2 1 2 1 2, ,2 2 2

x x y y z z

4.2 Scalar product cosv w v w , where is the angle between v andw

1 1 2 2 3 3v w v w v wv w , where

1

2

3

v

v

v

v ,

1

2

3

w

w

w

w

Angle between twovectors

1 1 2 2 3 3cosv w v w v w

v w

4.3 Vector equation of a line = + r a b

Parametric form of theequation of a line

0 0 0, ,x x l y y m z z n

Cartesian equations of aline

0 0 0x x y y z z

l m n

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4.5 Vector product2 3 3 2

3 1 1 3

1 2 2 1

v w v w

v w v w

v w v w

v w where

1

2

3

v

v

v

v ,

1

2

3

w

w

w

w

sinv w v w , where is the angle between v and w

Area of a triangle 1

2A v w where v and w form two sides of a triangle

4.6 Vector equation of a plane = +r a b + c

Equation of a plane(using the normal vector)

r n a n

Cartesian equation of aplane

ax by cz d

Topic 5: Statistics and probability

5.1 Population parametersLet

1

k

i

i

n f

Mean

1

k

i i

if x

n

Variance 2 2 2

2 21 1

k k

i i i i

i i

f x f x

n n

Standard deviation2

1

k

i i

i

f x

n

5.2 Probability of an event A ( )P( )

( )

n AA

n U

Complementary events P( ) P( ) 1A A

5.3 Combined events P( ) P( ) P( ) P( )A B A B A B

Mutually exclusive events P( ) P( ) P( )A B A B

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5.4 Conditional probability P( )P

P( )

A BA B

B

Independent events P( ) P( ) P( )A B A B

Bayes theorem P( )P |P |

P( )P | P( )P |

B A BB A

B A B B A B

1 1 2 2 3 3

( ) ( )( | )

( ) ( | ) ( ) ( | ) ( ) ( | )

i i

i

P B P A BP B A

P B P A B P B P A B P B P A B

5.5 Expected value of adiscrete random variableX

E( ) P( )X x X x

Expected value of acontinuous random

variableX

E( ) ( )dX x f x x

Variance 22 2Var( ) E( ) E( ) E( )X X X X

Variance of a discreterandom variableX

2 2 2Var( ) ( ) P( ) P( )X x X x x X x

Variance of a continuous

random variableX

2 2 2Var( ) ( ) ( )d ( )dX x f x x x f x x

5.6 Binomial distribution

Mean

Variance

~ B( , ) P( ) (1 ) , 0,1, ,x n xn

X n p X x p p x nx

E( )X np

Var( ) (1 )X np p

Poisson distribution

Mean

Variance

e~ Po( ) P( ) , 0,1, 2,

!

x mm

X m X x xx

E( )X m

Var( )X m

5.7 Standardized normal

variablex

z

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Topic 6: Calculus

6.1 Derivative of ( )f x

d ( ) ( )( ) ( ) lim

d h

y f x h f xy f x f x

x h

6.2 Derivative of nx 1( ) ( )n nf x x f x nx

Derivative of sinx ( ) sin ( ) cosf x x f x x

Derivative of cosx ( ) cos ( ) sinf x x f x x

Derivative of tanx 2( ) tan ( ) secf x x f x x

Derivative of ex ( ) e ( ) ex x

f x f x

Derivative of lnx 1( ) ln ( )f x x f x

x

Derivative of secx ( ) sec ( ) sec tanf x x f x x x

Derivative of cscx ( ) csc ( ) csc cotf x x f x x x

Derivative of cotx 2( ) cot ( ) cscf x x f x x

Derivative ofx

a ( ) ( ) (ln )x x

f x a f x a a

Derivative of loga x 1( ) log ( )ln

af x x f x

x a

Derivative of arcsinx

2

1( ) arcsin ( )

1f x x f x

x

Derivative of arccosx

2

1( ) arccos ( )

1f x x f x

x

Derivative of arctanx 2

1( ) arctan ( )

1f x x f x

x

Chain rule( )y g u , where

d d d( )

d d d

y y uu f x

x u x

Product rule d d d

d d d

y v uy uv u v

x x x

Quotient rule

2

d d

d d dd

u vv u

u y x xyv x v

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6.4 Standard integrals 1d , 1

1

nn x

x x C nn

1

d lnx x Cx

sin d cosx x x C

cos d sinx x x C

e d ex xx C

1d

ln

x xa x a C

a

2 2

1 1d arctan

xx C

a x a a

2 2

1d arcsin ,

xx C x a

aa x

6.5 Area under a curve

Volume of revolution(rotation)

db

aA y x or d

b

aA x y

2 2 d or d

b b

a aV y x V x y

6.7 Integration by parts d dd d

d d

v uu x uv v x

x xor d du v uv v u

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Options

Topic 7: Statistics and probabilityFurther mathematics HL topic3

7.1

(3.1)

Probability generatingfunction for a discreterandomvariableX

( ) ( ) ( )X x

x

G t E t P X x t

7.2

(3.2)

Linear combinations of twoindependent random

variables1 2,X X

1 1 2 2 1 1 2 2

2 2

1 1 2 2 1 1 2 2

E E E

Var Var Var

a X a X a X a X

a X a X a X a X

7.3

(3.3)

Sample statistics

Mean x

1

k

i i

i

f x

xn

Variance2

ns 2 2

2 21 1

( )k k

i i i i

i i

n

f x x f x

s xn n

Standard deviationn

s 2

1

( )k

i i

i

n

f x x

sn

Unbiased estimate of

population variance2

1ns

2 2

2 2 21 11

( )

1 1 1 1

k k

i i i i

i i

n n

f x x f xn n

s s xn n n n

7.5

(3.5)

Confidence intervals

Mean, with knownvariance x z n

Mean, with unknownvariance

1nsx tn

7.6

(3.6)

Test statistics

Mean, with knownvariance /

xz

n

Mean, with unknown

variance 1 /n

xt

s n

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7.7

(3.7)

Sample product momentcorrelation coefficient

1

2 2 2 2

1 1

n

i i

i

n n

i ii i

x y nx y

r

x nx y n y

Test statistic for H0: = 0 2

2

1

nt r

r

Equation of regression lineofx ony

1

2 2

1

( )

n

i i

i

n

i

i

x y nx y

x x y y

y n y

Equation of regression lineofy onx

1

2 2

1

( )

n

i i

i

n

i

i

x y nx y

y y x x

x nx

Topic 8: Sets, relations and groupsFurther mathematics HL topic4

8.1

(4.1)

De Morgans laws ( )

( )

A B A B

A B A B

Topic 9: CalculusFurther mathematics HL topic 5

9.5

(5.5)

Eulers method1 ( , )n n n ny y h f x y ; 1n nx x h , where h isa constant

(step length)

Integrating factor for

( ) ( )y P x y Q x

( )d

eP x x

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9.6

(5.6)

Maclaurin series 2( ) (0) (0) (0)

2!

xf x f x f f

Taylor series 2( )( ) ( ) ( ) ( ) ( ) ...2!

x af x f a x a f a f a

Taylor approximations

(with error term ( )nR x )( )( )

( ) ( ) ( ) ( ) ... ( ) ( )!

nn

n

x af x f a x a f a f a R x

n

Lagrange form ( 1)1( )( ) ( )

( 1)!

nn

n

f cR x x a

n, where c lies between a andx

Maclaurin series forspecial functions

2

e 1 ...

2!

x xx

2 3

ln(1 ) ...2 3

x xx x

3 5

sin ...3! 5!

x xx x

2 4

cos 1 ...2! 4!

x xx

3 5

arctan ...3 5

x xx x

Topic 10: Discrete mathematicsFurther mathematics HL topic 6

10.7

(6.7)

Eulers formula forconnected planar graphs

2v e f , where v is the number of vertices, e is the number

of edges,fis the number of faces

Planar, simple, connected

graphs

3 6e v for 3v

2 4e v if the graph has no triangles

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Formulae for distributions

Topics 5.6, 5.7, 7.1, further mathematics HL topic 3.1

Discrete distributionsDistribution Notation Probability mass

functionMean Variance

Geometric ~ GeoX p 1xpq

for 1,2,...x

1

p

2

q

p

Negative binomial ~ NB ,X r p 1

1

r x rx

p qr

for , 1,...x r r

r

p

2

rq

p

Continuous distributionsDistribution Notation Probability

density functionMean Variance

Normal 2~ N ,X 2

1

21 e

2

x

2

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Further mathematics

Topic 1: Linear algebra

1.2 Determinant of a 2 2

matrix deta b

ad bcc d

A A A

Inverse of a 2 2 matrix1 1 ,

det

a b d bad bc

c d c aA A

A

Determinant of a 3 3

matrix det

a b ce f d f d e

d e f a b ch k g k g h

g h k

A A