(PDF) Maths HL Formula Booklet - PDFSLIDE.NET (2023)

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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

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