HL formula booklet.pdf

(1)

Mathematics HL and

further mathematics HL

formula booklet

For use during the course and in the examinations First examinations 2014

(2)

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

Topic 9: Calculus 11

Topic 10: Discrete mathematics 12

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

(3)

Formulae

Prior learning

Area of a parallelogram

A

= ×

b h

, where

b

is the base,

h

is the height

Area of a triangle

1

(

)

2

A

=

b h

×

, where

b

is the base,

h

is the height

Area of a trapezium

1

(

)

2

A

=

a

+

b h

, where

a

and

b

are the parallel sides,

h

is the height

Area of a circle 2

A

= π

r

, where

r

is the radius

Circumference of a circle

C

= π

2

r

, where

r

is the radius

Volume of a pyramid

1

3

(

area of base vertical height

)

=

×

V

Volume of a cuboid

V

= × ×

l

w h

, where

l

is the length,

w

is the width,

h

is the height

Volume of a cylinder 2

V

= π

r h

, where

r

is the radius,

h

is the height

Area of the curved surface of a cylinder

2

A

= π

rh

, where

r

is the radius,

h

is the height

Volume of a sphere

4

3

V

= π

r

, where

r

is the radius

Volume of a cone

1

2

3

V

= π

r h

, where

r

is the radius,

h

is the height

Distance between two points

( ,

x y

₁ ₁

)

and

(

x y

₂

,

₂

)

2 2

1 2 1 2

(

)

(

)

d

=

x

−

x

+

y

−

y

Coordinates of the midpoint of a line segment with endpoints

1 1

( ,

x y

)

and

(

x y

₂

,

₂

)

1 2

,

1 2

2

x

+

x

y

+

y













Solutions of a quadratic

equation The solutions of

ax

2

+

bx

+ =

c

0

are

2

4

2

b

ac

x

a

− ±

−

(4)

Core

Topic 1: Algebra

1.1 The

n

th term of an

arithmetic sequence 1

(

1)

= +

−

n

u

n

d

The sum of

n

terms of an

arithmetic sequence

(

2

1

(

1)

)

(

1

)

2

=

+

−

=

+

n n

n

S

u

n

d

u

The

n

th term of a geometric sequence

1 1

n n

u

=

u r

−

The sum of

n

terms of a

finite geometric sequence 1

(

1)

1

(1

)

1

n n

n

u r

u

r

S

r

−

=

−

,

r

≠

1

The sum of an infinite

geometric sequence 1

1

u

S

r

∞

=

−

,

r

<

1

1.2 Exponents and logarithms x

log

a

= ⇔

b

x

=

b

, where

a

>

0,

b

>

0,

a

≠

1

ln

e

x x a

a

=

log

x ax

a

= =

x

a

log

c b

c

a

b

=

1.3

Combinations

!

!(

)!

n

r

r n

r

 

=

 

−

 

Permutations

!

(

)!

n

n P

r

=

n

−

r

Binomial theorem

(

)

1

n n

n

n r r n

a

b

a

b

a

b

r

− −

 

+

=

+

 

+ +

 

+ +

 



 



1.5 _{Complex numbers}

z

= + =

a

i

b

r

(cos

θ

+

i sin )

θ

=

r

e

iθ

=

r

cis

θ

(5)

Topic 2: Functions and equations

2.5 Axis of symmetry of the graph of a quadratic function

2

( )

2

axis of symmetry

b

f x

ax

bx

c

x

a

=

+

+ ⇒

= −

2.6 _Discriminant 2

4

b

ac

∆ =

−

Topic 3: Circular functions and trigonometry

3.1 Length of an arc

l

=

θ

r

, where

θ

is the angle measured in radians,

r

is the radius

Area of a sector

1

2

A

=

θ

r

, where

θ

is the angle measured in radians,

r

is the

radius

3.2 Identities

sin

tan

cos

θ

=

1

sec

cos

θ

=

1

cosec

sin

θ

=

Pythagorean identities 2 2

2 2

cos

sin

1

1 tan

sec

1 cot

csc

θ

+

=

+

=

+

=

3.3 Compound angle identities

sin (

A

±

B

)

=

sin

A

cos

B

±

cos sin

A

B

cos (

A

±

B

)

=

cos cos

A

B



sin

A

sin

B

tan

tan (

)

1 tan

tan

A

B

A

B

A

B

±

=



Double angle identities

sin 2

θ

=

2sin cos

θ

2 2 2 2

cos 2

θ

=

cos

θ

−

sin

θ

=

2 cos

θ

− = −

1 1 2sin

θ

2

2 tan

tan 2

1 tan

θ

=

(6)

3.7

Cosine rule

c

2

=

a

2

+

b

2

−

2

ab

cos

C

;

2 2 2

cos

2

a

b

c

C

ab

+

−

=

Sine rule

sin

a

b

c

A

=

B

=

C

Area of a triangle

1

sin

2

A

=

ab

C

Topic 4: Vectors

4.1

Magnitude of a vector 2 2 2

1 2 3

v

=

+

v

, where

1 2 3

v

 

 

=  

 

 

v

Distance between two points

( ,

x y z

₁ ₁

, )

₁ 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 the

midpoint of a line segment with endpoints

( ,

x y z

₁ ₁

, )

₁ ,

2 2 2

(

x y z

,

)

1 2

,

1 2

,

1 2

2

x

+

x

y

+

y

z

+

z













4.2 Scalar product

v w

⋅ =

v w

cos

θ

_{, where}

θ

_{is the angle between}

v

_and

w

1 1 2 2 3 3

v w

⋅ =

+

v w

, where

1 2 3

v

 

 

=  

 

 

v

,

1 2 3

w









=  









w

Angle between two

vectors

cos

θ

=

v w

1 1

+

v w

2 2

+

v w

3 3

v w

4.3 Vector equation of a line

r

=

a

+

λ

b

Parametric form of the

equation of a line

x

=

x

0

+

λ

l y

, ,

=

y

0

+

λ

m z

=

z

0

+

λ

n

Cartesian equations of a

line 0 0 0

x

y

z

l

m

n

(7)

4.5

Vector product

2 3 3 2 3 1 1 3 1 2 2 1

v w

−









× =



−





−







v w

where

1 2 3

v

 

 

=  

 

 

v

,

1 2 3

w









=  









w

sin

θ

×

=

v w

, where

θ

is the angle between

v

and

w

Area of a triangle

1

2

=

×

A

v w

where

v

and

w

form two sides of a triangle

4.6 Vector equation of a plane

r

=

a

+

λ

b +

µ

c

Equation of a plane (using the normal vector)

⋅ = ⋅

r n

a n

Cartesian equation of a plane

ax

+

by

+

cz

=

d

Topic 5: Statistics and probability

5.1

Population parameters Let

1

k

i i

n

f

=

∑

Mean

µ

1

k

i i i

f x

n

µ

=

∑

=

Variance

σ

2

(

)

2 2

2 1 1 2

k k

i i i i

i i

f x

n

µ

σ

= =

µ

−

=

∑

=

∑

−

Standard deviation

σ

(

)

2 1

k

i i i

f x

n

µ

σ

=

−

=

∑

5.2

Probability of an event

A

P ( )

( )

n A

A

n U

=

Complementary events

P ( )

A

+

P (

A

′

) 1

=

5.3 Combined events

P (

A

∪

B

)

=

P ( )

A

+

P ( )

B

−

P (

A

∩

B

)

(8)

5.4

Conditional probability

P (

)

P (

)

P ( )

A

B

A B

B

∩

=

Independent events

P (

A

∩

B

)

=

P ( ) P ( )

A

B

Bayes’ theorem

P ( | )

P ( ) P ( | )

P (

) P ( |

)

B

A B

B A

B

A B

B

A B

=

′

+

′

1 1 2 2 3 3

P(

) P( |

)

P (

| )

P(

) P( |

)

P(

) P( |

)

P(

) P( |

)

i i

i

B

A B

B A

B

A B

B

A B

B

A B

=

+

5.5 Expected value of a discrete random variable

X

E ( )

X

= =

µ

∑

x

P (

X

=

x

)

Expected value of a continuous random variable

X

E ( )

X

µ

∞

x f x

( ) d

x

−∞

= =

∫

Variance

Var ( )

X

=

E (

X

−

µ

)

2

=

E (

X

2

)

−

[

E ( )

X

]

2

Variance of a discrete random variable

X

2 2 2

Var ( )

X

=

∑

(

x

−

µ

) P (

X

=

x

)

=

∑

x

P (

X

=

x

)

−

µ

Variance of a continuous random variable

X

2 2 2

Var ( )

X

∞

(

x

µ

)

f x

( ) d

x

∞

x f x

( ) d

x

µ

−∞ −∞

=

∫

−

=

∫

−

5.6

Binomial distribution

Mean

Variance

~ B ( , )

P (

)

n

x

(1

)

n x

,

0, 1,

,

X

n p

X

x

p

x

n

x

−

 

⇒

=

 

−

=

 



E ( )

X

=

np

Var ( )

X

=

np

(1

−

p

)

Poisson distribution

Mean

Variance

e

~ Po ( )

P (

)

,

0, 1, 2,

!

x m

m

X

m

X

x

−

⇒

=



E ( )

X

=

m

Var ( )

X

=

m

5.7 Standardized normal

(9)

Topic 6: Calculus

6.1

Derivative of

f x

( )

0

d

(

)

( )

lim

d

h

y

f x

h

f x

y

f x

x

→

h

+

−





′

=

⇒

=









6.2 _{Derivative of} n

x

f x

( )

=

x

n

⇒

f x

′

( )

=

nx

n−1

Derivative of

sin

x

f x

( )

=

sin

x

⇒

f x

′

( )

=

cos

x

Derivative of

cos

x

f x

( )

=

cos

x

⇒

f x

′

( )

= −

sin

x

Derivative of

tan

x

f x

( )

=

tan

x

⇒

f x

′

( )

=

sec

2

x

Derivative of

e

x

f x

( )

=

e

x

⇒

f x

′

( )

=

e

x

Derivative of

ln

x

f x

( )

ln

x

f x

( )

1

x

′

=

⇒

=

Derivative of

sec

x

f x

( )

=

sec

x

⇒

f x

′

( )

=

sec tan

x

Derivative of

csc

x

f x

( )

=

csc

x

⇒

f x

′

( )

= −

csc cot

x

Derivative of

cot

x

2

( )

cot

( )

csc

f x

=

x

⇒

f x

′

= −

x

Derivative of

a

x

f x

( )

=

a

x

⇒

f x

′

( )

=

a

x

(ln )

a

Derivative of

log

_a

x

( )

log

( )

1

ln

a

f x

x

f x

x

a

′

=

⇒

=

Derivative of

arcsin

x

2

1

( )

arcsin

( )

1

f x

x

f x

x

′

=

⇒

=

−

Derivative of

arccos

x

2

1

( )

arccos

( )

1

f x

x

f x

x

′

=

⇒

= −

−

Derivative of

arctan

x

( )

arctan

( )

1

₂

1

f x

x

f x

x

′

=

⇒

=

+

Chain rule

y

=

g u

( )

, where

( )

d

y

u

f x

x

u

x

=

⇒

=

×

Product rule

d

y

v

u

y

uv

u

v

x

=

⇒

=

+

d

u

v

u

(10)

6.4 Standard integrals 1

d

,

1

n

x

C

n

+

=

+

≠ −

+

∫

1

d

x

ln

x

C

x

=

+

∫

sin d

x x

= −

cos

x

+

C

∫

cos d

x x

=

sin

x

+

C

∫

e d

x

=

e

x

+

C

∫

1

d

ln

x x

a

x

a

C

a

=

+

∫

2 2

1

d

x

arctan

x

C

a

x

a

 

=

 

+

 

∫

2 2

1

d

x

arcsin

x

C

,

x

a

x

 

=

 

+

<

 

−

∫

6.5

Area under a curve

Volume of revolution (rotation)

d

b

a

A

=

∫

y x

or b

d

a

A

=

∫

x y

2

π d

b

a

V

=

∫

y

x

or b

π d

2

a

V

=

∫

x

y

6.7

Integration by parts

d

v

u

x

uv

v

x

=

−

x

(11)

Options

Topic 7: Statistics and probability

Further mathematics HL topic 3

7.1 (3.1)

Probability generating function for a discrete random variable

X

( )

E ( )

x

P (

)

x x

G t

=

t

=

∑

X

=

x t

E ( )

X

=

G

′

(1)

(

)

2

Var ( )

X

=

G

′′

(1)

+

G

′

(1)

−

G

′

(1)

7.2 (3.2)

Linear combinations of two independent random variables

X

₁

,

X

₂

(

)

( )

(

)

( )

1 1 2 2 1 1 2 2

2 2

1 1 2 2 1 1 2 2

E

Var

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

x

n

=

∑

Variance

s

_n2

2 2

2 1 1 2

(

)

k k

i i i i

i i

n

f x

x

f x

s

x

n

= =

−

=

∑

=

∑

−

Standard deviation

s

_n

2 1

(

)

k i i i n

f x

x

s

n

=

−

=

∑

Unbiased estimate of population variance

s

_n2₋₁

2 2

2 2 1 1 2

1

(

)

1

k k

i i i i

i i

n n

f x

x

f x

n

s

x

n

= = −

−

=

−

∑

7.5 (3.5) Confidence intervals

Mean, with known

variance

x

z

n

σ

± ×

Mean, with unknown

variance n1

s

x

t

n

−

± ×

7.6 (3.6) Test statistics

(12)

Mean, with unknown variance 1

/

n

x

t

s

n

µ

−

=

7.7

(3.7) _{Sample product moment}

correlation coefficient 1

2 2 2 2

1 1 n i i i n n i i i i

x y

n x y

r

x

n x

y

n y

= − =

−

=



−



−















∑

Test statistic for

H

0:

ρ

= 0

2

1

n

t

r

−

=

−

Equation of regression line

of

x

on

y

1

2 2 1

(

)

n i i i n i i

x y

n x y

x

y

n y

= =



−











− =

−



−











∑

Equation of regression line

of

y

on

x

1

2 2 1

(

)

n i i i n i i

x y

n x y

y

x

n x

= =



−











− =

−



−











∑

Topic 8: Sets, relations and groups

8.1 (4.1)

De Morgan’s laws

(

)

(

)

A

B

A

B

A

B

A

B

′

∪

=

∩

′

∩

=

∪

Topic 9: Calculus

9.5 (5.5)

Euler’s method

1

(

,

)

n n n n

y

₊

=

y

+

h

×

f x y

;

x

_n₊₁

=

x

_n

+

h

, where

h

is a constant (step length)

Integrating factor for

( )

y

′ +

P x y

=

Q x

( )d

(13)

9.6

(5.6) Maclaurin series

2

( )

(0)

2!

x

f x

=

f

+

x f

′

+

f

′′

+



Taylor series

2

(

)

( )

(

)

( )

( ) ...

2!

x

a

f x

=

f a

+

x

−

a f a

′

+

−

f

′′

a

+

Taylor approximations

(with error term

R x

_n

( )

) ( )

(

)

( )

(

)

( ) ...

( )

!

n

x

a

f x

f a

x

a f a

f

a

R x

n

−

′

=

+

−

+ +

+

Lagrange form

( 1)

1

( )

(

)

(

1)!

+

=

−

+

n

n n

f

c

R x

x

a

n

, where

c

lies between

a

and

x

Maclaurin series for special functions

2

e

1

...

2!

x

= + +

x

+

2 3

ln (1

)

...

2

3

x

+

= −

+

−

3 5

sin

...

3!

5!

x

= −

x

+

−

2 4

cos

1

...

2!

4!

x

= −

+

−

3 5

arctan

...

3

5

x

= −

x

+

−

Topic 10: Discrete mathematics

10.7 (6.7)

Euler’s formula for connected planar graphs

2

v

− + =

e

f

, where

v

is the number of vertices,

e

is the number of edges,

f

is the number of faces

Planar, simple, connected graphs

3

6

≤

−

e

v

for

v

≥

3

2

4

≤

−

(14)

Formulae for distributions

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

Discrete distributions

Distribution Notation Probability mass

function

Mean Variance

Geometric

X

~ Geo ( )

p

x1

pq

−

for

x

=

1, 2,...

1

p

2

q

p

Negative binomial

X

~ NB( ,

r p

)

1

r x r

x

p q

r

−







−







for

x

=

r r

,

+

1,...

r

p

2

rq

p

Continuous distributions

Distribution Notation Probability

density function

Mean Variance

Normal

X

~ N ( ,

µ σ

2

)

2

1 2

1

e

2

π

x µ σ

σ

−  

(15)

Further mathematics

Topic 1: Linear algebra

1.2

Determinant of a 2 2×

matrix

det

a

b

ad

bc

c

d





=





⇒

=

−





A

Inverse of a 2 2× matrix 1

1

,

det

a

b

d

b

ad

bc

c

d

c

a

−









=





⇒

=





≠

−









A

Determinant of a 3 3×

matrix

det

a

b

c

e

f

d

f

d

e

d

e

f

a

b

c

h

k

g

k

g

h

g

h

k









=





⇒

=

−

+







