Diploma Programme

Mathematics HL and Further mathematics HL Formula booklet For use during the course and in the examinations First examinations 2014

©Mathematical International Baccalaureate Organization 2011 studies SL: Formula booklet

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CONTENTS Formulae

1

Prior learning

1

Topic 1—Core: Algebra

2

Topic 2—Core: Functions and equations

3

Topic 3—Core: Circular functions and trigonometry

4

Topic 4—Core: Vectors

5

Topic 5—Core: Statistics and probability

7

Topic 6—Core: Calculus

9

Topic 7—Option: Statistics and probability (further mathematics HL topic 3) 11 Topic 8—Option: Sets, relations and groups (further mathematics HL topic 4) 13 Topic 9—Option: Calculus (further mathematics HL topic 5)

14

Topic 10—Option: Discrete mathematics (further mathematics HL topic 6) 15 Formulae for distributions (topic 5.6, 5.7, 7.1, further mathematics HL topic 3.1) 16 Discrete distributions

16

Continuous distributions

17

Further Mathematics Topic 1—Linear algebra

18

2

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Mathematical studies SL: Formula booklet

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Formulae Prior learning Area of a parallelogram

A  b  h , where b is the base, h is the height

Area of a triangle

1 A  (b  h) , where b is the base, h is the height 2

Area of a trapezium

1 A  (a  b) h , where a and b are the parallel sides, h is the height 2

Area of a circle

A  r 2 , where r is the radius

Circumference of a circle

C  2r , where r is the radius

Volume of a pyramid

1 V  (area of base  vertical height) 3

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

V  r 2 h , where r is the radius, h is the height

Area of the curved surface of a cylinder

A  2rh , where r is the radius, h is the height

Volume of a sphere

4 V  r 3 , where r is the radius 3

Volume of a cone

1 V  r 2 h , where r is the radius, h is the height 3

Distance between two points ( x1 , y1 ) and ( x2 , y2 )

d  ( x1  x2 )2  ( y1  y2 )2

Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 )

 x1  x2 y1  y2  ,   2   2

Solutions of a quadratic equation

The solutions of ax 2  bx  c  0 are x 

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b  b2  4ac 2a

Topic 1—Core: Algebra 1.1

un  u1  (n  1)d

The nth term of an arithmetic sequence The sum of n terms of an arithmetic sequence

n n Sn  (2u1  (n  1)d )  (u1  un ) 2 2 un  u1r n 1

The nth term of a geometric sequence

The sum of n terms of a S n  finite geometric sequence

1.2

u1 (r n  1) u1 (1  r n )  , r 1 r 1 1 r

u1 , r 1 1 r

The sum of an infinite geometric sequence

S

Exponents and logarithms

a x  b  x  log a b , where a  0, b  0 a x  e x ln a

log a a x  x  aloga x log b a 

log c a log c b

Combinations

 n n!    r  r !(n  r )!

Permutations

nP

Binomial theorem

 n (a  b)n  a n    a n 1b  1

1.5

Complex numbers

z  a  ib  r (cos  isin  )  rei  r cis

1.7

De Moivre’s theorem

 r (cos  isin )

1.3

2

r



n! (n  r )!

n

 n    a nr br  r

 bn

 r n (cos n  isin n )  r n ein  r n cis n

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Topic 2—Core: Functions and equations 2.5

2.6

Axis of symmetry of the graph of a quadratic function

y  ax 2  bx  c  axis of symmetry x  

Discriminant

  b 2  4ac

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b 2a

Topic 3—Core: Circular functions and trigonometry 3.1

l   r , where  is the angle measured in radians, r is the radius

Length of an arc

1 A   r 2 , where  is the angle measured in radians, r is the 2 radius

Area of a sector

3.2

Identities

tan  

sin  cos

sec 

1 cos

cosec  = Pythagorean identities

1 sin 

cos 2   sin 2   1 1  tan 2   sec 2  1  cot 2   csc 2

3.3

sin( A  B)  sin A cos B  cos A sin B

Compound angle identities

cos( A  B)  cos A cos B sin A sin B

tan( A  B ) 

Double angle identities

tan A  tan B 1 tan A tan B

sin 2  2sin  cos

cos 2  cos 2   sin 2   2cos 2   1  1  2sin 2 

tan 2 

3.7

4

2 tan  1  tan 2 

Cosine rule

c 2  a 2  b 2  2ab cos C ; cos C 

Sine rule

a b c   sin A sin B sin C

Area of a triangle

1 A  ab sin C 2

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a 2  b2  c2 2ab

Topic 4—Core: Vectors 4.1

Magnitude of a vector

Distance between two points ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 , z1 ) , ( x2 , y2 , z2 ) 4.2

Scalar product

 v1    v  v  v2  v3 , where v   v2  v   3 2 1

2

2

d  ( x1  x2 )2  ( y1  y2 )2  ( z1  z2 )2

 x1  x2 y1  y2 z1  z2  , ,   2 2   2

v  w  v w cos , where  is the angle between v and w

 v1   w1      v  w  v1w1  v2 w2  v3w3 , where v   v2  , w   w2  v  w   3  3

4.3

4.5

v1w1  v2 w2  v3 w3 v w

Angle between two vectors

cos 

Vector equation of a line

r = a + λb

Parametric form of the equation of a line

x  x0  l , y  y0  m, z  z0  n

Cartesian equations of a line

x  x0 y  y0 z  z0   l m n

Vector product

 v2 w3  v3 w2   v1   w1        v  w   v3 w1  v1w3  where v   v2  , w   w2  v w v w  v  w   1 2 2 1  3  3 v  w  v w sin  , where  is the angle between v and w

4.6

1 v  w where v and w form two sides of a triangle 2

Area of a triangle

A

Vector equation of a plane

r = a + λb +  c

Equation of a plane (using the normal vector)

r n  an

Cartesian equation of a

ax  by  cz  d

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plane

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Topic 5—Core: Statistics and probability k

Let n   fi

5.1

i 1

Population parameters k

Mean 



fx

i i

i 1

n k

Variance  2

2 

 f x   i 1

i

Standard deviation 

5.2

5.3





n k

 f x   i 1

Probability of an event A P( A) 

i

k

2

i

fx i 1

i i

n

2

 2

2

i

n n( A) n(U )

Complementary events

P( A)  P( A)  1

Combined events

P( A  B)  P( A)  P( B)  P( A  B)

Mutually exclusive events

P( A  B)  P( A)  P( B)

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Topic 5—Core: Statistics and probability (continued) 5.4

P A B 

Independent events

P( A  B)  P( A) P( B)

Bayes’ Theorem

P  B | A 

P( Bi | A) 

5.5

P( A | Bi ) P( Bi ) P( A | B1 ) P( B1 )  P( A | B2 ) P( B2 ) 

P( A | Bn ) P( Bn )



E( X )     x f ( x)dx 

Var( X )  E( X   ) 2  E( X 2 )   E(X ) 

2

Variance of a discrete random variable X Variance of a continuous random variable X

8

P( B )P  A | B   P( B)P  A | B 

x

Variance

5.7

P( B )P  A | B 

E( X )     x P( X  x)

Expected value of a discrete random variable X Expected value of a continuous random variable X

5.6

P( A  B) P( B)

Conditional probability

Var( X )   ( x   )2 P( X  x)   x2 P( X  x)   2 







Var( X )   ( x   )2 f ( x)dx   x 2 f ( x)dx   2

Mean

 n X ~ B(n , p)  P ( X  x)    p x (1  p) n  x , x  0,1,  x E( X )  np

Variance

Var( X )  np(1  p)

Poisson distribution

X ~ Po (m)  P ( X  x) 

Mean

E( X )  m

Variance

Var( X )  m

Standardized normal variable

z

Binomial distribution

m x e m , x  0,1, 2, x!

x



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,n

Topic 6—Core: Calculus 6.2

dy  f ( x  h)  f ( x )   f ( x)  lim   h 0 dx h  

Derivative of f ( x)

y  f ( x) 

Derivative of x n

f ( x)  xn  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)  sec2 x

Derivative of e x

f ( x)  e x  f ( x)  e x

Derivative of ln x

f ( x)  ln x  f ( x) 

Derivative of sec x

f ( x)  sec x  f ( x)  sec x tan x

Derivative of csc x

f ( x)  csc x  f ( x)  csc x cot x

Derivative of cot x

f ( x)  cot x  f ( x)  csc2 x

Derivative of a x

f ( x)  a x  f ( x)  a x (ln a)

Derivative of loga x

f ( x)  log a x  f ( x) 

Derivative of arcsin x

f ( x)  arcsin x  f ( x) 

Derivative of arccos x

f ( x)  arccos x  f ( x)  

Derivative of arctan x

f ( x)  arctan x  f ( x) 

Chain rule

y  g (u ) , where u  f ( x) 

Product rule

y  uv 

Quotient rule

du dv u u dy y   dx 2 dx v dx v

1 x ln a

dy dv du u v dx dx dx v

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1 x

9

1 1  x2

1 1  x2

1 1  x2 dy dy du   dx du dx

Topic 6—Core: Calculus (continued) 6.4

n  x dx 

Standard integrals

x n 1  C , n  1 n 1

1

 x dx  ln x  C

 sinx dx   cos x  C  cosx dx  sin x  C e

a

6.5

10

Integration by parts

x

dx 

1 x a C ln a

1 1  x dx  arctan    C 2 x a a

a

2



 x dx  arcsin    C , x  a a a x 1

2

2

b

b

a

a

A   ydx or A   xdy

Area under a curve Volume of revolution (rotation)

6.7

dx  e x  C

x

b

b

a

a

V   πy 2 dx or V   πx 2 dy

dv

du

 u dx dx  uv   v dx dx or  udv  uv   vdu

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Topic 7—Option: Statistics and probability (further mathematics HL topic 3) Probability generating function for a discrete (3.1) random variable X 7.1

Linear combinations of 7.2 (3.2) two independent random variables X1 , X 2 7.3

G (t )  E (t X )   P( X  x)t x x

E  a1 X1  a2 X 2   a1E  X1   a2 E  X 2  Var  a1 X1  a2 X 2   a1 Var  X1   a2 Var  X 2  2

2

Sample statistics k

fx

(3.3) Mean x

x

i i

i 1

n k

Variance sn2

 f (x

sn2 

i

i 1

i



n k

Standard deviation sn

 f (x

sn 

i

i 1

k

fx

 x )2

i

2

i i

i 1

 x2

n

 x )2

n k

k

 f (x  x )  f x 2

Unbiased estimate of population variance sn21 7.5 Confidence intervals (3.5) Mean, with known variance Mean, with unknown variance

7.6 Test statistics (3.6) Mean, with known variance

n 2 sn21  sn  n 1

x  z

x t

z

 n

sn 1 n

x 

/ n

Mean, with unknown

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i 1

i

i

n 1



i 1

i i

n 1

2



n 2 x n 1

variance

7.7

t

Sample product moment

sn 1 / n n

correlation coefficient

r

Test statistic for H0:

tr

ρ=0

12

x 

x y i 1

i

i

 nx y

n  n 2  2 2    xi  nx   yi  ny 2   i 1  i 1 

n2 1 r2

Equation of regression line of x on y

 n    xi yi  nx y  ( y  y ) x  x   i n1  2 2    yi  ny   i 1 

Equation of regression line of y on x

 n    xi yi  nx y  ( x  x ) y  y   i n1  2 2    xi  nx   i 1 

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Topic 8—Option: Sets, relations and groups (further mathematics HL topic 4) De Morgan’s laws 8.1 (4.1)

( A  B)  A  B ( A  B)  A  B

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Topic 9—Option: Calculus (further mathematics HL topic 5) f ( x)  f (0)  x f (0) 

Maclaurin series 9.6 (5.6)

( x  a) 2  f (a )  ... 2!

Taylor series

f ( x)  f (a )  ( x  a) f (a ) 

Taylor approximations (with error term Rn ( x) )

f ( x)  f (a)  ( x  a ) f (a )  ... 

Lagrange form

Rn ( x) 

Maclaurin series for special functions

ex  1  x 

( x  a)n ( n ) f (a)  Rn ( x) n!

f ( n1) (c) ( x  a)n 1 , where c lies between a and x (n  1)!

x2  ... 2!

ln(1  x)  x 

x 2 x3   ... 2 3

sin x  x 

x3 x5   ... 3! 5!

cos x  1 

x2 x4   ... 2! 4!

arctan x  x 

x3 x5   ... 3 5

yn1  yn  h  f ( xn , yn ) ; xn1  xn  h , where h is a constant (steplength)

Euler’s method 9.5 (5.5) Integrating factor for y  P( x) y  Q( x)

14

x2 f (0)  2!

e

P ( x )dx

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Topic 10—Option: Discrete mathematics (further mathematics HL topic 6) 10.7 Euler’s formula for (6.7) connected planar graphs Planar, simple, connected graphs

v  e  f  2 , where v is the number of vertices, e is the number of edges, f is the number of faces e  3v  6 for v  3 e  2v  4 if the graph has no triangles

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Formulae for distributions (topic 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 

pq x 1

1 p

q p2

r p

rq p2

for x  1,2,... Negative binomial

X ~ NB  r , p 

 x  1 r x  r  p q  r  1

for x  r , r  1,...

16

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Continuous distributions Distribution

Notation

Normal

X ~ N   , 2 

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Probability density function 1  x   

  1 e 2  2π

17

2

Mean

Variance



2

Further Mathematics Topic 1—Linear algebra

1.2

18

Determinant of a 2  2 matrix

a b A   det A  A  ad  bc c d

Inverse of a 2  2 matrix

a b 1  d 1 A  A   det A  c c d

b   , ad  bc a

Determinant of a 3  3 matrix

a b  A d e g h 

f d b k g

c e  f   det A  a h k 

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f d c k g

e h

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