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
1
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
© International Baccalaureate Organization 2011
Mathematical studies SL: Formula booklet
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
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 
© International Baccalaureate Organization 2004
1
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
© International Baccalaureate Organization 2004
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
© International Baccalaureate Organization 2004
3
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
© International Baccalaureate Organization 2004
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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5
plane
6
© International Baccalaureate Organization 2004
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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7
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

© International Baccalaureate Organization 2004
,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
© International Baccalaureate Organization 2004
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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11
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

© International Baccalaureate Organization 2004
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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13
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
© International Baccalaureate Organization 2004
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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15
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
© International Baccalaureate Organization 2004
Continuous distributions
Distribution
Notation
Normal
X ~ N   , 2 
© International Baccalaureate Organization 2004
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 
© International Baccalaureate Organization 2004
f
d
c
k
g
e
h
© International Baccalaureate Organization 2004
19