Mathematical studies SL: Formula booklet 1ยฉ International Baccalaureate Organization 2011
Mathematics HL
and Further mathematics HL
Formula booklet
For use during the course and in the examinations
First examinations 2014
Diploma Programme
2 ยฉ International Baccalaureate Organization 2011
CONTENTS
Formulae 1
Prior learning 1
Topic 1โ€”Core: Algebra 2
Topic 2โ€”Core: Functions and equations 3
Topic 3โ€”Core: Circular functions and trigonometry 4
Topic 4โ€”Core: Vectors 5
Topic 5โ€”Core: Statistics and probability 7
Topic 6โ€”Core: Calculus 9
Topic 7โ€”Option: Statistics and probability (further mathematics HL topic
3) 11
Topic 8โ€”Option: Sets, relations and groups (further mathematics HL
topic 4) 13
Topic 9โ€”Option: Calculus (further mathematics HL topic 5) 14
Topic 10โ€”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 1โ€”Linear algebra 18
Mathematical studies SL: Formula booklet 3
ยฉ International Baccalaureate Organization 2004 1
Formulae
Prior learning
Area of a parallelogram A b h๏€ฝ ๏‚ด , where b is the base, h is the height
Area of a triangle
1
( )
2
๏€ฝ ๏‚ดA b h , where b is the base, h is the height
Area of a trapezium
1
( )
2
๏€ฝ ๏€ซA a b h , where a and b are the parallel sides, h is the height
Area of a circle
2
๏€ฝ ๏ฐA r , where r is the radius
Circumference of a circle 2๏€ฝ ๏ฐC r , where r is the radius
Volume of a pyramid
1
(area of base vertical height)
3
๏€ฝ ๏‚ดV
Volume of a cuboid ๏€ฝ ๏‚ด ๏‚ดV l w h , where l is the length, w is the width, h is the height
Volume of a cylinder 2
๏€ฝ ๏ฐV r h , where r is the radius, h is the height
Area of the curved surface of
a cylinder
2๏€ฝ ๏ฐA rh, where r is the radius, h is the height
Volume of a sphere 34
3
๏€ฝ ๏ฐV r , where r is the radius
Volume of a cone 21
3
๏€ฝ ๏ฐV r h, where r is the radius, h is the height
Distance between two
points 1 1( , )x y and 2 2( , )x y
2 2
1 2 1 2( ) ( )๏€ฝ ๏€ญ ๏€ซ ๏€ญd x x y y
Coordinates of the midpoint of
a line segment with endpoints
1 1( , )x y and 2 2( , )x y
1 2 1 2
,
2 2
๏€ซ ๏€ซ๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
x x y y
Solutions of a quadratic
equation The solutions of 2
0ax bx c๏€ซ ๏€ซ ๏€ฝ are
2
4
2
b b ac
x
a
๏€ญ ๏‚ฑ ๏€ญ
๏€ฝ
2 ยฉ International Baccalaureate Organization 2004
Topic 1โ€”Core: Algebra
1.1 The nth
term of an
arithmetic sequence
1 ( 1)๏€ฝ ๏€ซ ๏€ญnu u n d
The sum of n terms of
an arithmetic sequence
1 1(2 ( 1) ) ( )
2 2
๏€ฝ ๏€ซ ๏€ญ ๏€ฝ ๏€ซn n
n n
S u n d u u
The nth
term of a
geometric sequence
1
1
๏€ญ
๏€ฝ n
nu u r
The sum of n terms of a
finite geometric sequence
1 1( 1) (1 )
1 1
๏€ญ ๏€ญ
๏€ฝ ๏€ฝ
๏€ญ ๏€ญ
n n
n
u r u r
S
r r
, 1๏‚นr
The sum of an infinite
geometric sequence
1
, 1
1
๏€ฝ ๏€ผ
๏€ญ
u
S r
r
1.2 Exponents and
logarithms
log๏€ฝ ๏ƒ› ๏€ฝx
aa b x b, where 0, 0a b๏€พ ๏€พ
ln
e๏€ฝx x a
a
log
log ๏€ฝ ๏€ฝ a xx
a a x a
log
log
log
๏€ฝ c
b
c
a
a
b
1.3 Combinations
Permutations
!
!( )!
๏ƒฆ ๏ƒถ
๏€ฝ๏ƒง ๏ƒท
๏€ญ๏ƒจ ๏ƒธ
n n
r r n r
!
( )!
Pr
n n
n r
๏€ฝ
๏€ญ
Binomial theorem 1
( )
1
๏€ญ ๏€ญ๏ƒฆ ๏ƒถ ๏ƒฆ ๏ƒถ
๏€ซ ๏€ฝ ๏€ซ ๏€ซ ๏€ซ ๏€ซ ๏€ซ๏ƒง ๏ƒท ๏ƒง ๏ƒท
๏ƒจ ๏ƒธ ๏ƒจ ๏ƒธ
n n n n r r nn n
a b a a b a b b
r
1.5 Complex numbers i (cos isin )๏ฑ ๏ฑ๏€ฝ ๏€ซ ๏€ฝ ๏€ซz a b r e cis๏ฑ
๏ฑ๏€ฝ ๏€ฝi
r r
1.7 De Moivreโ€™s theorem ๏› ๏(cos isin ) (cos isin ) e cis๏ฑ
๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ ๏€ฝ
n n n in n
r r n n r r n
ยฉ International Baccalaureate Organization 2004 3
Topic 2โ€”Core: Functions and equations
2.5 Axis of symmetry of the
graph of a quadratic
function
2
axis of symmetry
2
b
y ax bx c x
a
๏€ฝ ๏€ซ ๏€ซ ๏ƒž ๏€ฝ ๏€ญ
2.6
Discriminant 2
4๏„ ๏€ฝ ๏€ญb ac
4 ยฉ International Baccalaureate Organization 2004
Topic 3โ€”Core: Circular functions and
trigonometry
3.1 Length of an arc ๏ฑ๏€ฝl r , where ๏ฑ is the angle measured in radians, r is the radius
Area of a sector 21
,
2
๏ฑ๏€ฝA r where ๏ฑ is the angle measured in radians, r is the
radius
3.2 Identities
sin
tan
cos
๏ฑ
๏ฑ
๏ฑ
๏€ฝ
1
sec
cos
๏ฑ
๏ฑ
๏€ฝ
cosec๏ฑ =
1
sin๏ฑ
Pythagorean identities 2 2
2 2
2 2
cos sin 1
1 tan sec
1 cot csc
๏ฑ ๏ฑ
๏ฑ ๏ฑ
๏ฑ ๏ฑ
๏€ซ ๏€ฝ
๏€ซ ๏€ฝ
๏€ซ ๏€ฝ
3.3 Compound angle
identities
sin( ) sin cos cos sin๏‚ฑ ๏€ฝ ๏‚ฑA B A B A B
cos( ) cos cos sin sin๏‚ฑ ๏€ฝA B A B A B
tan tan
tan( )
1 tan tan
๏‚ฑ
๏‚ฑ ๏€ฝ
A B
A B
A B
Double angle identities sin2 2sin cos๏ฑ ๏ฑ ๏ฑ๏€ฝ
2 2 2 2
cos2 cos sin 2cos 1 1 2sin๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ๏€ฝ ๏€ญ ๏€ฝ ๏€ญ ๏€ฝ ๏€ญ
2
2tan
tan2
1 tan
๏ฑ
๏ฑ
๏ฑ
๏€ฝ
๏€ญ
3.7 Cosine rule
2 2 2
2 2 2
2 cos ; cos
2
๏€ซ ๏€ญ
๏€ฝ ๏€ซ ๏€ญ ๏€ฝ
a b c
c a b ab C C
ab
Sine rule
sin sin sin
๏€ฝ ๏€ฝ
a b c
A B C
Area of a triangle
1
sin
2
๏€ฝA ab C
ยฉ International Baccalaureate Organization 2004 5
Topic 4โ€”Core: Vectors
4.1 Magnitude of a vector
2 2 2
1 2 3๏€ฝ ๏€ซ ๏€ซv v vv , where
1
2
3
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
v
v
v
v
Distance between two
points 1 1 1( , , )x y z and
2 2 2( , , )x y z
2 2 2
1 2 1 2 1 2( ) ( ) ( )๏€ฝ ๏€ญ ๏€ซ ๏€ญ ๏€ซ ๏€ญd x x y y z z
Coordinates of the
midpoint of a line
segment with endpoints
1 1 1( , , )x y z , 2 2 2( , , )x y z
1 2 1 2 1 2
, ,
2 2 2
๏€ซ ๏€ซ ๏€ซ๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
x x y y z z
4.2 Scalar product cos๏ฑ๏ƒ— ๏€ฝv w v w , where ๏ฑ is the angle between v and w
1 1 2 2 3 3๏ƒ— ๏€ฝ ๏€ซ ๏€ซv w v w v wv w , where
1
2
3
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
v
v
v
v ,
1
2
3
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
w
w
w
w
Angle between two
vectors
1 1 2 2 3 3
cos๏ฑ
๏€ซ ๏€ซ
๏€ฝ
v w v w v w
v w
4.3 Vector equation of a
line
= + ฮปr a b
Parametric form of the
equation of a line
0 0 0, ,๏ฌ ๏ฌ ๏ฌ๏€ฝ ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ ๏€ซx x l y y m z z n
Cartesian equations of
a line
0 0 0๏€ญ ๏€ญ ๏€ญ
๏€ฝ ๏€ฝ
x x y y z z
l m n
4.5 Vector product
2 3 3 2
3 1 1 3
1 2 2 1
v w v w
v w v w
v w v w
๏€ญ๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏‚ด ๏€ฝ ๏€ญ๏ƒง ๏ƒท
๏ƒง ๏ƒท๏€ญ๏ƒจ ๏ƒธ
v w where
1
2
3
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
v
v
v
v ,
1
2
3
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
w
w
w
w
sin๏ฑ๏‚ด ๏€ฝv w v w , where๏ฑ is the angle between v and w
Area of a triangle
1
2
๏€ฝ ๏‚ดA v w where v and w form two sides of a triangle
4.6 Vector equation of a
plane
= + ๏ญฮปr a b + c
Equation of a plane
(using the normal vector)
๏ƒ— ๏€ฝ ๏ƒ—r n a n
Cartesian equation of a ax by cz d๏€ซ ๏€ซ ๏€ฝ
6 ยฉ International Baccalaureate Organization 2004
plane
ยฉ International Baccalaureate Organization 2004 7
Topic 5โ€”Core: Statistics and probability
5.1
Population parameters
Mean ๏ญ
Variance 2
๏ณ
Standard deviation ๏ณ
Let
1๏€ฝ
๏€ฝ ๏ƒฅ
k
i
i
n f
1
๏ญ ๏€ฝ
๏€ฝ
๏ƒฅ
k
i i
i
f x
n
๏€จ ๏€ฉ
2 2
2 21 1
๏ญ
๏ณ ๏ญ๏€ฝ ๏€ฝ
๏€ญ
๏€ฝ ๏€ฝ ๏€ญ
๏ƒฅ ๏ƒฅ
k k
i i i i
i i
f x f x
n n
๏€จ ๏€ฉ
2
1
๏ญ
๏ณ ๏€ฝ
๏€ญ
๏€ฝ
๏ƒฅ
k
i i
i
f x
n
5.2 Probability of an event A
( )
P( )
( )
๏€ฝ
n A
A
n U
Complementary events P( ) P( ) 1๏‚ข๏€ซ ๏€ฝA A
5.3 Combined events P( ) P( ) P( ) P( )๏ƒˆ ๏€ฝ ๏€ซ ๏€ญ ๏ƒ‡A B A B A B
Mutually exclusive
events
P( ) P( ) P( )๏ƒˆ ๏€ฝ ๏€ซA B A B
8 ยฉ International Baccalaureate Organization 2004
Topic 5โ€”Core: Statistics and probability
(continued)
5.4 Conditional probability ๏€จ ๏€ฉ P( )
P
P( )
๏ƒ‡
๏€ฝ
A B
A B
B
Independent events P( ) P( ) P( )๏ƒ‡ ๏€ฝA B A B
Bayesโ€™ Theorem ๏€จ ๏€ฉ
๏€จ ๏€ฉ
๏€จ ๏€ฉ ๏€จ ๏€ฉ
P( )P |
P |
P( )P | P( )P |
๏€ฝ
๏‚ข๏€ซ ๏‚ข
B A B
B A
B A B B A B
1 1 2 2
( | ) ( )
( | )
( | ) ( ) ( | ) ( ) ( | ) ( )
i i
i
n n
P A B P B
P B A
P A B P B P A B P B P A B P B
๏€ฝ
๏€ซ ๏€ซ
5.5 Expected value of a
discrete random
variable X
E( ) P( )๏ญ๏€ฝ ๏€ฝ ๏€ฝ๏ƒฅx
X x X x
Expected value of a
continuous random
variable X
E( ) ( )d๏ญ
๏‚ฅ
๏€ญ๏‚ฅ
๏€ฝ ๏€ฝ ๏ƒฒX x f x x
Variance ๏› ๏
22 2
Var( ) E( ) E( ) E( )๏ญ๏€ฝ ๏€ญ ๏€ฝ ๏€ญX X X X
Variance of a discrete
random variable X
2 2 2
Var( ) ( ) P( ) P( )๏ญ ๏ญ๏€ฝ ๏€ญ ๏€ฝ ๏€ฝ ๏€ฝ ๏€ญ๏ƒฅ ๏ƒฅX x X x x X x
Variance of a continuous
random variable X
2 2 2
Var( ) ( ) ( )d ( )d๏ญ ๏ญ
๏‚ฅ ๏‚ฅ
๏€ญ๏‚ฅ ๏€ญ๏‚ฅ
๏€ฝ ๏€ญ ๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒX x f x x x f x x
5.6 Binomial distribution
Mean
Variance
~ B( , ) P( ) (1 ) , 0,1, ,๏€ญ๏ƒฆ ๏ƒถ
๏ƒž ๏€ฝ ๏€ฝ ๏€ญ ๏€ฝ๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
x n xn
X n p X x p p x n
x
E( ) ๏€ฝX np
Var( ) (1 )๏€ฝ ๏€ญX np p
Poisson distribution
Mean
Variance
o
e
~ P ( ) P( ) , 0,1, 2,
!
๏€ญ
๏ƒž ๏€ฝ ๏€ฝ ๏€ฝ
x m
m
X m X x x
x
E( ) ๏€ฝX m
Var( ) ๏€ฝX m
5.7 Standardized normal
variable
๏ญ
๏ณ
๏€ญ
๏€ฝ
x
z
ยฉ International Baccalaureate Organization 2004 9
Topic 6โ€”Core: Calculus
6.2 Derivative of ( )f x
0
d ( ) ( )
( ) ( ) lim
d ๏‚ฎ
๏€ซ ๏€ญ๏ƒฆ ๏ƒถ๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ฝ ๏ƒง ๏ƒท
๏ƒจ ๏ƒธh
y f x h f x
y f x f x
x h
Derivative of n
x 1
( ) ( ) ๏€ญ
๏‚ข๏€ฝ ๏ƒž ๏€ฝn n
f x x f x nx
Derivative of sin x ( ) sin ( ) cos๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x
Derivative of cos x ( ) cos ( ) sin๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x
Derivative of tan x 2
( ) tan ( ) sec๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x
Derivative of ex
( ) e ( ) e๏‚ข๏€ฝ ๏ƒž ๏€ฝx x
f x f x
Derivative of ln x
1
( ) ln ( )๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x
x
Derivative of sec x ( ) sec ( ) sec tan๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x x
Derivative of csc x ( ) csc ( ) csc cot๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x x
Derivative of cot x 2
( ) cot ( ) csc๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x
Derivative of x
a ( ) ( ) (ln )๏‚ข๏€ฝ ๏ƒž ๏€ฝx x
f x a f x a a
Derivative of loga x
1
( ) log ( )
ln
๏‚ข๏€ฝ ๏ƒž ๏€ฝaf x x f x
x a
Derivative of arcsin x
2
1
( ) arcsin ( )
1
๏‚ข๏€ฝ ๏ƒž ๏€ฝ
๏€ญ
f x x f x
x
Derivative of arccos x
2
1
( ) arccos ( )
1
๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญ
๏€ญ
f x x f x
x
Derivative of arctan x 2
1
( ) arctan ( )
1
๏‚ข๏€ฝ ๏ƒž ๏€ฝ
๏€ซ
f x x f x
x
Chain rule ( )๏€ฝy g u , where
d d d
( )
d d d
๏€ฝ ๏ƒž ๏€ฝ ๏‚ด
y y u
u f x
x u x
Product rule
d d d
d d d
๏€ฝ ๏ƒž ๏€ฝ ๏€ซ
y v u
y uv u v
x x x
Quotient rule 2
d d
d d d
d
๏€ญ
๏€ฝ ๏ƒž ๏€ฝ
u v
v u
u y x xy
v x v
10 ยฉ International Baccalaureate Organization 2004
Topic 6โ€”Core: Calculus (continued)
6.4 Standard integrals
1
d , 1
1
๏€ซ
๏€ฝ ๏€ซ ๏‚น ๏€ญ
๏€ซ๏ƒฒ
n
n x
x x C n
n
1
d ln๏€ฝ ๏€ซ๏ƒฒ x x C
x
sin d cos๏€ฝ ๏€ญ ๏€ซ๏ƒฒ x x x C
cos d sin๏€ฝ ๏€ซ๏ƒฒ x x x C
e d e๏€ฝ ๏€ซ๏ƒฒ
x x
x C
1
d
ln
๏€ฝ ๏€ซ๏ƒฒ
x x
a x a C
a
2 2
1 1
d arctan
๏ƒฆ ๏ƒถ
๏€ฝ ๏€ซ๏ƒง ๏ƒท
๏€ซ ๏ƒจ ๏ƒธ
๏ƒฒ
x
x C
a x a a
2 2
1
d arcsin ,
๏ƒฆ ๏ƒถ
๏€ฝ ๏€ซ ๏€ผ๏ƒง ๏ƒท
๏ƒจ ๏ƒธ๏€ญ
๏ƒฒ
x
x C x a
aa x
6.5 Area under a curve
Volume of revolution
(rotation)
d๏€ฝ ๏ƒฒ
b
a
A y x or d๏€ฝ ๏ƒฒ
b
a
A x y
2 2
ฯ€ d or ฯ€ d๏€ฝ ๏€ฝ๏ƒฒ ๏ƒฒ
b b
a a
V y x V x y
6.7 Integration by parts
d d
d d
d d
๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒ
v u
u x uv v x
x x
or d d๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒu v uv v u
ยฉ International Baccalaureate Organization 2004 11
Topic 7โ€”Option: Statistics and probability
(further mathematics HL topic 3)
7.1
(3.1)
Probability generating
function for a discrete
random variable X
( ) ( ) ( )X x
x
G t E t P X x t๏€ฝ ๏€ฝ ๏€ฝ๏ƒฅ
7.2
(3.2)
Linear combinations of
two independent random
variables 1 2,X X
๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ
๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ
1 1 2 2 1 1 2 2
2 2
1 1 2 2 1 1 2 2
E E E
Var Var Var
๏‚ฑ ๏€ฝ ๏‚ฑ
๏‚ฑ ๏€ฝ ๏€ซ
a X a X a X a X
a X a X a X a X
7.3
(3.3)
Sample statistics
Mean x
Variance 2
ns
Standard deviation ns
1
22
2 21 1
2
1
( )
( )
๏€ฝ
๏€ฝ ๏€ฝ
๏€ฝ
๏€ฝ
๏€ญ
๏€ฝ ๏€ฝ ๏€ญ
๏€ญ
๏€ฝ
๏ƒฅ
๏ƒฅ ๏ƒฅ
๏ƒฅ
k
i i
i
k k
i i i i
i i
n
k
i i
i
n
f x
x
n
f x x f x
s x
n n
f x x
s
n
Unbiased estimate of
population variance
2
1๏€ญns
22
2 2 21 1
1
( )
1 1 1 1
๏€ฝ ๏€ฝ
๏€ญ
๏€ญ
๏€ฝ ๏€ฝ ๏€ฝ ๏€ญ
๏€ญ ๏€ญ ๏€ญ ๏€ญ
๏ƒฅ ๏ƒฅ
k k
i i i i
i i
n n
f x x f x
n n
s s x
n n n n
7.5
(3.5)
Confidence intervals
Mean, with known
variance
Mean, with unknown
variance
๏ณ
๏‚ฑ ๏‚ดx z
n
1๏€ญ
๏‚ฑ ๏‚ด ns
x t
n
7.6
(3.6)
Test statistics
Mean, with known
variance
Mean, with unknown
/
๏ญ
๏ณ
๏€ญ
๏€ฝ
x
z
n
12 ยฉ International Baccalaureate Organization 2004
variance
1 /
๏ญ
๏€ญ
๏€ญ
๏€ฝ
n
x
t
s n
7.7 Sample product
moment
correlation coefficient
Test statistic for H0:
ฯ = 0
Equation of regression
line of x on y
Equation of regression
line of y on x
๏ƒท
๏ƒธ
๏ƒถ
๏ƒง
๏ƒจ
๏ƒฆ
๏€ญ๏ƒท
๏ƒธ
๏ƒถ
๏ƒง
๏ƒจ
๏ƒฆ
๏€ญ
๏€ญ
๏€ฝ
๏ƒฅ๏ƒฅ
๏ƒฅ
๏€ฝ๏€ญ
๏€ฝ
n
i
i
n
i
i
n
i
ii
ynyxnx
yxnyx
r
1
22
1
22
1
2
1
2
r
n
rt
๏€ญ
๏€ญ
๏€ฝ
)(
1
22
1
yy
yny
yxnyx
xx n
i
i
n
i
ii
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๏ƒง
๏ƒง
๏ƒง
๏ƒง
๏ƒจ
๏ƒฆ
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๏€ฝ๏€ญ
๏ƒฅ
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)(
1
22
1
xx
xnx
yxnyx
yy n
i
i
n
i
ii
๏€ญ
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๏ƒฅ
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๏€ฝ
๏€ฝ
ยฉ International Baccalaureate Organization 2004 13
Topic 8โ€”Option: Sets, relations and groups
(further mathematics HL topic 4)
8.1
(4.1)
De Morganโ€™s laws ( )
( )
๏‚ข ๏‚ข ๏‚ข๏ƒˆ ๏€ฝ ๏ƒ‡
๏‚ข ๏‚ข ๏‚ข๏ƒ‡ ๏€ฝ ๏ƒˆ
A B A B
A B A B
14 ยฉ International Baccalaureate Organization 2004
Topic 9โ€”Option: Calculus
(further mathematics HL topic 5)
9.6
(5.6)
Maclaurin series
2
( ) (0) (0) (0)
2!
๏‚ข ๏‚ข๏‚ข๏€ฝ ๏€ซ ๏€ซ ๏€ซ
x
f x f x f f
Taylor series
2
( )
( ) ( ) ( ) ( ) ( ) ...
2!
๏€ญ ๏‚ข๏‚ข๏‚ข๏€ฝ ๏€ซ ๏€ญ ๏€ซ ๏€ซ
x a
f x f a x a f a f a
Taylor approximations
(with error term ( )nR x )
Lagrange form
( )( )
( ) ( ) ( ) ( ) ... ( ) ( )
!
๏€ญ
๏‚ข๏€ฝ ๏€ซ ๏€ญ ๏€ซ ๏€ซ ๏€ซ
n
n
n
x a
f x f a x a f a f a R x
n
( 1)
1( )
( ) ( )
( 1)!
๏€ซ
๏€ซ
๏€ฝ ๏€ญ
๏€ซ
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!
๏€ฝ ๏€ซ ๏€ซ ๏€ซx x
x
2 3
ln(1 ) ...
2 3
๏€ซ ๏€ฝ ๏€ญ ๏€ซ ๏€ญ
x x
x x
3 5
sin ...
3! 5!
๏€ฝ ๏€ญ ๏€ซ ๏€ญ
x x
x x
2 4
cos 1 ...
2! 4!
๏€ฝ ๏€ญ ๏€ซ ๏€ญ
x x
x
3 5
arctan ...
3 5
๏€ฝ ๏€ญ ๏€ซ ๏€ญ
x x
x x
9.5
(5.5)
Eulerโ€™s method 1 ( , )๏€ซ ๏‚ด๏€ฝ ๏€ซn n n ny y h f x y ; 1๏€ซ ๏€ฝ ๏€ซn nx x h , where h is a constant (step-
length)
Integrating factor for
( ) ( )๏‚ข๏€ซ ๏€ฝy P x y Q x
( )d
e๏ƒฒP x x
ยฉ International Baccalaureate Organization 2004 15
Topic 10โ€”Option: Discrete mathematics
(further mathematics HL topic 6)
10.7
(6.7)
Eulerโ€™s formula for
connected planar
graphs
2๏€ญ ๏€ซ ๏€ฝv 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๏‚ฃ ๏€ญe v for 3v ๏‚ณ
2 4๏‚ฃ ๏€ญe v if the graph has no triangles
16 ยฉ International Baccalaureate Organization 2004
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 ๏€จ ๏€ฉ~ GeoX p 1๏€ญx
pq
for 1,2,...๏€ฝx
1
p 2
q
p
Negative binomial ๏€จ ๏€ฉ~ NB ,X r p 1
1
๏€ญ๏€ญ๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ญ๏ƒจ ๏ƒธ
r x rx
p q
r
for , 1,...๏€ฝ ๏€ซx r r
r
p 2
rq
p
ยฉ International Baccalaureate Organization 2004 17
Continuous distributions
Distribution Notation Probability
density function
Mean Variance
Normal ๏€จ ๏€ฉ2
~ N ,๏ญ ๏ณX
2
1
21
e
2ฯ€
๏ญ
๏ณ
๏ณ
๏€ญ๏ƒฆ ๏ƒถ
๏€ญ ๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
x
๏ญ 2
๏ณ
18 ยฉ International Baccalaureate Organization 2004
Further Mathematics Topic 1โ€”Linear algebra
1.2 Determinant of a 2 2๏‚ด
matrix
det
a b
ad bc
c d
๏ƒฆ ๏ƒถ
๏€ฝ ๏ƒž ๏€ฝ ๏€ฝ ๏€ญ๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
A A A
Inverse of a 2 2๏‚ด matrix 1 1
,
det
a b d b
ad bc
c d c a
๏€ญ ๏€ญ๏ƒฆ ๏ƒถ ๏ƒฆ ๏ƒถ
๏€ฝ ๏ƒž ๏€ฝ ๏‚น๏ƒง ๏ƒท ๏ƒง ๏ƒท
๏€ญ๏ƒจ ๏ƒธ ๏ƒจ ๏ƒธ
A A
A
Determinant of a 3 3๏‚ด
matrix
det
๏ƒฆ ๏ƒถ
๏ƒง ๏ƒท
๏€ฝ ๏ƒž ๏€ฝ ๏€ญ ๏€ซ๏ƒง ๏ƒท
๏ƒง ๏ƒท
๏ƒจ ๏ƒธ
a b c
e f d f d e
d e f a b c
h k g k g h
g h k
A A
ยฉ International Baccalaureate Organization 2004 19

Mathematics HL and Further mathematics HL Formula Booklet Zimsec Cambridge Zimbabwe

  • 1.
    Mathematical studies SL:Formula booklet 1ยฉ International Baccalaureate Organization 2011 Mathematics HL and Further mathematics HL Formula booklet For use during the course and in the examinations First examinations 2014 Diploma Programme
  • 2.
    2 ยฉ InternationalBaccalaureate Organization 2011 CONTENTS Formulae 1 Prior learning 1 Topic 1โ€”Core: Algebra 2 Topic 2โ€”Core: Functions and equations 3 Topic 3โ€”Core: Circular functions and trigonometry 4 Topic 4โ€”Core: Vectors 5 Topic 5โ€”Core: Statistics and probability 7 Topic 6โ€”Core: Calculus 9 Topic 7โ€”Option: Statistics and probability (further mathematics HL topic 3) 11 Topic 8โ€”Option: Sets, relations and groups (further mathematics HL topic 4) 13 Topic 9โ€”Option: Calculus (further mathematics HL topic 5) 14 Topic 10โ€”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 1โ€”Linear algebra 18
  • 3.
    Mathematical studies SL:Formula booklet 3
  • 5.
    ยฉ International BaccalaureateOrganization 2004 1 Formulae Prior learning Area of a parallelogram A b h๏€ฝ ๏‚ด , where b is the base, h is the height Area of a triangle 1 ( ) 2 ๏€ฝ ๏‚ดA b h , where b is the base, h is the height Area of a trapezium 1 ( ) 2 ๏€ฝ ๏€ซA a b h , where a and b are the parallel sides, h is the height Area of a circle 2 ๏€ฝ ๏ฐA r , where r is the radius Circumference of a circle 2๏€ฝ ๏ฐC r , where r is the radius Volume of a pyramid 1 (area of base vertical height) 3 ๏€ฝ ๏‚ดV Volume of a cuboid ๏€ฝ ๏‚ด ๏‚ดV l w h , where l is the length, w is the width, h is the height Volume of a cylinder 2 ๏€ฝ ๏ฐV r h , where r is the radius, h is the height Area of the curved surface of a cylinder 2๏€ฝ ๏ฐA rh, where r is the radius, h is the height Volume of a sphere 34 3 ๏€ฝ ๏ฐV r , where r is the radius Volume of a cone 21 3 ๏€ฝ ๏ฐV r h, where r is the radius, h is the height Distance between two points 1 1( , )x y and 2 2( , )x y 2 2 1 2 1 2( ) ( )๏€ฝ ๏€ญ ๏€ซ ๏€ญd x x y y Coordinates of the midpoint of a line segment with endpoints 1 1( , )x y and 2 2( , )x y 1 2 1 2 , 2 2 ๏€ซ ๏€ซ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ x x y y Solutions of a quadratic equation The solutions of 2 0ax bx c๏€ซ ๏€ซ ๏€ฝ are 2 4 2 b b ac x a ๏€ญ ๏‚ฑ ๏€ญ ๏€ฝ
  • 6.
    2 ยฉ InternationalBaccalaureate Organization 2004 Topic 1โ€”Core: Algebra 1.1 The nth term of an arithmetic sequence 1 ( 1)๏€ฝ ๏€ซ ๏€ญnu u n d The sum of n terms of an arithmetic sequence 1 1(2 ( 1) ) ( ) 2 2 ๏€ฝ ๏€ซ ๏€ญ ๏€ฝ ๏€ซn n n n S u n d u u The nth term of a geometric sequence 1 1 ๏€ญ ๏€ฝ n nu u r The sum of n terms of a finite geometric sequence 1 1( 1) (1 ) 1 1 ๏€ญ ๏€ญ ๏€ฝ ๏€ฝ ๏€ญ ๏€ญ n n n u r u r S r r , 1๏‚นr The sum of an infinite geometric sequence 1 , 1 1 ๏€ฝ ๏€ผ ๏€ญ u S r r 1.2 Exponents and logarithms log๏€ฝ ๏ƒ› ๏€ฝx aa b x b, where 0, 0a b๏€พ ๏€พ ln e๏€ฝx x a a log log ๏€ฝ ๏€ฝ a xx a a x a log log log ๏€ฝ c b c a a b 1.3 Combinations Permutations ! !( )! ๏ƒฆ ๏ƒถ ๏€ฝ๏ƒง ๏ƒท ๏€ญ๏ƒจ ๏ƒธ n n r r n r ! ( )! Pr n n n r ๏€ฝ ๏€ญ Binomial theorem 1 ( ) 1 ๏€ญ ๏€ญ๏ƒฆ ๏ƒถ ๏ƒฆ ๏ƒถ ๏€ซ ๏€ฝ ๏€ซ ๏€ซ ๏€ซ ๏€ซ ๏€ซ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ ๏ƒจ ๏ƒธ n n n n r r nn n a b a a b a b b r 1.5 Complex numbers i (cos isin )๏ฑ ๏ฑ๏€ฝ ๏€ซ ๏€ฝ ๏€ซz a b r e cis๏ฑ ๏ฑ๏€ฝ ๏€ฝi r r 1.7 De Moivreโ€™s theorem ๏› ๏(cos isin ) (cos isin ) e cis๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ ๏€ฝ n n n in n r r n n r r n
  • 7.
    ยฉ International BaccalaureateOrganization 2004 3 Topic 2โ€”Core: Functions and equations 2.5 Axis of symmetry of the graph of a quadratic function 2 axis of symmetry 2 b y ax bx c x a ๏€ฝ ๏€ซ ๏€ซ ๏ƒž ๏€ฝ ๏€ญ 2.6 Discriminant 2 4๏„ ๏€ฝ ๏€ญb ac
  • 8.
    4 ยฉ InternationalBaccalaureate Organization 2004 Topic 3โ€”Core: Circular functions and trigonometry 3.1 Length of an arc ๏ฑ๏€ฝl r , where ๏ฑ is the angle measured in radians, r is the radius Area of a sector 21 , 2 ๏ฑ๏€ฝA r where ๏ฑ is the angle measured in radians, r is the radius 3.2 Identities sin tan cos ๏ฑ ๏ฑ ๏ฑ ๏€ฝ 1 sec cos ๏ฑ ๏ฑ ๏€ฝ cosec๏ฑ = 1 sin๏ฑ Pythagorean identities 2 2 2 2 2 2 cos sin 1 1 tan sec 1 cot csc ๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ 3.3 Compound angle identities sin( ) sin cos cos sin๏‚ฑ ๏€ฝ ๏‚ฑA B A B A B cos( ) cos cos sin sin๏‚ฑ ๏€ฝA B A B A B tan tan tan( ) 1 tan tan ๏‚ฑ ๏‚ฑ ๏€ฝ A B A B A B Double angle identities sin2 2sin cos๏ฑ ๏ฑ ๏ฑ๏€ฝ 2 2 2 2 cos2 cos sin 2cos 1 1 2sin๏ฑ ๏ฑ ๏ฑ ๏ฑ ๏ฑ๏€ฝ ๏€ญ ๏€ฝ ๏€ญ ๏€ฝ ๏€ญ 2 2tan tan2 1 tan ๏ฑ ๏ฑ ๏ฑ ๏€ฝ ๏€ญ 3.7 Cosine rule 2 2 2 2 2 2 2 cos ; cos 2 ๏€ซ ๏€ญ ๏€ฝ ๏€ซ ๏€ญ ๏€ฝ a b c c a b ab C C ab Sine rule sin sin sin ๏€ฝ ๏€ฝ a b c A B C Area of a triangle 1 sin 2 ๏€ฝA ab C
  • 9.
    ยฉ International BaccalaureateOrganization 2004 5 Topic 4โ€”Core: Vectors 4.1 Magnitude of a vector 2 2 2 1 2 3๏€ฝ ๏€ซ ๏€ซv v vv , where 1 2 3 ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ v v v v Distance between two points 1 1 1( , , )x y z and 2 2 2( , , )x y z 2 2 2 1 2 1 2 1 2( ) ( ) ( )๏€ฝ ๏€ญ ๏€ซ ๏€ญ ๏€ซ ๏€ญd x x y y z z Coordinates of the midpoint of a line segment with endpoints 1 1 1( , , )x y z , 2 2 2( , , )x y z 1 2 1 2 1 2 , , 2 2 2 ๏€ซ ๏€ซ ๏€ซ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ x x y y z z 4.2 Scalar product cos๏ฑ๏ƒ— ๏€ฝv w v w , where ๏ฑ is the angle between v and w 1 1 2 2 3 3๏ƒ— ๏€ฝ ๏€ซ ๏€ซv w v w v wv w , where 1 2 3 ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ v v v v , 1 2 3 ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ w w w w Angle between two vectors 1 1 2 2 3 3 cos๏ฑ ๏€ซ ๏€ซ ๏€ฝ v w v w v w v w 4.3 Vector equation of a line = + ฮปr a b Parametric form of the equation of a line 0 0 0, ,๏ฌ ๏ฌ ๏ฌ๏€ฝ ๏€ซ ๏€ฝ ๏€ซ ๏€ฝ ๏€ซx x l y y m z z n Cartesian equations of a line 0 0 0๏€ญ ๏€ญ ๏€ญ ๏€ฝ ๏€ฝ x x y y z z l m n 4.5 Vector product 2 3 3 2 3 1 1 3 1 2 2 1 v w v w v w v w v w v w ๏€ญ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏‚ด ๏€ฝ ๏€ญ๏ƒง ๏ƒท ๏ƒง ๏ƒท๏€ญ๏ƒจ ๏ƒธ v w where 1 2 3 ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ v v v v , 1 2 3 ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ w w w w sin๏ฑ๏‚ด ๏€ฝv w v w , where๏ฑ is the angle between v and w Area of a triangle 1 2 ๏€ฝ ๏‚ดA v w where v and w form two sides of a triangle 4.6 Vector equation of a plane = + ๏ญฮปr a b + c Equation of a plane (using the normal vector) ๏ƒ— ๏€ฝ ๏ƒ—r n a n Cartesian equation of a ax by cz d๏€ซ ๏€ซ ๏€ฝ
  • 10.
    6 ยฉ InternationalBaccalaureate Organization 2004 plane
  • 11.
    ยฉ International BaccalaureateOrganization 2004 7 Topic 5โ€”Core: Statistics and probability 5.1 Population parameters Mean ๏ญ Variance 2 ๏ณ Standard deviation ๏ณ Let 1๏€ฝ ๏€ฝ ๏ƒฅ k i i n f 1 ๏ญ ๏€ฝ ๏€ฝ ๏ƒฅ k i i i f x n ๏€จ ๏€ฉ 2 2 2 21 1 ๏ญ ๏ณ ๏ญ๏€ฝ ๏€ฝ ๏€ญ ๏€ฝ ๏€ฝ ๏€ญ ๏ƒฅ ๏ƒฅ k k i i i i i i f x f x n n ๏€จ ๏€ฉ 2 1 ๏ญ ๏ณ ๏€ฝ ๏€ญ ๏€ฝ ๏ƒฅ k i i i f x n 5.2 Probability of an event A ( ) P( ) ( ) ๏€ฝ n A A n U Complementary events P( ) P( ) 1๏‚ข๏€ซ ๏€ฝA A 5.3 Combined events P( ) P( ) P( ) P( )๏ƒˆ ๏€ฝ ๏€ซ ๏€ญ ๏ƒ‡A B A B A B Mutually exclusive events P( ) P( ) P( )๏ƒˆ ๏€ฝ ๏€ซA B A B
  • 12.
    8 ยฉ InternationalBaccalaureate Organization 2004 Topic 5โ€”Core: Statistics and probability (continued) 5.4 Conditional probability ๏€จ ๏€ฉ P( ) P P( ) ๏ƒ‡ ๏€ฝ A B A B B Independent events P( ) P( ) P( )๏ƒ‡ ๏€ฝA B A B Bayesโ€™ Theorem ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ P( )P | P | P( )P | P( )P | ๏€ฝ ๏‚ข๏€ซ ๏‚ข B A B B A B A B B A B 1 1 2 2 ( | ) ( ) ( | ) ( | ) ( ) ( | ) ( ) ( | ) ( ) i i i n n P A B P B P B A P A B P B P A B P B P A B P B ๏€ฝ ๏€ซ ๏€ซ 5.5 Expected value of a discrete random variable X E( ) P( )๏ญ๏€ฝ ๏€ฝ ๏€ฝ๏ƒฅx X x X x Expected value of a continuous random variable X E( ) ( )d๏ญ ๏‚ฅ ๏€ญ๏‚ฅ ๏€ฝ ๏€ฝ ๏ƒฒX x f x x Variance ๏› ๏ 22 2 Var( ) E( ) E( ) E( )๏ญ๏€ฝ ๏€ญ ๏€ฝ ๏€ญX X X X Variance of a discrete random variable X 2 2 2 Var( ) ( ) P( ) P( )๏ญ ๏ญ๏€ฝ ๏€ญ ๏€ฝ ๏€ฝ ๏€ฝ ๏€ญ๏ƒฅ ๏ƒฅX x X x x X x Variance of a continuous random variable X 2 2 2 Var( ) ( ) ( )d ( )d๏ญ ๏ญ ๏‚ฅ ๏‚ฅ ๏€ญ๏‚ฅ ๏€ญ๏‚ฅ ๏€ฝ ๏€ญ ๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒX x f x x x f x x 5.6 Binomial distribution Mean Variance ~ B( , ) P( ) (1 ) , 0,1, ,๏€ญ๏ƒฆ ๏ƒถ ๏ƒž ๏€ฝ ๏€ฝ ๏€ญ ๏€ฝ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ x n xn X n p X x p p x n x E( ) ๏€ฝX np Var( ) (1 )๏€ฝ ๏€ญX np p Poisson distribution Mean Variance o e ~ P ( ) P( ) , 0,1, 2, ! ๏€ญ ๏ƒž ๏€ฝ ๏€ฝ ๏€ฝ x m m X m X x x x E( ) ๏€ฝX m Var( ) ๏€ฝX m 5.7 Standardized normal variable ๏ญ ๏ณ ๏€ญ ๏€ฝ x z
  • 13.
    ยฉ International BaccalaureateOrganization 2004 9 Topic 6โ€”Core: Calculus 6.2 Derivative of ( )f x 0 d ( ) ( ) ( ) ( ) lim d ๏‚ฎ ๏€ซ ๏€ญ๏ƒฆ ๏ƒถ๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ฝ ๏ƒง ๏ƒท ๏ƒจ ๏ƒธh y f x h f x y f x f x x h Derivative of n x 1 ( ) ( ) ๏€ญ ๏‚ข๏€ฝ ๏ƒž ๏€ฝn n f x x f x nx Derivative of sin x ( ) sin ( ) cos๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x Derivative of cos x ( ) cos ( ) sin๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x Derivative of tan x 2 ( ) tan ( ) sec๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x Derivative of ex ( ) e ( ) e๏‚ข๏€ฝ ๏ƒž ๏€ฝx x f x f x Derivative of ln x 1 ( ) ln ( )๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x Derivative of sec x ( ) sec ( ) sec tan๏‚ข๏€ฝ ๏ƒž ๏€ฝf x x f x x x Derivative of csc x ( ) csc ( ) csc cot๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x x Derivative of cot x 2 ( ) cot ( ) csc๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญf x x f x x Derivative of x a ( ) ( ) (ln )๏‚ข๏€ฝ ๏ƒž ๏€ฝx x f x a f x a a Derivative of loga x 1 ( ) log ( ) ln ๏‚ข๏€ฝ ๏ƒž ๏€ฝaf x x f x x a Derivative of arcsin x 2 1 ( ) arcsin ( ) 1 ๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญ f x x f x x Derivative of arccos x 2 1 ( ) arccos ( ) 1 ๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ญ ๏€ญ f x x f x x Derivative of arctan x 2 1 ( ) arctan ( ) 1 ๏‚ข๏€ฝ ๏ƒž ๏€ฝ ๏€ซ f x x f x x Chain rule ( )๏€ฝy g u , where d d d ( ) d d d ๏€ฝ ๏ƒž ๏€ฝ ๏‚ด y y u u f x x u x Product rule d d d d d d ๏€ฝ ๏ƒž ๏€ฝ ๏€ซ y v u y uv u v x x x Quotient rule 2 d d d d d d ๏€ญ ๏€ฝ ๏ƒž ๏€ฝ u v v u u y x xy v x v
  • 14.
    10 ยฉ InternationalBaccalaureate Organization 2004 Topic 6โ€”Core: Calculus (continued) 6.4 Standard integrals 1 d , 1 1 ๏€ซ ๏€ฝ ๏€ซ ๏‚น ๏€ญ ๏€ซ๏ƒฒ n n x x x C n n 1 d ln๏€ฝ ๏€ซ๏ƒฒ x x C x sin d cos๏€ฝ ๏€ญ ๏€ซ๏ƒฒ x x x C cos d sin๏€ฝ ๏€ซ๏ƒฒ x x x C e d e๏€ฝ ๏€ซ๏ƒฒ x x x C 1 d ln ๏€ฝ ๏€ซ๏ƒฒ x x a x a C a 2 2 1 1 d arctan ๏ƒฆ ๏ƒถ ๏€ฝ ๏€ซ๏ƒง ๏ƒท ๏€ซ ๏ƒจ ๏ƒธ ๏ƒฒ x x C a x a a 2 2 1 d arcsin , ๏ƒฆ ๏ƒถ ๏€ฝ ๏€ซ ๏€ผ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ๏€ญ ๏ƒฒ x x C x a aa x 6.5 Area under a curve Volume of revolution (rotation) d๏€ฝ ๏ƒฒ b a A y x or d๏€ฝ ๏ƒฒ b a A x y 2 2 ฯ€ d or ฯ€ d๏€ฝ ๏€ฝ๏ƒฒ ๏ƒฒ b b a a V y x V x y 6.7 Integration by parts d d d d d d ๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒ v u u x uv v x x x or d d๏€ฝ ๏€ญ๏ƒฒ ๏ƒฒu v uv v u
  • 15.
    ยฉ International BaccalaureateOrganization 2004 11 Topic 7โ€”Option: Statistics and probability (further mathematics HL topic 3) 7.1 (3.1) Probability generating function for a discrete random variable X ( ) ( ) ( )X x x G t E t P X x t๏€ฝ ๏€ฝ ๏€ฝ๏ƒฅ 7.2 (3.2) Linear combinations of two independent random variables 1 2,X X ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ ๏€จ ๏€ฉ 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 E E E Var Var Var ๏‚ฑ ๏€ฝ ๏‚ฑ ๏‚ฑ ๏€ฝ ๏€ซ a X a X a X a X a X a X a X a X 7.3 (3.3) Sample statistics Mean x Variance 2 ns Standard deviation ns 1 22 2 21 1 2 1 ( ) ( ) ๏€ฝ ๏€ฝ ๏€ฝ ๏€ฝ ๏€ฝ ๏€ญ ๏€ฝ ๏€ฝ ๏€ญ ๏€ญ ๏€ฝ ๏ƒฅ ๏ƒฅ ๏ƒฅ ๏ƒฅ k i i i k k i i i i i i n k i i i n f x x n f x x f x s x n n f x x s n Unbiased estimate of population variance 2 1๏€ญns 22 2 2 21 1 1 ( ) 1 1 1 1 ๏€ฝ ๏€ฝ ๏€ญ ๏€ญ ๏€ฝ ๏€ฝ ๏€ฝ ๏€ญ ๏€ญ ๏€ญ ๏€ญ ๏€ญ ๏ƒฅ ๏ƒฅ k k i i i i i i n n f x x f x n n s s x n n n n 7.5 (3.5) Confidence intervals Mean, with known variance Mean, with unknown variance ๏ณ ๏‚ฑ ๏‚ดx z n 1๏€ญ ๏‚ฑ ๏‚ด ns x t n 7.6 (3.6) Test statistics Mean, with known variance Mean, with unknown / ๏ญ ๏ณ ๏€ญ ๏€ฝ x z n
  • 16.
    12 ยฉ InternationalBaccalaureate Organization 2004 variance 1 / ๏ญ ๏€ญ ๏€ญ ๏€ฝ n x t s n 7.7 Sample product moment correlation coefficient Test statistic for H0: ฯ = 0 Equation of regression line of x on y Equation of regression line of y on x ๏ƒท ๏ƒธ ๏ƒถ ๏ƒง ๏ƒจ ๏ƒฆ ๏€ญ๏ƒท ๏ƒธ ๏ƒถ ๏ƒง ๏ƒจ ๏ƒฆ ๏€ญ ๏€ญ ๏€ฝ ๏ƒฅ๏ƒฅ ๏ƒฅ ๏€ฝ๏€ญ ๏€ฝ n i i n i i n i ii ynyxnx yxnyx r 1 22 1 22 1 2 1 2 r n rt ๏€ญ ๏€ญ ๏€ฝ )( 1 22 1 yy yny yxnyx xx n i i n i ii ๏€ญ ๏ƒท ๏ƒท ๏ƒท ๏ƒท ๏ƒธ ๏ƒถ ๏ƒง ๏ƒง ๏ƒง ๏ƒง ๏ƒจ ๏ƒฆ ๏€ญ ๏€ญ ๏€ฝ๏€ญ ๏ƒฅ ๏ƒฅ ๏€ฝ ๏€ฝ )( 1 22 1 xx xnx yxnyx yy n i i n i ii ๏€ญ ๏ƒท ๏ƒท ๏ƒท ๏ƒท ๏ƒธ ๏ƒถ ๏ƒง ๏ƒง ๏ƒง ๏ƒง ๏ƒจ ๏ƒฆ ๏€ญ ๏€ญ ๏€ฝ๏€ญ ๏ƒฅ ๏ƒฅ ๏€ฝ ๏€ฝ
  • 17.
    ยฉ International BaccalaureateOrganization 2004 13 Topic 8โ€”Option: Sets, relations and groups (further mathematics HL topic 4) 8.1 (4.1) De Morganโ€™s laws ( ) ( ) ๏‚ข ๏‚ข ๏‚ข๏ƒˆ ๏€ฝ ๏ƒ‡ ๏‚ข ๏‚ข ๏‚ข๏ƒ‡ ๏€ฝ ๏ƒˆ A B A B A B A B
  • 18.
    14 ยฉ InternationalBaccalaureate Organization 2004 Topic 9โ€”Option: Calculus (further mathematics HL topic 5) 9.6 (5.6) Maclaurin series 2 ( ) (0) (0) (0) 2! ๏‚ข ๏‚ข๏‚ข๏€ฝ ๏€ซ ๏€ซ ๏€ซ x f x f x f f Taylor series 2 ( ) ( ) ( ) ( ) ( ) ( ) ... 2! ๏€ญ ๏‚ข๏‚ข๏‚ข๏€ฝ ๏€ซ ๏€ญ ๏€ซ ๏€ซ x a f x f a x a f a f a Taylor approximations (with error term ( )nR x ) Lagrange form ( )( ) ( ) ( ) ( ) ( ) ... ( ) ( ) ! ๏€ญ ๏‚ข๏€ฝ ๏€ซ ๏€ญ ๏€ซ ๏€ซ ๏€ซ n n n x a f x f a x a f a f a R x n ( 1) 1( ) ( ) ( ) ( 1)! ๏€ซ ๏€ซ ๏€ฝ ๏€ญ ๏€ซ 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! ๏€ฝ ๏€ซ ๏€ซ ๏€ซx x x 2 3 ln(1 ) ... 2 3 ๏€ซ ๏€ฝ ๏€ญ ๏€ซ ๏€ญ x x x x 3 5 sin ... 3! 5! ๏€ฝ ๏€ญ ๏€ซ ๏€ญ x x x x 2 4 cos 1 ... 2! 4! ๏€ฝ ๏€ญ ๏€ซ ๏€ญ x x x 3 5 arctan ... 3 5 ๏€ฝ ๏€ญ ๏€ซ ๏€ญ x x x x 9.5 (5.5) Eulerโ€™s method 1 ( , )๏€ซ ๏‚ด๏€ฝ ๏€ซn n n ny y h f x y ; 1๏€ซ ๏€ฝ ๏€ซn nx x h , where h is a constant (step- length) Integrating factor for ( ) ( )๏‚ข๏€ซ ๏€ฝy P x y Q x ( )d e๏ƒฒP x x
  • 19.
    ยฉ International BaccalaureateOrganization 2004 15 Topic 10โ€”Option: Discrete mathematics (further mathematics HL topic 6) 10.7 (6.7) Eulerโ€™s formula for connected planar graphs 2๏€ญ ๏€ซ ๏€ฝv 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๏‚ฃ ๏€ญe v for 3v ๏‚ณ 2 4๏‚ฃ ๏€ญe v if the graph has no triangles
  • 20.
    16 ยฉ InternationalBaccalaureate Organization 2004 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 ๏€จ ๏€ฉ~ GeoX p 1๏€ญx pq for 1,2,...๏€ฝx 1 p 2 q p Negative binomial ๏€จ ๏€ฉ~ NB ,X r p 1 1 ๏€ญ๏€ญ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ญ๏ƒจ ๏ƒธ r x rx p q r for , 1,...๏€ฝ ๏€ซx r r r p 2 rq p
  • 21.
    ยฉ International BaccalaureateOrganization 2004 17 Continuous distributions Distribution Notation Probability density function Mean Variance Normal ๏€จ ๏€ฉ2 ~ N ,๏ญ ๏ณX 2 1 21 e 2ฯ€ ๏ญ ๏ณ ๏ณ ๏€ญ๏ƒฆ ๏ƒถ ๏€ญ ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ x ๏ญ 2 ๏ณ
  • 22.
    18 ยฉ InternationalBaccalaureate Organization 2004 Further Mathematics Topic 1โ€”Linear algebra 1.2 Determinant of a 2 2๏‚ด matrix det a b ad bc c d ๏ƒฆ ๏ƒถ ๏€ฝ ๏ƒž ๏€ฝ ๏€ฝ ๏€ญ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ A A A Inverse of a 2 2๏‚ด matrix 1 1 , det a b d b ad bc c d c a ๏€ญ ๏€ญ๏ƒฆ ๏ƒถ ๏ƒฆ ๏ƒถ ๏€ฝ ๏ƒž ๏€ฝ ๏‚น๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏€ญ๏ƒจ ๏ƒธ ๏ƒจ ๏ƒธ A A A Determinant of a 3 3๏‚ด matrix det ๏ƒฆ ๏ƒถ ๏ƒง ๏ƒท ๏€ฝ ๏ƒž ๏€ฝ ๏€ญ ๏€ซ๏ƒง ๏ƒท ๏ƒง ๏ƒท ๏ƒจ ๏ƒธ a b c e f d f d e d e f a b c h k g k g h g h k A A
  • 23.