Booklet 0.2
Formula List of
Applications and Interpretation
Higher Level
for IBDP Mathematics
λ
Intensive Notes
by Topics here
IBDP Maths Info.
& Exam Tricks here
Exam & GDC Skills
Video List here
More Mock Papers &
Marking Service here
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
1

Standard Form
Standard Form:
A number in the form ()a 10k , where 1  a  10 and k is an integer
2


Approximation and Error
Summary of rounding methods:
2.71828
Correct to 3
significant figures
Correct to 3
decimal places
Round off
2.72
2.718
Consider a quantity measured as Q and correct to the nearest unit d :
1
d : Maximum absolute error
2
1
1
Q  d  A  Q  d : Range of the actual value A
2
2
1
Q  d : Lower bound (Least possible value) of A
2
1
Q  d : Upper bound of A
2
Maximum absolute error
100% : Percentage error
Q
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Functions
The function y  f ( x) :
1.
f (a) : Functional value when x  a
2.
3.
Domain: Set of values of x
Range: Set of values of y
Properties of rational function y 
1.
2.
3.
ax  b
:
cx  d
1
: Reciprocal function
x
a
y  : Horizontal asymptote
c
d
x   : Vertical asymptote
c
y

f g ( x)  f ( g ( x)) : Composite function when g ( x) is substituted into f ( x)

Steps of finding the inverse function y  f 1 ( x) of f ( x) :

1.
Start from expressing y in terms of x
2.
Interchange x and y
3.
Make y the subject in terms of x
Properties of y  f 1 ( x) :
1.
f ( f 1 ( x))  f 1 ( f ( x))  x
2.
The graph of y  f 1 ( x) is the reflection of the graph of y  f ( x) about
yx

f 1 ( x) exists only when f ( x) is one-to-one in the restricted domain
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Summary of transformations:
f ( x)  f ( x)  k :
Translate upward by k units
f ( x)  f ( x)  k :
Translate downward by
k units
f ( x)  f ( x  h) :
Translate to the left by h units
f ( x)  f ( x  h) :
Translate to the right by
h units
f ( x)  kf ( x) :
Vertical stretch of
scale factor k
f ( x)  f (kx) :
kf ( x)
f ( x)
f (kx)
x
Horizontal compression of
scale factor k
f ( x)   f ( x ) :
Reflection about the x -axis
f ( x )  f (  x) :
Reflection about the y -axis

Variations:
1.
y  kx, k  0 : y is directly proportional to x
2.
y
k
, k  0 : y is inversely proportional to x
x
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Quadratic Functions
General form y  ax 2  bx  c , where a  0 :
a0
The graph opens upward
a0
The graph opens downward
c
y -intercept
h
b
2a
k  ah2  bh  c
xh

x -coordinate of the vertex
y -coordinate of the vertex
Extreme value of y
Equation of the axis of symmetry
Other forms:
1.
y  a( x  h)2  k : Vertex form
2.
y  a( x  p)( x  q) : Factored form with x -intercepts p and q
b
pq

2a
2

h

The x -intercepts of the quadratic function y  ax 2  bx  c are the roots of the
corresponding quadratic equation ax2  bx  c  0
5
Exponential and Logarithmic Functions

y  a x : Exponential function, where a  1

y  log a x : Logarithmic function, where a  0

y  log x  log10 x : Common Logarithmic function

y  ln x  log e x : Natural Logarithmic function, where e  2.71828... is an
exponential number
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Properties of the graphs of y = a x :
a >1
0 < a <1
y -intercept = 1
y increases as x increases
y decreases as x increases
y tends to zero as x tends to
y tends to zero as x tends to
negative infinity
positive infinity
Horizontal asymptote: y = 0

Laws of logarithm, where a , b , c , p , q , x > 0 :
1.
x = a y ⇔ y = log a x
2.
log a 1 = 0
3.
log a a = 1
4.
log a p + log a q =
log a pq
5.
p
log a p − log a q =
log a
q
6.
log a p n = n log a p
7.
log b a =

f ( x) =
L
: Logistic function, where L , C and k are positive constants
1 + Ce − kx

Semi-log model:

log c a
log c b
1.
y =k ⋅ a x ⇔ ln y =(ln a ) x + ln k : Semi-log model
2.
ln a : Gradient of the straight line graph on ln y -x plane
3.
ln k : Vertical intercept of the straight line graph on ln y -x plane
Log-log models:
1.
y =k ⋅ x n ⇔ ln y =n ln x + ln k : Log-log model
2.
n : Gradient of the straight line graph on ln y - ln x plane
3.
ln k : Vertical intercept of the straight line graph on ln y - ln x plane
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Systems of Equations

ax  by  c
: 2  2 system

dx  ey  f

ax  by  cz  d

ex  fy  gz  h : 3  3 system
ix  jy  kz  l


The above systems can be solved by PlySmlt2 in TI-84 Plus CE
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
Arithmetic Sequences
Properties of an arithmetic sequence un :
1.
u1 : First term
2.
d  u2  u1  un  un1 : Common difference
3.
un  u1  (n  1)d : General term ( n th term)
4.
Sn 
n
u  u  u  u 
r 1
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
n
n
 2u1  (n  1)d   u1  un  : The sum of the first n terms
2
2
r
1
2
3
 un 1  un : Summation sign
Geometric Sequences
Properties of a geometric sequence un :
1.
u1 : First term
2.
r  u2  u1  un  un1 : Common ratio
3.
un  u1  r n1 : General term ( n th term)
4.
Sn 
u1 (1  r n )
: The sum of the first n terms
1 r
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S 
9

u1
: The sum to infinity of a geometric sequence un , given that 1  r  1
1 r
Financial Mathematics
Compound Interest:
PV : Present value
r % : Interest rate per annum (per year)
n : Number of years
k : Number of compounded periods in one year
kn
r 

FV  PV 1 
 : Future value
 100k 
I  FV  PV : Interest

Inflation:
i % : Inflation rate
R% : Interest rate compounded yearly
( R  i)% : Real rate


Annuity:
1.
Payments at the beginning of each year
2.
Payments at the end of each year
Amortization:
1.
Payments at the beginning of each year
2.
Payments at the end of each year
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10 Coordinate Geometry
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


Consider the points P( x1 , y1 ) and Q( x2 , y2 ) on a x - y plane:
y2  y1
: Slope of PQ
x2  x1
1.
m
2.
d  ( x2  x1 )2  ( y2  y1 )2 : Distance between P and Q
3.
 x1  x2 y1  y2  : Mid-point of
PQ
,


2 
 2
Consider the points P( x1 , y1 , z1 ) and Q( x2 , y2 , z2 ) on a x - y - z plane:
1.
z -axis: The axis perpendicular to the x - y plane
2.
d  ( x2  x1 )2  ( y2  y1 )2  ( z2  z1 )2 : Distance between P and Q
3.
 x1  x2 y1  y2 z1  z2  : Mid-point of
PQ
,
,


2
2 
 2
Forms of straight lines with slope m and y -intercept c :
1.
y  mx  c : Slope-intercept form
2.
Ax  By  C  0 : General form
Ways to find the x -intercept and the y -intercept of a line:
1.
Substitute y  0 and make x the subject to find the x -intercept
2.
Substitute x  0 and make y the subject to find the y -intercept
11 Voronoi Diagrams

Elements in Voronoi Diagrams:
Site: A given point
Cell of a site: A collection of points which is closer to the site than other sites
Boundary: A line dividing the cells
Vertex: An intersection of boundaries

Related problems:
1.
Nearest neighbor interpolation
2.
3.
Incremental algorithm
Toxic waste dump problem
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12 Trigonometry

Consider a right-angled triangle ABC :
AB2  BC2  AC2 : Pythagoras’ Theorem
AB

sin   AC

BC

: Trigonometric ratios
cos  
AC

AB

 tan   BC


Properties of a general trigonometric function y  A sin B( x  C )  D :
1.
2.
3.
4.

A
ymax  ymin
: Amplitude
2
360
2
or
Period Period
y  ymin
D  max
2
C can be found by substitution of a point on the graph
B
Properties of graphs of trigonometric functions:
1.
2.
Amplitude  1
3.
1  sin x  1
1.
2.
Amplitude  1
3.
1  cos x  1
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Period  360 or 2
Period  360 or 2

Trigonometric identities:
sin 
1.
tan  
cos 
2
2.
sin   cos2   1

ASTC diagram
y
S (90    180)
sin   0
sin   0
cos  0
cos  0
tan   0
tan   0
T (180    270)
sin   0
C (270    360)
cos  0
cos  0
tan   0
tan   0
A (0    90)
x
sin   0
13 2-D Trigonometry

Consider a triangle ABC :
1.
sin A sin B
a
b
or
: Sine rule


sin A sin B
a
b
2.
a2  b2  c2  2bc cos A
3.

b2  c 2  a 2
or cos A 
: Cosine rule
2bc
1
ab sin C : Area of the triangle ABC
2
Consider a sector OPRQ with centre O , radius r and POQ   :
2 r 
 r2 
 r2 

360

360
: Arc length PRQ
: Area of the sector OPRQ

1
 r 2 sin  : Area of the segment PRQ
360 2
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Consider a triangle ABC :
sin A sin B
a
b
or
: Sine rule


sin A sin B
a
b
Note: The ambiguous case exists if two sides and
an angle are known, and the angle is opposite
to the shorter known side

x
y rad
: Method of conversions between degree and radian

180  rad

Consider a sector OPRQ with centre O , radius r and POQ   in radian:
1.
r : Arc length PQ
2.
1 2
r  : Area of the sector OPRQ
2
1 2
r (  sin  ) : Area of the segment PRQ
2
3.
14 Areas and Volumes


For a cube of side length l :
1.
6l 2 : Total surface area
2.
l 3 : Volume
For a cuboid of side lengths a , b and c :
1.
2(ab  bc  ac) : Total surface area
2.

For a prism of height h and cross-sectional area A :
1.

abc : Volume
Ah : Volume
For a cylinder of height h and radius r :
1.
2 r 2  2 rh : Total surface area
2.
2 rh : Lateral surface area
3.
 r 2 h : Volume
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For a pyramid of height h and base area A :
1
1.
Ah : Volume
3

For a circular cone of height h and radius r :
1.
l  r 2  h2 : Slant height
2.
 r 2   rl : Total surface area
 rl : Curved surface area
3.
4.


1 2
 r h : Volume
3
For a sphere of radius r :
1.
4 r 2 : Total surface area
2.
4 3
 r : Volume
3
For a hemisphere of radius r :
1.
3 r 2 : Total surface area
2.
2 r 2 : Curved surface area
3.
2 3
 r : Volume
3
15 Complex Numbers

Terminologies of complex numbers:
i  1 : Imaginary unit
z  a  bi : Complex number in Cartesian form
a : Real part of z
b : Imaginary part of z
z*  a  bi : Conjugate of z  a  bi
z  r  a 2  b2 : Modulus of z  a  bi
arg( z )    arctan
b
: Argument of z  a  bi
a
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

Properties of Argand diagram:
1.
Real axis: Horizontal axis
2.
Imaginary axis: Vertical axis
Forms of complex numbers:
1.
z  a  bi : Cartesian form
2.
z  r (cos  isin )  rcis : Modulus-argument form
3.
z  rei : Euler form
Properties of moduli and arguments of complex numbers z1 and z2 :
1.
z1 z2  z1 z2
2.
z
z1
 1
z2
z2
3.
arg( z1 z2 )  arg z1  arg z2
4.
z 
arg  1   arg z1  arg z2
 z2 
If z  a  bi is a root of the polynomial equation p( z )  0 , then z*  a  bi is
also a root of p( z )  0
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16 Matrices

Terminologies of matrices:
 a11

a
A   21


 am1
a12
a22
am 2
a1n 

a2 n 
: A m  n matrix with m rows and n columns


amn 
aij : Element on the i th row and the j th column
0
1 0


0 1
0

: Identity matrix
I




1
0 0
0
0 0


0 0
0

: Zero matrix
0




0
0 0
0 
 a11 0


0 
 0 a22
: Diagonal matrix




0
ann 
 0
A  det(A) : Determinant of A
A is non-singular if det(A)  0
A 1 : Inverse of A
A 1 exists if A is non-singular

a b 
For any 2  2 square matrices A  
:
c d
1.
A  det(A)  ad  bc : Determinant of A
2.
A 1 
1  d b 

 : Inverse of A
ad  bc  c a 
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Operations of matrices:
b1n   a11  b11
 

bmn   am1  bm1
1.
 a11


a
 m1
a1n   b11
 

amn   bm1
2.
 a11

k
a
 m1
a1n   ka11
 

amn   kam1
3.
cij  ai1b1 j  ai 2b2 j 
a1n  b1n 


amn  bmn 
ka1n 


kamn 
 ainbnj : The element on the i th row and the j th
a1n  b11
b1k 
 a11



column of C  AB  

 , where A , B and C
a
amn 
bnk 
 m1
 bn1
are m  n , n  k and m  k matrices respectively

ax  by  c
 x
A 2  2 system 
can be expressed as AX  B , where X    can be
dx  ey  f
 y
1
 a b  c 
solved by X  A B  
  
d e  f 
1

ax  by  cz  d
 x

 
A 3  3 system ex  fy  gz  h can be expressed as AX  B , where X   y  can
ix  jy  kz  l
z

 
a

1
be solved by X  A B   e
i



b
f
j
c

g
k 
1
d 
 
h
l
 
Eigenvalues and eigenvectors of A :
1.
det(A  I) : Characteristic polynomial of A
2.
Solution(s) of det(A  I)  0 : Eigenvalue(s) of A
3.
v : Eigenvector of A corresponding to the eigenvalue  , which satisfies
Av   v
Diagonalization of A :
1.
 1 0

0 2
D


0 0
0

0
: Diagonal matrix of the eigenvalues of A


n 
2.
V   v1
v n  : A matrix of the eigenvectors of A
3.
A  VDV1  An  VDn V 1
v2
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Two-dimensional transformation matrices:
1.
2.
3.
4.
5.
6.
7.
8.
9.
1 0 

 : Reflection about the x -axis
 0 1
 1 0 

 : Reflection about the y -axis
 0 1
 cos 2 sin 2 

 : Reflection about the line y  mx , where m  tan 
 sin 2  cos 2 
1 0

 : Vertical stretch with scale factor k
0 k 
 k 0

 : Horizontal stretch with scale factor k
 0 1
k 0

 : Enlargement about the origin with scale factor k
0 k 
 cos   sin  

 : Rotation with positive angle  anticlockwise about the
 sin  cos  
origin
 cos  sin  

 : Rotation with positive angle  clockwise about the origin
  sin  cos  
Area of the image  det(T )  Area of the object, where T is the
transformation matrix
17 Vectors

Terminologies of vectors:

AB : Vector of length AB with initial point A and terminal point B

OP : Position vector of P , where O is the origin


AB : Magnitude (length) of AB
vˆ 
1
v : Unit vector parallel to v , with vˆ  1
v
0 : Zero vector
i : Unit vector along the positive x -axis
j : Unit vector along the positive y -axis
k : Unit vector along the positive z -axis
17
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
 v1 
 
A vector v can be expressed as v  v1i  v2 j  v3k or v   v2 
v 
 3

Properties of vectors:
1.


3.
 u1   v1   u1  v1 
    

 u2    v2    u2  v2 
u  v  u  v 
 3  3  3 3
v and kv are in the same direction if k  0
4.
v and kv are in opposite direction if k  0
5.
 v1   kv1 
  

k  v2    kv2 
 v   kv 
 3  3
2.


AB  OB OA
 u1 
 v1 
 
 
Properties of the scalar product u  v of u   u2  and v   v2  where  is the
u 
v 
 3
 3
angle between u and v :
1.
u  v  u1v1  u2v2  u3v3  u v cos
2.
i  i  j j  k  k  1
3.
i  j  j k  k  i  0
4.
u and v are in the same direction if u  v  u v
5.
u and v are in opposite direction if u  v   u v
6.
u and v are perpendicular if u  v  0
7.
u  v  v u
8.
u u  u
2
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
 u1 
 v1 
 
 
Properties of the vector product u  v of u   u2  and v   v2  where  is the
u 
v 
 3
 3
angle between u and v :
2.
 u2v3  u3v2 


u  v   u3v1  u1v3   u v sin  nˆ , where nˆ //(u  v)
u v u v 
 1 2 2 1
i  i  j j  k  k  0
3.
i  j  k , j  k  i and k  i  j
4.
j  i  k , k  j  i and i  k   j
5.
u and v are parallel if u  v  0
6.
u and v are perpendicular if u  v  u v
7.
u  v  ( v  u)
1.





The area of the parallelogram with adjacent sides AB and AD is AB AD

The area of the triangle with adjacent sides AB and AD is

Forms of the straight line with fixed point A(a1 , a2 , a3 ) and direction vector


1  
AB AD
2
b  b1i  b2 j  b3k :

1.
 x   a1   b1 
     
 y    a2   t  b2  , t 
 z  a  b 
   3  3
2.
 x  a1  b1t

 y  a2  b2t : Parametric form
z  a b t
3
3

Vector components:
1.
2.
uv
: Vector component of u parallel to v
v
u v
v
: Vector component of u perpendicular to v
19
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
18 Graph Theory

Terminologies of graphs:
Vertex: A point on a graph
Edge: Arcs that connect vertices
Walk: A sequence of edges
Path: A sequence of edges that passes through any vertex and any edge at most
once
Degree of a vertex: Number of edges connecting the vertex
Connected graph: A graph that there exists at least one walk between any two
vertices
Unconnected graph: A graph that there exist at least two vertices that there is no
walk between them
Subgraph of a graph: A collection of some edges and vertices of the original
graph
Loop: An edge that starts and ends at the same vertex
Simple graph: A graph that has no loops and no multiple edges connecting the
same pair of vertices
Multiple graph: A graph that has multiple edges connecting at least one pair of
vertices
Cycle: A path that the starting vertex is the end vertex
Tree: A connected graph with no cycles
Spanning tree: A tree that connects all vertices in the graph

Directed graphs:
1.
2.
Directed graph: A graph that all edges are assigned with directions
In-degree of a vertex: Number of edges connecting and pointing towards
3.
the vertex
Out-degree of a vertex: Number of edges connecting and pointing away
from the vertex
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
Adjacency matrix M of a graph with n vertices:
1.
n  n : Order of M
2.
The entry mij  1 if there is an edge connecting the vertex i and the vertex
j , and mij  0 if otherwise
3.
4.
M p shows the number of walks of length p in the graph
p
 M shows the number of walks of length less than or equal to p in the
r
r 1
graph
5.

Algorithms of finding minimum spanning trees:
1.
Kruskal’s algorithm
2.



The column sum of a transition matrix of a directed graph must be equal
to 1
Prim’s algorithm
Eulerian trails and circuits:
1.
2.
Trail: A sequence of edges that passes through any edge at most once
Circuit: A trail that the starting vertex is the end vertex
3.
4.
Eulerian trail: A trail that passes through all edges of a graph
Eulerian circuit: A circuit that passes through all edges of a graph
5.
An Eulerian trail exists if there exists two and only two vertices of odd
degree
6.
An Eulerian circuit exists if all vertices are of even degree
7.
Chinese postman problem can be used to find the route of minimum
weight that covers all edges of a graph
Hamiltonian paths and cycles:
1.
Complete graph: A graph that there exists an edge for any pair of two
2.
vertices
Hamiltonian path: A path that passes through all vertices of a graph
3.
Hamiltonian cycle: A cycle that passes through all vertices of a graph
Travelling Salesman problem:
1.
Travelling Salesman problem can be used to find the cycle of minimum
weight that passes through all vertices of a graph
2.
Nearest neighbour algorithm can be used to find the upper bound of the
solution of a travelling salesman problem
3.
Deleted vertex algorithm can be used to find the lower bound of the
solution of a travelling salesman problem
21
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
19 Differentiation

dy
 f ( x) : Derivative of the function y  f ( x) (First derivative)
dx

Rules of differentiation:

1.
f ( x)  x n  f ( x)  nx n1
2.
f ( x)  p( x)  q( x)  f ( x)  p( x)  q( x)
3.
f ( x)  cp( x)  f ( x)  cp( x)
Relationship between graph properties and the derivatives:
1.
f ( x)  0 for a  x  b : f ( x) is increasing in the interval
2.
f ( x)  0 for a  x  b : f ( x) is decreasing in the interval
3.
f (a)  0 : (a, f (a)) is a stationary point of f ( x)
4.
f (a)  0 and f ( x) changes from positive to negative at x  a :
(a, f (a)) is a maximum point of f ( x)
5.
f (a)  0 and f ( x) changes from negative to positive at x  a :
(a, f (a)) is a minimum point of f ( x)

Tangents and normals:
1.
f (a) : Slope of tangent at x  a
2.
3.
4.

1
: Slope of normal at x  a
f (a )
y  f (a)  f (a)( x  a) : Equation of tangent at x  a
 1 
y  f (a )  
 ( x  a) : Equation of normal at x  a
 f (a) 
Derivatives of a function y  f ( x) :
1.
d 2 y d  dy 
    f ( x) : Second derivative
dx 2 dx  dx 
2.
dn y
 f ( n ) ( x) : n -th derivative
n
dx
22
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
More rules of differentiation:
f ( x)  sin x  f ( x)  cos x
f ( x)  cos x  f ( x)   sin x
1
f ( x)  tan x  f ( x) 
cos 2 x
f ( x)  e x  f ( x)  e x
f ( x)  ln x  f ( x) 

f ( x)  p(q( x))  f ( x)  p(q( x))  q( x)
f ( x)  p ( x) q ( x)
 f ( x)  p( x)q( x)  p( x)q( x)
p( x)
q( x)
p( x)q( x)  p( x)q( x)
 f ( x) 
(q( x)) 2
f ( x) 
1
x
f (a)  0 and f ( x) changes sign at x  a : (a, f (a)) is a point of inflexion of
f ( x)

dN dN dx

 : Rate of change of N with respect to the time t
dt
dx dt

Tests for optimization:

1.
First derivative test
2.
Second derivative test
Applications in kinematics:
1.
s(t ) : Displacement with respect to the time t
2.
v(t )  s(t ) : Velocity
3.
a(t )  v(t ) : Acceleration
20 Integration and Trapezoidal Rule

Integrals of a function y  f ( x) :
1.
 f ( x)dx : Indefinite integral of f ( x)
2.
 f ( x)dx : Definite integral of f ( x) from a to b
b
a
23
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Rules of integration:
1 n +1
n
1.
=
∫ x dx n + 1 x + C
2.
∫ ( p′( x) + q′( x))dx = p( x) + q( x) + C
3.

x)dx cp ( x) + C
∫ cp′( =
b
∫ f ( x)dx : Area under the graph of f ( x) and above the x -axis, between x = a
a
and x = b , where f ( x) ≥ 0

Trapezoidal Rule:
a , b ( a < b ): End points
n : Number of intervals
h=
b−a
: Interval width
n
1
b
∫ f ( x)dx can be estimated by 2 h [ f ( x ) + f ( x ) + 2( f ( x ) + f ( x ) +  + f ( x ))]
a

1
n
2
n −1
Estimation by Trapezoidal Rule:
1.
The estimation overestimates if the estimated value is greater than the
actual value of
2.
b
∫ f ( x)dx
a
The estimation underestimates if the estimated value is less than the
actual value of

0
b
∫ f ( x)dx
a
More rules of integration:
− cos x + C
∫ sin xdx =
∫ e d=x e + C
xdx sin x + C
∫ cos =
dx ln x + C
∫ x=
x
x
1
1
dx tan x + C
∫ cos x=
Integration by substitution
2
b
b
a
a
)dx [ f ( x=
)]
f (b) − f (a )
∫ f ′( x=

Areas on x - y plane, between x = a and x = b :
1.
2.
b
∫ f ( x) dx : Area between the graph of f ( x) and the x -axis
a
b
∫ f ( x) − g ( x) dx : Area between the graph of f ( x) and the graph of g ( x)
a
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

Areas on x - y plane, between y  c and y  d :
d
1.
 g ( y) dy : Area between the graph of g ( y) and the y -axis
2.
 g ( y)  f ( y) dy : Area between the graph of g ( y) and the graph of f ( y)
c
d
c
Volumes of revolutions about the x -axis, between x  a and x  b :
1.
b
V    ( f ( x))2 dx : Volume of revolution when the region between the
a
graph of f ( x) and the x -axis is rotated 360 about the x -axis
2.
b
V    (( f ( x))2  ( g ( x))2 )dx : Volume of revolution when the region
a
between the graphs of f ( x) and g ( x) is rotated 360 about the x -axis

Volumes of revolutions about the y -axis, between y  c and y  d :
1.
d
V    ( g ( y))2 dy : Volume of revolution when the region between the
c
graph of g ( y) and the y -axis is rotated 360 about the y -axis
2.
d
V    (( g ( y))2  ( f ( y ))2 )dy : Volume of revolution when the region
c
between the graphs of g ( y) and f ( y) is rotated 360 about the y -axis

Applications in kinematics:
1.
a(t ) : Acceleration with respect to the time t
2.
v(t )   a(t )dt : Velocity
3.
s(t )   v(t )dt : Displacement
4.
d   v(t ) dt : Total distance travelled between t1 and t 2
t2
t1
21 Differential Equations

dy
 f ( x, y ) : First order differential equation
dx

Solving
dy
 f ( x) g ( y ) by separating variables:
dx
dy
1
 f ( x) g ( y )  
dy   f ( x)dx
dx
g ( y)
25
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Coupled differential equations:
1.
 dx 
 dx
 dt   a b  x 
 dt  ax  by
can
be
expressed
as
 

 
 dy   c d  y 
 dy  cx  dy
 
 dt
 dt 
3.
a b
1 , 2 : Eigenvalues of 

c d
v1 , v 2 : Eigenvectors corresponding to 1 and 2 respectively
4.
x  Ae1t v1  Be2t v 2 : Solution of the system
5.
Stable equilibrium if 1 , 2  0 or 1  a  bi , 2  a  bi and a  0
6.
Unstable equilibrium if 1 , 2  0 or 1  a  bi , 2  a  bi and a  0
7.
Saddle point if 12  0
2.
dy
 f ( x, y ) by Euler’s method, with ( x0 , y0 ) and step length h :
dx
 xn 1  xn  h

 y  y  h dy
n
 n 1
dx ( xn , yn )


Solving

 dx
 dt  ax  by
Solving 
by Euler’s method, with (t0 , x0 , y0 ) and step length h :
 dy  cx  dy
 dt

dx
 xn 1  xn  h dt
( tn , xn , yn )

tn1  tn  h and 
 y  y  h dy
n
 n 1
dt (tn , xn , yn )


Predator-prey models:
 dx
 dt  (a  by ) x
, where a , b , c and d are positive constants

 dy  (cx  d ) y
 dt
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
The second-order differential equation
d2 x
dx
 a  bx  0 can be expressed as
2
dt
dt
 dv
 dt  av  bx

 dx  v
 dt
22 Statistics


Relationship between frequencies and cumulative frequencies:
Data
Frequency
10
20
30
f1
f2
f3
Data less than
or equal to
10
20
30
Measures of central tendency for a data set {x1 , x2 , x3 ,
Cumulative
frequency
f1
f1  f 2
f1  f 2  f3
, xn } arranged in
ascending order:
x  x  x3   xn
1.
: Mean
x 1 2
n
2.
The datum or the average value of two data at the middle: Median
3.

The datum appears the most: Mode
Measures of dispersion for a data set {x1 , x2 , x3 ,
, xn } arranged in ascending
order:
1.
xn  x1 : Range
2.
Two subgroups A and B can be formed from the data set such that all
data of the subgroup A are less than or equal to the median, while all
data of the subgroup B are greater than or equal to the median
3.
Q1  The median of the subgroup A: Lower quartile
4.
Q3  The median of the subgroup B: Upper quartile
5.
Q3  Q1 : Inter-quartile range (IQR)
6.
( x1  x )2  ( x2  x )2  ( x3  x )2 

n
 ( xn  x ) 2
: Standard deviation
27
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Box-and-whisker diagram:

A datum x is defined to be an outlier if x  Q1  1.5IQR or x  Q3  1.5IQR

Coding of data:
1.
Only the mean, the median, the mode and the quartiles will change
when each datum of the data set is added or subtracted by a value
2.
All measures of central tendency and measures of dispersion will
change when each datum of the data set is multiplied or divided by a
value
23 Probability


Terminologies:
1.
U : Universal set
2.
A : Event
3.
4.
x : Outcome of an event
n(U ) : Total number of elements
5.
n( A) : Number of elements in A
Formulae for probability:
1.
P( A  B)  P( A)  P( B)  P( A  B)
2.
P( A)  1  P( A)
3.
P( A | B) 
4.
P( A)  P( A  B)  P( A  B)
5.
P( A  B)  P( A  B)  1
6.
P( A  B)  P( A)  P( B) and P( A  B)  0 if A and B are mutually exclusive
7.
P( A  B)  P( A)  P( B) and P( A | B)  P( A) if A and B are independent
P( A  B)
P( B)
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

Venn diagram:
1.
Region I: A  B
2.
3.
Region II: A  B
Region III: A  B
4.
Region IV: ( A  B)
Tree diagram:
1.
Path I: P( A  B)  pq
2.
Path I + Path III:
 P( B)
 P( A  B)  P( A  B)
 pq  (1  p)r

Markov Chain with transition matrix T :
1.
det(T  I) : Characteristic polynomial of T
2.
Solution(s) of det(T  I)  0 : Eigenvalue(s) of T
3.
v : Steady state probability vector, which is the eigenvector of T
corresponding to the eigenvalue   1
4.
v 0 : Initial state probability vector
5.
v n  Tn v 0 : State probability vector after n transitions
6.
The column sum of T must be equal to 1
24 Discrete Probability Distributions

Properties of a discrete random variable X :
X
x1
x2
…
xn
P( X  x)
P( X  x1 )
P( X  x2 )
…
P( X  xn )
1.
P( X  x1 )  P( X  x2 ) 
 P( X  xn )  1
2.
E( X )  x1P( X  x1 )  x2 P( X  x2 ) 
3.
E( X )  0 if a fair game is considered
 xn P( X  xn ) : Expected value of X
29
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
25 Binomial Distribution


Properties of a random variable X ~ B(n, p) following binomial distribution:
1.
2.
Only two outcomes from every independent trial (Success and failure)
n : Number of trials
3.
p : Probability of success
4.
X : Number of successes in n trials
Formulae for binomial distribution:
2.
n
P( X  r )    p r (1  p)n r for 0  r  n , r 
r
E( X )  np : Expected value of X
3.
Var( X )  np(1  p) : Variance of X
1.
4.
np(1  p) : Standard deviation of X
5.
P( X  r )  P( X  r  1)  1  P( X  r  1)
26 Poisson Distribution


Properties of a random variable X ~ Po( ) following Poisson distribution:
1.
The expected number of occurrences of an event is directly proportional to
2.
the length of the time interval
The numbers of occurrences of the event in different disjoint time intervals
3.
4.
are independent
 : Mean number of occurrences of an event
X : Number of occurrences of an event
Formulae for Poisson distribution:
2.
e    r
for r  0 , r 
r!
E( X )   : Expected value of X
3.
Var( X )   : Variance of X
4.
5.
 : Standard deviation of X
1.
P( X  r ) 
P( X  r )  P( X  r  1)  1  P( X  r  1)
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27 Normal Distribution

Properties of a random variable X ~ N( ,  2 ) following normal distribution:
5.
 : Mean
 : Standard deviation
The mean, the median and the mode are the same
The normal curve representing the distribution is a bell-shaped curve
which is symmetric about the middle vertical line
P( X   )  P( X   )  0.5
6.
The total area under the curve is 1
1.
2.
3.
4.
28 Linear Combinations of Variables

Properties of linear transformations of independent random variables:
1.
E(aX  b)  aE( X )  b
2.
Var(aX  b)  a 2 Var( X )
3.
E(a1 X1  a2 X 2 
4.
Var(a1 X1  a2 X 2 
 an X n )  a1E( X1 )  a2E( X 2 ) 
 an E( X n )
 an X n )  a12 Var( X1 )  a22Var( X 2 ) 
 an2Var( X n )
29 Point Estimation

Central limit theorem:
1.
X i ~ N ( ,  2 )
2.
X
3.
 2 
X ~ N  ,
 when n is sufficiently large
n 

X1  X 2 
n
 Xn
: Sample mean
31
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Properties of point estimation:
n
1.
2
n 1
s
(X  X )
 i 1
i
n 1
2
: Sample variance
n
(X  X )
2
i
2.
sn2  i 1
3.
sn21 
4.
E( X )  X
5.
E( sn21 )   2
n
n 2
sn
n 1
30 Interval Estimation

(1   )% confidence interval for population mean  when the population
variance  2 is known:
X  X2   Xn
1.
: Sample mean
X 1
n


 

 
2.
, X  z 
 X  z 
 , where P  Z  z  
2
n
n


2
2 
2

3.
: Interval width
2z 
n
2

(1   )% confidence interval for population mean  when the population
variance  2 is unknown:
X  X2   Xn
1.
: Sample mean
X 1
n


 
sn 1
s 
2.
, X  t   n 1  , where P  X  t   
 X  tn 1,  
n 1,
n 1,
2
n
n

2
2

2 
s
3.
2t   n 1 : Interval width
n 1,
n
2
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31 Bivariate Analysis

Correlations:
Positive
Strong
0.75 < r < 1
Moderate
0.5 < r < 0.75
Weak
0 < r < 0.5
No
Negative
r =0
Weak
−0.5 < r < 0
Moderate
−0.75 < r < −0.5
Strong
−1 < r < −0.75
where r is the correlation coefficient

Linear regression:
=
y ax + b : Regression line of y on x

Correlation Coefficient for ranked data:
rs : Spearman’s Rank Correlation Coefficient

Coefficient of determination:
R 2 : Coefficient of determination
R 2 % of the variability of the data can be explained by the regression model

Sum of square residuals:
x
x1
x2
…
xn
y
y1
y2
…
yn
ŷ1
ŷ2
…
yˆ n
Predicted
value of y
=
SS res

n
∑ ( y − yˆ ) : Sum of square residuals
i =1
i
i
2
Non-linear regressions:
1.
2.
3.
Quadratic regression
Cubic regression
Quartic regression
4.
5.
Exponential regression
Logarithmic regression
6.
7.
Power regression
Logistic regression
33
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
32 Statistical Tests

Hypothesis test:
H 0 : Null hypothesis
H1 : Alternative hypothesis
C : Critical value in the hypothesis test
 : Significance level

 2 test for independence for a contingency table with r rows and c columns:
n  rc : Total number of data
Oi ( i  1, 2,
, n ): Observed frequencies
Ei ( i  1, 2,
, n ): Expected frequencies
v  (r  1)(c  1) : Degree of freedom
(Oi  Ei )2
:  2 test statistic
Ei
i 1
n
2
 calc

H 0 : Two variables are independent
H1 : Two variables are not independent
2
H 0 is rejected if calc
 C or the p -value is less than the significance level
2
H 0 is not rejected if calc
 C or the p -value is greater than the significance
level

 2 goodness of fit test for a contingency table with 1 row and c columns:
v  c  1 : Degree of freedom
H 0 : The data follows an assigned distribution
H1 : The data does not follow an assigned distribution
2
H 0 is rejected if calc
 C or the p -value is less than the significance level
2
H 0 is not rejected if calc
 C or the p -value is greater than the significance
level

Two sample t test:
1 ,  2 : The population means of two groups of data
H 0 : 1  2
H1 : 1  2 , 1  2 (for 1-tailed test), 1  2 (for 2-tailed test)
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level
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
More about  2 test for independence and  2 goodness of fit test:
All expected frequencies Ei should be greater than 5

Z test when the population variance  2 is known:
 : Population mean
H 0 :   0
H1 :   0 ,   0 (for 1-tailed test),   0 (for 2-tailed test)
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level

One sample t test when the population variance  2 is unknown:
 : Population mean
H 0 :   0
H1 :   0 ,   0 (for 1-tailed test),   0 (for 2-tailed test)
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level

Paired t test:
d  x  y : Difference between each pair of data from two variables x and y
H 0 : d  0
H1 : d  0 , d  0 (for 1-tailed test), d  0 (for 2-tailed test)
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level

Test involving binomial distribution:
X ~ B(n, p)
H 0 : p  p0
H1 : p  p0 , p  p0
x : Observed value of X
P( X  x) for H1 : p  p0 or P( X  x) for H1 : p  p0 : p -value under X ~ B(n, p0 )
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics

Test involving Poisson distribution:
X ~ Po( )
H 0 :   0
H1 :   0 ,   0
x : Observed value of X
P( X  x) for H1 :   0 or P( X  x) for H1 :   0 : p -value under X ~ Po(0 )
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level

Test involving bivariate normal distribution:
 : Product moment correlation coefficient
H0 :   0
H1 :   0 ,   0 (for 1-tailed test),   0 (for 2-tailed test)
H 0 is rejected if the p -value is less than the significance level
H 0 is not rejected if the p -value is greater than the significance level

Type I and type II errors:
 : Significance level
P(Reject H 0 | H 0 is true)   : Type I error
P(Not reject H 0 | H 0 is not true) : Type II error
33 Paper 3 Analysis

Nature of paper: Structured question

Time allowed: 60 minutes

Maximum mark: 55 marks

Number of questions: 2

Mark range per question: 25 marks to 30 marks

Weighting: 20% of the total mark
36
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
Ways of assessing:
1.
Identify a hypothesis test
2.
State
(a)
a condition
(b)
(c)
a reason
an assumption
3.
Write down
(a)
the value of a quantity
(b)
the formula of a quantity
4.
Find
(a)
the value of a quantity
(b)
the formula of a quantity
5.
Solve an equation
6.
7.
Perform a hypothesis test
Show
(a)
a quantity equals to a value
8.
(b)
the formula of a quantity
Estimate the value of a quantity
9.
10.
Predict the value of a quantity
Sketch a graph
11.
12.
Suggest an improvement
Explain a reason
13.
Describe a result
14.
Verify
(a)
the value of a quantity
15.
(b)
the expression of a quantity
Comment on
(a)
(b)
the validity of an argument
a statement
37
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
Notes
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39
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Your Practice Paper – Applications and Interpretation HL for IBDP Mathematics
40
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