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1
EF 2452 Mathematics for Economics and Finance
Tutorial 1
1. Announcement:
(a) Attendence of tutorial will also be counted towards 10% participation part of the
overall raw score. Being late for 15 minutes will be counted as absence. Leaving
early withou

Tutorial 3
1. Example 6.4 Textbook P63
A firm has cost function C(q) = 1500 + 15q 3q 2 + q 3 . Show that its marginal cost is always
positive.
2. Exercise 6.6 Textbook P66
Suppose that the demand function for a good is
q D (p) =
8000
p2 + 1
where q is the

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' AIMEE ~
Question No.
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LINEAR EQUATIONS II
Arbitrage portfolios and state prices
Suppose there are assets and states, then the
returns matrix is defined as each of its
entry being the return from investing in one unit
of asset when state occurs.
(Remember we had three assets a

INTRODUCTION TO
CALCULUS
Finding Derivatives
Rate of change of a function
We are interested in: how does the dependent
variable change responding to a change in the
independent variable?
Idea: Take a function : . If changes from
to + , then () changes f

MATHEMATICAL TERMS
AND NOTATIONS
Sets, functions and equations
brief introduction: why math?
Math vs. language
Concision and precision.
Example: John is always willing to exchange apples for bananas on
the basis of two apples for one banana.
The correspon

VECTORS AND MATRICES
Preferences and convexity
Vectors
An n-vector is a list of numbers. It can be
written either as a row-vector,
1 , 2 , , ,
Or a column-vector,
1
2
. .
.
.
Vectors
E.g. (1, 2) is a 2 dimensional row vector and
a 3 dimensional column vec

PARTIAL DERIVATIVES
Functions of several variables
A function of one variable: =
+1
.
2 +1
A function of two variables:
, = 2 2 + 3 2 ;
, =
;
+ 1
2 +
, =
;
, = 0.5 0.5 .
Use the notation : 2 for a function which
assigns to each , in 2 a value (, )

THE DERIVATIVE IN
ECONOMICS
Some examples
Global maximum/minimum
Global refers to the constraint set of our
optimization problem. A global maximum is
necessarily a local maximum, but a local maximum
may not be a global maximum.
We will focus on optimizati

MATHEMATICAL TERMS
AND NOTATIONS
A Model of Market, Sequences and Examples
A model of the market
The market of a single good, say Coke.
Care about price and quantity of Coke: (, ).
From producers point of view:
Supply
function: what is the quantity of Co

OPTIMIZATION IN TWO
VARIABLES
Profit maximization again
Suppose Apple produces two goods, iPhones and
iPads. Let be the unit of iPhones and be the unit
of iPads that Apple produces.
Assume the cost of producing a combination of
and is , = 5 + 2 + 2
Apple

MATRIX ALGEBRA
What is a matrix
A matrix is an array of numbers
1
11
21 2
.
Lets call this matrix . is an matrix, or it has
size and sometimes we denote it as . This
means has rows and columns. An element
of is the number at the th row and the column i

LINEAR EQUATIONS
Linear equations in matrix form
A system of linear equations in unknowns
1 , 2 , , is a set of equations of the form
11 1 + 12 2 + + 1 = 1
21 1 + 22 2 + + 2 = 2
1 1 + 2 2 + + = .
The s are the coefficients of the system.
1 , 2 , is a solu

INTRODUCTION TO
OPTIMIZATION
A first look
Some special functions: power
If is a natural number: 0.
= for times.
+ = .
0
= 1.
=
(
1
)
1
.
=
Since = 0 = 1.
+()
=
1
.
= . Call th root of .
1
= ( ) . Now the power function is defined for all
rational .