Lecture 5 ECON2050
Carlos Oyarzun
September 4, 2012
1
Table of contents
1
Handout Chapter 1. Unconstrained Optimisation via Calculus
1.1: Functions of One Variable
2
Chapter 8
8.6: The Extreme Value Theorem
2
Handout Chapter 1. Unconstrained Optimisation
Semester Two Mock In-Class Quiz Examinations, 2016
ECON2050 Mathematical Economics
School of Economics
EXAMINATION
Semester Two Mock In-Class Quiz Examinations, 2016
ECON2050 Mathematical Economics
This paper is for St Lucia Campus students.
Examination D
E CON 2050
S EMESTER 2, 2016
Tutorial 9
The problems below are for your September 22/23 and October 6/7 tutorials.
Part 1
In the last tutorial, you already maximized/minimized some functions whose domain was
an open set. Below some additional problems wit
Semester Two Mock In-Class Quiz Examinations, 2016
ECON2050 Mathematical Economics
School of Economics
EXAMINATION
Semester Two Mock In-Class Quiz Examinations, 2016
ECON2050 Mathematical Economics
This paper is for St Lucia Campus students.
Examination D
E CON 2050
S EMESTER 2, 2016
Tutorial 8
Problem 1
Analyze whether the following functions are concave or convex. (Try to apply
some of the different methods of showing concavity/convexity that we discussed
in the lectures.)
1. f : R2 R, (x, y) 7 x2 5y 2 4
Semester Two Mock Mid-Semester Examinations, 2016
ECON2050 Mathematical Economics
School of Economics
EXAMINATION
Semester Two Mock Mid-Semester Examinations, 2016
ECON2050 Mathematical Economics
This paper is for St Lucia Campus students.
Examination Dur
E CON 2050
S EMESTER 2, 2016
Tutorial 3
Problem 1
Let A ( Rk . We say that a function f : Rk \A R can be continuously
extended (to Rk ) if there is a continuous function g : Rk R such that f (x) =
g(x) for all x Rk \A. Prove that the following functions c
E CON 2050
S EMESTER 2, 2016
Tutorial 6
Problem 1
d(u(x,y)
dt
Calculate the derivative
if
1
1
u : R2+ R, (x, y) 7 x + 4x 4 y 2 3y,
x : R+ R+ , t 7 t3 , and
y : R+ R+ , t 7
1
t
Solution: The derivative
d(u(x,y)
dt
is the derivative of the composition
u
E CON 2050
S EMESTER 2, 2016
Tutorial 5
Problem 1 (compare SB Examples 14.2, 14.3, 14.9 and 14.11)
3
1
The production function of a firm is f : R2+ R, (K, L) 7 4K 4 L 4 . Currently, the firm is using 10,000 units of capital (K) and employing 625 workers
(
E CON 2050
S EMESTER 2, 2016
Tutorial 2
Problem 1
Draw the following sets. Explain whether they are open and whether they are
compact, and determine their sets of interior points and boundary points.
(a) S1 = cfw_(x, y) R2 : |x| + |y| 1
(b) S2 = cfw_(x, y
E CON 2050
S EMESTER 2, 2016
Tutorial 7
Problem 1
Classify each of the following matrices according to its definiteness.
3 1
0
8 2 2
5 4 0
A = 1 4 0 ; B = 2 6 1 ; C = 4 5 0
0
0 3
2 1 2
0 0 2
Solution:
1. The leading principal minors of A are
D3 =
D1 = 3
E CON 2050
S EMESTER 2, 2016
Tutorial 4
Problem 1
x
You are given the function f : D R, (x, y) 7 e y + ln(x2 + 1), where D is
the largest domain D R2 on which f is well-defined.
(a) Determine D.
Solution: The domain of f is D = cfw_(x, y) R2 : y 6= 0.
(b)
Semester Two Mock Mid-Semester Examinations, 2016
ECON2050 Mathematical Economics
School of Economics
EXAMINATION
Semester Two Mock Mid-Semester Examinations, 2016
ECON2050 Mathematical Economics
This paper is for St Lucia Campus students.
Examination Dur
Lecture 1-2 ECON2050
Carlos Oyarzun
September 11, 2012
1
Table of contents
1
Chapter 4
4.2: Basic Denitions
4.3: Graphs of Functions
4.4: Linear (and Ane) Functions
4.6-4.10: Other real functions you should know
2
Chapter 5
5.3: Inverse Functions
3
Chapte
Lecture 7 ECON2050
Carlos Oyarzun
September 10, 2012
1
Table of contents
1
Unconstrained Optimisation via Calculus
Functions of Several Variables
2
Unconstrained Optimisation via Calculus
Functions of Several Variables
Outline
In slides 5 we have two type
Lecture 3 ECON2050
Carlos Oyarzun
August 30, 2012
1
Chapter 7
7.1: Implicit Dierentiation
7.1: Implicit Dierentiation
Sometimes functions f are dened implicitly by an equation R(x, y ) = 0.
Such functions are called implicit functions.
Example
y x2 = 0
fo
Lecture 4 ECON2050
Carlos Oyarzun
August 30, 2012
1
Table of contents
1
Preliminaries
The Space R2
2
Chapter 11
11.1: Functions of Two Variables
11.2: Partial Derivatives with Two Variables
Directional derivative
3
Chapter 13
13.1: Two Variables: Necessar
Part 3: Constrained Optimization
Mathematics II
Vadym Lepetyuk
based on teaching materials by Josefa Toms
Departamento de Fundamentos del Anlisis Econmico
2011-12
Vadym Lepetyuk (UA)
Part 3: Constrained Optimization
2011-12
1 / 79
3.1. Constrained optimiz
E CON 2050
S EMESTER 2, 2016
Tutorial 1
1 Two-Valued Logic: A Very Brief Review
Every sentence expressing a proposition is either true (t) or false (f). Formally, such
sentences are all we deal with in ECON2050. For example: Anns demand for coffee is
3 cu
E CON 2050
S EMESTER 2, 2016
Tutorial 8
Problem 1
Analyze whether the following functions are concave or convex. (Try to apply
some of the different methods of showing concavity/convexity that we discussed
in the lectures.)
1. f : R2 R, (x, y) 7 x2 5y 2 4
E CON 2050
S EMESTER 2, 2016
Tutorial 5
Problem 1 (compare SB Examples 14.2, 14.3, 14.9 and 14.11)
3
1
The production function of a firm is f : R2+ R, (K, L) 7 4K 4 L 4 . Currently, the firm is using 10,000 units of capital (K) and employing 625 workers
(
E CON 2050
S EMESTER 2, 2016
Tutorial 6
Problem 1
Calculate the derivative
d(u(x,y)
dt
if
1
1
u : R2+ R, (x, y) 7 x + 4x 4 y 2 3y,
x : R+ R+ , t 7 t3 , and
y : R+ R+ , t 7
1
t
Problem 2
Suppose that it follows from the implicit function theorem that, i
E CON 2050
S EMESTER 2, 2016
Tutorial 7
Problem 1
Classify each of the following matrices according to its definiteness.
3 1
0
8 2 2
5 4 0
A = 1 4 0 ; B = 2 6 1 ; C = 4 5 0
0 0 2
0
0 3
2 1 2
Problem 2
Demonstrate that if g : R2 R is convex and b R then S
Lecture 8 ECON2050
Carlos Oyarzun
November 5, 2012
1
Table of contents
1
Unconstrained Optimisation via Calculus
Positive and Negative Denite Matrices and Optimisation
1.4: Coercive Functions and Global Minimisers
Unconstrained Optimisation via Calculus
P