Solution: Problem Set 4
Ruoyan Huang
(Due: 08/12/2014)
1. Find all partial derivatives , the tangent plane at the point (1, 1, 1), as well as the directional derivatives
at (1, 2, 3) in the direction (3, 2, 1) of the following functions f : R3 R:
z
(a) f

University of Rochester
Simon Business School
AEC 505: Mathematical Techniques in Econ
Zeroeth Summer 2014
Instructor: Ruoyan Huang
Office: CS 4-312
Phone: (919) 360-0500
Email:
ruoyan.huang@simon.rochester.edu

Solution: Problem Set 2
Instructor: Ruoyan Huang
July 9, 2014
1. A set X Rn may be open or not open, closed or not closed, bounded or not bounded,
connected or not connected, and convex or not convex. For which of the possible 32 combinations of these pro

Solution: Problem Set #3
Instructor: Ruoyan Huang
(Due: 07/29/2014)
1. Which of the following describes best the space of all functions from domain X to range Y :
(a) X Y
(b) X Y
(c) Y X
(d) RXY
Y X best describes the space of all functions from domain X

Problem Set 4
Ruoyan Huang
(Due: 08/14/2014)
1. Find all partial derivatives , the tangent plane at the point (1, 1, 1), as well as the directional derivatives
at (1, 2, 3) in the direction (3, 2, 1) of the following functions f : R3 R:
z
(a) f (x, y, z)

Problem Set #3
Instructor: Ruoyan Huang
(Due: 07/29/2014)
1. Which of the following describes best the space of all functions from domain X to range Y :
(a) X Y
(b) X Y
(c) Y X
(d) RXY
2. Consider the following functions f : R R:
0
if x = 0
(a) f (x) =
c

Lecture 11: Constrained Optimization
Instructor: Ruoyan Huang
July 24, 2014
1
1.1
Optimization subject to an equality constraint
Problem setup
Let f : X R and g : X R be differentiable functions form X Rn to the reals. We will also
assume that X is open i

Lecture 11: Constrained Optimization
Instructor: Ruoyan Huang
July 24, 2014
1
1.1
Optimization subject to an equality constraint
Problem setup
Let f : X R and g : X R be differentiable functions form X Rn to the reals. We will also assume
that X is open i

Lecture 10: Unconstrained Optimization
Instructor: Ruoyan Huang
July 24, 2014
1
Maxima and minima
Optimization plays an important role in economics:
In a typical economic optimization problem, the assumption contains two parts:
Objective Function:
Opt

Problem Set 2
Instructor: Ruoyan Huang
(Due: Tuesday, 07/22/2014)
1. A set X Rn may be open or not open, closed or not closed, bounded or not bounded, connected or not
connected, and convex or not convex. For which of the possible 32 combinations of these

Solution: Problem Set #1
Instructor: Ruoyan Huang
July 9, 2014
1. Let the universe be R. Let
A = cfw_x | x > 0
B = cfw_2, 4, 8, 16, 3
C = cfw_2, 4, 6, 8, 10, 12, 14
D = cfw_x | 3 < x < 9
E = cfw_x | x
1
Calculate the sets:
(a) B C = cfw_2, 4, 8,
(b) B C =

Theorem 0.1. Let f : X Y be a function. The following statements are equivalent: (i) f satises
Denition 1.1, (ii) f satises Denition 1.2, (iii) f satises Denition 1.6.
Proof. To prove the equivalence, Well show (iii) (ii), (ii) (i), and (i) (iii).
1) Firs

Problem Set #1
Instructor: Ruoyan Huang
July 1, 2014
(Due: Thursday, July 10th)
1. Let the universe be R. Let
A = cfw_x | x > 0
B = cfw_2, 4, 8, 16, 3
C = cfw_2, 4, 6, 8, 10, 12, 14
D = cfw_x | 3 < x < 9
E = cfw_x | x
Calculate the sets:
(a) B C,
(b) B C,

Solution: Midterm
Instructor: Ruoyan Huang
(07/24/2014)
The exam is closed books/closed notes. You have 150 minites to complete the test.
1. (10 points) A and B are nite sets:
(a) When is it true that |A B| = |A| + |B|?
if A B = (they are disjoint)
(b) Wh

Lecture 7: Multivariate Calculus
Instructor: Ruoyan Huang
July 19, 2014
1
1.1
Functions of several variables
Partial Derivatives
Denition 1.1. (Partial derivative and partially differentiable)
1
Notation 1. (Partial derivative) We use Di f (a) to denote t

Lecture 4: Limits and Convergence
Instructor: Ruoyan Huang
July 3, 2014
1
Limit points
Denition 1.1. (limit point or accumulation point)
First, Lets consider any point x (0, 1). Are these points limit points of X = (0, 1)?
1
Second, lets consider two sp

Lecture 8: Higher-order Derivatives
Instructor: Ruoyan Huang
July 24, 2014
1
1.1
Functions of one variable
General denitions
Denition 1.1. (Second-order derivative) The derivative of a C1 function f : R R is a function
f : R R. If f is also differentiable

Lecture 6: Differentiation
Instructor: Ruoyan Huang
July 14, 2014
1
Differentiability
Denition 1.1. (Derivative)
1
Proposition 1.1. The derivative of f at a measure the rate of change, or slope of f , at the point a.
Thus, a differentiable function f is :

Lecture 5: Continuity
Instructor: Ruoyan Huang
July 8, 2014
1
1.1
Continuous Functions
Continuity and Function limits
The rst denition of function continuity is derived from the limit of a function from previous
class. Lets formally dene the following:
De

Lecture 9: Integration and Differential Equations
Instructor: Ruoyan Huang
July 24, 2014
1
1.1
Integration
The Riemann Integral
Suppose we are given a continuous function f : R R and want to nd the area under the graph
of f between the two points a, b R.

Lecture 3: Topological set properties
Instructor: Ruoyan Huang
(07/08/2014)
1
1.1
Distance Measures on the vector space
Euclidean or Pythagorean distance
The Euclidean or Pythagorean distance between two points x, y Rn
1.2
Metric and Metric Properties
T

Supplemental materials
July 10, 2014
Proof of Lemma 2.2: let (xt ) be a Cauchy sequence in R. Then (xt )
converges to a point x R
Proof. Let (m ) be a strictly decreasing sequence of positive reals such that m 0. Since (xt ) is a Cauchy
sequence in R, for

Lecture 1: Sets, Numbers, Proofs
Instructor: Ruoyan Huang
07/01/2014
1
Sets
1.1
Denition
Vocabulary of Sets:
Universe:
Set:
Example 1.1. A = cfw_1, 2, 3, B = cfw_4, 5, 6, C = cfw_Robert, Jay, Jenni f er
R
= cfw_ (The empty set or null set)
1
Criterion

Class calendar
Class 1 (07/01): Introduction, Sets, numbers and Proof
Class 2 (07/03): Vector and Matrix
Class 3 (07/08): Topological set properties
Class 4 (07/10): Limits and Convergence I
Class 5 (07/15): Limits

Course Outline/ Class Calendar
Each of the class sessions is devoted to an individual topic, and covered in a separated note I will send out
in advance. The detailed schedule of classes is as follow:
1
Lecture 1: Sets, Numbers, proofs
Sets
Denitions
Vo