The rst order condition for the maximization in Bellman equation (12.13) is
s
@
ft (at ; st ) +
@at
t+1
@
gt (at ; st ) = 0; t = 0; 1; :; T
@at
1
which is precisely the Euler equation derived using the Lagrangian approach and the Maximum principle.
Exampl
Chapter 12
Dynamic Optimization
12.1
Introduction
A consumer maximizes a intertemporal utility function,
max u(c0 ) + u(c1 )
c0 ;c1
subject to c0 +
c1
=w
1+r
where represents the discount rate of future consumption. His/her constraint means that
given his
Chapter 10
Unconstrained Optimization
10.1
Denitions
Let F : U Rn ! R1 be a real-valued function of n variables. Then,
(1) A point x 2 U is a maximum of F on U if F (x ) F (x) for all x 2 U:
(2) x 2 U is a strict maximum if x is a maximum and F (x ) > F (
Chapter 8
Quadratic Forms and Denite
Matrices
8.1
Quadratic Forms
Denition 8.1 A quadratic Form on Rn is a real-valued function of the form
Q(x1 ; :; xn ) =
X
aij xi xj
(8.1)
ij
where each term is a monomial of degree two.
It is possible that (8.1) is tra
Chapter 6
Calculus of Several Variables
6.1
Preliminaries
Denition 6.1 (Graph of a Function) Let f : D
the subset of Rn+1 consisting of all the points
Rn ! R: Dene the graph of f to be
(x1 ; :; xn ; f (x1 ; :; xn )
in Rn+1 for (x1 ; :; xn ) in D: In symbo
Study Question #2
Show lim
x!4
p
x = 2:
Proof. Goal: We should show that there is a such that when we take > 0; x is in jf (x)
for all > 0 and for all element x 2 I = jx 4j < : We rewrite as
jf (x) 2j <
p
< x 2<
,
p
x
,2
2j <
2<
p
< x<2+
)2 < x < (2 + )2
Study Question #2
Show lim
x!4
p
x = 2:
Proof. Goal: We should show that there is a such that when we take > 0; x is in jf (x)
for all > 0 and for all element x 2 I = jx 4j < : We rewrite as
jf (x) 2j <
p
< x 2<
) (2
p
,
x
,2
2j <
2<
p
< x<2+
)2 < x < (2
Study Question #1
1. Consider points:
(i) P = (2; 1) and Q = (0; 8)
(ii) W = ( 1; 2; 5) and Z = ( 1; 0; 3)
!
!
(a) Use vectors P Q and W Z to depict displacements from P to Q and from W to Z:
!
!
P Q = ( 2; 9) and W Z = (0; 2; 8)
p
p
!
!
!
!
(b) Calculate
Study Question #1
1. Consider points:
(i) P = (2; 1) and Q = (0; 8)
(ii) W = ( 1; 2; 5) and Z = ( 1; 0; 3)
!
!
(a) Use vectors P Q and W Z to depict displacements from P to Q and from W to Z:
!
!
P Q = ( 2; 9) and W Z = (0; 2; 8)
p
p
!
!
!
!
(b) Calculate
Q1: study question #2 3 = 1/2 (1) |( ) 1| <
, 1 + = 1 + = 3/2 ?
1
2
A1: . open interval (1) , boundary 3/2
. study question #2 .
.
Q2: chapter 3 Fact 3.8 ?
A2: .
Fact 3.8 Any system of linear equations having A as its coefficient matrix will have at
Chapter 5
Calculus of One Variable
5.1
The Denitions
In univariate case, we dene the slope of a nonlinear function f at a point (x0 ; f (x0 ) on its
graph as the slope of the tangent line to the graph of f at that point. We call the slope of
the tangent l
Study Question #2
Show lim
x!4
p
x = 2:
Proof. Goal: We should show that there is a such that when we take > 0; x is in jf (x)
for all > 0 and for all element x 2 I = jx 4j < : We rewrite as
jf (x) 2j <
p
< x 2<
,
p
x
,2
2j <
2<
p
< x<2+
)2 < x < (2 + )2
Chapter 4
Analytic Background
4.1
Limit in R1
A sequence of real numbers is an assignment of a real number to each natural number.
Denition 4.1 (
interval) The
interval is dened as
I (r) = fs 2 Rj js
where
rj < g
is a small and positive real number.
Denit
Study Question #1
1. Consider points:
(i) P = (2; 1) and Q = (0; 8)
(ii) W = ( 1; 2; 5) and Z = ( 1; 0; 3)
!
!
(a) Use vectors P Q and W Z to depict displacements from P to Q and from W to Z:
!
!
P Q = ( 2; 9) and W Z = (0; 2; 8)
p
p
!
!
!
!
(b) Calculate
Chapter 2
The Geometry of Euclidean Space
2.1
Vectors
Consider R2 and R3 : A vector is to be a directed line segment beginning at the origin.
Example 2.1 Suppose a man goes 3 units to the right and 1 unit up from the origin. We
represent the man behavior
Chapter 1
Logics and Functions
1.1
1.1.1
Logic
And and Or
The statement
\A and B
means that both A is true and B is true.
The statement
\A or B
means that A is true or B is true or both are true.
Example 1.1 The statement
x > 5 and x < 7
is true for the