DF (x )(y x ) = 0. However,
F (1 t )x + t y) F (x )
t 0
t
(1 t )F (x ) + tF (y) F (x )
lim
t 0
t
= F (y) F (x ) > 0,
g (0) = lim
a contrdiction.
17.3 Calculus Criteria for Concavity (21.1)
Theorem 63
Denition 36 (p.161). An m m matrix A = (ai j ) is called an upper-triangular matrix
if ai j = 0 for i > j. A is called a lower-triangular matrix if ai j = 0 for i < j. A is called
a diagonal matrix if
Econ 205 Midterm, Fall Semester 2014, 14:0015:15
You have 75 minutes. Total score is 60. Points are indicated in the [square brackets].
Within each large question, you are allowed to use results in th
Econ 205 Midterm, Fall Semester 2014, 14:0015:15
[15 points] 1. Dierentiation and Critical Points
[3] (1) g (x) =
x (x
2) on (0; 2). as the function is dierentiable, all critical points occur at x0
wh
Econ 205 Midterm, Spring 2016
[9 points] 1. Continuity and Dierentiability
[3] (1) from
limh"0
1
h
1
h
(g (0 + h)
g (0) =
1
h
( h)3 = 0. therefore, g 0 (0) = 0.
[3] (2) if x > 0, limh!0
if x < 0, limh
Econ 205 Midterm, Spring 2016
You have 75 minutes. Total score is 60. Points are indicated in the [square brackets].
Using Theorems NOT Listed in the question may result in ZERO Point for that questio
Econ 205 Midterm, Spring 2015
You have 75 minutes. Total score is 60. Points are indicated in the [square brackets].
Questions that need to be answered are written in bold.
Within each large question,
Econ 205 Final, Spring 2016
You have 75 minutes. Total score is 60. Points are indicated in the [square brackets].
Using Theorems NOT Listed in the question will result in ZERO Point for that question
Econ 205 Final, Spring 2016
[6 points] 1. Sets
[3] (1) all (closed) sets containing T also contain 5. the intersections of them include 5.
[3] (2) consider B" (x) fy : jy xj < "g = (5 "; 5 + "). for a
Theorem 53 (Thm 11.2, p.243). Let A = (a1 , a2 , , am). If a1 , a2 , , am are linearly
independent if and only if det A = 0.
Proof. a) only if: If a1 , a2 , , am are linearly independent, then A is on
Example 6. Suppose that we have a production function Q = kxa yb . Then,
Q
= akxa1 yb ,
x
2Q
= abkxa1 yb1 ,
x y
Q
= bkxa yb1
y
2Q
= abkxa1 yb1
y x
14 Some Linear Algebra
14.1 Deniteness of Quadrat
Mathematics for Economists
1 Introduction
1.1 Motivation
Why do we need to know mathematics in order to learn economics?
What is economics?
In economics we learn how the economy works in various si
2.7 Differentiability and Continuity
A function f is differentiable at x0 if the following limit exists
lim
h0
f (x0 + h) f (x0 )
.
h
A function f is continuous at x0 if limh0 f (x0 + h) exists and
4.2 Inverse Functions
Denition 4 (p.76). For a given function f : E1 R1 , E1 R1 , we say a function
g : E2 R1 , E2 R1 , is an inverse of f if
g( f (x) = x for all x E1 and
f (g(y) = y for all y E2 .
1
6 Integration (A.4)
6.1 Indenite Integral
Consider a continuous function f (x), where f (x) > 0 for all x.
Consider the area under the graph of y = f (x) from a certain point a to another
point x an
9 Limits and Open Sets (ch.12)
9.1 Sequences in Real Numbers
A sequence of real numbers cfw_x1 , x2 , , xn , is an assignment of a real number xn
to each natural number n.
Denition 7 (p.255). Let cfw
12 Calculus of Several Variables (ch.14)
12.1 Partial Derivative
Denition 25 (p.300). Let F : Rk R. The partial derivative of f with respect to xi at
x0 = (x0 , , x0 ) is dened as
1
k
F (x0 , , x0 + h
Theorem 41 (Thm 30.5, p.828). Let f : R1 R1 be a C2 function. For any point a <
b R1 , there is a point c (a, b) such that f (b) = f (a) + f (a)(b a) + 1 f (c)(b
2
a)2 .
Proof. Dene
2
f (b) f (a) f (
Econ 205 Final, Fall 2015
[7 points] 1. Linear Function
[4] (1) let u = (u1 ; u2 ; u3 ) and v = (v1 ; v2 ; v3 ). then, G (ru + sv) = (ru1 + sv1 ) + (ru2 + sv2 ) (ru3 + sv3 ) =
r (u1 + u2 u3 ) + s (v1