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 (Thm 21.3(p.511) and Thm 21.5(p.513). Let F : U R1 be
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 one-to-one and
onto, hence has an inverse function. Call
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 Quadratic Forms (16.2)
Denition 27. Let A be an m m symmetric
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 (a)(b a) .
(b a)2
Suppose that f (x) = M for all x (a, b
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, , x0 ) F (x0 , , x0 , , x0 )
F 0
i
i
1
1
k
k
(x ) = l
Theorem 34 (Thm 13.3, p.291). The general quadratic form Q(x1 , , xk ) = i j ai j xi x j
can be written as
1
a11 2 a12 1 a1k
x1
2
1 a12 a22 1 a2k x2
2
2
x1 x2 xk .
.
. . ,
.
.
.
. .
.
.
.
.
.
1
1
xk
akk
2 a1 k
2 a2 k
or xT Ax, where A is a symmetric m
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_x1 , x2 , , xn , be a sequence of real numbers. A rea
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 and denote it by A(x; a).
What is the derivative of A(x;
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
Example: f (x) = 3 2x, g(y) = 2 (3 y)
If f has an in
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 limh0 f (x0 +
h) = f (x0 ).
Example 1 (Non-differentiab
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 situations.
An economy consists of various people (consu
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 ai j = 0 for i = j.
Theorem 56 (Fact 26.11, p.731). Th