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8/25/2010
1
Agenda
•
Course Overview
•
Marginal Analysis
•
Elasticities of Demand
•
Unconstrained optimization
•
Isosurfaces and constrained optimization
•
What To Take Away
Microeconomics
The Structure of the Course
Consumers and
Producers
Market
Interaction
Today’s lecture
Uncertainty
Perfect competition
Monopoly and Pricing strategies
Competitive Strategy
Auctions
Information in Markets and Agency
Game Theory
Introduction to Markets
Consumer Theory and Demand
Technology and Production
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Agenda
•
Course Overview
•
Marginal Analysis
•
Elasticities of Demand
•
Unconstrained optimization
•
Isosurfaces and constrained optimization
•
What To Take Away
Definition of Derivative
Often we are interested in how a given function of interest, say
f(x)
(which can represent demand, supply, costs, utility, etc…) varies
with changes in the underlying independent variable
x
.
One measure of how much a function varies between two points x
1
and x
2
is simply the ratio of the change in the function to the change
in the independent variable calculated between those points.
If we want to understand the variation of a function at a given point,
say x
1
, we could compute the limit of the previous quotient as the
point x
2
tends to x
1
. This gives us the derivative
f’(x
1
)
of the
function
f
at the point x
1
.
21
( )
( )
f x
f x
xx
1
( )
( )
lim
1
x=x
f x
f x
df
f'(x )=

=
dx
x
x
8/25/2010
3
Geometric interpretation
The derivative
gives us the
slope of the
tangent to the
function at a
given point
We can draw a straight line going through x
1
with slope equal to
the quotient
.
As we move the point x
2
closer to x
1
, this straight line becomes
tangent to the function
f(x)
at the point x
1
.
21
( )
( )
f x
f x
xx
x
1
f(x
1
)
x
2
f(x
2
)
x’
2
f(x’
2
)
The derivative as a first order approximation
To study the behavior of functions it is often convenient to
approximate a function by simpler functions.
For instance, we can approximate a function by a constant function
whose value is the value of the function at a point
x
1
,
f(x
1
)
. While
this approximation is rather accurate for points very close to
x
1
, the
error that we make with this approximation maybe extremely large
as we move away from
x
1
.
Moving to greater accuracy we could approximate a function by a
linear function. In this case if we use the following linear function
we can actually get a pretty good approximation. In particular, the
maximum possible error that we incur with this approximation
increases quadratically as
x
moves away from the point
x
1
.
1
1
1
f(x )+ f'(x )(x x )
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Some examples
The derivative of a constant is the zero function.
Linear functions:
Power functions:
Logarithmic functions:
Exponential functions:
()
d
ax
a
dx
1
d
x
x
dx
1
(ln )
d
x
dx
x
xx
d
e
e
dx
Rules of calculus
Linearity
: The derivative of a sum of functions is the sum of the
derivatives.
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This note was uploaded on 11/30/2010 for the course ECON 251 taught by Professor Tontz during the Fall '10 term at USC.
 Fall '10
 Tontz
 Microeconomics

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