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```Intermediate Microeconomics
Math Review
1
Functions and Graphs

Functions are used to describe the relationship between two
variables.

Ex: Suppose y = f(x), where f(x) = 2x + 4
 This means
 if x is 1, y must be 2(1) + 4 = 6
 if x is 2, y must be 2(2) + 4 = 8
* This relationship can also be described via a graph.
2
Rate-of-Change and Slope

We are often interested in rate-of-change of one variable
relative to the other.

For example, how does the amount of output (y) change as a firm
increases the quantity of an input (x)?

This is captured by the slope of a graph.
3
Rate-of-Change and Slope

For linear functions, this is constant and equal to “rise/run” or
Δy/Δx.
y
y
2
6
-4
1
4
-2
2
2
2
4
x
slope = rise/run
3
4
x
slope = rise/run = -2/1 = -2
=(change in y)/(change in x)
= -4/2 = -2
4
Slope

Positive Slope means that the
relationship between the two
variables is such that as one goes
up so does the other, and vice
versa.
y

Negative Slope means that the
relationship between the two
variables is such that as one goes
up the other goes down, and vice
versa.
y
x
x
5
Non-linear Relationships

Things gets slightly more complicated when relationships are “non-linear”.

Consider the functional relationship y = f(x), where f(x) = 3x2 + 1
y
y
slope = 24/2 = 12
slope = 9/1 = 9
28
24
13
9
4
1

4
2
3
x
1
1
2
x
For non-linear relationships, rise/run is a discrete approximation of the slope at any
given point.

This approximation is better the smaller the change in x we consider.
6
Non-linear Relationships

We can also approximate the slope analytically:

Consider again the relationship y = f(x), where f(x) = 3x2 + 1

Starting at x = 1, if we increase x by 2 what will be the
corresponding change in y?
f (1  2 )  f ( 2 )
( 3 ( 3 )  1)  ( 3 (1)  1)
2

2
2

28  4
2
 12
2
Similarly, starting at x = 1, if we increase x by 1 what will be the
corresponding change in y?
f (1  1)  f (1)
1


( 3 ( 2 )  1)  ( 3 (1)  1)
2

2
1

13  4
9
1
So this functional relationship between x and y means that how
much y changes due to a change in x depends on how big of a
change in x and where you evaluate this ratio.
7
The Derivative

As discussed before, we get a better approximation to the relative
rate-of-change the smaller the change in x we consider.

In particular, given a relationship between x and y such that y = f(x) for
some function f(x), we have been considering the question of “if x
increases by Δx, what will be the relative change in y?”, or
y
x

f ( x  x)  f ( x)
x
The derivative is just the limit of this expression as Δx goes to zero, or
df ( x )
dx


 lim
f ( x  x)  f ( x)
x  0
x
We will also sometimes express the derivative of f(x) as f’(x)
8
The Derivative

Given y = f(x), where f(x) = 3x2 + 1,
what is expression for derivative?
y

So what is slope of f(x) = 3x2 + 1
at x = 1?
28

slope = ?
What is slope of f(x) = 3x2 + 1 at
x = 3?
slope = ?
4

How do we interpret these
slopes?
1
3
x
9
Derivatives

Rules for calculating derivatives - See “Math Review”

Second Derivative - the derivative of the derivative.


Intuitively, if the first derivative gives you the slope of a function at
a given point, the second derivative gives you the slope of the slope
of a function at a given point.
In other words, second derivative is rate-of-change of the slope.
10
Derivatives
y
y
x
x
11
Finding maxima and minima

Often calculus methods are used for finding what value maximizes
or minimizes a function.

A necessary (but not sufficient) condition for an “interior” maximum or
minimum is where the first derivative equals zero.
y
y
f(x)
f(x)
x*
x
x*
x
12
Finding maxima and minima

This means that when trying to find where a function reaches its
maximum or minimum, we will often take the first derivative and
set it equal to zero.

Often referred to as “First Order Condition”

f(x) = 10x – x2
F.O.C.: 10 – 2x = 0
x* = 5

How do we know if this is a maximum or a minimum?

13
Partial Derivatives

Often we will want to consider functions of more than one variable.

For example: y = f(x, z), where f(x, z) = 5x2z + 2

We will often want to consider how the value of such function changes
when only one of its arguments changes.
For example, output is function of labor and capital. How does output
change as we increase labor but hold capital fixed?


This is called a Partial derivative.
14
Partial Derivatives

The Partial derivative of f(x, z) with respect to x, is simply the derivative of
f(x, z) taken with respect to x, treating z as just a constant.

Examples:

What is the partial derivative of f(x, z) = 5x2z3 + 2 with respect to x? With
respect to z?

What is the partial derivative of f(x, z) = 5x2z3 + 2z with respect to x? With
respect to z?
f ( x, z )

A partial derivatives of the function f with respect to x is denoted
z
15
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