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
12
4
8
4
2
1
-2
2
4
2
4
x
slope = rise/run
2
3
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 (kind of “average” of rates-of-change over a range).

This approximation for a given point 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) 2  1)  (3(1) 2  1) 28  4


 12
2
2
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) (3(2) 2  1)  (3(1) 2  1) 13  4


9
1
1
1

So the larger the change in x we consider, the more we kind of misstate the strength of the relationship between x and y at a given point
(as you are averaging over a bigger area).
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 f ( x  x)  f ( x)

x
x

The derivative is just the limit of this expression as Δx goes to zero, or
df ( x)
f ( x  x)  f ( x)
 lim x0
dx
x

We will also often 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?

slope = ?
What is slope of f(x) = 3x2 + 1 at
x = 3?

28
How do we interpret these
slopes?
slope = ?
4
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.
You can generally figure out the sign of both the first and second
derivatives just looking at the graph of the function.
10
Derivatives
y
y
f(x)
x
g(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 (e.g., q = f(L,K)).
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