Econ251x
Rahsan Akbulut
Math Review
This handout reviews rules of exponents, functions, slope, simple equations, and rules of
differentiation. Throughout this course, we will see plenty of applications of these mathematical
tools in economics.
Rules of Exponents
1.
x
a
x
b
= x
a+b
also: e
a
e
b
= e
a+b
2.
(x
a
)
b
= x
ab
also: (e
a
)
b
= e
ab
3.
x
-a
= 1/ (x
a
)
also: x
a
/ x
b
= x
a-b
4.
(xy)
a
= x
a
y
a
also: (x/y)
a
= x
a
y
-a
= x
a
/ y
a
5.
e
ln(a)
= a
6.
x
1
= x
7.
x
0
= e
0
= 1
Functions:
A function maps one set of values into another.
For example:
y = x
2
maps 0, 1, 2, .
.. into 0, 1, 4, .
..
We call
x
the argument of the function. In general terms, we write
y=f(x)
A function can be
univariate
(1 argument) or
multivariate
(many arguments).
Univariate functions can be easily represented in a two-dimensional graph. An increasing
function is upward sloping. A decreasing function is downward sloping.
A function of two variables would need a 3-D graph for a full representation
We draw such functions on a 2-D graph by holding one variable fixed.
For example: In the production function
y=f(k,l),
where
y
is output,
k
capital, and
l
labor, we
hold
k
fixed and plot
y
vs
l
, or
l
fixed and plot
y
vs
k
.
Slope:

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Slope is a measure of the “steepness” of the function. If
y = f(x),
the slope is a measure of how
much
y
increases for a unit increase in
x
. (In calculus, this is the derivative of
f
.)
For linear functions (straight lines), the slope is constant.
For example, if the functional form is

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