Mathematics for Economists
Ken Danger
Problem Set 1
1)
Use the definition of the derivative to determine the derivatives of the following
functions.
a)
b)
Mathematics for Economists
Ken Danger
Problem Set 2
1)
Use calculus techniques to establish the first and second derivatives of the following
functions and to graph the functions below by indicating 1) points where f(x) = 0, 2) the yintercept, 3) all rela
Mathematics for Economists
Ken Danger
Problem Set 8
1.
Partially differentiate the following equations with respect to x1 and x2. Note the
superscript stand for a power and the subscript stands for the variable index. For
example: x2 stands for x two; x2
Mathematics for Economists
Ken Danger
Problem Set 9
1.
Partially differentiate the following with respect to x1 and x2:
2.
Find the total differential of the following functions:
3.
Find the total derivative of V with respect to X.
Z = 4 + 5X
Mathematics for Economists
Ken Danger
Problem Set 6
1)
For the following system of equations,
a)
Find solutions to the values of x1, x2, and x3 using the matrix inversion technique.
b)
Find solutions to the values of x1, x2, and x3 using Cramers rule.
Mathematics for Economists
Ken Danger
Problem Set 3
1)
Solve the following equations for x:
2)
Solve the equation
3.
Differentiate the following functions:
hint: Factor out the term (x-1). Also, you
should find three distinct solutions.
3a)
3b)
4)
Graph t
Basic Rules of Differentiation;
First and Second Order Conditions;
and Graphing
1
Derivative of a constant
(1a) f(x) k constant
(1b)
df(x)
0
dx
Example of a constant function
2
Power Rule
(2a)
(2b)
f(x) ax k
df(x)
f' (x) akx k -1
dx
3
Power Rule Example
y
Exponents and Logarithms
Review and Look Forward
So far we have studied polynomials such
as
y = x2 + 2x + 3
y = (x2+5)4
Many processes naturally have the
variable in the exponent such as
Y = 2x
2
Example Exponential Functions
3
Economics of 2t
t
Value = 2
Solving a System of Equations
through Matrix Inversion and
Cramers Rule
Solving a System Using Matrix
Inversion
1.
2.
3.
4.
5.
6.
Find the determinant
Find all minors and cofactors for the
coefficient matrix
Form a matrix of cofactors
Find the adjoint of
Solutions to Systems of Linear
Equations and
Matrix Algebra
System of Equations
A system of linear equations is given by
(1a)
(1b)
2x + 3y = 7
x
+y =1
A nonlinear system is given by
(2a)
x2 y = 0
(2b)
x +y=5
Solutions: One, None, Infinite
No Solution
One
Integration
Review and Look Forward
Weve studied the derivatives of certain
types of functions
Integration (or anti derivatives) undo the
derivative to find the underlying function.
Power Rule
(1)
1
x dx
x n 1 C
n 1
n
Power Rule Example
f(x) 3x 6
Power R
Single Variable Calculus:
Foundations
Any Two Points on a Straight Line
Gives You the Slope
For a degree one polynomial, any two points can be used to calculate
straight line.
(1)
y 2 - y1
m
x2 x1
Example:- Freezing32 9boiling points on Celsius and Fahren