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NATIONAL OPEN UNIVERSITY OF NIGERIA
MATHEMATICS FOR ECONOMISTS II
ECO 256
SCHOOL OF ARTS AND SOCIAL SCIENCES
COURSE GUIDE
Course Developer/Writer:
OHIOZE, Wilson Friday - NOUN
Course Editor:
Dr
...
ECO 256: Mathematics for Economist II is a two-credit and one-semester
undergraduate course for Economics students
...
This course guide tells you how important research
is to students of economics, and how statistical tools can be applied in solving some
basic economic problems
...
It suggests some general guidelines for the amount of
time required of you on each unit in order to achieve the course aims and objectives
successfully
...
Course Content
This course is a continuation of mathematics for Economists I (ECO 255)
...
As it is, the course will take you through derivatives to matrix and
their applications
...
To familiarize students with the basic topics in the course
...
To expose students to the task ahead in their study of higher level mathematics for
economists as they progress in the course
...
The unit objectives are included at
the beginning of each unit; you should endeavor to read them before you start working
through the unit
...
You should always look at each unit objectives after
completing the unit to ascertain the level of achievements
...
In this way, you can be sure you have
done what was required of you by the units
...
At the end of the course period, students are expected to be able to:
Define and explain the concept of derivative
...
Define and understand integration
Differentiate between integration and differentiation
...
Define and explain linear/matrix algebra
...
Use matrix to solve complex systems of equations, and a lot more
...
Each unit contains self-assessment exercises called Student Assessment Exercises (SAE)
...
At the end of the course there is a final examination
...
Course Material
The major component of the course, what you have to do and how you should allocate
your time to each unit in order to complete the course successfully on time are listed
below:
4
1
...
3
...
5
...
Module 1
Unit 1:
Unit 2:
Unit 3
Unit 4
calculus
Derivate I
Derivate II
Integration
Differentiation and Integration in Economics
Module 2
Unit 1:
Unit 2:
Unit 3:
Unit 4:
Optimization
Introduction to Optimization
Functions of Several Variables
Optimization with Constraints
Differentials
Module 3
Unit 1
Unit 2
Unit 3
Unit 4
Linear Algebra
Matrix
Matrix operations
Matrix Inversion
Economics applications of matrix
Each study unit will take at least two hours, and it includes the introduction, objectives,
main content, self-assessment exercise, conclusion, summary and reference
...
Some of the selfassessment exercise will necessitate discussion, brainstorming and argument with some
of your colleagues
...
There are also textbooks under the references and other (on-line and off-line) resources
for further reading
...
You are required to study the materials; practice the selfassessment exercises and tutor-marked assignment (TMA) questions for greater and indepth understanding of the course
...
5
Textbook and References
For further reading and more detailed information about the course, the following
materials are recommended:
Adams, R
...
(2006)
...
,
Toronto, Ontairo, Canada
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
Lial, M
...
, Greenwill, R
...
P
...
Calculus with Applications (8th
Edition): Pearson Education, Inc
...
Sydsaeter, K
...
(2002)
...
Assignment File
Assignment files and marking scheme will be made available to you
...
The marks you
obtain from these assignments shall form part of your final mark for this course
...
There are three assignments in this course
...
Remember, you are required to submit all your assignments by the due dates
...
6
Assessment
There are two types of assessment for the course
...
In attempting the assignments, you are expected to apply information, knowledge and
techniques gathered during the course
...
The work you submit to your tutor for assessment will
count as 30 % of your total course mark
...
This examination will also count for 70% of your total course mark
...
You will submit all the
assignments
...
The TMAs
constitute 30% of the total score
...
You will be able to complete your assignments from the information and materials
contained in your set books, reading and study units
...
You should use other references to have a broad viewpoint of the subject and also to give
you a deeper understanding of the subject
...
Make sure that each assignment reaches your tutor on or before the deadline given
in the Presentation File
...
Extensions will not be granted after the due date unless there are exceptional
circumstances
...
The examination will consist of questions which reflect the
types of self-assessment practice exercises and tutor-marked problems you have
previously encountered
...
Revise the entire course material using the time between finishing the last unit in the
module and that of sitting for the final examination to do a thorough revision of the
course material
...
The final
examination covers information from all parts of the course
...
Assignment
Marks
Assignments (Best three assignments out of four that is
marked)
30%
Final Examination
70%
Total
100%
Course Overview
The Table presented below indicates the units, number of weeks and assignments to be
taken by you to successfully complete the course - Mathematics for Economists II (ECO
256)
...
Differentials
Module 3Linear Algebra
1
Matrix
2
Matrix Operations
3
Matrix Inversion
4
Economic applications of matrix
8
Assessment
(end of unit)
Week 1
Week 1
Week 2
Week 2
Assignment 2
Assignment 2
Assignment 3
Assignment 2
Week 3
Week 4
Week 5
Week 6
Assignment 2
Assignment 3
Assignment 3
Assignment 3
Week 7
Week 8
Week 9
Week 10
Assignment 3
Assignment 2
Assignment 3
Assignment 2
How to Get the Most from This Course
In distance learning, the study units replace the university lecturer
...
Think of it as reading the lecture instead of listening to a lecturer
...
Just as
a lecturer might give you an in-class exercise, your study units provides exercises for you
to do at appropriate points in time
...
The first item is an introduction to the
subject matter of the unit and how a particular unit is integrated with the other units and
the course as a whole
...
These objectives let you know
what you should be able to do by the time you have completed the unit
...
When you have finished the unit
you must go back and check whether you have achieved the objectives
...
The main body of the unit guides you through the required reading from other sources
...
Some units
require you to undertake practical overview of historical events
...
The purpose of the practical overview of some certain historical economic issues are in
twofold
...
Second, it
will give you practical experience and skills to evaluate economic arguments, and
understand the roles of history in guiding current economic policies and debates outside
your studies
...
Self-assessments are interspersed throughout the units, and answers are given at the ends
of the units
...
You should do each selfassessment exercises as you come to it in the study unit
...
The following is a practical strategy for working through the course
...
Remember that your tutor's job is to help you
...
9
1
...
2
...
Refer to the `Course overview' for more details
...
Important information, e
...
details of your tutorials, and the date of the
first day of the semester is available from study centre
...
Whatever method you choose to use, you should decide on and write in your own
dates for working breach unit
...
Once you have created your own study schedule, do everything you can to stick to
it
...
If you get into difficulties with your schedule, please let your tutor know
before it is too late for help
...
Turn to Unit 1 and read the introduction and the objectives for the unit
...
Assemble the study materials
...
You will also need both the study
unit you are working on and one of your set books on your desk at the same time
...
Work through the unit
...
As you work through the unit you will be instructed
to read sections from your set books or other articles
...
7
...
8
...
Keep in mind that you will
learn a lot by doing the assignments carefully
...
Submit all assignments no later than the due date
...
Review the objectives for each study unit to confirm that you have achieved them
...
10
...
Proceed unit by unit through the course and try to pace your
study so that you keep yourself on schedule
...
When you have submitted an assignment to your tutor for marking do not wait for
its return `before starting on the next units
...
When the
assignment is returned, pay particular attention to your tutor's comments, both on
the tutor-marked assignment form and also written on the assignment
...
12
...
Check that you have achieved the unit objectives (listed at the
beginning of each unit) and the course objectives (listed in this Course Guide)
...
You will be notified of the dates, times and location of these tutorials
...
Your tutor will mark and comment on your assignments, keep a close watch on your
progress and on any difficulties you might encounter, and provide assistance to you
during the course
...
They will be marked by your
tutor and returned to you as soon as possible
...
The following might be circumstances in which you would find help necessary
...
• You do not understand any part of the study units or the assigned readings
• You have difficulty with the self-assessment exercises
• You have a question or problem with an assignment, with your tutor's comments on an
assignment or with the grading of an assignment
...
This is the only chance to have face to
face contact with your tutor and to ask questions which are answered instantly
...
To gain the maximum benefit
from course tutorials, prepare a question list before attending them
...
Summary
The course, Mathematics for Economists II (ECO 256), exposes you to the basic concept
of Derivative, integration, derivative and integration in economics, introduction to
optimization, functions of several variables, optimization with constraints, differential,
matrix, matrix operations, matrix inversion and matrix in economics
...
Thereafter it shall enlighten you
about decision making as regard using mathematics for economics
...
However, to gain a lot from the course, please try to apply whatever you learn
in the course to term papers writing in other aspect of economics courses
...
11
MODULE 1:
CALCULUS
Unit 1: Derivatives I
Unit 2: Derivatives II
Unit 3: Integration
Unit 4: Economic applications of Derivatives and Integration
UNIT 1
DERIVATIVE I
CONTENTS
1
...
0
3
...
0
5
...
0
7
...
1
The Derivative
3
...
3
Rules of Differentiation
Conclusion
Summary
Tutor Marked Assignment
Reference and Further Readings
1
...
In fact, it is a prerequisite course
for ECO 256
...
Mathematics for economists I would have introduced
you to what you are expected to study in this advanced version at the rounding-off of the
course (ECO 255)
...
In this aspect (Mathematics for economists II), you will be
studying mathematical economics at a level higher than what you have studied in the
foundation class
...
We shall be treating topics like derivatives, integration, optimization with
constraint, functions of variables, linear algebra, and etcetera
...
It is important to
make it clear to students that, mathematical economics is not pure mathematics but an
application of mathematical concepts in explaining economic issues
...
12
2
...
0
MAIN CONTENTS
3
...
Comparative static has to do with the comparison of various
states of equilibrium that are related to kind of sets of values of parameters and
exogenous variables
...
If there
disequilibriumoccurs in the demand function or model in a way that there will be changes
in the exogenous variables (they are variables in which the values are determined outside
the model
...
In
fact, it is always on the left-hand of an equation)
...
’ The rate of change of the endogenous variables
with respect to the change in a certain exogenous variable is what the concept of
derivatives deals with in mathematics
...
One important
feature of a linear graph is its slope, a number which signifies the steepness of the line
...
The slope of a linear graph is oftenreferring to as “slope of a line
...
y
A
B
o
x
13
From the figure above, the slope of the linear graph AB can be interpreted to mean the
average amount of change in the vertical axis represented by ‘y’, per unit change in the
horizontal axis represented by ‘x’, fondly written as
...
If we have
b, is defined thus:
and
as coordinate points on a linear graph, the slope which is as
b=
...
In fact, the derivate at a point of a function of a single
variable is the slope of the tangent line to the graph of the function at that very point
...
At this point (see point c in the figure below), the slope of
the curve can be determined
...
At that point, as seen above, the slope of the
tangential line is equal to the derivative of the function at the point where the graph and
the line coincide
...
In order words,
the slope of a curve at a point on the curve is the same as the average rate of change
...
From this definition, we can infer one fundamental issue, that is, two points are
involved here (points A and B), see figure above
...
In this case,
we can find the average rate of change between the two points
...
Where f(a) and f(b) are functional values of the variables (a and b)
...
5, 20),b =(1, 48),c = (1
...
To determine
the average speed or average rate of change, we will apply the formula above
...
y
distance
104
or f(b) – f(a)
20
o
Meanwhile, recall that
0
...
5 to
t = 2
...
Delta is denoted thus ( ; it is synonymous to the
concept of limit
...
For instance if we have
, it means that as y moves from one stationary
position to another stationary point, x changes too
...
To calculate the rate of change using points a and f[(
...
...
15
Let us consider a scenario where the formula can be applicable to resolving practical
issue(s)
...
Find the average
rate of change or slope of cost per bag for producing between 1 and 6 bags
...
The cost of producing 1 bag is
C (1) = 150 + 10(1) - 12 = 159 or N159
...
Therefore, the average rate of
change in the cost of production of the bags is
C (6) C (1) 174 159
15
...
Given a function y = f(x), the derivative of the function f at x, written
f
(x) or dy/dx, is
define as
f’ (x) =
lim
x 0
f’(x) =
lim
h0
f ( x x) f ( x)
if the limit exists, or
x
f ( x1 h) f ( x )
h
Wheref’(x) is read “the derivative of fwith respect to x or f prime of x”
...
Meanwhile, a
function is differentiable at a point if the derivative exists at a given point
...
For simplicity, a function is continuous when a curve is drawn without any break
...
A function
fis continuous at a point x = a only if: i) f(x) is defined, that is x = aii) limit of f(x)
existsand iii) the limit of f(x) = the limit of f(a)
...
However, where a function is undefined, that is,
the denominator is equal to zero
...
2
Derivative Notations
Notations are generally used in mathematics and mathematical subjects alike
...
Notations are used in mathematics, physical sciences, engineering, and economics
...
In calculus, the notations of derivative of a function can be represented in many forms
...
dx
Let us use one or two examples worked based on all we have discussed concerning
derivative
...
If y = 2x2 + 7x – 18, write the derivative of the equation
y’
dy
dx
d
2
2 x 7 x 18
dx
or
Dx (2x2 + 7x – 18)
Example 2
...
3
...
When a function or a mathematical model is differentiable, we can observe
from the model or function how the endogenous variables change with respect to change
17
in the exogenous variables
...
Differentiation is
the process of determining the derivative of a function or a mathematical model
...
In discussing
the rules of differentiation, authors or writers also use functions such as g(x) and h(x),
where g and h are both unspecified functions of x
...
That is, any change
that is observed in the left hand variable(s)is dependent on the behaviours of the
variable(s) on the right hand of the model
...
The dependability of the y on x can be determined using differential calculus
called derivative as already explained
...
These rules are regard as rules for a function of
one variable
...
Note that k is constant
because it is a numerate standing alone without any variable attached to it, or multiplying
it
...
If we have a model or a
function g(x) = mx + dwhich is a linear function, the derivative will be equal to m(the
coefficient of x)
...
If g = mx + d, the derivative will be dg/dx or g’(x) equal to m
...
Find the derivative of the following functions
i)
f(x) = 5x – 3
ii)
f(x) = 10 +
iii)
g(t)= 13t
1
x
4
Solution: i) f’(x) = 5
ii) f’(x) =
iii)
Rule 3:
1
4
g’(t) = 12
The power Function Rule
The derivative of any power function is determined by multiplying the coefficient ofthe
function by the exponential and the alphabet variable to the power of the exponential
minus one
...
Now given that y = Mxn, to
differentiate this function (that is, the derivative)
dy
n-1
n-1
= m*n*x orf’(x) = mnx
...
Differentiate the followings using power function rule
...
Rule 4:
The product of a constant and aFunction
Considering this rule, to differentiate any function or any economic model that has the
combination of a constant term and a function, is the product of the constant term and the
derivate of the function
...
If y = h(x) exist, then
dy
= kh’(x) 0rDx[kh(x)] = kh’(x)
...
i)
ii)
8
x
iii)
(-6x)
i)
dy
3-1
2
2
= 7(3x ) = 7(3x ) = 21x
dx
ii)
Solution:
7x3
Applying one of the laws of indices,
f’(x) =8(-1x-1-1) = 8(-1x-2) = -8x-2 or
iii)
8
will now be 8x-1 then
x
8
x
2
Open the bracket, and you will have -6x, then
f’(x) = -6(1x1-1) = -6x0 = -6(1) = -6
...
Example 7: if z is a function of the function given below, determine the derivative
...
Solution:
dz
= (2k4 + 5) (15k4) + (3k5– 8) (8k3) – product rule (see unit 2, rule 2)
dk
= 30k8 + 75k4+ 24k8 – 64k3
= 54k8 + 75k4 - 64k3
...
Write and explain in your own way, the rules of differentiation you have studied
thus far
...
0
CONCLUSION
In this unit, you have learnt about derivative and some rules of differentiation
...
In fact, we have identified
that differentiation is the process of determining the derivative of a function or a
mathematical model
...
Most functions
used in economics are specific, and have the property that they can be differentiated at
every point on the curve
...
5
...
Recall that in the begin, we have said that the concept of derivative is a
significant aspect of comparative statics because it is the most basic part of mathematics
known as differential calculus, which is concerned about the rate of change
...
In-between these two points, there is either increase or decrease in speed
as time changes
...
For instance, distance and time are good examples of the said sets
of variable
...
For example, the study of an isolated market model in
economics will be made simpler with a good understanding of the concept of derivative
...
0
TUTOR MARKED ASSIGNMENT
Determine the derivatives of these functions using any of the rules already studied
...
p = 7q3(5q –150), R = (t4 + 12) (t6 + 8)
...
7
...
A
...
Calculus: A Complete Course (6th edition): Pearson Education Inc
...
Chinang, A
...
& Kevin, W
...
Fundamental Methods of mathematical Economics
21
(4th edition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
L, Greenwill, R
...
P (2005)
...
, Boston, Massachusetts, USA
...
0 Introduction
2
...
0
Main Content
3
...
)
3
...
3
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
The concept of derivative is directly concerned with the
notion of rate of change in mathematical sciences which is applicable to other fields of
study such as Economics
...
In applying this, some fundamental guidelines were given
...
”
In this unit, we shall be starting off by continuing our discussion on the rules of
differentiation
...
2
...
Apply these rules to solve problems
...
3
...
1
RULES OF DIFFERENTIATION (CONT
...
You have learnt some
earlier in the preceding unit
...
Rule 1:
The sums or differences rules
23
Now we want to consider a situation where two or more components of a function are
differentiated separately, and their products either added or subtracted
...
The derivative of functions in this category is equal to the
sum or difference of the individual components
...
Example 1: solve the following using the rules of sums and differences in derivatives
...
So, let u(x) = 4x2 and g(x) = 6x3; then y = u(x) +
g(x)
...
dx
ii)
This is a case of subtraction
...
It therefore follows that,
dp
2
3
= 45q – 12q
...
So, R’ = u’(t) – g’(t) + h’(t) which is the difference
and sum of the individual differentiated functions
...
(Since
dt
Rule 2:
of a constant (8) = 0)
The Product rule
This rule is about two or more components of a function that are being considered
wholly, neither added nor subtracted, but multiplied
...
Assuming we have two components of a
function, the derivative will be the first component times the derivative of the second
component plus the second component times the derivative of the first component
...
h(x),where g(x) and h(x)are functions that can be differentiated
...
h’(x) + h(x)
...
Solution:let 3x + 2 = g(x) and 4x3 = h(x) so, g’(x) = 3 and h’(x) = 12x2
...
dx
Note: In performing mathematical operations as regards the use of the product rule,
students should remember that the derivative of a product of two functions is not mere
multiplication of two individual derivatives
...
Rule 3:
The Quotient rule
Merely looking at the quotient rule, we can observe that this rule is the direct opposite of
the product rule just discussed
...
The derivative in this situation is the product of the
denominator and the differentiated numerator, minus the product of the numerator and
the differentiated denominator, all over the square of the denominator
...
g '( x) g ( x)
...
Find the derivative y’
x2
Solution: let 4x-2 = u and x+2 = v , therefore, u’ = 4 and v’= 1
...
u ' u
...
An inverse Function rule
25
Before now, we have been dealing with functions in which one variable is directly
dependent on the other
...
Recall that, y = f(x) is a function which represents one on one mapping
...
In this instance, the relationship
shows in this function is a direct one
...
The derivative of an inverse function as
already given is the reciprocal of the derivative of the direct function given
...
dx dy / dx
Example 4: q = 5p + 45, find the derivative of q-1
Solution:
f’(p) = 5
...
Also, find the derivative of the inverse function of q, if q = p5 + p
...
4
dp
5 p 1
The Chain rule
This rule comes up under the discussion of the derivative of a function of a function
...
In this, there is more than one function, and the
functions are dependent on one another
...
A good instance is where y is function of u and u in turn is a function of z so,
y = f(u) and u = h(z), then
y = f[h(z)]
...
dy du
dy
...
Differentiate the function
using chain rule
...
using chain rule,
dy dy du
...
Note, without the use of
chain’s rule, solving the problem will be very cumbersome
...
To solve this, let the variables in bracket equal z to create two
components’ effects in a single function
...
It then follows that
dy dy dz
...
Rule 5:
The Implicit Functions rule
This form of differentiation is somehow cumbersome
...
Meanwhile, implicit functions are the direct opposite of explicit functions
...
There are
times when y cannot be solved explicitly, so, we use the rule of implicit differentiation to
solve the problem
...
Here, y is seen
explicitly in the function, say f of x
...
on the
other hand, the function could take the form 4x + 2y = 8
...
so, solving for y in the equation will make it an explicit of x
...
However, there are times some functions become complicated to determine the explicit
function
...
27
Three basic steps are considered in determining the derivatives of an implicit function of
x
...
Solve the equation that resulted from the derivative
...
Find dy/dx
y3-2x2y2+x4 = 0
3
Solution:
d( y )
dx
d (2 x
2
dx
2
y ) d(x )
4
dx
= 0(Note:
d (2 x 2 y 2 )
was solved using the product
dx
rule)
3y2d/dx-4xy2-4yx2d/dx+4x3 = 0
3y2d/dx-4x2yd/dx = 4xy2-4x3
Factorize both sides of the equation, and we have
d
( y)(3 y 4 x 2 ) = 4 x( y 2 x 2 )
dx
=
dy
(3 y 4 x 2 ) = 4 x( y 2 x 2 )
dx
Makedy/dx the subject of the equation, we will have
2
dy/dx=
4 x( y x )
2
y (3 y 4 x )
2
...
dx 15 y 2
SELF ASSESSMENT EXERCISE 1
How can you differentiate a product rule from a quotient rule?
Differentiate the following functions: i) y =
3
...
HIGHER-ORDER DERIVATIVES
Thus far all we have discussed concerning derivatives of functions can be called ‘first
derivatives
...
’It is circumstance where the first derivative of the original function
is itself a differentiable function
...
If the function is further differentiable, a third derivative
exists and on like that
...
Higher-order
derivatives are estimated by the successive application of the guiding rules of
differentiation to the derivative of the preceding order
...
Suppose we have an equation of the
form
q = 4p3+ p2+ 20,
dq
= q ' = 8p2 + 2p + 0 1st differentiation (1st derivative)
dp
2
dq
dp
2
= q '' = 16p + 2
nd
nd
2 differentiation (2 derivative)
3
d q = q ''' = 16
dp
3
rd
rd
3 differentiation (3 derivative)
4
dq
dp
4
= q '''' = 0
th
th
4 differentiation (4 derivative)
...
Example 7: find the 5th derivative of the function y = 4x4 + 7x3 + 2x2
...
The way to go in this case is to
start the differentiation with the primitive or initial equation already given
...
Note that the notation used in this solution is not amongst the ones given above
...
Also, within the purview of higher-order derivatives, there is what is commonly referred
to as ‘higher-order partial derivatives
...
In calculus, the second derivative of any function would bring to fur the
relative extrema (maxima and minima) of that function
...
See unit two of module two for detailed account on partial differentiation
...
Which order of derivative is f’’’’’ in differential calculus?
4
...
You have equally learnt that there are higher version of
derivatives refer to as “higher-order derivatives
...
Higher-order
30
derivative is vital in the discussion optimization in calculus
...
5
...
0
Other rules of differentiations such as product rule, quotient rule, a function of
a function rule, and many more that were perceived more complicated than the
ones discussed in unit one
...
In this topic, you were exposed to the fact that an
equation can be differentiated more than one if circumstance surrounding it
demands it
...
x
dx
Find the derivative of the function q =
p
3
2 using thechain rule
...
0
y
If y = Ak , compute d
...
A
...
Calculus: A Complete Course (6th edition): Pearson Education Inc
...
Chiang, A
...
& Wainwright, K
...
Fundamental Methods of mathematical
Economics (4thedition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
L, Greenwill, R
...
P
...
Calculus with Applications (8th
Edition): Pearson Education, Inc
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0
2
...
0
Main Content
3
...
2 Notation of Integration
3
...
3
...
3
...
4 Definite Integrals
3
...
1
Properties of Definite Integral
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
Recall, we have
mentioned that, it (differential calculus) measures the rate of change of a function or an
equation
...
To determine the derivative of a particular function, the
function (i
...
the original) would have to be differentiated
...
Working from the rate of change to determine the original equation
amounts to revising the process of differentiation, this mathematical method iscalled
integration, antiderivative or antidifferentiation
...
0
OBJECTIVES
At the end of this unit, the student should be able to do the followings:
Define and explain integration
...
Know the difference between it and differentiation
...
0
MAIN CONTENTS
3
...
Here, F(x)
is the primitive function, and it is termed integral of F’(x)
...
To further understand the difference between integration
and differentiation, we consider this simple analogy
...
It is integration when the descendants were
able to locate their origin (which is the primitive function)
...
In a clearer form,
integration is movement backward while, the forward movement is differentiation
...
When a
function is specified say, y = f(x) = x4 this represents an area under a curve and is
between two distinct points b and a; which can be determined by integrating the function
...
For instance, ifx6 is differentiated, and it resulted to 6x5, it then means that
the derivative of
will bex4
...
So, if we have the primitive function added to a constant i
...
+
c in this case, the c is a constant that will be equal to zero when differentiated
...
2
f(x) =
+ c
...
Also in integral calculus, there is a notation
that symbolizes integration
...
For indefinite
integral, we use
...
We have said it is an ‘integral sign’, whereas
f(x) which is always part of it is termed ‘integrand’, and the dx aspect is the
differentiation operator
...
f ( x)dx
From the notation given above, three parts can be explained here
...
If we decide to make two parts out of the three to be one (i
...
f(x)dx), you will have an entity called differential of original function F(x) which mean d
F(x) = f(x) dx
...
The
notation can be interpreted thus, ‘the indefinite integral of f(x) with respect to x is F(x)
added to a constant
...
That is,
to reverse the own process of differentiation
...
Explain the inclusion of an arbitrary constant in integration notation
...
3
Rules of Integration
As we have rules in differential calculus that guide its operations, so it is integral
calculus
...
That is, if we work from the primitive function to get the derivative in the case of
differential calculus, for integration, we work from the derivatives to arrive at the initial
function (the primitive one)
...
Note that, the rules
of integration are dependent on the rules of derivatives
...
If we then substitute these in expression we had
earlier above, what we will have is integration rule
...
Let apply this to solve
some mathematical problems
...
x6 dx
...
xdx
...
Remember
that our n = 6, therefore
x6 dx =
1 7
x +C
7
ii) Note that
x
3
is same as x3/2, where n in this case is 3/2 and n+1 = 5/2,
therefore
x
3
dx =
x
5/ 2
+C=
5
2
2
5
x
5
C
(Note that 1/(5/2) = 2/5)
iii) The derivative function in this instance is x, our n is equal to 1
...
This is a way of checking results in integration
...
Solution: 10 and -3 are constant values and are expressed in the form of derivatives
...
The integral of a constant and a function rule
kf(x)dx = k
f(x) dx
36
In this case, outcome is constant value multiplied by the integral of the function
...
3and x2 are the constant value and functional variable
respectively
...
ii)
Applying the method in solving above (i), we will have:
Therefore, we have 9
Rule 4:
=
=2
...
The integral of x-1 rule
This is also known as logarithmic rule of integration
...
You will recall that under the power rule of integration i
...
rule
1, n = -1 is not accepted
...
1
dx ln x C
x
(x> 0)
The rule stated is applicable where x is positive that is x > 0
...
Also, under this same rule, we have
x
37
1
dx can sometimes be stated thus
x
f '( x)
dx ln f ( x) C ,where f(x) is positive or
f ( x)
= ln f ( x) C , in this case, f(x) not positive
...
Example 4: integrate thefunction
6
dx
...
Therefore, applying rule 3 and 4,
x
we will have
Rule 5:
6
dx = 6
x
1
dx = 6 ln x + c
...
Without much ado, the derivative of
exponential ex is the ex itself
...
More generally, (where e = exponential)
1 ax
C
...
ae
The sum or difference of two functions rule
The integral of sum or difference of two or more functions is the sum or difference of the
individual integral
...
3
Solution: To resolve this problem, we need the combined application of the rules we
have discussed thus far
...
3
...
Integration
by substitution is useful when the integral function is becoming too large and difficult to
handle
...
Example 7: Compute the integral
( x 2) dx
...
In doing this, the three basic steps stated above shall be considered
...
3
3
We shall now use rule 1 (i
...
power rule) to
evaluate this stage, then you have:
= u
4
4
c
c) We shall now put the value of u in the integral
...
3
...
4
Integration by Parts
Outside integration by substitution just discussed, one other method of solving
cumbersome mathematical problems is by using integration by parts
...
It is
basically used when an integrand is a product or quotient of differentiable function and
cannot be stated as a constant multiple
...
For instance, the second integral on the
right-hand side can be solved thus:
h( x) g '( x)dx h( x) g ( x) g ( x)h '( x)dx
Example 8: Integrate the following function x( x3) dx
...
This is because the problem is in the form ( vdu
), can only be solved using the method of integration by parts
...
Also, let u =
4/3
3
( x3) , sincedu = (x+3)2/3 dx
...
4
It is important to note that arriving at the required answer, combination of integral rules is
very essential
...
g
...
There are complicated functions in integration
...
These can be found in many available mathematical books and
tables
...
7
...
4
7 xe x
7
2
4dx ,
dx
4 x 2 and
8x
( x 3)
3
dx
...
It
is so, because the functions were dependent on a single variable and had no precise
numerical value
...
In this instance,
F(P) – F(q) is referred to as “definite integral” offover(p, q)
...
y
Y=f (x)
o
a
b
x
q
As such, the already familiar integral sign is somehow modified to the form p
...
we then have:
q
p
b
f ( x)dx Or f ( x)dx
...
b
b
a
is a command that b and a should be substituted for x in the outcome of
a
The notation
f ( x)dx =
b
a
the integration to determine F(b) and F(a), and subtract accordingly as indicated in the
symbol above
...
Example 9:Evaluate
7
2
3
4x dx
...
For instance, the indefinite integral for this very function is x4 + c
...
1 x
Example 10:Evaluate (
0
Solution: Note that the indefinite integral is ln 1 x x c , therefore the definite integral
will be
2
1
2
(
2 x)dx = ln 1 x x 2 (ln 3 4) (ln1 0) ln 3 4 (note
0
0 1 x
2
that
ln1is zero)
...
4
...
p
r
(where α is any arbitrary number)
q
f ( x)dx f ( x)dx f ( x)dx
p
r
q
p
f ( x)dx f ( x)dx
d
b
c
d
a
b
c
a
p
f ( x)dx f ( x)dx f ( x)dx f ( x)dx
...
These properties of definite integration are vital to solving definite integral problems
...
SELF ASSESSMENT EXERCISE 3
Evaluate the following integrals:
4
...
2
CONCLUSION
42
In this section, we have studied another aspect of calculus called integration
...
Dynamics is term which has to do with the study of
the specific time paths of variables
...
What integration as topic has
done in the body of mathematical science, is to fill the gap left by differential calculus,
which is the ‘time paths
...
5
...
We have said earlier that integral
calculus is the direct opposite of differential calculus
...
Basically, what we did here was to
study some vital areas in integration
...
All these we have fully
dealt with
...
0
TUTOR MARKED ASSIGNMENTS
Find the following:
(4cx d )(cx
2
ax)5dx , 3e2 x dx , and
4x
x
dx
7
2
Evaluate the followings:
5
0
7
...
A
...
Calculus: A Complete Course (6th edition): Pearson Education Inc
...
Chiang, A
...
& Wainwright, K
...
Fundamental Methods of mathematical
Economics (4thedition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
L, Greenwill, R
...
P (2005)
...
, Boston, Massachusetts, USA
...
(2002)
...
UNIT 4
ECONOMIC APPLICATIONS OF DERIVATIVES AND
INTEGRATION
CONTENTS
1
...
0 Objectives
3
...
1 Marginal Concept
3
...
3 Integration Applied
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
That is, we looked at it the way people in
the field of mathematical sciences will do
...
That, derivative is about the rate
of change between two or more variables
...
That the other
name of integration is antiderivative or antidifferentiation
...
e
...
Therefore, the understanding of all these mathematics, and it’s applications in economics
by Economists make the subject interesting and grey areas in it are easy to resolve
...
2
...
3
...
1
Marginal Concept
Theconcept of marginal in economics is divided into two
...
These concepts are best discussed
under profit maximization or cost minimization in either the principle or intermediate
economics
...
Therefore, theonus is on the management team to
pursue this noble objective
...
All this is about the concepts of total revenue and total cost
...
This is impossible
without a good understanding of the concept of marginal in economics
...
In the same vein, marginal cost is the
45
change in total cost owing to the production of an additional unit of any product
...
So if TC is total cost and TR is total revenue are both
linearly related positively to the production output Q, then their function will be
expressed thus:
If
Also, if
TR = f(Q),
then
MR
dTR
dQ
TC = f(Q),
then
MC
dTC
dQ
In a nutshell, the marginal concepts of any function in economics are determined by
differentiating the total function
...
Solution:Given the total cost function, the marginal cost function is an expression of the
derivative of the total function, thus we have:
MC
dTC
10Q 7
...
That is
At Q = 6, MC 10(6) 7 67
At Q= 9, MC 10(9) 7 97
...
Estimate themarginal revenue(MR) function base on the outputs level given above
...
Therefore
MR
dTR
15 2Q
dQ
Now, to estimate the marginal revenue function based on the level of outputs given,
substitute the output one after the other and you will get the marginal revenue at the
various levels of production
...
You will recall that the primary aim of any businessman is to make profit
...
That
gain is what is referred to as profit
...
We can from this, formulate a function called profit
function
...
e
...
Note, that the notation for profit in economics is represented by the sign , and
the function is expressed thus
(Q) TR TC or PQ CQ
What we have seen in the profit function is that, the amount any businessman would gain
or make as profit is dependent on the volume of sales or volume of production (Q)
...
This brings us to the issue of additional or
marginal gain/profit
...
Note that, P and Cin the model PQ CQ are per unit price and per unit cost
respectively
...
Solution:Given the profit function, the marginal gain function is an expression of the
derivative of the total profit/profit function, thus we have,
M
d
2Q 16
...
To evaluate the function as required based on
the outputs given, substitute the output one after the other and you will arrive at the
answers
...
What we have seen with marginal concept is to estimate the additional cost or gain per
unit of any product produced in any firm
...
This
concept estimate the total function to get either the cost of producing a unit of product or
the revenue per a unit of good sold
...
That is, if we are to
determine for instance the average revenue (AR), we will have:
47
AR
TR
Q
After you have determined the function, you can proceed to estimate the average function
by substituting for Q in the function
...
Hint: note that the TE = P*Q
...
88Y , evaluate the marginal
propensity to consume (MPC)
...
2
The Production Functions
The theory of production is about input and output analysis in the production of goods
and services in any firm
...
The basic inputs
considered in production theory are always factors of production that has been narrowed
down to Capital (K) and Labour (L), while the output is represented by (Q)
...
The transformation of these inputs into useful item is through a process called production
technique
...
Production function by any mean is a statement relating how inputs
can be combined to achieve various possible levels of output
...
From the information given above, we can determine the marginal and the average
products if the total production function is known
...
3
Integration Applied
Integrals are vital to analyzing issues in Economics
...
48
a) If net investment I is the rate of change in capital stock formation k over time
t(k(t))
...
That is, I(t) = dK(t)/dt =K’t
...
This we can do by integrating the
capital stock with respect to time of net investment:
Kt = I (t )dt K (t ) c K (t ) K 0
...
In the same vein, the total cost of a product can be determined from the marginal cost of
that product through the process of integration
...
Again in the expressionc (small lettered) is equal to fixed or initial cost
...
Example 1: Assuming that marginal cost (MC) is 50 + 60Q – 18Q2, if fixed cost is
75; determine the total cost (TC)
...
The best way to do that is apply
integration
...
Therefore,
TC = 50Q + 30Q2 – 6Q3 +75
...
e
...
The cost per unit (AC) will be:
AC =
However,
= 50 + 30Q – 6Q2+
...
SELF ASSESSMENT EXERCISE 2
49
Find the consumption function (C) if marginal propensity to consume
(MPC) is 0
...
Determine the capital function (K) if the rate of net investment (I) is
and stock of capital at tequal to zero is 50
...
0
,
CONCLUSION
In this unit, what we have done is to practically apply the principles of
differentiation and integration to issues as they relate to economics
...
5
...
You will recall that, in unit three of this module we stated that integral
calculus is the direct opposite of differential calculus (derivative)
...
We looked at marginal concept
...
However, in a reverse manner, the total cost/revenue can be
determined from the marginal cost/revenue using integration
...
0
TUTOR MARKED ASSIGNMENTS
Find the total revenue function and the per unit price given that marginal
revenue (MR) is 40 – 4Q – Q3
If marginal cost (MC) is 24e0
...
Find the marginal revenue (MR) function of the demand function Q = 72 –
4p
...
4Yd, where
Yd is Y – T, and T is 200, find marginal propensity to consume (MPC)
...
7
...
A
...
Calculus: A Complete Course (6th edition): Pearson Education Inc
...
Chiang, A
...
& Wainwright, K
...
Fundamental Methods of mathematical
Economics (4th edition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
L, Greenwill, R
...
P (2005)
...
, Boston, Massachusetts, USA
...
(2002)
...
MODULE 2:
OPTIMIZATION
Unit 1: Introduction to Optimization
Unit 2: Function of Variables
Unit 3: Optimization with Constraints
51
Unit 4: Differentials
UNIT 1
Introduction to Optimization
CONTENTS
1
...
0 Objectives
3
...
1 Optimization of Functions
3
...
1
A case of Concave and Convex functions
3
...
2
...
3 Fundamental Clues on Optimization Analysis
4
...
0 Summary
6
...
0 References and Further Readings
1
...
However, a lot
of economic models or functions involve more than one independent variable
...
In this case, we can define the
model as a function of two independent variables where Q is the endogenous variable; L
and K are the exogenous variables
...
By definition, equilibrium is the
state of optimum position for a given economic unit such as household, business entity,
or the entire economy, in which they strive to attain that equilibrium
...
In a quest to have the best as mentioned, economic
agents try to understand the effect of one exogenous variable on the endogenous variable
which is basically measured using partial derivative and others that are discussed below
...
0
OBJECTIVES
At the end of this unit, the Students should be able to:
Discuss and explain optimization problem
Understand what partial differentiation is all about
52
Apply optimization to solve economic problems
3
...
1
Optimization of Functions
Recall we stated that optimization is about the quest for the best
...
Without the
aid of a graph, this is done with the methods of relative extreme and inflection points and
a lot more
...
An economicmodel or function of the nature
is said to be rising or falling at x = a
...
Recall that in unit one of module one, we stated that derivative
(that is the differentiation of the initial/original function) measures the rate of change and
slope of a function
...
See figure below:
y
y
o
a
o
x
a
x
Slope < 0
Falling function at x = a
(b)
Slope > 0
Rising function at x = a
(a)
> 0: rising function at x =a
< 0: falling function at x = a
When an economic model or function is rising or falling over its entire area is referred to
as amonotonic function
...
1
...
On the other hand, a function is convex at x = aif in an area very close to
the graph of the function is wholly above the tangential line
...
It is important to note that, if
> 0 for all x in the region,
is convex
...
However, the sign of the first
differentiation does not matter for concavity
...
y
y
(b) g’(x) < 0
g”(x) >0
(a) g’(x) > 0
g”(x) >0
Convex at x = a
y
a
x
a
x
a
y
a
x
x
(d) g’(x) < 0
g”(x) < 0
Concave at x = a
Supposing we have two functions g = -2x3 + 4x2 + 9x – 15 and g = (5x2 – 8)2, test for their
convexity and concavity at x = 3
...
= 2(5x2 – 8)(10x) = 20x(5x2 – 8) = 100x3 – 160x
= 300x2 – 160
= 300(3)2 – 160 = 300*9 – 160 = 2700 – 160 = 2540 > 0 (convex)
...
State if the function g(x)= 2x2 – 11x + 3 is rising, falling or static at x = 6
...
2
Relative Extrema
Recall that graph and relation are not the right tools to solve optimality problems, except
with the methods of ‘relative extreme and inflection
...
To start with, what is relative extrema is all about? It is a point at which any
economic model or function as we shall soon observe is at a relative maximum or
minimum
...
e
...
If that is the case, the first derivative of the function at
ais either zero or be undefined function or equation
...
’ this point is a standstill position
...
The first test for Stationarity or
critical point can be established on either the peak or bottom of the business cycle (see
explanation below)
...
In
doing this, a mathematical approach will be required, and this is by taking the secondorderderivative of the initial economic model or function
...
It shows that the model or function is convexand that, the graph of the function or
equation stays wholly above the tangential line at x = a, the function or model is at a
relative minimum (like the bottom of a business cycle graph), b) If
< 0 at a, it
signifies that, the model or function is concave and that the graph of the function is
whollyunderneath the tangential line atx = a, the function or model is at a relative
maximum (like the peak of a business cycle graph), and c) If we have a situation
different from the two instances already discussed, that is where
= 0, it means that
the test is inconclusive
...
In other
words, if we subject it to further test the situation where the first derivative is zero, we
will arrived at any of the conclusionssummarized thus mathematically
=0
> 0:
55
relative minimum at x = a
=0
< 0:
y
y
0
relative maximum at x = a
...
2
...
’ Most economic models or functions that are been studied
are often convex or concave in an interval at the second derivative
...
The point at which a function or model changed
from being concave to convex or vice versa are referred to as inflection point
...
Or, an inflection point
for a function g(x) if there exists an interval about any point on the graph such that: i)
g’’(x)≥ 0 in say (a, c)and g’’(x) ≤ 0 in say (c, b), or ii) g’’(x) ≤ 0 in say (a, c) and g’’(x) ≥
0 in say (c, b)
...
Inflection point occurs only where the second derivative equals zero or is undefined
...
This time, it is to
determine either the convexity or concavity of the graph
...
In sum, for an inflection
point at c, as seen in Figure below, the followings are observed: 1)
= 0 at c or is
undefined, 2) concavity changes at x = c, and 3) graph crosses its tangent line at x = c
...
a) g(x) = -5x2 + 130x – 50
Solution:This problem will be solved by way of following some steps
...
i)
Differentiate the primitive equation or function (take the first derivative),
set it equal to zero, and then solve for x to determine the critical value(s)
...
g’’(x) = - 10
g’’(13) = - 10 < 0 this is concave, and is a relative
maximum
...
solve for x
h’(x) = 9x2 – 72x + 135 = 0
= 9(x2 – 8x + 15) = 0
= 9(x – 3) (x – 5) = 0
x =3 x = 5 these are the critical values
Take the second-order derivative and estimate based on the critical values
determined
...
3
convex, and is relative minimum
Fundamental Clues on Optimization Analysis
As we round-off this section on introduction to optimization, it is essential that students
should familiarize themselves with some facts on optimization which will assist them in
57
analyzing optimization problems
...
” Assuming you are given a primitive function, model or equation
that has an optimization problem and is differentiable, know these:
Differentiate the primitive function, model or equation (1st derivative), set it equal
to zero, and solve for the critical value(s)
...
It locates the points at which the function,
model, or equation is neither risingnor falling, but at a leveled ground
...
After the step above, you then again differentiate the first derivative (2 nd
derivative), estimate the result using the critical value(s), and check for the sign(s)
...
Where the necessary or first-order condition is met, the second step, calledthe
second-order derivative analysis is known as thesufficiency condition
...
In such cases, the continuous
differentiation analysis is helpful: (a) If the first non-zero value of a higher-order
differentiation, when evaluated at a critical value, is an odd-number say 3, 5, 7
etc
...
(b) Also, if the
first non-zero value of a higher-order differentiation, when evaluated at a critical
value is an even number say 2, 4, 6, etc
...
But, if the even number is negatively signed, it is shows that
the function or equation is concave and is relative maximum, and if vice versa it
shows that the equation is convex and is relative minimum
...
52 – 6p + 6, and test if it is
relative maximum or minimum or inflection points
...
0
CONCLUSION
Under differential calculus in module one, we saw a case of a function of a single
variable, how to find the first derivative and that of the second derivative treated under
higher-ordered derivatives
...
5
...
We also discussed convexity and concavity as a case in optimization
...
If the sign is positive, it shows
that the model or equation is convex but, if otherwise, it means that the model or
equation is concave
...
You will recall that
we said relative extrema and inflection points are modes of analyzing
optimization problems
...
At a particular instance in the life of a function or model, there
is always a transformation that is, minimum changing to maximum, or vice versa
...
6
...
Given that y = 2x3 – 3x2 + 11x – 30
...
59
Determine the critical values, relative extrema (i
...
maximum or minimum), or
possible inflection point of the equation z = - (a – 8)4
...
0
REFERENCES AND FURTHER READINGS
Adams, R
...
(2006)
...
,
Toronto, Ontairo, Canada
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
Lial, M
...
N, & Ritchey, N
...
Calculus with Applications (8th
edition): Pearson Education, Inc
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0 Introduction
2
...
0 Main Content
60
3
...
1
...
2 Rules of Partial Differentiation
3
...
4 Multivariable Functions and Optimization
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
From the equations we have seen so far, there is always one
independent variable on the right-hand side of the equation or function deciding the fate
of the dependent variable on the left-hand side
...
’ In this section, we shall be looking at a situation
where we shall be having more than one variable or parameter as independent variables
on the right-hand side of the function or equation
...
’When cases like
these are treated, any problem with multi-product organization can successfully be
resolved by applying it
...
0
OBJECTIVES
At the end of this section, the students should be able to:
See the difference between a single-variable function and a multi-variable
function
...
3
...
1
Functions of Several Variables
To restate the obvious, what you have done so far in the preceding units as mentioned in
the intro of this unit wererestricted to a function of a single variable (i
...
exogenous
61
variable) such as y =
...
For instance, in production
theory, the production function has more than one exogenous parameter, Q = g (L, K)
...
Q is the endogenous variable (variable that is determined within the model);
L and K are exogenous variables (variable that are determined outside the model)
...
’ In the main time, we will be discussing
more of this
...
This equation or function can be analyzed in a way that,
we can investigate the impact of the exogenous variables individually on the endogenous
variable
...
One basic mode of carrying out this
measurement is the use of a mathematical method called “Partial Differentiation
...
The partial differentiation of z with
respect to x measures the sudden rate of change of z with respect to x while the
exogenous variable y remains unchanged
...
1
...
Unlike what we saw when we had y = f(x), a situation of one-variable
function
...
Note, to find the partial differentiation of a function as
z
, assumes that the variable ydoes not exist in the function, and
x
z
differentiate the function with respect tox(w
...
t
...
The same operation goes for
...
Consider the examples below:
Example 1: The partial derivatives of a two-variable function such as z =5x2y4are
determined thus:
Solution:
(i)
When differentiating with respect to x, treat the y variable as a constant term,
then:
z
= 10xy4
x
(ii)
Now, differentiate with respect to y, treat the x variable as a constant term also,
and you will have:
z
=20x2y3
y
Example 2: To find the partial derivatives for this two-variable functionz = 7x3 – 3x2y2+
6y4:
Solution:
(i)
When differentiating with respect to x, let the variable y be held constant as in
example 1, then:
z
= 21x2 - 6xy2
x
(ii)
Now, differentiate with respect to y, treat the x variable as a constant term also,
and you will have
z
= - 6x2 + 24y3
y
SELF ASSESSMENT EXERCISE 1
Determine the partial differentiation of the followings:
g(w, x, y) = 4w3 + 10wxy2 – y2 + x4
h(p, n) = 10p3 + 6pn2 + 7n3
3
...
These rules are referred to as rules of partial differentiation, just as we have
in differentiation proper in unit one and two of the preceding module
...
Rule1:
Product Rule
This rule is similar to the earlier treated version in the previous module
...
Given z = g(x, y)
...
Solution:Using product rule as given above, first differentiate w
...
t to x, then w
...
t toy
...
Like we mentioned under rule one in this section, its application is
done with some measure of caution because of its multivariable nature
...
Given that z =
g ( x, y )
where h( x, y) is not equal to zero
...
Solution:Using the quotient rule as already indicated in the question; first differentiate
w
...
t to x, and then w
...
t to y thus:
Rule 3:
Power-Function Rule Universalized
This rule is similar in operation to the version already seen in the previous module
...
If we havez = [h(x, y)]n,
where y is held constant
where x is held constant
Example 3: Supposing, z = (x2 -5y3)5, find the partial derivatives?
Solution:We can only find the partial derivatives of this function except the powerfunction rule universalized is applied
...
r
...
r
...
thus
we have:
65
(Note that, the asterisk is
multiplication sign)
SELF ASSESSMENT EXERCISE 2
Use any of the rules just discussed to determine the 1st-order partial differentiation of the
following equations:
q = 4p2(4p + 9i)
q
3
...
2x 4 y
Second-Order Partial Derivatives
What we have done so far in this unit is the first-order derivative of the functions, models
or equations given
...
e
...
e
...
The differentiation of the first-order derivative’s
result is called second-order derivative
...
Also,
(
y, y ) is exactly opposite of the previous one
...
r
...
r
...
Let us see one or two worked examples
...
Solution:In tackling this problem, what is required on the part of the leaners (students) is
patience and sense of reasoning
...
4
...
e
...
In that scenario, we have been able to determine the relative extrema,
inflection point, etc
...
For such a function where z = g(x, y), to be at a relative extrema,
three conditions must be fulfilled
...
This shows that at the given
point, often referred to as critical, the function is neither rising nor falling but is at
a relative table like form
...
This
67
indicates that from that table-like form, the function is concave when the curve
bends downward in the case of relative maximum and convex when it bends
upward in the case of minimum
...
This
isrequired to rule out an inflection point
...
1
...
2
...
3
...
e) If
, the test is inconclusive
...
If the
function is simply concave (convex) in x and y on an interval, the critical point is a
relative or local maximum (minimum)
...
4
...
However, there are extreme cases where a single endogenous parameter is affected by
more than one exogenous parameter
...
With what we have discussed thus far in this topic, we
can confidently apply this to solve problems associated with multi-products ventures
...
0
SUMMARY
In sum, we have in this unit, considered the followings:
We have studied functions of several variables
...
This will assist researchers to tackle optimization problems in a multi-products
organization
We also have discussed Partial derivatives
...
In the application of partial derivatives, some
fundamental rules were studied
...
We have discussed the conditions to be met to be able to determine the
relative extrema and inflection points in a function that has more than one
exogenous parameter
...
0
TUTOR MARKED ASSIGNMENT
Find the 2nd-partial and crossed partials of the followings:
q = 3pi4 + 5p3i
q = p0
...
3
Findthe critical values, and determine if these values/points in the function is at relative
extrema, inflection point, or inconclusive
...
0
REFERENCES AND FURTHER READINGS
Adams, R
...
(2006)
...
,
Toronto, Ontairo, Canada
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
69
Ekanem, O
...
(2000)
...
Lial, M
...
N, & Ritchey, N
...
Calculus with Applications (8th
edition): Pearson Education, Inc
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0 Introduction
2
...
0 Main Content
3
...
2 Constrained Optimization with the Lagrange Multiplier
3
...
1 The Method of Lagrange Multiplier
3
...
0 Conclusion
5
...
0 Tutor Marked Assignment
7
...
0
INTRODUCTION
In this section, we shall still continue our discussion on the concept of optimization, but
with a difference
...
One basic feature of this form of optimization is that all
the choice variables were independent of one another
...
However, in the field of economics, certain problems needed to be optimized
...
For instance, the
amounts of different items demanded by a buyer must fulfill the budget constraint of the
buyer (in this case, the buyer’s income can be seen as a constraint)
...
In particular, the method of Lagrange
multipliers will be studied to understand how problems in optimization are resolved using
it
...
0
OBJECTIVES
At the end of this unit, the students should be able to:
Know the difference between unconstrained optimization and the constrained type
...
Study the usefulness of Lagrange multipliers in optimization discussion
...
0
MAIN CONTENT
3
...
No matter how bad any economic activity may look, there
will always be an optimum point, if we apply the concept of optimization
...
So, in concept
of optimization these limiting factors are duly recognized as constraints
...
This concept is about
the study of the satisfaction of consumer(s) subject to their income level, which is always
referred to as Budget constraint
...
The marginal utilities can
be determined by taking the partial derivatives of the total utility function (U) w
...
t x1
and x2
...
Which
means, the buyer is able to any of good he/she deems fit
...
The case as presented has very little or no
pragmatic relevance in economics
...
Assuming, the buyer intends to
expend a total amount of ₦ 120, on the two items (x1and x2), and if the prices of the items
are:x1 = 8 and x2 = 2, we can now express this limiting factors (income and prices) that
have been incorporated in a linear form thus:
8x1 + 2x2 = 120
With this constraint, the items x1 and x2 are now mutually dependent
...
In trying to observe the relative extrema of both the constrained and unconstrained
optimization, the difference between the two can be illustrated in a three-dimensional
graph (see graph below)
...
Also, in terms of values, the value of the unconstrained maximum is expected
to be bigger than that of the constrainedmaximum
...
However, the constrained maximum is not expected to surpass that
of the unconstrained maxima
...
SELF ASSESSMENT EXERCISE 1
The optimization of a consumer’s utility without constraint amount to little or no
economic sense, discuss
...
3
...
In sum, constrained optimization is the quest for the best in the midst of
available (limited) resources
...
There are many ways constrained problem
optimization can be resolved
...
However, some constraints could be complicated function, or when there are
several constraints to be considered, the methods of substitution and elimination of variables
become ineffective
...
3
...
1 TheMethod of Lagrange Multiplier
The essence of the Lagrange multiplier is to convert a constrained-extremum problem
into a form that can be resolved applying the 1st-order condition
...
In this situation the
consumer is faced with some limiting factors, which in this case is represented with
budget constraint (Px X + PyY = I)
...
Therefore, there is a problem of preference optimization
among the available items by the buyer subject to the limiting factor (budget constraint)
...
What we
have just seen, is a classic constrained maximization problem
...
When this happens, the problem becomes unconstrained
maximization which can be solved by the method of substitution and elimination
...
In such
circumstance, other systems should be employed
...
Essentially, the same mode is often employed by
economists even for problems that are quite easy to express as unconstrained problems
...
Besides that, the introduction of Lagrange multipliers can be
modified in a number of more complicated constrained optimization problems, such as
those expressed in terms of inequalities
...
From these, we can combine the objective function and the constraint
function to get a new function
...
(Note that the sign λ is a Greek letter often called lambdain mathematical sciences, and is
the Lagrange multiplier)
...
Note that the constraint function is
all the time equal to zero, with the multiplication of λ to form λ[k - h(x, y)] is equally set
to zero and its addition does not change the value of the original function
...
They (the values) are
determined by partly (partial) differentiating the functionFw
...
t all the three mutually
dependent variables (x0, y0, and λ0),setting them equal to zero, and resolve
simultaneously, that is:
74
Fx(x, y, λ) = 0
Fy(x, y, λ) = 0
Fλ (x, y, λ) = 0
...
Example 1: Optimize the function q = 8p2 + 6pi + 12i2subject to the constraint p + i =
112
...
(i)
Set the constraint equal to zero by subtracting the variables from the constant
value
p + i = 112, set this equal to zero thus, we have
112 – p– i = 0
Multiply the result in step (i) by λ and add the outcome of the two to the original function
to form the Lagrange function q
...
8i
Substitute p = 1
...
8i – i = 0
2
...
75
measures the effect of
Now that we have determined our critical values, substitute them in (a) we will
have,
q= 8(72)2 + 6(72)(40) + 12(40)2 + (1,392)(112 – 72 - 40)
= 8(5,184) + 6(2,880) + 12(1,600) + 1,392(0) = 77,952
...
(i)
Let the constraints be equal to zero
30 - x - 2y -z = 0
10 – 2x + y +3z = 0
Multiply the result in step (i) by λ and add the outcome of the two to the original function
to form the Lagrange function F
...
Solving this system by elimination gives
xo = 10,
yo = 10,
zo = 0,
SELF ASSESSMENT EXERCISE 2
76
λo1 = 12 and λo2 = 4
Optimize the function f(x, y) = 8x2 – 4xy + 12y2 subject to x + y = 36
Optimize the function f(x, y, z)=2xyz2 subject to x + y + z = 112
3
...
Some examples have seen solved using this tool (Lagrange multiplier)
...
In the worked examples,especially example one, our λ equals to
1,392
...
Lagrange multipliers are often referred to as shadow prices
...
Example 1: To prove that a unit change in the constant of the constraint will cause a
change of approximately 1,392 units in Q in the earlier worked example
...
Q = 8p2 + 6pi + 12i2 + λ(113 – p – i)
Qx = 16p + 6i – λ = 0
Qy= 6p + 24i – λ = 0
Qλ = 113 - p - i = 0
By solving simultaneously, this gives:
p0 = 73
...
72
λ0 = 1,416
...
6 which is 1,399
...
SELF ASSESSMENT EXERCISE 3
Discuss the impact of one unit change in constant constraint on the value of the
objective function
...
0
CONCLUSION
We have seen the case of constraint optimization, and how the Lagrange multiplier was
used to resolve the problem
...
In economics, indeed human wants are many, but, the resources to meet
them are limited (the limiting factors)
...
5
...
That, constraints are the limiting
factors that would not allow a consumer or buyer to purchase all he/she desire to
get
...
Also, we have discussed constrained optimization
...
However, we have found out that that is not practicable
...
Lastly, we studied a method often used by Economists to resolve constrained
optimization problems
...
It is about subjecting
the objective function to constant constraint, and by this mode, the variables are
made mutually independent with one another
...
6
...
0
REFERENCES AND FURTHER READINGS
Adams, R
...
(2006)
...
,
Toronto, Ontairo, Canada
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
78
Lial, M
...
N, & Ritchey, N
...
Calculus with Applications (8th
edition): Pearson Education, Inc
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0 Introduction
79
2
...
0 Main Content
3
...
2 Total and Partial Differentials
3
...
4 Derivatives of Implicit and Inverse Functions
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
In unit two, our study was mainly on partial differentiation and its modus
operand
...
That is,
with partial differentiation, there exists no relationship among the exogenous parameters
...
For instance, in a simple national-income
equation with two endogenous parameters (variables) Y & C:
Y = C + I o + Go
C = C(Y, To)
These equations can be reduced to a single equation
Y = C(Y, To) + Io + Go
...
Instead, total differential will be more
appropriate in resolving economic problem of this kind
...
0
OBJECTIVES
At the end of this aspect of the module, the Students should be able to:
80
Explain what differentials is all about
See the use of differentials in resolving economic issues
Distinguished between total and partial differentiation
...
0
MAIN CONTENT
3
...
The derivative dy/dxcan as
well be expressed as a ratio of differentials wherebydy is the differential of y and dx the
differential of x
...
Differentials as a mathematical method as the some fundamental characteristics that
would help our understanding of its operations
...
In this instance, dy is dependent on x and dx
...
(ii) Since dy is dependent on dx, it
therefore means that if dx = 0, then dy = 0
...
(iii) the differential dy can only be
expressed in terms of other differential(s) like dx
...
r
...
= 8p + 11
Then multiply this outcome by a specific change in p (dp) to get the actual change in
q(dq)
...
Let us see a worked example
...
r
...
Therefore,
dq
= 15p2 + 8p
dp
derivative of q w
...
t p,
Then multiply it with the little change in p denoted by dp, and we have
dq
= dq = (15p2 + 8p) dp
...
2
18 p 4
10 p
q (6 p 3)(5 p 12)
q
Total and Partial Differentials
Total differentials as a concept can be discussed in reference to a function of two or more
exogenous variables
...
Supposing we have an economic model of the form Q = g(K, L), that is a production
function, where Q is output, K is capital, L is labour
...
This gives the marginal impact of the variables K and L
on the endogenous variable Q
...
The same goes for the variable L
...
r
...
The partial derivatives have
played the role of a converter which helped in arriving at the total change in Q
occasioned by the individual change in the two exogenous variables
...
r
...
Example 2: Determine the total differential for production functions below:
(i)
Q (z1, z2) = xz1 + yz2 and (ii)Q (z1, z2) =
than zero
...
(ii)
Q
Q1 2 z1 z2
z1
again,
Q
2
Q2 3z2 z1
z2
Therefore,
2
dQ = Q1dz1 + Q2dz2 = (2 z1 z2 )dz1 (3z2 z1 )dz2
...
By definition, a partial differential measures the rate of influence on the endogenous
variable for a two or multivariable function occasioned by a small change in one of the
exogenous variables, assuming other right-hand variables are held constant
...
3
Total Derivatives
We have discussed instances of partial derivative, partial differential, and total
differential
...
For total derivative, its modules operand
is not in any form similar to that of total differential
...
This process is diagrammatically represented below for better understanding
...
The total derivative measures the direct effect of p on q, ,
plus the indirect influence of p on q through i, that is,
...
Solution:We will apply the total derivative methodology to resolve this problem
...
Substitute all these into the model
above, we have,
84
36p2 + 14(16p + 6) = 36p2 + 224p + 84
As a check, we substitute the function g into the function f, to get
q = 12p3 + 14(8p2 + 6p + 16) = 12p3 + 112p2 + 84p + 224
Thus:
36p2 + 224p +84
...
4
Derivatives of Implicit and Inverse Functions
Recall that in the beginning of this module, we treated the explicit function in which the
endogenous variable is on the left hand side of the equal sign, and the exogenous variable
is situated on the right hand side
...
However, as we advanced in the module often, we
came across implicit functions, in which the variables including the constant term are all
on the left hand side of the equal to sign (=)
...
If an implicit function f (p, q) =0 exists and fp ≠ 0 at the point around which the implicit
function is defined, the total differential is simply fpdp + fq dq= 0
...
Having said that, we can
then rearrange the total differentials to get the implicit function rule:
Notice that the derivative dq/dp is the negative of the reciprocal of the corresponding
partials
...
Assuming we have an inverse function, the
inverse function rule states that the derivative of the inverse function is the reciprocal of
the derivative of the original function
...
Example 4: Find the derivative of the implicit function: 14x2 – 2y = 0
Solution: Remember the implicit rule of derivative
...
Substitute this in the model above,
= 14x
...
Solution:know that the derivative of an inverse function is the reciprocal of the
derivative
...
Thus,
4
...
While derivative is the
ratio of differentials, differential is the differential of the variables
...
However, in a situation where an explicit
solution is not possible, partial derivative will be ineffective in its application
...
In sum,
differential is the mathematical method that assists economists to resolve economic issues
with implicit form
...
0
SUMMARY
Thus far in this section, we have treated the following:
86
Differentials and derivatives
...
The derivative of implicit and inverse functions
...
6
...
Where y = 1000 – 3x2
Find the derivative of the function f(x, y) =4x2 + 3xy + 6y3, using implicit rule
...
0
REFERENCES AND FURTHER READINGS
Adams, R
...
(2006)
...
,
Toronto, Ontairo, Canada
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
Lial, M
...
N, & Ritchey, N
...
Calculus with Applications (8th
edition): Pearson Education, Inc
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0
2
...
0
3
...
1 Systems of Equations (Linear)
3
...
2
...
3 Matrix and Vector Operations
3
...
1 Multiplication of Vectors
Laws in Matrix
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
Optimization is basically of two categories, the constrained and the unconstrained
...
While the latterlooks at optimization
with the view that the quest for the best can be gotten without recourse to any limiting
factor
...
We have
seen optimization in terms of two or more variables
...
In this part of the module, we shall be discussing matrix algebra
...
If eventually these equations are all
88
linear, the study of such systems of equations belongs to an area of mathematics called
linear (or matrix) algebra
...
2
...
State the laws in matrix
...
0
MAIN CONTENT
3
...
For the purpose of refreshing your memory, here is an example of such a
system, whereby the two unknowns are designated by x1 and x2,
2x + 4y = 4
4x –2y = 8
A solution to this system is a pair of numbers (x, y) which satisfies both equations
...
Take on
the first equation, and make xthe subject of the equation
...
Then
insert x into the second equation, and we have4(- 2y +2) – 2y = 8
...
The only
solution is therefore (x, y) = (2, 0)
...
However,
there are instances in matrix operations as we shall soon see in which we consider a large
number of equations with unknown parameters or variables, and then we shall be looking
at a notation that will be suitable for that instance
...
In such a system of
equations, themis always larger than, equal to, or less than n
...
+ b1nxn = c1
89
b12x1 + b22x2 +
...
, bmn are referred to ascoefficients, whilec1…
...
It is very
important we notice the subscripts used in the notation
...
At times, most of these coefficients are always zero (0)
...
3
...
1 of this unit that matrix and is another name for linear
algebra
...
The numbers (parameters or
variables) are called elements of the matrix
...
1)
...
The number of rows m and columns n shows the scopes of
the matrix (m × n),this is pronounced m by n, meaning m rows by n columns
...
That is what
we have seen by the term m by n matrix
...
Asquare matrix is a good example of such a case
...
Another instance in matrix is a case
of single column matrix where the dimension is such that it is m by 1
...
Also, we can have a case of a single row, with 1 by n
dimension;this is a case ofrow vector
...
Supposing in a matrix, we decide to make some
manipulations by way of transforming the rowsto columns and the columns to rows, what
we have just done is known astranspose matrix
...
Examples 1: Given
A=
B=
3×3
C=
2×3
D=
3×1
90
1×3
The matrices as shown above have given us some scopes or dimensions of matrices to be
exposed to, even larger ones not included in the scenario
...
By this, the matrix is thus a square matrix (see the
definition above)
...
The subscripts are the m n earlier discussed
...
For instance, b32 is an element
which is placed in row 3, and column 2; b13 is an element sited in row 1, and column 3
...
That is, it has two rows and three
columns
...
For example, its b23 element is 7;
its b11 element is 1, and many more
...
Notice that, the composition or the
numbers of element in a matrix can be determined via the dimension of the matrix
...
Recall that, we have mentioned that a matrix is transposed when the rows are transformed
into columns, and columns into rows
...
The notation for transpose matrix in the case of matrix B is written
B’ (or BT)
...
Instead of the initial 2 by 3, what we now have is
a matrix of 3 by 2 dimensions
...
Apart from a square matrix, the dimension of a
matrix changes anytime the matrix is transposed
...
3
...
1 Linear Algebra: Its Roles
Recall we have stated before now that, we shall bediscussing the importance of matrix or
Linear algebrain mathematical studies
...
As a result of this, matrix as a part
of mathematics fills the following gaps in mathematical studies:
91
a) It allows complicated system of equations expressed in a form that is
understandable
...
b) With matrix or linear algebra, one can easily determine if a particular
mathematical problem is resolvable before it is attempted
...
But with
the understanding of matrix, one could determine if the situation is solvable
...
It can only be
applied to systems of linear equations
...
A good instance where matrix is applicable is a situation where a dealer in all forms
of airtime or recharge cards has a lot of shops, say 5 where he/she sells
...
See details below:
SHOPS
GLO
MTN
AIRTEL ETISALAT
A
100
150
80
200
B
250
300
100
150
C
234
200
120
220
D
350
270
300
400
E
450
250
275
356
From the matrix above, the dealer can have an ideal of all the stock of each network
airtime he/she has in the five shops
...
Then down and up of a
column in the matrix, gives the total stock level of any item in the five shops
...
3
Matrix and Vector Operations
You will recall that matrix is about the arrangement of numbers horizontally and
vertically (that is, in rows and in column)
...
As a result, vectors are suitable for the application of all
the algebraic tasks already discussed
...
These are the
row and the column vectors
...
While the latter, is the arrangement of numbers in a matrix in vertical order
...
3
...
For example, supposing that:
92
a=
and b =
, going by the principle of multiplication,
we have matrix a times matrix b
...
Remember that matrix a is a2 by 1,
whilematrix bis a 1 by 3 matrix
...
In matrix
multiplication, the dimension/scope/magnitude of the resultant matrix is a combination of
the rows (m) and columns (n) of the individual matrix
...
It is important to note that for matrices to be conformable, the number of columns in the
lead matrix (matrix that comes before the other matrix in any matrix operation, e
...
Matrixa as in the example above) must be equal to the number of rows in the lag matrix
(a matrix that comes after the lead matrix, e
...
matrix b)
...
The scope of this resultant matrix is 1
by 1 as already indicated
...
SELF ASSESSMENT EXERCISE 2
What are vector matrices?
Distinguished between a vector matrix and a scalar matrix
...
4
Laws in Matrix
In matrix, there are laws that guide its operations
...
These laws are commutative, associative, and distributive in nature
...
It is important to note that both multiplication and
addition in matrix are done in line with commutative, associative, and distributive laws
...
However, since the
addition is merely the summation of the corresponding elements of the matrices involved,
the order of their summation is immaterial
...
But, in case of scalar multiplication, the law (i
...
commutative) is followed
(that is, cd = dc)
...
In the case of multiplication, the law(that is associative) is applied only when
the matrices are in order of conformability
...
Lastly, as we have seen
in the laws already discussed, in the case of multiplication, the distributive law is x(y + z)
= xy + xz
...
This is so because of the convertibility principle involved
...
Let us consider some worked examples applying those laws
...
Solution:Note the calculations are done in conformity with law of commutative
...
(i)
a+b
=
=b+a=
(ii)
a–b
=
=
=-b+a=
=
From the workings above, the matrix addition and subtraction in line with the
commutative law as discussed earlier has been shown using numbers
...
Example 2:Supposing we have:
a=
b=
...
For matrix a, it is a 2by3, while matrix b is 3by 2
...
In this
case the two matrices are conformable in that, the numbers of columns in matrix a is
equal to the numbers of rows in matrix b, therefore:
94
ab =
=
2by2
2by2
ba =
=
3by3
3by3
Considering these outcomes, we have been able to confirm that ab ≠ ba
...
In this instance (ab),
the matrix b is premultiplied by matrix a, while matrix a is postmultiplied by matrix b
...
See below the applicability of the law
with numeric example
...
95
SELF ASSESSMENT EXERCISE 3
Determine (i) a – b and (ii) – b + a if
a=
b=
Find (i) xy and (ii) yx if
x=
4
...
We further stated that, these numbers are located in a particular position in the matrix
...
This shows that the placement or location of number in a matrix is very vital in matrix
operation
...
0
SUMMARY
In this module which is basically about linear algebra, we started by treating matrix as
one of the units that forms part of the entire module
...
We discussed in this part simultaneous equation with
two models or functions, and how issues in it could be resolved by the method of
elimination
...
We have defined matrix and some basic terms in line with matrix operations
...
Also, we discussed the roles of matrix in economic studies
...
We equally considered the laws in matrix operations
...
And we also have discussed their
applicability to economic issues
...
0
TUTOR MARKED ASSIGNMENT
Discuss the rows and column in matrix
...
7
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0 Introduction
2
...
0 Main Content
3
...
2 Addition and Subtraction of Matrices
3
...
4 Identity and Null Matrices
4
...
0 Summary
6
...
0 Reference and Further Readings
1
...
These laws are
commutative, associative and distributive in nature
...
We have treated some
worked examples in line with these laws
...
Under this topic, we
shall be considering sub-topics such as linear dependence, scalar and vector operations,
transpose of matrix, and many more
...
That a matrix will always have an m-by-n(also written as m×n)
order or dimension
...
For instance, a 3-by-2 (3 × 2) ordermatrix has three rows and two columns
...
This brings us to equality of matrices
...
That is if we have two
matrices X and Y, and their orders are 3- by- 3 (i
...
X3×3and Y3×3), it then means that
matrix X is equal to matrix Y(X = Y)
...
0
OBJECTIVES
At the end of this unit the student should be able to
Discuss linear dependence amongst matrices
Explain and conduct operations as regard transpose of a matrix
Distinguish between identity matrix and null matrix
...
0
MAIN CONTENT
3
...
In other words, if two or
more vectors (either row or column) are examined mathematically to be equal to a
particular vector matrix, it means that there is linear dependence among the set of vectors
...
if:
x =
, y =
, and z=
...
I know for sure someone will ask how:
2x – y =
-
=
= z
...
How?
2x – y – z = 0 that is
...
Also, we can have row vector matrices say two that can be linearly dependent
...
The concept of linear dependence has a simple interpretation
...
When more than two vectors in
the 2-space are considered, there emerges this significance conclusion: once we have
found two linearly independentvectors in the 2-space (say, x and y) all the other vectors in
the space will be expressed as a linear combination of these (x and y)
...
Because of this, any set of three or more 2-vectors (three
or more vectors in a 2-space) must be linearly dependent
...
99
3
...
For instance two or three matrices to be added or subtracted must be
uniform in their scope (i
...
X3×3, Y3×3, etc
...
Therefore, this enables each entry of one matrix to be added to
(subtracted from) the corresponding element of the other matrix
...
Example 1: Given matrices x and y below, determine the sum of matricesx and y
...
x=
y=
3by3
then x+y =
3by3
=
3by3
3by3
Example 2: Given two matrices k and j of the same order, the entries of the two matrices
can be subtracted thus:
k=
j=
then k-j =
=
Note that in matrix operations for addition and subtraction, the order/dimension of the
matrices involved, and the place of the elements in the matrices are important
...
3
Matrices Multiplication
In matrix operations, the multiplication of two or more matrices with dimensions
(mbyn)1, (mbyn)2
...
That is, n1 = m2, or the number of columns in matrix 1, the lead matrix, equal the number
of rows in matrix 2, the lag matrix
...
Each row vector in the lead matrix is then multiplied by each
column vector of the lag matrix, in accordance with the rules for multiplying row and
column vectors already discussed in unit one Section 3
...
1
...
One quick way for test for conformability in matrix operations as regards multiplication,
before undertaking any operation, is to place the dimensions/scope in the order in which
matrices are to be multiplied, then see if the number of columns in the lead matrix is the
same with the number of rows in the lag matrix
...
e
...
Then, the multiplication can take place as
proposed
...
The first thing is to determine the conformability of the matrices
...
Therefore, they are conformable, and are said to be defined
...
These
forms of matrices are not right for multiplication, and not defined
...
4
...
x=
3
...
It is designated by the signI, orIn, where the subscript nassists to
show its row (as well as column) dimension (nbyn)
...
However, we can simply represent the
two by I instead of I2 and I3
...
For instance, for any number say x, we have 1(x) = x(1) =
x
...
Therefore,
IZ =
=
=
=Z
This has proved that special feature about identity matrix is right as stated
...
Identity matrix, because of its special nature, during process of multiplication, it can
beinsertedor deleted from the matrices, and the product matrix will not be affected
...
Observe that dimension conformability it preserved whether
or not an identity matrixappears in the product
...
Universally written as,
(In)k = In
(k = 1,2,…,n)
This is a case of idempotent matrix
...
Null Matrix
Just as seen in matrix operation with identity matrix where itplays the role of the number
1 in algebra, a null matrix or zero matrixrepresented by 0, also plays a role in matrix
operations
...
A
null matrix is simply a matrix whose entries are all zero
...
Therefore,
102
=
and
=
...
This is becausenull matrices
obey the rules of operation (subject to conformability) as stated below with regard to
addition and multiplication:
+
+
=
and
=
Note that, in multiplication, the null matrix to the left of the equals sign and the one to the
right may be of different dimensions
...
4
...
We have also looked at some special forms of matrix
such as identity and null matrices
...
5
...
We discovered that certain matrices are
somehow a product of the combinations of one or more matrices through a process
of some mathematical manipulations
...
This we did bearing
inmind the laws in matrix
...
The same applies to
multiplication in matrix
...
103
Furthermore, we treated identity and null matrices
...
On the other hand, a null matrix unlike an identity matrix has all the entries to be
zeros
...
In the case of transpose, it is
like having the direct opposite of the initial matrix
...
6
...
0
, and z =
, verify that
(x + y) + z = x + (y + z)
(x + y) – z = x + (y – z)
Given x =
(i)
,y=
,y=
, and z =
:
Determine: a) XI b) IXc) IZ, and d) ZI
...
REFERENCES AND FURTHER READINGS
Chiang, A
...
& Wainwright, K
...
Fundamental Methods of mathematical
Economics (4th edition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
(2002)
...
104
UNIT 3
MATRIX INVERSION
CONTENTS
1
...
0 Objectives
3
...
1 Determinants and Nonsingularity
3
...
1 Determinants of 3rd Order
3
...
3 Identity and Null Matrices
3
...
0 Conclusion
5
...
0 Tutor Marked Assignment
7
...
0
INTRODUCTION
In unit two, we discussed basically the mathematical operations (addition, subtraction,
and multiplication) involved in matrices
...
For instance, we saw the case of the special forms of matrix (that is, the
identity and null matrices) where their inclusion or exclusion from matrix operation make
the product matrix unchanged or changed
...
Matrix inversion in linear algebra is primarily about Determinants and Nonsingularity of
matrices
...
And it is (i
...
matrix inversion) required to put mathematical
operations right in linear algebra for easy application
...
2
...
Lastly, apply Cramer’s rule to matrix operations
...
0
MAIN CONTENT
3
...
For a given matrix,
say Z, The determinant of it has a notation
Determinant is only operational when we
have a square matrix
...
This is arrived at bymultiplying the two
entries on the principal diagonal and subtracting from it the product of the two entries off
the principal diagonal
...
Note that z11 and z22 elements are on the principal
diagonal while z12 and z21 are on the off the principal diagonal
...
If the determinant of a
matrix is zero, it means that the determinantvanishes and such matrix is referred to
assingular matrix
...
So, if the determinant of matrix is not
equal to zero, the matrix is said to be nonsingular and all matrix rows and columns are
not linearly dependent
...
Hence, our duty is to
see that linearly dependence does not arise in our models
...
Assuming we have a system of
equations with coefficient matrix Z, to test for linear dependence of the equation, find the
determinant, and observe the followings:
(i)
(ii)
If
= 0, the matrix is singular and there is linear dependence among the
equations
...
However, If
≠ 0, the matrix is nonsingular and there is no linear
dependence among the equations
...
This brings us to the issue of matrix ranking
...
The rank of a matrix can also be tested for too
...
If (Z)
Therefore, the matrix is
nonsingular, and there is no existence of linear dependence between any of its rows, or
columns
...
It therefore shows that
matrix Y is singular, and there is the existence of linear dependence between its rows and
columns
...
1
...
Now, we want to consider a3by3 matrix, commonly called
‘third-order determinant
...
These three
products are derived thus:
(i) Take the first entry of the first row, , and mentally erase the rest entries in the
row and column in which
is located
...
Then multiply
by the
determinant of the remaining entries, which is a 2 by 2 matrix
...
See (2) below
...
107
(iii)
Finally, take the third entry of the first row, , mentally delete as usual the
rest entries in the row and column in which
appears
...
(1)
(2)
(3)
From the outcome we have above the determinant of the matrix Y is estimated thus:
=
=
+
(- 1)
)–
(
+
(
)+
(
)
= a scalar
...
The same goes for the case of 4by4 and so on
...
Example 1: Given matrix X, find the determinant
X=
Solution:This a 3by3 matrix, the warning holds
...
Therefore, the determinant is determined thus below:
=
+
(- 1)
+
= 4[2(1) – 3(3)] – [6(1) – 3(2)] + 5[6(3) – 2(2)]
= 4(-7) – (0) + 5(14) = 42
...
SELF ASSESSMENT EXERCISE 1
Distinguished between a second-order matrix and a third-order matrix
...
1
...
We have
equally seen how determinants are estimated from a given matrix
...
These properties will provide us with the
ways in which matrices can be explored to simplify the entries to zero, before
determining the determinant:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
3
...
That is, if matrix Yis given, the determinant of matrix Yis
not different from the determinant of matrix Y transpose
...
Multiplying the entries of any column or row by a constant will alter the value
of the determinant by the constant
...
The determinant of a triangular matrix, that is a matrix with zero entries
everywhere above or below the principal diagonal, is the same as the product
of the entries on the principal diagonal
...
Minors and Cofactors
In section 3
...
1
...
The entries of the matrix remaining after the deletion form a
subdeterminant of the matrix often referred to asminor
...
Thus, a minor is the determinant of the
submatrix formed by deleting the ith row and jth column of the matrix
...
1 subsection 3
...
1 as our guide, we have:
Observe that,
is the minor of
,
the minor of
of
...
The notation for a cofactor
is
, and is a minor with a prescribed sign
...
If i + j is equal to an odd number,
to an odd power is negative
...
1 subsection 3
...
1 are as
1)
Since
=
= 1,
=
Note the superscript 1+1 above
...
2)
Since
=
= -1,
=
=
Also, the superscript 1+2, means row 1, column 2
...
110
SELF ASSESSMENT EXERCISE 2
Define a cofactor and a minor; state the main difference between the two
...
3
Cofactor and Adjoint Matrices
In section 3
...
At the end of the section, we saw how a cofactor was worked from a minor
...
Still we continue our
study on matric inversion; we shall be looking at cofactor and adjoint matrices
...
If
it happens that this matrix (the cofactor matrix) is transposed, whereby the rows are
transformed to columns, and the columns to rows, the new matrix so formed is known
asadjoint matrix
...
X=
Solution:
Hints:
In find the cofactor of a matrix, the allocation of the signs is very important
...
There is a short cut as regards signs distribution in a cofactor matrix
...
The sign
already determined will be used
...
Therefore, replacing the elements with their cofactors according to the laws of cofactors,
C=
=
111
The adjoint matrix Adj X is the transpose of C (
Adj X =
)
...
3
...
1 Transposes and Inverses Matrices
Recall we had mentioned in the starting of the module, precisely in the first unit a little
about matrix transpose, and we equally mentioned that it will be discussed fully in the
latter part of this module
...
Universally, the notation for any transposed matrix say A is
represented by or
...
See the transposed matrix below:
or
=
3by2
From the solution, the initial 2by3 matrix, after transposed, is now a matrix of dimension
3by2
...
For a square matrix, the dimensions remain the same
...
We shall continue our study in this part by
looking at the inverse of a matrix
...
In the same vein, the inverse of a matrix also a “derived” matrix, may or may
not exist
...
AA-1 = A-1 A = I
112
That is, whether A is pre- or postmultiplied by A-1, the product will be the same identity
matrix
...
4
...
In the start of this part, recall we
mentioned that in economics, certain models could be cumbersome, making evaluations
difficult
...
We have seen how determinants of matrices are
determined, minors of matrix estimated, cofactors, adjoint matrix, and many more
...
5
...
That a matrix is
nonsingular when there is no linear dependence between the rows or columns
...
We treated the concepts of minor and cofactors
...
Finally, we took a swift at cofactor matrix and adjoint matrix
...
6
...
113
7
...
C
...
(2005)
...
Dowling, E
...
(2001)
...
Ekanem, O
...
(2000)
...
Sydsaeter, K & Hammond, P
...
Essential Mathematics for Economic Analysis:
Pearson education ltd, Edinburgh gate, England
...
0 Introduction
2
...
0 Main Content
3
...
2 Linear Equations with Matrix Algebra
3
...
0 Conclusion
5
...
0 Tutor Marked Assignment
7
...
0
INRODUCTION
The just rounded off unit (unit three to be precise) was mainly on matrix inversion
...
It is in fact, the
climax in linear or matrix algebra discussion
...
We also
discussed Nonsingularity matrix, and discovered that, only a nonsingular matrix has
determinant
...
However, we shall continue our study in this last unit of this module by applying what we
have learnt so far in the module, most especially in unit three to solving some basic
economic issues
...
2
...
Apply it (matrix inversion) into solving matrix equation
...
3
...
1
Matrix Inversion
For a given matrix, A, an inverse of the matrix which is represented by
may or may
not be found
...
We have seen a case of necessary but not
sufficient condition for the existence of an inverse matrix
...
When a matrix has no inverse, the matrix is referred to as a singular matrix
...
The formula for deriving the inverse
of a matrix is
=
Adj A
...
It therefore shows
that a singular matrix has no inverse
...
AA-1 = A-1 A = I
That is, whether A is pre- or postmultiplied by A-1, the product will be the same identity
matrix I
...
About four (4) steps are involved in the estimation of an inverse matrix
...
Finally, divide the Adjoint matrix so derived in step three by the determinant in
step one to arrive at the inverse of the matrix in question (i
...
= A-1)
...
Example 1:Assuming Z =
determine
Solution:What we shall be doing here is to determining the inverse of the matrix by
following the steps stated above:
(i)
=4
-2
116
-2
= 4(12-0) -2(0-6) -2(0-4)
= 4(12) -2(-6) -2(-4)
= 48 + 12 + 8
= 68
(ii)
Minor
&
cofactor
matrix
=
=
(iii)
Adjoint matrixZ =
a
...
SELF ASSESSMENT EXERCISE 1
Find the inverse of the matrix below:
X=
...
3
...
One vital role of inverse matrix
in matrix algebra is that, it is used to resolve matrix equations
...
Remember that, the multiplication of an inverse matrix with the
matrix produces an identity matrix
...
Also recall that, an identity matrix multiply by a matrix equal
to that matrix
...
This implies that the coefficient of the
inverse matrix multiplied by the column vector of Y produces the equation’s solution
...
12x1 -12x2 +10x3 = 10
6x1+ 28x2 -12x3 = 14
-4x1 +4x2 + 8x3= 9
Therefore, in matrix form, we have
=
If the determinant is 68, therefore:
=
=
=
=
=
= 0
...
0588 and
118
= 1
...
3
...
It is also employed to solve n linear equations in n
unknowns
...
Example 1: Cramer’s rule is used below to solve the system of equations
6
+5
= 49
3
+4
= 32
1
...
Find the determinant of A
= 6(4) – 5(3) = 9
3
...
, with the vector of
=
Find the determinant of
,
= 49(4) – 5(32) = 196 – 160 = 36
and use the formula for Cramer’s rule,
=
=
=4
4
...
=
Take the determinant,
= 6(32) – 49(3) = 192 – 145 = 45
and use the formula
119
=
=
=5
SELF ASSESSMENT EXERCISE 2
Use Cramer’s Rule to solve the following equations
(i)
2x + y = 4
x + 2y = 8
(ii)
(iii)
4
...
We have seen that our
statement at the start of this module that, the essence of linear algebra is to proffer
solutions to cases where some systems of equations maybe complex and applying simple
elimination method to solving it may be impossible
...
That is basically what matrix inversion
and Cramer’s Rule is all about
...
0
SUMMARY
In summary, in this unit, we have done justice to the followings:
We started by treating inversion of matrix
...
We considered linear equation with matrix algebra
...
Finally, our discussion in this unit culminated in studying of Cramer’s rule
...
The rule
is a simplified method of solving a system of linear equations through the use of
determinants
...
’
6
...
0
REFERENCES AND FURTHER READINGS
Chiang, A
...
& Wainwright, K
...
Fundamental Methods of mathematical
Economics (4th edition): McGraw-Hill/Irwin, New York, NY, USA
...
T
...
Introduction to Mathematical Economics (3rd): Schaum’s outline
series, McGraw-Hill, USA
...
T
...
Mathematics for Economics and Business (2nd): Uniben Press,
Benin City, Nigeria
...
(2002)
...
121
122