# Elementary Mathematics for Economics

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ELEMENTARYMATHEMATICSFOR ECONOMICSR.C. Joshi NancyM.A., M.Phil. B. Tech.Formerly Head, P.G. Dept. of MathematicsDoaba College, JALANDHAR

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• VISHAL PUBLISHING CO.Future for WINNERS

JALANDHAR DELHI

Catering the need ofSecond year B.A./B.Sc. Students of Economics (Major)

Third Semester of Guwahati and other Indian Universities.

2nd Semester

ELEMENTARYMATHEMATICS

FOR ECONOMICS

R.C. Joshi NancyM.A., M.Phil. B. Tech.

Formerly Head, P.G. Dept. of MathematicsDoaba College, JALANDHAR

• CONTENTS

1. LINEAR EQUATIONS 1101. Introduction

1.1. Special Products1.2. Definition of an Equation1.3. Identity and Equations1.4. Linear EquationsExercise1

2. Economic application of linear equations inone variableExercise2

3. Economic ApplicationsExercise3Questions VSA and MCQ

2. SYSTEM OF EQUATIONS 11241. Introduction

1.1. Simulataneous Linear Equations1.2. Methods of Solving Simultaneous

Linear EquationsExercise1

2. Business application of Linear Equations inTwo VariablesExercise2

3. Market Equilibrium when demand andsupply of two commodities are givenExercise3

4. Economic Applications of Linear Equations4.1. Effect of Taxes and subsides in

Equilibrium Price and QuantityExercise4Questions VSA and MCQ

1.1. To solve the Standard QuadraticEquation

Exercise12. Equation reducible to Quadratic Equation

Exercise23. Equation of the form ax + b/x = c, where x

is an expression containing the variable,may be solved by putting x = y.Exercise3

4. Irrational Equations. An equation in whichthe unknown quantity occurs under a radicalis called an irrational equationExercise4

5. Equation which can be put in the formax2 + bx + p 2 ax bx c + k=0 be solved by

putting 2 ax bx c y .Exercise5

6. Equation of type2 2 ax bx k ax bx k = p, can be

solved by putting A = 2 ax bx k ,

B = 2 ax bx k .Exercise6

7. Reciprocal EquationsExercise7

9. Application in EconomicsExercise9Questions VSA and MCQ

4. FUNCTION, LIMIT AND CONTINUITYOF FUNCTIONS 43761. Introduction

1.1. Definition of Function1.2. Image and pre-image1.3. Domain1.4. Real Valued Function1.5. Types of Functions1.6. Linear Homogeneous Function1.7. Functions in Economics

1. Demand Function2. Supply Function3. Total Cost Function4. Revenue Function5. Profit Function6. Consumption Function7. Production Function

1.8. Value of a function at a pointExercise 1

2. Limit2.1. Left Limit2.2. Definition : Left Hand Limit2.3. Theorem on Limits2.4. Methods of Finding Limit of a Function

Type 1. Method of FactorsType 2. Method of Substitution

• Type 3. Use of Binomial Theorem for any index.Type 4. Rationlazing MethodType 5. Evaluation of limit when x

Exercise 23. Some Important Limit

Exercise34. Continuity

4.1. Continuity Definitions 24.2. Type of Discontinuity of a FunctionIllustrative ExamplesExercise4Questions VSA and MCQ

5. SETS 771051. Introduction

1.1. Definition1.2. Representation of Sets1.3. Some Standard SetsExercise1

2. Types of Sets2.1. Empty Set2.2. Finite and Infinite Sets2.3. Equal SetsExercise 2

3. Subset3.1. Proper Subset3.2. Singleton Set or Unit Set3.3. Power Set3.4. Comparable Sets3.5. Universal SetsExercise3

4. Venn Diagrams4.1. Operations on Sets4.2. Union of Sets

Illustrative Examples4.3. Definition4.4. Some Properties of the Operation of

Union4.5. Intersection of Sets4.6. Definition4.7. Disjoint Sets4.8. Some Properties of Operation of

Intersection4.9. Difference of Sets4.10. Symmetric Difference of two Sets4.11. Complement of a Set4.12. Complement LawsExercise 4

5. Number of Elements in a Set6. Economic Application of Sets

Exercise 5Question VSA and MCQ

6. MATRICES 1061351. Introduction

1.1. Matrix1.2. Types of MatricesExercise 1

2. Sum of Matrices2.1. Properties of Addition of Matrices2.2. Scalar Multiple of a Matrix2.3. Properties of Scalar MultiplicationExercise 2

3. Product of Two Matrices3.1. Zero Matrix as the Product of Two

non Zero Matrices3.2. Theorem3.3. Distributive Law3.4. Associative Law of Matrix

Multiplication3.5. Positive Integral Powers of a Square

Matrix A3.6. Matrix Polynomial

Exercise 34. Transpose of Matrix

4.1. Properties of Transpose of Matrices4.2. Special Types of MatricesExercise 4Questions VSA and MCQ

7. DETERMINANTS 1361741. Determinants

1.2. Determinant of a Matrix of order 3 31.3. Singular Matrix1.4. Minor1.5. CofactorsExercise 1

2. Properties of Determinants2.1. To Evaluate Determinant of Square

Matrices2.2. Type I2.3. Type II2.4. Type III2.5. Type IV2.6. Type VExercise - 2

3. Solution of a System of Linear Equations3.1. Homegeneous System of Linear

EquationsExercise - 3Questions VSA and MCQ

• 8. ADJOINT AND INVERSE OF A MATRIX175195

1.1. Theorem1.2. (a) The Inverse of a Matrix

(b) Singular and Non-Singular Matrix1.3. The Necessary and Sufficent Condition

for a Square Matrix to Possess itsInverse is That | A | 0.

Exercise 12. Elementary Transformation

2.1. Symbols for Elementary Transformation2.2. Equivalent Matrices2.3. Elementary Matrices2.4. Theorem2.5. Inverse of a Matrix by Elementary

Transformation2.6. Method to Compute the Inverse

Illustration ExamplesExercise 2

3. Rank of a Matrix3.1. Steps to Determine the Rank of a MatrixExercise 3Questions VSA and MCQ

9. SOLUTIONS OF SIMULTANEOUSLINEAR EQUATIONS 1962101. Introduction

1.1. To solve simultaneous linear equationswith the help of inverse of a matrix

1.2. Criterion of Consistency1.3. Type-I1.4. Type-II1.5. Type-III

Exercise 1Questions VSA and MCQ

10. NATIONAL INCOME MODEL 2112201. National Income Model

1.1. Solving National Income Model UsingInverse Method or Matrix Method

1.2. Partial Equilibrium Market Model1.3. Application of partial equilibrium market

modelExerciseQuestions VSA and MCQ

11. STRUCTURE OF INPUT OUTPUT TABLE221237

1. Introduction1.1. Characteristics of Input-Output

Analysis

1.2. Assumptions of Input-Output Analysis1.3. Types of Input-output Models1.4. Main Concept of Input-output Model1.5. Input-output Analysis Techniques1.6. Technological Coefficient Matrix1.7. Steps to determine Gross Level of

Output and Labour Requirements1.8. The Hawkins-Simon Conditions or

Viability Conditions of the Input-outputModelThree Sector Economy

ExerciseQuestions VSA and MCQ

12. DERIVATIVE 2382891. Introduction

1.1. Definition1.2. Another Definition1.3. Differentiation by delta methodExercise 1

2. Derivation of some standard functionsExercise 2

3. Differentiation of product of two functonsExercise 3

4. Differentiation of quotient of two functionsExercise 4

5. Differentiation of a function of a function :The chain ruleExercise 5

6. If y = un, where u is function of x, thendydx

= nun1 dudx

.Exercise 6

7. (a) If y = loga u, where u = f(x), thendydx

= 1u

loga e dudx

.

Exercise 78. (a) If y = au, where u is function of x, then

dydx

= au log a dudx

, where a is constant.

Exercise 89. Differentiation of Implicit Function

Exercise 910. Differentiation of Parametric Functions

Exercise 1011. Differentiation of a function w. r. to another

functionExercise 11

• 12. Logarithmic DifferentiationExercise 12

13. Higher DerivativesExercise 13Questions VSA and MCQ

13. PARTIAL AND TOTALDIFFERENTIATION 2903111. Function of Two Variables

1.1. Partial DerivativeExercise 1

2. Higher Order Partial Derivatives2.1. Change of order of differentiationExercise 2

3. Homogeneous Functions3.1. Linear Homogeneous functionExercise 3

4. Properties of homogeneous functions4.1. First property of homogeneous

function4.2. Second property of homogeneous

function4.3. Eulers Theorem (Property III)Exercise 4

5. Total Differential5.1. Method to Determine Total Differential5.2. Total DerivativeQuestions VSA and MCQ

14. INTEGRATION (WITH ECONOMICAPPLICATIONS) 3123601. Introduction

1.1. Constant of Integration1.2. Basic Rules of IntegrationExercise 1

2. Integration by substitutionExercise 2

3. To evaluate integrals of type ( )f x

dxax b .

Exercise 34. Definite Integral

Exercise 45. To integrate an expression which involves

linear , Method is, put linear = y.Exercise 5

6. To show that

1[ ( )]

[ ( )] ( ) , 11

nn f xf x f x dx c n

n

.Exercise 6

7. To show that ( )( )

f xdx

f x

= log | f (x) | +c.Exercise 7

8. To show that ( )

( ) ( )log

f xf x aa f x dx c

a ,

where a is constant.Exercise 8

9. Method of Partial FractionExercise 9

10. Case II. Partial FractionExercise 10

11. Case III. When the denominator containslinear repeated factorsExercise 11

12. Type IV. When denominator contains aquadratic factor of the type x2 + a.Exercise 12

13. Integration by parts13.1. TYPE I. When integral of one of the

function is not known.Exercise 13

14. Type II. The single function whose integralis not known can also be integrated byintegration by parts.Exercise 14

15. Type III. When integral of both thefunctions is known, then we take polynomialin x as first function.Exercise 15

16. Type IV.

[ ( ) ( ) ] ( )n x xf x f x e dx f x e Exercise 16

17. Application of Integration in Economics :Marginal cost. total cost.Exercise 17Questions