Applied Mathematics for Business and Economics

[Pages:87]Applied Mathematics for

Business and Economics

Norton University Year 2010

Lecture Note

Applied Mathematics for Business and Economics

Contents

Page Chapter 1 Functions 1 Definition of a Function (of one variable) .........................................................1

1.1 Definition ..................................................................................................1 1.2 Domain of a Function ...............................................................................1 1.3 Composition of Functions.........................................................................2

2 The Graph of a Function ....................................................................................3

3 Linear Functions ................................................................................................5 3.1 The Slope of a Line...................................................................................5 3.2 Horizontal and Vertical Lines ...................................................................6 3.3 The Slope-Intercept Form .........................................................................6 3.4 The Point-Slope Form...............................................................................6

4 Functional Models .............................................................................................8 4.1 A Profit Function ......................................................................................8 4.2 Functions Involving Multiple Formulas ...................................................8 4.3 Break-Even Analysis ................................................................................9 4.4 Market Equilibrium.................................................................................11

Chapter Exercises...................................................................................................12

Chapter 2 Differentiation: Basic Concepts 1 The Derivative

Definition .........................................................................................................19

2 Techniques of Differentiation ..........................................................................20 2.1 The Power Rule.......................................................................................20 2.2 The Derivative of a constant ...................................................................21 2.3 The Constant Multiple Rule....................................................................21 2.4 The Sum Rule .........................................................................................21 2.5 The Product Rule ....................................................................................21 2.6 The Derivative of a Quotient ..................................................................21

3 The Derivative as a Rate of change .................................................................22 3.1 Average and Instantaneous Rate of Change ...........................................22 3.2 Percentage Rate of Change .....................................................................23

4 Approximation by Differentials; Marginal Analysis .......................................23 4.1 Approximation of Percentage change .....................................................24 4.2 Marginal Analysis in Economics ............................................................25 4.3 Differentials ............................................................................................27

5 The Chain Rule ................................................................................................27

6 Higher-Order Derivatives ................................................................................29 6.1 The Second Derivative............................................................................29 6.2 The nth Derivative ...................................................................................30

7 Concavity and the Second Derivative Test ......................................................30

8 Applications to Business and Economics ........................................................34 8.1 Elasticity of Demand.....................................................................................34 8.2 Levels of Elasicity of Demand................................................................36 8.3 Elasticity and the Total Revenue ............................................................36

Chapter Exercises...................................................................................................38

Chapter 3 Functions of Two Variables 1 Functions of Two Variables.............................................................................49

2 Partial Derivatives............................................................................................50 2.1 Computation of Partial Derivatives ........................................................50 2.2 Second-Order Partial Derivatives ...........................................................52

3 The Chain Rule; Approximation by the Total Differential..............................53 3.1 Chain Rule for Partial Derivatives ..........................................................53 3.2 The Total differential ..............................................................................55 3.3 Approximation of Percentage Change ....................................................56

4 Relative Maxima and Minima .........................................................................56

5 Lagrange Multipliers........................................................................................59 5.1 Contrained Optimization Problems.........................................................59 5.2 The Lagrange Multiplier .........................................................................61

Chapter Exercises...................................................................................................62

Chapter 4 Linear Programming (LP) 1 System of Linear Inequalities in Two Variables..............................................72

1.1 Graphing a Linear Inequality in Two Variables .....................................72 1.2 Solving Systems of Linear Inequalities ..................................................73

2 Geometric Linear Programming ......................................................................74

Chapter Exercises....................................................................................................... 77

Bibliography ............................................................................................................. 81

This page is intentionally left blank.

Lecture Note

Function

Chapter 1

Functions

1 Definition of a Function 1.1 Definition

Let D and R be two sets of real numbers. A function f is a rule that matches each

number x in D with exactly one and only one number y or f ( x) in R . D is called the

domain of f and R is called the range of f . The letter x is sometimes referred to as independent variable and y dependent variable.

Examples 1:

Let f (x) = x3 - 2x2 + 3x + 100 . Find f (2) .

Solution:

f (2) =23 - 2 ? 22 + 3? 2 +100 = 106

Examples 2

A real estate broker charges a commission of 6% on Sales valued up to $300,000. For

sales valued at more than $ 300,000, the commission is $ 6,000 plus 4% of the sales

price.

a. Represent the commission earned as a function R.

b. Find R (200,000).

c. Find R (500,000).

Solution

a.

R

(

x)

=

0.06x 0.04x

+

6000

for 0 x 300, 000 for x > 300, 000

b. Use R ( x) = 0.06x since 200, 000 < 300, 000

R (200, 000) = 0.06 ? 200, 000 = $12, 000

c. Use R ( x) = 0.04x + 6000 since 500, 000 > 300, 000

R (500, 000) = 0.04 ? 500, 000 + 6000 = $26, 000

1.2 Domain of a Function

The set of values of the independent variables for which a function can be evaluated is

called the domain of the function.

D = {x \ / y \, y = f ( x)}

Example 3 Find the domain of each of the following functions:

a. f ( x) = 1 , b. g ( x) = x - 2

x-3 Solution

a. Since division by any real number except zero is possible, the only value of x

for which f ( x) = 1 cannot be evaluated is x = 3 , the value that makes the

x-3

denominator of f equal to zero, or D = \ - {3} .

1

Lecture Note

Function

b. Since negative numbers do not have real square roots, the only values of x for

which g ( x) = x - 2 can be evaluated are those for which x - 2 is nonnegative, that is, for which x - 2 0 or x 2 or D = [2, + ) .

1.3 Composition of Functions

The composite function g h ( x) is the function formed from the two functions g (u ) and h ( x) by substituting h ( x) for u in the formula for g (u ) .

Example 4

Find the composite function g h ( x) if g (u ) = u2 + 3u +1and h ( x) = x +1.

Solution Replace u by x+1 in the formula for g to get.

g h( x) = ( x +1)2 + 3( x +1) +1 = x2 + 5x + 5

Example 5 An environmental study of a certain community suggests that the average daily level

of carbon monoxide in the air will be C ( p ) = 0.5 p +1 parts per million when the

population is p thousand. It is estimated that t years from now the population of the

community will be P (t ) = 10 + 0.1t2 thousand.

a. Express the level of carbon monoxide in the air as a function of time. b. When will the carbon monoxide level reach 6.8 parts per million?

Solution a. Since the level of carbon monoxide is related to the variable p by the equation.

C ( p) = 0.5 p +1

and the variable p is related to the variable t by the equation.

P (t ) = 10 + 0.1t2

It follows that the composite function

( ) ( ) C P (t ) = C 10 + 0.1t2 = 0.5 10 + 0.1t2 +1 = 6 + 0.05t2

expresses the level of carbon monoxide in the air as a function of the variable t.

b. Set C P (t ) equal to 6.8 and solve for t to get

6 + 0.05t2 = 6.8 0.05t2 = 0.8 t2 = 16 t=4

That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.

2

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download