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´╗┐Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics

A Semester Course in Finite Mathematics for Business and Economics

Marcel B. Finan c All Rights Reserved

August 10, 2012

1

Contents

Preface

4

Mathematics of Finance

5

1. Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Discrete and Continuous Compound Interest . . . . . . . . . . . 12

3. Ordinay Annuity, Future Value and Sinking Fund . . . . . . . . 19

4. Present Value of an Ordinay Annuity and Amortization . . . . . 26

Matrices and Systems of Linear Equations

34

5. Solving Linear Systems Using Augmented Matrices . . . . . . . . 34

6. Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . 42

7. The Algebra of Matrices . . . . . . . . . . . . . . . . . . . . . . 53

8. Inverse Matrices and their Applications to Linear Systems . . . . 62

Linear Programming

69

9. Solving Systems of Linear Inequalities . . . . . . . . . . . . . . . 69

10. Geometric Method for Solving Linear Programming Problems . 77

11. Simplex Method for Solving Linear Programming Problems . . 86

12. The Dual Problem: Minimization with Constraints . . . . . . 97

Counting Principles, Permuations, and Combinations

106

13. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

14. Counting Principles . . . . . . . . . . . . . . . . . . . . . . . . 117

15. Permutations and Combinations . . . . . . . . . . . . . . . . . 123

Probability

129

16. Sample Spaces, Events, and Probability . . . . . . . . . . . . . 129

17. Probability of Unions and Intersections; Odds . . . . . . . . . . 141

18. Conditional Probability and Independent Events . . . . . . . . 148

19. Conditional Probability and Bayes' Formula . . . . . . . . . . . 154

20. Random Variable, Probabiltiy Distribution, and Expected Value 160

Statistics

168

21. Graphical Representations of Data . . . . . . . . . . . . . . . . 168

22. Measures of Central Tendency . . . . . . . . . . . . . . . . . . . 183

23. Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . 196

24. Binomial Distributions . . . . . . . . . . . . . . . . . . . . . . . 205

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25. Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . 217

Answer Key

236

Index

276

3

Preface

This book is a collection of lecture notes for a freshmen level course in mathematics designated for students in Business, Economics, Life Sciences and Social Sciences. The content is suitable for a one semester course. A college algebra background is required for this course. Marcel B. Finan Russellville, Arkansas

4

Mathematics of Finance

1. Simple Interest

Interest is a change of value of money. For example, when you deposit money into a savings account, the interest will increase your money based on the interest rate paid by your bank. In contrast, when you get a loan, the interest will increase the amount you owe based upon the interest rate charged by your bank. We can look at interest as the fee for using money. There are two types of interest: Simple interest and compound interest. In this section we discuss the former one and postpone the discussion of the later to the next section. Interest problems generally involve four quantities: principal(s), investment period length(s), interest rate(s), amount value(s). The money invested in financial transactions will be referred to as the principal or the present value, denoted by P. The amount it has grown to will be called the amount value or the future value and will be denoted by A. The difference I = A - P is the amount of interest earned during the period of investment. Interest expressed as a percent of the principal will be referred to as an interest rate. The unit in which time of investment is measured is called the measurement period. The most common measurement period is one year but may be longer or shorter (could be days, months, years, decades, etc.) Let r denote the annual fee per $100 and P denote the principal. We call r the simple interest rate. Thus, if one deposits P = $500 in a savings account that pays r = 5% interest then the interest earned for the two years period will be 500 ? 0.05 ? 2 = $50. In general, if I denotes the interest earned on an investment of P at the annual interest rate r for a period of t years then

I = P rt

and the amount of the investment at the end of t years is

A(t) = principal + interest = P (1 + rt).

Example 1.1 John borrows $1,500 from a bank that charges 10% annual simple interest rate. He plans to make weekly payments for two years to repay the loan. (a) Find the total interest paid for this loan.

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