[Steven t karris] mathematics for business, scien(bookfi org)

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Mathematics

for Business, Science,

and Technology

With MATLAB®and Spreadsheet Applications

Steven T. Karris

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Includes a

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and Statistics Illustrated

with Numerous

Real-World Examples

SECOND

EDITION

Orchard Publications

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For example, to go to page 4 of Chapter 2, enter 2-4 in the “page #” box at the top of the

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This text includes the following chapters and appendices:

• Numbers and Arithmetic Operations • Elementary Algebra • Intermediate Algebra

• Fundamentals of Geometry • Fundamentals of Plane Trigonometry • Fundamentals of Calculus •

Mathematics of Finance and Economics • Depreciation, Impairment, and Depletion

• Introduction to Probability and Statistics • Random Variables • Common Probability Distributions

and Tests • Curve Fitting, Regression, and Correlation • Analysis of Variance (ANOVA) • Introduction

to MATLAB • The Gamma and Beta Functions and Distributions • Introduction to Markov Chains

Each chapter contains numerous practical applications supplemented with detailed instructions for

using MATLAB and Microsoft Excel obtain quick answers.

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Mathematics

for Business, Science, and Technology

With MATLAB® and Spreadsheet Applications

SECOND EDITION

Students and working professionals will find that our Mathematics for Business, Science, and

Technology, Second Edition, is a concise and easy-to-read text for a variety of basic and advanced

mathematical topics. This book contains all necessary material for the successful completion of a

degree in business or technology.

FEATURES

• There are no prerequisites for the content of this book.

• Presents a methodological approach in learning the basic mathematical concepts through various

practical examples

• Presents a unique approach to verify lengthy computations with computer software packages.

ISBN 0-9744239-0-4

Steven T. Karris is the president and founder of Orchard Publications. He earned a bachelors degree

in electrical engineering at Christian Brothers University, Memphis, Tennessee, a masters degree in

electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has done post-master

work at the latter. He is a registered professional engineer in California and Florida. He has over 30

years of professional engineering experience in industry. In addition, he has over 25 years of teaching

experience that he acquired at several educational institutions as an adjunct professor. He is currently

with UC Berkeley Extension.

$39.95 U.S.A.

Mathematics

for Business, Science, and Technology

Second Edition

With MATLAB®and Spreadsheet Applications

Steven T. Karris

Orchard Publications

www.orchardpublications.com

Mathematics for Business, Science, and Technology, Second Edition

With MATLAB® and Spreadsheet Applications

Copyright 2003 Orchard Publications. All rights reserved. Printed in the United States of America. No part of this

publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval

system, without the prior written permission of the publisher.

Direct all inquiries to Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538, U.S.A.

URL: http://www.orchardpublications.com

Product and corporate names are trademarks or registered trademarks of the MathWorks, Inc., and Microsoft

Corporation. They are used only for identification and explanation, without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Library of Congress Control Number: Pending. Contact [email protected] for updated information.

Copyright Number TX-5-471-563

ISBN 0-9744239-0-4

Disclaimer

The publisher has used his best effort to prepare this text. However, the publisher and author makes no warranty of any

kind, expressed or implied with regard to the accuracy, completeness, and computer codes contained in this book, and

shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the

performance or use of these programs.

Preface

This book is different from others of the same subject. It goes from one extreme to another; starts

with junior high math material and ends with college graduate material.

It is written for

a. high school graduates preparing to take business or science courses at community colleges or

universities

b. working professionals who feel that they need a math review from the very beginning

c. young students and working professionals who are enrolled in continued education

institutions, and majoring in business related topics, such as business administration and

accounting, and those pursuing a career in science, electronics, and computer technology.

Chapter 1 begins with basic arithmetic operations, introduces the SI system of units, and discusses

different types of graphs.

Chapter 2 is an introduction to the basics of algebra.

Chapter 3 is a continuation of Chapter 2 and presents some practical examples with systems of

two and three equations.

Chapters 4 and 5 discuss the fundamentals of geometry and trigonometry respectively. These

treatments are not exhaustive; these chapters contain basic concepts that are used in science and

technology.

Chapter 6 is an abbreviated, yet a practical introduction to calculus.

Chapters 7 and 8 are new for this edition. They serve as an introduction to the mathematics of

finance and economics and the concepts are illustrated with numerous real-world applications

and examples.

Chapters 9 through 13 are devoted to probability and statistics. Many practical examples are

given to illustrate the importance of this branch of mathematics. The topics that are discussed,

are especially important in management decisions and in reliability. Some readers may find

certain topics hard to follow; these may be skipped without loss of continuity.

In all chapters, numerous examples are given to teach the reader how to obtain quick answers to

some complicated problems using computer tools such as MATLAB®and Microsoft Excel.®

Appendix A is intended to teach the interested reader how to use MATLAB. Many practical

examples are presented. The Student Edition of MATLAB is an inexpensive software package; it

can be found in many college bookstores, or can be obtained directly from

The MathWorks™ Inc., 3 Apple Hill Drive, Natick, MA 01760-2098

Phone: 508 647-7000, Fax: 508 647-7001

http://www.mathworks.com

e-mail: [email protected]

Appendix B introduces the gamma and beta functions. These appear in the gamma and beta

distributions and find many applications in business, science, and engineering. For instance, the

Erlang distributions, which are a special case of the gamma distribution, form the basis of queuing

theory.

Appendix C is an introduction to Markov chains. A few practical examples illustrate their

application in making management decisions.

All feedback for typographical errors and comments will be most welcomed and greatly

appreciated.

New to the Second Edition

This is an refined revision of the first edition. The most notable changes are the addition of the

new Chapters 7 and 8, chapter-end summaries, and detailed solutions to all exercises. The latter is

in response to many students and working professionals who expressed a desire to obtain the

author’s solutions for comparison with their own.

The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation

of presentation material.

The last major change is the improvement of the plots generated by the latest revisions of the

MATLAB® Student Version, Release 13.

Orchard Publications

www.orchardpublications.com

[email protected]

Mathematics for Business, Science, and Technology, Second Edition i

Orchard Publications

Table of Contents

Chapter 1

Numbers and Arithmetic Operations

Number Systems.......................................................................................................................... 1-1

Positive and Negative Numbers .................................................................................................. 1-1

Addition and Subtraction ........................................................................................................... 1-2

Multiplication and Division ........................................................................................................ 1-7

Integer, Fractional, and Mixed Numbers .................................................................................. 1-10

Reciprocals of Numbers............................................................................................................. 1-11

Arithmetic Operations with Fractional Numbers..................................................................... 1-12

Exponents .................................................................................................................................. 1-21

Scientific Notation .................................................................................................................... 1-24

Operations with Numbers in Scientific Notation ..................................................................... 1-26

Square and Cubic Roots............................................................................................................ 1-28

Common and Natural Logarithms ............................................................................................ 1-30

Decibel....................................................................................................................................... 1-32

Percentages................................................................................................................................ 1-32

International System of Units (SI) ............................................................................................ 1-33

Graphs ....................................................................................................................................... 1-37

Summary.................................................................................................................................... 1-41

Exercises .................................................................................................................................... 1-46

Solutions to Exercises................................................................................................................ 1-47

Chapter 2

Elementary Algebra

Introduction................................................................................................................................. 2-1

Algebraic Equations .................................................................................................................... 2-2

Laws of Exponents....................................................................................................................... 2-5

Laws of Logarithms...................................................................................................................... 2-8

Quadratic Equations.................................................................................................................. 2-11

Cubic and Higher Degree Equations......................................................................................... 2-13

Measures of Central Tendency ................................................................................................. 2-13

Interpolation and Extrapolation................................................................................................ 2-15

Infinite Sequences and Series.................................................................................................... 2-18

Arithmetic Series....................................................................................................................... 2-19

Geometric Series........................................................................................................................ 2-19

Harmonic Series ........................................................................................................................ 2-21

ii Mathematics for Business, Science, and Technology, Second Edition

Orchard Publications

Proportions .................................................................................................................................2-23

Summary.....................................................................................................................................2-24

Exercises .....................................................................................................................................2-28

Solutions to Exercises.................................................................................................................2-30

Chapter 3

Intermediate Algebra

Systems of Two Equations............................................................................................................3-1

Systems of Three Equations .........................................................................................................3-6

Matrices and Simultaneous Solution of Equations ......................................................................3-6

Summary.....................................................................................................................................3-25

Exercises .....................................................................................................................................3-29

Solutions to Exercises.................................................................................................................3-31

Chapter 4

Fundamentals of Geometry

Introduction .................................................................................................................................4-1

Plane Geometry Figures................................................................................................................4-1

Solid Geometry Figures ..............................................................................................................4-17

Using Spreadsheets to Find Areas of Irregular Polygons ...........................................................4-21

Summary.....................................................................................................................................4-24

Exercises .....................................................................................................................................4-29

Solutions to Exercises.................................................................................................................4-31

Chapter 5

Fundamentals of Plane Trigonometry

Introduction .................................................................................................................................5-1

Trigonometric Functions..............................................................................................................5-2

Trigonometric Functions of an Acute Angle...............................................................................5-2

Trigonometric Functions of an Any Angle..................................................................................5-3

Fundamental Relations and Identities .........................................................................................5-6

Triangle Formulas.......................................................................................................................5-12

Inverse Trigonometric Functions...............................................................................................5-14

Area of Polygons in Terms of Trigonometric Functions............................................................5-14

Summary.....................................................................................................................................5-16

Exercises .....................................................................................................................................5-18

Solutions to Exercises.................................................................................................................5-19

Mathematics for Business, Science, and Technology, Second Edition iii

Orchard Publications

Chapter 6

Fundamentals of Calculus

Introduction................................................................................................................................. 6-1

Differential Calculus.................................................................................................................... 6-1

The Derivative of a Function...................................................................................................... 6-3

Maxima and Minima ................................................................................................................. 6-11

Integral Calculus........................................................................................................................ 6-15

Indefinite Integrals .................................................................................................................... 6-16

Definite Integrals....................................................................................................................... 6-16

Summary.................................................................................................................................... 6-21

Exercises .................................................................................................................................... 6-23

Solutions to Exercises................................................................................................................ 6-24

Chapter 7

Mathematics of Finance and Economics

Common Terms........................................................................................................................... 7-1

Interest......................................................................................................................................... 7-6

Sinking Funds............................................................................................................................ 7-23

Annuities ................................................................................................................................... 7-28

Amortization.............................................................................................................................. 7-33

Perpetuities ................................................................................................................................ 7-36

Valuation of Bonds.................................................................................................................... 7-37

Spreadsheet Financial Functions .............................................................................................. 7-44

The MATLAB Financial Toolbox ............................................................................................ 7-58

Comparison of Alternate Proposals........................................................................................... 7-65

Kelvin’s Law .............................................................................................................................. 7-68

Summary.................................................................................................................................... 7-72

Exercises .................................................................................................................................... 7-75

Solutions to Exercises................................................................................................................ 7-78

Chapter 8

Depreciation, Impairment, and Depletion

Depreciation Defined .................................................................................................................. 8-1

Items that Can Be Depreciated................................................................................................... 8-2

Items that Cannot Be Depreciated ............................................................................................. 8-2

Depreciation Rules ...................................................................................................................... 8-2

When Depreciation Begins and Ends ......................................................................................... 8-3

Methods of Depreciation............................................................................................................. 8-3

iv Mathematics for Business, Science, and Technology, Second Edition

Orchard Publications

Straight-Line (SL) Depreciation Method .................................................................................... 8-4

Sum of the Years Digits (SYD) Method....................................................................................... 8-4

Fixed-Declining Balance (FDB) Method..................................................................................... 8-6

The 125%, 150%, and 200% General Declining Balance (GDB) Methods................................8-8

The Variable Declining Balance (VDB) method.........................................................................8-9

The Units of Production (UOP) method...................................................................................8-10

Depreciation Methods for Income Tax Reporting.....................................................................8-11

The Accelerated Cost Recovery System (ACRS) .....................................................................8-12

The Modified Accelerated Cost Recovery System (MACRS) ..................................................8-12

Section 179.................................................................................................................................8-16

Impairments................................................................................................................................ 8-18

Depletion....................................................................................................................................8-19

Valuation of a Depleting Asset ..................................................................................................8-20

Summary.....................................................................................................................................8-25

Exercises .....................................................................................................................................8-27

Solutions to Exercises.................................................................................................................8-28

Chapter 9

Introduction to Probability and Statistics

Introduction .................................................................................................................................9-1

Probability and Random Experiments.......................................................................................... 9-1

Relative Frequency.......................................................................................................................9-2

Combinations and Permutations.................................................................................................. 9-4

Joint and Conditional Probabilities ..............................................................................................9-7

Bayes’ Rule..................................................................................................................................9-10

Summary.....................................................................................................................................9-12

Exercises .....................................................................................................................................9-14

Solutions to Exercises.................................................................................................................9-15

Chapter 10

Random Variables

Definition of Random Variables................................................................................................. 10-1

Probability Function...................................................................................................................10-2

Cumulative Distribution Function.............................................................................................10-2

Probability Density Function......................................................................................................10-9

Two Random Variables............................................................................................................10-11

Statistical Averages .................................................................................................................. 10-12

Summary...................................................................................................................................10-19

Exercises ...................................................................................................................................10-22

Solutions to Exercises...............................................................................................................10-24

Mathematics for Business, Science, and Technology, Second Edition v

Orchard Publications

Chapter 11

Common Probability Distributions and Tests

Properties of Binomial Coefficients ........................................................................................... 11-1

The Binomial (Bernoulli) Distribution ..................................................................................... 11-2

The Uniform Distribution ......................................................................................................... 11-6

The Exponential Distribution ................................................................................................. 11-10

The Normal (Gaussian) Distribution...................................................................................... 11-13

Percentiles ............................................................................................................................... 11-32

The Student’s t-Distribution ................................................................................................... 11-36

The Chi-Square Distribution .................................................................................................. 11-41

The F Distribution................................................................................................................... 11-44

Chebyshev’s Inequality............................................................................................................ 11-46

Law of Large Numbers............................................................................................................. 11-47

The Poisson Distribution......................................................................................................... 11-47

The Multinomial Distribution................................................................................................. 11-52

The Hypergeometric Distribution........................................................................................... 11-53

The Bivariate Normal Distribution ......................................................................................... 11-56

The Rayleigh Distribution....................................................................................................... 11-57

Other Probability Distributions............................................................................................... 11-59

Sampling Distribution of Means.............................................................................................. 11-63

Z-Score .................................................................................................................................... 11-64

Tests of Hypotheses and Levels of Significance...................................................................... 11-65

The z, t, F, and tests ........................................................................................................... 11-72

Summary.................................................................................................................................. 11-78

Exercises .................................................................................................................................. 11-87

Solutions to Exercises.............................................................................................................. 11-89

Chapter 12

Curve Fitting, Regression, and Correlation

Curve Fitting ............................................................................................................................. 12-1

Linear Regression ...................................................................................................................... 12-2

Parabolic Regression.................................................................................................................. 12-7

Covariance............................................................................................................................... 12-10

Correlation Coefficient............................................................................................................ 12-12

Summary.................................................................................................................................. 12-17

Exercises .................................................................................................................................. 12-19

Solutions to Exercises.............................................................................................................. 12-21

F

2

vi Mathematics for Business, Science, and Technology, Second Edition

Orchard Publications

Chapter 13

Analysis of Variance (ANOVA)

Introduction ............................................................................................................................... 13-1

One-way ANOVA .....................................................................................................................13-1

Two-way ANOVA .....................................................................................................................13-8

Two-factor without Replication ANOVA.................................................................................13-8

Two-factor with Replication ANOVA .................................................................................... 13-14

Summary...................................................................................................................................13-25

Exercises ...................................................................................................................................13-29

Solutions to Exercises...............................................................................................................13-31

Appendix A

Introduction to MATLAB®

MATLAB® and Simulink®.......................................................................................................A-1

Command Window .....................................................................................................................A-1

Roots of Polynomials ...................................................................................................................A-3

Polynomial Construction from Known Roots .............................................................................A-4

Evaluation of a Polynomial at Specified Values..........................................................................A-6

Rational Polynomials...................................................................................................................A-8

Using MATLAB to Make Plots ................................................................................................A-10

Subplots .....................................................................................................................................A-19

Multiplication, Division and Exponentiation ...........................................................................A-19

Script and Function Files ..........................................................................................................A-26

Display Formats .........................................................................................................................A-31

Appendix B

The Gamma and Beta Functions and Distributions

The Gamma Function ..................................................................................................................B-1

The Gamma Distribution ...........................................................................................................B-15

The Beta Function .....................................................................................................................B-17

The Beta Distribution ...............................................................................................................B-20

Appendix C

Introduction to Markov Chains

Stochastic Processes .................................................................................................................... C-1

Stochastic Matrices ..................................................................................................................... C-1

Transition Diagrams.................................................................................................................... C-4

Regular Stochastic Matrices........................................................................................................ C-5

Some Practical Examples............................................................................................................. C-7

Mathematics for Business, Science, and Technology, Second Edition 1-1

Orchard Publications

Chapter 1

Numbers and Arithmetic Operations

his chapter is a review of the basic arithmetic concepts. It is intended for readers feeling

that they need a math review from the very beginning. It forms the basis for understanding

and working with relations (formulas) encountered in business, science and technology.

Readers with a fair mathematical background may skip this chapter. Others may find it useful as

well as a convenient source for review.

1.1 Number Systems

The decimal (base 10) number system uses the digits (numbers) 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This

is the number system we use in our everyday arithmetic calculations such as the monetary trans￾actions. Another number system is the binary (base 2) that uses the digits 0 and 1 only. The binary

system is used in computers and it is being taught in electronics courses. We will not be concerned

with the binary system in this text.

1.2 Positive and Negative Numbers

A positive number is a number greater than zero and it is understood to have a plus (+) sign in

front of it. The (+) sign in front of a positive number is generally omitted. Thus, any number

without a sign in front of it is understood to be a positive number. A negative number is less than

zero and it is written with a minus (–) sign*

in front of it. The minus () sign in front of a negative

number is a must; otherwise it would not be possible to distinguish the negative from the positive

numbers. Positive and negative numbers can be whole (integer) or fractional numbers. Several

examples will be presented in this chapter to illustrate their designation, how they are added, sub￾tracted, multiplied, and divided with other numbers. To avoid confusion between the addition

operation (+) and positive numbers, which are also denoted with the (+) sign, we will enclose

positive numbers with their sign inside parentheses whenever necessary. Likewise, we will enclose

negative numbers in parentheses to distinguish them from the subtraction () symbol. This will be

illustrated with the examples that follow.

Example 1.1

Joe Smith’s checking account shows a balance of $534.29. Thus, we can say that his balance is

+534.39 dollars but we normally omit the plus (+) sign, and we say that his balance is 534.39

dollars.

* The financial community, such as banks, usually enclose a negative number in parentheses without the minus

sign. Most often, this designation appears in financial statements.

T

Chapter 1 Numbers and Arithmetic Operations

1-2 Mathematics for Business, Science, and Technology, Second Edition

Orchard Publications

Example 1.2

Bill Jones, unaware that his checking account has a balance of only $78.31, makes a purchase of

$128.74. He pays this amount with a check. His new account balance is now dollars.

Here, the minus () sign is a must.

The absolute value of a number is that number without a positive or negative sign, and is enclosed

in small vertical lines. For example, the absolute value of X is written as |X|. The number 0 (zero)

is considered neither positive nor negative; it is the number that separates the negative from the

positive numbers. The positive and negative numbers that we are familiar with, are referred to as

the real numbers*

and are shown below on the so-called real axis of numbers**.

Figure 1.1. Representation of Real Numbers

In our subsequent discussion, we will only be concerned with real numbers and thus the word real

will not be used further.

1.3 Addition and Subtraction

The following rules apply for the addition of numbers.

Rule 1: To add numbers with the same sign, we add the absolute values of these numbers and

place the common sign (+ or –) in front of the result (sum). We can omit the plus sign in

the result if positive. We must not omit the minus sign if the result is negative.

Example 1.3

Perform the addition

Solution:

The plus sign between the given numbers indicates addition of three positive numbers whose sign

is positive and it is omitted. However, we can enclose these numbers in parentheses just to

emphasize that the numbers are positive. Addition of the absolute values of these numbers yield a

* The reader may have heard the expression “imaginary numbers”. The square root of minus 1, i.e, , is an

example of an imaginary number; it does not fit anywhere in the real axis of numbers. We will not be concerned

with these numbers in this text. There is a brief discussion in Appendix A in conjunction with MATLAB.

** Only whole numbers are shown on the real axis of Figure 1.1. However, it is understood that within each divi￾sion, there are numbers such as 1.5, 2.75 etc.

–128.74

–1

-6 -5 -4 -3 -2 -1 0 123 456

Real Axis

7 16 0.5 + +