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

1.5 Integer, Fractional, and Mixed Numbers A decimal point is the dot in a number that is written in decimal form.. For instance, where and are fractional numbers expressed in rational f

M T Mathematics for Business, Science,

and Technology

With MATLAB®and Spreadsheet Applications

Steven T Karris

x y

10 0 20 0 30 0 4 00 50 0 2,0 00

SECOND EDITION

Orchard Publications

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In this eBook, each chapter or section has its own page numbering scheme, made up of an identifier and page number, separated by a hyphen

For example, to go to page 4 of Chapter 2, enter 2-4 in the “page #” box at the top of the screen and click “Go” To go to page 4 of Appendix A, enter A-4, and so forth

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.

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.

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.

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 info@orchardpublications.com 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.

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 educationinstitutions, and majoring in business related topics, such as business administration andaccounting, 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 discussesdifferent 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 oftwo and three equations

Chapters 4 and 5 discuss the fundamentals of geometry and trigonometry respectively Thesetreatments are not exhaustive; these chapters contain basic concepts that are used in science andtechnology

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 offinance and economics and the concepts are illustrated with numerous real-world applicationsand examples

Chapters 9 through 13 are devoted to probability and statistics Many practical examples aregiven 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 findcertain 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 tosome 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 practicalexamples are presented The Student Edition of MATLAB is an inexpensive software package; itcan be found in many college bookstores, or can be obtained directly from

e-mail: info@mathwork.com

Appendix B introduces the gamma and beta functions These appear in the gamma and betadistributions and find many applications in business, science, and engineering For instance, theErlang distributions, which are a special case of the gamma distribution, form the basis of queuingtheory

Appendix C is an introduction to Markov chains A few practical examples illustrate theirapplication in making management decisions

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

New to the Second Edition

This is an refined revision of the first edition The most notable changes are the addition of thenew 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 theauthor’s solutions for comparison with their own

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

Table of Contents

Chapter 1

Numbers and Arithmetic Operations

Number Systems 1-1Positive and Negative Numbers 1-1Addition and Subtraction 1-2Multiplication and Division 1-7Integer, Fractional, and Mixed Numbers 1-10Reciprocals of Numbers 1-11Arithmetic Operations with Fractional Numbers 1-12Exponents 1-21Scientific Notation 1-24Operations with Numbers in Scientific Notation 1-26Square and Cubic Roots 1-28Common and Natural Logarithms 1-30Decibel 1-32Percentages 1-32International System of Units (SI) 1-33Graphs 1-37Summary 1-41Exercises 1-46Solutions to Exercises 1-47

Chapter 2

Elementary Algebra

Introduction 2-1Algebraic Equations 2-2Laws of Exponents 2-5Laws of Logarithms 2-8Quadratic Equations 2-11Cubic and Higher Degree Equations 2-13Measures of Central Tendency 2-13Interpolation and Extrapolation 2-15Infinite Sequences and Series 2-18Arithmetic Series 2-19Geometric Series 2-19Harmonic Series 2-21

Exercises 2-28Solutions to Exercises 2-30

Chapter 3

Intermediate Algebra

Systems of Two Equations 3-1Systems of Three Equations 3-6Matrices and Simultaneous Solution of Equations 3-6Summary 3-25Exercises 3-29Solutions to Exercises 3-31

Chapter 4

Fundamentals of Geometry

Introduction 4-1Plane Geometry Figures 4-1Solid Geometry Figures 4-17Using Spreadsheets to Find Areas of Irregular Polygons 4-21Summary 4-24Exercises 4-29Solutions to Exercises 4-31

Chapter 5

Fundamentals of Plane Trigonometry

Introduction 5-1Trigonometric Functions 5-2Trigonometric Functions of an Acute Angle 5-2Trigonometric Functions of an Any Angle 5-3Fundamental Relations and Identities 5-6Triangle Formulas 5-12Inverse Trigonometric Functions 5-14Area of Polygons in Terms of Trigonometric Functions 5-14Summary 5-16Exercises 5-18Solutions to Exercises 5-19

Chapter 6

Fundamentals of Calculus

Introduction 6-1Differential Calculus 6-1The Derivative of a Function 6-3Maxima and Minima 6-11Integral Calculus 6-15Indefinite Integrals 6-16Definite Integrals 6-16Summary 6-21Exercises 6-23Solutions to Exercises 6-24

Chapter 7

Mathematics of Finance and Economics

Common Terms 7-1Interest 7-6Sinking Funds 7-23Annuities 7-28Amortization 7-33Perpetuities 7-36Valuation of Bonds 7-37Spreadsheet Financial Functions 7-44The MATLAB Financial Toolbox 7-58Comparison of Alternate Proposals 7-65Kelvin’s Law 7-68Summary 7-72Exercises 7-75Solutions to Exercises 7-78

Chapter 8

Depreciation, Impairment, and Depletion

Depreciation Defined 8-1Items that Can Be Depreciated 8-2Items that Cannot Be Depreciated 8-2Depreciation Rules 8-2When Depreciation Begins and Ends 8-3Methods of Depreciation 8-3

Fixed-Declining Balance (FDB) Method 8-6The 125%, 150%, and 200% General Declining Balance (GDB) Methods 8-8The Variable Declining Balance (VDB) method 8-9The Units of Production (UOP) method 8-10Depreciation Methods for Income Tax Reporting 8-11The Accelerated Cost Recovery System (ACRS) 8-12The Modified Accelerated Cost Recovery System (MACRS) 8-12Section 179 8-16Impairments 8-18Depletion 8-19Valuation of a Depleting Asset 8-20Summary 8-25Exercises 8-27Solutions to Exercises 8-28

Chapter 9

Introduction to Probability and Statistics

Introduction 9-1Probability and Random Experiments 9-1Relative Frequency 9-2Combinations and Permutations 9-4Joint and Conditional Probabilities 9-7Bayes’ Rule 9-10Summary 9-12Exercises 9-14Solutions to Exercises 9-15

Chapter 10

Random Variables

Definition of Random Variables 10-1Probability Function 10-2Cumulative Distribution Function 10-2Probability Density Function 10-9Two Random Variables 10-11Statistical Averages 10-12Summary 10-19Exercises 10-22Solutions to Exercises 10-24

Chapter 11

Common Probability Distributions and Tests

Properties of Binomial Coefficients 11-1The Binomial (Bernoulli) Distribution 11-2The Uniform Distribution 11-6The Exponential Distribution 11-10The Normal (Gaussian) Distribution 11-13Percentiles 11-32The Student’s t-Distribution 11-36The Chi-Square Distribution 11-41The F Distribution 11-44Chebyshev’s Inequality 11-46Law of Large Numbers 11-47The Poisson Distribution 11-47The Multinomial Distribution 11-52The Hypergeometric Distribution 11-53The Bivariate Normal Distribution 11-56The Rayleigh Distribution 11-57Other Probability Distributions 11-59Sampling Distribution of Means 11-63Z-Score 11-64Tests of Hypotheses and Levels of Significance 11-65The z, t, F, and tests 11-72Summary 11-78Exercises 11-87Solutions to Exercises 11-89

Chapter 12

Curve Fitting, Regression, and Correlation

Curve Fitting 12-1Linear Regression 12-2Parabolic Regression 12-7Covariance 12-10Correlation Coefficient 12-12Summary 12-17Exercises 12-19Solutions to Exercises 12-21

F2

Analysis of Variance (ANOVA)

Introduction 13-1One-way ANOVA 13-1Two-way ANOVA 13-8Two-factor without Replication ANOVA 13-8Two-factor with Replication ANOVA 13-14Summary 13-25Exercises 13-29Solutions to Exercises 13-31

Appendix A

Introduction to MATLAB®

MATLAB® and Simulink® A-1Command Window A-1Roots of Polynomials A-3Polynomial Construction from Known Roots A-4Evaluation of a Polynomial at Specified Values A-6Rational Polynomials A-8Using MATLAB to Make Plots A-10Subplots A-19Multiplication, Division and Exponentiation A-19Script and Function Files A-26Display Formats A-31

Appendix B

The Gamma and Beta Functions and Distributions

The Gamma Function B-1The Gamma Distribution B-15The Beta Function B-17The Beta Distribution B-20

Appendix C

Introduction to Markov Chains

Stochastic Processes C-1Stochastic Matrices C-1Transition Diagrams C-4Regular Stochastic Matrices C-5Some Practical Examples C-7

Chapter 1

Numbers and Arithmetic Operations

his chapter is a review of the basic arithmetic concepts It is intended for readers feelingthat they need a math review from the very beginning It forms the basis for understandingand 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 aswell 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 concernedwith 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 negativenumber is a must; otherwise it would not be possible to distinguish the negative from the positivenumbers Positive and negative numbers can be whole (integer) or fractional numbers Severalexamples 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 additionoperation (+) and positive numbers, which are also denoted with the (+) sign, we will enclosepositive numbers with their sign inside parentheses whenever necessary Likewise, we will enclosenegative numbers in parentheses to distinguish them from the subtraction () symbol This will beillustrated 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

Example 1.2

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

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 thepositive 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 toemphasize 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 sion, there are numbers such as 1.5, 2.75 etc.

Addition and Subtraction

sum of 23.5, and this also represents an absolute value We should remember that the absolute

value of a number is that just that number without regard to being positive or negative Now,

since all three numbers are positive, the sum is +23.5 or simply 23.5 as shown below The final result, 23.5, does not represent an absolute number; it is a positive number whose sign has been

omitted, as it is customary Thus,

where the symbol Ÿ means conversion from signed numbers to absolute values and vice versa Ofcourse, these steps will be unnecessary after one becomes familiar with the rules

Example 1.5

Perform the addition

Solution:

In this example, we are asked to add three negative numbers The sum of the absolute values is 5.

Since all given numbers are negative, we place the minus sign in front of the sum The result then

is (5) or simply 5 The negative sign cannot be omitted Thus,

Consider the subtraction of number B from number A, that is, A–B The number A is called the

minuend and the number B is called the subtrahend The result of the subtraction is called the ference These definitions are illustrated with Examples 1.6 and 1.7 below.

1.25

– + –0.75 + –3

1.25

– + –0.75 + –3 Ÿ 1.25 + 0.75 + 3 = 5 Ÿ –5 = –5

find-Rule 2: To add two numbers with different signs, we subtract the number with the smaller

abso-lute value from the number with the larger absoabso-lute value, and we place the sign of the larger number in front of the result (sum).

248 857– = –609

Difference Subtrahend Minuend

37+ –15

37+ –15 = +37 ...

multiply the numerators to obtain the numerator of the product Then, we multiply the denominators to obtain the denominator of the product If the numbers are in decimal point form, we multiply the... obtain the numerator of the quotient; then we multiply the denominator of the divi-dend by the numerator of the divisor to get the denominator of the quotient in accordance withRule 12 Then,

a... class="text_page_counter">Trang 25

tomary to place the minus sign in front of the bar that separates them For instance,

We must not forget