M T H A Mathematics for Business, Science, and Technology With MATLAB®and Spreadsheet Applications Steven T. Karris x y 10 0 20 0 30 0 400 500 2,0 00 4,0 00 6,000 8,0 00 10,000 0 Unit s S old 12 ,00 0 14,000 Revenue Cost Break-Even Point Profit $ Includes a Comprehensive Treatment of Probability and Statistics Illustrated with Numerous Real-World Examples SECOND EDITION Orchard Publications www.orchardpublications.com How to go to your page: 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. Orchard Publications Visit us on the Internet www.orchardpublications.com or email us: info@orchardpublications.com 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 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. 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: info@mathwork.com 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 info@orchardpublications.com 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 [...]... rational form, by another number (divisor) of the same form, we multiply the numerator of the dividend by the denominator of the divisor to obtain the numerator of the quotient Then, we multiply the denominator of the dividend by the numerator of the divisor to obtain the denominator of the quotient If the numbers are in decimal point form, we first express them in rational form and convert to integers... follow the same steps as Example in 1.28, except that we perform subtraction instead of addition Then, 0.75 – 0.005 0.750 – 0.005 0.745 Rule 11: To multiply a number given in a rational form by another number of the same form, we 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,... is to illustrate how the sign of the result is obtained The same is true for the examples that follow Example 1.15 Perform the multiplication 53 – 18 Mathematics for Business, Science, and Technology, Second Edition Orchard Publications 1-7 Chapter 1 Numbers and Arithmetic Operations Solution: For this example, the multiplier is positive and the multiplicand is negative Since they have different signs,... same form, we first express the numbers in a common denominator; then, we subtract the numerator of the subtrahend from the numerator of the minuend to obtain the numerator of the difference The denominator of the difference is the common denominator If the numbers are in decimal point form, we first align the given numbers with the decimal point; then, we subtract the subtrahend from the minuend as it... 1.13 Perform the subtraction – 75 – – 125 Solution: Here, both the minuend and subtrahend are negative, and the minus sign between them indicates subtraction Therefore, we change the sign of the subtrahend from minus to plus, and at the same time we change the subtraction operation to addition Next, we must add these numbers which now have different signs For this addition we follow Rule 2, that is,... are positive numbers The minus sign between them indicates subtraction Therefore, we enclose them in parentheses with the positive (+) sign, and we change the sign of the subtrahend from plus to minus, while at the same time, we change the subtraction operation to addition Next, we need to add these numbers, one of which is positive and the other negative For this addition we follow Rule 2, that is,... 18480 This example required a considerable amount of work This was the procedure one would follow before the advent of scientific calculators and personal computers Let us now verify the result with Excel directly, that is, without first computing the LCM Start with a blank worksheet Click on the letter A of Column A and observe that the entire column is now highlighted, that is, the background on this... these steps when performing arithmetic operations Mathematics for Business, Science, and Technology, Second Edition Orchard Publications 1-5 Chapter 1 Numbers and Arithmetic Operations Example 1.12 Perform the subtraction – 86 – 37 Solution: Here, the minuend is negative, the subtrahend is positive, and the minus sign between them indicates subtraction Therefore, we enclose 37 in parentheses with the... white to black Click on Format>Cells>Fraction>Up to three digits>OK In A1 through A6, type 3/4, 2/5, 4/7, 11/12/ 7/16 and 19/22 respectively Observe that these numMathematics for Business, Science, and Technology, Second Edition Orchard Publications 1-15 Chapter 1 Numbers and Arithmetic Operations bers appear on the spreadsheet as entered After the last entry and pressing the key, the selected... change the subtraction operation to addition Next, we need to add these two positive numbers, and for this addition we follow Rule 1, that is, we add the absolute values and we place the plus sign in front of the result The steps are shown below 53 – – 18 = +53 – – 18 = +53 + +18 53 + 18 = 71 +71=71 Again, these steps indicate the “train of thought” and the reader is not expected to write down all these . parentheses without the minus sign. Most often, this designation appears in financial statements. T Chapter 1 Numbers and Arithmetic Operations 1-2 Mathematics. positive. The steps are shown below. Note 1.3 The reader can use a hand calculator to obtain the result. The intent here is to illustrate how the sign of the