Edward Dowling - Schaums Outline of Mathematical Methods for Business and Economics

SCHAUMS OUTLINE OF
Mathematical Methods for Business and Economics

Edward T. Dowling, Ph.D.Chair and Professor
Department of Economics
Fordham UniversitySchaums Outline Series Copyright 1993 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-170246-1 MHID: 0-07-170246-6 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-163532-5, MHID: 0-07-163532-7. All trademarks are trademarks of their respective owners.

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Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hills prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting there from. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. and M.A. degrees in economics today need a variety of mathematical skills to successfully complete their degree requirements and compete effectively in their chosen careers. degrees in economics today need a variety of mathematical skills to successfully complete their degree requirements and compete effectively in their chosen careers.

Unfortunately, the requisite mathematical competence is not the subject of a single course in mathematics such as Calculus I or Linear Algebra I, and many students, pressed with the demands from business and economics courses, do not have space in their schedules for a series of math courses. Mathematical Methods for Business and Economics is designed to cull the mathematical tools, topics, and techniques essential for success in business and economics today. It is suitable for a one- or two-semester course in business mathematics, depending on the previous background of the students. It can also be used profitably in an introductory calculus or linear algebra course by professors and students interested in the business connections and applications. The theory-and-solved-problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. No mathematical proficiency beyond the high-school level is assumed.

The learning-by-doing pedagogy will enable students to progress at their own rate and adapt the book to their own needs. Mathematical Methods for Business and Economics can be used by itself or as a supplement to other texts for undergraduate and graduate students in business and economics. It is largely self-contained. Starting with a basic review of high-school algebra in , the book consistently explains all the concepts and techniques needed for the material in subsequent chapters. This book contains 1,066 problems, all of them solved in considerable detail. To derive the most from the book, students should strive as soon as possible to work independently of the solutions.

This can be done by solving problems on individual sheets of paper with the book closed. If difficulties arise, the solution can then be checked in the book. For best results, students should never be satisfied with passive knowledgethe capacity merely to follow or comprehend the various steps presented in the book. Mastery of the subject and doing well on exams require active knowledgethe ability to solve any problem, in any order, without the aid of the book. Experience has proved that students of very different backgrounds and abilities can be successful in handling the subject matter of this text when the material is presented in the current format. I wish to express my gratitude for help with this book to Dr.

Dominick Salvatore and Dr. Timothy Weithers of Fordham University; Maria Cristina Cacdac-Ampil, a doctoral candidate at Fordham; Professor Henry Mark Smith, the reviewer; and John Carleo, John Aliano, Maureen Walker, Pat Koch, and Patty Andrews of McGraw-Hill. EDWARD T. DOWLTNG

Given a positive integer n, xn signifies that x is multiplied by itself n number of times. Here x is referred to as the base; n is called an exponent. By convention an exponent of one is not expressed: x(1) = x, 8(1) = 8. By definition any nonzero number or variable raised to the zero power is equal to 1: x0 = 1, 30 = 1, and 00 is undefined.

Assuming a and b are positive integers and x and y are real numbers for which the following exist, the rules of exponents are presented below, illustrated in Examples 1 and 2, and treated in Problems 1.1, 1.24, 1.26, and 1.27. EXAMPLE 1. In multiplication, exponents of the same variable are added; in division, exponents of the same variable are subtracted; when raised to a power, the exponents of a variable are multiplied, as indicated by the rules above and shown in the examples below followed by illustrations. (

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