Schaum's outlines college mathematics

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  • 1.SCHAUMS OUTLINE OFTheory and Problems ofCOLLEGE MATHEMATICS THIRD EDITION Algebra Discrete Mathematics Precalculus Introduction to Calculus FRANK AYRES, Jr., Ph.D. Formerly Professor and Head Department of Mathematics, Dickinson CollegePHILIP A. SCHMIDT, Ph.D. Program Coordinator, Mathematics and Science Education The Teachers College, Western Governors University Salt Lake City, UtahSchaums Outline Series McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

2. Copyright 1958 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. 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. 0-07-142588-8 The material in this eBook also appears in the print version of this title: 0-07-140227-6All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at or (212) 9044069.TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (McGraw-Hill) and its licensors reserve all rights in and to the work. 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 therefrom. 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. DOI: 10.1036/0071425888 3. PREFACEIn the Third Edition of College Mathematics, I have maintained the point-of-view of the rst two editions. Students who are engaged in learning mathematics in the mathematical range from algebra to calculus will nd virtually all major topics from those curricula in this text. However, a substantial number of important changes have been made in this edition. First, there is more of an emphasis now on topics in discrete mathematics. Second, the graphing calculator is introduced as an important problemsolving tool. Third, material related to manual and tabular computations of logarithms has been removed, and replaced with material that is calculator-based. Fourth, all material related to the concepts of locus has been modernized. Fifth, tables and graphs have been changed to reect current curriculum and teaching methods. Sixth, all material related to the conic sections has been substantially changed and modernized. Additionally, much of the rest of the material in the third edition has been changed to reect current classroom methods and pedagogy, and mathematical modeling is introduced as a problem-solving tool. Notation has been changed as well when necessary. My thanks must be expressed to Barbara Gilson and Andrew Littell of McGraw-Hill. They have been supportive of this project from its earliest stages. I also must thank Dr. Marti Garlett, Dean of the Teachers College at Western Governors University, for her professional support as I struggled to meet deadlines while beginning a new position at the University. I thank Maureen Walker for her handling of the manuscript and proofs. And nally, I thank my wife, Dr. Jan Zlotnik Schmidt, for putting up with my frequent need to work at home on this project. Without her support, this edition would not have been easily completed. PHILIP A. SCHMIDT New Paltz, NYiii 4. For more information about this title, click here.CONTENTSPART IReview of Algebra 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.PART IIElements of Algebra Functions Graphs of Functions Linear Equations Simultaneous Linear Equations Quadratic Functions and Equations Inequalities The Locus of an Equation The Straight Line Families of Straight Lines The CircleTopics in Discrete Mathematics 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.PART III1Arithmetic and Geometric Progressions Innite Geometric Series Mathematical Induction The Binomial Theorem Permutations Combinations Probability Determinants of Orders Two and Three Determinants of Order n Systems of Linear Equations Introduction to Transformational GeometryTopics in Precalculus 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.Angles and Arc Length Trigonometric Functions of a General Angle Trigonometric Functions of an Acute Angle Reduction to Functions of Positive Acute Angles Graphs of the Trigonometric Functions Fundamental Relations and Identities Trigonometric Functions of Two Angles Sum, Difference, and Product Formulas Oblique Triangles Inverse Trigonometric Functions Trigonometric Equations Complex Numbers vCopyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.3 8 13 19 24 33 42 47 54 60 6473 75 84 88 92 98 104 109 117 122 129 136153 155 161 169 178 183 189 195 207 211 222 232 242 5. viCONTENTS35. 36. 37. 38. 39. 40. 41. 42. 43. 44.PART IVIntroduction to Calculus 45. 46. 47. 48. 49. 50. 51. 52.APPENDIX A APPENDIX B APPENDIX C INDEXThe Conic Sections Transformation of Coordinates Points in Space Simultaneous Equations Involving Quadratics Logarithms Power, Exponential, and Logarithmic Curves Polynomial Equations, Rational Roots Irrational Roots of Polynomial Equations Graphs of Polynomials Parametric EquationsThe Derivative Differentiation of Algebraic Expressions Applications of Derivatives Integration Innite Sequences Innite Series Power Series Polar CoordinatesIntroduction to the Graphing Calculator The Number System of Algebra Mathematical Modeling254 272 283 294 303 307 312 319 329 336343 345 355 360 371 377 383 389 394410 414 421 424 6. PART IREVIEW OF ALGEBRACopyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 7. This page intentionally left blank. 8. Chapter 1Elements of Algebra IN ARITHMETIC the numbers used are always known numbers; a typical problem is to convert 5 hours and 35 minutes to minutes. This is done by multiplying 5 by 60 and adding 35; thus, 5 60 35 335 minutes. In algebra some of the numbers used may be known but others are either unknown or not specied; that is, they are represented by letters. For example, convert h hours and m minutes into minutes. This is done in precisely the same manner as in the paragraph above by multiplying h by 60 and adding m; thus, h 60 m 60h m. We call 60h m an algebraic expression. (See Problem 1.1.) Since algebraic expressions are numbers, they may be added, subtracted, and so on, following the same laws that govern these operations on known numbers. For example, the sum of 5 60 35 and 2 60 35 is 5 2 60 2 35; similarly, the sum of h 60 m and k 60 m is h k 60 2m. (See Problems 1.21.6.)POSITIVE INTEGRAL EXPONENTS. If a is any number and n is any positive integer, the product of the n factors a a a a is denoted by an . To distinguish between the letters, a is called the base and n is called the exponent. If a and b are any bases and m and n are any positive integers, we have the following laws of exponents: (1)am an amn(2)am n amn(3)am amn ; an(4) (5)a 6 0;m > n;am 1 ; an anma 6 0;m


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