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SCHAUM’S OUTLINE OF Theory and Problems of COLLEGE MATHEMATICS THIRD EDITION Algebra Discrete Mathematics Precalculus Introduction to Calculus FRANK AYRES, Jr., Ph.D. Formerly Professor and Head Department of Mathematics, Dickinson College PHILIP A. SCHMIDT, Ph.D. Program Coordinator, Mathematics and Science Education The Teachers College, Western Governors University Salt Lake City, Utah Schaum’s Outline Series McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

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ebook_copyright 8.5 x 11.qxd 5/30/03 10:39 AM Page 1 Copyright © 1958 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in the United States of America. Except as per- mitted 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: 0-07-140227-6 All 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 train- ing programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904- 4069. 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-Hill’s 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, spe- cial, 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

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PREFACE In the Third Edition of College Mathematics, I have maintained the point-of-view of the ﬁrst two editions.Students who are engaged in learning mathematics in the mathematical range from algebra to calculus will ﬁnd 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 problem- solving 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 reﬂect 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 reﬂect 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 ﬁnally, 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.S CHMIDT New Paltz, NY iii

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For more information about this title, click here. CONTENTS PART I Review of Algebra 1 1. Elements of Algebra 3 2. Functions 8 3. Graphs of Functions 13 4. Linear Equations 19 5. Simultaneous Linear Equations 24 6. Quadratic Functions and Equations 33 7. Inequalities 42 8. The Locus of an Equation 47 9. The Straight Line 54 10. Families of Straight Lines 60 11. The Circle 64 PART II Topics in Discrete Mathematics 73 12. Arithmetic and Geometric Progressions 75 13. Inﬁnite Geometric Series 84 14. Mathematical Induction 88 15. The Binomial Theorem 92 16. Permutations 98 17. Combinations 104 18. Probability 109 19. Determinants of Orders Two and Three 117 20. Determinants of Order n 122 21. Systems of Linear Equations 129 22. Introduction to Transformational Geometry 136 PART III Topics in Precalculus 153 23. Angles and Arc Length 155 24. Trigonometric Functions of a General Angle 161 25. Trigonometric Functions of an Acute Angle 169 26. Reduction to Functions of Positive Acute Angles 178 27. Graphs of the Trigonometric Functions 183 28. Fundamental Relations and Identities 189 29. Trigonometric Functions of Two Angles 195 30. Sum, Difference, and Product Formulas 207 31. Oblique Triangles 211 32. Inverse Trigonometric Functions 222 33. Trigonometric Equations 232 34. Complex Numbers 242 v Copyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

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vi CONTENTS 35. The Conic Sections 254 36. Transformation of Coordinates 272 37. Points in Space 283 38. Simultaneous Equations Involving Quadratics 294 39. Logarithms 303 40. Power, Exponential, and Logarithmic Curves 307 41. Polynomial Equations, Rational Roots 312 42. Irrational Roots of Polynomial Equations 319 43. Graphs of Polynomials 329 44. Parametric Equations 336 PART IV Introduction to Calculus 343 45. The Derivative 345 46. Differentiation of Algebraic Expressions 355 47. Applications of Derivatives 360 48. Integration 371 49. Inﬁnite Sequences 377 50. Inﬁnite Series 383 51. Power Series 389 52. Polar Coordinates 394 APPENDIX A Introduction to the Graphing Calculator 410 APPENDIX B The Number System of Algebra 414 APPENDIX C Mathematical Modeling 421 INDEX 424

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PART I REVIEW OF ALGEBRA Copyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

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Chapter 1 Elements of Algebra IN ARITHMETIC the numbers used are always known numbers; a typical problem is to con- vert 5 hours and 35 minutes to minutes.This is done by multiplying 5 by 60 and adding 35; thus, 5 · 60 þ 35 ¼ 335 minutes. In algebra some of the numbers used may be known but others are either unknown or not speciﬁed; 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 þ m ¼ 60h þ m.We call 60 h þ m an algebraic expression.(See Problem 11. ). 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 þ 35 and 2 · 60 þ 35 is ð5 þ 2Þ · 60 þ 2 · 35; similarly, the sum of h · 60 þ m and k · 60 þ m is ðh þ kÞ · 60 þ 2m.(See Problems 1.2–1.6.) 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 ¼ amþn (2) ðamÞn ¼ amn (3) a amn ¼ amÿn; a ¼6 0; m > n; amn ¼ an1ÿm ; a ¼6 0; m < n (4) ða · bÞn¼ anbn an an (5) b ¼ bn ; b ¼6 0 (See Problem 1.7.) LET n BE A POSITIVE INTEGER and a and b be two numbers such that bn ¼ a; then b is called an nth root of a.Every number a ¼6 0 has exactly n distinct nth roots. If a is imaginary, all of its nth roots are imaginary; this case will be excluded here and treated later. (See Chapter 35.) 3 Copyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

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4 ELEMENTS OF ALGEBRA [CHAP.1 If a is real and n is odd, then exactly one of the nth roots of a is real.For example, 2 is the real cube 3 5 root of 8, 2 ¼ 8 , and ÿ3 is the real ﬁfth root of ÿ243½ðÿ3Þ ¼ ÿ243. If a is real and n is even, then there are exactly two real nth roots of a when a > 0, but no real nth roots of a when a < 0.For example, þ3 and ÿ3 are the square roots of 9; þ2 and ÿ2 are the real sixth roots of 64. THE PRINCIPAL nth ROOT OF a is the positive real nth root of a when a is positive and the real nth pﬃﬃ n root of a, if any, when a is negative.The principal nth root of a is denoted by a, called a radical.The integer n is called the index of the radical and a is called the radicand.For example, pﬃﬃ pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ 6 5 9 ¼ 3 64 ¼ 2 ÿ243 ¼ ÿ3 (See Problem 1.8.) ZERO, FRACTIONAL, AND NEGATIVE EXPONENTS. When s is a positive integer, r is any n integer, and p is any rational number, the following extend the deﬁnition of a in such a way that the laws (1)-(5) are satisﬁed when n is any rational number. DEFINITIONS EXAMPLES 0 0 0 1 0 (6) a ¼ 1; a ¼6 0 2 ¼ 1; ¼ 1; ðÿ8Þ ¼ 1 100 r=s ps ﬃﬃﬃr ps ﬃﬃ r 1=2 pﬃﬃ 5=6 p6 ﬃﬃﬃ5 5 ÿ2=1 ÿ2 1 (7) a ¼ a ¼ a 3 ¼ 3; ð64Þ ¼ 64 ¼ 2 ¼ 32; 3 ¼ 3 ¼ 9 pﬃﬃ ÿp p ÿ1 1 ÿ1=2 (8) a ¼ 1=a ; a ¼6 0 2 ¼ ; 3 ¼ 1 3 2 pﬃ 2 p [ NOTE: Without attempting to deﬁne them, we shall assume the existence of numbers such as a ; a ; . . . ; in which the exponent is irrational.We shall also assume that these numbers have been deﬁned in such a way that the laws (1)–(5) are satisﬁed.] (See Problem 1.9–1.10.) Solved Problems 1.1 For each of the following statements, write the equivalent algebraic expressions: ðaÞ the sum of x and 2, ðbÞ the sum of a and ÿb, ðcÞ the sum of 5a and 3b, ðd Þ the product of 2a and 3a, ðeÞ the product of 2a and 5b, f the number which is 4 more than 3 times x, g the number which is 5 less than twice y, ðhÞ the time required to travel 250 miles at x miles per hour, ði Þ the cost (in cents) of x eggs at 65¢ per dozen. 2 ðaÞ x þ 2 ðd Þ ð2aÞð3aÞ ¼ 6a g 2y ÿ 5 ðbÞ a þ ðÿbÞ ¼ a ÿ b ðeÞ ð2aÞð5bÞ ¼ 10ab ðhÞ 250=x ðcÞ 5a þ 3b f 3x þ 4 ði Þ 65 x=12 1.2 Let x be the present age of a father. ðaÞ Express the present age of his son, who 2 years ago was one-third his father’s age. ðbÞ Express the age of his daughter, who 5 years from today will be one-fourth her father’s age. ðaÞ Two years ago the father’s age was x ÿ 2 and the son’s age was ðx ÿ 2Þ=3.Today the son’s age is 2 þ ðx ÿ 2Þ=3. 1 ðbÞ Five years from today the father’s age will be x þ 5 and his daughter’s age will be ðx þ 5Þ.Today the 4 1 daughter’s age is ðx þ 5Þ ÿ 5. 4

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