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 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. 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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 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 thosecurriculainthistext.However,asubstantialnumberofimportantchangeshave beenmadeinthisedition.First,thereismoreofanemphasisnowontopicsindiscrete mathematics. Second, the graphing calculator is introduced as an important problem- solvingtool.Third,materialrelatedtomanualandtabularcomputationsoflogarithms has been removed, and replaced with material that is calculator-based. Fourth, all materialrelatedtotheconceptsoflocushasbeenmodernized.Fifth,tablesandgraphs 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 alsomustthankDr.MartiGarlett,DeanoftheTeachersCollegeatWesternGovernors University, for her professional support as I struggled to meet deadlines while beginninganewpositionattheUniversity.IthankMaureenWalkerforherhandling ofthemanuscript andproofs.Andfinally,Ithankmywife, Dr.JanZlotnikSchmidt, 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, NY iii 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. 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 PART I REVIEW OF ALGEBRA Copyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. This page intentionally left blank. 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, 5·60þ35¼335minutes. Inalgebrasomeofthenumbersusedmaybeknownbutothersareeitherunknownornotspecified; thatis,theyarerepresentedbyletters.Forexample,converthhoursandmminutesintominutes.Thisis doneinpreciselythesamemannerasintheparagraphabovebymultiplyinghby60andaddingm;thus, h·60þm¼60hþm. We call 60hþm 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 5·60þ35 and 2·60þ35isð5þ2Þ·60þ2·35;similarly,thesumofh·60þmandk·60þmisðhþkÞ·60þ2m.(See Problems 1.2–1.6.) POSITIVEINTEGRALEXPONENTS. Ifaisanynumberandnisanypositiveinteger,theproduct ofthenfactorsa·a·a···aisdenotedbyan.Todistinguishbetweentheletters,aiscalledthebaseandn 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 ¼amþn (2) ðamÞn¼amn am am 1 (3) ¼am(cid:5)n; a6¼0; m>n; ¼ ; a6¼0; m<n an an an(cid:5)m (4) ða·bÞn¼anbn (cid:1) (cid:2) a n an (5) ¼ ; b6¼0 b bn (See Problem 1.7.) LETnBEAPOSITIVEINTEGERandaandbbetwonumberssuchthatbn¼a;thenbiscalledannth root of a. Every number a6¼0 has exactly n distinct nth roots. Ifaisimaginary,allofitsnthrootsareimaginary;thiscasewillbeexcludedhereandtreatedlater. (See Chapter 35.) 3 Copyright 1958 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use. 4 ELEMENTSOFALGEBRA [CHAP.1 Ifais(cid:3)realand(cid:4) nisodd,thenexactlyoneofthenthrootsofaisreal.Forexample,2istherealcube root of 8, 23¼8 , and (cid:5)3 is the real fifth root of (cid:5)243½ð(cid:5)3Þ5¼(cid:5)243(cid:7). Ifaisrealandniseven,thenthereareexactlytworealnthrootsofawhena>0,butnorealnth rootsofawhena<0.Forexample,þ3and(cid:5)3arethesquarerootsof9;þ2and(cid:5)2aretherealsixth roots of 64. THEPRINCIPALnthROOTOFaisthepositiverealnthrootofawhenaisppffioffisitiveandtherealnth rootofa,ifany,whenaisnegative.Theprincipalnthrootofaisdenotedby na,calledaradical.The integer n is called the index of the radical and a is called the radicand. For example, pffiffi pffiffiffi pffiffiffiffiffiffiffi 9¼3 664¼2 5(cid:5)243¼(cid:5)3 (See Problem 1.8.) ZERO, FRACTIONAL, AND NEGATIVE EXPONENTS. When s is a positive integer, r is any integer, and p is any rational number, the following extend the definition of an in such a way that the laws (1)-(5) are satisfied when n is any rational number. DEFINITIONS EXAMPLES (cid:6) (cid:7) (6) a0¼1;a6¼0 20¼1; 1 0¼1;ð(cid:5)8Þ0¼1 (7) ar=s¼psffiaffiffir¼(cid:3)psffiaffi(cid:4)r 31=2¼p1ffi3ffi0;0ð64Þ5=6¼(cid:6)p6ffi6ffi4ffi(cid:7)5¼25¼32;3(cid:5)2=1 ¼3(cid:5)2 ¼1 9 pffiffi (8) a(cid:5)p¼1=ap;a6¼0 2(cid:5)1¼1;3(cid:5)1=2¼1 3 2 ffiffi p [NOTE: Withoutattemptingtodefinethem,weshallassumetheexistenceofnumberssuchasa 2;ap;...;inwhich theexponentisirrational.Weshallalsoassumethatthesenumbershavebeendefinedinsuchawaythat thelaws(1)–(5)aresatisfied.](SeeProblem1.9–1.10.) Solved Problems 1.1 For each of the following statements, write the equivalent algebraic expressions: ðaÞ the sum of x and 2, (cid:3)ðbÞ(cid:4)thesumofaand(cid:5)b,ðcÞthesumof5aand3b,ð(cid:3)d(cid:4)Þtheproductof2aand3a,ðeÞtheproductof2aand5b, f the number which is 4 more than3 times x, g the number which is 5 less thantwice y, ðhÞ the time requiredtotravel250milesatxmilesperhour,ðiÞthecost(incents)ofxeggsat65¢perdozen. (cid:3) (cid:4) ðaÞ xþ2 ðdÞ ð2aÞð3aÞ¼6a2 g 2y(cid:5)5 ðbÞ aþð(cid:5)bÞ¼a(cid:5)b ðeÞ ð2aÞð5bÞ¼10ab ðhÞ 250=x (cid:3) (cid:4) (cid:3) (cid:4) ðcÞ 5aþ3b f 3xþ4 ðiÞ 65x=12 1.2 Letxbethepresentageofafather.ðaÞExpressthepresentageofhisson,who2yearsagowasone-thirdhis father’sage.ðbÞExpresstheageofhisdaughter,who5yearsfromtodaywillbeone-fourthherfather’sage. ðaÞ Two years ago the father’s age was x(cid:5)2 and the son’s age was ðx(cid:5)2Þ=3. Today the son’s age is 2þðx(cid:5)2Þ=3. ðbÞ Fiveyearsfromtodaythefather’sagewillbexþ5andhisdaughter’sagewillbe1ðxþ5Þ.Todaythe 4 daughter’sageis1ðxþ5Þ(cid:5)5. 4