AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 69 Exploring Mathematics with CAS Assistance Topics in Geometry, Algebra, Univariate Calculus, and Probability Lydia S. Novozhilova Robert D. Dolan a d Exploring Mathematics with CAS Assistance Topics in Geometry, Algebra, Univariate Calculus, and Probability AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 69 Exploring Mathematics with CAS Assistance Topics in Geometry, Algebra, Univariate Calculus, and Probability Lydia S. Novozhilova Robert D. Dolan Providence, Rhode Island ClassroomResourceMaterialsEditorialBoard CynthiaJ.Huffman,Editor BretJ.Benesh JudithCovington MariaFung RussellGoodman JoelK.Haack GizemKaraali HaseebA.Kazi TamaraJ.Lakins JessicaM.Libertini SarahLoeb CandicePrice 2020MathematicsSubjectClassification.Primary97-01,97D99,97E99,97F99,97G99,97H99, 97I99,97K99,97M99,97N99. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/clrm-69 LibraryofCongressCataloging-in-PublicationData Names:Novozhilova,LydiaS.,1948-author.|Dolan,RobertD.(RobertDomenic),1993-author. Title: ExploringmathematicswithCASassistance: topicsingeometry,algebra,univariatecalculus,and probability/LydiaS.Novozhilova,RobertD.Dolan. Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety, [2022]|Series:Classroomresourcematerials,1557-5918;volume69|Includesbibliographicalreferences andindex. 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VisittheAMShomepageathttps://www.ams.org/ 10987654321 272625242322 To Alice and Beatrice LydiaNovozhilova To Desirée BobbyDolan Contents Introduction xi Acknowledgments xv Part1 Algebra&Geometry 1 1 ComputerAlgebraSystemsandElementsofAlgorithmics 3 1.1 Termsandnotation 4 1.2 Aboutdatatypesanddatastructures 5 1.3 Elementsofalgorithmicsandalgorithmicproblemsolving 8 1.4 Glossary 16 2 TopicsinClassicalGeometry 19 2.1 Review: Matrices,vectors,andlines 19 2.2 Rigidtransformationsoftheplane 25 2.3 Complexnumbersinclassicalgeometry 27 2.4 Threecentersofatriangle 30 2.5 Glossary 33 3 MoreTopicsinClassicalGeometry 35 3.1 Lab2: TheEulerline 35 3.2 TheSimsonline 37 3.3 Conics 40 3.4 Glossary 45 4 TopicsinElementaryNumberTheory 47 4.1 NumberofprimesandtheRiemannHypothesis 49 4.2 Algorithmsfromelementarynumbertheory 50 4.3 Pythagoreantriples 56 4.4 Lab4: PlottinglegsofprimitivePythagoreantriples 58 4.5 LinearDiophantineequationsintwovariables 59 4.6 Lab5: IndustrialapplicationofanLDEinthreevariables 62 4.7 Glossary 63 5 TopicsinAlgebra: SolvingUnivariateAlgebraicEquations 65 5.1 Rootsofunivariatepolynomials 66 5.2 Geometryofcubicequations: Countingthenumberofrealroots 69 5.3 Lab6: SolvingcubicequationsusingVieta’ssubstitution 74 5.4 Nonnegativeunivariatepolynomials 76 vii viii Contents 5.5 Glossary 78 6 TopicsinAlgebra: BivariateSystemsofPolynomialEquations 79 6.1 Linearsystemsoftwoequations 80 6.2 Nonlinearsystemsofpolynomialequations: Motivatingexample 82 6.3 Solvingnonlinearpolynomialsystems 85 6.4 Implicitizationofplanecurves 94 6.5 Glossary 96 Part2 CalculusandNumerics 97 7 Derivatives 99 7.1 Review: Definitions,notation,andterminology 100 7.2 Convexityofaunivariatefunction 104 7.3 Somefactsaboutfunctionsandderivatives 105 7.4 Lab8: Constructingasquarecircumscribedaboutellipse 110 7.5 Glossary 112 8 DefiniteIntegrals 113 8.1 Review: Somebasicconceptsandfactsofunivariateintegralcalculus 114 8.2 Areaofaregionboundedbyasimpleclosedcurve 116 8.3 Lab9: Submergencedepthofabodyofrevolutioninequilibrium 121 8.4 Solvingsomeordinarydifferentialequations 123 8.5 Glossary 127 9 ApproximatingZerosofFunctionsbyIterationMethods 129 9.1 Fixedpointiterationmethod 130 9.2 Newton’smethod 135 9.3 Lab11: Kepler’sEquationandderivingKepler’sSecondLaw 137 9.4 Lab12: Explorationofthelogisticmaps 140 9.5 Glossary 141 10 PolynomialApproximations 143 10.1 Taylorpolynomials 145 10.2 InterpolatingpolynomialsintheLagrangeform 147 10.3 Piecewisepolynomialinterpolation: Splines 150 10.4 Approximatinglargedatasets: Regression 153 10.5 Tworeal-lifeapplicationsoftheLSmethod 156 10.6 Glossary 158 11 TrigonometricApproximation 159 11.1 Shortreviewoftrigonometricfunctions 160 11.2 Fourierseries 164 11.3 Abouttheaccuracyoftrigonometricapproximations 167 11.4 CelebratedclassicalapplicationofFourierseries 170 11.5 Glossary 173 12 FourierAnalysisinMusicandSignalProcessing 175 12.1 Introductionandbackground 175 Contents ix 12.2 Fourierseriesandperiodicsignals 177 12.3 TheFouriertransformfornon-periodicsignals 180 12.4 TheDiscreteFourierTransform 182 12.5 Fourierseriesinsignalprocessing 187 12.6 Glossary 188 Part3 ProbabilityandStatistics 191 13 ProbabilityandStatisticsBasics 193 13.1 Review: Somebasicconceptsofprobability 194 13.2 Somediscreteprobabilitydistributions 197 13.3 Aboutcontinuousprobabilitydistributions 201 13.4 LawofLargeNumbers 204 13.5 CentralLimitTheorem 206 13.6 Glossary 208 14 ComputerSimulationofStatisticalSampling 209 14.1 Randomnumbergeneration 209 14.2 Lab 17: CLT and LLN in action: Life expectancy in the world population 211 14.3 Samplingfromnon-uniformdistributions(optional) 213 14.4 MonteCarlomethodsforfindingintegralsandareas 215 14.5 Lab18: Buffon’sneedleproblem 222 14.6 Glossary 223 15 SimpleRandomWalks 225 15.1 Simplerandomwalksonintegers 226 15.2 Lab19: Thegambler’sruinproblem 231 15.3 Randomwalkonthesquarelattice 233 15.4 Lab20: Drunkensailorproblem 234 15.5 Glossary 235 A DataforLab17inChapter14 237 Bibliography 239 Index 241