Functions and change : a modeling approach to college algebra and trigonometry

737 Pages·2008·10.36 MB·English

by  Bruce Crauder

Functions and Change
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Functions and Change
A Modeling Approach to College
Algebra and Trigonometry
Bruce Crauder / Benny Evans / Alan Noell
Oklahoma State University
Thisprojectwassupported,inpart,
bythe
National Science Foundation
Opinionsexpressedarethoseoftheauthors
andnotnecessarilythoseoftheFoundation
Houghton Mifflin Company
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ISBN-13:978-0-618-85804-0
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Contents
Preface xi 1.4 Functionsgivenbywords 70
Comparingformulasandwords 70
Gettingformulasfromwords 73
Proportion 75
Prologue: Calculator Arithmetic 1
AnotherLook:Geometricconstructions 76
Chapter1Summary 89
Typingmathematicalexpressions 2
Chapter1ReviewExercises 91
Rounding 2
Parenthesesandgrouping 3
Minussigns 4
Specialnumbersπ ande 5 2
Graphical and Tabular
Chaincalculations 5
Analysis 95
Scientificnotation 7
AnotherLook:Orderofoperations 8
ChapterPSummary 16 2.1 Tablesandtrends 96
ChapterPReviewExercises 17 Gettingtablesfromformulas 96
Optimizingwithtablesofvalues 99
AnotherLook:Limits 101
1 2.2 Graphs 112
Functions 19
Hand-drawngraphsfromformulas 112
Graphingwiththecalculator 112
1.1 Functionsgivenbyformulas 20
Tracingthegraph 113
Functionsofonevariable 20
Choosingaviewingwindowinpractical
Functionsofseveralvariables 21
settings 113
AnotherLook:Definitionofafunction 23
Gettinglimitingvaluesfromgraphs 117
AnotherLook:Shiftingandstretching 119
1.2 Functionsgivenbytables 35
Readingtablesofvalues 35 2.3 Solvinglinearequations 132
Fillinggapsbyaveraging 35
Thebasicoperations 132
Averageratesofchange 36
Reversingtherolesofvariables 134
Spottingtrends 38
AnotherLook:Equationsthatarelinearinone
AnotherLook:Moreonaverageratesof
variable 137
change 41
2.4 Solvingnonlinearequations 148
1.3 Functionsgivenbygraphs 52
Thecrossing-graphsmethod 148
Readinggraphs 52
Thesingle-graphmethod 149
Concavityandratesofchange 54
AnotherLook:Solvingnonlinearequationsby
Inflectionpoints 56
factoring 153
Drawingagraph 57
AnotherLook:Secantlines 60
vi Contents
2.5 Optimization 163 4
Exponential Functions 269
Optimizingatpeaksandvalleys 163
Optimizingatendpoints 166
4.1 Exponentialgrowthanddecay 270
AnotherLook:Optimizingwithparabolas 168
Exponentialgrowth 270
Chapter2Summary 179
Exponentialdecay 271
Chapter2ReviewExercises 181
Constantproportionalchange 273
Unitconversion 275
Exponentialfunctionsanddailyexperience 277
3 AnotherLook:Elementarypropertiesof
Straight Lines and Linear
exponents 277
Functions 185
4.2 Modelingexponentialdata 286
3.1 Thegeometryoflines 186 Recognizingexponentialdata 286
Characterizationsofstraightlines 186 Constructinganexponentialmodel 287
Theslopeofaline 187 Growthanddecayfactorunitsinexponential
Gettingslopefrompoints 188 modeling 290
AnotherLook:Planegeometry:Paralleland AnotherLook:Testingunevenlyspaced
perpendicularlines 190 data 293
3.2 Linearfunctions 201 4.3 Modelingnearlyexponentialdata 301
Constantratesofchange 201 Exponentialregression 301
Linearfunctionsandstraightlines 203 AnotherLook:Nonlinearmodels 304
Linearequationsfromdata 205
4.4 Logarithmicfunctions 314
AnotherLook:Formsofequations 208
TheRichterscale 314
3.3 Modelingdatawithlinearfunctions 217 Howthecommonlogarithmworks 315
Testingdataforlinearity 217 Thelogarithmastheinverseoftheexponential
Linearmodels 218 function 316
Graphingdiscretedata 220 AnotherLook:Solvingexponential
AnotherLook:Unevenlyspaceddata 223 equations 318
3.4 Linearregression 233 4.5 Connectingexponentialandlinear
data 328
Theregressionline 233
Usesoftheregressionline:Slopeand Fromlineartoexponentialdata 328
trends 234 Fromexponentialtolineardata:The
AnotherLook:Linearregressionformula,the logarithm 329
error 238 Exponentialregressionusingthenatural
logarithm 330
3.5 Systemsofequations 250 AnotherLook:Applyingthelawsof
Graphicalsolutionsofsystemsof logarithms 333
equations 250 Chapter4Summary 343
Algebraicsolutions 253 Chapter4ReviewExercises 345
AnotherLook:Solutionbyelimination,
matrices 254
Chapter3Summary 263
Chapter3ReviewExercises 266
Contents vii
5 6
A Survey of Other Common The Geometry of Right Triangles
Functions 349 437
5.1 Powerfunctions 350 6.1 Sidesofrighttriangles 438
Homogeneitypropertyofpowerfunctions 351 ThetheoremofPythagoras 438
Comparingexponentialandpower Areaofatriangle 439
functions 355 AnotherLook:Ademonstrationofthe
AnotherLook:Homogeneity,moreonpower Pythagoreantheorem 441
andexponentialfunctions 357
6.2 Angles 447
5.2 Modelingdatawithpowerfunctions 364 Degreeandradianmeasure 447
Theconnectionbetweenpowerdataandlinear Slicesofpie 448
data 364 Similartriangles 450
Gettingapowermodelfromdata 364 AnotherLook:Moreonsimilartriangles 452
Almostpowerdata 367
Graphingonalogarithmicscale:Common 6.3 Righttriangletrigonometry 457
versusnaturallogarithms 369 Sine,cosine,andtangentfunctions 457
AnotherLook:Moreonpowerregression 370 Secant,cosecant,andcotangentfunctions 461
AnotherLook:Specialtriangles 461
5.3 Combininganddecomposingfunctions 382
Chapter6Summary 469
Sums,products,andlimitingvalues 382
Chapter6ReviewExercises 470
Compositionoffunctions 384
Piecewise-definedfunctions 386
AnotherLook:Chaoticbehavior 388
7
Periodic Functions 473
5.4 Quadraticfunctionsandparabolas 397
Linearratesofchangeandsecond-order
7.1 Trigonometricfunctionsasperiodic
differences 399
functions 474
Quadraticregression 402
Extendingthetrigonometricfunctions 474
Almostquadraticdata 403
Quadrants 478
Thequadraticformula 404
Periodandamplitudeofsineandcosine
AnotherLook:Completingthesquare,complex
functions 479
numbers 405
Adjustingperiodandamplitude 479
5.5 Higher-degreepolynomialsandrational AnotherLook:Referenceangles 481
functions 414
7.2 Modelingwithperiodicfunctions 485
Higher-degreepolynomialsandtheirroots 414
Periodandamplitudeforperiodicphenomena
Cubicandotherpolynomialmodels 415
485
Rationalfunctions 417
Soundwaves 487
AnotherLook:Factoringpolynomials,behavior
Usingthesinefunctiontomodelperiodic
atinfinity 420
phenomena 487
Chapter5Summary 432
AnotherLook:Otherperiodicfunctions 490
Chapter5ReviewExercises 434
viii Contents
7.3 Modelingperiodicdata 498 Constantvelocitymeanslineardirected
Verticalandhorizontalshifting 498 distance 569
Sinemodelsbyestimation 501 Whendistanceisgivenbyaformula 575
Sineregression 505
9.2 Ratesofchangeforotherfunctions 579
AnotherLook:Howcalculatorshandle
trigonometricfunctions 505 Examplesofratesofchange 579
Propertiesthatallratesofchangeshare 580
Chapter7Summary 513
Whentherateofchangeiszero 581
Chapter7ReviewExercises 514
Whentherateofchangeisconstant 583
9.3 Estimatingratesofchange 589
8 Ratesofchangefortabulateddata 589
Relationships Among
Ratesofchangeforfunctionsgivenby
Trigonometric Functions 517
formulas 591
8.1 Trigonometricidentities 518 9.4 Equationsofchange:Linearandexponential
Basicidentities 518 functions 597
Areaandsumformulas 520 Equationofchangeforlinearfunctions 597
Double-andhalf-angleformulas 522 Instantaneousratesofchange 598
AnotherLook:Clevertriangles 524 Equationofchangeforexponential
functions 599
8.2 Lawsofsinesandcosines 530 Whyequationsofchange? 602
Lawofsines 530
Lawofcosines 531 9.5 Equationsofchange:Graphical
AnotherLook:Acuriousrelationship 533 solutions 606
Equilibriumsolutions 606
8.3 Inversetrigonometricfunctions 540 Sketchinggraphs 609
Defininginversetrigonometricfunctions 540 Chapter9Summary 618
Makingpicturestorepresentinverse
Chapter9ReviewExercises 620
trigonometricfunctions 542
Theinversetangentfunction 543
AnotherLook:Machin’sformula 545
10
Mathematics of Population
8.4 ComplexnumbersandDeMoivre'stheorem
Ecology 623
550
Complexnumbers 550
10.1 Populationdynamics:Exponential
ThecomplexexponentialfunctionandtheEuler
growth 624
formula 550
DeMoivre'stheorem 554 Exponentialgrowth 624
AnotherLook:Polarcoordinates 558 Doublingtimes 626
Empiricaldata 627
Chapter8Summary 563
Discretegenerations 628
Chapter8ReviewExercises 566
10.2 Populationdynamics:Logisticgrowth 633
Logisticgrowthmodel 633
9 Applicationtoharvestingrenewable
Rates of Change 567
resources 634
Logisticequationofchange 635
9.1 Velocity 568
Integralformofthelogisticequation 637
Gettingvelocityfromdirecteddistance 568 Thevalueofthelogisticmodel 639
Contents ix
10.3 Populationstructure:Survivorship
curves 644
Lifetablesandsurvivorshipcurves 644
Classifyingsurvivorshipcurves 645
Usefulnessofsurvivorshipcurves 648
Chapter10Summary 655
Chapter10ReviewExercises 656
BriefAnswerstoSelectedExercises A-1
Index A-45