Functions Modeling Change: A Preparation for Calculus

Cover Page......Page 1
Title Page......Page 5
Copyright Page
......Page 6
Preface
......Page 7
Table of Contents......Page 13
1 LINEAR FUNCTIONS AND CHANGE......Page 17
Representing Functions: Words, Tables, Graphs, and Formulas......Page 18
Mathematical Models......Page 19
Function Notation......Page 20
When Is a Relationship Not a Function?......Page 21
1.2 RATE OF CHANGE......Page 26
Increasing and Decreasing Functions......Page 27
Function Notation for the Average Rate of Change......Page 29
Constant Rate of Change......Page 33
A General Formula for the Family of Linear Functions......Page 36
Tables for Linear Functions......Page 37
Warning: Not All Graphs That Look Like Lines Represent Linear Functions......Page 39
Finding a Formula for a Linear Function from a Table of Data......Page 43
Finding a Formula for a Linear Function from a Graph......Page 44
Finding a Formula for a Linear Function from a Verbal Description......Page 45
Alternative Forms for the Equation of a Line......Page 47
Interpreting the Parameters of a Linear Function......Page 51
Intersection of Two Lines......Page 52
Equations of Horizontal and Vertical Lines......Page 54
Slopes of Parallel and Perpendicular Lines......Page 55
Justification of Formula for Slopes of Perpendicular Lines......Page 56
Laboratory Data: The Viscosity of Motor Oil......Page 60
Interpolation and Extrapolation......Page 61
How Regression Works......Page 62
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER ONE......Page 66
CHECK YOUR UNDERSTANDING......Page 69
TOOLS FOR CHAPTER 1: LINEAR EQUATIONS AND THE COORDINATE PLANE......Page 71
2 FUNCTIONS......Page 77
Finding Output Values: Evaluating a Function......Page 78
Finding Input Values: Solving Equations......Page 79
Finding Output and Input Values From Tables and Graphs......Page 80
Choosing Realistic Domains and Ranges......Page 85
Using a Graph to Find the Domain and Range of a Function......Page 86
Using a Formula to Find the Domain and Range of a Function......Page 87
2.3 PIECEWISE DEFINED FUNCTIONS......Page 89
The Absolute Value Function......Page 92
Composition of Functions......Page 95
Inverse Functions......Page 96
Concavity and Rates of Change......Page 100
Summary: Increasing and Decreasing Functions; Concavity......Page 102
2.6 QUADRATIC FUNCTIONS......Page 104
Finding the Zeros of a Quadratic Function......Page 105
Concavity and Quadratic Functions......Page 106
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TWO......Page 110
CHECK YOUR UNDERSTANDING......Page 113
TOOLS FOR CHAPTER 2: QUADRATIC EQUATIONS......Page 115
3 EXPONENTIAL FUNCTIONS......Page 121
Growing at a Constant Percent Rate......Page 122
Growth Factors and Percent Growth Rates......Page 124
A General Formula for the Family of Exponential Functions......Page 125
Finding a Formula for an Exponential Function......Page 131
Similarities and Differences between Linear and Exponential Functions......Page 133
3.3 GRAPHS OF EXPONENTIAL FUNCTIONS......Page 138
Graphs of the Exponential Family: The Effect of the Parameter b......Page 139
Solving Exponential Equations Graphically......Page 141
Fitting Exponential Functions to Data......Page 142
3.4 CONTINUOUS GROWTH AND THE NUMBER e......Page 146
Exponential Functions with Base e Represent Continuous Growth......Page 147
3.5 COMPOUND INTEREST......Page 152
Continuous Compounding and the Number e......Page 153
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER THREE......Page 157
CHECK YOUR UNDERSTANDING......Page 161
TOOLS FOR CHAPTER 3: EXPONENTS......Page 162
4 LOGARITHMIC FUNCTIONS......Page 167
What is a Logarithm?......Page 168
Logarithmic and Exponential Functions are Inverses......Page 169
The Natural Logarithm......Page 170
Justification of log(a · b) = loga + logb and log(a/b) = loga − logb......Page 172
4.2 LOGARITHMS AND EXPONENTIAL MODELS......Page 175
Doubling Time......Page 177
Half-Life......Page 178
Exponential Growth Problems That Cannot Be Solved By Logarithms......Page 179
The Graph, Domain, and Range of the Common Logarithm......Page 183
Graph of Natural Logarithm......Page 185
Logarithms and Orders of Magnitude......Page 186
The Solar System and Beyond......Page 191
Logs of Small Numbers......Page 193
Using Logs to Fit an Exponential Function to Data......Page 195
Sales of Compact Discs......Page 196
Fitting An Exponential Function To Data......Page 197
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER FOUR......Page 201
CHECK YOUR UNDERSTANDING......Page 204
TOOLS FOR CHAPTER 4: LOGARITHMS......Page 205
5 TRANSFORMATIONS OF FUNCTIONS AND THEIR GRAPHS......Page 209
Vertical and Horizontal Shift: The Heating Schedule For an Office Building......Page 210
Formulas for a Vertical or Horizontal Shift......Page 211
Translations of a Function and Its Graph......Page 212
Inside and Outside Changes......Page 213
Combining Horizontal and Vertical Shifts......Page 214
5.2 REFLECTIONS AND SYMMETRY......Page 218
A Formula for a Reflection......Page 219
Symmetry About the y-Axis......Page 221
Symmetry About the Origin......Page 222
Combining Shifts and Reflections......Page 223
Vertical Stretch: A Stereo Amplifier......Page 227
Formula for Vertical Stretch or Compression......Page 228
Stretch Factors and Average Rates of Change......Page 230
Combining Transformations......Page 231
Horizontal Stretch: A Lighthouse Beacon......Page 235
Formula for Horizontal Stretch or Compression......Page 236
The Vertex of a Parabola......Page 241
Formulas for Quadratic Functions......Page 242
Applications of Quadratic Functions......Page 245
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER FIVE......Page 249
CHECK YOUR UNDERSTANDING......Page 253
Completing the Square......Page 255
Deriving The Quadratic Formula......Page 256
6 TRIGONOMETRIC FUNCTIONS......Page 259
Ferris Wheel Height As a Function of Time......Page 260
Graphing the Ferris Wheel Function......Page 261
Periodic Functions: Period, Midline, and Amplitude......Page 262
The Unit Circle......Page 266
Definition of Sine and Cosine......Page 267
Coordinates of a Point on a Circle of Radius r......Page 269
Definition of a Radian......Page 273
Arc Length......Page 276
Sine and Cosine of a Number......Page 277
Exact Values of the Sine and Cosine......Page 279
Graphs of the Sine and Cosine Functions......Page 280
Amplitude......Page 281
Midline......Page 282
Period......Page 285
Horizontal Shift......Page 287
Phase Shift......Page 288
Using the Transformed Sine and Cosine Functions......Page 289
The Tangent Function......Page 295
Relationships Between the Trigonometric Functions......Page 297
The Reciprocals of the Trigonometric Functions: Secant, Cosecant, Cotangent......Page 298
Solving Trigonometric Equations Using the Inverse Cosine......Page 301
The Inverse Cosine Function......Page 303
The Inverse Sine and Inverse Tangent Functions......Page 304
Solving Equations Using Reference Angles......Page 306
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER SIX......Page 311
CHECK YOUR UNDERSTANDING......Page 315
Special Angles: 30°, 45°, 60°......Page 317
7 TRIGONOMETRY......Page 323
The Law of Cosines......Page 324
The Law of Sines......Page 326
Equations Versus Identities......Page 329
Why We Need Identities......Page 330
Double-Angle Formula for Sine......Page 331
Double Angle Formulas for Cosine and Tangent......Page 332
Solving Trigonometric Equations......Page 333
7.3 SUM AND DIFFERENCE FORMULAS FOR SINE AND COSINE......Page 336
Rewriting sint + cost......Page 337
Rewriting a[sub(1)] sin Bt + a[sub(2)] cos Bt......Page 338
Justification of cos(θ − Φ) = cosθcosΦ + sinθsinΦ......Page 339
Obtaining Other Identities from the Identity for cos(Φ − θ)......Page 340
Justification of Other Sum and Difference Formulas......Page 341
Sums of Trigonometric Functions......Page 343
Damped Oscillation......Page 345
Oscillation With a Rising Midline......Page 347
Acoustic Beats......Page 349
Relation Between Cartesian and Polar Coordinates......Page 352
Graphing Equations in Polar Coordinates......Page 354
Extending the Definition of Functions......Page 357
Algebra of Complex Numbers......Page 358
Euler’s Formula......Page 360
Polar Form of a Complex Number......Page 361
Euler’s Formula and Trigonometric Identities......Page 362
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER SEVEN......Page 364
CHECK YOUR UNDERSTANDING......Page 367
8 COMPOSITIONS, INVERSES, AND COMBINATIONS OF FUNCTIONS......Page 369
The Effect of a Drug on Heart Rates......Page 370
Formulas for Composite Functions......Page 371
Decomposition of Functions......Page 373
Definition of Inverse Function......Page 378
Finding a Formula for an Inverse Function......Page 379
Evaluating an Inverse Function Graphically......Page 381
A Property of Inverse Functions......Page 383
Restricting the Domain......Page 385
The Difference of Two Functions Defined by Formulas: A Measure of Prosperity......Page 389
The Quotient of Functions Defined by Formulas and Graphs: Prosperity......Page 392
The Quotient of Functions Defined by Tables: Per Capita Crime Rate......Page 393
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER EIGHT......Page 398
CHECK YOUR UNDERSTANDING......Page 401
9 POLYNOMIAL AND RATIONAL FUNCTIONS......Page 403
Proportionality and Power Functions......Page 404
The Effect of the Power p......Page 405
Finding the Formula for a Power Function......Page 408
9.2 POLYNOMIAL FUNCTIONS......Page 412
A General Formula for the Family of Polynomial Functions......Page 413
The Long-Run Behavior of Polynomial Functions......Page 414
Zeros of Polynomials......Page 415
9.3 THE SHORT-RUN BEHAVIOR OF POLYNOMIALS......Page 418
Factored Form, Zeros, and the Short-Run Behavior of a Polynomial......Page 419
Finding the Formula for a Polynomial from its Graph......Page 421
The Average Cost of Producing a Therapeutic Drug......Page 425
The Long-Run Behavior of Rational Functions......Page 426
What Causes Asymptotes?......Page 429
The Zeros and Vertical Asymptotes of a Rational Function......Page 431
The Graph of a Rational Function......Page 433
Finding a Formula for a Rational Function from its Graph......Page 434
When Numerator and Denominator Have the Same Zeros: Holes......Page 435
Comparing Power Functions......Page 439
Comparing Exponential Functions and Power Functions......Page 440
Comparing Log and Power Functions......Page 441
The Spread of AIDS......Page 444
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER NINE......Page 451
CHECK YOUR UNDERSTANDING......Page 455
TOOLS FOR CHAPTER 9: ALGEBRAIC FRACTIONS......Page 457
Reducing Fractions: Canceling......Page 458
Splitting Expressions......Page 459
10 VECTORS AND MATRICES......Page 463
Adding Displacements Using Triangles......Page 464
Vector Notation......Page 465
Addition of Vectors......Page 466
Subtraction of Vectors......Page 467
Scalar Multiplication......Page 469
Properties of Vector Addition and Scalar Multiplication......Page 470
Unit Vectors......Page 472
Resolving a Vector in the Plane into Components......Page 473
Vectors in n Dimensions......Page 475
Population Vectors......Page 477
Economics......Page 479
Physics......Page 480
Computer Graphics: Position Vectors......Page 481
10.4 THE DOT PRODUCT......Page 484
Properties of the Dot Product......Page 485
Work......Page 487
10.5 MATRICES......Page 490
Addition, Subtraction, and Scalar Multiplication......Page 491
Multiplication of a Matrix and a Vector......Page 492
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TEN......Page 499
CHECK YOUR UNDERSTANDING......Page 502
11 SEQUENCES AND SERIES......Page 503
Notation for Sequences......Page 504
Arithmetic Sequences......Page 505
Geometric Sequences......Page 506
Geometric Sequences and Exponential Functions......Page 507
Domestic Death from AIDS n years after 1980......Page 509
The Sum of an Arithmetic Series......Page 510
Summation Notation......Page 512
Bank Balance......Page 516
Geometric Series......Page 517
Drug Levels in The Body......Page 519
Long-term Drug Level in the Body......Page 521
Present Value of a Series of Payments......Page 523
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER ELEVEN......Page 527
CHECK YOUR UNDERSTANDING......Page 529
12 PARAMETRIC EQUATIONS AND CONIC SECTIONS......Page 531
The Path of the Sojourner Robot......Page 532
Other Parametric Curves......Page 537
12.2 IMPLICITLY DEFINED CURVES AND CIRCLES......Page 541
Circles......Page 542
12.3 ELLIPSES......Page 545
12.4 HYPERBOLAS......Page 549
A General Formula for Hyperbolas......Page 550
Using Geometry in Mathematical Models: Orbits......Page 553
Geometric Definitions of Conics......Page 554
Parabolas......Page 562
Properties of Hyperbolic Functions......Page 568
Identities Involving cosh x and sinh x......Page 569
Parameterizing the Hyperbola Using Hyperbolic Functions......Page 570
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TWELVE......Page 572
CHECK YOUR UNDERSTANDING......Page 574
ANSWERS TO ODD NUMBERED PROBLEMS......Page 575
INDEX......Page 605