Functions Modeling Change A Preparation For Calculus, 5th Edition

Cover
Title Page
Copyright Page
PREFACE
Table of Contents
1 LINEAR FUNCTIONS AND CHANGE
1.1 FUNCTIONS AND FUNCTION NOTATION
Representing Functions: Words, Tables, Graphs, and Formulas
Mathematical Models
Function Notation
Functions Don’t Have to Be Defined by Formulas
When Is a Relationship Not a Function?
1.2 RATE OF CHANGE
Rate of Change of a Function
Increasing and Decreasing Functions
Function Notation for Average Rate of Change
1.3 LINEAR FUNCTIONS
Constant Rate of Change
A General Formula for the Family of Linear Functions
Tables for Linear Functions
Warning: Not All Graphs That Look Like Lines Represent Linear Functions
1.4 FORMULAS FOR LINEAR FUNCTIONS
Finding a Formula for a Linear Function from a Table of Data
Finding a Formula for a Linear Function from a Graph
Finding a Formula for a Linear Function from a Verbal Description
Alternative Forms for the Equation of a Line
Equations of Horizontal and Vertical Lines
Slopes of Parallel and Perpendicular Lines
Justification of Formula for Slopes of Perpendicular Lines
1.5 MODELING WITH LINEAR FUNCTIONS
Interpreting the Parameters of a Linear Function
The Effect of the Parameters on the Graph of a Linear Function
Intersection of Two Lines
Linear Inequalities in Two Variables
1.6 FITTING LINEAR FUNCTIONS TO DATA
Laboratory Data: The Viscosity of Motor Oil
Interpolation and Extrapolation
How Regression Works
Correlation
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER ONE
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER ONE: LINEAR EQUATIONS AND THE COORDINATE PLANE
Solving Linear Equations
Solving Exactly Versus Solving Approximately
Solving Linear Inequalities
Systems of Linear Equations
Intersection of Two Lines
2 FUNCTIONS
2.1 INPUT AND OUTPUT
Finding Output Values: Evaluating a Function
Finding Input Values: Solving Equations
Finding Output and Input Values From Tables and Graphs
2.2 DOMAIN AND RANGE
Choosing Realistic Domains and Ranges
Using a Graph to Find the Domain and Range of a Function
Using Formulas and Graphs to Find Domains and Ranges
2.3 PIECEWISE-DEFINED FUNCTIONS
The Absolute Value Function
Visualizing Absolute Value on the Number Line
2.4 PREVIEW OF TRANSFORMATIONS: SHIFTS
Vertical Shift: The Heating Schedule for an Office Building
Horizontal Shift: The Heating Schedule
Inside Versus Outside Changes
Combining Horizontal and Vertical Shifts
2.5 PREVIEW OF COMPOSITE AND INVERSE FUNCTIONS
Composition of Functions
Inverse Functions
Relation Between Composition and Inverses: A Function and Its Inverse Undo Each Other
2.6 CONCAVITY
Concavity and Rates of Change
Summary: Increasing and Decreasing Functions; Concavity
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TWO
STRENGTHEN YOUR UNDERSTANDING
3 QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
Finding the Zeros of a Quadratic Function
Concavity and Rates of Change for Quadratic Functions
Formulas for Quadratic Functions
3.2 THE VERTEX OF A PARABOLA
The Vertex Form of a Quadratic Function
Modeling with Quadratic Functions
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER THREE
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER 3: QUADRATIC EQUATIONS
Deriving the Quadratic Formula
4 EXPONENTIAL FUNCTIONS
4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS
Increasing at a Constant Percent Rate: Exponential Growth
Growth Factors and Percent Growth Rates
A General Formula for the Family of Exponential Functions
Projections into the Future
4.2 COMPARING EXPONENTIAL AND LINEAR FUNCTIONS
Identifying Linear and Exponential Functions From a Table
Finding a Formula for an Exponential Function
Formulas and Rates of Change: Linear Versus Exponential
4.3 GRAPHS OF EXPONENTIAL FUNCTIONS
Graphs of the Exponential Family: The Effect of the Parameter a
Graphs of the Exponential Family: The Effect of the Parameter b
Solving Exponential Equations Graphically
Finding an Exponential Function for Data
4.4 APPLICATIONS TO COMPOUND INTEREST
4.5 THE NUMBER e
Exponential Functions with Base e
Connection: The Number e and Compound Interest
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER FOUR
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER 4: EXPONENTS
5 LOGARITHMIC FUNCTIONS
5.1 LOGARITHMS AND THEIR PROPERTIES
What Is a Logarithm?
Logarithmic and Exponential Functions Are Inverses
Properties of Logarithms
The Natural Logarithm
Justification of log(a • b) = log a + log b and log(a/b) = log a – log b
Justification of log(bt) = t • log b
5.2 LOGARITHMS AND EXPONENTIAL MODELS
Doubling Time
Half-Life
Converting Between Q = abt and Q = aekt
Exponential Growth Problems That Cannot Be Solved by Logarithms
5.3 THE LOGARITHMIC FUNCTION AND ITS APPLICATIONS
The Graph, Domain, and Range of the Common Logarithm
Graph of Natural Logarithm: Rate of Change and Concavity
Applications of Logarithms
Vertical Asymptotes
5.4 LOGARITHMIC SCALES
The Solar System and Beyond
Logs of Small Numbers
Log-Log Scales
Using Logs to Fit Nonlinear Functions to Data
The iTunes Store
Summary: Fitting Exponential and Power Functions to Data
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER FIVE
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER: LOGARITHMS
6 TRANSFORMATIONS OF FUNCTIONS AND THEIR GRAPHS
6.1 SHIFTS, REFLECTIONS, AND SYMMETRY
Reflections
Combining Shifts and Reflections
Symmetry Under Reflection: Even and Odd Functions
6.2 VERTICAL STRETCHES AND COMPRESSIONS
Vertical Stretch: An Amplifier
Negative Stretch Factor
Formula for Vertical Stretch or Compression
Stretch Factors and Average Rates of Change
Combining Stretches and Shifts
6.3 HORIZONTAL STRETCHES AND COMBINATIONS OF TRANSFORMATIONS
Horizontal Stretch: A Lighthouse Beacon
Formula for Horizontal Stretch or Compression
Combining Transformations
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER SIX
STRENGTHEN YOUR UNDERSTANDING
7 TRIGONOMETRY AND PERIODIC FUNCTIONS
7.1 INTRODUCTION TO PERIODIC FUNCTIONS
The London Eye Ferris Wheel
Ferris Wheel Height as a Function of Time
Graphing the Ferris Wheel Function
Average Rate of Change: How Fast We Move Up Depends on Where We Are
Periodic Functions: Period, Midline, and Amplitude
7.2 THE SINE AND COSINE FUNCTIONS
Height on the FerrisWheel as a Function of Position
Height on the FerrisWheel as a Function of Angle
The Unit Circle
The Sine and Cosine Functions
Coordinates of a Point on a Circle of Radius r
7.3 RADIANS AND ARC LENGTH
Definition of a Radian
Arc Length in Circle of Radius r
Sine and Cosine of a Number
Exact Values of the Sine and Cosine
7.4 GRAPHS OF THE SINE AND COSINE
Amplitude and Midline: Vertical Stretches and Shifts of the Sine and Cosine
7.5 SINUSOIDAL FUNCTIONS
Period
Horizontal Shift
Summary of Transformations
Optional: Phase Shift
7.6 THE TANGENT FUNCTION
7.7 TRIGONOMETRIC RELATIONSHIPS AND IDENTITIES
The Pythagorean Identity
Relationships Between the Graphs of the Sine, Cosine, and Tangent
Relationships Involving Reciprocals of the Trigonometric Functions
Summarizing the Trigonometric Relationships
7.8 INVERSE TRIGONOMETRIC FUNCTIONS
The Inverse Cosine Function
The Inverse Sine and Inverse Tangent Functions
Solving Using the Unit Circle
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER SEVEN
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER 7: SPECIAL ANGLES
Other Values of the Sine and Cosine Functions
8 TRIANGLE TRIGONOMETRY AND POLAR COORDINATES
8.1 TRIG FUNCTIONS AND RIGHT TRIANGLES
The Sine and Cosine Functions in Right Triangles
The Tangent Function in Right Triangles
8.2 NON-RIGHT TRIANGLES
The Law of Cosines
The Law of Sines
8.3 POLAR COORDINATES
Relation Between Cartesian and Polar Coordinates
Graphing Equations in Polar Coordinates
Graphing Inequalities in Polar Coordinates
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER EIGHT
STRENGTHEN YOUR UNDERSTANDING
9 TRIGONOMETRIC IDENTITIES, MODELS, AND COMPLEX NUMBERS
9.1 TRIGONOMETRIC EQUATIONS
Finding Solutions Graphically
Finding Solutions Algebraically
Solving Trigonometric Equations
9.2 IDENTITIES, EXPRESSIONS, AND EQUATIONS
Equations Versus Identities
Double-Angle Formula for Sine
Double-Angle Formulas for Cosine and Tangent
9.3 SUM AND DIFFERENCE FORMULAS FOR SINE AND COSINE
Sums and Differences of Angles
9.4 TRIGONOMETRIC MODELS AND SUM IDENTITIES
Damped Oscillation
Oscillation With a Rising Midline
Sums and Differences of Sines and Cosines: Same Periods
Rewriting a1 sin Bt + a2 cos Bt
Modeling with Sums of Trigonometric Functions with the Same Period
9.5 SUMS WITH DIFFERENT PERIODS AND ACCOUSTIC BEATS
Sums and Differences of Sines and Cosines: Same Amplitudes
Acoustic Beats
9.6 COMPLEX NUMBERS AND DE MOIVRE’S THEOREM
Using Complex Numbers to Solve Equations
Algebra of Complex Numbers
The Complex Plane and Polar Coordinates
Euler’s Formula
Polar Form of a Complex Number
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER NINE
STRENGTHEN YOUR UNDERSTANDING
10 COMPOSITIONS, INVERSES, AND COMBINATIONS OF FUNCTIONS
10.1 COMPOSITION OF FUNCTIONS
Decomposition of Functions
10.2 INVERTIBILITY AND PROPERTIES OF INVERSE FUNCTIONS
Definition of Inverse Function
Notation and Formulas for Inverse Functions
Non-invertible Functions: Horizontal Line Test
Evaluating an Inverse Function Graphically
The Graph, Domain, and Range of an Inverse Function
A Property of Inverse Functions
Restricting the Domain to Create an Inverse
10.3 COMBINATIONS OF FUNCTIONS
The Difference of Two Functions Defined by Formulas: A Measure of Prosperity
Factoring a Function’s Formula into a Product
The Quotient of Functions Defined by Formulas and Graphs: Prosperity
The Quotient of Functions Defined by Tables: Per-Capita Crime Rate
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TEN
STRENGTHEN YOUR UNDERSTANDING
11 POLYNOMIAL AND RATIONAL FUNCTIONS
11.1 POWER FUNCTIONS AND PROPORTIONALITY
Graphs of Power Functions
The Effect of the Power p
Dominance
Finding the Formula for a Power Function
Asymptotes and Limit Notation
Power Functions and Proportionality
11.2 POLYNOMIAL FUNCTIONS
A General Formula for the Family of Polynomial Functions
The Long-Run Behavior of Polynomial Functions
Zeros of Polynomials
11.3 THE SHORT-RUN BEHAVIOR OF POLYNOMIALS
Factored Form, Zeros, and the Short-Run Behavior of a Polynomial
Finding the Formula for a Polynomial from its Graph
11.4 RATIONAL FUNCTIONS
The Average Cost of Producing a Therapeutic Drug
What Is a Rational Function?
The Long-Run Behavior of Rational Functions
What Causes Asymptotes?
11.5 THE SHORT-RUN BEHAVIOR OF RATIONAL FUNCTIONS
The Zeros and Vertical Asymptotes of a Rational Function
The Graph of a Rational Function
Rational Functions as Transformations of Power Functions
Finding a Formula for a Rational Function from its Graph
When Numerator and Denominator Have the Same Zeros: Holes
11.6 COMPARING POWER, EXPONENTIAL, AND LOG FUNCTIONS
Comparing Power Functions
Comparing Exponential Functions and Power Functions
Comparing Log and Power Functions
11.7 FITTING EXPONENTIALS AND POLYNOMIALS TO DATA
The Spread of AIDS
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER ELEVEN
STRENGTHEN YOUR UNDERSTANDING
SKILLS REFRESHER FOR CHAPTER 11: ALGEBRAIC FRACTIONS
12 VECTORS AND MATRICES
12.1 VECTORS
Distance Versus Displacement
Adding Displacements Using Triangles
Vectors
Vector Notation
Addition of Vectors
Subtraction of Vectors
Scalar Multiplication
Properties of Vector Addition and Scalar Multiplication
12.2 THE COMPONENTS OF A VECTOR
Unit Vectors
Resolving a Vector in the Plane into Components
Displacement Vectors
Vectors in n Dimensions
Lengths of Vectors and Deciding Whether Vectors Are Parallel
12.3 APPLICATIONS OF VECTORS
Alternate Notation for the Components
Population Vectors
Economics
Physics
Computer Graphics: Position Vectors
12.4 THE DOT PRODUCT
Properties of the Dot Product
Work
12.5 MATRICES
Addition, Subtraction, and Scalar Multiplication
Multiplication of a Matrix and a Vector
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER TWELVE
STRENGTHEN YOUR UNDERSTANDING
13 SEQUENCES AND SERIES
13.1 SEQUENCES
Notation for Sequences
Arithmetic Sequences
Geometric Sequences
Geometric Sequences and Exponential Functions
13.2 DEFINING FUNCTIONS USING SUMS: ARITHMETIC SERIES
Domestic Deaths from AIDS n years after 1980
Arithmetic Series
The Sum of an Arithmetic Series
Summation Notation
13.3 FINITE GEOMETRIC SERIES
Bank Balance
Geometric Series
Drug Levels in the Body
13.4 INFINITE GEOMETRIC SERIES
Long-term Drug Level in the Body
Present Value of a Series of Payments
CHAPTER SUMMARY
REVIEW EXERCISES AND PROBLEMS FOR CHAPTER THIRTEEN
STRENGTHEN YOUR UNDERSTANDING
ANSWERS TO ODD PROBLEMS
INDEX
FORMULA SUMMARY: ALGEBRA
TRIGONOMETRIC IDENTITIES
TABLE OF GREEK LETTERS
EULA