Pracalculus through Modeling and Visualization, 2e

Table of Contents

Gary Rockswold

Chapter 1 INTRODUCTION TO FUNCTIONS AND GRAPHS

1.1 Numbers, Data, and Problem Solving

• Sets of Numbers
• Scientific Notation
• Problem Solving

1.2 Visualization of Data

• One-Variable Data
• Two-Variable Data
• The Distance Formula
• Visualizing Data with a Graphing Calculator

1.3 Functions and Their Representations

• Basic Concepts
• Representations of Functions
• Formal Definition of a Function
• Graphing Calculators and Functions
• Identifying Functions

1.4 Types of Functions and Their Rates of Change

• Constant Functions
• Linear Functions
• Slope as a Rate of Change
• Nonlinear Functions
• Average Rate of Change

Chapter 2 LINEAR FUNCTIONS AND EQUATIONS

2.1 Linear Functions; Constructing Models

• Exact and Approximate
• Graphs of Linear Functions
• Modeling with Linear Functions

2.2 Equations of Lines

• Forms for Equations of Lines
• Determining Intercepts
• Horizontal, Vertical, Parallel, and Perpendicular Lines
• Direct Variation

2.3 Linear Equations

• Equations
• The Intersection-of-Graphs Method
• Solving Linear Equations Symbolically
• Problem-Solving Strategies

2.4 Linear Inequalities

• Inequalities
• Texhniques for Solivng Inequalities
• Compound Inequalities
• Interval Notation

2.5 Piecewise-Defined Linear Functions

• Evaluating and Graphing Piecewise-Defined Functions
• The Greatest Integer Functions
• The Absolute Value Function
• Equations and Inequalities Involving Absolute Values

2.6 Linear Approximation and Curve Fitting

• Midpoint Formula
• Interpolation and Extrapolation
• Linear Regression

Chapter 3 QUADRATIC FUNCTIONS AND EQUATIONS

3.1 Quadratic Functions and Models

• Basic Concepts
• Vertex Form
• Models and Applications
• Quadratic Regression

3.2 Quadratic Equations and Problem Solving

• Basic Concepts
• Solving Quadratic Equations
• Problem Solving

3.3 Quadratic Inequalities

• Graphical Solutions
• Symbolic and Numerical Solutions

3.4 Transformations of Graphs

• Vertical and Horizontal Translations
• Stretching and Shrinking
• Reflections of Graphs
• Dynamically Displayed Data: An Application

Chapter 4 NONLINEAR FUNCTIONS AND EQUATIONS

4.1 Nonlinear Functions and Their Graphs

• Polynomial Functions
• Increasing and Decreasing Functions
• Extrema of Nonlinear Functions
• Symmetry

4.2 Polynomial Functions and Models

• Graphs of Polynomial Functions
• Polynomial Regression
• Piecewise-Defined Polynomial Functions 

4.3 Real Zeros of Polynomial Functions

• Division of Polynomials
• Factoring Polynomials
• Graphs and Multiple Zeros
• Rational Zeros
• Polynomial Equations

4.4 The Fundamental Theorem of Algebra

• Complex Numbers
• Quadratic Equations with Complex Solutions
• Fundamental Theorem of Algebra
• Polynomial Equations with Complex Solutions

4.5 Rational Functions and Models

• Rational Functions
• Vertical Asymptotes
• Horizontal Asymptotes
• Identifying Asymptotes
• Rational Equations
• Variation

4.6 Polynomial and Rational Inequalities

• Polynomial Inequalities
• Rational Inequalities

4.7 Power Functions and Radical Equations

• Rational Exponents and Radical Notation
• Power Functions and Models
• Equations Involving Radicals
• Power Regression

Chapter 5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

5.1 Combining Functions

• Arithmetic Operations on Functions
• Composition of Functions

5.2 Inverse Functions

• Inverse Operations
• One-to-One Functions
• Symbolic Representations of Inverse Functions
• Other Representations of Inverse Functions

5.3 Exponential Functions and Models

• Linear and Exponential Growth
• Exponential Models
• Compound Interest
• The Natural Exponential Function

5.4 Logarithmic Functions and Models

• The Common Logarithmic Function
• Solving Equations
• Logarithms with Other Bases
• General Logarithmic Equations

5.5 Properties of Logarithms

• Basic Properties of Logarithms
• Change of Base Formula

5.6 Exponential and Logarithmic Equations

• Exponential Equations
• Logarithmic Equations

5.7 Constructing Nonlinear Models

• Exponential Model
• Logarithmic Model
• Logistic Model

Chapter 6 TRIGONOMETRIC FUNCTIONS

6.1 Angles and their Measure

• Angles
• Degree Measure
• Arc Length
• Area of a Sector

6.2 Right Triangle Trigonometry

• Basic Concepts of Trigonometric Functions
• Applications of Right Triangle Trigonometry
• Complementary Angles and Cofunctions

6.3 The Sine and Cosine Functions and Their Graphs

• Definitions
• The Unit Circle
• Representations of the Sine and Cosine Functions
• Modeling with the Sine and Cosine Functions

6.4 Other Trigonometric Functions and Their Graphs

• Definitions and Basic Identities
• Representations of Other Trigonometric Functions
• Modeling with Trigonometric Functions

6.5 Modeling with Trigonometric Functions

• Transformations of Trigonometric Functions
• Models Involving Trigonometric Functions
• Simple Harmonic Motion

6.6 Inverse Trigonometric Functions

• Review of Inverses
• The Inverse Sine Function
• The Inverse Cosine Function
• The Inverse Tangent Function
• Solving Triangles and Equations

Chapter 7 TRIGONOMETRIC IDENTITIES AND EQUATIONS

7.1 Fundamental Identities

• Reciprocal and Quotient Identities
• Pythagorean Identities
• Negative-Angle Identities

7.2 Verifying Trigonometric Identities

• Simplifying Trigonometric Expressions
• Symbolic Verification with graphical and Numerical Support

7.3 Trigonometric Equations

• Referene Angles
• Solving Trigonometric Equations

7.4 Sum and Difference Identities

• Sum and Difference Identities for Cosine
• Other Sum and Difference Identities

Multiple-Angle Identities

• Double Angle Identities
• Half-Angle Identities
• Solving Equations
• Product-to-Sum and Sum-to-Product Identities

Chapter 8 FURTHER TOPICS IN GEOMETRY

8.1 Law of Sines

• Oblique Triangles
• Solving Triangles
• The Ambiguous Case

8.2 Law of Cosines

• Derivation of Law of Cosines
• Solving Trianges
• Area Formulas

8.3 Vectors

• Basic Concepts
• Operations on Vectors
• The Dot Product
• Work

8.4 Parametric Equations

• Basic Concepts
• Applications of Parametic Equations

8.5 Polar Equations

• Polar Coordinate System
• Graphs of Polar Equations
• Solving Polar Equations

8.6 Trigonometric Form and Roots of Complex Numbers

• Trigonometric Form
• Products and Quotients of Complex Numbers
• De Moivre's Theorem
• Roots of Complex Numbers

Chapter 9 SYSTEMS OF EQUATIONS AND INEQUALITIES

9.1 Functions And Equations of More Than One Variable

• Functions of Two Variables
• Systems of Equations
• The Method of Substitution
• Graphical and Numerical Methods
• Joint Variation

9.2 Linear Systems of Equations and Inequalities in Two Variables

• Types of Linear Systems in Two Variables
• The Elimination Method
• Systems of Inequalities
• Linear Programming

9.3 Solutions of Linear Systems Using Matrices

• Representing Systems of Linear Equations with Matrices
• Row-Echelon Form
• Gaussian Elimination
• Solving Systems of Linear Equations with Technology

9.4 Properties and Applications of Matrices

• Matrix Notation
• Sums, Differences, and Scalar Multiples of Matrices
• Matrix Products
• Technology and Matrices

9.5 Inverses of Matrices

• Matrix Inverses
• Representing Linear Systems with Matrix Equations
• Solving Linear Systems with Inverses
• Finding Inverses Symbolically

9.6 Determinants

• Definitions and Calculation of Determinants
• Area of Regions
• Cramer’s Rule
• Limitations on the Method of Cofactors and Cramer’s Rule

Chapter 10 CONIC SECTIONS AND NONLINEAR SYSTEMS

10.1 Parabolas

• Equations and Graphs of Parabolas
• Reflective Property of Parabolas
• Translations of Parabolas

10.2 Ellipses

• Equations and Graphs of Ellipses
• Translations of Ellipses
• Reflective Property of Ellipses
• Circles
• Solving Systems of Equations and Inequalities

10.3 Hyperbolas

• Equations and Graphs of Hyperbolas
• Translations of Hyperbolas
• Reflective Property of Hyperbolas

Chapter 11 FURTHER TOPICS IN ALGEBRA

11.1 Sequences

• Sequences and Functions
• Representations of Sequences
• Arithmetic Sequences
• Geometric Sequences

11.2 Series

• Basic Concepts
• Arithmetic Series
• Geometric Series
• Summation Notation

11.3 Counting

• Fundamental Counting Principle
• Permutations
• Combinations

11.4 The Binomial Theorem

• Derivation of the Binomial Theorem
• Pascal’s Triangle

11.5 Probability

• Definition of Probability
• Compound Events
• Independent and Dependent Events

Chapter R REFERENCE: BASIC CONCEPTS FROM ALGEBRA AND GEOMETRY

R.1 Formulas from Geometry

• Geometric Shapes in a Plane
• The Pythagorean Theorem
• Three-Dimensional Objects
• Similar Triangles
• A Summary of Geometric Formulas

R.2 Circles

• Equations and Graphs of Circles
• Finding the Center and Radius of a Circle

R.3 Integer Exponents

• Bases and Positive Exponents
• Zero and Negative Exponents
• Product, Quotient, and Power Rules

R.4 Polynomial Expressions

• Addition and Subtraction of Monomials
• Addition and Subtraction of Polynomials
• Distributive Properties
• Multiplying Polynomials
• Some Special Products

R.5 Factoring Polynomials

• Common Factors
• Factoring by Grouping
• Factoring Trinomials
• Difference of Two Squares
• Perfect Square Trinomials
• Sum and Difference of Two Cubes

R.6 Rational Expressions

• Multiplication and Division of Rational Expressions
• Common Denominators
• Addition and Subtraction of Rational Expressions
• Clearing Fractions
• Complex Fractions

Appendix A  A Library of Functions

Appendix B  The TI-83/83 Plus Graphing Calculators

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