This is completed downloadable of Precalculus Enhanced with Graphing Utilities 7th Edition Sullivan Test Bank

Product Details:

ISBN-10 ‏ : ‎ 0134119282
ISBN-13 ‏ : ‎ 978-0134119281
Author: Michael Sullivan

Prepare, Practice, Review The Sullivan’s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Series has evolved to meet today’s course needs by integrating the usage of graphing calculators, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors. In the Seventh Edition, there are several new features that appear in both the text and MyMathLab. Retain Your Knowledge problems offer the type of “final exam material” that students can use to maintain their skills throughout each chapter.

Table of Content:

1 Graphs
Outline
1.1The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
PREPARING FOR THIS SECTION
Objectives
Rectangular Coordinates
Graphing Utilities
Use the Distance Formula
Example 1 Finding the Distance between Two Points
Solution
Proof of the Distance Formula
Example 2 Finding the Length of a Line Segment
Solution
Example 3 Using Algebra to Solve Geometry Problems
Solution
Use the Midpoint Formula
Example 4 Finding the Midpoint of a Line Segment
Solution
Graph Equations by Plotting Points
Example 5 Determining Whether a Point Is on the Graph of an Equation
Solution
Example 6 Graphing an Equation by Plotting Points
Step-by-Step Solution
Example 7 Graphing an Equation by Plotting Points
Solution
Graph Equations Using a Graphing Utility
Example 8 Expressing an Equation in the Form y = {expression in x}
Solution
Example 9 Graphing an Equation Using a Graphing Utility
Step-by-Step Solution
Use a Graphing Utility to Create Tables
Example 10 Creating a Table Using a Graphing Utility
Step-by-Step Solution
Find Intercepts from a Graph
Example 11 Finding Intercepts from a Graph
Solution
Use a Graphing Utility to Approximate Intercepts
Example 12 Approximating Intercepts Using a Graphing Utility
Solution
1.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
‘Are You Prepared?’ Answers
1.2 Intercepts; Symmetry; Graphing Key Equations
PREPARING FOR THIS SECTION
Objectives
Find Intercepts Algebraically from an Equation
Example 1 Finding Intercepts from an Equation
Solution
Test an Equation for Symmetry
Example 2 Symmetric Points
Example 3 Finding Intercepts and Testing an Equation for Symmetry
Solution
Seeing the Concept
Know How to Graph Key Equations
Example 4 Graphing the Equation y = x3 by Finding Intercepts and Checking for Symmetry
Solution
Example 5 Graphing the Equation x = y2
Solution
Example 6 Graphing the Equation y=1x
Solution
1.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
1.3 Solving Equations Using a Graphing Utility
PREPARING FOR THIS SECTION
Objective
Solve Equations Using a Graphing Utility
Example 1 Using ZERO (or ROOT) to Approximate Solutions of an Equation
Solution
Example 2 Using INTERSECT to Approximate Solutions of an Equation
Solution
1.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
1.4 Lines
Objectives
Calculate and Interpret the Slope of a Line
Example 1 Finding and Interpreting the Slope of a Line Given Two Points
Square Screens
Exploration
Exploration
Graph Lines Given a Point and the Slope
Example 2 Graphing a Line Given a Point and a Slope
Solution
Find the Equation of a Vertical Line
Example 3 Graphing a Line
Solution
Use the Point–Slope Form of a Line; Identify Horizontal Lines
Example 4 Using the Point–Slope Form of a Line
Example 5 Finding the Equation of a Horizontal Line
Solution
Write the Equation of a Line in Slope–Intercept Form
Seeing the Concept
Seeing the Concept
Example 6 Finding the Slope and y-Intercept
Solution
Find the Equation of a Line Given Two Points
Example 7 Finding an Equation of a Line Given Two Points
Solution
Graph Lines Written in General Form Using Intercepts
Example 8 Graphing an Equation in General Form Using Its Intercepts
Solution
Find Equations of Parallel Lines
Example 9 Showing That Two Lines Are Parallel
Solution
Example 10 Finding a Line That Is Parallel to a Given Line
Solution
Find Equations of Perpendicular Lines
Proof
Example 11 Finding the Slope of a Line Perpendicular to Another Line
Example 12 Finding the Equation of a Line Perpendicular to a Given Line
Solution
1.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
1.5 Circles
PREPARING FOR THIS SECTION
OBJECTIVES
Write the Standard Form of the Equation of a Circle
Example 1 Writing the Standard Form of the Equation of a Circle
Solution
Graph a Circle by Hand and by Using a Graphing Utility
Example 2 Graphing a Circle by Hand and by Using a Graphing Utility
Solution
Example 3 Finding the Intercepts of a Circle
Solution
Work with the General Form of the Equation of a Circle
Example 4 Graphing a Circle Whose Equation Is in General Form
Solution
Example 5 Finding the General Equation of a Circle
Solution
Overview
1.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Chapter Projects
2 Functions and Their Graphs
Outline
A Look Back
A Look Ahead
2.1 Functions
Preparing for this Section
OBJECTIVES
Determine Whether a Relation Represents a Function
Example 1 Maps and Ordered Pairs as Relations
EXAMPLE 2 Determining Whether a Relation Is a Function
Solution
EXAMPLE 3 Determining Whether a Relation Is a Function
Solution
EXAMPLE 4 Determining Whether an Equation Is a Function
Solution
EXAMPLE 5 Determining Whether an Equation Is a Function
Solution
Find the Value of a Function
EXAMPLE 6 Finding Values of a Function
Solution
EXAMPLE 7 Finding Values of a Function on a Calculator
Comment
Implicit Form of a Function
Find the Difference Quotient of a Function
EXAMPLE 8 Finding the Difference Quotient of a Function
Solution
Find the Domain of a Function Defined by an Equation
EXAMPLE 9 Finding the Domain of a Function
Solution
EXAMPLE 10 Finding the Domain in an Application
Solution
Form the Sum, Difference, Product, and Quotient of Two Functions
EXAMPLE 11 Operations on Functions
Solution
2.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
2.2 The Graph of a Function
Preparing for this Section
OBJECTIVES
Identify the Graph of a Function
EXAMPLE 1 Identifying the Graph of a Function
Solution
Obtain Information from or about the Graph of a Function
EXAMPLE 2 Obtaining Information from the Graph of a Function
Solution
EXAMPLE 3 Obtaining Information about the Graph of a Function
Solution
EXAMPLE 4 Average Cost Function
Solution
2.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
2.3 Properties of Functions
Preparing for This Section
Objectives
Determine Even and Odd Functions from a Graph
EXAMPLE 1 Determining Even and Odd Functions from the Graph
Solution
Identify Even and Odd Functions from an Equation
EXAMPLE 2 Identifying Even and Odd Functions
Solution
Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant
EXAMPLE 3 Determining Where a Function Is Increasing, Decreasing, or Constant from Its Graph
Solution
Use a Graph to Locate Local Maxima and Local Minima
EXAMPLE 4 Finding Local Maxima and Local Minima from the Graph of a Function and Determining Where the Function Is Increasing, Decreasing, or Constant
Solution
Use a Graph to Locate the Absolute Maximum and the Absolute Minimum
EXAMPLE 5 Finding the Absolute Maximum and the Absolute Minimum from the Graph of a Function
Solution
Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing
EXAMPLE 6 Using a Graphing Utility to Approximate Local Maxima and Minima and to Determine Where a Function Is Increasing or Decreasing
Solution
Find the Average Rate of Change of a Function
EXAMPLE 7 Finding the Average Rate of Change
Solution
The Secant Line
EXAMPLE 8 Finding the Equation of a Secant Line
Solution
2.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
2.4 Library of Functions; Piecewise-defined Functions
Preparing for This Section
Objectives
Graph the Functions Listed in the Library of Functions
EXAMPLE 1 Graphing the Cube Root Function
Solution
EXAMPLE 2 Graphing the Absolute Value Function
Solution
Seeing the Concept
Comment
Graph Piecewise-defined Functions
EXAMPLE 3 Graphing a Piecewise-defined Function
Solution
EXAMPLE 4 Analyzing a Piecewise-defined Function
Solution
EXAMPLE 5 Cost of Electricity
Solution
2.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
2.5 Graphing Techniques: Transformations
Objectives
Graph Functions Using Vertical and Horizontal Shifts
Exploration
Result
Example 1 Vertical Shift Down
Solution
Exploration
Result
Example 2 Combining Vertical and Horizontal Shifts
Solution
Graph Functions Using Compressions and Stretches
Exploration
Result
Exploration
Result
Example 3 Graphing Using Stretches and Compressions
Solution
Graph Functions Using Reflections about the x-Axis or y-Axis
Exploration
Result
Exploration
Result
Example 4 Determining the Function Obtained from a Series of Transformations
Solution
Example 5 Combining Graphing Procedures
Solution
Example 6 Combining Graphing Procedures
Solution
2.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
2.6 Mathematical Models: Building Functions
Objective
Build and Analyze Functions
Example 1 Finding the Distance from the Origin to a Point on a Graph
Solution
Example 2 Area of a Rectangle
Solution
Example 3 Close Call?
Solution
2.6 Assess Your Understanding
Applications and Extensions
Chapter Review
Library of Functions
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
3 Linear and Quadratic Functions
Outline
A Look Back
A Look Ahead
3.1 Properties of Linear Functions and Linear Models
Preparing for This Section
Objectives
1 Graph Linear Functions
Example 1 Graphing a Linear Function
Solution
2 Use Average Rate of Change to Identify Linear Functions
Proof
Example 2 Using the Average Rate of Change to Identify Linear Functions
Solution
3 Determine Whether a Linear Function Is Increasing, Decreasing, or Constant
Example 3 Determining Whether a Linear Function Is Increasing, Decreasing, or Constant
Solution
4 Build Linear Models from Verbal Descriptions
Example 4 Straight-line Depreciation
Solution
Example 5 Supply and Demand
Solution
3.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Mixed Practice
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
3.2 Building Linear Models from Data
Preparing for This Section
Objectives
1 Draw and Interpret Scatter Diagrams
Example 1 Drawing and Interpreting a Scatter Diagram
Solution
2 Distinguish between Linear and Nonlinear Relations
Example 2 Distinguishing between Linear and Nonlinear Relations
Solution
Example 3 Finding a Model for Linearly Related Data
Solution
3 Use a Graphing Utility to Find the Line of Best Fit
Example 4 Finding a Model for Linearly Related Data
Solution
3.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Mixed Practice
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
3.3 Quadratic Functions and Their Properties
Preparing for This Section
Objectives
Quadratic Functions
1 Graph a Quadratic Function Using Transformations
Example 1 Graphing a Quadratic Function Using Transformations
Solution
2 Identify the Vertex and Axis of Symmetry of a Quadratic Function
Example 2 Locating the Vertex without Graphing
Solution
3 Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts
Example 3 How to Graph a Quadratic Function by Hand Using Its Properties
Step-by-Step Solution
Example 4 Graphing a Quadratic Function Using Its Vertex, Axis, and Intercepts
Solution
Example 5 Graphing a Quadratic Function Using Its Vertex, Axis, and Intercepts
Solution
4 Find a Quadratic Function Given Its Vertex and One Other Point
Example 6 Finding the Quadratic Function Given Its Vertex and One Other Point
Solution
5 Find the Maximum or Minimum Value of a Quadratic Function
Example 7 Finding the Maximum or Minimum Value of a Quadratic Function
Solution
3.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
3.4 Build Quadratic Models from Verbal Descriptions and from Data
Preparing For This Section
Objectives
1 Build Quadratic Models from Verbal Descriptions
Example 1 Maximizing Revenue
Solution
Example 2 Maximizing the Area Enclosed by a Fence
Solution
Example 3 Analyzing the Motion of a Projectile
Solution
Example 4 The Golden Gate Bridge
Solution
2 Build Quadratic Models from Data
Example 5 Fitting a Quadratic Function to Data
Solution
3.4 Assess Your Understanding
Applications and Extensions
Mixed Practice
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
3.5 Inequalities Involving Quadratic Functions
Preparing For This Section
Objective
1 Solve Inequalities Involving a Quadratic Function
Example 1 Solving an Inequality
By Hand Solution
Graphing Utility Solution
Example 2 Solving an Inequality
Solution
Example 3 Solving an Inequality
Solution
3.5 Assess Your Understanding
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
4 Polynomial and Rational Functions
Outline
A Look Back
A Look Ahead
4.1 Polynomial Functions and Models
PREPARING FOR THIS SECTION
Objectives
Identify Polynomial Functions and Their Degree
Example 1 Identifying Polynomial Functions
Solution
Power Functions
Exploration
Exploration
Graph Polynomial Functions Using Transformations
Example 2 Graphing a Polynomial Function Using Transformations
Solution
Example 3 Graphing a Polynomial Function Using Transformations
Solution
Identify the Real Zeros of a Polynomial Function and Their Multiplicity
Example 4 Finding a Polynomial Function from Its Zeros
Solution
Seeing the Concept
Example 5 Identifying Zeros and Their Multiplicities
Example 6 Investigating the Role of Multiplicity
Solution
Turning Points
Exploration
Example 7 Identifying the Graph of a Polynomial Function
Solution
End Behavior
Example 8 Identifying the Graph of a Polynomial Function
Solution
Example 9 Writing a Polynomial Function from Its Graph
Solution
Analyze the Graph of a Polynomial Function
Example 10 How to Analyze the Graph of a Polynomial Function
Step-by-Step Solution
Example 11 How to Use a Graphing Utility to Analyze the Graph of a Polynomial Function
Step-by-Step Solution
Build Cubic Models from Data
Example 12 A Cubic Function of Best Fit
Solution
4.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
4.2 The Real Zeros of a Polynomial Function
PREPARING FOR THIS SECTION
Objectives
Use the Remainder and Factor Theorems
Example 1 Using the Remainder Theorem
Solution
Comment
Proof
Example 2 Using the Factor Theorem
Solution
Proof
Use Descartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function
Example 3 Using the Number of Real Zeros Theorem and Descartes’ Rule of Signs
Solution
Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function
Example 4 Listing Potential Rational Zeros
Solution
Find the Real Zeros of a Polynomial Function
Example 5 How to Find the Real Zeros of a Polynomial Function
Step-by-Step Solution
Example 6 Finding the Real Zeros of a Polynomial Function
Solution
Solve Polynomial Equations
Example 7 Solving a Polynomial Equation
Solution
Use the Theorem for Bounds on Zeros
Proof (Outline)
Example 8 Finding Upper and Lower Bounds of Zeros
Solution
Example 9 Obtaining Graphs Using Bounds on Zeros
Solution
Example 10 Finding the Real Zeros of a Polynomial Function
Solution
Use the Intermediate Value Theorem
Example 11 Using the Intermediate Value Theorem and a Graphing Utility to Locate Zeros
Solution
4.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Discussion and Writing
4.3 Complex Zeros; Fundamental Theorem of Algebra
PREPARING FOR THIS SECTION
Objectives
Proof
Use the Conjugate Pairs Theorem
Proof
Proof
Example 1 Using the Conjugate Pairs Theorem
Solution
Find a Polynomial Function with Specified Zeros
Example 2 Finding a Polynomial Function Whose Zeros Are Given
Solution
Proof
Find the Complex Zeros of a Polynomial Function
Example 3 Finding the Complex Zeros of a Polynomial Function
Solution
4.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Discussion and Writing
4.4 Properties of Rational Functions
Preparing for This Section
Objectives
Find the Domain of a Rational Function
Example 1 Finding the Domain of a Rational Function
Example 2 Graphing y=1×2
Solution
Example 3 Using Transformations to Graph a Rational Function
Solution
Asymptotes
Exploration
Result
Find the Vertical Asymptotes of a Rational Function
Example 4 Finding Vertical Asymptotes
Solution
Exploration
Result
Find the Horizontal or Oblique Asymptote of a Rational Function
Example 5 Finding a Horizontal Asymptote
Solution
Example 6 Finding a Horizontal or Oblique Asymptote
Solution
Example 7 Finding a Horizontal or Oblique Asymptote
Solution
Example 8 Finding a Horizontal or Oblique Asymptote
Solution
4.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
4.5 The Graph of a Rational Function
Preparing for This Section
Objectives
Analyze the Graph of a Rational Function
Example 1 How to Analyze the Graph of a Rational Function
Example 2 Analyzing the Graph of a Rational Function
Solution
Example 3 Analyzing the Graph of a Rational Function
Solution
Example 4 Analyzing the Graph of a Rational Function
Solution
Example 5 Analyzing the Graph of a Rational Function with a Hole
Solution
Example 6 Constructing a Rational Function from Its Graph
Solution
Solve Applied Problems Involving Rational Functions
Example 7 Finding the Least Cost of a Can
Solution
4.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
4.6 Polynomial and Rational Inequalities
Preparing for This Section
Objectives
Solve Polynomial Inequalities Algebraically and Graphically
Example 1 Solving a Polynomial Inequality Using Its Graph
Example 2 How to Solve a Polynomial Inequality Algebraically
The Role of Multiplicity in Solving Polynomial Inequalities
Solve Rational Inequalities Algebraically and Graphically
Example 3 Solving a Rational Inequality Using Its Graph
Example 4 How to Solve a Rational Inequality Algebraically
The Role of Multiplicity in Solving Rational Inequalities
4.6 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
5 Exponential and Logarithmic Functions
Outline
A Look Back
A Look Ahead
5.1 Composite Functions
Objectives
1 Form a Composite Function
Example 1 Evaluating a Composite Function
Solution
Comment
2 Find the Domain of a Composite Function
Example 2 Finding a Composite Function and Its Domain
Solution
Example 3 Finding the Domain of f ∘ g
Solution
Example 4 Finding a Composite Function and Its Domain
Solution
Example 5 Showing That Two Composite Functions Are Equal
Solution
Calculus Application
Example 6 Finding the Components of a Composite Function
Solution
Example 7 Finding the Components of a Composite Function
Solution
5.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
5.2 One-to-One Functions; Inverse Functions
Objectives
1 Determine Whether a Function Is One-to-One
Example 1 Determining Whether a Function Is One-to-One
Solution
Example 2 Using the Horizontal-line Test
Solution
2 Determine the Inverse of a Function Defined by a Map or a Set of Ordered Pairs
Example 3 Finding the Inverse of a Function Defined by a Map
Solution
Example 4 Finding the Inverse of a Function Defined by a Set of Ordered Pairs
Solution
Example 5 Verifying Inverse Functions
Solution
Example 6 Verifying Inverse Functions
Solution
3 Obtain the Graph of the Inverse Function from the Graph of the Function
Example 7 Graphing the Inverse Function
Solution
4 Find the Inverse of a Function Defined by an Equation
Example 8 How to Find the Inverse Function
Step-by-Step Solution
Procedure for Finding the Inverse of a One-to-One Function
Example 9 Finding the Inverse Function
Solution
Example 10 Finding the Inverse of a Domain-restricted Function
Solution
Summary
5.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.3 Exponential Functions
Objectives
1 Evaluate Exponential Functions
Example 1 Using a Calculator to Evaluate Powers of 2
Solution
Introduction to Exponential Growth
Proof
Example 2 Identifying Linear or Exponential Functions
Solution
2 Graph Exponential Functions
Example 3 Graphing an Exponential Function
Solution
Properties of the Exponential Function f(x) = ax, a > 1
Example 4 Graphing an Exponential Function
Solution
Properties of the Exponential Function f(x) = ax, 0 < a < 1
Example 5 Graphing Exponential Functions Using Transformations
Solution
3 Define the Number e
Example 6 Graphing Exponential Functions Using Transformations
Solution
4 Solve Exponential Equations
Example 7 Solving an Exponential Equation
Algebraic Solution
Graphing Solution
Example 8 Solving an Exponential Equation
Solution
Example 9 Exponential Probability
Solution
Summary
Properties of the Exponential Function
5.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.4 Logarithmic Functions
Objectives
Example 1 Relating Logarithms to Exponents
1 Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements
Example 2 Changing Exponential Statements to Logarithmic Statements
Solution
Example 3 Changing Logarithmic Statements to Exponential Statements
Solution
2 Evaluate Logarithmic Expressions
Example 4 Finding the Exact Value of a Logarithmic Expression
Solution
3 Determine the Domain of a Logarithmic Function
Example 5 Finding the Domain of a Logarithmic Function
Solution
4 Graph Logarithmic Functions
Properties of the Logarithmic Function f(x) = loga > 0, a ≠ 1
Example 6 Graphing a Logarithmic Function and Its Inverse
Solution
Example 7 Graphing a Logarithmic Function and Its Inverse
Solution
5 Solve Logarithmic Equations
Example 8 Solving Logarithmic Equations
Solution
Example 9 Using Logarithms to Solve an Exponential Equation
Solution
Example 10 Alcohol and Driving
Solution
Summary
Properties of the Logarithmic Function
5.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.5 Properties of Logarithms
Objectives
1 Work with the Properties of Logarithms
Example 1 Establishing Properties of Logarithms
Solution
Proof of Property (1)
Proof of Property (2)
Example 2 Using Properties (1) and (2)
Proof of Property (3)
Proof of Property (5)
Proof of Property (6)
2 Write a Logarithmic Expression as a Sum or Difference of Logarithms
Example 3 Writing a Logarithmic Expression as a Sum of Logarithms
Example 4 Writing a Logarithmic Expression as a Difference of Logarithms
Solution
Example 5 Writing a Logarithmic Expression as a Sum and Difference of Logarithms
Solution
3 Write a Logarithmic Expression as a Single Logarithm
Example 6 Writing Expressions as a Single Logarithm
Solution
4 Evaluate a Logarithm Whose Base Is Neither 10 Nor e
Example 7 Approximating a Logarithm Whose Base Is Neither 10 Nor e
Solution
Proof
Example 8 Using the Change-of-Base Formula
Solution
5 Graph a Logarithmic Function Whose Base Is Neither 10 Nor e
Example 9 Graphing a Logarithmic Function Whose Base Is Neither 10 Nor e
Solution
Summary
Properties of Logarithms
5.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.6 Logarithmic and Exponential Equations
Objectives
1 Solve Logarithmic Equations
Example 1 Solving a Logarithmic Equation
Algebraic Solution
Graphing Solution
Example 2 Solving a Logarithmic Equation
Algebraic Solution
Graphing Solution
Example 3 Solving a Logarithmic Equation
Algebraic Solution
Graphing Solution
2 Solve Exponential Equations
Example 4 Solving an Exponential Equation
Algebraic Solution
Graphing Solution
Example 5 Solving an Exponential Equation
Algebraic Solution
Graphing Solution
Example 6 Solving an Exponential Equation
Algebraic Solution
Graphing Solution
Example 7 Solving an Exponential Equation That Is Quadratic in Form
Algebraic Solution
Graphing Solution
3 Solve Logarithmic and Exponential Equations Using a Graphing Utility
Example 8 Solving Equations Using a Graphing Utility
Solution
5.6 Assess Your Understanding
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.7 Financial Models
Objectives
1 Determine the Future Value of a Lump Sum of Money
Example 1 Computing Compound Interest
Solution
Example 2 Comparing Investments Using Different Compounding Periods
Example 3 Using Continuous Compounding
2 Calculate Effective Rates of Return
Example 4 Computing the Effective Rate of Interest—Which Is the Best Deal?
Solution
3 Determine the Present Value of a Lump Sum of Money
Example 5 Computing the Value of a Zero-Coupon Bond
Solution
4 Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money
Example 6 Rate of Interest Required to Double an Investment
Solution
Example 7 Time Required to Double or Triple an Investment
Solution
5.7 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Inflation
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models
Objectives
1 Find Equations of Populations That Obey the Law of Uninhibited Growth
Uninhibited Growth of Cells
Example 1 Bacterial Growth
Solution
Example 2 Bacterial Growth
Solution
2 Find Equations of Populations That Obey the Law of Decay
Uninhibited Radioactive Decay
Example 3 Estimating the Age of Ancient Tools
Solution
3 Use Newton’s Law of Cooling
Newton’s Law of Cooling
Example 4 Using Newton’s Law of Cooling
Solution
4 Use Logistic Models
Logistic Model
Properties of the Logistic Model, Equation (5)
Example 5 Fruit Fly Population
Solution
Example 6 Wood Products
Solution
5.8 Assess Your Understanding
Applications and Extensions
Retain Your Knowledge
5.9 Building Exponential, Logarithmic, and Logistic Models from Data
Objectives
1 Build an Exponential Model from Data
Example 1 Fitting an Exponential Function to Data
Solution
2 Build a Logarithmic Model from Data
Example 2 Fitting a Logarithmic Function to Data
Solution
3 Build a Logistic Model from Data
Example 3 Fitting a Logistic Function to Data
Solution
5.9 Assess Your Understanding
Applications and Extensions
Mixed Practice
Retain Your Knowledge
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
6 Trigonometric Functions
Outline
A Look Back
A Look Ahead
6.1 Angles and Their Measure
Preparing for This Section
Objectives
Degrees
Example 1 Drawing an Angle
Solution
Convert between Decimal and Degree, Minute, Second Measures for Angles
Example 2 Converting between Degree, Minute, Second, and Decimal Forms
Algebraic Solution
Graphing Solution
Radians
Find the Length of an Arc of a Circle
Example 3 Finding the Length of an Arc of a Circle
Solution
Convert from Degrees to Radians and from Radians to Degrees
Example 4 Converting from Degrees to Radians
Solution
Example 5 Converting from Radians to Degrees
Solution
Example 6 Finding the Distance between Two Cities
Solution
Find the Area of a Sector of a Circle
Example 7 Finding the Area of a Sector of a Circle
Solution
Find the Linear Speed of an Object Traveling in Circular Motion
Example 8 Finding Linear Speed
Solution
6.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
6.2 Trigonometric Functions: Unit Circle Approach
Preparing for This Section
Objectives
The Unit Circle
Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle
Example 1 Finding the Values of the Six Trigonometric Functions Using a Point on the Unit Circle
Solution
Trigonometric Functions of Angles
Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
Example 2 Finding the Exact Values of the Six Trigonometric Functions of Quadrantal Angles
Solution
Example 3 Finding Exact Values of the Trigonometric Functions of Angles That Are Integer Multiples of Quadrantal Angles
Solution
Find the Exact Values of the Trigonometric Functions of π4=45∘
Example 4 Find the Exact Values of the Trigonometric Functions of π4=45∘
Solution
Example 5 Finding the Exact Value of a Trigonometric Expression
Solution
Find the Exact Values of the Trigonometric Functions of π6=30∘ and π3=60∘
Example 6 Finding the Exact Values of the Trigonometric Functions of π3=60∘
Solution
Example 7 Finding the Exact Values of the Trigonometric Functions of π6=30∘
Solution
Example 8 Constructing a Rain Gutter
Solution
Find the Exact Values of the Trigonometric Functions for Integer Multiples of π6=30∘,π4=45∘, and π3=60∘
Example 9 Finding Exact Values for Multiples of π4=45∘
Solution
Example 10 Finding Exact Values for Multiples of π6=30∘ or π3=60∘
Use a Calculator to Approximate the Value of a Trigonometric Function
Example 11 Using a Calculator to Approximate the Value of a Trigonometric Function
Solution
Use a Circle of Radius r to Evaluate the Trigonometric Functions
Example 12 Finding the Exact Values of the Six Trigonometric Functions
Solution
6.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
6.3 Properties of the Trigonometric Functions
Preparing for This Section
Objectives
Determine the Domain and the Range of the Trigonometric Functions
Determine the Period of the Trigonometric Functions
Example 1 Finding Exact Values Using Periodic Properties
Solution
Determine the Signs of the Trigonometric Functions in a Given Quadrant
Example 2 Finding the Quadrant in Which an Angle θ Lies
Solution
Find the Values of the Trigonometric Functions Using Fundamental Identities
Example 3 Finding Exact Values Using Identities When Sine and Cosine Are Given
Solution
Example 4 Finding the Exact Value of a Trigonometric Expression Using Identities
Solution
Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle
Example 5 Finding Exact Values Given One Value and the Sign of Another
Option 1 Using a Circle
Option 2 Using Identities
Example 6 Given the Value of One Trigonometric Function and the Sign of Another, Find the Values of the Remaining Ones
Option 1 Using a Circle
Option 2 Using Identities
Use Even–Odd Properties to Find the Exact Values of the Trigonometric Functions
Proof
Example 7 Finding Exact Values Using Even–Odd Properties
Solution
6.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
6.4 Graphs of the Sine and Cosine Functions*
Preparing for This Section
Objectives
The Graph of the Sine Function y = sin x
Graph Functions of the Form y = A sin(ωx) Using Transformations
Example 1 Graphing Functions of the Form y = A sin(ωx) Using Transformations
Solution
Example 2 Graphing Functions of the Form y = A sin(ωx) Using Transformations
Solution
The Graph of the Cosine Function y = cos x
Graph Functions of the Form y = A cos (ωx) Using Transformations
Example 3 Graphing Functions of the Form y = A cos (ωx) Using Transformations
Solution
Sinusoidal Graphs
Determine the Amplitude and Period of Sinusoidal Functions
Example 4 Finding the Amplitude and Period of a Sinusoidal Function
Solution
Graph Sinusoidal Functions Using Key Points
Example 5 Graphing a Sinusoidal Function Using Key Points
Example 6 Graphing a Sinusoidal Function Using Key Points
Solution
Example 7 Graphing a Sinusoidal Function Using Key Points
Solution
Find an Equation for a Sinusoidal Graph
Example 8 Finding an Equation for a Sinusoidal Graph
Solution
Example 9 Finding an Equation for a Sinusoidal Graph
Solution
6.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
Preparing for This Section
Objectives
The Graph of the Tangent Function y = tan x
Graph Functions of the Form y = A tan (ωx) + B and y = A cot (ωx) + B
Example 1 Graphing Functions of the Form y = A tan (ωx) + B
Solution
Example 2 Graphing Functions of the Form y = A tan (ωx) + B
Solution
The Graph of the Cotangent Function y = cot x
The Graphs of the Cosecant Function and the Secant Function
Graph Functions of the Form y = A csc (ωx) + B and y = A sec (ωx) + B
Example 3 Graphing Functions of the Form y = A csc(ωx) + B
Solution
6.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
6.6 Phase Shift; Sinusoidal Curve Fitting
Objectives
Graph Sinusoidal Functions of the Form y = A sin (ωx − ϕ)+ B
Example 1 Finding the Amplitude, Period, and Phase Shift of a Sinusoidal Function and Graphing It
Solution
Example 2 Finding the Amplitude, Period, and Phase Shift of a Sinusoidal Function and Graphing It
Solution
Build Sinusoidal Models from Data
EXAMPLE 3 Finding a Sinusoidal Function from Temperature Data
Solution
Example 4 Finding a Sinusoidal Function for Hours of Daylight
Solution
Example 5 Finding the Sine Function of Best Fit
Solution
6.6 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Discussion and Writing
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
7 Analytic Trigonometry
Outline
A Look Back
A Look Ahead
7.1 The Inverse Sine, Cosine, and Tangent Functions
Preparing For This Section
Objectives
The Inverse Sine Function
1 Find the Exact Value of an Inverse Sine Function
Example 1 Finding the Exact Value of an Inverse Sine Function
Solution
Example 2 Finding the Exact Value of an Inverse Sine Function
Solution
2 Find an Approximate Value of an Inverse Sine Function
Example 3 Finding an Approximate Value of an Inverse Sine Function
Solution
3 Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions
Example 4 Finding the Exact Value of Certain Composite Functions
Solution
Example 5 Finding the Exact Value of Certain Composite Functions
Solution
The Inverse Cosine Function
Example 6 Finding the Exact Value of an Inverse Cosine Function
Solution
Example 7 Finding the Exact Value of an Inverse Cosine Function
Solution
Example 8 Using Properties of Inverse Functions to Find the Exact Value of Certain Composite Functions
Solution
The Inverse Tangent Function
Example 9 Finding the Exact Value of an Inverse Tangent Function
Solution
4 Find the Inverse Function of a Trigonometric Function
Example 10 Finding the Inverse Function of a Trigonometric Function
Solution
5 Solve Equations Involving Inverse Trigonometric Functions
Example 11 Solving an Equation Involving an Inverse Trigonometric Function
Solution
7.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
7.2 The Inverse Trigonometric Functions (Continued)
Preparing For This Section
Objectives
1 Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions
Example 1 Finding the Exact Value of Expressions Involving Inverse Trigonometric Functions
Solution
Example 2 Finding the Exact Value of Expressions Involving Inverse Trigonometric Functions
Solution
Example 3 Finding the Exact Value of Expressions Involving Inverse Trigonometric Functions
Solution
2 Define the Inverse Secant, Cosecant, and Cotangent Functions
Example 4 Finding the Exact Value of an Inverse Cosecant Function
Solution
3 Use a Calculator to Evaluate sec−1 x, csc−1 x, and cot−1 x
Example 5 Approximating the Value of Inverse Trigonometric Functions
Solution
4 Write a Trigonometric Expression as an Algebraic Expression
Example 6 Writing a Trigonometric Expression as an Algebraic Expression
Solution
7.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
7.3 Trigonometric Equations
Preparing For This Section
Objectives
1 Solve Equations Involving a Single Trigonometric Function
Example 1 Checking Whether a Given Number Is a Solution of a Trigonometric Equation
Solution
Example 2 Finding All the Solutions of a Trigonometric Equation
Solution
Example 3 Solving a Linear Trigonometric Equation
Solution
Warning
Example 4 Solving a Trigonometric Equation
Solution
Warning
Example 5 Solving a Trigonometric Equation
Solution
2 Solve Trigonometric Equations Using a Calculator
Example 6 Solving a Trigonometric Equation with a Calculator
Solution
Warning
3 Solve Trigonometric Equations Quadratic in Form
Example 7 Solving a Trigonometric Equation Quadratic in Form
Solution
4 Solve Trigonometric Equations Using Fundamental Identities
Example 8 Solving a Trigonometric Equation Using Identities
Solution
Example 9 Solving a Trigonometric Equation Using Identities
Solution
5 Solve Trigonometric Equations Using a Graphing Utility
Example 10 Solving a Trigonometric Equation Using a Graphing Utility
Solution
7.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
7.4 Trigonometric Identities
Preparing For This Section
Objectives
1 Use Algebra to Simplify Trigonometric Expressions
Example 1 Using Algebraic Techniques to Simplify Trigonometric Expressions
Solution
2 Establish Identities
Example 2 Establishing an Identity
Solution
Example 3 Establishing an Identity
Solution
Example 4 Establishing an Identity
Solution
Example 5 Establishing an Identity
Solution
Example 6 Establishing an Identity
Solution
Example 7 Establishing an Identity
Solution
Example 8 Establishing an Identity
Solution
7.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
7.5 Sum and Difference Formulas
Preparing For This Section
Objectives
Proof
1 Use Sum and Difference Formulas to Find Exact Values
Example 1 Using the Sum Formula to Find an Exact Value
Solution
Example 2 Using the Difference Formula to Find an Exact Value
Solution
Proof
Proof
Example 3 Using the Sum Formula to Find an Exact Value
Solution
Example 4 Using the Difference Formula to Find an Exact Value
Solution
Example 5 Finding Exact Values
Solution
2 Use Sum and Difference Formulas to Establish Identities
Example 6 Establishing an Identity
Solution
Proof
Proof
Example 7 Establishing an Identity
Solution
Example 8 Establishing an Identity
Solution
3 Use Sum and Difference Formulas Involving Inverse Trigonometric Functions
Example 9 Finding the Exact Value of an Expression Involving Inverse Trigonometric Functions
Solution
Example 10 Writing a Trigonometric Expression as an Algebraic Expression
Solution
4 Solve Trigonometric Equations Linear in Sine and Cosine
Example 11 Solving a Trigonometric Equation Linear in Sine and Cosine
Option 1
Option 2
Example 12 Solving a Trigonometric Equation Linear in sin θ and cos θ
Solution
7.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
7.6 Double-angle and Half-angle Formulas
Objectives
1 Use Double-angle Formulas to Find Exact Values
Example 1 Finding Exact Values Using the Double-angle Formulas
Solution
2 Use Double-angle Formulas to Establish Identities
Example 2 Establishing Identities
Solution
Example 3 Establishing an Identity
Solution
Example 4 Solving a Trigonometric Equation Using Identities
Solution
Example 5 Projectile Motion
Solution
3 Use Half-angle Formulas to Find Exact Values
Example 6 Finding Exact Values Using Half-angle Formulas
Solution
Example 7 Finding Exact Values Using Half-angle Formulas
Solution
7.6 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
7.7 Product-to-Sum and Sum-to-Product Formulas
Objectives
1 Express Products as Sums
Example 1 Expressing Products as Sums
Solution
2 Express Sums as Products
Proof
Example 2 Expressing Sums (or Differences) as Products
Solution
7.7 Assess Your Understanding
Skill Building
Applications and Extensions
Retain Your Knowledge
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
8 Applications of Trigonometric Functions
Outline
A Look Back
A Look Ahead
8.1 Right Triangle Trigonometry; Applications
Preparing For This Section
Objectives
1 Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles
Example 1 Finding the Value of Trigonometric Functions from a Right Triangle
Solution
Example 2 Constructing a Rain Gutter
Solution
2 Use the Complementary Angle Theorem
Example 3 Using the Complementary Angle Theorem
3 Solve Right Triangles
Example 4 Solving a Right Triangle
Solution
Example 5 Solving a Right Triangle
Solution
4 Solve Applied Problems*
Example 6 Finding the Width of a River
Solution
Example 7 Finding the Inclination of a Mountain Trail
Solution
Example 8 Finding the Height of a Cloud
Solution
Example 9 Finding the Height of a Statue on a Building
Solution
Example 10 The Gibb’s Hill Lighthouse, Southampton, Bermuda
Solution
Example 11 Finding the Bearing of an Object
Solution
Example 12 Finding the Bearing of an Airplane
Solution
8.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
8.2 The Law of Sines
Preparing For This Section
Objectives
1 Solve SAA or ASA Triangles
Example 1 Using the Law of Sines to Solve an SAA Triangle
Solution
Example 2 Using the Law of Sines to Solve an ASA Triangle
Solution
2 Solve SSA Triangles
Example 3 Using the Law of Sines to Solve an SSA Triangle (No Solution)
Solution
Example 4 Using the Law of Sines to Solve an SSA Triangle (One Solution)
Solution
Example 5 Using the Law of Sines to Solve an SSA Triangle (Two Solutions)
Solution
3 Solve Applied Problems
Example 6 Finding the Height of a Mountain
Solution
Example 7 Rescue at Sea
Solution
Proof of the Law of Sines
8.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
8.3 The Law of Cosines
Preparing For This Section
Objectives
Proof
1 Solve SAS Triangles
Example 1 Using the Law of Cosines to Solve an SAS Triangle
Solution
2 Solve SSS Triangles
Example 2 Using the Law of Cosines to Solve an SSS Triangle
Solution
3 Solve Applied Problems
Example 3 Correcting a Navigational Error
Solution
8.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
8.4 Area of a Triangle
Preparing For This Section
Objectives
Proof
1 Find the Area of SAS Triangles
Example 1 Finding the Area of an SAS Triangle
Solution
2 Find the Area of SSS Triangles
Example 2 Finding the Area of an SSS Triangle
Solution
Proof of Heron’s Formula
8.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
8.5 Simple Harmonic Motion; Damped Motion; Combining Waves
Preparing For This Section
Objectives
1 Build a Model for an Object in Simple Harmonic Motion
Example 1 Build a Model for an Object in Harmonic Motion
Solution
2 Analyze Simple Harmonic Motion
Example 2 Analyzing the Motion of an Object
Solution
3 Analyze an Object in Damped Motion
Example 3 Analyzing a Damped Vibration Curve
Solution
Exploration
Result
4 Graph the Sum of Two Functions
Example 4 Graphing the Sum of Two Functions
Solution
Example 5 Graphing the Sum of Two Sinusoidal Functions
Solution
8.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
Chapter Review
Things to Know
Formulas
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
9 Polar Coordinates; Vectors
Outline
A Look Back, A Look Ahead
9.1 Polar Coordinates
Preparing for This Section
Objectives
1 Plot Points Using Polar Coordinates
Example 1 Plotting Points Using Polar Coordinates
Solution
Example 2 Finding Several Polar Coordinates of a Single Point
Example 3 Finding Other Polar Coordinates of a Given Point
Solution
2 Convert from Polar Coordinates to Rectangular Coordinates
Proof
Example 4 Converting from Polar Coordinates to Rectangular Coordinates
Solution
3 Convert from Rectangular Coordinates to Polar Coordinates
Example 5 How to Convert from Rectangular Coordinates to Polar Coordinates with the Point on a Coordinate Axis
Step-by-Step Solution
Example 6 How to Convert from Rectangular Coordinates to Polar Coordinates with the Point in a Quadrant
Step-by-Step Solution
Example 7 Converting from Rectangular Coordinates to Polar Coordinates
Solution
4 Transform Equations between Polar and Rectangular Forms
Example 8 Transforming an Equation from Polar to Rectangular Form
Solution
Example 9 Transforming an Equation from Rectangular to Polar Form
Solution
9.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
9.2 Polar Equations and Graphs
Preparing for This Section
Objectives
1 Identify and Graph Polar Equations by Converting to Rectangular Equations
Example 1 Identifying and Graphing a Polar Equation (Circle)
Solution
Example 2 Identifying and Graphing a Polar Equation (Line)
Solution
Example 3 Identifying and Graphing a Polar Equation (Horizontal Line)
Solution
2 Graph Polar Equations Using a Graphing Utility
Example 4 Graphing a Polar Equation Using a Graphing Utility
Solution
Example 5 Identifying and Graphing a Polar Equation (Vertical Line)
Solution
Example 6 Identifying and Graphing a Polar Equation (Circle)
Solution
Example 7 Identifying and Graphing a Polar Equation (Circle)
Solution
Exploration
3 Test Polar Equations for Symmetry
4 Graph Polar Equations by Plotting Points
Example 8 Graphing a Polar Equation (Cardioid)
Solution
Exploration
Example 9 Graphing a Polar Equation (Limaçon without Inner Loop)
Solution
Exploration
Example 10 Graphing a Polar Equation (Limaçon with Inner Loop)
Solution
Exploration
Example 11 Graphing a Polar Equation (Rose)
Solution
Exploration
Example 12 Graphing a Polar Equation (Lemniscate)
Solution
Example 13 Graphing a Polar Equation (Spiral)
Solution
Classification of Polar Equations
Sketching Quickly
Example 14 Sketching the Graph of a Polar Equation Quickly
Solution
Calculus Comment
9.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
9.3 The Complex Plane; De Moivre’s Theorem
Preparing for This Section
Objectives
1 Plot Points in the Complex Plane
Example 1 Plotting a Point in the Complex Plane
Solution
2 Convert a Complex Number between Rectangular Form and Polar Form
Example 2 Writing a Complex Number in Polar Form
Solution
Example 3 Plotting a Point in the Complex Plane and Converting from Polar to Rectangular Form
Solution
3 Find Products and Quotients of Complex Numbers in Polar Form
Proof
Example 4 Finding Products and Quotients of Complex Numbers in Polar Form
Solution
4 Use De Moivre’s Theorem
Example 5 Using De Moivre’s Theorem
Solution
Example 6 Using De Moivre’s Theorem
Algebraic Solution
Graphing Solution
5 Find Complex Roots
Theorem Finding Complex Roots
Proof (Outline)
Example 7 Finding Complex Cube Roots
Solution
Historical Problems
9.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
9.4 Vectors
Objectives
Geometric Vectors
Adding Vectors Geometrically
Multiplying Vectors by Numbers Geometrically
1 Graph Vectors
Example 1 Graphing Vectors
Solution
Magnitude of Vectors
2 Find a Position Vector
Example 2 Finding a Position Vector
Solution
3 Add and Subtract Vectors Algebraically
Example 3 Adding and Subtracting Vectors
Solution
4 Find a Scalar Multiple and the Magnitude of a Vector
Example 4 Finding Scalar Multiples and Magnitudes of Vectors
Solution
5 Find a Unit Vector
Proof
Example 5 Finding a Unit Vector
Solution
6 Find a Vector from Its Direction and Magnitude
Example 6 Finding a Vector When Its Magnitude and Direction Are Given
Solution
Example 7 Finding the Direction Angle of a Vector
Solution
7 Model with Vectors
Example 8 Finding the Actual Speed and Direction of an Aircraft
Solution
Example 9 Finding the Weight of a Piano
Solution
Example 10 Analyzing an Object in Static Equilibrium
Solution
9.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
9.5 The Dot Product
Preparing for This Section
Objectives
1 Find the Dot Product of Two Vectors
Example 1 Finding Dot Products
Solution
Proof
2 Find the Angle between Two Vectors
Example 2 Finding the Angle θ between Two Vectors
Solution
3 Determine Whether Two Vectors Are Parallel
Example 3 Determining Whether Two Vectors Are Parallel
4 Determine Whether Two Vectors Are Orthogonal
Theorem
Example 4 Determining Whether Two Vectors Are Orthogonal
5 Decompose a Vector into Two Orthogonal Vectors
Example 5 Decomposing a Vector into Two Orthogonal Vectors
Solution
Example 6 Finding the Force Required to Hold a Wagon on a Hill
Solution
6 Compute Work
Example 7 Computing Work
Solution
9.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
9.6 Vectors in Space
Preparing for This Section
Objectives
Rectangular Coordinates in Space
1 Find the Distance between Two Points in Space
Example 1 Using the Distance Formula
Solution
2 Find Position Vectors in Space
Example 2 Finding a Position Vector
Solution
3 Perform Operations on Vectors
Example 3 Adding and Subtracting Vectors
Solution
Example 4 Finding Scalar Products and Magnitudes
Solution
Example 5 Finding a Unit Vector
Solution
4 Find the Dot Product
Example 6 Finding Dot Products
Solution
5 Find the Angle between Two Vectors
Example 7 Finding the Angle between Two Vectors
Solution
6 Find the Direction Angles of a Vector
Example 8 Finding the Direction Angles of a Vector
Solution
Example 9 Finding a Direction Angle of a Vector
Solution
Example 10 Writing a Vector in Terms of Its Magnitude and Direction Cosines
Solution
9.6 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
9.7 The Cross Product
Objectives
1 Find the Cross Product of Two Vectors
Example 1 Finding a Cross Product Using Equation (1)
Solution
Example 2 Evaluating Determinants
Example 3 Using Determinants to Find Cross Products
Solution
2 Know Algebraic Properties of the Cross Product
Proof
3 Know Geometric Properties of the Cross Product
Proof of Property (8)
4 Find a Vector Orthogonal to Two Given Vectors
Example 4 Finding a Vector Orthogonal to Two Given Vectors
Solution
5 Find the Area of a Parallelogram
Example 5 Finding the Area of a Parallelogram
Solution
9.7 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Discussion and Writing
Retain Your Knowledge
Chapter Review
Objectives
Review Exercises
Chapter Test
Chapter Test Prep Videos
Cumulative Review
Chapter Projects
10 Analytic Geometry
Outline
A Look Back
A Look Ahead
10.1 Conics
Objectives
1 Know the Names of the Conics
10.2 The Parabola
Preparing for This Section
Objectives
1 Analyze Parabolas with Vertex at the Origin
Example 1 Finding the Equation of a Parabola and Graphing It
Solution
Example 2 Graphing a Parabola Using a Graphing Utility
Solution
Example 3 Analyzing the Equation of a Parabola
Solution
Example 4 Analyzing the Equation of a Parabola
Solution
Example 5 Finding the Equation of a Parabola
Solution
Example 6 Finding the Equation of a Parabola
Solution
2 Analyze Parabolas with Vertex at (h, k)
Example 7 Finding the Equation of a Parabola, Vertex Not at the Origin
Solution
Example 8 Using a Graphing Utility to Graph a Parabola, Vertex Not at Origin
Solution
Example 9 Analyzing the Equation of a Parabola
Solution
3 Solve Applied Problems Involving Parabolas
Example 10 Satellite Dish
Solution
10.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
10.3 The Ellipse
Preparing for This Section
Objectives
1 Analyze Ellipses with Center at the Origin
Example 1 Finding an Equation of an Ellipse
Solution
Example 2 Graphing an Ellipse Using a Graphing Utility
Solution
Example 3 Analyzing the Equation of an Ellipse
Solution
Example 4 Analyzing the Equation of an Ellipse
Solution
Example 5 Finding an Equation of an Ellipse
Solution
2 Analyze Ellipses with Center at (h, k)
Example 6 Finding an Equation of an Ellipse, Center Not at the Origin
Solution
Example 7 Using a Graphing Utility to Graph an Ellipse, Center Not at the Origin
Solution
Example 8 Analyzing the Equation of an Ellipse
Solution
3Solve Applied Problems Involving Ellipses
Example 9 A Whispering Gallery
Solution
10.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
10.4 The Hyperbola
Preparing for This Section
Objectives
1 Analyze Hyperbolas with Center at the Origin
Example 1 Finding and Graphing an Equation of a Hyperbola
Solution
Example 2 Using a Graphing Utility to Graph a Hyperbola
Solution
Example 3 Analyzing the Equation of a Hyperbola
Solution
Theorem Equation of a Hyperbola; Center at (0, 0) Transverse Axis along the y-Axis
Example 4 Analyzing the Equation of a Hyperbola
Solution
Example 5 Finding an Equation of a Hyperbola
Solution
2 Find the Asymptotes of a Hyperbola
Proof
Example 6 Analyzing the Equation of a Hyperbola
Solution
Example 7 Analyzing the Equation of a Hyperbola
Solution
3 Analyze Hyperbolas with Center at (h, k)
Example 8 Finding an Equation of a Hyperbola, Center Not at the Origin
Solution
Example 9 Analyzing the Equation of a Hyperbola
Solution
4 Solve Applied Problems Involving Hyperbolas
Example 10 Lightning Strikes
Solution
10.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Retain Your Knowledge
10.5 Rotation of Axes; General Form of a Conic
PREPARING FOR THIS SECTION
Objectives
1 Identify a Conic
Proof
Example 1 Identifying a Conic without Completing the Squares
Solution
2 Use a Rotation of Axes to Transform Equations
Example 2 Rotating Axes
Solution
3 Analyze an Equation Using a Rotation of Axes
Example 3 Analyzing an Equation Using a Rotation of Axes
Solution
Example 4 Analyzing an Equation Using a Rotation of Axes
Solution
4 Identify Conics without a Rotation of Axes
Example 5 Identifying a Conic without a Rotation of Axes
Solution
10.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
10.6 Polar Equations of Conics
PREPARING FOR THIS SECTION
Objectives
1 Analyze and Graph Polar Equations of Conics
Example 1 Analyzing and Graphing the Polar Equation of a Conic
Solution
Exploration
Example 2 Analyzing and Graphing the Polar Equation of a Conic
Solution
Example 3 Analyzing and Graphing the Polar Equation of a Conic
Solution
2 Convert the Polar Equation of a Conic to a Rectangular Equation
Example 4 Converting a Polar Equation to a Rectangular Equation
Solution
10.6 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
10.7 Plane Curves and Parametric Equations
PREPARING FOR THIS SECTION
Objectives
1 Graph Parametric Equations by Hand
Example 1 Graphing a Curve Defined by Parametric Equations
Solution
2Graph Parametric Equations Using a Graphing Utility
Example 2 Graphing a Curve Defined by Parametric Equations Using a Graphing Utility
Solution
Exploration
3 Find a Rectangular Equation for a Curve Defined Parametrically
Example 3 Finding the Rectangular Equation of a Curve Defined Parametrically
Solution
Example 4 Describing Parametric Equations
Solution
4 Use Time as a Parameter in Parametric Equations
Example 5 Projectile Motion
Solution
Exploration
Example 6 Simulating Motion
Solution
5 Find Parametric Equations for Curves Defined by Rectangular Equations
Example 7 Finding Parametric Equations for a Curve Defined by a Rectangular Equation
Solution
Example 8 Finding Parametric Equations for an Object in Motion
Solution
The Cycloid
Applications to Mechanics
10.7 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
11 Systems of Equations and Inequalities
Outline
A Look Back
A Look Ahead
11.1 Systems of Linear Equations: Substitution and Elimination
Objectives
Example 1 Movie Theater Ticket Sales
Solution
Example 2 Examples of Systems of Equations
Example 3 Solving a System of Linear Equations Using a Graphing Utility
Solution
1 Solve Systems of Equations by Substitution
Example 4 How to Solve a System of Linear Equations by Substitution
Step-by-Step Solution
2 Solve Systems of Equations by Elimination
Example 5 How to Solve a System of Linear Equations by Elimination
Step-by-Step Solution
Example 6 Movie Theater Ticket Sales
Solution
3 Identify Inconsistent Systems of Equations Containing Two Variables
Example 7 An Inconsistent System of Linear Equations
Solution
4 Express the Solution of a System of Dependent Equations Containing Two Variables
Example 8 Solving a System of Dependent Equations
Solution
5 Solve Systems of Three Equations Containing Three Variables
Example 9 Solving a System of Three Linear Equations with Three Variables
Solution
6 Identify Inconsistent Systems of Equations Containing Three Variables
Example 10 Identify an Inconsistent System of Linear Equations
Solution
7 Express the Solution of a System of Dependent Equations Containing Three Variables
Example 11 Solving a System of Dependent Equations
Solution
Example 12 Curve Fitting
Solution
11.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
11.2 Systems of Linear Equations: Matrices
Objectives
Definition
1 Write the Augmented Matrix of a System of Linear Equations
Example 1 Writing the Augmented Matrix of a System of Linear Equations
Solution
2 Write the System of Equations from the Augmented Matrix
Example 2 Writing the System of Linear Equations from the Augmented Matrix
Solution
3 Perform Row Operations on a Matrix
Example 3 Applying a Row Operation to an Augmented Matrix
Solution
Example 4 Finding a Particular Row Operation
Solution
4 Solve a System of Linear Equations Using Matrices
Definition
Example 5 How to Solve a System of Linear Equations Using Matrices
Step-by-Step Solution
Example 6 Solving a System of Linear Equations Using Matrices (Row Echelon Form)
Algebraic Solution
Graphing Solution
Example 7 Solving a Dependent System of Linear Equations Using Matrices
Solution
Example 8 Solving an Inconsistent System of Linear Equations Using Matrices
Solution
Example 9 Solving a System of Linear Equations Using Matrices
Solution
Example 10 Financial Planning
Solution
11.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
11.3 Systems of Linear Equations: Determinants
Objectives
1 Evaluate 2 by 2 Determinants
Example 1 Evaluating a 2 by 2 Determinant
Algebraic Solution
Graphing Solution
2 Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables
Example 2 Solving a System of Linear Equations Using Determinants
Algebraic Solution
Graphing Solution
3 Evaluate 3 by 3 Determinants
Example 3 Finding Minors of a 3 by 3 Determinant
Solution
Example 4 Evaluating a 3 by 3 Determinant
Solution
4 Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables
Example 5 Using Cramer’s Rule
Solution
5 Know Properties of Determinants
Theorem
Proof for 2 by 2 Determinants
Example 6 Demonstrating Theorem (11)
Proof
Example 7 Demonstrating Theorem (13)
Example 8 Demonstrating Theorem (14)
Example 9 Demonstrating Theorem (15)
11.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Retain Your Knowledge
11.4 Matrix Algebra
Objectives
Definition
Example 1 Arranging Data in a Matrix
Example 2 Examples of Matrices
1 Find the Sum and Difference of Two Matrices
Definition
Example 3 Adding and Subtracting Matrices
Algebraic Solution
Graphing Solution
Example 4 Demonstrating the Commutative Property
2 Find Scalar Multiples of a Matrix
Example 5 Operations Using Matrices
Algebraic Solution
Graphing Solution
3 Find the Product of Two Matrices
Definition
Example 6 The Product of a Row Vector and a Column Vector
Example 7 Using Matrices to Compute Revenue
Solution
Example 8 Multiplying Two Matrices
Solution
Algebraic Solution
Graphing Solution
Example 9 Multiplying Two Matrices
Solution
Example 10 Multiplying Two Square Matrices
Solution
Theorem
Example 11 Multiplication with an Identity Matrix
Solution
4 Find the Inverse of a Matrix
Definition
Example 12 Multiplying a Matrix by Its Inverse
Solution
Example 13 Finding the Inverse of a Matrix
Algebraic Solution
Graphing Solution
Example 14 Showing That a Matrix Has No Inverse
Algebraic Solution
Graphing Solution
Seeing the Concept
5 Solve a System of Linear Equations Using an Inverse Matrix
Example 15 Using the Inverse Matrix to Solve a System of Linear Equations
Solution
Algebraic Solution
Graphing Solution
11.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
11.5 Partial Fraction Decomposition
Objectives
1 Decompose PQ Where Q Has Only Nonrepeated Linear Factors
Example 1 Nonrepeated Linear Factors
Solution
2 Decompose PQ Where Q Has Repeated Linear Factors Case 2: Q has repeated linear factors.
Example 2 Repeated Linear Factors
Solution
Example 3 Repeated Linear Factors
Solution
3 Decompose PQ Where Q Has a Nonrepeated Irreducible Quadratic Factor
Example 4 Nonrepeated Irreducible Quadratic Factor
Solution
4 Decompose PQ Where Q Has a Repeated Irreducible Quadratic Factor Case 4: Q contains a repeated irreducible quadratic factor.
Example 5 Repeated Irreducible Quadratic Factor
Solution
11.5 Assess Your Understanding
Skill Building
Mixed Practice
Retain Your Knowledge
11.6 Systems of Nonlinear Equations
Objectives
1 Solve a System of Nonlinear Equations Using Substitution
Example 1 Solving a System of Nonlinear Equations
Algebraic Solution Using Substitution
Graphing Solution
2 Solve a System of Nonlinear Equations Using Elimination
Example 2 Solving a System of Nonlinear Equations
Algebraic Solution Using Elimination
Graphing Solution
Example 3 Solving a System of Nonlinear Equations
Algebraic Solution Using Substitution
Graphing Solution
Example 4 Solving a System of Nonlinear Equations
Algebraic Solution Using Elimination
Graphing Solution
Example 5 Solving a System of Nonlinear Equations
Algebraic Solution
Graphing Solution
Example 6 Running a Long-Distance Race
Solution
Historical Problem
11.6 Assess Your Understanding
Skill Building
Mixed Practice
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
11.7 Systems of Inequalities
Objectives
Example 1 Examples of Inequalities in Two Variables
1 Graph an Inequality by Hand
Example 2 Graphing an Inequality by Hand
Solution
Example 3 Graphing an Inequality by Hand
Solution
Example 4 Graphing Linear Inequalities by Hand
Solution
2 Graph an Inequality Using a Graphing Utility
Example 5 Graphing an Inequality Using a Graphing Utility
Solution
3 Graph a System of Inequalities
Example 6 Graphing a System of Linear Inequalities by Hand
Solution
Example 7 Graphing a System of Linear Inequalities Using a Graphing Utility
Solution
Example 8 Graphing a System of Linear Inequalities by Hand
Solution
Example 9 Graphing a System of Linear Inequalities by Hand
Solution
Example 10 Graphing a System of Nonlinear Inequalities by Hand
Solution
Example 11 Graphing a System of Four Linear Inequalities by Hand
Solution
Example 11 Financial Planning
Solution
11.7 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Retain Your Knowledge
11.8 Linear Programming
Objectives
1 Set Up a Linear Programming Problem
Example 1 Financial Planning
Solution
Definition
2 Solve a Linear Programming Problem
Example 2 Analyzing a Linear Programming Problem
Solution
Definition
Theorem Location of the Solution of a Linear Programming Problem
Example 3 Solving a Minimum Linear Programming Problem
Solution
Example 4 Maximizing Profit
Solution
11.8 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
Chapter Review
Things to Know
Systems of equations (pp. 724–726)
Matrix (p. 739)
Determinants and Cramer’s Rule (pp. 755, 757, 758–759, and 760)
Matrix (p. 765)
Linear programming problem (p. 810)
Location of solution (p. 812)
Objectives
Review Exercises
Chapter Test
Chapter Test Prep Videos
Cumulative Review
Chapter Projects
12 Sequences; Induction; the Binomial Theorem
Outline
A Look Back, A Look Ahead
12.1 Sequences
Preparing for This Section
Objectives
Write the First Several Terms of a Sequence
Example 1 Writing the First Several Terms of a Sequence
Algebraic Solution
Graphing Solution
Example 2 Writing the First Several Terms of a Sequence
Solution
Example 3 Writing the First Several Terms of a Sequence
Solution
Example 4 Determining a Sequence from a Pattern
The Factorial Symbol
Exploration
Write the Terms of a Sequence Defined by a Recursive Formula
Example 5 Writing the Terms of a Recursively Defined Sequence
Algebraic Solution
Graphing Solution
Example 6 Writing the Terms of a Recursively Defined Sequence
Solution
Use Summation Notation
Example 7 Expanding Summation Notation
Solution
Example 8 Writing a Sum in Summation Notation
Solution
Find the Sum of a Sequence Algebraically and Using a Graphing Utility
Example 9 Finding the Sum of a Sequence
Algebraic Solution
Graphing Solution
Solve Annuity and Amortization Problems
Example 10 Saving for Spring Break
Solution
Example 11 Mortgage Payments
Solution
12.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
12.2 Arithmetic Sequences
Objectives
Determine Whether a Sequence Is Arithmetic
Example 1 Determining Whether a Sequence Is Arithmetic
Example 2 Determining Whether a Sequence Is Arithmetic
Solution
Example 3 Determining Whether a Sequence Is Arithmetic
Solution
Find a Formula for an Arithmetic Sequence
Example 4 Finding a Particular Term of an Arithmetic Sequence
Solution
Example 5 Finding a Recursive Formula for an Arithmetic Sequence
Solution
Exploration
Find the Sum of an Arithmetic Sequence
Proof
Example 6 Finding the Sum of an Arithmetic Sequence
Solution
Example 7 Finding the Sum of an Arithmetic Sequence
Solution
Example 8 Creating a Floor Design
Solution
12.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
12.3 Geometric Sequences; Geometric Series
Objectives
Determine Whether a Sequence Is Geometric
Example 1 Determining Whether a Sequence Is Geometric
Example 2 Determining Whether a Sequence Is Geometric
Solution
Example 3 Determining Whether a Sequence Is Geometric
Solution
Find a Formula for a Geometric Sequence
Example 4 Finding a Particular Term of a Geometric Sequence
Solution
Exploration
Find the Sum of a Geometric Sequence
Proof
Example 5 Finding the Sum of the First n Terms of a Geometric Sequence
Solution
Example 6 Using a Graphing Utility to Find the Sum of a Geometric Sequence
Solution
Determine Whether a Geometric Series Converges or Diverges
Intuitive Proof
Example 7 Determining Whether a Geometric Series Converges or Diverges
Solution
Example 8 Repeating Decimals
Solution
Example 9 Pendulum Swings
Solution
12.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
12.4 Mathematical Induction
Objective
Prove Statements Using Mathematical Induction
Example 1 Using Mathematical Induction
Solution
Example 2 Using Mathematical Induction
Solution
Example 3 Using Mathematical Induction
Solution
Example 4 Using Mathematical Induction
Solution
12.4 Assess Your Understanding
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
12.5 The Binomial Theorem
Objectives
Evaluate (nj)
Example 1 Evaluating (nj)
Solution
Proof
Use the Binomial Theorem
Theorem Binomial Theorem
Example 2 Expanding a Binomial
Solution
Example 3 Expanding a Binomial
Solution
Example 4 Finding a Particular Coefficient in a Binomial Expansion
Solution
Example 5 Finding a Particular Term in a Binomial Expansion
Solution A
Solution B
Proof
12.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Cumulative Review
Chapter Projects
13 Counting and Probability
Outline
A Look Back
A Look Ahead
13.1 Counting
Preparing for This Section
Objectives
1 Find All the Subsets of a Set
Example 1 Finding All the Subsets of a Set
Solution
2 Count the Number of Elements in a Set
Example 2 Analyzing Survey Data
Solution
Example 3 Counting
Solution
3 Solve Counting Problems Using the Multiplication Principle
Example 4 Counting the Number of Possible Meals
Solution
Example 5 Forming Codes
Solution
13.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
13.2 Permutations and Combinations
Preparing for This Section
Objectives
1 Solve Counting Problems Using Permutations Involving n Distinct Objects
Example 1 Counting Airport Codes [Permutation: Distinct, with Repetition]
Solution
Example 2 Forming Codes [Permutation: Distinct, without Repetition]
Solution
Example 3 Lining People Up
Solution
Example 4 Computing Permutations
Solution
Example 5 The Birthday Problem
Solution
2 Solve Counting Problems Using Combinations
Example 6 Listing Combinations
Solution
Example 7 Using Formula (2)
Solution
Example 8 Forming Committees
Solution
Example 9 Forming Committees
Solution
3 Solve Counting Problems Using Permutations Involving n Nondistinct Objects
Example 10 Forming Different Words
Solution
Example 11 Arranging Flags
Solution
13.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
Explaining Concepts: Discussion and Writing
Retain Your Knowledge
13.3 Probability
Objectives
Example 1 Tossing a Fair Coin
1 Construct Probability Models
Example 2 Determining Probability Models
Solution
Example 3 Constructing a Probability Model
Solution
Example 4 Constructing a Probability Model
Solution
2 Compute Probabilities of Equally Likely Outcomes
Example 5 Calculating Probabilities of Events Involving Equally Likely Outcomes
Solution
Example 6 Computing Compound Probabilities
Solution
3 Find Probabilities of the Union of Two Events
Example 7 Computing Probabilities of the Union of Two Events
Solution
Example 8 Computing Probabilities of the Union of Two Mutually Exclusive Events
Solution
4 Use the Complement Rule to Find Probabilities
Example 9 Computing Probabilities Using Complements
Solution
Example 10 Birthday Problem
Solution
13.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Retain Your Knowledge
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Chapter Test Prep Videos
Cumulative Review
Chapter Projects
14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function
Outline
A Look Back
A Look Ahead
14.1 Finding Limits Using Tables and Graphs
Preparing for this Section
Objectives
Find a Limit Using a Table
Example 1 Finding a Limit Using a Table
Solution
Example 2 Finding a Limit Using a Table
Solution
Example 3 Finding a Limit Using a Table
Solution
Find a Limit Using a Graph
Example 4 Finding a Limit by Graphing
Solution
Example 5 A Function That Has No Limit at 0
Solution
Example 6 Using a Graphing Utility to Find a Limit
Solution
14.1 Assess Your Understanding
Concepts and Vocabulary
Skill Building
14.2 Algebra Techniques for Finding Limits
Objectives
Example 1 Using Formulas (1) and (2)
Find the Limit of a Sum, a Difference, and a Product
Example 2 Finding the Limit of a Sum
Solution
Example 3 Finding the Limit of a Difference
Solution
Example 4 Finding the Limit of a Product
Solution
Example 5 Finding Limits Using Algebraic Properties
Solution
Example 6 Finding the Limit of a Monomial
Solution
Find the Limit of a Polynomial
Example 7 Finding the Limit of a Polynomial
Solution
Find the Limit of a Power or a Root
Example 8 Finding the Limit of a Power or a Root
Solution
Find the Limit of a Quotient
Example 9 Finding the Limit of a Quotient
Solution
Example 10 Finding the Limit of a Quotient
Solution
Example 11 Finding Limits Using Algebraic Properties
Solution
Find the Limit of an Average Rate of Change
Example 12 Finding the Limit of an Average Rate of Change
Solution
14.2 Assess Your Understanding
Concepts and Vocabulary
Skill Building
14.3 One-sided Limits; Continuous Functions
Preparing for this Section
Objectives
Find the One-sided Limits of a Function
Example 1 Finding One-sided Limits of a Function
Solution
Determine Whether a Function Is Continuous
Example 2 Determining the Numbers at Which a Rational Function Is Continuous
Solution
Example 3 Determining Where a Piecewise-defined Function Is Continuous
Solution
14.3 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Explaining Concepts: Discussion and Writing
14.4 The Tangent Problem; The Derivative
Preparing For This Section
Objectives
The Tangent Problem
Find an Equation of the Tangent Line to the Graph of a Function
Example 1 Finding an Equation of the Tangent Line
Solution
Find the Derivative of a Function
Example 2 Finding the Derivative of a Function
Solution
Example 3 Finding the Derivative of a Function Using a Graphing Utility
Solution
Example 4 Finding the Derivative of a Function
Solution
Find Instantaneous Rates of Change
Example 5 Finding the Instantaneous Rate of Change
Solution
Find the Instantaneous Velocity of a Particle
Example 6 Finding the Instantaneous Velocity of a Particle
Solution
14.4 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Applications and Extensions
14.5 The Area Problem; The Integral
Preparing For This Section
Objectives
The Area Problem
Approximate the Area under the Graph of a Function
Example 1 Approximating the Area under the Graph of f(x) = 2x from 0 to 1
Solution
Example 2 Approximating the Area under the Graph of f(x) = x2
Solution
Definition of Area
Approximate Integrals Using a Graphing Utility
Example 3 Using a Graphing Utility to Approximate an Integral
Solution
14.5 Assess Your Understanding
Concepts and Vocabulary
Skill Building
Chapter Review
Things to Know
Objectives
Review Exercises
Chapter Test
Chapter Projects
A Review
B The Limit of a Sequence; Infinite Series
Answers
Credits
Subject Index
LIBRARY OF FUNCTIONS