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Tables of Contents for College Mathematics Through Applications

Chapter/Section Title

Page #

Page Count

Preface

xv

Linear Equations

0

62

The Mathematical Crystal Ball

2

4

Take a Walk with the CBL

6

4

Looking at the Data

8

2

Distance, Time, and Two Kinds of Rate

10

7

Velocity and Speed

10

7

Equation of a Line: Slope and y-Intercept

17

16

Slope of a Straight Line

18

1

Applications of Slope

19

4

Intercepts of a Graph

23

2

Fitting a Line to the Data---The Slope-Intercept Form of the Line

25

3

The Constant Velocity Equation

28

5

Three More Forms of the Linear Equation

33

16

Using the Slope and a Point

34

1

Using Two Points

35

1

Using the Angle of the Line

36

5

Decimal Precision and Decimal Accuracy

41

8

A Linear Model That Uses All the Points

49

13

Fitting a Regression Line to Data

51

6

Summary and Review

57

3

Test

60

2

Quadratic Functions

62

60

The Bouncing Ball

64

1

Velocity That Varies

65

5

Graphs That Tell a Story

65

1

Average Speed and Average Velocity

66

4

Follow the Bouncing Ball

70

5

CBL and the Bouncing Ball

70

2

Looking at One Bounce

72

1

Quadratic Regression

73

1

Thinking about Quadratic Regression

74

1

Introduction to Functions

75

8

Introduction to Functions

75

2

Naming a Function

77

2

How Functions Are Defined

79

1

Graphing Functions

80

1

Looking More Closely at a Graph

81

2

Working with Parabolas

83

14

Roots of a Quadratic Function

83

1

Roots and Factors

84

6

Maximum or Minimum of a Quadratic Function

90

1

Graphs of Quadratic Functions

91

2

Approximating Real Roots of Any Function

93

4

Projectile Motion

97

11

A Falling Object

97

2

Transforming Data

99

1

Giving it a Toss

100

2

Comparing Transformed Graphs

102

1

Another Look at Coefficients a and b

102

2

Thinking about the Quadratic Model

104

4

Velocity at an Instant

108

14

Return to the Bouncing Ball

115

2

Summary and Review

117

2

Test

119

3

Models of Periodic Data: Introducing Trigonometry

122

96

Analyzing the Touchtone Phone

124

1

Introduction to the Trigonometric Functions

125

12

Definitions of the Trigonometric Functions

126

1

Calculations with Triangles

127

10

Graphs of Trigonometric Functions

137

16

Angles Larger than 90°

137

4

Drawing the Graphs

141

1

The Peculiar Tangent Function

142

2

Negative Angles and Other Strange Things

144

4

Modeling Alternating Currents

148

1

Modeling Vibrations and Sound Waves

149

4

Period, Amplitude, Frequency, and Roots

153

17

Cycle and Period

153

3

Frequency

156

1

Amplitude

157

5

Solving Trigonometric Equations

162

5

Roots of Trigonometric Functions

167

3

Vertical and Horizontal Translations

170

12

Vertical Shifts or Translations of Functions

171

1

Phase Shift

172

1

Horizontal Shifts or Translations of Functions

173

6

Fitting Trigonometric Functions to Periodic Data

179

3

Modeling With Radian Measure

182

13

Radians and Degrees

182

4

Modeling the Motion of a Spring

186

3

Modeling Sounds and Music

189

6

Modeling Wave Combinations

195

23

More about Alternating Current

195

2

Alternating Current and the Addition of Trigonometric Functions

197

1

Combining Waves of Different Frequency

198

4

Modeling Musical Notes and Chords

202

10

Summary and Review

212

4

Test

216

2

Mathematical Models in Geometry: Images in Two and Three Dimensions

218

74

What's the Best Shape?

220

2

Volume and Surface Area

222

14

Volume

222

3

Surface Area

225

2

Exploring Max/Min Problems

227

4

First Attempt at the Minimum Area of a Cylinder

231

5

Triangles Are Everywhere

236

17

Why Is the Area of a Triangle Half the Base x Height?

237

1

Trigonometry and the Areas of Triangles

238

2

The Law of Sines

240

4

The Pythagorean Theorem

244

2

Distance on a Graph

246

2

The Law of Cosines

248

5

Pythagorean Theorem and Circles

253

13

Definition of a Circle and Some of Its Parts

254

3

Circles Not Centered at the Origin

257

1

Pythagorean Trigonometry Identities

258

2

Circumference and Area of a Circle

260

6

Definition and Use of Radians

266

12

Radians

266

4

Sectors and Arcs of Circles

270

2

Summary of Differences: Radians vs. Degrees

272

1

Moving in Circles

272

6

Prisms, Pyramids, and Other 3-D Figures

278

14

Prisms

278

1

Pyramids

278

2

Cylinders

280

1

Cones

281

1

Solution of the Chapter Project's Cylinder Problem

282

5

Summary and Review

287

3

Test

290

2

Modeling Motion in Two Dimensions

292

82

What's the Best Angle?

294

1

Prlimary Analysis

294

3

Flight Trajectories

297

8

Looking at Trajectories

297

1

Vectors and the Components of Velocity

297

3

The Influence of Gravity

300

5

Parametric Equations

305

16

Graphing Parametric Equations

306

2

Trigonometry, Circles, and Parametric Equations

308

2

Parametric Equations of Ellipses

310

2

Plotting Planetary Trajectories

312

2

Modeling Planetary Velocity

314

1

Modeling Other Gravity-Influenced Trajectories

315

6

Vectors

321

18

Vectors in Navigation

321

2

Modeling Vector Sums with Triangles

323

2

Modeling Vector Sums with Parallelograms

325

5

Vectors and Gravity: Motion on a Ramp

330

2

Modeling Impedance in RC Circuits

332

7

Vectors and Complex Numbers

339

20

Types of Numbers

339

5

Imaginary Numbers

344

4

Arithmetic of Complex Numbers

348

2

Geometry of Complex Numbers

350

4

Polar Form for a Complex Number

354

3

Phasors and Complex Numbers

357

2

Solving the Best-Angle Problem

359

15

Solution That Uses Algebra and Trigonometry

363

1

What If the Beginning and Ending Heights Are Different

364

3

Summary and Review

367

4

Test

371

3

Polar Graphing and Elementary Programming

374

66

Programming a Robot Arm

376

2

Review of Two-Dimensional Graphing

378

11

Connecting Cartesian and Parametric Graphing

378

7

Parametric Equations and the Oscilloscope

385

2

Lissajous Curves

387

2

Polar Coordinates

389

15

Plotting Points in Polar Coordinates

389

1

Plotting Polar Equations

390

6

Patterns in Polar Graphs

396

1

Patterns in Roses and Their Relatives

397

2

Connecting Polar and Rectangular Equations

399

5

Applications of Polar Graphing: Ellipses

404

12

Ellipses and Eccentricity

404

3

Plotting Orbits of Comets

407

1

Polar Equations of Ellipses

408

2

Other Applications of Ellipses

410

6

Understanding TI-83 Programs

416

5

Introduction to TI-83 Programming

417

2

How the Program XYGRABBR Works

419

2

Writing TI-83 Programs

421

7

Entering a Program

421

2

Editing an Existing Program

423

5

Solving the Chapter Project

428

12

Plotting Curves One Point at a Time

428

3

Plotting Connected Curves

431

5

Summary and Review

436

2

Test

438

2

Modeling With Sequences and Series

440

100

Estimating Cross-Sectional Areas

442

1

Preliminary Analysis

442

4

Sequences

446

15

Defining Sequences Directly

447

7

Defining Sequences Recursively

454

7

Limits of Sequences

461

11

Sequences and Chaos Theory

465

1

A Sequence of Areas

466

6

Looking More Closely at Limits

472

12

Three Important Sequences

472

5

The Algebra of Limits

477

7

Applications of Geometric Sequences

484

25

Interest on a Savings Account

484

7

Bouncing Ball

491

3

Charing and Discharging a Capacitor

494

5

Radioactivity and Half-Life

499

3

Exponents and Notes in a Musical Scale

502

2

Negative, Zero, and Fractional Exponents

504

5

Arithmetic and Geometric Series

509

13

A Legendary Bonus

510

4

Limits of Geometric Series

514

4

Charging a Capacitor

518

2

Arithmetic Series

520

2

Estimating Areas under Curves

522

18

Riemann Sums

523

2

Estimation with Middle Rectangles

525

3

A Program to Estimate Areas

528

2

How the Program Works

530

4

Summary and Review

534

4

Test

538

2

Modeling With Algebraic Functions

540

92

Seismographs and Pendulums

542

1

Preliminary Analysis

542

5

Power Functions: f (x) = kxn

547

14

Algebraic Functions

548

2

Exploring the Graphs of f (x) = kxn

550

7

Summary: Shapes of Power Function Graphs

557

4

Variation and Power Functions

561

14

A Legend about Gifts and Gold

562

1

Direct Variation

562

4

Variation of Light Intensity with Distance

566

2

Inverse Variation

568

7

Polynomial Functions

575

9

Roots of Polynomials

576

2

Factored Form of a Polynomial

578

1

Limits and Graphs of Polynomial Functions

579

5

Application of Polynomials

584

11

How do Calculators Compute Trigonometric Values?

584

2

The nth Term of a Sequence

586

9

Patterns in Slopes of Power Functions

595

11

Plotting a Tangent to a Curve

597

4

A Program that Draws Tangents

601

1

Modeling the Slope Function

602

4

Analyzing Rational Functions

606

16

Shifted Forms of Power Functions

606

8

Variations on Familiar Functions

614

4

Solving a Light Intensity Problem with Graph, Table, and Algebra

618

4

Solving the Chapter Project

622

10

Summary and Review

626

4

Test

630

2

Exponential and Logarithmic Functions

632

64

Modeling Temperature Changes

634

1

Preliminary Analysis

634

4

Exponential Functions

638

6

Graphs of Exponential Functions

640

2

Intercepts and Asymptotes in Exponential Functions

642

2

Logarithms and Inverse Functions

644

14

What Are Logarithms?

644

3

Properties of Logarithms

647

4

Why Study Logarithms?

651

3

Inverse Functions

654

4

Natural Logarithms and the Number e

658

15

Continuous Growth and the Number e

659

5

Solving Exponential Equations

664

3

Charging a Capacitor and the Number e

667

1

The Slope Function for Exponential and Logarithmic Functions

668

5

Using Logarithmic Scales

673

10

The Mathematics of the Logarithmic Scale

674

8

Semilog Graphing on the Graphing Calculator

682

1

Analysis of the Cooling Data

683

13

Exponential Regression

683

1

Slope Function of Data

684

1

Vertical Transformation of the Data

685

1

The Accuracy of the Model

686

5

Summary and Review

691

3

Test

694

2

Systems of Equations and Inequalities

696

66

Maximizing the Profit

698

1

Preliminary Analysis

698

2

Review: Equations in One Variable

700

9

Solving Equations Without a Calculator

701

1

When to Use the Calculator?

702

1

Solving Equations with a Calculator

703

6

Solving Two Equations in Two Variables

709

18

Solving Two Equations: The Addition Method

711

1

Solving Two Equations: The Graphical Method

712

1

Solving Two Equations: The Substitution Method

713

3

Questions About the Addition Method

716

2

Application: Kirchhoff's Laws of Circuit Analysis

718

3

How Many Solutions Are Possible?

721

6

Inequalities in One Variable

727

13

Representing the Solution of an Inequality with a Graph

728

1

Solving an Inequality with Algebra

729

4

Solving an Inequality with a Calculator

733

2

Solving Inequalities: Binary Logic and the Step Function

735

3

Binary Logic and the Manufacturing Problem

738

2

Inequalities in Two Variables

740

12

The Graph of a System of Inequalities

740

5

Graphing Systems of Inequalities on the Calculator

745

1

Graphing the Systems for the Chapter Project

746

6

Linear Programming

752

10

What Is Linear Programming?

753

2

When the Maximum Occurs at an Unacceptable Point

755

1

Adding a New Constraint

756

3

Summary and Review

759

2

Test

761

1

Matrices and 3-D Graphing

762

88

Scaling and Rotating Points

764

1

Preliminary Analysis

764

2

Introduction to the Transformation of Points

766

14

Scaling

767

3

Rotation

770

2

Rotation and Scaling Together

772

2

The Trigonometry of Rotation

774

6

Introduction to Matrix Algebra

780

19

Matrix Multiplication

781

4

Matrix Algebra on the TI-83

785

1

Which Matrices Can Be Multiplied?

786

1

Choosing the Dimensions of a Matrix

787

2

Matrices and Transformation of Points in Two Dimensions

789

10

Matrices and Equation Solving

799

11

Representing Systems with Matrices

799

2

Solving a Matrix Equation

801

9

Graphing in Three Dimensions

810

7

Cartesian Coordinates in Three Dimensions

810

1

Computing Distances in Space

811

4

Motion in 3-D

815

2

Plotting Equations in 3-D

817

10

Solving Systems in Three or More Dimensions

822

5

Curves and Surfaces in 3-D

827

10

Parametric Equations in 3-D

827

3

Cylindrical Coordinates

830

2

Spherical Coordinates

832

5

Rotations and Translations in Three Dimensions

837

13

Scaling in Three Dimensions

838

1

Rotation in Three Dimensions

838

3

Summary and Review

841

7

Test

848

2

Modeling with Probability and Statistics

850

65

Assessing Quality on the Assembly Line

852

1

Preliminary Analysis

852

4

Looking at Data

856

11

Data Analysis---Failure Rate

857

2

Data Analysis---Throwing Two Dice

859

2

Analyzing Data with the TI-83

861

1

Representing Data Graphically---The Histogram

862

5

Using Probability to Model Uncertainty

867

13

Translating English to Mathematics

869

1

How Many Ways Can It Happen---Adding Probabilities

870

2

Expected Number: How Well Can We Predict?

872

8

Probability of Binary Events

880

12

Independent Events---Multiplying Probabilities

880

3

Summary of Probability So Far

883

4

Counting Events

887

5

Analyzing Games of Chance

892

13

Card Games

892

1

Dice Games, Odds, and Expected Return

893

5

Roulette

898

2

State Lotteries

900

5

Solving the Chapter Project

905

10

Summary and Review

909

4

Test

913

2

Solutions to Selected Exercises

915

84

Index of Applications

999

2

Index

1001