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Fundamentals of Precalculus

by: Dugopolski, Mark

Fundamentals of Precalculus

by: Dugopolski, Mark

ISBN 13:
9780321122322
ISBN 10:
0321122321
Edition: 1st
Format: Hardcover
Copyright: 01/01/2004
Publisher: Addison Wesley
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Summary

Fundamentals of Precalculus is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text can be covered in a one-semester or one-term course with a minimum of deleting or skipping around. A student who is well acquainted with the material in this text will have the necessary skills, understanding, and insights required in calculus.

Preface

xiii

Supplements

xvii

Graphs and Functions

(105)

Real Numbers and Their Properties

(9)

The Real Numbers

(2)

Properties of the Real Numbers

(1)

Additive Inverses

(1)

Relations

(1)

Absolute Value

(3)

Equations Involving Absolute Value

(2)

Linear and Absolute Value Inequalities

(10)

Interval Notation

(1)

Linear Inequalities

(2)

Compound Inequalities

(2)

Absolute Value Inequalities

(2)

Modeling with Inequalities

(3)

Equations and Graphs in Two Variables

(11)

The Cartesian Coordinate System

(1)

The Distance Formula

(1)

The Midpoint Formula

(1)

The Circle

(3)

The Line

(2)

Using a Graph to Solve an Equation

(3)

Linear Equations in Two Variables

(11)

Slope of a Line

(2)

Point-Slope Form

(1)

Slope-Intercept Form

(1)

Using Slope to Graph a Line

(1)

The Three Forms for the Equation of a Line

(1)

Parallel Lines

(1)

Perpendicular Lines

(1)

Applications

(3)

Functions

(12)

The Function Concept

(1)

Identifying Functions

(3)

Domain and Range

(1)

Function Notation

(2)

The Average Rate of Change of a Function

(2)

Constructing Functions

(3)

Graphs of Relations and Functions

(12)

Graphing Equations

(3)

Semicircles

(1)

Piecewise Functions

(3)

Increasing, Decreasing, and Constant

(5)

Families of Functions, Transformations, and Symmetry

(12)

Translation

(3)

Reflection

(2)

Stretching and Shrinking

(2)

The Linear Family of Functions

(1)

Symmetry

(5)

Operations with Functions

(9)

Basic Operations with Functions

(3)

Composition of Functions

(3)

Applications

(3)

Inverse Functions

(18)

One-to-One and Invertible Functions

(1)

Determining Whether a Function Is One-to-One

(2)

Inverse Functions Using Function Notation

(2)

Graphs of f and f-1

(1)

Finding Inverse Functions Mentally

(5)

Highlights

(2)

Chapter 1 Review Exercises

101

(4)

Concepts of Calculus: Limits

105

(1)

Polynomial and Rational Functions

106

(74)

Quadratic Functions and Inequalities

107

(8)

Two Forms for a Quadratic Function

107

(2)

Opening, Vertex, and Axis of Symmetry

109

(1)

Intercepts

110

(1)

Quadratic Inequalities

111

(2)

Applications of Maximum and Minimum

113

(2)

Complex Numbers

115

(7)

Definitions

115

(2)

Addition, Subtraction, and Multiplication

117

(1)

Division of Complex Numbers

118

(1)

Roots of Negative Numbers

119

(1)

Imaginary Solutions to Quadratic Equations

120

(2)

Zeros of Polynomial Functions

122

(11)

The Remainder Theorem

122

(1)

Synthetic Division

123

(3)

The Factor Theorem

126

(1)

The Fundamental Theorem of Algebra

127

(1)

The Rational Zero Theorem

127

(6)

The Theory of Equations

133

(6)

The Number of Roots of a Polynomial Equation

133

(1)

The Conjugate Pairs Theorem

134

(1)

Descartes' Rule of Signs

135

(4)

Miscellaneous Equations

139

(10)

Factoring Higher-Degree Equations

139

(1)

Equations Involving Square Roots

140

(2)

Equations with Rational Exponents

142

(1)

Equations of Quadratic Type

143

(2)

Equations Involving Absolute Value

145

(4)

Graphs of Polynomial Functions

149

(11)

Drawing Good Graphs

149

(1)

Symmetry

149

(2)

Behavior at the x-Intercepts

151

(1)

The Leading Coefficient Test

151

(3)

Sketching Graphs of Polynomial Functions

154

(1)

Polynomial Inequalities

155

(5)

Rational Functions and Inequalities

160

(20)

Rational Functions and Their Domains

160

(1)

Horizontal and Vertical Asymptotes

161

(2)

Oblique Asymptotes

163

(1)

Sketching Graphs of Rational Functions

164

(4)

Rational Inequalities

168

(1)

Applications

169

(5)

Highlights

174

(1)

Chapter 2 Review Exercises

175

(4)

Concepts of Calculus: Instantaneous rate of change

179

(1)

Trigonometric Functions

180

(104)

Angles and Their Measurements

181

(10)

Degree Measure of Angles

181

(4)

Radian Measure of Angles

185

(3)

Arc Length

188

(3)

The Sine and Cosine Functions

191

(8)

Definition

191

(2)

Sine and Cosine of a Multiple of a 45°

193

(1)

Sine and Cosine of a Multiple of a 30°

194

(1)

Sine and Cosine of an Arc

195

(1)

Approximate Values for Sine and Cosine

196

(1)

The Fundamental Identity

196

(1)

Modeling the Motion of a Spring

197

(2)

The Graphs of the Sine and Cosine Functions

199

(14)

The Graph of y = sin(x)

199

(3)

The Graph of y = cos(x)

202

(1)

Transformations of Sine and Cosine

203

(2)

Changing the Period

205

(2)

The General Sine Wave

207

(2)

Frequency

209

(4)

The Other Trigonometric Functions and Their Graphs

213

(11)

Definitions

213

(2)

Graph of y = tan(x)

215

(2)

Graph of y = cot(x)

217

(2)

Graph of y = sec(x)

219

(1)

Graph of y = csc(x)

220

(4)

The Inverse Trigonometric Functions

224

(9)

The Inverse of the Sine Function

224

(2)

The Inverse Cosine Function

226

(1)

Inverses of Tangent, Cotangent, Secant, and Cosecant

227

(3)

Compositions of Functions

230

(3)

Right Triangle Trigonometry

233

(9)

Trigonometric Ratios

233

(1)

Right Triangles

234

(2)

Solving a Right Triangle

236

(1)

Applications

237

(5)

Identities

242

(11)

Pythagorean Identities

242

(1)

Odd and Even Identities

243

(1)

Sum and Difference Identities

243

(1)

Cofunction Identities

244

(1)

Double-Angle and Half-Angle Identities

245

(2)

Product and Sum Identities

247

(2)

The Function y = asinx + bcosx

249

(4)

Conditional Trigonometric Equations

253

(11)

Cosine Equations

253

(2)

Sine Equations

255

(1)

Tangent Equations

256

(1)

Equations Involving Multiple Angles

257

(1)

More Complicated Equations

258

(3)

Modeling Projectile Motion

261

(3)

The Law of Sines and the Law of Cosines

264

(20)

Oblique Triangles

264

(1)

The Law of Sines

264

(2)

The Ambiguous Case (SSA)

266

(3)

The Law of Cosines

269

(3)

Bearing

272

(5)

Highlights

277

(2)

Chapter 3 Review Exercises

279

(4)

Concepts of Calculus: Area of a circle and π

283

(1)

Exponential and Logarithmic Functions

284

(44)

Exponential Functions and Their Applications

285

(11)

Exponential Functions

285

(1)

Domain of Exponential Functions

285

(1)

Graphing Exponential Functions

286

(3)

The Exponential Family of Functions

289

(2)

Exponential Equations

291

(1)

The Compound Interest Model

291

(2)

Continuous Compounding and the Number e

293

(3)

Logarithmic Functions and Their Applications

296

(10)

Logarithmic Functions

296

(2)

Graphs of Logarithmic Functions

298

(1)

The Logarithmic Family of Functions

299

(1)

Logarithmic and Exponential Equations

300

(3)

Applications

303

(3)

Rules of Logarithms

306

(8)

The Product Rule for Logarithms

306

(1)

The Quotient Rule for Logarithms

307

(1)

The Power Rule for Logarithms

307

(1)

The Inverse Rules

308

(1)

Using the Rules

309

(2)

The Base Change Formula

311

(3)

More Equations and Applications

314

(14)

Logarithmic Equations

314

(1)

Exponential Equations

315

(2)

Strategy for Solving Equations

317

(1)

Radioactive Dating

317

(2)

Newton's Model for Cooling

319

(1)

Paying off a Loan

320

(3)

Highlights

323

(1)

Chapter 4 Review Exercises

324

(3)

Concepts of Calculus: Evaluating transcendental functions

327

(1)

Conic Sections, Polar Coordinates, and Parametric Equations

328

(58)

The Parabola

329

(9)

The Geometric Definition

329

(1)

Developing the Equation

330

(1)

The Standard Equation of a Parabola

331

(1)

Graphing a Parabola

332

(1)

Parabolas Opening to the Left or Right

333

(2)

Applications

335

(3)

The Ellipse and the Circle

338

(10)

Definition of Ellipse

338

(1)

The Equation of the Ellipse

339

(2)

Graphing an Ellipse Centered at the Origin

341

(2)

Translations of Ellipses

343

(1)

The Circle

344

(1)

Applications

344

(4)

The Hyperbola

348

(14)

The Definition

349

(1)

Developing the Equation

350

(1)

Graphing a Hyperbola Centered at (0,0)

351

(4)

Hyperbolas Centered at (h,k)

355

(2)

Finding the Equation of a Hyperbola

357

(1)

Classifying the Conics

358

(4)

Polar Equations

362

(10)

Polar Coordinates

362

(1)

Polar-Rectangular Conversions

363

(1)

Polar Equations

364

(3)

Converting Equations

367

(5)

Polar Equations of the Conics

372

(4)

Alternative Definition of the Conics

372

(1)

Polar Equations of the Conics

372

(2)

Equivalency of the Definitions

374

(2)

Parametric Equations

376

(10)

Graphs of Parametric Equations

376

(1)

Eliminating the Parameter

377

(1)

Writing Parametric Equations

378

(3)

Highlights

381

(1)

Chapter 5 Review Exercises

382

(3)

Concepts of Calculus: The reflection property of a parabola

385

(1)

APPENDIX A BASIC ALGEBRA REVIEW

386

A.1 Exponents and Radicals

386

(10)

Exponential Expressions

386

(1)

Negative Integral Exponents

387

(1)

Rules of Exponents

388

(1)

Roots

389

(1)

Rational Exponents

389

(2)

Radical Notation

391

(1)

The Product and Quotient Rules for Radicals

392

(1)

Simplified Form and Rationalizing the Denominator

393

(1)

Operations with Radical Expressions

394

(2)

A.2 Polynomials

396

(6)

Definitions

396

(1)

Naming and Evaluating Polynomials

397

(1)

Addition and Subtraction of Polynomials

397

(1)

Multiplication of Polynomials

398

(1)

Using FOIL

399

(1)

Special Products

399

(1)

Division of Polynomials

400

(2)

A.3 Factoring Polynomials

402

(5)

Factoring out the Greatest Common Factor

402

(1)

Factoring by Grouping

403

(1)

Factoring ax2 + bx + c

403

(1)

Factoring the Special Products

404

(1)

Factoring the Difference and Sum of Two Cubes

405

(1)

Factoring Completely

406

(1)

A.4 Rational Expressions

407

Reducing

407

(2)

Multiplication and Division

409

(1)

Building up the Denominator

410

(1)

Addition and Subtraction

411

Answers to Selected Exercises

(1)

Credits

(1)

Index

Amazon no longer offers textbook rentals. We do!

Fundamentals of Precalculus

Fundamentals of Precalculus

9780321122322

0321122321

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Fundamentals of Precalculus

Fundamentals of Precalculus

9780321122322

0321122321

Summary

Table of Contents

Supplemental Materials

Write a Review

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