Mathematics A Practical Odyssey

, by Johnson, David B.; Mowry, Thomas A.

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ISBN: 9780534350758 | 0534350755
Cover: Hardcover
Copyright: 12/30/1997

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1 LOGIC

(53)

1.1 Deductive vs. Inductive Reasoning

(10)

Problem Solving

(1)

Deductive Reasoning

(1)

Deductive Reasoning and Venn Diagrams

(1)

Historical Note: Aristotle

(4)

Inductive Reasoning

(1)

Exercises 1.1

(2)

1.2 Symbolic Logic

(8)

Statements

(1)

Compound Statements and Logical Connectives

(1)

The Negation XXXp

(1)

Historical Note: Gottfried Wilhelm Leibniz

(1)

The Conjunction p XXX q

(1)

The Disjunction p XXX q

(1)

The Conditional p XXX q

(1)

Exercises 1.2

(2)

1.3 Truth Tables

(14)

The Negation XXXp

(1)

The Conjunction p XXX q

(1)

The Disjunction p XXX q

(4)

The Conditional p XXX q

(4)

Historical Note: George Boole

(1)

Equivalent Expressions

(2)

De Morgan's Laws

(1)

Exercises 1.3

(1)

1.4 More on Conditionals

(7)

Variations of a Conditional

(2)

Equivalent Conditionals

(1)

The "Only If" Connective

(1)

The Biconditional p XXX q

(1)

Exercises 1.4

(2)

1.5 Analyzing Arguments

(10)

Valid Arguments

(4)

Tautologies

(1)

Historical Note: Charles Lutwidge Dodgson

(4)

Exercises 1.5

(1)

CHAPTER 1 REVIEW

(3)

2 SETS AND COUNTING

(58)

2.1 Sets and Set Operations

(11)

Notation

(2)

Universal Set and Subsets

(1)

Intersection of Sets

(1)

Mutually Exclusive Sets

(1)

Union of Sets

(1)

Complement of a Set

(2)

Historical Note: John Venn

(1)

Set Theory and Logic (prerequisite: Chapter 1)

(1)

Exercises 2.1

(3)

2.2 Applications of Venn Diagrams

(11)

Surveys

(4)

De Morgan's Laws

(2)

Historical Note: Augustus De Morgan

(1)

Exercises 2.2

(4)

2.3 Introduction to Combinatorics

(8)

The Fundamental Principle of Counting

(3)

Factorials

(2)

Exercises 2.3

(3)

2.4 Permutations and Combinations

(13)

With versus Without Replacement

(1)

Permutations

(2)

Combinations

(7)

Historical Note: Chu Shi-chieh

(1)

Exercises 2.4

(3)

2.5 Infinite Sets

(12)

One-to-One Correspondence

(2)

Historical Note: Georg Cantor

100

(1)

Countable Sets

101

(4)

Uncountable Sets

105

(1)

Points on a Line

106

(2)

Exercises 2.5

108

(2)

CHAPTER 2 REVIEW

110

(2)

3 PROBABILITY

112

(73)

3.1 History of Probability

113

(7)

Roulette

113

(2)

Dice and Craps

115

(1)

Cards

115

(1)

Historical Note: Blaise Pascal

116

(1)

Historical Note: Gerolamo Cardano

117

(1)

Exercises 3.1

118

(2)

3.2 Basic Terms of Probability

120

(15)

Finding Probabilities and Odds

121

(3)

Relative Frequency versus Probability

124

(1)

Mendel's Use of Probabilities

125

(1)

Historical Note: Gregor Johann Mendel

126

(2)

Probabilities in Genetics

128

(1)

Genetic Screening

129

(1)

Historical Note: Nancy Wexler

130

(2)

Exercises 3.2

132

(3)

3.3 Basic Rules of Probability

135

(9)

Mutually Exclusive Events

135

(1)

Pair-of-Dice Probabilities

136

(2)

More Probability Rules

138

(1)

Probabilities and Venn Diagrams

139

(1)

Exercises 3.3

140

(3)

Fractions on a Graphing Calculator

143

(1)

3.4 Combinatorics and Probability

144

(8)

Historical Note: Lotteries and Keno

147

(3)

Exercises 3.4

150

(2)

3.5 Expected Value

152

(7)

Why the House Wins

154

(1)

Decision Theory

155

(1)

Betting Strategies

155

(1)

Exercises 3.5

156

(3)

3.6 Conditional Probability

159

(11)

Probabilities and Polls

159

(3)

The Product Rule

162

(1)

Tree Diagrams

163

(3)

Exercises 3.6

166

(4)

3.7 Independence; Trees in Genetics

170

(12)

Dependent and Independent Events

170

(3)

Product Rule for Independent Events

173

(1)

Trees in Medicine and Genetics

173

(4)

Hair Color

177

(1)

Exercises 3.7

178

(4)

CHAPTER 3 REVIEW

182

(3)

4 STATISTICS

185

(91)

4.1 Population, Sample, and Data

186

(17)

Population versus Sample

186

(1)

Frequency Distributions

186

(2)

Grouped Data

188

(2)

Histograms

190

(2)

Histograms and Relative Frequency Density

192

(3)

Pie Charts

195

(2)

Exercises 4.1

197

(4)

Histograms on a Graphing Calculator

201

(2)

4.2 Measures of Central Tendency

203

(11)

The Mean

204

(4)

The Median

208

(1)

The Mode

209

(1)

Exercises 4.2

210

(4)

4.3 Measures of Dispersion

214

(14)

Deviations

214

(1)

Variance and Standard Deviation

215

(3)

Alternate Methods for Finding Variance

218

(4)

Exercises 4.3

222

(4)

Measures of Central Tendency and Dispersion on a Graphing Calculator

226

(2)

4.4 The Normal Distribution

228

(18)

Discrete versus Continuous Variables

228

(1)

Normal Distributions

229

(2)

Probability, Area, and Normal Distributions

231

(1)

Historical Note: Carl Friedrich Gauss

232

(2)

The Standard Normal Distribution

234

(6)

Converting to the Standard Normal

240

(4)

Exercises 4.4

244

(2)

4.5 Polls and Margin of Error

246

(12)

Sampling and Inferential Statistics

247

(1)

Sample Proportion versus Population Proportion

247

(2)

Margin of Error

249

(1)

Historical Note: George H. Gallup

250

(6)

Exercises 4.5

256

(2)

4.6 Linear Regression

258

(14)

Linear Trends and Line of Best Fit

259

(3)

Coefficient of Linear Correlation

262

(5)

Exercises 4.6

267

(2)

Linear Regression on a Graphing Calculator

269

(3)

CHAPTER 4 REVIEW

272

(4)

5 FINANCE

276

(79)

5.1 Simple Interest

277

(9)

National Debt

280

(1)

Add-On Interest

281

(1)

Credit Card Finance Charge

282

(1)

Historical Note: Credit Card History

283

(1)

Exercises 5.1

284

(2)

5.2 Compound Interest

286

(14)

Annual Yield

292

(3)

Exercises 5.2

295

(3)

Doubling Time on a Graphing Calculator

298

(2)

5.3 Annuities

300

(13)

Calculating Short-Term Annuities

301

(3)

Calculating Long-Term Annuities

304

(2)

Tax-Deferred Annuities

306

(1)

Sinking Funds

307

(1)

Present Value of an Annuity

308

(2)

Exercises 5.3

310

(3)

Annuities on a Graphing Calculator

313

(1)

5.4 Amortized Loans

313

(21)

Amortization Schedules

315

(5)

Finding an Unpaid Balance

320

(3)

Exercises 5.4

323

(5)

Amortization Schedules on a Computer

328

(6)

5.5 Annual Percentage Rate on a Graphing Calculator

334

(9)

Historical Note: The Truth in Lending Act

335

(2)

Finance Charges

337

(3)

Estimating Prepaid Finance Charges

340

(1)

Exercises 5.5

341

(2)

5.6 Payout Annuities

343

(9)

Calculating Short-Term Payout Annuities

344

(1)

Comparing Payout Annuities and Savings Annuities

345

(1)

Calculating Long-Term Payout Annuities

345

(2)

Payout Annuities with Inflation

347

(2)

Exercises 5.6

349

(3)

CHAPTER 5 REVIEW

352

(3)

6 GEOMETRY

355

(96)

6.1 Perimeter and Area

356

(15)

Polygons

357

(3)

Heron's Formula for the Area of a Triangle

360

(3)

Right Triangles

363

(2)

Circles

365

(2)

Exercises 6.1

367

(4)

6.2 Volume and Surface Area

371

(12)

Problem Solving

371

(2)

Surface Area

373

(2)

Spheres

375

(3)

Cones and Pyramids

378

(1)

Exercises 6.2

379

(4)

6.3 Egyptian Geometry

383

(13)

Units of Measurement

385

(1)

Empirical Geometry (If It Works, Use It)

386

(2)

The Great Pyramid of Cheops

388

(1)

The Rhind Papyrus

389

(2)

Pi and the Area of a Circle

391

(3)

Exercises 6.3

394

(2)

6.4 The Greeks

396

(12)

Thales of Miletus

396

(1)

Pythagoras of Samos

397

(2)

Euclid of Alexandria

399

(1)

Deductive Proof

400

(2)

Historical Note: Archimedes of Syracuse

402

(1)

Archimedes of Syracuse

402

(3)

Exercises 6.4

405

(3)

6.5 Right Triangle Trigonometry

408

(14)

Angle Measurement

409

(1)

Special Triangles

410

(2)

Trigonometric Ratios

412

(4)

Using a Calculator

416

(1)

Finding an Acute Angle

417

(1)

Angles of Elevation and Depression

418

(2)

Exercises 6.5

420

(2)

6.6 Conic Sections and Analytic Geometry

422

(13)

Historical Note: Hypatia

423

(1)

Conic Sections

423

(1)

The Circle

424

(1)

The Parabola

425

(3)

The Ellipse

428

(2)

The Hyperbola

430

(4)

Exercises 6.6

434

(1)

6.7 Non-Euclidean Geometry

435

(11)

The Parallel Postulate

435

(1)

Giralamo Saccheri

436

(1)

Carl Friedrich Gauss

437

(1)

Janos Bolyai

437

(1)

Nikolai Lobachevsky

438

(1)

Bernhard Riemann

439

(1)

Geometric Models

440

(1)

A Comparison of Triangles

441

(2)

Geometry in Art

443

(2)

Exercises 6.7

445

(1)

CHAPTER 6 REVIEW

446

(5)

7 MATRICES AND MARKOV CHAINS

451

(60)

7.0 Review of Matrices

452

(12)

Terminology and Notation

452

(1)

Matrix Multiplication

453

(3)

Historical Note: Arthur Cayley and James Joseph Sylvester

456

(2)

Properties of Matrix Multiplication

458

(2)

Identity Matrices

460

(1)

Exercises 7.0

460

(3)

Matrix Multiplication on a Graphing Calculator

463

(1)

7.1 Markov Chains

464

(14)

Transition Matrices

466

(1)

Probability Matrices

467

(1)

Using Markov Chains to Predict the Future

468

(2)

Historical Note: Andrei Andreevich Markov

470

(5)

Exercises 7.1

475

(3)

7.2 Systems of Linear Equations

478

(22)

Linear Equations

478

(1)

Systems of Equations

479

(3)

Solving Systems: The Elimination Method

482

(1)

Solving Systems: The Gauss-Jordan Method

483

(8)

Exercises 7.2

491

(2)

Technology and the Row Operations

493

(7)

7.3 Long-Range Predictions with Markov Chains

500

(9)

Exercises 7.3

506

(3)

CHAPTER 7 REVIEW

509

(2)

8 LINEAR PROGRAMMING

511

(61)

8.0 Review of Linear Inequalities

513

(12)

Systems of Linear Inequalities

516

(3)

Finding Corner Points

519

(1)

Exercises 8.0

520

(1)

Graphing Linear Inequalities on a Graphing Calculator

521

(4)

8.1 The Geometry of Linear Programming

525

(17)

Creating a Model

526

(1)

Historical Note: George Dantzig

527

(2)

Analyzing the Model

529

(3)

Some Graphing Tips

532

(1)

Why the Corner Principle Works

532

(7)

Exercises 8.1

539

(3)

8.2 Introduction to the Simplex Method

542

(11)

Comparison of the Gauss-Jordan and Simplex Methods

548

(1)

Karmarkar's New Method

549

(1)

Historical Note: Hindu Mathematics

550

(1)

Exercises 8.2

551

(2)

8.3 The Simplex Method: Complete Problems

553

(17)

Pivoting with the Simplex Method

554

(1)

When to Stop Pivoting

555

(3)

Why the Simplex Method Works

558

(5)

Exercises 8.3

563

(2)

Technology and the Simplex Method

565

(5)

CHAPTER 8 REVIEW

570

(2)

9 EXPONENTIALS AND LOGARITHMIC FUNCTIONS

572

(74)

9.0A Review of Exponentials and Logarithms

573

(12)

Functions

573

(1)

Exponential Functions

573

(2)

Rational and Irrational Numbers

575

(1)

The Natural Exponential Function

575

(3)

Logarithms

578

(1)

Common Logarithms

578

(1)

Historical Note: Leonhard Euler

579

(3)

The Natural Logarithm Function

582

(2)

Exercises 9.0A

584

(1)

9.0B Review of Properties of Logarithms

585

(11)

The Inverse Properties

585

(1)

Solving Exponential Equations

586

(2)

The Exponent-Becomes-Multiplier Property

588

(2)

Historical Note: John Napier

590

(2)

The Division-Becomes-Subtraction Property

592

(1)

The Multiplication-Becomes-Addition Property

593

(1)

Solving Logarithmic Equations

594

(1)

Exercises 9.0B

595

(1)

9.1 Exponential Growth

596

(16)

Delta Notation

597

(2)

The Exponential Model

599

(7)

Exponential Growth and Compound Interest (prerequisite: Chapter 5)

606

(2)

Exercises 9.1

608

(4)

9.2 Exponential Decay

612

(17)

Half-Life

616

(2)

Relative Decay Rate

618

(3)

Historical Note: Marie Curie

621

(1)

Radiocarbon Dating

622

(2)

Historical Note: Willard Frank Libby

624

(3)

Exercises 9.2

627

(2)

9.3 Logarithmic Scales

629

(15)

Earthquakes

629

(3)

The Richter Scale

632

(4)

Earthquake Magnitude and Energy

636

(1)

The Decibel Scale

636

(3)

Historical Note: Alexander Graham Bell

639

(2)

Exercises 9.3

641

(3)

CHAPTER 9 REVIEW

644

(2)

10 CALCULUS

646

10.0 Review of Ratios, Parabolas, and Functions

647

(13)

Ratios, Proportions, and Rates

647

(2)

Similar Triangles

649

(2)

Parabolas

651

(1)

Secant Lines and Their Slopes

652

(2)

Locating the Vertex

654

(3)

Functions

657

(1)

Exercises 10.0

658

(2)

10.1 The Antecedents of Calculus

660

(16)

Ancient Mathematics

660

(1)

Greek Mathematics

661

(2)

Hindu Mathematics

663

(1)

Arabic Mathematics

664

(4)

Historical Note: Arab Mathematics

668

(1)

Mathematics during the Middle Ages

669

(1)

Mathematics during the Renaissance

669

(4)

Historical Note: Rene Descartes

673

(1)

Exercises 10.1

674

(2)

10.2 Four Problems

676

(17)

Problem 1: Find the Distance Traveled by a Falling Object

677

(4)

Problem 2: Find the Trajectory of a Cannonball

681

(1)

Historical Note: Galileo Galilei

682

(3)

Problem 3: Find the Line Tangent to a Given Curve

685

(3)

Problem 4: Find the Area of Any Shape

688

(2)

Exercises 10.2

690

(3)

10.3 Newton and Tangent Lines

693

(9)

Cauchy's Reformulation of Newton's Method

695

(3)

Historical Note: Isaac Newton

698

(3)

Exercises 10.3

701

(1)

10.4 Newton on Falling Objects and the Derivative

702

(11)

Average Speed and Instantaneous Speed

702

(1)

Newton on Gravity

702

(4)

The Derivative

706

(3)

Interpreting a Derivative

709

(2)

Exercises 10.4

711

(2)

10.5 The Trajectory of a Cannonball

713

(14)

The Motion Due to the Explosion

714

(1)

The Motion Due to Gravity

715

(1)

The Equation for the Cannonball's Motion

716

(5)

The Generalized Equation for the Cannonball's Motion (prerequisite: Section 6.5)

721

(4)

Exercises 10.5

725

(2)

10.6 Newton and Areas

727

(11)

Find the Area of Any Shape

727

(1)

The Area Function A(x)

727

(3)

Newton and the Area Function

730

(1)

Antiderivatives

731

(2)

Antiderivatives and Areas

733

(3)

Exercises 10.6

736

(2)

10.7 Conclusion

738

(2)

Rates of Change

738

(1)

Sketching Curves

738

(1)

Finding Maximum and Minimum Values

739

(1)

Motion

739

(1)

Areas and Volumes

739

(1)

CHAPTER 10 REVIEW

740

APPENDIXES

A-1

(81)

Appendix I Using a Scientific Calculator

A-1

(10)

The Equals Button

A-1

(1)

The Clear Button

A-2

(1)

The Subtraction Symbol and the Negative Symbol

A-2

(1)

Order of Operations and Use of Parentheses

A-3

(3)

The Shift Button

A-6

(1)

Memory

A-7

(1)

Scientific Notation

A-8

(1)

Exercises

A-9

(2)

Appendix II Using a Graphing Calculator

A-11

(11)

The Enter Button

A-11

(1)

The 2nd and Alpha Buttons

A-11

(1)

Correcting Typing Errors

A-12

(1)

The Subtraction Symbol and the Negative Symbol

A-12

(1)

The Multiplication Symbol

A-13

(1)

Order of Operations and Use of Parentheses

A-13

(3)

Memory

A-16

(3)

Scientific Notation

A-19

(1)

Exercises

A-20

(2)

Appendix III Graphing with a Graphing Calculator

A-22

(3)

The Graphing Buttons

A-22

(1)

Graphing a Line

A-22

(1)

Exercises

A-23

(2)

Appendix IV Finding Points of Intersection with a Graphing Calculator

A-25

(2)

Exercises

A-26

(1)

Appendix V Dimensional Analysis

A-27

(5)

Exercises

A-29

(3)

Appendix VI Body Table for the Standard Normal Distribution

A-32

(1)

Appendix VII Answers to Selected Exercises

A-33

(49)

Credits

A-82

(1)

Index

A-83

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Mathematics A Practical Odyssey

Table of Contents

Supplemental Materials