Calculus in Context: Background, Basics, and Applications

Calculus in Context: Background, Basics, and Applications

by Alexander J. Hahn

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Overview

Calculus in Context: Background, Basics, and Applications by Alexander J. Hahn

Breaking the mold of existing calculus textbooks, Calculus in Context draws students into the subject in two new ways. Part I develops the mathematical preliminaries (including geometry, trigonometry, algebra, and coordinate geometry) within the historical frame of the ancient Greeks and the heliocentric revolution in astronomy. Part II starts with comprehensive and modern treatments of the fundamentals of both differential and integral calculus, then turns to a wide-ranging discussion of applications. Students will learn that core ideas of calculus are central to concepts such as acceleration, force, momentum, torque, inertia, and the properties of lenses.

Classroom-tested at Notre Dame University, this textbook is suitable for students of wide-ranging backgrounds because it engages its subject at several levels and offers ample and flexible problem set options for instructors. Parts I and II are both supplemented by expansive Problems and Projects segments. Topics covered in the book include:

• the basics of geometry, trigonometry, algebra, and coordinate geometry and the historical, scientific agenda that drove their development
• a brief, introductory calculus from the works of Newton and Leibniz
• a modern development of the essentials of differential and integral calculus
• the analysis of specific, relatable applications, such as the arc of the George Washington Bridge; the dome of the Pantheon; the optics of a telescope; the dynamics of a bullet; the geometry of the pseudosphere; the motion of a planet in orbit; and the momentum of an object in free fall.

Calculus in Context is a compelling exploration—for students and instructors alike—of a discipline that is both rich in conceptual beauty and broad in its applied relevance.

Product Details

ISBN-13: 9781421422305
Publisher: Johns Hopkins University Press
Publication date: 04/15/2017
Edition description: New Edition
Pages: 712
Product dimensions: 7.00(w) x 10.00(h) x 1.90(d)
Age Range: 18 Years

About the Author

Alexander J. Hahn is a professor of mathematics at the University of Notre Dame. He is the author of Basic Calculus: From Archimedes to Newton to Its Role in Science and Mathematical Excursions to the World’s Great Buildings.

Table of Contents

Preface ix

Part I

1 The Astronomy and Geometry of the Greeks 1

1.1 The Greeks Explain the Universe 2

1.2 Achieving the Impossible? 6

1.3 Greek Geometry 9

1.4 The Pythagorean Theorem 13

1.5 The Radian Measure of an Angle 15

1.6 Greek Trigonometry 19

1.7 Aristarchus Sizes Up the Universe 24

1.8 Problems and Projects 28

2 The Genius of Archimedes 47

2.1 The Conic Sections 48

2.2 The Question of Area 53

2.3 Playing with Squares 55

2.4 The Area of a Parabolic Section 58

2.5 The Method of Archimedes 62

2.6 Problems and Projects 67

3 A New Astronomy 81

3.1 A Fixed Sun at the Center 82

3.2 Copernicus's Model of Earth's Orbit 85

3.3 About the Distances of the Planets from the Sun 89

3.4 Tycho Brahe and Parallax 91

3.5 Kepler's Elliptical Orbits 93

3.6 The Studies of Galileo 98

3.7 The Size of the Solar System 103

3.8 Problems and Projects 109

4 The Coordinate Geometry of Descartes 123

4.1 The Real Numbers 124

4.2 The Coordinate Plane 128

4.3 About the Parabola 133

4.4 About the Ellipse 136

4.5 Quadratic Equations in x and y 139

4.6 Circles and Trigonometry 143

4.7 Problems and Projects 149

5 The Calculus of Leibniz 161

5.1 Straight Lines 162

5.2 Tangent Lines to Curves 166

5.3 The Function Concept 170

5.4 The Derivative of a Function 173

5.5 Fermat, Kepler, and Wine Barrels 176

5.6 The Definite Integral 181

5.7 Cavalieri's Principle 185

5.8 Differentials and the Fundamental Theorem 187

5.9 Volumes of Revolution 191

5.10 Problems and Projects 195

6 The Calculus of Newton 205

6.1 Simple Functions and Areas 206

6.2 The Derivative of a Simple Function 208

6.3 From Simple Functions to Power Series 211

6.4 The Mathematics of a Moving Point 216

6.5 Galileo and Acceleration 223

6.6 Dealing with Forces 226

6.7 The Trajectory of a Projectile 231

6.8 Newton Studies the Motion of the Planets 235

6.9 Connecting Force and Geometry 239

6.10 The Law of Universal Gravitation 245

6.11 Problems and Projects 249

Part II

7 Differential Calculus 271

7.1 Mathematical Functions 272

7.2 A Study of Limits 274

7.3 Continuous Functions 278

7.4 Differentiable Functions 285

7.5 Computing Derivatives 289

7.6 Some Theoretical Concerns 294

7.7 Derivatives of Trigonometric Functions 296

7.8 Understanding Functions 298

7.9 Graphing Functions: Some Examples 304

7.10 Exponential Functions 308

7.11 Logarithm Functions 312

7.12 Hyperbolic Functions 318

7.13 Final Comments about Graphs 321

7.14 Problems and Projects 325

8 Applications of Differential Calculus 343

8.1 Derivatives as Rates of Change 344

8.1.1 Growth of Organisms 344

8.1.2 Radioactive Decay 346

8.1.3 Cost of Production 347

8.2 The Pulley Problem of L'Hospital 348

8.2.1 The Solution Using Calculus 349

8.2.2 The Solution by Balancing Forces 351

8.3 The Suspension Bridge 353

8.4 An Experiment of Galileo 360

8.4.1 Sliding Ice Cubes and Spinning Wheels 361

8.4.2 Torque and Rotational Inertia 363

8.4.3 The Mathematics behind Galileo's Experiment 368

8.5 From Fermat's Principle to the Reflecting Telescope 370

8.5.1 Fermat's Principle and the Reflection of Light 372

8.5.2 The Refraction of Light 375

8.5.3 About Lenses 379

8.5.4 Refracting and Reflecting Telescopes 384

8.6 Problems and Projects 391

9 The Basics of Integral Calculus 411

9.1 The Definite Integral of a Function 412

9.2 Volume and the Definite Integral 416

9.3 Lengths of Curves and the Definite Integral 419

9.4 Surface Area and the Definite Integral 422

9.5 The Definite integral and the Fundamental Theorem 427

9.6 Area as Antiderivative 431

9.7 Finding Antiderivatives 435

9.7.1 Integration by Substitution 435

9.7.2 Integration by Parts 437

9.7.3 Some Algebraic Moves 439

9.8 Inverse Functions 440

9.9 Inverse Trigonometric and Hyperbolic Functions 443

9.9.1 Trigonometric Inverses 443

9.9.2 Hyperbolic Inverses 448

9.10 Trigonometric and Hyperbolic Substitutions 452

9.11 Some Integral Formulas 455

9.12 The Trapezoidal and Simpson Rules 458

9.13 One Loop of the Sine Curve 461

9.14 Problems and Projects 465

10 Applications of Integral Calculus 479

10.1 Estimating the Weight of Domes 480

10.1.1 The Hagia Sophia 480

10.1.2 The Roman Pantheon 483

10.2 The Cables of a Suspension Bridge 489

10.3 From Pocket Watch to Pseudosphere 492

10.3.1 Volume and Surface Area of Revolution of the Tractrix 495

10.3.2 The Pseudosphere 497

10.4 Calculating the Motion of a Planet 499

10.4.1 Determining Position in Terms of Time 501

10.4.2 Determining Speed and Direction 508

10.4.3 Earth, Jupiter, and Halley 511

10.5 Integral Calculus and the Action of Forces 512

10.5.1 Work and Energy, Impulse and Momentum 513

10.5.1 Analysis of Springs 517

10.5.2 The Force in a Gun Barrel 519

10.5.3 The Springfield Rifle 522

10.6 Problems and Projects 523

11 Basics of Differential Equations 547

11.1 First-Order Separable Differential Equations 548

11.2 The Method of Integrating Factors 551

11.3 Direction Fields and Euler's Method 553

11.4 The Polar Coordinate System 558

11.5 The Complex Plane 562

11.6 Second-Order Differential Equations 567

11.7 The Basics of Power Series 570

11.8 Taylor and Maclaurin Series 574

11.9 Solving a Second-Order Differential Equation 579

11.10 Free Fall with Air Resistance 582

11.10.1 Going Up 584

11.10.2 Coming Down 585

11.10.3 Bullets and Ping-Poug Balls 588

11.11 Systems with Springs and Damping Elements 589

11.11.1 The Family Sedan and the Stock Car 592

11.12 More about Hanging Cables 595

11.13 Problems and Projects 602

12 Polar Calculus and Newton's Planetary Orbits 623

12.1 Graphing Polar Equations 624

12.2 The Conic Sections in Polar Coordinates 627

12.3 The Derivative of a Polar Function 630

12.4 The Lengths of Polar Curves 633

12.5 Areas in Polar Coordinates 635

12.6 Equiangular Spirals 638

12.7 Centripetal Force in Cartesian Coordinates 642

12.8 Going Polar 644

12.9 From Conic Section to Inverse Square Law and Back Again 647

12.10 Gravity and Geometry 650

12.11 Spiral Galaxies 657

12.12 Problems and Projects 661

References 677

Image Credits and Notes 683

Index 685

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