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.

Table of Contents:
Preface Part I 1. The Astronomy and Geometry of the Greeks 1.1. The Greeks Explain the Universe 1.2. Achieving the Impossible? 1.3. Greek Geometry 1.4. The Pythagorean Theorem 1.5. The Radian Measure of an Angle 1.6. Greek Trigonometry 1.7. Aristarchus Sizes Up the Universe 1.8. Problems and Projects 2. The Genius of Archimedes 2.1. The Conic Sections 2.2. The Question of Area 2.3. Playing with Squares 2.4. The Area of a Parabolic Section 2.5. The Method of Archimedes 2.6. Problems and Projects 3. A New Astronomy 3.1. A Fixed Sun at the Center 3.2. Copernicus's Model of Earth's Orbit 3.3. About the Distances of the Planets from the Sun 3.4. Tycho Brahe and Parallax 3.5. Kepler's Elliptical Orbits 3.6. The Studies of Galileo 3.7. The Size of the Solar System 3.8. Problems and Projects 4. The Coordinate Geometry of Descartes 4.1. The Real Numbers 4.2. The Coordinate Plane 4.3. About the Parabola 4.4. About the Ellipse 4.5. Quadratic Equations in x and y 4.6. Circles and Trigonometry 4.7. Problems and Projects 5. The Calculus of Leibniz 5.1. Straight Lines 5.2. Tangent Lines to Curves 5.3. The Function Concept 5.4. The Derivative of a Function 5.5. Fermat, Kepler, and Wine Barrels 5.6. The Definite Integral 5.7. Cavalieri's Principle 5.8. Differentials and the Fundamental Theorem 5.9. Volumes of Revolution 5.10. Problems and Projects 6. The Calculus of Newton 6.1. Simple Functions and Areas 6.2. The Derivative of a Simple Function 6.3. From Simple Functions to Power Series 6.4. The Mathematics of a Moving Point 6.5. Galileo and Acceleration 6.6. Dealing with Forces 6.7. The Trajectory of a Projectile 6.8. Newton Studies the Motion of the Planets 6.9. Connecting Force and Geometry 6.10. The Law of Universal Gravitation 6.11. Problems and Projects Part II 7. Differential Calculus 7.1. Mathematical Functions 7.2. A Study of Limits 7.3. Continuous Functions 7.4. Differentiable Functions 7.5. Computing Derivatives 7.6. Some Theoretical Concerns 7.7. Derivatives of Trigonometric Functions 7.8. Understanding Functions 7.9. Graphing Functions 7.10. Exponential Functions 7.11. Logarithm Functions 7.12. Hyperbolic Functions 7.13. Final Comments about Graphs 7.14. Problems and Projects 8. Applications of Differential Calculus 8.1. Derivatives as Rates of Change 8.1.1. Growth of Organisms 8.1.2. Radioactive Decay 8.1.3. Cost of Production 8.2. The Pulley Problem of L'Hospital 8.2.1. The Solution Using Calculus 8.2.2. The Solution by Balancing Forces 8.3. The Suspension Bridge 8.4. An Experiment of Galileo 8.4.1. Sliding Ice Cubes and Spinning Wheels 8.4.2. Torque and Rotational Inertia 8.4.3. The Mathematics behind Galileo's Experiment 8.5. From Fermat's Principle to the Reflecting Telescope 8.5.1. Fermat's Principle and the Reflection of Light 8.5.2. The Refraction of Light 8.5.3. About Lenses 8.5.4. Refracting and Reflecting Telescopes 8.6. Problems and Projects 9. The Basics of Integral Calculus 9.1. The Definite Integral of a Function 9.2. Volume and the Definite Integral 9.3. Lengths of Curves and the Definite Integral 9.4. Surface Area and the Definite Integral 9.5. The Definite Integral and the Fundamental Theorem 9.6. Area as Antiderivative 9.7. Finding Antiderivatives 9.7.1. Integration by Substitution 9.7.2. Integration by Parts 9.7.3. Some Algebraic Moves 9.8. Inverse Functions 9.9. Inverse Trigonometric and Hyperbolic Functions 9.9.1. Trigonometric Inverses 9.9.2. Hyperbolic Inverses 9.10. Trigonometric and Hyperbolic Substitutions 9.11. Some Integral Formulas 9.12. The Trapezoidal and Simpson Rules 9.13. One Loop of the Sine Curve 9.14. Problems and Projects 10. Applications of Integral Calculus 10.1. Estimating the Weight of Domes 10.1.1. The Hagia Sophia 10.1.2. The Roman Pantheon 10.2. The Cables of a Suspension Bridge 10.3. From Pocket Watch to Pseudosphere 10.3.1. Volume and Surface Area of Revolution of the Tractrix 10.3.2. The Pseudosphere 10.4. Calculating the Motion of a Planet 10.4.1. Determining Position in Terms of Time 10.4.2. Determining Speed and Direction 10.4.3. Earth, Jupiter, and Halley 10.5. Integral Calculus and the Action of Forces 10.5.1. Work and Energy, Impulse and Momentum 10.5.2. Analysis of Springs 10.5.3. The Force in a Gun Barrel 10.5.4. The Springfield Rifle 10.6. Problems and Projects 11. Basics of Differential Equations 11.1. First-Order Separable Differential Equations 11.2. The Method of Integrating Factors 11.3. Direction Fields and Euler's Method 11.4. The Polar Coordinate System 11.5. The Complex Plane 11.6. Second-Order Differential Equations 11.7. The Basics of Power Series 11.8. Taylor and Maclaurin Series 11.9. Solving a Second-Order Differential Equation 11.10. Free Fall with Air Resistance 11.10.1. Going Up 11.10.2. Coming Down 11.10.3. Bullets and Ping-Pong Balls 11.11. Systems with Springs and Damping Elements 11.11.1. The Family Sedan and the Stock Car 11.12. More about Hanging Cables 11.13. Problems and Projects 12. Polar Calculus and Newton's Planetary Orbits 12.1. Graphing Polar Equations 12.2. The Conic Sections in Polar Coordinates 12.3. The Derivative of a Polar Function 12.4. The Lengths of Polar Curves 12.5. Areas in Polar Coordinates 12.6. Equiangular Spirals 12.7. Centripetal Force in Cartesian Coordinates 12.8. Going Polar 12.9. From Conic Section to Inverse Square Law and Back Again 12.10. Gravity and Geometry 12.11. Spiral Galaxies 12.12. Problems and Projects References Image Credits and Notes Index