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.

• PrefacePart I1. The Astronomy and Geometry of the Greeks1.1. The Greeks Explain the Universe1.2. Achieving the Impossible?1.3. Greek Geometry1.4. The Pythagorean Theorem1.5. The Radian Measure of an Angle1.6. Greek Trigonometry1.7. Aristarchus Sizes Up the Universe1.8. Problems and Projects2. The Genius of Archimedes2.1. The Conic Sections2.2. The Question of Area2.3. Playing with Squares2.4. The Area of a Parabolic Section2.5. The Method of Archimedes2.6. Problems and Projects3. A New Astronomy3.1. A Fixed Sun at the Center3.2. Copernicus's Model of Earth's Orbit3.3. About the Distances of the Planets from the Sun3.4. Tycho Brahe and Parallax3.5. Kepler's Elliptical Orbits3.6. The Studies of Galileo3.7. The Size of the Solar System3.8. Problems and Projects4. The Coordinate Geometry of Descartes4.1. The Real Numbers4.2. The Coordinate Plane4.3. About the Parabola4.4. About the Ellipse4.5. Quadratic Equations in x and y4.6. Circles and Trigonometry4.7. Problems and Projects5. The Calculus of Leibniz5.1. Straight Lines5.2. Tangent Lines to Curves5.3. The Function Concept5.4. The Derivative of a Function5.5. Fermat, Kepler, and Wine Barrels5.6. The Definite Integral5.7. Cavalieri's Principle5.8. Differentials and the Fundamental Theorem5.9. Volumes of Revolution5.10. Problems and Projects6. The Calculus of Newton6.1. Simple Functions and Areas6.2. The Derivative of a Simple Function6.3. From Simple Functions to Power Series6.4. The Mathematics of a Moving Point6.5. Galileo and Acceleration6.6. Dealing with Forces6.7. The Trajectory of a Projectile6.8. Newton Studies the Motion of the Planets6.9. Connecting Force and Geometry6.10. The Law of Universal Gravitation6.11. Problems and ProjectsPart II7. Differential Calculus7.1. Mathematical Functions7.2. A Study of Limits7.3. Continuous Functions7.4. Differentiable Functions7.5. Computing Derivatives7.6. Some Theoretical Concerns7.7. Derivatives of Trigonometric Functions7.8. Understanding Functions7.9. Graphing Functions7.10. Exponential Functions7.11. Logarithm Functions7.12. Hyperbolic Functions7.13. Final Comments about Graphs7.14. Problems and Projects8. Applications of Differential Calculus8.1. Derivatives as Rates of Change8.1.1. Growth of Organisms8.1.2. Radioactive Decay8.1.3. Cost of Production8.2. The Pulley Problem of L'Hospital8.2.1. The Solution Using Calculus8.2.2. The Solution by Balancing Forces8.3. The Suspension Bridge8.4. An Experiment of Galileo8.4.1. Sliding Ice Cubes and Spinning Wheels8.4.2. Torque and Rotational Inertia8.4.3. The Mathematics behind Galileo's Experiment8.5. From Fermat's Principle to the Reflecting Telescope8.5.1. Fermat's Principle and the Reflection of Light8.5.2. The Refraction of Light8.5.3. About Lenses8.5.4. Refracting and Reflecting Telescopes8.6. Problems and Projects9. The Basics of Integral Calculus9.1. The Definite Integral of a Function9.2. Volume and the Definite Integral9.3. Lengths of Curves and the Definite Integral9.4. Surface Area and the Definite Integral9.5. The Definite Integral and the Fundamental Theorem9.6. Area as Antiderivative9.7. Finding Antiderivatives9.7.1. Integration by Substitution9.7.2. Integration by Parts9.7.3. Some Algebraic Moves9.8. Inverse Functions9.9. Inverse Trigonometric and Hyperbolic Functions9.9.1. Trigonometric Inverses9.9.2. Hyperbolic Inverses9.10. Trigonometric and Hyperbolic Substitutions9.11. Some Integral Formulas9.12. The Trapezoidal and Simpson Rules9.13. One Loop of the Sine Curve9.14. Problems and Projects10. Applications of Integral Calculus10.1. Estimating the Weight of Domes10.1.1. The Hagia Sophia10.1.2. The Roman Pantheon10.2. The Cables of a Suspension Bridge10.3. From Pocket Watch to Pseudosphere10.3.1. Volume and Surface Area of Revolution of the Tractrix10.3.2. The Pseudosphere10.4. Calculating the Motion of a Planet10.4.1. Determining Position in Terms of Time10.4.2. Determining Speed and Direction10.4.3. Earth, Jupiter, and Halley10.5. Integral Calculus and the Action of Forces10.5.1. Work and Energy, Impulse and Momentum10.5.2. Analysis of Springs10.5.3. The Force in a Gun Barrel10.5.4. The Springfield Rifle10.6. Problems and Projects11. Basics of Differential Equations11.1. First-Order Separable Differential Equations11.2. The Method of Integrating Factors11.3. Direction Fields and Euler's Method11.4. The Polar Coordinate System11.5. The Complex Plane11.6. Second-Order Differential Equations11.7. The Basics of Power Series11.8. Taylor and Maclaurin Series11.9. Solving a Second-Order Differential Equation11.10. Free Fall with Air Resistance11.10.1. Going Up11.10.2. Coming Down11.10.3. Bullets and Ping-Pong Balls11.11. Systems with Springs and Damping Elements11.11.1. The Family Sedan and the Stock Car11.12. More about Hanging Cables11.13. Problems and Projects12. Polar Calculus and Newton's Planetary Orbits12.1. Graphing Polar Equations12.2. The Conic Sections in Polar Coordinates12.3. The Derivative of a Polar Function12.4. The Lengths of Polar Curves12.5. Areas in Polar Coordinates12.6. Equiangular Spirals12.7. Centripetal Force in Cartesian Coordinates12.8. Going Polar12.9. From Conic Section to Inverse Square Law and Back Again12.10. Gravity and Geometry12.11. Spiral Galaxies12.12. Problems and ProjectsReferencesImage Credits and NotesIndex

The depth of detail in each application [offered by Calculus in Context] provides an excellent structure for guiding students through the “why should we care” moments that every calculus class experiences.—Mathematical Association of America

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.

• PrefacePart I1. The Astronomy and Geometry of the Greeks1.1. The Greeks Explain the Universe1.2. Achieving the Impossible?1.3. Greek Geometry1.4. The Pythagorean Theorem1.5. The Radian Measure of an Angle1.6. Greek Trigonometry1.7. Aristarchus Sizes Up the Universe1.8. Problems and Projects2. The Genius of Archimedes2.1. The Conic Sections2.2. The Question of Area2.3. Playing with Squares2.4. The Area of a Parabolic Section2.5. The Method of Archimedes2.6. Problems and Projects3. A New Astronomy3.1. A Fixed Sun at the Center3.2. Copernicus's Model of Earth's Orbit3.3. About the Distances of the Planets from the Sun3.4. Tycho Brahe and Parallax3.5. Kepler's Elliptical Orbits3.6. The Studies of Galileo3.7. The Size of the Solar System3.8. Problems and Projects4. The Coordinate Geometry of Descartes4.1. The Real Numbers4.2. The Coordinate Plane4.3. About the Parabola4.4. About the Ellipse4.5. Quadratic Equations in x and y4.6. Circles and Trigonometry4.7. Problems and Projects5. The Calculus of Leibniz5.1. Straight Lines5.2. Tangent Lines to Curves5.3. The Function Concept5.4. The Derivative of a Function5.5. Fermat, Kepler, and Wine Barrels5.6. The Definite Integral5.7. Cavalieri's Principle5.8. Differentials and the Fundamental Theorem5.9. Volumes of Revolution5.10. Problems and Projects6. The Calculus of Newton6.1. Simple Functions and Areas6.2. The Derivative of a Simple Function6.3. From Simple Functions to Power Series6.4. The Mathematics of a Moving Point6.5. Galileo and Acceleration6.6. Dealing with Forces6.7. The Trajectory of a Projectile6.8. Newton Studies the Motion of the Planets6.9. Connecting Force and Geometry6.10. The Law of Universal Gravitation6.11. Problems and ProjectsPart II7. Differential Calculus7.1. Mathematical Functions7.2. A Study of Limits7.3. Continuous Functions7.4. Differentiable Functions7.5. Computing Derivatives7.6. Some Theoretical Concerns7.7. Derivatives of Trigonometric Functions7.8. Understanding Functions7.9. Graphing Functions7.10. Exponential Functions7.11. Logarithm Functions7.12. Hyperbolic Functions7.13. Final Comments about Graphs7.14. Problems and Projects8. Applications of Differential Calculus8.1. Derivatives as Rates of Change8.1.1. Growth of Organisms8.1.2. Radioactive Decay8.1.3. Cost of Production8.2. The Pulley Problem of L'Hospital8.2.1. The Solution Using Calculus8.2.2. The Solution by Balancing Forces8.3. The Suspension Bridge8.4. An Experiment of Galileo8.4.1. Sliding Ice Cubes and Spinning Wheels8.4.2. Torque and Rotational Inertia8.4.3. The Mathematics behind Galileo's Experiment8.5. From Fermat's Principle to the Reflecting Telescope8.5.1. Fermat's Principle and the Reflection of Light8.5.2. The Refraction of Light8.5.3. About Lenses8.5.4. Refracting and Reflecting Telescopes8.6. Problems and Projects9. The Basics of Integral Calculus9.1. The Definite Integral of a Function9.2. Volume and the Definite Integral9.3. Lengths of Curves and the Definite Integral9.4. Surface Area and the Definite Integral9.5. The Definite Integral and the Fundamental Theorem9.6. Area as Antiderivative9.7. Finding Antiderivatives9.7.1. Integration by Substitution9.7.2. Integration by Parts9.7.3. Some Algebraic Moves9.8. Inverse Functions9.9. Inverse Trigonometric and Hyperbolic Functions9.9.1. Trigonometric Inverses9.9.2. Hyperbolic Inverses9.10. Trigonometric and Hyperbolic Substitutions9.11. Some Integral Formulas9.12. The Trapezoidal and Simpson Rules9.13. One Loop of the Sine Curve9.14. Problems and Projects10. Applications of Integral Calculus10.1. Estimating the Weight of Domes10.1.1. The Hagia Sophia10.1.2. The Roman Pantheon10.2. The Cables of a Suspension Bridge10.3. From Pocket Watch to Pseudosphere10.3.1. Volume and Surface Area of Revolution of the Tractrix10.3.2. The Pseudosphere10.4. Calculating the Motion of a Planet10.4.1. Determining Position in Terms of Time10.4.2. Determining Speed and Direction10.4.3. Earth, Jupiter, and Halley10.5. Integral Calculus and the Action of Forces10.5.1. Work and Energy, Impulse and Momentum10.5.2. Analysis of Springs10.5.3. The Force in a Gun Barrel10.5.4. The Springfield Rifle10.6. Problems and Projects11. Basics of Differential Equations11.1. First-Order Separable Differential Equations11.2. The Method of Integrating Factors11.3. Direction Fields and Euler's Method11.4. The Polar Coordinate System11.5. The Complex Plane11.6. Second-Order Differential Equations11.7. The Basics of Power Series11.8. Taylor and Maclaurin Series11.9. Solving a Second-Order Differential Equation11.10. Free Fall with Air Resistance11.10.1. Going Up11.10.2. Coming Down11.10.3. Bullets and Ping-Pong Balls11.11. Systems with Springs and Damping Elements11.11.1. The Family Sedan and the Stock Car11.12. More about Hanging Cables11.13. Problems and Projects12. Polar Calculus and Newton's Planetary Orbits12.1. Graphing Polar Equations12.2. The Conic Sections in Polar Coordinates12.3. The Derivative of a Polar Function12.4. The Lengths of Polar Curves12.5. Areas in Polar Coordinates12.6. Equiangular Spirals12.7. Centripetal Force in Cartesian Coordinates12.8. Going Polar12.9. From Conic Section to Inverse Square Law and Back Again12.10. Gravity and Geometry12.11. Spiral Galaxies12.12. Problems and ProjectsReferencesImage Credits and NotesIndex

The depth of detail in each application [offered by Calculus in Context] provides an excellent structure for guiding students through the “why should we care” moments that every calculus class experiences.—Mathematical Association of America

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.

• PrefacePart I1. The Astronomy and Geometry of the Greeks1.1. The Greeks Explain the Universe1.2. Achieving the Impossible?1.3. Greek Geometry1.4. The Pythagorean Theorem1.5. The Radian Measure of an Angle1.6. Greek Trigonometry1.7. Aristarchus Sizes Up the Universe1.8. Problems and Projects2. The Genius of Archimedes2.1. The Conic Sections2.2. The Question of Area2.3. Playing with Squares2.4. The Area of a Parabolic Section2.5. The Method of Archimedes2.6. Problems and Projects3. A New Astronomy3.1. A Fixed Sun at the Center3.2. Copernicus's Model of Earth's Orbit3.3. About the Distances of the Planets from the Sun3.4. Tycho Brahe and Parallax3.5. Kepler's Elliptical Orbits3.6. The Studies of Galileo3.7. The Size of the Solar System3.8. Problems and Projects4. The Coordinate Geometry of Descartes4.1. The Real Numbers4.2. The Coordinate Plane4.3. About the Parabola4.4. About the Ellipse4.5. Quadratic Equations in x and y4.6. Circles and Trigonometry4.7. Problems and Projects5. The Calculus of Leibniz5.1. Straight Lines5.2. Tangent Lines to Curves5.3. The Function Concept5.4. The Derivative of a Function5.5. Fermat, Kepler, and Wine Barrels5.6. The Definite Integral5.7. Cavalieri's Principle5.8. Differentials and the Fundamental Theorem5.9. Volumes of Revolution5.10. Problems and Projects6. The Calculus of Newton6.1. Simple Functions and Areas6.2. The Derivative of a Simple Function6.3. From Simple Functions to Power Series6.4. The Mathematics of a Moving Point6.5. Galileo and Acceleration6.6. Dealing with Forces6.7. The Trajectory of a Projectile6.8. Newton Studies the Motion of the Planets6.9. Connecting Force and Geometry6.10. The Law of Universal Gravitation6.11. Problems and ProjectsPart II7. Differential Calculus7.1. Mathematical Functions7.2. A Study of Limits7.3. Continuous Functions7.4. Differentiable Functions7.5. Computing Derivatives7.6. Some Theoretical Concerns7.7. Derivatives of Trigonometric Functions7.8. Understanding Functions7.9. Graphing Functions7.10. Exponential Functions7.11. Logarithm Functions7.12. Hyperbolic Functions7.13. Final Comments about Graphs7.14. Problems and Projects8. Applications of Differential Calculus8.1. Derivatives as Rates of Change8.1.1. Growth of Organisms8.1.2. Radioactive Decay8.1.3. Cost of Production8.2. The Pulley Problem of L'Hospital8.2.1. The Solution Using Calculus8.2.2. The Solution by Balancing Forces8.3. The Suspension Bridge8.4. An Experiment of Galileo8.4.1. Sliding Ice Cubes and Spinning Wheels8.4.2. Torque and Rotational Inertia8.4.3. The Mathematics behind Galileo's Experiment8.5. From Fermat's Principle to the Reflecting Telescope8.5.1. Fermat's Principle and the Reflection of Light8.5.2. The Refraction of Light8.5.3. About Lenses8.5.4. Refracting and Reflecting Telescopes8.6. Problems and Projects9. The Basics of Integral Calculus9.1. The Definite Integral of a Function9.2. Volume and the Definite Integral9.3. Lengths of Curves and the Definite Integral9.4. Surface Area and the Definite Integral9.5. The Definite Integral and the Fundamental Theorem9.6. Area as Antiderivative9.7. Finding Antiderivatives9.7.1. Integration by Substitution9.7.2. Integration by Parts9.7.3. Some Algebraic Moves9.8. Inverse Functions9.9. Inverse Trigonometric and Hyperbolic Functions9.9.1. Trigonometric Inverses9.9.2. Hyperbolic Inverses9.10. Trigonometric and Hyperbolic Substitutions9.11. Some Integral Formulas9.12. The Trapezoidal and Simpson Rules9.13. One Loop of the Sine Curve9.14. Problems and Projects10. Applications of Integral Calculus10.1. Estimating the Weight of Domes10.1.1. The Hagia Sophia10.1.2. The Roman Pantheon10.2. The Cables of a Suspension Bridge10.3. From Pocket Watch to Pseudosphere10.3.1. Volume and Surface Area of Revolution of the Tractrix10.3.2. The Pseudosphere10.4. Calculating the Motion of a Planet10.4.1. Determining Position in Terms of Time10.4.2. Determining Speed and Direction10.4.3. Earth, Jupiter, and Halley10.5. Integral Calculus and the Action of Forces10.5.1. Work and Energy, Impulse and Momentum10.5.2. Analysis of Springs10.5.3. The Force in a Gun Barrel10.5.4. The Springfield Rifle10.6. Problems and Projects11. Basics of Differential Equations11.1. First-Order Separable Differential Equations11.2. The Method of Integrating Factors11.3. Direction Fields and Euler's Method11.4. The Polar Coordinate System11.5. The Complex Plane11.6. Second-Order Differential Equations11.7. The Basics of Power Series11.8. Taylor and Maclaurin Series11.9. Solving a Second-Order Differential Equation11.10. Free Fall with Air Resistance11.10.1. Going Up11.10.2. Coming Down11.10.3. Bullets and Ping-Pong Balls11.11. Systems with Springs and Damping Elements11.11.1. The Family Sedan and the Stock Car11.12. More about Hanging Cables11.13. Problems and Projects12. Polar Calculus and Newton's Planetary Orbits12.1. Graphing Polar Equations12.2. The Conic Sections in Polar Coordinates12.3. The Derivative of a Polar Function12.4. The Lengths of Polar Curves12.5. Areas in Polar Coordinates12.6. Equiangular Spirals12.7. Centripetal Force in Cartesian Coordinates12.8. Going Polar12.9. From Conic Section to Inverse Square Law and Back Again12.10. Gravity and Geometry12.11. Spiral Galaxies12.12. Problems and ProjectsReferencesImage Credits and NotesIndex

The depth of detail in each application [offered by Calculus in Context] provides an excellent structure for guiding students through the “why should we care” moments that every calculus class experiences.—Mathematical Association of America