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# Calculus in Context

Background, Basics, and Applications

Alexander J. Hahn

Publication Date

**A new approach to teaching calculus that uses historical examples and draws on applications from science and engineering.**

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…

**A new approach to teaching calculus that uses historical examples and draws on applications from science and engineering.**

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.

Related

Reviews

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.

Recommended.

Hahn's book is the perfect choice for college and university teachers who want to teach calculus with reference to its origins and applications.

Very well written in an engaging and enthusiastic style: it is very suitable for first year students, is perhaps not too demanding for students about to enter university, and it is particularly useful to those with more than a passing interest in astronomy. There is plenty to learn for the reader, and the massive text is also a good reference book on calculus. This labour of love from the author more than satisfies the high hopes for a good calculus book... and I highly recommend it.

About

Publication Date

Status

Available

Trim Size

7

x 10

Pages

712

ISBN

9781421422305

Illustration Description

645 line drawings

Subject

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

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

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

Resources

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