Differential Calculus: From Practice to Theory

Front Matter

• To the Instructor
  • What Do Students Need from Calculus?
  • Some (Possibly Startling) Choices We’ve Made
  • Some Practical Advice
  • The TRIUMPHS Project
  • Rantings from the Cranky Old Guys in the Back of the Room
  • A Plea for Help
• Preface: Calculus Is a Rock

Part I: From Practice…

• 1. Introduction: Some Advice on Problem Solving
  • Using Letters Instead of Numbers
  • Substitution, or Making Things “Easy on the Eyes”
  • An “Easy” Problem From Geometry
  • Our Advice, a Synopsis
• 2. Science Before Calculus
  • Apologia
  • Some Preliminaries
  • The Laziness of Nature
  • Fermat’s Method of Adequality
  • Descartes’s Method of Normals
  • Roberval, Conic Sections, and the Dynamic Approach
  • Snell’s Law and the Limitations of Adequality
• 3. Differentials, Differentiation, and the Derivative
  • Historical Introduction
  • The General Differentiation Rules
• 4. Slopes, Tangents, and Rates of Change
  • Slopes and Tangents
  • Defining the Tangent Line
  • The Vomit Comet
  • Galileo Drops the Ball
  • The Derivative
  • Thinking Dynamically
  • Newton’s Method of Fluxions
  • Self-Intersecting Curves and Parametric Equations
  • Bridges, Chains, Domes, and Telescopes
• 5. Calculus and Trigonometry
  • A Trigonometric Interlude
  • Polar Coordinates
  • The Differentials of the Sine and Cosine Functions
  • The Differentials of the Other Trigonometric Functions
  • The Inverse Tangent and Cotangent Functions
  • The Witch of Agnesi and the Inverse Tangent Function
  • The Differentials of the Inverse Tangent and Inverse Cotangent Functions
  • The Other Inverse Trigonometric Functions
  • Curvature
• 6. Approximation Methods
  • Root Finding: Two Pre-Calculus Approaches
  • Newton’s Method
  • Euler’s Method
  • Higher Derivatives, Lagrange, and Taylor
• 7. Exponentials and Logarithms
  • The Natural Exponential
  • Exponential Growth
  • The Natural Logarithm
  • The Derivative of the Natural Logarithm
  • General Logarithms and Exponentials
  • Leonhard Euler, Harmonic Oscillators, and Complex Numbers
• 8. Optimization: Going to Extremes
  • Introduction
  • Preliminaries: Some Simple Optimizations
  • Reflections, Refractions, and Rainbows
  • Global vs. Local Extrema
  • Optimization, the Abstract Problem
  • Concavity and the Second Derivative Test
  • Optimization Problems
• 9. Graphing with Calculus
  • Graphing with a Forumula for y(x)
  • Graphing Without Formulas
  • Graphing with a Formula for dy/dt: Initial Value Problems
  • Initial Value Problems
• 10. Modeling with Calculus
  • Population Dynamics
  • Selected Modeling Problems
• 11. Limits and L’Hôpital’s Rule
  • Horizontal Asymptotes as Limits “at Infinity”
  • The Squeeze Theorem
  • Vertical Asymptotes
  • Indeterminate Forms and L’Hôpital’s Rule

Part II: …To Theory

• 12. What’s Wrong with Differentials?
  • Calculus and Bishop Berkeley
  • Secants and Tangents
• 13. The Differentiation Rules via Limits
  • The Limit Rules (Theorems)
  • The General Differentiation Theorems, via Limits
  • The Chain Rule
  • The Product Rule
  • The Other General Differentiation Rules
  • Derivatives of the Trigonometric Functions, via Limits
  • Inverse Functions
• 14. The First Derivative Test, Redux
  • Fermat’s Theorem
  • Rolle’s Lemma and the Mean Value Theorem
  • The Proof of the First Derivative Test
• 15. When the Derivative Doesn’t Exist
  • One Sided Derivatives
• 16. Formal Limits
  • Getting Around Infinity
  • Limits at a Real Number
  • Limit Laws (Theorems)

Appendix A: TRIUMPHS: Primary Source Projects (PSPs)

• Fermat’s Method for Finding Maxima and Minima
  • Examples of Fermat’s Method
  • Resolution
  • Recommendations for Further Reading
• Fourier’s Proof of the Irrationality of e
  • Proof by Contradiction
  • Incommensurate Numbers
  • Some Fundamental Sets of Numbers
  • Fourier’s Proof of the Irrationality of e
  • What About e²?
• L’Hôpital’s Rule
  • Limits of Indeterminate Type
  • L’Hôpital’s Analyse and the Calculus of Differentials
  • L’Hôpital’s Rule: Determining Limits of Indeterminate Type
  • Conclusion
• Investigations Into d’Alembert’s Definition of Limit
  • Introduction
  • D’Alembert’s Limit Definition
  • A More Precise Definition of Limit
  • Conclusion
• The Radius of Curvature According to Christiaan Huygens
  • The Longitude Problem
  • Huygens and The Radius of Curvature
  • The Modern Equation for Curvature
• How to Calculate π: Machin’s Inverse Tangents
  • Introduction
  • Part 1: It All Started with Arctangent
  • Part 2: Addition and Subtraction Formulas for Tangent
  • Part 3: Choosing a Better Angle
• The Derivatives of the Sine and Cosine Functions
  • Introduction
  • Part 1: Introducing the Derivative
  • Part 2: Exploring the Derivative
  • Part 3: Trigonometric Functions

Appendix B: TRIUMPHS: Notes to the Instructor

• Fermat’s Method for Finding Maxima and Minima
• Fourier’s Proof of the Irrationality of e
• L’Hôpital’s Rule
• Investigations Into d’Alembert’s Definition of Limit
• The Radius of Curvature According to Christiaan Huygens
• How to Calculate π: Machin’s Inverse Tangents
• The Derivatives of the Sine and Cosine Functions