Milne Open Textbooks

**Author(s):**
Eugene Boman and Robert Rogers

**Publication Date:** August 25, 2023

**ISBN:** 978-1-942341-95-6

**Affiliation: **SUNY Fredonia

*Differential Calculus: From Practice to Theory* covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the foundational ideas (limits, continuity) are developed to replace infinitesimals, first intuitively then rigorously. This approach is more historically accurate than the usual development of calculus and, more importantly, it is pedagogically sound. The text also incorporates curated activities from the TRansforming Instruction in Undergraduate Mathematics Instruction via Primary Historical Sources (TRIUMPHS) project to provide students with ample opportunities to develop relevant competencies.

**To the Instructor**- 0.1 What Do Students Need From Calculus?
- 0.2 Some (Possibly Startling) Choices We’ve Made
- 0.3 Some Practical Advice
- 0.4 The TRIUMPHS Project
- 0.5 Rantings From the Cranky Old Guys in the Back of the Room
- 0.6 A Plea For Help

**Preface: Calculus is a Rock****I**.**From Practice…****1. Introduction: Some Advice on Problem Solving**- 1.1 Using Letters Instead of Numbers
- 1.2 Substitution, or Making Things “Easy on the Eyes”
- 1.3 An “Easy” Problem From Geometry
- 1.4 Our advice, a synopsis

**2. Science Before Calculus**- 2.1 Apologia
- 2.2 Some Preliminaries
- 2.3 The Laziness of Nature
- 2.4 Fermat’s Method of Adequality
- 2.5 Descartes’s Method of Normals
- 2.6 Roberval, Conic Sections, and the Dynamic Approach
- 2.7 Snell’s Law and the Limitations of Adequality

**3. Differentials, Differentiation, and the Derivative**- 3.1 Historical Introduction
- 3.2 The General Differentiation Rules

**4. Slopes, Tangents, and Rates of Change**- 4.1 Slopes and Tangents
- 4.2 Defining the Tangent Line
- 4.3 The Vomit Comet
- 4.4 Galileo Drops the Ball
- 4.5 The Derivative
- 4.6 Thinking Dynamically
- 4.7 Newton’s Method of Fluxions
- 4.8 Self-intersecting Curves and Parametric Equations
- 4.9 Bridges, Chains, Domes, and Telescopes

**5. Calculus and Trigonometry**- 5.1 A Trigonometric Interlude
- 5.2 Polar Coordinates
- 5.3 The Differentials of the Sine and Cosine Functions
- 5.4 The Differentials of the Other Trigonometric Functions
- 5.5 The Inverse Tangent and Cotangent Functions
- 5.6 The Witch of Agnesi and the Inverse Tangent Function
- 5.7 The Differentials of the Inverse Tangent and Inverse Cotangent Functions
- 5.8 The Other Inverse Trigonometric Functions
- 5.9 Curvature

**6. Approximation Methods**- 6.1 Root Finding: Two Pre-Calculus Approaches
- 6.2 Newton’s Method
- 6.3 Euler’s Method
- 6.4 Higher Derivatives, Lagrange, and Taylor

**7. Exponentials and Logarithms**- 7.1 The Natural Exponential
- 7.2 Exponential Growth
- 7.3 The Natural Logarithm
- 7.4 The Derivative of the Natural Logarithm
- 7.5 General Logarithms and Exponentials
- 7.6 Leonhard Euler, Harmonic Oscillators, and Complex Numbers

**8. Optimization: Going to Extremes**- 8.1 Introduction
- 8.2 Preliminaries: Some Simple Optimizations
- 8.3 Reflections, Refractions, and Rainbows
- 8.4 Global vs. Local Extrema
- 8.5 Optimization, the Abstract Problem
- 8.6 Concavity and the Second Derivative Test
- 8.7 Optimization Problems

**9. Graphing with Calculus**- 9.1 Graphing with a Formula for
*y***(***x***)** - 9.2 Graphing Without Formulas
- 9.3 Graphing with a Formula for d
*y*/d*t*: Initial Value Problems

- 9.1 Graphing with a Formula for
**10. Modeling with Calculus**- 10.1 Population Dynamics
- 10.2 Selected Modeling Problems

**11. Limits and L’Hôpital’s Rule**- 11.1 Horizontal Asymptotes as Limits “at Infinity”
- 11.2 The Squeeze Theorem
- 11.3 Vertical Asymptotes
- 11.4 Indeterminate Forms and L’Hôpital’s Rule

**II. …To Theory****12. What’s Wrong With Differentials?**- 12.1 Calculus and Bishop Berkeley
- 12.2 Secants and Tangents

**13. The Differentiation Rules via Limits**- 13.1 The Limit Rules (Theorems)
- 13.2 The General Differentiation Theorems, via Limits
- 13.3 The Chain Rule
- 13.4 The Product Rule
- 13.5 The Other General Differentiation Rules
- 13.6 Derivatives of the Trigonometric Functions, via Limits
- 13.7 Inverse Functions

**14. The First Derivative Test, Redux**- 14.1 Fermat’s Theorem
- 14.2 Rolle’s Lemma and the Mean Value Theorem
- 14.3 The Proof of the First Derivative Test

**15. When the Derivative Doesn’t Exist**- 15.1 One Sided Derivatives

**16. Formal Limits**- 16.1 Getting Around Infinity
- 16.2 Limits at a Real Number
- 16.3 Limit Laws (Theorems)

**A. TRIUMPHS: Primary Source Projects (PSPs)**- A.1 Fermat’s Method for Finding Maxima and Minima
- A.1.1 Examples of Fermat’s Method
- A.1.2 Resolution
- A.1.3 Recommendations for Further Reading

- A.2 Fourier’s Proof of the Irrationality of
*e*- A.2.1 Proof by Contradiction
- A.2.2 Incommensurate Numbers
- A.2.3 Some Fundamental Sets of Numbers
- A.2.4 Fourier’s Proof of the Irrationality of
*e* - A.2.5 What about
*e*2?

- A.3 L’Hôpital’s Rule
- A.3.1 Limits of Indeterminate Type
- A.3.2 L’Hôpital’s
*Analyse*and the Calculus of Differentials - A.3.3 L’Hôpital’s Rule: Determining Limits of Indeterminate Type
- A.3.4 Conclusion

- A.4 Investigations Into d’Alembert’s Definition of Limit
- A.4.1 Introduction
- A.4.2 D’Alembert’s Limit Definition
- A.4.3 A More Precise Definition of Limit
- A.4.4 Conclusion

- A.5 The Radius of Curvature According to Christiaan Huygens
- A.5.1 The Longitude Problem
- A.5.2 Huygens and The Radius of Curvature
- A.5.3 The Modern Equation for Curvature

- A.6 How to Calculate
*π*: Machin’s Inverse Tangents- A.6.1 Introductin
- A.6.2 Part 1: It all started with arctangent
- A.6.3 Part 2: Addition and Subtraction formulas for tangent
- A.6.4 Part 3: Choosing a better angle

- A.7 The Derivatives of the Sine and Cosine Functions
- A.7.1 Introduction
- A.7.2 Part 1: Introducing the derivative
- A.7.3 Part 2: Exploring the derivative
- A.7.4 Part 3: Trigonometric functions

- A.1 Fermat’s Method for Finding Maxima and Minima
**B. TRIUMPHS: Notes to the Instructor**- B.1 Fermat’s Method for Finding Maxima and Minima
- B.2 Fourier’s Proof of the Irrationality of
*e* - B.3 L’Hôpital’s Rule
- B.4 Investigations Into d’Alembert’s Definition of Limit
- B.5 The Radius of Curvature According to Christiaan Huygens
- B.6 How to Calculate
*π*: Machin’s inverse tangents - B.7 The Derivatives of the Sine and Cosine Functions