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Differential Calculus: From Practice to Theory

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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.

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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 dy/dt: Initial Value Problems

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 e2?

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

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

Eugene Boman

Eugene Boman is Professor, Emeritus at the Harrisburg campus of the Pennsylvania State University. He has taught college level mathematics since he began his graduate studies at the University of Connecticut in 1984.  In 2008 he won the Carl B. Allendorfer Award for excellence in expository mathematical writing from the editors of Mathematics Magazine for the article “Mom! There’s an Asteroid in My Closet” (Mathematics Magazine, Vol. 80 (2007), pp. 247-273). He earned his BA from Reed College in 1984. He earned his MA in 1986 and his Ph.D. in 1993, both from the University of Connecticut.

Robert Rogers

Robert Rogers is a SUNY Distinguished Teaching Professor of Mathematics at SUNY Fredonia where he has been on the faculty since 1987.  He received his BS in Mathematics with Certification in Secondary Education from SUNY – Buffalo State, his MS in Mathematics from Syracuse University, and his PhD in Mathematics from the University at Buffalo.  He is a past president of the Association of Mathematics Teachers of New York State and a former chair and governor of the Mathematical Association of America Seaway Section.  He is a former editor of the New York State Mathematics Teachers’ Journal and is an MAA – Seaway Section Distinguished Lecturer.  He is a recipient of the SUNY Fredonia President’s Award for Excellence in Teaching and the MAA Seaway Section’s Clarence Stephens Award for Teaching Excellence.  He is a recipient of the MAA Distinguished Service Award – Seaway Section and is a member of the New York State Mathematics Educators’ Hall of Fame.