: Publication

PREFACE

This textbook (Calclus:  A Rigorous, Yet Student-Friendly Approach)has grown from my lecture notes from teaching the calculus sequence, and hence, has a structure similar to them. The text also has a similar structure to the Precalculus text co-written with my colleague Jeffery VanHamlin. One of the original goals of writing this, was to create a hybrid of a textbook, study guide and solutions manual. In doing so, I hoped this would increase the likelihood that students would actually read it. When I first began classroom testing early versions of the manuscript for this text, I was pleasantly surprised to see many students spending time reading the sections, as opposed to only using it for the homework exercises. With the students that did a significant amount of reading, I observed that their mathematical maturity significantly improved from the beginning of the first semester to the end of the third semester.

The best way to learn calculus, or any other mathematical subject, is to work as many exercises as you can. It’s not just the number of exercises, but also the variety.  For this reason, the exercise sections contain different types of problems. You should also work through the examples found throughout the text. Treat them as homework exercises, and consult the solution if you get stuck. Additionally, reading the textbook, coming to and participating in class, discussing the material with friends and watching videos are all valuable study techniques. Ideally, your study habits should include a variety of these activities.

There is more material in this book than can reasonably be covered in a standard three semester sequence. The reason more material has been added is to increase the textbooks flexibility to instructors and to provide additional topics that interested students can explore on their own. In fact, in numerous sections I have introduced theorems and ideas which do not normally appear in an introductory calculus textbook, but which I feel are developmentally appropriate for students to learn about. These topics include the second difference quotient, the  derivative test, differentiation using a partial fraction expansion, Cauchy’s Mean Value Theorem, higher order derivatives of parametric curves, the total derivative, extrema in higher dimensions, Weierstrass substitution, integration using the even-odd decomposition of a function, the necessary algebra to compute the derivative of any radical function using the limit of a difference quotient, discrete derivatives, Feynman integration, quadruple integrals, integrals of order , Raabe’s test, the logarithm test and the curvature of a single variable function.

Regarding the subtitle, the textbook is mathematically rigorous, including proofs of many theorems. There are also theoretical exercises which will allow the student to practice with writing some basic proofs and using rigorous definitions. However, for a textbook at this level, there needs to be a balance between rigor and clarity. That is, an extremely rigorous book is generally not easy to read. On the other hand, a textbook which aims solely for clarity, may lack rigor. A goal with this text, is to have aspects of both. This can be seen in a variety of ways, for example, some theorems are “proved’ with an informal reasoning, since that may allow the student to see the basic idea more clearly, as opposed to worrying about a mathematically complete argument. Also, the textbook is student friendly in that it is written and formatted, primarily with the student in mind. Of course, the content also needs to be written for other mathematicians, but they are not the primary audience for reading the text. The textbook is also student friendly in that it contains a large number of worked examples, the majority of the answers appear in the back and all hand-drawn (by the author) sketches. I feel that the hand-drawn sketches help the student to see how to do the sketches better themselves. Certainly computer generated graphics look great, and help the student to visualize concepts, but again I feel that the hand-drawn sketches are useful for the students and is something different than all other calculus texts.

As is common with calculus textbooks, this text covers all three semesters of the introductory calculus sequence. Generally speaking, the first semester of the sequence corresponds to the first four chapters (excluding chapter 0), although I typically cannot finish all of chapter 4 by the end of calculus 1. The second semester of the sequence corresponds to chapters 5-7 (I generally begin with a review of chapter 4) and the third semester corresponds to chapters 8-12. The first chapter of the book, chapter 0, provides a very brief overview of some basic topics from precalculus. Ideally, that chapter is for reference only, and is not intended as a first exposure to the material.

Chapter 0. A Very Brief Review of Precalculus
§0.1      Algebra                                                                   
§0.2      Trigonometry

                                                              
Chapter 1. Limits
§1.1      Rates of Change                                                                
§1.2      One-Sided Limits                                                              
§1.3      Limits                                                                              
§1.4      Properties of Limits
§1.5      The Precise Definition of a Limit
§1.6      Infinite Limits and Limits at Infinity
§1.7      Continuity

Chapter 2. Differentiation
§2.1      Tangent Lines
§2.2      The Derivative
§2.3      Basic Differentiation Techniques
§2.4      Derivatives of Trigonometric Functions
§2.5      The Chain Rule
§2.6      Implicit Differentiation and Parametric Curves
§2.7      Derivatives of Inverse Functions
§2.8      Derivatives of Rational Functions using Partial Fractions (Optional)
§2.9      Differentials

Chapter 3. Applications of Derivatives
§3.1      Rates of Change
§3.2      Related Rates
§3.3      The Extreme Value Theorem
§3.4      The Mean Value Theorem
§3.5      The First Derivative Test
§3.6      The Second Derivative Test
§3.7      L'Ho ̂pital's Rule
§3.8      Optimization
§3.9      Newton’s Method (Optional)
§3.10        Antiderivatives

Chapter 4. Integration
§4.1      Riemann Sums
§4.2      The Definite Integral
§4.3      The Fundamental Theorem of Calculus
§4.4      The Substitution Rule and The Area Between Curves
§4.5      Logarithms and Hyperbolic Functions
§4.6      Basic Integration Techniques

Chapter 5. Applications of Integration
§5.1      Volume: The Disk Method and The Washer Method
§5.2      Volume: The Method of Cylindrical Shells
§5.3      The Center of Mass, Centroid and Work
§5.4      Arc Length and Surface Area

Chapter 6. Integration Techniques
§6.1      Integration by Parts
§6.2      Integration by Partial Fractions
§6.3      Integrals of Trigonometric Functions
§6.4      Trigonometric Substitution
§6.5      Integration Using the Even-Odd Decomposition of a Function (Optional)
§6.6      Improper Integrals

Chapter 7. Sequences and Infinite Series
§7.1      Sequences
§7.2      Infinite Series
§7.3      The Integral Test
§7.4      The Comparison Tests
§7.5      The Ratio and Root Tests
§7.6      The Alternating Series Test
§7.7      Power Series
§7.8      The Taylor and Maclaurin Series
§7.9      Other Tests (Optional)
§7.10      Fourier Series (Optional)

Chapter 8. Vectors
§8.1      Euclidean Space
§8.2      Vectors
§8.3      Multiplying Vectors
§8.4      Lines and Planes in Space
§8.5      Cylinders and Quadric Surfaces

Chapter 9. Vector Functions
§9.1      Vector Functions and Arc Length
§9.2      The TNB-frame
§9.3      The Curvature of a Space Curve

Chapter 10. Partial Derivatives
§10.1       Multivariable Functions and the Topology of R^n
§10.2       Limits and Continuity of Multivariable Functions
§10.3       Partial Derivatives
§10.4       The Chain Rule
§10.5       The Directional Derivative
§10.6       Tangent Planes
§10.7       Extreme Values
§10.8       Lagrange Multipliers
§10.9       The Total Derivative (Optional)
§10.10     Concavity, Curvature and Optimization in Higher Dimensions (Optional)

Chapter 11. Multiple Integrals
§11.1     Double Integrals
§11.2     Area, Center of Mass and Surface Area
§11.3     Polar Coordinates
§11.4     Triple Integrals and Higher Order Integrals
§11.5     Triple Integrals using Other Coordinate Systems
§11.6     Substitution in a Double Integral

Chapter 12. Vector Calculus
§12.1     Line Integrals
§12.2     Vector Fields and Flux
§12.3     The Fundamental Theorem of Line Integrals
§12.4     Green's Theorem
§12.5     Surface Integrals
§12.6     Parametrized Surfaces
§12.7     Stokes' Theorem
§12.8     The Divergence Theorem

Appendix
 A.1 The Derivative of a Radical Function
 A.2         Feynman Integration
 A.3         Discrete Derivatives
 A.4         The Topology of the Real Line
A.5         Symbols Glossary
A.6         A Table of Integrals

Answers to Selected Exercises
Index

Nick Goins received his B.S in mathematics from the University of Michigan – Dearborn and his M.A. in mathematics from the University of Toledo. He is a professor of mathematics at St. Clair County Community College. His primary mathematical interests include topology and real analysis in addition to teaching calculus courses. He is also the author of CALCULUS:  A Rigorous, Yet Student-Friendly Approach. In his spare time, he enjoys traveling, cooking, reading and spending time with his wife and family.