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## The ABC’s of Calculus (Volume 1)

Angelo B. Mingarelli

### About the Book

#### Single Variable or Volume 1

**Table of Contents**

**Functions and Their Properties**- The Meaning of a Function
- Function Values and the Box Method
- The Absolute Value of a Function
- A Quick Review of Inequalities
- The Triangle Inequalities

- Chapter Exercises
- Using Computer Algebra Systems (CAS)

**Limits and Continuity**- One-Sided Limits of Functions
- Two-Sided Limits and Continuity
- Important Theorems About Continuous Functions
- Evaluating Limits at Inﬁnity
- How to Guess a Limit
- Limits Equal to ±∞
- Chapter Exercises

**The Derivative of a Function**- Motivation
- Working with Derivatives
- The Chain Rule
- Implicit Functions and Their Derivatives
- Derivatives of Trigonometric Functions
- Important Results About Derivatives
- Inverse Functions
- Inverse Trigonometric Functions
- Derivatives of Inverse Trigonometric Functions
- L’Hospital’s Rule
- Chapter Exercises
- Challenge Questions
- Using Computer Algebra Systems

**Exponentials and Logarithms**- Exponential Functions and Their Logarithms
- Euler’s Number, e = 2.718281828
- Euler’s Exponential Function and the Natural Logarithm
- Derivative of the Natural Logarithm
- Differentiation Formulae for General Exponential Functions
- Differentiation Formulae for General Logarithmic Functions
- Applications
- Chapter Exercises
- Using Computer Algebra Systems

**Curve Sketching**- Solving Polynomial Inequalities
- Solving Rational Function Inequalities
- Graphing Techniques
- Chapter Exercises

**Integration**- Antiderivatives and the Indeﬁnite Integral
- Deﬁnite Integrals
- The Summation Convention
- Area and the Riemann Integral
- Chapter Exercises
- Using Computer Algebra Systems

**Techniques of Integration**- Trigonometric Identities
- The Substitution Rule
- Integration by Parts
- The Product of a Polynomial and a Sine or Cosine
- The Product of a Polynomial and an Exponential
- The Product of a Polynomial and a Logarithm
- The Product of an Exponential and a Sine or Cosine

- Partial Fractions
- Review of Long Division of Polynomials
- The Integration of Partial Fractions

- Products of Trigonometric Functions
- Products of Sines and Cosines
- Fourier Coeﬃcients
- Products of Secants and Tangents

- Trigonometric Substitutions
- Completing the Square in a Quadratic (Review)
- Trigonometric Substitutions

- Improper Integrals
- Chapter Exercises
- Using Computer Algebra Systems

**Applications of the Integral**- Motivation
- Finding the Area Between Two Curves
- The Volume of a Solid of Revolution
- Measuring the Length of a Curve
- Moments and Centers of Mass
- Chapter Exercises
- Using Computer Algebra Systems

**Appendix A: Review of Exponents and Radicals****Appendix B: The Straight Line****Appendix C: A Quick Review of Trigonometry**- The Right-angled Isosceles Triangle (RT45)
- The RT30 triangle
- The basic trigonometric functions
- Identities
- The Law of Sines
- The Law of Cosines
- Identities for the Sum and Difference of Angles

**Appendix D: The Natural Domain of a Function**

#### Teacher Resources/Student Resources

Teacher resources may include a question bank in the form of multiple choice questions covering every section of the text along with a Comprehensive Solutions Manual (also available to students).

#### Philosophy

The idea behind this text is to introduce students in the Sciences or Engineering to Calculus sequentially and by means of many solved examples. It has been used by tens of thousands of students over a period of 20 years and the text is reviewed and updated every year by students through in class comments and correspondence and their names are included in perpetuity in subsequent editions. The author simplifies the exposition of all the material to a point where it all appears quite normal and easy to read. Almost all the steps in reaching a solution to a given problem are outlined and this is one of the major strengths of this text.

#### Online Ebook

The ebook is available for lease only at https://books.mingarelli.com for a period of 120 days (the duration of a standard semester in North American colleges and universities) and all for a low price of $50 CAD.

#### A few student reviews (2019-2020)

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#### About the Author

Angelo B. Mingarelli, Ph.D. (Toronto), is Professor of Mathematics at Carleton University, Canada where he’s been an active teacher and researcher for the past 25 years. He is the author of over 100 scientific articles and 6 books two of which are educational, the rest being research oriented. He was awarded a Faculty of Science Teaching Award on two occasions as well as a University-wide Teaching Achievement Award. Prior to joining Carleton, he was Professor of Mathematics at the University of Ottawa for 12 years. As an academic he was President of the Carleton University Academic Staff Association (a certified trade union in Ontario) for five terms and served on its Board for 10 years. As well, he sat as a faculty member on the Carleton University Board of Governors for a three year term. Among his extracurricular interests one finds his passion in creating art for sentient AI beings.