# Calculus

**Author(s):**
Mylan Redfern
Betounes
,
David
Betounes

**Edition:
**
1

**Copyright:
**
2021

**Pages:
**
1028

The book presents the essential calculus material using a traditional approach and organization of the canonical collection of concepts. The writing style is clear and concise, oriented toward students’ understanding, and augmented by numerous figures, drawings, animations (movies), and lecture videos.

A major feature of the book is the abundance of new examples and exercises, both traditional and computational, which can be blended into the calculus sequence according to the instructor’s preferences.

*Features of the Book*

- Can be bundled with a detailed and elaborate electronic Solutions Manuals (SMs) for the odd-numbered exercises in Chapters 1–14. The SMs, written by the authors, contain all the steps in the progression toward the answer and many comments on the underlying algebra (and calculus). NOTE: These SMs are titled
and*Calculus Single Variable Solutions Manual*and may be purchased separately.*Calculus Multivariable Solutions Manual* - Includes unique sections: An Alternative to Trig Substitutions, A New Class of Arc Length Problems, A New Class of Surface Area Problems, An Overlooked Class of Solids of Revolution, and A Simplified, Rigorous Approach to the Theorems of Stokes, Green, and Gauss.
- Contains thoughtfully designed exercise sets, with exercises ranging from basic to hard, both computational and theoretical

Preface

**Chapter 1 Limits**

1.1 Limits: An Informal View

1.2 Limit Tools: The Graphical Method

1.3 Limit Tools: The Numerical Method

1.4 Limit Tools: The Algebraic Method

1.5 Limit Laws

1.6 One-Sided Limits

1.7 Continuous Functions

1.8 Limits Involving Infinity

1.9 The Definition of a Limit

**Chapter 2 Derivatives**

2.1 The Tangent Line Problem

2.2 The Derivative Function

2.3 Derivatives of Power Functions

2.4 Velocity

2.5 Differentials and Higher Derivatives

2.6 The Product and Quotient Rules

2.7 Derivatives of Trig Functions

2.8 The Chain Rule

2.9 Derivatives of Exponential and Logarithmic Functions

2.10 Implicit Differentiation

2.11 Derivatives of Inverse Functions

2.12 Hyperbolic Functions

2.13 Related Rates

**Chapter 3 Applications of Derivatives**

3.1 Curve Sketching: 1st Derivatives

3.2 Curve Sketching: 2nd Derivatives

3.3 Maximum and Minimum Values

3.4 Optimization

3.5 The Mean Value Theorem

3.6 L’ Hospital’s Rule

3.7 Antiderivatives and Integrals

**Chapter 4 Integrals**

4.1 Areas and Sums

4.2 Riemann Sums

4.3 Definite Integrals

4.4 The Fundamental Theorem of Calculus

4.5 The Substitution Method

**Chapter 5 Applications of Integrals**

5.1 Areas of Planar Regions

5.2 Solids with Known Sectional Area

5.3 Solids of Revolutions–Disk Method

5.4 The Washer and Shell Methods

5.5 Lengths of Curves

5.6 Surface Area of Solids of Revolutions

5.7 Force and Work

5.8 Moments and Center of Mass

**Chapter 6 Techniques of Integration**

6.1 Integration by Parts

6.2 Trigonometric Integrals

6.3 Trigonometric Substitutions

6.4 Partial Fractions

6.5 Integrals via a Computer Algebra System

6.6 Improper Integrals

**Chapter 7 Differential Equations**

7.1 Introduction to Differential Equations

7.2 Separable DEs and Modeling

7.3 First-Order Linear DEs

**Chapter 8 Sequences and Series**

8.1 Sequences

8.2 Infinite Series

8.3 The nth Term and Integral Tests

8.4 The Comparison Tests

8.5 The Ratio and Root Tests

8.6 Absolute and Conditional Convergence

8.7 Power Series

8.8 Taylor and Maclaurin Series

8.9 Operations on Power Series

**Chapter 9 Plane Curves and Polar Coordinates**

9.1 Parametric Equations

9.2 Parametric Calculus - Derivatives

9.3 Parametric Calculus - Integrals

9.4 Polar Coordinates

9.5 Polar Coordinate Calculus

9.6 Conic Sections

**Chapter 10 Vectors and Geometry**

10.1 Three-Dimensional Coordinate Systems

10.2 Vectors

10.3 The Dot Product

10.4 The Cross Product

10.5 Lines and Planes in Space

10.6 Cylinders and Quadric Surfaces

**Chapter 11 Partial Derivatives**

11.1 Functions of Two or More Variables

11.2 Multivariable Limits and Continuity

11.3 Partial Derivatives

11.4 The Chain Rule

11.5 Directional Derivatives & Gradients

11.6 Tangent Planes

11.7 Local Extrema and Saddle Points

**Chapter 12 Multiple Integrals**

12.1 Double Integrals and Iterated Integrals

12.2 Double Integrals over General Regions

12.3 Double Integrals in Polar Coordinates

12.4 Applications of Double Integrals

12.5 Triple Integrals

12.6 Triple Integrals in Cylindrical and Spherical Coordinates

**Chapter 13 Geometry of Curves and Surfaces**

13.1 Space Curves and Arc Length

13.2 Surfaces and Surface Area

**Chapter 14 Vector Analysis**

14.1 Vector Fields & Differential Operators

14.2 Line Integrals

14.3 The Fundamental Theorem of Line Integrals

14.4 Surface Integrals

14.5 Stokes’ Theorem - Green’s Theorem

14.6 The Divergence Theorem

Answers to Odd Exercises OA-1

Index I-1

**Mylan Redfern Betounes**

Mylan Redfern received her BS in mathematics from Augusta College and her PhD in mathematics from Louisiana State University. She has authored numerous refereed publications in stochastic analysis and has published two books on Mathematical Computing and Calculus. She has taught all levels of mathematics at the University of Southern Mississippi, Valdosta State University, and the University of Texas, Permian Basin.

**David Betounes**

David Betounes received his BArch in architecture from the University of Southern California and his PhD in mathematics from Florida State University. He has authored numerous refereed publications in differential geometry and mathematical physics and has published four books on DEs, PDEs, Mathematical Computing, and Calculus. He has taught all levels of mathematics at the University of Southern Mississippi, Valdosta State University, and the University of Texas, Permian Basin.

The book presents the essential calculus material using a traditional approach and organization of the canonical collection of concepts. The writing style is clear and concise, oriented toward students’ understanding, and augmented by numerous figures, drawings, animations (movies), and lecture videos.

A major feature of the book is the abundance of new examples and exercises, both traditional and computational, which can be blended into the calculus sequence according to the instructor’s preferences.

*Features of the Book*

- Can be bundled with a detailed and elaborate electronic Solutions Manuals (SMs) for the odd-numbered exercises in Chapters 1–14. The SMs, written by the authors, contain all the steps in the progression toward the answer and many comments on the underlying algebra (and calculus). NOTE: These SMs are titled
and*Calculus Single Variable Solutions Manual*and may be purchased separately.*Calculus Multivariable Solutions Manual* - Includes unique sections: An Alternative to Trig Substitutions, A New Class of Arc Length Problems, A New Class of Surface Area Problems, An Overlooked Class of Solids of Revolution, and A Simplified, Rigorous Approach to the Theorems of Stokes, Green, and Gauss.
- Contains thoughtfully designed exercise sets, with exercises ranging from basic to hard, both computational and theoretical

Preface

**Chapter 1 Limits**

1.1 Limits: An Informal View

1.2 Limit Tools: The Graphical Method

1.3 Limit Tools: The Numerical Method

1.4 Limit Tools: The Algebraic Method

1.5 Limit Laws

1.6 One-Sided Limits

1.7 Continuous Functions

1.8 Limits Involving Infinity

1.9 The Definition of a Limit

**Chapter 2 Derivatives**

2.1 The Tangent Line Problem

2.2 The Derivative Function

2.3 Derivatives of Power Functions

2.4 Velocity

2.5 Differentials and Higher Derivatives

2.6 The Product and Quotient Rules

2.7 Derivatives of Trig Functions

2.8 The Chain Rule

2.9 Derivatives of Exponential and Logarithmic Functions

2.10 Implicit Differentiation

2.11 Derivatives of Inverse Functions

2.12 Hyperbolic Functions

2.13 Related Rates

**Chapter 3 Applications of Derivatives**

3.1 Curve Sketching: 1st Derivatives

3.2 Curve Sketching: 2nd Derivatives

3.3 Maximum and Minimum Values

3.4 Optimization

3.5 The Mean Value Theorem

3.6 L’ Hospital’s Rule

3.7 Antiderivatives and Integrals

**Chapter 4 Integrals**

4.1 Areas and Sums

4.2 Riemann Sums

4.3 Definite Integrals

4.4 The Fundamental Theorem of Calculus

4.5 The Substitution Method

**Chapter 5 Applications of Integrals**

5.1 Areas of Planar Regions

5.2 Solids with Known Sectional Area

5.3 Solids of Revolutions–Disk Method

5.4 The Washer and Shell Methods

5.5 Lengths of Curves

5.6 Surface Area of Solids of Revolutions

5.7 Force and Work

5.8 Moments and Center of Mass

**Chapter 6 Techniques of Integration**

6.1 Integration by Parts

6.2 Trigonometric Integrals

6.3 Trigonometric Substitutions

6.4 Partial Fractions

6.5 Integrals via a Computer Algebra System

6.6 Improper Integrals

**Chapter 7 Differential Equations**

7.1 Introduction to Differential Equations

7.2 Separable DEs and Modeling

7.3 First-Order Linear DEs

**Chapter 8 Sequences and Series**

8.1 Sequences

8.2 Infinite Series

8.3 The nth Term and Integral Tests

8.4 The Comparison Tests

8.5 The Ratio and Root Tests

8.6 Absolute and Conditional Convergence

8.7 Power Series

8.8 Taylor and Maclaurin Series

8.9 Operations on Power Series

**Chapter 9 Plane Curves and Polar Coordinates**

9.1 Parametric Equations

9.2 Parametric Calculus - Derivatives

9.3 Parametric Calculus - Integrals

9.4 Polar Coordinates

9.5 Polar Coordinate Calculus

9.6 Conic Sections

**Chapter 10 Vectors and Geometry**

10.1 Three-Dimensional Coordinate Systems

10.2 Vectors

10.3 The Dot Product

10.4 The Cross Product

10.5 Lines and Planes in Space

10.6 Cylinders and Quadric Surfaces

**Chapter 11 Partial Derivatives**

11.1 Functions of Two or More Variables

11.2 Multivariable Limits and Continuity

11.3 Partial Derivatives

11.4 The Chain Rule

11.5 Directional Derivatives & Gradients

11.6 Tangent Planes

11.7 Local Extrema and Saddle Points

**Chapter 12 Multiple Integrals**

12.1 Double Integrals and Iterated Integrals

12.2 Double Integrals over General Regions

12.3 Double Integrals in Polar Coordinates

12.4 Applications of Double Integrals

12.5 Triple Integrals

12.6 Triple Integrals in Cylindrical and Spherical Coordinates

**Chapter 13 Geometry of Curves and Surfaces**

13.1 Space Curves and Arc Length

13.2 Surfaces and Surface Area

**Chapter 14 Vector Analysis**

14.1 Vector Fields & Differential Operators

14.2 Line Integrals

14.3 The Fundamental Theorem of Line Integrals

14.4 Surface Integrals

14.5 Stokes’ Theorem - Green’s Theorem

14.6 The Divergence Theorem

Answers to Odd Exercises OA-1

Index I-1

**Mylan Redfern Betounes**

Mylan Redfern received her BS in mathematics from Augusta College and her PhD in mathematics from Louisiana State University. She has authored numerous refereed publications in stochastic analysis and has published two books on Mathematical Computing and Calculus. She has taught all levels of mathematics at the University of Southern Mississippi, Valdosta State University, and the University of Texas, Permian Basin.

**David Betounes**

David Betounes received his BArch in architecture from the University of Southern California and his PhD in mathematics from Florida State University. He has authored numerous refereed publications in differential geometry and mathematical physics and has published four books on DEs, PDEs, Mathematical Computing, and Calculus. He has taught all levels of mathematics at the University of Southern Mississippi, Valdosta State University, and the University of Texas, Permian Basin.