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Softcover ISBN:  9781470455880 
Product Code:  TEXT/64 
List Price:  $99.00 
MAA Member Price:  $74.25 
AMS Member Price:  $74.25 
Sale Price:  $64.35 
eBook ISBN:  9781470463052 
Product Code:  TEXT/64.E 
List Price:  $99.00 
MAA Member Price:  $74.25 
AMS Member Price:  $74.25 
Sale Price:  $64.35 
Softcover ISBN:  9781470455880 
eBook ISBN:  9781470463052 
Product Code:  TEXT/64.B 
List Price:  $198.00 $148.50 
MAA Member Price:  $148.50 $111.38 
AMS Member Price:  $148.50 $111.38 
Sale Price:  $128.70 $96.53 

Book DetailsAMS/MAA TextbooksVolume: 64; 2020; 571 ppMSC: Primary 26; 00
Calculus from Approximation to Theory takes a fresh and innovative look at the teaching and learning of calculus. One way to describe calculus might be to say it is a suite of techniques that approximate curved things by flat things and through a limiting process applied to those approximations arrive at an exact answer. Standard approaches to calculus focus on that limiting process as the heart of the matter. This text places its emphasis on the approximating processes and thus illuminates the motivating ideas and makes clearer the scientific usefulness, indeed centrality, of the subject while paying careful attention to the theoretical foundations. Limits are defined in terms of sequences, the derivative is defined from the best affine approximation, and greater attention than usual is paid to numerical techniques and the order of an approximation. Access to modern computational tools is presumed throughout and the use of these tools is woven seamlessly into the exposition and problems. All of the central topics of a yearlong calculus course are covered, with the addition of treatment of difference equations, a chapter on the complex plane as the arena for motion in two dimensions, and a much more thorough and modern treatment of differential equations than is standard.
Dan Sloughter is Emeritus Professor of Mathematics at Furman University with interests in probability, statistics, and the philosophy of mathematics and statistics. He has been involved in efforts to reform calculus instruction for decades and has published widely on that topic. This book, one of the results of that work, is very well suited for a yearlong introduction to calculus that focuses on ideas over techniques.
ReadershipUndergraduate students interested in calculus.

Table of Contents

Cover

Title page

Copyright

Contents

Preface

Chapter 1. Sequences and Limits

1.1. Calculus: Areas and tangents

Problems 1.1

1.2. Sequences

Problems 1.2

1.3. The sum of a sequence

Problems 1.3

1.4. Difference equations

Problems 1.4

1.5. Nonlinear difference equations

Problems 1.5

Chapter 2. Functions and Their Properties

2.1. Functions and their graphs

Problems 2.1

2.2. Trigonometric functions

Problems 2.2

2.3. Limits and the notion of continuity

Problems 2.3

2.4. Continuous functions

Problems 2.4

2.5. Some consequences of continuity

Problems 2.5

Chapter 3. Derivatives and Best Affine Approximations

3.1. Best affine approximations

Problems 3.1

3.2. Derivatives and rates of change

Problems 3.2

3.3. Differentiation of rational functions

Problems 3.3

3.4. Differentiation of compositions

Problems 3.4

3.5. Differentiation of trigonometric functions

Problems 3.5

3.6. Newton’s method

Problems 3.6

3.7. The Mean Value Theorem

Problems 3.7

3.8. Finding maximum and minimum values

Problems 3.8

3.9. The geometry of graphs

Problems 3.9

Chapter 4. Integrals

4.1. The definite integral

Problems 4.1

4.2. Numerical approximations

Problems 4.2

4.3. The Fundamental Theorem of Calculus

Problems 4.3

4.4. Using the Fundamental Theorem

Problems 4.4

4.5. More techniques of integration

Problems 4.5

4.6. Improper integrals

Problems 4.6

4.7. More on area

Problems 4.7

4.8. Distance, position, and length

Problems 4.8

Chapter 5. Taylor Polynomials and Series

5.1. Polynomial approximations

Problems 5.1

5.2. Taylor’s theorem

Problems 5.2

5.3. Infinite series revisited

Problems 5.3

5.4. The comparison test

Problems 5.4

5.5. The ratio test

Problems 5.5

5.6. Absolute convergence

Problems 5.6

5.7. Power series

Problems 5.7

5.8. Taylor series

Problems 5.8

5.9. Some limit calculations

Problems 5.9

Chapter 6. More Transcendental Functions

6.1. The exponential function

Problems 6.1

6.2. The natural logarithm function

Problems 6.2

6.3. Models of growth and decay

Problems 6.3

6.4. Integration of rational functions

Problems 6.4

6.5. Inverse trigonometric functions

Problems 6.5

6.6. Trigonometric substitutions

Problems 6.6

6.7. Hyperbolic functions

Problems 6.7

Chapter 7. The Complex Plane

7.1. The algebra of complex numbers

Problems 7.1

7.2. The calculus of complex functions

Problems 7.2

7.3. Motion in the plane

Problems 7.3

7.4. The twobody problem

Problems 7.4

Chapter 8. Differential Equations

8.1. Numerical solutions

Problems 8.1

8.2. Separation of variables

Problems 8.2

8.3. Firstorder linear equations

Problems 8.3

8.4. Secondorder linear equations

Problems 8.4

8.5. Pendulums and massspring systems

Problems 8.5

8.6. Phase planes

Problems 8.6

8.7. Power series solutions

Problems 8.7

Appendix A. Answers to Selected Problems

A.1. Chapter 1

A.2. Chapter 2

A.3. Chapter 3

A.4. Chapter 4

A.5. Chapter 5

A.6. Chapter 6

A.7. Chapter 7

A.8. Chapter 8

Index

Back Cover


Additional Material

Reviews

Dan Sloughter's 'Calculus from Approximation to Theory' offers a refreshing, highlevel introduction to the subject.
John Ross, Southwestern University


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
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Calculus from Approximation to Theory takes a fresh and innovative look at the teaching and learning of calculus. One way to describe calculus might be to say it is a suite of techniques that approximate curved things by flat things and through a limiting process applied to those approximations arrive at an exact answer. Standard approaches to calculus focus on that limiting process as the heart of the matter. This text places its emphasis on the approximating processes and thus illuminates the motivating ideas and makes clearer the scientific usefulness, indeed centrality, of the subject while paying careful attention to the theoretical foundations. Limits are defined in terms of sequences, the derivative is defined from the best affine approximation, and greater attention than usual is paid to numerical techniques and the order of an approximation. Access to modern computational tools is presumed throughout and the use of these tools is woven seamlessly into the exposition and problems. All of the central topics of a yearlong calculus course are covered, with the addition of treatment of difference equations, a chapter on the complex plane as the arena for motion in two dimensions, and a much more thorough and modern treatment of differential equations than is standard.
Dan Sloughter is Emeritus Professor of Mathematics at Furman University with interests in probability, statistics, and the philosophy of mathematics and statistics. He has been involved in efforts to reform calculus instruction for decades and has published widely on that topic. This book, one of the results of that work, is very well suited for a yearlong introduction to calculus that focuses on ideas over techniques.
Undergraduate students interested in calculus.

Cover

Title page

Copyright

Contents

Preface

Chapter 1. Sequences and Limits

1.1. Calculus: Areas and tangents

Problems 1.1

1.2. Sequences

Problems 1.2

1.3. The sum of a sequence

Problems 1.3

1.4. Difference equations

Problems 1.4

1.5. Nonlinear difference equations

Problems 1.5

Chapter 2. Functions and Their Properties

2.1. Functions and their graphs

Problems 2.1

2.2. Trigonometric functions

Problems 2.2

2.3. Limits and the notion of continuity

Problems 2.3

2.4. Continuous functions

Problems 2.4

2.5. Some consequences of continuity

Problems 2.5

Chapter 3. Derivatives and Best Affine Approximations

3.1. Best affine approximations

Problems 3.1

3.2. Derivatives and rates of change

Problems 3.2

3.3. Differentiation of rational functions

Problems 3.3

3.4. Differentiation of compositions

Problems 3.4

3.5. Differentiation of trigonometric functions

Problems 3.5

3.6. Newton’s method

Problems 3.6

3.7. The Mean Value Theorem

Problems 3.7

3.8. Finding maximum and minimum values

Problems 3.8

3.9. The geometry of graphs

Problems 3.9

Chapter 4. Integrals

4.1. The definite integral

Problems 4.1

4.2. Numerical approximations

Problems 4.2

4.3. The Fundamental Theorem of Calculus

Problems 4.3

4.4. Using the Fundamental Theorem

Problems 4.4

4.5. More techniques of integration

Problems 4.5

4.6. Improper integrals

Problems 4.6

4.7. More on area

Problems 4.7

4.8. Distance, position, and length

Problems 4.8

Chapter 5. Taylor Polynomials and Series

5.1. Polynomial approximations

Problems 5.1

5.2. Taylor’s theorem

Problems 5.2

5.3. Infinite series revisited

Problems 5.3

5.4. The comparison test

Problems 5.4

5.5. The ratio test

Problems 5.5

5.6. Absolute convergence

Problems 5.6

5.7. Power series

Problems 5.7

5.8. Taylor series

Problems 5.8

5.9. Some limit calculations

Problems 5.9

Chapter 6. More Transcendental Functions

6.1. The exponential function

Problems 6.1

6.2. The natural logarithm function

Problems 6.2

6.3. Models of growth and decay

Problems 6.3

6.4. Integration of rational functions

Problems 6.4

6.5. Inverse trigonometric functions

Problems 6.5

6.6. Trigonometric substitutions

Problems 6.6

6.7. Hyperbolic functions

Problems 6.7

Chapter 7. The Complex Plane

7.1. The algebra of complex numbers

Problems 7.1

7.2. The calculus of complex functions

Problems 7.2

7.3. Motion in the plane

Problems 7.3

7.4. The twobody problem

Problems 7.4

Chapter 8. Differential Equations

8.1. Numerical solutions

Problems 8.1

8.2. Separation of variables

Problems 8.2

8.3. Firstorder linear equations

Problems 8.3

8.4. Secondorder linear equations

Problems 8.4

8.5. Pendulums and massspring systems

Problems 8.5

8.6. Phase planes

Problems 8.6

8.7. Power series solutions

Problems 8.7

Appendix A. Answers to Selected Problems

A.1. Chapter 1

A.2. Chapter 2

A.3. Chapter 3

A.4. Chapter 4

A.5. Chapter 5

A.6. Chapter 6

A.7. Chapter 7

A.8. Chapter 8

Index

Back Cover

Dan Sloughter's 'Calculus from Approximation to Theory' offers a refreshing, highlevel introduction to the subject.
John Ross, Southwestern University