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Calculus From Approximation to Theory
 
Dan Sloughter Furman University, Greenville, SC
Calculus From Approximation to Theory
Calculus From Approximation to Theory
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-5588-0
Product Code:  TEXT/64
List Price: $99.00
MAA Member Price: $74.25
AMS Member Price: $74.25
eBook ISBN:  978-1-4704-6305-2
Product Code:  TEXT/64.E
List Price: $99.00
MAA Member Price: $74.25
AMS Member Price: $74.25
Softcover ISBN:  978-1-4704-5588-0
eBook: ISBN:  978-1-4704-6305-2
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
Calculus From Approximation to Theory
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Calculus From Approximation to Theory
Calculus From Approximation to Theory
Dan Sloughter Furman University, Greenville, SC
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-5588-0
Product Code:  TEXT/64
List Price: $99.00
MAA Member Price: $74.25
AMS Member Price: $74.25
eBook ISBN:  978-1-4704-6305-2
Product Code:  TEXT/64.E
List Price: $99.00
MAA Member Price: $74.25
AMS Member Price: $74.25
Softcover ISBN:  978-1-4704-5588-0
eBook ISBN:  978-1-4704-6305-2
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
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 642020; 571 pp
    MSC: 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.

    Readership

    Undergraduate 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 two-body 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. First-order linear equations
    • Problems 8.3
    • 8.4. Second-order linear equations
    • Problems 8.4
    • 8.5. Pendulums and mass-spring 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
  • Reviews
     
     
    • Dan Sloughter's 'Calculus from Approximation to Theory' offers a refreshing, high-level introduction to the subject.

      John Ross, Southwestern University
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 642020; 571 pp
MSC: 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.

Readership

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 two-body 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. First-order linear equations
  • Problems 8.3
  • 8.4. Second-order linear equations
  • Problems 8.4
  • 8.5. Pendulums and mass-spring 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, high-level introduction to the subject.

    John Ross, Southwestern University
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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