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Exploratory Examples for Real Analysis
 
Exploratory Examples for Real Analysis
MAA Press: An Imprint of the American Mathematical Society
eBook ISBN:  978-1-4704-5835-5
Product Code:  CLRM/20.E
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
Exploratory Examples for Real Analysis
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Exploratory Examples for Real Analysis
MAA Press: An Imprint of the American Mathematical Society
eBook ISBN:  978-1-4704-5835-5
Product Code:  CLRM/20.E
List Price: $45.00
MAA Member Price: $33.75
AMS Member Price: $33.75
  • Book Details
     
     
    Classroom Resource Materials
    Volume: 202003; 160 pp

    This text supplement contains 12 exploratory exercises designed to facilitate students' understanding of the most elemental concepts encountered in a first real analysis course: notions of boundedness, supremum/infimum, sequences, continuity and limits, limit suprema/infima, and pointwise and uniform convergence. In designing the exercises, the authors ask students to formulate definitions, make connections between different concepts, derive conjectures, or complete a sequence of guided tasks designed to facilitate concept acquisition. Each exercise has three basic components: making observations and generating ideas from hands-on work with examples, thinking critically about the examples, and answering additional questions for reflection.

    The exercises can be used in a variety of ways: to motivate a lecture, to serve as a basis for in-class activities, or to be used for lab sessions, where students work in small groups and submit reports of their investigations. While the exercises have been useful for real analysis students of all ability levels, the authors believe this resource might prove most beneficial in the following scenarios:

    A two-semester sequence in which the following topics are covered: properties of the real numbers, sequences, continuity, sequences and series of functions, differentiation, and integration.

    A class of students for whom analysis is their first upper division course.

    A group of students with a wide range of abilities for whom a cooperative approach focusing upon fundamental concepts could help to close the gap in skill development and concept acquisition.

    An independent study or private tutorial in which the student receives a minimal level of instruction.

    A resource for an instructor developing a cooperative, interactive course that does not involve the use of a standard text.

    Ancillary materials, including Visual Guide Sheets for those exercises that involve the use of technology and Report Guides for a lab session approach are provided online at: http:www.saintmarys.edu/~jsnow.

    In designing the exercise, the authors were inspired by Ellen Parker's book, Laboratory Experiences in Group Theory, also published by the MAA.

  • Table of Contents
     
     
    • Chapters
    • 1. Boundedness of Sets
    • 2. Introducing the “Epsilon Definition” of Least Upper Bound
    • 3. Introduction to the Formal Definition of Convergence
    • 4. Experience with the Definition of the Limit of a Sequence
    • 5. Experience with the Negation of the Definition of Convergence
    • 6. Algebraic Combinations of Sequences
    • 7. Conditions Related to Convergence
    • 8. Understanding the Limit Superior and the Limit Inferior
    • 9. Continuity and Sequences
    • 10. Another Definition of Continuity
    • 11. Experience with the $\epsilon -\delta $ Definitions of Continuity and Limit
    • 12. Uniform Convergence of a Sequence of Functions
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 202003; 160 pp

This text supplement contains 12 exploratory exercises designed to facilitate students' understanding of the most elemental concepts encountered in a first real analysis course: notions of boundedness, supremum/infimum, sequences, continuity and limits, limit suprema/infima, and pointwise and uniform convergence. In designing the exercises, the authors ask students to formulate definitions, make connections between different concepts, derive conjectures, or complete a sequence of guided tasks designed to facilitate concept acquisition. Each exercise has three basic components: making observations and generating ideas from hands-on work with examples, thinking critically about the examples, and answering additional questions for reflection.

The exercises can be used in a variety of ways: to motivate a lecture, to serve as a basis for in-class activities, or to be used for lab sessions, where students work in small groups and submit reports of their investigations. While the exercises have been useful for real analysis students of all ability levels, the authors believe this resource might prove most beneficial in the following scenarios:

A two-semester sequence in which the following topics are covered: properties of the real numbers, sequences, continuity, sequences and series of functions, differentiation, and integration.

A class of students for whom analysis is their first upper division course.

A group of students with a wide range of abilities for whom a cooperative approach focusing upon fundamental concepts could help to close the gap in skill development and concept acquisition.

An independent study or private tutorial in which the student receives a minimal level of instruction.

A resource for an instructor developing a cooperative, interactive course that does not involve the use of a standard text.

Ancillary materials, including Visual Guide Sheets for those exercises that involve the use of technology and Report Guides for a lab session approach are provided online at: http:www.saintmarys.edu/~jsnow.

In designing the exercise, the authors were inspired by Ellen Parker's book, Laboratory Experiences in Group Theory, also published by the MAA.

  • Chapters
  • 1. Boundedness of Sets
  • 2. Introducing the “Epsilon Definition” of Least Upper Bound
  • 3. Introduction to the Formal Definition of Convergence
  • 4. Experience with the Definition of the Limit of a Sequence
  • 5. Experience with the Negation of the Definition of Convergence
  • 6. Algebraic Combinations of Sequences
  • 7. Conditions Related to Convergence
  • 8. Understanding the Limit Superior and the Limit Inferior
  • 9. Continuity and Sequences
  • 10. Another Definition of Continuity
  • 11. Experience with the $\epsilon -\delta $ Definitions of Continuity and Limit
  • 12. Uniform Convergence of a Sequence of Functions
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.