Introduction to Real Analysis An Educational Approach

ISBN-10: 0470371366
ISBN-13: 9780470371367
Edition: 2009
List price: $114.00
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Description: Along with presenting the many topics of analysis, this book guides readers to make the necessary connections between calculus and the secondary curriculum. In addition, this book provides readers with a solid foundation so that their future  More...

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Book details

List price: $114.00
Copyright year: 2009
Publisher: John Wiley & Sons, Incorporated
Publication date: 7/14/2009
Binding: Hardcover
Pages: 262
Size: 6.50" wide x 10.00" long x 0.75" tall
Weight: 1.100
Language: English

Along with presenting the many topics of analysis, this book guides readers to make the necessary connections between calculus and the secondary curriculum. In addition, this book provides readers with a solid foundation so that their future instruction and classroom teaching is based on fundamental understanding. The book begins with an outline of typical AP calculus topics. To help students review the discussed topics, sets of exercises are included and are similar to those in the upper range of a standard calculus text. The book moves on to analysis with more rigorous investigations. The topology of the line is discussed as well as limits and continuity, making sure to include unusual examples in order to direct a reader's thinking beyond intuitive reasoning and on to more rigorous understanding. The dichotomy of pointwise and uniform convergence is then addressed and is followed by differentiation and integration. In an effort to broaden the reader's perspective, Riemann-Stieltjes integrals are introduced. Also introduced is the Lebesgue measure as the generalization of the Euclidean length of an interval. The book concludes with a collection of different topics that an instructor can use to further extend and enhance the class. The author has chosen diverse subjects to allow for varied approaches to the subject.

Preface
Acknowledgments
Elementary Calculus
Preliminary Concepts
Limits and Continuity
Differentiation
Integration
Sequences and Series of Constants
Power Series and Taylor Series
Summary
Exercises
Interlude: Fermat, Descartes, and the Tangent Problem
Introduction to Real Analysis
Basic Topology of the Real Numbers
Limits and Continuity
Differentiation
Riemann and Riemann-Stieltjes Integration
Sequences, Series, and Convergence Tests
Pointwise and Uniform Convergence
Summary
Exercises
Interlude: Euler and the "Basel Problem"
A Brief Introduction to Lebesgue Theory
Lebesgue Measure and Measurable Sets
The Lebesgue Integral
Measure, Integral, and Convergence
Littlewood's Three Principles
Summary
Exercises
Interlude: The Set of Rational Numbers Is Very Large and Very Small
Special Topics
Modeling with Logistic Functions-Numerical Derivatives
Numerical Quadrature
Fourier Series
Special Functions-The Gamma Function
Calculus Without Limits: Differential Algebra
Summary
Exercises
Definitions & Theorems of Elementary Real Analysis
Limits
Continuity
The Derivative
Riemann Integration
Riemann-Stieltjes Integration
Sequences and Series of Constants
Sequences and Series of Functions
A Brief Calculus Chronology
Projects in Real Analysis
Historical Writing Projects
Induction Proofs: Summations, Inequalities, and Divisibility
Series Rearrangements
Newton and the Binomial Theorem
Symmetric Sums of Logarithms
Logical Equivalence: Completeness of the Real Numbers
Vitali's Nonmeasurable Set
Sources for Real Analysis Projects
Sources for Projects for Calculus Students
Bibliography
Index

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