1st Edition

CounterExamples From Elementary Calculus to the Beginnings of Analysis

    368 Pages 141 B/W Illustrations
    by Chapman & Hall

    This book provides a one-semester undergraduate introduction to counterexamples in calculus and analysis. It helps engineering, natural sciences, and mathematics students tackle commonly made erroneous conjectures. The book encourages students to think critically and analytically, and helps to reveal common errors in many examples.

    In this book, the authors present an overview of important concepts and results in calculus and real analysis by considering false statements, which may appear to be true at first glance. The book covers topics concerning the functions of real variables, starting with elementary properties, moving to limits and continuity, and then to differentiation and integration. The first part of the book describes single-variable functions, while the second part covers the functions of two variables.

    The many examples presented throughout the book typically start at a very basic level and become more complex during the development of exposition. At the end of each chapter, supplementary exercises of different levels of complexity are provided, the most difficult of them with a hint to the solution.

    This book is intended for students who are interested in developing a deeper understanding of the topics of calculus. The gathered counterexamples may also be used by calculus instructors in their classes.

    Introduction
    Comments
    On the structure of this book
    On mathematical language and notation
    Background (elements of theory)
    Sets
    Functions

    FUNCTIONS OF ONE REAL VARIABLE
    Elementary properties of functions
    Elements of theory
    Function definition
    Boundedness
    Periodicity
    Even/odd functions
    Monotonicity
    Extrema
    Exercises

    Limits
    Elements of theory
    Concepts
    Elementary properties (arithmetic and comparative)
    Exercises

    Continuity
    Elements of theory
    Local properties
    Global properties: general results
    Global properties: the famous theorems
    Mapping sets
    Weierstrass theorems
    Intermediate Value theorem
    Uniform continuity
    Exercises

    Differentiation
    Elements of theory
    Concepts
    Local properties
    Global properties
    Applications
    Tangent line
    Monotonicity and local extrema
    Convexity and inflection
    Asymptotes
    L’Hospital’s rule
    Exercises

    Integrals
    Elements of theory
    Indefinite integral
    Definite (Riemann) integral
    Improper integrals
    Applications
    Exercises

    Sequences and series
    Elements of theory
    Numerical sequences
    Numerical series: convergence and elementary properties
    Numerical series: convergence tests
    Power series
    Exercises

    FUNCTIONS OF TWO REAL VARIABLES
    Limits and continuity
    Elements of theory
    One-dimensional links
    Concepts and local properties
    Global properties
    Multidimensional essentials
    Exercises

    Differentiability
    Elements of Theory
    One-dimensional links
    Concepts and local properties
    Global properties and applications
    Multidimensional essentials
    Exercises

    Integrability
    Elements of theory
    One-dimensional links
    Multidimensional essentials
    Exercises
    Bibliography
    Symbol Description
    Index

    Biography

    Andrei Bourchtein, Ludmila Bourchtein