CounterExamples: From Elementary Calculus to the Beginnings of Analysis, 1st Edition (Hardback) book cover

CounterExamples

From Elementary Calculus to the Beginnings of Analysis, 1st Edition

By Andrei Bourchtein, Ludmila Bourchtein

Chapman and Hall/CRC

362 pages | 141 B/W Illus.

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pub: 2014-09-09
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Description

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.

Table of Contents

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

About the Series

Textbooks in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT028000
MATHEMATICS / Set Theory
MAT037000
MATHEMATICS / Functional Analysis