CounterExamples
From Elementary Calculus to the Beginnings of Analysis
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Book 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