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