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Full Description
This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. It can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting where lectures are organized around the presentation of problems and solutions. This book is appropriate for any student who has taken (or is concurrently taking) an introductory course in calculus. The book includes sixteen appendices that review some indispensable prerequisites on techniques of proof writing with special attention to the notation used the course.
Contents
Intervals
Toplogy of the real line
Continuous functions from $\mathbb{R}$ to $\mathbb{R}$
Sequences of real numbers
Connectedness and the intermediate value theorem;
Compactness and the extreme value theorem;
Limits of real valued functions;
Differentiation of real valued functions;
Metric spaces;
Interiors, closures, and boundaries;
The topology of metric spaces;
Sequences in metric spaces;
Uniform convergence;
More on continuity and limits;
Compact metric spaces;
Sequenctial characterization of compactness;
Connectedness;
Complete spaces;
A fixed point theorem;
Vector spaces;
Linearity;
Norms;
Continuity and linearity;
The Cauchy integral;
Differential calculus;
Partial derivatives and iterated integrals;
Computations in $\mathbb{R}^n$;
Infinite series;
The implicit function theorem;
Higher order derivatives;
Quantifiers;
Sets;
Special subsets of $\mathbb{R}$;
Logical connectives;
Writing mathematics;
Set operations;
Arithmetic;
Order propertiers of $\mathbb{R}$;
Natural numbers and mathematical induction;
Least upper bounds and greatest lower bounds;
Products, relations, and functions;
Properties of functions;
Functions that have inverses;
Products;
Finite and infinite sets;
Countable and uncountable sets;
Bibliography;
Index.
