About this textbook
This
book provides a self-contained and rigorous introduction to calculus of
functions of one variable. The presentation and sequencing of topics emphasizes
the structural development of calculus. At the same time, due importance is
given to computational techniques and applications. The authors have strived to
make a distinction between the intrinsic definition of a geometric notion and
its analytic characterization. Throughout the book, the authors highlight the
fact that calculus provides a firm foundation to several concepts and results
that are generally encountered in high school and accepted on faith. For
example, one can find here a proof of the classical result that the ratio of
the circumference of a circle to its diameter is the same for all circles.
Also, this book helps students get a clear understanding of the concept of an
angle and the definitions of the logarithmic, exponential and trigonometric
functions together with a proof of the fact that these are not algebraic
functions. A number of topics that may have been inadequately covered in
calculus courses and glossed over in real analysis courses are treated here in
considerable detail. As such, this book provides a unified exposition of
calculus and real analysis.
The
only prerequisites for reading this book are topics that are normally covered
in high school; however, the reader is expected to possess some mathematical
maturity and an ability to understand and appreciate proofs. This book can be
used as a textbook for a serious undergraduate course in calculus, while parts
of the book can be used for advanced undergraduate and graduate courses in real
analysis. Each chapter contains several examples and a large selection of
exercises, as well as "Notes and Comments" describing salient
features of the exposition, related developments and references to relevant
literature.