Throughout, the exposition makes a distinction between the intrinsic geometric definition of a notion and its analytic characterization, establishing firm foundations for topics often encountered earlier without proof. This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. From reviews: [The first edition is] a rigorous, well-presented and original introduction to the core of undergraduate mathematics — first-year calculus. It develops this subject carefully from a foundation of high-school algebra, with interesting improvements and insights rarely found in other books.

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The final prices may differ from the prices shown due to specifics of VAT rules 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.

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## A course in calculus and real analysis

It emerged from inv- tigations into such basic questions as? In the third century B. In the early seventeenth century, Fermat and Descartes studied the problem of? But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In part- ular, they showed that the geometric problems of? In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today,calculus occupies a centralplacein mathematics and is an essential component of undergraduate education.

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## A Course in Calculus and Real Analysis

Author by : Hugo D. The first part of the text presents the calculus of functions of one variable. This part covers traditional topics, such as sequences, continuity, differentiability, Riemann integrability, numerical series, and the convergence of sequences and series of functions. The second part focuses on functions of several variables. It introduces the topological ideas such as compact and connected sets needed to describe analytical properties of multivariable functions. This part also discusses differentiability and integrability of multivariable functions and develops the theory of differential forms on surfaces in Rn. The third part consists of appendices on set theory and linear algebra as well as solutions to some of the exercises.

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The final prices may differ from the prices shown due to specifics of VAT rules 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.