A Course in Calculus and Real Analysis

Description

Offering a unified exposition of calculus and classical real analysis, this textbook presents a meticulous introduction to single‐variable calculus. 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. Each chapter contains numerous examples and a large selection of exercises, as well as a “Notes and Comments” section, which highlights distinctive features of the exposition and provides additional references to relevant literature. This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. A new chapter discusses sequences and series of real‐valued functions of a real variable, and their continuous counterpart: improper integrals depending on a parameter. Two new appendices cover a construction of the real numbers using Cauchy sequences, and a self‐contained proof of the Fundamental Theorem of Algebra. In addition to the usual prerequisites for a first course in single‐variable calculus, the reader should possess some mathematical maturity and an ability to understand and appreciate proofs. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. The authors’ A Course in Multivariable Calculus is an ideal companion volume, offering a natural extension of the approach developed here to the multivariable setting. Front Matter ....Pages I-IX Numbers and Functions (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 1-40 Sequences (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 41-66 Continuity and Limits (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 67-104 Differentiation (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 105-148 Applications of Differentiation (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 149-180 Integration (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 181-232 Elementary Transcendental Functions (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 233-294 Applications and Approximations of Riemann Integrals (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 295-364 Infinite Series and Improper Integrals (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 365-424 Sequences and Series of Functions, Integrals Depending on a Parameter (Sudhir R. Ghorpade, Balmohan V. Limaye)....Pages 425-502 Back Matter ....Pages 503-538