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Über dieses Buch

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.

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. […] This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously. N.J. Wildberger, Aust. Math. Soc. Gaz.



1. Numbers and Functions

Let us begin at the beginning. When we learn the script of a language, such as the English language.
Sudhir R. Ghorpade, Balmohan V. Limaye

2. Sequences

The word sequence refers to a succession of certain objects. For us, these objects will be real numbers.
Sudhir R. Ghorpade, Balmohan V. Limaye

3. Continuity and Limits

In the previous chapter,we studied real sequences, that is, real-valued functions defined on the subset.
Sudhir R. Ghorpade, Balmohan V. Limaye

4. Differentiation

Differentiation is a process that associates to a real-valued function f another function.
Sudhir R. Ghorpade, Balmohan V. Limaye

5. Applications of Differentiation

The notion of differentiation is remarkably effective in studying the geometric properties of functions.
Sudhir R. Ghorpade, Balmohan V. Limaye

6. Integration

In this chapter, we embark upon a project that is of a very different kind as compared to our development of calculus.
Sudhir R. Ghorpade, Balmohan V. Limaye

7. Elementary Transcendental Functions

In this chapter, we shall use the theory of Riemann integration developed in Chapter 6 to introduce some classical functions.
Sudhir R. Ghorpade, Balmohan V. Limaye

8. Applications and Approximations of Riemann Integrals

In this chapter, we shall consider some geometric applications of Riemann integrals.
Sudhir R. Ghorpade, Balmohan V. Limaye

9. Infinite Series and Improper Integrals

If a1, . . . , an are any real numbers, then we can add them together and form their sum a1 + · · · + an..
Sudhir R. Ghorpade, Balmohan V. Limaye

10. Sequences and Series of Functions, Integrals Depending on a Parameter

In Chapter 9, we studied infinite series of real numbers as well as their continuous analogues, namely improper integrals of the first kind.
Sudhir R. Ghorpade, Balmohan V. Limaye


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