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Über dieses Buch

This book is intended for students familiar with a beginner's version of differential and integral calculus stressing only manipulation offormulas and who are now looking for a closer study of basic concepts combined with a more creative use of information. The work is primarily aimed at students in mathematics, engineering, and science who find themselves in transition from elementary calculus to rigorous courses in analysis. In addition, this book may also be of interest to those preparing to teach a course in calculus. Instead of exposing the reader to an excess of premature abstractions that so easily can degenerate into pedantry, I felt it more useful to stress instruc­ tive and stimulating examples. The book contains numerous worked out examples and many of the exercises are provided with helpful hints or a solution in outline. For further exercises the interested reader may want to consult a problem book by the author entitled Problems and Propositions in Analysis (New York: Marcel Dekker, 1979). For the history of calculus I recommend the book by C. B. Boyer, The Concepts of the Calculus (New York: Dover, 1949).

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Logarithmic and Exponential Functions

Abstract
The curve y = 1/x, for x > 0, is of special interest to us; it is located above the x-axis in the first quadrant of the x, y plane and it is seen to be symmetric with respect to the line y = x (because the equation xy = 1 remains unchanged when x and y are interchanged). See Figure 1.1 for a display of the curve under consideration.
Gabriel Klambauer

Chapter 2. Limits and Continuity

Abstract
Let x be any real number. By the absolute value of x, in notation |x|, we mean x if x ≥ 0 and — x if x ≤ 0. If we picture x as a point on the number line, then |x| can be viewed as the distance between the points 0 and x. It is obvious that |− x| = |x|.
Gabriel Klambauer

Chapter 3. Differentiation

Without Abstract
Gabriel Klambauer

Chapter 4. Applications of Differentiation

Without Abstract
Gabriel Klambauer

Chapter 5. Integration

Abstract
The problem of finding the area of a region in a plane bounded by a given curve has fascinated mathematicians for a long time. We shall consider a few of the celebrated examples that have come down to us from times past; the special methods of quadrature used in these examples are of a rather clever kind.
Gabriel Klambauer

Chapter 6. Additional Topics in Integration

Without Abstract
Gabriel Klambauer

Chapter 7. Infinite Series

Without Abstract
Gabriel Klambauer

Backmatter

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