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More calculus of a single variable / Peter R. Mercer.

By: Material type: TextTextSeries: Undergraduate texts in mathematicsPublication details: New York: Springer, 2014.Description: xvi, 411 p. : illustrations ; 24 cmISBN:
  • 9781493919253 (alk. paper)
Subject(s): DDC classification:
  • 515 23 M554
Contents:
1 The Real Numbers -- 2 Famous Inequalities -- 3 Continuous Functions -- 4 Differentiable Functions -- 5 The Mean Value Theorem -- 6 The Exponential Function -- 7 Other Mean Value Theorems -- 8 Convex Functions and Taylor's Theorem -- 9 Integration of Continuous Functions -- 10 The Fundamental Theorem of Calculus -- 11 Techniques of Integration -- 12 Classic Examples -- 13 Simple Quadrature Rules -- 14 Error Terms -- A The Proof of Theorem 9.1 --
Summary: This book goes beyond the basics of a first course in calculus to reveal the power and richness of the subject. Standard topics from calculus — such as the real numbers, differentiation and integration, mean value theorems, the exponential function — are reviewed and elucidated before digging into a deeper exploration of theory and applications, such as the AGM inequality, convexity, the art of integration, and explicit formulas for π. Further topics and examples are introduced through a plethora of exercises that both challenge and delight the reader. While the reader is thereby exposed to the many threads of calculus, the coherence of the subject is preserved throughout by an emphasis on patterns of development, of proof and argumentation, and of generalization.
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Includes bibliographical references and index.

1 The Real Numbers --
2 Famous Inequalities --
3 Continuous Functions --
4 Differentiable Functions --
5 The Mean Value Theorem --
6 The Exponential Function --
7 Other Mean Value Theorems --
8 Convex Functions and Taylor's Theorem --
9 Integration of Continuous Functions --
10 The Fundamental Theorem of Calculus --
11 Techniques of Integration --
12 Classic Examples --
13 Simple Quadrature Rules --
14 Error Terms --
A The Proof of Theorem 9.1 --

This book goes beyond the basics of a first course in calculus to reveal the power and richness of the subject. Standard topics from calculus — such as the real numbers, differentiation and integration, mean value theorems, the exponential function — are reviewed and elucidated before digging into a deeper exploration of theory and applications, such as the AGM inequality, convexity, the art of integration, and explicit formulas for π. Further topics and examples are introduced through a plethora of exercises that both challenge and delight the reader. While the reader is thereby exposed to the many threads of calculus, the coherence of the subject is preserved throughout by an emphasis on patterns of development, of proof and argumentation, and of generalization.

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