More Calculus of a Single Variable

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 pi. 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.
More Calculus of a Single Variable is suitable as a text for a course in advanced calculus, as a supplementary text for courses in analysis, and for self-study by students, instructors, and, indeed, all connoisseurs of ingenious calculations.

List of contents

Preface.- 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.- Index.

About the author

Peter R. Mercer is Professor of Mathematics at the State University of New York College at Buffalo.

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.
More Calculus of a Single Variable is suitable as a text for a course in advanced calculus, as a supplementary text for courses in analysis, and for self-study by students, instructors, and, indeed, all connoisseurs of ingenious calculations.

Additional text

“This book is a friendly, accessible treatment of advanced calculus of functions of one variable, intended for readers who have completed a full-year course in introductory single-variable calculus. … Each chapter is accompanied by a healthy number of exercises whose range provides both practice and additional conceptual and topical development. Summing Up: Highly recommended. Lower- and upper-division undergraduates.” (S. J. Colley, Choice, Vol. 52 (10), June, 2015)

"... when we teach a course in calculus ... we are forced to omit many beautiful ideas ... The book under review is a remarkable resource to fill this gap ... The book is very well motivated and well written, and is very suitable for readers to follow as self-study. I recommend it highly for undergraduate students who have completed a course in calculus, and also for students who have a course in mathematical analysis. It will also serve well for anyone who is interested in developing undergraduate research at the calculus level." 
—MAA Reviews

Report

"This book is a friendly, accessible treatment of advanced calculus of functions of one variable, intended for readers who have completed a full-year course in introductory single-variable calculus. ... Each chapter is accompanied by a healthy number of exercises whose range provides both practice and additional conceptual and topical development. Summing Up: Highly recommended. Lower- and upper-division undergraduates." (S. J. Colley, Choice, Vol. 52 (10), June, 2015)
"... when we teach a course in calculus ... we are forced to omit many beautiful ideas ... The book under review is a remarkable resource to fill this gap ... The book is very well motivated and well written, and is very suitable for readers to follow as self-study. I recommend it highly for undergraduate students who have completed a course in calculus, and also for students who have a course in mathematical analysis. It will also serve well for anyone who is interested in developing undergraduate research at the calculus level."
-MAA Reviews

Product details

• Authors: Peter R Mercer, Peter R. Mercer
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 31.08.2016
• EAN: 9781493946815
• ISBN: 978-1-4939-4681-5
• No. of pages: 411
• Dimensions: 165 mm x 237 mm x 25 mm
• Weight: 657 g
• Illustrations: XVI, 411 p. 60 illus.
• Series: Undergraduate Texts in Mathematics; Undergraduate Texts in Mathematics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis