# Counterexamples in Calculus

### By Sergiy Klymchuk

Catalog Code: CXC
Print ISBN: 978-0-88385-765-6
Electronic ISBN: 978-1-61444-109-0
112 pp., Paperbound, 2010
List Price: $41.00 MAA Member:$30.75
Series: Classroom Resource Materials

Counterexamples in Caculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed. Incorrect statements are grouped topically with sections devoted to Functions, Limits, Continuity, Differential Calculus and Integral Calculus.

This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. In that light it may well be useful for:

• high school teachers and university faculty as a teaching resource
• high school and college students as a learning resource
• calculus instructors as a professional development resource.

### Table of Contents

Foreward
Introduction
Counterexamples in Calculus
I. Statements
II. Suggested Solutions
References
About the Author

### Excerpt: Section I: Statements, Part 4: Differential Calculus (p. 20)

4.8 If a function is continuous and decreasing on (a, b) then its derivative is nonpositive on (a, b).

4.9 If a function has a positive derivative at every point in its domain, then the function is increasing everywhere in its domain.

4.10 If a function ƒ(x) is defined on [a, b] and has a local maximum at the point c ∈ (ab), then in a sufficiently small neighborhood of the point x = c, the function is increasing for all x < c and decreasing for all x > c.

4.11 If a function ƒ(x) is a differentiable for all real x and ƒ(0) = ƒ'(0) = 0, then ƒ(x) = 0 for all real x.

### About the Author

Sergiy Klymchuk is an Associate Professor of the School of Computing and Mathematical Sciences at the Auckland University of Technology, New Zealand. He has 29 years experience teaching university mathematics in different countries. His PhD (1988) was in differential equations and recent research interests are in mathematics education and epidemic modeling. He has more than 140 publications including several books on popular mathematics and science that have been, or are being, published in 11 countries.