Counterexamples in Calculus (Classroom Resource Materials) by Sergiy Klymchuk (PDF)

Description

Counterexamples in Calculus 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, for high school and college students as a learning resource, and as a professional development resource for calculus instructors.

User’s Reviews

Editorial Reviews: Review This brief work is exactly as described, a collection of counterexamples from calculus. The book is organized into two parts. The first part is a list of 90 short statements grouped into five categories: “Functions,” “Limits,” “Continuity,” “Differential Calculus,” and “Integral Calculus.” The second part is longer and includes counterexamples to each of the statements and well-known calculus theorems… Klymchuk provides an excellent reminder to calculus instructors and students of the need for rigor in calculus…In addition to helping calculus users to avoid complacency, this work provides wonderful illustrations of what can go wrong if one ignores the conditions placed on the theorems and rules of calculus. This book should be required reading for any long-term instructor of calculus. –CHOICE Magazine Book Description A resource for introductory calculus courses. It provides incorrect mathematical statements which require students to create counterexamples to disprove them. Book Description A resource to enhance the teaching of single variable calculus courses. This book features incorrect mathematical statements that require students to create counterexamples to disprove them. This book fills a gap in the literature, providing a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. From the Back Cover Counterexamples in Calculus 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 • a professional development resource for calculus instructors About the Author Dr 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 of experience teaching university mathematics in different countries. He was born in Ukraine in 1958. Since 1996 he has lived in New Zealand. His PhD (1988) from Odessa National University, Ukraine was in differential equations. At present his main research interests are in mathematics education. He is a member of the Royal Society of New Zealand. He is also a member of a number of international groups in mathematics education including International Group for the Psychology in Mathematics Education (PME) and International Community of Teachers of Mathematical Modelling and Applications (ICTMA). Apart from Counterexamples in Calculus he also has written Paradoxes and Sophisms in Calculus as a supplementary resource to enhance teaching and learning of introductory calculus. He has more than 140 publications including three books on popular mathematics and science that have been, or are being, published in 11 countries: Money Puzzles, Science Puzzles and Shape Puzzles. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a book of incorrect mathematical statements. Specifically, it’s a collection of statements that might reflect erroneous thinking in the beginning calculus student, followed by a counterexamples disproving each such statement. As a pedagogical tool, examples like those collected in this book are invaluable to the instructor of a first course in calculus. Having worked with students of many different levels of mathematical experience (ranging from those struggling with elementary arithmetic to those completing their degrees in mathematics at university), I’ve found that one of the most common pitfalls in mathematical education is the “almost correct understanding.” Students, particularly those racing against the clock to finish an exam or a homework problem, often take mental shortcuts and learn algorithms that generally produce the correct responses even though they are built on unsound mathematical reasoning and can lead to disaster if not corrected.This book collects several of these kinds of mistakes in a single volume, perfect for the calculus instructor to use as in-class examples, as homework problems, or as exam questions to help the students think about their mathematics more rigorously and develop into more skilled mathematicians. Divided into five main sections (functions, limits, continuity, differential calculus, and integral calculus), the book presents the reader first with some fairly simple erroneous statements, to which most students ought, with a bit of thought, to be able to construct a counterexample and then with some more interesting examples for the more advanced student. While none of the counterexamples should be beyond the ability of the first-year calculus student, several of the statements will provide enough of a challenge to help hone the students’ mathematical ability.The idea behind this book is a truly great one, and I wanted to be able to say I really loved the book. Unfortunately, it does have some flaws that tarnish its quality a bit.To begin with, the book only covers single-variable calculus. While this is no surprise as it is clearly disclosed in the advertising material, it would be a far better book if it included multivariate calculus (and perhaps even differential equations) as well, collecting a carefully curated selection of incorrect statements to be used throughout the entire introductory calculus experience. Indeed, many of the examples in the book can be readily found in a more comprehensive calculus textbook, so the advantage of this particular work is the curation: by collecting a particular type of example into such a single volume uninterrupted by other types of material, the instructor can conveniently access quality educational material.However, this brings us to the book’s greatest flaw: it is simply far too short. The first section of the book (excluding the introduction and front matter) consists of only ninety mathematical statements. The bulk of the book’s approximately 100 pages is devoted to providing suggested counterexamples for each statement. While these counterexamples are often charming and (quite helpfully for many students) illustrated with quality graphs, they can’t fully compensate for the relative lack of content. To put it bluntly, this stand-alone volume contains fewer exercises than did the first chapter alone of the textbook I used when I took my first calculus course. While we would be unreasonable to expect a supplementary book focused only on counterexamples to contain as many exercises as a complete textbook, I don’t think I’m out of line to suggest that, especially for the price, there should at least be more than ninety of them.At the end of the day, I did enjoy this book, I did find some of the counterexamples charming, and I have used examples from the book to good success with my students, but I can’t in good conscience award more than three stars to a book with a retail price of $41 that collects fewer than 100 exercises. Instructors who don’t mind the price will find it useful, but I would advise students to save their money and look elsewhere.

⭐I found this book extremely helpful in teaching Calculus. Thoughtfully selected examples highlight important details about Calculus theorems and distill student’s misunderstandings and difficulties.

⭐this is not a book. It is a joke. Or maybe a fraud.Offhand (I shall write a detail review at a later time) 95% of all the “counterexamples” should be easy to construct by the first time students. Why, I routinely include harder “true false, if false give an example” problems on my exams and the students do well on these problems.Again offhand (I shall do a careful count) 95% of all the “counterexamples” (excluding the completely trivial ones) are included in any good Calculus book. Spivak has many more interesting counterexamples! http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918I bought this book hoping to find a worked out example of mixed partials of f(x,y) = xy(x^2 – y^2)/(x^2 + y^2) at (0,0) DOn’t know what I was thinking. Maybe that calculus of several variables would also be covered. It is not. I read the entire book in some 25 minutes on a bus trip from my day job to my teaching assignment. Found absolutely nothing of there that I have not seen before.Several charts in the book appear to be empty. Just the axes. Come on!How can you have a book on counterexamples in Calculus without f(x) = 0 if x is irrational and = 1/b when x is rational a/b in a reduced form? Well this one managed. Like I said before. Just use Spivak, he has more counterexamples there.

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