Counter-Examples in Calculus
There are ninety incorrect statements in this book. Some are superficially plausible, but all are wrong. And they are not just wrong, but provably wrong. Examples:
- “The tangent to a curve at a point is the line which touches the curve at that point but does not cross it there.”
- “If a function is twice differentiable at a local minimum point, then its second derivative is positive at that point.”
- “If a function is continuous on R and the tangent line exists at any point on its graph then the function is differentiable at any point on R.”
The fascination of this wonderful 115-page book lies in trying to construct a function that disproves each statement. The pedagogical technique of asking for counter-examples has been found by Associate Professor Klymchuk to enhance critical thinking, and he rightly includes in the book his studies that provide some justification of this. Construction of counter-examples moves students beyond over-reliance on learned strategies, to an improvement in their appreciation of the properties of limits, continuity, functions, and the calculus.
The incorrect statements certainly sharpen awareness of definitions and notation, and prompt the reader to re-examine ‘certainties’ with greater scrutiny! Klymchuk acknowledges the well-known Counterexamples in Calculus (Gelbaum and Olmsted, 1964), which covers some of the same ground but is ‘well beyond the scope of first-year university Calculus courses’ (p.7). Klymchuk’s book develops the same approach for less-experienced students. One or two of the problems were too hard for me, but most of them are within the reach of our Calculus students.
Counter-examples have a distinguished lineage. From the Greek dialogues to the present, they engage the learner, demanding creativity and real understanding. Proofs and Refutations (Lakatos, 1976) is the classic use of disproof. From a gentle classroom dialogue about exceptions to rules in geometry, Lakatos presents a deep discussion about the nature of proof.
Perhaps asking for counter-examples is a technique that teachers can apply to other content areas, and, following Klymchuk’s lead, researchers might attempt to establish what deeper understanding ensues. I am reminded, for example, of Martin Gardner’s question ‘Is the circle the only closed curve of constant width?’ (2001, p.35), as he introduces the Reuleaux triangle and other exceptions.
It is good to see that, where possible, the solutions in Klymchuk’s book provide large, clear graphs of each function discussed. Some solutions are ingenious, some are beautiful, and some lead to profound mathematics. Klymchuk seeks to expand the ‘example set’ of his readers, and I believe he is successful in this endeavour.
Counter-Examples in Calculus unfortunately does not contain a glossary, nor an index, and there are a few trivial proofing errors. Even for such a small book, addressing these issues might help the inexpert intended audience. However, I found the book thought-provoking, even fun, and I heartily recommend it to you and your students.
So, what was wrong with those examples? For the first, try y equals x cubed at the origin. For the second, y equals the fourth power of x has at the minimum point a second derived function which is not positive; it is zero. For the third, graph y equals the cube root of x squared, and see for yourself.
References
Gardner, M. (2001). The colossal book of mathematics. New York: W.W. Norton.
Gelbaum, B.R. & Olmsted, J.M.H. (1964). Counterexamples in analysis. San Francisco: Holden-Day Inc.
Klymchuk, S. (2004). Counter-Examples In Calculus. Auckland: Maths Press.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.
Dr. Paul Brown teaches at Carmel School, Perth. He is the author of ‘Spreadsheet Explorations’ and ‘Squares’, both published by the Australian Association of Mathematics Teachers
Counter-Examples In Calculus
Sergiy Klymchuk
Australian Association of Mathematics Teachers
AU$20 for members, AU$25 otherwise
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