Calculus Counterexamples

Counterexamples are important tools in mathematical argumentation.  As a math instructor I naturally gravitate to refutation by counterexample when an erroneous statement arises.  My students however are less familiar with this type of argument, so first I give this example.  “My wife says that I never take her out to dinner.  I reply that I took her out to McDonald’s for a hamburger seven years ago.  Therefore I win the argument.”  Students generally laugh.  I philosophize about using the word “never” and then we get on with the mathematics.

Counterexamples in Calculus by Sergiy Klymchuk is intended for introductory calculus courses.  It has useful misstatements  and counterexamples that prove them so.  However many of the problems require more sophistication than a typical first year calculus student may have.

One case in point is the use of functions like f(x) = 1 if x is rational and f(x) is -1 if x is irrational.  Student may have a passing acquaintance with the different types of real numbers since most algebra books start with a classification of real numbers, but thinking about limits pertaining to f(x) requires knowledge of the concept of density of numbers which students have not seen before much less studied for understanding.  Most beginning calculus texts use graphical or data (with rational numbers) examples to motivate the concept of limits.  They do not start with a study of real numbers.  This gives a certain hand-waving quality to our discussion of limits and students do not find examples like f(x) very persuasive.

Another case is Dr. Kylmchuk’s use of the Fresnel integral.  This integral is indeed discussed in Stewart on page 380 as an example of defining a function as an integral.  Yet Stewart doesn’t have room to discuss exactly where the function comes from or that it is convergent.  I don’t think beginning students would find problem 5.12 in Counterexamples in Calculus persuasive since as Dr. Klymchuk states the integral’s convergence depends on complex analysis ideas.  And if student are supposed to come with their own counterexamples, they will never think of this example particularly given the incomplete discussion in Stewart.

The wonderful Weierstrass function is another example of a function beyond beginning calculus students abilities.  The problem here is that the function is defined as an infinite sum of functions. This is not covered in elementary calculus courses.

But first year calculus students need at least an exposure to using counterexamples and Counterexamples in Calculus has useful problems for this purpose.  For instance I would use problems 4.3, 4.4 and 4.5 to study the converse of the theorem that if a function is differentiable at a point then  it is continuous at that point.  It would be a story of trying to “fix” the converse statement so that it would be true.   I expect we would try to continuing our “fixing” process past problem 4.5 and arrive at having a differentiable function, that is, a dead end to our attempt.

Another way I would use counterexamples would be to show that every part of the statement of a theorem is necessary. As Dr. Klymchuk states,  there is a difference between brackets and parentheses.  I like examples 3.7 and 3.14 for this.

In sum Counterexamples in Calculus has useful problems and emphasizes an important method in mathematical thinking.  Some of the examples are too sophisticated for beginning calculus students.

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About jrh794

I am a sixty-five year old math instructor at Southern Oregon University. I taught at the College of the Siskiyous in Weed California for twenty-six years. Prior to that I worked as a computer programmer, carpenter and in various other jobs. I graduated from Rice University in 1967 and have a MS in Operations Research from Stanford. In the past I have hand-built a stone house and taken long solo bicycle tours. Now I ride my mountain bike and play golf for recreation.
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