Counterexamples in Calculus

One way I motivate critical thinking in my Intro to Proofs class is by using counterexamples. The book, Using Counter-examples in Calculus by Mason and Klymchuk, provides an accessible set of ideas to think about. Producing counterexamples is an important step to thinking about proofs in general, especially for students who are used to computations. What I really liked about the book were the graphs that accompanied many of the counterexamples. Here is one statement to think about:

True or false: The tangent to a curve at a point is the line which touches the curve at that point but does not cross it there.

The book is actually intended for first year calculus students and could work in that context if the course emphasizes concepts along with the standard skill set.

And the counterexample for the statement quoted above is x cubed at the origin.

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