Random examples

Some examples are better than others - most of the time.  For example, when first teaching the product rule, it is not a good idea to use $xe^x$: since the derivative of $e^x$ is $e^x$, students don't see the form of the product rule explicitly.  That said, there are times when the technique being taught are so general, that they would work equally well with any example.  In cases like these, I like to construct a random example, with the class's help.

Suppose I wanted to show my students that the Taylor series for any polynomial is simply the polynomial back again.  If students have not thought about this yet, the result can be rather surprising (after all, the Taylor series for the other standard examples look nothing like the starting function).  What I don't want to do is look down in my notes, carefully copy down a polynomial and start from there.  While this would definitely be an example worth sharing, I fear that students would not be impressed.  Of course I, the powerful math professor, could come up with an example of a function which is identical to it's Taylor series.  Just another example, they would think.

Instead, I write on the board "Find the Taylor series for $f(x) =$" and then dramatically point at a student and demand, "What's your favorite number!?"  After regaining his or her composure, the student will say, perhaps, 7.  I write $7 x^3 +$ and then repeat with another student.  Continuing in this fashion, the class and I together come up with a random polynomial.  And wouldn't it be amazing if this random polynomial happened be it's own Taylor polynomial?  Surely that cannot be a coincidence.

Random examples like these can be used all over the place, although it is important not to use them as a substitute for a well written lecture.  As with any interaction with individual students in class, this technique will keep students alert.  Most find it humorous (especially when a student can't remember their favorite number).  And most importantly, when used correctly, the random example can drive home the fact that the mathematical technique can be used in any case, not just the special cases the professor has prepared.