I must admit that allowing students to play with their graphing calculators in my classes sometimes leads them to questions which I myself am not always prepared to answer immediately. I almost make it a point to get back to these questions, but some of them take me a couple of days. The following two questions, however, have kept me busy thinking for an entire week, and today it's Saturday, I decided to post this question to this discussion list.
(1) Do all periodic functions contain sin x or cos x?
(2) If the nth derivative of f is periodic, is f also periodic?
Incidentally, my students also ask the converse to these questions. My response was there are so many functions containing sine and cosine which are not periodic. A simple example is sin x^2 or sin (1/x). Of course there are trivial examples, like 2 sin x csc x, which simplify to a real constant. Or the functions x + sin x or x sin x.
As for the second one, if f is periodic, then its slope at every point, the derivative of f at that point, must necessarily be periodic as well.
However for the above questions, my gut feeling is a yes to both questions. If I am wrong, can somebody please give a counterexample?
Jose Nilo G. Binongo Fukuoka International School, Japan