>> I'm a math team coach and homeschooling parent looking >> for some extra depth for the math learners in my care. >> Most of the young people on my math team get their >> main math instruction from the U. of MN Talented Youth >> Mathematics Program, using the Stewart Early Transcendentals >> textbook with an excellent supplementary materials. Even >> though they've got some good instruction already, I'm looking >> for suggestions of calculus problems that really make a >> learner THINK about calculus concepts and go beyond the >> usual school homework exercise. > > Browse and search the AP-Calculus forum > http://mathforum.org/kb/forum.jspa?forumID=63
I see your P.S. noting that you saw my posting there. (I posted there first, but you saw it later because of asynchronous posting to a moderated email list. I thought it would be a good idea to post that question there, Dave, because I had seen your name in the archives [smile].)
> Indiana University-Purdue University Fort Wayne > Honors Calculus web page (maybe some ideas here) > http://www.ipfw.edu/math/h-calculus/ > > MIT's Fall 2002 Honors Calculus Lecture Notes > http://tinyurl.com/ranuv > > Here's an example that I posted in the AP-Calculus > forum on 15 March 2006 at > > http://mathforum.org/kb/message.jspa?messageID=4542260 > > Let [[x]] denote the greatest integer less than > or equal to x, and define f(x) by > > f(x) = x * [[-(x^2)]]. > > Computing the derivative of f'(0) leads us to > compute the limit as h --> 0 of the expression > > [ f(0 + h) - f(0) / h ], > > which, for h not equal to 0, can be rewritten > as [[-(h^2)]]. > > Note that this is defined for h=0, as we were able > to cancel the h in the denominator. However, plugging > in h=0 gives an incorrect result! Even though we > can plug in h=0, _we're_supposed_ to take the limit > as h --> 0, and this gives -1. Thus, f'(0) = -1. > > The neat thing about this example is that if you > simplify sufficiently so that you can plug in h=0, > and then plug in h=0, you don't get the correct > result. > > This example is given in: > > David Sanders, "A cautionary counter-example", > Mathematical Gazette 59 #407 (March 1975), 44-45.
That's an interesting problem. Thanks for the reference.