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Topic: Seeking Beautiful Single-Variable Calculus Problems
Replies: 8   Last Post: Jul 16, 2006 2:35 PM

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Karl M. Bunday

Posts: 127
Registered: 12/6/04
Re: Seeking Beautiful Single-Variable Calculus Problems
Posted: Jul 16, 2006 2:35 PM
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Dave wrote, in reply to my questions:

>> 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].)

> Caltech Calculus Lecture Notes
> http://www.math.caltech.edu/classes/ma1a/index.html#lect
> http://www.math.caltech.edu/classes/ma_9/index1.html#lect


Very good. Thank you.

> 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.

--
Karl M. Bunday P.O. Box 1456, Minnetonka MN 55345
Learn in Freedom (TM) http://learninfreedom.org/
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