Here is my favorite "paradox", more accurately described as "find the error". It involves calculus, but so did some recent posts in this thread.

1. Define the function

   f(x) = 2 arcsin(x) + arcsin( 1 - 2 x^2 ).

   So f(x) is defined for x in the interval [-1, 1].

2. Differentiate and simplify: find f'(x) = ??
3. Check the unexpected answer for f'(x) by calculating f(0) and f(1). No problem so far -- these values do not contradict a fundamental theorem of calculus.
4. Recheck f'(x) by calculating f(-1). *NOW* we have a problem.
5. Find your error. Don't tell me that you didn't make a mistake in Step 2 -- everybody does.

Barry Wolk
Dept of Mathematics
University of Manitoba
Winnipeg Manitoba Canada

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