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AP Calculus Question of the Month: August
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by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut
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My favorite function is 
. Get out your graphing calculator and see if you can answer my favorite question about my favorite function: How many roots does
have in the half-open interval 
?
Try it now.
Okay, I misled you about the calculator. That was on purpose. There are an infinite number of roots in the interval. Your calculator can show you one, occasionally two, but never more than two at a time on a single screen. Don't throw out your calculator; even powerful, computer-based graphing programs cannot do much better.
So here's this month's real question:
Find all the roots of
in the interval
. Explain why any graphing calculator or computer program cannot produce a graph that shows more than two of these roots in any graphing window or computer screen.
I like to use this problem at the beginning of the year or when discussing graphing calculator use with teachers. A full explanation will lead you to review several different mathematical concepts. You can use this question at the beginning of the year to review some major topics from precalculus mathematics. This question is also useful to lead a discussion of how calculators and computers draw graphs and of the limitations of such graphs.
Complete the question before viewing the the answers and commentary!
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