Since this is not, and cannot be an AP question it is impossible to answer (1) with any validity. Flip a coin if you insist on an answer.
For (2) I would give this student a pat on the back and applaud his or her original approach to the problem. He or she is doing more original and inventive thinking than most students taking AP calculus. I probably would not award bonus points, but I would celebrate the novelty of the approach. I would also point out that s/he is making a number of huge assumptions about the calculus of complex valued functions since everything has been defined and proved for real-valued functions only.
I certainly wouldn't penalize this student for thinking outside the box. We already have far too many students who are afraid to think because they have had the idea of "there is only one correct way to approach a problem" beaten into them by such grading practices.
Alan Lipp 19 Payson Avenue Easthampton, MA 01026 413-529-3278
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-----Original Message----- From: Wes Loewer [mailto:email@example.com] Sent: Thursday, February 16, 2012 2:18 PM To: AP Calculus Subject: [ap-calculus] imaginary imagination
I realize that hyperbolic functions are not on the AB (or BC) list of topics, but I like to introduce them if time allows. I recently put the following integral on a test.
integral 1/(1-9x^2) dx
Since they haven't yet seen integrals evaluated using partial fractions, the answer I intended was:
(1/3)*arctanh(3x) + c
One student rewrote it as:
integral -1/(i^2+9x^2) dx
and ended up with
-1/(3i) * arctan(3x/i)
and since arctan(ix) = -i arctanh(x), his answer is equivalent.
I realize that on the AP exam functions are assumed to have real domains, and at the beginning of the year I mentioned that we would only be dealing with reals, but, as the student pointed out, my test paper instructions did not actually say anything about reals.
So: 1) Would this receive full credit if it were on an AP? 2) What kind of credit would you give if this were on your own test?
I'm torn between either taking off, or giving bonus for being creative and thinking outside the box. I'll probably just do both and call it even. :-)