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Topic: [ap-calculus] imaginary imagination
Replies: 2   Last Post: Feb 17, 2012 3:00 PM

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Alan Lipp

Posts: 1,091
Registered: 12/6/04
RE: [ap-calculus] imaginary imagination
Posted: Feb 16, 2012 4:23 PM
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Wes,

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

Alan Lipp
19 Payson Avenue
Easthampton, MA 01026
413-529-3278

The Williston Northampton School inspires students to live with passion,
purpose, and integrity.




-----Original Message-----
From: Wes Loewer [mailto:wjltemp-apedg@yahoo.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. :-)

-Wes Loewer
Rosslyn Academy
Nairobi, Kenya


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http://apcentral.collegeboard.com/calculusbc
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