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Re: if using Mathematica to solve an algebraic problem

• To: mathgroup at smc.vnet.net
• Subject: [mg109069] Re: if using Mathematica to solve an algebraic problem
• From: Richard Fateman <fateman at cs.berkeley.edu>
• Date: Mon, 12 Apr 2010 06:54:43 -0400 (EDT)
• References: <hps1co$6i3$1@smc.vnet.net>

Murray Eisenberg wrote:
> One thing you "missed" is that the integral is not "elementary" in the 
> technical sense of the term, or what many folks would call 
> "closed-form".  And that's one of the things we often discuss in a 
> calculus class -- that this or that function, although itself 
> elementary, does not have an elementary antiderivative (even though the 
> Fundamental Theorem of Calculus guarantees that it does have an 
> antiderivative, since it's continuous).
>

This is a subtle point, and I wonder if it is really taught so often in 
calculus classes.  If a student is given an integral to compute there 
are quite a few possibilities.
1. She can compute it in terms of elementary functions (and check it 
etc.) A winner.
2. She cannot figure it out, though it does exist in terms of elementary 
functions. (And this can be shown by typing it in to a computer).
3. She cannot figure it out, though it does exist. But the computer 
fails to do it for one of several reasons. (a) a bug, (b) she typed it 
in wrong. (c) failure in design.
4. She cannot figure it out and it does not exist. Neither does the 
computer find it.  But it is unclear if the computer has proved 
non-existence, encountered a bug, or what.
5. She can figure it out and the computer cannot. (fairly unlikely these 
days, but not impossible).
6. She thinks she has figured it out but the computer cannot. But she is 
wrong.

Now for students who are unaware that there are integrals without closed 
forms, and who don't have a computer or a smart friend, the options are 
fewer..

   (a) I can do it
   (b) I can't do it.

Now how is using Mathematica going to waste time?  I'm surprised that 
others have not encountered this, because I have, even when I was NOT 
teaching calculus.

Students taking the calculus class would visit me during office hours 
because they were using Mathematica [or alternatives] (on their own or 
in labs) and found that they got different answers. Or had other 
problems. Why?

1. Sometimes Mathematica got the wrong answer. Typically having the 
wrong sign of sqrt(1-cos(x)^2) kind of thing.  I don't know if it still 
screws up in later versions.
2. Sometimes the answers looked different but were the same because the 
output forms were not identical.
3. Sometimes the answers were definitely different. e.g. sin(x)^2+C
or -cos(x)^2+C.  Obviously different, but equally correct for integral 
of 2*sin(x)*cos(x).
4. Sometimes they had trouble setting up the program on their own 
computer and their instructor could not help because they were running 
some funny system. unix or non-unix or mac or ...
5. Sometimes they had trouble because they were befuddled by typing 
Cos[] instead of cos()  or did not understand the proper syntax
like they were  typing 3.1415d0 or 1,200.

This did not happen so often, but it may be that there were far more
puzzled students than I saw.  At least enough students found it 
disconcerting that this aspect of the course tended to be omitted (at 
instructor's option).

These examples were from UC Berkeley students.

In an earlier position, I taught a calculus course (with Paul Wang, now 
at Kent State) at MIT, using another computer algebra system and also
BASIC.  As I recall, the student survey suggested that (a) BASIC was a
waste of time; (b) Some people would have liked to learn what was inside
the integration program  (the Risch algorithm).  This is consistent with
the "meta" mind set that some techies have.  Instead of solving sudoku
puzzles, why not write a program to solve them.  or the meta-meta mind 
set which would be to write a program to produce sudoku puzzles of a 
specified difficulty.   But those are not the typical students.

RJF


RJF