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Re: if using Mathematica to solve an algebraic problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109244] Re: if using Mathematica to solve an algebraic problem
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Mon, 19 Apr 2010 02:48:33 -0400 (EDT)

Mathematical issue: I cannot imagine teaching integration without 
discussing the matter of what "elementary" means and that there is 
actually an algorithm for determining whether an elementary function has 
an elementary integral (and, if so, for finding it) -- and accordingly 
letting the students see some elementary functions that do not have 
elementary integrals.

As with so much mathematics as taught, the most time is spent on 
(sometimes algebraically elaborate) problems which can be "solved" 
exactly. E.g. finding antiderivatives in closed-form; solving ordinary 
differential equations exactly in closed-form.  But in practice, many 
elementary functions of great interest do NOT have elementary 
antiderivatives; many (most?) differential equations of interest do NOT 
closed-form solutions.

Mathematica issue:  It's VERY instructive for a student to obtain one 
answer for an indefinite integral and then see Mathematica obtain a 
seemingly different answer.  And just as instructive as the student 
obtaining one answer and the back-of-the-book answer seeming to be 
different. If nothing else, it helps drive home the point that the 
indefinite integral is unique only up to an additive constant.

(We face the same issue when, in first-year calculus, we use an on-line 
homework system where students type in a correct answer for an 
indefinite integral but the answer supplied by the system is different. 
Fortunately, the system finesses the issue by differentiating the 
student's answer, differentiates the problem author's supplied author, 
simplifies both, and then checks whether the results are the same. 
Unfortunately, the back-end engine doing that is not Mathematica but 
another CAS, which makes it much less pleasant to modify problems or 
author new ones, at least for me.)



On 4/12/2010 6:54 AM, Richard Fateman wrote:
> 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
>
>
>
>
>
>

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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