Re: when did lim(sin(x),x->inf) become interval [-1,1]?

Alec Mihailovs wrote:


Mathematics in CAS and mathematics taught in a calculus course is quite different. 

I think there are a few good reasons for this, and a few bad reasons.

Good reasons:

1. The calculus student is not mathematically sophisticated. Therefore
some answers that a CAS produces that are mathematically correct
from a more advanced standpoint, look to the calculus student as
wrong.  Things like log( negative value) require complex analysis,
unknown to calc students.

2. The mathematics in a CAS is "algorithmic" and the techniques used
can be quite different from the calculus student  [The phrase
"constructive mathematics" is quite different in meaning, and from
the algorithm perspective mostly irrelevant].

Bad reasons:

3. The CAS is mathematically incorrect. That is, there exists a correct
answer but it is not computed. Or there is no answer, but the CAS
returns one, anyway. I call these bugs; others defend them on the
grounds that "the program is behaving as it was programmed to behave."
This converts the reason into the next one:

4. The CAS is not designed correctly. That is, there is a "feature"
that is, mathematically speaking, erroneous. This can be "corrected"
only by violating the mathematical sense or logic that
might be part of the user's education or understanding of math.

5. (Often the cause of the previous two) The programmer/designer of
the CAS did not use "the best" mathematics in the program, perhaps by
laziness, ignorance, or a quest for efficiency instead of correctness.
Occasionally it is a deliberate battle for conciseness at the expense of rigor
(e.g. misusing notions or notations like "="  for infinity, as RHL
mentions).

So I would generally caution students that computers may be good
at manipulation of expressions, but were designed and written
by fallible humans within a context that might not exactly match
theirs.


RJF
.

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