Descartes' Rule of Signs, and synthetic division, were important tools that
simplified the process of finding roots of polynomials. There just wasn't
any other way, when all we had to work with was paper and pencil and of
course the slide rule. I think it can be safely said that no one whose job
requires finding roots now uses those techniques, for the same reason that I
use a calculator to balance my checkbook - speed and accuracy. I advise
students to know which equations are not worth picking up the calculator
for, and the list will vary among students. But they should know how to
solve ANY equation with iterative methods on the calculator, to ANY level of
accuracy up to the limit of the machine, which I bet is usually enough.
(And they should understand how many roots of a polynomial they are looking
for, and how the polynomial behaves for large abs(x); whether Descartes'
Rule helps there is a matter of personal preference - I never use it.)
The question of integrals has a similar answer. There are infinitely many
integrals the TI-92 cannot do. There are a relative few that can be done
with the tricks of calculus - substitution, parts, partical fractions. The
fact that a few of these few can be done by symbolic methods but not by
technology is surely not an argument against using technology.
But why do we want to do integrals? Most of the time, my guess is that
engineers require a numerical answer, which can be obtained for any integral
by the TI-83 quite easily. The hard part of calculus, and what we should
focus on, is understanding the problem well enough to get the right integral
and limits, and interpreting the answer correctly. (And by the way, on the
intellectual and historical bases of calculus.) After that, turning it over
to the technology is best, I think.
But numerical integration doesn't give the intellectual pleasure of say, the
problem of deriving the volume of a sphere by integration. I prefer to make
a big deal out of that derivation, and I use it as a reason to introduce
trig substitution, because in this case we don't seek a number, we want a
general result.
-Sandy
_______________________________
William J. (Sandy) Wagner
127 O'Connor St., Menlo Park, CA 94025
(650)328-8657 (voice) (650)323-1035 (fax)
sandyw@best.com
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