Re: opponents of taylor and l'hospital ?
- From: The World Wide Wade <aderamey.addw@xxxxxxxxxxx>
- Date: Mon, 05 May 2008 16:35:43 -0700
In article
<13974fd3-8184-4deb-8e37-6c7032f06c56@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
lwalke3@xxxxxxxxx wrote:
On May 4, 8:32 am, amy666 <tommy1...@xxxxxxxxxxx> wrote:
some while ago when talking about limits , i used l'hospitals rule or
taylor series.
i was then told by certain people that they dont like l'hospitals rule nor
taylor series.
they said l'hospitals rule was overrated !
can those people plz explain what is it that makes them opponents of
l'hospital rule and taylor series ??
i dont see a downside of l'hospitals rule ?
and i was quite amazed to read that on a mathforum from even relative
regular posters.
I think that tommy1729 raises a very interesting point here.
I know that I've seen some of these posts that actually recommend
that L'Hopital's rule not be taught in calculus courses. I kept trying
to find such posts using a Google search. I couldn't find them, but
I know I've seen them. (It may have been Dr. Herman Rubin who
discourages L'Hopital's rule, since he's definitely outspoken on
what should and shouldn't be taught in school, but since I can't
find the thread I can't be sure whether he's the one.)
I actually agree with tommy1729 here. I believe that we should
continue to teach L'Hopital's rule in calculus classes.
In this thread below, Dr. Gerry Myerson does give a valid reason
why L'Hopital's rule in certain situations is flawed:
Myerson:
There's an objection to using l'Hopital to do
limit as x goes to zero of (sin x) over x.
The objection is that to use l'Hopital, you need to know
the derivative of sin x --- but to know the derivative
of sin x, you have to know the limit as x goes to zero
of (sin x) over x, so using l'H is circular (no pun intended)
reasoning.
But I believe that in classes that are taken mainly by
students whose major is not mathematics, such as
freshman calculus, complete mathematical rigor is not
always necessary. The goal of the class for non-math
majors is _computation_, not _proof_.
No, the main goal is not computation (you can do these things on a
computer), and it's not rigorous proof. It's understanding and
intuition.
Save the proofs for
upper division classes taken only by math majors, not
engineers or physics majors.
Fine, but that has little to do with the dislike of many here for the
way LHR is invoked.
Suppose a high school senior taking the AP Calculus exam
this month sees the following problem:
lim (x->0) (sin x)/x =
A) 0
B) 1
C) -1
D) does not exist
If I were this student, I'd simply use L'Hopital's rule and find
the anwer to be (cos x)/1 = cos 0 = 1 in seconds. Why
should I care that the limit is used in the proof that
d/dx (sin x) = cos x? The question didn't ask to prove that
d/dx (sin x) = cos x -- it asked to find the limit, and the
fastest way to find the limit is L'Hopital's rule.
No, the fastest way is simply to know this result. It is one of the
fundamental bedrock limits of calculus. If you don't know that limit
like the back of your hand after first semester calc, something has
gone wrong.
I agree that there are cases, such as lim (x->0) P(x)/Q(x)
where P and Q are both polynomials, where there are
other methods (such as long division) that are more
appropiate than L'Hopital's rule. But whenever I see
lim (x->0) P(x)/Q(x) where P or Q is any function but a
polynomial, then I'm using L'Hopital (provided that it can
actually be used) -- despite the fact that limits such as
lim (x->0) (sin x)/x or lim (x->0) (e^x-1)/x are needed to
prove the differentiation rules for sin x and e^x.
In fact, to this day I don't remember how to prove the
differentiation rules for sin x and e^x -- yet I still recall
lim (x->0) (six x)/x = lim (x->0) (e^x-1)/x = 1, thanks to
L'Hopital's rule.
Forget about proofs, that's a red herring. The limits here are nothing
but derivatives themselves! lim (x->0) (e^x-1)/x is by definition the
derivative of e^x at 0. That's it. If a calc student doesn't know
that, then he has learned very little.
cannot differentiate any function from first principles only.
I still recall my multivariable calculus professor, who
referred to L'Hopital's rule as the one rule that every
student from calculus likes. (He jokingly called the Mean
Value Theorem "the much hated Mean Value Theorem.")
On the other hand, there are some teachers (thankfully
not any teacher I had, but someone who posts here at
sci.math) who would emphasize using the definition of
the derivative and the Riemann integral to differentiate or
integrate any function, and not teach the differentiation
rules for any function, not even polynomials. This is
definitely inappropriate for high school, or for any
class populated by non-math majors. Using the definition
of the derivative to differentiate a polynomial is an
error-prone process. Using the rules is much less likely
to result in mistakes.
But the rules for differentiating polynomials are something you can
teach an 8 year old. Such rules are trivial; any dull mind can learn
them. Again, your citation of some mad teacher's practices have little
to do with the main issue.
.
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