Re: opponents of taylor and l'hospital ?

WWW wrote

( forgive my shorthand notation )

In article
<13974fd3-8184-4deb-8e37-6c7032f06c56@xxxxxxxxxxxxxxxx
egroups.com>,
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.

understanding , intuition and computation often go hand in hand.



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.

sure , but users of l'hospital know this result from the back of their hand too.

they use l'hospital to show it to others , not because they dont know it themselves.



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.

sure , but thats exactly one of my points.

let me explain : when they say ( and they did in this thread ) you need the lim of sin(x) to compute sin(x)/x ( at 0 ) , that limit is just the derivative.

just as you said lim = derivative.

so in fact it does not give a circular or counterexample of l'hospital, since instead of LHR -> needs lim , we get , in a way , LHR = derivative = lim.

and btw we can also apply taylor or h =/= 0 , h^2 = 0 to solve in a more algebraic way.


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.

nostalgy i assume.

my prof often said : "everything (in the real world) is a wave"

to which i replied : " if you have a hammer , everything looks like a nail "

the matter is still unsettled , in fact it got more complicated over the years e.g. wave vs particle vs curvature in physics , discreet event vs continue wave vs attractors in certain economic models etc

regards

tommy1729
.

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