Asymptote
Rich Benedict
John Hanna
Hi, Rich,
You could graph the asymptote as another function since it is not part of the function. If the asymptote is a vertical line then you could construct it and show its equation.
More enticing: use the CAS capability of the propfrac( ) function to decompose a rational function into a polynomial and a ‘proper’ fraction (kind of like a mixed number). The poly part is the asymptote.
‘Holes’ in graphs (removable discontinuities) such as in y=(x^2 -4)/(x-2) at x=2 and ‘vertical asymptotes’ such as in y=1/(x-3) at x=3 are misleading and inappropriate and they do not appear on the TI-Nspire thanks to the more sophisticated graphing algorithms. But if you evaluate either of these functions at the points in question you’ll get ‘undef’ as a result.
John Hanna
T3 - Teachers Teaching with Technology
"Reality is an approximation."
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Joe
separately but a hole is part of the graph in that they represent a
place where the graph is undefined and should be shown. To graph
right thru the hole as though the function is defined there is a
mistake. Why do you say that they are "misleading and
inappropriate?"
On Mar 31, 7:19 am, John Hanna <johneha...@gmail.com> wrote:
> Hi, Rich,
>
> You could graph the asymptote as another function since it is not part of
> the function. If the asymptote is a vertical line then you could construct
> it and show its equation.
>
> More enticing: use the CAS capability of the propfrac( ) function to
> decompose a rational function into a polynomial and a 'proper' fraction
> (kind of like a mixed number). The poly part is the asymptote.
>
> 'Holes' in graphs (removable discontinuities) such as in y=(x^2 -4)/(x-2) at
> x=2 and 'vertical asymptotes' such as in y=1/(x-3) at x=3 are misleading and
> inappropriate and they do not appear on the TI-Nspire thanks to the more
> sophisticated graphing algorithms. But if you evaluate either of these
> functions at the points in question you'll get 'undef' as a result.
>
> John Hanna
>
>
> www.johnhanna.us
>
> T3 - Teachers Teaching with Technology
>
> "Reality is an approximation."
>
> _____
>
> From: tins...@googlegroups.com [mailto:tins...@googlegroups.com] On Behalf
> Of Rich Benedict
> Sent: Wednesday, March 31, 2010 9:45 AM
> To: tins...@googlegroups.com
> Subject: [tinspire] Asymptote
>
> I am a new CAS user and I am still trying to find how to show the asymptote
> on a graph of a function such as a rational function. I would love to know
> how. Thanks
>
> Rich
>
> --
> To post to this group, send email to tins...@googlegroups.com
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John Hanna
The 'hole' has dimension 0.
John Hanna
jeh...@optonline.net
www.johnhanna.us
T3 - Teachers Teaching with Technology
"Reality is an approximation."
-----Original Message-----
From: tins...@googlegroups.com [mailto:tins...@googlegroups.com] On Behalf
Of Joe
Sent: Wednesday, March 31, 2010 12:48 PM
To: tinspire
Subject: [tinspire] Re: Asymptote
Asymptotes are not part of the graph and should be constructed
separately but a hole is part of the graph in that they represent a
place where the graph is undefined and should be shown. To graph
right thru the hole as though the function is defined there is a
To unsubscribe, reply using "remove me" as the subject.
Joe
very important, and to ignore it is poor math. Perhaps from an
engineering point of view where all dimensions have tolerances, a
defect of zero width might(?) be ignored with no penalty. On the
other hand, how wide does a defect/discontinuity have to be, in order
to bring down the bridge, or crash the airplane? Actually, we are not
engineers we are mathematicians, which means our answers are not plus
or minus so much, our answers are exact and that is one of the
beauties of math. The square root of three is the square root of
three only, and has zero width on the real number line. Does that
mean that we should behave as if the tangent of 60 degrees doesn't
exist? And if we are to ignore a hole because it has zero width, do
we also ignore x=1 as the solution of x^2-2x+1=0? The only case I can
think of for ignoring such a discontinuity is the case for not
labeling it on the graph, because that would take a software change,
an expense that the company wants to avoid. Yeh, that might be a good
business reason, but is is still poor math!
On Mar 31, 9:57 am, John Hanna <johneha...@gmail.com> wrote:
> Re: Why do you say that they are "misleading and inappropriate?"
>
> The 'hole' has dimension 0.
>
> John Hanna
> jeha...@optonline.netwww.johnhanna.us
> For more options, visit this group athttp://groups.google.com.au/group/tinspire?hl=en-GB?hl=en-GB
> The tns documents shared by group members are archived athttp://lafacroft.com/archive/nspire.php
>
> To unsubscribe, reply using "remove me" as the subject.- Hide quoted text -
>
> - Show quoted text -
Joe
math!"
> > - Show quoted text -- Hide quoted text -
Wayne
This is not necessarily "poor" mathematics. The function that is
graphed by the calculator (without the "hole") is simply a
representative of the equivalence class of all functions that also
includes the subject function (with the "hole"). As John points out,
the set of points at which the subject function is discontinuous has
Lebesque measure zero, in fact the set is finite. John called it
"dimension 0", but the precise term is Lebesque measure zero. In
classical real analysis, two functions whose set of discontinuities
differ only by a set of Lebesque measure zero are in the same
equivalence class of functions and can be considered to be equivalent
for most advanced mathematical purposes. I refer you to any standard
graduate level text on real analysis, for example Royden,
http://www.amazon.com/Real-Analysis-3rd-Halsey-Royden/dp/0024041513,
if you are interested. In the classical function space L2, these
functions would be considered equivalent. This is just one of many
good reasons to graph the function without the "hole". There are also
some reasons not to do that, but those reasons do not make it poor
mathematics to graph the function without the "hole". In fact, many
mathemticians would consider it more mathematically correct to handle
these functions in the manner used by the Nspire.
Wayne
Wayne
differ only on a set of Lebesgue measure zero.
Wayne
On 1 Apr, 00:35, Wayne <waynep...@bellsouth.net> wrote:
> Joe,
>
> This is not necessarily "poor" mathematics. The function that is
> graphed by the calculator (without the "hole") is simply a
> representative of the equivalence class of all functions that also
> includes the subject function (with the "hole"). As John points out,
> the set of points at which the subject function is discontinuous has
> Lebesque measure zero, in fact the set is finite. John called it
> "dimension 0", but the precise term is Lebesque measure zero. In
> classical real analysis, two functions whose set of discontinuities
> differ only by a set of Lebesque measure zero are in the same
> equivalence class of functions and can be considered to be equivalent
> for most advanced mathematical purposes. I refer you to any standard
> graduate level text on real analysis, for example Royden,http://www.amazon.com/Real-Analysis-3rd-Halsey-Royden/dp/0024041513,
Joe
answer and my gut feeling is that I should let it go at that, but I
just cannot resist the temptation of pointing out that while if is
good theory to consider the earth as flat for short range surface map
making, it probably isn't wise to teach your class that the earth is
flat in general. Another consideration is that assuming that you have
taught your high school math class, a graduate level real analysis
course, it is perfectly understandable to suggest that their
calculator should ignore holes in the plots it presents. As for
argueing the applicability of the equivalence class to the nspire cas,
I'll pass thank you. As I said, I asked a question and got an answer
and I should probably let it go at that.
Wayne
Nspire. To reiterate, my point is that it often good mathematics to
ignore differences in functions when they differ only on a set of
measure zero. For example, in defining a metric (norm) and an inner
product on L2, one should do precisely that. A little study of the
book that I recommended will help you to understand why this is true.
That is the approach the Nspire has taken in treating functions with
removable singularities and that is not "bad" mathematics. I am not
sure what a flat earth or my teaching pedigree have to do with that
but if that is important to you, then I will confess that I do not
teach high school math courses. On the other hand, I do work with
classroom teachers often and they understand why this is a reasonable
way to treat the graphing of functions with removable singularities.
I hope that you understand that the choice made by developers when
building the graphing function for the Nspire has a significant base
of mathematics that makes that choice a good one. So the real point
is that the Nspire developers made a reasonable choice that is
supported mathematically and did not make a "bad" mathematics choice,
as you suggested.
Best regards,
Wayne
Eric Findlay
medium, it should be invisible to the naked eye. The only reason it was
visible on the previous calculators is because the pixels were so large,
if the point they measured for the pixel was undefined, the whole pixel
was off (which would make it appear as if there were several undefined
values around that point). In order to see that "hole" on a proper
graph, you'd have to zoom in an infinite number of times.
--
For I am convinced that neither death nor life, neither angels nor
demons, neither the present nor the future, nor any powers, neither
height nor depth, nor anything else in all creation, will be able to
separate us from the love of God that is in Christ Jesus our Lord.
- Romans 8:38-39 (NIV)
--
Eric Findlay
AKA Eagle-Man
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Nelson Sousa
Joe
height, the line graphing the equation should be of zero height,
making it invisible. The result is a mathematically advanced
calcuator that generates invisible graphs containing invisible holes.
Of course the testing Boards won't be able to see the invisible
graphs, so in their infinite wisdom, they will conclude that the new
graphing theory prevents cheating, and they can be convinced to outlaw
all other types of calculators for testing purposes. Then TI can
double and triple the price of these calculators on the basis of the
increased costs associated with making the graphs invisible. Of
course there might be one or two people who actually perfer to see the
graphs and holes but who needs them. They can just be ignored and
marginalized as outdated quacks. So if you are listening TI, lets get
on with it. This is going to be really exciting, expecially when you
product the models that generate invisible 3D graphs. Oops, I forgot,
we already have that feature. Sorry.
On Apr 2, 6:25 pm, Nelson Sousa <nso...@gmail.com> wrote:
> Recipe for disaster:
>
> - Take 1 TI-84 Plus, 1 TI-89 Titanium, 1 Voyage 200, or any other TI
> calculator (HPs and Casios do the same, I just don't know the settings to
> make this appear);
> - Define function (x-1)/(x-1)^2 and graph it using ZoomDec; you see the
> hole.
> - Now add 0.05 to xmin and xmax (e.g., on the TI-84 Plus choose -4.65 as
> xmin and +4.75 as xmax). See the graph. The hole disappears.
>
> Lets do it again with a different function, shall we? Plot
> (x-sqrt(2))/(x-sqrt(2))^2 with any one of the predefined zoom settings.
> There's no visible hole at x=sqrt(2).
>
> Not only holes appear just because pixels are large but you can't trust them
> to appear everytime. A hole is visible only if the discontinuity is exactly
> in the center of a pixel.
>
> Between erratic behaviour and not being able to see holes, I choose a
> coherent behaviour anytime.
>
> Nelson
>
> ...
>
> read more »- Hide quoted text -
Eric Findlay
We were just giving you various reasons why TI used the method that they
used, but you know what? YOU DON'T HAVE TO USE IT! That's the best
part about living in a free country. You have the choice! So go please
make up your mind and either use the Nspire (software) or don't, but
don't complain about every little thing that is "wrong" with the
calculator and then get all snippy when we give you valid arguments in
favour of said "errors."
--
For I am convinced that neither death nor life, neither angels nor
demons, neither the present nor the future, nor any powers, neither
height nor depth, nor anything else in all creation, will be able to
separate us from the love of God that is in Christ Jesus our Lord.
- Romans 8:38-39 (NIV)
--
Eric Findlay
AKA Eagle-Man
> No virus found in this incoming message.
> Checked by AVG - www.avg.com
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Sean Bird
I think the user (like Rich, the one who began this thread) will be pleased with the result of what is now displayed when you use trace for functions like this. Also try menu, Trace, Graph Trace for f2(x)=1/(x+1). Type in -1 and see what happens. In the bottom right (-1,undef) is displayed and a lovely vertical line shows up.
- Sean Bird
Covenant Christian High School
Indianapolis, IN
Joe
My post was meant to provoke some laughs.
BTW, the lack of the Gamma function for extending the factorial
operation to non-integers is significant, and available on at least
one other calculator that was designed a generation ago, but for some
it's probably not a big deal, so it's not something that I ever
brought up. On the other hand, providing 3D graphing strikes me as
the single most important improvement that TI can make to the nspire
cas, so I don't know why they continue to ignore that issue, while
working on other more trivial pursuits, but it does make me wonder if
the problem is simply that they choose to work on the easier stuff
while ignoring 3d graphing because they simply don't have the
necessary talent to produce a good 3d graphing program. In which case
it probably will never occur. About that I don't know, but yes I
appreciate living in a free country and the choices that I have,
because I and others like me chose to protect those freedoms by
putting our life on the line in defense of those rights. In fact, I
believe on the basis of my military service that I "earned" the right
to complain and don't need any lectures on patriotism, but to get back
to the point, yeah sure, having the location of holes in a graph
identified would be nice, but I would much rather have 3d graphing.
How about you?
Nelson Sousa
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lee kucera
lk
On Apr 3, 2010, at 9:38 AM, Joe wrote:
yeah sure, having the location of holes in a graph
identified would be nice, but I would much rather have 3d graphing.
How about you?
lee kucera
a.p. statistics
leek...@gmail.com
Joe
trivial improvements and only a few asking for 3D graphing, but you
need to replace the word "people" with "high school math teachers."
It is also clear that while TI needs a continueing flow of orders to
stay in that business (which BTW is in everyone's best interests
compared to getting stuck with a bunch of discontinued products), it
is also clear that after a school buys a set of nspires they won't be
making that kind of purchase again for many many years if ever. So
whether K-12 teachers realize it or not, it is in their best interests
for TI to use their resources to make the kind of improvements that
will significantly increase sales by making the product appeal to a
broader market, as opposed to wasting their limited resources on
trivial things that really don't effect sales. TI has fixed the power
sucking problem. They have fixed the input statement problem. They
have modernized with the touch pad and rechargable batteries. They
have done a terrific job of developing a more appealing keyboard. And
now it is time to make the nspire products appealing to the big
college, university, and professional market. It is time to catch up
with the capabilities offered by other high end calculators and go a
step beyond that. So whats in it for the high school math teacher?
The knowledge that they will recieve continued service for many years
to come because the nspire/nspire cas calculators are the best of the
best.
lee kucera
In many areas, it is not the k-12 school buying the units--the students are purchasing their own. So that IS a repeated purchase. Yes, I agree it would be nice to get more support at the higher ed level. However, most of the college folk I know don't use ANY handheld/graphing unit and many do not allow their students to use them (a whole different discussion).
lk
Wayne
I have a great idea. Please establish a new Google group to appeal to
this "broader market" and leave this group to us "K-12 teachers" who
don't "realize what's in our best interest". You have made the same
points here ad nauseam, so I am sure the users in this broader market
will have a much better understanding of what the Nspire needs to be.
You can then use the overwhelming responses from this broader market
to influence TI to build the calculator that you want.
Just a suggestion.
Wayne
Joe
confess that I do not teach high school math courses. On the other
classroom teachers is not the same as being a classroom teacher, I had
the impression that you are not a teacher, but now you say "us K-12
teachers" and since the word us would indicate that you are, which is
it?
> > best.- Hide quoted text -
Wayne
making fun of you?
Wayne
> > - Show quoted text -- Hide quoted text -
Joe
impression that you are a high school student. Am I right about
that?
hastern
mistaken impressions ...
Joe
not you, a question based on his inconsistent posts and I get an
insulting post from you. What is bothering you?
> > that?- Hide quoted text -
Kathleen Adams
Try to have a great day and Happy Easter to whom it applies... :-)
> wrote: > > > > > Thats ok Wayne. I forgive you but I am still curious. I have the > > impression that you are a high school student. Am I right about > > that?- Hide quoted text - > > - Show quoted text - -- To post to this group, send email to tins...@googlegroups.com
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Jenny Orton
I just did!
Jenny