Comprehensive Graphing (was: roots of x^12 = 2^x)
- From: David W. Cantrell <DWCantrell@xxxxxxxxxxx>
- Date: 12 Apr 2007 00:58:43 GMT
"Ioannis" <morpheus@xxxxxxxxxxxx> wrote:
"Ioannis" <morpheus@xxxxxxxxxxxx> wrote in message
news:1176304073.615854@xxxxxxxxxxx
"Phil Carmody" <thefatphil_demunged@xxxxxxxxxxx> wrote in message[snip]
news:871wirca90.fsf@xxxxxxxxxxxxxxxxxxxxxxx
I am aghast that you seemed to think that x^12 would dominate 2^x
as x increased.
It wasn't my fault. Honest :-) That was Maple V release 4's fault. To
make sure that I got the right behavior at +oo, I tried a quick:
plot(x^12-2^x,x=-infinity..infinity);
Very interesting. Mathematica, for example, balks at any request to graph
from -Infinity to Infinity.
Unfortunately, version V of Maple misbehaves on this graph, producing
slightly different variants each time. The time I checked, it produced
a spike at +oo for x->oo (erroneous), so I didn't check further. By
breaking the domain of the graph into smaller regions one can get the
correct behavior, but I stupidly relied on just that one glance I got
from the -infinity..infinity range, which was incorrect.
And indeed, to further my defense for being a careless putz (8*(, here
are the two graphs:
plot(x^12-2^x,x=-infinity..infinity);
Maple V release 4:
http://misc.virtualcomposer2000.com/graph1.gif
Maple 9:
http://misc.virtualcomposer2000.com/graph2.gif
The Maple V graph is clearly wrong. It's completely missing the third
root. That's the one I saw first and didn't even think twice about
exp(x) > x^n, which is an immediate givaway.
This probably deserves a new thread, so I've changed the title
to "Comprehensive Graphing".
I've used such graphs for many years (although I did not produce such a
graph for x^12 - 2^x until after you had posted the graphs produced by
Maple). I have called them "comprehensive graphs". Is there an "accepted"
name for them?
For comparison with your Maple 9 graph, here's a comprehensive graph which
I produced with Mathematica:
<http://img264.imageshack.us/img264/4936/comprehensivegraphmq6.gif>
The code I used was
f[x_] := x/(1 - Abs[x]); fInv[x_] := x/(1 + Abs[x]); g[x_] := x^12 - 2^x;
Plot[fInv[g[f[x]]], {x, -1, 1}, AspectRatio -> Automatic, Ticks -> False]
Above, you can see the scaling function f which I used. It's my "default"
for comprehensive graphing, but of course many other such functions could
be used (e.g. Tanh for f, ArcTanh for fInv). What scaling function does
Maple use by default for such graphs? Can the user specify alternative
scaling functions?
The Maple 9 graph also seems to be wrong, although "less wrong" than the
Maple V graph. I didn't check this one when I answered the question, but
to me it implies that the function reaches +infinity between the second
and third root,
It doesn't imply that to me. Rather, it just shows that the function gets
very large between the second and third roots.
and that the third root is very large in magnitude,
almost close to +infinity.
Well, that's presumably because it is! In my comprehensive graph, for
example, the distance between 0 and +oo is 1, while the distance between
that third root and +oo is merely 0.013...
David
It would be nice to see the corresponding graph Maple 10 produces..
- References:
- roots of x^12 = 2^x
- From: chapkovski
- Re: roots of x^12 = 2^x
- From: Ioannis
- Re: roots of x^12 = 2^x
- From: Gerry Myerson
- Re: roots of x^12 = 2^x
- From: David W . Cantrell
- Re: roots of x^12 = 2^x
- From: Ioannis
- Re: roots of x^12 = 2^x
- From: Phil Carmody
- Re: roots of x^12 = 2^x
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- Re: roots of x^12 = 2^x
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