| Science Forum Index » Mathematics Forum » "deceptive" graphs of functions |
|
Page 1 of 1 |
|
| Author |
Message |
| David Bernier |
 Posted: Sat Feb 19, 2005 7:02 pm |
|
|
|
Guest
|
For f: (0, oo) -> R defined by:
f(x) = cos(x)/(2+log(1+log(1+1/(x*x)))),
lim_{x->0} f(x) = 0.
When I plot the function from 10^(-7) to 10^(-6),
it's not "clear" that lim_{x->0} f(x) = 0.
It might be interesting to gather more
examples.
David Bernier |
|
|
| Back to top |
|
|
|
| pi |
| Posted: Sun Feb 20, 2005 7:57 am |
|
|
|
Guest
|
David Bernier wrote:
[quote:2e654a40d9]For f: (0, oo) -> R defined by:
f(x) = cos(x)/(2+log(1+log(1+1/(x*x)))),
lim_{x->0} f(x) = 0.
When I plot the function from 10^(-7) to 10^(-6),
it's not "clear" that lim_{x->0} f(x) = 0.
[/quote:2e654a40d9]
that is because the denominator takes an exceedingly small number many
many orders of magnitude less than the 10^-7 you listed. The double
logs cause this trouble. Mathematica does show that the graph goes
down around x=0 but it is not obvious from the graph that that is what
the limit is. This is just one of those functions you have to work by
hand and know that 0 is correct. |
|
|
| Back to top |
|
|
|
| Oscar Lanzi III |
| Posted: Mon Feb 21, 2005 6:34 am |
|
|
|
Guest
|
When calculators first came out they used 8-digit arithmetic but no
floating point -- making for some nice loss of significant digits when
numbers were much less than 1. I'd enter 99 and press x^2, 1/x,
sqrt(x), and 1/x in that order: 99.01. Then 99.06 on the next round.
The calculations did not stabilize until 99 became 100!
--OL |
|
|
| Back to top |
|
|
|
| David Bernier |
| Posted: Mon Feb 21, 2005 10:34 am |
|
|
|
Guest
|
Dave Rusin wrote:
[...]
[quote:1e5232421d]
1. Plot x^4 + log(1 + cos(x)) on [0, 4]. Anything interesting to report?
2. Describe the graph of 40 + log(x) - x/100 over its domain.
3. Plot x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1 on [0.99, 1.01].
How many times does it cross the axis? (The polynomial equals (x-1)^7
but you have to enter it the long way. Why?)
[/quote:1e5232421d]
I like number 3. Using WIMS "Function calculator", on [.99, 1.01],
the plotted graph crossed the x-axis a few dozen times.
The "mystery" begins to lift by changing to [.9, 1.1] where
one sees just how flat the graph is around 1, and
assuming one knows a bit about how computers do
floating point arithmetic (32-bit floats, 64-bit doubles, ...).
David Bernier
[quote:1e5232421d]I save examples of these types at http://www.math-atlas.org/99/calc_errors .
dave
[/quote:1e5232421d] |
|
|
| Back to top |
|
|
|
|
