Quiz: calculating a definite integral
01-03-2014, 09:04 PM (This post was last modified: 01-03-2014 09:05 PM by Bunuel66.)
Post: #41
 Bunuel66 On Vacation Posts: 29 Joined: Jan 2014
RE: Quiz: calculating a definite integral
(01-03-2014 08:39 PM)Thomas Klemm Wrote:
(01-03-2014 05:38 PM)Bunuel66 Wrote:  This seems to show that having the equality is not enough for keeping it directly after integrating.
The problem I see is that $$u=-\frac{1}{x}$$ is not defined for $$x=0$$. The Taylor-series of $$\exp(u)$$ is not defined for $$u=-\infty$$.

Cheers
Thomas

Don't get the point, $$\exp(-\infty)$$=0. The serie is converging whatever the sign of x (more and more slowly as you're closing to 0-....). Then we have two expressions who provides similar values whatever the sign of x, and after integration we have a new set of expressions with one which is no more defined on one side (x<0). And as you mention, this is not exactly a Taylor serie in the sense that the sum is not using the derivatives of u(x). The problem is maybe a little bit more subtle (at least for me) than it seems ;-(...

Regards
01-03-2014, 09:32 PM
Post: #42
 Thomas Klemm Senior Member Posts: 1,158 Joined: Dec 2013
RE: Quiz: calculating a definite integral
(01-03-2014 08:03 PM)W_Max Wrote:  My pocket HP30b (yes, not wp34s ), using simplest rectangle method and step 0.00005 give 1.29128599414 after 3 minutes. Enjoy simple methods
Did you try to calculate $$\sum_{n=1}^{10}n^{-n}$$?
Should take ~0.09s.

Cheers
Thomas
01-03-2014, 11:08 PM
Post: #43
 W_Max Junior Member Posts: 21 Joined: Dec 2013
RE: Quiz: calculating a definite integral
Not exactly, but close to. I wrote simple RPN program.

0 STO4
LBL00 RCL3 INPUT +/- Y^X
STO+4 RCL1 STO+3 RCL2 RCL3 ?> GT00
RCL4 RCL1 * Stop

0.00005 STO1, 1 STO2 0 STO3

It takes about 2+ min to complete (20000 cycles or 166cycles/sec! ). As HP30b is relatively fast machine - such a 'brute force' method gives acceptable result too.
01-06-2014, 10:28 AM
Post: #44
 Thomas Klemm Senior Member Posts: 1,158 Joined: Dec 2013
RE: Quiz: calculating a definite integral
(01-03-2014 09:04 PM)Bunuel66 Wrote:  Don't get the point, $$\exp(-\infty)$$=0.

The domain of $$\exp(x)$$ is $$\mathbb{R}$$, but $$-\infty \notin \mathbb{R}$$. Thus you can not just plug $$-\infty$$ into the Taylor-series of this function and expect everything works. You can calculate $$\lim_{x\to\infty}\exp(x)$$ but that's not the same as $$\exp(-\infty)$$. This expression is just not defined.

HTH
Thomas
01-07-2014, 06:12 PM
Post: #45
 Bunuel66 On Vacation Posts: 29 Joined: Jan 2014
RE: Quiz: calculating a definite integral
(01-06-2014 10:28 AM)Thomas Klemm Wrote:
(01-03-2014 09:04 PM)Bunuel66 Wrote:  Don't get the point, $$\exp(-\infty)$$=0.

The domain of $$\exp(x)$$ is $$\mathbb{R}$$, but $$-\infty \notin \mathbb{R}$$. Thus you can not just plug $$-\infty$$ into the Taylor-series of this function and expect everything works. You can calculate $$\lim_{x\to\infty}\exp(x)$$ but that's not the same as $$\exp(-\infty)$$. This expression is just not defined.

HTH
Thomas

Could have been rewriten as a limit to be more rigorous...;-) That said the serie gives the same value than the function also for x<0. Doesn't seems to be the point. And as you mention this is not a Taylor serie strictly speaking.

Regards.
01-09-2014, 07:45 AM
Post: #46
 Thomas Klemm Senior Member Posts: 1,158 Joined: Dec 2013
RE: Quiz: calculating a definite integral
(01-07-2014 06:12 PM)Bunuel66 Wrote:  Could have been rewriten as a limit to be more rigorous...;-)

Cheers
Thomas
01-14-2014, 02:31 PM
Post: #47
 Thomas Klemm Senior Member Posts: 1,158 Joined: Dec 2013
RE: Quiz: calculating a definite integral
01-14-2014, 02:53 PM
Post: #48
 walter b On Vacation Posts: 1,957 Joined: Dec 2013
RE: Quiz: calculating a definite integral
Nice quiz!

d:-)
01-16-2014, 08:21 PM
Post: #49
 Gerson W. Barbosa Senior Member Posts: 1,056 Joined: Dec 2013
RE: Quiz: calculating a definite integral
(12-31-2013 01:14 PM)Thomas Klemm Wrote:  It takes 2'27" to calculate this integral on a DM-15CC with FIX 9.
It takes 28" to do it with the RPN-15C emulator on my iPhone.
On a real HP-15C it takes probably much longer.
5 seconds on my outdated iPhone 4s with HP-15C emulator by HP:

Code:
001- f LBL A 002-   CHS 003-   y^x 004- g RTN

0 ENTER 1 f Integrate --> 1.291285997 (blinking, but that's another story)

Estimated +4 hours on a real HP-15C.

As a comparison the following return the same result (no blinking, of course!) in 13.7 and 13.4 seconds, respectively, on my 30-year old HP-15C:
Code:
 001- f LBL A            001- f LBL A 002-   0                002- f MATRIX 1 003-   STO 0            003-   8 004-   9                004-   STO I 005-   STO I            005- f LBL 0 006- f LBL 0            006-   RCL 1 007-   RCL I            007-   RCL+ I 008-   ENTER            008-   ENTER 009-   CHS              009-   CHS 010-   y^x              010-   y^x 011-   STO+ 0           011-   STO+ 0 012- f DSE I            012- f DSE I 013-   GTO 0            013-   GTO 0 014-   RCL 0            014-   RCL 0 015- g RTN              015- g RTN

Cheers,

Gerson.
 « Next Oldest | Next Newest »

User(s) browsing this thread: 1 Guest(s)