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M_SHIRAISHI ïŒ The_New_York_Academy_of_Sciences
M_SHIRAISHI wrote:
> 以äžã«åŒçšããã®ã¯ãæåãªã解ææŠè«ãïŒé«æšè²æ²»ïŒèïŒããã®ãã®ïŒåæž p.37ïŒã§ãã
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> 確ãã«ã解ææŠè«ãã®äžã§ãããã®éšåã®èª¬æã®ãŸããã¯
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Shinji KONO @ Information Engineering, University of the Ryukyus
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I wrote:
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M_SHIRAISHI wrote:
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> Yuzuru Hiraga <hir...@slis.tsukuba.ac.jp> wrote in message news:<4073A3DC...@slis.tsukuba.ac.jp>...
> >
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M_SHIRAISHI wrote:
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keizi kounoike
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hir...@ulis.ac.jp (Yuzuru Hiraga) stupidly wrote in message news:<c5h4vc$po2$1...@hagi.cc.tsukuba.ac.jp>...
> In article <800c7853.04041...@posting.google.com> eu...@apionet.or.jp (M_SHIRAISHI) writes:
>
> ãx ããèªèº«ãx ã®åœæ°ãšã¿ãã°ããšããã®ã¯ããããŠæžãã° x=x ãšãªããã©
ãå®æ°é
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# æ¥ãç¥ãïŒããããã«ã¿ã¬ãïŒïŒïŒ
> > ãx ããèªèº«ãx ã®åœæ°ãšã¿ãã°ããšããã®ã¯ããããŠæžãã° x=x ãšãªããã©
>
>
> ãå®æ°é
ãïŒã§âåŸãâãïŒã®âªäžæ¬¡åœæ°â«ã xïŒx ã§è¡šããããããšèšãã®ãïŒïŒïŒ
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> # æ¥ãç¥ãïŒããããã«ã¿ã¬ãïŒïŒïŒ
âŠã«ã€ãªããã®?
ãŠããããããããŸã®åé¡ã§âäžæ¬¡é¢æ°âã«äœã®çšããã?
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ãx ããèªèº«ãåãšããåœæ°ãã£ãŠã®ã¯ãå®æ°é
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ã£ãŠããšã«ä»ãªãã¬ã£ãŠããšã ã
# åãããã®ãããã®çšåºŠã®ããšãïŒ ããã¡ãã¡äººæ§ã®åŸ¡æåã°åãããšãïŒïŒïŒ
dyïŒïŒf'(x)ã»â³x ãšå®çŸ©ããŠããã®ã§ãyïŒf(x)ïŒx ãšãããç¹å¥ãªå Žåãã ãšã
f'(x)ïŒ(x)'ïŒïŒã§ããããã€ïŒyïŒx ãªã®ã ããïŒdyïŒdx ãæç«ããŸãã
åŸã£ãŠããã®ãç¹å¥ãªå Žåãã«éããdxïŒdyïŒ(x)'ã»â³x ïŒâ³x ãšãªããŸãã
ãã£ãŠãïŒãã®ãç¹å¥ãªå Žåãã«éãïŒïŒdxïŒâ³x ãæç«ããã£ãŠã¯ã±ã§ãã
<800c7853.04041...@posting.google.com>ã®èšäºã«ãããŠ
eu...@apionet.or.jpããã¯æžããŸããã
eurms> dyïŒïŒf'(x)ã»â³x ãšå®çŸ©ããŠããã®ã§ãyïŒf(x)ïŒx ãšãããç¹å¥ãªå Žåãã ãšã
eurms> f'(x)ïŒ(x)'ïŒïŒã§ããããã€ïŒyïŒx ãªã®ã ããïŒdyïŒdx ãæç«ããŸãã
eurms>
eurms> åŸã£ãŠããã®ãç¹å¥ãªå Žåãã«éããdxïŒdyïŒ(x)'ã»â³x ïŒâ³x ãšãªããŸãã
eurms> ãã£ãŠãïŒãã®ãç¹å¥ãªå Žåãã«éãïŒïŒdxïŒâ³x ãæç«ããã£ãŠã¯ã±ã§ãã
y = x+1 ã§ã dy = dx ã®ãããªæ°ããããã§ããã©, ç§ãæ··ä¹±ããŠãã
ã ãã§ãããã?
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å ±ç§åŠç ç©¶ç§ èšç®æ©æ°çç§åŠå°æ»
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ããããŸããé¢æ°å€ã y ã«ãããããã°ãããªãã ããããã©ã
å¥ã«ãåã®äžã ããããµã€ã¯ãã€ãæ²ç·äžã ãããçé¢äžã ãããäœåŠã ãããã
x = x ã§ãã?
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yïŒx+1 ã®å Žåã§ãã確ãã« dyïŒdx ãæç«ããŸãã
# äžè¬ã«ãc ãä»»æå®æ°ãšããŠãyïŒxïŒc ã®å Žåã«éããdyïŒdx ãæç«ããã®ã§ã
ãyïŒx ã®å Žåã«éããdyïŒdx ãæç«ããããšèšã£ãã®ã¯âªèª€ãâ«ãªã®ã§ããè©«ã³
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<800c7853.04041...@posting.google.com>ã®èšäºã«ãããŠ
eu...@apionet.or.jpããã¯æžããŸããã
eurms> yïŒx+1 ã®å Žåã§ãã確ãã« dyïŒdx ãæç«ããŸãã
eurms>
eurms> # äžè¬ã«ãc ãä»»æå®æ°ãšããŠãyïŒxïŒc ã®å Žåã«éããdyïŒdx ãæç«ããã®ã§ã
eurms> ãyïŒx ã®å Žåã«éããdyïŒdx ãæç«ããããšèšã£ãã®ã¯âªèª€ãâ«ãªã®ã§ããè©«ã³
eurms> ããŠãèšæ£ããŠãããŸãã
ããã¯éç®±ã®é
ã ããå®ã¯ã©ã¡ãã§ãããããšãªãã§ããã©,
<800c7853.04041...@posting.google.com>
ã§
eurms> ãããããããªããšãèš±ãããã®ã§ããã°ãyïŒx ã®å Žåã«ã¯ dy
eurms> ïŒdxïŒâ³x ã§ããã®ã ãããdyïŒâ³x ã (4) ã«ä»£å
¥ã§ããçã§
eurms> ããããããããšã
eurms> ããããããããããããâ³xïŒf'(x)ã»â³x
eurms> ããããããããããããâŽã f'(x)ïŒïŒ
ãšããã®ã¯, ãy = f(x) = x ãšããç¹æ®ãªå Žåã«ã¯ f'(x) = 1 ãšãªãã
ãšãã£ãŠããã ãã§ãã£ãŠ,
eurms> ãšãªã£ãŠããŸãããyïŒf(x) ã¯ïŒ(埮åå¯èœãª)ä»»æã®åœæ°ïŒã§ã
eurms> ãããšããä»®å®ã«åããäžåçã§ããã
ãšã¯é¢ä¿ãªãããããªãããª, ãšæããŸã.
y = x ãšç¹æ®åãããããã dy = dx = Îx ãšã§ããã®ã§ãã£ãŠ,
# ãã¡ãã y = x + c (c ã¯ä»»æã®å®æ°) ãªã dy = dx = Îx ãšã§ãã
# ãã©.
ç¹æ®åããç¶æ³ããã®ãŸãŸç¹æ®åããŠããªããšãã«ãŸã§åºãã¡ããŸãã
ããããªãããªã.
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ãããã M_SHIRAISHI ãããäœãèããã®ãã
ã²ããŠã¯ã©ãã§ã©ãééã£ãã®ããããããŸããã
ã# ä»ãŸã§ãããã€ãã£ããäœèããŠããã ïŒããšäžæè°ã ã£ãã
ãŸãåºçºç¹ãšããŠïŒ
ãM_SHIRAISHI ããã«ã¯ dx = Îx ãã©ãããŠåºãŠããã®ãããããªãã£ãã
ãå
ããã©ãã°ãx ããèªèº«ã x ã®åœæ°ããããããªãã£ãã
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ãã㧠dx=Îx ã®ã€ãã€ãŸãåãããããã次ã®ããã«èããã
ãã»y = x ãšããæçé¢æ°ãèããããã®ãšãå®çŸ©ã«ãã dy=Îx ã§ããã
ãã# ããã¯æ£ããã
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ãã# ã¡ãã£ãšåé¡ã¯ãããããŸããããæ£ããã
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ïŒã€ã¯ dx=Îx ã y=x ãšãããç¹å¥ãªå Žåãã«ã ãæãç«ã€ãšèããããšã
ããã¯æ£ãããšèšãã°æ£ãããã ãã©ãy ãªã©ãšããäœèšãªå€æ°ãå°å
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ãz = x, dz = Îx, dz = dx â dx = Îx
ãšããã£ãŠå
šãåãããšãz èªèº«ïŒy èªèº«ïŒã¯è¡šèå°ããæ¶ããŠããŸã£ãŠããã
ãããŠããã¯ç§ã x = t ãšç¬ç«å€æ°ã®åŽã«äœæ¥å€æ° t ãå°å
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y = x ãªã©ãšæžãããããåºçºç¹ã® y = f(x) ã® y ãšæ··åããŠããŸãã
dy = dx = Îx ã® dy ãš dy = f'(x)Îx ã® dy ãåããã®ã ãšæã£ãŠããŸã£ãã
äžèšã®ããã«ãy ã§ã¯ãªã z ã䜿ã£ãŠããã°ãããªæ··åã¯ããªãã£ãã§ãããã
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> å¥ã«ãåã®äžã ããããµã€ã¯ãã€ãæ²ç·äžã ãããçé¢äžã ãããäœåŠã ãããã
> x = x ã§ãã?
βακαΌÏÎœïŒ
âxïŒxâãªããŠãªãæçåŒãã§ãã£ãŠåœæ°ãè¡šããããã¯ã±ãç¡ãã¯ïŒ
In article <c5j90a$cue$1...@hagi.cc.tsukuba.ac.jp> I wrote:
>ãã»ãŸã y=x ã ãããdy=dx ã§ããã
>ãã# ã¡ãã£ãšåé¡ã¯ãããããŸããããæ£ããã
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dx ãšã¯äœãªã®ããããããå®çŸ©ãããŠãããäžæã ããã
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>
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ããããããš dx=Îx ãšã®éããåé¡ãªã®ã§ãã£ãŠã
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æ
ïŒé«æšè²æ²»(å士)ãšã Cauchy ãšã(?)ã¯ãyïŒxïŒc ãšããç¹æ®ãªå Žå
ã«ã ãæãç«ã€ dxïŒâ³x ãšããé¢ä¿ããäžè¬ã®å Žåã«é©çšããŠã·ããã¿
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埡倧ã£ãŠfj.sci.mathã§ã¯ããã眵åããŸãã£ãŠããã©ãsci.mathã§ã¯
ç«ããã¶ã£ãããã«äœå§¿å¢ã ãïœ
埡倧ã¯sci.mathã§ã±ã³ããªããžè¬æŒãã²ãé ãã«ããŠããããã ãã©ã
ã±ã³ããªããžè¬æŒãåŒãåãã«åºãããææé£äžãé»ãããããªãïŒïœ
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"M_SHIRAISHI" <eu...@apionet.or.jp> wrote in message
news:800c7853.0404...@posting.google.com...
> hir...@ulis.ac.jp (Yuzuru Hiraga) wrote in message news:<c5h4vc$po2$1...@hagi.cc.tsukuba.ac.jp>...
> > In article <800c7853.04041...@posting.google.com>
eu...@apionet.or.jp
(M_SHIRAISHI) writes:
> > > >ã解çã
> > >
> > > ãã£ã±ããšãã§ããªãæ··ä¹±ããŠãã¿ããã§ããã
> >
> > βακαΌÏÎœïŒ
> >
> > ãšãã§ããªãæ··ä¹±ããŠå±
ãã®ã¯è²Žæ§ã®ã»ãã ãïŒçç¬ïŒå²ç¬
>
> 埡倧ã£ãŠfj.sci.mathã§ã¯ããã眵åããŸãã£ãŠããã©ãsci.mathã§ã¯
> ç«ããã¶ã£ãããã«äœå§¿å¢ã ãïœ
>
> 埡倧ã¯sci.mathã§ã±ã³ããªããžè¬æŒãã²ãé ãã«ããŠããããã ãã©ã
> ã±ã³ããªããžè¬æŒãåŒãåãã«åºãããææé£äžãé»ãããããªãïŒïœ
â埡倧âã£ãŠã®ã¯ããã£ããã誰ã®ããšãããã¹ã³ãŽããã ãœ(ã)ã
"M_SHIRAISHI" <eu...@apionet.or.jp> wrote in message
news:800c7853.0404...@posting.google.com...
> hir...@ulis.ac.jp (Yuzuru Hiraga) wrote in message news:<c5h4vc$po2$1...@hagi.cc.tsukuba.ac.jp>...
> > In article <800c7853.04041...@posting.google.com>
eu...@apionet.or.jp
(M_SHIRAISHI) writes:
> > > >ã解çã
> > >
> > > ãã£ã±ããšãã§ããªãæ··ä¹±ããŠãã¿ããã§ããã
> >
> > βακαΌÏÎœïŒ
> >
> > ãšãã§ããªãæ··ä¹±ããŠå±
ãã®ã¯è²Žæ§ã®ã»ãã ãïŒçç¬ïŒå²ç¬
>
eu...@apionet.or.jp (M_SHIRAISHI) writes:
> æ
ïŒé«æšè²æ²»(å士)ãšã Cauchy ãšã(?)ã¯ãyïŒxïŒc ãšããç¹æ®ãªå Žå
> ã«ã ãæãç«ã€ dxïŒâ³x ãšããé¢ä¿ãã
dx = â³x ã®æç«ã« y ãªããé¢ä¿ãªãã£ãŠã°ã
å¡æã©ãã ---- ãœããã¡ã¯ãæªã ãäžåã«ãæãã¬ãïŒããœ(ã)ã
# ãŸããCauchy ãšã(?)ãé«æšè²æ²»(å士)ãšã ---- ãšèšãããã圌ã以éã®
æ®ãã©ãã¹ãŠã®æ°åŠè
ãã¡ ---- ãèŠèœãšããŠãããã¹ãªã®ã ããããœããã¡
ã®åŠã*å¡æ*ããã¹ã«æ°ãä»ãã¬ã®ã¯ãç¡çããã¬ãšããã§ã¯ããããã»ã»ã»ã
ã§ããœããã¡ã¯ããæçåœæ°ïŒf(x)ïŒx ããâxïŒxâãšããæçåŒã§è¡šããããã
ãšïŒæãã«ãïŒïŒäž»åŒµããŠãä»åŸããºãŒããšãäžéã®ç¬ããã®ã«ãªã£ãŠå±
ãç¶ã
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## ãããïœãããã§ãã²ãšã€ãâxïŒxâã®ã°ã©ãã§ãæžããŠã¿ããïŒïŒçç¬ïŒå²ç¬
âdyïŒïŒf'(x)ã»â³xâãªããå®çŸ©ãã¯ãç¬ç«å€æ°ã x ã§è¡šãããåŸå±å€æ°ã y
ã§è¡šãããšããåæã«ç«èãããã®ã§ãããšããããšãããœããã¡ã«ã¯ãäžåã«ã
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In article <68fzb6h...@somwinh.msi.co.jp>, NISHIZAWA Yutaka <yut...@msi.co.jp> writes
> å¥ã«ã©ãæã蟌ãã§ãåæã ãã©ãããããããšã«ããŠããªã«ãããäºãããã®
> ããªããããåéã®å€§å®¶ã倧埡æã®ãééãããææããŠæŠã«å
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> ã¯ééã£ãŠãããã·ãªãŒãºãšäŒŒããããªããã§ã¯ãããŸãããã©âŠã
ããã£ãŠæ¬åœã«ééããªã®ããªãã
ãã¡ãããæ®éã®å®çŸ©ãç¥ã£ãŠããèªãã ã®ã§ãå€ãã£ãŠããªãã
ããããæããªãã£ããã©ã
ã§ãã埮åã§éèŠãªã®ã¯ãé¢æ°ã®åç¹ã«ç·åååã察å¿ããããšãª
ãã§ãããããã®å¯Ÿå¿ãã
lim
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ãšã
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ãšãã§åãã®ã¯ãããã ãã©ãããèªäœã¯å®ã¯ãããŸãéèŠã§ã¯ãªããŠ
éèŠãªã®ã¯....
ã£ãŠãªè©±ãªããããªãããšã
---
Shinji KONO @ Information Engineering, University of the Ryukyus
æ²³éçæ²» @ çç倧åŠå·¥åŠéšæ
å ±å·¥åŠç§
>
> dyïŒïŒf'(x)ã»â³x ãšå®çŸ©ããŠããã®ã§ãyïŒf(x)ïŒx ãšãããç¹å¥ãªå Žåãã
ãšã
> f'(x)ïŒ(x)'ïŒïŒã§ããããã€ïŒyïŒx ãªã®ã ããïŒdyïŒdx ãæç«ããŸãã
>
ã¡ãã£ãšããåãããªãã®ã§ããïŒ
dyïŒïŒf'(x)ã»â³x ãšå®çŸ©ããŠãããšããã°ïŒ
yïŒf(x)ïŒx ã®å Žåã«ã¯ïŒå®çŸ©ããã¯
dyïŒ(x)'ã»â³x ïŒâ³x -------(1)
ãšãªãã¯åããã®ã§ããïŒãããã ïŒyïŒx ãªã®ã ããïŒdyïŒdx ãæç«ãšããã®ã
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yïŒx ããèšããããªã®ã¯
â³y ïŒâ³x --------(2) ã ãã®ãããªæ°ãããŸããã
㧠yïŒx ãªã®ã§ xïŒy ãšèŠãã°å®çŸ©ãã
dxïŒ(y)'ã»â³y ïŒâ³y -------(3)
(2)ãš(1)(3)ãããdyïŒdx
> åŸã£ãŠããã®ãç¹å¥ãªå Žåãã«éããdxïŒdyïŒ(x)'ã»â³x ïŒâ³x ãšãªããŸãã
> ãã£ãŠãïŒãã®ãç¹å¥ãªå Žåãã«éãïŒïŒdxïŒâ³x ãæç«ããã£ãŠã¯ã±ã§ãã
㧠yïŒf(x) ã«ããã x èªèº«ã¯ãã®ç¹å¥ãªå Žåã«ããããããªæ°ãããã®ã§ããã
ïŒxãããããã«å€åãããšã ïŒyïŒf(x)ã«ãããïŒyã«é¢ä¿ãªãïŒèªåèªèº«ãšåãã«
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çŽ äººèããªã®ã§ãºã¬ãŠãããç¥ããŸãããã
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> å ããã©ãã°ãx ããèªèº«ã x ã®åœæ°ããããããªãã£ãã
ãšããã®æ§ã«âæºå
ïŒïŒäžâã¯æã£ãŠããã®ã§ãã£ããããœ(ã)ã
# ãx ããèªèº«ã x ã®åœæ°ããªãè¡šçŸã¯ã(çŸä»£ç芳ç¹ããããã°)é©åã§ã¯
ãªãããx ããèªèº«ã x ã®åã§ãããããªåœæ°ããšèšãæããã¹ãã§ããã
> # ããã¯ç¢ºãã«å°ããããã«ãããšããã§ã¯ããã
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ïŒïŒäžâã¯æã£ãŠããã®ã§ããã®çºã«ãã解ææŠè«ãã®
該åœç®æã«ãã¹ãããããšã«å
šãæ°ãã€ããªãã£ãã®ã§ãã£ãããœ(ã)ã
# äœãããâæºå
ïŒïŒäžâã¯ãæçåœæ°ïŒf(x)ïŒx ã "xïŒx" ãªããæçåŒãã§
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## "xïŒx" ã®ã°ã©ããïŒããããããªãã®ãæžãããã®ãªãïœïŒæžããŠã¿ãŠ
ãããªãïŒããœ(ã)ã
M_SHIRAISHI wrote:
> dyïŒïŒf'(x)ã»â³x ãšå®çŸ©ããŠããã®ã§ãyïŒf(x)ïŒx ãšãããç¹å¥ãªå Žåãã ãšã
> f'(x)ïŒ(x)'ïŒïŒã§ããããã€ïŒyïŒx ãªã®ã ããïŒdyïŒdx ãæç«ããŸãã
ããã¯ãã¯ããã¡ã§ããã
ç¹°ãè¿ãã«ãªããŸãããdx ãæªå®çŸ©ã§ãã
代æ°ç圢åŒäž»çŸ©ã®ç«å Žããšã£ãŠïŒæ£ç¢ºã«ã¯æ¿«çšããŠïŒããy=x ã ãã dy=dxã
ãšããã®ã¯é»èªã§ããªãã§ã¯ãªãã§ããïŒãããæåã«è¿°ã¹ããã®ïŒã
ããã§ã奜æçã«èšã£ãŠå®çŸ©ãæ¡åŒµãããŠããŸã£ãŠããŸãã
ãšãããäžæ¹ã§ãæ¬äººã¯ïŒ
In <800c7853.04040...@posting.google.com>(7 Apr 2004 02:46:46 -0700)
> å®çŸ©ã¯ãdy:ïŒf'(x)ã»â³x ã ãã ã
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å
žåçãªèªçžèªçžã§ãããåç¬ãªããŸã ãç®ããŒãã«é ããããããããªã
ãšããããåããæã§äžæ¬ãšãããšããã§ãã
ãããç¹°ãè¿ãã§ãããäžã®å®çŸ©ã® dy ã§ã¯ y ã¯åŸå±å€æ°ã§ãã
ã ããç¬ç«å€æ° x ã«å¯Ÿãã dx ã¯å®çŸ©ãããŠããŸããã
ãããã ãªã€ãªåŸå±å€æ°ãšèŠãªãããšããã®ãã解ææŠè«ãã®è¶£æšã§ã
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ããããã ãªã€ãªãã§ããããšã¯ãç¶ãéšåã«ãã
ããx ãç¬ç«å€æ°ã§ãããšãã«ã¯ãäžèšã® dx=Îx ãšããããšã¯ã
ããããŸãã«çŽ°å·¥ãéããããã§ããã...ã
ãšããé«æšèªèº«ã®èšèã«ãè¡šããŠããŸãã
ã# ãããåœç¶ãªããããã ãªã€ãªããšãééã£ãŠãããã¯å¥åé¡ã§ãã
ãã㊠M_SHIRAISHI ããã¯äžè²«ããŠãããç解ã§ããŠããŸããããã
y = x ãªããŠãã®ãæã¡åºãããããããã§ãã
x ãç¬ç«å€æ°ã®ãŸãŸã«ããŠãããŠã¯èº«åãããšããŸããã®ã§ã
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kounoike ããïŒéŽ»æ± ããã§ããã®ããªïŒïŒãçåã«æãããã®ãåœç¶ã®ããšã§ãã
> åŸã£ãŠããã®ãç¹å¥ãªå Žåãã«éããdxïŒdyïŒ(x)'ã»â³x ïŒâ³x ãšãªããŸãã
> ãã£ãŠãïŒãã®ãç¹å¥ãªå Žåãã«éãïŒïŒdxïŒâ³x ãæç«ããã£ãŠã¯ã±ã§ãã
ãããã£ãŠããã¯è°è«ãå®å
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äžã¯ y = x ã®ãšãïŒ
ããdy = Îx ïŒããã¯å®çŸ©éãïŒ
ããdy = dx ïŒM_SHIRAISHI æµå€©äžãïŒ
ãã dx=Îx ãçµè«ãšããŠå°ããŠãŸãããåæãšçµè«ãéã§ïŒ
ããdy = Îx
ããdx = ÎxãïŒãããå®çŸ©ããå°ããïŒ
ãã dy=dx ãçµè«ãããã®ã§ãã
ãã£ãšãäžè¬ã®å Žåã®
ããdy = f'(x) dx
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ãããèšããç§ã®æïŒ
In <c5h4vc$po2$1...@hagi.cc.tsukuba.ac.jp>(Tue, 13 Apr 2004 16:37:00 +0000)
> ãx ããèªèº«ãx ã®åœæ°ãšã¿ãã°ããšããã®ã¯ããããŠæžãã° x=x ãšãªããã©ã
> ããã§ã¯æçåŒãšåºå¥ãã€ããªãã®ã§è£å©å€æ°ãå°å
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> ããx = x(t) = t
> ããy = f(x) = f(t)
> ãšæžããŸãããã
> ãããšãã ãã®å®çŸ©ãã
> ããdx = x'(t)Ît = Ît
> ããdy = f'(t)Ît
> äžãäžã«ä»£å
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> ããdy = f'(t)dx = f'(x)dx
> ãšããã ãã®ããšã§ããã
ã« M_SHIRAISHI ããã¯ãªããã²ãããèšåãé¿ããŠããŸããã
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ãM_SHIRAISHI ã®ææ«éèšäºãããïŒ
In <800c7853.0404...@posting.google.com>(14 Apr 2004 11:40:06 -0700)
> ã§ããœããã¡ã¯ããæçåœæ°ïŒf(x)ïŒx ããâxïŒxâãšããæçåŒã§è¡šããããã
> ãšïŒæãã«ãïŒïŒäž»åŒµããŠã
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In <800c7853.0404...@posting.google.com>(14 Apr 2004 00:35:57 -0700)
> hir...@ulis.ac.jp (Yuzuru Hiraga) stupidly wrote in message
news:<c5h4vc$po2$1...@hagi.cc.tsukuba.ac.jp>...
>> ãx ããèªèº«ãx ã®åœæ°ãšã¿ãã°ããšããã®ã¯ããããŠæžãã° x=x ãšãªããã©
>
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> y=x ã®å Žåã« f'(x)=1 ãšãªãã®ã®ã©ããäžåçãªãã§ããããïŒ
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yïŒf(x) ãïŒ(埮åå¯èœãª)ä»»æã®åœæ°ïŒã§ããã®ã«ãf'(x)=1 ã§ããããç¡ãã§ãããã
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eu...@apionet.or.jpããã¯æžããŸããã
eurms> ãããŠãf(x) ã xïŒc ã§ããæ§ãªãïŒç¹æ®ãªåœæ°ïŒãã§ã¯ãªããäžè¬ã®å Žåã ãšã
eurms> f'(x)ïŒïŒ ãšã¯éããªãã®ã§ãdyïŒf'(x)ã»â³x ïŒ â³x ãèšããªãã®ã§ãdxïŒâ³x ãšã¯
eurms> èšããªãã
ãã¡ãã y = f(x) ãäžè¬ã®é¢æ°ã®å Žåã«ã¯ f'(x) ãšã¯éããªãã®ã§
dy = Îx ã¯ãããŸãã. ãšã¯ãã, ãã®å Žå dy = dx ããããªãã¯ã
ãªã®ã§ãdx = Îx ãšã¯ãããªãããšããã®ã¯ãdx = Îx ãã©ããã¯ã
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eurms> å³ã¡ãdxïŒâ³xããæç«ããã®ã¯ãyïŒxïŒc ã§ããæ§ãªãïŒç¹æ®ãªåœæ°ïŒã®å Žåãã«éã£ãŠ
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# ã©ãã ã"ãã£ã¡(sci.math)"ã«è¡ã£ãŠãæçš¿ããŠããããïŒ
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## äžçäžã®ç¬ããã®ã«ãªããã ãœ(ã)ã
ããã last straw ã§ãããããã«ãã£ãŠ M_SHIRAISHI æ°ãèªåã§
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å
šç¶éããªãïŒç解ã§ããªãïŒã®ã§ã¯åŠäœãšãããããã
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ã# èªå·±ççŸã£ãã£ãŠããŸããã§ããã
======
y = 3x ãšããŸãããããããš
ãdy = d(3x) = 3dx
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ãdy = 3Îx
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ã3dx = 3Îxãã€ãŸã dx = Îx
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ãã¡ããå®éã«ã¯ y=3x ã®å Žåã§ãªããä»»æã®ïŒæ¬¡é¢æ° y = ax+b ã§ã
åãã ããããã«ã¯ä»»æã® y=f(x) ã«å¯ŸããŠã dx = Îx ã¯æãç«ã€ã
ïŒãšãããããããã dx=Îx ãæãç«ã€ããšã« f(x) ã¯ç¡é¢ä¿ãïŒ
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z=g(x)=x ãšãããš
dzïŒdxïŒg'(x)ã»â³xïŒ1ã»â³xïŒâ³x ãã£ãŠ dxïŒâ³x.
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dyïŒf'(x)ã»dx.
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ã»ãã£ïŒãhttp://www.age.ne.jp/x/eurms/Quiz_06iv2004-2.html
# ãŸããäžå¿ã®æŽåæ§ã¯ããããã ããLeibniz ã Euler ã®èãã
dx, dy ã«ç¡éå°ã®éã ã£ãã®ã«ãdyïŒf'(x)ã»â³x ãšãå®çŸ©ãããŠ
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# ïœè»žãè¡šããæ¹çšåŒã¯ yïŒ0 ã ã£ãŠããšããããäžåŠçã§ãç¥ã£ãŠããã¯ãïŒçç¬ïŒå²ç¬
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ãdxïŒâ³xããæç«ããã®ã¯ãïŒyïŒx ã ãã§ã¯ãªããŠïŒyïŒxïŒc ã§ããæ§ãªãïŒç¹æ®
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ããããããã«äž»åŒµããŠãããã®æžãæãã§ãã倧å€å€±ç€ŒèŽããŸããã
M_SHIRAISHI wrote:
> ãæçåœæ°ãæçåŒ xïŒx ã§è¡šããããŠããã®ã°ã©ããïœè»žèªäœã ã
> ãªããŠäž»åŒµããŠãæãã£ããæ¥ãæããŠãããœãã®æ§ãªã¢ãã®çºã«ã
...
ãªããæåŸããããããããšããããªããªã£ã¡ãã£ãã¿ããã ãã©ã
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ïŒM_SHIRAISHI ãããé€ããŠã¯ïŒïŒäººãããŸããã
ã ãããã㯠M_SHIRAISHI ããã«ããæªè³ªãªæé ã§ããããã ãã
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ããã³ãïŒãy = 1, å°ãæžãæã㊠y-1 = 0 ã¯äœãæå³ããã§ãããïŒ
ä»æŽãã·ã©ã¯åãããªããœ(ã)ã
ïŒã解ææŠè«ãã§ã¯ãx'=1 ã ããããšæžãããŠãããã§ããã
ïŒããããå¯ããã«ã
ïŒãx = x âãdx = x' Îx â dx = Îx
ïŒãšåãåãã®ãäžçªçŽ çŽãããªãã®ããªãã
ãšããã±ãªããšãæžããã®ã¯ãYuzuruã貎æ§ã ãããªã
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ãšããã±ãªããšãæžããã®ããYuzuruã貎æ§ã ãããªããœ(ã)ã
ãŸããŸãããããªããŠãã¿ãŸãããããã®ã©ãããããªãã®ã§ããããïŒ
y_2 = x_1 â dy_2 =(x_1)'ã»Îx_1
ã OK ãªãã§ãããã
ãªã x = x ã¯åžžã«æãç«ã€ã®ã§ïŒ
dx =x'ã»Îx
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ãªããåéãããŠããã®ã§ãããã(..)ïŒ
In <800c7853.04042...@posting.google.com> M_SHIRAISHI wrote:
> # ãŸããäžå¿ã®æŽåæ§ã¯ããããã ãã...
ãªãã ããã㯠M_SHIRAISHI ããã®æå宣èšã ã£ãã®ã§ããã
çŽ çŽãããªããªãã
> ã»ãã£ïŒãhttp://www.age.ne.jp/x/eurms/Quiz_06iv2004-2.html
ã§æ¹ããŠäžãèŠããšãäœã®ããšã¯ãªããïŒé±éãåã«æããŠãããããšïŒ
In <c5j90a$cue$1...@hagi.cc.tsukuba.ac.jp> (Wed, 14 Apr 2004 11:58:02 +0000) I wrote:
> ...
> å€æ°åã y ãã z ã«å€ããŠã
> ãz = x, dz = Îx, dz = dx â dx = Îx
> ãšããã£ãŠå
šãåãããšã
> ...
> ãããŠç¬¬ïŒã®ãã¹ããããèŽåœçãªã®ã§ããã
> y = x ãªã©ãšæžãããããåºçºç¹ã® y = f(x) ã® y ãšæ··åããŠããŸãã
> dy = dx = Îx ã® dy ãš dy = f'(x)Îx ã® dy ãåããã®ã ãšæã£ãŠããŸã£ãã
> äžèšã®ããã«ãy ã§ã¯ãªã z ã䜿ã£ãŠããã°ãããªæ··åã¯ããªãã£ãã§ãããã
ãããã¯æšæ¥ Y.N. ãããæžããããšïŒ
ãããå®è³ªçã«æžãåããã ãã®ããšãããªãã
ãâå¡©âãéããããªã«ããæé ã ãã§ãªãåœçªãŸã§ãããã§ããïŒ
ãããããã§ã¯èšŒæãšããŠäžå®å
šãšãŸã§æããŠãããã§ããïŒ
=====
ã§åé ã®ç¶ããèŠããšïŒ
> # ãŸããäžå¿ã®æŽåæ§ã¯ããããã ããLeibniz ã Euler ã®èãã
> dx, dy ã«ç¡éå°ã®éã ã£ãã®ã«ãdyïŒf'(x)ã»â³x ãšãå®çŸ©ãããŠ
> ã·ããã¿ã®ã§ã¯ãç¡éå°ãšã¯å
šãäœã®é¢ä¿ãç¡ããããã©ãæ¹äŸ¿ãšããŠã
> dyïŒf'(x)dx ãåŸãããã«éããªãã (ãÐŽã)
ç¡éå°ãšããéãéã¯èªåèªèº«ã§æœ°ãã¡ãã£ãããšãŸã§å¿ããã®ïŒ
ããããäžå¯§ã«ïŒåºŠãŸã§ãã
I wrote:
> ãŸãæ³åããã«ããå€æ° x ãé¢æ°ã®ããã«èšãã®ã¯ãããããããšãã
> ãç¡éå°ã§ãã dx ãæééã®Îx ãšçå€ã§ãããããªããããããã¯éã«
> ãÎxâ0 ã§ã¯ã©ããªããã ããšããã£ããšããã§
> ãæ°ã«å
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æºè§£æãåŒã£åŒµãåºãããããããªãããšã
ããã«å¯ŸããŠïŒ
Message-ID: <800c7853.04040...@posting.google.com>
> å
šéšãããºã¬ãããœ(^ã^)ã
Message-ID: <800c7853.04040...@posting.google.com>
> åŸã£ãŠãåœç¶ãªããã
...
> ãªããŠããšã¯ãå
šãã®âªèŠåœéãâ«ãšããããšã«ãªãã
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======
ã€ãã§ã«éŽ»æ± ããã®æžãããïŒ
In <c659bl$32j$1...@caraway.media.kyoto-u.ac.jp> keizi kounoike <koun...@mbh.nifty.com> wrote:
> ãããããªã蚌æãããããããŒã ããŒãžã«èŒãããããããã®ãªãïŒ
> M_SHIRAISHI[ãã]ããããã«äž»åŒµããŠãã
>
> ãdxïŒâ³xããæç«ããã®ã¯ãïŒyïŒx ã ãã§ã¯ãªããŠïŒyïŒxïŒc ã§ããæ§ãªãïŒç¹æ®
> ãªåœæ°ïŒã®å Žåãã«éã£ãŠã®ããšã§ããã
>
> ã«å¯ŸããŠïŒå¹³è³ãããæãã£ããééã£ãŠããŸãããšç€ºããå蚌ã«å¯Ÿããå蚌ïŒãã
> ã®ãªãïŒã瀺ãã¹ãã§ã¯ãªãã§ããããã
ããã¯ããããŸã éãéïŒãšæ¬äººã ããæããã®ïŒã¯ãããã§ããã
ãy=x ã ãã§ãªããy=x+c ã«èšæ£ããšããã®ãšåæ§ã«ã
ãy=x+c ã ãã§ãªããy=ax+b ã«èšæ£ãããšãããã
ãã¡ããããã¯ãã®å Žéãã«éããªããŠããããã x+c ãšããæç¹ã§
ããšããšã®è«ç¹ã厩å£ãax+b ã§ã¯å®å
šçŠè§£ã
ãŸã ax+b ãŸã§è¡ã£ã¡ãããšã埮åãã£ãŠããã ãããäžè¬ã®å Žåã«ã
dx=Îx ã§ããããšã¯ããæãå±ããã§ããããã
ãšæã£ãŠãããªããšïŒ
ãhttp://www.age.ne.jp/x/eurms/Quiz_06iv2004.html
ãããæå x+c ãšãªã£ãŠããã®ããã£ãã ax+c ã«å€æŽãããŠãŸããã
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ax+b ãããªã ax+c ã«ãªã£ãŠãã®ãã尻尟ãèŠããŠãããã§ãããããã
ã ãã©çžå€ãããäžè¬ã®å Žåã«ãæãç«ã€ããšãããã£ãŠãªãã®ã¯ïŒ
ãhttp://www.age.ne.jp/x/eurms/NB.html#00
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ã€ãã§ã ãã©ïŒ
ãhttp://www.age.ne.jp/x/eurms/A_Paradox_of_The_Central_Limit_Theorem.html
ãã®äžã®ã¹ãã«ãã¹ïŒTenorem, satates, numbetrs, infinitively) ã
çŽããŠããã°ããã®ã«ãã
sci.math ã®ãã¡ãã®ã¹ã¬ãã㯠4/7 ã§é絶ããïŒå°»å°Ÿå·»ããŠéããïŒã¿ããã§ããã
ïŒå¹³è³ïŒ
M_SHIRAISHI wrote:
> Yuzuru Hiraga <hir...@slis.tsukuba.ac.jp> wrote in message news:<40863398...@slis.tsukuba.ac.jp>...
>>M_SHIRAISHI wrote:
>>>ãæçåœæ°ãæçåŒ xïŒx ã§è¡šããããŠããã®ã°ã©ããïœè»žèªäœã ã
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I wrote:
> sci.math ã®ãã¡ãã®ã¹ã¬ãã㯠4/7 ã§é絶ããïŒå°»å°Ÿå·»ããŠéããïŒã¿ããã§ããã
ããæžããšã決ãŸãã®ãéããã®ã¯ Robin Chapman ã®ã»ãã ããšãã
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æçåŒâx = xâã¯ãããªãåœæ°ãè¡šãããŠã¯ããªãã®ã«ãããã
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M> ïŒãx = x âãdx = x' Îx â dx = Îx
ãšããè¡šçŸã«ãããŠãïŒã"x = x" ãé¢æ°ãè¡šããŠããããšã¯èªããªããšæãã®
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eu...@apionet.or.jp (M_SHIRAISHI) wrote in message news:<800c7853.04042...@posting.google.com>...
ãã£ãšâªèŽåœçâ«ãªã®ã¯ã埮å(differential) dy ã f'(x)ã»â³x ãš
ãå®çŸ©ãããŠã·ããã¿ã®ã§ã¯ãé«éã®åŸ®å d^2y, d^3y, d^2x etc. ã
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M_SHIRAISHI wrote:
> ãã£ãšâªèŽåœçâ«ãªã®ã¯ã埮å(differential) dy ã f'(x)ã»â³x ãš
> ãå®çŸ©ãããŠã·ããã¿ã®ã§ã¯ãé«éã®åŸ®å d^2y, d^3y, d^2x etc. ã
> å®çŸ©ã§ããªããªã£ãŠããŸããã
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ã解ææŠè«ãã® p.51 ãèŠãŠããããã
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> ã(dy)^2 ã¯é«äœã®ç¡éå°ãªã®ã§ãç¡èŠ
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ãããæã¡åºãã解çãããšãã誀çãšã¯ç¡é¢ä¿ã
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ã# p.37 ã ãšãŸã åæ埮åã®å
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dy ã f'(x)ã»â³x ãšå®çŸ©ããå Žåã«ã¯ãdy 㯠y ã x ã®ã¿ã®
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Stupid_Yuzuru repetedly and stupidly wrote:
>
>> ããŒããããŸã説æã§ã¯ãªããªãã
>> èŠããã« y=x ãš x=x ã®éãããšèšãã°ããã®ããªãã
>
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ç¬ãããããããªããããããã«ã¿ã¬ãïŒïŒïŒ
"xïŒx" ã«éãããæçåŒã«ã¯ã°ã©ããªã©ååšãã¬ã¯ïŒ
æçåŒã¯ãã¹ãŠã"0ïŒ0" ãšããçåŒãšåå€ã ããã ã
> ïŒé埮åã§æãåºãããã©ãæŒç¿åé¡ãšããŠã®ïŒ
>
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>
> ãšããã®ã¯æ¡å€é£ãããäœãããŸãããæ¹ãªãããªã
ãããªåé¡ã¯ãé äœãç¡ãããšã ïŒïŒ
lim_{hâ0}[{f(x+h)ïŒf(x-h)ïŒ2f(x)}ïŒh^2]
ïŒlim_{hâ0}[{f(x+h)ïŒf(x)}ïŒh ïŒ {f(x)ïŒf(x-h)}ïŒh]ïŒh
ïŒlim_{hâ0}[f'(x)ïŒf'(x-h)]ïŒh
ïŒlim_{hâ0}[f''(x-h)]
ïŒf''(x)
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M_SHIRAISHI wrote:
>>ïŒé埮åã§æãåºãããã©ãæŒç¿åé¡ãšããŠã®ïŒ
>>
>>ããlim_{hâ0} {f(x+h)+f(x-h)-2f(x)}/h^2
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>
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>
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> ïŒlim_{hâ0}[{f(x+h)ïŒf(x)}ïŒh ïŒ {f(x)ïŒf(x-h)}ïŒh]ïŒh
> ïŒlim_{hâ0}[f'(x)ïŒf'(x-h)]ïŒh
> ïŒlim_{hâ0}[f''(x-h)]
> ïŒf''(x)
>
> # ãšæžãããã©ãééã£ãŠããããŠãªãããœ(ã)ã
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ã(f(x+h)-f(x))/h = f'(x+ah) (0âŠaâŠ1)
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M_SHIRAISHI wrote:
> åæã¯ééã£ãŠããŠãããã®åž°çµã¯æ£ããããšãåšãåŸããš
> ããè«çæ³åã®ã¿ããšãªäŸãšèšãããã ãœ(ã)ã
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ã# é²æ£é§æ¶ããŠããŸããã ãã©ãã
In article <408DF9B1...@slis.tsukuba.ac.jp>
Yuzuru Hiraga <hir...@slis.tsukuba.ac.jp> writes:
> äžã®åé¡ã¯ãïŒæ¬¡ã®ããŒã©ãŒå±éã䜿ãã®ãæãæ£çµ±çãªè§£çãš
> æããŸãã
ããããª.
> å¥è§£ãšããŠããã¿ã«ã®å®çã䜿ããšããã®ããã£ãŠã
> ããã§ç¢ºãã«ã§ããã®ã ãã©ãäœãã£ãŠãã®ããèŠãã«ãããŠã
> ã¡ãã£ãšéªéã£ãœãæããããŠããŸãã
é¢æ°ã埮åå¯èœã§, ãã®å°é¢æ°ãäžç¹ x ã§åŸ®åå¯èœ, ãšããæ¡ä»¶ã§
蚌æã§ããŸãã.
--
å¡æ¬åç§@å¿çšæ°åŠ.é«åååŠç§.ç¹ç¶åŠéš.京éœå·¥èžç¹ç¶å€§åŠ
Tsukamoto, C. : chi...@kit.ac.jp
<408DF9B1...@slis.tsukuba.ac.jp>ã®èšäºã«ãããŠ
hir...@slis.tsukuba.ac.jpããã¯æžããŸããã
hiraga> ïŒæ¬¡ãŸã§ããšããããå¹³åå€ã®å®çãŸã§ã§åŠçã§ããããšèšãã°ïŒ
hiraga> ã(f(x+h)-f(x))/h = f'(x+ah) (0âŠaâŠ1)
hiraga> ã(f(x)-f(x-h))/h = f'(x-bh) (0âŠbâŠ1)
hiraga> ãšãªã£ãŠ
hiraga> ã(f(x+ah)-f(x-bh))/h
hiraga> ãŸã§ã¯ããã®ã§ããããã®ãŸãŸã§ã¯ a ãš b ã®é¢ä¿ãèŠããªããã
hiraga> å
ã«é²ããªãããããæ±ãããšããã®ã¯å®è³ªçã«ïŒæ¬¡ã®å±éãŸã§
hiraga> èŠãããšã«ãªã£ãŠããŸãã®ããªããããšãå¥æ³ãããã®ãããã
ã³ãŒã·ãŒã®å¹³åå€ã®å®ç:
F(x), G(x) ãåºé [a, b] ã§åŸ®åå¯èœãªãã°
[F(b) - F(a)] / [G(b) - G(a)] = F'(c) / G'(c)
ãæºãã c â (a, b) ãååšãã
ã«ãã㊠F(h) = f(x+h) + f(x-h) - 2f(x), G(h) = h^2 ãšããã°ã§ã
ããããªæ°ãããŸã.
--
åå€å±å€§åŠå€§åŠé¢ æ
å ±ç§åŠç ç©¶ç§ èšç®æ©æ°çç§åŠå°æ»
å°é åç·
é·æéèããŠãããããªããŠããã®ãšããå¿ãããŠãfj.sci.math
ãèŠãŠãªãã£ãã®ã ã èŠãã®ã¯ 27iv2004ãã€ãŸããä»æ¥ã ã
> ãšããããæåŸã®ãšããã ãã
>
> M_SHIRAISHI wrote:
> >>ïŒé埮åã§æãåºãããã©ãæŒç¿åé¡ãšããŠã®ïŒ
> >>
> >>ããlim_{hâ0} {f(x+h)+f(x-h)-2f(x)}/h^2
> >>
> >>ãšããã®ã¯æ¡å€é£ãããäœãããŸãããæ¹ãªãããªã
> >
> > ãããªåé¡ã¯ãé äœãç¡ãããšã ïŒïŒ
> >
> > lim_{hâ0}[{f(x+h)ïŒf(x-h)ïŒ2f(x)}ïŒh^2]
> > ïŒlim_{hâ0}[{f(x+h)ïŒf(x)}ïŒh ïŒ {f(x)ïŒf(x-h)}ïŒh]ïŒh
> > ïŒlim_{hâ0}[f'(x)ïŒf'(x-h)]ïŒh
> > ïŒlim_{hâ0}[f''(x-h)]
> > ïŒf''(x)
> >
> > # ãšæžãããã©ãééã£ãŠããããŠãªãããœ(ã)ã
>
> ã¯ããèŠäºã« 0 ç¹ã
> ãŸãçœçŽãæ¬åœã® 0 ç¹ãšããã°ãã¡ãã£ãšãããéšåç¹ã¯
> ã€ãããããã 0 ç¹ã¯ããããããã
>
> çžå€ãããèŠäºãªãŸã§ã« lim ã®æå³ãããã£ãŠãŸãããã
f''(x) ãšããçã¯ãå®ã¯ãlimit ãªããã䜿ã£ãŠåºãããã®
ã§ã¯ãªãã®ã ã limit ã䜿ã£ãŠèšç®ãããã«èŠããããŠããã
ã ãã®ããšã§ãªãããœ(ã)ã
df(x)ïŒdx ã lim_{hâ0}[{f(x+h)ïŒf(x)}ïŒh] ã§å®çŸ©ãã
Cauchyã®æµåã¯ãdf(x) ã dx èªäœã«æå³ãç¡ããªã£ãŠããŸã
ãšããæå³ã§ãéåžžã«ããºã€ãããããªã®ã ã
å æ¥ããã§ãã£ãããã«ãdy ã dx ã¯ç¡éå°ã®éãªã®ã ã
ã©ããã£ãããdy ã dx ã®æ¬æ¥ã®æå³ã埩掻ãããããšã
ã§ãããïŒ
# Robinson ã® Non-Standard Analysis ãªã©ãšãããã®ã¯ã
æã®éªšé ã§ããããããªãã®ã«å°æ¥æ§ã¯ç¡ãã
æ£ããã©ããã¯åãããŸãããïŒã¡ãã£ãšèããŠã¿ãŸããã
(1)ãåæãªã®ã§ïŒãããªãã§ããã®ããªãš(ïŒ)
ãŸãïŒ(1)â(1')ãããã®ãçåã ãã
f(x+Îx) = f'(x)Îx+f(x) ------(1)
f(x+2Îx) = f'(x+Îx)Îx+f(x+Îx) ------(1')
f'(x+Îx) = f''(x)Îx+f'(x) ------(2)
(1')ã®å³èŸºã®f'(x+Îx)ã«(2)ã代å
¥ãïŒ
f(x+2Îx) = (f''(x+Îx)+f'(x))Îx+f(x+Îx)
ããã«(1)ãã f'(x)Îx=f(x+Îx)-f(x)ã§ãããäžåŒã«ä»£å
¥ãæŽçãããš
f(x+2Îx) = f''(x)(Îx)^2+2f(x+Îx)-f(x)
ãããã
f(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2
{f(x+2Îx)+f(x)-2f(x+Îx)}/(Îx)^2=f''(x)
ããã§ïŒX=x+Îxãšãããš
{f(X+Îx)+f(X+Îx)-2f(X)}/(Îx)^2=f''(X-Îx)
Îxâ0ã§
f''(X-Îx)=f''(X)
ãšãªãã
ãããããæ¯æ¥ïŒïœïœã§èŠãèŠãã®èªäœèªæŒãã£ãŠãããã«ïœ
In article <0404271942...@flame.hirata.nuee.nagoya-u.ac.jp>
Takao Ono <ta...@hirata.nuee.nagoya-u.ac.jp> writes:
> ã³ãŒã·ãŒã®å¹³åå€ã®å®ç:
> F(x), G(x) ãåºé [a, b] ã§åŸ®åå¯èœãªãã°
> [F(b) - F(a)] / [G(b) - G(a)] = F'(c) / G'(c)
> ãæºãã c â (a, b) ãååšãã
> ã«ãã㊠F(h) = f(x+h) + f(x-h) - 2f(x), G(h) = h^2 ãšããã°ã§ã
> ããããªæ°ãããŸã.
ããã¿ã«ã®å®çãããã䜿ã£ãŠèšŒæãããŠããããšã¯åŸ¡åç¥ã§ããã?
ãã®æ¹æ³ã ãš f'' ã«ã€ããŠã¯ x ã§ã®ååšã ãããã°ããããã§ããã
ã ããå¡æ¬ãããèšã£ãŠããã®ããããªã®ããªã
ïŒãããã¯ããã§ã¯ãªããããããªããïŒ...(*)
=======
ããŒããå°ã£ããªã
ãããã¿ã«ã®å®çããšèšã£ãã®ã¯ãããã話ã§ãããšèšã£ããæãïŒ
ãŸãæãããããã£ãšãã§ãã
ã ãã©ããé ã話ãšããããã§ããªãã
ããã¿ã«ã¯ã³ãŒã·ãŒã®å¹³åå€ã®å®çã®æ¥µé圢ã§ãããã
ã§ãããã¿ã«ã䜿ã£ãŠïŒ
ãlim {f(x+h)+f(x-h)-2f(x)}/h^2 = lim {f'(x+h)-f'(x-h)}/2h
ããã§ããã¿ã«äžæ¬ã§è¡ããªãããïŒå䜿ã£ãŠïŒ
ã= lim {f''(x+h)+f''(x-h)}/2
ãšãªã£ãŠ x ã®è¿åã§ã® f'' ã®ååšãšããå¿
èŠã«ãªããã©ã
ããã¿ã«ã¯ããã§ãããŠããŸã£ãŠéåžžã®æ¥µéèšç®ã«æ»ãã°ã
å°éããã®èšãïŒ
ãlim {f'(x+c)-f'(x-c)}/2c (hâ0 ã®ãšã câ0)
ãšå€§å·®ãªãããšã«ãªããŸããããã¿ã«ãåãéããã ãã§ã
ã# ã¡ããã©ãã®çŽåã« lim {f(x+2h)-f(x)}/h ãšã lim {f(x)-f(x-h)}/h
ã# ïŒM_SHIRAISHI ããã®ééãã®ïŒã€ïŒãšãã®ç·Žç¿åé¡ãããããã ãã
ãã£ãšãããã¶ãããã¿ã«ã«åŒã£åŒµãããŠã§ãããã
ã³ãŒã·ãŒããçŽæ¥ãšããã®ãèŠèœãšããŠããã®ã¯ç¢ºãã§ãã
ããããšãããããŸãã
ïŒã©ã°ã©ã³ãžã¥ã®ïŒå¹³åå€ã®å®çã ãš f(x+h), f(x-h) ããã©ãã©ãªã®ãã
ãŸãšããŠé¢åã¿ãŠããŸãããã§ããã
======
ãã ããŸã åé¡ã¯ãããŸãã
ïŒã€ã¯èšŒæãšããŠæããããšããç¹ã
äžã§ããã¿ã«ã¯åãéãšæžãããã©ã蚌æéçšã¯ç¡èŠããŠ
å®çã¯æäžãšèããã°ãã³ãŒã·ãŒããã¯ããã¿ã«ã®ã»ããæ±ããããã
ïŒãã£ãšãããã¿ã«ãïŒå䜿ããšïŒåç®ãŸã§çªã£èµ°ãããã ãã©ãïŒ
ãããŠããã¿ã«ãåŒã£ãããã®ã¯ããh ã®é¢æ°ãšããŠåŸ®åããšããã®ã
çºæ³ã®è»¢æã䌎ãç¹ã§ãããã¯ã³ãŒã·ãŒã§ãåãããšã
çºæ³ã®é£ããã ãã§ãªãããïŒéå·®åã®æ¥µéã¯ïŒé埮åãããã£ãš
çŽæ¥çã«èŠããŠã»ããããã§ãã
(*) ãšãããŸã§æžãããå¡æ¬ããã®ïŒ
> ããã¿ã«ã®å®çãããã䜿ã£ãŠèšŒæãããŠããããšã¯åŸ¡åç¥ã§ããã?
ã®èšäºãæ¥ãã®ã§ããã¯ãå¡æ¬ããã¯éãèãã®ããã§ããã
ããŒããèŠããªãã
# ã€ãã§ã«ãïŒæ¬¡ã®ããŒã©ãŒå±éããšããããç¥ã£ãŠããã®ã¯
# ã³ãŒã·ãŒã䜿ãæµåã
======
éŽ»æ± ããã®ã¯ãïŒéå·®åã®æ¥µéã¯ïŒé埮åãã®ç²Ÿç¥ã«æ²¿ã£ãŠããã®ã§ããïŒ
> f(x+Îx) = f'(x)Îx+f(x) ------(1)
> f(x+2Îx) = f'(x+Îx)Îx+f(x+Îx) ------(1')
> f'(x+Îx) = f''(x)Îx+f'(x) ------(2)
ãããã¯ããããæéè¿äŒŒã ããã誀差é
ã䌎ããŸãããã
é ã« e1Îx, e2Îx, e3Îx ãšãããšã
e1, e2, e3 ã¯ãããã Îxâ0 ã®ãšããâ0ã
ãããããšïŒ
> f(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2
ã«ã¯å®éã«ã¯ïŒ
ãf(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2 + (-e1+e2+e3)Îx
ã{f(x+2Îx)+f(x)-2f(x+Îx)}/(Îx)^2 = f''(x)+ (-e1+e2+e3)/Îx
ãšãã誀差é
ãã€ããŠåãã
ãããå§æ«ã§ããªãããããå®éã«ã¯æ¶ãããã§ãããããæ¹ãããããªãã
ïŒå¹³è³ïŒ
In article <c6m0dj$3m4$1...@hagi.cc.tsukuba.ac.jp>
Yuzuru Hiraga <hir...@ulis.ac.jp> writes:
> ãããã¿ã«ã®å®çããšèšã£ãã®ã¯ãããã話ã§ãã
ã ãããšæã£ããã, å°éããã«ãããã followup ããããŸã§.
ãŸãæ®éäžåã ã䜿ããã®ã§ããã.
# lim (1 - cos x)/x^2
# xâ0
# ã§äºå䜿ããããšããã«ãªã.
ããããã
ãã£ãæžãå¿ãããã©ãããã¿ã«ã£ãŠåŸªç°è«æ³ã«ãªãå±éºã
ã€ããŠåãã®ãããŸãã¡ãªçç±ã®ïŒã€ã§ããããŸãã
ïŒå¹³è³ïŒ
ãã£ã±ãããã«è¡ã£ã¡ããïŒéãã¡ããïŒã®ãã
>f''(x) ãšããçã¯ãå®ã¯ãlimit ãªããã䜿ã£ãŠåºãããã®
>ã§ã¯ãªãã®ã ã limit ã䜿ã£ãŠèšç®ãããã«èŠããããŠããã
>ã ãã®ããšã§ãªãããœ(ã)ã
確ãã«ãèŠããããã ãããäžå³ã¯å šãã®ãã¿ã©ã¡ã ããã
ãããããã ãã©ãã©ãã M_SHIRAISHI ããã¯èªåã§æèããªã
ãšããã§ã¯ãããŸãããŠãŒã¢ã¢ãçºæ®ãããšããããã£ãŠïŒ
> ãããªåé¡ã¯ãé äœãç¡ãããšã ïŒïŒ
>
> lim_{hâ0}[{f(x+h)ïŒf(x-h)ïŒ2f(x)}ïŒh^2]
> ïŒlim_{hâ0}[{f(x+h)ïŒf(x)}ïŒh ïŒ {f(x)ïŒf(x-h)}ïŒh]ïŒh
> ïŒlim_{hâ0}[f'(x)ïŒf'(x-h)]ïŒh
> ïŒlim_{hâ0}[f''(x-h)]
> ïŒf''(x)
>
> # ãšæžãããã©ãééã£ãŠããããŠãªãããœ(ã)ã
ããæžããèšäºã®æåã®ã»ãã§ïŒ
> åæã¯ééã£ãŠããŠãããã®åž°çµã¯æ£ããããšãåšãåŸããš
> ããè«çæ³åã®ã¿ããšãªäŸãšèšãããã ãœ(ã)ã
ãšããã®ã¯ãèŠäºã«äžå·¡ããŠèªå·±å®çµããŠãŸããã
ãã£ãšããèžè¡æ§ããšããç¹ã§ã¯ãã®åæžãã â«dx/(x+a) ã«ã¯å£ãããªã
ééãæ¹ã®ãã©ãšãã£ã足ããªãã
ã¿ã¯ã±ïŒããªããšãããããŸãããããœ(ã)ã
ããããããããããåŠæ¥å°é
ããããããããããæºæ
§æ·±é
ããããããããããä¹
é»æ¯èŠ
ããããããããããäžåé説
> > dy ã f'(x)ã»â³x ãšå®çŸ©ããå Žåã«ã¯ãdy 㯠y ã x ã®ã¿ã®
> > åœæ°ã§ãããšãã«ããæå³ãæããªãã ãããã«ããã®å Žåã
> > dy 㯠x ãš x_1 ïŒäœããâ³xïŒx_1ïŒx ãšããïŒãšã®åœæ°ã§
> > ãã£ãŠãx ã®ã¿ã®åœæ°ã§ã¯ãªãã®ã ãããd(dy) ã¯æå³ãæã
> > ãªãã ã€ãŸãã(d^2)y ã¯æå³ãæããªããªã£ãŠããŸãã
>
> ãªããããããããããã£ã¡ãã£ãŠãã¿ããã§ããã
貎æ§ããæå³ãæŽããã«ããããããã£ã¡ãã£ãŠããã®ã ããœ(ã)ã
# ã©ãã ãäžèšã®æŒ¢æã®æå³ã¯æŽãããïŒããŸããç¡çãªæ³šæã ãããã
> ã©ãããæãã€ããŠããã®ãèããã®ãé¢åã ãã
ããã¯ã貎æ§ãããããããã£ã¡ãã£ãŠãã蚌æ ã ãªã
> >>>ããŒããããŸã説æã§ã¯ãªããªãã
> >>>èŠããã« y=x ãš x=x ã®éãããšèšãã°ããã®ããªãã
>
> >>>>ãããïœãããã§ãã²ãšã€ãâxïŒxâã®ã°ã©ãã§ãæžããŠã¿ããïŒïŒçç¬ïŒå²ç¬
> >>>
> >>>ç°¡åãããã
> >>>çŽç·ïŒæ¬æžããŠãããŸãã
>
> ãããèšã£ããã
>
> > ç¬ãããããããªããããããã«ã¿ã¬ãïŒïŒïŒ
> >
> > "xïŒx" ã«éãããæçåŒã«ã¯ã°ã©ããªã©ååšãã¬ã¯ïŒ
> > æçåŒã¯ãã¹ãŠã"0ïŒ0" ãšããçåŒãšåå€ã ããã ã
>
> ãããšäžã®åŒçšãšå
šç¶é¢ä¿ãªããããã
βακαΌÏÎœïŒ
äœããäžã®åŒçšãšå šç¶é¢ä¿ãªããã ïŒ
ãâxïŒxâã®ã°ã©ãã§ãæžããŠã¿ããããšããããã¡ãã®çªã£èŸŒã¿ã«ã
æãã«ãåµã£ãŠããâxïŒxâã®ã°ã©ãã¯çŽç·ã ããšçããŠããªããã
ä»æŽãã·ã©ãåãããšã§ãæã£ããïŒãïŒçç¬ïŒå²ç¬
ãã£ã±ãŸã¡ããã§ãããã»ãã«ãïŒx+Îx=Xãå€ã§ããã
F(x)={f(x+h)-f(x)}/h ----(1)
hâÎx ã§ãF(x)=f'(x)ã----(1')
{F(x)-F(x-h)}/h -----(2)
hâÎx ã§ãF'(x)=f''(x) ----(2')
(2)ã«(1)ã代å
¥ãããš
{(f(x+h)-f(x))/h- (f(x)-f(x-h))/h}/h={f(x+h)+f(x-h)-2f(x)}/h^2
hâÎx ã§(2')ãã
{f(x+h)+f(x-h)-2f(x)}/h^2=f''(x)
ãããªãã§ããã®ã ããããæãããŠ(2')ãèšããã®ã ãããïŒïŒïŒã
倧ãã¹ãã§ã€ã¯ã§ãããåãæ¶ããŸããM_SHIRAISHIããã®èšŒæãšå€ãããªãã§ã
ãã
ãšèšãããšã¯ïŒïŒç¹ã§ããããã£ã±ïŒæ
£ãããã®ã«æãã ããããããªãã
M_SHIRAISHI wrote:
>>>dy ã f'(x)ã»â³x ãšå®çŸ©ããå Žåã«ã¯ãdy 㯠y ã x ã®ã¿ã®
>>>åœæ°ã§ãããšãã«ããæå³ãæããªãã ãããã«ããã®å Žåã
>>>dy 㯠x ãš x_1 ïŒäœããâ³xïŒx_1ïŒx ãšããïŒãšã®åœæ°ã§
>>>ãã£ãŠãx ã®ã¿ã®åœæ°ã§ã¯ãªãã®ã ãããd(dy) ã¯æå³ãæã
>>>ãªãã ã€ãŸãã(d^2)y ã¯æå³ãæããªããªã£ãŠããŸãã
>>
>>ãªããããããããããã£ã¡ãã£ãŠãã¿ããã§ããã
>
> 貎æ§ããæå³ãæŽããã«ããããããã£ã¡ãã£ãŠããã®ã ããœ(ã)ã
ãã£ãšãŒããŒãªãŒã
ããdy ã f'(x)ã»â³x ãšå®çŸ©ããå Žåã«ã¯ãdy 㯠y ã x ã®ã¿ã®
ããåœæ°ã§ãããšãã«ããæå³ãæããªãã ãããã« f(x) = ag(x), y=f(x)
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Yuzuru Hiraga wrote:
> éŽ»æ± ããã®ã¯ãïŒéå·®åã®æ¥µéã¯ïŒé埮åãã®ç²Ÿç¥ã«æ²¿ã£ãŠããã®ã§ããïŒ
>
>>f(x+Îx) = f'(x)Îx+f(x) ------(1)
>>f(x+2Îx) = f'(x+Îx)Îx+f(x+Îx) ------(1')
>>f'(x+Îx) = f''(x)Îx+f'(x) ------(2)
>
> ãããã¯ããããæéè¿äŒŒã ããã誀差é
ã䌎ããŸãããã
> é ã« e1Îx, e2Îx, e3Îx ãšãããšã
> e1, e2, e3 ã¯ãããã Îxâ0 ã®ãšããâ0ã
> ãããããšïŒ
>
>>f(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2
>
> ã«ã¯å®éã«ã¯ïŒ
> ãf(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2 + (-e1+e2+e3)Îx
> ã{f(x+2Îx)+f(x)-2f(x+Îx)}/(Îx)^2 = f''(x)+ (-e1+e2+e3)/Îx
> ãšãã誀差é
ãã€ããŠåãã
äžã®æåŸã®ïŒåŒã¯ïŒ
> ãf(x+2Îx)+f(x)-2f(x+Îx)=f''(x)(Îx)^2 + (-e1+e2+e3Îx)Îx
> ã{f(x+2Îx)+f(x)-2f(x+Îx)}/(Îx)^2 = f''(x) + (-e1+e2)/Îx + e3
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