                        ୠ ࠡ  N1
                      6

            ⢮  窮 ⥬ Eureka.  襭
            ⥬   ࠢ.

                              ࠡ
                          6

     ਮ⥭ 몮 ࠡ  窮 ⥬  Eureka.  襭
⥬  ࠢ   ⥬ Eureka.

                        ᪮ .
    ) ⥣஢  ⥬ Eureka  ।祭  
⮬⨧樨  ᮢ ⥬᪨ ⮢  祭 -
 ᫮.    ꥤ:  ।,  ᫨⥫,  䨪
(஢ ࠢ쭮 ᫥),   ⮢  ⮩ -
䮯ந⥫. ⥬ ਥ஢    IBM PC XT  AT  -
 ࠧ      ᪥  ꥬ ࠭ ଠ樨 
360 .  ଠ쭠 ࠡ ⥬ ᯥ稢   512 
 . ⥬  ࠡ    ⥬᪮ ᮯ,
  ᯮ짮 ⥫쭮 蠥 ᪮ ࠡ.
     Eureka  ᫥騥 ࠭祭:
    - ᨬ쭠  䨪஢  40 ᨬ,   10 -
 ᭮묨;
    - ᫮ । 짮⥫ 㭪権   10;
    - ᫮ ᯮ㥬 ᫮ ⠭   200;
    - ᫮ ६   12;
    - ᫮ ⠭  ६  㣨  6.
    )  㧪 ⥬   ந⠫஢, .. ७
 䠩, 室騥  ,  ࠡ稩    ४. ᪠-
騬 䠩 ⥬  䠩 eureka.exe.
     ᫥ ᪠  ࠭   ⠡ 窨 -
. ࠭ 뢠 ࠧ   .
     Edit -    ।஢ ⥪ ;
     Solution -  뢮 १⮢;
     Report -  뢮   ᫥  ࠭,  ਭ  
䠩  ७ log;
     Verify -  ஢ન 筮 १.
       ᨢ ﭨ  ୮ ࠬ,    ⨢-
 - .  ᯮ  ⨢ . ஬ , ⠡
窨 ᮤন    ப  .    ப  
।⠢   ࠡ  祢묨 蠬 (hot keys).  Ee
ᮤন    ᨬ  ० ࠡ ⥬.  -
訩     ப  ।⠢  ० ।஢.  
⮬ 砥  । ᫥騥 :
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  );
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 ( ୮ ⨨)  室 ࠧࠬ.
     F6 - Next - ४祭 ⨢  ( 横);
     F7 - Beg.Bek - ⬥⪠ 砫 ;
     F8 - EndBek - ⬥⪠  
     SCROOL - Size/move -  ࠧ   .
     ⨥   Ctrl  Alt ਢ  ᢥ稢  祢
.     㦥  hotkeys ⠪ :
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     Alt+E - 室   ।஢;
     Alt+S -  饭 ;
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     Alt+X - 室  ⥬.
      孥 ப 窨   樨  ᭮  
⥬:
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     ᫨ ⨢஢  孥 ப  File,  ᫥ 
 Enter ஥   ᫥騬 㭪⠬:
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     OS shell - ६ 室  MS DOS (   Exit);
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     ᫨ ⨢஢   孥 ப   -
 Enter,      ।஢ .
     쥩 樥 孥 ப   Solve. ᫥ -
  ।஢  祭 㦭  Esc (  
  ப )  ⨢஢ 㭪  Solve,  -
   ⨥  Enter.  ᫨  ᠭ  訡  
窨 ७  ⥬ ,   筥  襭.   砭
⮣  १ ࠡ 㤥 ।⠢   Solution.
     ⢥  孥 ப - Commands.  ⨢樨 ⮩
樨  ⨨  Enter 뢠 ᫥饥 :
     Verify - ஢ઠ 襭 (१ ࠡ ⮩  뢮-
   );
     Calculate - 祭  ( 몫祭 - Esc);
     Find other -  㣮 襭 (.. 樮 ⮤ -
 ⮫쪮      襭,    宦 㣨
 ᪫     .      
 );
     Iterate -  権 ᫥ ⠭ 襭 (  ᯮ-
  筥  襭  ᫮,   -
  ⨣,  ६ ⢥   襭 祭).
       孥 ப (Report) 뢠 ᫥饥 :
     Go - ⠢  (१ ⮩    
Report);
     Output - ࠢ 뢮  (࠭, ਭ);
     Formatted - ଠ஢ ;
     Capture -    䠩 eureka.log (   EUREKA.LOG
EXIST.A TO ERASE  䠩    .  祭
  ப ४祭 㤥  ON,  OFF);
     Logfile name -   log-䠩.

      ⮩  樨 孥 ப (Gragh) ⮨  
㭪⮢.
     Plot -  ஥  䨪  (᫥  祭   ࠢ
࠭ 㬥,  ᫨,  筮,  㭪 ।⥫쭮 ᠭ 
   Function,  㤥  ஥ 䨪 㭪樨 騩 
⥪⮢ ᨬ ᥢ䨪.  ⨨ F5  䨪 -
  ᪮ ०  ᮪ ࠧ襭);
     Output - 뢮 䨪  ࠭  ਭ;
     List - 뢮 ⠡ (᫥ 뢮 砫쭮 祭, 蠣  -
᫥  ⢠ 祪    祭 㭪権 -
 ⠡    祭ﬨ  㬥  㭪樨,  ᫨,  筮,
㭪 ।⥫쭮 ᠭ    Function    
Edit);
     Function -  㭪樨,   ந.

    6襬 ᫥⥫쭮 ⢨ 室  ஥ -
6䨪 㭪樨  ஡.
    6ᮡ N 1
  7\\\\\\\\\\\\\
    6⨢ (..     Enter)  㭪 -
6    - Graph.   襬   ⨢
6㭪 - Function.     ᫥ ⮣  ப  
6襩 樨 (ਬ y(x)  ab)   Enter.   -
6 ப   । 襩 㭪樨  (ਬ sin(x)+x^2)
6  Enter. ᫥ ⮣ ⨢  㭪    
6- Plot.    ப  砫 ࢠ ஥ -
6䨪   Enter.   襥    ࢠ-
6 0 6 Enter.0 6  १  ᫥ ⢨  -
6   ᮤঠ饥 䨪 믮  ᨬ  ᥢ-
6䨪.  ᫨ ⥯ 0 6F5,   䨪       ࠭
6  ⨭ 䨪.0 6୮  ⨥  F5  ਢ    -
6饭  ࠭    ﭨ0 6 ⢮襥  ࢮ  ⮩
6.0 6䨪   ᮢ   ࠭  ᨢ ᥢ-
6䨪,0 6᫨  । F5 뫠   F4.0 6   ⮬,   ⮣
6⮡     ०  騩  ᯮ짮0  6⨭ 䨪
6室  F7.

    6ᮡ N 2
  7\\\\\\\\\\\\\
    6   Edit.    ।   ᪮-
6 㭪権 (ਬ:
                   6z(x)=sin(x)+x^2
                   6p(x)=deriv(deriv(5*cos(x),x),x)0
                   6m(x)=1/x      )
6   ᫨⥫  (ਬ t=z(1)).
6   ப    ⨢   㭪  Solve.
6᫥ ⮣,  ᫨⥫쭠  㤥 襭 ⨢ 㭪
6 Graph.   襬  ⨢ 㭪 Plot.   ⮬
6 , 饥    㭪  ( ᫠ । 
6 Edit)  ஥ 䨪.  롮  㭪樨  ⢫ 
6 .       㭪樨    Enter.
6  믮   ⢨,    I56 ᯮᮡ ᫥ ⨢-
6樨 㭪 Plot.
    6᫨  ॡ  ஥ 䨪 㣮 㭪樨 (
6᫠ ।    Edit),  室:       Edit,
6   ⮣    ( ⮬ ।஢   易⥫쭮),
6⨢஢ 㭪 Solve    ᠭ  ⢨.

     6ਬ砭
   7\\\\\\\\\\\\\
    6 뢮  ࠭ 㭪樨  ⠡筮  ਣ  ᠭ-
6  ᯮᮡ. ⫨稥  ⮫쪮 ,   㭪 Plot
6⨢ 㭪 List.   ⮬ Eureka ॡ :  砫
6ࢠ ᫥,  蠣 ᫥  ᫮ 祪,   -
6 祭 㭪樨.

   쬠  孥 ப (Options)  ᫥饥 :
     Variables -   祭 ६  宦  ।-
;
     Settings - ⠭ ⥬ ( ਬ:  accuracy - -
 譮 ᫥;  complex [5yes4no0] - ࠧ蠥 ᫥  
묨 ᫠; casefold [5yes4no0]  ࠬ஬ yes ⬥ -
饥  㬮砭 ࠧ  ய묨  묨 㪢;  di-
gits -  । ᫮   १⮢ ᫥; substlevel N -
 ⢮ ८ࠧ ६,  室   -
 ⮬᪨ ࠦ १ 㣨.  N=0 ⠪ ८ࠧ-
  믮.  ⨬ 祭 N  0   6.    
 ஢  ⮣  ࠬ    
१). ஬ ᫥   㭪  ᮤন    
⠭,  祭   㧭,  ᯮ짮 -
襩 F1 (.. help).
     olors - ⠭ ᪨ ; ࠬ  ⥪⮢;
     Directories - ⠭ ४ਨ (⥬  ⤥ 䠩 -
 ࠭  ࠧ ४.  ⮬ 砥 㦭 㪠 -
,  室  䠩  䠩  ਬࠬ ⮢.);
     LoadSETUP - 㧪 ⠭筮 䠩;
     WriteSETUP -  ⠭筮 䠩.

     쬠  孥 ப (Window) ⠪  :
     Open - 뢠 㪠 ;
     Close - 뢠 㪠 ;
     Next -  ⨢ ᫥饥 ;
     Zoom -  ⨢ ;
     Tile -  ࠧ  ࠢ묨;
     Stack - ᯮ    㣮;
     Goto - 室  ⨢   .

  ) ⥬   ࠨ᪨ ࠢ     -
  , ⠪    樮 ⮤. ..   -
⮤ 襭 ⠪ ,  ⮤ ࠬ  ⮤  , ᪠뢠-
   ⥬⨪,  ᬮਬ  ⮫쪮  -
 ⮤.  樮 ⮤ ।⠢      
⮬ ,  Eureka 蠥 ⥬  ࠢ (   -
 ) 樮묨 ⮤.  ⮬  ᯮ짮 ⠭-
  ६  㣨, ।  ᢮   筮 -
.
     樮 ⮤  ਬ  ࠪ⨪  訬 ⥬ 
ࠧ०묨 栬.  ࠡ⠭ 讥 ᫮ ࠧ 樮-
 ⮤,       ਥ஢  襭 ࠢ⥫쭮
㧪  . ᬮਬ      -
樮 ⮤.

     _⮤ ⮩ 樨
      ⮣ ⮡ ਬ ⮤ ⮩ 樨  襭 ⥬
 ࠢ
                               =b4                   0 (1)
c ⭮   ஦ 楩 ,  室 ८ࠧ 
⥬  
                                 4^  ^
                               =+b4                  0 (2)

      4^                                ^
  - ⭠  (nxn),   b -  ⮫  n. -
 ⮩ ᯮᮡ ਢ ⥬ (1)   (2) ࠧ 410  -
 ࠢ ⥬ (1)
                     4-1
                 6x416=a4116(-a4126x426-a4136x436-...-a41n6x4n6)
420  ண ࠢ  ..  १ 砥 ⥬6:
                           4^     ^         ^     ^
                 6x416=      -a4126x426-a4136x436-...-a41n6x4n6+b41
                     4^           ^         ^     ^
                 6x426=-a4216x416-4     6-a4236x436-...-a42n6x4n6+b42
                 6.....................................
                     4^     ^     ^     ^    6     4^
                 6x4n6=-a4n16x416-a4n26x426-a4n36x436-a4n46x446-...+b4n

                                         4^
 ன        室 㫥 .
⠫  ஦  㫠6 :
                  4^                     ^
                  6a4ij6=a4ij6/a4ii6          b4i6=b4i6/a4ii6 .
                                            4(0)   (0)  (0)       (0)
             롥६ 砫쭮 ਡ6 x4 6=4  6(x41 6,4 6x42 6,4 6...4 6,x4n  6).
   ⢥  砫60  ਡ  롨 ⮫ ᢮
童 (b410, b420,...b4n0). ⠢   ࠢ  ⥬ (2) -
 ࢮ ਡ
                            4(1) ^ 5 4(0) 5 4^
                           6x4 6=4  6A4 6x4  6+4  6b

    60த        ,   稬   ᫥⥫쭮
5(0)0,5 05(1)0,5 05(2)0,5 0...,5 05(k)0, ... 6 0ਡ, 塞  㫥

                            4(k+1)  ^  (k)  ^
                           6x4   6=4   6A4 6x4  6+4  6b ,  k=0,1,2,...

     6060 ᫥ ⥮६  室 ⮤ ⮩ 樨.

     _६
      믮   ᫮ :

           4n6                                    4n
         6Ŀ4                                6
 4M6 4A6 4X6    \      4^6  4 6                 4 6   4 6\  4 6    4  6
 617,6i7,6n  7 4 6/4    6 a4ij  6 = q < 1 <=>  a4ii6  >4 6 / 4   6 a4ii6 4 ,6i=1,...,n
         6                                
          6j=1                                 4 6j=1
                                               6i7-6j
     
          4n                                     n
        464                                 6
 4M6 4A6 4X6 4  6\4      ^                          6\4          
617,6j7,6n7  4  6/4      6a4ij   6=4 6q4 6< 14 6<=>4  6a4jj  6>4  6/4      6a4ij 6 ,j=1,...,n
        464                                 6
         6i=1                                   i=1
                                               6i7-6j
        :
(I)  襭  ⥬ (2)   ⢥;
(II)   ந쭮  砫쭮  ਡ 5(0)0 ࠢ 業
譮6:
                   4(k)   -        k            (0)  -
        4M A X   6 x4i  6-4  6x4i6  <= q4  M A X  6 x4i  6-4 6x4i6  .
        617,6i7,6n4                     7  617,6i7,6n

     砭 1
         ⥮६  室 ⮤ ⮩ -
樨  ⥬  ࠢ   㫨஢  
饬 .

     砭 2
      믮 ᫮ ⥮६  室 ⮤ ⮩ -
樨  ⥬  ࠢ ࠢ ᫥ ਮୠ
業6:
                  4(k)   -                          (k)    (k-1)
        4M A X   6x4i  6-4  6x4i  <=6 q/(1-q)4 6 4M A X  6 x4i  6-4  6x4i6 4   6
        617,6i7,6n4                    617,6i7,6n

     _⮤ 
      ⮤  䨪樥 ⮤ ⮩ 樨.  ᭮-
  䨪樨 ⮨   ⮬,      ᫥  ।
                                    4_
(+1)- ਡ  ⭮ 6x4i0  6i0>1 ᯮ 㦥 -
                                      4-       -
 (+1)- ਡ 6 0 6x410 ,...6,x4i-10 ,    -  ਡ6-
,   ।饬 ⮤.
      (+1)- 樨 i-        606:
    4(k+1)  ^    (k+1)      ^      (k+1)  ^      (k)    ^    (k)   ^
   6x4i  6=4  6-a4i,16x41   6-4 6...4 6-a4i,i-16x4      6-a4i,i+16x4  6-...-a4i,n6x4   6+4  6b

     筮 ᫮ 室 ⮤  ᮢ    ਢ-
 㫨஢  ᫮ 室 ⮤ ⮨ 樨.
    2) Eure6k0a   ⥬  ࠢ (   -
 㣨 )  ⠢ - ணࠬ.   ਬ 
襭  ⥬  ࠢ6:
                                             6                   
          62x416+3x426+5x436=31                      2  3  5        31 
          6-x416+3x436=11      ..  Ax=b,   A=-1  0  3     b= 11 
          6x416-7x426+5x436=0                        1 -7  5         0 
                                             6                   

60㦭 ᤥ   Edit    ਢ  ᥩ (Eu6-
re6k0a ਬ   ࠪ᪨ 6 ).

  6I)0                     6  II)
                         6
     62*X1+3*X2+5*X3=31      2*X1+2*X3+3*X2+3*X3-31=0
     6-1*X1+3*X3=11          -X1+3*X3=0
     6x1-7*X2+5*X3=0         X1-5*X2+5*X3-2*X2=0
                         6

        ᫥ 祣        ப  ( 60 ESC)
6 0 6  0  㭪 Solve 6 0 Enter. ᫨ 6 06 0 ⥬
஦,  ⪠ 6 0襭 6 0 6 0ॢ  ᯥ. 6 0 襬 6 0砥
6det A7-60 0 ⮬   襭 (Solution)   १, 6-
祭    筮:

      6X1=1.00000000    X2=3.000000000    X3=4.000000000

     Eure6k0a   ⥬  ࠢ    ⮫쪮  
⢨⥫묨,      묨 樥⠬.   襭 ⠪
ࠢ ᢮,  ਬ,   ᫥ 殮  ⮪
 -   ࠤ孨᪨ 楯   ࠡ  ६ ⮪.
 ਢ ਬ    Edit ⥬   ࠢ
 묨 䨥樥⠬6:

     6$ complex=yes
       6i^2=-1

     6(2+i)*X1+7*X2+(7-i)*X3=0
     6(5-i)*X1+i*X2+3*i*X3=2
     6(3-i)*X1+2*X2+5*X3=4


                               6
                             6
     ) ஢     ஥  Eure6k0a  
  襭  ⥬ ⮤ ⮩ 樨.
     ) ஢    Edit  㭪  Solve  
  ⥬ ஦.
     )   ⥬.  ஢ 襭   
Verify. 6060   襭   Report.
     )  ,     襩 ⥬.  ⮣,
ᯮ ࠢ⢮ 6AA5-16=E0,6 ⠢ n526 ࠢ   n526 묨,
6 n*n ࠧ 室 .
     6d) ᯮ  ࠢ⢮ AA5-16=E , ஢  0  6
 㭪 6(06)  ⭮  A.



                        ୠ ࠡ N2
                      6

           㭪60 ⥬ Eure6k0a. 襭 
         ࠢ. 6 0襭 ⥬    ࠢ.
         ᫥ ६60 㭪権   ६.

                            6 ࠡ
                          6

     ਮ⥭ 몮 襭  ࠢ  ⥬  -
 ࠢ   ⥬ Eure6k0a.

                        _᪮ 

     a) 䠢  ⥬  Eure6k0a  ᮤন ⠭  ᨬ.
 ⨭᪨ ய (   Z)   (    z)  㪢,  
⠪  ᯥ槭.
     : - ࠧ⥫  ࠦ ࠧ饭   ப;
     ; - ⬥砥 砫 ப ;
     {} -  ᪮ ࠧ頥 ਩;
     [] - ᯮ  ࠡ  ࠧ묨 ﬨ;
     $ - 㪠뢠,  ᫥饥 ᫮-४⨢;
     = -  ᢠ;
     := -  (।)  㭪樨  짮⥫    砫
祭 ६.
      ࠦ ᫥ ᨬ 60᪮ 樨  -
७   ப.
     ४⨢, ⭮騥  ⠭,         
Edit   .
     _ਬ
     $ settings
     acuracy=0.000001
     digits=5
     $ end
     Eureca  ந ᫥騥 樨:
     + ᫮;  - ⠭;  * 㬭; / ; ^  
⥯; ()    ਮ 権;  .  ⤥ 楫 
᫠  ஡;  ,⤥ ६   㣠    ᯨ᪠;  <
; > ; <=   ࠢ; >=   ࠢ.
     ਮ 権 ।    몠 ᨪ,  ᪠ 
..
     Eure6k0a  㭪樨 re(z)  im(z), 騥 ⢨⥫ 
  ᭮ ᫠ z=x+iy. । ਬ  㭪権
室   ४⨢:  $ complex=yes    60
i^2=-1.
                       ࠨ60 㭪6
     abs(z) -  ;  exp(z) - ᫥ e=2,71828...   ⥯ z;
floor(x) - 楫  ; ln(z) - ᫥ ࠫ쭮 䬠 z;
log 10(z) - ᫥ 筮 䬠 z;  sqrt(z) -  ᫥
 ⭮  z;  pos(x) - 頥   >0  0  ⨢
砥; sgn(x) - 頥: 1  >0, -1  <0  0  x=0
             _ਣ᪨  ࡮᪨ 㭪樨
     atan2(y,x) - ᫥ ⠭  न⠬  x      (㣮
祭     १,  ண -(0,0)  (,));
     polar(x,y) - ८ ⮢ न  ;
     sin(z), cos(z), tan(z) - ᫥ ᨭ, ᨭ  ⠭ z;
     sinh(z), cosh(z),  tanh(z) - ᫥  ࡮᪨  ᨭ,
ᨭ  ⠭ z.
     ஬ ᫥  㭪権 Eure6k0a     㭪権  
楤:
     deriv(f(x),x) - ᫥ ந -樨 f(x);
     integ (f(x),x,a,b) - ᫥ । ⥣ࠫ  f(x) 
।    b.
     fact(n) - ᫥ 䠪ਠ ᫠ n;
     ncum(x) -  ᯥ樠 㭪 訡 ()   ଠ-
 ।;
     poly(x,an,...,a0) -    祭    ⢨⥫  
 ୥  an*x^n+...a1*x+a0    㭪-
 P(x)  祭   窥 .
     sum(f(i),i,n,k) -  㬬 f(i)  6 060 i, 饬
 n  k.
      ⥬  Eure6k0a 짮⥫    -
  㭪樨 १ 騥 ஥. 㭪樨 짮⥫ -
  :
      -樨 (ᯨ᮪ ६)=ࠦ
     
      -樨 (ᯨ᮪ ६):=믠
      ଠ  ᯮ  ᫨   㭪樮쭠 -
 ᬠਢ  ਡ.
     ) Eureca     ந ( ࠫ)  ᪮
ଥ.    祭 ந  窥 ᫥  -
⮤.     ⥬  Eure6k0a  60  ந 
᮪ 浪. ਬ, ⮡ ᫨  ந 㭪樨
sin(x), 筮 :
     F=deriv(deriv(sin(x),x),x).
      ਢ     Edit.  ਨ  
 .
        ;᫥ ந
        $ settings   ; ⠭
        digits=12  ; ᫠ 
        $ end        ; १
        ;  㭪 d(x)=d(sin(x))/dx
        d(x)=deriv(sin(x),x)
        d1=d(4.3) ; ᫥ 㭪 d(x)=cos(x)
        ;  窥 x=4.3
     ᫥ ⮣  祭 襭     ப
  ⨢஢ 㭪  Solve.    ⮬  ᯮ  㭪  
6G0raph  ந 䨪 d(x).
     )  f(x) - 㭪, ।  १ [a,b]. ।-
                                                          4_
,     ⮬  १ ᮤন ⢥ 窠 x 쭮
                                                    4-
㬠 f(x), 祬 㭪 ண 뢠  7 0x7,0x  ண -
             4-
⠥ 7 0 7 0x7.0x.   㭪 뢠 㭨쭮.  ⨬,  
筮 ᬮ  樨 㭪樨 f(x), ⠪  -
 ᢮    樨          㭪樨
g(x)=-f(x).  ࠧ  襭  ਠ ६쭮  -
.
      ⮤ 樨 ᭮    ࠢ  祭  㭪樨
f(x), 塞  窠 x1,x2,...,x4n0.  ⮤  뢠 -
⮤ אַ ᪠,   窨 x4i0 - ஡묨 窠.   -
 䥪⨢ ⮤  ⮣ 鸞  ⮤ ⮣ 祭.
      祭 १ 뢠 ⠪  ࠧ  १  
 ࠢ6  , ⭮襭0  ᥣ १   6 0 6-
襩  ࠢ ⭮襭  6 0襩    襩 
१.
     ⮥ 祭  १ [a,b] ⢫    ᨬ-
筮 ᯮ ⭮⥫쭮 業 १ 祪:

              62                                2
    7a6=a +  ( b - a )7        b6=a +  ( b - a )
               7|\\\6                            7|\\\\
          63 + 7?6 5                         1 + 7?6 5

      ⮬  窠 7a0 ⢫  ⮥  祭  ⮫쪮 १
[a,b],   १ [a,7b0 ]. ஬ ⮣ 窠7 b0 ⢫ ⮥ 6-
祭  ⮫쪮 १ [a,b],   १ [ 7a0,b].
     । (+1)6 0 樨 6 0ந ᫥騬 ࠧ. 6 0窨
7a5(k)0  7b5(k)0 7 0室  㫠:

                    62                           5    62
    7a5(k)6=a5(k)6 + 7  D5(k)7     b5(k)6=a5(k)6 +   7D5(k)
                     7|\\\\6                       5    7|\\\\\
                63 + 7?6 5                   5      61 + 7?6 5

 60 7D5(k)0 -  १ 樨 ६㬠  50 樨.

     ᫨   6f(7a5(k)6)5 7,5 6f(7b5(k)6)  ,   
                            6                         
        6x5(k+1)0 6ਭ 0 60 6a5(k+1)4,6b5(k+1)65=60 6a5(k)4,7b5(k)60                                                   7`
                            6                         
                   6   x5(k+1)6=7a5(k)


     ᫨   6f(7a5(k)6) > f(7b5(k)6)  ,   

                           6                         
        6x5(k+1)6 ਭ  a5(k+1)6,b5(k+1)6= 7a5(k)6,b5(k)6
                           6                         

                   6   x5(k+1)=7b5(k)

   ⨬,  窠 6x5(k)0 ⮨  殢 १ [a5(k)0, b5(k)0] 
60稭,  ॢ        62
                            67  D5(k)
                                 7|\\\\6        .
                            61 + 7?6 5

     ⮬ ୠ 業:
                                  62
              6 x5(k) - 6x5* 7 ,6 7  D5(k)6 =7 D5(k+1)
                                   7|\\\\6                .
                              61 + 7?6 5

                                                          7|\\\
                                                         7?0 65   +  1
   ..     ᮪頥  १ 6 06   
                                                               62
6ࠧ,0  ࠢ ᫥ 業 譮:

                              6            Ŀ5k+1
                              6      2      
              6 x5(k)6 - x5* 7 , 6  5  (b - a)
                              6       7|\\\\6 
                              6  1 + 7?6 5   7 6

    ࠧ,  ⮤ ⮣ 祭 室  ᪮ -
᪮ ணᨨ, ⥫ ன

                                   52
                         g5 0=5 7  ~6 0.62
                                   7|\\\\
                              615 6+5 7?6 5

      ⮤,        䥪⨢묨,
᫨ 㥬 㭪 筮 .     
 䨪ﬨ ⮤ 襭  ࠢ ਬ⥫-
  ࠢ f(x)=0.
     ) Eure6k0a      ᪠ ६㬠 㭪樨 
  ४⨢:7 0$min  $max.  ⮬ ᫨ 㭪  -
쪮 ६㬮,     宦 ⮣  㦥  
ᮢ 䨪 㭪樨  室  ⮣ 䨪    砫쭮
ਡ   ࠭祭60   ᪠ ६㬠.   ⨢ 砥
 ६㬠 㤥 ந室  砫 祭  -
⥬ Eure6k0a  㬮砭   ਢ   ⮬ ६, -
 ⥫  .  ਢ ਬ 6   Edit. 
6   ६.

     $ max (T)
     V(x)=5*x*exp(-x/2)*(2+sin(3*x))
     x:=2
     V(x)>10
     T=V(x)

        १ 襭 砥: T=10.629942, x=2.58050146.

     ) ७  5*0 ࠢ f(x)=0 뢠 , ᫨ f(x5*0)=0 
f'(x5*0)7-00.  ⨢ 砥 ७ 뢠 .  ᫮  m
뢠 ⭮    x5*0, ᫨ f5(k)0(x5*0)=0  =0,1,2,...m-1 
f5(m)0(x5*0)7-00.7  0᪨7 0७ 7 0x ᮮ⢥ 7 0窥 7 0祭
䨪 7 0㭪樨  y=f(x)   O6x0.  襭 7 0 7 0᪠ ୥
 ࠢ ⢫    ⠯.   ⠯  뢠6-
 ⠯ 樨  ୥, ன  -  ⠯ 樮 6-
 ୥.6 0  6 0⠯ 㤮6 0 믮     ᪨
। ⥬ Eure6k0a. 6 0  ஬  ⠯   6 0᫥  
୥  筮 6e0>0 ᯮ -  樮 6 0⮤,
 5 0ந ᫥⥫쭮 6x5(0)0,6x5(1)0,..,6x5(n)0...5 0ਡ6-
,6 0室 5 05 0  6 x5*0. 6 0㫨㥬6 0  6  0 6 0⮤
6 0 ⥮६.

          6६0 61.0( 室 ⮤ ⮭)
        6

      6x5*0- ⮩ ⢥ ७ ࠢ  f(x)=0    
f'(x)7-00  ⭮ U4r0(x5*0)={x:60x-x5*60<r}. ।60,  f''(x) -
뢭  U4r0(x5*0) 
          0<m=inf50f'(x)5 6,5 0M=sup50f''(x)50 6,0 6 x 06.U4r6(x5*6)
      6祬
                  6M x5(0)6 - x5*
            6q = 56 < 1  .
                       62 m

    , ᫨ 5 0砫쭮 ਡ 5 6x5(0) 6ਭ 5 0U4r0(x5(*)0),6 0
⮤  ⮭
                      6f(x5(k)6)
      6x5(k+1)6=x5(k) 6-5 6   ,    k=0,1,2,...  室  x5(*)6,
                      6f'(x5(k)6)
祬6 06 0譮 6 0ࠢ 業

        6 x5(k) 6- x5*6 7 ,6 q5t-1  6x5(0)6 - x5* 6 ,  t=25k 6.

     _砭 1.
      ⥮६        
୥.
     _砭 2.
      ⭮,  ६  㭪樨  f(x) 室  窠,  
f'(x)=0. ⮬ ᫨  g(x)=f'(x) 믮  ᫮  ਢ
 ⥮६,   樮 , ਡ騩  窥 ६-
6 f(x)0, 㤥  :

                            6f'(x5(k)6)
            6x5(k+1)=6x5(k)6 -    ,    k=0,1,2,....
                            6f''(x5(k)6)

     )  ᪠ 襭 ⥬  N- ࠢ  N
60묨,  

                  f410 (6x410, 6x420, ..., 6x4n0)=0
                  f420 (6x410, 6x420, ..., 6x4n0)=0 6    0 (1)
                  ......................
                  6f0n (6x410, 6x420, ..., 6x4n0)=0

砥 6 0祭 6 0,  ..  6 0ॠ 6 0᫥ ।-
 6  0⭨ 6 0 6 0 6 0 6 0ࠬ஢.   6 0⠯6 0 襭
⠪6 0,6 06  0 6 06 060 6 0砥, 6 0 6 0 6 0襭
x5*0=(6x410,...,6x4n0)6 0,6 0..  ⢠ ᮤঠ饣 6x5*0.
     (    㤥    -
,  孨 -  樨).    ⢥  ⠪  ⢠
㯠 ࠫ      N - ୮ ࠭⢥.   
  襭  樨       -
 襭 㤮⢮⥫쭮, ᫨    ᮮࠦ 㤠6-
 6 0 襥 6 0砫쭮 ਡ 6x5(0)0. 6 0 6 0⥩ 6 0
( ⥬   ࠢ   묨)   ᯮ짮-
 ᪨ ⮤.
        ஬ 60  6᫥ 0襭   筮 -
   樮 ⮤.  ᬮਬ  ⢥  ਬ
⮤, 뢠 ⮤ ⮩ 樨.  ८ࠧ㥬 ⥬ (1) 
᫥饬 ⭮  (   㤮  権):

                   6x41=7F416(x41,6x42,....,6x4n6)
                   6x426=7F426(x416,x426,....,x4n6)
                   6....................       (2)
                   6x4n6=7F4n6(x416,x426,....,x4n6)


     ᫨  -㭪,6 7 F6=(7F416,7F426,....,7F4n6)5T6,  06 0 ⥬ (2)
 ⠪:
                             6x=7F6(x)           (3)

     砫쭮  ਡ 6x5(0)=(6x415(0)0,6x425(0)0,...,6x4n5(0)0)5T0 .
⠢6    0 ࠢ6  0 6 0⥬ (3),6 0 稬6  x5(1)6=7F6(x5(0)6)0.
⠢ 6 x5(1)0  6 0 ࠢ      (3),6 0  6x5(2)6=7F6(x5(1)6)  0..
த 6 0᫥ 6 0 㫠6  x5(k+1)6=7F6(x5(k)6) ,0  6k7.600 稬 6-
⥫쭮 6x5(0)0,6x5(1)0,...,6x5(k)0,...  ਡ  襭 6x5*0.

                6                             
                6  d7 F416(x)           7 6d7F416(x)  
                64 6 ......  
                6   dx416            7 6  7 6dx4n6      -    ( 
                6.............................
   7F6'(x) =0 6.............................   ந )  
                6                             
                6  7 6d7F4n6(x)           7 6d7F4n6(x)4 6    ᮮ⢥7 F6(x).
                6  .......  
                6   7 6dx416              7 6dx4n 6   
                6                             

60㫨㥬 ⥮६  室 ⮤  権.

                  6६ 2 .
               6
         ன 4 7s 0- ⭮ 4 0襭 4 6x5*0 㭪樨4  0 7F4i6(x)
(i=1,2,...,n) ७㥬  믮 ࠢ⢮

             6n
          6Ŀ           
   6m a x 0  6\        7 6d7F4i6(x) 
   617,6i7,6n0   6/       ĳ7 ,6 q5 6,   07,6q<1  q - ﭭ .
          6     7 6dx4j6  
            6j=1             

      ᨬ  롮 砫쭮 ਡ 6x5(0)0  㪠-
 7s 0- ⭮ 7 0 7 0樮 ᫥⥫쭮  室
 ⮩ ⭮,  ⮤ 室  ᪮ 60᪮ ண6-
ᨨ  ࠢ ᫥ 業 譮:


          6m a x  x4j5(k)6 - x4j5*6 7 ,6  q5k6 m a x  x4j5(0)6 - x4j5*6
          617,6j7,6n                       17,6j7,6n


     (      ⥮६ 2   㫨஢ 
 饬 ).
     _砭 3.
      ᫮ ⥮६ 2 ୠ ਮୠ 業 譮:

                                 6q
     6m a x  x4j5(k)6 - x4j5*6  7,6    m a x   x4j5(k)6 - x4j5(k-1)6 
     617,6j7,6n0              5       61 -  q   17,6j7,6n

    稨 筮 襣 砫쭮 ਡ 6x5(0)0 
, 
                                  6n
                               6Ŀ              5 6
                    6q7 ~6 m a x   \        5 7 6d7F4i6(x5(0)6) 
                        617,6i7,6n0   6/       56
                               6      5 7 6dx4j6   5 6    .
                                 6j=1    5             
     _ਬ
      १  㠫쭮  䨪 ਢ f1  f2  -
⥬  ࠢ

              6f(x416,x426) = x41534 6+ x42536 - 8x416x426 = 0
                                                      6(4)
              6f(x416,x426) = x416ln(x426) - x426ln(x416) = 0

㦨,5 05 0    襭4 0室  窨 5 0(3.86;02).
஢ 室 5 0⮤  権  5 0⥬ (4),5 0᫨ 
⢥ 5 0砫쭮 5 0ਡ 5 0 窠6  x415(0)6=3.8    x425(0)6=2.

     ८ࠧ㥬 ⥬   㡤  ਬ ⮤  -
 権.

                  437|\\\\\\\\\\\\
             6x416 = 7?6 8x416x426 - x4253  =7 F416(x416,x426)

                          6x426         x41
             6x42 6= x426 +  -  =7 F426(x416,x426)
                        6ln(x426)     ln(x416)

      ᫥  q  ᯮ㥬 ⥬ Eure6k0a.    
Edit ᫥ :
             x=3.8 : y=2
             a=deriv ((16*x-8)^(1/3),x)
             b=deriv ((30.4*y-y^3)^(1/3),y)
             c=deriv (2+2/ln(2)-x/ln(x),x)
             d=deriv (y+y/ln(y)-3.8/ln(3.8),y)
             K=abs(a)+abs(b) : p=abs(c)+abs(d)
     १ 襭 砥 p<k 7~0 0.815...<1.
  C⥫쭮, ⮤ ⮩ 樨   砥 室.
     )  ⮣, ⮡  ७  襭 ⥬ (4) 
 । Eure6k0a, 筮 ᤥ   Edit c 
                x^3+y^3=8*x*y
                x*ln(y)=y*ln(x)
                x:=3.8 : y:=2
      ⤠ ।  Solve.

                               6
                             6
     ) ஢   䨪  ⠡  । Eure6k0a 稥
 y ।  ࠢ f(x)=0  㪠 १  -
樨 .
     )    ⥬ Eure6k0a m  M.  ஢ 믮
᫮
                   6M  x5(0) 6- x5*6 
                  65 6  < 1
                         62 m

     ( ⥮६  室 ⮤  ⮭)    襣  ࠢ
f(x)=0.
     )   ࠢ   ⥬ Eure6k0a.
     ) ࠧ  ஢ 䨪 ਢ  । 
⥬  ࠢ   㪠  אַ㣮쭨  樨
. 롥 砫쭮 ਡ . ।⠢ ⥬  -
  6x=7F6(x).
     ) ᯮ ⥬ Eureka, ᫨
                  6n
               6Ŀ           5   6
                6\       7 6d7F4i6(x5(0)6) 
                6/     56
               6     4 7 6dx4j6 5   6
                 6j=1            5   6

  i- ப 6 0 (i=1,2, ..., 6n0). 6 0 ᭮ 6 0⮣
஢  믮 ᫮ ⥮६  室 ⮤ ⮩ 6-
樨.
     )   ⥬    Eureka.



                     ୠ ࠡ N3
                6

        ६ 㭪樨  ६.  
        ணࠬ஢.

                            ࠡ.
                         6

     ਮ⥭ 몮 ᫥ ६㬮 㭪樨   -
   襭   ணࠬ஢   ⥬
E6u0reka.
                      ᪮ .
                     6
     )  f(x)=6f0(x410,  ...,x4n0) - 㭪  n ⢨⥫  -
, 㥬  ஬ ⢥ . ᫨ =R5n0,  
  ᫮ 樨.   ⨢ 砥   
᫮ 6 0樨.   6 0  㭪樨 6 0 6 0६, 6 0
ᨬ樨 㭪樨  f(x)  ᢮         樨  㭪6-
樨 g(x)=-f(x).
       ⥬⨪ ⭮,   ࠤ 㭪樨 f(x), ।-
塞  窥 =(410, ..., 4n0)  

                6 7 6df(a)           7 6df(a) 
         6u(a) = ,....,  ĳ  ,  㪠뢠  ࠢ
                6  7 6dx416             7 6dx4n6  

᪮३襣 6 0⠭6 0 㭪樨 f(x)  窥 . 6 0 -u(a), 6-
뢠 ࠤ⮬, 㪠뢠 6 0ࠢ ᪮३襣6 0 뢠
㭪樨 f(x)  窥 . 窠  뢠 樮୮ 窮 㭪樨
f(x), ᫨  ⮩ 窥 믮 n ࠢ:

                    7(
                    726    df(a)
                    726   = 0
                    726    7 6dx41
                    7*6..............
                    72
                    726    df(a)
                    726   = 0
                    794     6dx4n

      f(x) 6 0 뢭 ७6㥬0.   
᫮ ⮣, ⮡ 樮ୠ 窠  뫠 窮 쭮 -
㬠,  ⥫쭠 । :

                       6                        4 6
                       6                        4 6
                       6  d526f(a)7 6        d526f(a)7 4 6
                       6 ,.....,4 6
                4^6        dx416dx416         dx416dx4n6  
                6(x) = .........................
                       6                         
                       6 7 6d526f(a)       7  6d526f(a)7  6
                       6 46,.....,46 
                       6  dx4n6dx416         dx4n6dx4n6  
                       6                         
             4^                                                  ^
       뢠 楩 . ,   4  06-
                              4^                     ^
⥫쭮 ।, ᫨  6(Ax,x)>00 6 06 x7-600 6 0 6 (Ax,x)=00 6 0 6 x=0,

  6x=(x416,...,x4n6)5T6.

      讥 ⢮ ࠧ ⮤ 宦  -
 㬠 㭪樨  ६. ᬮਬ  ⢥ -
   . ⮤ ⮤ 뢠 ⮤ ᪮३襣 ᪠ 
   ।⠢⥫ 讣 ᥬ⢠ 樮 -
.  5(k) 0- ਡ  窥 㬠 ,  u5(k)0=u(x5(k)0) -6-
祭 5 0ࠤ  窥 5(k)0. 5 0  ࠧ,    5 0-
 窨 5()0 ࠢ ᪮३襣 뢠 㭪樨 f(x) -
 ⨣ࠤ⮬ 5 0-u5(k)0. 室  ⮣ 樮  -
 ᪮३襣 ᪠ 뢠  :

                   6x5(k+1)6 = x5(k) 6-5 7a4k6u5(k)   4 6      4    6(1)

   - 7a40 蠣 ᪠, 롨ࠥ  ᮮࠦ 樨 㭪樨
  ᪠୮ ६ 7f4k6(7a6) = f(x5(k) 6-5 7a6u5(k)6)  08 5 7a0>0. 5 0..
6 0⢥ 7 a46 0롨ࠥ6 7 a0  ண 7 f4k6(7a4k6) =7 6min 7f4k6(7a6)0  6 0 7a6>0.

     ᬮਬ ਬ  ⮣  ⮤  樨 筮
㭪樨 f(x410, ..., x4n0)=f(x), 

                    6n     n4              6n
              61  Ŀ Ŀ     4     6Ŀ
      6f(x) =   \     \    a4ij6x4i6x4j6 - \    b4i6x4i6      (2)
              62   /     /          4    6/
                 6       4    6
                   6i=1   j=1            i=1

     樥 4ij0    ⠬  ᨬ筮 ⥫쭮
।  .  ᯮ  祭, 襬 f(x)
⠪:
                       61
               6f(x) = (Ax,x) - (b,x)      4       6 4 6(3)
                       62


     ᫨ ࠤ     㭪樨 (2).

              6   n      n        4    6'4     6   n       4  6 '
  6d f(x)0    610 6 Ŀ Ŀ     4    64      6 Ŀ    4  6 
 60 6= 46  \      \     a4ij6x4i6x4j6 4   6-4   6 \     b4i6x4i6  
   6d x4k6 0    62   /      /         4    64   6 4  6  /         4  6 =
              6       4    64      6      4  6
              6  i=1    j=1       4    6x4k    6  i=1      4  6 x4k

               4(6  k=1,....,n)

      6   n        4 6         n4 6                  n4  6       4  6'
    610 6 Ŀ    4 6  4  6    Ŀ          4   6 Ŀ    4    6
 6=0 6ĳ  \    a41j6x416x4j 6+..+  \     a4kj6x4k6x4j6 +..+  \     a4nj6x4n6x4j6 - b4k6 =
    620 6  /      4 6      4  6   /                4   6/        4    6
      6    4 6       4  6           4   6    4    6
      6  j=1        4 6      4 6j=1                 j=14  6      4  6x4k


    61                                     4           6    4   6
 6= ĳ a41k6x416 +..+ (a4k16x416 +..+ 2a4kk6x4k6 +..+ a4kn6x4n6) +..+ a4nk6x4n6  - b4k6 =
    62                                        4           6 4   6

    61               4     6       4   6            4     6
 6= ĳ (a41k6 + a4k16)x416 +..+ 2a4kk6x4k6 +..+ (a4kn6 + a4nk6)x4n6  - b4k6 =
    62                    4     6     4   6         4     6
             6( ..  A ᨬ筠 )
                                                    6n
    61      4   6                 4       6        4 6Ŀ
 6= ĳ 2a4k16x416 +..+ 2a4kk6x4k6 +..+ 2a4kn6x4n6  - b4k6 =   \     a4kj6x4j6 - b4k
    62             4  6          4        6       4 6  /
                                                 6
                                                  6j=1
     ⥫쭮 砥:
                              6n
                   6df(x)    Ŀ
                  6 =  \     a4kj6x4j6 - b4k          (4)
                    6dx4k6      /
                            6
                             6j=1

       筮 ଥ  :

                      6u(x) = Ax - b                    (5)


  ७   ࠢ⢠ (4)  40 (=1, ...n), 砥4 0:

                       6d526f(x)
                     6 = a4pk
                      6dx4p6dx4k
                              4^
  ࠧ,   ()      ࠢ .
60  㫥 (5)  (1)    :

             6x5(k+1)6 = x5(k)6 -7 a4k6(Ax5(k) 6-5 6b)             (6)

     ⨬, 5 6:

          61
 7f4k6(7a6) = (A(x5(k) 6-5 7a6u5(k)6),x5(k)6 -7 a6u5(k)6) -5 6(b,x5(k)6 -7 a6u5(k)6) =
          62
                                                       6(*)
   61                        5       6 5   61
6= (Au5(k)6,u5(k)6 )7a526 - (u5(k)6,u5(k)6)7a6 + (Ax5(k)6,x5(k)6) - (b,x5(k)6)
   62                          5         62

6( ⥫⢮   (*) .   ୮ ࠡ N 3 )

      㭪  筮 㭪樥 ࠬ8 7a0  ⨣-
 㬠  ⠪ 祭8 7a0=7a4,0  ண

          7f4k5'6(7a4k6) = (Au5(k)6,u5(k)6)7a4k6 - (u5(k)6,u5(k)6) = 0

     ࠧ,  ਬ⥫쭮   樨 筮 㭪樨
(3) ⮤ ᪮३襣 ᪠ ⥭     㫥  (6),

                        6(u5(k)6,u5(k)6)
                 7a4k6 = 56                  (7)
                        6(Au5(k)6,u5(k)6)

       ᫥ ⥮६.

     _६
      - ᨬ筠, ⥫쭮 । ,  -
 筠 㭪 (3).    롮 砫쭮
ਡ ⮤ ᪮३襣 ᪠ (6),  (7) 室  ୠ ᫥-
 業 譮
                        7|\\\\\\\6                 5n
                       7/  l4max6    7  l4max7 6-7 l4min6  
    6 x5(k)6 - x5* 67 ,6   7/ 6  ĳ5  6x5(0)6 - x5* 
                     7?    l4min6    7  l4max7 6+7 l4min6  
                                  6               

      7l4min7 0 7l4max7 0- 쭮  ᨬ쭮 ᮡ⢥ 6-
祭   .
     ) ⥬  Eureka  蠥      㬠 (ᨬ㬠)
㭪樨 6 0᪮쪨 6 0६.  6 0⮬      ࠭6-
祭  㤮  ᪠ 6 0砫 祭. 6 0 ஢ન 6 0ᯮᮡ6-
 ⥬ Eureka  ⨬樮  ࠧࠡ⠭  6-
⮢ ,  ᮤঠ .   ⠪  -  㬠
㭪樨 ப.  60  㭪 6 0 ६ ࠧ  嬥6-
 ࠭⢥ " ࠣ ",  㤭騩 .   ਢ 6-
  a Edit,  ஢ 㭪 ப.

                  6$ min (F)
                  6f(x,y)=100*(y-x^2)^2+(1-x)^2
                  6F=f(x,y)
                  6x:=-1.2
                  6y:=0
砫쭮 祭 ६   襭 =y=1  F=0.

   ⮢  ᮤন ࠭祭.

                  6$ min (F)
                  6f(x,y)=(x-2)^2+(y-1)^2
                  6F=f(x,y)
                  6-x^2+y>=0
                  6-x-y+2>=0

筮 襭: F=1, x=1, y=1.
      襭  ⨬樮   ࠭祭ﬨ Eure6k0a 뢮
  Solution ᮮ饭  ⮬, ᪮쪮  㤮⢮७ ࠭-
祭.  쭮 砥 뢮 100 %.  ᫨  ᫮ ⥫쭮
 祬 100 %,     㦨 ਧ 筮 宦
६㬠.    ணࠬ,  뫠  ᯮᮡ -
  ⨬樮 .  ⮬   ⮢   ⮬,
 Eureka    ࠢ  ।  .
     )  襭 鸞 孮᪨  ᪨ ஡  -
  :
    max Z(x)  min Z(x)
                                 6n
                               6Ŀ
        ᫨6            Z(x) =  \     c4j6x4j6 + c40
                                6/
                               6
                                6j=1
      ࠭祭
                        6n
                      6Ŀ
                       6\      a4ij6x4j7 ,6 b4i6     ( i=1,m416 )
                       6/
                      6
                       6j=1

                        6n
                      6Ŀ
                       6\      a4ij6x4j6 = b4i6     ( i=m416+1,m426 )
                       6/
                      6
                       6j=1

                        6n
                      6Ŀ
                       6\      a4ij6x4j7  .6 b4i6    ( i=m426+1,m )
                       6/
                      6
                       6j=1

                      6x4j7.6v4j7.60  4         6  4    6(4 6j=1,n41 6)

                      6x4j7,6w4j7,60 4           6 4    6(4 6j=n416+1,n4 6)

       뢠 砬  ணࠬ஢.  
襭   ᮧ ᯥ樠 ⮤, 뢠 ᨬ--
. 祭   ணࠬ஢  ।⮬ ᯥ-
樠쭮 ,  ⮬ ᬮਬ    砩.    n=2
(.. Z(x)=410410+420420+400 )   ⮬  ᫥騥 ࠭祭

                      2
                    Ŀ
                     \6     a4ij6x4j7 ,6 b4i6      ( i=1,m )
                     /
                    
                     6j=1

                       6x4j7.6v4j7.60 4  6          ( j=1,2 )

     ⮬ 砥 襭   來  60-
. 6 0室   ࠭祭 6 0ந 㣮쭨  -
⨬ 襭.  ,      窨  ᪮  㭪 Z(x)
ਭ 6 0䨪஢ 6 0祭 Z40. 6 0⢮6 0 6 0祪,  6-
, 6 0Z(x)=Z40  6 0ﬠ 6 0410410+420420+400=Z40 6 0௥ୠ 6 0
76
6C0=(410,420), 室饬 6 0  砫 6 0न.  ᫨   ।6-
 6 0ࠫ쭮 6 0ᠬ 6 0ᥡ 6 0 6 0ࠢ  ,  
㭪  Z(x) 㤥 ,   ⨢ ࠢ - 6-
                                                         76
.    אַ Z  ࠢ  6C0  
  㣮쭨 ⨬ 襭     設.
䨪㥬   אַ Z.  ⮩ 窥 㭪  Z(x)  ਬ
쭮 祭.
                                                                76
      쭥襬  אַ Z   ࠢ    6C0  
ன १  設,  室  㣮쭨 ⨬ -
襭.  ⮩ 窥 㭪 Z(x) ਬ ᨬ쭮 祭.
      ,  ﬠ Z    㣮쭨 ⨬
襭 ( 室   室 )     (設 -
쭨),   ᪮筮 ⢮ 祪 (஭ 㣮쭨).
᫨  ⨬ 襭 ,  㬠  () -
㬠 Z(x)    ᮢᥬ.
     ᬮਬ ⨯   ணࠬ஢.   -
    60⥫쭮 450 ⮭ த   ᯮᮡ -
  ࠧ ⮣ த. ᭮ ࠬ  -
 ⮢    40- ⮭ ࢮ,  60- ⮭ ன,  80-
⮭ 쥩 ࠧ த  .    ⮢  
ࠧ⥩ த  ᯮ  ਠ  ࠧ -
⭮襭.  ᯮ ᫥騬 ᠬ ਠ:  ࢮ -
100 ⮭,  ண - 150 ⮭,  쥣 - 120 ⮭  ⢥⮣ - 180
⮭.   室 ਠ  ந⢮  ⮭  
ࠧ த ᢥ  ⠡.

Ŀ
6Ŀ室 -0                                               
6  Ŀ 0 60                                               
6-0 6Ŀ⮭0   60     I-      II-      III-    IV-      
6-Ŀ0    60                                               
6 த0                                               
Ĵ
ࢠ             0.3 ⮭   0.2 ⮭  0.4 ⮭  0.4 ⮭  
Ĵ
             0.2 ⮭   0.1 ⮭  0.3 ⮭  0.6 ⮭  
Ĵ
             0.2 ⮭   0.5 ⮭  0.2 ⮭  0.3 ⮭  


     ॡ  ⨬쭮 ( ᫥ ᨬ樨 ਡ뫨) -
⢮   ⠢ த  ᫮,    ⮨-
 6 0ࠧ⥩ 6 0⮣ 6 0த 6 0ࠢ: 6 0ࢮ - 13.5, 6-
 -11.36 0 쥣 - 8.2    ⮭.
      襭  ਢ        ⥬ Eureka
㦭 ᤥ ᫥    Edit.

          6$ max (f)

          6Z(x,y,v)=13.5*x+11.3*y+8.2*v

          60.3*x+0.2*y+0.2*v 7,6 100
          60.2*x+0.1*y+0.5*v 7,6 150
          60.4*x+0.3*y+0.2*v 7,6 120
          60.4*x+0.6*y+0.3*v 7,6 180
          6x7 . 640 : y7 . 660 : v7 . 680
          6x+y+v 7, 6450

          6f=Z(x,y,v)

   ᫥ 襭 砥 ᫥騥 १.

   6f 7~6 4538.9983 , x 7~6 90.001006 , y 7~6 119.99876 , v 7~6 239.99985

                   6x+y+v 7~6 499.9996

   6c  ⠭  砥:

              60.3*x+0.2*y+0.2*v 7~6 99.000023
              60.2*x+0.1*y+0.5*v 7~6 150.00000
              60.4*x+0.3*y+0.2*v 7~6 120.00000
              60.4*x+0.6*y+0.3*v 7~6 179.99961

    _뢮  (*)

                61
               6(A(x - 7a6u),x -7 a6u) - (b,x -7 a6u) =
                62
        61                  1
     6= (A(x -7 a6u),x) - (A(x - 7a6u),7a6u) - (b,x) + (b,u)7a6 =
        62                  2
    61                                        
 6= ĳ (Ax,x) - (Au,x)7a6 - (Ax,u)7a6 +5 6(Au,u)7a526 - (b,x) + (b,u)7a6 =
    62                                        

6
       ⨬,6 0 (Au,x)6 0=6 0(Ax,u). 6  0⢨⥫쭮,6 0(Au6,0x)6 0=6 0(6u0,6Ax0)
6.. 0 - ᨬ筠 6  (u,Ax) = (Ax,u)  ᢮ ᪠୮-
6 ந .
6

    61 5                   
 6= ĳ (Ax,x) + (Au,u)7a52 6 - (Ax,u)7a6 - (b,x) + (b,u)7a6 =
    62 5                   
         6( .. u = Ax - b ,   Ax = u + b )

    61 5                   
 6= ĳ (Ax,x) + (Au,u)7a52  - 6(u + b,u)7a6 - (b,x) + (b,u)7a6 =
    62 5                   

    617 6                   
 6= ĳ (Ax,x) + (Au,u)7a526  - (u,u)7a6 - (b,u)7a6 - (b,x) + (b,u)7a6 =
    62                    

    61                      1
 6= (Au,u)7a52 -6 (u,u)7a6 + (Ax,x) - (b,x) .
    62                      2

                           6
                         6
     ) ⠢    ।  筮 㭪樨.
஢    좥 ⥫  ।
 . 
                         6(7l4max6 -4 7l4min6)
                   6q =  
                         6(7l4max6 -7 l4min6)

饥 ⥫ ᪮ ணᨨ  ᪮ ன
室 6 0⮤ 6 0᪮३襣 6 0᪠. ( ⮣ ⠢ ࠢ
6det (A -7 l6E) = 0    ᯮ 楤 4 6poly(x,a4n6,...,a406)4.


     ) ਤ㬠 ਬ   ணࠬ஢    
      ⥬ Eureka.   樥 楫
㭪樨  ࠭祭.    . ਤ㬠 ࠪ⮢ -
祭 १⠬.



                      ୠ ࠡ N4
                    4

           ਡ 㭪権.  ᫥ ।
          ࢠ. 襭 ७樠 ࠢ.

                           6 ࠡ
                         6

    ਮ⥭ 몮 ᫥ । ⥣ࠫ, -
 . ࠢ  ਡ 㭪権 ⮤  -
⮢   ⥬ Eure6k0a.

                     6᪮ 
                   6

           I)6  ᫥ । ⥣ࠫ
          7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
    ꥬ १  ⥣஢  [a,b]   १ [x4i-10,x4i0]
窠 a=x400<x410<...<x 4n0=b.  ⥣ࠫ ࠧ  ⮬  㬬
⥣ࠫ:
                              6n     x4i
                 6Ŀ      Ŀ7   !
              6I = \    I4i6 =  \7      26  f(x) dx         (1)
                  6/          /7      2
                 6      7   1
                             6i=1    x4i-1

6稬: f4i6=f(x4i6) , f4i-1/26=f(x4i-1/26)4 ,6  x4i-1/26=(x4i-1 6+ x4i6)/2
 h=x4i0- x4i-10 㤥  ﭭ.
    )  6 0ਡ 6 0 6 0⥣ࠫ     (1)  -
頤 אַ㣮쭨, ᭮ 6 0ண 6 0१6 0 [x4i-10,x4i0],  -
 ࠢ f4i-1/20.   稬 ਡ ࠢ⢮:

                          6n
                        6Ŀ
                  6I 7~6 h  \      f4i-1/26 = I46               4 6   (2)
                         6/
                        6
                         6i=1

    㫠 (2)  뢠 ⠢6 0 ⭮ 6 0㫮6 0 אַ-
쭨.
    )   ਡ    ⥣ࠫ6 0 6 0  (1)6 0 -
頤 4 0࠯樨, ஭ 4 0ன6 4 6 [x4i-16,x4i6] , [x4i-16,(x4i-16,f4i-16)] ,
6[(x4i-16,f4i-16),(x4i6,f4i6)] 0  6[x4i6,(x4i6,f4i6)] .0  6 0⮬  稬 ਡ6-
 ࠢ⢮:6
                                6n
              6               Ŀ    
              6 f406 + f4n6        \     f4i6 
        6I7 ~6 h   +    /          = I46               (3)
              6    2              
              6                i=1      

㫠 (3) 뢠 ⠢ ⭮ 㫮 ࠯権.
      )  ਡ 4 0 ⥣ࠫ 4 0    (1) 6-
頤 6 0䨣, ᯮ 6 0 ࠡ, 室饩 १ 窨:
6(x4i-16,f4i-16) , (x4i-1/26,f4i-1/26) 0 6(x4i6,f4i6)0. 6  0᫥6 4 6 0⥣஢6 4 0
ᮮ⢥ ८ࠧ 砥 ਡ ࠢ⢮:

                          6n            4     6 n-1
           6     4 6   4 6  Ŀ       4     6 Ŀ   
        6h0  6    4 6   4 6    \            4     6 \       
   6I7 ~6   f40 6+ f4n6 + 4 /     f4i-1/26  +  2 /    f4i6  = I46     (4)
        660  6      4 6    4 6       4     6    
           6       4 6    4 6i=1            4     6i=1    

    㫠 (4) 뢠 ⠢ ⭮ 㫮 ᮭ.
      ) 6060  ᯮᮡ 5 0᫥5 0 । ⥣-
 5 0 筮 5 0 5 0筮.  襭 5 0  ᯮᮡ
⠪:
                 6M42 6(b - a)                          M426 (b - a)
  6 I - I4 6 7,6 4 6h52 6      I - I46 7 ,6  h526 ,
                     624                                  12

               6 M426 = m a x  f5''6(x) 
                        5[a,b]

                 6M446 (b - a)
  6 I - I46 7 ,6  h546  ,  M446 = m a x  f5''''6(x) 
                   628805                     [a,b]

 )    ⥬ Eure6k0a   । ⥣6-
 ᫥ ⮤  ஫ 譮 १.  
 譮, ⥬  6 0 ᫥.6 0  ਢ6-
 ,  室 ᤥ   Edit  ᫥6
⥣ࠫ:
            49
            7!    |\\\\\\\\\
            72   /5 6   x53
            72  /5 7\\\\\\\\\\6 dx
            71 ?6   5 6x546 + 31
            57.3

              6(     Edit )

               6y(x)=sqrt(x^3/(x^4+31))
               6z=integ(y(x),x,7.3,9)

    ᫥ ⮣    祭  襭6 0  6 06 0  
ப   ⨢஢ 㭪 Solve.
     騩 6 0ਬ    6 0 㭪樨 6 0짮⥫,
ᮤঠ饩 6 0᫥ 6 0थ 6 0⥣ࠫ. 6 0  砥 -
 6 0,  ன ꥪ⮬6  0 6 0ﭭ ᪮७.  
⠪ 6 0 6 0 6 0ᯮ 㭪   Graph  6 0 6 0ந
䨪 ன .

               v(t)=7*t+5
               s1=integ(v(t),t,0,4)
               s(tp)=integ(7*t+5,t,0,tp)

    ᫨     㭪権  ᫥6 0 ⥣ࠫ 6 0 6  0ந
 6 0㭪樨6  0짮⥫,  6 0 㦭 6 0室6 0 㭪樮-
 6  0ᨬ    ᫥  뢠饩 6 0᪮ 6 0ᯨ᪠6 0 -
⢮.6 0  6 0⨢6 0 砥  6 0  6 0    譮
-  ᮡ  ६.
    6 0 6 0 ⥬ 6 0Eure6k0a 6 0 6 0   ⥣ࠫ.
6 0 ⮬    ॡ6 0 ᯮ짮 6 0 6 0Interate 6 0
筥 6 0᫥, 稢6  0- 60௠ 6 0६.

     ᫥  ⥣ࠫ:
          46 8                   6 42 4 6
          7! !                   6 7! ! !
          72 26  e5xy6 dxdy    7    6 72 2 26 xyz dxdydz
          71 1                   6 71 1 1
          55 7.3                 6 51 3 5
      ।஢   ᠭ ᫥ ଠ-
:
          6I=integ(integ(exp(x*y),x,7.3,8),y,5,6)
          6P=integ(x*integ(y*integ(z,z,1,2),y,3,4),x,5,6)


       II) ਡ 㭪権 ⮤  ⮢.
      7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
    )  7 0㭪 7 0y=f(x)  7 0⠡楩 7 0ਡ 祭
y4i7~0f(x4i0),7 0i=0,1,...,n4  0祭  訡  7e4i0=y4i5*0-y4i0, 6 y4i5*6=f(x4i6)
ᯮ㥬  பᨬ樨 㭪樨 f  :

          6y7 6=7 F4m6(x)7 =6 a407f406(x) + a417f416(x) +...+ a4m7f4m6(x)

     6 7f406(x),7f416(x),...,7f4m6(x) 0- 6⥬  㭤⠫  㭪権,
6a406,a416,...,a4m6 -0 6᪮  0ࠬ 6 0, 6 0騥 樥⠬
饭 童7  F4m6(x).
        ⥬ ਡ ࠢ:

              6a407f406(x406) + a417f416(x406) +...+ a4m7f4m6(x406) 7~6 y40
              6a407f406(x416) + a417f416(x416) +...+ a4m7f4m6(x416) 7~6 y41
              6.......................................
              6a407f406(x4n6) + a417f416(x4n6) +...+ a4m7f4m6(x4n6) 7~6 y4n

              60  筮  : Pa=y.

     6 0⢥ 6 0 6  0 6  0롮 6 0ࠬ஢ 6 0a400,a410,...,a4m  0
⮤  ⮢ ᯮ min S(a,y), 

               6n0   6   m                52
             6Ŀ  Ŀ              
    6S(a,y) =  \      \    a4j7f4j6(x4i6) - y4i6   ,   a=(a406,a416,...,a4m6)
              6/      /                 
             6                
              6i=00  6  j=0               

    ⥩訩6 0 ᯮᮡ 6 0襭 6 0⮩  6 0⮨  ᯮ짮
室 ᫮ ६㬠 6㭪樨  S 0:

                    6dS
                   6 = 0  ,    k=0,1,...,n
                    6da4k

    6   ந    冷 㬬஢  -
6室  ⥬  ࠨ᪨ ࠢ :

     6m0   6    n            4    6       n
  6Ŀ   Ŀ       4    6       Ŀ
   6\        \7     f4j6(x4i6)7f4k6(x4i6) a4j6 = \4 6    y4i7f4k6(x4i6) , (k=0,1,...,n)
   6/        /           4    6       4 6/
  6            4    6    4 6
   6j=00   6   i=0           4    6      i=0

 4 0뢠 ଠ쭮 ⥬ ⮤  4 0⮢.
 筮   ⥬   ⠪:

      6P5T6Pa = Py   5  6稬:5 6 P5T6P =    P5T6y = b , ⮣

                       6a = b                       (5)
᪮ ࠬ 6a406,a416,...,a4m0   襭 ⥬ (5).

     ᫨6 0   பᨬ樨6 0  㭪樨  y=f(x)  ᯮ6 0 
g(x,a), 6 0 g(x,a)6 0  6 0 6 0ࠬ஢  6 0a400,a410,...,a4m0,
 ਬ   ⮢ ਢ   -
। ᪮ ࠬ஢  ᫮ 㬠 㭪樨

                      6n
                    6Ŀ             4 652
           6S(a,y) =  \      g(x4i6,a) - y4i6
                     6/                 4 6
                    6             4 6
                     6i=0
    4 0  4 0쬠 㤭  襭    ॡ4  0ᯥ樠
⮤ 4 0樨   4 0宦 4 0ࠬ஢,   4 0 
 4 0  4 0 4 0ᢥ  4 0.  , -
, ᨬ y  x    y=ae5bx0,   a>0. 4 0䬨
 6 0ࠢ⢮, 6 0 室 6 0 6 0 6 0ᨬ 6  0ln6(0y6)0=ln6(a)+bx
稭 6 Y0=ln6(0y6) 0 6 0 x.
    ) ᬮਬ ᪮쪮 ਬ஢ 襭 㪠  6 0 
⥬ Eure6k0a.
    _ਬ1
       祪 6 0(x410,y410),(x420,y420),...(x4n0,y4n0)   6 06-
 樥 a  b  ᨬ:

           6y(x)=a+bx  ..7  f406(x)=1,7 f416(x)=x               (6)

⠪, ⮡ 6 0 ﬠ 6 0y(x)6 0 諠  "" 6 0祪 6 0 6 0訬
6騬  0। 6 0⪫祭  .  ᨬ (6) ਡ-
 6 0⮬ 6 0  6 0筮 ࠢ⢠ 6 0 ᯮ짮
 :=.6 0 ⮡ 6 0⤠ ⥬ 6 0 6 0 襭 ⮤ -
 6 0⮢ 6 0室 6 0㪠 6 0४⨢ $ substlevel=0.  
न 6 0  祪  ⠪6 0:6  0(7,46.05),(9,7),(11,8),(15,96.07),
⮣   Edit   ᤥ  ᫥ 

    f(x):=a+b6*x
    f(7)=4.5:f(9)=7:f(11)=8:f(15)=9.7
    $ substlevel=0

᫥ 6 0⮣ 6 0祭 襭6 0 60  6 0 6 0ப
  ⨢஢ 㭪 Solve.
    _ਬ 2
      ᨬ   ᫮ :

    y(x)=ax530+be5x0+c6(1/x)7 6 0 ..  7f400(x)=x530,  7f410(x)=e5x0,  7f420(x)=1/x

     ⮬    Edit 㤥  ᫥騩 :

    f(x):=a*f1(x)+b*f2(x)+c*f3(x)
    f1(x)=x^34 0:4 0f2(x)=exp(x)4 0:4 0f3(x)=1/x
    f(3)=74 0:4 0f(4)=5.74 0:4 0f(5)=4.74 0:4 0f(6.3)=6.4
    f(8.1)=7.544 0:4 0f(9)=8.74
    $ substlevel=0

    _ਬ 3
     6 0 6 0ᨬ6 06 0:6 0y(x)=e5t6,5 6 t=ax5n6+b
Ec   Edit ᤥ :
    f(x):=exp(a*x^n+b)
    f(1)=1.49:f(2)=2.35:f(3)=4.26
    f(4)=8.59:f(5)=19.01
    $ substlevel=0
,   ⢥ ⢥  祭 祭:
      a=0.25247859 6, 0b=0.14432727 6, 0 6n0=1.4951058
     6C0⥬7 0 Eure6k0ca  室    ࠬ    7 0 
᫮ ᨬ⥩ ਬ y(x)=ae5-bx0+ab.
    E  릭  பᨬ樮 ᨬ6 0-6 0쭠
ᨬ y(x)=a4m0x5m0+...a410x+a400. ⢮  60 祪 
ॢ6 0 m+1. ᫨ 6 0 6 0ࠢ 6 0⮩ 6 0稭,  6 0ॠ6 0 
ॣ,  筠 쭠 பᨬ.


           6III)  襭 ७樠 ࠢ.
          7\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
    ) ⥬ Eure6k0a 6 0 ᯮᮡ  6 0襭  . ࠢ.
,    ⥬    60 6 0⮤
60.   6 0⮬6 0਩6 0 ࠭60 ᪮쪨 (7-8) 窠,
᪮  6 0 ⨢6 0砥6  06 0 ⥬6 0 ८ࠧ뢠
६  ⠢    㣠   ௠.6 0⢨
 몥 ⥬ ஢6 ࣠樨 横 ᫮ . ஬
6⮣ ६  ⥬ Eureka ८। ,  -
6 ਬ஬.    Edit ᠭ ਢ  ଠ-
6.
            6y=7
            6p(y)=sin(y)*exp(y)
            6p1=p(4)
            6y1=y
            6y=847
            6y2=y
     6 ⮬   Solution ᫥ 襭  ᫥騥 -
6祭 ६ : y=7 , y1=7  y2=7. 祭 p1=p(4)  㤥  -
6 ୮.

    6) ਡ 襭    ⮤    砥
6 ਡ 襭  . ࠢ y4x5'6=f(x,y(x)), 㤮⢮饬
6砫쭮 ᫮ y(x406)=y406. 筮 襭   ⮨  -
6஥ 4  6⠡ 4  6ਡ 4 6祭 4  6y416,y426,...,y4n 6 4 64  6 窠
6x416,x426,...,x4n6.4  6 ᥣ x4i6=x406+ih4 6, i=1,2,...,n. 窨 x4i6 뢠
6㧫 ⪨, a h - 蠣 ⪨ (h>0).
   6 ⮤  y4i+16   㫥:

               6y4i+1 6=4 6y4i 6+4 6hf(x4i6,y4i6)   , i=0,1,.....

   6  ⮤ ⭮  㯯 蠣 ⮤,    
6 窨 (x4i+16,y4i+16) ॡ ଠ ⮫쪮  ᫥ -
6᫥ 窥 (x4i6,y4i6).
   6⮤    . ।, 
6⭠ 窠 (x4i6,y4i6)  ᪮  ⥣ࠫ쭮  ਢ.   -
6⥫쭠  ⮩ ਢ,  室  १  (x4i6,y4i6)  ।
6ࠢ:

  6z(x)=y4i 6+ y4x5'6(x4i6)(x-x4i6) , .. 5 6y4x5'6(x4i6)=f(x4i6,y4i6) 4 64 6 x4i+16-x4i6=h ,

              6   z(x4i+16) = y4i6+ hf(x4i6,y4i6)=y4i+1

    6) 襬  ⢥ ਬ ᫥饥 . ࠢ: y5'6-y = e5x
6  砫묨  ᫮ﬨ  x406=3 , y(x406)=y406=4e536 ,    e7 ~6 2.71828....
6 襭 ⮣ . ࠢ  㭪 5 6y = (x + 1)e5x6.
6롥६ 蠣 ⪨ h = 0.05 .   ⮬ 砥   襭 . ࠢ-
6 ⮤    Edit   ᤥ ᫥ :

         6f(x,y)=exp(x)+y
         6h=0.05 : x0=3 : y0=4*exp(3)

         6y1=y0+h*f(x0,y0)   :   x1=x0+h
         6y2=y1+h*f(x1,y1)   :   x2=x1+h
         6y3=y2+h*f(x2,y2)   :   x3=x2+h
         6y4=y3+h*f(x3,y3)   :   x4=x3+h
         6y5=y4+h*f(x4,y4)   :   x5=x4+h
         6y6=y5+h*f(x5,y5)   :   x6=x5+h
         6y7=y6+h*f(x6,y6)   :   x7=x6+h


                             6
                           6

       6)   筮  襭 ।   . ࠢ.
       6)        ⥬   Eureka   筮  襭
6. ࠢ ⮤ .
       6)  ਡ 襭  . ࠢ  -
6 ଥ ᯮ   ⮣ ।  பᨬ樮 -
6ᨬ.
       6) ।  頤,  祭   䨪 筮 
6䨪0  6ਡ 襭  १ 筮 襭.
       6) । 頤, 祭   䨪 筮 -
6襭      0x   १  筮  襭.
       6e) ।    業  ⭮襭  頤, ᫥ 
6㭪 (),  頤, ᫥    㭪  ().

