

                         EUREKA.

                              

    ⥣஢  ⥬ Eureka ।祭 
襭   祭 ᫮       ⥬᪨
.
       ⥬ Eureka   ᫥騥 :
               1) 襭  ࠢ;
               2) ᫥ ୥ ;
               3) ᫥ । ⥣ࠫ;
               4) ᫥ ந 㭪樨;
               5)  ६㬮 㭪権  
                   ६;
               6) 襭 ⥬  ࠢ;
               7) 襭 ⥬  ࠢ;
               8) பᨬ 㭪権;
               9) ௮ 㭪権;
               10)    ணࠬ஢;
⥬ ꥤ: ।, ᫨⥫, 䨪 (஢
ࠢ쭮 ᫥), ⮢  ⮩ 䮯-
⥫.⥬  ਥ஢    IBM PC XT  AT   
ࠧ    ᪥ ꥬ   360 .  ⥬
 ࠡ      ⥬᪮  ᮯ,  
 ᯮ짮 ⥫쭮 蠥 ᪮ ࠡ.

                          㧪 ⥬

    室 믮 䠩 eureka.exe.
    ᫥ ᪠  ࠭      ⠡ 窨
⥬. ࠭ 뢠 ࠧ   :
        Edit -     ।஢ ⥪ ;
        Solution -  뢮 १⮢;
        Report -  뢮   ᫥  ࠭,ਭ
                   䠩  ७ log;
        Verify -  ஢ન 筮 १.
      ᨢ ﭨ   ୮ ࠬ,  -
⨢ - . ᯮ  ⨢ .

.
                             - 2 -

                            ⥬

    ஬ , ⠡ 窨  ᮤন     ப
.
      孥 ப 窨  樨 ᭮
 ⥬:
     File - ࠡ  䠩;
     Edit - ।஢ ⥪饣 䠩;
     Solve -  ᫨⥫;
     Commands - 롮  ࠢ;
     Report - ⮢ ;
     Graph - 뢮 䨪  ⠡;
     Options -  権 ⥬;
     Window - ࠡ  .
室   ப  믮 襩 ESC.
      ப  뢠  ࠡ  祢묨
蠬  (hot keys). Ee ᮤন    ᨬ
 ० ࠡ ⥬.訩   ப  ।⠢-
  ० ।஢. ⮬ 砥  । ᫥-
騥 :
     F1 - Help -   ⥪ (     -
                 樨   );
     F2 - Save -  ⥪饣 䠩  ;
     F3 - Load - 㧪 䠩  ᪠;
     F5 - Zoom - ७  ⨢     ࠭  -
                 饭  ( ୮ ⨨)   室
                 ࠧࠬ;
     F6 - Next - ४祭 ⨢  ( 横);
     F7 - BegBek - ⬥⪠ 砫 ;
     F8 - EndBek - ⬥⪠  ;
     SCROOL - Size/move -  ࠧ   .
     ⨥  Ctrl  Alt ਢ  ᢥ稢  -
祢 .
     Esc - ⬥  (室  饥 );
     Alt+E - 室   ।஢;
     Alt+S -  襭 ;
     Alt+C -  ஥ ;
     Alt+X - 室  ⥬.

.
                             - 3 -

                        樨  䠩

     ᫨ ⨢஢  孥 ப   File, ᫥
  Enter ஥   ᫥騬 㭪⠬:
     Load - 㧪 䠩;
     New - ⮢    䠩 (⪠ );
     Save -  ⥪饣 䠩;
     Directory - ᬮ ४ਨ;
     Change dir - ᬥ ⥪饩 ४ਨ;
     New directory - ᮧ  ४ਨ;
     Rename - २ ⥪饣 䠩;
     OS shell - ६ 室  MS DOS (  
                Exit);
     Quit - 室  ⥬  砭 ࠡ.

                    ஢ ⥪ 

     ᫨ ⨢஢   孥 ப  
 Enter,      ।஢ .

                        襭 

    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 ⮤

.
                             - 4 -

                ਢ ⮫쪮      襭,
                  宦 㣨  ᪫ -
                    .    
                  .   ⮬ ࠤ  ᪠ 
                襭  ⠭:  radius = ⢨-
                ⥫쭮 ᫮.  㬮砭 ࠤ ࠢ .);
   Iterate -  権 ᫥ ⠭ 襭 ( -
                 筥   襭 
             ᫮,     筮   ⨣, 
             ६ ⢥   襭 祭).

                        ନ஢ 

     ᮤন: , ⨭ ணࠬ, १ 襭 
 䨪樨  䨪  㭪樨.
       孥 ப ( Report ) 뢠  ᫥饥
:
    Go - ⠢   (१ ⮩   
           Report);
    Output - ࠢ  뢮  (࠭, ਭ);
    Formatted - ଠ஢ ;
    Capture -    䠩 eureka.log (  
               EUREKA.LOG EXIST.A TO ADD,E TO ERASE
                䠩      . 
               祭      ப  ४祭
               㤥  ON,  OFF);
    Logfile name -   log-䠩.

                         ஥ 䨪

      ⮩  樨  孥 ப ( Gragh ) ⮨ 
 㭪⮢:
     Plot - ஥ 䨪 ;
     Output - 뢮 䨪  ࠭  ਭ;
     List - 뢮 ⠡ ;
     Function -  㭪樨,   ந.

     襬 ᫥⥫쭮 ⢨, 室  ஥-
 䨪 㭪樨  ஡.
     ᮡ N 1

.
                             - 5 -

    ⨢ (..      Enter)  㭪
孥    - Graph.   襬  -
 㭪 - Function.   ᫥ ⮣ ப -
   襩 㭪樨 (ਬ y(x)  ab)   En-
ter.     ப  । 襩 㭪-
樨 (ਬ sin(x)+x^2)   Enter.  ᫥ ⮣ ⨢-
 㭪    - Plot.   ப -
  砫  ࢠ  ஥  䨪   Enter.  
 襥    ࢠ   Enter. 
१   ᫥ ⢨  ᯫ  ,
ᮤঠ饥 䨪,  믮 ᨬ ᥢ䨪. ᫨ -
   F5,   䨪    ࠭  
⨭ 䨪. ୮ ⨥ F5 ਢ  饭 -
࠭   ﭨ,⢮襥  ࢮ  ⮩ .
䨪   ᮢ   ࠭  ᨬ ᥢ-
䨪,  ᫨ । F5 뫠   F4.   ⮬,   ⮣
⮡   ०, 騩 ᯮ짮 ⨭ -
, 室  F7.

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

.
                             - 6 -

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

                        ࠬ ⥬

    쬠  孥 ப (Options)  ᫥饥 -
:
     Variables -   祭  ६   宦 
                 ।;
     Settings -   ⠭  ⥬:
               accuracy -  譮 ᫥;
               complex [yes/no]  -    ࠬ஬  yes  ࠧ蠥
               ᫥    묨  ᫠;
               casefold [yes/no] -    ࠬ஬   yes  ⬥
               饥    㬮砭  ࠧ    ய-
               묨  묨 㪢;
               digits - । ᫮   १⮢ -
               ᫥;
               substlevel=n -    ⢮  ८ࠧ
               ६, 室   ६  ⮬-
               ᪨ ࠦ  १  㣨.  n = 0 ⠪
               ८ࠧ  믮. ⨬ 祭
               n: 0,1,2,....,6.   㬮砭  ⠭ ࠢ 
               .  ᫨    蠥  蠥 -
               ,   ஢ n  㪠 ।  拉 
               砥 蠥 . ,     N14   
               ᠬ⥫쭮  ࠡ  ४   ⢥
               ࢮ ப ⨭  $ substlevel=2 .
                   ஬ ᫥,  㭪  ᮤ-
                  ⠭,  祭   
               㧭, ᯮ짮 襩 F1 (.. Help).

     olors - ⠭ ᪨ , ࠬ  ⥪⮢;
     Directories - ⠭ ४ਨ (⥬  ⤥ 䠩-
                     ࠭  ࠧ ४. ⮬
                   砥 㦭 㪠 ⥬,  室 
                   䠩  䠩  ਬࠬ ⮢.);
     Load SETUP - 㧪 ⠭筮 䠩;
     Write SETUP -  ⠭筮 䠩.

.
                             - 7 -

                          

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

                          ⥬

     Eureka  ᫥騥 ࠭祭:
     - ᨬ쭠  䨪  40 ᨬ,  10
        ᭮묨;
     - ᫮ । 짮⥫ 㭪権   10;
     - ᫮ ᯮ㥬 ᫮ ⠭   200;
     - ᫮ ६   12;
     - ᫮ ⠭  ६  㣨  6.
      ⮬  ᯮ짮 ⠭  ६ 
     㣨, । ᢮   筮 襭.

     䠢 ⥬ Eureka ᮤ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 䬥᪮ 樨 -
 ७   ப.
     Eureka  ந ᫥騥 樨:
    + ᫮;
    - ⠭;

.
                             - 8 -

    * 㬭;
    / ;
    ^   ⥯;
    ()    ਮ 権;
    < ; > ; <=   ࠢ; >=   ࠢ.

                            㭪樨

     Eureka  㭪樨 re(z)   im(z),  騥  ⢨-
⥫    ᭮ ᫠ z=x+iy.  । ਬ-
  㭪権 室    ४⨢:   $ complex=yes
    i^2=-1  i = sqrt(-1).
abs(z) -  ;  exp(z) - ᫥  e=2,71828...  ⥯ z;
floor(x) - 楫   ;   ln(z) - ᫥  ࠫ쭮 -
䬠  z;  log10(z)  -  ᫥   筮  䬠  z;
sqrt(z) - ᫥  ⭮  z;  pos(x) - 頥
  >0  0  ⨢ 砥; sgn(x) - 頥: 1  >0,
-1   <0  0   x=0; atan2(y,x) - ᫥ ⠭  
न⠬  x      (㣮 祭     १,
  ண  (0,0)    (,));   polar(x,y)  - ८ࠧ
⮢ न  ;  sin(z), cos(z), tan(z) - -
: ᨭ, ᨭ  ⠭ z;  sinh(z), cosh(z),tanh(z) -
᫥   ࡮᪨:   ᨭ,  ᨭ    ⠭ z.
஬  ᫥    㭪権  Eureka    㭪権
  楤:
     fact(n) - ᫥ 䠪ਠ ᫠ n;
     ncum(x) -  ᯥ樠 㭪 訡 ()  -
쭮 ।;
     sum(f(i),i,n,k) -  㬬 f(i)   i, -
饬  n  k.

      ⥬ Eureka 짮⥫    -
室  㭪樨 १ 騥 ஥. 㭪樨 짮-
⥫   :
      㭪樨 (ᯨ᮪ ६) = ࠦ
     
      㭪樨 (ᯨ᮪ ६) := ࠦ
      ଠ ᯮ,  ᫨  㭪樮쭠 -
ᨬ ᬠਢ  ਡ.

.
                             - 9 -

 Ŀ
           ਬ  蠥 ⥬ EUREKA.            
        -------------------------------------------           
 

                  ਬ N1
                ------------

    ࠢ: e5(x^2)0-5x+1=0.
                  襭
  ࠥ    Edit:  exp(x^2)-5*x+1=0.  ந  ⢨ 
ᠭ  㭪 " 襭  " (     㤥  -
 "   ").

  訢  砥   Solution:

                Variables         Values
                     x    =    1.3086594

  ⤥    ஡  㣮  -
, ..    Edit: (exp(x^2)-5*x+1)/(x-1.3086594)=0 
  . ᪠  㣮  襭  ⠪  -
  㭪   Find other  ⠭  radius.

                 ਬ N2
               ------------
  ᫨ ୨  x560-x540-x530+3x520-1, ..  ࠢ:
x560-x540-x530+3x520-1=0.
                  襭
   ᫥ 祭,   ⠪  ⢨⥫  
୥   ⥬  Eureka  ᯥ樠쭠 㭪:
poly(x,an,......,a0).
  ࠥ   Edit:
   $ settings        ; 砫  ⠭
   complex=yes       ;   묨 ᫠
   accuracy=1.0e-9   ;  筮 ᫥
   digits=8          ; ⢮   १⮢ ᫥
   $ end             ;   ⠭
   i=sqrt(-1)        ; ।  
   p(x):=poly(x,1,0,-1,-1,3,0,-1)

   訢  砥   Solution:

.
                             - 10 -

                 Roots to the polynomial p

        #      Real part        Imaginary part
        1      0.69807525        0.0000000
        2     -0.54737816        0.0000000
        3      0.94982970        0.6507578
        4      0.94982970       -0.6507578
        5     -1.0251783         0.9608054
        6     -1.0251783        -0.9608054

   ᫥  宦  ୥ ᤥ 롮 ஢. -
 , ⨩  ⢥ ୨  .  ⮣ ᤥ
  Edit ᫥騥 :

      $ settings
      complex=yes
      accuracy=1.0e-9
      digits=8
      $ end
      i=sqrt(-1)
    a=0.69807525
    z1=a^6-a^4-a^3+3*a^2-1
    b=0.94982970+0.6507578*i
    c=0.94982970-0.6507578*i
    z2=b*b*b*b*b*b-b*b*b*b-b*b*b+3*b*b-1
    z3=c*c*c*c*c*c-c*c*c*c-c*c*c+3*c*c-1

   訢  㡥  ⮬,  祭   ࠭-
  窠  ࠪ᪨  ࠢ .    ᮦ  㣠 ଠ
  ࠡ  묨 ᫠   ⥬ Eureka  
ਢ    訡筮  १. ᫨  Eureka  뤠 ᮮ-
  " Error 5: too many formulas ", ஢塞  ୨   ।
ﬨ, 㯭묨  ࠡ⪨ ⥬.

                 ਬ N3
               -------------
                                                    4_____
    ᫨   ந  㭪樨   f(x)=3lg(x)-7?0(x/2)+x52 0  
窥 0,5.

.
                             - 11 -

                 襭
   ..  ⥬ Eureka   ࠡ⠥  㭪  
ࠫ ,  ࠧ    १ ⭮-
襭  ࠫ: lg(x)=ln(x)/ln(10).

    ࠥ   Edit:
          a=1/ln(10)
          f(x)=3*a*ln(x)-sqrt(x/2)+x^2
          x=0.5
          z=deriv(f(x),x)
    訢  砥   Solution:

          Variables       Values
               a    =     0.43429448
               x    =     0.50000000
               z    =     3.1057669

                  ਬ 4
                -----------
                                      lg(1+x)
  ᫨ ⥣ࠫ  㭪樨 f(x)=7 \\\\\\\ 0  ࢠ [0,1].
                                       (1+x52)

                 襭
   ࠥ   Edit:
          a=1/ln(10)
          f(x)=a*ln(1+x)/(1+x^2)
          z=integ(f(x),x,0,1)
   訢  砥   Solution:

          Variables       Values
               a    =     0.43429448
               z    =     0.11821420

                  ਬ 5
                -----------

   ஢,   a  <= 0,9 믮 ࠢ⢮:
         7p
         7!0             sin520(x)7                      p
         720  7\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 0 dx   =7   \\\
         710         1 + 2 a cos x + a520               2
         50

.
                             - 12 -

   ⢮ ஢  窠 a = -0,9;-0,45;0;0,45;0,9.

                 襭
   ࠥ   Edit:

    t=3.1415926/2
    f(a,x)=sin(x)^2/(1+2*a*cos(x)+a^2)
    t1=-0.9 : i1=integ(f(t1,x),x,0,3.1415926)
    t2=-0.45 : i2=integ(f(t2,x),x,0,3.1415926)
    t3=0 : i3=integ(f(t3,x),x,0,3.1415926)
    t4=0.45 : i4=integ(f(t4,x),x,0,3.1415926)
    t5=0.9 : i5=integ(f(t5,x),x,0,3.1415926)

  訢  砥   Solution:

          Variables       Values

              i1     =     1.5707963
              i2     =     1.5707963
              i3     =     1.5707963
              i4     =     1.5707963
              i5     =     1.5707963
              k      =     1.5707963
              t1     =    -0.9000000
              t2     =    -0.4500000
              t3     =     0.0000000
              t4     =     0.4500000
              t5     =     0.9000000

                  ਬ 6
                -----------
   Eureka      ᪠ ६㬠 㭪樨 
  ४⨢:  $min  $max.    ⮬,  ᫨  㭪 
 ᪮쪮 ६㬮,   宦 ⮣,   -
,   ᮢ 䨪 㭪樨  室  ⮣ -
䨪    砫  ਡ    ࠭祭    ᪠  
६㬠.  ⨢ 砥  ६㬠 㤥 ந室
 砫 祭,   ⥬  Eureka   㬮砭  
  ਢ   ⮬ ६,  ⥫  .

  ᫨  ᨬ  㭪樨 f(x)=5xe5(-x/2)0(2+sin(3x)),  祬
    10.

.
                             - 13 -

     ࠥ   Edit:

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

     訢  砥   Solution:

                 Variables           Values

                     T         =     10.629942
                     x         =     2.5805009

                  ਬ 7
                -----------
    ᫨  㭪樨  f(x)=x520+y520+z520-1.

    ࠥ   Edit:

       $ min (Fxyz)
       F(x,y,z) = x^2 +y^2 +z^2 -1
       Fxyz = F(x,y,z)

     訢  砥   Solution:

                 Variables           Values

                    Fxyz        =    -1.0000
                      x         =     6.1257e-13
                      y         =    -1.3030e-12
                      z         =    -5.9622e-14

                  ਬ 8
                -----------

          㬠    ஭ a.    
 ஡ ᫥騬  ࠧ:   㣫    १
      ஡  c   袠.  
 ஭ १ , ⮡ ஡  
⨬.     a=6.

.
                             - 14 -

    ࠥ   Edit:

     $ settings
       accuracy=1.0e-12
     $ end
     $ Max(Y)
         a=6
     G(x)=x*(a-2*x)^2
     Y=G(x) : 0<x<a/2

    訢  砥   Solution:

                 Variables           Values

                     a       =       6.0000000
                     x       =       1.0000000
                     Y       =      16.0000000

                  ਬ 9
                -----------7
                                        7(
                                        72 0 2x + 3y +  z =  11
    ⥬  ࠢ: 7 0  7*   0x +  y +  z =   4
                                        720  7x - 2y - 3z = -37
                  襭7               9
   ࠥ   Edit:

       2*x+3*y+z=11
       x+y+z=4
       7*x-2*y-3*z=-37

   訢  砥   Solution:

                 Variables           Values
                     x        =      -3.000000
                     y        =       5.000000
                     z        =       2.000000

.
                             - 15 -


                  ਬ 10
                ------------
                                             5   7 
   ᫨      A=        .
                                             2   3 

                  襭
   ⥬  Eureka    ᯥ樠쭮 㭪樨    ᫥
⭮ .      ⭮,  :  A*A5-10=E.  .. :

            5   7     a   b       1   0 
                   *          =                 
            2   3     c   d       0   1 


    5   7     a     1       5   7     b     0 
           *     =                *     =    
    2   3     c     0       2   3     d     1 

   ࠥ   Edit:

       5*a+7*c=1
       2*a+3*c=0
       5*b+7*d=0
       2*b+3*d=1

   訢  砥   Solution:

                 Variables           Values
                      a      =        3.000000
                      b      =       -7.000000
                      c      =       -2.000000
                      d      =        5.000000

                  ਬ 11
                ------------
                                        7(
                                        720 7 0e52x0 + sin(3x) - y520 = 0
    ⥬  ࠢ: 7*
                                        720 x530 + 7*y + tg(5*x520) = 0
                                        79
  砫  ᫮  x400=-1  y400=0,3.

.
                             - 16 -

   ࠥ   Edit:

     exp(2*x)+sin(3*x)-y^2=0
     x^3+7*y+tan(5*x^2)=0
       x:=-1  : y:=0.3

   訢  砥   Solution:

                 Variables           Values
                    x         =      -1.0414127
                    y         =       0.32744950

                  ਬ 12
                ------------
   祭  ᯥਬ⠫    ᨬ ⢥म  
ਭ Hb  ⥯ ଠ樨 e      ⮢ ⠫.
  ।⠢  ᫥饩 ⠡:

   Ŀ
     e   5     10  15    20  25    30  40   50   60  
   Ĵ
     Hb  130  141  152  163  170  180  194  206  213 
   

   ந  ᨬ :  Hb(e) = 118 + a*e^b.
᫨ Hb(e)  e=25  祭 ᨬ.

     ࠥ   Edit:

                  Hb(e) = 118 + a*e^b
                  Hb(5) = 130 : Hb(10) = 141
                  Hb(15) = 152 : Hb(20) = 163
                  Hb(25) = 170 : Hb(30) = 180
                  Hb(40) = 194 : Hb(50) = 206
                  Hb(60) = 213
                  t = 25 : y = Hb(t)

    訢  砥   Solution:

                 Variables           Values
                      a        =      5.2800065
                      b        =      0.70583936
                      t        =     25.00000
                      y        =    169.21009

.
                             - 17 -

                  ਬ 13
                ------------
   㭪        ⠡. ந  ௮樮
 ࠭  ᫨  祭  窠: 6,9  7,9.

         Ŀ
           x     2      4      6      8      10  
         Ĵ
           y   13,14  8,28   9,91   5,976  16,68 
         

    ᫨  ᪠  ᨬ          
⮬  ᫮ 祪    ⠡   筮  ࠢ  ⥯
  ,  ⥬ Eureka ⢫  ࠭
௮.

     ࠥ   Edit:

      L(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e
    L(2) = 13.14 : L(4) = 8.28
    L(6) = 9.91 : L(8) = 5.97
    L(10) = 16.68
    x = 6.9 : y = L(x)
    x1 = 7.9 : y1 = L(x1)

       Solution 砥

                           Solution

                 Variables           Values

                      a       =       0.08406250
                      b       =      -1.93250000
                      c       =      15.59500000
                      d       =     -51.97750000
                      e       =      68.83000000
                      x       =       6.90000000
                      x1      =       7.90000000
                      y       =       8.36504340
                      y1      =       6.11795090

.
                             - 18 -

    Ŀ
          ᠬ⥫쭮 ࠡ 㤥⮢.  
    

                    1
                 -----------

    ⥬  ࠢ:

      7(
      720           3x + 2z + v = 0
      7*0      -x + 5y + 4z - v = -14
      720          x +y + z + v = 1
      790  1,5x + 0,5y + z -7*v = -6,5



                    2
                 -----------

         A:


                   3   4   2 
                             
             A =   2   3   5 
                             
                   1   2   9 


                    3
                 -----------

   । 樥 a, b, c  ᪮ ᨬ

                    y(x) = ax520 + bx + c.

.
                             - 19 -

                     ᯥਬ⠫ 

       Ŀ
                                                  
         x   1     1,5   2    2,5    3    4    5  
       ĳ
                                                  
        y    0,1  0,225  0,4  0,625  0,9  1,6  2,5
       

    ஢  ᫥    祭  㫥    祪
x = 1,5;3  5.


                    4
                 -----------

   । 樥 a, b, c  d ᪮ ᨬ

  y(x) = aT400( t ) + bT410( t ) + cT420( t ) + dT430(t ) , :

   T400(t)=1 , T410(t)=t , T420(t)=2t520-1 , T430(t)=4t530-3t.

                   ᯥਬ⠫ 
           Ŀ
                                                      
             x    5     6     7     8      9     10   
           Ĵ
                                                      
            y    50,3  60,1  80,3  91,9  101,1  110,6 
           

     ஢  ᫥    祭  㫥    祪
x = 6;8  10.

                    5
                 -----------

    祭 㭪樨 f (x),   ⠡筮,  ᫥
窠:

.
                             - 20 -

            h         3h         5h         7h         9h
       x410+ 7\\\0 , x410+ 7\\\\0 , x410+ 7\\\\0 , x410+ 7\\\\0 , x410+ 7\\\\0 .
            2          2          2          2          2

                     室 :

                  h = 0,005   x410 = 1,335


                 㭪   ⠡楩
       Ŀ
                                                      
         x   1,335  1,340  1,345  1,350  1,355  1,360 
       Ĵ
                                                      
        f(x) 4,162  4,256  4,353  4,455  4,562  4,673 
       


                    6
                 -----------

    ⮤  ⭮  ௮஢,    祭 -
⮢ x,    祭  㭪樨 y=f(x)   ࠢ
4,21 ; 4,31 ; 4,41 ; 4,51 ; 4,61 .

                  㭪   ⠡筮
   Ŀ
                                                      
         x   1,335  1,340  1,345  1,350  1,355  1,360 
   Ĵ
                                                      
     y=f(x)  4,162  4,256  4,353  4,455  4,562  4,673 
   


                    7
                 -----------

    ।,      ୨  ࠢ

.
                             - 21 -

      x550 + 0,5x540 - 3x530 + 27x520 + 13,5x - 81 = 0

          ⢨⥫.


                    8
                 -----------

   ஢,      a  <= 0,9 믮 ࠢ⢮
         7p
         7!0             sin(x)
         720     __________________________ dx   =  2 .
         720       7|\\\\\\\\\\\\\\\\\\\\\\
         710      7?0   1 - 2 a cos x + a52
         50

   ⢮ ஢  窠 a = -0,9;-0,45;0;0,45;0,9.


                    9
                 -----------

   ஢ ࠢ⢮
             a                         a52
             7!0                      1  7!
             720  x530 sin(x520) dx   =  7\\\0 720  x  sin(x) dx
             720                      2  72
             710                         71
             500                         50
      5 0a, 饬  1  2  蠣 h = 0,2 .


                    10
                 ------------
    ।, ७      ࠢ,

            7|\\\\\\\0     1
           7?0  x + 1   - 7\\\0  = 07 0   x520 - sin(7p0x) = 0,
                         x

.
                             - 22 -

  ਭ騩  १   [ 0,7 ; 0,8 ],   ?
        ୨      筮  e = 0,00001.


                    11
                 ------------

   ।,  ७      ࠢ,
                                      7(           |\\\\\ )
               7                      720 0,7854- x7?0 1-x527 0 72
   lg(6x) - 7\\\\\\\\0=0    0,3+x-cos72\\\\\\\\\\\\\\\\\\20=0,
             6(2x+1)                  720  7   0 1-x527 0     7  2
                                      79        0        7  0
  ਭ騩  १  [ 0,5 ; 0,6 ],   ?
       ୨      筮  e = 0,00001 .


                    12
                 -------------

      襭  ⥬    ࠢ:
          7(    0                      7|\
          720  tg(xyz52 0+ 0,4) - x520  + 7?0z  - 1 = 0 ,
          72
          72
          7*0    0,5x520  + 2y520 + z520 - 2 = 0 ,
          72
          72
          720     x + y + z - 2,54732 = 0 ,
          79
    ਭ    砫쭮  ਡ   (1,1;0,5;0,95)
           筮 ᫥  = 0,00001 .



                    13
                 ------------

     襭 ⥬  ࠢ:

.
                             - 23 -

         7(0                  7|\\
         720   e5XY0 - x520z520 + y7?0z  - 1,5  = 0 ,
         72
         720          7|\
         7*0   ( x + 7?0z  - 0,5 ) + y520 - 1 = 0 ,
         72
         72
         720    x + y + z - 1,84643 = 0 ,
         79
    ਭ  砫쭮 ਡ   (0,4;0,5;0,95)
           筮 ᫥  = 0,00001 .


                    14
                 ------------

   ᯥਬ⠫  ᨬ ⭮⥫쭮 㤫-
    ⥬    ⠭     ।⠢
᫥騬 ⠡栬:

                        t   , . C
          Ŀ
                                                       
             27       327      402      477      627   
          Ĵ
                                                       
             777      927     1077     1227     1827   
          

                  ⭮⥫쭮 㤫  7d0 , %
          Ŀ
                                                       
             10       7,5      6        5,5      5     
          Ĵ
                                                       
             5        5        8       10 0      2     
          

   பᨬ஢ ᯥਬ⠫  ᨬ

.
                             - 24 -
        7d0(t) = a + b*t  + c/t + d*sin(t).

   믮  ᫥    祭  ᨬ  
   祪 t = 402; 627; 1077; 1827.


                    15
                 ------------

    祭 ⥣ࠫ쭮 ᨭ

                         x
                         7!0    sin(u)
              Si( x ) =  720     du
                         710      u
                         0
  ⠡:

    Ŀ
        x    0,000 0,500 1,000 1,500 2,000 2,500 3,000
    Ĵ
      Si(x)  0,000 0,493 0,946 1,325 1,605 1,779 1,849
    

      ᫨   Si ( 2,357 )       பᨬ饩
ᨬ.     ࠢ  ᫨  ⥣ࠫ    -
  ஥   ⥬  Eureka  㭪樨  integ(f(x),x,a,b).
 ᫥ ⥣ࠫ  ।  ࠢ 0,0000001.


                    16
                 ------------

    祭    ଠ쭮  ᪮  ⥣ࠫ
  ண  த

                   7p0/2
                    7!0   7|\\\\\\\\\\\\\\\\\\\
            y(t) =  720  7?0 1 - sin520(t)sin520(x)  dx
                    71
                    0

.
                             - 25 -

  ⠡:

   Ŀ
       t    0,000  7p0/36  7p0/18  7p0/12   7p0/9  57p0/36  7p0/6 
   Ĵ
     y(t)   1,571  1,568 1,559 1,544 1,524 1,498 1,467
   

      ⮤  ⭮  ௮஢,  ᫨7  0t
        y(t) = 1,46    1,56.
      ஢   믮   ࠢ⢠:

                     7p0/2
                      7!   |\\\\\\\\\\\\\\\\\\
               y(t) - 720  7?0 1 - sin520(t)sin520(x) dx = 0 .
                      71
                      0
        祭 祭.

.
                             - 26 -

       : ⮢  ᨫ쥢,
                 ࠨ쥢
        .  ___________. ࠦ 800 .

