_2#1. .

     21 _ , ,,
        _2 .

     2  0 뢠  ࢠ,ᮤঠ騩
      .
     2  .0 뢠 ⭮ .,
      ன 襭 ᠬ 窠.
     2 "+"  0뢠  -
     ᪮ ஬⮪  (;+  ).
      "-"  0뢠  -
     ᪮ ஬⮪  (-  ;b).
     2 0 뢠 ꥤ 
      ⭮⥩ +    -2 0 .
     㭪 f() 뢠2 ᪮筮 0  ⭮
     .,᫨   ᫠  >0  ப
     . . ⠪,   ᫠ ,ਭ饣
     ப... 믮 ࠢ⢮ f()< .
                   >0  U    U  => f(x)<
     ᫮2 0 뢠2 ।0 -樨 f()  .,᫨
      ன ப.. ⮩ 窨 - f() 
     ।⠢   f()=+ (),  ()-᪮筮
       ⭮ ..
                       limf(x)=

     - f() 뢠2 뢭0  .,᫨  ன
     ..  -  ।⠢    :f()=f( )+ (),
       ()-..  ...
     묨 ᫮,f()-뢭  .,᫨   ⮩ 窥
      ।   ࠢ 祭 -樨.
     2:0  -樨 뢭   窥
      ।.
        2奬0:1.- ୠ
              2. ।
              3. 뢭
              4. । ࠢ 祭 -樨
              5. 祭 -樨 ࠢ 0
              6.  ।⠢   ..
     2  :
     2६#1:0⢥ ⠭,- ..-0
     2६#2:0᫨  ()   () -..  .., 
     㬬 ⮦ ..  ⮩ .
     - f() 뢠2 ࠭祭0  ..,᫨ .
     ப ..  . ᫮ >0,⠪  f()<
       窥 ப....
                  U     M>0: f(x)<M x U
     2६#3:0᫨  () -..  ..,  ࠭祭
      ⮩ .
     2६#4: ந ..  ࠭祭:
     ᫨ -  () -.., f() -࠭祭  ..,
      ()*f() -..  ...
     2६#5: ஬筮 ..:
     ᫨  ()   () -..  ..   ()< ()< ()

                            - 2 -
      .. U ,  () -..  ...
      .. 뢠2 ࠢ묨0,᫨  । 
     ⭮襭.
     ..  ()   ()  .. 뢠2  浪0,
     ᫨ ।  ⭮襭  ᫮  ࠢ 0.
      ..  .. 뢠2 묨0,᫨
     ।  ⭮襭 ࠢ 1.
     2६#1:0᫨      - ..,  ࠧ
      ..  ᮪ 浪,祬    祬   .
     2६#2:0᫨ ࠧ  ..  ..  ᮪
     浪,祬    祬   ,        ..
     2 ᭮ ⮢ ..:
     0
     sin  
     -1  
     ln(1+)  
     (1+) -1   
     2ᨬ᪨ ।⠢:
     0
     sinx=x+0(x)
     e =1+x+0(x)
     ln(1+x)=+0(x)
     (1+x) =1+ x+0(x)
     2- ...:
     ᫨2 0 () 2 0 () -...  ..,2 0 () 2 0 () -...
      ..  . lim    =, ⮣ . lim      ࠢ .

     22 _    .

     ᫨  ()   () -..  ..  lim   =0,  ()
     뢠 2᪮筮      ᮪ 浪,0祬
      ().   ()=( ()).
     2砭:0᫨  ()- ᮪ 浪,祬  (),
       ()=(k ()),k=0
     2६ :0᫨  -७ 童, 童
       ⪠  (- ).

     23 _ - -, .

     2  業 -樨,饩 । ⫨  :
     ᫨ । -樨 f()  . ࠢ   >0,
     /2<f()<3/2  ன ப⮩ ...
     2砭:0᫨ । <0, 3/2<f()</2.
     2#1.室 ᫮ ࠭稭 -樨,
               2饩 ।:
     ᫨ - f()   窥 ।,  ࠭祭
      ⭮ ⮩ 窨.
     2#2.䬥᪨ 樨  -ﬨ,
               2 ।.
     ᫨ f()  f()  ।  .:
                      lim f()=

                      lim f()=B,

     ⮣ 1..।  㬬   ࠢ 㬬 ।.
           2..।  ந   ࠢ
             ந ।.
           3.᫨ =0, .। ⭮襭   ࠢ
             ⭮襭 ।.

                            - 3 -
     2,  :
       2.1:0᫨ - f(), ।  ., 0,
            f()>0  ப....
           ,᫨ f(), ।  ., 0,
            f()<0  ப....
       2.2:0᫨ - f()  ।  .  f()>0 
           ன ப...,  । f()>0  ..
       2.3:0᫨ -樨 f()  f()  ।  .:
                        lim f()=

                        lim f()= 

           f()<f()  ன ப..., 
           । <.
       2.4  । ஬筮 -樨:
           ᫨ -樨 f()  f()      ।
             .  - f()<f()<f()  ன ப.
           .., ⮣ .। f()   ࠢ .
     2  室  ।   뢭
               2-樨:
     ᫨ - f(u) 뢭  .U, - u= () 
     ।  ., । -樨  () ࠢ U, ⮣
     ᫮ - f[ ()]  ।  .   ।
     ࠢ f(U),.. । f[ ()] ࠢ 祭 -樨
      ।  .f[ ()]=flim ().

     24 _     e.

     2 0 뢠 -,
     । ன -ࠫ ᫠.
     㫠2 -:
                           2(a+b)=   a b
     2c=n!/k!(n-k)!
     2c0 2-0- ⠭  n  k.
     2n!=1*2*3*...*n
     20 뢠 ᥢ ⢠ 
     ⢠, ⭮ ᬠਢ ⠭ ⢠
      n-⮢  k-⮢.
     2砭: 0!=1
     2  樥⮢:
     2n=1           1   1
     2n=2         1   2   1
     2n=3       1   3   3   1
     2n=4     1   4   6   4   1
     2n=5    1  5   10  10  5  1
     2n=6   1  6  15  20  15  6  1

     lim(1+x) =e

     25_   -. - 
       _2..
     - f() 뢠2 ᪮筮 让0  ..,᫨
     1/f() 㤥 ..
     _2ᨬ:
     ﬠ  뢠2 ᨬ⮩0 ਢ L,᫨ ﭨ 
     .,饩  ਢ L, אַ  ६  0,

                            - 4 -
     .  ਢ 㤠  ᪮筮,.. 
     ﭨ   .  䨪஢ . ६  ᪮-
     筮.
     _2ᨬ 䨪 -樨:
     2६#1:0 ⮣,⮡ ﬠ kx+b 뫠 ᨬ⮩ 
     + ,室  筮,⮡ f()=kx+b+ () 
     + .
     2६#2:0 ⮣,⮡ ﬠ y=kx+b 뫠 -⮩ -
     -樨 f()  + ,室  筮 ⢮
     ।  +  f()/=k  .।  +
     [f()-kx]=b,..,᫨     ।  .,
     - .
     _2᫥  -樨  .窨
     _2ࠧ뢠.䨪 祪 ࠧ뢠:
     20:  -0窠,  ன - 
     ।,   뢭.
     21:   -0窠, ன - 
     । ᫥, । ࠢ,   ।  ࠢ.
     22:   0-窠,  
     窮 ࠭ ࠧ뢠  窮 ࠧ뢠 ࢮ த.

     26_    -  -.

     2 -0-- -樨 뢭   窥,
     .. .-  ।,  - ⠪ -権,
      ।,࠭  뢭.
     2⢠:0᫨ f() . .  f()>0, - 
       ன .. ;᫨ f()  f() .
      .,  㬬 ⮦ . ⮩ 窥.
     2 -:
     - f() 뢠2 뢭  .[a;b]0,᫨ 
     .  窥 ࢠ (a;b)  뢭 
     . ࠢ   . ᫥.
                    lim f(x)=f(a),lim f(x)=f(b)

     2 :
     2६#1:0᫨ - f() .  .[a;b]   
     १ ਭ 祭 ࠧ  (f()*f(b)<0),
      .窠   .[a;b],⠪  f()=0.
     2६#2:0᫨ - .  .[a;b]    .
     ਭ ࠧ 祭 (f(a)=f(b)), ⮣  
     ᫠ Q,饣  f()  f(b),..,ਭ..
     [a;b],⠪  f()=Q.

     2 :
     2६#1:0᫨ - f() .  .[a;b], .
     ᫠ m<f(x)<M   窥 ⮣ १ (..-
     ࠭祭)
     2६#2:0᫨ - f() . .[a;b], .
     窨 x  x  [a;b],⠪  f(x )<f(x)<f(x )  
     窥 ⮣ १.

_2#2: .

     21._  .       .0 _2-.


                            - 5 -
     १ AB 뢠 2ࠢ0,᫨ 㪠, 
     祪 A  B .砫,  殬.
      ࠢ १ 뢠 2ࠢ묨0,᫨  
         ࠫ ,-ࠢ  
      ,..᫨  砥  㣮  ࠫ.
     ७ᮬ.
     2஬0 뢠 ࠢ १.
      뢠 2묨0,᫨     אַ
       ࠫ. .
      뢠 2묨0,᫨     
     ࠫ. -.
     2㬬 ஢ a  b 0뢠 ,祭 a+b,砫
     ண ᮢ  砫  a,  - 殬 b,
      ᫮, 砫  b ᮢ饭  殬 .
     2ந   ᫮   0뢠 ,祭
      ,⠪ :
                  1. a= *a
                     a=0,᫨  =0
                  2. 
                     ,᫨  >0
                     ,᫨  <0
     2-  :

     21.⨢:
           b:+b=b+a
     2砭:0 ᫥, 㬬 ஢   b  ந
       ࠫࠬ,஥     b,
     祬 砫   ஢ ᮢ饭.
     22.樠⨢:
         ,b  :(+b)+=+(b+)
     2砭:0 ᫥, ⮡ ᫮   , ,...,
     㦭 ᫮   ,ᮢ 砫 ᫥饣 
      殬 ।饣,⮣  㬬 -몠.
     3. ,뢠 2-0,⠪    :
       +0=.
     4.   .,뢠 2⨢0,.-,
     ⠪  +(-)=0
     5.  :1*=
     6.     ᥫ      :( * )*= ( )= ( )
     7.     ᥫ      :( + )*= + 
     8.    b   ᫠  : *(+b)= + b
     2 ஢   b0 뢠  (+(-b))
     ᫨    , ,...,   ᫠   ,  ,...,  , 
        +   +...+    -뢠 2 樥 ஢
      , ,...,   樥⠬   ,  ,...,  .
     ⢮, ⮢ ண । 樨 (᫮
      㬭  ᫮),  ࠢ  ᥬ -
     (ᨮ) 뢠 2 ࠭⢮.

     22._⨥  ᨬ,ࠧ୮,  न樨.


                            - 6 -
     ⥬ ஢  , ,...,  뢠 2 ᨬ0,᫨
         ஢ ⮩ ⥬   
     ⠫ ஢ ⮩ ⥬.
       
      ⮣,⮡ ⥬ ஢  , ,...,  뫠 2  -
     20 室  筮,⮡ ⢮ ᫠   ,  ,...,  ,
      ࠢ 0,⠪       +   +...+  
     ࠢ﫠 -.
     ⥬ ஢ 뢠 2  ᨬ0,᫨    -
       ᨬ,..     ⮩ ⥬  -
       樥 ⠫  ࠢ⢮ 0  -
     樨 ஢ ⮩ ⥬  ⮫쪮  ⮬ 砥,
      樥 ࠢ 0.
     2୮  ࠭⢠0 뢠 ᨬ쭮 ᫮
       ᨬ ஢.
     2ᮬ 0뢠    ᨬ  ⥬  ஢,⠪,
      ன  ,ਭ騩 ⮬ ࠭,
      ࠦ    樨 ஢ ⮩ ⥬.
     2६ ⢥:
     ᫨    , , , ࠧ     ⮬
      ⢥:
                        =   +   +  
     ᫨      , , , 樥 ࠧ  
     ⮬  뢠2 न⠬0.
                        =(  ,  ,  )
     2砭:0   ⮣    ࠧ  ࠧ
     न.
     2᫮ ୮:
                              /  =  /  =  /
     2砭:0᫨    ஡  ⥫ 0, ࠢ⢮
     㦭  ⠪,  ᫨⥫ ⮦ 0.
     2᪮ - אַ:
                       x x /m=y-y /p=z-z /q

     23._     , .
        _2-   ...

     20   ࠬ (⫨묨  0) 뢠 訩
     㣮   砬,஢   窨 ࠭⢠ 
     ࠢ  ஢.
     2᫥ ஥樥0     b (b=0) 뢠 ᫮
     ࠢ ந    cos 㣫  .
                       =*cos a,b
           2-:0      (+b)= + b
                       (ka)=k 
     2஥樥   0 뢠  १  
     ᭮ﬨ ௥஢,饭  祪     .
     2-஬0 窨 ࠭⢠ 뢠 ,騩  
       ன 䨪஢ 窨,. ᮬ.
      ந0   b 뢠 ᫮ ࠢ ந
       ஢  cos 㣫  .

     2C-:
     1.᫮ ௥୮ ஢: (,b)=0 <=> 2_0b
     2.⨢: (,b)=(b,)
     3.:
                    3.1: ( + ;b)=( ,b)+( ,b)
                         (,b +b )=(,b )+(,b )
                    3.2: ( ,b)=(, b)= (,b)
     2ࠢ:0୮ ந ஢ ࠢ 㬬 ந
             ᮮ⢥ न.
                         (,b)=x x +y y +z z
     2ਫ:

                            - 7 -
     1.= (,) = x +y +z ,᫨ =(x,y,z)
     2.(,b)=0<=>2_0b
     3.cos ,b=(,b)/b
     4. =(,b)/b

     2ࠢ騬 ᨭᠬ 㣫0 뢠 cos 㣫,
      ࠧ  ࠬ  i,j,k.
                          cos  =x/a
                          cos  =y/a
                          cos  =z/a

     cos  +cos  +cos  =1,.. (x +y +z )/a=1.

     4._2   -.
     2楩 浪 m*n0 뢠 אַ㣮쭠 ⠡ ᥫ,
     ᮤঠ m ப  n ⮫殢.
     2⭮ 楩 n-浪0 뢠 , ன
     ᫮ ப ࠢ  ⮫殢  ࠢ n.
      . ⠢  ᮮ⢥⢨ ᫮ 뢠
     2।⥫ .
     2।⥫ . n-浪0 뢠 ᫮ ࠢ
     ࠨ᪮ 㬬 ᥢ ந n-⮢
     ,     ப   ⮫,
     祬 ।  ந  । ࠢ
     ⠢  "+"  "-".
     2ࠨ᪮ 㬬0 뢠 㬬, ன -
     ⠢ "+", - "-".
      ,  No ப ᮢ  No ⮫
     ࠧ2   .
       ப   ⮫栬  ᮮ⢥騬
     ࠬ 뢠2 ࠭ᯮ஢,0 稢 -
     2࠭ᯮ஢.

     2- :
     21.0 ࠭ᯮ஢   ।⥫  .
     22.0᫨    ⠬  ப (⮫), 
      ।⥫ 㬭  -1.
     23.0।⥫  ࠢ 㬬 ந ⮢
      - ப (⮫)   ࠨ᪨ .
     24.0᫨   - ப (⮫) 
      㬭  ᫮ k,   ।⥫ 㬭  k.
     25.0᫨   - ப (⮫) 
      ।⠢ ᮡ 㬬  ᫠, ।⥫ 
      ࠢ 㬬  ।⥫. ࢮ   ⮩ ப
        ᫠,  ண -,  ⠫ ப
         ।⥫ .
     26.0।⥫   ,᫨    ப
      (⮫) ਡ   ⠫ ப (⮫殢).
     27.0᫨   ப 㬭   ᮮ⢥騥
       ࠨ᪨  㣮 ப  ᫮,  0.
     28.0  ࠨ᪨  ⮢ -
        ப ࠢ ।⥫, ண   ⮩ ப
        ᮮ⢥騥 樥  樨, ⠫
       ப ᮢ  ப  ।⥫.

     2஬0,ᮮ⢥騬     ,뢠 ।⥫
     , ,᫨    ભ ப
      ⮫,  ⮨   .
     2ࠨ᪨ 0     뢠 ᫮ ࠢ
                               2  =  *(-1)

     2 ਧ
     2ࠢ⢠ 
     2।⥫:
     21.0᫨   - ப (⮫)  ࠢ
      , ।⥫ ࠢ 0.
     22.0᫨      ப (⮫), 
      ।⥫ ࠢ 0.
     23.0᫨  ᮤন  ப,ᮮ⢥騥 
      ன ய樮,  ।⥫ ࠢ 0.

     2室  筮
     2᫮ ࠢ⢠ 
     2।⥫:
      ⮣ ⮡ ।⥫   ࠢ 0,室 
     筮,⮡  ப (⮫) 뫨  ᨬ.

     25._    .

     2ன  ஢ a,b,c,0砫  ᮢ饭,
     뢠2 ࠢ,0᫨ 砩訩      
     b  ᮢ騬 ⨢ ᮢ ५    .
     ⨢ 砥 ன 뢠2 .
     2- ਥ஢ ஥ :
     21.0᫨ a,b,c -ࠢ, ன b,c,a  c,a,b  ⮦ ࠢ묨.
        ⠭ 뢠2 横᪮ ⠭.0.. 
       横.⠭ ਥ ன  .
     22.0᫨ a,b,c -ࠢ, ன b,a.c  a,c,b -...,᫨
        ⠬ -  , ਥ ன
       .
     2 ந0 a  b 뢠  ,⠪ :
       1.᫨   b  (b),  ୮ ந
         =[a,b]=0.
       2.᫨   b  , =[a,b] ௥७   _ b,
         ..[a,b] _ - ஢   b  [a,b] ࠢ  ⠪
         ஭, ன ஢ a,b,[a,b] -ࠢ. ୮
         ந ࠢ [a,b]=bsin ab=S ࠫࠬ,
         ஥     b.
     2- ୮ ந:
       21.0[a,b]=0 <=>ab.
       22.⨪⨢:
           [a,b]=[b,a], [a,b]=-[b,a].
       23.:
           3.1:[a +a ,b]=[a ,b]+[a ,b]
               [a,b +b ]=[a,b ]+[a,b ].
           3.2:[ a,b]=[a, b]= [a,b].

                                 i j k
                           [a,b]=x y z
                                 x y z

     2ଠ 0 -  ௥ -.
                   Ax+By+Cz+D=0 => n=(A,B,C)

     2     -ﬨ 0뢠 㣮   ଠ묨
     ࠬ.
     2   אַ  - 0뢠 㣮  אַ  
     ஥樥  -,sin ⮣ 㣫   ࠢ cos  ,   -㣮 
     ࠢ騬 ஬ אַ  ଠ ஬ -.
     2蠭 ந ஢0 a ,b ,c  뢠2 _0᫮.,ࠢ
     ᪠୮ ந ୮ ந ஢ a  b 
      .
                         ([a,b],c)
     2᪨ 
     2ᬥ蠭 ந:
     21.0᫨  a,b,c ,  ᬥ蠭 ந
     ࠢ 0.
     22.0᫨  a,b,c  , _ .ᬥ蠭 ந-
      ࠢ ꥬ ࠫ,஥   ,
     祬 ᬥ蠭 ந ⥫쭮,᫨ ன a,b,c --
     ,  ⥫쭮,᫨ ன  -.

     2- ᬥ蠭
     2ந:
     21.0([a,b],c)=(a,[b,c])
       ([a,b],c) -ᬥ蠭 ந a,b,c.
       (a,[b,c]) -ᬥ蠭 ந b,c,a.
      ᬥ蠭 ந ࠢ,.. ࠫ    
      ਥ樨 ஥  ( 横᪮ ⠭ ਥ-
      ஥  ).
      - 뢠,  ᪮   ⠢:
                         (a,b,c)=([a,b],c)
     22.0(a,b,c)=(b,c,a)=(c,a,b)=(c,a,b)=-(b,a,c)=-(a,c,b)=-(c,b,a)
     23.0 ⮣,⮡ a,b,c 뫨 묨 <=> (a,b,c)=0
     24.0 ⮣,⮡ a,b,c 뫨  ᨬ묨 <=> (a,b,c)=0
     25.ਫ:
                  5.1: (a+b,c,d)=(a,c,d)+(b,c,d)
                  5.2: ( a,b,c)=(a, b,c)=(a,b, c)= (a,b,c)
     2᫥ ᬥ蠭
     2ந:
                  a=(x ,y ,z )
                  b=(x ,y ,z )
                  c=(x ,y ,z )
                            x y z
                  ([a,b],c)=x y z
                            x y z
     26 _  .
     
     ﬠ  - - 砩 אַ  ࠭⢥.
      אַ  ࠭⢥   ଠ쭮 .
     2 樥⮬ 0אַ,   ࠫ-  y 뢠-
      ᫮ k,  ࠢ tg 㣫,   㦭 
     ⨢ ᮢ  ५ ⥫   ,  ⮡
      ⠫ ࠫ-  אַ.
                             2tg =(k -k )/1+k k
      ௥ : 1+k k=0
      ࠫ :k =k

_2#2:    .
     
     _21  . -,   .
     
     - f()  뢠  2७㥬 0 .,  ᫨ 
     饭 f( + )-f( )  ।⠢  
     Q( ) +( ), ( ) -..,     ,  Q( )
     -..  ᮪ 浪, 祬  .
                         Q( )=lim (f( + )-f( ))/ 
      । 뢠 2ந -樥  窥  0  砥
     f'( ).
     2ந -樥 f() 0 . 뢠 ।  ⭮-
      饭  -樨    饭  㬥 ,  
     0.
     ( )'= 
     (a )'=a lna, ((e )'=e )
     (log x)'=1/xlna, ((lnx)'=1/x)
     sin'x=cosx
     cos'x=-sinx
     tg'x=1/cos x
     ctg'x=-1/sin x
     arcsin'x=1/ 1-x
     arccos'x=-1/ 1-x
     arctg'x=1/1+x
     arcctg'x=-1/1+x
     sh'x=chx      (shx=e -e /2)
     ch'x=shx      (chx=e +e /2)
     th'x=1/ch x   (thx=shx/chx)
     cth'x=-1/sh x (cthx=chx/shx)
     
     f(x + x)-f(x )=f'(x ) x+o( x),
     ᫠ f'(x ) x -    ,    ᫨
     f'()=0,   ᫠ ..  浪   .
     ⮬  ᫠    ⮩ 㬬  
     뢠2 ७樠 -樨0  ..
     2७樠 0७㥬 -樨   .  뢠
       饭,     .
                          df=f'(x ) x
     ᨬ᪮ ।⠢:
                                f(x + x)=f(x )+f'(x ) x+o( x)
     f(x + x)=f(x )+df
     
     22_   .
     1. ᫨ - f(x) ⮦⢥ const,    ந
     ⮦⢥ 0.
                    (C)'=0
     2. ᫨ - u(x)  v(x) .  ., :
        1)    .  ⮩ 窥 
           ( u+ v)'= u'+ v'
        2)  ந .  .  (uv)'=u'v+uv'
                           (uvw)'=u'vw+uv'w+uvw'
        3) ᫨ ஬ ⮣ v(x )=0,  ⭮襭
                           (u/v)'=u'v-uv'/v
     3. ࠢ . ᫮ -樨.
        f(u) .  .Uo, u(x) .  ., u(x )=u =>
        f(u(x)) -.  .  (f(u(x)))'=f'(u ) u'(x )
