 . 14     
 


        ?          ,  ,    .   14    38    ,   .   :   ,   ,       .   ;      .  ,  ,      .     ,            .





 

 . 14     





  





   


     ,  ,    . -     ,      :    ,        ,        .      .

    : , ,  ,   .      .    ,       ,          .

     ,    .    ,        .       :      ,     .          .

  14    38  : 6 , 28    4 .    ,   .      ,       .       ,     .

      ,     .         ,     : 16  ,       , 14  .        ;        .




   


   ,  ,    ,     .   f(x)      x,    f  x.    ,       .

   ,    , ,     .       ,       .          .

           .       ,          .




  


x?  x?    ;  x^4   .  ?(x + 1)         .  (x + 1)/(x ? 2)    ,    .     ,    .

f?(x)        -; f?(a)      ,   .    . h  ?x   ; ?f     .    h    ,       .

  |h|  :   h  .  h ? 0,  |h| = h;  h < 0,  |h| = ?h. , |?0,2| = 0,2.          .  | ?  |    .

   ,         .    ,              .     :    h,       ,     .  ;      11.

    .         : (2; 5)        , [2; 5]   .  ?   -,    .  ?   ; =  .

            .      .       :    ,        .      .

  ,      .        x.    ,     ,      .




 





 1


1.  (2 + h)? ? 4   h  .        h?

   1 (#litres_trial_promo)

   1 (#litres_trial_promo)




 2


2.  (3/4 ? 1/2)/(1/8),   .

   2 (#litres_trial_promo)

   2 (#litres_trial_promo)




 3


3.    x,    ?(12 ? 3x).    x = 1.

   3 (#litres_trial_promo)

   3 (#litres_trial_promo)




 4


4.     (3; ?2)    4.   .

   4 (#litres_trial_promo)

   4 (#litres_trial_promo)




 5


5.  x? ? 5x + 6     .

   5 (#litres_trial_promo)

   5 (#litres_trial_promo)




 6


6.   x  (x? ? 16)/(x ? 4)?  ,    .

   6 (#litres_trial_promo)

   6 (#litres_trial_promo)




  


1.    (#case-1)

2.     (#case-2)

3.    (#case-3)

4.    (#case-4)

5.      (#litres_trial_promo)

6.    (#litres_trial_promo)

7.    (#litres_trial_promo)

8.     (#litres_trial_promo)

9.    (#litres_trial_promo)

10.       (#litres_trial_promo)

11.     (#litres_trial_promo)

12.     (#litres_trial_promo)

13.     (#litres_trial_promo)

14.     (#litres_trial_promo)




   





 1.   


:    ,     ,     .


 

   

    

  


  

  ,         .   f(x)  x  ,    f(x)   .     f  x. ,   f(3)     3      .

   ,   :   .      .       ,          .     ,       .

    a,   b.    ?x = b ? a.  ?    ,    .    :  ,      .     .

   : ?f = f(b) ? f(a).               .  f(?x)    :     ?x?      ?f    ,    .


 

 f(x) = x? ? 2x + 5    3  5/2.   ,      .    .

1.    ,   . ,      ;    .     .

2.    f(3) = 9 ? 6 + 5 = 8.  3  ,   8  ;      .

3.  : f(5/2) = 25/4 ? 5 + 5 = 25/4.     ,    .

4.    ?x = 5/2 ? 3 = ?1/2.          .

5.    ?f = 25/4 ? 8 = ?7/4.    7/4,         .

6.     ,       .    ,     :   .

:     ; f(3) = 8, f(5/2) = 25/4, ?x = ?1/2, ?f = ?7/4.

:    f(x) = (x ? 1)? + 4.   4 + 4 = 8  9/4 + 4 = 25/4.       .


  

 : : ?f = f(?x),    f(?1/2) = 25/4.

     ?1/2,        .          .

 :    :     .   25/4 ? 8,     .


   

     :  a  b,  f(a)  f(b).       .

    (#navigation)




 1.1


 g(x) = 6/(x ? 2).    4  5.   ,  ?x  ?g  ,        .

   1.1 (#litres_trial_promo)

   1.1 (#litres_trial_promo)




 1.2


 q(x) = ?(9 ? x)    5  0.      .         10?

   1.2 (#litres_trial_promo)

   1.2 (#litres_trial_promo)




 2.    


:    ,               .


 

   

 

 ,   


  

         .        a  b  (f(b) ? f(a))/(b ? a),  a ? b.     ,     .

     .             .       .     ,    ,      ,    .

 a < b  ,     .   ,     ;   .       ,         .       .

      .    ,           .     .        ,     :        .


 

    s(t) = t? + 2t,  t  , s  , 0 ? t ? 5.                   .

1.         .       1,   4;    3 .

2.  : s(1) = 1 + 2 = 3 ; s(4) = 16 + 8 = 24 .     ,      .

3.    24 ? 3 = 21 .       : 21/3 = 7 /.

4.    s(2) = 4 + 4 = 8 .   8 ? 3 = 5   2 ? 1 = 1 ,   5 /.

5.  ,    .       ,         .

6.  7 /  ,           .          .

:    1  4       7 /;  1  2   5 /.

:    a  b  s(b) ? s(a)   (b ? a)(a + b + 2).    a + b + 2:  7  5     .


  

 : :         s(4)/4 = 6 /.

         .                ,  s(0) = 0.

 :     .            .


   

      .    ,          .

    (#navigation)




 2.1


   V(t) = 30 ? 2t ? t?    ,    , 0 ? t ? 4.          .          .

   2.1 (#litres_trial_promo)

   2.1 (#litres_trial_promo)




 2.2


   y(t) = (t ? 2)? + 1,     ,    , 0 ? t ? 4.       1  3 .   ,       ?

   2.2 (#litres_trial_promo)

   2.2 (#litres_trial_promo)




 3.   


:       ,       .


 

  

   

  


  

      a,    a + h.  h   .      (f(a + h) ? f(a))/h.     h ? 0,    .  h    a,   .

,   h      .            ,       a   f?(a).    ,    .        .

        ,  ,        ,    h  .         :       ,   .

   h   ,        .       .          ,      .       .


 

     f(x) = 2x? + x ? 3   a = ?1.    , ,   .

1.      ,  ?1    .  h    ,   .  f(?1) = 2 ? 1 ? 3 = ?2.

2.   x  ?1 + h: f(?1 + h) = 2(?1 + h)? + (?1 + h) ? 3.     .

3.   1 ? 2h + h?.       f(?1 + h) = ?2 ? 3h + 2h?.

4.   : f(?1 + h) ? f(?1) = ?3h + 2h?.   ,        .

5.  h ? 0    h   ?3 + 2h.      ;     - .

6.   h    2h      .  ,    ?3  2|h|;        .

7.       ?3. ,    f?(?1) = ?3.   ,    f(?1),   ?2.

: f?(?1) = ?3.

:   f(x) ? f(?1) = 2x? + x ? 1 = (x + 1)(2x ? 1).  x ? ?1   2x ? 1    ?3  x ? ?1.


  

 : :  h = 0  0/0,    .

       .         .

 :     h ? 0.     ,     .


   

      .     ;      .

    (#navigation)




 3.1


   g?(2),  g(x) = 5 ? 4x. ,           .

   3.1 (#litres_trial_promo)

   3.1 (#litres_trial_promo)




 3.2


 p?(2)  p(x) = x?   .     ,       .

   3.2 (#litres_trial_promo)

   3.2 (#litres_trial_promo)




 4.   


:                 .


 

  

 

   


  

      . Ÿ    ,       ,     .        .   ,       .

     a   (a; f(a))    f?(a).       y = f(a) + f?(a)(x ? a).     :    .      ,     .

      .      a + h    ,   h,   . ,      f?(a). ,     ,      .

     .    ,            ,         .    .       ,     .


 

   f(x) = x? ? 4x  a = 1.          ,      .

1.   .    f(1) = 1 ? 4 = ?3,      (1; ?3).

2.    f(1 + h) = (1 + h)? ? 4(1 + h) = ?3 ? h + 3h? + h?.

3.  f(1)     h  ?1 + 3h + h?.    ?1, , f?(1) = ?1.

4.  : y = ?3 ? (x ? 1),   y = ?x ? 2.   x :      .

5.     : x? ? 4x = ?x ? 2.  x? ? 3x + 2 = 0.

6.     (x ? 1)?(x + 2).     x = 1  x = ?2;      .

7.  x = ?2    .      (?2; 0).  ,  ?2  (x ? 1)? ,  x + 2  :    .




  .


   .

   ,     (https://www.litres.ru/book/raznoe/poniat-proizvodnuiu-14-razborov-s-ob-iasneniiami-i-uprazhneniiami-74418063/)  .

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