Ejercicios Resueltos de Funciones de Varias Variables PDF

Title Ejercicios Resueltos de Funciones de Varias Variables
Course Cálculo II
Institution Universidad de Castilla La Mancha
Pages 50
File Size 2.2 MB
File Type PDF
Total Downloads 71
Total Views 141

Summary

Ejercicios Prácticos con respuestas...


Description



 1  

  

 !"





 # $ %#&

   '   '  

%'&

 

' '   ( )

* %& +, %& - %& +,. $ *   

  /01      ' 

2







 

 !" 

 01    '   

  



S

3

 !" 



,0%&$    '  '

  2  



4





 

 !" 

%'&*3 



' /01      )





S

5

 !" 



045  

 ' '   ( )

 '    ,01   (

 '

+$   6   



6



' 

 '    





 !"

 



   

  6     ' 7           26

   

6/06/08 /06 $ 

  ' ' 9 *$   6         '  9    6  6   6   6  26# 6        '       

 '6 #  6   26#  6    '6 #      * 

#       '  '  9     



       9   '  9      '



'   '

'

 '    '    '  9   



 ' #         : '  '  9       '



  26#  #  #  '   9   '  ' 

 ( $

  (   (    % 6  &    9   9   2 

  

 

  

 

&4   26 2     26 2  



&    6     6  



&  26 2     26 2  ; 



S

7

 !" 





*  &  %262&  

 2

 %2  62&   %262&  

%2   &2(  %2   &( 2   

 2

%2   &9  29 

2  

2

 2  (

2

  %262   &   %26 2&  2  2%2   &(  2( %2   & 2 29  %2   &9      2  2    2  2  (   %26 2&  

&* 3.   6       6



 



 

%(  (  9 &% 9  9 &  %(   (  &%9' & % 9   9& ' %( 9   (&% 9   9&  % (   ( &%9 ' & % 9   9 &'

 

  6     % 6 &    2

8



 %  6  &   % 6  &  



 !"

 

(

 



(

                2 9 9       

 





(

 

   

(

  

 9 9               ' < !/   2   2

 2



 &

      %262&     





 %262  &   %262&      %26 2&     %26 2&           %%2   &(  (29 % 9  %2   &9 &  %2%2   &(  2( %2   &&%2&       %29  %2   &9 &'  =   #    =      %26 2&      





  %2   6 2&    %26 2&     %26 2&    %26 2&            %(%2   &29  %2   &( &%%2   &9  29  %%2   &2(  %2   &( 2&%2&      %29  %2   &9 &'   =    #    =

 > $.$/%262&0$/%262&    $                 $ $ ' 

                       N                      N 

 

  $ .               

     26 2  #  26 2  6   9 ' 

 

1              / .       ,        

18





 !"

 



 # 26 2   9 ' 



 

 ' '  #  6 #  26 2    ' '  # 26 2 

 

  



  #   26 2       

 

..

  '   '  9 '

     ' '   6     6  9 ' 9 '   ' ' 



  

 

16$

     9   6   6        .       #   %#6#6#&   '  $   6 6     1#2     %#6 #6 #&  $ $  $      6  

   #6#6#   # #6#6# 6  

'

       #6#6#   # 

      #6#6#   # 

#      '   #6#6#   2 

   #6#6#  

 #6 #6 2 6  6 6  



'

 * 



S

19

 !"





$   %  '   '   '  #  &     

'

    ' '  #  ' '  ' '  



'  #



'

$ %   2   &

 A>   ' #' $      6  $  .     6  

%#6 '&  '   9   (   #  ?*          $   .%#6'&%96(&;O$   >.    9  #6 (  '    '6' 

 

 

 # #  6     ' ' 

 %#6'& '  9  (  # '

# 9 #        %& ' ( (

.    #6 '   #   #6'   '      #6'   %&   ..  #  #6 '  ' 

 9  #6'    ( 

. $/%&%& >    6 #6'   

    #6' 6    #6' 6     

' #9 *    (    $

20



# 9   # #  < < '      6  6  6 (   ' '  B ' ( '



 !"

 

  9       6     ''  '     2   



 6     26 2 



 6     26 2 

 &$$ D&!$        '  &$$  &   26 2      26 2   &1%'68#&.$:2P -K 

 *  &$%/6&$.. '  '  '    2 ' '    '  '  2 '  

 )

  '  B '     '  '

 6$$$0/08'/  D&! 

 6  262     '

  6     2

9 ''     '

'



     '







S

21

 !"







 2

9 ' '    '   (  ' 9    '  9 

9

 

(  9

 2

 #

 &*!  

 6  262 

  6   

 2

    #6 '

9

 

'

'

 2

'       

 '  ' 

 2

 @.$ .. ! I&8  16$6.$$ 5  &  N  26 2  

  6 2     26 2  

 2

9 ' #   '    2  '

1 NN  26 2   

  6 2     26 2  

 2



 N

   26 2 

  N

 6 2 

 •

N  26 2  



N % 6 & 

  26      26 2 

 2

 

 2

2 2  2 

  



 9  '   



'

' '   '   

 N  6 2  

( '

 '  '



 9  '    '

 ' '   '   

 6    26 2  

#  (

36 N  26 2   

 N 6 2    N   26 2 

 2

22





# 2   (  )*    2





 !"

 



 NN  /   26 2  

 &  1 %'68#&6 $%/6 &  $6  .     $    

 16     '6 #   #  '6 # 6  

 •

 # '6 #  

  N  6   

' (  9 ' '  ' 9

 N  '6 #  

 ' '   '    <

'

 N   6   

9

'  9! (  '! ' (  B# )

9  '   

 '

' '   '   

 N   '6 #  

B '  '  B#

  2

 •

      # 9      6    6   9 9   ' ' 

  6     '6 #   

 (   # 9  '   6 2 6  6      )  )  ' ' 

 H "5      $   6      #( *$  9 9     6    26 2  6          '   '   2    6     26 2    %& 6$6.5/$  6    26 2  3 %&

? $ ;

%&

$ 

%&

%&/    26 2  6$ .     26 2  /





S

23

 !"





 * %&*     6   ,$    26 2   

   6       26 2  

 2



 9   9    9 '

 

2   9    9 



 2

 %&

    #  26 2   26 2   26 2    

   9

2  6 2    26 2  '   26 2          #         2 2 2      

 26 2     2

  26     26 2



24



 

 2

 

  16 #   26 2     

2 

9  '  #  #   2



%&

 !"

 



> $   $     .     $ 

!.

        26 2  #   26 2  6      6            

%&    26 2    9    9   %& >       26 2  9    9     $ 

 6

    9   9 6  .  6   6      ,   '

N     9     9 '      9            2     #  2 '     2      9   N    2 '  9  (     2  ' '                  '   '   '            '    '  '     '        '  '   '           '   '   '   

  26 2     2 

  2   6 2    26 2

  26 2     2 

  26 2      26 2



 !   2   

 2



 

 2

' 

2 !   2  ' 

2 

 2

2 

2 2   2

 2 

2





S

27

 !"





1/ '    %& .    6   ,

$     26 2  

 2    6 2      26 2  

 2



  !   !   '     '

 

'

'

'

      

 !  !  '   

2



 2

'

 

2     '  



 2

  > 6$$      6  

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