Upload
alexandra-costache
View
227
Download
0
Embed Size (px)
Citation preview
8/13/2019 Sinteze Matematica Aplicata in Economie
1/69
=
( ) =
( ) =
! !
( ) = ( ) = = ! """ #
$" # $$" ! #
! "#!$ " " $" % & %$" % "
%
$' ( )( )$ + ( " " $"
) $* ) *)" +$"
( )+
=
& & &
& &
& & &
"
$" " '
+
=
==
8/13/2019 Sinteze Matematica Aplicata in Economie
2/69
$"% " " $" %" % $"( & " % $") $"* & & $"
" $" " $"* , - $* - *
) ))
= = =
=
$"% $ "- -" ( +# , # '%-" %
- %-" % -) + #$" - + " &
'+ +
$' ( )( )$ + ( . "
' $* $* ( * *
$* * $*
$"%
" ( ) ( ) $
=
," ( ) ( ) $ + = %
" ( )) ) )
= ( ) $
" ( ) ( ) $
=
& "# '(%) "# '( ( ) $ -
( ) $
$. + = =
/ ' 0 - ( ) $ &
= =
1 , 2
*('3!# 3
1& ( ) $ , '
" ( )
=
," ( ) ( )
=
8/13/2019 Sinteze Matematica Aplicata in Economie
3/69
(
" ( )
=
+ ( ) $ , , "111
*) # "#! ( ) $ ( ) $ # /
#/1 4# ' / ( ) $ ( ) $
#/ 1( 4# ' / - ( )$ ( )- $ #/
1-1
#) '( "# ,'-"%% . , '
"0 ( ) $ + !'
$ 01 = $ ! & =
2 $2 $" 1
$"0$"$" 2 .1 #'1+ ( )+ $ + 5 ,# '
$2 %0 + $ %0 312 % 3 1 % +
#'2 $ 4 011"1 12014112 1111
+ .! "#" %
'" ( 5 ' (( #
!" & $% ,4 $* 5
( ' & '' , ' ' & ( "
*(' / $ ! "
4"4" # % $" + ,! !
' ! # % $ 4"4" ,## 67
+ ! # % . ! 8
$" $" "
& 0
8/13/2019 Sinteze Matematica Aplicata in Economie
4/69
)
+ $1 $# & ,#
/ $1 $# & ,#
0 $1 9, $ -
,# *(': # "
'' " ,' * 6( * ,'. 6
* , 4 ' #
) * , " 7
* . ' 4 ( ( "
% $* " 5' !! """ ( 6 *"
$% 3 ( ( ( ( (
$(#%8 !
*(' #
1 !
3 9 , !" ! 9
/ "# $"$.$!%3 +$"%
1&
" " !
8/13/2019 Sinteze Matematica Aplicata in Economie
5/69
*
11 2
!"% # '
" + 5 /
= = =
=
&
"
' $0" # $0" , # " /$0" .
5 " & $ ! - &"
7 ' * 5 5 5
&& & , ' , 5 5 5
. .( $ #" # # + !
, # + !
# # "
' ' ' 4 "
1(" , # ! $$. # # 3#$(#" .! !
'
(" 5
= =
=
# (" ,# $ $ ,
$% 4# $8 ' *9 ' " " ' '
)" 5
= =
*(' (" , " $ #
&" ,#'
/
5
) ))
) ))
= =
=
=
=
=
;
& 1(" , # ! $$. # # 3 #$(#" $! $#56
8/13/2019 Sinteze Matematica Aplicata in Economie
6/69
&$' ' ( ' ' ( - ' ' ()
5# #
+ 1(" , # ! $$. # # 3 #$(#"
# ' 5 /
= = =
=
& &
$% : ' ' ' ( ' (' ; "
5# # #
*(' # $ 4 " ,# . .
! '
( )5 5 5
& & &= 5
&
( ) ( )= =
5 5 5 5 5
&
& & & &
! " # $%8 ' 4"'
4 " 3 ' ( ' ' ' 4 " ,'' . % 1'' ; ."
*(' - ) & ,#
5)" 5) &
8/13/2019 Sinteze Matematica Aplicata in Economie
7/69
?
/ 1(" , # ! #
*" 5
=
=
5 *" ! #'
8/13/2019 Sinteze Matematica Aplicata in Economie
8/69
D
& &
&
&
'
'
'
. , $ $ $ - 1 ,!# E . $9 + ! $ , & ( . , ( "
* + , ! 5, . - C
. 5, $ # 9 # (
# ) . $ # 9 - ( , ' %
- - # - # $ , % ' "
) $ # " 1 $ % $ % $ $ $
*% & '
" $. % $ ; % , $ $ % 1 >"
;
," 1 >"
=
! $ !# % % "
" 1 >(" $ % "
$# , %" ! $ !# %
+ ,! ! ,#' 8 *% 6 4 ( (
' ( *%6 ( 5 . *%&"
+ , ' * & & ' ?, . # ' , ? 5 *%& . - . * ."
%" # - $ # ,
% *% ' ' . . ,4 ( 5 *%"
3' . . ( # ,
8/13/2019 Sinteze Matematica Aplicata in Economie
9/69
F
. ' . ,4 ' . . .
' . ., . ' "
# %"'5 )
= = &
/. - 9 ,#'
( ) ( ) + 5 5 5) ) . ) ' ( 5 ) = ! # %" )=
3 $ ! ( ) $ ) , ,# . 59 '
( ) ( ) + +
! " #$! %&'()
* 3 ' . 7 *
. @" A&B 7 7 7 ( ,; *7 & (
.#@@" 5,'
A "A '6 BB $.'" =5 5 5 7 #" & 5 5 & 5# 5 7(" ( ) ( ) 5 5= 5 7#
? ( ) 5 C 5 C+ = + 5C 7
4 # 4 # 7 1 7" * E # $ -
# 9 $# # 9 "
+ #
( ){ } 5 5 5 5 = = # '
@%A' ( ) ( )5 C 5 C 5 C 5 5 5 C + = + + = = ( )= C C @ A' ( ) ( ))5 )5 )5 5 5 5 = = )
5# 6 + ' ( ){ } 0 0 5 5 5 0 = = = 1 ( )5 5 $ 5 5
, 0# $" # $# ! #
-0# # # $ 0 ! # 0
8/13/2019 Sinteze Matematica Aplicata in Economie
10/69
&
+ . /
7 # { }% = # 7
7 . 5 7 *(' , . 5 (
,; .5 = + + + #
# ,# % . ' 7(
' 7 . 7 ' ": # #
"
* 5 { } ( ) ( ) ( ) & & & & & & = = = !
#0
+0# { } 5 5 5 ! # # $0
,+ # $0"
{ }+ = = + !# " & $ ! #
8 { }% = . ' . 7"$'
( ) ( )1 % 1
= = =
=
%"'41%"71%" 7 # %! 7
( ) ( ) ( ). . 1 1 . . 1
" ( ) ( ) ( )5 1 1 1 5 = + ( ) ( )1 1 & =
7 # % . '. 7
. ' ( + + + = = . ; &= = = =
8/13/2019 Sinteze Matematica Aplicata in Economie
11/69
+B { } 5 5 # # $0 !
, B { }+ = = # ( )$ 0 !
7 # *0 { } # 7 #
{ } +05 { } 7 .,0 -0 ,
10 % { } # # 7 % - # ,# #
# 0*" # 0 # G +" E# " # 0
#
'
H " # 0
, $ .
8 7 ' . [ ] { } = ' . 7" % ' [ ] ' . 7
[ ] ( , 7# [ ] 7
* [ ] { } = ! ,! # 0 (
+ [ ] { } + = = = ! ,! $ # $0"
* 7 # /
7 - ,!++1 # 1 ,!
,8 7 ' . [ ] { } = ' . 7"
[ ] , 7 [ ] 7# 3 5 7 5 5 5 '( ,;
5 5
=
=
"
8/13/2019 Sinteze Matematica Aplicata in Economie
12/69
% 5 5 5 * '
5 , [ ] { } = "'
8/13/2019 Sinteze Matematica Aplicata in Economie
13/69
(
57 5 5
=
=
5 C .
=
=
[ ] ( ) [ ]/ 1 /.
=
[ ] [ ]/ / . 5 $ ,! IJ
I.J1 ,! IJ ,! I.J
11 ,! I.J ,! IJ
7# : 7 :7 ' : ' .
7 : '. , ' (' 7 ' .
D ' : 7 ( ' . ,'
: 4 :#
!" : ) ) :" 7# :,# : ' :
7"
: IJ7 ,# 7 ,# # 0 5,#'
( ){ }0 5 5 0 = & & ( ){ }& & = 0 5 5 0 ,# # 0
0# # # ! ,# # 8
7 # " ! ,# # 7
! "
7 #
[ ] { } = ,! $ 7 7 - $ ,! [ ] ' = + + +
#'
[ ] { }. = + ,! $ 7 & '5 5 5 5 = + + +
5 C . C . C . = + + + ( )5 7 '
C 5
C 5
5
= = = +
+ ! $1 , .
5$ ,! [ ] $ ; ,# $ $ ,! [ ] $ $ ,! [ ] 5 ,# 5$ ,! [ ]. /'
8/13/2019 Sinteze Matematica Aplicata in Economie
14/69
)
[ ]
[ ]
5
5
5 5
5
5
5
. 5
5 5
5 5
5 5
5
5 5
5 5 5
&
&
&
&
&
+ + +
+ + +
# '
( ) 5
= =
=
&
' ( )D 0 = $ ,! [ ] { } 0 = ( )D 0 & = $ ,! [ ] , '
5 D 5 D 5 D D &+ + =
/. ,# $ ,! [ ] ,!
{ }D D E D& $ ,! { }D D # , ( ) &
- $ 1 D '
" , D& . ,
," $ ! ,
7E # : " , ' . 7 E (' E7( '
'' """ C(5(C5( +=+ 7C5 "" 5)()5( = 75 ) # " " # ! # (
3 (' . " #
8/13/2019 Sinteze Matematica Aplicata in Economie
15/69
*
B "5(C = $#. C E ! 7 (C 5 ( "
( 5% C" ( 5" % ( C" 5C73 # E7( ' '
" 7((
=
==
( 7 E' '"( 7" E,"( 1 " 1( " 7
' ( ) { }K 7 E ( 7 E ( '= + B L KBL" 4B"
( 0 0 ' --"-%--(-"D # D D '
( ) D ) ))
)
=
=
D ))
)
=
= "/#' ' 77 5" F 57 # "/#' 7' 77 7" 7 # " 7
!"! # $ %" !"!
( )( 7 E 7 E K # '
& @ % A ' ( )K7 E
( )K7 E ( )K7 E ' ( ) +( 7 E ( 'K( )( ) ( ) ( )7 E ( 5 ( 5 5 5 7 + = +
0 @ A ' ) ( )K7 E ( )G 7 E '
( ) ( 7 EK ( ) ( )( ) ( )) )( 7 E )( 5 )( 5 = ' 5 7 5 ' ( )( )(+
!'$' ( )( )K 7 E + ( ' . ( "
( )( 7 EK
( )
{ }6( 5 7 ( 5 F= =
( ){ }M( C E C ( 5 5 7= = ("
6( ,# # 7 M( ,# # E
4 7" $
8/13/2019 Sinteze Matematica Aplicata in Economie
16/69
&B " ( $ ! #'B" {-=-8
||-1||
8/13/2019 Sinteze Matematica Aplicata in Economie
39/69
(F
.8#% 0 ' ' ( 7* * ' 7 ' . 4 @ ' MB"
/8 /B - B
# $ / B/" /& M/ # /
/ /
&
/ / # 9 . -4B" 8
# 98 # " # 9
# 9: 8 B
# # - / B/ "
# # /
, * " / 9 $ # / . -4L" 8
# $9: 8 " / B #
. / . E/ 0# / " : 8 # / /8"
B # # - -" /% /
# / 8 - B&" # #
/ - 0>& $. ||-||
8/13/2019 Sinteze Matematica Aplicata in Economie
40/69
)&
# ' 8 8 -S" -S" # ' V 88 " # [-S"-S"] " " " " 2
# ' 8 8 !-S"B ' -! S! 2 , ! $ - S"
('
-! -%-S" 1 -S" S! -S%S" 1 -S" 2 #('! -%-S%S" 1 -S" ,# $ ' ! -! % S!M ! #
# ,
) 8# 9 8# ## 8# 8 3&," # ' 8 !-S".8# " " 9 ; >&
5 *>6 ( ,; , |54|
8/13/2019 Sinteze Matematica Aplicata in Economie
41/69
)
"5
( 5
"C
( C
," #
," ""5 ' "5 C
( 5 C
"C 5
( 5 C
#
"( 2# ##
'8 8
!-S" 3&,".8#8' ( 9 ; " "5
C
( 5 C (
=
3 (' ( , ( "
"!, -%- S,%S # # '
5
C
=&
&
&
.8#8 ( 0 0! D6*" (' *0 *5(*5 , (' ( ( , 5 , D6" ( (, . C , D6
* ., 4.8#(., 4 '88 !-S" 3," .8# 8' ( 9 % * : ,
D* (' *5(*5*. *C(*C . , *. 5
" "
" "
"5 C
( 5 (
5 .
( C (
C
("1 , 9 % ( :(' (*5C , D*
5L*(5L*
(
5
5
"
"
CL*(CL*
(
C
C
"
"
# # ,#'
-V3" "
5
5( D
5&SV3"
"
C
C( D
C&
# -S" $ - S" / , # $ - S" $ -S" /
"!, # . ' (5L ! . C
. # , -/ # $ S ' 8 8( Y-S!" 3,"
# '
-V3" "
5
5( D
5& SV3"
"
C
C( D
C&
!V3" "
( D
&
"(., 4 # !4
8/13/2019 Sinteze Matematica Aplicata in Economie
42/69
)
'8 8 # 9" !-S" # $." -VSV - #
, , -# # $. # -V SV
- # $. # -V SV # /'
= = = = ( (5 ( ( (5 C (5 ( 5 5 5C ( 5 C
" "
= = = = ( (
C 5 ( (
(
C (C5 ( C 5 C ( C C
"
"
#' W-S WS- % + # #
(" , # ( # # 1"
' 8 8 !-S" 3,"
K3"
( D ( D
( D ( D5 5C
C5 C
" "
" "
# $ 3,"1 &;7 # ' 8 8 !-S"
#' ( ( ( (5 C 5C C5 $ 3-S" $
' = ( D ( D5C C5 " "
"!, !# E 59Y!
# (( .8# # 8# ," # ' 8 8 !-S"
# # $ - 3,
- ! # # -VSV
.8# % "(*5C ", D*5C ! , '
! -V- % SVS % - % S & . -S &
# #" # " !" #'
! -V- % SVS ! !
5 5
C C+
# # # $ 3-S" "E $ ,$ '
" "
(
5 5
(
C C
5 5
C C (+ = +
' 8 8( Y -S!" #'
(
5 5
(
C C
(
+ +
8/13/2019 Sinteze Matematica Aplicata in Economie
43/69
)(
# ' 8 8( .8# # !4 # ' 8 8 ! -S" $
# # # -S" #'
" "
5 5
C C (+
+ ! '
( ( 5 ( 5 C ( C5 5C C
= + +
+ # ' 8 8 Y --" # . '
" "
5 5
5 5
5 5 (
+ + +
+ ( '
+ + + + + ( 5 ( C ( ( 5C ( 5 ( C5 C 5C 5 C
( ., # 8# 4! # ' 8 8 -S"-S" # # 'V 8 V8 !" # V# ' ! [-"-S"] , - S # 8 '
= +
= +
( ( .
( ( .5 5 . 5
C C . C
+ -"-" # ![-"-"] $ ,
- ' = + ( ( . .
"!,3 # , Y --"
- -S" ,#'
= =
= =
F F
5
5
5 F
F
5
5
C5
C
+ - -"
=
=
F F
5
5
5
+ 2# 4 ' 8 8 !-S" # -S"& # '/ 8/ 8
S-" $. -/ --"" 9 : #-S"& # --"" & -/
.8# 8' C(*5 ( ' 8*5C6 "" "
:8 8*5C (' ( /H0! * "
"8*6#!"8*5C 85L*5C 8CL*5C . :7 **:7/H#
8/13/2019 Sinteze Matematica Aplicata in Economie
44/69
))
9"8CL*6
@"5 . :6 . 767 (' C(*5:676( ,;
(* 8*5(*5 6 5:6#@@"(' (*5 . :6
=
C 8 5 C
8 5 C5
C
" "
"!, # -S!"& ,# # !-S"
# $ ! !V &'
=
=
8
8
8
855
C
C
8$-( # ' 8 8 S-" .
% $1 - 3, S3-" $ - # -" . $ ! # -"'
3"" 37"" 7"" 7
-# # -"-" 3-" % 8-"
8-" E. '3-" &% -1" % -1"
% % -1"
# & # # ,#'
-" " %
"
_ " "
"5
( 0 5
+
=
0 5 5
(
"
"
"_ " "=
+
+
+
-" %-1" &
8/13/2019 Sinteze Matematica Aplicata in Economie
45/69
)*
8
"_ " " "
"
5
5 C
C ( 2
+ +
+
%-1" ],%-1," & 6 *.(6
4 5 ( D5
"
8/13/2019 Sinteze Matematica Aplicata in Economie
46/69
)
% " . + . %&>
" >%.. +=+=+
! % ) . ( ) 0 )"%# ,
%))) == "#" " " $%3 .
>65 * ( ,; ( >* ( .
|&& &!& """" & &|
8/13/2019 Sinteze Matematica Aplicata in Economie
51/69
*
. ! - $. > ' '
.
.
.
! $. '
.
.
.
.
+ + +
+
'
.
. 8 5 !
.
- ! ,
$ ( $: !/'
- $. )
8 5 !'
)
= -"
7 7
$ ( 3 ! - $. '
)
+
8 5 !'
)
+
= -"
7 7
"$ # 2 7 ! - $. '
)
+
"
>
8/13/2019 Sinteze Matematica Aplicata in Economie
52/69
*
)
"+
)
65 *5" ( ,; .|(*54(*5 |*5" M (' *( " (' (
>6 5 * ( ,; >* (5
|(*5 4 (*5"|
8/13/2019 Sinteze Matematica Aplicata in Economie
53/69
*(
3 ( ( 5 " " * ( "
,# ( " $ 5"" #
% (' ( ,4
( " ."
% (' ( $
( " ."
$' 5 ( .
( "
% (' ( ( (' ( ' *% . ( 5
"% (' ( " ( (' *%
( . ( (' ( ( "$ " ( # #
! > - '
|-" |
8/13/2019 Sinteze Matematica Aplicata in Economie
54/69
*)
0
) ) (
)
)
=
=
=
&
&
&
=
!"
3 5 " ,# S-1,#
C - 18%8" $
# $ , " # $% $ HS 01 # 8-""
! ,# HS # $ - P
-" " % "
_ " "
5
(
=
$ -" &,# ! # $ 01 $ ;
&
&'()** *+&&,)*(%&
- , # # # #
, $ . # . . - # # # , # # # # # + # , !. ,! 3! $ . ! # #
# # ( & = =558
# 5 5* ""=
55 = 8
# #093 # # ! #
@(5 = "= # 5#
@(55
+= ""&
&
. +' . 5 # #
@5(5 = ""= ( # M I--J4# ,
(5
5"
" =
. & 6 , @
8/13/2019 Sinteze Matematica Aplicata in Economie
55/69
**
=
5
5
@(
&&
""
"
. =5
5
5N
&"
"
[ ] 555 - #
@(N5
= ""
&
5 ! #
@55 += " = %' E # '8b8 " -"%-cG& # --" # #
=+
5
5
05
&&
&&
"
-5 +=
" 0--5 += +' # '
=
55 =
9 # I99J 3. * O I99J . ,#5 # , ,
= "= # 9 ,# ## , , 5 ! #
@5555 ++= "=""=
5# 4# 9 @5555 +++= "="===
E ,# 5 = """== ++=
8!. # ,# #
" 0-5 - = (+' ('
#'
""= 55 += # IJ 3 ! # $#
-
&
" 6 IJ" ,#
=
5
&&
"-"""- =
#5
8/13/2019 Sinteze Matematica Aplicata in Economie
56/69
* 0 > U
5 9 { }( )>$
$$ ' # #' " $=& "
> =+
0# 0 > - $ # ! - # Q ##" > 0 $ > U $ + > #
8/13/2019 Sinteze Matematica Aplicata in Economie
60/69
55 =
! 5& M # $$>55 '" &
" 5& # # ! -5 , # , # !
3 , $ # $ !$ $ # ! 3 $ H ! ! !
3 $5 -" { }( )$ >55 = " : # ! 9, 1 " 9, 15 - >& G & $. ""
&55 > =+ > E >&
1 $9 -
$ # ! # # " $9 ! ! $ !"
,# 5 0 { }( )$ %5 " %5 >
%% " " - 3*5& ,# 5& 9,
00(5(5 =
'" (*5& h, ($ 5& 5
/5(5 " &=
! $ 9,5&: { }( )0 $1 0: { }( 0 $1 07 -
9 00 ' *: 7 $. , : , 1 7
/ $# #$9 0 # * P #
-& 9, i & % # ! ! (*5&3 5& & & 49,
9, &+
: 9,5& #
{ }
{ }&&&
&&&
"&"""
"&"""
55:5:5:5E5E
55:5:5:5E5E
%
%%
===
===
8/13/2019 Sinteze Matematica Aplicata in Economie
61/69
0
&8 > 0
,"
,"
8/13/2019 Sinteze Matematica Aplicata in Economie
62/69
8/13/2019 Sinteze Matematica Aplicata in Economie
63/69
(= ! &