Trandafir Matematici Pentru Economisti

  • View
    66

  • Download
    9

Embed Size (px)

DESCRIPTION

Manual de matematica pentru economisti. Pentru studiu universitar.

Text of Trandafir Matematici Pentru Economisti

  • !"## $%%&

    '

    ! " ()

    #)##))*+#,- . ##/) ) "# # )$%%&0'1$)234*56&7 6& 80 %29 %()#)#)#),- )*+#

    28 &:%&2728&%&27286$%&27

    '#5* ;#(53 )'(3 /'3

  • #$"">"> .>

    % &'(

    $)#*+

    " ""#$%$"C)$%%&

  • /

    AAAAAAAAAAAAAAAAAAAAAAA 33 3 AAAAAAAAAAAAAAA 3,

    33>0" #AAAAAAAAAAAAAAA 3,3>0" #AAAAAAAAAAAAAAA 353,- $#AAAAAAAAAAAAAAAAAAA *3 2# AAAA +5"#B-20# AAAAAAAAAAAA +56-# B0AAAAAAAAAAAAAAAAA 5,4C1#0>-AAAAAAAAAAAAAAAAA 5+

    , AAAAAAAAAAAAA 6+,3" $#AAAAAAAAAAAAAAAAAA 6+,0AAAAAAAAAAAAAAAAAAAA 4*,," " #$#AAAAAAAAAAAA 43,

  • +

    ,36)" 02AAAAAAAAAAAAAAAAAA 33,34)" "1##" 02AAAAAAAAAA 33,,*)" #"#B%AAAAAAAAAAAAA 33

  • 5

    0 AAAAAAAAAAAAAAAAA 330 >>!(1>#"!

    -#>!-#>AAAAAAAAAAAAAA 3""2 C- #2 "#AAAAA AAAAAAAAAAAAAAAAAA /-AAAAAAAAAAAAAA 5*+/G0-"0"""" >!-# 1#

    #> %("#"#"->-AAAAAAAAAAA 53+/,G0-"0"""" >"1%

    -# #B-""#"#"-AAAAAAAA 52" AAAAAA 55+//G0-"0"""0>02#AAAAAAAA 56

    5AAAAA 6*53>2" ->AAAAAAAAAAAAAA 6*

    533C11#$ (0AAAAAAAAA 653>2"">"0B&>"-$ "#B0-#

    0"#">2"AAAAAAAAAAAAA 6/53," $2->-0AAAAAAA 6+53

  • 6

    5>2"->AAAAAAAAAAAAAA 665,>2">AAAAAAAAAAAAAA 6452" 0;AAAAAAAAAAAAAA 43

    6AAAAAA 4,63 "2#AAAAAAAAAAAAAAAA 4,69#0>AAAAAAAAAAAAAAA 4,

    639#>""-">"> !- > -$&( #-> > 4,

    69#>""-">"> !- > -$&("->1##->>AAAAAAAAAAAAAAAAA 4/

    6,9#>""-">"> !- > -$&! #-> >!#"%B>(>"#8& >"-" AAAAAAAAAAAAAAAAAAA 44

    600"#-">AAAAAAA ,**6,9#-11#>AAAAAAAAAAAAAAAA ,*

    6,39#>""-">"#!"-B&"#>"-#">"#">!"8#"#>"-#0B&"##->>(">>H2#H1#AAAAA ,*

    6,33&"#>AAAAAAAAAAAAAAA ,*6,3&"#""AAAAAAAAAAAAAA ,*/

    6,9#>""-">"#!">>(>-#&-"#-> > ,*+

    6,3&"#>AAAAAAAAAAAAAAA ,*56,&"#""AAAAAAAAAAAAAA ,3*

    4 AAAAAAAAAAAAA ,343""1& (>#"">>0(-A ,349 >"- 0 #>>0"#"(-#

    1& AAAAAAAAAAAAAAAAAAAAAAA ,3,4,2-11#>#>>#(#>$#AAA ,3/4-"" 0(-B

    20> AAAAAAAAAAAAAAAA ,3+4

  • 4

    4/9#(-AAAAAAAAAAAAAA ,344/39#" #0 ::73:: AA ,*4/9#" #0 ::73:: AA ,,4/,9"#(-">>->>!0->$

    ;- #!">> !--"# ::7:: AAAAAAAAAA ,/

    4/>->>!0->$;- #!"> !--"# :3:73:: AAAAAAAAA ,4

    4//9#(-">>->>!0->;- #!">> !--"# ::7:: AAAAAAAA ,,

    AAAAAAAAAAAAAAAAAA ,,/

  • 3*

  • ! " # # # "" # # # #

    $% $

    &

  • 3,

    3((, (

    33"&

    >>0 " #" ">"

    !!! 3 !>>>"1 07

    33

    33

    333333

    333

    " ( " !!!3 ( !!!3 >">#!

    3

    ! !!!3 33

    >">"1 0 0# 7 33,

    "7

    3

    3

    3333

    3

    3

    9 > "0( ! 1 > "0( !I

    ""#>>0#" " ( ">"->"-"0"">>0$#"0">"#> $ >0"#" #>>0"#"

    >0"#" -">2 >(">0>>0"0!>"00"#>0>>0!> $ >0"#" #>>0"#">"0( >>0 0-1# " 0! >-$! >>0 0-1#0 G &"# B " ;> " >>0 "0 " > --!>>0"#>$"0>>00-1#

    -.$333

    J

    !7

    3

    3

    33333

    J

  • 3>>0"0

    J

    22 ! " >>0"# 333 > >>00-1#0

    ""#>>0#>-#&(-0#0 >">$9#"E">>!-#%2 #0">"#"

    9#"E">>> B> 0 #0>">$#>>0"#"B">>08$#!$#0-%%$1# " #>>0"#"";- ">2"" B "#$1#$ 2#""

    *33 !"$1# 3 -0" -$ "#3 > B0-> " - 33 )#0"# 33 >$"0C0" $$7

    33

    3

    33

    3,

    33

    3,

    33

    33

    33" 3 " # !!,! !" 33

    B0"# (-%" 3,33 !!! (>>" !-" , (01 " #7" 7

    333

    33

    33

    3

    ,333

    3,,3

    33

    3

    " 7

    333

    33

    33

    3

    ,333

    3,,3

    33

    3

    1 > #">>08$#">>0"# #B">"

    3 > # B-0" !" ""

  • 3/

    333

    3

    333

    33

    33

    3

    333

    3

    333

    33

    33

    3

    33

    3

    33

    3,

    33

    3,

    33

    33

    33/

    G-"0 ! "#"# B" "!>$

    #2>-$(!-(0 !>$"0 #0">" " #";- " "$$ "#""

    #20"#$"-% %"$00-"#0"- -"#0>">$1# >0"#33/8$#">>0"#333>-#"#(>80"H""#0-"28"#"

    >" "" ">"# ( 0 #1 >>0"#" #"#"# "" >>0 8$# > 1 > #7 # B% >B0--#0"# *33 ! 33 -$"#>B& )#0##B% >" & ### #0 ### # > #"#& 0% "-"282# >20"#"(#"##0"#"#"#(-$"#"# $ 2#" -">"# #- 2# -$"#" ( -">"# # - ## 2# ! >B0-#-$

    801 07

    33 3 3 3

    3

    3

    3 3 3 3

    * *

  • 3+

    "7

    33

    33

    -" !3 !

    33

    3333

    -" !3 ! !3 !

    33

    3333

    -" ! !

    33

    33

    G0>0#!B-#"0 >1 >>08$#">>0"# #

    G- >1 73 * * 33!3

    33

    33

    * 3 * 33!

    3

    3

    * * 3 3 3!

    3

    3

    #" >>0"#">(7

    33

    33!

    3

    333

    33!3

    333

    ( J

    22 ! >>0"# > 0-1#0

    /$ >$>>0"#7

    3 , ">!1 07

  • 35

    , < 3

    3 3 / 3 < 3 4 3 :, :3 3

    * :3 5 :, ,

    * :3 5 :, , 3 * :34 >0 #0# ,, ! ," "#!

    #20"#"0-">0"#>0-1#0

    J

    22 0"#0;0"#>- 0>">"--#>" 3 (

    #" >>0"#">7

    3 , "

    !!! 3 >>>"1 07

    33 3 3 3 3

    3 3

    3 3

    33

    ">0"#K -& ""#>0#K L>"K L>0"#" %" ">0"#K L%(K L- ">#">>0> "0""#>>0- B0"# " " "3-1 (">" #7

  • 36

    33

    33

    333333

    3

    >"

    33

    33

    333333

    3,

    ""#>>0#" 3>"3,>"#>""#"">>0 " - "! >-$ > ! # " " ">"";#!-&$"#2#&!("07

    33

    33

    3333333

    3"

    33

    33

    3333333

    3/

    " * -" !3 0 "0 >#" >>0"#" " 3! >-$ 3,! "

    >>0$#$ >0"#" #>>0"#" $

    33

    3>0"#" 3>>>"1 0 0#

    ( * >#" >">>0 *

    >0"# " > > 0 B >>0 " 3> >"1 0 0# 7

    >" " * !" ** ! >>#" >>0"#"

    ***

  • 34

    K L ** ! >#" -" >>0"# 3#" >>0"#"

    " /$ >$>>0"#" 7

    0"#" >> 0 B">>0"

    7

    *!!/

    3

    /

    3

    /

    3

    /

    3/

    5

    /

    3

    3/

    3/

    3 !>#" >>0"#"" >7

    /

    3/

    3#2"# " >-"#"0## >"-"#"0#0-#; C0"# 0! > >" - 7

    3 - " K L #2 0-& ! >& -8#0 !! ! "#0>"0 !

    - B0"# " ># KML #2 0-& ; ! >& ! -8#0 !" ! "#0 !

    1 $ # ! # " !! $ #% 7

    33 ! ! .3 && !& !! .3, ! ! !* !! * % '

    !! ** 3

  • 3

    1 (# " #### !!! 3 ! #! # !# # ### !!! 3

    1 ( # " #### !!! 3 !

    *33 ### 3,3& *3

    $)2 3$4$3$!

    !### !!! 3

    # K L ### !!! 3 $ # - " ;> >#

    !!! 3 !" "#!> #B%7*33 ###

    "*3 !"-"0>7

    #

    #

    #

    #

    3,

    3

    ,

    3

    3

    $"# 3# >>01 # ## $

    K L C>"-"0! $"# "0% $! 3# > > 01 $ ### !!! , " ;> ># !!! , " "#> #B%7

    #### ,,3 >"

    !#### *,,3 $ ### !!! 3 >"#-

    1

    ! # " ( #!* $ # ! %+

    # ) # &

    2 3$4$5 ! # " " #* !!! 3 & ! ! # !#

    #& 7 C>"-"0 $"# !# > - ;-0 B "

    0"B " $1& * ("07

    # 33

    # !!! 33

  • &%#" # 1 07 ! !!! * 333

    1& !!! 3 ! 0 !>"#- !

    ># 01 >" "# 7 33 ! ! ! ! A! ! G> !"$>>01 # " $1&

    1 ,%

    !!! 3 &

    # !# & * #! & *

    # !!! 3 >0"# 07

    3

    !

    % & '

    -> >- #7

    33

    3

    3

    (

    3

    3

    2 3$4$4$#7

    *

    *

    3

    3 !

    *

    3

    *

    !A!

    3

    *

    *

    %& & # 0 G>- "# ;> 1&2 3$4$6$( # !### !!! 3 #

    # #)!

    # #

  • ,

    7$

    #

    3

    33

    3 !

    #

    3

    !A!

    # 3

    )>"#- ("0 ;> ># !!! 3 !" "#!> #B%7

    33 3 3

    3

    3

    *

    *

    >8$#">>0"#02" ">"( " 7

    *

    *

    33

    33333

    >0"#020>#" >#" 1# ("0 2 "0 "# ">"# 2 ! >>0"# > "

    0 ( 0 >#" 1# $ >" #-

    9 >0$#7

    3

    3

    3333

    3$4$7$- # #7

    #

    3

    33

    3 !

    #

    3

    !A!

    # 3

    %& & #

  • "0(

    -"-# # . >$#--( 3$8$3$ 0 0

    !!. )7 -# # "0-# . ! B3,,!

    B1& " #* !!3 # 3+3>0>7 *( ##.

    >" !. #/ *( 3+

    "

    .

    3

    333

    !

    3*

    *3

    / (

    #

    #

    # 3

    # 3+"#>>0"#7

    (

    ((

    *

    *

    *

    33

    33

    33333

    ###

    ###

    ###

    3+,

    G> !#$"#"--" # "#>">#" #>>0"#"023+,#" #>>0"#"023+,