1o.meros μαθηματικα γενικης Κ.Ε

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    0.* A1#.!210"f, g !2../*1.1*-

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    1. * .!&"f(x0) .".!.&1"1#!21"f

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    3. * .!&".".!.&1"1#!21"f1.

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    9. * .!&" 2" 1#!21" f (x) = [ 0.

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    g (x) = 1#x 0..212$.fx) = x)1#x .

    gx) = 1#x)x. 2. * .!&2&1#.!210&x) = e

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    6. * 0/!1*2"1#!21"f (x) = [ 0.

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    21*R 2.x A: f (x) 0 +2.x A: g (x) 0

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    11. * .1#!21f0.1#0$"1210x02#0/#!1*2".

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    3. * 212$120202*2"1#!21"2"12"02!.3

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    5. * 212.!#12.!.3".!.1210"/*1#.!210&

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    6. * 212.!#12.!.3".!.1210"/*1#.!210&

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    7. * 212$12002*1#!21"#0.1212022*

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    8.* 212$12002*1#!21"#0.1212022*

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    !&210"1#!&1"1*2".21"

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    f (x) =

    [

    A = ..................................

    f (x) =[

    +

    A = ..................................

    /f (x) = [ + A = ..................................

    0f (x) =[

    +A = ..................................

    2. *

    [[OLP

    f (x) = - 2.!020..1#!+10202.

    [[OLP

    g (x))2.

    .g (x) = 3 f (x) - 1

    [[OLP

    g (x) = ..............

    g (x) = 2 - 4 f (x)[[

    OLP

    g (x) = ..............

    g (x) = (2 f (x))2

    [[OLP

    g (x) = ..............

    /g (x) =[I

    [I

    [[OLP

    g (x) = ..............

    0g (x) = [I +

    [[OLP

    g (x) = ..............

    3. * .1#!+10202..!.2&)!.

    .[

    OLP

    (2x2

    + 6x - 1) = .....................................

    [

    OLP

    [

    [+= ................................................

    [

    OLP

    (5 [ ) = ........................................

    /[

    OLP

    [(3x + 2) (5x - 3)]2

    = .............................

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    0

    [

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    [x + 31#x] = ....................................

    12[

    OLP

    [2x - 41#x] = .................................

    4. * .1#!+10202..!.2&)!.

    .[

    OLP

    [

    [= ................................................

    [

    OLP

    [[

    += .............................................

    [

    OLP

    [

    [[

    +++

    = .....................................

    /[

    OLP

    [[

    [

    += ........................................

    5.* .1#!+10202"2"2&.!.+&2&.!.2&1#.!210&

    12..212$.10.

    .f (x) = x2 f (0) =............................

    f (x) = x2

    + 1 f (1) =............................

    f (x) = 2x2

    - 3 f (- 1) = .........................

    /f (x) = x f (

    ) = .........................

    0f (x) =[

    f (0) =............................

    6.* .!0202"01+10"2&03.2&2&!.3+.!.1210&2&

    .!.2&1#.!210&12..212$.10.

    .f (x) = x2

    - 1 A (0, f (0)) y = ............................

    f (x) = 2x2 - 1 A (1, f (1)) y = ............................

    f (x) = 3x2

    - 2 A (- 1, f (- 1)) y = ............................

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    7.* +.0!.3.!12.12"y = f (x) $.!202.212$!.3

    .!12.12"!+2".!.+#2"

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    8.* 212/2.2*1#.!210!+12012122#"

    .212$#"2*#"2&!+2&.!.+&2#"

    2

    f (x)

    2

    f (x)

    x - 1

    (x - 1)2

    (x2

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    (x - 1)

    [

    [

    [

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    9.* 212/2.2*1#.!210!+12012122#"

    .212$#"2*#"2&!+2&.!.+&2#"

    2

    f (x)

    2

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    [

    [

    [

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    10. * 212/2.2*1#.!210!+12012122#"

    .212$#"2*#"2&!+2&.!.+&2#"

    2

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    ln [

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    11.* 2.!.2&../2.2*2011!&1#.!210&.1#-

    !+102021202.212$0/!1*2#"212+0

    2!+2.!&)2#".212.2/0*20!.!&)2#"

    2 2

    0/!1*

    2+

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    &"

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    h (x) =[

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    [

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    [

    [

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    !&210".2#"

    1. ** 02.1#!21f0f (x) = x2

    - 3x + 2. .!020

    .20/!1*2"

    .0"2"2#x $#0f (x) = 0

    20/!1*2"1#!21"g (x) =[[

    [ +

    2. ** 02.1#!21g 0g (x) = x2 + 2.

    .+.0"2"2#x R $#0g (x) = 0;

    .!020i) 20/!1*2"1#!21"f (x) =[

    [ +

    ii) 20/!1*2"1#!21"h (x) = [ +

    3. ** 02.1#!21g 0g (x) = x2

    - 1.

    .+.0"2"2#x R $#0g (x) = 0;

    +.0"2"2#x R 1#!21g (x) 0.02

    .!020i) 20/!1*2"1#!21"f (x) =[

    [

    ii) 20/!1*2"1#!21"h (x) = [

    iii) 20/!1*+2"1#!21"3(x) =[

    [

    4. ** 02.1#!21g0g (x) = x - 4.

    .+.0"2"2#x R $#0g (x) = 0;.!02020/!1*2"1#!21"f (x) =

    [

    [

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    5. ** 2.1#.!210"f, g 0f (x) = x2 - 4x - 2 .g (x) = 3x - 2, x R..!020

    .22*2"1#!21"f (x) + g (x) ..!1/!102020/!-

    1*2",

    22*2"1#!21"f (x) - 2g (x) ..!1/!102020/-

    !1*2",

    22*2"1#!21"f (x) g (x) ..!1/!102020/!-1*2",+

    /22*2"1#!21"[J

    [I..!1/!102020/!1*

    2",

    6. ** 2.1#.!210"f, g 0f (x) = 2x2

    - 3x + 1, g (x) = 5x2

    - 1, x R.

    .!020

    .2[

    OLP

    f (x).2[

    OLP

    g (x)

    2[

    OLP

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    7. ** 02.1#!21303(x) =[

    [

    +.!020

    .20/!1*2"

    2[

    OLP

    3 (x)

    2[

    OLP

    >3 (x)]3

    8. ** 02.1#!21f0f (x) = [

    . .!020.20/!1*2"

    2

    [

    OLP

    f (x)

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    9. ** 2.1#.!210"f, g 0f (x) = 6x3 + 5x - 1, g (x) = 2x - 1, x R..!020

    .2.[

    OLP

    f (x),[

    OLP

    g (x)

    2[

    OLP

    [J

    [I

    10. **

    [[OLP

    f (x).!0202

    [[OLP

    3(x), )2.

    .3x) = 3 f (x)3(x) = 3 f (x) - 2

    3(x) =[I

    [I

    /3(x) = [I

    11. ** 02.1#!21f0f (x) =[

    [

    +.!020

    .20/!1*2",2

    [OLP

    f (x)

    12. ** 02.1#!21f0f (x) =[

    [

    +.!020

    .20/!1*2",

    2

    [

    OLP

    f (x)

    13. ** 02.1#!21f0f (x) =[

    [.!020

    .20/!1*2",

    2[

    OLP

    f (x)

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    14. ** +.0"2"2#. R 1#!21f (x) =.[

    [ +

    $00/!1*

    21*R 2&!..2+.!+

    15. ** +.0"2"2#. R 1#!21f (x) =.[[

    [ ++

    $00/

    !1*21*R 2&!..2+.!+

    16. ** 02.1#!21f0f (x) =[

    [

    +. .!020

    .20/!1*2",

    2[

    OLP

    f (x).

    .0021020,.f (x) 0.1#0$"121x0 = 1.

    17. ** 02.1#!21f0f (x) =

    =

    +

    [.

    [[[

    .

    .+.x 3 0.1#0$"1#!21

    +..22#. R 1#!21f (x) 0.1#0$"1210x0 = 3;

    18. ** 02.1#!21f0f (x) =

    =

    [.

    [[

    [[

    .!020

    .2[

    OLP

    f (x)

    222#. R+1201#!21f.0.1#0$"1210x0 = 2.

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    19. ** 02.1#!21f0f (x) =

    =

    +

    [.

    [[

    [[

    .!020

    .20/!1*2",

    2[

    OLP+

    [

    [[ +

    222#. R+120f.0.1#0$"1210x0 = 1

    20. ** 02.1#!21 f0 f (x) =

    =

    +

    [.

    [[

    [[

    . .!020 2

    22#. R, +1201#!21f.0.1#0$"1210x0 = 2.

    21. ** /.+"0)"202!.+#0./.03!1020&"1#!212"

    /./

    .20!02!)2# 20./)2#

    22. ** 020"0#!"+0)"!!+#+ )02.2. 21+120 2 0./) 2#..!.012.0!). 10

    12 m2. .03!10202"x 2"0#!"&"1#!21 2#-

    #"y 2"0#!"+

    23. ** ."#)"2.".2."r $00./)cm2. .03!10202

    0!02!)2#&"1#!212".2."r.

    24. ** 02.1#!21 f0f (x) =[

    , x R. .!020

    .2f(3)

    21#20012/0*#1"2"03.2"2".*"2"1#!2

    1"f, 12100x = 3

    201&12".!.&03.2"

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    25. ** 02.1#!21f0f (x) = .x2, x R. R...!0202f(2).

    .!1/!10202.+1201#20012"/0*#1"2"03.2"

    2".*"2"f1210f (2)) .0.

    26. ** 02.1#!21f0f (x) = x2

    + 1, x R...!0202f(0).

    .!1/!102021#20012/0*#1"2"03.2"2".*

    "2"f12100x = 0.

    .!020201&12"03.2"2".*"2"f1210

    (0, f (0)).

    27. ** 02.1#!21f0f (x) = x2

    - 5x + 6, x R. .!020.2f(x)

    201&12"03.2"2"!.3".!12.1"2"1#!21"

    f, #0..!12.xx.

    28. ** 02.1#!21f0f (x) = 2x2 - .x, x R, . R...!0202f(2).

    .!1/!10202.+12003.22"!.3".!12.1"2"

    1#!21"f 1210f.1$.2002. xx&.

    45.

    29. ** .!00 &.#1$.20 0 2. xx 03.2 2"

    .*"#0.!.3.!12.12"1#!21"f (x) = - 2x2

    + x - 3

    1210

    , f (

    )).

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    30. ** 1 0)"2*#02. 0#*!../02.1#.!210 2#

    $!)#.)22*S (t) = 2t + t2)#2t 02!2.10sec .2S 10

    2!..!020

    .212.$*22.2#2*12$!)/12.>] sec

    212..2.$*22.2#2*)2.t = 1 sec (1 sec0220

    12#

    31.** 10)"2*#02000#*!.1/02.1#.!210

    2#$!)#t 10sec) .)22*S (t) = 3t2 - t. .!020

    .212.$*22.2#2*12$!)/12.[2, 4] sec

    212..2.$*22.2#2*)2.t = 3 sec (3 sec 0220

    12#

    32. ** 2.$*22. 0)" 2* # 02. 0#*!.. 1#.!210 2#

    $!)#t (10 sec), /02..)22*#(t) = 3t2 - 5.

    ..03!10202!#)02."2"2.$*22."02$#12#-

    2*&"!"t, )2.t = t0.

    ) .#10202!#)02."2"2.$*22."02$#12#-

    2*&"!"t, )2.t = 10 sec (10 sec 022012#.

    33. ** ."#1)"!&02.02.1#.!2102#$!)#t 10

    +!0"1*3&.022* (t) = 103 - 5 102 (1 + t)-1...!0202.!$).!)!&(t = 0).

    .!0202.!)2&!&)2.t = +!0".

    .!0202!#)02."2##1*2&!&&"!"2

    $!))2.t = 9+!0".

    34. ** #1)"."0!$"/02.1#.!2102#$!)#t 102

    .)22*t) = 10 e004t10$/0".!0202!#)02."2##1*.#2"2"0!$"&"!"2$!)*120!..)2

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    35.** 2.1#.!210"f, g 0f (x) =[

    H

    [, g (x) = e

    xx2. .!020

    .!+2.!&i) 2"f .ii) 2"g.

    ".!.+#"i) f. ii) g

    36. ** . !020 #+# (x) 2!2# .* 22 +120

    37. ** 02.1#!21f0f (x) = 2x - x2.

    ..!020i) 2f (x) ii) 2f(x)

    ../0$0)2(1 - x) f(x) + f(x) = 0, .0x R.

    38. ** 02.1#!21f0f (x) = e2x

    .

    ..!020i) 2f (x) ii) 2f(x)

    ./0020)2f(x) - f(x) = 0, .0x R.

    39. ** 02.1#!21f0f (x) = e.x, . R. .!020.f(x)

    f(x)

    "2"2#.+120.1$*01$1f(x) + 2f(x) = 3f (x), .-

    0x R.

    40. ** 02.1#!21f0f (x) = (3x - 2) [+ .!020

    .f(x)

    f

    41. ** 02.1#!21f0f (x) =H

    H[

    [

    +.!020

    .0/!1*2"

    f(x).

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    42. ** 02.1#!21f0f (x) =H

    [[

    .!020

    .0/!1*2"

    f(x).

    43. ** 02.1#!21f0f (x) =1#[

    [.!020

    .0/!1*2"

    f(x).

    44. ** 02.1#!21f0f (x) =

    x

    3+ 2x

    2+ 3x + 1, x R. .!020

    .f(x)

    . 10. 2" .*" 2" 1#!21" 12. . 03.2)00"

    1.#20..!0"12.xx.

    45. ** 02.1#!21f0f (x) = (x + 1)2, x R. .!020

    .f(x)

    1#20012 /0*#1" 2" 03.2" 2" .*" 2" f12

    100202

    46. ** 02.1#!21f0f (x) = - x2 + 3x - 1, x R. .!020 .f (x)

    01&12"03.2"2".*"2"f, #1$.2002

    .xx&..

    47. ** 02.1#!21f0f (x) = .x + 1)2, x R. R.

    ..!0202f (x).

    .!1/!10202.+1201#20012"/0*#1"2"03.2

    "2".*"2"f1210f(1)).0.

    .!020201&12".!.&03.2"0#0."

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    48. ** 02.1#!21f0f (x) = x2 - 4x + 2, x R...!0202f(x)

    .!1/!10202102"!.3".!12.1"2"1#!21"

    f, 1203.22"1$.20&.02.xx.

    49. ** 02.1#!21 f 0 f (x) = 2x2 .x . R . 0#0.

    y = 3x - 1, x R. .#10202..+1200#0.y = 3x - 1 .0.03.22"!.3".!12.1"2"f12100202

    50. ** 02.1#!21f0f (x) =

    x

    3+ x

    2- 2x + 1, x R. .!020

    .f(x).

    ) "01+10"2&03.2&2"!.3".!12.1"2"f, #0-

    ..!0"120#0.y = x + 3.

    51. ** 02.1#!21f0f (x) =[

    , x R, x 0.

    ../0020)2f.

    .

    .0. R. 0.

    . !1/!1020 2 01&1 2" 0#0." # 03202. 12 10

    .

    .

    2"!.3".!12.1"2"f.

    52. ** 02.1#!21f 0 f (x) = x3 - 9x2 + 15x - 3, x R...!0202f(x).

    .002102022.2".!1/!10202..!)2.22".#!$#

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    53. ** 2. 1#.!210" f, g 0 2*#" f (x) = 2x2

    - 4x - 1 .

    g (x) = 4x - x2

    + 2, x R. .!020.i) 2f(x) .ii) 2g (x).

    "10".2"0"1#.!210".!#1#.!)2.2

    "2"2&.!22&.#2+

    54. ** 02.1#!21f0f (x) =

    x

    3- 2x

    2- 5x - 2, x R. .!020

    .f(x)

    +.0"2"2#x $#0f(x) = 0

    0".)2".!.&2"2&x 0.10".!22&.2f

    /"2"2&.!22&

    55. ** 02.1#!21f0f (x) = x2x + 3, x R R...!0202.+120f.$0121x = 1 2).!)2.210

    0/#".!)2.2.!#101#!21121x = 1;

    56. ** 02.1#!21f0f (x) = x3

    - 3x, x R. .!0*2./.12.2.#f0.

    .*#1.

    -#1.

    57. ** 02.1#!21f0f (x) = x2e-x...!0*f (x), f (x).

    .00201#!21f&"!"22.2"

    .!1/!12*2..!)2.22".#!$#

    58. ** 02.1#!21f0f (x) = (2x - x2) e

    x, x R.

    ..!0*i) 20/!1*2"ii) f(x).f (x).

    .0020f&"!"i) 22.2",

    ii) 2. .!)2.2 2" .. 0212* .#2

    .#!$#.

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    59. ** 02.1#!21f0f (x) = x3x2 + 3x - 1, x R R...!0202f (x).

    .!1/!10202.+120f.$02.!)2.2.12.10.

    02020"x1 = 2, x2 = - 2.

    .!0202"2"2&.!22&

    60. ** )). 2. !+..!.)!..0 2 /.0!02!

    0.00#$02120./)

    61. ** )).2.!+..!.)!..00./)m2, .!020

    2"/.1210"00##$02!)20!0!02!

    62. ** ../0020)2.)).2.1102!&.#0.00!..

    10*.2."R, 21)0#!$00.*20!0./)

    63. ** .!0*/*.!x, y 012.0!)!1.#.$#2

    0.*20!)0

    64. ** 2+1"0)"$.*0.!2.2"0./!$)12"

    2# 1#.!210 2# $!)# .2.10#" 10+!0" !1002. .) 2

    2*2"1#!21"

    (t) = t2

    + 250t-1

    .)20!..20212!/"

    )10..#2)

    65. ** 0!0.#.2..+0."!!.1)"#02.1.12

    ..0)".10*"02.$*22.#!1002..)22*2"1#!-

    21"

    ##

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    66. ** 0!0.W (t), # .//02. .) . 02.02. 0 2

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    1$*2##2$!12t = t0.

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    67. ** 2012!#2&.12+0&3!0&0.12.0!2.200#2..

    $!).12"/!$)12"02.3!".0212/!0.2&

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    (t) = t2

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    )#t @$!)"..!1/!120$!12.22.!..202

    12!/"

    )10..#2)2!/"

    68. ** 02.2/!.10)"!.1*1.3!.0!!302./02.

    .)22*2"1#!21"f (x) = x2.x.!12.0!.x 0!-

    1./)12#3.!#10mg..0.0/0/01)22./)1"2#

    3.!#+120.$#020.*20!02.2/!.12#!.1*

    69. ** .0!121.$.!.12".!.10#002.*&2.%.

    ..2*!0#120)2.!.10#x 2.%+20//.

    1200!#[

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    .1#!21f0..!.&11./12.

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    x = x0.)200$12.x = x0;

    2 /.) 1$. .!#102.

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    !21"f. .$.!.2!10200

    2"!210"#.#* 2 1

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    y

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    0/!1*2"f0.>@ 0/!1*2"f0.@ 2/12.1#!21f.!#102-)12.x = 3.

    1$*0)2f 0. 1$*0)2fx) > 0 .x (2, 3) .

    fx) > 0 .x (3, 4).

    2/12.f0.1&".*#1..12/12.0.1&"3#1.

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    2".*"2"1#!21"f, 1210f(1)).0.

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    4. 23. 42. 5. 24. 43.

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    7. 26. 45.

    8. 27. 46.

    9. 28. 47. i)

    10. 29. ii)

    11. 30. iii)

    12. 31. iv)

    13. 32. v)

    14. 33. vi)

    15. 34. vii)

    16. 35. viii)

    17. 36. ix)

    18. 37. x)

    19. 38.

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    4. 14. + 23.

    5. + 15. + 24. +

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    8. 18. 27. +

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    .210"12"0!&210".212$1"

    1. 2 2

    f (x) = 2x R

    f (x) =[

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    f (x) = [

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    f (x) = [ [1, )

    f (x) =[

    [

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    2. 2 2

    f (x) = [ [0, + )

    f (x) = [+ [- 2, + )

    f (x) =[

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    f (x) =[

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    3. 1 4. 1 5. 1 6. 1

    2 2 2 + 2 +3 + 3 +

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    7.2

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    (f(x) g (x) I[J[I[J[

    [J

    [I=

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    [J[I[J[I

    [f (g (x)@ IJ[J[

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    [[OLP

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    /J[[I[I

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    3..[

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    [

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    [

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    [(3x + 2) (5x - 3)]2

    = 64

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    12[

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    6..\

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    [

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    [

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    [

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    + R [

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    [[

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    A = R B = R

    3. . x = 1 J[!)2.[ (- , - 1) (1, + )LA = R - {- 1, 1}

    ii) B = (- , - 1] [1, + )

    iii) +, - 1) (1, + )

    4. . x = 4 A = R - {4}

    5. .f (x) + g (x) = x2

    - x - 4 5

    3f (x) - 2g (x) = 3x2

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    f (x) g (x) = 3x3

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    29.

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    .# = 6 m/sec #PVHF0WVHF

    .# = 17 m/sec #PVHF0WVHF

    32. .#W0) = 6t0 m/sec2 #PVHF2

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    37. .LI[[ ii) I[

    .LI[H2x

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    39. .I[.H.[ I[.2e.

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    .LI[[ LLJ[[

    I.[.!#10PLQ J.[.!#10PD[

    f (1) = - 3 min g (2) = 6 max

    .I[[2

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    +.[1I.!#10PLQ +.[2I.!#10PD[

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    57. .I[[H-x - x2e-x = e-x (2x - x2) I[H-x (x2 - 4x + 2)

    1#!21I0.1&"3#1.12, 0], [2, .1&".*#1.12/12.>@

    +.[.!#100$1210I

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    .LA = R ii) I[Hx

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    LI0.1&"3#1.12., - ], [ , + )

    . 1&".*#1.12> , ]ii) +.[ $#01202012#2I ) = 3, 4

    +.[ $#00$12020.$12#2I ) = - 1,174

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    +.[I.!#10PD[I

    +.[I.!#10PLQI

    60. .2202!&0[\

    )#0!02!"

    61. 2*00"/.1210"0.2#202!.+#0[\P

    63. x = y = 6.

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    1#$)22.2#0.

    4..L2"/.&.20*#1LL12".)".#2!#

    /.!2!)".#1+/.!2!"1#0$"

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    5.

    0/00"

    .!*#$)22.i

    !12

    1#$)22.i

    1 5 5

    2 6 11

    3 4 15

    4 5 20

    5 6 26

    6 4 3030

    6.

    C i i

    21 4 4

    22 5 9

    23 6 15

    24 4 19

    25 3 22

    26 6 28

    27 2 30

    30

    L!0" LL!0" LLL!0"

    7.

    .2" i fi % Fi %

    27 3 19 19

    28 3 19 38

    29 5 31 69

    30 2 12 81

    31 3 19 100

    16 100

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    8..

    xi . i i

    0 1 1

    2 1 2

    3 1 3

    4 3 6

    5 4 10

    6 2 12

    7 3 15

    8 3 18

    9 2 20

    10 3 23

    13 1 24

    15 1 25

    16 1 26

    26

    )220"

    10..212$".."0.

    1

    02.3!",..11&" /!/!+" /+" 0!!+"

    .10

    0.2/!$ 48.500 13.500 28.000 7.000 97.000

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    .

    0!" 2.1 fi %

    0! 20,8 18

    1. 44 38

    3! 30,5 27

    #!+ 10,5 9

    0.. 9 8

    114,8 100

    +.2#)/!.. 0! 0!#1.: 1383! 96#!+ 330.. 28

    12. ,..11&" 180/!/!+" 38,8

    /+" 116,5

    0!!+" 24,7

    13.

    .)" i fi %

    0 0 0

    1 10 3

    2 20 6

    3 20 6

    4 30 9

    5 40 13

    6 50 16

    7 50 16

    8 40 13

    9 30 9

    10 30 9

    320 100

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    141

    14. .00" 00"

    00" /00" 000"

    17.

    !)"

    ./+[i)

    !)"i)

    00+fi fi % ixi

    0 135 0,338 34 0

    1 220 0,55 55 220

    2 8 0,02 2 16

    3 15 0,037 4 45

    4 12 0,03 3 48

    5 10 0,025 2 50

    400 1 100 379

    L [ 0,947 LL/

    18.0./)2#!+2#!. 1 = 10.

    0./)2#2!2#!. 3 = 30.

    0./)2#22.!2#!. 4 = 20.

    !.20./)2#/0*20!#!. 2 = 40.

    0/22"2#0.2*%"2#.0.

    19../.2#02...12.2..2.*#1.2!1#0)2/

    01"0..!.2!1.0./FP

  • 8/7/2019 1o.meros .

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    12.

    10FPi

    !12

    1#$)22.i

    [150 - 155) 1 1

    [155 - 160) 2 3

    [160 - 165) 3 6

    [165 - 170) 8 14

    [170 - 175) 10 24

    [175 - 180) 7 31[180 - 185) 6 37

    [185 - 190) 3 40

    [190 - 195) 1 41

    41

    02 2#1) !.3 2" /.1# .) 2 /!.. .!

    12+ 1#$22& 2." ./1 $#0 . .)1FP

    .)2!..222"/.1#

    20..0"

    >.i

    [20 - 25) 3

    [25 - 30) 15

    [30 - 35) 12

    [35 - 40) 9

    [40 - 45) 5

    [45 - 50) 3

    [50 - 55) 2

    [55 - 60) 1

    50

    L# LL#

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    143

    21.

    !0.&"10

    +!0"i fi % i Fi %

    [400, 500) 15 3,75 15 3,75

    [500, 600) 45 11,25 60 15

    [600, 700) 60 15 120 30

    [700, 800) 75 18,75 195 48,75

    [800, 900) 70 17,5 265 66,25

    [900, 1000) 60 15 325 81,25[1000, 1100) 50 12,5 375 93,75

    [1100, 1200) 25 6,25 400 100

    400 100

    22. [ .2"

    23. [ $!).

    24..!0..

    25. .L [ = 184,25 LL/ iii) R = 23

    0!2&1i) [ = 186 LL/ iii) R = 20

    0!2&1i) [ = 185,66 LL/ iii) R = 25

    0!2&1i) [ = 184,62 LL/ iii) R = 26

    26. [ w = 59,25

    28. . [ = 11 V2 = 26,2 V 5,11// 041 = 7,5 Q2 = 11,5 Q3 = 14,5

    12 5 &9

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    144

    29.

    10"2!

    1"[i)i xii xi - [ (xi - x)

    2 i (xi - [ )2

    [4, 6) 5 7 35 - 6 36 252

    [6, 8) 7 13 91 - 4 16 208

    [8, 10) 9 17 153 - 2 4 68

    [10, 12) 11 18 198 0 0 0

    [12, 14) 13 29 377 2 4 116

    [14, 16) 15 11 165 4 16 176

    [16, 18) 17 5 85 6 36 180

    100 1104 1000

    L [ 11 ii) s2 = 10 iii) s 3,16 iv) CV 29%

    30.

    /$"10

    $/0"/!$

    02!"

    2"[i

    i xi2

    xii xi2i

    [30, 35) 32,5 8 1056,25 260 8450

    [35, 40) 37,5 10 1406,25 375 14062,5

    [40, 45) 42,5 16 1806,25 680 28900

    [45, 50) 47,5 15 2256,25 712,5 33843,75

    [50, 55) 52,5 10 2756,25 525 27562,5

    [55, 60) 57,5 8 3306,25 460 26450

    [60, 65) 62,5 3 3906,25 187,5 11718,75

    70 3200 150987,5

    L [ $ ii) s2 = 67 iii) s 8,5 iv) CV = 17,39%

  • 8/7/2019 1o.meros .

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    145

    31.

    !)".2$"

    10+!0"!)"1#10#+

    !12

    1#$)22.

    [1000, 1200) 8 8

    [1200, 1400) 16 24

    [1400, 1600) 28 52

    [1600, 1800) 32 84

    [1800, 2000) 12 96

    [2000, 2200) 4 100

    [2200, 2400) 0 100

    100

    / 1#10#" $# /!0..2$"!)20!.) 2121#$)

    22.

    32.

    #10" .2" i xi

    10 70 140

    20 5 5

    20 5 15

    80

    33. [ 0.122"2)20V2 =

    =

    L

    (ti - [ )2.

    VV2 =

    L

    (ti - [ )2

    (t1 - [ )2

    + (t2 - [ )2

    + + (t - [ )2

    = 0 t1 = t2 = = t = [ .

    34.. ) 173

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    146

    35. . \ =

    [

    \ L

    LL

    L

    +=

    == = [ = 2, s

    \ = s

    [ .

    \ = [ + c, s\ = s

    [ .

    2"."02.2".#*.2F/0"2)20/.*.1

    0.022.12.#02..2F

    36. . [+ =

    W

    W

    W ++++++ = = [

    [ = [ [ [

    /

    [

    =

    [ 0) 0 [ + [

    37. .

    !+2

    02.1[

    0*20!

    02.1\x

    2xy y

    2

    6 8 36 48 645 7 25 35 49

    8 7 64 56 49

    8 10 64 80 100

    7 5 49 35 25

    6 8 36 48 64

    10 10 100 100 100

    4 6 16 24 36

    9 8 81 72 64

    7 6 49 42 36[ \ [2 = 520 [\ \2 = 587

    \ = 4 + 0,5x

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    147

    38. .

    x y x2

    xy y2

    1 1 1 1 1

    3 2 9 6 4

    4 4 16 16 16

    6 4 36 24 16

    8 5 64 40 25

    9 7 81 63 49

    11 8 121 88 64

    14 9 196 126 81

    [ \ [2

    = 524 [\ \2

    = 256

    \ [$01#20012/0*#1"./!$02..)

    210

    \ = 8,18

    39. .

    0!.

    (x)

    +0&02!.

    (y)x

    2xy y

    2

    75 82 5625 6150 6724

    80 78 64000 6240 6084

    93 86 8649 7998 7396

    65 72 4225 4680 5184

    87 91 7569 7917 8281

    71 80 5041 5680 6400

    98 95 9604 9310 9025

    68 72 4624 4896 5184

    84 89 7056 7476 7921

    77 74 5929 5698 5476

    [ [\ [2

    = 64722 [\ \2

    = 67675

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    148

    \ = 29,23 + 0,66x

    [2)2012+0&02!..!0

    40. .U [\0.02!.1#1$0210"

    41.

    x y x - [ y - \ (x - [ )2

    (y - \ )2

    (x - [ ) (y - \ )

    35 102 - 9 5 81 25 - 45

    39 100 - 5 3 25 9 - 15

    41 97 - 3 0 9 0 0

    43 97 - 1 0 1 0 0

    40 98 - 4 1 16 1 - 4

    53 100 9 3 81 9 27

    50 97 6 0 36 0 0

    49 91 5 - 6 25 36 - 30

    40 94 - 4 - 3 16 9 12

    50 94 6 - 3 36 9 - 18

    *. 440 970 326 98 - 73

    .U

    .[.\0..!2!.1#1$021.

    0/#!$0 .!2 1#1$21 02.* 2" 2" 2& $!0!3&

    .2"2"2&+1.0)2)2.22&$!0!3&32022&+.0.0

  • 8/7/2019 1o.meros .

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    149

    42..

    x y x2

    xy y2

    2 18 4 36 324

    4 12 16 48 144

    5 10 25 50 100

    6 8 36 48 64

    8 7 64 56 49

    11 5 121 55 25

    [ [\ [2 = 266 [\ \2 = 706

    r = - 0,92.

    [[.\\2)20

    [ \ [2 [\ \2

    10 39 100 390 1521

    14 21 196 294 441

    16 15 256 240 225

    18 9 324 162 81

    22 6 484 132 36

    28 0 784 0 0

    [ [\ [2 = 2144 [\ \2 = 2304

    r = - 0,92.

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    150

    43.

    .

    (x)

    01

    ..2"\x

    2xy y

    2

    56 17 3136 952 289

    42 12 1764 504 144

    72 14 5184 1008 196

    36 10 1296 360 100

    63 13 3969 819 169

    47 09 2209 423 81

    55 11 3025 605 121

    49 08 2401 392 64

    38 11 1444 418 121

    60 15 3600 900 225

    [ [\ [2

    = 28028 [\ \2

    = 1510

    .U

    \ = 4,852 + 0,138x

    +01#.."."02+0.0!#

    44. .

    x y x - [ y - \ (x - [ )2

    (y - \ )2

    (x - [ ) (y - \ )

    1 2,5 - 2 - 1,5 4 2,25 3

    1 4,0 - 2 0 4 0 0

    2 3,0 - 1 - 1 1 1 1

    3 4,5 0 0,5 0 0,25 0

    3 4,0 0 0 0 0 0

    4 3,5 1 - 0,5 1 0,25 - 0,5

    7 6,5 4 2,5 16 6,25 10

    *. 21 28 0 0 26 10 13,5

    U 0,83

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    151

    45. .

    !+2

    02.1[

    0*20!

    02.1\x

    2xy y

    2

    6 8 36 48 64

    5 7 25 35 49

    8 7 64 56 49

    8 10 64 80 100

    7 5 49 35 25

    6 8 36 48 6410 10 100 100 100

    4 6 16 24 36

    9 8 81 72 64

    7 6 49 42 36

    [ \ [2

    = 520 [\ \2

    = 587

    U.0.02!.1#1$0210".2

  • 8/7/2019 1o.meros .

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    20!/32101202!.2&2$.)#

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    3)&! 2#&" 2 2#0!120&"$.)"

    12 #2 12 .2.10# 2& 3!.2& Assuan . Assiout.

    1&" 0202..0 120! 0./) 1) 21#/

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    ..!2#..$&!0.2Gttingen!*!2)202&..2+0!0#+.

    2)1#2!&1" /.1& ..2+ (Klein - Hilbert - Minkowski 02.

    #32"12.0122#Gttingen02.2.2)".2"12#20$02#)0!# 0+ 2 02. .2" 12#20$0 2" Breslaw. 02.

    2.2)" .2" 12 .012 2# Gttingen . /./$02. 2 Felix Klein 12

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