416
ΑΛΓΕΒΡΑ Β´ ΓΕΝΙΚΟΥ ΛΥΚΕΙΟΥ

Άλγεβρα Β' Λυκείου Γενικής Παιδείας Βιβλίο Μαθητή

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Άλγεβρα Β' Λυκείου Γενικής Παιδείας Βιβλίο Μαθητή

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  • :

    . .

    : 1991 : 1992, 1993, 1994, 1995, 1996, 1997, 1998, 2012 .

  • . , . , . . , .

  • . (2010) , - . , . , . , . . . : , (2010), - . , . , . , . . . - (2010), - (2010).

    :

    1 , -, . , .

    2 , , , - .

    3 - . , -, . - .

    4 . 5 , .

  • 1: 1.1 .......................................................................9 1.2 ..............................................................24

    2: 2.1 -- ............................30 2.2 - ...........................40

    3: 3.1 ..................................................49 3.2 ...........................................60 3.3 1 ......................................................65 3.4 ..................................................73 3.5 ............................................83 3.6 ...........................89 3.7 2 .......................................97 3.8 ...................103 3.9 f(x)=x+x ..................................................108 3.10 .......................................................................114

    4: - 4.1 ..................................................................................128 4.2 ................................................................132 4.3 .....................................140 4.4 .......150

    5: 5.1 ....................................................................160 5.2 ...................................................................................173 5.3 ..............................................................181

    .........................................................193 - ...................................195

  • 1

    1.1

    x + y =

    x + y = , 0 0, , . : :

    0, , :

    x y y x

    y x ,

    + = = +

    = +

    -

    =

    y'y

    .

    :

    9 0, (. ),

    x+y=, =0

    0 x

    y

    x+y=, 0

    x

    y

    0

  • 10 1:

    9 = 0 , y =

    x'x y'y

    (. ).

    = 0 ( 0 ),

    - y'y

    x'x

    .

    :

    9 x 2y 2 =

    1

    2 =

    y'y 1 .

    9 y=2 x'x - y'y 2 .

    9 x=2 y'y - x'x 2.

    .

    1y x 1

    2=

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x+y=, =0

    x

    y

    0

    y= x1y

    0 2-1

    x

    12

    y=2

    x

    y

    2

    0

    x=2

    x

    y

    0 2

    x x .

    = =

  • 1.1 11

    , (4,-1) x 2y 6, = 4 2( 1) 4 2 6. = + = , , (16,5), ( 10, 8)

    1k, k 3 , k2

    .

    2 x 2 x+y= x+y= , - , , 2 x 2

    x y

    'x ' y '

    + = + =

    . . . . : .

    () (') , .. (),

    ( ) ' ( ') + , () - 0, (') '.

    ( ) ' ( ') + () (').

    - 1o :

    9 0, : = = 9 = = , + = +.

    , , :

    x 2y 6 (1)

    3x 4y 8 (2)

    = + =

  • 12 1:

    , ( )

    , .. (1) x.

    x

    3(2y 6) 4y 8 6y 18 4y 8

    10y 10

    y 1

    + + = + + = = =

    y :

    x 2( 1) 6 4= + =

    (4, 1).

    ,

    .

    - , .

    , x = 4 y 1,= :1 : 4 2(1) = 6

    2 : 3 4 4( 1) 12 4 8 + = =

    ( ) , :

    x 2y 6

    3x 4y 8

    = + + =

    x 2y 6

    y 1

    = + =

  • 1.1 13

    , - , : . :

    x 2( 1) 6 x 2 6 x 4 = + = = . (4, 1) ( - ).

    2 x 2

    . , - .

    ,

    , , . 2 x 2 . , 2 x 2 , , , .

    3x 6y 18

    3x 4y 8

    + = + =

    x 2y 6 ( 3)

    3x 4y 8 1

    = + =

    3x 6y 3x 4y 18 8 10y 10 y 1 + + + = + = =

    x 2y 6

    3x 4y 8

    = + =

    x y

    'x ' y '

    + = + =

    A(4, 1)

    3x+4y=8 x2y=6y

    O x

  • 14 1:

    2 x 2 -.

    :

    9 2x 3y 3

    4x 9y 1

    + = =

    2y x 1

    34 1

    y x9 9

    = + =

    ,

    , .

    ,

    (1, 0,3) .

    ,

    - 1(1, )

    3.

    9 x 2y 3

    2x 4y 5

    + = + =

    1 3y x

    2 21 5

    y x2 4

    = + =

    , ,

    .

    9

    y 1 2x

    4x 2y 2

    + = =

    y 2x 1y 2x 1

    = =

    , .

    (k,2k 1), k .

    y

    x0

    21

    1

    1

    y

    x0

    1

    2

    y

    x0

    2

    1

    ,

  • 1.1 15

    , 2 x 2 :9 9 9 .

    2 x 2

    - 2 x 2 .

    0 0. - :

    1 2 -

    1

    =

    2

    '

    '

    =

    '

    '

    , ' ' 0 , 1 2

    ,

    ' ' ' 'x x x

    ' ' ' '

    ( ' ' )x ' '

    ' 'x

    ' '

    + = + =

    =

    =

    :

    ' ' ' ' ' 'y

    ' ' ( ' ' )

    ( ' ' )

    ( ' ' )

    + + = + =

    =

    x y

    'x ' y '

    + = + =

    1

    2

    y x ( )

    ' 'y x ( )

    ' '

    = + = +

  • 16 1:

    ' 'y

    ' '

    =

    , ,

    '

    '

    =

    , - =0 , 1 2

    , . . =0 ' =0.

    :

    ' ' 0

    ' ' 0 = .

    ' ' ,

    :

    .

    D, x , : .

    , D, - y , :

    .

    ' ' ' '(x, y) ,

    ' ' ' '

    =

    x y

    'x ' y '

    + = + =

    ' ' ' '(x, y) ,

    ' ' ' '

    =

    D' '

    =

    D ' '' '

    = =

    xD ' '' '

    = =

    yD ' '' '

    = =

  • 1.1 17

    - , , :

    :

    9

    ,

    .

    : .

    , (x, y) (4, 1)= .

    9

    - . 2,

    2x 3y 40. =

    .

    x 2y 6

    3x 4y 8

    = + =

    1 2D 1 4 3( 2) 4 6 10 0

    3 4

    = = = + =

    x

    6 2D 24 16 40

    8 4

    = = + =

    yD 10y 1D 10

    = = =

    y

    1 6D 8 18 10

    3 8= = =

    xD 40x 4D 10

    = = =

    2x 402x 3y 40 y

    3

    = =

    D0, , (x,y)

    D=0, .

    xDxD

    =

    x y

    'x ' y '

    + = + =

    yDyD

    =

    2 3D 2( 6) 4( 3) 12 12 0

    4 6

    = = = + =

    2x 3y 40

    4x 6y 80

    = =

    2x 3y 40,

    2x 3y 40

    = =

  • 18 1:

    9

    .

    .

    1

    - , . , . , . , .

    D, Dx, D

    y. :

    2k 40k, , k3

    3x y 11

    9x 3y 6

    + = + =

    3 1D 9 9 0

    9 3= = =

    3x y 11

    3x y 2

    + = + =

    2

    x y 1

    x 2y

    = =

    22

    x

    2 2 2 2y 2

    1D 2 ( 2)

    2

    1 1D 2( 1) 2

    2

    1D ( 1) (1 1) (2 )

    = = + =

    = = + =

    = = = + =

  • 1.1 19

    , D = 0 . :

    :9 D0, 0 2, (x, y), :

    .

    , 0 2 ,

    .

    9 D = 0 , = 0 = 2 , . : = 0 ,

    . = 2 ,

    . , ,

    3 x 3 x+y+z=0 , - , , , . . 2x+3y+z=6 - (2, 1,5) , 22+3(1)+5=6.

    D 0 ( 2) 0= =

    xD 2 ( 2) 1xD ( 2) ( 2)

    = = = =

    2 2

    yD (2 ) ( 2)yD ( 2) ( 2)

    = = = =

    0x y 1 y 1

    0x 2y 0 y 0

    = = = =

    2 =0 =

    2x y 1 2x y 1

    4x 2y 2 2x y 1

    = = = =

    .( )k,2k 1 , k

    1 ,

    2x y 1 y 2x 1 = =

  • 20 1:

    :

    1 1 1 1x y z + + = , 2 2 2 2x y z + + = 3 3 3 3x y z + + = , - , , 3 x 3

    - 2 x 2 .

    . , .. (3) z, - . : (4)

    (1) (2) :

    9 2x y 3z 9 2x y 3(3x y 8) 9 11x 2y 15 + = + + = + = (5)9 (6)

    (5), (6)

    x=1 y=2 . x y (4) z 3.= (1,2, 3).

    x 3y z 10 x 3y (3x y 8) 10 x y 1+ = + + = + =

    1 1 1 1

    2 2 2 2

    3 3 3 3

    x y z

    x y z

    x y z

    + + = + + = + + =

    2x y 3z 9 (1)

    x 3y z 10 (2)

    3x y z 8 (3)

    + = + = + =

    3x y z 8 z 3x y 8+ = = +

    3 x 3, ,

    2 x 2, 3 x 3

    .

    11x 2y 15,

    x y 1

    + = + =

  • 1.1 21

    1.

    i) ii) .

    2. i) ii)

    3. :

    i) ii)

    4. : i) ii)

    5. : i) ii)

    6. , .

    i) ii) iii)

    7. :

    i) ii)

    x y

    7 8x y 45

    = + =

    x 3y 3

    xy 2

    3

    =

    =

    2y x 2

    1x y 1 0

    2

    = +

    + =

    2x y 7

    3x 5y 4

    + = =

    2x 5y 4

    6x 7y 100

    = + =

    ( 3 1)x 4y 7

    1x ( 3 1)y 1

    2

    + + =

    + =

    2x 3y 40

    4x 6y 80

    = =

    x 1 y 2

    3 44x 3y 8

    = + =

    x 5 2y 1 2 02 7

    x 6 y 6 83 2

    + + + = + =

    2y 3x 8

    x 3y 1 0

    = + + =

    x y 4

    x y 2

    = + =

    ( 3 1)x 2y 2

    x ( 3 1)y 1 3

    + =

    + + =

    3x y 11

    9x 3y 2

    + = =

    2x 1 y 243 4

    x 3 x y32 3

    + = + =

  • 22 1:

    8. :

    i) ii) iii)

    1. i) 1 2 .

    ii) 1 2 ;

    2. 26 , 68 . ;

    3. 1,5 4. 2200 - 5050 . .

    4. R T R =T + . 20 C 0,4 80 C 0,5 , .

    5. , 50% 80% . 100 ml - 68% ;

    6. 1 : 2x 4y 3 + = 2 : x 2y + = , . i) 1 2 .ii) -

    ; iii) ;

    3x 2y 11

    2x 5y 2 3

    5x y 2 33

    = = + =

    5x y 3 4

    x 3y 2

    3x 2y 2 2

    + = + = + =

    yx 2 3

    23x

    y 52

    5x 3y 2 16

    + = + + =

    + =

    y

    x1 41

    1 12

    O

  • 1.1 23

    7. :

    i) 21 : x y + = 2 : x y 1 + = .

    ii) 3 : x y = 2 : x y 1 + = .

    8. :

    i) ii)

    9. 1 2 6 = , 1 3 5=O O

    2 3 7=O O . .

    10. , . N = x , B = y = z , - , .

    11. . 50% , 10% 30% . 52 lit 32% , - - -. .

    12. -, 2y x x= + + . - .

    ( 1)x 2y 1,

    4x ( 1)y 2

    = + =

    ( 2)x 5y 5,

    x ( 2)y 5

    + = + + =

    1 2

    3

    x+y=, 0

    x

    y

    0

    y

    x1

    K(1,4)

    y=g(x)

    O

    y

    x

    y=h(x)

    O 2 4

    4

    y

    x

    y=f(x)

    OK(2, 1)

    3

  • 24 1:

    1.2

    , -. , 13 89.

    x, y , x y 13+ = 2 2x y 89+ = . -

    , :

    : (1), , .. x , (2).

    x y 13 y 13 x+ = = (3).

    2 = 9 . :

    (3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , .

    1

    2 2

    x y 13 (1)

    x y 89 (2)

    + =

    + =

    2 2 2 2

    2

    2

    x (13 x) 89 x 169 26x x 89

    2x 26x 80 0

    x 13x 40 0.

    + = + + =

    + =

    + =

    8

    5

    13 3x

    2

    = =

    x y 5 (1).

    xy 6 (2)

    + =

    =

  • 1.2 25

    (1) x (2). :

    (3) x=2 y=3 , x=3 y=2 . (2, 3) (3, 2) .

    5 6. , Vieta

    2 5 6 0 + = . 2 3 (2,3) (3,2).

    x+y=5 , xy=6

    6

    yx

    = . -

    .

    x y 5 y 5 x+ = =

    x 2 x 3. = =

    y

    xO 11

    AB

    (3).

    2

    2

    5x x 6

    x 5x 6 0

    =

    + =

    xy 6 x(5 x) 6= =

  • 26 1:

    (2,3) (3,2) . (2,3) (3,2) .

    2

    (1) x (2).

    (2) :

    . 2x = ,

    2 13 36 0 + = , =9 =4 .

    9 =9 (1) x=3 y=2 x 3= .9 = 4

    (1) x=2 y=3 x 2= y 3= .

    (3,2), ( 3, 2), (2,3) ( 2, 3).

    xy 6= 6

    yx

    = , -

    2 2x y 13+ = (0,0) 13 = .

    6xy 6 y

    x= =

    22 2 2

    22

    4 2

    4 2

    6x y 13 x 13

    x

    36x 13

    x

    x 36 13x

    x 13x 36 0

    + = + =

    + =

    + =

    + =

    2x 9 x 3= = x 3=

    2x 4 x 2= = x 2=

    2 2

    xy 6 (1)

    x y 13 (2)

    =

    + =

  • 1.2 27

    1. :

    2. :

    i) ii) iii) .

    3. 201

    S t t2

    = + 0 t = + , 0S t

    2+

    = .

    1. o - .

    2 2x y xy 3.

    x y 1

    + + =

    + =

    2y 3x

    12x 3y 4

    =

    =

    2 2x y 9

    x y 0

    + =

    =

    2 2x y 5

    xy 2

    + =

    =

    .

    2

    2 2

    x 2y 10

    x y 25

    = +

    + =

    y

    x 1

    AB

    A B

  • 28 1:

    2. :

    3. 120cm2 . 3cm , 2cm , . -.

    4. 2y x= y 2x k, k= + . - k .

    5. o 22y x

    y x

    =

    = +

    .

    1

    I. :

    , , ,

    , , .

    ) , ) , ) .

    II. , , .

    1. , . A

    2. D = 0 , .

    3. xy 1

    x y 0

    = + =

    .

    4. 2 2x y 1+ = 2y x 1= +

    . A

    1

    x 2y 1( ) :

    2x 4y 0

    + = + =

    2

    x y 1( ) :

    x 2y 4

    + = + =

    3

    2x 4y 2( ) :

    x 2y 1

    + = + =

    4 2

    x y 1( ) :

    x y 1

    =

    + =

    (1) (2) (3) (4)

    2

    2

    2xy y 5y 0

    y x 4x 3

    =

    = +

  • 2

    ) [4,16] .

    . .

    2.1 --

    T = f (t) t (t =0) (t =24).

    T=f(t)

    5

    11T(C)

    3

    4 16 24 t(h)

    f(t1)

    T=f(t)

    f(t2)

    5

    11T(C)

    4 t 16 24 t(h)1 t2

  • 2.1 -- 31

    , , - , 1 2t , t [4,16] 1 2t t< :

    1 2f (t ) f (t ) 0 .

    ) , [16,24] .

    f - , 1 2x , x 1 2x x< :

    1 2f (x ) f (x ) +

    >

    f , 1 2x , x 1 2x x< :

    1 2f (x ) f (x )>

    T=f(t)

    5

    11T(C)

    3

    O 4 16 24 t(h)

  • 2.1 -- 33

    :) 1t 4= , f (4) 3= . :

    f (t) f (4) 3 = , t [0, 24] T = f (t) t = 4 -, f (4) 3= . :

    0x A , 0f (x ) f min f (x) .

    , 4f (x) 3x 1= + .

    4x 0 , x,

    43x 0 , x,

    43x 1 1+ , x.:

    f (x) f (0) , x.

    , f 0x 0= , f (0) 1=

    ) 2t 16= , T(16)=11 . :

    f (t) f (16) 11 = , t [0, 24] T =f (t) t =16

    , f (16)=11. :

    f, , 0x A () :

    0f (x) f (x ) , x A

    f, , 0x A () :

    0f (x) f (x ) , x A

  • 34 2:

    0x A , 0f (x ) - f max f ( x) . , 4f (x) 3x 1= + .

    4x 0 , x,

    43x 0 , x,

    43x 1 1 + , x.:

    f (x) f (0) , x

    , f 0x 0= , f (0) 1= . () () .

    (. ) (. )

    (. ) (. ).

    x

    y

    1

    1 y=f(x)

    y

    x

    1

    11

    y=f(x)

    y

    x

    1y=f(x)

    1

    y

    x

    1

    1

    y=f(x)

  • 2.1 -- 35

    ) - C

    f f -

    . Cf

    y'y, C

    f

    y'y Cf .

    , , M(x, y) Cf

    y'y M '( x, y) M(x, y) M '( x, y) C

    f , y = f (x) y = f (x) , :

    f . :

    yy.

    , 4 2f (x) 2x x 1= + , x :

    4 2 4 2f ( x) 2( x) ( x) 1 2x x 1 f (x) = + = + =

    , y'y .

    ) - C

    f f

    . C

    f

    , - C

    f -

    Cf .

    f, , , x A :

    x A f ( x) f (x) =

    f ( x) f (x) =

    y

    xx

    y

    x

    M M

    O

    f(x) f(x)

    Cf

    y

    y

    y

    xxx

    M

    M

    Of(x)

    f(x)

    Cf

  • 36 2:

    , , M(x, y) Cf

    M '( x, y) M(x, y)

    M '( x, y) Cf , y = f ( x) y f ( x), =

    :

    f ( x) f (x) = f . :

    .

    , 3f (x) 2x x= , x : 3 3f ( x) 2( x) ( x) 2x x f (x) = = + = , .

    : 2y x= ,

    4y x= , 6y x= ., , y'y , , - y x= , 3y x= , 5y x= ., , , .

    - f [6,6]. f :

    f, , , x A :

    x A f ( x) f (x) =

  • 2.1 -- 37

    ) f : i) , ii) , iii) .) f , - .

    f , y'y . , y'y f, - f, ().

    :) f : i) [0,2] [5,6],

    ii) [ 2,0] [ 6, 5], [0,2] [5,6] f .

    iii) [ 5, 2] [2,5] .

    y

    x 1

    4

    652256 x

    A

    A

    B

  • 38 2:

    ) f 4 x 6 6. :

    max f (x) f ( 6) f (6) 4= = = f 0 x

    0. :min f (x) = f (0) = 0.

    1) - :

    ) ) .

    2) - , .

    3) :

    i) 2f (x) x 6x 10= + x=3.

    ii) 22x

    g(x)x 1

    =+

    x=1.

    4) :

    i) 2 41f (x) 3x 5x= + ii) 2f (x) 3 x 1= + iii) 3f (x) x 1= + iv) 3 54f (x) x 3x= v)

    2

    5x

    f (x)1 x

    =+

    vi) 6 22x

    f (x)x 1

    =+

    .

    1

    y=f(x)

    x

    y

    2

    2

    y=g(x)

    x

    y

    1

    y=h(x)

    x

    y

    1

    2

  • 2.1 -- 39

    5) :

    i) 11

    f (x)x

    =

    ii) 2f (x) x 2= iii) 3f (x) x 1 x 1= +

    iv) 4 2

    1x

    xf (x)x 1

    +=

    + v) 5f (x) x= vi)

    26f (x) 1 x= .

    6) .

    7)

    8) -

    ) ) .

    y=f(x) y=g(x) y=h(x)

    x x x

    y y y

    y=f(x) y=g(x) y=h(x)

    x x x

    y y y

    C1 C2 C3

    x x x

    y y y

  • 40 2:

    2.2 -

    ) f (x) x 1= + .

    x 1, x 0

    f (x) ,x 1, x 0

    +

  • 2.2 - 41

    ) f (x) x 1= .

    x 1, x 0f (x) ,

    x 1, x 0

    0 , c ( )

    y

    y=|x|1

    y=|x|

    1 1

    1

    1

    1 1

    O x

    x

    c

    y=(x)

    y=(x)+c

    c cc

    y

    c

    O

  • 42 2:

    ) f (x) x 1= .

    x 1, x 1

    f (x) ,x 1, x 1

    +

  • 2.2 - 43

    f :

    f (x) (x c)= , c > 0 , c ( ).

    :

    f (x) (x c)= , f x x c , c x. , f c ( ).

    ) f (x) x 1= + .

    x 1, x 1

    f (x) ,x 1, x 1

    < = +

    f (x) x 1= + , 9 y x 1= , x 1 9 y x 1= + , x 1 , 1 x'x - x 'Oy xOy (x) x = ()., (x) x = - 1 , -

    Ox

    c

    c

    c

    c

    xxc

    (x-c)

    C Cf

    f(x)

    y

    y

    x

    y=|x+1|

    y=|x|

    1 1

    11

    11

    O

  • 44 2:

    f , :

    f (x) (x c)= + , c > 0 , c ( ).

    f (x) x 1= + . , , ,

    f (x) (x 1)= + , x, f (x) x 1= + x

    (x) x = x +1.

    :

    f (x) (x c)= + , f x x + c , c x. , f c ( ).

    f (x) x 3 2= + + .

    y x 3= + , - y x= 3 . - y x 3 2= + + , y x 3= + 2 .

    xx x+c

    yc

    c

    O

    c

    c

    =(x+c)f(x)

    Cf C

  • 2.2 - 45

    , -

    f (x) x 3 2= + + -

    -

    y x= , - 3

    - 2

    ().

    : , - :

    f (x) (x c) d= , c,d > 0, - .

    A

    1. -:

    (x) x = , f (x) x 2= + g(x) x 2= .

    2. :

    (x) x = , h(x) x 2= + q(x) x 2= .

    3. :

    (x) x = , F(x) x 2 1= + + G(x) x 2 1= .

    y

    x

    y=|x+3|y=|x+3|+2

    y=|x|

    13

    2

    O

  • 46 2:

    4. i) 2f (x) 2x 4x 5= + 2f (x) (x p) q= + -

    2g(x) 2x= f.

    ii) 2f (x) 2x 8x 9= + , - g 2g(x) 2x= .

    5. 1 O(0,0) A(2,0).

    -:

    i) f (x) (x) 2= + g(x) (x) 2= ii) h(x) (x 3)= + q(x) (x 3)= iii) F(x) (x 3) 2= + + G(x) (x 3) 2= .

    6. 2(x) 2x 1 = . - f :

    i) 2 1 . ii) 3 2 .iii) 2 1 .iv) 3 2 .

    C

    O x

    y

  • 2.2 - 47

    I) - , , - .

    1. f , f .

    2. .

    3. (1,2) , (2,1) (3,3) .

    4. f 1, f (0) < 0 .

    5. f - (1,2) (2,5) , f .

    6. f 1, f (x) = 2 .

    7. f :[ 1, 2] 2f (x) 3x= .

    8. , .

    9. f , f - .

    10. f , f .

    II) f.

    f,

    4(x) 3x , = 1 2 , :

    ) 4f (x) 3(x 1) 2= + ) 4f (x) 3(x 1) 2=

    ) 4f (x) 3(x 1) 2= + + ) 4f (x) 3(x 1) 2= +

    2

  • 3

    1 1(MM ) (NN )

    (OM) (ON)= 1 1

    (OM ) (ON )

    (OM) (ON)= 1 1

    1 1

    (MM ) (NN )

    (OM ) (ON )=

    3.1

    . - - MM1 NN1 , 1O M M

    1O N N

    , :

    ,

    ,

    , 1(MM )

    (OM)1(OM )

    (OM)1

    1

    (MM )

    (OM )

    , - . , , , , , .

    , 1M O M

    , :

    1(MM )

    (OM) = ( )

    11

  • 50 3:

    , - ,

    , 00 3600

    Oxy , Ot Ox - Ot (. , ). Ox , Ot .

    (x, y) M1 x'x (. ). (. ), , , :

    , ,

    1( ) x = , 1( M) y = 2 2(OM) x y 0= + = > . ,

    :

    , , , .

    1(MM )

    (OM) = 1

    ( M )

    (OM)

    = 1

    1

    (MM )

    (OM ) = 1

    1

    ( M )

    (M )

    =

    y =

    x

    =

    y

    x =

    x

    y = 2 2x y 0 = + >

    1( M )

    (OM)

    =

    1

    1

    (MM )

    (OM ) =

    1

    1

    ( M )

    (M )

    =

    ( ) ( )

    ( )

    2

    1 1

    2

    x xx

    t

    y

    t

    (x,y)(x,y)

    yy

  • 3.1 51

    1

    1

    ( M )

    (M )

    =

    2 2x y 0 = + >

    , - ( ).

    :

    (x, y) ( )

    2 2x y 0 = + >

    .

    360

    Ox - Oxy - . 300 , Ox

    360 30 390 . = + =

    360 , - :

    360 , = + N* 00 < 3600 Ox, , - 30 , Ox - 360 30 390+ = :

    , ( x0) ,

    , ( y0)

    y =

    y

    x =

    x =

    x

    y =

    2 2x y 0 = + >

    (360 30 ) 390 . = + =

    y

    x

    390

    30

    0

    0

    0

    y

    t

    t

    x 30

    390

    0

  • 52 3:

    - :

    ( 360 ), = + N 0 360 .

    ( ) Ox .

    Ox , , - , 360 , + . Ox , , , 360 , + - .

    , k 360 , + k , .

    , k :

    , ( x0) ,

    , ( y0)

    y =

    y

    x =

    x =

    x

    y =

    2 2x y 0 = + >

    (k 360 ) , + = (k 360 ) + =

    (k 360 ) , + = (k 360 ) + =

  • 3.1 53

    (0,0) - =1 - . - .

    - , .. - 35 , = (, ).

    35 =

    =1

    35 0,57. =

    , , , . - , - .

    , 35 =

    =1, 35 0,82. =

    , (x, y), :

    x'x , y'y .

    :

    1. ' , 1. :

    = x = = y =

    1 1 1 1

    7060

    5040

    20

    10

    350

    340

    330320

    310300

    290280270260

    250240

    220210

    200

    190

    180170

    160

    150

    140130

    120110

    100 90 80

    y

    x

    (x,y)

    N(,)

    x

    y

    35

    0,1

    0,1 0

    o

    o

    o

    o

    o

    o

    o

    o o

    o

    o o o o

    o

    o

    o

    o

    o

    o

    o

    o

    o

    o

    o o

    o o

    o

    o

    o o

    o

  • 54 3:

    2. , , .

    10 20 30 40

    + + + + + + + +

    - M(x, y). - . 1 ,

    ( ) ( )( )

    ( ) 1

    = = =

    yE , (AE)= yE ,

    = yE.

    .

    :

    = yE =

    , x =1 , .

    AO

    B

    B

    M

    A

    t

    x

    y

    E(1,y )E

  • 3.1 55

    . , AB (, ) ( 1rad), . , ( rad) S .= - :

    ( 1 rad ) , -, ( 1 rad).

    - , :

    , rad . - 2,

    360 2 rad.

    ,

    1 rad 3602

    ,

    ,

    rad 180

    .

    , 180 =

    , :

    180 =

    :9 60 ,

    180 =

    = 600

    180 3 = =

    603

    = rad.

    A

    1rad

    B

    O

  • 56 3:

    9 56

    rad ,

    180 =

    56

    =

    556 150

    180 6 180

    = = =

    56

    rad=150 .

    - .

    rad

    0 0 0 1 0

    30 6 1

    23

    23

    33

    45 4 2

    22

    21 1

    60 3 3

    212

    3 33

    90 2 1 0

    0

    , x rad x,

    (x rad), ( x rad), (x rad) (x rad),

    x, x, x x.

    , .. rad3

    3

    (100rad ) 100 .

  • 3.1 57

    1 h , . .

    :

    h48 , =

    h48

    =

    h70 , =

    h70

    =

    20m = =

    h h 20,48 70

    =

    20 70 48

    h .70 48

    =

    70 2,75 48 1,11.

    61,05h 371,64

    37m .

    2 750o .

    750 360 2 30,

    1750 (2 360 30 ) 30

    2 = + = =

    3750 302

    = =

    3750 303

    = = 750 30 3 = =

    750 2 360 30= +

    (1) :

  • 58 3:

    3 793

    rad.

    79 79

    2 .3 6=

    79 6

    13 1. 79 79 12 13 2 13 2 ,

    3 6 6 3 = = + = +

    :

    79 313 23 3 3 2 = + = =

    79 13 3 2

    = =

    79 33 3

    = =

    79 33 3 3

    = =

    1. x, y .

    2. .

    3. S = 6cm. , :

    i) = 1cm ii) = 2cm iii) = 3cm .

    4. rad i) 30 ii) 120 iii) 1260 iv) 1485 .

    5. :

    i) 10

    rad ii) 5

    6

    rad iii) 91

    3

    rad iv) 100rad .

    6. i) 1830 ii) 2940 iii) 1980 iv) 3600 .

    A

    x

    260

    3

    B

    B

    A

    y

    0

    6

    300

    300

    4

  • 3.1 59

    1. - , - 1000 - (1 0,3 m) .

    .

    i) h =30 ,45 60 .

    ii) , h=1000 ;

    2. :

    i) : () = () = 2 45 = 2.

    ii) ( ) 4 22,5 . =

    iii) ().

    iv) ,

    , ( ) 2 2 2 . = v) 22,5 . vi)

    ;3. .

    4. 1mm . - .

    1.000 pi

    N (N)

    () (pi)

    h

    700

    A B

    1 1

    22,5o 3 345o

    A

    B

    6

    300

    300

  • 60 3:

    3.2

    . .

    :

    1. 2 2 1 + =

    M (x, y) , :

    x = y =

    ,(OM) =1

    2 22 2 2( ) x y x y = + = + :

    2 2x y 1,+ =

    :

    2 2 1 + =

    2. = =

    :

    y

    x

    = =

    ( x 0= )

    x

    y

    = =

    ( y 0= ).

    AA

    t

    y

    y

    xx

    1

    |x|

    |y|

    (x,y)

    O

  • 3.2 61

    (1) (2), - .

    3. . = 1

    :

    =

    =

    ( 0 0)

    :

    1 = =

    .

    4.

    22 2

    2 2

    11 1

    = =

    + +

    i) 2 2 1 + = 2 0 :

    2 22 2

    2 2 2 2 2

    1 1 111

    + = + = =

    + .

    2 21

    .1

    =+

    ii) 2 2 1 + = 2 21 ,

    1 =

    +

    :2

    2 2 22 2 2

    1 11 1 .

    1 1 1

    + = = =+ + +

    2

    22 .1

    =

    +

  • 62 3:

    1 513

    = 90 180 ,<

  • 3.2 63

    1. 3x5

    = x ,2< < -

    x rad.

    2. 2x3

    = 3

    x ,2

    < < -

    x rad.

    3. 3x3

    = 3

    x 2 ,2< < -

    x rad.

    4. 2 5x5

    = 0 x ,

    2

    < < -

    x rad.

    5. x 2 = 3 x 2 ,2< < -

    2 x x

    .1 x

    +

    6. , x :

    i) x = 0 x = 0.

    ii) x = 1 x = 1.

    iii) 3

    x5

    =

    4x .

    5 =

    7. , M ( x, y) x = 3 y = 3, O(0,0) = 3.

    8. x = 2 y = 3 , 9x2 + 4y2=36.

    9. x = r , y = r z = r , x2 + y2 + z2 = r2.

    10. :

    i) 11

    =+

    ii) 4 4 22 1 = .

  • 64 3:

    11. :

    i) 1 21 +

    + =+

    ii) x x 2 .1 x 1 x x

    + =

    +

    12. :

    i) + =+

    ii) 2 2 2 2 . =

    13. :

    i) x x x x1 x 1 x

    + = +

    ii) 1(1 x) 1 x x

    x + =

    iii) 1 x xx x

    = +

    iv) 1 1x x x x.x x

    =

    B

    1. x + x = , -:

    i) x x ii) 1 1x x+

    iii) x x + iv) 3 3x x. +

    2. :

    i) 4 4 2 2x x 1 2 x x + = ii) 6 6 2 2x x 1 3 x x. + =

    iii) 6 6 4 42( x x) 3( x x) + + - x , .

    3. x ,2 2

    < < 1 x 1 x

    2 x.1 x 1 x

    + =

    +

    4. 0 x ,2

    < 1 x 1 x 1 x x

    .x 1 x1 x 1 x

    + + + = =

    +

  • 3.3 1 65

    3.3 10

    , , 0 90 .

    ' ' .

    ' , ' = , , - , x'x. . , - :

    1( 30 ) (30 )

    2 = =

    3( 30 ) (30 )

    2 = =

    3( 30 ) (30 )

    3 = =

    ( 30 ) (30 ) 3 = =

    9 , :

    24 4 2 = =

    24 4 2 = =

    14 4 = =

    1

    4 4 = =

    : . :9 :

    ( ) ( )

    ( ) ( )

    = = = =

    A

    (x,y)

    (x,y)

    A

    y

    x

    t'

    t

  • 66 3:

    180

    ' -

    180 , ' 180 , = , , ' y'y . . , - :

    :

    180o . :

    9 150 180 30 ,= :

    9 2 ,3 3 = :

    2 33 3 3 2 = = =

    (180 ) (180 )

    (180 ) (180 )

    = =

    = =

    1150 (180 30 ) 30

    2 = = =

    3150 (180 30 ) 30

    2 = = =

    3150 (180 30 ) 30

    3 = = =

    150 (180 30 ) 30 3 = = =

    tt

    y

    x

    A A

    (x,y) (x,y)180

    B

    B

    0

  • 3.3 1 67

    180

    '

    180 , ' 180 , = + , , - ' . - . - , :

    2 13 3 3 2 = = =

    2 33 3 3 = = =

    2 33 3 3 3 = = =

    :

    180o -, . :

    9 210 180 30 ,= + : 1

    210 (180 30 ) 302

    = + = =

    3210 (180 30 ) 30

    2 = + = =

    3210 (180 30 ) 30

    3 = + = =

    210 (180 30 ) 30 3 = + = =

    (180 ) (180 )

    (180 ) (180 )

    + = + =

    + = + =

    y

    A A

    B

    (x,y)

    (x,y)

    t

    t

    y

    x

    180 +0

  • 68 3:

    9 4 ,3 3 = + :

    4 33 3 3 2 = + = =

    4 13 3 3 2 = + = =

    4 33 3 3 = + = =

    4 33 3 3 3 = + = =

    90

    ' - 90 , ' 90 , = , , ' xOy . . , - :

    ,

    90o , - .

    , 60 90 30 ,= :

    360 30 ,2

    = =

    160 30 ,2

    = =

    60 30 3 = = 360 30

    3 = =

    (90 ) (90 )

    (90 ) (90 )

    = =

    = =

    t

    t

    (x,y)

    (y,x)B

    A

    B

    A90

    y

    xO

    y=x

    0

  • 3.3 1 69

    , 0 90 .

    1 1 536 .4+

    =o

    54 .o

    54 90 36 ,=

    2 2 1 + =

    2 254 54 1, + = : 22 2 1 5 6 2 5 10 2 554 1 54 1 1 ,

    4 16 16 + +

    = = = =

    :

    :

    2 - :

    ) 90 ,+ ) 270 ) 270 +

    i) 90 90 ( ),+ = :

    (90 ) (90 ( )) ( ) . + = = =

    90 .+

    54 10 2 554 .

    54 1 5

    = = +

    1 554 364+

    = =

    10 2 5544

    =

    54 1 55454 10 2 5

    + = =

  • 70 3:

    ii) 270 180 (90 ) = + , : (270 ) (180 (90 )) (90 ) = + = = .

    270 .

    iii) 270 360 90 360 ( 90 ),+ = + = + : (270 ) ( 90 ) (90 ) + = = = .

    270 .+

    1. : i) 1200 ii) 2850 .

    2.

    i) 1876

    rad ii) 21

    4

    rad.

    3. : i) ( ) = + ii) ( ) 0+ + =

    iii) 2 2 +

    = iv) .2 2 +

    =

    4.

    5. :

    9( x) (2 x) x

    21.

    21(13 x) ( x) x

    2

    + + = +

    6. :2 2( x) ( x) (2 x) 2 x .

    2 + +

    ( ) (180 ).

    ( ) (90 )

    + +

  • 3.3 1 71

    1. :

    2. :

    2

    5 7(5 ) (7 )

    2 21.

    5 7(5 ) (7 )

    2 2

    + + =

    + +

    3. x x 5,3 6 + + =

    -

    :

    2 2x x .

    3 6 + +

    4. :( x)

    0 1.x ( x)

    +< < + +

    I) - , , - .

    1. =1, = 0 . 2. = 0 , = 1. 3. + =2 . 4. 21 = 5. 2 220 70 1 + =

    6. x (x ) x =

    7. x 2 2x x =

    8. (x ) x 0,2

    + = x 0 =

    9. x (x ) ( x) 06 3

    + =

    495 120 495 ( 120 ).

    ( 120 ) 495

    + +

  • 72 3:

    II. .

    III. .

    1. ( =900 ) , :

    ) 2 2 1, + = ) 2 2 1, + = ) =1. 2. : ) ( + ) = , ) ( + ) = , ) ( + ) = . 3. :

    ) ( ) ,2 2

    + = ) ( ) ,2 2

    + = ) ( ) .

    2 2+

    =

    1 1200

    2 1500

    3 2100

    4 3000

    5 2100

    6 3000

    7 3000

    8 2100

    3

    32

    33

    12

    12

    33

    32

    3

  • 3.4 73

    - -

    112

    - -

    . t,

    1

    t 12

    + 1

    t 12

    . -

    , t, -

    t, 1

    t 12

    + , 1t 12

    .

    112

    .

    t.

    , t, t + 2 sec t2 sec.

    3.4

  • 74 3:

    ( t) 2 sec.

    :

    , , 1 1. -

    . .

    - , , , .,

    f (t) t,= t .

    - .

    :

    x (x rad) .

    x (x rad) =

    ( 360 ) ( 360 ) , + = = x :(x 2 ) (x 2 ) x + = =

    2. .

    f , >0 , x A :

    i) x T A,+ x T A

    ii) f (x T) f (x T) f (x)+ = =

    f.

  • 3.4 75

    x (x rad). =

    2.

    , : x

    xx

    =

    :

    { }1 x x 0= x 1

    (x ) (x ) x, + = = .

    , , :

    xx

    x

    =

    :

    { }2 x x 0= .

    f(x) = x

    f(x) = x 2, 2, . [0,2]. x xrad . - , , . :

    x 0

    ,2

    .

    , - f(x) = x

    0, .2

    f(x) = x :

    y

    xx rad

    B

    A

    M

    M

    O

  • 76 3:

    ,2

    3,2

    3 ,22

    x ,2

    = 12

    =

    3

    x ,2

    = 3

    1.2

    =

    :

    . [0, 2]:

    . :

    x 0 2

    x 0 1 0,71 0 0,71 1 0,71 02 0,712

    4

    2 3

    4 5

    4 3

    2 7

    4

    x

    x

    2 3

    2

    20

    0 1 0 1 0. .

    y

    x

    0x2pi

    O

    1

    12pipi

    4

    3pi2

    7pi4

    y = x

    43pi4

    pi 2pi5pi

  • 3.4 77

    f(x) = x , 2, [2, 4], [4, 6] . [ 2 ,0],[ 4 , 2 ] . , .

    . - x ( x) x. = f(x) = x 0(0,0) .

    f(x) = x

    f(x) = x 2, 2, .. [0, 2]. :

    - :

    x

    x

    2 3

    2

    20

    1 0 1 0 1. . .

    x 0 2

    x 1 0,71 0 0,71 1 0,71 0 0,71 1

    4

    2 5

    43

    2 3

    2 7

    4

    22

    pi/2pi/2

    1.5

    0.5

    0.5

    1

    1.5

    0

    1

    pi 2pi3pi/2 5pi/2

    x

    y

  • 78 3:

    y = x

    0 x 2 p .

    f(x) = x 2, :

    y

    x

    pipi 0

    1

    1

    2pi 3pi 4pipi2

    pi2

    5pi2

    7pi2

    y = x

    3pi2

    . x (x) = x. f(x) = x - y'y.

    f(x) = x

    f(x) = x ,

    , .. , .2 2

    ( , ,2 2

    ).

    x rad - . x .: x

    2

    2

    ' , . f(x) = x

    , .2 2

    y

    x

    0x2pi

    0

    1

    1

    pi

    2pipi

    43pi2

    7pi4

    y = x

    5pi4

    3pi4

    2pi

    y

    O

    x

  • 3.4 79

    x 2

    x .

    x2

    = -

    f. x 2

    x .+ x2

    = f.

    f(x)=x , - , :

    - . f(x)=x .

    f(x)=x , ( 5.3: (x)=x .

    x 0

    x 1 0 1

    2

    3

    4

    6

    6

    4

    3

    2

    3 1,7 3 1,73 0,6

    3

    3 0,63

    x2pipi pi 2pi

    23pi

    2pi

    23pi

    2pi

    y = x

    yy

    O O

    BE

    MB

    B

    A AA A

    B

    x x

    yy

    O O

    BE

    MB

    B

    A AA A

    B

    x x

  • 80 3:

    1 f(x)=3x .

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

    f (x 2 ) 3 (x 2 ) 3 x f (x)+ = + = = , x . f (x 2 ) 3 (x 2 ) 3 x f (x), = = = x .

    - f(x)=3x.

    2 f(x)=2x

    f(x)=2x , 2x 2, , x - . , f(x)=2x . :

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

    3 f(x)=32x .

    3, 3 .

    x 0 2

    x 0 1 0 1 03x 0 3 0 3 0

    2 3

    2

    x 0

    2x 0 1 0 1 0

    4

    2 3

    4

    y = xy = 3x

    y

    2pi pi123

    321 2pi 3pi 4pi

    xpi

    y = x

    y = 2x

    x3pi2 2 2 2

    pi 3pi 2pipi pi

    1

    1

    y

  • 3.4 81

    , f(x)= x, ,>0: (i) ,

    . (ii) 2 .

    f(x)= x, ,>0

    1. ,

    i) f (x) 2 x,= g(x) 0,5 x,= h(x) 2 x,= 0 x 2 ii) f (x) 2 x,= g(x) 0,5 x,= h(x) 2 x,= 0 x 2 .2.

    f (x) x=

    g(x) 1 x= + h(x) 1 x= + 3.

    f (x) x= g(x) 3x,= 0 x 2 .4.

    f (x) x= g(x) 3x,= 0 x 2 .

    5. xf (x) 2 .2

    = -

    ; ; f .

    x 0

    32x 0 3 0 3 0

    2

    4 3

    4

    f(x)=32x :

    - .

    y

    x

    3

    3

    pi 2pi3pi3pi0pi pi pi2

    pi4 2 2 2

    y = 32x

  • 82 3:

    6. xf (x) 22

    = .

    7.

    i) f (x) x= ii) g(x) 1 x= + iii) h(x) 1 x= + .

    8. f (x) 2x.=

    9. f (x) x.=

    B

    1. :

    2.

    y 3 t ,6 =

    y

    t .i) -

    .ii) 0 t 12.

    pi

    pi

    pi

    pi

    pi

    pi

    pi

    2pipi

    pi pi 2pi

    pi pi 2pi

    pi pi 2pi

    2pi

    2pi

    2pi1

    1

    1

    1

    3

    2.5

    0.5

    x

    x

    x

    x

    x

    x

    x

    y

    y

    y

    y

    y

    y

    y

  • 3.5 83

    3. 1m. ,

    1

    h 1 3t,3

    = +

    t . i) -

    . ii) iii) -

    0 t 2 .4. H x

    - x(t) =0,1+0,1.3t, t . i) .ii) 0 t 2 . 0,15m;

    3.5

    x=

    1

    x .2

    =

    y = x

    1

    y .2

    =

    x, f (x) x= -

    1

    .2

    2,

    x, (. ), 2 . 2, .

    xy=x

    pipiO

    2pi

    yy=

    pi6

    12

    5pi6

    pi1m

  • 84 3:

    -

    1

    x2

    = [0, 2], 6

    5 ,

    6 6

    =

    5 1.

    6 6 2

    = =

    1x

    2 =

    , x = , = , :

    ,

    x 2= +

    x 2 ( )= +

    x 23

    ,

    x 23

    = = + +

    1 3x2

    =

    3 ,

    3 2

    = 3

    ( ) .3 2

    =

    x ( ),3

    = :

    2 12x4 2 + =

    1 ,

    6 2

    = 2x4 6 + =

    ,

    y

    M M

    x5pi/6

    pi/6

    12

    AOx 2 6,

    5x 2

    6

    = + = +

  • 3.5 85

    2x 2 2x 2 x4 6 6 4 24

    + = + = + =

    72x 2 2x 2 x

    4 6 6 4 24 + = + = + = +

    x =

    ,

    .. 1

    x .2

    =

    -

    1

    x2

    = [ , ] 3

    ,3

    1.

    3 3 2 = =

    1

    x2

    =

    ,

    x24

    ,7

    x24

    = = +

    2x 24 6

    ,

    2x 24 6

    + = + + = +

    x 23

    x 23

    = + =

    y

    x

    M

    M

    pi3pi3

    12

  • 86 3:

    ,

    , x = , = ,

    1 2x2

    =

    2 ,

    4 2

    = x ,4

    =

    :

    2 32x2

    =

    3 ,

    6 2

    = 3

    6 2 =

    5 3.

    6 2

    =

    5

    2x ,6

    =

    x 2

    x 2

    = +

    =

    x 24

    ,

    x 24

    = + =

    52x 2

    6,

    52x 2

    6

    = + =

    5x

    12,

    5x

    12

    = + =

  • 3.5 87

    x =

    x 3. = - . , , , ..

    ,2 2

    , .

    , x 3 =

    . ,3

    3.3

    =

    x 3 = : x ,3

    = + .

    , x = , x = , :

    x ,= +

    x . =

    1 x 1 =

    1,4

    = 1.4 =

    x ,

    4 =

    x ,

    4

    =

    2 x 3 =

    x ,6

    = +

    3,6

    = x ,6

    =

    pi xpi pi

    y

    pi22

    3

  • 88 3:

    1.

    i) x 0 = ii) 2x2

    = iii) x 0 = iv) 2

    x2

    =

    2.

    i) 1

    x2

    =

    ii) x 1 = iii) 2

    x2

    =

    iv) x 1 =

    3.

    i) x 0 =

    ii) 3

    x3

    = iii) x 1 = iv) x 3 =

    4.

    i) 3

    x3

    =

    ii) 3

    x3

    =

    5. i) (1 x)(2 x 3) 0 =

    ii) (2 x 2)(1 x) 0 + =

    6.

    i) ( 3 x)(1 x) 0+ =

    ii) 2(2 x 1)( x 3) x 0 + = 7. -

    i) x 0,951 =

    ii) x 0,809 = iii) x 28,636 =

    8. i) 2 3x 3 =

    ii)

    x 1 05

    + = iii) 2x3 3 07

    =

    9.

    i) x 13 + =

    ii) 2 3x 1

    4 =

    iii) 5x 3

    4 =

    10. i) 22 1 0 + = ii)

    22 x 3 x 2 0 + = iii) 23 t 3 2 3 t = +

    11. i) 2 2x 5 x 4 + = ii) x 2x 1 = 12. x,

    :

    i) f (x) 3 x ,2 =

    0 x 2 , < ii) g(x) 7 x ,2

    =

    0 x 2 <

  • 3.6 89

    13. ( -

    ) tS 75 50 ,

    6

    = +

    t o t = 1 . i) 100000 , ii)

    . 1.

    i) x x 04 + =

    ii) 2x 3x 0

    3 + =

    2. i) x x 1 x x + = + ii) 2

    12 x 4

    x =

    3. x 1 = (3,4).

    * 4. 1+x=x [0,2).

    5. : x x3 = +

    [0, 2).

    3.6

    , - 1 2M , M (. 1). , - . (. 2).

    O O

    y y

    x x -

    -

    M (, )1M (, ) M((-), (-))2

    (1,0)

    . (1) . (2)

    (1,0)

  • 90 3:

    , 1 2M ,M , :

    1: 2: : 1 0 : ( ) ( )

    2 1 M OM AOM ,= = 2 1(M M ) (AM).= 2 2

    2 1(M M ) (AM)=

    :2 2

    1 2 2 1 2 1(P P ) (x x ) (y y ) ,= +

    1 1 1P (x , y ) 2 2 2P (x , y ), :

    2 2( )= +

    (2)

    2 2 ( ).=

    2 22 1( ) ( ) =

    2 2( ) 2 2 ( ) + =

    ( )+ =

    :

    :

    (1)( ) = +

    , , 00 360 , < < , .

    , :( ( )) ( ) ( ) = + =

    ( ) + =

    22 1(M M )

    2 2

    2 2 2 2

    ( ) ( )

    2 2= +

    = + + +

    2( ) 2 2

    2 2

    [ ( ) 1] [ ( ) 0]

    ( ) 1 2 ( ) ( )

    = +

    = + +

  • 3.6 91

    (1) (2) - , - . , :

    (1), , - .

    , :

    x x2 =

    x x,

    2 =

    :

    ( ) ( ) ( )2 2

    2 2

    + = + = = + = +

    :

    (3)

    :

    ( ) + = +

    ( ) =

    15 (45 30 ) 45 30 45 30

    2 3 2 1 2( 3 1)

    2 2 2 2 4

    = = +

    += + =

    75 (45 30 ) 45 30 45 30

    2 3 2 1 2( 3 1)

    2 2 2 2 4

    = + =

    = =

    15 (45 30 ) 45 30 45 30

    2 3 2 1 2( 3 1)

    2 2 2 2 4

    = =

    = =

    75 (45 30 ) 45 30 45 30

    2 3 2 1 2( 3 1)

    2 2 2 2 4

    = + = +

    += + =

  • 92 3:

    : (5)

    , :

    (6)

    :

    (7) (8)

    ( )1 +

    + =

    ( )1

    =+

    1( )

    + =

    +1

    ( )+

    =

    , :

    3145 30 3 3315 (45 30 )1 45 30 3 3 31

    3(3 3)(3 3) 12 6 3 2 3

    6(3 3)(3 3)

    = = = =

    + ++

    = = =

    +

    ( )( )

    ( )

    1

    + + + = =

    +

    + + = =

    - + , , .

    , : ( ), + , ( ) 0, + 0 0. :

    0

  • 3.6 93

    -

    1 3 ,5

    = 3 22< <

    12 ,13

    = 3 ,2

    < <

    +.

    (+) = + ( ) + = . :

    2 2 9 161 1 ,25 25

    = = = 3 22< <

    2 2 144 251 1 ,169 169

    = = =25 5 ,

    169 13 = = 3

    2

    < 1 , x =1 x

  • 144 4: -

    0 2x x 6 0 (x 3)(x 2) 0 x 3+ + x 2 2x x 6+ x 3< x > 2 , x 3= x = 2 3 x 2. < 2 , x 3< 1< x < 2 . x 3,= x =1 x = 2 .

    A(x) . B(x). ... . (x)>0 ( <

    2 2(x 1)(x x 6)(2x x 1) 0 + + + < x 2 2P(x) (x 1)(x x 6)(2x x 1)= + + + .

    P(x) x ( , 3) (1,2).

    3 2x 3x x 2 0 + + >

    1,

    3 2P(x) x 3x x 2= + + :

    1 (2, )+ (x) ,

    2(x 2)(x x 1) 0 >1 5 1 5

    (x 2) x x 0.2 2

    + >

  • 4.3 145

    . 1 5 ,22

    +

    (x) ,

    , x2, . , (x) .

    x , 1 5 1 5x2 2 +

    < < x>2.

    , . , - , :

    : f (, f()) (,f()) x'x, .

    11 1 0f (x) x x ...... x

    = + + + +

    ,

  • 146 4: -

    3f (x) x 3x 1= +

    1o :

    , , (1,2)

    2o : 1,1, 1,2, ... 1,9 :

    (1,5 , 1,6).

    3o : (1,5 , 1,6) :

    (1,53 , 1,54) 1,53 < < 1,54. = 1,5.

    f (1,5) 0,13 0

    f (1,6) 0,30 0

    f (1,53) 0,01 0

    f (1,54) 0,03 0

    1. . i) 4 25x 6x= ii) 3 2x 2x 9x 18 0+ = iii) 5 4 3 23x 5x 3x 5x+ = + iv) 6x 64 0 = v) 3 2x x 2 0+ = vi) 3x 7x 6 0 + =vii) 3(x 1) 1 0+ + = viii)

    2 2 37(3x 2) (1 x) (3x 2)(1 x) 0+ + =

    ix) 3 2 2x 8 7(x 5x 6) 9x 36+ = + + + x)

    4 3x 3x 6x 4 0 + =

    3x 3x 1 0 + = - 1 2. .

    f (1) 1 0

    f (2) 3 0

    =

    1 2

    B(2,3)

    A(1,1)

    1,5

  • 4.3 147

    2. i) 3 2x 3x x 2 0 + + = ii) 3 23x 8x 15x 4 0+ + = iii) 3x 10x 12 0 = iv) 3 2x 2x 7x 6 0+ + + =

    3. . i) 4x 3x 2 0+ = ii) 4 3 22x 3x 6x 24x 5 0 + + =

    4. , x.

    i) 2 2P(x) (2 3x)(x x 2)(x x 1)= + ii) 2 2 2Q(x) ( x 4)(x 3x 2)(x x 1)= + + + +

    5. : i) 52x 162x 0 ii) 3 2 2(x x 2x 2)(x 9) 0 + > iii) 3 22x 5x 6x 9 0 + > iv) 3 2x 4x 3x 18 0 +

    6. : i) 3 2x 2x 3x 6 0+ + + > ii) 4 3 2x 6x 22x 30x 13 0 + + iii) 3x 3x 2 0 + < iv) 4 3 2x x x 3x 6 0 +

    7. x'x :

    i) 3 2f (x) 3x 3x 5x 2= ii) 3g(x) 4x 3x 1=

    8. , - 4 3 2f (x) x 5x 3x x= + + x'x.

    9. :

    i) 8 4x 15x 16 0 = ii) 6 3(x 1) 9(x 1) 8 0 + =

    iii) 2

    x x6 5 6 0x 1 x 1

    + = + +

    10. 3x 5x 3 0+ = (0,1) .

  • 148 4: -

    '

    1. :

    i) 3 21 1 1 4

    x x x 0,10 2 5 5

    + + = ii) 3 25 22 5

    x x x 06 3 2

    + =

    2. , 4 3 2P(x) x x x 16x 12= + + x+1 x 2. (x) = 0.

    3. k , 3 2x x kx 3 0 + + = .

    * 4. x 2 x 2 0, + = N, *2, ,

    .

    5. 6 4 2P(x) x 5x 10x k= + k x 1 (x). k (x) = 0.

    6. 5dm 9dm, . , - dm 21dm3.

    7. t

    2

    3

    3t tc .

    t 50

    +=

    + ,

    4 33t 2t 300t 200 0.+ = t .

    8. 36m3, x.

    3 x 3 x

    1m

    xm

    xmxm

  • 4.3 149

    9. . V,

    .

    10. t , mm - 2d 15t(t 6t 9).= - ;

    11. , , 108cm ( ). , 11664 cm3.

    12. i) (1,2)

    ii)

    y=x3+x2 x - .

    iii) .

    2 3500V (2000 100 20 ),3

    = +

    1 1B( , ).

    2 2

    3 2x x 5x 3 0+ + =

    3 33x xy

  • 150 4: -

    13. . :

    ) 200ml , 1cm, .

    3 2x 3x 2x 200 0.+ + = x mm.

    ) - 1lit, 10cm .

    3 2r 10r 318 0+ = r

    mm. ( 1000 318 ).

    ) - , 250ml, 5cm .

    h ( -).

    4.4

    , , . .

    1o 2 2 1x 02x 1 x(2x 1)

    + =

    x x 0 1

    x .2

    :

    xcm

    x+2 cmx+1 cm

    r+10 cm

    r cm

    h+5cmh+5cm

  • 4.4 151

    3o 2x 7 x 2.+ =

    x 7x .2

    ' x :

    ( )

    3 1. - x=1 .

    2 2

    4 3

    3

    3

    2 1 2 1x 0 x(2x 1)x x(2x 1) x(2x 1) 0

    2x 1 x(2x 1) 2x 1 x(2x 1)

    2x x 2x 1 0

    x (2x 1) 2x 1 0

    (2x 1)(x 1) 0

    + = + =

    + =

    + =

    + =

    , , . - - .

    2x 7 x 2+ = +

    2 2( 2x 7) (x 2)+ = +

    22x 7 x 4x 4+ = + +2x 2x 3 0+ =

    12

    1. - x 1.=

    2 x x 2.=

    x 0. -, 2x x 4x 4,= + 2x 5x 4 0 + = x1 = 4 x2 = 1. x, - x 0 . , :

    x = 4 : 4 4 2= x = 1 : 1 1 2= . x=4.

    ( )

  • 152 4: -

    4 2x 6 x 4 1.+ + =

    x, 2x 6 0+ x 4 0,+ x 3. ' x :

    3 5. x = 5 .

    2x 6 1 x 4+ = + +2 2( 2x 6) (1 x 4)+ = + +

    2x 6 1 2 x 4 x 4+ = + + + +

    x 1 2 x 4+ = +2(x 1) 4(x 4)+ = +

    2x 2x 15 0 =

    ( )

    ( )

    ( )

    (x)

    > 0 (< 0)B(x)

    A(x)0 A(x) B(x) 0

    B(x)> >

    A(x)0 A(x) B(x) 0,

    B(x)< <

    , A(x) B(x) 0 > A(x) B(x) 0 < (x).

    , 2

    2

    (x 1)(2x x 1)0.

    x x 6

    + +>

    +

    2 2(x 1)(x x 6)(2x x 1) 0 + + + > , P(x)> 0 , , P(x) -

    x ( 3,1) (2, ). +

    .:

    2, 3 4, x , .

  • 4.4 153

    2

    2

    x 4x 30.

    x 3x 4

    +

    + :

    2

    2 22

    x 4x 30 (x 4x 3)(x 3x 4) 0

    x 3x 4

    + + +

    + 2x 3x 4 0.+

    2x 4x 3 + 1 3 , 2x 3x 4+ 1 4 .

    :2 2P(x) (x 4x 3)(x 3x 4)= + +

    x - +x2- 4x + 3 + + - +x2 + 3x-4 + - + +

    P(x) + - - +

    -4

    00 0

    100

    30

    0

    x ( , 4) [3, ). +

    1. :

    i) 2 2

    2

    3x 1 2 x 3x 2

    x 1 x x x

    + =

    ii)

    2

    2

    x 2 4

    x 1 x 1 x 1 =

    +

    2. : 3 22 x x 2 x 2 0 + + =

    3. :

    i) 3x 4x=

    ii) 3x 2 4 = iii) 5x 1 4 =

    iv) x 3 x 1+ = + v) x 3 10 x 1+ = + vi) x x 20 10+ =

    vii) x 8x 32 x

    = +

    viii)

    1 2 x x 1+ = +

    4. :

    i)

    x 2 0x 1

    >

    + ii)

    2x 1 0x 3

    +

    iii)

    2

    2

    x x 2 0x x 2

    +

    A(x) 0B(x)

    -

    x

    A(x) B(x) 0 B(x) 0.

  • 154 4: -

    ( )

    5. :

    i) 2x 3 4x 1

    +>

    ii) x 2 4

    3x 5

    + iii)

    2x 3x 10 2 0x 1

    +

    iv) x 23x 5 x 1

    v) x 32x 1 x 2

    +

    6. : 2 2 1x2x 1 x(2x 1)

    +

    B

    1. :

    i) 2x 3 1 3x+ < ii) x 3 x 5 > 2. :

    i) x 3 x 10 0+ = ii) 3 2 3x x 6 0+ =3. :

    i) 2 2x x 4 x x 2+ = + ii)

    4. :

    i) x 1 = ii) 24x 1 2x+ =

    5. 4 3 22 x 3 x 3 x 3 x 4 0 + =

    1. 2 3(x x 1)(x 1) x 1,+ + = - 3 3 1 3 2P(x) x x x + += + + x

    2+x+1 (, , ).

    2. : i) 1f (x) x ( 1)x 1+ = + +

    2(x 1) . - .

    ii) 1g(x) ( 2)x x x 2n n-= n- -n + n -n + 3(x 1) .

    3. : i) 4 3x 2x 2x 1 0+ + + = ii) 4 3 2x x 4x x 1 0+ + + =

    ( 4 3 2x x x x 0, + + + + = 0.

    ).

    x 1 x 4 x 4 + = +

  • 4.4 155

    : x2

    4. i) 4 3 2x x 16x 2x 4 0+ + = ii) 4 3 2x 8x 13x 8x 1 0+ + + =5. 2 2 2(x 2x 1) 3(x 2x 3) 14 0+ + + + =

    : 2x 2x 1 y.+ =6. , 5 2x 3x x+ + + , -

    2x 2, 5x+8.

    7. 17 16 15 14P(x) x 12x 12x 12x ... 12x 1,= + + + (11) (13).

    8. - . - C , x 3 210x 100x 270x 180. = +

    i) ;

    ii) ;

    1x y.

    x+ =

  • 156 4: -

    2 1,414213,= 2,

    ,

    2

    2 2.

    =

  • 157

    2

    1,24

    x2

    =

    3 (*)x x+ =

    3 2 3 2

    3 3x2 2 2 2 2 2 = + + + +

    * 3 : 3 2x x x 0+ + + = - x y

    3

    = 3x Ax B.+ =

  • 158 4: -

  • . : , :

    :31 1 1 1 1

    2 2 2 2 8 = =

    :0 1 = 1 1 ,

    = = 0 N*

    - . :

    2 22 3 93 2 4

    = =

    122 ,

    145

    , 0, .

    . O

    , . T - ..

    253 ; Av p q pq( ) =

    , :

    4-

    5

    5.1

    .... , 1,

    , 1

    > = =

    N*

    2 2 55 25 5(3 ) 3 3

    = =

  • 5.1 161

    253 5 2x 3 ,= 5 23 .

    2

    5 253 3 .=

    Av >0, , :

    =

    , , , , 0 0. =

    ..

    , - .To . ..

    - :

    23 2 338 8 64 4= = =

    43 43 3

    4 43 4

    1 1 127 2727 327

    = = = =

    1 1 11 771234 3 4 34 12

    + = = = =

    21,4

    1,43,421

    5

    2

    1.4

    5

    1.4

    3.21

    7 373

    xy

    xy

    xy ( )

    =

    +/ =

    =

    : x x , - ;

    23 ;

    ( B' . 104)

    2 , , . 1, 1,4, 1,41, 1,414, 1,4142, 1,41421, 1,414213,.... (1)

  • 162 5:

    2 3:31, 31,4, 31,41, 31,414, 31,4142, 31,41421, 31,414213, (2) :

    1

    1,4

    1,41

    1,414

    1,4142

    1,41421

    1,414213

    4,4,74,724,7

    3 33 65553673 0

    284,728

    696503 769503 73393 78393 0154,7288

    =

    Av : (1) , (2) , . - 23 . - 23 4,7288.

    :

    Av >0, x x ,

    + , ( ) , ( ).To x x. :

    , x>0, 0x=0.O :

    3 2

    2

    x lim

    =

    23

    2

    xy

    xy

    x

    exp

    =

    =

  • 5.1 163

    , ' , - .

    :

    , 1 2x, x , x , :

    1 2 1 2x x x x+ = 1 2 1 2x x x x: =

    x x x( ) = x

    xx

    ( )

    =

    1 2 1 2x x x x( ) =

    . x x . x x, :

    f : xf (x) ,= , 1, .

    = 1, f(x) = 1.

    f(x) = 2x . :

    x -5 -4 -3 -2 -1 0 1 2 3 4 5

    1 2 4 8 16 3212

    14

    18

    116

    132

    y=2x

    (x, y) - . , -

    xf (x) = >1, : 0

    1

    1x

    y

    y=2x

  • 164 5:

    .

    (0, )+ .

    . 1 2x , x :

    1 2x x ,< 1 2x x <

    y'y (0,1) - x.

    x

    x

    y

    x1

    2x

    1x

    2

    y= , >1x

    A(0,1)

    x1

    g(x) .2

    =

    :

    x -5 -4 -3 -2 -1 0 1 2 3 4 5

    32 16 8 4 2 112

    14

    18

    116

    132

    x12

    (x, y) - . , -

    xf (x) = 0

  • 5.1 165

    f(x) = 2x x1

    g(x)2

    =

    x :

    - y'y.

    xx

    x

    1 1g(x) 2 f ( x)

    2 2 = = = =

    f(x)=x, 0 1,< : 1 2x x , 1 2

    x x

    , , :

    1 2x x , = 1 2x x .=, :

    , . .

    1 2x x1 2x x = =

    xx 1

    1

    x

    y

    y=2y=2 xx

    x2

    1

    x2

    x

    1 2

    1x

    x

    yy

    xO Ox1x 2x x

  • 166 5:

    i) :

    ii) 2x x x 2 x3 8 3 9 0 (3 ) 8 3 9 0 = =

    3x = y, y28y9 = 0 1 9. :

    x3 1= x3 9=' , 3x> 0, 3x = 32 x=2, .

    2o :

    3x = 2y = :

    x y

    x y

    2 3 3 2 22( )15 3 2 9

    2

    =

    + =

    = 1 = 8,

    x = 0 y = 3.

    2 3 2215 92

    = + =

    x

    y

    3 12 8

    =

    =

    x 0

    y 3

    3 32 2

    =

    =

    ( )

    3x 3x 612 2 264

    3x 6

    x 2

    = =

    = =

    1 :

    i) 3x 1264

    = ii) x x9 8 3 9 0 =

    11

  • 5.1 167

    i)

    ii)

    4o :

    i) xf (x) 2 3= + ii) x 3g(x) 2 = iii) x 3h(x) 2 2= +

    i) f - (x) = 2x 3 . ii) g (x) = 2x 3 .iii) h (x) = 2x 3 2 .

    3o :

    i) ii)

    3 > 1

    2x 3x 139

    >

    2x x1 1( )2 4

    + >2

    2

    x 3x 2

    x 3x 2 0

    >

    + >

    2 2x x x x 21 1 1 1( ) ( ) ( )2 4 2 2

    + +<

    + >

    x

    y

    4

    O

    1

    1

    3

    y=2

    y=2x

    x+3

    x1O

    y=2

    y=2x3

    y

    3

    x

    1 x

    y

    1

    1

    O

    y=2

    y=2x3+2

    32A(3,3)

    x

  • 168 5:

    e . - 1 , 100% 1, 2, 3, ... , , . -:

    0 (1 )

    = + , .100

    =

    =1, =1 11 1(1 1) 2 = + = .

    =2, = 12

    2

    211 1 2,252

    = + =

    .

    =3, = 13

    3

    311 1 2,443

    = + =

    .

    ............................................................................................................................

    =, = 1

    1 11 1 1

    = + = +

    .

    :

    1 2 4 12 52

    2 2,25 2,441406 2,613035 2,70481311

    +

    ( ) ( )

    ( )

    ( ) ( )

    1000 10000 100000 1000000

    2,716923 2,718145 2,718268 2,718280

    , , 11

    +

    . - e. , Leohard Euler (1707-1783). e e = 2,71828.

    1e lim 1

    = +

  • 5.1 169

    . , , .

    - e. - f(x) = ex. - .

    e

    c t

    0Q(t) Q e= (1) , t. To Qo - Q ( t = 0) Q0 > 0, c . - . c > 0 Q , c

  • 170 5:

    5 ,

    t5

    0Q(t) Q 2

    =

    5 , c t0Q(t) Q e= :

    c 5 5 c00

    1c 55

    Q 1Q e e

    2 21

    e 22

    = =

    = =

    t

    50Q(t) Q 2 .

    =

    1. :

    i) xf (x) 3= x11

    f (x) ( )3

    =

    ii) xf (x) 3 ,= x

    2f (x) 3 2= + x

    3f (x) 3 3= iii) xf (x) 3 ,=

    x 24f (x) 3

    = x 25f (x) 3+=

    iv) xf (x) 3= x 26f (x) 3 1= +

    v) xg(x) e= , x 21g (x) e ,+= x2g (x) e

    = x3g (x) e 2= +

    2. : i) x2 64= ii) iii) iv) x 13

    81 =

    v) x3 64( )4 27

    = vi) 4x x 127 9 += vii) x 1 x32 16 = viii)

    2x x 23 1 =

    3. : i) 2x 1 x2 4 2 0+ = ii)

    x x2 4 5 2 2 0 + =iii) 2x 1 x3 26 3 9 0+ =4. : i)

    2x 5x 65 1 + < ii) 2x 4 x 17 7 +> iii) x 1 2x 41 1( ) ( )2 2

    + ' :

    y

    x

    A(1,0)1x 2x

    y=log x, 0

  • 5.3 183

    , , : 1 2log x log x , = 1 2x x=

    , :

    1 2 1 2log x log x x x = = .. 22log (x 1) 3, = :

    2 2 32 2 2

    22 2

    2

    2

    log (x 1) 3 log (x 1) log 2

    log (x 1) log 8

    x 1 8

    x 9

    = =

    =

    =

    =

    x 3 = x 3=

    (x, y) - - (x)=lnx.

    f(x)=lnx+1 - (x)=lnx 1 , g(x) ln(x 2)= - (x)=lnx 2 .

    , .

    -

    1 i) (x) ln x = ii) f (x) ln x 1= + iii) g(x) ln(x 2)=

    x 0,2 0,3 0,5 0,7 1 2 3 4 5

    y=lnx 1,6 1,2 0,7 0,4 0 0,7 1,1 1,4 1,6

    (x) = lnx :

    x

    1

    1

    3O

    y

    y=Inx

    y=In(x-2)

    y=1+Inx

    2

  • 184 5:

    2 : 2>1 :

    .

    2>1 log0,5 x 1 0. > :

    x x=2 . - , x = 2.

    1. :

    2f (x) log x= 12

    g(x) log x=

    ; .

    2. :

    f (x) log x= , g(x) log x 1= h(x) log(x 1)= 3. xf (x) =

    g(x) log x= :

    i) (2,4) ii) B( 2,4) iii) (2, 4) i) ( 2, 4) 4. ASA

    DIN. x ASA y DIN y = 1+101ogx, x= 50, 100, 200, 400, 800, 1600 ASA. ; ( log2 = 0,3).

    0,25 0,5,>

    x 1 = x 2=

    2

    2log0,5 1 log0,5

    log0,5 log0,5

    >

    >log0,25 log0,5>

    2 22 2 2 2 2log (x x) log 2 log (x 1) log (x x) log [2(x 1)] = + =

    2x x 2(x 1) =

  • 5.3 185

    5. : i) log(x 1) log(x 1) log 2+ + = ii) log(x 1) log x 1 log5 + = iii) 2 2log x (log x)= iv) 2log(x 1) log x log 2+ =6. : i) x 1 x5 2 = ii) x 1 x 13 2 +=7. : i) 3log 2 3log 5 ii) 0,3log 5 0,3log 7

    iii) 2log(x 1)+ log 2x

    8. 7[H ] 10+ > 7[H ] 10 .+ < p.

    1. :

    i) f (x) ln x= ii) 21

    f (x) ln x2

    = iii) f (x) ln x= iv) f (x) log(10x 20)= 2. :

    i) 2f (x) ln(x x 1)= + + ii) 1 x

    f (x) ln1 x

    =

    +3. x log178, x xlog 81(2 2 3 ),+ x log3

    ;

    * 4. log log log = = 1

    =

    5. : i) log x log x= ii) 4 2ln x 5ln x 4 0 + =

    6. log5 log xx 5=

    2log x log55 5 4 x= +

    7. :

    i) 2log(xy) 4log 2

    log x log y 3(log 2)

    =

    =ii)

    xy 8

    log y 2log x

    = =

    iii) y 2x

    2log y log x log 2

    = = +

    8. : i) 2 2log x (log x)> ii) 2log(x 4) log3x < iii)

    log xx 10>* 9. 2 6log 3 log 9>10. , > 0 :

    >

  • 186 5:

    1. : i) 2 3x 5(x 3x 1) 1 + = ii)

    2x 3x 1x x+ + =

    2. :

    log log 2log log+ + + = ( + , 1 )

    3. log 2( ) , = log , log log

    (0 , , 1, 0)< > .

    4. , , , :

    5. log5 1 log 2= log(2x)x 5=

    6. (0, )2

    :

    x x x xlog 2 log 2 log 2 log 2 0 + + =

    7. (0, )2

    : x x( x) ( x) =

    8. : x x x27 12 2 8 0+ >

    ()

    log log log

    log log log

    =

    0 , , , 1<

  • 187

    3

    4

    4096 :128 32 (12 7 5)

    16 4096 (4 3 12)

    4096 8 (12 : 4 3)

    = =

    = =

    = =

    *

  • 188 5:

    1 4

    11 ,10+

    = +

    4

    11 1,000110

    = + =

    0

    =

    0

    1

    100.000.000

    10.000

    +

    = = +

    0 100.000.000 =

    1 100.010.000 =

    00 =

    110 =

    220 =

    84

    110 110

    = +

    23027 999.999.779 =

    10 =

    23027230.270 =

    .....................................................................................................

    84

    110 1 ,10

    = +

    2 100.020.001 =

  • 189

    2 4 8 16 32 64....

    1 2 4 8 16 32= = = = = =

  • 190 5:

    8x 10= 5y 10

    =

    yx e ,= y ln x.=

    3ln(1,00984768) 9,7995075 10 0,0097995075 0,0098= =

    84

    110 110

    = +

    4 4 44 5 510

    10 10 1010 10 10

    8 4 4 4

    1 1 11 1 110 10 10 10

    = + = + = +

    410

    4

    11 2,71814592710

    + =

    4 y10

    4

    1x 1

    10

    = +

  • 191

    1y ,x

    =

    x

    H

    y

    x1y=

  • 192 5:

    A log E=

  • (' ) 193

    1. : 2 2x 2 3 x x x 3 =

    2. i) x x + = 2 2 2. +

    ii) (1 t) x t x 2+ + = - t ( , ).

    3. 3

    2

    331 3

    =

    ,12

    3 2x 3x 3x 1 0. + =

    4. ( 9) 198 22 9= 9. 1+9+8 = 18

    9. 17397 1933 9= 1+7+3+9+7 = 27 -

    9. : 9, +++ -

    9. : 3 210 10 10>= + + + . -

    3 2f (x) x x x= + + + f (x) (x 1) (x) f (1)= + x=1 x=10.

    5. ( ). .

    : 11 1 0x x .... x 0

    + + + + =

    . 0

    -

    , 0

    . :

    i) : 3 22x x x 1 0+ + = 4 3 26x 29x 27x 9x 1 0+ + + + = ii) 2 12 .6. x x x3 4 5+ = .7.

    ( )

    xx2 4 2

    3 3 + =

  • 194 (' )

    8. 2log(x 3) log( x)+ = ; 9.

    4

    x log x 2 + = (0,) - .

    10. : x x3log (16 2 12 ) 2x 1 + 11. i)

    f (x) ln x= g(x) 1 x= ln x 1 x.

    ii) f (x) ln x= 2g(x) x 1= + 2ln x x 1. +

    12. , , , , :

    , , > 0, ++ =

    = =

    :

    13. , , , , :

    , , > 0, ++ =

    = =

    () = , (KM) = , () = , = , = , =

    14. , , , , :

    0 < , , <

    :

    ++ = .

    15. , , , , :

    0 < , , <

    , (M) = , (KM) = , () = , = A, = , = .

    = =

    2 2 2

    2 2 2

    2 2 2

    22 B2

    = +

    = +

    = +

    2 2 2

    2 2 2

    2 2 2

    222

    = + = + = +

    2 2 2

    2 2 2

    2 2 2

    222

    = + = + = +

  • - 195

    (3, 1) (21,24) 3 5, .4 3

    (3, 4) (5, 2).

    2, ,2

    +

    (2, 1).

    )( 3 1)( 1), , + +

    (4,3, 5)(10 2, 16 2, ) , + +

    11: y x 2,2

    = + 2 : y x 1 =

    1 11R T .600 30

    = +

    11 ,2

    = 212

    = 3 .2

    1,

    ii)

    3, 3, =

    3 3, =

    10. x ,= y ,= z .=

    11.

    2f (x) x 4x 3,= +2g(x) x 2x 3,= + +

    2h(x) 0,5x 3x 4.= +

    ( 1, 2), (2, 1).

    2. 2 4,3 3

    ( 1, 2), (1, 2), ( 2, 1), (2,1).2 1 , ,

    1 1 + + + +

    -

    3 2 3 2,2 2

    3 2 3 2, ,2 2

  • 196 -

    (4,3), ( 4,3), (0, 5).(1,0), (3,0), (2, 1), (4,3).

    1. ( ,1], [1, ),+ ( ,0], [2, ),+ ( , 1],

    [ 1,0], [1, ).+

    f (1) 1=

    h( 1) 2, = h(1) 2=

    f (x) f (3)g(x) g(1)

    4. 2f (x) 2 (x 1) 3= +2f (x) 2 (x 2) 1.=

    22(x 2) 22(x 3) 3 22(x 2)+ 22(x 3) 3+

    1. x 3,= y 3 2,= 0 45 . =

    2. 1, = 3. =

    3.

    4. 6 2

    3

    334

    5460

    18000 .

    6. 1 3 3, , , 32 2 3

    3 1 3, , 3,2 2 2

    0, 1,0,

    0,1,0,

    478, 733, 1062 58 .

    2 2 2 2 .2

    12 8 3, 12 3+

  • 197

    4 3 4x , x , x .5 4 3

    = = =

    2. 5 5 2 5x , x , x .3 2 5

    = = =

    3. 1 3x 3, x , x .2 2

    = = =

    4. 5 5 2x , x , x .2 3 3

    = = =

    5. 8 5 20 .5

    6.

    2 12

    2

    21

    2

    21

    2(3 ) .2

    3.

    2.

    2.

    y x,= xy ,2

    = y 3x=

    y x,= y 3 x,= y 0,5 x,= y 2,5 x.=

    323

    23

    x ,=

    x ,2

    = +

    x 2 ,4

    =

    187 1 ,6 2

    = 187 36 2

    =

    187 3 ,6 3

    =187

    3.6

    =

    21 2 ,4 2

    = 21 24 2

    =

    21 211.

    4 4

    = =

    3x 2 x 2 ,

    4 2

    = + = +

    3 11200 , 1200 ,2 2

    = =

    31200 3, 1200 .

    3 = =

    1 3( 2850 ) , ( 2850 ) ,

    2 23

    ( 2850 ) , ( 2850 ) 3.3

    = =

    = =

    3.3

    ( ) 180 ,+ + =

    90 .

    2 2 +

    + =

    7x 2 x 2 ,

    6 6

    = = +

    ii)

  • 198 -

    x 2 ,2

    =

    3x 2 ,4

    =

    x 2 , .= + x ,= x ,

    6

    = +

    x ,4

    = +

    x , .6

    = +

    x ,3

    =

    5x 2 ,6

    =

    24 7 24+

    12x , .60

    =

    10.

    t , .6

    =

    12.3.

    2x 2 x

    3 3

    x ,2

    = =

    = +

    x x ,3 4

    = = +

    24

    x ,36

    =

    5x 2 x 2

    4 4x 2 ,

    = = +

    =

    2 3x 2 x 2 ,

    5 5

    = + = +

    3x 2 ,

    5

    = +

    22

    2652 ,6

    =

    = +

    =

    52 ,6

    =

    2x 2 ,

    3

    =

    x ,6

    =

    x 22

    = + x 23

    = +

    2x 2 ,

    3

    = +

    6 6 2x x ,

    9 9+ +

    = =

    22x , .

    45

    = +

    6 2x ,

    9+

    =

    x 10 5 ,= +

    42 7x ,

    12+

    =

    4x 2 ,

    5

    =

    x 2 ,3

    =

    2x 2 x 2 ,

    3 3

    = =

  • 199

    x ,2

    = g 72 =

    3x2

    =

    3g 7.2 =

    x ,8

    =

    x ,5 30

    = +

    x ,4

    = +

    13 .4

    , .2

    7 13 19, , ,12 2 12 12

    12

    12

    x2

    2

    3

    3x 12

    x

    x

    7. 2(1 3)105 ,

    4+

    =

    2(1 3)105 ,4

    =

    105 2 3, =

    105 2 3, = +

    2(1 3)195 ,4

    =

    2(1 3)195 ,4

    + =

    195 2 3, =

    195 2 3. = +

    33( ) ,65

    + =

    56( ) ,65

    + =

    33( ) ,56

    + =

    56( ) .33

    + =

    x ,6

    = +

    x ,3

    =

    x ,4

    = +

    ( ) =

    2x x ,

    4 5

    = = +

    2x ,

    5

    = +

  • 200 -

    ( 2 ) (( ) ) .... + = + + =

    3. x4

    =5x .4

    =

    4. ,4

    =

    ....4 = =

    8. x6

    =2x .3

    =

    1 323

    3

    4 2 6

    242 ,25

    =72 ,25

    =

    242 ,7

    =72 .24

    =

    16( 2 )13

    + =

    x 2 x 2 ,= = + 2x 2 x 2 ,3 2

    = = +

    2 2 2 ,16 2 +

    =

    2 2 2 ,16 2 + +

    =

    2 2 2 .16 2 2 2

    + + =

    +

    4 ,2 13

    =9 ,

    2 13

    =

    2 ,2 3

    =3 .

    2 2

    =

    1 ,2 5

    =4 ,

    2 5

    =

    1 ,2 2

    = 2.2

    =

    2x 2 x 2 ,2 3

    = =

    x 2 ,3

    =

    x 2 ,2

    =

    x 2 ,= +

    3. 3 ,8 8 2 + =

    6. 5x 2 x 2 x 2 ,2 6 6

    = = + = +

    x x ,3 3

    = + =

    1 ,4

    3 24+

    2 2 2 ,16 2 2 2

    + =

    + +

    3...

    8 8

    =

  • 201

    14

    14

    3x x

    2x 6x 2x 8x +

    1 ( 8x 4x)2

    1 2x2

    2x x ,3 10 +

    = =

    2 2x x ,8 2

    + = =

    62

    22

    5. 2 3x x 2 4x x

    2 2x 2 x 2 x2

    4 2 +

    2 x2 .

    2

    x x 24 6

    = = +

    5x 2 ,6

    = +

    ++ =

    11f (x) 2 x6 = +

    3f (x) 2 x4 = +

    4f (x) 2 x3 = +

    f (x) 2 x4 =

    4x 2 ,3

    =

    x 22

    =

    x 2 ,=

    7x 212

    =

    11x 2 ,12

    = +

    512 12

    = =

    2. ( ) ( ) 2 24 + = +

    ,4

    =

    ( ) ( ) 2 2. + =

    7x x

    12 12

    = = +

    x ,3

    =

    x6

    =6 2

    x9

    + =

    6 2x ,

    9

    =

  • 202 -

    3. 102 2 2,+ 2 2 2. +

    4. 7x x ,24 24

    = + = +

    5. ,4

    =

    h 40( 2 1)=

    6. ,8

    =

    P 1 2= +

    1. AB 284 m2. B 1,58 m 5. 0115 6. 51,2 m

    7. 60,25

    1.-5.

    6.-7.

    8.-10. 12

    =

    11. 0 30 . =

    3.

    4. x 2 ,3

    =

    x 26

    = +

    5x 2 ,6

    = +

    6. 133

    7 (10 2 2)m,+

    (10 2 2)m

    (t 1)10 44

    + +

    9. 6 2 2 6 2, ,4 2 4 +

    12. x 24

    = +

    2x ,3 4

    = +

    13. S 200 2 2 2004 = + +

    ,8

    = maxS 200( 2 1)= +

    15. B 60 ,= 45 =

    1.

    2.

    3. 12

    =

    4. 1 =

    5. 1 1,

  • 203

    1. 10, =

    2. 2, = 19 =

    3. 3, = 5 =

    4. 2 20, ,3 3

    23

    =23

    =

    5. 2P(x) x 5x 1.= +

    1. 2(x) 3x 3x 8, = 3 2(x) x 3x 9x 27, = + + +

    3 16(x) 4x ,6

    = 5(x)3

    =

    2(x) 2x 1, = +

    (x) x 3, = + 2(x) 6x 8x 3 = +2(x) x , = 2(x) x 7 = +

    2. 3. k 4=

    4. 2(x) x 10x 25, 0 = + =2(x) x 8x 64, 0 = + =

    4 3 2(x) x x x x 1, 2 = + + + + =3 2(x) 3x 6x 12x 24, 48 = = 2(x) 4x 14x 30, 0 = + =

    5. P( 11) 4840 =6.

    7. P(x) x y = P( y) 0 =8.

    9. P(x) x 1= + P( 1) 0. =10.

    1. x x = 2. P(x) ( x ) (x)= + +

    0 = = =

    3. x 1 x 2

    4. P(0) 0,= P( 1) 0, =1P( ) 02

    =

    5. , = 1 =

    1. 6 60, ,5 5 2,3, 3

    50,1, 1,3

    2, 2

    1,2, 3 22 13,1,3 22

    2,9 4 5,9 4 5 +

    2, 2 , 1, 2

  • 204 -

    2. 1, 4 2 1

    3. 1, 2

    1, 56. x 2> x 1= x 2<

    x 1 x 2

    7. 1( ,0)2

    8. 2 5 x 0 1 x 2 5 < < < < +

    9. 2, 2 3 ,25

    1. 1, 2, 4 3,2, 15

    2. 1,2, 2, 3

    3. 3,1, 7, 11 4.

    5. 1, 1, 2 3 2 , 2 3 2 + +6. 7. t 4,6 c 0, 458. 9. 10.

    11. y 4x 108+ 2x y 11664,=

    12. y 5x 3=

    1x 1= 2x 3= ( 3, 18)

    1. x 3= 2, 2

    2.

    3. 59 1

    6. x 1 x 0>

    1. 3 2x2 5

    <

    x 5 0,

    f , 1 2x , x 1 2x x< :

    1 2f (x ) f (x )>

    T=f(t)

    5

    11T(C)

    3

    O 4 16 24 t(h)

  • 2.1 -- 33

    :) 1t 4= , f (4) 3= . :

    f (t) f (4) 3 = , t [0, 24] T = f (t) t = 4 -, f (4) 3= . :

    0x A , 0f (x ) f min f (x) .

    , 4f (x) 3x 1= + .

    4x 0 , x,

    43x 0 , x,

    43x 1 1+ , x.:

    f (x) f (0) , x.

    , f 0x 0= , f (0) 1=

    ) 2t 16= , T(16)=11 . :

    f (t) f (16) 11 = , t [0, 24] T =f (t) t =16

    , f (16)=11. :

    f, , 0x A () :

    0f (x) f (x ) , x A

    f, , 0x A () :

    0f (x) f (x ) , x A

  • 34 2:

    0x A , 0f (x ) - f max f ( x) . , 4f (x) 3x 1= + .

    4x 0 , x,

    43x 0 , x,

    43x 1 1 + , x.:

    f (x) f (0) , x

    , f 0x 0= , f (0) 1= . () () .

    (. ) (. )

    (. ) (. ).

    x

    y

    1

    1 y=f(x)

    y

    x

    1

    11

    y=f(x)

    y

    x

    1y=f(x)

    1

    y

    x

    1

    1

    y=f(x)

  • 2.1 -- 35

    ) - C

    f f -

    . Cf

    y'y, C

    f

    y'y Cf .

    , , M(x, y) Cf

    y'y M '( x, y) M(x, y) M '( x, y) C

    f , y = f (x) y = f (x) , :

    f . :

    yy.

    , 4 2f (x) 2x x 1= + , x :

    4 2 4 2f ( x) 2( x) ( x) 1 2x x 1 f (x) = + = + =

    , y'y .

    ) - C

    f f

    . C

    f

    , - C

    f -

    Cf .

    f, , , x A :

    x A f ( x) f (x) =

    f ( x) f (x) =

    y

    xx

    y

    x

    M M

    O

    f(x) f(x)

    Cf

    y

    y

    y

    xxx

    M

    M

    Of(x)

    f(x)

    Cf

  • 36 2:

    , , M(x, y) Cf

    M '( x, y) M(x, y)

    M '( x, y) Cf , y = f ( x) y f ( x), =

    :

    f ( x) f (x) = f . :

    .

    , 3f (x) 2x x= , x : 3 3f ( x) 2( x) ( x) 2x x f (x) = = + = , .

    : 2y x= ,

    4y x= , 6y x= ., , y'y , , - y x= , 3y x= , 5y x= ., , , .

    - f [6,6]. f :

    f, , , x A :

    x A f ( x) f (x) =

  • 2.1 -- 37

    ) f : i) , ii) , iii) .) f , - .

    f , y'y . , y'y f, - f, ().

    :) f : i) [0,2] [5,6],

    ii) [ 2,0] [ 6, 5], [0,2] [5,6] f .

    iii) [ 5, 2] [2,5] .

    y

    x 1

    4

    652256 x

    A

    A

    B

  • 38 2:

    ) f 4 x 6 6. :

    max f (x) f ( 6) f (6) 4= = = f 0 x

    0. :min f (x) = f (0) = 0.

    1) - :

    ) ) .

    2) - , .

    3) :

    i) 2f (x) x 6x 10= + x=3.

    ii) 22x

    g(x)x 1

    =+

    x=1.

    4) :

    i) 2 41f (x) 3x 5x= + ii) 2f (x) 3 x 1= + iii) 3f (x) x 1= + iv) 3 54f (x) x 3x= v)

    2

    5x

    f (x)1 x

    =+

    vi) 6 22x

    f (x)x 1

    =+

    .

    1

    y=f(x)

    x

    y

    2

    2

    y=g(x)

    x

    y

    1

    y=h(x)

    x

    y

    1

    2

  • 2.1 -- 39

    5) :

    i) 11

    f (x)x

    =

    ii) 2f (x) x 2= iii) 3f (x) x 1 x 1= +

    iv) 4 2

    1x

    xf (x)x 1

    +=

    + v) 5f (x) x= vi)

    26f (x) 1 x= .

    6) .

    7)

    8) -

    ) ) .

    y=f(x) y=g(x) y=h(x)

    x x x

    y y y

    y=f(x) y=g(x) y=h(x)

    x x x

    y y y

    C1 C2 C3

    x x x

    y y y

  • 40 2:

    2.2 -

    ) f (x) x 1= + .

    x 1, x 0

    f (x) ,x 1, x 0

    +

  • 2.2 - 41

    ) f (x) x 1= .

    x 1, x 0f (x) ,

    x 1, x 0

    0 , c ( )

    y

    y=|x|1

    y=|x|

    1 1

    1

    1

    1 1

    O x

    x

    c

    y=(x)

    y=(x)+c

    c cc

    y

    c

    O

  • 42 2:

    ) f (x) x 1= .

    x 1, x 1

    f (x) ,x 1, x 1

    +

  • 2.2 - 43

    f :

    f (x) (x c)= , c > 0 , c ( ).

    :

    f (x) (x c)= , f x x c , c x. , f c ( ).

    ) f (x) x 1= + .

    x 1, x 1

    f (x) ,x 1, x 1

    < = +

    f (x) x 1= + , 9 y x 1= , x 1 9 y x 1= + , x 1 , 1 x'x - x 'Oy xOy (x) x = ()., (x) x = - 1 , -

    Ox

    c

    c

    c

    c

    xxc

    (x-c)

    C Cf

    f(x)

    y

    y

    x

    y=|x+1|

    y=|x|

    1 1

    11

    11

    O

  • 44 2:

    f , :

    f (x) (x c)= + , c > 0 , c ( ).

    f (x) x 1= + . , , ,

    f (x) (x 1)= + , x, f (x) x 1= + x

    (x) x = x +1.

    :

    f (x) (x c)= + , f x x + c , c x. , f c ( ).

    f (x) x 3 2= + + .

    y x 3= + , - y x= 3 . - y x 3 2= + + , y x 3= + 2 .

    xx x+c

    yc

    c

    O

    c

    c

    =(x+c)f(x)