C Gymnasiou Mathimatika Methodologia & Askiseis

download C Gymnasiou Mathimatika Methodologia & Askiseis

of 28

  • date post

    04-Apr-2015
  • Category

    Documents

  • view

    32.994
  • download

    4

Embed Size (px)

Transcript of C Gymnasiou Mathimatika Methodologia & Askiseis

1.5 : . : :

1

( + ) = 2 + 2 + 22

: 2 ( + ) = ( + )( + )= 2 + + + 2 = 2 + 2 + 2 :

2

( ) = 2 2 + 22

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

3

( + )( ) = 2 2: ( + )( ) = 2 + 2

= 2 2 :

4

( + ) = 3 + 3 2 + 3 2 + 33

:

1

( + ) = ( + )( + )3

2

= ( + )( 2 + 2 + 2 ) = 3 + 2 2 + 2 + 2 + 2 2 + 3 = 3 + 3 2 + 3 2 + 3 :

5

( ) = 3 3 2 + 3 2 33

: 2 , ( ). 4 :

: . ( + ) ( ) .2 2

+ .

!!! 2 ( + ) 2 + 2

:

2 + 2 + 2 2 + 2

2

) = , : 1 , , .

2 , , := , , : = (1) (2) : = ( ) 3 =0 = :

1. : + ) = 2 2 2 2

) ( 2 + 4)( 2 + 9) ( + 6) = (3 2)2

2

) (1 ): 2 2 2 2 + = ( + ) ( ) = 2 2 4 4

( + ) ( )2

2

= = 4 4 2 + 2 + 2 2 + 2 2 2 + 2 4 = = = 4 4 42 2

2 + 2 + 2 ( 2 2 + 2 )

+ = 2 2 .

3

) : ( 2 + 4)( 2 + 9) ( + 6)2 =

2 2 + 9 2 + 4 2 + 36 ( 2 2 + 12 + 36) = 2 2 + 9 2 + 4 2 + 36 2 2 12 36 = 9 2 + 4 2 12 + 2 2 2 2 + 36 36 = 9 2 12 + 4 2 = (3 ) 2(3)(2) + (2) =2 2

(3 2)

2

2. x , y , :2 2 2 ( x + y + ) + ( x y + ) = 2 y 2 + ( x + )

2 . :

( x + y + ) + ( x y + ) =2 2

= ( x 2 + y 2 + 2 + 2 xy + 2 y + 2x) + ( x 2 + y 2 + 2 2 xy 2 y + 2 x) = 2 x 2 + 2 y 2 + 22 + 4x

( x + y + ) + ( x y + ) = 2 x 2 + 2 y 2 + 22 + 4x2 2

(1) (2)

: 2 2 y 2 + ( x + ) = 2( y 2 + x 2 + 2 + 2 ) = 2 x 2 + 2 y 2 + 22 + 4x (1) (2) : 2 2 2 ( x + y + ) + ( x y + ) = 2 y 2 + ( x + )

) , ( ). , . 3 . :

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

4

, :

2( x 2 + y 2 ) = ( x + y ) 2

2 x 2 + 2 y 2 = x 2 + y 2 + 2 xy 2 x 2 + 2 y 2 x 2 y 2 2 xy = 0 x 2 + y 2 2 xy = 0

( x y) = 02

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

4. + + = 0, :3 + 3 + 3 = 3 . + + = 0 = ( + ) . : 3 + 3 + 3 = 3 + 3 + ( + ) = 3 + 3 ( + ) =3 3

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

(1)(2)

5

, : ( , ..) ( , ..). ( , , ..) . : 1) :

1 ) x > 0 , x + 2 , x 1 ) x < 0 , x + 2 , x ) x, y 0 , x + y 2 xy . : ) x > 0 : 1 x+ 2 x 1 xx + 2x xx2 +1 2 x x2 2 x +1 0

( x 1) 02

. ) x < 0 : 1 x + 2 x 1 x x + 2 x x x 2 + 1 2 x x + 2 x +1 0 2

( )

( x +1) 02

. ) : x + y 2 xy x + y 2 xy 0 x 2 xy + y 0

(6

x y

)

2

0

7

1) : 2 2 ) (2 x + 1) (3x 2) (2 x + 5)(5 2 x )) (2 x 2 + x 1)(2 x 2 x + 1) + ( x 3)( x + 1) 4 ( x 1)( x + 1)( x 2 + 1): ) x 2 + 16 x 28 , ) 0

2) : ) (12 x5 6 x 4 3 x3 ) : (3 x 3 ) ) (2 x 3 y 2 6 x 2 y 3 + 3 x 2 y 2 ) : (3 x 2 y 2 ) 3) : 2 ) (43 + __ ) = __ + __ + 25 2 ) (3x + __ ) = __ + 4 y 2 + __2

) ( __ + __ ) = x 2 + x + __2

6 5

) ( __ + __ ) = __ + 6 x 2 y 3 + __2

1 ) __ + = __ + x + __ 22

1) : 2 2 3 2 2 x ) 4 ( x y ) 3 2y : y2 x102 3

)

x 0 ( x 3 y ) ( x 2 ) y 2 ( y 2 x )3

:

x 4 y : x3 y5 x4

: y 2

2) = 2x + 3 B = - 4x + 1 = 3 : 24 x 2 30 x + 9

3) : 3 3 9 x 2 + 3) x 3 x + 3 = ( ) 2 2 43 3 2 x x 2 + 27) x 3 x + 3 + = ( ) 3 3 27

) ( + ) ( ) ( + ) = 4 ( + )3 2

8

4) ) ( x 2 + 3 xy 2 ) =2

) 2 2 = 50 + = 10 = ) 10 2 2 5 3 15 = 5) 3 ) 4 4 ( ) ( + ) = 2 ( 2 2 ) )

( 2 + 4)( 2 + 9) ( + 6)2 = (3 2)2i.

6) :

(2 + 3 2 )

3

ii. iii. iv. v. vi. vii.

( x + y z) 1 3x + 32 2

2

(x 1) ( x 8)(8 + x)

( (

x y

)(

x+ y

)

2 x 1 x + 3

)

2

7) : 2 2 i. ( x + 5) = (3 x) + 24 ii. iii.

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

2 2

(3x 2) + (2 x + 3)(2 x 3) = 2 x (7 x 22) ( x + 3)

8) : i. 2 + 2 2 ii.

( ) = 2 + 2 + 2 2 + 2 22

9) x = 3 2 y = 3 2 + 2 3 , :x2 + y2 xy 10) x

1 = , x

x2 +

1 1 = x4 + 4 2 x x

11) x, y, x + y + = 2 x 2 + y 2 = 4 . 2xy = 2 4

9

12) x + y = i. ii.

7 3 x y = . : 2 2 2 2 x +y = ( x + 2)( y + 2) =

13) 2 2 = 20 + = 5 14) ( + ) = 20 , 2 + 2 = 8 . 2

15) x + y = 4 x y = 3 : i. ii. iii. iv. x 2 + y2 = x 3 + y3 =

( x y) =2

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

16) : 2 2 ( + 1) ( 1) = 3 i. 2 2 2 2 + = 4 ii. 2

10

1.6 : ( ) .

; , - . .

, . . + = ( + ) : : xy xz xw .. xy + xz xw = x + = x ( y + z + w) x x x .. i. 2 + 2 + 2 2 2 , 2 , 2 : 2 + 2 + 2 = 2( + + ) ii. 2 x + 4 xy + 6 xz 2x : 2 x + 4 xy + 6 xz = 2 x (1 + 2 y + 3 z ) . : , . . . .. iii. 2 x + 4 yx 2 y = 2 x yx + 4 2 y = x (2 y ) + 2(2 y ) = (2 y )( x + 2) 2 x, yx x 4, 2 y 2. (2 y ) x (2 y ) + 2(2 y ) .

11

2 2 . 2 2 = ( + )( ) . . .. x 2 4 = ( x 2)( x + 2)

, 2 : 2 2 ( + ) = 2 + 2 + 2 ( ) = 2 2 + 2 ..i.

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

2

ii.

x 2 + 4 xy + 4 y 2 = ( x + 2 y ) 2

2

:

f ( x ) = x 2 + x + 0 , 2

. : A. = 1 x 2 + x + . : . .. , : x 2 + x + = ( x + )( x + ) , . B. 0 x 2 + x + x2 + ' x + ' . .

12

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

3 + 3 = ( + )( 2 + 2 )

3 3 = ( )( 2 + + 2 ) . , : i. , ii. , iii. , iv. , v. , , . , : i. , ii. , , iii. , iv. , . , . . , .

13

1. :3 2 ) 3 x + 15 x

2 x5 y 2 + 4 x3 y 4 6 x5 y 3 ): ) 3 x . 3x 2 . 2 : 1 : 3 15 x 2 3 2 2 3x 3 x + 15 x = 3 x 2 + 2 = 3 x 2 ( x + 5) 3x 3x 2 : 3 x 3 + 15 x 2 = 3 x 2 x + 3 x 2 5 = 3 x 2 ( x + 5) ) . 2 x 3 y 2 . : 2 x5 y 2 4 x3 y 4 6 x5 y 3 2 x5 y 2 + 4 x3 y 4 6 x5 y 3 = 2 x3 y 2 3 2 + 3 2 3 2 = 2x y 2x y 2x y = 2 x 3 y 2 (2 x53 y 22 + 2 x 33 y 42 3 x 53 y 32 ) = = 2 x 3 y 2 (2 x 2 + 2 y 2 3 x 2 y ) . .

2. :

x 2 ( ) + y 2 ( ) )) 3 ( x y ) x + y ) (3 + 2)( x y ) + (4 2 )( y x): 14

) . . : x 2 ( ) + y 2 ( ) = ( )( x 2 + y 2 ) ) . x y y x . x + y x + y = ( x y ) , x y. :

3 ( x y ) x + y = 3 ( x y ) + ( x y ) = = 3 ( x y ) ( x y ) = 3 ( x y ) 1 ( x y ) = = ( x y )(3 1) ) y x = ( x y ) , x y . : (3 + 2)( x y ) + (4 2)( y x) = (3 + 2)( x y ) + (4 2) ( x y ) = = (3 + 2)( x y ) (4 2)( x y ) = ( x y ) (3 + 2) (4 2 ) = = ( x y )(3 + 2 4 + 2) = ( x y )(3 + 2 + 2 4) = = ( x y )(5 2) : , , , .

3. : ) x y + x y x3 + x 2 + x + 1 )2 2 ) x y x xy + x + y 1: ) . . . : x y x y = (x y ) + (x y ) =

= ( x y ) + ( x y ) = ( x y )( + )) . . . : x3 + x 2 + x + 1 = ( x 3 + x) + ( x 2 + 1) = ( xx 2 + x) + ( x 2 + 1) =

= x ( x 2 + 1) + ( x 2 + 1) = ( x 2 + 1)( x + 1)15

) . :

x 2 y x 2 xy + x + y 1 = ( x 2 y x 2 ) ( xy x ) + ( y 1) =

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

x 2 y x 2 xy + x + y 1 = ( x 2 y xy + y ) x 2 + x 1 = = y ( x 2 x + 1) ( x 2 x + 1) = ( x 2 x + 1)( y 1)

4. : ) x 4 y 4 ) x 2v y 2v3 3 ) 8 x: ) : :

(x )2

v

= x v2

x 4 y 4 = ( x 2 ) ( y 2 ) = ( x 2 + y 2 )( x 2 y 2 ) = = ( x 2 + y 2 )( x + y )( x y )) :

x 2 v y 2 v = ( x v ) ( y v ) = ( x v + y v )( x v y v )2 2

) : :

3 3 = ( )( 2 + + 2 )

3 2 83 x 3 = (2 ) x3 = (2 x) (2) + (2 ) x + x 2 = 2 2 = (2 x)(4 + 2x + x )

: , , . 3 .50. 16

3 3 = ( )( 2 + + 2 ) 3 + 3 = ( + )( 2 + 2 )

5. :x2 2 x y 2 +1: . . . :x 2 2 x y 2 = ( x 2 2 x + 1) y 2 = ( x 1) y 2 =2

= ( x 1 + y )( x 1 y ) . : .

6. : ) x 2 + 7 x + 12 ) x 2 + 7 x 60: ) 12 7. : 4 3 3 4 = 12 3 + 4 = 7 . x 2 + 7 x + 12 = ( x + 3)( x +