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### Transcript of Διαγωνίσματα Α και...

• 2017 1 2

2016-2017

:

:

A 15+10=25

A1. 2 x x 0 1 , , 0 .

1 2x , x (1), :

1 2

S x x

1 2

P x x

A2. ,

1. 0 2 x x 0 , ,

.

2. 0 >0 : x d(x, ) .

3. x y 0 x y 0 .

4. , , , :

.

5. 2 .

B 8+9+8=25

m : d(,30) 5

1. 25

35 .

2. 100 m.

;

.

3. 50 15 .

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• 2017 2 2

5+5+7+8=25

: f (x) (x x 1) (x x 1) x 1

1. : 2f (x) x x 1 x 1 .

2. f (x) 0 1 2

x , x

1 2

1 x 0 x .

3. 2

x , 2

x

2

x 1 .

4. 2 1 1 2

2 x 2 x

2 1 2

2 x x 3 .

9+8+8=25

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

.

1. (1).

2.

x 1 .

3.

x 1 .

16 - 01 - 2017

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• 1

1

-

A 15+10=25

1. : 2 2 1 , R

2.

.

1.

.

2. R

2

3.

y ' y

4. A f R , f (2) f (3) f

5. f , ox A ,

ox A , f .

6+7+12=25

f (x) 8 x 8 x

1. .

2. .

3. f .

,

f .

A B

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• 2

4+(6+9+6)=25

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

3.

1. ,

2. =2 =1

A. f [0,]

. 5

y2

f [0,]

, . (

).

. f (x) k

[0,] kR .

5+7+6+7=25

f

0,2

3f (x) x x (x x 5) x .

1. f

2. f (x) 5 x

3. , :

25

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• . . 1

. . , .

, . .

.

.

: 1 x x , x [0,2)

1

x 0,2 . :

2 2

2 2

2 2 2 2 2

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

x [0,2)

x 1 3 x x x 2 2

x , x 2

.

.

.

.

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• . . 2

2

1 x 0 x 0,2 ,

x 0

:

2 2

2 2

2 2

2

1 x x1 x xx 0

1 2x x x

x 0,

1 2x x 1 x

x 0,

2x 2 x 0

x 0,

2x 1 x 0

x 0,

x 0 x=-1

x 0,

3x= x x= 2 2

x 0,

x x= 2

, :

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• . . 3

3

2

2 2

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

x [0,2)

x xx 0 x(1 x) (1 x) 0 (1 x) (x 1) 0 x 1 x 1

x x 2

4

2 2

22

2 2

x x 11 x x

1 x x

x 1 x 1 1 x x

x 1 2 x x 1

1 x x2x (x 1) 0

1 x x

x 0,2

x 0 x 1

1 x x

x x 2

.

.

R

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• . . 4

5

x X x Y

2 2 2 2

1 x x 1 X Y x x 1 Y X 1

H . . .

x2

x

6

f (x) x g(x) x 1 .

1 x x , x [0,2)

[0, 2)

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• . . 5

7

:

1 x x x x 1 x x 144 x x 14

x x 4 4 4

2 (x )4 2

x [0,2)

(x ) 4 4

x 2 x 2 4 4 4 4

x 2 x 2 2

x x 2

8

:

x x 4 4 4f (x) x x x x 2 (x ) 4 4 4

f y=1.

:

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• . . 6

9 1 x 1 1 x 1 0 1 x 2

x 0,2 1 x x :

0 1 x x 1 :

x [0,2) 3x ,0 1 x 1 1 x 0 2 2 x ,

0 x 1 0 x 1 2x 0,

x , 2

, , :

, ,

: x 1 x ,

x x2

10

x , 2

f (x) 1 x x , x , 2

:

f '(x) x x f "(x) x x 0 x , 2

f ' 3x4

,

x 2

34

f x - +

f(x)

f max x2

x f f () 02

,

1 x x f (x) 0 x2

x .

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• . . 7

11

s(x) f (x) g(x) x x . y=s(x) y=1, :

12 1 x x , 0,2 ,

, 2

f x 1 x x , x , 2

f f 02

f , , 2

Rolle , 2

f 0

f x x x 0 x , 2

f(x) =0 , 2

x , x 2

- 2016

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_2017-----2016 (1)------