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### Transcript of Mathematica gr μαθ θετ κατ...

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18 MA 2016

1 (18/05/2016, 23:00)

• 2

mathematica.gr mathematica

http://www.mathematica.gr/forum/viewtopic.php?f=133&t=54288

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mathematica.gr

http://www.mathematica.gr/forum/viewtopic.php?f=133&t=54288

• 3

( )

18 MA 2016

:

()

1. f (, ), 0x ,

f .

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

f . 7

2. f, g .

4 3. -

. 4

4. , , , , , , .

) f :[, + IR , G f [,+ ,

f(x)dx G() G() .

) f,g 0x f(x) g(x) 0x ,

o ox x x xlim f(x) lim g(x)

.

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

0 0(,x ) (x ,) .

) f 1 1 , y ,

y f(x) x .

) f *, +, f [,+ M m.

10

1. . 262, ( i)

2. . 141

3. . 246

• 4

4. ) , . 334

) , . 166 ) , . 252

) , . 152 ) , . 195

B

2

2

xf(x)

x 1

, x IR .

1. f , - f .

6 2. f , f

. 9

3. f . 7

4. 1, 2, 3 f . ( )

3

:

1. f IR 2 2

2xf (x)

(x 1)

.

2 2

2xf (x) 0 0 x 0

(x 1)

.

x 0 f (x) 0 f x0 = 0, f

[0, ) .

x 0 f (x) 0 f x0 = 0, f

( ,0] .

f x 0 f(0) 0 .

2. f R 2

2 3

6x 2f (x)

(x 1)

.

2 3

2 3

2 3

2 (x 1) 02 2

2 3

2 (x 1) 02 2

2 3

2 (x 1) 02 2

2 3

6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x x

(x 1) 3 3 3 3

6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x

(x 1) 3 3 3 3

6x 2 1 3 3 3f (x) 0 0 6x 2 0 x x x x

(x 1) 3 3 3 3

• 5

f 3

,3

,

3 3,

3 3

3

,3

.

3 3

A ,f3 3

3 3

B ,f3 3

3 1A ,

3 4

3 1B ,

3 4

.

3. f , R.

2 2

2 2x x x

x xlim f(x) lim lim 1

x 1 x

xlim f(x) 1

.

f y 1

. 4. , , f :

x

f(x)

f(x)

0- 33

0

0 0

+ +

f(x)

33

14

14

+ +

11

1, 2, 3, f.

: f .

• 6

1. 2x 2e x 1 0 , x IR .

4 2. f :IR IR

2 22 x 2f (x) e x 1

x IR . 8

3. 2x 2f(x) e x 1 , x IR , f .

4 4. f 3 ,

f x 3 f x f x 3 f x x [0, ) . 9

:

1. 2x 2f x e x 1 , IR

2xf x 2x e 1 .

2 22 x xx 0 e 1 e 1 0 x IR ,

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

,

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

f x f 0 f x 0 ,

x = 0.

2x 2e x 1 0 , x 0 .

2. 2x 2e x 1 0 , x IR :

2x 2f(x) e x 1 , x IR .

, 2 22 x 2 2 x 2f(x) 0 f (x) 0 (e x 1) 0 e x 1 0 x 0 .

, f x 0 .

f (0, ) .

f (0, ) .

• 7

:

f(x) 0 (0, ) , 2x 2f(x) e x 1,

f(x) 0 (0, ) , 2x 2f(x) e x 1.

, f(0) 0 , : 2x 2f(x) e x 1 , x [0, )

2x 2f(x) e x 1 , x [0, ) .

, f ( ,0) .

f ( ,0) .

:

f(x) 0 ( ,0) , 2x 2f(x) e x 1,

f(x) 0 ( ,0) , 2x 2f(x) e x 1 .

, f(0) 0 , 2x 2f(x) e x 1 , x ( ,0]

2x 2f(x) e x 1 , x ( ,0] .

,

f :

) 2x 2f(x) e x 1, x IR )

2x 2f(x) (e x 1), x IR

)

2

2

x 2

x 2

e x 1, x 0

f(x)

(e x 1), x 0

)

2

2

x 2

x 2

e x 1, x 0

f(x)

(e x 1), x 0

3. H f IR,

2 2x xf (x) 2xe 2x 2x e 1

2 2 2 22 x x 2 x xf (x) 4x e 2e 2 4x e 2 e 1 .

2x 0 x R 2xe 1 .

22 x4x e 0 x IR , f (x) 0 x IR f (x) 0

x 0 .

, f x 0 , f .

4. h x f x 3 f x 0, .

h 0, , :

h x f x 3 f x .

f IR, f 0, IR .

x 0, x 3 x f -

:

• 8

f x 3 f x f x 3 f x 0 h x 0 .

h 0, , 1 1 .

f x 3 f x f x 3 f x , x 0, :

1 1

h x h x x x

.

x x , x IR , x 0 .

x 0 x x , x 0 .

x 0 .

f R , ,

:

0

f(x) f (x) xdx

f(IR) IR x 0

f(x)lim 1

x

f(x) xe x f(f(x)) e x IR .

1. f() ( 4) f (0) 1 ( 3)

7 2. ) f IR ( 4) ) f IR . ( 2)

6

3. x

x xlim

f(x)

6

4.

e2

1

f(lnx)0 dx

x

6 :

1.

0

0 0

0 00 0

f(x) f (x) xdx

f x x dx f (x) xdx

f x x f x xdx f x x f x xdx

f f 0 0 f f 0 0

f f 0 .

• 9

, x 0 ,

x 0 x 0

f(x)limf(x) lim x 10 0.

x

f x0 = 0, ( ), f(0) 0 , f() = .

x 0 x 0

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

x 0 x x

f (0) 1 .

2.

) f 0x IR .

f 0x , 0x 0x -

. Fermat 0f (x ) 0 .

( f -

R, f(x)e IR ( ), f(f(x))

IR ( ) xe IR) -

.

:

f(x) xe f (x) 1 f (f(x))f (x) e

0x x :

0 0f(x ) x

0 0 0e f (x ) 1 f (f(x ))f (x ) e

0x

01 e x 0 ,

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

) f x 0 x IR f , f -

IR

f 0 1 0 f x 0 x IR

f IR.

3. f(x) IR, x x

f(x),f(IR) lim lim f(x)

R (), , x

f(lim x)

x

2lim 0

f(x) .

x(, ), > 0, :

x x |x| |x| 2 2 x x 2

f(x) f(x) f(x) f(x) f(x) f(x)

x x

2 2lim lim 0

f(x) f(x)

, :

• 10

x

x xlim 0

f(x)

.

4.

e

1

f lnxdx

x

1u lnx du dx

x . :

x 1 u 0 x e u .

:

e

1 0

f lnxdx f u du: 1

x .

f IR *0, + f(0) = 0, f() = ,

0 u f 0 f u f 0 f u

, , (

),

e2 2

0 0 1

f lnx0 f u du du 0 dx

x

.

:

1. ye y 1 y = 0, 2x 2e x 1

x = 0. 2x 2e x 1 0 , x 0 .

1. lnx x 1 x 0 , x 1 .

, yx e , y IR , y ylne e 1 , ye y 1 , y IR , -

y 0 .

, 2x 2e x 1 , x IR , 2x 0 ,

x 0 .

, 2x 2e x 1 0 , x 0 .

1. 2g(x ) 0 xg(x) e x 1 xg (x) e 1 .

x 0 g (x) 0 , x 0 g (x) 0 x 0 g (0) 0 .

g R ( ,0]

[0, ) 0 g(0) 0 .

g(x) 0 x 0 . 2 2g(x ) 0 x 0 x 0 .

: g(x) 0 0

xe x 1 0 , .

• 11

3. 2 2x 2 xf (x) (e x 1) 2x(e 1), x R .

2 2 2 21 2 1 2x x x x2 2

1 2 1 20 x x 0 x x 1 e e 0 e 1 e 1 (1)

1 20 2x 2x (2) (1),(2)

2 21 2x x

1 2 1 22x (e 1) 2x (e 1) 0 f (x ) f (x )

f [0, ) .

2( x)f ( x) 2x e 1 f (x), x R

1 2 1 2x x 0 x x 0

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

f ( , 0] f x 0 f

IR f R.

4. x 0 .

, , x 0x 0 .

x 0 0|x | x ( |x| |x| x 0 )

0 0|x | |x | 3 0 0x x 3 .

:

0 0|x | 3 x 0 0 0 0|x | |x | 3 x x 3 x

0 0 0 0|x |,|x | 3 , x ,x 3 x

1 0 0 2 0 0 |x |,|x | 3 , x ,x 3 :

1 23f ( ) 3f ( )

1 2f ( ) f ( ) f ( ) 1-1

1 2 , 1 2 , .

0 0x |x | 3 0 0 0 0|x | x |x | 3 x 3 .

:

0 0 0 0f(x ) f |x | f(x 3) f |x | 3

x

0 0 0 0|x |,x , |x | 3,x 3 x 1 0 0 2 0 0 (|x |,x ), |x | 3,x 3