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### Transcript of S4E ±„ 1.µ‰¯± [Edit by Lisari Team 2015-16]

• l i s a r i . b l o g s p o t . g r

2015-16

study4exams.gr

lisari team

• lisari team / lisari.blogspot.gr study4exams.gr

K

1

1: - -

1: - .

I - -

[ - .1.2 ].

. "" "" .

(A R R )A O / .

, ( )

f x A ( )x A () y ( )yR R .

:

f : A R x y R ( )y f x=

x f yR f x y.

i. x ( ) . y , , f x .

ii. ( )f x ( )y f x= .

iii. xR f f x , ( ){ }fA D x : y f x= = = R R

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2

iv. ( )f A f R ( )y f A x A ( )y f x= , {f (A) y := R x A }y f (x)=

v. (x,y) x =

y = x f f

fC

. x fC y

.

:

f: :

) .

) f (x) x A .

:

f (x) f , R f (x) .

2 / 230

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3

1.

fC f, f fC

xx', fC .

[ ]fD ,=

2. f ( )y f x= , :

( ){ }fA D x : f x= = R R

:

( )f x R :

i. ( ) 11 1 0f x x x ... x = + + + + , i R, i 0,1,..., = 0, N ( ( )f x

fA D= = R

), :

ii. P(x)f (x)Q(x)

= ( P(x), Q(x), ), :

{ }fA D x : Q(x) 0= = R .

iii. f (x) g(x)= { }fA D x : g(x) 0= = R 2, N .

iv. g(x)f (x)h(x)

= ( )( )f

g xA D x : h(x) 0 0

h x = =

R 2, N .

v. [ ]f (x) ln P(x)= { }fA D x : P(x) 0= = >R .

vi. [ ]f (x) P(x)= fA D= = R .

vii. [ ]f (x) P(x)= fA D= = R .

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viii. [ ]f (x) P(x)= ( )fA D x : P x , Z2 = = +

R .

ix. [ ]f (x) P(x)= ( ){ }fA D x : P x , Z= = R .

:

" " .

( , , ..) .

fC f, f fC

yy' ( )f A fC .

( ) [ ]f A ,=

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1. fC ( )y f x= , :( ) ( )0 0 f 0 0M x , y C y f x = .

2. x A yR

fC .

fC , ..

3. f 0x 0x x= fC .

4. ( )f x 0=

'.

5. ( ) fC yy'

( )f 0 y= .

5 / 230

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6

6. ( )f x 0> fC xx', ( )f x 0< fC xx'.

7. ( ) ( )f x g x= fC

gC , f, g .

8. ( ) ( )f x g x> ( ) ( )g x f x> fC gC gC fC .

o g f> [ ) ( ), ,

o f g> ( ) ( ], ,

o [ ]f gD D , =

6 / 230

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7

.

9. ( )f x fC xx'.

10. ( )f x fC ( ) fC xx' ( ) xx'.

: f : A B R ,

o ( ) ( )1 2f x f x 1 2x x 1 2x , x A . o 1 2x x= ( ) ( )1 2f x f x= 1 2x , x A

:

yy'.

(0,0) .

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8

:

f : A R g : B R .

f g ( )f g= :

1. ( )A B= . 2. x A ( ) ( )f x g x= .

: f gD D A

f gf (x) g(x) x A

= = = =

1. f ,g

.

2. f g f gD D x

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

( ) ff x x , D= = R ( ) gg x x, D= = R

[ )0, = + . ( ) ( )f x x x g x= = = x . . ( )

0x R ( ) ( )0 0f x g x , ( ) ( )f 8 g 8 . 3.

( )f g , . ' ( ) ( )f x g x= x A .

4. f g= , .

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5. f g "" R . ' ""

R , , f, g. f gE D D ( ) ( )f x g x= x E . f g= .

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:

f, g , A B O / .

f g+ ,

f g ,

fg

fg

f, g :

f g+ ( )( ) ( ) ( )f g x f x g x+ = + A B

f g ( )( ) ( ) ( )f g x f x g x = A B

fg ( )( ) ( ) ( )fg x f x g x= A B

fg

( ) ( )( )f xf x

g g x

=

( ){ }A B x A B : g x 0 =

1.

A B . A B O = / , . A B O / .

2. f g+ , f g , fg , fg

, .

.

3. ' f . :

( )( ) ( )f x f x = fD A = R ( )( ) ( )f x f x = fD A = *N

10 / 230

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f : A , g : B R R

. :

( ) ( )( )gf

x f x g f x (1)

f (A) B O / .

:

1

f : A , g : B R R , ( )f A B .

gof : A R , ( )( )x g f x

f g.

:

2

f : A R g : B R

( )f A B , ( )f A B O / .

' ,

11 / 230

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( )f x B , ( ){ }x A : f x B = .

:

gof : R ( )( )x g f x , {x A : f (x) B} =

f g.

:

: fog 1. g R , x A ( )f x R .

f : A R g A .

2. , .

3. f : R R ( )g x x= . fog gof= , : () gof , fog R () xR :

( )( ) ( )( ) ( )fog x f g x f x= =

( )( ) ( )( ) ( )gof x g f x f x= =

12 / 230

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' g I .

4. fog gof . . . :

( ) ( ) ( )( )fog oh x fo goh x=

5. . .

6. .

: 14/2/2012

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14 / 230

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1

1: - -

2:

[. 1.3: - ].

:

f :

,

1 2x , x 1 2x x< :

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

,

1 2x , x 1 2x x< :

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

15 / 230

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2

f , f .

f :

, 1 2x , x

1 2x x< :

( ) ( )1 2f x f x

,

1 2x , x 1 2x x< :

( ) ( )1 2f x f x

16 / 230

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3

:

f A :

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

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

f () () f .

17 / 230

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4

-

( )f x x= + , 0 > = =fA D R

( )( )f =A R .

3.

( )f x x= + , 0 < = =fA D R

( )( )f =A R .

4.

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( ) 3f x x= , 0 > = =fA D R

( )( )f =A R .

5.

( ) 3f x x= , 0 < = =fA D R

( )( )f =A R .

6.

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

,2

,2 +

.

x2

=

f2 4 =

.

7.

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

,2

,2 +

.

x2

=

f2 4 =

.

8.

20 / 230

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( ) xf x = , 1 > = =fA D R ( ) ( )( )f 0,= +A .

,

( ) xf x e= , 9.

( ) xf x = , 0 1< < = =fA D R ( ) ( )( )f 0,= +A .

10.

21 / 230

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( )f x log x= , 1 > ( )0,= = +fA D

( )( )f =A R .

11.

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

( )0,= = +fA D ( )( )f =A R .

12.

22 / 230

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9

( )f x x= , =A R ,

0,2

,

3,2 2

3 , 22

T 2= f

2 , 22 +

,

32 , 2

2 2 + +

( )32 , 2