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! " , " [3], ' 1, . 15-25 ' +.

, .

, -

. -

-

.

. + X Y , , x y -

. ! ' +-

x X

y Y , '-

, + X f -

 

 

 

 

y = f ( x), x X .

. + X "# ( ) / 0

f ( x) "

D( f ) ; x ' / 0. .+

f ( X )

y Y , , x X ,

 

 

& / 0.

! +

f ( X ) + - , C ( . .

x X

+ C ), / 0 -

, , -

 

 

 

 

y = f ( x) = C = const .

'

 

y = f (x) ' -

Oxy

+ Oxy

(x, f ( x)) , x X .

( :

116

1. : ": / 0

/ . , / 0

F (x, y) = 0, ', / 0 y(x) .

! " " -, ', + '-, / .

2.< / ": ' / / 0.

3.= " ": / 0 " 0, -

' / 0.

* .

1. > " / 0

y =

 

2

 

.

 

 

 

 

x2

4

 

 

 

 

 

+ ,. + + " 0,

+ . = ", "

: x2 4 > 0 . @ x ( ; 2) (2;+ ) .

: D( f ) = {( ; 2) (2;+ )} .

2. > " / 0 y = 3x + log2 (x + 1) .

+ ,. ' -

, ' / + . C, " -

x + 1 > 0, x ( 1;+ ) .

: D( f ) = ( 1;+ ).

-

1. D / 0

. E 0 y = f (x) " X

&., " ' x X f ( x) = f (x) .

E 0 y = f (x) " X &., -

" ' x X f ( x) = f (x) .

E 0, , ,

/ 0 / 0 . F , :

1.@" G G / 0 -

.

2.< / G / 0 .

3.< / G / 0 -

.

* .

f (x) = x4 2x2 , . . ( x)4 = x4 ,( x)2 = x2 ,

1. E 0

f ( x) = f (x) .

 

2. E 0

f (x) =

x3 2x

. H,

x2

 

 

 

117

f ( x) =

( x)3 2( x)

=

x3 + 2x

=

x3 2x

= f (x).

( x)2

x2

x2

 

 

 

 

3.E 0 f (x) = 5x + 3 / 0 " ' , -

x x ', , "

, , / 0.

2./ 0

. E 0 y = f (x) "

X

&, T > 0 , :

 

 

1)

x X , x T X x + T X ;

 

 

2)

" ' x X f (x) = f (x + T ) ;

 

3)

T - . J - T -

/ 0 y = f (x) .

 

 

* .

 

 

E 0 f (x) = 3sin 2x + 1 T = ,

 

' , / 0 y = sin x

2 ,

-

sin 2x .

3.@' / 0

. E 0 y = f (x) " X

0 & + X0 , ( X0 X ), -

M , " ' x X0

f (x) M . E 0

y = f (x) 0 & +

X0 ,

m , " ' x X0 f (x) m .

E 0 y = f (x) 0 & + X0 , -

L > 0 , " ' x X0 | f (x) | L

.

@, / 0 ' ' ', ' '-

' .

@ +, ' / 0 +

' / 0.

* .

E 0

f (x) =

1

 

'

. =

x2 +1

 

 

1

 

 

 

 

 

x2 0, 0

 

 

1,

, '.

x2

+ 1

 

 

 

 

 

 

 

4. . / 0

 

.

E 0 y = f (x) #/ -

+

X ,

" x1, x2 X

x1 < x 2 ,

f (x1 ) < f (x 2 ) , "- ' "- -

/ 0.

118

E 0

y = f (x)

#/

X , "

x1, x2 X x1 < x2 ,

f (x1 ) > f (x2 ) ,

"- -

' - / 0.

! / 0 y = f (x) X " X , -

0

X .

 

 

 

 

X ,

E 0 #/ +

x1 < x2 ,

f (x1 ) f (x2 ) , " x1, x2 X ,

#/

+ X ,

 

x1 < x2 ,

f (x1 ) f (x2 ) , "

x1, x2 X .

 

 

 

X

, ", "

/ 0 / 0,

X .

 

 

* .

 

 

 

 

 

 

E 0 y = x3 . H, -x1 < x2 , x13 < x23 .

119

, , 1

.

 

 

 

 

 

/ 0 y = f ( x) , X - + ,

Y0 = f ( X ) - + . , / 0 "

X ( . 4.1 / 0). = ' + -

y Y0 " -

x X , . .

 

+ Y0

/-

y

 

 

0 x = ( y) .

J / 0 -

 

 

 

 

 

/ 0 y = f ( x) ,

Y0 y

 

 

/ 0 - "-

 

 

. H " / 0 Y0 - " -

 

 

 

+ , X - + -

 

x

 

. ! / 0 ,

O

x

 

/ x = ( y)

 

X

 

- - y = f ( x) -

M. 4.1

 

x .

 

 

 

 

" " ' / 0, "

- " / 0 y = ( x) . F -'

, ' / " / 0 y = f ( x) y = ( x)

" ' ' .

! / 0 y = f ( x) " + X ,

' y Y0 , " '

x X . , " / 0 / 0 y = f ( x) . > ' " + X0 +-

X , y = f ( x) + ". M /-

0 y = f ( x) ,

x X0 , / 0 x = ( y) , " .

* .

y = x3 ,

 

E 0

( . . X = ), -

+ Y0

= . C, , / 0 " / 0

y = 3x , X = .

(

M /:

y = f (u), u U , u = ( x), x X .

120

/ / 0 y = f (u) + U ,

/ 0 u = ( x) + X . @" , /, -

, # #, #

.

. + X * + / 0

x X , u = ( x) - + + U / 0 f (u) .

 

u & 0 + /-

0,

x , + X * ,

 

.

 

 

 

 

 

 

 

 

 

*

.

/ 0: / 0 y = f (u) = lg u,

 

M

+ U = (0, + ) ,

/ 0 u = ( x) = 1 x2 , +-

X =

 

. ,

 

(

X

)

=

(

,1

 

 

 

 

] .

. +

( X )

U " , (

+), ' / +

/ 0 + X * x , u = 1 x2 > 0 . M- -

' " 1 < x < 1. F, / -

+ / 0

y= lg (1 x2 ), ' x X * = ( 1,1)

@ , / 0, , -, " ' .

O , ' " : +, -

, +, 0 . 3 0 & / 0 / 0,

' ' "

' .

P ' " / 0:

1)0 0 / 0 (' " '); -

/ 0

y = a0 xn + a1xn 1 + ... + an 1x + an ,

' ak - , n - 0 0 ;

2)" 0 / 0 0 - 0 / 0;

3)0 / 0; ' " /- 0, ' -

. / 0:

121

 

 

 

 

 

 

 

2x + 1

 

 

y =

3x

2

+ 2,

y =

.

 

5x + 3

Q " / 0, ' ",

/ 0. - 0 / 0 -

:

1)/ 0 y = xa , ' a - 0 x > 0 ;

2)/ 0 y = ax (a > 0,a 1) ;

3)' / / 0 y = loga x(a > 0,a 1);

4)

' / 0 y = sin x , y = cos x , y = tgx ';

5)

" ' / 0 y = arcsin x , y = arccos x

'.

O + , / 0 ' "

- 0 / 0, + / 0,

/ (+, ,

+ ) 0 / 0 / 0,

. >, ,

/ 0

 

(

 

 

)

1 x2

 

 

 

 

 

y = lg

 

x +

1 + x2

 

, y = arctg

2x4

, y = tg

 

 

 

.

 

 

 

x

x

. " , / 0 / 0, -

+ " /,

+ /,

/ . =, , / 0

x + 2, x 0,

 

 

 

 

y =

 

 

 

 

x2 , x > 0

 

 

 

 

, .

 

 

 

 

+ , &

 

 

 

 

* 1. > " / 0

y =

 

3

.

2

x

 

 

 

+ ,: + + , . . x 2 , , , " / 0 +

x x = 2 .

* 2. > " / 0 y =

x 2

+

5 x

.

+ ,: O + + + " 0-

, . .

+

x 2 0

5 x 0 ,

, , + " x 2 x 5 . ,

,

x

2;5 .

 

 

 

 

 

[ ]

 

 

 

 

122

* 3. , " / 0 y = x3 + sin x -

.

+ ,: / 0 x x ,-

0. F ' , y = sin x - / 0 -

, ,

y( x) = ( x)3 + sin( x) = x3 sin x = (x3 + sin x) = y(x),

, , / 0 .

* 4. , " / 0 y = x2 + cos 2x -

.

+ ,: / 0 x x ,-

0. F ' , / 0 y = cos x -

, ,

y( x) = ( x)2 + cos 2( x) = x2 + cos 2x = y(x) ,

, , / 0 .

 

3x

 

 

* 5. , " / 0 y =

 

 

.

x +

5

+ ,: E 0 y

 

 

 

x = 5 . C, -

+ y( x)

x = 5 . @

x = 5 . ,

/ 0 + " , , . . / 0 / 0 " ' .

* 6. , " / 0 y = x2 + 3x + 7 -

.

+ ,: / x x : y( x) = ( x)2 + 3( x) + 7 = x2 3x + 7 .

=. . y( x) = y(x) y( x) = y(x) .

= ", / 0 , , . . , / 0 " ' .

* 7. , / 0 y = 3x2 + 7 '

, ' .

+ ,: 3x2 + 7 7 , , ' . = x2 + ' "- , / 0 '-

.

* 8. , / 0 y = 3x3 + 2x2

+

[0;+ ).

 

 

 

 

 

+ ,: 0 x1 < x2 . = ' x13 < x23

x12 < x22 . C,

y(x ) = 3x2

+ 2x2

< 3x3 + 2x2 = y(x ),

1

1

1

2

2

2

. . / 0 +.

123

* 9. , / 0 y = x2 3x " +

( ;0] .

+ ,: x1 < x2 0 . H 0 x x1 < x2 -

, " , ' "

x12 > x22 . T + " x1 < x2 0 ,

3x1 > 3x2 . P , "

x12 3x1 > x22 3x2 , , , y(x) " - +.

; 4.1

1.H / 0 " / 0.

2.H ' / / 0.

3.H G / 0, G

/ 0, / 0 " ' . T +, "-

' /.

4.H / 0. .

5.P/ / 0, ' - +. .

6.P/, / 0 .

7.H " / 0. T + G -

.

8.O " ' / 0 ? T + " + .

9., / 0 ,.

10.H " / 0.

11.H / 0.

4.2.* ". * .

-

:

* " .

".

; , .

< & & ", .

.

! " , " [3], ' 2, . 26-42 ' – .

* "

124

/ 0

, ' ' + . .

-' ' – -

.

. E 0, + ,

& " "#.

F, f (n),(n = 1, 2,...) . C /- 0 " G + " " :

f(1) = x1, f (2) = x2 , f (3) = x3,L, f (n) = xn ,L.

+- 0 x1, x2 ,L, xn ,L + -

& " {xn}.

,

 

xn , +

xn = f (n)

- /

&

" {xn }.

D

'

/,

 

 

 

1

 

 

n

" xn . >, {xn} =

 

,

{xn} = ncos

 

.

 

 

 

 

 

n

 

2

. {xn} 0 &,

 

M > 0,

, n

xn M.

: ', {xn} 0 &

( ), ' M , 0 &-

( ), - ' m . @, ' ' ,

. ! + ' ,

+ " '. @, '

( 0 ') " :

. {xn} 0 &, -

" ' M > 0, n0 , -

xn0 > M .

* .

1. {xn} = {1n} ', n -

xn 1.

2. {xn} = {n 5} ' , xn 4

n .

3.{xn} = { 1 n} ' ; xn 1.

4.{n 5} { 1 n} '.

. {xn} #/,

125

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