Диференціальне числення ФОЗ

e = 1 +1 +

1

+

1

+ ... +

1

+ R

(1),

(16.1)

2!

3!

n!

n

Rn (1) =

ec

, 0 < c < 1.

(n +1)!

):

Rn (1)

≤

e1

<

3

.

(n + 1)!

(n + 1)!

% ! n :

3

< 0, 001.

(n + 1)!

# (n + 1)! >

3

= 3000 , n = 6 . # " " "

0, 001

! (16.1) n = 6 , !:

e ≈ 2 +

1

+

1

+

1

+

1

+

1

= 2 +

1

+

1

+

1

+

1

+

1

≈ 2, 718 .

2!

3!

4!

5!

6!

2

6

24

120

720

! '' * " ! + " 3 (.

., " y = sin x " * +

! + , " ' * " / " , , " ! , , :

		x3		x3			x5
	sin x ≈ x , sin x ≈ x −		, sin x ≈ x −		+			.
		6		6		120

y = sin x # ( P ( x) = x ,					P ( x) = x −	x3	,
				1	3	6
P ( x) = x −	x3	+	x5	, " ! + ' * ' ', ! +
5	6	120

. 26.

62

$ . 26.

II. )! " * (.

. ( ! ! '

lim f ( x )

x→ x0 g ( x)

0 . %, y = f ( x)

0

y = g ( x) ' * x0 " " n '. # -

y = f (x ) y = g ( x ) x0 ' * ' # (,

' 3 ( %. & , ", " ! '- , * " 3.

#.

1. 6 ( lim ex − e− x − 2x .

x→		0		x − sin x
6 ' / ,:
ex = 1 + x +	x2		+	x3	+ o (x3 ), e− x = 1 − x +	x2	−	x3	+ o (x3 ),
				6
2					2		6

sin x = x − x3 + o (x3 ). 6

6 ,:

63

x

− x

1 + x +

x2

+

x3

− 1 + x −

x2

+

x3

+ o (x3 ) − 2x

e

− e

− 2x

lim

= lim

2

6

2

6

=

x − sin x

x − x + x

3

x→ 0

x→ 0

+ o (x3 )

o (x3 )

6

x3

+ o (x3 )

1

+

= lim

3

= lim

3

x3

= 2 .

o (x3 )

x→ 0 x3

+ o (x3 )

x→ 0 1

+

6

6

x

3

# " , lim

o (x3 )

= 0 .

x

3

x→ 0

$" * ∞ − ∞ .

2. lim

1

−

1

.

x→ 0

x2

sin 2 x

/ ,:

x

3

2

+ o (x3 ) − x2

x −

sin2 x − x

2

1

1

6

lim

−

= lim

= lim

=

x2 (x + o ( x))2

x→ 0

x2

sin 2

x

x→ 0

x2 sin 2 x

x→ 0

x

3

x

6

x

3

x2 − 2x

+

+ 2

x −

o (x3 ) + o (x6 ) − x2

36

= lim

6

6

=

x

2

(x

2

+ 2x o (x ) + o (x

2

))

x→ 0

−

x4

+ o (x4 )

−

x3

+ o (x4 )

−

1

+

o (x4 )

= lim

3

= lim

4

= lim

3

x

4

= −

1

.

(x2 + o (x2 ))

(x4 )

x→ 0 x2

x→ 0 x4 + o (x4 )

x→ 0

o

3

1 +

x4

# : (o (x2 ))2

= o (x4 ),

x2 o (x2 ) = o (x4 ),

xo ( x) = o (x2 ).

17.&

" & .

* " , +

(, , ! (. # *

" ' * " " .

64

I. 0 .

. ,, , ', " + *

: ', , ', ( . -

«& »). ) ( + " + !

' ', 3 '. # + ' , " " -

, ! " ! " .

! ( ( ) ). & %

y = f ( x ) (a,b) x (a,b ) -

f ′( x ) > 0 ( f ′( x) < 0) , y = f ( x )	$ ( ) -
(a,b) .
. % " ,	f ′( x ) > 0 x (a,b) . $ -

" * " x1, x2 (a,b) , x1 < x2 . 6 -

' 2 +, + x1, x2 ( * " c , ! -

*:

f ( x2 ) − f ( x1 ) = f ′(c )( x2 − x1 ) .

) * f ′(c ) > 0 x2 − x1 > 0 , f ( x2 ) − f ( x1 ) > 0 , ! f ( x2 ) > f ( x1 ) , !

" y = f ( x ) , (a,b) , ( ! ! .

". )! + " , ! , -

" y = f ( x ) , (,) (a,b) , ,, f ′( x ) > 0 ( f ′( x) < 0) x (a,b) . (, " ' y = x3

(−1;1) . & , ' ' * ( ( ( "-

(), y′	x=0	= 3x2	= 0 .
			x=0

! ( ( ) ). & % -

y = f (x ) (a,b) $ ( ) (a,b), f ′(x ) ≥ 0 ( f ′( x) ≤ 0) x (a,b ).

. % " , " y = f (x ) ,

(a,b). & * * x (a,b) x x , ! x + x (a,b ) . #, " x > 0 , f (x + x ) > f ( x ) , " x < 0 ,

f (x +		x ) < f (x ). & ! 3 "
	y	=	f ( x + x) − f (x )	> 0 .
	x		x

6 ' ! " :

f ′(x ) = lim	y	≥ 0 ,
x→0	x

( ! ! .

65

6 ,, (! -

" ") + * " " -

, " ! ', ', ! ,. #

' * " I . ) + " + "

!:

1)( I ;

2)( " (, ! 3 "-

;

3)+ ;

, , " ,, ’, – -

" ,.

#. 6 ( + " "

y = (3x − 1) 3( x − 4)2 .

6 (:

	2		′	2
f ′(x ) =	(3x −1)( x − 4)			= 3( x − 4)		+ (3x
		3			3

= 33 (x − 4)2 + 2( x − 1) = 9( x − 4) + 2 (3x −1) =
33 x − 4	3 3 x − 4

2 −1

−1) ( x − 4) 3 =

3

15x − 38

3 3 x − 4

.

: x =		38	( ( f ′(x ) = 0 )
1	15

& ( " (:

$ . 27.

) . & f ′(x )

			38
1)		− ∞;		: f ′( x ) > 0 – " ,;
			15

2)38 ; 4 : f ′( x ) < 0 – " ,;

15

3)(4; + ∞): f ′( x) > 0 – " ,.

II. # .

x2 = 4 ( ( f ′(x ) ,).

+ :

66

6 + ’" + "- ", " . & ".

. # x0 , * " y = f ( x ) ,

" , ( ( x0 − δ, x0 + δ) , , x ( x0 − δ, x0 + δ) , x ≠ x0 * f ( x ) < f ( x0 ) .

. # x0 , * " y = f ( x ) ,

" , ( ( x0 − δ, x0 + δ) , , , x ( x0 − δ, x0 + δ) , x ≠ x0 *:

f ( x ) > f ( x0 ) .

# ' * "

.

6 * ,, " " * (

. & " , * " (! *3 ! (-

3 " , ( ! ", 3 "-, +, * , . , 3-

+ * , " + ( ! *3 (3) -

", + ( ). #

" + *, * * * ( . 28).

$ . 28.

# x1, x3 , x5 – , x2 , x4 , x6 – . 8

, " + ! ! *3, + - " ( f ( x4 ) > f ( x1 ) ).

$3 " , ' + -

. # ! , " + +, ,

' . 5 '' * " " .

! ( ). & % x0

y = f ( x ) , f ′( x0 ) = 0 .

67

. ) * x0 – , ,

( x0 − δ, x0 + δ) (, x0 " , * " (! *3 ! ( 3

* ". # 7 ( . . 12) f ′( x0 ) = 0 .

". )! + " , , !

, " ( ', ', , "-

* ( .

#. $" ' y = x3 . % , y′ = 3x2

x = 0 ', '. $ x = 0 " "

, ( . 29).

$ . 29.

6 3 ! + , ,. 8 ,, x = 0 " y = x , . (

,. + , , ,, - " , . . , " y = 3 x , ( '

x = 0 , ( ( . 5).

# , " ', ', ! ,

(! I ) * " .

" , ! ", ( , , ! -

.

! (* ). ( x0 –

I y = f ( x ) , , -

x0 , , , . ), %

$ x0 f ′( x ) $ $ , x0

. + , % $ $ " $ -

(" f ′( x ) > 0 x < x0 , f ′( x ) < 0 x > x0 ), , % $ $ " $ (" f ′( x ) < 0

x < x0 , f ′( x ) > 0 x > x0 ).

68

& % $ $ $ "-

, .

. % " , " " δ > 0 -:

f ′( x ) > 0		x ( x0 − δ, x0 ) ,
f ′( x ) < 0		x ( x0 , x0 + δ) .

# ( x0 − δ, x0 ) " y = f ( x ) ,,

( x0 , x0 + δ) – ,. # x ( x0 − δ, x0 ) ( x0 , x0 + δ) + , * " -

*: f ( x ) < f ( x0 ) . ( ,, x0 – .

6 , ( + " -

.

1.6 ( I .

2.. 8 ! , * "

, " y = f ( x ) ( , " ,

' . 8 ! , * ", ( -

.

#.

6 ( +

y = 3 x2 ex .

2

−

1

2

2

2

2 + 3x

6 (:

+ (3 x )

y′ =

x 3 ex + x 3 ex =

ex =

ex .

3

3 3 x

3 3 x

%

f ′(x )

', ' x = − 2 3 , x = 0 . ) + -

x1 = − 2 3,

x2 = 0 . ! '.

x

2

−

2

2

−∞; −

−

; 0

0

(0; + ∞)

3

3

3

y

max

min

y′

+

0

–

,

+

#

+ ", – -

+ " . ) + x1 = − 23 , x2 = 0 – -

.

2. 6 ( +	y =	x + 1	,
		( x − 2)2

x (− ∞; 2) (2; + ∞).

6 (:

y′ =	( x − 2)2	− ( x + 1)2( x − 2)	=	x − 2	− 2( x + 1)	=	−x − 4	= −	x + 4	.
		(x − 2)4		(x − 2)3			( x − 2)3		( x − 2)3
					69

/ , : x1 = −4 ( y′ = 0 ) x2 = 2 ( y′ ,).

! ':

x	(− ∞; − 4)	−4	(−4; 2)	2	(2; + ∞)
y		min
y′	–	0	+	,	–

) + x1 = −4 , ' , x2 = 2 '

" ", * ", * ( " , '.

3. 6 ( +

y= 3 x (1 − x) .

6 (:

	2					1					1 − x												1 −
	y′ =	1	x−		(1 − x ) − x			=									− 3				=					4x		.
																				x
				3			3
															)2
	3							3( 3															33 x2
													x
/ , : x1 = 0 ( y′																									,) x2 = 1 4 ( y′ = 0 ).
! ':
x	(− ∞; 0)					0					1						1						1
											0;															; + ∞
																	4
											4												4
y																	max
y′	+					,					+						0							–

) + x2 = 1 4 , ' , x1 = 0 , ' -

, * ! , * " .

! (& ). ( x0

y = f (x ) 1– 2– , f ′(x0 ) = 0 ,

f ′′( x0 ) ≠ 0 . ) x0 y = f (x ) . + -

, % f ′′( x0 ) < 0 , , % f ′′( x0 ) > 0 .

. % " , f ′′( x0 ) > 0 . #

f ′′( x) , ( ( x0 − δ, x0 + δ) , " f ′′( x) > 0 . )-

+ " f ′(x )	, ' ' * . f ′(x ) < f ′(x0 ) = 0
x ( x0 − δ, x0 ),	f ′(x ) > f ′(x0 ) = 0 x ( x0 , x0 + δ ) , !

y = f (x ) x0 ', ( '. 0

70

+ * ' ' x0 " y = f ( x ) " , -. # .

#. 6 ( y = x2 ln x , x (0; + ∞) . 6 (:

y′ = 2x ln x + x2 1 = 2x ln x + x . x

'' ( ", , ' x > 0 , ,:

2 ln x + 1 = 0 , x = e−					1	. ) + , x = e−						1
					2							2 .
											0
(:
y′′ = 2 ln x + 2x	1		+ 1 = 2 ln x + 3 .
	x
% "' ' x0 , ,:
−		1					1
y′′(x0 ) = 2 ln e 2				+ 3 =	2 −				+ 3 = 2	> 0	,
							2

+ x0 3 " " , .

18.&

" & (").

III. 0 .

$" :

$ . 30.

+ ' '? ), ,.

+ " (. x0 " ,

3, " *. # " , , "

: " ! " ! + ! " 3-

, " *, ! * " , "

! ". # ! " -

71