Диференціальне числення ФОЗ

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Одесский национальный университет им. И.И. Мечникова

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[НЕСОРТИРОВАННОЕ]

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d (uv) = vdu + udv ,

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<<< < Предыдущая 1 2 34 / 104 5 6 7 8 9 10 > Следующая >>>

’" y′ . % " + , * " - + x ' y .

#. 6 ( y′			, " x3 − 4x2 y + 3xy2	− y3	= 3 .
		x
& * ! , :
3x2 − 8xy − 4x2 y′ + 3 y2 + 3x 2 yy′ − 3 y2 y′ = 0 .
!:
(6xy − 4x2 − 3 y2 ) y′ = 8xy − 3x2 − 3 y2 .
6:
y′ =	8xy − 3x2	− 3 y2	.
y′ =	6xy − 4x2	− 3 y2	.
	6xy − 4x2	− 3 y2

*3 * " " ' " - " * " «* " ( ! * -

».

10. , & " . ' & .

% , " ( . , " -


( x , "			y = f ( x + x ) − f (x ) ( +
! " ( . .4):
y = A( x) x + α( x, x) x ,
α →0	x →0 . , A( x) = f ′( x ) .
% 3 ( A( x)		x " y , (			x , !
( (	x , ( A( x) = f ′( x ) . ( -
α( x, x)	x , ' ", +			x x →0 .
& , (		x . 8	x , ( !
3, + 3 (. # 3 (, ( (					x ,
' * . %				x

! ", , . 5 ( -

, * "	dy . # ,
":
dy = df = y′ x = f ′( x ) x .
% ( y = x , y′ = x′ = 1, dx =	x , !

+ ! , * " . 6 " *: dy = y′dx = f ′( x )dx .

# (! *3 3. 6 ,

y′ = f ′(x ) = dy . dx

$" :

1. 6 ( y = x arctg

x .

/ ,:

′

dy = y′dx = x′arctg

x + x (arctg

x ) dx =

arctg

x + x

dx =

1 + x

2(1 + x )arctg

x +

dx .

2 (1 + x )

2. 6 ( y = ln sin 2x

) * x

!) x = π 8,

x = 0,1 .

/ ,:

) dy = y′dx = (ln sin 2x )′ dx =

2 cos 2x dx

= 2 ctg 2x dx = 2 ctg 2x

x .

sin 2x

!) dy

2 ctg

0,1

= 2 0,1 = 0, 2 .

x=0,1

& ' ! ' ( . 7) + " ! ' -

dc = 0

d (cos x ) = −sin x dx

d (xn ) = nxn−1dx

d (tg x ) =

cos2 x

d (a x ) = a x ln a dx

d (ctg x) = −

sin 2 x

d (ex ) = ex dx

d (arcsin x)

1 − x2

d (loga x ) =

d (arccos x ) = −

x ln a

1 − x2

d (ln x) =

d (arctg x ) =

1 + x2

d (sin x ) = cos x dx

d (arcctg x )

= −

1 + x2

& " + :

d (u ± v ) = du ± dv ,

5 ! * ' * ( -

. , ! +, * ,

' " . , "

' y = f (x ) = f (ϕ(t )) ( :

dy = yt′ dt = f ′(x )ϕ′(t )dt = f ′(x )dx .

# ! 1- " ! , ' +

, , x + ' ', , x ,' 3 . 5"

* , * " ( . 6 " ! + ! ". $" ( y = f (x ) x :

y = f ( x + x ) − f ( x) = f ′( x ) x + α ( x, x ) x ,

α → 0 x → 0 . !: y = dy + α ( x, x) x .

8 + ", α x x 3 (, + -

dy . # + , , - ! + *:

y = f ( x + x) − f ( x) ≈ dy = f ′(x ) x .
% 3: x0 = x, x1 = x + x , ' * + :
f (x1 ) ≈ f (x0 ) + f ′( x0 )(x1 − x0 ).	(10.1)

5" , ' " ! + ! *. ' * " ' :

( ! ! + ( " y = f (x ) x = x1 , !-f (x1 ). 9 ' * 3 x = x0 , " + ", * "

x = x1 (! x1 − x0 ), " ( " -

y = f (x ), + " , . 0 "

f (x1 ) ! + " * ' (10.1).

	1
#. $" ' y = x . # y′ =			. %



2		x

x0 = 1, x1 = 1 + h , h – " . # (10.1) - ! , ":

1 + h ≈ 1 + h . 2

5' + " ! + " ! " -

. ' * ', ( ! + 4, 02 . / ,:

4, 02 = 4 + 0, 02 = 4 (1 + 0, 005) = 4 1 + 0, 005 = 21 + 0, 005 =

≈2 1 + 0,005 = 2(1 + 0, 0025) = 2 1, 0025 = 2, 005 .

. ! + ' 5- " ": 4, 02 = 2,00499 .

2. $" ' y = arctg x; y′ = 1 + x2 . 7 (10.1) ,:

arctg ( x ) ≈ arctg ( x		) +	x1 − x0	.

1	0		1 + x2
		0

)!, , arctg 0,97 . & *: x0 = 1, x1 = 0,97 , :

0, 03 π

arctg 0,97 ≈ arctg1 − 1 + 12 = 4 − 0, 015 ≈ 0,77 .

. ! + ' 5- " ": 0,77017. 7 ' (10.1) ' * ", " , , ! +

* ! *. & + ' * " 3

, " ! ' * *. & " "- ' * " * .

( ’" '

. $" ( x0 y = f (x )

( . 19).

$ . 19.

% x0 . $" " ,

, ":

y= kx + b ,

k = f ′(x0 ) (, ( ). ) * " "

* M (x0 , f ( x0 )), " " ! , ":

y − f ( x0 ) = f ′(x0 )(x − x0 ).

(10.2)

5 ( , " " .

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

y = f ( x0 + x) − f ( x0 ) . 2 ( " (10.2), " , " " -

, + , :

f ( x0 ) + f ′( x0 )( x0 + x − x0 ) − f ( x0 ) − f ′( x0 )( x0 − x0 ) = f ′( x0 ) x ,

3, " y = f ( x ) x0 . . .19

+ KT " TMK . * + -

:	MK	=

			x, tgTMK = f ′( x0 ).

# , y = f (x )

x0 , ,

y = f (x ) x0 .

6’" , ( . .( *

M , * " + ". % x (t ) – -

M t ( .3, .1). #

x′(t ) – , 3 * t . ! x′(t )	t ,
! dx (t ) , , 3 ", " ( (3 ! M +	t ,

" ! " " ( ' 3 ' v (t ) = x′(t ) .

5 ( , ( . 7 + 3 " x , " ( (

' + t , ", * " dx (

t → 0 ) ", +	t . , "	t * , 3 *
v (t ) , , ", + [t,t + t ]			, (+
.
#.
1. " " y = x3 x			= 2 .
		0
' * " " (10.2), ,:
f (x ) = x3 , f ′( x ) = 3x2 , f (x	) = 23 = 8, f ′(x	) = 3 22 = 12 ,
0	0

3 " " , ": y − 8 = 12( x − 8).

y = 12x − 88 .

2. M t > 0 + , * " ':

x (t ) = 2 ln t . 6 ( ! + 3 ", " ( ( ' -t0 = 2 t1 = 2,5 .

/ ,: x (t ) = 2 ln t,			x′(t ) =			2	,
/ ,: x (t ) = 2 ln t,			x′(t ) =				,
						t
x ≈ dx (t ) = x′(t )	t =	2	t	=	2(t1 − t0 )			=	2(2,5	− 2)	= 0,5 .
x ≈ dx (t ) = x′(t )	t =		t	=		t0		=	2		= 0,5 .
			t			t0			2

+ ( 3 " ',:

x = 2 ln (t	) − 2 ln t		= 2 ln	t1	= 2 ln	2,5	= 2 ln1, 25 ≈ 0, 446 .
		0
1				t0	2

11. # .

. ( " y = f (x ) ( (a,b). > + , ,' x . 8 " " + ( (a,b),

+ + " , ! ( ( f ′(x ))′ . 5" , * "

y = f (x ) , * " f ′′( x).

8 f ′′( x) ' ( (a,b), + -

+ " ( f ′′(x ))′ , " , * " f ′′′( x) , * "

y = f (x ). :

f(IV ) ( x) = ( f ′′′( x ))′ – 4- ",

f(V ) ( x) = ( f (IV ) ( x))′ – 5- ",

…

f (n) ( x ) = ( f (n−1) ( x ))′ – n – ".

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

1. 6 ( f (n) ( x ) , " f (x ) = akx (k = const, a > 0, a ≠ 1). / ,:

f′(x ) = akx ln a k = kakx ln a = (k ln a )akx ,

f′′( x ) = kakx ln a k (ln a ) = (k ln a )2 akx ,

f′′′( x) = (k ln a )2 akx ln a k = (k ln a )3 akx .

0 + ' , ,: f (n) ( x ) = (k ln a )n akx .

6: (ekx )(n) = k nekx .

2. 6 ( f (n) ( x ) , " f (x ) = sin x . / ,:

f′(x ) = cos x = sin x + π ,

f ′′( x ) = cos x +

= sin x +

= sin x + 2

f ′′′(x ) = cos x + 2

= sin x + 2

= sin x + 3

f(n) ( x ) = sin x + n π .

3. 6 ( f ′′′( x), " f (x ) = x arcsin x .

/ ,:

										x													−	1
										x													−
	f ′(x ) = arcsin x +											= arcsin x + x (1 − x2 )												2 ,


									1 − x2
			1								1				x						3	(2x ) = 2(1 − x2 )−								1	+ x2 (1 − x2 )−	3
	f ′′( x ) = (1 − x2 )−						+ (1 − x2 )−							−		(1 − x2 )−
						2							2								2									2		2 ,
						2							2								2									2		2 ,
											2
								3									3
	f ′′′(x ) = − (1 − x2 )−								(−2x ) + 2x (1 − x2 )−										−
	f ′′′(x ) = − (1 − x2 )−							2	(−2x ) + 2x (1 − x2 )−									2	−
	3x	2	5									4x					6x			3					2x		3	+ 4x
−			(1 − x2 )−		(−2x ) =											+							=						.
				2
				2

2											(1 − x2 )3						(1 − x2 )5									(1 − x2 )5

% " + ( ( (, -

( . .(:

x= ϕ(t ), y = ψ (t ) (α ≤ t ≤ β) ,

ϕ, ψ ( (α,β) ,

ϕ′(t ) ≠ 0 t (α,β) . # ( . . 9):

y′ = ψ′(t ) . x ϕ′(t )

8 ϕ, ψ ( (α,β) ,

ψ′(t ) ′ ψ′′(t )ϕ′(t ) − ψ′(t )ϕ′′(t )

) ′

( ′(

))2

′

ψ′

(

ϕ′(t )

′′

(

′

)

yxx

ϕ′(t )

ψ′′(t )ϕ′(t ) − ψ′(t )ϕ′′(t )

(ϕ′(t ))3

+ ".

#. 6 ( y′′ , " x = a (cos t + t sin t ),

y = a (sin t − t cos t )

(0 ≤ t ≤ 2π).

/ ,:

ϕ′(t ) = a (−sin t + sin t + t cos t ) = at cos t ,

ϕ′′(t ) = a (cos t − t sin t );

ψ′(t ) = a (cos t + t sin t − cos t ) = at sin t , ψ′′(t ) = a (sin t + t cos t ) .

) +:

y′′ =

a (sin t + t cos t ) at cos t − at sin t a (cos t − t sin t )

a3t 3 cos3 t

at cos3 t

( " * ().

0 3 " , ".

#. 6 ( y′′ , " 2x2 + xy − y3 = 1 .

/ ,: 4x + y + xy′ − 3 y2 y′ = 0 , :

y′ =

4x + y

3 y2 − x

y′′ =

(4 + y′)(3 y2 − x ) − (4x + y )(

6 yy′ −1)

(3 y2 − x)2

12 y2

+ 3 y2 y′ − 4x − xy′ − 24xyy′ − 6 y2 y′ + 4x + y

(3 y2 − x)2

12 y2 + y − (24xy + 3 y2 + x ) y′

(3 y2 − x )2

12 y2 + y − (24xy + 3 y2 + x )

4x + y

3 y2 − x

(3 y2 − x)2

36 y4

− 48xy2 − 96x2 y − 4x2 − 2xy

(3 y2 − x)3

% " , ( ( . 8 x (t ) – -

* t , x′(t ), " -

3, ', 3 ( : x′(t ) = v (t ) .

x′′(t ) = v′(t ) ' 3 * 3 - ', , ' w(t ) t : x′′(t ) = w(t ).

#. 6 ( " t = 3 , "

! , * " : x (t ) = t 2e−t .

/ ,: v (t ) = x′(t ) = 2te−t − t 2e−t ;

w(t ) = x′′(t ) = 2e−t − 2te−t − 2tet + t 2e−t = (t 2 − 4t + 2)e−t .

6: w(3) = −e−3 .

!. & % u v x n - ,

uv x n - , :

(uv)( n) = ∑Cnk u(k )v( n−k ) .

(11.1)

k =0

7 (11.1) , * " ".

2 (!. 6 , .

% n = 1 (11.1) + , * ", *

(uv)′ = u′v + uv′ .

%, (11.1) , ' " n = m . # ! , * ":

(uv)( m) = ∑Cmk u( k )v( m−k ) .

k =0

* " n = m + 1. $":

(uv)

( m+1)

= (

(uv)

( m)

′

( k )

( m−k ) ′

( k )

( m−k )

′

)

∑Cmu

∑Cm

) =

k =0

= ∑Cmk

(u( k +1)v( m−k ) + u( k )v( m−k +1) ) = ∑Cmk u( k +1)v( m−k ) + ∑Cmk u(k )v( m−k +1) =

k =0

m−1

= u( m+1)v + ∑Cmk u( k +1)v(m−k ) +

∑Cmk u(k )v( m−k +1) + uv( m+1) =

k =0

k =1

= u( m+1)v + ∑Cmk −1u( k )v( m+1−k ) + ∑Cmk u( k )v( m−k +1) + uv( m+1) =

k =1

= u( m+1)v + ∑(Cmk −1 + Cmk )u( k )v( m+1−k ) + uv( m+1) = u( m+1)v + ∑Cmk +1u( k )v( m+1−k ) + uv( m+1) =

k =1

m+1

= ∑Cmk +1u( k )v( m+1−k ) , k =0

! (11.1) + , * " ( " n = m + 1. # " -

' " ! * ,:

C k	= C k −1	+ C k .
m+1	m	m

# 2 (! . > " ( "

! .*':

(a + b)n = ∑Cnk ak bn−k . k =0

+ + - ".

. (

y = f ( x ) , * " 1– " - , .

% , * " 2– " d 2 y . # !: d 2 y = d (dy ) .

) * dx + * x , dx , " , +

, ,:

d 2 y = d ( f ′( x )dx ) = ( f ′(x )dx )′ dx = ( f ′( x ))′ dx2 = f ′′( x)dx2 .

f ′′( x ) = d 2 y . dx2

' * " ":

d 3 y = d (d d 4 y = d (d

d n y = d (d

2 y ) = f ′′′( x)dx3 ,

3 y ) = f (IV ) ( x )dx4 ,

n−1 y ) = f (n) (x )dxn .

6 * ,:

f (n) ( x) = d n y . dxn

6 +, , 1– ",

2– ! *3 " + ' * * , !

* d n y = f (n) (x )dxn n > 1 + , * " 3 , x " "- , * " + ' '. 8 + x ' " ", * " ,' - 3 , " * , ". (, ( x = x (t ), dx = = x′(t )dt . ) +

d 2 y = d (dy ) = d ( f ′( x )dx ) = d ( f ′( x))dx + f ′(x )d (dx ) =

= f ′′(x )dx2 + f ′( x)d 2 x = f ′′( x)dx2 + f ′′(x ) x′′(t )dt 2 ,

! ! , * ". #. 6 ( d 3 y , " y = x2 sin 2x .

/ ,: f ′(x ) = 2x sin 2x + 2x2 cos 2x ,

f′′( x ) = 2sin 2x + 4x cos 2x + 4x cos 2x − 4x2 sin 2x = (2 − 4x2 )sin 2x + 8x cos 2x ,

f′′′(x ) = −8x sin 2x + 2(2 − 4x2 )cos 2x + 8cos 2x −16x sin 2x =

= −24sin 2x + (2 − 8x2 )cos 2x .

# :

d 3 y = ((12 − 8x2 )cos 2x − 24x sin 2x )dx3 .

<<< < Предыдущая 1 2 34 / 104 5 6 7 8 9 10 > Следующая >>>

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