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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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матан Бесов - весь 2012

.pdf

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<<< < Предыдущая 1 2 3 45 / 295 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 > Следующая >>>

y = f (x)

U (x0) f (x0) = 0

x = f −1(y) y0 =

= f (x0) (f	−1		1
= f (x0) (f		) (y0) =	f (x0)

f −1

U (y0) y0

f x0

x = x − x0 y = f (x0 + x) − f (x0)

y = (f (x0) + ε(Δx))Δx,

ε(Δx) → 0 x → 0

f "

x y #

y x = ϕ(Δy) $ ϕ(0) = = 0 ϕ

U (0) % &

y = (f (x0) + ε(ϕ(Δy)))Δx.

ε(ϕ(Δy)) → 0 y → → 0 &

x	=		1	→	1	y → 0,
y			(x0) + ε(ϕ(Δy))
		f			f (x0)

" "

lim

(Δx)

lim

y→0

x=ϕ(Δy)

y→0

(ϕ(Δy))

= lim

y→0

(Δx) x=ϕ(Δy)

x→0

(Δx)

(x0)

§
(	y	(Δx) !	y

	x		x

f (y0), ϕ (x0) y0 = ϕ(x0)

(f (ϕ)) (x0) = f (y0)ϕ (x0)

) f (y0) ϕ (x0)

f ϕ * y0 x0

z = F (x) = f (ϕ(x))


U (x0) x0 )
		z y
f ϕ
z = f (y0)Δy + ε(Δy)Δy,	ε(Δy) → 0		y → 0,
y = ϕ (x0)Δx + ε1(Δx)Δx,	ε1(Δx) → 0		x → 0.

+ ε % " ε(0) = 0

$* "		y = 0
	, $* y
x z		x
	y

z = F (x0) = f (y0)[ϕ (x0)Δx + ε1(Δx)Δx] + ε(Δy)Δy = = f (y0)ϕ (x0)Δx + f (y0)ε1(Δx)Δx + ε(Δy)Δy.

$ x


	z	= f (y0)ϕ (x0) + f (y0)ε1(Δx) + ε(Δy)			y
						.
	x				x	.
- y → 0			y	→ ϕ (x0)		x → 0
- y → 0			x	→ ϕ (x0)		x → 0
*						x → 0

t0 (α, β)

f (x) f (a, b) → R x	(α, β)	→
→ (a, b) f (x0) x (t0) x0 =	x(t0)

df (x)(t0) = f (x(t0))x (t0) dt = f (x(t0)) dx(t0).

df (x) = f (x) dx, x : (α, β) → (a, b).

! dx " # $

df (x) x

% &

' ( y = xα (0, ∞) → R α R % !

y = eα ln x = eu u = α ln x

) &&

(xα) = (eα ln x) = eα ln xα x1 = αxα−1

* f ϕ $

+ &+ U (y0) U (x0) $

% ! , $

f ϕ

-! &+ y0 x0

. & & / $

(f (ϕ)) (x0) = f (y0)ϕ (x0) $ !

+ !

) . ) ! & f U (y0 −0) → → R f−(y0) 0$

f U (y0 + 0)

f (y) = f (y0) + f−(y0)(y − y0) y > y0.

/ & f U (y0) → R ! $

f (y0) = f−(y0)

1 ! U (x0) ϕ

-! x0 )

+ f ϕ $

( & -! ! *

) + !

y(x)

x = ϕ(t),

y= ψ(t).

U (t0) x0 = ϕ(t0)

ϕ (t0) ψ (t0) t = ϕ−1(x) y = ψ(t) = ψ(ϕ−1(x))

					1		ψ (t0)
	dy
		(x0) = ψ	(t0)			=		.
dx					(t0)
				ϕ			ϕ	(t0)

F (x, y) = 0 ! xU (x0) # y = f (x)

F (x, f (x)) = 0 x U (x0).

$ f

F (x, y) = 0

f U (x0)

! F (x, f (x)) ≡ 0 %

$ !

& '$

F ( ! ! %

# f ) f *

F (x, y) = x2 + y2 − 1 y = = f (x) + # x2 + y2 − 1 = 0

x (−1, 1) x2 + (f (x))2 − 1 ≡ 0 !

(−1, 1) f

(−1, 1) $ !

2x + 2f (x)f (x) = 0 f (x) = − f (xx)

f U (x0)

f (x) x U (x0) f (x)

x x0

(f ) (x0)

f x0 f (x0)

! " n f



	(n)		(n−1)
f	(n)	(x0) = (f	(n−1)		, n N.
f		(x0) = (f		(x))	, n N.

x=x0

# "

f (n)(x0) f (n−1) $

U (x0) x0

n f (n)(x0)

dnf (x0)

dxn

% f (0)(x) f (x)

	f (n)(x0) g(n)(x0)
x0
&◦ (f ± g)(n) = f (n) ± g(n)
'◦
	(f g)(n)		= f (n)g + Cn1f (n−1)g(1) + . . . + f g(n) =
	n		(n−k)		(k)			n!
		k	(n−k)		(k)	k		n!
	=	Cnf		g		Cn	=	k!(n − k)!
	k=0

(cf )(n)(x0) = cf (n)(x0), f (n)(x0).

( ) *

! n = 1

+ ! $

n * n + 1

Cnkf (n−k)g(k)

(f g)(n+1) = ((f g)(n)) =

k=0

= Cnk f (n−k+1)g(k) + f (n−k)g(k+1) =

k=0

(n+1−k)

n+1

(n+1−j)

(k)

j−1

(j)

= Cnf

+ Cn

k=0

j=1

= Cn0f (n+1)g +

(Cnk + Cnk−1)f (n+1−k)g(k) + Cnnf (0)g(n+1).

k=1

, Cnk + Cnk−1 = Cnk+1 #

k−1

+ Cn

k!(n − k)!

(k − 1)!(n − k + 1)!

n!(n − k + 1 + k)

(n + 1)!

k!(n − k + 1)!

k!(n + 1 − k)!

= Cn+1.

! * +-

f * f "

U (x0) x0

df (x) = f (x) dx, x U (x0),

x .

dx / f

x0 . f (x0)/ $

df (x) δ(df (x)) 0

df 1

$ δ δx

x0


d2f (x0) δ(df )(x0)	= δ(f (x) dx)(x0)				=
δx=dx				δx=dx
							2
					= f	(x0)(dx)	2	.
	= (f	(x) dx)	(x0)δx		= f	(x0)(dx)		.

δx=dx

f (x0)

			n f
x0
	n		n−1
d					.
		f (x0) δ(d		f )(x0)

δx=dx
! " " " " "
# f (n)(x0) #
dnf (x0) = f (n)(x0)(dx)n.	$

% " n 2 n = 1

& x ' " " % " n = 2 ( )"

f (x)

f " x0 ) " x

" t0

x = x(t) " " t x0 = x(t0) *" "

d2f (x) = (f (x))tt(dt)2 = (f (x)x )t(dt)2 =

= (f (x)(x )2 + f (x)x )(dt)2 = f (x)(dx)2 + f (x) d2x.

* d2f (x) = f (x)(dx)2 + f (x)d2x

+ $ n = 2 !

" ! "

* " ,

(f ± g)(n) (f g)(n) " #- $ "

dn(f ± g) dn(f g)

& " ! -#

. f

) f

% " ,

/ " " f f (n) n N

" n

U (x0) x0

U (x0)

f (x0) f (x0) = 0

f (x0) = min f f 0 x > 0 f 0

U (x0) x x

x < 0 x → → 0 f (x0) 0 f (x0) 0

f (x0) = 0

f
◦	[a, b]
!◦	(a, b)
"◦	f (a) = f (b)

ξ (a, b): f (ξ) = 0

#$ f ≡ const

% f ≡ const &$

' ( [a, b] ) * f

$ $

+ (a, b)

min f < max f , - f

[a,b] [a,b]

$ .

◦ [a, b]

!◦ (a, b)

ξ (a, b): f (b) − f (a) = f (ξ)(b − a)

§

) * F (x) = f (x) − λx $

λ (. .( F 3◦

( / 0 F (a) = F (b)

	f (b) − f (a)
f (a) − λa = f (b) − λb, λ =	f (b) − f (a)
f (a) − λa = f (b) − λb, λ =	b − a	.	1 2

F ( ( 1◦ 2◦ (

/ / ) * F


ξ (a, b) : F (ξ) = 0,			f (ξ) − λ = 0,	1!2
	f (b) − f (a)
λ =	b − a	+ (
3 +
-			ξ (a, b),
	f (b) − f (a) = f (ξ)(b − a),		ξ (a, b),	1"2

(

f (ξ) =	f (b) − f (a)	, ξ (a, b),
	b − a

$ ( +

( 3 + 0 $ ξ (a, b)

) ) * f (ξ, f (ξ))

4$ (a, f (a)) (b, f (b))

x0 ) * f 4 lim f (x) 4 x→x0

f (x0) = lim f (x) 5 +

x→x0

6 7 + 4 4

U (x0) f x06

) *

x2 sin 1 , x = 0, f (x) = x

0x = 0

(−∞, +∞) x0 = = 0

f, g

◦ [a, b]

◦ (a, b)

◦ g = 0 (a, b)

ξ (a, b)

f (b) − f (a)		f (ξ)
g(b) − g(a)	=	g (ξ)	.

g(b) == g(a) g

! (a, b) "

3◦

# F (x) = f (x) − λg(x) x [a, b] $ λ

F (a) = F (b) f (a) − λg(a) = f (b) − λg(b) %

λ= f (b) − f (a) . g(b) − g(a)

# λ & ' F "

( ξ(a, b): F (ξ) = 0 #

	f (ξ)
f (ξ) − λg (ξ) = 0,		= λ,
	g (ξ)

) * +

,- + g(x) = x

. ,- "

& / ! * + & " ' f & ' g &

$ 1 + & n N / "

/ n = 0

# f (n)(x0) ) + ! "

U (x0)

f (x) = n f (k)(x0) (x − x0)k + rn(f, x) = Pn(f, x) + rn(f, x), k!

k=0

2 3

& ' f

x0 # 1	f (k)(x0)	(x −x0)k k & "
	k!

) ! Pn(f, x) 4 rn(f, x) 4

& ) ! 2 n"+ 3

Pn(f, x)

rn(f, x)

Pn(x) rn(x)

(n)

(x0) f

U (x0)

x U (x0)

(rn(f, x)) = rn−1(f , x).

(rn(f, x)) = f (x) −

f (k)(x0)

(x − x0)k

k=0

k−1

f (k)(x0)

= f

(x) −

(k − 1)!

(x − x0)

= rn−1

, x).

k=1

n N

f (n)(x0) 2 3

rn(f, x) = o((x − x0)n) x → x0

# n = 1 $

f x0

f (x) − f (x0) = f (x0)(x − x0) + o(x − x0), x → x0,

n − − 1 1 n

! " #$ %

& %

x > x0

rn(f, x) = rn(f, x) − rn(f, x0) = rn−1(f , ξ)(x − x0),

$ x0 < ξ < x

' rn−1(f , ξ) = o((ξ − − x0)n−1) = o((x − x0)n−1) x → x0

rn(f, x) = o((x − x0)n) x → x0.

( "

x > x0 x < x0 n N0 f (n) [x0, x] [x, x0]

f (n+1) (x0, x) (x, x0)

)

rn(f, x) = f (n+1)(x0 + θ(x − x0)) (x − x0)n+1 = (n + 1)!

f (n+1)(ξ)

= (n + 1)! (x − x0)n+1,

0 < θ < 1

* " (

x > x0 n = 0

θ (0, 1)

f (x) = f (x0) + f (x0 + θ(x − x0))(x − x0).

+ "

% & #$

n −1 n N

! " ,-

rn(f, x)			rn(f, x) − rn(f, x0)
(x − x0)n+1	=	(x − x0)n+1 − (x0 − x0)n+1				=
			rn−1(f , ξ)		(f )(n)(η)			f (n+1)(η)
		=	(n + 1)(ξ − x0)n	=			=		,
					(n + 1)n!			(n + 1)!

$ x0 < η < ξ < x

U (x0)

f (x) = a0 + a1(x − x0) + . . . + an(x − x0)n + o((x − x0)n), f (x) = b0 + b1(x − x0) + . . . + bn(x − x0)n + o((x − x0)n)

x → x0 a0 = b0 a1 = b1 an = bn

* " .

f " $ $

" "

c0+c1(x−x0)+ . . .+cn(x−x0)n = o((x−x0)n) x → x0 /

c0 = c1 = . . . = cn = 0

% / x → x0

c0 = 0 0 /

x − x0

c1 +c2(x−x0)+ . . .+cn(x−x0)n−1 = o((x−x0)n−1) x → x0.

% x → x0

c1 = 0

0 ( $

x − x0 % c2 = 0

- % c0 = = c1 = . . . = cn = 0 ( "

f (n)(x0)
f (x) = a0 + a1(x − x0) + . . . + an(x − x0)n+
+o((x − x0)n) x → x0.	1
+o((x − x0)n) x → x0.

! f ! !

n = 3 ! "# $ % &

f (x0) f (x0)				f (x0)
					\|x\| < 1 1 + x + x2 + . . . + xn =
	1 − xn+1				1		2		n
=	1 − x				1 − x	= 1 + x + x		+ . . . + x		+ rn(x) '
			−xn+1		n
rn(x) =			1 − x	= o(x ) x → 0 (
					" f (x) =
	1
=	1 − x	! ! !

)" $ '

" '

! f U (x0 + 0) U (x0 − − 0) $ f (k)(x0) ( ! *

◦ f (x) = ex x0 = 0 + f (k)(x) = ex

ex = 1 + x +

+ . . . +

+ rn(x),

eθx

n+1

' rn(x) =

= O(x

) = o(x ) x →

(n + 1)!

→ 0 0 < θ < 1

,◦ f (x) = sin x

x0 = 0 +

{f (k)(0)}k∞=0 = {0, 1,

0, −1, 0, 1, 0, −1, . . .},

x2n+1

sin x = x −

− · · ·

+ (−1)n

+ o(x2n+2)

(2n + 1)!

x → 0

) " "

' ( 2n + 2*' "

◦ f (x) = cos x x0

= 0 -' " ( !

sin x "

x2n

cos x = 1 −

−

+ . . . + (−1)n

+ o(x2n+1)

(2n)!

x → 0

.◦ f (x) = ln(1 + x)

x0 = 0 +

f (x) =

= (1 + x)−1,

1 + x

f (k)(x) = (−1)k−1(k − 1)!(1 + x)−k,

ln(1 + x) = x −

− . . . + (−1)n−1

+ rn(x),

rn(x) =

(−1)n

xn+1

= o(xn) x → 0,

n+1

n + 1 (1 + θx)

' 0 < θ < 1

x0 = 0 α R +

/◦ f (x) = (1 + x)α

f (k)(x) = α(α − 1) . . .(α − k + 1)(1 + x)α−k,

(1 + x)α = 1 + αx + α(α − 1) x2 + α(α − 1)(α − 2) x3 + . . .

2! 3!

+ α(α − 1) . . .(α − n + 1) xn + o(xn) x → 0. n!

			n	n
	n				n!	k
	n		k k		n!	k
(1 + x)		=	Cnx =	k=0	k!(n − k)!	x	,
			k=0	k=0

! (1 + x)n *

" " 0 '

1
lim	ex − e−x − 2x	,	lim	1		−	1	,
	x − sin x				x2
x→0			x→0				sin2 x

! ! *

" "

x0 = 0

f(x)

g(x)

x → x0

	! " #
		0	∞
$ %			∞
		0

& ! " ' ( #

) *

% %

" ' + ,-)

" $

( $

. / #

0◦

f g (a, b)

◦

b − a < ∞

lim f (x) =

lim

g(x) = 0

x→a+0

◦ g = 0 (a, b)

◦

lim

(x)

x→a+0 g (x)

f (x)

lim

f (x)

lim

x→a+0 g(x)

x→a+0 g (x)

f g a

f (a) = g(a) = 0 f g

[a, b) ! "
	f (x)		f (x) − f (a)		f (ξ)
		=	g(x) − g(a)	=	g (ξ)	, a < ξ < x < b.
	g(x)

= A

lim

(x)

ε > 0

x→a+0 g

(x)

δ = δ(ε) > 0, δ < b − a :

f (x)

Uε(A)

f (ξ)

g(x) g (ξ)

a < x < a + δ < b

f (x)

lim

= A

x→a+0 g(x)

$◦

f g (c, +∞) c > 0

◦

lim f (x) =

lim

g(x) = 0

x→+∞

◦ g = 0 (c, +∞)

◦

lim

f (x)

x→+∞ g (x)

f (x)

lim

f (x)

lim

x→+∞ g(x)

x→+∞ g (x)

= A

lim

(x)

x→+∞ g (x)

t 0,

lim

−

= A.

t→0+0

t→0+0 g

−

lim

= A

t→0+0 g

& ' lim

f (x)

= A

(

x→+∞ g(x)

) *

+ , +

) + § -'

. * '/

, . & *

* . .

◦

f g (a, b)

−∞ < a < b < ∞

g(x) = ±∞

◦

lim f (x) = ±∞

lim

x→a+0

◦ g = 0 (a, b)

◦

lim

(x)

x→a+0 g

f (x)

lim

f (x)

lim

x→a+0 g(x)

x→a+0 g (x)

= A

lim

(x)

x→a+0 g (x)

xε (a, b)

f (x)

Uε(A) x (a, xε).

g (x)

a < x < xε

f (x) − f (xε)

f (ξ)

g(x) − g(xε)

g (ξ)

δ > 0 a < x < a +

+ δ < xε < b

f (xε)

g(xε)

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

< ε,

< ε.

f (x)

g(x)

" # $ %

f (x)

g(x)

! %

1 −

g(xε)

f (x)

g(x) f

(ξ)

, a < x < a + δ.

g x

f (xε)

(

)

( )

−

f (x)

& % & ! '

$ (

# %

(1 −

(ξ)

−3ε, 1 + 3ε) & %

(ξ) & '

% Uε(A) ) $ ! $ &

§ !

$ % Uη (A) # η = η(ε) = η(A, ε) > 0

xUδ (a + 0)

* % η(ε) $ #

ε > 0 ) '

lim f (x) = A x→a+0 g(x)

# $ $ +

% η(A, ε) ε , '

$ 0 < A < +∞ " # & % '

& ! % (A − ε, A + ε)

& ! (1 − 3ε, 1 + 3ε) )- %

(A − 3Aε − ε + 3ε2, A + 3Aε + ε + 3ε2) Uη (A), # η = η(A, ε) = 3Aε + ε + 3ε2

. " / 0$ '

$- - - x → a + 0 x → a − 0 x → a x → +∞ x → −∞ 01 $ -

lnεx

3 &

lim

ε > 0 4

x→+∞ x

ln x

lim

ε−1

lim

= 0.

x→+∞ x

x→+∞

εx

x→+∞ εx

. '

# & 1 $

1$ * #

3 &

lim

xxn

n N a > 1 4

x→+∞ a

n(n − 1)xn−2

lim

= lim

nxn−1

= lim

= . . .

ax ln2 a

x→+∞ ax

x→+∞ ax ln a

x→+∞

= lim

= 0.

x→+∞ ax ln

" / 0 $ $ '

+ & 5 $ $ $0$ -

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