матан Бесов - весь 2012

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Московский государственный физико-технический университет (МФТИ)

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[НЕСОРТИРОВАННОЕ]

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(a − 1) 2 (a − 1) < 2r(a − 1),

ar − 1 a n − 1 1

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f [a, b]

m = min f M = max f f

[a,b] [a,b]

[m, M ]

f : X → Yf

X Yf

x1, x2 X x1 = x2 f (x1) = f (x2) !

" f #

X ↔ Yf y Yf

x X " !

f (x) = y $

f −1 : Yf → X.

% " f −1 & " $ ' f ( ) ! $"

y = f (x) x = f −1(y), f −1(f (x)) = x x X, f (f −1(y)) = y y Yf .

f : X → Yf

X f −1: Yf → X

! f f −1 & $" y = x

	f : [a, b] → R
[a, b]
[A, B] =
= min f, max f
[a,b] [a,b]

	§
* +					Yf
f A f (x) B x [a, b] Yf
[A, B] , ! & $ -' C [A, B]
c [a, b]: f (c) = C [A, B] Yf			.
Yf = [A, B]
. ! f −1 &
/	$& f −1			y0
(A, B) x0 = f −1(y0) (a, b) ε > 0
		[x0 − ε, x0 + ε] [a, b].
y1 f (x0 − ε) y2 f (x0 + ε)
y
B		y = f (x)
		y = f (x)
y2
y0	2δ
y1
A
0	a	x0 − ε x0 x0 + ε	b	x
		0 1

% " f #

[x0 − ε, x0 + ε] [y1, y2] [A, B]

) y1 < y0 < y2 ( δ > 0 & (y0 −

− δ, y0 + δ) (y1, y2) !

f −1(Uδ (y0)) f −1((y1, y2)) = Uε(x0).

. f −1 $& y0

$ y0 = A y0 = B ! #

"" $& f −1 y0 & " #

! $ 2

f : (a, b) → R

(a, b)

(A, B) = inf f, sup f
(a,b)	(a,b)
		Yf
f ! " #
A < f (x) < B x (a, b).		$%&

' # # # # f (x0) B

x0 (a, b)# (

f # f (x) > B x (x0, b)#

# B = sup f

(a,b)

! " #
y0 (A, B) x0 (a, b) : f (x0) = y0.	$)&

* + " #

x1, x2 (a, b) : f (x1) < y0, f (x2) > y0.

! " f[x1,x2] f

[x1, x2] ,- " (

# #

x0 [x1, x2] : f (x0) = y0.

. # $)&

* $% $)& # f (a, b) = (A, B)

/ # f −1 (

" y0 (A, B) 0 " #

% .

)

# " (

! n N 1 f (x) = xn2 [0, +∞) → [0, +∞)#

[0, +∞) !

f −1#

√

n x = x

[0, +∞)

# m, n N

a > 0

n #

1 m

, a−

. ar a (0, +∞)# r Q

' 3 r, ρ (

4 a > 0

5 (

ar r

%◦ r1 < r2 ar1

< ar2 a > 1# ar1

> ar2 0 < a <

< 16

)◦ ar1 ar2 = ar1+r2 6

7◦ (ar1 )r2 = ar1r2 6

8◦ a0 = 16

9◦ (ab)r = arbr

* 3 + #

a−rar = a0 = 1 a−r =

> 0 r Q. $%&

a > 1 r Q |r| 1

|ar − 1| 2|r|(a − 1).

$)&

! r =

# n N

! " λ a

− 1 > 0 . a

= 1 + λ# a 1 + nλ#

a − 1

n #

− 1 1 (a − 1).

a n

$7&

0 < r 1	1	< r	1
	n + 1		n

n N ar

−1 r < 0

|ar − 1| = ar|a−r − 1| ar2(−r)(a − 1)

! " #$ ar < 1$

% &

a > 0$ x R$ lim rn = x n→∞

ax lim arn .

n→∞

' " (

) lim arn *

n→∞

lim arn & "+ {rn} n→∞

rn → x*

x = r & ar

! ) a > 1$ rn → x n → ∞

|rn − rm| 1 n, m n1 -

{rn} . # n, m n1(

|arn − arm | = arm |arn−rm − 1| arm 2|rn − rm|(a − 1). /

0$ {rn}

# # -# # #$ M > 0 arm M m N.

1 - {rn} # 2 "

3 4 (

ε > 0 nε N : |rn − rm| < ε n, m nε.

5 & / 0 < ε 1

|arn − arm | M 2(a − 1)ε n, m nε.

' &$ # {arn } "

3 4	1 # 3 4 -#$
lim arn 6& )		1◦				$
n→∞		1
lim arn > 0
n→∞
0 < a < 1 arn =
0 < a < 1 arn =		1			rn $
		a
	lim arn & ,
	n→∞					rn
# , lim				1
# , lim			a
	n→∞		a
7 a = 1

! a > 1$ rn → x$ rn → x n → ∞

rn − rn → 0 n → ∞$

|arn −arn | = arn |arn−rn −1| M 2|rn −rn|(a−1) → 0 (n → ∞).

5 $

lim arn − lim arn = lim (arn − arn ) = 0,

n→∞ n→∞ n→∞

+ &

7 0 < a < 1 #

arn = 11 rn a

! 8 #

{rn}$ rn = r n N

a > 0 # x → ax$ x

(−∞, +∞) &" # a

)◦ ax > 0 x (−∞, +∞)

◦ a > 1 ax 0 < a < 1

◦ axay = ax+y

◦ (bc)x = bxcx

◦ (ax)y = axy

◦ ax (−∞, +∞)

1◦ 2◦

2◦ a > 1 x < y r, ρ

! x < r < ρ < y

rn → x ρn → y n → ∞ ! rn r ρn ρ

n N " # $ ! %

& ! ! % '

( !

ax = lim arn ar < aρ lim aρn = ay ,

n→∞ n→∞

ax < ay

% 0 < a < 1 !$ #

3◦ rn → x ρn → y n → ∞ " #

axay = lim arn lim aρn = lim (arn aρn ) =		lim arn+ρn = ax+y .
n→∞ n→∞	n→∞	n→∞

) $ ! * axa−x =

	0	−x	1
= a = 1 a			=	ax
	4◦ !$
	) $ !
			ax > bx		a > b,	x > 0,
$ # 4◦ $ c > 1 bc = a
ρ	5◦		a > 1 x > 0		y > 0	rn ↑ x rn ↓ x ρn ↑ y
ρ	↓ y
n

" #

axy ← arnρn = (arn )ρn (ax)ρn (ax)y (ax)ρn (arn )ρn = = arnρn → axy ,

(ax)y = axy

# ( ( x y !$ #

% 0 < a < 1 $ * a > 1 ! + *

,$ ax = 11 x a

6◦ -! ! . '

*+ / /+ 0

|ax − 1| 2|x|(a − 1) a > 1, |x| 1.

1# ! 2 3 $ rn ! r # rn → x n → ∞ %$ 4 ! '

5 ! & ax %

x0 (−∞, +∞) a > 1 " #

|ax0+Δx − ax0 | = ax0 |a x − 1| ax0 2| x|(a − 1) → 0

x → 0 /

% 0 < a < 1 $ * a > 1 ! + *

,$ ax = 11 x a

6$ /$ & y = ax

a > 0 a = 1 $

/$ y = loga x ) a = e /$ ln x loge x

loga x : (0, +∞) → (−∞, +∞)

(0, +∞)

(−∞, +∞)


					a > 1 " # A =
=		inf		ax = 0 B = sup	ax = +∞
	(−∞,+∞)			(−∞,+∞)
		) ! ! an = (1 + α)n > 1 + nα → +∞ a−n <
<	1		→ 0 n → ∞
<		1 + nα	→ 0 n → ∞

7 $ $ ! % / / % & '

% 0 < a < 1 !$ #

a = 1

a	loga x	= x, loga a	x	= x.

1◦ loga xy = loga x + loga y x, y > 0

! aloga xy = xy aloga x+loga y = aloga xaloga y = xy

" 1◦ #$ %

2◦ loga xα = α loga x x > 0 α R

! aloga xα = xα aα·logα x = (aloga x)α = xα "

2◦

3◦ loga b · logb a = 1 a > 0 b > 0 a = 1 b = 1

! aloga b·logb a = (aloga b)logb a = blogb a = a a1 = a

" 3◦

& α R ' x → xα : (0, +∞) → (0, +∞)

α !

xα = (eln x)α = eα ln x.

& "

(0, +∞)

& α > 0 (

( ) [0, +∞)

* "

+ , "

| sin x| |x| x R.

#-%

§
. & 0 < x < π2			/
0

& AOB	y
	1
x ) AB x	1
sin x = \|BC\| 3 [BC]		B

sin x = \|BC\| < \|BA\| < x.	x
sin x = \|BC\| < \|BA\| < x.	O	C A	x
	O	C A	x
4 #-% 0 <	/	1 2
< x < π
2 5 6 "
#-% − π < x < 0	* 6
2
#-% x = 0 \|x\| 1
sin x cos x tg x ctg x

. & y = = sin x x0 R

	x) − sin x0					x		x
y = sin(x0 +			= 2 cos		x0 +	x	sin	x	.
y = sin(x0 +			= 2 cos		x0 +	2	sin	2	.
						2		2
5 #-%
			x
			x
\|	y\| 2	sin	2	\|	x\|,

y → 0 x → 0

	−							,
arcsin x : [−1, 1] →		π		,		π
					2
	2
arccos x : [−1, 1] → [0, π] ,
arctg x : (−∞, ∞) → −			π		,		π		,

	2				2

arcctg x

: (−∞, ∞) → (0, π)

sin x

−

cos x [0, π]

tg x

−

ctg x (0, π)

arcsin x arccos x arctg x arcctg x

arcsin x arccos x

" #$ % & ! %

sin x <

x <

tg x,

sin x

cos x <

< 1,

0 < x <

' (

sin x

cos x ! "

% 0 < |x| < π2 ) % % x → 0 ! cos x → cos 0 = 1 %

cos x! %

2◦ lim

tg x

= 1 ' % cos x

x→0

lim

tg x

= lim

sin x

lim

= 1 · 1 = 1.

x→0

x→0 cos x

3◦ lim

arcsin x

= 1

x→0

arcsin x

* !

sin yy y=arcsin x

y %

sin y

arcsin x

+ !

arcsin x

% %

' %

% arcsin x x = 0!

% % % ! ,

-(
%	y	y = 0
	sin y

% % % %

arctg x

4◦

lim

arctg x

)

x→0

! % " 3◦

tg y

y=arctg x

5◦

lim (1 + x)

= e ) " !

x→0

lim

(1 + x)

= e.

x→0+0

n+1

/ % !

lim

1 +

= lim 1 +

= e

n→∞

% 0 %

1 +

n+1

) 0 < x < 1! nx N!

< x

nx + 1

−2

nx+2

= 1 +

1 +

nx + 1

nx+1

(1 + x) x 1 +

. 1

nx+1

n+1

lim

1 +

= lim

1 +

= e.

x→0+0

n→∞

lim

f (x)

x→0+0

lim

f (xn)

n→∞

{xn} xn → 0 + 0 n

→ ∞

! xn =

" #$% e

x → 0 + 0

& ' #$% (

#$% #)%

(

lim

(1 + x)

= e.

#*%

x→0−0

−1 < x < 0

( y −x

− y

−x

> 0

1 + x

= (1−y)

−

(1+x) x

y =

1 − y

= 1 +

1 − y

= (1+z) z

(1 + x)

= (1 + z)

+1 z=

−x

, −1 < x < 0,

1+x

(1 + x)

(f ◦ ϕ)(x)

f : (0, +∞) → R,

ϕ : (−1, 0) → (0, +∞),

lim

ϕ(x) = 0

lim

f (z) = e

x→0−0

z→0+0

lim

(1 + x)

= e.

#-%

x→0−0

" #*% #-% 5◦

./ + 0

f ◦ ϕ ' f, ϕ

' ' '

( '

. ) ' 0

(

' ' '

◦ ϕ˜

z 0,

f (z) =

(1 + z)

+1 z > 0,

−x

− 1 < x < 0,

ϕ˜(x) =

1 + x

x 0.

6◦ lim

loga(1 + x)

= loga e #a > 0 a = 1%

x→0

loga(1+x) = loga(1+x)

ϕ(x) = (1 + x) x

, 0

5◦

7◦	lim	ln(1 + x)			= 1 # ' 6◦%

	x→0		x
8◦	lim	ax − 1		= ln a #a > 0 a = 1%

	x→0	x
y = ax − 1						ax−1		y ln a
			ln(1 + y)
+ x =						x	=		y=ax−1
				ln a				ln(1 + y)
, 0
7◦
9◦	lim	ex − 1		= 1 # ' 8◦%
		x
	x→0

" ' x → 0 x sin x tg x arcsin x arctg x ln(1 + x) ex − 1.

10◦ lim	(1 + x)α − 1	= α # 0
10◦ lim	x	= α # 0
x→0	x
9◦%

f U (x0)

	x0 R	f (x) − f (x0)
	lim
		x − x0
	x→x0

f x0

f (x0)

(sin x) = cos x

(cos x)

= − sin x (xn) =

= nxn−1 "n N# (ax) = ax ln a

f (x0) g (x0)

$◦ (f ± g) (x0) = f (x0) ± g (x0)

%◦

(f g) (x0)

= f (x0)g(x0) + f (x0)g (x0)

(cf ) (x0) = cf (x0) c

&◦

g(x0) = 0

(x0) =

f (x0)g(x0) − f (x0)g (x0)

g(x0)2

' (

)

x = x − x0

f = f (x0 +

x) − f (x0)

g = g(x0 +

x) − g(x0) + )

f (x)

f (x0)

−

(f (x0) + f )g(x0) − f (x0)(g(x0) + g)

g(x)

g(x0)

x g(x0 + x)g(x0)

			§
	f			g
		g(x0) − f (x0)			→	f (x0)g(x0) − f (x0)g (x0)
=	x	g(x0) − f (x0)		x		f (x0)g(x0) − f (x0)g (x0)
=	g(x0		+ x)g(x0)				g(x0)2
	g(x0		+ x)g(x0)				g(x0)2

x → x0

f U (x0) x0 R ' x0 *

f (x0) = f (x0 + x) − f (x0) = A x + o(Δx) "$#

x → 0 ) A R

+ ) f
x0	−∞ < x < ∞,
df (x0) = A x,	−∞ < x < ∞,	"%#

, f x0

f x0

f (x0)

A = f (x0)

1◦ f

x0 + ) "$#

) x

f (x0) = A + o(1). x

- ! x → 0

f (x0) = A

2◦

		f (x0) = lim		f (x0)	.

			x→0	x
	(x0) −	f (x0)	= o(1)	x → 0
+ ) f
		x				x
. *
	f (x0) = f (x0)Δx + o(Δx) x → 0.
						"

A = f (x0) f

	f
x0 f x0
!
f (x0)		"
f (x0) → 0	x → 0 #
f x0
f (x) = \|x\| x0 = 0		!

$ % & $ !

# &

x0 &

f (x0) ' # & ( ) !

& x0

(

* + ( , f

x0
df (x0) = f (x0)Δx, −∞ < x < ∞.
+ &	x !
% dx =	x -. !
df (x0)

df (x0) = f (x0) dx, −∞ < dx < +∞.

# dx

df (x0) ' / !

dxdf % f

. ( ) !

f g

x0 f ± g f g g(x0) = 0

g x0

◦ d(f ± g) = df ± dg

,◦ d(f g) = g df + f dg
f			g df − f dg
*◦ d	g	=
				g2

)

) 0 2◦ 1 !

& (f g) = f g + f g (

d(f g) = (f g) dx = g f dx + f g dx = g df + f dg.

$ M0Mh M0 = (x0, f (x0))Mh = (x0 + h, f (x0 + h)) . y = f (x) . h = 0 2 *

y = f (x)

x0 + h

3 2 *

0 & M0Mh

y = k(h)(x − x0) + y0,

. y0 = f (x0),

k(h) =

f (x0 + h) − f (x0)

0 h

-. Mh %

M0 ' & . & !

k(h)

f (x0) k(h) → k0 = = f (x0)

(x0, f (x0))

! " # $

f (x0)

f (x0, f (x0))

y = y0 + f (x0)(x − x0), y0 = f (x0).

f U (x0)

f (x0)

(x0, f (x0)) y = λ(x − x0) + y0 y0 = f (x0)

f (x) − y = o(x − x0) x → x0.

% f $

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

f (x) − y = [f (x0) − λ](x − x0) + o(x − x0).

o(x − x0) x → x0

λ = f (x0)	y =
= λ(x − x0) + y0
%

" !' " # $

f (x0)

tg α α ( " '

' $

) ' '

' %

df (x0) = f (x)Δx			x $

f
x0	f	→ +∞ *−∞ ∞+	x → 0 ,
	x

f x0 f (x0) = +∞

*−∞, ∞+ f (x0, f (x0))

x = x0

- $

f (x0)


	* +
f x0
f+(x0) lim		f (x0 +	x) − f (x0)	,

	x→0+0		x
f−(x0)	lim	f (x0 +	x) − f (x0)	,
			x
	x→0−0

. ! #
f+(x0) f−(x0) (

f (x0)

" #

f−(x0) f+(x0) f−(x0) = f+(x0)

% " "

f+(x0) f x0

% " " .

/ 0 1 $

- $

f (x) = | sin x|

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