матан Бесов - весь 2012

# # $

0

% 0

∞

∞ % % #

# &

lim xx

x→0+0

§

f

(a, b)

'◦ f 0 f 0 (a, b)

f

(a, b)

(◦ f > 0 f < 0 (a, b)

f

(a, b)

) &

! # %

f (x2) − f (x1) = f (ξ)(x2 − x1), a < x1 < ξ < x2 < b.

f (a, b) x0, x

, ! f > 0 (a, b) $

# &- f (a, b)

. f (x) = x3 x (−1, 1)

* ! x0

/ 0 &- f

U (x0) &- f

˚

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

1 . %

! x0

/ 0 &- f

* ! / 0 !

/ 0

/ 0

! " #

$%

!

x0 f

f (x0) f (x0) = 0

&§ '!(!

) f (x0) = 0

f (x) = x3 x0 = = 0!

f x0

x0 x0

f

& ! * f > 0

- f −

+ , x0! 0 x0

f !

) ,

+ x0

$ % f ,

+ x0 = 0

f ! ! " # f (x0) > 0 x0

f !

$

% &

& '

f (x0) = . . . = f (n−1)(x0) = 0 f (n)(x0) = 0

(◦ " n = 2k x0

# f (2k)(x0) > 0

f (2k)(x0) < 0$ f %

1◦ " n = 2k + 1 x0

# $ f f (2k+1)(x0) > 0 #

f (2k+1)(x0) < 0$

& ! + x

˚

U (x0)

f (x) − f (x0) = f (n)(x0) (x − x0)n + ε(x − x0)(x − x0)n = n!

f (n)(x0) sgn(f (x) − − f (x0)) = sgn f (n)(x0) sgn(x − x0)n

(x − x0)n x0

n

− ˚

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

f (x0) > 0 x0

f (x0) < 0 x0 f

f (x0) = 0 f (x0) > 0 x0 f (x0) < 0 x0

f

! " ! # $ % & f : [a, b] → R

' ( )

' (a, b) * ) !

$ % & f $ f (a) f (b)

§

* % &$ f (a, b) +$ !

[α, β] (a, b) ! % % & f

$' (α, f (α)) (β, f (β)) *

, &$ f $

" # (a, b) $ α, β: a < α < < β < b f (x) lα,β (x) " f (x) lα,β (x)#

x (α, β)

- !

! % &$ f $

" #

(a, b)

* ) (a, b) $

% & f

. % &

* $ $ $

! )%% & &

(x, f (x))

$ ! )%% &

& (x0, f (x0))

% &$ f

( *

) $

f−(x0) f+(x0) f−(x0 + 0).

% &$

(

!

* % & f (x) = 1 − |x| x (−1, 1)

$ $ '

0 % &

( 1 ! (

◦ f < 0 > 0 (a, b) f

(a, b)

a < α < x < β < b

! " #$

f (x) − lα,β (x) = (β − x)f (ξ)(x − α) + (x − α)f (η)(x − β) =

β − α

# a < α < ξ < ζ < η < β < b

% & $ 1◦

$ 2◦

' 1◦

($ ) $ !

! ! ' $

* x0

f (x0, f (x0)) +

, f

◦ f (x0) '

-

◦ x0 , # !

, # !

. 1◦ , f

x0

x0 f f

x0 f (x0) = 0

/ #

f (x0) = 0 / f (x0) > 0 * #

f > 0 ! U (x0) !

x0 U (x0) # !

$ ' ! # '

! f (x0) f "

x0

x0 f

# ' &

# ! ,

f (x0) = 0 f (x0) = 0

x0 f

! " " #

(a, +∞) y = kx + b

f x → +∞

f (x) = kx + b + o(1) x → +∞.

x →

→ −∞

f

x → +∞ y = kx + b

! x → −∞

" x = x0 !

f # $!

f (x0 + 0) f (x0 −0) +∞

−∞

%! !#

!# f (x) = x −

− 2 arctg x f (x) = ln(1 + x)

§

1◦.

! 2◦. !

3◦.

4◦. "! !

5◦. ! # !

!

6◦. $

7◦. !

# 8◦. ! ! % &

9◦.

Ox ! #

10◦. '

( ) *

§

t T R

r = r(t)

T

r

!

! r(t) = = (x(t), y(t), z(t)) x(t) y(t) z(t) " #

# r(t) T

$ % T #

$

& |r| % r

'r0 # !

r = r(t) t → t0 ( r0 = lim r(t)

t→t0

lim |r(t) −r0| = 0

t→t0

) * !

$! r = r(t)

˚

U (t0)

+ $

% $ f (t) = |r(t) − −r0| , ε δ

! $

$! r = r(t)

˚

U (t0) ' r0 # ! r = r(t)

t → t0

ε > 0 δ = δ(ε) > 0 : |r(t) −r0| < ε t : 0 < |t −t0| < δ.

§

- r(t) = (x(t), y(t), z(t)) r0 = (x0, y0, z0)

|r(t) −r0| = (x(t) − x0)2 + (y(t) − y0)2 + (z(t) − z0)2.

. * /

4 5 #

# $ ! %/

!

5

r

lim r(t) =

t→t0−0

=r(t0 − 0)

1◦!4◦ "

r

U (t0) # t0$

lim r(t) = r(t0)

t→t0

% $

" " &

r1 r2

f t0 r1±r2 fr1 (r1,r2) [r1r,2] t0

'

( )

$ *

r = r(t)$

U (t0)$

) +

, r(t) = (x(t), y(t), z(t))$ $ $

r (t) = (x (t), y (t), z (t)).

#

+

+ + *

§

r = r(t)$

U (t0)$ t0$

t0

r1 r2 f

t0

0◦ (r1 +r2) = r1 +r2

1◦ (rf1) = f r1 + rf1

2◦ (r1,r2) = (r1,r2) + (r1,r2) 3◦ [r1,r2] = [r1,r2] + [r1,r2]

' * 4◦

r1(t0 + t) ×r2(t0 + t) −r1(t0) ×r2(t0) =

= (r1(t0 + t) −r1(t0)) ×r2(t0 + t)+

+r1(t0) × (r2(t0 + t) −r2(t0)).

4 ) t "

t → 0$ * 4◦

* 5

r(t(τ ))$ τ U (τ0)

% r(t(τ )) = x(t(τ )), y(t(τ )), z(t(τ ))

* *

dtd r(t(τ )) = x (t(τ ))t (τ ), y (t(τ ))t (τ ), z (t(τ ))t (τ ) =

=r (t(τ ))t (τ ).

dr = r t dτ = r dt.

dr

dr = r dt t

!

" #$

#$ %

& $

r (t) = (r (t)) &' r(n)(t) = (r(n−1)(t))

&'

r(n)(t0) U (t0)

˚

t U (t0)

r(t) = n r(k)(t0) (t − t0)k +ε(t − t0)(t − t0)n, k!

k=0

− → →

ε(t t0) 0 t t0

( ) %

r(t) = (x(t), y(t), z(t)) *

+ " r(t) #

'$

+

,-

+ ) o((t − t0)n) t → t0

§

! % r %

. /

. /

"

) . )/ $

! % r %

. / r

. /

0 $

1 &

$ ' 2 !

) r(t) = (cos t, sin t, 0) 0 t 2π + r (t) = = (− sin t, cos t, 0) |r (t)| = 1

ξ

3

%' 2

r

[a, b] (a, b) ξ (a, b)

|r(b) −r(a)| = (r(b) −r(b)e,) = (r(b)e,) − (r(a)e,).

5 % f (t) = (r(t),e) ( *

2 "

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

=f (ξ)(b − a) = (r (ξ)e,)(b − a).

-%

R3

Γ R3

Γ= {(x(t), y(t), z(t)) : a t b},

x y z [a, b] !

" #

$ % & ! '

( ! R3) * !+

, ! (

-

R3 (

. / Γ !

Γ = {r(t) : a t b} Γ = {rˆ(t) : a t b},

r(t) (x(t), y(t), z(t)) rˆ(t)

(x(t), y(t), z(t)) Γ ! / {t, rˆ(t)}

. M R3 ! "

# & Γ t1, t2 [a, b] t1 = t2 rˆ(t1) = rˆ(t2) = M

0 ( ! !

" # 0 Γ !

rˆ(a) = rˆ(b) 0 !

! a t1 < t2 b rˆ(t1) = rˆ(t2) t1 = a t2 = b

1 ! t

rˆ(t) & " &

-, Γ

/ / ! /

rˆ(a) rˆ(b)

§

1 & & Γ - t0 t0 + + t [a, b] - % / ! rˆ(t0) rˆ(t0 +

4 (t0, rˆ(t0)) %

!

r = r(t0) +r (t0)τ, −∞ < τ < +∞.

! &

1

t !

& Γ

t0 = a t0 = b t0

0 r

Γ = {r(t) a t b}

r(t) !

! [a, b]

" (t0, rˆ(t0)) !

# !! !

$ % ! &

" $ (

Γ = {rˆ(t) : a t c}, Γ = {rˆ(t) : c t b}

Γ

* !$

Γ % &%

) + ! Γ

$ ,

!

-! ! % !

! )

Γ = {r(t) : a t b}, t = g(τ ), ρ(τ ) = r(g(τ )),

˜ { }

Γ = ρ(τ ) : α τ β .

. ! & ˜ Γ $ Γ

! ! ! t = g(τ )

) + ! ! ! ! !

§

1◦ g [α, β] ↔ [a, b] !

[α, β]

/% ! ! + ! ˜ Γ !$

Γ $ ! & ! ! ! g−1 % g

) 0 ! 1 ! !

2 ! $! (

! " ! ! ( ! (

& % & 1◦

%

1◦◦ g [α, β]

) & g(α) = a g(β) = b

Γ 3 !

! ! ! Γ

% ! ! t = g(τ ) & & ! !

1◦ , !

2◦ g ! !

[α, β]4

3◦ g (τ ) = 0 α τ β

) + ! !

! Γ (

! & ! & & ˜

Γ

) 1◦ 2◦ 3◦ % g

g−1 % ! $ !

˜

Γ Γ + ! $& (

& $ ! % !

!

) ! !

& ! 1◦ 2◦ 3◦ !

! !

5 % ( % &4

6 % ( 4 7 ( & &