матан Бесов - весь 2012
.pdflim |
x + sin x |
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x |
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∞ |
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+∞ − (+∞), (+0)0, (+∞)0, |
1∞. |
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0 · ∞, |
# # $
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
(a, b) * |
f (x) − f (x0) |
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x − x0 |
0 + f |
(x0) 0 |
, ! 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
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! |
U (x0) f |
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x0 x0
f
& ! * f > 0
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< 0 |
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+ 0)! + |
U (x0 − 0) f |
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U (x0 |
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, - . / f (x) − f (x0) = f (ξ)(x − x0) |
- f −
+ , x0! 0 x0
f !
) ,
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2x2 + x2 sin |
1 |
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x = 0, |
x |
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f (x) = |
x = 0. |
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0 |
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+ x0
$ % f ,
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, U (x0 − 0) U (x0 + 0) x0 |
− 0), |
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˚ |
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f (x) − f (x0) < 0 (> 0) x U (x0 |
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+ 0). |
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f (x) − f (x0) > 0 (< 0) x U (x0 |
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x = 0, |
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1 |
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x2 sin |
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x |
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f (x) = |
x = 0. |
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0 |
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+ 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!
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= |
f (n)(x0) |
+ ε(x − x0) (x − x0)n, |
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n! |
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ε(x − x0) → 0 x → x0! |
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2 ˚ |
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U (x0) |
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(n) |
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(x0) |
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− x0)| |
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! + / , |
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2 |
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n! |
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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
$' (α, f (α)) (β, f (β)) *
y = lα,β (x), x [α, β], ! lα,β (x) = |
β − x |
x − α |
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β − α |
f (α) + |
β − α |
f (β). |
, &$ 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
. % &
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β − x |
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x − α |
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f (x) − lα,β (x) = |
β − α |
(f (x) − f (α)) + |
β − α |
(f (x) − f (β)) 0 |
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"/# |
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f (x) − f (α) |
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f (β) − f (x) |
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x − α |
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β − x |
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* $ $ $
! )%% & &
(x, f (x))
$ ! )%% &
& (x0, f (x0))
% &$ f
( *
) $
f−(x0) f+(x0) f−(x0 + 0).
% &$
(
!
* % & f (x) = 1 − |x| x (−1, 1)
$ $ '
0 % &
( 1 ! (
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f f |
(a, b) |
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/◦ f 0 0 (a, b) ! " |
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! |
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f |
(a, b)# |
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◦ 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
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(x − α)(β − x)f (ζ)(ξ − η) |
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= |
β − α |
0 (> 0 f |
(ζ) > 0), |
# 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) # !
$ ' ! # '
§ |
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! f (x0) f "
x0
x0 f
# ' &
# ! ,
f (x0) = 0 f (x0) = 0
x0 f
! " " #
$ $% |
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< 0 U (x0)$ |
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◦ # f (x0) > 0 f (x0) |
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y = f (x) % & % |
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˚ |
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' y = f (x0) + f |
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(x0)(x −x0) x U (x0) |
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◦ # f (x0) |
= 0 f (x0) |
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= 0 U (x0)$ |
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y = f (x) ( |
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˚ |
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x < x0 x > x0 x U (x0) % |
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# |
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0 1 2$ 1◦ * ! |
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f (x)−(f (x0)+f (x0)(x−x0)) = |
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f |
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(x0) |
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+ ε(x − x0) (x−x0)2 |
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2! |
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$ 2◦ + * ! |
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f (x)−(f (x0)+f (x0)(x−x0)) = |
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(x0) |
+ ε(x − x0) (x−x0)3 |
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3! |
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˚ |
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x Uδ (x0) # δ > 0 |
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, f |
(a, +∞) y = kx + b
f x → +∞
f (x) = kx + b + o(1) x → +∞.
x →
→ −∞
f
x → +∞ y = kx + b
lim |
f (x) |
= k, |
lim (f (x) − kx) = b. |
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x |
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x→+∞ |
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x→+∞ |
! 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◦. $
§ |
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7◦. !
# 8◦. ! ! % &
9◦.
Ox ! #
10◦. '
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f (x) = |
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x2 |
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x(x − 2) |
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f |
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(x) = |
(x − 1)2 |
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(x) = |
(x − 1)3 |
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x |
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(−∞, 0) |
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(0, 1) |
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(1, 2) |
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(2, +∞) |
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f |
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− |
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y |
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4 |
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1 |
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+ . |
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асимпт |
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. |
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x = 1 |
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асимпт |
+ |
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( ) *
§
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.
. * /
lim r(t) = r0 / % |
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t→t0 |
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# $ |
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lim x(t) = x0, |
lim y(t) = y0, |
lim z(t) = z0. |
t→t0 |
t→t0 |
t→t0 |
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lim r1(t), |
lim r2(t), |
lim f (t), |
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t→t0 |
t→t0 |
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t→t0 |
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f |
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0◦ |
lim (r1(t) ±r2(t)) = lim r1(t) ± lim r2(t) |
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1◦ |
t→t0 |
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t→t0 |
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t→t0 |
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lim f (t)r1 |
(t) = lim f (t) |
lim r1(t) |
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t→t0 |
t→t0 |
t→t0 |
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2◦ lim (r1(t)r,2(t)) = |
lim r1(t), lim r2(t) |
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3◦ |
t→t0 |
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t→t0 |
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t→t0 |
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lim [r1(t),r2(t)] = |
lim r1(t), lim r2(t) |
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t→t0 |
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t→t0 |
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t→t0 |
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4 5 #
# $ ! %/
!
5
! |
$ 6 |
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! 4◦ lim r1 |
(t) = r10 lim r2(t) = |
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t→t0 |
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t→t0 |
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= r20 r1(t) = r10 + α(t) r2(t) = r20 + β(t) |
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α(t), β(t) → |
0 = (0, 0, 0) t → t0 |
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.r1(t) ×r2(t) −r10 ×r20 = (r10 +α(t)) × |
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(r20 + β(t)) − |
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−r10 ×r20 |
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= r10 × β(t) + α(t) ×r20 + α(t) × β(t) → 0 |
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t → t0 |
|α(t) ×r20| |α(t)| |r20| → 0, |
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|r10 × β(t)| |
|r10| |β(t)| → 0, |
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|α(t) × β(t)| |
|α(t)| |β(t)| → 0. |
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r
˚ |
+ 0) r0 |
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U (t0 |
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t0 |
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r0 |
= lim r(t) = r(t0 + 0), |
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t→t0+0 |
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lim |r(t) −r0| = 0 |
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t→t0+0 |
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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 (t0) lim r0(t0 + |
t) −r(t0) |
, |
|
t |
|||
t→0 |
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) +
, r(t) = (x(t), y(t), z(t))$ $ $
r (t) = (x (t), y (t), z (t)).
#
+
+ + *
§
r = r(t)$
U (t0)$ t0$
t = t0 + |
˚ |
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t U (t0) |
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r(t0 + t) −r(t0) = A t +ε(Δt)Δt, |
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ε(Δt) → |
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0 t → 0 |
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- " |
$ $ |
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+ r |
(t0) * r |
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(t0) |
t0 & ) A = r |
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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))
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d2r(t) = δ(dr(t)) δt=dt = δ(r (t) dt) δt=dt = r (t)dt2, |
&'
n |
n−1 |
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d r(t) = δ(d |
r(t)) |
δt=dt |
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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
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(ξ)(2π − 0) |
0 = r(2π) −r(0) = r |
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ξ
3
%' 2
r
[a, b] (a, b) ξ (a, b)
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|r(b) −r(a)| |r (ξ)|(b − a). |
.4/ |
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3 r(b) = r(a) |
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|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 +
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4 (t0, rˆ(t0)) %
!
r = r(t0) +r (t0)τ, −∞ < τ < +∞.
5 |
1 r t > 0 |
rˆ(t0) |
rˆ(t0 + t) (6 7 ! |
-, ! r |
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t |
! &
1
t !
& Γ
t0 = a t0 = b t0
0 r
Γ = {r(t) a t b}
r(t) !
! [a, b]
" (t0, rˆ(t0)) !
Γ r |
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! |
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# !! !
$ % ! &
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' ! |
% % ( |
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) Γ = {rˆ(t) a t b} c (a, b) |
" $ (
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 ( & &