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Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
y
y
ɩɪɢɜɵɩɨɥɧɟɧɢɢɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɫɜɨɞɢɬɫɹɤɪɟɲɟɧɢɸɫɢɫɬɟɦɵ
ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɡɥɨɜɵɯ ɡɧɚ
ɱɟɧɢɣɢɫɤɨɦɨɣɮɭɧɤɰɢɢ Ɉɩɪɟɞɟɥɢɜɨɫɧɨɜɧɵɟɧɟɢɡɜɟɫɬɧɵɟ ɞɚɥɟɟ
ɱɟɪɟɡɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢɨɩɪɟɞɟɥɹɸɬɭɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹ
2Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɣɨɩɟɪɚɬɨɪɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹɢɡɝɢɛɚ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɬɪɨɟɧɢɟɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯ ɨɩɟɪɚɬɨɪɨɜ ɄɊɈ
ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɮɭɧɤɰɢɢ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ w (x, y).
ɇɚɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɨɛɥɚɫɬɶ ɧɚɥɨɠɢɦɫɟɬɤɭ ɭɡɥɨɜɫ ɪɚɜɧɵɦɢ ɢɧ
ɬɟɪɜɚɥɚɦɢ ¨
x
ɢ ¨
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɨ ɨɫɹɦ x ɢ y ɪɢɫ Ɍɚɤɢɦ
y
ɨɛɪɚɡɨɦɫ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣɬɨɱɤɢ ɡɪɟɧɢɹ ɦɵ ɩɟɪɟɯɨɞɢɦɨɬ ɧɟɩɪɟ
ɪɵɜɧɨɣɨɛɥɚɫɬɢɤɞɢɫɤɪɟɬɧɨɣɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣɫɟɬɤɟ. ɇɚɷɬɨɣ
ɫɟɬɤɟ ɛɭɞɟɦ ɫɬɪɨɢɬɶ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɩɪɨɢɡɜɨɞ
ɧɵɯɢɞɪɭɝɢɯɱɥɟɧɨɜɜɯɨɞɹɳɢɯɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟ
¨
x ¨x
l
gbh
i a k c m
¨
¨
f d e
n
x
w'w
y
k
–1
11
1
u
'
y
2
w
w
2
'
y
k
1
–2
1
u
2
'
y
y
w'w
–1
x
k
2
w
w
1 –2
2
'
x
k
1
u
1
'
x
1
u
1
2
'
x
1
2
w''w
yx
k
–1
–1
1
''u4
yx
1
Ɋɢɫ 2.1. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɚɹ ɫɟɬɤɚɢɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵ
41
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ȼɩɪɟɞɩɨɥɨɠɟɧɢɢɦɚɥɨɫɬɢɪɚɡɦɟɪɨɜɲɚɝɨɜɫɟɬɤɢ ¨xɢ¨yɯɨɬɹ
ɷɬɨɧɟɫɨɜɫɟɦɬɚɤɡɚɩɢɲɟɦɩɟɪɜɵɟɩɪɨɢɡɜɨɞɧɵɟɨɬɮɭɧɤɰɢɢ w ɞɥɹ
ɭɡɥɚk ɜɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣɮɨɪɦɟɩɪɢ¨
= const, ¨y= const):
x
w
w
|
x
w
k
ww
2
'
x
w
w
ac
;
|
y
w
k
ww
bd
(2.1)
.
2
'
y
ɉɨɥɭɱɟɧɧɵɟ ɨɩɟɪɚɬɨɪɵ ɩɪɨɢɡɜɨɞɧɵɯ ɜ ɫɯɟɦɚɬɢɱɧɨɦ ɜɢɞɟ
ɢɡɨɛɪɚɠɟɧɵɧɚɪɢɫ ɩɨɞɫɟɬɤɨɣɭɡɥɨɜɢɫɛɨɤɭɈɧɢɫɢɦɦɟɬɪɢɱ
ɧɵɨɬɧɨɫɢɬɟɥɶɧɨɭɡɥɚ kɩɨɷɬɨɦɭɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɢɦɜɵɪɚɠɟɧɢɹ
ɧɚɡɵɜɚɸɬɫɹɰɟɧɬɪɚɥɶɧɵɦɢɤɨɧɟɱɧɵɦɢɪɚɡɧɨɫɬɹɦɢ.
ɉɪɚɜɵɟ ɢɥɟɜɵɟɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɞɥɹɭɡɥɚk ɡɚɩɢɲɭɬɫɹɬɚɤ
ww
w
w
x
w
c
|
'
k
x
w
w
k
|
y
w
k
ww
'
y
w
w
kd
x
w
k
ww
a
k
|
'
x
ɥɟɜɥɟɜɩɪɩɪ
w
w
w
y
k
ww
|
'
y
ȼɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɮɭɧɤɰɢɢ w ɜ ɭɡɥɟ k ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɪɢ
ɦɟɧɹɹɨɩɟɪɚɬɨɪɵɩɟɪɜɨɣɩɪɨɢɡɜɨɞɧɨɣɤɩɟɪɜɵɦɠɟɩɪɨɢɡɜɨɞɧɵɦ
2
w
w
x
w
2
w
w
y
w
2
w
w
yx
ww
§
1
w
w
¨
|
¨
x
w
'
x
k
©
§
1
w
w
¨
|
¨
y
w
'
y
k
©
§
1
w
w
¨
|
¨
x
w
'
y
k
©
ɥɟɜɩɪ
·
w
w
¸
x
w
w
w
y
w
|
¸
kk
¹
ɥɟɜɩɪ
·
¸
|
¸
kk
¹
·
w
w
¸
|
¸
x
w
¹
bd
2
www
cka
22
'
x
2
22
'
fe
42
;
www
dkb
y
''
yx
;
wwww
(2.2)
hg
.
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵɬɚɤɠɟɩɪɢɜɟɞɟɧɵɧɚɪɢɫ 2.1.
ɋɤɥɚɞɵɜɚɹ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɢɦ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɜɵɪɚɠɟɧɢɹɞɥɹɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨɪɚɅɚɩɥɚɫɚɜɭɡɥɟk:
2
w
w
k
w
ɢɥɢ
2
w
w
k
w
2
w
2
x
2
w
2
x
2
w
w
y
w
a
|
kk
)1(2
kb
22
'
x
wwwww
PPP
c
d
(2.3)
2
w
w
y
w
a
|
kk
)1(2
kb
22
'
y
wwwww
KKK
c
d
,
(2.4)
bk
.;;;
42
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
2
'
ɝɞɟ
x
;
P
'
K
2
y
Ȼɢɝɚɪɦɨɧɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ
ɩɨɥɭɱɢɬɶ ɞɜɚɠɞɵ ɩɪɢɦɟɧɢɜ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ
2
'
1
y
.
2
'
P
x
ɞɥɹ ɭɡɥɚ k ɦɨɠɧɨ
w4
ɡɚɩɢ
w2
ɫɚɧɧɵɣ ɜ ɜɢɞɟ ɢ ɋɧɚɱɚɥɚ ɢɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ
ɩɪɟɞɫɬɚɜɢɦɨɩɟɪɚɬɨɪ
224
|
ww
kk
ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
w4
22222
PPP
)1(2
2
'
x
wwwww
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨɞɫɬɚɜɢɜ ɫɸɞɚ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɨɩɟɪɚɬɨɪɵ
2
j
ɜɵɪɚɠɟɧɧɵɟɫ ɩɨɦɨɳɶɸ (2.4), ɢɩɪɢɜɟɞɹ
,,,,,,
dckbajw
ɩɨɞɨɛɧɵɟɱɥɟɧɵɩɨɥɭɱɢɦ
4
w
|
k
22
''
yx
2222)1(4
22
''
yx
wwww
)1(4)1(4)1(4)668(
KPKKP
cbak
(2.5)
wwwwwwwww
PKPKP
nmlihgfed
.
Ɋɚɡɧɨɫɬɧɚɹ ɫɯɟɦɚɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɩɟɪɚɬɨɪɚ Ʌɚɩɥɚɫɚɢɡɨɛɪɚ
ɠɟɧɚɧɚɪɢɫ 2.2, ɚȼɬɨɦɫɥɭɱɚɟɟɫɥɢɢɫɩɨɥɶɡɭɟɬɫɹɤɜɚɞɪɚɬɧɚɹɫɟɬɤɚ
= ¨y= ¨ȝ = Ș = ɨɩɟɪɚɬɨɪɩɪɢɧɢɦɚɟɬɜɢɞɪɢɫ 2.2, ɛ)
¨
x
dckba
.
wwwwwwwwwwwww
4
|
w
k
4
'
)(2)(820
ɇɟɤɨɬɨɪɵɟɨɞɧɚɤɨɜɩɨɥɧɟɪɚɡɪɟɲɢɦɵɟ ɫɥɨɠɧɨɫɬɢ ɩɪɟɞɫɬɚɜ
ɥɹɟɬ ɩɨɫɬɪɨɟɧɢɟ ɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɩɟɪɚɬɨɪɚ ɜ ɫɥɭɱɚɟ ɧɟɪɚɜɧɨ
ɦɟɪɧɨɝɨɲɚɝɚɫɟɬɤɢ¨
cRQVW¨yFRQVWɚɬɚɤɠɟɞɥɹɤɨɫɨɭɝɨɥɶ
x
ɧɨɣɢɥɢɩɨɥɹɪɧɨɣɫɢɫɬɟɦɵɤɨɨɪɞɢɧɚɬ>4].
ȼɤɨɧɟɱɧɨɦɫɱɟɬɟɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɣɨɩɟɪɚɬɨɪɞɢɮɮɟɪɟɧ
ɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨɭɡɥɚ k ɡɚ
ɩɢɲɟɬɫɹɜɜɢɞɟ
4
q
w
k
k
D
43
(2.6)
,,...,1,
Nk
nmlihgfedcbak
.
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɝɞɟN – ɨɛɳɟɟ ɱɢɫɥɨɜɧɭɬɪɟɧɧɢɯ ɭɡɥɨɜ ɫɟɬɤɢ q
ɫɢɜɧɨɫɬɶɧɚɝɪɭɡɤɢ ɩɪɢɯɨɞɹɳɟɣɫɹɧɚ ɩɥɨɳɚɞɤɭ ¨
– ɫɪɟɞɧɹɹ ɢɧɬɟɧ
k
ɩɪɢɦɵɤɚɸ
xרy
ɳɭɸɤɭɡɥɭ k. ȿɫɥɢɜɭɡɥɟ k ɩɪɢɥɨɠɟɧɚɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹɫɢɥɚ F
ɨɧɚ ɭɱɢɬɵɜɚɟɬɫɹ ɜ ɜɢɞɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ
ɚ)
2
4
Ș
k
–4(1+Ș)
2
ɛ)
4
1 20
k
Ɋɢɫ 2.2. ȻɢɝɚɪɦɨɧɢɱɟɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚ
ȝ
–ȝ
8+6(ȝȘ)
–ȝ
ȝ
2
–812
–8
–82
1
–8
2
–4(1+Ș)
1
¨
¨
2
2
¨
1
u
4
'
Fq ''
1
u
Ș
¨
y
x
22
''
yx
ɬɨ
k
.
)(
yxkk
Ɉɩɟɪɚɬɨɪ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɩɥɚɫɬɢɧɵ ɥɟ
ɠɚɳɟɣɧɚɭɩɪɭɝɨɦɜɢɧɤɥɟɪɨɜɫɤɨɦɨɫɧɨɜɚɧɢɢ (ɩɪɢɞɟɣɫɬɜɢɢɪɚɫɩɪɟ
ɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢɢɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯɫɢɥ), ɡɚɩɢɲɟɬɫɹɬɚɤ
k
4
w
y
k
w
D
§
1
¨
q
kk
¨
D
©
·
F
k
¸
¸
''
yx
¹
(2.7)
.,...,1,
Nk
ɇɚɤɥɚɞɵɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɩɟɪɚɬɨɪ ɢɥɢ ɧɚ ɜɫɟ
ɭɡɥɵ ɫɟɬɤɢ ɜ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɪɨɝɢɛɵ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ
ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɡɥɨɜɵɯ ɧɟɢɡ
ɜɟɫɬɧɵɯ w
Ɉɞɧɚɤɨ ɩɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɭɡɥɨɜ ɥɟɠɚ
k
ɳɢɯɜɛɥɢɡɢɤɨɧɬɭɪɚɩɥɚɫɬɢɧɵɜɧɢɯɜɨɣɞɭɬɧɟɢɡɜɟɫɬɧɵɟɩɪɨɝɢɛɵ
ɢɜɡɚɤɨɧɬɭɪɧɵɯɭɡɥɚɯ ɫɟɬɤɢɉɨɷɬɨɦɭɞɥɹɪɟɲɟɧɢɹɫɢɫɬɟɦɵɭɪɚɜ
44
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɧɟɧɢɣ ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɤɪɚɹɯ ɩɥɚɫɬɢɧɵ
ɬɚɤɠɟɡɚɩɢɫɚɧɧɵɟɱɟɪɟɡɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢ
2.1.2. ɍɱɟɬɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
Ɋɚɫɫɦɨɬɪɢɦ ɯɚɪɚɤɬɟɪɧɵɟ ɫɥɭɱɚɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɧɚɩɪɢ
ɦɟɪɞɥɹɩɪɚɜɨɝɨɤɪɚɹɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵx=a.
Ɂɚɞɟɥɚɧɧɵɣ ɤɪɚɣ ɪɢɫ 2.3, ɚ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɹ ɭɡɥɚ k ɧɚ
ɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵɜɫɨɨɬɜɟɬɫɬɜɢɢɫɢɢɦɟɟɦ:
ww
w
w
k
w
;0
x
w
|
k
ac
2
'
x
ɨɬɫɸɞɚ
,0
(2.8)
.
ww
ac
ɒɚɪɧɢɪɧɨɨɩɟɪɬɵɣɤɪɚɣ ɪɢɫɛȼɫɨɨɬɜɟɬɫɬɜɢɢɫ
ɢɞɥɹɭɡɥɚk ɢɦɟɟɦ
2
w
w
k
w
;0
x
w
a
|
k
www
2
c
k
22
'
x
ɨɬɫɸɞɚ
,0
ww
ac
(2.9)
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɥɹ ɡɚɞɟɥɚɧɧɨɝɨ ɢ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɝɨ ɤɪɚɹ
ɩɥɚɫɬɢɧɵɭɡɥɨɜɵɟɧɟɢɡɜɟɫɬɧɵɟɧɚɤɨɧɬɭɪɟɜɡɚɤɨɧɬɭɪɧɵɯɭɡɥɚɯ
ɦɨɠɧɨɢɫɤɥɸɱɢɬɶɫɩɨɦɨɳɶɸɪɚɜɟɧɫɬɜɢ
ɋɜɨɛɨɞɧɵɣɤɪɚɣ ɪɢɫ 2.3, ɜ). ɇɚɷɬɨɦɭɱɚɫɬɤɟɤɨɧɬɭɪɚɞɥɹɭɡ
ɥɚk ɜɫɨɨɬɜɟɬɫɬɜɢɢɫɢɦɟɟɦ
2
w
w
w
x
2
w
w
Q
2
2
w
k
2
§
w
w
w
¨
2
¨
w
w
x
xy
©
2
·
w
w
Q
¸
2
¸
w
y
¹
k
(2.10)
.0)2(;0
ɂɫɩɨɥɶɡɭɹ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟ ɜɵɪɚɠɟɧɢɹ ɢ ɩɪɢɦɟɧɢɜ ɨɩɟ
ɪɚɬɨɪɩɟɪɜɨɣɩɪɨɢɡɜɨɞɧɨɣɤɨɜɬɨɪɵɦɩɪɨɢɡɜɨɞɧɵɦɩɨɫɥɟɩɪɢ
ɜɟɞɟɧɢɹɩɨɞɨɛɧɵɯɱɥɟɧɨɜɢɢɫɤɥɸɱɟɧɢɹɜɟɥɢɱɢɧ¨
wwwww
;0)()()1(2
PQPQ
dbcak
ɢ¨yɩɨɥɭɱɢɦ
x
wwwwwwww
KQKQQK
mihgfeac
(2.11)
.0))(2()()3(2
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɯɟɦɵ ɨɩɟɪɚɬɨɪɨɜ ɞɥɹ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
ɢ ɢɡɨɛɪɚɠɟɧɵɧɚ ɪɢɫ Ⱥɧɚɥɨɝɢɱɧɨɫɨɫɬɚɜɥɹɸɬɫɹ
ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɨɩɟɪɚɬɨɪɵɞɥɹɥɟɜɨɝɨɤɪɚɹɩɥɚɫɬɢɧɵx=ɜɷɬɨɦ
ɫɥɭɱɚɟɩɪɢɜɟɞɟɧɧɵɟɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵɪɚɡɜɟɪɧɭɬɫɹɧɚɝɪɚɞɭɫɨɜ.
Ⱦɥɹɜɟɪɯɧɟɝɨ ɢ ɧɢɠɧɟɝɨɤɪɚɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɨɩɟɪɚɬɨɪɵɪɚɡɜɟɪ
45
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
)
y
)
)
)
ɧɭɬɫɹɧɚ ɝɪɚɞɭɫɨɜɤɪɨɦɟ ɬɨɝɨ ɜɨɩɟɪɚɬɨɪɚɯ ɞɥɹɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ
ɤɨɷɮɮɢɰɢɟɧɬɵȝɢȘɩɨɦɟɧɹɸɬɫɹɦɟɫɬɚɦɢɪɢɫɝ).
ɚ
x=a
x
ak c
¨
x ¨x
y
w
a
1
ɜ
x=a
w
1
ɛ
x=a
x
ak c
¨
x ¨x
y
w
c
a
w
c
–1 1
x
gh
i a k c m
f e
¨
¨
x ¨x
ȝȞ
–1 ȝȞ –1
ȝȞ
y
w
i w
a
w
k
w
c
2 – Ȟ Ȟ– 2
Ș Ș(Ȟ–6) 0 Ș(6–Ȟ) –ȘȞ
2 – Ȟ Ȟ– 2
w
m
ɝ
ȝ
2 – Ȟȝ(Ȟ–6) 2 – Ȟ
0
– 2 ȝ(6–2Ȟ) Ȟ – 2
Ȟ
–ȝȞ
Ɋɢɫ 2.3. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɨɩɟɪɚɬɨɪɵɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
46
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Ʉɚɤ ɜɢɞɢɦ ɜ ɫɥɭɱɚɟ ɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ ɜ ɪɚɫɱɟɬ ɜɜɨɞɹɬɫɹ ɞɜɚ
ɫɥɨɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɯ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡɥɨɜ – ɫɬɨɥɶɤɨ ɠɟ ɫɥɨɟɜ
ɫɤɨɥɶɤɨ ɢɩɪɢɧɚɥɨɠɟɧɢɢɨɩɟɪɚɬɨɪɚɧɚɭɡɥɵɧɟɡɚɤɪɟɩɥɟɧɧɨɝɨ
ɤɨɧɬɭɪɚɩɥɚɫɬɢɧɵɉɪɢ ɷɬɨɦ ɤɚɠɞɨɦɭɬɚɤɨɦɭ ɭɡɥɭ ɧɚ ɤɨɧɬɭɪɟ ɨɬ
ɜɟɱɚɸɬ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ȼ ɪɟɡɭɥɶɬɚɬɟ ɨɛɳɟɟ ɱɢɫɥɨ ɞɨɩɨɥɧɢ
ɬɟɥɶɧɵɯɡɚɤɨɧɬɭɪɧɵɯɧɟɢɡɜɟɫɬɧɵɯɛɭɞɟɬɪɚɜɧɨ ɨɛɳɟɦɭ ɱɢɫɥɭɞɨ
ɩɨɥɧɢɬɟɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɵɪɚɠɚɸɳɢɯ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ
ɜɫɟɯɭɡɥɨɜɧɚɫɜɨɛɨɞɧɨɦɤɪɚɟɩɥɚɫɬɢɧɵ
ɗɬɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫɨɫɬɚɜɥɟɧɧɵɟ ɫ ɩɨɦɨɳɶɸ
ɪɚɜɟɧɫɬɜ ɞɨɛɚɜɥɹɸɬɫɹ ɤ ɭɪɚɜɧɟɧɢɹɦ ɜɵɪɚɠɚɸɳɢɦ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵȼɪɟɡɭɥɶɬɚɬɟ ɢɦɟ
ɟɦɩɨɥɧɭɸ ɪɚɡɪɟɲɚɸɳɭɸ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜ
ɧɟɧɢɣɢɡɪɟɲɟɧɢɹɤɨɬɨɪɨɣɧɚɯɨɞɹɬɫɹɜɟɥɢɱɢɧɵɩɪɨɝɢɛɨɜɜɨɜɫɟɯ
ɭɡɥɚɯɫɟɬɤɢ ɜɤɥɸɱɚɹ ɡɚɤɨɧɬɭɪɧɵɟ. Ɂɧɚɱɟɧɢɹ w ɜ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡ
ɥɚɯɜɞɚɥɶɧɟɣɲɟɦɢɫɩɨɥɶɡɭɸɬɫɹɩɪɢɜɵɱɢɫɥɟɧɢɢ ɜɧɭɬɪɟɧɧɢɯɭɫɢ
ɥɢɣɜɭɡɥɨɜɵɯɬɨɱɤɚɯ ɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ
2.1.3. ȼɵɱɢɫɥɟɧɢɟ ɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣɢɧɚɩɪɹɠɟɧɢɣ
ɉɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɨɝɢɛɨɜ ɜɨ ɜɫɟɯ ɭɡɥɚɯ ɫɟɬɤɢ ɧɚɥɨɠɟɧ
ɧɨɣ ɧɚ ɩɥɚɫɬɢɧɭ ɫ ɩɨɦɨɳɶɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯɨɩɟɪɚɬɨɪɨɜɜɵɱɢɫɥɹɟɦɡɧɚɱɟɧɢɹɢɡɝɢɛɚɸɳɢɯɢɤɪɭɬɹɳɟɝɨɦɨ
ɦɟɧɬɨɜɜɷɬɢɯɭɡɥɚɯɉɨɞɫɬɚɜɢɜɜɮɨɪɦɭɥɵɜɵɪɚɠɟɧɢɹɜɬɨɪɵɯ
ɩɪɨɢɡɜɨɞɧɵɯɢɩɪɢɜɟɞɹɩɨɞɨɛɧɵɟɱɥɟɧɵɩɨɥɭɱɢɦ
M
M
H
x
y
2
k
'
x
D
2
k
'
y
Q
D
k
)1(
''
4
yx
k
e
f
ɫa
dbk
.
wwww
g
h
PQPQ
wwwww
;)()1(2
db
wwwww
ca
(2.12)
;)()1(2
KQKQ
D
ɋɯɟɦɵɨɩɟɪɚɬɨɪɨɜɞɥɹɜɵɱɢɫɥɟɧɢɹɦɨɦɟɧɬɨɜɩɪɢɜɟɞɟɧɵɧɚɪɢɫ
47
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
)
)
M
x
k
–ȝȞ
M
y
k
–1–1 2ȝȞ)
D
u
–ȝȞ
¨
x
H
k
2
'
x
1
0
¨
¨
–1 1
x
–1
y
u
2(1+ȘȞ)
D''Q
4
–1
–1
¨
y
–ȘȞ–ȘȞ
D
u
2
'
y
)1(
yx
Ɋɢɫ 2.4. Ɉɩɟɪɚɬɨɪɵɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣ
ɇɚɢɛɨɥɶɲɢɟɡɧɚɱɟɧɢɹɧɚɩɪɹɠɟɧɢɣɞɟɣɫɬɜɭɸɳɢɯɧɚɧɢɠɧɟɣ
ɢɜɟɪɯɧɟɣɩɨɜɟɪɯɧɨɫɬɹɯ ɩɥɚɫɬɢɧɵɨɩɪɟɞɟɥɹɟɦɩɨɮɨɪɦɭɥɚɦ
6
6
M
x
x
;
h
M
y
y
;
h
6
H
.
r Wr Vr V
22 2
h
(2.13)
ɉɪɢɦɟɪɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɨɣɩɥɚɫɬɢɧɵɆɄɊ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɪɚɫɱɟɬɚ ɠɟɥɟɡɨɛɟɬɨɧɧɨɣ ɩɥɢɬɵ ɪɚɡɦɟɪɚ
ɦɢa=ɦb=ɦɢɬɨɥɳɢɧɨɣh=ɦɧɚɝɪɭɠɟɧɧɨɣɪɚɜɧɨɦɟɪ
ɧɨɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸq=ɤɇɦ
2
ɢɞɜɭ
ɦɹɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢɫɢɥɚɦɢF=ɤɇɄɪɚɹɩɥɢɬɵɩɚɪɚɥɥɟɥɶɧɵɟ
ɨɫɢx – ɲɚɪɧɢɪɧɨɨɩɟɪɬɵɩɚɪɚɥɥɟɥɶɧɵɟɨɫɢy – ɡɚɞɟɥɚɧɵɪɢɫɚ).
Ɍɪɟɛɭɟɬɫɹɨɩɪɟɞɟɥɢɬɶɩɪɨɝɢɛɵɜɭɡɥɚɯɧɚɥɨɠɟɧɧɨɣɧɚɩɥɢɬɭɫɟɬɤɢ
î ɩɨɫɬɪɨɢɬɶ ɷɩɸɪɵ ɩɪɨɝɢɛɨɜ w ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ M
ɢ M
ɞɥɹ ɫɪɟɞɧɢɯ ɫɟɱɟɧɢɣ ɩɥɢɬɵ ɩɪɢ x=a ɢ y=b ɷɩɸɪɭ
y
ɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜH ɞɥɹɝɪɚɧɢɱɧɨɝɨɫɟɱɟɧɢɹy = 0.
ɍɱɢɬɵɜɚɹ ɫɢɦɦɟɬɪɢɸ ɩɥɚɫɬɢɧɵ ɫ ɡɚɤɪɟɩɥɟɧɢɹɦɢ ɢ ɩɪɢɥɨ
ɠɟɧɧɨɣɧɚɝɪɭɡɤɨɣɨɬɧɨɫɢɬɟɥɶɧɨɨɫɟɣx=aɢy=bɧɭɦɟɪɚɰɢɸ
ɭɡɥɨɜ ɫɟɬɤɢ ɜɵɩɨɥɧɢɦ ɬɚɤɠɟ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɢɯ
ɨɫɟɣȼɟɥɢɱɢɧɵɲɚɝɨɜɫɟɬɤɢ¨
ɤɨɷɮɮɢɰɢɟɧɬɵ
22
yx
=a/4 = ɦ¨y=b/4 = ɦɬɨɝɞɚ
x
P K '' P
48
.553,01;778,1
x
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
)
)
ɚ
RPOP’ Rƍ
2Ǝ S 2 1 2
3Ǝ T 3 4 3
2Ǝ 6֤ 2 1 2
y
¨
2ƍ 1ƍ 2ƍ
ɛ
x
1,771
2 –11,11 2
0,563
–6,2522,04–6,25
0,563
2 –11,11 2
b/2 = 3 ɦb/2 = 3 ɦ
1,771
a/2 = 4 ɦa/2 = 4 ɦ
y
F q F
¨
x
¨
x
¨
x
¨
x
Ɋɢɫ 2.5. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɚɹɫɟɬɤɚɞɥɹɪɚɫɱɟɬɚɩɥɢɬɵ
22
ɉɟɪɟɧɟɫɟɦɡɧɚɦɟɧɚɬɟɥɶ
ɢɡɥɟɜɨɣ ɜɩɪɚɜɭɸ ɱɚɫɬɶ ɞɢɮ
''
yx
ɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹɢɡɝɢɛɚɢɩɨɞɫɬɚɜɢɜɜɛɢɝɚɪɦɨɧɢ
ɱɟɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚɫɦ ɪɢɫ 2.2, ɚ) ɤɨɷɮɮɢɰɢɟɧɬɵ ȝ Ș ɢ Ȟ
ɡɚɩɢɲɟɦɷɬɨɬɨɩɟɪɚɬɨɪɜɱɢɫɥɨɜɨɦɜɢɞɟɪɢɫ 2.5, ɛ).
ɉɪɢɦɟɦ ɞɥɹ ɠɟɥɟɡɨɛɟɬɨɧɚ: E=3,2 Â10
4
Ɇɉɚ =3,2Â107ɤɇɦ2,
Ȟ = 0,15, ɬɨɝɞɚɰɢɥɢɧɞɪɢɱɟɫɤɚɹɠɟɫɬɤɨɫɬɶɩɥɢɬɵ:
3
hED
2
)1(12
Q
37
2,0102,3
2
)0,15(112
3
.ɦɤɇ1082,21
ɉɨɞɫɱɢɬɚɟɦ ɜɟɥɢɱɢɧɵ ɜɯɨɞɹɳɢɟ ɬɟɩɟɪɶ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ
ɭɪɚɜɧɟɧɢɣ
22
q
''
D
22
5,1230
3
1082,21
3
;ɦ1037,12
22
F
''
yxyx
D
''
yx
5,12120
3
1082,21
3
.ɦ105,16
Ɂɚɩɢɲɟɦɪɚɜɟɧɫɬɜɚ ɜɵɬɟɤɚɸɳɢɟ ɢɡ ɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ ɪɚɫ
ɫɦɚɬɪɢɜɚɟɦɨɣ ɩɥɢɬɵ ȼɨ ɜɫɟɯ ɭɡɥɚɯ ɧɚ ɤɨɧɬɭɪɟ ɩɥɢɬɵ ɡɧɚɱɟɧɢɹ
ɩɪɨɝɢɛɨɜ ɪɚɜɧɵ ɧɭɥɸ Ⱦɥɹ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡɥɨɜ ɜɞɨɥɶ ɲɚɪɧɢɪɧɨ
49
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
)
)
ɨɩɟɪɬɵɯ ɫɬɨɪɨɧ w1ƍ=–w1; w2ƍ=–w
w
2Ǝ=w2
, w3Ǝ=w
.
3
, ɜɞɨɥɶ ɡɚɳɟɦɥɟɧɧɵɯɫɬɨɪɨɧ
2
ɂɫɩɨɥɶɡɭɹɨɩɟɪɚɬɨɪɪɢɫ 2.5, ɛɫɭɱɟɬɨɦɩɪɢɜɟɞɟɧɧɵɯɜɵɲɟ
ɪɚɜɟɧɫɬɜ ɫɨɫɬɚɜɥɹɟɦ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɜɧɭɬɪɟɧɧɢɯ ɭɡɥɨɜ ɫɟɬɤɢ
1, 2, 3, 4.
ɍɡɟɥ1: 22,04 Âw
+1,78(w
1–w1
ɍɡɟɥ2: 22,04 Âw
) = 12,37 Â10
– 6,25 (w2+ w2) – 11,11 Âw4+ 2 (w3+ w
1
–3
.
– 6,25 Âw1– 11,11 Âw3+2Âw4+ 0,563 (w2+w
2
)+
3
2
+
+1,78(w
2–w2
ɍɡɟɥ3: 22,04 Âw
+ 0,563 (w
ɍɡɟɥ4: 22,04 Âw
+2(w
+w2+w2+w2) = 12,37 Â10
2
) = 12,37 Â10
) = (12,37 Â10
3+w3
–3
.
– 6,25 Âw4– 11,11 (w2+w2) + 2 (w1+w1) +
3
4
–3
+ 16,5 Â10
–3
) = 28,87 Â10
– 6,25 (w3+w3) – 11,11 (w1+w
–3
.
1
–3
) +
.
ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ ɥɢɧɟɣ
ɧɵɯɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜɧɟɧɢɣ:
22,04 Âw
– 12,5 Âw2+4Âw3– 11,11 Âw4= 12,37 Â10
1
–6,25 Âw1+ 23,17 Âw2– 11,11 Âw3+2Âw4= 12,37 Â10
4 Âw1– 22,22 Âw2+ 23,17 Âw3– 6,25 Âw
– 22,22 Âw1+8Âw2– 12,5 Âw3+ 22,04 Âw
=28,87Â10
4
= 12,37 Â10
4
–3
;
–3
;
–3
;
–3
.
)
ȼɦɚɬɪɢɱɧɨɣɮɨɪɦɟɫɢɫɬɟɦɚɭɪɚɜɧɟɧɢɣɢɦɟɟɬɜɢɞ
ª
«
«
«
«
¬
11,110,45,1204,22
0,211,1117,2325,6
25,617,2322,220,4
04,225,120,822,22
w
½
º
1
°
°
»
w
°
°
2
»
u
¾
®
w
»
3
°
°
»
¼
°
°
w
¿
¯
4
37,12
½
°
°
37,12
°
°
®
87,28
°
°
37,12
¯
3
.10
¾
°
°
¿
Ɂɚɦɟɬɢɦ ɱɬɨ ɦɚɬɪɢɰɭ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɫɢɦ
ɦɟɬɪɢɱɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɟɫɥɢ ɜɬɨɪɨɟ ɭɪɚɜɧɟ
ɧɢɟɭɦɧɨɠɢɬɶɧɚɚɱɟɬɜɟɪɬɨɟ ɪɚɡɞɟɥɢɬɶ ɧɚ
Ɋɟɲɢɜ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɩɨɥɭɱɢɦ ɜɟɤɬɨɪ ɭɡɥɨɜɵɯ ɩɟɪɟɦɟ
ɳɟɧɢɣɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣɩɥɢɬɵ
50
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