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ȽɅȺȼȺ 5
ɑɂɋɅȿɇɇɈȿ ȾɂɎɎȿɊȿɇɐɂɊɈȼȺɇɂȿ
ɂ ɂɇɌȿȽɊɂɊɈȼȺɇɂȿ
5.1. Ɇɟɬɨɞɵ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ
ȼɵɱɢɫɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɦɟɬɨɞɚɦɢ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ
ɢɦɟɟɬ ɫɦɵɫɥ, ɤɨɝɞɚ ɚɧɚɥɢɬɢɱɟɫɤɢɣ ɪɚɫɱɟɬ ɩɪɨɢɡɜɨɞɧɨɣ ɧɟɜɨɡɦɨɠɟɧ ɢɥɢ ɮɭɧɤɰɢɹ ɡɚɞɚɧɚ ɧɚɛɨɪɨɦ ɬɨɱɟɤ ɜ ɬɚɛɥɢɱɧɨɣ ɮɨɪɦɟ.
Ʉɪɨɦɟ ɬɨɝɨ, ɱɢɫɥɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɟɬɫɹ ɩɪɢ ɪɟɲɟɧɢɢ ɦɧɨɝɢɯ ɡɚɞɚɱ (ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɨɢɫɤ ɪɟɲɟɧɢɣ
ɧɟɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɨɢɫɤ ɬɨɱɟɤ ɷɤɫɬɪɟɦɭɦɚ ɮɭɧɤɰɢɢ ɢ ɞɪ.).
Ɇɟɬɨɞɵ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ
Ɇɟɬɨɞɵ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɨɫɧɨɜɚɧɵ ɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ
ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ
ɩɪɨɢɡɜɨɞɧɨɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɱɢɫɥɟɧɧɨ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ
ɪɚɜɧɚ ɬɚɧɝɟɧɫɭ ɭɝɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɣ, ɩɪɨɜɟɞɟɧɧɨɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ ɤ
ɞɚɧɧɨɣ ɤɪɢɜɨɣ, ɢ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ ɯ (ɪɢɫ. 22).
Ɋɢɫ. 22. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɟɬɨɞɚ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ
ɉɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ f(x) ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
)x(f)dxx(f
dx
−+
00
.
′
Ɂɚɦɟɧɹɹ ɩɪɢɪɚɳɟɧɢɟ dx ɧɚ ɤɨɧɟɱɧɭɸ ɜɟɥɢɱɢɧɭ Δɯ (ɲɚɝ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ), ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɭ [11]
df
)x(f
0
dx
lim
==
→
0dx
81
)x(f)xx(f
−Δ+
00
′
)x(f
=
0
ȿɫɥɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ ɡɚɞɚɧɚ ɜ ɜɢɞɟ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɢ,
ɬɨ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɡɧɚɱɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɧɟɨɛɯɨɞɢɦɨ ɩɨɥɭɱɢɬɶ ɡɧɚɱɟɧɢɟ
ɮɭɧɤɰɢɢ f(x) ɜ ɬɨɱɤɟ x
ɉɗȼɆ ɪɚɫɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ
ȿɫɥɢ ɮɭɧɤɰɢɹ ɡɚɞɚɧɚ ɬɚɛɥɢɱɧɵɦ ɦɟɬɨɞɨɦ, ɬɨ ɟɫɬɶ ɧɚɛɨɪɨɦ ɡɧɚɱɟɧɢɣ
ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɚɯ, ɬɨ ɮɨɪɦɭɥɚ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ (ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ x ɨɛɪɚɡɭɸɬ ɜɨɡɪɚɫɬɚɸɳɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ
ɜɢɞɟ
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɷɬɢɯ ɜɵɪɚɠɟɧɢɣ, ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɜ ɬɨɱɤɟ xi ɨɰɟɧɢɜɚɟɬɫɹ ɩɨ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ ɜ ɷɬɨɣ ɢ ɜ ɫɥɟɞɭɸɳɟɣ x
ɭɫɥɨɜɧɨ ɧɚɡɜɚɬɶ ɩɪɚɜɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɶɸ. ɇɟɬɪɭɞɧɨ ɡɚɩɢɫɚɬɶ ɮɨɪɦɭɥɭ ɞɥɹ
ɥɟɜɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ
ɢɥɢ
ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɬɨɱɧɨɫɬɢ ɦɟɬɨɞɵ ɥɟɜɨɫɬɨɪɨɧɧɟɣ ɢ ɩɪɚɜɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɟɣ ɪɚɜɧɨɡɧɚɱɧɵ. Ȼɨɥɟɟ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɞɚɟɬ ɦɟɬɨɞ
ɫɬɢ
(ɱɬɨ ɨɫɨɛɟɧɧɨ ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɝɥɚɞɤɢɯ ɮɭɧɤɰɢɣ).
ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɛɨɥɟɟ ɬɨɱɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɤ ɢɫɤɨɦɨɦɭ ɡɧɚɱɟɧɢɸ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x
ɞɜɭɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ
ɢɥɢ ɞɥɹ ɮɭɧɤɰɢɣ, ɡɚɞɚɧɧɵɯ ɜ ɜɢɞɟ ɜɵɛɨɪɤɢ [11]:
ɢ ɜ ɬɨɱɤɟ x0 + Δx. ɉɨɫɥɟ ɱɟɝɨ ɦɨɠɧɨ ɩɪɨɫɬɨ ɫ ɩɨɦɨɳɶɸ
0
′
)x(f
=
i
[11]
′
)x(f
=
0
′
)x(f
=
i
Ɇɟɬɨɞ ɞɜɭɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ
ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ
0
′
)x(f
=
0
′
)x(f
=
i
x
Δ
−
+
xx
−
+
00
x
Δ
−
xx
−
−
x2
Δ
−
xx
−
.
′
.
)x(f
0
)x(f)x(f
i1i
.
i1i
. Ɍɚɤɨɣ ɫɩɨɫɨɛ ɦɨɠɧɨ
i+1
)ɯx(f)x(f
Δ−−
,
)x(f)x(f
1ii
−
.
1ii
ɞɜɭɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨ-
)ɯx(f)xx(f
Δ−−Δ+
00
1i1i
−+
)x(f)x(f
1i1i
−+
.
82
ɇɚɝɥɹɞɧɨ ɫɪɚɜɧɢɬɶ ɨɞɧɨɫɬɨɪɨɧɧɸɸ ɢ ɞɜɭɫɬɨɪɨɧɧɸɸ ɪɚɡɧɨɫɬɢ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɜ ɩɪɨɢɡɜɨɞɧɭɸ, ɬɚɤ ɤɚɤ ɬɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɮɭɧɤɰɢɢ
ɜ ɬɨɱɤɟ x
. Ɍɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɧɚ ɪɢɫ. 23, ɚ ɨɛɨɡɧɚɱɟɧɨ ɤɚɤ tgĮ1 . ȼ
i
ɦɟɬɨɞɟ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ (ɪɢɫ. 23, ɚ) ɜɦɟɫɬɨ ɤɚɫɚɬɟɥɶɧɨɣ ɩɪɨɜɨɞɢɬɫɹ
ɢ x
ɩɪɹɦɚɹ ɱɟɪɟɡ ɬɨɱɤɢ x
ɬɨ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ (tgĮ
ɜɪɟɦɹ ɤɚɤ ɜ ɦɟɬɨɞɟ ɞɜɭɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ, ɩɪɨɜɟɞɹ ɩɪɹɦɭɸ ɱɟɪɟɡ ɬɨɱɤɢ
ɢ x
(ɪɢɫ. 23, ɛ), ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɡɧɚɱɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɩɪɚɤɬɢɱɟɫɤɢ ɫɨɜɩɚ-
i+1
1
. ȿɫɥɢ ɜ ɨɤɪɟɫɬɧɨɫɬɹɯ ɬɨɱɤɢ xi ɮɭɧɤɰɢɹ ɧɟ ɝɥɚɞɤɚɹ,
i
i+1
) ɛɭɞɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɬɨɱɧɨɝɨ. ȼ ɬɨ
2
x
ɞɚɸɳɟɟ ɫ ɬɨɱɧɵɦ.
i-
Ɋɢɫ. 23. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ:
− ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɪɚɡɧɨɫɬɶ; ɛ − ɞɜɭɫɬɨɪɨɧɧɹɹ ɪɚɡɧɨɫɬɶ
ɚ
ɑɚɫɬɧɨɟ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɢ
ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ
Ɉɬɞɟɥɶɧɨ ɧɟɨɛɯɨɞɢɦɨ ɨɬɦɟɬɢɬɶ ɱɢɫɥɟɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɱɚɫɬɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɨɜ ɮɭɧɤɰɢɣ ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɫɟ ɚɪɝɭɦɟɧɬɵ ɮɭɧɤɰɢɢ
ɫɬɚɧɨɜɹɬɫɹ ɤɨɧɫɬɚɧɬɚɦɢ, ɤɪɨɦɟ ɚɪɝɭɦɟɧɬɚ, ɩɨ ɤɨɬɨɪɨɦɭ ɩɪɨɜɨɞɢɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ, ɚ ɬɪɟɛɭɟɦɵɣ ɩɨɪɹɞɨɤ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨɥɭɱɚɟɬɫɹ ɩɭɬɟɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɵɯ, ɜɩɥɨɬɶ ɞɨ ɬɪɟɛɭɟɦɨɝɨ ɩɨɪɹɞɤɚ [11]:
df
dx
=
i
−Δ+
x
Δ
i
ɉɪɨɢɡɜɨɞɧɵɟ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ
ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɩɪɨɢɡɜɨɞɧɵɯ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ ɩɪɨɢɡɜɨɞɧɚɹ (n)-ɝɨ ɩɨɪɹɞɤɚ ɫɱɢɬɚɟɬɫɹ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ (n-1)-ɝɨ ɩɨɪɹɞɤɚ. Ɍɚɤ, ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ:
2
d
dx
fd
=
2
dx
′′
=
′′
))x(f()x(f
ɢɥɢ
,...)x(...,f,...)xx(...,f
iii
.
df
·
§
.
¸
¨
dx
¹
©
83
−
Ɍɨɝɞɚ ɮɨɪɦɭɥɚ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɩɪɢɦɟɬ ɜɢɞ [11]
ff
−
2002
−
−
x2
Δ
x2
Δ
=
ff2f
+−
202
−
.
2
)x2(
Δ
dx
ff
2
fd
2
df
d
·
§
=
dx
=
¸
¨
dx
¹
©
′
′
−
ff
−
x2
Δ
−
11
x2
Δ
=
5.2. Ɇɟɬɨɞɵ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ
Ɇɟɬɨɞɵ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
Ɇɧɨɝɢɟ ɢɧɠɟɧɟɪɧɵɟ ɡɚɞɚɱɢ ɬɪɟɛɭɸɬ ɜɵɱɢɫɥɟɧɢɣ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚ-
ɥɨɜ ɜɢɞɚ
ɝɞɟ f(x) − ɩɨɞɵɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ, ɧɟɩɪɟɪɵɜɧɚɹ ɧɚ ɨɬɪɟɡɤɟ [a, b].
Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɜɵɱɢɫɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ-Ʌɟɣɛɧɢɰɚ:
b
³
ɂɧɬɟɝɪɚɥ ɮɭɧɤɰɢɢ – ɚɧɚɥɨɝ ɫɭɦɦɵ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɫɥɚɝɚɟɦɵɯ. ȼ ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ ɢɦɟɟɬɫɹ ɜ ɜɢɞɭ ɪɚɡɛɢɟɧɢɟ ɨɛɥɚɫɬɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɨɬɪɟɡɤɢ ɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɪɨɢɡɜɟɞɟɧɢɣ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɚɪɝɭɦɟɧɬɚ, ɩɪɢɧɚɞɥɟɠɚɳɟɝɨ ɤɚɠɞɨɦɭ ɨɬɪɟɡɤɭ, ɢ ɞɥɢɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɨɬɪɟɡɤɚ (ɪɢɫ. 24).
ɂɧɬɟɝɪɚɥ
Ɍɚɤ, ɫ ɩɨɦɨɳɶɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɨɩɪɟɞɟɥɹɸɬ ɨɛɴɟɦɵ ɫɥɨɠɧɵɯ ɬɟɥ, ɷɧɟɪɝɢɸ,
ɪɚɛɨɬɭ, ɞɚɜɥɟɧɢɟ, ɦɚɫɫɭ, ɷɥɟɤɬɪɢɱɟɫɤɢɣ ɡɚɪɹɞ ɢ ɦɧɨɝɢɟ ɞɪɭɝɢɟ ɜɟɥɢɱɢɧɵ.
ȿɫɥɢ ɢɧɬɟɝɪɚɥ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɜɵɱɢɫɥɟɧ ɚɧɚɥɢɬɢɱɟɫɤɢ, ɢɥɢ ɮɭɧɤɰɢɹ ɡɚɞɚɧɚ ɝɪɚɮɢɱɟɫɤɢ ɥɢɛɨ ɬɚɛɥɢɱɧɵɦ ɦɟɬɨɞɨɦ, ɬɨ ɩɪɢɦɟɧɹɸɬ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ
ɜɵɱɢɫɥɟɧɢɣ, ɬɚɤɢɟ ɤɚɤ [11]:
ɦɟɬɨɞ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ;
−
ɦɟɬɨɞ ɬɪɚɩɟɰɢɣ;
−
ɦɟɬɨɞ Ɇɨɧɬɟ-Ʉɚɪɥɨ ɢ ɞɪ.
−
ȼɫɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɨɫɧɨɜɚɧɵ ɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɦ ɫɦɵɫɥɟ ɢɧɬɟɝɪɚɥɚ. ȿɫɥɢ f(x) ɧɚ ɨɬɪɟɡɤɟ [a, b], ɬɨ ɢɧɬɟɝɪɚɥ
ɷɬɨ ɨɞɢɧ ɢɡ ɨɫɧɨɜɧɵɯ ɢɧɫɬɪɭɦɟɧɬɨɜ ɪɚɛɨɬɵ ɫ ɮɭɧɤɰɢɹɦɢ.
a
b
dx)x(f ,
³
a
)a(F)b(Fdx)x(f
−=
b
dx)x(f
³
a
.
84
ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ ɩɥɨɳɚɞɢ ɮɢɝɭɪɵ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɝɪɚɮɢɤɨɦ ɮɭɧɤɰɢɢ ɭ = f(x), ɨɬɪɟɡɤɨɦ ɨɫɢ ɚɛɫɰɢɫɫ, ɩɪɹɦɨɣ ɯ = ɚ ɢ ɩɪɹɦɨɣ ɯ = b. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɱɢɫɥɟɧɢɟ
ɢɧɬɟɝɪɚɥɚ ɪɚɜɧɨɫɢɥɶɧɨ ɜɵɱɢɫɥɟɧɢɸ ɩɥɨɳɚɞɢ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ.
Ɋɢɫ. 24. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ
Ɇɟɬɨɞ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ
Ɋɚɡɞɟɥɢɦ ɨɬɪɟɡɨɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ [a, b] ɧɚ n ɪɚɜɧɵɯ ɱɚɫɬɟɣ (n ɷɥɟɦɟɧɬɚɪ-
abh−
ɧɵɯ ɨɬɪɟɡɤɨɜ). Ⱦɥɢɧɚ ɤɚɠɞɨɝɨ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɨɬɪɟɡɤɚ
ɧɢɹ ɛɭɞɭɬ ɯ
ɷɬɢɯ ɬɨɱɤɚɯ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭ
= ɚ, ɯ1 = ɚ + h, x2 = a + 2h, …xn = b. ȼɵɱɢɫɥɢɦ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɜ
0
= f(ɚ), y1 = f(ɚ + h), y2 = f(a + 2h), …yn = f(b). ɉɪɢ
0
=
n
. Ɍɨɱɤɚɦɢ ɞɟɥɟ-
ɦɟɬɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ ɩɥɨɳɚɞɶ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ ɩɪɢɛɥɢɠɟɧɧɨ ɡɚɦɟɧɹɟɬɫɹ ɩɥɨɳɚɞɶɸ ɦɧɨɝɨɭɝɨɥɶɧɢɤɚ, ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ n ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ
(ɪɢɫ. 25). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɱɢɫɥɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫɜɨɞɢɬɫɹ ɤ ɧɚɯɨɠɞɟɧɢɸ ɫɭɦɦɵ n ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ [11]:
b
.
)y...yyy(hhy...hyhyhydx)x(fS
³
a
b
³
a
++++≈+++≈≈
1n2101n210
−−
.
)y...yy(hhy...hyhydx)x(fS
+++≈++≈≈
n21n21
Ɋɢɫ. 25. Ɇɟɬɨɞɵ ɥɟɜɵɯ (ɚ) ɢ ɩɪɚɜɵɯ (ɛ) ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ
85
Ɍɚɤɠɟ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɟɬɫɹ ɮɨɪɦɭɥɚ ɫɪɟɞɧɢɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ [11]
(ɪɢɫ. 26)
1n
¦
−
0i
=
h
.
)
x(yhS
+≈
i
2
Ɋɢɫ. 26. Ɇɟɬɨɞɵ ɫɪɟɞɧɢɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ
Ɇɟɬɨɞ ɬɪɚɩɟɰɢɣ
ɉɪɢ ɷɬɨɦ ɦɟɬɨɞɟ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ
ɩɥɨɳɚɞɶ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ ɡɚɦɟɧɹɟɬɫɹ ɩɥɨɳɚɞɶɸ ɦɧɨɝɨɭɝɨɥɶɧɢɤɚ, ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɢɡ n ɬɪɚɩɟɰɢɣ (ɪɢɫ. 27). ɉɪɢ ɷɬɨɦ ɤɪɢɜɚɹ ɡɚɦɟɧɹɟɬɫɹ ɥɨɦɚɧɵɦɢ
ɭɱɚɫɬɤɚɦɢ.
Ɋɢɫ. 27. Ɇɟɬɨɞ ɬɪɚɩɟɰɢɣ
86
ɉɥɨɳɚɞɶ ɷɥɟɦɟɧɬɚɪɧɨɣ ɬɪɚɩɟɰɢɢ (ɪɢɫ. 28) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
1
S
0
2
.
h)ba(
⋅+=
Ɋɢɫ. 28. ɗɥɟɦɟɧɬɚɪɧɚɹ ɬɪɚɩɟɰɢɹ
Ɏɨɪɦɭɥɚ ɦɟɬɨɞɚ ɬɪɚɩɟɰɢɣ ɢɦɟɟɬ ɜɢɞ
b
³
a
≈≈
yy
+
(hdx)x(fS
n0
++++
2
.
)y...yy
1n21
−
Ɇɟɬɨɞ Ɇɨɧɬɟ-Ʉɚɪɥɨ
ɗɬɨɬ ɦɟɬɨɞ ɩɪɢɦɟɧɹɸɬ ɞɥɹ ɪɟɲɟɧɢɹ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɡɚɞɚɱ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɢ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɝɪɚɥɨɜ.
ɋɭɬɶ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɧɚ ɨɬɪɟɡɤɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
[a, b] ɜɵɛɢɪɚɸɬ N ɫɥɭɱɚɣɧɵɯ ɬɨɱɟɤ x
, x2, x3,..xN, ɹɜɥɹɸɳɢɯɫɹ ɡɧɚɱɟɧɢɹɦɢ ɫɥɭ-
1
ɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɯ ɫ ɪɚɜɧɨɦɟɪɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɧɚ ɨɬɪɟɡɤɟ. Ⱦɥɹ ɤɚɠɞɨɣ ɬɨɱɤɢ ɜɵɱɢɫɥɹɸɬ ɩɥɨɳɚɞɶ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ, ɨɞɧɚ ɫɬɨɪɨɧɵ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ (b – ɚ),
ɚ ɜɬɨɪɚɹ − ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ f(x
):
i
−=
. Ƚɟɨɦɟɬɪɢɱɟɫɤɚɹ ɢɧ-
)x(f)ab(S
ii
ɬɟɪɩɪɟɬɚɰɢɹ ɦɟɬɨɞɚ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 29.
Ɋɢɫ. 29. Ɇɟɬɨɞ Ɇɨɧɬɟ-Ʉɚɪɥɨ
87
ȼɫɥɟɞɫɬɜɢɟ ɫɥɭɱɚɣɧɨɝɨ ɜɵɛɨɪɚ x
ɡɧɚɱɟɧɢɟ ɩɥɨɳɚɞɟɣ Si ɬɚɤɠɟ ɛɭɞɟɬ ɧɨ-
i
ɫɢɬɶ ɫɥɭɱɚɣɧɵɣ ɯɚɪɚɤɬɟɪ. ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɢɧɬɟɝɪɚɥɚ ɩɪɢɧɢɦɚɸɬ ɪɟɡɭɥɶɬɚɬ ɭɫɪɟɞɧɟɧɢɹ ɩɥɨɳɚɞɟɣ S
[11]
i
b
≈≈
dx)x(fS
³
a
+++
S...SS
N21
=
N
N
−
ab
¦
N
=
.
)x(f
i
1i
5.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 5.1.
ɮɭɧɤɰɢɢ
Ɋɚɫɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɹ ɩɪɨɢɡɜɨɞɧɵɯ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
2
ɜ ɡɚɞɚɧɧɨɣ ɬɨɱɤɟ ɯ
ɯ6ɯ3y
+=
= 1, ɩɪɢɪɚɳɟɧɢɟ ɮɭɧɤɰɢɢ ǻɯ = 0,1.
0
Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɹɟɦ ɡɚɜɢɫɢɦɨɫɬɶ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ ɦɟɬɨɞɨɦ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ:
)x(f)xx(f
−Δ+
00
′
)x(f
=
0
x
Δ
.
Ⱦɚɥɟɟ ɧɚɯɨɞɢɦ
,
23,10)1,1(f)xx(f
0
==Δ+
9)1(f)x(f
==
0
,
923,10
′
)1(f =
−
=
1,0
3,12
.
ɉɪɨɜɟɪɢɦ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɩɭɬɟɦ:
′
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɚɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɪɚɫɱɟɬɚ ɫɨɫɬɚɜɥɹɟɬ 0,3.
′
,
6ɯ6)x(f +=
0
.
126ɯ6)x(f
=+=
Ⱦɚɥɟɟ ɪɚɫɫɱɢɬɚɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɩɨ ɡɚɜɢɫɢɦɨɫɬɢ
−
ff
ff
−
−
Δ
2002
Δ
x2
x2
=
+−
ff2f
−
202
,
2
Δ
)x2(
dx
2
fd
2
df
d
·
§
=
dx
¸
¨
dx
¹
©
ff
−
=
Δ
x2
−
′−′
11
Δ
x2
=
,
52,11)2,1(f)ɯ2ɯ(ff
02
−
02
′′
)x(f
==
0
⋅
==Δ+=
,
72,6)8,0(f)ɯ2ɯ(ff
==Δ−=
72,69252,11
+⋅−
2
)1,02(
6
=
.
88
ɗɬɨ ɪɟɲɟɧɢɟ ɫɨɜɩɚɞɚɟɬ ɫ ɚɧɚɥɢɬɢɱɟɫɤɢɦ, ɬɚɤ ɤɚɤ ɚɧɚɥɢɬɢɱɟɫɤɢ
′′
.
6)x(f
=
0
Ɂɚɞɚɱɚ 5.2. Ɋɚɫɫɱɢɬɵɜɚɬɶ ɡɧɚɱɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɚɧɚɥɢɬɢɱɟ-
ɫɤɢ ɢ ɫ ɩɨɦɨɳɶɸ ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ
1
1
.
dx
³
+
2ɯ
0
Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɹɟɦ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɬɨɞɚ ɩɪɚɜɵɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ
(ɩɪɢɧɢɦɚɟɦ ɲɚɝ h = 0,2):
b
³
a
b
1
S
≈
³
2ɯ
+
a
ɉɪɨɜɟɪɢɦ ɧɚɣɞɟɧɧɨɟ ɪɟɲɟɧɢɟ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɩɭɬɟɦ:
1
1
³
2ɯ
+
0
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɚɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɪɚɫɱɟɬɚ ɫɨɫɬɚɜɥɹɟɬ 0,017.
Ɂɚɞɚɱɚ 5.3. Ɋɚɫɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫ ɩɨɦɨɳɶɸ
ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ
π
Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɹɟɦ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɬɨɞɚ ɩɪɚɜɵɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ (ɩɪɢ-
423,0)357,0385,0417,0455,05,0(2,0dx
≈++++≈
1
0
2/
dxxcos2
−
³
0
=−=+=
.
.
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Ʉɨɧɬɪɨɥɶɧɵɟ ɜɨɩɪɨɫɵ
ɉɟɪɟɱɢɫɥɢɬɟ ɦɟɬɨɞɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢ
1.
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ ɞɥɹ ɱɢɫɥɟɧɧɨ-
2.
ɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ?
3.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɝɪɚɮɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɦɟɬɨɞɚ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɪɚɡ-
ɧɨɫɬɢ?
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɞɜɭɫɬɨɪɨɧɧɟɣ ɪɚɡɧɨɫɬɢ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ
4.
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ?
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɞɥɹ ɩɪɨɢɡ-
5.
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɱɢɫɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɞɥɹ ɱɚɫɬɧɵɯ
6.
ɩɪɨɢɡɜɨɞɧɵɯ ɮɭɧɤɰɢɣ ɦɧɨɝɢɯ ɩɟɪɟɦɟɧɧɵɯ?
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟ-
7.
ɝɪɢɪɨɜɚɧɢɹ?
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɬɪɚɩɟɰɢɣ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟɝɪɢɪɨɜɚ-
8.
ɧɢɹ?
ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ Ɇɨɧɬɟ-Ʉɚɪɥɨ ɞɥɹ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟɝɪɢ-
9.
ɪɨɜɚɧɢɹ?
90
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