Математика для юристов - Д.А. Ловцова
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ȼɨɩɪɨɫɵ ɢ ɡɚɞɚɱɢ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1.Ⱦɚɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɢ ɩɨɹɫɧɢɬɶ ɟɟ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ.
2.ɇɚɣɬɢ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɣ:
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ɝ) y |
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ɞ) y |
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3.ɉɨɹɫɧɢɬɶ, ɤɚɤ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɨɢɡɜɨɞɧɚɹ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɮɭɧɤɰɢɣ.
4.ɂɫɫɥɟɞɨɜɚɬɶɮɭɧɤɰɢɸ y x3 3ux2 9ux 14 ɢɩɨɫɬɪɨɢɬɶɟɟɝɪɚɮɢɤɧɚ ɨɬɪɟɡɤɟ[ 2, 4].
5.Ⱦɚɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɮɭɧɤɰɢɢ ɢ ɩɨɹɫɧɢɬɶ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ.
6.Ɂɚɩɢɫɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɵ ɮɭɧɤɰɢɣ:
ɚ) y |
(x 1)4, |
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sin2(x), |
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ɝ) y |
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7.ɉɨɹɫɧɢɬɶ, ɤɚɤ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ɜ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɹɯ. Ɏɨɪɦɭɥɚ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɨɝɪɟɲɧɨɫɬɢ ɮɭɧɤɰɢɢ ɢɡ-ɡɚ ɩɨɝɪɟɲɧɨɫɬɢ ɚɪɝɭɦɟɧɬɚ. Ɏɨɪɦɭɥɚ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɧɨɜɨɝɨ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɩɨ ɟɟ ɫɬɚɪɨɦɭ ɡɧɚɱɟɧɢɸ ɢ ɩɪɢɪɚɳɟɧɢɸ ɚɪɝɭɦɟɧɬɚ.
8.ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɥɢɱɢɧ:
ɚ) 4 272 , |
ɛ) e1.05, |
ɜ) ln(e 0.272), ɝ) sin(290).
9.ɉɨɹɫɧɢɬɶ ɩɪɢɦɟɧɟɧɢɟ ɮɨɪɦɭɥɵ Ɍɟɣɥɨɪɚ ɞɥɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɮɭɧɤɰɢɢ. Ʉɚɤ ɨɰɟɧɢɜɚɟɬɫɹ ɩɨɝɪɟɲɧɨɫɬɶ ɬɚɤɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ?
10.Ɏɭɧɤɰɢɸ y e-x ɪɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ a 0 ɢ ɞɥɹ ɷɬɨɝɨ ɪɚɡɥɨɠɟɧɢɹ:
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ɚ) ɧɚɣɬɢ ɬɨɬ ɞɢɚɩɚɡɨɧ ɡɧɚɱɟɧɢɣ x!0, ɞɥɹ ɤɨɬɨɪɨɝɨ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɢɛɥɢɠɟɧɢɹ ɮɭɧɤɰɢɢ ɬɪɟɦɹ ɫɥɚɝɚɟɦɵɦɢ ɧɟ ɩɪɟɜɵɫɢɬ ɜɟɥɢɱɢ-
ɧɵ H 0.1;
ɛ) ɧɚɣɬɢ ɬɨ ɤɨɥɢɱɟɫɬɜɨ ɫɥɚɝɚɟɦɵɯ ɜ ɪɚɡɥɨɠɟɧɢɢ ɮɭɧɤɰɢɢ, ɤɨɬɨɪɨɟ ɨɛɟɫɩɟɱɢɬ ɜɵɱɢɫɥɟɧɢɟ ɟɟ ɡɧɚɱɟɧɢɹ ɜ ɬɨɱɤɟ x 0.25 ɫ ɬɨɱɧɨɫɬɶɸ
H 0.1.
Глава 6. Основы интегрального исчисления
6.1.Определенный интеграл
Ʉɩɨɧɹɬɢɸ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɩɪɢɜɟɥɢ ɡɚɞɚɱɢ ɜɵɱɢɫɥɟɧɢɹ ɭɫɪɟɞɧɟɧɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɫɥɨɠɧɵɯ ɩɪɨɰɟɫɫɨɜ (ɫɤɚɠɟɦ, ɬɚɤɢɯ ɤɚɤ ɞɢɧɚɦɢɤɚ ɩɪɟɫɬɭɩɧɨɫɬɢ, ɞɟɦɨɝɪɚɮɢɢ ɢ ɞɪ.). Ɇɵ ɠɟ ɩɨɹɫɧɢɦ ɫɦɵɫɥ ɷɬɨɝɨ ɩɨɧɹɬɢɹ ɧɚ ɩɪɨɫɬɨɣ ɡɚɞɚɱɟ.
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ɉɨɥɨɠɢɦ, ɱɬɨ ɬɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɢɬɶ |
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f(x) |
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[a,b] (ɪɢɫ. 6.1) (ɬɨ ɟɫɬɶ ɩɥɨɳɚɞɶ ɤɪɢɜɨ- |
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ɥɢɧɟɣɧɨɣ |
ɬɪɚɩɟɰɢɢ |
aABb). |
Ɋɟɲɟɧɢɟ |
Aɷɬɨɣ ɡɚɞɚɱɢ ɨɩɢɫɵɜɚɟɬɫɹ ɬɚɤɨɣ ɤɨɧɫɬ-
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ɉɪɚɜɚɹ ɱɚɫɬɶ ɷɬɨɣ ɡɚɩɢɫɢ ɱɢɬɚɟɬɫɹ ɬɚɤ: |
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Ɋɢɫ.6.1 |
ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɨɬ a ɞɨ |
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b ɨɬ f(x) ɩɨ dx. Ɂɞɟɫɶ: |
f(x) – ɩɨɞɵɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ, a – ɧɢɠɧɢɣ ɩɪɟɞɟɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ,
b – ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
Ɉɛɵɱɧɨ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɩɨɥɶɡɭɸɬɫɹ ɮɨɪɦɭɥɨɣ ɇɶɸɬɨɧɚɅɟɣɛɧɢɰɚ:
bb
F FNL ³f(x)udx F(x) F(b) F(a), |
(6.1) |
aa
ɝɞɟ FNL – ɡɧɚɱɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ F, ɩɨɥɭɱɟɧɧɨɟ ɩɨ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ,
F(x) – ɩɟɪɜɨɨɛɪɚɡɧɚɹ ɞɥɹ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ f(x).
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Ɏɭɧɤɰɢɹ F(x) ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɞɥɹ ɮɭɧɤɰɢɢ f(x), ɟɫɥɢ f(x) ɟɫɬɶ ɪɟɡɭɥɶɬɚɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ F(x):
f(x) F’(x).
ɇɚɩɪɢɦɟɪ, ɞɥɹ f(x) 3ux2 ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɛɭɞɟɬ F(x) x3. ȼ ɫɚɦɨɦ ɞɟɥɟ, (x3)’ 3ux2. Ɉɞɧɚɤɨ ɥɟɝɤɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɢ (x3 1000)’ 3ux2, ɢ (x3 0.001)’ 3ux2, ɢ ɜɨɨɛɳɟ (x3 C)’ 3ux2. ɉɨɷɬɨɦɭ, ɟɫɥɢ F(x) – ɩɟɪɜɨɨɛ-
ɪɚɡɧɚɹ ɞɥɹ f(x), ɚ C – ɥɸɛɚɹ ɤɨɧɫɬɚɧɬɚ, ɬɨ ɢ ɮɭɧɤɰɢɹ F(x) C ɬɚɤɠɟ ɛɭɞɟɬ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɞɥɹ f(x), ɬɨ ɟɫɬɶ
f(x) (F(x) C)’. (6.2)
Ɉɬɵɫɤɚɧɢɟ ɩɟɪɜɨɨɛɪɚɡɧɨɣ F(x) ɞɥɹ f(x) ɧɚɡɵɜɚɸɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɮɭɧɤɰɢɢ f(x) ɢ ɡɚɞɚɸɬ ɬɚɤɨɣ ɡɚɩɢɫɶɸ:
³f(x)udx F(x).
Ʌɟɜɚɹ ɱɚɫɬɶ ɷɬɨɣ ɮɨɪɦɭɥɵ ɱɢɬɚɟɬɫɹ ɬɚɤ: ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɨɬ f(x) ɩɨ dx. Ƚɨɜɨɪɹɬ, ɱɬɨ ɞɥɹ ɨɬɵɫɤɚɧɢɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ F(x) ɞɥɹ f(x) ɧɭɠɧɨ ɜɡɹɬɶ ɢɧɬɟɝɪɚɥ ɨɬ ɮɭɧɤɰɢɢ f(x) ɢɥɢ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɮɭɧɤɰɢɸ f(x).
Ɉɬɦɟɬɢɦ, ɱɬɨ ɇɶɸɬɨɧ ɢ Ʌɟɣɛɧɢɰ ɩɪɚɤɬɢɱɟɫɤɢ ɜ ɨɞɧɨ ɢ ɬɨ ɠɟ ɜɪɟɦɹ ɢɡɨɛɪɟɥɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɢ ɢɧɬɟɝɪɚɥɶɧɨɟ ɢɫɱɢɫɥɟɧɢɹ. Ɉɞɧɚɤɨ ɸɪɢɫɬ Ʌɟɣɛɧɢɰ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɮɢɡɢɤɚ ɇɶɸɬɨɧɚ, ɩɨɧɢɦɚɥ ɜɚɠɧɨɫɬɶ ɷɮɮɟɤɬɢɜɧɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɢɦɜɨɥɢɤɢ, ɛɥɚɝɨɞɚɪɹ ɤɨɬɨɪɨɣ «ɩɨɪɚɡɢɬɟɥɶɧɵɦ ɨɛɪɚɡɨɦ ɫɨɤɪɚɳɚɟɬɫɹ ɪɚɛɨɬɚ ɦɵɫɥɢ». ɂɦɟɧɧɨ
Ʌɟɣɛɧɢɰ ɩɪɢɦɟɧɢɥ ɫɢɦɜɨɥ d ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɢ ɫɢɦɜɨɥ ³ |
ɞɥɹ |
ɢɧɬɟɝɪɚɥɚ. |
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Ɏɨɪɦɭɥɨɣ (6.1) ɤɨɞɢɪɭɸɬ ɬɚɤɭɸ ɞɜɭɯɷɬɚɩɧɭɸ ɩɪɨɰɟɞɭɪɭ ɜɵɱɢɫ- |
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ɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ. |
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ɗɬɚɩ 1. ȼɵɱɢɫɥɹɸɬ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ³f(x)udx ɢ ɧɚɯɨ- |
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ɞɹɬ ɜɵɪɚɠɟɧɢɟ F(x) ɞɥɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ. |
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ɗɬɚɩ 2. ȼ ɩɨɥɭɱɟɧɧɭɸ ɮɨɪɦɭɥɭ F(x) ɩɨɞɫɬɚɜɥɹɸɬ ɫɧɚɱɚɥɚ x |
b ɢ |
ɧɚɯɨɞɹɬ ɱɢɫɥɨ F(b), ɩɨɬɨɦ ɜ ɮɨɪɦɭɥɭ F(x) ɩɨɞɫɬɚɜɥɹɸɬ x a ɢ ɧɚɯɨɞɹɬ ɱɢɫɥɨ F(a), ɚ ɡɚɬɟɦ ɩɭɬɟɦ ɜɵɱɢɬɚɧɢɹ F(a) ɢɡ F(b) ɧɚɯɨɞɹɬ ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ F FNL.
ȼ ɷɬɨɣ ɩɪɨɰɟɞɭɪɟ ɧɚɢɛɨɥɶɲɭɸ ɫɥɨɠɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɹɟɬ ɷɬɚɩ 1. ɉɨɷɬɨɦɭ ɨɫɬɚɧɨɜɢɦɫɹ ɩɨɞɪɨɛɧɨ ɧɚ ɦɟɬɨɞɚɯ ɜɵɱɢɫɥɟɧɢɹ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ.
Ɉɬɦɟɬɢɦ ɞɥɹ ɧɚɱɚɥɚ, ɱɬɨ ɜ ɦɚɬɟɦɚɬɢɤɟ (ɫɬɪɨɝɨɫɬɢ ɪɚɞɢ) ɪɟɡɭɥɶɬɚɬ ɜɵɱɢɫɥɟɧɢɹ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɡɚɩɢɫɵɜɚɸɬ ɫ ɭɱɟɬɨɦ ɮɚɤɬɚ (6.2):
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³f(x)udx F(x) C
ɢɤɨɧɫɬɚɧɬɭ C ɧɚɡɵɜɚɸɬ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ. ɉɨɞɫɬɚɜɢɦ ɜ ɮɨɪɦɭɥɭ ɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ ɜɦɟɫɬɨ F(x) ɫɭɦɦɭ F(x) C:
b |
b |
(F(x) C) (F(b) C) (F(a) C) F(b) F(a) F(x) .
a |
a |
Ʉɚɤ ɜɢɞɢɦ, ɞɥɹ ɧɚɲɢɯ ɰɟɥɟɣ ɨɬɦɟɱɟɧɧɚɹ ɫɬɪɨɝɨɫɬɶ ɧɟ ɧɭɠɧɚ, ɢ ɪɟɡɭɥɶɬɚɬ ɜɵɱɢɫɥɟɧɢɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ³f(x)udx ɞɥɹ ɢɫ-
ɩɨɥɶɡɨɜɚɧɢɹ ɟɝɨ ɜ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ ɞɨɫɬɚɬɨɱɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɬɨɥɶɤɨ ɮɨɪɦɭɥɨɣ F(x) (ɛɟɡ ɫɥɚɝɚɟɦɨɝɨ C).
ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɩɟɪɜɨɨɛɪɚɡɧɨɣ F(x) ɢɫɩɨɥɶɡɭɸɬ ɫɜɨɣɫɬɜɚ ɢɧɬɟɝɪɚɥɨɜ, ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɨɜ ɨɬ ɬɢɩɨɜɵɯ ɮɭɧɤɰɢɣ, ɬɟ ɢɥɢ ɢɧɵɟ ɩɪɢɟɦɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
ɋɜɨɣɫɬɜɚ ɢɧɬɟɝɪɚɥɨɜ. ɋɧɚɱɚɥɚ ɫɜɨɣɫɬɜɚ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ.
1. ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɜɫɟɝɞɚ ɛɟɪɟɬɫɹ ɨɬ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ F(x):
³f(x)udx ³F'(x)udx ³dF(x) F(x).
2. Ʉɨɧɫɬɚɧɬɭ ɦɨɠɧɨ ɜɵɧɨɫɢɬɶ ɡɚ ɡɧɚɤ ɢɧɬɟɝɪɚɥɚ:
³A uf(x)udx Au³f(x)udx , |
A const. |
3. ɂɧɬɟɝɪɚɥ ɨɬ ɫɭɦɦɵ ɮɭɧɤɰɢɣ ɪɚɜɟɧ ɫɭɦɦɟ ɢɧɬɟɝɪɚɥɨɜ ɨɬ ɮɭɧɤ- ɰɢɣ-ɫɥɚɝɚɟɦɵɯ:
³(f(x) g(x))udx ³f(x)udx ³g(x)udx .
Ⱥ ɬɟɩɟɪɶ ɫɜɨɣɫɬɜɚ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ.
4. ȿɫɥɢ a c b, ɬɨ
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³f(x)udx |
³f(x)udx ³f(x)udx , |
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ɬɨ ɟɫɬɶ ɩɥɨɳɚɞɶ ɩɨɞ ɤɪɢɜɨɣ f(x) ɧɚ ɨɬɪɟɡɤɟ [a,b] ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɤɚɤ ɫɭɦɦɭ ɩɥɨɳɚɞɟɣ ɩɨɞ ɧɟɸ ɧɚ ɨɬɪɟɡɤɚɯ [a,c] ɢ [c,b]. ɂɧɨɝɞɚ ɫɥɚɝɚɟɦɵɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜɵɱɢɫɥɢɬɶ ɩɪɨɳɟ, ɱɟɦ ɟɝɨ ɥɟɜɭɸ ɱɚɫɬɶ.
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5. ɉɨɧɹɬɢɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɪɚɫɲɢɪɹɟɬɫɹ ɬɚɤ:
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³f(x)udx . |
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Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, |
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F(b) F(a) (F(a) F(b)) ³f(x)udx . |
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Ɂɧɚɱɢɬ, ɢ |
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F(x) F(x) .
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Ɍɚɛɥɢɰɚ ɢɧɬɟɝɪɚɥɨɜ. ȼ ɦɚɬɟɦɚɬɢ- |
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ɤɟ ɪɚɡɪɚɛɨɬɚɧɵ ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɨɜ ɨɬ |
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ɬɵɫɹɱ ɮɭɧɤɰɢɣ. Ɇɵ ɠɟ ɨɝɪɚɧɢɱɢɦɫɹ ɬɚɛ- |
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ɥɢɰɟɣ ɢɧɬɟɝɪɚɥɨɜ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɬɢɩɨɜɵɯ |
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1(!) ɢ ɫɬɪɨɤɭ b ɬɚɛɥ. 6.1 |
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ɩɪɢɦɟɧɢɬɶ ɧɟɥɶɡɹ. |
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ln(a)
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ɩɥɨɳɚɞɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɤɪɢɜɨɣ |
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(ɪɢɫ.6.2). Ɉɬɦɟɬɢɦ, |
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ɧɚ ɨɬɪɟɡɤɟ « |
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ɱɬɨ ɤɪɢɜɚɹ f(x) ɧɚ ɷɬɨɦ ɨɬɪɟɡɤɟ ɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢ- |
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ɬɟɥɶɧɨ ɨɫɢ x |
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¢ɫɜɨɣɫɬɜɨ 5 ɢɧɬɟɝɪɚɥɨɜ² |
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3uS
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³cos(x)udx ¢ɫɢɦɦɟɬɪɢɹ (ɪɢɫ.6.2)² |
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2u³cos(x)udx 2usin(x) |
2u(0 1) 2. |
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ɗɬɨɬ ɩɪɢɦɟɪ ɢɥɥɸɫɬɪɢɪɭɟɬ ɜɚɠɧɭɸ ɨɫɨɛɟɧɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧ-
b
ɬɟɝɪɚɥɚ. ȼ ɡɚɩɢɫɢ ³f(x)udx ɨɛɴɟɤɬɵ x, f(x), dx, a, b – ɜɟɥɢɱɢɧɵ ɚɥɝɟɛɪɚɢ-
a
ɱɟɫɤɢɟ, ɪɚɡɧɨɫɬɶ F(b) F(a) ɜ ɮɨɪɦɭɥɟ ɇɶɸɬɨɧɚ–Ʌɟɣɛɧɢɰɚ — ɬɨɠɟ ɜɟɥɢɱɢɧɚ ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ. Ɂɧɚɱɢɬ, ɢ FNL – ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ (ɢ, ɤɚɤ ɜ ɧɚɲɟɦɩɪɢɦɟɪɟ, ɦɨɠɟɬɩɪɢɧɢɦɚɬɶɨɬɪɢɰɚɬɟɥɶɧɨɟɡɧɚɱɟɧɢɟ).
ȼ ɦɚɬɟɦɚɬɢɤɟ ɪɚɡɪɚɛɨɬɚɧɨ ɨɝɪɨɦɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɩɪɢɟɦɨɜ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɟ ɩɪɢɦɟɧɹɸɬɫɹ ɧɚ ɩɟɪɜɨɦ ɷɬɚɩɟ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɢ ɩɨɡɜɨɥɹɸɬ ɥɸɛɨɣ ɢɧɬɟɝɪɚɥ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɤ ɬɚɛɥɢɱɧɨɦɭ. Ɇɵ ɠɟ ɨɩɢɲɟɦ ɬɨɥɶɤɨ ɞɜɚ ɤɥɚɫɫɢɱɟɫɤɢɯ ɩɪɢɟɦɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ. ɗɬɨɬ ɩɪɢɟɦ ɩɪɢɦɟɧɹɟɬɫɹ ɬɨɝɞɚ, ɤɨɝɞɚ ɡɚɞɚɧɧɵɣ ɢɧɬɟɝɪɚɥ ɢɦɟɟɬ ɬɚɤɨɣ ɜɢɞ:
b
F ³f(M(x))uMc(x)udx ,
a
76
ɬɨ ɟɫɬɶ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɫɬɨɢɬ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɢ f(M(x)) ɧɚ ɞɢɮɮɟɪɟɧɰɢɚɥ ɜɥɨɠɟɧɧɨɣ ɮɭɧɤɰɢɢ M’(x)udx.
ɇɚ ɩɟɪɜɨɦ ɷɬɚɩɟ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ F ɜɵɩɨɥɧɹɸɬ ɡɚɦɟɧɭ t M(x). Ɍɨɝɞɚ dt dM(x) M’(x)udx ɢ
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³f(t)udt F(t). |
³f(M(x))uM (x)udx |
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɮɨɪɦɭɥɭ F(t) ɦɨɠɧɨ ɛɵɥɨ ɛɵ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɚ ɜɬɨɪɨɦ ɷɬɚɩɟ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ, ɧɟɨɛɯɨɞɢɦɨ ɢ ɩɪɟɞɟɥɵ a ɢ b ɩɨ ɩɟɪɟɦɟɧɧɨɣ x ɡɚɦɟɧɢɬɶ ɩɪɟɞɟɥɚɦɢ at ɢ bt ɩɨ ɩɟɪɟɦɟɧɧɨɣ t: at M(a), bt M(b). Ɍɨɝɞɚ
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ȼɵɱɢɫɥɢɦ, ɤ ɩɪɢɦɟɪɭ, |
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ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ. ɇɚɩɨɦɧɢɦ ɮɨɪɦɭɥɭ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ:
(u(x)uv(x))’ u’(x)uv(x) u(x)uv’(x).
əɫɧɨ, ɱɬɨ ɩɟɪɜɨɨɛɪɚɡɧɚɹ ɞɥɹ ɩɪɚɜɨɣ ɱɚɫɬɢ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɪɚɜɧɚ ɩɟɪɜɨɨɛɪɚɡɧɨɣ ɞɥɹ ɟɝɨ ɥɟɜɨɣ ɱɚɫɬɢ:
u(x)uv(x) ³uc(x)uv(x)udx ³u(x)uvc(x)udx .
ɉɟɪɟɩɢɲɟɦ ɷɬɨ ɪɚɜɟɧɫɬɜɨ ɬɚɤ:
77
³u(x)uv'(x)udx u(x)uv(x) ³u'(x)uv(x)udx .
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɥɢ ɮɨɪɦɭɥɭ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɱɚɫɬɹɦ. ȿɟ ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɛɨɥɟɟ ɤɨɦɩɚɤɬɧɨ, ɩɨɥɨɠɢɜ |
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u(x) u, v(x) v, v (x) u dx dv |
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ɂɬɚɤ, ɟɫɥɢ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɫɬɨɢɬ ɩɪɨɢɡɜɟɞɟɧɢɟ ɨɞɧɨɣ ɮɭɧɤɰɢɢ u ɧɚ ɞɢɮɮɟɪɟɧɰɢɚɥ dv ɨɬ ɞɪɭɝɨɣ ɮɭɧɤɰɢɢ v, ɬɨ, ɩɪɢɦɟɧɢɜ
ɮɨɪɦɭɥɭ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɱɚɫɬɹɦ, ɩɨɥɭɱɢɦ ɢɧɬɟɝɪɚɥ
ɪɵɣ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɬɚɛɥɢɱɧɵɦ. ȼɵɱɢɫɥɢɦ, ɧɚɩɪɢɦɟɪ,
S
F ³sin(x)ux udx
0
u sin(x), dv x dx, du cos(x)
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dx,
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ȼɬɨɪɨɣ ɢɧɬɟɝɪɚɥ ³x2 ucos(x)udx ɩɨɥɭɱɢɥɫɹ ɧɟ ɬɚɛɥɢɱɧɵɦ ɢ
ɫɥɨɠɧɟɟ ɢɫɯɨɞɧɨɝɨ ³sin(x)ux udx . Ɂɧɚɱɢɬ, ɧɚɲ ɜɵɛɨɪ u ɢ dv ɨɤɚɡɚɥɫɹ
ɧɟɭɞɚɱɧɵɦ. ɉɨɩɪɨɛɭɟɦ ɞɪɭɝɨɣ ɜɚɪɢɚɧɬ ɧɚɡɧɚɱɟɧɢɹ u ɢ dv.
S
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ɑɬɨɛɵ ɩɪɢɦɟɧɢɬɶ ɩɪɢɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɱɚɫɬɹɦ, ɧɭɠɧɨ ɭɦɟɬɶ ɪɚɡɛɢɬɶ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɧɚ ɞɜɚ ɦɧɨɠɢɬɟɥɹ, ɚ ɢɦɟɧɧɨ, ɧɚ u ɢ ɧɚ dv. ɇɢɤɚɤɢɯ ɨɛɳɢɯ ɩɪɚɜɢɥ ɞɥɹ ɷɬɨɝɨ ɧɟɬ, ɤɪɨɦɟ ɫɥɟɞɭɸɳɢɯ:
78
dx ɞɨɥɠɟɧ ɛɵɬɶ ɜɫɟɝɞɚ ɱɚɫɬɶɸ dv, ɧɭɠɧɨ ɭɦɟɬɶ ɛɪɚɬɶ ɢɧɬɟɝɪɚɥ ɨɬ dv,
ɤɨɝɞɚ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɟɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɮɭɧɤɰɢɣ, ɛɨɥɟɟ ɫɥɨɠɧɭɸ ɢɡ ɧɢɯ ɧɭɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɱɚɫɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɚ dv.
ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɨɞɢɧ ɢɥɢ ɨɛɚ ɩɪɟɞɟɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ ɛɟɫɤɨɧɟɱɧɵ. ɂɧɬɟɝɪɚɥɵ ɫ ɬɚɤɢɦɢ ɩɪɟɞɟɥɚɦɢ ɧɚɡɵɜɚɸɬ ɧɟɫɨɛɫɬɜɟɧɧɵɦɢ. ɉɨɫɤɨɥɶɤɭ ɨɩɟɪɚɰɢɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢ ɜɡɹɬɢɹ ɩɪɟɞɟɥɚ ɥɢɧɟɣɧɵ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɫ ɛɟɫɤɨɧɟɱɧɵɦ, ɫɤɚɠɟɦ, ɜɟɪɯɧɢɦ ɩɪɟɞɟɥɨɦ ɜɵɱɢɫɥɹɸɬ ɬɚɤ:
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Ʉɚɤ ɜɢɞɢɦ, ɩɪɹɦɚɹ ɩɨɞɫɬɚɧɨɜɤɚ b |
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ɩɪɟɞɟɥɚ ɞɚɟɬ ɛɟɫɤɨɧɟɱɧɨɫɬɶ ɢ ɜ ɱɢɫɥɢɬɟɥɟ ɞɪɨɛɢ, ɢ ɜ ɟɟ ɡɧɚɦɟɧɚɬɟɥɟ. Ɋɚɫɫɦɨɬɪɢɦ ɷɬɭ ɫɢɬɭɚɰɢɸ ɩɨɞɪɨɛɧɟɟ.
ȿɫɥɢ lim f(x) |
lim g(x) |
f , ɬɨ ɜɵɱɢɫɥɟɧɢɟ lim |
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ɦɨɣ ɩɨɞɫɬɚɧɨɜɤɢ ɩɪɢɜɨɞɢɬ ɤ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɜɢɞɚ f . Ɋɚɫɤɪɵɜɚɸɬ f
ɷɬɭ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɩɨ ɩɪɚɜɢɥɭ Ʌɨɩɢɬɚɥɹ:
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79
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Ɍɨɱɧɨ ɬɚɤ ɡɚɩɢɫɵɜɚɟɬɫɹ ɩ ɪ ɚ ɜ ɢ ɥ ɨ |
Ʌ ɨ ɩ ɢ ɬ ɚ ɥ ɹ |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɨɬɧɨɲɟɧɢɟ |
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ɩɨɞ ɡɧɚɤɨɦ ɩɪɟɞɟɥɚ ɡɚɦɟɧɹɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɢɯ ɩɪɨɢɡ- |
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, ɢ ɭɠɟ ɜ ɷɬɨ ɧɨɜɨɟ ɨɬɧɨɲɟɧɢɟ ɞɟɥɚɸɬ ɩɪɹɦɭɸ ɩɨɞɫɬɚ- |
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Ɂɧɚɱɢɬ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ |
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§b |
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¨§ b 1 c ¸· |
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lim ¨ |
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lim ¨ |
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lim ¨ |
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bof© eb |
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¹ f bof©¨ eb c ¹¸ bof©eb |
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ɂɬɚɤ, |
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x 1 |
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³x ue |
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udx |
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1 lim ¨ |
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1 |
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¸ 1 0 1. |
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bof© eb |
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bof©eb |
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Ⱥɧɚɥɨɝɢɱɧɨ ɜɵɱɢɫɥɹɸɬ ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɫ ɛɟɫɤɨɧɟɱɧɵɦ ɧɢɠɧɢɦ ɩɪɟɞɟɥɨɦ:
b |
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§b |
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³ |
f(x)udx |
lim |
¨ |
³ |
f(x)udx |
¸ |
F(b) lim F(a) , |
¨ |
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aof¨ |
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ao f |
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f |
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©a |
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¹ |
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ɚ ɬɚɤɠɟ ɫ ɛɟɫɤɨɧɟɱɧɵɦɢ ɜɟɪɯɧɢɦ ɢ ɧɢɠɧɢɦ ɩɪɟɞɟɥɚɦɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:
f
³f(x)udx
f
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§b |
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lim |
¨ |
³ |
f(x)udx |
¸ |
¨ |
¸ |
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bof ¨ |
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ao f© a |
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lim F(b) |
lim F(a) . |
bof |
ao f |
80