Математика для менеджеров. Часть I. Учебное пособие
.pdfɌȿɆȺ ɊəȾɕ 12 Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ
ɉɭɫɬɶ ɡɚɞɚɧɚ ɛɟɫɤɨɧɟɱɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɢɫɟɥ U1,U2, Un Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ
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U1 U2 Un Un |
(1) |
n 1
ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɜɵɦ ɪɹɞɨɦ Ɂɞɟɫɶ Un – ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ
ɉɪɢɦɟɪɵ
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ɋɭɦɦɵ |
ɜɢɞɚ |
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S U1, S2 U1 U 2, ,Sn U1 U 2 U n |
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ɧɚɡɵɜɚɸɬɫɹ ɱɚɫɬɢɱɧɵɦɢ ɫɭɦɦɚɦɢ ɪɹɞɚ
ȿɫɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ S1,S2, ,Sn ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɢɦɟɟɬ
ɩɪɟɞɟɥ ɬɨ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ ɗɬɨɬ ɩɪɟɞɟɥ ɧɚɡɵɜɚ ɟɬɫɹ ɫɭɦɦɨɣ ɪɹɞɚ
12.2. ɉɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ
ɇɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ ȿɫɥɢ ɪɹɞ ɫɯɨɞɢɬɫɹ ɬɨ
limU n 0 Ɉɛɪɚɬɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɧɟɜɟɪɧɨ ɬɨ ɟɫɬɶ ɞɚɧɧɨɟ ɭɫɥɨ
n
ɜɢɟ ɦɨɠɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɨ ɪɹɞ ɛɭɞɟɬ ɪɚɫɯɨɞɢɬɶɫɹ
Ⱦɨɫɬɚɬɨɱɧɵɟ ɩɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ
1. ɉɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ ɂɦɟɟɦ ɞɜɚ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ
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U1 U2 Un Un ; |
(2) |
n 1
121
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1 2 n n . |
(3) |
n 1
ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɬɚɤɨɣ ɧɨɦɟɪ N ɱɬɨ ɞɥɹ ɜɫɟɯ ɱɥɟɧɨɜ ɪɹɞɚ ɭ ɤɨ ɬɨɪɵɯ n N ɜɵɩɨɥɧɹɟɬɫɹ U n n Ɍɨɝɞɚ ɢɡ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
ɫɥɟɞɭɟɬ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɚ ɢɡ ɪɚɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɫɥɟɞɭɟɬ ɪɚɫɯɨɞɢɦɨɫɬɶɪɹɞɚ
2. ɉɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ ɉɭɫɬɶ ɞɚɧ ɪɹɞ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ
U1 U 2 U n ɢ ɫɭɳɟɫɬɜɭɟɬ lim U n 1 Ɍɨɝɞɚ ɩɪɢ 1
n U n
ɪɹɞ ɫɯɨɞɢɬɫɹ ɚ ɩɪɢ 1 ɪɚɫɯɨɞɢɬɫɹ ɚ ɩɪɢ 1 ɜɨɩɪɨɫ ɨɫɬɚɟɬ ɫɹ ɨɬɤɪɵɬɵɦ
12 Ɂɧɚɤɨɩɟɪɟɦɟɧɧɵɟ ɪɹɞɵ
Ɋɹɞ ɫɨɞɟɪɠɚɳɢɣ ɤɚɤ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɬɚɤ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɱɥɟɧɵ ɧɚɡɵɜɚɟɬɫɹ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɦ.
Ɂɧɚɤɨɱɟɪɟɞɭɸɳɢɦɫɹ ɧɚɡɵɜɚɟɬɫɹ ɪɹɞ ɜɢɞɚ
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U1 U 2 ( 1)n 1U n |
( 1)n 1U n ɝɞɟ Un 0 (n 1,2, ) . |
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Ɋɹɞɵ ɜɢɞɚ U1 |
U 2 ( 1)nU n ( 1)nU n ɬɚɤɠɟ |
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ɧɚɡɵɜɚɸɬɫɹ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɦɢɫɹ |
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ɉɪɢɡɧɚɤ ɚɛɫɨɥɸɬɧɨɣ ɫɯɨɞɢɦɨɫɬɢ
Ɂɧɚɤɨɩɟɪɟɦɟɧɧɵɣ ɪɹɞ
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( 1)nU n U1 U 2 U3 |
( 1)nU n (4) |
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ɫɯɨɞɢɬɫɹ ɟɫɥɢ ɫɯɨɞɢɬɫɹ ɪɹɞ |
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Un U1 |
U2 |
Un |
(5) |
n 1
Ɋɹɞ ɧɚɡɵɜɚɟɬɫɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɦɫɹ. ȿɫɥɢ ɪɹɞ ɫɯɨɞɢɬɫɹ ɚ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɬɨ ɪɹɞ ɧɚɡɵɜɚɟɬ
122
ɫɹ ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɢɦɫɹ ɉɪɢ ɷɬɨɦ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɦɨɠɧɨ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ ɭɫɬɚɧɨɜɢɬɶ ɛɟɡ ɢɫɫɥɟɞɨɜɚɧɢɹ ɪɹɞɚ
ɉɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ Ʌɟɣɛɧɢɰɚ
ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ
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( 1)n 1U n U1 U 2 ( 1)n 1U n , |
U n 0 (n 1,2,...) . |
n 1 |
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ȿɫɥɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɞɜɚ ɭɫɥɨɜɢɹ
1) U1 U 2 U n ,
2) limU n 0 ɬɨ ɬɚɤɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɢ ɟɝɨ ɫɭɦɦɚ ɧɟ ɩɪɟɜɨɫɯɨ
n
ɞɢɬ ɩɟɪɜɨɝɨ ɱɥɟɧɚ S U1 .
12 ɋɬɟɩɟɧɧɵɟ ɪɹɞɵ
Ɋɹɞ ɜɢɞɚ an xn a0 a1x a2x2 an xn ɧɚɡɵɜɚɟɬɫɹ ɫɬɟ
n 0
ɩɟɧɧɵɦ ɪɹɞɨɦ Ɂɞɟɫɶ ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ a1, a2, …, ak ,… – ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɹɞɚ a0 – ɫɜɨɛɨɞɧɵɣ ɱɥɟɧ ɋɬɟɩɟɧɧɵɟ ɪɹɞɵ ɹɜ ɥɹɸɬɫɹ ɨɞɧɢɦ ɢɡ ɜɢɞɨɜ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ ɜɢɞɚ
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Un (x) U0(x) U1(x) Un (x) |
(6) |
n 0
Ⱦɥɹ ɥɸɛɨɝɨ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɢɦɟɟɬɫɹ ɢɧɬɟɪɜɚɥ –R, R ɧɚɡɵ ɜɚɟɦɵɣ ɢɧɬɟɪɜɚɥɨɦ ɫɯɨɞɢɦɨɫɬɢ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɤɨɬɨɪɨɝɨ ɪɹɞ ɫɯɨɞɢɬɫɹ ɚ ɜɧɟ ɢɧɬɟɪɜɚɥɚ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɇɚ ɝɪɚɧɢɰɚɯ ɢɧɬɟɪɜɚ ɥɚ ɪɹɞ ɦɨɠɟɬ ɥɢɛɨ ɫɯɨɞɢɬɫɹ ɥɢɛɨ ɪɚɫɯɨɞɢɬɫɹ ɑɢɫɥɨ R ɧɚɡɵɜɚ ɟɬɫɹ ɪɚɞɢɭɫɨɦ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɨɧ ɧɚɯɨɞɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
R lim |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɢɫɤ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɟɝɨ ɪɚɞɢɭɫɚ ɫɯɨɞɢɦɨɫɬɢ R ɢ ɢɫɫɥɟ ɞɨɜɚɧɢɹ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɧɚ ɝɪɚɧɢɰɚɯ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢɩɪɢ x R ).
123
ɉɪɢɦɟɪ ɇɚɩɢɫɚɬɶ ɬɪɢ ɩɟɪɜɵɯ ɱɥɟɧɚ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɩɨ ɡɚ ɞɚɧɧɨɦɭ ɨɛɳɟɦɭ ɱɥɟɧɭ U n an xn ɧɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ
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ɪɹɞɚ U n , ɟɫɥɢ Un |
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Ɋɟɲɟɧɢɟ ɉɟɪɜɵɟ ɬɪɢ ɱɥɟɧɚ ɪɹɞɚ ɛɭɞɭɬ U1 |
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62 x2 |
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63 x3 |
.ɂɦɟɟɦ an |
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Ɉɩɪɟɞɟɥɹɟɦ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ |
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ɂɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɢɦɟɟɬ ɜɢɞ |
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ɉɭɫɬɶ x |
5 ɉɨɥɭɱɚɟɦ ɱɢɫɥɨɜɨɣ ɪɹɞ |
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ɉɪɢɦɟɧɹɟɦ ɤ ɷɬɨɦɭ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɦɭɫɹ ɪɹɞɭ ɩɪɢɡɧɚɤ Ʌɟɣ ɛɧɢɰɚ
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Ɉɛɚ ɭɫɥɨɜɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɹɞ ɩɪɢ x 56
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ɩɨɷɬɨɦɭ ɩɨ ɩɪɢɡɧɚɤɭ ɫɪɚɜɧɟɧɢɹ ɪɹɞ ɪɚɫ |
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ɯɨɞɢɬɫɹ ɬɚɤ ɤɚɤ ɪɚɫɯɨɞɢɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨ
ɫɬɢɪɹɞɚ 56; 56 .
12 Ɋɹɞ Ɇɚɤɥɨɪɟɧɚ
Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ f(x) ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ
ɜɢɞ f(x) f(0) |
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ɇɚɢɛɨɥɟɟ ɭɩɨɬɪɟɛɢɬɟɥɶɧɵ ɪɚɡɥɨɠɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɮɭɧɤɰɢɣ |
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ɉɪɢɦɟɪ ȼɵɱɢɫɥɢɬɶ 4 17 ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɢɫɩɨɥɶɡɭɹ ɪɚɡɥɨɠɟɧɢɟ f (x) (1 x)m ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ
Ɋɟɲɟɧɢɟ |
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21 0,01562 |
0,00037 0,00001 |
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ɉɨɥɭɱɟɧɧɵɣ ɪɹɞ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɢ ɟɝɨ ɱɥɟɧɵ ɭɛɵɜɚɸɬ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɩɨɷɬɨɦɭ ɩɨɝɪɟɲɧɨɫɬɶ ɧɟ ɩɪɟɜɡɨɣɞɟɬ ɩɟɪɜɨɝɨ ɨɬɛɪɨɲɟɧɧɨɝɨ ɱɥɟɧɚ Ɉɱɟɜɢɞɧɨ ɱɬɨ Â
ɋɥɟɞɨɜɚɬɟɥɶɧɨ 4 17 2(1 0,01562 0,00037) 2,0305 .
ɉɪɢɦɟɪ ȼɵɪɚɡɢɬɶ ɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ , ݔ ɜ ɜɢɞɟ
ɫɯɨɞɹɳɟɝɨɫɹ ɪɹɞɚ ɢɫɩɨɥɶɡɭɹ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɞɥɹ ɩɨɞɵɧɬɟɝɪɚɥɶ ɧɨɣ ɮɭɧɤɰɢɢ ɇɚɣɬɢ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɷɬɨɝɨ ɢɧɬɟɝɪɚɥɚ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
Ɋɟɲɟɧɢɟ ȼ ɪɚɡɥɨɠɟɧɢɢ ɮɭɧɤɰɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɤɨɬɨɪɨɟ ɢɦɟɟɬ ɜɢɞ
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ɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɩɪɢɡɧɚɤɚ Ʌɟɣɛɧɢɰɚ Ɍɪɟɬɢɣ ɱɥɟɧ ɷɬɨɝɨ ɪɹɞɚ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɦɟɧɶɲɟ 0,001 ɉɨɷɬɨɦɭ ɞɥɹ ɨɛɟɫɩɟɱɟ ɧɢɹ ɬɪɟɛɭɟɦɨɣ ɬɨɱɧɨɫɬɢ ɧɭɠɧɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɥɢɲɶ ɩɟɪɜɵɟ ɞɜɚ ɱɥɟɧɚ ɪɹɞɚ ɨɤɪɭɝɥɢɜ ɪɟɡɭɥɶɬɚɬ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɞɚɧɧɨɣ
ɬɨɱɧɨɫɬɶɸ , ݔ ݔ 0,048.
126
ȼɈɉɊɈɋɕ Ʉ ɗɄɁȺɆȿɇɍ ɉɈ ɆȺɌȿɆȺɌɂɄȿ
ɇɚɩɪɚɜɥɟɧɢɟ ɩɨɞɝɨɬɨɜɤɢ ©Ɇɟɧɟɞɠɦɟɧɬ»ɫɟɦɟɫɬɪ
Ɋɚɡɞɟɥ , ɈɋɇɈȼɕ ɅɂɇȿɃɇɈɃ ȺɅȽȿȻɊɕ
Ɍɟɦɚ Ɇɚɬɪɢɰɵ ɢ ɨɩɪɟɞɟɥɢɬɟɥɢ
1.ɉɨɧɹɬɢɟ ɦɚɬɪɢɰɵ ȼɢɞɵ ɦɚɬɪɢɰ Ɉɩɟɪɚɰɢɢ ɧɚɞ ɦɚɬɪɢɰɚɦɢ ɋɜɨɣɫɬɜɚ ɨɩɟɪɚɰɢɣ ɧɚɞ ɦɚɬɪɢɰɚɦɢ
2.Ɉɩɪɟɞɟɥɢɬɟɥɢ ɤɜɚɞɪɚɬɧɵɯ ɦɚɬɪɢɰ Ɏɨɪɦɭɥɵ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɢɬɟɥɟɣ ɦɚɬɪɢɰ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɉɪɚɜɢɥɨ ɋɚɪɪɭɫɚ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɢɬɟɥɟɣ ɦɚɬɪɢɰ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ ɋɜɨɣɫɬɜɚ ɨɩɪɟɞɟɥɢɬɟɥɟɣ
3.Ɇɢɧɨɪ ɢ ɚɥɝɟɛɪɚɢɱɟɫɤɨɟ ɞɨɩɨɥɧɟɧɢɟ ɷɥɟɦɟɧɬɚ ɤɜɚɞɪɚɬɧɨɣ ɦɚɬɪɢɰɵ Ɍɟɨɪɟɦɚ Ʌɚɩɥɚɫɚ ɢ ɫɯɟɦɚ ɟɟ ɩɪɢɦɟɧɟɧɢɹ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɢɬɟɥɟɣ ɤɜɚɞɪɚɬɧɵɯ ɦɚɬɪɢɰ ɥɸɛɨɝɨ ɩɨɪɹɞɤɚ
4.Ɉɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɇɟɨɛɯɨɞɢɦɨɟ ɢ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ Ⱥɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ
5.Ɋɚɧɝ ɦɚɬɪɢɰɵ Ɍɟɨɪɟɦɚ ɨ ɧɟɢɡɦɟɧɧɨɫɬɢ ɪɚɧɝɚ ɦɚɬɪɢɰɵ ɩɪɢ ɟɟ ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ Ɍɟɨɪɟɦɚ ɨ ɪɚɧɝɟ ɦɚɬɪɢɰɵ
Ɍɟɦɚ ɋɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ
6.ɋɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢ ɮɨɪɦɵ ɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ Ɉɩɪɟɞɟɥɢɬɟɥɶ ɫɢɫɬɟɦɵ Ɍɟɨɪɟɦɚ Ʉɪɨɧɟɤɟɪɚ-Ʉɚɩɟɥɥɢ ɋɨɜɦɟɫɬɧɚɹ ɧɟɫɨɜɦɟɫɬɧɚɹ ɧɟɨɩɪɟɞɟɥɟɧɧɚɹ ɢ ɨɩɪɟɞɟɥɟɧɧɚɹ ɫɢɫɬɟɦɵ
7.Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ
8.Ɍɟɨɪɟɦɚ Ʉɪɚɦɟɪɚ Ɇɟɬɨɞ Ƚɚɭɫɫɚ
Ɍɟɦɚ ɗɥɟɦɟɧɬɵ ɜɟɤɬɨɪɧɨɝɨ ɚɧɚɥɢɡɚ
9.ȼɟɤɬɨɪɵ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ ȼɟɤɬɨɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ Ʌɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɤɬɨɪɨɜ
10.Ʌɢɧɟɣɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ Ɋɚɡɦɟɪɧɨɫɬɶ ɢ ɛɚɡɢɫ ɜɟɤɬɨɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪɚ ɜ ɪɚɡɧɵɯ ɛɚɡɢɫɚɯ
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11.ȿɜɤɥɢɞɨɜɨ ɩɪɨɫɬɪɚɧɫɬɜɨ ɇɨɪɦɚ ɜɟɤɬɨɪɚ Ɉɪɬɨɧɨɪɦɢɪɨɜɚɧɧɵɣ ɛɚɡɢɫ
12.Ʌɢɧɟɣɧɵɟ ɨɩɟɪɚɬɨɪɵ ɢ ɨɩɟɪɚɰɢɢ ɧɚɞ ɧɢɦɢ ɋɜɹɡɶ ɦɟɠɞɭ ɦɚɬɪɢɰɚɦɢ ɨɩɟɪɚɬɨɪɚ ɜ ɪɚɡɧɵɯ ɛɚɡɢɫɚɯ ɋɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɨɩɟɪɚɬɨɪɚ ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɣ ɦɧɨɝɨɱɥɟɧ ɥɢɧɟɣɧɨɝɨ ɨɩɟɪɚɬɨɪɚ
13.Ʉɜɚɞɪɚɬɢɱɧɚɹ ɮɨɪɦɚ ɉɪɢɜɟɞɟɧɢɟ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɨɪɦɵ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ Ɂɚɤɨɧ ɢɧɟɪɰɢɢ ɤɜɚɞɪɚɬɢɱɧɵɯ ɮɨɪɦ
Ɍɟɦɚ ɗɥɟɦɟɧɬɵ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɝɟɨɦɟɬɪɢɢ
14.ɍɪɚɜɧɟɧɢɟ ɥɢɧɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɍɪɚɜɧɟɧɢɟ ɩɭɱɤɚ ɩɪɹɦɵɯ Ɉɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɍɫɥɨɜɢɟ ɩɚɪɚɥɥɟɥɶɧɨɫɬɢ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɫɬɢ ɩɪɹɦɵɯ Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ ɞɨ ɩɪɹɦɨɣ
15.Ʉɪɢɜɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ Ɉɤɪɭɠɧɨɫɬɶ ɢ ɷɥɥɢɩɫ
16.Ƚɢɩɟɪɛɨɥɚ ɢ ɩɚɪɚɛɨɥɚ
17.ɍɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɢ ɩɪɹɦɨɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ Ɉɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɍɫɥɨɜɢɟ ɩɚɪɚɥɥɟɥɶɧɨɫɬɢ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɫɬɢ ɩɥɨɫɤɨɫɬɟɣ
Ɋɚɡɞɟɥ ,, ɈɋɇɈȼɕ ɆȺɌȿɆȺɌɂɑȿɋɄɈȽɈ ȺɇȺɅɂɁȺ
Ɍɟɦɚ Ɏɭɧɤɰɢɹ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
18.ɉɨɧɹɬɢɟ ɦɧɨɠɟɫɬɜɚ Ɉɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɑɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ Ɇɨɞɭɥɶ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ Ɉɤɪɟɫɬɧɨɫɬɶ ɬɨɱɤɢ
19.Ɉɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ ɋɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɮɭɧɤɰɢɢ ɋɜɨɣɫɬɜɚ ɮɭɧɤɰɢɣ
20.Ɉɛɪɚɬɧɚɹ ɮɭɧɤɰɢɹ ɋɥɨɠɧɚɹ ɮɭɧɤɰɢɹ ɗɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ Ƚɪɚɮɢɤɢ Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɮɭɧɤɰɢɣ
21.ɂɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɣ
22.ɉɪɢɦɟɧɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɷɤɨɧɨɦɢɤɟ ɢ ɭɩɪɚɜɥɟɧɢɢ Ɏɭɧɤɰɢɹ ɩɨɥɟɡɧɨɫɬɢ ɉɪɨɢɡɜɨɞɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ Ɏɭɧɤɰɢɢ ɢɡɞɟɪɠɟɤ ɫɩɪɨɫɚ ɩɨɬɪɟɛɥɟɧɢɹ ɢ ɩɪɟɞɥɨɠɟɧɢɹ
128
Ɍɟɦɚ ɉɪɟɞɟɥɵ ɢ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɣ
23.ɑɢɫɥɨɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɉɪɟɞɟɥ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɟɞɟɥɚ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
24.ɉɪɟɞɟɥ ɮɭɧɤɰɢɢ ɜ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɢ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɉɪɟɞɟɥ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ ɢ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ
25.Ȼɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɜɟɥɢɱɢɧɵ ɢ ɢɯ ɫɜɹɡɶ ɫ ɩɪɟɞɟɥɚɦɢ ɮɭɧɤɰɢɣ ɋɜɨɣɫɬɜɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɜɟɥɢɱɢɧ Ȼɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɟ ɜɟɥɢɱɢɧɵ ɂɯ ɫɜɨɣɫɬɜɚ ɋɜɹɡɶ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɢ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɯ ɜɟɥɢɱɢɧ
26.Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɨ ɩɪɟɞɟɥɚɯ ɉɪɢɡɧɚɤɢ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɩɪɟɞɟɥɚ Ɂɚɦɟɱɚɬɟɥɶɧɵɟ ɩɪɟɞɟɥɵ
27.ɇɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ Ɍɨɱɤɢ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ ɢ ɢɯ ɤɥɚɫɫɢɮɢɤɚɰɢɹ
Ɋɚɡɞɟɥ ,,, ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȿ ɂɋɑɂɋɅȿɇɂȿ
Ɍɟɦɚ ɉɪɨɢɡɜɨɞɧɚɹ ɢ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ɨɞɧɨɣ ɩɟɪɟ ɦɟɧɧɨɣ
28.Ɉɩɪɟɞɟɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɢ ɦɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ ɋɯɟɦɚ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ Ɂɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɧɟɩɪɟɪɵɜɧɨɫɬɶɸ ɮɭɧɤɰɢɢ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɶɸ
29.Ɉɫɧɨɜɧɵɟ ɩɪɚɜɢɥɚ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɉɪɨɢɡɜɨɞɧɵɟ ɨɫɧɨɜɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɣ ɩɨɤɚɡɚɬɟɥɶɧɨɣ ɫɬɟɩɟɧɧɨɣ ɢ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ ɉɪɨɢɡɜɨɞɧɚɹ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ
30.ɉɪɨɢɡɜɨɞɧɵɟ ɜɵɫɲɢɯ ɩɨɪɹɞɤɨɜ Ɇɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɗɤɨɧɨɦɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ
31.Ⱦɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ȿɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɋɜɨɣɫɬɜɚ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɮɭɧɤɰɢɢ ɉɪɢɦɟɧɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɜ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɹɯ
Ɍɟɦɚ ɉɪɢɥɨɠɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ
32. Ⱦɨɫɬɚɬɨɱɧɨɟ ɢ ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɹ ɜɨɡɪɚɫɬɚɧɢɹ ɮɭɧɤɰɢɢ Ⱦɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɭɛɵɜɚɧɢɹ ɮɭɧɤɰɢɢ ɗɤɫɬɪɟɦɭɦɵ ɮɭɧɤɰɢɢ
129
ɇɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɷɤɫɬɪɟɦɭɦɚ ɋɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ Ⱦɨɫɬɚɬɨɱɧɵɟ ɭɫɥɨɜɢɹ ɷɤɫɬɪɟɦɭɦɚ ɮɭɧɤɰɢɢ ɋɯɟɦɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɮɭɧɤɰɢɢ ɧɚ ɷɤɫɬɪɟɦɭɦ ɇɚɢɛɨɥɶɲɟɟ ɢ ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɧɚ ɨɬɪɟɡɤɟ ɫɯɟɦɚ ɢɯ ɨɬɵɫɤɚɧɢɹ
33.ȼɵɩɭɤɥɨɫɬɶ ɮɭɧɤɰɢɢ Ɍɨɱɤɢ ɩɟɪɟɝɢɛɚ ɇɟɨɛɯɨɞɢɦɨɟ ɢ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɹ ɩɟɪɟɝɢɛɚ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ ɋɯɟɦɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɚ ɜɵɩɭɤɥɨɫɬɶ ɜ ɬɨɱɤɟ ɩɟɪɟɝɢɛɚ
34.Ⱥɫɢɦɩɬɨɬɵ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ ȼɟɪɬɢɤɚɥɶɧɚɹ ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɢ ɧɚɤɥɨɧɧɚɹ ɚɫɢɦɩɬɨɬɵ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ ɉɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ ɢ ɟɝɨ ɩɪɢɦɟɧɟɧɢɟ ɤ ɜɵɱɢɫɥɟɧɢɸ ɩɪɟɞɟɥɨɜ ɮɭɧɤɰɢɣ
Ɍɟɦɚ Ɏɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ Ɏɇɉ
35.Ɉɩɪɟɞɟɥɟɧɢɟ Ɏɇɉ Ɉɛɥɚɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ Ʌɢɧɢɹ ɭɪɨɜɧɹ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɉɪɟɞɟɥ ɢ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ
36.ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɢ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɉɪɨɢɡɜɨɞɧɚɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɢ ɝɪɚɞɢɟɧɬ
37.ɗɤɫɬɪɟɦɭɦɵ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɇɟɨɛɯɨɞɢɦɨɟ ɢ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɹ ɷɤɫɬɪɟɦɭɦɚ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɋɯɟɦɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɧɚ ɷɤɫɬɪɟɦɭɦ
Ɋɚɡɞɟɥ IV ɂɇɌȿȽɊȺɅɖɇɈȿ ɂɋɑɂɋɅȿɇɂȿ ȾɂɎɎȿ ɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə ɂ ɊəȾɕ
Ɍɟɦɚ 10 ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
38.ɉɟɪɜɨɨɛɪɚɡɧɚɹ ɮɭɧɤɰɢɢ ɢ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɋɜɨɣɫɬɜɚ ɧɟɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ
39.ɂɧɬɟɝɪɚɥɵ ɨɬ ɨɫɧɨɜɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ Ɇɟɬɨɞɵ ɧɚɯɨɠɞɟɧɢɹ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ Ɇɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ
40.Ɇɟɬɨɞ ɡɚɦɟɧɵ ɩɟɪɟɦɟɧɧɨɣ Ɇɟɬɨɞ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ ɱɚɫɬɹɦ
Ɍɟɦɚ 11 Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
41. Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɢ ɷɤɨɧɨɦɢɱɟɫɤɢɣ ɫɦɵɫɥ Ⱦɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɢɧɬɟɝɪɢɪɭɟɦɨɫɬɢ
130