Планирование эксперимента. Учебное пособие-1
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ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ
ȼɨɦɧɨɝɢɯɫɥɭɱɚɹɯɩɪɢɩɪɨɜɟɞɟɧɢɢɷɤɫɩɟɪɢɦɟɧɬɚɩɟɪɟɞɢɫɫɥɟɞɨɜɚɬɟɥɟɦ ɜɨɡɧɢɤɚɟɬ ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɬɚɤɢɯ ɡɧɚɱɟɧɢɣ ɮɚɤɬɨɪɨɜ ɩɪɢ ɤɨɬɨɪɵɯ ɨɬɤɥɢɤ ɜɵɯɨɞɧɨɣ ɩɚɪɚɦɟɬɪ ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɷɤɫɬɪɟɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɦɚɤɫɢɦɭɦɚ ɢɥɢ ɦɢɧɢɦɭɦɚ Ɍɚɤɨɣ ɷɤɫɩɟɪɢɦɟɧɬ ɧɚɡɵɜɚɟɬɫɹ ɷɤɫɬɪɟɦɚɥɶɧɵɦ ɢ ɪɟɲɚɟɬɫɹ ɨɩ ɬɢɦɢɡɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ ɤɨɬɨɪɚɹ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ x x xn ɬɨɱɤɢ ɮɚɤɬɨɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɤɨɬɨɪɚɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɷɤɫɬɪɟɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ ɨɬɤɥɢɤɚ y f x x xn
ɉɨɫɬɪɨɟɧɢɟ ɦɨɞɟɥɢ ɧɚɯɨɠɞɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ ɢ ɨɩɪɟɞɟɥɟɧɢɟ
ɨɩɬɢɦɭɦɚ ɨɬɤɥɢɤɚ ɫ ɩɨɦɨɳɶɸ ɦɨɞɟɥɢ ± ɨɞɢɧ ɢɡ ɩɨɞɯɨɞɨɜ ɤ ɪɟɲɟɧɢɸ ɷɬɨɣ ɡɚ
ɞɚɱɢ Ⱦɪɭɝɨɣ ɩɨɞɯɨɞ ± ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɢɬɟɪɚɰɢɨɧɧɵɯ ɩɪɨɰɟɞɭɪ ɤɨɝɞɚ ɧɚ ɤɚɠ
ɞɨɦ ɲɚɝɟ ɞɟɥɚɟɬɫɹ ɪɹɞ ɨɩɵɬɨɜ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɧɭɠɧɨ ɢɡɦɟɧɹɬɶ ɮɚɤɬɨɪɵ
ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɉɪɢ ɷɬɨɦ ɢɧɮɨɪɦɚɰɢɹ ɩɨɥɭ
ɱɚɟɦɚɹ ɧɚ ɨɱɟɪɟɞɧɨɦ ɲɚɝɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɫɥɨɜɢɣ ɩɪɨɜɟɞɟɧɢɹ
ɨɩɵɬɨɜ ɧɚ ɫɥɟɞɭɸɳɟɦ ɲɚɝɟ
Ɍɚɤɚɹ ɩɨɲɚɝɨɜɚɹ ɨɩɬɢɦɢɡɚɰɢɹ ɨɫɧɨɜɚɧɧɚɹ ɧɚ ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɚɯ ɦɚɬɟɦɚ
ɬɢɤɢ ɜɟɫɶɦɚ ɷɮɮɟɤɬɢɜɧɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɦ ɷɤɫɩɟɪɢɦɟɧɬɟ
Ɇɟɬɨɞ ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ
ɉɭɫɬɶɬɪɟɛɭɟɬɫɹɧɚɣɬɢɦɢɧɢɦɭɦɨɬɤɥɢɤɚ y f x x xn ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɬɨɪɨɝɨ ɧɟɢɡɜɟɫɬɧɨ
ȼɵɛɟɪɟɦ ɩɪɨɢɡɜɨɥɶɧɨ ɧɚɱɚɥɶɧɭɸ ɬɨɱɤɭ M ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x x xn ɢ ɡɚɮɢɤɫɢɪɭɟɦ ɡɧɚɱɟɧɢɹ ɜɫɟɯ ɮɚɤɬɨɪɨɜ ɤɪɨɦɟ ɩɟɪɜɨɝɨ Ɍɨɝɞɚ ɨɬɤɥɢɤ y f x x xn ɩɪɟɜɪɚɬɢɬɫɹ ɜ ɮɭɧɤɰɢɸ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ Ȼɭɞɟɦ ɢɡɦɟɧɹɬɶ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɚ x x ɜ ɫɬɨɪɨɧɭ ɭɛɵɜɚɧɢɹ ɨɬɤɥɢɤɚ ɫ ɧɟɤɨɬɨɪɵɦ ɲɚɝɨɦ ɩɨɤɚ
ɧɟ ɞɨɣɞɟɦ ɞɨ ɡɧɚɱɟɧɢɹx |
x ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɦɢɧɢɦɭɦɭ ɨɬɤɥɢɤɚ ɩɨɫɥɟ ɤɨɬɨ |
ɪɨɝɨ ɮɭɧɤɰɢɹ ɧɚɱɢɧɚɟɬ |
ɜɨɡɪɚɫɬɚɬɶ ɉɨɥɭɱɢɦ ɬɨɱɤɭ M ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ |
x x xn ɉɪɢ ɷɬɨɦ f x x xn f x x xn
Ɍɟɩɟɪɶ ɮɢɤɫɢɪɭɟɦ ɮɚɤɬɨɪɵ x x x |
x x |
x xn xn ɢ ɢɡɦɟ |
ɧɹɹ x x ɜ ɫɬɨɪɨɧɭ ɭɛɵɜɚɧɢɹ ɨɬɤɥɢɤɚ y |
f x x xn ɞɜɢɠɟɦɫɹ ɞɨ ɬɟɯ |
|
ɩɨɪ ɩɨɤɚ ɧɟ ɞɨɣɞɟɦ ɞɨ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ ɩɪɢ x |
x ɉɨɥɭɱɢɦ ɬɨɱɤɭ M |
|
ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x x xn ȼɵɩɨɥɧɢɜ ɬɚɤɢɟ ɠɟ ɞɟɣɫɬɜɢɹ ɫɨ ɜɫɟɦɢ ɨɫɬɚɥɶɧɵɦɢ ɮɚɤɬɨɪɚɦɢ ɞɨxn ɜɤɥɸ
ɱɢɬɟɥɶɧɨ ɫɧɨɜɚ ɜɟɪɧɟɦɫɹ ɤ x ɢ ɩɨɜɬɨɪɢɦ ɩɪɨɰɟɫɫ ɉɨɥɭɱɢɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶ ɧɨɫɬɶ ɬɨɱɟɤ M M M M ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɦ ɡɧɚɱɟɧɢɣ ɨɬɤɥɢɤɨɜ ɨɫɬɚɧɚɜɥɢɜɚɟɦ ɩɪɨɰɟɫɫ ɢ ɩɪɢɧɢɦɚɟɦ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɜ ɩɨɫɥɟɞɧɟɣ ɬɨɱɤɟ M k ɡɚ ɦɢɧɢɦɚɥɶɧɨɟ
Ɍɚɤɨɟ ɩɨɨɱɟɪɟɞɧɨɟ ɞɜɢɠɟɧɢɟ ɩɨ ɤɚɠɞɨɣ ɤɨɨɪɞɢɧɚɬɟ ɧɚɡɵɜɚɸɬ ɟɳɟ ɦɟɬɨ
ɞɨɦ Ƚɚɭɫɫɚ Ɂɟɣɞɟɥɹ
ȿɫɥɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɬɤɥɢɤɚ y f x x xn ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚ ɛɵɥɨ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɚɣɞɟɧɨ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ ɬɨ ɜɵɱɢɫɥɹɹ ɱɚɫɬɧɵɟ ɩɪɨ ɢɡɜɨɞɧɵɟ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɢ ɜɟɥɢɱɢɧɭ ɢɡɦɟɧɟɧɢɹ ɨɬɤɥɢɤɚ ɩɨ ɤɚɠɞɨɦɭ ɮɚɤɬɨɪɭ
Ɇɟɬɨɞ ɩɪɨɫɬ ɧɨ ɬɪɟɛɭɟɬ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɨɩɵɬɨɜ ɢ ɷɬɨ ɱɢɫɥɨ ɪɚɫɬɟɬ ɫ ɪɨ
ɫɬɨɦ ɱɢɫɥɚ ɮɚɤɬɨɪɨɜ Ʉɪɨɦɟ ɬɨɝɨ ɜ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ ɤɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɮɚɤɬɨɪɚ
ɜ ɥɸɛɭɸ ɫɬɨɪɨɧɭ ɜɵɡɵɜɚɟɬ ɭɦɟɧɶɲɟɧɢɟ ɨɬɤɥɢɤɚ ɦɟɬɨɞ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɤ ɧɟ
ɩɪɚɜɢɥɶɧɨɦɭ ɪɟɡɭɥɶɬɚɬɭ
ɉɪɨɰɟɫɫ ɩɨɢɫɤɚ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ ɦɟɬɨɞɨɦ ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚ
ɰɢɢ ɜ ɝɪɚɮɢɱɟɫɤɨɦ ɜɢɞɟ ɞɥɹ ɞɜɭɦɟɪɧɨɝɨ ɫɥɭɱɚɹ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ
Ɋɢɫɉɨɢɫɤ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ ɦɟɬɨɞɨɦ ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɝɪɚɮɢɱɟɫɤɨɦ ɜɢɞɟ ɞɥɹ ɞɜɭɦɟɪɧɨɝɨ ɫɥɭɱɚɹ
ɉɨ ɷɬɨɦɭ ɦɟɬɨɞɭ ɜɵɛɢɪɚɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɚɹ ɬɨɱɤɚ M ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɟɟ ɤɨɨɪɞɢɧɚɬɵ ɉɨɢɫɤ ɨɩɬɢɦɭɦɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨɨɱɟɪɟɞɧɵɦ ɜɚɪɶɢɪɨɜɚɧɢɟɦ ɤɚɠɞɨɝɨ ɢɯ ɮɚɤɬɨɪɨɜ ɉɪɢ ɷɬɨɦ ɫɧɚɱɚɥɚ ɢɡɦɟɧɹɸɬ ɨɞɢɧ ɮɚɤɬɨɪ x ɩɪɢ ɮɢɤɫɢ
ɪɨɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɮɚɤɬɨɪɚ x ɞɨ ɬɟɯ ɩɨɪ ɩɨɤɚ ɧɟ ɩɪɟɤɪɚɳɚɟɬɫɹ ɩɪɢɪɨɫɬ ɮɭɧɤɰɢɢɨɬɤɥɢɤɚɬɨɱɤɚ M ȼɞɚɥɶɧɟɣɲɟɦɢɡɦɟɧɹɟɬɫɹɞɪɭɝɨɣɮɚɤɬɨɪ x ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɩɟɪɜɨɝɨ ɮɚɤɬɨɪɚ x ɢ ɞɚɥɟɟ ɩɪɨɰɟɞɭɪɚ ɩɨɜɬɨɪɹ ɟɬɫɹ ɉɪɨɰɟɫɫ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɤɨɝɞɚ ɥɸɛɨɟ ɢɡɦɟɧɟɧɢɟ ɮɚɤɬɨɪɚ ɜ ɬɭ ɢɥɢ ɞɪɭ ɝɭɸ ɫɬɨɪɨɧɭ ɧɟ ɩɪɢɜɨɞɢɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɮɭɧɤɰɢɢ ɢɥɢ ɤɨɝɞɚ ɷɬɢ ɢɡɦɟɧɟɧɢɹ ɫɬɚ ɧɨɜɹɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɵɦɢ
ɉɪɢ ɢɡɦɟɧɟɧɢɢ ɮɚɤɬɨɪɨɜ ɩɨ ɦɟɬɨɞɭ ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɧɟɨɛɯɨ
ɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɞɢɚɩɚɡɨɧɵ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ ȼ ɦɟɬɨɞɟ
ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɚɤ ɧɚɬɭɪɚɥɶɧɵɟ ɬɚɤ ɢ ɛɟɡ
ɪɚɡɦɟɪɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɨɜ
Ɇɟɬɨɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɹ
ȼ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜɵɲɟ ɩɥɚɧɚɯ ɉɎɗ n ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɬɨɱɤɢ ɜ ɡɚɜɢ ɫɢɦɨɫɬɢ ɨɬ ɱɢɫɥɚ ɮɚɤɬɨɪɨɜ ɪɚɫɩɨɥɚɝɚɥɢɫɶ ɜ ɜɟɪɲɢɧɚɯ ɤɜɚɞɪɚɬɚ ɤɭɛɚ ɝɢɩɟɪɤɭ ɛɚ ȼ ɩɥɚɧɚɯ ɦɨɠɧɨ ɬɚɤɠɟ ɪɚɫɩɨɥɚɝɚɬɶ ɬɨɱɤɢ ɜ ɜɟɪɲɢɧɚɯ ɫɢɦɩɥɟɤɫɚ ɋɢɦɩɥɟɤ ɫɨɦ ɜ n ɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɧɚɡɵɜɚɟɬɫɹ ɜɵɩɭɤɥɵɣ ɦɧɨɝɨɝɪɚɧɧɢɤ ɢɦɟɸɳɢɣ n ɜɟɪɲɢɧɭ ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɟɪɟɫɟɱɟɧɢɟɦn ɝɢɩɟɪɩɥɨɫɤɨ ɫɬɟɣ ɞɚɧɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɋɢɦɩɥɟɤɫ ɧɚɡɵɜɚɟɬɫɹ ɩɪɚɜɢɥɶɧɵɦ ɢɥɢ ɪɟɝɭɥɹɪɧɵɦ ɟɫɥɢ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɜɫɟɦɢ ɟɝɨ ɜɟɪɲɢɧɚɦɢ ɪɚɜɧɵ ɉɪɢɦɟɪɚɦɢ ɪɟɝɭɥɹɪɧɵɯ ɫɢɦɩɥɟɤɫɨɜ ɹɜɥɹɸɬɫɹ ɩɪɚɜɢɥɶɧɵɣ ɬɪɟɭɝɨɥɶɧɢɤ ɜ ɞɜɭɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢ ɬɟɬɪɚɷɞɪ ɜ ɬɪɟɯɦɟɪɧɨɦ
ɉɥɚɧɢɪɨɜɚɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɟɝɭɥɹɪɧɵɯ ɫɢɦɩɥɟɤɫɨɜ ɩɨɡɜɨɥɹɟɬ ɩɨɫɬɪɨɢɬɶ ɩɥɚɧ ɫɨɞɟɪɠɚɳɢɣ ɦɢɧɢɦɚɥɶɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɨɩɵɬɨɜ ɞɨ ɫɬɚɬɨɱɧɨɟ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ
ȼɵɛɨɪ ɪɚɡɦɟɪɨɜ ɫɢɦɩɥɟɤɫɚ ɢ ɟɝɨ ɧɚɱɚɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɢɡɜɟɫɬɧɨɣ ɫɬɟ
ɩɟɧɢ ɩɪɨɢɡɜɨɥɟɧ ɋɢɦɩɥɟɤɫ ɜ ɮɚɤɬɨɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɭɞɨɛɧɨ ɪɚɫɩɨɥɚɝɚɬɶ ɫɥɟ
ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ɐɟɧɬɪ ɫɢɦɩɥɟɤɫɚ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ ɨɞɧɚ ɢɡ ɜɟɪɲɢɧ ɥɟɠɢɬ ɧɚ ɤɨɨɪɞɢɧɚɬɧɨɣ ɨɫɢ ɚ ɨɫɬɚɥɶɧɵɟ ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶ ɧɨ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɩɥɨɫɤɨɫɬɟɣ ɢ ɝɢɩɟɪɩɥɨɫɤɨɫɬɟɣ ɜ ɦɧɨɝɨɦɟɪɧɨɦ ɫɥɭ ɱɚɟ ɉɪɢ n ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ ɫɢɦɩɥɟɤɫɚ ɦɨɝɭɬ ɡɚɞɚɜɚɬɶɫɹ ɦɚɬɪɢ ɰɟɣ
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ɑɢɫɥɨ ɫɬɨɥɛɰɨɜ ɦɚɬɪɢɰɵ ɪɚɜɧɨ ɱɢɫɥɭ ɮɚɤɬɨɪɨɜ ɱɢɫɥɨ ɫɬɪɨɤ ɧɚ ɟɞɢɧɢɰɭ
ɛɨɥɶɲɟ ɱɢɫɥɚ ɫɬɨɥɛɰɨɜ ɋ ɞɨɛɚɜɥɟɧɢɟɦ ɧɨɜɨɝɨ ɮɚɤɬɨɪɚ ɱɢɫɥɨ ɫɬɨɥɛɰɨɜ ɢ ɫɬɪɨɤ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɧɚ ɟɞɢɧɢɰɭ ɇɚɩɪɢɦɟɪ ɩɪɢn ɩɨɥɭɱɢɦ ɦɚɬɪɢɰɭ
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ȼ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɢ ɤɚɤ ɩɪɚɜɢɥɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɛɟɡɪɚɡɦɟɪɧɵɟ ɮɚɤ
ɬɨɪɵ ɚ ɢɯ ɤɨɧɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫ ɭɱɟɬɨɦ ɜɵɛɪɚɧɧɵɯ ɞɢɚɩɚɡɨɧɨɜ
ɜɚɪɶɢɪɨɜɚɧɢɹ Ɉɛɵɱɧɨ ɷɬɢ ɡɧɚɱɟɧɢɹ ɡɚɪɚɧɟɟ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɧɨɦɟɪɚ ɮɚɤɬɨɪɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
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ɝɞɟi ɧɨɦɟɪ ɮɚɤɬɨɪɚ |
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Ɍɨɝɞɚ ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ |
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ɩɪɢ ɦɢɧɢɦɚɥɶɧɨɦ ɱɢɫɥɟ ɨɩɵɬɨɜN ɉɪɢ ɬɚɤɨɦ ɜɵɛɨɪɟ ɱɢɫɥɨɜɵɯ ɡɧɚɱɟɧɢɣ ɮɚɤ
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ɬɨɪɨɜ ¦xiu i |
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Ɍɚɤɨɣ ɫɢɦɩɥɟɤɫ ɩɥɚɧ ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɷɤɨɧɨɦɧɵɦ ɩɨ ɱɢɫɥɭ ɨɩɵɬɨɜ ɞɥɹ
ɩɨɫɬɪɨɟɧɢɹ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ
ɇɚ ɪɢɫ ɩɨɤɚɡɚɧɵ ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ ɫɢɦɩɥɟɤɫɚ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ
ɢɦ ɧɨɦɟɪɚ ɨɩɵɬɨɜ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɫɢɦɩɥɟɤɫ ɩɥɚɧɚ ɞɥɹ ɞɜɭɯ ɮɚɤɬɨɪɨɜ ɐɟɧɬɪ
ɫɢɦɩɥɟɤɫɚ ɪɚɡɦɟɳɚɟɬɫɹ ɜ ɰɟɧɬɪɟ ɤɨɨɪɞɢɧɚɬ ɩɟɪɜɚɹ ɢ ɜɬɨɪɚɹ ɜɟɪɲɢɧɚ ɪɚɫɩɨɥɚ ɝɚɸɬɫɹ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɨɪɞɢɧɚɬɧɨɣ ɨɫɢx ɚ ɬɪɟɬɶɹ ɜɟɪɲɢɧɚ ɧɚ ɨɫɢx
Ɋɢɫɋɢɦɩɥɟɤɫɧɵɣ ɩɥɚɧ ɩɪɢn
ɋɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɟ ɦɨɠɟɬ ɩɪɢɦɟɧɹɬɶɫɹ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚ
ɰɢɢ ɢ ɩɨɡɜɨɥɹɟɬ ɩɪɢ ɩɪɟɞɟɥɶɧɨ ɦɚɥɨɦ ɤɨɥɢɱɟɫɬɜɟ ɨɩɵɬɨɜ ɜɵɣɬɢ ɧɚ ɨɛɥɚɫɬɶ
ɦɚɤɫɢɦɭɦɚ ɢɥɢ ɦɢɧɢɦɭɦɚ ɨɬɤɥɢɤɚ ɛɟɡ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɩɨɫɬɪɨɟɧɢɹ ɮɭɧɤɰɢɢ
Ⱥɥɝɨɪɢɬɦ ɩɨɢɫɤɚ ɨɛɥɚɫɬɢ ɨɩɬɢɦɭɦɚ ɜɤɥɸɱɚɟɬ ɫɥɟɞɭɸɳɢɟ ɲɚɝɢ
ɋɬɪɨɢɬɫɹ ɢɫɯɨɞɧɵɣ ɫɢɦɩɥɟɤɫɉɪɨɜɨɞɹɬɫɹ ɨɩɵɬɵ ɜ ɬɨɱɤɚɯ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɪɲɢɧɚɦ ɢɫɯɨɞɧɨɝɨ
ɫɢɦɩɥɟɤɫɚ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɪɲɢɧɚ ɜ ɤɨɬɨɪɨɣ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɹɜɥɹ
ɟɬɫɹ ɧɚɢɯɭɞɲɢɦ ɜ ɫɦɵɫɥɟ ɨɩɬɢɦɭɦɚɋɬɪɨɢɬɫɹ ɧɨɜɵɣ ɫɢɦɩɥɟɤɫ ɇɨɜɵɣ ɫɢɦɩɥɟɤɫ ɨɛɪɚɡɭɟɬɫɹ ɢɡ ɜɟɪɲɢɧ ɢɫ
ɯɨɞɧɨɝɨ ɫɢɦɩɥɟɤɫɚ ɧɨ ɧɚɢɯɭɞɲɚɹ ɬɨɱɤɚ ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɧɨɜɭɸ ɇɨɜɚɹ ɬɨɱɤɚɜɵɛɢɪɚɟɬɫɹɫɢɦɦɟɬɪɢɱɧɨɨɬɧɨɫɢɬɟɥɶɧɨɝɪɚɧɢɫɢɦɩɥɟɤɫɚɧɚɯɨ ɞɹɳɟɣɫɹ ɩɪɨɬɢɜ ɧɚɢɯɭɞɲɟɣ ɬɨɱɤɢ
ȼ ɧɨɜɨɣ ɬɨɱɤɟ ɜɵɩɨɥɧɹɟɬɫɹ ɨɩɵɬ ɢ ɨɩɹɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɢɯɭɞɲɚɹ ɬɨɱ ɤɚ ɫɢɦɩɥɟɤɫɚ ɤɨɬɨɪɚɹ ɡɚɦɟɧɹɟɬɫɹ ɧɚ ɧɨɜɭɸ ɢ ɨɛɪɚɡɭɟɬɫɹ ɧɨɜɵɣ ɫɢɦ ɩɥɟɤɫ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɢɫɯɨɞɧɵɣ ɫɢɦɩɥɟɤɫ ɩɨɫɬɟɩɟɧɧɨ ɫɦɟɳɚɟɬɫɹ ɜ ɨɛ ɥɚɫɬɶ ɨɩɬɢɦɭɦɚ
Ʉɪɢɬɟɪɢɹɦɢ ɨɤɨɧɱɚɧɢɹ ɩɪɨɰɟɫɫɚ ɦɨɝɭɬ ɹɜɥɹɬɶɫɹ ɫɥɟɞɭɸɳɢɟ ɫɢɬɭɚɰɢɢɊɚɡɧɨɫɬɶ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɜ ɜɟɪɲɢɧɚɯ ɫɢɦɩɥɟɤɫɚ ɫɬɚɧɨɜɢɬɫɹ
ɧɟɡɧɚɱɢɬɟɥɶɧɨɣɗɬɨɨɡɧɚɱɚɟɬɜɵɯɨɞɜɩɨɱɬɢɫɬɚɰɢɨɧɚɪɧɭɸɨɛɥɚɫɬɶ ɛɥɢɡɤɭɸ ɤ ɨɩɬɢɦɭɦɭ ɥɢɛɨ ɞɨɫɬɢɠɟɧɢɟ ɭɱɚɫɬɤɚ ɜ ɜɢɞɟ ©ɩɥɚɬɨª ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟɨɛɯɨɞɢɦɵ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɨɩɵɬɵ ɜ ɫɬɨɪɨɧɟ ɨɬ ɫɢɦɩɥɟɤɫɚ ɩɨɞɬɜɟɪɠɞɚɸɳɢɟ ɜɵɯɨɞ ɜ ɨɛɥɚɫɬɶ ɨɩɬɢɦɭɦɚ
Ɋɟɡɭɥɶɬɚɬ ɨɩɵɬɚ ɜ ɧɨɜɨɣ ɜɟɪɲɢɧɟ ɫɢɦɩɥɟɤɫɚ ɬɪɟɛɭɟɬ ɜɨɡɜɪɚɬɚ ɜ ɩɪɟɠ ɧɟɟ ɩɨɥɨɠɟɧɢɟ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɨɱɤɚ ɨɩɬɢɦɭɦɚ ɧɚɯɨɞɢɬɫɹ ɜɧɭɬɪɢ ɫɢɦ ɩɥɟɤɫɚ
ɇɨɜɵɟ ɬɨɱɤɢ ɫɢɦɩɥɟɤɫɚ ɧɚ ɩɪɨɬɹɠɟɧɢɢ ɧɟɫɤɨɥɶɤɢɯ ɲɚɝɨɜ ɩɪɢɜɨɞɹɬ ɤ ɰɢɤɥɢɱɟɫɤɨɦɭ ɜɪɚɳɟɧɢɸ ɫɢɦɩɥɟɤɫɚ ɜɨɤɪɭɝ ɨɞɧɨɣ ɢɡ ɟɝɨ ɜɟɪɲɢɧ ɫ ɧɚɢɥɭɱɲɢɦ ɡɧɚɱɟɧɢɟɦ ɨɬɤɥɢɤɚ
ɇɚ ɪɢɫ ɩɪɟɞɫɬɚɜɥɟɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɜɵɩɨɥɧɟɧɢɹ ɨɩɵɬɨɜ ɜ ɩɥɚɧɟ ɞɥɹ ɞɜɭɯ ɮɚɤɬɨɪɨɜ ɩɪɢ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɢ ɂɫɯɨɞɧɵɣ ɫɢɦɩɥɟɤɫ ± ɬɪɟ ɭɝɨɥɶɧɢɤ ɬɪɢ ɜɟɪɲɢɧɵ ɤɨɬɨɪɨɝɨ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɨɦɟɪɚɦ ɨɩɵɬɨɜ ɚ ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ ɧɚɩɪɢɦɟɪ ± ± ± ɡɧɚɱɟɧɢɹɦ ɮɚɤ ɬɨɪɨɜ ɜ ɩɟɪɜɵɯ ɬɪɟɯ ɨɩɵɬɚɯ
Ɋɢɫɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ɤ ɨɛɥɚɫɬɢ ɨɩɬɢɦɭɦɚ ɩɪɢ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɢ
ɉɭɫɬɶ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɜ ɬɨɱɤɟ ɨɤɚɡɚɥɨɫɶ ɧɚɢɯɭɞɲɢɦ ɜ ɫɦɵɫɥɟ ɨɩɬɢɦɭ
ɦɚ Ɍɨɝɞɚ ɬɨɱɤɚ ɨɬɨɛɪɚɠɚɟɬɫɹ ɡɟɪɤɚɥɶɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɝɪɚ
ɧɢ ɢɫɯɨɞɧɨɝɨ ɫɢɦɩɥɟɤɫɚ ɢ ɩɨɥɭɱɚɟɦ ɬɨɱɤɭ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ ɫɨɨɬɜɟɬ
ɫɬɜɭɸɳɭɸ ɱɟɬɜɟɪɬɨɦɭ ɨɩɵɬɭ ɢ ɭɫɥɨɜɢɹɦ ɟɝɨ ɩɪɨɜɟɞɟɧɢɹ
Ɍɟɩɟɪɶ ɟɫɥɢ ɢɡ ɬɪɟɯ ɜɟɪɲɢɧ ɬɨɱɤɢ ɧɚ ɪɢɫ ɧɨɜɨɝɨ ɫɢɦɩɥɟɤɫɚ
ɧɚɢɯɭɞɲɟɣ ɨɤɚɡɚɥɚɫɶ ɜɟɪɲɢɧɚ ɬɨ ɨɧɚ ɨɬɨɛɪɚɠɚɟɬɫɹ ɡɟɪɤɚɥɶɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɜɟɪɲɢɧɵ ɫɢɦɩɥɟɤɫɚ ɢ ɩɨɥɭɱɚɟɦ ɧɨɜɭɸ ɬɨɱɤɭ ɜ ɤɨɬɨɪɨɣ ɜɵ
ɩɨɥɧɹɟɬɫɹ ɨɩɵɬ Ʉɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɨɩɪɟɞɟɥɹɸɬ ɭɫɥɨɜɢɹ ɩɪɨɜɟɞɟɧɢɹ ɩɹɬɨɝɨ ɨɩɵɬɚ x x ɉɪɨɰɟɫɫ ɩɨɜɬɨɪɹɟɬɫɹ ɢ ɧɚɯɨɞɢɬɫɹ ɫɥɟɞɭɸɳɚɹ ɬɨɱ ɤɚ
Ɉɞɧɚɤɨ ɬɨɱɤɢ ɚ ɬɟɦ ɛɨɥɟɟ ɬɨɱɤɚ ɧɚ ɪɢɫ ɭɠɟ ɜɵɯɨɞɹɬ ɡɚ ɩɪɟɞɟɥɵ ɨɛɥɚɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ȼ ɫɥɭɱɚɟ ɟɫɥɢ ɧɨɜɚɹ ɜɟɪɲɢɧɚ ɫɢɦɩɥɟɤɫɚ ɜɵɯɨɞɢɬ ɡɚ ɩɪɟɞɟɥɵ ɨɛɥɚɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɬɨ ɧɟɨɛɯɨɞɢɦɨ ɜɟɪɧɭɬɶɫɹ ɤ ɢɫ ɯɨɞɧɨɦɭ ɫɢɦɩɥɟɤɫɭ ɢ ɜɵɛɪɚɬɶ ɞɥɹ ɨɬɨɛɪɚɠɟɧɢɹ ɞɪɭɝɭɸ ɬɨɱɤɭ ± ɩɪɟɞɵɞɭɳɭɸ ɧɚɢɯɭɞɲɟɣ ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɩɪɢ ɩɟɪɜɨɦ ɠɟ ɨɬɨɛɪɚɠɟɧɢɢ ɧɨɜɚɹ ɬɨɱɤɚ ɜɵɯɨɞɢɬ ɡɚɩɪɟɞɟɥɵɨɛɥɚɫɬɢɩɥɚɧɢɪɨɜɚɧɢɹɉɨɷɬɨɦɭɧɟɨɛɯɨɞɢɦɨɭɦɟɧɶɲɢɬɶɪɚɡɦɟɪɵ ɢɫɯɨɞɧɨɝɨ ɫɢɦɩɥɟɤɫɚ ɜ ɧɟɫɤɨɥɶɤɨ ɪɚɡ ɧɚɩɪɢɦɟɪ ɜ ɪɚɡɚ ɫɨɯɪɚɧɹɹ ɰɟɧɬɪ ɫɢɦ ɩɥɟɤɫɚ ɜ ɰɟɧɬɪɟ ɩɥɚɧɚ ɢ ɬɨɥɶɤɨ ɩɨɫɥɟ ɷɬɨɝɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɨɞɜɢɝɚɬɶɫɹ ɤ ɨɛɥɚɫɬɢ ɨɩɬɢɦɭɦɚ
ȿɫɥɢ ɧɨɜɚɹ ɜɟɪɲɢɧɚ ɫɢɦɩɥɟɤɫɚ ɞɚɟɬ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɯɭɠɟ ɱɟɦ ɜ ɜɟɪɲɢ
ɧɚɯ ɩɪɟɞɵɞɭɳɟɝɨ ɫɢɦɩɥɟɤɫɚ ɬɨ ɬɚɤɠɟ ɞɟɥɚɟɬɫɹ ɜɨɡɜɪɚɬ ɤ ɩɪɟɞɵɞɭɳɟɦɭ ɫɢɦ ɩɥɟɤɫɭɢɜɵɛɢɪɚɸɬɜɧɟɦɬɨɱɤɭɛɥɢɠɚɣɲɭɸɤɧɚɢɯɭɞɲɟɣɞɥɹɨɬɨɛɪɚɠɟɧɢɹ ɜ ɧɨɜɭɸ ɜɟɪɲɢɧɭ
ɉɨɫɥɟ ɜɵɩɨɥɧɟɧɢɹ ɩɟɪɜɵɯ ɬɪɟɯ ɨɩɵɬɨɜ ɜ ɬɨɱɤɚɯ ɢɫɯɨɞɧɨɝɨ ɫɢɦɩɥɟɤɫɚ ɪɢɫ ɦɨɠɧɨ ɭɦɟɧɶɲɢɬɶ ɪɚɡɦɟɪ ɫɢɦɩɥɟɤɫɚ ɜ Dɪɚɡ D ɫɬɹɝɢɜɚɹ ɟɝɨ ɜ ɫɬɨɪɨɧɭ ɧɚɢɥɭɱɲɟɣ ɜɟɪɲɢɧɵ ɍɦɟɧɶɲɟɧɧɵɣ ɫɢɦɩɥɟɤɫ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɧɚɢɥɭɱɲɭɸ ɬɨɱɤɭ ɞɜɟ ɞɪɭɝɢɟ ɬɨɱɤɢ ɛɭɞɭɬ ɩɨɞɬɹɝɢɜɚɬɶɫɹ ɤ ɧɚɢɥɭɱɲɟɣ ɩɪɢ ɷɬɨɦ ɰɟɧɬɪ ɫɢɦɩɥɟɤɫɚ ɛɭɞɟɬ ɬɚɤɠɟ ɫɦɟɳɚɬɶɫɹ ɜ ɫɬɨɪɨɧɭ ɧɚɢɥɭɱɲɟɣ ɬɨɱɤɢ Ʉɨ ɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ ɢɡɦɟɧɹɟɦɵɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɬɹɝɢɜɚɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɥɟɞɭ ɸɳɢɦ ɨɛɪɚɡɨɦ
xiu D xis Dxiu
ɝɞɟxis i ɹ ɤɨɨɪɞɢɧɚɬɚ ɧɚɢɥɭɱɲɟɣ ɜɟɪɲɢɧɵ
xiu i ɹ ɤɨɨɪɞɢɧɚɬɚ ɢɡɦɟɧɹɟɦɨɣ ɜɟɪɲɢɧɵ ɫ ɧɨɦɟɪɨɦu
ɇɚɩɪɢɦɟɪɧɚɪɢɫ ɩɨɤɚɡɚɧɨ ɫɬɹɝɢɜɚɧɢɟɢɫɯɨɞɧɨɝɨɫɢɦɩɥɟɤɫɚ ɩɨɫɥɟ ɩɪɨɜɟɞɟɧɢɹ ɩɟɪɜɨɣ ɫɟɪɢɢ ɨɩɵɬɨɜ ɤ ɜɟɪɲɢɧɟ ɢ ɭɦɟɧɶɲɟɧɢɟ ɪɚɡɦɟɪɨɜ ɫɢɦɩɥɟɤ ɫɚ ɜ ɪɚɡɚD
Ɋɢɫɋɬɹɝɢɜɚɧɢɟ ɫɢɦɩɥɟɤɫɚ ɤ ɩɟɪɜɨɣ ɜɟɪɲɢɧɟ
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Ʉɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧɵ x |
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Ʉɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧɵ |
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Ɋɚɡɦɟɪɵ ɫɢɦɩɥɟɤɫɚ ɢɧɨɝɞɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɭɦɟɧɶɲɚɬɶ ɤɚɠɞɵɣ ɪɚɡ ɫɬɹɝɢɜɚɹ ɯɭɞɲɢɟ ɜɟɪɲɢɧɵ ɤ ɧɚɢɥɭɱɲɟɣ ɩɪɨɢɡɜɨɞɢɬɶ ɢɡɦɟɪɟɧɢɹ ɜ ɧɨɜɵɯ ɜɟɪɲɢɧɚɯ ɜɟɪɲɢɧɵ cɢ cɧɚ ɪɢɫ ɢ ɬɨɥɶɤɨ ɩɨɫɥɟ ɷɬɨɝɨ ɜɵɛɢɪɚɬɶ ɯɭɞɲɭɸ ɬɨɱɤɭ ɢɡ ɬɪɟɯ ɬɨɱɤɢ c cɢ ɨɬɨɛɪɚɠɚɬɶ ɟɟ ɜ ɧɨɜɭɸ ɉɪɢ ɷɬɨɦ ɦɨɠɧɨ ɤɚɠɞɵɣ ɪɚɡ ɭɜɟɥɢɱɢɜɚɬɶ ɜɟɥɢɱɢɧɭ D ɩɪɢɛɥɢɠɚɹ ɟɟ ɤ ɟɞɢɧɢɰɟ ɢ ɬɟɦ ɫɚɦɵɦ ɩɨɫɬɟ ɩɟɧɧɨ ɡɚɦɟɞɥɹɬɶ ɭɦɟɧɶɲɟɧɢɟ ɫɢɦɩɥɟɤɫɚ ɩɨ ɦɟɪɟ ɩɪɢɛɥɢɠɟɧɢɹ ɤ ɨɩɬɢɦɭɦɭ ɪɢɫ
Ɋɢɫɉɨɫɬɟɩɟɧɧɨɟ ɭɦɟɧɶɲɟɧɢɟ ɪɚɡɦɟɪɨɜ ɫɢɦɩɥɟɤɫɚ ɫ ɩɨɦɨɳɶɸ ɫɬɹɝɢɜɚɧɢɹ ɟɝɨ ɜ ɫɬɨɪɨɧɭ ɧɚɢɥɭɱɲɟɣ ɬɨɱɤɢ
ɂɡɜɟɫɬɧɨ ɦɧɨɠɟɫɬɜɨ ɦɨɞɢɮɢɤɚɰɢɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨ
ɜɚɧɢɹ ɫ ɪɚɡɥɢɱɧɵɦɢ ɜɚɪɢɚɧɬɚɦɢ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ ɫɢɦɩɥɟɤɫɚ Ɍɚɤɢɟ ɦɟɬɨɞɵ ɩɨ ɢɫɤɚ ɨɩɬɢɦɭɦɚ ɨɛɥɚɞɚɸɬ ɩɨɜɵɲɟɧɧɨɣ ɝɢɛɤɨɫɬɶɸ ɩɨɡɜɨɥɹɸɳɟɣ ɭɱɢɬɵɜɚɬɶ ɨɫɨɛɟɧɧɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɢ ɨɬɤɥɢɤɚ
ɉɪɢ ɞɜɢɠɟɧɢɢ ɤ ɨɛɥɚɫɬɢ ɨɩɬɢɦɭɦɚ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢ ɧɟɪɟɝɭɥɹɪɧɵɟ ɫɢɦɩɥɟɤɫɵɦɨɠɧɨɭɱɢɬɵɜɚɬɶɡɧɚɱɟɧɢɹɨɬɤɥɢɤɚɜɨɜɫɟɯɬɨɱɤɚɯɩɪɟɞɵɞɭɳɟɝɨ ɫɢɦɩɥɟɤɫɚ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɧɨɜɨɣ ɬɨɱɤɢ ȼɨɡɦɨɠɧɨ ɨɬɪɚɠɟɧɢɟ ɧɚɢɯɭɞɲɟɣ ɬɨɱ ɤɢ ɫ ɧɟɤɨɬɨɪɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɨɬɪɚɠɟɧɢɹ ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɤɨɨɪɞɢɧɚɬ ɧɨɜɨɣ ɬɨɱɤɢ ɦɨɠɧɨ ɭɱɢɬɵɜɚɬɶ ɡɧɚɱɟɧɢɹ ɨɬɤɥɢɤɚ ɜ ɞɪɭɝɢɯ ɬɨɱɤɚɯ ɩɥɚɧɚ
ɉɨɥɟɡɧɵɦ ɜɚɪɢɚɧɬɨɦ ɫɢɦɩɥɟɤɫ ɩɥɚɧɢɪɨɜɚɧɢɹ ɭɫɤɨɪɹɸɳɢɦ ɞɜɢɠɟɧɢɟ ɤ ɨɩɬɢɦɭɦɭ ɦɨɠɟɬ ɫɬɚɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ ɩɪɟɞɵɞɭɳɟ ɝɨ ɫɢɦɩɥɟɤɫɚ ɤɨɨɪɞɢɧɚɬ ɧɟ ɨɞɧɨɣ ɚ ɧɟɫɤɨɥɶɤɢɯ ɧɨɜɵɯ ɬɨɱɟɤ ɇɚɩɪɢɦɟɪ ɞɥɹɞɜɭɯɮɚɤɬɨɪɧɨɝɨɷɤɫɩɟɪɢɦɟɧɬɚɢɫɯɨɞɧɵɣɫɢɦɩɥɟɤɫɞɚɥɯɭɞɲɭɸɬɨɱɤɭ ɬɨɱɤɚ ɧɟɫɤɨɥɶɤɨ ɥɭɱɲɟ Ɍɨɝɞɚ ɧɨɜɵɟ ɬɨɱɤɢ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɧɚ ɩɪɨɞɨɥɠɟ ɧɢɢ ɫɬɨɪɨɧ ɢ ɩɪɢɱɟɦ ɜɟɥɢɱɢɧɚ ɷɬɢɯ ɩɪɨɞɨɥɠɟɧɢɣ ɬɟɦ ɛɨɥɶɲɟ ɱɟɦ ɯɭɠɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɥɨɯɚɹ ɬɨɱɤɚ ɪɢɫ
ɊɢɫɈɩɪɟɞɟɥɟɧɢɟ ɧɟ ɨɞɧɨɣ ɚ ɞɜɭɯ ɧɨɜɵɯ ɬɨɱɟɤ ɞɥɹ ɞɜɭɯɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ
Ɇɟɬɨɞ ɫɢɦɩɥɟɤɫɧɨɝɨ ɩɥɚɧɢɪɨɜɚɧɢɹ ɩɨɡɜɨɥɹɟɬ ɛɟɡ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɩɨ ɫɬɪɨɟɧɢɹ ɦɨɞɟɥɢ ɢ ɢɡɭɱɟɧɢɹ ɜɥɢɹɧɢɹ ɮɚɤɬɨɪɨɜ ɧɚɣɬɢ ɨɛɥɚɫɬɶ ɨɩɬɢɦɭɦɚ ȼ ɷɬɨɦ ɦɟɬɨɞɟ ɧɟ ɬɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɟɧɢɹ ɝɪɚɞɢɟɧɬɚ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɩɨɷɬɨɦɭ ɨɧ ɤɚɤ ɢ ɦɟɬɨɞ ɩɨɤɨɨɪɞɢɧɚɬɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɨɬɧɨɫɢɬɫɹɤ ɛɟɡɝɪɚɞɢɟɧɬɧɵɦ ɦɟɬɨɞɚɦ ɩɨɢɫɤɚ ɨɩɬɢɦɭɦɚ
Ƚɪɚɞɢɟɧɬɧɵɟ ɦɟɬɨɞɵ ɩɨɢɫɤɚ ɨɩɬɢɦɭɦɚ ɨɬɤɥɢɤɚ
Ƚɪɚɞɢɟɧɬɧɵɣɦɟɬɨɞɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬɞɜɢɠɟɧɢɟɤɨɩɬɢɦɭɦɭɩɨɧɚɢɤɪɚɬ ɱɚɣɲɟɦɭ ɩɭɬɢ ɬɨ ɟɫɬɶ ɷɬɨ ɞɜɢɠɟɧɢɟ ɤ ɷɤɫɬɪɟɦɭɦɭ ɩɨ ɝɪɚɞɢɟɧɬɭ
Ƚɪɚɞɢɟɧɬ ± ɷɬɨ ɜɟɤɬɨɪ ɤɨɬɨɪɵɣ ɧɚɩɪɚɜɥɟɧ ɜ ɫɬɨɪɨɧɭ ɧɚɢɛɵɫɬɪɟɣɲɟɝɨ ɭɜɟ ɥɢɱɟɧɢɹ ɮɭɧɤɰɢɢ y f x x xn
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ɋɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɜɚɪɢɚɧɬɨɜ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɝɪɚɞɢɟɧɬɧɵɯ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢɫɨɜɦɟɫɬɧɨɫɦɟɬɨɞɚɦɢɩɥɚɧɢɪɨɜɚɧɢɹɷɤɫɩɟɪɢɦɟɧɬɚɊɚɫɫɦɨɬɪɢɦ ɦɟɬɨɞȻɨɤɫɚ ɍɢɥɫɨɧɚɦɟɬɨɞɤɪɭɬɨɝɨɜɨɫɯɨɠɞɟɧɢɹɩɨɩɨɜɟɪɯɧɨɫɬɢɨɬɤɥɢɤɚ ɩɨɥɭɱɢɜɲɢɣ ɧɚɢɛɨɥɶɲɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɜ ɩɪɚɤɬɢɤɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫ ɫɥɟɞɨɜɚɧɢɣ ɢ ɨɛɴɟɞɢɧɹɸɳɢɣ ɥɭɱɲɢɟ ɱɟɪɬɵ ɝɪɚɞɢɟɧɬɧɵɯ ɦɟɬɨɞɨɜ ɢ ɦɟɬɨɞɚ Ƚɚɭɫɫɚ Ɂɟɣɞɟɥɹ
Ɇɟɬɨɞ ɜɤɥɸɱɚɟɬ ɫɥɟɞɭɸɳɢɟ ɲɚɝɢȼɵɛɢɪɚɸɬɫɹ ɞɢɚɩɚɡɨɧɵ ɢɡɦɟɧɟɧɢɹ ɮɚɤɬɨɪɨɜ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɛɥɚɫɬɶ
ɩɥɚɧɢɪɨɜɚɧɢɹ