Математика для менеджеров. Часть I. Учебное пособие
.pdfɆɨɫɤɨɜɫɤɢɣ ɝɭɦɚɧɢɬɚɪɧɨ-ɷɤɨɧɨɦɢɱɟɫɤɢɣ ɢɧɫɬɢɬɭɬ
ȼɨɪɨɧɟɠɫɤɢɣ ɮɢɥɢɚɥ
ȿ Ɉ Ɉɤɭɧɟɜɚ C ɂ Ɇɨɢɫɟɟɜ
ɆȺɌȿɆȺɌɂɄȺ
ȾɅə ɆȿɇȿȾɀȿɊɈȼ
ɍɑȿȻɇɈȿ ɉɈɋɈȻɂȿ
ɑȺɋɌɖ
ȼɨɪɨɧɟɠ 15
ɍȾɄ 51 Ȼ Ɉ
Ɉɤɭɧɟɜɚ ȿ Ɉ Ɇɚɬɟɦɚɬɢɤɚ ɞɥɹ ɦɟɧɟɞɠɟɪɨɜ ɭɱɟɛɧɨɟ ɩɨ ɫɨɛɢɟ ɑɚɫɬɶ , ȿ Ɉ Ɉɤɭɧɟɜɚ ɋ ɂ Ɇɨɢɫɟɟɜ. – ȼɨɪɨɧɟɠ ȼɎ ɆȽɗɂ –157 ɫ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɢɡɭɱɟɧɢɹ ɞɢɫɰɢ ɩɥɢɧɵ ©Ɇɚɬɟɦɚɬɢɤɚª ɫɬɭɞɟɧɬɚɦɢ ɧɚɩɪɚɜɥɟɧɢɹ ɩɨɞɝɨɬɨɜɤɢ 080200.62 «Ɇɟɧɟɞɠɦɟɧɬ» ȼ ɩɨɫɨɛɢɢ ɢɡɥɨɠɟɧ ɤɪɚɬɤɢɣ ɬɟɨɪɟɬɢ ɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɩɪɢɜɟɞɟɧɵ ɩɪɢɦɟɪɵ ɡɚɞɚɱ ɜɨɩɪɨɫɵ ɞɥɹ ɩɨɞ ɝɨɬɨɜɤɢ ɤ ɷɤɡɚɦɟɧɭ ɢ ɤɨɧɬɪɨɥɶɧɵɟ ɡɚɞɚɧɢɹ ɜ ɜɚɪɢɚɧɬɚɯ Ɋɚɫ ɫɦɨɬɪɟɧɵ ɫɥɟɞɭɸɳɢɟ ɬɟɦɵ ɦɚɬɪɢɰɵ ɢ ɨɩɪɟɞɟɥɢɬɟɥɢ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɷɥɟɦɟɧɬɵ ɜɟɤɬɨɪɧɨɝɨ ɚɧɚɥɢɡɚ ɷɥɟɦɟɧɬɵ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɝɟɨɦɟɬɪɢɢ ɨɫɧɨɜɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɩɪɨɢɡɜɨɞɧɚɹ ɢ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɨɩɪɟɞɟɥɟɧɧɵɣ ɢ ɧɟɨɩɪɟɞɟ ɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɪɹɞɵ Ɇɚɬɟɪɢ ɚɥɵ ɩɨɫɨɛɢɹ ɪɟɤɨɦɟɧɞɨɜɚɧɵ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɜɫɟɯ ɮɨɪɦ ɨɛɭɱɟɧɢɹ ɚ ɬɚɤɠɟ ɞɥɹ ɩɪɨɜɟɞɟɧɢɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɧɹɬɢɣ ɢ ɤɨɧɬɪɨɥɶɧɵɯ ɮɨɪɦ ɩɪɨɜɟɪɤɢ ɡɧɚɧɢɣ ɫɬɭɞɟɧɬɨɜ
ɉɟɱɚɬɚɟɬɫɹ ɩɨ ɪɟɲɟɧɢɸ ɭɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɣ ɤɨɦɢɫɫɢɢ ɮɚɤɭɥɶɬɟɬɚ ɷɤɨɧɨɦɢɤɢ ɢ ɭɩɪɚɜɥɟɧɢɹ ȼɨɪɨɧɟɠɫɤɨɝɨ ɮɢɥɢɚɥɚ Ɇɨɫɤɨɜɫɤɨɝɨ ɝɭɦɚɧɢɬɚɪɧɨ-ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɢɧɫɬɢɬɭɬɚ
Ɋɟɰɟɧɡɟɧɬɵ
Ɂɨɧ Ȼ Ⱥ ɞ ɮ -ɦ ɧ ɩɪɨɮ ɡɚɜ ɤɚɮɟɞɪɨɣ ɦɚɬɟɦɚɬɢɱɟ ɫɤɨɣ ɮɢɡɢɤɢ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢ ɬɟɬɚ Ƚɥɭɯɨɜ Ⱦ Ⱥ. ɤ ɬ ɧ ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɩɪɢɤɥɚɞɧɨɣ ɢɧɮɨɪ
ɦɚɬɢɤɢ ɂɧɫɬɢɬɭɬɚ ɦɟɧɟɞɠɦɟɧɬɚ ɦɚɪɤɟɬɢɧɝɚ ɢ ɮɢ ɧɚɧɫɨɜ
Ɉɤɭɧɟɜɚȿ.Ɉ., Ɇɨɢɫɟɟɜ ɋ ɂ ȼɎɆȽɗɂ, 2015
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ɋɈȾȿɊɀȺɇɂȿ |
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ɌȿɆȺ ɆȺɌɊɂɐɕ ɂ ɈɉɊȿȾȿɅɂɌȿɅɂ............................................. |
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ȼɢɞɵ ɦɚɬɪɢɰ................................................................................... |
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Ⱦɟɣɫɬɜɢɹ ɧɚɞ ɦɚɬɪɢɰɚɦɢ............................................................... |
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Ɉɩɪɟɞɟɥɢɬɟɥɢ............................................................................... |
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Ɉɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ........................................................................ |
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Ɋɚɧɝ ɦɚɬɪɢɰɵ................................................................................ |
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Ɍɟɨɪɟɦɚ ɨ ɧɟɢɡɦɟɧɧɨɫɬɢ ɪɚɧɝɚ ɦɚɬɪɢɰɵ ɩɪɢ ɟɟ |
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ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ ........................................................ |
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ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɦɚɬɪɢɰ ɜ ɷɤɨɧɨɦɢɤɟ........................................... |
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ɌȿɆȺ ɋɂɋɌȿɆɕ ɅɂɇȿɃɇɕɏ ɍɊȺȼɇȿɇɂɃ............................... |
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ɋɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢ ɮɨɪɦɵ ɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ |
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ɩɪɟɞɫɬɚɜɥɟɧɢɹ...................................................................................... |
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Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɨɣ |
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ɦɚɬɪɢɰɵ ............................................................................................... |
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Ɍɟɨɪɟɦɚ Ʉɪɚɦɟɪɚ.......................................................................... |
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Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ ........... |
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Ɍɟɨɪɟɦɚ Ʉɪɨɧɟɤɟɪɚ - Ʉɚɩɟɥɥɢ ɋɨɜɦɟɫɬɧɚɹ ɧɟɫɨɜɦɟɫɬɧɚɹ |
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ɨɩɪɟɞɟɥɟɧɧɚɹ ɢ ɧɟɨɩɪɟɞɟɥɟɧɧɚɹ ɫɢɫɬɟɦɚ.......................................... |
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ɌȿɆȺ ɗɅȿɆȿɇɌɕ ȼȿɄɌɈɊɇɈȽɈ ȺɇȺɅɂɁȺ.............................. |
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ȼɟɤɬɨɪɵ ɧɚ ɩɥɨɫɤɨɫɬɢ ɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.................................... |
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Ⱦɟɣɫɬɜɢɹ ɧɚɞ Q-ɦɟɪɧɵɦɢ ɜɟɤɬɨɪɚɦɢ........................................... |
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ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ ȼɟɤɬɨɪɧɨɟ ɢ ɥɢɧɟɣɧɨɟ |
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ɩɪɨɫɬɪɚɧɫɬɜɨ........................................................................................ |
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Ʌɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɢ ɧɟɡɚɜɢɫɢɦɨɫɬɶ ɜɟɤɬɨɪɨɜ.................... |
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Ȼɚɡɢɫ Ɋɚɡɥɨɠɟɧɢɟ ɜɟɤɬɨɪɨɜ ɩɨ ɛɚɡɢɫɭ....................................... |
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Ʉɜɚɞɪɚɬɢɱɧɵɟ ɮɨɪɦɵ.................................................................. |
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Ɍɟɦɚ ɗɅȿɆȿɇɌɕ ȺɇȺɅɂɌɂɑȿɋɄɈɃ ȽȿɈɆȿɌɊɂɂ................... |
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Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɜ ɞɟɤɚɪɬɨɜɵɯ ɩɪɹɦɨɭɝɨɥɶɧɵɯ |
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ɤɨɨɪɞɢɧɚɬɚɯ......................................................................................... |
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Ʌɢɧɢɢ ɢ ɢɯ ɭɪɚɜɧɟɧɢɹ ................................................................. |
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ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɥɢɧɢɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ 5 ............................ |
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4 ɍɝɨɥ ɦɟɠɞɭ ɞɜɭɦɹ ɩɪɹɦɵɦɢ....................................................... |
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Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ ɞɨ ɩɪɹɦɨɣ.................................................. |
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ɍɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ ɧɚ ɩɥɨɫɤɨɫɬɢ................................................. |
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Ʉɪɢɜɵɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɡɚɞɚɧɧɵɟ ɜ ɤɚɧɨɧɢɱɟɫɤɨɦ ɜɢɞɟ....... |
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ɉɪɢɜɟɞɟɧɢɟ ɨɛɳɟɝɨ ɭɪɚɜɧɟɧɢɹ ɥɢɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɤ |
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ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ ........................................................................... |
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ɉɥɨɫɤɨɫɬɶ ɢ ɩɪɹɦɚɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ........................................... |
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ɌȿɆȺ ɎɍɇɄɐɂə ɈȾɇɈɃ ɉȿɊȿɆȿɇɇɈɃ ..................................... |
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ɉɨɧɹɬɢɟ ɦɧɨɠɟɫɬɜɚ...................................................................... |
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Ɉɩɟɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ........................................................ |
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ɑɢɫɥɨɜɵɟ ɦɧɨɠɟɫɬɜɚ ................................................................... |
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Ɇɨɞɭɥɶ ɞɟɣɫɬɜɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ.................................................. |
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Ɉɤɪɟɫɬɧɨɫɬɶ ɬɨɱɤɢ....................................................................... |
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Ɉɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ................................................................. |
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ɋɜɨɣɫɬɜɚ ɮɭɧɤɰɢɣ........................................................................ |
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ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɝɪɚɮɢɤɨɜ ɮɭɧɤɰɢɣ........................................... |
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ɂɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟ ɮɭɧɤɰɢɢ....................................................... |
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Ɏɭɧɤɰɢɢ ɜ ɷɤɨɧɨɦɢɤɟ................................................................ |
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ɌȿɆȺ ɉɊȿȾȿɅɕ ɂ ɇȿɉɊȿɊɕȼɇɈɋɌɖ ɎɍɇɄɐɂɃ..................... |
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ɑɢɫɥɨɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ..................................................... |
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ɉɪɟɞɟɥ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ....................................... |
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ɉɪɟɞɟɥ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ.............................................................. |
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Ȼɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɮɭɧɤɰɢɢ........................................................ |
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Ȼɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɟ ɮɭɧɤɰɢɢ ɢ ɢɯ ɫɜɹɡɶ ɫ ɛɟɫɤɨɧɟɱɧɨ |
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ɦɚɥɵɦɢ................................................................................................. |
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ɋɪɚɜɧɟɧɢɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɮɭɧɤɰɢɣ..................................... |
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Ɂɚɦɟɱɚɬɟɥɶɧɵɟ ɩɪɟɞɟɥɵ............................................................... |
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6. ɇɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ............................................................. |
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Ɍɨɱɤɢ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ ɢ ɢɯ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ........................... |
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Ɂɚɞɚɱɚ ɨ ɧɟɩɪɟɪɵɜɧɨɦ ɧɚɱɢɫɥɟɧɢɢ ɩɪɨɰɟɧɬɨɜ........................ |
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ɌȿɆȺ ɉɊɈɂɁȼɈȾɇȺə ɂ ȾɂɎɎȿɊȿɇɐɂȺɅ ɎɍɇɄɐɂɂ |
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ɈȾɇɈɃ ɉȿɊȿɆȿɇɇɈɃ.......................................................................... |
87 |
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Ɉɩɪɟɞɟɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣɢ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɮɭɧɤɰɢɢ............. |
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Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ......................................... |
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Ɇɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ............................................ |
88 |
ɗɤɨɧɨɦɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ.......................................... |
89 |
Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɢ ɬɚɛɥɢɰɚ |
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ɩɪɨɢɡɜɨɞɧɵɯ........................................................................................ |
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ɉɪɨɢɡɜɨɞɧɚɹ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ................................................... |
93 |
ɌȿɆȺ ɉɊɂɅɈɀȿɇɂə ɉɊɈɂɁȼɈȾɇɈɃ ......................................... |
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Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ................ |
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ɂɫɫɥɟɞɨɜɚɧɢɟ ɮɭɧɤɰɢɢ ɢ ɩɨɫɬɪɨɟɧɢɟ ɟɟ ɝɪɚɮɢɤɚ..................... |
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ɉɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ........................................................................ |
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ɌȿɆȺ ɎɍɇɄɐɂɂ ɇȿɋɄɈɅɖɄɂɏ ɉȿɊȿɆȿɇɇɕɏ..................... |
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Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ...................................................................... |
101 |
ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɢ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ -ɝɨ ɩɨɪɹɞɤɚ102 |
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Ƚɪɚɞɢɟɧɬ ɮɭɧɤɰɢɢ ɉɪɨɢɡɜɨɞɧɚɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ................. |
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ɗɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ.................................... |
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ɌȿɆȺ ɈɉɊȿȾȿɅȿɇɇɕɃ ɂ ɇȿɈɉɊȿȾȿɅȿɇɇɕɃ ɂɇɌȿȽɊȺɅ105 |
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ɇɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ...................................................... |
105 |
Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɨɣ ɜ ɧɟɨɩɪɟɞɟɥɟɧɧɨɦ ɢɧɬɟɝɪɚɥɟ................. |
106 |
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ...................................................... |
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ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɪɚɰɢɨɧɚɥɶɧɵɯ ɞɪɨɛɟɣ................................. |
108 |
Ɉɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ.......................................................... |
109 |
Ɏɨɪɦɭɥɚ ɇɶɸɬɨɧɚ – Ʌɟɣɛɧɢɰɚ............................................... |
110 |
Ƚɟɨɦɟɬɪɢɱɟɫɤɢɟ ɩɪɢɥɨɠɟɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ...... |
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ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɢɧɬɟɝɪɚɥɨɜ ɜ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ......... |
112 |
ɌȿɆȺ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕȿ ɍɊȺȼɇȿɇɂə........................... |
113 |
Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ.................................................................... |
113 |
ɍɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ........................ |
114 |
Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ 115 |
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Ʌɢɧɟɣɧɵɟ ɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ |
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ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ........................ |
116 |
5
Ʌɢɧɟɣɧɵɟ ɧɟɨɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ |
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ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ........................ |
118 |
ɌȿɆȺ ɊəȾɕ..................................................................................... |
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Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ.................................................................... |
121 |
12.2 ɉɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ..... 121 |
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Ɂɧɚɤɨɩɟɪɟɦɟɧɧɵɟ ɪɹɞɵ............................................................ |
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ɋɬɟɩɟɧɧɵɟ ɪɹɞɵ....................................................................... |
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Ɋɹɞ Ɇɚɤɥɨɪɟɧɚ.......................................................................... |
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ȼɈɉɊɈɋɕ Ʉ ɗɄɁȺɆȿɇɍ ɉɈ ɆȺɌȿɆȺɌɂɄȿ................................ |
127 |
ɄɈɇɌɊɈɅɖɇȺə ɊȺȻɈɌȺ ɉɈ ȾɂɋɐɂɉɅɂɇȿ ©ɆȺɌȿɆȺɌɂɄȺª132 |
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ɅɂɌȿɊȺɌɍɊȺ........................................................................................ |
156 |
6
ɌȿɆȺ ɆȺɌɊɂɐɕ ɂ ɈɉɊȿȾȿɅɂɌȿɅɂ
ȼ ɪɚɡɞɟɥɟ ³Ʌɢɧɟɣɧɚɹ ɚɥɝɟɛɪɚ´ ɨɫɧɨɜɧɨɟ ɜɧɢɦɚɧɢɟ ɭɞɟɥɹɟɬɫɹ ɦɚɬɪɢɰɚɦ ɨɩɪɟɞɟɥɢɬɟɥɹɦ ɢ ɫɢɫɬɟɦɚɦ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɨ ɫɤɨɥɶɤɭ ɜ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɦɚɬɪɢɱɧɵɟ ɦɨɞɟɥɢ - ɦɟɠɨɬɪɚɫɥɟɜɨɝɨ ɛɚɥɚɧɫɚ ɜ ɩɥɚ ɧɨɜɵɯ ɪɚɫɱɟɬɚɯ ɩɪɢ ɪɚɫɱɟɬɚɯ ɮɨɧɞɚ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɵ ɢ ɬ ɞ
1.1. ȼɢɞɵ ɦɚɬɪɢɰ
Ɍɚɛɥɢɰɚ m n ɱɢɫɟɥ ɚij ɜɢɞɚ
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ɫɨɫɬɨɹɳɚɹ ɢɡ m ɫɬɪɨɤ ɢ n ɫɬɨɥɛɰɨɜ ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɟɣ. ɑɢɫɥɚ aij ɫɬɨɹɳɢɟ ɧɚ ɩɟɪɟɫɟɱɟɧɢɢ i-ɣ ɫɬɪɨɤɢ ɢ j-ɝɨ
ɫɬɨɥɛɰɚ ɧɚɡɵɜɚɸɬɫɹ ɷɥɟɦɟɧɬɚɦɢ ɦɚɬɪɢɰɵ
ɆɚɬɪɢɰɵȺ ɚij ɢ ȼ bij ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ ɟɫɥɢ ɨɧɢ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɪɚɡɦɟɪɵ ɢ ɞɥɹ ɤɚɠɞɨɣ ɩɚɪɵ ɢɧɞɟɤɫɨɜ ɜɵɩɨɥɧɹ ɟɬɫɹɪɚɜɟɧɫɬɜɨaij= bij.
ɇɭɥɟɜɨɣ ɦɚɬɪɢɰɟɣ ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɚ ɜɫɟ ɷɥɟɦɟɧɬɵ
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ɤɨɬɨɪɨɣ ɪɚɜɧɵ ɧɭɥɸ 0 |
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Ɇɚɬɪɢɰɚ ɭ ɤɨɬɨɪɨɣ m=n ɧɚɡɵɜɚɟɬɫɹ ɤɜɚɞɪɚɬɧɨɣ ɦɚɬɪɢɰɟɣ n-ɝɨɩɨɪɹɞɤɚ
ȼ ɤɜɚɞɪɚɬɧɨɣ ɦɚɬɪɢɰɟ ɷɥɟɦɟɧɬɵ a11 ɚ22 ɚnn ɫɨɫɬɚɜɥɹ ɸɬ ɝɥɚɜɧɭɸ ɞɢɚɝɨɧɚɥɶ.
7
Ʉɜɚɞɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɭ ɤɨɬɨɪɨɣ ɤɚɠɞɵɣ ɷɥɟɦɟɧɬ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɪɚɜɟɧ ɟɞɢɧɢɰɟ ɚ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɪɚɜɧɵ ɧɭ
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ɥɸ ɧɚɡɵɜɚɟɬɫɹ ɟɞɢɧɢɱɧɨɣ:ȿ |
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1.2. Ⱦɟɣɫɬɜɢɹ ɧɚɞ ɦɚɬɪɢɰɚɦɢ |
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ɍɦɧɨɠɟɧɢɟ ɦɚɬɪɢɰɵ ɧɚ ɱɢɫɥɨ |
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ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɦɚɬɪɢɰɵ A |
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ɧɚ ɱɢɫɥɨ Ȝ |
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a21 a22... a2n |
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ɧɚɡɵɜɚɟɬɫɹɦɚɬɪɢɰɚ A |
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ɉɪɢɦɟɪ ɇɚɣɬɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɚɬɪɢɰɵ A |
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2. ɋɥɨɠɟɧɢɟ ɜɵɱɢɬɚɧɢɟ ɦɚɬɪɢɰ
ɋɭɦɦɨɣ (ɪɚɡɧɨɫɬɶɸ ɦɚɬɪɢɰȺ ɢ ȼ ɜ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ m ɫɬɪɨɤ ɢ nɫɬɨɥɛɰɨɜ ɧɚɡɵɜɚɟɬɫɹ ɦɚɬɪɢɰɚ ɋɫ ɷɥɟɦɟɧɬɚɦɢ ɪɚɜ ɧɵɦɢ ɫɭɦɦɚɦ ɪɚɡɧɨɫɬɹɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɷɥɟɦɟɧɬɨɜ ɫɥɚɝɚɟ ɦɵɯ
8
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Ɋɟɲɟɧɢɟ A B |
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ɍɦɧɨɠɟɧɢɟ ɦɚɬɪɢɰ |
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ɉɭɫɬɶ |
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ɞɚɧɵ ɦɚɬɪɢɰɚ |
A |
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21a22...a2n |
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ɪɚɡɦɟɪɧɨɫɬɢ |
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k n ɢ |
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ak1ak2 ...akn |
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b |
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1m |
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ɦɚɬɪɢɰɚ B |
b |
21b22 ...b2m |
ɪɚɡɦɟɪɧɨɫɬɢ n |
m . |
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bn1bn2 ...bnm |
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ɉɭɫɬɶ ɱɢɫɥɨ ɫɬɨɥɛɰɨɜ ɦɚɬɪɢɰɵȺ ɫɨɜɩɚɞɚɟɬ ɫ ɱɢɫɥɨɦ ɫɬɪɨɤ ɦɚɬɪɢɰɵ ȼ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɚɬɪɢɰɭ Ⱥ ɧɚɡɵɜɚɸɬ ɫɨɝɥɚɫɨ ɜɚɧɧɨɣ ɫ ɦɚɬɪɢɰɟɣ ȼ
9
ɉɪɨɢɡɜɟɞɟɧɢɟɦ ɦɚɬɪɢɰɵ Ⱥ ɧɚ ɦɚɬɪɢɰɭ ȼ ɧɚɡɵɜɚɟɬɫɹ ɬɚ
c |
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1m |
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c |
21c22...c2m |
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ɪɚɡɦɟɪɧɨɫɬɢk m ɤɚɠɞɵɣ |
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ɤɚɹ ɦɚɬɪɢɰɚ AB C |
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ck1ck2 ...ckm |
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ɷɥɟɦɟɧɬ ɤɨɬɨɪɨɣ cij (i 1,...,k; j 1,...,m)ɧɚɯɨɞɢɬɫɹ ɤɚɤ ɫɭɦɦɚ
ɩɪɨɢɡɜɟɞɟɧɢɣ ɷɥɟɦɟɧɬɨɜ ɜɡɹɬɵɯ ɩɨ ɩɨɪɹɞɤɭ ɢɡ i-ɣ ɫɬɪɨɤɢ ɦɚɬ ɪɢɰɵ Ⱥ ɢ M-ɝɨ ɫɬɨɥɛɰɚ ɦɚɬɪɢɰɵ ȼ
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ɉɪɢɦɟɪ |
ɇɚɣɬɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɚɬɪɢɰ |
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Ɋɟɲɟɧɢɟ.AB 1 |
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2 2 |
0 1 1 ( 3) |
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2 1 0 5 1 2 |
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1 2 3 1 2 ( 3) |
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1 1 4 ( 3) |
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11 13 |
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ɉɪɢɦɟɪ |
ɇɚɣɬɢ |
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ɩɪɨɢɡɜɟɞɟɧɢɟ |
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ɦɚɬɪɢɰ |
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ɢ B 3 |
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Ɋɟɲɟɧɢɟ: AB |
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10