Пакет MATHEMATICA: Первые уроки

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ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

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MATHEMATICA: Ы

ɄȺɁȺɇɖ-2001

ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

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ɞɨɰ. ɤɚɮɟɞɪɵ ɝɟɨɦɟɬɪɢɢ ɄȽɍ, ɤ.ɮ.-ɦ.ɧ.

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ɐɟɥɶ ɢɡɞɚɧɢɹ – ɜɵɪɚɛɨɬɤɚ ɧɚɱɚɥɶɧɵɯ ɧɚɜɵɤɨɜ ɪɚɛɨɬɵ ɫ ɩɚɤɟɬɨɦ MATHEMATICA. ȼ ɨɫɧɨɜɭ ɢɡɥɨɠɟɧɢɹ ɩɨɥɨɠɟɧɵ ɞɜɚ ɫɸɠɟɬɚ ɧɚ ɩɨɫɬɪɨɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ, ɯɨɪɨɲɨ ɢɥɥɸɫɬɪɢɪɭɸɳɢɯ ɜɨɡɦɨɠɧɨɫɬɢ ɩɪɢɦɟɧɟɧɢɹ ɩɚɤɟɬɚ, ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɚɦɨɫɬɨɹɬɟɥɶɧɵɣ ɢɧɬɟɪɟɫ. Ⱦɥɹ ɜɨɫɩɪɢɹɬɢɹ ɦɚɬɟɪɢɚɥɚ ɩɟɪɜɨɝɨ ɪɚɡɞɟɥɚ ɩɨɫɨɛɢɹ ɧɭɠɧɵ ɥɢɲɶ ɫɚɦɵɟ ɷɥɟɦɟɧɬɚɪɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɡɧɚɧɢɹ (ɫɜɟɞɟɧɢɹ ɨ ɮɭɧɤɰɢɹɯ ɢ ɩɪɟɞɟɥɚɯ). ȼɬɨɪɨɣ ɪɚɡɞɟɥ ɬɪɟɛɭɟɬ ɡɧɚɧɢɹ ɨɫɧɨɜ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ. ɉɨɫɨɛɢɟ ɪɚɫɫɱɢɬɚɧɨ ɤɚɤ ɧɚ ɫɬɭɞɟɧɬɨɜ, ɬɚɤ ɢ ɧɚ ɫɩɟɰɢɚɥɢɫɬɨɜ, ɧɭɠɞɚɸɳɢɯɫɹ ɜ ɛɵɫɬɪɨɦ ɨɫɜɨɟɧɢɢ ɩɚɤɟɬɚ MATHEMATICA.

: ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ

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(ɜ ɪɚɦɤɚɯ ɫɨɜɦɟɫɬɧɨɝɨ ɩɪɨɟɤɬɚ Ʉɚɡɚɧɫɤɨɝɨ ɢ Ƚɢɫɫɟɧɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɨɜ). Ɋɟɤɨɦɟɧɞɭɟɦ ɟɝɨ ɜɫɟɦ, ɤɬɨ ɡɚɢɧɬɟɪɟɫɨɜɚɧ ɜ ɫɨɜɦɟɫɬɧɨɦ ɩɪɢɦɟɧɟɧɢɢ ɩɚɤɟɬɨɜ MATHEMATICA ɢ STATISTICA ɜ ɫɨɰɢɚɥɶɧɨ-ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɫɮɟɪɟ. ȼ ɨɫɧɨɜɟ ɢɡɥɨɠɟɧɢɹ – ɪɟɚɥɶɧɵɟ ɢ ɢɝɪɨɜɵɟ ɡɚɞɚɱɢ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɬɟ, ɱɬɨ ɪɚɫɫɦɨɬɪɟɧɵ ɜ ɧɚɫɬɨɹɳɟɣ ɛɪɨɲɸɪɟ.

© ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 5

“ – ”, – ɝɥɚɫɢɬ ɧɚɪɨɞɧɚɹ ɦɭɞɪɨɫɬɶ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, (ɤɚɤ ɧɟ ɪɚɡ ɩɨɞɬɜɟɪɠɞɚɥɚ ɢɫɬɨɪɢɹ), ɨɬ ɩɪɚɜɢɥɶɧɨ ɫɞɟɥɚɧɧɵɯ ɩɟɪɜɵɯ ɲɚɝɨɜ ɩɨɪɨɣ ɡɚɜɢɫɢɬ ɭɫɩɟɯ ɜɫɟɝɨ ɦɟɪɨɩɪɢɹɬɢɹ. ɑɬɨ ɠɟ ɤɚɫɚɟɬɫɹ ɨɫɜɨɟɧɢɹ ɩɚɤɟɬɚ MATHEMATICA, ɬɨ ɡɞɟɫɶ, ɧɚ ɧɚɲ ɜɡɝɥɹɞ, ɧɚɢɥɭɱɲɢɦ ɫɬɚɪɬɨɦ ɛɭɞɟɬ ɨɡɧɚɤɨɦɥɟɧɢɟ ɫ ɩɪɢɦɟɧɟɧɢɹ ɩɚɤɟɬɚ ɤ ɪɟɲɟɧɢɸ ɧɟɤɨɬɨɪɵɯ ɡɚɞɚɱ. ɗɬɨ ɩɨɦɨɠɟɬ ɱɢɬɚɬɟɥɸ ɧɚɤɨɩɢɬɶ ɬɭ ɤɪɢɬɢɱɟɫɤɭɸ ɦɚɫɫɭ ɡɧɚɧɢɣ, ɤɨɬɨɪɚɹ ɡɚɬɟɦ ɩɨɡɜɨɥɢɬ ɟɦɭ ɛɨɥɟɟ ɝɥɭɛɨɤɨ ɢɡɭɱɢɬɶ ɩɚɤɟɬ. Ɇɵ ɧɟ ɫɬɪɟɦɢɦɫɹ ɦɚɤɫɢɦɚɥɶɧɨ ɨɯɜɚɬɢɬɶ ɡɞɟɫɶ ɜɨɡɦɨɠɧɨɫɬɢ ɩɚɤɟɬɚ ɢ ɧɟ ɨɛɴɹɫɧɹɟɦ ɟɝɨ ɜɧɭɬɪɟɧɧɸɸ ɥɨɝɢɤɭ – ɞɥɹ ɷɬɨɝɨ ɪɟɤɨɦɟɧɞɭɟɦ ɧɚɲɟɦɭ ɱɢɬɚɬɟɥɸ ɪɭɫɫɤɨɹɡɵɱɧɵɟ ɢɫɬɨɱɧɢɤɢ: [1] (ɚ ɬɚɤɠɟ [2] ɢ [3]). Ɉɞɧɚɤɨ, ɩɨ ɧɚɲɟɦɭ ɭɛɟɠɞɟɧɢɸ, ɱɢɬɚɬɶ ɷɬɢ ɤɧɢɝɢ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɭɠɟ ɬɨɝɞɚ, ɤɨɝɞɚ ȼɵ ɫɬɚɥɤɢɜɚɟɬɟɫɶ ɫ ɤɚɤɨɣ-ɬɨ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɟɣ ɢ ɭɠɟ ɢɦɟɟɬɟ ɧɟɤɨɬɨɪɵɟ ɧɚɱɚɥɶɧɵɟ ɧɚɜɵɤɢ ɪɚɛɨɬɵ ɫ ɩɚɤɟɬɨɦ. ɇɚɱɚɬɶ ɠɟ ɦɵ ɩɪɟɞɥɚɝɚɟɦ ɫ ɩɚɪɵ ɧɚɝɥɹɞɧɵɯ ɩɪɢɦɟɪɨɜ, ɯɨɪɨɲɨ ɢɥɥɸɫɬɪɢɪɭɸɳɢɯ ɜɨɡɦɨɠɧɨɫɬɢ ɩɪɢɦɟɧɟɧɢɹ ɩɚɤɟɬɚ, ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɚɦɨɫɬɨɹɬɟɥɶɧɵɣ ɢɧɬɟɪɟɫ. Ɇɵ ɪɚɫɫɱɢɬɵɜɚɟɦ, ɱɬɨ ɧɚɲɢɦɢ ɱɢɬɚɬɟɥɹɦɢ ɛɭɞɭɬ ɤɚɤ ɫɬɭɞɟɧɬɵ, ɬɚɤ ɢ ɫɩɟɰɢɚɥɢɫɬɵ, ɧɭɠɞɚɸɳɢɟɫɹ ɜ ɛɵɫɬɪɨɦ ɩɪɢɨɛɪɟɬɟɧɢɢ ɨɫɧɨɜɧɵɯ ɧɚɜɵɤɨɜ ɪɚɛɨɬɵ ɫ ɩɚɤɟɬɨɦ MATHEMATICA. Ⱦɥɹ ɜɨɫɩɪɢɹɬɢɹ ɦɚɬɟɪɢɚɥɚ ɩɟɪɜɨɝɨ ɪɚɡɞɟɥɚ ɩɨɫɨɛɢɹ ɧɭɠɧɵ ɥɢɲɶ ɫɚɦɵɟ ɷɥɟɦɟɧɬɚɪɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɡɧɚɧɢɹ (ɫɜɟɞɟɧɢɹ ɨ ɮɭɧɤɰɢɹɯ ɢ ɩɪɟɞɟɥɚɯ). ȼɬɨɪɨɣ ɪɚɡɞɟɥ ɬɪɟɛɭɟɬ ɡɧɚɧɢɹ ɨɫɧɨɜ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ. ȼ ɤɨɧɰɟ ɛɪɨɲɸɪɵ ɩɪɢɜɟɞɟɧ ɫɩɢɫɨɤ ɢɫɩɨɥɶɡɨɜɚɧɧɵɯ ɝɨɪɹɱɢɯ ɤɥɚɜɢɲ ɢ ɤɨɦɚɧɞ ɩɚɤɟɬɚ. ɉɨɞɪɨɛɧɟɟ ɨɧɢ ɨɩɢɫɚɧɵ ɜ Help’ɟ. ɂɫɩɨɥɶɡɨɜɚɧɧɵɟ ɲɪɢɮɬɵ ɧɟɫɭɬ ɨɩɪɟɞɟɥɟɧɧɭɸ ɫɦɵɫɥɨɜɭɸ ɧɚɝɪɭɡɤɭ: Ш “Tim es New Roman, , ” , . “Courier New, ” MATHEMATICA. “Times New Roman, ”– . “Times New Roman, , ”– . “Times New Roman, , ”– MATHEMATICA. Ɇɵ ɛɥɚɝɨɞɚɪɧɵ ɧɚɭɱɧɨɦɭ ɫɨɬɪɭɞɧɢɤɭ Ʉɚɡɚɧɫɤɨɝɨ ɎɢɡɢɤɨɌɟɯɧɢɱɟɫɤɨɝɨ ɂɧɫɬɢɬɭɬɚ Ю К ɡɚ ɩɨɦɨɳɶ, ɨɤɚɡɚɧɧɭɸ ɧɚɦ ɜ ɨɫɜɨɟɧɢɢ ɩɚɤɟɬɚ MATHEMATICA, ɚ ɬɚɤɠɟ ɡɚ ɰɟɧɧɵɟ ɡɚɦɟɱɚɧɢɹ ɢ ɩɪɨɮɟɫɫɨɪɭ Ƚɢɫɫɟɧɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ (Armin Bohnet) – ɡɚ ɩɪɢɨɛɪɟɬɟɧɢɟ ɩɚɤɟɬɚ. Э ([email protected]), ([email protected])

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ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

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ɉɪɟɞɫɬɚɜɶɬɟ ɫɟɛɟ, ɱɬɨ ȼɵ ɩɨɥɨɠɢɥɢ 1 ɦɥɧ. ɪɭɛɥɟɣ ɜ ɛɚɧɤ “Ɂɨɥɨɬɵɟ ɝɨɪɵ” ɩɨɞ 100% ɝɨɞɨɜɵɯ. Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɷɬɨ ɛɵɥɨ 1 ɹɧɜɚɪɹ 2000 ɝɨɞɚ. ɉɪɢɞɹ ɱɟɪɟɡ ɝɨɞ, ȼɵ ɩɨɥɭɱɢɬɟ ɜ 2 ɪɚɡɚ ɛɨɥɶɲɭɸ ɫɭɦɦɭ, – ɬɨ ɟɫɬɶ ɧɚ 1 ɹɧɜɚɪɹ 2001 ɝɨɞɚ ȼɵ ɛɭɞɟɬɟ ɢɦɟɬɶ 2 ɦɥɧ. ɪɭɛɥɟɣ. ȿɫɥɢ ɠɟ ȼɵ ɩɪɢɞɟɬɟ ɧɟ ɱɟɪɟɡ ɝɨɞ, ɚ ɱɟɪɟɡ ɩɨɥɝɨɞɚ, ɬɨ ɟɫɬɶ 1 ɢɸɥɹ 2000 ɝɨɞɚ, ɬɨ ɩɨɥɭɱɢɬɟ ɫɭɦɦɭ ɜ 1,5 ɪɚɡɚ ɛɨɥɶɲɭɸ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ, ɬɨ ɟɫɬɶ 1,5 ɦɥɧ. ɪɭɛɥɟɣ. ȿɳɟ ɱɟɪɟɡ ɩɨɥɝɨɞɚ ɨɧɚ ɭɜɟɥɢɱɢɬɫɹ ɜ 1,5 ɪɚɡɚ ɢ ɫɨɫɬɚɜɢɬ 2,25 ɦɥɧ. ɪɭɛɥɟɣ. Ⱥ ɟɫɥɢ ɩɪɢɯɨɞɢɬɶ ɜ ɛɚɧɤ ɤɚɠɞɵɣ ɦɟɫɹɰ? 1 ɮɟɜɪɚɥɹ 2001 ɝɨɞɚ ɫɭɦɦɚ ȼɚɲɟɝɨ ɜɤɥɚɞɚ ɫɨɫɬɚɜɢɬ 1 ɦɥɧ., ɭɜɟɥɢɱɟɧɧɵɣ ɜ 1 121 ɪɚɡ, ɬɨ ɟɫɬɶ 1 121 ɦɥɧ. ɪɭɛɥɟɣ. ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɤɚɠɞɵɣ ɦɟɫɹɰ ɦɵ ɛɭɞɟɦ ɭɜɟɥɢɱɢɜɚɬɶ ɫɭɦɦɭ ɜɤɥɚɞɚ ɜ 1 121 ɪɚɡ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, 1 ɹɧɜɚɪɹ 2001 ɝɨɞɚ 12

ɫɭɦɦɚ ɜɤɥɚɞɚ ɫɨɫɬɚɜɢɬ (1 121 ) , ɬɨ ɟɫɬɶ, ɤɚɤ ɧɟɬɪɭɞɧɨ ɩɨɫɱɢɬɚɬɶ ɧɚ ɤɚɥɶ-

ɤɭɥɹɬɨɪɟ, ɩɪɢɦɟɪɧɨ 2 ɦɥɧ. 613 305 ɪɭɛɥɟɣ. ȼɵ ɜɢɞɢɬɟ, ɱɬɨ ɷɬɨ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ, ɱɟɦ ɫɭɦɦɚ, ɩɨɥɭɱɟɧɧɚɹ ɜ ɩɟɪɜɨɦ ɢɥɢ ɜɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ. ɇɟ ɨɲɢɛɤɚ ɥɢ ɷɬɨ? ȼɟɞɶ ɩɪɢ ɪɚɛɨɬɟ ɫ ɞɪɨɛɹɦɢ ɤɚɥɶɤɭɥɹɬɨɪ ɩɪɨɢɡɜɨɞɢɬ ɨɤɪɭɝɥɟɧɢɹ. ɉɪɨɜɟɪɢɦ ɧɚɲɢ ɪɚɫɱɟɬɵ ɜ ɩɚɤɟɬɟ MATHEMATICA. Ɇɵ ɫɱɢɬɚɟɦ, ɱɬɨ ɩɚɤɟɬ MATHEMATICA ɜɟɪɫɢɢ 3.0 ɢɥɢ ɜɵɲɟ ɭɠɟ ɭɫɬɚɧɨɜɥɟɧ ɧɚ ȼɚɲɟɦ ɤɨɦɩɶɸɬɟɪɟ. ȿɫɥɢ ɟɟ ɮɢɪɦɟɧɧɨɝɨ ɡɧɚɱɤɚ

ɟɳɟ ɧɟɬ ɧɚ ȼɚɲɟɦ ɪɚɛɨɱɟɦ ɫɬɨɥɟ,

ɧɚɣɞɢɬɟ ɮɚɣɥ Mathematica.exe ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɩɢɤɬɨɝɪɚɦɦɨɣ ɢ ɩɟɪɟɬɚɳɢɬɟ ɟɟ ɧɚ ɪɚɛɨɱɢɣ ɫɬɨɥ. Ⱦɜɨɣɧɨɣ ɳɟɥɱɨɤ ɩɨ ɩɢɤɬɨɝɪɚɦɦɟ – ɢ ɩɚɤɟɬ ɡɚɩɭɳɟɧ. ɉɟɪɟɞ ȼɚɦɢ ɧɟɧɚɞɨɥɝɨ ɩɨɹɜɥɹɟɬɫɹ ɡɚɫɬɚɜɤɚ, ɚ ɡɚɬɟɦ – ɱɢɫɬɵɣ ɥɢɫɬ (Notebook, ɡɚɩɢɫɧɚɹ ɤɧɢɠɤɚ) ɢ ɦɟɧɸ ɫ ɩɚɧɟɥɶɸ ɢɧɫɬɪɭɦɟɧɬɨɜ ɧɚɜɟɪɯɭ. ɉɪɨɫɬɨ ɩɢɲɟɦ ɧɚ ɧɟɦ (13/12)^12 ɢ ɧɚɠɢɦɚɟɦ Shift+Enter. . ȼ ɩɚɤɟɬɟ MATHEMATICA ɫɭɳɟɫɬɜɭɸɬ ɦɧɨɠɟɫɬɜɨ ɩɪɢɟɦɨɜ, ɨɛɥɟɝɱɚɸɳɢɯ ɜɜɨɞ ɮɨɪɦɭɥ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɜɨɡɜɟɞɟɧɢɹ ɜ ɫɬɟɩɟɧɶ ɧɟɤɨɬɨɪɨɝɨ ɜɵɪɚɠɟɧɢɹ ɟɝɨ ɧɭɠɧɨ ɜɵɞɟɥɢɬɶ ɢ ɧɚɠɚɬɶ Ctrl+^. ɋɤɨɛɤɢ ɪɢɫɭɸɬɫɹ ɫɚɦɢ ɫɨɛɨɣ ɢ ɤɭɪɫɨɪ ɩɟɪɟɯɨɞɢɬ ɜ ɜɟɪɯɧɢɣ ɢɧɞɟɤɫ, ɤɭɞɚ ɫɥɟɞɭɟɬ ɜɜɟɫɬɢ ɩɨɤɚɡɚɬɟɥɶ ɫɬɟɩɟɧɢ. Ctrl+ ɜɨɡɜɪɚɳɚɟɬ ɤɭɪɫɨɪ ɜ ɫɬɪɨɤɭ. ȼɨɨɛɳɟ, ɦɧɨɝɢɟ ɤɥɚɜɢɲɢ, ɛɭɞɭɱɢ ɧɚɠɚɬɵ ɜɦɟɫɬɟ ɫ Ctrl, ɫɨɡɞɚɸɬ ɲɚɛɥɨɧ ɮɨɪɦɭɥɵ – ɩɨɷɤɫɩɟɪɢɦɟɧɬɢɪɭɣɬɟ.

ȼɵ ɭɜɢɞɢɬɟ ɫɥɟɞɭɸɳɭɸ ɤɚɪɬɢɧɭ, ɢɡɨɛɪɚɠɟɧɧɭɸ ɧɚ ɪɢɫ.1.1. ɇɭ ɤɚɤ ɬɭɬ ɧɟ ɜɫɩɨɦɧɢɬɶ ɚɧɟɤɞɨɬ ɩɪɨ ɦɚɬɟɦɚɬɢɤɚ, ɤɨɬɨɪɵɣ ɧɚ ɜɨɩɪɨɫ “Ƚɞɟ ɦɵ?”, ɡɚɞɚɧɧɵɣ ɟɦɭ ɫ ɜɨɡɞɭɲɧɨɝɨ ɲɚɪɚ ɡɚɛɥɭɞɢɜɲɢɦɢɫɹ ɩɭɬɟɲɟɫɬɜɟɧɧɢɤɚɦɢ, ɧɟɦɧɨɝɨ ɩɨɞɭɦɚɜ, ɞɚɥ ɚɛɫɨɥɸɬɧɨ ɬɨɱɧɵɣ, ɧɨ ɫɨɜɟɪɲɟɧɧɨ ɛɟɫɩɨɥɟɡɧɵɣ ɨɬɜɟɬ. (Ⱦɨɝɚɞɚɣɬɟɫɶ, ɤɚɤɨɣ).

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 7

1.1. Ɏɪɚɝɦɟɧɬ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ ɩɚɤɟɬɚ MATHEMATICA. ɇɟ ɭɞɢɜɥɹɣɬɟɫɶ – ɩɟɪɟɞ ȼɚɦɢ – ɞɪɨɛɶ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɬɨɱɧɵɦ ɡɧɚɱɟɧɢɟɦ ɫɭɦɦɵ ȼɚɲɟɝɨ ɜɤɥɚɞɚ.

ɇɨ ɧɟ ɫɩɟɲɢɬɟ ɨɛɜɢɧɹɬɶ ɩɚɤɟɬ MATHEMATICA – ȼɵ ɡɚɞɚɥɢ ɜɨɩɪɨɫ ɜ ɬɟɪɦɢɧɚɯ ɰɟɥɵɯ ɱɢɫɟɥ – ɜ ɧɢɯ ɠɟ ɢ ɜɵɪɚɠɟɧ ɨɬɜɟɬ. ȿɫɥɢ ɛɵ ɩɨɫɥɟ ɱɢɫɥɚ 13 ȼɵ ɩɨɫɬɚɜɢɥɢ ɬɨɱɤɭ, ɨɬɜɟɬ ɛɵɥ ɛɵ ɜ 10-ɢɱɧɵɯ ɞɪɨɛɹɯ: 2.61304, ɬɨ ɟɫɬɶ ɩɪɢɦɟɪɧɨ ɬɨ ɠɟ, ɱɬɨ ɢ ɛɵɥɨ ɩɨɥɭɱɟɧɨ ɫ ɩɨɦɨɳɶɸ ɤɚɥɶɤɭɥɹɬɨɪɚ. ɇɨ ɦɵ ɩɨɦɧɢɦ, ɱɬɨ ɷɬɨ ɱɢɫɥɨ ɩɨɥɭɱɟɧɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɚɫɱɟɬɨɜ ɫ ɨɤɪɭɝɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ. Ʉɚɤ ɠɟ ɩɪɟɞɫɬɚɜɢɬɶ ɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ (ɞɪɨɛɶ) ɜ ɭɞɨɛɨɜɚɪɢɦɨɦ ɜɢɞɟ? ȼ ɩɚɤɟɬɟ MATHEMATICA ɟɫɬɶ ɜɨɡɦɨɠɧɨɫɬɶ ɡɚɩɢɫɚɬɶ ɪɟɡɭɥɶɬɚɬɵ ɜɵɱɢɫɥɟɧɢɣ ɫɨ ɫɤɨɥɶ ɭɝɨɞɧɨ ɜɵɫɨɤɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ (ɨɧɚ ɨɝɪɚɧɢɱɢɜɚɟɬɫɹ ɥɢɲɶ ɜɨɡɦɨɠɧɨɫɬɹɦɢ ɤɨɦɩɶɸɬɟɪɚ). Ɂɞɟɫɶ ɧɭɠɧɨ ɢɦɟɬɶ ɜ ɜɢɞɭ ɫɥɟɞɭɸɳɟɟ: ɜ ɩɚɤɟɬɟ MATHEMATICA ɯɪɚɧɢɬɫɹ ɩɪɨɬɨɤɨɥ ɜɫɟɯ ɫɞɟɥɚɧɧɵɯ ɜ ɩɪɨɰɟɫɫɟ ɞɚɧɧɨɝɨ ɫɟɚɧɫɚ ɪɚɛɨɬɵ ɜɵɱɢɫɥɟɧɢɣ. (Ʉɚɤ ɝɨɜɨɪɢɥ ɨɞɢɧ ɢɡɜɟɫɬɧɵɣ ɩɟɪɫɨɧɚɠ ɂɥɶɮɚ ɢ ɉɟɬɪɨɜɚ, “ɍ ɦɟɧɹ ɜɫɟ ɯɨɞɵ ɡɚɩɢɫɚɧɵ”). In[ ] ɨɛɨɡɧɚɱɚɟɬ ɜɜɨɞ, Out[ ] – ɜɵɜɨɞ ɩɪɢ ɨɛɪɚɳɟɧɢɢ ɫ ɭɤɚɡɚɧɧɵɦ ɧɨɦɟɪɨɦ. Ɍɚɤ ɤɚɤ ɦɵ ɨɛɪɚɳɚɥɢɫɶ ɤ ɫɢɫɬɟɦɟ ɥɢɲɶ ɨɞɧɚɠɞɵ, Out[1] ɫɨɞɟɪɠɢɬ ɩɨɥɭɱɟɧɧɭɸ ɧɚɦɢ ɞɪɨɛɶ. Ɉɛɪɚɳɟɧɢɟ: N[Out[1]] ɛɭɞɟɬ ɨɡɧɚɱɚɬɶ, ɱɬɨ ɪɟɡɭɥɶɬɚɬ ɩɟɪɜɨɝɨ ɜɵɱɢɫɥɟɧɢɹ ɩɟɪɟɜɨɞɢɬɫɹ ɜ ɞɟɫɹɬɢɱɧɭɸ ɞɪɨɛɶ. Ɍɨɱɧɨɫɬɶ ɩɪɢ ɷɬɨɦ ɜɵɛɢɪɚɟɬɫɹ ɚɜɬɨɦɚɬɢɱɟɫɤɢ. Ɉɛɪɚɳɟɧɢɟ: N[Out[1],10] ɛɭɞɟɬ ɨɡɧɚɱɚɬɶ, ɱɬɨ ɦɵ ɯɨɬɢɦ ɩɨɥɭɱɢɬɶ ɜ ɪɟɡɭɥɶɬɚɬɟ 10 ɡɧɚɤɨɜ (ɜɫɟɝɨ). Ɍɚɤɨɣ ɠɟ ɪɟɡɭɥɶɬɚɬ ɛɵɥ ɩɨɥɭɱɟɧ ɧɚɦɢ ɫ ɩɨɦɨɳɶɸ ɤɚɥɶɤɭɥɹɬɨɪɚ. Ʉɚɤ ɢ ɩɪɢ ɪɚɛɨɬɟ ɫ ɤɚɥɶɤɭɥɹɬɨɪɨɦ, ɦɵ ɦɨɠɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ ɩɪɟɞɵɞɭɳɟɝɨ ɜɵɱɢɫɥɟɧɢɹ. Ɉɛɪɚɳɟɧɢɟ ɤ ɩɪɟɞɵɞɭɳɟɣ ɜɵɯɨɞɧɨɣ ɹɱɟɣɤɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɡɧɚɤɚ ɩɪɨɰɟɧɬɚ: %. Ɍɚɤ, ɜɜɨɞ N[%,10] ɧɚ ɜɬɨɪɨɦ ɲɚɝɟ ɷɤɜɢɜɚɥɟɧɬɟɧ N[Out[1],10]. . ɑɬɨɛɵ ɡɚɫɬɚɜɢɬɶ ɧɟɫɤɨɥɶɤɨ ɤɨɦɚɧɞ ɜɨɫɩɪɢɧɢɦɚɬɶɫɹ ɜɦɟɫɬɟ (ɢ ɬɟɦ ɫɚɦɵɦ ɢɡɛɟɠɚɬɶ ɜɵɜɨɞɚ ɪɟɡɭɥɶɬɚɬɨɜ ɜɵɩɨɥɧɟɧɢɹ (Out’ɨɜ) ɤɚɠɞɨɣ ɢɡ ɧɢɯ), ɧɭɠɧɨ ɪɚɡɞɟɥɹɬɶ ɢɯ ɫɢɦɜɨɥɨɦ “;” . Ɇɵ ɧɟ ɭɜɢɞɟɥɢ ɩɨɤɚ ɩɪɟɢɦɭɳɟɫɬɜ ɩɚɤɟɬɚ MATHEMATICA ɩɟɪɟɞ ɤɚɥɶɤɭɥɹɬɨɪɨɦ, ɤɪɨɦɟ ɜɨɡɦɨɠɧɨɫɬɢ ɡɚɞɚɜɚɬɶ ɱɢɫɥɨ ɰɢɮɪ ɞɥɹ ɜɵɜɨɞɚ ɪɟɡɭɥɶɬɚɬɚ. ɇɨ ɧɟ ɛɭɞɟɦ ɫɩɟɲɢɬɶ ɫ ɜɵɜɨɞɚɦɢ. ȼɟɪɧɟɦɫɹ ɜ ɧɚɲ ɛɚɧɤ “Ɂɨɥɨɬɵɟ ɝɨɪɵ”. ɑɬɨ ɟɫɥɢ ɩɟɪɟɫɱɢɬɵɜɚɬɶ ɜɤɥɚɞ ɤɚɠɞɵɣ ɞɟɧɶ, ɚ ɟɳɟ ɥɭɱɲɟ – ɤɚɠɞɵɣ ɱɚc, ɤɚɠɞɭɸ ɦɢɧɭɬɭ, ɤɚɠɞɭɸ ɫɟɤɭɧɞɭ? Cɨɡɞɚɟɬɫɹ ɜɩɟɱɚɬɥɟɧɢɟ, ɱɬɨ ɟɫɥɢ ɩɟɪɟɫɱɟɬ ɛɭɞɟɬ ɩɪɨɢɡɜɨɞɢɬɶɫɹ ɨɱɟɧɶ ɱɚɫɬɨ, ɬɨ ɱɟɪɟɡ ɝɨɞ ɫɭɦɦɚ ɜɨɡɪɚɫɬɟɬ ɜ ɨɱɟɧɶ ɦɧɨɝɨ ɪɚɡ. ɏɜɚɬɢɬ ɥɢ ɩɪɢɧɟɫɟɧɧɵɯ ɧɚɦɢ ɦɟɲɤɨɜ, ɱɬɨɛɵ ɭɧɟɫɬɢ ɜɫɟ ɞɟɧɶɝɢ? ɉɪɢɤɢɧɟɦ: ɩɪɢ ɟɠɟɫɟɤɭɧɞɧɨɦ ɩɟɪɟɫɱɟɬɟ ɜɤɥɚɞɚ ɢɬɨɝɨɜɚɹ ɫɭɦɦɚ ɫɨɫɬɚɜɢɬ 2 ɦɥɧ. 718 281 ɪɭɛɥɶ. ɏɦ, ɦɵ ɞɭɦɚɥɢ, ɛɭɞɟɬ ɛɨɥɶɲɟ. ȼɨɡɧɢɤɚɟɬ ɫɦɭɬɧɨɟ ɩɨɞɨɡɪɟɧɢɟ –

8

ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɥɢ ɩɪɟɞɟɥɚ ɷɬɨɣ ɫɭɦɦɵ ɩɪɢ ɫɤɨɥɶ ɭɝɨɞɧɨ ɱɚɫɬɨɦ ɩɨɫɟɳɟɧɢɢ ɛɚɧɤɚ? Ɋɚɡɛɟɪɟɦɫɹ: ɩɭɫɬɶ n – ɤɨɥɢɱɟɫɬɜɨ ɧɚɲɢɯ ɜɪɟɦɟɧɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɜ ɝɨɞɭ (ɞɧɟɣ, ɦɢɧɭɬ, ɫɟɤɭɧɞ,…). ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɮɨɪɦɭɥɵ ɢɦɟɸɬ ɜɢɞ: (1 + n1 ) . ɋɭɳɟɫɬɜɭɟɬ ɥɢ lim (1 + n1 ) ɩɪɢ n

n

(1 +

1 n

)

n

? ɋɩɪɨɫɢɦ ɨɛ ɷɬɨɦ

ɩɚɤɟɬ MATHEMATICA. ɋɧɚɱɚɥɚ ɜɵɹɫɧɢɦ, ɡɧɚɟɬ ɥɢ MATHEMATICA ɩɪɟɞɟɥɵ. ɗɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɫ ɩɨɦɨɳɶɸ ɷɥɟɤɬɪɨɧɧɨɝɨ ɭɱɟɛɧɢɤɚ, ɜɫɬɪɨɟɧɧɨɝɨ ɜ ɩɚɤɟɬ ɢ ɜɵɡɵɜɚɟɦɨɝɨ ɩɪɢ ɩɨɦɨɳɢ ɨɩɰɢɢ Help . ɇɭɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ MATHEMATICA – ɨɱɟɧɶ ɞɪɭɠɟɫɬɜɟɧɧɵɣ ɩɚɤɟɬ. ȼ ɧɟɦ ɫɭɳɟɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɫɩɨɫɨɛɨɜ ɩɨɢɫɤɚ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɢɧɮɨɪɦɚɰɢɢ. Ɍɚɤ, ɞɨɫɬɚɬɨɱɧɨ ɧɚɛɪɚɬɶ ??Limit, (ɢɥɢ, ɧɚɩɪɢɦɟɪ, ɬɨɥɶɤɨ ??Li* ɢɥɢ ??*imit), ɢ ɩɟɪɟɞ ɧɚɦɢ ɩɨɹɜɢɬɫɹ ɫɩɢɫɨɤ ɜɫɟɯ ɤɥɸɱɟɜɵɯ ɫɥɨɜ, ɫɨɞɟɪɠɚɳɢɯ ɞɚɧɧɭɸ ɩɨɞɫɬɪɨɤɭ. ȼɵɞɟɥɢɦ ɦɵɲɤɨɣ ɧɭɠɧɨɟ ɧɚɦ ɫɥɨɜɨ ɢ ɧɚɠɦɟɦ F1 – ɨɬɤɪɨɟɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɫɬɪɚɧɢɰɚ Help. Ɇɨɠɧɨ ɛɵɥɨ ɫɞɟɥɚɬɶ ɢ ɬɚɤ: Help – Help – ɜɜɟɫɬɢ ɫɥɨɜɨ Limit ɜ ɩɪɟɞɥɚɝɚɟɦɨɟ ɩɨɥɟ ɢ ɧɚɠɚɬɶ Enter. Help ɫɨɞɟɪɠɢɬ ɧɟ ɬɨɥɶɤɨ ɨɩɢɫɚɧɢɟ ɫɢɧɬɚɤɫɢɫɚ ɨɛɪɚɳɟɧɢɹ ɤ ɮɭɧɤɰɢɢ, ɧɨ ɢ, ɱɬɨ ɨɱɟɧɶ ɭɞɨɛɧɨ, ɩɪɢɦɟɪɵ ɟɟ ɢɫɩɨɥɶɡɨɜɚɧɢɹ. Ɍɚɤ, ɦɵ ɭɜɢɞɢɦ ɩɪɢɦɟɪ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɚ, ɨɛɳɢɣ ɜɢɞ ɨɛɪɚɳɟɧɢɹ ɤ ɮɭɧɤɰɢɢ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɚ: Limit[expr, x → x0 ], ɨɩɢɫɚɧɢɟ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ ɢ (ɜ ɪɭɛɪɢɤɟ Futher Examples) ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɪɢɦɟɪɵ. . ȼ ɩɚɤɟɬɟ MATHEMATICA ɫɬɪɟɦɥɟɧɢɟ ɚɪɝɭɦɟɧɬɚ ɨɛɨɡɧɚɱɚɟɬɫɹ ɡɧɚɤɨɦ “ − > ” (ɞɟɮɢɫ ɢ “ɫɬɪɨɝɨ ɛɨɥɶɲɟ”), ɛɟɫɤɨɧɟɱɧɨɫɬɶ – ɫɥɨɜɨɦ Infinity. ɋɬɪɟɥɨɱɤɭ ɢ ɛɟɫɤɨɧɟɱɧɨɫɬɶ ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɩɨɤɪɚɫɢɜɟɟ, ɟɫɥɢ ɞɨ ɢ ɩɨɫɥɟ “ − > ” ɢ “inf” ɧɚɠɚɬɶ Esc. ȿɫɥɢ ɨɞɧɢ ɢ ɬɟ ɠɟ ɨɩɟɪɚɰɢɢ ɜɵɡɵɜɚɸɬɫɹ ɧɟɨɞɧɨɤɪɚɬɧɨ, ɭɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜɯɨɞɹɳɢɟ ɩɚɧɟɥɢ ɢɧɫɬɪɭɦɟɧɬɨɜ ɩɚɤɟɬɚ (Palettes). ɂɯ ɜɵɡɨɜ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɱɟɪɟɡ ɦɟɧɸ: Files – Palettes. (ɋɦ. ɪɢɫ. 1.2.) ȼ ɩɚɥɢɬɪɟ 2 Basic Calculations – Calculus – Common Operations ɫɭɳɟɫɬɜɭɟɬ ɲɚɛɥɨɧ ɞɥɹ ɡɚɞɚɧɢɹ ɩɪɟɞɟɥɚ, ɜ 5 Complete Characters – ɡɧɚɤɢ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɢ ɫɬɪɟɦɥɟɧɢɹ. . Ⱦɥɹ ɜɜɨɞɚ ɫ ɩɨɦɨɳɶɸ ɲɚɛɥɨɧɚ ɥɭɱɲɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɡɚɞɚɬɶ ɫɬɢɥɶ ɜɯɨɞɧɨɣ ɹɱɟɣɤɢ ɤɚɤ Standard: Cell – Convert To – StandartForm.

ɇɚ ɪɢɫɭɧɤɟ 1.2 ɩɨɤɚɡɚɧ ɪɟɡɭɥɶɬɚɬ ɩɪɢɦɟɧɟɧɢɹ ɞɜɭɯ ɫɩɨɫɨɛɨɜ ɧɚɩɢɫɚɧɢɹ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɚ: , ɬɪɟɛɭɸɳɟɝɨ ɩɨ ɫɥɨɜɚɦ ɢɡɜɟɫɬɧɨɝɨ ɮɢɡɢɤɚ Ɋ. Ƀɨɫɬɚ [4], “ɪɭɞɢɦɟɧɬɚɪɧɨɝɨ ɜɥɚɞɟɧɢɹ ɥɚɬɢɧɫɤɢɦ ɢ ɝɪɟɱɟɫɤɢɦ ɚɥɮɚɜɢɬɚɦɢ”, ɢ . Ɍɚɦ ɠɟ ɢɡɨɛɪɚɠɟɧɵ ɞɜɟ ɢɧɫɬɪɭɦɟɧɬɚɥɶɧɵɟ ɩɚɥɢɬɪɵ – ɨɬɵɳɢɬɟ ɧɚ ɧɢɯ ɲɚɛɥɨɧɱɢɤ ɞɥɹ ɩɨɤɚɡɚɬɟɥɹ ɫɬɟɩɟɧɢ ɢ ɫɢɦɜɨɥ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. (ȿɳɟ ɛɨɥɟɟ ɧɚɝɥɹɞɧɵɣ ɩɪɢɦɟɪ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɲɚɛɥɨɧɨɜ ȼɵ ɭɜɢɞɢɬɟ ɧɚ ɫɬɪɚɧɢɰɟ 18.)

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 9

1.2. Ɏɪɚɝɦɟɧɬ ɪɚɛɨɱɟɝɨ ɞɨɤɭɦɟɧɬɚ ɢ ɞɜɟ ɩɚɥɢɬɪɵ.

ɂɬɚɤ, ɫɤɨɥɶ ɛɵ ɱɚɫɬɨ ɦɵ ɧɢ ɩɟɪɟɫɱɢɬɵɜɚɥɢ ɫɭɦɦɭ ɧɚɲɟɝɨ (ɢɡɜɢɧɢɬɟ, ɤɨɧɟɱɧɨ ɠɟ, ȼɚɲɟɝɨ) ɜɤɥɚɞɚ, ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ ɟɟ ɭɜɟɥɢɱɟɧɢɹ. Ɉɧ ɪɚɜɟɧ ɱɢɫɥɭ , ɫɜɨɣɫɬɜɚ ɤɨɬɨɪɨɝɨ ɛɵɥɢ ɭɫɬɚɧɨɜɥɟɧɵ ɜ 18 ɜɟɤɟ ɜɟɥɢɤɢɦ ɦɚɬɟɦɚɬɢɤɨɦ Ʌɟɨɧɚɪɞɨɦ ɗɣɥɟɪɨɦ. (ɉɨɞɪɨɛɧɨɫɬɢ – ɫɦ., ɧɚɩɪɢɦɟɪ, ɜ [5]). Ɂɞɟɫɶ ɪɟɱɶ ɲɥɚ ɨ 100% ɝɨɞɨɜɵɯ. Ʉ ɫɨɠɚɥɟɧɢɸ, ɬɚɤɚɹ ɫɬɚɜɤɚ ɧɟɪɟɚɥɶɧɚ ɞɚɠɟ ɞɥɹ ɛɚɧɤɚ “Ɂɨɥɨɬɵɟ ɝɨɪɵ”. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɩɪɨɰɟɧɬɧɚɹ ɫɬɚɜɤɚ ɪɚɜɧɚ x. (Ɋɚɧɟɟ x=1.) Ȼɭɞɟɬ ɥɢ ɫɭɳɟɫɬɜɨɜɚɬɶ ɩɪɟɞɟɥ ɫɭɦɦɵ ɜɤɥɚɞɚ ɜ ɷɬɨɦ ɫɥɭɱɚɟ? Ɉɩɹɬɶ ɨɛɪɚɬɢɦɫɹ ɤ ɩɚɤɟɬɭ MATHEMATICA. Ɋɟɡɭɥɶx

ɬɚɬɨɦ ɜɵɱɢɫɥɟɧɢɹ: Limit[(1+x/n)^n,n->Infinity] ɹɜɥɹɟɬɫɹ e . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɩɪɨɰɟɧɬɧɨɣ ɫɬɚɜɤɢ ɫɭɦɦɚ ɜɤɥɚɞɚ ɛɭɞɟɬ ɫɬɚɛɢɥɢɡɢɪɨɜɚɬɶɫɹ ɩɪɢ ɫɤɨɥɶ ɭɝɨɞɧɨ ɛɨɥɶɲɨɦ ɡɧɚɱɟɧɢɢ n. ɉɪɢ ɤɚɠɞɨɦ ɡɧɚɱɟɧɢɢ x ɦɵ ɛɭɞɟɦ ɩɨɥɭɱɚɬɶ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɩɪɟɞɟɥɚ. ɗɬɨ ɢ ɛɭɞɟɬ ɬɚ ɩɪɟɞɟɥɶɧɚɹ ɫɭɦɦɚ, ɤɨɬɨɪɭɸ ȼɵ ɩɨɥɭɱɢɥɢ ɛɵ ɱɟɪɟɡ ɝɨɞ ɩɨɫɥɟ ɨɮɨɪɦɥɟɧɢɹ ɜɤɥɚɞɚ, ɟɫɥɢ ɛɵ ɜɨɨɛɳɟ ɧɟ ɩɨɤɢɞɚɥɢ ɫɬɟɧ ɛɚɧɤɚ ɢ ɧɟɩɪɟɪɵɜɧɨ ɨɫɭɳɟɫɬɜɥɹɥɢ ɩɟɪɟɫɱɟɬ ɜɤɥɚɞɚ. ɑɬɨ ɠɟ ɞɟɥɚɬɶ ɬɟɦ ɜɤɥɚɞɱɢɤɚɦ, ɭ ɤɨɝɨ ɧɟɬ ɩɨɞ ɪɭɤɨɣ ɤɨɦɩɶɸɬɟɪɚ ɫ ɩɚɤɟɬɨɦ MATHEMATICA ɢ ɞɚɠɟ ɧɟɬ ɤɚɥɶɤɭɥɹɬɨɪɚ? Ⱦɚɜɚɣɬɟ ɫɞɟɥɚɟɦ ɞɥɹ ɧɢɯ ɫɥɟɞɭɸɳɟɟ: ɪɚɫɩɟɱɚɬɚɟɦ ɜ ɩɚɤɟɬɟ MATHEMATICA ɝɪɚɮɢɤ ɩɪɟɞɟɥɶɧɨɣ ɫɭɦɦɵ ɤɚɤ ɮɭɧɤɰɢɢ ɨɬ ɩɪɨɰɟɧɬɧɨɣ ɫɬɚɜɤɢ x. ɋ ɷɬɢɦ ɝɪɚɮɢɤɨɦ ɨɧɢ ɞɥɹ ɥɸɛɨɝɨ ɡɧɚɱɟɧɢɹ x ɥɟɝɤɨ ɭɡɧɚɸɬ ɩɪɟɞɟɥ ɫɜɨɢɯ ɮɢɧɚɧɫɨɜɵɯ ɜɨɡɦɨɠɧɨɫɬɟɣ. ɉɨɫɬɪɨɟɧɢɟ ɝɪɚɮɢɤɨɜ (ɮɭɧɤɰɢɹ Plot) – ɨɞɧɚ ɢɡ ɧɚɢɛɨɥɟɟ ɢɫɩɨɥɶɡɭɟɦɵɯ ɜɨɡɦɨɠɧɨɫɬɟɣ ɩɚɤɟɬɚ MATHEMATICA. (ȿɟ ɩɚɪɚɦɟɬɪɵ ɨɩɢɫɚɧɵ ɧɚ ɫɬɪ.8.) ɇɚɛɢɪɚɟɦ: Plot[E^x,{x, 0,1.2}] ɢ ɩɨɥɭɱɚɟɦ ɤɚɪɬɢɧɤɭ, ɢɡɨɛɪɚɠɟɧɧɭɸ ɧɚ ɪɢɫɭɧɤɟ 1.3.

10

ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

1.3. Ƚɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɟɞɟɥɶɧɨɝɨ ɝɨɞɨɜɨɝɨ ɞɨɯɨɞɚ (ɬ.ɟ. ɞɨɯɨɞɚ, ɩɨɥɭɱɟɧɧɨɝɨ ɜ ɤɨɧɰɟ ɝɨɞɚ ɩɪɢ ɛɟɫɤɨɧɟɱɧɨ ɱɚɫɬɨɦ ɩɟɪɟɫɱɟɬɟ ɫɭɦɦɵ ɜɤɥɚɞɚ) ɨɬ ɩɪɨɰɟɧɬɧɨɣ ɫɬɚɜɤɢ. ɉɪɨɰɟɧɬɧɚɹ ɫɬɚɜɤɚ ɢɡɦɟɧɹɟɬɫɹ ɡɞɟɫɶ ɨɬ 0 ɞɨ 120%.

Ƚɪɚɮɢɤ ɯɨɪɨɲ, ɧɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɟɝɨ ɟɳɟ ɛɨɥɟɟ ɩɨɧɹɬɧɵɦ ɢ ɥɟɝɤɨ ɱɢɬɚɟɦɵɦ. Ⱦɥɹ ɷɬɨɝɨ ɫɭɳɟɫɬɜɭɟɬ ɦɧɨɠɟɫɬɜɨ ɩɚɪɚɦɟɬɪɨɜ ɮɭɧɤɰɢɢ Plot. ɂɦ ɦɨɠɧɨ ɩɪɢɫɜɚɢɜɚɬɶ ɡɧɚɱɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɡɧɚɱɤɚ “->” (rule, ɩɪɚɜɢɥɨ): Plot[ а ]. ȼ ɬɚɛɥɢɰɟ 1.1 ɨɩɢɫɚɧɵ ɧɟɤɨɬɨɪɵɯ ɢɡ ɩɚɪɚɦɟɬɪɨɜ ɮɭɧɤɰɢɢ Plot. 1.1. ɉɚɪɚɦɟɬɪɵ ɮɭɧɤɰɢɢ Plot. PlotLabel->"INCOME DEPENDING ON X RATE" Ɂɚɝɨɥɨɜɨɤ ɝɪɚɮɢɤɚ

PlotRange->{0,3.5} Ⱦɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ ɩɟɪɟɦɟɧɧɨɣ y ɧɚ ɝɪɚɮɢɤɟ. (ɉɨ ɭɦɨɥɱɚɧɢɸ ɨɧ ɜɵɛɢɪɚɟɬɫɹ ɬɚɤ, ɱɬɨɛɵ ɩɨɤɚɡɚɬɶ ɯɚɪɚɤɬɟɪ ɩɨɜɟɞɟɧɢɹ ɮɭɧɤɰɢɢ ɧɚ ɜɫɟɦ ɡɚɞɚɧɧɨɦ ɢɧɬɟɪɜɚɥɟ ɢɡɦɟɧɟɧɢɹ ɚɪɝɭɦɟɧɬɚ.) ȿɫɥɢ ɬɨɱɤɚ y=0 ɧɟ ɩɨɩɚɞɚɟɬ ɜ ɷɬɨɬ ɞɢɚɩɚɡɨɧ, ɨɫɶ ɚɛɫɰɢɫɫ ɩɟɪɟɧɨɫɢɬɫɹ ɜ ɧɢɠɧɸɸ ɢɥɢ ɜɟɪɯɧɸɸ (ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɮɭɧɤɰɢɢ) ɟɝɨ ɝɪɚɧɢɰɭ.

AxesLabel->{"x","E^x"} ɇɚɞɩɢɫɢ ɧɚ ɨɫɹɯ ɚɛɫɰɢɫɫ ɢ ɨɪɞɢɧɚɬ.

Ticks->{{0.0,0.2,0.4,0.6,0.8,1.0,1.2,1.4}, {0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5}} Ɋɚɡɦɟɬɤɚ ɨɫɟɣ ɚɛɫɰɢɫɫ (ɩɟɪɜɵɣ ɫɩɢɫɨɤ) ɢ ɨɪɞɢɧɚɬ (ɜɬɨɪɨɣ ɫɩɢɫɨɤ). Ɍɨɱɤɢ, ɧɟ ɩɨɩɚɞɚɸɳɢɟ ɜ ɞɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ x, y, ɧɟ ɨɬɨɛɪɚɠɚɸɬɫɹ.

GridLines –> {0.0,0.4,0.8,1.2},{0.0,0.5,1.0,1.5,2.0,2.5}} Ʌɢɧɢɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɪɟɲɟɬɤɢ – ɚɛɫɰɢɫɫɵ ɟɟ ɭɡɥɨɜ ɡɚɞɚɧɵ ɤɨɦɩɨɧɟɧɬɚɦɢ ɩɟɪɜɨɝɨ ɫɩɢɫɤɚ, ɨɪɞɢɧɚɬɵ – ɜɬɨɪɨɝɨ. Ɉɬɨɛɪɚɠɚɸɬɫɹ ɥɢɲɶ ɬɟ ɭɱɚɫɬɤɢ ɥɢɧɢɣ ɪɟɲɟɬɤɢ, ɤɨɬɨɪɵɟ ɩɨɩɚɞɚɸɬ ɜ ɞɢɚɩɚɡɨɧ ɢɡɦɟɧɟɧɢɣ ɩɟɪɟɦɟɧɧɵɯ x, y. Ɂɚɦɟɬɢɦ, ɱɬɨ ɡɧɚɱɟɧɢɹ ɫɜɨɣɫɬɜ Ticks GridLines ɧɟ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 11 ɞɪɭɝɨɦ.

PlotStyle->{Thickness[0.009],RGBColor[1,0,0]} ɋɬɢɥɶ ɥɢɧɢɢ ɝɪɚɮɢɤɚ. Ɂɞɟɫɶ ɡɚɞɚɧɵ ɩɚɪɚɦɟɬɪɵ: ɬɨɥɳɢɧɚ ɥɢɧɢɢ ɤɚɤ 0.009 ɲɢɪɢɧɵ ɪɢɫɭɧɤɚ ɢ ɰɜɟɬ ɝɪɚɮɢɤɚ (ɤɪɚɫɧɵɣ).

AxesStyle->{Thickness[0.005],RGBColor[0,1,0]} ɋɬɢɥɶ ɨɫɟɣ. Ɂɞɟɫɶ ɡɚɞɚɧ ɨɞɢɧ ɬɨɥɶɤɨ ɫɩɢɫɨɤ ɩɚɪɚɦɟɬɪɨɜ, ɨɧ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɨɛɟɢɯ ɨɫɟɣ. Ɇɨɠɧɨ ɛɵɥɨ ɡɚɞɚɬɶ ɪɚɡɧɵɟ ɫɬɢɥɢ ɞɥɹ ɨɫɢ ɚɛɫɰɢɫɫ ɢ ɨɫɢ ɨɪɞɢɧɚɬ, (ɫɨɡɞɚɜ ɞɥɹ ɷɬɨɝɨ ɞɜɚ ɫɩɢɫɤɚ ɩɚɪɚɦɟɬɪɨɜ), ɧɨ ɦɵ ɩɨɥɟɧɢɥɢɫɶ.

ɉɨɫɤɨɥɶɤɭ ɱɟɪɧɨ-ɛɟɥɚɹ ɩɟɱɚɬɶ ɧɟ ɩɨɡɜɨɥɹɟɬ ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɬɶ ɜɫɸ ɩɨɥɧɨɬɭ ɝɪɚɮɢɱɟɫɤɢɯ ɜɨɡɦɨɠɧɨɫɬɟɣ ɩɚɤɟɬɚ MATHEMATICA, ɦɵ ɧɟ ɩɨɦɟɳɚɟɦ ɡɞɟɫɶ ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɜɢɞ ɧɚɲɟɝɨ ɝɪɚɮɢɤɚ – ɩɨɫɬɪɨɣɬɟ ɟɝɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ. ɋɨɯɪɚɧɢɦ ɡɚɩɢɫɧɭɸ ɤɧɢɠɤɭ ɫ ɩɨɦɨɳɶɸ ɨɩɰɢɢ Save ɦɟɧɸ File. ɉɪɢ ɷɬɨɦ ɫɨɡɞɚɟɬɫɹ ɮɚɣɥ ɫ ɪɚɫɲɢɪɟɧɢɟɦ .nb. ɉɚɤɟɬ MATHEMATICA ɩɨɡɜɨɥɹɟɬ ɪɚɛɨɬɚɬɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɮɚɣɥɚɦɢ (ɡɚɩɢɫɧɵɦɢ ɤɧɢɠɤɚɦɢ). ɉɪɢ ɷɬɨɦ ɧɭɦɟɪɚɰɢɹ ɜɫɟɯ “ɯɨɞɨɜ” ɹɜɥɹɟɬɫɹ ɫɤɜɨɡɧɨɣ. ɇɟ ɩɭɝɚɣɬɟɫɶ, ɟɫɥɢ ȼɵ ɧɟɧɚɪɨɤɨɦ ɡɚɤɪɵɥɢ ɜɫɟ ɡɚɩɢɫɧɵɟ ɤɧɢɠɤɢ ɢ ɜɢɞɢɬɟ ɦɟɧɸ MATHEMATICɢ ɩɚɪɹɳɢɦ ɩɪɹɦɨ ɧɚ ɪɚɛɨɱɟɦ ɫɬɨɥɟ ɢɥɢ ɧɚ ɮɨɧɟ ɨɤɧɚ ɤɚɤɨɝɨ-ɧɢɛɭɞɶ ɞɪɭɝɨɝɨ ɩɪɢɥɨɠɟɧɢɹ ɧɚɩɨɞɨɛɢɟ ɭɥɵɛɤɢ ɑɟɲɢɪɫɤɨɝɨ ɤɨɬɚ. Ɉɬɤɪɨɣɬɟ ɧɨɜɭɸ ɡɚɩɢɫɧɭɸ ɤɧɢɠɤɭ ɨɩɰɢɟɣ: File – New. ɉɨɥɭɱɢɦ ɧɚɤɨɧɟɰ ɦɚɬɟɪɢɚɥɢɡɨɜɚɧɧɵɣ ɢɬɨɝ ɧɚɲɟɣ ɪɚɛɨɬɵ – ɪɚɫɩɟɱɚɬɚɟɦ ɝɪɚɮɢɤ. ɉɟɱɚɬɶ ɜ ɩɚɤɟɬɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢ ɜɵɛɨɪɟ ɨɩɰɢɢ Print ɦɟɧɸ File, ɧɨ ɩɪɟɠɞɟ ɭɫɬɚɧɨɜɢɦ ɫɬɢɥɶ ɩɟɱɚɬɢ (ɨɫɧɨɜɧɵɟ ɩɚɪɚɦɟɬɪɵ ɫɬɪɚɧɢɰɵ) ɨɩɰɢɟɣ Printing Settings ɦɟɧɸ File. ɇɟ ɛɭɞɟɦ ɩɟɪɟɱɢɫɥɹɬɶ ɩɚɪɚɦɟɬɪɵ ɩɟɱɚɬɢ – ɨɧɢ ɬɪɚɞɢɰɢɨɧɧɵ ɞɥɹ Windows. ȼ ɩɚɤɟɬɟ MATHEMATICA ɫɭɳɟɫɬɜɭɸɬ ɬɚɤɠɟ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɫɬɢɥɟɜɵɟ ɬɚɛɥɢɰɵ (stylesheets), ɡɚɞɚɸɳɢɟ ɪɟɠɢɦ ɩɟɱɚɬɢ: ɪɚɛɨɱɢɣ, ɱɟɪɧɨɜɨɣ, ɪɟɠɢɦ ɩɪɟɡɟɧɬɚɰɢɢ ɢ ɩɪ. Ɋɟɠɢɦɵ ɪɚɡɥɢɱɚɸɬɫɹ ɪɚɡɦɟɪɨɦ ɲɪɢɮɬɚ ɢ ɩɥɨɬɧɨɫɬɶɸ ɩɟɱɚɬɢ. ȼɵɛɨɪ ɪɟɠɢɦɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɩɰɢɢ Printing Style Environment ɦɟɧɸ Format. Ɍɟɩɟɪɶ ȼɵ ɦɨɠɟɬɟ ɩɨɥɭɱɢɬɶ ɩɪɨɬɨɤɨɥ ɜɫɟɝɨ ȼɚɲɟɝɨ ɞɢɚɥɨɝɚ ɫ ɩɚɤɟɬɨɦ. Ɉɞɧɚɤɨ, ɜɫɹ “ɩɨɪɬɹɧɤɚ”, ɤɚɤ ɩɪɚɜɢɥɨ, ɛɵɜɚɟɬ ɧɟ ɧɭɠɧɚ. ɇɭɠɧɨ ɭɛɪɚɬɶ ɢɡ ɞɢɚɥɨɝɨɜɨɝɨ ɨɤɧɚ ɥɢɲɧɢɟ ɮɪɚɝɦɟɧɬɵ, ɧɚɩɢɫɚɬɶ ɡɚɝɨɥɨɜɨɤ, ɤɨɦɦɟɧɬɚɪɢɢ, ɩɪɢɜɟɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɮɨɪɦɭɥɵ. Ɉɛɪɚɬɢɬɟ ɜɧɢɦɚɧɢɟ: ɫɩɪɚɜɚ ɟɫɬɶ ɜɟɪɬɢɤɚɥɶɧɵɟ ɥɢɧɢɢ, ɜɵɞɟɥɹɸɳɢɟ ɧɚ ɷɤɪɚɧɟ ɨɬɞɟɥɶɧɵɟ ɹɱɟɣɤɢ ɢ ɢɯ ɝɪɭɩɩɵ. ȼɵ ɦɨɠɟɬɟ ɞɟɥɚɬɶ ɫ ɹɱɟɣɤɚɦɢ ɜɫɟ, ɱɬɨ ɯɨɬɢɬɟ – ɭɞɚɥɹɬɶ ɢɯ, ɫɜɟɪɬɵɜɚɬɶ, ɨɛɴɟɞɢɧɹɬɶ, ɭɩɪɚɜɥɹɬɶ ɢɯ ɫɬɢɥɟɦ – ɬɟɦ ɫɚɦɵɦ ɫɨɞɟɪɠɢɦɨɟ ɷɤɪɚɧɚ ɦɨɠɟɬ ɛɵɬɶ ɨɮɨɪɦɥɟɧɨ ɬɚɤ, ɤɚɤ ȼɵ ɬɨɝɨ ɩɨɠɟɥɚɟɬɟ. ɉɨɤɚ ɦɵ ɢɦɟɥɢ ɞɟɥɨ ɫ ɹɱɟɣɤɚɦɢ ɬɢɩɚ Input, Output, Standard (ɫɦ. ɨɱɟɧɶ ɜɚɠɧɨɟ ɡɚɦɟɱɚɧɢɟ ɧɚ ɫɬɪ. 6). ɋɭɳɟɫɬɜɭɸɬ ɢ ɞɪɭɝɢɟ ɬɢɩɵ: ɡɚɝɨɥɨɜɤɢ (Title, Subtitle,...), ɬɟɤɫɬ, ɮɨɪɦɭɥɵ ɢ ɩɪ. Ɉɧɢ ɜɵɩɨɥɧɹɸɬ ɮɭɧɤɰɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɩɰɢɣ ɬɟɤɫɬɨɜɵɯ ɪɟɞɚɤɬɨɪɨɜ, ɩɨɡɜɨɥɹɹ ɤɪɚɫɢɜɨ ɨɮɨɪɦɢɬɶ

12

ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

ɫɨɞɟɪɠɢɦɨɟ ɡɚɩɢɫɧɨɣ ɤɧɢɠɤɢ. Ɉɫɧɨɜɧɵɟ ɜɨɡɦɨɠɧɨɫɬɢ ɪɚɛɨɬɵ ɫ ɹɱɟɣɤɚɦɢ ɜ ɩɚɤɟɬɟ MATHEMATICA ɦɵ ɩɟɪɟɱɢɫɥɢɥɢ ɜ ɬɚɛɥɢɰɟ 1.2. 1.2. Ɋɚɛɨɬɚ ɫ ɹɱɟɣɤɚɦɢ Ч ȼɵɞɟɥɢɬɶ

ɍɞɚɥɢɬɶ ɋɝɪɭɩɩɢɪɨɜɚɬɶ

ɉɨɞɜɟɫɬɢ ɤɭɪɫɨɪ (ɨɧ ɢɡɦɟɧɢɬ ɫɜɨɸ ɮɨɪɦɭ ɧɚ ɬɨɧɤɭɸ ɫɬɪɟɥɤɭ ɫ ɜɟɪɬɢɤɚɥɶɧɨɣ ɱɟɪɬɨɣ ɜɩɟɪɟɞɢ) ɤ ɫɤɨɛɤɟ ɹɱɟɣɤɢ ɢ ɳɟɥɤɧɭɬɶ ɦɵɲɶɸ. ɇɚɠɚɬɶ Delete. ȼɵɞɟɥɟɧɧɚɹ ɹɱɟɣɤɚ ɛɭɞɟɬ ɭɞɚɥɟɧɚ. ɋɭɳɟɫɬɜɭɟɬ ɞɜɚ ɪɟɠɢɦɚ ɝɪɭɩɩɢɪɨɜɤɢ: ɚɜɬɨɦɚɬɢɱɟɫɤɢɣ ɢ ɪɭɱɧɨɣ. ȼɵɛɨɪ ɪɟɠɢɦɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɨɩɰɢɟɣ Cell – Cell Grouping. ɉɪɢ ɚɜɬɨɦɚɬɢɱɟɫɤɨɦ ɪɟɠɢɦɟ (Automatic Grouping) ɝɪɭɩɩɢɪɭɸɬɫɹ ɜɯɨɞɧɚɹ ɢ ɜɵɯɨɞɧɚɹ ɹɱɟɣɤɢ. ɉɪɢ ɪɭɱɧɨɦ ɪɟɠɢɦɟ (Manual Grouping) ɦɨɠɧɨ ɫɝɪɭɩɩɢɪɨɜɚɬɶ ɥɸɛɨɟ ɤɨɥɢɱɟɫɬɜɨ ɫɦɟɠɧɵɯ ɹɱɟɟɤ. Ⱦɥɹ ɷɬɨɝɨ ɢɯ ɫɥɟɞɭɟɬ ɜɵɞɟɥɢɬɶ ɢ ɜɵɛɪɚɬɶ ɨɩɰɢɸ Cell – Cell Grouping – Group Cells.

ɋɜɟɪɧɭɬɶ/Ɋɚɡɜɟɪɧɭɬɶ (ɞɥɹ ɝɪɭɩɩɵ ɹɱɟɟɤ)

Ⱦɜɨɣɧɨɣ ɳɟɥɱɨɤ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɤɨɛɤɟ.

Ɂɚɞɚɬɶ ɫɬɢɥɶ ɹɱɟɣɤɢ

Format – Style – ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɨɩɰɢɹ.

ɂɬɚɤ, ɦɵ ɪɚɫɫɦɨɬɪɟɥɢ ɨɞɢɧ ɩɪɢɦɟɪ ɩɪɢɦɟɧɟɧɢɹ ɩɚɤɟɬɚ MATHEMATICA. Ɉɬɦɟɬɢɦ ɧɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ ɩɚɤɟɬɚ, ɧɚ ɤɨɬɨɪɵɟ ȼɵ, ɞɨɥɠɧɨ ɛɵɬɶ, ɭɠɟ ɨɛɪɚɬɢɥɢ ɜɧɢɦɚɧɢɟ: ɚ) ȼɫɟ ɜɵɪɚɠɟɧɢɹ (ɢɦɟɧɚ ɮɭɧɤɰɢɣ, ɩɚɪɚɦɟɬɪɨɜ, ɨɛɨɡɧɚɱɟɧɢɹ ɢɡɜɟɫɬɧɵɯ ɤɨɧɫɬɚɧɬ) ɩɢɲɭɬɫɹ ɫ ɛɨɥɶɲɨɣ ɛɭɤɜɵ. ɇɚɩɪɢɦɟɪ: (ɧɚɲ ɡɥɨɩɨɥɭɱɧɵɣ ɩɪɟɞɟɥ), Infinity (ɛɟɫɤɨɧɟɱɧɨɫɬɶ), Pi (ɱɢɫɥɨ π ), GoldenRatio (ɨɬɧɨɲɟɧɢɟ ɡɨɥɨɬɨɝɨ ɫɟɱɟɧɢɹ: ɤɫɬɚɬɢ, ɢɦɟɧɧɨ ɬɚɤɢɦ ɹɜɥɹɟɬɫɹ ɩɪɢɧɹɬɨɟ ɩɨ ɭɦɨɥɱɚɧɢɸ ɫɨɨɬɧɨɲɟɧɢɟ ɞɥɢɧɵ ɢ ɲɢɪɢɧɵ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɨɛɥɚɫɬɢ, ɧɚ ɤɨɬɨɪɨɣ ɪɚɡɦɟɳɚɟɬɫɹ ɝɪɚɮɢɤ). ɛ) ȼɫɟ ɚɪɝɭɦɟɧɬɵ ɮɭɧɤɰɢɣ ɡɚɤɥɸɱɟɧɵ ɜ ɤɜɚɞɪɚɬɧɵɟ ɫɤɨɛɤɢ. ɋɬɪɨɤɢ ɡɚɤɥɸɱɟɧɵ ɜ ɤɚɜɵɱɤɢ. ɇɟɤɨɬɨɪɵɟ ɩɚɪɚɦɟɬɪɵ ɮɭɧɤɰɢɣ ɨɩɪɟɞɟɥɹɸɬɫɹ (ɫɦ., ɧɚɩɪɢɦɟɪ, PlotStyle ɜ ɬɚɛɥɢɰɟ 1.1), ɡɧɚɱɟɧɢɹ ɤɨɬɨɪɵɯ ɦɨɝɭɬ ɡɚɞɚɜɚɬɶɫɹ ɜ ɩɪɨɢɡɜɨɥɶɧɨɦ ɩɨɪɹɞɤɟ. ɜ) ȼɜɟɫɬɢ ɨɞɧɭ ɢ ɬɭ ɠɟ ɮɨɪɦɭɥɭ ɦɨɠɧɨ ɪɚɡɧɵɦɢ ɫɩɨɫɨɛɚɦɢ: ɧɚɛɨɪɨɦ ɫɥɨɜ ɫ ɤɥɚɜɢɚɬɭɪɵ ɢɥɢ ɫ ɩɨɦɨɳɶɸ ɲɚɛɥɨɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɩɚɧɟɥɢ (Pallete). (ɋɦ., ɧɚɩɪɢɦɟɪ, ɪɢɫɭɧɨɤ 1.2). Ɇɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɥɢɲɶ ɦɚɥɭɸ ɱɚɫɬɶ ɜɨɡɦɨɠɧɨɫɬɟɣ ɩɚɤɟɬɚ MATHEMATICA – ɞɥɹ ɚɧɚɥɢɡɚ ɧɚɲɟɝɨ ɩɪɨɫɬɨɝɨ ɩɪɢɦɟɪɚ ɩɨɬɪɟɛɨɜɚ-

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 13

ɥɢɫɶ ɥɢɲɶ ɫɚɦɵɟ ɨɫɧɨɜɧɵɟ ɟɝɨ ɫɪɟɞɫɬɜɚ. ɉɨɡɞɧɟɟ ɦɵ ɨɫɬɚɧɨɜɢɦɫɹ ɟɳɟ ɧɚ ɧɟɤɨɬɨɪɵɯ ɜɨɡɦɨɠɧɨɫɬɹɯ ɩɚɤɟɬɚ, ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɦɵɯ ɩɪɢ ɩɪɨɜɟɞɟɧɢɢ ɪɚɡɥɢɱɧɵɯ ɜɵɱɢɫɥɟɧɢɣ, ɢɯ ɜɢɡɭɚɥɢɡɚɰɢɢ, ɨɮɨɪɦɥɟɧɢɢ ɪɟɡɭɥɶɬɚɬɨɜ.

2.

(

).

ȼɵ ɫɥɵɲɢɬɟ ɩɨ ɦɟɫɬɧɨɦɭ ɪɚɞɢɨ, ɤɚɤ ɦɷɪ ȼɚɲɟɝɨ ɝɨɪɨɞɤɚ ɭɬɜɟɪɠɞɚɟɬ, ɱɬɨ ɫɪɟɞɧɢɣ ɞɨɯɨɞ ɟɝɨ ɠɢɬɟɥɟɣ ɫɨɫɬɚɜɥɹɟɬ 1 000 ɭɫɥɨɜɧɵɯ ɟɞɢɧɢɰ. ȼɵ ɜɨɡɦɭɳɟɧɵ! ɋɚɦɵɦ “ɩɨɩɭɥɹɪɧɵɦ” ɫɪɟɞɢ ȼɚɲɢɯ ɦɧɨɝɨɱɢɫɥɟɧɧɵɯ ɪɨɞɫɬɜɟɧɧɢɤɨɜ, ɡɧɚɤɨɦɵɯ, ɡɧɚɤɨɦɵɯ ɜɚɲɢɯ ɪɨɞɫɬɜɟɧɧɢɤɨɜ ɢ ɪɨɞɫɬɜɟɧɧɢɤɨɜ ȼɚɲɢɯ ɡɧɚɤɨɦɵɯ ɹɜɥɹɟɬɫɹ ɞɨɯɨɞ ɜ 500 ɭ.ɟ. ɏɨɬɹ, ɱɭɬɶ ɩɨɪɚɡɦɵɫɥɢɜ, ȼɵ ɩɨɧɢɦɚɟɬɟ, ɱɬɨ ɬɚɤɨɟ ɜɨɡɦɨɠɧɨ. ȿɫɥɢ ɩɪɟɞɫɬɚɜɢɬɟɥɢ ɜɵɫɲɟɝɨ ɫɜɟɬɚ (ɬ.ɟ. 1% ɧɚɫɟɥɟɧɢɹ) ɩɨɥɭɱɚɸɬ, ɫɤɚɠɟɦ, ɩɨ 100 000 ɭ.ɟ., ɬɨ ɞɚɠɟ ɩɪɢ ɧɭɥɟɜɨɦ ɞɨɯɨɞɟ ɨɫɬɚɥɶɧɨɝɨ ɧɚɫɟɥɟɧɢɹ (99%), ɫɪɟɞɧɢɣ ɞɨɯɨɞ ɛɭɞɟɬ ɫɨɫɬɚɜɥɹɬɶ: 0.01 ⋅ 100 000=1 000 (ɭ.ɟ.). Ƚɨɜɨɪɹ ɹɡɵɤɨɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ, ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɞɨɯɨɞɚ – , – ɧɟ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɪɟɞɧɢɦ ɟɝɨ ɡɧɚɱɟɧɢɟɦ, ɢɥɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ. ɑɟɪɟɡ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ȼɵ ɩɨɥɭɱɚɟɬɟ ɩɨɜɵɲɟɧɢɟ ɩɨ ɫɥɭɠɛɟ, ɢ ȼɚɲ ɞɨɯɨɞ ɜɵɪɚɫɬɚɟɬ ɚɠ ɞɨ 800 ɭ.ɟ. ȼɵ ɡɚɦɟɱɚɟɬɟ, ɱɬɨ ɩɨɥɨɜɢɧɚ ȼɚɲɢɯ ɫɨɫɥɭɠɢɜɰɟɜ ȼɚɦ ɡɚɜɢɞɭɟɬ, ɚ ɩɨɥɨɜɢɧɚ ɫɦɨɬɪɢɬ ɫɜɵɫɨɤɚ. Ɂɧɚɱɢɬ, ȼɵ – ɬɢɩɢɱɧɵɣ ɩɪɟɞɫɬɚɜɢɬɟɥɶ ɫɪɟɞɧɟɝɨ ɤɥɚɫɫɚ! ɇɨ ɞɨɯɨɞ-ɬɨ ɭ ȼɚɫ ɝɨɪɚɡɞɨ ɧɢɠɟ ɫɪɟɞɧɟɝɨ! ɗɬɨ ȼɚɦ ɧɟ ɞɚɟɬ ɩɨɤɨɹ, ɢ ȼɵ ɞɨɫɬɚɟɬɟ ɭɱɟɛɧɢɤɢ, ɝɞɟ ɢɡɥɚɝɚɸɬɫɹ ɨɫɧɨɜɧɵɟ ɫɜɟɞɟɧɢɹ ɢɡ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɟ. (ɏɨɪɨɲɨ ɟɫɥɢ ɭ ȼɚɫ ɟɫɬɶ ɤɧɢɝɢ ɩɨ ɩɪɢɤɥɚɞɧɨɣ ɫɬɚɬɢɫɬɢɤɟ, ɧɚɩɪɢɦɟɪ, [6], – ɩɪɨɱɢɬɚɣɬɟ ɞɥɹ ɧɚɱɚɥɚ ɩɟɪɜɵɟ ɞɜɟ ɝɥɚɜɵ. ȿɫɥɢ ɠɟ ȼɵ ɨɞɨɥɟɟɬɟ ɝɥɚɜɵ 9 ɢ 10 ɤɧɢɝɢ [7], ɬɨ ɜɨɫɩɪɢɹɬɢɟ ɩɨɫɥɟɞɭɸɳɟɝɨ ɦɚɬɟɪɢɚɥɚ ɷɬɨɣ ɛɪɨɲɸɪɵ ɧɟ ɫɨɫɬɚɜɢɬ ɞɥɹ ȼɚɫ ɬɪɭɞɚ.) Ɍɟɩɟɪɶ ɦɵ ɭɜɟɪɟɧɵ ɜ ɬɨɦ, ɱɬɨ ȼɵ ɜɥɚɞɟɟɬɟ ɨɫɧɨɜɧɵɦɢ ɩɨɧɹɬɢɹɦɢ ɢɡ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ (ɧɟɩɪɟɪɵɜɧɵɟ ɢ ɞɢɫɤɪɟɬɧɵɟ (ɫ.ɜ.), ɢ , ɱɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ.ɜ.: , , , ), ɚ ɬɚɤɠɟ ɢɡ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ ( , , ɩɚɪɚɦɟɬɪɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ). ɂɬɚɤ, 800 ɭ.ɟ. – ɷɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɨɯɨɞɚ, ɢɥɢ 50% . ɉɟɪɟɞ ɧɚɦɢ ɬɪɢ (!) ɪɚɡɥɢɱɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫ.ɜ., ɩɪɢɱɟɦ ɜɫɟ ɨɧɢ ɜ ɧɟɤɨɬɨɪɨɦ ɫɦɵɫɥɟ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɫɪɟɞɧɢɣ ɟɟ ɭɪɨɜɟɧɶ. (Ⱥɧɚɥɨɝɢɱɧɚɹ ɫɢɬɭɚɰɢɹ ɨɩɢɫɚɧɚ ɜ [8, ɫ.146]). ɗɬɨ ɜɫɟ ɩɨɧɹɬɧɨ. ɇɨ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɞɨɯɨɞɚ ɜ ɨɛɳɟɫɬɜɟ – ɷɬɨ ɜɫɟ ɪɚɜɧɨ, ɱɬɨ ɫɪɟɞɧɹɹ ɬɟɦɩɟɪɚɬɭɪɚ ɩɚɰɢɟɧɬɨɜ ɜ ɛɨɥɶɧɢɰɟ. ɑɬɨɛɵ ɨɰɟɧɢɬɶ ɫɨɫɬɨɹɧɢɟ ɨɛɳɟɫɬɜɚ, ɧɭɠɧɨ ɡɧɚɬɶ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɨɯɨɞɚ, ɚ ɧɟ ɬɨɥɶɤɨ ɟɝɨ ɨɬɞɟɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ. , .

14

ɗ.ɘ. Ʌɟɪɧɟɪ, Ɉ.Ⱥ. Ʉɚɲɢɧɚ

Ɉɞɧɢɦ ɢɡ ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɟɣ ɹɜɥɹɟɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ , ɢɥɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ. ɉɥɨɬɧɨɫɬɶ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫ ɩɚɪɚɦɟɬɪɚɦɢ (a, σ) ɢɦɟɟɬ ɜɢɞ: pη ( x ) =

1

σ 2π

−

e

( x −a )2 2σ 2

.

ɉɨ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ ɪɚɫɩɪɟɞɟɥɟɧɵ ɦɧɨɝɢɟ ɫ.ɜ., ɫɨɜɟɪɲɟɧɧɨ ɪɚɡɧɵɟ ɩɨ ɫɜɨɟɣ ɮɢɡɢɱɟɫɤɨɣ ɫɭɬɢ. ɇɟ ɩɨɞɨɣɞɟɬ ɥɢ ɨɧɨ ɞɥɹ ɨɩɢɫɚɧɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɞɨɯɨɞɨɜ? Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɦɚɬ. ɨɠɢɞɚɧɢɟ, ɦɨɞɚ ɢ ɦɟɞɢɚɧɚ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɨɜɩɚɞɚɸɬ ɫ ɩɚɪɚɦɟɬɪɨɦ a. (ȼɵ ɦɨɠɟɬɟ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ ɜ ɷɬɨɦ, ɢɫɩɨɥɶɡɭɹ ɨɱɟɜɢɞɧɨɟ ɪɚɜɟɧɫɬɜɨ pη ( a + t ) = pη ( a − t ) .) Ⱦɥɹ ɧɚɲɟɝɨ ɠɟ ɫɥɭɱɚɹ ɦɚɬ. ɨɠɢɞɚɧɢɟ, ɦɨɞɚ ɢ ɦɟɞɢɚɧɚ ɞɨɯɨɞɚ ɪɚɡɥɢɱɧɵ. Ɂɧɚɱɢɬ, ɞɨɯɨɞ ɜ ɧɚɲɟɦ ɝɨɪɨɞɟ ɹɜɧɨ ɪɚɫɩɪɟɞɟɥɟɧ ɧɟɧɨɪɦɚɥɶɧɨ! ɇɟ ɛɭɞɟɦ ɩɪɢɡɵɜɚɬɶ ɤ ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɸ ɞɨɯɨɞɨɜ, ɚ ɡɚɣɦɟɦɫɹ ɥɭɱɲɟ ɢɫɫɥɟɞɨɜɚɧɢɟɦ ɫɜɨɣɫɬɜ ɞɪɭɝɢɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɧɨɪɦɚɥɶɧɵɦ ɹɜɥɹɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɭɦɦɵ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɧɟɡɚɜɢɫɢɦɵɯ ɮɚɤɬɨɪɨɜ, – ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɝɨɜɨɪɹɬ ɨɛ ɯɚɪɚɤɬɟɪɟ ɢɯ ɜɨɡɞɟɣɫɬɜɢɹ ɧɚ ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɫ.ɜ. ȼ ɷɤɨɧɨɦɢɤɟ ɠɟ ɯɚɪɚɤɬɟɪ ɫɨɜɦɟɫɬɧɨɝɨ ɞɟɣɫɬɜɢɹ ɦɧɨɝɨɱɢɫɥɟɧɧɵɯ ɫɥɭɱɚɣɧɵɯ ɮɚɤɬɨɪɨɜ ɧɟɪɟɞɤɨ ɹɜɥɹɟɬɫɹ ɧɟ ɚɞɞɢɬɢɜɧɵɦ, ɚ : ɡɧɚɱɟɧɢɟ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɩɪɢɡɧɚɤɚ η, ɞɨɫɬɢɝɧɭɬɨɟ ɡɚ ɫɱɟɬ ɞɟɣɫɬɜɢɹ ɫɥɭɱɚɣɧɨɝɨ ɮɚɤɬɨɪɚ ξ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɬɟɤɭɳɟɦɭ ɡɧɚɱɟɧɢɸ η ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ (1+ξ). (Ɍɚɤ, ɜ ɩɪɢɦɟɪɟ ɢɡ ɪɚɡɞɟɥɚ 1, ɫɭɦɦɚ ɜɤɥɚɞɚ ɜ ɛɚɧɤɟ “Ɂɨɥɨɬɵɟ ɝɨɪɵ” ɧɚ 1 ɮɟɜɪɚɥɹ 2000 ɝɨɞɚ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɫɭɦɦɟ ɜɤɥɚɞɚ ɧɚ 1 ɹɧɜɚɪɹ 2000 ɝɨɞɚ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ (1+ξ), ɝɞɟ ξ ɪɚɜɧɚ 1/12 ɫɬɚɜɤɢ ɩɪɨɰɟɧɬɚ ɧɚ 2000 ɝɨɞ.) ȿɫɥɢ ɱɢɫɥɨ ɫɥɭɱɚɣɧɵɯ ɮɚɤɬɨɪɨɜ ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɨ, ɜɨɡɞɟɣɫɬɜɢɟ ɤɚɠɞɨɝɨ ɢɡ ɧɢɯ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɢ ɢɦɟɟɬ ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɵɣ ɯɚɪɚɤɬɟɪ, ɬɨ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɢɦɟɬɶ ɧɟ ɧɨɪɦɚɥɶɧɨɟ, ɚ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɧɨɪɦɚɥɶɧɨɟ (ɥɨɝɧɨɪɦɚɥɶɧɨɟ) ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɉɥɨɬɧɨɫɬɶ ɥɨɝɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: pη ( x ) =

1

σ x 2π

e

−

(ln x −ln a ) 2 2σ 2

(2.1)

Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ:

Fη ( x ) =

1

σ 2π

∫e

ln x

0

− ( t −ln a2 ) 2σ

2

dt

(2.2)

ɉɚɤɟɬ MATHEMATICȺ: ɩɟɪɜɵɟ ɭɪɨɤɢ 15

2.1. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɥɨɝɚɪɢɮɦ ɫ.ɜ., ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɥɨɝɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ ɫ ɩɚɪɚɦɟɬɪɚɦɢ (a, σ), ɢɦɟɟɬ ɧɨɪɦɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ ɩɚɪɚɦɟɬɪɚɦɢ (ln a, σ). Ʉɪɨɦɟ ɬɨɝɨ, ɩɨɤɚɠɢɬɟ ɫ ɩɨɦɨɳɶɸ ɩɚɤɟɬɚ MATHEMATICA (ɫɦ. ɭɤɚɡɚɧɢɟ ɤ ɞɚɧɧɨɦɭ ɭɩɪɚɠɧɟɧɢɸ), ɱɬɨ ɩɥɨɬɧɨɫɬɶ ɥɨɝɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɟɫɬɶ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ (2.2) ɢ ɧɚɱɟɪɬɢɬɟ ɟɟ ɝɪɚɮɢɤ ɩɪɢ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɨɜ a=800 ɢ σ=0.685568 ɢ ɫɪɚɜɧɢɬɟ ɟɝɨ ɫ ɝɪɚɮɢɤɨɦ ɧɚ ɪɢɫɭɧɤɟ 2.2. ( ): MATHEMATICA – ɜɵɫɨɤɨ ɢɧɬɟɥɥɟɤɬɭɚɥɶɧɵɣ ɩɚɤɟɬ, ɭɦɟɸɳɢɣ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɢ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɢ ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɨɦɚɧɞɵ: а

D[ Integrate[ ,

а

, а

а

], , {а

,

-

}].

. ȿɫɥɢ ȼɚɦ ɥɟɧɶ ɧɚɛɢɪɚɬɶ ɤɨɦɚɧɞɵ ɧɚ ɤɥɚɜɢɚɬɭɪɟ, ɢɫɩɨɥɶɡɭɣɬɟ ɲɚɛɥɨɧɵ ɢɡ ɩɚɧɟɥɢ: 2 Basic Calculations (ɫɦ. ɪɢɫ. 1.2). ȼɩɢɫɵɜɚɬɶ ɮɨɪɦɭɥɵ ɜ ɤɜɚɞɪɚɬɢɤɢ ɲɚɛɥɨɧɚ – ɪɚɛɨɬɚ ɩɨɱɬɢ ɸɜɟɥɢɪɧɚɹ. Ɉɛɥɟɝɱɢɬɶ ɟɟ ɦɨɠɧɨ ɡɚ ɫɱɟɬ “ɜɧɟɞɪɟɧɢɹ” ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ ɜ placeholder. Ⱦɥɹ ɷɬɨɝɨ ɧɭɠɧɨ ɜɵɞɟɥɢɬɶ ɮɨɪɦɭɥɭ ɢ ɧɚɠɚɬɶ ɲɚɛɥɨɧ ɢɧɬɟɝɪɚɥɚ. ɉɟɪɟɦɟɳɟɧɢɟ ɤɭɪɫɨɪɚ ɦɟɠɞɭ ɧɢɠɧɢɦ ɢ ɜɟɪɯɧɢɦ ɩɪɟɞɟɥɚɦɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɥɚɜɢɲ: Ctrl+% (Ctrl+5). :

ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ (ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ): M (η )

Ɇɨɞɚ:

= ae

Mode(η )

2

= ae −σ

Median (η )

Ɇɟɞɢɚɧɚ:

σ2

2

(2.3)

=a

ɂɡ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɜɵɲɟ ɮɨɪɦɭɥ ɜɢɞɧɨ, ɱɬɨ ɤɭɛ ɦɟɞɢɚɧɵ ɥɨɝɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɤɜɚɞɪɚɬɚ ɫɪɟɞɧɟɝɨ ɧɚ ɦɨɞɭ. ȼ ɧɚɲɟɦ ɩɪɢɦɟɪɟ ɷɬɨ ɪɚɜɟɧɫɬɜɨ ɜɵɩɨɥɧɹɟɬɫɹ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ: 800 ≈ 1000 ⋅ 500 . 3

2

.

ȼ ɩɚɤɟɬɟ MATHEMATICA ɟɫɬɶ ɨɬɞɟɥɶɧɵɟ ɦɨɞɭɥɢ (packages) ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɫɩɟɰɢɚɥɶɧɵɯ ɜɵɱɢɫɥɟɧɢɣ. ɑɬɨɛɵ ɜɵɩɨɥɧɢɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɪɚɫɱɟɬɵ, ɩɨɞɤɥɸɱɢɦ “ɦɚɫɬɟɪɚ”, ɤɨɬɨɪɵɣ ɛɭɞɟɬ ɫɚɦ ɨɫɭɳɟɫɬɜɥɹɬɶ ɩɨɢɫɤ ɧɭɠɧɨɝɨ ɦɨɞɭɥɹ: