ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ ȽɈɋɍȾȺɊɋɌȼȿɇɇɈȿ ɈȻɊȺɁɈȼȺɌȿɅɖɇɈȿ ɍɑɊȿɀȾȿɇɂȿ ȼɕɋɒȿȽɈ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə «ȼɈɊɈɇȿɀɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ»
ȼ.ɇ. Ƚɪɭɡɞɟɜ
ɆȺɌȿɆȺɌɂɑȿɋɄɂȿ ɍɊȺȼɇȿɇɂə ȼ ȽȿɈɎɂɁɂɄȿ ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɤ ɥɚɛɨɪɚɬɨɪɧɵɦ ɪɚɛɨɬɚɦ ɞɥɹ ɜɭɡɨɜ
ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ 2007
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɝɟɨɥɨɝɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ 17 ɦɚɹ 2007 ɝ., ɩɪɨɬɨɤɨɥ ʋ 6
Ɋɟɰɟɧɡɟɧɬ Ⱥ.ȼ. ɇɢɤɢɬɢɧ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɝɟɨɮɢɡɢɤɢ ɝɟɨɥɨɝɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɝɟɨɥɨɝɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ, ɨɛɭɱɚɸɳɢɯɫɹ ɧɚ 2 ɤɭɪɫɟ ɞ/ɨ. Ⱦɥɹ ɫɩɟɰɢɚɥɶɧɨɫɬɢ: 011200 (020302) – Ƚɟɨɮɢɡɢɤɚ
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ɇɚ ɝɟɨɥɨɝɢɱɟɫɤɨɦ ɮɚɤɭɥɶɬɟɬɟ ȼȽɍ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 2 ɤɭɪɫɚ ɞɧɟɜɧɨɝɨ ɨɬɞɟɥɟɧɢɹ ɫɩɟɰɢɚɥɢɡɚɰɢɢ «Ƚɟɨɮɢɡɢɤɚ» ɩɨ ɞɢɫɰɢɩɥɢɧɟ «Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɜ ɝɟɨɮɢɡɢɤɟ» ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɟɞɟɪɚɥɶɧɵɦ ɫɬɚɧɞɚɪɬɨɦ ɢ ɭɬɜɟɪɠɞɟɧɧɨɣ ɭɱɟɛɧɨɣ ɩɪɨɝɪɚɦɦɨɣ ɞɚɸɬɫɹ ɨɛɳɢɟ ɫɜɟɞɟɧɢɹ ɩɨ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɹɦ, ɢɫɩɨɥɶɡɭɟɦɵɦ ɜ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɨɛɨɫɧɨɜɚɧɢɹɯ ɝɟɨɮɢɡɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɪɚɡɜɟɞɤɢ. Ⱦɚɧɧɨɟ ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɫɨɞɟɪɠɢɬ ɡɚɞɚɧɢɹ ɤ ɥɚɛɨɪɚɬɨɪɧɵɦ ɪɚɛɨɬɚɦ ɩɨ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɹɦ, ɬɚɤɠɟ ɜ ɧɟɦ ɢɡɥɚɝɚɟɬɫɹ ɩɨɪɹɞɨɤ ɢɯ ɜɵɩɨɥɧɟɧɢɹ. Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 1 (3 ɱɚɫɚ) Ɂɚɞɚɧɢɟ 1. ɇɚɯɨɠɞɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɥɢɧɢɢ. Ⱦɚɧɵ ɞɜɟ ɬɨɱɤɢ Ⱥ(ɯ1, ɭ1) ɢ B(ɯ2, ɭ2). ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɥɢɧɢɢ, ɪɚɫɫɬɨɹɧɢɟ ɬɨɱɟɤ ɤɨɬɨɪɨɣ ɨɬ ɬɨɱɤɢ B ɜ k ɪɚɡ ɛɨɥɶɲɟ, ɱɟɦ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ Ⱥ. Ɂɧɚɱɟɧɢɹ ɯ1, ɯ2, ɭ1, ɭ2, k ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 1 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ Ⱥ(ɯ1, ɭ1) ɢ B(ɯ2, ɭ2) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: d = ( x2 x 1 ) 2 ( y 2 y1 ) 2 . Ɍɚɛɥɢɰɚ 1 ɇɨɦɟɪ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ɜɚɪɢɚɧɬɚ ɯ1 2 4 3 4 2 3 –4 4 –6 7 –3 2 –1 0 ɯ2 4 5 4 3 1 –4 3 –3 7 –3 4 –1 0 –5 ɭ1 0 –7 4 5 5 2 1 –4 4 –6 7 –3 2 –1 ɭ2 4 5 5 2 1 –4 4 –6 7 –3 2 –1 0 2 k 2 2 3 3 4 4 2 2 3 3 4 4 5 5 ɇɨɦɟɪ 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ɜɚɪɢɚɧɬɚ ɯ1 –3 –4 3 4 1 –1 –6 –3 –9 8 3 –3 5 –5 ɯ2 –4 –3 4 3 –1 –6 9 –4 8 3 –3 5 –5 6 ɭ1 3 4 1 –1 –6 9 –9 8 3 –3 5 –5 6 0 ɭ2 4 1 –1 –6 9 –9 8 3 –3 5 –5 6 0 –6 k 2 3 3 4 4 5 5 1 2 2 3 3 4 4 Ɂɚɞɚɧɢɟ 2. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ, ɫɜɹɡɚɧɧɵɯ ɫ ɪɚɫɩɨɥɨɠɟɧɢɟɦ ɥɢɧɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ. Ɂɚɞɚɱɚ 1. Ɂɚɞɚɧɨ ɭɪɚɜɧɟɧɢɟ ɥɢɧɢɢ ɯ2 + ɭ2 = 25 ɢ ɤɨɨɪɞɢɧɚɬɵ ɯ1 ɢ ɭ1 ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ Ɇ, ɨɩɪɟɞɟɥɢɬɶ, ɥɟɠɢɬ ɥɢ ɬɨɱɤɚ Ɇ ɧɚ ɞɚɧɧɨɣ ɥɢɧɢɢ. Ɂɧɚɱɟɧɢɹ ɯ1, ɯ2 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 1 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ.
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Ɂɚɞɚɱɚ 2. Ⱦɚɧɵ ɭɪɚɜɧɟɧɢɹ ɞɜɭɯ ɥɢɧɢɣ: ɭ = kɯ2 + ɭ2 ɢ ɭ = (x1 + x2)y1. Ɉɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɷɬɢɯ ɥɢɧɢɣ. Ɂɧɚɱɟɧɢɹ ɯ1, ɯ2, ɭ1, ɭ2, k ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 1 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɚɜɢɥɨ ɪɟɲɟɧɢɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ: ɧɭɠɧɨ ɜ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɥɢɧɢɢ ɩɨɞɫɬɚɜɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ Ɇ. ȿɫɥɢ ɩɪɢ ɷɬɨɦ ɭɪɚɜɧɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬɫɹ, ɬ. ɟ. ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɬɫɹ ɬɨɠɞɟɫɬɜɨ, ɬɨ ɬɨɱɤɚ ɥɟɠɢɬ ɧɚ ɞɚɧɧɨɣ ɥɢɧɢɢ. Ɂɚɞɚɱɚ 3. ɇɚɣɬɢ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɨɤɪɭɠɧɨɫɬɢ ɯ2 + ɭ2 = k2 ɫ ɨɫɹɦɢ ɤɨɨɪɞɢɧɚɬ. Ɂɧɚɱɟɧɢɟ k ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 1 ɫɨɝɥɚɫɧɨ ɡɚɞɚɧɧɨɦɭ ɜɚɪɢɚɧɬɭ. ɉɪɚɜɢɥɨ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 3: ɱɬɨɛɵ ɧɚɣɬɢ ɚɛɫɰɢɫɫɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɞɚɧɧɨɣ ɥɢɧɢɢ ɫ ɨɫɶɸ Ɉɏ, ɧɟɨɛɯɨɞɢɦɨ ɜ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɥɢɧɢɢ ɩɨɥɨɠɢɬɶ ɭ = 0 ɢ ɪɟɲɢɬɶ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɯ, ɱɬɨɛɵ ɧɚɣɬɢ ɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɞɚɧɧɨɣ ɥɢɧɢɢ ɫ ɨɫɶɸ Ɉɍ, ɧɭɠɧɨ ɜ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɥɢɧɢɢ ɩɨɥɨɠɢɬɶ ɯ = 0 ɢ ɪɟɲɢɬɶ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭ. ɉɪɢɦɟɪ 1. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɥɢɧɢɢ, ɪɚɫɫɬɨɹɧɢɟ ɬɨɱɟɤ ɤɨɬɨɪɨɣ ɨɬ ɬɨɱɤɢ ȼ (12, 16) ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟ, ɱɟɦ ɨɬ ɬɨɱɤɢ Ⱥ (3, 4). ɉɭɫɬɶ ɧɚ ɡɚɞɚɧɧɨɣ ɥɢɧɢɢ ɥɟɠɢɬ ɬɨɱɤɚ Ɇ(ɯ, ɭ). ɉɨ ɭɫɥɨɜɢɸ: 2ȺɆ = ȼɆ. ɇɚɣɞɟɦ ɪɚɫɫɬɨɹɧɢɹ ȺɆ ɢ ȼɆ. ȺɆ = ( x 3) 2 ( y 4) 2 , ȼɆ = ( x 12) 2 ( y 16) 2 . Ɍɨɝɞɚ 2 ( x 3) 2 ( y 4) 2 = ( x 12) 2 ( y 16) 2 . ɉɪɟɨɛɪɚɡɭɟɦ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜ ɜɢɞɟ: 4((ɯ – 3)2 + (ɭ – 4)2) = (ɯ – 12)2 + + (ɭ – 16)2 ɢɥɢ 4ɯ2 – 24ɯ + 36 + 4ɭ2 – 32ɭ + 64 = ɯ2 – 24ɯ + 144 + ɭ2 – 32ɭ + 256. ɍɩɪɨɫɬɢɦ ɞɚɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɢ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɢɫɤɨɦɨɣ ɥɢɧɢɢ: 3ɯ2 + 4ɭ2 = 156. ɉɪɢɦɟɪ 2. ɇɚɣɬɢ ɬɨɱɤɭ ɩɟɪɟɫɟɱɟɧɢɹ ɩɚɪɚɛɨɥɵ ɭ = ɯ2 ɢ ɩɪɹɦɨɣ ɭ = 4. Ⱦɨɫɬɚɬɨɱɧɨ ɫɨɜɦɟɫɬɧɨ ɪɟɲɢɬɶ ɫɢɫɬɟɦɭ ɢɡ ɭɪɚɜɧɟɧɢɣ ɷɬɢɯ ɥɢɧɢɣ: y ® ¯y
x2 4
; ɯ2 = 4, ɨɬɤɭɞɚ ɯ1 = 2, ɭ1 = 4, ɯ2 = –2, ɭ2 = 4. Ⱥ(–2, 4) ɢ ȼ(2, 4) – ɬɨɱɤɢ
ɩɟɪɟɫɟɱɟɧɢɹ ɞɚɧɧɵɯ ɥɢɧɢɣ. ȿɫɥɢ ɫɢɫɬɟɦɚ ɧɟ ɢɦɟɟɬ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɪɟɲɟɧɢɣ, ɬɨ ɥɢɧɢɢ ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ. Ɂɚɞɚɧɢɟ 3. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ, ɫɜɹɡɚɧɧɵɯ ɫ ɭɪɚɜɧɟɧɢɟɦ ɩɪɹɦɨɣ ɥɢɧɢɢ. Ɂɚɞɚɱɚ 4. Ⱦɚɧɵ ɤɨɨɪɞɢɧɚɬɵ ɞɜɭɯ ɬɨɱɟɤ Ⱥ(ɯ1, ɭ1) ɢ ȼ(ɯ2, ɭ2) ɢ ɤɨɷɮɮɢɰɢɟɧɬ k. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ: ɚ) ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɥɢɧɢɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ⱥ(ɯ1, ɭ1) ɫ ɡɚɞɚɧɧɵɦ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ k; ɛ) ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɥɢɧɢɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɜɟ ɡɚɞɚɧɧɵɟ ɬɨɱɤɢ Ⱥ(ɯ1, ɭ1) ɢ ȼ(ɯ2, ɭ2); ɜ) ɡɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɵɯ, ɩɨɥɭɱɟɧɧɵɯ ɜ ɩɭɧɤɬɚɯ ɚ) ɢ ɛ) ɜ ɨɛɳɟɦ ɜɢɞɟ, ɜ ɜɢɞɟ ɭɪɚɜɧɟɧɢɹ ɫ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ, ɜ ɜɢɞɟ ɭɪɚɜɧɟɧɢɹ ɜ «ɨɬɪɟɡɤɚɯ»; ɝ) ɧɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɩɪɹɦɵɯ ɤ ɥɢɧɢɹɦ, ɩɨɥɭɱɟɧɧɵɦ ɜ ɩɭɧɤɬɚɯ ɚ) ɢ ɛ), ɢ ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɬɨɱɤɭ ȼ(ɯ2, ɭ2). Ɂɧɚɱɟɧɢɹ ɯ1, ɯ2, ɭ1, ɭ2, k ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 1 ɫɨɝɥɚɫɧɨ ɡɚɞɚɧɧɨɦɭ ɜɚɪɢɚɧɬɭ. 4
ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɚɧɧɭɸ ɬɨɱɤɭ Ⱥ(ɯ1, ɭ1) ɫ ɡɚɞɚɧɧɵɦ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: ɭ – ɭ1 = = k(ɯ – ɯ1). ɍɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɜɟ ɡɚɞɚɧɧɵɟ ɬɨɱɤɢ Ⱥ(ɯ1, ɭ1) ɢ ȼ(ɯ2, ɭ2), ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: ɜ ɨɬɪɟɡɤɚɯ ɢɦɟɟɬ ɜɢɞ:
y y1 y 2 y1
=
x x1 . ɍɪɚɜɧɟɧɢɟ x 2 x1
x y + =1. ɉɭɫɬɶ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɥɢɧɢɢ ɡɚɞɚɧɨ ɜ a b
ɨɛɳɟɦ ɜɢɞɟ: Ⱥɯ + ȼɭ + ɋ = 0. Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɤ ɷɬɨɣ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɝɨ ɱɟɪɟɡ ɬɨɱɤɭ Ɇ(ɯ1, ɭ1) ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɩɨ ɮɨɪɦɭɥɟ: Ⱥ(ɭ – ɭ1) – ȼ(ɯ – ɯ1) = 0. ɉɪɢɦɟɪ 3. Ⱦɚɧɨ Ⱥ(2, 3), ȼ(–1, 4), k = 4: ɚ) ɭ – 3 = 4(ɯ – 2); ɭ = 4ɯ – 5 – ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ⱥ(2, 3) ɫ ɡɚɞɚɧɧɵɦ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ k = 4; ɛ)
x2 1 2 1 11 ; ɭ – 3 = x ; ɭ = x – ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ 1 2 3 3 3 3
y 3 43
ɥɢɧɢɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɞɜɟ ɡɚɞɚɧɧɵɟ ɬɨɱɤɢ Ⱥ(2, 3) ɢ ȼ(–1, 4); ɜ) 4ɯ – ɭ – 5 = 0; ɯ + 3ɭ – 11 = 0 – ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɣ ɩɪɹɦɵɯ; 1 11 – ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɵɯ ɫ ɭɝɥɨɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ; 3 3 x y 1 – ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɵɯ ɜ ɨɬɪɟɡɤɚɯ; 11 11 3
ɭ = 4ɯ – 5; ɭ = x x y 5 5 4
1;
ɝ) 4(ɭ – 4) + (ɯ + 1) = 0; ɭ = 1/4ɯ + 15/4 – ɭɪɚɜɧɟɧɢɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɤ ɩɪɹɦɨɣ 4ɯ – ɭ – 5 = 0; (ɭ – 4) – 3(ɯ + 1) = 0; ɭ = 3ɯ + 7 – ɭɪɚɜɧɟɧɢɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ ɤ ɩɪɹɦɨɣ ɯ + 3ɭ – 11 = 0. Ɂɚɞɚɧɢɟ 4. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ, ɫɜɹɡɚɧɧɵɯ ɫ ɭɪɚɜɧɟɧɢɹɦɢ ɤɪɢɜɵɯ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ. Ɍɚɛɥɢɰɚ 2 ɇɨɦɟɪ 1 2 3 4 5 6 7 8 9 10 11 12 13 ɜɚɪɢɚɧɬɚ Ⱥ 2 4 3 4 2 3 4 3 6 7 3 2 1 C 8 7 4 5 5 2 1 4 4 6 7 3 2 D 4 5 5 2 1 4 4 3 1 4 3 3 7 E 2 2 3 3 4 4 2 2 3 3 4 4 5 F – – – – – – – – – – – – – 13 10 14 13 13 7 13 14 10 13 10 14 13 ɇɨɦɟɪ ɜɚɪɢɚɧɬɚ A C D E F
14
15 16 17 18 19 20
21 22
23 24 25 26 27
4 3 3 4 3 4 1 1 6 3 9 8 3 3 1 2 4 5 1 1 6 9 9 8 3 3 5 5 2 5 4 1 1 6 9 9 8 3 3 5 5 6 5 1 2 3 3 4 4 5 5 1 2 2 3 3 –13 –7 –8 –7 –6 –9 –14 –8 –14 –8 –9 –6 –9 –9 5
Ɂɚɞɚɱɚ 5. Ⱦɚɧ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɨɤɪɭɠɧɨɫɬɢ: Ⱥɯ2 + Ⱥy2 + Dx + + Ey + F = 0. ɋɨɫɬɚɜɢɬɶ ɧɨɪɦɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɤɪɭɠɧɨɫɬɢ, ɧɚɣɬɢ ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɢ ɪɚɞɢɭɫ. ɇɚɱɟɪɬɢɬɶ ɨɤɪɭɠɧɨɫɬɶ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉɏɍ. Ɂɧɚɱɟɧɢɹ Ⱥ, D, E, F ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 2 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɇɨɪɦɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɨɤɪɭɠɧɨɫɬɢ ɢɦɟɟɬ ɜɢɞ: (ɯ – ɯ0)2 + (ɭ – ɭ0)2 = = R2, ɝɞɟ ɯ0, ɭ0 – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɨɤɪɭɠɧɨɫɬɢ, R – ɪɚɞɢɭɫ ɨɤɪɭɠɧɨɫɬɢ. Ɂɚɞɚɱɚ 6. Ⱦɚɧ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɷɥɥɢɩɫɚ: Ⱥɯ2 + ɋy2 + Dx + Ey + + F = 0. ɋɨɫɬɚɜɢɬɶ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɷɥɥɢɩɫɚ, ɧɚɣɬɢ ɩɨɥɭɨɫɢ ɷɥɥɢɩɫɚ, ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ, ɮɨɤɭɫɨɜ, ɰɟɧɬɪɚ ɢ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ. ɇɚɱɟɪɬɢɬɶ ɷɥɥɢɩɫ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉɏɍ. Ɂɧɚɱɟɧɢɹ Ⱥ, ɋ, D, E, F ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 2 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ʉɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɷɥɥɢɩɫɚ ɢɦɟɟɬ ɜɢɞ:
( x x0 ) 2 ( y y0 ) 2 + = 1, a2 b2
ɝɞɟ ɯ0, ɭ0 – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɷɥɥɢɩɫɚ, ɚ, b – ɩɨɥɭɨɫɢ ɷɥɥɢɩɫɚ. Ɍɨɱɤɢ Ⱥ1(ɚ, 0), ȼ1(0, b), Ⱥ2(–ɚ, 0), B2(0, –b) – ɜɟɪɲɢɧɵ, ɚ F1(c, 0) ɢ F2(–c, 0) – x2 a2
ɮɨɤɭɫɵ ɷɥɥɢɩɫɚ, ɨɩɢɫɵɜɚɟɦɨɝɨ ɭɪɚɜɧɟɧɢɟɦ
y2 b2
+
= 1. ȼɟɥɢɱɢɧɚ
c – ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɷɥɥɢɩɫɚ. a
ɫ = a 2 b 2 – ɮɨɤɭɫɧɨɟ ɪɚɫɫɬɨɹɧɢɟ. H
Ɂɚɞɚɱɚ 7. Ⱦɚɧ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɝɢɩɟɪɛɨɥɵ: Ⱥɯ2 – ɋy2 + Dx + Ey + + F = 0. ɋɨɫɬɚɜɢɬɶ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɝɢɩɟɪɛɨɥɵ, ɧɚɣɬɢ ɩɨɥɭɨɫɢ ɝɢɩɟɪɛɨɥɵ, ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ, ɮɨɤɭɫɨɜ ɢ ɰɟɧɬɪɚ, ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ, ɭɪɚɜɧɟɧɢɹ ɚɫɢɦɩɬɨɬ. ɇɚɱɟɪɬɢɬɶ ɝɢɩɟɪɛɨɥɭ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉɏɍ. Ɂɧɚɱɟɧɢɹ Ⱥ, ɋ, D, E, F ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 2 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ʉɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɝɢɩɟɪɛɨɥɵ:
( x x0 ) 2 a2
( y y0 ) 2 b2
–
= 1 ɢ
( x x0 ) 2 ( y y0 ) 2 – = –1, ɝɞɟ ɯ0, ɭ0 – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɝɢɩɟɪɛɨɥɵ; ɚ – 2 a b2
ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɩɨɥɭɨɫɶ ɞɥɹ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɦɧɢɦɚɹ ɩɨɥɭɨɫɶ ɞɥɹ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ; b – ɦɧɢɦɚɹ ɩɨɥɭɨɫɶ ɞɥɹ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɩɨɥɭɨɫɶ ɞɥɹ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ. ȼɟɥɢɱɢɧɚ ɫ = a 2 b 2 ɧɚɡɵɜɚɟɬɫɹ ɮɨɤɭɫɧɵɦ ɪɚɫɫɬɨɹɧɢɟɦ. Ɍɨɱɤɢ Ⱥ1(ɚ, 0), Ⱥ2(–ɚ, 0) – ɜɟɪɲɢɧɵ, F1(c, 0) ɢ
c – ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɝɢɩɟɪɛɨɥɵ, a
F2(–c, 0) – ɮɨɤɭɫɵ, ɚ H
ɨɩɢɫɵɜɚɟɦɨɣ ɭɪɚɜɧɟɧɢɟɦ ɜɟɪɲɢɧɵ, F1(0, ɫ) ɢ
x2 a2
–
y2 b2
= 1. Ɍɨɱɤɢ ȼ1(0, b), B2(0, –b) –
F2(0, –ɫ) – ɮɨɤɭɫɵ, ɚ H 2
ɝɢɩɟɪɛɨɥɵ, ɨɩɢɫɵɜɚɟɦɨɣ ɭɪɚɜɧɟɧɢɟɦ
c b
– ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ
2
x y – 2 = –1. y a2 b
r
b x – ɭɪɚɜɧɟɧɢɹ a
ɚɫɢɦɩɬɨɬ ɝɢɩɟɪɛɨɥɵ. Ɂɚɞɚɱɚ 8. Ⱦɚɧ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɣ ɩɚɪɚɛɨɥ: Ⱥɯ2 + Dx + Ey + F = 0. 2 ɢ ɋy + Dx + Ey + F = 0. ɋɨɫɬɚɜɢɬɶ ɤɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥ, ɧɚɣɬɢ 6
ɨɫɢ ɫɢɦɦɟɬɪɢɢ, ɤɨɨɪɞɢɧɚɬɵ ɮɨɤɭɫɨɜ ɢ ɭɪɚɜɧɟɧɢɟ ɞɢɪɟɤɬɪɢɫɵ. ɇɚɱɟɪɬɢɬɶ ɩɚɪɚɛɨɥɵ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ Ɉɏɍ Ɂɧɚɱɟɧɢɹ Ⱥ, ɋ, D, E, F ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 2 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ʉɚɧɨɧɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɩɚɪɚɛɨɥɵ: (ɯ – ɯ0)2 = 2ɪ(ɭ – ɭ0) (ɫ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɶɸ ɫɢɦɦɟɬɪɢɢ ɯ = ɯ0) ɢ (ɭ – ɭ0)2 = 2ɪ(ɯ – ɯ0) (ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɶɸ ɫɢɦɦɟɬɪɢɢ ɭ = ɭ0), ɝɞɟ: ɪ – ɩɚɪɚɦɟɬɪ ɩɚɪɚɛɨɥɵ, ɬɨɱɤɚ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ ɯ0, ɭ0 – ɜɟɪɲɢɧɚ ɩɚɪɚɛɨɥɵ. Ⱦɥɹ ɩɚɪɚɛɨɥɵ, p 2
ɨɩɢɫɵɜɚɟɦɨɣ ɭɪɚɜɧɟɧɢɟɦ ɯ2 = 2ɪɭ, ɬɨɱɤɚ F(0, ) ɧɚɡɵɜɚɟɬɫɹ ɮɨɤɭɫɨɦ, ɚ ɩɪɹɦɚɹ ɭ = –
p – ɞɢɪɟɤɬɪɢɫɨɣ. Ⱦɥɹ ɩɚɪɚɛɨɥɵ, ɨɩɢɫɵɜɚɟɦɨɣ ɭɪɚɜɧɟɧɢɟɦ ɭ2 2
=
2ɪɯ,
ɬɨɱɤɚ
p p F( , 0) ɧɚɡɵɜɚɟɬɫɹ ɮɨɤɭɫɨɦ, ɚ ɩɪɹɦɚɹ ɯ = – – ɞɢɪɟɤɬɪɢɫɨɣ. 2 2
ɉɪɢɦɟɪ 4. Ⱦɚɧ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɷɥɥɢɩɫɚ: 2ɯ2 + 4y2 + 4x + 8y + + 5 = 0. ɋɨɫɬɚɜɢɬɶ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɷɥɥɢɩɫɚ, ɧɚɣɬɢ ɩɨɥɭɨɫɢ ɷɥɥɢɩɫɚ, ɤɨɨɪɞɢɧɚɬɵ ɜɟɪɲɢɧ, ɮɨɤɭɫɨɜ, ɰɟɧɬɪɚ ɢ ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ. 2ɯ2 + 4y2 + 4x + 8y – 6 = 0 2(ɯ2 + 2 ǜ ɯ ǜ 1 + 1 - 1) + 4(ɭ2 + 2 ǜ ɭ ǜ 1 + + 1 – 1) – 6 = 0 2(ɯ + 1)2 – 2 + 4(ɭ + 1)2 – 4 – 6 = 0 2(ɯ + 1)2 + 4(ɭ + + 1)2 = 12
( x 1) 2 ( y 1) 2 6 3
1 – ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɷɥɥɢɩɫɚ; ɚ =
6,
3 – ɩɨɥɭɨɫɢ ɷɥɥɢɩɫɚ, Ɉ(–1, –1) – ɤɨɨɪɞɢɧɚɬɵ ɰɟɧɬɪɚ ɷɥɥɢɩɫɚ, ɋ = 63 3 – ɮɨɤɭɫɧɨɟ ɪɚɫɫɬɨɹɧɢɟ, F1( 3 , 0), F2(– 3 , 0) – ɮɨɤɭɫɵ ɷɥɥɢɩɫɚ, Ⱥ1( 6 , 0), Ⱥ2(– 6 , 0), B1(0, 3 ), B2(0, – 3 ) – ɜɟɪɲɢɧɵ ɷɥɥɢɩɫɚ.
b =
H
3 6
0,5 – ɷɤɫɰɟɧɬɪɢɫɢɬɟɬ ɷɥɥɢɩɫɚ.
Ɂɚɞɚɧɢɟ 5. Ɉɩɟɪɚɰɢɢ ɫ ɜɟɤɬɨɪɚɦɢ. Ʉɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɨɜ ɚ, b ɢ ɫ ɜɵɛɪɚɬɶ ɢɡ ɬɚɛɥɢɰɵ 3 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ɍɚɛɥɢɰɚ 3 ɇɨɦɟɪ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ɜɚɪɢɚɧɬɚ ɚɯ 2 4 3 4 2 3 –4 4 –6 7 –3 2 –1 0 ɚɭ 4 5 4 3 1 –4 3 –3 7 –3 4 –1 0 –5 ɚz 0 –7 4 5 5 2 1 –4 4 –6 7 –3 2 –1 bx 4 5 5 2 1 –4 4 –6 7 –3 2 –1 0 2 by 2 2 3 3 4 4 2 2 3 3 4 4 5 5 bz 5 –4 2 1 –6 –8 9 4 0 –2 8 4 3 5 cx 3 –4 4 –6 7 1 –4 3 –3 7 –4 3 –3 7 cy –4 3 –3 7 –3 3 –4 4 –6 7 7 –3 2 –1 cz 7 –3 2 –1 0 –4 3 –3 7 –3 3 –4 4 –6 7
ɇɨɦɟɪ ɜɚɪɢɚɧɬɚ ɚɯ ɚɭ ɚz bx by bz cx cy cz
15 16 17 18 19 20 21 22 23 24
25
26
27
28
–3 –4 3 4 1 –1 –6 –4 –3 4 3 –1 –6 9 3 4 1 –1 –6 9 –9 4 1 –1 –6 9 –9 8 2 3 3 4 4 5 5 9 4 0 –2 8 4 3 –1 4 –6 9 –4 8 3 4 1 –1 –6 9 –3 –4 –3 –4 3 4 4 1 –1
3 –3 5 –5 3 –4 –6 8 –4
–3 5 –5 6 3 2 9 3 9
5 –5 6 0 4 1 –3 –3 –4
–5 6 0 –6 4 –6 –4 –4 8
–3 –9 8 –4 8 3 8 3 –3 3 –3 5 1 2 2 5 1 5 4 1 –1 –6 9 –4 –6 9 –3
Ɂɚɞɚɱɚ 9. Ɉɩɪɟɞɟɥɢɬɶ ɞɥɢɧɭ ɜɟɤɬɨɪɨɜ ɚ, b, ɫ, ɢɯ ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɢ ɡɚɩɢɫɚɬɶ ɤɨɨɪɞɢɧɚɬɧɭɸ ɮɨɪɦɭ ɜɟɤɬɨɪɨɜ. Ⱦɥɢɧɚ ɜɟɤɬɨɪɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ: |a| = ɚ = a x2 a y2 a z2 . ɇɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɜɟɤɬɨɪɚ a ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɣ: ɚɯ = |a| cos Į, ɚɭ = |a| cos ȕ, ɚz = |a| cos Ȗ. Ʉɨɨɪɞɢɧɚɬɧɚɹ ɮɨɪɦɚ ɜɟɤɬɨɪɨɜ ɢɦɟɟɬ ɜɢɞ: ɚ = ɚɯi + ɚɭj + ɚzk. Ɂɚɞɚɱɚ 10. ɍɦɧɨɠɢɬɶ ɜɟɤɬɨɪɵ ɚ, b, ɫ ɧɚ ɫɤɚɥɹɪ Ȝ = – 3. ɇɚɣɬɢ: ɫɭɦɦɭ ɜɟɤɬɨɪɨɜ ɚ, b, ɫ; ɪɚɡɧɨɫɬɶ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɚ ɢ b; ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨɪɨɜ ɚ ɢ b; ɫɤɚɥɹɪɧɵɣ ɤɜɚɞɪɚɬ ɜɟɤɬɨɪɚ ɫ; ɤɨɫɢɧɭɫ ɭɝɥɚ ɦɟɠɞɭ ɜɟɤɬɨɪɚɦɢ ɚ ɢ b, ɚ ɢ ɫ, b ɢ ɫ. ɉɪɢ ɭɦɧɨɠɟɧɢɢ ɜɟɤɬɨɪɚ ɧɚ ɫɤɚɥɹɪ, ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɚ ɭɦɧɨɠɚɸɬɫɹ ɧɚ ɷɬɨɬ ɫɤɚɥɹɪ, ɬ. ɟ. Ȝɚ = Ȝɚɯi + Ȝɚɭj + Ȝɚzk. ɉɪɢ ɫɥɨɠɟɧɢɢ (ɢɥɢ ɜɵɱɢɬɚɧɢɢ) ɜɟɤɬɨɪɨɜ ɢɯ ɨɞɧɨɢɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɤɥɚɞɵɜɚɸɬɫɹ (ɢɥɢ ɜɵɱɢɬɚɸɬɫɹ), ɬ. ɟ. ɚ ± b = (ax ± bx)i + (ay ± by)j + (az ± bz)k. ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɚ ɢ b ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: ɚ ǜ b = ɚɯbɯ + ɚɭbɭ + ɚzbz. ɋɤɚɥɹɪɧɵɣ ɤɜɚɞɪɚɬ ɜɟɤɬɨɪɚ ɪɚɜɟɧ ɤɜɚɞɪɚɬɭ ɦɨɞɭɥɹ ɷɬɨɝɨ ɜɟɤɬɨɪɚ, ɬ. ɟ. ɚ2 = ɚ2. Ʉɨɫɢɧɭɫ ɭɝɥɚ ɦɟɠɞɭ ɞɜɭɦɹ ɧɟɧɭɥɟɜɵɦɢ ɜɟɤɬɨɪɚɦɢ ɚ ɢ b ɪɚɜɟɧ: cos ij = G G a b . ab
Ɂɚɞɚɱɚ 11. ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɟ ɤɨɥɥɢɧɟɚɪɧɨɫɬɢ, ɨɩɪɟɞɟɥɢɬɶ, ɤɨɥɥɢɧɟɚɪɧɵ ɥɢ ɜɟɤɬɨɪɵ ɚ, b, ɫ ɩɨɩɚɪɧɨ ɦɟɠɞɭ ɫɨɛɨɣ. ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɫɬɢ ɜɟɤɬɨɪɨɜ, ɨɩɪɟɞɟɥɢɬɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɥɢ ɜɟɤɬɨɪɚ ɚ, b, ɫ ɩɨɩɚɪɧɨ ɦɟɠɞɭ ɫɨɛɨɣ. ɇɚɣɬɢ ɜɟɤɬɨɪɚ ɢ ɢɯ ɦɨɞɭɥɢ, ɤɨɬɨɪɵɟ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɟɤɬɨɪɨɜ ɚ ɢ b, ɚ ɢ ɫ, b ɢ ɫ. ɇɚɣɬɢ ɫɦɟɲɚɧɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɬɪɟɯ ɜɟɤɬɨɪɨɜ ɚ, b ɢ ɫ. Ⱦɜɚ ɜɟɤɬɨɪɚ ɤɨɥɥɢɧɟɚɪɧɵ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɢɯ ɨɞɧɨɢɦɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ, ɬ. ɟ.
ax bx
ay by
az . Ⱦɜɚ ɜɟɤɬɨɪɚ bz
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɫɭɦɦɚ ɩɚɪɧɵɯ ɩɪɨɢɡɜɟɞɟɧɢɣ ɢɯ ɨɞɧɨɢɦɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɪɚɜɧɚ ɧɭɥɸ, ɬ. ɟ. ɚɯbɯ + ɚɭbɭ + ɚzbz = 0. ɉɨɞ 8
ɜɟɤɬɨɪɧɵɦ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɚ ɢ b ɩɨɧɢɦɚɟɬɫɹ ɜɟɤɬɨɪ ɫ = ɚ ɯ b = [ɚ, b], ɦɨɞɭɥɶ ɤɨɬɨɪɨɝɨ ɪɚɜɟɧ ɩɥɨɳɚɞɢ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ, ɩɨɫɬɪɨɟɧɧɨɝɨ ɧɚ ɞɚɧɧɵɯ ɜɟɤɬɨɪɚɯ, ɬ. ɟ. |c| = ab sin ij, ɷɬɨɬ ɜɟɤɬɨɪ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɩɥɨɫɤɨɫɬɢ, ɩɨɫɬɪɨɟɧɧɨɣ ɧɚ ɧɢɯ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦɚ ɢ ɜɟɤɬɨɪɵ ɚ, b ɢ ɫ ɨɛɪɚɡɭɸɬ ɩɪɚɜɭɸ ɬɪɨɣɤɭ ɜɟɤɬɨɪɨɜ. ȿɫɥɢ ɞɜɚ ɜɟɤɬɨɪɚ ɡɚɞɚɧɵ ɫɜɨɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ, ɬ. ɟ. ɚ = ɚɯi + ɚɭj + ɚzk ɢ b = bɯi + bɭj + bzk, ɬɨ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɬɚɤɢɯ ɜɟɤɬɨɪɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ ɨɩɪɟɞɟɥɢɬɟɥɹ
G i ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ ɩɨ ɮɨɪɦɭɥɟ: ɚ ɯ b = a x bx
G j
ay by
G k a z . ȿɫɥɢ ɜɟɤɬɨɪɵ ɡɚɞɚɧɵ bz
ɫɜɨɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ, ɬ. ɟ. ɚ = ɚɯi + ɚɭj + ɚzk, b = bɯi + bɭj + bzk ɢ ɫ = ɫɯi + + ɫɭj + ɫzk ɬɨ ɫɦɟɲɚɧɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɬɚɤɢɯ ɜɟɤɬɨɪɨɜ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ ax
ay
az
ɨɩɪɟɞɟɥɢɬɟɥɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ ɩɨ ɮɨɪɦɭɥɟ: ɚbɫ = bx b y bz . cx
cy
cz
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 2 (3 ɱɚɫɚ) Ɂɚɞɚɧɢɟ 1. Ɂɚɞɚɱɢ ɧɚ ɭɪɚɜɧɟɧɢɹ ɩɥɨɫɤɨɫɬɢ ɢ ɩɪɹɦɨɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ. Ɂɚɞɚɱɚ 1. ɇɚɣɬɢ: ɚ) ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɡɚɞɚɧɧɭɸ ɬɨɱɤɭ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (ɯ1, ɭ1, z1), ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɜɟɤɬɨɪɭ N(A,B,C); ɛ) ɧɨɪɦɢɪɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɟɧɧɨɣ ɩɥɨɫɤɨɫɬɢ; ɜ) ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɪɢ ɬɨɱɤɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (ɯ1,ɭ1,z1), (ɯ2,ɭ2,z2) ɢ (ɯ3,ɭ3,z3); ɝ) ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɬɨɱɤɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (ɯ2,ɭ2,z2) ɢ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɞɨ ɩɥɨɫɤɨɫɬɢ, ɭɪɚɜɧɟɧɢɟ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɟɧɨ ɜ ɩɭɧɤɬɟ ɚ); ɞ) ɞɜɭɝɪɚɧɧɵɣ ɭɝɨɥ ɦɟɠɞɭ ɞɜɭɦɹ ɩɥɨɫɤɨɫɬɹɦɢ, ɭɪɚɜɧɟɧɢɹ ɤɨɬɨɪɵɯ ɩɨɥɭɱɟɧɵ ɜ ɩɭɧɤɬɚɯ ɚ) ɢ ɜ). ȼɫɟ ɧɟɨɛɯɨɞɢɦɟɟ ɞɚɧɧɵɟ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɜɵɛɢɪɚɸɬɫɹ ɢɡ ɬɚɛɥɢɰɵ 4 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɍɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ɇ1(ɯ1, ɭ1, z1) ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɜɟɤɬɨɪɭ N(Ⱥ, ȼ, ɋ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: Ⱥ(ɯ – ɯ1) + + ȼ(ɭ – ɭ1) + ɋ(z – z1) = 0. ɇɨɪɦɢɪɨɜɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɥɨɫɤɨɫɬɢ ɢɦɟɟɬ ɜɢɞ: Ax By Cz D A2 B 2 C 2
= 0. Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ Ɇ2(ɯ2, ɭ2, z2) ɞɨ ɩɥɨɫɤɨɫɬɢ,
ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: d =
Ax1 By1 Cz1 D A2 B 2 C 2
. Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɧɚɱɚɥɚ
ɤɨɨɪɞɢɧɚɬ ɞɨ ɩɥɨɫɤɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ: d0 =
D 2
A B2 C 2
. ɉɭɫɬɶ
ɞɚɧɵ ɞɜɟ ɩɥɨɫɤɨɫɬɢ Ⱥ1ɯ + ȼ1ɭ + ɋ1z + D1 = 0 ɢ Ⱥ2ɯ + ȼ2ɭ + ɋ2z + D2 = 0 ɫ ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɜɟɤɬɨɪɚɦɢ N1(A1, B1, C1) ɢ N2(A2, B2, C2), ɬɨɝɞɚ
9
ɞɜɭɝɪɚɧɧɵɣ ɭɝɨɥ ɦɟɠɞɭ ɧɢɦɢ ɪɚɜɟɧ ɭɝɥɭ, ɨɛɪɚɡɨɜɚɧɧɨɦɭ ɜɟɤɬɨɪɚɦɢ N1 ɢ G G N N
N2 ɢ ɤɨɫɢɧɭɫ ɟɝɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: cos ij = G 1 G2 . N1 N 2
10
11
12
Ɍɚɛɥɢɰɚ 4 13 14
4 –6 7 –3 7 –3 –4 4 –6 –6 7 –3 2 3 3 4 0 –2 3 –3 7 4 –6 7 –3 7 –3 4 2 3 –3 7 –3 4 1 –4
–3 4 7 2 4 8 –4 7 3 –4 4 3
2 –1 –3 –1 4 4 3 –3 –4 4 5 –3
–1 0 2 0 5 3 –3 2 4 –6 5 7
0 –5 –1 2 5 5 7 –1 –6 4 1 –3
16 17 18 19 20 21 22 23 24
25
26
27
28
–4 –3 4 1 3 4 4 1 –4 4 3 4
3 –3 5 –5 3 –4 –6 8 –4 –5 5 4
–3 5 –5 6 3 2 9 3 9 6 5 3
5 –5 6 0 4 1 –3 –3 –4 –3 1 –1
–5 6 0 –6 4 –6 –4 –4 8 4 2 –6
ɇɨɦɟɪ 1 ɜɚɪɢɚɧɬɚ 2 Ⱥ 4 B 0 C x1 4 y1 2 z1 5 x2 3 y2 –4 z2 7 x3 1 y3 4 z3 4
2
ɇɨɦɟɪ 15 ɜɚɪɢɚɧɬɚ –3 Ⱥ –4 B 3 C x1 4 y1 2 z1 9 x2 –1 y2 4 z2 –3 x3 4 y3 8 z3 –3
3
4
5
6
7
4 3 4 2 3 –4 5 4 3 1 –4 3 –7 4 5 5 2 1 5 5 2 1 –4 4 2 3 3 4 4 2 –4 2 1 –6 –8 9 –4 4 –6 7 1 –4 3 –3 7 –3 3 –4 –3 2 –1 0 –4 3 –4 3 –3 7 –3 1 2 3 –4 1 –4 3 5 5 4 2 3 –4
3 4 1 –1 3 0 –6 –1 3 5 –3 3
4 3 –1 –6 4 –2 9 –6 4 5 5 8
1 –1 –6 9 4 8 –4 9 4 1 –5 3
–1 –6 9 –9 5 4 8 –3 1 2 6 –3
–6 9 –9 8 5 3 3 –4 –1 8 –3 5
8
9
–3 –9 8 –4 8 3 8 3 –3 3 –3 5 1 2 2 5 1 5 4 1 –1 –6 9 –4 –6 9 –3 3 –3 5 4 4 4 –5 6 –3
Ɂɚɞɚɱɚ 2. ɇɚɣɬɢ: ɚ) ɤɚɧɨɧɢɱɟɫɤɨɟ ɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ ɫ ɧɚɩɪɚɜɥɹɸɳɢɦ ɜɟɤɬɨɪɨɦ s(m, n, p) ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ (ɯ0, ɭ0, z0); ɛ) ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɜɟɤɬɨɪɚ. Ɂɧɚɱɟɧɢɹ m, n, p, x0, y0, z0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 5 ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɞɚɧɧɵɦ ɜɚɪɢɚɧɬɨɦ.
10
ɉɚɪɚɦɟɬɪɢɱɟɫɤɢɟ x ° ®y °z ¯
ɭɪɚɜɧɟɧɢɹ
ɩɪɹɦɨɣ
ɥɢɧɢɢ
ɜ
ɩɪɨɫɬɪɚɧɫɬɜɟ:
x0 mt y 0 nt . ȿɫɥɢ ɢɡ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɢɫɤɥɸɱɢɬɶ ɩɚɪɚɦɟɬɪ t, ɬɨ z 0 pt
ɩɨɥɭɱɢɦ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ ɤɚɧɨɧɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɹɦɨɣ ɥɢɧɢɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜ ɜɢɞɟ:
x x0 m
y y0 n
z z0 . ɑɢɫɥɚ m, n ɢ p ɧɚɡɵɜɚɸɬɫɹ p
ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɩɪɹɦɨɣ ɥɢɧɢɢ. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Į, ȕ, Ȗ ɭɝɥɵ, ɨɛɪɚɡɨɜɚɧɧɵɟ ɩɪɹɦɨɣ ɥɢɧɢɟɣ ɫ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ, ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ cos Į, cos ȕ, cos Ȗ ɹɜɥɹɸɬɫɹ ɧɚɩɪɚɜɥɹɸɳɢɦɢ ɤɨɫɢɧɭɫɚɦɢ ɜɟɤɬɨɪɚ s, ɩɨɥɭɱɢɦ: m = s ǜ cos Į, n = s ǜ cos ȕ, p = s ǜ cos Ȗ, ɝɞɟ s = m 2 n 2 p 2 – ɞɥɢɧɚ ɜɟɤɬɨɪɚ s. Ɉɬɫɸɞɚ ɩɨɥɭɱɢɦ: cos Į =
m n p , cos ȕ = , cos Ȗ = . s s s
2
3
4
5
10
11
12
Ɍɚɛɥɢɰɚ 5 13 14
4 5 –7 5 2 –4
3 4 4 5 3 2
4 3 5 2 3 1
2 3 –4 4 –6 7 1 –4 3 –3 7 –3 5 2 1 –4 4 –6 1 –4 4 –6 7 –3 4 4 2 2 3 3 –6 –8 9 4 0 –2
–3 4 7 2 4 8
2 –1 –3 –1 4 4
–1 0 2 0 5 3
0 –5 –1 2 5 5
ɇɨɦɟɪ 15 16 17 18 19 20 21 22 23 24 25 26 27 ɜɚɪɢɚɧɬɚ m –3 –4 3 4 1 –1 –6 –3 –9 8 3 –3 5 n –4 –3 4 3 –1 –6 9 –4 8 3 –3 5 –5 p 3 4 1 –1 –6 9 –9 8 3 –3 5 –5 6 x0 4 1 –1 –6 9 –9 8 3 –3 5 –5 6 0 y0 2 3 3 4 4 5 5 1 2 2 3 3 4 z0 9 4 0 –2 8 4 3 5 1 5 –4 2 1
28
ɇɨɦɟɪ 1 ɜɚɪɢɚɧɬɚ m 2 n 4 p 0 x0 4 y0 2 z0 5
6
7
8
9
–5 6 0 –6 4 –6
Ɂɚɞɚɧɢɟ 2. Ɉɩɟɪɚɰɢɢ, ɫɜɹɡɚɧɧɵɟ ɫ ɜɟɤɬɨɪ-ɮɭɧɤɰɢɟɣ. Ɂɚɞɚɱɚ 3. Ⱦɚɧɚ ɜɟɤɬɨɪ - ɮɭɧɤɰɢɹ: r(t) = x(t)i + y(t)j + z(t)k, ɝɞɟ t – ɩɚɪɚɦɟɬɪ, x(t) = nt2, y(t) = t2 + m, z(t) = pt + m – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ r(t) ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɬɢ: ɚ) ɩɪɨɢɡɜɨɞɧɭɸ ɜɟɤɬɨɪ-ɮɭɧɤɰɢɢ r = r(t)ɢ ɟɟ ɦɨɞɭɥɶ ɜ ɬɨɱɤɟ t0 = 2; ɛ) ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ r(t) ɜ ɬɨɱɤɟ t0 = 2; ɜ) ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɤ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ r(t) 11
ɜ ɬɨɱɤɟ t0 = 2. Ɂɧɚɱɟɧɢɹ m, n, p ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 5 ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɚɪɢɚɧɬɨɦ. ɉɨɞ ɩɪɨɢɡɜɨɞɧɨɣ ɜɟɤɬɨɪ – ɮɭɧɤɰɢɢ r(t) = x(t)i + y(t)j + z(t)k G dr
G 'r ( t ) lim . ȿɫɥɢ ɮɭɧɤɰɢɢ x(t), y(t), z(t) – ɩɨɧɢɦɚɸɬ ɜɟɤɬɨɪ, ɪɚɜɧɵɣ dt 't o0 't G dr dx G dy G dz G 't o 0 i j k. ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɟ, ɬɨ ɩɪɢ ɧɚɯɨɞɢɦ dt dt dt dt
G dr ȼɟɤɬɨɪ
dt
ɧɚɩɪɚɜɥɟɧ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ ɜ ɬɨɱɤɟ Ɇ (ɜ ɫɬɨɪɨɧɭ
ɜɨɡɪɚɫɬɚɧɢɹ ɮɭɧɤɰɢɢ). Ɇɨɞɭɥɶ ɩɪɨɢɡɜɨɞɧɨɣ ɜɟɤɬɨɪ-ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ G dr
ɜ ɜɢɞɟ:
(
dt
dx 2 dy 2 dz 2 ) ( ) ( ) . dt dt dt
ɍɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɤ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ r(t) = x(t)i + y(t)j + z(t)k ɜ ɬɨɱɤɟ Ɇ0(ɯ0, ɭ0, z0) ɢɦɟɟɬ ɜɢɞ: x x0 x0/
y y0 y 0/
z z0 , ɝɞɟ ɯ0 = ɯ(t0), y0 = y(t0), z0 = z(t0), x0/ = x/(t0), y0/ = z 0/
= y/(t0), z0/ = z/(t0). ɍɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɬ. ɟ. ɩɥɨɫɤɨɫɬɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɤɚɫɚɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤɚɫɚɬɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɢɦɟɟɬ ɜɢɞ: x0/(ɯ – ɯ0) + y0/(ɭ – ɭ0) + z0/(z – z0) = 0. Ɂɚɞɚɧɢɟ 3. ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɢ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ. Ɂɚɞɚɱɚ 4. ɇɚɣɬɢ: ɚ) ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɡɚɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ; ɛ) ɩɪɨɢɡɜɨɞɧɭɸ 2
2 2
wz wl
ɮɭɧɤɰɢɢ
z = 2(ɯ + ɭ ) ɜ ɧɚɩɪɚɜɥɟɧɢɢ l, ɡɚɞɚɧɧɨɝɨ ɜɟɤɬɨɪɨɦ ɚ = mi + nj ɜ ɬɨɱɤɟ Ɇ0(ɯ0, ɭ0). Ⱦɥɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ ɚ) ɮɭɧɤɰɢɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɩɪɟɞɨɫɬɚɜɥɹɸɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ, ɚ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɹ ɛ) ɡɧɚɱɟɧɢɹ m, n, x0, y0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 5. ɑɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɩɨ ɨɞɧɨɣ ɢɡ ɷɬɢɯ ɩɟɪɟɦɟɧɧɵɯ ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɱɚɫɬɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ ɤ ɩɪɢɪɚɳɟɧɢɸ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɧɟɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɩɨɫɥɟɞɧɟɟ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ. 12
ȿɫɥɢ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɯ, ɬɨ ɜɬɨɪɚɹ ɧɟɡɚɜɢɫɢɦɚɹ ɩɟɪɟɦɟɧɧɚɹ ɭ ɫɱɢɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɢ ɧɚɨɛɨɪɨɬ. Ɋɚɫɫɦɨɬɪɢɦ ɨɬɧɨɲɟɧɢɟ ɱɚɫɬɧɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ z ɩɨ ɩɟɪɟɦɟɧɧɨɣ ɯ ɤ ɩɪɢɪɚɳɟɧɢɸ ǻɯ ɷɬɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɬ. ɟ. 'xz 'x
f (x 'x, y) f (x, y) 'x
. ɉɪɟɞɟɥ ɷɬɨɝɨ ɩɪɢɪɚɳɟɧɢɹ ɩɪɢ ǻɯ ĺ 0, ɟɫɥɢ ɬɚɤɨɜɨɣ
ɫɭɳɟɫɬɜɭɟɬ, ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɮɭɧɤɰɢɢ z
=
f(x,
ɩɨ
y)
ɯ
ɢ
ɨɛɨɡɧɚɱɚɟɬɫɹ
wz wx
ɢɥɢ
fx/(x,
ɬ. ɟ.
y),
f (x 'x, y) f (x, y) wz = 'xlim . Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ o 0 'x wx
wz
f (x, y 'y) f (x, y)
wz ɢɥɢ fɭ/(x, y) ɨɬ ɮɭɧɤɰɢɢ z = f(x, y) ɩɨ ɭ: = lim . 'y wy 'yo0 wy wz ɮɭɧɤɰɢɢ ɜ ɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ l ɩɨɧɢɦɚɟɬɫɹ ɉɨɞ ɩɪɨɢɡɜɨɞɧɨɣ wl
ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɤ ɜɟɥɢɱɢɧɟ ɩɟɪɟɦɟɳɟɧɢɹ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɩɨɫɥɟɞɧɟɟ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ, ɬ. ɟ. ' z wz wz wz wz = lim l = cos Į + cos ȕ. ɉɪɨɢɡɜɨɞɧɚɹ ɞɚɟɬ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ 'l o0 'l wy wl wx wl wz ɮɭɧɤɰɢɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ l, ɮɭɧɤɰɢɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ l. ɉɪɨɢɡɜɨɞɧɚɹ wl
ɡɚɞɚɧɧɨɝɨ ɜɟɤɬɨɪɨɦ wz ɮɨɪɦɭɥɟ: wl
wz M ( x, y ) = wx
ɚ = ɚɯi + ɚɭj, ɜ ɬɨɱɤɟ Ɇ(ɯ, ɭ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ
M ) x, y )
cosĮ +
wz wy
M ( x y )
a
a a
cosȕ, ɝɞɟ: cosĮ = Gx , cosȕ = Gy . a
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 3 (3 ɱɚɫɚ) Ɂɚɞɚɧɢɟ 1. ȼɵɱɢɫɥɟɧɢɟ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ, ɪɟɡɭɥɶɬɚɬɵ ɩɪɨɜɟɪɢɬɶ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ɍɚɛɥɢɰɚ ɩɪɨɢɡɜɨɞɧɵɯ ɨɫɧɨɜɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ. 1. (xn)/ = nxn-1, (ɫ)/ = 0, (x)/ = 1, (x2)/ = 2x, ( x )/ = 2. (ɚɯ)/ = ax ln a, (ɟɯ)/ = ɟɯ. 3. (logax)/ =
1 1 , (ln x)/ = . x ln a x
4. (f(x)M(x))/ = f(x)M(x)(M(x)f/(x)+M/(x)lnf(x)). 5. (ln y)/ =
yc . y
13
1 , 2 x
1 x
( )/ = –
1 . x2
6. (sin x)/ = cos x, (cos x)/ = –sin x, (tg x)/ = 7. (arcsin x)/ =
1
1 x2 1 . (arcctg x)/ = – 1 x2
, (arccos x)/
1 1 , (ctg x)/ = – 2 . cos 2 x sin x 1 /
= –
1 x2
, (arctg x) =
1 , 1 x2
ȿɫɥɢ ɮɭɧɤɰɢɹ ɡɚɞɚɧɚ ɧɟɹɜɧɨ, ɬ. ɟ. F(x, y) = 0, ɬɨ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ ɬɚɤɨɣ ɮɭɧɤɰɢɢ ɧɭɠɧɨ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ, ɪɚɫɫɦɚɬɪɢɜɚɹ ɭ ɤɚɤ ɮɭɧɤɰɢɸ ɨɬ ɯ, ɚ ɡɚɬɟɦ ɢɡ ɩɨɥɭɱɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɭɸ. ɂɧɬɟɝɪɚɥɵ ɨɬ ɨɫɧɨɜɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ. x n 1 x2 + C, n z –1, ³ dx = ɯ + ɋ, ³ xdx = + ɋ, n 1 2 2 x3 1 1 1 ³ xdx = 3 + ɋ, ³ x dx = 2 x + ɋ, ³ x 2 dx = – x + ɋ. ax 1 2. ³ dx = ln|x| + C, ³ a x dx = + C, a > 0, a z 1, ³ e x dx = ex + C. ln a x 3. ³ sin xdx = –cos x + C, ³ cos xdx = sin x + C.
1.
³ 0dx =
4.
³
1 1 x2
1
³1 x
2
³ x dx =
dx = arcsin x + C = –arccos x + C,
= –arccos 5.
n
C,
1
³
a2 x2
7. 8. 9.
dx = arctg x + C = -arcctg x + C,
³a
2
x 1 1 dx = arctg + C = x2 a a
x 1 , a z 0. a a xa 1 1 1 2 ³ x 2 a 2 dx = 2a ln x a + C, a z 0, ³ x 2 r a dx = ln| x + x r a | + C, a z 0. x 1 1 1 2 2 ³ cos2 x dx = tg x + C, ³ sin 2 x dx = –ctg x + C, ³ a 2 r x 2 dx = r 2 ln|a r x | + C. x 2 a2 x x 2 2 2 2 2 ³ a 2 r x 2 dx = r a r x + C, ³ a x dx = 2 a x 2 arcsin a + C. x 2 a2 2 2 2 2 2 ³ x r a dx = 2 x r a r 2 ln x x r a + C.
ɉɪɢɦɟɪ 1. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ.
³(
x +C= a
x + C, a > 0, –a < x < a. a
= – arcctg 6.
dx = arcsin
x3
1
x
1 x 4 )dx . x8
14
³(
x3
1
x
1 x 4 )dx = x8
³x
1 2
dx +
³x
1 3
dx +
³x
8
dx +
³x
4
dx =
2
3
x 2 x 3 x 7 2 x 3 33 x 2 1 7 4x3 C . 4 x3 C = 3 2 7 3 2 7x 2 3
ɉɪɢɦɟɪ 2. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
1
³ 1 2 x dx .
Ɂɚɦɟɧɚ ɩɟɪɟɦɟɧɧɵɯ dt 1 dt 1 1 t = = = ln | t | C . dx 1 2 ɯ, dt 2dx = ³ ³ 1 2x 2t 2³ t 2 dt dx 2 ɉɪɢɦɟɪ 3. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ³ xe2 x dx . ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ 2 x ³ xe dx = u
x, du
dx, dv
e 2 x dx, = u ǜ v –
³ vdu
e 2 x 2 e 2 x xe 2 x e 2 x –³ dx = C. 2 2 4
=x
e 2 x – 2
v
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɱɚɫɬɹɦ ɜɵɱɢɫɥɹɸɬɫɹ ɢɧɬɟɝɪɚɥɵ ɜɢɞɚ:
³ x ln x dx,
³ x sin xdx , ³ x cos xdx , ³ x arcsin xdx , ³ x arctg dx ɢ ɞɪ. ɉɪɢɦɟɪ 4. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
³x
2
dx . 4x 8
ȿɫɥɢ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɫɬɨɢɬ
³x
ɤɜɚɞɪɚɬɧɵɣ ɬɪɟɯɱɥɟɧ, ɬɨ
2
dx = = 4 x 8 ɧɟɨɛɯɨɞɢɦɨ ɜɵɞɟɥɢɬɶ ɩɨɥɧɵɣ
ɤɜɚɞɪɚɬ dx 1 x2 = arctg C. = ³ 2 2 ( x 2) 2 4 dx
³
x 2 7 x 15
³ (x
2
dx = 2 x 2 4) 4 8
Ⱥɧɚɥɨɝɢɱɧɨ ɜɵɱɢɫɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥ ɜɢɞɚ:
.
ɉɪɢɦɟɪ 5. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
2x 1
³ ( x 1) ( x 2) dx . 2
ɂɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɧɟɨɩɪɟɞɟɥɟɧɧɵɯ A dx + =³ ( x 1) ɤɨɷɮɮɢɰɢɟɧɬɨɜ B C 1 1 1 +³ dx + ³ dx = ³ dx + ³ dx = dx + ³ ( x 2) ( x 1) ( x 2) ( x 1) 2 ( x 1) 2
2x 1
³ ( x 1) ( x 2) dx = 2
15
= –ln|x ––1| +
1 + ln|x – 2| + C. x 1
ɇɚɯɨɞɹɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɵ Ⱥ, B, C: =
A B C = x 1 ( x 1) 2 x 2
Ax 2 3 Ax 2 A Bx 2 B Cx 2 2Cx C ( x 1) 2 ( x 2)
x2 A C
0 A C A 1 A( x 1)( x 2) B( x 2) C ( x 1) 1 ° ° x A B C 3 2 0 C B 0 ® ®C 1 . ( x 1) 2 ( x 2) ° ° ¯ C 2 B 1 ¯ B 1 x 0 2 A 2B C 1 2
ɉɪɢɦɟɪ 6. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ: ³ sin 2 xdx .
³ sin
2
xdx =
³
1 cos 2 x dx = 2
1
³ 2 dx
–
³
cos 2 x dx = 2
1 x 2
–
1 sin 2 x C 4
(ɩɪɢ
ɜɵɱɢɫɥɟɧɢɢ ɚɧɚɥɨɝɢɱɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɢɫɩɨɥɶɡɭɸɬɫɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ 1 cos 2 x 1 cos 2 x ɢ cos2 x = ). 2 2 sin x ɉɪɢɦɟɪ 7. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ: ³ 3 dx . cos x sin x 1 d (cos x) cos 2 x ³ cos 3 x dx = ³ cos 3 x = 2 C = 2 cos 2 x C . ɉɪɢɦɟɪ 8. ȼɵɱɢɫɥɢɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɵɣ ɢɧɬɟɝɪɚɥ: ³ sin x cos xdx .
ɮɨɪɦɭɥɵ ɩɨɧɢɠɟɧɢɹ ɫɬɟɩɟɧɢ: sin2x =
³ sin x cos 3xdx
=
1 1 1 1 sin 2 xdx + ³ sin 4 xdx = cos(2 x) – (sin(2 x) sin( 4 x))dx = 2 ³ 2³ 2 4
1 sin( 4 x) C . ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɚɧɚɥɨɝɢɱɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɢɫɩɨɥɶɡɭɸɬɫɹ 8 1 ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ: sin x ǜ cos y = [sin(x – y) + sin(x + y)]; 2 1 1 sin x ǜ sin y = [cos(x – y) – cos(x + y)]; cos xǜcos y = [cos(x – y) + cos(x + y)]. 2 2
–
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɟɧɢɟ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ. ȼɵɱɢɫɥɢɬɶ ɨɩɪɟɞɟɥɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɪɟɲɟɧɢɟ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ b
ɜɢɞɟ ɮɨɪɦɭɥɵ ɇɶɸɬɨɧɚ – Ʌɟɣɛɧɢɰɚ: ³ f ( x)dx = F(x)
b a
= F(b) – F(a).
a
Ɂɚɞɚɧɢɟ 3. ȼɵɱɢɫɥɟɧɢɟ ɧɟɫɨɛɫɬɜɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ. ȼɵɱɢɫɥɢɬɶ ɧɟɫɨɛɫɬɜɟɧɧɵɟ ɢɧɬɟɝɪɚɥɵ. Ɂɚɞɚɧɢɟ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ.
ɜɵɞɚɟɬɫɹ
f
ɇɟɫɨɛɫɬɜɟɧɧɵɦ
ɢɧɬɟɝɪɚɥɨɦ
³ f ( x)dx
ɨɬ
ɮɭɧɤɰɢɢ
f(x)
ɧɚ
a
ɩɨɥɭɢɧɬɟɪɜɚɥɟ [a, +f) ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥ ɮɭɧɤɰɢɢ ɩɪɢ t, ɫɬɪɟɦɹɳɟɦɫɹ 16
f
t
ɤ + f, ɬ. ɟ. ³ f ( x)dx = lim
t o f
a
ȿɫɥɢ ɩɪɟɞɟɥ ɫɭɳɟɫɬɜɭɟɬ ɢ ɤɨɧɟɱɟɧ, ɬɨ
³ f ( x)dx . a
ɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɪɚɫɯɨɞɹɳɢɦɫɹ. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɧɟɫɨɛɫɬɜɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɹɬɶ ɩɥɨɳɚɞɶ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɣ ɢɥɢ ɛɟɫɤɨɧɟɱɧɨɣ ɮɢɝɭɪɵ. ɉɨ ɚɧɚɥɨɝɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɧɚ b
ɩɨɥɭɢɧɬɟɪɜɚɥɟ (–f, b] ɜ ɜɢɞɟ:
b
f ( x)dx =
³
f
³ f ( x)dx .
lim
b o f
ɇɟɫɨɛɫɬɜɟɧɧɵɣ
t
f
f
a
ɢɧɬɟɝɪɚɥ ɧɚ ɢɧɬɟɪɜɚɥɟ (–f, + f) ɢɦɟɟɬ ɜɢɞ: ³ f ( x)dx = f
+
³ f ( x)dx
f
³ f ( x)dx . a
f
ɉɪɢ ɷɬɨɦ ɢɧɬɟɝɪɚɥ
³ f ( x)dx
ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ. ȿɫɥɢ ɯɨɬɹ ɛɵ ɨɞɢɧ ɢɡ
f
f
a
ɢɧɬɟɝɪɚɥɨɜ
³ f ( x)dx , ³ f ( x)dx
f
ɪɚɫɯɨɞɢɬɫɹ, ɬɨ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
a
f
ɧɚɡɵɜɚɟɬɫɹ ɪɚɫɯɨɞɹɳɢɦɫɹ.
³ f ( x)dx
f
f
ɇɟɫɨɛɫɬɜɟɧɧɵɣ
ɢɧɬɟɝɪɚɥ
³e
x2 2
2S ɧɚɡɵɜɚɟɬɫɹ
dx =
ɢɧɬɟɝɪɚɥɨɦ
f
ɗɣɥɟɪɚ – ɉɭɚɫɫɨɧɚ. b
ɇɟɫɨɛɫɬɜɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ
³ f ( x)dx ɨɬ
ɮɭɧɤɰɢɢ y = f(x) ɧɚ
a
b G
ɩɨɥɭɢɧɬɟɪɜɚɥɟ [a, b) ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥ lim
G o0
³ f ( x)dx ,
ɝɞɟ G > 0, ɬ. ɟ.
a
b G
b
=
³ f ( x)dx
lim
G o0
a
³ f ( x)dx .
ȿɫɥɢ
ɩɪɟɞɟɥ
ɫɭɳɟɫɬɜɭɟɬ
ɢ
ɤɨɧɟɱɟɧ,
ɬɨ
a
ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ – ɪɚɫɯɨɞɹɳɢɦɫɹ. Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɧɹɬɢɟ ɧɟɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ ɨɬ b
ɮɭɧɤɰɢɢ y = f(x) ɧɟɩɪɟɪɵɜɧɨɣ, ɧɨ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɧɚ (ɚ, b]:
³ f ( x)dx
=
a
b
= lim
G o0
³G f ( x)dx . ȿɫɥɢ ɮɭɧɤɰɢɹ ɧɟ ɨɝɪɚɧɢɱɟɧɚ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ ɫ (ɚ, b),
a
b
ɬɨ ɢɧɬɟɝɪɚɥ
³ f ( x)dx
ɧɚɡɵɜɚɟɬɫɹ ɧɟɫɨɛɫɬɜɟɧɧɵɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɧɬɟɝɪɚɥ
a
c
b
³ a
f ( x)dx =
³ a
b
f ( x)dx +
³ f ( x)dx
ɫɱɢɬɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ, ɟɫɥɢ ɫɯɨɞɹɬɫɹ ɞɜɚ
c
17
b
ɧɟɫɨɛɫɬɜɟɧɧɵɯ ɢɧɬɟɝɪɚɥɚ, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɢɧɬɟɝɪɚɥ
³ f ( x)dx
ɛɭɞɟɬ
a
ɪɚɫɯɨɞɹɳɢɦɫɹ. f
ɉɪɢɦɟɪ 1. ȼɵɱɢɫɥɢɬɶ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ ³ 1
f
1 dx . x2
b
1 1 1 1 dx = lim( ) 1b = lim( 1) 1 . ³1 x 2 dx = blim b of of ³ x 2 b of x b 1 1
ɉɪɢɦɟɪ 2. ȼɵɱɢɫɥɢɬɶ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
³ 0
1
³ 0
1 dx . x
1
1 1 dx = lim ³ dx = lim(2 x ) 1H = lim(2 1 2 H ) = 2. H o0 H o0 H o0 x x H
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 4 (3 ɱɚɫɚ) Ɂɚɞɚɧɢɟ 1. ȼɵɱɢɫɥɟɧɢɹ ɝɪɚɞɢɟɧɬɚ ɮɭɧɤɰɢɢ. ȿɫɥɢ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ ɡɚɞɚɧ ɧɟɤɨɬɨɪɵɣ ɫɤɚɥɹɪ, ɬɨ ɜ ɞɚɧɧɨɣ ɨɛɥɚɫɬɢ ɛɭɞɟɬ ɨɩɪɟɞɟɥɟɧɨ ɫɤɚɥɹɪɧɨɟ ɩɨɥɟ. ɉɥɨɫɤɨɟ ɫɤɚɥɹɪɧɨɟ ɩɨɥɟ ɦɨɠɧɨ ɡɚɜɢɫɚɬɶ ɜ ɜɢɞɟ u = f(x, y). Ⱦɥɹ ɨɛɥɚɫɬɢ, ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɈɏɍZ, ɫɤɚɥɹɪɧɨɟ ɩɨɥɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ: u = f(x, y, z). ȿɫɥɢ ɞɥɹ ɤɚɠɞɨɣ ɬɨɱɤɢ Ɇ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɧɟɤɨɬɨɪɨɣ ɨɛɥɚɫɬɢ, ɡɚɞɚɧ ɜɟɤɬɨɪ ɚ = F(Ɇ), ɬɨ ɝɨɜɨɪɹɬ ɜ ɷɬɨɣ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɨ ɜɟɤɬɨɪɧɨɟ ɩɨɥɟ, Ⱦɥɹ ɫɥɭɱɚɹ ɩɥɨɫɤɨɝɨ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ, ɬ. ɟ. ɨɛɥɚɫɬɶ ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɥɨɫɤɨɫɬɢ Ɉɏɍ, ɜɟɤɬɨɪɧɚɹ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ ɚ = F(ɯ, ɭ). ɉɟɪɟɯɨɞɹ ɤ ɤɨɨɪɞɢɧɚɬɚɦ ɜɟɤɬɨɪɚ ɚ, ɩɨɥɭɱɢɦ ɚɯ = Fx(ɯ, ɭ) ɢ ɚɭ = Fy(ɯ, ɭ) Ⱦɥɹ ɫɥɭɱɚɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɢɦɟɟɦ: ɚ = F(ɯ, ɭ, z) ɢɥɢ ɜ ɤɨɨɪɞɢɧɚɬɚɯ: ɚɯ = Fx(ɯ, ɭ, z), ɚɭ = Fy(ɯ, ɭ, z), ɚz = Fz(ɯ, ɭ, z). Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɬɨɱɟɤ Ɇ, ɞɥɹ ɤɨɬɨɪɵɯ ɫɤɚɥɹɪɧɨɟ ɩɨɥɟ ɫɨɯɪɚɧɹɟɬ ɩɨɫɬɨɹɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɬ. ɟ. f(M) = const, ɧɚɡɵɜɚɟɬɫɹ ɩɨɜɟɪɯɧɨɫɬɶɸ (ɢɥɢ ɥɢɧɢɟɣ) ɭɪɨɜɧɹ ɫɤɚɥɹɪɧɨɝɨ ɩɨɥɹ (ɷɤɜɢɩɨɬɟɧɰɢɚɥɶɧɚɹ ɩɨɜɟɪɯɧɨɫɬɶ). ɉɭɫɬɶ u = f(x, y) – ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɟ ɩɥɨɫɤɨɟ ɫɤɚɥɹɪɧɨɟ ɩɨɥɟ. Ɍɨɝɞɚ wu wu , } ɧɚɡɵɜɚɟɬɫɹ ɝɪɚɞɢɟɧɬɨɦ ɩɨɥɹ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ wx wy G wu G wu ɮɨɪɦɭɥɟ: grad u = i j , ɝɞɟ i ɢ j – ɟɞɢɧɢɱɧɵɟ ɜɟɤɬɨɪɵ, ɧɚɩɪɚɜɥɟɧɧɵɟ wx wy
ɜɟɤɬɨɪ grad u = {
ɩɨ ɨɫɹɦ ɤɨɨɪɞɢɧɚɬ Ɉɏ ɢ Ɉɍ. Ⱥɧɚɥɨɝɢɱɧɨ, ɞɥɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɫɤɚɥɹɪɧɨɝɨ ɩɨɥɹ u = f(x,y,z) ɟɝɨ ɝɪɚɞɢɟɧɬ ɟɫɬɶ ɜɟɤɬɨɪ grad u = { G wu
ɨɩɪɟɞɟɥɹɟɦɵɣ ɩɨ ɮɨɪɦɭɥɟ: grad u = i
wx
wu wu wu , , }, wx wy wz
G wu G wu j k . Ƚɪɚɞɢɟɧɬ ɫɤɚɥɹɪɧɨɝɨ wz wy
ɩɨɥɹ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ Ɇ ɩɨ ɦɨɞɭɥɸ ɢ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɜɟɧ ɦɚɤɫɢɦɚɥɶɧɨɣ 18
ɫɤɨɪɨɫɬɢ
ɢɡɦɟɧɟɧɢɹ ɩɨɥɹ ɜ ɷɬɨɣ ɬɨɱɤɟ, ɬ. ɟ. 2
2
=
G n0
gradu gradu
l
wu wl
gradu
=
2
§ wu · § wu · § wu · ¨ ¸ ¨¨ ¸¸ ¨ ¸ . © wx ¹ © wy ¹ © wz ¹
ɟɞɢɧɢɱɧɵɣ
max
ɜɟɤɬɨɪ
ɂɫɩɨɥɶɡɭɹ G n0 ɤ
ɧɨɪɦɚɥɢ
wu G wu G wu G i j k wx wy wz 2
2
§ wu · § wu · § wu · ¨ ¸ ¨¨ ¸¸ ¨ ¸ © wx ¹ © wy ¹ © wz ¹
ɝɪɚɞɢɟɧɬ,
ɦɨɠɧɨ
ɷɤɜɢɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɨɩɪɟɞɟɥɢɬɶ ɩɨɜɟɪɯɧɨɫɬɢ:
. 2
Ɉɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ ɝɪɚɞɢɟɧɬɚ ɜɵɪɚɠɚɸɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɮɨɪɦɭɥɚɦɢ: grad(u + v) = gradu + gradv; grad(u ǜ v) = uǜgradv + vǜgradu; grad(c ǜ u) = = c ǜ gradu. Ɂɚɞɚɱɚ 1. ɇɚɣɬɢ: ɚ) ɝɪɚɞɢɟɧɬ ɮɭɧɤɰɢɢ u =
xp
ɜ ɬɨɱɤɟ
2
mx ny 2
Ɇ0(ɯ0, ɭ0); ɛ) ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɧɨɪɦɚɥɢ ɤ ɷɤɜɢɩɨɬɟɧɰɢɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ; ɜ) ɦɚɤɫɢɦɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɩɨɥɹ ɜ ɬɨɱɤɟ M0. Ɂɧɚɱɟɧɢɹ m, n, p, x0,y0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɇɨɦɟɪ 1 ɜɚɪɢɚɧɬɚ m 2 n 4 p 0 x0 4 y0 2 z0 5
2
11
12
7 –3 2 –3 3 –2
3 4 1 2 4 8
2 –1 2 –1 4 4
1 0 1 0 5 3
0 –5 1 2 5 5
ɇɨɦɟɪ 15 16 17 18 19 20 21 22 23 24 ɜɚɪɢɚɧɬɚ m 3 4 3 4 1 1 6 3 9 8 n –4 –3 4 3 –1 –6 9 –4 8 3 p 3 1 1 1 2 3 3 4 3 3 x0 4 1 –1 –6 9 –9 8 3 –3 5 y0 2 3 3 4 4 5 5 1 2 2 z0 9 4 0 –2 8 4 3 5 1 5
25
26
27
28
3 –3 2 –5 3 –4
3 5 2 6 3 2
5 –5 1 0 4 1
5 6 0 –6 4 –6
3 4 4 5 3 2
4 4 3 2 2 3 1
5
6
2 3 1 –4 1 2 1 –4 4 4 –6 –8
7 4 3 1 4 2 9
8 4 –3 4 –6 2 4
9
Ɍɚɛɥɢɰɚ 6 13 14
10
4 5 1 5 2 –4
3
6 7 3 7 3 0
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɟɧɢɟ ɩɨɬɨɤɚ ɜɟɤɬɨɪɚ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɞɢɜɟɪɝɟɧɰɢɢ. ɉɪɢɦɟɧɟɧɢɟ ɮɨɪɦɭɥɵ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ. ɉɨɬɨɤ Q ɜɟɤɬɨɪɚ ɚ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ S ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɨɥɢɱɟɫɬɜɨ ɫɢɥɨɜɵɯ ɥɢɧɢɣ, ɩɪɢɧɢɡɵɜɚɸɳɢɯ ɷɬɭ ɩɨɜɟɪɯɧɨɫɬɶ. ɉɨɬɨɤ 19
ɜɟɤɬɨɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ: Q =
G
G
³³ a dS
ɢɥɢ Q =
S
Q =
³³ (a dydz a x
y
dxdz a z dxdy ) ɢɥɢ Q =
S
³³
³³ a dS n
ɢɥɢ
S
(ax ǜ cos(n,x) + ay ǜ cos(n,y) +
S
+ az ǜ cos(n,z))dS, ɝɞɟ ɚn – ɩɪɨɟɤɰɢɹ ɜɟɤɬɨɪɚ ɚ ɧɚ ɧɨɪɦɚɥɶ n ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: ɚn = ax ǜ cos(n,x) + ay ǜ cos(n,y) + az ǜ cos(n,z), cos(n,x), cos(n,y), cos(n,z) – ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ S. ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ S ɡɚɦɤɧɭɬɚɹ ɢ Q > 0, ɬ. ɟ. ɢɡ ɨɛɴɟɦɚ ɜɵɯɨɞɢɬ ɫɢɥɨɜɵɯ ɥɢɧɢɣ ɛɨɥɶɲɟ, ɱɟɦ ɜɯɨɞɢɬ, ɢ ɩɨɬɨɤ ɧɚɡɵɜɚɟɬɫɹ ɨɛɢɥɶɧɨɫɬɶɸ ɢɫɬɨɱɧɢɤɚ. ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ S ɡɚɦɤɧɭɬɚɹ ɢ Q < 0, ɬ. ɟ. ɫɢɥɨɜɵɯ ɥɢɧɢɣ ɜɵɯɨɞɢɬ ɦɟɧɶɲɟ, ɱɟɦ ɜɯɨɞɢɬ (ɨɬɪɢɰɚɬɟɥɶɧɚɹ ɨɛɢɥɶɧɨɫɬɶ ɢɥɢ ɫɬɨɤ). ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ S ɡɚɦɤɧɭɬɚɹ ɢ Q = 0, ɪɚɜɟɧɫɬɜɨ ɜɵɯɨɞɹɳɢɯ ɢ ɜɯɨɞɹɳɢɯ ɫɢɥɨɜɵɯ ɥɢɧɢɣ. ɉɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɩɨɬɨɤɚ ɜɟɤɬɨɪɚ ɱɟɪɟɡ ɡɚɦɤɧɭɬɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɤ ɜɟɥɢɱɢɧɟ ɨɬɝɪɚɧɢɱɟɧɧɨɝɨ ɨɛɴɟɦɚ, ɤɨɝɞɚ ɨɧ ɫɬɹɝɢɜɚɟɬɫɹ ɜ ɬɨɱɤɭ, ɧɚɡɵɜɚɟɬɫɹ ɞɢɜɟɪɝɟɧɰɢɟɣ ɢɥɢ ɪɚɫɯɨɞɢɦɨɫɬɶɸ ɜɟɤɬɨɪɚ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ G
div a, ɬ. ɟ. div a(p) = lim
G
³³ a ds S
'W o P
'W
.
Ⱦɢɜɟɪɝɟɧɰɢɹ – ɫɤɚɥɹɪɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ. Ɉɧɚ ɨɩɪɟɞɟɥɹɟɬ ɤɨɥɢɱɟɫɬɜɨ ɜɟɤɬɨɪɧɵɯ ɥɢɧɢɣ, ɧɚɱɢɧɚɸɳɢɯɫɹ ɜ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɨɛɴɟɦɟ, ɬ. ɟ. ɞɢɜɟɪɝɟɧɰɢɹ ɢɦɟɟɬ ɫɦɵɫɥ ɩɥɨɬɧɨɫɬɢ ɢɫɬɟɱɟɧɢɹ ɜɟɤɬɨɪɧɵɯ ɥɢɧɢɣ ɢɡ ɬɨɱɤɢ. Ɉɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ (ɜ ɬɨɱɤɟ ɧɚɯɨɞɢɬɫɹ ɢɫɬɨɱɧɢɤ ɩɨɥɹ), ɨɬɪɢɰɚɬɟɥɶɧɨɣ (ɧɚɯɨɞɢɬɫɹ ɫɬɨɤ) ɢɥɢ ɪɚɜɧɨɣ ɧɭɥɸ (ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ ɧɟɬ). ɑɢɫɥɟɧɧɚɹ ɜɟɥɢɱɢɧɚ ɞɢɜɟɪɝɟɧɰɢɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɥɨɬɧɨɫɬɶ ɢɫɬɨɱɧɢɤɚ. ɇɚ ɩɪɚɤɬɢɤɟ ɞɢɜɟɪɝɟɧɰɢɸ ɭɞɨɛɧɟɟ ɧɚɯɨɞɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: div a =
wa x wa y wa z . wx wy wz
ɉɨɥɟ, ɜ ɤɨɬɨɪɨɦ div a = 0, ɧɚɡɵɜɚɟɬɫɹ
ɫɨɥɟɧɨɢɞɚɥɶɧɵɦ. ɋɜɨɣɫɬɜɚ ɞɢɜɟɪɝɟɧɰɢɢ: div (a + b) = div a + div b; div (ua) = u ǜ div a + + gradu ǜ a$ div (ca) = c ǜ div a, ɝɞɟ ɫ – ɤɨɧɫɬɚɧɬɚ. Ⱦɢɜɟɪɝɟɧɰɢɹ ɜɟɤɬɨɪɚ ɢ ɩɨɬɨɤ ɜɟɤɬɨɪɚ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɮɨɪɦɭɥɨɣ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ. Ɉɧɚ ɝɥɚɫɢɬ: ɩɨɬɨɤ ɜɟɤɬɨɪɚ ɚ ɱɟɪɟɡ ɡɚɦɤɧɭɬɭɸ ɩɨɜɟɪɯɧɨɫɬɶ S ɪɚɜɟɧ ɢɧɬɟɝɪɚɥɭ ɨɬ ɞɢɜɟɪɝɟɧɰɢɢ, Gɜɡɹɬɨɦɭ ɩɨ G G ɨɛɴɟɦɭ V, ɨɝɪɚɧɢɱɟɧɧɨɦɭ ɞɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ S, ɬ. ɟ. ³³ a dS ³³³ divadW . S
V
Ɏɨɪɦɭɥɚ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ ɩɪɢɦɟɧɹɟɬɫɹ ɞɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢɧɬɟɝɪɚɥɚ ɩɨ ɨɛɴɟɦɭ ɜ ɢɧɬɟɝɪɚɥ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɨɝɪɚɧɢɱɢɜɚɸɳɟɣ ɷɬɨɬ ɨɛɴɟɦ. Ɂɚɞɚɱɚ 2. ɇɚɣɬɢ ɡɧɚɱɟɧɢɟ Fn ɢ ɩɨɬɨɤ ɜɟɤɬɨɪɚ ɫɢɥɵ ɩɪɢɬɹɠɟɧɢɹ G F
f
M G r ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɫɮɟɪɵ ɪɚɞɢɭɫɚ R ɫ ɰɟɧɬɪɨɦ ɜ ɧɚɱɚɥɟ r3 20
ɤɨɨɪɞɢɧɚɬ. f = 6,67 ǜ 10
cɦ 3 9 3 ɝ ɫ 2 . M = |m| ǜ 10 ɝ, r = 10 · |p|, ɫɦ. Ɂɧɚɱɟɧɢɹ, m
–8
ɢ p ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 1. ɉɨɬɨɤ ɧɚɣɞɟɦ ɩɨ ɮɨɪɦɭɥɟ: Q =
³³ a dS n
, ɝɞɟ an = Fn.
S
Q =
ɇɨɪɦɚɥɶ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɫɮɟɪɵ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɞɢɭɫɨɦ, ɢ
³³ F dS . n
S
ɩɪɨɟɤɰɢɹ M = f 3 R
ɪɚɜɧɚ
Fn
ɦɨɞɭɥɸ
G F,
ɜɟɤɬɨɪɚ
ɬ. ɟ.
Fn
=
|F|
=
M x y z f 2 , ɬɚɤ ɤɚɤ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɫɮɟɪɵ r = R. Ɍɟɩɟɪɶ R M M ɧɚɯɨɞɢɦ ɩɨɬɨɤ: Q = f 2 ³³ dS = f 2 4SR 2 = 4ʌfM. R S R 2
2
2
Ɂɚɞɚɱɚ 3. ȼɵɱɢɫɥɢɬɶ ɞɢɜɟɪɝɟɧɰɢɸ ɟɞɢɧɢɱɧɨɝɨ ɪɚɞɢɭɫɚ-ɜɟɤɬɨɪɚ ɜ ɬɨɱɤɟ Ɇ0(ɯ0, ɭ0, z0). Ɂɧɚɱɟɧɢɹ ɯ0, ɭ0, z0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 2. ȼɵɱɢɫɥɢɬɶ ɞɢɜɟɪɝɟɧɰɢɸ ɟɞɢɧɢɱɧɨɝɨ ɪɚɞɢɭɫɚ-ɜɟɤɬɨɪɚ ɜ ɬɨɱɤɟ Ɇ0(0, 3, 4). G G r G , ɝɞɟ r r
G
ȿɞɢɧɢɱɧɵɣ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɢɦɟɟɬ ɜɢɞ: r 0 G r
G G G xi yj zk ,
x2 y2 z2 .
Ⱦɥɹ
ɧɚɯɨɠɞɟɧɢɹ
ɞɢɜɟɪɝɟɧɰɢɢ
ɜɨɫɩɨɥɶɡɭɟɦɫɹ
ɫɜɨɣɫɬɜɨɦ
G G G r 1 div (ua) = u ǜ div a + gradu ǜ a, ɩɨɥɚɝɚɹ, ɱɬɨ u = G ɢ ɚ = r ; div ( r 0 ) = div G = r r G G G G G G wx wy wz 1 1 1 = G ǜdiv r + grad G ǜ r ; div r = div ( xi yj zk ) = = 3; grad G = wx wy wz r r r
1 2
3
G 1 2
3
G
= grad(x2 + y2 + z2)–1/2 = – ( x 2 y 2 z 2 ) 2 2 xi – ( x 2 y 2 z 2 ) 2 2 yj – 3 G 1 2 ( x y 2 z 2 ) 2 2 zk = 2
grad
G G G xi yj zk
–
3
ɇɚɣɞɟɦ
.
ɫɤɚɥɹɪɧɨɟ
ɩɪɨɢɡɜɟɞɟɧɢɟ
(x 2 y 2 z 2 ) 2
G 1 G 1 ǜ r ; grad G ǜ r = r r
xx yy zz
=–
3 2
r2 r3
1 . ɉɨ ɧɚɣɞɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ r
(x 2 y 2 z 2 ) G 3 1 2 ɨɩɪɟɞɟɥɹɟɦ div r 0 = G G G . ɉɨɞɫɬɚɜɥɹɹ ɤɨɨɪɞɢɧɚɬɵ ɡɚɞɚɧɧɨɣ ɬɨɱɤɢ, r r r G 2 2 ɩɨɥɭɱɢɦ div r 0 ( M ) = . 0 2 32 4 2 5
G a
Ɂɚɞɚɱɚ 4.
ȼɵɱɢɫɥɢɬɶ
ɞɢɜɟɪɝɟɧɰɢɸ
ɩɨɥɹ
G G G (mx 2 ny 2 )i ( py 2 mz 2 ) j (nz 2 px 2 )k ɜ ɬɨɱɤɟ Ɇ(ɯ0, ɭ0, z0). Ɂɧɚɱɟɧɢɹ m, n,
p, ɯ0, ɭ0, z0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ.
21
Ɂɚɞɚɱɚ 5. ɋ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ ɧɚɣɬɢ G G G G ɩɨɬɨɤ ɜɟɤɬɨɪɚ a (mx 2 ny 2 )i ( py 2 mz 2 ) j (nz 2 px 2 )k ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɤɭɛɚ ɫɨ ɫɬɨɪɨɧɚɦɢ 0 ɯ 1, 0 ɭ 1, 0 z 1. ɉɪɢɦɟɪ 3. ɋ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ ɧɚɣɬɢ G G G G ɩɨɬɨɤ ɜɟɤɬɨɪɚ a ( x 2 y 2 )i ( y 2 z 2 ) j ( z 2 x 2 )k ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɤɭɛɚ ɫɨ ɫɬɨɪɨɧɚɦɢ 0 ɯ 1, 0 ɭ 1, 0 z 1. ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ – Ƚɚɭɫɫɚ G
G
G
G
³³ a dS ³³³ diva dW ɧɚɣɞɟɦ div a S
wa x wa y wa z G . div a = wx wy wz
G
ɩɨ ɮɨɪɦɭɥɟ div a =
V
= 2(ɯ + ɭ + z ). Ɍɟɩɟɪɶ ɧɚɯɨɞɢɦ ɩɨɬɨɤ: Q = 2³³³ ( x y z )dW = V
= 2³³³ ( xdW ydW zdW ) . Ʉɚɠɞɵɣ ɢɡ ɬɪɟɯ ɢɧɬɟɝɪɚɥɨɜ ɪɟɲɚɟɦ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ: V
1
ɝɞɟ S – ɩɨɜɟɪɯɧɨɫɬɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ ɨɫɢ ɯ. ȿɟ
³³³ xdW ³ xdx ³³ dydz , V
S
0
ɩɥɨɳɚɞɶ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɫɬɨɪɨɧ, ɞɥɢɧɵ ɤɨɬɨɪɵɯ ɪɚɜɧɵ 1, ɬ. ɟ. x2 2
1 . Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɧɚɣɬɢ ɢ ɞɜɚ 2 1 1 1 ɞɪɭɝɢɯ ɢɧɬɟɝɪɚɥɚ ɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ: Q = 2( ) = 3. 2 2 2
³³ dydz
1 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
S
³³³ xdW V
1 0
Ɂɚɞɚɧɢɟ 3. ȼɵɱɢɫɥɟɧɢɟ ɰɢɪɤɭɥɹɰɢɢ ɜɟɤɬɨɪɚ, ɪɨɬɨɪɚ. ɉɪɢɦɟɧɟɧɢɟ ɮɨɪɦɭɥɵ ɋɬɨɤɫɚ. Ɋɚɛɨɬɚ ɜɟɤɬɨɪɚ ɩɨ ɩɟɪɟɦɟɳɟɧɢɸ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜɞɨɥɶ ɤɪɢɜɨɣ G G ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɪɢɜɨɥɢɧɟɣɧɵɦ ɢɧɬɟɝɪɚɥɨɦ: u = ³ a dr . ɗɬɨɬ ɢɧɬɟɝɪɚɥ L
ɧɚɡɵɜɚɸɬ ɬɚɤɠɟ ɥɢɧɟɣɧɵɦ ɢɧɬɟɝɪɚɥɨɦ. ɋɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ, ɫɬɨɹɳɟɟ ɩɨɞ ɟɝɨ ɡɧɚɤɨɦ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɱɟɪɟɡ ɩɪɨɟɤɰɢɢ. ȿɫɥɢ ɤɪɢɜɚɹ L ɡɚɦɤɧɭɬɚɹ, ɬɨ ɩɪɨɢɡɜɨɞɢɦɚɹ ɜɟɤɬɨɪɨɦ ɪɚɛɨɬɚ ɧɚɡɵɜɚɟɬɫɹ ɰɢɪɤɭɥɹɰɢɟɣ. Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɰɢɪɤɭɥɹɰɢɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɥɸɛɭɸ ɢɡ ɮɨɪɦɭɥ: G G G G u = ³ a dr ɢɥɢ u = ³ aW dr ɢɥɢ u = ³ (a x dx a y dy a z dz ) , ɢɥɢ u = ³ a W 0 ds , L
L
L
G
L
ɝɞɟ ɚIJ – ɩɪɨɟɤɰɢɹ ɜɟɤɬɨɪɚ ɚ ɧɚ ɤɚɫɚɬɟɥɶɧɭɸ ɤ ɤɪɢɜɨɣ L, W 0 – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ L. ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɪɚɛɨɬɵ ɞɨɥɠɧɨ ɛɵɬɶ ɡɚɞɚɧɨ ɧɚɩɪɚɜɥɟɧɢɟ ɨɛɯɨɞɚ ɩɨ ɤɪɢɜɨɣ. Ⱦɥɹ ɩɪɚɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɟɝɨ ɩɪɢɧɢɦɚɸɬ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ. Ɋɨɬɨɪɨɦ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪ, ɩɪɨɟɤɰɢɢ ɤɨɬɨɪɨɝɨ ɧɚ ɤɚɠɞɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɜɧɵ ɩɪɟɞɟɥɭ ɨɬɧɨɲɟɧɢɹ ɰɢɪɤɭɥɹɰɢɢ ɩɨɥɹ ɩɨ ɤɨɧɬɭɪɭ ɩɥɨɫɤɨɣ ɩɥɨɳɚɞɤɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɷɬɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ, ɤ ɜɟɥɢɱɢɧɟ ɩɥɨɳɚɞɤɢ, ɤɨɝɞɚ ɪɚɡɦɟɪɵ ɟɟ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ, ɬ. ɟ. G
rotna = lim
'S o0
G
³ a dr L
'S
. ɋɚɦ ɜɟɤɬɨɪ rot a, ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɠɟ ɜɢɯɪɟɜɵɦ ɩɨɥɟɦ ɢ ɟɝɨ 22
ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ § wa z wa y · G ¸i + wz ¸¹ © wy
rot a = ¨¨
ɜ
ɜɢɞɟ
§ wa x wa z · G ¨ ¸j+ wx ¹ © wz
ɨɩɪɟɞɟɥɢɬɟɥɹ:
G rota
G i w wx ax
G j w wy ay
G k w wz az
ɢɥɢ
§ wa y wa x · G ¨¨ ¸k . wy ¸¹ © wx
Ɋɨɬɨɪ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɤɬɨɪɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ. ȼɟɤɬɨɪɧɨɟ ɩɨɥɟ, ɜ ɤɨɬɨɪɨɦ rot a = 0, ɧɚɡɵɜɚɟɬɫɹ ɛɟɡɜɢɯɪɟɜɵɦ ɢɥɢ ɩɨɬɟɧɰɢɚɥɶɧɵɦ. Ɉɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ ɪɨɬɨɪɚ: rot(a + b) = rot a + rot b; rot(ua) = uǜrot a + + gradu x a; div(rot a) = 0; div(a x b) = b ǜ rot a – a ǜ rot b; rot(rot a) = graddiv a – – ¨a. ɐɢɪɤɭɥɹɰɢɹ ɜɟɤɬɨɪɧɨɝɨ ɩɨɥɹ ɫɜɹɡɚɧɚ ɫ ɟɝɨ ɪɨɬɨɪɨɦ ɮɨɪɦɭɥɨɣ G G G G ɋɬɨɤɫɚ: ³ a dr ³³ rota ds . ɐɢɪɤɭɥɹɰɢɹ ɜɟɤɬɨɪɚ ɩɨɥɹ ɩɨ ɤɨɧɬɭɪɭ L ɪɚɜɧɚ L
S
ɩɨɬɨɤɭ ɪɨɬɨɪɚ ɩɨɥɹ ɱɟɪɟɡ ɥɸɛɭɸ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɷɬɢɦ ɤɨɧɬɭɪɨɦ. Ɏɨɪɦɭɥɚ ɋɬɨɤɫɚ ɩɨɡɜɨɥɹɟɬ ɜɵɱɢɫɥɢɬɶ ɰɢɪɤɭɥɹɰɢɸ, ɧɟ ɩɪɨɜɨɞɹ ɩɪɹɦɵɯ ɜɵɱɢɫɥɟɧɢɣ, ɚ ɬɚɤɠɟ ɡɚɦɟɧɢɬɶ ɤɨɧɬɭɪɧɵɟ ɢɧɬɟɝɪɚɥɵ ɩɨɜɟɪɯɧɨɫɬɧɵɦɢ ɢ ɧɚɨɛɨɪɨɬ. G G G G Ɂɚɞɚɱɚ 6. ɇɚɣɬɢ rot a , ɟɫɥɢ a = (mxpy2z + nxp) i + nxpymzn j + (x3yp + G + pz2) k . Ɂɧɚɱɟɧɢɹ m, n, p ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. G G G G ɉɪɢɦɟɪ 4. ɇɚɣɬɢ rot a , ɟɫɥɢ a = (3x2y2z + 3x) i + 2x3yz j + (x3y2 + 2 G + 3z ) k . G k § w ( x 3 y 2 3z 2 ) w (2 x3 yz ) · G G w = ¨¨ rota ¸¸i + wy wz wz © ¹ 3 2 2 x y 3z G § w (3 x 2 y 2 z 3 x) w ( x 3 y 2 3 z 2 ) · G § w (2 x 3 yz ) w (3 x 2 y 2 z 3 x) · G 3 3 ¨¨ ¸¸ j + ¨¨ ¸¸k = (2x y – 2x y) i + wz wx wx wy © © ¹ ¹ G G + (3x2y2 – 3x2y2) j + (6x2yz – 6x2yz) k = 0. G G G G Ɂɚɞɚɱɚ 7. ȼɵɱɢɫɥɢɬɶ ɪɚɛɨɬɭ ɜɟɤɬɨɪɚ a yzi xzj xyk ɜɞɨɥɶ ɨɬɪɟɡɤɚ G G i j w w wx wy 3 x 2 y 2 z 3 x 2 x 3 yz
ɩɪɹɦɨɣ ɨɬ ɬɨɱɤɢ Ⱥ(m, n, p) ɞɨ ɬɨɱɤɢ B(x0, y0, z0). Ɂɧɚɱɟɧɢɹ m, n, p, x0, y0, z0 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɡɚɞɚɧɧɨɦɭ ɜɚɪɢɚɧɬɭ. G G G G ɉɪɢɦɟɪ 7. ȼɵɱɢɫɥɢɬɶ ɪɚɛɨɬɭ ɜɟɤɬɨɪɚ a yzi xzj xyk ɜɞɨɥɶ ɨɬɪɟɡɤɚ ɩɪɹɦɨɣ ɨɬ ɬɨɱɤɢ Ⱥ(1, 2, 3) ɞɨ ɬɨɱɤɢ B(6, 1, 1). G G G G ɉɪɢɦɟɧɢɦ ɮɨɪɦɭɥɭ u = ³ a dr , ɪɚɫɩɢɫɚɜ ɟɟ ɜ ɜɢɞɟ: ³ a dr = L
=
³ (a dx a x
AB
y
dy a z dz ) =
AB
³ ( yzdx xzdy xydz) . ȼɵɪɚɡɢɦ ɡɧɚɱɟɧɢɹ ɭ ɢ z, ɚ ɬɚɤɠɟ
AB
dy ɢ dz ɱɟɪɟɡ ɯ ɢ dx. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɤɚɧɨɧɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɟɦ ɩɪɹɦɨɣ:
23
x xA y yA z zA = = . ɉɨɞɫɬɚɜɢɦ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ Ⱥ ɢ ȼ ɢ ɫɨɫɬɚɜɢɦ x B x A yBy A z B z A x 1 y2 z 3 = = ɢ –x + 1 = 5y – 10, y = –1/5x + ɞɜɟ ɩɚɪɵ ɭɪɚɜɧɟɧɢɣ: 6 1 1 2 1 3
+ 11/5, dy = –1/5dx, –2x + 2 = 5z – 15, z = –2/5x +17/5, dz = –2/5dx. ɉɨɞɫɬɚɜɢɦ ɩɨɥɭɱɟɧɧɵɟ ɞɚɧɧɵɟ ɜ ɢɫɯɨɞɧɨɟ ɜɵɪɚɠɟɧɢɟ ɢ ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ G G ɪɚɛɨɬɭ: ³ a dr = ³ ( yzdx xzdy xydz ) = AB
AB
6
x 11 2 x 11 2 17 2 17 1 = ³ (( )( x ) x( x )( ) x( )( ))dx 5 5 5 5 5 5 5 5 5 5 1 6
=
1 1 1 (6 x 2 78 x 187)dx = (432 – 1404 +1122 – 2 + (2 x 3 39 x 2 187 x) 16 = 25 25 25 ³1
+ 39 – 187) = 0. Ɂɚɞɚɱɚ 8. G ɉɨ ɮɨɪɦɭɥɟ ɋɬɨɤɫɚ ɧɚɣɬɢ ɰɢɪɤɭɥɹɰɢɸ ɜɟɤɬɨɪɚ G G G 2 2 2 a mx 2 y p i nj pzk ɜɞɨɥɶ ɨɤɪɭɠɧɨɫɬɢ x + y = R . Ɂɧɚɱɟɧɢɹ m, n, p ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 6 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ G 8. ɉɨ ɮɨɪɦɭɥɟ ɋɬɨɤɫɚ ɧɚɣɬɢ ɰɢɪɤɭɥɹɰɢɸ ɜɟɤɬɨɪɚ G G G 2 2 2 a x 2 y 3 i j zk ɜɞɨɥɶ ɨɤɪɭɠɧɨɫɬɢ x + y = R . G G G G ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ ɋɬɨɤɫɚ ³ a dr ³³ rota ds , ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɩɨɬɨɤ L
S
ɪɨɬɨɪɚ ɜɟɤɬɨɪɚ ɱɟɪɟɡ ɡɚɦɤɧɭɬɭɸ ɩɨɜɟɪɯɧɨɫɬɶ S, ɨɝɪɚɧɢɱɟɧɧɭɸ ɡɚɞɚɧɧɨɣ G G G ɨɤɪɭɠɧɨɫɬɶɸ, ɬ. ɟ. ³³ rota dS . Ⱦɥɹ ɷɬɨɝɨ ɜɧɚɱɚɥɟ ɧɚɯɨɞɢɦ rota ɩɨ ɮɨɪɦɭɥɟ: S
wa § wa G rot a = ¨¨ z y w wz y © 2 2 G 3x y ) k . Ⱦɥɹ
· G § wa x wa z ¸¸i + ¨ wx © wz ¹
·G ¸j+ ¹
G G § wa y wa x · G ¨¨ ¸¸k = (0 - 0) i + (0 - 0) j + (0 – w w x y © ¹
ɭɞɨɛɫɬɜɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜɜɟɞɟɦ ɰɢɥɢɧɞɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ: x = rcos ij, y = rsin ij, dS = rdrd ij. Ɍɟɩɟɪɶ ɩɪɨɜɨɞɢɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɢ ɧɚɯɨɞɢɦ ɩɨɬɨɤ ɪɨɬɨɪɚ, ɤɨɬɨɪɵɣ G
ɢ ɛɭɞɟɬ ɪɚɜɟɧ ɰɢɪɤɭɥɹɰɢɢ ɜɟɤɬɨɪɚ a :
G
G
2S
³³ rota dS = S
3R 6 = 6
2S
3R 6 ³0 sin M cos MdM = 6 4 2
2
2S
3R 6 ³0 sin 2MdM = 6 4 2
2S
–
R6 1 1 cos 4Md (4M ) ) = (M sin(4M )) 02S ³ 16 4 4 0
R
–3 ³ sin 2 M cos 2 MdM ³ r 5 dr = 0
2S
0
2S
R6 1 cos 4M ³0 2 dM = 16 ( ³0 dM –
R 6S . 8
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 5 (3 ɱɚɫɚ) Ɂɚɞɚɧɢɟ 1. Ɉɩɪɟɞɟɥɟɧɢɟ ɩɚɪɚɦɟɬɪɨɜ ɭɪɚɜɧɟɧɢɹ ɩɪɹɦɨɣ ɥɢɧɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ ɦɟɬɨɞɨɦ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ. ȼ ɝɟɨɮɢɡɢɤɟ ɱɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɩɨɥɶɡɨɜɚɬɶɫɹ ɷɦɩɢɪɢɱɟɫɤɢɦɢ ɮɨɪɦɭɥɚɦɢ, ɫɨɫɬɚɜɥɟɧɧɵɦɢ ɧɚ ɨɫɧɨɜɟ ɧɚɛɥɸɞɟɧɢɹ ɡɚ ɨɩɪɟɞɟɥɟɧɧɵɦɢ 24
ɜɟɥɢɱɢɧɚɦɢ. Ɉɞɢɧ ɢɡ ɧɚɢɥɭɱɲɢɯ ɫɩɨɫɨɛɨɜ ɩɨɥɭɱɟɧɢɹ ɬɚɤɢɯ ɮɨɪɦɭɥ – ɷɬɨ ɦɟɬɨɞ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɞɜɭɦɹ ɜɟɥɢɱɢɧɚɦɢ ɯ ɢ ɭ. Ɂɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɭ ɢ ɯ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɨɣ: ɭ = ɚɯ + b, ɝɞɟ ɚ ɢ b ɧɟɤɨɬɨɪɵɟ ɩɨɫɬɨɹɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɩɨɞɥɟɠɚɳɢɟ ɨɩɪɟɞɟɥɟɧɢɸ. ɋɩɨɫɨɛ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɨɛɪɚɬɶ ɤɨɷɮɮɢɰɢɟɧɬɵ ɚ ɢ b ɬɚɤ, ɱɬɨɛɵ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɩɨɝɪɟɲɧɨɫɬɟɣ ɛɵɥɚ ɛɵ ɜɨɡɦɨɠɧɨ ɦɟɧɶɲɟɣ. n
Ɉɛɨɡɧɚɱɢɦ: x
¦x
n
i
i 1
n
, y
¦y
n
i
i 1
n
_ 2
, x
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ:
¦x
n
2 i
i 1
_
,
xy
¦x i 1
i
yi
.
n n °b ax y Ɋɟɲɢɦ _ ® _ . °¯bx a x 2 xy
ɞɚɧɧɭɸ
_
ɫɢɫɬɟɦɭ ɢ ɩɨɥɭɱɢɦ ɪɚɫɱɟɬɧɵɟ ɮɨɪɦɭɥɵ: ɚ =
xy x y _
, b = y x a .
x 2 (x ) 2
Ɋɚɫɱɟɬɵ ɥɭɱɲɟ ɩɪɨɜɨɞɢɬɶ ɜ ɬɚɛɥɢɰɟ yi x i2 ʋ ɩ. ɩ. ɯi 1 2 … n
xi yi
n
¦ i 1
_
_
y
x
xy
x2
Ɂɚɞɚɱɚ 1. Ɇɟɬɨɞɨɦ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɨɩɪɟɞɟɥɢɬɶ ɩɚɪɚɦɟɬɪɵ ɚ ɢ b ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɭ = ɚɯ + b. ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɭ ɨɬ ɯ ɢ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɩɪɹɦɨɣ ɭ = ɚɯ + b. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɚɪɢɚɧɬɨɦ ɜɵɛɪɚɬɶ ɡɧɚɱɟɧɢɹ ɯ ɢ ɭ ɜ ɬɚɛɥɢɰɟ 7. ȼɚɪɢɚɧɬ 1 ȼɚɪɢɚɧɬ 2 ȼɚɪɢɚɧɬ 3 ȼɚɪɢɚɧɬ 4 ȼɚɪɢɚɧɬ 5
xi yi xi yi xi yi xi yi xi yi
0,0 0,9 1,6 1,7 3,2 2,5 4,8 3,3 6,4 4,0
0,2 1,1 1,8 1,8 3,4 2,6 5,0 3,4 6,6 4,1
0,4 1,2 2,0 1,9 3,6 2,7 5,2 3,5 6,8 4,2
0,6 1,3 2,2 2,0 3,8 2,8 5,4 3,6 7,0 4,3
Ɍɚɛɥɢɰɚ 7 0,8 1,0 1,4 1,5 2,4 2,6 2,1 2,2 4,0 4,2 2,9 3,0 5,6 5,8 3,7 3,8 7,2 7,4 4,4 4,5 25
ȼɚɪɢɚɧɬ 6 ȼɚɪɢɚɧɬ 7 ȼɚɪɢɚɧɬ 8 ȼɚɪɢɚɧɬ 9 ȼɚɪɢɚɧɬ 10 ȼɚɪɢɚɧɬ 11 ȼɚɪɢɚɧɬ 12 ȼɚɪɢɚɧɬ 13 ȼɚɪɢɚɧɬ 14 ȼɚɪɢɚɧɬ 15 ȼɚɪɢɚɧɬ 16 ȼɚɪɢɚɧɬ 17 ȼɚɪɢɚɧɬ 18 ȼɚɪɢɚɧɬ 19 ȼɚɪɢɚɧɬ 20 ȼɚɪɢɚɧɬ 21 ȼɚɪɢɚɧɬ 22 ȼɚɪɢɚɧɬ 23 ȼɚɪɢɚɧɬ 24 ȼɚɪɢɚɧɬ 25 ȼɚɪɢɚɧɬ 26
xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi xi yi
8,0 4,8 0,1 1,0 1,7 1,8 3,3 2,6 4,9 3,3 4,9 4,8 3,3 1,0 5,8 1,8 0,5 2,6 5,2 3,3 2,0 0,9 2,6 1,7 5,2 2,5 5,8 3,3 7,4 4,0 2,0 7,9 2,6 9,1 5,2 0,5 5,8 0,3 7,4 2,0 2,0 1,7
8,2 4,9 0,3 1,1 1,9 1,9 3,5 2,7 5,1 3,4 5,1 4,9 3,4 1,1 6,0 1,9 0,6 2,7 5,4 3,4 2,2 1,1 2,8 1,8 5,4 2,6 6,0 3,4 7,6 4,1 2,2 8,1 2,8 9,2 5,4 0,6 6,0 0,4 7,6 2,1 2,2 1,9
8,4 5,0 0,5 1,2 2,1 2,0 3,7 2,7 5,3 3,5 5,3 5,0 3,5 1,2 6,2 2,0 0,7 2,7 5,6 3,5 2,4 1,2 3,0 1,9 5,6 2,7 6,2 3,5 7,8 4,2 2,4 8,2 3,0 9,3 5,6 0,7 6,2 0,5 7,8 2,2 2,4 2,1
8,6 5,1 0,7 1,3 2,3 2,1 3,9 2,8 5,5 3,6 5,5 5,1 3,6 1,3 6,4 2,1 0,8 2,8 5,8 3,6 2,6 1,3 3,2 2,0 5,8 2,8 6,4 3,6 8,0 4,3 2,6 8,3 3,2 9,4 5,8 0,8 6,4 0,6 8,0 2,3 2,6 2,3
8,8 5,2 0,9 1,4 2,5 2,2 4,1 2,9 5,7 3,7 5,7 5,2 3,7 1,4 6,6 2,2 0,9 2,9 6,0 3,7 2,8 1,4 3,4 2,1 6,0 2,9 6,6 3,7 8,2 4,4 2,8 8,4 3,4 9,5 6,0 0,9 6,6 0,7 8,2 2,4 2,8 2,5 26
9,0 5,3 1,1 1,5 2,7 2,3 4,3 3,0 5,9 3,8 5,9 5,3 3,8 1,5 6,8 2,3 1,0 3,0 6,2 3,8 3,0 1,5 3,6 2,2 6,2 3,0 6,8 3,8 8,4 4,5 3,0 8,5 3,6 9,6 6,2 1,0 6,8 0,8 8,4 2,5 3,0 2,7
ȼɚɪɢɚɧɬ 27 xi yi ȼɚɪɢɚɧɬ 28 xi yi
2,6 3,3 5,2 4,9
2,8 3,5 5,4 5,1
3,0 3,7 5,6 5,3
3,2 3,9 5,8 5,5
3,4 4,1 6,0 5,7
3,6 4,3 6,2 5,9
Ɂɚɞɚɧɢɟ 2. Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ Ʉɪɚɦɟɪɚ. Ⱦɚɧɚ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɪɢɱɟɦ m = n. a11 x1 a12 x2 ... a1n xn b1 , °a x a x ... a x b ° 21 1 22 2 2n n 2, ® ..............................................., ° °am1 x1 am 2 x2 ... amn xn bm . ¯ ɉɭɫɬɶ ' – ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ Ⱥ, ɫɨɫɬɚɜɥɟɧɧɨɣ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɧɟɢɡɜɟɫɬɧɵɯ, '1, '2, . . . , 'n – ɨɩɪɟɞɟɥɢɬɟɥɢ ɦɚɬɪɢɰ, ɩɨɥɭɱɚɟɦɵɟ ɢɡ ɦɚɬɪɢɰɵ Ⱥ ɡɚɦɟɧɨɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɫɬɨɥɛɰɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɯ ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ. Ɍɨɝɞɚ, ɟɫɥɢ ' z 0, ɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ, ɨɩɪɟɞɟɥɹɟɦɨɟ ɩɨ ɮɨɪɦɭɥɚɦ: ɯ1 =
'1 ' , ɯ2 = 2 , ' '
. . . , ɯn =
'n . Ⱦɚɧɧɵɟ '
ɮɨɪɦɭɥɵ ɢ ɹɜɥɹɸɬɫɹ ɮɨɪɦɭɥɚɦɢ Ʉɪɚɦɟɪɚ. Ɂɚɞɚɱɚ 2. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ Ʉɪɚɦɟɪɚ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ: a11 x1 a12 x2 a 13 x3 b 1 ° ®a21 x1 a22 x2 a23 x3 b2 . Ɂɧɚɱɟɧɢɹ ɚ11, ɚ12, ɚ13, ɚ21,ɚ22, ɚ23, ɚ31, ɚ32, ɚ33, b1, b2, b3 °a x a x a x b ¯ 31 1 32 2 33 3 3
ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 8 ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 1. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ Ʉɪɚɦɟɪɚ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ: x1 x2 x3 3 ° ®2 x1 x2 x3 11 . °x x 2x 8 3 ¯ 1 2
Ɇɚɬɪɢɰɚ Ⱥ, ɫɨɫɬɚɜɥɟɧɧɚɹ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɧɟɢɡɜɟɫɬɧɵɯ ɢɦɟɟɬ: 1 1 1 §1 1 1· ¸ ¨ Ⱥ= ¨ 2 1 1 ¸ . ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ Ⱥ: ' = 2 1 1 = 5, ɬ. ɤ. ¨1 1 2¸ 1 1 2 ¹ ©
' z 0, ɬɨ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ. ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɢɬɟɥɶ '1 ɦɚɬɪɢɰɵ, ɩɨɥɭɱɟɧɧɨɣ ɢɡ ɦɚɬɪɢɰɵ Ⱥ ɩɭɬɟɦ 3
1 1
ɡɚɦɟɧɵ ɩɟɪɜɨɝɨ ɫɬɨɥɛɰɚ ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ: '1 = 11 1 8
1
1 = 2
= 3(2 – 1) + 1(22 – 8) + 1(11 – 8) = 3 + 14 + 3 = 20. ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɢɬɟɥɶ '2 ɦɚɬɪɢɰɵ, ɩɨɥɭɱɟɧɧɨɣ ɢɡ ɦɚɬɪɢɰɵ Ⱥ ɩɭɬɟɦ ɡɚɦɟɧɵ ɜɬɨɪɨɝɨ ɫɬɨɥɛɰɚ 27
1
3
1
ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ: '2 = 2 11 1 = 1(22 – 8) – 3(4 – 1) + 1(16 – 1
8
2
– 11) = 14 – 9 + 5. ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɢɬɟɥɶ '3 ɦɚɬɪɢɰɵ, ɩɨɥɭɱɟɧɧɨɣ ɢɡ ɦɚɬɪɢɰɵ Ⱥ ɩɭɬɟɦ ɡɚɦɟɧɵ ɬɪɟɬɶɟɝɨ ɫɬɨɥɛɰɚ ɫɬɨɥɛɰɨɦ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ: '3 1 1
= 2 1
1 1
3 11 = 1(8 – 11) + 1(16 – 11) + 3(2 – 1) = –3 + 5 + 3 = 5. Ɍɨɝɞɚ 8
ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɩɨɥɭɱɢɦ ɜ ɜɢɞɟ: ɯ1 =
' '1 ' = 20/5 = 4, ɯ2 = 2 = 10/5 = 2, ɯ3 = 3 = 5/5 = 1. ' ' '
ɇɨɦɟɪ ɜɚɪɢɚɧɬɚ 1 2 ɚ11 1 2 ɚ12 –1 –3 ɚ13 1 4 ɚ21 2 3 ɚ22 1 4 ɚ23 1 –2 ɚ31 1 4 ɚ32 1 2 ɚ33 2 3 b1 3 20 b2 11 –11 b3 8 9
3 4 1 1 1 2 –2 3 2 2 3 3 –7 –4 5 3 2 1 1 1 6 2 16 –5 16 3
5 5 8 1 3 –2 6 2 1 –1 2 –7 –5
6 7 8 1 7 2 2 2 1 1 3 –1 –2 5 3 3 –3 4 –3 2 0 3 10 1 –4 –11 0 5 5 1 8 15 0 –5 15 –6 10 36 1
9 1 1 1 3 2 1 1 3 –1 22 47 18
10 2 –3 –1 3 4 3 1 1 1 –6 –5 –2
Ɍɚɛɥɢɰɚ 8 11 12 13 –2 1 1 1 2 –1 0 3 1 1 2 2 –2 3 1 –1 –1 1 3 3 1 4 1 1 –2 –4 2 –6 6 1 5 4 4 13 0 4
ɇɨɦɟɪ ɜɚɪɢɚɧɬɚ 14 15 16 17 18 19 20 21 22 23 24 25 26 ɚ11 1 5 1 7 2 1 2 –2 1 1 2 1 1 ɚ12 2 8 2 2 1 1 –3 1 2 –1 –3 1 2 ɚ13 3 1 1 3 –1 1 –1 0 3 1 4 –2 3 ɚ21 2 3 –2 5 3 3 3 1 2 2 3 2 2 ɚ22 3 –2 3 –3 4 2 4 –2 3 1 4 3 3 ɚ23 –4 6 –3 2 0 1 3 –1 –1 1 –2 –7 –4 ɚ31 3 2 3 10 1 1 1 3 3 1 4 5 3 ɚ32 1 1 –4 –11 0 3 1 4 1 1 2 2 1 ɚ33 1 –1 5 5 1 –1 1 –2 –4 2 3 1 1 b1 6 14 4 12 2 3 –2 –1 3 2 6 0 12 b2 1 7 –2 4 7 6 10 –2 5 8 10 –4 2 b3 5 2 4 4 2 3 3 5 4 8 18 16 10
28
Ɂɚɞɚɧɢɟ 3. Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ. Ⱦɚɧɚ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɪɢɱɟɦ m = n: a11 x1 a12 x2 ... a1n xn b1 °a x a x ... a x b ° 21 1 22 2 2n n 2 ® . Ɂɚɩɢɲɟɦ ɞɚɧɧɭɸ ɫɢɫɬɟɦɭ ɜ ɦɚɬɪɢɱɧɨɦ ................................................... ° °¯am1 x1 am 2 x2 ... amn xn bm
ɜɢɞɟ: Ⱥ ɏ = ȼ. Ɉɩɪɟɞɟɥɹɟɬɫɹ ɨɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ Ⱥ–1. Ɍɨɝɞɚ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɜ ɦɚɬɪɢɱɧɨɦ ɜɢɞɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ɏ = Ⱥ–1 ȼ, ɢɥɢ ɜ ɪɚɡɜɟɪɧɭɬɨɦ ɜɢɞɟ: § A11 § x1 · ¨ ¨ ¸ x 2 ¨ ¸ = ¨ A12 ¨ ... ¸ ¨ ... ¨ ¸ ¨ x © n¹ © A1n
A 21 ... A22 ... ... ... A2 n ...
An1 · ¸ An 2 ¸ ... ¸ ¸ Ann ¹
§ b1 · ¨ ¸ ¨ b2 ¸ . ¨ ... ¸ ¨ ¸ © bn ¹
Ɇɚɬɪɢɰɚ Ⱥ–1 ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɬɧɨɣ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɜɚɞɪɚɬɧɨɣ ɦɚɬɪɢɰɟ Ⱥ, ɟɫɥɢ ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɷɬɨɣ ɦɚɬɪɢɰɵ ɧɚ ɞɚɧɧɭɸ ɤɚɤ ɫɩɪɚɜɚ, ɬɚɤ ɢ ɫɥɟɜɚ ɩɨɥɭɱɚɟɬɫɹ ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ, ɬ. ɟ. Ⱥ–1 Ⱥ = Ⱥ Ⱥ–1 = ȿ. Ɉɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ Ⱥ–1 ɫɭɳɟɫɬɜɭɟɬ ɢ ɟɞɢɧɫɬɜɟɧɧɚɹ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɨɩɪɟɞɟɥɢɬɟɥɶ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ ɨɬɥɢɱɟɧ ɨɬ ɧɭɥɹ, ɬ. ɟ. _Ⱥ_ z 0. Ⱥɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ. 1. ɇɚɯɨɞɢɬɫɹ ɨɩɪɟɞɟɥɢɬɟɥɶ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ _Ⱥ_. ȿɫɥɢ _Ⱥ_ z 0, ɬɨ ɨɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɫɭɳɟɫɬɜɭɟɬ. 2. ɇɚɯɨɞɢɬɫɹ ɦɚɬɪɢɰɚ ȺɌ, ɬɪɚɧɫɩɨɧɢɪɨɜɚɧɧɚɹ ɤ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɟ Ⱥ. 3. ɇɚɯɨɞɹɬɫɹ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɞɨɩɨɥɧɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɬɪɚɧɫɩɨɧɢɪɨɜɚɧɧɨɣ ɦɚɬɪɢɰɵ ȺijɌ ɢ ɫɨɫɬɚɜɥɹɟɬɫɹ ɢɡ ɧɢɯ ɩɪɢɫɨɟɞɢɧɟɧɧɚɹ ɦɚɬɪɢɰɚ Ⱥɩ, ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɪɚɜɧɵ: ɚijɩ = ȺijɌ. 4. ȼɵɱɢɫɥɹɟɬɫɹ ɨɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɩɨ ɮɨɪɦɭɥɟ: Ⱥ–1 = 1/_Ⱥ_ Ⱥɩ 5. ɉɪɨɜɟɪɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ Ⱥ–1, ɢɫɯɨɞɹ ɢɡ ɟɟ ɨɩɪɟɞɟɥɟɧɢɹ: Ⱥ–1 Ⱥ = Ⱥ Ⱥ–1 = ȿ. Ɂɚɞɚɱɚ 3. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ a11 x1 a12 x 2 a 13 x3 b1
ɭɪɚɜɧɟɧɢɣ: °®a 21 x1 a 22 x2 a 23 x3 b2 . Ɂɧɚɱɟɧɢɹ ɚ11, ɚ12, ɚ13, ɚ21,ɚ22, ɚ23, ɚ31, ɚ32, °a x a x a x 32 2 33 3 ¯ 31 1
b3
ɚ33, b1, b2, b3 ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 8 ɫɨɝɥɚɫɧɨ ɡɚɞɚɧɧɨɦɭ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 2. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ x1 x2 x3 °
3
ɭɪɚɜɧɟɧɢɣ: ®2 x1 x2 x3 11 . °x x 2x 3 ¯ 1 2
8
Ɇɚɬɪɢɰɚ Ⱥ, ɫɨɫɬɚɜɥɟɧɧɚɹ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɧɟɢɡɜɟɫɬɧɵɯ ɢɦɟɟɬ: 29
1 1 1 §1 1 1· ¸ ¨ Ⱥ= ¨ 2 1 1 ¸ . ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ Ⱥ: ' = 2 1 1 = 5. ɇɚɣɞɟɦ ¨1 1 2¸ 1 1 2 ¹ ©
ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɭ ɤ ɞɚɧɧɨɣ ɦɚɬɪɢɰɟ. _Ⱥ_ = 5, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ ɫɭɳɟɫɬɜɭɟɬ. §1 ¨
2 1· ¸
¨1 ©
1 2 ¸¹
ɇɚɯɨɞɢɦ ɦɚɬɪɢɰɭ ȺɌ: ȺɌ = ¨ 1 1 1 ¸ . ɇɚɯɨɞɢɦ ɚɥɝɟɛɪɚɢɱɟɫɤɢɟ ɞɨɩɨɥɧɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ȺɌ: Ⱥ11Ɍ = (–1)1+1
1 1 1 1 = 2 – 1 = 1, Ⱥ12Ɍ = (–1)1+2 = –(–2 – 1) = 3; 1 2 1 2 1 1 2 1 = –1 – 1 = –2, Ⱥ21Ɍ = (–1)2+1 = –(4 – 1) = –3; 1 1 1 2
Ⱥ13Ɍ = (–1)1+3 Ⱥ22Ɍ = (–1)2+2
1 1 1 2 = 2 – 1 = 1, Ⱥ23Ɍ = (–1)2+3 = –(1 – 2) = 1; 1 2 1 1
Ⱥ31Ɍ = (–1)3+1
2 1 1 1 = 2 – 1 = 1, Ⱥ32Ɍ = (–1)3+2 = –(1 + 1) = –2; 1 1 1 1
Ⱥ33Ɍ = (–1)3+3
1 2 = 1 + 2 = 3. 1 1
2· ¸ 1 ¸ . ȼɵɱɢɫɥɹɟɦ ¨ 1 2 3 ¸ ¹ © 3 2 · § 1/ 5 3 / 5 2 / 5· § 1 ¸ ¨ ¸ ¨ ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɭ: Ⱥ–1 = 1/_Ⱥ_ Ⱥɩ =1/5 ¨ 3 1 1 ¸ = ¨ 3 / 5 1 / 5 1/ 5 ¸ . ¨ 1 2 3 ¸ ¨ 1/ 5 2 / 5 3 / 5 ¸ ¹ © ¹ © § 1 ¨
ɇɚɯɨɞɢɦ ɩɪɢɫɨɟɞɢɧɟɧɧɭɸ ɦɚɬɪɢɰɭ: Ⱥɩ = ¨ 3
3
1
ɉɪɨɜɟɪɹɟɦ ɩɪɚɜɢɥɶɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɹ ɨɛɪɚɬɧɨɣ ɦɚɬɪɢɰɵ: § 1/ 5 ¨
Ⱥ–1 Ⱥ = ¨ 3 / 5 ¨ 1/ 5 ©
2 / 5· § 1 1 1 · §1 0 0· ¸ ¨ ¸ ¸ ¨ 1/ 5 ¸ ¨ 2 1 1 ¸ = ¨ 0 1 0 ¸ . ¨0 0 1¸ 2 / 5 3 / 5 ¸¹ ¨© 1 1 2 ¸¹ ¹ © 3/ 5
1/ 5
ɂɫɩɨɥɶɡɭɹ ɨɛɪɚɬɧɭɸ ɦɚɬɪɢɰɭ, ɧɚɯɨɞɢɦ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɜɢɞɟ: ɏ = Ⱥ–1 ȼ Ɂɚɩɢɲɟɦ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɜ ɪɚɡɜɟɪɧɭɬɨɦ ɜɢɞɟ: 3 / 5 2 / 5· § 3 · § x1 · § 1/ 5 ¸ ¨ ¸ ¨ ¸ ¨ = 3 / 5 1 /5 1 / 5 ¸ ¨11¸ . x ¸ ¨ 2 ¨ ¨x ¸ ¨ 1/ 5 2 / 5 3 / 5 ¸ ¨ 8 ¸ ¹ © ¹ © 3¹ ©
Ɋɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɛɭɞɭɬ ɢɦɟɬɶ ɜɢɞ: ɯ1 = 1/5 3 + 3/5 11 – 2/5 8 = 4, ɯ2 = –3/5 3 + 1/5 11 + 1/5 8 = 2, ɯ3 = 1/5 3 – 2/5 11 + 3/5 8 = 1. 30
Ɂɚɞɚɧɢɟ 4. Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ. Ⱦɚɧɚ ɫɢɫɬɟɦɚ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɪɢɱɟɦ m = n. a11 x1 a12 x2 ... a1n xn b1 °a x a x ... a x b ° 21 1 22 2 2n n 2 ® . ............................................... ° °¯am1 x1 am 2 x2 ... amn xn bm
Ɇɟɬɨɞ Ƚɚɭɫɫɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɫ ɩɨɦɨɳɶɸ ɷɥɟɦɟɧɬɚɪɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɩɪɢɜɨɞɢɬɫɹ ɤ ɪɚɜɧɨɫɢɥɶɧɨɣ ɫɢɫɬɟɦɟ ɫɬɭɩɟɧɱɚɬɨɝɨ ɜɢɞɚ. ɉɟɪɟɯɨɞ ɫɢɫɬɟɦɵ ɤ ɪɚɜɧɨɫɢɥɶɧɨɣ ɟɣ ɫɢɫɬɟɦɟ ɧɚɡɵɜɚɟɬɫɹ ɩɪɹɦɵɦ ɯɨɞɨɦ Ƚɚɭɫɫɚ. ɇɚɯɨɠɞɟɧɢɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɢɡ ɫɢɫɬɟɦɵ ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɬɧɵɦ ɯɨɞɨɦ Ƚɚɭɫɫɚ. ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ƚɚɭɫɫɚ ɭɞɨɛɧɨ ɩɪɨɜɨɞɢɬɶ ɧɟ ɫ ɫɚɦɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ, ɚ ɫ ɪɚɫɲɢɪɟɧɧɨɣ ɦɚɬɪɢɰɟɣ § a11 ¨ a ɜɢɞɚ: Ⱥ = ¨¨ 21 ... ¨¨ © am1
a12 a22 ... am 2
a1n b1 · ¸ ... a2 n b2 ¸ ... ... ¸ ¸ ... amn bn ¸¹ ...
Ɂɚɞɚɱɚ 4. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ: a11 x1 a12 x2 a 13 x3 b 1 ° ®a21 x1 a22 x2 a23 x3 b2 . Ɂɧɚɱɟɧɢɹ ɚ11, ɚ12, ɚ13, ɚ21,ɚ22, ɚ23, ɚ31, ɚ32, ɚ33, b1, b2, b3 °a x a x a x b ¯ 31 1 32 2 33 3 3
ɜɡɹɬɶ ɢɡ ɬɚɛɥɢɰɵ 8 ɩɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 3. Ɋɟɲɢɬɶ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ: x1 x2 x3 3 §1 1 ¨ ° ®2 x1 x2 x3 11 . Ɋɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɢɦɟɟɬ ɜɢɞ: Ⱥ = ¨ 2 1 °x x 2x 8 ¨1 1 3 © ¯ 1 2 §1 1 1 ¨ ɉɪɟɨɛɪɚɡɭɟɦ ɞɚɧɧɭɸ ɦɚɬɪɢɰɭ ɤ ɫɬɭɩɟɧɱɚɬɨɦɭ ɜɢɞɭ: ¨ 2 1 1 ¨1 1 2 © §1 1 ¨ 3 ¨0 2 © §1 1 ¨ ¨0 3 ¨0 0 ©
¨0
1 | 3· §1 ¸ ¨ 1 | 5¸ ¨ 0 ¨0 1 | 5 ¸¹ © 0 | 2· 1 § ¸ ¨ 0 | 6¸ ¨0 ¨0 1 | 1 ¸¹ ©
1 1 | 3· §1 ¸ ¨ 3 1 | 5¸ ¨ 0 ¨0 0 5 | 5 ¸¹ © 1 0 1 0 | 2· § ¸ ¨ 1 0 | 2¸ ¨ 0 1 ¨0 0 0 1 | 1 ¸¹ ©
ɉɟɪɟɣɞɟɦ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ: x ° ® x2 °x ¯ 3
4, 2, 1.
31
1 |
3· ¸ 1 | 11¸ . 2 | 8 ¸¹ | 3· ¸ | 11¸ | 8 ¸¹
1 1 | 3· ¸ 3 1 | 5¸ 0 1 | 1 ¸¹ 0 | 4· ¸ 0 | 2¸ . 1 | 1 ¸¹
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 6 (3 ɱɚɫɚ). Ɋɟɲɟɧɢɟ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɨɜ Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɢɫɤɨɦɭɸ ɮɭɧɤɰɢɸ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ, ɷɬɢ ɩɟɪɟɦɟɧɧɵɟ ɢ ɩɪɨɢɡɜɨɞɧɵɟ ɪɚɡɥɢɱɧɵɯ ɩɨɪɹɞɤɨɜ. ȿɫɥɢ ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ ɡɚɜɢɫɢɬ ɨɬ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɨɛɵɤɧɨɜɟɧɧɵɦ, ɟɫɥɢ ɨɬ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɟɦɟɧɧɵɯ, ɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. Ɉɛɵɤɧɨɜɟɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ n-ɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ: F(ɯ, ɭ, ɭ/, y//, ... y( n)) = 0. ɉɨɪɹɞɨɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɪɚɜɟɧ ɩɨɪɹɞɤɭ ɫɬɚɪɲɟɣ ɩɪɨɢɡɜɨɞɧɨɣ. ɋɬɟɩɟɧɶɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɫɬɟɩɟɧɶ ɫɬɚɪɲɟɣ ɩɪɨɢɡɜɨɞɧɨɣ. Ɋɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɥɸɛɚɹ ɮɭɧɤɰɢɹ y = M(x), ɤɨɬɨɪɚɹ, ɛɭɞɭɱɢ ɩɨɞɫɬɚɜɥɟɧɚ ɜ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɨɛɪɚɳɚɟɬ ɟɝɨ ɜ ɬɨɠɞɟɫɬɜɨ. Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ n-ɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ ɟɝɨ ɪɟɲɟɧɢɟ ɭ = M(ɯ, ɋ1, ɋ2, . . . , ɋn) ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɨɣ ɯ ɢ n ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ ɋ1, ɋ2, . . . , ɋn. ɑɚɫɬɧɵɦ ɪɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɪɟɲɟɧɢɟ, ɩɨɥɭɱɚɟɦɨɟ ɢɡ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɤɨɧɤɪɟɬɧɵɯ ɡɧɚɱɟɧɢɹɯ ɩɨɫɬɨɹɧɧɵɯ ɋ1, ɋ2, . . . , ɋn. ȿɫɥɢ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɡɚɞɚɧɨ ɧɟɹɜɧɨ, ɬ. ɟ. ɜ ɜɢɞɟ: M(ɯ, ɭ) = 0, ɬɨ ɨɧɨ ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. Ɂɚɞɚɧɢɟ 1. Ɋɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ, ɪɚɡɪɟɲɟɧɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɞɧɨɣ ɢɦɟɸɬ ɜɢɞ: y/ = f(x, y). Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ – ɷɬɨ ɬɚɤɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ: dy/dx = f(x)g(x) ɢɥɢ ɜ ɜɢɞɟ: M(x)N(y)dx + + P(x)Q(y)dy = 0, ɝɞɟ: f(x), M(x), P(x) – ɧɟɤɨɬɨɪɵɟ ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ ɯ, ɚ g(y), N(y), Q(y) – ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ ɭ. Ɉɞɧɨɪɨɞɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ – ɷɬɨ ɬɚɤɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ: y/ = g(y/x), ɝɞɟ: g – ɧɟɤɨɬɨɪɚɹ ɮɭɧɤɰɢɹ. ȼɜɟɞɟɧɢɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɢ z = y/x ɩɨɡɜɨɥɹɟɬ ɫɜɟɫɬɢ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɤ ɭɪɚɜɧɟɧɢɸ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ. Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ – ɷɬɨ ɬɚɤɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɟ ɢɦɟɟɬ ɜɢɞ: y/ + f(x)y = g(x), ɝɞɟ f(x) ɢ g(x) – ɧɟɤɨɬɨɪɵɟ ɧɟɩɪɟɪɵɜɧɵɟ ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ ɯ. Ɋɟɲɟɧɢɟ ɥɢɧɟɣɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɳɟɬɫɹ ɜ ɜɢɞɟ: y(x) = u(x) ǜ v(x). Ɉɞɧɚ ɢɡ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɜɵɛɢɪɚɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɨ, ɚ ɞɪɭɝɚɹ – 32
ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɹ y/ + f(x)ɭ = g(x). ɉɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɜ ɪɟɲɟɧɢɟ ɭ = uv, ɩɨɥɭɱɢɦ: y/ = u/v + uv/. ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɹ y ɢ y/ ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ: u/v + uv/ + f(x)uv = g(x). ɋɝɪɭɩɩɢɪɨɜɚɜ ɫɥɚɝɚɟɦɵɟ ɜ ɥɟɜɨɣ ɱɚɫɬɢ, ɛɭɞɟɦ ɢɦɟɬɶ: (u/v + f(x)uv) + uv/ = g(x). ȼɵɧɨɫɹ ɨɛɳɢɣ ɦɧɨɠɢɬɟɥɶ v ɡɚ ɫɤɨɛɤɢ, ɩɨɥɭɱɢɦ: (u/ + f(x)u)v + uv/ = g(x). ɇɚɯɨɞɢɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ u = u(x) ɢɡ ɭɪɚɜɧɟɧɢɹ u/ + f(x)u = 0, ɤɨɬɨɪɨɟ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ. ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ u ɜ ɭɪɚɜɧɟɧɢɟ (u/v + f(x)uv) + uv/ = g(x), ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ, ɢɡ ɤɨɬɨɪɨɝɨ ɧɚɯɨɞɢɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ v. Ⱦɚɥɟɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ ɜ ɜɢɞɟ: y = uv. ɉɪɢɦɟɪ 1. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ y 2 1 dx = xydy. Ɋɚɡɞɟɥɢɦ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɧɚ ɜɵɪɚɠɟɧɢɟ ɯ (ɩɪɢ ɯ z 0), ɩɪɢɯɨɞɢɦ ɤ ɪɚɜɟɧɫɬɜɭ
³
dx x
³
ydy y2 1
ɢɥɢ
ln_x_ =
dx x
ydy y2 1
y2 1
. ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɢɦ:
y 2 1 + C1. Ɂɚɩɢɲɟɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ:
ɯ = r eC1 e y 1 ɢɥɢ ɯ = ɋ e y 1 . ɉɪɢɦɟɪ 2. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ (ɯ + 2y)y/ = 1. ɉɪɢ ɪɟɲɟɧɢɢ ɬɚɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɨɧɨ ɩɪɢɜɨɞɢɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ ɡɚɦɟɧɨɣ ɩɟɪɟɦɟɧɧɨɣ z = ax + by + c, ɝɞɟ ɚ, b ɢ ɫ – ɧɟɤɨɬɨɪɨɟ ɱɢɫɥɨ. ɉɨɥɨɠɢɦ z = x + 2y. Ɍɨɝɞɚ z/ = 1 + 2y./, ɨɬɤɭɞɚ 2
y/ =
2
1 / (z – 1). ɂɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɜɨɞɢɬɫɹ ɤ ɜɢɞɭ: 1/2z(z/ – 1) = 1, 2
ɤɨɬɨɪɵɣ ɞɨɩɭɫɤɚɟɬ ɪɚɡɞɟɥɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ. ɉɪɢɦɟɪ 3. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ y/ = Ɍɚɤ ɤɚɤ
x 2y . x
x 2y = 1 + 2y/x, ɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ: y/ = 1 + 2y/x. x
ɉɨɥɨɠɢɦ z = y/x, ɬɨɝɞɚ y = zx ɢ y/ = z/x + z. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ z/x + z = 1 + + 2z ɢɥɢ z/x = 1 + z – ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ. Ɋɚɡɞɟɥɹɹ ɩɟɪɟɦɟɧɧɵɟ, ɩɨɥɭɱɢɦ:
dz 1 z
dx . ɂɧɬɟɝɪɢɪɭɹ ɩɨɱɥɟɧɧɨ ɩɨɫɥɟɞɧɟɟ x
ɪɚɜɟɧɫɬɜɨ, ɩɨɥɭɱɢɦ: ln_1 + z_ = ln_Cɯ_, ɨɬɤɭɞɚ 1 + z = Cx. ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɩɟɪɜɨɧɚɱɚɥɶɧɵɦ ɩɟɪɟɦɟɧɧɵɦ, ɩɨɥɭɱɢɦ: 1 + y/x = Cx, ɨɬɤɭɞɚ y = (Cx – 1)x – ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɉɪɢɦɟɪ 4. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ: xy/ – 2y = 2x4. Ɋɚɡɞɟɥɢɦ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɱɚɫɬɢ ɧɚ ɯ, ɩɨɥɭɱɢɦ ɥɢɧɟɣɧɨɟ 2 y = 2x3. ɉɭɫɬɶ y = uv, ɬɨɝɞɚ x 2 y/ = u/v + uv/. ɂɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ: u/v + uv/ – uv = 2x3, ɢɥɢ x
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ y/ –
33
2 2 2 uv) + uv/ = 2x3. Ɉɬɤɭɞɚ (u/ – u)v + uv/ = 2x3. ɉɨɥɨɠɢɦ u/ – u = x x x du 2dx = , ɨɬɤɭɞɚ, ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, 0, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, u x
(u/v –
ɩɨɥɭɱɢɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɟ, ɬ. ɟ. ln_u_ = 2ln_x_. Ɉɬɤɭɞɚ u = x2. ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɟ u ɜ ɭɪɚɜɧɟɧɢɟ (u/v –
2 uv) + uv/ = 2x3, ɩɨɥɭɱɢɦ x
v/x2 = 2x3, ɢɥɢ v/ = 2x. Ɋɟɲɚɹ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ, ɩɨɥɭɱɚɟɦ u = x2 + C. Ɍɨɝɞɚ ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ y = uv = (x2 + C)x2 = x4 + Cx2. Ɂɚɞɚɧɢɟ 2. Ɋɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɞɨɩɭɫɤɚɸɳɢɟ ɩɨɧɢɠɟɧɢɹ ɩɨɪɹɞɤɚ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɚɪɢɚɧɬɨɦ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɞɨɩɭɫɤɚɸɳɟɟ ɩɨɧɢɠɟɧɢɟ ɩɨɪɹɞɤɚ, ɦɨɠɟɬ ɛɵɬɶ ɫɜɟɞɟɧɨ ɤ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɦɭ ɪɟɲɟɧɢɸ ɞɜɭɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. ȿɫɥɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ y// = f(x), ɬɨ ɨɧɨ ɪɟɲɚɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ. ȿɫɥɢ ɜ ɡɚɩɢɫɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɟ ɜɯɨɞɢɬ ɢɫɤɨɦɚɹ ɮɭɧɤɰɢɹ ɭ, ɬ. ɟ. ɨɧɨ ɢɦɟɟɬ ɜɢɞ G(x,y/,y//) = 0, ɬɨ ɬɚɤɨɟ ɭɪɚɜɧɟɧɢɟ ɪɟɲɚɟɬɫɹ ɩɭɬɟɦ ɜɜɟɞɟɧɢɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɢ z = y/. ȿɫɥɢ ɜ ɭɪɚɜɧɟɧɢɟ ɧɟ ɜɯɨɞɢɬ ɩɟɪɟɦɟɧɧɚɹ ɯ, ɬ. ɟ. ɨɧɨ ɢɦɟɟɬ ɜɢɞ: G(y,y/,y//) = 0, ɬɨ ɩɨɪɹɞɨɤ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɩɨɧɢɡɢɬɶ, ɟɫɥɢ ɡɚ ɧɟɡɚɜɢɫɢɦɭɸ ɩɟɪɟɦɟɧɧɭɸ ɜɡɹɬɶ y, ɚ ɡɚ ɧɟɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ ɜɡɹɬɶ z = y/. ɉɪɢɦɟɪ 5. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ xy// + y/ = 0. ɉɨɥɨɠɢɦ z = y/, ɬɨɝɞɚ y// = z/. ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɹ y/ ɢ y// ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ: xz/ + z = 0. Ɉɬɤɭɞɚ
dz dx = - . z x
ɂɧɬɟɝɪɢɪɭɹ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɪɢɯɨɞɢɦ ɤ ɪɟɲɟɧɢɸ: z = C1/x. ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ ɮɭɧɤɰɢɢ, ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ y/ = C1/x ɢɥɢ C dy = 1 . ɂɧɬɟɝɪɢɪɭɹ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɚɟɦ: y = C1ln_x_ + C2. dx x
ɉɪɢɦɟɪ 6. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ 2yy// = (y/)2 + 1. ɉɨɥɨɠɢɦ z = z(y) = y/, ɬɨɝɞɚ y// =
dz dz dy = = z/(y)z. ɂɫɯɨɞɧɨɟ dx dy dx
ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ: 2ɭzz/ = z2 + 1. ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ. ɉɪɟɨɛɪɚɡɭɟɦ ɟɝɨ ɤ ɜɢɞɭ:
2 zdz z2 1
dy y
ɢɥɢ
d ( z 2 1) z2 1
dy . y
ȼɵɩɨɥɧɹɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɩɨɥɭɱɚɟɦ: ln(z2 + 1) = ln_yC_ ɢɥɢ z2 + 1 = yC. Ɉɬɤɭɞɚ ɫɥɟɞɭɟɬ z = r yC 1 . Ɍɚɤ ɤɚɤ z = y/, ɬɨ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɦɭ ɭɪɚɜɧɟɧɢɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭ(ɯ): y/ = r yC 1 ɢɥɢ
dy yC 1
r dx . ȼɵɩɨɥɧɹɹ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɩɨɥɭɱɚɟɦ r yC 1 = ɋ(ɯ + ɋ1)/2 ɢɥɢ ɋy – 1 = C2(x + C1)2/4. 34
Ɉɬɤɭɞɚ ɭ =
C 2 ( x C1 ) 2 1 4C C
– ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ. Ɂɚɞɚɧɢɟ 3. Ɋɟɲɟɧɢɟ ɥɢɧɟɣɧɵɯ ɨɞɧɨɪɨɞɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɚɪɢɚɧɬɨɦ. Ʌɢɧɟɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɢɦɟɟɬ ɜɢɞ: ɭ// + py/ +qy = f(x), ɝɞɟ p, q – ɧɟɤɨɬɨɪɵɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ, f(x) – ɧɟɤɨɬɨɪɚɹ ɮɭɧɤɰɢɹ. ȿɫɥɢ f(x) = 0, ɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɭ// + py/ +qy = 0 ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦ, ɟɫɥɢ f(x) 0, ɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɭ// + py/ +qy = f(x) ɧɚɡɵɜɚɟɬɫɹ ɧɟɨɞɧɨɪɨɞɧɵɦ. Ɋɚɫɫɦɨɬɪɢɦ ɪɟɲɟɧɢɟ ɥɢɧɟɣɧɵɯ ɨɞɧɨɪɨɞɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɬ. ɟ. ɭɪɚɜɧɟɧɢɣ ɜɢɞɚ: ɭ// + py/ +qy = 0. ɋɨɫɬɚɜɥɹɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ: Ȝ2 + p Ȝ + q = 0. ȿɫɥɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɞɜɚ ɪɚɡɥɢɱɧɵɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɹ Ȝ1 ɢ Ȝ2, ɬɨɝɞɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: ɭ = ɋ1ɟȜ1ɯ + ɋ2ɟȜ2ɯ, ɝɞɟ ɋ1 ɢ ɋ2 – ɧɟɤɨɬɨɪɵɟ ɱɢɫɥɚ. ȿɫɥɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɞɜɚ ɨɞɢɧɚɤɨɜɵɯ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɤɨɪɧɹ Ȝ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: ɭ = ɟȜɯ(ɋ1 + ɯɋ2). ȿɫɥɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɤɨɦɩɥɟɤɫɧɵɟ ɫɨɩɪɹɠɟɧɧɵɟ ɤɨɪɧɢ Į + iȕ ɢ Į – iȕ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: ɭ = ɟĮɯ(ɋ1sinȕɯ + ɋ2cosȕx). ɉɪɢɦɟɪ 7. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: ɭ// – 3y/ + 2y = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ O2 – 3O + 2 = 0. ɇɚɯɨɞɢɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = 1, O2 = 2. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ ɪɚɡɧɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: y = C1ex + C2e2x. ɉɪɢɦɟɪ 8. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: ɭ// – 2y/ + y = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ: O2 – 2O + 1 = 0. ɇɚɯɨɞɢɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = O2 = 1. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ ɨɞɢɧɚɤɨɜɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: y = ex(ɋ1 + C2ɯ). ɉɪɢɦɟɪ 9. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: ɭ// – 2y/ + 2y = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ: O2 – 2O + 2 = 0. ɇɚɯɨɞɢɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = 35
2 48 2
2 2i 2
1 i
ɢ O2 =
2 48 2
2 2i = 1 + i. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ 2
ɭɪɚɜɧɟɧɢɹ ɤɨɦɩɥɟɤɫɧɵɟ, ɫɨɩɪɹɠɟɧɧɵɟ ɱɢɫɥɚ, ɩɪɢ ɱɟɦ D = 1 ɢ E = 1, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: y = ex(ɋ1sinx + C2cosx). Ɂɚɞɚɧɢɟ 4. Ɋɟɲɟɧɢɟ ɥɢɧɟɣɧɵɯ ɨɞɧɨɪɨɞɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ, ɬ. ɟ. ɭɪɚɜɧɟɧɢɣ ɜɢɞɚ: ɭ// + py/ +qy = f(x). Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɦɨɠɧɨ ɪɟɲɢɬɶ ɦɟɬɨɞɨɦ ɜɚɪɢɚɰɢɢ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ. ȼɧɚɱɚɥɟ ɧɚɯɨɞɢɬɫɹ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɭ// + py/ +qy = 0 ɜ ɜɢɞɟ: ɭ = ɋ1ɭ1 + ɋ2ɭ2. Ɂɚɬɟɦ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɹɬ ɜ ɜɢɞɟ: ɭ = ɋ1(ɯ)ɭ1 + ɋ2(ɯ)ɭ2, ɬ. ɟ. ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɨɫɬɨɹɧɧɵɟ ɋ1 ɢ ɋ2 ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɩɟɪɟɦɟɧɧɨɣ ɯ. ɉɪɢ ɷɬɨɦ ɋ1(ɯ) ɢ ɋ2(ɯ) ɦɨɝɭɬ ɛɵɬɶ °C1/ ( x) y1 C2/ ( x) y2 / / / / ¯°C1 ( x) y1 C2 ( x) y2
ɧɚɣɞɟɧɵ ɤɚɤ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ: ®
0, f ( x).
ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧ ɞɪɭɝɨɣ ɩɨɞɯɨɞ ɩɪɢ ɪɟɲɟɧɢɢ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. ȼɧɚɱɚɥɟ ɧɚɯɨɞɹɬ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ( y ), ɚ ɡɚɬɟɦ ɩɨ ɜɢɞɭ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ, ɧɚɯɨɞɹɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ( ~y ). Ɋɚɫɫɦɨɬɪɢɦ ɩɨɪɹɞɨɤ ɡɚɩɢɫɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ( ~y ) ɩɨ ɜɢɞɭ ɩɪɚɜɨɣ ɱɚɫɬɢ ɢɫɯɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ȿɫɥɢ ɩɪɚɜɚɹ ɱɚɫɬɶ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: ɟDɯɊ(ɯ), ɝɞɟ Ɋ(ɯ) – ɦɧɨɝɨɱɥɟɧ n-ɨɣ ɫɬɟɩɟɧɢ ɢ D ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɢɞɚ: ~y = ɟDɯɆ(ɯ), ɝɞɟ Ɇ(ɯ) = = Ⱥ0 + Ⱥ1ɯ + Ⱥ2ɯ2 + . . . + Ⱥnxn. ȿɫɥɢ D ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɪɚɬɧɨɫɬɢ ɤ (ɤ = 1 ɢɥɢ ɤ = 2), ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɢɞɚ: ~y = ɯɤɟDɯɆ(ɯ). ȿɫɥɢ ɩɪɚɜɚɹ ɱɚɫɬɶ ɥɢɧɟɣɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ: ɟDɯ(Ɋn(ɯ)cosEx + + Pm(x)sinEx), ɝɞɟ Ɋn(ɯ) ɢ Ɋm(ɯ) – ɦɧɨɝɨɱɥɟɧɵ ɫɬɟɩɟɧɢ n ɢ m ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢ z = D + iE ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɢɞɚ: ~y = ɟDɯ(Mk(ɯ)cosEx + Nk(x)sinEx), ɝɞɟ Mk(x) = Ⱥ0 + Ⱥ1ɯ + Ⱥ2ɯ2 + . . . + Ⱥkxk, Nk(x) = B0 + B1ɯ + B2ɯ2 + . . . + Bkxk, k = max(n, m). ȿɫɥɢ ɠɟ z = D + iE ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɢɞɚ: Dɯ ~ y = ɯɟ (Mk(ɯ)cosEx + Nk(x)sinEx). 36
ɉɪɢɦɟɪ 10. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ: ɭ// – 3y/ + 2y = ɟɯ. Ɋɟɲɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɨɞɧɨɪɨɞɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɭ// – 3y/ + 2y = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ O2 – 3O + 2 = 0. ɇɚɯɨɞɢɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ O1 = 1, O2 = 2. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ ɪɚɡɧɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: y = C1ex + C2e2x. ɇɚɯɨɞɢɦ: y1 = ex, y2 = e2x, y1/ = ex, y2/ = 2e2x. ɉɨɥɚɝɚɹ, ɱɬɨ ɋ1 ɢ ɋ2 ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɧɧɨɣ ɯ, ɧɚɯɨɞɢɦ ɩɟɪɜɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɷɬɢɯ ɮɭɧɤɰɢɣ, ɪɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ: °C1/ ( x)e x C 2/ ( x)e 2 x 0, °C1/ e x C 2/ e 2 x 0, . ® / ® / x °¯C1 ( x)e x C 2/ ( x)2e 2 x e x °¯C1 e C 2/ 2e 2 x e x
ȼɵɱɢɬɚɟɦ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɟ, ɩɨɥɭɱɢɦ: ɋ2/ɟ2ɯ = ɟɯ, ɋ2 = ex/e2x = e–x. ɇɚɣɞɟɦ ɋ1/ ɜ ɜɢɞɟ: ɋ1/ex + e–xe2x = 0; (ɋ1/ + 1)ɟɯ = 0; ɋ1/ = –1. ɉɨɥɭɱɢɥɢ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ: ɋ2/ = ɟ-ɯ ɢ ɋ1/ = –1. Ɋɟɲɢɦ ɷɬɢ ɭɪɚɜɧɟɧɢɹ. dC2/dx = e–x, dC2/ = e–xdx, ɨɬɤɭɞɚ: C2 = –e–x + C3, dC1/dx = –1, dC1 = –dx, ɨɬɤɭɞɚ: C1 = –x + C4, ɝɞɟ ɋ3 ɢ ɋ4 – ɧɟɤɨɬɨɪɵɟ ɩɨɫɬɨɹɧɧɵɟ ɱɢɫɥɚ. Ɉɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ: ɭ = (–ɯ + ɋ4)ɟ–ɯ + (–ɟ–ɯ + ɋ3)ɟ2ɯ = ɋ4ɟ–ɯ+ɋ3ɟ2ɯ – ɯɟ–ɯ – ɟɯ – ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ 2-ɝɨ ɩɨɪɹɞɤɚ. ɉɪɢɦɟɪ 11. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ: y// – 3y/ = 1 + 6x. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ: ɭ = y ~y . ɇɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: y// – 3y/ = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ O2 – 3O = 0 ɢ ɧɚɣɞɟɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = 0, O2 = 3. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ ɪɚɡɧɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: 0x 3x 3x y = C1e + C2e = C1 + C2e . ɂɫɯɨɞɹ ɢɡ ɩɪɚɜɨɣ ɱɚɫɬɢ ɢɫɯɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɡɚɩɢɲɟɦ ɩɨɥɧɵɣ ɜɢɞ ɩɪɚɜɨɣ ɱɚɫɬɢ: 1 + 6ɯ = ɟ0ɯ(1 + 6ɯ). O = 0 – ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ Ɋ1(ɯ) = 1 + 6ɯ – ɦɧɨɝɨɱɥɟɧ 1-ɣ ɫɬɟɩɟɧɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = ɯ(Ⱥ + ȼɯ). ɇɚɣɞɟɦ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ Ⱥ ɢ ȼ. ~y / = Ⱥ + ȼɯ + ȼɯ = Ⱥ + 2ȼɯ. ~ y // = 2ȼ. ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɹ ~ y/ ɢ ~ y // ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ: 2ȼ – – 3Ⱥ – 6ȼɯ = 1 + 6ɯ, ɢɥɢ (2ȼ – 3Ⱥ) – 6ȼɯ = 1 + 6ɯ. Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ /
2 B 3 A 1, Ɋɟɲɚɹ ɷɬɭ ɫɢɫɬɟɦɭ, ɧɚɯɨɞɢɦ, ¯6 B 6.
ɪɚɜɧɨɫɢɥɶɧɨ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ: ® ɱɬɨ Ⱥ = ȼ = –1.
37
Ɍɨɝɞɚ ɢɫɤɨɦɨɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = ɯ(–1 – ɯ). Ⱥ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ɭ = C1 + C2e3x + ɯ(–1 – ɯ). ɉɪɢɦɟɪ 12. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: y// – 3y/ + 2ɭ = 2ɟ3ɯ. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ: ɭ = y ~y . ɇɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: y// – 3y/ + 2ɭ = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ: O2 – 3O + 2 = 0. ɇɚɣɞɟɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = 1, O2 = 2. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ ɪɚɡɧɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: x 2x y = C1e + C2e . ɂɫɯɨɞɹ ɢɡ ɩɪɚɜɨɣ ɱɚɫɬɢ ɢɫɯɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɡɚɩɢɲɟɦ ɩɨɥɧɵɣ ɜɢɞ ɩɪɚɜɨɣ ɱɚɫɬɢ: 2ɟ3ɯ, ɝɞɟ O = 3 ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, Ɋ0(ɯ) = 2 – ɦɧɨɝɨɱɥɟɧ ɧɭɥɟɜɨɣ ɫɬɟɩɟɧɢ. Ɍɨɝɞɚ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = Ⱥɟ3ɯ. ɇɚɣɞɟɦ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ Ⱥ. ~y / = 3Ⱥɟ3ɯ. ~y // = 9Ⱥɟ3ɯ. ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɹ ~y / ɢ ~y // ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ: 9Ⱥɟ3ɯ – 9Ⱥɟ3ɯ + 2Ⱥɟ3ɯ = = 2ɟ3ɯ, ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ: 2Ⱥ = 2, ɢɥɢ Ⱥ = 1. ɂɫɤɨɦɨɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = ɟ3ɯ. Ɍɨɝɞɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ɭ = C1ex + + C2e2x + ɟ3ɯ. ɉɪɢɦɟɪ 13. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: y// – 2y/ + 5ɭ = sin2x. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ: ɭ = y ~y . ɇɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: y// – 2y/ + 5ɭ = 0. ɋɨɫɬɚɜɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ: O2 – 2O + 5 = 0. ɇɚɣɞɟɦ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ: O1 = 1 + 2i, O2 = 1 – 2i, D = 1, E = 2. Ɍɚɤ ɤɚɤ ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɦɩɥɟɤɫɧɵɟ ɢ ɫɨɩɪɹɠɟɧɧɵɟ ɱɢɫɥɚ, ɬɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ: y = ex(ɋ1ɫos2x + C2sin2x). ɂɫɯɨɞɹ ɢɡ ɩɪɚɜɨɣ ɱɚɫɬɢ ɢɫɯɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ sin2x, ɡɚɩɢɲɟɦ ɩɨɥɧɵɣ ɜɢɞ ɩɪɚɜɨɣ ɱɚɫɬɢ: ɟ0ɯ(0 cos2x + 1 sin2x), ɝɞɟ O = 0 + 2i – ɧɟ ɹɜɥɹɟɬɫɹ ɤɨɪɧɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, Ɋ0(ɯ) = 0, Q0(x) = 1 – ɦɧɨɝɨɱɥɟɧɵ ɧɭɥɟɜɨɣ ɫɬɟɩɟɧɢ. Ɍɨɝɞɚ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = ɟ0ɯ(Acos2x + Bsin2x) = Acos2x + Bsin2x. 38
ɇɚɣɞɟɦ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ Ⱥ ɢ ȼ. ~y / = –2Ⱥsin2x + 2Bcos2x, ~ y // = –4Ⱥcos2x – 4Bsin2x. ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɹ ~ y/ ɢ ~ y // ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɢɦ: –4Ⱥcos2x – 4Bsin2x + 4Ⱥsin2x – 4Bcos2x + 5Acos2x +5Bsin2x = sin2x, ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ Ⱥcos2x + Bsin2x + 4Asin2x – 4Bcos2x = = sin2x, ɢɥɢ (Ⱥ – 4B)cos2x + (B + 4A)sin2x = sin2x. ɉɟɪɟɣɞɟɦ ɤ ɪɚɜɧɨɫɢɥɶɧɨɣ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ:
A 4 B 0, Ɋɟɲɚɹ ɞɚɧɧɭɸ ɫɢɫɬɟɦɭ, ® ¯ B 4 A 1.
ɩɨɥɭɱɢɦ: Ⱥ = 4/17, B = 1/17. ɂɫɤɨɦɨɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ~y = (4/17)cos2x + (1/17)sin2x. Ɍɨɝɞɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: ɭ = ex(ɋ1ɫos2x + C2sin2x) + (4/17)cos2x + (1/17)sin2x. Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 7 (3 ɱɚɫɚ). ɑɢɫɥɨɜɵɟ ɪɹɞɵ Ɂɚɞɚɧɢɟ 1. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɑɢɫɥɨɜɵɦ ɪɹɞɨɦ ɧɚɡɵɜɚɟɬɫɹ ɪɹɞ ɜɢɞɚ: u1 + u2 + u3 + . . . =
f
¦u
n
, ɝɞɟ
n 1
u1, u2, u3, . . . ɱɥɟɧɵ ɪɹɞɚ, un – ɨɛɳɢɦ ɱɥɟɧɨɦ ɪɹɞɚ. ɋɭɦɦɚ n = ɩɟɪɜɵɯ ɱɥɟɧɨɜ ɪɹɞɚ Sn ɧɚɡɵɜɚɟɬɫɹ n-ɨɣ ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɨɣ ɪɹɞɚ. Ɋɹɞ ɧɚɡɵɜɚɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ, ɟɫɥɢ Sn ɩɪɢ n ĺ ɫɬɪɟɦɢɬɫɹ ɤ ɤɨɧɟɱɧɨɦɭ ɩɪɟɞɟɥɭ, ɬ. ɟ. lim Sn = S, ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ n of
ɪɚɫɯɨɞɹɳɢɦɫɹ. ɑɢɫɥɨ S ɧɚɡɵɜɚɸɬ ɫɭɦɦɨɣ ɪɹɞɚ. ɉɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɨɜ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ. ɉɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ. ɉɭɫɬɶ ɞɚɧɵ ɞɜɚ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ: f
¦u n 1 f
¦v
n
n
= u1 + u2 + u3 + . . . (1),
= v1 + v2 + v3 + . . . (2),
n 1
ɩɪɢɱɟɦ ɤɚɠɞɵɣ ɢɡ ɱɥɟɧɨɜ ɪɹɞɚ (1) ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ (2), ɬ. ɟ. un vn, ɬɨɝɞɚ ɢɡ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (2) ɫɥɟɞɭɟɬ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ (1), ɚ ɢɡ ɪɚɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (1) ɫɥɟɞɭɟɬ ɪɚɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ (2). Ⱦɥɹ ɫɪɚɜɧɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬɫɹ ɷɬɚɥɨɧɧɵɟ ɪɹɞɵ: 1. ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ
f
¦ aq
n 1
– ɫɯɨɞɢɬɫɹ ɩɪɢ _q_ < 1, ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ _q_ t
n 1
1; 2. ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ
f
1
¦n
– ɪɚɫɯɨɞɢɬɫɹ;
n 1
39
3. ɨɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ
f
1
¦ nD
– ɫɯɨɞɢɬɫɹ ɩɪɢ D > 1, ɪɚɫɯɨɞɢɬɫɹ
n 1
ɩɪɢ D d 1. ɉɪɢɦɟɪ 1. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ
f
1
¦ n3
n 1
.
n 1
ɋɪɚɜɧɢɦ ɞɚɧɧɵɣ ɪɹɞ ɫɨ ɫɯɨɞɹɳɢɦɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɪɹɞɨɦ
f
¦3
1
n 1
,
n 1
1 < 1. Ɍɚɤ ɤɚɤ ɱɥɟɧɵ ɢɫɯɨɞɧɨɝɨ ɪɹɞɚ, ɧɚɱɢɧɚɹ ɫɨ ɜɬɨɪɨɝɨ, ɦɟɧɶɲɟ 3 1 1 1 1 , , . . . , ɱɥɟɧɨɜ ɫɯɨɞɹɳɟɝɨɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɹɞɚ, ɬ. ɟ. 6 3 27 9 1 1 , ɬɨ ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɢɡɧɚɤɚ ɫɪɚɜɧɟɧɢɹ ɢɫɯɨɞɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ. n 3n 1 3n 1
ɟɝɨ q =
ɉɪɟɞɟɥɶɧɵɣ ɩɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ. ɉɭɫɬɶ ɞɚɧɵ ɞɜɚ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ:
f
¦u
n
= u 1 + u 2 + u3 + . . . ɢ
n 1
f
¦v
n
= v1 + v2 + v3 + . . .
n 1
ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ ɢ ɨɬɥɢɱɧɵɣ ɨɬ ɧɭɥɹ ɩɪɟɞɟɥ lim(
un ) = k, vn
ɬɨ ɨɛɚ ɪɹɞɚ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫɯɨɞɹɬɫɹ ɢɥɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɫɯɨɞɹɬɫɹ. ɉɪɢɦɟɪ 2. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ:
f
2n 2 5 . n3 1
¦ n
ɋɪɚɜɧɢɦ ɞɚɧɧɵɣ ɪɹɞ ɫ ɪɚɫɯɨɞɹɳɢɦɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɪɹɞɨɦ
f
1
¦n . n 1
u Ɍɚɤ ɤɚɤ lim n n of v n
2n 2 5 1 lim ( : ) = 2 z 0, ɬɨ ɢɫɯɨɞɧɵɣ ɪɹɞ, ɬɚɤ ɠɟ ɤɚɤ ɢ n of n3 n
ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ, ɪɚɫɯɨɞɢɬɫɹ. ɉɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ. ɉɭɫɬɶ ɞɥɹ ɪɹɞɚ
f
¦u
n
ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ
n 1
ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ (n + 1)-ɝɨ ɱɥɟɧɚ ɤ n-ɱɥɟɧɭ, ɬ. ɟ. nlim of
un 1 = k, un
ɬɨ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɟɫɥɢ k < 1, ɢ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ, ɟɫɥɢ k > 1, ɟɫɥɢ ɠɟ k = 1, ɬɨ ɜɨɩɪɨɫ ɨ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɨɫɬɚɟɬɫɹ ɧɟɪɟɲɟɧɧɵɦ. ɉɪɢɦɟɪ 3. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ: ɚ)
f
n
¦2 n 1
n
, ɛ)
f
3n n! . n 1 n
¦ n
n 1 n u n 1 1 = < 1, ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ ɚ) ɬɚɤ ɤɚɤ lim n 1 = lim ( n 1 : n ) = lim nof 2 n o f 2n n of u 2 2 n
Ⱦɚɥɚɦɛɟɪɚ ɢɫɯɨɞɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ;
40
ɛ) ɬɚɤ ɤɚɤ = 3 lim ( n of
n n ) = n 1
un 1 3n 3n!(n 1)n n 3n 1 (n 1)! 3n n! = lim ( : n ) = lim = n o f ( n 1) n 1 n o f ( n 1) n ( n 1)3n n! un n 3 3 ! 1 , ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ Ⱦɚɥɚɦɛɟɪɚ ɪɹɞ 1 n e lim (1 ) nof n
lim
n of
ɪɚɫɯɨɞɢɬɫɹ. ɂɧɬɟɝɪɚɥɶɧɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ. ɉɭɫɬɶ ɞɚɧ ɪɹɞ
f
¦u
n
, ɱɥɟɧɵ ɤɨɬɨɪɨɝɨ
n 1
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɢ ɧɟ ɜɨɡɪɚɫɬɚɸɬ, ɬ. ɟ. u1 t u2 t . . . t un t . . ., ɚ ɮɭɧɤɰɢɹ f(x), ɨɩɪɟɞɟɥɟɧɧɚɹ ɩɪɢ ɯ t 1, ɧɟɩɪɟɪɵɜɧɚɹ, ɧɟ ɜɨɡɪɚɫɬɚɸɳɚɹ ɢ f(1) = u1, f(2) = u2, . . . , f(n) = un . . . Ɍɨɝɞɚ ɞɥɹ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
f
¦u
n
ɧɟɨɛɯɨɞɢɦɨ ɢ
n 1
f
ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɫɯɨɞɢɥɫɹ ɧɟɫɨɛɫɬɜɟɧɧɵɣ ɢɧɬɟɝɪɚɥ
³
f ( x)dx .
1
ɉɪɢɦɟɪ 4. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ f
1
¦ nD .
ɪɹɞɚ
n 1
ɉɭɫɬɶ f(x) = ɜɨɡɪɚɫɬɚɸɳɚɹ.
1 . Ɏɭɧɤɰɢɹ f(x) ɩɪɢ ɯ t 1 ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɢ ɧɟ xD
ɉɨɷɬɨɦɭ
ɫɯɨɞɢɦɨɫɬɶ f
ɧɟɫɨɛɫɬɜɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ b
lim
b of
dx
³ xD 1
= lim ( b of
dx ³1 xD . ɂɦɟɟɦ
ɪɹɞɚ f
ɪɚɜɧɨɫɢɥɶɧɚ
ɫɯɨɞɢɦɨɫɬɢ
b
dx dx . ȿɫɥɢ D = 1, ɬɨ ³1 xD = blim o f ³ xD 1 b
= lim (ln_x_) 1b = lim ( ln_b_ – ln1) = v. ȿɫɥɢ D > 1, ɬɨ lim ³ b of
b of
b of
1
dx = xD
x D 1 b 1 1 1-D )1 = lim (b – 1) = . ɂɬɚɤ, ɞɚɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ D D 1 1 D b of D 1
> 1 ɢ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ D d 1. Ɂɚɞɚɧɢɟ 2. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɯɫɹ ɪɹɞɨɜ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɨɞ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɦɫɹ ɪɹɞɨɦ ɩɨɧɢɦɚɟɬɫɹ ɪɹɞ, ɜ ɤɨɬɨɪɨɦ ɱɥɟɧɵ ɩɨɩɟɪɟɦɟɧɧɨ ɬɨ ɩɨɥɨɠɢɬɟɥɶɧɵɟ, ɬɨ ɨɬɪɢɰɚɬɟɥɶɧɵɟ. Ɍɟɨɪɟɦɚ Ʌɟɣɛɧɢɰ (ɉɪɢɡɧɚɤ Ʌɟɣɛɧɢɰɚ). ȿɫɥɢ ɱɥɟɧɵ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ ɭɛɵɜɚɸɬ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ, ɬ. ɟ. u1 > u2 > . . . > un > . . ., ɢ ɩɪɟɞɟɥ ɟɝɨ ɨɛɳɟɝɨ ɱɥɟɧɚ ɩɪɢ n o f ɪɚɜɟɧ ɧɭɥɸ, ɬ. ɟ. lim un = 0, ɬɨ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɚ ɟɝɨ ɫɭɦɦɚ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɩɟɪɜɨɝɨ n of
ɱɥɟɧɚ, ɬ. ɟ. S d u1. ɉɨɝɪɟɲɧɨɫɬɶ ɩɪɢ ɩɪɢɛɥɢɠɟɧɧɨɦ ɜɵɱɢɫɥɟɧɢɢ ɫɭɦɦɵ ɫɯɨɞɹɳɟɝɨɫɹ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ Ʌɟɣɛɧɢɰɚ, ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɵ ɩɟɪɜɨɝɨ ɨɬɛɪɨɲɟɧɧɨɝɨ ɱɥɟɧɚ ɪɹɞɚ. 41
Ɂɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɦɫɹ, ɟɫɥɢ ɫɯɨɞɢɬɫɹ ɤɚɤ ɫɚɦ ɪɹɞ, ɬɚɤ ɢ ɪɹɞ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɟɝɨ ɱɥɟɧɨɜ. Ɂɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɢɦɫɹ, ɟɫɥɢ ɫɚɦ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɚ ɪɹɞ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɟɝɨ ɱɥɟɧɨɜ, ɪɚɫɯɨɞɢɬɫɹ. Ⱥɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ ɨɫɧɨɜɧɨɦ ɜ ɫɢɥɭ ɬɨɝɨ, ɱɬɨ ɢɯ ɱɥɟɧɵ ɛɵɫɬɪɨ ɭɛɵɜɚɸɬ. ɍɫɥɨɜɧɨ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɨɝɨ, ɱɬɨ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɫɥɚɝɚɟɦɵɟ ɭɧɢɱɬɨɠɚɸɬ ɞɪɭɝ ɞɪɭɝɚ. ɉɪɢɦɟɪ 5. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ:
f
(1) n 1 . n2 1
¦ n
Ɍɚɤ ɤɚɤ ɱɥɟɧɵ ɞɚɧɧɨɝɨ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ ɭɛɵɜɚɸɬ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ, ɬ. ɟ. 1 > ɱɥɟɧɚ ɪɚɜɟɧ 0. ɬ. ɟ. lim
n of
1 1 1 > 2 > . . . > 2 > . . . , ɢ ɩɪɟɞɟɥ ɨɛɳɟɝɨ n 22 3
1 = 0, ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ ɪɹɞ ɫɯɨɞɢɬɫɹ. n2
Ɍɚɤ ɤɚɤ ɪɹɞ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɞɚɧɧɨɝɨ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ, ɬ. ɟ. ɪɹɞ
f
1
¦n
2
, ɫɯɨɞɢɬɫɹ, ɬɨ ɢɫɯɨɞɧɵɣ
n 1
ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ. Ɂɚɞɚɧɢɟ 3. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ɋɹɞɵ, ɱɥɟɧɚɦɢ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɫɬɟɩɟɧɧɵɟ ɮɭɧɤɰɢɢ, ɧɚɡɵɜɚɸɬɫɹ ɫɬɟɩɟɧɧɵɦɢ ɪɹɞɚɦɢ. Ɉɧɢ ɢɦɟɸɬ ɜɢɞ: ɫ0 + ɫ1ɯ + ɫ2ɯ2 + . . . + ɫnxn + . . ., ɝɞɟ ɫ0, ɫ1, ɫ2, …, cn – ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ. ɋɨɜɨɤɭɩɧɨɫɬɶ ɬɟɯ ɡɧɚɱɟɧɢɣ ɯ, ɩɪɢ ɤɨɬɨɪɵɯ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɧɚɡɵɜɚɟɬɫɹ ɨɛɥɚɫɬɶɸ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ. Ɍɟɨɪɟɦɚ Ⱥɛɟɥɹ. ȿɫɥɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ ɡɧɚɱɟɧɢɢ ɯ = ɯ0 z 0, ɬɨ ɨɧ ɫɯɨɞɢɬɫɹ, ɩɪɢɬɨɦ ɚɛɫɨɥɸɬɧɨ, ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ ɯ ɬɚɤɢɯ, ɱɬɨ _ɯ_ < _ɯ0_. ȿɫɥɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ ɯ = ɯ1, ɬɨ ɨɧ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ ɯ ɬɚɤɢɯ, ɱɬɨ _ɯ_ > _ɯ1_. ɂɡ ɬɟɨɪɟɦɵ Ⱥɛɟɥɹ ɫɥɟɞɭɟɬ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɱɢɫɥɨ R t 0, ɱɬɨ ɩɪɢ _ɯ_ < R ɪɹɞ ɫɯɨɞɢɬɫɹ, ɚ ɩɪɢ _ɯ_ > R – ɪɚɫɯɨɞɢɬɫɹ. ɑɢɫɥɨ R ɧɚɡɵɜɚɟɬɫɹ ɪɚɞɢɭɫɨɦ ɫɯɨɞɢɦɨɫɬɢ, ɚ ɢɧɬɟɪɜɚɥ (–R; R) – ɢɧɬɟɪɜɚɥɨɦ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ. ɇɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ, ɬ. ɟ. ɩɪɢ ɯ = –R ɢ ɯ = R, ɪɹɞ ɦɨɠɟɬ ɤɚɤ ɫɯɨɞɢɬɫɹ, ɬɚɤ ɢ ɪɚɫɯɨɞɢɬɫɹ. Ɋɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: R = lim nof
cn . ȿɫɥɢ R = 0, ɬɨ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɜɵɪɨɠɞɚɟɬɫɹ ɜ ɬɨɱɤɭ. cn 1
ȿɫɥɢ R = f, ɬɨ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɨɯɜɚɬɵɜɚɟɬ ɜɫɸ ɱɢɫɥɨɜɭɸ ɨɫɶ. 42
ɉɪɢɦɟɪ 6. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ: 1 + ɯ + + ɯ 2 + . . . + ɯn + … Ⱦɚɧɧɵɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɪɹɞɨɦ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ q = ɯ. Ɋɹɞ ɫɯɨɞɢɬɫɹ, ɟɫɥɢ _ɯ_ < 1. Ɉɬɫɸɞɚ, ɨɛɥɚɫɬɶɸ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɹɜɥɹɟɬɫɹ ɢɧɬɟɪɜɚɥ (–1; 1). ɉɪɢɦɟɪ 7. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ:
2n x n
f
¦ (2n 1) n 1
ɇɚɯɨɞɢɦ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ: R = lim nof = lim
n of
2n (2n 1)
2
n
3
:
2 n 1 (2(n 1) 1)
2
n 1
(2n 3) 2 3 lim n o f 2 (2n 1) 2
=
3
ɨɬɤɭɞɚ ɩɨɥɭɱɢɦ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ: (–
=
2
3n
.
cn = cn 1
3 . Ɍɨɝɞɚ |x| < R, 2
3 3 ; ). 2 2
ɂɫɫɥɟɞɭɟɦ ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ ɧɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ. 3 ɞɚɧɧɵɣ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɩɪɢɦɟɬ ɜɢɞ 2 1 1 ɫɥɟɞɭɸɳɟɝɨ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ: 1 – 2 + 2 – . . . + 3 5 (1) n + . . . ɗɬɨɬ ɪɹɞ ɫɯɨɞɢɬɫɹ ɩɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ. ɉɪɢɱɟɦ, (2n 1) 2
ɇɚ ɥɟɜɨɦ ɤɨɧɰɟ ɩɪɢ ɯ = -
ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ. 3 2
ɇɚ ɩɪɚɜɨɦ ɤɨɧɰɟ ɩɪɢ ɯ = ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ: 1 +
1 32
+
1 52
ɩɨɥɭɱɚɟɦ ɱɢɫɥɨɜɨɣ ɪɹɞ ɫ – . . . +
1 + . . ., (2n 1) 2
ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɫɨɛɨɣ ɨɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ ɩɪɢ D = 2, ɭ ɤɨɬɨɪɨɝɨ ɜɫɟ ɱɥɟɧɵ ɫ ɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ ɪɚɜɧɵ ɧɭɥɸ. ɗɬɨɬ ɪɹɞ ɫɯɨɞɢɬɫɹ. ɂɬɚɤ, ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɞɚɧɧɨɝɨ ɪɹɞɚ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: [–
3 3 ; ]. 2 2
ɉɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɧɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ ɩɪɢɦɟɧɹɬɶ ɩɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ ɧɟ ɢɦɟɟɬ ɫɦɵɫɥɚ, ɬɚɤ ɤɚɤ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɫɟɝɞɚ ɛɭɞɟɦ ɩɨɥɭɱɚɬɶ
lim
n of
u n 1 un
= 1. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɤɨɦɟɧɞɭɟɬɫɹ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɪɭɝɢɟ ɩɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ. Ɂɚɞɚɧɢɟ 4. Ɋɹɞ Ɇɚɤɥɨɪɟɧɚ ɢ ɟɝɨ ɩɪɢɦɟɧɟɧɢɟ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ, ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɚ. ȿɫɥɢ ɮɭɧɤɰɢɹ f(x) ɨɩɪɟɞɟɥɟɧɚ ɢ n ɪɚɡ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ ɯ = 0, ɬɨ ɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ ɫɭɦɦɵ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ, ɬ. ɟ. ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɜɢɞɚ:
43
f(x) = c0 + c1x + c2x2 + c3x3 + . . . + cnxn + . . . ȼɵɪɚɡɢɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɹɞɚ ɱɟɪɟɡ f(x): c0 = f(0), c1 = f/(0), c2 =
f // (0) f /// (0) , c3= . 2! 3!
Ɋɹɞɨɦ Ɇɚɤɥɨɪɟɧɚ ɧɚɡɵɜɚɟɬɫɹ ɪɹɞ ɜɢɞɚ: f(x) = f(0) + f/(0)x +
f // (0) 2 f /// (0) 3 f ( n ) (0) n x + x +...+ x +... 2! 3! n!
Ɋɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɧɟɤɨɬɨɪɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ. x2 x3 xn + +...+ + . . . Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (–f; f). 2! 3! n! 3 5 n 1 2 n 1 x x (1) x + . . . Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (–f; f). 2. sin x = ɯ– + – . . . + 3! 5! (2n 1)! ɯ
1. ɟ = 1 + ɯ +
x2 x4 (1) n x 2 n + –...+ + . . . Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (–f; f). 2! 4! (2n)! m(m 1) 2 m(m 1)(m 2) 3 m(m 1)(m n 1) n x 4. (1 + ɯ)m = 1 + mx + x + x +...+ 2! 3! n!
3. ɫos x = 1–
+ . . ., ɝɞɟ m – ɥɸɛɨɟ ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ ɱɢɫɥɨ. ɂɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (–1; 1) (ɧɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ ɩɪɢ ɯ = 1 ɢ ɯ = –1 ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɡɚɜɢɫɢɬ ɨɬ ɤɨɧɤɪɟɬɧɵɯ ɡɧɚɱɟɧɢɣ m). x2 x3 (1) n x n 1 + –...+ + . . .Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ (–1; 1). 2 3 n 1 2 1 e x ɉɪɢɦɟɪ 8. Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ ɮɭɧɤɰɢɸ ɭ = . x2 2 x x3 xn + + . . . + + . . . ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ: ɟɯ = 1 + ɯ + 2! 3! n! 4 6 2 x x – +...+ Ɂɚɦɟɧɢɦ ɜ ɷɬɨɣ ɮɨɪɦɭɥɟ ɯ ɧɚ (–ɯ2), ɩɨɥɭɱɢɦ: e x = 1 – ɯ2 + 2! 3! 2n 4 6 2n 2 x x x x + . . . , ɬɨɝɞɚ 1 – e x = ɯ2 – + + . . . + (1) n 1 +..., ɢ + (1) n n! 2! 3! n! 2 1 e x x2 x4 x2n 2 =1– + + . . . + (1) n 1 +... ɧɚɤɨɧɟɰ ɩɨɥɭɱɢɦ: 2 x 2! 3! n! 1 x . ɉɪɢɦɟɪ 9. Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ ɮɭɧɤɰɢɸ: ɭ = ln 1 x 2 x x3 (1) n x n 1 + –...+ + ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɚɡɥɨɠɟɧɢɟɦ: ln(1 + x) = ɯ – 2 3 n 1 x2 x3 x n 1 – –...– – . . ., + ... Ɂɚɦɟɧɢɦ ɯ ɧɚ (–ɯ), ɩɨɥɭɱɢɦ: ln(1–x) = –ɯ – 2 3 n 1 x2 x3 1 x (1) n x n 1 = ln(1+x) – ln(1–x) = (ɯ – + –...+ + . . .) – (–ɯ – ɬɨɝɞɚ ln 1 x 2 3 n 1 x2 x3 x3 x5 x n 1 x 2 n 1 – –...– – . . .) = 2(x + + +... + . . . ). – 2 3 n 1 3 5 2n 1
5. ln(1 + x) = ɯ –
44
1
ɉɪɢɦɟɪ 10. ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɚ
³
x e x dx ɫ
0
ɬɨɱɧɨɫɬɶɸ ɞɨ 0,0001. x2 xn + ...+ + . . ., ɡɚɦɟɧɢɦ ɯ ɧɚ (–ɯ), 2! n! x2 (1) n x n =1–ɯ+ –...+ + . . . ɍɦɧɨɠɚɹ ɞɚɧɧɵɣ ɪɹɞ ɧɚ x , 2! n!
ȼ ɪɚɡɥɨɠɟɧɢɢ: ɟɯ = 1 + ɯ + ɬɨɝɞɚ: ɟ–ɯ
–ɯ
ɩɨɥɭɱɢɦ:
x ɟ
1
1
1
ɢɧɬɟɝɪɢɪɭɹ ɪɹɞ, ɩɨɥɭɱɢɦ: 3
5
1 0
2 x2 5
1 0
. . . +
1
(1) n x n!
n
1 2
3 2
+ . . . ɉɨɱɥɟɧɧɨ 1
(1) n x x e dx = ³ x dx – ³ x dx + . . . + ³ n! 0 0 0
³ x
1
1 2
1
x
0
2 = x2 3
3
= x 2 ɟ–ɯ = x 2 – x 2 + . . . +
n
3 2
1 0
3 (n ) n! 2
. . . =
n
1 2
dx + . . .
(1) n 2 2 2 ... = 0,66667 – ... (2n 3)n! 3 5
– 0,40000 + 0,14286 – 0,03704 + 0,00758 – 0,00128 + 0,00018 –… | 0,37897 | | 0,3790. Ɂɚɞɚɧɢɹ ɤ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɟ 7 ȼɚɪɢɚɧɬ 1. Ɂɚɞɚɧɢɟ 1. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ:
f
5n 6
f
¦ 100n 1 , ¦ n 1
n 1
f
1
¦
,
2
n(1 n )
n 1
n(n 1) . 3n
Ɂɚɞɚɧɢɟ 2. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɯɫɹ ɪɹɞɨɜ:
f
n
(1) n n , ¦ 3 n 1 n 4
(1) n 1 , 2 1 (3n)
¦
(1) n n 2 . ¦ 3 n 1 n 2
f
f
3n (2) n ( x 1) n . n n 1 ex 1 Ɂɚɞɚɧɢɟ 4. Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɮɭɧɤɰɢɸ: ɭ = . x f
Ɂɚɞɚɧɢɟ 3. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ: ¦
Ɂɚɞɚɧɢɟ 5. ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɫ ɬɨɱɧɨɫɬɶɸ 0,001: 1/ 2
³ o
dx
1 x2
.
ȼɚɪɢɚɧɬ 2. Ɂɚɞɚɧɢɟ 1. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ:
f
7n 2
f
¦ 17n 1 , ¦ n 1
n 1
f
1 3
2
n (2 n )
,
¦ n 1
n(n 5) 6n
45
Ɂɚɞɚɧɢɟ 2. ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɯɫɹ ɪɹɞɨɜ:
f
n
(3) n , ¦ n n 1 ( 4n 1) f
(1) n 1 , 2 1 ( 2n)
¦
(1) n 4 . ¦ 2 n 1 n 1 ( 2n 1) 2 f
Ɂɚɞɚɧɢɟ 3. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ:
2 n 2n x . 1 5
f
¦ n
Ɂɚɞɚɧɢɟ 4. Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɮɭɧɤɰɢɸ: y = cos2x. Ɂɚɞɚɧɢɟ 5. ȼɵɱɢɫɥɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɫ ɬɨɱɧɨɫɬɶɸ 0,001: 1/ 2
³e
x2
dx .
0
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 8 (3 ɱɚɫɚ). Ɋɹɞɵ ɢ ɢɧɬɟɝɪɚɥɵ Ɏɭɪɶɟ Ɂɚɞɚɧɢɟ 1. Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɪɹɞ Ɏɭɪɶɟ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɭɫɬɶ f(x) ɟɫɬɶ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɫ ɩɟɪɢɨɞɨɦ 2ʌ. Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ ɜɢɞɚ: f(x) =
a0 f ¦ (a n cos nx bn sin nx) ɧɚɡɵɜɚɟɬɫɹ 2 n1
ɪɹɞɨɦ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f(x). Ʉɨɷɮɮɢɰɢɟɧɬɵ ɚ0, ɚn, bn ɧɚɡɵɜɚɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f(x) ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ: ɚ0 =
1
S
S
³S f ( x)dx , ɚn =
1
S
S
³S f ( x) cos nxdx , bn =
1
S
S
³ f ( x) sin nxdx .
S
ȿɫɥɢ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɬɨ ɟɝɨ ɫɭɦɦɚ ɟɫɬɶ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ f(x) ɜ ɢɧɬɟɪɜɚɥɟ (ʌ, ʌ) ɫ ɩɟɪɢɨɞɨɦ 2ʌ. ɋɭɦɦɚ ɩɨɥɭɱɟɧɧɨɝɨ ɪɹɞɚ S(x) ɪɚɜɧɚ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ f(x) ɜ ɬɨɱɤɚɯ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɮɭɧɤɰɢɢ. ȼ ɬɨɱɤɚɯ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ f(x) ɫɭɦɦɚ ɪɹɞɚ ɪɚɜɧɹɟɬɫɹ ɫɪɟɞɧɟɦɭ ɚɪɢɮɦɟɬɢɱɟɫɤɨɦɭ ɩɪɟɞɟɥɨɜ ɮɭɧɤɰɢɢ f(x) ɫɩɪɚɜɚ ɢ ɫɥɟɜɚ, ɬ. ɟ. ɟɫɥɢ ɯ = ɫ – ɬɨɱɤɚ ɪɚɡɪɵɜɚ f (c 0) f (c 0) . 2
ɮɭɧɤɰɢɢ, ɬɨ S(x)x=c =
ȿɫɥɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɪɚɡɥɚɝɚɟɬɫɹ ɧɟɱɟɬɧɚɹ ɮɭɧɤɰɢɹ, ɬɨ ɚ0 = 0, ɚn = 0, 2
bn = ɚ0 =
S 2
S
S
³ f ( x) sin nxdx .
ȿɫɥɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɪɚɡɥɚɝɚɟɬɫɹ ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ, ɬɨ
0
2
S
f ( x)dx , ɚn = ³ f ( x) cos nxdx , bn = 0. S³ S 0
0
ɉɪɢɦɟɪ 1. Ⱦɚɧɚ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ f(x) = x, –ʌ < x ʌ. Ɋɚɡɥɨɠɢɬɶ ɮɭɧɤɰɢɸ f(x) ɜ ɪɹɞ Ɏɭɪɶɟ. ɉɨ ɮɨɪɦɭɥɟ ɚ0 =
1
S
S
³S f ( x)dx , ɧɚɯɨɞɢɦ ɚ0 =
46
1
S
S
³S xdx
=
1 x2 S 2
S S
= 0.
ɉɨ ɮɨɪɦɭɥɟ =
1
S
sin nx n
[x
ɚn =
1
S
[ x
S
S
³S f ( x) cos nxdx ,
ɧɚɯɨɞɢɦ ɚn =
1
S
S
³ x cos nxdx
=
S
S
S S
1 sin nxdx] = 0. n ³S
ɉɨ ɮɨɪɦɭɥɟ bn = =
1
cos nx n
S S
1
S
S
³S f ( x) sin nxdx ,
ɧɚɯɨɞɢɦ bn =
1
S
S
³ x sin nxdx
=
S
2 S 1 n+1 . = (–1) cos nxdx ] n n ³S
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɚɟɦ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ: f(x) = 2[
sin x sin 2 x sin 3x sin nx – + – … (–1)n+1 +…]. 1 2 3 n
ɉɭɫɬɶ f(x) ɟɫɬɶ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɫ ɩɟɪɢɨɞɨɦ 2l. Ɋɹɞ Ɏɭɪɶɟ ɞɥɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɫ ɩɟɪɢɨɞɨɦ 2l ɢɦɟɟɬ ɜɢɞ: f(ɯ) =
l l a0 f nS nS S 1 1 ¦ (a n cos x) , ɝɞɟ: ɚ0 = ³ f ( x)dx , ɚn = ³ f ( x) cos n xdx , x bn sin 2 n1 l l l l l l l l
bn =
S 1 f ( x) sin n xdx . l ³l l
ɉɪɢɦɟɪ 2. Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɟɪɢɨɞɢɱɟɫɤɭɸ ɮɭɧɤɰɢɸ f(x) = |x| c ɩɟɪɢɨɞɨɦ 2l ɧɚ ɨɬɪɟɡɤɟ [–l; l]. l
Ɍɚɤ ɤɚɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɮɭɧɤɰɢɹ ɱɟɬɧɚɹ, ɬɨ ɚ0 = ɚn =
0, ɩɪɢ k ɱɟɬɧɨɦ, l S nSx 2 2l ° x cos dx = x cos nxdx = ® 4l l ³0 l S 2 ³0 ° 2 2 , ɩɪɢ k ɧɟɱɟɬɧɨɦ, ¯ S k
2 xdx l ³0
l,
bn = 0.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɏɭɪɶɟ ɢɦɟɟɬ ɜɢɞ: S 3S (2 p 1)S ª º x x cos x cos cos » l 4l « l l l |x| = 2 « ... ...» . 2 S « 1 32 (2 p 1) 2 » ¬ ¼
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɨɜ Ɏɭɪɶɟ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɭɫɬɶ ɮɭɧɤɰɢɹ f(x) ɨɩɪɟɞɟɥɟɧɚ ɧɚ ɛɟɫɤɨɧɟɱɧɨɦ ɢɧɬɟɪɜɚɥɟ (-; +) ɢ f
ɚɛɫɨɥɸɬɧɨ ɢɧɬɟɝɪɢɪɭɟɦɚ ɧɚ ɧɟɦ, ɬ. ɟ. ɫɭɳɟɫɬɜɭɟɬ ɢɧɬɟɝɪɚɥ:
³
f ( x) dx
Q . (1)
f
ȿɫɥɢ ɮɭɧɤɰɢɹ f(x) ɤɭɫɨɱɧɨ-ɦɨɧɨɬɨɧɧɚɹ ɧɚ ɤɚɠɞɨɦ ɤɨɧɟɱɧɨɦ ɢɧɬɟɪɜɚɥɟ ɨɝɪɚɧɢɱɟɧɚ ɧɚ ɛɟɫɤɨɧɟɱɧɨɦ ɢɧɬɟɪɜɚɥɟ ɢ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ (1), ɬɨ ɩɪɢ l o f : f(ɯ)=
1
f
f
( f (t ) cos D (t x) dt)dĮ. (2) S³ ³ 0
f
47
ɋɬɨɹɳɟɟ ɫɩɪɚɜɚ ɜɵɪɚɠɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ Ɏɭɪɶɟ ɞɥɹ ɮɭɧɤɰɢɢ f(x). Ⱦɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ ɢɦɟɟɬ ɦɟɫɬɨ ɞɥɹ ɜɫɟɯ ɬɨɱɟɤ, ɝɞɟ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ. ȼ ɬɨɱɤɚɯ ɪɚɡɪɵɜɚ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ 1
f
f
( f (t ) cos D (t x) dt)dĮ = S³ ³ f
0
f ( x 0) f ( x 0) . 2
ɉɪɟɨɛɪɚɡɭɟɦ ɢɧɬɟɝɪɚɥ, ɫɬɨɹɳɢɣ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (2), ɪɚɫɤɪɵɜɚɹ cosĮ(t – x): cosĮ(t – x) = cos Įt cos + sin Įt sin Įx. ȼɵɧɨɫɹ cos Įx ɢ sinĮx ɡɚ ɡɧɚɤ ɢɧɬɟɝɪɚɥɨɜ, ɝɞɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨ ɩɟɪɟɦɟɧɧɨɣ t, ɩɨɥɭɱɢɦ: f(ɯ)=
1
f
f
1
f
f
( f (t ) cos Dt dt)cos Įx dĮ + ³ ( ³ f (t ) sin Dt dt)sin Įx dĮ. (3) S³ ³ S f
0
f
0
Ʉɚɠɞɵɣ ɢɡ ɢɧɬɟɝɪɚɥɨɜ ɩɨ t, ɫɬɨɹɳɢɣ ɜ ɫɤɨɛɤɚɯ, ɫɭɳɟɫɬɜɭɟɬ, ɬɚɤ ɤɚɤ ɮɭɧɤɰɢɹ f(t) ɚɛɫɨɥɸɬɧɨ ɢɧɬɟɝɪɢɪɭɟɦɚ ɜ ɢɧɬɟɪɜɚɥɟ (–; +), ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɚɛɫɨɥɸɬɧɨ ɢɧɬɟɝɪɢɪɭɟɦɚ ɢ ɮɭɧɤɰɢɹ f(t)cos Įt ɢ f(t)sinĮt. Ɋɚɫɫɦɨɬɪɢɦ ɱɚɫɬɧɵɟ ɫɥɭɱɚɣ ɮɨɪɦɭɥɵ (3): 1. ɉɭɫɬɶ f(x) – ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ f(t)cos Įt – ɮɭɧɤɰɢɹ ɱɟɬɧɚɹ, ɚ f
f(t)sinĮt – ɧɟɱɟɬɧɚɹ. Ɍɨɝɞɚ ɩɨɥɭɱɚɟɦ:
f
³ f (t ) cos Dt dt
f
= 2 ³ f (t ) cos Dt dt,
ɚ
0
f
³ f (t ) sin Dt dt
=
0.
Ɏɨɪɦɭɥɚ
(3)
ɜ
ɷɬɨɦ
ɫɥɭɱɚɟ
ɩɪɢɦɟɬ
ɜɢɞ:
f
f(ɯ)=
2
f
f
( f (t ) cos Dt dt)cos Įx dĮ . S³ ³ 0
(4)
0
2. ɉɭɫɬɶ f(x) – ɧɟɱɟɬɧɚɹ ɮɭɧɤɰɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ f(t)cos Įt – ɮɭɧɤɰɢɹ f
ɧɟɱɟɬɧɚɹ, ɚ f(t)sin Įt – ɱɟɬɧɚɹ. Ɍɨɝɞɚ ɩɨɥɭɱɚɟɦ:
f
³ f (t ) sin Dt dt = 2 ³ f (t ) sin Dt dt,
f
0
f
ɚ
³ f (t ) cos Dt dt
=
0.
Ɏɨɪɦɭɥɚ (3)
ɜ
ɷɬɨɦ ɫɥɭɱɚɟ
ɩɪɢɦɟɬ
ɜɢɞ:
f
f(ɯ)=
2
S
f
³ 0
f
( ³ f (t ) sin Dt dt)sin Įx dĮ . (5) 0
ȼ ɬɨɱɤɚɯ ɪɚɡɪɵɜɚ ɜɦɟɫɬɨ ɜɵɪɚɠɟɧɢɹ f(x) ɜ ɥɟɜɵɯ ɱɚɫɬɹɯ ɪɚɜɟɧɫɬɜ ɫɥɟɞɭɟɬ ɩɢɫɚɬɶ
f ( x 0) f ( x 0) . 2
ȼ ɜɵɪɚɠɟɧɢɢ (3) ɢɧɬɟɝɪɚɥɵ, ɫɬɨɹɳɢɟ ɜ ɫɤɨɛɤɚɯ, ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɨɬ Į. ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ: ȼ(Į) =
1
S
Ⱥ(Į) =
1
S
f
³ f (t ) cos Dt dt,
f
f
³ f (t ) sin Dt dt. Ɍɨɝɞɚ ɮɨɪɦɭɥɭ (3) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
f
f
f(ɯ)= ³ (Ⱥ(Į)cos Įx + ȼ(Į )sin Įx) dĮ. (6) 0
48
Ɏɨɪɦɭɥɚ (6) ɞɚɟɬ ɪɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɢ f(x) ɧɚ ɝɚɪɦɨɧɢɤɢ ɫ ɧɟɩɪɟɪɵɜɧɨ ɦɟɧɹɸɳɟɣɫɹ ɨɬ 0 ɞɨ ɱɚɫɬɨɬɨɣ Į. Ɂɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɚɦɩɥɢɬɭɞ ɢ ɧɚɱɚɥɶɧɵɯ ɮɚɡ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɱɚɫɬɨɬɵ Į ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɮɭɧɤɰɢɢ Ⱥ(Į) ɢ ȼ(Į). 2
ȼ ɮɨɪɦɭɥɟ (4) ɩɨɥɨɠɢɦ F(Į) = 2
ɮɨɪɦɭɥɚ (4) ɩɪɢɦɟɬ ɜɢɞ: f(ɯ)=
S
f
³ f (t ) cos Dt dt.
(7)
Ɍɨɝɞɚ
0
f
F(Į)cos Įx dĮ. (8)
³
S
0
Ɏɭɧɤɰɢɹ F(Į) ɧɚɡɵɜɚɟɬɫɹ ɤɨɫɢɧɭɫ-ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ ɞɥɹ ɮɭɧɤɰɢɢ f(x). ȿɫɥɢ ɜ ɪɚɜɟɧɫɬɜɟ (7) ɫɱɢɬɚɬɶ F(Į) ɡɚɞɚɧɧɨɣ, ɚ f(t) ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɟɣ, ɬɨ ɨɧɨ ɹɜɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɞɥɹ ɮɭɧɤɰɢɢ f(t). Ɏɨɪɦɭɥɚ (8) ɞɚɟɬ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɇɚ ɨɫɧɨɜɚɧɢɢ ɮɨɪɦɭɥɵ (5) ɦɨɠɟɦ ɧɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɟ ɪɚɜɟɧɫɬɜɚ: Ɏ(Į) =
2
S
f
³ f (t ) sin Dt dt. (9)
2
f(ɯ)=
S
0
f
³
Ɏ(Į)sin Įx dĮ. (10)
0
Ɏɭɧɤɰɢɹ Ɏ(Į) ɧɚɡɵɜɚɟɬɫɹ ɫɢɧɭɫ-ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ. ɉɪɢɦɟɪ 3. ɉɭɫɬɶ f(x) = e–ȕx, ȕ > 0, x 0. ɉɨ ɮɨɪɦɭɥɟ (7) ɨɩɪɟɞɟɥɹɟɦ ɤɨɫɢɧɭɫ-ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ: 2
F(Į) =
S
f
³
e–ȕt cos Dt dt =
0
E 2 . ɉɨ ɮɨɪɦɭɥɟ (9) ɨɩɪɟɞɟɥɹɟɦ ɫɢɧɭɫS E 2 D 2
ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ: Ɏ(Į) =
2
S
f
³
e–ȕt sin Dt dt =
0
D 2 S E 2 D 2
ɉɨ ɮɨɪɦɭɥɚɦ (8) ɢ (10) ɧɚɯɨɞɢɦ ɜɡɚɢɦɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ 2E
f
cos Dx dD S ³0 E 2 D 2
f
e Ex , x 0,
2 D sin Dx dD S ³0 E 2 D 2
e Ex , x > 0.
ɉɪɢɦɟɪ 4. ɉɪɟɞɫɬɚɜɢɬɶ ɢɧɬɟɝɪɚɥɨɦ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ: f ( x)
1, 0 x 1, ° °1 ® , x 0 ɢ x 1, °2 °¯0, x 0 ɢ x ! 1.
Ⱦɚɧɧɚɹ ɮɭɧɤɰɢɹ ɹɜɥɹɟɬɫɹ ɤɭɫɨɱɧɨ-ɝɥɚɞɤɨɣ, ɬɚɤ ɤɚɤ ɨɧɚ ɫɨɫɬɨɢɬ ɢɡ ɬɪɟɯ ɝɥɚɞɤɢɯ ɱɚɫɬɟɣ: ɭ = 0 ɧɚ (– f , 1), ɭ = 1 ɧɚ (0, 1) ɢ ɭ = 0 ɧɚ (1, f ) ɢ ɢɦɟɟɬ ɞɜɟ ɬɨɱɤɢ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɩɪɢ ɯ = 0 ɢ ɯ = 1. ɗɬɚ ɮɭɧɤɰɢɹ ɚɛɫɨɥɸɬɧɨ ɢɧɬɟɝɪɢɪɭɟɦɚ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ, ɬɚɤ ɤɚɤ ɜɧɟ ɨɬɪɟɡɤɚ [0, 1] ɨɧɚ ɪɚɜɧɚ ɧɭɥɸ, ɢ ɢɧɬɟɝɪɚɥ ɨɬ ɧɟɟ ɩɨ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ ɫɜɟɞɟɬɫɹ ɤ ɢɧɬɟɝɪɚɥɭ ɩɨ ɨɬɪɟɡɤɭ [0, 1]. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɚɤɚɹ ɮɭɧɤɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɢɧɬɟɝɪɚɥɨɦ Ɏɭɪɶɟ.
ɉɨ
ɮɨɪɦɭɥɟ
f(ɯ)=
1
f
f
( f (t ) cos D (t x) dt)dĮ S³ ³ 0
49
f
ɢɦɟɟɦ:
f(ɯ)
1
=
f
f
=
f
sin D (t x)
³
S
D
0
1 0
1
dĮ =
S
f
³
1
0
sin D (1 x) sin Dx
=
0
sin
f
2
dĮ =
D
0
f
( 1 · cos D (t x) dt)dĮ. S³ ³
f
0
1
1
( f (t ) cos D (t x) dt)dĮ= S³ ³
D
³
S
2
cos
D (1 2 x) 2
D
0
dĮ
. ȼ ɬɨɱɤɚɯ ɯ = 0 ɢ ɯ = 1, ɝɞɟ f(x) ɬɟɪɩɢɬ ɪɚɡɪɵɜ, ɩɨɥɭɱɟɧɧɨɟ 1 f ( x 0) f ( x 0) = = 2 2
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɫɨɯɪɚɧɹɟɬɫɹ, ɬɚɤ ɤɚɤ ɜ ɷɬɢɯ ɬɨɱɤɚɯ = f(x).
1 1 ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ ɯ = 0 ɩɨɥɭɱɢɦ: f(0) = = 2 S
=
1
f
S³ 0
sin D
D
f
dĮ, ɱɬɨ ɪɚɜɧɨɫɢɥɶɧɨ ɪɚɜɟɧɫɬɜɭ ɉɪɟɞɫɬɚɜɢɬɶ
5.
=
D
0
ɉɪɢɦɟɪ f ( x)
sin D
³
S 2
ɢɧɬɟɝɪɚɥɨɦ
2 sin
f
D 2
³
D
0
cos
D 2
dĮ =
. Ɏɭɪɶɟ
ɮɭɧɤɰɢɸ
x °1 , 0 d x d 2, ɩɪɨɞɨɥɠɢɜ ɟɟ ɱɟɬɧɵɦ ɨɛɪɚɡɨɦ ɞɥɹ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ® 2 °0, x ! 2. ¯
ɡɧɚɱɟɧɢɣ. Ɂɚɞɚɧɧɚɹ ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ, ɩɨɷɬɨɦɭ ɤ ɧɟɣ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɮɨɪɦɭɥɭ f(ɯ) =
2
f
f
( f (t ) cos Dt dt)cos Įx dĮ, ɜ ɤɨɬɨɪɨɣ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɮɭɧɤɰɢɸ f(t) S³ ³ 0
0
ɧɚɞɨ ɬɨɥɶɤɨ ɩɨ ɨɬɪɟɡɤɭ [0, 2], ɬɚɤ ɤɚɤ ɜɧɟ ɷɬɨɝɨ ɨɬɪɟɡɤɚ ɨɧɚ ɪɚɜɧɚ ɧɭɥɸ. ɂɦɟɟɦ f(ɯ) =
2
f
2
t
( (1 ) cos Dt dt)cos Įx dĮ. 2 S³ ³ 0
0
ȼɵɱɢɫɥɢɦ
ɨɬɞɟɥɶɧɨ
2
t ³0 (1 2 ) cos Dt dt = (1 –
ɜɧɭɬɪɟɧɧɢɣ
t sin Dt ) 2 D
ɢɧɬɟɝɪɚɥ
ɩɨ
ɱɚɫɬɹɦ:
2
2 0
1 1 sin Dtdt = (0 – 0) - 2 cos Dt 02 = 2D ³0 2D
+
f
=
1 cos 2D sin 2 D 2 sin 2 D = , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, f(ɯ) = ³ 2 cos Dx d D , ɜ ɱɚɫɬɧɨɫɬɢ, 2 2 S 0 D 2D D f
ɩɪɢ ɯ = 0 ɢɦɟɟɦ: f(0) = 1 =
2 sin 2 D
S³ 0
D
2
f
d D , ɬ. ɟ.
³
sin 2 D
0
D2
dD =
S 2
.
Ɂɚɞɚɧɢɟ 3. ȼɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɨɜ Ɏɭɪɶɟ ɜ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɟ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. f(ɯ)=
1 2S
f
³
f
f
( ³ f (t ) eiĮ(t-x) dt)dĮ – ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ ɜ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɟ. f
ɗɬɭ ɮɨɪɦɭɥɭ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɬɚɤ: f(ɯ) = 50
1 2S
f
³
f
(
1 2S
f
iĮt
³ f (t ) e
f
dt)e-iĮxdĮ.
ɇɚ
ɨɫɧɨɜɚɧɢɢ 1
F*(Į) =
2S
f
ɩɨɫɥɟɞɧɟɝɨ
ɪɚɜɟɧɫɬɜɚ 1
iĮt
dt, (1) ɬɨɝɞɚ: f(x) =
³ f (t ) e
2S
f
f
³
ɦɨɠɧɨ
ɧɚɩɢɫɚɬɶ:
F*(Į)e–iĮxdĮ.
f
Ɏɭɧɤɰɢɹ F*(Į) ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ ɞɥɹ ɮɭɧɤɰɢɢ f(t) ɢɥɢ ɫɩɟɤɬɪɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ɮɭɧɤɰɢɢ f(t). Ɇɨɞɭɥɶ |F*( Į)| ɧɚɡɵɜɚɟɬɫɹ ɚɦɩɥɢɬɭɞɧɵɦ ɫɩɟɤɬɪɨɦ ɮɭɧɤɰɢɢ f(t). Ɏɭɧɤɰɢɹ f(t) ɧɚɡɵɜɚɟɬɫɹ ɨɛɪɚɬɧɵɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ ɞɥɹ ɮɭɧɤɰɢɢ F*(Į). ɗɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɨɬɥɢɱɚɸɬɫɹ ɡɧɚɤɨɦ ɩɪɢ i. ɉɪɢɦɟɪ 6. ɉɪɟɞɫɬɚɜɢɬɶ ɮɭɧɤɰɢɸ f(x) ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɨɣ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ. ɇɚɣɬɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɮɭɧɤɰɢɸ ɢ ɚɦɩɥɢɬɭɞɧɵɣ ɫɩɟɤɬɪ ɮɭɧɤɰɢɢ f(x). e 0.2 x , ɩɪɢ x ! 0,
f(x) = °®
°¯0, ɩɪɢ x 0.
ɉɨ ɮɨɪɦɭɥɟ (1) ɧɚɯɨɞɢɦ ɫɩɟɤɬɪɚɥɶɧɭɸ
ɩɥɨɬɧɨɫɬɶ ɮɭɧɤɰɢɢ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɩɪɢ t < 0 f(t) = 0: F*(Į) = =
1 2S
f
³e
0.2 t iDt
1 e (0.2iD )t 2S (0, 2 iD )
dt =
0
f 0
=
1 2S
f
³e
0.2 t
eiĮt dt
0
1 . 2S (0, 2 iD )
Ⱥɦɩɥɢɬɭɞɧɵɣ ɫɩɟɤɬɪ |F*( Į)| ɨɩɪɟɞɟɥɹɟɦ ɜ ɜɢɞɟ: |F*( Į)| = |
1 |= 2S (0, 2 iD )
1 2S 0, 2 iD
1 2S (0, 04 D 2
.
Ʉɨɦɩɥɟɤɫɧɚɹ ɮɨɪɦɚ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ ɢɦɟɟɬ ɜɢɞ: f(x) =
1 2S
f
1
–iĮx
³ 0.2 iD ɟ
dĮ.
f
Ɂɚɞɚɧɢɹ ɤ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɟ 8 ȼɚɪɢɚɧɬ 1. Ɂɚɞɚɧɢɟ 1. Ɋɚɡɥɨɠɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɜ ɪɹɞ Ɏɭɪɶɟ: f(x) = x + 1, ɧɚ (–ʌ, ʌ). f
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ: ³ e 4 x sin 3x cos 2 xdx . 0
Ɂɚɞɚɧɢɟ 3. ɉɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɨɣ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ, ɧɚɣɬɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɢ ɚɦɩɥɢɬɭɞɧɵɣ 1, x (0, a ), ɫɩɟɤɬɪ: f(x) = °®
°¯0, x (0, a ).
51
ȼɚɪɢɚɧɬ 2. Ɂɚɞɚɧɢɟ 1. Ɋɚɡɥɨɠɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɜ ɪɹɞ Ɏɭɪɶɟ: f(x) =
S x 2
, ɧɚ
(0, 2ʌ). f
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ:
³e
3 x
cos 3x cos 4 xdx .
0
Ɂɚɞɚɧɢɟ 3. ɉɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɨɣ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ, ɧɚɣɬɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɢ ɚɦɩɥɢɬɭɞɧɵɣ 2e ax x (0, f),
ɫɩɟɤɬɪ: f(x) = ®
¯0, x (f, 0),
a ! 0.
ȼɚɪɢɚɧɬ 3. Ɂɚɞɚɧɢɟ 1. Ɋɚɡɥɨɠɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɜ ɪɹɞ Ɏɭɪɶɟ: f(x) = |x+1|, ɧɚ (–ʌ, ʌ). 1, 0 x 2, Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ: f(x) = °® °¯0, x ! 2.
Ɂɚɞɚɧɢɟ 3. ɉɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɨɣ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ, ɧɚɣɬɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɢ ɚɦɩɥɢɬɭɞɧɵɣ 2 x 2 , 1 d x d 1, ° ɫɩɟɤɬɪ: f(x) = °®1, 1 x d 2, ° °¯0, x ! 2.
ȼɚɪɢɚɧɬ 4. Ɂɚɞɚɧɢɟ 1. Ɋɚɡɥɨɠɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɜ ɪɹɞ Ɏɭɪɶɟ: f(x) = 0, S x d 0, . ® ¯ x, 0 x S . cos x, 0 x 2, ¯0, x ! 2.
Ɂɚɞɚɧɢɟ 2. ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ: f(x) = ®
Ɂɚɞɚɧɢɟ 3. ɉɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɭɸ ɮɭɧɤɰɢɸ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɨɣ ɢɧɬɟɝɪɚɥɚ Ɏɭɪɶɟ, ɧɚɣɬɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɢ ɚɦɩɥɢɬɭɞɧɵɣ 1, x [1,1], . ¯0, x [1,1].
ɫɩɟɤɬɪ: f(x) = ®
Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 9 (3 ɱɚɫɚ). ɍɪɚɜɧɟɧɢɹ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ Ɂɚɞɚɧɢɟ 1. ɉɪɨɫɬɟɣɲɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. ɉɪɢɦɟɪ 1. ɇɚɣɬɢ ɮɭɧɤɰɢɸ z = z(x, y), ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
wz wx
1.
52
ɂɧɬɟɝɪɢɪɭɟɦ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ ɯ, ɩɨɥɭɱɢɦ: ³ wz ³ wx z = x + + ij(y), ɝɞɟ ij(y) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ. ɗɬɨ ɟɫɬɶ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɉɪɢɦɟɪ 2. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ
wz 2 wy 2
6 y , ɝɞɟ z = z(x, y).
Ⱦɜɚɠɞɵ ɢɧɬɟɝɪɢɪɭɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɨ ɭ. ɉɨɥɭɱɚɟɦ
wz wy
3 y 2 M ( x) ,
z = y3 + y ij(x) + ȥ(x), ɝɞɟ ij(x) ɢ ȥ(x) – ɩɪɨɢɡɜɨɥɶɧɵɟ ɮɭɧɤɰɢɢ. ɉɪɢɦɟɪ 3. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ
wz 2 wxwy
0 , ɝɞɟ z = z(x, y).
ɂɧɬɟɝɪɢɪɭɟɦ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ ɯ, ɩɨɥɭɱɢɦ
wz wy
ɂɧɬɟɝɪɢɪɭɟɦ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ ɩɨ ɭ, ɩɨɥɭɱɢɦ z =
f ( y) .
³ f ( y)dy + ij(x).
Ɂɚɞɚɧɢɟ 2. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɥɢɧɟɣɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ɋɚɫɫɦɨɬɪɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ X
wz wz Y wx wy
Z , ɝɞɟ X, Y ɢ
Z – ɮɭɧɤɰɢɢ x, y, z. ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɪɟɲɢɦ ɫɢɫɬɟɦɭ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
dx X
dy Y
dz . ɉɭɫɬɶ ɪɟɲɟɧɢɟ ɷɬɨɣ ɫɢɫɬɟɦɵ Z
ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ Ȧ1(x, y, z) = C1, Ȧ2(x, y, z) = C2. Ɍɨɝɞɚ ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɢɫɯɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: Ɏ(Ȧ1(x, y, z), Ȧ2(x, y, z)) = 0, ɝɞɟ Ɏ(Ȧ1, Ȧ2) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ. ɉɪɢɦɟɪ 4. ɇɚɣɬɢ ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ x
wz wz y wx wy
z.
Ɋɚɫɫɦɨɬɪɢ ɫɢɫɬɟɦɭ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ dx dy dz dx dy . Ɋɟɲɢɦ ɭɪɚɜɧɟɧɢɟ , ɩɨɥɭɱɢɦ ln|y| = ln|xC1| ɢɥɢ ɭ = ɯɋ1, x y z x y dx dz y z C1 . Ɋɟɲɢɦ ɭɪɚɜɧɟɧɢɟ , ɩɨɥɭɱɢɦ ln|z| = ln|xC2| ɢɥɢ z = ɯɋ2, C2 . x z x x y z Ɉɛɳɢɣ ɢɧɬɟɝɪɚɥ ɡɚɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: Ɏ( , ) = 0 ɢɥɢ x x z y y = ij( ), ɬ. ɟ. z = x ij( ), ɝɞɟ ij – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ. x x x wz wz 0. ɉɪɢɦɟɪ 5. ɇɚɣɬɢ ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ ( x 2 y 2 ) 2 xy wx wy
53
dz . ȼɨɫɩɨɥɶɡɭɟɦɫɹ 0 dx dy ɫɜɨɣɫɬɜɨɦ ɩɪɨɩɨɪɰɢɢ ɢ ɩɪɟɞɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɜ ɜɢɞɟ x 2 y 2 2 xy dx dy dx dy d ( x y) d ( x y) ɢɥɢ ɜ ɜɢɞɟ . ɂɧɬɟɝɪɢɪɭɟɦ x 2 2 xy y 2 x 2 y 2 2 xy ( x y) 2 ( x y) 2 1 1 1 1 2y C , C, 2 C. ɞɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ, ɩɨɥɭɱɢɦ x y x y x y x y x y2 y ɉɨɫɥɟɞɧɟɟ ɪɚɜɟɧɫɬɜɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ 2 2 C1 . x y
Ɂɚɩɢɲɟɦ
ɫɢɫɬɟɦɭ
ɭɪɚɜɧɟɧɢɣ
dx x2 y2
dy 2 xy
ȼɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɜɢɞ dz = 0. Ɉɬɫɸɞɚ z = C2. Ɉɛɳɢɣ ɢɧɬɟɝɪɚɥ ɢɦɟɟɬ ɜɢɞ: Ɏ(
y y ,z) = 0 ɢɥɢ z = ij( 2 2 ). x2 y2 x y
Ɂɚɞɚɧɢɟ 3. ɍɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɹ ɤɨɧɟɱɧɨɣ ɫɬɪɭɧɵ. Ɂɚɞɚɧɢɟ ɜɵɞɚɟɬɫɹ ɩɪɟɩɨɞɚɜɚɬɟɥɟɦ ɫɨɝɥɚɫɧɨ ɜɚɪɢɚɧɬɭ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɦɚɥɵɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɨɞɧɨɪɨɞɧɨɣ ɫɬɪɭɧɵ ɫ ɡɚɤɪɟɩɥɟɧɧɵɦɢ ɤɨɧɰɚɦɢ ɢɦɟɟɬ ɜɢɞ w 2u wt 2
w 2u ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ u(0, t) = u(l, t) = 0 (– f < t < + f ) ɢ wx 2 ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ u(x, 0) = ij(x), u/t(x, 0) = ȥ(x), (0 d x d l ). a2
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ u(x, t), ɞɚɸɳɟɟ ɨɬɤɥɨɧɟɧɢɟ ɬɨɱɟɤ ɫɬɪɭɧɵ ɫ ɚɛɫɰɢɫɫɨɣ ɯ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t, ɜɵɪɚɠɚɟɬɫɹ ɪɹɞɨɦ Ɏɭɪɶɟ u(x, t) = f
=
¦(A
k
k 1
cos
kSat kSat kSx Bk sin ) sin , l l l
1
ɝɞɟ
Ak=
kSx 2 M ( x) sin dx , l ³0 l
1
kSx 2 Bk = \ ( x) sin dx , k = 1, 2, 3, … kSa ³0 l
w 2u w 2u a2 2 ɫ 2 wx wt ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ u(x, 0) = ij(x), u/t(x, 0) = ȥ(x), (– f d x d f ).
ɍɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɣ ɛɟɫɤɨɧɟɱɧɨɣ ɫɬɪɭɧɵ ɢɦɟɟɬ ɜɢɞ
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɨɬɫɭɬɫɬɜɭɸɬ, ɬɚɤ ɤɚɤ ɫɬɪɭɧɚ ɛɟɫɤɨɧɟɱɧɚɹ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɜ ɫɥɭɱɚɟ ɛɟɫɤɨɧɟɱɧɵɯ ɫɪɟɞ ɩɪɢɦɟɧɹɸɬ ɢɧɬɟɝɪɚɥɶɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ. ɉɪɢ ɷɬɨɦ ɡɚɦɟɧɹɸɬ ɞɚɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɭɪɚɜɧɟɧɢɟɦ ɞɥɹ Ɏɭɪɶɟ – ɨɛɪɚɡɨɜ ɟɝɨ ɱɚɫɬɟɣ. ɇɚɯɨɞɹɬ ɢɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ (ɨɛɵɱɧɨ ɛɨɥɟɟ ɩɪɨɫɬɨɝɨ) Ɏɭɪɶɟ – ɨɛɪɚɡ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ, ɚ ɡɚɬɟɦ ɫ ɩɨɦɨɳɶɸ ɨɛɪɚɬɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɏɭɪɶɟ ɩɨɥɭɱɚɸɬ ɫɚɦɨ ɪɟɲɟɧɢɟ. ɉɪɢɦɟɪ 6. Ɉɞɧɨɪɨɞɧɚɹ ɫɬɪɭɧɚ, ɡɚɤɪɟɩɥɟɧɧɚɹ ɧɚ ɤɨɧɰɚɯ ɯ = 0 ɢ ɯ = l, ɢɦɟɟɬ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɮɨɪɦɭ ɩɚɪɚɛɨɥɵ ij(x) = ɯ(l – x). Ɉɩɪɟɞɟɥɢɬɶ ɫɦɟɳɟɧɢɟ ɬɨɱɟɤ ɫɬɪɭɧɵ ɨɬ ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɧɚɱɚɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ. 54
Ɂɚɞɚɱɚ ɫɜɨɞɢɬɫɹ ɤ ɪɟɲɟɧɢɸ ɭɪɚɜɧɟɧɢɹ
w 2u wt 2
a2
w 2u ɩɪɢ ɝɪɚɧɢɱɧɵɯ wx 2
ɭɫɥɨɜɢɹɯ u(0, t) = u(l, t) = 0 ɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ u(x, 0) = ij(x) = x(l – x), wu ( x,0) = ȥ(x) = 0. wt
ȼɵɱɢɫɥɹɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ Ak ɢ Bk: 1
kSx 2 kSx 2 Ak= ³ (lx x 2 ) sin (lx x 2 ) cos dx = kS l l 0 l
=
2l 2
k S
2
(l 2 x) sin
kSx l
l 0
+
1
4l 2
k S
2
³ sin 0
1
l 0
+
kSx 2 (l 2 x) cos dx = kS ³0 l
kSx 4l 2 kSx dx = 3 3 cos l l k S
l 0
=
4l 2 (1 cos kS ) = k 3S 3
0, ɩɪɢ k ɱɟɬɧɨɦ, 4l = 3 3 (1 (1) k ) = °® 8l 2 k S ° 3 3 , ɩɪɢ k ɧɟɱɟɬɧɨɦ. ¯S k 1 kSx 2 Bk = 0 sin dx = 0. ɉɨɷɬɨɦɭ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ: kSa ³0 l 2
u(x, t) =
8l 2
S3
f
cos
¦ k 1
(2k 1)Sat (2k 1)Sx sin l l . (2k 1) 3
ɉɪɢɦɟɪ 7. Ⱦɚɧɚ ɫɬɪɭɧɚ, ɡɚɤɪɟɩɥɟɧɧɚɹ ɧɚ ɤɨɧɰɚɯ ɯ = 0 (ɬɨɱɤɚ 0) ɢ ɯ = l (ɬɨɱɤɚ B). ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɮɨɪɦɚ ɫɬɪɭɧɵ ɢɦɟɟɬ ɜɢɞ ɥɨɦɚɧɨɣ ɥɢɧɢɢ ɈȺB, ɬɨɱɤɚ Ⱥ(l/2,h). ɇɚɣɬɢ ɮɨɪɦɭ ɫɬɪɭɧɵ ɞɥɹ ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t, ɟɫɥɢ ɧɚɱɚɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ. ȼɵɩɢɫɚɬɶ ɧɟɫɤɨɥɶɤɨ ɱɥɟɧɨɜ ɪɹɞɚ Ɏɭɪɶɟ. h 2h , ɬ. ɟ. , l/2 l 2h x. ɉɪɹɦɚɹ AB ɨɬɫɟɤɚɟɬ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɩɪɹɦɨɣ ɟɫɬɶ u = l
ɍɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɹɦɨɣ ɈȺ ɪɚɜɟɧ
ɧɚ ɨɫɹɯ ɤɨɨɪɞɢɧɚɬ ɨɬɪɟɡɤɢ l ɢ 2h, ɡɧɚɱɢɬ ɭɪɚɜɧɟɧɢɟ ɷɬɨɣ ɩɪɹɦɨɣ ɢɦɟɟɬ ɜɢɞ 2h (l – x). ɂɬɚɤ, M ( x) 1 ɢɥɢ u = l
x u l 2h
l
l 2h °° l x, ɩɪɢ 0 d ɯ d 2 , ® ° 2h (l x), ɩɪɢ l d x d l , °¯ l 2
kSx 2 4h M ( x) sin dx = 2 l ³0 l l
ɇɚɯɨɞɢɦ Ak ɢ Bk. Ak =
l/2
³ x sin 0
\ ( x)
kSx dx + l
l
+
kSx 4h (l x) sin dx = | ɢɧɬɟɝɪɢɪɭɟɦ ɩɨ ɱɚɫɬɹɦ | = l l 2 l ³/ 2
=
kSx 4h x cos kSl l
l/2 0
+
4h kSl
l/2
³ cos 0
kSx 4h kSx (l x) cos dx – kSl l l
55
l
l l/2
–
kSx 4h cos dx = kSl l ³/ 2 l
0.
kSx l / 2 2h kSx 4h 4h 2h kS kS cos cos + 2 2 sin – 2 2 sin 0 + 2 l 2 l kS kS k S k S 4h 8h kS kS = 2 2 sin . Bk = 0. sin 2 2 k 2S 2 k S kS kSx kSDt 8h f 1 cos . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, u(x, t) = 2 ¦ 2 sin sin l l 2 S k 1k
=
ȼɵɩɢɲɟɦ ɧɟɫɤɨɥɶɤɨ ɱɥɟɧɨɜ ɪɹɞɚ: u(x,t) =
l l/2
8h
S2
=
4h kS sin + 2 k 2S 2
(sin
Sx
· cos
SDt
l l 1 1 1 3Sx 3SDt 5Sx 5SDt 7Sx 7SDt · cos + 2 sin · cos – 2 sin · cos + …). – 2 sin l l l l l l 3 5 7
–
ɉɪɢɦɟɪ 8. ɉɭɫɬɶ ɧɚɱɚɥɶɧɵɟ ɨɬɤɥɨɧɟɧɢɹ ɫɬɪɭɧɵ, ɡɚɤɪɟɩɥɟɧɧɨɣ ɜ ɬɨɱɤɟ ɯ = 0 ɢ ɯ = l, ɪɚɜɧɵ ɧɭɥɸ, ɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
wu wx
°v0 (const ), ɩɪɢ | x l / 2 | h / 2, . Ɉɩɪɟɞɟɥɢɬɶ ɮɨɪɦɭ ɫɬɪɭɧɵ ɞɥɹ ® °¯0, ɩɪɢ | x l / 2 |! h / 2.
ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t. Ɂɞɟɫɶ ij(x) = 0, ɚ ȥ(x) = v0 ɜ ɢɧɬɟɪɜɚɥɟ ((l – h)/2, (l + h)/2) ɢ ȥ(x) = 0 ɜɧɟ ɷɬɨɝɨ ɢɧɬɟɪɜɚɥɚ. (l h ) / 2
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Ak = 0, Bk =
2 kSx v0 sin dx = kSa (l ³h ) / 2 l
2v 0 l 2v 0 l kSx (l h ) / 2 kS (l x) kS (l h) [cos cos cos ] = (l h ) / 2 = 2l 2l kSD kS l k 2S 2 a 4v l f 1 4v l kS kSh kSDt kSx kS kSh . Ɉɬɫɸɞɚ, u(x, t) = 20 ¦ 2 sin sin sin sin . = 2 02 sin sin 2 2l l l 2 2l k S D S a k 1k 4v l Sh SDt Sx 1 sin sin – 2 ȼɵɩɢɲɟɦ ɧɟɫɤɨɥɶɤɨ ɱɥɟɧɨɜ ɪɹɞɚ: u(x, t) = 20 (sin 2l l l S a 3 3Sh 5Sh 3SDt 3Sx 5SDt 5Sx 1 sin sin + 2 sin sin sin – …). sin 2l l l 2l l l 5
=
Ɂɚɞɚɧɢɹ ɤ ɥɚɛɨɪɚɬɨɪɧɨɣ ɪɚɛɨɬɟ 9
1. ɇɚɣɬɢ ɮɭɧɤɰɢɸ z = z(x, y), ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
wz wx
2.
2. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ
wz 2 wy 2
6 y , ɝɞɟ z = z(x, y).
3. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ
wz 2 wxwy
1 , ɝɞɟ z = z(x, y).
wz wz y 2z . wx wy wz wz 5. ɇɚɣɬɢ ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ ( x 2 y 2 ) 2 xy wx wy
4. ɇɚɣɬɢ ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ x
0.
6. Ɉɞɧɨɪɨɞɧɚɹ ɫɬɪɭɧɚ, ɡɚɤɪɟɩɥɟɧɧɚɹ ɧɚ ɤɨɧɰɚɯ ɯ = 0 ɢ ɯ = 2, ɢɦɟɟɬ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɮɨɪɦɭ ɩɚɪɚɛɨɥɵ ij(x) = ɯ(2 – x). Ɉɩɪɟɞɟɥɢɬɶ 56
ɫɦɟɳɟɧɢɟ ɬɨɱɟɤ ɫɬɪɭɧɵ ɨɬ ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɧɚɱɚɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ. 7. Ⱦɚɧɚ ɫɬɪɭɧɚ, ɡɚɤɪɟɩɥɟɧɧɚɹ ɧɚ ɤɨɧɰɚɯ ɯ = 0 (ɬɨɱɤɚ 0) ɢ ɯ = 2 (ɬɨɱɤɚ B). ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɮɨɪɦɚ ɫɬɪɭɧɵ ɢɦɟɟɬ ɜɢɞ ɥɨɦɚɧɨɣ ɥɢɧɢɢ ɈȺB, ɬɨɱɤɚ Ⱥ(1, 0, 2). ɇɚɣɬɢ ɮɨɪɦɭ ɫɬɪɭɧɵ ɞɥɹ ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t, ɟɫɥɢ ɧɚɱɚɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɨɬɫɭɬɫɬɜɭɸɬ. ȼɵɩɢɫɚɬɶ ɧɟɫɤɨɥɶɤɨ ɱɥɟɧɨɜ ɪɹɞɚ Ɏɭɪɶɟ 8. ɉɭɫɬɶ ɧɚɱɚɥɶɧɵɟ ɨɬɤɥɨɧɟɧɢɹ ɫɬɪɭɧɵ, ɡɚɤɪɟɩɥɟɧɧɨɣ ɜ ɬɨɱɤɟ ɯ = 0 ɢ ɯ = l, ɪɚɜɧɵ ɧɭɥɸ, ɚ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
wu wx
°1, ɩɪɢ | x 1/ 2 | 0,1, Ɉɩɪɟɞɟɥɢɬɶ ɮɨɪɦɭ ɫɬɪɭɧɵ ɞɥɹ ɥɸɛɨɝɨ ® °¯0, ɩɪɢ | x 1/ 2 |! 0,1.
ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t. Ʌɢɬɟɪɚɬɭɪɚ 1. Ⱦɟɦɢɞɨɜɢɱ Ȼ.ɉ. Ʉɪɚɬɤɢɣ ɤɭɪɫ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ / Ȼ.ɉ. Ⱦɟɦɢɞɨɜɢɱ, ȼ.Ⱥ. Ʉɭɞɪɹɜɰɟɜ. – Ɇ. : ȼɫɢɪɟɥɶ ; ȺɋɌ, 2001. – 656 ɫ. 2. ɀɞɚɧɨɜ Ɇ.ɋ. ɗɥɟɤɬɪɨɪɚɡɜɟɞɤɚ : ɭɱɟɛɧɢɤ ɞɥɹ ɜɭɡɨɜ / Ɇ.ɋ. ɀɞɚɧɨɜ. – Ɇ. : ɇɟɞɪɚ, 1986. – 316 ɫ. 3. ɋɦɢɪɧɨɜ ȼ.ɂ. Ʉɭɪɫ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ / ȼ.ɂ. ɋɦɢɪɧɨɜ. – Ɇ. : ɇɚɭɤɚ, 1974. – 656 ɫ. 4. Ɍɢɯɨɧɨɜ Ⱥ.ɇ. ɍɪɚɜɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ / Ⱥ.ɇ. Ɍɢɯɨɧɨɜ, Ⱥ.Ⱥ. ɋɚɦɚɪɫɤɢɣ. – M. : ɇɚɭɤɚ, 1977. – 736 ɫ.
57
ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ
Ƚɪɭɡɞɟɜ ȼɥɚɞɢɫɥɚɜ ɇɢɤɨɥɚɟɜɢɱ ɆȺɌȿɆȺɌɂɑȿɋɄɂȿ ɍɊȺȼɇȿɇɂə ȼ ȽȿɈɎɂɁɂɄȿ ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɤ ɥɚɛɨɪɚɬɨɪɧɵɦ ɪɚɛɨɬɚɦ ɞɥɹ ɜɭɡɨɜ
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 06.08.07. Ɏɨɪɦɚɬ 60×84/16. ɍɫɥ. ɩɟɱ. ɥ. 3,5. Ɍɢɪɚɠ 50 ɷɤɡ. Ɂɚɤɚɡ 1465. ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɩɥ. ɢɦ. Ʌɟɧɢɧɚ, 10. Ɍɟɥ. 208-298, 598-026 (ɮɚɤɫ) http://www.ppc.vsu.ru; e-mail:
[email protected] Ɉɬɩɟɱɚɬɚɧɨ ɜ ɬɢɩɨɝɪɚɮɢɢ ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɨɝɨ ɰɟɧɬɪɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɭɥ. ɉɭɲɤɢɧɫɤɚɹ, 3. Ɍɟɥ. 204-133. 58
