Математика для юристов - Д.А. Ловцова
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Ɍɚɛɥɢɰɚ 4.2 |
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Ɂɚɞɚɱɚ. Ɍɚɛɥ.4.2 ɫɨɞɟɪɠɢɬ ɞɚɧɧɵɟ |
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ɨ ɤɨɥɢɱɟɫɬɜɟ ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧɧɵɯ ɩɪɟ- |
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ɫɬɭɩɥɟɧɢɣ ɉ ɜ ɝɨɪɨɞɟ Ȼ. ɡɚ ɩɨɫɥɟɞɧɢɟ |
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ɝɨɞɵ ɝ.. ȼɨɫɩɨɥɧɢɬɶ ɧɟɞɨɫɬɚɸɳɢɟ ɫɜɟɞɟɧɢɹ ɨ ɤɨɥɢɱɟɫɬɜɟ ɉ ɡɚ 2003 ɝɨɞ ɢ ɧɚɣɬɢ ɨɠɢɞɚɟɦɨɟ ɡɧɚɱɟɧɢɟ ɉ ɜ 2006 ɝɨɞɭ.
Ɋɟɲɟɧɢɟ. ɉɨ ɬɚɛɥ. 4.2 ɫɬɪɨɢɦ ɝɪɚɮɢɤ (ɪɢɫ. 4.7).
ɉɨ ɮɨɪɦɭɥɟ (4.3) ɞɥɹ ɝ1 2003 ɧɚɯɨɞɢɦ, ɱɬɨ k |
1. ɉɨɞɫɬɚɜɥɹɟɦ ɷɬɨ |
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ɡɧɚɱɟɧɢɟ k ɜ ɮɨɪɦɭɥɭ (4.2) ɢ ɜɵɱɢɫɥɹɟɦ |
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ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ ɉ1: |
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2004 2002
ɉɪɨɜɟɪɢɦ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɡɚɞɚɜ |
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ɬɨɱɤɭ (ɝ1,ɉ1) ɧɚ ɝɪɚɮɢɤɟ (ɪɢɫ. 4.7). |
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Ɂɧɚɱɟɧɢɟ ɉ2 ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɪɹ- |
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ɝ2 2006 ɧɚɨɫɢ ɚɛɫɰɢɫɫ ɢ ɧɚɣɬɢ ɬɨɱɤɭɉ2 |
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ɦɨ ɩɨ ɝɪɚɮɢɤɭ, ɞɥɹ ɱɟɝɨ ɡɚɞɚɬɶ ɬɨɱɤɭ |
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ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɩɪɹɦɨɣ (ɪɢɫ. 4.7). |
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Ɋɢɫ.4.7 |
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Ɍɟɦ |
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ɝ2 2006, ɢ |
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ɉ2 ɉ2 ɉ3 ɉ2 u(ɝ2 ɝ2)
ɝ3 ɝ2
274 285 274 u(2005 2004) 285.
2005 2004
4.3. Предел функции
Ⱦɥɹ ɧɚɱɚɥɚ ɭɤɚɠɟɦ ɧɚ ɬɨ, ɱɬɨ ɦɨɞɭɥɶ ɪɚɡɧɨɫɬɢ |
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°p q° |
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ɱɢɫɟɥ p ɢ q ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɸ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ |
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ɷɬɢɦɢ ɱɢɫɥɚɦɢ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ (ɪɢɫ. 4.8). |
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Ⱥ ɬɟɩɟɪɶ ɩɨɹɫɧɢɦ, ɱɬɨ ɨɡɧɚɱɚɟɬ ɮɪɚɡɚ «x ɫɬɪɟ- |
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ɦɢɬɫɹ ɤ a». ȼɨ-ɩɟɪɜɵɯ, ɷɬɨ ɡɧɚɱɢɬ, ɱɬɨ x ɦɟɧɹɟɬɫɹ. ɇɨ |
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ɦɟɧɹɟɬɫɹ ɬɚɤ, ɱɬɨ ɤɚɠɞɨɟ ɟɝɨ ɫɥɟɞɭɸɳɟɟ ɡɧɚɱɟɧɢɟ |
Ɋɢɫ. 4.8 |
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ɛɥɢɠɟ ɤ ɱɢɫɥɭ ɚ, ɱɟɦ ɩɪɟɞɵɞɭɳɟɟ. ɂ ɤɚɤɨɟ ɛɵ ɦɚɥɨɟ |
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G!0 ɧɢ ɧɚɡɧɚɱɢɬɶ, ɧɚɫɬɚɧɟɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɛɭɞɟɬ ɜɵɩɨɥɧɟɧɨ ɭɫɥɨɜɢɟ ~x a~ G, ɬɨ ɟɫɬɶ ɪɚɫɫɬɨɹɧɢɟ ɨɬ x ɞɨ a ɛɭɞɟɬ ɦɟɧɶɲɟ, ɱɟɦ G. ɂ ɩɨɫɥɟ-
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ɞɭɸɳɢɟ ɢɡɦɟɧɟɧɢɹ x ɧɢɤɨɝɞɚ ɧɟ ɧɚɪɭɲɚɸɬ ɷɬɨ ɭɫɥɨɜɢɟ. Ɏɚɤɬ «x ɫɬɪɟɦɢɬɫɹ ɤ a» ɨɛɨɡɧɚɱɚɸɬɤɨɧɫɬɪɭɤɰɢɟɣ xoa.
ɑɢɫɥɨ A ɧɚɡɵɜɚɟɬɫɹ ɩ ɪ ɟ ɞ ɟ ɥ ɨ ɦ ɮɭɧɤɰɢɢ y f(x) ɩɪɢ x, ɫɬɪɟɦɹɳɟɦɫɹ ɤ a, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ H ɧɚɣɞɟɬɫɹ ɬɚɤɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ G, ɱɬɨ ɪɚɫɫɬɨɹɧɢɟ ɨɬ y ɞɨ A ɛɭɞɟɬ ɦɟɧɶɲɟ, ɱɟɦ H, ɚ ɪɚɫɫɬɨɹɧɢɟ ɨɬ x ɞɨ a — ɦɟɧɶɲɟ, ɱɟɦ G.
Ɂɚɩɢɫɵɜɚɟɬɫɹ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɬɚɤ:
lim f(x) A ,
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ɚ ɫɦɵɫɥ ɷɬɨɣ ɡɚɩɢɫɢ ɩɨɹɫɧɹɟɬ ɪɢɫ.4.9. |
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ɉɪɢɦɟɪ. Ɋɚɫɫɦɨɬɪɢɦ ɛɟɫɤɨɧɟɱɧɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɬɨɱɟɤ ɧɚ |
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ɱɢɫɥɨɜɨɣ ɨɫɢ: |
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x0 2 1 |
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~xk a~ |
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1.75, …, |
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xk 2 1 |
,… |
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Ʌɸɛɨɟ ɢɡ ɷɬɢɯ ɱɢɫɟɥ ɦɟɧɶɲɟ |
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Ɋɢɫ.4.9 |
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ɇɨ ɩɨ ɦɟɪɟ ɪɨɫɬɚ ɡɧɚɱɟ- |
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ɧɢɹ k ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɢ ɤɨɥɢɱɟ- |
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ɫɬɜɨ ɬɚɤɢɯ ɱɢɫɟɥ, ɤɨɬɨɪɵɟ ɜɫɟ ɬɟɫɧɟɟ ɢ ɬɟɫɧɟɟ ɫɤɚɩɥɢɜɚɸɬɫɹ ɨɤɨɥɨ |
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ɬɨɱɤɢ a. Ɋɚɫɫɬɨɹɧɢɟ ~xk a~ ɭɦɟɧɶɲɚɟɬɫɹ ɢ ɫ ɤɚɠɞɵɦ ɧɨɜɵɦ k ɦɨɠɟɬ |
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ɫɬɚɬɶ ɦɟɧɶɲɟ ɥɸɛɨɝɨ ɧɚɩɟɪɟɞ ɡɚɞɚɧɧɨɝɨ ɱɢɫɥɚ G!0. |
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Ʉɚɠɞɨɦɭ ɡɧɚɱɟɧɢɸ x ɢɡ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ x0, xk, |
, xk, ɨɬɜɟ- |
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ɱɚɟɬ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ y f(x). ɇɚɩɪɢɦɟɪ, ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ y |
x2 |
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ɫɨɫɬɚɜɹɬ ɬɚɤɭɸ ɛɟɫɤɨɧɟɱɧɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ: |
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y0 12 1, |
y1 1.52 |
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3.0625, |
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yk ¨2 |
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ɜɫɟ ɛɥɢɠɟ ɤ A |
4 ɢ ɫ ɤɚɠ- |
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Ʉɚɤ ɜɢɞɢɦ, ɤɚɠɞɨɟ ɫɥɟɞɭɸɳɟɟ ɡɧɚɱɟɧɢɟ yk |
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ɞɵɦ ɧɨɜɵɦ k ɪɚɡɧɨɫɬɶ ~yk A~ ɦɨɠɟɬ ɫɬɚɬɶ ɦɟɧɶɲɟ ɥɸɛɨɝɨ ɧɚɩɟɪɟɞ |
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ɡɚɞɚɧɧɨɝɨ ɱɢɫɥɚ H!0. |
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ɂɡɥɨɠɟɧɧɨɟ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ |
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lim x2 4 .
xo2
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Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɨɥɨɠɢɦ H 2 3#0.13 ɢ ɧɚɣɞɟɦ G. ɂɡ ɭɫɥɨɜɢɹ
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ɧɚɯɨɞɢɦ, ɱɬɨ kt5. Ⱦɚɥɟɟ ~xk 2~ |
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ɉɪɢ kt5 ɩɨɥɭɱɢɦ, ɱɬɨ G 2 5 ɢ ~xk 2~d2 5#0.03.
Ƚɨɜɨɪɹɬ, ɱɬɨ xo f, ɟɫɥɢ x ɦɟɧɹɟɬɫɹ ɬɚɤ, ɱɬɨ ɤɚɠɞɨɟ ɫɥɟɞɭɸɳɟɟ ɟɝɨ ɡɧɚɱɟɧɢɟ ɫɬɚɧɨɜɢɬɫɹ ɛɨɥɶɲɟ ɩɪɟɞɵɞɭɳɟɝɨ. ɂ ɤɚɤɨɟ ɛɵ ɛɨɥɶɲɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ M ɧɢ ɧɚɡɧɚɱɢɬɶ, ɧɚɫɬɭɩɢɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ x ɨɤɚɠɟɬɫɹ ɛɨɥɶɲɟ M, ɢ ɞɚɥɟɟ ɷɬɨ ɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɧɢɦɢ ɧɟ ɦɟɧɹɟɬɫɹ.
Ɍɨɝɞɚ ɭɬɜɟɪɠɞɟɧɢɟ lim f(x) A ɨɡɧɚɱɚɟɬ ɫɥɟɞɭɸɳɟɟ: ɞɥɹ ɥɸɛɨɝɨ H!0
xof
ɧɚɣɞɟɬɫɹ ɬɚɤɨɟ M!0, ɱɬɨ~f(x) A~ H ɩɪɢ x!M.
Ƚɨɜɨɪɹɬ, ɱɬɨ xo f, ɟɫɥɢ x ɦɟɧɹɟɬɫɹ ɬɚɤ, ɱɬɨ ɤɚɠɞɨɟ ɫɥɟɞɭɸɳɟɟ ɟɝɨ ɡɧɚɱɟɧɢɟ ɫɬɚɧɨɜɢɬɫɹ ɦɟɧɶɲɟ ɩɪɟɞɵɞɭɳɟɝɨ. ɂ ɤɚɤɨɟ ɛɵ ɛɨɥɶɲɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ M ɧɢ ɧɚɡɧɚɱɢɬɶ, ɧɚɫɬɭɩɢɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ x ɨɤɚɠɟɬɫɹ ɦɟɧɶɲɟ M, ɢ ɞɚɥɟɟ ɷɬɨ ɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɧɢɦɢ ɧɟ ɦɟɧɹɟɬɫɹ.
Ɍɨɝɞɚ ɭɬɜɟɪɠɞɟɧɢɟ lim f(x) A ɨɡɧɚɱɚɟɬ ɫɥɟɞɭɸɳɟɟ: ɞɥɹ ɥɸɛɨɝɨ
xo f
H!0 ɧɚɣɞɟɬɫɹ ɬɚɤɨɟ M!0, ɱɬɨ~f(x) A~ H ɩɪɢ x M.
ɉɭɫɬɶ ɢɦɟɸɬɫɹ ɮɭɧɤɰɢɢ f(x) ɢ g(x), ɤɨɬɨɪɵɟ ɢɦɟɸɬ ɤɨɧɟɱɧɵɟ ɩɪɟɞɟɥɵ ɜ ɬɨɱɤɟ x a (a – ɜɟɥɢɱɢɧɚ ɤɨɧɟɱɧɚɹ ɢɥɢ ɛɟɫɤɨɧɟɱɧɚɹ):
lim f(x) A , lim g(x) B .
xoa xoa
ɉɪɢ ɜɵɱɢɫɥɟɧɢɹɯ ɩɪɟɞɟɥɨɜ ɢɫɩɨɥɶɡɭɸɬ ɫɥɟɞɭɸɳɢɟ ɢɯ ɫɜɨɣɫɬɜɚ:
1) |
lim C |
C , C const, 2) |
lim (f(x) r g(x)) |
A rB, |
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lim (f(x)ug(x)) |
A uB , |
4) lim (Cuf(x)) |
CuA , |
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lim ¨ |
f(x) |
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ɢ g(x) z0 ɞɥɹ a Gdxda G. |
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ɉɪɟɞɥɨɠɢɦ ɱɢɬɚɬɟɥɸ ɞɚɬɶ ɫɥɨɜɟɫɧɨɟ ɬɨɥɤɨɜɚɧɢɟ ɤɚɠɞɨɦɭ ɢɡ ɷɬɢɯ ɫɜɨɣɫɬɜ.
ɉɪɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɩɪɢ ɜɵɱɢɫɥɟɧɢɢ ɩɪɟɞɟɥɚ ɮɭɧɤɰɢɢ ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɞɟɥɚɸɬ ɩ ɪ ɹ ɦ ɭ ɸ ɩ ɨ ɞ ɫ ɬ ɚ ɧ ɨ ɜ ɤ ɭ x a ɜ ɮɨɪɦɭɥɭ ɞɥɹ f(x) ɫ ɬɟɦ, ɱɬɨɛɵ ɧɚɣɬɢ A f(a), ɬɨ ɟɫɬɶ
lim f(x) f(a) A.
xoa
ɉɪɢ ɷɬɨɦ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɬɚɤɢɟ ɪɟɡɭɥɶɬɚɬɵ:
53
1) |
ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɤɨɧɟɱɧɨɣ ɬɨɱɤɟ, ɧɚɩɪɢɦɟɪ, lim x2 |
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2) |
ɛɟɫɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɤɨɧɟɱɧɨɣ ɬɨɱɤɟ, ɧɚɩɪɢɦɟɪ, |
lim |
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xo0 x |
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3) |
ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɛɟɫɤɨɧɟɱɧɨɣ ɬɨɱɤɟ, ɧɚɩɪɢɦɟɪ, |
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xof x |
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4) |
ɛɟɫɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɜ ɛɟɫɤɨɧɟɱɧɨɣ ɬɨɱɤɟ, |
ɧɚɩɪɢɦɟɪ, |
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lim x2 |
f . |
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xof |
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Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɨɜ ɨɬ ɨɬɧɨɲɟɧɢɹ ɮɭɧɤɰɢɣ.
ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɮɭɧɤɰɢɹ f(x) – ɪɚɰɢɨɧɚɥɶɧɚɹ ɞɪɨɛɶ
g(x) , ɭ ɤɨɬɨɪɨɣ g(x) ɢ h(x) – ɩɨɥɢɧɨɦɵ ɫɬɟɩɟɧɟɣ x: h(x)
g(x) anuxn an-1uxn-1 |
a0, |
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h(x) |
bmuxm bm-1uxm,- |
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1 b0. |
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ȼɵɱɢɫɥɢɦ ɩɪɟɞɟɥ ɷɬɨɣ ɪɚɰɢɨɧɚɥɶɧɨɣ ɞɪɨɛɢ ɩɪɢ xof: |
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§g(x)· |
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uxn a |
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¢ɜɵɧɟ- |
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bm 1 ux |
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xof©h(x) ¹ |
xof©bm ux |
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ɫɟɦ ɡɚ ɫɤɨɛɤɢ xn ɜ ɱɢɫɥɢɬɟɥɟ ɢ xm ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɜɨɣɫɬɜɚɦɢ ɩɪɟɞɟɥɨɜ²
an |
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u lim xn m . |
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bm |
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bm xof |
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xof© x |
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Ⱥ ɬɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ n ɢ m.
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ɉɪɢ n m ɩɨɥɭɱɢɦ n m 0 ɢ lim xn m |
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xof |
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§ a |
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Ɂɧɚɱɢɬ, ɩɪɢ n m |
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ɉɪɢ n m ɩɨɥɭɱɢɦ n m 0 ɢ lim xn m |
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xof |
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§ a |
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Ɂɧɚɱɢɬ, ɩɪɢ n m |
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ɉɪɢ n!m ɩɨɥɭɱɢɦ n m!0 ɢ lim xn m |
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xof |
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Ɂɧɚɱɢɬ, ɩɪɢ n!m |
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ɇɚɩɪɢɦɟɪ, |
xof©bm ux |
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lim |
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2, n m² |
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xof 2ux2 1 |
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lim |
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2, n m, an 1, bm |
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xof 2ux2 1 |
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lim |
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xof 2ux2 1 |
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Ɉɬɦɟɬɢɦ, ɱɬɨ |
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ɩɪɢ n m ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɷɬɢɦ ɠɟ ɩɪɚɜɢɥɚɦ, ɚ ɩɪɢ n!m ɩɨɥɭɱɢɦ |
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f ɩɪɢ ɱɟɬɧɨɦ (n m), |
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® |
m). |
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¯ f ɩɪɢɧɟɱɟɬɧɨɦ (n |
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ɉɭɫɬɶ f(x) ɢ g(x) ɬɚɤɨɜɵ, ɱɬɨ ɢ |
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lim f(x) |
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lim g(x) |
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xoa |
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xoa |
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ɝɞɚ ɩɪɹɦɚɹ ɩɨɞɫɬɚɧɨɜɤɚ x |
a ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɚ |
lim |
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xoa g(x) |
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ɪɟɡɭɥɶɬɚɬ |
0 |
, ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɸɬ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶɸ ɜɢɞɚ |
0 |
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ɩɨɫɤɨɥɶɤɭ ɢ ɜ ɱɢɫɥɢɬɟɥɟ, ɢ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜɟɥɢɱɢɧɵ ɫɤɨɥɶ ɭɝɨɞɧɨ ɛɥɢɡɤɢɟ ɤ ɧɭɥɸ, ɧɨ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ ɪɚɜɧɵɟ ɧɭɥɸ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɭɠɧɨ ɜɵɩɨɥɧɢɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɪɚɠɟɧɢɣ ɞɥɹ f(x) ɢ g(x), ɢ ɜɵɱɢɫɥɢɬɶ ɩɪɟɞɟɥ ɨɬ ɨɬɧɨɲɟɧɢɹ ɪɟɡɭɥɶɬɚɬɨɜ ɷɬɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɬɨ ɟɫɬɶ
ɪɚɫɤɪɵɬɶ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ 0 . ɉɭɫɬɶ, ɧɚɩɪɢɦɟɪ, f(x) ɢ g(x) – ɩɨɥɢ- 0
ɧɨɦɵ. Ɍɨɝɞɚ ɤɚɠɞɵɣ ɢɡ ɩɨɥɢɧɨɦɨɜ f(x) ɢ g(x) ɫɥɟɞɭɟɬ ɪɚɡɥɨɠɢɬɶ ɧɚ ɦɧɨɠɢɬɟɥɢ. Ɉɞɢɧ ɢɡ ɷɬɢɯ ɦɧɨɠɢɬɟɥɟɣ ɢ ɜ ɪɚɡɥɨɠɟɧɢɢ f(x), ɢ ɜ ɪɚɡɥɨɠɟɧɢɢ
g(x) ɨɛɹɡɚɬɟɥɶɧɨ ɪɚɜɟɧ (x a). ɉɨɷɬɨɦɭ ɱɢɫɥɢɬɟɥɶ ɢ ɡɧɚɦɟɧɚɬɟɥɶ f(x) g(x)
ɫɨɤɪɚɬɢɬɫɹ ɧɚ (x a). Ɋɟɡɭɥɶɬɚɬɨɦ ɛɭɞɟɬ ɧɨɜɚɹ ɞɪɨɛɶ M(x) , ɢ ɨɫɬɚɟɬɫɹ
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J(x) |
ɜɵɱɢɫɥɢɬɶ ɩɪɟɞɟɥ |
§M(x)· |
ɩɪɹɦɨɣ ɩɨɞɫɬɚɧɨɜɤɨɣ. ȼɵɱɢɫɥɢɦ, |
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xoa© |
J(x) ¹ |
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55
ɧɚɩɪɢɦɟɪ: |
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4ux 21 |
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4ux 21 (x 7)u(x 3) |
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ȼ ɬɟɨɪɢɢ ɩɪɟɞɟɥɨɜ ɨɛɹɡɚɬɟɥɶɧɨ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɞɜɚ ɩɪɟɞɟɥɚ, ɧɚɡɵɜɚɟɦɵɯ ɡɚɦɟɱɚɬɟɥɶɧɵɦɢ:
§sin(x)·
ɩɟɪɜɵɣ ɡɚɦɟɱɚɬɟɥɶɧɵɣ ɩɪɟɞɟɥ lim ¨ ¸ 1,
xo0© x ¹
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ȼɧɢɦɚɧɢɟ ɤ ɷɬɢɦ ɩɪɟɞɟɥɚɦ ɨɛɴɹɫɧɹɟɬɫɹ, ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ, ɤɪɚɫɨɬɨɣ ɢ ɢɡɹɳɟɫɬɜɨɦ ɢɯ ɞɨɤɚɡɚɬɟɥɶɫɬɜ (ɤɨɬɨɪɵɟ ɦɵ ɡɞɟɫɶ, ɤ ɫɨɠɚɥɟɧɢɸ, ɜɵɧɭɠɞɟɧɵ ɨɩɭɫɬɢɬɶ), ɚ ɬɚɤɠɟ ɬɟɦ, ɱɬɨ ɫ ɢɯ ɩɨɦɨɳɶɸ ɨɬɵɫɤɢɜɚɸɬɫɹ ɦɧɨɝɢɟ ɞɪɭɝɢɟ ɩɪɟɞɟɥɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɜɬɨɪɨɝɨ ɢɡ ɷɬɢɯ ɡɚɦɟɱɚɬɟɥɶɧɵɯ ɩɪɟɞɟɥɨɜ. ȼɨ ɜɪɟɦɟɧɚ ɪɚɫɰɜɟɬɚ ɜ Ɋɨɫɫɢɢ ɮɢɧɚɧɫɨɜɵɯ ɩɢɪɚɦɢɞ ɤɨɧɰɟɪɧ «Ȼɟɬɬɢ» ɩɪɢɧɢɦɚɥ ɜɤɥɚɞɵ ɧɚ ɬɪɢ ɦɟɫɹɰɚ ɩɨɞ 100% (ɬɨ ɟɫɬɶ ɨɛɟɳɚɥ ɭɞɜɨɢɬɶ ɫɭɦɦɭ ɜɤɥɚɞɚ ɡɚ ɬɪɢ ɦɟɫɹɰɚ). Ⱥ ɩɪɢ ɜɤɥɚɞɟ ɧɚ 1.5 ɦɟɫɹɰɚ ɫɬɚɜɤɚ ɭɦɟɧɶɲɚɥɚɫɶ ɞɨ 50%, ɩɪɢ ɜɤɥɚɞɟ ɧɚ ɬɪɢ ɧɟɞɟɥɢ – ɞɨ 25%, ɬɨ ɟɫɬɶ ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɫɪɨɤɚ ɜɤɥɚɞɚ ɜ n ɪɚɡ ɜɨ ɫɬɨɥɶɤɨ ɠɟ ɪɚɡ ɭɦɟɧɶɲɚɟɬɫɹ ɢ ɫɬɚɜɤɚ. Ɉɬɫɸɞɚ ɫɥɟɞɨɜɚɥɨ, ɱɬɨ ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ ɞɟɧɶɝɢ,
ɫɤɚɠɟɦ, ɧɚ ɱɟɬɜɟɪɬɶ ɫɪɨɤɚ, ɩɨ ɢɫɬɟɱɟɧɢɢ ɤɨɬɨɪɨɝɨ ɫɧɹɬɶ ɭɠɟ |
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ɜɥɨɠɟɧɧɨɝɨ, ɫɧɨɜɚ ɩɨɥɨɠɢɬɶ ɜɫɟ ɞɟɧɶɝɢ ɧɚ ɱɟɬɜɟɪɬɶ ɫɪɨɤɚ ɢ ɬ.ɞ. Ɍɨɝɞɚ
ɡɚ ɬɪɢ ɦɟɫɹɰɚ ɩɟɪɜɨɧɚɱɚɥɶɧɚɹ ɫɭɦɦɚ ɭɜɟɥɢɱɢɬɫɹ ɜ |
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ȿɫɥɢ ɠɟ ɨɩɢɫɚɧɧɭɸ ɩɪɨɰɟɞɭɪɭ ɩɨɜɬɨɪɹɬɶ ɜɞɜɨɟ ɱɚɳɟ, ɬɨ ɫɭɦɦɚ ɩɟɪɜɨ-
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ɧɚɱɚɥɶɧɨɝɨ ɜɤɥɚɞɚɡɚ ɬɪɢɦɟɫɹɰɚ ɭɜɟɥɢɱɢɬɫɹ ɭɠɟ ɜ ¨1 |
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ɇɚɣɞɟɦ ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɩɪɟɞɟɥ ɭɜɟɥɢɱɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɜɤɥɚɞɚ.
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ |
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ɬɨɦ, ɱɬɨ ɩɪɢ ɪɚɡɛɢɟɧɢɢ ɬɪɟɯɦɟɫɹɱɧɨɝɨ ɫɪɨɤɚ ɧɚ n ɱɚɫɬɟɣ ɩɨɥɭɱɢɦ
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¸ . Ⱥɬɟɩɟɪɶɭɫɬɪɟɦɢɦ n ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ: |
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Ʉɚɤ ɜɢɞɢɦ, ɫɥɨɠɧɵɟ ɩɪɨɰɟɧɬɵ ɩɪɢɜɟɥɢ ɧɚɫ ɤ ɱɢɫɥɭ e – ɨɫɧɨɜɚɧɢɸ ɧɚɬɭɪɚɥɶɧɵɯ ɥɨɝɚɪɢɮɦɨɜ.
ȼɨɩɪɨɫɵ ɢ ɡɚɞɚɱɢ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1.Ɉɩɪɟɞɟɥɢɬɶ ɩɨɧɹɬɢɟ «ɮɭɧɤɰɢɹ». Ʉɚɤɢɟ ɛɵɜɚɸɬ ɫɩɨɫɨɛɵ ɡɚɞɚɧɢɹ ɮɭɧɤɰɢɣ?
2.ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɮɭɧɤɰɢɣ:
ɚ) y e-x, xt0; |
ɛ) y ln(x), x!0; |
ɜ) y x2 1, x R. |
3.ɉɨɹɫɧɢɬɶ ɩɨɧɹɬɢɟ «ɨɛɪɚɬɧɚɹ ɮɭɧɤɰɢɹ». Ʉɚɤ ɢɡɨɛɪɚɡɢɬɶ ɩɪɹɦɭɸ ɢ ɨɛɪɚɬɧɭɸ ɮɭɧɤɰɢɢ ɜ ɨɞɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ?
4.ɇɚɣɬɢ ɮɭɧɤɰɢɸ, ɨɛɪɚɬɧɭɸ ɤ y 2 x. ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɩɪɹɦɨɣ
ɢɨɛɪɚɬɧɨɣ ɮɭɧɤɰɢɣ ɜ ɨɞɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ.
5.ɉɨɹɫɧɢɬɶ ɩɨɧɹɬɢɟ «ɫɥɨɠɧɚɹ ɮɭɧɤɰɢɹ».
6.Ɂɚɩɢɫɚɬɶ ɫɭɩɟɪɩɨɡɢɰɢɢ f(g) ɢ g(f) ɞɥɹ ɮɭɧɤɰɢɣ:
ɚ) f(x) 2ux, g(x) x2; |
ɛ) f(x) |
2x, g(x) 2ux. |
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ɉɨɫɬɪɨɢɬɶ ɝɪɚɮɢɤɢ ɤɚɠɞɨɣ ɢɡ ɫɥɨɠɧɵɯ ɮɭɧɤɰɢɣ. |
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7. ȼɵɞɟɥɢɬɶ ɜɥɨɠɟɧɧɭɸ ɢ ɜɧɟɲɧɸɸ ɮɭɧɤɰɢɢ |
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ɚ) h(x) e cos(x) 1 ; |
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8.ɉɨɹɫɧɢɬɶ ɫɦɵɫɥ ɢ ɧɚɡɧɚɱɟɧɢɟ ɩɪɨɰɟɞɭɪɵ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɬɚɛɥɢɱɧɵɯ ɮɭɧɤɰɢɣ.
9.ɉɪɢɜɟɫɬɢ ɮɨɪɦɭɥɵ ɥɢɧɟɣɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɬɚɛɥɢɱɧɨɣ ɮɭɧɤɰɢɢ ɢ ɩɨɹɫɧɢɬɶ ɩɪɨɰɟɞɭɪɭ ɢɯ ɩɪɢɦɟɧɟɧɢɹ.
10.Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɥɢɱɟɫɬɜɚ D ɞɨɪɨɠɧɵɯ ɩɪɨɢɫɲɟɫɬɜɢɣ ɫɨ ɫɦɟɪɬɟɥɶɧɵɦ ɢɫɯɨɞɨɦ ɜ ɝɨɞ ɨɬ ɱɢɫɥɚ v ɚɜɬɨɦɨɛɢɥɟɣ ɧɚ ɨɞɧɭ ɬɵɫɹɱɭ ɧɚɫɟɥɟɧɢɹɨɩɢɫɵɜɚɟɬɫɹɫɨɨɬɧɨɲɟɧɢɟɦɜɢɞɚ
D(v) 3 v . |
(Ⱦɉ) |
ɉɨɫɬɪɨɢɬɶ ɬɚɛɥɢɰɭ ɩɨ ɮɨɪɦɭɥɟ (Ⱦɉ) ɧɚ ɨɬɪɟɡɤɟ [a,b], a 0, b 80, n 4. ȼɵɩɨɥɧɢɬɶ ɥɢɧɟɣɧɭɸ ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɷɬɨɣ ɬɚɛɥɢɰɵ ɢ ɧɚɣɬɢ ɡɧɚɱɟɧɢɹ D ɞɥɹ v1 10 ɢ v2 120 ɩɨ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɦ ɮɨɪɦɭɥɚɦ. Ɉɰɟɧɢɬɶ ɩɨɝɪɟɲɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɚɰɢɢ.
57
11. Ɂɚɞɚɧɚ ɬɚɛɥɢɱɧɚɹ ɮɭɧɤɰɢɹ (ɬɚɛɥ. Ⱥɩɩɪɨ). |
Ɍɚɛɥɢɰɚ Ⱥɩɩɪɨ |
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ȼɵɩɨɥɧɢɬɶ ɥɢɧɟɣɧɭɸ ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɷɬɨɣ |
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ɮɭɧɤɰɢɢ. ɇɚɣɬɢ ɡɧɚɱɟɧɢɹ y ɞɥɹ x1 1 ɢ x2 3. |
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12. Ⱦɚɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɩɪɟɞɟɥɭ ɮɭɧɤɰɢɢ. ɉɨɹɫɧɢɬɶ ɜɵɱɢɫɥɟɧɢɟ ɩɪɟɞɟɥɚ ɩɭɬɟɦ ɩɪɹɦɨɣ ɩɨɞɫɬɚɧɨɜɤɢ. ɉɟɪɟɱɢɫɥɢɬɶ ɜɨɡɦɨɠɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɪɹɦɨɣ ɩɨɞɫɬɚɧɨɜɤɢ.
13.ɉɟɪɟɱɢɫɥɢɬɶ ɫɜɨɣɫɬɜɚ ɩɪɟɞɟɥɨɜ.
14.ɉɪɟɞɟɥ ɨɬ ɨɬɧɨɲɟɧɢɹ ɩɨɥɢɧɨɦɨɜ ɩɪɢ x ɫɬɪɟɦɹɳɟɦɫɹ ɤ f,
ɤf.
15.ɉɨɹɫɧɢɬɶ, ɤɨɝɞɚ ɜɨɡɧɢɤɚɟɬ ɢ ɤɚɤ ɪɚɫɤɪɵɜɚɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧ-
ɧɨɫɬɶ ɜɢɞɚ 0 . 0
16.Ɂɚɩɢɫɚɬɶ ɨɩɪɟɞɟɥɟɧɢɟ ɞɥɹ ɤɚɠɞɨɝɨ ɢɡ ɡɚɦɟɱɚɬɟɥɶɧɵɯ ɩɪɟɞɟ-
ɥɨɜ.
17.ȼɵɱɢɫɥɢɬɶ:
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18. ȼɵɩɨɥɧɢɬɶ ɤɨɧɬɪɨɥɶɧɨɟ ɞɨɦɚɲɧɟɟ ɡɚɞɚɧɢɟ (Ɍɟɫɬ 4. ɎɍɇɄɐɂɂ).
Глава 5. Основы дифференциального исчисления
5.1. Производная
ɉɭɫɬɶ ɞɚɧɚ ɮɭɧɤɰɢɹ y f(x) (ɪɢɫ.5.1). Ɂɚɮɢɤɫɢɪɭɟɦ ɧɟɤɨɬɨɪɭɸ ɬɨɱɤɭ x ɢɡ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ Df ɷɬɨɣ ɮɭɧɤɰɢɢ. Ⱥ ɞɪɭɝɚɹ ɬɨɱɤɚ x 'x ɨɬɫɬɨɢɬ ɨɬ x ɧɚ ɜɟɥɢɱɢɧɭ 'x. ȼɟɥɢɱɢɧɚ 'x ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɪɚɳɟɧɢɟɦ ɚɪɝɭɦɟɧɬɚ, ɚ ɪɚɡɧɨɫɬɶ 'y f(x 'x) f(x) – ɩɪɢɪɚɳɟɧɢɟɦ ɮɭɧɤɰɢɢ.
ɉɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ y f(x) ɜ ɬɨɱɤɟ x – ɷɬɨ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ ɤ ɩɪɢɪɚɳɟɧɢɸ ɚɪɝɭɦɟɧɬɚ, ɤɨɝɞɚ ɩɪɢɪɚɳɟɧɢɟ ɚɪɝɭɦɟɧɬɚ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:
58
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ɉɨɫɤɨɥɶɤɭ y f(x), ɩɪɨɢɡɜɨɞɧɭɸ f’(x) ɨɛɨɡɧɚɱɚɸɬ ɟɳɟ ɢ ɬɚɤ: y’.
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ɉɪɨɰɟɫɫ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ |
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ɨɬ ɮɭɧɤɰɢɢ f(x) ɧɚɡɵɜɚɟɬɫɹ ɟɟ ɞɢɮ- |
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ɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ. |
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Ɋɢɫ.5.1 ɩɨɹɫɧɹɟɬ ɝɟɨɦɟɬɪɢɱɟ- |
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ɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ. Ɉɬɧɨɲɟɧɢɟ |
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ɟɫɬɶ ɬɚɧɝɟɧɫ ɭɝɥɚ E ɧɚɤɥɨɧɚ ɫɟɤɭɳɟɣ |
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AB. ȿɫɥɢ ɬɟɩɟɪɶ ɭɫɬɪɟɦɢɬɶ 'x ɤ ɧɭɥɸ, ɬɨ |
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ɬɨɱɤɚ B ɧɚ ɤɪɢɜɨɣ f(x) ɫɬɚɧɟɬ ɩɪɢɛɥɢ- |
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ɠɚɬɶɫɹ ɤ ɬɨɱɤɟ A, ɭɝɨɥ E ɭɫɬɪɟɦɢɬɫɹ ɤ |
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ɡɧɚɱɟɧɢɸ D. Ⱥ D ɟɫɬɶ ɭɝɨɥ ɧɚɤɥɨɧɚ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ f(x) ɜ ɬɨɱɤɟ A. Ɂɧɚɱɢɬ,
ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ f(x) ɜ ɬɨɱɤɟ x – ɷɬɨ ɬɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ (ɢɥɢ ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ) ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɤɪɢɜɨɣ f(x) ɜ ɬɨɱɤɟ ɫ ɚɛɫɰɢɫɫɨɣ x:
f’(x) tg(D).
ɉɪɨɢɡɜɨɞɧɚɹ ɢɦɟɟɬ ɢ ɮɢɡɢɱɟɫɤɨɟ ɬɨɥɤɨɜɚɧɢɟ, ɦɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ. ɂɫɚɚɤ ɇɶɸɬɨɧ ɧɚɲɟɥ ɨɩɪɟɞɟɥɟɧɢɟ ɞɥɹ ɦɝɧɨɜɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ, ɤɨɬɨɪɨɟ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ, ɧɨ ɧɟ ɪɚɜɧɨɦɟɪɧɨ (ɬɚɤ, ɤɚɤ ɞɜɢɝɚɥɫɹ ɤɚɦɟɧɶ, ɨɬɜɟɫɧɨ ɛɪɨɲɟɧɧɵɣ Ƚɚɥɢɥɟɟɦ ɫ ɜɟɪɲɢɧɵ ɉɢɡɚɧɫɤɨɣ ɛɚɲɧɢ). ɉɭɫɬɶ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɬɟɥɨ ɩɪɨɲɥɨ ɩɭɬɶ s(t), ɚ ɤ ɦɨɦɟɧɬɭ t 't – ɩɭɬɶ s(t 't). Ɍɨɝɞɚ ɫɪɟɞɧɹɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɧɚ ɨɬɪɟɡɤɟ ɜɪɟɦɟɧɢ [t, t 't] ɛɭɞɟɬ ɬɚɤɨɣ:
vɫɪ |
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ȿɫɥɢ ɬɟɩɟɪɶ ɭɫɬɪɟɦɢɬɶ 't ɤ ɧɭɥɸ, ɬɨ ɩɪɟɞɟɥɨɦ ɞɥɹ vɫɪ ɢ ɛɭɞɟɬ ɦɝɧɨɜɟɧɧɚɹ ɫɤɨɪɨɫɬɶ v(t) ɬɟɥɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t:
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Ɂɧɚɱɢɬ, ɫ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ f’(x) – ɷɬɨ ɦɝɧɨɜɟɧɧɚɹ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɮɭɧɤɰɢɢ f(x) ɜɛɥɢɡɢ ɬɨɱɤɢ x.
Ɇɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ ɢɥɥɸɫɬɪɢɪɭɟɬ ɬɨɬ ɜɚɠɧɵɣ ɮɚɤɬ, ɱɬɨ ɞɥɹ ɮɭɧɤɰɢɢ y f(x), ɭ ɤɨɬɨɪɨɣ y ɢ x ɹɜɥɹɸɬɫɹ ɪɚɡɦɟɪɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ, ɪɚɡɦɟɪɧɨɫɬɶ ɩɪɨɢɡɜɨɞɧɨɣ ɟɫɬɶ ɪɚɡɦɟɪɧɨɫɬɶ y, ɞɟɥɟɧɧɚɹ ɧɚ ɪɚɡɦɟɪɧɨɫɬɶ x (ɧɚɩɪɢɦɟɪ, ɪɚɡɦɟɪɧɨɫɬɶ ɫɤɨɪɨɫɬɢ ɟɫɬɶ ɦ/ɫ ɢɥɢ ɤɦ/ɱɚɫ).
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Ⱦɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ x Df ɩɪɨɢɡɜɨɞɧɚɹ f’(x) ɫɚɦɚ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɨɬ x, ɢ ɟɟ ɦɨɠɧɨ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ:
(f’(x))’ f’’(x).
ɉɨɥɭɱɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ f(x) ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ – ɭɫɤɨɪɟɧɢɟ ɮɭɧɤɰɢɢ f(x) ɜɛɥɢɡɢ ɬɨɱɤɢ x. ȼɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɬɨɠɟ ɟɫɬɶ ɮɭɧɤɰɢɹ ɨɬ x. ɉɪɨɞɨɥɠɚɹ ɷɬɨɬ ɩɪɨɰɟɫɫ, ɧɚɣɞɟɦ f(n)(x) – ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ f(x) n-ɝɨ ɩɨɪɹɞɤɚ. ȿɫɥɢ ɜɫɟ n ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ f(x) ɫɭɳɟɫɬɜɭɸɬ, ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ f(x) ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ n ɪɚɡ. ɇɨ ɞɥɹ ɥɸɛɨɝɨ n
f(n)(x) f(n-1)(x) c,
ɬɨ ɟɫɬɶ ɩɪɨɢɡɜɨɞɧɚɹ f(n)(x)ɩɨɪɹɞɤɚ n – ɷɬɨ ɩɟɪɜɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ ɩɪɨɢɡɜɨɞɧɨɣ (n 1)-ɝɨ ɩɨɪɹɞɤɚ f(n-1)(x), ɩɨɷɬɨɦɭ ɞɨɫɬɚɬɨɱɧɨ ɢɡɭɱɢɬɶ ɩɨɞɪɨɛɧɨ ɩɪɨɢɡɜɨɞɧɭɸɩɟɪɜɨɝɨɩɨɪɹɞɤɚ f’(x).
ɋɮɨɪɦɭɥɢɪɭɟɦ ɩɪɚɜɢɥɚ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɢɡɜɨɞɧɨɣ. Ɉɧɢ ɛɚɡɢɪɭɸɬɫɹ ɧɚ ɫɥɟɞɭɸɳɢɯ ɫɜɨɣɫɬɜɚɯ ɩɪɨɢɡɜɨɞɧɵɯ (ɤɨɬɨɪɵɟ ɨɫɧɨɜɚɧɵ ɧɚ ɫɜɨɣɫɬɜɚɯ ɩɪɟɞɟɥɨɜ, ɩɨɫɤɨɥɶɤɭ ɩɪɨɢɡɜɨɞɧɚɹ ɨɩɪɟɞɟɥɟɧɚ ɱɟɪɟɡ ɩɪɟɞɟɥ).
Ɍɚɛɥɢɰɚ 5.1
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1. ȿɫɥɢ C,D const, ɬɨ
(Cuf(x) Dug(x))’ (Cuf(x))’ (Du g(x))’ Cuf’(x) Dug’(x)
(ɩɪɨɢɡɜɨɞɧɚɹ ɫɭɦɦɵ ɟɫɬɶ ɫɭɦɦɚ ɩɪɨɢɡɜɨɞɧɵɯ, ɤɨɧɫɬɚɧɬɭ ɦɨɠɧɨ ɜɵɧɨɫɢɬɶ ɡɚ ɡɧɚɤ ɩɪɨɢɡɜɨɞɧɨɣ).
2. ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ:
(f(x)ug(x))’ f’(x)ug(x) f(x)ug’(x).
3. ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɨɬɧɨɲɟɧɢɹ ɮɭɧɤ-
ɰɢɣ:
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4. ɉɪɨɢɡɜɨɞɧɚɹ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɢ f(s(x)) f(s):
(f(s(x)))’ f’(s)us’(x) fs' us'x
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