Математика для менеджеров. Часть I. Учебное пособие

ɂɫɩɨɥɶɡɭɹ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɮɭɧɤɰɢɣ, ɢɦɟɟɦ sin (x) ~ (x), ɚ ɬɚɤɠɟ

tg (x) ~ (x), arctg (x) ~ (x) .

ɉɪɢɦɟɪ ɇɚɣɬɢ ɩɪɟɞɟɥ lim tg5x x 0 sin7x

Ɍɚɤ ɤɚɤ tg5x ~ 5x ɢ sin7x ~ 7x ɩɪɢ ɯ ɬɨ ɡɚɦɟɧɢɜ ɮɭɧɤɰɢɢ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɦɢ ɩɨɥɭɱɢɦ

ȼɬɨɪɨɣ ɡɚɦɟɱɚɬɟɥɶɧɵɣ ɩɪɟɞɟɥ

Ɉɧ ɢɦɟɟɬ ɜɢɞ lim1 1x x lim 1 1 e ,

x 0

ɝɞɟ ɟ – ɢɪɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɧɨɟɅɨɝɚɪɢɮɦɵ ɫ ɨɫɧɨɜɚɧɢɟɦ ɟ ɧɚɡɵɜɚɸɬɫɹ ɧɚɬɭɪɚɥɶ ɧɵɦɢ ɢ ɨɛɨɡɧɚɱɚɸɬɫɹ logex=ln x ɋ ɩɨɦɨɳɶɸ ɷɬɨɝɨ ɩɪɟɞɟɥɚ

ɪɚɫɤɪɵɜɚɸɬ ɬɚɤ ɠɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɜɢɞɚ ^ }.

x 3 x 3

ɉɪɢɦɟɪ. ɇɚɣɬɢ ɩɪɟɞɟɥ lim . x x 1

81

Ɋɟɲɟɧɢɟ

ɢɥɢ

ln(1 (x)) ~ (x), a (x) 1~ (x)lna,(1 (x))m 1~ m (x).

82

ɝɞɟP(x) a0xn a1xn 1 ... an , Q(x) b0xm b1xm 1 ... bm .

6.8 ɇɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ

Ɏɭɧɤɰɢɹ ɭ f(x) ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɜ ɬɨɱɤɟ ɯ0,

ɟɫɥɢ

lim f(x) f(x0 ) .

x x0

ɂɡ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɨɫɧɨɜɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɢ ɨɫ ɧɨɜɧɵɯ ɬɟɨɪɟɦ ɨ ɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤɰɢɹɯ ɫɥɟɞɭɟɬ ɱɬɨ ɥɸɛɚɹ ɷɥɟɦɟɧɬɚɪɧɚɹ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ ɜɨ ɜɫɹɤɨɣ ɬɨɱɤɟ ɜ ɤɨɬɨɪɨɣ ɨɧɚ ɨɩɪɟɞɟɥɟɧɚ ɩɪɢ ɷɬɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɤɨɧɟɱɧɨ ɱɬɨ ɮɭɧɤɰɢɹ ɨɩɪɟɞɟɥɟɧɚ ɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ

Ɏɭɧɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɧɚ ɦɧɨɠɟɫɬɜɟX ɟɫɥɢ ɨɧɚ ɧɟɩɪɟɪɵɜɧɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɷɬɨɝɨ ɦɧɨɠɟɫɬɜɚ

ɉɪɢɦɟɪɵ ɋɥɟɞɭɸɳɢɟ ɮɭɧɤɰɢɢ ɧɟɩɪɟɪɵɜɧɵ ɧɚ ɭɤɚɡɚɧɧɵɯ ɦɧɨɠɟɫɬɜɚɯ ɚ ɮɭɧɤɰɢɹ y 2x3 7x 1 ɧɟɩɪɟɪɵɜɧɚ ɧɚ 5

Ɍɨɱɤɢ ɜ ɤɨɬɨɪɵɯ ɧɚɪɭɲɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɫɬɶ ɮɭɧɤɰɢɢ, ɧɚɡɵɜɚɸɬɫɹ ɬɨɱɤɚɦɢ ɪɚɡɪɵɜɚ ɷɬɨɣ ɮɭɧɤɰɢɢ

83

ȼɫɟ ɬɨɱɤɢ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ ɪɚɡɞɟɥɹɸɬɫɹ ɧɚ ɬɨɱɤɢ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɢ ɬɨɱɤɢ ɭɫɬɪɚɧɢɦɨɝɨ ɪɚɡɪɵɜɚ

Ɍɨɱɤɚ ɪɚɡɪɵɜɚ x0 ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɨɣ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɮɭɧɤɰɢɢ y f x ɟɫɥɢ ɜ ɷɬɨɣ ɬɨɱɤɟ ɫɭɳɟɫɬɜɭɸɬ ɤɨɧɟɱɧɵɟ

Ɋɟɲɟɧɢɟ Ɉɱɟɜɢɞɧɨ ɱɬɨ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ ɧɚ ɤɚɠɞɨɦ ɢɡ ɬɪɟɯ ɢɧɬɟɪɜɚɥɨɜ x 1, 1 x ɢ x 3. Ɍɨɱɤɢx ɢ x = 3 ɹɜɥɹɸɬɫɹ ɩɨɞɨɡɪɢɬɟɥɶɧɵɦɢ ɧɚ ɧɚɥɢɱɢɟ ɪɚɡɪɵɜɚ ɇɚɣɞɟɦ ɨɞɧɨ ɫɬɨɪɨɧɧɢɟ ɩɪɟɞɟɥɵ

Ɂɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x ɪɚɜɧɨ y( 1 )= 12+ 1=2. ɉɪɟɞɟɥ ɫɥɟɜɚ ɪɚɜɟɧ ɩɪɟɞɟɥɭ ɫɩɪɚɜɚ ɢ ɪɚɜɟɧ ɡɧɚɱɟɧɢɸ

ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜ ɬɨɱɤɟ x ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ

ɂɫɫɥɟɞɭɟɦ ɩɨɜɟɞɟɧɢɟ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ x = 3:

84

ɉɪɟɞɟɥ ɫɥɟɜɚ ɧɟ ɪɚɜɟɧ ɩɪɟɞɟɥɭ ɫɩɪɚɜɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜ ɬɨɱɤɟ xɮɭɧɤɰɢɹ ɬɟɪɩɢɬ ɪɚɡɪɵɜ ɩɟɪɜɨɝɨ ɪɨɞɚ ɬɢɩɚ ©ɫɤɚɱɨɤª Ƚɪɚ ɮɢɤ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɟɧ ɧɚ ɪɢɫɭɧɤɟ

6

4

2

Ɂɚɞɚɱɚ ɨ ɧɟɩɪɟɪɵɜɧɨɦ ɧɚɱɢɫɥɟɧɢɢ ɩɪɨɰɟɧɬɨɜ

ɉɪɢɜɟɞɟɦ ɩɪɢɦɟɪ ɷɤɨɧɨɦɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɜ ɤɨɬɨɪɨɣ ɜɨɡ ɧɢɤɚɟɬ ɱɢɫɥɨ e ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɜ ɛɚɧɤ ɩɨɦɟɳɟɧɚ ɫɭɦɦɚ B0 ɩɨɞ p % ɝɨɞɨɜɵɯ Ɍɨɝɞɚ ɱɟɪɟɡ ɝɨɞ ɫɭɦɦɚ ɜɤɥɚɞɚ ɫɨɫɬɚɜɢɬ

B1 B0 p 100B0 B0 1 ,

ɝɞɟ ɜɜɟɞɟɧɨ ɨɛɨɡɧɚɱɟɧɢɟ 100p .

ɉɪɟɞɩɨɥɨɠɢɦ ɱɬɨ ɜɤɥɚɞ ɦɨɠɧɨ ɫɧɹɬɶ ɩɨ ɢɫɬɟɱɟɧɢɢ ɥɸɛɨ ɝɨ ɫɪɨɤɚ ɜ ɬɟɱɟɧɢɟ ɝɨɞɚ ɢ ɧɚɱɢɫɥɟɧɢɟ ɧɚ ɜɤɥɚɞ ɩɪɨɩɨɪɰɢɨɧɚɥɶ

ɧɨ ɷɬɨɦɭ ɫɪɨɤɭ ɬ ɟ ɡɚ ɩɨɥɝɨɞɚ ɛɭɞɟɬ ɧɚɱɢɫɥɟɧɨ 2p % ɡɚ ɦɟɫɹɰ -

ɱɢɬɶ ɞɨɯɨɞ ɛɨɥɶɲɢɣ ɱɟɦ B1 ɞɟɣɫɬɜɭɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ȿɫɥɢ ɧɚɩɪɢɦɟɪ ɜ ɫɟɪɟɞɢɧɟ ɝɨɞɚ ɡɚɤɪɵɬɶ ɫɱɟɬ ɢ ɩɨɥɭɱɟɧɧɭɸ

ɬɨ ɜ ɤɨɧɰɟ ɝɨɞɚ ɫɭɦɦɚ ɜɤɥɚɞɚ ɫɨɫɬɚɜɢɬ

85

ȿɫɥɢ ɩɨɜɬɨɪɹɬɶ ɨɩɟɪɚɰɢɸ ɡɚɤɪɵɬɢɹ-ɨɬɤɪɵɬɢɹ ɫɱɟɬɚ ɱɚɳɟ ɧɚɩɪɢɦɟɪ ɤɚɠɞɵɣ ɦɟɫɹɰ ɬɨ ɤ ɤɨɧɰɭ ɝɨɞɚ ɛɭɞɟɦ ɢɦɟɬɶ

ɥɢ ɩɪɟɞɩɨɥɨɠɢɬɶ ɱɬɨ ɨɩɟɪɚɰɢɹ ɡɚɤɪɵɬɢɹ-ɨɬɤɪɵɬɢɹ ɫɱɟɬɚ ɩɪɨɢɡ ɜɨɞɢɬɫɹ n 1 ɪɚɡ ɜ ɝɨɞɭ ɱɟɪɟɡ ɪɚɜɧɵɟ ɩɪɨɦɟɠɭɬɤɢ ɜɪɟɦɟɧɢ ɬɨ ɜ

ɫɬɚɜɢɬɶ ɱɬɨ ɩɪɨɰɟɧɬɵ ɧɚɱɢɫɥɹɸɬɫɹ ɧɟɩɪɟɪɵɜɧɨ ɱɢɫɥɨ ɨɩɟɪɚ ɰɢɣ ɡɚɤɪɵɬɢɹ-ɨɬɤɪɵɬɢɹ ɫɱɟɬɚ ɧɟɨɝɪɚɧɢɱɟɧɧɨ ɪɚɫɬɟɬ ɬɨ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɚɤɫɢɦɚɥɶɧɨɟ ɱɢɫɥɨ ɩɪɨɰɟɧɬɨɜ ɧɚ ɤɨɬɨ ɪɨɟ ɝɢɩɨɬɟɬɢɱɟɫɤɢ ɦɨɠɟɬ ɭɜɟɥɢɱɢɬɶɫɹ ɜɤɥɚɞ ɩɪɢ ɞɚɧɧɨɣ ɫɯɟɦɟ

ɧɚɥɶɧɨɣ ɫɬɚɜɤɟ p 100)ɦɚɤɫɢɦɚɥɶɧɚɹ ɷɮɮɟɤɬɢɜɧɚɹ ɫɬɚɜɤɚ ɫɨɫɬɚɜɢɬ e 1 100% 171%.

86

ɌȿɆȺ 7 ɉɊɈɂɁȼɈȾɇȺə ɂ ȾɂɎɎȿɊȿɇɐɂȺɅ ɎɍɇɄɐɂɂ ɈȾɇɈɃ ɉȿɊȿɆȿɇɇɈɃ

Ɉɩɪɟɞɟɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɨɣɢ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɮɭɧɤɰɢɢ

ɂɦɟɟɦ y/ lim y x ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɩɪɟɞɟɥɚ ɮɭɧɤ

x 0

ɰɢɢ y x y/ ɝɞɟ 0ɩɪɢ x 0.

Ɉɬɫɸɞɚ ¨y=y ā¨x Įā¨x.

ɉɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ¨x ɢ ɩɪɢy 0 ɢɦɟɟɦ y y x .

Ƚɥɚɜɧɚɹ ɱɚɫɬɶ y¨x ɩɪɢɪɚɳɟɧɢɹ ¨y ɮɭɧɤɰɢɢ ɥɢɧɟɣɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ¨x ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɨɦ ɮɭɧɤɰɢɢ ɢ ɨɛɨ ɡɧɚɱɚɟɬɫɹ dy=y ¨x.

ɉɨɥɨɠɢɜ ɭ ɯ ɩɨɥɭɱɢɦ dx=(x ¨x ā¨x ¨x ɢ ɩɨɷɬɨɦɭ dy=y'dx.

ɗɬɚ ɮɨɪɦɭɥɚ ɜɟɪɧɚ ɢ ɜ ɬɨɦ ɫɥɭɱɚɟ ɟɫɥɢ ɯ ɟɫɬɶ ɮɭɧɤɰɢɹ ɧɨɜɨɣ ɩɟɪɟɦɟɧɧɨɣ t.

Ɍɟɨɪɟɦɚ ɨ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɧɟɩɪɟɪɵɜɧɨɫɬɶɸ ɢ ɞɢɮɮɟɪɟɧ ɰɢɪɭɟɦɨɫɬɶɸ ɮɭɧɤɰɢɢ ȿɫɥɢ ɮɭɧɤɰɢɹݕ = (ݔ) ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɜ ɬɨɱɤɟ ݔ , ɬɨ ɨɧɚ ɜ ɷɬɨɣ ɬɨɱɤɟ ɧɟɩɪɟɪɵɜɧɚ Ɉɛɪɚɬɧɚɹ ɬɟɨɪɟɦɚ ɜɨ ɨɛɳɟ ɝɨɜɨɪɹ ɧɟɜɟɪɧɚ ɬɟ ɟɫɥɢ ɮɭɧɤɰɢɹ ɧɟɩɪɟɪɵɜɧɚ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ ɬɨɨɧɚɧɟɨɛɹɡɚɬɟɥɶɧɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɜɷɬɨɣɬɨɱɤɟ

Ɍɚɤ ɧɚɩɪɢɦɟɪ ɮɭɧɤɰɢɹ ݕ = |ݔ| ɧɟɩɪɟɪɵɜɧɚ ɜ ɬɨɱɤɟ ݔ = 0 ɧɨ ɧɟ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɜɷɬɨɣɬɨɱɤɟ

87

7.2. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ

ݕ = (ݔ )+ (ݔ )(ݔ ݔ ) – ɨɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ

ݔ = 1, (ݔ ) = 1 = 1,(ݔ) = 2ݔ,

(ݔ ) = (1) = 2·1 = 2.

ݕ= 1 +2(ݔ 1) = 2ݔ 1 – ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɤɚɫɚɬɟɥɶɧɨɣ

ݕ= (ݔ ) ( ) (ݔ ݔ ) – ɨɛɳɟɟ ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɢ

ݕ= 1 (ݔ 1) = ݔ + – ɭɪɚɜɧɟɧɢɟ ɧɨɪɦɚɥɢ

ݐ = (1) = 2, = ݐ2, 63° – ɭɝɨɥ ɧɚɤɥɨɧɚ ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɨɫɢ Ɉx.

7.3. Ɇɟɯɚɧɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ

ɉɪɨɢɡɜɨɞɧɚɹ ɩɭɬɢ ɩɨ ɜɪɟɦɟɧɢ (t ) ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɜ ɦɨɦɟɧɬ

t ɬɨɟɫɬɶ (ݐ ) = (ݐ ) Ɂɚɞɚɱɚ Ɍɟɥɨɞɜɢɠɟɬɫɹɩɪɹɦɨɥɢɧɟɣɧɨɩɨɡɚɤɨɧɭ (ݐ) = ݐ

2ݐ ݐ Ɉɩɪɟɞɟɥɢɬɶɫɤɨɪɨɫɬɶɢɭɫɤɨɪɟɧɢɟɬɟɥɚɩɪɢݐ = 2. Ɋɟɲɟɧɢɟ ɋɨɝɥɚɫɧɨ ɦɟɯɚɧɢɱɟɫɤɨɦɭ ɫɦɵɫɥɭ ɩɪɨɢɡɜɨɞɧɨɣ

(ݐ ) = (ݐ ). ɂɦɟɟɦ (ݐ) = 3ݐ 4ݐ 1

88

ȿɫɥɢ ݐ = ݐ = 2 ɬɨ (2) = 3,֜ (2) = 3(ˏΤ˔)(ݐ ) = (ݐ )

(ݐ) = ( (ݐ)) = (3ݐ 4ݐ 1) = 6ݐ 4(2) = 6 ·2 4 = 8,֜ (2) = 8(ˏΤc ).

7.4. ɗɤɨɧɨɦɢɱɟɫɤɢɣ ɫɦɵɫɥ ɩɪɨɢɡɜɨɞɧɨɣ

ɉɭɫɬɶ ݕ – ɢɡɞɟɪɠɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ ɚ ݔ - ɤɨɥɢɱɟɫɬɜɨ ɜɵɩɭɫ ɤɚɟɦɨɣ ɩɪɨɞɭɤɰɢɢ

Ɍɨɝɞɚ ݔ – ɩɪɢɪɨɫɬ ɩɪɨɞɭɤɰɢɢݕ – ɩɪɢɪɚɳɟɧɢɟ ɢɡɞɟɪɠɟɤ ɩɪɨɢɡɜɨɞɫɬɜɚ

– ɫɪɟɞɧɟɟ ɩɪɢɪɚɳɟɧɢɟ ɢɡɞɟɪɠɟɤ ɩɪɨɢɡɜɨɞɫɬɜɚ ɧɚ ɟɞɢɧɢ ɰɭ ɩɪɨɞɭɤɰɢɢ

ɞɟɪɠɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɪɢɛɥɢɠɟɧɧɨ ɞɨɩɨɥɧɢ ɬɟɥɶɧɵɟ ɡɚɬɪɚɬɵ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɨ ɟɞɢɧɢɰɵ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɩɪɨɞɭɤɰɢɢ

Ɂɚɞɚɱɚ Ɏɭɧɤɰɢɹ ɢɡɞɟɪɠɟɤ ɩɪɨɢɡɜɨɞɫɬɜɚ ɧɟɤɨɬɨɪɨɣ ɮɢɪ ɦɵ ɢɦɟɟɬ ɜɢɞ ݕ(ݔ) = 0,1ݔ 1,2ݔ + 5ݔ + 250 ɞɟɧ ɟɞ ɇɚɣɬɢ ɩɪɟɞɟɥɶɧɵɟ ɢ ɫɪɟɞɧɢɟ ɢɡɞɟɪɠɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ ɢ ɜɵɱɢɫɥɢɬɶ ɢɯ ɡɧɚɱɟɧɢɟ ɩɪɢ ݔ = 10.

Ɋɟɲɟɧɢɟ

ݕ(ݔ) = 0,3ݔ 2,4ݔ +5 ɟɞ ɦɟɫ ɩɪɟɞɟɥɶɧɵɟ ɢɡɞɟɪɠɤɢ

ݕ(10) = 11

ɋɪɟɞɧɢɟ ɢɡɞɟɪɠɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ ɪɚɜɧɵ

ȼɵɜɨɞ ɉɪɢ ɞɚɧɧɨɦ ɭɪɨɜɧɟ ɩɪɨɢɡɜɨɞɫɬɜɚ ɤɨɥɢɱɟɫɬɜɟ ɜɵ ɩɭɫɤɚɟɦɨɣ ɩɪɨɞɭɤɰɢɢ ɫɪɟɞɧɢɟ ɡɚɬɪɚɬɵ ɧɚ ɩɪɨɢɡɜɨɞɫɬɜɨ ɨɞɧɨɣ ɟɞɢɧɢɰɵ ɩɪɨɞɭɤɰɢɢ ɫɨɫɬɚɜɥɹɸɬ ɞɟɧ ɟɞ ɚ ɭɜɟɥɢɱɟɧɢɟ ɨɛɴɟ ɦɚ ɧɚ ɨɞɧɭ ɟɞɢɧɢɰɭ ɩɪɨɞɭɤɰɢɢ ɨɛɨɣɞɟɬɫɹ ɮɢɪɦɟ ɩɪɢɛɥɢɠɟɧɧɨ ɜɞɟɧ ɟɞ

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7.5. Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɢ ɬɚɛɥɢɰɚ ɩɪɨɢɡɜɨɞɧɵɯ

Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ

ɉɭɫɬɶ ɋ – ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ ɱɢɫɥɨ U=U(x ɢ ȣ=ȣ(x) – ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɟ ɮɭɧɤɰɢɢ

ɋ '=0; ɋÂȣ '=ɋȣ';

3.(U ȣ)'=U ȣ 4. (Uāȣ)'=U ȣ Uȣ ;

5.U U 2 U .

Ɍɚɛɥɢɰɚ ɩɪɨɢɡɜɨɞɧɵɯ

3.loga U U 1lna U .ɋɥɟɞɫɬɜɢɟ lnU 1U U .

4.sinU cosU U .5. cosU sinU U .

Ɋɟɲɟɧɢɟ ɉɪɢɦɟɧɢɦ ɮɨɪɦɭɥɭ Un n Un 1 U ,

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