y = f(x) P(x1, y1)
y = y1 + f'(x1)(x - x1),
f'(x1) df/dx x = x1 (. 6.1).
P , . , g(x, y) = 0. :
gx(x1, y1)(x - x1) + gy(x1, y1)(y - y1) = 0,
gx(x1, y1) gy(x1, y1) g/ x g/ y P.
. x2 + y2 - 1 = 0 (1, 0) .
g(x, y) = x2 + y2 - 1 = 0, gx = 2x gy = 2y, , gx(1, 0) = 2 gy(1, 0) = 0. , 2(x - 1) + 0(y - 0) = 0, x = 1. , , .
, P, :
y = y1 - (x - x1)/f '(x1).
, P. :
gy(x1, y1)(x - x1) - gx(x1, y1)(y - y1) = 0.
, .
2. | |||||
, = f () ∆ ∆ ∆ 0: , . | |||||
1. . | |||||
f(x) = C = const. : 1: 0: | |||||
2. . | |||||
2: u = u(x) v = v(x) . u΄(x) v΄(x). | |||||
w(x) = u(x) + v(x). Ÿ : ∆w(x) = w(x+∆x) w(x) = [u(x+∆x) + v(x+∆x)] [u(x) + v(x)] = =[u(x+∆x) u(x)] + [v(x+∆x) v(x)] = ∆u(x) + ∆v(x). | |||||
.. : ∆w = ∆u + ∆v. ∆ ≠ 0: . , Δ, ∆ : | |||||
w(x) = u(x) v(x). Ÿ : ∆w(x) = w(x+∆x) w(x) = [u(x+∆x) v(x+∆x)] [u(x) v(x)] = =[u(x) + ∆u(x)]∙[v(x) + ∆v(x)] [u(x) ∙ v(x). : ∆u(x) = u(x+∆x) u(x) => u(x+∆x) = u(x) + ∆u(x). : ∆w = (u+∆u)(v+∆v) uv = uv +u∆v + v∆u+∆u∆v uv => ∆w = u∆v + v∆u +∆u∆v. ∆ ∆ → 0: : , , .. , : , . .. | |||||
: , . : w΄ = u΄ + v΄, . .. | |||||
:
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3. . | ||||
3: u = u(x) v = v(x) . | ||||
: | ||||
w(x) = u(x) v(x). Ÿ : ∆w(x) = w(x+∆x) w(x) = [u(x+∆x) v(x+∆x)] [u(x) v(x)] = =[u(x) + ∆u(x)]∙[v(x) + ∆v(x)] [u(x) ∙ v(x). : ∆u(x) = u(x+∆x) u(x) => u(x+∆x) = u(x) + ∆u(x). : ∆w = (u+∆u)(v+∆v) uv = uv +u∆v + v∆u+∆u∆v uv => ∆w = u∆v + v∆u +∆u∆v. ∆ ∆ → 0: : , , .. , : , . .. | ||||
1: : 2: | ||||
4. . | ||||
4: u = u(x) v =v(x ≠0 . | ||||
u΄(x) v΄(x). | ||||
: | ||||
. | ||||
Ÿ : : ∆u(x) = u(x+∆x) u(x) => u(x+∆x) = u(x) + ∆u(x). : ∆ ∆ → 0: , , , .. | ||||
, : | ||||
5. . 5: , ( ). = f(u), a u = g(x) . f΄(u) g΄(x). : (, u(x) Δx → 0 Δu → 0). : 6. | ||||||||||
: | ||||||||||
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x y = f (x). x y M 0(x, y). x D x, x + D x y+ D y = f (x + D x). M 1(x + D x, y + D y). M 0 M 1 j, Ox, , .
D x , M 1 , M 0, j D x. Dx 0 j a , M 0 a, . :
.
, f ´(x) = tga
.. f ´(x) x , f (x) M 0(x, y) Ox.