2.
1.1
f 0 , 0 , , 0 . f 0 . , , .
. (.. ), f 0.
f 0 .
,
,
.
. . S(t) , t= 0 t > 0. , t0 t, 0 < < t0 < t, S(t) - S(t0). t0 t, - t0 . , t0 . , y = f(x), =0.
-0 0 , h: - 0 = = h. f(x) f(x0) f 0 , (h). = =0 + = 0 + h,
= ,. = ,
:
= , == .
, .
.
1. 0, 0 , : , 0 = = 0; = 0, .. = 0.
2. f(x) = a , a> 0, a ≠ 1. , 0 - - . : (h) = f(x0+h) f(x0) = a - a = = a (a - 1). , = = a = a lna.
3. f(x) = cosx, a 0 - . (h) = cos(x0+h) - cosx0 = -2 sin(x0+ sin . :
= = - sin(x0+ = - sinx0 .
4. f(x) = sinx, a 0 - . (h) = sin(x0+h) - sinx0 = 2 cos(x0+ sin . :
= = cos(x0+ = cosx0 .
5. f(x) = , μ , 0 > 0. : (h) = f(x0+h) f(x0) = (0+h) - 0 = 0 . :
= = 0 = μ 0 .
. μ , f(x) = x (, μ ), , , x0 < 0: = μ 0 .
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.
6. f(x) = . , 5 x0 ≠ 0 = 0 . x0 = 0. :
(h) = f( 0 +h) f( 0 ) = ( 0 +h) - 0 =h . C, = = . , , .
7. f(x) = | | = x0 = 0: (h) = f( 0 +h) f( 0 ) = |0+ h | - |0| = | h |. :
= = 1; = = -1. , , .. .
1.2. ,
. 0, 0 ,
1)
2) , (h)
0 :
(h) = h + o(h). (1)
(h) = f(x0+h) f(x0), (1) :
f(x0+h) = f(x0) + h + o(h) (2) ; h = x x0. x = x0 + +h (2) :
f(x) = f(x0) + (x x0) + o(x x0) = x+B+ o(x x0), B = f(x0) - x0. , f f(x) = x+B+ o(x x0), . : x0 f(x) l(x) = Ax+B. - , f(x), , l(x) = Ax+B.
1. ( ) , f 0, 0 , , .
► . (1).
= =
, .
. = . h, , .. α(h):
h (h) = h + h α (h). = , α(h)→ 0 h → 0. : , .. h α (h) = o(h). , (h) (1), = , 0. ◄
1. 0, (1) , , = .
.
2. , 0, h,
(h) = h + h α (h), (3) α (h) , : α(h)→ 0 h → 0 α(0)= 0.
(3) .
, 2,3 4 0 ; 1 . f(x) = , μ , 0 , 0 ≠0, - (. 5 ). f(x) = f(x) = | | 0 = 0, , (. 5 7).
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2. ( )
, .
► f 0. (1), : (h) → 0 h → 0. (. 1, . 5.1) f 0 . ◄
. 2 . : , , . , f(x) = | | 0 = 0, (. 7).
1.3. ,
3. ( ) f g 0. :
1. F =f+g a 0 ,
2. F =f g a 0 ,
3. , F = a 0 ,
.
► f g 0, (. (2)):
f(x0+h) = f(x0) + h + o(h);
g(x0+h) = g(x0) + h + o(h). (4)
1. F =f+g. (4), :
= h + o(h), = . , F 0, =
2. F =f g. (4), :
= h + o(h), = + , F 0,
3. , F = . g , 0. (.1. .5.1) , . , F . :
= , , (h) = h + o(h). :
(h) = (4), :
:
(h) = .
: α(h) = . , α(h)→ 0 h → 0 α(h). :
(h) = (1 + α(h)) h + o(h) = Ah + hα (h) + o(h)). α(h) h = o(h) o(h)+ o(h)= o(h), (h) = Ah + o(h).
, F = . ◄
8. f(x)= sinx, g(x) = cosx, F(x) = 3) , 0, 0 ≠ , n , :
= F(x) = 0, 0 ≠ , n , : =
4. ( ) f 0, g 0, 0 = f(x0). 0, . .
► f g , 0 0 . (. 1, .5.2) 0.
h , 0+ h . :
. f(x0+h) = y0 + ; .. .
(3) , α(Δ) , α(Δ) → 0 Δ → 0 α (0) = 0. f , h + o(h). , :
=
= = , . , . :
. : , . , - , . α(Δ) → 0 Δ → 0 (. ), , α(Δ) → 0 . , ; , , .. .
, = h + o(h), = , , . ◄
5. ( ) f , . f 0, , 0, 0 = f(0), .
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► (. 1, .5.5) (, d), 0 = f(0) . , 0 + (, d). : . 0 = f(0), , g(y0) = = x0. , g(y0+ ) = g(y0) + = x0 + . g(y0+ ) = = x0 + : y0+ =f(x0 + ), = f(x0 + ) 0 = f(x0 + ) -- f(0). , . ≠ 0, g , , :
. . , , (. ). , . , = , .. . ◄
9. f(x) = a , a> 0, a ≠ 1. , (. 2) - . - y0 = a , . 0 , 0 - .
10. f(x) = sinx. , cos . , g(y) = arcsiny y0 = = sinx0, a . - , y0 - (-1,1).
11. f(x) = tgx. , . , g(y) = arctgy y0 = tgx0, a . - , y0 - .
1.4.
f () = (α, β), α < < β, . = f(x0 +h) f(x0) h, h, + h (α, β), .. h = (α - , β - ); h Δ . (1) , , , 0, , h (: = ) (. 1, .). = 0, .
. f h, h , 0: .
df df(h):
df(h) h. (1) : h = (α - , β - )
= df(h) + (h) (5)
, df(h) - . , h df(h) = 0, .. . h, (. (5)); , df(h). , , .
12. . f(x)= = , 0 = 8, h = 0,12. = f(x0+h), f(x0) = 2. : (h) = f(x0+h) f(x0); f(x0+h) = = f(x0) + (h) ≈ f(x0) + df(h). , h = 0,12, : ≈ 2 + + df (0,12). df(h) 0 = 8 h = 0,12. : = , = , , df(h)= h = h =. 0,12 : ≈ 2 + df( 0,12 ) = 2 + 0,12 = 2 + 0,01 = 2,01.
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