:
:
15.
, , .
y=f(x),- .x₀, - - . (1) (1) 3- :
1)-
2)- f(x) lim x
3)lim f(x) .=
y=f(x) (a;b) .o . x , .
, -
2
(1) :
(2)
. - y=f(x), ., . (2),. -
- y=f(x),- (a;b), . .
y=f(x),- [a;b], (a;b) . X=a , .x=b
- - -. x= x₀,-. , 3 ..
16.
=, -, = -, 3- .
-
. - . 1- 2- .
. x₀-. 1 , - y=f(x), . , :
) ₁=₂, x₀ - .
) A₁≠A₂, x₀ - . . - |₂-₁|, -, . 1 .
., x₀ - . 2 - y=f(x) 1 =∞ .
17. -.
1. 2 .
2. - u= . x₀, - y=f(u) . u₀=ϕ(x₀) - y=(ϕ(x))- . x₀
3. - .
18.
.
.
t OM=S. s=s(t). .
.
, . . t( )
L 2 M . . L ., . . . , . , . .
|
|
- y=f(x).
,
,. ., . ,
=
19.
, . .
- y=f() (a;b)
1. ͼ(a;b) , 2. . - y. ∆y=f(x+∆x)-f(x)
3...
4. . ∆->0
, - - f(x) f, f (x),y(x) ..
= f (x)
- y=f(x) . , - , .
-. - y=f(x) .(a;b), (;b)
. .y`(Xo);
v`(t)=
t S t- .
.
K=f`(x)
- .= - .- .?????
.
: - - ,
: - .
20.
.
- u=u(x),v=v(x)
1.(vu)ˈ=(uˈvˈ)
2.(uv)ˈ=uˈv+vˈu (cuˈ)cuˈ -
3.()ˈ=
4.
y=f(u) u=ϕ(x)
yᵪˈ=yˈᵤ Uₓ
5. - -:
y=f(x), x=ϕ(y)
yₓˈ=
1.(c)ˈ=0 c-const
2.()ˈ=
2. (a u)' = a u lna ×u'.
3. (e u)' = e u u'.
4. (log a u)' = u' /(u ln a).
5. (ln u)' = u'/u.
6. (sin u)' = cos u ×u'.
7. (cos u)' = - sin u ×u'.
8. (tg u)' = 1/ cos 2 u ×u'.
9. (ctg u)' = - u' / sin 2 u.
10. (arcsin u)' = u' / .
11. (arccos u)' = - u' / .
12. (arctg u)' = u' /(1 + u 2).
13. (arcctg u)' = - u' /(1 + u 2).
21.
, - - y=f(x), . , - . - - . . . - F(x;y)=0 . - . - .