.. - |Xn|¥ ( xn=1/n, n0, xn¥)
-:
- ... . ... (1/¥=0; 1/0=¥)
- ... ( ) ...
- 2 ..=...
- 2 .. =
- -:
:
1. - y=f(x) [a,b] f(a)*f(b)<0, .. f(a) f(b) , (a,b) =, f(c)=0 ()- -.
2. - y=f(x) [a,b], .
3. - y=f(x) [a,b], min m max M ( ).
:
1. - f(x) g(x) 0, , , ( j(0)¹0) - -, 0
2. - y=f(x) 0, f(x0)>0, 0, f(x)>0
3. y=f(U) U0, U=j(x) U0=j(x0), - y=f[j(x)] 0.
, . .
1. ncp.=DS/Dt, n=lim(DS/Dt), Dt0
2. pcp.=Dm/Dl, pT=lim(Dm/Dl), Dl0
Dy=f(x+Dx)-f(x), y=f(x)
lim(Dy/Dx)=lim((f(x+Dx)-f(x))/Dx)
Dx0 Dx0
- - .
y=f(x+Dx)-f(x), y=f(x). - :
lim(Dy/Dx)=lim((f(x+Dx)-f(x))/Dx)=dy/dx
Dx0 Dx0
: lim(Dy/Dx)=y` Dx0
1) y=x, Dy=Dx, y`=x=lim(Dy/Dx)=1.
2) y=x2, Dy=(x+Dx)2-x2=x2+2xDx+Dx2-x2=Dx(2x-Dx),
(x2)`=lim((Dx(2x+Dx))/Dx)=lim(2x+Dx)=2x
x0 Dx0
.
KN=Dy, MK=Dx
DMNK/tg2=Dy/Dx
:
limtga=lim(Dy/Dx) Dx0
tga0=y`
aa0
Dx0 MN M(tga0=y`, aa0)
, tg , x0.
.
: f(x) g(x) . , :
. :
y(x)=f(u(x)) f(u) u(x), y(x)=f(u(x))u(x).
. .
:
-:
- = - .
y`=f(x)*U`, yx`=yU`*Ux`, dy/dx=dy/dU=dU/dx
:
-.
y=f(x), x=j(y) - -.
- , = 0, - = -, .. xy`=1/yx`.
Dy/Dx=1/(Dy/Dx) - , , = :
|
|
lim(Dy/Dx)=1/(lim(Dy/Dx), .. yx`=1/xy f`(x)=1/j`(x)
: