25. 26.
α() ~ β() , α() - β() .
α() - β() α() β() α() ~ β().
α() β() , .
α() β() γ(), .
→0 α() , β1(), β2(),, βn(), γ()= α()+ β1()+β2()++βn() α().
27. y=f(x) 0 . y=f(x) 0, :
:
- 0 ;
- →0
- 0 , .. .
, f(x) , f(x) 0
29. .
0 1 y=f(x), ( )
, :
- 1=2 0 ;
- 1≠2 0 .
|A1 A2| .
0 2 y=f(x), ( ) , .
30. . 1. , . : , . 2. y=f(x) [a, b] f(a)=A, f(b)=B, . : y=f(x) [a, b] , [a, b] , 0.
31. y=f(x) 0 . , 0.
y=f(x) - , y . .
|
|
f(x) x y=f(x) , . . .
32. , .
33. : f(x) g(x) . , :
. :
y(x)=f(u(x)) f(u) u(x), y(x)=f(u(x))u(x).
. .
:
34. y=xn,nN
y=ax, a>0, a≠1
y=logax, a>0, a≠1
y=cosx, y=sinx, y=tgx, y=ctgx (cosx)=-sinx (tgx)= (ctgx)=
y=arccosx, y=arcsinx, y=arctgx, y=arcctgx:(arccosx)= (arcsinx)= (arctgx)= (arcctgx)= -
35. . . .
36. .
37. - , F(x,y)=0, .
y=f(x), y=x2-1 -
F(x,y)=0, a2=x2+y2 - -.
1)a2=x2+y2 - , , - - .
y`=2x+2y=0, .. -
y*y`=-x, y`=-x/y
2) x3-3xy+y3=0
3x3-3(xy)`+3y2*y`=0 //:3
x2-(x`y+y`x)+y2*y`=0
y`y2-xy`=y-x2
y`=(y-x2)/(y2-x)
38. {x=ϕ(t), y=ᴪ(t), tT}
ϕ'(t), ᴪ(t) .! ϕ'(t)≠0
yx=yt\xt
, yx , .
39. - , .
.
- , U=const, , V=const.
40. limy=A, y=A+a
limDy/Dx=y`, Dy/Dx=y`+a, Dy=y`Dx+aDx
Dx0
Dy=y`Dx+e, e-..., ,, Dx(a), .
dy=y`Dx
- . , .. D - ... , D.
y=x, dy=dx=x`Dx=Dx, dx=Dx
y¹x, dy=y`dx, y`=dy,dx
: - , - (x0,f(x0)) x0 Dx
41. dy= f ′()* d; y=f(U); U=ϕ(x)
dy= f ′(U)* dU . .
42. ᴧ y=f(x) ᴧ= f ′(x)*ᴧ+α*ᴧ, α→0 ᴧ→0 ᴧ= dy+ α*ᴧ. α*ᴧ , ᴧ, ᴧ~ dy, , ᴧ. .
|
|
43. y=f(x), . . y=f(x). , f(x) ()=(f(x)). f(x) . . - . 2 d2y\dx2(d2y dx ). a) y=f(x) . . . S(t)=V(t); V(t)=a; a=S(t). b) F(x, y)=0. . , .
c) {x=ϕ(t), y=ᴪ(t), tT} yx=yt\xt; yxx= (yx)t\xt
44. y=f(x) , . dy=f ′(x)dx , . .
, .
n- , (n-1)- , .. , .
45. 1. .
f(x) / [a,b] . (a,b) f(a)=f(b) . . (a,b), , f(c)=0.
2. .
f(x) [a,b] (a,b), .
. (a,b), , : f(b)-f(a)=f(c)(b-a).
: ., g(x)=x, g(x)=1¹0.
3. .
f(x), g(x) . :
1). f(x), g(x) . [a,b]
2). f(x), g(x) . (a,b)
3). g(x)¹0 . (a,b), . .
g(b)¹g(a) ( ).
1). F(x) [a,b]
2). F(x) (a,b)
3). F(a)=0; F(b)=0
. Î(a,b); F()=0
4. f(x), (a,b) - , =0
46.
, 0 / 0.