y=f(x) (-1) (-1), () ( ) (-1), .. () = ((-1))′. , =(y)- , y=(y) ..
, .
. - :
f(x) | fn(x) |
Xa Ex Ekx Akx Lnx Logax Sinkx Cos kx | A(a-1)*(a-2)**(-n+1)* a-n Kn*ekx (K* Lna)n*akx (-1)n-1*(n-1)!/xn (-1)n-1*(n-1)!/(xn*lna) kn*sin(kx+n*π/2) kn*cos (kx+n*π/2) |
- y=f(v) :
d2y=d(dy) - 2-
d3y=d(d2y)
dny=d(d n-1 y) - - n-
- y=f(v), v - v=+ , d2y=y(dv)2, d3y=y(dv)3,, dny=y(n)(dv)n.
y=f(v), v=g(x)≠+, d2y=f(v)*(dv)2+ f(v)d2v .. (.. - ).
.
f(x) n x0.
T(x) = f(x0) + ((f(x0))/1!)(x x0)1 + (f (x0))/2!(x x0)2 ++ (f (n)(x0))/n!(x x0)n
n- f(x) x0.
f(x) ε x0 (n + 1) . , 0,
F(x) = T(x) + (f(n+1)(c) / (n + 1)!)(x x0)n+1 ,
(x) n- f(x) 0,
rn(x) = (f(n+1)(c) / (n + 1)!)(x x0)n+1 .
, (n+1)- f(x) 0. rn(x) , (-0)n 0. (lim (rn(x)/(-0)n) = lim [((f(n+1)(c))/(n+1)!)(x-x0)] = 0
f(n+1) (c) 0.) f(x) Tn(x) (*) , ( 0)n, 0.
(*) .
(*) , ( 0):
1) (1+x)a 1 + (a/1!)x + (a(a-1)/2!)x2 ++ (a(a-1)(a-n+1)/n!)xn,
2) ex 1 + x/1! + x2/2! ++ xn/n!,
3) ln(1+x) x x2/2 + x3/3 x4/4 ++(-1)n+1xn/n
4) sin x x x3/3! + x5/5! x7/7! ++(-1)kx2k+1/(2k+1)!,
5) cos x 1 x2/2! + x4/4! x6/6! ++(-1)kx2k/(2k)!,
n.
.
=f(x) [a,b] , =f(x)-const, , f¢(x)=0 "'[a,b]. =f(x), y=g(x) (a,b) f¢(x)=g¢(x), f(x)=g(x)+C.
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y=f(x) , 1,2', 1<x2Þ f(x1)<f(x2), x1<x2Þ f(x1)>f(x2).
. , (a,b) (a,b) , =f(x) .
f¢(x)>0 Þ y=f(x) (a,b) ( )
.
(a,b) 1 2, 1<2. , , f(x2)-f(x1)= f¢(c)(x2-x1). 1<c<x2, . f¢(c)³0 f(x2)³f(x1). , , f(x) .
.
. , f(x) 0 , , f¢(x0)=0.
.
0 , (0-e, 0+e), f(x0) . f¢(x0)=0.
, , .
. 0 f(x) , 0 f(x), , 0 .
. ( , , )
f(x) . f¢(x) + -. Î (0 -D, 0] f¢(x)>0 Þ Þ f(x0)³f(x) "CÎ(x0-D, x0]
"CÎ[0,0+D) f¢(x)<0, , Î[0,0+D) Þf(x0)³f(x) Î[0,0+D).
: Î (0-D, 0+D) 0 =f(x). ...