Θεώρημα Taylor για 1 & 2 μεταβλητές
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Transcript of Θεώρημα Taylor για 1 & 2 μεταβλητές
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1 Taylor
Taylor . .
1.1
I R , n 2 N0; f : I ! R n + 1 I: x0 2 I:
, x 2 I; # = #(x) x0 x;
(1) f (x) =nX
i=0
(x x0)ii !
f (i)(x0) +(x x0)n+1(n+ 1)!
f (n+1)(#):
Taylor Taylor f x0:
1.1 (1):
n = 0 (1) -, f (x) = f (x0) + (x x0)f 0(#):
# x # = x + (1 )x0 2 (0; 1); x x0.
pn(x) (1),
pn(x) :=
nXi=0
(x x0)ii !
f (i)(x0):
pn , n; Taylor f x0. f pn (1) f (x) pn(x); Taylor. . p(i)n (x0) = f (i)(x0); i = 0; : : : ; n; n pn f x0 pn f x0:
# x; (1) n+1: , (1), f; n + 1: f (n+1) .
(1) x 2 I; f - n:
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1.2
R2 , . n 2 N0; f : ! R n+ 1 ; f n+ 1 . (x0; y0) 2 :
, (x; y) 2 ; (; #) = ((x; y); #(x; y)) (x; y) (x0; y0);
(2)
f (x; y) =
nXi+j=0i;j0
(x x0)i(y y0)ji !j !
@i+jf
@xi@yj(x0; y0)
+
n+1Xi=0
(x x0)i(y y0)n+1ii !(n+ 1 i)!
@n+1f
@xi@yn+1i(; #):
1.2 (2):
(; #) (x; y) #
=
xy
+ (1
)x0y0
; 2 (0; 1):
pn(x; y) (2),
pn(x; y) :=
nXi+j=0i;j0
(x x0)i(y y0)ji !j !
@i+jf
@xi@yj(x0; y0):
pn , n; Taylor f (x0; y0). f pn (2) f (x; y) pn(x; y):
@i+jpn
@xi@yj(x0; y0) =
@i+jf
@xi@yj(x0; y0); i; j = 0; : : : ; n;
i + j n; n pn f (x0; y0) pn f (x0; y0):
(; #) (x; y); (2) n+ 1: , (2), f; n + 1:
@n+1f
@xi@yn+1i .
(2) (x; y) 2 ; f n: