. .
.
[ a, b ], M m .
.
1. [ a, b ] (.2.3 ).
ࠠ ᠠ ⠠
. 2.3. .
2. , (. 2.3 ).
3. (. 7.3).
= f (x):
1. [ a, b ],
, . . (a, b),
2. f () = f (b).
(a, b) ,
a < c < b f `(c) = 0.
. . f() = f(b) =0, : .
. = f (x) [ a, b ], , , . , , .. . , , a < c < b = f (), . , .
1 > c, 2< c (. 2.4).
1:D x = 1 , D x > 0, D f (x) = f ( 1) - f () = f ( 1) - < 0.
2:D x = 2 , D x < 0, D f (x) = f ( 2) - f () = f ( 2) - < 0.
, f `(c) = 0.
. 2.3. .
. .
( ). = f (x) [ a, b ], , . . (a, b), , a < c < b : , ,
f (b) - f () = f `(c)(b - ). (2.18)
MN (. . 2.4).
. 2.4. .
. f (x) g (x) [ a,b ], g (x) ≠ 0, , . . (a, b), , a < c < b
|
|
(2.19)
. f (x) g (x) [ a, b ] f () = g () = 0 (a < c < b), →, ,
(2.20)
. .
. .
. - = 1/,
.
. = f (x) (a,b) (n +1) ˗ . n ˗ , n x 0
(2.21)
0 (2.21) .
(2.22)
a < x < b, a<x 0 <b, x 0 <c<x.
堠
(2.23)
. Rn (x). , n (2.23). ½ Rn (x)½.
,
Δ = 0,
Δ f (x) = f (x) - f (x 0),
d n f (x)= f n(x)Δ ,
(2.24)
- , x 0 = 0.
(2.25)
. .
. f (x) = ex 0 = 0. , :
f (x) = ex, f ¢(x) = ex, f ¢¢(x) = ex,..., f (k)(x) = ex.
f (0) = f ¢(0) = f ¢¢(0) =... = f (k)(0) = 1.
(2.25) f (x) = ex
ex. = 1,
e.
n = 2 2.5, 0.23,
n = 3 2.667, 0.052,
n = 10 2.7182818 , 4.3 ּ 10-7.