. f(x) , - , , >0 D>0,
ïf(x)ï>M
,
0 < ïx - aï < D
.
, ïf(x)ï>M f(x)>M, :
f(x)<M, :
:
. , ¥, +¥ -¥, , ¥, +¥ -¥.
.
. f(x)0 ( ¥) ,
.
a(), b() g() . a, b g . , .. .
, f(x) = x10 , f(x) = x.
. , a , b.
. , a b .
. a b . a ~ b.
. 0 f(x) = x10 f(x) = x.
.. f(x) = x10 , f(x) = x.
. a k b, .
, . , , .
. , 0 , .. a - 2 b.
. , 0 , .. a b .
.
1) a ~ a,
2) a ~ b b ~ g, a ~ g,
3) a ~ b, b ~ a,
4) a ~ a1 b ~ b1 , .
: ) a ~ a1 ,
) b ~ b1 ,
4 , .. , . , .
.
|
|
tg5x ~ 5x sin7x ~ 7x 0, , , :
. .
1 cosx = 0, .
.
a b - , b - , a, g = a + b - , a. .
, a - g.
. 2 + 0, . , a = 2, b = ,
.
.
. , P(x) = a0xn + a1xn-1 ++an,
Q(x) = b0xm + b1xm-1 ++bm - .
:
.
.
, .
, , :
. .
. .
. .
. .
. .
. .
.
x2 6x + 8 = 0; x2 8x + 12 = 0;
D = 36 32 = 4; D = 64 48 = 16;
x1 = (6 + 2)/2 = 4; x1 = (8 + 4)/2 = 6;
x2 = (6 2)/2 = 2; x2 = (8 4)/2 = 2;
. .
: =
= .
. .
. .
.
x2 3x + 2 = (x 1)(x 2)
x3 6x2 + 11x 6 = (x 1)(x 2)(x 3), ..
x3 6x2 + 11x 6 x - 1
x3 x2 x2 5x + 6
- 5x2 + 11x
- 5x2 + 5x
6x - 6
6x - 6 0
x2 5x + 6 = (x 2)(x 3)
. .
.
. f(x), 0, 0, , ..
:
. f(x) 0, 0, , 0 .
:
y
f(x0)+e
f(x0)
f(x0)-e
0 x0-D x0 x0+D x
:
y
f(x0)+e
f(x0)
f(x0)-e
x0 x
. f(x) 0, e>0 D>0, ,
.
. f(x) = 0, 0 .
|
|
f(x) = f(x0) + a(x)
a() 0.
.
1) , 0 , 0.
2) , g(x) 0.
3) .
:
u = f(x), v = g(x) = 0, v = g(f(x)) .
, .
.
1) f(x) = C, C = const .
2) , , . , .
3) .
3 y = sinx.
Dy = sin(x + Dx) sinx, :
, . D0 , ..
, D0.
, , , .. D . , = sinx = 0 , .. .
.
, .
.
f(x), 0, . , = 0 , , .
, . .
(. ) , .
0
(. ) , .
. 0 f(x), f(x) 0 .
. 0 1- , f(x) , .
, = 0, , .
, 1 . 1 , .
. 0 2 , f(x) .
. ( (1805-1859) , - 1837)
0.
. f(x) = 0 = 0 2 , ..
.
. f(x) =
= 0, , .. = 0 1 . , .. :
|
|
:
. f(x) = =
y
sign(x) . = 0 . .. , 1 . = 0, f(0) = 1, , f(0) = -1, , f(x) - , 1 1, , , = 0 1 . 1 .
, , 1 , , , .
.
. f(x) (), ().
, .
, .
1: ( ( (1815-1897)- )). , , , .. [a, b] M £ f(x) £ M.
, , 0, , [a, b] , 0, 0.
2: , [a, b], .
.. 1 2, f(x1) = m, f(x2) = M,
m £ f(x) £ M
( f(x) = sinx).
.
3: ( ). , [a, b], .
4: f(x) = 0, 0, .
5: ( (1781-1848) ). f(x)- [a, b] , , f(x) = 0.
.. sign(f(a)) ¹ sign(f(b)), $ 0: f(x0) = 0.
. f(x) [a, b], e>0 D>0 , 1Î[a,b] x2Î[a,b] ,
ï2 1ï< D
ïf(x2) f(x1)ï < e
, e D, , D e .
6: ( (1845-1918)- ). , , .
( , .)
|
|
.
(0, ), , .. D>0 , 1 2 , ïf(x1) f(x2)ï>e, e - , 1 2 .
7: f(x) , , = g(y) , .
. , .
= -1 = 1 1
. , .
= 0 = 1 1