.
. f(x) , - , , >0 D>0,
ïf(x)ï>M
,
0 < ïx - aï < D
.
, ïf(x)ï>M f(x)>M, :
f(x)<M, :
:
a x a x a x
. , ¥, +¥ -¥, , ¥, +¥ -¥.
.
. f(x)0 ( ¥) ,
5. .
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 .
6. .
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 = ,
.
7. .
.
.
, .
, , :
. .
. .
. .
. .
. .
. .
.
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)
. .
:
1)
2)
3)
4)
5)
6)
7)
8)
3.
1. .
. 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.
2. .
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 , .. .
.
, .
3. .
f(x), 0, . , = 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
0 x
-1
sign(x) . = 0 . .. , 1 . = 0, f(0) = 1, , f(0) = -1, , f(x) - , 1 1, , , = 0 1 . 1 .
, , 1 , , , .
4. .
. 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
-4 -1 0 1
. , .
= 0 = 1 1
-p -p/2 0 1 x
4.
1. , .
. f(x) = 0 , .
f(x)
f(x0 +Dx) P
Df
f(x0) M
a b Dx
0 x0 x0 + Dx x
f(x) (a, b). .
,
a - f(x) (x0, f(x0)).
, - .
:
: .
, .
f(t), t- , f(t)- ( ) .
, - , .. .
2. .
. () f(x) = 0 () , .
f(x) = 0, . , . - 0, - , 0, .
: f(x) = ïxï- = 0 , , , .
. ( ) f(x) 0, .
, .
3. .
f(x) = u, g(x) = v - , .
1) (u v)¢ = u¢ v¢
2) (u×v)¢ = u×v¢ + u¢×v
3) , v ¹ 0
.
.
1)¢ = 0; 9)
2)(xm)¢ = mxm-1; 10)
3) 11)
4) 12)
5) 13)
6) 14)
7) 15)
8) 16)
4. .
. y = f(x); u = g(x), u f.
.
( , Dx0, Du0, .. u = g(x) )
.
5. - .
, , , . , .
u = f(x) v = g(x) , , f(x)>0.
y = uv. , :
lny = vlnu
. .
:
:
:
6. .
= f(x) , x = g(y) , .
|
|
x = g(y) :
.. g¢(y) ¹ 0
.. .
. arctg.
arctg , tg, .. :
,
:
.. :
, , .
7. .
y = f(x) :
: , a0, D0.
: .
aDx- , f¢(x)Dx, .. f¢(x)Dx- D.
. f(x) .
dy df(x).
, dy = f¢(x)Dx
dy = f¢(x)dx.
:
8. .
y
f(x)
K
dy
M Dy
L
a
x x + Dx x
DMKL: KL = dy = tga×Dx = y¢×Dx
, f(x) .
9. .
u = f(x) v = g(x)- , , :
1) d(u v) = (u v)¢dx = u¢dx v¢dx = du dv
2) d(uv) = (uv)¢dx = (u¢v + v¢u)dx = vdu + udv
3) d(Cu) = Cdu
4)
10. .
.
y = f(x), x = g(t), . - .
dy = f¢(x)g¢(t)dt = f¢(x)dx.
, dy , - , .
, - ,
dx = Dx,
t, D ¹ dx.
dy = f¢(x)Dx .
. .
:
. .
.
.
.
11. .