Similar presentations:
Основные теоремы дифференциального исчисления. Теорема Ферма
1.
f ( x0 ) 02.
ylim
0
x 0
x
и
y
lim
0
x 0
x
x 0
y
y
lim
lim
0
x 0
x x 0 x
f ( x0 ) 0
3.
4.
f ( ) 05.
6.
yy f (x)
a
1
2
b
x
7.
y f (x)y
f (b)
f (a)
a
b
x
8.
y f (x)y
f (a)
a
f (b)
b
x
9.
f (a) f (b)f (b)
y
f (a )
y f (x)
a
b
x
10.
11.
f (b) f (a)f ( )
b a
12.
f ( ) (b a) f (b) f (a)13.
yB
y f (x)
A
a
b
x
14.
( a, b )15.
16.
17.
18.
19.
yy
x
x
20.
y x 4x 32
21.
y ( x 4x 3) 2x 42
y 2 x 4 0 при
x 2
y 2 x 4 0
x 2
при
( 2 ; )
( ; 2)
22.
f ( x) f ( x0 )23.
f ( x) f ( x1 )24.
yy f (x)
x1 x2
x3
x
25.
f ( x0 ) 026.
27.
28.
y x2
29.
y ( x ) 2 xy 2 x 0 при x 0
2
x 0
y 0
30.
yx 0
y x
2
x
31.
y x 13
32.
y ( x 1) 3x2
y 3x 0 при x 0
3
x 0
y 1
2
33.
yy x
2
y 1
x
34.
35.
y f (x)36.
37.
y x( x 1)3
38.
y ( x( x 1) ) ( x 1) 3x ( x 1)3
3
2
( x 1) ( x 1 3x) ( x 1) (4 x 1)
2
2
39.
( x 1) (4 x 1) 02
x1 1
1
x2
4
40.
yy
1
4
1
x
41.
271
f min
256
4
42.
43.
44.
45.
x1 x2 f ( x1 ) f ( x2 )f
2
2
46.
yy f (x)
f ( x1 ) f ( x2 )
2
x1 x2
f
2
x1
x1 x2
2
x2
x
47.
x1 x2 f ( x1 ) f ( x2 )f
2
2
48.
yx1 x2
f
2
y f (x)
f ( x1 ) f ( x2 )
2
x1
x1 x2
2
x2
x
49.
50.
51.
52.
f ( x0 ) 053.
54.
55.
56.
y x ( x 1)3
57.
y x ( x 1)3
( x 1)
3
3x ( x 1)
2
( x 1) ( x 1 3x) ( x 1) (4 x 1)
2
2
y ( x 1) (4x 1) 2( x 1) (4x 1) 4( x 1)
2
2
( x 1) 8x 2 4x 4 ( x 1) 12x 6
y ( x 1) 12 x 6 0
58.
1x1
2
x2 1
y
y
1
2
1
1
f
16
2
1
f (1) 0
x
59.
60.
yy f (x)
x
61.
yy f (x)
x
62.
yy f (x)
x
63.
x x0 0x x0 0
(слева )
(справа )
64.
lim f ( x)x x0 0
lim f ( x)
x x0 0
65.
lim f ( x) f ( x0 )x x0
66.
lim f ( x) bx
67.
f ( x)lim
k
x
x
lim f ( x) k x b
x
68.
3x
y 2
x 1
69.
3x
lim 2
x
x 1
70.
f ( x)x3
x2
lim
lim 2 : x lim 2
1
x
x
x
x
x 1
x 1
k 1
x3
lim f ( x) kx lim 2
x
x
x x 1
x x x
x
lim
lim
0
2
2
x
x
x 1
x 1
3
3
b 0
y x
71.
72.
73.
74.
1 xy
2
1 x
2
75.
x 1( ; 1) ( 1;1) (1; )
1 ( x) 1 x
f ( x)
f ( x)
2
2
1 ( x) 1 x
2
2
76.
x 177.
x 1x 1
1 x2
lim
2
x 1 0 1 x
1 x2
lim
2
x 1 0 1 x
78.
1 x21
lim
2
x 1 x
lim
x
1 x2
1
lim
2
x 1 x
f ( x)
1 x2
lim
2
x x (1 x )
x
1 x 2 x(1 x 2 ) ( 2 x)(1 x 2 )
y
2
2 2
(1 x )
1 x
2
79.
2 x 2 x3 2 x 2 x34x
2 2
(1 x )
(1 x 2 ) 2
y
y
0
f min (0) 1
x
80.
( ; 1) ( 1;0)(0;1) (1; )
2 2
2 2
4 x (4 x) (1 x ) 4 x (1 x )
y
2 2
2 4
(1 x )
(1 x )
4 (1 x 2 ) 2 4 x 2(1 x 2 ) ( 2 x) 4 4 x 2 16 x 2
2 4
2 3
(1 x )
(1 x )
81.
4(1 3x 2 )(1 x 2 )3
y
y
1
0
x
82.
( 1 ; 1)( ; 1) (1; )
x 0
1 0
y
1
1 0
83.
y1 x2
y
2
1 x
1
1
1
1
x