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W
:
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2
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3
W
W
J 4 J 5
6
8
10
J J 12
13
14
n 16
18 20
22
24
26
28
31
32
34
36
38
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4
K
↖ Wax b+ ( )0a ≠
+∞ b
a − −∞ x
a a ax b+
↖ W²ax bx c + + ( )0a ≠
W( ) ²x ax bx c Ρ = + +
WWWW
( ) 0x x ∈ Ρ = ( )x Ρ (
)x Ρ
0∆ < S = ∅
+∞ −∞ x
a ( )x Ρ
0∆ = { }2
bS
a
−=
+∞ ba
− −∞ x
a
a
( )x Ρ ( )
²
2
bx a x
a
Ρ = +
= b² - 4ac∆
0∆ >
{ };1 2S x x =
W
1 2
bx
a − − ∆=
2 2
bx
a
− + ∆=
+∞2x 1x −∞
x
a
a
a ( )x Ρ
F W1 2 x x
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5
K
↖ W
a b
( )2 2 22a b a ab b+ = + +
( )2 2 22a b a ab b− = − + ( )( )2 2a b a b a b− = −
+
( )3 3 2 2 33 3a b a a b ab b+ = + + +
( )3 3 2 2 33 3a b a a b ab b− −− = +
( )( )3 3 2 2a b a b a ab b− = − + +
( )( )3 3 2 2a b a b a ab b−+ = + +
↖
W
P Q
f x WWWW
f WWWW
( ) ( ) f x x = Ρ f D =
( )( )
( )
x f x
Q x
Ρ= ( ){ }/ 0 f D x Q x = ∈ ≠
( ) ( ) f x x = Ρ ( ){ }/ 0 f D x
x = ∈ Ρ ≥
( )( )
( )
x f x
Q x
Ρ= ( ){ }/ 0 f D x Q x = ∈
>>>>
( )( )
( )
x f x
Q x
Ρ= ( ) }0Q x >>>> ( ){
/ 0 f D x x = ∈ Ρ ≥
( )( )
( )
x f x
Q x
Ρ= ( ) }0Q x ≠
( )
( )
/ 0 f x
D x
Q x
Ρ= ∈ ≥
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6
)( K
↖ ( )* n n x x ∈ x
x W
0lim 0n x
x →
= 0
lim 0x
x →
=>>>>
1lim 0n
x x →−∞=
1lim 0n
x x →+∞=
limx
x →+∞
= +∞
1lim 0
x x →+∞=
n WWWW n WWWW
lim n
x x →+∞ = +∞
lim n x
x →−∞
= +∞
0
1lim n
x x →= +∞
>>>>
0
1lim n
x x →= +∞
>
0
1lim n
x x →= −∞
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7
↖
W
( ) ( )
( )( )
00
limlim 0
x x x x
f x V x
f x V x
→→
− ≤ ⇒ ==
l
l
( ) ( ) ( )
( )
( )
( )
0 0
0
lim lim
limx x x x
x x
u x f x V x
u x f x
V x → →
→
≤ ≤ = ⇒ =
=
l l
l
( ) ( )
( )( )
00
limlim x x x x
u x f x
f x u x →
→
≤ ⇒ = +∞= +∞
( ) ( )
( )( )
00
limlim x x x x
u x V x
f x V x →
→
≤ ⇒ = −∞= −∞
0x
0x
+∞
−∞
↖ W
W
+∞ +∞ −∞ l l l (
)
0
limx x
f x →
−∞ +∞ −∞ +∞ −∞ 'l ( )
0
limx x
g x →
+∞ −∞ +∞ −∞ l + l' (
) ( )[ ]0
limx x
g x f x → +
W
0 +∞ −∞ −∞ 0>>>>l 0>>>l 0
-
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8
K
↖ W
W ( ) ( )0
0limx x
f f x f x →
⇔ = 0x
– W •( ) (
)
00lim
x x f f x f x
→>
⇔ = 0x
•( ) ( )0
0limx x
f f x f x →<
⇔ = 0x
f
0 f x ⇔ 0x
↖
W
f ] [,a b f ] [,a
b f [ ],a b f ] [,a
b
a b
↖
W
f g I k
• f g + f
g × kf I
• g I 1
g f
g I
W •
•
• x x +
• sinx x cosx x
• tanx x
{ }/
2
k k π
π− + ∈
↖ W
f
I g J W( ) f I J ⊂
Wg f ο I
↖
W
•
•
W f I
( ) f I
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9
( )f I
I f
I f I
[ ],a b ( ) ( )[ ]; f a f b ( ) (
)[ ]; f b f a
[ [,a b ( ) ( ); limx b
f a f x
−→
( ) ( )lim ;x b
f x f a
−→
] ],a b ( ) ( )lim ;x a
f x f b+→
( ) ( ); limx a
f b f x +→
] [,a b ( ) ( )lim ; limx bx a
f x f x −+ →→
( ) ( )lim ; limx b x a
f x f x − +→ →
[ [,a +∞ ( ) ( ); limx
f a f x →+∞
( ) ( )lim ;x
f x f a →+∞
] [,a +∞ ( ) ( )lim ; limx x a
f x f x + →+∞→
( ) ( )lim ; limx x a
f x f x +→+∞ →
] ],a −∞ ( ) ( )lim ;x
f x f a →−∞
( ) ( ); limx
f a f x →−∞
] [,a −∞ ( ) ( )lim ; limx x a
f x f x −→−∞ →
( ) ( )lim ; limx x a
f x f x − →−∞→
( ) ( )lim ; limx x
f x f x →−∞ →+∞
( ) ( )lim ; limx x
f x f x →+∞ →−∞
↖
W
f [ ],a b β ( ) f a (
) f b
α [ ],a bW( ) f
α β =
W
f [ ],a b ( ) ( ) 0 f a f
b×
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10
K
↖ W
f 0x W( ) ( )0
0 0
lim
x x
f x f x
x x →
−
−
f 0x W(
)0' f x
↖ J W
f
0x
f 0x W( )( )
( )0 0 0'y f x x x f x = − +
u W( ) ( )( ) ( )0 0 0'u x f x x x f x
= − +
f
0x
f
0x
↖ J W
f
0x ( ) ( )
0
0
0
limx x
f x f x
x x →
−−
>
f
0x W( )0' f x d
f 0x W( )
( )
0
0
0
limx x
f x f x
x x →
−−
<
f 0x W(
)0' f x g
f 0x f
0x ( ) ( )0 0' ' f x f x g
d =
↖ W
f 0x f 0x
↖
W
( )f x ′ ( )f x 0 k (
)k ∈ 1 x 1
²x
−
1
x
1r rx − r x { }( )*
1r ∈ −
1
2 x
x
cos x sin x
sin x − cos x 2
2
11 tan
cosx
x + = tan x
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11
↖
-
J
W
( )u v u v ′ ′ ′+ = + ( )u v u v ′ ′′− =
− ( ) ( ) ( )k ku k u ′ ′∈ =
( )uv u v uv ′ ′′ = + ( ) 1.n n u nu
u −′ ′=
( )
1
²
v
v v
′ ′−=
( ) ²u u v uv
v v
′ ′ ′−=
( )u v u v v ο ο ′ ′′ = × ( ) 2u
u u
′′ =
↖ W
f I
( ) 0 f x I f x ′⇔ ∀ ∈ ≥
I
( )' 0 f x I f x ⇔ ∀ ∈ ≤ I
( )' 0 f x I f x ⇔ ∀ ∈ =
I ↖ W
( )f C WWWW
( ) ( )
( )0
0
0
lim0x x
f x f x a
x x a →
−=
− ≠ ( )( )0 0;A x f x
a
( ) ( )
0
0
0
lim 0x x
f x f x
x x →
−=
−
f
0x ( )( )0 0;A x f x
( ) ( )( )
0
00lim
0
f x f x a
x x x x a +
− =−→ ≠
( )( )0 0;A x f x
a
( ) ( )000
lim 0 f x f x
x x x x +
−=
−→
f
0x ( )( )0 0;A x f x
( ) ( )000
lim f x f x
x x x x +
−= −∞
−→
( )( )0 0;A x f x ( ) (
)0
00
lim f x f x
x x x x +
−= +∞
−→
f
0x
( )( )0 0;A x f x ( ) ( )
( )
0
0 0lim
0
f x f x a
x x x x a −
−=
→ − ≠
( )( )0 0;A x f x
a
( ) ( )00 0
lim 0 f x f x
x x x x −−
=→ −
f
0x ( )( )0 0;A x f x
( ) ( )00 0
lim f x f x
x x x x −−
= −∞→ −
( )( )0 0;A x f x ( ) (
)0
0 0lim
f x f x
x x x x −−
= +∞→ −
f
0x
( )( )0 0;A x f x
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12
–
K
↖ W
x a = ( ) f C
W •( )2 f f x D a x D ∀ ∈ −
∈ •( ) ( )2 f x D f a x f x ∀ ∈ −
=
↖ W
( ),I a b
( ) f C
W
•( )2 f f x D a x D ∀ ∈ − ∈ •( ) (
)2 2 f x D f a x f x b∀ ∈ − + =
↖ – - W
W( ) 0x I f x ′′∀ ∈ ≤
W (
) f C I
W( )
0x I f x ′′∀ ∈ ≥
W (
) f C I
f ′′ 0x
( ) f C 0x
f ′ 0x
( ) f C 0x
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13
K
( ) ( )[ ]lim 0x
f x ax b→∞
− + = ( )
( )0lim
x a
f x a
x →∞ ≠=
( )limx
f x →∞
=∞
( )lim
x
f x
x →∞= ∞
( )[ ]limx
f x ax b
→∞
− =
( )[ ]limx
f x ax
→∞
− = ∞
W
( ) f C W
∞
( ) f C W
y ax =
∞
( ) f C W
W
y ax b= +
∞
( ) f C W
W y a =
∞
( )limx
f x a →∞
=
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14
K
↖W f I
f ( ) f
I I
W1 f −
W •
( ) ( )
( )
1 f x y f y x
x I y f I
− = = ⇔ ∈ ∈
• ( )( )1x I f f x x ο−∀ ∈ =
•( ) ( )( )1y f I f f y y ο −∀ ∈ =
↖ W f
I
x ( ) f I y
I
W( ) ( )1 f x y f y x − = ⇔
=
y x ( )1 f x − x ( ) f
I
↖ W
f I
1 f − ( ) f
I
↖ W
f I
0x ( ) f I ( )0 0y f
x =
f 0x ( )0' 0
f x ≠
1 f − 0y
W ( ) ( )( )
10
0
' 1
' f y
f x
− =
f I
f I f ′
I
1 f −
( ) f I
W ( ) ( ) ( )
( )
11
' 1
'x f I f x
f f x
−−∀ ∈ =
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15
↖
W
f I
1 f −
f
↖
W
↖ W
(
)f C (
)1f C −
( ) ( ), f A a b C ∈ ( ) (
)1' , f A b a C −∈
Wx a =
Wy a =
Wy b=
Wx b=
Wy ax b= +
W1 by x a a
= +
W
x ay b= +
F E
F E
F E
F E
f I
f 1 f −
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16
( )*n n ∈
K
↖ W
Wn x x +
n
Wn n
n x x
+→
++++
::::
( ) 2; n n x y x y x y +∀ ∈ = ⇔ =
W
•2x x =• W3 x x
↖W
( )
( )
2; *
n n
n n
n n
n n
x y n
x x
x x
x y x y
x y x y
+∀ ∈ ∀ ∈
=
=
= ⇔ => ⇔ >
( ) ( ) ( )
( )
( )
22 *; ;
0
n n n
m m n n
n
n n
n m n m
x y m n
x y x y
x x
x x y y y
x x
+
×
∀ ∈ ∀ ∈
× = ×
=
= ≠
=
W
x y x y
x y
−− =
+ 3 3
3 3 33² ²
x y x y
x x y y
−− =
+ +
↖ W
f
WWWW
f WWWW ( ) n f x x = [
[0; f D = +∞
( ) ( )n f x u x = ( ) }0u
x ≥ { / f u D x x D = ∈
∈
↖ W
0x 0x +∞
−∞
( )
0
limx x
u x →
( )
0
lim n
x x u x
→
0≥l n l
+∞ +∞
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17
↖ W
n x x +
u I
u I ( )n x u x
I
↖ W
n x x ] [0;+∞
W
] [ ( )1
10; n
n n x x
n x −′∀ ∈ +∞ =
u I
u I
( )n x u x I
W( )( ) ( )
( )[ ] 1n
n n
u x x I u x
n u x −
′′∀ ∈ =
↖ W( ) n a x x a ∈ ∈ =
n n
0a > { }n S a = { };n n S
a a = − 0a = { }0S = {
}0S =
0a { / f u D x x D = ∈
∈
•( )( ) ( )( ) ( ) ( )[ ]1 1
11 'n n n u x u x u x u x n
−′ ′ = = × ×
x y *+ r r ′*
•( ) ' 'r r r r x x
×= •' 'r r r r x x x +× =
•
r r
r x x
y y
=
•( )r r r x y x y × = ×
•'
'1 r r
x x
−= •r
r r
r
x x
x
′−′
=
-
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18
K
↖ – W
1n n u u r + = +
r
1n n u q u + = ×
q
( )n pu u n
p r = + −
( )p n ≤
n pn pu u q
−×=
( )p n ≤
1 1...
1
n p
p n p
q u u u
q
− + − + + = × −
1 1...
1
n p
p n p
q u u u
q
− + − + + = × −
( )1q ≠
a b c
2b a c = + ²b a c = ×
↖
–
:
( )n n I u ∈
•( ) n n n I u n I u M ∈
⇔ ∀ ∈ ≤ M
•( ) n n n I u n I u m ∈
⇔ ∀ ∈ ≥ m
•( )n n I u ∈ ( )n n
I u ∈ ⇔
↖ W
( )n n I u ∈
•( ) 1n n n n I u n I u u +∈
⇔ ∀ ∈ ≤
•
( ) 1n n n n I u n I u u +∈
⇔ ∀ ∈ ≥
•( ) 1n n n n I u n I u u
=+∈ ⇔ ∀ ∈
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19
↖ W
( )n αW*α ∈ W
0α > 0α 1q = 1
1q − <
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20
K
↖ W
W
f I
F f I
W
•F I
•( ) ( )'x I F x f x ∀ ∈ =
W
f I
F f I W
f I W (
) ( )x F x k k + ∈
f I
0x I 0y
F f
I
W( )0 0F x y =
↖ W - W
W
f g
I k
F G
f g I
W
•F G + f
g + I •kF
kf I
-
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21
↖ W
( )F x ( )f x
ax k + a ∈
1 ²2
x k + x
1k
x
−+
1
²x
2 x k + 1
x
1
1
r x k
r
++
+ r x { }( )* -1r ∈ −
cos x k − + sin x
sin x k + cos x
tan x k + 1
1 tan ²cos²
x x
+ =
ln x k + 1
x
( )k ∈ x k e
+ x e
↖ W
( )F x ( )f x
( )2 u x k + ( )
( )
'u x
u x
( )
1k
v x +
( )
( )[ ]
'
²
v x
v x
−
( )[ ]
1
1
r
u x k r
+
++
( ) ( )[ ]' r u x u x × { }( )*
-1r ∈ −
( )ln u x k + ( )
( )
'u x
u x
( )u x k e + ( )
( )' u x u x e ×
( )1
sin ax b k a
+ + ( )cos ax b+ ( )0a ≠
( )k ∈ ( )1 cos ax b k
a − + + ( )sin ax b+ (
)0a ≠
-
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-
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23
f [ ],a b
f C
W
x a = x b=
W( ) . .b f x dx u Aa
∫
f g [ ],a b
f C g C
W
x a = x b=W
W( ) ( ) . .b f x g x dx u Aa
− ∫
W
WWWW
f
[ ],a b ( ) . .
b f x dx u A
a
∫
f
[ ],a b ( ) . .
b f x dx u A
a
− ∫
• f
[ ],a c
• f [ ],c b
( ) ( ) . .c b
f x dx f x dx u A
a c
+ − ∫ ∫
( ) f C ( )g C
[ ],a b ( ) ( )( ) . .
b f x g x dx u A
a
− ∫
•( ) f C ( )g C
[ ],a c
•( )g C ( ) f C
[ ],c b
( ) ( )( ) ( ) ( )( ) . .c b
f x g x dx g x f x dx u A
a c
− + − ∫ ∫
↖ :
(
) f C
[ ];a b
W ( )( )² .b
V f x dx u v a π
= ∫
uv W
-
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K
↖
W
1
x x
] [0; +∞
1 Wln
W
ln 1e = ln 1 0=
] [ ] [0; 0;x y ∀ ∈ +∞ ∀ ∈ +∞ ln lnx y x y = ⇔
= ln lnx y x y > ⇔ >
] [0;
ln y
x y
x y x e
∀ ∈ +∞ ∀ ∈
= ⇔ =
] [ ] [
( )
( )
0; 0;
ln ln ln
ln ln
1ln ln
ln ln ln
r
x y
xy x y
x r x
x x
x x y
y
∀ ∈ +∞ ∀ ∈ +∞
= +
=
= − = −
( )r ∈
n W( )* ln lnn x x n x ∀ ∈ =
W
f
WWWW
f WWWW
( ) ( )[ ]ln f x u x = ( ) }0u x
>{ / f u D x x D = ∈ ∈ ( ) (
)( )2ln f x u x =
( ) ( )ln f x u x = ( ) }0u
x ≠{ / f u D x x D = ∈
∈
W
( )lim lnx
x →+∞
= +∞ lnlim 0n x
x
x →+∞=
( )0lim lnx x → = −∞>
( )0lim ln 0n
x x x → =>
1
lnlim 1
1x
x
x →=
−
( )0
ln 1lim 1x
x
x →
+=
( )n *∈
W
lnx x ] [0;+∞
u I u I (
)[ ]lnx u x I
