-
001-126_update.indd 1 10/5/2013 8:48:43 pi
-
:
. .
: 1991 : 1992, 1993, 1994, 1995, 1996, 1997, 1998, 2012 .
& , - / & / .
.
001-126_update.indd 2 21/2/2014 4:36:43
-
. , . , . . , .
& , - / & / .
.
001-126_update.indd 3 21/2/2014 4:37:04
-
001-126_update.indd 4 10/5/2013 8:48:43 pi
-
. (2010) , - . , . , . , . . . : , (2010), . , . , . , . . .
(2010), - (2010).
:
1 , -, . , .
2 , , , .
3 - . , , - . - .
4 - - . 5 , .
001-126_update.indd 5 10/5/2013 8:48:43 pi
-
001-126_update.indd 6 10/5/2013 8:48:43 pi
-
1: 1.1
.........................................................................9
1.2
................................................................24
2: 2.1 -- ..............................30 2.2 -
.............................40
3: 3.1 ....................................................49
3.2 ............................................60 3.3 1
........................................................65 3.4
....................................................73 3.5
..............................................83 3.6
.............................89 3.7 2
.........................................97 3.8
....................103 3.9 f(x)=x+x
....................................................108 3.10
.........................................................................
114
4: - 4.1
....................................................................................128
4.2
..................................................................132
4.3 .......................................140 4.4 .........150
5: 5.1
.....................................................................160
5.2
.....................................................................................173
5.3
...............................................................181
................................................................193
- ............................195
001-126_update.indd 7 10/5/2013 8:48:43 pi
-
001-126_update.indd 8 10/5/2013 8:48:43 pi
-
1
1.1
x + y =
x + y = , 0 0,
, . : :
0, :
x y y x
y x
+ = = +
= + ,
-
= y'y
.
:
9 0, (. ),
x+y=, =0
O x
y
x+y=, 0
x
y
O
001-126_update.indd 9 10/5/2013 8:48:44 pi
-
10 1:
9 = 0 , y =
x'x y'y
(. ).
= 0 ( 0 ), .
- y'y
x'x
.
:
9 x y =2 2
=12
y'y 1.
9 y = 2 x'x - y'y 2 .
9 x = 2 y'y - x'x 2.
.
y x= 12
1
x x= =
001-126_update.indd 10 10/5/2013 8:48:44 pi
-
1.1 11
, (4, 1) x y =2 6, 4 2 1 4 2 6 = + =( ) . , , (16,5), ( , ) 10
8
.
2 x 2 x+y= x+y= , , , 2 x 2
x yx y+ =+ =
' ' '
. . - . . :
. () (') , .. (),
+ ( ) ' ( ') , () - 0, (') '.
+ ( ) ' ( ') () (').
- 2o :
9 0, : = =9 = = , + = +.
, , :
x yx y =+ =
2 6 13 4 8 2
( )( )
001-126_update.indd 11 10/5/2013 8:48:45 pi
-
12 1:
, ( )
, .. (1) x.
x
3 2 6 4 8 6 18 4 8
10 101
( )y y y y
yy
+ + = + + =
= =
y :
x = + =2 1 6 4( )
( , ).4 1
,
.
- , .
, x = 4 y = 1, :1 : 4 2(1) = 6
2 : 3 4 4 1 12 4 8 + = =( )
( ) , :
x yx y= ++ =
2 63 4 8
x yy= +=
2 61
001-126_update.indd 12 10/5/2013 8:48:46 pi
-
1.1 13
, , : . :
x x x = + = =2 1 6 2 6 4( ) . ( , )4 1 ( - ).
2 x 2
. , - .
,
, , . 2 x 2 . , 2 x 2 , , , .
+ = + =
3 6 183 4 8
x yx y
x yx y =+ =
2 63 4 8
+ + + = + = = 3 6 3 4 18 8 10 10 1x y x y y y
x yx y =+ =
2 63 4 8
x yx y+ =+ =
' ' '
A(4, 1)
3x+4y=8 x2y=6y
O x
. (3)
. 1
001-126_update.indd 13 10/5/2013 8:48:47 pi
-
14 1:
2 x 2 .
:
9 2 3 34 9 1x yx y+ = =
y x
y x
= +
=
23
1
49
19
,
, .
,
(1, 0,3) .
,
- 1
13,
.
9 x yx y+ =+ =
2 32 4 5
y x
y x
= +
=
12
32
12
54
, ,
.
9
y xx y+ = =
1 24 2 2
y xy x= =
2 12 1
, .
.
y
x
21
1
1
O
y
x
1
2O
x
y
2
1
O,
001-126_update.indd 14 10/5/2013 8:48:48 pi
-
1.1 15
, 2 x 2 :9 9 9 .
2 x 2
- 2 x 2 .
0 0. - :
1 2
1
= 2
= ''
.
'
', ' ' 0 , 1 2
,
+ = +
=
=
''
''
''
''
( ' ' ) '
x x x
x
'' '' '
=
x
:
x yx y+ =+ =
' ' '
y x
y x
= +
= +
( )
''
''
( )
1
2
y =
+ =
+ +
' '' '
' ' ' '( ' '
)
( ' ' )( ' ' )
=
001-126_update.indd 15 10/5/2013 8:48:49 pi
-
16 1:
y =
' '' '
, ,
=
'
', =0 , 1 2
, . . =0 ' =0.
:
' ' 0
' ' = 0 . ' ' ,
:
.
D, x , : .
, D, y , :
.
( , ) ' '' '
, ' '' '
x y =
x yx y+ =+ =
' ' '
( , ) ' '' '
, ' '' '
x y =
D = ' '
D = =
' '' '
Dy = =
' ' ''
001-126_update.indd 16 10/5/2013 8:48:50 pi
-
1.1 17
, , :
:
9
,
.
: .
, ( , ) ( , )x y = 4 1 .
9
. 2,
2 3 40x y = .
.
x yx y =+ =
2 63 4 8
D =
= = + = 13 4
21 4 3 2 4 6 10 0( )
Dx =
= + =68 4
224 16 40
yDDy= =
=
1010
1x DDx= = =
4010
4
2 3 40 2 403
x y y x = =
D0, , (x,y)
D=0, .
x yx y+ =+ =
' ' '
yDDy=
2 3 404 6 80x yx y = =
2 3 402 3 40x yx y = =
,
x DDx=
001-126_update.indd 17 10/5/2013 8:48:51 pi
-
18 1:
9
.
.
1
- , . , . , . , .
D, Dx, Dy. :
k k k, ,2 403
3 119 3 6x yx y+ =+ =
3 113 2x yx y+ =+ =
x yx y =
=
122
D
D
D
x
y
=
= + =
= = + =
=
1
11
2
2
2 2
2 1 2
1
22
2
( )
( )
== = + = 2 2 2 21 1 1 2( ) ( ) ( )
001-126_update.indd 18 10/5/2013 8:48:52 pi
-
1.1 19
, D = 0 . :
:9 D 0, 0 2, (x, y), :
.
, 0 2 ,
.
9 D = 0 , = 0 = 2 , . : = 0 ,
. = 2 ,
. , ,
3 x 3 x+y+z=0 , - , , , . . 2x+3y+z=6 - ( , , )2 1 5 ,
22+3(1)+5=6.
D = = = =0 2 0 0 2 ( )
x DDx= =
=
=
22
22
1
( )( )( )
yDD
y= =
=
=
2 22
22
2( )( )
( )( )
0 10 2 0
10
x yx y
yy
= =
==
2 14 2 2
2 12 1
x yx y
x yx y
= =
= =
.
1
,
2 1 2 1x y y x = =
001-126_update.indd 19 7/5/2014 4:36:51
-
20 1:
:
1 1 1 1x y z+ + = , 2 2 2 2x y z+ + = 3 3 3 3x y z+ + = , - , ,
3 x 3
- 2 x 2 .
. , .. (3) z, - . : (4)
(1) (2) :
9 2 3 9 2 3 3 8 9 11 2 15x y z x y x y x y + = + + = + =( ) (5)9
(6)
(5), (6)
x=1 y=2 . x y (4) z = 3. ( , , ).1 2 3
x y z x y x y x y+ = + + = + =3 10 3 3 8 10 1( )
1 1 1 1
2 2 2 2
3 3 3 3
x y zx y zx y z
+ + =+ + =+ + =
2 33 10 2
3 8 3
x y zx y zx y z
+ = + =+ =
( )( )
3 8 3 8x y z z x y+ = = +
3 x 3, ,
2 x 2, 3 x 3
.
11 2 151
x yx y+ =
+ =
,
001-126_update.indd 20 10/5/2013 8:48:54 pi
-
1.1 21
1.
i) ii) .
2. i) ii)
3. :
i) ii)
4. : i) ii)
5. : i) ii)
6. , .
i) ii) iii)
7. :
i) ii)
x y
x y7 8
45
=
+ =
x yx y
=
=
3 3
32
2 212
1 0
y x
x y
= +
+ =
2 73 5 4x yx y+ = =
2 5 46 7 100x yx y =+ =
( )
( )
3 1 4 712
3 1 1
+ + =
+ =
x y
x y
2 3 404 6 80x yx y = =
x y
x y
=
+ =
13
24
4 3 8
x y
x y
+
++ =
+
=
52
2 17
2 0
63
62
8
2 3 83 1 0
y xx y
= + + =
x yx y =+ =
42
( )
( )
3 1 2 2
3 1 1 3
+ =
+ + =
x y
x y
3 119 3 2x yx y+ =
=
2 13
4 24
32
33
x y
x x y
=
+
+ =
001-126_update.indd 21 10/5/2013 8:48:55 pi
-
22 1:
8. :
i) ii) iii)
1. i) 1 2 .
ii) 1 2 ;
2. 26 , 68 . ;
3. 1,5 4. 2200 - 5050 . .
4. R T R =T + . 20C 0,4 80C 0,5 , .
5. , 50% - 80% . 100 ml 68% ;
6. 1 2 4 3: x y+ = 2 2: x y+ = , . i) 1 2 .
ii) -;
iii) ;
3 2 112 5 2 35 2 33
x yx yx y
= =+ =
5 3 43 2
3 2 2 2
x yx yx y
+ = + = + =
x y
x y
x y
+ =
+ + =
+ =
22 3
32
5
5 3 2 16
y
x1 41
1 22
O
001-126_update.indd 22 10/5/2013 8:48:56 pi
-
1.1 23
7. :
i) 12: x y+ = 2 1: x y+ = .
ii) 3 : x y = 4 1: x y+ = .
8. :
i) ii)
9. 1 2 6= , O O1 3 5=
O O2 3 7= . .
10. , . N = x , B = y = z , - , .
11. . 50% , 10% 30% . 52 lit 32% , - . .
12. -, y x x= + + 2 . .
( )( )
,
= + =
1 2 1
4 1 2x y
x y( )
( ),
+ =+ + =
2 5 5
2 5x y
x y
1 2
3
x+y=, 0
x
y
0
y
x1
K(1,4)
y=g(x)
O
y
x
y=h(x)
O 2 4
4
y
x
y=f(x)
OK(2, 1)
3
001-126_update.indd 23 10/5/2013 8:48:58 pi
-
24 1:
1.2
, -. , 13 89.
x, y , x y+ =13 x y2 2 89+ = .
, :
: (1), , .. x , (2).
x y y x+ = = 13 13 (3).
2 = 9 . :
(3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , .
1o
x yx y+ =
+ =
1389 22 2 ( )
x x x x x
x xx x
2 2 2 2
2
2
13 89 169 26 89
2 26 80 013 40 0
+ = + + =
+ =
+ =
( )
.
x = 13 32
8
5
001-126_update.indd 24 10/5/2013 8:48:59 pi
-
(1) x (2). :
(3) x=2 y=3 , x=3 y=2 . (2, 3) (3, 2) .
- 5 6. , - Vieta
2 5 6 0 + = . 2 3 (2,3) (3,2).
x+y=5 , xy = 6
yx
=6
. -
.
x y y x+ = = 5 5
y
xO 11
AB
(3).xy x x
x xx x
= = = + =
6 5 65 6
5 6 0
2
2
( )
=x 2 x = 3.
1.2 25
001-126_update.indd 25 7/5/2014 4:37:35
-
26 1:
(2,3) (3,2) . (2,3) (3,2) .
2
(1) x (2).
(2) :
. x2 = , 2 13 36 0 + = , =9 =4 .
9 =9 x2= 9 x = 3 x = 3 (1) x=3 y=2 x = 3 y = 2 .9 = 4
.
(1) x=2 y=3 x = 2 y = 3. (3,2), ( , ), 3 2 (2,3) ( , ). 2 3
xy = 6 yx
=6
, -
x y2 2 13+ = (0,0) = 13 .
xy yx
= =6 6
x y xx
xx
x xx x
2 2 22
22
4 2
4 2
13 6 13
36 13
36 1313 36
+ = +
=
+ =
+ =
+ = 00
x x2 4 2= = x = 2
xyx y
=
+ =
6 113 22 2
( )( )
001-126_update.indd 26 10/5/2013 8:49:01 pi
-
1.2 27
1. :
2. :
i) ii) iii) .
3. S t t= + 021
2 = +0 t ,
S t= + 02
.
1. o - .
x y xyx y
2 2 31
+ + =+ =
.
y xx y= =
312 3 4
2 x yx y
2 2 90
+ = =
x yxy
2 2 52
+ ==
.
x yx y
2
2 2
2 1025
= +
+ =
y
x 1
AB
A B
001-126_update.indd 27 10/5/2013 8:49:02 pi
-
28 1:
2. :
3. 120cm2. - 3cm , 2cm , - . .
4. y x= 2 y x= + . k .
5. o 22y x
y x== +
.
1
I. :
, , ,
, , .
) , ) , ) .
II. , , .
1. , . A
2. D = 0 , .
3. xyx y
=+ =
10
.
4. x y2 2 1+ = y x= +2 1
. A
( ) :12 1
2 4 0x yx y+ =+ =
( ) :21
2 4x yx y+ =+ =
( ) :32 4 2
2 1x y
x y+ =+ =
( ) :4 21
1x yx y =
+ =
(1) (2) (3) (4)
2 5 04 3
2
2
xy y yy x x
=
= +
001-126_update.indd 28 10/5/2013 8:49:03 pi
-
( ) :4 21
1x yx y =
+ =
001-126_update.indd 29 10/5/2013 8:49:03 pi
-
2
) [4,16] .
. .
2.1 --
T = f (t) t (t =0) (t =24).
T=f(t)
5
11T(C)
3
4 16 24 t(h)
f(t1)
T=f(t)
f(t2)
5
11T(C)
4 t 16 24 t(h)1 t2
001-126_update.indd 30 10/5/2013 8:49:03 pi
-
2.1 -- 31
, , - , t t1 2 4 16, [ , ] t t1 2< :
f t f t( ) ( )1 2 0 .
) , [16,24] .
f , x x1 2, x x1 2< :
f x f x( ) ( )1 2 + > + >( ) ( )
f , x x1 2, x x1 2< :
f (x1) > f (x2)
T=f(t)
5
11T(C)
3
O 4 16 24 t(h)
001-126_update.indd 32 10/5/2013 8:49:05 pi
-
2.1 -- 33
:) t1 4= , f ( )4 3= . :
f t f( ) ( ) =4 3 , t[ , ]0 24 T = f (t) t = 4 , f ( )4 3= .
:
x A0 , f x( )0 f min f (x) .
, f x x( ) = +3 14 . x4 0 , x,
3 04x , x,
3 1 14x + , x.:
f x f( ) ( ) 0 , x.
, f x0 0= , f ( )0 1=
) t2 16= , T(16)=11 . :
f t f( ) ( ) =16 11 , t[ , ]0 24 T = f (t) t =16
, f (16)=11. :
f, , x A0 () :
f x f x( ) ( ) 0 , x A
f, , x A0 () :
f x f x( ) ( ) 0 , x A
001-126_update.indd 33 10/5/2013 8:49:08 pi
-
34 2:
x A0 , f x( )0 - f max f ( x) . , f x x( ) = +3 14 .
x4 0 , x,
3 04x , x,
+ 3 1 14x , x.:
f x f( ) ( ) 0 , x
, f x0 0= , f ( )0 1= . () () .
(. ) (. )
(. ) (. ).
x
y
1
1 y=f(x)
y
x
1
11
y=f(x)
y
x
1y=f(x)
1
y
x
1
1
y=f(x)
001-126_update.indd 34 10/5/2013 8:49:09 pi
-
2.1 -- 35
) - Cf f . Cf y'y, - Cf y'y Cf .
, , M(x, y) Cf y'y M x y'( , ) M(x, y) M x y'( , ) Cf , y = f
(x) y = f (x), :
f . :
yy.
, f x x x( ) = +2 14 2 , x :
f x x x x x f x( ) ( ) ( ) ( ) = + = + =2 1 2 14 2 4 2
, y'y.
) - Cf f . Cf , - Cf - Cf .
f, , , x A :
x A f x f x( ) ( ) =
f x f x( ) ( ) =
y
xx
y
x
M M
O
f(x) f(x)
Cf
y
y
y
xxx
M
M
Of(x)
f(x)
Cf
001-126_update.indd 35 10/5/2013 8:49:10 pi
-
36 2:
, , M(x, y) Cf
M x y'( , ) M(x, y) M x y'( , ) Cf , y = f ( x) = y f x( ),
:
f x f x( ) ( ) = f . :
.
, f x x x( ) = 2 3 , x : f x x x x x f x( ) ( ) ( ) ( ) = = + =
2 23 3
, .
: y x= 2 ,y x= 4 , y x= 6 ., , y'y , , y x= , y x= 3 , y x= 5 .,
, , .
- f [6,6]. f :) f :
f, , , x A :
x A f x f x( ) ( ) =
001-126_update.indd 36 10/5/2013 8:49:11 pi
-
2.1 -- 37
i) , ii) , iii) .) f , - .
f , y'y . , y'y f, - f, ().
:) f : i) [0,2] [5,6],
ii) [ , ]2 0 [ , ], 6 5 [0,2] [5,6] f .
iii) [ , ] 5 2 [2,5] .
y
x 1
4
652256 x
A
A
B
001-126_update.indd 37 10/5/2013 8:49:12 pi
-
38 2:
) f 4 x 6 6. :
max ( ) ( ) ( )f x f f= = =6 6 4 f 0 x
0. :min f (x) = f (0) = 0.
1) - :
) ) .
2) - , .
3) :
i) f x x x( ) = +2 6 10 x=3.
ii) g x xx
( ) =+
212
x=1.
4) :
i) f x x x12 43 5( ) = + ii) f x x2 3 1( ) = + iii) f x x3 1( )
= +
iv) f x x x4
3 53( ) = v) f xxx5
2
1( ) =
+ vi) f x x
x6 22
1( ) =
+.
1
1
y=f(x)
x
y
2
2
y=g(x)
x
y
1
y=h(x)
x
y
1
2
001-126_update.indd 38 10/5/2013 8:49:14 pi
-
2.1 -- 39
5) :
i) f xx11( ) =
ii) f x x2 2( ) = iii) f x x x3 1 1( ) = +
iv) f xx
xx4 2
1
1( ) =
+
+ v) f x x5( ) = vi) f x x6
21( ) = .
6) .
7)
8) -
) ) .
y=f(x) y=g(x) y=h(x)
x x x
y y y
y=f(x) y=g(x) y=h(x)
x x x
y y y
C C C
x x x
y y y
1 32
001-126_update.indd 39 10/5/2013 8:49:15 pi
-
40 2:
2.2 -
) f x x( ) = +1 .
f xx xx x
( ),,
,= + 0 , c ( )
y
y=|x|1
y=|x|
1 1
1
1
1 1
O x
x
c
y=(x)
y=(x)+c
c cc
y
c
O
f xx xx x
( ),,
,= 2 2 0 . ,
:
, , , .
=( )( )MMOM
1 =( )( )MOM
1 =( )( )MMOM
1
1
=( )( )M
M1
1
=y
=x
=yx
=xy
= + >x y2 2 0
=( )( )MOM
1
=( )( )MMOM
1
1
=( )( )M
M1
1
2
1 1
2
x xx
t
y
t
(x,y)(x,y)
yy
( )( )
( )
001-126_update.indd 50 10/5/2013 8:49:28 pi
-
3.1 51
=( )( )M
M1
1
= + >x y2 2 0
, - ( ).
:
(x, y) ( )
= + >x y2 2 0 .
360
Ox - Oxy - . 30 , Ox
= + =360 30 390 . 360 , - :
= +360 , N* 0o < 360o Ox, , - - 30 , Ox 360 + 30 = 390 :
, ( x0) ,
, ( y0)
=y
=yx
=x
=xy
= + >x y2 2 0
= + = ( )360 30 390
y
x
390
30
0
0
0
y
t
t
x 30
390
0
001-126_update.indd 51 10/5/2013 8:49:29 pi
-
52 3:
:
= +( ),360 N 0 360 x y2 2 0 .
( ) Ox .
Ox , , - , +360 , . Ox , , , + 360 , - .
, k +360 , k , .
, k :
,
=y
=yx
=x
=xy
= + >x y2 2 0
( ) ,k + =360 ( )k + =360
( ) ,k + =360 ( )k + =360
( x0)
, ( y0) ,
001-126_update.indd 52 10/5/2013 8:49:30 pi
-
3.1 53
(0,0) - = 1 - . - .
- , .. - = 35 , (, ).
35 = =1
35 0 57 = , .
, , , . - , .
,
35 = =1, 35 0 82 = , .
, - (x, y), :
x'x , y'y .
:
1. , 1. :
= x = = y =
1 1 1 1
7060
5040
20
30
10
1350
340
330320
310300
290280270260
250240
220230
210
200
190
180170
160
150
140130
120110
100 90 80
y
x
(x,y)
N(,)
x
y
35
0,1
0,1 0
o
o
o
o
o
o
o
o
o o
o
o o o o
o
o
o
o
o
o
o
o
o
o
o
o o
o o
o
o
o o
o
001-126_update.indd 53 10/5/2013 8:49:31 pi
-
54 3:
2. , , .
+ + + + + + + +
- M(x, y). - . 1 ,
yE , (AE)= yE ,
= yE.
.
:
= yE =
, x = 1 , -.
AO
B
B
M
A
t
x
y
E(1,y )E
1 2 3 4
= = =( )( )
( ) ( )
1
001-126_update.indd 54 10/5/2013 8:49:32 pi
-
3.1 55
. , AB (, ) ( 1rad), . , ( rad) S = . - :
( 1 rad ) , , ( 1 rad).
- , :
rad . - 2,
360 2 rad.
,
1 rad 3602pi
,
,
rad pi
180
.
, pi
= 180
, :
:9 60 ,
= 60 pi
pi
= =60180 3
603
=pi
rad.
A
1rad
B
O
001-126_update.indd 55 7/5/2014 4:40:07
-
56 3:
9 56pi
rad ,
pi
=180
pi
=56
56
18056 180
150
pi
pi
= = =
56pi rad = 150.
- .
rad
0 0 0 1 0
30pi6
12
32
33 3
45pi4
22
22
1 1
60pi3
32
12
3 33
90pi2 1 0
0
, x rad x,
(x rad), (x rad), (x rad) (x rad),
x, x, x x.
, .. pi3rad
pi3
(100rad ) 100 .
001-126_update.indd 56 10/5/2013 8:49:36 pi
-
3.1 57
1 h . .
:
48 = h
, = h48
70 = h
, = h70
h h
48 7020
= , h =
20 70 4870 48
.
70 2 75 , 48 111 , .
h 61 051 64
37,,
37m .
2 750 .
750 360 2 30,
750 2 360 30 30 12
= + = =( )
750 30 32
= =
750 30 33
= = 750 30 3 = =
750 2 360 30 = +
(1) :
A = = 20m
K
h
4 704820 m
001-126_update.indd 57 10/5/2013 8:49:38 pi
-
58 3:
3 793pi
rad.
793
796
2pi pi= . 79 6
13 1. 793
796
2 13 16
2 13 23
pipi pi pi
pi= = +
= + ,
:
pi
pipi
pi79
313 2
3 332
= +
= =
pi
pi793 3
12
= =
pi
pi79
3 33= =
pi
pi793 3
33
= =
1. x, y .
2. .
3. S = 6cm. , :
i) = 1cm ii) = 2cm iii) = 3cm .
4. rad i) 30 ii) 120 iii) 1260 iv) 1485.
5. :
i) pi10
rad ii) 56pi
rad iii)
913pi
rad iv) 100rad .
6. i) 1830 ii) 2940 iii) 1980 iv) 3600.
A
x
260
3
B
B
A
y
0
6
300
300
4
001-126_update.indd 58 10/5/2013 8:49:40 pi
-
3.1 59
1. - , - 1000 (1 0,3 m) - .
.
i) h = 60. ii) , h=1000 ;
2. :
i) : () = () = 2 45 = 2.
ii) ( ) , . = 4 22 5
iii) ().
iv) ,
, ( ) . = 2 2 2 v) 22 5, .
vi) ;
3. .
4. 1mm . - .
1.000 pi
N (N)
() (pi)
h
700
A B
1 1
22,5o 3 345o
A
B
6
300
300
001-126_update.indd 59 10/5/2013 8:49:41 pi
-
60 3:
3.2
. .
:
1.
M (x, y) , :
x = y =
,(OM) =1 ( ) 2 2 2 2 2= + = +x y x y :
x y2 2 1+ = ,
:
2 2 1+ =
2.
:
= =yx
( x = 0 )
= =xy
( y = 0 ).
AA
t
y
y
xx
1
|x|
|y|
(x,y)
O
001-126_update.indd 60 10/5/2013 8:49:42 pi
-
3.2 61
(1) (2), - .
3. . = 1
:
=
= ( 0 0)
:
= =1 .
4.
i) 2 2 1+ = 2 0 :
2
2
2
2 22
22
2
1 1 1 11
+ = + = =+ .
22
11
=+
.
ii) 2 2 1+ =
22
11
=+
,
:
22
22
22
2
11
1 1 11 1
++
= = +
=+
.
22
21=
+.
22
11
=+
22
21=
+.
001-126_update.indd 61 10/5/2013 8:49:43 pi
-
62 3:
1 = 513
90 180 <
-
3.2 63
1. x = 35
pi
pi2<
-
64 3:
11. :
i)
1
1 2+
++
=
ii)
xx
xx x1 1
2
++
= .
12. :
i)
++
= ii) 2 2 2 2 = .
13. :
i)
xx
xx
x x1 1
+
= + ii) ( )1 1 1 +
=
xx
x x
iii)
iv)
B
1. x + x = , -:
i) x x ii) 1 1 x x
+
iii) x x+ iv) 3 3x x+ .
2. :
i) 4 4 2 21 2x x x x+ = ii) 6 6 2 21 3x x x x+ = .
iii) 2 36 6 4 4( ) ( ) x x x x+ + - x , .
3. <
-
3.3 1 65
3.3 1o
, , 0 90. ' ' .
' ,
' = , , - , x'x. - . , - :
( ) ( ) = = 30 30 12
( ) ( ) = =30 30 3
2
( ) ( ) = = 30 30 33
( ) ( ) = = 30 30 3
9 , :
pi
pi
= = 4 4
22
pi
pi
= =4 4
22
pi
pi
= = 4 4
1
pi
pi
= = 4 4
1
: . :9 :
( ) ( )( ) ( ) = =
= =
A
(x,y)
(x,y)
A
y
x
t'
t
001-126_update.indd 65 10/5/2013 8:49:50 pi
-
66 3:
180o
' - 180 , ' ,= 180 , , ' y'y . -. , - :
:
180 . :
9 150 180 30 = , :
9 23 3pi
pipi
= , :
pi
pipi
pi2
3 3 332
=
= =
150 180 30 30 12
= = =( )
150 180 30 30 32
= = = ( )
150 180 30 30 33
= = = ( )
150 180 30 30 3 = = = ( )
tt
y
x
A A
(x,y) (x,y)180
B
B
0
001-126_update.indd 66 10/5/2013 8:49:51 pi
-
3.3 1 67
180o
' 180 ' ,= +180 , - , ' . -. - , :
pi
pipi
pi2
3 3 312
=
= =
pi
pipi
pi2
3 3 33
=
= =
pi
pipi
pi2
3 3 333
=
= =
:
180 -, .
:
9 210 180 30 = + , :
210 180 30 30 12
= + = = ( )
210 180 30 30 3
2 = + = = ( )
210 180 30 30 3
3 = + = =( )
210 180 30 30 3 = + = =( )
y
A A
B
(x,y)
(x,y)
t
t
y
x
180 +0
001-126_update.indd 67 10/5/2013 8:49:53 pi
-
68 3:
9 43 3pi
pipi
= + , :
pi
pipi
pi4
3 3 332
= +
= =
pi
pipi
pi4
3 3 312
= +
= =
pi
pipi
pi4
3 3 33
= +
= =
pi
pipi
pi4
3 3 333
= +
= =
90o
' 90 , ' ,= 90 , , ' xOy . . - , - :
,
90 , - .
, 60 90 30 = , :
60 30 32
= = ,
60 30 12
= = ,
60 30 3 = = 60 3033
= =
( ) ( )( ) ( )90 9090 90
= =
= =
t
t
(x,y)
(y,x)B
A
B
A90
y
xO
y=x
0
001-126_update.indd 68 10/5/2013 8:49:54 pi
-
3.3 1 69
, 0 90.
1 36 1 54
=+ . -
54.
54 90 36 = ,
2 2 1+ =
2 254 54 1 + = , :
2 22
54 1 54 1 1 54
1 6 2 516
10 2 516
= = +
=
+=
,
:
:
2 - :
) 90 +, ) 270 ) 270 +
i) 90 90 + = ( ), :
( ) ( ( )) ( ) .90 90 + = = = 90 +.
54 5454
10 2 51 5
= =+
.
54 36 1 54
= =+
54 10 2 54
=
54 5454
1 5
10 2 5
= =+
001-126_update.indd 69 10/5/2013 8:49:56 pi
-
70 3:
ii) 270 180 90 = + ( ) , : ( ) ( ( )) ( )270 180 90 90 = + = =
.
270 .
iii) 270 360 90 360 90 + = + = + ( ), : ( ) ( ) ( )270 90 90 + =
= = .
270 +.
1. : i) 1200 ii) 2850.
2.
i) 1876pi
rad ii)
214pi
rad.
3. : i) = +( ) ii) + + =( ) 0
iii) 2 2=
+ iv) 2 2=
+ .
4.
5. : pi pi
pi
pi pi
( ) ( )
( ) ( )
+ +
+
x x x
x x x
2 92
13 212
= 1.
6. :
pi pi pi pi2 22 22
( ) ( ) ( ) . + +
x x x x
( ) ( )( ) ( )
. + +
18090
001-126_update.indd 70 10/5/2013 8:49:58 pi
-
3.3 1 71
1. :
2. :
pi pi pi
pi
pi
( ) ( )
( )
5 7 52
72
5
+
+
+ (( ).
7 52
72
12
pi pi
pi
+
=
3. pi pi3 6
5
+ +
=x x , -
:
pi
pi2 23 6
+ +
x x .
4. :0 1< +
+ +0 , xA : i) x T A+ , x T A
ii) f x T f x T f x( ) ( ) ( )+ = =
f.
001-126_update.indd 74 10/5/2013 8:50:04 pi
-
3.4 75
x xrad= ( ).
2.
, , :
x xx
=
:
1 0= { }x x x1
pi pi ( ) ( ) ,x x x+ = = .
, , :
x xx
=
:
2 0= { }x x .
f(x) = x
f(x) = x 2, 2, . [0,2]. x xrad . - , , . :
x 0 pi2, .
, - f(x) = x
02
, .pi
f(x) = x :
y
xx rad
B
A
M
M
O
001-126_update.indd 75 10/5/2013 8:50:05 pi
-
76 3:
pi
pi2,
pipi, 32
32
2pi pi,
x = pi2, pi
21=
x = 32pi , pi3
21= .
:
. [0, 2]:
. :
x 0 2
x 0 1 0,71 0 0,71 1 0,71 022
0 71 ,
pi4
pi2
34pi 5
4pi 3
2pi 7
4pi
x
x
pi2
32pi
20
0 1 0 1 0. .
y
x
0x2pi
O
1
12pipi
4
3pi2
7pi4
y = x
43pi4
pi 2pi5pi
001-126_update.indd 76 10/5/2013 8:50:07 pi
-
3.4 77
f(x) = x , 2, [2, 4], [4, 6] . [ , ],[ , ] 2 0 4 2pi pi pi . ,
.
. - x ( ) . = x x f(x) = x (0,0) .
f(x) = x
f(x) = x 2, 2, .. [0, 2]. :
- :
54pi
x
x
pi2
32pi
20
1 0 1 0 1. . .
x 0 2
x 1 0,71 0 0,71 1 0,71 0 0,71 1
pi4
pi2
54pi 3
2pi 7
4pi
pi/2pi/2
1.5
0.5
0.5
1
1.5
0
1
pipi 2pi2pi 3pi/2 7pi/23pi/25pi/2 4pi3pi5pi/2 x
y
y = x
34pi
001-126_update.indd 77 10/5/2013 8:50:08 pi
-
78 3:
y = x
0 2 x pi .
f(x) = x 2, :
y
x
pipi o
1
1
2pi 3pi 4pipi2
pi2
5pi2
7pi2
y = x
3pi2
. x ( ) . =x x f(x) = x y'y.
f(x) = x
f(x) = x ,
, ..
pi pi2 2
, .
( , pi pi2 2, ).
x rad - .
x .:
x pi2
pi2
' , . f(x) = x
pi pi2 2
, .
y
x
0x2pi
o
1
1
pi
2pipi
43pi2
7pi4
y = x
5pi4
3pi4
2pi
y
O
x
001-126_update.indd 78 10/5/2013 8:50:09 pi
-
3.4 79
x pi2
x .
x = pi2
-
f. x pi2
x +. x = pi2
f.
f(x)=x , , :
- . f(x)=x .
f(x)=x , ( 3.3: (x) = x ).
x 0
x 1 0 1
pi2
pi3
pi4
pi6
pi6
pi4
pi3
3 1 7 , 3 1 7 ,33
0 6 , 33
0 6 ,
x
y
2pipi pi 2pi
23pi
2pi
23pi
2pi
y = x
yy
O O
BE
MB
B
A AA A
B
x x
yy
O O
BE
MB
B
A AA A
B
x x
pi2
001-126_update.indd 79 10/5/2013 8:50:11 pi
-
80 3:
1 f(x)=3x.
f(x)=3x - (x)=x. 2, :
f x x x f x( ) ( ) ( )+ = + = =2 3 2 3pi pi , x . f x x x f x( )
( ) ( ), = = =2 3 2 3pi pi x .
- f(x)=3x.
2 f(x)=2x.
f(x)=2x , 2x 2, , x . , f(x)=2x . :
f x x x x f x( ) ( ) ( ) ( ),+ = + = + = =pi pi pi 2 2 2 2 x f x
x x x f x( ) ( ) ( ) ( ), = = = =pi pi pi 2 2 2 2 x. , -
f(x)=2x.
3 f(x)=32x.
3, 3 .
x 0 2
x 0 1 0 1 03x 0 3 0 3 0
pi2
32pi
x 0
2x 0 1 0 1 0
pi4
pi2
34pi
y = xy = 3x
y
2pi pi123
321 2pi 3pi 4pi
xpi
y = x
y = 2x
x3pi2 2 2 2
pi 3pi 2pipi pi
1
1
y
001-126_update.indd 80 10/5/2013 8:50:12 pi
-
3.4 81
f(x)= x, ,>0: (i) , ,
. (ii) 2pi
.
f(x)= x, ,>0
1. ,
i) f x x( ) ,= 2 g x x( ) , ,= 0 5 h x x( ) ,= 2 0 2 x pi . ii)
f x x( ) ,= 2 g x x( ) , ,= 0 5 h x x( ) ,= 2 0 2 x pi .2.
f x x( ) =
g x x( ) = +1 h x x( ) = +1 3.
f x x( ) = g x x( ) ,= 3 0 2 x pi .4.
f x x( ) = g x x( ) ,= 3 0 2 x pi .
5. -
; -; f - .
x 0
32x 0 3 0 3 0
pi2
34pi
f(x)=32x :
- .
y
x
3
3
pi 2pi3pi3pi0pi pi pi2
pi4 2 4 2
y = 32xpi4
001-126_update.indd 81 10/5/2013 8:50:14 pi
-
82 3:
6. f x x( ) = 2 .2
.
7.
i) f x x( ) = ii) g x x( ) = +1 iii) h x x( ) = +1 .
8. f x x( ) .= 2
9. f x x( ) .=
B
1. :
2.
y t=
3 6
pi , y
t .i) -
.
ii) 0 12 t .
pi
pi
pi
pi
pi
pi
pi
2pipi
pi pi 2pi
pi pi 2pi
pi pi 2pi
2pi
2pi
2pi1
1
1
1
3
2.5
0.5
x
x
x
x
x
x
x
y
y
y
y
y
y
y
i)
ii)
001-126_update.indd 82 10/5/2013 8:50:16 pi
-
3.5 83
3. 1m. ,
h t= +1 13
3 , t
. i) -
. ii) iii) -
0 t 2.4. H x
- x(t) =0,1+0,1.3t, t . i) .ii) 0 t 2. 0,15m;
3.5
x=
x = 12.
y = x
y = 12.
x, f x x( ) = -
12. 2,
x, (. ), 2 . 2, .
xy=x
pipiO
2pi
yy=
pi6
12
5pi6
pi1m
001-126_update.indd 83 10/5/2013 8:50:16 pi
-
84 3:
-
x = 12
[0, 2], pi6
pipi pi
=6
56
,
pi
pi
656
12
= = .
x = 12
, x = , = , :
,x = +2pi
x = + 2pi pi ( )
1 x = 32
pi3
32
= , pi
= 3
32
.
pix =
3, :
2 pi24
12
x +
=
pi6
12
= , pi pi24 6
x +
= ,
y
M M
x5pi/6
pi/6
12
AOx
x
= +
= +
26
2 56
pipi
pipi,
x
x
=
= + +
23
23
pipi
pi pipi,
001-126_update.indd 84 10/5/2013 8:50:18 pi
-
3.5 85
24
26
2 26 4 24
x x x+ = + = + = pi pi pi pi pi pi pi pi
24
26
2 26 4
724
x x x+ = +
= + = +
pipi pi
pipi pi
pi pipi
pi
x =
, ,
.. x = 12.
-
x = 12
[ , ]pi pi pi3
pi3,
pi
pi
3 312
=
= .
x = 12
x
x
=
= +
pipi
pipi
24
724
,
24
26
24
26
x
x
+ = +
+ = +
pipi
pi
pipi pi
pi,
x
x
= +
=
23
23
pipi
pipi,
y
x
M
M
pi3pi3
12
001-126_update.indd 85 10/5/2013 8:50:20 pi
-
86 3:
, x = , = ,
1 x = 22
pi4
22
= , pix =4,
:
2 2 32x =
pi6
32
= , pipi
= 6
32
pi56
32
= .
pi2 56
x = ,
x
x
= +
=
24
24
pipi
pipi,
2 2 56
2 2 56
x
x
= +
=
pipi
pipi,
x
x
= +
=
pipi
pipi
512
512
,
x = 2 +
x = 2
001-126_update.indd 86 10/5/2013 8:50:22 pi
-
3.5 87
x =
x = 3 . - . , ,
, ..
pi pi2 2,
, .
, x = 3
. pi3,
pi3
3= .
x = 3 : x = +pi pi3, .
, x = , x = , :
x = .
1 x = 1
pi4
1= , pi
= 4
1. pix =
4,
x = pi pi
4,
2 x = 3
x = +pi pi6,
pi6
3= , pix =6,
pi xpi pi
y
pi22
3
x = + ,
001-126_update.indd 87 10/5/2013 8:50:24 pi
-
88 3:
1.
i) x = 0 ii) x = 22
iii) x = 0 iv) x = 22
2.
i) x = 12
ii) x = 1 iii) x = 22
iv) x = 1
3.
i) x = 0
ii) x = 33
iii) x =1
iv) x = 3
4.
i) x = 33
ii) x = 33
5. i) ( )( )1 2 3 0 = x x
ii) ( )( )2 2 1 0 x x+ =
6.
i) ( )( )3 1 0+ = x x
ii) ( )( )2 1 3 02 x x x+ = 7. -
i) x = 0 951,
ii) x = 0 809, iii) x = 28 636,
8. i) .
ii)
x5
1 0+ = iii) 327
3 0 x =
9.
i) pix +
= 3
1
ii) 2 34
1 pix
= iii)
pi4
5 3
=x
10. i) 2 1 02 + = ii) 2 3 2 0
2 x x+ = iii) 3 3 2 32 t t= +
11. i) 2 25 4x x+ = ii) x x =2 1 12. x,
:
i) . 0 2
-
3.6 89
13. ( )
S t= + 75 506
pi , t o
t = 1 . i) 100000 . ii)
. 1.
i) pix x+
=4
0 ii) pi23
3 0x x +
=
2. i) x x x x + = +1 ii) .
3. x =1 (3,4).
* 4. 1+x=x [0,2).
5. : pix x= +
3
[0, 2).
3.6
, M M1 2, (. 1).
, - . (. 2).
O O
y y
x x
M (, )1M (, ) M((), ())2
(1,0)
. (1) . (2)
(1,0)
001-126_update.indd 89 10/5/2013 8:50:28 pi
-
90 3:
, M M1 2, , :
1: 2: : 1 0 : ( ) ( )
M OM AOM2 1 ,= = ( ) ( ).M M AM2 1 = : ( ) ( )M M AM2 1
2 2=
:
( ) ( ) ( ) ,PP x x y y1 2 2 12
2 12= +
P x y1 1 1( , ) P x y2 2 2( , ), :
= +2 2( )
(2)
= 2 2 ( ).
( ) ( ) 2 12 2=
2 2 2 2 + = ( ) ( )
+ = ( ):
:
(1)
, , 0 3600 <
-
3.6 91
(1) (2) - , . , :
(1), , - .
, :
pi
2
=x x
pi
2
=x x, :
: (3)
(4)
:
= = +
= + =
15 45 30 45 30 45 30
22
32
22
12
2
( )
(( )3 14+
= + =
= =
75 45 30 45 30 45 30
22
32
22
12
2
( )
(( )3 14
= =
= =
15 45 30 45 30 45 30
22
32
22
12
2 3
( )
( 14
)
= + = +
= + =
75 45 30 45 30 45 30
22
32
22
12
2 3
( )
( ++14
)
(+) = +
( ) =
pi
=
2
pi
+
= +2
001-126_update.indd 91 10/5/2013 8:50:31 pi
-
92 3:
: (5)
, :
(6)
, :
(7) (8)
( )+ = +1
( ) = +1
( )+ = +
1
( ) = +
1
, :
= =
+=
+=
+
15 45 30 45 30
1 45 30
1 33
1 33
3 33 3
( )
== +
=
= ( )( )( )( )3 3 3 33 3 3 3
12 6 36
2 3
+ , , -.
, : ( ),+ , ( ) ,+ 0 0 0. :
0[ ]
( ) ( )( )
+ =++
=+
=
+
=+
1
001-126_update.indd 92 10/5/2013 8:50:33 pi
-
3.6 93
-
1 = 35, 3
22pi pi< < =
1213
, pi pi< < 32
,
+.
(+) = + ( )+ = . :
2 21 1 925
1625
= = = , 32
2pi pi< 1 , x =1 x
-
144 4: -
0 x x x x x2 6 0 3 2 0 3+ + ( )( ) x 2, x x2 6+ x < 3 x >
2 , x = 3 x = 2 < 2
1 52
1 52
0
127-208_update.indd 144 10/5/2013 8:53:09 pi
-
4.3 145
. 1 52
2+
, (x) ,
, x 2, . , (x) .
x , 1 52
1 52
< 2.
, . , - , - :
: f (, f()) (,f()) x'x, .
,
-
146 4: -
f x x x( ) = +3 3 1
1o :
, , (1,2).
2o : 1,1, 1,2, ... 1,9 :
, (1,5 , 1,6).
3o : (1,5 , 1,6) :
, (1,53 , 1,54) 1,53 < < 1,54. = 1,5.
ff( , ) ,( , ) ,1 5 0 13 01 6 0 30 0
ff( , ) ,( , ) ,1 53 0 01 01 54 0 03 0
1. .
i) 5 64 2x x= ii) x x x3 22 9 18 0+ =
iii) 3 5 3 55 4 3 2x x x x+ = + iv) x6 64 0 = v) x x3 2 2 0+ =
vi) x x3 7 6 0 + =vii) ( )x + + =1 1 03 viii) 7 3 2 1 3 2 1 0
2 2 3( ) ( ) ( )( )x x x x+ + =
ix) x x x x3 2 28 7 5 6 9 36+ = + + + ( ) x) x x x
4 33 6 4 0 + =
x x3 3 1 0 + = - 1 2. .
ff( )( )1 1 02 3 0=
1 2
B(2,3)
A(1,1)
1,5x
127-208_update.indd 146 10/5/2013 8:53:11 pi
-
4.3 147
2.
i) x x x3 23 2 0 + + = ii) 3 8 15 4 03 2x x x+ + =
iii) x x3 10 12 0 = iv) x x x3 22 7 6 0+ + + =
3. .
i) x x4 3 2 0+ = ii) 2 3 6 24 5 04 3 2x x x x + + =
4. , x.
i) P x x x x x x( ) ( )( )( )= +2 3 2 12 2 ii) Q x x x x x x( )
( )( )( )= + + + +2 2 24 3 2 1
5. :
i) 2 162 05x x ii) ( )( )x x x x3 2 22 2 9 0 + > iii) 2 5 6 9
03 2x x x + > iv) x x x3 24 3 18 0 +
6. :
i) x x x3 22 3 6 0+ + + > ii) x x x x4 3 26 22 30 13 0 + +
iii) x x3 3 2 0 + < iv) x x x x4 3 2 3 6 0 +
7. x'x :
i) f x x x x( ) = 3 3 5 23 2 ii) g x x x( ) = 4 3 13
8. , - f x x x x x( ) = + +4 3 25 3 x'x.
9. :
i) x x8 415 16 0 = ii) ( ) ( )x x + =1 9 1 8 06 3
iii) 61
51
6 02x
xx
x+
+ +
=
10. x x3 5 3 0+ = (0,1) .
127-208_update.indd 147 10/5/2013 8:53:13 pi
-
148 4: -
1. :
i) 110
12
15
45
03 2x x x+ + = , ii) x x x3 256
223
52
0 + =
2. , P x x x x x( ) = + + 4 3 2 16 12 x+1 x 2. (x) = 0.
3. k , x x kx3 2 3 0 + + = .
* 4. x x + =2 2 0, N, 2, .
5. P x x x x k( ) = +6 4 25 10 , k x 1 (x). k (x) = 0.
6. - 5dm 9dm, . , - dm 21dm3.
7. t
c t tt
=+
+3
50
2
3 . ,
3 2 300 200 04 3t t t+ = . t .
8. 36m3, x.
3 x 3
3
3
x
x
x
1m
xm
xmxm
127-208_update.indd 148 7/5/2014 4:46:42
-
4.3 149
9. . V,
.
10. t , mm - d = 15t(t2 6t 9). ;
11. , , 108cm ( ). - , 11664 cm3.
12. i) (1,2)
ii) y = x
3+x2 x - .
iii) .
V = + 5003
2000 100 20 2 3pi ( ),
B 12
12
, .
3 33x xy
x3 + x2 5x + 3 = 0
127-208_update.indd 149 10/5/2013 8:53:16 pi
-
150 4: -
13. . :
) 200ml , 1cm, - . x x x3 23 2 200 0+ + = . x mm.
) - 1lit 10cm . r r3 210 318 0+ = r
mm. ( 1000 318pi
).
) - , 250ml, 5cm .
h ( -).
4.4
, , . .
1 xx x x
2 22 1
12 1
0+
=( )
x x 0 x 12. -
:
xcm
x+2 cmx+1 cm
r+10 cm
r cm
h+5cmh+5cm
127-208_update.indd 150 10/5/2013 8:53:17 pi
-
4.4 151
3 2 7 2x x+ = .
x x 72. x :
( )
3 1. x=1 .
xx x x
x x x x xx
x xx x
2 222 1
12 1
0 2 1 2 1 22 1
2 1 12 1
+
= +
( )( ) ( ) ( )
( )==
+ =
+ =
+ =
0
2 2 1 02 1 2 1 0
2 1 1 0
4 3
3
3
x x xx x xx x( )
( )( )
, , . - - .
2 7 2x x+ = +
( ) ( )2 7 22 2x x+ = +
2 7 4 42x x x+ = + +
x x2 2 3 0+ =
12
1. - x = 1.
2 x x= 2.
x 0. -, x x x= +2 4 4, x x2 5 4 0 + = x1 = 4 x2 = 1. x, - x 0 .
, :
x = 4 : 4 4 2= x = 1 : 1 1 2= . x=4.
( )
127-208_update.indd 151 10/5/2013 8:53:19 pi
-
152 4: -
4 2 6 4 1x x+ + = .
x, 2 6 0x + x + 4 0, x 3. ' x :
3 5. x = 5 .
2 6 1 4x x+ = + +
( ) ( )2 6 1 42 2x x+ = + +
2 6 1 2 4 4x x x+ = + + + +
x x+ = +1 2 4
( ) ( )x x+ = +1 4 42
x x2 2 15 0 =
( )
( )
( )
(x)
> 0 (< 0)B(x)
A xB x
A x B x( )( )
( ) ( )> >0 0 A xB x
A x B x( )( )
( ) ( ) ,< 0 A x B x( ) ( ) < 0 (x).
, ( )( ) .x x x
x x + +
+ >
1 2 16
02
2
( )( )( )x x x x x + + + >1 6 2 1 02 2 , P(x)> 0 , ,
P(x)
x +( , ) ( , ).3 1 2
.:
2, 3 4, x , .
127-208_update.indd 152 10/5/2013 8:53:21 pi
-
4.4 153
x xx x
2
2
4 33 4
0 ++
. :
x xx x
x x x x2
22 24 3
3 40 4 3 3 4 0 +
+ + + ( )( ) x x2 3 4 0+ .
x x2 4 3 + 1 3 , x x2 3 4+ 1 4 . :
P x x x x x( ) ( )( )= + + 2 24 3 3 4
x - +x2- 4x + 3 + + - +x2 + 3x-4 + - + +
P(x) + - - +
-4
0
1
00
30
x +( , ) [ , ).4 3
1. :
i) 3 1
12 3 22
2
2xx x x
x xx
= +
ii)
xx x x
2
2121
41
+
=
2. : 2 2 2 03 2 x x x+ + =
3. :
i) x x3 4=
ii) 3 2 4x = iii) 5 1 4x =
iv) x x+ = +3 1 v) x x+ = +3 10 1 vi) x x+ =20 10
vii) x xx
=
+8
23
viii)
1 2 1+ = +x x
4. :
i)
xx+
>21
0
ii) 2 1
30x
x+
iii) x xx x
2
2
22
0 +
A xB x( )( )
0 -
x
A x B x( ) ( ) 0 B x( ) . 0
127-208_update.indd 153 10/5/2013 8:53:23 pi
-
154 4: -
( )
5. :
i) 2 3
14x
x+
>
ii) xx+
2
3 54 iii)
x xx
2 3 101
2 0
+
iv) xx x3 5
21
v) xx x2 1
32
+
6. : xx x x
2 22 1
12 1
+
( )
B
1. :
i) 2 3 1 3x x+ < ii) x x > 3 52. :
i) x x+ =3 10 0 ii) x x23 3 6 0+ =3. :
i) x x x x2 24 2+ = + ii)
4. :
i) x =1 ii) 4 1 22x x+ = 5. 2 3 3 3 4 04 3 2 x x x x + =
1. ( )( ) ,x x x x2 31 1 1+ + = - P x x x x( ) = + ++ +3 3 1 3 2
x
2+x+1 (, , ).
2. :
i) f x x x( ) ( )= + ++ 1 1 1 ( ) .x 12 -
.
ii) g x x x x( ) ( )= + + 2 21 ( ) .x 13
3. :
i) x x x4 32 2 1 0+ + + = ii) x x x x4 3 24 1 0+ + + =
( x x x x4 3 2 0+ + + + = , 0. ).
x x x + = +1 4 4
127-208_update.indd 154 10/5/2013 8:53:25 pi
-
(' ) 155
: x2
4.
i) x x x x4 3 216 2 4 0+ + = ii) x x x x4 3 28 13 8 1 0+ + + =5.
( ) ( )x x x x2 2 22 1 3 2 3 14 0+ + + + =
: x x y2 2 1+ = .6. , x x x5 23+ + + , -
x2 2 , 5x+8.
7. P x x x x x x( ) ... ,= + + + 17 16 15 1412 12 12 12 1 (11)
(13).
8. - . - C , x = + 10 100 270 1803 2x x x .
i) ;
ii) ;
xx
y+ =1 .
127-208_update.indd 155 10/5/2013 8:53:27 pi
-
156 4: -
127-208_update.indd 156 10/5/2013 8:53:29 pi
-
157
* 3 : x x x3 2 0+ + + = - x y=
3 x Ax B3 + = .
127-208_update.indd 157 10/5/2013 8:53:31 pi
-
158 4: -
127-208_update.indd 158 10/5/2013 8:53:33 pi
-
127-208_update.indd 159 10/5/2013 8:53:33 pi
-
. : , :
:
=
=
12
12
12
12
18
3
:
0 1=
= =
1 1 , 0 *
- . :
=
=
23
32
94
2 2
212 , 5
14 -
, 0, .
. O ,
. T -
.. 325 ; Av ( ) p q pq=
, :
4
-
5
5.1
*
( )3 3 325 5
255 2= =
127-208_update.indd 160 7/5/2014 4:49:33
-
5.1 161
325 x5 23= , 325 .
3 325 25= .
Av >0, , :
=
, , , , 0 0 = .
..
, .To . ..
- :
8 8 64 423 23 3= = =
27 27 127
127
13
43 43
43
43 4
= = = =
4 314
13
14
13
712 712 = = = =
+
21,4
1,43,21
2
1.4
5
1.4
3.21
7 3
xy
xy
xy ( )
=
+/ =
=
: x x , - ;
3 2 ; ( B' . 45)
2 , , .
1, 1,4, 1,41, 1,414, 1,4142, 1,41421, 1,414213,.... (1)
573
127-208_update.indd 161 10/5/2013 8:53:36 pi
-
162 5:
2 3:31, 31,4, 31,41, 31,414, 31,4142, 31,41421, 31,414213, (2)
:
3 33 65553673 0696503 769503
44 74 724
1
1 4
1 41
1 414
1 4142
=,
,
,
,
,,,
,,,,
,
,
7284 7284 7288
73393 78393 015
1 41421
1 414213
Av : (1) , - (2) , - .
3 2 . 3 4 72882 , .
:
Av >0, x x , + , ( ) , - ( ).
To x x. :
, x > 0, 0x= 0.O :
3 2
2
x =lim
3 2
2
=
=expxy
xy x
127-208_update.indd 162 10/5/2013 8:53:36 pi
-
5.1 163
, , - .
:
, x x x, ,1 2, :
x x x x1 2 1 2 = +
( ) = x x x
=x x
x
( ) x x x x1 2 1 2=
. x x . x x, :
f : f x x( ) ,= , 1, .
= 1, f(x) = 1.
f(x) = 2x . - :
x -5 -4 -3 -2 -1 0 1 2 3 4 5
1 2 4 8 16 3212
14
18
116
132y=2
x
(x, y) - . , -
f x x( ) = >1, : 0
1
1x
y
y=2x
x x x x1 2 1 2: =
127-208_update.indd 163 10/5/2013 8:53:38 pi
-
164 5:
.
( , )0 + .
. x x1 2, :
x x1 2< , x x1 2<
y'y (0,1) - x.
x
x
y
x1
2x
1x
2
y= , >1x
A(0,1)
g xx
( ) .=
12
:
x -5 -4 -3 -2 -1 0 1 2 3 4 5
32 16 8 4 2 112
14
18
116
132
12
x
(x, y) - . , -
f x x( ) = 0
-
5.1 165
f(x) = 2x g xx
( ) =
12
x :
y'y.
f(x)=x, 0 1< ,
: x x1 2 , x x1 2
, , :
x x1 2= , x x1 2= .
, :
, . .
x x x x1 2 1 2= =
xx 1
1
x
y
y=2y=2 xx
x2
1
x2
x
1 2
1x
x
yy
xO Ox1x 2x x
g x f xx
xx( ) ( )=
= = =
12
12
2
127-208_update.indd 165 10/5/2013 8:53:41 pi
-
166 5:
i) :
ii)
3x = y, y28y9 = 0 1 9.
: 3 1x = 3 9x =
' , 3x> 0, 3x = 32 x=2, .
2 :
3x = 2y = :
2 3 3 2 22
5 3 12
2 9
=
+ =
x y
x y
= 1 = 8,
x = 0 y = 3.
2 3 22
5 12
9
=
+ =
3 12 8
x
y
=
=
3 32 2
0
3
x
y
=
=
( )
1 :
i) 2164
3x = ii) 9 8 3 9 0x x =
11
2 164
2 2
3 62
3 3 6x x
xx
= =
= =
[ ]32x 8 . 3x 9 = 0 (3x)2 8. 3x9 = 0
127-208_update.indd 166 10/5/2013 8:53:42 pi
-
5.1 167
i)
ii)
4 :
i) f x x( ) = +2 3 ii) g x x( ) = 2 3 iii) h x x( ) = +2 23
i) f - (x) = 2x 3 .
ii) g (x) = 2x 3 .
iii) h (x) = 2x 3 2 .
3 :
i) ii)
3 > 1
3 19
2 3x x >
x
y
4
O
1
1
3
y=2
y=2x
x+3
x1O
y=2
y=2x3
y
3
x
1x
y
1
1
O
y=2
y=2x3+2
32A(3,3)
x
3 19
3 3
3 23 2 0
2 23 3 2
2
2
x x x x
x xx x
> >
>
+ >
x < 1 x > 2
12
14
12
12
22 0
2 2 2
2
2
<
+ >
+ +x x x x
x xx x
< x 2
12
1 0, c . . c > 0 Q - , c
-
170 5:
5 ,
Q t Qt
( ) =
052 .
5 , Q t Q ec t( ) = 0 :
Q t Qt
( ) .=
052
1. :
i) f x x( ) = 3 f xx
113
( ) =
ii) f x x( ) ,= 3 f xx
2 3 2( ) = + f xx
3 3 3( ) = iii) f x x( ) ,= 3 f x
x4
23( ) = f x x523( ) = +
iv) f x x( ) = 3 f x x623 1( ) = +
v) g x ex( ) = , g x ex12( ) ,= + g x e x2 ( ) =
g x e x3 2( ) = +
2. :
i) 2 64x = ii) iii) iv) 3 181
=x
v) 34
6427
=x
vi) 27 94 1x x= + vii) 32 16
1x x= viii) 3 12 2x x =
3. :
i) 2 4 2 02 1x x+ = ii) 2 4 5 2 2 0 + =x x
iii) 3 26 3 9 02 1x x+ =4. :
i) 5 12 5 6x x + < ii) 7 72 4 1x x +> iii) 1
212
1 2 4
0 :
*7. : i) log log =1 ii) log log
2 3 6 =
iii) log log log =1
*8. : i) log log
+ =1 0
ii) log ( ) log ( ) log ( ) log ( ) + =
log log x x= 22
5.3
.
4.2, x>0 log . x ,
x +( , )0 log , x
f : ( , )0 + f x x( ) log= .
, , f x x( ) log .=
log , x y xy= =
(,) y x= log , (,) y = x . , , (,) (,) xy x y.' '
:
y = x
xy x y.' 'y x= log
127-208_update.indd 181 7/5/2014 4:52:44
-
182 5:
f x x( ) = :
> 1, g x x( ) log := ( , )0 + .
,
x x1 2< , log log x x1 21 :
.
2>1 log0,5 x >1 0. :
x x=2 . , x = 2.
1. :
f x x( ) log= 2 g x x( ) log= 12
; .
2. :
f x x( ) log= , g x x( ) log= 1 h x x( ) log( )= 13. f x x( ) =
-
g x x( ) log= :
i) (2,4) ii) B( , )2 4 iii) ( , )2 4 i) ( , ) 2 44. ASA
DIN. x ASA y DIN y = 1+101ogx, x= 50, 100, 200, 400, 800, 1600
ASA. ; ( log2 = 0,3).
0 25 0 5, , ,>
2 0 5 1 0 50 5 0 50 25 0 5
2
log , log ,log , log ,log , log ,
>
>>
log ( ) log log ( ) log ( ) log [ ( )](
22
2 2 22
22
2 1 2 12
x x x x x xx x
= + =
= xx 1)
=x 1 x = 2
127-208_update.indd 184 10/5/2013 8:54:12 pi
-
5.3 185
5. : i) log x log x( ) ( ) log+ + =1 1 2 ii) log x x( ) log log
+ = 1 1 5 iii) log (log )x x2 2= iv) log( ) log logx x2 1 2+ =6. :
i) 5 21x x= ii) 3 21 1x x +=7. : i) log3 2 log3 5 ii) log ,0 3 5
log ,0 3 7
iii) log( )x2 1+ log2x
8. [ ]H+ >10 7 [ ] .H+ ii) log( ) logx x2 4 3 < iii) x
xlog >10* 9. log log2 63 9>10. , > 0 :
>
127-208_update.indd 185 7/5/2014 4:53:33
-
186 5:
1. : i) ( )x x x2 3 53 1 1 + = ii) x xx x
2 3 1+ + =
2. :
log log log log + + + = 2 ( + , 1 )
3. ( ) ,log = 2 log , log log
.
4. , , , :
5. log log5 1 2= x xlog( )2 5=
6. 02
, pi
:
log log log logx x x x 2 2 2 2 0+ + =
7. 02
, pi
: ( ) ( )
x xx x=
8. : 27 12 2 8 0x x x+ >
()
log loglog log
loglog
= 0 1<
, , ,
127-208_update.indd 186 10/5/2013 8:54:17 pi
-
187
*
127-208_update.indd 187 10/5/2013 8:54:19 pi
-
188 5:
0
1
100 000 000
10 000
=
= +
+
. .
.
0 100 000 000= . .
1 100 010 000= . .
0 0=
10 1=
20 2=
= +
10 1
110
84
23027 999 999 779= . .
10 =
230 270 23027. =
.....................................................................................................
= +
10 1
110
84 ,
2 100 020 001= . .
127-208_update.indd 188 10/5/2013 8:54:21 pi
-
189
21
42
84
168
3216
6432
= = = = = = ....
127-208_update.indd 189 10/5/2013 8:54:23 pi
-
190 5:
= +
10 1
110
84
101 1
101 1
108 410 10
4
101
4 4 4
= +
= +
0010
4
10 105 4 5
1 110
= +
1 110
2 7181459274104
+
= ,
xy
= +
1 1104
104
127-208_update.indd 190 10/5/2013 8:54:25 pi
-
191
x
H
y
x1y=
127-208_update.indd 191 10/5/2013 8:54:27 pi
-
192 5:
127-208_update.indd 192 10/5/2013 8:54:29 pi
-
(' ) 193
1. : 2 22 3 3x x x x =
2. i) x x+ = 2 2 2+ .
ii) t ( , ).pi pi
3.
3 3
1 3
3
2=
pi12
,
4. ( 9) 198 22 9= 9. 1+9+8 = 18
9. 17397 1933 9= 1+7+3+9+7 = 27 -
9. : 9, +++ -
9. : >= + + + 10 10 103 2 . -
f x x x x( ) = + + + 3 2 f x x x f( ) ( ) ( ) ( )= +1 1pi x=1
x=10.
5. ( ).
.
:
x x x+ + + + =
11
1 0 0....
. 0
, 0 . :
i) :
2 1 03 2x x x+ + = 6 29 27 9 1 04 3 2x x x x+ + + + = ii) 2 12
.6. 3 4 5x x x+ = .7.
( )
23
43
2
+ =x
x
( )1 2+ + = t x t x
x x x3 23 3 1 0 + = .
127-208_update.indd 193 10/5/2013 8:54:31 pi
-
194 (' )
8. 2 3log( ) log( )x x+ = ; 9. pix x+ =log
4
2 (0,) - .
10. : log ( )3 16 2 12 2 1x x x +
11. i) f x x( ) ln= g x x( ) = 1 ln .x x 1
ii) f x x( ) ln= g x x( ) = +2 1 ln .x x +2 1
12. , , , , :
, , > 0, ++ =
= =
:
13. , , , , :
, , > 0, ++ =
= =
() = , (KM) = , () = , = , = , =
14. , , , , :
0 < , , <
:
++ = .
15. , , , , :
0 < , , <
, (M) = , (KM) = , () = , = A, = , = .
= =
2 2 2
2 2 2
2 2 2
222
= +
= +
= +
B
2 2 2
2 2 2
2 2 2
222
= +
= +
= +
2 2 2
2 2 2
2 2 2
222
= +
= +
= +
127-208_update.indd 194 10/5/2013 8:54:33 pi
-
- 195
2.
8.
9.10.
12.
2.
2 11 1
+ ++
+
, ,
-
3 22
3 22
,
3 22
3 22
,
127-208_update.indd 195 10/5/2013 8:54:37 pi
-
196 -
1.
4.
1.
2.
3.
4.
5.
6.
1.
2.
3.
4.
127-208_update.indd 196 10/5/2013 8:54:41 pi
-
197
2.
3.
4.
5.
6.
3.
pi187
612
= , pi187
632
= ,
pi187
633
= , pi187
63= .
pi21
422
= , pi214
22
= ,
pi
pi21
4214
1= = .
1200 32
1200 12
= = , ,
1200 3 1200 33
= = , .
( ) , ( ) ,
( ) , ( ) .
= =
= =
2850 12
2850 32
2850 33
2850 3
3.3
( ) , + + =180
. 2 2
90+ +
=
ii)
,x x= + = 24
2 32
pipi
pipi
Z+
x x= = + 26
2 76
pipi
pipi
Z,
127-208_update.indd 197 10/5/2013 8:54:45 pi
-
198 -
3.
4.
5.
10.
12.
2 25
2 35
x
x
= +
= +
pipi
pipi Z,
x = pi pi 6, Z
.,x = + pi pi 2245
Z
x = +10 5pi pi Z,
x = + 42 712pi pi
, Z
x = 2 45
pipi
Z,
22
23
2 23
,
x
x
x= + = +
= +
pipi
pipi
pipi
Z
,
xx
x
= = +
=
24
2 54
2
pipi
pipi
pi Z
x x= = 2 23 3
pipi
pipi
.
x x ,= = + pi pi3 4
pipi
Z
69
6 29
x
x = +
=+
pi pi
,pi pi
Z
x
x = +
=
24 73624
36
pi pi
pi pi Z,
x = 1260pi pi
x = 2 56
pipi
,
22
26
2 56
,
pipi
pipi
pipi
=
= +
=
Z
127-208_update.indd 198 10/5/2013 8:54:48 pi
-
199
7.
x
x =
= +
pipi
pi pi
425
Z, . ( ) ,+ =
6533
127-208_update.indd 199 10/5/2013 8:54:52 pi
-
200 -
3.
4.
8.
9.
6.
1.
pi16
2 2 2
2 2 2=
+
+ +,
pi
pi3
8 8= ...
,
,
127-208_update.indd 200 10/5/2013 8:54:57 pi
-
201
5.
2.
x = pi 3
,
x = pi6
x = +6 29
pi pi
x = 6 29
pi pi, .Z
x x= = +pi pi pi pi12
712
127-208_update.indd 201 10/5/2013 8:55:01 pi
-
202 -
3.
4.
5.
6.
1. 2. 5. 6.
7.
1.-5.
6.-7.
8.-10.
11.
3.
4.
6.
7
9.
12.
13.
15.
1.
2.
3.
4.
5.
x
x ,
= +
= +
26
2 56
pipi
pipi
Z.
24
x = +pi pi
23 4
, x = + pi pi
Z
127-208_update.indd 202 10/5/2013 8:55:04 pi
-
203
1.
2.
3.
4.
5.
1.
2. 3.
4.
5. 6.
7. 8.
9. 10.
1. 2.
3.
4.
5.
1.
.
127-208_update.indd 203 10/5/2013 8:55:09 pi
-
204 -
2.
3.
6.
7.
8.
9.
1.
2.
3. 4.
5. 6. 7. 8. 9. 10.
11.
12.
1.
2.
3.
6.
1.
2. 3. 4.
5.
1.
2.
26
2 16
( ) ,
x
x
= +
= +
pipi
pipi
.
P x x x( ) ( ) + + =2 1x x x x x + + + +3 1 3 21 3
127-208_update.indd 204 10/5/2013 8:55:14 pi
-
205
3.
4.
5.
7.
8.
9.
2. x = 2
3.
4.
5.
1.
1
2.
3.
6.
7.
8.
9.
10.
11.
f x x x x x P x( ) ( ) ( ) .... ( ) ( )= = = 1 1 1
P x x x( ) ( ) ( )= 1 pi
127-208_update.indd 205 10/5/2013 8:55:17 pi
-
206 -
1.
2.
3. 5. 6. 7. 8.
1.
7.
8.
3.
4.
5.
6.
127-208_update.indd 206 10/5/2013 8:55:20 pi
-
- 207
1. x x= + = + pi pi pi pi 12 4
, Z .
2. ii)
3.
pi12
2 3= .
5. i x112
) =
i2
12
13
2 3 2 3) , , , +
ii) 2
x2 2 0 = .
6. x = 2.
7. x = 1.
8. < 0 =12.
9. x = pi4
.
10. 2 4094 3 8188, ,< x .
11. i) x ( , ]0 1 .
ii) x +[ , )1 .
( )
x x= + = + pi pi pi pi 12 4
, Z .
x = + 22
pipi
, Z .
x x x= = = +1 2 3 2 3 .
x x x= = = +1 2 3 2 3 .
x x x= = = +1 2 3 2 3 .
127-208_update.indd 207 10/5/2013 8:55:22 pi
-
127-208_update.indd 208 21/2/2014 4:42:47
-
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