: . .
: 1991 : 1992, 1993, 1994, 1995, 1996, 1997, 1998, 2012 .
. , . , . . , .
. (2010) , . , . , . , . . . : , (2010), . , . , . , . . .
(2010), (2010). : 1 , , . , . 2 , , , . 3 . , , . . 4 . 5 , .
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
1
1.1 x + y = x + y = , 0 0, , . : : 0, , :
x + y = y = x + y= x+ , = y'y . y 0 x+y=, =0 x
y
0
x+y=, 0 x
: 9 0, (. ),
10
1:
x'x y'y (. ). y = 0 ( 0 ), x+y=, =0 x = x = . 0 x y'y x'x . y y
: 1 y= x1 2 9 x 2y = 2 1 x+y=, =0 y x 1 = 2 1 y y = 0 2 1 2 0 x y=
x1 2 -1 y'y 1 . 9 = 0 , y =
x
1 y= x1 0 2 x 2 9 y=2 -1 x'x y'y 2 . 0 2 x -1
y
2 y
y=2
0 2
y=2
x
0 y
x
9 x=2 y'y x'x 2.
y 0
x=2
2 x=2
x
0 2 .
x
1.1
11
, (4,-1) x 2y = 6, 4 2(1) = 4 + 2 = 6. , , (16,5), (10, 8) 1 k,
k 3 , k . 2 2 x 2 x+y= x+y= , , , 2 x 2 x + y = ' x + ' y = ' . . .
. : . () (') , .. (), () + ' ( ') , () 0, (') '. () + ' ( ') ()
('). 1o : 9 0, : = = 9 = = , + = +. , , : 6 x 2y =(1) 8 3x + 4y
=(2)
12
1:
, ( ) , .. (1) x. x = 2y + 6 8 3x + 4y = x 3(2y + 6) + 4y = 6y +
18 + 4y = 8 8 10y = 10 y =1 x = 2y + 6 y = 1 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 ( ) ,
:
1.1
13
= x 2y 6 (3) 3x + 6y =18 8 3x + 4y = 8 1 3x + 4y = , , : 3x + 6y
+ 3x + 4y = 18 + 8 10y = 10 y = 1 . : x 2(1) = 6 x + 2 = 6 x = 4 .
(4, 1) ( ). 2 x 2 6 x 2y = 8 3x + 4y = . , .y 3x+4y=8 O x2y=6 A(4,
1) x
, x + y = ' x + ' y = ' , , . 2 x 2 . , 2 x 2 , , , .
14
1:
2 x 2 . : 2 y = x +1 3 2x + 3y = 3 , 9 4x 9y = 1 4 1 y x = 9 9 ,
. y , 2 1 1 (1, 0,3) . , x 1 0 (1, 1 ) . 3 3 x + 2y = 9 5 2x + 4y =
1 3 y= x+ 2 2 , , 1 5 y = x 2 4 . 2x y + 1 = 9 4x 2y = 2 y = 2x 1 ,
y = 2x 1 , . (k, 2k 1), k .y 1 0 x 2
y
1 2
0
x
1.1
15
, 2 x 2 : 9 9 9 . 2 x 2 2 x 2 . x + y = ' x + ' y = ' 0 0. : y =
x + (1 ) y =' x + ' ( ) 2 ' ' 1 2 ' 1 = 2 = ' ' , ' ' 0 , 1 2 ' , '
' ' ' x + = x + x = ' ' ' '
( ' ' )x = ' ' ' ' x= ' ' : ' ' '+ '+ ' ' y = + = ' ' ( ' ' ) =
( ' ' ) ( ' ' )
16
1:
y =
' ' ' '
, , ' ' ' ' (x, y) = , ' ' ' ' ' = , - =0 , 1 2 ' , . . =0 ' =0.
x + y = ' x + ' y = ' : ' ' 0 ' ' ' ' (x, y) = , ' ' ' ' ' ' = 0 .
' ' , ' ' : D= = ' ' . ' ' D= D, x , : = ' ' . ' ' , D, y , : Dx =
Dy = = ' ' . ' '
1.1
17
, , : x + y = ' x + ' y = '
D Dx y = y D D D=0, . D0, , (x,y) x = : 6 x 2y = 9 8 3x + 4y =
=1 4 3(2) = 4 + 6 =10 0, 3 4 . Dx = : = x 1 6 6 2 8 10 = 24 + 16 =
40 D y = = 18 = 3 8 8 4 D y 10 D x 40 = = 1 . = = 4 y = D 10 D 10
D= 1 2
, (x, y) (4, 1) . = 40 2x 3y = 9 80 4x 6y = 2 3 D= = 6) 4(3) =12
+ 12 = 2( 0 4 6 . 2, 40 2x 3y = , 40 2x 3y = 40. 2x 3y = 2x 40 2x
3y = 40 y = . 3
18
1:
2k 40 k, , k 3 11 3x + y = 9 6 9x + 3y = 3 1 = 99 = 0 9 3 . 11
3x + y = 2 3x + y = . D= 1 x y = 1 2 x 2y =
, . , . , . , . D, Dx, Dy. : D= 2
1 2
= 2 + 2 = ( 2)
Dx = Dy =
1 1 = 2( 1) + = 2 2 1 2
= 2 2 ( 1) = 2 (1 + 1) = 2 (2 )
1.1
19
, D = 0 . :D = 0 ( 2)= 0 =0 =2
: 9 D0, 0 2, (x, y), : Dx ( 2) 2 1 = = = = x D ( 2) ( 2) y= Dy D
= 2 (2 ) 2 ( 2) = = . ( 2) ( 2)
, 0 2 , 1 , . 9 D = 0 , = 0 = 2 , . : = 0 , 1 0x y = 1 y = 2y 0x
= 0 = 0 y . = 2 , = = 2x y 1 2x y 1 = = 4x 2y 2 2x y 1 . , 2x y =1
y = 2x 1 ,
( k, 2k 1) ,
k .
3 x 3 x+y+z=0 , , , , . . 2x+3y+z=6 (2, 1,5) , 22+3(1)+5=6.
20
1:
: 1x + 1 y + 1z = 1 , 2 x + 2 y + 2 z = 2 3 x + 3 y + 3 z = 3 ,
, , 3 x 3 1x + 1 y + 1z = 1 2 x + 2 y + 2 z = 2 x + y + z = 3 3 3 3
2 x 2 . (1) 2x y + 3z = 9 10 (2) x + 3y z = 3x + y z = 8 (3) . , ..
(3) z, . : 3x + y z = 8 z = 3x + y 8 (4) (1) (2) : 9 2x y + 3z = 9
2x y + 3(3x + y 8) = 9 11x + 2y = 15 9 x + 3y z = 10 x + 3y (3x + y
8) = 10 x + y = 1 (5), (6) 15 11x + 2y = , 1 x + y = x=1 y=2 . x y
(4) z = 3. (1, 2, 3). 3 x 3, , 2 x 2, 3 x 3 .
(5) (6)
1.1
21
4 x y = 1. 2 x + y = i) ii) . 2. x y = i) 7 8 x + y = 45 3. : x
5 2y + 1 0 2 + 7 +2= i) x + 6 y 6 = 8 3 2 4. : 3 x 3y = i) x 3 y =2
y+2 2x 1 3 = 4 4 ii) x x +3 3 = y 2 3 2y= x + 2 ii) 1 0 2 x y +1
=
x 1 y 2 = ii) 3 4 4x + 3y = 8
5. : 7 2x + y = i) 4 3x 5y = = 2y 3x 8 ii) 0 x + 3y + 1 =
6. , . 4 2x 5y = i) ii) 100 6x + 7y = 7. : 40 2x 3y = 80 4x 6y =
11 3x + y = iii) 2 9x 3y =
2 ( 3 1)x + 2y = i) x + ( 3 + 1)y =1 3
( 3 + 1)x + 4y = 7 ii) 1 1 x + ( 3 1)y = 2
22
1:
8. : 3x 2y =11 i) 2x 5y 2 =3 5x + y 2 =33
y x + 2 = 3 2 5x y + 3 = 4 ii) x 3y + = 2 iii) 3x + y + = 5 3x
2y + 2 = 2 2 5x + 3y 2 =16 y 1 2 O 1 1 4 1
1. i) 1 2 . ii) 1 2 ;
x
2. 26 , 68 . ; 3. 1,5 4. 2200 5050 . . 4. R T R =T + . 20 C 0,4
80 C 0,5 , . 5. , 50% 80% . 100 ml 68% ; 3 , 6. 1 : 2x + 4y = 2 : x
+ 2y = . i) 1 2 . ii) ; iii) ;
1.1
23
7. : i) 1 : x + y = 2 ii) 3 : x y = 2 : x + y =1. 2 : x + y =1.
( 2)x + 5y =5 ii) , x + ( + 2)y =53
8. : ( 1)x 2y =1 i) , 4x ( + 1)y = 2
9. 1 2 =6 , O1O3 = 5 O2O3 = 7 . . 10. , . N = x , B = y = z , ,
.
1
2
11. . 50% , 10% 30% . 52 lit 32% , . . 12. , y = x 2 + x + . .y
3
yy=f(x)
K(1,4) y=g(x)4
y
y=h(x)
O
K(2, 1)
x
1
O
x
O
2
4
x
24
1:
1.2 , . , 13 89. x, y , x + y = x 2 + y 2 =. 13 89 , : 13 (1) x
+ y = 2 2 89 x + y = (2) : (1), , .. x , (2). x + y = 13 y = 13 x x
2 + (13 x) 2 = 89 x 2 + 169 26x + x 2 = 89 0 2x 2 26x + 80 = 0. x 2
13x + 40 = 2 = 9 . : = x 13 3 = 28 5
(3).
(3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , . 5 x + y = (1)
1 . (2) xy = 6
1.2
25
(1) x (2). : x + y =5 y =5 x (3). xy =6 x(5 x) =6 5x x 2 = 6 x 2
5x + 6 = 0 x 2 = 3. = x (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 6 y = . x
.
y
A 1 O 1
B x
26
1:
(2,3) (3,2) . (2,3) (3,2) . 2 (1) xy = 6 2 2 13 x + y = (2)
(1) x (2). xy = 6 y = (2) :2 2
6 x2 2
6 x + y = 13 x + = 13 x 36 x2 + 2 = 13 x x 4 + 36 = 2 13x x 4
13x 2 + 36 = 0
. x 2 = , 2 13 + 36 = 0 , =9 =4 . 9 =9 2 x =9 x =3 x = 3 (1) x=3
y=2 x = 3 . 9 = 4 x 2 = 4 x = 2 x = 2 (1) x=2 y=3 x = 2 y = 3 .
(3,2), (3, 2), (2,3) (2, 3).
6 , x 2 2 x + y = (0,0) = 13 . 13
xy = 6 y =
1.2
27
. y
A
B x
1 B A
x 2 + y 2 + xy = 3 1. : . 1 x + y = 2. : y = 3x 2 i) 4 12x 3y =
x 2 + y2 = 9 ii) 0 x y = x 2 + y2 = 5 iii) xy = 2
. 1 3. S = 0 t + t 2 = 0 + t , 2 + 0 = S t . 2 x 2 2y + 10 = 1.
o 2 2 25 x + y = .
28
1:
2xy y 2 5y = 0 2. : 2 y = x 4x + 3 3. 120cm2 . 3cm , 2cm , . .
2x 4. y = x 2 y = + k, k . k . 2y = x 2 5. o y= x + . 1 I. :1 x y =
1 1 2 x + 2y = x + y = 2x + 4y = ( 4 ) : (1 ) : ( 3 ) : ( 2 ) : 2
1, 4, 0, x + 2y = x + 2y = 2x + 4y = x + y =1
, , . ) , ) , ) . (1) (2) (3) (4) II. , , . 1. , . A 2. D = 0 ,
. xy = 1 . 3. 0 x + y = 4. x 2 + y 2 = = x 2 + 1 1 y . A
2 . . 2.1 -- T = f (t) t (t =0) (t =24).11 T(C)
T=f(t) 5 3
4
16
24 t(h)
) [4,16] .11 T(C)
T=f(t) 5 f(t1) f(t2)
4
t1
t2
16
24 t(h)
2.1 --
31
, , , t1 , t 2 [4,16] t1 < t 2 : f (t1 ) < f (t 2 ) T = f
(t ) [4,16] . : f , x1 , x 2 x1 < x 2 : f (x1 ) < f (x 2 ) f
f . , f (x) 2x 3 . = x1 , x 2 R , x1 < x 2 . : x1 < x 2 2x1
< 2x 2 2x1 3 < 2x 2 3 : f (x1 ) < f (x 2 )
f (x) = x + , > 0 . ) , [16,24] .T(C)
T=f(t) 5 f(t1) f(t2)
4
16
t1
t2
24 t(h)
32
2:
, , , t1 , t 2 [16, 24] t1 < t 2 : f (t1 ) > f (t 2 ) T =
f (t ) [16,24]. : f , x1 , x 2 x1 < x 2 : f (x1 ) > f (x 2 )
f f , f (x) = + 5 . 2x x1 , x 2 , x1 < x 2 . : x1 < x 2 2x1
> 2x 2 2x1 + 5 > 2x 2 + 5 f (x1 ) > f (x 2 ) : f (x) = x +
, < 0 . . T = f (t).T(C)
11
T=f(t) 5 3
O
4
16
24 t(h)
2.1 --
33
: ) t1 = 4 , f (4) = 3 . : f (t) f (4) = t [0, 24] 3 , T = f (t)
t = 4 , f (4) = 3 . : f, , x 0 A () : f (x) f (x 0 ) , x A x 0 A ,
f (x 0 ) f min f (x) . , f (x) 3x 4 + 1 . = : x 4 0 , x , 3x 4 0 ,
x , 3x 4 + 1 1 , x . f (x) f (0) , x . , f x 0 = 0 , f (0) = 1 ) t
2 = 16 , T(16)=11 . : f (t) f (16) = t [0, 24] 11 , T =f (t) t =16
, f (16)=11. : f, , x 0 A () : f (x) f (x 0 ) , x A
34
2:
x 0 A , f (x 0 ) f max f ( x) . , f (x) = 4 + 1 . 3x : x 4 0 , x
, 3x 4 0 , x , 3x 4 + 1 1 , x . f (x) f (0) , x
, f x 0 = 0 , f (0) = 1 . () () . (. ) (. ) (. ) (. ). y y=f(x)
1 1 x 1 1
y y=f(x)
x
y 1 y=f(x) 1 x
y 1 y=f(x) 1 x
2.1 --
35
) Cf f . Cf y'y, Cf y'y Cf .M
y yf(x) f(x)
M x x
x OCf
, , M(x, y) Cf y'y M '( x, y) M(x, y) M '( x, y) Cf , y = f (x)
y = f (x) , : f ( x) = f (x) f . : f, , , x A : f (x) x A f ( x) =
yy. , f (x) = 2x 4 x 2 + 1 , x : f ( x) = 2( x) 4 ( x) 2 + 1 = 2x 4
x 2 + 1 = f (x) , y'y . ) Cf f . Cf , Cf Cf .y y xf(x) Cf
Mf(x)
O y
x
x
M
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 A : x A f ( x) =f (x) . = ,
f (x) 2x 3 x , x : f ( x) = x)3 ( x) =2x 3 + x =f (x) 2( , . : y =
x 2 , y = x 4 , y = x 6 ., , y'y , , y = x , y = x 3 , y = x 5 ., ,
, . f [6,6]. f :
2.1 --
37
) f : i) , ii) , iii) . ) f , .y
1
x
f , y'y . , y'y f, f, ().A A
4
B
6 5
2
2
5
6
x
: ) f : i) [0,2] [5,6], ii) [2, 0] [6, 5], [0,2] [5,6] f . iii)
[5, 2] [2,5] .
38
2:
) f 4 x 6 6. : max f (x) = f (6) = f (6) = 4 f 0 x 0. : min f
(x) = f (0) = 0.
1) : ) ) .y y=f(x) y y=g(x) x y y=h(x) x 1 2
1
2
2
1
x
2) , . 3) : i) f (x) = x 2 6x + 10 x=3. 2x x=1. ii) g(x) = 2 x
+1 4) : i) f1 (x) 3x 2 + 5x 4 = iv) f 4 (x) x 3x =3 5
ii) f 2 (x) 3 x + 1 = x2 v) f 5 (x) = 1+ x
iii) f 3 (x) x + 1 = vi) f 6 (x) = 2x . x2 +1
2.1 --
39
5) : i) f1 (x) = 1 x x+ ii) f 2 (x) = x2 iii) f 3 (x) = x 1 x +
1
1 iv) f 4 (x) = 2 x x +1
v) f 5 (x) =
x
vi) f 6 (x) =
1 x2 .
6) .y y=f(x) y y=g(x) x y y=h(x)
x
x
7) y y=f(x) y y=g(x) y y=h(x)
x
x
x
8) ) y C1
y C2
) .y C3 x
x
x
40
2:
2.2 - = ) f (x) x + 1 . x + 1, x < 0 f (x) = , x + 1, x 0 y f
(x) x + 1 , = y=|x|+1 1 9 y = x + 1 , x 0 9 y= x + 1 , x 0 , 1 y'y
x' 1 y=|x| 1 O 1 x 1
1
x 'Oy xOy , , (x) =(). x , (x) = x (1) 1 , f (x) x + 1 . , , , =
: f (x) = + 1 , x , (x) x f (x) 1 (x) . : f , : f (x) = + c , c
> 0 , (x) c ( )(1)
y'y .
2.2 -
41
y c c c O
y=(x)+c c y=(x) c x
= ) f (x) x 1 . x 1, x < 0 f (x) = , x 1, x 0 f (x) x 1 , = 9
y = x 1 , x 0 y=|x| 1 1 1
y 1
9 y= x 1 , x 0 , O 1 y=|x|1 x 1 1 y'y x 'Oy xOy (x) =(). x x ,
(x) = 1 , = f (x) x 1. , , , : f (x) = 1 , x , (x) x f (x) 1 (x). :
f , : f (x) = c , c > 0 , (x) c ( )
42
2:
y y=(x) O c c c y=(x)c c c x
) f (x) x 1 . = x + 1, x < 1 f (x) = , x 1, x 1 f (x) x 1 , =
9 y = x + 1 , x 1 9 y= x 1 , x 1 ,y 1 1 y=|x| 1 1 y=|x|1
1 x'x O x x 'Oy xOy (x) =(). x , (x) = x (2) 1 , f (x) x 1 . , ,
, = f (x) =(x 1) , x , = f (x) x 1 x (x) = x 1. x(2)
x'x .
2.2 -
43
: f : f (x) =(x c) , c > 0 , c ( ). f (x) =(x c) , f x x c ,
c x. , f c ( ).y c C (x-c) f(x) c c Cf
c xc x O x
= ) f (x) x + 1 . x 1, x < 1 f (x) = , x + 1, x 1 f (x) x + 1
, = y=|x| 1 9 y = x 1 , x 1 1 9 y= x + 1 , x 1 , y=|x+1| 1 x'x 1 x
'Oy xOy
y 1 1
O
1
x
(x) =(). x , (x) = x 1 , -
44
2:
f (x) x + 1 . , , , = f (x) =(x + 1) , x , f (x) x + 1 x = (x) =
x +1. x : f , : f (x) =(x + c) , c > 0 , c ( ). f (x) =(x + c) ,
f x x + c , c x. , f c ( ).c y c
Cfc
C f(x) =(x+c)
c O x x+c x
f (x) = x + 3 + 2 .
= y x + 3 , y = x 3 . y = x + 3 + 2 , y x + 3 2 = .
2.2 -
45
, f (x) = x + 3 + 2 y = x , 3 2 ().
y
y=|x+3|+2
y=|x+3|
y=|x| 2 3 O 1 x
: , : f (x) =(x c) d , c,d > 0 , .
A 1. : (x) = x, f (x) x + 2 = g(x) x 2 . =
2. : (x) = x, h(x) x + 2 = q(x) x 2 . =
3. : (x) = x, F(x) = x + 2 + 1 G(x) = x 2 1 .
46
2:
4. i) f (x) = 2x 2 4x + 5 f (x) =(x p) 2 + q g(x) = 2x 2 f. ii)
f (x) = 2 + 8x 9 , 2x 2 g g(x) = 2x . 5. 1 O(0,0) A(2,0).y C
O
x
: (x) (x) i) f (x) = + 2 g(x) = 2 ii) h(x) =(x + 3) q(x) =(x 3)
iii) F(x) =(x + 3) + 2 G(x) =(x 3) 2 . 6. (x) = 2x 2 1 . f : i) 2 1
. ii) 3 2 . iii) 2 1 . iv) 3 2 .
2.2 -
47
2 I) , , . 1. f , f . 2. . 3. (1,2) , (2,1) (3,3) . 4. f 1, f
(0) < 0 . 5. f (1,2) (2,5) , f . 6. f 1, f (x) = 2 . 7. f :[1,
2] f (x) = 3x 2 . 8. , . 9. f , f . 10. f , f .
II) f. f, (x) = , 1 3x 4 2 , : ) f (x) = 3(x 1) 4 + 2 ) f (x) =
3(x + 1) 4 + 2 ) f (x) = 3(x 1) 4 2 ) f (x) = 3(x + 1) 4 2
3
3.1 . MM1 NN1 , O M M1 O N N1 , : (MM1 ) (NN1 ) (OM1 ) (ON1 ) ,
= = (OM) (ON) (OM) (ON) , (MM1 ) (OM1 ) , (OM) (OM) (MM1 ) (OM1 ) 1
1
(MM1 ) (NN1 ) = (OM1 ) (ON1 )
, . , , , , , . , M1 O M , :(MM1 ) (OM)
=
(
)
50
3:
= =
(M1 ) (OM)
(MM1 ) (OM1 )
( ( (
) ) )
, , (M1 ) = (M )1
, 00 3600 Oxy , Ot Ox Ot (. , ). Ox , Ot .y t 2 (x,y) (x,y) y x
1 x 1 2 x t y
(x, y) M1 x'x (. ). (. ), , , : = (MM1 ) (M1 ) (MM1 ) (M1 ) , =
, = = (OM) (OM) (OM1 ) (M1 )
y (1 ) = x , (1M) = (OM) = x 2 + y 2 = > 0 . , : = y x x y ,
= , = = , = y xx 2 + y2 > 0 .
3.1
51
, ( ). : = y, x ,
= =
y xx y
( x0) ( y0)
,=
x 2 + y2 > 0
=
(x, y) ( ) = . 360 Ox Oxy . 300 , Ox = 360 + 30 390. = 360 , : =
360 + , N* 00 < 3600 Ox, , 30 , Ox 390 360 + 30 = : = (360 + 30
) = 390.y 390 0 300 x t y t 300
x 2 + y 2 > 0
x
390 0
52
3:
: = ( 360 + ), N 0 < 360 360 , , 0 360. , , , : = y, x ,
= =
y xx y
( x0) ( y0)
,=
x 2 + y2 > 0
=
(x, y) ( ) = .x 2 + y 2 > 0
( ) Ox . Ox , , , 360 + , . Ox , , , 360 + , . , k 360 + , k , .
, k : (k 360 + ) = , (k 360 + ) = , (k 360 + ) = (k 360 + ) =
3.1
53
, , , . , . (0,0) =1 . . , .. =35 , (, ). =1 35 = 0,57. 35 = ,
35 =110 120 o 130 o 140o 150o 160o 170o 180o 190 o 200o 210 o 220
oo 100o 90 80 o o
y
70 o
60 o
50o 40o
N(,)20 o
x
0,1
3500,1
10 o
x350 o 340o
(x,y)
330 o 320 o y 310o 300o 240o o 290o 250 o o 280o 260 270
=1, 35 = 0,82. , (x, y), : = x = = y = x'x , y'y . : 1. ' , 1.
:1 1 1 1
54
3:
2. , , .10 20 30 40
+ + + +
+
+ +
+
M(x, y). . 1 , ( ) ( ) = = ( ) () 1y B
t E(1,yE )
MA O
A
x
B
=
yE , (AE)= yE , = yE. . : = yE = , x =1 , .
3.1
55
. , AB (, ) ( 1rad), . , ( rad) S = . :
B O 1rad A
( 1 rad ) , , ( 1 rad). , : , rad . 2, 360 2 rad. , 1 rad ,
360 , 2 180 . 180 , :
rad
, =
: 9 60 , = 180 = 600 = = 180 3 60 = rad. 3
= 180
56
3:
9
5 rad , 6
5 = = 6 180 5 5 6 = = = 150 180 6 180 5 rad=150. 6 . 0 30 45 60
90 rad 0 0 13 2 2 2
03 3
3 13 3
6 4 3 2
1 22 2 3 2
1 3
1 20
1
0
, x rad x, (x rad), ( x rad), (x rad) (x rad), x, x, x x. , ..
rad 3 3 (100rad ) 100 .
3.1
57
1 h , . .
:
48 = 70 =
h h , = 48 h h , = 70
= = 20mh h 2070 48 = 20, h = . 48 70 70 48 70 2, 75 48 1,11. (1)
: 61, 05 37 1, 64 37m . h 2 750o .
750 360 2 30, 750 =360 + 30 2
750 = (2 360 + 30 ) = 30 =750 = 30 = 3 3
1 2
750 = 30 =750 = 30 = 3
3 2
58
3:
3
79 rad. 3
79 79 = 2. 79 6 3 6 79 79 1 = 2 13 + 2 13 2 + , = = 13 1. 3 6 6
3 : 79 3 = 13 2 + = = 3 3 3 2 79 = = 3 3 3
79 1 = = 3 3 2
79 3 = = 3 3 3
1. x, y .B 6 300
A x y 3
4A 600
2. .B
2
30
0
3. S = 6cm. , : i) = 1cm ii) = 2cm iii) = 3cm . 4. rad i) 30 ii)
120 iii) 1260 iv) 1485.
5. : i)
rad 10
ii)
5 rad 6
iii)
91 rad 3
iv) 100rad .
6. i) 1830 ii) 2940 iii) 1980 iv) 3600.
3.1
59
1. N (N ) h , 70 ( ) ( ) 1000 1.000 (1 0,3 m) . . i) h = 30 , 45
60. ii) , h=1000 ;0
2. : i) : () = () = 2 45 = 2. ii) ()= 4 22,5. iii) (). iv) ,
, () 2 2 2 . = v) 22,5. vi) ; A 3. 30 . 60
4. 1mm . .
B
3 330 0
A
22,5 o 1
45o 1
B
60
3:
3.2 . . : 1. M (x, y) , : x = y = t y
2 + 2 = 1
, 2 2 2 (OM) =1 ( )= x + y= x 2 + y 2 : x +y = 1, : 2 + 2 = 1 2.
= 2 2
(x,y) |y| A 1
y
x |x| O
A
x
=
:
= =
y = ( x = 0 ) x x = ( y = 0 ). y
3.2
61
(1) (2), . 3. : . = 1
=:
= ( 0 0) = 1.
=
4.
= 2
1 2 2 = 1 + 2 1 + 2
i) 2 + 2 = 1 2 0 : 2 2 1 1 1 2 + = = +1 2 = . 2 2 2 2 1 + 2 2 =
1 . 1 + 2 1 , 1 + 2
ii) 2 + 2 = 1 2 = : 2 + 2 =
1 1 2 = 1 2 = 1 2 = . 1 + 2 1 + 2 1 + 2
2 . 1 + 2
62
3:
5 1 = 90o < < 180o, 13 .
2 + 2 = 1 2 = 1 2 . 5 : 13 2 25 169 25 144 5 2 = 1 = 1 = = . 169
169 169 13 90 < < 180 , < 0 , : 144 12 = 169 13 = = , : =
5 12 5 12 = 13 = = 13 = . 12 5 12 5 13 13 2 4 4 2 2 i) + = 1 2 4 4
2 ii) = 2 1
i) : 4 + 4 = ( 2 ) 2 + ( 2 ) 2= ( 2 + 2 ) 2 2 2 2
= 1 2 2 2 , ii) : 4 4 = ( 2 ) 2 ( 2 ) 2
( 2 + 2 = 1 )
= ( 2 + 2 )( 2 2 ) ( 2 + 2 = 1 ) = 2 2 = 2 (1 2 ) = 2 2 1.
3.2
63
3 < x < , 5 2 x rad. 2 3 2. x = < x < , 3 2 x rad.1.
x = 3. x =
3 3 < x < 2, 2 3 x rad. 2 5 0 < x < , 2 5 x rad. 3
< x < 2, 2 2xx . 1 + xi) x = 0 x = 0.
4. x =
5. x = 2
6. , x : ii) x = 1 x = 1. 3 4 iii) x = x = . 5 5 7. , M ( x, y)
x = 3 y = 3, O(0,0) = 3. 8. x = 2 y = 3 , 9x2 + 4y2=36. 9. x = r ,
y = r z = r , x2 + y2 + z2 = r2. 10. : 1 i) = 1 +
4 4 2 ii) = 2 1 .
64
3:
11. : i) 1 + 2 x x 2 + =ii) + = . 1 x 1 + x x 1 +
12. : i) + = + ii) 2 2 = 2 2 .
13. : x x 1 + = x + x ii) (1 x) 1 + i) = x x 1 x 1 x x iii) 1 1
1 = x x iv) x x = x x. x + x x x
B 1. x + x = , : 1 1 i) x x ii) + x x iii) x + x 2. : i) 4 x + 4
x = 1 2 2 x 2 x ii) 6 x + 6 x = 1 3 2 x 2 x. iii) 2( 6 x + 6 x) 3(
4 x + 4 x) x , . 3. iv) 3 x + 3 x.
< x < , 2 2
1 + x 1 x = 2x. 1 x 1 + x
4. 0 x < , 2
1 + x + 1 x 1 + x x = = . 1 + x 1 x x 1 x
3.3 1
65
3.3 10 , , 0 90. ' ' . ' , ' = , , , x'x. . , :y (x,y) t
A
A
x
(x, y) t'
() = () =
() = () =
: . : 9 :
1 2 3 (30 ) = (30 ) = 3 (30 ) = (30 ) = 9 , : 2 = = 4 2 4 = = 1
4 4
(30 ) = (30 ) =
3 2
(30 ) = (30 ) = 3
2 = = 4 2 4 = = 1 4 4
66
3:
180 ' 180 , = 180 , ' t , , ' (x,y) y'y . A . , : (180 ) = (180
) = : 180o . : = 9 150 180 30 , :y B t 180 0
(x,y)
A x
B
(180 ) = (180 ) =
150 = (180 30 ) = 30 =
1 23 2
150 = (180 30 ) = 30 = 150 = (180 30 ) = 30 = 3 3
150 = (180 30 ) = 30 = 3
9
2 = , : 3 3
3 2 = = = 3 3 2 3
3.3 1
67
1 2 = = = 3 3 2 3 2 = = = 3 3 3 3
3 2 = = = 3 3 3 3 180 ' 180 , = 180 + , , ' , ' . . , : (180 + )
= (180 + ) = y B 1800+ A (x,y) t t (x,y) A x
(180 + ) = (180 + ) =
: 180o , . : 9 210 180 + 30 , : =
210 = (180 + 30 ) = 30 =
1 23 2
210 = (180 + 30 ) = 30 = 210 = (180 + 30 ) = 30 =210 = (180 + 30
) = 30 =
3 33
68
3:
9
4 = + , : 3 3
3 4 = + = = 3 3 2 3 1 4 = + = = 3 3 2 3 4 = + = = 3 3 3 3
3 4 = + = = 3 3 3 3 90 ' 90 , ' 90 , , = , ' xOy . . , : (90 ) =
(90 ) = , 90o , . , 60 90 30 , : =60 = 30 =60 = 30 =y B t (y,x) y=x
t (x,y) A O0 90
A
x
B
(90 ) = (90 ) =
3 , 23
1 , 2 3 60 = 30 = 3 60 = 30 =
3.3 1
69
, 0 90.
1 36o = 54o.
1+ 5 . 4
54 90 36 , =54 = 36 =
1+ 5 4
2 + 2 = 1 2 54 + 2 54 = 1, : 2 1+ 5 6 + 2 5 10 2 5 2 2 54 = 1 54
= 1 = , = 1 4 16 16 : 10 2 5 54 = 4 : 54 = = 54 541+ 5 10 2 5
54 =
54 = 54
10 2 5 . 1+ 5
2 : ) 90 + ,
) 270
) 270 +
i) 90 += 90 (), : (90 + ) = (90 ()) = () = . 90 + .
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)
187 rad 6
ii)
21 rad. 4
3. : i) = ( + ) iii) ii) + ( + ) = 0 iv)
+ = 2 2
+ = . 2 2
4. () (180 + ) . () (90 + )
9 ( x) (2 + x) + x 2 = 1. 5. : 21 (13 + x) ( x) x 2 6. : 2 ( x)
+ ( x)(2 x) + 2 2 x . 2
3.3 1
71
1. : 495 120 + 495 (120 ) . (120 ) + 495 2. : 5 7 (5 + ) (7 ) +
2 2 = 2 1. 5 7 (5 + ) (7 ) + 2 2
3. x + + x = 5, 3 6 : 2 x + 2 + x . 3 6
4. : 00 , x A : i) ii)
x + T A, x T A f (x + T) = f (x T) = f (x)
f.
. .
1 1. -
, ,
, , , ., f (t) = t, t . . : x (x rad) .
x = (x rad) ( + 360 ) = ( 360 ) = , x :
(x + 2) = (x 2) = x
2. .
3.4
75
x = (x rad). 2. , : x x = x :1 =
{x x 0}
x 1 (x + ) = (x ) = x, . , , : x x = x : 2 =
{x x 0}
. f(x) = x f(x) = x 2, 2, . [0,2]. x xrad . , , . : y B x 0 M ,
. 2 M x , x rad O A f(x) = x 0, . 2 f(x) = x :
76
3:
, 2 3 , 2 3 , 2 2
, = 1 2 2 3 3 = 1. x = , 2 2 : x = x x 0 0 2 1 . 0 3 2 1 . 2
0
. : x x 0 0 4 2 0, 71 2 2 1 3 4 0,71 0 5 4 3 2 7 4 0,71 2 0
0,71 1
. [0, 2]:y 1 O 1 4 3 2 4 y= x 5 3 7 4 2 4 2 0x2 x
3.4
77
f(x) = x , 2, [2, 4], [4, 6] . [2, 0],[4, 2] . , .y 1.5 1 0.5 /2
0 0.5 1 1.5 /2 3/2 2 5/2 x
. x ( x) = x. f(x) = x 0(0,0) . f(x) = x f(x) = x 2, 2, .. [0,
2]. : x x 0 1 . 2 0 1 . 3 2 0 2 1 .
: x x 4 2 2 0,71 2 0 3 2 0,71 5 4 0,71 3 2 0 7 4 0,71
0 1
1
2 1
78
3:
y = x 0 x 2p .y 1 0 1 4 2 y = x 3 5 4 4 3 7 2 2 4 0x2 x
f(x) = x 2, :y 1 2 0 1 2 3 2 2 5 2 y = x x 3 7 2 4
. x (x) = x. f(x) = x y'y. f(x) = x f(x) = x , , .. , . 2 2 ( ,
, ). 2 2 x rad . x . : y x 2 2 O x ' , . f(x) = x , . 2 2
3.4
79
x . 2 x = 2 f. x 2 x +. x = 2 f. x y y B B A A O O B B A A x M x
M E E A A y yB B
O O B B
A A x x
f(x)=x , , :x x 2 3 4 6 0 6 4 3 2
3 0, 6 0 3 1, 7 1 3
3 0, 6 1 3
3 1, 7
. f(x)=x .y = x
2
3 2
2
x 2 3 2 2
f(x)=x , ( 5.3: (x)=x .
80
3:
1 f(x)=3x .
f(x)=3x (x)=x. 2, : f (x + 2)= 3 (x + 2)= 3 x = f (x) , x . f (x
2)= 3 (x 2)= 3 x = f (x), x . f(x)=3x.x x 3x 0 0 0 2 1
0 0
3 2 1
2 0 02
y 3 2 1
y = 3 x
y= x
3
3
1 2 3
2
3
x 4
2 f(x)=2x
f(x)=2x , 2x 2, , x . , f(x)=2x . : f (x + ) = 2(x + ) = (2x +
2) = 2x = f (x), x
f (x ) = 2(x ) = (2x 2) = 2x = f (x), x . , f(x)=2x.x 2x 0 0 4 1
2 0 3 4 1y
03 2y= x
1 2 1 2 3 2
y = 2x
x 2
3 f(x)=32x .
3, 3 .
3.4
81
f(x)=32x :x 32x 0 0 4 3 2 0 3 4 3y
0 2
y = 3 2x
3 x 0 3 4 2 2 3 2 2
.
3
, f(x)= x, ,>0: (i) , . (ii) 2 . f(x)= x, ,>0
1. , = i) f (x) = 2x, g(x) 0,5 x, h(x) = 2 x, 0 x 2 = ii) f (x)
= 2x, g(x) 0,5 x, h(x) = 2 x, 0 x 2 . 2. f (x) = x
g(x) = 1 + x h(x) = 1 + x 3. f (x) = x g(x) = 3x, 0 x 2 . 4. f
(x) = x g(x) = 3x, 0 x 2 . x 5. f (x) = 2 . 2 ; ; f .
82
3:
x . 2 7. 6. f (x) = 2 i) f (x) = x ii) g(x) = 1 + x . iii) h(x)
= 1 + x
8. f (x) = 2x. 9. f (x) = x. B 1. :y 1 1 1 y y 2 x 2 2 x x
1 3
y y 2
x
x y y 2.5 2 x 2 2 x
0.5
2. y= 3 t , y 6 t . i) . ii) 0 t 12.
3.5
83
3. 1m. , 1 h = 1 + 3t, 3 t . i) . 1m 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
1 x = . 2 y = x 1 y = . 2y y= 1 2 6 5 6 2 y= x x
x=
O
x , f (x) = x 1 . 2, 2 x, (. ), 2 . 2, .
84
3:
1 5 [0, 2], = , x = 2 6 6 6 5 1 = = . 6 6 2 y
x =
1 2 x= 2 + 6 , 5 x= 2 + 6
M
1 2 O5/6 /6
M x A
, x = , = , :
x= 2 + , x = 2 + ( ) 1 x =
3 2
3 3 = , ( ) = . 3 2 3 2 x = ( ), : 3 x= 2 3 , x = 2 + + 3 1 2 2x
+ = 4 2
1 = , 2x + = , 6 2 4 6
3.5
85
2x + 4 = 2 + 6 , 2x + = 2 + 4 6
2x + = 2 + 2x= 2 + x= 4 6 6 4 242x + 7 = 2 + 2x = 2 + x = + 4 6
6 4 24
x = 24 , 7 x = + 24 x = , 1 .. x = . 2 x = = 3
1 [, ] , 2 3 3 1 = . 3 2y M 3 1 2 3 M x
1 x = 2 x= 2 + 3 , x= 2 3
86
3:
, x = , = , x= 2 + , x = 2
1 x =
2 2
2 = , x = , 4 4 2 : x= 2 + 4 , x= 2 4 2 2x =
3 2
3 3 5 3 = , = = . 6 2 6 2 6 2 5 2x = , 6 5 2x= 2 + 6 , 5 2x= 2 6
5 x = + 12 , 5 x = 12
3.5
87
x = x = 3. . , , , .. , 2 2 , .y 3 x
2
2
, x = 3 .
, = 3. 3 3 x = 3 : x = + , . 3 , x = , x = , :
x = + , x = . 1 x = 1
= 1, = 1. x = , 4 4 4 x = , 42 x = 3
= 3, x = , 6 6 x = + , 6
88
3:
1. i) x = 0 ii) x =
2. 1 i) x = ii) x = 1 2 3. 3 i) x = 0 ii) x = 3 4. 3 i) x = 3 5.
i) (1 x)(2x 3) = 0 6.
2 2
iii) x = 0 iii) x = iii) x = 1 ii) x = 3 3 2 2
iv) x =
2 2
iv) x = 1 iv) x = 3
ii) (2x + 2)(1 x) = 0
i) ( 3 + x)(1 x) = 0 ii) (2x + 1)(2 x 3)x = 0 7. ii) x = 0,809
iii) x = 28,636 i) x = 0,951 8. x 2x ii) + 1 = 0 iii) 3 i) 23x = 3
3=0 5 7 9. i) x + = 1 ii) 2 3x = 1 iii) 5x = 3 3 4 4 10. i) 2 2 + 1
= 0 ii) 2 2 x + 3x 2 = 0 iii) 32 t= 3 + 2 3t 11. ii) x 2x = 1 i) 2
x + 5 2 x = 4 12. x, : i) f (x) = 3 x , 0 x < 2, ii) g(x) = 7 x
, 0 x < 2 2 2
3.6
89
13. ( t ) = 75 + 50 , S 6 t o t = 1 . i) 100000 , ii) . 1. i) x
+ x = 0 ii) 2x + 3x = 0 3 4 2. 1 i) x x + 1 = x + x ii) 2x = 4 2 x
3. x = 1 (3,4). * 4. 1+x=x [0,2). 5. : x = + x [0, 2). 3
3.6 , M1 , M 2 (. 1). , . (. 2).y M 1 (, )-
M 2 (, )
y
M((-), (-))-
x (1,0) O
x (1,0)
O
. (1)
. (2)
90
3:
, M1 , M 2 , : 1: 2: : : 1 ( ) 0 ( )
M 2 OM1 = AOM = , (M 2 M1 ) = (AM). (M 2 M1 ) 2 = (AM) 2
:
(P1P2 ) =
(x 2 x1 ) 2 + (y 2 y1 ) 2 ,
P1 (x1 , y1 ) P2 (x 2 , y 2 ), : (M 2 M1 )2 = ( ) 2 + ( ) 2 = 2
+ 2 2 + 2 + 2 2= 2 2( + )
( )2
= [( ) 1]2 + [( ) 0]2 = 2 ( ) + 1 2( ) + 2 ( )= 2 2( ).
( 2 1 ) 2 = ( ) 2 2 2( + )= 2 2( )
:
+ = ( ) ( ) = +
(1)
, , 0 < < 3600 , , . , :( ()) = () + () = : ( + ) =
(2)
3.6
91
(1) (2) , . , : 15 = (45 30 ) = 45 30 + 45 30 = 2 3 2 1 + = 2 2
2 2 2 3 2 1 = 2 2 2 2 2( 3 + 1) 4 2( 3 1) 4
75 = (45 + 30 ) = 45 30 45 30 =
(1), , . x = x x = x, : 2 2 ( + ) = ( + ) = ( ) 2 2 = + = + 2
2
:( + ) = +
(3)
:( ) =
, : 15 = (45 30 ) = 45 30 45 30 = 2 3 2 1 = 2 2 2 2 2( 3 1)
4
75 = (45 + 30 ) = 45 30 + 45 30 = 2 3 2 1 + = 2 2 2 2 2( 3 + 1)
4
92
3:
+ , , . , : ( + ), , ( + ) 0, 0 0. :
= ( + )
( + ) + = 0 ( + ) + + = = 1 : ( + ) = + 1 (5)
, : ( ) = 1 + (6)
: ( + ) = 1 + (7) ( ) = + 1 (8)
, :45 30 15 = (45 30 ) = = 1 + 45 30 = 3 3 = 3 3 3 3+ 3 1+ 3
1
(3 3)(3 3) 12 6 3 = = 2 3 6 (3 + 3)(3 3)
3.6
93
3 45 + 30 3 = 3+ 3 75 = (45 + 30 ) = = 1 45 30 3 3 3 1 3
1+
=
(3 + 3)(3 + 3) 12 + 6 3 = = 2+ 3 6 (3 3)(3 + 3)
- 3 3 12 3 1 = , < < 2 = , < < , 5 2 13 2 +.
(+) = + ( + ) = . : 2 = 1 2 = 1 2 = 1 2 = 1 9 16 = , = 25 25
16 4 3 < < 2 = , 2 25 5
25 5 144 25 3 = , < < = , = 169 13 169 169 2
3 12 4 5 16 ( + ) = + = 5 13 5 13 65 4 12 3 5 63 = , 5 13 5 13
6516 63
( + ) =
:( + ) =
( + ) =
63 16
2 : ( + ) ( ) = 2 2
( + )( ) = ( + )( ) = 2 2 2 2 = 2 (1 2) (1 2 ) 2 = 2 2 2 2 + 2 2
= 2 2
94
3:
3 : 2x = x + 6
2x = x + 2x = x + x 6 6 6 2x = x 4= 3= x x = 3 3 x = + , 3 x = 3
1 x + x 2 2 3x + x 3x
[ x 0 ]
4 : + + =
, , , . + = , ( + ) : 2( + ) = ( )
+ = 1 + = + + + =
5 2 4 1 = t, 2 = (t + ) 3 = (t + ). 3 3 = 1 + 2 + 3 .
3.6
95
= t + (t + 2 ) + (t + 4 )
3 3 2 2 4 4 = t + t + t + t + t 3 3 3 3 3 3 1 1 = t + t + t 2 +
t 2 + t 2 2 1 1 = t t t = 0 2 2
1. , , : i) 12 4 12 4 ii) 170 50 + 170 50
iii) 110 70 110 70
iv)
2. : i) 3x(2x) 3x(2x) 3. :) i) (x + ) + (x = 4 4 2x
7 7 + 12 12 12 12 4 4
ii) (x + )x + (x + )x ii) 2 (x ) 2 (x + ) = 2xx 4 4
4. , , : i) 17 4 17 4 18 9 18 9
ii) iv)
70 20 + 70 20165 + 15 1 165 15
5. : i) 2xx + 2xx iii)x 2x 1 + x2x
7 12 4 iii) 7 1 + 12 4
ii) x + x x + x 6 6
+ 2x + x 3 6 iv) 1 + 2x x 3 6
96
3:
6. : i) x + + x = x 3 3 ii) ( + )( + ) = ( + ) + ( )
7. , , 105 195. 8. : i) + =( + )
ii) + =
( + )
9. , : i) ( + ) = ii)63 655 13 5
16 ( + ) = 65 3 5
3
10. +, : i) = , ii) = ,3 5 = 5 , 13 4 = , 5 6 0 2 , x < 3
1< x < 2 . x = 3, x =1 x = 2 . A(x) . B(x). ... . (x)>0 (
0 (< 0) , (x 1)(x 2 + x 6)(2x 2 + x + 1) < 0 x P(x) = (x 1)(x
2 + x 6)(2x 2 + x + 1) . P(x) x (, 3) (1, 2). x 3 3x 2 + x + 2 >
0
1, 1 5 1+ 5 (x 2) x x > 0. 2 2 P(x) = x 3 3x 2 + x + 2 :
(x 2)(x 2 x 1) > 0
1 (2, +) (x) ,
4.3
145
1+ 5 . (x) , 2 ,2 , x2, . , (x) .
1 5 2
+ 1+ 5 2
2
+ +
x , 1 5 < x < 1 + 5 x>2.2 2
, . , , :
f (x) = x + 1x 1 + ...... + 1x + 0 , 0 (< 0) . : A(x) A(x)
> 0 A(x) B(x) > 0 < 0 A(x) B(x) < 0, B(x) B(x) , A(x)
B(x) > 0 A(x) B(x) < 0 (x). , (x 1)(2x 2 + x + 1) > 0. x2
+ x 6 (x 1)(x 2 + x 6)(2x 2 + x + 1) > 0 , P(x)> 0 , , P(x) x
(3,1) (2, +).
4.4
153
A(x) 0 B(x) x
A(x) B(x) 0 B(x) 0.
x 2 4x + 3 2 0 (x 2 4x + 3)(x 2 + 3x 4) 0 x + 3x 4 0. 2 x + 3x 4
x 2 4x + 3 1 3 ,
x 2 4x + 3 0. : x 2 + 3x 4
x 2 + 3x 4 1 4 . :P(x) = (x 2 4x + 3)(x 2 + 3x 4) x x - 4x + 3
x2 + 3x-4 P(x)2
-
-4
+ + +
+ 0 0-
1 0 0 0
-
3 0 0
+ + + +
+-
x (, 4) [3, +).
1. : 3x 2 1 2 x 2 3x + 2 x2 2 4 i) 2 = ii) = 2 x 1 x +1 x 1 x 1
x x x 3 2 2. : 2 x + x + 2x 2 = 0 3. : i) iv)
x 3 = 4x
ii) v)
3x 2 = 4 x +3 =
iii)
5x 1 = 4 x + x 20 = 10
x + 3 = x +1
10 x + 1 vi)
x 8 + 3 viii) 1 + 2 x = x + 1 2 x 4. : 2x + 1 x2 0 ii) i) >0
x 3 x +1 x vii) =
iii)
x2 x 2 0 x2 + x 2
154
4: -
5. : 2x + 3 x2 i) ii) >4 4 x 1 3x + 5 x 2 x 3 iv) v) 2x 1 x +
2 3x 5 x 16. : x 2 + B 1. : i) 2x + 3 < 1 3x 2. : ii) ii) ii)
ii)3
iii)
x 2 3x 10 +20 x 1
2 1 2x 1 x(2x 1)
x 3 > x 5x2 + 3 x 6 = 0
0 i) x + 3 x 10 = 3. :i) x 2 + x 4 = i)
x2 + x 2
x 1 + x 4=4x 2 + = 2x 1
x+4
4. :
x 1 =
5. 2 4 x 33 x 3 2 x 3x + 4 = 0 ( )
1. (x 2 + x + 1)(x 1) = x 3 1, P(x) = 3 + x 3+1 + x 3+ 2 x2+x+1
(, x , ). 2. : i) f (x) = x +1 ( + 1)x + 1 (x 1) 2 . . ii) g(x) =
(n - 2)x n -nx n-1 + nx -n + 2 (x 1)3 . 3. :
0 i) x 4 + 2x 3 + 2x + 1 =
0 ii) x 4 + x 3 4x 2 + x + 1 =
( x 4 + x 3 + x 2 + x + = 0, 0. ).
4.4
155
: x2 x +
1 = y. x
4.
0 i) x 4 + x 3 16x 2 2x + 4 =: x 2 + 2x 1 = y.
0 ii) x 4 + 8x 3 + 13x 2 8x + 1 =
5. (x 2 + 2x 1) 2 3(x 2 + 2x + 3) + 14 = 0 6. , x 5 + 3x 2 + x +
, x 2 2, 5x+8. 7. P(x) =17 12x16 + 12x15 12x14 + ... + 12x 1, x
(11) (13). 8. . C , x = 10x 3 100x 2 + 270x 180. i) ; ii) ;
156
4: -
2 = 1, 414213,
,
= 2. 2
2
2,
157
x1,2 =
2 4 2
x 3 + x = (*) + + + 3 + x= 2 2 2 2 2 2 3 3 2 3 2
* 3 : x 3 + x 2 + x + = 0 x= y x 3 + Ax = B. 3
158
4: -
-
4 5
5.1 . : , : .... >1 , = , N* =1 , : 3 1 1 1 1 1 = = 8 2 2 2
2
: 1 1 0 = = 1 = , 0 N* . :9 2 3 = = 4 3 2 1 1 2 2 , 5 4 2 2
, 0, . . O , . T 2 .. 35 ; Av ( p )q = pq , :(3 = 3= 32 )2 5 5 2
5 5
5.1 2 5
161
3 x 5 = 32 , 5 32 . 3 5 = 5 32 . Av >0, , : =
2
, , , , 0 = 0. ..= 83 4 32 3
= 823
3
= 4 643
27 =
27 4 =
1 = 27 4
3
1 1 = 4 4 3 27
, . To . ..4
3 = 4 3 = 4
1
1
1 1 + 3
= 12 = 12 7
7
: 21,4 1,43,421 57 3
x 1.4y y
2 1.4 5
= = =
x 3.21 +/ xy
( 7 3 )
: x x , ; 3 2 ; ( B' . 104) 2 , , . 1, 1,4, 1,41, 1,414, 1,4142,
1,41421, 1,414213,.... (1)
162
5:
2 3: 3 1, 31,4, 31,41, 31,414, 31,4142, 31,41421, 31,414213, (2)
: 31 = 3 31,4 4, 6555367 31,41 4, 7069650 31,414 4, 7276950 31,4142
4, 7287339 31,41421 4, 7287839 31,414213 4, 7288015 Av : (1) , (2)
, . 3 2 . 3 2 4, 7288. : Av >0, x x , + , ( ) , ( ). To x x. : x
lim =
, x>0, 0x=0. O : 3 2xy xy
3
2
2
xexp
= =
2
5.1
163
, ' , . : , x, x1 , x 2 , : x1 x 2 = x1 + x 2
x1 : x 2 = x1 x 2 ( x1 ) x 2 =2 x1x x ( )x = x
( ) x = x x
. x x . x x, :
f : f (x) = x , , 1, . = 1, f(x) = 1. f(x) = 2x . : x y=2x-5
1 32
-4
1 16
-3
1 8
-2
1 4
-1
1 2
0 1
1 2 y
2 4
3 8
4 16
5 32
(x, y) . , f (x) = x >1, :
y=2x
1 0 1
x
164
5:
. (0, +) . . x1 , x 2 : x1 < x 2 , x < x y'y (0,1) x.1
2
y x2
x1 A(0,1) x
y=x , >1x x1 x2
1 g(x) = . 2 :
x -5 x 1 32 2
-4
-3
-2
-1
0 1
16
8
4
2
1 1 2
2 1 4
3 1 8y
4 1 16
5 1 32
(x, y) . ,
y=( 1 ) 21
x
f (x) = x 0 0 1 > 0 , f(x) = x . log . , > 0 1 > 0, : a
x = q x = loga q : O log o :
log 2 8 = 3 1 log 4 2 = 2 log10 0, 001 = 3log 0,5 0, 25 = 2
, , , ,
8 = 232=4
0, 001 = 103 0, 25 = 0,52
, > 0 1 , x > 0 : log x = log = x , 1 = 0 = 1, : log 1 = 0
log = 1 = , . , , , , .
5.2
175
>0 1, 1 , 2 , > 0 k : = 1. log (12 ) log 1 + log 2
2 3. log = k log k
1 2. log = log 1 log 2
1. : log 1 = 1 x x1 =1 x1 x 2 = 12 ,
log 2 = 2 x x 2 =2
(1)
x1 + x 2 = 12 , log (12 ) x1 + x 2 = , (1), :
log (12 ) log 1 + log 2 =2. . 3. : (2) log = x x = : kx =k , log
k = kx , (2), : log k = k log > 0
.
= , 1 log = log = log 1
1
176
5:
:A = 1 log 2 256 + 2 log 2 3 log 2 18 2 1 log 2 256 + 2 log 2 3
log 2 18 2
:A =
= log 2 256 + log 2 32 log 2 18
[ 3]
= log 2 16 + log 2 9 log 2 18 16 9 = log 2 18
[ 1,2]
= log 2 8 log 2 23 3 = =
, 10. . , log log10. : log = x 10 x = , : log 213 log 0,325 213
0.325 log log = = 2.3283796030.488116639
e . e. . , ln, loge. : ln = x e x =
5.2
177
, , : ln 325 ln0,37 325 0,37 ln ln = =
5.7838251720.994252273
10 e, . , . ,>0 , 1 , >0 : log =log log
*
log = x. = x , :log= log = x log= log log x
= ( x log )log log
: log log= log , log =
: log = log log log = ln ln
log log log, ln ln. :
log 2 17 =
log17 = 4.087462841 log 2
log > 0 1 > 0, , > 0 1 > 0 .
178
5:
1 Richter R I I R = log I 0 . i) R I = 1000I 0 ii) I R I 0 iii)
1 Richter.
I0
= i) 1000 0 R = log
I : I0
1000I0 = log= log1000 3 R = I0ii)
R = log
I I = 10R I = I0 10R I0 I0
(1)
iii) I , I ' R, R' . ' R= R + 1 (1) :
I ' I 10R ' 10R +1 ' = 0 = = 10, I= 10 I I I0 10R 10R I ' 10 I 1
Richter. 2 p . ' pH = log[H + ] , [H + ] H + . i) pH : ~ : [H + ] -
6, 3 103 ~ : [H + ] - 5,0 109 ii) [H + ] : ~ : pH- 4, 2 ~ : pH -
6,6
5.2
179
i) pH log(6,3 103 ) 2, 2 pH log(5, 0 109 ) 8,3 ii) pH 4, 2 , 4,
2 = + ] log[H + ] = [H + ] = [H + ] =105 log[H 4, 2 104,2 6,3 pH 6,
6 , 6, 6 = + ] log[H + ] = [H + ] = [H + ] =107 log[H 6, 6 106,6
2,5 3o 32 N(t) N 0 e 0,0495t , t , = 32.
t , N(t) =N 1 N 0 e 0,0495t =0 e 0,0495t = 2 2
N0 . : 2
1 2 0, 0495t = 0, 69314718 0, 0495t =ln
t = 14
1. , , : ii) log 1 10 iii) log 1 32 i) log10 0, 00110 2
iv) log 9
2. x : 1 ii) log 4 x = i) log10 x = 3 3. : 3 ii) log 8 = i) log
16 = 4
27 3
v) log 2 16
vi) log 32
8 272
2
iii) log
x=
2 3
4. : 2 ii) 3log 1 i) log 2 3 + 2 log 2 4 log 2 12 = 10 2 + log10
5 log10 4 = iii)
2
iii) log 0,1 = 3
1 1 1 log10 25 + log10 8 log10 32 = 2 1 log10 2 3 5
iv) 2log 2 6 2log 2
3
=2
180
5:
v) 2 log 2 (2 + 2) + log 2 (6 4 2) = 2 5. t Q(t) = Q0 e0,34t ,
Q0 . ; 6. , p ( Pascals), h ( ) i) k, 3050m 68900 Pascals. ii)
1000m; 7. () . L0 6. L m : L
= 101300 e kh p
m = 2,5 log 6
i) m L 5 100 L0 . = ii) 1 6 ; 8. S(t) ( ) t S(t) = 100(1 e kt ).
i) k, 15000 . ii) 5 ;
L0
1. : i) 41 1 log 2 3 2
ii) 9 2
1
log3 18 1
2. 1, 2, 3, ... , log1, log2, log3, ... . 3. log2 log8. - .
2
= log 2
10 log log 10 10 ....10 10 = 4. :
5.3
181
5. :
* 6. x> 0 : *7. :
1 1 1 1 log(1 ) + log(1 ) + log(1 ) + .... + log(1 ) = log 2 3 4
log x = log 2 x 2ii) log 2 log 3 = 6
i) log log =1 iii) log log log =1 *8. : i) log + log 1 = 0
ii) log () + log (= log () log () )
5.3 . 4.2, x>0 log x. , x (0, +) log x,
f : (0, +) f (x) = log x . , , f (x) = log x. log x = y y = x,
(,) y = log x, (,) y = x . , , (,) (,) xOy x 'Oy '. :
y = log x y = x
xOy x 'Oy '.
182
5:
y (,) y=x 1 y=x
y= x
y
(,)
1 1 (,) x 1
(,) x
>1 y=logx y=x
y=logx 0 0, 0= 103 + 102 + 10 + . f (x) = x 3 + x 2 + x + f (x)
= (x 1)(x) + f (1) x=1 x=10. 5. ( ). . : x + 1x 1 + .... + 1x + 0 =
0 . 0 , 0 . : i) : 2x 3 + x 2 + x 1 = 0 6x 4 + 29x 3 + 27x 2 + 9x +
1 = 0 ii) 2 12 . 6. 3x + 4 x = . 5x 7.
2 4 2x + = 3 3
x
194
(' )
8. 2 log(x + 3) = log(x) ; 9. x + log x = 2 (0,) 4 . 10. : log 3
(16x 2 12x ) 2x + 1 11. i) f (x) = ln x g(x) = 1 x ii) f (x) = ln x
g(x) =x 2 + 1 2 ln x x + 1. 12. , , , , : = = , , > 0, ++ = : 2
= 2 + 2 2ln x 1 x.
2 = 2 + 2 2 B 2 = 2 + 2 2 13. , , , , : = = , , > 0, ++ = ()
= , (KM) = , () = , = , = , =
14. , , , , : 2 = 2 + 2 2 2 2 2 0 < , , < = + 2 2 = 2 + 2
2 : = =
++ = .
15. , , , , : 2 = 2 + 2 2 2 2 2 0 < , , < = + 2 2 = 2 + 2
2 , (M) = , (KM) = , () = , = A, = , = .
-
195
-
(3, 1) (3, 4)
(21, 24) (5, 2).
3 5 , . 4 3
ii)
3, = 3, 3 = 3,
+2 , , 2 (2, 1).
( 3 + 1)( + 1), ,
)
10. x = , y = , z = . 11. f (x) = x 2 4x + 3, g(x) =x 2 + 2x +
3, h(x) 0,5x 2 3x + 4. =
(4,3, 5)(10 + 2, 16 + 2, ), 1 x 1 : y = x + 2, 2 : y = 1 2
1 11 R = T + . 600 301 1 = , 2 2 = 1 2
(1, 2), (2, 1).
2.3 . 2
2 4 , 3 3
3 2 3 2 2 , 2 , 3 2 3 2 , 2 2 (1, 2), (1, 2), (2, 1), (2,1).
1,
2 + + 1 , , +1 +1
196
-
(4,3), (4,3), (0, 5). (1, 0), (3, 0), (2, 1), (4,3). 1. x = 3, y
= 3 2, = 450.
2. = 1, = 3. 3. 1.(,1],
[1, +),[2, +),
(, 0],(, 1],
4.
6
2 3 33 4
[1, 0], f (1) = 1 h(1) = 2, h(1) = 2 f (x) f (3) g(x) g(1)
[1, +).
5460 18000 . 6.1 3 3 , , , 3 2 2 3 3 1 3 , , 3, 2 2 20, 1, 0,
0,1, 0,
478, 733, 1062
58.
4.
f (x) =2 (x 1) 2 + 3 f (x) =2 (x 2) 2 1. 2(x 2) 2 2(x + 2) 2 2(x
3) 2 3 2(x + 3) 2 3
2 212 + 8 3, 12 3
2 2 . 2
197
3. + ( + ) =180 ,4 3 4 x = , x = , x = . 5 4 3 5 5 2 5 , x = , x
= . x = 3 2 5 1 3 x = 3, x = , x = . 2 2
+ + 90. = 2 2
2. 3.
4. x =5
5 , x = 2
5 2 , x = . 3 3
5. 8 5 20 . 6. 2 1 2 2 2 1 (3 2 ) . 2 3.31200 = 3 1 , 1200 = , 2
2
2.
2 2 1
2.x y = x, y = , y = 3x 2 y = x, y = 3x, = 0,5x, yy = 2,5x.
3 1200 = 1200 = . 3, 3
3 2 3
2 3
ii)
1 3 (2850 ) = , (2850 ) = , 2 2 3 (2850 ) = , (2850 ) = 3. 3
187 1 = , 6 2
187 3 = 6 2
x = , x= 2 +
187 3 = , 6 3
187 = 3. 6
21 2 21 2 = , = 4 2 4 2
x = + , 2 x= 2 , 4x= 2 6 x= 2 +
4
x= 2 +
3 , 2
21 21 = = 1. 4 4
7 , 6
198
-
x = x = , = x x = + , 6 x = + , 4 x= x = + , . 6 x = , 6 x = x =
, 3 = x x= 2 + x= 2 + 3 2 10. = 2 x= 2 + , 3 5 =x= 2 + 4 4 x= 2 , x
= x = + , 3 4 2 x = 2 x = 3 3 x = + , 2 x= 2
x= 2 , 2 3 x= 2 , 4 x= 2 + , .
22 , . 45 6 + 6 + 2 = x = x , 9 9 x = +x = 6 + 2 , 9
10 + 5, 42 + 7 , 12 2 5 , 6 24 + 7 24
24 , 36
12 , . 60 2 2 2 + 6 55 2 == 2 , , 66 x= 2 , 3
t = , . 6
2 x= 2 x= 2 , 2 3 3 3 x= 2 + x= 2 + , 5 5 2 x= 2 , 3 3 x= 2 + ,
54 , 512.
x= 2
3.
199
x=
, 2
g = 7 2
3x
3 g = 7. 2
3 x= 2
x 7. 105 =
1 2
x
2(1 + 3) , 4 2(1 3) 105 = , 4 105 = 2 3,
105 = 2 + 3, x = , 2(1 3) 8 195 = , 4 = x + , 5 30 2(1 + 3) 195
= , 4 x = + , 4 195= 2 3, 2 x = x = + , 4 5 195= 2 + 3. 2 x = + ,
33 ( + ) = , 5 65 13 . 56 4 ( + ) = , 65 , . 33 2 ( + ) = , 56 7 13
19 56 , , , ( + ) = . 12 2 12 12 33 x = + , 6 x = , 1 1 3 2 2 x = +
, 4 2 x 2 ( )
3
=
200
-
( + 2) = (( + ) + ) = ....
3. x = 4.
4 =
x=
5 . 4
2 2+ 2 = , 16 2+ 2+ 2 2+ 2+ 2 = . 16 2 2+ 2 9 4 , = , = 2 13 2
13 2 3 = , = . 2 3 2 2
, 4 = = .... 4 6 x= 2 . 3
8. x =
1 4 = , = , 2 5 2 5
1
3 3 4 6
3 2 2
1 = , = 2. 2 2 2 2 x= 2 , 2 3
x= 2
x= 2
2 =
24 7 , 2 = , 25 25 24 7 2 = , 2 = . 7 24 16 13
, 3 x= 2 , 2 x= 2 + ,
3.
( + 2) =
3 + = , 8 8 2 3 = ... 8 86 6 x = , x = + 3 3 2
x= 2 x= 2 + , x= 2 2 x= 2 + , 3 2
6. x= 2 x= 2 + x= 2 + 5 ,
2 2+ 2 = , 16 2 = 16 2+ 2+ 2 , 2
1 , 4
3+2 4
201
1 4
1 4
3x x
++ =
2x 6x 2x + 8x 1 (8x 4x) 2 1 2x 2
2 + x = = x , 3 10= x6 2
2 + 2 x = , 8 2
2 22 4xx
5.
23xx
x 2 2 + 4 22 2
2 2x 2x
11 f (x) = 2 x + 6 3 f (x) = 2 x + 4 4 f (x) = 2 x + 3 f (x) = 2
x 4 4 , 3 x= 2 2 x= 2 x= 2 ,
x . 2 x = x = 2 + 4 6 5 x= 2 + , 6
x= 2
x = = xx=
, 3
7 x = + 12 12
7 12 11 x= 2 + , 12
6 + 2 x= 9 6 6 2 x , = 9
=
5 = 12 12 = () + () 2 2 + 4
2.
= , 4 () + () = 2 2.
202
-
3.
10 2 2 + 2, 2 2 + 2. 7 x = + , 24 24
4. x = +
6 5 x= 2 + , 6 x= 2 + 13 3 (10 + 2 2)m, 7
5. = , 4 = 40( 2 1) h
6.
6.
= , 8 P= 1+ 2
(10 2 2)m
(t + 1) 10 + 4 4 2 6+ 2 , 2 4 12. x= 2 + 4 2 = x + , 3 4
9. 1. AB 284 m 2. B 1,58 m 5. 1150 6. 51, 2 m 7. 60, 25 1.-5.
6.-7. 8.-10.1 2 11. = 300. =
6 2 , 4
13. = 200 2 2 + + 200 S 4
= = 200( 2 + 1) , Smax 8
15. B = 60 , =45
1. 2.1 2 4. =1
3. 4. x= 2 , 3
3. =
5.
1
1,
203
5. P(11) = 4840 6. 1. 7. = 10,
P(x) x y =P( y) = 0
2. = 2, = 19 3. = 3, = 5 4.2 2 0, , 3 3 2 = 3
8. 9. 10.= 2 3
P(x) x + 1 =
P(1) = 0.
5. P(x) = x 2 5x + 1.
1. 2. 3.
x x =P(x) = (x + )(x) +
= 0 = =
1.
(x) = 3x 3x 8, (x) = x + 3x + 9x + 27,(x) = 4x 3 16 5 , (x) = 6
3 2 (x) = 2x + 1,3 2
2
x 1
x2
4.
P(0) = 0, P(1) = 0, 1 P( ) = 0 2 5. = , = 1
(x) = + 3, (x) = 6x 2 8x + 3 x
(x) = (x) = x 2 + 7 x2 , 2. 3. 4.k = 4(x) = x 2 + 10x 25, =
0
1.
0,
6 6 , 5 5 5 0,1, 1, 3 1, 2, 32 13 ,1, 3 22
2,3, 3 2, 22
(x) x 2 8x + 64, 0 = =(x) x 4 + x 3 + x 2 + x + 1, 2 = = (x) =
3x 3 6x 2 12x 24, = 48 (x) 4x 2 + 14x 30, = 0 =
2,9 + 4 5,9 4 5 2, 2 , 1, 2
204
-
2. 3. 6.
1, 4 1, 2 1, 5
2
1
x > 2x 1
x =1
x < 2
1.
x = 3
2,
2
x2
2. x = 2 + 3.
x = (2 + 1) , 6 6
1 ( , 0) 2 8. 2 5 < x < 0 1 < x < 2 + 5
7.
596. x 1
1
9.
2, 2
3 ,2 5
x>0
1. 1.1, 2, 4 3, 2, 15
2. 1, 2, 2, 3 3. 3,1, 7, 11 4. 5. 1, 1, 2 + 3 2 , 6. 7. t 4, 6 c
0, 45 8. 9. 10. 11. y + 4x 108 12.= 5x 3 y
3 2 x
