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Άλγεβρα Β' Λυκείου Γενικής Παιδείας Βιβλίο Μαθητή
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:
. .
: 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
.
:
9 0, (. ),
x+y=, =0
0 x
y
x+y=, 0
x
y
0
10 1:
9 = 0 , y =
x'x y'y
(. ).
= 0 ( 0 ),
- y'y
x'x
.
:
9 x 2y 2 =
1
2 =
y'y 1 .
9 y=2 x'x - y'y 2 .
9 x=2 y'y - x'x 2.
.
1y x 1
2=
x+y=, =0
x
y
0
y= x1y
0 2-1
x
12
y=2
x
y
2
0
x=2
x
y
0 2
x+y=, =0
x
y
0
y= x1y
0 2-1
x
12
y=2
x
y
2
0
x=2
x
y
0 2
x+y=, =0
x
y
0
y= x1y
0 2-1
x
12
y=2
x
y
2
0
x=2
x
y
0 2
x+y=, =0
x
y
0
y= x1y
0 2-1
x
12
y=2
x
y
2
0
x=2
x
y
0 2
x x .
= =
1.1 11
, (4,-1) x 2y 6, = 4 2( 1) 4 2 6. = + = , , (16,5), ( 10, 8)
1k, k 3 , k2
.
2 x 2 x+y= x+y= , - , , 2 x 2
x y
'x ' y '
+ = + =
. . . . : .
() (') , .. (),
( ) ' ( ') + , () - 0, (') '.
( ) ' ( ') + () (').
- 1o :
9 0, : = = 9 = = , + = +.
, , :
x 2y 6 (1)
3x 4y 8 (2)
= + =
12 1:
, ( )
, .. (1) x.
x
3(2y 6) 4y 8 6y 18 4y 8
10y 10
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 + = =
( ) , :
x 2y 6
3x 4y 8
= + + =
x 2y 6
y 1
= + =
1.1 13
, - , : . :
x 2( 1) 6 x 2 6 x 4 = + = = . (4, 1) ( - ).
2 x 2
. , - .
,
, , . 2 x 2 . , 2 x 2 , , , .
3x 6y 18
3x 4y 8
+ = + =
x 2y 6 ( 3)
3x 4y 8 1
= + =
3x 6y 3x 4y 18 8 10y 10 y 1 + + + = + = =
x 2y 6
3x 4y 8
= + =
x y
'x ' y '
+ = + =
A(4, 1)
3x+4y=8 x2y=6y
O x
14 1:
2 x 2 -.
:
9 2x 3y 3
4x 9y 1
+ = =
2y x 1
34 1
y x9 9
= + =
,
, .
,
(1, 0,3) .
,
- 1(1, )
3.
9 x 2y 3
2x 4y 5
+ = + =
1 3y x
2 21 5
y x2 4
= + =
, ,
.
9
y 1 2x
4x 2y 2
+ = =
y 2x 1y 2x 1
= =
, .
(k,2k 1), k .
y
x0
21
1
1
y
x0
1
2
y
x0
2
1
,
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
' '
+ = + =
=
=
:
' ' ' ' ' 'y
' ' ( ' ' )
( ' ' )
( ' ' )
+ + = + =
=
x y
'x ' y '
+ = + =
1
2
y x ( )
' 'y x ( )
' '
= + = +
16 1:
' 'y
' '
=
, ,
'
'
=
, - =0 , 1 2
, . . =0 ' =0.
:
' ' 0
' ' 0 = .
' ' ,
:
.
D, x , : .
, D, - y , :
.
' ' ' '(x, y) ,
' ' ' '
=
x y
'x ' y '
+ = + =
' ' ' '(x, y) ,
' ' ' '
=
D' '
=
D ' '' '
= =
xD ' '' '
= =
yD ' '' '
= =
1.1 17
- , , :
:
9
,
.
: .
, (x, y) (4, 1)= .
9
- . 2,
2x 3y 40. =
.
x 2y 6
3x 4y 8
= + =
1 2D 1 4 3( 2) 4 6 10 0
3 4
= = = + =
x
6 2D 24 16 40
8 4
= = + =
yD 10y 1D 10
= = =
y
1 6D 8 18 10
3 8= = =
xD 40x 4D 10
= = =
2x 402x 3y 40 y
3
= =
D0, , (x,y)
D=0, .
xDxD
=
x y
'x ' y '
+ = + =
yDyD
=
2 3D 2( 6) 4( 3) 12 12 0
4 6
= = = + =
2x 3y 40
4x 6y 80
= =
2x 3y 40,
2x 3y 40
= =
18 1:
9
.
.
1
- , . , . , . , .
D, Dx, D
y. :
2k 40k, , k3
3x y 11
9x 3y 6
+ = + =
3 1D 9 9 0
9 3= = =
3x y 11
3x y 2
+ = + =
2
x y 1
x 2y
= =
22
x
2 2 2 2y 2
1D 2 ( 2)
2
1 1D 2( 1) 2
2
1D ( 1) (1 1) (2 )
= = + =
= = + =
= = = + =
1.1 19
, D = 0 . :
:9 D0, 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= =
xD 2 ( 2) 1xD ( 2) ( 2)
= = = =
2 2
yD (2 ) ( 2)yD ( 2) ( 2)
= = = =
0x y 1 y 1
0x 2y 0 y 0
= = = =
2 =0 =
2x y 1 2x y 1
4x 2y 2 2x y 1
= = = =
.( )k,2k 1 , k
1 ,
2x y 1 y 2x 1 = =
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 2x y 3z 9 2x y 3(3x y 8) 9 11x 2y 15 + = + + = + = (5)9 (6)
(5), (6)
x=1 y=2 . x y (4) z 3.= (1,2, 3).
x 3y z 10 x 3y (3x y 8) 10 x y 1+ = + + = + =
1 1 1 1
2 2 2 2
3 3 3 3
x y z
x y z
x y z
+ + = + + = + + =
2x y 3z 9 (1)
x 3y z 10 (2)
3x y z 8 (3)
+ = + = + =
3x y z 8 z 3x y 8+ = = +
3 x 3, ,
2 x 2, 3 x 3
.
11x 2y 15,
x y 1
+ = + =
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
7 8x y 45
= + =
x 3y 3
xy 2
3
=
=
2y x 2
1x y 1 0
2
= +
+ =
2x y 7
3x 5y 4
+ = =
2x 5y 4
6x 7y 100
= + =
( 3 1)x 4y 7
1x ( 3 1)y 1
2
+ + =
+ =
2x 3y 40
4x 6y 80
= =
x 1 y 2
3 44x 3y 8
= + =
x 5 2y 1 2 02 7
x 6 y 6 83 2
+ + + = + =
2y 3x 8
x 3y 1 0
= + + =
x y 4
x y 2
= + =
( 3 1)x 2y 2
x ( 3 1)y 1 3
+ =
+ + =
3x y 11
9x 3y 2
+ = =
2x 1 y 243 4
x 3 x y32 3
+ = + =
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 + . 20 C 0,4 80 C 0,5 , .
5. , 50% 80% . 100 ml - 68% ;
6. 1 : 2x 4y 3 + = 2 : x 2y + = , . i) 1 2 .ii) -
; iii) ;
3x 2y 11
2x 5y 2 3
5x y 2 33
= = + =
5x y 3 4
x 3y 2
3x 2y 2 2
+ = + = + =
yx 2 3
23x
y 52
5x 3y 2 16
+ = + + =
+ =
y
x1 41
1 12
O
1.1 23
7. :
i) 21 : x y + = 2 : x y 1 + = .
ii) 3 : x y = 2 : x y 1 + = .
8. :
i) ii)
9. 1 2 6 = , 1 3 5=O O
2 3 7=O O . .
10. , . N = x , B = y = z , - , .
11. . 50% , 10% 30% . 52 lit 32% , - - -. .
12. -, 2y x x= + + . - .
( 1)x 2y 1,
4x ( 1)y 2
= + =
( 2)x 5y 5,
x ( 2)y 5
+ = + + =
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
24 1:
1.2
, -. , 13 89.
x, y , x y 13+ = 2 2x y 89+ = . -
, :
: (1), , .. x , (2).
x y 13 y 13 x+ = = (3).
2 = 9 . :
(3), x=8 y=5 , x=5 y=8. (8, 5) (5, 8). 5 8. , , .
1
2 2
x y 13 (1)
x y 89 (2)
+ =
+ =
2 2 2 2
2
2
x (13 x) 89 x 169 26x x 89
2x 26x 80 0
x 13x 40 0.
+ = + + =
+ =
+ =
8
5
13 3x
2
= =
x y 5 (1).
xy 6 (2)
+ =
=
1.2 25
(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
6
yx
= . -
.
x y 5 y 5 x+ = =
x 2 x 3. = =
y
xO 11
AB
(3).
2
2
5x x 6
x 5x 6 0
=
+ =
xy 6 x(5 x) 6= =
26 1:
(2,3) (3,2) . (2,3) (3,2) .
2
(1) x (2).
(2) :
. 2x = ,
2 13 36 0 + = , =9 =4 .
9 =9 (1) x=3 y=2 x 3= .9 = 4
(1) x=2 y=3 x 2= y 3= .
(3,2), ( 3, 2), (2,3) ( 2, 3).
xy 6= 6
yx
= , -
2 2x y 13+ = (0,0) 13 = .
6xy 6 y
x= =
22 2 2
22
4 2
4 2
6x y 13 x 13
x
36x 13
x
x 36 13x
x 13x 36 0
+ = + =
+ =
+ =
+ =
2x 9 x 3= = x 3=
2x 4 x 2= = x 2=
2 2
xy 6 (1)
x y 13 (2)
=
+ =
1.2 27
1. :
2. :
i) ii) iii) .
3. 201
S t t2
= + 0 t = + , 0S t
2+
= .
1. o - .
2 2x y xy 3.
x y 1
+ + =
+ =
2y 3x
12x 3y 4
=
=
2 2x y 9
x y 0
+ =
=
2 2x y 5
xy 2
+ =
=
.
2
2 2
x 2y 10
x y 25
= +
+ =
y
x 1
AB
A B
28 1:
2. :
3. 120cm2 . 3cm , 2cm , . -.
4. 2y x= y 2x k, k= + . - k .
5. o 22y x
y x
=
= +
.
1
I. :
, , ,
, , .
) , ) , ) .
II. , , .
1. , . A
2. D = 0 , .
3. xy 1
x y 0
= + =
.
4. 2 2x y 1+ = 2y x 1= +
. A
1
x 2y 1( ) :
2x 4y 0
+ = + =
2
x y 1( ) :
x 2y 4
+ = + =
3
2x 4y 2( ) :
x 2y 1
+ = + =
4 2
x y 1( ) :
x y 1
=
+ =
(1) (2) (3) (4)
2
2
2xy y 5y 0
y x 4x 3
=
= +
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
2.1 -- 31
, , - , 1 2t , t [4,16] 1 2t t< :
1 2f (t ) f (t ) 0 .
) , [16,24] .
f - , 1 2x , x 1 2x x< :
1 2f (x ) f (x ) +
>
f , 1 2x , x 1 2x x< :
1 2f (x ) f (x )>
T=f(t)
5
11T(C)
3
O 4 16 24 t(h)
2.1 -- 33
:) 1t 4= , f (4) 3= . :
f (t) f (4) 3 = , t [0, 24] T = f (t) t = 4 -, f (4) 3= . :
0x A , 0f (x ) f min f (x) .
, 4f (x) 3x 1= + .
4x 0 , x,
43x 0 , x,
43x 1 1+ , x.:
f (x) f (0) , x.
, f 0x 0= , f (0) 1=
) 2t 16= , T(16)=11 . :
f (t) f (16) 11 = , t [0, 24] T =f (t) t =16
, f (16)=11. :
f, , 0x A () :
0f (x) f (x ) , x A
f, , 0x A () :
0f (x) f (x ) , x A
34 2:
0x A , 0f (x ) - f max f ( x) . , 4f (x) 3x 1= + .
4x 0 , x,
43x 0 , x,
43x 1 1 + , x.:
f (x) f (0) , x
, f 0x 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)
2.1 -- 35
) - C
f f -
. Cf
y'y, C
f
y'y Cf .
, , M(x, y) Cf
y'y M '( x, y) M(x, y) M '( x, y) C
f , y = f (x) y = f (x) , :
f . :
yy.
, 4 2f (x) 2x x 1= + , x :
4 2 4 2f ( x) 2( x) ( x) 1 2x x 1 f (x) = + = + =
, y'y .
) - C
f f
. C
f
, - C
f -
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
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 . :
.
, 3f (x) 2x x= , x : 3 3f ( x) 2( x) ( x) 2x x f (x) = = + = , .
: 2y x= ,
4y x= , 6y x= ., , y'y , , - y x= , 3y x= , 5y x= ., , , .
- f [6,6]. f :
f, , , x A :
x A f ( x) f (x) =
2.1 -- 37
) f : 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
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) - :
) ) .
2) - , .
3) :
i) 2f (x) x 6x 10= + x=3.
ii) 22x
g(x)x 1
=+
x=1.
4) :
i) 2 41f (x) 3x 5x= + ii) 2f (x) 3 x 1= + iii) 3f (x) x 1= + iv) 3 54f (x) x 3x= v)
2
5x
f (x)1 x
=+
vi) 6 22x
f (x)x 1
=+
.
1
y=f(x)
x
y
2
2
y=g(x)
x
y
1
y=h(x)
x
y
1
2
2.1 -- 39
5) :
i) 11
f (x)x
=
ii) 2f (x) x 2= iii) 3f (x) x 1 x 1= +
iv) 4 2
1x
xf (x)x 1
+=
+ v) 5f (x) x= vi)
26f (x) 1 x= .
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
C1 C2 C3
x x x
y y y
40 2:
2.2 -
) f (x) x 1= + .
x 1, x 0
f (x) ,x 1, x 0
+
2.2 - 41
) f (x) x 1= .
x 1, x 0f (x) ,
x 1, x 0
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
42 2:
) f (x) x 1= .
x 1, x 1
f (x) ,x 1, x 1
+
2.2 - 43
f :
f (x) (x c)= , c > 0 , c ( ).
:
f (x) (x c)= , f x x c , c x. , f c ( ).
) 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 , 1 x'x - x 'Oy xOy (x) x = ()., (x) x = - 1 , -
Ox
c
c
c
c
xxc
(x-c)
C Cf
f(x)
y
y
x
y=|x+1|
y=|x|
1 1
11
11
O
44 2:
f , :
f (x) (x c)= + , c > 0 , c ( ).
f (x) x 1= + . , , ,
f (x) (x 1)= + , x, f (x) x 1= + x
(x) x = x +1.
:
f (x) (x c)= + , f x x + c , c x. , f c ( ).
f (x) x 3 2= + + .
y x 3= + , - y x= 3 . - y x 3 2= + + , y x 3= + 2 .
xx x+c
yc
c
O
c
c
=(x+c)f(x)
Cf C
2.2 - 45
, -
f (x) x 3 2= + + -
-
y x= , - 3
- 2
().
: , - :
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= .
y
x
y=|x+3|y=|x+3|+2
y=|x|
13
2
O
46 2:
4. i) 2f (x) 2x 4x 5= + 2f (x) (x p) q= + -
2g(x) 2x= f.
ii) 2f (x) 2x 8x 9= + , - g 2g(x) 2x= .
5. 1 O(0,0) A(2,0).
-:
i) f (x) (x) 2= + g(x) (x) 2= ii) h(x) (x 3)= + q(x) (x 3)= iii) F(x) (x 3) 2= + + G(x) (x 3) 2= .
6. 2(x) 2x 1 = . - f :
i) 2 1 . ii) 3 2 .iii) 2 1 .iv) 3 2 .
C
O x
y
2.2 - 47
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] 2f (x) 3x= .
8. , .
9. f , f - .
10. f , f .
II) f.
f,
4(x) 3x , = 1 2 , :
) 4f (x) 3(x 1) 2= + ) 4f (x) 3(x 1) 2=
) 4f (x) 3(x 1) 2= + + ) 4f (x) 3(x 1) 2= +
2
3
1 1(MM ) (NN )
(OM) (ON)= 1 1
(OM ) (ON )
(OM) (ON)= 1 1
1 1
(MM ) (NN )
(OM ) (ON )=
3.1
. - - MM1 NN1 , 1O M M
1O N N
, :
,
,
, 1(MM )
(OM)1(OM )
(OM)1
1
(MM )
(OM )
, - . , , , , , .
, 1M O M
, :
1(MM )
(OM) = ( )
11
50 3:
, - ,
, 00 3600
Oxy , Ot Ox - Ot (. , ). Ox , Ot .
(x, y) M1 x'x (. ). (. ), , , :
, ,
1( ) x = , 1( M) y = 2 2(OM) x y 0= + = > . ,
:
, , , .
1(MM )
(OM) = 1
( M )
(OM)
= 1
1
(MM )
(OM ) = 1
1
( M )
(M )
=
y =
x
=
y
x =
x
y = 2 2x y 0 = + >
1( M )
(OM)
=
1
1
(MM )
(OM ) =
1
1
( M )
(M )
=
( ) ( )
( )
2
1 1
2
x xx
t
y
t
(x,y)(x,y)
yy
3.1 51
1
1
( M )
(M )
=
2 2x y 0 = + >
, - ( ).
:
(x, y) ( )
2 2x y 0 = + >
.
360
Ox - Oxy - . 300 , Ox
360 30 390 . = + =
360 , - :
360 , = + N* 00 < 3600 Ox, , - 30 , Ox - 360 30 390+ = :
, ( x0) ,
, ( y0)
y =
y
x =
x =
x
y =
2 2x y 0 = + >
(360 30 ) 390 . = + =
y
x
390
30
0
0
0
y
t
t
x 30
390
0
52 3:
- :
( 360 ), = + N 0 360 .
( ) Ox .
Ox , , - , 360 , + . Ox , , , 360 , + - .
, k 360 , + k , .
, k :
, ( x0) ,
, ( y0)
y =
y
x =
x =
x
y =
2 2x y 0 = + >
(k 360 ) , + = (k 360 ) + =
(k 360 ) , + = (k 360 ) + =
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
10
350
340
330320
310300
290280270260
250240
220210
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
54 3:
2. , , .
10 20 30 40
+ + + + + + + +
- M(x, y). - . 1 ,
( ) ( )( )
( ) 1
= = =
yE , (AE)= yE ,
= yE.
.
:
= yE =
, x =1 , .
AO
B
B
M
A
t
x
y
E(1,y )E
3.1 55
. , AB (, ) ( 1rad), . , ( rad) S .= - :
( 1 rad ) , -, ( 1 rad).
- , :
, rad . - 2,
360 2 rad.
,
1 rad 3602
,
,
rad 180
.
, 180 =
, :
180 =
:9 60 ,
180 =
= 600
180 3 = =
603
= rad.
A
1rad
B
O
56 3:
9 56
rad ,
180 =
56
=
556 150
180 6 180
= = =
56
rad=150 .
- .
rad
0 0 0 1 0
30 6 1
23
23
33
45 4 2
22
21 1
60 3 3
212
3 33
90 2 1 0
0
, x rad x,
(x rad), ( x rad), (x rad) (x rad),
x, x, x x.
, .. rad3
3
(100rad ) 100 .
3.1 57
1 h , . .
:
h48 , =
h48
=
h70 , =
h70
=
20m = =
h h 20,48 70
=
20 70 48
h .70 48
=
70 2,75 48 1,11.
61,05h 371,64
37m .
2 750o .
750 360 2 30,
1750 (2 360 30 ) 30
2 = + = =
3750 302
= =
3750 303
= = 750 30 3 = =
750 2 360 30= +
(1) :
58 3:
3 793
rad.
79 79
2 .3 6=
79 6
13 1. 79 79 12 13 2 13 2 ,
3 6 6 3 = = + = +
:
79 313 23 3 3 2 = + = =
79 13 3 2
= =
79 33 3
= =
79 33 3 3
= =
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) 10
rad ii) 5
6
rad iii) 91
3
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
3.1 59
1. - , - 1000 - (1 0,3 m) .
.
i) h =30 ,45 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
60 3:
3.2
. .
:
1. 2 2 1 + =
M (x, y) , :
x = y =
,(OM) =1
2 22 2 2( ) x y x y = + = + :
2 2x y 1,+ =
:
2 2 1 + =
2. = =
:
y
x
= =
( x 0= )
x
y
= =
( y 0= ).
AA
t
y
y
xx
1
|x|
|y|
(x,y)
O
3.2 61
(1) (2), - .
3. . = 1
:
=
=
( 0 0)
:
1 = =
.
4.
22 2
2 2
11 1
= =
+ +
i) 2 2 1 + = 2 0 :
2 22 2
2 2 2 2 2
1 1 111
+ = + = =
+ .
2 21
.1
=+
ii) 2 2 1 + = 2 21 ,
1 =
+
:2
2 2 22 2 2
1 11 1 .
1 1 1
+ = = =+ + +
2
22 .1
=
+
62 3:
1 513
= 90 180 ,<
3.2 63
1. 3x5
= x ,2< < -
x rad.
2. 2x3
= 3
x ,2
< < -
x rad.
3. 3x3
= 3
x 2 ,2< < -
x rad.
4. 2 5x5
= 0 x ,
2
< < -
x rad.
5. x 2 = 3 x 2 ,2< < -
2 x x
.1 x
+
6. , x :
i) x = 0 x = 0.
ii) x = 1 x = 1.
iii) 3
x5
=
4x .
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. :
i) 11
=+
ii) 4 4 22 1 = .
64 3:
11. :
i) 1 21 +
+ =+
ii) x x 2 .1 x 1 x x
+ =
+
12. :
i) + =+
ii) 2 2 2 2 . =
13. :
i) x x x x1 x 1 x
+ = +
ii) 1(1 x) 1 x x
x + =
iii) 1 x xx x
= +
iv) 1 1x x x x.x x
=
B
1. x + x = , -:
i) x x ii) 1 1x x+
iii) x x + iv) 3 3x x. +
2. :
i) 4 4 2 2x x 1 2 x x + = ii) 6 6 2 2x x 1 3 x x. + =
iii) 6 6 4 42( x x) 3( x x) + + - x , .
3. x ,2 2
< < 1 x 1 x
2 x.1 x 1 x
+ =
+
4. 0 x ,2
< 1 x 1 x 1 x x
.x 1 x1 x 1 x
+ + + = =
+
3.3 1 65
3.3 10
, , 0 90 .
' ' .
' , ' = , , - , x'x. . , - :
1( 30 ) (30 )
2 = =
3( 30 ) (30 )
2 = =
3( 30 ) (30 )
3 = =
( 30 ) (30 ) 3 = =
9 , :
24 4 2 = =
24 4 2 = =
14 4 = =
1
4 4 = =
: . :9 :
( ) ( )
( ) ( )
= = = =
A
(x,y)
(x,y)
A
y
x
t'
t
66 3:
180
' -
180 , ' 180 , = , , ' y'y . . , - :
:
180o . :
9 150 180 30 ,= :
9 2 ,3 3 = :
2 33 3 3 2 = = =
(180 ) (180 )
(180 ) (180 )
= =
= =
1150 (180 30 ) 30
2 = = =
3150 (180 30 ) 30
2 = = =
3150 (180 30 ) 30
3 = = =
150 (180 30 ) 30 3 = = =
tt
y
x
A A
(x,y) (x,y)180
B
B
0
3.3 1 67
180
'
180 , ' 180 , = + , , - ' . - . - , :
2 13 3 3 2 = = =
2 33 3 3 = = =
2 33 3 3 3 = = =
:
180o -, . :
9 210 180 30 ,= + : 1
210 (180 30 ) 302
= + = =
3210 (180 30 ) 30
2 = + = =
3210 (180 30 ) 30
3 = + = =
210 (180 30 ) 30 3 = + = =
(180 ) (180 )
(180 ) (180 )
+ = + =
+ = + =
y
A A
B
(x,y)
(x,y)
t
t
y
x
180 +0
68 3:
9 4 ,3 3 = + :
4 33 3 3 2 = + = =
4 13 3 3 2 = + = =
4 33 3 3 = + = =
4 33 3 3 3 = + = =
90
' - 90 , ' 90 , = , , ' xOy . . , - :
,
90o , - .
, 60 90 30 ,= :
360 30 ,2
= =
160 30 ,2
= =
60 30 3 = = 360 30
3 = =
(90 ) (90 )
(90 ) (90 )
= =
= =
t
t
(x,y)
(y,x)B
A
B
A90
y
xO
y=x
0
3.3 1 69
, 0 90 .
1 1 536 .4+
=o
54 .o
54 90 36 ,=
2 2 1 + =
2 254 54 1, + = : 22 2 1 5 6 2 5 10 2 554 1 54 1 1 ,
4 16 16 + +
= = = =
:
:
2 - :
) 90 ,+ ) 270 ) 270 +
i) 90 90 ( ),+ = :
(90 ) (90 ( )) ( ) . + = = =
90 .+
54 10 2 554 .
54 1 5
= = +
1 554 364+
= =
10 2 5544
=
54 1 55454 10 2 5
+ = =
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) 1876
rad ii) 21
4
rad.
3. : i) ( ) = + ii) ( ) 0+ + =
iii) 2 2 +
= iv) .2 2 +
=
4.
5. :
9( x) (2 x) x
21.
21(13 x) ( x) x
2
+ + = +
6. :2 2( x) ( x) (2 x) 2 x .
2 + +
( ) (180 ).
( ) (90 )
+ +
3.3 1 71
1. :
2. :
2
5 7(5 ) (7 )
2 21.
5 7(5 ) (7 )
2 2
+ + =
+ +
3. x x 5,3 6 + + =
-
:
2 2x x .
3 6 + +
4. :( x)
0 1.x ( x)
+< < + +
I) - , , - .
1. =1, = 0 . 2. = 0 , = 1. 3. + =2 . 4. 21 = 5. 2 220 70 1 + =
6. x (x ) x =
7. x 2 2x x =
8. (x ) x 0,2
+ = x 0 =
9. x (x ) ( x) 06 3
+ =
495 120 495 ( 120 ).
( 120 ) 495
+ +
72 3:
II. .
III. .
1. ( =900 ) , :
) 2 2 1, + = ) 2 2 1, + = ) =1. 2. : ) ( + ) = , ) ( + ) = , ) ( + ) = . 3. :
) ( ) ,2 2
+ = ) ( ) ,2 2
+ = ) ( ) .
2 2+
=
1 1200
2 1500
3 2100
4 3000
5 2100
6 3000
7 3000
8 2100
3
32
33
12
12
33
32
3
3.4 73
- -
112
- -
. t,
1
t 12
+ 1
t 12
. -
, t, -
t, 1
t 12
+ , 1t 12
.
112
.
t.
, t, t + 2 sec t2 sec.
3.4
74 3:
( t) 2 sec.
:
, , 1 1. -
. .
- , , , .,
f (t) t,= t .
- .
:
x (x rad) .
x (x rad) =
( 360 ) ( 360 ) , + = = x :(x 2 ) (x 2 ) x + = =
2. .
f , >0 , x A :
i) x T A,+ x T A
ii) f (x T) f (x T) f (x)+ = =
f.
3.4 75
x (x rad). =
2.
, : x
xx
=
:
{ }1 x x 0= x 1
(x ) (x ) x, + = = .
, , :
xx
x
=
:
{ }2 x x 0= .
f(x) = x
f(x) = x 2, 2, . [0,2]. x xrad . - , , . :
x 0
,2
.
, - f(x) = x
0, .2
f(x) = x :
y
xx rad
B
A
M
M
O
76 3:
,2
3,2
3 ,22
x ,2
= 12
=
3
x ,2
= 3
1.2
=
:
. [0, 2]:
. :
x 0 2
x 0 1 0,71 0 0,71 1 0,71 02 0,712
4
2 3
4 5
4 3
2 7
4
x
x
2 3
2
20
0 1 0 1 0. .
y
x
0x2pi
O
1
12pipi
4
3pi2
7pi4
y = x
43pi4
pi 2pi5pi
3.4 77
f(x) = x , 2, [2, 4], [4, 6] . [ 2 ,0],[ 4 , 2 ] . , .
. - x ( x) x. = f(x) = x 0(0,0) .
f(x) = x
f(x) = x 2, 2, .. [0, 2]. :
- :
x
x
2 3
2
20
1 0 1 0 1. . .
x 0 2
x 1 0,71 0 0,71 1 0,71 0 0,71 1
4
2 5
43
2 3
2 7
4
22
pi/2pi/2
1.5
0.5
0.5
1
1.5
0
1
pi 2pi3pi/2 5pi/2
x
y
78 3:
y = x
0 x 2 p .
f(x) = x 2, :
y
x
pipi 0
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 ,
, .. , .2 2
( , ,2 2
).
x rad - . x .: x
2
2
' , . f(x) = x
, .2 2
y
x
0x2pi
0
1
1
pi
2pipi
43pi2
7pi4
y = x
5pi4
3pi4
2pi
y
O
x
3.4 79
x 2
x .
x2
= -
f. x 2
x .+ x2
= f.
f(x)=x , - , :
- . f(x)=x .
f(x)=x , ( 5.3: (x)=x .
x 0
x 1 0 1
2
3
4
6
6
4
3
2
3 1,7 3 1,73 0,6
3
3 0,63
x2pipi 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
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.
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.
3 f(x)=32x .
3, 3 .
x 0 2
x 0 1 0 1 03x 0 3 0 3 0
2 3
2
x 0
2x 0 1 0 1 0
4
2 3
4
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
3.4 81
, f(x)= x, ,>0: (i) ,
. (ii) 2 .
f(x)= x, ,>0
1. ,
i) f (x) 2 x,= g(x) 0,5 x,= h(x) 2 x,= 0 x 2 ii) f (x) 2 x,= 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 .
5. xf (x) 2 .2
= -
; ; f .
x 0
32x 0 3 0 3 0
2
4 3
4
f(x)=32x :
- .
y
x
3
3
pi 2pi3pi3pi0pi pi pi2
pi4 2 2 2
y = 32x
82 3:
6. xf (x) 22
= .
7.
i) f (x) x= ii) g(x) 1 x= + iii) h(x) 1 x= + .
8. f (x) 2x.=
9. f (x) x.=
B
1. :
2.
y 3 t ,6 =
y
t .i) -
.ii) 0 t 12.
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
3.5 83
3. 1m. ,
1
h 1 3t,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=
1
x .2
=
y = x
1
y .2
=
x, f (x) x= -
1
.2
2,
x, (. ), 2 . 2, .
xy=x
pipiO
2pi
yy=
pi6
12
5pi6
pi1m
84 3:
-
1
x2
= [0, 2], 6
5 ,
6 6
=
5 1.
6 6 2
= =
1x
2 =
, x = , = , :
,
x 2= +
x 2 ( )= +
x 23
,
x 23
= = + +
1 3x2
=
3 ,
3 2
= 3
( ) .3 2
=
x ( ),3
= :
2 12x4 2 + =
1 ,
6 2
= 2x4 6 + =
,
y
M M
x5pi/6
pi/6
12
AOx 2 6,
5x 2
6
= + = +
3.5 85
2x 2 2x 2 x4 6 6 4 24
+ = + = + =
72x 2 2x 2 x
4 6 6 4 24 + = + = + = +
x =
,
.. 1
x .2
=
-
1
x2
= [ , ] 3
,3
1.
3 3 2 = =
1
x2
=
,
x24
,7
x24
= = +
2x 24 6
,
2x 24 6
+ = + + = +
x 23
x 23
= + =
y
x
M
M
pi3pi3
12
86 3:
,
, x = , = ,
1 2x2
=
2 ,
4 2
= x ,4
=
:
2 32x2
=
3 ,
6 2
= 3
6 2 =
5 3.
6 2
=
5
2x ,6
=
x 2
x 2
= +
=
x 24
,
x 24
= + =
52x 2
6,
52x 2
6
= + =
5x
12,
5x
12
= + =
3.5 87
x =
x 3. = - . , , , ..
,2 2
, .
, x 3 =
. ,3
3.3
=
x 3 = : x ,3
= + .
, x = , x = , :
x ,= +
x . =
1 x 1 =
1,4
= 1.4 =
x ,
4 =
x ,
4
=
2 x 3 =
x ,6
= +
3,6
= x ,6
=
pi xpi pi
y
pi22
3
88 3:
1.
i) x 0 = ii) 2x2
= iii) x 0 = iv) 2
x2
=
2.
i) 1
x2
=
ii) x 1 = iii) 2
x2
=
iv) x 1 =
3.
i) x 0 =
ii) 3
x3
= iii) x 1 = iv) x 3 =
4.
i) 3
x3
=
ii) 3
x3
=
5. i) (1 x)(2 x 3) 0 =
ii) (2 x 2)(1 x) 0 + =
6.
i) ( 3 x)(1 x) 0+ =
ii) 2(2 x 1)( x 3) x 0 + = 7. -
i) x 0,951 =
ii) x 0,809 = iii) x 28,636 =
8. i) 2 3x 3 =
ii)
x 1 05
+ = iii) 2x3 3 07
=
9.
i) x 13 + =
ii) 2 3x 1
4 =
iii) 5x 3
4 =
10. i) 22 1 0 + = ii)
22 x 3 x 2 0 + = iii) 23 t 3 2 3 t = +
11. i) 2 2x 5 x 4 + = ii) x 2x 1 = 12. x,
:
i) f (x) 3 x ,2 =
0 x 2 , < ii) g(x) 7 x ,2
=
0 x 2 <
3.6 89
13. ( -
) tS 75 50 ,
6
= +
t o t = 1 . i) 100000 , ii)
. 1.
i) x x 04 + =
ii) 2x 3x 0
3 + =
2. i) x x 1 x x + = + ii) 2
12 x 4
x =
3. x 1 = (3,4).
* 4. 1+x=x [0,2).
5. : x x3 = +
[0, 2).
3.6
, - 1 2M , M (. 1). , - . (. 2).
O O
y y
x x -
-
M (, )1M (, ) M((-), (-))2
(1,0)
. (1) . (2)
(1,0)
90 3:
, 1 2M ,M , :
1: 2: : 1 0 : ( ) ( )
2 1 M OM AOM ,= = 2 1(M M ) (AM).= 2 2
2 1(M M ) (AM)=
:2 2
1 2 2 1 2 1(P P ) (x x ) (y y ) ,= +
1 1 1P (x , y ) 2 2 2P (x , y ), :
2 2( )= +
(2)
2 2 ( ).=
2 22 1( ) ( ) =
2 2( ) 2 2 ( ) + =
( )+ =
:
:
(1)( ) = +
, , 00 360 , < < , .
, :( ( )) ( ) ( ) = + =
( ) + =
22 1(M M )
2 2
2 2 2 2
( ) ( )
2 2= +
= + + +
2( ) 2 2
2 2
[ ( ) 1] [ ( ) 0]
( ) 1 2 ( ) ( )
= +
= + +
3.6 91
(1) (2) - , - . , :
(1), , - .
, :
x x2 =
x x,
2 =
:
( ) ( ) ( )2 2
2 2
+ = + = = + = +
:
(3)
:
( ) + = +
( ) =
15 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= = +
+= + =
75 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= + =
= =
15 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= =
= =
75 (45 30 ) 45 30 45 30
2 3 2 1 2( 3 1)
2 2 2 2 4
= + = +
+= + =
92 3:
: (5)
, :
(6)
:
(7) (8)
( )1 +
+ =
( )1
=+
1( )
+ =
+1
( )+
=
, :
3145 30 3 3315 (45 30 )1 45 30 3 3 31
3(3 3)(3 3) 12 6 3 2 3
6(3 3)(3 3)
= = = =
+ ++
= = =
+
( )( )
( )
1
+ + + = =
+
+ + = =
- + , , .
, : ( ), + , ( ) 0, + 0 0. :
0
3.6 93
-
1 3 ,5
= 3 22< <
12 ,13
= 3 ,2
< <
+.
(+) = + ( ) + = . :
2 2 9 161 1 ,25 25
= = = 3 22< <
2 2 144 251 1 ,169 169
= = =25 5 ,
169 13 = = 3
2
< 1 , x =1 x
144 4: -
0 2x x 6 0 (x 3)(x 2) 0 x 3+ + x 2 2x x 6+ x 3< x > 2 , x 3= x = 2 3 x 2. < 2 , x 3< 1< x < 2 . x 3,= x =1 x = 2 .
A(x) . B(x). ... . (x)>0 ( <
2 2(x 1)(x x 6)(2x x 1) 0 + + + < x 2 2P(x) (x 1)(x x 6)(2x x 1)= + + + .
P(x) x ( , 3) (1,2).
3 2x 3x x 2 0 + + >
1,
3 2P(x) x 3x x 2= + + :
1 (2, )+ (x) ,
2(x 2)(x x 1) 0 >1 5 1 5
(x 2) x x 0.2 2
+ >
4.3 145
. 1 5 ,22
+
(x) ,
, x2, . , (x) .
x , 1 5 1 5x2 2 +
< < x>2.
, . , - , :
: f (, f()) (,f()) x'x, .
11 1 0f (x) x x ...... x
= + + + +
,
146 4: -
3f (x) x 3x 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.
f (1,5) 0,13 0
f (1,6) 0,30 0
f (1,53) 0,01 0
f (1,54) 0,03 0
1. . i) 4 25x 6x= ii) 3 2x 2x 9x 18 0+ = iii) 5 4 3 23x 5x 3x 5x+ = + iv) 6x 64 0 = v) 3 2x x 2 0+ = vi) 3x 7x 6 0 + =vii) 3(x 1) 1 0+ + = viii)
2 2 37(3x 2) (1 x) (3x 2)(1 x) 0+ + =
ix) 3 2 2x 8 7(x 5x 6) 9x 36+ = + + + x)
4 3x 3x 6x 4 0 + =
3x 3x 1 0 + = - 1 2. .
f (1) 1 0
f (2) 3 0
=
1 2
B(2,3)
A(1,1)
1,5
4.3 147
2. i) 3 2x 3x x 2 0 + + = ii) 3 23x 8x 15x 4 0+ + = iii) 3x 10x 12 0 = iv) 3 2x 2x 7x 6 0+ + + =
3. . i) 4x 3x 2 0+ = ii) 4 3 22x 3x 6x 24x 5 0 + + =
4. , x.
i) 2 2P(x) (2 3x)(x x 2)(x x 1)= + ii) 2 2 2Q(x) ( x 4)(x 3x 2)(x x 1)= + + + +
5. : i) 52x 162x 0 ii) 3 2 2(x x 2x 2)(x 9) 0 + > iii) 3 22x 5x 6x 9 0 + > iv) 3 2x 4x 3x 18 0 +
6. : i) 3 2x 2x 3x 6 0+ + + > ii) 4 3 2x 6x 22x 30x 13 0 + + iii) 3x 3x 2 0 + < iv) 4 3 2x x x 3x 6 0 +
7. x'x :
i) 3 2f (x) 3x 3x 5x 2= ii) 3g(x) 4x 3x 1=
8. , - 4 3 2f (x) x 5x 3x x= + + x'x.
9. :
i) 8 4x 15x 16 0 = ii) 6 3(x 1) 9(x 1) 8 0 + =
iii) 2
x x6 5 6 0x 1 x 1
+ = + +
10. 3x 5x 3 0+ = (0,1) .
148 4: -
'
1. :
i) 3 21 1 1 4
x x x 0,10 2 5 5
+ + = ii) 3 25 22 5
x x x 06 3 2
+ =
2. , 4 3 2P(x) x x x 16x 12= + + x+1 x 2. (x) = 0.
3. k , 3 2x x kx 3 0 + + = .
* 4. x 2 x 2 0, + = N, *2, ,
.
5. 6 4 2P(x) x 5x 10x k= + k x 1 (x). k (x) = 0.
6. 5dm 9dm, . , - dm 21dm3.
7. t
2
3
3t tc .
t 50
+=
+ ,
4 33t 2t 300t 200 0.+ = t .
8. 36m3, x.
3 x 3 x
1m
xm
xmxm
4.3 149
9. . V,
.
10. t , mm - 2d 15t(t 6t 9).= - ;
11. , , 108cm ( ). , 11664 cm3.
12. i) (1,2)
ii)
y=x3+x2 x - .
iii) .
2 3500V (2000 100 20 ),3
= +
1 1B( , ).
2 2
3 2x x 5x 3 0+ + =
3 33x xy
150 4: -
13. . :
) 200ml , 1cm, .
3 2x 3x 2x 200 0.+ + = x mm.
) - 1lit, 10cm .
3 2r 10r 318 0+ = r
mm. ( 1000 318 ).
) - , 250ml, 5cm .
h ( -).
4.4
, , . .
1o 2 2 1x 02x 1 x(2x 1)
+ =
x x 0 1
x .2
:
xcm
x+2 cmx+1 cm
r+10 cm
r cm
h+5cmh+5cm
4.4 151
3o 2x 7 x 2.+ =
x 7x .2
' x :
( )
3 1. - x=1 .
2 2
4 3
3
3
2 1 2 1x 0 x(2x 1)x x(2x 1) x(2x 1) 0
2x 1 x(2x 1) 2x 1 x(2x 1)
2x x 2x 1 0
x (2x 1) 2x 1 0
(2x 1)(x 1) 0
+ = + =
+ =
+ =
+ =
, , . - - .
2x 7 x 2+ = +
2 2( 2x 7) (x 2)+ = +
22x 7 x 4x 4+ = + +2x 2x 3 0+ =
12
1. - x 1.=
2 x x 2.=
x 0. -, 2x x 4x 4,= + 2x 5x 4 0 + = x1 = 4 x2 = 1. x, - x 0 . , :
x = 4 : 4 4 2= x = 1 : 1 1 2= . x=4.
( )
152 4: -
4 2x 6 x 4 1.+ + =
x, 2x 6 0+ x 4 0,+ x 3. ' x :
3 5. x = 5 .
2x 6 1 x 4+ = + +2 2( 2x 6) (1 x 4)+ = + +
2x 6 1 2 x 4 x 4+ = + + + +
x 1 2 x 4+ = +2(x 1) 4(x 4)+ = +
2x 2x 15 0 =
( )
( )
( )
(x)
> 0 (< 0)B(x)
A(x)0 A(x) B(x) 0
B(x)> >
A(x)0 A(x) B(x) 0,
B(x)< <
, A(x) B(x) 0 > A(x) B(x) 0 < (x).
, 2
2
(x 1)(2x x 1)0.
x x 6
+ +>
+
2 2(x 1)(x x 6)(2x x 1) 0 + + + > , P(x)> 0 , , P(x) -
x ( 3,1) (2, ). +
.:
2, 3 4, x , .
4.4 153
2
2
x 4x 30.
x 3x 4
+
+ :
2
2 22
x 4x 30 (x 4x 3)(x 3x 4) 0
x 3x 4
+ + +
+ 2x 3x 4 0.+
2x 4x 3 + 1 3 , 2x 3x 4+ 1 4 .
:2 2P(x) (x 4x 3)(x 3x 4)= + +
x - +x2- 4x + 3 + + - +x2 + 3x-4 + - + +
P(x) + - - +
-4
00 0
100
30
0
x ( , 4) [3, ). +
1. :
i) 2 2
2
3x 1 2 x 3x 2
x 1 x x x
+ =
ii)
2
2
x 2 4
x 1 x 1 x 1 =
+
2. : 3 22 x x 2 x 2 0 + + =
3. :
i) 3x 4x=
ii) 3x 2 4 = iii) 5x 1 4 =
iv) x 3 x 1+ = + v) x 3 10 x 1+ = + vi) x x 20 10+ =
vii) x 8x 32 x
= +
viii)
1 2 x x 1+ = +
4. :
i)
x 2 0x 1
>
+ ii)
2x 1 0x 3
+
iii)
2
2
x x 2 0x x 2
+
A(x) 0B(x)
-
x
A(x) B(x) 0 B(x) 0.
154 4: -
( )
5. :
i) 2x 3 4x 1
+>
ii) x 2 4
3x 5
+ iii)
2x 3x 10 2 0x 1
+
iv) x 23x 5 x 1
v) x 32x 1 x 2
+
6. : 2 2 1x2x 1 x(2x 1)
+
B
1. :
i) 2x 3 1 3x+ < ii) x 3 x 5 > 2. :
i) x 3 x 10 0+ = ii) 3 2 3x x 6 0+ =3. :
i) 2 2x x 4 x x 2+ = + ii)
4. :
i) x 1 = ii) 24x 1 2x+ =
5. 4 3 22 x 3 x 3 x 3 x 4 0 + =
1. 2 3(x x 1)(x 1) x 1,+ + = - 3 3 1 3 2P(x) x x x + += + + x
2+x+1 (, , ).
2. : i) 1f (x) x ( 1)x 1+ = + +
2(x 1) . - .
ii) 1g(x) ( 2)x x x 2n n-= n- -n + n -n + 3(x 1) .
3. : i) 4 3x 2x 2x 1 0+ + + = ii) 4 3 2x x 4x x 1 0+ + + =
( 4 3 2x x x x 0, + + + + = 0.
).
x 1 x 4 x 4 + = +
4.4 155
: x2
4. i) 4 3 2x x 16x 2x 4 0+ + = ii) 4 3 2x 8x 13x 8x 1 0+ + + =5. 2 2 2(x 2x 1) 3(x 2x 3) 14 0+ + + + =
: 2x 2x 1 y.+ =6. , 5 2x 3x x+ + + , -
2x 2, 5x+8.
7. 17 16 15 14P(x) x 12x 12x 12x ... 12x 1,= + + + (11) (13).
8. - . - C , x 3 210x 100x 270x 180. = +
i) ;
ii) ;
1x y.
x+ =
156 4: -
2 1,414213,= 2,
,
2
2 2.
=
157
2
1,24
x2
=
3 (*)x x+ =
3 2 3 2
3 3x2 2 2 2 2 2 = + + + +
* 3 : 3 2x x x 0+ + + = - x y
3
= 3x Ax B.+ =
158 4: -
. : , :
:31 1 1 1 1
2 2 2 2 8 = =
:0 1 = 1 1 ,
= = 0 N*
- . :
2 22 3 93 2 4
= =
122 ,
145
, 0, .
. O
, . T - ..
253 ; Av p q pq( ) =
, :
4-
5
5.1
.... , 1,
, 1
> = =
N*
2 2 55 25 5(3 ) 3 3
= =
5.1 161
253 5 2x 3 ,= 5 23 .
2
5 253 3 .=
Av >0, , :
=
, , , , 0 0. =
..
, - .To . ..
- :
23 2 338 8 64 4= = =
43 43 3
4 43 4
1 1 127 2727 327
= = = =
1 1 11 771234 3 4 34 12
+ = = = =
21,4
1,43,421
5
2
1.4
5
1.4
3.21
7 373
xy
xy
xy ( )
=
+/ =
=
: x x , - ;
23 ;
( B' . 104)
2 , , . 1, 1,4, 1,41, 1,414, 1,4142, 1,41421, 1,414213,.... (1)
162 5:
2 3:31, 31,4, 31,41, 31,414, 31,4142, 31,41421, 31,414213, (2) :
1
1,4
1,41
1,414
1,4142
1,41421
1,414213
4,4,74,724,7
3 33 65553673 0
284,728
696503 769503 73393 78393 0154,7288
=
Av : (1) , (2) , . - 23 . - 23 4,7288.
:
Av >0, x x ,
+ , ( ) , ( ).To x x. :
, x>0, 0x=0.O :
3 2
2
x lim
=
23
2
xy
xy
x
exp
=
=
5.1 163
, ' , - .
:
, 1 2x, x , x , :
1 2 1 2x x x x+ = 1 2 1 2x x x x: =
x x x( ) = x
xx
( )
=
1 2 1 2x x x x( ) =
. x x . x x, :
f : xf (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
132
y=2x
(x, y) - . , -
xf (x) = >1, : 0
1
1x
y
y=2x
164 5:
.
(0, )+ .
. 1 2x , x :
1 2x x ,< 1 2x x <
y'y (0,1) - x.
x
x
y
x1
2x
1x
2
y= , >1x
A(0,1)
x1
g(x) .2
=
:
x -5 -4 -3 -2 -1 0 1 2 3 4 5
32 16 8 4 2 112
14
18
116
132
x12
(x, y) - . , -
xf (x) = 0
5.1 165
f(x) = 2x x1
g(x)2
=
x :
- y'y.
xx
x
1 1g(x) 2 f ( x)
2 2 = = = =
f(x)=x, 0 1,< : 1 2x x , 1 2
x x
, , :
1 2x x , = 1 2x x .=, :
, . .
1 2x x1 2x x = =
xx 1
1
x
y
y=2y=2 xx
x2
1
x2
x
1 2
1x
x
yy
xO Ox1x 2x x
166 5:
i) :
ii) 2x x x 2 x3 8 3 9 0 (3 ) 8 3 9 0 = =
3x = y, y28y9 = 0 1 9. :
x3 1= x3 9=' , 3x> 0, 3x = 32 x=2, .
2o :
3x = 2y = :
x y
x y
2 3 3 2 22( )15 3 2 9
2
=
+ =
= 1 = 8,
x = 0 y = 3.
2 3 2215 92
= + =
x
y
3 12 8
=
=
x 0
y 3
3 32 2
=
=
( )
3x 3x 612 2 264
3x 6
x 2
= =
= =
1 :
i) 3x 1264
= ii) x x9 8 3 9 0 =
11
5.1 167
i)
ii)
4o :
i) xf (x) 2 3= + ii) x 3g(x) 2 = iii) x 3h(x) 2 2= +
i) f - (x) = 2x 3 . ii) g (x) = 2x 3 .iii) h (x) = 2x 3 2 .
3o :
i) ii)
3 > 1
2x 3x 139
>
2x x1 1( )2 4
+ >2
2
x 3x 2
x 3x 2 0
>
+ >
2 2x x x x 21 1 1 1( ) ( ) ( )2 4 2 2
+ +<
+ >
x
y
4
O
1
1
3
y=2
y=2x
x+3
x1O
y=2
y=2x3
y
3
x
1 x
y
1
1
O
y=2
y=2x3+2
32A(3,3)
x
168 5:
e . - 1 , 100% 1, 2, 3, ... , , . -:
0 (1 )
= + , .100
=
=1, =1 11 1(1 1) 2 = + = .
=2, = 12
2
211 1 2,252
= + =
.
=3, = 13
3
311 1 2,443
= + =
.
............................................................................................................................
=, = 1
1 11 1 1
= + = +
.
:
1 2 4 12 52
2 2,25 2,441406 2,613035 2,70481311
+
( ) ( )
( )
( ) ( )
1000 10000 100000 1000000
2,716923 2,718145 2,718268 2,718280
, , 11
+
. - e. , Leohard Euler (1707-1783). e e = 2,71828.
1e lim 1
= +
5.1 169
. , , .
- e. - f(x) = ex. - .
e
c t
0Q(t) Q e= (1) , t. To Qo - Q ( t = 0) Q0 > 0, c . - . c > 0 Q , c
170 5:
5 ,
t5
0Q(t) Q 2
=
5 , c t0Q(t) Q e= :
c 5 5 c00
1c 55
Q 1Q e e
2 21
e 22
= =
= =
t
50Q(t) Q 2 .
=
1. :
i) xf (x) 3= x11
f (x) ( )3
=
ii) xf (x) 3 ,= x
2f (x) 3 2= + x
3f (x) 3 3= iii) xf (x) 3 ,=
x 24f (x) 3
= x 25f (x) 3+=
iv) xf (x) 3= x 26f (x) 3 1= +
v) xg(x) e= , x 21g (x) e ,+= x2g (x) e
= x3g (x) e 2= +
2. : i) x2 64= ii) iii) iv) x 13
81 =
v) x3 64( )4 27
= vi) 4x x 127 9 += vii) x 1 x32 16 = viii)
2x x 23 1 =
3. : i) 2x 1 x2 4 2 0+ = ii)
x x2 4 5 2 2 0 + =iii) 2x 1 x3 26 3 9 0+ =4. : i)
2x 5x 65 1 + < ii) 2x 4 x 17 7 +> iii) x 1 2x 41 1( ) ( )2 2
+ ' :
y
x
A(1,0)1x 2x
y=log x, 0
5.3 183
, , : 1 2log x log x , = 1 2x x=
, :
1 2 1 2log x log x x x = = .. 22log (x 1) 3, = :
2 2 32 2 2
22 2
2
2
log (x 1) 3 log (x 1) log 2
log (x 1) log 8
x 1 8
x 9
= =
=
=
=
x 3 = x 3=
(x, y) - - (x)=lnx.
f(x)=lnx+1 - (x)=lnx 1 , g(x) ln(x 2)= - (x)=lnx 2 .
, .
-
1 i) (x) ln x = ii) f (x) ln x 1= + iii) g(x) ln(x 2)=
x 0,2 0,3 0,5 0,7 1 2 3 4 5
y=lnx 1,6 1,2 0,7 0,4 0 0,7 1,1 1,4 1,6
(x) = lnx :
x
1
1
3O
y
y=Inx
y=In(x-2)
y=1+Inx
2
184 5:
2 : 2>1 :
.
2>1 log0,5 x 1 0. > :
x x=2 . - , x = 2.
1. :
2f (x) log x= 12
g(x) log x=
; .
2. :
f (x) log x= , g(x) log x 1= h(x) log(x 1)= 3. xf (x) =
g(x) log x= :
i) (2,4) ii) B( 2,4) iii) (2, 4) i) ( 2, 4) 4. ASA
DIN. x ASA y DIN y = 1+101ogx, x= 50, 100, 200, 400, 800, 1600 ASA. ; ( log2 = 0,3).
0,25 0,5,>
x 1 = x 2=
2
2log0,5 1 log0,5
log0,5 log0,5
>
>log0,25 log0,5>
2 22 2 2 2 2log (x x) log 2 log (x 1) log (x x) log [2(x 1)] = + =
2x x 2(x 1) =
5.3 185
5. : i) log(x 1) log(x 1) log 2+ + = ii) log(x 1) log x 1 log5 + = iii) 2 2log x (log x)= iv) 2log(x 1) log x log 2+ =6. : i) x 1 x5 2 = ii) x 1 x 13 2 +=7. : i) 3log 2 3log 5 ii) 0,3log 5 0,3log 7
iii) 2log(x 1)+ log 2x
8. 7[H ] 10+ > 7[H ] 10 .+ < p.
1. :
i) f (x) ln x= ii) 21
f (x) ln x2
= iii) f (x) ln x= iv) f (x) log(10x 20)= 2. :
i) 2f (x) ln(x x 1)= + + ii) 1 x
f (x) ln1 x
=
+3. x log178, x xlog 81(2 2 3 ),+ x log3
;
* 4. log log log = = 1
=
5. : i) log x log x= ii) 4 2ln x 5ln x 4 0 + =
6. log5 log xx 5=
2log x log55 5 4 x= +
7. :
i) 2log(xy) 4log 2
log x log y 3(log 2)
=
=ii)
xy 8
log y 2log x
= =
iii) y 2x
2log y log x log 2
= = +
8. : i) 2 2log x (log x)> ii) 2log(x 4) log3x < iii)
log xx 10>* 9. 2 6log 3 log 9>10. , > 0 :
>
186 5:
1. : i) 2 3x 5(x 3x 1) 1 + = ii)
2x 3x 1x x+ + =
2. :
log log 2log log+ + + = ( + , 1 )
3. log 2( ) , = log , log log
(0 , , 1, 0)< > .
4. , , , :
5. log5 1 log 2= log(2x)x 5=
6. (0, )2
:
x x x xlog 2 log 2 log 2 log 2 0 + + =
7. (0, )2
: x x( x) ( x) =
8. : x x x27 12 2 8 0+ >
()
log log log
log log log
=
0 , , , 1<
187
3
4
4096 :128 32 (12 7 5)
16 4096 (4 3 12)
4096 8 (12 : 4 3)
= =
= =
= =
*
188 5:
1 4
11 ,10+
= +
4
11 1,000110
= + =
0
=
0
1
100.000.000
10.000
+
= = +
0 100.000.000 =
1 100.010.000 =
00 =
110 =
220 =
84
110 110
= +
23027 999.999.779 =
10 =
23027230.270 =
.....................................................................................................
84
110 1 ,10
= +
2 100.020.001 =
189
2 4 8 16 32 64....
1 2 4 8 16 32= = = = = =
190 5:
8x 10= 5y 10
=
yx e ,= y ln x.=
3ln(1,00984768) 9,7995075 10 0,0097995075 0,0098= =
84
110 110
= +
4 4 44 5 510
10 10 1010 10 10
8 4 4 4
1 1 11 1 110 10 10 10
= + = + = +
410
4
11 2,71814592710
+ =
4 y10
4
1x 1
10
= +
191
1y ,x
=
x
H
y
x1y=
192 5:
A log E=
(' ) 193
1. : 2 2x 2 3 x x x 3 =
2. i) x x + = 2 2 2. +
ii) (1 t) x t x 2+ + = - t ( , ).
3. 3
2
331 3
=
,12
3 2x 3x 3x 1 0. + =
4. ( 9) 198 22 9= 9. 1+9+8 = 18
9. 17397 1933 9= 1+7+3+9+7 = 27 -
9. : 9, +++ -
9. : 3 210 10 10>= + + + . -
3 2f (x) x x x= + + + f (x) (x 1) (x) f (1)= + x=1 x=10.
5. ( ). .
: 11 1 0x x .... x 0
+ + + + =
. 0
-
, 0
. :
i) : 3 22x x x 1 0+ + = 4 3 26x 29x 27x 9x 1 0+ + + + = ii) 2 12 .6. x x x3 4 5+ = .7.
( )
xx2 4 2
3 3 + =
194 (' )
8. 2log(x 3) log( x)+ = ; 9.
4
x log x 2 + = (0,) - .
10. : x x3log (16 2 12 ) 2x 1 + 11. i)
f (x) ln x= g(x) 1 x= ln x 1 x.
ii) f (x) ln x= 2g(x) x 1= + 2ln x x 1. +
12. , , , , :
, , > 0, ++ =
= =
:
13. , , , , :
, , > 0, ++ =
= =
() = , (KM) = , () = , = , = , =
14. , , , , :
0 < , , <
:
++ = .
15. , , , , :
0 < , , <
, (M) = , (KM) = , () = , = A, = , = .
= =
2 2 2
2 2 2
2 2 2
22 B2
= +
= +
= +
2 2 2
2 2 2
2 2 2
222
= + = + = +
2 2 2
2 2 2
2 2 2
222
= + = + = +
- 195
(3, 1) (21,24) 3 5, .4 3
(3, 4) (5, 2).
2, ,2
+
(2, 1).
)( 3 1)( 1), , + +
(4,3, 5)(10 2, 16 2, ) , + +
11: y x 2,2
= + 2 : y x 1 =
1 11R T .600 30
= +
11 ,2
= 212
= 3 .2
1,
ii)
3, 3, =
3 3, =
10. x ,= y ,= z .=
11.
2f (x) x 4x 3,= +2g(x) x 2x 3,= + +
2h(x) 0,5x 3x 4.= +
( 1, 2), (2, 1).
2. 2 4,3 3
( 1, 2), (1, 2), ( 2, 1), (2,1).2 1 , ,
1 1 + + + +
-
3 2 3 2,2 2
3 2 3 2, ,2 2
196 -
(4,3), ( 4,3), (0, 5).(1,0), (3,0), (2, 1), (4,3).
1. ( ,1], [1, ),+ ( ,0], [2, ),+ ( , 1],
[ 1,0], [1, ).+
f (1) 1=
h( 1) 2, = h(1) 2=
f (x) f (3)g(x) g(1)
4. 2f (x) 2 (x 1) 3= +2f (x) 2 (x 2) 1.=
22(x 2) 22(x 3) 3 22(x 2)+ 22(x 3) 3+
1. x 3,= y 3 2,= 0 45 . =
2. 1, = 3. =
3.
4. 6 2
3
334
5460
18000 .
6. 1 3 3, , , 32 2 3
3 1 3, , 3,2 2 2
0, 1,0,
0,1,0,
478, 733, 1062 58 .
2 2 2 2 .2
12 8 3, 12 3+
197
4 3 4x , x , x .5 4 3
= = =
2. 5 5 2 5x , x , x .3 2 5
= = =
3. 1 3x 3, x , x .2 2
= = =
4. 5 5 2x , x , x .2 3 3
= = =
5. 8 5 20 .5
6.
2 12
2
21
2
21
2(3 ) .2
3.
2.
2.
y x,= xy ,2
= y 3x=
y x,= y 3 x,= y 0,5 x,= y 2,5 x.=
323
23
x ,=
x ,2
= +
x 2 ,4
=
187 1 ,6 2
= 187 36 2
=
187 3 ,6 3
=187
3.6
=
21 2 ,4 2
= 21 24 2
=
21 211.
4 4
= =
3x 2 x 2 ,
4 2
= + = +
3 11200 , 1200 ,2 2
= =
31200 3, 1200 .
3 = =
1 3( 2850 ) , ( 2850 ) ,
2 23
( 2850 ) , ( 2850 ) 3.3
= =
= =
3.3
( ) 180 ,+ + =
90 .
2 2 +
+ =
7x 2 x 2 ,
6 6
= = +
ii)
198 -
x 2 ,2
=
3x 2 ,4
=
x 2 , .= + x ,= x ,
6
= +
x ,4
= +
x , .6
= +
x ,3
=
5x 2 ,6
=
24 7 24+
12x , .60
=
10.
t , .6
=
12.3.
2x 2 x
3 3
x ,2
= =
= +
x x ,3 4
= = +
24
x ,36
=
5x 2 x 2
4 4x 2 ,
= = +
=
2 3x 2 x 2 ,
5 5
= + = +
3x 2 ,
5
= +
22
2652 ,6
=
= +
=
52 ,6
=
2x 2 ,
3
=
x ,6
=
x 22
= + x 23
= +
2x 2 ,
3
= +
6 6 2x x ,
9 9+ +
= =
22x , .
45
= +
6 2x ,
9+
=
x 10 5 ,= +
42 7x ,
12+
=
4x 2 ,
5
=
x 2 ,3
=
2x 2 x 2 ,
3 3
= =
199
x ,2
= g 72 =
3x2
=
3g 7.2 =
x ,8
=
x ,5 30
= +
x ,4
= +
13 .4
, .2
7 13 19, , ,12 2 12 12
12
12
x2
2
3
3x 12
x
x
7. 2(1 3)105 ,
4+
=
2(1 3)105 ,4
=
105 2 3, =
105 2 3, = +
2(1 3)195 ,4
=
2(1 3)195 ,4
+ =
195 2 3, =
195 2 3. = +
33( ) ,65
+ =
56( ) ,65
+ =
33( ) ,56
+ =
56( ) .33
+ =
x ,6
= +
x ,3
=
x ,4
= +
( ) =
2x x ,
4 5
= = +
2x ,
5
= +
200 -
( 2 ) (( ) ) .... + = + + =
3. x4
=5x .4
=
4. ,4
=
....4 = =
8. x6
=2x .3
=
1 323
3
4 2 6
242 ,25
=72 ,25
=
242 ,7
=72 .24
=
16( 2 )13
+ =
x 2 x 2 ,= = + 2x 2 x 2 ,3 2
= = +
2 2 2 ,16 2 +
=
2 2 2 ,16 2 + +
=
2 2 2 .16 2 2 2
+ + =
+
4 ,2 13
=9 ,
2 13
=
2 ,2 3
=3 .
2 2
=
1 ,2 5
=4 ,
2 5
=
1 ,2 2
= 2.2
=
2x 2 x 2 ,2 3
= =
x 2 ,3
=
x 2 ,2
=
x 2 ,= +
3. 3 ,8 8 2 + =
6. 5x 2 x 2 x 2 ,2 6 6
= = + = +
x x ,3 3
= + =
1 ,4
3 24+
2 2 2 ,16 2 2 2
+ =
+ +
3...
8 8
=
201
14
14
3x x
2x 6x 2x 8x +
1 ( 8x 4x)2
1 2x2
2x x ,3 10 +
= =
2 2x x ,8 2
+ = =
62
22
5. 2 3x x 2 4x x
2 2x 2 x 2 x2
4 2 +
2 x2 .
2
x x 24 6
= = +
5x 2 ,6
= +
++ =
11f (x) 2 x6 = +
3f (x) 2 x4 = +
4f (x) 2 x3 = +
f (x) 2 x4 =
4x 2 ,3
=
x 22
=
x 2 ,=
7x 212
=
11x 2 ,12
= +
512 12
= =
2. ( ) ( ) 2 24 + = +
,4
=
( ) ( ) 2 2. + =
7x x
12 12
= = +
x ,3
=
x6
=6 2
x9
+ =
6 2x ,
9
=
202 -
3. 102 2 2,+ 2 2 2. +
4. 7x x ,24 24
= + = +
5. ,4
=
h 40( 2 1)=
6. ,8
=
P 1 2= +
1. AB 284 m2. B 1,58 m 5. 0115 6. 51,2 m
7. 60,25
1.-5.
6.-7.
8.-10. 12
=
11. 0 30 . =
3.
4. x 2 ,3
=
x 26
= +
5x 2 ,6
= +
6. 133
7 (10 2 2)m,+
(10 2 2)m
(t 1)10 44
+ +
9. 6 2 2 6 2, ,4 2 4 +
12. x 24
= +
2x ,3 4
= +
13. S 200 2 2 2004 = + +
,8
= maxS 200( 2 1)= +
15. B 60 ,= 45 =
1.
2.
3. 12
=
4. 1 =
5. 1 1,
203
1. 10, =
2. 2, = 19 =
3. 3, = 5 =
4. 2 20, ,3 3
23
=23
=
5. 2P(x) x 5x 1.= +
1. 2(x) 3x 3x 8, = 3 2(x) x 3x 9x 27, = + + +
3 16(x) 4x ,6
= 5(x)3
=
2(x) 2x 1, = +
(x) x 3, = + 2(x) 6x 8x 3 = +2(x) x , = 2(x) x 7 = +
2. 3. k 4=
4. 2(x) x 10x 25, 0 = + =2(x) x 8x 64, 0 = + =
4 3 2(x) x x x x 1, 2 = + + + + =3 2(x) 3x 6x 12x 24, 48 = = 2(x) 4x 14x 30, 0 = + =
5. P( 11) 4840 =6.
7. P(x) x y = P( y) 0 =8.
9. P(x) x 1= + P( 1) 0. =10.
1. x x = 2. P(x) ( x ) (x)= + +
0 = = =
3. x 1 x 2
4. P(0) 0,= P( 1) 0, =1P( ) 02
=
5. , = 1 =
1. 6 60, ,5 5 2,3, 3
50,1, 1,3
2, 2
1,2, 3 22 13,1,3 22
2,9 4 5,9 4 5 +
2, 2 , 1, 2
204 -
2. 1, 4 2 1
3. 1, 2
1, 56. x 2> x 1= x 2<
x 1 x 2
7. 1( ,0)2
8. 2 5 x 0 1 x 2 5 < < < < +
9. 2, 2 3 ,25
1. 1, 2, 4 3,2, 15
2. 1,2, 2, 3
3. 3,1, 7, 11 4.
5. 1, 1, 2 3 2 , 2 3 2 + +6. 7. t 4,6 c 0, 458. 9. 10.
11. y 4x 108+ 2x y 11664,=
12. y 5x 3=
1x 1= 2x 3= ( 3, 18)
1. x 3= 2, 2
2.
3. 59 1
6. x 1 x 0>
1. 3 2x2 5
<
x 5 0,
f , 1 2x , x 1 2x x< :
1 2f (x ) f (x )>
T=f(t)
5
11T(C)
3
O 4 16 24 t(h)
2.1 -- 33
:) 1t 4= , f (4) 3= . :
f (t) f (4) 3 = , t [0, 24] T = f (t) t = 4 -, f (4) 3= . :
0x A , 0f (x ) f min f (x) .
, 4f (x) 3x 1= + .
4x 0 , x,
43x 0 , x,
43x 1 1+ , x.:
f (x) f (0) , x.
, f 0x 0= , f (0) 1=
) 2t 16= , T(16)=11 . :
f (t) f (16) 11 = , t [0, 24] T =f (t) t =16
, f (16)=11. :
f, , 0x A () :
0f (x) f (x ) , x A
f, , 0x A () :
0f (x) f (x ) , x A
34 2:
0x A , 0f (x ) - f max f ( x) . , 4f (x) 3x 1= + .
4x 0 , x,
43x 0 , x,
43x 1 1 + , x.:
f (x) f (0) , x
, f 0x 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)
2.1 -- 35
) - C
f f -
. Cf
y'y, C
f
y'y Cf .
, , M(x, y) Cf
y'y M '( x, y) M(x, y) M '( x, y) C
f , y = f (x) y = f (x) , :
f . :
yy.
, 4 2f (x) 2x x 1= + , x :
4 2 4 2f ( x) 2( x) ( x) 1 2x x 1 f (x) = + = + =
, y'y .
) - C
f f
. C
f
, - C
f -
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
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 . :
.
, 3f (x) 2x x= , x : 3 3f ( x) 2( x) ( x) 2x x f (x) = = + = , .
: 2y x= ,
4y x= , 6y x= ., , y'y , , - y x= , 3y x= , 5y x= ., , , .
- f [6,6]. f :
f, , , x A :
x A f ( x) f (x) =
2.1 -- 37
) f : 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
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) - :
) ) .
2) - , .
3) :
i) 2f (x) x 6x 10= + x=3.
ii) 22x
g(x)x 1
=+
x=1.
4) :
i) 2 41f (x) 3x 5x= + ii) 2f (x) 3 x 1= + iii) 3f (x) x 1= + iv) 3 54f (x) x 3x= v)
2
5x
f (x)1 x
=+
vi) 6 22x
f (x)x 1
=+
.
1
y=f(x)
x
y
2
2
y=g(x)
x
y
1
y=h(x)
x
y
1
2
2.1 -- 39
5) :
i) 11
f (x)x
=
ii) 2f (x) x 2= iii) 3f (x) x 1 x 1= +
iv) 4 2
1x
xf (x)x 1
+=
+ v) 5f (x) x= vi)
26f (x) 1 x= .
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
C1 C2 C3
x x x
y y y
40 2:
2.2 -
) f (x) x 1= + .
x 1, x 0
f (x) ,x 1, x 0
+
2.2 - 41
) f (x) x 1= .
x 1, x 0f (x) ,
x 1, x 0
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
42 2:
) f (x) x 1= .
x 1, x 1
f (x) ,x 1, x 1
+
2.2 - 43
f :
f (x) (x c)= , c > 0 , c ( ).
:
f (x) (x c)= , f x x c , c x. , f c ( ).
) 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 , 1 x'x - x 'Oy xOy (x) x = ()., (x) x = - 1 , -
Ox
c
c
c
c
xxc
(x-c)
C Cf
f(x)
y
y
x
y=|x+1|
y=|x|
1 1
11
11
O
44 2:
f , :
f (x) (x c)= + , c > 0 , c ( ).
f (x) x 1= + . , , ,
f (x) (x 1)= + , x, f (x) x 1= + x
(x) x = x +1.
:
f (x) (x c)= + , f x x + c , c x. , f c ( ).
f (x) x 3 2= + + .
y x 3= + , - y x= 3 . - y x 3 2= + + , y x 3= + 2 .
xx x+c
yc
c
O
c
c
=(x+c)f(x)