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ΑΛΓΕΒΡΑ Β´ ΓΕΝΙΚΟΥ ΛΥΚΕΙΟΥ

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Άλγεβρα Β' Λυκείου Γενικής Παιδείας Βιβλίο Μαθητή

### Transcript of Άλγεβρα Β' Λυκείου Γενικής Παιδείας Βιβλίο Μαθητή

• :

. .

: 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 .= - :

- , :

,

,

,

.

, 180 =

, :

180 =

:9 60 ,

180 =

= 600

180 3 = =

603

A

B

O

• 56 3:

9 56

180 =

56

=

556 150

180 6 180

= = =

56

- .

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, x, x x.

3

• 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

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

6

3

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< < -

2. 2x3

= 3

x ,2

< < -

3. 3x3

= 3

x 2 ,2< < -

4. 2 5x5

= 0 x ,

2

< < -

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

4

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 .

- .

:

( 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

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

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)