2014

1 - -

2014

2 - -

1. () .

() ( )() = (), - ()

().

() :

() = () +

1. :

( () + ()) = () + ()

1. ,

(), . [, ]

()

.

.

.

2.

e. Ba :

() = ()

() = () +

2014

3 - -

( )

+

++

+

+

+

+

+

+

|| +

+

+

+ +

+

+

+

+

+

:

a) (8 + 4 3 5) e) 2

2+1

b) (1 + )3

f)

sin2 cos2

c) (+1)(23)

3 g) tan2

d) ()(1+)

3

2.1. 2.1.1

:

a)

+=

+

= { =

=

} =

+=

+=

+ =

+

b) :

2+32.

c)

=

(

)

= {

=

=

} =

= + =

+

d)

2014

4 - -

e)

()

f) :

=

|

+ | +

=

|

+

| +

g)

=

=

=

= {

= =

} =

=

()(+)=

{ } =

(

+) =

(| | | + |) + =

|

+| + =

|

+| + =

|

+

| + =

|

+

+

| + =

|(

)| + = |

| +

h)

i) ( ) j) k) ( + )

l)

m)

-

sin

sin

cos

cos

n)

o)

---------------------------------------------------------------------------------------------------------------

:

2 2 = sin ( = cos ).

2 + 2 = sinh .

2 2 = cosh .

2++

.

:

2 + + = ( +

2)

2

+4 2

4

----------------------------------------------------------------------------------------------------------------

2014

5 - -

:

a) sin1 +

12

b) 2 + 2 = { = sinh

= cosh } = 1 + sinh2 cosh = 2

cosh2 = 2 1+cosh 2

2 = 2 (

2+ sinh 2) +

c)

22

d) ln

ln ln(ln )

e)

1+cos =

2cos

2

2.1.2

=

, . .

:

a) ln = { = ln

=1

| =

= } = ln

1

= ln +

b) sin1 tan1

c) 72

d) sin cos e) (2 + 5 + 6) cos 2

f) sin1 = { = sin1

=1

12| =

2

2=

} =1

2 2 sin1

1

2

2

12=

{ = sin

= cos } =

1

2 2 sin1

1

2

sin2 cos

cos =

1

22 sin1

1

4(1

cos 2) =1

22 sin1

1

4 +

1

8sin 2 + =

1

22 sin1

1

4sin1 +

1

4

1 2 + g) tan1

h) ln( + 1 + 2)

i) cos

sin2

j) sin2

k) cos2 ln tan2 2

l) tan

1

(1+2)32

m) sin ln

2.1.3.

:

2014

6 - -

a)

(+1)(+2)(+3)

b)

(1)2(+1)2

c)

3+1

d)

2+2+5

e) 3+1

352+6 - ,

.

f) 10

2+2

----------------------------------------------------------------------------------------------------------------

2.1.4.

a) (1

11 , 2

22 , 3

33 , ) , ( = 1,2, . ) ,

= , = (1, 2, ).

,

:

a)

+ 3 = {

= 6

= 65} =

65

3+2

b) 1+

1

3

c)

(1+2+ 3

)

d) 2+

3

+ 2+3

----------------------------------------------------------------------------------------------------------------

2.1.5.

(sin , cos ) =

tan

2, : cos =

12

1+2, sin =

2

1+2.

:

a) (tan ) = tan , =1

1+2

b) (sin , cos ) (sin , cos ) = (sin , cos ) =

tan , sin =2

1+2, cos =

1

1+2

c) (sin , cos ) ( sin , cos )=- (sin , cos ) = cos d) (sin , cos ) (sin , cos ) = (sin , cos ) =

sin

2014

7 - -

:

1) () = 0

2) () = ()

3) (() + ()) = () + ()

4) [, ] ( [, ]) () = ()

+

()

:

a)

sin2 = {

= = sin1 } =

1

212

12=

2(12)

b) sin2 cos3

c)

tan

d) sin2

cos3

e)

2 sin cos +5

f)

cos4 +sin4

g) sin2

sin +2 cos

h)

sin3

i) sin cos

cos4 +sin4

3.

1. [, ] .

2. : - :

() = () () = () |

:

a) (2 2 + 3)2

1

b) 1+

2

4

1

c) sin(ln )

1

d) tan

4

4

e) sin 5

60

= sin !!!

2014

8 - -

:

1. () = ( )

0

0

2. [, ]

() = 2 ()

0

3. [, ] :

() = 0

f) ln(1+)

1+2 = {

= tan = tan1

=

1+2} = ln(1 + tan ) = ln(1 +

40

40

1

0

tan(

4 )) = ln (1 +

1tan

1+tan ) = ln

2

1+tan = (ln 2 ln(1 + tan ) =

40

40

40

4ln 2

2 =

4ln 2 =

8ln 2

g) sin

20

h) sin1 2

0

4.

:

() = lim

()

() = lim

()

()

= lim

()

:

a)

2= lim

2= lim

(

1

|1

) = lim

(1

) + 1 = 1

1

1

b)

1

c)

1

0

d)

ln

2

1

e)

1+2

f) tan1

(1+)2

0

2014

9 - -

5.

= (), :

1 + 2()

:

()

:

|()|

2 () () :

(() ())

= () - . :

= 2 ()1 + 2()

:

= 2()