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2

O

y

fC

( ,0)x

x

f , , = , . f f 0 , : = ft

dt .

, .

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F = ft dt 3

O

y

1 4xx

f( ) 1 x x +

(1,0) B( ,0)x

( , 1)x x

(1,2)

()

= + , . , , .

, 0 1 4. , ,

= 1 , : = ft

1

dt = F (1)To

: = = +

2 =

2 + + 1 12

=1

22 + 2 3 > 0. (2)

1 2 : F = 12

2 + 2 3.

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4

O

y

x

fC f( ) 2x

1 4

2

f( ) 2x x

F = 12

2 + 2 = + 1 = f ,

F,

f

1,4

.

= , < 1, , , . f :F = 2, < 1

2

1,

1

F = f , 0,4. : f

, o ft

dt , F f .

:

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F = ft dt 5

f

f , :

F = ft

dt ,

F = ft

dt

= f .

O

y

x hx

x

fC

, , : F: F = lim

h0F + h F

h(1)

: = Z BEZ

=

f

t

+h

dt

f

t

dt = F

+ h

F

. (2)

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6

h , , : = . = h. f . (3)

2 3 F + h F = h. f .

(1) :F = lim

h0h. f

h= lim

h0 f = f.

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F = ft dt 7

: =

: f.

F

: . = 1 2 ,

1, 2 .

:F = ft

dt = f.

=

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8

=

..

, .

.

.. f = ln , = 0, +.

1 0, +, F = 0, +.

F : F = ln , > 0 .

F1 = lnt

t

1

1d t = 0 F1 =

ln1

1 = 0 ,

: = 0.. F : F = ln =

1 ln2 , > 0.

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F = ft dt 9

: F (0,e e, +). = e , F Fe = lnt

t

e

1

dt = lnt lnte1

dt =1

2 ln2t1e = 12

e, 1/2.. : lim0F = lim0 ft

dt , 0 DF .

F , , : lim0F = F0 = ft

0

dt.

. = F

d , , : = F

d = F F

d =

F

f

d

,

f

d

.

+ + 0 12

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10

: =

g F = ft dt ,

: = Fg. : F g , : = g .

= D

H,

H

, : , . = 1 2 , : , , . :

= ftg

dt = fgg.

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F = ft dt 11

=

+

.

: f = , , 0 0, +.

F:

1,

2 + 2

, 0 0, +. 1, 2 + 2 , 0 , 1 , 0, :2 + 2 , 0 2 + 2 > 0 0,2. : DF = 0,2. F :

F = tt2

+21

dt = 2 + 22 + 2 2 + 2

=2 + 2

2 + 2 2 + 2 =2 22 2

2 2 ,

0,2

.

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12

: =

: f. :

g1

g2

,

.

f 1 2, :g1, g2 1 g1, g2 2.

H : .

: = ftg1 dt + ftg2

dt =

ftg2

dt ftg1

dt .

:

= f

g2

g2

f

g1

g1

.

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F = ft dt 13

f , F : F = f (1) = Fg1g2 = Fg2 Fg1 (2)

2 , :

= Fg2 g2 F g1 g1 =

fg2 g2 fg1 g1 .: , F, , .

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14

: G = et1 2

lnt 12dt

f

= e ln

1

2,

< 1

> 1.

: = , 1 1, + :

2 < 1 1 < 1 2 > 1 1 > 1 0,1 , 1 , G = , 1 0,1. G: :

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F = ft dt 15

1 = , 1 , G :

G

=

et

2ln

t

1

2dt +

et

1

ln

t

1

2dt =

et1

lnt 12dt et2

lnt 12dt (1)

G :

G = et1

lnt 12dt et2

lnt 12dt

= e1 ln

1

1

2

1

e 2 ln

2

1

2

2

=

e1 ln2 2 e 2 ln2 12. (2) 2 = 0,1 , (2).

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16

:

=

,

.

,

,

:

= , .

.. .

= ,

..

f ,g,h . : g, t = u u t

, :

=

( )

1 2

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F = ft dt 17

1 : = + , g =

, . :

.

.. =

= .

.

g

= 2

f

2

t

2

1

dt

6

2

2

,

.

2 f2t21

dt , 2t = u . 2 dt = du.

2, 4, 2 f2t2

1

dt :

f2t21

2 dt = fu42

du.

1 2 2 4

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18

g , :g

=

f

u

4

2du

6

2

2

=

fu2

du + fu4

du 62 2

g = fu4

du fu2

du 62 2, . (1)

:g = 4 f4 2 f2 12 2 =22 f4 f2 12 2 = 26 + 1 12 2 = 0 .

c , g = c g0 = 0 , : g = 0.. g = 0 , :

fu42

du 62 2 = 0 fu42

d u = 62 + 2 (1)

= 1 = 2, 1 : fu4

2

du = 8 , fu84

du = 28 , :

fu42

du + fu84

du = 8 + 28 f82

d = 36.

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F = ft dt 19

2 : , +) : = > , +).. : , .

.

+ + , , , + .

.

+ = + ,

,

.

. : fe 1 , 0.

g =f

e 0, +) 0, +g = e f e f

e2 =f f

e > 0.

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20

0, +), 0 :g g0 f

e f0e0

fe 1 f e

. :h = ft

0

d t + 2 + e 0.

h0 = ft00

d t + 20 + e0 = 3 , : h h0. h0:

0,

:

h = ft0

d t + 2 + e = f + 2 ln + e .: h0 = f0 + 2ln + 1 = 1 + 2 l n + 1 = 2ln + 1.

h

Fermat , :h0 = 0 2ln + 1 = 0 ln + 1 = 0 ln = 1 = 1/e .

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F = ft dt 21

. : ft0

d t + e 2 = 0

: = ft

0

d t + e 2 , 0,1. 0,1.

0 = ft0

0

d t + e0 2 = 1 2 = 1 < 0 .

1 = ft10

d t + e 1 2 = ft10

d t + e 3.

.

: f

e

f

e

0 ,

f e 10

d 0 f10

d e10

d

f10

d e 01 0 f1

0

d e 1 0

f10

d e 1 +e3 f10

d + e 3 2e 4 > 0 1 > 0. : 0. 1 < 0 , Bolzano 0,1: = 0.

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22

3 f , : = = . f . 0, e , : f = 0 .

F

=

f

u

0

du ,

0, e

.

F 0, e F = f , 0, e. F 0, e

0, e :

F = Fe F0e 0 =

fue0

du fu00

du

e=

e

e= 1,

f = 1. (1)

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F = ft dt 23

fe = 1. (2) 1 2 : f = fe. (3) f , e Rolle , , e f = 0 .

f

. 4 : :

=

,

, , . f.

= 0 , : ft0

dt 1

2f0 f f

0

=1

ft0

dt 12

1 f ft0

dt 12

f 1

f

t

0

dt

1

2

f

1

(1)

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24

= 0 , :

f

t

0

dt

1

2

f

f

0

f

t

0

dt

1

2

f

1

ft0

dt 12

f 1 (2)

A 1 2 : ft0

dt =1

2f 1 (3)

3 :f = 1

2f f = 2f f 2f = 0 e2

f e2 2e2 f = 0 e2 f = 0 . .. e2 f = c, c . (4) = 0, 4 : f0 = c c = 1 . 4 :e2 f = 1 f = e2 , .

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F = ft dt 25

5 = , .. f

.. , :

+

+

.

. : + +

.

. f 0, + :f = 1

2e

2 + 1

2 e

2 =1 2 e

2 , > 0.

- f:

f = 0 1 2 e2 = 0e2>0 = 1

f > 0 1 2 e

2 > 0

, e2>0 < 1

f

< 0

> 1.

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26

= 1, f1 = 1e .. 1, +) f t + 1 , : f + 1 ft f

f + 1+1

dt ft

+1

dt f

+1

dt

f + 1 ft+1

dt f. (1). : lim

+

f , lim

+

f + 1:

lim+f = lim+e

2

++ , De L =

lim+

1

21

2e

2

= lim+1

e2

= 0.

: lim+f + 1 = 0 , 1 :

lim+ ft+1

dt = 0.

+ + 0 e1/

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F = ft dt 27

6. : : = + ,

= .

.

= .. :

( ) .

. : + = .. t = u dt = du : f = 1 fu0

1du = 1 + fu1

0

du (1)

f = fu10

du + f 1, . (2) 2 :f1 = f0 = 1, = 45.

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28

. : lim1 f 1( 1)3 = lim1

fu10

du

( 1)3 0

0De L =

lim1 fu10 du + f 1

3( 1)2 = +.. g = e fe 1:: fe = 1 + e fue

10

du

fe 1 = e fue 1

0

du , g :

g = e fe 1 = fue 1

0

du (3)

lim+g = lim+ fue 1

0

du (4)

, e = , 0 :lim+g = lim0 fu1

0

du

= fu10

du = fu01

du = 2 = 2.