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• c 20

16

.

- .

.

16 2016

1 - 16 2016

• - .

1

1. x0 .

2. .

3. - .

() f - A f (x) 6= 0 x A, f A.

() f (x) = ln(x) g(x) = ex (f g

)(x) = x.

() limx10

(f (x) g(x)) = f (10) g(10).() f x0 Df , .

() f [, ] - (, ) f () = 0, f () f () 0

3. - ;

6 + 7 + 12 = 25

3 - 16 2016

• - .

3

1. f : R R :

x, y R, |f (x) f (y)| |(x) (y)|

() f 2.

() f R.() f

2 -

f (

2

).

3 + 5 + 7 = 15

2. f [, ], . x (, ) :

f (x) f ()x

f () f ()

=1

2(x)f (), (, )

10

- 16 2016 4

• - .

4

f : R R f (0) = > 0 :

x, y R (xy + 1 6= 0) f (x)f (y) = f( x + y

1 + xy

)(1)

, 1 8, f .

1. f (1) = f (1) = 0 f (0) = 1.2. f (x) = 0 1

1.3. x (1, 1) f f (x) >

0.4. x1 R 6= 1, 0, 1,

() t |t| 1 & |y| > 1

) x + y1 + xy

< 1(2)

5 - 16 2016

• - . (|x| < 1 & |y| > 1

)(|x| > 1 & |y| < 1

) x + y1 + xy

> 1(3)

8. g R :

g(x) =

{x [1, 1] g(x) = f (x)x / [1, 1] g(x) = f (x)

g(x) .

9. x 6= 1, f (x) =

1 + x1 x/2

.

10. : f (x), - 9, f (1) = 0, .

11. , .

(1) (2) = 3.5, (3) = 3.5, (4) (5) =5 (6) (7) (8) = 6 (9) (10) (11) = 7

- 16 2016 6

• c 20

16

.

- .

1 :

1. : ( . 217 ) x 6= x0 :

f(x) f(x0) =f(x) f(x0)

x x0 (x x0)

:

limxx0

[f(x) f(x0)] = limxx0

(f(x) f(x0)

x x0 (x x0)

)= lim

xx0

f(x) f(x0)x x0

limxx0

(x x0)= f (x0) 0 = 0

, limxx0

f(x) = f(x0), f x0.

2. ( . 194) f , [, ]. :

() f [, ] () f() 6= f(), f() f() , x0 (, ) : f(x0) = .

3. () () () () ()

7 - 16 2016

• - .

2 :

1. v . - 2v.

AB

A= () A = 4

(), t1 =

A

v=

4

v()

B

AB= () B = 4(), t2 =

B

2v+B

2v=

7

2v+

4()2v

2. t2 > t1, :

7 + 4()2v

>4

v() 7 + 4()

2 4

()> 0

f() =7 + 4()

2 4

()> 0

f(x) =7 + 4()

2 4

()=

7() + 4() 82()

, .

3. .

f () = 21 2()2()

f (x) = 0 x =

6. 0 f(x) . >

6

f () < 0 f(x) . x = 6

. ,

f(

6

) 0, 03 > 0

f(0) = 0, 5 < 0

lim

2f() =

f() = 0 1 2. (1, 2) f() > 0.: 1 2. ( : 1 = 0, 39 rad 2 = 0, 64 rad).

- 16 2016 8

• - .

3 :

1. () x R, |f(x) f(x+ 2)| |(x) (x+ 2)| = 0

, x R, f(x) = f(x+ 2), f 2.() f x0 R

:

0 |f(x) f(x0)| |(x) (x0)|

:0 lim

xx0|f(x) f(x0)| lim

xx0|(x) (x0)|

: limxx0

|f(x) f(x0)| = 0

limxx0

f(x) = f(x0)

() :

x, y R,f(x) f(

2

) (x) (2

)

x 2

x R {2

}:

f(x) f(

2

)x

2

(x)

(2

)x

2

limx

2

(x)

(2

)x

2

= (

2

)= 0

,

limx

2

f(x) f

(2

)x

2

0

= 0 : f

(2

)= 0.

2. x (, )

=

f(x) f()x

f() f()

x (4)

f(x) :

f(x) = f() +f() f()

(x ) + (x )(x ) (5)

9 - 16 2016

• - .

:

h(t) = f(t)

(f() +

f() f()

(t ) + (t )(t )

)

h(x) [, x]. h() = 0 h(x) = 0. Rolle 1 (, x)

h(1) = 0 (6)

[x, ]. Rolle 2 (x, )

h(2) = 0 (7)

h(x) , [1, 2] [, ], h(1) = h

(2) = 0. Rolle (1, 2)

h() = 0 (8)

, h(x) = f (t) 2 h() = f () 2 8= 0, , = 12f ().

: 4 :

1

2f () =

f(x) f()x

f() f()

x

- 16 2016 10

• - .

4 :

1. () (1): x = x y = 0. : f(x)f(0) = f(x) f(x))f(0) 1) = 0. f(x) = 0 f (0) = 0 = , . :f(0) = 1.

() (1): x = x y = 1.

f(x)f(1) = f(x+ 1

1 + x

) f(x)f(1) = f(1) f(1)(f(x) 1) = 0

f(x) = 1 f (x) = 0 . , f(1) = 0.() (1): x = x y = 1.

f(x)f(1) = f(x 1

1 x

) f(x)f(1) = f(1) f(1)(f(x) 1) = 0

f(x) = 1 f (x) = 0 . , f(1) = 0.

2. / 1, 1 f() = 0. (1) x = y = , f()f() = f(0) = 1. f() = 0, f()f() = 0, . , 1 1.

3. f [1, 1]! (1) : x = y. , x R : f(x)2 = f

( 2x1 + x2

). ,

f( 2x

1 + x2

)> 0. x =

2 (, ).

2x

1 + x2= (). , f(()) > 0, (, ).

, f(x) > 0, x (1, 1).

4. x1 6= 1, 0, 1.

()

x1 + y

1 + x1y= x1 + t

x1 + t = (x1 + t)(1 + x1y)y 6= 1x1

y(1 (x+ 1 + t)x1) = x1 + t x1 = ty 6= 1x1

(9)

x1 + t =1

x1 t = 1 x

2

x16= 0 t = 0 .

.

, y = t1 x21 x1t

.

11 - 16 2016

• - .

:t

1 x21 x1t6= 1

x1:

t

1 x21 x1t6= 1

x1 tx1 + 1 x1(x1 + t)

x1(1 x1(x1 + t))6= 0

. . .

x21 1

x1(1 x1(x1 + t))6= 0

, t =t

1 x21 x1t.

() : limt0

f(x1 + t) f(t)t

.

f(x1 + t) f(t)t

9=

f(x1+t1+x1t

) f(x1)

t(1)=

f(x1)f(t) f(x1)t

=f(x1)

(f(t) 1

)t

= f(x1) f(t) 1

t tt

,tt

= = 11 x21 x1t

, t 0 11 x21

.

, limt0

f(t) 1t

= f (0) = , t 0 t 0.

: limt0

f(x1 + t) f(t)t

=f(x1)

1 x21. : f -

x1 f (x1) =f(x1)

1 x21 x1 6= 1, 0, 1.

5. f (x) =f(x)

1 x2 f(x) > 0 x 6= 1, 0, 1, :

f (x)

f(x)=

1 x2=

2

((1 + x)

1 + x (1 x)

1 x

)

=

2 ln

(1 + x

1 x

)

, ln(f(x)

)=

2 ln

(1 + x

1 x

)+ c. , f(0) = 1, , c = 0 x

(1, 1), f(x) =(1 + x

1 x

)/2.

6. f(x) = 0 x 6=1, 1. 1 1. ,

- 16 2016 12

• - .

(,1) (1,+). x = x (1), f(x)f(x) = f(0) = 1. f(x) f(x) . : f (,1) (1,+).

7.1 x+ y

1 + xy= = (1 x)(1 y)

1 + xy(10)

1 +x+ y

1 + xy= = (1 + x)(1 + y)

1 + xy(11)

11 (10) x = x y = y. (10) (11), :

() |x| < 1 & |y| < 1 1 x > 0 1 y > 0 1 + xy > 0. :

1 x+ y1 + xy

> 0 & 1 +x+ y

1 + xy< 0

x+ y1 + xy

< 1()

|x| > 1 & |y| > 1

1x

+ 1y

1 + 1xy

< 1 x+ y1 + xy < 1()

|x| < 1 & |y| > 1 1 >

x+1y

1 + x 1y

= 1 + xyx+ y x+ y1 + xy > 1() |x| > 1 & |y| < 1 .

8. [1, 1] f(x) = g(x) g(0) = > 0, (1) g(x) xy 6= 1.

() |x| < 1 & |y| < 1

x+ y1 + xy < 1. , f(x) = g(x).

() |x| > 1 & |y| > 1

x+ y1 + xy < 1. ,

g(x)g(y) = (f(x))(f(y)) = f(x)f(y) = f

(x+ y

1 + xy

)= g

(x+ y

1 + xy

)

() |x| < 1 & |y| > 1

x+ y1 + xy > 1. ,

g(x)g(y) = f(x)(f(y)) = f(x)f(y) = f

(x+ y

1 + xy

)

=

[ g

(x+ y

1 + xy

)]= g

(x+ y

1 + xy

)

13 - 16 2016

• - .

() |x| > 1 & |y| < 1 .: g(x) .

9. x 6= 1 f (1, 1)

f(x) =

(1 + x

1 x

)/2=

1 + x1 x/2

1 + x

1 x> 0, x (1, 1).

x A = (,1) (1,+) f(x) > 0, 6,

: f(x) =

1 + x1 x/2

.

, :f (x)

f(x)=

1 x2 A :

ln(f(x)

)=

2 ln

1 + x1 x

f(x) > 0 1 + x

1 x< 0 x A. ,

ln(f(x)

)= ln

1 + x1 x/2

+ ln c f(x)2 = c

1 + x1 x

(1), f 2(x) = f

(2x

1 + x2

).

limx+

f 2(x) = limx+

f

(2x

1 + x2

)(12)

X =2x

1 + x2 x + X 0. , f(x)

0 f(0) = 1 limx+

f(x) = 1. , 12 :

1 = c 1 c = 1. : x 6= 1, f(x) =

1 + x1 x/2

.

10. f(x) =

1 + x1 x/2

x 6= 1 f(1) = 0

.

f

(x+ y

1 + xy

)=

x+y1+xy

+ 1x+y1+xy

1

/2

=

(1 + x)(1 + y)(1 x)(1 y)/2

= f(x)f(y)

- 16 2016 14

• - .

x = 1 y = 1 f(x)f(y) = f

(x+ y

1 + xy

)= 0. :

ln f(x) =

2 ln 1 + x

1 x

(1, 1), f (x) = f(x)1 x2

f (0) = .

: f(x), 9, f(1) = 0, .

11. 2 :

() f(x) =

1 + x1 x/2

, x 6= 1, f(1) = 0.

() g(x) ={x [1, 1] g(x) = f(x)x / [1, 1] g(x) = f(x)

15 - 16 2016