2013

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| | 2013 | |

|

1. 2. 3. 4. / / / /

|

1. 0=-2|2-z|+|2-z|2=|2-z|+|2-z|2=|2-z|+)2-z2)(-z(2=|2-z|+2)-z2)(-z( 22 ()

|z 2| = 1 (2,0) = 1

pi , pipi |z|

3

2. O z1, z2 w2+w+ = 0 Im(z1) = - Im(z2) pi 1=)zIm(1=|)zIm(|2=|)zIm(2|2=|)Im(z-)zIm(| 11121 Rmi,-m=z,i+m=z 21

z1 |z 2| =1 2=m1=1+)2-m(1=|i+2)-(m|1=|2-i+m|1=|2-z| 221 Rmi,-2=z,i+2=z 21 Vieta -=z+z=S 21 -4=

5==zz=P 21

,

2=|)Im(z-)zIm(| 21 z1, z2 ( pi ) pi (z1), B(z2) 2 = 2 (=1)

1)0,2(K

30

2013

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3. o1

22

3 a-va-v-a=v

3+|v|3+|v|3|a|+|va|+|va||a-va-v-a|=|v| 2o122o1223 pi

1-3+|v|3+|v|3|v|3+|v|3+|v|3|v| 2323 -1 -1 1-)1+|v|+|v(|3|v| 23

-1

1+|v|+|v|1

-31+|v|+|v|

|v|22

3

)-1)(

1+|v|+|v|1

-31+|v|+|v|

1+|v|+|v||v(|22

2

-1|v1+|v|+|v|

1-3| 2

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H K(x) = (0, )+ () = ( )x(Klim

0x)x(Klim

+x) = (-2, + )

0 )+-2,( (x) = 0 = (0, )+

pi f(g(x)) = 1

3.

04pi

-x

dt)t(fx=)x(h [0, pi/4), pi (0, pi/4)

)4pi

-f(xx-dt)t(fx=)x(h 04pi

-x

h(0) = h(pi/4) = 0 pi .Rolle pi )

4pi

,0(x o 0=)x(h o

0=)4pi

-f(xx-dt)t(fx oo0

4pi

-x

o

o

)4pi-f(xx=dt)t(fx oo04pi

-x

o

o

)

4pi

-f(xx

x=dt)t(f o

o

o0

4pi

-xo

)4pi-f(xx=dt)t(f oo04pi

-xo

|

1.

0=h

h)-f(1-f(1)+f(1)-5h)+f(1lim0=

hh)-f(1-5h)+f(1

lim0h0h

0(1)f0(1)f60(1)f(-1)-(1)f5

===0=)h

f(1)-h)-f(1-

hf(1)-5h)+f(1(lim

0h

:

)x(f a=h

)f(x-)ah+x(flim o

oo

0h

)x(f

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www.lazaridi.info Page 4 of 4

2.

0>1-x

1-)x(f=)x(g * x > 1 g (1, + )

* .1.

1+xx

du)u(g=)x(h , x > 1 g(x)-)1+x(g=)x(h > 0 h (1,+ )

: < + 1 g

g() < g(+1)

g(+1) g() > 0

)5+x2(h>)5+x8(h 42 1>5+x2,1>5+x8 42 h

(1,+

) (0,2)(-2,0)x0>)x->x-x4>x2-x8x2>x85+x2>5+x8 242424242 (4x00 2

3.

f [1, x] pi )x,1(

1-x1-)x(f

=

1-xf(1)-)x(f

=)(f

pi 01)-(f(x)-1)-(x)(xf1)-(x)(xf1-f(x) >