.olzano

. olzano

0x

0x x0

y

B(,f ())

(, f()) f()

f ()

O

x

- -

http://www.perikentro.blogspot.gr/ :

1

1.1.

2xx

2x1x3

1xx

)x(f2

IR

, IR 1.2.

0x0

0xx1

x)x(f

0x0

0xx1

x)x(g2

1.3.

0x2

0xx1

x24x

)x(f2

IR f IR 1.4. f ,g: IR),0[ : f2(x) +2g2(x) x34x ),0[x . f , g =0

1.5. f x0=1 71x

8x)x(flim

1x

1x

)1(f)x(flim

1x

1.6. IRIR:f xf(x)3x IRx f(0) 1.7.

*N,IRx,xe

1xelim)x(f

2tx

tx

t

: 1

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http://www.perikentro.blogspot.gr/ :

2

2.1. f , IR :

5x2

1e)x(flim

x2

0x

. f(0).

2.2. f: IRIR :

2xxx

x3)x(fxxlim

3

222

0x

. Cf (0,2)

f x0=0 2.3. f: IRIR : 2f3(x)+3f(x)=x2 x IR . f x0=0 2.4. :

5xe)1(2)e5xln()(

5x016x8x)x(f

x522

2

. )x(flim5x

).x(flim5x

. , IR f x0=5 . , ). )x(flim

x

2.5. f : f(x-y)=f(x)+f(y)+5xy x,y IR . : . f(0)=0 . f x0=0 , f IR. 2.6. f: IRIR : f(xy)=f(x)+f(y) x,y IR . . f =1 IR . f 1x0 IR

: 2

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http://www.perikentro.blogspot.gr/ :

3

3.1.

1x03xx2

0x13x3x)x(f

2

3

f(x)=0 [-1,1].

3.2.

1xxx4

1x3x2x)x(f

2

2

Bolzano [0,3]. [0,3] , xx . 3.3. f(x)=x3+x2 g(x)=5x-3 )2,4(x0 3.4. ]4,2[]4,2[:g,f . f(2)=2 ,g(4)=4 ]4,2[ , : 3f()+5g()=8 3.5. f: IR]2,1[ 1-1 f(1)+f(2)=0. )2,1( f()=0. 3.6. f: IR]2,0[ f(0)=2, f(1)=4, f(2)=-4. f2(x)=9 (0,2). 3.7. f [, ] 2f2()+f()[f()-3]+5=0 ),( f()=0. 3.8. >0 ++12 Ii)2(f2i)5(f5

z

f(x)=x (2, 5)

: Bolzano 3

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http://www.perikentro.blogspot.gr/ :

4

4.1. f(x)=0 (0,1) f(x)=(x-1)2x-1+ln(x+1). 4.2. (x+1)2x+1=1 (-1,0) 4.3. x3-(-2)x+1=0 (-1,0) +=2 4.4. f:[,] IR f()+f()=0,.>0. f(x)=0 [,]. 4.5. f(x)=x3+2x-1. . f . f(x)=0 (0,1). 4.6. f,g x , f(x)-g(x)=cx , c 0 . f(x)=0 1 , 2 10 )x(f . x0[,] :

0

0 x)x(f

4.8. f xx . 4.9. f(x)= 5x4xx3 23 . )1,0( f()=-3,2456. 4.10. f IR. )2015,1( f(1)+f(2)++f(2015)=2015f()

: Bolzano-... 4

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5

5.1. f:[-2,2] IR : x3+f10(x)=8 x[-2,2]. f (-2,2). 5.2. f: IRIR f(1)+2f(2)+3f(3)=0 f(x) 0 x IR . f 5.3. f:[0,2] IR .

x0(0,3) : 10

)2(f4)1(f5)0(f)x(f 0

.

5.4. f:[0,1] IR 4

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http://www.perikentro.blogspot.gr/ :

6

6.1. f(x)=lnx+ex-1 i. f

ii. x0>0 : 1exln 0x

0 6.2. f:(0,3) IR (0,1] [1,3). f(1)=2 1)x(flim

0x

2)x(flim

3x

. f . f(x)=0 (0,3).

6.3. 1x1

x2

6.4. f(x)=ex+lnx-3. . f . . ex+lnx=3

.

x1

flimx

6.5. z=ix

i2i3x

x IR

. Re(z) m(z) x. . IRx0 z . . f(x)=Re(z)(x2+1). 6.6. IRIR:f IR. f(8)=7 2))x(f(f)x(f IRx , f(7) f(1). 6.7. f:IR IR f3(x)+3f(x)=e2x-1, xIR. . f(0).

. 1e)x(f x2 , xIR.

. f x0=0.

: Bolzano - 6

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http://www.perikentro.blogspot.gr/ :

7

: 7.1. f x0 7.2. f . 7.3. f x0 0)x(f 0 , x0 f f(x0). 7.4. f , f-1 f(). 7.5. f x0 g x0 f+g x0 7.6. f , g x0 f+g x0 7.7. f [, ] f()f() 0 , f(x)=0 [, ] 7.8. f IR , . 7.9. f [, ] f()f()>0 , f(x)=0 (, ) 7.10.

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

: 7

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8

7.11. f [, ] f()f()0 f(x)