g lukeiou papagrigorakis 2016 17

2016 - 2017

M

333

: - &

: - 16.08

M.ed. 2016

: http://users.sch.gr/mipapagr

email: [email protected]

3

. 2016-2017

1 1.01 2

1f(x) n 1x

A)

B) f x 1 0

) f x 0

) 21 1f f( 2 ) f

x2 x 1

1.02 2f x x x 1 .

f x f x 1

1.03 x x1f x 2 .

f x y f x y 2f x f y , x, y R

1.04 f

R x

x4f x

4 2

. Y f x f 1 x

1 2 2002 2003f f ...f f2004 2004 2004 2004

1.05

3 2 2y x

:

) 2y x 5

) y 4x 1

) 3, 1

)

)

) 135- xx

1.01

,

1.02 o R

2 2

2

x 2x 5, x 3 2f x

x 4, 2- x

1.03

v km/h, 36 0,0001v

.

1000 km v

1.04 1 lt.

, 3cm

0,02 euro

1.05 10

.

x

1.06

x ,

E B .

1 BE x

f g R

f x g y x y x, y R

4 -

http://users.sch.gr/mipapagr

1.07 N

f(x) ln x

6f x 10

1.08

f x x g x x 1 h x 1 x

1k xx

1m xx 1

1n xx 1

1.09

f(x) ln( x), x 0 g(x) ln( x), x 0

k(x) ln x m(x) ln x t(x) ln x

1.10 N

) f(x) x x , x 0,2

) t(x) 2 2x ) 2f(x) x

1.11 :

) f x 2x x B) 2x 1, x 1

f(x)3 x , x 1

)

x

2

e , x 0g(x) lnx , 0

5

. 2016-2017

1.21

2f x x ln x 2(x) x x x 1g(x)

x x-3x+2 x

t(x) 1

x 2 x 1

h(x) 2

x 2

1 x 1 x

24 - xk(x)(x - 1) x 1

1.22 :

2 xf x2x 1

2xg xx 2x

21t x

2 x 5x 3

2

2x 2xx 1(x)

e e

xe er x

2x 1 ln x

, 1m x ln xx 1

p(x) xe - 1 + 1 - lnx 2q x ln 1 x

1.23 R

2f(x) ln x 4x 1 R

1.24

2

x x 1x x 1f x

9 4.3 27

x 2

1h x4 x

k(x) 2x 1 xm(x) (e 1)ln(x 1)

1t x ln xx 1

1r x ln xx 1

3 2p(x) x x 1

lnxq x x

1.25

2

2x 2xx 1k(x)

e e

, xt x ln xx 1

2

2x x 2r(x)

x x

, 3 x 2

k x2x 4 x 1

m x ln(x 1) f(x) xe - 1 + 1 - lnx

2r x ln x x 1 , 1t x x 1 , x 0, 2

1.26

: ) 1 xf x e 3 x 1, 2

) f x 3 ln 1 2x 1 , x 2, 1 /2

) 2f x x 4x 3

1.27

: ) x 1f xx 1

x 2, 5

) 2

6f xx 4 2

x , 2

1.28 :

2x 2 2 x 3f(x)x 1 3 x 5

, g(x) 3 2 x 1

1.29

f(x) 1log 1x

, x

x 15 eg x

5 e

2

2x 2xt(x)x 4

2r x x 4x 3 x 2, 5

1.30 y f x .

:

) f x 2

) f x 0

) f x 1

) f x 2

) f x , 3,3

6 -


1.31 f(x) x 1 .

)

f . 2

1x - 1f (x)x - 1

3

2 2x 1f (x)

x - x 1

23f (x) x 1 4 1f (x) x 1x

x 15f x ln e ln(x 1)6f (x) e

) R

.

1.32

1 xf(x)x

xg(x)

1 x

1.33 E

x xf x 1 2 2 1 g x 0

1.34 E

2f(x) x x 1 2

1g(x)x x 1

1.35 .

) ln xf(x) e ln xg(x) e

) f(x) 1ln 2x

g x ln 1 2x ln x

1.36 R

3

2x 3x 4f(x)

x x 4

g(x) x 1

1.37 f,g : R R

x R

2 2f x g x 1 2 x f x x g x

1.38 f g , gf

f(x) 4 |x|] g(x) x 1

1.39 f g , gf

f(x) 2x 1, x 2

x , x 2

lnx, 0 x 3g x

-2x 3, x 3

1.40 f,g : R R

2 2g x f x 2f x x 3 , x R .

gC Oy

1.41 N f : R R

: 2 2f x x 1 , x R

1.42 N f g ,

2 2f g (x) 2 f g (x) 2 x R

1.43

f : R R 2 2f (x) 2f x x

x R f x 1 x R

1.44 f : R R x R

f x 1 f x 2 0

1.45 f : R R

: f x x , x R

1.46 f : R R

2 x xf x 4e f x e , x R

1.47 f x, y R f(x)f(y) 1 f(x) f(y) xy

7

. 2016-2017

1.48 N

2g(x) ln x x 1 , 2x 3 x 0 3 x 0f(x)2x 3 x 0

1.49 f : R R x f(x) f( x) 2 2f( x) 0, x R .

f .

1.50 ** f : R R f(x y) f(x y) 2f(x) f(y)

x, y R . : ) f 0 0

) f

) f x f(x) x R

1.51 f : R R

2f x x 2 2x x R .

1.52 f,g : R R

: 2f x f x f x 2g x g x g x

x R . f

g

1.53 f(x y) f(x) f(y) , x, y R

f

1.54 f : R R f(x) 0 x R

1.55 f : R R g(x) f(x) + f( x)

1.56 f,g

f gA A R : f,g

f g f g ,

f /g , (g(x) 0 )

1.57 f ( )

, :

f(x) ln(1 x) f(x) 2 1 x

2f(x) ln(x 1) ln x 4f(x) (3x 5)

2 2f(x) ln x 1 ln x 3 xf(x) x

1.58 2f x 1 x , g x 3 x 2

f g g f

1.59 f g

) f(x) 1 x g(x) ln x

) f(x) x 1 x (0, 2)x 1 x [2, 4)

, g(x) x 1

1.60 f(x) 2ln(x x 1)

g(x) x x1 e e2

(f g)(x) (g f)(x) x , x R

1.61

h 2h(x) f(x 4) f(x 1) fD [0, 5)

1.62 f :

) f ln(2x) x 3 , x e ,

) 2(f g)(x) x x 1 g(x) x 1

) (g f)(x) 2 x 2g(x) x

8 -


1.63 f : R R

(1 x)f(x 1) f(1 x) x 1 , x R

1.64 ff : A R ,

gg : A R f gf A A . :

A) f , gof .

B) f , g f

.

1.65 f(x) f(2 x) x , x R

1.66 B f : 0, R

xf ln x f x 1e

x 0 .

1.67 f

2f x x x 1 f x 1 x x R

1.68 2xf f(x) e x R ,

f 2014

1.69 f f (x) 2x 1

x R f 1

1.70 f : ) (1 x)f(x 1) f(1 x) x 1 , x R

) 212f(x) f xx

, x R *

1.71 f(x) x 32 x

R ,

: f f (x) x x 2

1.72 f : R R

f f(x) 2x 1 , x R .

f x 1

1.73 * *f : R R

f(2004) 1 x, y 0

2004 2004f(x)f(y) f f 2f(xy)x y

,

1.74 *f : R R f(y ) 1f(x y) f(x) e x, y R .

f(x) 1f(x) e 1 f( x)f(x) e x R

f

1.75 f : R R

x ye f x f y f x y , x, y R .

f 0 1 , 1f x

f x , x R

f

1.76 f : R R

2f x 2 f 3x 0 , x R . fC x x

.

1.77 f : 0, R

xf x f y lny

x, y 0,

f 1 0 . f x ln x , x 0

1.78

f : R 0 R 1 1f x f xx x

x R 0

1.79 ** f : R R ,

f(x) f( x) x 3

x ,1 f(x) f( x) x 3

x 0, . A f(3) 4, f( 3) 2 ,

f(x) (www.mathematica.gr)

9

. 2016-2017

1.80

f(x) 5 5 x xg x ln e x

3t x x 1 2 4 xm x e 3

r(x) x 3x 2

(x) ln x 0 x 2

1 2x x 2

1.81

k x ln x , g(x) 2x 1

x 1

m x ln x

1.82 f : 0, 0,

. 1 1g x f

f x x

0,

1.83 f

R 2, 1 1,2 1 . A

1.84 3f(x) x 3x 1 , x R , :

) f

)

33 2 2x 1 3 x 1 1 x 1 3 x 1 1

1.85 5 3f(x) x xx

, x 0, , :

f

53 3

33x 2 x 2 1

x 2

1.86

xf x 2 x .

23x x 6 2x 22 2 x 5x 6

1.87 ) x x3 4f x 1

5 5

, x R

f . .

) x x x3 4 5 .

1.88 f : (0, ) R

xf x -f y =fy

x, y 0 .

1 f 0 .

f . 0,

1.89 f,g

, , ,

g x f x , x , f

. f g(x) g f(x) x ,

1.90 f : R R R f(f(x )) x x R ,

f(x) x , x R

1.91 f : R R ,

2 22f (x ) 2xf(6x 8) 4 3x, x R

1.92 f : 0, 0,

31 f x 1 x xf x

x 0, . f

1.93 * f

x R : 2x 3f(x)f x

5

,

f(x) x , x R

10 -


1.94 f : R R R .

5 2 2 5f x x 1 f 1 x x 2 x x

1.95 N :

) ln x 1 x ) x 1 xe1 x

1.96 3 xf x x 5 1

) .

)

7x 12 3x 183 3

7x 12 3x 18

5 5(3x 18) (7x 12)5 1 5 1

..

1.97 f : R R ,

1, 1 1, 2

)

) f 2x 1 1

) 2f x 2 f x 2014

1.98 f : R R

f f(x) 3x 0 x R .

1.99 f : R R ,

g x f f(x) 1,0

(mathematica.gr)

1.100 f 0, .

2 3f x f x f x f x

1.101 f : 0, R

1f f x 0x

x 0 .

g x f h(x)

1 xh x1 x

. N h

1,1

x 2x 11x 111xh e h e h e h e 1,1

1.102 * f : R R

: x

23ef(x)

2 f (x)

, x R

. f(x) 0 x R .

. f

.

. : ln f(x) 0 .

1.103 f(x) ln x x 1

: ) x2x e ln 1 x 1

) 2

22

2x x 3ln 4x x 33x x

1.104 )

f x ln x x 1

) 2 x 2x ln x

1.105 f(x) x ln x ,

:

) 2f(x 1) 2x ln(2x 2) 2

) 3x 2 ln x x

) x 1ln 1 2 x x2 x

1.106 * f : R R

f x xe f x 1 e

x R

2f f(x 2x f f(x 2)

11

. 2016-2017

1.107

g(x) 4 x 2 43t(x) 4 x 4x 2r x x 4x 5 f : [ 1, 4) R f(x) 2x 1

x 1 x 2

x3x 1 x 2

1.108

) f x 1 2ln x 1 , x 2, 3

) f : [ 1, 4) R f(x) 2x 1

1.109

) 2f x x 4x 5

) 2x xf x e 2e 3

1.110

) 2x xf x 2e x 4 2e x 5 ) 2x xf x e 2e 3

1.111 ) 1x 2x

x 0

. x xf(x) 9 80 9 80 . N f(x) 2 x R f

1.112 f : R R f(0) 1

)

22f(x)g(x)

1 f (x)

1 .

)

x

2x2e(x) 2013

1 e

1.113 R , 2f(x) x ( 1)x 2 2

1.114 f,g : R R

22(f(x) g(x) 1 x R .

h x f x f 1 x g x g 1 x

(mathematica)

1.115 2f x x 6x 8 , x R ,

) f x

)

) f 2x 3 0 ) f f x x 2 0

1.116 2f x x 6x 8 , x R ,

) f x

)

) f 2x 3 0 ) f f x x 2 0

1.117 2

2x x 2f xx x 1

A) f 3

B)

3 43 3f x f x 3x 6 02 2

) , R

f 1 f 2 1 6 0

1.118 2x3f x 5 x e .

) f

.

) f .

) 23 2 2x 4x 105 x 2x 5 e

23 2 2x 8x 85 x 4x 4 e

) 2 2 2 2 2 23 35 1 e 5 1 e 2 0

12 -


1:1

1.119 , 1 1 :

) x 1f x 3e 2 ) xf x e x 1

) f x x ln x 2 ) 5 3f x x x x

1.120 f : R R 2015f f x 3f x x 0 , x R

1-1

1.121 f : [1, ) R

2f(f(x)) 2x 3x 2 ,

x [1, ) . f 1 1

1.122 f : R R 1 1 . 3F x f x 2f x 3

1 1 .

1.123 R 1 1

f(x) 24 x x 0

x 8 x 0

1.124 1 1

f 2 26f x f (x) 9 x R

1.125 f x 2 x ln x

) N f

) 1 x ln x 0

) x ln x 1

1.126 .

) x 1e ln x 2 x )

7 5 3x 2x 3x x 6

) 2x x 2x 1 2e e x x 1 )

x x x6 8 10

1.127 N

442 2log 1 log 5 5 5 5 1

1.128 f,g : R R

2f f (x) x 5x 9 2g x x xf x 3 ,

x R . N f 3 3 g

1 1

1.129 2f x 2x ln x 1 , x 0 f 1-1

:

2

24

3x - 2 +12 x - 3x +2 = ln

x +1

2,

1.130 f : R g : R , B f(A)

g f 1 1 g 1 1

1.131 yxx e y e , x, y R

) x y .

) 22 3x x 2x 3x 2 e e

1.132 3 3e e , R

1.133 x

x2 4f(x) 23 3

:

x x x3 2 4 3 3 6

1.134 x 3f x e x x 1

) N 1 1

) : 2x -x 2 3 2 x+3 3e +(x - x) + x - 2x = e +(x +3) +3

1.135 f : R R :

52f (x) f x 3x x R .

) f

) 2f x x 1 1

13

. 2016-2017

1.136 f : R R

f xf x 2e x 2 x R

) f

)

xg x x 2e

) f 1

) f

1.137 f R ,

31 2f x 1 x

f x x R .

) f

f 0

) 5 3f x x x 3 1

1.138 f , R

3f x 3 xf x e x e , x R .

f 3f x x

1.139 )

5 3h x x x x , x R .

) f R

5 3f x f x f x x

x R . f

) h x 3

f 3

1.140 f R , : f 1 f(x) 2x 6 f x

. .

:

( 1)

1.141 f : (0, ) R

: xf x -f y =fy

x, y 0

f x 0 ,

) f 1 1

)

2 2f x f x 3 f x 1 f x 1

1.142 f : R R

: x xf(e x) 8f(x 1) 2008 e

x R . f

: : f x 223

x xf e 2 e 223 .

1.143 f : R R

f xe f x x x R

) f 1-1

) f 1 .

) x 4 2x 1e e x 5

1.144 *** 1 1

f : 0,1 R f 0 f 1 1 .

1 2x , x 0,1

1 24f x 2f x 1 .

1.145 f

0, f (x )f e lnx

x 0, . f x ln x ,

x 0,

1.146 f : 0, R

f(x)f e lnx . f .

x R

f 1 1 f(3) 2

2f(1 2f(x x 1)) f(1 f(5)) 4

14 -


1.147

) 3f(x) x 1 ) f(x) 5 x 2

) xf(x) log 3 10 ) xf(x) ln(2 e ) x

) 2f x 2 x 3 , x 3

1.148

) f(x) 2x 3 .x 4

) f(x) 3x

3x3 e3 e

) xf(x) log1 x

) xf(x)1 x

) 2x 1 , x 0

f(x)9x , x 0

) 3 2f(x) x 3x 3x ) 2f(x) ln x 1 x

1.149 1fC fC

f(x) 1 x , x 1,0

1.150 f f(f(f(x))) 2x 7 x R .

f(1) 3, f(3) 9 . f

1 1 1f (x) 9 .

1.151 f 3f(x) 2x x 2 .

) f .

) 1f(x) f (x) .

) 1f (5x 6) 1 .

1.152 f x x ln x

) 1f e 1

) 2

22

2 1ln 4 5

1.153 f , g : R R ,

1f g 1g f ,

1g 1f

1.154 3f(x) x x 2

)

) f(x) 12 , 1f (x) 2

) 1fC

y x

) 2 3 3 2(2 x) x x x 2

: 1f (x) 3 , 1f (x 1) x 5

1.155 f : R R f R R

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

f , 1f

fC 1fC

1.156 f,g : R R

R

f g g f , 1 1f g g f

1.157

y x

1.158 f : R R 5f(x) x x 1 .

A) 11 13f f

B) 1x f (x)

1.159

5x 2f x2x 5

y x

15

. 2016-2017

1.160 f : R R

f f f ... x ... 2x 1

f 1

1.161 1 1 f : R (0, ) f(x y) f(x) f(y)

x, y R . : 1 1 1f (xy) f (x) f (y) , x, y f(R)

1.162 f : R R 1, x R 2 2xf (x) 2f(x) e 1 . f .

1.163 f : R R f x y f x f y , x, y R .

x R : x 0 f x 0 .

) f

) 2 2f 4x 2005 f 4x 2005 2f 8x 4

1.164 x

xe 1f xe 1

1 xg x ln

1 x

.

f 1f g

g f

g

f , 1f

: f 1 1 1f .

2 2x g x x g x

1.165 ) f . R ox R , o o o of f(x x f x x

B) 34x 1f x3

, 1f

fC 1fC .

1.166 f : R R f(x y) f(x)f(y) x, y R R ,

f() 0 . :

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

f(x)f(x y)f(y)

, x R

16 -


1.167 H f : R R , R fC

A 5,9 B 2, 3 :

) f .

) 1 2f 3 f (x 2x) 9 1 2f x ln 1 2x

) 2f x 12f x 27 f x ln x 4 9

1.168 3

3 3x 1 2x 12 3

1.169 f : 0, 0, f 1 1 g x xf x 1

.

) f .

B) N f x ln e x 0

) 7 5 9f x f x f x f x

1.170 f : R R : f(x)f(x) e x , x R

f ..

fC x x , 1

: f(x) x x R

: f(x)2f(2x 1) e x .

1.171 f : R g : B R ,

)

)

)

1.172 ) f, g R . :

) g R g f R , f

R

) g R f g R , f

R

) f ,g x

xe 1f x

e

g x ln x 1

) g f g f(x) x x R

) 2 2x 1 x 1 x 1 x 1e 1 e e 1 e

f(A) B g f 1 1 f 1 1

B f(A) g f 1 1 g 1 1

f(A) B g f 1 1 f g 1 1

17

. 2016-2017

X0

1.173

) 2 3x 12 3lim

x 1 x 1

) 1

x 1

x ( 1)x 1lim *x 1

1.174

) x 2lim

3 6 x x 6x 2

) 3 2x 0lim x x

) 2

x 9

x 81lim2x x 6x 3 x 9

1.175

) x 1lim

3 2 3x 2 x 1x 2 x 1

) x 1lim

2x 3 x 4x 3x 1

) x 1lim

2x 1 x 1x 1

, )

x 1

x 1 x 1lim

2 x 1

1.176 x 0lim f(x)

x 2 x 2 x 0f(x) 4x

0 x 0

1.177 :

) x 0

6x 3xlimx 2x

) 2

1lim

) 2 2

2x 0

x 2 x 2limx 4 2

1.178

) x 1

x 1lim 1x 1 x

) x 1

1 x 1lim 1 x 2

1.179 :

x 3

xlim

x 1 2

x 0

2x xlim

x 1 x

1.180 :

)

x 1

x 3 2lim

(x 1)

)

22x 0

xlim

x

) 2x 0

1 1 xlim

x

1.181 ( )

x 0lim

1x x

2

2x 0

1x xlim

x x

1.182 N

x 0

x 2x ... xlim 28x

1.183 x 1

f(x) 5lim 0f(x) 2

x 1lim f(x) 5

1.184 x 0lim

f(x) 2x

x 0lim 2 2

xf(3x)-f(-x) 2x3x x

1.185 x 2lim g(x) 7

,

2

2x 2

2 g(x) g(x) x xf(x)lim

x x 4

18 -


1.186 x 1lim f(x)g(x)

x 1 x 1

g(x)lim f(x)(x 1) lim 3

x 1

1.187 2xlim f x 0

,

0xlim f x

.

1.188 f R

x 3lim f(x) 2x 5 4

.

x 3lim f(x)

1.189 2x 3lim 12f(x) 4f (x) 9

,

x 3limf(x)

1.190 x 2

f(x) 4lim 1x-2

f x f 1 x , x R x 1lim f(x)

1.191

1xlim f(x)

1x

lim g(x) ,

15

xlim f(x) g(x)

1

2 4xlim f(x) g(x)

1.192 f : R R

x 1lim

f(x) x 2x-1

,

x 1lim 2 2

f(x) 1f (x) x

x 1lim

2

2

f (x) f(x) 2

f (x) 3 2x

1.193 A 2 2f x 2f x x 0

x R x 0lim f(x)

=1.

1.194 f: R R

x 1lim f(x) 2

x 0lim[f(x-1)-f(1-x)]

1.195 x 3

f(x) xlim 2

x-3

, x 3lim f(x)

1.196 f x 0

f(x)lim 3x

) x 0

f(vx)lim 3vx

, v 0

) 2 2f (vx) x 2f(vx) x

x R 3v 1

1.197 f

f x y f x f y , x, y R .

) f

) x 0lim f x 0

,

x lim f x f

R

)

x 0

f xlim 2

x

x 2f x 2

lim 2x 2

x 0

f (x) f(x)lim 0

x

1.198 f : R R *

3 2 3f x 2x f x 3 x ,

x R * . x 0

f(x)lim Rx

,

1

x 0

f(x)limx

x 0

f(f(x))limx

2

2x 1

f(x x)limx 3x 2

.

1.199 *f : R R

: xf(x) f(y) fy

x, y 0

) f(x) 0

1 f 1 1

) x 1

f(x)lim 2x 1

x4

f(x) f(x)limx x

1.200 f

x 0

f(4x)lim 3f x

. N

x 0

f(64x)lim 3f x

19

. 2016-2017

1.201 ( )

) x 4

2 xlimx 3 x 2

) 3

3

x 1

x 1lim(x 1)

) x 4

5 2xlimx x 2x 2 x 4

) 2

x

5 xlimx

1.202 ( )

) 2x 1

x 1lim

x 2 x 1

)

x 0

x 16 4limx x

) 2

x 1

x 5xlimx x x x 1

)

2

x 0

3x 2limx 1

) 2x 1x 5 x 3limx 1 (x 1)

)2

3x 0

3 2xlim1 x

1.203 A x 1

h(x)lim|x 2|

x 1limg(x)

1.204 2x 1lim (x 4)f(x) 3x 2

x 1lim f(x)

1.205 x 2

2x 3 5limf(x)

x 2lim f(x)

1.206 x 1limg(x) 3

x 1

g(x) 2x g(x) 6 4xlim

x x x x 1

1.207 2

2

x 2 , x 1x 1f(x)

x x 5 , x 1x 1

,, R

x 1lim f(x)

1.208 2

(x) x 0xf(x)

x x x 0x 2 x

x 0lim f(x)

R

1.209 22x 1 x

f(x) x x x

x 4lim f(x)

R

1.210 R 2 2x 9x 5lim

(x )

1.211 , R : 3 2

3x 1

x ( )x (2 1)x 3 lim Rx 3x 2

1.212 , R x 1

x x 2lim 8

x 3 2

1.213 R

2

3 2x -x 2f(x)

x -3x 3x-1

1 .

1.214 N R :

A) 2

2x 2

x x 6limx x

` B) 3x 1

x limx 7 2

1.215 N 2

x 4

x lim(x 4)( x 2)

, R

1.216 ,, R 3

x 2

x x 6lim 4x 2

20 -


1.217

) xlim x x x x

) 2 2xlim x x 3 x

) 2 2xlim x x x 2 2x

1.218

) 2

x

x 2 xlimx x 2

) x x 1

x 2 x 3x

e 3lime 3

) 2

2x

x 2x 3 4lim

x 2x 5

) x 0

3 2 log xlim

1 2 log x

1.219

) 2 2x

x 2xlimx x 3

) 2

3x

xx xlimx

1.220 x

xx

ln(1 3 )limln(1 2 )

1.221 2

xlim x 1 x

1.222

2

t

ln(t t 1)limln t

1.223 f : 0, R

x

f

xlim 3

x

x

ln f xlim

ln x

1.224 o

2xx

x 1xlim e

1.225 3 2( 1)x ( )x x 3f(x)

x 1

xlim f(x)

, R

1.226 2x 2x 3f(x) -x-

x 1

, R xlim f(x) 3 11

1.227 2f(x) x 2x 3 x

xlim f(x)

R

1.228 2f(x) x 4 x

0 , , , x

3lim f(x)2

1.229 , R :

2xlim x 2x 3 x 12

1.230 0 ,

x -lim

x x

x x 2 1 -2 1

,

1.231 0 ,

x lim

x x 1

x 1 x 2 2

1.232 xlim f(x)

0 ,, R

2 2 2f(x) x 1 x 2 x 3

1.233 2 2x f x ln

x

0

x 0lim f x

, x lim f x

x lim f(x) ln(x)

.

21

. 2016-2017

1.234 f

: 2 2x f(x) x 2 , x 0 .

A) x 2

f(x)-4limx 2

B) x 2

f(x)-4limx 2 2

) 2

x 2

f (x)-16limx 2

) 2x 2f(x)-1 -3

limx 4

1.235 x 0lim f x

, x 0limg x

2 2f x g x 2f x 4g x 5 x , x R .

1.236

o

2 2x xlim f x g x 0 ,

ox x

lim f x

ox x

lim g x 0

1.237 A 2 2f x 2f x x 0

x R , x 0lim f(x) 1

1.238 f ox 2 x R

2x 2 f x x 7x 10 . x 2lim f(x)

1.239 2

2x 0

1x xlim

x x

1.240 ** f 2f x f x x x R . N

x

f x 1limx 2

1.241 2

x

x xlim 1x x

1.242 2

xlim x x 1 x

1.243 x

x xlimx x

1.244

xlim ln x 1 ln x x

1.245

2

x

x 3lim3 x x

1.246

2 2xx 2xlim

x x 3

1.247 x

x 1xxlim

x 1

1.248 x 2lim f(x) 4

f(x) 4

x R , :

2

2x 2

2f (x) 7f(x) 4lim

f (x) 16

x 2

f(x) 2lim

3f(x) 12

1.249 f R

xlim f(x) 5x-7 3

. :

xlim f(x)

, x

f(x)limx

, 2x2f(x)-3x 5lim5x 2x 1

1.250 f

x

f(x) xlimf(x) x

xlim f(x)

1.251 x

f(x)lim Rx

xlim f(x)-x

2f(x) 9x 1 3x

1.252 f : ,0 0,

x 2

xlim

1 x x f x

f x 0 , x R . Y xlim f x

1.253 f : 0, R

x

f xlim l 0,

x

.

x

ln f xlim

ln x

22 -

1.254 f

2xf(x) 5x 2 x (x 1)(x 2) , x R

1.255 x 2

f(x)-2xlim 1x-2

f

, 2

x 2

xf(x)-2x 3f(2)-6xlimx-2

1.256 x R

2 2 2 x 2xf(x) f (x) x x x 2f(x) .

f ox 0 .

1.257 f : R R

5f x f x x x R .

ox 0 .

1.258 H f 0

xxf(x) e 1 , x R . B f(0)

1.259 1f x ln xx

1f

,

1 2

1x 0

f x xlim 1

x f x

1.260 f : R R

3 2xf x 3f x e 1 , x R

) 2xf(x) e 1 , x R

) f

) x 0

1lim xfx

1.261 2

3

1 1x , x 2 xf xx x, x

) 0 f

ox .

) f 0

1.262 f : R R ,

2 2f(x) 2f(x) x 0 , x R

) f(x) 1 x

) f 0

) x 0

1lim xfx

1.263

1x

1x

2 2 , x 0f x2 1 , x 0

) :

x lim f x

, x -lim f x

, x 0lim f x

,

x 0lim f x

) f

;

1.264 f 0 f(x y) f(x) f(y)

x, y R R

1.265 f 1 f(xy) f(x) f(y) x, y 0, .

0,

1.266 f :

f x y f x f y 1 , x, y R .

f R ,

R .

23

. 2016-2017

1.267

2x 1 x

x 2x

1 ,2 2

1.268 22 2 0

x x 1 x - 1

, , 0

, 1 2 , 1,1

2 2

21 2

-1 1

1.269 3 2x x 0 ,

, R , 0 , 1 0 .

1,1

1.270 ' f : , R , ,

2f() 2f() .

ox [,] , 2

o of(x ) x .

1.271 f : [0,] R ,

f(0) f() .

ox 0, , o of(x ) f x2

.

1.272 f R x R f(x) f(x 2) 0

:

A) f

B) R f() f( 2)

1.273 xf(x) 20 x e , g(x) 21 (x x) .

(,) 21y 20x ,

(,) 0,0 . fC , gC .

ox 0,1

1.274 f : R R

R f 4 f 4 0

f x 0 *x R .

f x f x 0 *x R f 0

1.275 f

1, 2 f 1 f 2 ,

ox 1,2 o3f( 1) 4f(2) 7f(x )

1.276 E

f : , R , f f , , .

ox (,) ,

07f x f 2f 4f

1.277 , 0 ,

x x ( )

.

1.278 f R f x f 2 x 0 x ,

f x 0

R .

1.279 f

1,2 f 2 6 , f 1 f 2 8 ,

, ox 1, 2

2o o of x x x .

1.280 f : 0, 0,

0

f f f 1

. ox

2016o of x x

24 -


1.281 ' f : 0,1 0,1

. ox 0,1

o of x x (S. Banach)

1.282 f ,

R f x f xe 4x 4e 0 x R

f 0 ln 2

1.283 1 2 1994 , ,..., 0,1 .

, ox 0,1

o 1 o 2 o 1994x x ..... x 997.

1.284 f : R R

2f x 2f x x 1 , x R

1.285 * f : R R

2f x 9 x R f 0 3 .

f x 3 x R

1.286 N

f x 4 x 2 x

f x 0

1.287 f

224x 9 f(x) 36

x 3, 3 . f 0 2

1.288

) 21 xf x , 0 x 1

x

) 3f x x x , x 0, /2

1.289 f : R R : f 1 f 2 f 3 f 4 .

f ;

1.290 ,, R

0,2

2 2 2 3 2

1.291 f x x

g x 2x

0,4

.

1.292 f,g : 0,1 0,1

f g g f x 0,1 .

f

0,1 . () ox 0,1

o of x x o og x x

1.293

f 2 2f x 4x 5x x , 0 x 2

1.294 f : R Z

f 1 2 , f x 2 , x R .

1.295 f (0, + )

x 0lim f x R

x lim f x R

,

x 0

ox 1o of x e ln x 1 .

1.296 f R f f(x) x x R .

R f

1.297 f : R R f 10 9

x R f x f f x 1 .

f 5

25

. 2016-2017

1.298 ) ln x 2 x 1 0

) 3 2

2

x 2 1 x 1 x 1f x

x x ln 1 x 1

R . f

R , R

1.299 2g x x xx 3 2x x x , g(x) 0

f x g(x) g(x) 0

R f

1.300 f : R R , 2 2 2f x x x , x R .

) f 0

) f R f f 0 0 .

1.301 f 1, 4 : f x 0 x 1, 4 , f 1 0

f 1 f 2 f 3 f 4 :

) f x 0 x 1, 4 ,

) 2g x f x f 1 f 2 1,2 .

) f .

1.302 - g(x) x5

, . 4 f g f()5

f

R , x [,] f(x ) f(g(x ))

1.303 f,g : 0, R g 0 1 x f(x) g(x) x 3f(x) g(x)

) f 0

) f x 0 x 0, 4 : ) x 2 f x xf x 2 x x 2

0, 2 .

) g x 0 x 0, 4 .

1.304 f : (0,1) R

x 0

f(x) 3lim 3x

22(x 1) (x 1)f(x) x 1 x (0,1)

) h(x) f(x) ln x 3

) f(x) 3g(x) e y x

0x (0,1)

26 -


1.305 A) f 1 1 . ,, ,

f() f() f() f() f() f()

B) f 1 1 o ,

.

1.306 f , R 3 2 3 2f x 4f x 6f x x 2x 6x 1

x R . f x 0 0,1

1.307 f R , f f(x) x 4 2f x x R .

:

. f 1 1

. A f

. ox R o of x x

. f 1 1

1.308 - - 6 . 6 6 .

6 , 6 , ,

.

1.309 f , f x 0 , x , . 1 2 3 vx , x ,x , ..., x ,

1 2 , , 1 2 3 v

1f x f x f x ... f x

f v

2 1 2 3 vf f x f x f x ... f x

1.310 f : IR IR 2f x f f(x) 4 x IR f 2 1

A) f 0x 1 , x 11lim f(x) 3

x 1

B) f 1, 2

1.311 f f x ln x ln x 1

) f 1f

) xg x f x e

) xf x e 1 0

27

. 2016-2017

1.312 f : 0, IR 3 3f x xf x x 0 , x 0,

.

) 3x f x 0

)

2

f x2f x 1 , x 0x x

g x 1 , x 0f x

, x 0x

1.313 f, : , .

, .

1.314 f R R ,

3f x 3f x x x R .

) f 1 1 .

) f . ) 2

xf x

f x 3

x R

) f

) x 0

f x 1limx 3

1.315 ** f , f f . f

0,1 f 0 f 1 0 .

) f 12

.

) N f 1

, 1, 2,3...

1.316 f g :

f R f x 0 , x R , f 0 1 , f 2009 2009

f f(x) f f(x) 1f f(x) 2 f f(x) 1

, x R .

g 2g x f x f 1 f 2 , x R . ' :

) f x 0 , x R ) R f 2

) f 1 f 2 f 3 f 4 ) g 1, 2

) f g .

63 x2 x 4 2x 8 xf(x) x6 x 4

f(0) (0,1] 6f 6

28 -


1.317 f,g : R R 2g x 7x 5x 2 , : 32 f x + 3f x = x + 5 ,

x R f R

1. , g , f

1f

2 f x x , y y

3. f 3x < x (_3) 1 xf x > g

4 x R 1x x g x h = xf

, , R {0} 2 2 2

4 4 4 2 2 2

1 1 1

+

1.318 1 : x

1 - x = 12

.

f R

f x

1 - f x = x + 12

x R

2 f 0 = 0

3 f R

4 : f x2 f x + 6 > 1 f xf 2 16f x - 3 > f 16 5 2 2x f x x f x

1.319 f : R R g :R R

x R f(f(x)) 2g(x) x

1 g R .

2 h(x) f(x) g(x)

0x R 0 0f(x ) x

3 fC gC .

4 0 0 0f(f(x x 2)) x x 2f(x x 2) 2

5 0 0f(f(lnx x 1)) ln x 1 x

1.320 f R f R R . f

fC A 1,5 B 3, 1 ,

1 f

2 1 1 2f 2 f (x x 3) 3

3 : 1 1 x x 1f 2 f (5 4 2 ) 3

4 2f x 5 4f x

29

. 2016-2017

1.321 f : 0, R 0, , R

f xf e ln x x 0, . x ln x f x t x ln x x 0,

1) t 1 1

2) f x ln x

3) 2

2x x 1ln 1 xx 2

4) x e2 2

x e 2x ef f e 1 f xe f 2e2

1.322 f R , f x y f x f y

x,y R

1 f 0 f x f x 0 x R

2 f x y f x f y x,y R .

f x 0

3 f 1 1 .

4 1 1 1f (x y) f (x) f (y) x, y f(R)

5 2f 10 x f x f x 1

1.323 , f : R R ,

: f 2 , f 2 f(x) 2004 x R .

1 2x f x f 0, .

2 f , :

) ( , ) f .

) fC f y 2x

ox ,

3 2xxf(x) 4xlim

x 1

4 h : R R f(x) h(x) 2004x , x R .

f x 0 1 2, .

h x 0 1 2,

30 -


1.324 f : R R xe 1 xe

xe 1

e ef(x)e e

1 f R f

2 f

3 1 xfe ( 1)e

, 0

4 3 2 4f x f x f x f x 0,

5 2 1

1 11

2 11

e elim fe e

1.325 f 0,1

2 2f (0) f (1) 13 6f(0) 4f(1) .

1 N f .

2 N 1x 2x (0,1)

f y 3x 1x

21 1 112f(x ) 3f 4f 5fe 2

3 1f(f (lnx 4) 1) 3

1.326 f : R R : f(1 ln x) 1 x lnx , x 0

1 : 1 xf(x) x e x R .

2 o f

3 xxe e 0,

4 xlim f(x)

, xlim f(x)

.

5 2 3f x f x f x 0,

1.327 f : R R 3f x f x 2x 0 x R

) f R

B) f R

) , R 2 2f f 2 2 5 0

) 3f x 3x 0

E) N 3 22x f x f x 1 0

31

. 2016-2017

1.328 f : R R 3 2 3f (x) xf (x) 2 x x R ,

x 0

f(x)limx

x x

f(x) f(x)lim limx x

, , R 0 .

1 ,

2 g(x) f(1 x) 1 , x R x 1limg(x) 1

x 1

xg(x) 2 3 x g(x)lim

x 1

3 x

f(x) x 2007limf( x) x 2007

1.329 *f : R R 1 92f(x) f 6xx x

x R 0

1) 4f(x) xx

.

2) f .

3) 2x 1|f(x)| 3lim2x x 1

.

4) x 2

5lim f(x)x 2

.

5) fC 0,0

1.330 2f(x) ln x x 1 ln (1 )x , x 0 0 . 1 0 ,

xlim f(x)

A 0 :

2 f .

3 f

4 f

B5 2 7ln2 f x f x f x 0,

1.331 f : 1, R : 2f x 1 2ln x f x

x 1 3

2x

f 1 x x 1lim

x x 1

.

1 f 1 0

2 2f x lnx 1 ln x , x 1,

3 xlim f x

2x 0lim ln x 1 ln x 4 1, f

5 , 1, f f 3 4 2

32 -


1.332 f : R R , x 0

f(x) 2lim 0x

1 f(0) 2

2 : 2

2x 0

x f(x)limx

f : 2 x 2xf (x) e f(x) e 1, x R

3 : x xf(x) e e , x R

4 xlim f(x)

, xlim f(x)

x

1f 2xlim

2 x

.

5 f ox 0

2016 - 2017

M

333

:

: 16.08

M.Ed. 2016

: http:users.sch.gr/mipapagr

33

. 2016-2017

2 - 2.01

2f(x) x -5x 6 2

2.02

0 1xxe x 0f(x)

1 x x 0

2.03 x x 2

f(x) x 2 2 x 2x 2

, R f 2

2.04 x 3

f(x)lim 7x 3

f

0x 3 , f 3

2.05 f 0.

x f(x) x x x , x 0 .

f ox 0

2.06 f : R R 2 2f(x) x (x 1) x R ,

f ox 1

2.07 f,g : R R

R f g 3 .

:

) x

f(x) f()limx

)

2 2

x

(f(x)) (f())limx

) 2 2

x

f(x) x f()limx

)

x

g()f(x) f()g(x)lim

x

2.08 f

ox of(x ) 3 , of (x ) 2 . B x x

2f(x)-6limx-x

2.09 f : R R 0 1 f(0) f(1) .

1f(2x) x2g(x)1f(2x-1) x2

12

f (0) f (1)

2.10 f : R R ox 0

x 0

f(2x) f(x)lim 3x

. f 0 3

2.11 f : R R 1 f (1) 2 .

x

xlim (x 1) f(1) f 2x 1

2.12 f : R R ox R .

f(x) x xg x

f (x )(x-x ) f(x ) x x

ox

2.13 f : R R 0

f x y f x f y xy x, y R ,

f R .

2.14 f

x 0

f x 2lim 3

x

:

A) f

2 f (2) 3

B) :

i) 2

2x 2

f (x) f(x)limx 4

ii)

x

2x 1lim x fx 2

34


2.15 f : R R , 0 .

2 2

x 0

f (3x) f (2x)lim 2f(0)f (0)x

2.16 f 0x 1 f (1) :

f xy xf y yf x x, y 0, .

000

f(x )f x +x

00 x 1

2.17 ** f : R (0, )

3 4f (x) 2x f(x) 8 , x R

f

0x 0 f (0) 0

2.18 f R x R ,

h 0

f(x 3h) f(x 2h)lim 5f xh

-

2.19

) xef(x)

1 x

) 2

1f xx 4

) ln xg(x) xxx 1

) xln xf xe

). x xg(x)1 x

) ln xg(x)x 2

) 2 xf x1 x

) 2xf(x)

ln x

) x2x 1h(x)

e

) 1 xf(x)1 x

2.20 1 h

h 0

e elimh

,

x 1

x 0

e elimx

, x 0

x 12lim

x

2.21 P

2P x P x x R .

2.22 f : R R ox e .

x

x 1

f(e ) xf(e)lim ef (e) f(e)x 1

2.23 h

h 0

e 1limh

2.24

A) x

x 0

e 1lim 1x

B) 5 5

x 2

x 2lim 80x 2

2.25 g

R , g e 1 g e 2 .

2

2 xf x x g xln x

f e

2.26 f :

yxf x y e f y e f x xy

x, y R :

A) f 0 B) f 0 0

) R

oxo o of x f x f 0 e x , ox R .

) f 0

R

oxo o of x f x f 0 e x ox R

2.27 f : R R 0x , 0 ,

:

) x

f(x)ln x f()ln f()lim f ()ln x

) 2x f(x) xf() f()lim f ()

x x

35

. 2016-2017

2.28 : 2 2f(x) x 3x ,

2 3f(x) (4x 1)

3 2f x ln x 3x ln 3

3f x ln 2x 2

x xf x 2 3 t , t R

4 32 3 2f x x 3 x 5 y , y R

2.29 :

) f x ln x , x 1

) x xf x log 2 3

) 4 32 3f x x 3 2x 5

2.30 :

) 2 1x x 0

f x x0 x 0

) 2f x x x 3 2

) log xf x x , x 0

) xf x x , x 0,2

) xf x 2

2.31 x 3f x e x x , x R .

) f

1f

) 1f 1fD ,

1 1f 1 2 .

2.32 f

0x 0 : 3 2 2f (x) x f(x) 2x x ,

x R f 0 .

2.33 ) f(x) c(x )(x )(x )

c,,, R x ,, :

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

) f 2 3 4 2

2

(x 5) (1 x )f(x)1 x

2.34 f R f x 0 x R .

) y f x

R .

) f 2 5 f 2 4

f 2 4

2.35 f

R 2 2f(x ) 3f 2x 1 5ln x 3x x 0 . f '(1) .

2.36 f : R R 0x , 0 ,

:

) x

f(x)ln x f()ln f()lim f ()ln x

) 2x f(x) xf() f()lim f ()

x x

2.37 f(x) x,x (0,)

) 1f

) 1f

,

1 21f (x) , x ( 1,1)

1 x

2.38 f f(x y) f(x)f(y) f(x) 0

x,y R . x 0

f(x) 1lim Rx

f R

36


2.39 f 0 x R

f f(x) f x 2x . f 0 1 f 0 2

2.40 f,g

R x R 2f(x )g x e ,

f 1 0 , g 1 2g(1)f 1

2.41 P x

2P x P x

2.42

2 2x x , x 0

f x x0, x 0

f x ox 0

2.43 f R .

) f f

) f f

) f

:

) fC 0,0

) f x f x

) f 0 0

) f 2g(x) (x 1)f(x) 3x g (0) 3

2.44 f : 0, R

xf x x x e , x 0 . f 0,

2/ xf (x) x xx 2xe

2x 0f(x) 1lim

x

2.45 f : R R R .

0x R

fC 0x 1fC

0f x .

2.46 f

R . x 1

f (4 x)lim 5x 1

f 3 5

2.47 f R. :

) h 0

f (x 2h) f (x)lim 2f (x)h

, xR

) h 0

f (x h) f (x)lim f (x)h

, xR

) h 0

4f (x 2h) 6f (x h) 10f (x)lim 2f (x)h

x R

2.48 :

A) 2xy ln e 1 x y 1 y 1 y B) y ln x ln x

2x y xy y 0

2.49 f R x R

2f x xf x , f 1 0 .

2.50 :

) f x x , () f x x2

) xf x xe () xf x e x

37

. 2016-2017

2.51 fC

0x 0 2

3

1x x 0xf(x)

x x 0

2.52 f

ox 1 2

x 1

f(x) xlim 7x 1

.

fC A 1,f 1

x 9y 5 0

2.53 3f(x) x .

fC

M( 2, 8)

2.54 f

: 2xln x f x x x x .

ox 1

fC

1,f(1) .

2.55 f : 0, R 1f xx

0 ,

Ox,Oy

ox

.

2.56 f gC , C

2f(x) x 2 21 1g(x) x8 2

y y .

2.57 2f x ln x x 3 ,

, R : 2x y 4 0

fC A 1,f 1 .

2.58 f f 2 x f 2 x 2x , x R .

2,f(2)

y x .

2.59 2x 2x x 2

f x x 2

x 1

,

,, R Cf

A(2, f(2))

2x y 1 0

2.60 2f x 4 x 2g x x 8x 20 .

fC gC .

2.61 0

2f x x 3x 1, f(1)

g xx

2.62 x xe ef

2(x)

x xe eg x( ) x2

.

2.63 f R f (x) 0

x R . gC g f(x)g(x)f (x)

x x ,

, x x

o45

2.64 **

4 2f x x 4x 3x .

f . (mathematica)

38


2.65 f : R R

: 2f x 2 x 3x 2 f x 3 2x 4 ,

x R .

1M ,02

fC

.

f

fC .

2.66 f x 2 ln x ,

x 0 , R .

fC M 1, f 1

R.

2.67 y 2x 0

y f(x) ,

ox 1 ,

gC 21g(x) f

x

1x 1

2.68 * 1f(x)x

xg(x) e ,

fC gC .

2.69

xg(x) e 2f(x) 2x ,

2.70 R f

xf(x) , y x .

2.71 f :

23f x 1 2f x 2 x 14x 5 , x R

) f .

) fC

1A 1,4

, .

2.72 ** f R , f ln x x ln x x , x 0 .

fC

ox 1 x x y y

2.73 ** ln (x)f xx

, x 0

)

fC x , f(x ) .

) A

x , f(x ) , ,

.

2.74

f : (0, ) R , 2f(x ) f(x) 3 ln x 4

)

f 1, f(1)

) : 2x 1x f(x) - 2lim

x - 1 .

2.75 x *f x e x, R

) B M fC

.

)

M R

2.76

21f(x) x 2x - 2(1 - ), R2

A)

.

B) fC

x x , y x .

) 0 ,

f

39

. 2016-2017

2.77 x, y fC , f(x) x .

.

2.78 Oxy

xf x e , x 0 . M A, B

M Ox Oy . M

1 m /sec . ot 1,e , :

) OAM ) AB

) fC M , x x

2.79 A Ox Oy ,

. Oy

Ox

A v 10 km/min .

) ; ( )

) A O 0, 0

Oy .

) A O 0, 0

3 km ;

2.80 4m , . 1m . 2m

1m /sec . 6m , :

A) y t

6y t 3x(t)

, 2 x t 6

B)

2m .

2.81 *** O 2m/sec

().

( ),( ) , 2m

( ),( ) AB .

AB OAB

( www.mathematica.gr)

40


. Rolle ... 2.82 . Rolle

f x x 1 x x 0,1

2.83 2x x x 0f(x)

3 ( )x x 0

,, R

Rolle 1,1 1,1

f 0 .

2.84 f

3,2 2

3,2 2

.

0 3x ,2 2

o o of (x ) f(x )x .

2.85 f , , > 0

(, ) f f

.

, f f

2.86 f

, (, ).

,

2 3 3f f

3 f

2.87 f ,g

, , f() f()g() g()

g(x)g x 0 x (,) .

, f () f()g () g()

2.88 f

R f(x) 0 x R f(2) ef(1)

.

f (x) f(x)

1,2 .

2.89 f : [,] R , : 2 2 2 2f () f () .

, : f f

2.90 f 1,1 , 1,1 ,

42f 5 f(1) f( 1) .

2.91 f ,g

[,] (,)

f x 0 x [,]

ln f() ln f() g() g() .

(,) f () f() g () 0

2.92 f : R R .

f 1 f 0 f 0 f 0 0 . N

x 0,1 3f x 0 .

2.93 ) 3 2f x x x 3x 1

x R R 1 1 .

) . Rolle

f x 2g x e x x 3

2.94

xf(x) e 2x

x 3g(x) e x

y y

41

. 2016-2017

2.95 N x x 11 2 3 0

2.96 xln 1 xe x

2.97 x x2 5 2 5x

2.98 x x5 x 4

2.99 ln x x 1 0

2.100 x xxe 1 e

2.101 : 2x x ln x 2

2.102 96 96 24x 3 x 1 16

2.103 5x 3x 0 R

2.104

2012 20112013x 2012 1 x 0 0,1 R

2.105 x 2e x x R

2.106 3 2x x x 0 2 , 0

R

2.107 8x 7x 6 R

2.108

xe x 1

xe x 1

2.109 N 3 2 ln x ln x ln x 0 , ,,, R

3 2 4 0

21,e

2.110 4 3 2x x 3x x 0

,,, R

, 2 8

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

2.111 51 km .

0,6 .

85 km /h .

2.112 f 1, 5 f 1 2

f x 2 , x (1, 5) 10 f(5) 6

2.113 f 0,5

f 5 f 0 1 .

, 0,5 2f 3f 1

2.114 f

1, 4 x R f 4x 4f x

25f 1100

1 2 3 , , 1, 4 1 2 3f f f 12

2.115 f x log x .

1,20 19 loge1 log2

.

2.116

xx

x 0

2 2limx x .

42


2.117 1 x 1 1lnx 1 x x

, x 0

2.118 2

2 1ln 1

, , R

2.119 , , R

2.120 x x1 x e 1 xe , 0 x 1

2.121 N :

) 1 1

x 1 xxe x 1 xe x 0 .

) e 2 ln e

2.122 N :

) x ln(x 1) xx 1

x>0

) x 1x e 1 (x 1)e x 1,2

2.123 x x 11 11 e 1 , x 0

x x

2.124 0 4

2

1 2 1 4 ( )4

2.125 f R R .

: f 1999 f 2002 f 2000 f 2001

***************************************************************

2.126 f R

fC , R

f 0 .

2.127 f

2, 2 2, 2

f( 2)= f(2)= 2 . f (x) 1 , x 2, 2

f(x) x , x 2, 2

2.128 f : R R .

f 1 f 0 f 0 f 0 0 . N

x 0,1 (3)f x 0 .

2.129 2 6 4 2f(x) x x x , *,, , R 2 23 5 .

.

2.130 R f f(ln) f(ln ) .

ln ln ln , ,, 0

2 e , 1 2 , R

1 2f ( ) f ( ) 0

2.131 f , , f f 0 .

ox ,

f x f x f x .

2.132 f : , R ,

, ,

f f 0 . :

) x , of x 0 ,

, f 0 ,

) x , of x 0 ,

, f 0 .

43

. 2016-2017

2.133 34x 2 x 1 0

0,1 R .

2.134 f , R f( 1) 1 , f(1) 1 .

) 1 21 1 1 2f f 2

) 1 21 1 1 2

1 1 2f '( ) f'( )

2.135 f

0, > 1 f(0) = 0 2f(x ) 2f(x), x [0,] .

1 2 , 0, 1 2f()f ( ) f ( )

2 - 1 .

2.136 f

2, 20

Rolle, :

) 1 2 , 2, 20

1 2 1 2f f 0 .

) 1 2 , (2,20) 1 2

1 23f ( )+ 2f ( ) = 0

) f (x) f(x) - f()

2, 20 .

) , , 2 20

2f 3f 4f 0

2.137 f : R R f 2 0 . R

,

f , f() , x x

P 2,0

2.138 f : 1, 4 R

f 1 2

f 4 8 .

fC .

2.139 f : , R

, , f 2 , f 2

) f x = 2x

, .

)

1 2 , , 1 2f f 4 .

2.140 04 3 2 ,

3 2f x x x x

0,1

2.141 N 3 2 ln x ln x ln x 0 , ,,, R

3 2 4 0

21,e .

2.142 f x x 1 ln 2x .

:

) 1 ,12

fC , f() x x

) x 2 2x2x e

1 ,12

2.143 f R .

f 0 f 10

f x2

.

ox 0,10

of x 0

44


2.144 f : R R , :

f x 2f x f x 2f x , x R

f 0 f 0 f 0 1 . :

) xh x f x e

2 2g x f x f x 2 f x f x

) f .

2.145 f : R R

: f x f y x y 1

x,y R . f

2.146 f f 1 2x 7 12x ,

x R f 1 2

2.147 f 21f x

x , x R *

f 1 f 1 2

2.148 : ) f (x) f(x) x R

f(0) f (0) 1 xf(x) e , x R ,

) (x) (x) 5x x R ,

(0) 1 (0) 4 , x(x) e 5x , x R

2.149 f R

*R f 0 0 ,

. f

2,1 2,1

2.150 f R . f x 2x 1 f x , x R

c R 2x xf x ce

2.151 f , x R f (x) f(x) x x f(0) 1 .

2.152 f : 0, R

f 02

f x f x

x 0, f x x ,

R .

2.153 f : R R

: 2x 2 f x 2x 5x 2 , x R

f 3 7

2.154

f : 0, 0,

f (x) f(x) ln f(x) x 0 f (1) 0

2.155 N , , f R * x R *

f(x) xf (x) , f(1) 1 f( 1) 2 .

2.156 f f 0 2 ,

x xf(x) e f (x) e 0 , xR

2.157 M(0, 3)

24

4 1

2.158

f : 0, 0, , f (1) 0

f (x) f(x) ln f(x) x 0

2.159 f ,

R 2x[f (x) f(x)]e f(x) f (x)

x R f 0 1 .B f

45

. 2016-2017

2.160 f : 0, R

f 02

f x f x

x 0, , f x x , R .

2.161 f R f x 0 , x R , f 1 9

M x, f(x)

4x f(x) , x R

2.162 f R , fC

O 0, 0 fC

O 0, 0

-2x y 3 0

2x 1 f (x) 4x f (x) 2f(x) 0, x R

2.163 f g

R f x 0 x R

f x g x f x g x

x R , f x g x

2.164 ) f : R R f x f x 0 , x R

f 0 f 0 0 . f

.

) g : R R

g x g x 0 x R g 0 0 ,

g 0 1 . g x x .

2.165 * f R f 0 0

f(x) 22xf x e 1 2x x 1 x R . f .

2.166 f : R R N .

2 f x f y x y , x,y R

f .

2.167 f , R ,

x, f(x)

x2xe f x 2A 1,e

f

2.168 * f : 0, + R

f xy f x f y x, y 0, +

f e e . f ox 1 .

f 0, +

f x e ln x , x 0, + .

2.169 f : R R ,

2f x y xy y f x

x,y R , f 1 1 , f 2 2 . f

R

2.170 f R

f 0 2 2xyf y x f y f x e ,

x,y R . :

) f x 0 x R f 0 1

) f R

) f 2x 2xf x e

2.171 f : 0, R ,

1 f 1 1

2 2f xy x f y y f x . x,y 0 .

) f

x 0

) 2f x x ln(x) x 0

46


- 2.172

) xf xln x

) f x x x, x [0,2)

2.173 N

) x

2

e ex 1 x 0f x

x ln x x 0

) 2 xf(x) ln(1 x ) e 1

2.174

2f x x x 1 20,3

2.175 N R ,

2 xf x x x 1 e , R .

2.176 f : R R f 0 0 f

,

f(x)g xx

, x 0 , .

2.177 O f g R f(0) g(0)

x R f (x)g(x) f(x)g (x)

g(x) 0 . : f(x) g(x)

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

2.178 * f

[0,+ ) 5 3f(x) 2 f(x) 3f(x)

2 3x xx 1 ln x 1 x 1

2 6. f

.

-

2.179 ln(x 1)f xln x

, x 2

) f

) :

) 2ln e 1 ln e 1 . ) 2ln x 1 ln x 1 ln x , x 2

2.180

2ln(x 1) x x 6 0

2.181 2xln(x 1) x 0x 2

2.182 x 1e 2x e 0

2.183 x 2

x 1 x 1x 1 x

2.184 22ln(x) x , x 0,

2.185 f : R R

f 1 x f 1 x x R .

f x 0 , x R ,

f x 0

2.186 f : R R , f x 0

f x3 3 2f x ln f(x) e x x 2x 1

x R . 2f ln x f 1 x

2.187 xg x x g x x R ,

xg(x)x

x 0 .

47

. 2016-2017

2.188 :

) 2f x x lnx ) xf x 2 , 0 x 2

) 2nxf xx

)

xef x2x

E) 2f x x 4 x

2.189 :

) x, x 0

f x 1 , x 0x

B) x 11 e , x 1f x

ln(1-x), x 1

2.190 22 xf x x e e .

2.191 , R

f x ln 2x x

x 1 2 ln 2 .

2.192

f : R R 2 2(f(x)) x 1 2xf(x) , x R .

f .

2.193 2f x x ln x .

fC f

.

2.194 R

3 2f x x 1 x 5 x 2

.

2.195 f : 0,1 R ,

f x 0

x 0,1 . 1 2x , x 0,1

1 2x x , 1 2f x f x 0

0,1 3f 0 .

2.196 R

2 xf x xe e .

2.197 f [,].

1 2x ,x (,) 1 2f(),f() f(x ), f(x ) ,

1 2 ( , ) f () 0 .

2.198 f,g : R R

:

f x x 1 g(x) xf x e e x x R .

fC A 0,1 ,

fC gC

ox 0

2.199 0 0(x , f(x )) , ox

f(x) x ln x x , R

R

2.200 f

0,3 f (x) 0 f(1) 1 , f(2) 1 .

2f(x)g(x)

1 f (x)

, 0 x 3 ,

g

2.201 f R . R

f() f () f () 0 f (x) 0 x ,

f (x) 0 ,

f (x) 0 f(x) 0 .

2.202 ** 2x 4x f x f x 0 , x 0, 4

f x 0 x 0, 4 .

48


2.203 x 2 x 0,

2.204 f

R 3f x f x x , x 0, .

f x 0

0,

2.205

22 ln x x 1 0

2.206

28x x x 1 0

R

2.207 R

3 2x x 4x 0

2.208 0 x 22e 2 2x x R

2.209 ** 2f x x 1 ln(x) ,

: ) 2f ln(x 1) f 6 x x 0

) 17 3 2008f x f x f x f x

2.210

x 1 ln x x 1 ,

.

2.211 * , 0 ,

2 3x x

2.212 f : R R f x 0 x R .

xf e f x

2.213 f : R R

f x 2f x f 0 1 .

2xf x e x 0 .

2.214 f : R R f 0 0 f x f x 0

x R , xf x 0 x 0

2.215 f , 0,1 ,

0,1 f x f 1 f 0

x 0,1

2.216 f R , : f 0 1

2xe f x 1 0 , x R .

fC A(0,1)

2.217 x 2e x x 0 , R R

2.218 ln x x

, x 0 ,

2.219 , 0 ln x x 1

x x 2

x 0 , 1 .

2.220 x f x x , x>0 ,

>0 f x 0 , x 0 . e

f e, .

2.221 f R f 0 f 0 0 f x 2x x R .

1f 13

49

. 2016-2017

2.222 f : R R ,

f x f x , x R

ox 0

f 0 0 . : x f x f x 0

2.223 xxf x 1 e

, R

)

f x 0 x R .

) 1 1e

xx 1g x 1 xe

.

2.224 f(x) x ln x , 0

)

x ln x x 0

)

f .

2.225 f R f 0 2 , f 0 0 ,

f x x 0 x R ,

3xf x 2 1

12

x R

2.226 f R x R

: 2 2f (3x 1) 4 4f(2x x 1) .

:

A (1, 4) : f () 0

B f

f (1) f (4)

f (x) 0

R

2.227

-

2.228 x x , x 0, 2 .

2.229 36x 6x x x 0

2.230 A)

x

vef(x) , N *x

B) v

x exev

, x (0, )

2.231 ) ee

) :

1821 1821 1

2.232 )

3 2f x 2x 2x x lnx , x 0

) ,, 0, 1 ,

3 3 3 2 2 22 3 2

2.233 f

2 xf x ln 1 x e

2.234 ** 2 xf(x) ln(1 x ) e 1

R 2f ln x f 1 x

2.235 A(1,1)

xf(x) e

2.236 0 , 0

ln ln ln2

2.237 x x 1x e , x 1,

50


- -

2.238 .

) 2 8h(x) xx

) 5 3g(x) 3x 5x

). 2g(x) ln x x 1 ) xf(x) xe

2.239

xx ln e x , x IR f

2.240 N

2 2g x ln x 2x ln x x 3

2.241

5 4 3 2f x x 5x 10x x , x R , , R

, 2 .

2.242 f : 0, R

f x x xf xx f(x)

x 0 . N f

0, .

2.243

f x x ln x x , R ,

A 1,3

) 4 1 :

) fC

4 x ln x x 3 , x 1 .

2.244 f : R R

f(x)2(x x 1)f (x) xe 0 xR

fC

.

2.245 f R .

f ox

.

2.246 N IR

4 3 2 2f x 2x 4x 3 2 4 5 x x 1 x R , .

2.247 f : 0,1 R

2f x x 4 f x x 0 x 0,1 .

fC

2.248 f xf (x) 2x 0 , x R .

A(0, f(0))

fC

2.249 f R f R f x 0 ,

x R , f 1 0

g x f x f 2 x , x R .

g , g

gC

2.250 f : [0, ) R

f(0) 0 .

f(x)g(x)x

(0, ) .

2.251 x 2 22f(x) 2e x , 0.

f , (0, )

51

. 2016-2017

2.252 f R

f(x) 2 xf(x) e 1 x x e x R .

)

) .

2.253 ) f .

1 2x ,x

1 21 2 f x f xx xf2 2

(Jensen)

B) :

2e 1 e 1 e 1

R

) ln ln ln2

, f,

2.254 x 0 , y 0 , 1 x y 1 ,

11 1 5x yx y 2

2.255 f R f x f x

f 0 f 0 0 . f

R

2.256 f R

f

, x R

3x3f x 4f4

2.257 f

1, f f 1 0

f x x 1 f x x 1, .

2.258 f

1,

f 1 1

f x x 1 f x 1 0 x 1, .

2.259 f : R R f x 0 f x 0

x R

2.260 f : R R x 1

x 1ef(x)

e 1

x R .

)

, .

) x 1

f(ln x) f (x 1) f(x 1) f (ln x)

2.261 f

1,

f 1 0 f x x 1 f x

x 1, .

2.262 ,, R ,

2.263 f

1,

f 1 1

f x x 1 f x 1 0 x 1, .

2.264 x 2f x e x x 1

) fC

.

) x 2e x x 1 .

) xe 1 x 1 x , x R

52


DE L HOSPITAL

2.265 )

xlim (x ln x)

) x 1lim lnx ln(lnx)

) 1x

xlim x e 1

)

1x

xlim x

2.266 :

) xx 0

x xxlimx e 1 x

)

x

x 1lim xlnx 1

) x

x 3limx 2

) x

xx

e 2x 1lim4e x 3

2.267

x ln x , 0 x 1

f x 1 x-1 , x=1

f 1 0, 5 .

2.268 N o 2x 0

1 1lim x x x

2.269 N 1x

x 0

elimx

1x 0

x

xlim

e

2.270 3 2

6 4x 0

1 x ln xlim

x ln x

2.271

x

xx

e 2x 1lim4e x 3

2.272

x x

x xx

3x ln x e elim

x ln x e e

2.273 N o x

ln x ln xlim

2 ln x

ln x

2.274 x 0

2x xlim1 x

2.275

4 x

2x 0

x e x 1lim 18

sinx x

2.276 x 2

22e 2x 2 xf x

x

f x

f xx 0

elime

2.277 f : R R

x xxf x e f x x e

x R . f 0 .

2.278 f : R R , , 1 x f x ln 1 x x

x 1 . f 0 . f

2 R 3f 02

f 0 f 0 0 . : x 0

f(x) f( x)lim 31 x

2.279 f : R R

R . x R

2h 0

f(x 4h) 2f(x 2h) f(x)lim 24x 8h

fC M 1, f(1)

y 5x 8 , f

2.280 1x

xln x x , x 0f x 1 , x=0

e ln( x) , x 0

, R f ox 0

2.281

,, x x

2x 0

e e lim 1x

53

. 2016-2017

2.282 N

x

3eh(x)x

, ln xf(x)x 1

, 21

xk x xe

x x x ln e 1

2.283 f,g : 0, R

g x f x x ln x 1 ln x x 0 .

y x 3 fC

, gC .

2.284 N y 2x 2 ln 2

xf x 2 ln e 1 2 ln 2

2.285 f : R R g

xg x xf e . y 2x 1 fC 0 , gC

.

2.286 f : R R ,

x

1lim f x 1x

2

x

xf x xlim 2

ln x x

.

y x 2

f

2.287 ,, R

f 2( 1)x x 5f(x)

3x

x 2 y 3 .

2.288 f

21f(x)

x x

x 3 x=5

A) .

B) y 0

fC +.

2.289 f : 0, R

xe xf(x) 1 x 0 .

x x

fC .

2.290 f(x) xln x, x 0

.

2.291 f y 2x 3 .

B 2 x

x 2 2

f(x) x x elim 1xf(x) 2x ln x x x

.

2.292 ) 3f(x) x 12x ) x 1f(x)x 1

) f(x) x x , x [ , ] ) ln xf xx

) ln x, x 1

f x1 x, x 1

) 2

2xf xx 1

2.293 2x 1

2 1f x e 2

( 1 0 )

54


2.294 f f x xln x x 3

, OM OM .

2.295 90 ,

. 4,0 , A [0, 4]

xx B 2y 4x x .

B AB ;

2.296 2000 , 5000 . ' 500

, 1000 .

.

2000 . , ;

2.297 . 70 . 30

30 . .

30 , 30 .

) ;

) ;

2.298 100 km x km/h .

0,8 2x2

400 lt/h. 9 /

) x ,

) , ,

) ;

2.299 : 2(t 1)P(t)

(t 1)

n , t 0 . :

A) , .

B) .

2.300 y 2x 3 .

A 9,4 .

2.301 82 . .

55

. 2016-2017

2.302 3 2f x x x 12x , , R ,

ox 2 A 1,f(1) 3, 5 .

A) , R f .

) f x 0 .

) f x 2004 .

) xf(x)limx

, xf(x)lim , Zx

2.303 f R : f(-1)=0

f x =3f x -2x+1 x

x+13f x =3e -2x-5 , x R

. f (f )-1 .

. : 2x 2f e 1 =f x 2 R . -1(f ) (x) + 1 = n(x) ;

2.304 f(x) ln x x

x 0 . x 0 f(x) 0

) = 1 , ) x x 1x e , x 0

) 2 22 21 1ln 2x 2 ln x 3x 3 2x 2

2.305 f x f(x) 2ln x f(x)f x e e x 0,

f 0 0 . :

) f 0, .

) Rolle o0, x .

) f x 33e xf x ln3

x 0,

) f .

) () : 3e 1y x 13e 3

fC ox 1

56


2.306 f(x) x ln x , R

) f ) xln xe

x 0

) x 0lim f(x)

xlim f(x)

) R xxe e

2.307 x xf(x) ln 1

>0

) f . ) xxe e

>0

) xxln

x 0 , =1 .

2.308 * f , R , f xf x e x 1 , x R

f 0 0

. f f f R

. x f(x) xf(x) x2 , x 0 .

. f .

2.309 * g : 0, R 0, , g 1 2 , g 1 8 ,

2g x x 4x

1g x 4 6x4x

x 0

)

) gC 2x

g x x 4lim

xg x 6x ln x

2.310 f , 4 :

f xe 3f ' x f x x 4 , f ' x 0 x 4 f 1 0 , f 1 1

) f , 4 , f

fC x 'x .

) 23f '' x f ' x fC , 4

) ox 0,1 0 0 0f x x f ' x (3)

) f x 4

) f x , 4 R

) f .

) f .

57

. 2016-2017

2.311 f R f x 0 x R .

) g x f x

) 0 f x 1 x R f 1 e , f 1 1 :

) g x ln f(x)

) g x ln f(x)

0x 1

2.312 f : R R IR .

2 3h 02f(x 3h) 5f(x) 3f(x 2h) 60lim

h x

2xlim f x 4x 9 2004 , ) 3

4f (x)x

) y 2x 2004 fC

) 2f x 2x 2004x

x 0,

2.313 ** f : R R : f(0) 0 21f (x)

3f (x) 1

x R .

. 3h(x) f (x) f(x) x , x IR .

) f 0,

) f ox 0

) 3xf(x)lim 0x

2.314 **H f 0, f f (x) f x 0

x 0 . :

) f 1 1 ) f f (x) x x 0 ) f 1 1 f x ln x .

2.315 f , , ) f , f

) f 1

) 1 2 , , 1 2 1 22f f 3

) ox , o2 f x

3

.

) f x 0 x , 1 2x , x , 1 2x x

1 2

1 2 3f '(x ) f '(x )

58


2.316 * f , ,

, . f f :

) 0x , : 0f( ) f( )f(x )

2

.

) 1x , 2x , 1 2x x : 1 21 1 2

f x f x f( ) f( )

) ox () .

2.317 f 1,e f 1 2 , f e e 1

1,4 . :

) 1 2x ,x 1,e 1 2x x 1 2f x f x 0

) 1,e f 0

) x 1,e 4o o o of x f (x ) 4f (x ) x ) y x e 2 f

1,e

) 1 2, 1,e 1 2 1 2f f 1

2.318 f R x R : 2 2f (3x 1) 4 4f(2x x 1) . :

A (1,4) : f ( ) 0

B f

f (1) f (4)

f (x) 0 R

2.319 x x 3 2x t t 6t 9t 4

t sec . :

A) ot 0

) ,

)

) .

2.320 f : 0, , :

f 0,

f 2 2

h 0

f 2 2016h f 2 2015hlim 0

h

59

. 2016-2017

2.321 f f [0,3] . :

31 2f 0 0 , E 32

. f

. f

. x 0

f xlim

x

B4. , f fC 1,2 .

( , )

2.322 f 2x ln x, x 0f x

, x 0

. 0

. i) f .

ii) f .

. f .

. x 1 : 2f(x 1) f(x) 1 ln xx

.

. : 2 2x

f(x 1) f(x)lim x(x 1) x

. ( )2f 1 h f 2h(1 ) , h 0 .

2.323 f : :

2 6f x x x f 2 0 f 2

. 3f x x ,x

. f 1f

. :

) fC , f 0 fC , . ) fC fC

. x, y x 0 3f x x

xx. A (0,0)

cm1 s . 0t 2 .

)

)

60


2.324 f 2x ln x , x> 0f x , x=0

. =0

. f

. f

. x 1 : 2f x 1 f x 1 ln xx

.

2 2x

f x 1 f xlim 3x

xx 1

2.325 G : 2,

f x 2

g x , x>2 x 2

:

. f 2 0 f 0x 2

. G 2,

. x x x xG 3 2 G 2 2 G e 2 G 2016 2 . G

. 3,4 2 22 11 14 G 7 12 2 f

2.326 f : 2x f x 3 x x , x

. 3 xx , x 0

f x x3 , x=0

.

B. xlim f x

. xf x e .

2.327 22xf x

x 1

, x

. f

1,0

. g : g x 3x 2 f x x

. A y 3x 2 g

. :

g x g x

g x g x 1x x 2

2g x x3 5lim lim14 3 xg x x 3x

61

. 2016-2017

2.328 f xx 1f(x)

e x

.

. f

. 1 1x ,f(x ) , 2 2x ,f(x ) 1 2x 1 x ,

f x x .

. , f( )

f x x .

. : 1 3xf(x)L lim

f (x)

, 2 x ln xxx x xL lime e

2.329 f : IR IR : f xe f x x 1 x IR R .

. f 1f

. f 0 0 f

M 0,f(0)

. f 2f x x , x IR

. 0,2 , 1 ef2

) f 1f

f 1f x

f x ,

2.330 2x x , x 2

f x4 x 2 2, -2 x 2

g x ln x , f

0x 2

. 3 8

. f

2,f 2 g N - f

0,7 .

62


2.331 f,g : 0, R

y x 1 0x 0 .

:

x 0, g x 0 , g x 0 xf x g x e x 1

g 0,

f 0 g 0 f 0 g 0 1

x 0, g x 0 g x 1

x 0, xe x 1 f x 1

f 0, f x x 1 x 0,

x

2x 0

f x elim

x

2.332 f :

2t 0lim f 1 t 1

t 0

f 1 2t f 1lim 4

t

. f 1,f 1

: y 2x 1

. d M (0,1).

,

t d.

. g :

2g f x x g 2x 5x ln x 6 x 0

) g B 2,g 2 xx yy

) x 2

f x 1 xg x 7lim

x 2

2.333 f A 0,1 1, 2xf x

ln x

x 0,1 1,

) f , f 0

) f

) 2x ln x , R f x

x 0,1 1, .

) 37f 125 61f 27 98f 64

2016 - 2017

M

&

222

: &

: 16.07

M.Ed. 2016

63

. 2016-17

3 3.01 f : R R f 2 e 2

xf 2x e x x R . f 1 0 .

3.02 f : R R F f R . f 1 1

f x F 2 x 1 x R ,

f

3.03 * F f : R R , :

2F x F x F x x R , 0 .

: ) F 0 F ,

) f x 0

R .

3.04 B f : R R

2xf x f x e x , x R f 0 0

3.05 f : R R F R . F x 0 , x R

F x F 2 x , x R ,

f x 0

3.06 N f x 2 x 1 x R

3.07

xln x e , x 1

f xe , x 1

3.08 f f x y f x f y , x, y R ,

f x cx x R

3.09 1980

R(t) = ke(ln2)t, t

1980. 1980 14

. . :

) t

1980,

)

1980-1990. ( ln2 0,7)

3.10

20,015x 2x 80 , x

.

1000 , :

) x

,

) , 30

60

.

3.11 t min

8t 5 dm3/min.

;

3.12 ' cm/sec t

(t) t(t 2) .

t 0

2cm ,

t 3

64


TO

3.13 H f R 3

0

f x dx 2

, 3

4

f x dx 1

10

4

f x dx 5 : 0

3

f u du ,

4

0

f x dx , 10

0

f x dx

3.14 2

1

f(x)dx 5 , 2

1

g(x)dx 3 , : ) 2

1

[2f(x) 3g(x)]dx ) 1

2

[g(x) 3f(x) 5]dx

3.15 4 22

2 22 4

2x 5 1dx 3 dx 4x 4 x 4

3.16 f R . ,, , R :

f x dx f x dx

f x dx f x dx

3.17 1

0f(x)dx 2

3

2g(x)dx 5 .

1 3

0 2f x g t dt dx

3.18 f R . ,, , R :

f x dx f x dx f x dx f x dx

f x dx f x dx 0

3.19 f R . ,, , R :

f x dx f x dx

f x dx f x dx

3.20 f R . ,, , R :

2 2

x x

f t e dx dt f t e dt dx

3.21 1

0f(x)dx 2

3

2g(x)dx 5 .

1 3

0 2f x g t dt dx

65

. 2016-17

x

F(x) f(t)dt

3.22 o :

x

1G x x ln tdt

2x 3x2

1K x t ln tdt

x 1 2

1

tF x dtln t t

2x

x

1N x dtln t

3.23 o :

2x

x

1M(x) dtln t

F x t x dt

, x x 1

5 2

2xH(x) t 4t dt

, 4x

x

ln t 1G x dt

2t 1

3.24 f R ,

1

0F x xf x t dt

12

0G x x tf xt dt

2

1

xH x f dtt

x 0,

3.25 o

2

t

G x xe dt

1x 2 t

0G x x e du

t

G xe dt

12 2

0K x x t xt dt

3.26 x y

2 21 2

1F(x) dt dy1 t t

3.27 N :

1

(2t) (2t)

1e dt e dt

, , R

3.28 F x

) x ln x

t

e 1F x ln tdt e dt

)

x xyu

6 6e du e dy

2 2

19 61F x tdt tdt

3.29 f R , :

) x t x

2 2

0 0 0

1t f u du dt x u f u du2

) x x u

0 0 0f u x u du u f(t)dt du

3.30 1 x t

0 1

e 1 dt dxt

3.31 :

x2

0F(x) 1 t dt

x4 2

0G(x) t dt

3.32 f ,

R :

y

x

f xf t dt

f y ,

x, y R

66


f x dx F x dx F F :

3.33 ) 1

2

0(1 x) dx )

2

1

1 dxx 5 x 2 )

3 23

26

x x 2 dx x )

1

50

dx dx6 x

3.34 ) 26

20

1 x dxx

)

2 2

21

x x x 1dxx )

2 3

1

x 2x 5dxx

) 2

1

x 1 dxx

3.35 A) 2

21

x x 2 dxx 2x

B) 1

2

0(2x 1) dx )

332

6

x 4 dxx

) 3 3

2

x 1 dxx 1

3.36 ) e

21

1 ln xdxx B) 2 20 2x x x x dx

) 2 x0 e ( x x)dx

3.37 ) e

x1

1 xdxe )

4 x 1x

1

2 2 x ln 2x

) 2

20

x x xdxx

g

g

f g(x) g (x)dx f u du - : u g x

3.38 A) 1

2 3

0x(x 1) dx B)

1 122

04x 2x 3 dx )

2e

e

ln(ln x) dxx ln x )

13

03x 1dx

3.39 ) 2

21

3 dxx x

) 1

340

1 dx(1 2x) )

1

20

2x 1 dxx x 2

)

3 3

0x xdx

3.40 )1

20

1 x dxx

)

1 2x x

x0

e 7e 6 dxe 3 )

5

2

e

e

1 dxx ln x ln(ln x) )

x1

2 xln 2

0e dx

3.41 ) 1

20

x 1 dxx 2x 3

) 1

x x0

1 dxe 1 e )

1

0

x dxln( x)

)

1

0xdx

3.42 ) 1

40

4x 5 dx(x 2)

)

1 x

0

3 dxx )

e

1

(ln x) dxx

) x1

x e

0e dx

3.43 ) 3

6

ln( x) dxx x

) 21

1x dx

x

) 2

13 x

0xe dx )

1

20

x dxx

67

. 2016-17

:

f x, ln x dx : u ln x

3.44 ) e

1

ln x dxx )

2e

e

1 dxxln x )

e

1

ln x 1 dxx )

e 2

1

(1 ln x) dxx

) e 3

1

1 ln x dxx

:

f x xdx

f x x dx u x u x

3.45 ) 3 26

x dx1 x

) 3

6

x 1 xdx

) 3

6

1 x xdx

2 x

) 1

3

0xdx

:

f P x , x dx . P x u x ,

3.46 ) 71

2 3

0x (2x 1) dx )

1 2

20

2x 3 dx(3x 1)

)

1

0(x 2) 3x 1dx )

12 3

0(x 2) (2x 1) dx

: P x < Q x .

Q x P x

P(x) Q (x)dx dx ln Q(x) ln Q() ln Q()

Q(x) Q(x)

= Q'(x) dxQ(x) = ln Q(x) c :

2

x dxx x

, 2 4 0 . :

3.47 2

21

2x 1 dxx 5x 6

.

22x 1f(x)

x 5x 6

fA R {2,3} 2x 1f(x)

(x 2)(x 3)

. A,B R ,

2x 1 A B(x 2)(x 3) x 2 x 3

, x R {2,3} , (A B 2)x 3A 2B 1 , x R {2,3} .

x R {2,3} , A B 2 0

3A 2B 1 0

,

A 5B 7

.

, 2 2 2

2 2

2 1 11 1 1

2x 1 5 7dx dx dx ... 5 ln x 2 7 ln x 3 ...x 5x 6 x 2 x 3

1 2 Q x x x ... x , : 1 2 1 2

P(x) A ...Q(x) x x x

.

3.48 )1

20

x 1 dxx 2x 7

) ) 4

23

x 1 dxx x 2

) 3

22

x 2 dxx 6x 7

) 3 2

32

x 2x 1 dxx x

:

2

P(x)dx

x x P(x) 2 2 4 0 ,

68


3.49 ) 1 2

20

x dxx 4 )

1 2

0

x x 2 dxx 3 )

1 2

20

x 2x 1 dxx 4x 3

)

3 3 2

22

x x 2x 1 dxx x

:

xx

f e , e dx : xu e , x ln u , xdu e dx ( )

3.50 ) 1 x

x0

e 1 dxe 2

)

1 x

2x0

e dxe 1 )

1

x0

1 dxe 1 )

1 2x

x0

e 1dxe

3.51 ) 1 x

2x0

e 1 dxe 4

)

1

2x0

2 dx1 e )

1 x

x x0

e dx(e 1)ln(e 1) )

1 x

x x0

e dxe e

f(x)g '(x)dx f(x)g(x) f (x)g(x)dx

1

x

P(x) dx P x x . x

x ln

.

e

3.52 ) 1

2 2x

0x e dx )

12 x

0x 3x e dx

2

P(x)(x )dx

P(x)(x )dx x ( x )

3.53 ) 1

2

0(3x x) ( 2x)dx )

1

02x (3x 1)dx )

2 2

0x 2xdx

3

x

I (x )dx ,

x

I (x )dx . x

, I . I .

3.54 ) 1

x

0e (3x 1)dx )

1x

0e xdx )

1x

0e 2xdx )

1x

02 xdx

4

f(x)ln(x )dx . f x ln x .

x .

3.55 ) e

1ln xdx )

e

1ln(2x 3)dx )

e2

1ln (2x 1)dx )

e

1

ln x dxx

3.56 ) e

1x ln(3x 4)dx )

e

1x ln(x 1)dx )

52

4ln( x x 9)dx

69

. 2016-17

:

f x, x dx : u x , u 0 , u x , 1u du dx v 1vudx du

3.57 ) 1

30

g lukeiou papagrigorakis 2016 17

Education