1. MNOŽINY

1. Jsou dány množiny A = {2; 4; 6}, B = {4; 6}. Urcete A u B, A n B, A - B, B - A a B~.

[Au B = {2;4; 6},AnB = {4;6},A-B = {2},B-A=0, B~ = {2}]2. Jsou dány množiny A = {x E N, 6 < x < IO}, B = {x E N, 8 ::; x ::; 9}. Urcete A u B,

AnB,A-B,B-Aa B~.

[Au B = {7;8; 9},AnB = {8;9},A-B = {7},B-A = 0,B~ = {7}]3. Jsou dány množiny S = {x E N; x::; 4} a T = {x E Z; -2 < x < I}. Urcete S u T, S n T.

rSuT = {-I. 0'1. 2' 3' 4} Sn T = 0 ]~ , , , , , ,4. Jsou dány množiny W = {x E Z; -2 ::;x < 4}, A = {x E No; x < 3}, B = {3; -2}. Urcete

AuB,AnB, A~, B~.

[A u B = {- 2; O;1;2;3}, AnB = 0, A~ = {- 2;-I;3},B~ = {-I; O;I;2}]5. Urcete sjednocení a prunik množin:

a) A = {-5. O. 3' 7} B = {O. l' 2. 3}, ", , , ,

b) A = {x E Z; X ~ O}, B = {x E Z; X::;O}

c) A = {x E Z; Ixl ~ 2}, B = {x E Z; Ixl ~ I}

[A u B = {x E Z;Jxl ~I},AnB = {xE Z;lxl ~2}]

d) A = {x E Z; X < -3}, B = {x E Z; Ixl ~ I}

[AUB= {xEZ;lxl ~I},AnB = {XEZ; x <-3}]

6. Urcete doplnek množiny A v množine B:

a) A = {3; 4; 5; 6; 7}, B = {x E No; x ::; IO} [A~ = {O;1; 2; 8; 9; lO}]

b) A = {x E Z; Ix! ~ 2}, B = Z [A~ = {-I; O;I}]

c)A= {x E No;x~2},B=No [A~ = {O;I}]

d) A = {x E Z; X~ 5}, B = {x E Z; X > 3} [A~ = {4}]

e) A = {x E R; x ::;-3}, B = {x E R; Ixl ~ 3} [A~ = {xER; x ~3}]

f) A = {x E R; Ixl ::;-1 },B = {x E R; x ::;-1 } [A~ = {xER; x ::;- 1}]7. Jsou dány množiny F = {xER; -2 < x < 2}, G = {x ER; -1 ::; x ::; 2}. Urcete FuG,

FnG, F-G.[F u G = {x E R; - 2 < x ::;2} , F n G = {x E R; - 1::;x < 2}, F - G = {x E R; - 2 < x < - I}]

8. Urcete všechny podmnožiny X množiny {I; 2; 3; 4}, pro než platí:a){I;3;4}nX={I;4}

b) {3} nX=0

c) {I;3} nX= {2;4}9. Urcete všechny množiny X, pro než je A u X = B, je-li:

a)A= {x E N;x::;2},B= {x E No;x<4} [X={0;I;2;3},{0;I;3},{0;2;3},{0;3}]

b) A = Z, B = No [nelze]

c)A=0,B={0} [X= {O}]

d) A = {I}, B = {2} [nelze]10. Urcete doplnkymnožinA=N, B = {x E N;x> I},C= {I} vmnožineN.

[A~ =0,B~ ={I},C~ ={xEN;x>I}]

[AuB= {-S.0.I'2'3'7 }AnB= {0'3 }], , , , , , ,

[A u B = Z,AnB = {O}]

[X = {I; 2; 4}, {I; 4}]

[X = {I}, {2}, {4}, {I; 2}, {I; 4}, {2; 4}, {I; 2; 4}, 0]

[nelze]

2. INTERVALY

1. Dané množiny zapište jako intervaly:a) A = {x E R; -4 < x ~ O}

b) B = {x E R; x < -Ji}

c)C= {x E R; 4 ~x~ J3}S

d) D = {x E R; x > 10}

e)E= {x E R;x~2,S}

f) F = {x E R; -8 < x < -S}

g) G = {x E R; x ~ -6,8}

h) H = {x E R; Ixl ~ 17}

i)I= {x E R; Ixl ~O}

[(-4;0)]

[(- 00;- Ji)]

[(: ;F3)][(10;00)]

[(- 00; 2,S)]

[(- 8; - S)]

[(- 6,8; 00)]

[(-17; 17)]

[(- 00; 00)]

[(-~; ~)]j) J = {x E R; x2 < 7}2. Dané intervaly zapište jako množiny:

a) (-6; 2)

b) (-00; 4)

c) (O;00)

d) (-8; O)

e) (-00;-3)

f)(~; 2~)

g) (-12,S; -10)3. Urcete i) sjednocení a ii) prunik intervalu:

a) (-00; 3), (2; 00) [i)(- 00;00),ii) (2; 3)]

b) (1; 00),(2; 00) [i)(1;00),ii) (2; 00)]

c)(2; 3), (1;00) [i)(1;oo),ii)(2;3)]

d) (-3; 2), (2; 4) [i)(-3; 2)u (2; 4),ii) 0]

e)(-oo; O),(O;I) [i)(- 00;1),ii) {O}]

f) (O;1), (O; I) [i)(O;1),ii) (0;1)]4. Rozhodnete, která z následujících množin je interval, a pak príslušný interval zapište:

a) {x E Z; X> O} [není]

b) {x E R; x > O} [(0;00)]

c){xER;-I~x~3} [(-1;3)]

d) Q [není]

e) R [(- 00;00)]

f) (3; 4) u (3; 4) [(3;4)]

g) {x E R; 1 < Ixl < 2} [není]

h) {x E R; Ixl ~ S} [není]

[{x E R; - 6 ~ x ~ 2}]

[{xER;x~4}]

[{xER;x>O}]

[{x E R; - 8 < x ~ O}]

[{x E R; x < - 3}]

[{XER;~~X<Ji}][{x E R; -12,S < x < -lO}]

3. MOCNINY S CELÝM EXPONENTEM

1 15a-3b2e 16a-2b-Ie315 . 8ab-3e-7 25a6b4e-4

a 3x+4b 4x+S a x-I b x+62. .

(j a2x-3b3x-2 a2x-3b3x-1

[2.3.5-la-12eIS]

[a2X+Sb3x]

3. e2~,27n 2;)'

4.(~

J3 .

(

2aSb-2

)

-2

e-3 d2 e-4 d3

5. 0,000048. 500008000 0,00060

6. 0,00001: 1000000,01 0,001

7. 35000 :70000000,00025 0,00001

8. 0,000000009. 2700003000 0,09

( J

2

( J

38a3b-2 ge2

9. 15a-2e3 : 4a4b-4

(2 . 92

)3 84

(3)

4

10. 33.42 '272' 2"

3y2 93.2r211. I 2:lY .5 27

2n-3 3n+4 3n+S 4n-S

12. xy. x y-n+1 n-2 3-2n I-Sn

X y X Y

[2 .3-I ]

[24a -4 b 7e]

[5.10-1]

[10-11]

[5-1 .10-3]

[32.10-6]

[212.3-8.5-2a22b-16e-12]

[rl.3]

[5-3]

[X8n-2 ylln ]

( J

-3

( J

-3a-3b e-2d3

13 - .-. e-Id2 a-IbS

15.36.73

(21,10

)

2

14. 2 :9.8 180.14

(-ut .(-45Y .70215.( )

3( )

4- 60 .182. - 75

16. (x::)' =(~2r

[a6b12e3 d-3]

[23 .3.5.73]

[22.3-1.5-6.72]

~3kZ-7k]

(

aSb-4

J

-3 .(

a-2b-3

J

-2

17. e-3d 2 . e4d -S

18. 0,002.400000: 90000,00008 0,003

[a-19b6e-17 d16]

[TI .10]

4. ODMOCNINY. MOCNINY S RACIONÁLNÍM EXPONENTEM

-l. 5WX 4 Y 7 X2 21. Yxy.j;Y [

5 15

]X 4 Y 28

1?í7. 5 1

. 6. ~X5 Y (5y-23

X -4 Vx2y

. 7. I a.Vb .~a..fbVa. Jh -;;-

[2M]

[2::]

[~][2f]

[)]

[a ~b -~][2-i ]

[3-&5~]

~ 2. J3(J-š+ F6)+ J-š(J3+ F6)- F6(J3+ Fs)1 1

~ 3. 22.if4.ifl6.3212

a2B 4. ~19a11-~

16

1 1 1 1- - - -

. 5. 2.82 -7.182 +5.722 -502

~ 3 1

~ 8. (%)4.(~)8 .(%)2 .18i

(15~ '27-~ ). 9. . jij9

(

1 1

)

-2 ',~254 . 98 ?v3ifi7

(

1 1

)

-3

103.8-2

. 10. . fiiJ4

(

1 1

)

-2"~

254 . 48 ~2Vs

3

[

1

, 11. 54 (25a-2b-2)ir[(5aWir~

[2~]

. 12. -J2ab?J4a2b4 V8a3b5 ~ha5b61{j4a2b2

[5&a'b' ]

[

9 35 17

]

24 a 12b 4

[a~ ]

~ 1

. 13a. a 2 2-1

. . a a4

(

2

J

~ .~

as a3 5 a3

5. MNOHOCLENY

1. Vypocítejte s využitím vzorcu:

a)(-a+~)'

b) (O,2x+O,4yY

c)(-ab2-c3Y

d) (3x + 2y Y + (3x - 2y Y - S(X2+ y2 )

e) (9p-2Y -(2p-1Y +4p

f) (3x - 2y)3

g)(x-;y)'

h) (O,2ab+10yY

i) (2X2 y3 Z - 3xy2 Z3y

2. Delte mnohocleny:

a) (- 4Sb5 -12b3 + 36b2 ): (- 6b2 )

b) (21x3 -31x2 +39x-6):(7x-1)

. c) (X4 -7X2 -9):(x-2)

[a2-a+~]

[O,O4X2 + O,16xy + O,16y2]

[a2b4 + 2ab2 c3 + C6]

[1Ox2]

[77p2 -2Sp+3]

[27x3 - 54x2y + 36xy2 - Sy3]

[3 2 16 2 64 3

]x -4x y+-xy --y

3 27

[O,OOSa3b3+ 1,2a2b2y + 60aby2 + 1000y3]

[Sx6y9z3 -36x5y8z5 + 54x4y7z7 -27x3y6Z9]

[Sb3+ 2b - 6]

[3X2 -4X+5-~

]7x-1

[

3 2 21

]x +2x -3x-6--

x-2

[5m2+ 3m -10]

[4X2 +2X+10-~

]x-3

[

3 a+2

]a +a-3+~ a -1

[

6 5 4 3 2 11

]a -a +a -a +a -a+ +-

a+1

. d) (15m4 -m3 -m2 +41m-70):(3m2 -2m+7)

( e) (4X3 -10x2 +4x-40):(x-3)

f) (a 5 - 3a 2 + 5): (a 2 - 1)

g) (a7 +2):(a+1)

3. Rozložte na soucin:

a) 1O(y-1)m+1- yb) rs - 6r + 24 - 4s

c) 2nz + ky + kz + 2ny

d) 3ac+ 2d - 3ad - 2ce) 6a3 -15a2 + 15b2 - 6ab2

f) (I -2Y - 12Z2

g) (2c+dY -(3d -lY

h) a4-b4

i) (y-1Y-4

[(y -lX10m -1)]

[(r - 4Xs - 6)]

[(z + y X2n + k)]

[(3a - 2Xc - d)]

[3(a - bXa + bX2a - 5)]

[(I - 2 + ft XI - 2 - ft)]

[(2c+ 4d -lX2c - 2d + 1)]

[(a - b Xa + bXa2 + b2)]

[(y + 1Xy -3)]

6. ÚPRAVY VÝRAZU

. 3.

1 a+6 .(

a+3 - a-2)7a+6 a+6 a-6

, 2.( 2X2 2 + 1)

:(1-~

)y -X x-y1

m+l+-m-l1

1+-m2-1

11'4. (1+1+~ ):(I-~

)1-1 1-1

[6~a]

[y:x]

[m+ 1]

[2~t']

~ 8.

x+y x-y-+-x-y x+y1 1-+-

(x+yr (x- yr1-1

1--6 1+1.. I(1- 1)

1+1+1

, 7. (a' :b' +bHC1, + b1,}:: :::]2 b 2 2 b 2

a ~+2a 2b- a +b + a1 1 1 1-+-b a

[X2- y2 ]. 5.

[1]

[

ab 2

]a-b

[a 2 + b 2 ]b a

1 [x' ~:+ 1]~ 9.

X

x-X

x--I-x

.. 10. a-e . a3 -e3 .(I+~_I+e

): e(l+e)-a

a2 + ae+ e2 a2b -be2 a -e e be

11 2X2 -2x+2. x3 +18 . 2 . 2x -25 x -4x-5

~ 12[(

~+ 3a .a2 +ab+b2)

. 2a+b

]

.~. a - b a 3 - b3 a + b . a 2 + 2ab + b 2 a + b

r4 - S4

[(

S2

J(

2r r2

J]13. r2s2 : 1+7 1--;+7

14 X .[(

X+2

)3. X(X2 +4X+4)

]. 3x + 6 x - 2 . 3x2 -12x + 12

[a:e]

[X~5]

[a~b]

[~

]r-s

[x~2]

58. LIMITA FUNKCE

1 1. 3x + 4

, 1mx42 x2 + 1

2 I' sm x

, 1mX4~ 1+ cos x

4

[2]

[F2-1]

x2 -93, lim 2

x43 x - 2x - 3

4 1. x2 + x - 2

, 1mx4-2 x2 + 5x + 6

5 I' x + 3

, 1m ~x4-3 "1/ x + 4 - 1

6 1. 2 - .Jx - 3

, 1mx47 x2 - 49

7 1. .Jx + 13 - 2.Jx + 1

, 1mx43 X 2 - 9

[~]

[-3]

[2]

[- 516]

[- /6]

[~]

[2]

[8]

8, lim sin 2xX40 3x

9 I' 1- cos2x

, 1mX40 x2

10, lim sin 4xX40 ~"I/x+l-l

11. lim 1- cos 2x + tg 2XX40 'xsmx

12, lim 2x3 - x2 + 5X4OC> X

2+x-2

3

13,limx-3x+lX4OC> 2 - 2 3X -x

[3]

[00]

2

14, lim x -2x+5X4OC> 2x 3 2-x +4

15, lim 2x3 + 5X4-OC>x2-5

2

16, lim x + 7x - 44x44 X

26- x+8

17,~ 9-x2éYFx -3

18 I' 3tg2 x,lm-X40 2X2

[-1]

[O]

19 I' cos x - smx

, 1mX4~ cos 2x

4

[- 00]

[1;]

~E./

[%]

[~]