Výsledky některých cvičení jsou uvedeny za příklady v ...Sudá funkce Funkce ( ) je sudá...

Vyacutesledky některyacutech cvičeniacute jsou uvedeny za přiacuteklady v zaacutevorkaacutech [ ] ( )

Funkce 2

Funkce definičniacute obor obor hodnot graf funkce 3

Lineaacuterniacute funkce 6

Kvadratickeacute funkce 8

Nepřiacutemaacute uacuteměrnost a exponenciaacutelniacute funkce 10

Logaritmickaacute funkce 12

Goniometrickeacute funkce 15

Vztahy mezi goniometrickyacutemi funkcemi 18

Goniometrickeacute rovnice 18

Posloupnosti 19

Aritmetickaacute posloupnost 19

Geometrickaacute posloupnost 21

Užitiacute GP složeneacute uacuterokovaacuteniacute 22

Mgr Pavel Viskup str 2

Funkce

Definice

funkce

Na množině čiacutesel 119863 je definovanaacute funkce je-li daacuten předpis podle ktereacuteho je

každeacutemu x naacuteležiacuteciacutemu do množiny D přiřazeno praacutevě jedno čiacuteslo y

Značiacuteme 119962 = 119943(119961)

Proměnnaacute 119961 se označuje jako argument funkce (nezaacutevisle proměnnaacute)

Proměnnaacute 119962 je funkčniacute hodnota (zaacutevisle proměnnaacute) 119863(119891) nazyacutevaacuteme definičniacutem

oborem funkce Pokud neniacute při zadaacuteniacute funkce uveden definičniacute obor pak se za definičniacute

obor obvykle považuje množina všech reaacutelnyacutech čiacutesel pro něž maacute funkce smysl Množinu

všech čiacutesel 119910 nazyacutevaacuteme oborem hodnot daneacute funkce 119867(119891)

119909 isin 119863(119891) 119910 isin 119867(119891)

Zadaacuteniacute

funkce

1 Analyticky ndash rovniciacute 119910 = 119891(119909) 2 Tabulkou ndash vyacutečtem hodnot 3 Graficky ndash křivkou přiacutemkou body ndash v pravouacutehleacute soustavě souřadnic ndash 119874119909119910

Body funkce zapisujeme [119909 119910] Parita

funkce

V matematice se některeacute funkce označujiacute jako sudeacute některeacute jako licheacute funkce Takoveacute

funkce vykazujiacute jisteacute druhy symetrie Tato symetrie se nazyacutevaacute parita funkce Existuje však

mnoho funkciacute ktereacute nejsou ani licheacute ani sudeacute

Sudaacute funkce Funkce 119891(119909) je sudaacute funkce pokud pro všechna x platiacute

119943(119961) = 119943(minus119961) To praacutevě znamenaacute že graf sudeacute funkce je osově

souměrnyacute podle osy y Mezi sudeacute funkce patřiacute všechny mocninneacute

funkce se sudyacutem mocnitelem a takeacute y = cos x

Lichaacute funkce

Funkce 119891(119909)je lichaacute funkce pokud pro všechna x platiacute

119943(minus119961) = minus 119943(119961) To praacutevě znamenaacute že graf licheacute funkce je středově

souměrnyacute podle počaacutetku soustavy souřadnic Mezi licheacute funkce patřiacute

všechny mocninneacute funkce s lichyacutem mocnitelem takeacute sin x tg x cotg x

Monotonie Monotonie je vlastnost označujiacuteciacute zda je funkce v bodě či na daneacutem intervalu monotoacutenniacute

tzn zda je konstantniacute rostouciacute klesajiacuteciacute přiacutep nerostouciacute či neklesajiacuteciacute

Tato vlastnost byacutevaacute někdy označovaacutena jako monotoacutennost

Funkce je definovaacutena v intervalu 119868 pokud pro všechna 1199091 lt 1199092 z tohoto intervalu platiacute

119891(1199091) lt 119891(1199092) je funkce rostouciacute

119891(1199091) gt 119891(1199092) je funkce klesajiacuteciacute

119891(1199091) ge 119891(1199092) je funkce nerostouciacute

119891(1199091) le 119891(1199092) je funkce neklesajiacuteciacute

119891(1199091) = 119891(1199092) je funkce konstantniacute

Periodickaacute

funkce

V matematice je periodickou funkciacute takovaacute kteraacute opakuje sveacute hodnoty po určiteacute konečneacute periodě

jejiacute proměnneacute 119961 Pro funkce to znamenaacute že celyacute graf lze vytvořit pomociacute kopiacuterovaacuteniacute určiteacute čaacutesti

opakovaneacute v pravidelnyacutech intervalech Přesněji řekneme že funkce f je periodickaacute s periodou t

jestliže 119891(119909 + 119905) = 119891(119909)

U každeacute funkce zadaneacute tabulkou určete paritu monotonii a periodičnost funkce

x ndash3 ndash2 ndash1 0 1 2 3

y ndash5 ndash3 ndash1 1 3 5 7

x ndash2 ndash1 0 1 2 3

y 10 15 20 10 15 20


y 5 3 1 0 1 3 5




Pravouacutehlyacute systeacutem souřadnic Oxy funkce definičniacute obor obor hodnot graf funkce

1 Naryacutesujte souřadnicovyacute systeacutem Oxy a pak body A[ndash1 2] B[2 ndash1] C[0 ndash3] D[ndash15 0]

2 Rozhodněte ve ktereacutem kvadrantu se nachaacuteziacute body A[02radic2] B[ndash1 ndash5] C[2 ndash1] D[ndash2 3] 119896119907119886119889119903119886119899119905119910 119860 1 119861 3 119862 4 119863 2

3 Přiřaďte každeacutemu bodu spraacutevnou variantu odpovědi

A [ndash2 ndash1] 1 ležiacute v I kvadrantu A

B [ndash2 0] 2 ležiacute v II kvadrantu B

C [2 1] 3 ležiacute v III kvadrantu C

D [0 ndash1] 4 ležiacute v IV kvadrantu D

E [2 ndash1] 5 ležiacute na ose x E

F [ndash2 1] 6 ležiacute na ose y F

4 Vypište

souřadnice

všech bodů

z obraacutezku

5 Urči ve

ktereacutem intervalu je

funkce rostouciacute a

ve ktereacutem klesajiacuteciacute

Pokud maacute funkce

extreacutem urči jeho

souřadnice Ve

ktereacutem intervalu

D(f) maacute funkce

zaacuteporneacute hodnoty

6 Na obraacutezku je daacutena funkce

a) Jakou maacute funkce paritu

b) Určete zda naacutesledujiacuteciacute body ležiacute na grafu pod či nad

grafem A[05 ndash2] B[ndash1 ndash2] C[2 1]

D[ndash2 0] E[ndash05 ndash15] F[ndash3 0]

c) Určete definičniacute obor funkce

d) Určete obor hodnot funkce

e) Určete intervaly monotonie teacuteto funkce

f) Určete ve ktereacutem intervalu definičniacuteho oboru jsou

hodnoty funkce kladneacute

7 Odpoviacutedejte na naacutesledujiacuteciacute otaacutezky ndash ANO NE

Funkce je sudaacute

Funkce je konstantniacute

Pro 119909 isin langminus2 minus1rang maacute funkce kladneacute hodnoty

Pro 119909 isin langminus2 minus1rang je funkce klesajiacuteciacute

Pro 119909 isin langminus1 1rang maacute funkce zaacuteporneacute hodnoty

Pro 119909 isin lang0 1rang je funkce rostouciacute

Funkce protiacutenaacute osu x v jednom bodě

Funkce protiacutenaacute osu y v jednom bodě


Funkce je sudaacute

Pro 119909 isin langminus2 1rang maacute funkce kladneacute hodnoty

Pro 119909 isin lang0 4rang je funkce klesajiacuteciacute

Pro 119909 isin lang3 6rang maacute funkce zaacuteporneacute hodnoty

Pro 119909 isin langminus5 minus4rang je funkce rostouciacute




9 Určete ktereacute z bodů A B C D jsou body funkce

a) 2

42

x

xy A[ndash2 ndash1] B[7 3] C[ndash1 1] D[3 radic5]

b) 32

102

x

xy A[0 radic3] B[4 radic12] C[1 3] D[2 radic6]

c) 2

22

x

xy A[ndash2 15] B[5 3] C[1 radic3] D[3 radic11]

d) 22 xxy A[0 radic2] B[ndash1 0] C[2 0] D[ndash2 3]

e) 2

123

xy A[0 ndash6] B[ndash2 9] C[2 ndash3] D[4 0]

f) 1522 xxy A[0 ndash15] B[ndash3 0] C[5 0] D[4 ndash1]

10 U všech funkciacute zjistěte funkčniacute hodnotu v bodě f (ndash1) a f (1)

a) 5

42

xy 21)1(40)1( ff

b) 8)5(2 2 xy 24)1(64)1( ff

c) 12

3

xy 3)1(1)1( ff

d) 4)3( 2 xy 12)1(0)1( ff

e) 2)2(

273

xy 29)1(3)1( ff

f) 1)2( 3 xy 28)1(2)1( ff

g) 9)1( 2 xy 9)1(5)1( ff

h) 11

4

xy 0)1(0)1( ff

i) 2)2(

1

xy

1)1(9

1)1( ff

j) 22

3

xy 1)1(5)1( ff

11 Zjisti zda u všech zadanyacutech funkciacute patřiacute čiacuteslo 10 do oboru hodnot

Definičniacute obor u všech funkciacute 119863(119891) = ⟨2 infin)

a) 2

3

xy )(10 fH

b) 5

42

xy )(10 fH

c) 8)5(2 2 xy )(10 fH

d) 12

3

xy )(10 fH

e) 6)3( 2 xy )(10 fH

f) 2)1( 3 xy )(10 fH

g) 9)1( 2 xy )(10 fH

h) 111

4

xy )(10 fH

i) 9)1(

12

xy )(10 fH

j) 22

3

xy )(10 fH


12 Zjisti souřadnice průsečiacuteků grafu funkce s osami x a y

a) 2

3

xy 51003 YX

b) 5

42

xy

5

4002 YX

c) 8)5(2 2 xy 4200703 21 YXX

d) 11

2

xy 1001 YX

e) 4)3( 2 xy 500105 21 YXX

f) 2)2(

163

xy 4004 YX

g) 1)2( 3 xy 7001 YX

h) 9)1( 2 xy 800402 21 YXX

i) 1)1(

14

xy 000002 21 YXX

j) 2)2(

1

xy

4

10YneniacuteX

k) 22

4

xy 0000 YX

13 Určete D(f) ndash definičniacute obor funkce Nutno znaacutet řešeniacute lineaacuterniacutech nerovnic v podiacuteloveacutem a součinoveacutem tvaru řešeniacute kvadratickyacutech nerovnic

a) 1

56 8

x

xy [D(f) = R-1]

b) x

xy [D(f) = (0 infin)]

c) 2

206

xy [D(f) = R]

d) 246

6

x

xy [D(f) = R4]

e) 245

12

xx

y [D(f) = R3 ndash8]

f) 408 xy [D(f) = ⟨5 infin)]

g) 98

12

xx

y [D(f) = R ndash9 1]

xy 7 [D(f) = (minusinfin 7⟩]

h) 1

1

xy [D(f) = (ndash1)]

i) 2

5


Konstantniacute

funkce

119962 = 119939

119863(119891) = 119877 119867(119891) = 119887 Funkce jejiacutež hodnota je na celeacutem oboru hodnot stejnaacute

tedy konstantniacute

b ndash je libovolneacute reaacutelneacute čiacuteslo Konstantniacute funkce neniacute ani

rostouciacute ani klesajiacuteciacute Konstantniacute funkce je sudaacute

Konstantniacute funkce je omezenaacute shora i zdola

Grafem konstantniacute funkce je přiacutemka rovnoběžnaacute s osou x

1 Na obraacutezku je znaacutezorněna konstantniacute funkce Určete definičniacute obor teacuteto

funkce obor hodnot Zapište funkci rovniciacute Rozhodněte zdali je funkce sudaacute

nebo lichaacute Určete kolik maacute funkce průsečiacuteků s osami x y


Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877

Lineaacuterniacute funkce je rostouciacute pro 119886 gt 0 a klesajiacuteciacute pro 119886 lt0

Lineaacuterniacute funkce neniacute ani sudaacute ani lichaacute

Lineaacuterniacute funkce neniacute omezenaacute ani shora ani zdola

Grafem lineaacuterniacute funkce je přiacutemka

Přiacutemaacute

uacuteměrnost Zvlaacuteštniacute přiacutepad lineaacuterniacute funkce 119962 = 119948119961

k je koeficient přiacutemeacute uacuteměrnosti či konstanta uacuteměrnosti

Lineaacuterniacute funkce

1 Určete koeficienty a b a zapište naacutezvy funkciacute určenyacutech rovnicemi uveďte zda je funkce určenaacute danou

rovniciacute rostouciacute klesajiacuteciacute nebo konstantniacute

a) y = 3x + 5 b) y = 05x c) y = ndash2 d) y = 1 ndash 3x e) y = 6x ndash 1

f) y = ndash3x + 2 g) y = ndash05x h) y = 5 i) y = 3

2x + 1

2 Rovnice funkciacute převeďte na tvar převeďte rovnici funkce f na tvar y = ax + b Zapište koeficienty a b

a) 2x + y = 0 b) x ndash y = 0 c) y ndash x + 5 = 0 d) 3

y ndash 1 = 0 e) ndash3x + y = 1

f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x

3 Sestrojte grafy funkciacute určenyacutech rovnicemi

a) y = x b) y = ndash x c) y = x + 1 d) y = x ndash 2 e) y = ndash x + 3 f) y = ndash x ndash 2

4 Sestrojte grafy funkciacute nebo jejich čaacutestiacute pro uvedeneacute definičniacute obory K daneacutemu definičniacutemu oboru určete

měřeniacutem v grafu obor hodnot funkce H(f) Uveďte zda je funkce rostouciacute klesajiacuteciacute nebo konstantniacute Funkce

jsou určeny rovniciacute

a) 119910 = minus119909 119863(119891) = (minusinfin 0rang b) 119910 = 119909 + 2 119863(119891) = 119877

c) 119910 =119909

2 119863(119891) = (0 4rang d) 119910 = 3 minus 119909 119863(119891) = langminus3 1)


měřeniacutem v grafu obor hodnot funkce H(f) Uveďte zda je funkce rostouciacute klesajiacuteciacute nebo konstantniacute

Funkce jsou určeny rovnicemi

a) y = 3x ndash 1 55)( fD b) y = ndash 2x + 2 3)( fD

c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3

xy RfD )( f) xy 51 RfD )(

6 Funkce je určena rovniciacute y = ndash4x + 3

Vypočtěte funkčniacute hodnoty f(0) f(1) f(3) f(10) f(ndash1) f(ndash3) f(minus05) f(ndash15) f(05)

7 Funkce f(x) s D(f) = R je daacutena rovnici y = 2x + 5

a) vypočiacutetejte hodnoty funkce f (2) f(0) f (ndash1)

b) doplňte naacutesledujiacuteciacute tabulku

c) Vypočiacutetejte souřadnice průsečiacuteku grafu funkce f se souřadnicovyacutemi osami (pokud existujiacute)

d) Sestrojte graf funkce f

e) Určete pro kteraacute Rx maacute funkce f nezaacuteporneacute hodnoty

x 3 05

y 7 1


8 Funkce f(x) s 119863(119891) = langminus1 5rang je daacutena rovnici 119962 =minus119961+120785

120784

a) vypočiacutetejte hodnoty funkce f (2) f (ndash1)




e) Určete pro kteraacute Rx maacute funkce f zaacuteporneacute hodnoty

9 Zjistěte vyacutepočtem jakaacute je vzaacutejemnaacute poloha bodů a grafů funkciacute určenyacutech rovnicemi (zda bod je bodem

funkce) 4

3 xy

A3 0 Bndash3 1 C6 ndash05


funkce) 2

1x

y A10 6 B2 2 C05 35


funkce) 119910 = 4119909 minus 5 A[0 3] B[1 ndash1] C[ndash2 ndash13] D[ndash3 17]


funkce) y = 05x + 1 A[2 3] B[ndash2 7] C[0 1]


funkce) y = ndash x + 3 A[ndash1 7] B[ndash2 7] C[1 4]

14 V rovnici funkce určete

a) čiacuteslo a tak aby bod M 1 ndash1 ležel na grafu funkce určeneacute rovniciacute y = ax + 4

b) čiacuteslo b tak aby graf funkce určeneacute rovniciacute 119962 =120785119961

120784+ 119939 prochaacutezel bodem Pndash4 2

15 Funkce je určena rovniciacute y = ax + 6 Vypočtěte čiacuteslo a tak aby graf funkce prochaacutezel danyacutem bodem

a) G 5 11 b) H ndash25 9 c) J 05 ndash5 d) M 2 0

16 Bod A [1 5] je bodem funkce y = ax + 4 Vypočiacutetejte koeficient a

17 Bod Z [2 5] je bodem funkce y = 3x + b Vypočiacutetejte koeficient b

18 Bod X [4 7] je bodem funkce y = ax ndash 3 Vypočiacutetejte koeficient a

19 Bod M [8 12] je bodem funkce y = ax + 4 Vypočiacutetejte koeficient a

20 Bod A [ndash1 5] je bodem funkce y = ndash 3x + b Vypočiacutetejte koeficient b

21 Zjisti rovnici lineaacuterniacute funkce kteraacute je zadaacutena dvěma body A[2 3] B[ndash2 4]

22 Zjisti rovnici lineaacuterniacute funkce kteraacute je zadaacutena dvěma body X[0 8] Y[ndash4 0]

23 Zjisti rovnici lineaacuterniacute funkce kteraacute je zadaacutena dvěma body M[5 5] N[1 1]

24 Vyjaacutedřete naacutesledujiacuteciacute zaacutevislosti jako funkce a zapište je rovniciacute funkce

a) Zaacutevislost obvodu čtverce na deacutelce jeho strany

b) Zaacutevislost deacutelky draacutetu na teplotě jestliže se draacutet o deacutelce 120 m při ohřaacutetiacute o 1 degC prodloužiacute o 0014 m

c) Zaacutevislost stavu krmiva na čase jestliže se ze zaacutesoby 10 tun denně zkrmiacute 280 kg

d) Zaacutevislost ujeteacute draacutehy vlaku na čase jestliže při vyacutejezdu ze stanice měl již za sebou ujetyacutech 60 km a daacutele

jel průměrnou rychlostiacute 30 kmh

27 V zemědělskeacutem zaacutevodě je zaacutesoba 2 000 litrů nafty Denně se z niacute pro provoz vozidel spotřebuje 150 litrů

Zapište rovniciacute zaacutevislost stavu zaacutesoby nafty na počtu dniacute Sestrojte graf teacuteto zaacutevislosti Z grafu určete

Na kolik dnů nafta vystačiacute Jakaacute bude zaacutesoba po osmi dnech Kolikaacutetyacute den musiacute byacutet objednaacutena novaacute nafta

objednaacutevaacute-li se při poklesu zaacutesoby na čtvrtinu původniacuteho množstviacute

28 Průměrnaacute spotřeba Škoda Superb je 7 litrů benzinu na 100 kilometrů Před cestou maacute řidič v naacutedrži 38 litrů

a) Sestavte rovnici zaacutevislosti množstviacute benzinu v naacutedrži (v litrech) na počtu ujetyacutech kilometrů

b) Po kolika ujetyacutech kilometrech zbyacutevaacute řidiči v naacutedrži ještě 5 litrů

x 4

y 025 1


29 U všech funkciacute zjistěte rovnice lineaacuterniacutech funkciacute Určete zda jsou rostouciacute či klesajiacuteciacute

Vypište průsečiacuteky funkciacute s osami

a) b) c)

d)

a) klesajiacuteciacute y = ndash2x + 2 [10] [02] b) rostouciacute y = 2x ndash 1

[050] [0ndash1] c) klesajiacuteciacute y = minus3119909+1

2 [

1

30] [005]

d) rostouciacute119910 =119909minus3

4 [30] [0

minus3

4]

Kvadratickaacute rovnice ndash opakovaacuteniacute Obecnaacute rovnice 119938119961120784 + 119939119961 + 119940 = 120782

Koeficienty rovnice 119886 119887 119888 isin 119877 119886 ne 0

Neznaacutemaacute je 119909 Kořeny rovnice jsou pak 119961120783 119961120784

1198861199092 ndash kvadratickyacute člen 119887119909 ndash lineaacuterniacute člen 119888 ndash absolutniacute člen

Diskriminant kvadratickeacute rovnice

119915 = 119939120784 minus 120786119938119940 119863 gt 0 ndash rovnice maacute 2 kořeny (2 řešeniacute)

119863 = 0 ndash rovnice maacute 1 dvojnaacutesobnyacute kořen (1 řešeniacute) 119863 lt 0 ndash rovnice nemaacute kořen (nemaacute řešeniacute)

Kořeny kvadratickeacute rovnice

119961120783120784 =minus119939 plusmn radic119915

120784119938

119961120783 =minus119939+radic119915

120784119938 Pokud je D = 0 použijeme 119961 =

minus119939

120784119938

119961120784 =minus119939minusradic119915

120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877

Kvadratickaacute funkce je sudaacute

Kvadratickaacute funkce je pro 119886 gt 0 omezenaacute zdola

Kvadratickaacute funkce je pro 119886 lt 0 omezenaacute shora

Grafem kvadratickeacute funkce je parabola

119938 gt 120782

119938 lt 120782


Kvadratickaacute funkce nepřiacutemaacute uacuteměrnost 1 Stanovte průsečiacuteky grafu funkce s osami x y

a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy

2 Zjistěte průsečiacuteky grafu funkce s osami x y pak upravte rovnici kvadratickeacute funkce tak abyste určili

souřadnice vrcholu Poteacute graf funkce načrtněte

a) y = x2 ndash 10x + 28 b) y = x2 ndash 8x + 18

c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12

3 Načrtni graf funkce a u všech funkciacute zjisti funkčniacute hodnotu f(2) a průsečiacutek s osou y

2)1( 2 xy 4)2( 2 xy 3)4( 2 xy

4 Načrtni graf funkce a u všech funkciacute zjisti funkčniacute hodnotu f(ndash1) a průsečiacutek s osou y

1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy

5 Vypočiacutetejte souřadnice průsečiacuteků kvadratickeacute a lineaacuterniacute funkce

a) 1198911 119910 = 1199092 minus 4119909 + 3 119886 1198912 119910 = 2119909 minus 5 A[4 3] B[2 minus1]

b) 1198911 119910 = 1199092 + 2119909 minus 3 119886 1198912 119910 = minus2119909 + 2 A[1 0] B[minus5 12]

c) 1198911 119910 = 1199092 minus 119909 119886 1198912 119910 = 3 minus 3119909 A[minus3 12] B[1 0]

d) 1198911 119910 = 1199092 minus 1 119886 1198912 119910 = minus119909 + 1 A[minus2 3] B[1 0]

e) 1198911 119910 = minus1199092 + 2 119886 1198912 119910 = 119909 A[1 1] B[minus1 minus1]

6 Určete souřadnice průsečiacuteků s osami napište rovnice kvadratickyacutech funkciacute ve tvaru y = (x ndash a)2 + b


Nepřiacutemaacute

uacuteměrnost 119962 =

119948

119961 119896 isin 119877 minus koeficient funkce 119896 ne 0

119863(119891) = 1198770 (všechna čiacutesla kromě nuly)

Funkce je lichaacute Grafem funkce je hyperbola kteraacute maacute dvě

větve

Pro 119896 gt 0 ležiacute větve hyperboly v I a III kvadrantu a je

klesajiacuteciacute

Pro 119896 lt 0 ležiacute větve hyperboly v II a IV kvadrantu a je

rostouciacute

Nepřiacutemaacute uacuteměrnost

1 Načrtněte graf funkce a u všech funkciacute zjisti funkčniacute hodnotu f(2) a průsečiacutek s osou x a y

2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy

3 Určete koeficient k funkce pokud viacutete že bodem funkce je A [2 4] a zjistěte průsečiacutek s osou x a y

1

x

ky Funkci načrtněte

4 Určete koeficient k funkce pokud viacutete že bodem funkce je A [ndash1 3] a zjistěte průsečiacutek s osou x a y

1x


Exponenciaacutelniacute

funkce

119962 = 119938119961

119886 isin 119877 minus koeficient 119886 gt 0 119886 ne 0 1

119863(119891) = 119877

119867(119891) = (0 infin) Exponenciaacutelniacute funkce neniacute ani sudaacute ani lichaacute

a je zdola omezenaacute Prochaacuteziacute vždy bodem

[0 1] protože 1198860 = 1

Funkce je pro 119886 isin (1 infin) rostouciacute

Funkce je pro 119886 isin (0 1) klesajiacuteciacute

Grafem funkce je exponenciaacutela

119886 isin (1 infin)

119886 isin (0 1)

119948 gt 120782

119948 lt 120782


Exponenciaacutelniacute funkce

1 Na zaacutekladě naacutečrtku grafu exponenciaacutelniacute funkce rozhodni o pravdivosti tvrzeniacute

a) 05ndash2 gt1 ano b) 5025 lt1 ne c) 0603 lt1 ano d) 35ndash05 gt1 ne

e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne

2 Využijte vlastnostiacute exponenciaacutelniacutech funkciacute a porovnejte exponenty p a r ndash načrtněte graf

a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]

3 Pro kteraacute a je exp funkce rostouciacute

xa

y

2

13

3

1a

4 Pro kteraacute a je exp funkce klesajiacuteciacute

xa

y

5

6 16 a

5 Řešte exponenciaacutelniacute rovnice proveďte zkoušku

27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x

6 Řeš exponenciaacutelniacute rovnice proveďte zkoušku (použijte substituci např 2x = y)

a) 2x+2 ndash 2x = 24 [x = 3] b) 52x ndash 35x = 10 [x = 1]

c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]

e) 32x+1 ndash 3 3x+2 = 3x ndash 9 [x1 = ndash1 x2 = 2] f) 4x + 7 2xndash2 = 05 [x = ndash2]

g) 25x = 02 ndash 4 5xndash1 [x = ndash1]

7 Přiřaďte rovnici k odpoviacutedajiacuteciacutemu grafu

I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce

119962 = 119845119848119840119938 119961 ndash čteme logaritmus 119961 při zaacutekladu 119938


119863(119891) = (0 infin)

119867(119891) = 119877 Tato funkce je INVERZNIacute k exponenciaacutelniacute funkci

pokud zaměniacuteme x za y a dostaacutevaacuteme definici

logaritmu

Logaritmus je exponent na kteryacute musiacuteme umocnit

zaacuteklad abychom ziacuteskali argument x



Dekadickyacute

logaritmus

Funkce dekadickyacute logaritmus rarr logaritmus při zaacutekladu 10

Dekadickyacute logaritmus značiacuteme log 119909 Zaacutepis log10 119909 se pro dekadickyacute logaritmus neužiacutevaacute

x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961

Vyacutepočet nedekadickeacuteho

logaritmu pomociacute

dekadickyacutech logaritmů

log119886 119909 =log 119909

log 119886

Některeacute vztahy vyplyacutevajiacuteciacute z definice logaritmu

1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909

Logaritmickaacute funkce

1 Určete logaritmus

a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]

c) 25log10000log 5 [2] d) 27log144log 312 [ndash1]

e) 64

1log010log2 8 [ndash6] f)

32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)


3 Určete čiacuteslo x y a (definice logaritmu)

a) log5 x = 2 b) 3log 2 x c) 416

1log a d) 1log3 x e) 3

1000

1log a

f) 38log a g) log11 11 = y h) log x = ndash3 i) y100

1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]

4 Určete definičniacute obor logaritmickeacute funkce

a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]

5 Načrtni graf log funkce zjisti zda body A B C ležiacute na grafu funkce

a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]

d) 3)102log( xy A[45 ndash1] B[0 ndash2] C[ndash45 ndash3]

6 Zjisti průsečiacuteky logaritmickeacute funkce s osami x y

a) 1log3 xy X[3 0] Y[neex]

b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]

7 Na zaacutekladě grafu log funkce rozhodni o pravdivosti tvrzeniacute

a) log2 7 log2 6 ano b) log05 3 log05 4 ne c) log02 075 0 ne

d) log5 8 log5 14 ano e) log3 11 0 ano

8 Určete logaritmus vyacuterazu (zlogaritmujte)

a) 3xy [ yx loglog3log ] b) b

a2 [ ba loglog2log ]

c) 3

4

b

a [ ba log3log4 ] d)

x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]


10 Vypočiacutetejte (nejprve odlogaritmujte použijte věty o logaritmech)

a) 9log4log 66 b) 5log40log 22 c) 25log40log

d) 4log2log54log 333 e) 12log6log 22 f) 100log25log 22

g) 8log500log2log 555 h) 5

1log

5

9log3

9

5log2

3

1

3

1

3

1

[119886) 2 119887) 3 119888) 3 119889) 3 119890) ndash 1 119891) ndash 2 119892) 3 ℎ) ndash 2 ] 11 Vyřešte rovnice

a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]

12 Je daacutena funkce 119910 = log1

2

(119909) Zjistěte z grafu

funkce zdali jsou vyacuteroky pravdiveacute či ne a) Pro 119909120598(05 1)jsou hodnoty funkce kladneacute

b) Pro 119909 gt 20 jsou 119891(119909) gt minus1

c) 119891(4) = 2

d) 119891(12) lt minus2

13 Je daacutena funkce 119910 = log2 119909 Zjistěte z grafu


b) Pro 119909120598(0 1) jsou 119891(119909) gt minus1

c) 119891(4) = 2

d) 119891(12) gt minus2

14 Je daacutena funkce 119910 = log 119909

Zjistěte z grafu funkce zdali jsou

vyacuteroky pravdiveacute či ne

a) log 3 lt log 5

b) log 300 gt log 250

c) log3

4lt log

1

5

d) log 03 lt log1

3


GONIOMETRICKEacute FUNKCE Jednotkovaacute

kružnice

kvadranty

V pravouacutehleacute soustavě souřadnic naryacutesujeme kružnici o poloměru 1 Rovina je

poloosami rozdělena na 4 kvadranty čiacuteslovaneacute v kladneacutem směru (proti směru

hodinovyacutech ručiček)

Obloukovaacute miacutera

radiaacuten

1 radiaacuten je středovyacute uacutehel kteryacute přiacuteslušiacute oblouku o stejneacute

deacutelce jako je poloměr kružnice Je to jednotkovyacute uacutehel při

měřeniacute v obloukoveacute miacuteře

Převod mezi miacuterou stupňovou a obloukovou

120654 je velikost uacutehlu v radiaacutenech a 120630 ve stupniacutech

Plnyacute uacutehel maacute 2π radiaacutenů ndash to je 360 stupňů

120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801

314asymp 57deg 1deg =

120587

180(119903119886119889)

Funkce

sinus

Sinus se definuje na jednotkoveacute kružnici (kružnici se

středem v počaacutetku a s poloměrem 1) Je-li α uacutehel

kteryacute maacute počaacutetečniacute rameno v kladneacute poloose x

je sin α roven y-oveacute souřadnici průsečiacuteku teacuteto

kružnice s koncovyacutem ramenem uacutehlu α jinak řečeno

rovnaacute se deacutelce kolmice spuštěneacute z tohoto bodu na

osu x

Grafem sinu v reaacutelneacutem oboru je sinusoida

Funkce je lichaacute 119956119946119951(ndash 119961) = ndash 119956119946119951 119961

Funkce je periodickaacute s periodou 360deg (2120587)

119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus

Kosinus se definuje na jednotkoveacute kružnici (kružnici se

středem v počaacutetku a s poloměrem 1) Je-li α uacutehel kteryacute

maacute počaacutetečniacute rameno v kladneacute poloose x

je cos α roven x-oveacute souřadnici průsečiacuteku teacuteto kružnice s

koncovyacutem ramenem uacutehlu α Grafem sinu v reaacutelneacutem

oboru je sinusoida posunutaacute po ose x o 90deg

Funkce je sudaacute 119940119952119956(ndash 119961) = 119940119952119956 119961


119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961

119915(119943) = 119929 minus 120791120782deg + 119948 120783120790120782deg pro uacutehly 90deg 270deg hellip neniacute funkce definovaacutena

119919(119943) = (minusinfin infin) V celeacutem definičniacutem oboru je funkce tangens rostouciacute Grafem je tangentoida

Funkce je lichaacute 119957119944(ndash 119961) = minus119957119944 119961

Funkce je periodickaacute s periodou 180deg (120587) 119905119892(119909 + 119896 180deg) = 119905119892 119909

Funkce

kotangens

119962 = 119940119952119957119944 119961

119915(119943) = 119929 minus 119948 120783120790120782deg pro uacutehly 0deg 180deg hellip neniacute funkce definovaacutena

119919(119943) = (minusinfin infin) V celeacutem definičniacutem oboru je funkce kotangens klesajiacuteciacute Grafem je kotangentoida


Funkce je periodickaacute s periodou 180deg (120587) 119888119900119905119892(119909 + 119896 180deg) =119888119900119905119892 119909

119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV

(0deg 90deg) (90deg 180deg) (180deg 270deg) (270deg 360deg)

sin x + + ndash ndash

cos x + ndash ndash +

tg x + ndash + ndash

cotg x + ndash + ndash X α = x α = 180deg ndash x α = x ndash 180deg α = 360deg ndash x

Hodnoty funkciacute uacutehlu x vyjaacutedřeneacute pomociacute hodnot uacutehlu α z I kvadrantu

α 0deg 30deg 45deg 60deg 90deg 180deg 270deg 360deg

0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0

Goniometrickeacute funkce 1 Zjistěte z tabulek bez použitiacute kalkulaacutetorů

sin 225deg = cos 135deg = tg 135deg = cotg 225deg = sin (ndash1080deg) = cos 855deg = tg (ndash45deg) = cotg (ndash765deg) =

sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =

2 Vypočtěte bez použitiacute kalkulaacutetorů

sin225degcos(ndash45deg) sin120587 ndash sin 240degcos135deg+ 4

6 [0]

cos135degsin225deg + cos210degsin120degndash cos240deg sin150deg= [0]

sin30degcos30deg ndash 2 sin45deg+ tg(ndash60deg) ndash 6cos360deg= [ )74

33 ]

3(cos45deg)2 ndash (sin60deg + tg30deg) ndash 2cos2

+ sin2120587 = [

12

7 ]


tg 45deg+ cotg 300deg+ tg 210deg+ cotg 225deg+ tg 135deg= [1]

3tg 120deg+ cotg 135degndash 2tg 225deg+ 2cos 300deg= [ndash2 ndash 3 3 ]

tg 210deg cotg 210degndash sin 240deg tg 240deg= [25]

119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]


Vztahy mezi goniometrickyacutemi funkcemi

119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783

1 Zjednodušte vyacuterazy

a) 1 minus 1198881199001199042119909 1199051198922119909 = [cos2x] b) 1199051198922119909 minus1

1198881199001199042119909= [ndash1]

c) (119904119894119899119909 minus 119888119900119904119909)2 + 2 119904119894119899119909 119888119900119904119909 minus 119905119892119909 119888119900119905119909 = [0] d) 119904119894119899119909minus1199041198941198993119909

1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]

2 Dokažte že platiacute

a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +

cos2119909 minussin2119909

sin 119909minuscos 119909= minus sin 119909

Goniometrickeacute rovnice

3 sin 119909 + 2 = 3 minus sin 119909 [30deg + k360deg 180deg + k360deg]

4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]

5 minus (cos 119909 + 4) = 3 cos 119909 [180deg + k180deg nebo + k ]

6 cos(minus119909) = 3 cos 119909 + radic2 [135deg + k360deg 225deg + k360deg]

7 3 cos 119909 = 2 sin2119909 [60deg + k360deg 300deg + k360deg]

8 2 sin 119909 = 2 sin2119909 minus 3 cos2119909 [90deg + k360deg 217deg + k360deg 323deg + k360deg]

9 sin 2119909 = (sin 119909 minus cos 119909)2 [15deg + k360deg 75deg + k360deg]

10 2 sin2119909 + 7 cos 119909 minus 5 = 0 [60deg + k360deg 300deg + k360deg]

11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]

12 cos2119909 minus cos 119909 = 0 [0deg 90deg 270deg + k360deg]

13 cos2 x ndash 2sinx +2 = 0 [90deg + k360deg]

14 6 1199041198941198992119909 + cos 119909 minus 5 = 0 [60deg 300deg 109deg 251deg + k360deg]

15 radic2 sin 119909 minus 2 sin 119909 cos 119909 = 0 [0deg + k180deg 45deg + k360deg 315deg + k360deg]

16 cos 119909 tg119909 + cos 119909 = 0 [90deg + k180deg 135deg + k180deg]

17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]

18 sin2119909 + 3 sin 119909 minus 2 = 2 [90deg + k360deg]


Posloupnosti

1 Určete členy posloupnosti a2 a3 a10

119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin

[ndash4 ndash9 ndash100] 119886119899 = (1198992minus4

2119899)

119899=1

infin

[05

6

24

5]

2 Určete prvniacute 3 členy posloupnosti (2119899 +119899

4) [225 45 875]

3 Určete kolikaacutetyacute člen posloupnosti maacute hodnotu 10

a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]

4 Vypočtěte hodnoty danyacutech členů posloupnosti daneacute rekurentně

119886119899+1 = minus1

2119886119899 minus 2 1198861 = 2 a2 a3 a4 = [ndash3 ndash05 ndash175]

119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost

119938119951 ndash n-tyacute člen posloupnosti

119938120783 ndash prvniacute člen posloupnosti

119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941

119941 minus diference neboli rozdiacutel dvou po sobě jdouciacutech členů aritmetickeacute posloupnosti

119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro

n-tyacute člen 119938119951 = 119938120783 + (119951 minus 120783) 119941

119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro

r-tyacute člen 119938119955 = 119938119956 + (119955 minus 119956) 119941

Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)

Aritmetickaacute posloupnost

1 V aritmetickeacute posloupnosti určete prvniacutech 5 členů je-li daacuteno

a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]

e) a7 = 22 a14 = ndash13 [ndash08 ndash03 02 07 12]

2 Určete 10 člen v AP je-li daacuteno a1 = ndash 20 a20 = ndash 675 [minus425]

3 V aritmetickeacute posloupnosti je daacuteno

a) a12 = 10 d = ndash2 určete prvniacute 4 členy [32 30 28 26]

b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]

c) a14 = 3 a23 =21 určete S15 [S15 = ndash135]


4 V aritmetickeacute posloupnosti je a1 = 6 S10 = 195 Určete a10 d a31 [a10 = 33 d = 3 a31 = 96]


6 Vypočiacutetejte součet členů aritmetickeacute posloupnosti když je daacuteno

a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]

c) a1 = ndash14 a6 = ndash15 S10 = [ndash149] d) a6 = ndash20 a16 = ndash60 S20 = [ndash760]

7 Vypočiacutetejte S8 když je daacuteno

a) a8 = 54 a4 = 26 (236) b) a7 = ndash100 a9 = ndash130 (ndash500)

8 Vypočiacutetejte a) S10 když je daacuteno a3 + a6 = 6 a5 + a7 = 0 [S10 = 10]

b) S5 když je daacuteno a1 + a5 =30 a3 + a4 =36 [S5 = 75]

9 Kolik členů maacute posloupnost pro kterou je daacuteno

a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]

10 V AP platiacute d = minus12 an = 15 Určete kolik prvniacutech členů maacute součet 456 [n = 8]

11 Vypočtěte součet všech členů konečneacute aritmetickeacute posloupnosti 2

3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]

12 Kolik prvniacutech členů AP daacutevaacute součet

a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]

13 Mezi čiacutesla 7 a 51 vložte tolik čiacutesel aby vznikla AP Součet danyacutech a vloženyacutech čiacutesel je 348 Určete vloženaacute

čiacutesla [a1 = 7 d = 4 vloženaacute čiacutesla 11 15 19 23 27 31 35 39 43 47]

14 V deviacutetičlenneacute aritmetickeacute posloupnosti je prostředniacute člen 25 a součet dvou posledniacutech je 85 Určete součet

všech členů posloupnosti [225]

15 Vypočtěte součet všech sudyacutech dvoucifernyacutech čiacutesel [2430]

16 Vypočtěte součet všech lichyacutech trojcifernyacutech čiacutesel [247 500]

17 Zeď je sestavena z krychliacute Uprostřed je největšiacute

krychle s hranou deacutelky 200 cm Vpravo i vlevo

od niacute se souměrně přidaacutevajiacute dalšiacute krychle jejichž

hrany se postupně zkracujiacute o 5 cm Zeď maacute na

obou konciacutech nejmenšiacute krychle s hranou deacutelky

20 cm Jak dlouhaacute je zeď v metrech

18 Posloupnost tvořiacute sedmnaacutect po sobě jdouciacutech přirozenyacutech lichyacutech čiacutesel seřazenyacutech vzestupně od nejmenšiacuteho

k největšiacutemu Prostředniacute člen a9 je čiacuteslo 23 O každeacutem z naacutesledujiacuteciacutech tvrzeniacute rozhodněte je-li pravdiveacute nebo

nepravdiveacute a) Rozdiacutel mezi dvěma sousedniacutemi členy je 1

b) a12 = 29

c) Všechny členy jsou většiacute než 5

d) Součet čtyř nejmenšiacutech členů je 40

19 Všechny schody majiacute šiacuteřku 45 cm Nejvyššiacute je prvniacute

schod každyacute naacutesledujiacuteciacute schod je o 05 cm nižšiacute

Prvniacute schod maacute vyacutešku 42 cm posledniacute jen 05 cm

20 Na rovneacutem draacutetě je navlečeno celkem 61 koraacutelků

Uprostřed řady je největšiacute s průměrem 20 mm Vedle něj jsou z každeacute strany dva koraacutelky s průměrem 19

mm potom dva koraacutelky s průměrem 18 mm atd V každeacute naacutesledujiacuteciacute dvojici se průměr koraacutelků o 1 mm

zmenšiacute Mezi koraacutelky nejsou žaacutedneacute mezery Jak dlouhaacute je řada koraacutelků

Jak dlouheacute a jak vysokeacute je

schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954

119954 ndash kvocient neboli podiacutel dvou po sobě jdouciacutech členů geometrickeacute posloupnosti

119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro

n-tyacute člen 119938119951 = 119938120783 119954119951minus120783

119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro

r-tyacute člen 119938119955 = 119938119956 119954119955minus119956

Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783

Geometrickaacute posloupnost

1 V GP je daacuteno a1 =16 q = 2

1 an =

8

1 Určete n Sn [n = 8 S8 =31

7

8]

2 a2 = 03 a4 = 0108 Vypočiacutetejte S5 [S5 = 11528]


4 a1 = ndash10 a4 = 80 Vypočiacutetejte S4 [S4 = 50]

5 Zjisti a1 a q pokud platiacute 2a1 ndash a3 = 20 2a2 ndash a4 = ndash40 [a1 = ndash10 q = ndash2]

6 a1 = 1 a4 = 0125 Kolik členů daacutevaacute součet Sn = 175 [n = 3]



9 V GP platiacute a) a1 + a2 + a3 = 35 a4 + a5 + a6 = 280 Urči a1 q [a1 = 5 q = 2]

b) a2 ndash a4 = 60 a1 ndash a3 = 15 Určete prvniacutech 5 členů [q = 4 a1 = ndash1 ndash4 ndash16 ndash64hellip]

c) a3 ndash a1 +16 = 0 a4 ndash a2 +48 = 0 Vypočti S3 [a1 = ndash2 q = 3 S3 = ndash26]

postup řešeniacute každyacute člen vyjaacutedřiacuteme pomoci 1 členu a kvocientu q řešiacuteme pak soustavu rovnic o dvou neznaacutemyacutech

10 Mezi čiacutesla 3

2 a 162 vložte 4 čiacutesla tak aby s danyacutemi tvořila po sobě jdouciacute členy GP [2 6 18 54]

11 Prvniacute člen šestičlenneacute GP je 5 posledniacute 160 Vypočtěte součet členů GP [315]

12 V GP je daacuteno a) a5 = 027 q = ndash3

1 určete a8 [a8 = ndash001]

b) a4 = ndash16 a5 = 32 určete prvniacutech 5 členů [2 ndash4 8 ndash16 32]

c) a1 = 6 q = 025 určete S4 [S4=255

32]

d) a1 = 9 q = 01 určete a8 S5 [a8 = 910ndash7 S5 = 99999]

e) a2 = ndash12 a5 = 96 určete q a8 S6 [q = 2 a8 = ndash768 S6 = 378]


13 GP tvořiacute 5 čiacutesel předposledniacute člen je 5 a prostředniacute 10 Vypočtěte součet

prvniacutech třiacute členů [70]

14 Obrazec je sestaven z podobnyacutech rovnoramennyacutech trojuacutehelniacuteků Sousedniacute

trojuacutehelniacuteky majiacute vždy jeden společnyacute bod a jejich vyacutešky na zaacutekladnu ležiacute na teacuteže

přiacutemce Nejmenšiacute trojuacutehelniacutek maacute deacutelku zaacutekladny 2 cm a velikost vyacutešky na

zaacutekladnu 1 cm Každyacute dalšiacute trojuacutehelniacutek maacute uvedeneacute rozměry dvakraacutet většiacute než

předchoziacute trojuacutehelniacutek

Složeneacute uacuterokovaacuteniacute

Na počaacutetku uacuterokovaciacuteho obdobiacute maacuteme hodnotu 119938120782

Hodnota vzroste (nebo klesne) na konci uacuterokovaciacuteho obdobiacute o uacuterok 119953

Na konci uacuterokovaciacuteho obdobiacute dostaneme hodnotu 119938119951

119951 ndash počet uacuterokovaciacutech obdobiacute Uacuterokovaciacutem obdobiacutem může byacutet rok čtvrtletiacute měsiacutec hellip

Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899

Pokles hodnoty 119886119899 = 1198860 (1 minus

119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899

Daň z uacuteroků Při vyplaacuteceniacute uacuteroků je z konečneacute sumy odečtena daň Proto musiacuteme před dosazeniacutem do vzorce sniacutežit uacuterok 119953 o přiacuteslušnou čaacutest Daň z uacuteroků 119941

Vyacuteslednyacute uacuterok 119901acute = 119901100minus119889

100= 119901 minus

119901119889

100

Užitiacute GP složeneacute uacuterokovaacuteniacute

1 Počet obyvatel města vzrostl za 12 let ze 75 000 na 94 800 O kolik se průměrně zvyšoval za jeden rok

2 Původniacute cena stroje byla 800 000 Kč Jakou cenu maacute stroj po 20 letech při 10 amortizaci [97 262 Kč]

3 Na uacutečtu s 6 uacuterokem maacuteme uloženo 10 000 Kč Na uacutečet již nebudeme nic uklaacutedat ani vybiacuterat

a) Kolik Kč bude na uacutečtu za 5 let [13 382Kč]

b) Za kolik let se čaacutestka 10000 Kč zdvojnaacutesobiacute [asi 12 let]

c) Na kolik by se musel vklad uložit aby se zdvojnaacutesobil za 5 let [p = 1487 ]

4 Deacutelky hran kvaacutedru tvořiacute po sobě jdouciacute členy geom posloupnosti Nejkratšiacute hrana měřiacute 2 cm Objem kvaacutedru

je jeden litr Zjisti deacutelku ostatniacutech hran [2 10 50]

5 Na VŠ se hlaacutesiacute 1920 uchazečů Přijiacutemaciacute řiacutezeniacute se konaacute v několika kolech tak že do dalšiacuteho kola postupuje

vždy polovina uchazečů Kolik musiacute škola uskutečnit kol aby přijala pouze 30 uchazečů [6]

6 Za kolik hodin se bakterie množiacuteciacute se děleniacutem rozmnožily z 5 000 na 1 280 000 Počet bakteriiacute se

zdvojnaacutesobiacute za hodinu [8]

7 Ve vesnici se za dva roky zvyacutešil počet obyvatel ze 100 na 121 Jakyacute byl průměrnyacute procentuaacutelniacute ročniacute

přiacuterůstek [10 ]

8 Ročniacute uacuterok je 4 Uacuterokovaciacute obdobiacute jsou 3 měsiacutece Za kolik let se naacutem vklad zdvojnaacutesobiacute [205 let]

9 Plat se zvyacutešil z 10 000 Kč na 13 400 za 6 let Jakyacute byl ročniacute naacuterůst [10 ]

10 Za kolik let se Janovi zvyacutešil vklad z 250 000 Kč na 388 000 Kč když ročniacute uacuterok byl 35 a zuacuteročovaniacute

obdobiacute bylo 1 rok (Daň z uacuteroku je 15 ) [15 let]

11 Za kolik let vzrostla cena tuny obiliacute z 3 000 Kč na 5 125 Kč když bereme ročniacute naacuterůst 55 [10 let]

12 Jakyacute byl ročniacute uacuterok v bance když za 3 roky jsme uspořili ze 50 000 Kč na 56 650 Kč [5 ]


Funkce

Definice

funkce

Na množině čiacutesel 119863 je definovanaacute funkce je-li daacuten předpis podle ktereacuteho je

každeacutemu x naacuteležiacuteciacutemu do množiny D přiřazeno praacutevě jedno čiacuteslo y

Značiacuteme 119962 = 119943(119961)

Proměnnaacute 119961 se označuje jako argument funkce (nezaacutevisle proměnnaacute)

Proměnnaacute 119962 je funkčniacute hodnota (zaacutevisle proměnnaacute) 119863(119891) nazyacutevaacuteme definičniacutem

oborem funkce Pokud neniacute při zadaacuteniacute funkce uveden definičniacute obor pak se za definičniacute

obor obvykle považuje množina všech reaacutelnyacutech čiacutesel pro něž maacute funkce smysl Množinu

všech čiacutesel 119910 nazyacutevaacuteme oborem hodnot daneacute funkce 119867(119891)

119909 isin 119863(119891) 119910 isin 119867(119891)

Zadaacuteniacute

funkce

1 Analyticky ndash rovniciacute 119910 = 119891(119909) 2 Tabulkou ndash vyacutečtem hodnot 3 Graficky ndash křivkou přiacutemkou body ndash v pravouacutehleacute soustavě souřadnic ndash 119874119909119910

Body funkce zapisujeme [119909 119910] Parita

funkce

V matematice se některeacute funkce označujiacute jako sudeacute některeacute jako licheacute funkce Takoveacute

funkce vykazujiacute jisteacute druhy symetrie Tato symetrie se nazyacutevaacute parita funkce Existuje však

mnoho funkciacute ktereacute nejsou ani licheacute ani sudeacute

Sudaacute funkce Funkce 119891(119909) je sudaacute funkce pokud pro všechna x platiacute

119943(119961) = 119943(minus119961) To praacutevě znamenaacute že graf sudeacute funkce je osově

souměrnyacute podle osy y Mezi sudeacute funkce patřiacute všechny mocninneacute

funkce se sudyacutem mocnitelem a takeacute y = cos x

Lichaacute funkce

Funkce 119891(119909)je lichaacute funkce pokud pro všechna x platiacute

119943(minus119961) = minus 119943(119961) To praacutevě znamenaacute že graf licheacute funkce je středově

souměrnyacute podle počaacutetku soustavy souřadnic Mezi licheacute funkce patřiacute

všechny mocninneacute funkce s lichyacutem mocnitelem takeacute sin x tg x cotg x

Monotonie Monotonie je vlastnost označujiacuteciacute zda je funkce v bodě či na daneacutem intervalu monotoacutenniacute

tzn zda je konstantniacute rostouciacute klesajiacuteciacute přiacutep nerostouciacute či neklesajiacuteciacute

Tato vlastnost byacutevaacute někdy označovaacutena jako monotoacutennost

Funkce je definovaacutena v intervalu 119868 pokud pro všechna 1199091 lt 1199092 z tohoto intervalu platiacute

119891(1199091) lt 119891(1199092) je funkce rostouciacute

119891(1199091) gt 119891(1199092) je funkce klesajiacuteciacute

119891(1199091) ge 119891(1199092) je funkce nerostouciacute

119891(1199091) le 119891(1199092) je funkce neklesajiacuteciacute

119891(1199091) = 119891(1199092) je funkce konstantniacute

Periodickaacute

funkce

V matematice je periodickou funkciacute takovaacute kteraacute opakuje sveacute hodnoty po určiteacute konečneacute periodě

jejiacute proměnneacute 119961 Pro funkce to znamenaacute že celyacute graf lze vytvořit pomociacute kopiacuterovaacuteniacute určiteacute čaacutesti

opakovaneacute v pravidelnyacutech intervalech Přesněji řekneme že funkce f je periodickaacute s periodou t

jestliže 119891(119909 + 119905) = 119891(119909)

U každeacute funkce zadaneacute tabulkou určete paritu monotonii a periodičnost funkce



x ndash2 ndash1 0 1 2 3

y 10 15 20 10 15 20


y 5 3 1 0 1 3 5














4 Vypište

souřadnice

všech bodů

z obraacutezku

5 Urči ve






souřadnice Ve


D(f) maacute funkce













Funkce je sudaacute









Funkce je sudaacute









a) 2

42

x


b) 32

102

x


c) 2

22

x



e) 2

123




a) 5

42

xy 21)1(40)1( ff

b) 8)5(2 2 xy 24)1(64)1( ff

c) 12

3

xy 3)1(1)1( ff

d) 4)3( 2 xy 12)1(0)1( ff

e) 2)2(

273

xy 29)1(3)1( ff

f) 1)2( 3 xy 28)1(2)1( ff

g) 9)1( 2 xy 9)1(5)1( ff

h) 11

4

xy 0)1(0)1( ff

i) 2)2(

1

xy

1)1(9

1)1( ff

j) 22

3

xy 1)1(5)1( ff



a) 2

3

xy )(10 fH

b) 5

42

xy )(10 fH

c) 8)5(2 2 xy )(10 fH

d) 12

3

xy )(10 fH

e) 6)3( 2 xy )(10 fH

f) 2)1( 3 xy )(10 fH

g) 9)1( 2 xy )(10 fH

h) 111

4

xy )(10 fH

i) 9)1(

12

xy )(10 fH

j) 22

3

xy )(10 fH



a) 2

3

xy 51003 YX

b) 5

42

xy

5

4002 YX

c) 8)5(2 2 xy 4200703 21 YXX

d) 11

2

xy 1001 YX

e) 4)3( 2 xy 500105 21 YXX

f) 2)2(

163

xy 4004 YX

g) 1)2( 3 xy 7001 YX

h) 9)1( 2 xy 800402 21 YXX

i) 1)1(

14

xy 000002 21 YXX

j) 2)2(

1

xy

4

10YneniacuteX

k) 22

4

xy 0000 YX


a) 1

56 8

x

xy [D(f) = R-1]

b) x


c) 2

206

xy [D(f) = R]

d) 246

6

x

xy [D(f) = R4]

e) 245

12

xx


f) 408 xy [D(f) = ⟨5 infin)]

g) 98

12

xx



h) 1

1


i) 2

5


Konstantniacute

funkce

119962 = 119939











Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877





Přiacutemaacute








2x + 1




f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x







c) 119910 =119909






c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3










x 3 05

y 7 1



120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100

































4 Vypište

souřadnice

všech bodů

z obraacutezku

5 Urči ve






souřadnice Ve


D(f) maacute funkce













Funkce je sudaacute









Funkce je sudaacute









a) 2

42

x


b) 32

102

x


c) 2

22

x



e) 2

123




a) 5

42

xy 21)1(40)1( ff

b) 8)5(2 2 xy 24)1(64)1( ff

c) 12

3

xy 3)1(1)1( ff

d) 4)3( 2 xy 12)1(0)1( ff

e) 2)2(

273

xy 29)1(3)1( ff

f) 1)2( 3 xy 28)1(2)1( ff

g) 9)1( 2 xy 9)1(5)1( ff

h) 11

4

xy 0)1(0)1( ff

i) 2)2(

1

xy

1)1(9

1)1( ff

j) 22

3

xy 1)1(5)1( ff



a) 2

3

xy )(10 fH

b) 5

42

xy )(10 fH

c) 8)5(2 2 xy )(10 fH

d) 12

3

xy )(10 fH

e) 6)3( 2 xy )(10 fH

f) 2)1( 3 xy )(10 fH

g) 9)1( 2 xy )(10 fH

h) 111

4

xy )(10 fH

i) 9)1(

12

xy )(10 fH

j) 22

3

xy )(10 fH



a) 2

3

xy 51003 YX

b) 5

42

xy

5

4002 YX

c) 8)5(2 2 xy 4200703 21 YXX

d) 11

2

xy 1001 YX

e) 4)3( 2 xy 500105 21 YXX

f) 2)2(

163

xy 4004 YX

g) 1)2( 3 xy 7001 YX

h) 9)1( 2 xy 800402 21 YXX

i) 1)1(

14

xy 000002 21 YXX

j) 2)2(

1

xy

4

10YneniacuteX

k) 22

4

xy 0000 YX


a) 1

56 8

x

xy [D(f) = R-1]

b) x


c) 2

206

xy [D(f) = R]

d) 246

6

x

xy [D(f) = R4]

e) 245

12

xx


f) 408 xy [D(f) = ⟨5 infin)]

g) 98

12

xx



h) 1

1


i) 2

5


Konstantniacute

funkce

119962 = 119939











Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877





Přiacutemaacute








2x + 1




f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x







c) 119910 =119909






c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3










x 3 05

y 7 1



120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























a) 2

42

x


b) 32

102

x


c) 2

22

x



e) 2

123




a) 5

42

xy 21)1(40)1( ff

b) 8)5(2 2 xy 24)1(64)1( ff

c) 12

3

xy 3)1(1)1( ff

d) 4)3( 2 xy 12)1(0)1( ff

e) 2)2(

273

xy 29)1(3)1( ff

f) 1)2( 3 xy 28)1(2)1( ff

g) 9)1( 2 xy 9)1(5)1( ff

h) 11

4

xy 0)1(0)1( ff

i) 2)2(

1

xy

1)1(9

1)1( ff

j) 22

3

xy 1)1(5)1( ff



a) 2

3

xy )(10 fH

b) 5

42

xy )(10 fH

c) 8)5(2 2 xy )(10 fH

d) 12

3

xy )(10 fH

e) 6)3( 2 xy )(10 fH

f) 2)1( 3 xy )(10 fH

g) 9)1( 2 xy )(10 fH

h) 111

4

xy )(10 fH

i) 9)1(

12

xy )(10 fH

j) 22

3

xy )(10 fH



a) 2

3

xy 51003 YX

b) 5

42

xy

5

4002 YX

c) 8)5(2 2 xy 4200703 21 YXX

d) 11

2

xy 1001 YX

e) 4)3( 2 xy 500105 21 YXX

f) 2)2(

163

xy 4004 YX

g) 1)2( 3 xy 7001 YX

h) 9)1( 2 xy 800402 21 YXX

i) 1)1(

14

xy 000002 21 YXX

j) 2)2(

1

xy

4

10YneniacuteX

k) 22

4

xy 0000 YX


a) 1

56 8

x

xy [D(f) = R-1]

b) x


c) 2

206

xy [D(f) = R]

d) 246

6

x

xy [D(f) = R4]

e) 245

12

xx


f) 408 xy [D(f) = ⟨5 infin)]

g) 98

12

xx



h) 1

1


i) 2

5


Konstantniacute

funkce

119962 = 119939











Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877





Přiacutemaacute








2x + 1




f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x







c) 119910 =119909






c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3










x 3 05

y 7 1



120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























a) 2

3

xy 51003 YX

b) 5

42

xy

5

4002 YX

c) 8)5(2 2 xy 4200703 21 YXX

d) 11

2

xy 1001 YX

e) 4)3( 2 xy 500105 21 YXX

f) 2)2(

163

xy 4004 YX

g) 1)2( 3 xy 7001 YX

h) 9)1( 2 xy 800402 21 YXX

i) 1)1(

14

xy 000002 21 YXX

j) 2)2(

1

xy

4

10YneniacuteX

k) 22

4

xy 0000 YX


a) 1

56 8

x

xy [D(f) = R-1]

b) x


c) 2

206

xy [D(f) = R]

d) 246

6

x

xy [D(f) = R4]

e) 245

12

xx


f) 408 xy [D(f) = ⟨5 infin)]

g) 98

12

xx



h) 1

1


i) 2

5


Konstantniacute

funkce

119962 = 119939











Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877





Přiacutemaacute








2x + 1




f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x







c) 119910 =119909






c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3










x 3 05

y 7 1



120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Lineaacuterniacute

funkce 119962 = 119938119961 + 119939

119863(119891) = 119877

119867(119891) = 119877





Přiacutemaacute








2x + 1




f) 6x + 4y = 12 g) 3

4y ndash 5 = 0 h) 5 +

4

3y =0 i) y + 2 = 3x







c) 119910 =119909






c) 105 xy 41)( fD d) 2

12

xy 3)( fD

e) 4

3










x 3 05

y 7 1



120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























120784







funkce) 4

3 xy



funkce) 2

1x

y A10 6 B2 2 C05 35


































x 4

y 025 1




a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100

























a) b) c)

d)



2 [

1

30] [005]


4 [30] [0

minus3

4]










120784119938

119961120783 =minus119939+radic119915


minus119939

120784119938


120784119938

Kvadratickaacute

funkce

119962 = 119938119961120784 + 119939119961 + 119940

119886 119887 119888 isin 119877 minus 119896119900119890119891119894119888119894119890119899119905119910 119886 ne 0

119863(119891) = 119877





119938 gt 120782

119938 lt 120782



a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























a) 592 2 xxy b) 10192 2 xxy

c) 3710 2 xxy d) 295 2 xxy

e) 232 xxy f) 302 xxy




c) y = x2 + 6x + 5 d) y = x2 + 14x + 44

e) y = x2 ndash 4x + 3 f) y = x2 + 8x + 12


2)1( 2 xy 4)2( 2 xy 3)4( 2 xy


1)2( 2 xy 2)1( 2 xy 5)2( xy 1)1( 2 xy









Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Nepřiacutemaacute


119948




větve


klesajiacuteciacute


rostouciacute



2)5( 1 xy 2

1

xy 1

1

xy


5

1

xy 1

3

1

xy 2

2

1

xy


1

x



1x



funkce

119962 = 119938119961


119863(119891) = 119877



[0 1] protože 1198860 = 1





119886 isin (0 1)

119948 gt 120782

119948 lt 120782





e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100


























e)

246

7

3

gt1 ne f)

246

3

7

gt1 ne


a) 15p 15r [p r] b)

rp

13

2

13

2 [p r] c) 012p 012r [p r]


xa

y

2

13

3

1a


xa

y

5

6 16 a


27

1

3

1)3( 3 x 149

7

17 xx 250)2(

2

1 4x 148

12 xx

1648

18 xx 16250)4( 2 x

15

1525

3x

25

120)5( 3 x



c) 121x = 22 + 911x [x = 1] d) 2x+3 ndash 112 = 2x [x = 4]




I II III IV

A 119910 = (1

2)

119909

+ 2

B 119910 = 3119909 minus 2

C 119910 = (1

4)

119909

minus 5

D 119910 = 2119909 minus 1


Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Logaritmickaacute

funkce



119863(119891) = (0 infin)



logaritmu





Dekadickyacute

logaritmus



x 100 000 10 000 1 000 100 10 1 01 001 0001

log x 5 4 3 2 1 0 -1 -2 -3

Věty o

logaritmech

119845119848119840119938(119961 119962) = 119845119848119840119938 119961 + 119845119848119840119938 119962

119845119848119840119938 (119961

119962) = 119845119848119840119938 119961 minus 119845119848119840119938 119962

119845119848119840119938(119961)119951 = 119847 119845119848119840119938 119961




log119886 119909 =log 119909

log 119886


1198860 = 1 rarr log119886 1 = 0 1198861 = 119886 rarr log119886 119886 = 1

log119886 119886119909 = 119909



a) 36

1log

6

1 b) 144log12 c) 1log 7 d) 12

1log12 e) 125log

5

1 f) 7log 49

[119886) 2119887) 2119888) 0119889) ndash 1119890) ndash 3119891) 05 ]


a) 16log100log 4 [4] b) 16log81log 29 [6]


e) 64


32

1log3

125

1log

2

15 [12]

g) 25log37

1log5

5

17 [ndash11]

119886 isin

(1 infin)

119886 isin

(0 1)



a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























a) log5 x = 2 b) 3log 2 x c) 416


1000

1log a


1log

10

1 j) 216

1log a

k) 28log a l) 2log3

x m) 2

1log 3 x n) 2

25

1log a

[a) 25 b)1

8 c)

1

2 d)

1

3 e) 10 f)

1

2 g)

1

2 h)

1

1000 i) 2 j) 4 k) 8 l) 3 m)

1

radic3 n) 5 ]


a) y = log5

x [ 0)( fD ] b) y = log

5

2x [ 0)( fD ]

c) y = log1

4

x [ 1)( fD ] d) y = log

4)2( x [ RfD )( ]

e) y = log52

23

x

x [ )

2

5

2

3()( fD ] f) y = log

5

32 x [ 51)( fD ]

g) y = log1x

x [ 10)( fD ] h) 12log 2 xxy [ RfD )( ]


a) 2log3 xy A[1 ndash2] B[9 0] C[27 2]

b) )2(log 2 xy A[1 ndash2] B[4 1] C[10 3]

c) 1)12(log 4 xy A[-20 2] B[4 1] C[52 2]




b) )8(log 2 xy X[ndash7 0] Y[0 3]

c) 1)4(log 2 xy X[ndash2 0] Y[0 1]







c) 3

4

b


x

a3 4

[ xa log2

1log

3

4 ]

9 Odlogaritmujte

a) 7loglog5log xa

[7

log

5xa]

b) zyx log2

15loglog2log37log

[

z

yx

5

7log

23

]

c) zyx 88888 log3

1log6loglog

5

39log

[

3

5 3

8

54log

zy

x ]






1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100



























1log

5

9log3

9

5log2

3

1

3

1

3

1


a) 1198971199001198925119909 minus 11989711990011989254 = 1 [20] b) 1198971199001198927119909 minus 4 = 1198971199001198921

4

16 [49]

c) 11989711990011989232 + 1198971199001198923119909 = 2 [45] d) 119897119900119892210 minus 1198971199001198922119909 = minus2 [40]

e) 1198971199001198928119909 minus 5 = 1198971199001198921

2

8 [64] f) 119897119900119892119909 minus 3 = 1198971199001198921

10 [100]

g) 1198971199001198922119909 minus 1 = 1198971199001198921

100 [005] h) 1198971199001198922(119909 minus 2) = 1198971199001198923125 [10]

i) 1198971199001198927119909 minus 11989711990011989273 = 11989711990011989272 [6] j) 1198971199001198922(1199093) = log 1000000 [4]


2




c) 119891(4) = 2

d) 119891(12) lt minus2




c) 119891(4) = 2

d) 119891(12) gt minus2




a) log 3 lt log 5


c) log3

4lt log

1

5

d) log 03 lt log1

3



kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























kružnice

kvadranty





radiaacuten







120572 =180120596

120587 120596 =

120572120587

180

1 (119903119886119889) =1801


120587

180(119903119886119889)

Funkce

sinus







osu x




119904119894119899(119909 + 119896 360deg) = sin 119909


Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Funkce

kosinus









119888119900119904(119909 + 119896 360deg) = 119888119900119904 119909

Funkce

tangens

119962 = 119957119944 119961 = 119853119834119847 119961





Funkce

kotangens

119962 = 119940119952119957119944 119961





119957119944 119961 119957119944 119961 119940119952119957119944 119961

119940119952119957119944 119961


kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























kvadranty

I II III IV








0 6

4

3

2

2

3 2

sin α 0 2

1

2

2

2

3 1 0 ndash1 0

cos α 1 2

3

2

2

2

1 0 ndash1 0 1

tg α 0 3

3 1 3 0 0

cotg α 3 1 3

3 0 0



sin2

3 = cos

6

5 = tg

2

= cotg

2

19 =

119904119894119899 (minus7120587

6) = 119888119900119904 (minus

4120587

3) = 119905119892 (minus

9120587

4) = 119888119900119905119892 (minus

7120587

4) =



6 [0]



33 ]


+ sin2120587 = [

12

7 ]





119905119892120587

4+ 119888119900119905119892

5120587

3+ 119905119892

3120587

4+ 119888119900119905119892

5120587

4+ 119905119892

7120587

6= [1]



119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100
























119853119840 120630 =119852119842119847 120630

119836119848119852 120630 119852119842119847 120784120630 = 120784 119852119842119847 120630 119836119848119852 120630 119956119946119951120784120630 + 119940119952119956120784120630 = 120783

119836119848119853119840 120630 =119836119848119852 120630

119852119842119847 120630 119836119848119852 120784120630 = 119940119952119956120784120630 minus 119956119946119951120784120630 119853119840 120630 119836119848119853119840 120630 = 120783



1198881199001199042119909= [ndash1]


1198881199001199043119909minus119888119900119904119909 119905119892119909 = [ndash1]


a) 2119905119892119909

1+1199051198922119909= 119904119894119899 2119909 b)

1198881199001199051198922119909119905119892119909+1199051198922119909

119905119892119909=

1

1199041198941198992119909

c) 1198881199001199051198922119909+1

1+1199051198922119909= 1198881199001199051198922119909 d) 1199051198922119909 1198881199001199042119909 + 1 minus 1198881199001199042119909 = 2 1199041198941198992119909

e) 1minus1199051198922119909

1+1199051198922119909= 1198881199001199042119909 f)

119904119894119899119909

1+119888119900119904119909+

119888119900119904119909+1

119904119894119899119909=

2

119904119894119899119909

g) 1minuscos 2119909+sin 2119909

1+cos 2119909+sin 2119909= 119905119892 119909 i) cos 119909 +





4 4 119905119892119909 = 119905119892119909 + radic3 [30deg + k180deg]







11 2 cos 119909 = 1 + 4 cos 119909 [2120587

3+ 2119896120587

4120587

3+ 2119896120587]






17 cos2 119909 + 7 cos 119909 + 6 = 0 [180deg + k360deg]



Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Posloupnosti


119886119899 = (1198992minus1198993

119899minus1)

119899=2

infin


2119899)

119899=1

infin

[05

6

24

5]


4) [225 45 875]


a) (2119899 minus 6) b) (1198992minus5

2) c) (1198992 minus 5119899 + 16) [a)8 b)5 c) 2 3]


119886119899+1 = minus1


119886119899 = 2119886119899minus1 minus 1 1198863 = 7 a1 a5 = [25 25]

119886119899+2 = 3119886119899+1 minus 119886119899 1198863 = minus1 1198864 = 4 a1 a5 = [20 13]

Aritmetickaacute

posloupnost



119938120784 = 119938120783 + 119941 119938120785 = 119938120784 + 119941 119938120786 = 119938120785 + 119941 119938120787 = 119938120786 + 119941


119938120784 minus 119938120783 = 119941 119938119951+120783 minus 119938119951 = 119941

Vzorec pro


119938120784 = 119938120783 + 119941 119938120785 = 119938120783 + 120784119941 119938120786 = 119938120783 + 120785119941 119938120787 = 119938120783 + 120786119941

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 =119951

120784(119938120783 + 119938119951)



a) a1 = 5 d = 3 [5 8 11 14 17]

b) a3 = ndash6 d = 8 [ndash6 2 10 18 26]

c) a2 = 15 d = ndash03 [15 12 09 06 03]

d) a4 = 5 a9 = 15 [71 64 57 5 43]





b) a18 = 4 d = minus1

5 určete a8 a33 [a33= 1 a8 = 6]






a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100


























a) a4 = 29 a10 = 71 S15 = [855] b) a3 = 29 a11 = 49 S5 = [145]







a) Sn = 28 a2 = 2 a5 = 32 [n = 4]

b) Sn = 392 a3 = 45 a8 = 10 [n = 7]

c) Sn = 0 a8 = 0 a10 = 1 [n = 15]



3

5

6 1 hellip 3

2

3 [n = 19 S19 =

247

6]


a) 250 je-li a3 = 5 d = 1 [n = 20]

b) 87 je-li a5 = 22 a6 = 27 [n = 6]

c) 130 je-li a1 = 4 d=2 [n = 10]
















b) a12 = 29











schodiště


Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100























Geometrickaacute

posloupnost



119938120784 = 119938120783 119954 119938120785 = 119938120784 119954 119938120786 = 119938120785 119954 119938120787 = 119938120786 119954


119938120784 119938120783 = 119954 119938119951+120783 119938119951 = 119954

Vzorec pro


119938120784 = 119938120783 119954 119938120785 = 119938120783 119954120784 119938120786 = 119938120783 119954120785 119938120787 = 119938120783 119954120786

Vzorec pro


Součet n členů 119878119899 = 1198861 + 1198862 + 1198863 + 1198864 + 1198865 + ⋯ + 119886119899

119930119951 = 119938120783119954119951 minus 120783

119954 minus 120783



1 an =

8


7

8]


















c) a1 = 6 q = 025 určete S4 [S4=255

32]
















Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100



































Vzrůst hodnoty 119886119899 = 1198860 (1 +

119901

100)

119899

119886119899 = 1198860 (1 + 001 119901)119899


119901

100)

119899

119886119899 = 1198860 (1 minus 001 119901)119899



100= 119901 minus

119901119889

100






















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