diff --git a/notes/.obsidian/plugins/obsidian-activity-history/data.json b/notes/.obsidian/plugins/obsidian-activity-history/data.json index ca22d1d..47acf5b 100644 --- a/notes/.obsidian/plugins/obsidian-activity-history/data.json +++ b/notes/.obsidian/plugins/obsidian-activity-history/data.json @@ -13,7 +13,7 @@ { "path": "/", "date": "2022-01-20", - "size": 735530 + "size": 767464 } ], "activityHistory": [ @@ -330,7 +330,7 @@ }, { "date": "2022-01-20", - "value": 251 + "value": 32257 } ] } diff --git a/notes/.obsidian/plugins/obsidian-admonition/data.json b/notes/.obsidian/plugins/obsidian-admonition/data.json index 0a4e9df..216b2f1 100644 --- a/notes/.obsidian/plugins/obsidian-admonition/data.json +++ b/notes/.obsidian/plugins/obsidian-admonition/data.json @@ -1,5 +1,18 @@ { - "userAdmonitions": {}, + "userAdmonitions": { + "sentence": { + "type": "sentence", + "color": "253, 13, 13", + "icon": { + "name": "sticky-note", + "type": "font-awesome" + }, + "command": true, + "title": 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09.25.49.excalidraw]] - [[data/Kótování koule_2022-01-07 12.14.15.excalidraw|Kótování koule_2022-01-07 12.14.15.excalidraw]] - [[data/Kótování koule_2022-01-07 12.54.53.excalidraw|Kótování koule_2022-01-07 12.54.53.excalidraw]] - [[data/Kótování na strojařský výkres_2021-11-26 12.00.18.excalidraw|Kótování na strojařský výkres_2021-11-26 12.00.18.excalidraw]] @@ -70,6 +71,20 @@ imagePrefix: 'data/' - [[data/Pasted image 20211220095727.png|Pasted image 20211220095727.png]] - [[data/Pasted image 20211220095757.png|Pasted image 20211220095757.png]] - [[data/Pasted image 20211220100010.png|Pasted image 20211220100010.png]] +- [[data/Pasted image 20220103085740.png|Pasted image 20220103085740.png]] +- [[data/Pasted image 20220116205304.png|Pasted image 20220116205304.png]] +- [[data/Pasted image 20220117084537.png|Pasted image 20220117084537.png]] +- [[data/Pasted image 20220117090110.png|Pasted image 20220117090110.png]] +- [[data/Pasted image 20220117090135.png|Pasted image 20220117090135.png]] +- [[data/Pasted image 20220117090249.png|Pasted image 20220117090249.png]] +- [[data/Pasted image 20220117090519.png|Pasted image 20220117090519.png]] +- [[data/Pasted image 20220117092741.png|Pasted image 20220117092741.png]] +- [[data/Pasted image 20220117092833.png|Pasted image 20220117092833.png]] +- [[data/Pasted image 20220117092859.png|Pasted image 20220117092859.png]] +- [[data/Pasted image 20220117093001.png|Pasted image 20220117093001.png]] +- [[data/Pasted image 20220117093236.png|Pasted image 20220117093236.png]] +- [[data/Pasted image 20220117093345.png|Pasted image 20220117093345.png]] +- [[data/Pasted image 20220117093451.png|Pasted image 20220117093451.png]] - [[data/tcpudp.jpg|tcpudp.jpg]] - [[data/tek-preruseni-dva.excalidraw|tek-preruseni-dva.excalidraw]] - [[data/Text Elements|Text Elements]] diff --git a/notes/dej/moderní/2. světová válka/2. světová válka.md b/notes/dej/moderní/2. světová válka/2. světová válka.md index 0c0d7b4..c18dc09 100644 --- a/notes/dej/moderní/2. světová válka/2. světová válka.md +++ b/notes/dej/moderní/2. světová válka/2. světová válka.md @@ -4,5 +4,6 @@ tags: [MOC, generated, index, dej] # 2. světová válka %% Zoottelkeeper: Beginning of the autogenerated index file list %% - [[dej/moderní/2. světová válka/Mnichovská dohoda|Mnichovská dohoda]] +- [[dej/moderní/2. světová válka/Noc dlouhých nožů|Noc dlouhých nožů]] - [[dej/moderní/2. světová válka/Norimberské zákony|Norimberské zákony]] %% Zoottelkeeper: End of the autogenerated index file list %% diff --git a/notes/dej/moderní/2. světová válka/Noc dlouhých nožů.md b/notes/dej/moderní/2. světová válka/Noc dlouhých nožů.md new file mode 100644 index 0000000..71f9e37 --- /dev/null +++ b/notes/dej/moderní/2. světová válka/Noc dlouhých nožů.md @@ -0,0 +1,8 @@ +# Noc dlouhých nožů +- 30. června až 2. července +- Operace kolibřík +- Rohmův puč +- Cílem byl SA +- SA přišla o moc +- Osamotnění a nárůst na moci jednotky SS +- Hitler se stal nejvýznamnější osobou v Německu \ No newline at end of file diff --git a/notes/dej/moderní/První republika/Politika první republiky.md b/notes/dej/moderní/První republika/Politika první republiky.md index f3d96ae..016f321 100644 --- a/notes/dej/moderní/První republika/Politika první republiky.md +++ b/notes/dej/moderní/První republika/Politika první republiky.md @@ -7,8 +7,8 @@ tags: [dej, dej/první-republika] 3. **V jaké budově sídlila za první republiky poslanecká sněmovna? Kde sídlí dnes?** Sídlil v Thunovském paláci, dnes ve Wallensteinském paláci 4. **Jak T. G. Masaryk oslavil tzv. Husův den roku 1925?** Vyvěsil znak kalichu na hrad. 5. **Jak se jmenoval nejvýznamnější politik z řad agrárníků? Jak k němu přistupoval T. G. Masaryk?** Antonín Švehla; respektoval ho, k ostatním měl výhrady -6. **Z jakého důvodu šéfové demokratických stran vytvořili tzv. Pětku? V jakých letech působila?** Snaha bránit extrémismu. +6. **Z jakého důvodu šéfové demokratických stran vytvořili tzv. Pětku? V jakých letech působila?** Snaha bránit extrémismu. 1921-1925 7. **Kolik procen získala sociální demokracie v prvních parlamentních volbách roku 1920? Jaké byly příčiny tohoto úspěchu?** 25.7% - krize dělníků -8. **Jaký z mužů 28. října byl členem národních socialistů? Charakterizujte osobnost tohoto politika.** E. Beneš +8. **Jaký z mužů 28. října byl členem národních socialistů? Charakterizujte osobnost tohoto politika.** E. Beneš - Prezident 9. **Jakou funkci vykonával prvorepublikový leader lidovců v době druhé světové války?** Jan Šrámek - Exilový předseda vládý 10. **Které členy "Mafie" byste našli v řadách národních demokratů?** Karel Kramář, A. Rašín \ No newline at end of file diff --git a/notes/ech/ech.md b/notes/ech/ech.md index 2be4515..328fd04 100644 --- a/notes/ech/ech.md +++ b/notes/ech/ech.md @@ -16,6 +16,7 @@ imagePrefix: 'data/' - [[ech/Prvky/Prvky|Prvky]] - [[ech/prvky|prvky]] - [[ech/radioaktivita|radioaktivita]] +- [[ech/Ředění|Ředění]] - [[ech/Směsi|Směsi]] - [[ech/Vazby|Vazby]] - [[ech/Veličiny|Veličiny]] diff --git a/notes/ech/Ředění.md b/notes/ech/Ředění.md new file mode 100644 index 0000000..b9ed518 --- /dev/null +++ b/notes/ech/Ředění.md @@ -0,0 +1,21 @@ +# Ředění +## Příklad +- koncentrát 2t 45% $\odot$ +- voda ? +- ředění ? 15% +- nemíchá s v kotli, ale v trubkách + +1. konc $m_1=2t$ $w_1 = 0.45$ +2. voda $m_2 = ?$ $w_1 = 0$ +-> obojí do směšovače (zapisuje se šipkou pod každou rovnicí, která jde do směšovače, který je značený čtvercem a popisem směšovač) + +*(podtržené neznáme)* +$m_1+ \underline{m_2} = \underline{m_3}$ +$W_{ci} = \frac{m_{ci}}{m_i}$ +$m_{ci} = W_{ci} * m_i$ + +$w_1m_1+w_2m_2 = w_3*\underline{m_3}$ +$2+m_2=m_3$ +$0.45*2+ 0*m_2=0.15*m_3$ +$m_3=\frac{0.9}{0.15} 6t$ +$m_2 = m_3-m_1 = 6 - 2 = 4t$ diff --git a/notes/mat/Funkce/Funkce.md b/notes/mat/Funkce/Funkce.md new file mode 100644 index 0000000..040fcba --- /dev/null +++ b/notes/mat/Funkce/Funkce.md @@ -0,0 +1,4 @@ +# Funkce +%% Zoottelkeeper: Beginning of the autogenerated index file list %% +- [[mat/Funkce/Kartézský součin|Kartézský součin]] +%% Zoottelkeeper: End of the autogenerated index file list %% diff --git a/notes/mat/Funkce/Kartézský součin.md b/notes/mat/Funkce/Kartézský součin.md new file mode 100644 index 0000000..06dc226 --- /dev/null +++ b/notes/mat/Funkce/Kartézský součin.md @@ -0,0 +1,13 @@ +# Kartézský součin +$$A = \{\circ;x;\triangle\}$$ +$$B = \{a;b;c;d\}$$ +$$A \times B = \{\forall[x;y]: x \in A, y \in B\}$$ +$$A \times B = \{[\circ; a]; [\circ; b]; [\circ; c]; [\circ; d]; [x; a]; [x; b]; [x; c]; [x; d]; [\triangle; a]; [\triangle; b]; [\triangle; c]; [\triangle; d]\}$$ + +## Binární relace +*A do B* +**Libovolná podmnožina $A \times B$.** + +*například:* +$$R_{AB} = \{[\circ; b]; [x; c]; [\triangle; a] \}$$ +![[data/Kartézský součin_2022-01-19 09.25.49.excalidraw.md]] \ No newline at end of file diff --git a/notes/mat/Lomené výrazy.md b/notes/mat/Lomené výrazy.md index 0040158..740c2d5 100644 --- a/notes/mat/Lomené výrazy.md +++ b/notes/mat/Lomené výrazy.md @@ -12,150 +12,6 @@ $$ $$ x \neq -1; x \neq 3; x \neq - \frac{1}{2} $$ -## Krácení a rozšiřování lomených výrazů -Lze krátit jen násobení a dělení, ne sčítání a odčítání. - -Pro libovolné výrazy $v_1$, $v_2$, $v_3$ -$$ -\frac{v_1 * v_3}{v_2 * v_3} = \frac{v_1}{v_2} -$$ -$$ -\xRightarrow{krácení} -$$ -$$ -\xLeftarrow{rozšiřování} -$$ -## Sčítání -- musí se převést na společného jmenovatele -$$\frac{7}{5} + \frac{4}{7} = \frac{7}{5}^{*7}_{*7} + \frac{4}{7}^{*5}_{*5*} = \frac{49+20}{35} = \frac{69}{35}$$ -- musí se stanovit podmínky -$$ -\frac{7}{6} + \frac{1}{x-3} [x\neq3] -$$ - -```ad-error -title: Věta - -Pro libovolné výrazy $V_1;V_2;V_3;V_4$ a pro všechny hodnoty proměnných, pro něž je $V_2 \neq 0; V_4 \neq 0$ platí $\frac{V_1}{V_2} + \frac{V_1}{V_4} = \frac{V_1V_4 + V_3V_2}{V_2V_4}$ -``` ---- -$$ -\frac{1}{x+1} + \frac{2}{x+3} = \frac{x+3}{(x+1)(x+3)} + \frac{x+1}{(x+1)(x+3)} = \frac{(x+3)+2(x+1)}{(x+1)(x+3)} = \frac{3x+5}{x^2+6x+9} -$$ ---- -$$ -[x\neq-1] [x\neq-3] -$$ -$$ -\frac{3x}{x+2}^{*(x-3)}_{*(x-30)} + \frac{1}{x-3}^{*(x+2)}_{*(x+2)} = \frac{3x^2 - 9x + x + 2}{(x+2)(x-3)} = \frac{3x^2-8x+2}{(x+2)(x-3)} -$$ -$$ -[x\neq-2][x\neq3] -$$ ---- -$$ -\frac{2}{x}+\frac{x}{3} + 4 = \frac{6+x^2+12x}{3x} -$$ -$$ -[x\neq0] -$$ ---- -$$ -\frac{3}{x} + \frac{y}{3x} + \frac{4}{y+1} = \frac{9x(y+1)+y(y+1)+12x}{3x(y+1)} = \frac{(y+1)(9x + y) + 12x}{3x(y+2)} -$$ -$$ -[x\neq0][y\neq-1] -$$ ---- -$$ -\frac{x+1}{x} + \frac{x}{x+1} = \frac{(x+1)(x+1)}{x(x+1)} + \frac{x^2}{x(x+1)} = \frac{x^2 + (x+1)(x+1)}{x(x+1)} = \frac{2x^2 + 2x + 1}{x(x+1)} -$$ -$$ -[x\neq0] [x\neq-1] -$$ ---- -$$ -\frac{x}{3} - \frac{x}{x+2} - 2 = \frac{x(x+2) - 3x - 3*2(x+2)}{3(x+2)} = \frac{x^2 + 2x - 3x - 6x - 12}{3(x+2)} = \frac{x^2-7x-12}{3x+6} -$$ -$$ -[x\neq-2] -$$ ---- -$$ -\frac{x+1}{x} + \frac{x-2}{2x} - 2x + 1 = \frac{2(x+1) + x - 2 - 2x2x + 2x}{2x} = \frac{5x-4x^2}{2x} = \frac{x(5-4x)}{2x} = \frac{5-4x}{2} = - 2x + \frac{5}{2} -$$ -$$ -[x\neq0] -$$ ---- -$$ -\frac{x}{x+2}-\frac{x+1}{x-3} = \frac{x(x-3) - (x+2)(x+1)}{(x+2)(x-3)} = \frac{x^2-3x-x^2-x-2x-2}{x^2-3x+2x-6} = \frac{-6x-2}{x^2-x-6} -$$ -$$ -[x\neq-2] [x\neq3] -$$ ---- -$$ -\frac{2-3x^2}{x-1} - \frac{2x-1}{2x} - 2 + 3x = \frac{2x(2-3x^2) - (x-1)(2x-1) - 2x(x-1)(3x-2)}{2x(x-1)} = \frac{4x-6x^3 - 2x^2 + x + 1 - 6x^3 + 4x^2 + 6x^2 - 2}{2x(x-1)} = \frac{-12x^2+11x+2}{2x(x-1)} -$$ -$$ -[x\neq1][x\neq0] -$$ ---- -$$ -\frac{y}{y^2 - x^2} - \frac{x}{x-y} = \frac{y}{(y-x)(y+x)} + \frac{x}{y-x} = \frac{y + x(x+y)}{(x+y)(y-x)} = \frac{x^2+xy+y}{y^2-x^2} -$$ -$$ -[x\neq0][y\neq0][x\neq y] -$$ ---- -$$ -\frac{a+b}{a}-\frac{a}{a-b}+\frac{b^2}{aa-ab} = \frac{(a+b)(a-b) - a^2 + b^2}{a(a-b)} = \frac{a^2-b^2 - a^2 + b^2}{a(a-b)} = 0 -$$ -$$ -[a\neq0][a\neq b] -$$ ---- -$$ -\frac{8-5x}{8+2x-x^2} - \frac{2x+2}{x^2-3x-4} = \frac{8-5x}{(x+2)(-x+4)} - \frac{2x+2}{(x+1)(x-4)} = \frac{(8-5x)(x+1) + (2x+2)(x+2)}{(x+2)(x+1)(x-4)} = \frac{8x+8-5x^2-5x + 2x^2+4x+4+2x}{(x+2)(x+1)(x-4)} = \frac{-3x^2+9x+12}{(x+2)(x+1)(x-4)} -$$ -$$ -\frac{3(x+1)(x-4)}{(x+2)(x+1)(x-4)} = \frac{3}{x+2} -$$ -$$ -[x\neq4][x\neq-2][x\neq-1] -$$ ---- -$$ -8+2x-x^2 = (x+2)(-x+4) -$$ ---- -### Úkol 17.1. ---- -$$ -x-y-\frac{x}{x+y} = \frac{(x-y)(x+y)-x}{x+y} = \frac{x^2-y^2-x}{x+y} -$$ -$$ -[x\neq-y] -$$ ---- -$$ -1-\frac{4xy}{(x+y)(x+y)} = \frac{(x+y)(x+y)-4xy}{(x+y)(x+y)} = \frac{x^2-2xy+y^2}{(x+y)(x+y)} = \frac{(x-y)(x-y)}{(x+y)(x+y)} -$$ -$$ -[x\neq-y] -$$ ---- -$$ -\frac{x-2}x - \frac{4x+1}{x-3} = \frac{(x-2)(x-3) - x(4x+1)}{x(x-3)} = \frac{-3x(x - 2)}{x(x-3)} -$$ -$$ -[x\neq0] [x\neq3] -$$ ---- -$$ -\frac{3x+8}{x-2} * \frac{x^2 - 4x + 4}{x^2 - 2x + 4} = \frac{(3x+8)(x-2)(x-2)}{(x-2)(x^2 - 2x + 4)} = \frac{3x^3 - 6x^2 - 6x^2 + 12x + 8x^2 - 16x - 16x + 32}{x^3 - 2x^2 + 4x - 2x^2 + 4x + 8} = \frac{3x^3 - 4x^2 - 12x + 32}{x^3 - 4x^2 + 8x + 8} = \frac{x(x(3x-4)-12)+32}{x(x(x-4)+8)+8} -$$ -$$ -[x\neq2] -$$ +![Krácení a rozšiřování lomených výrazů](Krácení%20a%20rozšiřování%20lomených%20výrazů.md) +![Sčítání a odčítání](Sčítání%20a%20odčítání.md) +![Dělení a umocňování lomených výrazů](Dělení%20a%20umocňování%20lomených%20výrazů.md) diff --git a/notes/mat/Lomené výrazy/Dělení a umocňování lomených výrazů.md b/notes/mat/Lomené výrazy/Dělení a umocňování lomených výrazů.md new file mode 100644 index 0000000..83b41f1 --- /dev/null +++ b/notes/mat/Lomené výrazy/Dělení a umocňování lomených výrazů.md @@ -0,0 +1,25 @@ +# Dělení a umocňování lomených výrazů +```ad-sentence +Pro libovolné výrazy $V_1;V_2;V_3;V_4$ a pro všechny hodnoty proměnných, pro něž je $V_2 \neq 0; V_4 \neq 0$, platí: $\frac{V_1}{V_2}*\frac{V_3}{V_4} = \frac{V_1*V_2}{V_3*V_4}$ +``` +```ad-sentence +Pro libovolné výrazy $V_1;V_2$ a pro libovolné přirozené číslo $k$ a pro všechny hodnoty proměnných, pro něž $V_2 \neq 0$, platí: $(\frac{V_1}{V_2})^k = \frac{V_1^k}{V_2^k}$ +``` +```ad-sentence +Pro libovolné výrazy $V_1;V_2;V_3;V_4$ a pro všechny hodnoty proměnných, pro něž je $V_2 \neq 0; V_3 \neq 0; V_4 \neq 0$, platí $\frac{V_1}{V_2} / \frac{V_3}{V_4} = \frac{V_1}{V_2} * \frac{V_4}{V_3} = \frac{V_1 * V_4}{V_2 * V_3}$ +``` +## Příklady +--- +$$ +\frac{12a^2b^2}{14x^2y^3}:\frac{18a^2b}{21x^3y^2} = \frac{12a^2b^2 * 21x^3y^2}{14x^2y^3 * 18a^2b} = \frac{bx}{y} +$$ +$$ +[x\neq0][y\neq0][a\neq0][b\neq0] +$$ +--- +$$ +\frac{(a+b)^2}{a^2-b^2}:\frac{a^2+b^2}{(a-b)^2} = \frac{a+b}{a-b}*\frac{a^2+2ab-b^2}{a^2+b^2} = \frac{(a+b)(a-b)(a-b)}{(a-b)(a^2+b^2)} = \frac{(a+b)(a-b)}{a^2+b^2} = \frac{a^2-b^2}{a^2+b^2} +$$ +$$ +[a\neq b][a^2 \neq b^2] +$$ \ No newline at end of file diff --git a/notes/mat/Lomené výrazy/Krácení a rozšiřování lomených výrazů.md b/notes/mat/Lomené výrazy/Krácení a rozšiřování lomených výrazů.md new file mode 100644 index 0000000..045fda2 --- /dev/null +++ b/notes/mat/Lomené výrazy/Krácení a rozšiřování lomených výrazů.md @@ -0,0 +1,13 @@ +# Krácení a rozšiřování lomených výrazů +Lze krátit jen násobení a dělení, ne sčítání a odčítání. + +Pro libovolné výrazy $v_1$, $v_2$, $v_3$ +$$ +\frac{v_1 * v_3}{v_2 * v_3} = \frac{v_1}{v_2} +$$ +$$ +\xRightarrow{krácení} +$$ +$$ +\xLeftarrow{rozšiřování} +$$ diff --git a/notes/mat/Lomené výrazy/Lomené výrazy.md b/notes/mat/Lomené výrazy/Lomené výrazy.md new file mode 100644 index 0000000..f553e64 --- /dev/null +++ b/notes/mat/Lomené výrazy/Lomené výrazy.md @@ -0,0 +1,6 @@ +# Lomené výrazy +%% Zoottelkeeper: Beginning of the autogenerated index file list %% +- [[mat/Lomené výrazy/Dělení a umocňování lomených výrazů|Dělení a umocňování lomených výrazů]] +- [[mat/Lomené výrazy/Krácení a rozšiřování lomených výrazů|Krácení a rozšiřování lomených výrazů]] +- [[mat/Lomené výrazy/Sčítání a odčítání|Sčítání a odčítání]] +%% Zoottelkeeper: End of the autogenerated index file list %% diff --git a/notes/mat/Lomené výrazy/Sčítání a odčítání.md b/notes/mat/Lomené výrazy/Sčítání a odčítání.md new file mode 100644 index 0000000..07c00dc --- /dev/null +++ b/notes/mat/Lomené výrazy/Sčítání a odčítání.md @@ -0,0 +1,133 @@ +# Sčítání a odčítání +## Sčítání +- musí se převést na společného jmenovatele +$$\frac{7}{5} + \frac{4}{7} = \frac{7}{5}^{*7}_{*7} + \frac{4}{7}^{*5}_{*5*} = \frac{49+20}{35} = \frac{69}{35}$$ +- musí se stanovit podmínky +$$ +\frac{7}{6} + \frac{1}{x-3} [x\neq3] +$$ + +```ad-sentence +Pro libovolné výrazy $V_1;V_2;V_3;V_4$ a pro všechny hodnoty proměnných, pro něž je $V_2 \neq 0; V_4 \neq 0$ platí $\frac{V_1}{V_2} + \frac{V_1}{V_4} = \frac{V_1V_4 + V_3V_2}{V_2V_4}$ +``` +--- +$$ +\frac{1}{x+1} + \frac{2}{x+3} = \frac{x+3}{(x+1)(x+3)} + \frac{x+1}{(x+1)(x+3)} = \frac{(x+3)+2(x+1)}{(x+1)(x+3)} = \frac{3x+5}{x^2+6x+9} +$$ +--- +$$ +[x\neq-1] [x\neq-3] +$$ +$$ +\frac{3x}{x+2}^{*(x-3)}_{*(x-30)} + \frac{1}{x-3}^{*(x+2)}_{*(x+2)} = \frac{3x^2 - 9x + x + 2}{(x+2)(x-3)} = \frac{3x^2-8x+2}{(x+2)(x-3)} +$$ +$$ +[x\neq-2][x\neq3] +$$ +--- +$$ +\frac{2}{x}+\frac{x}{3} + 4 = \frac{6+x^2+12x}{3x} +$$ +$$ +[x\neq0] +$$ +--- +$$ +\frac{3}{x} + \frac{y}{3x} + \frac{4}{y+1} = \frac{9x(y+1)+y(y+1)+12x}{3x(y+1)} = \frac{(y+1)(9x + y) + 12x}{3x(y+2)} +$$ +$$ +[x\neq0][y\neq-1] +$$ +--- +$$ +\frac{x+1}{x} + \frac{x}{x+1} = \frac{(x+1)(x+1)}{x(x+1)} + \frac{x^2}{x(x+1)} = \frac{x^2 + (x+1)(x+1)}{x(x+1)} = \frac{2x^2 + 2x + 1}{x(x+1)} +$$ +$$ +[x\neq0] [x\neq-1] +$$ +--- +$$ +\frac{x}{3} - \frac{x}{x+2} - 2 = \frac{x(x+2) - 3x - 3*2(x+2)}{3(x+2)} = \frac{x^2 + 2x - 3x - 6x - 12}{3(x+2)} = \frac{x^2-7x-12}{3x+6} +$$ +$$ +[x\neq-2] +$$ +--- +$$ +\frac{x+1}{x} + \frac{x-2}{2x} - 2x + 1 = \frac{2(x+1) + x - 2 - 2x2x + 2x}{2x} = \frac{5x-4x^2}{2x} = \frac{x(5-4x)}{2x} = \frac{5-4x}{2} = - 2x + \frac{5}{2} +$$ +$$ +[x\neq0] +$$ +--- +$$ +\frac{x}{x+2}-\frac{x+1}{x-3} = \frac{x(x-3) - (x+2)(x+1)}{(x+2)(x-3)} = \frac{x^2-3x-x^2-x-2x-2}{x^2-3x+2x-6} = \frac{-6x-2}{x^2-x-6} +$$ +$$ +[x\neq-2] [x\neq3] +$$ +--- +$$ +\frac{2-3x^2}{x-1} - \frac{2x-1}{2x} - 2 + 3x = \frac{2x(2-3x^2) - (x-1)(2x-1) - 2x(x-1)(3x-2)}{2x(x-1)} = \frac{4x-6x^3 - 2x^2 + x + 1 - 6x^3 + 4x^2 + 6x^2 - 2}{2x(x-1)} = \frac{-12x^2+11x+2}{2x(x-1)} +$$ +$$ +[x\neq1][x\neq0] +$$ +--- +$$ +\frac{y}{y^2 - x^2} - \frac{x}{x-y} = \frac{y}{(y-x)(y+x)} + \frac{x}{y-x} = \frac{y + x(x+y)}{(x+y)(y-x)} = \frac{x^2+xy+y}{y^2-x^2} +$$ +$$ +[x\neq0][y\neq0][x\neq y] +$$ +--- +$$ +\frac{a+b}{a}-\frac{a}{a-b}+\frac{b^2}{aa-ab} = \frac{(a+b)(a-b) - a^2 + b^2}{a(a-b)} = \frac{a^2-b^2 - a^2 + b^2}{a(a-b)} = 0 +$$ +$$ +[a\neq0][a\neq b] +$$ +--- +$$ +\frac{8-5x}{8+2x-x^2} - \frac{2x+2}{x^2-3x-4} = \frac{8-5x}{(x+2)(-x+4)} - \frac{2x+2}{(x+1)(x-4)} = \frac{(8-5x)(x+1) + (2x+2)(x+2)}{(x+2)(x+1)(x-4)} = \frac{8x+8-5x^2-5x + 2x^2+4x+4+2x}{(x+2)(x+1)(x-4)} = \frac{-3x^2+9x+12}{(x+2)(x+1)(x-4)} +$$ +$$ +\frac{3(x+1)(x-4)}{(x+2)(x+1)(x-4)} = \frac{3}{x+2} +$$ +$$ +[x\neq4][x\neq-2][x\neq-1] +$$ +--- +$$ +8+2x-x^2 = (x+2)(-x+4) +$$ +--- +### Úkol 17.1. +--- +$$ +x-y-\frac{x}{x+y} = \frac{(x-y)(x+y)-x}{x+y} = \frac{x^2-y^2-x}{x+y} +$$ +$$ +[x\neq-y] +$$ +--- +$$ +1-\frac{4xy}{(x+y)(x+y)} = \frac{(x+y)(x+y)-4xy}{(x+y)(x+y)} = \frac{x^2-2xy+y^2}{(x+y)(x+y)} = \frac{(x-y)(x-y)}{(x+y)(x+y)} +$$ +$$ +[x\neq-y] +$$ +--- +$$ +\frac{x-2}x - \frac{4x+1}{x-3} = \frac{(x-2)(x-3) - x(4x+1)}{x(x-3)} = \frac{-3x(x - 2)}{x(x-3)} +$$ +$$ +[x\neq0] [x\neq3] +$$ +--- +$$ +\frac{3x+8}{x-2} * \frac{x^2 - 4x + 4}{x^2 - 2x + 4} = \frac{(3x+8)(x-2)(x-2)}{(x-2)(x^2 - 2x + 4)} = \frac{3x^3 - 6x^2 - 6x^2 + 12x + 8x^2 - 16x - 16x + 32}{x^3 - 2x^2 + 4x - 2x^2 + 4x + 8} = \frac{3x^3 - 4x^2 - 12x + 32}{x^3 - 4x^2 + 8x + 8} = \frac{x(x(3x-4)-12)+32}{x(x(x-4)+8)+8} +$$ +$$ +[x\neq2] +$$ \ No newline at end of file diff --git a/notes/mat/mat.md b/notes/mat/mat.md index 676f4e5..d8e2000 100644 --- a/notes/mat/mat.md +++ b/notes/mat/mat.md @@ -10,7 +10,9 @@ imagePrefix: 'data/' - [[mat/Číselné obory|Číselné obory]] - [[mat/Číselné soustavy|Číselné soustavy]] - [[mat/Druhá odmocnina|Druhá odmocnina]] +- [[mat/Funkce/Funkce|Funkce]] - [[mat/Intervaly|Intervaly]] +- [[mat/Lomené výrazy/Lomené výrazy|Lomené výrazy]] - [[mat/Lomené výrazy|Lomené výrazy]] - [[mat/Matematika|Matematika]] - [[mat/Mnohočlen|Mnohočlen]] diff --git a/notes/notes.md b/notes/notes.md index 8c1a68d..38eee72 100644 --- a/notes/notes.md +++ b/notes/notes.md @@ -7,6 +7,7 @@ noteOnly: true %% Zoottelkeeper: Beginning of the autogenerated index file list %% - [[ang/ang|ang]] - [[cjl/cjl|cjl]] +- [[conflict-files-obsidian-git|conflict-files-obsidian-git]] - [[data/data|data]] - [[dej/dej|dej]] - [[ech/ech|ech]]