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Mozo: /* Kettősintegrál */
2009-05-14T19:57:44Z
<p><span class="autocomment">Kettősintegrál</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. május 14., 19:57-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">30. sor:</td>
<td colspan="2" class="diff-lineno">30. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\int\limits_{x=0}^1\int\limits_{y=0}^{1-x} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{x=0}^1\frac{1}{7}[x^2y^7]_{y=0}^{1-x}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{7}x^2(1-x)^7\,\mathrm{d}x=?</math>  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\int\limits_{x=0}^1\int\limits_{y=0}^{1-x} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{x=0}^1\frac{1}{7}[x^2y^7]_{y=0}^{1-x}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{7}x^2(1-x)^7\,\mathrm{d}x=?</math>  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>:<math>\int\limits_{y=0}^1\int\limits_{x=0}^{1-y} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{y=0}^1\frac{1}{3}[x^3y^6]_{x=0}^{1-y}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{3}y^6(1-y)^3\,\mathrm{d}y=...</math>  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>:<math>\int\limits_{y=0}^1\int\limits_{x=0}^{1-y} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{y=0}^1\frac{1}{3}[x^3y^6]_{x=0}^{1-y}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{3}y^6(1-y)^3<ins class="diffchange diffchange-inline">\,\mathrm{d}y=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">=\int\limits_{x=0}^1\frac{1}{3}(y^6-3y^7+3y^{8}-y^{9} </ins>\,\mathrm{d}y=...</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A2a_2008/11._gyakorlat&diff=5240&oldid=prev
Mozo: /* Totális és folytonos parciális deriválhatóság */
2009-05-14T19:45:56Z
<p><span class="autocomment">Totális és folytonos parciális deriválhatóság</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. május 14., 19:45-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">19. sor:</td>
<td colspan="2" class="diff-lineno">19. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ugyanis  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ugyanis  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{(x^2+x^2)^{\frac{3}{2}}}=\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{2^{\frac{3}{2}}x^3}=\lim\limits_{(x,x)\to (0,0)}\frac{1}{2^{\frac{3}{2}}}(x+1)\to \frac{1}{2^{\frac{3}{2}}}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{(x^2+x^2)^{\frac{3}{2}}}=\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{2^{\frac{3}{2}}x^3}=\lim\limits_{(x,x)\to (0,0)}\frac{1}{2^{\frac{3}{2}}}(x+1)\to \frac{1}{2^{\frac{3}{2}}}</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Tehát nem folyt diff, mert ha az lenne, akkor deriválható is lenne (egy környzetben létezik a derivált!). Jacobi van, gradiens nincs, mert az a differenciál leképezés skalárinvariánssa lenne ( az az ''m'' vektor, melyet az ''Ax''=''m''<math>\cdot</math>''x'' definiál, de nincs alkalmas ''A'', így nincs alkalmas ''m'').  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Tehát nem folyt diff, mert ha az lenne, akkor deriválható is lenne (egy környzetben létezik a derivált!). Jacobi van, gradiens nincs, mert az a differenciál leképezés skalárinvariánssa lenne ( az az ''m'' vektor, melyet az ''Ax''=''m''<math>\cdot</math>''x'' definiál, de nincs alkalmas ''A'', így nincs alkalmas ''m'').</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">==Kettősintegrál==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">''T'': a (0,0), (0,1), (1,0) csúcspontú háromszög.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>f(x,y)=x^2y^6\,</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\int\limits_{x=0}^1\int\limits_{y=0}^{1-x} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{x=0}^1\frac{1}{7}[x^2y^7]_{y=0}^{1-x}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{7}x^2(1-x)^7\,\mathrm{d}x=?</math> </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math>\int\limits_{y=0}^1\int\limits_{x=0}^{1-y} x^2y^6\,\mathrm{d}x\mathrm{d}y=\int\limits_{y=0}^1\frac{1}{3}[x^3y^6]_{x=0}^{1-y}\,\mathrm{d}x=\int\limits_{x=0}^1\frac{1}{3}y^6(1-y)^3\,\mathrm{d}y=...</math> </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A2a_2008/11._gyakorlat&diff=5239&oldid=prev
Mozo: /* Totális és folytonos parciális deriválhatóság */
2009-05-14T19:32:36Z
<p><span class="autocomment">Totális és folytonos parciális deriválhatóság</span></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. május 14., 19:32-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">8. sor:</td>
<td colspan="2" class="diff-lineno">8. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0, & \mathrm{ha} & (x,y)=(0,0)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>0, & \mathrm{ha} & (x,y)=(0,0)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{matrix}\right.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{matrix}\right.</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>függvény az origóban?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>függvény az origóban<ins class="diffchange diffchange-inline">? Folyt. diff.-e, létezik-e a gradiens, létezik-e a Jacobi-mátrix ott</ins>?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\partial_xf(0,0)=\lim\limits_{x\to 0}\frac{x^4-0}{x(x^2+0)}=0</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\partial_xf(0,0)=\lim\limits_{x\to 0}\frac{x^4-0}{x(x^2+0)}=0</math></div></td></tr>
<tr><td colspan="2" class="diff-lineno">19. sor:</td>
<td colspan="2" class="diff-lineno">19. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ugyanis  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>ugyanis  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{(x^2+x^2)^{\frac{3}{2}}}=\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{2^{\frac{3}{2}}x^3}=\lim\limits_{(x,x)\to (0,0)}\frac{1}{2^{\frac{3}{2}}}(x+1)\to \frac{1}{2^{\frac{3}{2}}}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math>\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{(x^2+x^2)^{\frac{3}{2}}}=\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{2^{\frac{3}{2}}x^3}=\lim\limits_{(x,x)\to (0,0)}\frac{1}{2^{\frac{3}{2}}}(x+1)\to \frac{1}{2^{\frac{3}{2}}}</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Tehát nem folyt diff, mert ha az lenne, akkor deriválható is lenne (egy környzetben létezik a derivált!). Jacobi van, gradiens nincs, mert az a differenciál leképezés skalárinvariánssa lenne ( az az ''m'' vektor, melyet az ''Ax''=''m''<math>\cdot</math>''x'' definiál, de nincs alkalmas ''A'', így nincs alkalmas ''m''). </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A2a_2008/11._gyakorlat&diff=5238&oldid=prev
Mozo, 2009. május 14., 19:06-n
2009-05-14T19:06:07Z
<p></p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr valign='top'>
<td colspan='2' style="background-color: white; color:black;">←Régebbi változat</td>
<td colspan='2' style="background-color: white; color:black;">A lap 2009. május 14., 19:06-kori változata</td>
</tr><tr><td colspan="2" class="diff-lineno">1. sor:</td>
<td colspan="2" class="diff-lineno">1. sor:</td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:''Ez az szócikk a [[Matematika A2a 2008]] alszócikke.''</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:''Ez az szócikk a [[Matematika A2a 2008]] alszócikke.''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">==Totális és folytonos parciális deriválhatóság==</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Deriválható-e az </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>f(x,y)=\left\{\begin{matrix}\frac{x^4-y^3}{x^2+y^2} & \mathrm{ha} & (x,y)\ne (0,0)\\ \\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">0, & \mathrm{ha} & (x,y)=(0,0)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\end{matrix}\right.</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">függvény az origóban?</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\partial_xf(0,0)=\lim\limits_{x\to 0}\frac{x^4-0}{x(x^2+0)}=0</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\partial_yf(0,0)=\lim\limits_{y\to 0}\frac{0-y^3}{y(0+y^2)}=-1</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">A deriválhatóság:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\lim\limits_{(x,y)\to (0,0)}\frac{\frac{x^4-y^3}{x^2+y^2}-0- 0\cdot x +y}{\sqrt{x^2+y^2}}=\lim\limits_{(x,y)\to (0,0)}\frac{x^4-y^3+y(x^2+y^2)}{(x^2+y^2)^{\frac{3}{2}}}=</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">=\lim\limits_{(x,y)\to (0,0)}\frac{x^4+yx^2}{(x^2+y^2)^{\frac{3}{2}}}\not\to 0</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">ugyanis </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">:<math>\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{(x^2+x^2)^{\frac{3}{2}}}=\lim\limits_{(x,x)\to (0,0)}\frac{x^4+x^3}{2^{\frac{3}{2}}x^3}=\lim\limits_{(x,x)\to (0,0)}\frac{1}{2^{\frac{3}{2}}}(x+1)\to \frac{1}{2^{\frac{3}{2}}}</math></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>{| class="wikitable" style="text-align:center"</div></td></tr>
</table>
Mozo
http://wiki.math.bme.hu/index.php?title=Matematika_A2a_2008/11._gyakorlat&diff=5237&oldid=prev
Mozo, 2009. május 14., 18:29-n
2009-05-14T18:29:26Z
<p></p>
<p><b>Új lap</b></p><div>:''Ez az szócikk a [[Matematika A2a 2008]] alszócikke.''<br />
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{| class="wikitable" style="text-align:center"<br />
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||[[Matematika A2a 2008/10. gyakorlat |10. gyakorlat]] || [[Matematika A2a 2008/11. gyakorlat |12. gyakorlat]]<br />
|}<br />
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[[Kategória:Matematika A2]]</div>
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