http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?action=history&feed=atom&title=Derivada_direcional 2026-07-03T21:13:45Z Histórico de edições para esta página nesta wiki MediaWiki 1.35.2 http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Derivada_direcional&diff=4394&oldid=prev 2018-04-06T12:48:32Z

<p></p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="pt"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisão anterior</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisão das 12h48min de 6 de abril de 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27" >Linha 27:</td> <td colspan="2" class="diff-lineno">Linha 27:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>D)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>D)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Para obter o zip que contém as instâncias deste exercício clique aqui<del class="diffchange diffchange-inline">(derivadaDirRnRm)</del></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Para obter o zip que contém as instâncias deste exercício clique aqui<ins class="diffchange diffchange-inline">[https://drive.tecnico.ulisboa.pt/api/drive/file/851498741309848/download]</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt</div></td></tr> </table>

Ist178052 http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Derivada_direcional&diff=4392&oldid=prev 2018-04-06T12:38:00Z

<p></p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="pt"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisão anterior</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisão das 12h38min de 6 de abril de 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7" >Linha 7:</td> <td colspan="2" class="diff-lineno">Linha 7:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*ANO: 1</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*ANO: 1</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*LINGUA: pt</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*LINGUA: pt</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*AUTOR: <del class="diffchange diffchange-inline">Equipa Calculo diferencial </del>e <del class="diffchange diffchange-inline">integral 2</del></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*AUTOR: <ins class="diffchange diffchange-inline">Ana Moura Santos </ins>e <ins class="diffchange diffchange-inline">Miguel Dziergwa</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*MATERIA PRINCIPAL: Derivadas parciais</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*MATERIA PRINCIPAL: Derivadas parciais</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*DESCRICAO:  </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*DESCRICAO: <ins class="diffchange diffchange-inline">Cálculo de derivada direcional</ins></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*DIFICULDADE: <del class="diffchange diffchange-inline">easy</del></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*DIFICULDADE: <ins class="diffchange diffchange-inline">**</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*TEMPO MEDIO DE RESOLUCAO: 15 mn</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*TEMPO MEDIO DE RESOLUCAO: 15 mn</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*TEMPO MAXIMO DE RESOLUCAO: 30 mn</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*TEMPO MAXIMO DE RESOLUCAO: 30 mn</div></td></tr> </table>

Ist178052 http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Derivada_direcional&diff=3864&oldid=prev 2018-02-25T15:51:24Z

<p></p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="pt"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisão anterior</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisão das 15h51min de 25 de fevereiro de 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8" >Linha 8:</td> <td colspan="2" class="diff-lineno">Linha 8:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*LINGUA: pt</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*LINGUA: pt</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*AUTOR: Equipa Calculo diferencial e integral 2</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*AUTOR: Equipa Calculo diferencial e integral 2</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*MATERIA PRINCIPAL:  </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*MATERIA PRINCIPAL: <ins class="diffchange diffchange-inline">Derivadas parciais</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*DESCRICAO:  </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*DESCRICAO:  </div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*DIFICULDADE: easy</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*DIFICULDADE: easy</div></td></tr> </table>

Ist12543 http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Derivada_direcional&diff=1962&oldid=prev 2016-08-31T12:24:01Z

<p></p> <table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="pt"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Revisão anterior</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revisão das 12h24min de 31 de agosto de 2016</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17" >Linha 17:</td> <td colspan="2" class="diff-lineno">Linha 17:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/div&gt;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/div&gt;</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Seja \(f: D \<del class="diffchange diffchange-inline">[Subset] \</del>\mathbb{R^2} <del class="diffchange diffchange-inline">\</del>\to <del class="diffchange diffchange-inline">\</del>\mathbb{R^3} \) uma função dada por \(f\)\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) = \(\left(\begin{array}{c}-2\log\left(x^2y^2\right)\\\sqrt{\frac{x^2}{16}-\frac{y^2}{9}+9}\\2\log\left(x^e+y^2\right)\\\end{array}\right)\). Então a derivada direcional de \(f\), nos pontos interiores do domínio \(D\), na direção do vector\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) é igual a:</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Seja \(f: D \<ins class="diffchange diffchange-inline">subset </ins>\mathbb{R^2} \to \mathbb{R^3} \) uma função dada por \(f\)\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) = \(\left(\begin{array}{c}-2\log\left(x^2y^2\right)\\\sqrt{\frac{x^2}{16}-\frac{y^2}{9}+9}\\2\log\left(x^e+y^2\right)\\\end{array}\right)\). Então a derivada direcional de \(f\), nos pontos interiores do domínio \(D\), na direção do vector\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) é igual a:</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{27x-16y}{12\sqrt{9x^2-16\left(y^2-81\right)}}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{27x-16y}{12\sqrt{9x^2-16\left(y^2-81\right)}}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)</div></td></tr> </table>

Ist178052 http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Derivada_direcional&diff=1961&oldid=prev 2016-08-31T12:20:08Z

<p>Criou a página com &quot;&lt;div class=&quot;toccolours mw-collapsible mw-collapsed&quot; style=&quot;width:420px&quot;&gt; &#039;&#039;&#039;Metadata&#039;&#039;&#039; &lt;div class=&quot;mw-collapsible-content&quot;&gt; *CONTEXTO : Primeiro ciclo universitário *AREA:...&quot;</p> <p><b>Página nova</b></p><div>&lt;div class=&quot;toccolours mw-collapsible mw-collapsed&quot; style=&quot;width:420px&quot;&gt;<br /> '''Metadata'''<br /> &lt;div class=&quot;mw-collapsible-content&quot;&gt;<br /> *CONTEXTO : Primeiro ciclo universitário<br /> *AREA: Matemática<br /> *DISCIPLINA: Calculo diferencial e integral 2<br /> *ANO: 1<br /> *LINGUA: pt<br /> *AUTOR: Equipa Calculo diferencial e integral 2<br /> *MATERIA PRINCIPAL: <br /> *DESCRICAO: <br /> *DIFICULDADE: easy<br /> *TEMPO MEDIO DE RESOLUCAO: 15 mn<br /> *TEMPO MAXIMO DE RESOLUCAO: 30 mn<br /> *PALAVRAS CHAVE: <br /> &lt;/div&gt;<br /> &lt;/div&gt;<br /> <br /> Seja \(f: D \[Subset] \\mathbb{R^2} \\to \\mathbb{R^3} \) uma função dada por \(f\)\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) = \(\left(\begin{array}{c}-2\log\left(x^2y^2\right)\\\sqrt{\frac{x^2}{16}-\frac{y^2}{9}+9}\\2\log\left(x^e+y^2\right)\\\end{array}\right)\). Então a derivada direcional de \(f\), nos pontos interiores do domínio \(D\), na direção do vector\(\left(\begin{array}{c}x\\y\\\end{array}\right)\) é igual a:<br /> <br /> A)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{27x-16y}{12\sqrt{9x^2-16\left(y^2-81\right)}}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)<br /> <br /> B)\(\left(\begin{array}{c}-\frac{4(3x+y)}{xy}\\\frac{3x-16y}{4\sqrt{9x^2-16\left(y^2-81\right)}}\\\frac{2\left(ex^e+6xy\right)}{x\left(x^e+y^2\right)}\\\end{array}\right)\)<br /> <br /> C)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{3x-16y}{4\sqrt{9x^2-16\left(y^2-81\right)}}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)<br /> <br /> D)\(\left(\begin{array}{c}-\frac{4(x+3y)}{xy}\\\frac{6ex^e+4xy}{x^{1+e}+xy^2}\\\end{array}\right)\)<br /> <br /> Para obter o zip que contém as instâncias deste exercício clique aqui(derivadaDirRnRm)<br /> <br /> Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt</div>

Ist178052