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<Input><Text-field style="Text" layout="Normal" alignment="centred"><Font size="26" bold="true" encoding="UTF-8">Cap\303\255tulo 9-Verificando a Diferenciabilidade</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Dizemos que uma fun\303\247\303\243o em 2 vari\303\241veis \342\200\213\342\200\213\303\251 diferenci\303\241vel em um ponto se o gr\303\241fico pr\303\263ximo a esse ponto puder ser aproximado pelo plano tangente. Uma pergunta mais dif\303\255cil \303\251 como saber quando uma fun\303\247\303\243o dada por uma f\303\263rmula \303\251 diferenci\303\241vel. Esta folha de trabalho analisa como verificar se uma fun\303\247\303\243o \303\251 diferenci\303\241vel em um ponto.

<Font size="16" bold="true">1-Cuidando dos pontos f\303\241ceis - fun\303\247\303\243o suave</Font></Font></Text-field>
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</Font><Font bold="false" encoding="UTF-8">Um teorema padr\303\243o afirma que uma fun\303\247\303\243o \303\251 diferenci\303\241vel em um ponto se ambas as derivadas parciais s\303\243o definidas e cont\303\255nuas nesse ponto. Lembre-se de que os polin\303\264mios s\303\243o fun\303\247\303\265es cont\303\255nuas. Da mesma forma, obtemos uma fun\303\247\303\243o cont\303\255nua se considerarmos a soma, diferen\303\247a, produto ou composi\303\247\303\243o de duas fun\303\247\303\265es cont\303\255nuas. Tamb\303\251m podemos tomar quocientes se o denominador n\303\243o for 0 e ra\303\255zes se a fun\303\247\303\243o n\303\243o for negativa. Lembramos tamb\303\251m que a derivada n\303\243o \303\251 definida quando obtemos uma raiz de 0 ou um negativo. Assim, com uma boa fun\303\247\303\243o definida em um \303\272nico caso alg\303\251brico, notamos facilmente que a fun\303\247\303\243o \303\251 diferenci\303\241vel em todos os pontos, exceto talvez nos pontos em que dividimos por 0 ou obtemos uma raiz de 0.

Exerc\303\255cio 1:

As seguintes fun\303\247\303\265es s\303\243o dos exerc\303\255cios para a se\303\247\303\243o sobre diferenciabilidade no livro de C\303\241lculo Multivari\303\241vel de McCallum. Para cada fun\303\247\303\243o, o racioc\303\255nio acima permite dizer que a fun\303\247\303\243o \303\251 &quot;agrad\303\241vel&quot; em todos os lugares, exceto talvez em um pequeno conjunto de pontos. </Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Para cada fun\303\247\303\243o (1-7 e 10), identifique os pontos &quot;dif\303\255ceis&quot;. Escreva no Maple!!!!</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal">(a) f (x, y) = x / y + y / x se x.y <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVErJk5vdEVxdWFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVErJk5vdEVxdWFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv</Equation>0 e f (x, y) = 0 se x.y = 0.
(b) f (x, y) = 2x.y / (<Equation executable="true" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGWC1GLDYlLUYvNiVRInlGJ0YyRjVGOEZALUYsNiUtRkQ2LVEiKUYnRj4vRkhGNEZKL0ZNRjRGTkZQRlJGVC9GV1EsMC4xNjY2NjY3ZW1GJy9GWkZib0Y4RkAvJStleGVjdXRhYmxlR0ZJRj4=</Equation> se (x, y) <Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation>0 e f (0, 0) = 0.

(c) f (x, y) = x * y / sqrt (x ^ 2 + y ^ 2) se (x, y) <Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation>0 e f (0, 0) = 0.

(d) f (x, y) = x ^ 2 * y / (x ^ 4 + y ^ 2) se (x, y) <Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation>0 e f (0, 0) = 0.

(e) f (x, y) = x * y ^ 2 / (x ^ 2 + y ^ 2) se (x, y)<Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation>0 e f (0, 0) = 0.

(f) f (x, y) = x * y ^ 2 / (x ^ 2 + y ^ 4) se (x, y)<Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation> 0 e f (0, 0) = 0.

(g) f (x, y) = sqrt (abs (x.y)).

(h) f (x, y) = x * y * (x ^ 2 - y ^ 2) / (x ^ 2 + y ^ 2) se (x, y)<Equation executable="true" style="2D Math" input-equation="" display="LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA">LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LVElJm5lO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZA</Equation> 0 e f (0, 0) = 0.


<Font size="16" bold="true" encoding="UTF-8">2-Pontos realmente ruins - Onde a fun\303\247\303\243o n\303\243o \303\251 cont\303\255nua.</Font><Font encoding="UTF-8">

Uma fun\303\247\303\243o s\303\263 pode ser diferenci\303\241vel onde \303\251 cont\303\255nua. Pode ser necess\303\241rio revisar a se\303\247\303\243o sobre continuidade. Lembre-se de que uma fun\303\247\303\243o \303\251 cont\303\255nua em um ponto se o limite nesse ponto for igual ao valor da fun\303\247\303\243o naquele ponto. Como estamos preocupados apenas com pontos dif\303\255ceis, a fun\303\247\303\243o ser\303\241 definida por uma regra especial no ponto e por uma f\303\263rmula pr\303\263xima.

Considere a fun\303\247\303\243o definida no exemplo 4, f (x, y) = x * y / (x ^ 2 + y ^ 2), a menos que (x, y) seja a origem, caso em que usamos a regra especial f (0 , 0) = 0. O problema est\303\241 na origem. A primeira maneira de verificar a continuidade \303\251 sempre olhar um gr\303\241fico para ver se a fun\303\247\303\243o salta.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">\303\211 claro, olhando para a figura, que esta fun\303\247\303\243o n\303\243o \303\251 cont\303\255nua na origem. Se olharmos para o eixo z, fica claro que todos os contornos diferentes se estendem at\303\251 a origem.

(Dica t\303\251cnica - Ao examinar fun\303\247\303\265es que podem ser descont\303\255nuas, geralmente \303\251 bom especificar a visualiza\303\247\303\243o ou o intervalo z. Caso contr\303\241rio, o gr\303\241fico de uma fun\303\247\303\243o que vai at\303\251 o infinito pode eliminar os outros detalhes do gr\303\241fico. Tamb\303\251m \303\251 \303\272til use a op\303\247\303\243o shading = z para que o esquema de cores seja determinado pelo valor z. Em pontos descont\303\255nuos, muitas cores tentam se unir.)

Tamb\303\251m gostar\303\255amos de uma maneira alg\303\251brica de olhar para a continuidade. Para fazer isso, escolhemos dois caminhos para o ponto problem\303\241tico que parecem seguir contornos diferentes. Podemos ent\303\243o pegar limites antiquados nos caminhos e ver que eles t\303\252m limites diferentes. (Podemos encontrar um caminho que n\303\243o tem limite.) No nosso exemplo, escolhemos a reta y = x e y = -x para aproximar-se da origem. Maple nos permite avaliar os limites simbolicamente ou graficamente.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Como os limites \303\240 medida que nos aproximamos da origem dos 2 caminhos s\303\243o diferentes, a fun\303\247\303\243o n\303\243o pode ser cont\303\255nua na origem.

<Font size="16" bold="true">Exerc\303\255cio:</Font>

2) Use o MAPLE para mostrar que a fun\303\247\303\243o definida por

f (x, y) = x ^ 2 * y / (x ^ 4 + y ^ 2) quando (x, y) </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVErJk5vdEVxdWFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVErJk5vdEVxdWFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv</Equation><Font encoding="UTF-8">(0, 0) e f (0, 0) = 0

n\303\243o \303\251 cont\303\255nuo na origem. Encontre 2 caminhos para a origem, para que os limites dos caminhos sejam diferentes um do outro.

&gt;

<Font size="16" bold="true">3-Pontos dif\303\255ceis diferenci\303\241veis \342\200\213\342\200\213- parciais cont\303\255nuos</Font>

Depois de lidar com os pontos positivos e os pontos realmente ruins, estamos prontos para come\303\247ar a trabalhar desde o final. Normalmente, estamos analisando pontos em que a f\303\263rmula principal teria divis\303\243o por 0; portanto, a fun\303\247\303\243o \303\251 definida por uma segunda regra nesses pontos. Se a fun\303\247\303\243o for cont\303\255nua nesse ponto, o valor correto foi escolhido para preencher a lacuna.

Usaremos a fun\303\247\303\243o definida por f (x, y) = x.y (x ^ 2 - y ^ 2) / (x ^ 2 + y ^ 2) longe da origem e f (0, 0) = 0 A origem \303\251 o ponto dif\303\255cil desse problema. Uma olhada no gr\303\241fico mostra que corrigimos o furo na origem com o valor correto para que a fun\303\247\303\243o seja cont\303\255nua.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Observando os gr\303\241ficos, fica claro que as parciais s\303\243o cont\303\255nuas na origem. Pelo teorema do livro, a fun\303\247\303\243o \303\251 diferenci\303\241vel na origem, portanto em toda parte.

Exerc\303\255cio:

3) Use a t\303\251cnica descrita acima para mostrar a fun\303\247\303\243o definida por f (x, y) = x ^ 3 * y ^ 3 / ((x ^ 2 + y ^ 2) ^ 2) se [x, y] &lt;&gt; [ 0, 0] ef (0,0) = 0 \303\251 diferenci\303\241vel em qualquer lugar.

&gt;

<Font size="16" bold="true">4-Cont\303\255nuo, mas n\303\243o diferenci\303\241vel por falta de derivadas parciais</Font></Font><Font bold="true">

</Font><Font encoding="UTF-8">Quando come\303\247amos a trabalhar em fun\303\247\303\265es que s\303\243o cont\303\255nuas, mas n\303\243o diferenci\303\241veis, as mais f\303\241ceis s\303\243o aquelas em que as derivadas parciais n\303\243o est\303\243o definidas. Estas s\303\243o fun\303\247\303\265es que n\303\243o s\303\243o diferenciadas quando fazemos uma se\303\247\303\243o transversal em x ou y. Os exemplos mais f\303\241ceis envolvem valores e ra\303\255zes absolutos.

Seja f (x, y) = abs (x + y). A fun\303\247\303\243o \303\251 cont\303\255nua em todos os lugares. Quando tomamos as se\303\247\303\265es transversais x = 0 e y = 0, obtemos abs (y) e abs (x), respectivamente. Ambas as curvas t\303\252m cantos, portanto n\303\243o s\303\243o diferenci\303\241veis de 0. Portanto, as derivadas parciais de f (x, y) n\303\243o existem.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Exerc\303\255cio 4

Mostre que a fun\303\247\303\243o f (x, y) = (x ^ 2 + y ^ 2) ^ (1/5) n\303\243o \303\251 diferenci\303\241vel na origem.

&gt;

<Font size="16" bold="true">5-O caso dif\303\255cil - mostrando a n\303\243o diferenciabilidade de uma fun\303\247\303\243o cont\303\255nua.</Font>

O texto indica que uma fun\303\247\303\243o pode ser diferenci\303\241vel mesmo que as parciais n\303\243o sejam cont\303\255nuas. Queremos alguma maneira de mostrar que uma fun\303\247\303\243o n\303\243o \303\251 diferenci\303\241vel.

O truque \303\251 perceber que, para uma fun\303\247\303\243o diferenci\303\241vel, todos os vetores tangentes em um ponto est\303\243o em um plano.

Suponha que F (x, y) seja diferenci\303\241vel em (a, b) com </Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation> = c e <Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font encoding="UTF-8"> = d.

Seja U = [u_1, u_2] um vetor unit\303\241rio. Ent\303\243o a derivada direcional na dire\303\247\303\243o U deve ser a mesma, calculando-a a partir da defini\303\247\303\243o da cervivativa direcional ou tomando o produto escalar de U com o vetor gradiente.

Considere a fun\303\247\303\243o definida por f (x, y) = (x ^ 2 * y) / (x ^ 2 + y ^ 2).

Os coment\303\241rios sobre pontos f\303\241ceis nos dizem que essa fun\303\247\303\243o \303\251 diferenci\303\241vel em qualquer lugar, exceto talvez na origem. Veja o gr\303\241fico da fun\303\247\303\243o e suas derivadas parciais.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Vemos nos gr\303\241ficos que a fun\303\247\303\243o \303\251 cont\303\255nua e que as parciais s\303\243o descont\303\255nuas.

<Font size="16" bold="true">6-O m\303\251todo da for\303\247a bruta</Font>

Podemos simplesmente fazer com que o Maple calcule as derivadas direcionais pelos dois m\303\251todos e compare-as.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Como as duas formas de calcular derivadas direcionais nem sempre concordam (e de fato discordam em todos os lugares, exceto nos eixos), conclu\303\255mos que a fun\303\247\303\243o n\303\243o \303\251 diferenci\303\241vel.

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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Exerc\303\255cios:

5) Mostre que a fun\303\247\303\243o no exerc\303\255cio 3 n\303\243o \303\251 diferenci\303\241vel na origem, mesmo que seja cont\303\255nua l\303\241.

&gt;

<Font size="16" bold="true">6-Uma abordagem n\303\243o gr\303\241fica</Font>

N\303\243o precisamos calcular a derivada direcional pelos dois m\303\251todos em todas as dire\303\247\303\265es para mostrar que uma fun\303\247\303\243o n\303\243o \303\251 diferenci\303\241vel. Simplesmente precisamos mostrar que os dois m\303\251todos d\303\243o respostas diferentes em alguma dire\303\247\303\243o.

Considere a fun\303\247\303\243o que envia (x, y) para x ^ 2 * y / (x ^ 2 + y ^ 2) + sin (x) -2 * y e envia a origem para 0. Mais uma vez, os coment\303\241rios sobre pontos f\303\241ceis diga-nos que essa fun\303\247\303\243o \303\251 diferenci\303\241vel em qualquer lugar, exceto talvez na origem. Calculamos as parciais na origem.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Agora calculamos a derivada direcional ao longo da linha y = x (theta = Pi / 4) usando os dois m\303\251todos.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Como obtemos respostas diferentes, conclu\303\255mos que a fun\303\247\303\243o n\303\243o \303\251 diferenci\303\241vel na origem.

6) Determine o status da fun\303\247\303\243o definida por

f (x, y) = x * y ^ 2 / (x ^ 2 + y ^ 4) se (x, y) (0, 0) ef (0, 0) = 0

(exerc\303\255cio 6 da se\303\247\303\243o 13.10) na origem. N\303\243o \303\251 cont\303\255nuo, cont\303\255nuo, mas n\303\243o diferenci\303\241vel ou diferenci\303\241vel na origem?

&gt;

</Font><Font size="16" bold="true">7-A maneira mais eficiente de resolver esses problemas - convertendo em coordenadas polares</Font><Font encoding="UTF-8">

Os exerc\303\255cios para esta se\303\247\303\243o foram criados com o ponto dif\303\255cil na origem e com as fun\303\247\303\265es sim\303\251tricas sobre a origem. Tais fun\303\247\303\265es s\303\243o frequentemente mais f\303\241ceis de descrever em termos de coordenadas polares. Podemos usar o Maple para converter fun\303\247\303\265es em coordenadas polares.</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">A fun\303\247\303\243o csgn (r) \303\251 uma fun\303\247\303\243o de sinal que \303\251 1 ou -1. Observando a forma polar, fica claro que a fun\303\247\303\243o \303\251 cont\303\255nua na origem. Tamb\303\251m podemos ver que a fun\303\247\303\243o \303\251 zero em qualquer lugar no eixo x ou y. O tempo verifica a diferenciabilidade \303\240 medida que r se aproxima de 0. O problema que encontramos \303\251 o csgn (r), que \303\251 1 em uma dire\303\247\303\243o e -1 na outra.
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Assim, para a maioria dos valores de teta, obtemos um canto, n\303\243o uma linha tangente. (Os limites da m\303\243o direita e da esquerda para a derivada existem, mas s\303\243o diferentes um do outro.) A fun\303\247\303\243o claramente n\303\243o \303\251 diferenci\303\241vel na origem.</Font></Text-field>
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Exerc\303\255cio:

7) Converta as fun\303\247\303\265es nas partes (b) - (h) do exerc\303\255cio 1 na forma polar. Comente sobre continuidade e diferenciabilidade na origem.</Font></Text-field>
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<Group view="presentation" inline-output="false" labelreference="L85" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L86" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L87" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L88" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L89" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L90" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L93" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L94" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L95" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" inline-output="false" labelreference="L96" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input><Text-field style="Text" size="14" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block>
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