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\begin{document}

\centerline{\bf MAT 0147 - C\'alculo 2 para Economia} 

\centerline{\bf Prova Substitutiva - 7 de dezembro de 2016}

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\noindent
{\bf Quest\ao\ 1} (1 pt) Seja $f(x,y)=xe^{-y^2}+y$. Calcule o m\'aximo valor da derivada direcional $D_{\mathbf u}f(1,0)$
de $f$ no ponto $(1,0)$ na dire\cao\ de um vetor unit\'ario ${\mathbf u}$. 

\newpage
\noindent
{\bf Quest\ao\ 2} (2,5 pts) Seja $f:\R\to\R$ uma fun\cao\ possuindo derivadas de primeira e de segunda ordem, $f'$ e $f''$. 
Seja $u:\R^2\to\R$ definida por $u(x,y)=f(\sqrt{x^2+y^2})$. Mostre que 
\[
u_{xx}(x,y)+u_{yy}(x,y)=f''(\sqrt{x^2+y^2})+\frac{f'(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}.
\]
{\sc Dica}: Voc\^e s\'o precisa da Regra da Cadeia do C\'alculo 1 para resolver este exerc\'icio. 

\newpage
\noindent
    {\bf Quest\ao\ 3} (2 pts) Mostre que todo plano tangente \`a superf\'icie $x^2+y^2=2z^2$ passa pela origem. 

\newpage
\noindent {\bf Quest\ao\ 4} (2 pts) Mostre que o \'unico ponto cr\'itico de $f(x,y)=x^2+y^2(1+x)^3$ \'e um ponto de 
m\'inimo local em $(0,0)$.

\newpage
\noindent
    {\bf Quest\ao\ 5}. (2,5 pts) Considere $f(x,y)=x^3+y^2$ e $g(x,y)=x^2+2y^2$.
\\(a) Encontre todos os pontos da curva $g(x,y)=1$ nos quais $\nabla f$ \'e paralelo a $\nabla g$.
\\(b) Encontre os pontos de m\'aximo e os pontos de m\'inimo 
de $f(x,y)$ sujeita \`a restri\cao\ $g(x,y)=1$


    

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