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\begin{document}
Universidade Federal da Paraiba

Centro de Ci\^{e}ncias Exatas e da Natureza

P\'os-Gradua\c{c}\~ao em Matem\'{a}tica

Disciplina: An\'{a}lise II- Prof. Milton

\begin{center}
\bigskip4$^{\underline{a}}\;\;$Lista de Exerc\'{\i}cios

\bigskip
\end{center}

\begin{enumerate}
\item Seja $f:\mathbb{R}^{2} \rightarrow\mathbb{R}$ definida por $f(0)=0$ e
\[
f(x,y)=xy/(x^{2}+y^{2}) \quad\mbox{se}\quad(x,y)\neq(0,0).
\]


\begin{enumerate}
\item Para quais vetores $u\neq0$ existe a derivada direcional $\frac{\partial
f}{\partial u}(0)$?

\item Existe $D_{1} f$ e $D_{2} f$ em 0?

\item $f$ \'e diferenci\'avel em 0?

\item $f$ \'e cont\'{\i}nua em 0?
\end{enumerate}

\item Seja $f:\mathbb{R}\rightarrow\mathbb{R}^{2}$ , $f\left(  t\right)
=\left(  \cos t,\sin t\right)  $. Fixado $a\in\mathbb{R}^{2}$ , encontre
$z\in\left(  0,2\pi\right)  $ tal que\bigskip\ $a\left[  f\left(  y\right)
-f\left(  x\right)  \right]  =a\left\{  f^{\prime}\left(  z\right)  \left(
y-x\right)  \right\}  $, com $x=0$ e $y=2\pi.\bigskip$

\item Seja $f:\mathbb{R}\rightarrow\mathbb{R}$ , $f\left(  x\right)  =\left\{
%
\begin{array}
[c]{c}%
e^{-x^{-2}},x\neq0\\
0,x=0
\end{array}
\right.  .$ Prove que f \'{e} classe $C^{\infty}.\bigskip$

\item Seja $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ uma fun\c{c}\~{a}o tal que
$\left\vert f\left(  x\right)  \right\vert \leq\left\Vert x\right\Vert ^{2}$ ,
para todo x $\in\mathbb{R}^{n}.$ Mostre que\bigskip\ f \'{e} diferenci\'{a}vel
na origem. Ache $df\left(  0\right)  .\bigskip$

\item Seja $f:\mathbb{R}\rightarrow\mathbb{R}^{n}$ um caminho
diferenci\'{a}vel tal que $\left\Vert f\left(  t\right)  \right\Vert
=1$,$\forall t\in R.$ Mostre\bigskip\ que $\left\langle f^{\prime}\left(
t\right)  ,f\left(  t\right)  \right\rangle =0.\bigskip$

\item Mostre que a fun\c{c}\~ao $f(x,y)=|xy|$ \'e diferenci\'avel em 0, mas
n\~ao \'e de classe $C^{1}$ em qualquer vizinhan\c{c}a de 0.

\item Seja $f:\mathbb{R}^{3} \rightarrow\mathbb{R}^{2}$ satisfazendo as
condi\c{c}\~oes $f(0)=(1,2)$ e
\[
Df(0)=\left[
\begin{array}
[c]{lcr}%
1 & 2 & 3\\
0 & 0 & 1
\end{array}
\right] .
\]
Seja $g:\mathbb{R}^{2} \rightarrow\mathbb{R}^{2}$ definida por
\[
g(x,y)=(x+2y+1,3xy).
\]
Achar $D(g\circ f)(0).$

\item Seja $f:\mathbb{R}^{2} \rightarrow\mathbb{R}^{3}$ e $g:\mathbb{R}^{3}
\rightarrow\mathbb{R}^{2}$ dada pelas equa\c{c}\~oes
\[
f(x)=(e^{2x_{1}+x_{2}}, 3x_{2}-\cos x_{1}, x_{1}^{2}+x_{2}+2),
\]
\[
g(x)=(3y_{1}+2y_{2}+y_{3}^{2}, y_{1}^{2}-y_{3}+1).
\]


\begin{enumerate}
\item Se $F(x)=g(f(x))$ achar $DF(0).$

\item Se $G(x)=f(g(y))$ achar $DG(0).$
\end{enumerate}

\item Seja $f:\mathbb{R}^{2} \rightarrow\mathbb{R}^{2}$ definida por
\[
f(x,y)=(x^{2}-y^{2}, 2xy).
\]


\begin{enumerate}
\item Mostre que $f$ \'e injetora sobre o conjunto $A$ consistindo de todos os
$(x,y)$ com $x>0.$

\item Quem \'e $B=f(A)$.

\item Se $g$ \'e a fun\c{c}\~ao inversa, achar $DG(0,1).$
\end{enumerate}

\item Seja $f:\mathbb{R}^{n} \rightarrow\mathbb{R}^{n}$ dada pela
equa\c{c}\~ao $f(x)=\|x\|^{2} x.$ Mostre que $f$ \'e de classe $C^{\infty}$ e
que $f$ leva a bola unit\'aria $B(0,1)$ sobre ela mesma de um modo injetor.
Mostre, contudo, que a fun\c{c}\~ao inversa n\~ao \'e diferenci\'avel em 0.

\item Sejam $\xi:I\rightarrow\mathbb{R}$ cont\'{\i}nua e $f:\mathbb{R}%
^{2}\rightarrow\mathbb{R}$ de classe $C^{1},$ com $\dfrac{\partial f}{\partial
y}\neq0$ em todos os pontos. Se $f(x,\xi(x))=0$ para todo $x\in I,$ prove que
$\xi$ \'{e} de classe $C^{1}.$

\item Seja $f:U\rightarrow\mathbb{R}$ definida no aberto $U\subset
\mathbb{R}^{2},$ tal que $(x^{2}+y^{2})f(x,y)+f(x,y)^{2}=1$ para qualquer
$(x,y)\in U.$ Prove que $f\in C^{\infty}.$

\item Seja $f:U\rightarrow\mathbb{R}$ definida no aberto $U\subset
\mathbb{R}^{n}.$ Se a fun\c{c}\~{a}o $g:U\rightarrow\mathbb{R},$ dada por
$g(x)=\int_{0}^{f(x)}(t^{2}+1)dt,$ for de classe $C^{\infty},$ ent\~{a}o $f$
tamb\'{e}m ser\'{a} $C^{\infty}.$

\item Seja $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ de classe $C^{\infty},$ com
$f(x,0)=f(0,y)=0$ para quaisquer $x,y\in\mathbb{R}.$ Mostre que existe
$g:\mathbb{R}^{2}\rightarrow\mathbb{R}$ de classe $C^{\infty}$ tal que
$f(x,y)=g(x,y)\cdot x\cdot y$ para qualquer $(x,y)\in\mathbb{R}^{2}.$

\item Seja $f:U\rightarrow\mathbb{R}$ de classe $C^{k}$ ($i\leq k\leq\infty)$
no aberto convexo $U\subset\mathbb{R}^{2},$ contendo a origem. Suponha que $f$
e todas as suas derivadas parciais de ordem $\leq i$ se anulam na origem.
Prove que existem fun\c{c}\~{o}es $a_{0},$ $a_{1},$ $\cdots,$ $a_{i}%
:U\rightarrow\mathbb{R}$ de classe $C^{k-i},$ tais que $f(x,y)=\sum
\limits_{j=0}^{i}a_{j}(x,y)x^{j}y^{i-j}$ para todo ponto $(x,y)\in U.$

\item Seja $f\left(  x,y\right)  =\left\{
\begin{array}
[c]{c}%
2xy\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\text{ , }x^{2}+y^{2}\neq0\\
0\text{ , }\left(  x,y\right)  =\left(  0,0\right)
\end{array}
\right.  $. Mostre que $\frac{\partial^{2}f}{\partial x\partial y}\left(
0,0\right)  \neq\frac{\partial^{2}f}{\partial y\partial x}\left(  0,0\right)
.$

\item Seja $f:U\times\left[  a,b\right]  \rightarrow\mathbb{R}$ uma
fun\c{c}\~{a}o cont\'{\i}nua, com derivadas parciais $\frac{\partial
f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}\bigskip$
cont\'{\i}nuas em $U\times\left[  a,b\right]  .$ Seja $g:U\rightarrow\left[
a,b\right]  $ de classe $C^{1}$, $U\subset\mathbb{R}^{n}$ aberto. Defina
a\bigskip\ seguinte fun\c{c}\~{a}o:$\varphi:U\rightarrow\mathbb{R}$ ,
$\varphi\left(  x\right)  =\int_{a}^{g\left(  x\right)  }f\left(  x,t\right)
dt.\bigskip$

\begin{enumerate}
\item Mostre que $\varphi$ \'{e} de classe $C^{1}.\bigskip$

\item Mostre que $\frac{\partial\varphi}{\partial_{x}i}\left(  x\right)
=\int_{a}^{g\left(  x\right)  }\frac{\partial f}{\partial x_{i}}\left(
x,t\right)  dt+\frac{\partial g}{\partial x_{i}}\left(  x\right)  f\left(
x,g\left(  x\right)  \right) . $
\end{enumerate}
\end{enumerate}


\end{document}
