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%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
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%	\mbox{}
%	\title{Limits and Derivatives (MATH 53)}
%	
%	\maketitle


\section*{Partial Derivatives}
\bigskip

%TODO: add to surfaces problem: Is the surface $x^2+4xy+3yz+z^2=0 x 2 + 4 x y + 3 y z + z 2 = 0$  some kind of cone (a union of lines through the origin)? 

\begin{problem}
	Is the following function continuous.
	\[
		f(x,y) = \twopf{y \, \sin \big(\frac{1}{x^2 + y^2} \big)}{(x,y) \neq (0,0)}{0}{(x,y) = (0,0)}.
	\]
	Find the partial derivatives of $f$ at $(0,0)$.
\end{problem}

\begin{problem}
	Find the partial derivatives at $(0,0)$ of the following function:
	\[
		f(x,y) = \twopf{\frac{xy}{x^2 + y^2}}{(x,y) \neq (0,0)}{0}{(x,y) = (0,0)}.
	\]
	Is the function $f$ continuous at $(0,0)$. [Answer: $f_x(0,0) = f_y(0,0) = 0$, but $f$ is not continuous.]
\end{problem}


\begin{problem} [Stewart, Exercise~14.3.101]
	Use Clairaut's Theorem to show that if the third-order partial derivatives of $f$ are continuous, then $f_{xyy} = f_{yxy} = f_{yyx}$.
\end{problem}

\begin{problem} [Stewart, Exercise~14.3.103]
	If
	\[
		f(x,y) = x(x^2 + y^2)^{-3/2} e^{\sin(x^2y)},
	\]
	find $f_x(1,0)$. [Hint: Can we simplify the function evaluating on certain constant one of the variables?]
\end{problem}
\begin{problem}
	If $f \colon \rr^2 \to \rr$ is differentiable and constant on the line $x=y$, argue that $f_x(t,t) = -f_y(t,t)$ for all $t \in \rr$.
\end{problem}

	
\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}