\documentclass[12pt]{letter}
\usepackage{latexsym}
\usepackage{amsmath, amssymb}
\oddsidemargin 0.0in
\textwidth 6.5in
\headheight -0.75in
\topmargin 0.5in
\headsep 0.0in
\textheight 8.7in
\parskip 3mm
\pagestyle{empty}
\begin{document}
{\bf HW 5\\Math 261, F18}
Please see the course syllabus for details on how to turn in your homework assignments. This one is due at the beginning of your class on \underbar{\bf Friday, October 5}.
\begin{enumerate}
\item Find the derivative of $h(x,y,z) = x+2y^2+3z^3$ at the point $(2, 0, \sqrt{2})$ in the direction of the vector $\mathbf{v}=\langle 1,1,0\rangle$.
\vspace{0.1in}
\item Find the equation for the tangent plane to the surface $x^2-xy-y^2-z=0$ at the point $(1,1,-1)$. Please give your answer in the form $Ax+By+Cz=D$.
\vspace{0.1in}
\item Give the best possible upper bound (using the technique from class, i.e. using a linear approximation) for the error in approximating $f(x,y)=x^2+3xy-2y^2$ at the point $(1,1)$, over the rectangle $|x-1|\leq 0.1$, $|y-1|\leq 0.3$. It is OK to leave your answer as a numerical expression (i.e., not simplified down to a number).
\vspace{0.1in}
\item Let $f$ be some function of the plane, such that $(1,1)$ and $(1,-1)$ are critical points. Suppose $f_{xx}=x+2$, $f_{xy}=x+y-2$, and $f_{yy}=y+1$. Classify (min/max/SP) the critical points $(1,1)$ and $(1,-1)$, clearly indicating any computed values you used to make your decision.
\vspace{0.1in}
\end{enumerate}
Be sure to study up on Lagrange multipliers, too, since we won't be able to give you homework on that before the exam!
\end{document}