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{\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 $\mathbf{v}=\langle 1,1,0\rangle$.

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\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 $Ax+By+Cz=D$.

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\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).

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\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.

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\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!

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