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\noindent {\bf 21-259} \hfill {\bf Calculus in 3-D} \hfill {\bf Spring 2003}

\centerline{\bf Answers to Review Problems for Test 2}

\begin{enumerate} 
\item[1.] \begin{enumerate} \item $\vec{\nabla}f(x,y,z) = \left<y,x+2yz,y^2\right>$ 

\item ${\displaystyle \frac{-37}{\sqrt{29}}}$ 
\end{enumerate} 

\item[2.] $\vec{\nabla} f(0,0,0) ={\displaystyle \frac{-5}{\sqrt{41}}} \left<6,-1,2\right>$ 

\item[3.] Tangent Plane:  $-2(x+1)+16(y-2)-(z-17)=0$

Normal Line:  ${\displaystyle \frac{x+1}{-2}} = {\displaystyle \frac{y-2}{16}} = {\displaystyle \frac{z-17}{-1}}$

\item[4.] $\vec{\nabla}f(2,1) = \left<-6,10\right>$

\item[5.] $(2,1,-2)$ and $(-2,-1,2)$

\item[6.] $ 3(rs)^2(r^2-s^2)^3r-6s\left[(rs)^3+2(r^2-s^2)^3\right](r^2-s^2)^2 - 12(2r+s)^2 $ 

\item[9.] ${\displaystyle 4u \frac{d^2w}{du^2}} + {\displaystyle  4 \frac{dw}{du}}$ 

\item[10.] \begin{enumerate} \item relative min at $(-1,1)$ 

\item relative min at $(0,0)$; saddle points at $\left({\displaystyle \frac{1}{2}},-1\right)$ and $(2,2)$
\end{enumerate} 

\item[11.] Absolute min occurs at (2,1).  Absolute max occurs at (0,4). $f(2,1)=-4$ and $f(0,4)=9$


\item[12.] \begin{enumerate} \item Possible maxima and minima at $(0,1,-3),\ (0,-1,3)$ $\left[{\rm corresponding\ to\ \lambda}\ = \pm {\displaystyle \frac{1}{2}}\right]$ and at $\left(\lambda,\ {\displaystyle \frac{1}{2 \lambda}},\ {\displaystyle \frac{-3}{2 \lambda}} \right)$ with $\lambda = \pm \sqrt{{\displaystyle \frac{10+\sqrt{90}}{2}}}$ and $\lambda = \pm \sqrt{{\displaystyle \frac{10-\sqrt{90}}{2}}}$. 

\item Possible maxima and minima at $
\left( {\displaystyle \frac{\sqrt{2}}{3}},\ \sqrt{2} \right), \
 \left({\displaystyle \frac{-\sqrt{2}}{3}},\  -\sqrt{2} \right),\
  \left({\displaystyle - \frac{\sqrt{2}}{3}},\  \sqrt{2}  \right),$\newline 
  $\left({\displaystyle \frac{\sqrt{2}}{3}},\ -\sqrt{2} \right)$, 


\end{enumerate}

\item[13.] \begin{enumerate} \item ${\displaystyle \int^1_0} {\displaystyle \int^x_{x^2}}f(x,y)dydx$

\item ${\displaystyle \int^{e^4}_1}\ {\displaystyle \int^2_{\frac{1}{2}lny}} f(x,y)dxdy$ 

\item ${\displaystyle \int^4_0}\ {\displaystyle \int^{y/2}_0}f(x,y)dxdy$

\end{enumerate} 

\item[14.] \begin{enumerate} \item ${\displaystyle \frac{2}{9}}$ \hspace{.50in}  \item 
${\displaystyle \frac{(2^{3/2}-1)}{4}}$ 

\end{enumerate}
\end{enumerate}

\end{document} 