\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} The figure below shows some level
curves of an unspecified function $f(x,y)$. These are curves of the
form $f(x,y)=k$ for integer values of $k$ from 1 to 11.  The
tick-marks on the axes are one unit apart.  The questions below refer
only to the region {\it inside}\/ the $k=0$ curve. To help understand
the diagram, note that the curves for $k=4$ and $k=8$ each intersect
the $x$-axis {\it twice}\/ at tick-marks.)

\medskip

\noindent a) Where is the quantity $\|\nabla f\|$ greatest?
Approximately what is its value there?  What is the direction of
$\nabla f$ there?

\medskip

\noindent b) Where is the quantity $\|\nabla f\|$ smallest?
Approximately what is its value there?  What is the direction of
$\nabla f$ there?
 
\medskip

\noindent c) Is $\nabla f(0,4)$ more nearly parallel to ${\bf i}+{\bf
j}$, ${\bf i}-{\bf j}$, or $-{\bf i}+{\bf j}$?
 

\bigskip
\input pictex
\line{\hss  \beginpicture
\setcoordinatesystem units <0.20truein,0.20truein> point at 0 0
\setplotarea x from -14 to 14, y from -13 to 13
 \axis bottom shiftedto y=0 ticks in from -13 to 13 by 1 /
 \axis left shiftedto x=0 ticks in from -12 to 12 by 1 /
 \put {$x$} [tl] at 14 -0.4
 \put {$y$} [tl] at 0.4 13
 \circulararc 145 degrees from 12.25 0 center at 0.25 0.0
 \circulararc -210 degrees from 12.25 0 center at 0.25 0.0
 \circulararc 145 degrees from 11.25 0 center at 0.25 0.0
 \circulararc -210 degrees from 11.25 0 center at 0.25 0.0
 \circulararc 144 degrees from 10.25 0 center at 0.25 0.0
 \circulararc -210 degrees from 10.25 0 center at 0.25 0.0
 \circulararc 143 degrees from 9.5 0 center at 0.5 0.0
 \circulararc -210 degrees from 9.5 0 center at 0.5 0.0
 \circulararc 143 degrees from 9.0 0 center at 1.0 0.0
 \circulararc -210 degrees from 9.0 0 center at 1.0 0.0
 \circulararc 142 degrees from 8.75 0 center at 1.75 0.0
 \circulararc -210 degrees from 8.75 0 center at 1.75 0.0
 \circulararc 141 degrees from 8.5 0 center at 2.5 0.0
 \circulararc -210 degrees from 8.5 0 center at 2.5 0.0
 \circulararc 139 degrees from 8.25 0 center at  3.25 0.0
 \circulararc -210 degrees from 8.25 0 center at  3.25 0.0
 \circulararc 137 degrees from 8.0 0 center at 4.0 0.0
 \circulararc -210 degrees from 8.0 0 center at 4.0 0.0
 \circulararc 135 degrees from 7.5 0 center at 4.5 0.0
 \circulararc -210 degrees from 7.5 0 center at 4.5 0.0
 \circulararc 125 degrees from 6.75 0 center at 4.75 0.0
 \circulararc -210 degrees from 6.75 0 center at 4.75 0.0
 \circulararc 100 degrees from 5.75 0 center at 4.75 0.0
 \circulararc -210 degrees from 5.75 0 center at 4.75 0.0
 \put {$\scriptstyle k=0$} [bl] at -10.542  6.200
 \put {$\scriptstyle1$} [bl] at  -9.276  5.700
 \put {$\scriptstyle2$} [bl] at  -8.310  5.200
 \put {$\scriptstyle3$} [bl] at  -7.194  4.700
 \put {$\scriptstyle4$} [bl] at  -5.828  4.200
 \put {$\scriptstyle5$} [bl] at  -4.212  3.700
 \put {$\scriptstyle6$} [bl] at  -2.596  3.100
 \put {$\scriptstyle7$} [bl] at  -0.980  2.600
 \put {$\scriptstyle8$} [bl] at  0.636  2.100
 \put {$\scriptstyle9$} [bl] at  2.002  1.600
 \put {$\scriptstyle10$} [bl] at  3.018  1.100
 \put {$\scriptstyle11$} [bl] at  3.884  0.600
 \endpicture\hss}
 








\vfil\eject\end

