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\centerline{MATH 241 -- Multivariable Calculus}
\centerline{Discussion Problems -- A Hike in the Mountains}
\centerline{October 4, 2013}
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\noindent
{\it Background}
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A contour plot for a function $f(x,y)$ on a set $U \subset {\bf R}^2$ is 
similar to the {\it contour maps} that you may have seen or used if you
have ever gone on a wilderness hiking trip.  A contour map is really 
just the contour plot for the {\it elevation function} describing the 
landscape of a region of the Earth's surface -- usually the elevation above
the reference level of sea level.  Today we will try to practice visualizing
what contour plots tell us about the shape of a graph $z = f(x,y)$ by looking
at an actual contour map and trying to describe the landscape it represents.
Here is the map, from Exercise 4 in Section 3.2 of our text: 

\vglue 5.0truein

The map shows a portion of a state park at Mount Moosilauke in New Hampshire.
The darker contours are marked with elevations that are multiples of 200 feet.
The lighter contours in between those come with each increase in elevation of
40 feet.  {\it Assume horizontal distances are according to a $1$ inch = $1$ mile
scale.}

The dotted lines on the map are hiking trails.  We will study 
a hike that follows this path:
\item{1)} The path starts from the lower left corner of the map. 
\item{2)} Then it follows the Beaver Brook trail right up to 
the right boundary of the map
\item{3)}  Finally, the trail takes a right turn and 
follows the unmarked trail into Jobildunk Ravine and continues
to the point near the lower right corner.  
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\noindent
Imagine that your group has been asked to write a description 
of the hiking trail for a guidebook.  Write up a verbal description
of about two medium length paragraphs, incorporating the answers
to the following questions.
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\noindent
{\it Questions}
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\item{A)}  What are the approximate elevations of the starting point, 
the place where you turn off the Beaver Brook trail, and the 
final point.
\item{B)}  Carefully describe the first part of the hike along the 
Beaver Brook trail.  Where is the trail the steepest and where is it
flatter (less steep)?  Approximately how long is this section of the 
hike?  (Note that it is not level, so you will need to figure out
how to carry out an estimate.  Describe the method you arrived at in 
your answer.) 
\item{C)}   Now, carefully describe the second part of the hike along
the unmarked Trail.  Be sure to describe any interesting features the trail
passes through.  
\bye
