MATH 241 – Multivariable Calculus Discussion Problems – A Hike in the Mountains October 4, 2013

Background

A contour plot for a function f(x, y) on a set U ⊂ R^{2} is similar to the 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 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:

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. 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:

1) The path starts from the lower left corner of the map.

2) Then it follows the Beaver Brook trail right up to the right boundary of the map 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.

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.

Questions

A) What are the approximate elevations of the starting point, the place where you turn

off the Beaver Brook trail, and the final point. 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.)

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.