Hiking and Climbing in Yosemite
Part 2: Climbing
If you don't have your contour map
window open and you haven't printed it yet, click here
to open the map window again, and print the map page.
The map shows some of the level curves
of a function z = f(x,y), representing the elevation
in feet at longitude x and latitude y, where x and y
are measured relative to the axes on the map, with the origin on the North shore
of Mirror Lake. The scale is approximately 1000 ft/cm. You will use the map
in the rest of the module to describe a hiking, rock climbing, and canoeing
trip through the park.
- You
begin a climb from Mirror Lake. Ambitiously, you decide to head off from (0,0)
in the direction of steepest ascent. Draw a unit vector u0i + v0j
from (0,0) pointing in this direction. What are the components of this
vector?
- Follow u0i + v0j
from its tail at (0,0) to its tip at the new point P1 = (x1,y1).
You've traveled 1000 feet -- at least in the plane of the map. Puffing and
panting at P1, you tell yourself that it seemed longer
than that. Of course it was -- you moved a distance
s = 1000
feet on the map, but you moved a distance w
on the surface. Read the change in elevation z
off the map, and approximate the distance w
you actually climbed.
- How steep was the climb in step
3? You can measure this by finding the rate of change of elevation at (0,0)
in the direction you moved. How would you approximate this? What value do
you get? What if you had moved away from (0,0) in a different direction
-- what other rates of change are possible? Find values for a few of these.
How are these rates of change related to the directional derivatives of the
function f?
- Recall that the vector grad f
at the point (x0,y0)
is the vector in the direction of greatest increase of the function f at
that point with magnitude given by the value of the directional derivative
in that direction of greatest increase. To gain some experience with this
definition, answer the following questions.
- What are the components of
grad f(0,0)?
- How can we write u0i + v0j
in terms of grad f(0,0)?
- How does the direction of
grad f(0,0) compare with the direction of (i.e., the direction
tangent to) the level curve through (0,0)?
- Tiring of intellectual diversions,
you decide to get some additional physical exercise by continuing your climb.
Full of ambition again, you head off in the direction of grad f(x1,y1),
climbing for another unit distance, s = 1000
feet, and arriving at P2 = (x2,y2).
Mark P2 on your map. What was z?
What was w?
How steep was the climb?
- Continue on to P3,
P4, P5, by moving, at each Pn,
s = 1000
feet in the direction of grad f(xn,yn).
When does this process terminate? Mark your path on the map, and compile a
table with the following entries for each interval from Pn
to Pn+1: n, s,
z, w,
grad f(xn,yn), and |grad f(xn,yn)|.
- Write a general formula giving
Pn+1 in terms of Pn and grad f(xn,yn).
(For this formula, identify Pn and Pn+1
with the vectors from the origin to these points.)
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