Yosemite, Part 2

Hiking and Climbing in Yosemite

Part 2: Climbing

If you don't have your contour map window open, click here to open it.

The map shows level curves of a function z = f(X,Y), representing the elevation in feet at (X,Y) as measured relative to the axes on the map. You will use the map in the rest of the module to describe a hiking, rock climbing, and canoeing trip through the park.

Use the pixel location of the origin (North shore of Mirror Lake) to write formulas that enable you to calculate X and Y (in pixels) from the applet-measured coordinates at any x and y.

Find and record the pixel location where Tenaya Creek crosses the 5500 foot contour. What is the horizontal distance from Mirror Lake to this point? Answer first in pixel units, and then convert your answer to feet.

If you hiked northeast from Mirror Lake along Tenaya Creek, what would be the average slope of your path? Suppose there were a sign at the start of the trail that told hikers “__% grade.” What number would be in the blank?

You may have noticed that, if you "mark" points on the map, then each point is joined to the previous one by a line segment. Also, if you mark a sequence of points and then click on "List points", a window will pop up with a list of the pixel coordinates of your points. This list can be pasted into a word processor or spreadsheet file. If you have any marked points now, click "Clear points" before starting the next step.

You begin a climb from Mirror Lake. Ambitiously, you decide to head off from (0,0) in the direction of steepest ascent. Draw a vector u₀i + v₀j from (0,0) approximately 1000 feet long pointing in this direction. What are the components of this vector? Answer first in pixel units and then in feet. We will think of 1000 feet as our "unit" of distance.

Follow u₀i + v₀j from its tail at (0,0) to its tip at the new point P₁ = (X₁,Y₁). You've traveled about 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 5? 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 horizontal distance, s, of about 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 P_n, about 1000 feet in the direction of grad f(X_n,Y_n). When does this process terminate? Mark your path on the map, and compile a table with the following entries for each interval from P_n to P_n+1: n, s, z, w, grad f(X_n,Y_n), and |grad f(X_n,Y_n)|.

Write a general formula giving P_n+1 in terms of P_n and grad f(X_n,Y_n). (For this formula, identify P_n and P_n+1 with the vectors from the origin to these points.)

Your browser will not provide a way to print from an applet. However, you can do a "screen capture" and then paste the map window into the word processor or spreadsheet file in which you are recording your data. When you are ready to move on to Part 3, click "Clear points".