Mathematical Modeling 2001

Problems 0703

1. For a 3-step random walk in the x direction, followed by a 3-step random walk in the y direction, at what average distance from the origin does a random walker end up?

• How might you figure out the average distance from the origin by a 10-step random walk in each the x and the y direction?

2. A string with a beads located at x = 0, 3 cm, 6 cm, ..., 18 cm is originally stretched out along the x axis. The end beads are 'nailed down' so they can't move. The other beads are free to move a little ways up and down; then they do so they stretch the string out a bit and the string stretches a bit tends to pull them back toward the x axis—unless of course a neighboring bead is further out, in which case the string tends to pull closer beads away from the x axis.

The beads are pulled to positions +2 mm (for the bead at 3 cm), -1 mm, -4 mm, +1 mm, +3 mm (that's the bead at 15 cm). Sketch a graph of this configuration, and determine the slopes of the six segments between the beads.

Each bead is inially stationary, and each bead experiences an rate of velocity change which is equal to the slope to its right, minus the slope to its left, with the acceleration in cm / s / s.

Where will each bead be, and what will be the velocity of each, 2 seconds later?

Starting from the velocities and positions you just found, determine the new slopes, and use these new slopes to get the new rates of velocity change. Calculate the velocities and positions after another 2 seconds.

3. Repeat the problem with the Christmas trees using 9 trees in a circle instead of 10. How does the behavior of this system differ from that of the 10-tree system for different rules, and why?

4. Make up the transition rule that gives the Christmas tree systems the most interesting behavior.

5. Sketch and label the trapezoidal graph for y = x^2 + 1 for x varying from 0 to 1 by increment .5—that is, evaluate y for x = 0, .5 and 1. What is your total area? Do you think this area is greater or less than the area under the actual y = x^2 + 1 curve?

Repeat for the same function on the same interval, but use increment .25

• Do you think your area is closer or less close to the actual area under the y = x^2 + 1 curve than before?
• Write your slopes as a sequence and predict the next slope.
• If your next slope is correct, how could you use it to predict the next value of y?