In-class questions:

1.  Each set of three points lies on the graph of a function which is obtained by stretching and/or shifting from one of the functions y = x^2 or y = 2^x.  Sketch the graph corresponding to each set of points:

(5, 3), (6, 4) and (7, 3).

(2, 1/2), (8, 2), (11, 4).

2.  Mama whale is 40 feet long and baby whale is 20 feet long.  Mama whale weighs 42 000 pounds.  If baby whale is geometrically similar to Mama whale, how much does he weigh?  Hint:  If you were to double baby whale's length without changing any of its other dimensions, you would double its weight get very long skinny whale.  If you were to half Mama whale's length without changing any of her other dimensions you would halve her weight but you would have a very short stubby fat whale.  In neither case would you get a geometrically similar whale.

3.  Sketch a trapezoidal graph for the function y = 2^x, for x values -1, 1, 3 and 5.  Note that the values -1, 1, 3, 5 can be specified as 'x values on the interval from -1 to 5 with increment 3'.  Find and label slopes and areas.

4.  For a random walk with 5 coin flips we can get -5 in one way, -3 in 5 ways, -1 in 10 ways, 1 in 10 ways, 3 in 5 ways and 5 in one way. 

Sketch this distribution and the curve that appears to fit it.

What are the mean, average deviation and standard deviation of one -5, five -3's, ten -1's, ten 1's, five 3's and one 5? 

Homework:

1.  Problem 15 from Second Night Problems was about 5 trees in a row, with 100 bugs initially on each.  At the beginning of every hour 10% of the bugs on each tree travel to each of the tree's immediate neighbors.  You found how many bugs were on each of the trees after 1, 2 and 3 hours.

Sketch a graph of the populations after 1 hour.  Then sketch another graph of the populations after 2 hours, and another for the populations after 3 hours.

Break each graph into trapezoids and find the slope of each trapezoid on each graph.  Label your graphs.

There is no need to find the areas.

Now find the rate of slope change for each interior tree, on each graph.

2.  A chain of 7 beads is created by connecting each bead to the next with a rubber band.  The chain is stretched between two points, with the end beads attached respectively to these points.  So the end beads don't move.

The remaining 5 beads are all free to move.  When undisturbed, the beads all lie along a straight line, which we will call the x axis.  Assume that each bead is 1 centimeter from each of its neighbors.

We now pull the bead right next to one of the ends back 1 millimeter and release it.

Sketch a graph showing the positions of the seven beads, represented as a trapezoidal approximation graph.  Find each slope, and find the rate of slope change for each of the interior beads (i.e., for each of the 5 beads not attached at the ends).

Now the rule that governs the motion of this system is simple:

• The rate of change of the velocity of a bead, with respect to clock time, is 40 cm/s^2 multiplied by the rate of slope change at the position of that bead.

Assume a time interval of 1 millisecond (that's .001 second) and figure out what happens to the system from now until the end of the universe.

OK, maybe from now until the end of the universe is too long.  Figure it for the next three milliseconds.  Caution:  be sure to keep track of the velocities.

Then when you get a chance use a spreadsheet to figure out the next tenth of a second (that would be 100 milliseconds).

3.  Suppose we have seven trees in a row. 

The trees on the ends are 'bug sinks'.  Bugs can migrate to those trees, but they never come back; the end trees accept bugs but don't share their bugs with their neighbors.  Nobody knows what happens to those bugs; maybe happy, maybe tragic, but they go and never come back. 

Each of the interior trees but the one in the middle has 50 bugs on it; the middle tree has 300 bugs.  The distance between the trees is taken to be 1.

Graph the populations and break your graph into trapezoids.  Find slopes and rate of slope change.

Now apply the following rule:

The population of an interior tree changes during a transition by .05 * rate of slope change.

What will be the populations of the seven trees after the first transition?

Graph those populations, and figure out what happens after another transition.

4.  Using the same initial populations as in the preceding, try the following transition rule:

For each transition, every bug flips a coin four times.  If it comes up all Heads the bug moves to the tree on its right.  All Tails moves the bug to the tree on its left.

How many bugs do we expect on each tree after the first transition?

5.  We build a model of Mama whale out of 1-centimeter cubes.  We then build an exact half-scale replica of Mama whale out of 1/2-centimeter cubes.  All cubes are made of identical material. 

How many times more massive is Mama whale than Baby whale?

If we paint both models and require 4 gallons of paint for Mama whale, how many gallons will we need for Baby whale, assuming the paint goes onto both with the same thickness.

6.  We start out with 20 rabbits in a fertile field.  The rabbits are also fertile, and half are male, half female.  The field is capable of supporting 100 rabbits.

If the rabbit population grows by 20% every month, without restriction, how many months will it take before the population is 100?

That doesn't really happen.  As the rabbits approach the 100-rabbit carrying capacity of the field, they reproduce more and more slowly.  Here's a more realistic rule:

In any month the number of rabbits grows by amount  .025 * P * (100 - P).

How many rabbits will we have by the time unrestricted growth has increased the population to 100?