Do all the problems, points as marked . You get to use your notes, and can clutch but not use your calculator. If things go wrong, explain what you were trying to do and what went wrong.

1. 1. (25 points) Populations of brine shrimp in lakes throughout the Great Basin follow discrete-time dynamical systems of the form
   

   Bt+1 = 100 - aBt,

   where the value of a is different in different lakes (and can be positive or negative).
   1. Find the equilibrium as a function of a. For which values of a is the equilibrium reasonable?
   2. For which values of a is the equilibrium stable?

2. 2. (25 points) A professor writes a textbook and wants to make a lot of money. Suppose that the number of books sold as a function of the price p is N(p) = 30000-p², where p is measured in dollars.
   1. How many books could this professor give away (if he set the price to 0)?
   2. What is the total revenue (price times number sold) as a function of p?
   3. What price maximizes the revenue?
   4. What is the maximum revenue? If the professor gets a %15 royalty, how much does he make?

3. 3. (25 points) Consider the function G(x) = [( x+x²)/( e^x-1)].
   1. Find G_¥(x), the leading behavior at infinity.
   2. Find lim_{x ® 0} G(x).

4. 4. (25 points) You are asked to find a solution of the equation -x³+3x = -1.
   1. Find an interval that must include a solution.
   2. Choose a reasonable guess of the solution.
   3. Use Newton's method to refine your guess once. Did it work?