## Microsoft word - supplements16.doc

MATH 232 – Scientific Calculus II
Homework Supplement S16
1. By writing out the first several terms (like we did with the example from class), try to derive an with k ≠ 0 , is known as the discrete exponential model (DEM). a. Use the analytical tool demonstrated in class (i.e. look for a pattern) to derive an explicit formula for the solution of the DEM, with initial condition a . b. Evaluating the long-term behavior of the DEM is easy. Use the result of part a. to show that the solution goes to zero if − 2 < k < 0 , and blows up if either − ∞ < k < −2 or 0 < k < ∞ . 3. The DEM is popular for modeling unconstrained growth. For limited growth (because of restrictions on, e.g., food or living space), a fancier model is needed. One possibility is the Ricker model:
where K represents the maximum sustainable population size (called the carrying capacity). This is
another one of those dfEs for which no explicit solution formula can be found, so numerical and
graphical tools are needed. For this exercise, set r = 0.2 and K = 1000. Starting with a = 1, use Excel