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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
to plot the first 100 terms in the solution sequence. Based on your plot, make an educated guess at the long-term behavior of the population being modeled with this recurrence relation. Does your answer make sense to you, in the context of population growth? Explain. 4. In class, we (quickly) derived a solution formula for the general linear dfE a a. Using this solution formula, compute the value of a , for the dfE a (HINT: To handle the summation part, use the formula ∑−m = when we first learned about geometric series.) b. Challenge: For the dfE in part a., evaluate lim a . (HINT: In the limit, the summation becomes an infinite (geometric) series. We know all about when a geometric series converges, and what it converges to.) 5. In the previous homework assignment, you constructed a model of the serum ibuprofen concentration of a person with a headache. Letting a denote the concentration of ibuprofen in her blood immediately before she takes the nth dose of ibuprofen, you built (or “should have built”, whichever is appropriate) the dfE model a a. Using Excel, compute the value of a and a . b. Using Excel, plot enough terms in the solution sequence to make an educated guess at the long-term concentration of ibuprofen in this person’s blood, assuming she never stops taking it. (Note: Don’t try this (i.e. never-ending ibuprofen) at home.) c. Noting that this is a linear dfE, use the solution formula derived in class to write an explicit d. Use the result of part b. to determine the long-term concentration of ibuprofen in this person’s blood, assuming she never stops taking it. 6. (A little chaos.) Even relatively simple-looking difference equations can have solutions that behave in bizarre ways. For example, consider the discrete logistic model (DLM): − a = ra (K a ) , which has a basic, seemingly harmless quadratic right-hand side. (Note: The preceding sentence is the equivalent of the statement, “He was quiet and kept to himself – I would have never expected him to do something this horrible,” often made by neighbors of serial killers.) In this model, r is a growth rate constant, and K is a carrying capacity. The reasoning behind the mathematical form of this model is that if either 0 a = or a = K , then the right-hand side is zero, so that the value of a no longer changes. a. Use Excel to compute the value of a . b. In Excel, plot the first 100 terms of the solution of the DLM, starting with a = 5 . Use the values r = 0.01 and K = 100 . c. Repeat part a., but use r = 0.02 . Describe what is different about this plot, compared to the d. Repeat part a. once more, but use r = 0.03 . Describe what is different about this plot, compared to the ones from parts a and b. (The seemingly patternless behavior seen in part c. is an example of the scientific concept of chaos.)


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