Math 472/572 - Assignment 3
Due: Tuesday,October 3. Nothing accepted after Thursday, October 5. 10% off for being late. Please work by yourself. See me if you need help.
Suppose t denotes time in years measured from September 1, 2002 and N = N(t) is the number of whitefish in Lake Huron measured in 10,000's. So N = 3 means 30,000 whitefish. Suppose the change in N is due to the following two types of things.
1. Births and deaths.
2. Immigration.
Suppose the net rate of change of whitefish due to births and deaths is proportional to N, i.e. it is equal to rN for some constant r. Furthermore, suppose the immigration rate is a constant, call this v. Thus, the rate of change of N with respect to time is the sum of rN and v, i.e.
(1) = rN + v.
If one solves this differential equation one obtains
(2) N = Noert +
where No = N(0) is the number of whitefish at time t = 0. Suppose one makes the following measurements.
1. On September 1, 2002 there were 30,000 whitefish in Lake Huron, i.e. No = 3
2. The net rate of immigration of whitefish into Lake Huron is 20,000 per year, i.e, v = 2
3. On September 1, 2003 there were 33,000 whitefish in Lake Huron, i.e. N(1) = 3.3
Substituting into (2) one obtains the equation f(r) = 3.3 where
(3) f(r) = 3er + .
1. (2 points) Solve the differential equation (1) to obtain the formula (2). The formula (2) is not valid for a certain value of r. What is that value of r and how should (2) and (3) be modified in that case?
2. (1 points) What is the physically relevant range of values of r? In answering this, don't assume that you have been told that on September 1, 2003 there were 33,000 whitefish in Lake Huron. I.e. you don't have any information on the change in the population from one year to the next.
3. (2 points) Use Mathematica, MATLAB, or your favorite Mathematics software to make a plot of f(r) for r varying over some appropriate range. Based on the graph, approximately what is the value of r that is the solution to f(r) = 3.3? Make a printout of the plot to turn in with your answer.
4. (3 points) Without using the graph you made in part 3, show how to apply the intermediate value theorem to conclude that there is a solution r to f(r) = 3.3 . Explain why the hypotheses of the intermediate value theorem are satisfied. For what values of r is f(r) continuous? Give reasons for your answer.
5. (3 points) Without using the graph you made in part 3, show that f(r) is increasing so that the solution r of f(r)= 3.3 is unique.
6. (2 points) Use Mathematica, MATLAB, or your favorite Mathematics software to find an approximation to the solution r of f(r)=3.3. Please turn in a printout of your computer input and output.
7. (2 points) Use the secant method to find another approximation r* to the solution r which has a relative error less than 10-5. How many iterations did it take? If you do this with software, see me if you need help with the programming.
8. (3 points) Use the error bound formula for the secant method to show how to estimate the number of iterations that would be needed in question 7 before you actually do question 7.
9. (2 points) Find f(r*) and use it and f’(r*) to estimate the error in r*, where r* is the value obtained in part 7. You should do this differently from the estimate in part 8 and you shouldn't use the value you found in part 6.