Problem 1 f(x)=x²-1 g(x) = -x + 2, x=0, x=2 a) Sketch the graphs of the equations given above in the same Cartesian Plane. b) Find the area of the region bounded by the graphs of the given equations

Problem 2 For each of the questions a), b), c) and d) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines and display the graph of the solid as seen in class. g(x)=√x y=0, x=3 a) x-axis b) y-axis c) x = 3 d) y = -2

Question 2 Shown in the figure below is a house with three rooms, A, B and C. Rooms B and C are the same size and shape. There is a furnace F in room C which heats room C. Let A(t), B(t), C(t) be the temperature at time t in rooms A, B, C respectively and let Am be the outside temperature. Shown in the figure are the cooling rate constants k₁, k₁, ką - k₁, k₁ for how heat moves through the walls of the house.

Question 3 Shown in the figure below is lake Erie and lake Ontario and the main rivers flowing through them (the arrows). Google says that the volume of lake Erie is about 500 km³ and lake Ontario is about 1500 km³ and the amount of water flowing through the lakes is about 60 km³ per year. Yes, that's kilometers cubed. The goal is to model the amount of pollution in the two lakes at time t (years). We will assume that initially, there is no pollution in either lake and that the river flowing into lake Erie is polluted and is bringing in 30 tons of pollutant per year. Let Er (t) be the amount of pollutant (in tons) in lake Erie at time t and let On (t) be the amount of _pollutant (in tons) in lake Ontario at time t (years). Part (a) Set up two differential equations, one for the amount of pollution in lake Erie at time t and the other for the amount of pollution in lake Ontario at time t. This problem is very much like the tank problem in the last assignment, except that here we have two tanks (two lakes). Parb (b) Solve the differential equations and plot the solutions for a suitable time domain. You should see that the amount of pollutant in each lake increases from 0 to a maximum. What are the _maximums? Part (c) Using the DEplot command in the DEtools package, generate a field plot with solution curves for initial values Er (0) = 0, On (0) = 0 and Er(0) = 500, On (0) = 0 and Er (0) = 500, On(0) = 1500 on the same plot.

Question 4 The Kermack-McKendrick virus spread model (where we partition the individuals in a population into those which are susceptible, infected, and recovered), is given by

Question 5 Let M(t) be the amount owed on a 30 year mortgage of $200,000 at time t years. Suppose the annual interest rate on the mortgage is 7=4%. Suppose the term of the mortgage is 30 years, i.e., M(30) should be 0. Suppose we pay $P per year. We can model the change in what we owe the bank with the differential equation M (t) = r.M(t) - P dollars per year and initial values M(0) =$200000. This model assumes the interest is charged continuously (banks usually charge interest daily which is approximately continuous) and if we assume we make the payments continuously (banks usually require us to pay monthly or weekly which is approximately continuous over 30 years). So the values we obtain with it will be approximations.

Question 6 In Question 5 the 30 year mortgage was $200,000 and the interest rate was 4% per year compounded daily. We modelled the change in what we owe the bank with the differential equation M (t) = r-M(t) - P dollars per year We solved the differential equation with initial value M(0) = 200000. We worked out if we set P= $11448.10 per year then M(30) = 0. But the differential equation assumes a continuous process in which we are continually being charged interest and we are continually paying down the mortgage. In reality the bank charges us interest once a day and we make payments once every two weeks or once a month at the end of the month. Let's assume monthly payments. Assuming no leap years for simplicity, over a 30 year period there will be 30-365 = 10,950 interest charges which is almost continuous. But there are only 30-12 = 360 payments, less continous.

Question 7 The following is a solution to the random walk exercise in Assignment 4 by a student. It is correct but very slow for large values of n.

Question 8 The study of Bohemian matrices is a current research topic. A family of Bohemian matrices is a set of matrices where the entries of the matrices are restricted to a finite set of values, for example, {0,1} or {-1,0,1}. One is interested in the set of eigenvalues of such a family of matrices. In the Maple code below I've constructed all 2 by 2 matrices whose entries are 0 or 1. Since a 2 by 2 matrix has 4 entries and each entry can be a 0 or a 1, there are 24 = 16 such matrices.