X 1 = = 1 2, X 2 = = 1 4.

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1 Solutions: 5: Choose an initial value for X and use values of r that are less than, greater than, and equal to one to test the above statements by computing two values of X for each value of r Choose X, r : r : r /: X, X X X, X X, X 5: An inedible alga is growing on a pond in a city park Only a small part of the pond is now covered by the algae, but the area covered is doubling each day The city decides to remove the algae once it covers half the pond If the pond will be completely overgrown in thirty days, on what day will it be half covered? Hint: Try working backward The pond will be completely overgrown in thirty days Since the algae doubles each day the pond will be half overgrown with algae on the twenty ninth day On the twenty eighth day it will be less than half covered So the city will remove the algae on the twenty ninth day 5: A rabbit population is growing at % a year If there are rabbits this year and time is discrete, how many will there be in years? Use a loop in SageMath to check your answer In this case r and X So X 597 Round one way or the other to get the actual rabbit population Sage code: init r for i in range,: x r*init init x init

2 55: While we have been working with r >, representing growth, r can be less than, representing a quantity that decreases over time The half-life of a radioactive element is the amount of time needed for half the element to decay What fraction of the initial amount of such an element will remain after ten half-lives? Each half life we lose of the element So after ten iterations we get 5: When money in a bank account accrues compound interest, the interest earned in one time period is added to the principal, and then the sum is used as the base for the next time period If you start off with $ and earn % interest that is compounded annually, how much money will you have in 5 years? In years? In years? How long will it take you to accumulate $,? In this case r and X So X X , X To get to we solve, n Simplifying, we get n Solving this we that after 7 years we will have 57: If r and X, what is X? X? X 9 and X : Similarly, for another function g : R R, choose vectors for ge, ge and work through the reasoning above to find the matrix representation of g What are the dimensions of this matrix? a ge b c d ge e f

3 So let X be a vector written as X e + X e, then a d gx X b + X e c f ax + dx bx + ex cx + fx a d b e X X c f 7: What are the matrices representing the following systems of equations? X N+ X N + Y N and Y N+ X N + 8Y N X N+ 5X N and Y N+ X N + Y N Z N+ 8Z N + 5W N and W N+ 7Z N + W N d a N+ a N and b N+ b N e a N+ b N and b N+ a N d e XN+ Y N+ XN+ Y N+ ZN+ W N+ an+ b N+ an+ b N+ [ 8 [ 5 ] XN Y N ] XN Y N [ ] 8 5 ZN 7 W N [ [ ] XN Y N ] XN Y N

4 8: What systems of equations are represented by the following matrices? You can use X and Y as your variables d e [ ] [ ] [ ] 5 [ ] 5 7 X N+ X N + 5Y N and Y N+ 7X N + 9Y N X N+ X N + Y N and Y N+ X N + Y N X N+ Y N and Y N+ 5X N d X N+ X N and Y N+ 5Y N e X N+ Y N and Y N+ 7X N + Z n and Z N+ X N + Z N 9: Use the method we used here to find the next years population if this years population consists of 5 juveniles and 8 adults J A 5 8 Multiply this by M, [ ]

5 : Evaluate [ ] [ ] 5 X Y Z X + Z X + Y + Z X + Y + Z : For the following functions, can fgx exist? f : R R 5 and g : R R f : R R and g : R R f : R 7 R 8 and g : R R 7 Yes No Yes 5

6 : If the matrices A and B have the following dimensions, does AB exist? A is a 5 matrix and B is a matrix A is a matrix and B is a matrix A is a 8 7 matrix and B is a 7 matrix Yes No Yes : Multiply: [ ] [ ] 5 5 [ ] [ ] [ ] 5 [ ] 7 [ ] 7 5 [ ]

7 : Verify that this calculation is correct by applying the good-year matrix M to the initial condition, and then applying the bad-year matrix M bad to the result How does your result compare to the above calculation? First apply M to the intial condition, [ ] Now apply M bad to the result, [ ] The same answer! : What does the matrix MM bad represent? A bad year followed by a good year : What matrix product represents a sequence of two good years, followed by two bad years, followed by a good year? Be careful about the order of multiplication MM bad M bad MM FE 7: Assume that f is a linear function Without using matrices, do the following: 5 If f and f, find f 7 7 If f 5 and f, find f 9 If f, f 5 and f 9 8, find f f

8 7 9 f f FE 8: Could the functions described below be linear? Justify your answers f f 5 and f 5, f 5 and f No, but 5 No, 5 + but + 5 : Come up with a Leslie matrix model for a fictional species with two life stages and describe the meaning of its entries, as above Let, M [ ] 5 5 This model has half the juveniles staying juveniles each year and % of the juveniles mature into adulthood The adults have and average of offspring each year so they reproduce once every ten years and half remain adults each year : Explain why the entries in each column of a transition matrix such as equation must add up to one Hint: Label the rows and columns, writing from and to where appropriate In this model we have two states, S and I The first column has two entries, the first entry says what percent leaves S and the second entry says what percent stays at S This governs all of S so the two entries must add up to % The same logic applies to the second column that governs I 8