MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, Exam Scores. Question Score Total

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1 MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, 2016 Exam Scores Question Score Total 1 10 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during the exam. You may use a calculator, but not one that has symbolic manipulation capabilities or a QWERTY keyboard. Free Response Questions: Show all your work on the page of the problem. Show all your work. Clearly indicate your answer and the reasoning used to arrive at that answer Grade 100 Bonus 10 Total 110

2 1. (10 points) Find the area of the region bounded by the curves f(x) = x2 2 and g(x) = x on the interval [0, 2]. y f g as illustrated in the following figure x 2

3 2. (7 points) Let N(t) denote the size of the population at time t. Assume that the population evolves according to the logistic equation. Assume also that the intrinsic growth rate is 1.7 and that the carrying capacity is 60. A. (4 points) Find a differential equation that describes the growth of this population. B. (3 points) Without solving the differential equation in (a), sketch solution curves of N(t) as a function of t when (i)n(0) = 2, (ii)n(0) = 50, and (iii)n(0) = 70. 3

4 3. (8 points) Allee Effect: Let N(t) denote the size of the population at time t. Assume that the population evolves according to the following equation. dn(t) dt = 0.3N(N 20)(1 N 100 ). A. (4 points) Sketch the solution graph when N(0) = 10. B. (4 points) Sketch the solution graph when N(0) = 30. 4

5 4. (15 points) Discrete Population Model: Consider an animal species that has two life stages: juvenile and adult. Suppose that we count the number of members of this population on a weekly basis. Let J t = number of juveniles at week t and A t = number of adults at week t. The relation between the population during two consecutive weeks can reasonably be described as follows J t+1 = J t mj t gj t + fa t = (1 m g)j t + fa t A t+1 = A t µa t + gj t = (1 µ)a t + gj t (1) where m is the fraction of juveniles that dies, g is the fraction of juveniles that becomes adult, f accounts for the newborns, and µ is the fraction of adults that dies. A. (3 points) Write the matrix form of the system of equations in (1) when m = 1/3, g = 1/3, f = 3, and µ = 2/3. B. (3 points) Find the eigenvalues of the matrix associated to the system in part A. What is the eventual growth rate of the population. 5

6 Problem 4 (continuation) C.(6 points) Find the eigenvectors of the matrix associated to the system in part A. D. (3 points) Compute the eventual ratio of juveniles to adults. Give your answer in percentages. 6

7 5. (10 points) Least-square approximation: The table below is the estimated population of the United States (in millions). Suppose there is a linear relationship between time t and population P (t). year population A. (8 points) Find the least squares solution to the linear system that arises from this data. B. (2 points) Use the linear model to predict the U.S. population in

8 6. (10 points) Consider the function f(x, y) = ln x 2 + y 2. A. (8 points) Compute the second order partial derivatives 2 f x 2 and 2 f y 2. B. (2 points) Compute 2 f x f y 2 8

9 [ ] 0 7. (10 points) Show that the equilibrium of 0 is unstable. x 1 (t + 1) = x 2 (t + 1) = x 2 (t) 1+x 2 1 (t) x 1 (t) 2+2x 2 2 (t) 9

10 8. (10 points) Use the graphical approach to classify the following Lotka-Volterra model of interspecific competition according to coexistence, founder control, species 1 excludes species 2, or species 2 excludes species dn 1 dt dn 2 dt ( = 3N 1 1 N N 2 25 ( = N 2 1 N N 1 30 ) ) The figure on the right shows the direction field of the system. 10

11 9. (15 points) Consider the following predator-prey model, ( dn dt =N 1 N ) 5P N K dp =2P N 8P dt A. (5 points) Draw the zero isoclines of the system for K = 20 and K = 3. B. (5 points) When K = 20, the zero isoclines intersect, indicating the existence of a nontrivial equilibrium. Analyze the stability of this nontrivial equilibrium. 11

12 Problem 9 (continuation) C. (5 points) Is there a minimum carrying capacity required in order to have a nontrivial equilibrium? 12

13 10. (5 points) Consider the Kermack-McKendrick model for the spread of an infectious disease in a population of fixed size N. If S(t) denotes the number of susceptibles at time t, I(t) the number of infectives at time t, and R(t) the number of immune individuals at time t, then ds dt = bsi di =bsi ai dt and R(t) = N S(t) I(t). Suppose that a = 200, b = 0.1, and N = Can the disease spread if, at time 0, there are 1000 infected individuals? Justify your answer. 13

14 11. Bonus Problem: (10 points) Solve the given initial-value problem [ dx1 ] [ ] [ ] dt 2 2 x1 (t) dx 2 = 2 3 x dt 2 (t) with x 1 (0) = 1 and x 2 (0) = 4. 14