Final_MAT1332_Fall_2021

Universit´e d’Ottawa University of Ottawa 1 Facult´e des sciences Math´ematiques et statistique Faculty of Science Mathematics and Statistics 613–562–5864 613–562–5776 www.uOttawa.ca STEM 336 Ottawa ON K1N 6N5 Marker’s use only: Question Marks 1-5 (/14) 6 (/5) 7 (/5) 8 (/5) 9 (/10) 10 (/13) Total (/52) MAT1332 A: Instructor Catalin Rada Final Examination December , 2021 Duration: 3 hours for writting the exam + 15 minutes, the time to upload in Brightspace. Family name: First name: Student number : • You have 3 hours to complete this exam. AFTER 3 hours HOURS STOP WRITITNG AND SCAN, AND UPLOAD YOUR EXAM IN BRIGHTSPACE. Please do not wait until the last minute to do this. When you are fi nished, scan the pages into a single document and upload it in the ”Assign- ments” tab on Brightspace (the same place you found this). You may use a scanner or your phone or any other device. • SHOW ALL WORK. MARKS ARE INDICATED FOR EACH EXERCISE. • SHOW ALL YOUR WORK! We shall Mark Your Paper Only If It is In Brightspace! In Assignments! Very important: WE ACCEPT ONLY BRIGHTSPACE SUBMISIONS. ONLY ONE FILE. ONLY ONE SUBMISSION. RESPECT THE DUE DATE, AND UPLOAD IN TIME: SIGNATURE .................................................................................................

Universit´e d’Ottawa University of Ottawa 2 1. (2 points) (a) Find the equation of the tangent plane to the graph of f ( x, y ) = e sin( x ) + e x + e y + x + y + 1 + sin(2 x ) + cos(6 x ) at the point: (0 , 0). Solution: (b) Compute the linearization at (0 , 1) and then use it to approximate g ( - 0 . 1 , 1 . 1) if g ( x, y ) = y ln( x 2 + x + 1) + e xy + 3 x + y + 1. Solution: 2. (2 points) (a) If z = 4 + 3 i and v = - 2 - i , then find z - 2 i v + 1 + 2 i . Express answer in a + bi form. Solution: (b) If z = 2 ⇥ cos( - ⇡ 12 ) + i sin( - ⇡ 12 ⇤ , find z 6 . Express answer in a + bi form. Solution:

Universit´e d’Ottawa University of Ottawa 4 4. (3 points) Given that y (0) = 2 e , solve: dy dx = e 2 x ye e 2 x Solution: 5. a) ( 2 points ) Find and draw the domain of f ( x, y ) = p y ln( - y + x ). Indicate if boundaries are included or not.

Universit´e d’Ottawa University of Ottawa 5 b) ( 1 point ) Find the range of g ( x, y ) = 66 p - 6 x + 5 y + 7 - 8 c) ( 2 points ) On the graph below, sketch two level curves of the function k ( x, y ) = p - 3 x + 3 y 2 + 3 . Label each level curve with the height of the corresponding part of the graph of f .

Universit´e d’Ottawa University of Ottawa 7 7. (5 points) Consider the following system of linear equations in the variables x , y , with parameter a : 2 x + 3 ay = - 4 a x + 3 y = 6 - 3 x - 9 y = - 18 Write down the augmented matrix of this linear system, then use row reduction to determine all values of a for which this system has (a) a unique solution, (b) no solution or (c) infinitely many solutions. Justify your answers. Solution:

Universit´e d’Ottawa University of Ottawa 8 8. (5 points) The autonomous di ↵ erential equation dP dt = e 2 P ( P 2 - 9 P + 14) . models the dynamics of a town population, with immigration and emigration. (a) (3 points) Identify all equilibria of the di ↵ erential equation and draw the phase line diagram corresponding to this system. Classify all equilibrium points based on the derivative test. Label each equilibrium, and add the appropriate arrows on your diagram. Solution: (b) (2 points) If initially there are 5 persons in the town, give your long-term predictions for this population, with explicit reference to your answer in (a): your phase line diagram in (a). Explain and justify your answer succinctly and clearly. Solution:

Universit´e d’Ottawa University of Ottawa 10 (c) (1 points) Determine/Write the general solution to the system of di ↵ erential equations. (d) (2 points) Solve the system, if x (0) = 1, y (0) = 4. USE ONLY row operations (take the augmented matrix to RREF)! Show all work. (e) (1 points) Classify the equilibrium point! Justify your answer! (f) (2 points) Sketch solutions using the 2 eigenvectors you obtained in b).

Universit´e d’Ottawa University of Ottawa 11 10. (2+4+2+4+1=13 points, over two pages: show all work!) A disease propagates through an isolated community. The population consists of x susceptible individuals and y infected individuals. There is a constant immigration into the community, and at any time, a certain percentage of susceptible individuals who are not infected leave the community. Infected individuals sometimes die, and sometimes live but remain infected. A system of nonlinear di ↵ erential equations describing the dynamics of this disease is given by dx dt = 4 y - 2 xy = f ( x, y ) dy dt = - 18 x + 6 xy = g ( x, y ) . (a) (2 points) Calculate the nullclines and equilibria that are biomeaningful of this system of di ↵ erential equations. (b) (4 points) Sketch the nullclines in the phase plane, below. Draw the direction arrows (in the first quadrant) along each nullcline as well as in each of the points: (1 , 1) and (3 , 1). Circle the equilibria (use bullets!). Label the axes.

Universit´e d’Ottawa University of Ottawa 12 10, continued. Recopy your equilibria from the previous page here: (c) (2 points) Compute the Jacobian J f,g ( x, y ), if: f ( x, y ) = 4 y - 2 xy and g ( x, y ) = - 18 x + 6 xy . (d) (4 points) For each of the equilibria in the box above, find the eigenvalues of the corre- sponding Jacobian matrix, and use these to classify the equilibria according to their stability. Show your work and explain how you decided the stability. (e) (1 points) For the initial condition given below, determine the populations in the long term: If initially x (0) = 6, y (0) = 3, then for t very large we have x ( t ) ⇡ , y ( t ) ⇡

Universit´e d’Ottawa University of Ottawa 13 Bonus (2 marks) FIND area of The Region Bounded by f ( x ) = p x and g ( x ) = x 2 . SHOW all work! Solution: