MATH133 TUTORIAL EXERCISES WEEK 10 1 a 1/3 (1) Consider the dynamical system given by xk+1 = Axk with A = and a > . For which 1/3 a 3 values of a is the origin an attractor? For which values is it a repellor? A saddle point? (2) The tuna population Tk and the anchovy population Ak are related by the equations Tk+1 = .3Tk + .7Ak Ak+1 = −pTk + 1.2Ak where p > 0 is the predation rate (the number of anchovies eaten by an average tuna in a month, measured in thousands). For which values of p does the model predict unbounded growth of the two populations? For which values of p does the model predict both populations will crash (that is, approach zero)? For what value of p does the model predict the populations will stabilize at a non-zero level? (3) What is the definition of a stochastic matrix? Let P be an n × n stochastic matrix. This exercise will show that P ~x = ~x has a non zero solution. Jusify each step below. (a) If all the other rows of P − I are added to the bottom row, we obtain a row of zeros. (b) The rows of P − I are linearly dependent. (c) The rank of P − I is less than n. (d) The nullity of P − I is at least 1. What can you conclude? (4) Verify that S = {(1, −2, 3, −4), (2, 1, −4, −3), (−3, 4, 1, −2), (4, 3, 2, 1)} forms an orthogonal basis for R4 . Write (−1, 2, 3, 7) as a linear combination of the elements of S; you don’t need to do any row reduction. (5) Find an othogonal basis for span((−1, 1, 5, 1), (−2, 2, 3, 2)). (6) Use the method of Lagrange multipliers in the following exercises. (a) Find the points on the curve x2 + xy + y 2 = 1 in the xy-plane that are nearest to and farthest from the origin. (b) Find the dimensions of the rectangle of largest perimeter that can be inscribed in the ellipse x2 /a2 + y 2 /b2 = 1 with sides parallel to the coordinate axes. What is the largest perimeter? (7) Find the global maximum and minimum of the function f (x, y) = x2 + 2x − y 2 with domain (x, y) : x2 + 4y 2 ≤ 4 . 2 + 4y 2 < 4 and also extreme values on the Check all stationary points in the interior (x, y) : x boundary (x, y) : x2 + 4y 2 = 4 . (8) Use Lagrange multipliers to find maximum and minimum values of the function subject to the given constraint: (a) f (x, y) = x2 y; x2 + 2y 2 = 6; [±4] (b) f (x, y) = exy ; x3 + 2y 3 = 16; (c) f (x, y, z) = z when (x, y, z) is constrained to be in the intersection of the plane x + y + z = 2 and the ellipsoid 2x2 + 2y 2 + z 2 = 4. [±2] (9) Find and classify all the stationary points of the function f (x, y) = x3 y + 12x2 − 8y. (10) Design a closed cylindrical container which holds 100 cm3 and has the minimal possible surface area. What is its radius and height? Test 2 will cover the material from Week 7 to Week 10 inclusive. It will be held in tutorials in Week 11. Date: October 2014. 1

© Copyright 2019 ExploreDoc