### Week 10

```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
```