(A) As noted earlier, ( 4 , 3 ) is a solution of the equation 3 x − 2 y = 6 Find three more solutions of this equation. Plot these solutions in a Cartesian coordinate system . What familiar geometric shape could be used to describe the solution set of this equation? (B) Repeat part (A) for the equation x = 2. (C) Repeat part (A) for the equation y = − 3.
(A) As noted earlier, ( 4 , 3 ) is a solution of the equation 3 x − 2 y = 6 Find three more solutions of this equation. Plot these solutions in a Cartesian coordinate system . What familiar geometric shape could be used to describe the solution set of this equation? (B) Repeat part (A) for the equation x = 2. (C) Repeat part (A) for the equation y = − 3.
(A) As noted earlier,
(
4
,
3
)
is a solution of the equation
3
x
−
2
y
=
6
Find three more solutions of this equation. Plot these solutions in a Cartesian coordinate system. What familiar geometric shape could be used to describe the solution set of this equation?
(B) Repeat part (A) for the equation
x
=
2.
(C) Repeat part (A) for the equation
y
=
−
3.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
10. In the general single period market model with = {W1, W2, W3}, one risky asset, S, and
a money market account, we have So = 4 for the risky asset. Moreover, the effective
rate of interest on the money market account is 5% and at time t = 1 we have
W1
W2 W3
S₁
100 50 40
21
21
21
(a) Calculate all risk-neutral probability measures for this model. [4 Marks]
(b) State if the model is arbitrage-free. Give a brief reason for your answer. [2 Marks]
(c) A large bank has designed an investment product with payoff X at time t = 1.
Given
W₁
W2
W3
X
0
1
1.5
show that X is an attainable contingent claim. [4 marks]
Question 1. (10 points)
A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by
dV
=
1.45V(2 In(V+1)).
dt
(a) (4 pts) Find all the equilibria and determine their stability using the stability condition.
(b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of
f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable.
(c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain
in biological terms what happens to the size of each of these tumours at time progresses.
For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.
Chapter 1 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
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