Cramers Rule
Suppose want to solve the following system of equations:
ax + by = e
cx + dy = f
where ad bc = 0. Cramers Rule says the solution is
x=
e
f
a
c
b
d
b
d
and y =
a
c
a
c
e
f
,
b
d
where | | represents the determinant of a matrix. This means
x=
e
Empirical Modeling
In the previous chapter, we used our prior knowledge to help construct the models. We then used
that knowledge to determine the constants within the model. These are called analytical models.
In this chapter, we will just use data (and
4.3 Growth of a Bacteria Population
When we looked at bacterial growth earlier, we considered an = an+1 an . Lets do that for the
following set of data:
n
an
an
n
an
an
0
10.3
6.9
10
440
80.4
1
17.2
9.8
11
520.4
40
2
27
18.3
12
560.4
40.1
3
45.3
34.9
13
6
4.4 A Linear Predator-Prey Model
Consider the ecosystem of rabbits and foxes. Well make the following assumptions about the
ecosystem.
1. Suppose foxes only prey are rabbits, and rabbits only predator are foxes. What are the
implications of this?
2.
3.
4.
2.3 Modeling with Proportionality
Homework pg. 27 28 #1, 2, 5, 6
Although we have studied proportional relationships so far, it is not always the case two variables
are proportional. For example, one variable may be proportional to the change in the other
Chapter 4
Discrete Dynamical Systems
A dynamical system is one that changes over time (as opposed to a static system). When time
is measured in discrete units (as opposed to continuously), then we have a discrete dynamical
system.
Examples of discrete dyn
In-class work and Homework for Tuesday 5/17
Do EXAMPLE 6.2.3 (on pages 171-173) and answer the question of Should you play the
game?. Please explain when and why you think you should. Extra credit if you can nd the
actual expected value of playing this g
4.5 A Nonlinear Predator-Prey Model
Again, well consider the ecosystem of rabbits and foxes. We will again assume
Fn = population of foxes at the end of month n and
we will assume Rn = population of rabbits at the end of month n.
Well use a similar model
4.6 Epidemics
Strep throat is certainly going around right now. I believe I had it this weekend even. We would
like to know how quickly such a bacterial infection can spread in a community of 5000.
We will consider the following assumptions:
1. No one ent
6.3 The Birthday Problem
Based on your reaction earlier in class, it seems you are very familiar with the Birthday Problem,
which asks, How many people do you need in a room to have (at least) a 50% chance of having
(at least) two people sharing a birthda
Chapter 7 - Optimization
In Calculus, when we wanted to maximize (or minimize) a function, we usually took the derivative
and set it equal to zero. This is an optimization problem (OP). Sometimes however, we have
OPs, where we are trying to maximize a fun
7.5 Solving Linear Programs
Recall, from 7.2, we used Excel to solve the following problem:
Suppose you have two types of baskets of fruit you sell. An Apple Tree Plus Orange basket has 3
apples and 1 orange and nets you a prot of $3. The Two by Two baske
7.6 Simplex Method
In the last section, we used a graphical approach to solve linear programs. In this chapter, well
use something called the Simplex Method. It was invented by George Dantzig in 1947.
We will use it to solve the following problem:
Example
Chapter 6
We have spent most of our time working with deterministic systems, where the behavior was known
once the parameters were known. In this chapter, well focus on probabilistic systems, in particular
Monte Carlo simulations. These are used to study
5.6 Eigenvalues in Systems of Dierential Equations
Recall from yesterday an eigenvalue-eigenvector of a square matrix matrix A is a constant and
nonzero vector x such that Ax = x. Eigenvalues can help analyze the long-term behavior of a
system of dierenti
5.1 Newtons Law of Cooling
Recall from earlier that Newtons Law of Cooling is:
dy
= k (y T ),
dt
where
y (t) = temperature of an object at time t
T = room temperature (or refrigerator or oven)
k = constant of proportionality
Lets nd the general solution o
5.2 Mixing Problem
Suppose you have a tank that contains 50 gallons of a solution composed of 90% water and 10%
alcohol. A second solution containing 50% water and 50% alcohol is added to the tank at the rate
of 2 gallons per minute, while at the same tim
Homework Due 4/12: pg. 34 35#2, 3, 4, 5
2.4 Fitting Straight Lines Analytically
If you have a set of points on a graph and feel the two variables are proportional, then how do
you go about nding the best-t line? More importantly, what do we even mean when