Announcements
Topics:
- finish section 2.2; work on sections 2.3, 3.1, and 3.2
* Read these sections and study solved examples in your
textbook!
Work On:
- Practice problems from the textbook and assignments
from the coursepack as assigned on the course web
page (under the link “SCHEDULE + HOMEWORK”)
Semilog graphs
Definition:
A semilog graph plots the logarithm of the
output against the input.
The semilog graph of a function has a reduced
range making the key features of the function
easier to distinguish.
Semilog graphs
Example:
Compare the graphs and semilog graphs for
2x
3x
g(x)

3e
and
f (x)  2e

Semilog graphs
Original Graphs
Semilog Graphs
ln y

Exponential Models
When the change in a measurement is
proportional to its size, we can describe the
measurement as a function of time by the
formula
S(t)  S(0)e t
where
S(t) is the value of the measurement at time t
S(0) is the initial value of the measurement, and

 is a parameter which describes the rate at
which the measurement changes

Doubling Time
.
Example:
A bacterial culture starts with
100 bacteria and after 3 hours
the population is 450 bacteria.
Assuming that the rate of growth of the
population is proportional to its size, find the
time it takes for the population to double.
Half-Lives of Drugs
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
2
3
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
2
3
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
3
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
3
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
12.5
4
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
12.5
4
0.54M(0)
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
12.5
4
0.54M(0)
6.25
5
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
12.5
4
0.54M(0)
6.25
5
0.55M(0)
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Half-Lives of Drugs
Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
0
M(0)
100
1
0.5M(0)
50
2
0.52M(0)
25
3
0.53M(0)
12.5
4
0.54M(0)
6.25
5
0.55M(0)
3.125
** Many drugs are not effective when less than 5% of
their original level remains in the body.
Trigonometric Functions
Trigonometric functions are used to model
quantities that oscillate.
Trigonometric Models
Example: Seasonal Growth
A population of river sharks in New Zealand
changes periodically with a period of 12 months. In
January, the population reaches a maximum of
14, 000, and in July, it reaches a minimum of 6, 000.
Using a trigonometric function, find a formula
which describes how the population of river sharks
changes with time.
Trigonometric Models
Example: (A40, #2.)
Graphs of Trigonometric Functions
Example:
Inverse Trigonometric Functions

Since the 3 main trigonometric functions are not
one-to-one on their natural domains we must first
restrict their domains in order to define inverses.
Inverse of Sine
Restrict the domain of f (x)  sin x to

[ 2 , 2 ].

Now the function is
one-to-one on this
interval so we can
define an inverse.
Inverse of Sine
The inverse of the restricted sine function is
denoted by f 1(x)  sin1 x or f 1(x)  arcsinx.
Cancellation equations:
arcsin(sin
x)  x

sin(arcsin x)  x
Calculate:
arcsin(12 )
x  [ 2 , 2 ]

(domain of sin x)
x  [1, 1]
(domain of arcsin x)


sin(arcsin(57 ))
arcsin(sin(  ))
Graphs of Sine and Arcsine
y = sin x
y = arcsin x
domain:
x  [ 2 , 2 ]
domain:
x  [1, 1]
range:
y  [1, 1]
range:
y  [ 2 , 2 ]

Inverse of Tangent
Restrict the domain of f (x)  tan x to ( 2 , 2 ).


This portion of tangent
passes the HLT so tangent
is one-to-one here
Inverse of Tangent
The inverse of the restricted tangent function is
denoted by f 1(x)  tan1 x or f 1(x)  arctanx.
Cancellation equations:
x  ( 2 , 2 )
arctan(
t
anx)

x


tan(arctanx)  x
x  (,)
Calculate:
arctan(1)
(restricted domain of tan x)
(domain of arctan x)


tan(arctan
10)
arctan( 3)
Graphs of Tangent and Arctangent
y = tan x
y = arctan x
y = cos x
y = arccos x
domain:
x  ( 2 , 2 )
domain: x  (,)
range:
y  (,)
range:

y  ( 2 , 2 )
Real-life Use of Arctangent
Example: Model for World Population
One of the many models used to analyze human
population growth is given by

2007 t 
P(t)  4.42857  arctan

2
42 
where t represents a calendar year and P(t) is
the population in billions.

Dynamical Systems
• Discrete-time dynamical systems describe a
sequence of measurements made at equally
spaced intervals
• Continuous-time dynamical systems, usually
known as differential equations, describe
measurements that are collected continuously
Dynamical Systems
• Discrete-time dynamical systems describe a
sequence of measurements made at equally
spaced intervals
• Continuous-time dynamical systems, usually
known as differential equations, describe
measurements that are collected continuously
Discrete-Time Dynamical Systems
A discrete-time dynamical system consists of an
initial value and a rule that transforms the
system from the present state to a state one
step into the future.
Discrete-Time Dynamical Systems
Example:
Consider a bacterial colony growing under
controlled conditions. The initial value “the
present population is 1.5 million” and the
dynamical rule “the population doubles every
hour” constitute a discrete-time dynamical
system.
Discrete-Time Dynamical Systems
and Updating Functions
m represent the measurement of some quantity.
The relation between the initial measurement m t and the
final measurement m t 1 is given by the discrete-time
Let
dynamical system
m t 1  f (m t )
 initial value
The updating function f accepts the
 the final value m t 1 as output.
and returns
mt as input
Note: 
t represents current time and t 1 represents one time-step
into the future



Discrete-Time Dynamical Systems
and Updating Functions
m represent the measurement of some quantity.
The relation between the initial measurement m t and the
final measurement m t 1 is given by the discrete-time
Let
dynamical system
m t 1  f (m t )
 initial value
The updating function f accepts the
 the final value m t 1 as output.
and returns
mt as input
Note: 
t represents current time and t 1 represents one time-step
into the future



Discrete-Time Dynamical Systems
and Updating Functions
m represent the measurement of some quantity.
The relation between the initial measurement m t and the
final measurement m t 1 is given by the discrete-time
Let
dynamical system
m t 1  f (m t )
 initial value
The updating function f accepts the
 the final value m t 1 as output.
and returns
mt as input
Note: 
t represents current time and t 1 represents one time-step
into the future



Discrete-Time Dynamical Systems
and Updating Functions
m represent the measurement of some quantity.
The relation between the initial measurement m t and the
final measurement m t 1 is given by the discrete-time
Let
dynamical system
m t 1  f (m t )
 initial value
The updating function f accepts the
 the final value m t 1 as output.
and returns
mt as input
Note: 
t represents present time and t 1 represents one time-step
into the future



Example:
A Discrete-Time Dynamical System for
a Bacterial Population
Data:
Colony
Initial Population bt
(millions)
Final Population bt+1
(millions)
1
0.47
0.94
2
3.30
6.60
3
0.73
1.46
4
2.80
5.60
5
1.50
3.00
6
0.62
1.24
Example:
A Discrete-Time Dynamical System for
a Tree Growth
Data:
Tree
Initial Height, ht
Final Height, ht+1
(m)
(m)
1
23.1
23.9
2
18.7
19.5
3
20.6
21.4
4
16.0
16.8
5
32.5
33.3
6
19.8
20.6
Example:
A Discrete-Time Dynamical System for
Absorption of Pain Medication
A patient is on methadone, a medication used to
relieve chronic, severe pain (for instance, after
certain types of surgery). It is known that every
day, the patient’s body absorbs half of the
methadone. In order to maintain an appropriate
level of the drug, a new dosage containing 1 unit
of methadone is administered at the end of
each day.
Solutions
Definition:
The sequence of values of mt for t  0, 1, 2, …
is the solution of the discrete-time dynamical
system m t 1  f (m t ) starting from the initial
condition m 0 .




Solutions
Definition:
The sequence of values of mt for t  0, 1, 2, …
is the solution of the discrete-time dynamical
system m t 1  f (m t ) starting from the initial
condition m 0 .


The graph of a solution is a discrete set of points

with the time t on the horizontal axis and the

measurement mt on the vertical axis.
Finding Solutions
Example 1:
Find a solution of the bacterial discrete-time
dynamical system bt 1  2bt .
Example 2:
Find a solution of the tree height discrete-time

dynamical system ht 1  ht  0.8.


Summary of Solutions
Basic Exponential Discrete-time Dynamical System
If bt 1  rbt with initial condition b 0 , then bt  b0 r t .
Basic Additive Discrete-time Dynamical System
 condition
If ht 1  ht  a with initial
 h0 ,
then ht  h0  at.

Manipulating Updating Functions
• All of the operations that can be applied to
ordinary functions can be applied to updating
functions, but with special interpretations
Composition
The updating function f updates the measurement
by one time step.
Compute f o f .

( f o f )(mt )  f ( f (mt ))

 f (mt 1)
 mt 2

The composition of an updating function with itself
corresponds
to a two-step updating function.

Composition
The updating function f updates the measurement
by one time step.
Compute f o f .

( f o f )(mt )  f ( f (mt ))

 f (mt 1)
 mt 2

The composition of an updating function with itself
corresponds
to a two-step updating function.

Composition
The updating function f updates the measurement
by one time step.
Compute f o f .

( f o f )(mt )  f ( f (mt ))

 f (mt 1)
 mt 2

The composition of an updating function with itself
corresponds
to a two-step updating function.

Composition
The updating function f updates the measurement
by one time step.
Compute f o f .

( f o f )(mt )  f ( f (mt ))

 f (mt 1)
 mt 2

The composition of an updating function with itself
corresponds
to a two-step updating function.

Composition
Example:
Compute the composition of the drug
concentration updating function with itself.
1
Mt 1  Mt 1
2
If M 0 
1, compute the concentration of
methadone in the patient’s blood every other
day for
4 days.
Inverse
The inverse function f
the updating function.
1
undoes the action of
1

f (mt 1 )  mt
The inverse function allows us to go backwards
one time-step and see what happened in the

past.
Inverse
Example:
If the concentration of methadone in patient’s
body on Wednesday is 4 units, was was the
concentration on Tuesday?
Cobwebbing
Cobwebbing is a graphical technique used to
determine the behaviour of solutions to a DTDS
without calculations.
This technique allows us to sketch the graph of
the solution (a set of discrete points) directly
from the graph of the updating function.
Cobwebbing
Cobwebbing is a graphical technique used to
determine the behaviour of solutions to a DTDS
without calculations.
This technique allows us to sketch the graph of
the solution (a set of discrete points) directly
from the graph of the updating function.
Cobwebbing
Algorithm:
1. Graph the updating function and the diagonal.
2. Plot the initial value m0 on the horizontal axis. From this point,
move vertically to the updating function to obtain the next value of
the measurement. The coordinates of this point are (m0,m1).
3. Move horizontally to the point (m1,m1) on the diagonal. Plot the
value m1 on the horizontal axis. This is the next value of the solution.
4. From the point (m1,m1) on the diagonal, move vertically to the
updating function to obtain the point (m1,m2) and then horizontally
to the point (m2,m2) on the diagonal. Plot the point m2 on the
horizontal axis.
5. Continue alternating (or “cobwebbing”) between the updating
function and the diagonal to obtain a set of solution points plotted
along the horizontal axis.
Cobwebbing
Example:
Starting with the initial condition b0  1 , sketch
the graph of the solution to the system bt 1  2bt
by cobwebbing 3 steps.


Cobwebbing
Cobwebbing
(b0 ,b1 )
b0


Cobwebbing
(b0 ,b1 )


 
b 0 b1
(b1,b1 )
Cobwebbing
(b1,b2 )
(b0 ,b1 )



 
b 0 b1
(b1,b1 )
Cobwebbing
(b1,b2 )
(b0 ,b1 )

(b2 ,b2 )
(b1,b1 )



 
b 0 b1

b2
Cobwebbing
(b2 ,b3 )

(b1,b2 )
(b0 ,b1 )

(b2 ,b2 )
(b1,b1 )



 
b 0 b1

b2
Cobwebbing
(b2 ,b3 )



(b2 ,b2 )
(b1,b2 )
(b0 ,b1 )
(b3 ,b3 )
(b1,b1 )



 
b 0 b1

b3
b2

A Solution From Cobwebbing
(b2 ,b3 )


(b2 ,b2 )
(b1,b2 )
(b0 ,b1 )
(b3 ,b3 )
(b1,b1 )



b 0 b1

b3
b2

Cobwebbing
Example:
Consider the DTDS for the methadone
concentration in a patient’s blood:
1
Mt 1  Mt 1
2
Cobweb for 3 steps starting from
(i) M 0  1
(ii) M 0  5
(iii) M
0 2
Cobwebbing
Cobwebbing
M0

Cobwebbing
M 0 M1
 
Cobwebbing
M 0 M1
 
Cobwebbing
M 0M1M 2
 
Cobwebbing
M 0M1M 2
 
Cobwebbing
M 0 MM
1 M
2 3
 
Cobwebbing
M0

Cobwebbing
M1


M0
Cobwebbing
M1


M0
Cobwebbing
M 2 M1
 

M0
Cobwebbing
M 2 M1
 

M0
Cobwebbing
M 3M 2M1
  

M0
Cobwebbing
M0

Equilibria
Definition:
A point m * is called an equilibrium of the DTDS
m t 1  f (m t )
if f (m* )  m* .


Geometrically,
the equilibria correspond to points
where the updating function intersects the
diagonal.
Equilibria
Equilibria
b*  0
Equilibria
Equilibria
M*  2
Solving for Equilibria
Algorithm:
1. Write the equation for the equilibrium.
2. Solve for m * .
3. Think about the results.

Solving for Equilibria
Examples:
Find the equilibria, if they exist, for each of the
following systems.
axt
(b) x t 1 
1 x t
1
(a) Mt 1  Mt 1
2

Cobwebbing
Example:
Consider the DTDS for a population of codfish
nt 1  0.6nt  5.3
where n t is the number of codfish in millions and t
is time.
 that initially there are 1 million codfish.
Suppose
 of the
 Determine the equilibria and the behaviour
population over time by cobwebbing.
Cobwebbing
Example:
Consider the DTDS for a population of codfish
nt 1  0.6nt  5.3
where n t is the number of codfish in millions and t
is time.
 that initially there are 1 million codfish.
Suppose
 of the
 Determine the equilibria and the behaviour
population over time by cobwebbing.
Cobwebbing
Cobwebbing
n0

Cobwebbing
n0


n1
Cobwebbing
n0


n1
Cobwebbing
n0 n2
 

n1
Cobwebbing
n0 n2
 

n1
Cobwebbing
n 0 n 2 n 3 n1
  
Cobwebbing
n 0 n 2 n 3 n1
  
Cobwebbing
n0 n2n 4n3 n1
  

A Solution From Cobwebbing
Solution:
n 0 n 2 n 3 n1
 
Stability of Equilibria
An equilibrium is stable if solutions that start
near the equilibrium move closer to the
equilibrium.
An equilibrium is unstable if solutions that start
near the equilibrium move away from the
equilibrium.
Stability of Equilibria
An equilibrium is stable if solutions that start
near the equilibrium move closer to the
equilibrium.
An equilibrium is unstable if solutions that start
near the equilibrium move away from the
equilibrium.