1. Announcements Topics: Work On:
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”)
Test 1 in one week!! Information posted soon on the webpage.

2. 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.

3. Semilog graphs
Example: Compare the graphs and semilog graphs for and

4. Semilog graphs
Original Graphs
Semilog Graphs

5. 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 where is the value of the measurement at time is the initial value of the measurement, and is a parameter which describes the rate at which the measurement changes

6. 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.
1.38 hours = 83 mins = 1 hour and 23 minutes

7. Half-Lives of Drugs How would you define the half-life of a drug?
BIG DIFFERENCE BETWEEN THE TIME YOU FEEL EFFECTS AND THE TIME IT IS ELIMINATED FROM YOUR BODY.
Anything missing from this list?

8. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

9. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

10. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

11. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

12. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

13. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

14. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

15. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

16. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

17. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

18. Half-Lives of Drugs Example: Thinking in Half-Lives
# of half-lives
amount left in body
% amount left in body
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.

19. Trigonometric Functions
Trigonometric functions are used to model quantities that oscillate.

20. 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.
Let S(t)=# sharks at time t. Sketch the graph. Label Jan (t=0) July (t=6) Jan (t=12).
S(t)=AcosB(t-C)+D where |A|=amplitude, period=2pi/B, C=phase shift (horizontal shift), D=average value (vertical shift)

21. Trigonometric Models
Example: (A40, #2.)

22. Graphs of Trigonometric Functions
Example:

23. 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.

24. Inverse of Sine Restrict the domain of to Now the function is
one-to-one on this
interval so we can
define an inverse.

25. Inverse of Sine
The inverse of the restricted sine function is denoted by or Cancellation equations: Calculate:
(domain of sin x)
(domain of arcsin x)

26. Graphs of Sine and Arcsine
y = sin x
y = arcsin x
domain:
range:
domain:
range:

27. Inverse of Tangent Restrict the domain of to This portion of tangent
passes the HLT so tangent
is one-to-one here

28. Inverse of Tangent
The inverse of the restricted tangent function is denoted by or Cancellation equations: Calculate:
(restricted domain of tan x)
(domain of arctan x)

29. Graphs of Tangent and Arctangent
y = tan x
y = arctan x
y = cos x
y = arccos x
domain:
range:
domain:
range:

30. Real-life Use of Arctangent
Example: Model for World Population One of the many models used to analyze human population growth is given by where t represents a calendar year and P(t) is the population in billions.
In this case, the formula works both ways, by modelling past population as well as trying to predict future growth.

31. 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

32. 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

33. 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.

34. 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.

35. Discrete-Time Dynamical Systems and Updating Functions
Let represent the measurement of some quantity. The relation between the initial measurement and the final measurement is given by the discrete-time dynamical system The updating function accepts the initial value as input and returns the final value as output. Note: represents current time and represents one time-step into the future

36. Discrete-Time Dynamical Systems and Updating Functions
Let represent the measurement of some quantity. The relation between the initial measurement and the final measurement is given by the discrete-time dynamical system The updating function accepts the initial value as input and returns the final value as output. Note: represents current time and represents one time-step into the future

37. Discrete-Time Dynamical Systems and Updating Functions
Let represent the measurement of some quantity. The relation between the initial measurement and the final measurement is given by the discrete-time dynamical system The updating function accepts the initial value as input and returns the final value as output. Note: represents current time and represents one time-step into the future

38. Discrete-Time Dynamical Systems and Updating Functions
Let represent the measurement of some quantity. The relation between the initial measurement and the final measurement is given by the discrete-time dynamical system The updating function accepts the initial value as input and returns the final value as output. Note: represents present time and represents one time-step into the future

39. Example: A Discrete-Time Dynamical System for a Bacterial Population
Data:
Colony
Initial Population bt
(millions)
Final Population bt+1 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

40. Example: A Discrete-Time Dynamical System for a Tree Growth
Data:
Tree
Initial Height, ht
(m)
Final Height, ht+1 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

41. 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.

42. Solutions
Definition: The sequence of values of for 0, 1, 2, … is the solution of the discrete-time dynamical system starting from the initial condition

43. Solutions
Definition: The sequence of values of for 0, 1, 2, … is the solution of the discrete-time dynamical system starting from the initial condition The graph of a solution is a discrete set of points with the time on the horizontal axis and the measurement on the vertical axis.

44. Finding Solutions
Example 1: Find a solution of the bacterial discrete-time dynamical system Example 2: Find a solution of the tree height discrete-time dynamical system

45. Summary of Solutions
Basic Exponential Discrete-time Dynamical System If with initial condition , then Basic Additive Discrete-time Dynamical System If with initial condition , then

46. Manipulating Updating Functions
All of the operations that can be applied to ordinary functions can be applied to updating functions, but with special interpretations

47. Composition
The updating function updates the measurement by one time step. Compute The composition of an updating function with itself corresponds to a two-step updating function.

48. Composition
The updating function updates the measurement by one time step. Compute The composition of an updating function with itself corresponds to a two-step updating function.

49. Composition
The updating function updates the measurement by one time step. Compute The composition of an updating function with itself corresponds to a two-step updating function.

50. Composition
The updating function updates the measurement by one time step. Compute The composition of an updating function with itself corresponds to a two-step updating function.

51. Composition
Example: Compute the composition of the drug concentration updating function with itself. If compute the concentration of methadone in the patient’s blood every other day for 4 days.

52. Inverse
The inverse function undoes the action of the updating function. The inverse function allows us to go backwards one time-step and see what happened in the past.

53. Inverse
Example: If the concentration of methadone in patient’s body on Wednesday is 4 units, was was the concentration on Tuesday?

54. 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.

55. 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.

56. Cobwebbing Algorithm: Graph the updating function and the diagonal.
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).
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.
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.
Continue alternating (or “cobwebbing”) between the updating function and the diagonal to obtain a set of solution points plotted along the horizontal axis.

57. Cobwebbing
Example: Starting with the initial condition , sketch the graph of the solution to the system by cobwebbing 3 steps.

58. Cobwebbing
Graph updating function and diagonal

59. Cobwebbing
Put initial condition on the horizontal axis and go up to the updating function to find the next measurement.

60. Cobwebbing
Head over to the diagonal to get the point (b1,b1) now, b1 becomes your new “initial condition” on the horizontal axis

61. Cobwebbing
Back up to the updating function to get b2 from b1

62. Cobwebbing
Head over to the diagonal so you can reflect this point down to the horizontal axis giving you another ‘initial condition”

63. Cobwebbing
Go up to the updating function to get b3 from b2

64. Cobwebbing
Back over to the diagonal and then drop down to plot b3 on the horizontal axis.

65. A Solution From Cobwebbing

66. Cobwebbing
Example:
Consider the DTDS for the methadone concentration in a patient’s blood:
Cobweb for 3 steps starting from

67. Cobwebbing

68. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

69. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

70. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

71. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

72. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

73. Cobwebbing

74. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

75. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

76. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

77. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

78. Cobwebbing
Cobweb for the drug concentration example, starting with two different initial conditions… if the initial condition is above the diagonal vs if its below

79. Cobwebbing

80. Cobwebbing

81. Equilibria
Definition: A point is called an equilibrium of the DTDS if Geometrically, the equilibria correspond to points where the updating function intersects the diagonal.

82. Equilibria

83. Equilibria

84. Equilibria

85. Equilibria

86. Solving for Equilibria
Algorithm:
Write the equation for the equilibrium.
Solve for
Think about the results.

87. Solving for Equilibria
Examples: Find the equilibria, if they exist, for each of the following systems. (a) (b)

88. Cobwebbing
Example: Consider the DTDS for a population of codfish where is the number of codfish in millions and is time. Suppose that initially there are 1 million codfish. Determine the equilibria and the behaviour of the population over time by cobwebbing.

89. Cobwebbing
Example: Consider the DTDS for a population of codfish where is the number of codfish in millions and is time. Suppose that initially there are 1 million codfish. Determine the equilibria and the behaviour of the population over time by cobwebbing.

90. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31.

91. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

92. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

93. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

94. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

95. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

96. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

97. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

98. Cobwebbing
EQUILIBRIA ALGEBRAICALLY: N*=3.31… ALSO NOTE THE POINT (3.31, 3.31) IN THE DIAGRAM.

99. A Solution From Cobwebbing

100. 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.

101. 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.