1. Announcements Topics: -roughly sections 2.3, 3.1, 3.2, and 3.3 * 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”)

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

3. Trigonometric Models Example: (A41, #2.)

4. Graphs of Trigonometric Functions Example:

5. 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.

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

7. 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)

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

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

10. 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)

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

12. 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.

13. 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

14. 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

15. 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.

16. 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.

17. 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

18. 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

19. 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

20. 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

21. Example: A Discrete-Time Dynamical System for a Bacterial Population ColonyInitial Population b t (millions) Final Population b t+1 (millions) 10.470.94 23.306.60 30.731.46 42.805.60 51.503.00 60.621.24 Data:

22. Example: A Discrete-Time Dynamical System for a Tree Growth TreeInitial Height, h t (m) Final Height, h t+1 (m) 123.123.9 218.719.5 320.621.4 416.016.8 532.533.3 619.820.6 Data:

23. 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.

24. 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

25. 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.

26. 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

27. 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

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

29. 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.

30. 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.

31. 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.

32. 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.

33. 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.

34. 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.

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

36. 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.

37. 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.

38. Cobwebbing Algorithm: 1.Graph the updating function and the diagonal. 2.Plot the initial value m 0 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 (m 0,m 1 ). 3.Move horizontally to the point (m 1,m 1 ) on the diagonal. Plot the value m 1 on the horizontal axis. This is the next value of the solution. 4.From the point (m 1,m 1 ) on the diagonal, move vertically to the updating function to obtain the point (m 1,m 2 ) and then horizontally to the point (m 2,m 2 ) on the diagonal. Plot the point m 2 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.

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

40. Cobwebbing

47. A Solution From Cobwebbing

48. Cobwebbing Example: Consider the DTDS for the methadone concentration in a patient’s blood: Cobweb for 3 steps starting from (i) (ii) (iii)

49. Cobwebbing

63. 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.

64. Equilibria

68. Solving for Equilibria Algorithm: 1.Write the equation for the equilibrium. 2.Solve for 3.Think about the results.

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

70. 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.

71. 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.

72. Cobwebbing

81. A Solution From Cobwebbing Solution:

82. 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.

83. 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.

84. MODELLING WITH DTDSs Bacterial Population Growth: The parameter is called per capita production. It represents the number of new bacteria produced per bacterium.

85. MODELLING WITH DTDSs Bacterial Population Growth: r=2 each bacterium divides into two daughter bacteria and each daughter has a 2/2=1=100% chance of survival r=1.5 each bacterium divides into two daughter bacteria and each daughter has a 1.5/2=3/4=75% chance of survival r=0.5 each bacterium divides into two daughter bacteria and each daughter has a 0.5/2=1/4=25% chance of survival

86. Bacterial Population Growth in General Solution: (Unrealistic) Assumption: r is constant Reality: r depends on the size of the population (resources are limited) small populations less competition higher r large populations more competition lower r

87. Bacterial Population Growth in General Solution: (Unrealistic) Assumption: r is constant Reality: r depends on the size of the population (resources are limited) small populations less competition higher r large populations more competition lower r

88. Bacterial Population Growth in General Solution: (Unrealistic) Assumption: r is constant Reality: r will depend on the size of the population (resources are limited) small populations less competition higher r large populations more competition lower r

89. Bacterial Population Growth in General Solution: (Unrealistic) Assumption: r is constant Reality: r will depend on the size of the population (resources are limited) small populations less competition higher r large populations more competition lower r

90. MODELLING WITH DTDSs Model for Limited Bacterial Population Growth: Replace the constant r by a function which matches natural observations:.

91. MODELLING WITH DTDSs Model for Limited Bacterial Population Growth: Replace the constant r by a function which matches natural observations:.

92. MODELLING WITH DTDSs Model for Limited Bacterial Population Growth: Example:

93. MODELLING WITH DTDSs Model for Limited Bacterial Population Growth: Example: Determine equilibria and behaviour of nearby solutions by cobwebbing.

94. elimination of chemicals *** filtration by kidneys (kidneys break down constant amount per hour … caffeine) *** breaking down the chemicals using enzymes from the liver (amount of chemical broken down depends on the amount present … alcohol)

95. Substance Absorption (Elimination) and Replacement (Consumption) Models Absorption of Caffeine: Our bodies eliminate caffeine at a constant rate of 13% per hour. DTDS: * Similar to “methadone” example amount of caffeine (mg) 1 hour later amount of caffeine now amount of “new” caffeine consumed at time t+1

96. Substance Absorption (Elimination) and Replacement (Consumption) Models Absorption of Caffeine: Our bodies eliminate caffeine at a constant rate of 13% per hour. DTDS: * Similar to “methadone” example amount of caffeine (mg) 1 hour later amount of caffeine now amount of “new” caffeine consumed at time t+1

97. Substance Absorption (Elimination) and Replacement (Consumption) Models Absorption of Caffeine: Our bodies eliminate caffeine at a constant rate of 13% per hour. DTDS: * Similar to “methadone” example amount of caffeine (mg) 1 hour later amount of caffeine now amount of “new” caffeine consumed at time t+1

98. Substance Absorption (Elimination) and Replacement (Consumption) Models Absorption of Caffeine: Our bodies eliminate caffeine at a constant rate of 13% per hour. DTDS: * Similar to “methadone” example amount of caffeine (mg) 1 hour later amount of caffeine now amount of “new” caffeine consumed at time t+1

99. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: The amount of alcohol that is broken down by the liver depends on the amount of alcohol present in the body. The larger the amount, the smaller the proportion of alcohol being eliminated. *Similar to the limited growth population model

100. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: The amount of alcohol that is broken down by the liver depends on the amount of alcohol present in the body. The larger the amount, the smaller the proportion of alcohol being eliminated. *Similar to the limited growth population model

101. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: The amount of alcohol that is broken down by the liver depends on the amount of alcohol present in the body. The larger the amount, the smaller the proportion of alcohol being eliminated. *Similar to the limited growth population model

102. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: The amount of alcohol that is broken down by the liver depends on the amount of alcohol present in the body. The larger the amount, the smaller the proportion of alcohol being eliminated. *Similar to the limited growth population model

103. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: The amount of alcohol that is broken down by the liver depends on the amount of alcohol present in the body. The larger the amount, the smaller the proportion of alcohol being eliminated. *Similar to the limited growth population model

104. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: DTDS: amount of alcohol (g) 1 hour later amount of alcohol now amount of “new” alcohol consumed at time t+1 rate of elimination

105. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: Rate of Elimination: DTDS:

106. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: Rate of Elimination: DTDS:

107. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: A standard drink contains 14g of alcohol. Compare what happens over time for the following situations: (a)You consume two drinks right away and continue to have half of a drink every hour (a)You consume one drink every hour

108. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: A standard drink contains 14g of alcohol. Compare what happens over time for the following situations: (a)You consume two drinks right away and continue to have half of a drink every hour (a)You consume one drink every hour

109. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: A standard drink contains 14g of alcohol. Compare what happens over time for the following situations: (a)You consume two drinks right away and continue to have half of a drink every hour (a)You consume one drink every hour

110. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: Example: A standard drink contains 14g of alcohol. Compare what happens over time for the following situations: (a)You consume two drinks right away and continue to have half of a drink every hour (a)You consume one drink every hour

111. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: (a) You consume two drinks right away and continue to have half of a drink every hour

112. Substance Absorption (Elimination) and Replacement (Consumption) Models Elimination of Alcohol: (b) You consume one drink every hour