Math 155 Notes on Lectures

Transcription

1 Math 155 Notes on Lectures Day 1 Preliminaries: Go over course policies. Note the exam dates and times (Thursday evenings) and state that no make-up exams are given, with a few exceptions if an alternate exam time request form is submitted at least a week in advance. Tutoring and office hours are held in the Great Hall of the TILT building. There will (likely) be a Math 155 table, and students are encouraged to meet there to work on practice problems. 1.1 Biology and Dynamics: 1. Examples of biomath: There are a number of examples in this section of how mathematics is used in biology. There s no need to go through all of them, but some motivational example is good to show them to where we re headed. Malaria is a good one since you can draw pictures of infected people. Early in the 20th century, it was discovered that malaria is transmitted by mosquitoes. (It is only female mosquitoes, by the way, that transmit the disease.) Question: Can the spread of disease be controlled by killing mosquitoes, even if not every mosquito is killed? Some argued that it couldn t. (You may have them discuss if it could or not and maybe see why it s not so obvious.) I really enjoyed reciting (with an emphasis on think) the following poem Sir Ronald Ross wrote as he was researching malaria: In this, O Nature, yield to me. I pace and pace, and think and think, and take with fever d hands, and note down all I see. That some distant light may haply break. The painful faces ask, can we not cure? We answer, No, not yet; we seek the laws. O reveal through all this thy obscure The unseen, small, million-murdering cause. (You don t have to read it.) Based on the asumptions that i) An infected person becomes infected when bitten by an infected mosquito and ii) an uninfected mosquito becomes infected after biting an infected person, Sir (i.e., future Sir) Ronald Ross wrote down a mathematical model, thought about how population sizes change with time, and showed that the disease could be eradicated even without killing every mosquito. 2. Dynamical Systems: Dynamical systems encode how things change. i) Discrete-time dynamical systems arise if measurements are made at equally spaced intervals. Examples: a) The population is a given year as a function of the population in the previous year. b) The number of mutant alleles present in one generation as a function of the number of mutant alleles present in the previous generation. You may wish to dwell on discrete-time dynamical systems, and even have them do a handout. *Handout: Hilary s handout. ii) Continuous-time dyamical systems arise if measurements are made continuously. 1.2 Variables, Parameters, and Functions in Biology: This section is mostly review, but most (all?) of them have not thought about variables and parameters, so define those and give examples. The bacterial population growth example comes back a lot and is a very good example to build intuition, so take some time to go through this with some data as in the tables on pages 6 and 7. You won t have time to go through all of these concepts, but it s a good idea to point out to them that they should know

2 i) functions: relations (although we won t emphasize this), range and domain, the vertical line test. I d spend a bit of time on something like Example The new feeling to get across to them here is that of patterns of change. Up to this point, they have thought of functions as formulas. But, in this class, we will discover the life in functions, how their rate of change drives them makes them go. Have they every thought about linear functions? Remember the slope? Wouldn t it be nice if you could read the rate of change off of the formulas for all functions so easily? You can have them do exercises such as problems 37, 39. ii) combining functions: adding, multiplying, composition. Composition is hard for them, particularly within applications. Something like Example , or problem 53 would be good. iii) inverse functions: They should be able to find inverses that exist. Suggestion: Dole out a number of problems, and have them work on them and discuss them with neighbors. It s a good opportunity to get them used to working and thinking and talking in class. 1.3 The Units and Dimensions of Measurements and Functions: 1. Converting between functions: Stress the importance of units with the examples of the chemical formula and conversion factor between inches and cm. Explain how to convert by doing Example (or something similar). Here s another example you can work out (or have students work out given the conversion factors): Example: Convert 1 C (cup) to ml given that 1 pint = liters and that 1 pint = 2 C. Answer: 1 C = ml. There s an algorithm on page 26. You may just point out that there is one there without explicity going over it. 2. Dimensions: Dimensions describe underlying quantities while units describe standards of measure. Give them several examples from Table such as the quantities length, area, and speed by giving them the dimensions and units. Ask them for some examples. 3. Fundamental relations: Fundamental relations are formulas for translating between dimensions. 1.4 Linear Functions and their Graphs It should be easy to follow the text in this section. 1. Proportionality relations: Cover proportionality relations as the book does, with some examples. Example provides their first example of an updating function, a concept that will we will use extensively later. Define the slope of a line. 2. The equation of a line: This section should be review from (junior?) high school. But, do cover it carefully as it is important and most students need to review it. Cover the point-slope and slope-intercept forms for a line. The example with Farenheit and Celcius is good. 3. Finding equations and graphing lines. Show them how to find the equation of a line from data (as in exercises 43-54) and discuss the concept of whether all the data points lie on a line. Be sure to introduce the terminology interpolation, as in example Discrete-Time Dynamical Systems *Handout: Writing DTDS s given English descriptions, and doing iteration on a calculator. Goals: Understaning the concept of updating functions, and how to iterate and graph them. Understanding subscript notation (b t ) Understanding the difference between the closed-form solution to an updating function and the updating function itself. Being able to find the solution to an updating function. 1. Population growth of bacteria: We learn here how to iterate the updating function. Be sure to define the initial condition and the index. Explain index/subscript notation (as in b t ). The key is that if we know

3 i) where (at what value) the quantity starts, and ii) how the quantity changes, then we can iterate find its values at later times. Work examples 1.5.1, 1.5.2, 1.5.3, Modeling: Help them understand math notation in terms of English sentences, and to able to put English sentences into math notation. See the handout. (3. Manipulation updating functions: One example of the compositon of an updating function with itself if good, but you won t have time to dwell on this. No need to do an example with an inverse.) (4. Units and Dimensions: Do Example and/or Example ) 5. Finding solutions: Do Example to teach them how to find a solution. Some students have great difficulties understanding how this is different from the updating function itself, so be sure to explain this slowly and surely. Also do Examples and Show them that the lines that the solutions corresponding to different initial conditions lie on parallel lines in Example Discuss why this is the case. Work Example , but omit unless you have time (not likely). Note: This is an important section with new ideas and lots of examples that are good to do. Points 1., 2., and 5. above are the ones to emphasize. 1.6 Analysis of Discrete-Time Dynamical Systems *Handout: Cobwebbing. It is not straightforward to find solutions to DTDS s. Most any DTDS you give me, I can t find find a solution for. Cobwebbing is a graphical technique for finding solutions. Goals: How to cobweb. What does a cobweb diagram tell us? What are equilibrium points and how do we find them graphically and algebraically? 1. Cobwebbing. Explain how to do this by example. Have them do the two handouts with you in class. You can use the dot-cam or the overhead to show them how. When you work examples, label the axes as well as the points (m 0, m 1 ), (m 1, m 2 ), (m 2, m 3 ), etc. on the appropriate places on the diagram, as in Fig Tell them that they must also label these points as such on homework, exams, and quizzes. Explain that the points on the updating function represent iterates; that is, successive values of the solution corresponding to each time step. Do examples and Leave these examples on the board, if possible, for the following discussion of equilibrium points. 2. Equilibria Explain graphically what an equilibrium point is. Give them definition Talk about whether the iterates are approachig the equilibrium in the examples. 3. Finding equilibria algebraically. Warning! Example is misleading! Adler solves for both the parameter r and the equilibrium b. Do NOT instruct them to solve for r. Many of them will solve for parameters anyway all the time, and it will be very annoying. Do examples 1.6.8, 1.6.9, and (or similar examples). It is important that they can solve for equilibria that depend on parameters. 1.7 Expressing Solutions with Exponential Functions Goals: Converting to exponential form. Solving doubling-time, half-life problems, and related problems (without memorizing a formula). Understanding and plotting semilog plots. 1. Bacterial population growth in general: Start with b t+1 = rb t. Explain why r (1.27) can have values different from 2. Go over how to find the solution (even though you did almost the same thing in section 1.6. By the way, they really should get to know this solution well. I was really upset that my students refused to even remember that we had a solution.). Give the results in teh table on page 80 and the graphs. You could make a handout on this to save some class time or use the dot-cam or overhead. Go over the graphs on page 80 carefully.

4 2. Laws of exponents and logs. Yes, M124 is a prerequisite, but don t assume that the students know these laws. I do find though that it takes a lot of time to write out all the laws on the board. You may put them on a handout (perhaps with the data for part 1) and then concentrate on examples and understanding the rules (for example, the rule a α+β = a α a β actually makes sense). Introduce the exponential function with base e and the natural log as its inverse. We ll use base e in this course. No need to dwell on base Expressing results with exponentials: Show them how to write b t = b 0 r t as b t = b 0 e (ln r)t. The students do need to know how to change from base a to base e. Questions requiring them to do that appear regularly on exams. Explain the usefulness of this expression while/by doing examples such as and Doubling time: Do examples such as and Do not encourage them to memorize the formula. 5. Half-life: Derive the general formula, but do not encourage them to memorize it. Do an example or two on half-life. 6. Semilog plots: Explain what they are and how to plot them. Do examples such as and Oscillations and Trigonometry *Handouts: 1. Graphing Trigonometric Functions (posted on the web; 2U if you want to hand it or something like it out). 2. Helpful handout on graphs of trig functions; on web (homework page as well as study resources page). This section may be covered concisely. Indeed, you may need this day to catch up and hardly cover this section at all. The key concepts: Oscillations are expressed by trigonometric functions (some biological examples: circadian rhythm, swinging arm,...) In this class, we use radians, not degrees. Make sure that calculators are in radian mode!!! Conversion factor: 1 = π radians 180 ; 1 = 180 π radians Off of the formula f(t) = A + B cos ( 2π T (t φ)) for a trigonometric function, one should be able to read off the average value A, amplitude B (and the related min and max values), phase φ, and period T. One may have to rewrite the formula to read off φ and T (e.g. if the formula is written as f(t) = cos(2t 4)). Those four values can also be read off of graphs and used to graph. The general picture is 1.9 A Model of Gas Exchange in the Lung * Handout: Premade charts that the students can fill in during the lecture can be handed out. Goals: Understanding the model of gas exchange in the lungs. Understanding updating functions involving one or more parameters. Understanding the dependence of the equilibrium on these parameters. Being able to write down a discrete-time dynamical system to describe a situation. Modeling: the exercises in this section have them write DTDS s given the English description of a situation. 1. A model of the lungs: Explain the lung model with Fig and do the calculations of the concentration using the five steps of the table on page 101. It s a good idea to show your calculations in each column;

5 then, you can omit the What we did column. Then, replace the concentration in the lung before a breath by c t to find the updating function, as done in the second table. Be sure to write out the final form of the updating function. Note: There are some wrong number values in the text. 2. The lung system in general: Now replace all of the various quantities by the parameters V, W, γ, and q = W/V = fraction of air exchanged, and derive the updating function, again using a table. (I just leave enough room in the table when I first make it, and then write the general quantities in a different color.) Also discuss the weighted average and give them definition Do examples like as time allows (and point out those examples to the students if there is not enough time to cover them all). 3. Lung dynamics with absorbtion. You are likely not to have much time for this, but please do try to cover it. In any case, point it out to them as something to read. 4. Writing down a DTDS given the description of a problem. Students should be able to convert simple descriptions of a situation into a DTDS. The prototypical example is population growth with harvesting; exercises in Section 1.9 are useful. The Spring 2009 Exam 1 has an infamous problem about seals that is similar to the lung model. You should point it out to the students as something to study, but we won t torture ourselves with a problem so difficult An Example of Nonlinear Dynamics Goals: Formulating and understanding a more complex example of a discrete-time dynamical system. Understanding and identifying stable and unstalbe equilibria. 1. A model of selection. Begin with the example in the book of a wild and a mutant-type population; b t+1 = 1.5b t ; m t+1 = 2m t, (you may twiddle with the numbers if you so desire), and define p t = number of mutants total population size = m t. m t + b t Follow Example to derive the updating function for p t ; p t+1 = 2p t 2p t + 1.5(1 p t ). Discuss why this updating function is better than the one on line Find the equilibria, and discuss their meaning (see page 114). 2. Now do the general case b t+1 = rb t ; m t+1 = sm t ; sp t p t+1 = sp t + r(1 p t ). You probably won t have time to do the whole derivation again for the general case, but it makes sense to replace the numbers by the variables for the p formula. It is important to spend time on the cobweb diagrams for the cases s > r, s < r and discuss the case s = r. Find the equilibria algebraically, notice that the stability is different, and formally define these terms as in Definition 1.16 if you havn t done so already.

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13 Math 155 Section 1.5 Discrete-Time Dynamical Systems 1. The situation: A tree starts at height 2m in year 0 and grows 5 meters per year. a) Write down a discrete-time dynamical system (DTDS) and initial condition describing this situation. Graph the updating function. b) Iterate your DTDS to find the height of the tree in years 1 through 7. Graph your values as a function of time. c) Write down a solution for this DTDS. d) Repeat a-c if the tree starts at height 20m and loses 50cm per year. e) Repeat a-c if the tree starts at height 20m and loses 10% of its height each year.

14 Iteration on a Calculator (TI-83, TI-84, etc.): Enter the initial condition (10 ENTER) Enter the dynamics using Ans for the measurement m t : (if the DTDS is m t+1 = 2m t 1, enter 2*Ans-1) Press ENTER repeatedly to iteratate. 2. The situation: A patient starts with a concentration of medicine in his bloodstream equal to 10 milligrams per liter. Each day, the patient uses up half of the medication in his bloodstream. However, the doctor gives him enough medication to increase the concentration of medication in the bloodstream by 4 milligrams per liter. a) Write down a discrete-time dynamical system (DTDS) and initial condition describing this situation. Graph the updating function. b) Iterate your DTDS to find the concentration of medicine in the bloodstream in days 1 through 7. Graph your values as a function of time. c) Write down a solution for this DTDS. d) Repeat a-c if the patient starts at concentration 10 milligrams per liter and uses 60% of the medication in the bloodstream each day.

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19 Step Volume Total Chemical Concentration Before Breath Exhaled After Exhale Inhaled After Inhale Step Volume Total Chemical Concentration Before Breath Exhaled After Exhale Inhaled After Inhale

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27 Math 155, Section 1.11 Let V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = , V c = 25, u = 8. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5. Math 155, Section 1.11 Let V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = , V c = 25, u = 8. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5. Math 155, Section 1.11 Let V t+1 represent the voltage of the AV node in the Heart Model. { e V t+1 = ατ V t + u if V t e ατ V c e ατ V t if V t > e ατ V c A. Suppose that e ατ = , V c = 25, u = 8. Does this system have an equilibrium? Why or why not? Justify your answer. If it has an equilibrium, find it algebraically. B. Cobweb from an initial value of V t = 50. Diagnose this heart as healthy, having a 2:1 block, or as having the Wenkebach Phenomenon. C. Repeat the above, replacing V c = 20 with V c = 5.