Math 141 Mathematical Modeling

Assignments

Unless stated otherwise, all homework assignments are from the textbook.

Class Reading Assignments Due
1 Chapter 0 and Chapter 1, Sections 1.1, 1.2 and 1.3 (pp. 1-19). (1) Peruse the links on the Course homepage.
(2) See Project 1.2 on pp 39-40. Do parts 1-6. Your work should consist of two parts: (i) A typed document, turned in at the beginning of class, that answers the questions posed in the problem. Include graphs and a printout of your Excel spreadsheet. Explain your work. Focus on good writing and clarity; and (ii) Your actual Excel spread sheet file should be saved as HW1.xls in your subfolder in the Student Work folder Under Courses -> w08 -> math -> math141-01-w08. Mon, Feb 11

2 Read Section 1.4 (Read the statement, but not the proof, of Theorem 1.2) (1) Page 38: 3, 4.
(2) Project #1, which is due on Monday, Feb. 18, is the Great Lakes Pollution problem on page 45. Start looking this over. You may work with one other student on this project, submitting a joint paper. Wed, Feb 13

3 Finish reading Chapter 1.4 (1) See here. (If you can't read the math symbols on the word document sheet (beginning of problem #2) please let me know.) Fri, Feb 15

4 (1) See Homework #4 handed out in class today
(2) Reminder: Project #1 due Wednesday, Feb 20 Mon, Feb 18

5 Read Sections 3.1-3.5 See HW handout. You are welcome to use Mathematica but do problems (a) - (f) and (h)-(i) by hand. Wed, Feb 20

6 Read Sections 3.6 (although you can skip the more mathy parts) and 3.7 (1) See HW #6 handout. You may use Mathematica but include well labeled output if you do.
(2) Changed date for the midterm. It's will be on Monday, March 3. So you won't feel like there's a void in your life on Monday, Feb. 25, there will be a short quiz on that day. Fri, Feb 22

7 See the HW#7 handout for exercises and reading. Mon, Feb 25

8 Read all of Sections 3.8, 3.9 and 3.10. (We will not be covering the material in 3.11.) (1) Page 138: 7
(2) These questions pertain to the Scot Pines example of Section 3.10. (i) How long do we expect a tree to take on average to move from girth size 3 to girth size 4? (ii) How long do we expect for a tree to move from girth size 2 to the largest girth size?
(3) Reminder: Project 2 is due Friday, Feb 29. The midterm exam is Monday, March 3. Wed, Feb 27

9 (1) Review the material on Markov chains.
(2) And for your enjoyment, read In Shuffling Cards, 7 is a Winning Number. (1) Using the example that was done in class as a guide, how many coin flips does it take, on average, until the pattern Tail-Head-Tail appears? (Hint: Model this using a state space with eight states.) Include relevant Mathematica output with your solutions.
(2) A high school is tracking its retention rates. It has compiled statistics on the the numbers of students who move up from one grade to another, who drop out, and who graduate. Among frosh, 10% repeat the grade, 10% drop out, and 80% become sophomores. Among sophomores, 10% repeat their grade, and 85% become juniors. Among juniors, 5% repeat their grade and 5% drop out. And among seniors, 90% graduate and 5% drop out. What's the probability that someone who begins high school actually graduates? What's the probability that a junior does not graduate? Show your work.
(3) Recall the simple random walk of a frog on the clock. (At each time step, the frog either hops to the left or to the right or stays where it is with equal probability.) If the frog starts at 12 o'clock, how many steps will it take on average until the frog first lands on 6 o'clock? Fri, Feb 29

10 Test #2 on Monday

11 (1) Read Chapter 5 up to page 245.
(2) Play with this Least Squares applet (1) A glucose solution is administered intravenously into the bloodstream at a constant rate k. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Show how to set up a model for the concentration of the glucose solution in the bloodstream that uses differential equations. Suppose the concentration at time 0 is C₀. Solve your differential equation to find the concentration at any time t. What happens to the concentration of glucose in the bloodstream over time?
(2) The air in a room with volume 180 cubic meters contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 cubic meters per minute and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run? Show how to use compartment analysis to set up a model using differential equations.
(3) Find a moderate size dataset of two variables (on the Web, in a research journal, in the newspaper) for which a linear model (straight line) might be appropriate. Use Excel to obtain a scatterplot of your data and insert the least square line. Find the equation of your line. Write a sentence assessing how appropriate you think a linear model is for these data. Wed, March 5

11 Read pages 245-250. (1) Here is data on the midyear population in Spain, in thousands, from 1955 to 2000: 29,319, 30,641, 32,085, 33,876, 35,564, 37,488, 38,535, 39,351, 39,750, and 40,016. (Thus the 1960 population was 30,641, and the 1990 population was 39,351. You will model these data four ways: (i) Using a simple exponential growth model; (ii) using the technique we used today in class, letting r=r(t) be a linear function of t; (iii) fitting a logistic model (You will need to estimate a carrying capacity (maximum) and a proportionality constant. Use your best judgement and the fact that when t is "small" the process behaves like a simple exponential model.) Note that the solution to the logistic differential equation is given on page 245.; and (iv) using the ideas we used in class today, but letting r(t) be a quadratic function. When you graph your r(t)'s, in Excel click on Chart and then pick Add Trendline. Now choose a second order polynomial (quadratic) function. The differential equation that you will need to solve will be different than in (ii), above, but it should still be straightforward. For all your models write clearly what you did, with an emphasis on explanation. Don't just put a lot of numbers and equations on the paper. Write in complete sentences. Finally, assess your four models using appropriate, and well labeled graphs. Which model is the best? Which model gives the best prediction, in your opinion, for the 2005 population? What is your best guestimate for the 2005 population? Fri, March 7

12 Read pp. 253-255 and 259. You are not responsible for Sections 5.2.3 or 5.2.4 or the material on complex eigenvalues. (1) Consider the system of differential equations (df/dt) = f - 3g and (dg/dt) = -2f + 4g, with initial conditions f(0)=g(0) = 10. Solve the system and discuss the long term behavior of f(t) and g(t).
(2) Project 5.1 on pages 309-310. Note that the solution in part (4) will be an implicit solution.
(3) The quiz on Monday will cover material from pages 239-251, 254-255, and 259.
(4) I will post solutions to this homework later in the weekend. Mon, March 10

Class	Reading	Assignments	Due
1	Chapter 0 and Chapter 1, Sections 1.1, 1.2 and 1.3 (pp. 1-19).	(1) Peruse the links on the Course homepage. (2) See Project 1.2 on pp 39-40. Do parts 1-6. Your work should consist of two parts: (i) A typed document, turned in at the beginning of class, that answers the questions posed in the problem. Include graphs and a printout of your Excel spreadsheet. Explain your work. Focus on good writing and clarity; and (ii) Your actual Excel spread sheet file should be saved as HW1.xls in your subfolder in the Student Work folder Under Courses -> w08 -> math -> math141-01-w08.	Mon, Feb 11
2	Read Section 1.4 (Read the statement, but not the proof, of Theorem 1.2)	(1) Page 38: 3, 4. (2) Project #1, which is due on Monday, Feb. 18, is the Great Lakes Pollution problem on page 45. Start looking this over. You may work with one other student on this project, submitting a joint paper.	Wed, Feb 13
3	Finish reading Chapter 1.4	(1) See here. (If you can't read the math symbols on the word document sheet (beginning of problem #2) please let me know.)	Fri, Feb 15
4		(1) See Homework #4 handed out in class today (2) Reminder: Project #1 due Wednesday, Feb 20	Mon, Feb 18
5	Read Sections 3.1-3.5	See HW handout. You are welcome to use Mathematica but do problems (a) - (f) and (h)-(i) by hand.	Wed, Feb 20
6	Read Sections 3.6 (although you can skip the more mathy parts) and 3.7	(1) See HW #6 handout. You may use Mathematica but include well labeled output if you do. (2) Changed date for the midterm. It's will be on Monday, March 3. So you won't feel like there's a void in your life on Monday, Feb. 25, there will be a short quiz on that day.	Fri, Feb 22
7		See the HW#7 handout for exercises and reading.	Mon, Feb 25
8	Read all of Sections 3.8, 3.9 and 3.10. (We will not be covering the material in 3.11.)	(1) Page 138: 7 (2) These questions pertain to the Scot Pines example of Section 3.10. (i) How long do we expect a tree to take on average to move from girth size 3 to girth size 4? (ii) How long do we expect for a tree to move from girth size 2 to the largest girth size? (3) Reminder: Project 2 is due Friday, Feb 29. The midterm exam is Monday, March 3.	Wed, Feb 27
9	(1) Review the material on Markov chains. (2) And for your enjoyment, read In Shuffling Cards, 7 is a Winning Number.	(1) Using the example that was done in class as a guide, how many coin flips does it take, on average, until the pattern Tail-Head-Tail appears? (Hint: Model this using a state space with eight states.) Include relevant Mathematica output with your solutions. (2) A high school is tracking its retention rates. It has compiled statistics on the the numbers of students who move up from one grade to another, who drop out, and who graduate. Among frosh, 10% repeat the grade, 10% drop out, and 80% become sophomores. Among sophomores, 10% repeat their grade, and 85% become juniors. Among juniors, 5% repeat their grade and 5% drop out. And among seniors, 90% graduate and 5% drop out. What's the probability that someone who begins high school actually graduates? What's the probability that a junior does not graduate? Show your work. (3) Recall the simple random walk of a frog on the clock. (At each time step, the frog either hops to the left or to the right or stays where it is with equal probability.) If the frog starts at 12 o'clock, how many steps will it take on average until the frog first lands on 6 o'clock?	Fri, Feb 29
10		Test #2 on Monday
11	(1) Read Chapter 5 up to page 245. (2) Play with this Least Squares applet	(1) A glucose solution is administered intravenously into the bloodstream at a constant rate k. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Show how to set up a model for the concentration of the glucose solution in the bloodstream that uses differential equations. Suppose the concentration at time 0 is C₀. Solve your differential equation to find the concentration at any time t. What happens to the concentration of glucose in the bloodstream over time? (2) The air in a room with volume 180 cubic meters contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 cubic meters per minute and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run? Show how to use compartment analysis to set up a model using differential equations. (3) Find a moderate size dataset of two variables (on the Web, in a research journal, in the newspaper) for which a linear model (straight line) might be appropriate. Use Excel to obtain a scatterplot of your data and insert the least square line. Find the equation of your line. Write a sentence assessing how appropriate you think a linear model is for these data.	Wed, March 5
11	Read pages 245-250.	(1) Here is data on the midyear population in Spain, in thousands, from 1955 to 2000: 29,319, 30,641, 32,085, 33,876, 35,564, 37,488, 38,535, 39,351, 39,750, and 40,016. (Thus the 1960 population was 30,641, and the 1990 population was 39,351. You will model these data four ways: (i) Using a simple exponential growth model; (ii) using the technique we used today in class, letting r=r(t) be a linear function of t; (iii) fitting a logistic model (You will need to estimate a carrying capacity (maximum) and a proportionality constant. Use your best judgement and the fact that when t is "small" the process behaves like a simple exponential model.) Note that the solution to the logistic differential equation is given on page 245.; and (iv) using the ideas we used in class today, but letting r(t) be a quadratic function. When you graph your r(t)'s, in Excel click on Chart and then pick Add Trendline. Now choose a second order polynomial (quadratic) function. The differential equation that you will need to solve will be different than in (ii), above, but it should still be straightforward. For all your models write clearly what you did, with an emphasis on explanation. Don't just put a lot of numbers and equations on the paper. Write in complete sentences. Finally, assess your four models using appropriate, and well labeled graphs. Which model is the best? Which model gives the best prediction, in your opinion, for the 2005 population? What is your best guestimate for the 2005 population?	Fri, March 7
12	Read pp. 253-255 and 259. You are not responsible for Sections 5.2.3 or 5.2.4 or the material on complex eigenvalues.	(1) Consider the system of differential equations (df/dt) = f - 3g and (dg/dt) = -2f + 4g, with initial conditions f(0)=g(0) = 10. Solve the system and discuss the long term behavior of f(t) and g(t). (2) Project 5.1 on pages 309-310. Note that the solution in part (4) will be an implicit solution. (3) The quiz on Monday will cover material from pages 239-251, 254-255, and 259. (4) I will post solutions to this homework later in the weekend.	Mon, March 10

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