# Math 244 Fall 2003 Sections 1-3 and 7 Professor Bumby

• main course page.
• semester course page.
• (Sections 1, 2, and 3) Recitation Instructor's Page for this course.
• A Maple worksheet that produces the slope fields shown at the start of the lecture on September 08. The worksheet also shows some solutions and includes comments on the equations.
• Notes on hanging cables from lecture on September 08.
• Notes prepared for Math 252 on Euler's method and its application to proving a version of the existence and uniqueness theorems.
• Notes on Matrix exponentials from lecture on October 27. There is also an alternative version prepared for Math 252 that concentrates on the discovery of this approach in the case of complex eigenvalues. Prior experience shows that this approach gives an easier solution than the method described in the textbook, and that it is also more likely to lead to a correct answer. In particular, this means that exam questions will be written with the expectation that this method will be used.
• Notes on topics from chapter 9 slightly edited from last semester. Although not ready in time for lectures on this material, it should be useful when reviewing for the final exam. Definitions are collected in one place and techniques are described for the efficient treatment of examples.
• An overview of chapter 5 indicating differences between notation in the textbook and in the solutions shown in lecture. The book's emphasis on formulas requiring a fixed notation may make it more difficult to understand the simple principles used in these methods. In particular, the exam will emphasize finding the initial terms in a series rather than the recurrence formula connecting the terms, so the notation for general terms is less significant than finding the coefficient of a given power of the independent variable.
• A look back at chapter 2 noting its connection to things appearing later in the course. Although added after the final exam, it will be useful to anyone with a temporary grade who needs to prepare for a special exam.

## Special Announcements

Because of the large demand for this course, a fourth section (Section 07) meeting Thursday 6* in Beck 011 (Livingston Campus) has been added. All applicants for the course will be directed to this section. The class no longer fits in the lecture room originally assigned, and has been relocated; see Calendar entries for September 08 and 10 for details. No change is required in the meeting places of the original recitation classes.

This lecture section has registered with the Maple Adoption Program. This means that the Maple laboratory projects are a required part of the course. This is a new program that includes some benefits.

1. We are looking into the possibility of a tutorial in the use of Maple conducted (remotely) by a representative of Maplesoft.
2. Students registered in this lecture section are entitled to purchase a copy of the Student Edition of Maple9 for \$75US. Contact the lecturer for more information.

There were reports of difficulty printing Lab 3. I have also noticed printing problems on the systems that I use. In order to develop a workaround and report the problem to Maple to allow it to be corrected, details of printing problems should be reported to bumby@math.rutgers.edu. Use "Maple Printing" as the subject line of your message.

Grades for the Maple Labs have been entered in the FAS Gradebook. The "comments" column contains the scores recorded on the individual labs (including lab 0 even though that did not contribute to the grade). Please report any discrepancies to bumby@math.rutgers.edu.

## Calendar

• Wednesday, September 03. First Lecture (in SEC-210): Begin segment on chapters 1 (Generalities) and 2 (First order equations). Maple Lab 0 (Practice Lab) assigned. Here are links to description and seed file in both classic format and xml format for Lab 0. Since Maple is used in the Multivariable Calculus course, it is assumed that you have prior experience with the program. If you have any difficulty with Lab 0, you should visit the lecturer's office for a guided tour of Maple.
• Thursday, September 04. First Recitation class.
• Monday, September 08. Second Lecture. Starting with this class, lecture is in HIL-114 on Monday. Slides showing some slope fields of differential equations were shown at the start of the lecture. A Maple worksheet that produces the slope fields has been prepared. Notes on hanging cables were included in the lecture.
• Wednesday, September 10. Third Lecture. Starting with this class, lecture is in PH-115 on Wednesday.
• Wednesday, September 17. Maple Lab 0 due.
• Monday, September 22. Lecture included a segment based on Notes on Euler's method. Although this material is not suitable for exams, it is an important part of understanding the subject. The treatment in the notes emphasizes the connection between showing that solutions exist and computing them. It is not the strongest form of the theorem, but it avoids special technical constructions and helps to explain how numerical methods work.
• Wednesday, September 24. Exam 1, covering chapters 1 (Generalities) and 2 (First order equations). Some lecture time will have been devoted to Chapter 8 (Numerical methods) because the topic is related to Maple Labs, but this material is not suitable for exams. Exam questions will emphasize methods for finding solutions in closed form, but the interpretation of differential equations that model some processes will also appear. The process of actually constructing an equation from a description of the application will not appear. You may use a calculator on exams, but no books or papers are allowed.
• Wednesday, October 08. Maple Lab 2 due. Here are links to description and seed file in both classic format and xml format for Lab 2.
• Monday, October 20. Exam 2, covering chapters 3 (Second order linear equations), 4 (higher order linear equations). Most effort will be devoted to second order equations, but the extension of each topic to higher order equations will be illustrated when that topic is introduced. The exam will only include higher order equations of the types shown in lecture. Initially, the treatment of chapter 3 will follow the order of the text, but the applications from sections 8 of chapter 3 will be introduced before the inhomogeneous equations of section 6, and section 7 will be done last. The method of "variation of parameters", treated in section 7, is too tedious to make a good examination problem. It is included mainly to help you understand some solutions found by Maple. It is unlikely that you will ever need to use this method for hand computations.
• Wednesday, October 22. Maple Lab 3 due. Here are links to description and seed file in both classic format and xml format for Lab 3.
• Monday, October 27. Lecture on examples of matrix exponentials. The approach taken in the course, including exam questions will follow notes for this lecture and not the textbook. This topic is a good example of the ability to treat special cases by methods that are much more efficient than general methods used in the first description of a problem. The notes on this topic include four exercises.
• Wednesday, November 05. Maple Lab 4 due. Here are links to description and seed file in both classic format and xml format for Lab 4.
• Wednesday, November 12. Exam 3, covering Chapter 7 (linear systems) and an introduction to concepts of "nullcline", "equlibrium points" and "linearization" from Chapter 9. The "Fundamental Matrices" described in section 7.7 are a central idea. Notes will be provided in connection with the lecture of Monday, October 27 to show how to find them directly in many cases, saving a lot of effort. The nonlinear systems that we study are autonomous, i.e., time does not appear explicitly on the right sides of the equations. The trajectories in a phase plane are the main objects of study, and we will concentrate on determining information about the general appearance of these curves. In small regions, such a system of differential equations can be approximated by a linear system with constant coefficients. The behavior of these special systems plays a large role in this work. The qualitative information that is obtained can be used to guide the choice of numerical methods for finding details.
• Wednesday, November 19. Maple Lab 5 due. Here are links to description and seed file in both classic format and xml format for Lab 5. A student noted that something is missing in the formula for "dh1" in section 1b. Both seed files were corrected just before 11AM on Thursday, Nov. 13, and a correct lab description was posted at 9:40AM on Friday, Nov. 13. The correction is minor: if you have an earlier version and want to use it, you can find the correct value by looking at the expression "de1y" and finding the expression for y in terms of x that makes it zero. An error in section 1e of the seed files was also corrected. This time, the lab description was correct. Later, an error was noted in part 2b, which has been corrected in the seed files around 6:20 PM on Monday, Nov. 17 and in the lab description around 10:35AM on Tuesday, Nov. 18. Again, only a slight modification of existing files was required: the given expression dh2 uses a parametric description of the set where there is no vertical component of the slope field to plot the nullcline. A parametric description is used because one part of this set is a vertical line with parametric description [4,t] (instead of [4,4] as written). The Maple instruction includes a third field giving the values of t that need to be plotted. The ease with which these corrections can be described should allow work to be completed on the original schedule.
• Monday, December 08. Exam 4, covering Chapter 5 on Series solutions, including sections 5.6-8 on Regular Singular Points using Bessel's Equation as an example. In all cases, emphasis will be on finding initial terms of the series in powers of x. The tricks needed to get recurrence formulas are a distraction: if the recurrence is simple, you can guess it from the first few cases; if it isn't, it is unlikely to help identify the general term of the series. Similarly, the use of a base point other than zero can be handled by a simple change of variable, so it contributes nothing new.
• ***time line***
• Wednesday, December 17, 8-11 AM, Final Exam in Hill 114. See Lecturer's Home Page and Recitation Instructor's Course Page for office hours during Reading days, December 11 and 12.

## Lecture details

The textbook exercises done as examples in lecture will be listed here.

Date Section Exercises
Sep. 03 1.3 7, 9, 14
Sep. 08 2.2 2, 7, 10, 27(a)
Sep. 10 2.5 9, 16.
2.1 14, 16, 25(part).
Sep. 15 2.6 1, 2, 15.
2.3 7.
2.4 3 (as example of linear equation).
Sep. 17 2.6 21,
after extending previous discussion of 2.1#25 and 2.2#27.
Sep. 22 2.6 22.
Sep. 29 3.1 3.
3.2 4, 23.
4.1 13.
4.2 35.
Oct. 01 3.2 7, 13, 17.
3.3 1, 15, 17.
4.2 29.
Oct. 06 3.2 25.
3.3 9.
3.4 7, 11, 19, 25.
3.5 6, 11.
4.1 21.
4.2 32
Oct. 08 3.5 9, 20, 23.
3.8 A family of examples with different damping terms.
Oct. 13 3.6 1, 6, 7, 13.
4.3 3.
3.9 17.
Oct. 15 3.7 7, 15.
4.4 2
Oct. 22 7.2 1
and examples of calculating eigenvalues and eigenvectors for 2 by 2 systems with integer eigenvalues and 3 by 3 triangular matrices.
Oct. 27 7.6; 7.7 examples from notes
Oct. 29 7.6 1
7.7 7,11
7.8 7
7.9 1
Nov. 03 7.9 3
9.1 5
9.2 5
Nov. 05 9.1 13
9.2 7
9.3 5
Nov. 10 9.4 1.
9.5 1, 3.
9.6, 9.7, 9.8 General discussion only.
Nov. 17 5.2 2, 5, 7, 15, 17, 19.
Nov. 19 5.3 22.
5.5 1, 13.
Nov. 24 5.6 1, 3, 9, 11, 16.
Dec. 01 5.2 6.
5.5 4, 7.
5.6 2, 6.
Dec. 03 notes general examples, including Bessel functions.

## Maple

As new versions of the Maple assignments are prepared, they will be linked here as well as on the course page and the semester page.

The course grades will be based on a ranking on a 700 point scale composed of the following items:

• Four class exams, 80 points each, total 320. Expected time for each class exam will be 60 minutes. This allows time before the exam for last minute questions and a preview of the next segment of the course. This buffer will protect the exam from being disrupted by students arriving a little late.
• One three hour final exam, total 200.
• Four graded Maple Labs, 20 points each, total 80.
• Recitation grade, typically graded homework and quizzes (details will be announced by recitation instructor), total 100.

Exam 1 has been graded. The average score was 44.098 and median was 46. Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed my dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight (although all questions on this exam had the same weight).

Exam 1
Distribution
Range Count
75 - 80 0
70 - 74 7
65 - 69 5
60 - 64 13
55 - 59 9
50 - 54 10
45 - 49 10
40 - 44 6
35 - 39 10
30 - 34 6
25 - 29 6
20 - 24 8
15 - 19 8
below 15 3
Problems
Prob. # Scaled Avg.
1 6.43
2 6.98
3 6.43
4 4.81
5 2.97

Exam 2 has been graded. The average score was 66.112. A scatter plot shows the comparison of grades on this exam with the score on exam 1. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). There is also a line of slope -1 showing sum of 100, which roughly indicates a satisfactory combined score for determining "warning grades". Individual grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed my dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight.

Exam 2
Distribution
Range Count
80 13
75 - 79 13
70 - 74 19
65 - 69 13
60 - 64 12
56 - 59 10
50 - 54 10
47 - 48 4
37 - 44 4
Problems
Prob. # Scaled Avg.
1 9.34
2 9.82
3 9.00
4 5.95
5 8.69
6 8.18
7 5.86

Exam 3 has been graded. The average score was 51.465, and the median was 55. A scatter plot shows the comparison of grades on this exam with the sum of scores on exams 1 and 2. Only individuals who took all exams are represented in this plot. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). There are also lines of slope -1 showing sums of 135, 150, 165, 180, 195, 210, and 225 that indicate a tendency for the total grades to be forming gaps at intervals of 15 points. Course grades will be based on a total that also includes Maple, recitation work and the final exam. All of these component should be present before attempting to make qualitative distinctions between grades. Individual exam grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed my dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight.

Exam 3
Distribution
Range Count
80 2
71 - 75 5
66 - 70 15
61 - 64 9
56 - 60 18
51 - 55 8
47 - 50 10
41 - 45 9
38 - 40 5
31 - 35 6
20 - 28 11
below 20 1
Problems
Prob. # Scaled Avg.
1 2.87
2 7.61
3 6.34
4 8.19
5 6.39
6 6.07

Exam 4 has been graded. The average score was 58.389, and the median was 63. A scatter plot shows the comparison of grades on this exam with the sum of scores on exams 1, 2 and 3. Only individuals who took all exams are represented in this plot. The line of positive slope in the figure is a regression line (the line that minimizes the sum of the squares of the vertical distances of the points). There are also lines of slope -1 showing sums of 140, 170, 200, 225, 260, and 275 that indicate a tendency for the total grades to be forming clusters. Course grades will be based on a total that also includes Maple, recitation work and the final exam. All of these component should be present before attempting to make qualitative distinctions between grades. Individual exam grades have been entered in the FAS Gradebook. There is also a distribution of scores (but no attempt to assign letter grades) and scaled averages (formed my dividing by the maximum possible score [or base score ] and multiplying by 10) for each problem. Scaling allows easy comparison of the difficulty of problems of different weight.

Exam 4
Distribution
Range Count
80 6
75 - 79 8
70 - 74 18
65 - 69 11
60 - 64 12
55 - 59 10
50 - 53 5
45 - 49 8
below 45 17
Problems
Prob. # Scaled Avg.
1 3.59
2 7.47
3 8.54
4 6.98
5 8.37

The final exam has been graded and individual exam grades entered in the FAS Gradebook. Median score was 147 out of 200. Other information will be posted when it is available.

Some scatter plots show comparisons between components of the grade: (1) Maple Labs and class exams; (2) Recitation grades and class exams; (3) Class exams and the final exam.

Each problem was closely related to a problem on a class exam. In some cases, the ability to prepare for specific problems gave a better average score (problem 5 from exam 1, problem 1 from exam 3 and problem 1 from exam 4); but in other cases, grade were better the first time (problem 3 from exam 2, problems 2 and 3 from exam 3, problem 2 from exam 4). In the table, the "source" column contains a number of the form m.n where m is the exam number and n is the problem number on that exam.
Final Exam
Distribution
Range Count
190 - 199 1
180 - 189 10
170 - 179 14
160 - 169 9
150 - 159 9
140 - 149 8
130 - 139 4
120 - 129 9
110 - 119 3
100 - 109 4
90 - 99 5
80 - 89 5
70 - 79 3
below 70 9
Problems
Source Prob. # Scaled Avg.
1.1 A1, B9, C5, D13 6.83
1.3 A12, B1, C8, D5 6.54
1.4 A5, B13, C1, D9 4.99
1.5 A9, B5, C13, D1 4.88
2.3 A10, B3, C12, D6 7.94
2.4 A2, B10, C11, D14 7.11
2.6 A6, B14, C9, D2 7.91
2.7 A13, B6, C2, D10 6.08
3.1 A14, B11, C3, D7 6.78
3.2 A11, B7, C14, D3 5.85
3.3 A7, B4, C10, D15 5.01
3.6 A3, B15, C7, D11 6.17
4.1 A8, B2, C6, D4 5.45
4.2 A15, B8, C15, D12 6.07
4.4 A4, B12, C4, D8 6.32

Here is a scatter plot showing the relation between all other components and the final exam. In addition to the trend line, there are lines showing totals of 620, 560, 510, 460, and 380 dividing into the grades of A, B+, B, C+, C, and "other".

A summary of all components of the grade has been entered in the FAS Gradebook. These numbers represent the content of my records at the time that course grades were determined even if earlier entries in the FAS gradebook were not updated.