# Math 244 Fall 2004 Sections 1-3: Professor Bumby

• main course page.
• semester course page.
• Recitation Instructor's Course Page
• Notes on hanging cables from lecture on September 01.
• Notes prepared for Math 252 on Euler's method and its application to proving a version of the existence and uniqueness theorems as mentioned in the lecture on September 20. Theory will not play a big role in this course, and computation will only appear in the Maple labs, but it is important to know that initial value problems usually have unique solutions since this assures you that any expression that satisfies the conditions is the only right answer. Correct answers, not methods, are the focus of all exercises and exam problems in this course.
• Notes on Matrix exponentials from lecture on October 25. 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. (Revised version, correcting some misprints and summarizing the computation in the 2 by 2 case, posted October 30, 2004)
• Notes on Phase planes for nonlinear systems. Definitions from chapter 9 are collected in one place and techniques are described for the efficient treatment of examples.
• Notes on Series methods 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. Some exercises are included with instructions resembling those that you can expect on the exam. The notes include a brief discussion of nonhomogeneous equations, but the notes were made available so late that this was not included in the exercises and will not be part of the exam.

## Special Announcements

This lecture section has registered with the Maple Adoption Program. This is a special program that Maplesoft offers because the Maple laboratory projects are a required part of the course. One benefit of the program is that students in this section may purchase a copy of our Maple 9.5 Student Edition for the reduced price of \$75.00 US (download only). You will need a Special Access Code that I can give you.

Homework for this course can be found on the Recitation Instructor's Course Page. It should be submitted in lecture on Wednesday to allow the recitation instructor to evaluate it before class. You should be sure include your neatly written name and section number on your paper.

The lecturer's regular office hours are to be found on his home page, but special office hours for this course will be announced here. On Tuesday, September 21, Prof. Bumby will be available in Hill 438 from 4PM to 7PM for questions related to the first exam in this course.

Change of due dates for Maple Labs. The grader was unable to work on Lab 1 when it was collected, but promises to have results by Monday, October 11 (instead of the expected date of October 4). To allow time to review results of graded labs, the due date of later labs will be changed. In particular, Lab 2 will certainly not be due on October 13. The calendar will be modified after confirming that the grader will have no conflicts with the new schedule.

The lecturer's special office hour before the first exam was so successful that special office hours for the second exam will be held on Friday, October 15 from 2 to 4 PM in Hill 438.

By popular demand, the lecturer will hold special office hours in Hill 438 from 11AM to 1PM on Tuesday, November 9 to aid in preparation for exam 3.

Other obligations have delayed the revision of Lab 5 for this semester, but it is now ready (Monday, November 22, 6:45 PM). Changes from last year's version mostly involve moving some steps to the supplementary worksheet, so that the main worksheet will be left with the nicer graphs and discussion sections to allow you to express your pleasure at not having to include unsatisfactory graphs. The supplementary worksheet also includes computations related to Liapunov's analysis of stability of stationary points (section 9.6).

A Maple worksheet based on the fragments shown in lecture on Wednesday, November 17 is now available. It concentrates on exercise 5.3.1, but shows how Maple can be forced to produce the coefficients of a series solution in a way that mimics hand computation. A second worksheet on the solution of Legendre's equation, both with the usual coordinate and with the independent variable shifted to the singular point (which is regular), is also available.

There appears to be a "conflict" (three exams in one calendar day) for ECE majors. A count of the curriculum field on the roster shows 15 majors in this program in the course, though not all may have this conflict. Several have spoken to me, and one has already obtained a form from the Dean. I have contacted Dean Bernath to verify the conflict, and found that 8 met the requirements for a separate exam. A mutually acceptable time for this exam has been found: it is Monday, December 20, 8 - 11 AM. This is the general Mathematics Department make-up time. Current plans are to have students in 244 take the exam in SEC-208. There should be signs on the outer doors of Hill Center on the morning of the exam giving the latest information. This is only for this group of students.

Office hours prior to the (regular) final exam will be held on Tuesday, December 21, from 4 to 7 PM in Hill 438. No other scheduled office hours for this course will be held this year, but those who get an early start preparing for the exam and have questions that cannot be answered by e-mail should ask for an appointment. In addition, last minute questions can be entertained in the exam room (SEC-117) from 11:15 AM until the exam begins at noon.

## Calendar

• Wednesday, September 01. First Lecture. Begin segment on chapters 1 (Generalities) and 2 (First order equations). Maple Lab 0 (Practice Lab) assigned. Here are links to the general instructions, Lab0 description and seed file. 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 02. First Recitation class.
• Wednesday, September 15. Maple Lab #0 due
• Wednesday, September 22. Exam #1. The exam will emphasize methods for solving differential equations: linear equations (section 2.1); separable equations (section 2.2); exact equations (section 2.6). Another important theme in the course concerns features of solutions that can be found from the equation itself without solving the equation. Many of these features can be seen in the slope field, but there isn't much that can be done with slope fields on an exam. However, the key applications - annuities (section 2.3) and population dynamics (section 2.5) - include studies of equilibrium points and the shape of the graphs of solutions that are easier to see in the equation than in formulas for the solution. Problems involving this aspect of applications are also likely to appear. Problems will resemble those done in lecture or assigned for homework. Note that no formula sheets are allowed on exams, but you are expected to have a graphing calculator.
• Wednesday, September 29. Maple Lab #1 due
• Monday, October 18. Exam #2. All of chapters 3 and 4 except variation of parameters (sections 3.7 and 4.4) will be covered (you are not barred from using variation of parameters, but there will be no problems for which it will be required). In particular, you can expect questions dealing with the applications in sections 3.8 and 3.9, but only to the extent that they illustrate the methods developed in sections 3.1 through 3.6 for solving the equations that appear in the applications. In addition to finding general solutions of equations, both homogeneous and inhomogeneous, you will also need (in some cases) to find solutions that satisfy given initial conditions.
• Wednesday, October 20. Maple Lab #2 due (revised)
• Wednesday, November 03. Maple Lab #3 due (revised)
• Wednesday, November 10. Exam #3. All of chapter 7 together with sections 1 through 4 of chapter 9 will be covered. However, the questions dealing with solutions of constant coefficient systems will be composed in the expectation that they will be solved by the method in the notes. Nonhomogeneous systems covered in Section 7.9 of the text are included in the syllabus for this exam. While exercise 3 from this section was rather tedious, exercise 7 is more typical of a problem that can be solved using undetermined coefficients on an exam. The general considerations in sections 9.1 through 9.3 are illustrated quite well by the competing species model discussed in section 9.4. Indeed, several examples of this type appeared in earlier sections without revealing their significance. Although a similar claim could be made for the predator-prey model of section 9.5, this did not appear in lecture until the class before the exam, so the exam will concentrate on systems of the competing species type. The general concepts of nullcline, equilibrium point, and linearization will be emphasized. There are notes that reflect the point of view of the person composing the exam.
• Wednesday, November 17. Maple Lab #4 due (revised)
• Wednesday, December 01. Maple Lab #5 due (revised)
• Monday, December 06. Exam #4
• Wednesday, December 22, 12:00PM To 3:00PM. Final Exam in SEC-117 (regular lecture room)

## Lecture details

The textbook exercises done as examples in lecture will be listed here. For many of these, Maple's solution will also be shown and interpreted. These Maple worksheets will be available for download so you can step through them. Although the exams will emphasize pencil-and-paper solutions, you may find Maple's solutions useful in refining your approach to the exercises. These worksheets are primarily for viewing as a partial record of the lecture, but your copy may be modified for your own use. In some cases, sections were revisited in later lectures. Examples done on later visits will be added to original list (in parentheses).

Date Section Exercises Maple
Sep. 01 1.3 5, 14, 17.
Sep. 08 2.2 2, 5, 10, 21. yes
2.5 16, 17. yes
Sep. 13 2.6 1, 11, 32. yes
Sep. 15 2.1 14, 16, 25.
2.3 8.
2.4 7 (22).
Sep. 20 2.8 NONE
2.7(8.1-3) NONE
2.X (p. 131) 1
Sep. 22 Exam 1 problem 5
Sep. 27 3.1 3, 12, (15). #3
3.2 1, 7, (23). #1
4.1 13, (20). #13
Sep. 29 3.3 9.
4.2 16, 23, 35, (32).
Oct. 04 3.4 7, 11, 19,
3.5 6, 12.
Oct. 06 3.8 Extra yes
Oct. 11 3.6 1, 2, 5, 13, 20 . yes
3.9 17, 18.
4.3 3, (11).
Oct. 13 3.7 15.
4.4 2. yes
Oct. 18 Exam 2
Oct. 20 7.2 1.
7.3 1, 2, 3.
Oct. 25 alternate approach
to 7.5 - 7.8
examples from notes yes
Oct. 27 7.4 6.
7.5 7.
7.8 3.
7.9 3, (7). 3
Nov. 01 9.1 5, 13.
9.2 5, (7).
Nov. 03 9.3 5, (7).
9.4 1, (4).
Nov. 08 9.5 1.
selections
from 9.6 - 9.8
NONE
Nov. 10 Exam 3
Nov. 15 5.2 1, 2, 5, 7, 15, 17.
Nov. 17 5.3 1, (5), (6). yes
Nov. 22 5.5 1, 13.
5.6 1, 3, 11. 11
Nov. 29 5.7 1, 3, 14, 15.
5.8 5.
Dec. 01 review
Dec. 06 Exam 4

## 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. If time permits, there will be five graded labs, and the 100 point total will be scaled to 80 points.
• 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 58.184 and the median was 58. 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, since the maximum score is always 10.

Exam 1
Distribution
Range Count
71 - 80 18
61 - 70 17
51 - 60 21
41 - 50 11
0 - 40 9
Problems
Prob. # Scaled Avg.
1 9.01
2 7.80
3 7.87
4 7.15
5 5.22

This distribution of grades between 50 and 80 was almost flat. Grades below 40 indicate a serious weakness.

Here is the Maple worksheet showing an approach to one version of problem 5 that was shown in lecture on September 27.

Exam 2 has been graded. The average score was 69.34 and the median was 71. 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). 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, since the maximum score is always 10.

Exam 2
Distribution
Range Count
77 - 80 20
73 - 76 9
69 - 72 16
66 - 68 7
60 - 64 9
55 - 58 5
36 - 43 4
Problems
Prob. # Scaled Avg.
1 9.59
2 9.25
3 7.44
4 8.81
5 7.54
6 9.17

Exam 3
Distribution
Range Count
75 - 80 9
70 - 73 8
68 - 69 6
60 - 64 16
55 - 59 14
44 - 51 10
33 - 39 4
below 25 4
Problems
Prob. # Scaled Avg.
1 5.04
2 8.17
3 8.56
4 8.45
5 8.04
6 5.10

Exam 4
Distribution
Range Count
76 - 80 10
71 - 75 15
66 - 70 15
61 - 65 9
56 - 60 6
51 - 53 4
48 - 50 5
below 45 5
Problems
Prob. # Scaled Avg.
1 8.70
2 8.58
3 8.85
4 8.36
5 5.61

Grades for the Maple labs are also available in the FAS Gradebook. The grade should be the total of the best four of the five grades labs. The comment field should show (L0) L1 + L2 + L3 + L4 + L5 - min, where Li is the score on Lab i, and min is the smallest of the graded labs. Information for Lab 0 is included for completeness although its score was not used in completing the grade. The rest of the line should show the details of the computation of the grade. A missing lab is indicated by the absence of a number between two pluses. Any discrepancies should be reported to bumby@math.rutgers.edu

The first graph shows the relation between Maple grades and exam scores; the second between recitation grades and exam scores. Correlation lines have been added.

The final exam has been graded and course grade submitted. The scatter plot shows the relation between the total of previous coursework and the exam. As usual. there is a regression line. There are also lines showing totals of 630, 590, 530, 490, 430 marking the divisions between letter grades (more detail is in a table below). Your score on the exam and grade for the course are available in the FAS Gradebook. The average exam score was 140 with a median of 149. The tables show the distribution of exam scores and of course grades.

When grades were listed in order, a clear minimal level of competence, corresponding to the lowest passing score, identified itself. Scores significantly below this received a grade of F, that is always a prescription that the course should be retaken in order to acquire a useful knowledge of the subject. Others fell short of a passing score for a clearly identified reason. These were given a grade of TD. This allows the option of doing nothing and getting a D for the course, or doing some extra work to get a grade of C. The entry in the FAS Gradebook has a brief comment indicating the nature of the extra work in these cases.

Summary
Exam
Range Count
190 - 199 6
180 - 189 5
170 - 179 9
160 - 169 9
150 - 159 6
140 - 149 5
130 - 139 4
120 - 129 7
110 - 119 6
100 - 109 6
below 100 7
A 639 - 694 13
B+ 593 - 627 10
B 532 - 584 18
C+ 494 - 525 13
C 433 - 483 8
TD 405 - 413 3
F below 380 5