Math 244 Fall 2005 Sections 13: Professor Bumby
Some links

main course page.

semester course page.

Lecturer's Home Page.

Recitation Instructor's
Home Page.

Recitation Instructor's
Course Page
 A Java Applet for Studying Ordinary Differential Equations (written by M. Rychlik of U. of Ariz., maintained at Rutgers by E. Sontag)

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 26. 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 Fall 2004 (except that a significant
typographical error was corrected on Oct. 15, 2005), that is the basis
of a lecture on October 10, 2005. 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. For exam 2, only 2by2
matrix examples will appear. The extension to the 3by3 triangular
case will be revisited in part 3 of the course. Maple Lab 4 includes
routines for using this method in Maple, so it will be the next lab
done in this section. It should be ready early in the week of
Oct. 10, but preparation was actually completed on Oct. 15.
 Notes on Inhomogeneous equations and
systems that should have been prepared before
exam 3 are now available. They give details underlying the approach
taken in lecture. Some exercises are included. An attempt was made
to avoid fractions in these exercises. The attempt was mostly
successful, but there may be one or two exercises that don't have
answers that are as simple as intended.
 Notes on Series
methods from Fall 2004 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 course will use Sakai
to enhance exchanges of information in the course. Follow the link in
the previous sentence to reach the Welcome screen (that looks like this screen shot). Enter your eden
username and password in the boxes (in the real welcome screen, not
the screen shot), and press the login button. You should be
automatically enrolled in any site that has been created for one of
your courses (this one is 640:244:[0103] F05).
Those sites will be shown in a row of tabs at the top of the
window. Your use of Sakai is coordinated through a site called
My Workspace.
In particular, other sites that can be joined can be found using the
Membership menu item in My
Workspace. Problems using Sakai should be reported to the
student help desk, 732445HELP. I can also demonstrate the use of
Sakai during my office hours (see my personal home
page for a schedule.
In many ways, resources are easier to make available through Sakai
than through a public web page, and easier to organize there. In
addition, Sakai allows for submission of assignments and grading of
quizzes. It is planned to use these features to facilitate submission
and grading of the Maple Labs. The process for doing this may evolve
during the semester as techniques for working with Sakai improve.
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 the Maple 10 Student Edition for the reduced price of $75.00
US (download only). You will need a Special Access Code
available from the Maple Adoption folder in the
Resources section of the Sakai site for this section.
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.
Calendar
 Thursday, September 01. First Recitation class. A slope field of a nonlinear
equation, with graphs of
numerically computed solutions was featured. Other images that
may have been shown include: a slope field whose solutions are tangents to a parabola; the slope field of a linear equation,
with graphs of exact solutions;
equation X together with some graphs of solutions; and a discontinuous slope field whose
solutions can easily be seen.
 Wednesday, September 07. 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 two Maple
worksheets: a seed file and a supplementary worksheet. 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.
 Monday, September 12. A quiz was added to the
Sakai site consisting of five TrueFalse questions on
definitions from chapter 1. The quiz will be available from 6PM
Monday, September 12 to 6PM Friday, September 16. The quiz is
required. The grade on the quiz will be counted toward your
grade. Moreover, the quiz will also give valuable information about
the quiz tool that will be used to allow that tool to aid in the
grading of Maple labs.
 Wednesday, September 21. Maple Lab #0 due. Rather than
collect printed versions of your work on the project, the
"Assignments" tool in Sakai will be used. It will give you a place to
upload the file. I will execute the file as part of grading it, so
you should use the "Remove Output" tool in the Edit Menu, save the
file, and upload that saved file.
 Wednesday, September 28. Exam #1. The exam will emphasize
methods for finding exact solutions of 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. A question
dealing with recognizing a given slope field and recognizing features
of the equation visible in the equation is likely (production of the
image in a form that can appear on the exam still needs to be tested).
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.
 Tuesday, October 11 through Friday, October 14. Maple Lab #1 due Changed (again)
from original date because of delay in grading lab 0. Once
again, this will be collected through the Sakai site.
 Monday, October 24. Exam #2. The main parts of chapters 3 and 7.
Since higher order equations and systems are two ways of looking at
the same thing, there are benefits to treating them together. The
emphasis will be on the linear, homogeneous, constant
coefficient case. Applications, as well as less special
types of equations will be considered in later segments of the course.
Linear algebra is useful for organizing work with systems, but there
isn't enough time to give a full treatment of that subject. The
sections at the beginning of chapter 7 that describe the way in which
a standard development of linear algebra applies to the study of
systems will be downplayed in favor of special methods for solving
systems of differential equations. These methods also apply to
finding eigenvalues and eigenvectors wherever they arise, but they do
not depend on a thorough understanding of those problems as linear
algebra. It is easier to guess solutions of a higher order equation,
so this will be used to introduce the subject. However, systems are a
better way to approach the solution of the equations.
 Tuesday, October 25 through Monday October 31. Maple lab 4 will
be collected through the Sakai site.
 Wednesday, November 16. Exam #3. This will cover chapter 4 and
the remainder of chapters 3 and 7. In addition to systems with more
than two dependent variables, or single equations of order more than
two, this segment will consider inhomogeneous equations (and systems)
and applications. The lectures in this segment include an
introduction to the phase plane, but this topic will
not be tested on the exam. A quick tour of Maple Lab 3 will be
useful, since the lab provides an introduction to computations that
illustrate both the free motion and forced
motion of simple oscillators. This will illustrate some of the
applications in sections 3.8 and 3.9.
 Tuesday, November 22 follows a Wednesday schedule, so the lecture
will meet, but there will be no recitation class this week.
 Monday, November 28 through Friday November 02. Maple lab 3 will
be collected through the Sakai site. The "optional"
Note at the end of project description that refers to
Section 3 of the supplementary worksheet is a
required part of the project in this section.
 Wednesday, December 07. Exam #4. The main topic will be chapter
5 on series solutions. Brief mention will be made of
the use of a phase plane to investigate qualitative
aspects of autonomous systems with two dependent
variables.
 Wednesday, December 21, 12:00PM To 3:00PM. Final Exam
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
pencilandpaper solutions, you may find Maple's solutions useful in
refining your approach to the exercises. In some cases, the Maple
worksheet will be shown in lecture, but Maple versions will be posted
whenever they reflect the lecture, however the exercise was done in
lecture. These worksheets are primarily for working through as a
partial record of the lecture, since they are posted in bare
form with output removed. You should download it and step through the
worksheet in Maple 10. You may also modify this copy to investigate
related questions. In some cases, sections were revisited in later
lectures. Examples done on later visits will be added to original
list (in parentheses).
3.8, 4.2
Date

Section

Exercises

Maple

Sep. 07

2.2

2, 5, (6), (7), 10, (14), 21, (23).

yes

Sep. 12

2.3

8, 12.


Sep. 14

2.6

1, 11, 13, 32.

yes

Sep. 19

2.4

7, 16.

yes

Sep. 21

2.1

14, 16.


2.5

15.

Sep. 26

2.7, 2.8, 8.*

.


Sep. 28

Exam 1


Oct. 03

3.1

3, 12, 15.


7.3

17.

Oct. 05

7.5



Oct. 10

3.4; 7.6

Examples from notes


Oct. 12

3.3; 7.7



Oct. 17

3.5

11, 15.


7.8

7.

Oct. 19

review


Oct. 24

Exam 2


Oct. 26




Oct. 31

3.7

3, (15).

yes

4.4


7.9 (part)

3, 7.

Nov. 02

3.6, 4.3, 7.9 (part)



Nov. 07

3.9



Nov. 09

9.1, 9.2



Nov. 14

review


Nov. 16

Exam 3


Lectures leading up to Exam 4 are not shown. While they could be
linked to sections of the textbook, the plan of the lectures was given
in the notes on series solutions.
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.
Grading
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.
An effort will be made to respect any clustering of grades in
assigning course grades.
Exam 1 has been graded. The average score was 55.6267 and the median
was 56. 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 by
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 
76  80 
11 
71  75 
7 
66  70 
3 
61  65 
11 
56  60 
6 
51  55 
7 
46  50 
10 
41  45 
8 
36  40 
4 
0  35 
8 

Problems
Prob. # 
Scaled Avg. 
1 
9.32 
2 
8.22 
3 
6.91 
4 
5.90 
5 
5.05 

Remember that these are grades out of 80.
Grades below 40 indicate a serious weakness.
Exam 2 has been
graded. The average score was 62.59 and the median was 65.5. 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). (Some lines of slope 1 may be
added to show the beginnings of clusters in the total scores.)
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 by
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 
76  80 
6 
71  75 
15 
66  70 
14 
61  65 
10 
56  60 
9 
51  55 
6 
46  50 
2 
41  45 
2 
0  40 
6 

Problems
Prob. # 
Scaled Avg. 
1 
8.27 
2 
6.13 
3 
8.63 
4 
7.60 
5 
8.57 

Exam 3 has been graded. The average score was 53.74 (with a median of
57).
A
scatter plot shows the comparison of grades on this exam with the sum
of the scores on exam 1 and exam 2. 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 by
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 3
Distribution
Range 
Count 
75  80 
4 
70 
3 
65  69 
6 
60  64 
11 
55  59 
12 
50  54 
10 
45  49 
5 
40  44 
6 
35  99 
5 
below 35 
5 

Problems
Prob. # 
Scaled Avg. 
1 
6.59 
2 
7.90 
3 
8.30 
4 
1.59 
5 
8.35 
6 
5.82 

There were difficulties with problem 4 that made assignment of
partial credit difficult. The fairest approach seemed to be to assign
credit only for complete or nearly complete solutions. The few with
credit will have a nice bonus, but other grades will be judged as
though this were a 70 point exam.
Exam 4 has been graded. The average score was 65.33 (with a median of 69). A
scatter plot shows the comparison of grades on this exam with the sum
of the scores on exam 1 and exam 2. 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 by 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 4
Distribution
Range 
Count 
76  80 
15 
71  75 
15 
66  70 
8 
61  65 
9 
56  60 
5 
51  55 
4 
46  50 
3 
0  45 
5 

Problems
Prob. # 
Scaled Avg. 
1 
9.41 
2 
8.61 
3 
8.76 
4 
5.98 
5 
7.90 

This graph shows
the relation between recitation grades and exam scores. A correlation
line has 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 575, 530, 475, 445, 395 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 151.5 with a median of
156. 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. The Sakai drop box for each of the students
with a lower score contains additional information about grades in
various parts of the course.
Summary
Exam
Range 
Count 
190  199 
7 
180  189 
10 
170  179 
5 
160  169 
8 
150  159 
10 
140  149 
8 
130  139 
2 
120  129 
5 
below 120 
10 

Letter Grades
Grade 
Range 
Count 
A 
578  653 
18 
B+ 
531  570 
13 
B 
492  525 
13 
C+ 
458  479 
9 
C 
400  433 
6 
other 
432  387 
6 

Comments on this page should be sent to: bumby@math.rutgers.edu
Last updated: January 02, 2005