Math 244 Fall 2003 Sections 13 and 7 Professor Bumby
Some links

main course page.

semester course page.

Lecturer's Home Page.

(Sections 1, 2, and 3) Recitation Instructor's
Home Page.

(Sections 1, 2, and 3) Recitation Instructor's
Page for this course.

(Section 7) Recitation Instructor's
Home Page.
 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.
 We are looking into the possibility of a tutorial in the use of
Maple conducted (remotely) by a representative of Maplesoft.
 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 SEC210): 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 HIL114 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 PH115 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 5.68 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, 811 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.
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.
 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 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.
Comments on this page should be sent to: bumby@math.rutgers.edu
Last updated: January 12, 2004