Schedule of Lectures -- This schedule has
the dates of all of the exams, and a rough schedule indicating which
lecture (from the syllabus) will be given on each class day.
Students are urged to read the appropriate section(s) of the book BEFORE
the lecture, so that they will have a general idea of what problems
will be addressed and what approaches will be taken so that they can
make better use of class time. Note that very often, there will
be material in the book that we will not have time to present in class,
so it is also important to read the appropriate section of the book
AFTER the lecture as well.
Note, the schedule of lectures is just an approximation; we may find
ourselves ahead of or behind schedule, depending on some
unknowns. Your regular attendance in class will ensure that you
will be aware of such changes.
Exam 1 Solutions -- There is an
error in the solution to problem 6 -- a square root sign was left off.
This did not change the eventual answer to the question.
Exam 2 Solutions
Exam 3 Solutions
Values of
Sine and Cosine at Standard Angles
-- All students are strongly advised to know the values of the six trig
functions at the standard angles (all multiples of pi/4 and pi/6); this
short commentary and the linked diagram will help you both remember
these values, and understand the fundamental relationship between
trigonometry and the unit circle.
The Remainder Theorem and the Factor Theorem
-- These are fundamental facts about polynomial algebra, and they are
simply stated and easy to prove; but for some reason they are not
covered in most high school curricula any more. I strongly
encourage students to look at this brief description and ensure an
understanding of these theorems and their proofs.
Some
comments about Single Variable
Integrals -- This is a document that I wrote in the Fall of 2004
after teaching the section about integrals of parametric functions.
Diagram, (Vector) Calculus in R^1 --
This diagram is an expression of the Fundamental Theorem of Calculus,
using the same format as is used for other theorems in the following
two diagrams for R^2 and R^3. Using this diagram and your
previous understanding of the FTC, you can see the analogies between
FTC and the other vector calculus theorems, and hopefully then
extrapolate to a better understanding of those vector calculus theorems.
Diagram, Vector Calculus in R^2 --
Students should use this diagram to help see the symmetries and
relationships between the Fundamental Theorem of Line Integrals (in
R^2) and Green's Theorem.
Diagram, Vector Calculus in R^3 --
Students
should use
this to
help mentally organize the constructions and theorems that we have
talked about in Chapter 15 (except for Green's Theorem). Note the
symmetrical relationships between the three boundary theorems, and how
the lifetime theorems can help suggest what type of integral is
appropriate for a given problem.
My other Course Websites
-- Students can find here all of the websites for every course I have
ever taught. This includes several Math 103 courses, each website
of which includes the solutions to each of the midterms I gave that
semester. Students should find this to be a very helpful study
resource.
Note however that in different terms, the exams fall at different
points in our movement through the syllabus. So for example when
studying for Exam 1 of this term, note that there might be some
problems on previous first exams that involve material we have not yet
covered -- and there might also be some problems on some of the second
exams that involve material that we have already covered. Keep
these possibilities in mind.
Also be aware that for some of these previous Math 103 courses I used
an experimental curriculum that is substantially different from that of
this current course. In particular, these are the Summer 2005 and
Fall 2005-06 courses. You are still encouraged to use the
materials on those pages as study resources, but make sure to ignore
any questions that involve material we are not studying in this
semester's course.
Blank
Old Exams -- Students can find here blank copies of my old Math 103
Exams. Use these if you would like to take the exam first and
then look at the solutions later.
Java Applets --
This is a site that was created by a colleague at Stanford; several of
the tools there could be useful to you for visualizing multivariable
functions. In particular, be sure to try the "3D Graph" tool, and
enter the following details into the controls: f(x,y) = (3x^2y -
y^3)/(x^2 + y^2), xsteps = 61, ysteps = 60... It is easy to check
that f = r sin(3(theta)) and thus that at the origin all of the
directional derivatives exist -- but, when you play with the
perspective controls (labelled "theta" and "phi"), you will see that
this function still has a serious irregularity at the origin. In
particular, this function is not differentiable at the origin.
Here is a picture of the graph from one
perspective.
Homework
Homework problems are assigned for every lecture, and students should
ideally complete each assignment on the day of the lecture. The
assigned problems for each lesson are listed on the syllabus.
Make sure you staple your homeworks! We cannot give credit to
students
for work that was lost as a result of not being stapled. Also,
make
sure to put at the top of the front page your name, the section
number(s) for those problems, and the course information (Math 103,
Clark Bray)
In order to give flexibility to students, the assignments for the
previous three lectures will be picked up in class every Friday at the
beginning
of class, and will be graded and returned as soon as possible.
No late homework will be accepted without the
Short Term
Illness Notification.
In calculating homework grades, the lowest of your homework scores will
be dropped. The purpose of this policy is to handle exceptional
circumstances.
Please do
not request to have late
homework accepted. Also, it is inadvisable to skip a homework
unless absolutely necessary, since only one homework will be dropped.
Working together in groups on homeworks is
strongly encouraged!
You will find that the people you are working with either (1)
understand something you don't, in which case they can explain it to
you; (2) understand something that you do understand, but from a
different point of view -- these additional perspectives can prove to
be very useful; or (3), don't understand something that you do
understand -- in which case you have the opportunity to explain it to
them... I think you will find that in the process of explaining
something, very often you will achieve a better understanding yourself.
Of course, it goes without saying that even though you may work in
groups, the homeworks you turn in must be your own work. You may
share ideas, perspectives, and approaches to problems, but copying is not
allowed. Furthermore, keep in mind that the homeworks are
primarily a learning tool, and count for a fairly low percentage of
your grade. Do not deprive yourself of this invaluable learning
opportunity!
Grading and
Exams
Final grades for the class will be determined by the total number of
points earned in the class. These points are given based on
performance on the items below, with the following maximum possible
scores:
Homework average: 50 possible points (the lowest homework
score is dropped)
Tests:
300 possible points
(3 exams x 100 points each)
Final Exam:
200 possible points
------------------------------------------------------------------------------------------------
Total:
550 possible
points
The student should be prepared for the fact that the grading system for
these exams is NOT the same as the one most students became accustomed
to in high school. There are two main properties in particular of the
high school system that will not be used in this class :
1) In most high school grading systems, there are fixed, arbitrary
numbers that determine the cutoffs between different letter grades --
these cutoffs were invariant, and independent of the exam. The problem
with this that it forces the instructor to create exams that are always
the same difficulty; in other words, the instructor must make sure that
all exams will yield the same mean score. Furthermore, it requires that
the distribution of scores also be roughly constant. Achieving both of
these goals is not only difficult, but impossible to perform perfectly.
This system ties the instructor's hands severely, and is totally
unnecessary! Of course it is important that final letter grades for a
class follow a prescribed plan, so that those letter grades have some
meaning outside of the context of that class. Ensuring that is actually
easier if the instructor chooses the cutoff numbers after having seen
the distribution of student scores. The cutoffs can then be chosen
while incorporating important considerations such as the difficulty of
the exam, or any other points about the exam that may be relevant.
2) The class average on exams in most high schools was usually
expected to be somewhere in the mid-eighties. While this is reasonable
considering the nature of high school, it is not always appropriate for
a college setting.
In this class, certainly, there are expectations for the student
that are much more demanding than those of most high schools. We expect
that the student will achieve much more than the mere ability to
reproduce what he or she has seen in class. In particular, we expect
that the student will achieve an understanding of the ideas that are at
the foundation of the methods -- and thereby gain the ability to apply
those ideas to situations that he or she has not already been exposed
to.
Since the expectations of this class are more difficult than those
of high school, it stands to reason that the exams, designed to test
the students mastery of these more lofty goals, must involve more
difficult questions; and therefore, the exams must be more difficult.
Clearly this will result in class averages that are lower than what one
would expect if the exams were more like those of high school. It will
also tend to result in score distributions that are more broad, since
the students responses can be expected to be more varied.
The student should be fully aware of these points before taking an exam
in this class.
It is very dangerous to associate letter grades with performances on
individual exams, because it is very difficult to predict how the
distributions for those exams will interact when the total score
distribution is formed. Therefore, the class will usually be informed
only of the class median and mean for a given exam -- letter grades
will not be assigned.
I will periodically distribute a list of all student total scores in
the class (with names removed), so that students can see exactly how
their own total scores compare to the rest of the class.
Calculators
We will not be using calculators in any aspect of this course.
You may use a calculator on a homework problem if you feel that it will
help you understand the concepts, but you may not make any reference to
the use of a calculator on the homework you turn in, except in the case
of a problem that requires substantial and nontrivial arithmetic.
Calculators will NOT be allowed on the final exam or on the in-class
exams. I
also generally discourage their use on the homeworks, since students
may develop a dependence that will hurt them on the exams.
You are encouraged though to take advantage of the online resources
linked to from this page, including the Java Applets (there are many
tools there that can help you visualize and develop a geometric
intuition for some of the ideas in this course).
Getting help
There are several resources that students should be aware of; make sure
to read the
Sources
of Help for First-Year Students.
You are highly encouraged to make good use of the Help
Room.
Details of the Math 103 Help Room schedule can be found at the above
link. You can also come to my office hours, or just
swing by sometime to see if I'm available. If you need to make a
special appointment to see me, send me an email.
I'd also like to emphasize that classmates can be an excellent resource
as well. I refer you above to my comments on this in the homework
section.
Be sure to realize that you are encouraged to use these resources for
more than just help on the homework... Ask questions about
general ideas you are having trouble with, specific parts of the
lectures that you did not understand... Of course you should also
seek help with homework if you find yourself stuck on a problem for an
extended period of time.
Honor Code
The Honor Code is taken very seriously on Duke campus, and you are all
reminded to make certain you are familiar with it.
In this course some collaboration is allowed and encouraged, but of
course your work must all be your own. Here are some specific
comments about the graded items in this class:
Homeworks -- You are encouraged to work in groups to exchange ideas and
help each other understand how to approach problems, but the student's
work must be his or her own. Copying and dictating are not
allowed.
Exams and Quizzes -- Students are not allowed to have any outside help
during exams or quizzes.
Attendance
Attendance at all lectures is required . If you miss a
lecture, it is your responsibility to catch up on the
topics that you missed. You should keep in mind that in this course,
the material builds on itself; if you miss some of the material,
subsequent lectures will seem much more difficult to you.
Absences from exams will be excused only for reasons such
as serious illness or appropriate official university activities.
In either case, a written notification from the dean is required.
In the case of illness, this must be done with the
Short-term
Illness Notification form. In the case that an absence from
an exam is excused, the grade will be determined based on your
performance on the final exam for the course, and relative to the
performance of the rest of the class on that exam.
Students should note that use of the Short-term Illness Notification
form is subject to the Duke Community Standard, discussed and linked
above. In particular, it is expected every student will take
reasonable responsibility for his/her own health at least to the extent
that such health is needed to be able to participate in classes.
Here are some examples:
-- If on the day of an exam in this course you have a debilitating
headache caused by a virus, then it would be appropriate to use the
short term illness form. However, if you have a debilitating
headache caused by a hangover, it would not be appropriate to use the
short term illness form.
-- If on the day of an exam in this course you have a severe cold
caught two days earlier while camping outside for basketball tickets,
then it would be appropriate to use the
short term illness form. However, if you have had the cold for
two weeks and are still camping outside for basketball tickets, it
would not be appropriate to use the short term
illness form.
Additional Comments
As was discussed in the section
on Grading and Exams, the goals of this course are higher than those of
high school math courses, and even higher than most single variable
calculus courses. Specifically, students in this course will be
expected not only to be able to perform computations, but also to be
able to demonstrate comprehension of the ideas behind those
computations.
We expect students to demonstrate this comprehension by showing the
process used to perform the computation, with sufficient commentary to
allow the reader to conclude the thought process the student was using
in working that problem.
The correct numerical answer does not guarantee that the student will
get full or even majority credit for the problem. Rather,
students will be graded on the validity of the method of computation,
the extent to which the computation and commentary demonstrate the
student understands the underlying ideas, and the clarity of those
explanations.
It is very likely that this system of evaluation is very different from
the ones the students might have become accustomed to in high school.
To a great extent, this is a writing class.
This might sounds like an outrageous statement, but the fundamental
point here is that this course, just like a history course or a
political science course, is about the comprehension and communication
of ideas. Obviously the types of ideas in question are different,
but like many or even most other courses on a college campus, this
course requires students to understand concepts, and then to
communicate that understanding through writing. In this course
the writing comes in the form of the answers that the student gives to
homework and exam problems, rather than essays.
Students should take very seriously the idea that the solutions they
write for the homework and the exam problems should reflect this
perspective. All too often students think of the writing they do
on an exam to be simply a personal convenience for them -- that is, a
tool to keep them from having to work the problem entirely in their
heads. This is NOT the right attitude, and it will not lead to
the desired credit on the exam.
Instead, think of your writing on homework and exam problems as being
documentation of your thought process. You are communicating to
another person, namely the grader, and your goal is to communicate that
you understand the tools needed to solve the given problem and that you
know how to implement those tools. Of course in communicating
that to the grader, you will also write down the necessary algebra to
allow you to arrive at the final answer.
Write neatly!
The all-too-common attitude in high school mathematics courses is that
the answer is the only thing that matters; if the grader can read the
answer, then the clarity of the rest of the work is not relevant.
As per the above comments, this is not the case in this course.
Everything that is written on the exam is going to be read and
considered for the contribution it makes to demonstrating comprehension
of the ideas. So it must all be legible.
The same standards of neatness and legibility appropriate to, say, a
history class should be applied in this mathematics class.
Because of the types of notation used we do not expect that the
student will type the solutions to homework problems; but, we expect
that the solutions will be written neatly and legibly, the papers
should not be crumpled or stained, and there should not be large areas
of the paper scratched out. (If you are not sure how to work a
problem and if you are going to put its solution on the same page as
that of another problem, you should do your scratch work on another
piece of paper, and then write out the solution neatly on the paper you
will turn in.)
Furthermore, note that the flow of ideas over the page should be
reasonable. Ideally, it should be "top down", or perhaps two or
three column if the student prefers. Either way, the grader
should be able to look at the page and effortlessly identify the
location of the beginning of the argument, and easily follow each
successive step until arriving at the conclusion.
Note, explanations that bounce around the page in a seemingly random
pattern will be more difficult for the grader to follow, which
invariably leaves the grader finding less clarity. This can lead
to the awarding of fewer points.
