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 Math 103 section about integrals of parametric functions.  When we get to the integrals portion of this course, the first part of this could be useful to you.

My other Course Websites -- Students can find here all of the websites for every course I have ever taught. 

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. 

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Exam Solutions

2008-2009 Fall:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions

2007-2008 Spring:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions
Exam 3
Exam 3 Solutions

2007-2008 Fall:
Exam 1
Exam 1 Solutions
Exam 2
Exam 2 Solutions (Note, there is a computational error in the bottom left element of the solution matrix in #2a.)
Exam 3
Exam 3 Solutions

2006-07 Spring:
Exam 1
Exam 2
Exam 3
Exam 1 Solutions
Exam 2 Solutions (Note, there are computational errors in the solution to problem 1a)
Exam 3 Solutions (Note, there is an arithmetic error on number 5; the answer should be 11, not 7.)

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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 will be listed on the syllabus.  (Note, we might find ourselves behind or ahead of the posted schedule; if so, you should do the problems as we actually finish the sections.)

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, and the section number(s) for those problems.

In order to give flexibility to students, the assignments for the previous three lectures will be picked up in section every Tuesday, 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 without filling out the Short Term Illness Notification.  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, 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!

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Here is the procedure we will use this semester for regrades:

(1) Write a clear and complete description of why you feel your paper deserves more points than you originally received.
(2) Attach that description to your homework paper.
(3) Put that paper into the pile in the following section, when your TA is collecting the next week's homework.
(4) The grader will receive your note and original paper, will give it fair consideration, will consult with me or your TA if necessary, and then will make a change to the score if that is deemed appropriate.  He will then also make the change on the homework gradesheet.
(5) The grader will put the paper back in the pile and it will be returned to you along with those other homeworks.

Here are a few thoughts to keep in mind about regrades:

(a) It is entirely possible and reasonable that the grader might have misread your paper, and with your explanation realize that you do indeed deserve more points.  In such a case, he will be very happy to award more points.

(b) It is also very common for a student to feel simply that too many points were taken off for a given error.  In these cases, the student should be prepared for the likely conclusion that no additional points will be awarded.  The point here is that this is a subjective situation, and a choice has to be made.  The grader makes the decision based on his feeling about the importance of a given aspect of the problem, and the grader's opinion on this question is the standard.

Common examples of these types of disagreements involve the amount of explanation that should be given, and the relative importance of different parts of the problem.  These are highly subjective questions, and reasonable people will come to different conclusions.

Remember that this is a curved class.  So, when it comes to questions about too many or too few points being taken off, it is far more important that the grader's scheme be applied consistently across the board for all students than that it be something other people might or might not agree with. 

(c) When you submit your paper for a regrade, the grader might possibly come to the conclusion that too many points were awarded in the first place.  In such a circumstance, your score could go down.  Of course the grader will always make such decisions dispassionately and fairly, but certainly you should only submit for a regrade in a situation where you feel you have a comfortably strong claim.

(d) The grader is a very reasonable and intelligent person, and absolutely deserving of being addressed politely and treated with respect.  Make sure to phrase your requests calmly and reasonably.  And of course, always be prepared for the possibility that the grader might have a different point of view than you on a given question, and that his fair and reasonable consideration of your request might yield no additional credit. 

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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
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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. 

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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).

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Getting help

There are several resources that students should be aware of; make sure to read the Sources of Help for First-Year Students.

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.

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Duke Community Standard

The Duke Community Standard 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. 

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Attendance

Attendance at all lectures and sections 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.

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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.  

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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.

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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.

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Sometimes there might be material that is fair game for an exam, but for which the homework problems are not due until after the exam due to the way the schedule and due dates are set up.  Make sure to consider this possibility when you are studying for the exam, and if this should be the case you are strongly encouraged to do the corresponding homework exercises before the exam in question, so that you can get the needed practice for the exam.

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There are many resources available to students in this class, and we expect students to avail themselves of all of them.

• Yourself -- By this I mean that you are ultimately the one responsible for ensuring that you learn the material.  You cannot wait for help to come to you, but rather you must seek it out independently and aggressively.  You must make sure that you are learning the material to a sufficiently deep level, and you must be responsible for finding the ways to improve your understanding if you are not certain that your understanding is sufficient.  Think creatively about the different ways that you can learn!
  
  
• Book/notes -- I strongly recommend that you read the appropriate sections in the book before class!  Of course with a first reading it is not expected that you will understand everything, but a rough idea of what the topic is and how the problems will be approached is extremely valuable when you come to lecture; it will allow you to spend more time thinking about the higher level ideas that we discuss there, and to minimize the distractions of thinking about where things are going, and tedious minutia that you could have picked up by reading.
  
  I also strongly recommend reading the appropriate sections of the book after class!  This will allow you to solidify the things you learned in class.
  
  Note also that in class, we will NOT cover all of the material in the book for which you are responsible.  There are far too many things in this course to be able to spend sufficient class time on all of them.  So, in class I will discuss the ideas that I think are the most important, and also the ideas that I think are most difficult to get from the book.  I will also do as many examples as time permits, but the priority must be on the concepts.  
  
  
• Instructor -- Of course this is a given, but the point here is that you must come to class in order to get any use from me as the instructor.  It is a tragic error to skip classes, thinking that your time is better spent doing other things.  You simply never know what will come up in class, and if you miss the presentation of an idea that you are not familiar with, it will be very difficult to make up for that loss.
  
  Note, in this course and most other math courses at Duke, the level at which the material is presented is dramatically higher than that in even the strongest high schools.  It is at that higher level that you will be tested...  So, while you may have done very well in your calculus courses in high school (note, almost every student in this class has already had a course in calculus in high school!), that is no guarantee that you will do well in this course.
  
  
• Help Room -- The help room, linked above, is a wonderful opportunity for students in this class to get assistance at a broad variety of times.  If you are having trouble with a particular problem then of course you are welcome to ask for help there.  But, more importantly, this is also a time when you can get help with concepts you are having trouble with.  Go in with broad questions, like, "I'm having trouble with problems that involve work," or "How can I improve in making good guesses on substitutions to use for computing antiderivatives?"  The people in the help room should be able to help you.
  
  
• Classmates -- As we discussed already in the homework section, you are encouraged to work in groups on the homework, to share ideas and exchange assistance on problems you are having trouble with.  In general I think that your classmates can be very useful resources for you.  Again, don't limit your use of this resource merely to specific problems.  Try to initiate broad conversations on topics you are not sure about, and share your different points of view.  If you find that neither you nor any of your classmates understands a particular point, then seek out help either from the Help Room, or from me.

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As previously discussed, the expectations in this class are very high.

• Memorization/Execution -- Students in this course were almost certainly made very familiar in high school with the idea that they should be able to remember needed formulas and techniques, and work a problem out to its conclusion.  That is, that they should be able to execute skills that are discussed in class.  Students in this class are generally very good at this sort of thing, because of that familiarity and the predominance of that in high school classes.
  
  However, because most high schools emphasize ONLY this skill, many students in this class expect that this is the only expectation that will be made of them.  This is most certainly NOT the case.
  
  
• Understanding -- Very often a student can be sufficiently familiar with an algorithm that he or she can execute the algorithm, and manipulate symbols in a way that yields an answer; but still this student mightnot understand either why the algorithm works, or the meaning and interpretation behind each individual step in the algorithm.
  
  Students in this course will be expected to move beyond this level.  That is, they will be expected to understand what is really happening with the algorithm, when it can be applied and why, and all of the appropriate interpretations.
  
  Problems will be constructed for the exams that will test these sorts of understanding.  Students will also  be expected to demonstrate such understanding by their written annotations and explanations of their work on their exam papers.  Solutions that do not have sufficient or communicative explanations will not receive full credit, even if the final numerical answer is correct.
  
  
• Creative Application -- If a student truly understands the tools that we discuss and how and why they work, then he or she should be able to apply those tools and ideas to new situations.  That is, students should be able to use those tools TO SOLVE TYPES OF PROBLEMS THAT THEY HAVE NOT SEEN BEFORE, EITHER IN CLASS, THE BOOK, OR THE HOMEWORK.  
  
  This will seem difficult at first, as students adjust to the needed higher level of understanding that they must demand of themselves in this course.  But with sufficient practice, students will become comfortable with this level of expectation.  
  
  To prepare for these sorts of problems, students can pick a particular idea or tool from the course, and then try to think themselves of a new sort of problem or circumstance in which the tool might apply.  Then the student should work out those details. 
  
  Of course students can also look at all of the exams I have given in previous semesters in this course.  

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Students in this course will need to "raise the bar" on their personal expectations on understanding, in order to do well in this course.

Note, understanding is not a binary thing.  That is, it is not the case that someone either understands something, or they don't.  Rather, there is a continuous spectrum of comprehension, and study continuously increases the level at which someone understands an idea.  (One might compare this scale to those for skills in a sport, say, basketball -- certainly one cannot say that a player either does or does not have a certain skill in the game; but rather, for each skill, there is a continuous spectrum of levels of ability.)

In high school, there is a "bar" that is implicitly set on this spectrum by the types and difficulties of the problems that students are expected there to be able to solve.  This bar is set at a level that, while it might be appropriate in a high school setting, is much lower than is appropriate at Duke.

This presents a kind of psychological challenge to students in this course.  That is, students must find a way to motivate themselves to understand things at a level higher than they are familiar with, and furthermore they must then also find ways to gauge their understanding.

This is a difficult task, but one that every student here must undertake.  

I recommend the following exercise to help gauge your understanding on any particular topic.  That is, ask yourself,

This hypothetical question has two things going for it.  First, note that the question forces a student to consider his or her ability to communicate on a particular topic.  My experience has been that in order to communicate effectively on a topic, the most fundamental requirement is a clear and comfortable understanding of the topic in question.  Furthermore, the process of preparing to communicate on a given topic forces the mind to organize the ideas in a way that helps understanding.

Second, public speaking is something that most people are naturally apprehensive about.  Often this apprehension is based in some sort of fear of the possibility that one might be exposed for not understanding something.  By considering a public speaking scenario, the possible resulting feelings of apprehension might be a clue that you do not actually understand things as well as you might have thought.

Note, some students take this to an even higher level by actually giving such presentations, in groups that they form themselves.  I think that this is a wonderful idea, and both presenting and listening can be useful learning tools for participating students in such groups.