Please visit me in my office during these times if possible. I am often in my office at other times and do not mind at all if you drop by to talk, though I cannot guarantee that I will always be there or have time to meet. If you are unable to see me during my office hours and cannot find me at other times, just contact me by phone or e-mail and I will find an alternative time to meet with you.

NOTE: As part of a policy that has proven to be very effective, I will require every student in the class to meet me in my office for a short discussion at least once between the first test and the last day of class. The meeting can be during any one of my regular office hours or at another time either by appointment or drop-in. We can discuss the class or anything else you'd like to talk about. This meeting is not optional: you must meet with me in order to pass the class.

• 26 August 2009 (Preliminaries / 1.1 Completeness Axiom )
• 28 August 2009 (1.2 Distribution of Integers and Rational Numbers / 1.3 Inequalities and Identities)
• 31 August 2009 (2.1 Convergence and Sequences)
• 4 September 2009 (Pause: Discuss Homework, Proofs and Set Notation)
• 7 September 2009 (2.1 Making New Convergent Sequences from Old Ones / 2.2 Boundedness of Sequences)
• 9 September 2009 (Two Re-Explanations / 2.3 Monotone Convergence Theorem)
• 11 September 2009 (2.4 The Sequential Compactness Theorem)
• 14 September 2009 (3.1 Continuous Functions)
• 16 September 2009 (3.1 Finishing Up / 3.2 The Extreme Value Theorem)
• 18 September 2009 (3.3 The Intermediate Value Theorem)
• 28 September 2009 (First Test Results / 3.4 Uniform Continuity)
• 30 September 2009 (3.5 ε-δ definition of continuity)
• 2 October 2009 (3.6 monotonic and inverse functions / 3.7 limits of functions)
• 5 October 2009 (4.1 DERIVATIVES!)
• 7 October 2009 (4.2 Derivatives of Inverses)
• 9 October 2009 (4.2 Chain Rule / 4.3 The Mean Value Theorem)
• 14 October 2009 (4.3 Geometric Implications of The Mean Value Theorem)
• 21 October 2009 (4.4 Cauchy's Mean Value Theorem)
• 23 October 2009 (6.1 Darboux Sums and Partitions)
• 26 October 2009 (6.1 Lower and Upper Integrals)
• 28 October 2009 (6.2 The Archimedes-Riemann Theorem)
• 30 October 2009 (6.2 Using the Archimedes-Riemann Theorem)
• 4 November 2009 (6.3 Properties of the Integral)
• 6 November 2009 (6.4 Continuous Functions are Integrable)
• 9 November 2009 (6.5 The First Fundamental Theorem of Calculus!)
• 11 November 2009 (6.6 The Second Fundamental Theorem of Calculus!)

Tests: There will be three tests in this class, each taking place on a Friday during the regularly scheduled class time. The questions on the test will either test your knowledge of a definition, your ability to determine the truth of a statement (e.g. true/false question), your ability to perform a computation or (most importantly) your ability to rigorously prove a mathematical statement. (See first day's lecture notes for additional information about the concept of proof and its importance.) The tests will be given on: September 25, October 19 (note changed date!) and November 20

Homework: Homework is due two classes after it is assigned. (So, homework assigned on Monday is due on Friday, problems assigned on Wednesday are due the next Monday, etc.) Questions will be chosen from the problems at the ends of the sections in the textbook and will be similar to the test questions discussed above. You are expected to do the homework yourself. As much as possible, you should try to do it without any assistance from anyone, as this is how you will learn to do well on the tests. However, of course, you can come to discuss homework problems with me if you need assistance. Furthermore, I offer the following unusual rule: you can discuss homework problems with other students or anyone else as well, so long as neither of you writes anything down (or type it or any other method of recording it) during the discussion!

Final Exam: The final exam for this course will be held on Monday December 14th from 8:00-11:00AM in our usual classroom. Note that this is the time scheduled for this course based on the time that we meet. There should not be any conflicts with this time since only exams for courses that meet at noon on Monday can be given at that time. This is the only time at which you will be able to take the exam for this course (not earlier or later).

The final exam for this class will be approximately as long as two of the class tests. Questions on the final can and will test material covered throughout the semester.

The titles of the chapters we will be covering should sound very familiar to you: Convergent Sequences, Continous Functions, Differentiation, Integration, etc. In fact, we will really be going back and relearning calculus here. Some of the details will be new, but many of the results we learn will actually already be known to the students in the class from previous calculus courses. The difference is that we will be doing it rigorously. In particular, the focus will not be on learning what is true, but on the techniques used by mathematicians to determine and demonstrate truth. This is the method of mathematical proof, which is the technique mathematicians have used since the time of the ancient Greeks to explore the abstract world of mathematics. By applying this technique carefully to the subject of calculus, which you already know, we will both be able to better understand calculus itself and have an opportunity to further develop your abilities with these methods used to prepare you for the future. In the future, you will be taking courses on very different sorts of abstract math or (if you choose to do math research) may even be exploring an area of the mathematical universe that has never been visited before. Prior to these sorts of excursions into the unknown, it makes sense to have practiced with these techniques for examining the abstract in a more familiar setting, as we will be doing here.

• It will be difficult for you to remember that the main focus here is not on knowing what the facts are, but in being able to derive those facts from a few basic axioms. In particular, almost as if it were a game, you will be prohibitted from making use of facts that you have learned in previous math classes, even if they are perfectly correct, unless we have proved them rigorously together or you have written a formal proof of them on your own.
• Even though I am giving you some freedom in discussing homework with other students as well as with me, do not forget that the purpose of the homework is to prepare you for the exams. If you abuse the help of others in order to get high homework grades, this may prevent you from doing well on the tests and exams, which make up much more of your final class score.
• I will not have time to discuss all of the details in class. This does not reflect poorly on my teaching; it is simply a fact that in the chapters we are covering there is a lot of material. I will do my best to select the most important topics and make sure that those are fully discussed during lecture. In fact, I think you can probably pass the class knowing only those things I discuss in lecture. However, it is expected that you will be reading the book as we go through it. Reading the book (a difficult task in itself, which might require some learning on your part) will be necessary if you plan to do well in the class.
• Expect to have to develop some new ways of working on and thinking about mathematics in this class. In many of your previous classes, it was probably possible to think of math as a series of steps to be memorized and performed. That method stops working at the higher levels of mathematics and this course is one in which many students first encounter this difficulty. Simply put, I believe the only way to succeed in this class is to learn the subject not as a list of techniques to remember, but as objects and deep facts about them that you must understand.