Math 321 reading, homework, and exam information

• You are expected both to attend lecture (and take notes!) and read the textbook. The lectures and textbook partly reinforce each other and are partly complementary. In particular, I will sometimes cover material in lecture which is not in the textbook, and you will sometimes be responsible for material in the textbook which is not covered in lecture.
• All homework assignments are due by 2 p.m. on the listed date, at my office. Late homework will not be accepted. If unusual circumstances arise and you contact me in a timely manner, then we can discuss alternative arrangements.
• Throughout this class, you need to explain your answers even when the problem doesn't explicitly ask for a proof; this typically means writing in complete English sentences. When deciding how much detail to include, here's the standard to keep in mind: your solution to a problem should be complete and clear enough that one of your classmates, who has paid attention in class but hasn't thought about that specific problem yet, could read your solution and understand exactly how it works. If you only try to convince the grader that you understand the solution, then you almost certainly won't write enough.

Final exam information

For good advice about studying, you should reread the article about study habits linked to from the syllabus. (Of course I know all of you already read it because it said to on the syllabus, but read it again to refresh your memory.) As a reminder, the most important point is to study actively — don't just read notes, do problems! But there's lots of other good information there, too.

About three quarters of the exam will focus explicitly on material covered since the midterm (sections 3.1–3.3, 4.1–4.5, and 4.7 of the textbook, and other topics covered in class since the midterm); the rest will be material from the first half of the course. (This is necessarily rough since newer material builds on older material.)

Definitions

You will be asked for complete, precise definitions of about ten terms, including context (e.g., what kind of thing might be open?). Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a metric space to be complete, I need to be able to use it, in principle anyway, to decide whether or not any metric space I ever meet is complete. The terms for you to know include all the terms listed below for the midterm, as well as the following terms on more recent material:

• differentiable
• derivative
• global Lipschitz condition
• class C^k (written C^r in the textbook)
• n^th order Taylor polynomial
• partition
• mesh
• Riemann integral
• Riemann integrable
• lower/upper sum
• lower/upper integral
• Darboux integrable
• characteristic function
• measure zero
• antiderivative
• improper integral
• converge (for a series)
• converge absolutely
• converge conditionally
• converge pointwise (for a sequence or series of functions)
• converge uniformly (for a sequence or series of functions)
• equicontinuous
• dense
• subalgebra of C[a,b]
• separates points
• fixed point
• contraction (the strong sense in which this term is used in the textbook)
• nowhere dense
• L^p norm
• l^p

If this list seems like a lot to memorize to you, keep in mind that you can't work with any of these concepts in a mathematically rigorous way without knowing the definitions — this is a list of the most important terms that you should know by heart anyway!

The rest

The rest of the exam will consist of several problems similar to (some of the more difficult) homework problems. Rather than suggest new problems for review, I recommend you start with the homework and homework solutions, including problems that you got right and those that weren't graded. All the same important ideas you need to know for the exam were needed on the homework. After you've done the homework problems again, work on the dozens of exercises in the book which I didn't assign.

Finally, if you ask me how many problems there will be on the exam, I will point out that without knowing exactly how long and hard the individual problems are, the answer would not tell you anything. I expect most students to need most of the allowed time for the exam. (I also expect that by the time of the exam you should be able to do a given problem more quickly than you could have done it on the homework, when you were first working with the relevant ideas.)

Midterm exam information

For good advice about studying, you should reread the article about study habits linked to from the syllabus. (Of course I know all of you already read it because it said to on the syllabus, but read it again to refresh your memory.) As a reminder, the most important point is to study actively — don't just read notes, do problems! But there's lots of other good information there, too.

The exam will cover all the course material from the Introduction, Sections 1.1–1.4 and Sections 2.1–2.4 of the textbook, as well as anything else covered in lecture up to October 10.

Definitions

You will be asked for complete, precise definitions of about six of the following terms, including context (e.g., what kind of thing might be open?). A few terms were defined first for real numbers, then for metric spaces. For these you should give the more general definition (i.e., the one for metric spaces). Some other terms were defined as something satisfying several conditions which were previously proved to be equivalent. For such terms, any one of the equivalent conditions is acceptable as a definition.

Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a metric space to be complete, I need to be able to use it, in principle anyway, to decide whether or not any metric space I ever meet is complete.

• cut
• Cauchy sequence
• converge
• lim sup
• inner product
• norm
• countable
• metric space
• complete metric space
• continuous
• homeomorphism
• open
• closed
• closure
• interior
• boundary
• limit point
• cluster point of a set
• cluster point of a sequence
• accumulation point
• sequentially compact
• covering compact
• totally bounded
• connected
• path connected

If this list seems like a lot to memorize to you, keep in mind that you can't work with any of these concepts in a mathematically rigorous way without knowing the definitions — this is a list of the most important terms that you should know by heart anyway!

The rest

The rest of the exam will consist of several problems similar to (some of the more difficult) homework problems. Rather than suggest new problems for review, I recommend you start with the homework and homework solutions, including problems that you got right and those that weren't graded. All the same important ideas you need to know for the exam were needed on the homework. After you've done the homework problems again, work on the dozens of exercises in the book which I didn't assign.

Finally, if you ask me how many problems there will be on the exam, I will point out that without knowing exactly how long and hard the individual problems are, the answer would not tell you anything. I expect most students to need most of the allowed time for the exam. (I also expect that by the time of the exam you should be able to do a given problem more quickly than you could have done it on the homework, when you were first working with the relevant ideas.)

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