Math 3321 -- Test page

Instructions for Test 1: No calculators. Time: 80 minutes. You should have read through all your classnotes and understood them by now (you are supposed to do this after each class, asking questions about things you dont understand). In preparation for the test read them through again (this will be much quicker).

The tests will have basically 4 kinds of problems.

1) Some questions will be taken more or less directly from the turned in Homework. So go through the keys of those carefully, trying to understand them, and go carefully through your graded work so you dont make the same mistakes again. Some questions will be very similar to questions from the Homework (assigned or turned in). So again review the homeworks.

2) Some questions may ask you to define something, or test if you know what certain words mean (like ``What is the Cauchy product of two series?"). So memorize the definitions in the classnotes.

3) Some questions may ask you to state a theorem or result from the classnotes, or fill in the blanks in the statement of the theorem or result from the classnotes. Do read through your classnotes again, making sure everything makes sense. And try to memorize the statements of the most important results, so that you can write them sensibly if you are asked for them. Make a list of things you don't understand and ask me or the grader.

4) Some questions may ask you to prove a theorem or result from the classnotes from a list of proofs that I will add to this page below. I recommend that you write out each proof 7 times closed-book (although some people may need less than 7), to be sure you can write it quickly and without errors.

Go carefully through the mock exam below.

Put in the time to study--and don't underestimate the time needed!!
Mock Exam for Test 1 . This was a mock exam from a previous year, and some questions may not be appropriate (like 0(a), 2b, 2d, 3b, 4a, and 6. In particular, there are more `unseen problems' here that require thought, whereas I intend the nontheory problems on the actual test to be either from the homework, or similar to homework problems youve seen.
List of proofs to study for the test from the first chapter: The Cauchy test (first bullet on p. 6), FACT on p. 7, the divergence test (p. 8), first bullet on p. 9, absolutely convergent series converge (p.12) and first bullet on p. 13 (note the latter proof is messed up, the inequality at the end of that proof should start the proof).

Numbers above refer to the current version of the online notes.)

List of proofs to study for the test from the second chapter: 1.17, 1.18, 1.21, 1.28, 1.54-1.55 (or their shorter versions from class). These numbers refer to the Paulsen notes, my versions in class are in some cases a bit shorter or stated slightly differently.

During the test review I noticed that there was one important class of problems that could be on the test that was not adequately covered on the homework, namely those involving derivatives of series (it was covered in class, e.g. when we worked example 1.29 in class, or when we discussed derivatives of power series). Thus for this topic of derivatives of series I am going to make a single exception to my rule that the only `unseen problems’ on the test would either be Homework problems or very similar to the homework. Thus you can expect a problem like 0(d) or 7(c) on the mock test, even though there was not one similar on Homeworks. Thus to prepare for this kind of question, review 0(d) or 7(c) on the mock test, and also example 1.29 in class, and what we discussed in class on derivatives of power series. You could also work some similar Problems in Paulsens notes.
It will be the same test for grad and undergrads. The syllabus ends with the use of the Stone-Weierstrass theorem to prove the (polynomial) weierstrass theorem.
Test 1
Key to Test 1 with detailed point allocation
Key to Test 1
Test 2 (on Fourier series) will be on April 9. (Note that Monday 6 April is the last day to drop a course or withdraw with a ‘W’.)
Test 2 instructions: all the instructions and advice above for Test 1 apply, except the syllabus begins at the March 3 class notes, i.e. after the Stone-Weierstrass theorem in the classnotes, with trig polynomials and the periodic Weierstrass theorem, and continuing to the end of Fourier series. Begin by doing the default instructions (which should go without saying): be reviewing through all your classnotes, understanding every line. You are supposed to do this after each class, asking questions about things you dont understand, but in preparation for the test read them through again (this will be much quicker). The test again will have basically the same 4 kinds of problems as on Test 1, and you were instructed above how to study carefully for each. Eg. Some questions may ask you to define something, or test if you know what certain words mean (like ``What is a Fourier series?"). So memorize the definitions in the classnotes. Some questions may ask you to state a theorem or result from the classnotes, or fill in the blanks in the statement of the theorem or result from the classnotes. So when you read through your classnotes, making sure everything makes sense, try to memorize the statements of the most important results, so that you can write them sensibly if you are asked for them. Make a list of things you don't understand and ask me or the grader. Some questions will ask you to prove a theorem or result from the classnotes from the list of proofs below. It is very difficult to memorize proofs you do not understand, so when reading your classnotes on a daily basis be sure to go through and understand all proofs so you dont have to figure them out from scratch the week of the test. Most people might have to write the proofs on the list out 7 times or so with closed books so that you can be sure not to forget their logic on the test. Review the HW keys and graded HW and understand them and your mistakes. Go through the mock exam carefully. Put in the time to study--and don't underestimate the time needed!!
Final list of proofs to study for Test 2: Why ||.||_2 is a seminorm on the Riemann integrable functions on [a,b], and is a norm on the continuous functions on [a,b]. In particular, the proof of the triangle inequality for ||.||_2 (also called Minkowski, see 1.106 (2)). 1.99, 1.100, 1.101, 1.102, 1.108 (these numbers refer to the Paulsen notes, my versions in class are in some cases a bit shorter or stated slightly differently). Corollaries 1, 2, 3, 4, and 4.5, 4.6, 4.9.
By the way, when writing proofs on the test, I prefer you write them in your own words (but with all the logic right). You are being graded there on if you have all the logic right, not on if you have the same words.
Mock Exam for Test 2 .
Test 2
Key to Test 2 Note the pi squared in 0.f key should be pi^4, and 2a is out of [3], and 6 has points [3+6+4+8].
Test 3/Part 1 of final will be on April 28. Part 2 of final will be on April 30. Start your study for Test 3 as usual by rereading your classnotes making sure you understand everything, ask any questions, and memorize definitions and statements of the important results. To study for the final, study just as for Tests 1, 2, 3. More instructions coming soon.
Test 3 instructions: all the instructions and advice above for Test 1 and Test 2 apply, except that syllabus is the chapter on multivariable calculus.
List of proofs to study for Test 3: 2.6, 2.7, 2.15, 2.20, 2.25, the proof of (i) of the Inverse Function Theorem in my notes (parts 1 and 3 in Paulsens statement of the theorem), the Corollary after the Inverse Function Theorem.
Mock Exam for Test 3. There will be no Mock Exam for the Final exams, since you can use the Mock Exams for Tests 1,2,3, and the actual Tests 1,2,3, as a Mock Exam for the final exams.
Part 2 of final will not include anything from the multivariable calculus chapter. Part 1 of the final will be dedicated to the multivariable calculus chapter (although that may be rolled into the Test 3 portion of the hour), and the material in my 16 page set of notes from the first weeks of class about series of NUMBERS (not functions). Thus there will be nothing on Part 1 of the final on series of FUNCTIONS (this will be in Part 2 of the final). List of proofs to study for Test 3/Part 1 of the final is the list of proofs to study for Test 3 above, plus the list of proofs I gave above from that 16 page set of notes, minus the Cauchy test (first bullet on p. 6). The list of proofs to study for Part 2 of the final is 1.17, 1.18, 1.21, 1.28, 1.54-1.55 in the notes (or their shorter versions from class), plus the list of proofs to study for Test 2 above minus 1.99, 1.100, 1.101, 1.102, and also minus Corollary 1 from my typed notes on Fourier series.
ADDED APRIL 23 at 1 pm: As it says on the course structure page on our website, I am usually happy to "drop the lowest of each type of grade (e.g.\ lowest test, lowest few homeworks, etc) for students who have not been showing gross irresponsibility in the class. By `gross irresponsibility' I mean for example who have quit turning in homeworks etc). The instructor may change the `formula' for the grades above at his discretion if doing so will benefit the class as a whole." NOTE: there will be some Test 3 material on to Final 1. Then I will drop the lowest one of Test 1, Test 2, Test 3, Final 1, and Final 2, assuming you meet the conditions in the last paragraph. This even allows a person who is happy with their Test 1 AND Test 2 grades to miss ONE of the Finals or Test 3 if they really must; assuming that they meet the conditions in the last paragraph.
I said in class that I plan to base most of the Finals Parts 1 and 2 on Test 1 and Test 2, and Mock Test 1 and Mock Test 2, although of course the Final Part 1 will also have a little Test 3 stuff as I just said. As I explained in class, this was to make the last week of class less brutal: we were not allowed by the College to have the final at the time on the University Exam calendar because of NSM graduation, which meant that we had to have the final in the last week of class along with Test 3. Although the class knew this for months, this still made the last week brutal without some concessions.