Math 244 - Spring 2007 - Course Information

Math 244 - Spring 2007 - Course Information

Here is the takehome part of the final exam.

What is real analysis? The very short answer is that real analysis is the mathematical theory behind elementary calculus. A slightly longer answer is that mathematical analysis is the branch of mathematics that deals with properties of functions. (In the same drastically oversimplified sense, the other two main branches of mathematics are algebra - having to with equations - and arithmetic - having to do with numbers . The ``real'' in real analysis has to do with the real numbers: the functions studied in real analysis are usually real-valued functions of a real number. One important purpose of the course is to revisit - a lot more carefully than on the first pass - important ideas, definitions, and theorems from elementary calculus.

Instructor: Paul Zorn, Old Music Hall 203, phone x3414, e-mail zorn@stolaf.edu

Office Hours, Spring 2006:

At other times, make an appointment or look for me in my office.

Keeping in touch: This is our class web page. I'll update it frequently with homework assignments, solutions, hints, etc.

Text: Russell A. Gordon, Real Analysis, A First Course, Second Edition .

What we'll cover: We'll cover most of Chapters 1-5 (omitting a few sections); and perhaps add a few selections from later chapters.

Grades, tests, important dates, etc.: There will be a midterm (75 pts, in class), three short quizzes (25 pts each), and a final exam (100 pts; part take-home, part in-class). Homework will also contribute heavily to your grade (up to 150 points over the semester). Here are some dates; all are tentative except for the final.

Quizzes: February 23 (Friday), March 16 (Friday), April 23 (Monday)
Midterm: March 21 (Wednesday)
Final: May 21 (Monday), 9-11 am

About learning disabilities : If you have a documented disability for which accommodations may be required in this class, please contact Ruth Bolstad or Connie Ford in the Academic Support Center (x3288) located at the very back of The Village. If you already have documentation on file in the Academic Support Center you are required to present your letters to the professor within the first two weeks of class.

Samples, solutions, etc: Here is a sample quiz. Here is a sample midterm test.

Homework policy: Homework will be assigned at most classes; a first draft is due on the second following class day. Late first-draft homework is not accepted, but it will usually be possible to revise homework.

I strongly encourage you to work with classmates on homework, but everyone should hand in his or her final paper. Homework assignments are listed below.

About homework drafts: The first draft response from the student grader is not a guarantee of correctness. (Sometimes I'll correct papers; in that case you should take my word for things, though of course I could be wrong.) It is up to you, not the grader, to be sure your second draft is as good as you can make it.

For your second draft, turn in a completely clean copy of every problem that you see any need to revise. Marks on your paper made by me or by a student reader on your paper are not ``corrections'' in the sense that you can hand them in and receive credit.

Think very hard about whether what you write makes sense by the strict standards of mathematical writing. For instance, the sentence

For all Q we have x^2 > 0.
technically makes no sense (even though it's possible to guess what the writer means). A statement that makes no sense cannot be true (or false, for that matter). It is possible, on the other hand, for a statement to make sense and be either true or false. Your first job is to write sentences that make sense. Your second job is to write sentences that are true.

Using LaTeX: LaTeX is a wonderful tool for mathematical word processing --- especially if (as in this course) you're producing multiple drafts of things. Here is some information on using LaTeX for word processing.

On writing proofs: There's a lot of writing in this course. Here are some hints and tips on coping with mathematical writing.

About homework:

Unless otherwise stated, assignments are from our text. Unstarred problems need not be handed in, but you should know how to do them! Only the starred problems will be graded, but you should think about all the problems. Each assignment is worth six points. (The two dates in each case represent first draft/final draft.)

Homework assignments--final drafts, especially--should be written up carefully, using full sentences (which may involve mathematical symbols), proper punctuation and grammar, etc. I'll say more about this in class.

Double-star problems: Problems marked with a double star are optional, but worth two points extra credit. They can be handed in at any time. Here are several.

1**. Section 2.2: 9
2**. Section 2.2: 24
3**. Section 2.2: 27
4**. Give a careful proof of part (e) of Theorem. 2.7.
5**. #29, page 126. (For extra glory, replace all 2's in the problem with 3's.)
6**. Suppose that f: R -> R is a function, and that f(x) is rational for all x. Show that f(x) is either constant or somewhere discontinuous.

Regular homework problems

Assignment 1: Due Friday 9 February/no second draft
Section 1.1: 1, 2, 8*, 9*, 10*
hintsassn1.pdf
solutionsassn1.pdf

Assignment 2: Due Monday 12 February/19 February
Section 1.1: 11, 12, 13*, 17*, 18*
hintsassn2.pdf
solutionsassn2.pdf

Assignment 3: Due Wednesday 14 February/21 February
Fun with proof mechanics --- get it here.

Assignment 4: Due Friday 16 February/23 February
Section 1.2: 1, 2, 6*, 9*, 13*
hintsassn4.pdf
solutionsassn4.pdf

Assignment 5: Due Monday 19 February/26 February
Section 1.2: 14, 15, 16*, 17*
Section 1.3: 7, 8*
hintsassn5.pdf
solutionsassn5.pdf

Assignment 6: Due Wednesday 21 February/28 February
Section 1.3: 9, 10, 11, 12*, 15*, 16, 20*
hintsassn6.pdf
solutionsassn6.pdf

Assignment 7: Due Monday 26 February/5 March
Section 1.4: 1*(OK to omit part g, but see the hints), 2, 3*, 4*, 8, 11
One more problem*: Show that any two open intervals (a,b) and (c,d) have the same cardinality. (Note: Assuming that both intervals are open and of finite length simplifies things a bit. But the result is actually true for any two intervals, open or closed, finite or infinite in length.)
hintsassn7.pdf
solutionsassn7.pdf

Assignment 8: Due Wednesday 28 February/7 March
Section 1.5: 21, 22*, 24, 25*, 26*
hintsassn8.pdf
solutionsassn8.pdf

Assignment 9: Due Friday 2 March/9 March
Section 2.1: 7, 8, 9*, 10*, 11*
hintsassn9.pdf
solutionsassn9.pdf

Assignment 10: Due Monday 5 March/12 March
Section 2.1: 5, 12*, 15*, 17*, 18
hintsassn10.pdf
solutionsassn10.pdf

Assignment 11: Due Wednesday 7 March/14 March
Section 2.1: 19*, 20a*b*, 22, 23*, 24, 28
hintsassn11.pdf
solutionsassn11.pdf

Assignment 12: Due Friday 9 March/16 March
Notes: You'll need to read just the first part of 2.2 (pp. 61 and 62) to get going here. The problems do not mention Cauchy sequences. Section 2.2: 1* (skip part c ... it's silly), 2*, 3, 7, 8*
hintsassn12.pdf
solutionsassn12.pdf

Assignment 13: Due Monday 12 March/19 March
Section 2.2: 12*, 14*, 15, 16*
hintsassn13.pdf
solutionsassn12.pdf

Assignment 14: Due Wednesday 14 March/21 March
Section 2.2: 20*, 24** (optional, but see definition of ** above), 36*, 37, 38*
hintsassn14.pdf
solutionsassn14.pdf

Assignment 15: Due Friday 16 March/23 March
Section 2.3: 1*, 2*, 3*
No hints seem needed this time ... these are pretty easy.

Assignment 16: Due Monday 19 March/April 2
Section 2.3: 5, 6*, 9*, 14*
hintsassn16.pdf
solutionsassn16.pdf
Assignment 17: Due Wednesday 4 April/18 April
Section 3.1: 2, 4a*d*, 5, 6*, 11*, 13, 15
hintsassn17.pdf
solutionsassn17.pdf

Assignment 18: Due Wednesday 11 April/18 April
Section 3.1: 18*, 20* (use Theorem 3.3 in c and d), 25*
hintsassn18.pdf
Note: We haven't (formally, in class) given the definitions involved in problem 25, but see Definition 3.6 on page 87 and give it a try. We'll discuss this further on Wednesday.
solutionsassn18.pdf

Assignment 19: Due Friday 13 April/20 April
Section 3.2: 2*, 4, 7*, 8, 11*, 25*
hintsassn19.pdf
solutionsassn19.pdf

Assignment 20: Due Monday 16 April/23 April
Section 3.3: 1*, 5, 7*, 9*
hintsassn20.pdf
Note: Some of these problems use the Intermediate and Extreme Value Theorems, but not the proofs. We'll give the proofs on Monday.
solutionsassn20.pdf

Assignment 21: Due Friday 20 April/27 April
Section 3.4: 2a*, b*, c, d, e*, 3, 4*, 5*
hintsassn21.pdf
solutionsassn21.pdf

Assignment 22: Due Monday 23 April/30 April
Problems on induction; get 'em here
hintsassn22.pdf
solutionsassn22.pdf

Assignment 23: Due Wednesday 25 April/2 May
Section 4.1: 1*, 2, 4*, 6*
hintsassn23.pdf
solutionsassn23.pdf

Assignment 24: Due Friday 27 April/4 May
Section 4.1: 7*, 13*, 16*, 30
hintsassn24.pdf
solutionsassn24.pdf

Assignment 25: Due Monday 30 April May/7 May
Section 4.2: 1*, 10*, 13*, 14
hintsassn25.pdf
solutionsassn25.pdf

Assignment 26: Due Wednesday 2 May/9 May
Section 4.2: 18**, 20*, 23*, 24*
No hints for now; can you get started?
solutionsassn26.pdf

Assignment 27: Due Friday 4 May/11 May
Section 5.1: 1, 5*, 6, 7, 8*, 9*
hintsassn27.pdf
solutionsassn27.pdf

Assignment 28: Due Monday 7 May/14 May
Section 5.1: 11*, 13*, 14*
hintsassn28.pdf
solutionsassn28.pdf

Assignment 29: Due Friday 11 May/no rewrite
Section 5.1: 19a*
Section 5.2: 12*, 13*
Notes: We'll discuss these ideas in class on Wednesday, 9 May, but have a look in advance. Problem 19a in Section 5.1 is not difficult once you understand what's being asked. To do Problem 13 in 5.2, use Theorem 5.10 and the result of Problem 19a in 5.1.
solutionsassn29.pdf

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