Math 444, Section Q13, Fall 2009

Homework assignments  Approximate Schedule, Score Reports

Partial solution of hw1(average 16.45/20) Solution of Quiz1,2(average 15.5/20,14.8/20)  Solution Exam1(average 40.2/50)

Old quiz1(solution) Old quiz2 (solution)  old exams (solution)

Old quiz3(solution) Old quiz4 (solution)  old exam2  old final

Solution of Quiz3 (average 10.9/20) Solution of Quiz4 (average 13.8/20) Solution of ex2 (average 38.9/50)

Solution of final (average 79/100) See Score Reports ex0 for your overall grade (1=A+,2=A,3=A-,4=B+…)

.       Lectures: TR 12:30-1:50 in 165 NOYES LAB

.       Instructor: Tao Mei, mei@math.uiuc.edu

.       Course homepage: www.math.uiuc.edu/~mei/math444.html

.       Office & Telephone: ALTGELD Hall 329, 217-244-4132

.       Office Hours: 5:00-6:00pm Monday or by appointment.

.       Textbook (required): R. B. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd Edition, John Wiley & Sons, 2000.

Course Description: This course is an introduction to e - d analysis on the real line for students who do not plan graduate study in mathematics. Students should leave the course not only with a basic understanding of the fundamental concepts of real analysis, but also an improved ability at reading and writing mathematical arguments. Math347 is a prerequisite. Regular homework is an important aspect of the course. See the Approximate Schedule for information of which section will be covered.

 

Tentative exam dates: There will be two hour-long exams during the semester and a final exam.  Books, notes, calculators are not allowed on exams. 

Exam #1 Thursday, Oct. 1st in class.

 

Exam #2 Tuesday, Nov. 17th in class.

Final Exam- 7:00-10:00 PM, Friday, Dec. 11 in our regular classroom. Please arrange any travel, etc., so that you can take the final on this date.   As a "non-combined" course, there will be no conflict exam given except for those few individuals who meet the official university criteria given in the student code. 

Homework and Quiz: Practice is essential for this course. We will have a homework assignment each week. Written homework is due at the beginning of class on the due date.  We will also have practicing quizzes. 3 poorest grades of HWs and quizzes will be dropped from the final score in the class. As a result HW extension or make-up quizzes will not be allowed.

Homework Rules: (1)Print your name, section at the top of the paper. (2) Staple the papers together if your homework takes more than one page. (3) Work out the problems, the answer only will not be accepted. (5)Be neat and legible.

You will be able to check your homework and exam grades at Score Reports, which is the Math Department's gradebook program.  This will be available beginning approximately two weeks into the semester.  All your h/work, quiz and exam grades will be posted there. You will be prompted for your NetID and NetID password after following the Score Reports link. Please check score reports regularly to make sure your grades have been correctly reported and tell me promptly about any errors.  You are responsible for keeping all of your graded homework and exams so that any discrepancies in recorded grades can be settled.

Grading: two midterm  (20%  each), final exam (30%),  Homework+Quizzes (30%)

Approximate Course Schedule

 

Chapter

 

Class Hours

1.

Preliminaries
(We will treat Sections 1.1, 1.2 lightly. Cover Section 1.3.)

3

2.

The Real Numbers
(Sections 2.1, 2.2 will be covered quickly, with emphasis placed
on 2.3 and 2.4. We may omit the discussion of decimals in Sec. 2.5.)

5

3.

Sequences
(We may omit Section 3.6.)

9

4.

Limits
(We will omit Section 4.3)

3

5.

Continuous Functions
(We will Omit Sections 5.5, 5.6 and omit approximation in (p. 140-144) Section 5.4)

6

6.

Differentiation
(We will omit Sections 6.3, 6.4.)

3

7.

The Riemann Integral
(We will omit Section 7.4)

6

8.

Sequences of Functions
(We sill cover 8.1 and as much of 8.2 as time permits.)

 

Homework Assignments (An complete assignment ends with a .. An assignment end with a ; is incomplete, additional problems will be added to it later. Only part (ii) problems need to be handed in.  Each team need to hand in only one homework)

Course hotline (email to students and instructor):  Share your questions on homework, course material. I encourage you to work with other groups. But each group must write up its own solution individually, to turn in. As a sample of writing hws, read my solution to prob2 of my old exam1)

 

Assignment #1 (due by the beginning of the class on Tuesday Sep. 1st):

(i) Section 1.1, #3, 13, Read the proof of Theorem 1.3.13.

(ii) Section 1.1, #5, 9,15, Section 1.3, #4,9, Construct a surjection from the set of real numbers R to the set P(N) of all subsets of natural numbers.  Use Theorem 1.3.13 to prove (i) P(N) is uncountable (ii) R is uncountable.

.

(Hint: Consider the function f from R to P(N) with f(x) being the set of all n such that 1 appears in x at the n-th position after the decimal point. For example, for real number x=2.137516899241, we set f(x)={1,5,12}. If 1 does not appear in x after the decimal point, we let f(x)=empty set. Easy fact: every bijection is a surjection, the composition of a surjection and a bijection is still a surjection).

 

Assignment #2 (due by the beginning of the class on Thursday Sep. 10):

(i) Section 1.2, #6, Find why the “proof” of “Claim” on page 15 is false, Given a<b in R, find a bijection from (a,b)={a<x<b} to R so we can conclude that (a,b) is uncountable, Section 2.3, #4, Section 2.4 #10.

(ii) Section 1.2, #7,  Section 2.2 #14, 15; Section 2.3 #4,9, Section 2.4, #2,4,18, Construct a bounded countable subset S of real numbers such that Sup S is in S but Inf S is not in S.

Assignment #3 (due by the beginning of the class on Tuesday Sep. 15):

(i)                  Section 2.5 #7, 16, Section 3.1# 1(c), 2(c), 9.

(ii)                Section 2.5 #3, 8, 12, Section 3.1 #3(b), 5(b), 8.

Assignment #4 (due by the beginning of the class on Tuesday Sep. 22):

(i)                  Section 3.1, # 17, Section 3.2, #5, 18, read Theorem 3.2.11 and its proof, Section 3.3, #3.

(ii)                Section 3.1, # 10, 14, Section 3.2 #2, 13(a), 15(b)(c), 17, Section 3.3 # 2, Bonus question: Construct a sequence (a_n) such that the first a few terms can be easy computed and a_n converges to square root of 5. Prove it really converges to root 5 by monotone convergence theorem. Compare your a_4 with the real value of root 5.

Assignment #5 (due by the beginning of the class on Tuesday Sep. 29):

(i)                  Section 3.4, #4(a), read second proof of Th3.4.8 on page 79.

(ii)                Section 3.4, #4(b), 9, 16.

 

Assignment #6 (due by the beginning of class on Tuesday Oct. 13)

(i)                  Section 3.5, #1, 2(a), 7, Section 3.7, #4, 8, 9.

(ii)                Section 3.5, #5, 9, 11,13, Section 3.7 #3(a), 6, 10, 1. Bonus question: 14(a) and 15(a) (hint: use the “calculus” method we used in class. Note the anti-derivative of 1/xlnx is ln(lnx)…).

 

Assignment #7 (due by the beginning of class on Tuesday Oct. 20)

(i)                  Section 4.1, #9 (use sequencial criteria), Section 4.2, #10, Section 5.1 #11, 15, Section #11.

(ii)                Section 4.1, #11(c), 14, Section 4.2,#4, 11,  Section 5.1 #10, 13, Section 5.2 #7, 9 (f is continuous on R means that f is continuous at any point), 15.

 

Assignment #8 (due by the beginning of class on Tuesday Oct. 27)

(i)                  Section 5.3, #7, 11, Section 5.4 #4,

(ii)                Section 5.3, #3, 5, 6, 13, Section 5.4, #1, 3, 14, Find examples of f, g both uniformly continuous on R (=the real line) but the product fg is not uniformly continuous on R.

 

Assignment #9 (due by the beginning of class on Tuesday Nov. 3rd)

(i)                  Section 6.1,

(ii)                Section 6.1, #1(b)(d), 5(a)(b)(c), 10, Section 6.2, #3(a)(d), 6,11, 14.

 

Assignment #10 (turn in at the beginning of class on Tuesday Nov. 10th  or Thursdsy Nov.12th)

(i) Section 7.1, #5, 7.

(ii) Section 7.1, #2(a)(d), 6(a),8. Bonus question, #10.

 

 

Assignment #11 (turn in at the beginning of class on Tuesday Dec. 8th ) (two more problems added)

(i)                  Section 7.2, #8,

(ii)                Section 7.2, #2, 7,10, 12, 16, Section 7.3, #4,8, 16(b)(d).