Math 311, Advanced Calculus I

New Announcements

Dec 7 Solutions to Exam 2 are posted. The link is at the end of this page.

Dec 7 The typed notes for the material on the exponential series are posted here

Course Information for all Sections

Textbook: Introduction to Analysis, 5th edn. by E. D. Gaughan, Published now by the American Mathematical Society. Available both in hardcover and in paperback. Check the publisher's website for a direct order option.
http://rutgersmath.thomsonsites.com/newtexts/cluster.cfm?cluster_id=1979
Prerequisites: Calc 4 and Math 300 or permission of the department
This course is required of all mathematics majors (except those taking Math 411). It is useful to students in mathematical fields who intend to do graduate work in those fields. It is not equivalent to the course "Advanced Calculus for Engineers."
Catalog copy: Introduction to the language and fundamental concepts of analysis. The real numbers, sequences, limits, continuity, differentiation in one variable.
Note that the catalog defers the Riemann integral to Math 312. Math 312 may be offered during Spring 2010 if there is sufficient student interest.
Format: Three 80-minute classes each week.
Learning goals: strengthening a student's
- understanding of the results of calculus and the basis for their validity
- understanding of the reasons for deductive reasoning,
- ability to understand definitions and proofs,
- ability to analyze conjectures, to find counter-examples to false statements, and to construct proofs of true statements
- mathematical communication skills

Course Information for Section 03

Lecturer: Amy Cohen
Contact Information:
- Office: Hill 530 on Busch
- E-mail: acc@math.rutgers.edu
- Web-page: math.rutgers.edu/~acc
- Phone: 732-445-2390 ext 3098
Course Meetings:
- Tuesdays, 6:10-7:30 AND 7:40-9:00pm in Hardenburgh A-4 on CAC
- Thursdays, 6:10-7:30pm in Hardenburgh A-4 on CAC
Office Hours:
- Thursdays: 7:30-8:30pm in HH-A4 (or elsewhere in HH if necessary)
- also by appointment (Send me email. We'll work out time and place)
Usual Format: lectures, discussion of homework, and workshops. two midterm exams and a final exam.
Content: The subject matter of this course is the elementary analysis of sequences or real numbers and of real-valued functions of a real variable. The topics include convergence of sequences and series, continuity of functions, and differentiability of functions. I will conclude the semester with a careful study of the power series giving the exponential function.
This course begins with the axioms for the real number system as a complete ordered field. It proves rigorously many of the major theorems stated with informal proof or even without proof in the first two semesters of calculus. Students will see why the axioms are needed, how the deductions build up a coherent theory, and how the results are used.

OLD Announcements

Nov 18 Sample solutions for Chapters 2 and 3 are posted. These are edited versions of my solutions from last year. So there are some "extra" solutions included. I forgot to change the heading so please ignore the "2008" at the top.

Nov 18 The lecture schedule and homework schedule have been updated.

Oct 27 There is a typo in my solution to problem 2.4 on exam 1. The line with estimate (1) should start

|a_j a_k - b_j b_k|=|(a_j - a_k)b_j +a_k(b_j - b_k)|

where the _j indicates that j is a subscript and _k indicates that k is a subscript.

Oct 20 Solutions to the first midterm are posted. The link follows the links to homework below.

Oct 20 The homework assignments have been updated. Oct 7 You do not have to turn in problems #21, 47 from tomorrow's homework set.

Oct 2:

There was an error in the homework assignment table. The set I had intended you to submit yesterday was Set 2 from Chapter 1, not Set 3. Those of you who turned in the wrong set should turn in the set "Ch.1, Set 2" on Oct 8.

I have corrected the due dates below.

I am dissapointed that none of you emailed me to inquire why the assignment for Oct 1 covered topics I had not yet lectured on. Someone should have asked!

GENERAL INFORMATION AND ADVICE

A tentative syllabus is available below. Beware of frequent adjustments.

Read the assigned sections once before each lecture. Read them again after each lecture before starting on the homework. Re-read them as often as necessary! Additional material may be posted on this web page.

Attendance is crucial. I will accept late homework only in special cases and even then only if I have not yet returned the graded set.

Make-up exams will be offered only if there is adequate reason to do so. A student's lack of preparation or lack of confidence is not an adequate reason. In most cases, if you must miss an exam you will know in time to discuss the matter with me (in person, by phone, or by email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.

EXAMS: There will be two midterm exams and a final exam.

WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.

The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the term the goal is to ensure that groups can make use of the ideas offered repeatedly earlier.

At the end of each workshop session, one problem will be assigned to be written up and submitted at the next workshop. While students are encouraged to work together outside of class, the write-ups should be individual work. These write-ups will be graded on two scales: 0-6 for content and 0-4 for exposition. Good reasoning and good mathematical exposition may be more vasuccess, I may permit the student to revise the write-up and resubmit it. In such cases I will replace the original score luable in the long run than any particular piece of mathematical technique.

If a student has made an honest effort but not achieved much success, I may permit the student to revise the write-up and resubmit it. In such cases I will replace the original score by the average of the original score and the score on the revised write-up.

Directions for workshop write-ups:

Each write-up is a short technical report. It should be a well-considered professional document; not a first draft.
Write on one side only. Put your name on each page. Number pages. Maintain at least a one-inch margin on all four sides.
Edit carefully. Use correct spelling and grammar.
Begin each problem, or major part of a problem, with the caption "Task" and a statement of the task. This statement should be self-contained.
Next, using the caption "Result", state the result of the task. Do not use any notation in the result that is not defined in the statement of the task. This step may be omitted if the task asks for a proof.
Finally, using the caption "Work" or "Proof" justify the result. Include diagrams or graphs if these will help the reader follow the work. Include special cases if that will help. Lay out your computations and your reasoning clearly. Where your work relies on results already proved in this course, be sure to cite them. You do not need to cite reasons for basic algebraic manipulations.
Think of your proofs as explaining why a result is valid.

xposition. Good reasoning and good mathematical exposition may be more vasuccess, I may permit the student to revise the write-up and resubmit it. In such cases I will replace the original score

HOMEWORK:

I will assign about 5 to 10 exercises a week for you to work on. I will assign about 3 to 5 of these to be turned in. These will be due one week after they are assigned. Of these, some (usually not all) will be graded. Other homework (whether turned in to not) may be discussed in class.

Homework will be due one week after it is assigned, usually on Thursdays.

Use the same format for writing up homework as for writing up workshop problems.

Each homework problem will be graded on a scale of 0-5. Remember, the grader cannot grade your mathematics unless its exposition makes it clear what is going on.

I have been known to put homework problems on exams. Obviously, it is to your benefit to learn from doing the homework and to learn more from the reader's comments and from the class discussions.

TERM GRADES:

Each midterm exam counts for 100 points. The final exam counts for 200 points. The best ten workshop write-ups count for 100 points. The homework sum will be rescaled to count for 50 points.

Homework is intended to help you learn the material. Poor performance on homework will not necessarily lower your term grade.

Because of the opportunity to revise and resubmit write-ups, poor performance on the workshop write-ups may lower your term grade from that suggested by exam grades alone.

An extremely weak final exam may lowe r a term grade below that suggested by the point-total taken by itself.

Tentative Syllabus

xposition. Good reasoning and good mathematical exposition may be more vasuccess, I may permit the student to revise the write-up and resubmit it. In such cases I will replace the original score email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.
EXAMS: There will be two midterm exams and a final exam.
WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.
The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the
email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.
EXAMS: There will be two midterm exams and a final exam.
WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.
The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the
EXAMS: There will be two midterm exams and a final exam.
WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.
The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the >L18; W9
email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.
EXAMS: There will be two midterm exams and a final exam.
WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.
The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the
email) IN ADVANCE. If we have not discussed the matter in advance, then I will need evidence of an emergency.
EXAMS: There will be two midterm exams and a final exam.
WORKSHOPS: We will usually have one workshop session each week, usually on Wednesdays. Workshops are essential for learning the course material. Students will work in small groups on specially constructed problem sets. Most problems will deepen understanding of recently presented material. Some problems will connect recent material to earlier material in the course. Some problems will provide motivation for upcoming material.
The lecturer will circulate among the groups coaching, but not demonstrating solutions. The goal at first is to offer ideas for analyzing the problem. Later in the

Date Lecture Topic Lecture or Workshop # Homework due

9/1 Ch. 0 L1; W1

9/3 Ch. 0 L2 Ch. 0, Set 1

9/8 Runs on a Monday Schedule!

9/10 Ch. 0 L3 Ch. 0, Set 2

9/15 Ch. 0 L4; W2

9/17 Ch. 0 L5 Ch. 0, Set 3

9/22 Ch. 1 L6; W3

9/24 Ch. 1 L7 Ch. 1, Set 1

9/29 Ch. 1 L8;W4
10/1 Ch. 1 L9 Ch. 1, Set 2

10/6 Ch. 1 L10; W5

10/8 Ch. 1 L11 Ch. 1; Set 3

10/13 Ch. 1 L12; W6

10/15 Ch.0-1 L13 = Exam #1

10/20 Ch. 2 L14; W7

10/22 Ch 2 L15 Ch 2, Set 1

10/27 Ch 2 L16; W8

10/29 Ch.3 L17 Ch 2, Set 2

11/3 Ch.3

11/5 Ch. 3 L19 Ch. 3, Set 1

11/10 Ch.3 L20; W10

11/12 Ch. 3 L21 Ch. 3. Set2

11/17 Ch. 4 L22; W11

11/19 Ch. 4 L23 Review after lecture

11/24 Ch. 2-3 Exam #2

11/26 Thanksgiving Holiday

12/1 Handout L25; W13 Ch.4, Set 1

12/3 Handout L26

12/8 Handout L27; W14 Ch.4, Set 2
12/10 Handout L28

Homework on the material in the Handout is embedded in the Handout itself.

Date	Lecture Topic	Lecture or Workshop #	Homework due
9/1	Ch. 0	L1; W1
9/3	Ch. 0	L2	Ch. 0, Set 1
9/8	Runs on a Monday Schedule!
9/10	Ch. 0	L3	Ch. 0, Set 2
9/15	Ch. 0	L4; W2
9/17	Ch. 0	L5	Ch. 0, Set 3
9/22	Ch. 1	L6; W3
9/24	Ch. 1	L7	Ch. 1, Set 1
9/29	Ch. 1	L8;W4
10/1	Ch. 1	L9	Ch. 1, Set 2
10/6	Ch. 1	L10; W5
10/8	Ch. 1	L11	Ch. 1; Set 3
10/13	Ch. 1	L12; W6
10/15	Ch.0-1	L13 = Exam #1
10/20	Ch. 2	L14; W7
10/22	Ch 2	L15	Ch 2, Set 1
10/27	Ch 2	L16; W8
10/29	Ch.3	L17	Ch 2, Set 2
11/3	Ch.3
11/5	Ch. 3	L19	Ch. 3, Set 1
11/10	Ch.3	L20; W10
11/12	Ch. 3	L21	Ch. 3. Set2
11/17	Ch. 4	L22; W11
11/19	Ch. 4	L23	Review after lecture
11/24	Ch. 2-3	Exam #2
11/26	Thanksgiving Holiday
12/1	Handout	L25; W13	Ch.4, Set 1
12/3	Handout	L26
12/8	Handout	L27; W14	Ch.4, Set 2
12/10	Handout	L28

Textbook Homework Problems

TENTATIVE HOMEWORK SETS -- Be alert for modifications!

Due Date Section Problems to do Problems to turn in

Sept 3 Ch. 0, Set 1 2, 4, 6, 8, 2, 6,

Sept 10 Ch. 0, Set 2 10, 12b, 14, 16, 18, 20 10, 16, 20

Sept 17 Ch. 0, Set 3 22, 24, 26, 32, 33, 38, 41, 44 24, 32, 38, 41, 44

Sept 24 Ch 1, Set 1 1, 3, 4, 6, 7, 9, 1, 4, 6b,d; 9

Oct 1 Ch 1, Set 2 25, 27, 29, 32, 34, 37, 38, 39, 40 27, 32a,c,f; 34, 37, 39

Oct 8 Ch 1, Set 3 14, 16, 21, 24, 45, 46, 47 14, 21, 45, 46, 47

Oct 20 Ch 2, Set 1 1, 2, 4, 5, 7, 8, 12, 13 2, 5, 8, 13

Oct 29 Ch 2, Set 2 11, 14, 16, 17, 19, 21, 22, 25 11, 16, 19, 22, 25

Nov 5 Ch. 3, Set 1 1, 2, 5, 6, 7, 9 2, 6, 9

Nov 12 Ch. 3, Set 2 13, 14, 31, 42, 13, 14, 31, 42

Dec 1 Ch. 4, Set 1 1, 2, 4, 5, 7, 11, 13, 14 2, 5, 11, 13, 14

Dec 8 Ch. 4, Set 2 16, 17, 19, 20, 26, 28, 29 16, 17, 19, 28

Sample Solutions to Homework Problems

Chapter 0
Chapter 1
Chapter 2(ignore the 2008 date)
Chapter 3 (ignore the 2008 date)
Chapter 4 (ignore the 2008 date)