MA 182, Fall 2005
HONORS MATHEMATICAL ANALYSIS II

General information

 Description: This is a beginner's course in mathematical analysis. The topics covered will be approximately those of a standard Calculus I course, plus a few typically from Calculus II. The style and format, though, will be rather different from the standard Calculus sequence. A higher level of mathematical rigor, and a deeper understanding of the few fundamental concepts involved, will be preferred to a more extensive syllabus. (In other words, precise definitions and proofs will be our main tools and we won't care if we don't cover yet another calculation technique--students in this class will be more than able to study that by themselves if and when they need it.)
The course is intended for math majors and students with a serious mathematical interest. Admittance is by invitation only (based on a high AP Calculus score or similar qualifications). This does not mean, however, that all the students invited or pre-enrolled should necessarily take the class. Although the Department of Mathematical Sciences hopes for a large enrollment, the student who does not like this type of class is advised to enroll in MA 115 or MA 116. (Note: A high AP Calculus score allows one to enroll in MA 116 without the MA 115 prerequisite.)

Time & Place: Monday, Tuesday, Wednesday 9:00-9:50, Pierce 120.

Lecturer: Marco Lenci. Office: Kidde 219B. Tel.: x-5453. E-mail: mlenci@math.stevens.edu.

Grader and Teaching Assistant: Ilona Murynets. Office: Kidde 105. Tel.: x-5431. E-mail: imurynet@stevens.edu.

Office hours: Prof. Lenci's official office hours are posted here (but students are welcome to visit at other times as well, by appointment). Ms. Murynets's office hours are Monday, Thursday 5:00-6:00.

Textbook: T. M. Apostol, Calculus, Volume I, 2nd ed., Wiley.

Grades and policy

Attendance: Attendance is very strongly recommended but not enforced. It is every teacher's experience that students who attend the lectures tend to do significantly better in a course than student who don't. However, students in this course will be considered responsible for managing their own time. (Besides, enrolling in an honors class and not attending seems contradictory.) As for missing quizzes or exams, see the policy on make-ups. (See also the last remark in the Final grade section.)

Homework: The syllabus contains a list of suggested homework problems from the textbook, for every week. The student is strongly adviced to work out as many as they can from the list. However, no homework is due or counts for the final grade. Nonetheless, students who want their homework graded and commented on can turn it in or, even better, discuss it with Prof. Lenci or Ms. Murynets during their office hours.

Quizzes: There will be a 10-minute quiz (almost) every week, on topics up to the previous week. The quiz will take place at the beginning of class on Tuesday (or Wednesday, if Tuesday is the first school day of the week; see syllabus). Students can miss up to 2 quizzes with no justification required (the worst 2 quiz grades will be discarded in the final average). Beyond that, see the policy on make-ups.

Exams: There will be three 55-minutes midterm exams and a 4-hour final exam. Dates will be announced during the semester. The midterm exams will take place on Wednesday during class time.

Make-ups: Make-up quizzes and exams are allowed only in exceptional cases, which must be motivated (possibly with the proper documentation) and communicated to Prof. Lenci as soon as possible. Failure to address any of these points will result in the request being denied and a grade of 0 being given for the corresponding test.

Final grade: The final average will be computed as follows

   25%   Quizzes (worst two grades discarded)
   40%   Midterms (worst grade discarded)
   35%   Final

and then converted to a letter grade using the following scale

Grade F D C- C C+ B- B B+ A- A
% Avg 0-50 50-60 60-64 64-70 70-74 74-78 78-82 82-86 86-90 90-100

In exceptional cases, the resulting final grade may be raised based on special considerations, such as a marked improvement of the student during the semester, a significant health problem or similar distressing situation affecting an exam, etc. In these cases, the diligence displayed by the student during the semester (including attendance and class participation) makes it easier for Prof. Lenci to be generous.

Syllabus

Note: The topic and homework lists are not considered definitive until after Wednesday of the current week. (Data in italics is merely the plan, and are subject to change.)

Week Topics Book Sections Homework Notes
8/29-31 Primitive concepts (sets, elements, functions, operations).
First 9 axioms of the reals. 		I 2.1 - I 3.4 I 2.5 #6,9
I 3.3 #5-10 		 
9/6-7 First 9 axioms (cont'd).
Min, max, inf and sup. 		I 3.4, I 3.8 I 3.5 #3-7,9 No class on 9/5 (Labor Day)
Quiz on Wed 9/7
9/12-14 The completeness axiom. Existence of roots.
Inductive sets. Naturals, integers, rationals.
Mathematical induction. 		I 3.9-11, I 3.13
I 3.6
I 4.1-2 		I 3.12 #1,2,6-9,12
I 4.4 #1-3,8,10,12 		 
9/19-21 More examples of proofs by induction.
Sigma notation. Absolute values. 		I 4.6-8 I 4.7 #1-4,11
I 4.9 #1,2
Lenci out
Substitute in
9/26-28 Limits of sequences (finite and infinite). 10.2 HW Sheet #1
10.4 #1,2,6,7,8,10,23,24,30 		 
10/3-5 Monotonic sequences.
Review. 		10.3 HW Sheet #1 Midterm #1 on Wed 10/5
10/11-12 Bolzano-Weierstrass Theorem.
Limits of functions. 		3.1-2 HW Sheet #2 No class on 10/10 (Columbus Day)
Quiz on Wed 10/12
10/17-19 Continuity. Composite functions.
Trigonometric functions (unrigorous). 		3.3-5, 3.7
2.5, 2.7
3.6 #1-5,11-14,21,22,27
3.8 #2-8,11-13,15-17,19 		 
10/24-26 Continuity and neighborhoods.
Exponential (unrigorous). Continuity of n-th root.
The Intermediate Value Theorem.
Limits involving infinity. 		3.9-10
7.14-15 (parts) 		3.11 #1,4,5
7.17 #10,12,13,28
HW Sheet #3 		 
10/31 - 11/2 Extrema of a continuous function.
Review. 		3.16   Midterm #2 on Wed 11/2
11/7-9 Derivatives and basic theorems.
Geometric interpretation.
Chain rule. 		4.1-5
4.7-8
4.10 		4.6 #2,4-10,14-20,30-34,36,38
4.9 #2,4,7,9,13
4.12 #1-7,14,17 		 
11/14-16 Inverse functions and their derivatives.
Logarithms.
Mean Value Theorem. First derivative test. 		3.12-13, 6.20
4.13-14, 4.16 		HW Sheet #4
3.15 #1-5
4.15 #2,8
4.19 #1-3,7-8,11-12 (only (a),(b)) 		 
11/21-22 L'Hopital's Rule (no proof).
Applications to exponentials, powers and logs.
Step functions. 		7.12, 7.14, 7.16
1.8-10 		7.13 #2,4-7,11,13,17
7.17 #1,3-6,11-13,18,19,21-22
1.11 #1,8 		No class on 11/23
(Thanksgiving recess)
11/28-30 Integrals of step functions.
Integrals of general functions.
Integrability of monotonic functions.
Basic properties of integrals.
Review. 		1.12-14
1.16-17
1.20-21
1.24 		1.15 #1,2,5,15
Understand Thm 1.12 		Midterm #3 on Thu 12/1
12/5-7 Proof of basic properties of integrals.
Primitives.
Fundamental Theorems of Calculus.
Substitution. 		1.27
5.1-3
5.6-7 		5.5 #1-11,14-15,20(a,b),22(a,b)
5.8 #1-18
HW Sheet #5 		 

Final Exam: Wed Dec 14, 1--5pm, E-230.   (NOTE: Change is not official, registrar will keep displaying old date.).

Communications to the class

Dec 16: Grades posted. Have a good break!
Dec 9: Class agreed to new date for the final: Wed Dec 14, 1--5pm, E-230.
Dec 6: Review session on Mon Dec 12, 2-4pm, in M-203.
Oct 31: Midterm #3 on Thu Dec 1, 2:00pm, in B-124.
Oct 19: Midterm #2 on Wed Nov 2 in class.
Sep 29: Date & location of final exam announced: Dec 17, 8am--12am, in E-329.
Sep 13: Midterm #1 on Wed Oct 5 in class.
Sep 13: Prof. Lenci will be substituted next week (Sep 19-21). Wed Sep 21's lecture might be canceled if enough material is covered on Monday and Tuesday.
Sep 7: Syllabus changed. Check the new homework list.

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