Analysis and Optimization (MATH V2500, Spring 2007)

Day & Time: Monday and Wednesday 1:10pm — 2:25pm Location: 207 Math Instructor: Joël Bellaïche Office Hours: Tuesdays, 9-11am, in 525 Mathematics Teaching assistants : Yao Liu, Peter Insley (undergraduate TAs), Tung To, Chenxu Li (graduate TAs) Recitations: Peter Insley will have recitations twice a week, to help you with the homework (due or not) and to answer questions. There will be one recitation every Mondays on 5-6PM in room 621 Math, and one evry Thursday 5-6PM in room 622 Math. Textbook: Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, Further Mathematics for Economic Analysis Prentice Hall (Financial Times) . ISBN 02736555760. It is available at the University bookstore Additional student support is available on Sydsaeter's web site Examinations: There will be one in class test. It will be on Wednesday, March 7 . The final exam is projected on Monday, May 07, 1:10pm-4pm Program: The course is a continuation of the calculus sequence developping the various analytic tools needed to solve optimizations's problems. Although the course will be mainly problems-solving oriented, a more important part than in the calculus course will be given to proofs, both in class and in the exercises or tests. A strong emphasis will be given to the applications to economics, with many examples developped in class or to be read in the book. The modelization of an economic question into a well-posed mathematical problem will sometimes be part of the exercises. Three mains topics will be studied in this class: (1) optimization with constraints (chapters 2 and 3 of the book), (2) calculus of variations and control theory (chapters 8, 9 and 10), (3) point-fix theorems and the theory of equilibrium (chapters 13 and 14). The theory of differential equations (chapters 5 to 7) will not be covered in this class. For that material, see the course Ordinary Differential Equations (Math E1210) Prerequisites: Calculus I, II, III, and Linear Algebra. (or the equivalent). The course will go at a fast rate and masterizing the prerequisites is a necessary condition of success. More precisely : The material of calculus I is basic and is to be known perfectly. The notions of local and global minima and maxima, and of critical points, and the methods to find them, will be the starting point for the first part of this course. From Calculus II, we will mainly need the "integration by parts" technics and the notion of improper integrals (chapter 7 of James Stewart's Calculus book) From Calculus III, it is extremely important that you know well the "calculus for functions of two or three variables" part (chapter 14 of Stewart's book). In Linear algebra, the content of chapter 1 (up to section 1.7) of our book (Sydsaeter, Hammond, Seierstad and Strom) should be known. A review of that material (including exercises) before the class begins could be a good investment.

Policies & Procedures

Goals: At the end of the course, students should be able to

• make calculations with agility, accuracy, intelligence and flexibility;
• explain the basic concepts of the course clearly and reason logically with them;
• solve extended problems, with good judgment in the choice of tools and in checking answers.

Expectations: To achieve these goals, students are advised to

• read each section of the textbook before or soon after the material is presented in class;
• attend the lectures;
• complete all homework assignments;
• discuss mathematics with other students.

Assessment: The course grades will be computed as follows:

20% Homework

30% Midterm test

50% Final exam

Homework: There will be two kinds of homework:

• Exercises give you a chance to practice and refine your basic calculus skills. They will not be collected.
• Written Problems sets consist of approximately 3 questions. They are not intended to be harder than the exercises although some may well be. Instead, they may ask you to apply a technique in a mildly novel context or combine concepts that you have only seen in isolation before. Problems sets are posted in PDF format; your browser can be trained to automatically open these files with the free program Acrobat Reader. There will be one problems set each week, for a total of 12 problems set. Your best ten problems sets grades will determine your homework grade Problems sets will be collected in class and late homework will not be accepted. Alternatively, you can leave your problems sets in my mailbox (in the math building, between second and third floor) before calss time on the due date. Graded homework can be found on a box on a table near the office 521. The grades will also be available on courseworks.

Written work: We write to communicate. Please bear this in mind as you complete assignments and take exams. You must explain your work in order to obtain full credit; an assertion is not an answer. For specific suggestions see A guide to writing in mathematics classes.

Academic honesty: It is the obligation of each student to understand the University's policies regarding academic honesty and to uphold these standards. Students are encouraged to talk about the problems, but should write up the solutions individually. Students should acknowledge the assistance of any book, software, student or professor.

Help: Help is available if you have trouble with homework or lecture material. My office hours are a good place to start. You may also take advantage of the Mathematics Help Room (333 Milbank Hall, on the Barnard campus). You may drop by whenever the Help Room is open; no appointment is necessary.

Calculators: Calculators — in particular graphing calculators — are not required for this course. If you have one, you are welcome to use it when you do your homework. However, calculators will not be allowed during any tests or exams.

Disabilities: Students with disabilities who will be taking this course and may need disability-related classroom accommodations are encouraged to make an appointment to see the instructor as soon as possible. Also, stop by the Office of Disability Services (BC, CC) to register for support services.

Schedule of Lectures