MAT 319-320 Introduction to Analysis, Spring 2016.

The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements. MAT 319 and MAT 320 are taught as one class for 6 weeks and then split. Here is a detailed syllabus.
  • References :
    Kenneth Ross, Elementary Analysis: The Theory of Calculus , 2nd edition.
    This is the required text; parts of the homework will be assigned from it. The book contains a lot more than will be covered in the course.

  • Exams: there will be two midterm exams and a final exam. The final exam will be held on Wednesday, May 11, 5:30-8:00 pm, in Library W4525 (the usual MAT 319 classroom).
    The final exam covers all material of the semester. A checklist of exam topics can be obtained by combining the midterm checklists (for the first and second exams) with the differentiation checklist for the material covered after Midterm II.
    Please pay special attention to all definitions and theorems; you will be asked to state several definitions and theorems and to reproduce the proofs on the exam.

    Midterm I was held on Tuesday, March 1, in class. The exam covered all material discussed in class until then (including the lecture on 2/15 and HW 5). A checklist of exam topics (along with some practice questions from textbook) is here. Here are solutions to Midterm I.

    Midterm II was held on Tuesday, April 19, in class. The exam covered all material discussed in class between midterm I and Midterm II (and also included series).
    A checklist of exam topics (along with some practice questions from textbook) is here.
    Solutions to midterm 2.

    Writing requirement: as part of math major requirements for graduation, every math major must submit 3 pieces of writing (2 expository and 1 proof). You can satisfy the *proof* part of the writing requirement in MAT 319. (Doing so will not affect your grade for MAT 319 and is not required for the completion of the course. The writing requirement can be completed through other 300-level courses, any time before you graduate. The proof writing is only required of math majors and not necessary for students in other majors/academic programs.)
    Here's how to satisfy the proof requirement: you need to bring me a proof, typically for a homework or exam question; you can take any question with a fairly substantial proof (not two lines please; you need a math argument, not a quick calculation). Choose a question you understand well (so that your proof is mathematically correct) and try to rewrite your proof nicely, in complete sentences with a lot of details. The goal is to produce and easily readable, textbook-worthy proof that could be understood by someone familiar with previous course material but not this particular question. After you submit the proof, I will read it and return it to you with my comments. Usually the writing is less than perfect at the first attempt, so you will have to rewrite your proof (sometimes more than once), following my comments. Once I am satified with the quality of writing, you get the proof credit. Please submit your own work, not posted solutions or excerpts from the book.

  • Schedule and homework:

    Week 12 (04/25 – 04/29) The mean value theorem and related results (section 29).
    Homework 12, due Wednesday, May 4: 29.2, 29.7 (read corollaries 29.4, 29.5 first), 29.8, 29.10, 29.13, 29.14, and one additional question:
    In class, we showed that if f attains maximum at x0, then f'(x0)=0, provided that f is differentiable at x0 and is defined on some interval containing x0. Prove the analogous statement for minimum, arguing in two ways: (a) Mimic the proof for maximum, (b) use a quick trick to relate min of f to max of some other function, as in previous homeworks.

    Week 11 (04/18 – 04/22) Differentiability (section 28).
    Homework 11, due Wednesday, Apr 27: pdf Solutions

    Week 10 (04/4 – 04/8) Continue continuity (section 18). Begin limits of functions (section 20).
    Homework 10, due Wednesday, Apr 13: pdf Solutions

    Week 9 (03/28 – 04/1) Important theorems about continuous functions on closed interval (section 18). Read p.133-136. (We did a different proof of the intermediate value theorem, see notes.)
    Homework 9, due Wednesday, Apr 6: pdf Homework 9 Solutions

    Week 8 (03/21 – 03/25) Continuous functions (section 17).
    Homework 8, due Wednesday, Mar 30: 17.4, 17.7 (please give a direct proof in (a)), 17.9, 17.10, 17.12. Homework 8 Solutions

    Week 7 (03/07 – 03/11) Integral test, alternating series test (section 15).
    Homework 7, due Wednesday, Mar 23: 15.2, 15.4, 15.6(c) (note that 15.6(a,b) were assigned in previous homework). Homework 7 Solutions

    Week 6 (02/29 – 03/4) Midterm on March 1. More on series: Ratio Test (see notes in Week 5).
    Homework 6, due Wednesday, Mar 9: pdf Homework 6 Solutions

    Week 5 (02/22 – 02/26) Subsequences (section 11, p.66-73 only). Series (sections 14-15). We will focus on series with positive terms only (this simplifies many of the proofs); here are some notes on series.
    We will follow these notes to take shortcuts for section 14. (The notes are from 2012; I might make some updates later on).
    Some notes from this week: More on limits; Proof of Thm 11.2(ii) (the book doesn't offer a lot of details).
    Homework 5, due Feb 29: pdf Homework 5 Solutions

    Week 4 (02/15 – 02/19) Monotone sequences and their convergence (section 10, p.56-60 only, skip Definition 10.6); subsequences (section 11, p.66-73 only).
    Homework 4, due Feb 22: 10.2, 10.5, 10.10, 11.4 (a,b,d,e only), 11.11.
    For 11.11, a solution is included at the end of the book, but that solution has a lot of unnecessary steps. You should be able to come up with a much easier and more direct proof. (Hint: just think what supremum means; see proof of Theorem 10.2 for inspiration. The question does not require any of the section 11 material.)
    Please also do this additional question: suppose that the sequence (an) converges to 5, the sequence (bn) converges to 2. Prove from definition that the sequence (cn), cn= 2 an - bn, converges to 8.
    Homework 4 Solutions
    Solutions for the quiz given on Monday

    Week 3 (02/08 – 02/12) Sequences and their limits: neighborhoods and tails, epsilon-definition. Arithmetics of limits (limit laws). Proof of limit laws, their applications.
    Supplementary notes on limits.
    Homework 3, due Feb 15: 8.2, 8.4, 8.6, 9.2, 9.4, 9.6 Solutions

    Week 2 (02/1 – 02/02) Sup and Inf, Completeness Axiom, idea of limits (sections 4, 5, 7).
    Homework 2, due Feb 8 4.2, 4.6, 4.10, 4.12, 4.14, 5.6, 7.2 Solutions

    Week 1 (01/25 – 01/29) Natural, rational, real numbers (sections 1, 2, 3).
    Homework 1, due Feb 1: 1.2, 1.4, 1.7, 1.10, 2.2, 2.4, 3.4, 3.6 Solutions
    Read them. Please also read sections 7 and 8 of the textbook.

  • Homework policy: Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed. Late homework will not be accepted.

  • Prerequisites: MAT 200 is a required prerequisite (together with completion of a one-variable calculus sequence).

    Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.