MAA 4211 Real Analysis and Advanced Calculus 1
University of Florida, Fall 2021
Architecture 215, MWF5 (11:45–12:35)
Instructor information:
Office: Little Hall 412
Office hours: Mondays and Tuesdays 10:00–11:00 and by appointment
Office phone: (352) 294-2338
Email: vatter@ufl.edu
Tentative schedule
(But the dates of assignments are unlikely to change.)
| Lec | Date | Topics |
|---|---|---|
| 1 | Wednesday 8/24 |
Introduction and √2 Section 1.1 |
| 2 | Friday 8/26 |
Sets, functions, and proofs Section 1.2 Suggested exercises: 3, 7, 8, 9, 10 |
| 3 | Monday 8/29 |
The axiom of completeness (suprema and infima) Section 1.3 Suggested exercises: 1, 3, 5, 6, 7, 8, 9 |
| 4 | Wednesday 8/31 |
Consequences of completeness Section 1.4 Homework 1 due before class |
| 5 | Friday 9/2 |
Consequences of completeness continued Section 1.4 Suggested exercises: 2, 4, 5 |
| Monday 9/5 | Labor day (no class) | |
| 6 | Wednesday 9/7 |
Cardinality Section 1.5 Suggested exercises: 4, 5, 6 |
| 7 | Friday 9/9 |
Sequences Section 2.2 Suggested exercises: 1, 2, 3, 6 Homework 2 due before class |
| 8 | Monday 9/12 |
Sequences Section 2.2 |
| Wednesday 9/14 | Exam 1 during class | |
| 9 | Friday 9/16 |
Limit theorems Section 2.3 Suggested exercises: 1, 3, 5, 7, 8, 10 |
| 10 | Monday 9/19 |
Limit theorems, continued Section 2.3 |
| 11 | Wednesday 9/21 |
Monotone sequences Section 2.4 Quiz 1 due on Canvas |
| 12 | Friday 9/23 |
Subsequences Section 2.5 Suggested exercises: 1, 2, 3 |
| 13 | Monday 9/26 |
Bolzano–Weierstrass Section 2.5 Suggested exercises: 5, 8, 9 |
| Wednesday 9/28 | Hurricane Ian (no class) | |
| Friday 9/30 | Hurricane Ian (no class) | |
| Monday 10/3 | No class | |
| 14 | Wednesday 10/5 |
Cauchy sequences Section 2.6 Suggested exercises: 2, 3, 4 Homework 3 due before class Quiz 2 due on Canvas before class |
| Friday 10/7 | Homecoming (no class) | |
| 15 | Monday 10/10 |
The Cantor set Section 3.1 |
| Wednesday 10/12 | Exam 2 during class | |
| 16 | Friday 10/14 |
Open and closed sets Section 3.2 Suggested exercises: 6, 13 |
| 17 | Monday 10/17 |
Open and closed sets, continued Section 3.2 Suggested exercises: 2, 3, 4, 5, 8 |
| 18 | Wednesday 10/19 |
Sequential compactness (but not open covers) Section 3.3 Suggested exercises: 1, 2, 3, 4, 5 Quiz 3 due on Canvas |
| 19 | Friday 10/21 |
Limits of functions Section 4.2 Suggested exercises: 1, 2, 5, 6, 7 |
| 20 | Monday 10/24 |
Continuous functions Section 4.3 Suggested exercises: 1, 3, 4, 5, 6, 11 |
| 21 | Wednesday 10/26 |
Continuous functions on compact sets Section 4.4 Suggested exercises: 1, 2, 3, 5, 9, 11 (9(a) will come up in our proof of the fundamental theorem of calculus, and 11 is important for the second semester of this course) Homework 4 due before class |
| 22 | Friday 10/28 |
The intermediate value theorem Section 4.5 Suggested exercises: 2, 3, 6, 7 |
| 23 | Monday 10/31 |
Derivatives Section 5.2 Suggested exercises: 1, 2, 3, 4 |
| Wednesday 11/2 | No class | |
| 24 | Friday 11/4 | Exam 3 during class |
| 25 | Monday 11/7 |
Derivatives continued and the chain rule Section 5.2 Suggested exercises: 1, 2, 3, 4 |
| 26 | Wednesday 11/9 |
The mean value theorems (but not L’Hôpital’s rule) Section 5.3 Suggested exercises: 1, 2, 3, 4, 7 Quiz 4 due on Canvas |
| Friday 11/11 | Veterans Day (no class) | |
| 27 | Monday 11/14 |
A continuous nowhere-differentiable function Section 5.4 Suggested exercises: 5, 6 |
| 28 | Wednesday 11/16 |
Introduction to integration Section 7.1 |
| 29 | Friday 11/18 |
The Riemann integral Section 7.2 Suggested exercises: 1, 2 Homework 5 due before class |
| 30 | Monday 11/21 |
Uniform continuity and the integrability of continuous functions Section 4.4 & 7.2 Suggested exercises: 3, 4, 7 (from Section 7.2) |
| Wednesday 11/23 | Thanksgiving break (no class) | |
| Friday 11/25 | Thanksgiving break (no class) | |
| 31 | Monday 11/28 |
Integrating functions with discontinuities Section 7.3 Suggested exercises: 1, 3, 4, 7 |
| Wednesday 11/30 | Exam 4 during class | |
| 32 | Friday 12/2 |
Properties of the integral Section 7.4 Suggested exercises: 1, 2, 3 |
| 33 | Monday 12/5 |
The fundamental theorem of calculus Section 7.5 Suggested exercises: 1, 4, 7 |
| 34 | Wednesday 12/7 |
Thomae’s function and integrability Sections 7.6 and 4.1 Quiz 5 due on Canvas |
Text
Abbott’s Understanding Analysis, 2nd edition.
Course content, goals, and objectives
Advanced treatment of limits, differentiation, integration and series.
In addition to mastery of the course content, course objectives include reading, writing, and discovering proofs and constructing proofs and counterexamples in analysis.
Expectations and grading rubric
Work submitted for a grade in this course will be graded in a rigorous fashion and should be prepared with a good deal of thought and care.
Most of the work required in this course will consist of writing proofs. For a problem worth 10 points, scores will be based on the following guidelines.
0 points.
The work contains no original steps toward a correct solution. This includes work that simply consists of relevant definitions or theorems without interpretation.
3 points.
The work contains some original steps toward a correct solution but does not contain a workable outline of the full solution. This grade is also used if the student has misunderstood the question or made an unwarranted simplifying assumption that makes the problem trivial.
6 points.
The work contains an outline of a correct solution and several steps toward this solution. However, the writing may be unclear, or there may be holes in the argument.
8 points.
The work resembles a full, complete proof, but it has some deficiencies. These may include incomplete sentences, abbreviating words with logical symbols such as those for “for all” or “implies”, imprecise definitions, or overlooking trivial cases.
10 points.
The work consists of a full, complete proof and is reasonably well written in complete sentences, without logical symbols. There may be minor typos or clumsy writing that could be improved, but no important steps of the solution are omitted or incorrect.
Evaluation
There will be 4 non-comprehensive exams, 5 homework assignments, and 5 quizzes throughout the semester, as indicated in the schedule.
The exams will collectively count for 50% of your grade, the homework assignments for 30%, and the quizzes for 20%.
Final letter grades will be assigned on a curve, which will be no tougher than the 10-point scale: 90%–100% will be some form of A, 80–90% will be at least some form of B, etc.
Exams will be in-person in the classroom. Homework may be submitted either online or physically, so long as it is before the start of class on the due-date (handwritten or typed are both fine). Quizzes will be untimed, multiple choice, and on Canvas. Exams will be closed-book and without notes. For quizzes and homework assignments, you are encouraged to consult your notes and textbook.
Late assignments will only be accepted by prior agreement or in the case of an excused absence.
If you have a disagreement with the grading of one of your solutions, I ask that you submit a written request for reconsideration within one month.
The UF grading policy for assigning grade points may be found at
https://catalog.ufl.edu/UGRD/academic-regulations/grades-grading-policies/
Honor code
UF students are bound by The Honor Pledge which states, “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code. On all work submitted for credit by students at the University of Florida, the following pledge is either required or implied: “On my honor, I have neither given nor received unauthorized aid in doing this assignment.” The Honor Code specifies a number of behaviors that are in violation of this code and the possible sanctions. Furthermore, you are obligated to report any condition that facilitates academic misconduct to appropriate personnel. If you have any questions or concerns, please consult with the instructor.
Class attendance
Attendance is strongly encouraged but is not a component of grades. Course materials will be provided to you with an excused absence, and you will be given a reasonable amount of time to make up work.
Requirements for class attendance and make-up exams, assignments, and other work in this course are consistent with university policies that can be found at https://catalog.ufl.edu/ugrad/current/regulations/info/attendance.aspx.
Accommodations for students with disabilities
Students with disabilities who experience learning barriers and would like to request academic accommodations should connect with the disability Resource Center (352-392-8565, https://disability.ufl.edu/). It is important for students to share their accommodation letter with their instructor and discuss their access needs, as early as possible in the semester.
Online evaluations
Students are expected to provide professional and respectful feedback on the quality of instruction in this course by completing course evaluations online via GatorEvals. Guidance on how to give feedback in a professional and respectful manner is available at https://gatorevals.aa.ufl.edu/students/. Students will be notified when the evaluation period opens, and can complete evaluations through the email they receive from GatorEvals, in their Canvas course menu under GatorEvals, or via https://ufl.bluera.com/ufl/. Summaries of course evaluation results are available to students at https://gatorevals.aa.ufl.edu/public-results/.
U matter, we care
Your well-being is important to the University of Florida. The U Matter, We Care initiative is committed to creating a culture of care on our campus by encouraging members of our community to look out for one another and to reach out for help if a member of our community is in need. If you or a friend is in distress, please contact umatter@ufl.edu so that the U Matter, We Care Team can reach out to the student in distress. A nighttime and weekend crisis counselor is available by phone at 352-392-1575. The U Matter, We Care Team can help connect students to the many other helping resources available including, but not limited to, Victim Advocates, Housing staff, and the Counseling and Wellness Center. Please remember that asking for help is a sign of strength. In case of emergency, call 9-1-1.
In-class recording
Students are allowed to record video or audio of class lectures. However, the purposes for which these recordings may be used are strictly controlled. The only allowable purposes are (1) for personal educational use, (2) in connection with a complaint to the university, or (3) as evidence in, or in preparation for, a criminal or civil proceeding. All other purposes are prohibited. Specifically, students may not publish recorded lectures without the written consent of the instructor.
A “class lecture” is an educational presentation intended to inform or teach enrolled students about a particular subject, including any instructor-led discussions that form part of the presentation, and delivered by any instructor hired or appointed by the University, or by a guest instructor, as part of a University of Florida course. A class lecture does not include lab sessions, student presentations, clinical presentations such as patient history, academic exercises involving solely student participation, assessments (quizzes, tests, exams), field trips, private conversations between students in the class or between a student and the faculty or lecturer during a class session.
Publication without permission of the instructor is prohibited. To “publish” means to share, transmit, circulate, distribute, or provide access to a recording, regardless of format or medium, to another person (or persons), including but not limited to another student within the same class section. Additionally, a recording, or transcript of a recording, is considered published if it is posted on or uploaded to, in whole or in part, any media platform, including but not limited to social media, book, magazine, newspaper, leaflet, or third party note/tutoring services. A student who publishes a recording without written consent may be subject to a civil cause of action instituted by a person injured by the publication and/or discipline under UF Regulation 4.040 Student Honor Code and Student Conduct Code.