**PLEASE NOTE:** All information in this syllabus is subject to
change in response to the resurging Covid epidemic. Changes will be
noted right here at the top of the syllabus, and will be announced in class.

The first change is already in place: we must me remotely through Jan 28. Access the class via Zoom: https://utexas.zoom.us/j/95227936925 but then starting Jan 31 (I hope) we will be in the classroom exclusively.

UPDATE: we are in fact meeting exclusively in RLP1.108 starting Jan 31 2022. If for some reason we need to return to Zoom sessions, I will use the meeting number above.

Unique ID: 54005 Instructor: Dave Rusin (rusin@math.utexas.edu) Office hrs: (1) Tuesdays noon-2pm and Wednesdays 4pm-5pm Feel free to drop in during those hours to my office, PMA 9.140. (2) I am also an academic advisor and hold advising hours Tuesdays 10:30-noon and Wednesdays 10-11:45; if you show up in my office I would be happy to talk to you if I am not in an advising session. (3) I can also arrange additional times to meet; send me email if you'd like to meet with me. Class meets MWF at 1pm in RLP1.108 Text: Introduction to Real Analysis, 4th Ed., by Bartle and Sherbert I don't want to use our class time to hammer out every detail of every proof: that's important but that's what books are for. So I will be trying to give the broader picture of why things work the way they do and why we care; please read the book to get the details afterwards. The book I have listed is a standard choice but you could just as easily get the details from another book, or another edition; just *read something*! This particular book has the advantage that there are copies that are shall we say "readily available". Teaching assistant: TBA Please note the (UPDATED! CORRECT!) time of your final exam: Saturday, May 14, 7:00 pm-10:00 pm There is no provision for taking the final exam earlier or later.

Course webpage: http://www.ma.utexas.edu/~rusin/361K/ There is also a Canvas page for this course.

(Odds and ends may be added here during the semester.)

- Real number system: order and completeness properties.
- Sequences and series: limit theorems, subsequences and accumulation points, monotone sequences, inferior and superior limits, Bolzano-Weierstrass theorem.
- Limits and continuity of functions: definition and elementary properties of limits of functions, continuity, Intermediate Value Theorem, Maximum-Minimum Theorem, continuity of inverse functions, uniform continuity on closed intervals.
- Differentiation: definition and geometric significance of the derivative, differentiation rules, Mean Value Theorem and its consequences, Taylor's Theorem, L'Hôpital's Rules, convexity.
- Riemann integration: definition and elementary properties of the Riemann integral, integrability of continuous and monotone functions, Fundamental Theorems of Calculus.

Either consent of mathematics advisor, or two of M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C-. May not be counted by students with credit for M365K with a grade of C or better.

Your semester grade will be based on a number of components. This structure is designed to encourage you to stay actively involved in the course all the way through the semester. Any adjustments to the schedules or policies will be announced multiple times in lecture and via email and on the course website shown above.

** Homework**: There will be homework due weekly.
I will drop the lowest homework grade and average the rest
to give you a "Homework Score" of up to 100 points for the semester.

**Exams**: There will be 2 mid-term exams, to be held
during the usual class period, and a comprehensive final exam.

Because my exams tend to be hard, I have a way to curve the exam scores. I will compute both your raw score and your curved score and whichever is higher for you will be the score I use when computing semester grades. The curving method is simple: I compute the mean, mu, and the standard deviation, sigma, of the class's raw scores, and then a person with a raw score of X will get a curved score of

85 + 10 (X-mu) / sigma(with a maximum curved score of 105). This way the mean curved score will be 85 and the standard deviation of the curved scores will be 10. In effect, a person with the average grade gets a "B", and the person whose exam grade is higher or lower by one standard deviation gets an "A" or a "C".

Textbooks, notes, and electronic devices (including phones and calculators) are not permitted during exams. The penalty for violating these conditions is a score of ZERO on the exam.

I expect the dates of the midterms to be February 25 and April 8 but this is subject to change. The last day of class is May 6. Please mark on your calendars now the time and date of the final exam.

**Semester grades**: Your numerical grade for the semester is a
weighted average of the components above, specifically it is the
dot product

< 0.30, 0.20, 0.20, 0.30 > . < HW, T1, T2, Final >This number is converted to a letter grade according to the following scale: https://xkcd.com/2329/ Just kidding. I use a pretty standard conversion formula:

97.0-100 | A+ |

93.0-96.9 | A |

90.0-92.9 | A- |

87.0-89.9 | B+ |

83.0-86.9 | B |

80.0-82.9 | B- |

77.0-79.9 | C+ |

73.0-76.9 | C |

70.0-72.9 | C- |

67.0-69.9 | D+ |

63.0-66.9 | D |

60.0-62.9 | D- |

0 -59.9 | F |

Class meetings: Our "lecture" meeting times together are very short so we must make the most of them. Attend daily, with all the materials you need to take notes and work problems. Per university policy, we will hold classes by Zoom for two weeks and then return to in-person classes

I usually can't follow the chat box during the lecture. You should feel free to interrupt me directly! I will usually see a raised-hand icon if you activate it, but I don't mind at all if you simply speak up. (Within reason, of course.)

Please note that when the class meets live, there will be NO opportunity
to attend class via Zoom. There will be NO recordings of the class. I wish
I could be more accommodating to students with special needs, but I sadly
I have found that many more students did *worse* in the class in recent
semesters because they stayed away from the classroom, and to prevent that
this semester I am offering no alternative to the in-person classes.
Obviously I will accommodate students who become ill or otherwise must be absent
during the semester, but if you know now that you must or prefer to have
an online class experience, you must sign up for a different section of M427L.

Attendance is not mandatory. But who are we kidding? We're going to work through the entire textbook in just 45 class meetings. At the pace of this class, any time you miss a class, you are at least a whole section behind! That's hard to make up. Plus, if you are (to take a common example) paying in-state tuition for 12 undergraduate credit-hours, the tuition alone costs you $50 for every one of those 45 class meetings. Are you really going to throw away $50 you have already paid, so that you can stay home and take a nap?

Covid safety: I will wear a mask and try to stay 6' away from people; I have gotten my shots including a booster. I request that you also do these things. Don't be a jerk: we ALL want this pandemic to end!

Make-ups:
It is in general not possible to make up missing quizzes or homework
assignments after the due date. If you believe you will have to miss
a graded event, please notify me *in advance*; I will try to arrange
for you to complete the work early.

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Religious holidays: If you are unable to participate in a required class activity (such as an exam) because it conflicts with your religious traditions, please notify me IN ADVANCE and I will make accommodations for you. Typically I will ask you to complete the required work before the religious observance begins.

Academic Integrity. Please read the message about Academic Integrity from the Dean of Students Office. I very much prefer to treat you as professionals whose honesty is beyond question. But this has proven difficult, especially with students taking exams remotely. I have reported students to the Dean for cheating and they have had to bear the consequences; I will do so again if I must. Don't be that person!

Campus safety: Please familiarize yourself with the Emergency Preparedness instructions provided by the university's Campus Safety and Security office. In the event of severe weather or a security threat, we will immediately suspend class and follow the instructions given. You may wish to sign up with the campus alert programs.

Counseling: Students often encounter non-academic difficulties during the semester, including stresses from family, health issues, and lifestyle choices. I am not trained to help you with these but do encourage you to take advantage of the Counselling and Mental Health Center, Student Services Bldg (SSB), 5th Floor, open M-F 8am-5pm. (512 471 3515, or www.cmhc.utexas.edu

Add dates: If you enroll within the first four class days of the semester, and have missed any graded material, I will adjust the weighting of your graded sections accordingly so that you are not penalized. No such accommodation is made for students who enroll on the 5th day or later. (Such students must enroll through the MPAA advising center in PMA, and ordinarily I do not admit students who ask to enroll then if they have missed any graded activities).

Drop dates: Jan 21 is the last day to drop without approval of the department chair; Feb 2 is the last day to drop the course for a possible refund; Apr 4 is the last day an undergraduate student may, with the dean's approval, withdraw from the University or drop a class except for urgent and substantiated, nonacademic reasons. For more information about deadlines for adding and dropping the course under different circumstances, please consult the Registrar's web page, http://registrar.utexas.edu/calendars/21-22/

Sharing of Course Materials is Prohibited: No materials used in this class, including, but not limited to, lecture hand-outs, videos, assessments (quizzes, exams, papers, projects, homework assignments), in-class materials, review sheets, and additional problem sets, may be shared online or with anyone outside of the class unless you have my explicit, written permission. Unauthorized sharing of materials promotes cheating. It is a violation of the University’s Student Honor Code and an act of academic dishonesty. I am well aware of the sites used for sharing materials, and any materials found online that are associated with you, or any suspected unauthorized sharing of materials, will be reported to Student Conduct and Academic Integrity in the Office of the Dean of Students. These reports can result in sanctions, including failure in the course.

We have a LOT to do this semester --- we'll work through the entire book together! So expect to fly through about three sections every week.

- Jan 19-21 : sections 1.1, 1.2
- Jan 24-28 : sections 1.3, 2.1, 2.2
- Jan31-2/4 : sections 2.3, 2.4, 2.5
- Feb 7-11 : sections 3.1, 3.2, 3.3
- Feb 14-18 : sections 3.4, 3.5, 3.6
- Feb 21-25 : sections 3.7; ** EXAM 1**
- Feb28-3/4 : sections 4.1, 4.2
- Mar 7-11 : sections 4.3, 5.1--5.3
- Mar 14-18 : *** SPRING BREAK ***
- Mar 21-25 : sections 5.4, 5.6, 6.1
- Mar 28-4/1: sections 6.1, 6.2, 6.3
- Apr 4- 8 : sections 6.4; ** EXAM 2 **
- Apr 11-15 : sections 7.1, 7.4, 7.2
- Apr 18-22 : sections 7.3, 8.1, 8.2
- Apr 25-29 : sections 8.3, 8.4
- May 2- 6 : further topics, review
- May 14: FINAL EXAM

You may have spent most of your mathematical life working on problems by yourself. This is a good thing; you become self-reliant. However, I strongly encourage you to work with one or two other students in this class on a regular basis. Challenge each other to solve the problems, to explain the concepts, and to ask each other for help. This is the way mathematics is done in the real world, and practicing this now can help you this semester and beyond.

Since you are adults, I leave it to you to monitor your level of understanding on your own, and to seek help when you need it. But please allow me to share my experience. Every student who starts this class has met the pre-requisites and has the expectation that he or she will succeed. Nonetheless, every semester, a nontrivial fraction of this group of bright, hard-working students ends up with a D or F, or withdraws. No one likes this outcome. Please be attentive to your progress on homeworks and quizzes and midterms. If you find you are always asking other people for help while studying; if you find that it takes you hours and hours to complete every homework set; if your quiz grades are low, or you score less than half the possible points on a midterm exam: in these cases, you CAN succeed, but ONLY if you change your patterns immediately. Optimism is a wonderful thing but it alone cannot bring the results you may want. Please see me early in the semester if you think you may have trouble during this course. I can try to help you with the material, or with your study habits, or else advise you to withdraw. Let's make this the first-ever 100% successful Math 427L class!

One more suggestion: have fun this semester! Some of us think math is so cool that we end up doing it for a living. I will try to convey to you some of what's kewl, and invite you to consider majoring (or minoring) in math, joining the math club, or simply taking more math classes. In my office I am always happy to talk about mathematics topics beyond what we discuss in class.