## Welcome!

This page is designed to give you an overview of the Advanced Calculus Sequence - Math 2141-2142-2143-2144. This two year sequence gives students interested in the "how and why" of calculus and it's related courses a chance to explore those questions.  While other math courses you've taken might emphasize tricks and recipes, this sequence will focus on seeing patterns and helps to provide a solid conceptual understanding of how math works instead of just gaining computation skills.

If you think you might be interested in this sequence, read on! If you'd like more information, our contact information is at the bottom of the page.

## Overview

The Mathematics Department has developed a mathematics sequence which may be of interest to a relatively small group of well-prepared and highly motivated students. Each course carries four credits and meets three times a week for 75 minutes. The sequence covers single-variable calculus, multi-variable calculus, linear algebra and differential equations from a more theoretical (as opposed to purely computation oriented) point of view.

### What material is covered?

In the first year of the sequence (Math 2141-2142), students are introduced to mathematical proofs and we balance the traditional approach of "how" questions (such as, how to find a particular integral or solve an optimization problem) with "what" and "why" questions (such as, what is an integral, what is a limit and why is it the foundation of calculus, and why are certain classes of functions integrable). The material covered in this year includes the standard single variable topics (roughly the equivalent of Math 1131-1132) as well as some of basic notions used in multi-variable calculus and the introduction to topics in linear algebra. The introduction to the main techniques and ideas of mathematical reasoning and proofs is roughly equivalent to Math 2710, Transition to Advanced Mathematics. The second year of this sequence (Math 2143-2144) covers material from multi-variable calculus, linear algebra and differential equations, again from a theoretical point of view although with an increased level of computation (as these topics are not included in either the AB or BC Calculus curricula).

Taking the entire Advanced Calculus sequence satisfies the prerequisites for any course that requires Math 1131Q (Calc 1), 1132Q (Calc 2), 2110Q(Calc 3), 2210Q (Linear Algebra), 2410Q(Differential Equations) and 2710 (Transitions to Advanced Mathematics).  The table below gives the equivalence (in terms of requirements) for each of the courses.

 Completion of: Satisfies the same the requirements as taking: 2141Q 1131Q 2142Q 1132Q and 2710 2143Q 2110Q 2144Q 2210Q and 2410Q

### How does 2141Q/2142Q differ from Math 1131-1132 or AP Calculus AB or BC?

This course will feel very different from math classes you've seen before. Everything is presented from a theoretical point of view. The theoretical point of view is essential to advanced study in mathematics and is a fundamental need for people planning careers involving theoretical science and engineering. Indeed, the emphasis on theory is accompanied by practice in how to "learn from one's mistakes." For example, since most of the continuous functions one initially considers (such as polynomials and trig functions) are differential, it is natural to conjecture that all continuous functions are differentiable. This conjecture turns out to be false, and discovering counter-examples leads to a deeper understanding of the relationship between continuity, differentiability and integrability.

For the material covered by the AP Calculus AB exam, very little routine drill will be included. The order of the material covered will also be substantively different from a standard calculus sequence such as our 1131-1132 sequence. Starting with methods of integration (and other topics included in the Calculus BC syllabus), drill will be included but will emphasize thinking rather than memorization. Students will be assumed to have learned most standard single variable calculus computations (such as computing basic derivatives and a few integrals) in high school or college courses; for that reason, students taking Math 2141 retain their AP calculus or ECE calculus credit. Although enrollment is not restricted to Honors students, students who are in the Honors Program automatically receive Honors credit for the courses in this sequence.

### How does it compare to the Honors sections of 1131/32 or 1151/51?

Math 1151/1152 (Honors Calc 1 and 2) are not longer offered at UConn.  Instead, you can take honors discussion sections for 1131 and 1132.  This means you'll still have the same lectures and cover the same material as the regular 1131/32 classes but have extra assignments in your discussion sections, often in the form of worksheets that dive a little deeper into the material. On the other hand, Advanced Calculus is a completely different course designed to present math from a much more rigorous perspective.

### What are some of the advantages for students taking the Advanced Calculus sequence?

There are a number of advantages for students taking the Advanced Calculus sequence. Completion of the four semester sequence with grades of C or higher satisfies the requirements for a minor in mathematics. (Typically a minor requires five mathematics courses after the year long single variable calculus sequence.) In addition, completing the sequence fulfills the prerequisites for the upper level mathematics courses which typically require a year of single variable calculus plus the four semester sequence Math 2110-2210-2410-2710. Therefore, students taking Advanced Calculus can move more quickly to (and are more prepared for) the upper level undergraduate, even some graduate, mathematics courses.

In addition, many students enjoy the small class environment.  You will be surrounded by a small group of like-minded students. Collaboration is stressed and many students quickly form working groups. You'll also have a great chance to get to know your instructor since the class often has around 20 students compared to 300+ in the 1131/1132/2110 lectures.

### What book is used?

The text for the first two semesters (Math 2141-2142) is the second edition of Apostol's Calculus -- Volume 1. Information about the text, hence also about the course, is available using the links below.

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051

https://www.amazon.com/Calculus-Vol-1-Apostol/dp/8126515198

For a book essentially written over fifty years ago, it is certainly overpriced; however it is the standard text nationwide, if not worldwide, for the highest level introductory math courses.

A description of the Advanced Calculus Sequence (written by Professor Bill Abikoff who started this program) can be found at

and the links on that page.

### What are past students saying?

I asked students who have started taking the sequence in both August 2019 (who have completed 2141 and 2142) and August 2018 (who completed all four courses) if there were things they wished they knew coming into the sequence and things they thought incoming students should know about the sequence. Here is what they said:

• Coming into this course I knew it was going to be proof-based math, but I wasn't really sure what it meant. I don't think it's common to work with the proofs that we have in this sequence prior to college. Most if not all of the proofs I learned from middle to high school were algebraic proofs, or theorems that you just need to memorize for the test. In this sequence, we focus more on logic and will require a lot more thinking. I would say that the advanced calculus sequence is an entirely different approach to mathematics, which can be overwhelming in the first few weeks but mind-opening in the long run. It is hard and demanding, and I would be lying if I say that I never considered dropping the class, however I am glad that I stuck with it. For the students who are used to excelling in math, there will be times where you feel like you're the dumbest person in the class, or that you're not "smart enough", but I promise you're not alone and you can do well in this sequence if you're willing to put in the time and effort. It's important to not only attend lectures and do the homework, but also to seek help from your professor and/or classmates sooner rather than later. Unlike math courses that focus on calculations, collaboration is key to exchange ideas. I learned a lot from this sequence and it has made me a more curious and inquisitive person overall. I would recommend anyone who is interested in math, theory and logic to give it a try. As a bonus, you will also gain the ability to understand top tier math memes
• As for what I wished I knew, I wished I had a better understanding of what the course entailed. Not just 'an in depth look at calculus' but more of how the course is a mix of intro to proofs, calc 3/4 etc. etc. Also highlighting the workload and commitment, either to get the minor or major would've been nice. It wasn't especially heavy, but it was a substantial commitment and time out of my week/weekend for two years. Even talking about how just one or two semesters is enough to obtain (relatively) intermediate analytical skills would be nice.
• A few things I think incoming students might want to know/things I learned in 2141:
• Stuff might feel really hard at first and you might spend hours thinking about problems, but that's totally normal and you will eventually find yourself really enjoying that process
• Make friends! You might be used to doing math by yourself from high school, but with long problem sets or proofs it's nice to be able to discuss stuff with other people. Plus you're in class with the same people for two whole years!
• Go to office hours--even if you think you might be able to figure something out yourself if you just stare at it long enough, it's extremely helpful to see how other people think and also to practice explaining math out loud while working through something with others that you might not fully understand yet. It's also a fun way to get to know other people in your class and the professor.
• One thing I wished I knew coming in would be how time consuming the work is and that the best way to ensure you do well is to make use of office hours and work together with other students. I think it's important for incoming students to understand that the class is going to be hard, especially since a lot of students that take advanced calc probably never struggled with math before (like myself at least). It might be a little bit of a wake up call to realize that this material is going to be difficult at first. But overall I think the little community and valuable study skills you obtain from these classes are worth it.
• When I was in high school, I was taught calculus but I always found myself wondering why what we are taught works, rather than just knowing how to use theorems. These courses have completely filled those gaps, and now I know even more than what I originally wanted to know, and plan to learn even more in the remaining two classes.This sequence is very, very difficult, but nowhere near impossible. Yes, you will have a few nights where you believe you cannot solve a problem that appears too complex, but if you try to tackle the problem from several different angles (not all at once), you will eventually find the answer, or at least find the path to the answer. If the answer does not come to you soon, you can seek help. Your classmates are more than willing to help you, as they have/are experiencing the same problems as you. I see the Advanced Calculus sequence as a community. Everyone struggles, but we help each other when we need it, and there are never any dumb questions.I feel that these courses greatly help in developing necessary skills for college. From the first two courses alone, I've gotten much better at managing my time, tackling problems from different angles, asking for help, and communicating complex issues.
• Coming into the Advanced Calc sequence, I wish that I had known that it was going to be a proof-based math course.  Maybe it was mentioned during Honors orientation, but so much information was discussed during that day that it probably flew over my head.  Additionally, I wish that I knew what “proof-based” or what “proofs” meant.  I struggled during the first half of Advanced Calculus 1 because I kept showing by example instead of proving.The way I try to explain proofs to others is asking them to recall two-column proofs from Geometry (Vertical Angle Theorem, Interior Angle Theorem, Congruency, all that jazz).  Now, instead of doing it in two-columns, imagine explaining your thought process in a paragraph.  That paragraph right there is a proof.
• See this Letter to Future Advanced Calc Students written in June 2019.
• See this longer response: Advanced Calculus from July 2020.

### What does it mean to learn proof based math? Do you have some sample lectures?

To give you an idea of what the course covers, there are two sample lectures provided. As lectures are generally given "live," even in the online learning setting, you can expect more interactivity in the actual lectures. These samples are meant to give you an idea of what it means to do proof based mathematics.

### Lecture 1

This is what Katie Hall's Fall 2020 class will start with. It is similar to where we started in Fall 2018 and 2019. The lecture below covers an introduction to sets and notation. Once you've watched it, you should try to worksheet posted after it.

https://youtu.be/qfmJh3E8288

Worksheet: Sets Activity

(Note: This worksheet is meant as a guided practice sheet, not something to be completed on one's own so don't feel itimidated if you can't answer all of the questions right away.)

### Lecture 10ish

To give you an idea of what the class looks like as we move deeper into the material, a sample lecture from content in the middle of 2141 is provided below. In it, we recall the definition of integrability (when the definite integral of a function exists) and then see examples of functions with an infinite number of discontinuities which may or may not be integrable.