**Subject Code/Name:** MAST10009 Accelerated Mathematics 2 **Workload:** For each week: 4x 1-hour lecture, 1x 1-hour tutorial

**Assessment:** 2x Written Assignments (5% each), Mid-Semester Test (10%), Final Exam (80%)

**Lectopia Enabled:** Yes, with screen capture. However, Barry writes on the whiteboard which isn’t recorded, so lecture attendance is necessary (or alternatively you can borrow someone else’s notes).

**Past exams available:** Yes, Final Exam 2009-2018. No short answers/solutions given, but the 2017 and 2018 Final Exam were discussed on the last two lectures.

**Textbook Recommendation:** Barry's Accelerated Mathematics 2 Printed Notes, which is available at UniMelb’s Co-op. The book contains all the lecture slides and exercises with short answers and is realistically necessary.

**Lecturer(s):** Prof. Barry Hughes

**Year & Semester of completion:** 2019 Sem 2

**Rating:** 4.5 out of 5

**Your Mark/Grade:** 97

**Comments: **This subject is part of the accelerated stream and covers the rest of Real Analysis and Calculus 2 (roughly 90% and 70%, respectively) that aren’t covered in its sem 1 counterpart, Accelerated Mathematics 1. Despite being a first-year subject, this subject puts a lot more emphasis on RA than Calc 2 so expect the difficulty to be of at least a second-year subject. Moreover, this subject covers the content of pretty much two subjects (although there are some overlaps) in the span of one semester so the pace will be very fast, in fact much faster than AM1.

If I were to describe this subject with one word, it would be

*rigour*. Many computations we used to do straight away are not allowed in this subject. Nearly every step requires proper justification and any assumptions made must be stated clearly (even when doing a simple integration by substitution). Visual proofs are also pretty much worthless in this subject, so some theorems that are obvious will take about half a page to prove. One of my mates described this subject as a mathematical essay, which I think is why this subject is a hit or miss (albeit much more likely to be a miss).

I find one of Barry’s quotes to be fitting in describing this subject:

"A common emotional response to my treatment is: this stuff was so easy at school, why does Barry make it so hard? Well, the reason why I make it so hard, is either that your teachers lied, or probably more likely, they have carefully protected you from things that might be troubling. However, most of you are over 18 now, so you can deal with

*R-rated* Mathematics"

Nonetheless, if you find the pace to be bearable and you’re willing to commit extra effort to appreciate the content, this can be a very eye-opening subject. You get to encounter numerous

*R-rated* concepts that are crucial and interesting in mathematical analysis but are often neglected.

The topics covered in this subject are (in order):

Content

1. Sequences (RA)

You’ll learn what a sequence is and the definition of a limit of a sequence using the ε-N definition. From this definition then several common limit theorems are proven. You’ll encounter a lot of limits, fractions and inequalities here, and you must get comfortable working with them (decreasing the denominator increases the fraction, etc) as they will be used a lot throughout the subject and will help a lot for the MST and the final exam.

You’ll also learn about bounds in sequences, in particular, the supremum (least upper bound) and infimum (greatest lower bound). You’ll find several unique theorems/properties possessed by real numbers which many seem visually obvious but are damn hard to prove. Towards the end, you’ll encounter the “order of hierarchy”. Make sure to remember this by heart as they will be extremely useful in Improper Integrals and Infinite Series.

*This topic was discussed in the first six lectures (which is before the last self-enroll date) and I’d say is a good representation of how rigorous this subject can get. My suggestion is that this is a good time to decide whether or not to continue if you’re trying out this subject.*

2. Limits and Continuity of Functions (RA)

Similar to sequences, you’ll learn what a function is and the definition of the limit and continuity of a function, this time with the ε-δ definition. You’ll also be introduced with the first major theorem in this subject: The Intermediate Value Theorem, which again, visually obvious but not so easy to prove.

3. Differential Calculus (RA)

Remember the first principle of derivative? Well, this subject tells that the first principle doesn’t happen to be the “first principle” after all. You’ll start by defining what it *truly *means when a function is differentiable at a point, and from there you’ll derive the first principle and prove several common rules eg. chain rule. You’ll also be introduced with some major theorems: First and Second-Order Mean Value Theorem, Taylor’s Theorem with Lagrange’s Remainder and L’Hôpital’s Rule.

4. Integral Calculus (RA + Calc 2)

You’ll first be taken into a journey of how the early 19th-century Mathematicians progressively attempted to find the area under a curve. Here, you’ll be introduced with the terms Riemann Sum, Upper/Lower Sum, Upper/Lower Darboux Integral and finally Riemann Integral which is the integral we learned in high school. After all that *then *connections are made to differential calculus, also known as the Fundamental Theorem of Calculus. (Unlike in most high schools where integration was directly taught to us as the “inverse” of differentiation). This subject then gradually shifted into the Calc 2 part as you’ll learn numerous techniques of integrations.

5. Differential Equations (Calc 2)

This is probably the easiest topic as almost everything is computational. You’ll learn numerous concepts surrounding 1st/2nd order ODE and how to solve them. You’ll also touch on the application of ODEs in Mathematical Modelling: Malthusian/Logistic Population Growth Model, “Solute-Solvent Mixing” Model, Motion with Drag, and RC/RL/RLC electric circuit.

6. Improper Integral (RA)

After taking a stroll through computational ODEs, you’ll be pulled back to the rigorous side of this subject as you’ll cover this topic. This topic covers the possibilities of standard (Riemann) integral going wrong eg. integrand goes unbounded or interval of integration is infinite, and whether the integral can still exist as an improper integral or is forever a *bad* integral. You’ll learn various tests to determine the fate (convergence) of such integrals. Try to develop the intuition in quickly determining which convergence test to use for certain scenarios, as these questions will definitely be on the exam and are worth quite a lot.

7. Infinite Series (RA)

Ever heard of the controversial Grandi’s Series (1-1+1-1… = 1/2) or perhaps the notorious Ramanujan Summation (1+2+3+... = -1/12)? This topic starts by rigorously defining what it means when an infinite series converges, and how deplorable (or what Barry likes to call, *antisocial*) it is to treat infinity as a number. Similar to Improper Integral, you’ll learn various tests to investigate the convergence of a series, and it’ll be very advantageous for the exam to develop such intuition. Towards the end, you’ll encounter some known types of series: Taylor Series, Power Series, Complex Series, Fourier Series (the last two I reckon are non-examinable).

LecturesEverything examinable on the lecture slides is in the textbook and are organised really well, so there’s no need to copy down the lecture content. However, when you read the textbook, there will be example problems in every lecture but often with no solution. This is where Barry would write down the solutions to these questions on the whiteboard during the lecture, which is not recorded and is one major downside of this subject.

Aside from that, Barry is an old fashioned yet fantastic lecturer. He has taught this subject since 2009 and his experience does show in how well he explains the concepts. He often uses simple analogies or hand gestures which I find helpful. He shows a lot of passion for this subject, and I generally find his explanations

~~and his quotes~~ to be intriguing. If you’re curious about how Barry explains stuff, here’s a video where Barry was interviewed about the Indian JEE exam:

https://youtu.be/0h_x13xHjVs?t=376Due to a large amount of content covered, it is crucial to stay up to date with the content by reading the textbook after each lecture and immediately come to consultation sessions when in doubt. Otherwise, it’ll be very difficult to catch up on the content since there are four lectures per week. Just as Barry said, “If you do no work in your time in this subject for about three consecutive days, you will be

*completely destroyed*.”

Tutorials / Problem SolvingA standard maths tute. You’ll sit in groups of 3 or 4 and solve the exercise questions in the textbook which you can actually do beforehand. The tutor will go around the class to check on your workings and give feedback. It’d also be wise to use this time to ask your tutor if you have any problems with the content.

AssignmentsThere are only two written assignments throughout the semester but they are very lengthy and worth 50 marks each. Barry recommended not spending over 8 hours on each assignment, although personally, I spent around 12-15 hours. The questions seemed intimidating at first, but after giving some thoughts I don’t find them to be too bad. The 1st assignment covers topic 1-2 (100% RA), while the 2nd assignment covers topic 3-5 (25% RA, 75% Calc2). The 1st assignment involves more analysis and proofs with limits, while the 2nd assignment involves more computations with ODEs (although the integrations are very nasty). I think most find the 2nd assignment to be easier but I prefer the 1st one as I’m more comfortable working with proofs and I dislike working with long and nasty algebra.

Also, make sure that all notations have been used properly as the markers are very scrupulous about that. Getting one simple misuse of notation is like bringing water through airport screening and will be an instant deduction. “Please, we don’t want any notational abuse. If you abuse notation, I will abuse you” - Barry

Mid-Semester TestThe MST will cover topic 1-3 up to the Mean Value Theorem, so it will be purely RA. Furthermore, it will be very fast-paced as the duration is only 45 minutes. Time management is paramount here. Fortunately, a lot of marks are allocated to simply stating definitions or theorems, so you can save a lot of time for the harder questions. Just be sure to include the important keywords, or memorise word by word if necessary. I lost 4 marks (out of 40, which equates to 1% of the final grade) simply because I didn’t include a few keywords I thought was unnecessary/obvious. Also, it’d be a good idea to familiarise with the example and exercise problems in the textbook as Barry may put some of them in the test.

Final ExamOk, this one’s definitely scary. A 3-hour, 80% weighted exam covering two subjects (one being a 2nd year subject), and to make matters worse, Barry persists that no formula/cheat sheet is given/allowed.

However, I think that the exam was fairly doable

*if* you have understood the content and done several past papers. Just like the MST, there’ll be a lot of “state the definition/theorem” questions and a couple of problems taken directly from the textbook, both are very handy to save time. Also, roughly 35% of the total mark is allocated to solving ODEs or integrals. They should be straightforward so it’d be great to work on your agility/techniques in integrating and in dealing with nasty algebra, and beware of miscalculations.

The scarier part is probably where you need to apply appropriate convergence tests to random improper integrals and series. This is purely analysis and will require more thinking as there are many tests to choose from. But after doing enough exercise questions you can notice some kind of pattern and develop the intuition to solve them instantly.

ConclusionDue to the fast-paced nature of this subject and the four lectures per week (where attendance is necessary), it can get overwhelming trying to keep up with the content. A substantial amount of effort and commitment will be required through consistent studying and attending lectures in person. Instead of taking this subject, you can actually take Calc 2 and RA separately and you’ll still cover the same content without having to struggle too much. Or if you have no interest in doing RA, it’s definitely better to just take Calc 2.

However, if you’re up for a challenge, completing this subject can be a rewarding experience, especially as a first-year. As far as I know, this subject can also replace both Calc 2 and RA as a prerequisite to numerous other subjects, which I think is pretty cool.