MA2002 Module Review

AY24/25 Sem 1
Lecturers: Dr. Wang Fei and Prof. Bao Huanchen
Tutor friend: Lou Yi

Assesment:
- Top 18 in-class quizzes out of 21: 20%
- 5 homework assignments: 30%
- Final examination: 50%

Content:
- Chapter 1: Intuitive Definition of Limits
- Chapter 2: Precise Definition of Limits
- Chapter 3: Continuous Functions
- Chapter 4: Derivatives
- Chapter 5: Applications of Differentiation
- Chapter 6: Integrals
- Chapter 7: Inverse Functions and Transcendental Functions
- Chapter 8: Techniques of Integration
- Chapter 9: Applications of Definite Integrals
- Chapter 10: First-Order Differential Equations

*Final examination was closed-book during this iteration and no help sheet was permitted.

A blended learning approach was adapted during this semester, where students could watch online videos (recorded by Dr. Wang Fei and uploaded to Canvas) before attending lectures. As such, there were no tutorial classes. The first one hour of each lecture was used to discuss problems from the exercise sheets. In fact, these were mainly tutorial questions from past iterations. During this iteration, it is crucial that one works with past year questions that were set by Dr. Wang Fei as there are a number of difficult and interesting questions that generally have a nice blend of Real Analysis.

Albeit a notable jump from H2 Mathematics, students who have taken HL Mathematics AA or H2 Further Mathematics would see some similarities between MA2002 and previously-learnt content. In particular, topics in Chapter 9 an Chapter 10 were previously taught in H2 FM, i.e. arc length, surface area of solid of revolution, shell method when finding volume of solid of revolution, and integrating factor method.

Students generally struggle with the \(\varepsilon\)-\(\delta\) definition of a limit, or more generally, the precise definition of a limit.

Dr. Wang Fei mentioned some interesting extension problems on top of the usual ones that were provided in class. A notable example involved proving the existence of a polynomial that interpolates the integer points defined by the function \(1^k+2^k+\ldots+n^k\) (known as Faulhaber sum). The homework questions were easy — I remember that a few were in the O-Level Additional Mathematics syllabus. There was one interesting question in particular, which was to use the precise definition of a limit to prove the following result: \[\lim_{x\rightarrow \infty} \left(x^3+ax+1\right)=\infty\] In the above question, \(a\) denotes an arbitrary real number. Many people sought help for this question. Anyway, I believe that most people scored very high marks for the homework assignments so the deciding factor was the final examination, which as mentioned, constituted 50% of the overall grade.

As expected, the final examination was difficult (for most students). I found it quite easy and finished the paper in about 1 hour 10 minutes out of the given duration of 2 hours. Also, I only slept for around 2-3 hours before the examination as I was chilling with my friends on campus and we even had supper at Al Amaan at 1 am! Anyway, a few questions from the final examination caught me by surprise. They are as follows.

First, there was this integral \[\int_{0}^{\pi/2}\frac{1}{5+3\operatorname{cos}\theta}\text{ }d\theta\] for which the only way out is by using the universal trigonometric substitution, also known as the tangent half-angle substitution or how I like to call it, the Weierstras substitution. This involves setting \(t=\operatorname{tan}\left(\theta/2\right)\) and I'll let you figure out the rest on your own. Also, you can't just geh kiang (Singlish for pretending to be smart) and perform somme Complex Analysis trick to it because I don't think it is allowed in the first place and although it does look similar to some integrals that I have encountered, the method will only work if the upper limit is changed to \(2\pi\). I almost forgot that this technique was covered in MA2002.

Also, there was this differential equation which I believe many were stuck as it was not the typical separable or integrating factor method kind of question. It was a Bernoulli differential equation so one has to use a substitution. Lastly, there was this question which looked exactly the same as one which I encountered during my Junior College days. It is a classic problem in Analysis, stating that if \(f\) is a function that is concave up, prove that for all \(a < b\), we have the inequality \[\frac{1}{b-a}\int_{a}^{b}f\left(x\right)\text{ }dx\le \frac{f\left(a\right)+f\left(b\right)}{2}.\] Expected grade: A+
Final grade: A+