A level H2 Math preparation tip #2: Integration

  • Integration Techniques

There are lots of formula to remember, but the ones commonly tested are:

a) Trigo: \displaystyle \int{{{{{\sin }}^{2}}{{\theta }_{{}}}d\theta }} (double angle formula)

b) Given in MF 15:

\displaystyle {{\int{{\frac{1}{{{{a}^{2}}+{{x}^{2}}}}}}}_{{}}}dx=\frac{1}{a}{{\tan }^{{-1}}}\left( {\frac{x}{a}} \right)+c

\displaystyle {{\int{{\frac{1}{{\sqrt{{{{a}^{2}}-{{x}^{2}}}}}}}}}_{{}}}dx={{\sin }^{{-1}}}\left( {\frac{x}{a}} \right)+c

int 1

int 2

c) Integration by substitution (same for parametric equations)

int sub 1

d) Integration by parts: \displaystyle \int{{uv'}}=uv-\int{{vu'}}

int parts

e) Partial Fractions

partial frac

  • Area / Volume

It is quite common for questions to calculate area or volume through substitution.

vol

area

  • Differential Equations

Two main types: Story or Non – story. Two general methods: Direct integration or variable separation approach.

It is also common for them to ask for sketches of solution curves and determine if the solution curves approaches or converges to some asymptote.

DE1 DE2 DE3

A level H2 Math preparation tips #1: Differentiation

Calculus (Differentiation, Integration and applications)

  • Calculus is the heaviest weightage among all the pure math “modules” in the exam. Over the years the total marks for this module is between 50 to 60. Hence I always emphasise to my students that this module is a MUST to master.
  • A level exams will not test directly on techniques of differentiation. They will ask application questions only. So far over the years they have been testing on tangents / normals and maximum / minimum problems. No rate of change questions (yet).
  • Tangents / Normals

For the past 8 years, 7 years have questions on tangents / normals asked in the form of parametric equation (since implict differentiation can be asked through Maclaurin’s series). Furthermore, the recent years’ questions has been algebraic intensive. E.g. they ask you to find equation of a tangent at a point with parameter “p” instead of say “2” to make the workings nicer. Also it is quite common for them to merge graphing techniques (sketch parametric curves) and integration (area under curve through parametric form) as 1 big question.

tangent

tangent2

  • Maximum / minimum problem

It is quite common that they will give you an object such as a box of various shapes and find the value of one of the lengths for maximum volume or surface area. Students may find the first part on formulating the equation to be harder than the differentiation itself. My suggestion is: if the equation is mentioned in the question as a “show” question, just use the answer to continue with the maximum / minimum computation which will usually have more marks. DO NOT give up the entire say 9 marks question just because you could not form the equation which is just approx 3 marks.

max

  • Maclaurin’s series
    For Macluarin’s series there are two main approaches: successive differentiation and using standard series from MF 15. Over the years, binomial expansion has been tested quite frequently. Also if you see the phrase “sufficiently small” and the question involves a triangle and angle –> small angle approx (using sine or cosine rule). Usually small angle approx questions is followed by binomial expansion too.

small angle

mac

Prelim Papers or A level papers?

Less than 3 weeks and A level exams will commence.

At this stage students and parents asked me: Should the students practice prelim papers or A level exam papers?

My answer: Prelim papers should come after a student finds A level papers easy.

If a student is struggling with A level papers, doing prelim papers (harder) may not help in building the foundation, but reduces the level of confidence and the drive to study.

This is why I created this A level Crash course specifically for students who struggle to get an “OK” grade for prelims. To show them A level papers are very different in nature and we should be prepared for that, not prelim papers.

Our prep course has completed 3 lessons successfully. We will be moving on to the 4th lesson this Wednesday. It is not too late to seek help, but time is seriously running out.

Wait and regret? Or take the first step now!

http://alevelmathtuition.sg
94385929

Another happy news for the day

I remembered last year she got a U for year 1 but managed to promote as she got other H2 passes. It was not easy as she could not repeat year 1 even if she wanted and that meant she has to do alot of catching up as the school continues with the year 2 syllabus (quicker pace).

I recalled how she scored E at the year 2 Jan test, B for March test and A for June exam. She whatsapp me telling me her cohort only 13 As and 9Bs for Prelims and I was nervous too for I hope she can get A (she is capable of it now). Then today she whatsapp me her results I was really happy. It showed that my method of coaching has proven effective for her.

She is from RI by the way (the school exam papers are of a certain standard).

Does your child need help? Sometimes it is about the correct coaching method (which school need not necessary provide) to see significant improvement.

Many kids are going for crash course. Is your child in the race too?

It is common for students who have not performed well in their Prelims to be seeking for help in this final chance before the national exams. Hence many tutors, including myself have launched crash courses to help the kids. However not all courses are conducted in the same way. Here’s the article and below is how my crash course differs from some of the other courses.

http://www.straitstimes.com/singapore/education/students-turn-to-crash-courses-to-boost-grades

1) Fees: Out to help or out to chop?

Some charges as much as $800 for 4 lessons (taking the chance to absorb money for desperate students). However I only charge $600 for 12 lessons. My aim is simple: the fees are used mostly to cover rental, only minimal goes in my pocket. However I hope to help as many students, especially those underperforming to improve in the limited time. This is also a way to test out the effectiveness of my teaching method.

2) Sit down and drill or learn something useful?

Several tuition centres just pass the students exam papers to practice for 2 hours while collecting fees over $100 for that piece of worksheet. Drilling helps if the student is able to cope the examination independently for which there is little usefulness of tuition. Here I opt to teach some exam techniques and suggestions on what students should focus in revision, how are the marks awarded, etc. Examination strategies are important, especially when a student may not be as academically strong as other student. Drilling can be done at home, a piece of worksheet cost $1 at most, not $100. Exam strategies are worth much more.

3) Big class vs small groups

Some courses have 20 students or more, which is the same as a class size in Junior College. Over there students listen to the tutor more than the tutor attends to the students. If your child needs serious help, then you will your child to have more attention from the tutor. Let’s face it, in a 2 hour session, how much time can a tutor coach each student if there are 20 or beyond? Small groups are more effective, and here I am keeping my class size up to 6 only. I focus more on producing results rather than earning the money.

Time is running out, if you think you or your child needs the help, sign up as soon as possible, but do find out more on the courses before hand. It is better to join slightly later than to join a course that is not productive at all.