If 
,
(i) find the Maclaurin’s series of f(x) up to and including the term in 
.
(ii) Hence deduce the series expansion of the function 
up to and including the term in 
Solution
(i) is routine, can easily work out to be
(ii) requires Higher Order Thinking (HOT)
We differentiate the series obtained in (i) to get 
Principle used: Differentiate one series expansion to get the series expansion for another series.
In Maclaurin’s series expansion, where up to f”(0) need to be evaluated, can be done quickly using the GC.
Example
Expand 
up to powers of 2.
Using TI 84 Plus (OS 2.55)
From G.C,
f(0)= 3/2
f'(0) = -3/4
f”(0)= 5/4
Therefore using Maclaurin’s series expansion,
If powers higher than 2 such as 3 are required, then do differentiation to find the first derivative. Then enter into Y1. Y2 and Y3 will then give the second and third derivative.
Solve the inequality
From G.C, 0.5<x< 2.81, x not equal to 2 (This is the easy part)
Hence solve
This is the HOT (Higher Order Thinking) part.
Method 1
Let f(x) =
Therefore f(x+2) =
f(x+2) is translation of f(x) by 2 units in the negative X direction.
Therefore, the solution is -1.5< x<0.81, x not equal to 0
This solution combines the concept of inequality and transformation of graph.
Method 2
Replace x by x+2.
Therefore 0.5<x+2< 2.81, x+2 not equal to 2
-1.5<x<0.81, x not equal to 0
Question: 6 adults and 3 children are sitting in a round table. Find the number of possible seating arrangement if none of the children sit together.
Solution: Choose 3 out of the 6 possible slots for the children to be seated between 2 adults, then multiply by 3! for the children permutation, then multiply by (6-1)! for the 6 adult circular permutation. Ans: 14400
Principle used: Slotting principle.
Q: There are five buildings. There are 4 different colours of paint. Each building must be painted with only 1 colour. Find the number of ways to paint all the 5 buildings if all 4 colours must be used.
Solution: 5 Choose 2 to have the same colour, then multiply by 4! for the colour permutation. Ans = 240
Principle used: Grouping principle.
Questions on factorial can appear in H2 math binomial expansion, partial fractions, method of difference, mathematical induction and probability. Knowing how to simplify and manipulate factorials are crucial to doing well.
Two commonly used techniques
1. Replace n! with n(n-1)!
2. Multiply by k/k or k! / k!
Example 1
Synthetic division can be used to divide a polynomial by a linear divisor, which is commonly required in H2 math and Amath.
It is faster than the traditional long division.
Partial fraction is commonly used in binomial expansion, method of difference and integration.
When the divisor consists only of linear factors, cover up rule can be used to find the constants in the partial fraction quickly.
Example
Express 
in partial fraction.
By cover up rule,
Therefore
0 = A+C
Therefore,
Some questions on functions require students to understand the difference between ff-1 and f-1f. Though both = x, they are two different functions because their domains might be different.
while 
Therefore,
while 
Example f(x) = x+2, x> 0
H2 math syllabus 9740 did not state that periodic functions are included in the syllabus. However, questions on periodic functions have appeared in A level 2009 and 2013.
So it is good to learn periodic, even and odd functions.
Periodic functions
A periodic function has a graph with a basic pattern that repeats at regular intervals.
For example, sin x is periodic and its period is 2 pi.
sin(x) = sin(x+ 2pi)
For a periodic function with period a
f(x) = f(x+a) = f(x+2a) = ..
Simiarly
f(x) = f(x-a) = f(x-2a) = …
i.e. f(x) = f(x+ka) where k is an integer
Even functions
A function is said to be even if f(x) = f(-x) for all values of x. The graphs of all even functions are symmetrical about the vertical axis.
Odd functions
A function is odd if f(x) = -f(-x) for all values of x. The graphs of all odd functions are symmetrical about the origin. One section of the graph can be rotated about the origin through 180 degrees to give the other section.
JC A Level H2 Math Tuition Singapore 
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