Elementary Mathematics - distance-time graphs & speed-time graphs

Below are the various types of distance-time graphs and its corresponding speed-time graphs.
Constant speed
At rest
Accleration at a constant rate
Deceleration at a constant rate
Deceleration at varying rate

Additional Mathematics - Inverse function

The inverse function is a one-to-one function. Each object in the domain is map to each object in the range. To satisfy this requirement, the function f(x), should not have any turning point. That is, the gradient cannot be equal to zero. This can be verified by computing the the differentiated results - dy/dx) should not be equal to zero.

Eg one-to-one function
Domain---------Range
1---------------3
2---------------5
3---------------7
4---------------9

Eg not one-to-one function
Domain----------Range
1
2---------------5
3---------------7
4---------------9

You can imagine the Range to be the mirror image of the Domain. The mirror is the line
y = x.

Inverse function of a linear equation
Eg, f(x) = x - 5
y = x - 5
exchange x to y and y to x,
x = y - 5
rearrange the equation,
x + 5 = y
y = x + 5
Inverse function of a quadratic equation
Eg, f(x) = x2 + 2x + 5
y = x2 + 2x + 5
y = (x + 1)2 + 4
exchange x to y and y to x
x = (y + 1)2 + 4
rearrange the equation
x - 4 = (y + 1)2
sq root(x - 4) = y + 1
[sq root(x - 4)] -1 = y
y = [sq root(x - 4)] - 1
Inverse function of a cubic equation
Eg f(x) = x3 + x - 1
y = x3 + x - 1
exchange x to y and y to x
x = y3 + y - 1
For cubic equation, it is impossible to re-arrange the equation to isolate y on the left hand side and the various x on the right hand side.
f-1(9), 9 = y3 + y - 1
using trial and error method, determine the value of y.
if y = 1, 1 + 1 - 1 not equal to 9
if y = 2, 8 + 2 - 1 = 9
Therefore, f-1(9) = 2

Oops! sorry for the untidy graphs!

Additional Mathematics - Integration

Integration is the reverse process of differentiation.
For eg,
when we differentiate 'A', we get 'B'.
when we integrate 'B', we get 'A'

To solve integration problems, we need a copy of the formulaes with us. I find the integration formulae list in the Longman A Maths guide comprehensive.

But there are some integration questions that I am unable to solve, even with the formulae.
For eg, Integrate (x)/[sq root(x - 1)].
The answer is y = (x + 2)[sq root(x - 1)]
I know the answer because this is a guided question.
Part 1 of the question requires me to differentiate y = (x + 2)[sq root(x - 1)]. The answer is (3/2)(x)/[sq root(x - 1)]
And Part 2 of the question requires me to integrate (x)/[sq root(x - 1)], which is the original function y.

Thankfully, I have not come across a question in the past 10 years of Additional Mathematics GCE 'O' level examination that requires integration of a complicated function without providing any guides. ^o^

In the integration formulaes provided, integrate (1/x) is (ln x + C). But, it does not say what is the answer for integrating (ln x). Luckily, I found an answer through a guided question. And, I thought I should record in this space for my readers.

Integrate (ln x) = (x)(ln x) - x + C
Integrate (x)(ln x) = (1/2)(x2)(ln x) - (1/4)(x2) + C
Integrate (x2)(ln x) = (1/3)(x3)(ln x) - (1/9)(x3) + C
follow the pattern.

Happy Integration ^o^

Poems - Jul 2009

现今社会追名利，我的偶像是明义。明义毕业莱佛仕,出家修行在佛寺.掌管仁慈帮病人,电视筹款当超人.气车洋房会员卡,宠物养匹澳洲马.还好养马非马子,难到最爱是鸭子? 为了五万帮志恒,如今法庭被质问.杜莱事件刚刚过,国人又气又难过. 人非圣贤谁无过, 认错最重要改过.

My friend's poem collection

I have a friend. He is like the ancient poet whom I have watched on television drama when I was a little girl. He sings a poem over everything he comes across. This space is dedicated to this friend - My friend's poem collection.

Preschool - Mathematics

In mainstream education, the Ministry of Education detailed the syllabus for a child's learning. For preschoolers, the MCYS do not provide the answers on what preschoolers should learn. The preschools are free to provide their own syllabus.

I have done a comparison of the syllabus of a few preschools, references and primary school textbooks and this is my list of 'must learn' for Primary 1 readiness on Mathematics.

1. Counting 1 to 150
2. Spelling the numbers in words 1 to 100
3. Patterns
4. Addition
5. Subtraction
6. Multiplication 1 to 50
7. Clock
8. Dollars and Cents

I am in favour of the Kumon workbook, available in Popular bookstore at $10.90 each. I like their techniques on imparting the Mathematics skills to preschoolers. The workbooks are also printed on quality papers and the layouts are attactive. However, I find the workbooks voluminous for a young child. There are many repetition to allow sufficient practices of each topic. My child takes advantage of these repetition and choose to copy the answers instead. So, I used the workbooks selectively to sustain a child's interest on the subject.

Additional Mathematics - Combinations and Permutations (no repetition)

Questions on Combinations and Permutations gives the stories and requires you to provide the solutions. Comparison and studying the reverse gives better understanding of combinations and Permutations. Read on. It helps.

Eg A, B, C
1. Combination: 3C2 = 3 = AB, AC, BC
The order is not important. It does not matter who comes first. The order is interchangeable ie AB = BA, AC = CA, BC = CB

2. Permutation: 3P2 = 6 = AB, BA, AC, CA, BC, CB
The order is important. It shows who comes first and who is next.

3. Factorial:
2! = 2P2 = 2 = AB, BA
3! = 3P3 = 6 = ABC, ACB, BAC, BCA, CAB, CBA
The order is important.

(1), (2) and (3) above looks at 1 group of object only. They look simple. Complication comes in when there are 2 or more groups of objects. When this happens, we need to use a mixture of the above 3 rules. The following scenarios are important as they are the trends of GCE 'O' Level examination questions.

4. Multiplication of Permutation and Factorial
Group 1: A, B, C
Group 2: 1

3P2(Group 1) = 6 = AB, AC, BC, BA, CA, CB
2! (both groups) = 2 = G1G2 or G2G1
3P2(Group 1) x 2!(both groups) = 6 x 2 = 12
= AB1, 1AB, AC1, 1AC, BC1, 1BC, BA1, 1BA, CA1, 1CA, CB1, 1CB

5. Multiplication of Permutation with Permutation and Factorial
Group 1: A, B, C
Group 2: 1, 2, 3, 4

3P2(Group 1) = 6 = AB, AC, BC, BA, CA, CB
4P2(Group 2) = 12 = 12, 21, 13, 31, 14, 41, 23, 32, 24, 42, 34, 43
2!(Group 1 & Group 2) = 2 = G1G2, G2G1

3P2(Group 1) x 4P2(Group 2) x 2!(Group 1 & Group 2) = 6 x 12 x 2 = 144

AB12 12AB AB21 21AB AB13 31AB AB14 14AB AB41 AB41 AB23 23BA AB32 32AB AB24 24AB
AC12 12AC AC21 21AC AC13 31AC AC14 14AC AC41 AC41 AC23 23AC AC32 32AC AC24 24AC
BC12 12BC BC21 21BC BC13 31BC BC14 14CB BC41 41BC BC23 23BC BC32 32BC BC24 24BC
BA12 12BA BA21 21BA BA13 31BA BA14 14AB BA41 41BA BA23 23BA BA32 32BA BA24 24BA
CA12 12CA CA21 21CA CA13 31CA CA14 14CA CA41 41CA CA23 23CA CA32 32CA CA24 24CA
CB12 12CB CB21 21CB CB13 31CB CB14 14CB CB41 41CB CB23 23CB CB32 32CB CB24 24CB

AB42 42AB AB34 34AB AB43 43AB AB31 31AB
AC42 42AC AC34 34AC AC43 43AC AC31 31AC
BC42 42BC BC34 34BC BC43 43BC BC31 31BC
BA42 42BA BA34 34BA BA43 43BA BA31 31BA
CA42 42CA AC34 34AC AC43 43AC AC31 31AC
CB42 42CB CB34 34CB CB43 43CB CB31 31CB Phew!

Now that you have mastered the patterns and properties of Combination, Permutation and Factorial and their multiplication. The challenges lies with the identification on what are the groups. Practice makes perfect.

If you like the above or you have any comments, pls let me know at SwallowTuition@gmail.com. I would like to hear from you. Thank You.