ADDITIONAL MATHEMATICS PROJECT WORK 1/2013
Name: Class: IC Number: School:
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CONTENT Page
No. 1.
Content page
2.
Actknowledgement
3.
Objectives
4.
Introduction
5.
Task Specification Specification
-
Part 1
-
Part 2
-
Part 3
-
Further exploration 1
-
Further exploration 2
6.
Conclusion
7.
Reflection
8.
Bibliography
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Objectives - The aims of carrying out this project work are: - To apply and adapt a variety of problem-solving problem- solving strategies to solve problems.
- To improve thinking skills. - To promote effective mathematical communication. - To develop mathematical knowledge through problem solving in a way that increases student’s interest and confidence.
- To use the language of mathematics to express mathematical ideas precisely.
- To provide learning environment that stimulates and enhances effective learning.
- To develop positive attitude towards mathematics.
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Acknowledgement I would like to express gratitude to Allah for His guidance and giving me strength and patience to finish my additional mathematics project work. Finally this project was completed successfully with all His blessed and guidance. Firstly I would like to thank my Additional Mathematics teacher, ____________________as he gives us important guidance and commitment commitment during this project project work. He has been a very supportive figure throughout the whole project. Next, I would like to thank the principal,_________________ for giving me the permission to bring the laptop to the school for making this project. And also my school, _________________________ for giving me the chance to create create this project work. work. School also provide me spaces to discuss and carry out this project work. Besides that, I would like to thank my beloved parents who provided everything needed in this project work, such as money, internet, reference books, laptops and so on. They contribute their time and spirit on sharing their experience with me. Their support may raise the spirit in me to do this project work smoothly and eficiently. Last but not least, I would also like to thank all the teachers and my friends for helping me collect much needed data and statistics for this project. Not forget to thank all the people who were involved directly or indirectly towards making this project a reality.
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INTRODUCTION The concept of function is one of the most important in Mathematics. However, However, it's history is relatively short. M.Kline credits [K. line p.388] Galileo (1564-1642) with the first statement of dependency of one quality on another, eg, ' the times of descent a long inclined planes of the same height, but off diffrent slopes ,are to each other as the height are slopes'. In a 1673 manuscript Leibniz used the word 'function' to mean any quality varying from point to point of a curve, like the length of the tangent or the normal. The curve itself was said to be given by an equation. But in 1714,he already used the word 'function' to mean quantities that depend on the variable. The notation f(x) was introduced by Euler in 1734.Still,in the 1930s, a well known Russian mathematician M. Luzin wrote: The function concept is one of the fundemental concept of modern mathematics. If did not arise suddenly. It arose more than 200 years ago out of the famous debate on the vibrating sting and underwent profoud changes in the very course of that heated polemic. From that time, on this continously, and this twin process continous to this very day. That is why no single formal definition can include the full content of the function concept. This content can be understood only by a study of the main lines of the development of science general and of mathematical physics in particular. Functions especially of the numeric variety, are given often confused with formula by means of which they are defined. In one of the discrete mathematics textbooks, the author fling a particular in ept remark to the effect that 'whereas classical mathematics is about formulae'. Charitably, I interpret the maxim as the authors attempt to emphasize the importance of functions in mathematics in general and discrete mathematics in particular.
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Examples of Functions In a compact form a function is a well defined def ined unidirectional dependency, a concept of which I shall provide several examples. Functions that are not numeric
The set of fingerprints is uniquely defined f or every person. That is to say, there is a function (call it f) from the set of people to the set of fingerprint sets. The function is not defined for every person, but only for those who were fingerprinted. Any fingerprint set uniquely defines a person. This function is inverse to f.
For every person, there is a unique DNA molecule whose copies are carried by every cell in human body. Thus there is a function from the set of people to the set of molecular structures known as the DNA. This function does not have an inverse; for, say, a pair of identical twins share the DNA structure.
Every triangle has the barycenter, the incenter, the orthocenter, the circumcenter, thesymmedian thesymmedian point, the isogonic center and many other remarkable points. These all are various functions defined on a set of triangles. Every triangle center is defined by a homogeneous function of in, say, barycentric coordinates, which is symmetric with respect to its arguments.
Every point in a plane of a Mandelbrot set defines a Julia set.
For every object, its optimal appearance occurs at a certain distance, which would make this distance a function of the appearance. But this may not be true: an object may look equally well from several distinct distances.
In a topological space, every set has a boundary, an interior, and a closure. The correspondence defines three functions on the set of all subsets of a topological space.
Any coloring in a game of Y can be uniquely reduced to a coloring of a smaller board in a manner that preserves the winning chains.
Function addition and various function multiplications serve additional examples of functions of two variables.
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Numeric functions The distinction between numeric and non numeric function is rather nebulous. It's neither standard nor common. I would think of a function as numeric provided that, on top of establishing a correspondence between number sets, the function gains some properties from that fact. If the numbers are used merely as tags or convenience symbols, house numbers for example, there is probably no point in thinking of the function as numeric. The house numbers may still be used to indicate the proximity of a house from a point of departure. More generally, we talk of a distance function, which I feel comfortable to think of as numeric. The most pedestrian definition of a function is the one common in high school textbooks, see for example [Jacobs, p. 122]: A function is a pairing of two sets of numbers so that to each element in i n the first set there corresponds exactly one number in the second set. So, what makes a function numeric? As I already mentioned, the distinction is loose, but mostly one thinks of a function as numeric if it is defined by means of an algebraic formula. A more broad convention that only requires the two paired sets in the definition to consist of (whatever) numbers , allows functions that only nominally relate numbers to one another and do not gain any essential properties from their being defined for number sets.
The function x
2
2
x relates a number to its square. We would commonly write f(x) = x , 2
but, to denote the result as, say, (x) is as consistent with Euler's notations as could be. In fact, for thebinomial thebinomial coefficient "n choose k", which is a function of two variables, n
several different notations - C(n, k), nCk, C k - are in common use.
A more general quadratic function is defined by 2
f(x) = ax + bx + c
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polynomial functions. For the simplicity title, I think, and is the second simplest of the polynomial
compete two functions: the constant function f(x) = const , and the identity function f(x) = x.
Some function, because of their importance and frequent use in mathematics and applications, have, with time, received special names. Such are the trigonometric x
functions sin(x), cos(x), and others, exponential function a and the logarithm log(x). Having a name does not mean that the values of the function can be easily calculated. Virtually for all x, sin(x) has to be approximated. But the same is true for apparently simpler functions, like x
1/2
.
New functions can be constructed as the combinations (in many senses) of other functions. Some combinations yield surprising results. For example, incomprehensible at first sight expression -½
is just a representation of the floor function: - 1/2
(I thank Andrew Newton for this example and for the link to an interesting discussion.) discussion.)
In a similar vein, Mohamed Al-Dabbagh who discovered the representation for the floor function in 1996, has also found that 2
)))·(1 + x 2
2 2
)))·(1 + x
2
H(x) = (sign(x) - 1) /4.
where H(x) is the Heaviside Step Function: H(x) = 1, 1/2, 0 depending as whether x > 0, x = 0,or x < 0.
Natural as it appears to be and common as it is, using formulas to define functions may be quite treacherous. Iain Stewart tells about his experience with offering the students to find the derivative of the function
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f(x) = log(log(sin(x))). By applying standard rules of the calculus, most students derive the answer.Curiously, answer.Curiously, the derivative makes sense for the values of x where sin(x) > 0. But having said that, we should also note that log(sin(x)) is never positive, so that log(log(sin(x))) does not make sense for any x.
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SOLUTION FOR PART 1
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1. (a) Primary School : { 1, 2, 3, 4, 5, 6 } Lower Secondary School : { α, β, λ } (b)
1· 2· 3· 4· 5· 6·
i. Domain : { 1, 2, 3, 4, 5, 6 } ii. Codomain : { α, β, λ } iii. Objects : 1, 2, 3, 4, 5, 6 iv. Images : α, β, λ v. Range : { α, β, λ } λ } vi. Relation : Many to one o ne relation
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α
β
λ
α· β· λ·
i. Domain : { α, β, λ } ii. Codomain : { B, C } iii. Objects : α, β, λ iv. Images : B, C v. Range : B, B, C } vi. Relation : Many to many relation
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B
C
1) Cartesian graph:
Arrow Diagram 1 :
1
2
3
4
5
6
Arrow Diagram 2 : B C
(b) Based on (a)(ii), determine whether each of the relations is a function? Give your reason. If yes, state whether its inverse function exist or not. Explain why. Answer: Relations in Arrow Diagram 1 is a function because the objects in P maps at only one image. Relation in Arrow Diagram 2 is not a function because the object in L has more than one image. Relation in Arrow Diagram 1 does not have inverse function because the object have two images.
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SOLUTION FOR PART 2
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Diagram 1 shows arrow diagrams for function f function f and and g g . f
g
-3
9
3
0
6
2
3
3
1
(a) Find (i)
Functions f Functions f and and g. g.
ANSWERS: f(x)=6-x g(x)=
(ii) The inverse functions of f and f and g g .
ANSWER: inverse function of f. f.
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ANSWER: inverse function of g. g.
(iii) The composite functions fg, functions fg, gf ,
and
ANSWERS:
)
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and What is in terms of f f , , g
(iv) Composite functions your conclusion? Express or/and .
ANSWERS:
() ( ) (b) Find the following composite functions:
and
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ANSWERS:
,
Hence,find (i)
and composite function
Function p Function p,if ,if function
ANSWER:
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(ii)
function h,if function
and composite function
ANSWER:
()
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(a) function of h of h is defined by h : x , x , find the
expression for the following functions:
(i) h 2
(ii) h 3
(iii) h 4
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(iv) h 5
h
30
h Based on my own calculation, when the highest power of the h 31
is even number, the answer will probably be x be x meanwhile when
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the highest power of the h is odd number, the answer will probably be
.
(b) function of g is g is defined by g by g :: x
, , find
Hence, deduce the inverse function of each of the
following functions. Check your answers using other method.
Answer:
g (x)
Let g(x) Let g(x) u
-1
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(i) g : g : x ,
g (x) Let g(x) Let g(x) u
-1
=
Other method: g(x)
=
,
⟹
g -1(x)
=
(ii) g : g : x , Let g(x) Let g(x)u ~ 24 ~
, g (x) -1
=
Other method: g(x)
=
,
⟹
g -1(x)
=
, Let g(x) Let g(x)u (iii) g : g : x
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-1
g (x)
=
,
Other method: g(x)
=
,
⟹
g -1(x)
=
,
where || || || Hence, determine (Show (Hints:- Recognize the pattern of the coefficients of and the (c) Given function find your workings.)
pattern of constant terms.)
ANSWERS:
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FURTHER EXPLORATION 1
The height of a light on a Ferris wheel is given as ,where H ,where H is is the height in metres above the ground,and t is t is taken in minutes.
(a) Find
the initial height of the light.
(b) At
what time is the light at its lowest position in the first revolution of the wheel?
(c) How
long does the wheel take to complete one revolution?
(d) Make
a conjecture of the speed of the ferris wheel by increasing the value of 120 in .
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ANSWERS: (a)
H(t) = 100 + 95 sin (120t)
(b) t H(t)
0 100
0.5 1.0 182.27
1.5 100
2.0 2.5 3.0 17.73 17.73 100
(a) From that graph, it can be shown clearly that H(t) min occurs between 20 minutes and 2.5 minutes. H(t) is minimum at axis of symmetry of 2.0 minutes and 2.5 minutes. Time taken when light is at the lowest position:
minutes
(c) Based on the graph, it can be shown that the initial height is 100m. After 1.5 minutes, the height of light is 100 metres from the ground but it is in the opposite side. After second 1.5 minutes, the height of light is 100 metres from the ground and is at the original position.
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Time taken for one complete revolution is at 3rd H(t) = 100 m at t = 3 minutes
(d) Let us fix the position of the light after t minutes which are 195 metres. Function 100 + 95 sin (120t)° 100 + 95 sin (135t)° 100 + 95 sin (150t)° 100 + 95 sin (165t)° 100 + 95 sin (180t)°
Value of t 0.75 0.6667 0.6 0.5455 0.5
From the table, it is shown that the increasing value of 120 in the function H(t) decreases the value of t. since speed is inversely proportional to t. , speed increases by the increasing value of 120.
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FURTHER EXPLORATION 2 a) To get length and breadth of the photograph, we need to find the length of the shorter part of the world that has been cut. Perimeter: 35 – 35 – 2(5) 2(5) = 25 cm
Area: (16.5 -) ( – 4) – 4)
Breadth: ( - 4) cm
Length: =
= 16.5 – 66 – 66 - 2 + 4 = - 2 + 20.5 - 66
= (16.5 - ) cm b) Answer. i. Determine:
Equation of symmetry, at AOS, = = =
= - ( - ) Maximum point: ( , )
Completing the square, A = - 2 + 20.5 – 66 – 66 = - [ 2 – 20.5 – 20.5 ] – 66 – 66
= - [ + 2( ) ] – 66 – 66 – (- ) ] – = - [( - ) – ( 2
2
2
Differentiation, A = - 2 + 20.5 – 66 – 66
2
= - 2 + 20.5
66
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max =
At max, = 0
, 0 = - 2 + 20.5 1 2 3 4 5 6 - -29 - 0 11.5 21 46.5
7 8 28.5 34
9 10 37.5 39
13.5
, It can be shown clearly that the maximum value of is happens between 10 and 11. 10 11 39 39.063 39 38.81 38.5 =
A/
A/
maximum
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11 38.5
c) Based on graph, determine.
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d) Determine the length and the breadth of the photograph when its area is 21cm3 and 37.96cm3 respectively.
√
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b=breadth, λ=length
b=2 b=2 cm b λ λ b (1) when
when
b = 6-4
λ=
b and perimeter = 25
λ
(2): when b and perimeter = 23
when
λ = 7.3 cm b = 5.2 cm
b = 1.13-4 = -2.87
b = 9.2-4 = 5.2 λ
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Conclusion
After I accomplished this project, I have found that the Additional Mathematics is fun and very useful in our daily l ife to make our life easier. I also have learnt the important i mportant of perseverance as time will be inverted to ensure the completion and excellence of this project. On the other hands, I have learnt the virtue to make together as I have helped and received help from my fellow peers in the completion of this project. Besides, I can adapt with everyday problems by applying the mathematics thinking s kills that I’ve had. I realised the importance to be thankful and appreciative during completing this task. This is because I able to apply my mathematical knowledge in daily life and appreciate the beauty of Additional Mathematics. Mathematics. This project is a several training stage for me to prepare myself for the demands of my future undertaking in the university and work life.
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Reflection While I conducting this project, a lot of information that I found. I have learnt how function appear in our daily life. Apart from that, this project encourages students to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication. Not only that, I had learned some moral values that I practice.this projecthad taught taught me to be more responsible on the works that are given to me to be completed. This project also had made me felt more confidence to do works and not to give easily when we could not find the solution for the question. I also learned to be more discipline on time, which I was given about only three weeks to complete this project and pass up just in time. I also enjoy doing this project because I can spend my time with fr iends and tighten our friendship by the time. Last but no least, I proposed this project should be continue because it brings a lot of moral values to the students and also teach the understanding of the student in Additional Mathematics.
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Bibliography
Khan Academy: Functions, free online micro lectures
Hazewinkel, Michiel, ed. (2001), "Function", Encyclopedia of Mathematics, Springer,ISBN Springer,ISBN 978-1-55608-010-4
Weisstein, Eric W., "Function", "Function", MathWorld .
The Wolfram Functions Site gives formulae and visualizations of many mathematical functions.
Shodor: Function Flyer, interactive Java applet for graphing and exploring functions.
xFunctions, a Java applet for exploring functions graphically.
Functions from cut-the-knot.
Function at ProvenMath.
Comprehensive web-based function graphing & evaluation tool.
Abstractmath.org Abstractmath.org articles on functions
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