The maximum value of y occurs when x = 4. Plugging x = 4 into the equation for y gives: y = -4sin60° + 80(4)sin60° y = -4√3 + 320√3 y = 316√3 So the maximum area is 316√3 square units.

1. OBJECTIVES
We students taking Additional Mathematics are required to carry out
project work while we are in Form 5. This Year the Curriculum
Development Division, Ministry Education has prepared two task for
us. We are to choose and complete only ONE task based on our area of
interest. This project can be done in groups or individually, but each of
us are expected to submit an individually written report. Upon
completion of the Additional Mathematics Project Work, we are to gain
valuable experiences and able to :
I. Apply and adapt a variety of problem solving strategies to solve
routine and non-routine problems.
II. Experience classroom environments which are challenging,
interesting and meaningful and hence improve their thinking
skills.
III. Experience classroom environments where knowledge and skills
are applied in meaningful ways in solving real-life problems.
IV. Experience classroom environment where expressing ones
mathematical thinking, reasoning and communication are highly
encouraged and expected.
V. Experience classroom environments that stimulates and enhances
effective learning.
VI. Acquire effective mathematical communication through oral and
writing and to use the language of mathematics to express
mathematical idea correctly and precisely.
VII. Enhance acquisition of mathematical knowledge and skills
through problem-solving in ways that increase interest and
confidence.
VIII. Prepare ourselves for the demand of our future undertakings and
in workplace.
1

2. IX. Realise that mathematics is an important and powerful tool in
solving real-life problems and hence develop positive attitude
towards mathematics.
X. Train ourselves not only to be independent learners but also to
collaborate, to cooperate, and to share knowledge in an engaging
and healthy environment.
XI. Use technology especially the ICT appropriately and effectively.
XII. Train ourselves to appreciate the instrinsic values of mathematics
and to become more creative and innovative.
XIII. Realize the importance and the beauty of mathematics.
We are expected to submit the project work within three weeks
from the first day is being administered to us. Failure to submit
the written report will result in us not receiving certificate.
2

3. APPRECIATION
First of all, I would like to say Alhamdulillah, thank you to Allah for
giving me the strength and health to do this project work and finish it
on time. Secondly, I would like to thank the principle of Sekolah
Menengah Kebangsaan Tun Ismail, Tuan Hj Tokijan Bin Hj. Abd Halim
for giving the permission to do my Additional Mathematics Project
Work. Not Forgetten to my parents for providing everything, such as
money, to buy anything that are related to this project work, their
advise, which is the most needed for this project and facilities such as
internet, books, computers, and all that. They also supported me and
and encouraged me to complete this task so that I will not
procrastinate in doing it.
Then I would like to thank to my Additional Mathematics teacher,
Puan Faridah for guiding me through out this project. Even I had
difficulties in doing this task, but she taught me patiently until we knew
what to do. She tried and tried to teach me until I understand what I’m
suppose to do with the project work.
Besides that, my friends who always supporting me. Even this
project is individually but we are cooperate doing this project especially
in discussion and sharing ideas to ensure our task will finish completely.
Last but not least, any party which involved either directly or indirect
in completing this project. Thank you everyone.
3

4. INTRODUCTION
[History of Functions]
In the 18th and 19th centuries, scientists discovered that the
elementary functions--powers, roots, trigonometric functions and their
inverses--had their limitations. They found that solutions for some
important physical problems--like the orbital motion of planets, the
oscillatory motion of suspended chains, and the calculation of the
gravitational potential of nearly spherical bodies--could not always be
described in a closed form using only elementary functions. Even in the
realm of pure mathematics, some quantities--such as the
circumference of an ellipse--were also impossible to discuss in such
terms. Functions describing solutions to these problems were often
expressed as infinite series, as integrals, or as solutions to differential
equations.
On further investigation, scientists noted that a relatively small number
of these special functions turned up over and over again in different
contexts. What's more, they noted that many other problems could be
solved in the form of a comparatively simple combination of these
newer functions with the elementary functions known to the ancients.
Functions that cropped up most frequently in scientific calculations
were given names and notations which have come into common usage:
Bessel functions, Struve functions, Mathieu functions, the spherical
harmonics, the Gamma function, the Beta function, Jacobi functions,
and most of the others appearing on this website.
In the second half of the 19th century mathematicians also started to
investigate these special functions from a purely theoretical
perspective. Alternative representations--as differential equations,
4

5. series, integrals, continued fractions, or other forms--were found for
many. Important publications on the topic at the turn of the century
include the four-volume masterpiece on the elliptic functions by J.
Tannery and J. Molk (published 1893-1902), containing hundreds of
pages of collected formulas; I. Todhunter's treatise on Laplace, Lamé
and Bessel functions (1875); E. Heine's treatise on spherical harmonics
(1881) and A. Wangerin's work on the same topic (1904).
Large tables with numerical values for the special functions also began
to appear, along with three-dimensional "graphs" made of wood or
plaster--masterpieces of precision sculpting--showing the behavior of
functions such as P and the Jacobi functions. Many of these models are
still on display in math departments throughout the world, and the
graphics on this website can be thought of as their modern, computer-
drawn counterparts.
Charles Babbage, who designed but was unable to build the "difference
engine," planned a printing device allowing the machine to generate
large tables automatically. A Swedish publisher named Georg Scheutz
and his son Edvard later built a difference engine that could set type. In
1857, the Scheutzes produced a mechanically generated table of
common logarithms to five decimal places for the integers from 1 to
10,000; each value took about thirty seconds to calculate.
Funktionentafeln mit Formeln und Kurven, the first modern handbook
of special functions--that is, one containing graphs, formulas, and
numerical tables--was published in 1909 by Eugene Jahnke and Fritz
Emde. The first text dealing comprehensively with most of the named
special functions was E. T. Whittaker and G. N. Watson's A Course of
Modern Analysis, 2nd Edition (1915).
This popular text consisted of two parts; Part I is a textbook of complex
analysis, while Part II is a handbook of special functions.
5

6. In 1939, orthogonal polynomials were given their first detailed
treatment by Gábor Szegö. This work was followed in 1943 by Wilhelm
Magnus and Fritz Oberhettinger's Formeln und Lehrsätze für die
speziellen Funktionen der Mathematischen Physik, the most complete
collection of formulas involving special functions yet prepared.
The massive Bateman Manuscript Project--the editing for posthumous
publication of Harry Bateman's accumulated notes on special functions,
which he stored in shoe boxes--was carried out by Arthur Erdélyi,
Wilhelm Magnus, Fritz Oberhettinger, and Francesco Tricomi,
culminating in 1953 with the classic three-volume work Higher
Transcendental Functions. This monumental collection contains not
only formulas, but also derivations, proofs, and historical remarks.
(Along with the contents of Higher Transcendental Functions,
Bateman's shoe boxes held enough material for a two-volume set of
integral transform tables.)
In Higher Transcendental Functions, Erdélyi introduced a new emphasis
on the importance of hypergeometric functions as an underlying,
unifying basis for the development of many of the categories of special
functions. Although 2F1 had been extensively studied since the time of
Gauss, mathematicians were slow to appreciate the importance of the
generalized hypergeometric pFq and to recognize the many relations
between hypergeometric functions and the special functions
encountered most frequently. Another innovation in Higher
Transcendental Functions was the inclusion of number-theoretical
functions.
6

7. Parallel to the handbooks dealing with series expansions, differential
equations, functional identities, and so forth of special functions, many
integral tables were developed in the 20th century.
Among these, especially the tables of W. Gröbner, N. Hofreiter, A.
Erdelyi, W. Magnus, F. Oberhettinger, I. S. Gradshteyn, I.H. Ryshik, H.
Exton, H. M. Srivastava, A. P. Prudnikov, Ya. A. Brychkov, and O. I.
Marichev are noteworthy.
Application of the electronic computer resulted in many massive
volumes containing hundreds of pages of tables for Bessel functions,
elliptic integrals, Legendre functions, and so on. An important
handbook containing graphs, formulas, and compute-generated
numerical data was assembled by Milton Abramowitz and Irene Stegun.
This work was published in 1964 by the National Bureau of Standards as
the Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables. Individual chapters were compiled by various
authors, leading to a certain unevenness in the quality of the material
and its presentation. Nevertheless, the Handbook of Mathematical
Functions remains a standard reference and is still in widespread use.
Ironically, the computer, that led to the creation of such mammoth
numeric tables is now eliminating the need for them. The ready
availability of computer processing time and technical software now
allows technical users to calculate the values of any needed function
without recourse to reference works. Mathematica can calculate every
special function on this website to any desired precision for any real or
complex values of the arguments and parameters.
Additionally, Mathematica can symbolically and numerically calculate
values for integrals or other operations and transformations involving
these functions, providing far more information than any single
handbook could possibly tabulate.
7

8. Thereby the need for an even more comprehensive collection of special
functions persists. And the Wolfram Functions Site is the most
complete such resource today. With tens of thousands of identities,
some extending over multiple pages if printed. The website is virtually
arbitrarily extensible and not bound to the limitations of a printed
book. Updating is easy, so new information can be quickly incorporated
into the website.
8

9. PART 1
(a)
Equation 1 : Axis of symmetry, x = 0.
General Form ,with c = 175
y
175
•
(-50,100)• •(50,100)
0 x
General Form
2
y ax bx c, c 175
2
ax bx 175
Passing through (50,100) ,
2
100 a ( 50 ) b ( 50 ) 175
2500 a 50 b 75 ........ (1)
Passing through (-50,100) ,
2
100 a ( 50 ) b ( 50 ) 175
2500 a 50 b 75 .......... ( 2 )
100 b 0
b 0.
2500 a 75
a 0 . 03
Quadratic Equation
2
y 0 . 03 x 175
9

10. Equation 2 : Axis of symmetry, x = 50.
Method 1: General Form, with c =100
(50,175) •
(0,100)• •(100,100)
0 50 100
Completing the square
2
y a(x b) c , b 50 , c 175
2
y a(x 50 ) 175
Passing through (0,100)
2
100 a (0 50 ) 1`75
75 2500 a
a 0 . 03
Quadratic Equation :
2
y 0 . 03 x 175
10

11. Equation 3 : Axis of symmetry, x = 0.
General Form, with c =75
y
75
•
. 0
• x
(-50,100) (50,100)
General Form
2
y ax bx c, c 75
2
ax bx 75
Passing through (-50,0) ,
2
0 a ( 50 ) b ( 50 ) 75
2500 a 50 b 75 ........ (1)
Passing through (50,0) ,
2
0 a ( 50 ) b ( 50 ) 75
2500 a 50 b 75 .......... ( 2 )
5000 a 150
a 0 . 03
2500 ( 0 . 03 ) 50 b 75
50 b 150
b 3
Quadratic Equation
2
y 0 . 03 x 75
11

12. (b)
Region A
Region B
50
Area of region A = 50
0 . 03 x
2
75 dx (refer to equation 3)
50
3
0 . 03 x
75 x
3 50
2
5000 cm
Area of region B = 100 x 100
= 10000 cm2 .
Total surface area = 10 000 + 5 000
= 15 000 cm2 .
12

13. PART B
(a) Cost of building
Structure 1
V = 1.5 x 0.13
= 0.195
Cost = 0.195 x 960
= RM 187.20
13

14. Structure 2
V=( + x 100 x 75) x 13
= 178 750
=0.17875
Cost = 0.17875 x 960
= RM 171.60
14

15. Structure 3
V=( +65(75) + (40)(75)) x 13
= 208 000
=0.208
Cost = 0.208 x 960
= RM 199.68
15

16. Sturture 4
V=( (80)(75)) x 13
= 188 500
= 0.1885
Cost = 0.1885 x 960
= RM 180.96
The structure built with minimum cost is structure 2
(b) If I am asked to choose the shape of sculpture to be built, I will
prefer to the structure 1. Of course, the main reason for me to choose
the shape is due to its beautifulness. For a, a parabolic shape give us a
sence of smoothness compare to other edges shape. Besides, most of
the memorial poles are made of this shape. Although use this shape is
not as cheap as using the shape structure 2 and 4, but for me RM 10
does not become a burden to our society.
16

17. PART C
The triangle ACE, ABD and BCF are equilateral triangle. (all angle = 60° )
y is the area of BDEF.
Using formula A = ab sin C,
y = [( x 80 x 80 sin 60°)-( x x sin 60°)-( x (80-x)(80-x) sin 60°)]
= (3200 - - ) sin 60°
= (3200 - - + 80x - 3200) sin 60°
= (- + 80x) sin 60°
= - sin 60° + 80x sin 60°
17

18. (b) y=- sin 60° + 80x sin 60°
÷ = -x sin 60° + 80 sin 60°
- sin 60° = m
x= X
80 sin 60°= c
x 1 2 3 4 5 6 7 8
68.42 67.55 66.68 65.82 64.95 64.09 63.22 62.35
The table above is used to plot the graph = -x sin 60° + 80 sin 60°.
The graph is shown on the graph paper next page.
From the graph,
When x = 5.5, = 64.5
y = 64.5 x 5.5
= 354.75
18

19. (c)Determine the maximum area of BDEF.
1st method,
Completing square of function,
y=a
y = (- + 80x) sin 60°
y = -1( - 80x + - ) sin 60°
y = -1 sin 60° + 1600 sin 60°
the maximum value of y
=q
= 1600 sin 60°
= 1385.64
2nd method,
Differentiation
y =- sin 60° + 80x sin 60°
= -2x sin 60° + 80 sin 60°
At turning point, =0
-2x sin 60° + 80 sin 60° = 0
-2x sin 60° = -80 sin 60°
x=
x = 40
When x = 40,
y=- sin 60° + 80(40) sin 60°
y = 1385.64
19

20. CONCLUSION
After doing research, answering questions, drawing graphs and some
problem solving, I saw that the usage othe people who Functions is
important. Functions is commonly used to help to measure. Especially
in measure the area of the building. In conclusion, Functions is a daily
life nessecities. Without it, it is harder to measure something. So, we
should thankful of the people who contribute in the idea of making
Functions.
REFLECTION
After spending countless hours, days and night to finish this project and
also sacrifing my time on my hobby, there are several things that I can
say..
Additional Mathematics.
The hardest thing that I had to face.
But without you, my life will never complete.
It is been about 1 year and half since I found you.
I still trying to understand you,
Always & Forever
Trying to know you, eventhough I had to sacrifice my whole life.
Your Name FOREVER the name on my LIPS.
Additional Mathematics.
20

21. REFERENCE
Additional Mathematics FORM 5. 2005. Selangor : Pustaka Kamza.
© 1998-2012 Wolfram Research, Inc. : Internet
ADDITIONAL MATHEMATICS . 2011. PETALING JAYA : SASBADI SDN.
BHD.
TOPGEAR ADDITIONAL MATHEMATICS . 2012.
BANDAR BARU BANGI : PENERBITAN PELANGI
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