22300 KUALA BESUT, BESUT, TERENGGANU DARUL IMAN.
===============
PROJECT WORK FOR ADDITIONAL MATHEMATICS 2010
PROJECT WORK 1
=============== NAME NAME :
NUR NUR ILI FA FARHANA RHANA BT SHU SHUHAIMI
I/C NO:
930609-02-5920
CLA CL ASS:
5 IBNU IBNU TAIMIYYA IYYAH
TEA TEACHER: PN. PN. MASRIZ MASRIZA A BT SA SAID
CONTENTs No
Contents
1 2 3 4 5 6 7
Introduction Appreciation History of Calculus Procedure and Findings Further Exploration Conclusion Reflection
Page
INTRODUCTION
We
students taking Additional Mathematics are required to carry out a project work while we are in Form 5.This year the Curriculum Development Division, Ministry of Education has prepared four tasks 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, and I gladly choose to do this individually.
The aim of carrying out this t his project work are:-
To apply and adapt a variety of problem-solving strategies to solve problems
To promote effective mathematical communication
To improve thinking skills
To develop mathematical knowledge through problem solving so lving in way that increases student¶s interest and confidence
To provide learning environment that t hat stimulates and enhances effective learning
To use language of mathematics to express mathematical ideas precisely
To develop positive attitude towards mathematics
Appreciation
Alhamdullilah, thank you to Allah SWT for giving the will to me to complete this Additional Additional Mathematics project. First of all, I would like to say thanks to my parents for providing everything such as money to buy anything that are related to this t his project work. They also supported me and encouraged me to complete this task. Secondly, I would like to thank the principal of Sekolah Menengah Kebangsaan Agama Nurul Ittifaq, Madame. Hjh. Wan Sabriah binti Wan Bakar for giving me the permission to do my task of Additional Mathematics Project Work. I also like to thank my Additional Mathematics teacher, Pn. Masriza binti Said for the guide and giving useful and important information for me to complete this project work.
What is ¶CALCULUS·????? Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related r elated by the fundamental theorem of calculus. Calculus is study of change, in the same sa me way that geometry is the study of shape and algebra is the study of operation s and their application to solving equations. A course in calculus is a gateway to other, more advance courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra algebr a alone is insufficient. Historically, calculus was called ¶the calculus of infinitesimals·, or ¶infinitesimal calculus·. More generally, calculus may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known propositional calculus, variational calculus, lambda calculus, pi calculus and join jo in calculus.
History
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems pro blems of mathematical physics. In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating ro tating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, many other problems discussed in his Principia Mathematica. In other work, he developed series expensions expensions for functions, including including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. These ideas were systematized into a true calculus of infinitesimals infi nitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of o f and contributor to calculus. His contribution was to provide prov ide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz paid a lot of attention to the formalism ² he often ofte n spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general ge neral physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton·s Newto n·s time, the fundamental theorem of calculus was known.
Procedure & finding ADDITIONAL MATHEMATICS PROJECT WORK 1/2010
The diagram below shows the gate of an art gallery. A concrete structure is built built at the upper part of the gate and the words µART GALLERY¶ is written on it. The top of the concrete structure is flat whereas the bottom is parabolic in shape. The concrete structure is supported by two vertical pillars at both ends. The distance between the two pillars is 4 metres and the height of the pillar is 5 metres. The height of the concrete structure structure is 1 metre. The shortest shortest distance from point A of the concrete structure to point B, that is the highest point on the parabolic shape, is 0.5 metres. A
0.5 m 1m
B
5m
4m
(a)
The parabolic shape of the concrete structure can be represented by various functions depending on the point of reference. Based on different points of reference, obtain at least three different functions which can be used to represent the curve of this concrete structure.
Solution:
Function 1
Maximum point ( 0 , 0.5 ) and pass through point ( 2 , 0 ) b)² + c y = (G b)²
b = 0 , c = 0.5 y = (G² y = G² substitute ( 2 , 0 ) into (1) 0 = (2)² + 0.5 0 = 4 + 0.5 4 = - 0.5 = - 0.5 4 = - 0.125
@y = - 0.125G²
Function 2
Maximum point ( 2 , 4.5 ) and pass through point ( 0 , 4 ) y = (G b)² b)² + c b = 2, c = 4.5 y = (G² Substitute ( 0 , 4 ) into (2) 4 = (0 ± 2)² + 4.5 4 = (-2)² + 4.5 4 = 4 + 4.5 4 = 4 ± 4.5 4 = - 0.5 = - 0.5 4 = - 0.025
@y = - 0.125(G²
Function 3
Maximum point ( 0 , 4.5 ) and pass through point ( 2 , 4 ) y = (G b)² b)² + c b = 0 , c = 4.5 y = (G² + 4.5
y = G² + 4.5
(3)
Substitute ( 2 , 4 ) into (3) 4 = (2)² + 4.5 4 = 4 + 4.5 4 = 4 ± 4.5 4 = - 0.5 = - 0.5 4 = - 0.125
@y = - 0.125G² + 4.5
(b)
The front surface of this concrete structure will will be painted before the words µART GALLERY¶ is written on it. Find the area to be painted.
Solution:
Area to be painted = Area of rectangle ± Area under the curve =
Further
exploration (a)
You are given four different shapes of concrete structures as shown in the diagrams below. All the structures have the same same thickness of 40 cm and are symmetrical. symmetrical. Structure
Structure
1
2 .5 m
1m 5m
4m
Structure
Structure
3 0.5 m
1m
0.5 m
1m
1m 5m
4m
4
2m
4m
5m
(i)
Given that the cost to construct 1 cubic metre of concrete is RM 840.00,
determine which structure will cost the minimum to construct.
Solution : Structure 1 Area =
Thickness = 40 cm = 0.4 m
Volume = Area x Thickness =
=
Cost =
= RM
m² x 0.4 m
m³
m³ x RM 840
Structure 2
Area = Area of Rectangle ± Area of Triangle
=
=
=
Volume = Area x Thickness =
=
Cost =
=
Structure 3
Area = Area of Rectangle ± Area of Trapezium = = =
Volume = Area x Thickness = =
Cost
= =
Structure 4
Area = Area of Rectangle ± Area of Trapezium = = = Volume = Area x Thickness = = Cost = =
@Structure
will cost the minimum to construct, that is RM
.
(ii)
As the president of the Arts Club, you are given the opportunity to decide on the shape of the gate to be constructed. Which shape would you choose? Explain and elaborate on your reasons for choosing the shape.
Answer: As the president preside nt of Arts Club, I will decide Structure Structur e as the shape f the gate to be constructed. It is because structure will cost the minimum and it is easier to be constructed compared to structure which is a curve.
(b)
The following questions refer to the concrete structure in the diagram below. If the value of k increases with a common difference of 0.25 m; (i)
complete Table 1 by finding the values of k and the corresponding areas of the concrete structure to be painted. 4m 0.5 m
1m k
Answer : (m) k (m)
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Area to
(correct
be painted (m )
to 4 decimal
places)
(4 x 1) - 0 + 4 x 0.5 = 3 2 (4 x 1) - 0.25 + 4 x 0.5 = 2.9375 2 (4 x 1) - 0.50 + 4 x 0.5 = 2.875 2 (4 x 1) - 0.75 + 4 x 0.5 = 2.8125 2 (4 x 1) - 1.00 + 4 x 0.5 = 2.75 2 (4 x 1) - 1.25 + 4 x 0.5 = 2.6875 2 (4 x 1) - 1.50 + 4 x 0.5 = 2.625 2 (4 x 1) ± 1.75 + 4 x 0.5 = 2.5625 2 (4 x 1) - 2.00 + 4 x 0.5 = 2.5 2
(ii)
observe the values of the area to be painted from Table 1. Do you see any pattern? Discuss.
Answer : The area to be paint decreases as the numbers :
k increases
0.25 m and form a series of
3 , 2.9375 , 2.875 , 2.8125 , 2.75 , 2.6875 , 2.625 , 2.5625 , 2.5 We
can see that the difference between each term and the next term is the same. 2.9375 ± 3 = -0.0625 2.875 ± 2.9375 = -0.0625 2.8125 ± 2.875 = -0.0625 2.75 ± 2.8125 = -0.0625 2.6875 ± 2.75 = -0.0625 2.625 ± 2.6875 = -0.0625 2.5625 ± 2.625 = -0.0625 2.5 ± 2.5625 = -0.0625
num bers in an Arithmetic @ we can deduce that is series of numbers Progression ( AP), AP), with a common difference, d = -0.0625 In conclusion, when k increases 0.25m, the area to be painted decreases by -0.0625m².
(c)
Express the area of the concrete structure to be painted in terms of k . Find the area a
k approaches
the value of 4 and predict the shape of the
concrete structure.
Answer : The area of the concrete structure s tructure to be painted = = =
@
Area of concrete structure to be painted = =
The shape of the concrete structure will be a rectangle with length 4m and breadth 0.5m, which may look like this :
Conclusion
After doing research, answering answering question, drawing graphs and some problem solving, I saw that the usage of calculus is important in daily life. It is not just j ust widely used in science, economics but also in engineering. Without it, marvelous buildings can¶t be built, human beings will not lead a luxurious life and many more. So, we should be thankful of the people who contribute cont ribute in the idea of calculus.
REFLECTION While you were conducting the project, what have you learnt? What moral values did you practise? Represent your opinions or feelings creatively through usage of symbols, illustrations, drawings or even in a song.
When things go wrong,as they sometimes will, When the road you·re tudging seems all uphill, When the funds are low and the dept are high, And you want to smile but you have to sigh, When care is pressing you down a bit, Rest,if you must, but DON·T QUIT!! Success is failure turned inside out, The silver tint of the cloud of doubt, And you never can tell how close you·re, It may be near when it seems a far. So,stick to the fight, When you are hardest hit, It·s when things go wrong, THAT I MUST NOT QUIT!!!!! It·s all about you add math«« Love you, my dear«..
