Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Additional mathematics
Additional mathematics
project work 1
project work 1
Nama kt sini
Nama kt sini
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Contents
Contents
Appreciation
Appreciation
Objectives
Objectives
Introduction
Introduction
Procedure and Findings
Procedure and Findings
6-9
6-9
Further Exploration
Further Exploration
10-14
10-14
Conclusion
Conclusion
Reflection
Reflection
References
References
Appendix
Appendix
18
18
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Appreciation
Appreciation
Firstly, I would like to
Firstly, I would like to give a big thanks give a big thanks to my parents for providingto my parents for providing everything, such as money, to buy
everything, such as money, to buy anything that are related to anything that are related to this project work,this project work, their advice and support. Then, I want to thank my teacher; Pn. Cikgu for teaching their advice and support. Then, I want to thank my teacher; Pn. Cikgu for teaching me Additional Mathematics form 5 and guiding me
me Additional Mathematics form 5 and guiding me throughout this project.throughout this project.
Last but not least, my friends who were doing this project with me and Last but not least, my friends who were doing this project with me and sharing our ideas and knowledge. They were helping each other so we can sharing our ideas and knowledge. They were helping each other so we can complete our project without any problems.
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Objectives
Objectives
The objectives of this project work are: The objectives of this project work are:
Apply and adapt a variety of problem-solving strategies to solve problems.Apply and adapt a variety of problem-solving strategies to solve problems.
Develop mathematical knowleDevelop mathematical knowledge through problem solving in a dge through problem solving in a way thatway that increases students’ interest and
increases students’ interest and confidence.confidence.
Develop positive attitude towards Develop positive attitude towards mathematics.mathematics.
Improve thinking skills and Improve thinking skills and creativity.creativity.
Promote efficiency of mathematical communicatPromote efficiency of mathematical communication.ion.
Provide learning environment that stimulates and Provide learning environment that stimulates and enhances effectiveenhances effective learning.
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Introduction
Introduction
Around AD 1000, the
Around AD 1000, the Islamic mathematicianIslamic mathematician Ibn al-Haytham (Alhacen)Ibn al-Haytham (Alhacen) waswas the first to derive the formula for
the first to derive the formula for the sum of the fourth powers of the sum of the fourth powers of anan arithmeticarithmetic progression
progression, using a method that , using a method that is readily generalizable to finding the formulais readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform
for the sum of any higher integral powers, which he used to perform integrationintegration.. In the 11th century, the Chinese
In the 11th century, the Chinese polymathpolymathShen KuoShen Kuodeveloped 'packing'developed 'packing' equations that dealt with
equations that dealt with integration.integration. In the 12th century, the In
In the 12th century, the Indian mathematician,dian mathematician,Bhāskara IIBhāskara II, developed an, developed an
early derivative representing infinitesimal change
early derivative representing infinitesimal change, and he , and he described an early formdescribed an early form of Rolle's Theorem. Also in the 12th century,
of Rolle's Theorem. Also in the 12th century, the Persian mathematicianthe Persian mathematicianSharaf al-Sharaf
al-Dīn al
Dīn al--Tūsī Tūsī discovered the derivative of discovered the derivative of cubic polynomialscubic polynomials, an important result in, an important result in
differential calculus differential calculus..
In the 14th century, Indian mathematician
In the 14th century, Indian mathematician Madhava of SangamagramaMadhava of Sangamagrama,, along with other
along with other mathematician-amathematician-astronomers of the Kerala stronomers of the Kerala School of astronomySchool of astronomy and mathematics, described special cases of Taylor series, which are treated in and mathematics, described special cases of Taylor series, which are treated in the text Yuktibhasa.
the text Yuktibhasa.
In the 19th century,
In the 19th century, calculuscalculuswas put on a much more rigorous footing bywas put on a much more rigorous footing by mathematicians such as
mathematicians such as CauchyCauchy,, RiemannRiemann, and, and WeierstrassWeierstrass. It was also during. It was also during this period that the ideas of
this period that the ideas of calculus were generalized to Euclidean space and thecalculus were generalized to Euclidean space and the complex plane.
complex plane. LebesgueLebesguegeneralized the notion of the integral so that virtuallygeneralized the notion of the integral so that virtually any function has an
any function has an integralintegral, while, whileLaurent SchwartzLaurent Schwartz extendedextendeddifferentiationdifferentiation inin much the same way.
much the same way.
Calculus
Calculus is a ubiquitous topic in most modern high schools and universitiesis a ubiquitous topic in most modern high schools and universities around the world.
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Procedure and Findings
Procedure and Findings
(A) Function I (A) Function I
Maximum point (0,4.5) and pass through point (2,4) Maximum point (0,4.5) and pass through point (2,4)
==−−
22 ++
b=0, c=4.5 b=0, c=4.5
==−−
22 ++ 44..55
==
22 ++ 44..55--- (1)--- (1) Substitute (2,4) into (1) Substitute (2,4) into (1) 4 = 4 =
22
22 ++ 44..55 4 4
++ 44.5 .5 = 4= 4 4 4
==−−
0.50.5
==−
−
..
++
..
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Function IIFunction II
Maximum point (0,0.5) and pass through point (2,0) Maximum point (0,0.5) and pass through point (2,0)
==−−
22 ++
b=0, c=0.5 b=0, c=0.5
==−−
00
22 ++ 00..55
==
22 ++ 00..55--- (2)--- (2) 0 = 0 =
22
22 ++ 00..55 4 4
==−−
0.50.5
==−−
0.1250.125
==−
−
..
++
..
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Function IIIFunction III
Maximum point (2,4.5) and pass through point (0,4) Maximum point (2,4.5) and pass through point (0,4)
==−−
22 ++
b=2, c=4.5 b=2, c=4.5
==−−
22
22 ++ 44..55 --- (3)--- (3) Substitute (0,4) into (3) Substitute (0,4) into (3) 4 = 4 =
00−−
22
22 + 4.5+ 4.5 4 = 4 = 44
++ 44..55 4 4
==−−
0.50.5
==−−
0.1250.125
==−
−
..
−−
++
..
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
(b) (b) = = 44 ×× 11−−
22−−
0.1250.125
22 + 0.5+ 0.5ⅆⅆ
2 2 0 0 = 4 = 4−−
22−−
0.1250.125
3 3 3 3 ++ 00..55
0 0 2 2 = 4 = 4−−
22
2 2 3 3−−
00
= 4= 4−−
4 4 3 3 = =
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Further Exploration
Further Exploration
(a)
(a) (1) (1) Structure Structure
Area = Area =2222 3 3
2 2 Volume = Volume = 2222 3 3 ×× 00..44 = =1616 15 15
3 3 Cost = Cost = 1616 15 15 ×× 884400 = =RM896.00RM896.00Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Structure 2 Structure 2 Area = Area =4 ×4 × 11−−
11 2 2 ×× 4 4 ×× 00..55 = = 33
22 Volume = Volume = 3 3 ×× 00.4.4 = =1.21.2
33 Cost = Cost = 1.1.2 2 ×× 848400 = =RM1008.00RM1008.00Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Structure 3 Structure 3 Area = Area =4 ×4 × 11−−
11 2 2 ××
11 ++ 44
× 0.5× 0.5 = =2.752.75
22 Volume = Volume = 22..7755 ×× 00..44 = =1.11.1
33 Cost = Cost = 1.1.1 1 ×× 848400 = =RM924.00RM924.00Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Structure 4 Structure 4 Area = Area =4 ×4 × 11−−
11 2 2 ××
22 ++ 44
× 0.5× 0.5 = =2.52.5
22 Volume = Volume = 2.2.5 5 ×× 0.0.44 = =11
33 Cost = Cost = 1 ×1 × 884040 = =RM840RM840 Conclusion ConclusionThe minimum cost to construct is
The minimum cost to construct is Structure 4.Structure 4.
(2)
(2) As the president of the Arts Club, I would like to choose structure 4As the president of the Arts Club, I would like to choose structure 4 because it is the cheapest and I
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
(B) (1) (B) (1)
k
k (m) (m) Area Area to to be be painted painted (m(m22))
0 0 33 0.25 2.9375 0.25 2.9375 0.5 2.875 0.5 2.875 0.75 2.8125 0.75 2.8125 1 2.75 1 2.75 1.25 2.6875 1.25 2.6875 1.5 2.625 1.5 2.625 1.75 2.5625 1.75 2.5625 2 2.5 2 2.5
Area to be painted was
Area to be painted was calculated using this formula:calculated using this formula: 44
−−
112
2
44 ++
0.50.5
(B) (2) (B) (2)
Areas to be
Areas to be painted are: 3, painted are: 3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625,2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, and 2.5 This is
and 2.5 This is an Arithmetic Progression (AP) with Common difference,an Arithmetic Progression (AP) with Common difference, d = 2.9375 - 3 = d = 2.9375 - 3 = -0.0625-0.0625 or or d = 2.875 - 2.9375 = d = 2.875 - 2.9375 = -0.0625-0.0625
When k increases by 0.25m, the area to
When k increases by 0.25m, the area to be painted decreases by 0.0625 mbe painted decreases by 0.0625 m
(C) (C)
Area
Area of of concrete concrete structure structure == 4 ×4 × 11
−−
112 2
44 ++
0.50.5
= = 44−−
11 4 4
44 ++
= = 44−−
11−−
11 4 4
= =
33−−
4 4
2 2 When When →→
44Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
CONCLUSION
CONCLUSION
I notice that quadratic function and integration so close in our daily life. Solving I notice that quadratic function and integration so close in our daily life. Solving problems will be easy by using calculus as well as quadratic function. As the result, we can problems will be easy by using calculus as well as quadratic function. As the result, we can calculate and identify problems involving integration (calculus) and quadratic function.
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Reflection
Reflection
I found a lot of information while conducting this project. Moreover, this project I found a lot of information while conducting this project. Moreover, this project encourages the student to think critically to identify and solve problems. It is also encourage encourages the student to think critically to identify and solve problems. It is also encourage student to gather information using the technologies such as the internet, improve thinking skills student to gather information using the technologies such as the internet, improve thinking skills and promote effective mathematical communication.
and promote effective mathematical communication.
Lastly, I proposed this project should be continue because it brings a lot of advantages to Lastly, I proposed this project should be continue because it brings a lot of advantages to the student and also test the student’s understanding in
Additional Mathematics Project Work 1
Additional Mathematics Project Work 1
Reference
Reference
Additional Mathematics TextbooksAdditional Mathematics Textbooks
http://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculus
