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Additional Mathematics Project Work 1Additional mathematics project work 1Nama kt sini 480123-3912-1329 5sc1 Pn. Cikgu Sekolah Menengah Kebangsaan ……….1Additional Mathematics Project Work 1ContentsAppreciation Objectives Introduction Procedure and Findings Further Exploration Conclusion Reflection References Appendix3 4 5 6-9 10-14 15 16 17 182Additional Mathematics Project Work 1AppreciationFirstly, I would like to give a big thanks to my parents for providing everything
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Additional Mathematics Project Work 1
1
Additional mathematics
project work 1
Nama kt sini
480123-3912-1329
5sc1
Pn. Cikgu
Sekolah Menengah Kebangsaan ……….
Additional Mathematics Project Work 1
2
Contents
Appreciation 3
Objectives 4
Introduction 5
Procedure and Findings 6-9
Further Exploration 10-14
Conclusion 15
Reflection 16
References 17
Appendix 18
Additional Mathematics Project Work 1
3
Appreciation
Firstly, I would like to give a big thanks to my parents for providing
everything, such as money, to buy anything that are related to this project work,
their advice and support. Then, I want to thank my teacher; Pn. Cikgu for teaching
me Additional Mathematics form 5 and guiding me throughout this project.
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
complete our project without any problems.
Additional Mathematics Project Work 1
4
Objectives The objectives of this project work are:
Apply and adapt a variety of problem-solving strategies to solve problems.
Develop mathematical knowledge through problem solving in a way that
increases students’ interest and confidence.
Develop positive attitude towards mathematics.
Improve thinking skills and creativity.
Promote efficiency of mathematical communication.
Provide learning environment that stimulates and enhances effective
learning.
Additional Mathematics Project Work 1
5
Introduction
Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was
the first to derive the formula for the sum of the fourth powers of an arithmetic
progression, using a method that is readily generalizable to finding the formula
for the sum of any higher integral powers, which he used to perform integration.
In the 11th century, the Chinese polymath Shen Kuo developed 'packing'
equations that dealt with integration.
In the 12th century, the Indian mathematician, Bhāskara II, developed an
early derivative representing infinitesimal change, and he described an early form
of Rolle's Theorem. Also in the 12th century, the Persian mathematician Sharaf al-
Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in
differential calculus.
In the 14th century, Indian mathematician Madhava of Sangamagrama,
along with other mathematician-astronomers of the Kerala School of astronomy
and mathematics, described special cases of Taylor series, which are treated in
the text Yuktibhasa.
In the 19th century, calculus was put on a much more rigorous footing by
mathematicians such as Cauchy, Riemann, and Weierstrass. It was also during
this period that the ideas of calculus were generalized to Euclidean space and the
complex plane. Lebesgue generalized the notion of the integral so that virtually
any function has an integral, while Laurent Schwartz extended differentiation in
much the same way.
Calculus is a ubiquitous topic in most modern high schools and universities
around the world.
Additional Mathematics Project Work 1
6
Procedure and Findings
(A) Function I
Maximum point (0,4.5) and pass through point (2,4) 𝑦 = 𝑎 𝑥 − 𝑏 2 + 𝑐 b=0, c=4.5 𝑦 = 𝑎 𝑥 − 𝑂 2 + 4.5 𝑦 = 𝑎𝑥2 + 4.5 ---------- (1) Substitute (2,4) into (1) 4 = 𝑎 2 2 + 4.5 4𝑎 + 4.5 = 4 4𝑎 = −0.5 𝒚 = −𝟎. 𝟏𝟐𝟓𝒙𝟐 + 𝟒. 𝟓
Additional Mathematics Project Work 1
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Function II
Maximum point (0,0.5) and pass through point (2,0) 𝑦 = 𝑎 𝑥 − 𝑏 2 + 𝑐 b=0, c=0.5 𝑦 = 𝑎 𝑥 − 0 2 + 0.5 𝑦 = 𝑎𝑥2 + 0.5 ---------- (2) 0 = 𝑎 2 2 + 0.5 4𝑎 = −0.5 𝑎 = −0.125 𝒚 = −𝟎. 𝟏𝟐𝟓𝒙𝟐 + 𝟎. 𝟓
Additional Mathematics Project Work 1
8
Function III
Maximum point (2,4.5) and pass through point (0,4) 𝑦 = 𝑎 𝑥 − 𝑏 2 + 𝑐 b=2, c=4.5 𝑦 = 𝑎 𝑥 − 2 2 + 4.5 ---------- (3) Substitute (0,4) into (3) 4 = 𝑎 0 − 2 2 + 4.5 4 = 4𝑎 + 4.5 4𝑎 = −0.5 𝑎 = −0.125 𝒚 = −𝟎. 𝟏𝟐𝟓 𝒙 − 𝟐 𝟐 + 𝟒. 𝟓
Additional Mathematics Project Work 1
9
(b)
= 4 × 1 − 2 −0.125𝑥2 + 0.5 ⅆ𝑥
2
0
= 4 − 2 −0.125𝑥3
3+ 0.5𝑥
0
2
= 4 − 2 2
3− 0 = 4 −
4
3
= 𝟐𝟐
𝟑𝒎𝟐
Additional Mathematics Project Work 1
10
Further Exploration
(a) (1) Structure 1
Area = 22
3𝑚2
Volume = 22
3× 0.4
= 16
15𝑚3
Cost = 16
15× 840
= RM896.00
Additional Mathematics Project Work 1
11
Structure 2
Area = 4 × 1 −1
2× 4 × 0.5
= 3𝑚2 Volume = 3 × 0.4 = 1.2𝑚3 Cost = 1.2 × 840 = RM1008.00
Additional Mathematics Project Work 1
12
Structure 3
Area = 4 × 1 −1
2× 1 + 4 × 0.5
= 2.75𝑚2
Volume = 2.75 × 0.4 = 1.1𝑚3 Cost = 1.1 × 840 = RM924.00
Additional Mathematics Project Work 1
13
Structure 4
Area = 4 × 1 −1
2× 2 + 4 × 0.5
= 2.5𝑚2 Volume = 2.5 × 0.4 = 1𝑚3 Cost = 1 × 840 = RM840 Conclusion The minimum cost to construct is Structure 4. (2) As the president of the Arts Club, I would like to choose structure 4 because it is the cheapest and I can minimize the construction cost.
Additional Mathematics Project Work 1
14
(B) (1)
k (m) Area to be painted (m2)
0 3
0.25 2.9375 0.5 2.875
0.75 2.8125 1 2.75
1.25 2.6875
1.5 2.625
1.75 2.5625
2 2.5
Area to be painted was calculated using this formula: 4 −1
2 4 + 𝑘 0.5
(B) (2) Areas to be painted are: 3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, and 2.5 This is an Arithmetic Progression (AP) with Common difference, d = 2.9375 - 3 = -0.0625 or d = 2.875 - 2.9375 = -0.0625 When k increases by 0.25m, the area to be painted decreases by 0.0625 m (C)
Area of concrete structure = 4 × 1 −1
2 4 + 𝑘 0.5
= 4 −1
4 4 + 𝑘
= 4 − 1 −1
4𝑘
= 3 −𝑘
4 𝑚2
When 𝑘 → 4 1
4𝑘 → 1
∴ Area → 2𝑚2
Additional Mathematics Project Work 1
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CONCLUSION
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
calculate and identify problems involving integration (calculus) and quadratic function.
Additional Mathematics Project Work 1
16
Reflection
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
student to gather information using the technologies such as the internet, improve thinking skills
and promote effective mathematical communication.
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.
Additional Mathematics Project Work 1
17
Reference
Additional Mathematics Textbooks
http://en.wikipedia.org/wiki/Calculus
http://en.wikipedia.org/wiki/Integration_(mathematics)
Additional Mathematics Project Work 1
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Appendix