PROJECT WORK 1 FOR ADDITIONAL MATHEMATICS 2010

calculus in Our Life

NAME :

TEACHER :

I/C :

SEKOLAH :

CONTENT

No Contents Page

1 Introduction

2 Procedure and Findings

3 Further Exploration

4 Conclusion

5 Reflection

INTRODUCTION

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 by the fundamental theorem of

calculus. Calculus is the study of change, in the same way that geometry is the study of shape

and algebra is the study of operations and their application to solving equations. A course in calculus is a

gateway to other, more advanced 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 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 calculi are propositional calculus, variational

calculus, lambda calculus, pi calculus, and join 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 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 fluid, the oblateness of the

earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica.

In other work, he developed series expansions for functions, 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 by Gottfried Wilhelm Leibniz, who was originally

accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His

contribution was to provide 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 – he often 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 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 time, the fundamental theorem of calculus was known.

Acknowledgement

First of all, I would like to say thank you 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 encouraged me to complete this task so that I will

not procrastinate in doing it.

Then I would like to thank to my teacher, Pn Lui for guiding me throughout this

project. Even I had some 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

supposed to do with the project work.

Besides that, my friends who always supporting me. Even this project is

individually but we are cooperated doing this project especially in disscussion 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 work. Thank you everyone.

Objectives

The aims of carrying out this project work are:-

to apply and adapt a variety of 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 students’ 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.

Procedure and Findings

Solution:(a)Function 1

Maximum point (0,4.5) and pass through point (2,4)y=a¿

b=0 , c=4.5

y=a¿

y=a x2+4.5−−−(1)

Substitute (2,4 ) into(1)

4=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125 x2+4.5

Function 2

Maximum point (0, 0.5) and pass through point (2, 0)y=a¿

b=0 , c=0.5

y=a¿

y=a x2+0.5−−−(2)

Substitute (2 ,0 )into (2)

0=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125 x2+0.5

Function 3

Maximum point (2, 4.5) and pass through point (0, 4)y=a¿

b=2 , c=4.5

y=a¿

Substitute (0 ,4 )into (3)

4=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125¿

(b)

Area to be painted= Area of rectangle - Area under the curve¿4 x1−2∫

0

2

(−0.125 x2+0.5 )dx

¿4−2 [−0.125 x3

3+0.5 x ]

0

2

¿4−2( 23−0)

¿223m2

Further Exploration

Solution:

(a)(i) Structure 1

Area=223m2

Volume=Area xThickness

¿223m2 x 0.4m

¿ 1615m3

Cost=1615m3 x RM 840

¿ RM 896

Structure 2

Area=Area of Rectangle−Area of Triangle

¿1mx 4m−12x 4mx0.5m

¿4m2−1m2

¿3m2

Volume=Area xThickness

¿3m2 x 0.4m

¿1.2m3

Cost=1.2m3 x RM 840

¿ RM 1008

Structure 3

Area=Area of Rectangle−Area of Trapezium

¿1mx 4m−(4m+1m)

2x 0.5m

¿4m2−54m2

¿2.75m2

Volume=Area xThickness

¿2.75m2 x 0.4m

¿1.1m3

Cost=1.1m3 x RM 840

¿ RM 924

Structure 4

Area=Area of Rectangle−Area of Trapezium

¿1mx 4m−(2m+4m)

2x 0.5m

¿4m2−1.5m2

¿2.5m2

Volume=Area xThickness

¿2.5m2 x 0.4m

¿1m3

Cost=1m3 x RM 840

¿ RM 840

∴Structure4will cost theminimum¿construct , that is RM 840.

(ii) As the president of the Arts Club, I will decide Structure 4 as the shape of the gate to be constructed. It is because Structure 4 will cost the minimum and it is easier to be constructed compared to Structure 1 which is a curve.

(b) (i)

k (m) Area to be painted(m2)0.00 4 x1−0+4

2x0.5=3

0.25 4 x1−0.25+42

x0.5=2.9375

0.50 4 x1−0.5+42x0.5=2.875

0.75 4 x1−0.75+42

x0.5=2.8125

1.00 4 x1−1+42x 0.5=2.75

1.25 4 x1−1.25+42

x 0.5=2.6875

1.50 4 x1−1.5+42x 0.5=2.625

1.75 4 x1−1.75+42

x 0.5=2.5625

2.00 4 x1−2+42x 0.5=2.5

(ii) There is a pattern in the area to be painted.

The area to be painted decreases as the k increases 0.25m and form a series of numbers:

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=−0.06252.875−2.9375=−0.06252.8125−2.875=−0.06252.75−2.8125=−0.06252.6875−2.75=−0.06252.625−2.6875=−0.0625

2.5625−2.625=−0.06252.5−2.5625=−0.0625

∴ We can deduce that this series of numbers is an Arithmetic Progression (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) Thearea of the concrete structure¿be painted

¿4 x1−(k+4 )

2x0.5

¿4− k4+1

¿3− k4

∴ k→4

k4→1

Areaof concrete structure ¿be painted→3−1

→2m2

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 questions, drawing graphs and some

problem solving, I saw that the usage of calculus is important in daily life.It

is not just widely used in science, economics but also in engineering. In

conclusion, calculus is a daily life nessecities. 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 in the

idea of calculus.

Reflection

After spending countless hours,days and night to finish this project and also sacrificing my time for chatting and movies in this mid year holiday,there are several things that I can say...

Additional Mathematics...From the day I born...From the day I was able to holding pencil...From the day I start learning...And...From the day I heard your name...

I always thought that you will be my greatest obstacle and rival in excelling in my life...But after countless of hours...Countless of days...Countless of nights...

After sacrificing my precious time just for you...Sacrificing my play Time..Sacrificing my Chatting...Sacrificing my Facebook...Sacrificing my internet...

Sacrifing my Anime... Sacrificing my Movies...I realized something really important in you...

I really love you...You are my real friend...You my partner...You are my soulmate...

I LOVE U ADDITIONAL MATHEMATIC