is there life after calculus

IS THERE LIFE AFTER CALCULUS?

IS THERE LIFE AFTER CALCULUS?

ACKNOWLEDGEMENT First and foremost, I would like to thank my Additional Mathematics teacher, Cikgu Tang Bet Ti as she gives us important guidance and commitment during this project work. She has been a very supportive figure throughout the whole project. I also would like to give thanks to all my friends for helping me and always supporting me to help complete this project work. They have done a great job at surveying various shops and sharing information with other people including me. Without them this project would never have had its conclusion. For their strong support, I would like to express my gratitude to my beloved parents. Also for supplying the equipment and money needed for the resources to complete this project. They have always been by my side and i hope they will still be there in the future. Last but not least, i would also like to thank all the nice shopkeepers, staffs, and citizens for helping me collect the much needed data and statistics for this. Not forgetting too all the other people who were involved directly or indirectly towards making this project a reality. I thank you all.

INTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK1/2011The aims of carrying out this project work are to enable students to:A. Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems;

B. Experience classroom environments which are challenging, interesting and meaningful and hence improve their thinking skills.

C. Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems

D. Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected

E. Experience a classroom environment that stimulates and enhances effective learning.

F. Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely

G. Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increase interest and confidence

H. Prepare ourselves for the demand of our future undertakings and in workplace

I. Realize that mathematics is an important and powerful tool in solving real-life problems and hence develop positive attitude towards mathematics.

J. Realize the importance and the beauty of mathematics

INTRODUCTION OF CALCULUS

Introduction Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the parabolic. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers. Leibniz (1646-1716) and Newton (1642-1727) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy. Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music. Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lvesque (1875-1941) put definite integration on a firm logical foundation. Lowville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm. In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms. Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev. In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts have led to an increasingly complete algorithmization of the Risch scheme. In the 1980s some progress was also made in extending his method to certain classes of special functions. The capability for definite integration gained substantial power in Mathematica, first released in 1988. Comprehensiveness and accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its integration code. Besides being able to replicate most of the results from well-known collections of integrals (and to find scores of mistakes and typographical errors in them), Mathematica makes it possible to calculate countless new integrals not included in any published handbook.

PART 1

PART 2A car travels along a road and ots velocity- time function is illustrated in Diagram 1. The straight line PQ is parallel to the straight line RS.Diagram 1A. From the graph, findi. The acceleration of the car in the first hour

Solution :

ii. The average speed of the car in the first two hours

=

=

=

B. What is the significance of the position of the graph

i. Above the t-axis

A positive slope (starting point at low time and low position) moves away from the base position.

ii. Bellow the t-axis

A negative slope (starting point at low time but low position) will move back towards to the base position.

C. Using two different methods, find the total distance travelled by the car.

i. Method 1 (Calculate the area under the graph)

Total distance travelled by the car ====

ii. Method 2 (Integration)

1st step: Finding all the equation from the graph

iii. Total distance travelled by the car =

=

=

==190 km

D. Based on the graph, write an interesting story of the journey in not more than 100 words.

Ali was driving his car at 20 km/h on the highway from PJ to Subang. After he started to time his journey, he drove with an acceleration of 60 km/h for one hour, then he drove at 80 km/h for 30 minutes, then he decelerated with a deceleration of 160 km/h for 30 minutes until he reached Puchong where he found that he missed the exit of Subang. Dismayed, he spent 30minutes to calm down at Puchong. He drove back to Subang with a deceleration of 160 km/h for 30 minutes. Then, he drove at 80 km/h for another 30 minutes and completed his journey with an acceleration of 160 km/h for 30 minutes.

PART 3Diagram 3 shows a parabolic satellite disc which is symmetrical a y-axis. Given the diameter of the dish is 8 m and the depth is 1 m.

Diagram 2A. Find the equation of the curve .

Substitute (4, 5) into the equation

B. To find the approximate area under a curve, we can divide the region into several vertical strips and then we add up the areas of all the strips. Using a scientific calculator or any suitable computer software, estimate the area bounded by the curve at (a), the x-axis, x=0 and x=4.

Conjecture: Approximate area obtained in Diagram (3)(iii) will give the closest answer to the actual value of area under curve when compared with the approximate area obtained in Diagram (3)(i) and Diagram (3)(iii).

0

yy = f(x)x1234Diagram 3 (i)

Area of first strip 4.000 0.5=2.000Area of the second strip4.016 0.5=2.008Area of the third strip4.063 0.5=2.032Area of the fourth strip4.141 0.5=2.071Area of the fifth strip4.250 0.5=2.125Area of the sixth strip4.391 0.5=2.196Area of the seventh strip4.563 0.5=2.282Total area =2.000+2.008+2.032+2.071+2.125+2.196+2.282+2.383=17.097 m2

Area of the eighth strip4.766 0.5=2.383y

y = f(x)x01234Diagram 3 (ii)

Area of first strip 4.016 0.5=2.0078Area of the second strip4.063 0.5=2.032Area of the third strip4.141 0.5=2.071Area of the fourth strip4.250 0.5=2.125Area of the fifth strip4.391 0.5=2.196Area of the sixth strip4.563 0.5=2.282Area of the seventh strip4.766 0.5=2.383Total area =2.008+2.032+2.071+2.125+2.196+2.282+2.383+2.500=17.597 m2

Area of the eighth strip 5.000 0.5 =2.500

yy = f(x)x01234Diagram 3 (iii)

Area of first & second strip4.016 1=4.016Area of the third & fourth strip4.141 1=4.141Area of the fifth & sixth strip4.391 1=4.391Area of the seventh & eighth strip4.766 1=4.766

Total area=4.016+4.141+4.391+4.766=17.314 m2

C.

i. Calculate the area under the curve using integration.

Area=====

ii. Compare answer in c (i) with the value obtained in (b). Hence, discuss which diagram gives the best approximate area.

Approximate area obtained in Diagram (3)(i), 17.097 m2 is too small to the area under curve in Answer (C)(i). Approximate area obtained in Diagram (3)(ii), 17.597 m2 is too big to the area under curve in Answer (C)(i). Approximate area obtained in Diagram (3)(iii), 17.314 m2 gives the closest answer to area under curve in Answer (C)(i). Therefore, conjecture is accepted.

iii. Explain how you can improve the value in (c) (ii).

We can improve the value in c (ii) by having more strips from till .

D. Calculate the volume of the satellite dish.

Volume=====

FURTHER EXPLORATION

A gold ring in Diagram 4(a) has the same volume as the solid of the revolution obtained when the shaded region in Diagram 4(b) is rotates about the x-axis. yx0Diagram 4(a)

-0.200.2xyDiagram 4(b)

Find A. The volume of the gold ring

Volume======

B. The cost of the gold needed for the ring(Gold density is . The price of gold is RM 155 per gram)

Mass of the gold==

CONCLUSION

I have done many researches throughout the internet and discussing with a friend who have helped me a lot in completing this project. Through the completion of this project, I have learned many skills and techniques. This project really helps me to understand more about the uses of calculus in our daily life.

This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found. Apart from that, this project encourages the student to work together and share their knowledge.

It is also encourage student to gather information from the Internet, improve thinking skills and promote effective mathematical communication. Last but not least, I proposed this project should be continue because it brings a lot of moral values to the student and also test the students understanding in Additional Mathematics.

Also while I was conducting the project, I have learnt and practiced lots of moral values. Some of them are that we should be patient when doing any work or project. This is to ensure our work is completed by time. We also should be calm when any source that we are trying to find comes out fruitless.

REFLECTION

After spending countless hours, day and night to finish this Additional Mathematics Project, here is what I got is,

Doing this project makes me realize how important Additional Mathematics is. Also, completing this project makes me realize how fun it is and likable is Additional Mathematics

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