1. OBJECTIVES We students taking Additional Mathematics are required to carry out project work while we are in Form 5. This Year the Curriculum Development Division, Ministry Education has prepared two task 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, but each of us are expected to submit an individually written report. Upon completion of the Additional Mathematics Project Work, we are to gain valuable experiences and able to : I. Apply and adapt a variety of problem solving strategies to solveroutine and non-routine problems.II. Experience classroom environments which are challenging,interesting and meaningful and hence improve their thinkingskills. III. Experience classroom environments where knowledge and skillsare applied in meaningful ways in solving real-life problems. IV.Experience classroom environment where expressing onesmathematical thinking, reasoning and communication are highlyencouraged and expected.V.Experience classroom environments that stimulates and enhanceseffective learning. VI.Acquire effective mathematical communication through oral andwriting and to use the language of mathematics to expressmathematical idea correctly and precisely.VII.Enhance acquisition of mathematical knowledge and skillsthrough problem-solving in ways that increase interest andconfidence.VIII. Prepare ourselves for the demand of our future undertakings andin workplace.1

2. IX.Realise that mathematics is an important and powerful tool insolving real-life problems and hence develop positive attitudetowards mathematics.X.Train ourselves not only to be independent learners but also tocollaborate, to cooperate, and to share knowledge in an engagingand healthy environment.XI. Use technology especially the ICT appropriately and effectively.XII.Train ourselves to appreciate the instrinsic values of mathematicsand to become more creative and innovative.XIII. Realize the importance and the beauty of mathematics.We are expected to submit the project work within three weeksfrom the first day is being administered to us. Failure to submitthe written report will result in us not receiving certificate.2 3. APPRECIATION First of all, I would like to say Alhamdulillah, thank you to Allah forgiving me the strength and health to do this project work and finish iton time. Secondly, I would like to thank the principle of SekolahMenengah Kebangsaan Tun Ismail, Tuan Hj Tokijan Bin Hj. Abd Halimfor giving the permission to do my Additional Mathematics ProjectWork. Not Forgetten to my parents for providing everything, such asmoney, to buy anything that are related to this project work, theiradvise, which is the most needed for this project and facilities such asinternet, books, computers, and all that. They also supported me andand encouraged me to complete this task so that I will notprocrastinate in doing it.Then I would like to thank to my Additional Mathematics teacher,Puan Faridah for guiding me through out this project. Even I haddifficulties in doing this task, but she taught me patiently until we knewwhat to do. She tried and tried to teach me until I understand what Imsuppose to do with the project work. Besides that, my friends who always supporting me. Even thisproject is individually but we are cooperate doing this project especiallyin discussion and sharing ideas to ensure our task will finish completely. Last but not least, any party which involved either directly or indirectin completing this project. Thank you everyone. 3 4. INTRODUCTION [History of Functions]In the 18th and 19th centuries, scientists discovered that theelementary functions--powers, roots, trigonometric functions and theirinverses--had their limitations. They found that solutions for someimportant physical problems--like the orbital motion of planets, theoscillatory motion of suspended chains, and the calculation of thegravitational potential of nearly spherical bodies--could not always bedescribed in a closed form using only elementary functions. Even in therealm of pure mathematics, some quantities--such as thecircumference of an ellipse--were also impossible to discuss in suchterms. Functions describing solutions to these problems were oftenexpressed as infinite series, as integrals, or as solutions to differentialequations.On further investigation, scientists noted that a relatively small numberof these special functions turned up over and over again in differentcontexts. Whats more, they noted that many other problems could besolved in the form of a comparatively simple combination of thesenewer functions with the elementary functions known to the ancients.Functions that cropped up most frequently in scientific calculationswere given names and notations which have come into common usage:Bessel functions, Struve functions, Mathieu functions, the sphericalharmonics, the Gamma function, the Beta function, Jacobi functions,and most of the others appearing on this website.In the second half of the 19th century mathematicians also started toinvestigate these special functions from a purely theoreticalperspective. Alternative representations--as differential equations, 4 5. series, integrals, continued fractions, or other forms--were found formany. Important publications on the topic at the turn of the centuryinclude the four-volume masterpiece on the elliptic functions by J.Tannery and J. Molk (published 1893-1902), containing hundreds ofpages of collected formulas; I. Todhunters treatise on Laplace, Lamand Bessel functions (1875); E. Heines treatise on spherical harmonics(1881) and A. Wangerins work on the same topic (1904).Large tables with numerical values for the special functions also beganto appear, along with three-dimensional "graphs" made of wood orplaster--masterpieces of precision sculpting--showing the behavior offunctions such as P and the Jacobi functions. Many of these models arestill on display in math departments throughout the world, and thegraphics on this website can be thought of as their modern, computer-drawn counterparts.Charles Babbage, who designed but was unable to build the "differenceengine," planned a printing device allowing the machine to generatelarge tables automatically. A Swedish publisher named Georg Scheutzand his son Edvard later built a difference engine that could set type. In1857, the Scheutzes produced a mechanically generated table ofcommon logarithms to five decimal places for the integers from 1 to10,000; each value took about thirty seconds to calculate.Funktionentafeln mit Formeln und Kurven, the first modern handbookof special functions--that is, one containing graphs, formulas, andnumerical tables--was published in 1909 by Eugene Jahnke and FritzEmde. The first text dealing comprehensively with most of the namedspecial functions was E. T. Whittaker and G. N. Watsons A Course ofModern Analysis, 2nd Edition (1915).This popular text consisted of two parts; Part I is a textbook of complexanalysis, while Part II is a handbook of special functions.5 6. In 1939, orthogonal polynomials were given their first detailedtreatment by Gbor Szeg. This work was followed in 1943 by WilhelmMagnus and Fritz Oberhettingers Formeln und Lehrstze fr diespeziellen Funktionen der Mathematischen Physik, the most completecollection of formulas involving special functions yet prepared.The massive Bateman Manuscript Project--the editing for posthumouspublication of Harry Batemans accumulated notes on special functions,which he stored in shoe boxes--was carried out by Arthur Erdlyi,Wilhelm Magnus, Fritz Oberhettinger, and Francesco Tricomi,culminating in 1953 with the classic three-volume work HigherTranscendental Functions. This monumental collection contains notonly formulas, but also derivations, proofs, and historical remarks.(Along with the contents of Higher Transcendental Functions,Batemans shoe boxes held enough material for a two-volume set ofintegral transform tables.)In Higher Transcendental Functions, Erdlyi introduced a new emphasison the importance of hypergeometric functions as an underlying,unifying basis for the development of many of the categories of specialfunctions. Although 2F1 had been extensively studied since the time ofGauss, mathematicians were slow to appreciate the importance of thegeneralized hypergeometric pFq and to recognize the many relationsbetween hypergeometric functions and the special functionsencountered most frequently. Another innovation in HigherTranscendental Functions was the inclusion of number-theoreticalfunctions. 6 7. Parallel to the handbooks dealing with series expansions, differentialequations, functional identities, and so forth of special functions, manyintegral tables were developed in the 20th century.Among these, especially the tables of W. Grbner, N. Hofreiter, A.Erdelyi, W. Magnus, F. Oberhettinger, I. S. Gradshteyn, I.H. Ryshik, H.Exton, H. M. Srivastava, A. P. Prudnikov, Ya. A. Brychkov, and O. I.Marichev are noteworthy.Application of the electronic computer resulted in many massivevolumes containing hundreds of pages of tables for Bessel functions,elliptic integrals, Legendre functions, and so on. An importanthandbook containing graphs, formulas, and compute-generatednumerical data was assembled by Milton Abramowitz and Irene Stegun.This work was published in 1964 by the National Bureau of Standards asthe Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables. Individual chapters were compiled by variousauthors, leading to a certain unevenness in the quality of the materialand its presentation. Nevertheless, the Handbook of MathematicalFunctions remains a standard reference and is still in widespread use.Ironically, the computer, that led to the creation of such mammothnumeric tables is now eliminating the need for them. The readyavailability of computer processing time and technical software nowallows technical users to calculate the values of any needed functionwithout recourse to reference works. Mathematica can calculate everyspecial function on this website to any desired precision for any real orcomplex valuesof the argumentsand parameters.Additionally, Mathematica can symbolically and numerically calculatevalues for integrals or other operations and transformations involvingthese functions, providing far more information than any singlehandbook could possibly tabulate. 7 8. Thereby the need for an even more comprehensive collection of specialfunctions persists. And the Wolfram Functions Site is the mostcomplete such resource today. With tens of thousands of identities,some extending over multiple pages if printed. The website is virtuallyarbitrarily extensible and not bound to the limitations of a printedbook. Updating is easy, so new information can be quickly incorporatedinto the website. 8 9. PART 1(a)Equation 1 : Axis of symmetry, x = 0.General Form ,with c = 175y175 (-50,100) (50,100)0 xGeneral Form 2y axbxc, c 175 2axbx175Passing through (50,100) ,2100a ( 50 ) b ( 50 )1752500 a 50 b 75 ........ (1)Passing through (-50,100) ,2100a ( 50 )b ( 50 )1752500 a 50 b75.......... ( 2 )100 b0b 0.2500 a 75a 0 . 03Quadratic Equation2y 0 . 03 x1759 10. Equation 2 : Axis of symmetry, x = 50.Method 1: General Form, with c =100 (50,175) (0,100) (100,100) 0 50100Completing the square 2ya(x b) c ,b50 , c 175 2ya(x 50 )175Passing through (0,100) 2100a (050 ) 1`75752500 aa 0 . 03Quadratic Equation : 2y 0 . 03 x 175 10 11. Equation 3 : Axis of symmetry, x = 0.General Form, with c =75y75. 0 x(-50,100)(50,100)General Form2yax bxc, c752axbx75Passing through (-50,0) ,20 a ( 50 )b ( 50 )752500 a50 b 75........ (1)Passing through (50,0) ,20 a ( 50 )b ( 50 ) 752500 a50 b 75.......... ( 2 )5000 a150a 0 . 032500 ( 0 . 03 ) 50 b75 50 b150b 3Quadratic Equation2y 0 . 03 x7511 12. (b) Region A Region B50Area of region A =50 0 . 03 x275 dx (refer to equation 3) 503 0 . 03 x75 x 3502 5000 cmArea of region B = 100 x 100= 10000 cm2 .Total surface area = 10 000 + 5 000 = 15 000 cm2 .12 13. PART B(a) Cost of buildingStructure 1V = 1.5 x 0.13= 0.195Cost = 0.195 x 960 = RM 187.20 13 14. Structure 2V=(+ x 100 x 75) x 13= 178 750=0.17875Cost = 0.17875 x 960 = RM 171.6014 15. Structure 3V=(+65(75) + (40)(75)) x 13 = 208 000 =0.208Cost = 0.208 x 960 = RM 199.6815 16. Sturture 4 V=((80)(75)) x 13 = 188 500 = 0.1885 Cost = 0.1885 x 960= RM 180.96 The structure built with minimum cost is structure 2(b) If I am asked to choose the shape of sculpture to be built, I willprefer to the structure 1. Of course, the main reason for me to choosethe shape is due to its beautifulness. For a, a parabolic shape give us asence of smoothness compare to other edges shape. Besides, most ofthe memorial poles are made of this shape. Although use this shape isnot as cheap as using the shape structure 2 and 4, but for me RM 10does not become a burden to our society. 16 17. PART CThe triangle ACE, ABD and BCF are equilateral triangle. (all angle = 60 )y is the area of BDEF.Using formula A = ab sin C,y = [( x 80 x 80 sin 60)-( x x sin 60)-( x (80-x)(80-x) sin 60)] = (3200 -- ) sin 60 = (3200 -- + 80x - 3200) sin 60 = (- + 80x) sin 60 = - sin 60 + 80x sin 6017 18. (b) y=- sin 60 + 80x sin 60 = -x sin 60 + 80 sin 60- sin 60 = m x= X 80 sin 60= cx1 2 3 4 5 6 7 8 68.42 67.55 66.68 65.82 64.95 64.09 63.22 62.35The table above is used to plot the graph = -x sin 60 + 80 sin 60.The graph is shown on the graph paper next page.From the graph,When x = 5.5, = 64.5 y = 64.5 x 5.5 = 354.75 18 19. (c)Determine the maximum area of BDEF.1st method,Completing square of function,y=a y = (-+ 80x) sin 60 y = -1(- 80x + -) sin 60 y = -1 sin 60 + 1600 sin 60 the maximum value of y =q = 1600 sin 60 = 1385.642nd method,Differentiation y =-sin 60 + 80x sin 60= -2x sin 60 + 80 sin 60At turning point, =0-2x sin 60 + 80 sin 60 = 0 -2x sin 60 = -80 sin 60x=x = 40When x = 40,y=- sin 60 + 80(40) sin 60y = 1385.64 19 20. CONCLUSIONAfter doing research, answering questions, drawing graphs and someproblem solving, I saw that the usage othe people who Functions isimportant. Functions is commonly used to help to measure. Especiallyin measure the area of the building. In conclusion, Functions is a dailylife nessecities. Without it, it is harder to measure something. So, weshould thankful of the people who contribute in the idea of makingFunctions. REFLECTIONAfter spending countless hours, days and night to finish this project andalso sacrifing my time on my hobby, there are several things that I cansay..Additional Mathematics.The hardest thing that I had to face.But without you, my life will never complete.It is been about 1 year and half since I found you.I still trying to understand you,Always & ForeverTrying to know you, eventhough I had to sacrifice my whole life.Your Name FOREVER the name on my LIPS.Additional Mathematics. 20 21. REFERENCEAdditional Mathematics FORM 5. 2005. Selangor : Pustaka Kamza. 1998-2012 Wolfram Research, Inc. : InternetADDITIONAL MATHEMATICS . 2011. PETALING JAYA : SASBADI SDN.BHD.TOPGEAR ADDITIONAL MATHEMATICS . 2012.BANDAR BARU BANGI : PENERBITAN PELANGI 21 22. 22