Additional Mathematics Project Work 2015

Name : Danesh VettriyapanClass : 5 Kreatif Teacher : PN.Chi Wai FenSchool : SMK Seri Serdang

Contents

A word of gratitude Objective What is function Type of function Extrema of function Pierre De Fermat Fermats Theorem

Part 1 Mathematical optimization Global and Local Extrema Methods to find Extrema 1st Derivative test 2nd Derivative test

Part 2 Encik Shahs sheep pen Rezas box The mall Linear programming Application in real life How it started Reflection

A Word of Gratitude

I would like to express my deepest gratitude to God for guiding me and providing me the will to complete this project. Without his blessings and guidance ,this project would have never been possible to accomplish.

First and foremost I would like to express my thanks to my Additional Mathematics teacher, Puan Seleyza Binti Sharir for guiding me all the way through the process of accomplishing this project. Her patience and advice helped me accomplish this project flawlessly.

Secondly, I would like to thank my parents as they provided me with the funds that I need to accomplish this project without any hiccups. Other than that, their never ending support gave me motivation to continue on with this project. Last but not least, I would like to thank all my friends that had helped me along the way. The group discussions had made this project richer with information. Also,thank you to those who were involved directly or indirectly during the making of this project.

Objective

Every form 5 student taking additional mathematics is required to carry out a project work. Upon completion of the project, it is hoped that students will gain some valuable experiences and able to: Apply and adapt a variety of problem solving strategies to solve routine and non-routine problems. Experience classroom environments which are challenging, interesting and meaningful hence improve their thinking skills. Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems. Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected. Experience classroom environments that stimulate and enhance effective learning. Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely. Enhance acquisition of mathematical knowledge and skills through problem-solving in ways that increases interest and confidence. Prepare students for the demands of their future undertakings and in workplace. Realise that mathematics is an important and powerful tool in solving real life problems and hence develop positive attitude towards mathematics. Train themselves not only to be independent learners but also to collaborate, to cooperate, and to share knowledge in engaging and healthy environment. Use technology especially the ICT appropriately and effectively. Train themselves to appreciate the intrinsic values of mathematics and to become more creative and innovative. Realise the importance and the beauty of mathematics.

What is Function?

Inmathematics, afunctionis arelationbetween asetof inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real numberxto its squarex2. The output of a function f corresponding to an inputxis denoted byf(x) (read "fofx"). In this example, if the input is 3, then the output is 9, and we may writef(3) = 9. Likewise, if the input is 3, then the output is also 9, and we may writef(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The inputvariable(s)are sometimes referred to as the argument(s) of the function.Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by aformulaoralgorithmthat tells how to compute the output for a given input. Others are given by a picture, called thegraph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as theinverseto another function or as a solution of a differential equation.The input and output of a function can be expressed as anordered pair, ordered so that the first element is the input (ortupleof inputs, if the function takes more than one input), and the second is the output. In the example above,f(x) =x2, we have the ordered pair (3, 9). If both input and output arereal numbers, this ordered pair can be viewed as theCartesian coordinatesof a point on the graph of the function.

Type of function

In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known collectively as EXTREMA (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, ADEQUALITY, for finding the maxima and minima of functions As defined in set theory, the maximum and minimum of a set the greatest and least elements in the set. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.The function x2 has a unique global minimum at x = 0. Extrema of Fucntion

The function x3 has no global minima or maxima. Although the first derivative (3x2) is 0 at x = 0, this is an ininflection point. The function x-x has a unique global maximum over the positive real numbers at x = 1/e. The function x3/3 x has first derivative x2 1 and second derivative 2x. Setting the rst derivative to 0 and solving for x gives stationary points at 1 and +1. From the sign of the second derivative we can see that 1 is a local maximum and +1 is a local minimum. Note that this function has no global maximum or minimum. The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. The function cos(x) has infinitely many global maxima at 0, 2, 4, , and infinitely many global minima at , 3, . The function 2 cos(x) x has infinitely many local maxima and minima, but no global maximum or minimum.

The function cos(3x)/x with 0.1 x 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. The function x3 + 3x2 2x + 1 defined over the closed interval (segment) [4,2] has a local maximum at x = 1153, a local minimum at x = 1+153, a global maximum at x = 2 and a global minimum at x = 4. FERMAT'S THEOREM.

Pierre De Fermat

PIERRE DE FERMAT ; 17 August 1601 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to innitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of nding the greatest and the smallest ordinates of curved lines, which is analogous to that of the dierential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica

Fermats Theorem

PIERRE DE FERMAT developed the technique of adequality (adaequalitas) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in mathematical analysis. According to Andr Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word (parisots) to refer to an approximate equality. Claude Gaspard Bachet de Mziriac translated Diophantus's Greek word into Latin as adaequalitas.[citation needed] Paul Tannery's French translation of Fermats Latin treatises on maxima and minima used the words adquation and adgaler.

Fermat used adequality first tofi nd maxima of functions, and then adapted it to find tangent lines to curves. To find the maximum of a term p(x), Fermat equated (or more precisely adequated) p(x) and p(x+e) and after doing algebra he could cancel out a factor of e, and then discard any remaining terms involving e. To illustrate the method by Fermat's own example, consider the problem of fi nding the maximum of p(x)=bx-x^2. Fermat adequated bx-x^2 with b(x+e)-(x+e)^2=bx-x^2+be-2ex-e^2.

Part one

In mathematics, computer science, operations research, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or a set of constraints), including a variety of different types of objective functions and different types of domains.

A real-valued function f defined on a domain X has a global maximum point at x if f(x*) _ f(x) for all x in X. Similarly, the function has a global (absolute) minimum point at x if f(x*) _ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.Global and Local Extrema

If the domain X is a metric space then f is said to have a local ( relative) maximum point at the point x if there exists some _ > 0 such that f(x*) _ f(x) for all x in X within distance _ of x*. Similarly, the function has a local minimum point at x if f(x*) _ f(x) for all x in X within distance _ of x*. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Note that a global maximum point is always a local maximum point, and similarly for minimum points. In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if, for all x in X with x* _ x, we have f(x*) > f(x), and x is a strict local maximum point if there exists some _ > 0 such that, for all x in X within distance _ of x with x* _ x, we have f(x*) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed (and bounded) interval of real numbers

Methods to find extrema

Methods to find Extrema

1st Derivative test2nd Derivative test

1st Derivative Test

The first derivative of the function f(x), which we write as f(x) or as df/dx is the slope of the tangent line to the function at the point x. To put this in non-graphical terms, the first derivative tells us how whether a function is increasing or decreasing, and by how much it is increasing or decreasing. This information is reflected in the graph of a function by the slope of the tangent line to a point on the graph, which is sometimes describe as the slope of the function. Positive slope tells us that, as x increases, f(x) also increases. Negative slope tells us that, as x increases, f(x) decreases. Zero slope does not tell us anything in particular: the function may be increasing, decreasing, or at a local maximum or a local minimum at that point. Writing this information in terms of derivatives, we see that:

if df/dx (p) > 0, then f(x) is an increasing function at x = p. if df/dx (p) < 0, then f(x) is a decreasing function at x = p. if df/dx (p) = 0, then x = p is called a critical point of f(x), and we do not know anything new about the behaviour of f(x) at x = p.

2nd Derivative Test

In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.The test states: if the function f is twice differentiable at a critical point x (i.e. f'(x) = 0), then:If f (x) < 0 then \ f has a local maximum at \ x.If f (x) > 0 then \ f has a local minimum at \ x.If f (x) = 0 the test is inconclusive.

i-Think Map

FURTHER EXPLORATION (A)

Linear programming

Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of infinitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.

Application in real life

Crew Scheduling

An airline has to assign crews to its flights. Make sure that each flight is covered. Meet regulations, eg: each pilot can only fly a certain amount each day. Minimize costs, eg: accommodation for crews staying overnight out of town, crews deadheading.Would like a robust schedule. The airlines run on small profit margins, so saving a few percent through good scheduling can make an enormous difference in terms of profitability. They also use linear programming for yield management.

Portfolio OptimizationMany investment companies are now using optimization and linear programming extensively to decide how to allocate assets. The increase in the speed of computers has enabled the solution of far larger problems, taking some of the guesswork out of the allocation of assets.

History

LEONID KANTOROVICHThe problem of solving a system of linear inequalities dates back at least as far as Fourier, who in 1827 published a method for solving them, and after whom the method of FourierMotzkin elimination is named. The first linear programming formulation of a problem that is equivalent to the general linear programming problem was given by Leonid Kantorovich in 1939, who also proposed a method for solving it. He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. About the same time as Kantorovich, the Dutch-American economist T. C. Koopmans formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 Nobel prize in economics. In 1941, Frank Lauren Hitchcock also formulated transportation problems as linear programs and gave a solution very similar to the later Simplex method; Hitchcock had died in 1957 and the Nobel prize is not awarded posthumously. During 1946-1947, George B. Dantzig independently developed general linear programming formulation to use for planning problems in US Air Force. In 1947, Dantzig also invented the simplex method that for the first time efficiently tackled the linear programming problem in most cases. When Dantzig arranged meeting with John von Neumann to discuss his Simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent. Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948. Postwar, many industries found its use in their daily planning. Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the observable universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the simplex algorithm. The theory behind linear programming drastically reduces the number of possible solutions that must be checked. The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems.

Ive found a lot of information while conducting this Additional Mathematics project. Ive learnt the uses of function in our daily life.Reflection

Apart from that, Ive learnt some moral values that can be applied in our daily life. This project has taught me to be responsible and punctual as I need to complete this project in a week. This project has also helped in building my confidence level. We should not give up easily when we cannot find the solution for the question.

Then, this project encourages students to work together and share their knowledge. This project also encourages students to gather information from the internet, improve their thinking skills and promote effective mathematical communication.

Lastly, I think this project teaches a lot of moral values, and also tests the students understanding in Additional Mathematics. Let me end this project with a poem;

Additional mathsYou are like a very bright star..Smiling at me..Every time teacher teaches me bout youAdditional mathsWho says you so difficult?Who says you kill them off?Who says you hurtparents feeling?You, one ofthe greatest creature of God..You, open my eyes..Telling me little by little bout The Knower of AllKnow, I feel so gratefulThat He has given me the chance to get to know youAdditional Maths, I love you

FURTHER EXPLORATION (B)

Cabinet X Cabinet Y

Cost(RM)100200

Space(m)0.60.8

Volume(m)0.81.2

Let When (0,7)When

Let When When

Let When When

a).The 3 inequalities which satisfy all the constraints,(i) (ii) (iii)

b(i). 024681214

7654310

(ii). 024681012

97.564.531.50

(ii). Maximum storage volumeMethod 1Test using corner point of Linear Programming Graph (8, 3), (0, 7), and (12, 0) Volume = Coordinate 1 - (8,3) Volume = 0.8(8) + 1.2(3) Volume = 10 cubic meter

Coordinate 2 - (0,7) Volume = 0.8(0) + 1.2(7) Volume = 8.4 cubic meter

Coordinate 3 - (12,0) Volume = 0.8(12) + 1.2(0) Volume = 9.6 cubic meter

Therefore, the maximum storage volume is 10 cubic meter.

Method 2 Using simultaneous equation.

Substitute equation 2 into 1

Apply the values of and into the formula , volume =Therefore, the maximum storage volume is 10m.

Cabinet x Cabinet y Total cost (Rm)

4 51400

5 41300

6 41400

7 31300

8 31400

9 21300

Justification

En. Shahs sheep pen(a) x

200m fencing y

Let the height be x ,width to be y

Total amount of fencing required,

Area of the pen =

When , maximum Area

(b).Let the side of the square to be cut off = cm

30cm

cm1.The volum of the open box is, 2 2) Domain of V= 02x

2. To find maximum value :

x=15 is rejected ,because it does not belong to domain of VTherefore,the maximum volume,

2

t/hoursP/number of people

00

1241

2900

31800

42700

53359

63600

73359

82700

91800

10900

11241

120

13241

Part 3(i) Based on the equation, a table has been constructed where t represents the number of hours starting from 0 hours to 23 hours and P represents the number of people.

Based on the table a graph was sketched using Microsoft Excel

(ii) The peak hours with 3600 people in the mall is after 6 hours of the mall opening , 9.30a.m. + 6hours =3.30p.m.(iii) 7.30p.m. is 10 hours after the mall opens.Based on the graph, the number of people at eh mall at 7.30p.m. 900 people.(iv) By using formula, t=3.84 hours and t=8.16 hours t=3 hours 50 minutes and t=8 hours 10 minutes Therefore,9.30 a.m.+3 hours 50 minutes = 1.20p.m. 9.30 a.m. +8 hours 10 minutes = 5.40 p.m.