APPRECIATION

Hello and greetings . I am grateful to be able to successfully complete this folio . I express my deepest appreciation and gratitude to the principal of SMK Bandar Baru Salak Tinggi , Puan Norma Binti Daud . Also, lots of appreciation to my Additional Mathematics teacher, namely Encik Azharudin for giving guidance to me . Next , thank you also to my parents for their support and encouragement to me during this assignment is completed .Thanks also to my friends from grade 5 ST 1 especially Adilah, Daryl, Azura, Izzati and Nadhirah which has helped me a lot. Finally, thank you to those who have helped me directly or indirectly through doing this project.

What is differentiation?

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable). It is a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced. The derivative measures the instantaneous rate of change of the function, as distinct from its average rate of change, and is defined as the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero.

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point.

The notion of derivative may be generalized to functions of several real variables. The generalized derivative is a linear map called the differential. Its matrix representation is the Jacobian matrix, which reduces to the gradient vector in the case of real-valued function of several variables.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus

Type of function

Extrema 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. 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 tofind 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.

Mathematical Optimization

MATHEMATICAL OPTIMIZATIONIn 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.

Global & Local Extrema

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

Methods to Find Extrema

Methods to find Extrema

2nd Derivative test1st 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: ifdf/dx (p) > 0, then f(x) is an increasing function at x = p. ifdf/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

En Shahs Sheep Pen

Let X be height Y be width

Total amount of fancing required :-X + X + X + X + Y + Y = 4X + 2Y = 200 --- first equation

Area of the pen :-XY ( height x width)A= XY --- second equationY = 100+2X --- third equation

Substitute equation 3 into equation 2A= x(100-2x)A= 100x2xDifferentiate the equationDA/DX= 100-4X100-4X= 04(25x-x)= 025-x= 0X= 25So when X = 25, Y= 100-2(25) Y= 50The dimension

Max area = 50m x 25m = 1250mRezas Box

Let the side of the square to be cut off be h cmThe volume of open boxV= h(30-2h)V= h(900-120h+4h)V= 900h-120h+4h

Find the maximum valueDV/DH = 900-240h+12h = 12(75-20h+h) = 12(h-15)(h-5)

H-15= 0 , H-5= 0H= 15 H= 5( X is rejected,not belong to domain of V)

The maximum volume/longest possible volumeV=h(30-2h) So, substitute H=5V= 2000cm

The Mall

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.

t/hoursP/number of people

00

1241

2900

31800

42700

53359

63600

73359

82700

91800

10900

11241

120

13241

II) When shopping it up to the peak and the number of visitors at this time

The peak hours with 3600 people in the mall is after 6 hours the mall opens 9:30 a.m. + 6 hours = 3:30 p.m.

III) Estimate the number of visitors in the shopping center at 7:30 pm

7:30 p.m. is 10 hours after the malls opens. Based on the graph, the number of people at the mall at 7:30 p.m. is 900 people IV) Specify the time when the number of visitors in the shopping center has reached 2570

By using formula, The time when the number of people reaches 2570 is at 1.20 pm

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 Optimization

Many 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.

How it started?

LEONID KANTOROVICH

The 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. 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'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.

I) Using all of the imformation given,

Part A Write all of the inequalities which meet the aforementioned constraints.

I. Cost : 100x + 200y 1400II. Space : 0.6x + 0.8y 7.2III. Volume = 0.8x + 1.2y

Part B Construct and shade the region which satisfies all the above constraints

I. Y= -1/2x +7X024681214

Y7654310

II. Y= -3/4x +9X024681012

Y97.564.531.50

II) By using two different methods , find the maximum amount of storage space.

Method 1 Test using corner point of Linear Programming Graph (8, 3), (0, 7), and (12, 0) Volume = 0.8x + 1.2y

1. Coordinate 1 - (8,3) Volume = 0.8(8) + 1.2(3) Volume = 10 cubic meter

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

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

Thus the maximum storage volume is 10 cubic meter.

Method 2 Using simultaneous equation

1) Y= -1/2x + 7 ---first equation2) Y= -3/4x + 9 --- second equation

Substitute Equation 2 into 1

-3/4x +9 = -1/2x +7

X=8Y=3

Applying the value of x and y in formula, Volume = 0.8x + 1.2yThus, the maximum storage volume is 10 cubic meter.

III) The list for all of the cabinet combined and the cost of it.

Cabinet XCabinet YTotal cost (RM)

461600

551500

641400

731300

831400

921300

IV) I would choose Cabinet X and Cabinet Y combined because it has the most total value of cabinets. It also only cost RM1300 which is the cheapest price compared to others. I also choose it because I can get 11 cabinets combined.

Reflection

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

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;

In math you can learn everything,Like maybe youll like comparing,You have to know subtraction,a.k.a brother of addition,You might say I already simplified,so now your work aintjankedified,So now dont think negative,Its better to think positive,Dont stab yourself with a fork,But its better to show your work,My math grades are fat,But not as fat as my cat,Lets get typical,And use a pencil,Add Math is fun!

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