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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