Interpretation of videos

INVERSE FUNCTION

Before we know about the definition of inverse function,we will suppose that a function F(x,y)=0 where function y=f(x) as the Vertical Line T (VLT) and also x=g(y) as the Horizontal Line T (HLT) are have the relation.

And we can solve the function by using function y=f(x) where x=g(x). If we looking for the function of y=x² and then x=g(y) is a horizontal line which intersects the graph in two points.

Let write y= 2x-1

and look at the graph of that function

So, in the x-intersect we find point of (½,0).

look at the line of y=x and we can substitute y=x into the function y = 2x - 1, so : x = 2x - 1

1 + x = 2x

1 = x

And , we have intersection point between line y = 2x – 1 and y = x, of course in point of (1,1).

Now, from these relation we want to solve the equation

2x – 1 = y

2x = y + 1

x = ½(y + 1)

x = ½y + ½

then exchange x into y,and y into x ,so obtain :

y = ½x + ½

We looking back on the graph then we get the other line. Let we graph a line containing of point (1,1) and (0,-1).

We have f(x) = 2x – 1 and g(x) = ½x + ½, so

f(g(x)) = 2(g(x)) – 1

= 2 (½x + ½) – 1

= x + 1 – 1

= x

On the other hand

g(f(x)) = ½(f(x)) + ½

= ½(2x – 1) + ½

= x - ½ + ½

= x

So the important problem of two functions is g = f^-1

f(g(x)) = f(f^-1(x))

= x

g(f(x)) = f^-1(f(x))

= x

The example :

Write the function of y = (x-1 )/(x+2)f

The x-intercept is gonna be equal to 1, and y-intercept is gonna be -1.

The solution is :

y = (x-1 )/(x+2)

y ( x + 2 ) = x - 1

yx + 2y = x – 1

yx – x = -1 – 2y

(y - 1) x = -1 – 2y

x^-1 = (-1-2y) / (y-1) ;

Then exchange x into y,and y into x;so we get:

y^-1 = (-1-2x) / (x-1)

Let; x = 0, y = -1

y = 0, x = -½

There are vertical asymtot x = 1 and horizontal asymtot y = -2

We can see from the figure of function y = (x-1 )/(x+1) and y^-1 = (-1-2x) / (x-1)that the two functions are reflected each other. So, the favorite function to look for the inverses are two function .

Video 1.” Solving the problem”

Question (1)

Figure above shows the graph of y=g(x).If the function h is define by h(x)=g(2x)+2, what is the value of h(1)??

Solution:

First,make the graph,then write the function h(x)=g(2x)+2. To get the value of of g(2),we can look the graph from absis=2,we make an straight line that intersect the function y=g(x),so from here we get g(2)=1.

S0,h(1)=1+2=3;

The solution is 3.

Question (2):

Let function f be define by f(x)=x+1, if 2f(p)=20, what is the value of f(3p)?

Solution:

Looking for f(3p) what is f when x=3p.

Write function f(x)=x+1,

Then write 2 f(p)=20; dividing by 2

f(p)=10

f(p)=f(x)

10=x+1

X=9

After get x=9,substitute to x=3p, x=3.9=27

Then f(3p)=x+1

=27+1=28.

So, value of f(3p) is 28.

Video 2. “Finding the roots of third equation by algebraic long division”.

Example:

Let x-3 a factor of x³-7x-6,

So we can write (x³-7x-6)/ (x-3)

We can find the other factor from that equation by using long division. We still using the method like in the elementary school.

Method:

find a partial quotient of x² by dividing x into x³ to get x²,then multiply x² by the divisor and sustract the product from the dividend. And repeat the process until you either “clear it out” or reach a remainder.

After do this method,we get no remainder. And the solution of (x³-7x-6)/ (x-3) is x²+3x +2.

Since (x³-7x-6)/ (x-3) is no remainder and x-3 is a factor of x³-7x-6, so, x²+3x +2 also a factor of x³-7x-6. Factors of x²+3x +2 are (x+2) and (x+1), and can be written:

x³-7x-6 =(x-3)(x+2)(x+1); setting the factor x³-7x-6=0 ; so

0=(x-3)(x+2)(x+1)

Thus x-3=0 ; x=3

X+2=0 ; x=-2

X+1=0 ; x=-1

So,roots are 3,-2,-1.

Roots of equation:

· 3 roots for this third degree equation.

· Quadratic equation always have at most 2 roots.

· A fourth degree equation

Video 3

“PRECALCULUS”.

The graph of rational function can have discontinuities,because has a polynomial in the denominator.

Example:

f(x)=(x+2) /(x-1) ; when x=1 we will get 3/0, so it will break in graph.

Let, insert x=0 ; f(0)=-2

Insert x=1; f(1)=???(it’s imposible)

But, the rational function don’t always work this way:

Let f(i)=1 / (i +1), this function never 0 because of the denominator is i+1.

But, rational function, the denominator can be zero and polynomial also have smooth / broken rational function.

For example:

Y= (x² -x-6) / (x-3) ; when x=3 so we’ll find 0/0????

So,to get the value of function y we can simplify the numerator.

My Experience..

Tell my friend about Limit Function

On 9th january 2009, I try to explain the material of mathematic to my friend. My friend is Alin. She is my classmate. I tell her about Limit. In senior high school ,we have got this material. So ,she know less about limit. First, I tell her the definition limit function. I tell,” Function f is defined at opened interval that having a, maybe at a have no value definition. Limit f(x) is L for x approach a,can be written:

Lim f(x) =L “.

x-->a

then I tell her about the postulates of limit, I just tell her a part of them. It’s just a introduction to learn about limit. So I don’t explain her of all the limit. In process I explain the limit, we always laughing together. And I try to be serious. I try to think that my friend is the students, and I am the teacher. I can explain easily, because she can catch my explain fast. After I tell her, I give her some exercises. I give her five exercises, and she can do three well, but not of two numbers. Then I give the explaining for her to get the correct answer. So, she can catch my explaining clearly now. I try not to be the real teacher for her, I just try to explain her, and we enjoy it. We learn together. Learn about limit.
From this moment, I can do the activity that can make my friend be clearly in limit. From she know about limit, now she more be professional (hehehe..just kidding..). And this is my experience. Try to be the great teacher (without angry…hehehe..).
This is my experience, maybe I’m not a perfect person..sorry I mean that I’m not the great teacher. And thanks for viewing my blog or thanks for reading.

The concept of limit:
Limit concept of limit is fundamental in understanding the differential calculus and integral calculus,that one of kinds of mathematics.
Definition
Function f is defined at opened interval that having a, maybe at a have no value definition. Limit f(x) is L for x approach a,can be written:
Lim f(x) =L
x-->a
If for every positive number ε, however so tidy will be got positive number δ, so
|f(x) – L| < ε is followed by 0 < |x – a| < δ

The Postulates
Postulate 1

If m and b are constanta, so

Lim (mx+b) = ma +b
x-->a

Postulate 2
If c is constanta, so for every arbitrary number a
Lim c =c
x-->a

Postulate 2
Lim x = a
x-->a

Postulate 4
If lim f(x) =L and lim g(x)= M, so
x-->a x-->a
lim [f(x) + g(x)] = L+M
x-->a

Postulate 5
If lim f(x) =L and lim g(x)= M, so
x-->a
lim [f(x) . g(x)] = L.M
x-->a

Postulate 6
If lim f(x) =L and n is arbitrary positive number, so
x-->a
lim [f(x)]^n = L^N

Postulate 7
If lim f(x) = L and lim g(x) = M, and M is not 0, so
x-->a x-->a
lim [f(x) / g(x)]= L/M
x-->a

The Right Limit and Left Limit

Definition
Function f is defined for every number at interval (a,c). So limit f(x) for x approach a from the right is R. can be write:

Lim f(x) = R
x-->a+

if for every positive number ε, however so tidy will be got positive number δ . so ;
|f(x) – R| < ε is followed by 0 < |x – a| < δ

Definition
Function f is defined for every number at interval (d,a). So limit f(x) for x approach a from the left is L. can be write:

Lim f(x) = L
x-->a-

if for every positive number ε, however so tidy will be got positive number δ . so ;
|f(x) – L| < ε is followed by 0 < |a-x| < δ

By the definition of left and right limit, so, all of postulates that have learn before is used for right limit and left limit.

Source: “Kalkulus I.by Drs.Soemoenar”

TRANSLATE THE MATHEMATIC ARTICLES

1. Translate an English mathematics article into Indonesian

Linear Functions

Linear function are functions that have x as the input variable, and x is raised only to the first power. Such functions look like the ones in the above graphic. Notice that x is raised to the power of 1 in each equation. Functions such as these yield graphs that are straight lines, and, thus, the name linear.

Equation of a Line-Slope-Intercept Form

y = mx + b

Above is a program that will help you visualize how changing the values for the slope, m , and the y-intercept, b , will affect the graph of the equation y = mx + b .

Notice that when the slope, m , is positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right.

When the slope is negative, the line slants downward to the right, and, as the slope becomes more and more negative, the line will slant downward steeper and steeper to the right.

Also, notice that when the y-intercept, b , is positive, the line crosses the y-axes above y = 0. When b is negative, the line crosses the y-axis somewhere below y = 0. In fact, b is the value on the y-axis where the line passes through this axis.The line intercepts, or crosses, the y-axis here, and, therefore, b is called the y-intercept.

Summary of Details

This linear function:

f(x) = mx + b

May be graphed on the x, y plane as this equation

y = mx + b

· This equation is called the slope-intercept form for a line.

· The graph of this equation is a straight line.

· The slope of the line is m .

· The line crosses the y-axis at b .

· The point where the line crosses the y-axis is called the y-intercept.

· The x, y coordinates for the y-intercept are (0, b ).

Equation of a Line-Point-slope form

y = m(x - x ₁ ) + y ₁ y

Above is a program that will help you visualize how changing the values for the point, ( x ₁ , y ₁ ), and for the slope, m , will affect the graph of the equation y = m(x - x ₁ ) + y ₁

Summary of Details

This linear function:

f(x) = m(x - x ₁ ) + y ₁

May be graphed on the x, y plane as this equation:

y = m(x - x ₁ ) + y ₁

This equation is often also written as:

y - y ₁ = m(x - x ₁ )

· This equation is called the point-slope form for a line.

· The graph of this equation is a straight line.

· A known point on the line is (x ₁ , y ₁ ) .

· The slope of the line is m .

Equation of a Line-General Form

Ax + By + C = 0 or y = (-A/B)x + (-C/B)

Above is a program that will help you visualize how changing the values for the variables A , B , and C will affect the graph of the equation y = (-A/B)x + (-C/B) . Notice that the slope is equal to the opposite of A divided by B , that is, (-A/B) . Also, notice that the y-intercept is equal to the opposite of C divided by B , that is, (-C/B) .

Summary of Details

This linear function:

f(x) = (-A/B)x + (-C/B)

May be graphed on the x, y plane as this equation:

y = (-A/B)x + (-C/B)

This equation is often also written as:

Ax + By + C = 0

· This equation is called the general form for a line.

· The graph of this equation is a straight line.

· The slope of the line is (-A/B) .

· The y-intercept is (-C/B) .

Translate:

Fungsi Linear

Fungsi linear adalah fungsi yang memiliki x sebagai variavel, dan x dimunculkan hanya sebagai suatu pangkat berderajat satu.
Fungsi tersebut disajikan dalam grafik garis lurus dan dinamakan linear.

Persamaan Garis-Gradien-Bentuk Intersept

y= mx+b

program diatas yang akan membantu anda memvisualisasikan bagaimana mengubah nilai untuk gradient,m, dan intersept_y, b, akan mempengaruhi grafik dari persamaan y= mx +b.

perhatikan bahwa jika slope(gradient),m, adalah positif, garis akan naik ke kanan. Bila m semakin besar (positif) maka garis akan semakin miring ke kanan ke atas. Jika slopenya negative, garis akan semakin ke bawah ke kanan,dan jika slopenya semakin negative garis akan semakin condong ke bawah dank e arah kanan.

Jika diperhatikan,jika intersept-y,b, itu positif, garis yang melewati sumbu y diatas y=0. Bila b negative, garis yang melewati sumbu y disembarang tempat di bawah y=o.b adalah nilai pada sumbu y diamna garis melewati sumbu ini. Garis memotong,atau melewati sumbu y disini,oleh sebab itu b dinamakan intersept-y.

Ringkasan rincian

Fungsi linear ini:

f(x)= mx+b

dapat digambarkan pada bidang x,y sebagai persamaan

y= mx+b

· Persamaan ini disebut persamaan garis dalam bentuk intersept-gradien.

· Grafik dari persamaan tersebut adalah garis lurus.

· Kemiringan garis adalah m.

· Garis melewati sumbu-y di b.

· Titik dimana garis melewati sumbu-y disebut y-intersept.

· Koordinat (x,y) yang memotong y adalah (0,b).

PERSAMAAN GARIS-TITIK-BENTUK GRADIEN.

y= m(x-x₁)+ y₁ atau y-y₁= m(x-x₁)

diatas adalah sebuah program yang akan membantu anda memvisualisasikan bagaimana mengubah nilai untuk titik (x₁,y₁) dan gradien m akan mempengaruhi grafik persamaan y= m(x-x₁)+ y₁

Ringkasan rincian

Fungsi linear ini

f(x)= m(x-x₁)+ y₁

pada bidan (x,y) dapat digambarkan sebagai persamaan

y= m(x-x₁)+ y₁

juga dapat ditulis

y-y₁= m(x-x₁)

· Persamaan garis ini dinamakan persamaan garis dalam bentuk gradient titik.

· Grafik dari persamaan garis ini adalah garis lurus.

· Diketahui titik pada garis adalah (x₁,y₁).

· Kemiringan garis adalah m.

BENTUK UMUM PERSAMAAN GARIS

Ax + By + C = 0 or y = (-A/B)x + (-C/B)

Diketahui bahwa gradient dari garis tersebut = -A/B, juga dapat diketahui bahwa intersept-y = -C/B.

Ringkasan rincian

Fungsi linear ini

f(x) = (-A/B)x + (-C/B)

dapat digambarkan pada bidang (x,y) sebagai:

y = (-A/B)x + (-C/B)

persamaan ini juga sering ditulis:

Ax + By + C = 0

· Persamaan ini disebut bentuk umum dari persamaan garis.

· Grafik persamaan adalah garis lurus.

· Kemiringan garis adalah (-A/B)

· Intersept-y adalah (-C/B).

1. Translate an Indonesian mathematics article into English

Geometri Euclid

.

Geometri Euclid merupakan sebuah sistem matematik yang disumbangkan oleh seorang ahli bernama Euclid dari Alexandria. Teks Euclid, Elements merupakan sebuah kajian sistematik yang terawal mengenai geometri. Ia sudah menjadi salah satu buku-buku yang paling berpengarh di dalam sejarah, sama banyaknya dengan kaedahnya yang mempunyai isi kandungan matematik. Kaedah cara yang mengandungi andaian satu set aksiom secara intuitif yang sangat menarik, dan kemudiannya membuktikan banyak usul (teorem-teorem) daripada aksiom-aksiom berkenaan. Walaupun banyak daripada keputusan-keputusan oleh Euclid sudah dinyatakan oleh ahli-ahli matematik Yunani sebelumnya, Euclid merupakan orang yang pertama untuk menunjukkan bagaimana usul-usul ini diletakkan secara sempurna membentuk satu deduksi dan sistem logik yang komprehensif.

Buku Elements ini bermula dengan geometri satah, yang masih lagi diajar di sekolah menengah sebagai satu sistem aksioman dan contoh-contoh pembuktian formal yang pertama. Kemudiannya, Elements merangkumi geometri pepejal dalam tiga dimensi, dan seterusnya geometri Euclid telah dipanjangkan kepada satu bilangan dimensi yang terhingga. Kebanyakan daripada Elements menyatakan keputusan-keputusan dalam apa yang kini disebut sebagai teori nombor, yang boleh dibuktikan menerusi kaedah geometri.

Selama dua ribu tahun, kata adjektif "Euclid" tidak diperlukan kerana pada masa itu tiada geometri lain dapat dibayangkan. Aksiom-aksiom Euclid nampak seperti sangat jelas sehinggakan apa-apa teorem lain yang dibuktikan daripadanya dianggap benar secara mutlak. Hari ini, bagaimanapun, banyak geometri bukan Euclid sudah diketahui, yang pertamanya telah dijumpai pada awal abad ke-19. Ia juga tidak boleh diambil mudah bahawa geometri Euclid hanya menggambarkan ruang fizikal. Satu implikasi daripada teori Einstein mengenai teori kerelatifan umum bahawa geometri Euclid merupakan satu anggaran yang baik kepada sifat-sifat ruang fizikal hanyak sekiranya medan graviti tidak terlalu kuat.

Pendekatan aksioma

Geometri Euclid merupakan satu sistem aksioman, yang mana semua teorem ("penyataan benar") adalah diambil daripada satu bilangan aksiom-aksiom yang terhingga. Pada permulaan buku Elements yang pertama, Euclid memberikan lima postulat (aksiom):

1. Apa-apa dua titik boleh dihubungkan dengan satu garis lurus.
2. Apa-apa tembereng garis lurus boleh dipanjangkan di dalam satu garis lurus.
3. Satu bulatan boleh dilukis dengan menggunakan satu garis lurus sebagai jejari dan satu lagi titik hujung sebagai pusat.
4. Semua sudut serenjang adalah kongruen.
5. Postulat selari. Jika dua garis bersilangan dengan yang ketiga dalam satu cara yang jumlah sudut dalaman adalah kurang daripada satu lagi, maka dua garis ini mesti bersilangan di atas satu sama lain sekiranya dipanjangkan secukupnya.

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.

The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.

Axiomatic approach

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a finite number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms):

1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

How to Express Mathematics

• Engaging

definition: take a part / be busy in something
sentence: Engaging the pythagoras theorem to get the solution of this questions

• sustainable

definition: keep alive/ in existing
sentence: Mr.Dani using the algorithmic sustainable design to solve his job.

• nationality

definition: membership of a particular nation
sentence : Lagrange who find the mecanique analytique is a person with French nationality.

• Jut Out

definition: stand out(from the surrounding surface )
sentence: the building which jut out to the river are broken because of the flood.