We can get mathematical object from the existent and possibility of existent in this world.
How to get mathematical object?
1. Idealization
To consider something is perfect. Just to consider because there is no anything in this world can be perfect.
2. Abstraction
Only to learn some characteristic from object. Look and analysis something from certain side.
Kind of Mathematical Object :
1). Mathematical thinking
Think consistently according first certainty.
2). Mathematical logic
Meaning of mathematical logic :
1. Can differentiate between two object
2. Can to order
Addition, Subtraction, Multiplication, etc are the example of mathematical logic.
Connection between mathematical thinking and Scientific work :
1. The Scientific work must be impersonal
Not concerned with personal.
2. The Scientific work must have a criteria.
Depend on the purpose.
3. The Scientific work must be objective.
The example of Scientific work, they are like proposal, text book, etc.
Selasa, 28 April 2009
Minggu, 05 April 2009
Quadratic Equation, Trigonometry, Limit and My Exercise in Lesson English
Last Meeting, I get some homework from English lesson. I have to posting about quadratic equation, trigonometry, and statistics. And I will try to explain about those.
1.Quadratic Equation
General formula of quadratic equation is a x square plus b x plus c equals zero, where a, b, and c is real number and a is not equals zero.
The value of x which to fulfill quadratic equation are called the roots of quadratic equation.
There are three ways to finish quadratics equation, they are:
1)with factorization
2)with completing perfect quadratics
3)with a formula
I will to try explaining these one by one
1) With factorization
For example there is a quadratic equation: x square plus ten x plus twenty one. And we will find the factors of that quadratic equation.
x square plus ten x plus twenty one
factors of x square are x and x
factors of twenty one which its sum is ten are seven and three
So, factors of x square plus ten x plus twenty one are x plus seven and x plus three.
2) With completing perfect quadratic
x square plus two a x plus a square is form of perfect quadratic, because x square plus two a x plus a square equals x plus a in bracket square. Whereas x square plus two ax is not perfect quadratics, because x square plus two a x is not equals x plus a in bracket square.
For example we will finish quadratic equation: x square plus six x plus two equals zero.
Answer:
x square plus six x plus two equals zero
x square plus six x equals negative two
x square plus six x plus a half of six in bracket square equals a half of six in bracket square minus two
x plus three in bracket square equals seven
x plus three equals plus minus the square root of seven
x equals the square root of seven, minus three, or
x equals negative the square root of seven, minus three
3) With Formula
The roots of quadratic equation a x square plus b x plus c equals zero, can be finished with formula.
x1 equals negative b plus the square root of open bracket b square minus four a c close bracket, all over two a
x2 equals negative b minus the square root of open bracket b square minus four a c close bracket, all over two a
Formula above usually called with abc formula
I have explained about three ways to finish a quadratic equation. And now, I will give example about quadratic equations.
Example:
Derry has a park which its shape is rectangle. Width of rectangle is three less than of the length. If the area of the park are twenty eight meters square. Find the length and width of the park.
Answer:
Given: x is the length of rectangle (park) in meter
x minus three is the width of rectangle (park) in meter
Twenty eight is the area of rectangle (park) in meter square
Equation
The area equals the length times the width
Twenty eight equals x times x minus three in bracket
Twenty eight equals x square minus three x
x square minus three x minus twenty eight equals zero
x minus seven in bracket times x plus four in bracket equals zero.
x equals seven or
x equals negative four (unacceptable)
So the length of the park equals seven meters and the width of the park equals four meters.
2.The Function of Trigonometry
At the plane coordinate, each point P(x,y) determinant the angle of XOP equals alpha. Between alpha, x, y and OP, there are connection as like this :
Sinus alpha equals y over OP
Cosines alpha equals x over OP
Tangents alpha equals y over x
Cotangents alpha equals x over y
Secants alpha equal OP over x
Co secants alpha equal OP over y
If we look the position of point P(x.y) at the coordinate plane, absis and ordinate can get the value negative, positive, and zero. Combination from the value absis and ordinate of point P, will determinant the value for the function of trigonometry.
Function of Trigonometry
1). Formula of cosines alpha plus beta in bracket
Cosines alpha plus beta in bracket equals cosines alpha times cosines beta in bracket minus sinus alpha times sinus beta in bracket.
Example: Cosines seventy five degrees
Answer:
Cosines seventy five degrees equals cosines forty five degrees plus thirty degrees in bracket.
Equals cosines forty five degrees times cosines thirty degrees in bracket minus sinus forty five degrees times sinus thirty degrees in bracket.
Equals a half the square root of two times a half the root square of three in bracket minus a half the square root of two times a half.
Equals a quarter the square root of six minus a quarter the square root of two
Equals a quarter times open bracket the square root of six minus the square root of two close bracket.
2). Formula of Cosines alpha minus beta in bracket
Cosines alpha minus beta in bracket equals cosines alpha times cosines beta in bracket plus sinus alpha times sinus beta in bracket.
Example: Cosines fifteen degrees
Answer:
Cosines fifteen degrees equals cosines forty five degrees minus thirty degrees in bracket.
Equals cosines forty five degrees times cosines thirty degrees in bracket plus sinus forty five degrees timer sinus thirty degrees in bracket.
Equals a half the root square of two times a half the square root of three in bracket plus a half the square root of two times a half in bracket.
Equals a quarter the square root of six plus a quarter the root square of two.
Equals a quarter open bracket the square root of six plus the root square of two close bracket.
3). Formula of Sinus alpha plus beta in bracket
Sinus alpha plus beta in bracket equals sinus alpha times cosines beta in bracket plus cosines alpha times sinus beta in bracket.
Example: Sinus one hundred and five degrees
Answer:
Sinus one hundred and five degrees equals sinus sixty degrees plus forty five degrees in bracket.
Equals sinus sixty degrees times cosines forty five degrees plus cosines sixty degrees times sinus forty five degrees in bracket.
Equals a half the square root of three times a half the square root of two in bracket plus a half times a half the square root of two in bracket.
Equals a quarter the square root of six plus a quarter the square root of two.
Equals a quarter open bracket the square root of six plus the square of two close bracket.
4). Formula of Sinus alpha minus beta in bracket.
Sinus alpha minus beta in bracket equals sinus alpha times cosines beta minus cosines alpha times sinus beta in bracket.
Example : Sinus fifteen degrees
Answer :
Sinus fifteen degrees equals sinus sixty degrees minus forty five degrees in bracket.
Equals sinus sixty degrees times cosines forty five degrees in bracket minus cosines sixty degrees times sinus forty five degrees in bracket.
Equals a half the square root of three times a half the square root of two in bracket minus a half times a half the square root of two in bracket.
Equals a quarter the square root of six minus a quarter the square root of two.
Equals a quarter open bracket the square root of six minus the square root of two close bracket.
5). Formula of tangents alpha plus beta in bracket
Tangents alpha plus beta in bracket equals tangents alpha plus tangents beta all over one minus open bracket tangents alpha times tangents beta close bracket.
6). Formula of tangents alpha minus beta in bracket
Tangents alpha minus beta in bracket equals tangents alpha minus tangents beta all over one plus open bracket tangents alpha times tangents beta close bracket.
My Exercise in Learning English
1.Explain how to proof that the square root of two is irational number!
Answer:
If the square root of two is rational number, so the square root of two equals a over b.Given a and b are prime integer number
a equals the square rot of two times b
a square equals two times b square
Because a square is two times any integer number, so a square is even, so that a is even too.
If a equals two times c
Two times c in bracket square equals two times b square
Four times c square equals two times b square
B square equals two times c square
So that, if b square is even, so is even too. But this is impossible because a and b bilangan bulat prima. So, can be proved that the square root of two is irrational number.
2.Explain how to show or to indicate that the sum angles of triangle is equal to one hundred and eighty degrees!
Answer:
There is a triangle ABC.
We can prove that the sum angles of triangle is equals to one hundred and eighty degrees by drawing a line through one vertex of triangle (through point B), parallel to the side opposite the vertex (parallel to AC). Note that the measure of straight angle at B equals the sum of the measure of the angles of triangle ABC.
Angle A plus angle B plus angle C equals one hundred and eighty degrees.
3.Explain how you are able to get phi!
Answer:
We can get phi from the circle. In a circle, we find the diameter and perimeter of that circle. And if we cut the circle, we can get the perimeter of the circle.So, we can get phi from the perimeter divided by diameter of the circle.
4.Explain how you are able to find the area of region boundered by the graph of y equals x square and y equals x plus two.
Answer:
First we must find the intersection point of y equals x square and y equals x plus two. We can use substitution method to find them.
X square equals x plus two
X square minus x minus two equals zero
X minus two in bracket times x plus one in bracket equals zero
X equals two or x equals negative one.
If x equals two, so y equals four. And if x equals negative one, so y equals one.
So that intersection point of curve y equals x square and the curve y equals x plus two are (two, four) and (negative one, one).
After we know about the intersection point of them, we must draw the curves.
And we can find the area of the region boundered by the graph of y equals x square and y equals x plus two with integral.
The area of the region is A
A equals definite integral x plus two minus x square dx from x equals negative one to x equals two.
A equals a half x square plus two x minus one third x cube, from x equals negative one to x equals two.
A equals a half times four plus two times two minus one third times eight in bracket minus a half plus two times negative one minus one third times negative one in bracket.
A equals two plus four minus eight third in bracket minus a half minus two plus one third in bracket.
A equals six minus eight thurd minus a half lus two minus one third.
A equals eight minus a half of seven.
A equals a half of nine.
5.Explain how you are able to determine the intersection point between the circle x square plus y square equals twenty and y equals x plus one!
Answer:
To determine the point between the circle x square plus y square equals twenty and y equals x plus one, wecan use substitution method.
X square plus y square equals twenty.
And we know that y equals x plus one, so this equation can be substitutioned in the equation above.
So,
X square plus open bracket x plus one close bracket square equals twenty.
X square plus x square plus two x plus one equals twenty.
Two times x square plus two x plus one minus twenty equals zero.
Two times x square plus two x minus nineteen equals zero.
Quadratic equation above can be finished by abc formula
x1 equals negative b plus the square root of open bracket b square minus four a c close bracket, all over two a
x2 equals negative b minus the square root of open bracket b square minus four a c close bracket, all over two a
So,
x1 equals negative two plus the square root of open bracket two square minus four times two times negative nineteen close bracket, all over two times two.
x1 equals negative two plus the square root of one hundred sixty six, all over four.
x1 equals negative two plus the square root of one hundred sixty six, all over four.
And,
x2 equals negative two minus the square root of open bracket two square minus four times two times negative nineteen close bracket, all over two times two.
x2 equals negative two minus the square root of one hundred sixty six, all over four.
If x1 equals negative two plus the square root of one hundred sixty six, all over four, So y1 equals two plus the square root of one hundred sixty six, all over four.
If x2 equals negative two minus the square root of one hundred sixty six, all over four, So y2 equals two minus the square root of one hundred sixty six, all over four.
So, we can find the intersection point of x square plus y square equals twenty and y equals x plus one.
1.Quadratic Equation
General formula of quadratic equation is a x square plus b x plus c equals zero, where a, b, and c is real number and a is not equals zero.
The value of x which to fulfill quadratic equation are called the roots of quadratic equation.
There are three ways to finish quadratics equation, they are:
1)with factorization
2)with completing perfect quadratics
3)with a formula
I will to try explaining these one by one
1) With factorization
For example there is a quadratic equation: x square plus ten x plus twenty one. And we will find the factors of that quadratic equation.
x square plus ten x plus twenty one
factors of x square are x and x
factors of twenty one which its sum is ten are seven and three
So, factors of x square plus ten x plus twenty one are x plus seven and x plus three.
2) With completing perfect quadratic
x square plus two a x plus a square is form of perfect quadratic, because x square plus two a x plus a square equals x plus a in bracket square. Whereas x square plus two ax is not perfect quadratics, because x square plus two a x is not equals x plus a in bracket square.
For example we will finish quadratic equation: x square plus six x plus two equals zero.
Answer:
x square plus six x plus two equals zero
x square plus six x equals negative two
x square plus six x plus a half of six in bracket square equals a half of six in bracket square minus two
x plus three in bracket square equals seven
x plus three equals plus minus the square root of seven
x equals the square root of seven, minus three, or
x equals negative the square root of seven, minus three
3) With Formula
The roots of quadratic equation a x square plus b x plus c equals zero, can be finished with formula.
x1 equals negative b plus the square root of open bracket b square minus four a c close bracket, all over two a
x2 equals negative b minus the square root of open bracket b square minus four a c close bracket, all over two a
Formula above usually called with abc formula
I have explained about three ways to finish a quadratic equation. And now, I will give example about quadratic equations.
Example:
Derry has a park which its shape is rectangle. Width of rectangle is three less than of the length. If the area of the park are twenty eight meters square. Find the length and width of the park.
Answer:
Given: x is the length of rectangle (park) in meter
x minus three is the width of rectangle (park) in meter
Twenty eight is the area of rectangle (park) in meter square
Equation
The area equals the length times the width
Twenty eight equals x times x minus three in bracket
Twenty eight equals x square minus three x
x square minus three x minus twenty eight equals zero
x minus seven in bracket times x plus four in bracket equals zero.
x equals seven or
x equals negative four (unacceptable)
So the length of the park equals seven meters and the width of the park equals four meters.
2.The Function of Trigonometry
At the plane coordinate, each point P(x,y) determinant the angle of XOP equals alpha. Between alpha, x, y and OP, there are connection as like this :
Sinus alpha equals y over OP
Cosines alpha equals x over OP
Tangents alpha equals y over x
Cotangents alpha equals x over y
Secants alpha equal OP over x
Co secants alpha equal OP over y
If we look the position of point P(x.y) at the coordinate plane, absis and ordinate can get the value negative, positive, and zero. Combination from the value absis and ordinate of point P, will determinant the value for the function of trigonometry.
Function of Trigonometry
1). Formula of cosines alpha plus beta in bracket
Cosines alpha plus beta in bracket equals cosines alpha times cosines beta in bracket minus sinus alpha times sinus beta in bracket.
Example: Cosines seventy five degrees
Answer:
Cosines seventy five degrees equals cosines forty five degrees plus thirty degrees in bracket.
Equals cosines forty five degrees times cosines thirty degrees in bracket minus sinus forty five degrees times sinus thirty degrees in bracket.
Equals a half the square root of two times a half the root square of three in bracket minus a half the square root of two times a half.
Equals a quarter the square root of six minus a quarter the square root of two
Equals a quarter times open bracket the square root of six minus the square root of two close bracket.
2). Formula of Cosines alpha minus beta in bracket
Cosines alpha minus beta in bracket equals cosines alpha times cosines beta in bracket plus sinus alpha times sinus beta in bracket.
Example: Cosines fifteen degrees
Answer:
Cosines fifteen degrees equals cosines forty five degrees minus thirty degrees in bracket.
Equals cosines forty five degrees times cosines thirty degrees in bracket plus sinus forty five degrees timer sinus thirty degrees in bracket.
Equals a half the root square of two times a half the square root of three in bracket plus a half the square root of two times a half in bracket.
Equals a quarter the square root of six plus a quarter the root square of two.
Equals a quarter open bracket the square root of six plus the root square of two close bracket.
3). Formula of Sinus alpha plus beta in bracket
Sinus alpha plus beta in bracket equals sinus alpha times cosines beta in bracket plus cosines alpha times sinus beta in bracket.
Example: Sinus one hundred and five degrees
Answer:
Sinus one hundred and five degrees equals sinus sixty degrees plus forty five degrees in bracket.
Equals sinus sixty degrees times cosines forty five degrees plus cosines sixty degrees times sinus forty five degrees in bracket.
Equals a half the square root of three times a half the square root of two in bracket plus a half times a half the square root of two in bracket.
Equals a quarter the square root of six plus a quarter the square root of two.
Equals a quarter open bracket the square root of six plus the square of two close bracket.
4). Formula of Sinus alpha minus beta in bracket.
Sinus alpha minus beta in bracket equals sinus alpha times cosines beta minus cosines alpha times sinus beta in bracket.
Example : Sinus fifteen degrees
Answer :
Sinus fifteen degrees equals sinus sixty degrees minus forty five degrees in bracket.
Equals sinus sixty degrees times cosines forty five degrees in bracket minus cosines sixty degrees times sinus forty five degrees in bracket.
Equals a half the square root of three times a half the square root of two in bracket minus a half times a half the square root of two in bracket.
Equals a quarter the square root of six minus a quarter the square root of two.
Equals a quarter open bracket the square root of six minus the square root of two close bracket.
5). Formula of tangents alpha plus beta in bracket
Tangents alpha plus beta in bracket equals tangents alpha plus tangents beta all over one minus open bracket tangents alpha times tangents beta close bracket.
6). Formula of tangents alpha minus beta in bracket
Tangents alpha minus beta in bracket equals tangents alpha minus tangents beta all over one plus open bracket tangents alpha times tangents beta close bracket.
My Exercise in Learning English
1.Explain how to proof that the square root of two is irational number!
Answer:
If the square root of two is rational number, so the square root of two equals a over b.Given a and b are prime integer number
a equals the square rot of two times b
a square equals two times b square
Because a square is two times any integer number, so a square is even, so that a is even too.
If a equals two times c
Two times c in bracket square equals two times b square
Four times c square equals two times b square
B square equals two times c square
So that, if b square is even, so is even too. But this is impossible because a and b bilangan bulat prima. So, can be proved that the square root of two is irrational number.
2.Explain how to show or to indicate that the sum angles of triangle is equal to one hundred and eighty degrees!
Answer:
There is a triangle ABC.
We can prove that the sum angles of triangle is equals to one hundred and eighty degrees by drawing a line through one vertex of triangle (through point B), parallel to the side opposite the vertex (parallel to AC). Note that the measure of straight angle at B equals the sum of the measure of the angles of triangle ABC.
Angle A plus angle B plus angle C equals one hundred and eighty degrees.
3.Explain how you are able to get phi!
Answer:
We can get phi from the circle. In a circle, we find the diameter and perimeter of that circle. And if we cut the circle, we can get the perimeter of the circle.So, we can get phi from the perimeter divided by diameter of the circle.
4.Explain how you are able to find the area of region boundered by the graph of y equals x square and y equals x plus two.
Answer:
First we must find the intersection point of y equals x square and y equals x plus two. We can use substitution method to find them.
X square equals x plus two
X square minus x minus two equals zero
X minus two in bracket times x plus one in bracket equals zero
X equals two or x equals negative one.
If x equals two, so y equals four. And if x equals negative one, so y equals one.
So that intersection point of curve y equals x square and the curve y equals x plus two are (two, four) and (negative one, one).
After we know about the intersection point of them, we must draw the curves.
And we can find the area of the region boundered by the graph of y equals x square and y equals x plus two with integral.
The area of the region is A
A equals definite integral x plus two minus x square dx from x equals negative one to x equals two.
A equals a half x square plus two x minus one third x cube, from x equals negative one to x equals two.
A equals a half times four plus two times two minus one third times eight in bracket minus a half plus two times negative one minus one third times negative one in bracket.
A equals two plus four minus eight third in bracket minus a half minus two plus one third in bracket.
A equals six minus eight thurd minus a half lus two minus one third.
A equals eight minus a half of seven.
A equals a half of nine.
5.Explain how you are able to determine the intersection point between the circle x square plus y square equals twenty and y equals x plus one!
Answer:
To determine the point between the circle x square plus y square equals twenty and y equals x plus one, wecan use substitution method.
X square plus y square equals twenty.
And we know that y equals x plus one, so this equation can be substitutioned in the equation above.
So,
X square plus open bracket x plus one close bracket square equals twenty.
X square plus x square plus two x plus one equals twenty.
Two times x square plus two x plus one minus twenty equals zero.
Two times x square plus two x minus nineteen equals zero.
Quadratic equation above can be finished by abc formula
x1 equals negative b plus the square root of open bracket b square minus four a c close bracket, all over two a
x2 equals negative b minus the square root of open bracket b square minus four a c close bracket, all over two a
So,
x1 equals negative two plus the square root of open bracket two square minus four times two times negative nineteen close bracket, all over two times two.
x1 equals negative two plus the square root of one hundred sixty six, all over four.
x1 equals negative two plus the square root of one hundred sixty six, all over four.
And,
x2 equals negative two minus the square root of open bracket two square minus four times two times negative nineteen close bracket, all over two times two.
x2 equals negative two minus the square root of one hundred sixty six, all over four.
If x1 equals negative two plus the square root of one hundred sixty six, all over four, So y1 equals two plus the square root of one hundred sixty six, all over four.
If x2 equals negative two minus the square root of one hundred sixty six, all over four, So y2 equals two minus the square root of one hundred sixty six, all over four.
So, we can find the intersection point of x square plus y square equals twenty and y equals x plus one.
Rabu, 01 April 2009
My Interpretations of the Videos in My English Lesson
In Last meeting, My friends and me watch some videos in English lesson. And from those videos we get something that will be important for us.
Video 1
In video 1, teach for us so that we not only look something from one side, but also we have to look something from another side. Because we will find different something from that another side. Don't afraid to do something, because we can’t know before we do it.
Video 2
In this life, we must have some dreams and wish. We have to believe in ourselves to reach that our dreams. Because self confidence is one of the modals for us to be better in the future. Although our self confidence is good and very important, but it’s not enough. So, we must believe to other people, like parents, teachers, friends etc. Because they can help us to reach our wish and what we want.
Video 3
What you know about Mathematics?
In Mathematics, there are symbols, graphic and then there are many lessons that is trigonometry, geometry, calculus, etc. For me, Mathematics is too difficult if only imagined. So I have to use calculator, graphic, table to ease for us to learn Mathematics lesson.
Video 4
Solving this Differential Equation!
(dy dx equals four x square)
(dy equals four x square dx)
(Integrate definite dy equals integrate four x square dx)
(y equals four third x cube plus constanta)
Video 5
1. (seven equals four a minus one)
(seven plus one equals four a minus one plus one)
(eight equals four a)
(a equals two)
2. (two third x equals eight)
(three second times two third x equals three second times eight)
(x equals twelve)
Video 1
In video 1, teach for us so that we not only look something from one side, but also we have to look something from another side. Because we will find different something from that another side. Don't afraid to do something, because we can’t know before we do it.
Video 2
In this life, we must have some dreams and wish. We have to believe in ourselves to reach that our dreams. Because self confidence is one of the modals for us to be better in the future. Although our self confidence is good and very important, but it’s not enough. So, we must believe to other people, like parents, teachers, friends etc. Because they can help us to reach our wish and what we want.
Video 3
What you know about Mathematics?
In Mathematics, there are symbols, graphic and then there are many lessons that is trigonometry, geometry, calculus, etc. For me, Mathematics is too difficult if only imagined. So I have to use calculator, graphic, table to ease for us to learn Mathematics lesson.
Video 4
Solving this Differential Equation!
(dy dx equals four x square)
(dy equals four x square dx)
(Integrate definite dy equals integrate four x square dx)
(y equals four third x cube plus constanta)
Video 5
1. (seven equals four a minus one)
(seven plus one equals four a minus one plus one)
(eight equals four a)
(a equals two)
2. (two third x equals eight)
(three second times two third x equals three second times eight)
(x equals twelve)
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