Assignment 5.
Square Equation ( translate book )
Reference: Erlangga, page 182 – 193
7.1 Definition of square equation
Look at some equation below
a) P^2-4 = 0
b) q^2 – 9q = 0
Every equation above only have one variable, and the highest power of variable is two. Equation
a) 2x^2 + 3x -2 = 0; a=2, b=3, c=-2
b) x^2-6x+9=0; a=1 b=-6 c=9
solving or roots of square equation of x are changer of x such that equation become correct.
Example 7.1
I equation x square minus 16 times equals zero
If x change by 4 so 4^2-16=0,
16 – 16=0 ( is correct )
If x change by -4 so -4^2-16=0,
16 – 16=0 ( is correct )
And so roots of equation x^2 – 16=0 is correct for 4 or -4
If x change by 2 so (2^2)-16=0,
4-16=-12(wrong)
If x change by 2 so (2^2)-16=0,
4-16=-12(wrong)
So 2 or (-2) are not roots of equation x^2-16=0. Actually roots of equation x^2-16=0 onli 4 or (-4). Lets try to other value of x!
There are three ways to solve square equation, there are by factoring , by complete the perfect square, and by using formula.
7.2 Solve Square equation by Factoring
We have known that multiply by zero will be product zero. On the contrary, a multiply if product zero, surely one its number that multiplied is zero.
Example:
If 3x=0 so that x=0
If 5(x-4)=0 so x-4=0
If pq=0 so p=0 or q=0
Example 7.2
1. Solve the square equation below :
a. x^2 +2x -3 =0
b. 3x^2=5x+2
c. 2x^2+6x=0
answers :
a. . x^2 +2x -3 =0
(x-1) (x+3)=0
X=1 or x=-3
b. 3x^2=5x+2
3x^2-5x-2=0
(3x+1)(x-2)=0
X=-1/3 or x=2
c. 2x^2+6x=0
2x(x+3)=0
2x=0 or x+3=0
X=0 or x=-3
2.Determine solving set of 9x^2 -4 =0
ANSWERS :
(3x-2)(3x+2)=0
X=-2/3 or x=2/3
Exercise 1
1. determine changer variable at every equation !
a. 3(3x-2)=0
b. p(p-4)=0
c. 2p(p+3)=0
d. (t-3)(t+1)=0
e. 5(t-2)(t+3)=0
f. (3n-4)^2=0
g. (-7n+1)(6n-5)=0
h. –(2m-5)(2m+5)=0
2. Solve equations by factoring!
a. 2x^2-x=0
b. 4x-6x^2=0
c. 9y^2-25=0
d. 16t^2-49=0
e. 3x^2-17x-6=0
3. Determine the solving sets of these equations
a. x^2+x=0
b. x^2-25=0
c. 9x^2-1=0
d. x^2+12x+35=0
4. Known 4 is one of solving roots of equation x^2+5=a(3x-a-2). Determine value of a.
5. Determine value of b if 5 is one of solving of equation 2x^2+x+b=0. Determine also the other solving.
7.2.1 Square equation in form of fraction
Oftentimes square equation writes by fraction form. To making to come easy, solve the equation, initially we must omit the denominator, by multiply by LCM (Least Common Multiply) of denominator.
Conclusion.
Steps to solving the square equation by complete prefect square are:
1. Place coefficient that have variable at left side and constant at right side.
2. Coefficient of x^2 must one
3. Add two side by square of half coefficient of x so that left side come perfect square
Kamis, 15 Januari 2009
Sabtu, 10 Januari 2009
final assignment( How difficult to explain math )
On Friday, 9th 2009 at 10.30pm, I try to explain about trigonometry to riza Aritara ( my lovely classmate) I explain about trigonometry function, and abot table trigonometry. First I explain about trigonometry function. This is my explanation:
> Trigonometry is study of right triangle
to remember you can use:
SOH is defined by sine equals opposite over hypotenuse
CAH is defined by cosine equals adjacent over hypotenuse
TOA is defined by tangent equals opposite over adjacent
Trigonometry function is only needed to know the values of side to find measure of an angle, figure out figure of all part of triangle.
Riza can understand may explanation very fast, and I give her a problem like this :
There is a right-triangle, with the length of opposite is 3, the length of adjacent is 4, and the length of hypotenuse is 5. And there is angle-x in front of the right-angle, and there is angle-m above the right-angle. What is the value of sinus of x, cosines of x, and tangent of x?
Riza can answer it correctly, although sometimes she is wrong to use the function of basic trigonometry, this is her correct answers :
SOH is defined by “Sinus is Opposite over Hypotenuse”, CAH is defined by “Cosine is Adjacent over Hypotenuse”, and TOA is defined by “Tangent is Opposite over Adjacent”.
So we get,
Sin x = Opp/Hyp = 4/5
Cos x = Adj/Hyp = 3/5
Tan x = Opp/Adj = 4/3
second, I explain about table trigonometry, there is:
sin 0=0, sin30=half, sin 45=half square root of two, sin 60=half square root of three and sin 90=1
cos 0=1, cos30=half square root of three, sin 45=half square root of two, cos 60=half and cos 90=0
for this table, riza has any problem to remember the table, I teach riza to remember faster by using our fingers. To sinus table, we start from thumb finger to little finger, there are sin 0=0, sin30=half, sin 45=half square root of two, sin 60=half square root of three and sin 90=1
To cosine table, we start from little finger to thumb finger.
for conclusion my problem to explain math to Riza aritara are: its very difficult to remember the table, second the function trigonometry has difficult to remember.
> Trigonometry is study of right triangle
to remember you can use:
SOH is defined by sine equals opposite over hypotenuse
CAH is defined by cosine equals adjacent over hypotenuse
TOA is defined by tangent equals opposite over adjacent
Trigonometry function is only needed to know the values of side to find measure of an angle, figure out figure of all part of triangle.
Riza can understand may explanation very fast, and I give her a problem like this :
There is a right-triangle, with the length of opposite is 3, the length of adjacent is 4, and the length of hypotenuse is 5. And there is angle-x in front of the right-angle, and there is angle-m above the right-angle. What is the value of sinus of x, cosines of x, and tangent of x?
Riza can answer it correctly, although sometimes she is wrong to use the function of basic trigonometry, this is her correct answers :
SOH is defined by “Sinus is Opposite over Hypotenuse”, CAH is defined by “Cosine is Adjacent over Hypotenuse”, and TOA is defined by “Tangent is Opposite over Adjacent”.
So we get,
Sin x = Opp/Hyp = 4/5
Cos x = Adj/Hyp = 3/5
Tan x = Opp/Adj = 4/3
second, I explain about table trigonometry, there is:
sin 0=0, sin30=half, sin 45=half square root of two, sin 60=half square root of three and sin 90=1
cos 0=1, cos30=half square root of three, sin 45=half square root of two, cos 60=half and cos 90=0
for this table, riza has any problem to remember the table, I teach riza to remember faster by using our fingers. To sinus table, we start from thumb finger to little finger, there are sin 0=0, sin30=half, sin 45=half square root of two, sin 60=half square root of three and sin 90=1
To cosine table, we start from little finger to thumb finger.
for conclusion my problem to explain math to Riza aritara are: its very difficult to remember the table, second the function trigonometry has difficult to remember.
tugas 4( explain the 8 video )
grammar
Tata bahasa adalah bagaimana bagian-bagian dari bahasa dihubungkan untuk membemtuk sebuah kalimat. Tipe kalimat yang paling dasar adalah simple sentence (kalimat sederhana). Dikatakan sederhana karena semua elemen di dalam kalimat tersebut terdiri dari subjek dan predikat. Subjek menunjukkan kegiatan dari kata kerja utama. Subjek utama adalah kata benda khusus yang menunjukkan sebuah kegiatan.
Contoh:
The happy litle child kicked the gnome over fence.
The happy litle child merupakan subjek.
Happy litle menghubungkan subjek sederhana child.
Child menunjukkan kata kerja “kick”
Predikat dari sebuah kalimat terdiri dari:
Main verb (kata kerja utama) + apapun yang mengikutinya.
Gabungan keduanya disebut predikat yang lengkap. Pada kalimat “The happy litle child kicked the gnome over fence” kita dapat mengidentifikasikan kata “kicked” sebagai kata kerja (predikat sederhana). Karena “the gnome” dan “over the fence” menunjukkan lebih jauh lagi tentang apa yang ditendang dan bagaimana tendangan itu. “Kicked the gnome over fence” merupakan predikat yang lengkap.
Kalimat sederhana dapat diperoleh tanpa subjek dan predikat. Kalimat perintah merupakan kalimat yang ditunjukkan langsung pada orang kedua yang adalah “kamu” atau memerintahkan seseorang untuk melakukan sesuatu.
Contoh:
Kicked the gnome over fence.
Kalimat diatas merupakan pedikat yang lengkap, tidak memiliki subjek. Dan siapa sebenarnya yang melakukan kegiatan tersebut? Jawabannya adlah “kamu”. Kalimat tersebut dapat ditulis menjadi “hey you, kick the gnome over fence!”. Dalam kalimat tersebut kata “you” tidak perlu ada karena sudah tersirat.
Kata kerja
Kata keja menunjukkan sebuah kegiatan atau untuk menjelaskan sebuah kegiatan atau menunjukkan apa yang sedang dilakukan oleh suatu benda atau seseorang. Kata kerja sangat penting ada dalam suatu kalimat.
Contoh:
Dave runs
“Run” adalah kata kerja yang menunjukkan apa yang sedang dilakukan oleh Dave.
Dalam bahasa inggris, kata kerja berubah bentuk untuk menunjukkan suatu kegiatan.
Misalnya:
I do, you do, we do, he does, she does, they do, it does.
Contoh:
Dave runs
Jika subjeknya diganti menjadi “i”, maka kalimatnya akan berubah menjadi “i run”. Kata kerja berubah karena perbedaan subjek. Dalam hal ini, subjek “i, you, we, they” menggunakan kata kerja “run”, sedangkan subjek “he, she, it” menggunakan kata keja “runs”.
Kata kerja dalam bentuk “to be” adalah:
“I am, you are, she is, he is, it is” merupakan singular subjek atau kata ganti tunggal. Sedangkan “we are, they are” merupakan plural subjek atau kata ganti jamak. Kata gan6ti tunggal diikuti oleh akta kerja tunggal, dan kata ganti jamak diikuti oleh kata kerja jamak.
Contoh:
Ms. Midori yodels
“Ms. Midori” merupakan kata ganti tunggal, dan “yodels” merupakan kata kerja tunggal.
Ms. Midori’s sister: Else, Gretel, Heidi.
“Else, Gretel, Heidi” merupakan kata ganti jamak
Else, Gretel, Heidi yodel
“yodel” merupakan kata kerja jamak
Dengan kata lain kita dapat menuliskannya sebagai berikut:
I yodel, you yodel, we yodel, they yodel, she yodels, he yodels, it yodels.
Kata keterangan
Adverb (kata keterangan) adalah kata yang menjelaskan verb (kata kerja), adjective (kata sifat), and adverb-adverb lainnya. Adverb digunakan untuk menjawab pertanyaan dengan kata Tanya “how?, how often?, when?, to what extend?”.
Sangat mudah untuk membentuk sebuah adverb. Caranya adalah adjective (kata sifat) dan diakhiri dengan akhiran –ly.
Contoh:
Simon might be ‘slow”, but he talks ‘slowly”.
Kata “slow” menjelaskan Simon. “Slow” = adjective.
Kata “slowly” menjawab pertanyaan “how does Simon talk? (bagaimana Simon berbicara)”. “Slowly” = adverb.
The color of these mushrooms is slightly different.
Slightly = slight + (-ly)
Sekarang mari kita pelajari tentang adverb yang diikuti oleh adverb-adverb lainnya.
Contoh:
This mushroom is very definitely poisonous.
“Poisonous” (adjective) dimodifikasikan oleh kata “definitely” (adverb).
Sedangkan “definitely” dijelaskan oleh adverb lain, yaitu kata “very”.
Pada umumnya kata “very” adalah adverb yang digunakan untuk menjelaskan adverb lainnya. “very” digunakan sebagai adverb dan tidak diakhiri dengan akhiran (-ly).
Sekarang kita akan mempelajari tenteng pengecualian dalam penggunaan akhiran “-ly” dalam adverb. Misal, untuk kata sifat (adjective) “ good” dan “fast” bila diubah dalam bentuk adverb, maka kita tidak boleh sembarang mengubahnya menjadi “goodly” dan “fastly” karena kedua kata itu salah.
Untuk kata “good”, bila kita akan menjawab pertanyaan “how?” maka kita menggunakan kata “well” sebagai adverb.
Contoh:
Candace can play the accordion very well.
Kata “well” adalah adverb yang dibentuk dari “good” (adjective).
Dan untuk pertanyaan “how Candace can play?” maka jawabannya adalah “candace’s playing is good”. Sedangkan good adalah kata sifat (adjective) yang menjelaskan gerund “playing”.
Ingat, gerund dapat dibentuk dari “verb (kata keja) + (-ing)”, dan digunakan sebagai kata benda (noun).
Basic trigonometry
Trigonometry comes from greek, they are trigon and metron. Trigonometry is really study of rectangle and the relationship between the side and the angle of rectangle. There is a right-triangle, with the length of opposite is 3, the length of adjacent is 4, and the length of hypotenuse is 5. And there is angle-x in front of the right-angle, and there is angle-m above the right-angle. What is the value of sinus of x, cosines of x, and tangent of x?
To solve them, use the simple trigonometry, they are SOH CAH TOA.
SOH is defined by “Sinus is Opposite over Hypotenuse”, CAH is defined by “Cosine is Adjacent over Hypotenuse”, and TOA is defined by “Tangent is Opposite over Adjacent”.
So we get,
Sin x = Opp/Hyp = 4/5
Cos x = Adj/Hyp = 3/5
Tan x = Opp/Adj = 4/3
If the angle is m, so we get tan m = ¾, the inverse of tangent x.
Kalimat majemuk
“it is the end of the world as we know it and I feel fine”. Dalam kalimat tersebut terdapat 2 klausa yang dihubungkan dengan 1 kata penghubung, yaitu “and”. Ketika suatu kalimat digunakan sebagai bagian-bagian dalam kalimat yang lebih besar, maka kalimat yang lebih kecil disebut klausa. Ketika sebuah klausa dapat berdiri sendiri dalam sebuah kalimat, maka klausa tersebut disebut “ndependent clause”. Dan jika kita memiliki 2 klausa dalam sebuah kalimat, maka kalimat tersebut disebut kalimat majemuk.
Untuk menggabungkan 2 independent clause, kita dapat menggunakan:
Tanda titik dua (:), ketika klausa ke-2 menjelaskan klausa ke-1.
Contoh:
I love my two sisters, they bake me pie
Untuk menggabungkan kalimat menjadi kalimat majemuk kita gunakan titik dua (:).
Kalimat di atas menjadi: “I love my two sisters: they bake me pie”
Titik koma (;), untuk menggantikan konjungsi.
Contoh:
“it is the end of the world as we know it and I feel fine”
Kalimat di atas dapat dipersingkat menjadi:
“it is the end of the world; I feel fine”
Garis penghubung (-)
Menggunakan garis penghubung karena klausa ke-2 dihubungkan dengan klausa ke-1.
Dengan demikian, dapat disimpulkan bahwa untuk menggabungkan kalimat ada 4 cara, yaitu:
Kata penghubung
Tanda titik dua (:)
Tanda titik koma (;)
Garis penghubung
Kalimat fragmen dapat didefinisikian jika kita mendapat porsi dalam suatu kalimat yang tidak dapat berdiri sendiri sebagai kalimat lengkap.
Contoh:
“My pet komodo dragon is as gentle as a lamb” merupakan kalimat lengkap.
“because he has no teeth” merupakan kalimat fragmen.
Dependent clause adalah klausa yang tidak dapat berdiri sendiri, terdapat pada klausa independent, dan bukan merupakan kalimat lengkap.
Kalimat kompleks terdiri dari:
Klausa dependent + klausa independent
Contoh:
Although Tom sleeps regulery, he is constantly tired.
“Although Tom sleeps regulery” merupakan klausa dependen.
“He is constantly tired” merupakan klausa independent.
“Although Tom sleeps regulary, he is constantly tired” merupakan kalimat kompleks.
Limit by inspection
There are two conditions, they are:
X goes to positive or negative infinity
Limit involves a polynomial divided by a polynomial
For example:
Limit of the function x cube plus four divided by x square plus x plus one as x approaches infinity. This problem is caused of two conditions, they are:
Polynomial over polynomial
X approaches infinity
The key to determine the limits by inspection is in looking at powers of x in the numerator and the denominator.
Remember!
To apply this rules, the things which must we do are:
We must be dividing by polynomials
X has to be approaching infinity
The first shortcut rule
If the highest power of x is greater in numerator, then limit is positive or negative infinity. X cube plus four divided by x square plus x plus one as x approaches infinity equals infinity equals positive or negative infinity. From that function we get three at x cube is the highest power of x in numerator. Two at x square is the highest power of x in denominator. Limit of function x cube plus four divided by x square plus x plus one as x approaches infinity equals infinity equals positive or negative infinity. It can be happened Since all the number are positive and x is going to positive infinity.
If you cannot tell it, the answer is positive or negative. And the rule is:
Substitute a large number for x
See if you end up with a positive or negative number
Whatever sign you get is the sign of infinitive for the limit
The second shortcut rule
If the highest power of x is in the denominator, then the limit is zero let limit of x square plus three divided by x cube plus one as x approaches infinitive equals zero. To at x square is highest power of x in numerator, and three x cube is the highest power of x in denominator
The third shortcut rule
This rule is used when the highest power of x in numerator is same as highest power of x in denominator. Limit of a function as x approach positive or negative infinity has the quotient on the coefficients at the two highest powers.
Remember!
Coefficient is the number that goes with a variable.
Example : two is the coefficient of two times x square, seventy five is the coefficient of seventy five times x fourth
The last shortcut rule
Limit of four time x cube plus x square plus one divided by three times x cube plus four as x approaches infinity has three as highest power at numerator and denominator
According to this rule, with conclude that the coefficient of x cube is over each other for example :
Limit of four time x cube plus x square plus one divided by three times x cube plus four as x approaches infinity equals four over three.
Pre-calculus
Graph of a rational function can be discontinuities because it has polynomial in the denominator. Is possible value x divide by zero?
Example: there is a function f, it is f of x equals x plus two divided by x minus one. If we insert x equals one, we get the value of f of one equals one plus two divided by one minus one equals three divided by zero. We know that three divided by zero is bad idea and the graph f of x equals x plus two divided by x minus one will break in function graph. If we insert x equals zero, we will get the function f of zero equals zero plus two divided by zero minus one equals negative two.
We will draw graph f of x equals x plus two divided by x minus one in the x y-coordinate plane. First we draw x y-coordinate plane, then we draw the graph f of x equals x plus two divided by x minus one as we know that this graph intersects axis y at point (0, -2), then we can draw the graph which through point (0, -2) and approach axis x in axis positive y. So we can say that graph f of x equals x plus two divided by x minus one is break. On the contrary, if there is line x equals one, so the graph approaches this line and axis positive x is discontinue.
Rational function does not always work this way!
Not all rational functions will give zero in denominator. Take the graph f of x equals one over x square plus one, it’s denominator never zero because of plus one and of course this graph is not break in function graph.
Do not forget!
Rational function denominator can be zero if the polynomial have smooth and unbroken curve and for rational function x equals zero in denominator, because that is impossible situation. It is impossible because there is not value for the function, so make break in function graph.
The break is showed up by two ways!
First, it can break in graph function because of the missing point in the graph. For example the graph f of x equals x square minus x minus six over x minus three has missing point at line x equals three. It can be happened because if we insert x equals three to the graph, so we get f of three equals three square minus three minus three over three minus three equals zero over zero. It can break because the value zero over zero is not possible, not feasible and not allowed. Typical f of three equals three square minus three minus three over three minus three equals zero over zero named missing point syndrome. If the result is zero over zero, to solve this problem we can make the factor top and button to be simplify. For example, y equals x square minus x minus six over x minus three. We get the top factor are x minus three and x plus two, the button factor is x minus three. So we can write y equals (x minus three) times (x plus two) over x minus three, and we can divide numerator and nominator with x minus three so we get y equals x plus two. Now, it is not problem if we insert x equals three to the new function y equals x plus two.
Trigonometry function
Now we will describe aright-triangle. Take x as the angle in front of the right-angle. The length of opposite is four, the length of adjacent is three and the length of hypotenuse is five.
To remember it you can see:
SOH is defined by sine equals opposite over hypotenuse
CAH is defined by cosine equals adjacent over hypotenuse
TOA is defined by tangent equals opposite over adjacent
Trigonometry function is only needed to know the values of side to find measure of an angle, figure out figure of all part of triangle. Things is included in trigonometry function are sine, cosine, tangent, cosecant , secant, cotangent. There are six basic of trigonometry function :
Sides of a triangle
Angel being measured
So we conclude that opposite equals side opposite to x, adjacent equals side adjacent to x.
Sin x equals opposite over hypotenuse, cosine x equals over hypotenuse, tangent x equals opposite over adjacent, cosecant x equal hypotenuse over opposite, secant x equals hypotenuse over adjacent, cotangent adjacent over opposite.
Tata bahasa adalah bagaimana bagian-bagian dari bahasa dihubungkan untuk membemtuk sebuah kalimat. Tipe kalimat yang paling dasar adalah simple sentence (kalimat sederhana). Dikatakan sederhana karena semua elemen di dalam kalimat tersebut terdiri dari subjek dan predikat. Subjek menunjukkan kegiatan dari kata kerja utama. Subjek utama adalah kata benda khusus yang menunjukkan sebuah kegiatan.
Contoh:
The happy litle child kicked the gnome over fence.
The happy litle child merupakan subjek.
Happy litle menghubungkan subjek sederhana child.
Child menunjukkan kata kerja “kick”
Predikat dari sebuah kalimat terdiri dari:
Main verb (kata kerja utama) + apapun yang mengikutinya.
Gabungan keduanya disebut predikat yang lengkap. Pada kalimat “The happy litle child kicked the gnome over fence” kita dapat mengidentifikasikan kata “kicked” sebagai kata kerja (predikat sederhana). Karena “the gnome” dan “over the fence” menunjukkan lebih jauh lagi tentang apa yang ditendang dan bagaimana tendangan itu. “Kicked the gnome over fence” merupakan predikat yang lengkap.
Kalimat sederhana dapat diperoleh tanpa subjek dan predikat. Kalimat perintah merupakan kalimat yang ditunjukkan langsung pada orang kedua yang adalah “kamu” atau memerintahkan seseorang untuk melakukan sesuatu.
Contoh:
Kicked the gnome over fence.
Kalimat diatas merupakan pedikat yang lengkap, tidak memiliki subjek. Dan siapa sebenarnya yang melakukan kegiatan tersebut? Jawabannya adlah “kamu”. Kalimat tersebut dapat ditulis menjadi “hey you, kick the gnome over fence!”. Dalam kalimat tersebut kata “you” tidak perlu ada karena sudah tersirat.
Kata kerja
Kata keja menunjukkan sebuah kegiatan atau untuk menjelaskan sebuah kegiatan atau menunjukkan apa yang sedang dilakukan oleh suatu benda atau seseorang. Kata kerja sangat penting ada dalam suatu kalimat.
Contoh:
Dave runs
“Run” adalah kata kerja yang menunjukkan apa yang sedang dilakukan oleh Dave.
Dalam bahasa inggris, kata kerja berubah bentuk untuk menunjukkan suatu kegiatan.
Misalnya:
I do, you do, we do, he does, she does, they do, it does.
Contoh:
Dave runs
Jika subjeknya diganti menjadi “i”, maka kalimatnya akan berubah menjadi “i run”. Kata kerja berubah karena perbedaan subjek. Dalam hal ini, subjek “i, you, we, they” menggunakan kata kerja “run”, sedangkan subjek “he, she, it” menggunakan kata keja “runs”.
Kata kerja dalam bentuk “to be” adalah:
“I am, you are, she is, he is, it is” merupakan singular subjek atau kata ganti tunggal. Sedangkan “we are, they are” merupakan plural subjek atau kata ganti jamak. Kata gan6ti tunggal diikuti oleh akta kerja tunggal, dan kata ganti jamak diikuti oleh kata kerja jamak.
Contoh:
Ms. Midori yodels
“Ms. Midori” merupakan kata ganti tunggal, dan “yodels” merupakan kata kerja tunggal.
Ms. Midori’s sister: Else, Gretel, Heidi.
“Else, Gretel, Heidi” merupakan kata ganti jamak
Else, Gretel, Heidi yodel
“yodel” merupakan kata kerja jamak
Dengan kata lain kita dapat menuliskannya sebagai berikut:
I yodel, you yodel, we yodel, they yodel, she yodels, he yodels, it yodels.
Kata keterangan
Adverb (kata keterangan) adalah kata yang menjelaskan verb (kata kerja), adjective (kata sifat), and adverb-adverb lainnya. Adverb digunakan untuk menjawab pertanyaan dengan kata Tanya “how?, how often?, when?, to what extend?”.
Sangat mudah untuk membentuk sebuah adverb. Caranya adalah adjective (kata sifat) dan diakhiri dengan akhiran –ly.
Contoh:
Simon might be ‘slow”, but he talks ‘slowly”.
Kata “slow” menjelaskan Simon. “Slow” = adjective.
Kata “slowly” menjawab pertanyaan “how does Simon talk? (bagaimana Simon berbicara)”. “Slowly” = adverb.
The color of these mushrooms is slightly different.
Slightly = slight + (-ly)
Sekarang mari kita pelajari tentang adverb yang diikuti oleh adverb-adverb lainnya.
Contoh:
This mushroom is very definitely poisonous.
“Poisonous” (adjective) dimodifikasikan oleh kata “definitely” (adverb).
Sedangkan “definitely” dijelaskan oleh adverb lain, yaitu kata “very”.
Pada umumnya kata “very” adalah adverb yang digunakan untuk menjelaskan adverb lainnya. “very” digunakan sebagai adverb dan tidak diakhiri dengan akhiran (-ly).
Sekarang kita akan mempelajari tenteng pengecualian dalam penggunaan akhiran “-ly” dalam adverb. Misal, untuk kata sifat (adjective) “ good” dan “fast” bila diubah dalam bentuk adverb, maka kita tidak boleh sembarang mengubahnya menjadi “goodly” dan “fastly” karena kedua kata itu salah.
Untuk kata “good”, bila kita akan menjawab pertanyaan “how?” maka kita menggunakan kata “well” sebagai adverb.
Contoh:
Candace can play the accordion very well.
Kata “well” adalah adverb yang dibentuk dari “good” (adjective).
Dan untuk pertanyaan “how Candace can play?” maka jawabannya adalah “candace’s playing is good”. Sedangkan good adalah kata sifat (adjective) yang menjelaskan gerund “playing”.
Ingat, gerund dapat dibentuk dari “verb (kata keja) + (-ing)”, dan digunakan sebagai kata benda (noun).
Basic trigonometry
Trigonometry comes from greek, they are trigon and metron. Trigonometry is really study of rectangle and the relationship between the side and the angle of rectangle. There is a right-triangle, with the length of opposite is 3, the length of adjacent is 4, and the length of hypotenuse is 5. And there is angle-x in front of the right-angle, and there is angle-m above the right-angle. What is the value of sinus of x, cosines of x, and tangent of x?
To solve them, use the simple trigonometry, they are SOH CAH TOA.
SOH is defined by “Sinus is Opposite over Hypotenuse”, CAH is defined by “Cosine is Adjacent over Hypotenuse”, and TOA is defined by “Tangent is Opposite over Adjacent”.
So we get,
Sin x = Opp/Hyp = 4/5
Cos x = Adj/Hyp = 3/5
Tan x = Opp/Adj = 4/3
If the angle is m, so we get tan m = ¾, the inverse of tangent x.
Kalimat majemuk
“it is the end of the world as we know it and I feel fine”. Dalam kalimat tersebut terdapat 2 klausa yang dihubungkan dengan 1 kata penghubung, yaitu “and”. Ketika suatu kalimat digunakan sebagai bagian-bagian dalam kalimat yang lebih besar, maka kalimat yang lebih kecil disebut klausa. Ketika sebuah klausa dapat berdiri sendiri dalam sebuah kalimat, maka klausa tersebut disebut “ndependent clause”. Dan jika kita memiliki 2 klausa dalam sebuah kalimat, maka kalimat tersebut disebut kalimat majemuk.
Untuk menggabungkan 2 independent clause, kita dapat menggunakan:
Tanda titik dua (:), ketika klausa ke-2 menjelaskan klausa ke-1.
Contoh:
I love my two sisters, they bake me pie
Untuk menggabungkan kalimat menjadi kalimat majemuk kita gunakan titik dua (:).
Kalimat di atas menjadi: “I love my two sisters: they bake me pie”
Titik koma (;), untuk menggantikan konjungsi.
Contoh:
“it is the end of the world as we know it and I feel fine”
Kalimat di atas dapat dipersingkat menjadi:
“it is the end of the world; I feel fine”
Garis penghubung (-)
Menggunakan garis penghubung karena klausa ke-2 dihubungkan dengan klausa ke-1.
Dengan demikian, dapat disimpulkan bahwa untuk menggabungkan kalimat ada 4 cara, yaitu:
Kata penghubung
Tanda titik dua (:)
Tanda titik koma (;)
Garis penghubung
Kalimat fragmen dapat didefinisikian jika kita mendapat porsi dalam suatu kalimat yang tidak dapat berdiri sendiri sebagai kalimat lengkap.
Contoh:
“My pet komodo dragon is as gentle as a lamb” merupakan kalimat lengkap.
“because he has no teeth” merupakan kalimat fragmen.
Dependent clause adalah klausa yang tidak dapat berdiri sendiri, terdapat pada klausa independent, dan bukan merupakan kalimat lengkap.
Kalimat kompleks terdiri dari:
Klausa dependent + klausa independent
Contoh:
Although Tom sleeps regulery, he is constantly tired.
“Although Tom sleeps regulery” merupakan klausa dependen.
“He is constantly tired” merupakan klausa independent.
“Although Tom sleeps regulary, he is constantly tired” merupakan kalimat kompleks.
Limit by inspection
There are two conditions, they are:
X goes to positive or negative infinity
Limit involves a polynomial divided by a polynomial
For example:
Limit of the function x cube plus four divided by x square plus x plus one as x approaches infinity. This problem is caused of two conditions, they are:
Polynomial over polynomial
X approaches infinity
The key to determine the limits by inspection is in looking at powers of x in the numerator and the denominator.
Remember!
To apply this rules, the things which must we do are:
We must be dividing by polynomials
X has to be approaching infinity
The first shortcut rule
If the highest power of x is greater in numerator, then limit is positive or negative infinity. X cube plus four divided by x square plus x plus one as x approaches infinity equals infinity equals positive or negative infinity. From that function we get three at x cube is the highest power of x in numerator. Two at x square is the highest power of x in denominator. Limit of function x cube plus four divided by x square plus x plus one as x approaches infinity equals infinity equals positive or negative infinity. It can be happened Since all the number are positive and x is going to positive infinity.
If you cannot tell it, the answer is positive or negative. And the rule is:
Substitute a large number for x
See if you end up with a positive or negative number
Whatever sign you get is the sign of infinitive for the limit
The second shortcut rule
If the highest power of x is in the denominator, then the limit is zero let limit of x square plus three divided by x cube plus one as x approaches infinitive equals zero. To at x square is highest power of x in numerator, and three x cube is the highest power of x in denominator
The third shortcut rule
This rule is used when the highest power of x in numerator is same as highest power of x in denominator. Limit of a function as x approach positive or negative infinity has the quotient on the coefficients at the two highest powers.
Remember!
Coefficient is the number that goes with a variable.
Example : two is the coefficient of two times x square, seventy five is the coefficient of seventy five times x fourth
The last shortcut rule
Limit of four time x cube plus x square plus one divided by three times x cube plus four as x approaches infinity has three as highest power at numerator and denominator
According to this rule, with conclude that the coefficient of x cube is over each other for example :
Limit of four time x cube plus x square plus one divided by three times x cube plus four as x approaches infinity equals four over three.
Pre-calculus
Graph of a rational function can be discontinuities because it has polynomial in the denominator. Is possible value x divide by zero?
Example: there is a function f, it is f of x equals x plus two divided by x minus one. If we insert x equals one, we get the value of f of one equals one plus two divided by one minus one equals three divided by zero. We know that three divided by zero is bad idea and the graph f of x equals x plus two divided by x minus one will break in function graph. If we insert x equals zero, we will get the function f of zero equals zero plus two divided by zero minus one equals negative two.
We will draw graph f of x equals x plus two divided by x minus one in the x y-coordinate plane. First we draw x y-coordinate plane, then we draw the graph f of x equals x plus two divided by x minus one as we know that this graph intersects axis y at point (0, -2), then we can draw the graph which through point (0, -2) and approach axis x in axis positive y. So we can say that graph f of x equals x plus two divided by x minus one is break. On the contrary, if there is line x equals one, so the graph approaches this line and axis positive x is discontinue.
Rational function does not always work this way!
Not all rational functions will give zero in denominator. Take the graph f of x equals one over x square plus one, it’s denominator never zero because of plus one and of course this graph is not break in function graph.
Do not forget!
Rational function denominator can be zero if the polynomial have smooth and unbroken curve and for rational function x equals zero in denominator, because that is impossible situation. It is impossible because there is not value for the function, so make break in function graph.
The break is showed up by two ways!
First, it can break in graph function because of the missing point in the graph. For example the graph f of x equals x square minus x minus six over x minus three has missing point at line x equals three. It can be happened because if we insert x equals three to the graph, so we get f of three equals three square minus three minus three over three minus three equals zero over zero. It can break because the value zero over zero is not possible, not feasible and not allowed. Typical f of three equals three square minus three minus three over three minus three equals zero over zero named missing point syndrome. If the result is zero over zero, to solve this problem we can make the factor top and button to be simplify. For example, y equals x square minus x minus six over x minus three. We get the top factor are x minus three and x plus two, the button factor is x minus three. So we can write y equals (x minus three) times (x plus two) over x minus three, and we can divide numerator and nominator with x minus three so we get y equals x plus two. Now, it is not problem if we insert x equals three to the new function y equals x plus two.
Trigonometry function
Now we will describe aright-triangle. Take x as the angle in front of the right-angle. The length of opposite is four, the length of adjacent is three and the length of hypotenuse is five.
To remember it you can see:
SOH is defined by sine equals opposite over hypotenuse
CAH is defined by cosine equals adjacent over hypotenuse
TOA is defined by tangent equals opposite over adjacent
Trigonometry function is only needed to know the values of side to find measure of an angle, figure out figure of all part of triangle. Things is included in trigonometry function are sine, cosine, tangent, cosecant , secant, cotangent. There are six basic of trigonometry function :
Sides of a triangle
Angel being measured
So we conclude that opposite equals side opposite to x, adjacent equals side adjacent to x.
Sin x equals opposite over hypotenuse, cosine x equals over hypotenuse, tangent x equals opposite over adjacent, cosecant x equal hypotenuse over opposite, secant x equals hypotenuse over adjacent, cotangent adjacent over opposite.
explain video Do You Believe in Me
Assignment 2.
Do You Believe in Me
A boy stand on the stage. He is a student from Dallas. He say all of his friends, and all of his friends, and all of people in that room. He ask them about himself. He can do anything, be anything, create anything, become anything. His name is Dalton Sherman. He very brave and confident, He enthusiasm to tell it.
Although Dallas has many student from some region, but Dallas can do anything. Dallas can produced useful to the city. He say that every single one of us can be graduate, ready for college, and can get workplace. We are ( Charles Rice Learning Center ) all showing up in their school. He ask the to believe him about we can reach a highest potential.
Whether they are a counselor, on librarian, a teacher assistant or work in the front office, whether you serve up meals in the cafeteria, Dalton needs themselves. What they doing is most important job in the city today. There’s probably easier ways to make a living. They must believe that Dallas student can achieve. He trust that all of people in Dallas can become a success people.
Do You Believe in Me
A boy stand on the stage. He is a student from Dallas. He say all of his friends, and all of his friends, and all of people in that room. He ask them about himself. He can do anything, be anything, create anything, become anything. His name is Dalton Sherman. He very brave and confident, He enthusiasm to tell it.
Although Dallas has many student from some region, but Dallas can do anything. Dallas can produced useful to the city. He say that every single one of us can be graduate, ready for college, and can get workplace. We are ( Charles Rice Learning Center ) all showing up in their school. He ask the to believe him about we can reach a highest potential.
Whether they are a counselor, on librarian, a teacher assistant or work in the front office, whether you serve up meals in the cafeteria, Dalton needs themselves. What they doing is most important job in the city today. There’s probably easier ways to make a living. They must believe that Dallas student can achieve. He trust that all of people in Dallas can become a success people.
Minggu, 14 Desember 2008
Represented The video
The Video
Video 1 : Factoring Polynomials
One way to find factors of polynomials is to formed the algebraic long division. For example lets see x minus 3 is a factor of x cube minus seven x minus six? When dividing x minus 3 into x cube minus seven x minus six. First step the problem make a long division problem. There is you dividing x – 3 into x cube plus zero x square minus seven x minus six. Zero because there is no second degree term. Now you must ask yourself what times x give you x cube? Of, course x square, so you multiply x minus 3. By x square, which give you x cube minus 3 x square to get 3 x square. Bring it down next term negative seven x. Dividing x minus 3 into 3 x square minus seven x. Just looking at the first time 3 x square dividing x is 3x. Multiply x minus 3 by 3x. We can get 3 x square minus nine x. Subtracting you have 2x minus 6. Dividing 2 x minus 6 by x-3 which equals 2 and without a remainder. So the solution for a long division problem is x square plus 3 x plus 2. Since x minus 3 divide in x cube minus seven x minus 6. We now know x cube minus 7 x minus 6 equals ( x minus 3) times ( x square plus 3 x plus 2 ). The quadratic expression x square plus 3 plus 2 can be factored into ( x plus 1) times ( x plus 2). So, x cube minus 7 x minus 6 equals ( x minus 3 ) times ( x plus 1) times ( x plus 2 ). Substitution x cube minus 7x minus 6 to zero we get 0=(x-3)(x+1)(x+2) Thus either x-3=0 or x+1=0 or x +2=0 Solving of x we get x=3, x=-1, x=-2. The roots of x cube minus 7 x minus 6 are 3, -1, -2.
Conclusion:
*) 3 roots for this 3rd degree equation
*) 2nd degree equation always have at most 2 roots
*) 4th degree equation would have 4 or fewer roots, and so on.
*) The degree of polynomials equation always limits the number of roots.
Long division process for 3rd order Polynomial:
1. Find a partial quotient of x square, by dividing x into x cube to get x square
2. Multiply x square by the divisor and subtract the product from the dividend.
3. Repeat the process until you either “ clear it out “ or reach a remainder.
Video 2 : Solve The Problems
The next problem is question 13. The figure shows of graph of y equals g of x. If the function h is defined by h(x) = g(2x)+2, what is the value of h(1)? And we are looking for h of 1.
The information of the graph h(x)=g(2x)+2 now we are looking for h(1). We substitute h(1) into this equation. H(1)=g(2)+2 now we get g(2), see when x equals 2, y equals 1. So g(2) is 1. H(1)=g(2)+2, h(1)=1+2, so h(1)=3.
Next is the function with no function problem. This question 13 page 534. Let the function f be defined by F(x)=x+1, if 2f(p)=20, what is the value of f(3p)? We are looking f(3p) what is f when x = 3p?
First information is F(x)=x+1, 2f(p)=20 The figure of 3p start with this equation down here:
2f(p)=20, the, if we are divide by 2 so : f(p)=10. Then f(p) is just what is function f(x) of f(p) f(p)=p+1=10, then p=9. Is this right answer? No, is not. We looking for x=3p, x=27. We have an equation f(x)=x+1, f(27)=27+8=28. The last answer is 28.
Question 17.
In the x y – coordinate plane, the graph of x equals y square minus 4 intersect line l at ( 0,p ) and ( 5,t ). What is the greatest possible value of the slope of l? we’ll be looking for greatest m. The graph intersect in x = 4 line l intersect at ( 0,p ) and ( 5,t ). X is zero, y is p and when x = 5, y=t what is the possible slope for line l. What we are doing now for get the slope? m
equals (y-y1) over (x2-x1) The slope is going to be m=(t-p) over 5. Numerator is t-p we have x=y square minus 4. We can play again the point of intersect ( 0,p ) and ( 5,t ) to the equation x= x square minus 4.
Video 3 : Pre Calculus
Graph of a rational function which can have discontinuities because has polynomial in the denominator.
Is possible value x divide by 0
Example : f(x) equals (x plus 2) over (x minus 1) F(1)= The value become 1+2 over 1-1 equals 3 over zero. That is bad idea.
Graph f(1)=1 plus 2 over zero : Break in finction graph.
F(x) = ( x+ 2) over ( x-1) : insert 0
F(0)=0+2 over 0-1 equals -2. Insert 1 F(1)=1+2 over 1-1 equals 3 over 0 is impossible.
Rational functions don’t always work in this way! Take graph f(x) = 1 over ( x square plus 1 ). Not all rational functions will give zero in denominator because of the +1 ( never zero ).
Rational functions denominator can be zero.
Polynomial have smooth and unbroken curve and for rational function x : zero in the denominator. That impossible situation. A break can show up in two ways. A simply type break is missing point on the graph. Y = ( x square minus x minus 6 ) over ( x minus 3). The graph loose like this if x = 3 ( 3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. That is not possible, not feasible, and not allowed. So that is no way if x = 3. This is a typical example to the missing point syndrome. Y = ((3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. When you see result of 0 over 0 and also tell you direction by possible factor top and bottom of rational function and simplify. For example. Y = ( x square minus x minus 6 ) over ( x minus 3), Equals ( x minus 3 ) times ( x plus 2) over ( x minus 3) so, y = x+2.
video4.
Now we talking about the inverse function.
we can write this function becomes relation of f(x,y)=o, then function y=f(x). is a straight line with y intersect (-1) and x intersect (1/2). Look at the line y=x. That line intersect the graph of y=2x-1. So, we get
x=2x-1
1+x=2x
then 1=x
So, the intersection of the line y=x and y=2x-1 with x=1 is (1,1).
Video 1 : Factoring Polynomials
One way to find factors of polynomials is to formed the algebraic long division. For example lets see x minus 3 is a factor of x cube minus seven x minus six? When dividing x minus 3 into x cube minus seven x minus six. First step the problem make a long division problem. There is you dividing x – 3 into x cube plus zero x square minus seven x minus six. Zero because there is no second degree term. Now you must ask yourself what times x give you x cube? Of, course x square, so you multiply x minus 3. By x square, which give you x cube minus 3 x square to get 3 x square. Bring it down next term negative seven x. Dividing x minus 3 into 3 x square minus seven x. Just looking at the first time 3 x square dividing x is 3x. Multiply x minus 3 by 3x. We can get 3 x square minus nine x. Subtracting you have 2x minus 6. Dividing 2 x minus 6 by x-3 which equals 2 and without a remainder. So the solution for a long division problem is x square plus 3 x plus 2. Since x minus 3 divide in x cube minus seven x minus 6. We now know x cube minus 7 x minus 6 equals ( x minus 3) times ( x square plus 3 x plus 2 ). The quadratic expression x square plus 3 plus 2 can be factored into ( x plus 1) times ( x plus 2). So, x cube minus 7 x minus 6 equals ( x minus 3 ) times ( x plus 1) times ( x plus 2 ). Substitution x cube minus 7x minus 6 to zero we get 0=(x-3)(x+1)(x+2) Thus either x-3=0 or x+1=0 or x +2=0 Solving of x we get x=3, x=-1, x=-2. The roots of x cube minus 7 x minus 6 are 3, -1, -2.
Conclusion:
*) 3 roots for this 3rd degree equation
*) 2nd degree equation always have at most 2 roots
*) 4th degree equation would have 4 or fewer roots, and so on.
*) The degree of polynomials equation always limits the number of roots.
Long division process for 3rd order Polynomial:
1. Find a partial quotient of x square, by dividing x into x cube to get x square
2. Multiply x square by the divisor and subtract the product from the dividend.
3. Repeat the process until you either “ clear it out “ or reach a remainder.
Video 2 : Solve The Problems
The next problem is question 13. The figure shows of graph of y equals g of x. If the function h is defined by h(x) = g(2x)+2, what is the value of h(1)? And we are looking for h of 1.
The information of the graph h(x)=g(2x)+2 now we are looking for h(1). We substitute h(1) into this equation. H(1)=g(2)+2 now we get g(2), see when x equals 2, y equals 1. So g(2) is 1. H(1)=g(2)+2, h(1)=1+2, so h(1)=3.
Next is the function with no function problem. This question 13 page 534. Let the function f be defined by F(x)=x+1, if 2f(p)=20, what is the value of f(3p)? We are looking f(3p) what is f when x = 3p?
First information is F(x)=x+1, 2f(p)=20 The figure of 3p start with this equation down here:
2f(p)=20, the, if we are divide by 2 so : f(p)=10. Then f(p) is just what is function f(x) of f(p) f(p)=p+1=10, then p=9. Is this right answer? No, is not. We looking for x=3p, x=27. We have an equation f(x)=x+1, f(27)=27+8=28. The last answer is 28.
Question 17.
In the x y – coordinate plane, the graph of x equals y square minus 4 intersect line l at ( 0,p ) and ( 5,t ). What is the greatest possible value of the slope of l? we’ll be looking for greatest m. The graph intersect in x = 4 line l intersect at ( 0,p ) and ( 5,t ). X is zero, y is p and when x = 5, y=t what is the possible slope for line l. What we are doing now for get the slope? m
equals (y-y1) over (x2-x1) The slope is going to be m=(t-p) over 5. Numerator is t-p we have x=y square minus 4. We can play again the point of intersect ( 0,p ) and ( 5,t ) to the equation x= x square minus 4.
Video 3 : Pre Calculus
Graph of a rational function which can have discontinuities because has polynomial in the denominator.
Is possible value x divide by 0
Example : f(x) equals (x plus 2) over (x minus 1) F(1)= The value become 1+2 over 1-1 equals 3 over zero. That is bad idea.
Graph f(1)=1 plus 2 over zero : Break in finction graph.
F(x) = ( x+ 2) over ( x-1) : insert 0
F(0)=0+2 over 0-1 equals -2. Insert 1 F(1)=1+2 over 1-1 equals 3 over 0 is impossible.
Rational functions don’t always work in this way! Take graph f(x) = 1 over ( x square plus 1 ). Not all rational functions will give zero in denominator because of the +1 ( never zero ).
Rational functions denominator can be zero.
Polynomial have smooth and unbroken curve and for rational function x : zero in the denominator. That impossible situation. A break can show up in two ways. A simply type break is missing point on the graph. Y = ( x square minus x minus 6 ) over ( x minus 3). The graph loose like this if x = 3 ( 3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. That is not possible, not feasible, and not allowed. So that is no way if x = 3. This is a typical example to the missing point syndrome. Y = ((3 square minus 3 minus 6 ) over (3 minus 3) equals 0 over 0. When you see result of 0 over 0 and also tell you direction by possible factor top and bottom of rational function and simplify. For example. Y = ( x square minus x minus 6 ) over ( x minus 3), Equals ( x minus 3 ) times ( x plus 2) over ( x minus 3) so, y = x+2.
video4.
Now we talking about the inverse function.
we can write this function becomes relation of f(x,y)=o, then function y=f(x). is a straight line with y intersect (-1) and x intersect (1/2). Look at the line y=x. That line intersect the graph of y=2x-1. So, we get
x=2x-1
1+x=2x
then 1=x
So, the intersection of the line y=x and y=2x-1 with x=1 is (1,1).
English To Indonesia
SISTEM ANGKA
1. Perhitungan aritmatika dengan jari
System ini diperoleh dari menghitung jari dengan angka yang ditulis seluruhnya dengan kata – kata. Perhitunga dengan jari ini digunakan oleh komunitas bisnis. Ahli matematika seperti Abul’s – Wafa ( Lahir 940) menulis beberapa acuan mengenai penggunaan system ini. Abul’s – Wafa sebagai ahli dalam penggunaan angka – angka Indian tetapi :
…. Tidak ada aplikasi fina di lingkaran bisnis dan diantara populasi kalifah timur untuk waktu yang lama.
Oleh karena itu Ia menulis teksnya menggunakan perhitungan aritmatika dengan jari sejak system ini digunakan oleh komunitas bisnis.
2. System sexagesimal
System yang kedua dari tiga system yang ada adalah sexagesimal system, dengan angka yang ditandai oleh huruf alphabet arab. Sistem ini berasal dari orang – orang babilonia dan sering digunakan oleh ahli matematika arab dalam pekerjaan astronomi.
3. Sistem angka Indian
System yang ketiga perhitungan angka Indian dan pecahan dengan system nilai decimal. Angka – angka yang digunakan diambel dari India, tetapi tidak ada symbol standard yang ditetapkan. Bagian lain pada dunia Arab menggunakan sedikit perubahan bentuk pada angka – angkanya. Awalnya metode Indian digunakan oleh orang Arab dengan sebuah papan debu. Papan debu dibutuhkan karena metode ini membutuhkan perpindahan angka – angka dalam perhitungan dan menghapus beberapa proses perhitungan. Penggunaan papan debu akan memilki fungsi yang sama juga dengan menggunakan papan tulis, kapur, dan penghapus papan tulis. BAgaimanapun, Al – Uqlisidi ( lahir 920 ) menunjukkan bagaimana mengenbangkan metode ini dengan menggunakan pulpen dan kertas. Al – Baghdadi juga mengkontribusikan untuk pengembangan dalam system decimal.
Sistem perhitungan yang ketiga ini paling banyak digunakan dalam metode angka eloeh orang – orang Arab. Sistem ini juga digunakan dalam penarikan akar oleh ahli matematika seperti Abul’l Wafa dan Omar khayam sebagai factor utama dalam pengembangan analisis dasar angka pada system decimal. Al – Kashi ( lahir 1380 ) mengkontribusi untuk perkembangan pada pecahan decimal tidak hanya untuk memperkirakan bilangan – bilangan aljabar, tetapi juga untuk bilangan real seperti π. Kontribusinya untk pecahan decimal adalah yang utama selama beberapa tahun , beliau dipertimbangkan sebagai penemu dari pecahan – pecahan decimal. Walaupun bukan yang pertama juga, al – Kashi memberikan algoritma untuk menghitung akar – n yang merupakan kasus khusus pada metode di beberapa abad terakhir oleh Ruffini dan Horner.
Walaupun ahli matematika Arab lebih terkenal dengan pekerjaan mereka dalam alajabar, teori bilangan, dan system angka, mereka juga berperan dalam sumbangan goemetri, trigonometri dan matematika astronomi. Ibrahin Ibn Sinan ( Lahir 908), yang mengenalkan metode integral yang lebih umum daripada Archimedes, dan Al – Quhi ( LAhir 940 ) yang menjadi figure pemimpin dalam sebuah perkembangan ilmu dan kelanjutan pada geometri Yunani tingkat tinggi di dunia islam. Para ahli matematika, dan teristimewa al - Haytam, mempelajari optikm dan penyelidikan alat – alat lensa opyik dibuat dari bagian – bagian yang berbentuk kerucut. Omar Kahayam menggabungkan penggunaan trigonometri dari teori pendekatan untuk meyediakan metode dalam persamaan aljabar dengan pengertian geometri.
Astronomi, perbedaan waktu, dan geografi menyediakan dorongan lain untuk penelitian geometrid an trigonometri. Sebagai contoh Ibrahim Ibn Sinan dan kakeknya Thabitt ibn Qurra keduanya mempelajari kurva permintaan dalam konstruksi jam matahari. Abul – Wafa dan Abu Nasr Mansur keduanya mengaplikasikan geometri lingkaran untuk astronomi dan juga rumus sinus dan tangent. Al Biruni ( Lahir 973 ) menggunakan rumus sinus dalam astronomi dan perhitungan garis bujur dan lintang pada beberapa kota. Lagi – lagi keduanya astronomi dan geografi mendorong al – Birunis memperluas studinya pada proyek setengah lingkaran dalam bidang.
Thabit ibn Qurra mengerjakan keduanya teori dan penelitian dalam astronomi al – Batahani ( lahir 850 ). Membuat kepastian penelitian yang mengijinkannya untuk mengembangkan pada prolemy’s data untuk matahari dan bulan. Nasir al – Din dan al – Tusi ( Lahir 1201), seperti kebanyakan ahli matematika Arab, berdasarkan teori astronominya pada kerja prolemy’s tetapi al – Tusi membuat perkembangan yang lebih berarti pada model prolemy’s dalam system planet yang lebih dikembangkan pada model heliosentris dalam waktu Copernicus.
Kebanyakan ahli matematika Arab menghasilkan table fungsi trigonometri sebagai bagian pembelajarannya pada astronomi termasuk di sini Ulugh Begh ( Lahir 1393) dan al – Kashi. Konstruksi instrument astronomi seperti astrolabe juga dikususkan pada Arab. Al – Mahani menggunakan astrolabe ketika Ahmed ( Lahir 835) , Al Kahzin ( Lahir 900) , Ibrahum Ibn Sinan , Al- Quhi, Abu Nasr Mansur ( Lahir 965 ), Al – Biruni, dan lainnya, semuanya menulis mengenai ketertarikan pada astrolabe. Sharaf al – Din , al – Tusi menemukan linear astrolabe.
Sumber :
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Arabic_matematics.html
1. Perhitungan aritmatika dengan jari
System ini diperoleh dari menghitung jari dengan angka yang ditulis seluruhnya dengan kata – kata. Perhitunga dengan jari ini digunakan oleh komunitas bisnis. Ahli matematika seperti Abul’s – Wafa ( Lahir 940) menulis beberapa acuan mengenai penggunaan system ini. Abul’s – Wafa sebagai ahli dalam penggunaan angka – angka Indian tetapi :
…. Tidak ada aplikasi fina di lingkaran bisnis dan diantara populasi kalifah timur untuk waktu yang lama.
Oleh karena itu Ia menulis teksnya menggunakan perhitungan aritmatika dengan jari sejak system ini digunakan oleh komunitas bisnis.
2. System sexagesimal
System yang kedua dari tiga system yang ada adalah sexagesimal system, dengan angka yang ditandai oleh huruf alphabet arab. Sistem ini berasal dari orang – orang babilonia dan sering digunakan oleh ahli matematika arab dalam pekerjaan astronomi.
3. Sistem angka Indian
System yang ketiga perhitungan angka Indian dan pecahan dengan system nilai decimal. Angka – angka yang digunakan diambel dari India, tetapi tidak ada symbol standard yang ditetapkan. Bagian lain pada dunia Arab menggunakan sedikit perubahan bentuk pada angka – angkanya. Awalnya metode Indian digunakan oleh orang Arab dengan sebuah papan debu. Papan debu dibutuhkan karena metode ini membutuhkan perpindahan angka – angka dalam perhitungan dan menghapus beberapa proses perhitungan. Penggunaan papan debu akan memilki fungsi yang sama juga dengan menggunakan papan tulis, kapur, dan penghapus papan tulis. BAgaimanapun, Al – Uqlisidi ( lahir 920 ) menunjukkan bagaimana mengenbangkan metode ini dengan menggunakan pulpen dan kertas. Al – Baghdadi juga mengkontribusikan untuk pengembangan dalam system decimal.
Sistem perhitungan yang ketiga ini paling banyak digunakan dalam metode angka eloeh orang – orang Arab. Sistem ini juga digunakan dalam penarikan akar oleh ahli matematika seperti Abul’l Wafa dan Omar khayam sebagai factor utama dalam pengembangan analisis dasar angka pada system decimal. Al – Kashi ( lahir 1380 ) mengkontribusi untuk perkembangan pada pecahan decimal tidak hanya untuk memperkirakan bilangan – bilangan aljabar, tetapi juga untuk bilangan real seperti π. Kontribusinya untk pecahan decimal adalah yang utama selama beberapa tahun , beliau dipertimbangkan sebagai penemu dari pecahan – pecahan decimal. Walaupun bukan yang pertama juga, al – Kashi memberikan algoritma untuk menghitung akar – n yang merupakan kasus khusus pada metode di beberapa abad terakhir oleh Ruffini dan Horner.
Walaupun ahli matematika Arab lebih terkenal dengan pekerjaan mereka dalam alajabar, teori bilangan, dan system angka, mereka juga berperan dalam sumbangan goemetri, trigonometri dan matematika astronomi. Ibrahin Ibn Sinan ( Lahir 908), yang mengenalkan metode integral yang lebih umum daripada Archimedes, dan Al – Quhi ( LAhir 940 ) yang menjadi figure pemimpin dalam sebuah perkembangan ilmu dan kelanjutan pada geometri Yunani tingkat tinggi di dunia islam. Para ahli matematika, dan teristimewa al - Haytam, mempelajari optikm dan penyelidikan alat – alat lensa opyik dibuat dari bagian – bagian yang berbentuk kerucut. Omar Kahayam menggabungkan penggunaan trigonometri dari teori pendekatan untuk meyediakan metode dalam persamaan aljabar dengan pengertian geometri.
Astronomi, perbedaan waktu, dan geografi menyediakan dorongan lain untuk penelitian geometrid an trigonometri. Sebagai contoh Ibrahim Ibn Sinan dan kakeknya Thabitt ibn Qurra keduanya mempelajari kurva permintaan dalam konstruksi jam matahari. Abul – Wafa dan Abu Nasr Mansur keduanya mengaplikasikan geometri lingkaran untuk astronomi dan juga rumus sinus dan tangent. Al Biruni ( Lahir 973 ) menggunakan rumus sinus dalam astronomi dan perhitungan garis bujur dan lintang pada beberapa kota. Lagi – lagi keduanya astronomi dan geografi mendorong al – Birunis memperluas studinya pada proyek setengah lingkaran dalam bidang.
Thabit ibn Qurra mengerjakan keduanya teori dan penelitian dalam astronomi al – Batahani ( lahir 850 ). Membuat kepastian penelitian yang mengijinkannya untuk mengembangkan pada prolemy’s data untuk matahari dan bulan. Nasir al – Din dan al – Tusi ( Lahir 1201), seperti kebanyakan ahli matematika Arab, berdasarkan teori astronominya pada kerja prolemy’s tetapi al – Tusi membuat perkembangan yang lebih berarti pada model prolemy’s dalam system planet yang lebih dikembangkan pada model heliosentris dalam waktu Copernicus.
Kebanyakan ahli matematika Arab menghasilkan table fungsi trigonometri sebagai bagian pembelajarannya pada astronomi termasuk di sini Ulugh Begh ( Lahir 1393) dan al – Kashi. Konstruksi instrument astronomi seperti astrolabe juga dikususkan pada Arab. Al – Mahani menggunakan astrolabe ketika Ahmed ( Lahir 835) , Al Kahzin ( Lahir 900) , Ibrahum Ibn Sinan , Al- Quhi, Abu Nasr Mansur ( Lahir 965 ), Al – Biruni, dan lainnya, semuanya menulis mengenai ketertarikan pada astrolabe. Sharaf al – Din , al – Tusi menemukan linear astrolabe.
Sumber :
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Arabic_matematics.html
Indonesia to English
Applying Trigonometry as a Basic Trilliteration for Global Positioning System ( GPS )
Long time ago, actually our great – grandparents have a unique ways in order to they not being lost. They build the easy looking monument, draw detail maps which need many energy, learned to read star in the sky. Then now with the development of technology, use of maps and compass replaced by other navigator tool named Global Positioning System ( GPS ). This system improve since 1970 by USA military department. In its grow, GPS used for other necessary. Actually that modern tool based from mathematics science named triliteration.
Global Positioning System is navigation system which exploiting satellite. GPS receiver get signal from some satellite around world. With the special formation, GPS can give position and time with very high height. GPS satellite around the earth twice a day. This satellite will transmission signal to the world, then other GPS will be differentiating time that transmit to calculate position.
GPS Principal, GPS must has at least 3 satellite to calculate 2 dimension position. With 4 satellites, GPS can calculate 3 dimension position. GPS position based by mathematics measurement named trilliteration which is need 3 basic point.
Determination point of GPS :
• Assuming you don’t know about your position, and you only know, you have 7 km from reference point “A”. Make circle with 7 cm of radian.
• Others, you know , you have 14 km from reference point “B”. draw, and you can see, that circle proportion on 2 points.
• Impossible you on 2 points at the same time, so you must draw one circle again, assumed you know, you have 3 km from reference point “C”. So you have 3 circle which are produced 1 point, this point your position.
Delivery data process by satellite
When GPS satellite sending signal to GPS receiver, that signal containing about position information satellite and time of that delivery signal. Because of there is a distance between satellite position and GPS receiver so there is difference time at the first time of signal transmitted with the time of accepted signal by the GPS component.
Speed of signal transmitted assumed like a velocity of light in a vacuum. With a difference time equal to T, so distance obtained as :
Distance ( S ) = time ( T ) x velocity of light ( V )
( S ) is the radian which used before.
Reason for using 4 satellites
GPS satellite sending position data ( X, Y, Z ) to receiver. Assumed there are 3 satellites that can accepted by our GPS, so there are 3 variables in 3 equations which accepted by GPS . That three equations will be produced a calculation that shows where is our location.
x1 + x2 + x3 = y1 + y2 + y3 =z1 + z2 + z3
Because every point ( X, Y, Z ) in satellite difference each other so in order to can produced the same calculation, we must add 3 variable that used by all of information, named ( a, b, c ) so the formula become :
ax1+ by1 + cz1 = ax2 +by2 +cy3 =ax3 + ay3 +cz3
Variable ( a, b, c) are information about our height. If the process delivery information to the receiver GPS in prefect condition, so with three information we can know about our position and how high our position.
There is one again variable that unknown by us that is “ Time ( T )”. That means we have 4 variable which unknown there are X, Y, Z, T. For calculate that variable, needed 4 different mathematics equation, because of that we need position information from satellite. So with 4 variable from 4 different equation, mathematically we can know about our position.
When we get information from 3 GPS satellite we know our position in 2 dimension, not 3 dimension, because height variable disregarded. We calculate data of X, Y, and T. Height data disregarded because GPS device majoring 2 dimension position.
Useful of GPS to life
• Military
• Navigation
• Geography information system
• Watcher earthquake
Reference :
Epsilon Edisi21/th.XIX/Oktober 2007/ hal.12 - 13
Long time ago, actually our great – grandparents have a unique ways in order to they not being lost. They build the easy looking monument, draw detail maps which need many energy, learned to read star in the sky. Then now with the development of technology, use of maps and compass replaced by other navigator tool named Global Positioning System ( GPS ). This system improve since 1970 by USA military department. In its grow, GPS used for other necessary. Actually that modern tool based from mathematics science named triliteration.
Global Positioning System is navigation system which exploiting satellite. GPS receiver get signal from some satellite around world. With the special formation, GPS can give position and time with very high height. GPS satellite around the earth twice a day. This satellite will transmission signal to the world, then other GPS will be differentiating time that transmit to calculate position.
GPS Principal, GPS must has at least 3 satellite to calculate 2 dimension position. With 4 satellites, GPS can calculate 3 dimension position. GPS position based by mathematics measurement named trilliteration which is need 3 basic point.
Determination point of GPS :
• Assuming you don’t know about your position, and you only know, you have 7 km from reference point “A”. Make circle with 7 cm of radian.
• Others, you know , you have 14 km from reference point “B”. draw, and you can see, that circle proportion on 2 points.
• Impossible you on 2 points at the same time, so you must draw one circle again, assumed you know, you have 3 km from reference point “C”. So you have 3 circle which are produced 1 point, this point your position.
Delivery data process by satellite
When GPS satellite sending signal to GPS receiver, that signal containing about position information satellite and time of that delivery signal. Because of there is a distance between satellite position and GPS receiver so there is difference time at the first time of signal transmitted with the time of accepted signal by the GPS component.
Speed of signal transmitted assumed like a velocity of light in a vacuum. With a difference time equal to T, so distance obtained as :
Distance ( S ) = time ( T ) x velocity of light ( V )
( S ) is the radian which used before.
Reason for using 4 satellites
GPS satellite sending position data ( X, Y, Z ) to receiver. Assumed there are 3 satellites that can accepted by our GPS, so there are 3 variables in 3 equations which accepted by GPS . That three equations will be produced a calculation that shows where is our location.
x1 + x2 + x3 = y1 + y2 + y3 =z1 + z2 + z3
Because every point ( X, Y, Z ) in satellite difference each other so in order to can produced the same calculation, we must add 3 variable that used by all of information, named ( a, b, c ) so the formula become :
ax1+ by1 + cz1 = ax2 +by2 +cy3 =ax3 + ay3 +cz3
Variable ( a, b, c) are information about our height. If the process delivery information to the receiver GPS in prefect condition, so with three information we can know about our position and how high our position.
There is one again variable that unknown by us that is “ Time ( T )”. That means we have 4 variable which unknown there are X, Y, Z, T. For calculate that variable, needed 4 different mathematics equation, because of that we need position information from satellite. So with 4 variable from 4 different equation, mathematically we can know about our position.
When we get information from 3 GPS satellite we know our position in 2 dimension, not 3 dimension, because height variable disregarded. We calculate data of X, Y, and T. Height data disregarded because GPS device majoring 2 dimension position.
Useful of GPS to life
• Military
• Navigation
• Geography information system
• Watcher earthquake
Reference :
Epsilon Edisi21/th.XIX/Oktober 2007/ hal.12 - 13
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