Jumat, 09 Januari 2009
about me
My name is Yogawati wulandariI live in Demangan Bangunharjo Sewon Bantul. I was born on the 30th of January 1990. I study at Yogyakarta State University, especially I studying mathematic. I have parents, they are my mother and my father. My father name is Mujiyono.My mother name is NgadilahI have two sistersMy old sister name is Hindun Puji Winarsih. She studies in Ahmad Dahlan UniversityMy young sister is Novia Feri Rahmadani. She studies in Junior High School.
englishII, cerpen, selasa,jam ke-3
Friday, 26th December 2008, I explain about polynomials to my friends. Her name is Indah Pertiwi, her number student is 07305144059. I am share about polynomials to my friends in the my home, at Demangan Bangunharjo Sewon Bantul, on 02.00 pm. First, I explain about polynomials to him. Indah have understood about it whitout return repetition. Second, I give some problems to him, and she can finish this problems. The conclution, Indah Pertiwi can get well what I submit. But any little problem when I explain about polynomial to him that is, her attention is less. Because she is being canker, else she pays me while play hand phone. But no problem, all go well. This is about polynomial that i explain to Indah Pertiwi.
Polynomials
Polynomial have variables and the power of the same variables in a series. This about polynomials that I take away from http: /library. thinkquest. org/20991/alg2/polyf. html.
When terms of a polynomial have the same variables raised to the same powers, the terms are called similar, or like terms. Like terms can be combined to make the polynomial easier to deal with. Example:
1. Problem: Combine like terms in the following
equation: 3x2 - 4y + 2x2.
Solution: Rearrange the terms so it is easier
to deal with.
3x2 + 2x2 - 4y
Combine the like terms.
5x2 - 4y
Probably the most important kind of polynomial multiplication that you can learn is the multiplication of binomials (polynomials with two terms). An easy way to remember how to multiply binomials is the FOIL method, which stands for first, outside, inside, last. Example:
1. Problem: Multiply (3xy + 2x)(x^2 + 2xy^2).
Simplify the answer.
Solution: Multiply the first terms of each bi-
nomial. (F)
3xy * x2 = 3x3y
Multiply the outside terms of each bi-
nomial. (O)
3xy * 2xy2 = 6x2y3
Multiply the inside terms of each bi-
nomial. (I)
2x * x2 = 2x3
Multiply the last terms of each bi-
nomial. (L)
2x * 2xy2 = 4x2y2
You now have a polynomial with four terms.
Combine like terms if you can
to get a simplified answer.
There are no like terms, so you have your final answer.
3x3y + 6x2y3 + 2x3 + 4x2y2
Although it would be nice if all you ever had to do was multiply binomials by other binomials, that isn't even close to reality. A perfect example of this is when you have to cube a binomial. Example:
2. Problem: Multiply (A + B)3 out.
Solution: Rewrite so you have something you can
actually multiply out.
(A + B)(A + B)(A + B)
Multiply the first two binomials together.
(A + B)(A + B)
A2 + AB + BA + B2
After combining like terms, you have
A2 + 2AB + B2
You now have a binomial and a
trinomial to multiply together.
(A2 + 2AB + B2)(A + B)
This is a slightly more complicated
situation than multiplying a binomial
by another binomial. Multiply the
first term of the binomial by each of
the terms in the trinomial and then
multiply the last term of the binomial
by each term in the trinomial.
A3 + 2A2B + AB2 + BA2 + 2AB2 + B3
Combine like terms if possible to
simplify the answer.
A3 + 3A2B + 3AB2 + B3
Factoring is the reverse of multiplication. When factoring, look for common factors. Example:
1. Problem: Factor out of a common factor of
4y2 - 8.
Solution: 4 is a common factor of
both terms, so pull it out and write
each term as a product of factors.
4y2 - (4)2
Rewrite using the distributive law of
multiplication, which says that
a(b + c) = ab + ac.
4(y2 - 2)
Sometimes, you will come across a special situation where both terms of a binomial are squares of another number, such as (x2 + 9). (x2 is the square of x and 9 is the square of 3.) There is a special formula for this situation, so you don't have to factor the binomial. The difference of squares formula is listed below. A2 - B2 = (A + B)(A - B) Example:
2. Problem: Factor y2 - 4.
Solution: Since y2 is the square
of y, and 4 is the square of 2,
this binomial fits the difference
of squares formula.
y2 - 4 = (y + 2)(y - 2)
Since trinomials are the most common polynomial you will be asked to factor, we will try to help you better understand how to factor quadratic trinomials, or trinomials whose highest power is two. Also, we assume you know how to multiply binomials (we use the "FOIL" method). Using a multiplication problem consisting of two binomials, we will show some important things to remember when factoring trinomials, which is the reverse of multiplying two binomials. Example: (x - 6)(x + 3) = x2 - 6x + 3x - 18 = x2 - 3x - 18 1. The first term of the trinomial is the product of the first terms of the binomials. 2. The last term of the trinomial is the product of the last terms of the binomials. 3. The coefficient of the middle term of the trinomial is the sum of the last terms of the binomials. 4. If all the signs in the trinomial are positive, all signs in both binomials are positive. Keeping these important things in mind, you can factor trinomials. Example:
3. Problem: Factor: x2 - 14x - 15.
Solution: First, write down two sets of parentheses to indicate the
product.
( )( )
Since the first term in the trinomial is the product of the
first terms of the binomials, you enter x as the first
term of each binomial.
(x )(x )
The product of the last terms of the binomials must equal
-15, and their sum must equal -14, and one of the
binomials' terms has to be negative. Four different pairs of
factors have a product that equals -15.
(3)(-5) = -15 (-15)(1) = -15
(-3)(5) = -15 (15)(-1) = -15
However, only one of those pairs has a sum of -14.
(-15) + (1) = -14
Therefore, the second terms in the binomial are -15 and
1 because these are the only two factors whose product
is -15 (the last term of the trinomial) and whose sum
is -14 (the coefficient of the middle term in
the trinomial).
(x - 15)(x + 1) is the answer.
Trinomials and binomials are the most common polynomials, but you will sometimes see polynomials with more than three terms. Sometimes, when you are dealing with polynomials with four or more terms, you can group the terms in such a way that common factors can be found. Example:
4. Problem: Factor 4x2 - 3x + 20x - 15.
Solution: Rearrange the terms so common
factors can be more easily found.
4x2 + 20x - 3x - 15
The first two terms have a common factor
in 4x. The last two terms have a
common factor in 3. Factor those
terms out.
4x(x + 5) - 3(x + 5)
Now you have a binomial. Each term
has a factor of (x + 5). Factor
that out for the final answer.
(x + 5)(4x - 3)
My activity with Indah Pertiwi
Friday, 26th December 2008, I explain about polynomials to my friends. Her name is Indah Pertiwi, her number student is 07305144059. I am share about polynomials to my friends in the my home, at Demangan Bangunharjo Sewon Bantul, on 02.00 pm. First, I explain about polynomials to him. Indah have understood about it whitout return repetition. Second, I give some problems to him, and she can finish this problems. The conclution, Indah Pertiwi can get well what I submit. But any little problem when I explain about polynomial to him that is, her attention is less. Because she is being canker, else she pays me while play hand phone. But no problem, all go well.
Polynomials
Polynomial have variables and the power of the same variables in a series. This about polynomials that I take away from http: /library. thinkquest. org/20991/alg2/polyf. html.
When terms of a polynomial have the same variables raised to the same powers, the terms are called similar, or like terms. Like terms can be combined to make the polynomial easier to deal with. Example:
1. Problem: Combine like terms in the following
equation: 3x2 - 4y + 2x2.
Solution: Rearrange the terms so it is easier
to deal with.
3x2 + 2x2 - 4y
Combine the like terms.
5x2 - 4y
Probably the most important kind of polynomial multiplication that you can learn is the multiplication of binomials (polynomials with two terms). An easy way to remember how to multiply binomials is the FOIL method, which stands for first, outside, inside, last. Example:
1. Problem: Multiply (3xy + 2x)(x^2 + 2xy^2).
Simplify the answer.
Solution: Multiply the first terms of each bi-
nomial. (F)
3xy * x2 = 3x3y
Multiply the outside terms of each bi-
nomial. (O)
3xy * 2xy2 = 6x2y3
Multiply the inside terms of each bi-
nomial. (I)
2x * x2 = 2x3
Multiply the last terms of each bi-
nomial. (L)
2x * 2xy2 = 4x2y2
You now have a polynomial with four terms.
Combine like terms if you can
to get a simplified answer.
There are no like terms, so you have your final answer.
3x3y + 6x2y3 + 2x3 + 4x2y2
Although it would be nice if all you ever had to do was multiply binomials by other binomials, that isn't even close to reality. A perfect example of this is when you have to cube a binomial. Example:
2. Problem: Multiply (A + B)3 out.
Solution: Rewrite so you have something you can
actually multiply out.
(A + B)(A + B)(A + B)
Multiply the first two binomials together.
(A + B)(A + B)
A2 + AB + BA + B2
After combining like terms, you have
A2 + 2AB + B2
You now have a binomial and a
trinomial to multiply together.
(A2 + 2AB + B2)(A + B)
This is a slightly more complicated
situation than multiplying a binomial
by another binomial. Multiply the
first term of the binomial by each of
the terms in the trinomial and then
multiply the last term of the binomial
by each term in the trinomial.
A3 + 2A2B + AB2 + BA2 + 2AB2 + B3
Combine like terms if possible to
simplify the answer.
A3 + 3A2B + 3AB2 + B3
Factoring is the reverse of multiplication. When factoring, look for common factors. Example:
1. Problem: Factor out of a common factor of
4y2 - 8.
Solution: 4 is a common factor of
both terms, so pull it out and write
each term as a product of factors.
4y2 - (4)2
Rewrite using the distributive law of
multiplication, which says that
a(b + c) = ab + ac.
4(y2 - 2)
Sometimes, you will come across a special situation where both terms of a binomial are squares of another number, such as (x2 + 9). (x2 is the square of x and 9 is the square of 3.) There is a special formula for this situation, so you don't have to factor the binomial. The difference of squares formula is listed below. A2 - B2 = (A + B)(A - B) Example:
2. Problem: Factor y2 - 4.
Solution: Since y2 is the square
of y, and 4 is the square of 2,
this binomial fits the difference
of squares formula.
y2 - 4 = (y + 2)(y - 2)
Since trinomials are the most common polynomial you will be asked to factor, we will try to help you better understand how to factor quadratic trinomials, or trinomials whose highest power is two. Also, we assume you know how to multiply binomials (we use the "FOIL" method). Using a multiplication problem consisting of two binomials, we will show some important things to remember when factoring trinomials, which is the reverse of multiplying two binomials. Example: (x - 6)(x + 3) = x2 - 6x + 3x - 18 = x2 - 3x - 18 1. The first term of the trinomial is the product of the first terms of the binomials. 2. The last term of the trinomial is the product of the last terms of the binomials. 3. The coefficient of the middle term of the trinomial is the sum of the last terms of the binomials. 4. If all the signs in the trinomial are positive, all signs in both binomials are positive. Keeping these important things in mind, you can factor trinomials. Example:
3. Problem: Factor: x2 - 14x - 15.
Solution: First, write down two sets of parentheses to indicate the
product.
( )( )
Since the first term in the trinomial is the product of the
first terms of the binomials, you enter x as the first
term of each binomial.
(x )(x )
The product of the last terms of the binomials must equal
-15, and their sum must equal -14, and one of the
binomials' terms has to be negative. Four different pairs of
factors have a product that equals -15.
(3)(-5) = -15 (-15)(1) = -15
(-3)(5) = -15 (15)(-1) = -15
However, only one of those pairs has a sum of -14.
(-15) + (1) = -14
Therefore, the second terms in the binomial are -15 and
1 because these are the only two factors whose product
is -15 (the last term of the trinomial) and whose sum
is -14 (the coefficient of the middle term in
the trinomial).
(x - 15)(x + 1) is the answer.
Trinomials and binomials are the most common polynomials, but you will sometimes see polynomials with more than three terms. Sometimes, when you are dealing with polynomials with four or more terms, you can group the terms in such a way that common factors can be found. Example:
4. Problem: Factor 4x2 - 3x + 20x - 15.
Solution: Rearrange the terms so common
factors can be more easily found.
4x2 + 20x - 3x - 15
The first two terms have a common factor
in 4x. The last two terms have a
common factor in 3. Factor those
terms out.
4x(x + 5) - 3(x + 5)
Now you have a binomial. Each term
has a factor of (x + 5). Factor
that out for the final answer.
(x + 5)(4x - 3)
My activity with Indah Pertiwi
Friday, 26th December 2008, I explain about polynomials to my friends. Her name is Indah Pertiwi, her number student is 07305144059. I am share about polynomials to my friends in the my home, at Demangan Bangunharjo Sewon Bantul, on 02.00 pm. First, I explain about polynomials to him. Indah have understood about it whitout return repetition. Second, I give some problems to him, and she can finish this problems. The conclution, Indah Pertiwi can get well what I submit. But any little problem when I explain about polynomial to him that is, her attention is less. Because she is being canker, else she pays me while play hand phone. But no problem, all go well.
tugas englishII, ahead motion, selasa jam ke-3
Ahead Motion
We must be university student that active position. For that, information not only got from when at class. But also from other source like; book, internet, blog, friends, patner communication, other blogger, participasing lecture, download video from u tube, etc. The most importantly in learn how do we communicate with another person, how do we give expression important development. Can by write book, essay, scientific work, etc. If this case to do, then competence the greater becomes. So that we progress, we must has competition soul
Blog is one of the form of education communication in learn english mathematic. Blog use make easier communication in learn English mathematic. In English mathematic, blog must be given photo and the identity be clear and original. In English mathematic, blog as activity of university student substitute porto folio.
Kamis, 08 Januari 2009
tugas englishII video pre-calculus, selasa jam ke-3
Nama: Yogawati Wulandari
Nim:07305144006
Prodi:Mat,NR
Video I
PRE-CALCULUS: GRAPH OF A RATIONAL FUNCTION
Pre – calculus, an advanced form of secondary school algebra, is a foundational mathematical dispcipline. It is sometimes considered to be an honors course. Courses and textbooks in precalculus are intended to prepare students for the study of calculus. Precalculus typically includes a review of algebra and trigonometry, as well as an introduction to exponential, logarithmic and trigonometric fungtions, vectors, complex numbers, conic sections, and analytic geometry. Equivalent college algebra and trigonometry.
A rational function is any function which can be written as the ratio of two polynomial functions. In the case of one variable, x, a rational function is a function of the form
Where P and Q are polynomial function in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q(x) is not zero.
Example:
The rational function is not defined at .
Video II
Limits by inspection
if f(x) representing rational function, for example by p(x) and q(x) representing poliynom, so to finish infinity limit dividedly numerator p(x) and denominator q(x) by the highest power of x that happened.
For example:
(1)
If the highest power of x in numerator more small than power of x in denominator, so the product from above is .
(2)
If the highest power of x in numerator and denominator is same, so the product from above is 3/2.
Video III
English Solving Problem Graph Math
Y=g(x)
Function h is defined by h(x)=g(2x)+2
The value of h(1)
h(x)=9(2x)+2
h(1)=9(2.1)+2 on curve g(x), if x=2 so y=1
=1+2=3
(13) Let the function f be defined by f(x)=x+1, if 2f(p)=20, the value of f(3p)?
Solve:
f(3p) when x=3p
f(x) = x+1
2f(p)=20
f(p)=10
f(p)=p+1=10
p=9
x=3p
x=3(9)
x=27
f(27)= 27 +1
= 28
(17) In the xy= coordinate plane, the graph of x=y+4 intersect line l at (0,p) and (5,t). what is the greathest possible valueof the slopeof l ?
Solve:
x=y-4
slope of line l =m=
x=0y=p
x=5y=t
m=
Video IV
f(p,q)=0
function y=f(x)=VLT
function
x=g(y):HLT=invertible
y=2x-1
x=0 y=-1
x=1/2 y=0
y=x
y=2x-1
2x=y+1
x=½ (y+1)
x=½y + ½
y=½y + ½
f(x)=2x-1
g(x) =½y + ½
f(g(x))=2(½y + ½)-1
=x+1-1=x
g(f(x))= ½(2x-1)+ ½
for example:
a)f(x)=3x+6 b)f(x)=27x
y=3x+6 y= 27x
f(x)=
Nim:07305144006
Prodi:Mat,NR
Video I
PRE-CALCULUS: GRAPH OF A RATIONAL FUNCTION
Pre – calculus, an advanced form of secondary school algebra, is a foundational mathematical dispcipline. It is sometimes considered to be an honors course. Courses and textbooks in precalculus are intended to prepare students for the study of calculus. Precalculus typically includes a review of algebra and trigonometry, as well as an introduction to exponential, logarithmic and trigonometric fungtions, vectors, complex numbers, conic sections, and analytic geometry. Equivalent college algebra and trigonometry.
A rational function is any function which can be written as the ratio of two polynomial functions. In the case of one variable, x, a rational function is a function of the form
Where P and Q are polynomial function in x and Q is not the zero polynomial. The domain of f is the set of all points x for which the denominator Q(x) is not zero.
Example:
The rational function is not defined at .
Video II
Limits by inspection
if f(x) representing rational function, for example by p(x) and q(x) representing poliynom, so to finish infinity limit dividedly numerator p(x) and denominator q(x) by the highest power of x that happened.
For example:
(1)
If the highest power of x in numerator more small than power of x in denominator, so the product from above is .
(2)
If the highest power of x in numerator and denominator is same, so the product from above is 3/2.
Video III
English Solving Problem Graph Math
Y=g(x)
Function h is defined by h(x)=g(2x)+2
The value of h(1)
h(x)=9(2x)+2
h(1)=9(2.1)+2 on curve g(x), if x=2 so y=1
=1+2=3
(13) Let the function f be defined by f(x)=x+1, if 2f(p)=20, the value of f(3p)?
Solve:
f(3p) when x=3p
f(x) = x+1
2f(p)=20
f(p)=10
f(p)=p+1=10
p=9
x=3p
x=3(9)
x=27
f(27)= 27 +1
= 28
(17) In the xy= coordinate plane, the graph of x=y+4 intersect line l at (0,p) and (5,t). what is the greathest possible valueof the slopeof l ?
Solve:
x=y-4
slope of line l =m=
x=0y=p
x=5y=t
m=
Video IV
f(p,q)=0
function y=f(x)=VLT
function
x=g(y):HLT=invertible
y=2x-1
x=0 y=-1
x=1/2 y=0
y=x
y=2x-1
2x=y+1
x=½ (y+1)
x=½y + ½
y=½y + ½
f(x)=2x-1
g(x) =½y + ½
f(g(x))=2(½y + ½)-1
=x+1-1=x
g(f(x))= ½(2x-1)+ ½
for example:
a)f(x)=3x+6 b)f(x)=27x
y=3x+6 y= 27x
f(x)=
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