NAMA : RANI RAHAYU P.
NIM : 07305144001
PRODI : MATEMATIKA NR’07
PROGRAM : BHS. INGGRIS II
EMAIL : ranny_15sept@yahoo.com
ranny.mathuny@gmail.com
CP : 08562544575


I. Pre-Calculus : Graph of a rational function.
Graph of a rational functionàcan have discontinues has a polynomial in the denominator.
Example :
f(x) = (x+2) / (x-1), if we insert x = 1, we get
f(1) = (1+2) / (1-1)
= 3 / 0àbecause the denominator is 0 is not possible, so we don’t insert x = 1, so x = 1 is bad choice.
If we insert x = 0, we get
f(0) = (0+2) / (0-1)
= 2 / (-1)
= -2 àwe can write (0,-2), so x = 0 is not bad choice.
Rational function denominator can be zero, not all rational function will give the not possible solution. Polynomialàx zero in the denominator (break in the graph, missing point, feasible, and not allowed).

Example :
y = (x2-x-6) / (x-3) if we insert x=3
= [(3)2-3-6] / (3-3)
= 0 / 0àis not possible.
y = (x2-x-6) / (x-3), we can write
= (x-3) (x+2) / (x-3), the numerator get factor (x-3) and (x-2) we can delete with the denumerator (x-3),so the equation become
y = x+2, if we insert x = 3 we can get
y = 5
(3,5)
The conclusion is if we insert x = 3 and the numerator can not factor we get not possible solution, but if we get the factor from the numerator we can get the posibble solution.

II.Limit by Inspection
Limit by inspection there is two condition
1. positive and negative invinitif
2. ………………………………
Example :
§ lim (x3+x4) / (x2+x+1) = ¥
xà¥
· polynomial over polynomial
· x approaches invinity
à if degree of the numerator f(x) > degree of the denumerator g(x) and the highest power of the numerator f(x) is positive, so :
lim f(x) / g(x) = ¥
xà¥
à if degree of the numerator f(x) > degree of the denumerator g(x) and the highest power of the numerator f(x) I negative, so :
lim f(x) / g(x) = -¥
xà¥
§ lim (x2+3) / (x3+1) = 0
xà¥
à if the degree of the numerator f(x) < degree of the denunerator g(x), so :
lim f(x) / g(x) = 0
xà¥
§ lim (4x3+x2+1) / (3x3+4) = 4 / 3
xà¥
à if the degree of the numeratoe f(x) = degree of the denumerator g(x), so :
lim f(x) / g(x) = coefficient the highest power from f(x) / coefficient the higgest xॠpower from g(x)


III. EXERCISE
1. The figure above shows the graph of y=g(x) in the function h is definicied by h(x)=g(2x)+2, what is the value is h(1)…..
h(x)=g(2x)+2,insert x=1
=g(2.1)+1
=g(2)+2,because g(2)=1 (from the graph), so we can write
=1+2
h(x) = 3
à so the value h(1)=3
2. Let the function f be definiced b f(x)=x+1 if 2f(p)=20, what is the value of f(3p)..
f(3p) àwhat is f when x=3p
f(x)=x+1
2f(p)=20
f(p)=10
f(x)=x+1,insert x=p
f(p)=p+1, because f(p)=10 so,
10=p+1
p=9
f(x)=x+1
f(3p)=3p+1, insert x=3p
because p=9, so we insert at f(3p)=3p+1, we get
f(3p)=3.9+1
=27+1
=28
3. In the xy coordinate plane, the graph of x=y2-4 intersecte line that (0,p) and (5,t). What is the greatest possible value of the slope f….
Greatest m
Curve x=y2-4 through (0,p) and (5,t) we get line g through (0,p) and (5,t) the slope from line g is (y2-y1) / (x2-x1) = (t-p) /(5-0), so the slope from line g is (t-p) / 5.
So the gretest value possible value of the slope f is m= (t-p) / 5

IV.Inverse Function
y=f(x)
x=g(y) àinvertible
Example :
f(x)=2x-1
g(x)=1/2x+1/2
f(g(x))=2(1/2x+1/2)-1, insert x=g(x)
g(f(x))=1/2(2x-1)+1/2, insert x=f(x)
àf(g(x))=gof composition of function
àg(f(x))=fog
y=(x-1) / (x+2)
y(x+2)=x-1
yx+2y=x-1
yx-x=-1-2y
(y-1)x=-1-2y
x=(-1-2y) / (y-1), inverse from this function is
y=(-1-2x) / (x-1)
from the two function above we can find the value from x and y
Example: cx=0, so 0=y-1
y=1
cy=0, so 0=-1-2x
x= -1/2
2x-1=y
2x=y+1
x=1/2 (y+1)
x=1/2y+1/2, inverse from this function is
y=1/2x+1/2

TUGAS 12
Diffrensial Equation
Definition: The simple equation from differensial eqation is a derivative of the function of an unknown, the function itself, the variables with the name of the function is defined, and constant. If the unknown function depends on a real variable, diffrensial equation called equality differensialusual.
Diffrensial equation consists of 5 types,namely:
1. Differensial exact equation
2.Diffrensial homogeneous equation
3.Diffrensial equation that can be separated
4.Diffrensial linier equation
5.DifferensialBernoulliequation.
Diffrensial Homogeneouse Equation
Definition: Diffrenesial order equation M-1 (x, y) dx + N (x, y) = 0 dy homogeneouse said, if written in the form of derivatives (dy / dx) = f (x, y), there such a function to f ( x, y) can be expressed in the form of g (y / x).
Step-step settlement:
1. There Persamaan Diffrensial in the form of F (x, y) dx + N (x, y) dy = 0 2. Then we let y = vx 3. Persamaan form Diffrensial become dy / dx = N (x, y) / m (x, y) 4. We substitution on the value of y diffrensial with vx 5. Complete there so that only variable in the equation v diffrensial is 6. Substitution with the value v = y/x 7. So there are similarities end diffrensial results in the form of dy / dx = g (y / x) 8. To find solutions we generally diffrenetiation y = vx become so dy / dx = v + x dv / dx 9. Substitution value of dy / dx on diffrensial of dy / dx = g (y / x) with v + x dv / dx 10.Complete with integral, so generally obtained a settlement. Here I will explain about the homogeneous Persamaan Diffrensial to my friends Apriliani Erni. This activity we do on 7 January 2009. The first thing I do is first to prepare the material (material) along with sample problems and solutions. Then I tried to explain to my friends with it. I do not use the media to explain the board to my friends. My them with capital sheet of paper containing materials and sample problems, and a bolpoin. At the beginning of the explanation on the definition of a common homogeneous differensial my friend does not have difficulties, then I try to explain the steps it differensial homogeneous equality through example problems. Through examples of problems that I have given the difficulty to step-4. He felt confused why should the value of y disubstitusikan with vx. I explain why the value of y substitute vx. With the explanation that I gave my friend began to understand because this step is a requirement that equality is a form of g (y / x). I went back to continue to the next step, and my friend did not experience problems until the last step. He was so enthusiastic and the spirit of listening to the explanation I gave. After that I give to my friend's question to see if my friend already understand what I described earlier. I wait until I finished the problem and in fact he did not have difficulties during the matter. After that we return the matter Persamaan Diffrensial homogeneous together, and the end result we match together. We give each other explanations when there is a different answer.

TUGAS 11
UTILIZATION BLOG TO DEVELOPMATH
Blog used to make it easier for us in touch or communicate with friends, teachers or other people. In addition, blogs can also be used to find the information we need. Formally, blogs can also be used for media such as discussion on a matter or can also be used to express something (learning, studying and writing). Blogs can also be used for learning mathematics in English. We can get the information we need and can also write things or something and then we write the show in the blog. Blog is an academic communication in the blog we can list or show academic identity (profile), displays photos, post / write, etc.. Constraints if we do not have a blog is easy to manage the FOLLOWER.

TUGAS 10
Intensive Language Via Blog
English is the language that provides a different aspect of culture. With the English language which is the international language we can easily communicate with other nations. The most important in the English language is how we can each do our improve speaking, how we communicate and develop the communication. Unlike the language we use language that reflects our culture that includes our local ingenuity that include things such as emotional and spiritual. Language / communication can be done with the various ways one of them is to take advantage of the blog. We can write the results of our minds using either English or Indonesian Language. Most important is that people who read the blog can we understand and understand what we write, so they can provide feedback, or responses to criticism. Blog is an informal means of communication. We can communicate with others across the blog. As a human being we should know that we can not be separated with each other. Should we talk to know with whom we spoke (the opponent we speak) and in what circumstances. For example, if we speak with our teachers or lecturers, we must know how to order the way we communicate in order to create a formal atmosphere. The level communicate classified into four, namely:
1. lower levelà material communication (blog, send message)
2. higher levelà formulaize communication (something good, bad, wrong)
3. lowest levelà normatif communication (good, bad)
4. highest levelà spiritual communication (believe in God / implementation of religion)
There are two kinds of unhealthy communication active vocal and univocal. That is can destroy the meaning of language.