CONCEPTS not MAGIC ( ahh :(

1.- What is the DIFFERENCE between finding the limit of a function at x = c and actually plugging in the number x = c? When are the two cases the SAME?




----OK when finding the limit of a function as x=c, your are looking for an output that gets closer and super close to that number c. And this is shown as lim f(x) as x-->c . On the other hand, when you actually plug in the number c into a determined function, you're looking for exactly a certain output and you are more concerned about what's actually happening at that point c and not at its surroundings.


For example, the function:


f(x)= 2x + 4
by plugging in exactly a number c in this case let's say 3,




f(2)= 2(3) + 4
= 6+ 4
= 10


we get 10 as the exact output at the point 3.


---- They both are the same when the lim of f(x) as x-->c it's equals to f(c). This means that there is not a hole in that part of the function. Therefore, they both are the same when a function is continuous.



2.- What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?




------The similarities between these two are that for both you are looking for the slope of a line. However, when you are looking for a derivative, you are looking for the slope of a tangent line that touches a certain point at a curve. The slope of that tangent line will give you the slope of the curve at that certain point.

Looking at the graph, the tangent line is the green one that touches that blue curve at a certain point.

And when looking for the slope of a line, you are just looking for the slope of a common line.

LIMITSSS and ideas..

I don't know why but I feel I kind of like limits after all.. because I have to picture some of the problem in my head specially when the limit of x approaches to +- infinity and it makes me understand it better...
HOWEVER....

1.) sometimes when the problems involve sin(x) I get confused. This is because it takes me time to understand how sin(x) as it approaches infinity it's equals 0 .I remember that's the answer but correct me if I'm wrong. I know sin(x) it's always between 1 and -1 but it's not like f(x)= 1/x where the function gets very close to 0 as it goes to infinity...ITS WEIRD.

2.) I also have trouble determining the limits when we have to look at graphs and there is that one filled dot by itself either above or under other open dot that is part of a line. That messes my head up. :-S


3.) Other concept that eludes me it's when we have to find the Right end behavior model and Left end behavior model of a problem like this one y=x^2+e^-x.
First I graph both lines and then I try to figure out to which one each side it's gonna look like but it's really hard. I try not use the calculator but I have toooo...it's sad :(

Majors and Colleges!!

1-The major category in which I'm interested is Biological and Biomedical Sciences - Physiology, Pathology and related sciences.
From that category I have these four areas of interest:

1.) Pathology: It's the study of disease with its causes and effects by examining damaged organs and tissue. It also studies how these diseases deteriorate the human body. There is Anatomical Pathology and Clinical Pathology. I'm interested in pursuing Anatomical Pathology, for I'm going to be able to perform autopsies and examine different tissues. Clinical Pathology is mainly working with body fluids at a laboratory while Anatomical Pathology deals with the gross examination of the organs.

2.) Cell Physiology: It's the study of the physiological processes of cells such as their respiration and digestion and how they keep us alive.

3.) Molecular Physiology: This is the study of the organs of the body and how they communicate between cells and other organs.

Biological and Biomedical Sciences - Animal Physiology.

4.) Animal Physiology: It's the study of the processes and systems that keep animals alive such as the digestive system and respiratory system.



2.-Colleges: I couldn't find any college here in CA but here are some of the colleges that offer the majors that I'm interested in.

The colleges that offer a major in Pathology are

1.) University of Maine:
This is a public university where the majority of first year students (38%) have a GPA ranging from
3.00 - 3.24. It is located in a rural setting at a large town (10,000 - 49,999)
They accept 77% of freshman applicants. However, the tuition for out-of-state students which is $23,876 doubles the one for in-state students. The university offers Bachelor's Master's and Doctoral degrees.


2.) University of Connecticut:
This University also offers a major in Animal Physiology.
This is a public four-year university that offers an Associate, Bachelor's, Master's, Doctoral and First Professional degrees.
A 42% of undergraduates are awarded with scholarships or grants.
They have ESL (English as a Second Language) programs as special study options.
The university seems to accept only those with SAT scores above 540.
The most popular major for an associate degree is agriculture with a 100% of preference for this degree.

- University that offers a major in Animal Physiology:
3.) Texas State University -San Marcos:
It is located in a suburban setting in a large town (10,000 - 49,999).This is a four-year public university that offers Bachelor's, Master's, Doctoral and First Professional degrees.
It has only 1% of out-of-state students.
The minimum for the ACT is 26 and SAT is 1180 (exclusive of writing) and an admission essay is required.
It guarantees on-campus housing for first-year students :)
They have nearly 300 social, service, religious, political, and professional organizations.

TIP$ AND HINT$

1.- Regarding transformations I try to do them by remembering the parent function. After that I see which part of the function has been "transformed" either the input or the output.
To know if the input is the one altered, I look for any number inside the parenthesis where x is like (2x) or (x+2) . Then the changes for input I know they can only be horizontally or from left to right or right to left because we are playing with the input, which is x, the x-axis.
Sometimes, there might not be a parenthesis and there'll be something like this: f(x)= x+1 so in that case I know that the output is the one being changed because it's not inside a parenthesis.

To know if the output is the one changed I look for numbers around the parenthesis, meaning outside of it
for example: -f(x)=(x), y= 3sin(3x) or y= 2sin(x)+3
The changes for the output I know they'll be vertically because we are playing with the output which is y, the y-axis.
However, there'll be times where there is no ( ) and just something like this : y= 4sin, and still the output is the altered one because it's not inside a ( ) with the x.


2.- Regarding trigonometry, I remember it because I memorized the unit circle. I kind of get confused with the points of π/6 and π/3 but I tried to remember them just by memorizing cosine of one of them, so I'll know that, for example the cosine of π/3 , which is 1/2 will be the sine of π/6 because they switch. Hope I didn't confuse some of you..but that's how I get to remember it,, :).
Also for tanx and cotx, at first I had trouble remembering which was on top so I just said that tanx is the original one so it obeys the rise/run for the slope , therefore tanx will sine/cosine (y/x) and since cotx is the "fake one" (for me) , it'll be the opposite cosine/sine (x/y).

3.- One thing that I still have a hard time doing is the graphing of the trigonometric functions and it gets worst when they have all of the vertical and horizontal transformations because I just get stuck and I don't know where to start. Also, when they have all this transformations I have to have certain points on the x-axis in order to graph them and I don't know which are the right ones to put. I need a tip for this in order to make it easier. I would like to know how to start graphing this and which points I should put on the x-axis :

y= -3tan(3x+π)+2 Help please!

InVeRsEs :) and logarithms :(

OK I'm being honest here I understood more to the inverses topic than logs because logs have been a pain everytime I take math :-s.

INVERSES.-

1-To find the inverse f^-1(x) of a function f(x) we simply change the x and y of the function and the input will now become the output and the output will become the input.

FOR EXAMPLE:

f(x)= 3x-9

y=3x-9--------switching------------x=3y-9

- Likewise, graphically, the domain of f(x) will become the range of f^-1(x) and the range of f(x) will become the domain of f^-1(x).

-The points of both graphs switch with one another because the graph of f(x) and its inverse f^-1(x) are equally divided by the y=x line, meaning that they are both symmetrical about the y=x line.

FOR EXAMPLE:

f(x)= x^3 + 2


In this graph the point (0,2) lies on f(x), so its inverse f^-1(x) (here its represented by g(x))

will have the point (2,0). Notice the x switches with the y.

2.- To know if a function has an existing inverse (meaning that the inverse is also a function) we use the horizontal line test to verify. If a function does have an existing inverse then we say the function is one to one.

FOR EXAMPLE:

By using the Vertical Line Test we can prove that f(x)=x^2 is a function because it touches the graph just once!

HOWEVER

If we use the Horizontal Line Test on

f(x)=x^2 again, we can see that it doesn't have an existing inverse because it touches the graph more than once!

The inverse of f(x)=x^2 looks like this :

is the x=y^2

Notice that if we do the Vertical Line Test on the inverse function it'll touch it twice so it's not a function!

SO, AS MISS HWANG SAID, THE HORIZONTAL LINE TEST IS DONE IN THE ORIGINAL FUNCTION, THIS CASE

F(X)=X^2 SO WE ARE ABLE TO KNOW IF IT'S GOING TO HAVE AN EXISTING INVERSE BEFORE WE ACTUALY GRAPH THE INVERSE.

In this way , F(X)=X^2 is not a ONE TO ONE function.

LOGARITHMS.- :-(

OK logs are hard for me to quickly understand! But this is what I understand so far:

3.- The inverse of an exponential function f(x)= 2^x , is the logarithmic function of x, f^-1(x)=log2x because if we recall in order to find the inverse of a function we change the y with the x in the function like this:

f(x)=2^x

y=2^x

x=2^y------------------------ When we exchange the x and y , it looks like this being

the inverse!

So, if the inverse of it is f^-1(x)= log2x, it has to look like that also.

f^-1(x)= log2x

y=log2x Remember y is the exponent, 2 is the

base and x is the answer.

2^y=x------------ Yeah it does look like the one before!!!!!

THEREFORE, log2x is the

inverse of f(x) =2^x

4.- From the topic of logarithms I also know that there is this number "e" which is equals to 2.718281....and that its inverse is the natural logarithm ln.

HOWEVER::::::

I don't really understand how ln can be its inverse. First of all ...what is ln besides being the "natural logarithm". Where did it come from???...HOw Can It Be the inverse of 2.718281??

Can someone give me an example on how to solve a problem with ln AND e ?

YEAH EVERYTIME THERE IS ln AND THE THE NUMBER e INVOLVED IN A PROBLEM I GET FRUSTRATED!! :(

EvEn anD oDD FuNcTiONs

Let's talk about functions. To start off, a function which can be

represented by f(x) is said to be a function because for every input x (DOMAIN) that we put in a function there is always just one single output which is y (RANGE). So, the domain of a function represents all the possible inputs for the x value and the range represents the outputs of the function, which is the value of y or f(x).



There are two types of functions even and odd. An even function is a function that is symmetrical about the y-axis meaning that one side of the function reflects across the y-axis. This type of function is represented by f(-x)=f(x). This means that if you plug -x for every x in the function you end up having the same function that you started with.

For example:

f(x)= -3x^2 + 4

f(-x)= -3(-x)^2 + 4

= -3(x^2) + 4

= -3x^2 + 4 -------you end up with the same

function you started with :)

The graph of that even function is the one below

- We can see that the right side of the function on quadrant 1 reflects on the other, quadrant 2. A characteristic of even functions is that a point (x,y) is on the graph only if the point

(-x, y) it's on it too. In this case we can see that the points (1,1) and (-1,1) are on the graph.

An odd function is a function that is symmetrical about the origin meaning that the function reflects across the y-axis and across the x-axis at the third quadrant. This type of function is represented by f(-x)= -1f(x). This means that if you plug -x for every x in the function the function will end up looking with all its signs opposite to what they were in the original function due to the -1.

For example:

f(x)= 2x^3 - 4x

f(-x)= 2(-x)^3 - 4(-x)

= 2(-x^3) + 4x

= -2x^3 + 4x ------------------- we end up having the opposite signs. It's

like putting a -1 in front ( -1f(x)) and

multiply everything by it.

The graph of the function above is this one

- For odd functions it's kind of hard to tell the reflection across the y- axis and at the third quadrant but it's there. In this kind of functions a point (x, y) is on the graph only if the point (-x, -y), which corresponds to the third quadrant, is there too. In this case the point (2, 8) and (-2, -8) are on the graph.

The odd function below shows the reflections across the y-axis and at the third quadrant.

If you would like to prove these reflections you can do it by tracing the graph of the function on butter paper.

You fold the paper in half at the y-axis of the graph and you can see that the function reflects across the y-axis having a mirror image and the reflection across the x-axis at the third quadrant.

This is f(x)=sinx

NOTE: A function can also be neither even nor odd. This happens when you plug in -X for every X in the function and at the end you end up neither with the same function you started with nor a function with all the signs opposite to the original.

TRYING TO BE HAPPY LIKE HIM!!!

About Me !!!

Hi everyone,
My name is Miriam del Rosario Ramos Rodriguez but everyone knows me just by Miriam Ramos. I'm 17 years old, and I'm in 12 grade "C" track. I'm a responsible person. I always like to do things right away and seek for help if I don't understand. Sometimes I may ask too much but that's the way I learn and try not to be ignorant. I like doing homework (that may seem weird but yeahh I do) because I can reinforce what I've learned and know even more. I say this because sometimes when I'm doing homework I go so deep in it that I end up having many windows open in my computer about the subject in my homework and that leads me to learn new things.

I'm a catlover, I really love them they are my favorite pet then the dogs. Right now I'm attending to a class at an animal shelter where we can learn more about domestic animals especially dogs. So far we haven't touched a cat just dogs, roosters and rabbits and I'm really excited to get to groom a cat. It's good to see when these animals get adopted and that they've got a new family and that makes me happy.
My favorite color is light purple, I really like chocolate especially Ferrero Rocher mmm. :).. My favorite subject so far has been physiology where I got to dissect a mink ( it was super cool). I'm interested in the medical field and I hope one day to become a pathologist and perform autopsies.

I'm the kind of person who likes to adhere to the rules and thus I struggle to always earn an A in my classes because I feel I'm taking advantage of this great gift that my parents are giving me, which is an education. And this education will also be my parents' only inheritance that will prepare me for my future.


This is the first AP class I'm taking in high school and I now it's a really good investment in my future since we use math everyday so why not enhance the little knowledge I have so far. I didn't take AP classes earlier because I was more focused on high school level classes and I kind of regret that because now these classes are of great consideration in the applications for universities. However, I feel I've improved and worked really hard to keep up with my English because this year I got to have three years living here in the United States and they had not been that easy. First of all when I came from El Salvador, my math was very poor since what students were seeing in middle school was very strange for me and I was so sad. In that only month that I went to middle school, I could learn what know seems pretty easy, which was how to graph a line (this is actually what we saw today in class :)). After that I came to this high school and my math has really improved a lot. I'm not a math whiz but I like to learn new concepts in math and expand my knowledge in this science. Sometimes my mind goes blank and I'll need some help to understand and then I can start doing my work very carefully not to mess up, for a number or symbol can change everything. I'm very cautious when doing math and that's why I take a while when solving problems.

I feel I will do well in this class and I hope to get much out of it. I'm very confident and I'm willing to help people (if I understand) when they get confused and I hope I can count on my classmates to ask them for help when I don't understand. I wish good luck to everyone and encouragement to strive to pass the class.