1.) "Finding the limit of a function" is basically saying, "What is the output approaching to as x gets closer and closer to a number?" while "plugging in c for x" is saying "what is the output when x = c". The limit may be different from when you plug in c for x.
When these two concepts are the same is when you are finding the limit of a continuous function.
For example,
f(x) = 4 - x^2 at x = 2
As a continuous function, the domain has no restrictions thus x can be any number therefore
the limit as x -> 2 of f(x) = f(2)
2.) Finding the derivative is basically finding the slope of the tangent line thus for both, you use your change of y over your change of x.
The difference is that finding the derivative is finding the slope of a curve at a point while finding the slope of a line, you are only finding the slope of that specified line.
Friday, December 18, 2009
Monday, December 7, 2009
Limits
3 ideas that elude me:
1.) Discontinuity when x approaches some number with a piece wise function
2.) Finding the extended function of the original equation
3.) The concept of integer x
1.) Discontinuity when x approaches some number with a piece wise function
2.) Finding the extended function of the original equation
3.) The concept of integer x
Tuesday, November 24, 2009
Colleges?
Majors
1.) Game and Simulation Programming
You learn how to make video games! With this major you can also figure out how to create programs or technology that can provide reconstruction on crime scenes or provide simulations such as flying a jet for the Air Force. This requires a massive knowledge on mathematics. But more importantly, you learn how to make video games.
2.) Biomedical Engineering
With this major you learn how to craft artificial limbs or organs and learn how to create, fix or improve medical equipment to ensure stability of the success rate for a treatment. It requires a lot of mathematics on how to even create a device that would work with organic material.
3.) Electrical Engineering
With this major you learn how to work primarily with electricity. So like you learn how to fix the electrical cables in any house, build or fix a electronic device and more.
Colleges/Universities
1.) DeVry University
This university circles around with updated factors meaning that they use up to date methods on teaching such as using the internet for lecturing, sending assignments, or turning in assignments or the student works together with the professor. It provides flexibility on whether taking your classes online, in campus or both. If you thinking of taking a engineering major, you might need a laptop that has the minimum requirement for that major.
2.) UCLA
Nice school. It requires a minimum of 3.0 GPA. Provides free internet access to all students. You can house there. The university holds 419 acres of land...
3.) UC Berkeley
Known for being in the top engineering schools. 1232 acre campus. 32% tuition increase for all UCs...
1.) Game and Simulation Programming
You learn how to make video games! With this major you can also figure out how to create programs or technology that can provide reconstruction on crime scenes or provide simulations such as flying a jet for the Air Force. This requires a massive knowledge on mathematics. But more importantly, you learn how to make video games.
2.) Biomedical Engineering
With this major you learn how to craft artificial limbs or organs and learn how to create, fix or improve medical equipment to ensure stability of the success rate for a treatment. It requires a lot of mathematics on how to even create a device that would work with organic material.
3.) Electrical Engineering
With this major you learn how to work primarily with electricity. So like you learn how to fix the electrical cables in any house, build or fix a electronic device and more.
Colleges/Universities
1.) DeVry University
This university circles around with updated factors meaning that they use up to date methods on teaching such as using the internet for lecturing, sending assignments, or turning in assignments or the student works together with the professor. It provides flexibility on whether taking your classes online, in campus or both. If you thinking of taking a engineering major, you might need a laptop that has the minimum requirement for that major.
2.) UCLA
Nice school. It requires a minimum of 3.0 GPA. Provides free internet access to all students. You can house there. The university holds 419 acres of land...
3.) UC Berkeley
Known for being in the top engineering schools. 1232 acre campus. 32% tuition increase for all UCs...
Friday, November 20, 2009
Tips & Hints
Transformations
How I usually get through transformations is to remember what the original graph looks like.
For example,
f(x)=1+2cos(x)
The original equation is f(x)=cos(x).
Since f(x)= cos(x) looks like this:
I can assume that the entire graph moves up by one and the upper loop heightens by one and the lower loop lowers by one. Making the graph look like this:
Trigonometry
Probably the best tip is to just memorize all those flash cards. Yet, looking at the book for some guidance sometimes help. *Note: The book knows all.*
For the csc and sec are easy to remember. Sec sort of looks like cos but reverse so sec=1/cos. Csc is the only 1 that has a 's' in the middle of the 2 letters so I can say that 's' is the same for sin so csc=1/sin. Cot is the only 1 that has a 't' so cot=1/tan. Remembering the graphs are some what harder to remember. Just remember that "csc is next to y and pi and sec is the 1 on them." Since we know the graphs are just a bunch of parabolas, 'csc is next to y and pi' means that the parabola is next to these two attributes and it cannot land on y or pi. So what is between y and pi?Its π/2. Therefore, the vertex of the parabolas will be on every xπ/2 where x represents any number that is not divisible by 2. So π/2, jot down a dot, 3π/2 jot down another dot, -π/2, jot a dot, -π/2 jot another dot. To remember which parabolas goes upward and others go downward, just remember DUD or down up down (this also applies to sec). Meaning that the first parabola from your left will point downward, the 2nd will point upward and the third parabola will point downward. This ONLY applys to when your graph scale goes up to 2π. 'sec is the 1 on them' means that the vertex of the parabolas will touch the y-axis and every whole pi. That '1' represents that every vertex from both csc and sec will be 1 unit away from the x-axis. So at (0,1) jot down a dot, (π,1) jot down another dot and so on.
Probably the most worrisome thing from trigonometry for me would be using the unit circle for solving problems like (4,13) Find sinΘ or Find arctan(1).
How I usually get through transformations is to remember what the original graph looks like.
For example,
f(x)=1+2cos(x)
The original equation is f(x)=cos(x).
Since f(x)= cos(x) looks like this:
I can assume that the entire graph moves up by one and the upper loop heightens by one and the lower loop lowers by one. Making the graph look like this:
Trigonometry
Probably the best tip is to just memorize all those flash cards. Yet, looking at the book for some guidance sometimes help. *Note: The book knows all.*
For the csc and sec are easy to remember. Sec sort of looks like cos but reverse so sec=1/cos. Csc is the only 1 that has a 's' in the middle of the 2 letters so I can say that 's' is the same for sin so csc=1/sin. Cot is the only 1 that has a 't' so cot=1/tan. Remembering the graphs are some what harder to remember. Just remember that "csc is next to y and pi and sec is the 1 on them." Since we know the graphs are just a bunch of parabolas, 'csc is next to y and pi' means that the parabola is next to these two attributes and it cannot land on y or pi. So what is between y and pi?Its π/2. Therefore, the vertex of the parabolas will be on every xπ/2 where x represents any number that is not divisible by 2. So π/2, jot down a dot, 3π/2 jot down another dot, -π/2, jot a dot, -π/2 jot another dot. To remember which parabolas goes upward and others go downward, just remember DUD or down up down (this also applies to sec). Meaning that the first parabola from your left will point downward, the 2nd will point upward and the third parabola will point downward. This ONLY applys to when your graph scale goes up to 2π. 'sec is the 1 on them' means that the vertex of the parabolas will touch the y-axis and every whole pi. That '1' represents that every vertex from both csc and sec will be 1 unit away from the x-axis. So at (0,1) jot down a dot, (π,1) jot down another dot and so on.
Probably the most worrisome thing from trigonometry for me would be using the unit circle for solving problems like (4,13) Find sinΘ or Find arctan(1).
Thursday, November 12, 2009
Inverses and Logs
Inverses
Its really easy to understand this subject.
Since a inverse is the reflection of a function in respect to the line y=x,
f(f-1(x))= x.
Example: If f(x)= x^3 then to find its inverse...
f(f-1(x))= (f-1(x))^3. Since you would replace the input. Instead of x, you placed f-1(x). It would be easier to write a different variable to represent f-1(x) because it takes too long to write down. Lets have ? = f-1(x). Since f(f-1(x)) has to equal x in order to find its inverse, we can replace f(f-1(x)) with x. Now your equation is x = (?)^3.
Solve for ?.
x = (?)^3 Take the cube root of both sides.
3NTHROOT x = ?
Since we replaced f-1(x) with ?, we can replace ? with f-1(x) hence
3NTHROOT x = f-1(x).
Logs
I understand this subject pretty well.
Example
Solve for x.
log232=x.
Step 1: If you remember the story Ms. Hwang illustrated, the log disappears and what ever number that is subscripted in the log in this case 2 is transferred over the equals sign and becomes the base of what ever number on the other side. So log232=x becomes 32=2x.
From this point on there are two ways that I discovered to figure this out.
Way #1: Fast Way
Step 2: You ask yourself the question, "2 to the power of what number will give me 32?"
The answer is 5. 2 to the power of 5 will give you 32. Thus x = 5.
Way #2: Reasonable way
Step 2: You start by changing 32 into the base of 2 so that you have the same base in both sides of the equation. So you get 2^x=2^5.
Step 3: Next you remove the base 2 and make the exponents into their own bases. Since taking the log2 into both sides of the equation will cancel out the bases of 2 leaving the exponents as their own base. Now you get x = 5.
Its really easy to understand this subject.
Since a inverse is the reflection of a function in respect to the line y=x,
f(f-1(x))= x.
Example: If f(x)= x^3 then to find its inverse...
f(f-1(x))= (f-1(x))^3. Since you would replace the input. Instead of x, you placed f-1(x). It would be easier to write a different variable to represent f-1(x) because it takes too long to write down. Lets have ? = f-1(x). Since f(f-1(x)) has to equal x in order to find its inverse, we can replace f(f-1(x)) with x. Now your equation is x = (?)^3.
Solve for ?.
x = (?)^3 Take the cube root of both sides.
3NTHROOT x = ?
Since we replaced f-1(x) with ?, we can replace ? with f-1(x) hence
3NTHROOT x = f-1(x).
Logs
I understand this subject pretty well.
Example
Solve for x.
log232=x.
Step 1: If you remember the story Ms. Hwang illustrated, the log disappears and what ever number that is subscripted in the log in this case 2 is transferred over the equals sign and becomes the base of what ever number on the other side. So log232=x becomes 32=2x.
From this point on there are two ways that I discovered to figure this out.
Way #1: Fast Way
Step 2: You ask yourself the question, "2 to the power of what number will give me 32?"
The answer is 5. 2 to the power of 5 will give you 32. Thus x = 5.
Way #2: Reasonable way
Step 2: You start by changing 32 into the base of 2 so that you have the same base in both sides of the equation. So you get 2^x=2^5.
Step 3: Next you remove the base 2 and make the exponents into their own bases. Since taking the log2 into both sides of the equation will cancel out the bases of 2 leaving the exponents as their own base. Now you get x = 5.
Friday, November 6, 2009
Even and Odd Functions
Even Functions
If f(x)= f(-x) then the function is even.
This is basically saying that every time you input a number for x, that number will change its sign into the opposite. For example, you input a 1 it will change into -1. If you input -2, it will change to 2. Graphing wise, the function is symmetrical in respect of the y-axis. Every input will have a output that is a reflection of the original graph. The input is changed into negative hence it is (-x,y) instead of (x,y)
Look at the equation: f(x)= x^2
When x = 1, then f(1) = 1. Also when placed into f(-x) since f(x) = f(-x), f(-(1)) = 1. The point (-1,1) is the reflection of the point (1,1) from the original equation f(x)=x^2. Therefore the equation f(x)=x^2 is a even function.
Odd Functions
If f(- x)= - f(x) then the function is odd.
This is saying that the original function f(x) equals the negative function of itself. For example, if f(x) = x+3 then - f(x) = - (x+3). Graphing wise, the function is symmetrical according to the origin. Not only the output is changed to negative but the input as well. Instead of (x,y) it becomes (-x,-y)
Look at the equation: f(x) = x^3
When x = 2, then f(- 2) = - 8 and - f(2) = - 8. The point (-2,-8) is the reflection of (-2,8) in respect of the x-axis. Look back at f(x) = x^2. Its negative domain (x) is the reflection of the positive domain. In f(x) = x^3, its negative domain is the reflection of the negative domain of f(x) = x^2. The negative x portion of the graph is the reflection of the y-axis as well as the x-axis but the graph involves the point (o,o), the origin hence the function is odd.
If f(x)= f(-x) then the function is even.
This is basically saying that every time you input a number for x, that number will change its sign into the opposite. For example, you input a 1 it will change into -1. If you input -2, it will change to 2. Graphing wise, the function is symmetrical in respect of the y-axis. Every input will have a output that is a reflection of the original graph. The input is changed into negative hence it is (-x,y) instead of (x,y)
Look at the equation: f(x)= x^2
When x = 1, then f(1) = 1. Also when placed into f(-x) since f(x) = f(-x), f(-(1)) = 1. The point (-1,1) is the reflection of the point (1,1) from the original equation f(x)=x^2. Therefore the equation f(x)=x^2 is a even function.Odd Functions
If f(- x)= - f(x) then the function is odd.
This is saying that the original function f(x) equals the negative function of itself. For example, if f(x) = x+3 then - f(x) = - (x+3). Graphing wise, the function is symmetrical according to the origin. Not only the output is changed to negative but the input as well. Instead of (x,y) it becomes (-x,-y)
Look at the equation: f(x) = x^3
When x = 2, then f(- 2) = - 8 and - f(2) = - 8. The point (-2,-8) is the reflection of (-2,8) in respect of the x-axis. Look back at f(x) = x^2. Its negative domain (x) is the reflection of the positive domain. In f(x) = x^3, its negative domain is the reflection of the negative domain of f(x) = x^2. The negative x portion of the graph is the reflection of the y-axis as well as the x-axis but the graph involves the point (o,o), the origin hence the function is odd.
Monday, October 26, 2009
About me.
Hi my name is Eric. I really enjoy solving problems that are particularly associated with mathematical content (except word problems...). I spend most of my time in front of the computer playing computer games based on shooting or some mmorpgs here and there. I'm incredibly bored though; can't find any good mmos to play so I just watch some anime with my free time. I'm rarely at home during the weekends since I have places to go and people to see. I play the piano when there is a song I desperately want to play or just play for the heck of it. (got nothing else to do so why not). I'm also composing some songs or at least I think I am. Not quite sure if the songs I'm making are from another song. I'm able to play 21 guns by Green Day, most of Kiss the Rain by Yiruma, Ballade pour Adeline by Richard Clayderman, One by Big Bang, Corazon de Nino by Raul di Blasto and Open Arms by Journey through memorization. (There are probably more but these are the ones that I can recall) There are also some songs that I can play also by memory but only for like 30 seconds... Heaven by D.J Sammy, He's a Pirate from Pirates of the Carribean, Springtime by Yiruma, True Light from D.N Angel, New Soul by Yael Naim, Fall for You by Second Serenade, All my Life by K-Ci and Jojo, Viva la Vida and Clocks by Cold Play. (There are probably more. Gave some thought thinking of these songs.) There is one particular song I heard when I was playing Counter-Strike (1st person shooter game). Unfortunately, I couldn't figure out what song it was or who made it... It had such a soothing melody that I want to see myself playing in the future.
So like I said before, I spend most of my time sitting in front of the computer. I used to be a hardcore gamer in middle school. (spent like 12 hrs straight playing a game) Now I just don't care much about playing mmorpgs. I just watch some anime that seem really good. If there were any anime I would recommend watching would be... GTO (really funny but perverted) , Bleach (fictional action series) , Trinity Blood (those who seem to adore vampires), Onegai Teacher and Onegai Twins (really funny series on school life), Love Hina (somewhat like the Onegai series) , Tenjou Tenge (those who like fist fighting), Dragon Drive (virtual reality action based series), Seto no Hanayome (LOL, you will laugh your *** off watching this series) and D.N. Angel (really nice action and romance series) There are more series but these are the ones that seem to come to mind.
Anyways, I think this is it on what I wanted to say so... if you have any songs that I can learn, anime I can watch, or some game you want my to try just post.
So like I said before, I spend most of my time sitting in front of the computer. I used to be a hardcore gamer in middle school. (spent like 12 hrs straight playing a game) Now I just don't care much about playing mmorpgs. I just watch some anime that seem really good. If there were any anime I would recommend watching would be... GTO (really funny but perverted) , Bleach (fictional action series) , Trinity Blood (those who seem to adore vampires), Onegai Teacher and Onegai Twins (really funny series on school life), Love Hina (somewhat like the Onegai series) , Tenjou Tenge (those who like fist fighting), Dragon Drive (virtual reality action based series), Seto no Hanayome (LOL, you will laugh your *** off watching this series) and D.N. Angel (really nice action and romance series) There are more series but these are the ones that seem to come to mind.
Anyways, I think this is it on what I wanted to say so... if you have any songs that I can learn, anime I can watch, or some game you want my to try just post.
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