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?
Difference- when you are finding the limit you are finding the y value that wouldbe at x=c, if there was no discontinuity in the function. But, when you actually plug in the number x=c, you will get the output.
For example if there is a removable discontinuity at x=2 in a function, the limit might turn out to be 1, while if you plug in 2 into the function you might get an output of 3.
Same- the two are the same when there is no discontinuity at x=c and the limit is the same as the output of the function.
2. What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?
Similarities- the derivative of a line and the slope of the line are basically the same thing: the change in y over the change in x.
Differences- I'm fairly sure that the differences in the two are in where the derivative and the slope are used.
For example, finding the slope of a parabola, would give you the value of the slope of the parabola's secant line, while finding ther derivative of that same parabola would give you the slope of the parabola's tangent line.
Saturday, December 19, 2009
Wednesday, December 9, 2009
Limits...
Okay well... limits...
Let's see... I think I understood most of the concept of limits. After going through the homework assignments of the past two weeks, I couldn't find any problems that I had left blank... Well, there was one, but after looking at it, I realized I could've just plugged in the number to find the limit. (It was #8 on pg 66, I think the large exponent threw me off haha)
I don't think there was much else...
Wait, one thing: Is there a difference between a limit being undefined and a limit not existing? I thought you could use the terms interchangabley but now I'm not so sure... Anyways, if there is a difference, what is it?
On a seperate note: How can you tell what a graph looks when the equation is in the form of a polynomial divided by a polynomial?
Ex: How can I know what
4x³-2x²+5
3x-4
looks like without having to use a calculator?
Let's see... I think I understood most of the concept of limits. After going through the homework assignments of the past two weeks, I couldn't find any problems that I had left blank... Well, there was one, but after looking at it, I realized I could've just plugged in the number to find the limit. (It was #8 on pg 66, I think the large exponent threw me off haha)
I don't think there was much else...
Wait, one thing: Is there a difference between a limit being undefined and a limit not existing? I thought you could use the terms interchangabley but now I'm not so sure... Anyways, if there is a difference, what is it?
On a seperate note: How can you tell what a graph looks when the equation is in the form of a polynomial divided by a polynomial?
Ex: How can I know what
4x³-2x²+5
3x-4
looks like without having to use a calculator?
Tuesday, November 24, 2009
Colleges
Majors
Preveterinary studies- not offered in many schools so you usually have to take other classes like biology and whatnot in order to be accepted into a veterinary school. Obviously, help prepare for the path to becoming a veterinarian. Need people skills...
Zoology- study animals. All kinds, not just vertebrates so probably not a good choice for those that a bit squeamish around bugs and worms and such. Study the animal and its anatomy and physiology like the cells and the organs. (Good references I bet...) A bit like ecology in that you study the animals' environment and how they adapt to it.
Wildlife biology- Study wildlife. The animals, the plants, the environment and how they all interact. Study the ecosystems from urban, suburban, rural areas and in the wild. Focus on vertebrates and their environments.
Colleges
UC Davis- In northern California, 15 miles from Sacramento. It has an equestrian and raptor center (raptor as in birds of prey, not the dinosaurs)
American University- Private school in Washington D.C. 80% of the students admitted are from out of state and offers AH DAMN IT! It's religiously affiliated....
Kalamazoo College- In Michigan. Only 32% of students from out of state were admitted and all first year students live in on-campus housing. But don't you just love the name. Kalamazoo. Haha...
Northern Arizona University- rural setting, 74% of applicants are admitted, 30% are from out of state. First-time first-year students are allowed to have a car. Located 140 miles from Phoenix, Arizona and there's a 400 acre forest close by as well as an observatory.
Stupid Amercan University... it was doing so well too... Oh well. The others are just as well.
Preveterinary studies- not offered in many schools so you usually have to take other classes like biology and whatnot in order to be accepted into a veterinary school. Obviously, help prepare for the path to becoming a veterinarian. Need people skills...
Zoology- study animals. All kinds, not just vertebrates so probably not a good choice for those that a bit squeamish around bugs and worms and such. Study the animal and its anatomy and physiology like the cells and the organs. (Good references I bet...) A bit like ecology in that you study the animals' environment and how they adapt to it.
Wildlife biology- Study wildlife. The animals, the plants, the environment and how they all interact. Study the ecosystems from urban, suburban, rural areas and in the wild. Focus on vertebrates and their environments.
Colleges
UC Davis- In northern California, 15 miles from Sacramento. It has an equestrian and raptor center (raptor as in birds of prey, not the dinosaurs)
American University- Private school in Washington D.C. 80% of the students admitted are from out of state and offers AH DAMN IT! It's religiously affiliated....
Kalamazoo College- In Michigan. Only 32% of students from out of state were admitted and all first year students live in on-campus housing. But don't you just love the name. Kalamazoo. Haha...
Northern Arizona University- rural setting, 74% of applicants are admitted, 30% are from out of state. First-time first-year students are allowed to have a car. Located 140 miles from Phoenix, Arizona and there's a 400 acre forest close by as well as an observatory.
Stupid Amercan University... it was doing so well too... Oh well. The others are just as well.
Saturday, November 21, 2009
Tips and Hints
I'm reallly starting to get tired of this...
but ANYWAYS...
1. How do I remember transformations? How DO I remember transformations? Well in order to understand transformations you have to know what the parent functions look like. If you don't your pretty much screwed. I mean, you can always plug in points but that takes a while and is WAY too much work. Now I'm a lazy person. I don't like doing work. I look for shortcuts.
For the sake of simplicity I'm gonna use the parent function
f(x)=sin x
(We all remember what that looks like right, because I'm not using pictures)
I see it this way: It all depends on where the number or change is
In f(x)= sin x + 1 the change is outside of the x. This means that the graph moves up or down, in this case it moves up.
In f(x)=sin (x + 1) the change is linked to the x. This means that the graph moves left or right. Here, the graph moves to the left, because when the change is linked to the x (by parentheses) it becomes like Opposite World. You go the other way. If the number is positive, you shift to the left. If it is negative, you shift to the right.
In f(x)=2sin x the change is... for lack of a better explanation... before the x. Therefore the graph's height increases. Here, the graph's highest point will now be 2 instead of 1.
In f(x)=sin 2x the change is directly in the x. This means that the graph either lengthens or shortens. In this case it shortens (once again, we're in Opposite World; if it's a fraction it gets longer, if it's a whole number it gets shorter)
In f(x)=sin -x the change is the sign of the x. The graph then flips vertically. So with this equation, the point (π/2, 1) would now be (π/2, -1) and so on.
There's more I think but I forget what they are at the moment...
2. Trigonometry... Trigonometry is triangles right? For this I can't really give advice. I don't use tricks, I just memorized the equations and rules... most of them anyway.
Besides, the equations are kind of like the tricks in themselves. Like the ones to find the legs of a 30-60-90 triangle.[If you don't remember: hyp is the hypotenuse, SL is the short leg and LL is the long leg]
It was hyp= 2SL. So the hypotenuse is double the size of the short leg of the triangle and LL=√3SL so the long leg is equal to the short leg times the square root of 3.
And then there's that SOH CAH TOA thing. I believe most of us are taught this when we're first introduced to the idea of finding sine, cosine and tangent of a triangle.
There's other stuff I'm sure but I don't remember it all so (and this is one of the only times I'll ever say this) go back to the notes from math analysis. Study your damn head off if you have to. Or hope other people have hints for this stuff (I know I will... Hell I'll do better than hope I'll PRAY that others have hints for remembering this stuff)
3. Still confused about... well I don't remember much from trig so I can't really be confused. Guess I'll have to relearn it all... Not looking forward to that. Then again, it'll be review so it should be easier the second time around, right? Other than that, no worries and no confusion.
but ANYWAYS...
1. How do I remember transformations? How DO I remember transformations? Well in order to understand transformations you have to know what the parent functions look like. If you don't your pretty much screwed. I mean, you can always plug in points but that takes a while and is WAY too much work. Now I'm a lazy person. I don't like doing work. I look for shortcuts.
For the sake of simplicity I'm gonna use the parent function
f(x)=sin x
(We all remember what that looks like right, because I'm not using pictures)
I see it this way: It all depends on where the number or change is
In f(x)= sin x + 1 the change is outside of the x. This means that the graph moves up or down, in this case it moves up.
In f(x)=sin (x + 1) the change is linked to the x. This means that the graph moves left or right. Here, the graph moves to the left, because when the change is linked to the x (by parentheses) it becomes like Opposite World. You go the other way. If the number is positive, you shift to the left. If it is negative, you shift to the right.
In f(x)=2sin x the change is... for lack of a better explanation... before the x. Therefore the graph's height increases. Here, the graph's highest point will now be 2 instead of 1.
In f(x)=sin 2x the change is directly in the x. This means that the graph either lengthens or shortens. In this case it shortens (once again, we're in Opposite World; if it's a fraction it gets longer, if it's a whole number it gets shorter)
In f(x)=sin -x the change is the sign of the x. The graph then flips vertically. So with this equation, the point (π/2, 1) would now be (π/2, -1) and so on.
There's more I think but I forget what they are at the moment...
2. Trigonometry... Trigonometry is triangles right? For this I can't really give advice. I don't use tricks, I just memorized the equations and rules... most of them anyway.
Besides, the equations are kind of like the tricks in themselves. Like the ones to find the legs of a 30-60-90 triangle.[If you don't remember: hyp is the hypotenuse, SL is the short leg and LL is the long leg]
It was hyp= 2SL. So the hypotenuse is double the size of the short leg of the triangle and LL=√3SL so the long leg is equal to the short leg times the square root of 3.
And then there's that SOH CAH TOA thing. I believe most of us are taught this when we're first introduced to the idea of finding sine, cosine and tangent of a triangle.
There's other stuff I'm sure but I don't remember it all so (and this is one of the only times I'll ever say this) go back to the notes from math analysis. Study your damn head off if you have to. Or hope other people have hints for this stuff (I know I will... Hell I'll do better than hope I'll PRAY that others have hints for remembering this stuff)
3. Still confused about... well I don't remember much from trig so I can't really be confused. Guess I'll have to relearn it all... Not looking forward to that. Then again, it'll be review so it should be easier the second time around, right? Other than that, no worries and no confusion.
Saturday, November 14, 2009
Logarithms and Inverses
1.Recap:
- Inverse functions- basically looks like the original function after it has been flipped over and rotated 90 degrees
- Logs- have to do with finding exponents. I'd explain but it seems many of my fellow classmates have already done so and quite well if I'm not mistaken.
- Natural logs- same as logs but uses "e" instead of actual numbers (even though e stands for a number...)
- Trigonometric functions- I actually remember this, ha! Sine and cosine are wavy graphs, secant and cosecant are parabolas and have asymptotes, tangent and cotangent graphs look like the graph of x³ and -x³ and have asymptotes. Let's see... oh and those trigonometric identities like secθ=1/cosθ, cscθ=1/sinθ, cotθ=1/tanθ, and those... others (forgot what they were called) the equations, cos²θ+sin²θ=1 and them others... (but that's a little too much innit? I only remember because we used them so much in math analysis with Gapac. On an unrelated note... I can also recite pi to the 17th place... Don't blame me though... I was bored in those few minutes before the bell rang and I noticed that banner/poster thing in her room so I decided Why not memorize pi?! Ha... how fun am I you guys, really? x]
2. What I did not understand completely:
Think I understood most things pretty well except for one or two things... I have to go find them so... found them!
Okay... first thing: finding the inverse equation of f(x). Like in the homework C2 #43, 44 and 45. I got it when Hwang explained it in class but then I tried to do these and... nothing. Completely clueless...
And homework C3 #60. The hell does that mean!? Stupid question annoyed me because it was so confusing...
Yeah I think that's it so... Help? Please?
3. Answered questions: seems most things have already been explained and rather well I might add.
Saturday, November 7, 2009
Even and Odd Functions
I almost forgot about this... Oh well... better late than never...
Ok so, from what I understand from the formulas:
f(x) -- This is what we start with.
-f(x) -- This is what we have to find.
f(-x) -- And this is what we have to get.
The graph/function is even if: f(x) = f(-x)
The graph/function is odd if: -f(x) = f(-x)
Note (for my examples)
In an even function points are across from the y-axis, in quadrants I & II
In and odd function points are diagonal from the origin, in qudrants I & III
Examples are needed maybe? Well then... for the sake of simplicity I'll use
1. f(x) = x
2. f(x) = x²
For the first equation (after some very simple calculations...):
f(x) = x
-f(x) = -x
f(-x) = -x
The latter two equation are the same causing -f(x) = f(-x)
This means that the graph is odd.
For the second equation (after equally simple calculations...):
f(x) = x²
-f(x) = x²
f(-x) = -x²
In this case, the first two equations are the same causing f(x) = f(-x)
This means that the graph is even.
Pictures too? Well then... the two graphs:
2. f(x) = x²
EVEN FUNCTION
Ok so, from what I understand from the formulas:
f(x) -- This is what we start with.
-f(x) -- This is what we have to find.
f(-x) -- And this is what we have to get.
The graph/function is even if: f(x) = f(-x)
The graph/function is odd if: -f(x) = f(-x)
Note (for my examples)
In an even function points are across from the y-axis, in quadrants I & II
In and odd function points are diagonal from the origin, in qudrants I & III
Examples are needed maybe? Well then... for the sake of simplicity I'll use
1. f(x) = x
2. f(x) = x²
For the first equation (after some very simple calculations...):
f(x) = x
-f(x) = -x
f(-x) = -x
The latter two equation are the same causing -f(x) = f(-x)
This means that the graph is odd.
For the second equation (after equally simple calculations...):
f(x) = x²
-f(x) = x²
f(-x) = -x²
In this case, the first two equations are the same causing f(x) = f(-x)
This means that the graph is even.
Pictures too? Well then... the two graphs:
1. f(x) = x
ODD FUNCTIONHere you can see that the points have been
rotated 180 degrees from the origin
so that they lie in quadrants I & III
2. f(x) = x²EVEN FUNCTION
In this graph the points are
directly across the y-axis
as if quadrant II is a mirror image of 
quadrant I
quadrant ISaturday, October 31, 2009
About Me
Well, one thing you should know about me is that I don't really like to talk about myself much...
Other than that.... hmm... well what should I say? What should you know? I suppose it would be a good idea to tell you that I am a procrastinator which is why I'm writing this on Saturday morning when I could have written it sometime before. In fact, the only reason I'm writing this so early in the morning is because I'm pretty sure I will forget to do this at all if not now. Oh yeah one final important thing, since I get this question almost every single freaking time I get a new class: I'm quiet because I usually don't have much to say. Why go around spouting useless nonsense if it's not going to help anyone? I won't waste anyone's time with stupid comments (unless the person's my friend of course, I love wasting their time!) Of course, there's always a time and a place for that. Don't speak unless you are spoken too is usually my rule of thumb and ow my cat's sitting on my leg, he's heavy...
So now on to the random facts about myself.
As said, I have a kitty, he's white and his name's Snoopy
Two dogs, Blackie and Osa, three guesses what color their fur is x] (osa means "bear" in Spanish for those that don't know)
Three birds with no names so I'll name them right now, let's see... One will be Bluey, the other will be Colorful and the last will be Rain Cloud (he's grey)
I'm VERY CREATIVE when it comes to names and titles as you can see...
Random phrases will remind me of songs (ex. if someone says "before I forget" in a sentence, I get reminded of the song by Slipknot)
I'm a bit of an Assassin's Creed fanatic. Can't wait for ACII now that they finally picked a release date: Nov. 17th. It's almost here YES!
And for those of you that actually did guess on my dog's fur color here's the answers:
Blackie and Osa
Oh one last thing! I have been to THE TOP OF THE WORLD!!! Don't believe
me? Here's some pictures:
If you want to know anything else feel free to ask.
Other than that.... hmm... well what should I say? What should you know? I suppose it would be a good idea to tell you that I am a procrastinator which is why I'm writing this on Saturday morning when I could have written it sometime before. In fact, the only reason I'm writing this so early in the morning is because I'm pretty sure I will forget to do this at all if not now. Oh yeah one final important thing, since I get this question almost every single freaking time I get a new class: I'm quiet because I usually don't have much to say. Why go around spouting useless nonsense if it's not going to help anyone? I won't waste anyone's time with stupid comments (unless the person's my friend of course, I love wasting their time!) Of course, there's always a time and a place for that. Don't speak unless you are spoken too is usually my rule of thumb and ow my cat's sitting on my leg, he's heavy...
So now on to the random facts about myself.
As said, I have a kitty, he's white and his name's Snoopy
Two dogs, Blackie and Osa, three guesses what color their fur is x] (osa means "bear" in Spanish for those that don't know)
Three birds with no names so I'll name them right now, let's see... One will be Bluey, the other will be Colorful and the last will be Rain Cloud (he's grey)
I'm VERY CREATIVE when it comes to names and titles as you can see...
Random phrases will remind me of songs (ex. if someone says "before I forget" in a sentence, I get reminded of the song by Slipknot)
I'm a bit of an Assassin's Creed fanatic. Can't wait for ACII now that they finally picked a release date: Nov. 17th. It's almost here YES!
And for those of you that actually did guess on my dog's fur color here's the answers:
Blackie and Osa
Oh one last thing! I have been to THE TOP OF THE WORLD!!! Don't believe
me? Here's some pictures:
If you want to know anything else feel free to ask.
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