1. When you're finding the limit of a function at x=c, you're finding the value that f(x) approaches as x approaches some constant. When plugging in x=c into f(x), you're finding what the actual value of f(c) is, even if f(x) doesn't necessarily approach f(c). The two cases are the same when f(x) is continuous at x=c, as given by the definition of continuity at a point lim f(x) x->c=f(c).
2. The similarities between finding the derivative and the slope of a line is that you essentially use the same formula (m=change in y/change in x), the only slight difference being when finding the derivative, you bring the two points infinitely closer to each other so that they become the same point. Mathematically, for the derivative, you use the formula lim (change in y/change in x) h->0, h being the distance between the two points. The difference is that while they are both the slope of a line, one is the slope of a specific line while the other is the slope of any line. When finding the derivative, you are finding the slope of a line, but you are finding the slope of a tangent line to a point on a curve. When saying you are finding the slope of a line, it generally refers to any ordinary line. In other words, when saying you're finding the slope of a line, it does not tell us any of the line's special properties, if it has any.
Friday, December 18, 2009
Wednesday, December 9, 2009
Limits
Eh, the last post was better. Anyway, for the most part, I understand limits very well. There's just one or two things that elude me:
1. The first would be the limit of the basic trig functions (sin x, cos x, tan x, csc x, sec x, cot x) as x-> positive or negative infinity. The problem is that unlike most functions, who reach a specific number or positive or negative infinity, the trig functions don't. They simply alternate between [-1,1] (sin x and cos x), (negative infinity,-1] and [1, infinity), (csc x and sec x), or (negative infinity, infinity) (tan x and cot x). Does that mean that the limit simply does not exist?
2. #13 and #14 on page 92 also confuse me. I know what they're asking and how to generally find what they're asking for, but when I try to find it algebraically, I get something that can't be simplified. Because they're absolute value problems, you can figure out the slope from the graph, but how would you find it algebraically?They are:
Find the slope of the curve at the indicated point:
13.f(x)=absolute value (x) at: a)x=2 b)x=-3
14.f(x)=absolute value (x-2) at x=1
Hm....yea, that's about it, I don't have a third thing: that's it. If anyone has answers to these questions, I would greatly appreciate it.
1. The first would be the limit of the basic trig functions (sin x, cos x, tan x, csc x, sec x, cot x) as x-> positive or negative infinity. The problem is that unlike most functions, who reach a specific number or positive or negative infinity, the trig functions don't. They simply alternate between [-1,1] (sin x and cos x), (negative infinity,-1] and [1, infinity), (csc x and sec x), or (negative infinity, infinity) (tan x and cot x). Does that mean that the limit simply does not exist?
2. #13 and #14 on page 92 also confuse me. I know what they're asking and how to generally find what they're asking for, but when I try to find it algebraically, I get something that can't be simplified. Because they're absolute value problems, you can figure out the slope from the graph, but how would you find it algebraically?They are:
Find the slope of the curve at the indicated point:
13.f(x)=absolute value (x) at: a)x=2 b)x=-3
14.f(x)=absolute value (x-2) at x=1
Hm....yea, that's about it, I don't have a third thing: that's it. If anyone has answers to these questions, I would greatly appreciate it.
Monday, November 23, 2009
Colleges
Wow, this wasn't as annoying as I thought it would be...
1.For majors, I actually found like 10 of them that I liked (all science-related). These are just some of them.
Astrophysics: This major basically deals with how stars came to be, what they're made of, etc. (I don't now why that sounds interesting, it just does). It also requires you take many math classes (stars and math, Yes!).
Biophysics: Biophysics is putting concepts from physics into biology. You would study things like energy flow within living organisms, the muscles within living organisms, the chemistry of the body, etc. This combines 2 subjects I'm interested in: biology and physics. Again, what more could one ask for?
Toxicology: Toxicology is the study of toxins (poisons). It deals with knowing all sorts of poisons, from inorganic hazardous products to bacteria harmful to humans and the environment. I would probably also throw forensic toxicology into this, as that deals with poisons dealt with in crimes and what not, but still involves poison, which is good enough for me.
Colleges: This was a bit more challenging than the majors but I picked these three based on how many majors they offered of those I liked or was interested in.
1.UCLA: This had about half of the ones I liked, the most out of any college. However, I know getting into this school isn't exactly the simplest thing to do, which is the only thing I'm not so fond of: the competition.
2.UC Berkeley: This school didn't have as much of the majors (only like 2, I think), but still, it had some. I haven't heard too much about this school (or wasn't paying attention when people were talking about it), so this one I need to investigate more.
3.UC San Diego: Again, this one only had like 2 of the major I was interested in. This I didn't even know existed, so this one I need to investigate this the most out of all of them .
Note: Yes, I know I haven't exactly investigated them, but now I have clue, unlike before. Once I thouroughly (spelling?) investigate them all, I'll edit this post and share what I found.
1.For majors, I actually found like 10 of them that I liked (all science-related). These are just some of them.
Astrophysics: This major basically deals with how stars came to be, what they're made of, etc. (I don't now why that sounds interesting, it just does). It also requires you take many math classes (stars and math, Yes!).
Biophysics: Biophysics is putting concepts from physics into biology. You would study things like energy flow within living organisms, the muscles within living organisms, the chemistry of the body, etc. This combines 2 subjects I'm interested in: biology and physics. Again, what more could one ask for?
Toxicology: Toxicology is the study of toxins (poisons). It deals with knowing all sorts of poisons, from inorganic hazardous products to bacteria harmful to humans and the environment. I would probably also throw forensic toxicology into this, as that deals with poisons dealt with in crimes and what not, but still involves poison, which is good enough for me.
Colleges: This was a bit more challenging than the majors but I picked these three based on how many majors they offered of those I liked or was interested in.
1.UCLA: This had about half of the ones I liked, the most out of any college. However, I know getting into this school isn't exactly the simplest thing to do, which is the only thing I'm not so fond of: the competition.
2.UC Berkeley: This school didn't have as much of the majors (only like 2, I think), but still, it had some. I haven't heard too much about this school (or wasn't paying attention when people were talking about it), so this one I need to investigate more.
3.UC San Diego: Again, this one only had like 2 of the major I was interested in. This I didn't even know existed, so this one I need to investigate this the most out of all of them .
Note: Yes, I know I haven't exactly investigated them, but now I have clue, unlike before. Once I thouroughly (spelling?) investigate them all, I'll edit this post and share what I found.
Thursday, November 19, 2009
Tips and Hints (my version anyway...)
Eh, I wasn't so fond of last week's post, and this one isn't all that much better...Enough of my useless ranting.
1.How do I remember transformations? First I start with the input (anything regarding the x). If the x is multiplied by something, I take the advice of a murderer who once said "Whatever your mind tells you to do, I implore you to do the opposite" (yes, bizare to take the advice of a murderer, but it works in this case). For example, when I see something like f(2x), i normally want to stretch the graph, but I do the opposite and compress ("shrink") it. If I see it has something added to it, like f(x+2), i want to shift the graph 2 units to the positive side (to the right), but instead I shift towards the negative side (to the left). In the case of subtraction, vice versa. If I see the output is multiplied by something, like 2 sin x, I just stretch it vertically according to the number (if the number is less than 1, compress it according to the number). When adding or subtracting to the output, add means up, subtraction means down. In short, for me, the trick is when changing the input, do the opposite of what you initially think and when changing the output, listen to your initial thoughts.
2.Trigonometry is simpler. When asked to find anything about the unit circle (coordinates, sin or cos of an angle, etc.), I just visualize how the triangle looks like in the unit circle and determine the needed information from that: the short side is 1/2 and long side is root of 3/2 for angle multiples of 30 degrees and all sides are root of 2/2 for multiples of 45 degrees (no, I don't actually have all the values memorized and can recall them in a second, I need the triangle for that). For graphs... I know how the curves for sin and cos look. All I need to know is where the hit the y-axis: if it's the origin, it's sin; if it's (0,1) it's cos. At that point, I just continue the curve. Tan, cot, csc, and sec , I don't have a trick, I just know those for some reason. Inverse graphs....those I actually need to memorize better before I get a trick to them (video games or memorizing the flash cards? damn procrastination...)
3. What worries me? Inverse graphs. I haven't memorized them and I need to know them (and the domain and range, now that I think about it...). Oh joy, more memorizing...
Note: Yes, it seems like they actually aren't tricks and I just memorized everything without tricks, but they are "tricks" to me.
1.How do I remember transformations? First I start with the input (anything regarding the x). If the x is multiplied by something, I take the advice of a murderer who once said "Whatever your mind tells you to do, I implore you to do the opposite" (yes, bizare to take the advice of a murderer, but it works in this case). For example, when I see something like f(2x), i normally want to stretch the graph, but I do the opposite and compress ("shrink") it. If I see it has something added to it, like f(x+2), i want to shift the graph 2 units to the positive side (to the right), but instead I shift towards the negative side (to the left). In the case of subtraction, vice versa. If I see the output is multiplied by something, like 2 sin x, I just stretch it vertically according to the number (if the number is less than 1, compress it according to the number). When adding or subtracting to the output, add means up, subtraction means down. In short, for me, the trick is when changing the input, do the opposite of what you initially think and when changing the output, listen to your initial thoughts.
2.Trigonometry is simpler. When asked to find anything about the unit circle (coordinates, sin or cos of an angle, etc.), I just visualize how the triangle looks like in the unit circle and determine the needed information from that: the short side is 1/2 and long side is root of 3/2 for angle multiples of 30 degrees and all sides are root of 2/2 for multiples of 45 degrees (no, I don't actually have all the values memorized and can recall them in a second, I need the triangle for that). For graphs... I know how the curves for sin and cos look. All I need to know is where the hit the y-axis: if it's the origin, it's sin; if it's (0,1) it's cos. At that point, I just continue the curve. Tan, cot, csc, and sec , I don't have a trick, I just know those for some reason. Inverse graphs....those I actually need to memorize better before I get a trick to them (video games or memorizing the flash cards? damn procrastination...)
3. What worries me? Inverse graphs. I haven't memorized them and I need to know them (and the domain and range, now that I think about it...). Oh joy, more memorizing...
Note: Yes, it seems like they actually aren't tricks and I just memorized everything without tricks, but they are "tricks" to me.
Saturday, November 14, 2009
How to graph Logarithmic Functions
Ok, while reading through some of my classmates' blogs, I realized that some don't know how to graph log functions. So here is my guide on how to do so.
Graphing average Logarithmic Functions (this means a log function without transformations): First, know how the general graph of an exponential function looks like, as it will come into play later. Ok, first you will need 2 or 3 points (this is for general logarithmic functions without transformations). One of these points is going to be (1,0) because anything taken to the power of 0 is 1 (if you don't believe this is a point, test it yourself). The second point is going to be (x,1), where is x is the base (the base taken to the first power is itself). The third point is (x,2), where x in this case is the base taken to the second power (if you have any doubts on these points, test them yourself). These 2 or three points should tell you what one part of the graph is going to look like. Now for the part of the graph involving fractions. If you know what the general exponential function looks like and know that logarithmic functions are simply exponential functions reflected across the line of y=x, then you might be able to figure out what the part of the graph looks like where x is a fraction. If you can't do so, as x is a fraction and the fraction gets smaller, the graph will get closer and closer to the y-axis (line x=0), but never actually touch it (if you can't visualize this, test fractions into logarithmic functions and you will see very soon what I mean). In the end, it should look like its exponential counterpart reflected across the line y=x (if you need more points than those suggested above, plot as many points you need until you yourself can see what the graph looks like).
Graphing the Natural Log: If you are graphing the natural log, f(x)=ln x, don't panic. It is not as different from graphing a logarithmic function because it is one as well. All you need to know is that the base is e which approximates to 2.71828282.... or something along those lines (please check the actual value yourself just to be sure). (1,0) is still a point, (e,1) becomes one point, and (e^2,2) becomes the next point. Then, create the general graph by connecting the dots with a curve (for the part of the graph where x is a fraction, see my above explanation). For the actual values of e and e^2, please use a calculator and approximate on the graph there positions. Again, nothing too hard.
Logarithmic Functions with Transformations: I have no trick to these. The only advice I can offer is to take the transformations slowly and one step at a time.
This is pretty much how it's done. If you have your own special method for graphing these, by all means do so, this is for those having difficulties. I hope this helps all of you who are struggling with graphing.
Graphing average Logarithmic Functions (this means a log function without transformations): First, know how the general graph of an exponential function looks like, as it will come into play later. Ok, first you will need 2 or 3 points (this is for general logarithmic functions without transformations). One of these points is going to be (1,0) because anything taken to the power of 0 is 1 (if you don't believe this is a point, test it yourself). The second point is going to be (x,1), where is x is the base (the base taken to the first power is itself). The third point is (x,2), where x in this case is the base taken to the second power (if you have any doubts on these points, test them yourself). These 2 or three points should tell you what one part of the graph is going to look like. Now for the part of the graph involving fractions. If you know what the general exponential function looks like and know that logarithmic functions are simply exponential functions reflected across the line of y=x, then you might be able to figure out what the part of the graph looks like where x is a fraction. If you can't do so, as x is a fraction and the fraction gets smaller, the graph will get closer and closer to the y-axis (line x=0), but never actually touch it (if you can't visualize this, test fractions into logarithmic functions and you will see very soon what I mean). In the end, it should look like its exponential counterpart reflected across the line y=x (if you need more points than those suggested above, plot as many points you need until you yourself can see what the graph looks like).
Graphing the Natural Log: If you are graphing the natural log, f(x)=ln x, don't panic. It is not as different from graphing a logarithmic function because it is one as well. All you need to know is that the base is e which approximates to 2.71828282.... or something along those lines (please check the actual value yourself just to be sure). (1,0) is still a point, (e,1) becomes one point, and (e^2,2) becomes the next point. Then, create the general graph by connecting the dots with a curve (for the part of the graph where x is a fraction, see my above explanation). For the actual values of e and e^2, please use a calculator and approximate on the graph there positions. Again, nothing too hard.
Logarithmic Functions with Transformations: I have no trick to these. The only advice I can offer is to take the transformations slowly and one step at a time.
This is pretty much how it's done. If you have your own special method for graphing these, by all means do so, this is for those having difficulties. I hope this helps all of you who are struggling with graphing.
Thursday, November 12, 2009
Logarithms and Inverses
What we understand??? Man, I really hate questions like this...Anyway, on to the topic at hand.
What do I understand about logarithms? Well, lets see. The way I view logarithms is this: in exponential form, the base from the logarithm stays the same. The other two numbers just switch places. That is to say, the number to the left of the equals sign in logarithmic form goes to the right in exponential form and the number all by itself (to the right of the equals sign) in logarithmic form becomes the exponent in exponential form. Essentially, all they do is switch places (it's a little mind trick I developed to remind myself how to do those problems, so if you can't understand it, that's understandable). What else?...Oh, right. In problems involving logarithms, to "undo" (cancel out, get rid of) a log sign, you make everything a power of the logarithmic base. If you have a base that needs to be gotten rid of, you simply make everything as part of a logarithm with a base the same as the exponential base (that gets a little confusing, I don't really now how to phrase that correctly to make more sense). In logarithms involving e, the same applies, only you use e as the base or ln, the natural logarithm (whichever the occasion calls for). Writing this and solving problems involving them made me see problems with exponents and logarithms aren't always as easy as they appear (like the problem in one of the homeworks e^x + e^-x=3, which I found out after some time is just treated like a quadratic equation in the end).
On to inverses (joy...). I understand that to find inverse equations, you just switch the x and y and solve for the new y. I also understand that the graphs of an equation and its inverse are symmetrical about the line y=x, so you can somewhat already know and graph the inverse without ever actually having seen it or graphed it (which is good for lazy people like me: less work!).
There is one thing I don't exactly understand. With some inverse problems, when the original function f(x) is given, they give you some sort of condition (like x>0 or something like that). Now, I understand that the condition is necessary otherwise the inverse won't be one-to-one. However, when I do those problems, I either get something different from the book or something that is in no way the inverse (when I plug it back into the original function, I don't get x as the output). I guess what I'm really trying to as is this: when given a function with a condition, do you have to do something different when finding the inverse or do you go about it normally? Those problems have a tendency to throw me off...
Again, if any of my explanations weren't clear, please let me know.
What do I understand about logarithms? Well, lets see. The way I view logarithms is this: in exponential form, the base from the logarithm stays the same. The other two numbers just switch places. That is to say, the number to the left of the equals sign in logarithmic form goes to the right in exponential form and the number all by itself (to the right of the equals sign) in logarithmic form becomes the exponent in exponential form. Essentially, all they do is switch places (it's a little mind trick I developed to remind myself how to do those problems, so if you can't understand it, that's understandable). What else?...Oh, right. In problems involving logarithms, to "undo" (cancel out, get rid of) a log sign, you make everything a power of the logarithmic base. If you have a base that needs to be gotten rid of, you simply make everything as part of a logarithm with a base the same as the exponential base (that gets a little confusing, I don't really now how to phrase that correctly to make more sense). In logarithms involving e, the same applies, only you use e as the base or ln, the natural logarithm (whichever the occasion calls for). Writing this and solving problems involving them made me see problems with exponents and logarithms aren't always as easy as they appear (like the problem in one of the homeworks e^x + e^-x=3, which I found out after some time is just treated like a quadratic equation in the end).
On to inverses (joy...). I understand that to find inverse equations, you just switch the x and y and solve for the new y. I also understand that the graphs of an equation and its inverse are symmetrical about the line y=x, so you can somewhat already know and graph the inverse without ever actually having seen it or graphed it (which is good for lazy people like me: less work!).
There is one thing I don't exactly understand. With some inverse problems, when the original function f(x) is given, they give you some sort of condition (like x>0 or something like that). Now, I understand that the condition is necessary otherwise the inverse won't be one-to-one. However, when I do those problems, I either get something different from the book or something that is in no way the inverse (when I plug it back into the original function, I don't get x as the output). I guess what I'm really trying to as is this: when given a function with a condition, do you have to do something different when finding the inverse or do you go about it normally? Those problems have a tendency to throw me off...
Again, if any of my explanations weren't clear, please let me know.
Wednesday, November 4, 2009
Even and Odd Functions
Even Functions: For even functions, first we need to know what f(-x)=f(x). In words, that means that for every input value of x, its output is the same as its opposite input. Simply stated, both x and -x have the same y value. Graphically, whatever shape the graph takes on one quadrant will be reflected across the y-axis and onto the other quadrant. Naturally, the graph will look symmetrical, divided along the y-axis. My explanation is a little confusing, so a good example to understand even functions would be be the function f(x)=x^2. No matter what x value is entered, its opposite x has the same y value. If you look at its graph, it is clearly symmetrical about the y-axis.
Odd Functions: Odd functions are harder to explain, so bare with me in my not-so-great-explanation. Again, we must first understand the mathematical meaning before we can understand its graphical meaning. The mathematical meaning, f(-x)=-f(x), means that if a point (x,y) lies on a graph, then the point (-x,-y) also lies on the graph. Or, if (x,-y) lies on a graph, the (-x,y) does too. Yes, I know, not the greatest explanation in the world. An easier way to think about it would be to say that if a point lies on a graph, then the opposite of that point (opposite x and y value) also lie on that point. Graphically, it's easier to understand. Graphically, a function is odd if the graph remains the same after rotating it around the origin 180 degrees (half a circle). Yes, that's really as simple as I can explain it. A perfect example would be the function f(x)=x. No matter what point you input, its opposite point is also a solution. It also looks exactly the same if rotated 180 degrees around the origin as well.
Odd Functions: Odd functions are harder to explain, so bare with me in my not-so-great-explanation. Again, we must first understand the mathematical meaning before we can understand its graphical meaning. The mathematical meaning, f(-x)=-f(x), means that if a point (x,y) lies on a graph, then the point (-x,-y) also lies on the graph. Or, if (x,-y) lies on a graph, the (-x,y) does too. Yes, I know, not the greatest explanation in the world. An easier way to think about it would be to say that if a point lies on a graph, then the opposite of that point (opposite x and y value) also lie on that point. Graphically, it's easier to understand. Graphically, a function is odd if the graph remains the same after rotating it around the origin 180 degrees (half a circle). Yes, that's really as simple as I can explain it. A perfect example would be the function f(x)=x. No matter what point you input, its opposite point is also a solution. It also looks exactly the same if rotated 180 degrees around the origin as well.
(Note: If any of my explanations were confusing, please let me know. I need to get better at explaining things to people.)
Tuesday, October 27, 2009
About Me
My name is Jesus Tejeda and I'm 16 years old. Sadly to say, my life is boring. Other than the basic nessecities of life, I only go to school, do my homework, watch anime, and play video games. Rarely do I ever leave my house unless it's school. The anime I enjoy watching either have to be comedic, like Ouran High School Host Club, have action, like Bleach or Inuyasha, or just be generally interesting, like Vampire Knight or Code Geass. For video games, I mostly play fantasy RPG's (Kingdom Hearts being my favorite series). Concerning the future, I know I'm going to college, but that's about as much as I've bothered to figure out. Everything else I'll deal with later. And that's about as much as you're ever going to need to know (seriously, my life is just that boring)...
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