The function f(x) from the graph f '(x)

1. The function f(x) is increasing at (-2,0) and (0,2) and decreasing at (negative infinity,-2) and (2, infinity). A function is increasing when f '(x)>0 and decreasing when f '(x)<0. Because the given graph is that of f '(x), we only have to check the output of the graph to find out when f '(x) is positive and negative. It can be told from looking at the graph that f '(x) is positive at (-2,0) and (0,2), meaning the function f(x) is increasing at those intervals. Similarly, it can be told that f '(x) is negative at (negative infinity, -2) and (2, infinity), so the function f(x) is decreasing at those intervals.

2. There is a local minimum at x=-2 and a local maximum at x=2. Extrema can only occur at critical points, or when f '(x)=0 or is undefined. In this case, from looking at the graph, those points would be at x=-2, x=0, and x=2. For there to be an extrema, the sign of f '(x) must change. Because it fails to do so at x=0, it isn't an extrema. At x=-2, the sign of f '(x) changes from negative to positive, so it must be a local minimum. At x=2, the sign of f '(x) changes from positive to negative, so it must be a local maximum.

3. The function f(x) is concave up at (negative infinity,-1.5) and (0, 1.5) and is concave down at (-1.5, 0) and (1.5, infinity). (Note: the values -1.5 and 1.5 are approximations, not the exact values). A function is concave up when f "(x)>0 and concave down when f "(x)<0.>0) and is concave down when the graph of f '(x) is decreasing (f "(x)<0). Just by looking at the graph, f '(x) is increasing at (negative infinity, -1.5) and (0, 1.5), so that is when the graph of f(x) is concave up. In the same manner, f '(x) is decreasing at (-1.5, 0) and (1.5, infinity), so that is when the graph of f(x) is concave down.

4. I would say f(x) is a fifth power polynomial equation. Normally, the graph of a derivative function is one degree less that the original function. The derivative of looks as if it is fourth power polynomial equation, so one can guess f(x) is a fifth power polynomial equation.

Mindsets

Wow, I didn't like this post...at all. With that being said...

1. When it comes to intelligence, I would say I'm the growth mindset...somewhat. I'm not quite sure why, but I have two conflicting answers. Based on the survey questions, I'm the growth mindset. According to the article, however, I'm of the fixed mindset. I do believe people can change their intelligence, but I try to avoid rediculously hard challenges and don't put in as much effort into things as I can. If the challenge seems doable, I'll take it. If I foresee disaster, I probably won't even attempt it (like I did with AP World History, which I dropped in a week). For those wondering how I've made it this far in math without 100% effort on my part, it's because I understand it the first time something is explained to me (usually), so I never really have problems. The times I do have problems, someone else usually asks them because they are also wondering it, so I myself never have to do the asking. I wonder if it's just that I'm lazy and laziness is what preventing me from putting in 100% effort? Well, moving on...

2. To be honest, I wouldn't know how my mindset has helped or hurt me in math. The reason being is that because I usually understand it the first time, I haven't really had problems. I think that for me to answer this question, I would need a math class that loses me completely to see what I'd do.

3. It doesn't really affect me. If it ever comes down to me actually having to train myself to study for my classes (if it has any relevance, I don't study for my classes), then I would be extremely shocked. Until I actually have to train my brain to do something, that really has no impact on me.

4. This will affect me when I come across a math class (any class for that matter) that I actually have to put in 100% effort. When such a time comes, I'll remember this article, and hopefully I'll give it everything I can. Until then, this topic will be sitting in the back of my mind.

Algebra vs. Calculus

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.

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.

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.

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.

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.