I can't even come up with mediocre puns for my titles now because derivatives are becoming an empty tumor in brain that is slowly consuming my soul. I have this basic understanding of derivatives, but all of this new information keeps being thrown at me, and I just can't keep up (thus giving me a giant lump in my head filled with nothing). I don't remember what day we got our quizzes back, but I do remember getting another B. I've never felt so defeated in my life fam. I've never performed this poorly with this kind of consistency; it's pretty crazy (and sad). I'm absolutely dreading the test and exam that are inevitably on their way. I really hope I can figure out what's going on by then.
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Initially, I had no idea what I was looking at in the first graph. At some point down the road, whether it was by divine intervention or some other superficial method, I began to more easily see and understand the relation between the point and slope. The biggest problem for me was developing a working graph. I don't exactly excel at open-ended math problems; the close-ended problems are hard enough. I wouldn't necessarily I overcame my difficulty, I just plugged and chugged until I got the desired answer. The second graph was essentially a more generalized version of the first. Variables replaced exact points. The third graph was very easy to create after the first two; just a few values and variables were changed from the original formula(s). As the secant lines approach each other, they come closer and closer to forming a tangent line.
A great man once said, "The derivative is the slope of the tangent at any point". I'm still not entirely sure how any of this applies to real life, but it doesn't matter because I actually understand what's going on! This handy-dandy 4 Step Process right here:
1. Find f(x+h) 2. Write f(x+h)-f(x) and simplify 3. Write f(x+h)-f(x) h 4. Take lim f(x+h)-f(x) h→0 h was extremely helpful and easy to use. Of course we're just in the baby phases of derivatives, but any math that I can comprehend is considered good math. There was also those neat little lab we did where looked at graphs and found derivatives with rulers and estimated points. As simple as it was, it felt pretty important. By that, I mean it actually felt applicable to real life (although I still don't know where). I found the algebra covered this week to be fairly simple, however the labs where we had to make gifs ate away at my soul. Having a 4 Step Process is something that is easy to follow; all you have to do is plug and chug. When I'm instructed to make graphs look a certain way with no further instruction... that's where things get dicey. I simply don't excel at open ended answers where I actually have to use my brain (I do consider it to be a very common trait though). I don't know where any of this is going, but I'm certain this week has been fairly easy because all h*ck is about to break loose in the time to come. I've only ever heard horror stories of derivatives; never anything nice. It is October after all, the most sp00ky time of the year. Bonus sp00ky link! This week we concluded the lessons over limits and took a quiz. I felt pretty confident with all of my answers with the exception of the 3 Prong Test problem. I know it's not a terribly complicated concept, I just hadn't payed enough attention to it during class and ended up paying for it. Anyway, this week as whole went much better than last week. I actually understood what a limit was, and I was also much more comfortable finding limits with algebra. It made reminisce the days of old where all we has to do was find x. Throwing in all the cos, sin, and tan garbage is kind of killing my vibe. Honestly, there was a shining moment of hope that I may actually succeed in this class... but I won't get ahead of myself. It appears that limits will be tied in with our next topic: continuities. And as always, new math always means hard math.
My music of choice this week has been from A Tribe Called Quest. They are a 3 man rap group that personifies would hip-hop SHOULD be all about. They have a smooth, relaxed flow which is always backed by some A1 jazz music. It's very easy to listen to, making it an optimal choice for studying to. While content may vary, nearly all of their lyrics are clean, meaning that you can listen to it with your mom. Most mainstream music today is garbage; all of this trap/pop rap with no lyrical depth is simply obnoxious. A Tribe Called Quest is hip-hop at its simplest, most pure form (and they're from the 90's as an added bonus). Jazz (We've Got) is one of their more popular songs, and is well worth the listen. It's safe to say that my emotions have ridden the roller coaster a few times this week (pictures above). I finally got the hang of the review sheet, so I was feeling pretty good for a little while. After the quiz, I was still feeling pretty good. Then we tried learning about "limits"; this is where the tilt began. At first I was all like, "bro what is this junk?", and was back to questioning why I was even in this class. It took me a little while just to understand the basic definition of a limit. I was really thrown off by the random points on the graph that rested on x values that were being shared with lines. The lab was a bit of a struggle at first, but by the time I got to the homework assignment, I understood everything pretty well. Solving problems algebraically was ezpz. Simplifying expressions and plugging in the approaching number for x was probably the easiest part and/or the part I best comprehended. So here I was feeling pretty good again - confident in my ability to find limits - and then I got my quiz back and had my heart ripped from my body. My score was lower than anticipated, but it doesn't really matter because I'm sure this base knowledge of limits will carry me into a bright future. What I learned this week will probably seem like child's play pretty soon. It's my guess that a large portion of our future lesson will involve a basic understanding of limits.
The summary of this week in AP Calc: I tilted. To lower stress levels, I listened to Lana Del Rey's new album "Honeymoon" a couple times over. It's a solid album, but it's pretty average compared to her previous releases. It's just lacking a certain uniqueness- a distinct signature sound - to the album. I'm not necessarily complaining, I love Lana, I'd just like a little more variety in her music. Still worth a listen, even if her last 2 albums were much, much better.
Like I said before, humans can't physically perceive the 4th dimension, but that's why nerds have the ability to draw geometric shapes from their imagination. A few examples are shown below. The picture on top displays where a 4th coordinate point would be located. The lower left-side picture is a 4th dimensional cube. You can actually buy a 4-D Rubik's cube that resembles the picture above if you have no life. Lastly, the gif in the lower right corner is another 8 cell tesseract in motion. The best way I've found to describe the 4th dimension is to imagine a tennis ball. Take the ball, and drop it. If you were to snapshot every frame of the ball's decent, then put all of those images together, you'd see a tube. It's a 3-D object occupying space over a set of time (a small visualization of the 4th dimension). A more mathematical way to visualize the 4th dimension is through the use of 90 degree angles. Look at the 1st dimension, a line measured by x. If you make a 90 degree angle to the line you get the 2nd dimension, measured by x and y. 90 degrees from a 2-D plane will put you into the 3rd dimension. What's 90 degrees from the 3rd dimension? The space an object has taken up over time: the 4th dimension. Understanding dimensions outside of the 3rd is very difficult and can be downright frustrating (much like a math class). Maintaining the brainpower to comprehend these complexities is something only mathematicians, physicists, and super nerds are capable of. Before theories of any other dimension can be formed, a basic understanding of math and coordinate planes is required.
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September 2015
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