So this week turned out to be pretty rough. In the past couple of weeks I have been feeling pretty confident about acing tests and feeling fine with all the topics. Although for some reason this unit was weird. I remember looking down at the quiz and feeling totally stupid because I felt like I did not get any of these concepts. I ended up not doing that well on a few of the quizzes which still isn't really that bad. I guess I'm just feeling worn out and math is becoming to repetitive and it is hard to stay awake in class. I guess I know I just need to pay more attention, concentrate more on doing homework, and making sure to really understand the topics before I take the tests. We are doing disks of revolution which is actually super simple and its nice to have a good balance of easy and hard topics. I am really glad its the end of the trimester and all my classes will focus on AP test studying soon because I am really getting tired of this whole school thing.
This week we had a quiz over section 6.1 and 6.2 which I thought was pretty easy. I honestly haven't been studying that much and I really am slacking on the homework but I think its because the topics have been pretty easy and I understand the tests well. Thats pretty cool since I have so much going on. I'm glad this is a class I don't really have to worry about. Honestly it is hard to write these blogs when I feel like I know the topics really well because its only when I am doing badly or really confused that I have a lot to say or write.
For example, today we did our first AP level FRQ and it was pretty rough, so I have a lot to say about it. The most confusing thing was that it was worded really weird and then I forgot to answer it like we learned the previous week and tried using the p=nrt formula?? I do not know why even thought to do it like that because it literally had nothing to do with the questions. I guess it wasn't a total train wreck but I was still confused.
In learning about the fundamental theorem of calculus I think I used both inductive and deductive reasoning to come to different conclusions on what I thought about it. Mainly for most things I think about I start off by being general and use inductive reasoning but once I become more involved in a topic and truly understand it my deductive reasoning kicks in. As of now I only am familiar with this theorem in a broad aspect so I can only infer what is going on based on graphs of antiderivatives and integrals. I think I don't fully understand the topic so I really am unable to figure out myself how this really works and why it could be useful. Sometimes in math I think the most coincidental things have a huge impact on a bunch of other random calculus related things so basically I think this theorem falls heavily on that assumption. It is very useful in finding the area under curves which probably relates to so many other things that we will eventually learn about. Calculus makes a lot of sense to me but it is really hard to look at things in a broad picture aspect. I wish my deductive reasoning kicked in a little earlier than usual so I could be the first to actually assess what is going on but sadly that is not the case.
So I thought this week would be really stressful because we started off with two review assignment days and literally I did not understand a single thing about the assignments. Actually I take that back, I knew what I was doing but the questions were phrased so weird and some of them took so long and were so drawn out that I really just wanted to give up. I would say that I am really good at optimization and the related rate problems, but messing with the first half of the assignment and the first and second derivatives was pretty confusing. On top of that, I really have stopped caring that much, which ya know is cool, and was just going to kind of wing the test.
Luckily, the test was really easy and I think I did well, so yeah, that is even cooler. I consider myself pretty good at math so I really don't know why I worry about it sometimes, I guess it is just kind of what I do. Basically wrapping up the chapter is all we did this week and I am not really looking forward to learn more new things! Love being a senior!!:)!!!!!
This week's new topic was taking what we knew from derivatives and optimization, but now finding rates of how parts and pieces change throughout a problem. I was expecting this lesson to be a harder than last week's but once you were able to understand the problem well, it turned out to be as tough as last week, a little easier even. I think what helped through this was that there were actual drawn out steps that worked to complete each problem. I think the hardest part of these problems was figuring out what equation to use. Once you were able to find the equation, the rest was easy because all you have to do is take the derivative and plug in what you have. The thing you had to keep in mind was using implicit differentiation with variable that were changing. All in all, I thought this week was pretty simple. The abstract problems were what made it a little tough, especially ones that consisted of a shadow, in the picture shown. After thinking about these kind of problems for a while they turned out to be not so bad. The trick for many of these was to just use similar triangles which helped you so the small amount of information that came with the variable would not complicate anything.
This week was all about optimization and I honestly thought it was going to be way worse than it actually was. Last week was harder for me because I kept getting all the derivatives wrong so when I tried finding critical points it just turned into a huge mess. However, this week was much easier since most of the time you didn't even have to find the derivative and you could just graph to find critical points. The only thing that was difficult this week was the conceptual idea of all the problems. Some required a lot of planning out in order to evaluate how to optimize something. The overall idea is pretty easy as well and I feel as if we did something similar back in 8th grade. I literally had like this deja vu moment because I remember doing a similar problem years ago with the same people in class explaining it. I thought this week's quiz was pretty simple too, although I'm not entirely sure how I did. Since there were only four problems, I wasn't too worried and I thought they were pretty easy. I have nothing else to say. I wish calculus was more riveting than this. Don't get me wrong, I love math, and think that there are really awesome equations and problems and just really abstract way of thinking. However, this class is very boring and not interesting enough for me to write weekly blogs about.
It is getting hard to write these things since we all we do every single week is derivatives. But hey, I'm not totally complaining. I don't think derivatives are really that difficult. I think they are just a little difficult and thats why they seem hard. I think I have been doing pretty well on these past quizzes though. We took a quiz Thursday that encompassed the chain real, u substitution, stuff with y that I can't remember the actual name of and am too lazy too look up. I honestly though it was all pretty simple. The initial concept is easy but the tricky part is doing problems that have so many rules intertwined within.
Friday we started finding derivatives of inverse trig function which is also a little difficult but really not all that bad. The thing I am a little worried about is the test. Not because it will be that hard but mainly because there has been so much we have learned leading up to it that it might seem a little daunting. There has been so many rules to keep track of that I feel like I could accidentally make a lot of little errors. I'm really not that worried though, because lets be honest, we have three weeks left of the trimester and I have four more retakes saved. ¯\_(ツ)_/¯
This week was all about the chain rule. I was very confused the first day we learned it but the days after that I was able to see more clearly how it works. I honestly am still slightly confused because when doing the mini quiz I kind of forgot a lot of simple rules to follow, although I did end up getting a good score. You would think that the finding anti derivatives using the chain rule would be harder but I actually though it was the easiest question on the quiz.
I think the main reason why derivatives can be confusing is because when you need to use these crazy rules in order to figure out a problem you forget why you are doing this to begin with. For me I tend to do this thing in math where I just concentrate on trying to solve one problem that I forget why I’m doing it in the first place. It all makes sense to me right now, I just think it is a common thing for people to fall into doing math without any purpose. In itself derivatives are pretty simple, there are just a ton of different ways to solve them and that’s why it makes calculus a little difficult. You have to concentrate on what rule you need to use for each problem.
This week we dove further into rules for derivatives. We learned the product rule, quotient rule, and others in order to calculate the derivative faster. We also were shown rules for trig functions which in my opinion were easier to use than the other rules. This whole lesson leading up to the quiz was actually pretty easy in my opinion. As long as you know the specific rules it is pretty simple to figure out any problem. Especially when you have functions that are long and to the seventh power for example. It makes life a little easier when you do not have to use f(x+h) – f(x)/h for problems such as those.
We also continued doing anti derivatives and finding higher order derivatives which are both super simple as well. This week we had a couple different homework assignments and worksheets but I was able to get everything done fast and I knew how to do a lot of what was on there. I am glad we are doing something fairly easy because it makes me not have to worry so much over this class and can concentrate on fun things like college applications. Im kidding obviously. In other exciting calc news, I redid my sheet of cardstock for tests and it looks nice. That is all.
This week we went more in depth with derivatives and took our first quiz on them. I actually do not thing the concept is too hard but I still was a little skeptical about the quiz. I think I did well but on the other hand I could be totally wrong about everything.
I though lab 6 was pretty interesting. It was all about the correlation between the graphs of f(x) and f'(x). When the graph of f(x) is increasing the graph of f'(x) is above the x-axis. The opposite goes for when f(x) is decreasing at any point because the derivative will be below the x-axis. This also means that the minimum and maximum values of f(x) are the x intercepts on f'(x). I thought this was cool how it all works out. Math is awesome like that sometimes. When you finally see that one way of doing something, why it works, and how it all intertwines, it can be sort of eye-opening.
Lastly at the end of the week we learned some rules for finding the derivative without using a calculator. For example, the power rule helps a lot without having to plug in a long number into (f(x-h)-f(x))/h. These rules make finding the derivative more simple without huge calculations where there is more of a chance for error.
ALSO, funny side note. We had a cross country meet on Friday and this particular course had a lot of curves through the woods. Well, as I was running, a girl and I were rounding a corner and some coach yelled out something about running the tangent line. It was like she was saying instead of rounding the corner, we should take a straight line in order to run the shortest possible race. I already knew that it was something you should do but I though it was funny that the coach phrased it like that.