My approach to a cross section problem would be to first identify the cross section in question. Using this I would find an equation that would find the area of the cross section. This would allow me to figure out my integrand. For instance, in a square, my radius for the circle would be the area shaded. Then I would square the radius to get the area for a square. To visualize the solid, I would just have to imagine the figure of the cross section jutting out of the graph. This activity allowed me to understand how to find volumes of cross sections because it caused me to picture the solid in question and go from there. Without these visualizations, it would have been much more difficult. This idea is useful in the real world because it allows for a model of something before actually doing it, and it allows for the creation of obscure 3D solids using unique cross sections.
A new trimester and the class is already deep in a very important subject - solids of revolutions. There are many different methods to find the area of a 2D shape when revolved around an axis. However, since there are more methods, it is harder for me to remember and apply each and every one properly. For instance, I cannot really differentiate between the shell and washer method. It seems easy, but then, I start to second guess and confuse myself. To help me figure it out, I utilize the representative rectangle. When the rectangle is perpendicular to the axis of rotation, I use washer, and when it is parallel, I use shell. This allows me to differentiate between the two and solve these area problems effectively.
It is small strategies like this that allow me to understand certain concepts fundamentally. For me to understand them completely, I need other learning strategies that I have not developed yet. This is why I think my current understanding of solids of revolution is lackluster. It just has not clicked for me. After missing a day of notes on definite and indefinite integrals, my understanding of the idea has been lackluster. I think I know most of it, but I do not think I actually do. There are small details that I am missing in the process of solving certain problems using U-sub that have been blurred out for me. Even during the test this week, I completely blanked out on what I specifically had to accomplish. What I need to do is either teach this to myself using outside sources, or I have to just ask more questions in class. Either way I think I should solidify my understanding of this concept and move to the next chapter confidently.
The exploration that we did Friday was interesting to say the least. It felt like the instructions were not clear, but the general message stuck through. I may have forgot some details, but I remember that somehow everything connected. Recently, I feel like our current state in calculus has been molded by connections. With these connections, I plan on understanding more and struggling less. After many weeks, I need to get back in the swing of things. I have these tendencies to hold off assignments, so I have to motivate myself if I want to finish off the trimester strong. During this week especially, I lost way too much motivation, and I am realizing my mistakes. I need to preform badly first for me to actually regain full motivation and come back strong.
However, I do feel as if I understood chapter 5, and my quiz score reflects that. I may not have finished the assignments on time, but I did listen carefully and take the necessary notes to do well on that quiz. This might bite me in the back later on, but I think I can easily re-learn it and move forward. There is not many concepts that I really need to grasp for this week, as it is just a continuation of the fundamental theorem of calculus. I think I understand the material enough to do the problems that I am asked. When learning about the fundamental theorem of calculus, I used both deductive and inductive reasoning to come to a conclusion. For instance, I used inductive when figuring out the derivative of F(x) would be f(x). I used deductive when analyzing that the integral on a certain interval would be F(x) - F(a) which would eventually lead up to F(x) would be the integral from a to x. This process helped me determine the fundamentality within this theorem, which was the fact that everything we had learned so far, connected together in this theorem (anti-derivative, derivative, area under a curve, etc).
The implications of this theorem allows for everything to be connected to each other in one way or another, allowing for solving issues multiple ways. The way in which it fits into calculus is based on how the problem goes in circle and states things like: the anti-derivative is equal to the area under a curve, the derivative of an integral is equal to the original function, etc. These are very broad statements that allow for the solving of many calculus related problems differently. This week's class introduced an expansion on integrals and the mechanics behind how they work. My initial thought is quite positive regarding integrals because they are not a difficult concept to understand. However, I do see the potential for them to get increasingly difficult as the class moves forward.
A friendly reminder that I received this week were the homework assignments. There were a good amount of assignments this week that I have to remember to finish and do on time to truly learn the material. Previously, I had done this, but coming back from break, I lost some motivation to do them. Although this is the case, I still did a majority of the assignments with some technical issues. This issue is the fact that I lost my calculator last week. It has honestly held me back regarding actually doing homework at home. I do not have a calculator that I can bring with me to do my the assignments. I need to find a replacement, so I can regain my motivation for this class. Being completely honest, this optimization section has been lackluster for me. I feel like I actually do not know what is going on, partly due to my lack of focus in class. However, I did miss one day of class because of the two-hour delay which frustrated me because it was an important day to grasp the idea of optimization.
Anyways, even though I did not understand the idea completely, I had a very vague understanding. I knew that the one had to find the critical points, derivative, etc to find the optimal values, but the equations from the problem were the main issue for me. I either misinterpreted the question or just flat out messed it up. I can attribute this to my low participation in and out of class. I mean, although I did not understand a focal point in solving optimization problems, I still did decent on the quiz. I could have done way worse if I did not get guidance on my question misinterpretation. I should probably do the homework problems with utmost seriousness, in hopes of actually understanding this concept completely. I still do not even know why finding the derivative even works (something to do with the extreme values?). I am just waiting for something to click in my head, and then I'll understand it. This week was a breath of fresh air because we finally finished off chapter 4, the remaining parts of trimester one, and moved onto new things: optimization. This application of what we learned is finally allowing us to dive into the meaning of calculus and new though processes. I find it very interesting that we can apply derivatives to basic things, allowing us to find answers to questions we previously could not figure out. For example, finding the perfect volume of a rectangle, etc.
These ideas are ideal for math because they are simple, yet complicated. They benefit those who prefer in depth thinking over people who plug and chug into equations. This way of thinking is great for moving onto harder subjects in calculus, as these are quite basic. In reference to these problems, I do not know exactly if graphing is the only approach, but if these problems could be solved otherwise, it would be interesting to learn. Other than that, I have been exposed to these problems previously, but not in the light of derivatives. This optimization is not necessarily new, but with derivatives, it is. I am intrigued to learn more about exactly what can be accomplished with what we learned in chapter four. This week allowed for the exploration of position, speed, and velocity through derivatives. It is a very interesting topic because its applications seem extraordinary. The class went through activities to understand more concepts to find a deeper meaning to derivatives. It is a very important step for us because it will most likely be a large portion of the rest of the class and even the AP test.
However, I still feel like I do not know what's happening in class anymore. I could not tell what we did on the individual days of class, but I could somewhat explain what we learned. Whatever we did, it allowed me to retain information on the derivatives of derivatives. Position, speed, and velocity became a major portion of this week through group activities or worksheets. Understanding how the graphs of position, velocity, and speed look for a function was very important. It seems to be based on things we have already learned, but the ideas became more refined. I am hoping to keep all of this information, and like the substitute teacher said on Friday, I hope the concepts begin to click. It has been close to a month now, and derivatives are still coming our way. This section seems rather long and consists of the same things. I mean at this point, I feel as if I have wired my way of thinking to understand derivatives, but then again, they are getting quite boring. I understand that this information needs to be learned before we move on; however, that does not stop me from losing motivation. The last two months consisted of me trying to stop procrastinating and understanding every concept, but now, I have gone back to my roots. My mindset is centered around the next quiz, if I can even remember when it is. Although, I have not completely gone haywire. I am still slightly intrigued.
With this in mind, a new obstacle has come our way in class: word problems. These, coupled with derivatives, are going to be difficult if I do not start practicing them. Luckily, we have started to incorporate them into our homework. I have started to apply derivatives to real-world problems and allow myself to think more in-depth which will definitely benefit me later on. However, I am still questioning when we can get out of the introduction, and completely move on to the application of derivatives. |
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May 2016
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