When learning about the Fundamental Theorem of Calculus, I used mostly Inductive Learning, but a little of Deductive Learning. By looking at the theorem alone, I didn't understand it much and I didn't really know how to use it. However, after playing around with the theorem and using it to answer a few problems in the book, I grasped the concept to the point where I believe I understand it. I think the Fundamental Theorem of Calculus is fundamental because it connects the use of integrals and differentiation, and by doing so opens up more possibilities that I probably don't know about yet. In my mind, this theorem means basically that it's possible to find integrals and derivatives in a way that I wouldn't normally be able to find them, but it will probably mean more to me once I use it more. To me, it implies that all that we learned and all that we will learn will be connected in some way. Knowing this, or even just thinking about this, it's interesting and also scary. Both integrals and differentiation are mildly difficult things alone. Putting them together make them even harder, and if this theorem is put with something else we will learn later I can't imagine how hard it will be to use.
After this trimester of AP Calculus, I feel like I've comes to understand the gist of how this class will flow throughout the school year. I can't tell of what is to come, but this trimester we went through Limits and Derivatives, mainly Derivatives, and ended with the beginning of applying those derivatives to other problems. Limits were relatively easy to understand because I took the third trimester of Pre-Calc, but Derivatives were a new thing altogether, causing me to have a few problems here and there. I still have problems figuring out complicated derivatives that involve strange situations with the chain rule and such, but I'm pretty confident that I could solve most things given enough time and maybe a little nudge in the right direction. As for how the class goes, I think the group activities we do really help with learning new material, or even just making older material concrete. The CCCs are helpful because not only can you get help with problems that other people got, but you can also work together with other people that didn't get the problem and solve it together. Along with the group activities, the CCC time helped me learn to fix the mistakes I made. This week in AP Calculus, we came to the end of chapter 3, finishing with learning about inverse trigonometric derivatives , exponential derivatives, and logarithmic derivatives. These rules are pretty simple, inverse trigonometric derivatives being the only ones that I will probably have trouble remembering because of how similar they all are. These are also pretty simple derivatives to be solved separately, but, as always, these just add more to the mess of things to have to take into account when doing more complex derivatives. I think the biggest problem I've had this week is making sure I properly make use of multiple rules of derivatives. Solving a derivative that only requires one rule to be used is easy, but combining a few of them, such as the chain rule with an inverse trigonometric derivative, it becomes a mess to try to simplify. A lot of the time I end up looking at a problem and wondering if it can be simplified further or not, or maybe if I just simplified it wrong. If I do this I end up being too worried about it, and I just change it back to something only partially simplified. This week in AP Calculus we worked more on the chain rule, but we also were introduced to "u" substitution and implicit differentiation. With "u" substitution, the chain rule becomes a lot less of a mess, and it's easier to differentiate, and anti-differentiate because you can clearly see the "u" and "du". I still make mistakes when doing the chain rule with "u" substitution, but my understanding of it and ability to do it is improving. I think my main problem with "u" substituting comes from just basic problems with anti-differentiating (forgetting the +c and other things) and sometimes basic problems with the chain rule (choosing the wrong inside function), but I'm happy that it's getting better. The other thing that was introduced that uses the chain rule is Implicit differentiation. Implicit differentiation is still a difficult topic for me. I only understand the process of it a little bit, which is the basic steps of treating one variable as a function of the other, so I can do basic problems involving it, but once it gets to a slightly more complex problem, I feel completely lost. Therefore, if I had to rate my understanding on the topic based on a scale of 5, I would give myself a 2, and a result of this was my struggle with the homework we were assigned on Thursday and Friday. However, because of this lack of understanding on the topic, I'm looking to study the subject a lot more because of the quiz on Monday. If left untended to, these problems are going to stack on top of each other and then finally crush me when the test day comes, so it's best to sort it out as early as possible. |
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February 2015
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