This week in A.P. Calculus, we started chapter seven and looked at some applications of anti - derivatives. It is pretty cool to be able to pull topics from calculus and relate them to real life. Most of the problems we worked through were position, velocity, and acceleration functions which I have come across lots this year. It has been so helpful having physics before this unit because it was there that I began to understand and familiarize with all three of these functions. This year, my physics, chemistry and calculus have, in some way, been connected. It is awesome to feel that all these tough concepts that I face on a daily basis are loosely interconnected. I have learned more this year than multiple school years combined. I am forced to push myself and constantly think outside the box. Some things move too fast for me, but I have always caught up and feel confident in my ability to face the toughest problems.
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This week in AP Calculus, U - Substitution made a return to the lives of Calculus students. This time around, in integrals. Also, we were introduced to slope fields. I can already tell that I miss chapter 5. 6.1 is easy to understand but when we move into taking complex anti - derivatives things get a little tougher. I definitely need more practice with antiderivatives before test day. I think U - Sub part two is easier than when we were first hit with the concept. Whether that is because of previous exposure or the little adjustment in problem types this time, I feel confident trying to get better. Sometimes, the problem will give you a designated 'U' which is so helpful in solving the problem. The other concept we covered was slope fields. Basically, At every x and y point you graph what the slope, or answer, to the equation is. It is a pretty basic idea that isn't too tough to solve.
ALSO, the debate and search for a Calculus shirt has officially begun!! Personally, I like the 'screenshot of Mr. Cresswell's Remind text message' idea the best. “In learning about the fundamental theorem of calculus, what type of learning did you primarily rely on, deductive or inductive? Or did you rely on both? Be specific! Also, explain why you believe the fundamental theorem of calculus is so fundamental? In your mind, what does it mean, what are its implications, and how does it fit in the context of calculus broadly? How does the notation used throughout the lesson connect to what you’ve already learned up until now in calculus?”
I used both inductive and deductive reasoning to understand the Fundamental This week in A.P. Calculus, we wrapped up the early sections in chapter five. We were introduced to the fundamental theorem of calculus, which covered evaluating integrals and finding areas under a curve. Like anything I have seen thus far in the year it is brand new but ties together concepts from earlier chapters. We discovered how derivatives and integrals are linked (fundamental theorem of calculus part 1). Basically, integrals and derivatives are inverses of each other. In many aspects I am still trying to wrap my head around the relationship of the two. I know that to evaluate an integral, I find the anti-derivative of the function and plug in the interval values and subtract b-a. There are some missing pieces conceptually that I have, but I hope that these concepts make more sense in the later sections of this chapter. There have been huge, fundamental, important ideas in Calculus tossed at us and most I have made sense of but some still remain blurry. I would like to know to know some real - life applications for this kind of work. I saw that in section 5.1 measuring cardiac output is one example. But, when would these concepts be used in the real world. I don't often question math topics in general and how they might be applicable, and I don't mean to be one of those annoyed students sick of doing math that's over their head but I think it would be interesting hearing an engineer, scientists or anyone that runs into this type of work in their jobs and lives.
These last two weeks in A.P. Calc, we covered Optimization and Related Rates. These two topics are some of the toughest I have come across this year. As a kid, like most others, I dreaded answering 'story problems' mainly because they always have been a weakness, and are today. Understanding and visualizing all parts to a real - world problem in actual words is hard to dissect and conquer. These problem types require you to set up a mathematical model from a drawing which I find to be the hardest part. Once you do that, taking the derivative and plugging in what you know is very straightforward. Setting up a mathematical model is difficult because that requires you to understand what the problem is asking you entirely and be able to envision it in your head. All this makes it a very complicated and grueling task. With any road block I have come across in math, extra practice is the best remedy. Cue the long hours!
Week number nine in A.P. Calculus consisted of a couple new sections in chapter three. This chapter has brought lots of trials and tribulations. Blood, sweat and tears have all been exhibited over the last couple weeks (not really). The day that we move on from this chapter will be like graduating from high school. But anyway, this week we looked at section 3.9 which was about deriving exponentials and logarithms. When I first saw that we were dealing with BOTH exponentials and logarithms I thought to myself 'ohh great' and 'just what I needed'. The good news about the whole thing is in my mind, it turned out to be one of the lighter sections in the chapter. When we were given the actual derivatives of e and ln and all we have to do is plug in U and solve is very straightforward. As always, there are exceptions but I am good with handling extra stuff like that.
This week we also looked at taking derivatives of inverse sin, cos, tan etc.. Just like I mentioned before, I think it is a pretty simple process (for the most part). There are always a few things that I struggle to pick up but I am understanding the bulk of everything this chapter has thrown at me. That being said is a major shift from how I have felt in the last few weeks. Most concepts are coming full circle now, just in time for the big test this week. This week in A.P. Calculus, we started the hardest unit that I've encountered thus far this year. That was U - Substitution. I think this unit is particularly challenging because I have a tough time using and thinking about multiple variables, not to mention anti - differentiating equations. Going through the notes, most of the material went over my head but after a few days' practice, I believe I was able to nail the bulk of the concept down. Topics that are completely unfamiliar which U-Sub is, tend to take more time to understand than topics that I have either seen before or can be worked out. There is not all that much to U - Sub to begin with, and I think that is why it can be so challenging. Really all there is to it is identifying U (which is easy) taking the derivative of U, messing with the original function to make it look nice, and then inputting the original equation back in for the evaluated U. There are lots of variations to this process, all messing with any set rules that can be used to solve the problem. With any difficult problem, I think the best approach is to attack it from multiple directions and multiple angles, if at all possible.
This week in A.P. Calculus, we focused our attention toward taking derivatives of regular functions and composite functions. These two topics are not all that hard for me to understand. For most of the functions you can just follow the chain rule or follow basic rules of algebra to get to the bottom of a problem. Years from now I'm not sure I will remember rules like the product rule or quotient rule but I think it is extremely helpful to understand the process of taking a derivative, for which reasons I am a little unsure of. One thing that I do know that I can get out of learning this material is continuing to build my critical thinking skills. That is one of the things math has done for me, and I am forever grateful for that.
The toughest part of this unit (so far) is deriving composite functions. Functions inside of one another have always been super hard to picture in my head and separate and solve. It usually helps if there is a systematic method to solve those functions like a set of rules to follow to get to the bottom of the problem and come up with a solution. Other than that, I feel like I am doing just fine. Peace, love, and prosperity.
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This week in A.P. Calculus, we were quizzed over Limits and began studying continuity. Personally, I think continuity is sort of an abstract idea, which makes it hard to wrap my head around some of the problems that appear. I like to visualize problems that exist and go forward with solving them. Seeing these problems in my head is difficult because they can be so abstract, just like the Salt and Pepper functions we saw today. Instead, today I relied on my ability to solve based on algebraic manipulation and graphing.
I do not especially enjoy this type of math, I prefer doing lots of algebra with values being changed and constantly moving. With continuity, I feel like we are merely evaluating functions, not really changing them or doing any math to them, although I know that that is wrong. All the same, math is math and very different problem types are always thrown at you and it is a matter of how one responds. |
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