This week in AP-Calculus, we learned about solids of revolution. We also took the 7.1-7.2 quiz, and I did pretty good on it. Solids of revolution using the disk method has been pretty easy so far. The concept took some explaining since it involved picturing a three dimensional shape, only using its two dimensional profile. Once I wrapped my head around that, the process was fairly quick. The equation for finding the volume of the shape just uses the equation for the volume of a cylinder, but replaces some parts of the equation with parts of integrals. The equation is pretty basic and can be used for the majority of the problems, even those in terms of y. Basically, the volume is equal to pi times the integral of the function, but you square the entire function. You use the bounds like normal and it will output the volume of the 3D shape created by the function.
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This week in AP-Calc, we learned about integrals as net change and finding area under two curves using integrals. This was the beginning of chapter 7, and like every chapter, there was a lot of new concepts to learn about. Integrating net change was not that hard since we already know about acceleration, velocity, and position. This was one of the first times we actually went into depth about acceleration, velocity, and position. I am glad we are going into depth on these topics since acceleration is an important topic for us to learn about since we all use acceleration to get to school. We also learned about areas between curves and how to find the area using integrals. To find the area between the curves, you subtract the integral of the bottom curve from the integral of the top curve. This was not a difficult topic to learn since we knew everything leading up to it. It uses integral rules that we learned in previous chapters. This is a nice start to a chapter since it is not too difficult and is a nice way to ease into the chapter.
This week in AP-Calc, we learned about slope fields and separation of variables. A slope field is a graph of the slope at all the points on your interval. This will give you a general idea of what the original function will look like. These are not that hard since you plug in the points and graph the slope at that point. Sometimes, you might have to draw a slope field for an equation with both x and y. You still do the same thing and plug the point in and graph the slope at that point. Separation of variables is slightly harder. For that, you need to separate the variables from each other. you need to get all of the on kind of variable on one side, and the other variable on the other side. After that, you can anti-derive and get the original function, but don't forget about the +C. After that you can solve it explicitly or leave it. If you are given a point on the original function, this is the time to plug it in and solve for the +C.
This week in AP Calculus, we learned U Substitution for definite and indefinite integrals. This week was not all that difficult as it was just a combination of topics that we have learned throughout the year. We basically are taking what we learned the last couple weeks, and applying the concept of U Substitution to it. By using U Substitution, it makes evaluating the integral easier as you are replacing part of the function with just a variable. I believe that this makes doing these problems without a calculator so much easier. The one new part of this that we learned was how to find the bounds when using U Substitution. We did an exploration on it and I think that it really helped explain why the bounds changed. Other than that, everything was review of topics or combining old topics with new topics. We also had a Chapter 5 test this week that I did really well on. Me understanding integrals really helped me this week because you need to have a good understanding of the basics before you can jump into the difficult things.
While learning the Fundamental Theorem of Calculus, I believe that I used a mix of both deductive reasoning, and inductive reasoning. I think I used more deductive reasoning than inductive reasoning because the topic felt more complex for me. It was easier for me to do a couple problems, and conclude that the Theorem was true, because by plugging in the numbers into the Theorem, I got an answer that made sense in the context of the problem. I started with the Theorem, and understood it by using data to show that the Theorem did in fact work.
I believe that this is a fundamental part of Calculus because it links derivatives and integrals together. Without it, we would have no truly accurate way to figure out the area under a function. This helps with evaluating the area, not just estimating it. I think that the notation of the integrals was not that difficult because all of the notation except the actual integral sign has been already used. We know what a function is, we have seen dx at the end of functions, and we know about intervals. The only thing that confused me at first was when you find the antiderivative and you used the line to show that you evaluate at the bounds. This week in AP Calc, we learned about integrals. An integral is used to find the exact area under a function on a set interval. This makes finding the area much easier and more accurate than using LRAM, MRAM, and RRAM. We also learned the rules to definite integrals and anti-derivatives. These rules were not hard and just seemed to be like the rules we learned for derivatives.
One big concept that we learned that I thought I would struggle with was the Fundamental Theorem of Calculus. This sounds like a scary thing, but once you use it a couple times, it becomes easy. It is a way to find the area under the function without having to use a calculator. All you have to do is find the anti-derivative of the function, evaluate it at the endpoints, and subtract them. Using the Fundamental Theorem of Calculus is way easier than doing the rectangle approximation and it also gives you the exact answer, not just an approximation. Another concept that I found interesting and useful was when we did the Desmos activity to figure out the equation to find the average height of the function on the interval. I think that that will come in handy in the future whenever I need to find an average area. This week in AP-Calc., we learned related rates. Surprisingly, I understood related rates quite a bit. They were not as hard as I expected them to be. I think the reason for my success were the six steps to solving the related rates problems. The steps made it really easy to organize my information and see the how to solve the question. Without those steps, I would have had a hard time trying to figure out the steps that they wanted me to take to get to the answer. I believe that the hardest step was the first one. Making sure you get your drawing right is so crucial to getting the problem right. If your drawing is wrong, your equation will be wrong, making your derivative wrong, but you will get the wrong answer. It is really nice when the book gives you a drawing to go off of because it makes things so easy. You can basically skip step one and half of step 4 because it gives you a lot of information straight up. At first, I did not think that related rates would be useful in the real world, but after seeing some of the problems that we had to do, they seem really useful for a lot of different scenarios. The hiking in the woods problem would be really useful for when I go on backpacking trips because I could find out the fastest route to go.
This week in AP-Calc........I didn't learn much of anything? I was not in class much do to how all my field trips fell on the calendar, but for the one day I was here, we took our Chapter 3 test. Finally finished what seemed like an eternity of constant derivative rules. I felt that chapter 3 should have been split up into two chapters as it is a very long, but very important chapter. I think that the test that we took on Wednesday went fairly well, but I did forget some of the older topics because they extend so far back in the class. Overall, for missing so many days leading up to the test, I'm proud of myself for remembering as much as I did. I definitely was glad when it was over though. All these Power rules, and Chain rules, and rules for taking derivatives of a logarithmic functions, it was getting a little repetitive. I am ready to move on to the next chapter and learn some more applications for derivatives and the rest of the fancy calculus stuff.
This week in AP-Calculus, we learned implicit differentiation and how to find the derivatives of exponential, logarithmic, and the inverses of sin/cos/tan. It seemed like a lot to learn in one week, but it was basically just applying what we already knew about derivatives to solve for these kinds of problems. Those parts were easy during the week, but the implicit differentiation was a little different than what we been doing recently. Implicit functions are different than what we have been using because they contain y's and x's on both sides of the equation. Because of this, you have to solve these problems a little differently. First, you have to differentiate both sides with respect to x, so basically do what we have already been doing. Next, you have to collect the terms of dy/dx to one side of the equation. Just simple algebra to move things around to get all the dy/dx terms on one side. After that, you factor out the dy/dx from the one side. Then finally, you finish with solving for dy/dx. Super easy once you get the hang of it, because it just becomes the basics of derivatives, plus some basic algebra to rearrange the equation. This is finally the last bit of chapter 3 that we have to learn, since we have been working on this chapter for almost a month now. It will be nice to finally move on to the next chapter.
This week in APCalc, we learned about u substitution. U substitution is an easier way to think about doing the Chain Rule. It's easier because you can replace the inside of a function with a variable, then just plug that part of the equation back in where the variable is. I find this useful in more complex equations, but I do not think I need to do it on every problem. It is definitely very useful on problems that have to do a double derivative problem. This is easier because it cleans up a lot of the numbers and replaces it with variables, making the work easier since everything is less cluttered. We also learned about finding the derivative with respect to y when it is not a simple y= problem. You have to take the derivative of each side, using derivative rules when needed. Then you do algebra to get all of the dy/dx's on the same side, then solve for dy/dx. Once you have that equation, you plug in your point to find the slope. What's different about this, is that you have to plug in both x and y into the equation. These problems are the collection of everything we've learned up to this point with derivatives.
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February 2018
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