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.
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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. |
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February 2018
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