This week we continued to work on different methods to find the volume of an integral when it is revolved. We learned how to do the shell method this week, which was a bit more intricate but not more difficult than the washer and disk method. I actually thought it was easier because I would often forget to square the functions. It's pretty easy for the most part but since there are so many details it's easy to forget one part of the equation and throw off your answer. Sketching the graph and the rectangle definitely helps you to remember. The hard part is not being able to double check your answer unless you have an answer key. If I get a negative volume then I definitely know that I did something wrong but other than that I just hope for the best.
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I think for me it made the most sense to do deductive reasoning. Knowing the rules gives me guidelines as I work on solving the problem. I can feel a bit lost when given a problem before the generalization. I do think both are important for different reasons and mixing it up, using both interchangeably is good because it forces you to think a little harder instead of just following a formula. It helps you actually understand versus memorizing the rule. Knowing that the antiderivative of the function is the area under the curve made things all connect. The fundamental theorem of calculus will help a lot as we work on problems about the area under the curve. It's fundamental and can be manipulated forwards and backwards since the antiderivative and the area relate to one another. This week we reviewed and took the Chapter 4 tests. In the beginning of the week it was hard to get going because I forgot so much over break. Towards the end of the week I realized that I hadn't just lost some of my knowledge but also all of my Chapter 4 book assignments. I have a seperate notebook for all my book work and I hadn't seen it in awhile. I figured it was somewhere at home so I wasn't too worried about it and did my review assignment on a sheet of paper instead of in my notebook. I had been on the look out for it all week but then Thursday night I was like okay I should really find for this notebook because it is due tomorrow. So I searched literally everywhere and went into slight panic mode telling my family to check any black notebook they have or see. I finally came to peace with the fact that it was most likely up north and long gone. My new plan was to dedicate my weekend to Chapter 4 of Calculus and turn the assignments in on Monday. So, THANK YOU Cresswell for being so kind! As the week progressed the problems got easier and I remembered more and more of how to go about solving them. I am hoping I did good on the test, it felt fairly easy but I already know I made a few little mistakes so we shall see! This week we learned how to optimize areas, volumes, side lengths, etc based on limited informations. I liked this week because it was applicable to real life and showed derivatives do serve a purpose. We had to combine all of our prior knowledge in order to solve these problems. The quiz was only four questions but I didn't even finish it. That's one thing that does kind of stink about these, they take a very long time to complete one problem. Also, although i think they are simple I have struggled this week. Since it combines so much prior knowledge I get stuck at some points and have to ask my classmates what to do from there, once they give me a little hint it immediately clicks but I'm still struggling to reach that point on my own. This week we learned more about derivatives, which I'm assuming we'll be doing again next week and the week after that. The functions we learned to derive this week were implicit functions and inverse trig functions. We also had a quiz over the "u" substitution and other stuff. I don't remember it completely but I think I was confident about it? I really can't remember but I feel pretty good about derivatives in general. Since there are so many dang rules it can sometimes get frustrating I'm gonna be honest. Like individually all the rules make sense and when you combine them it makes sense too I just don't really enjoy spending ten minutes on one question. Luckily it is not conceptually difficult for me, just physically. This week we learned how the function, the first derivative and the second derivative all relate. It understood the concept but I made way too many plans outside of school this week so I was bad and did about half of each assignment. I know I can understand it, I just honestly didn't put forth much effort so that's on me. The quiz I felt okay about but I didn't know two questions at all (clearly the sections I didn't complete). Most likely will be retaking this quiz.
This week we learned how to find the derivative of more complex functions and the antiderivative. I found it to be fairly easy, sometimes it was scary to look at but once you break down the problem it's not too bad. The hardest part for me at first was when we switched to antiderivatives. I obviously remembered how to find the antiderivative but it took some practice before I was able to do it quickly again. The other thing I struggled with was figuring out what the u value would be. Once I understood the chain rule better and realized it was the inside of the composite function then it helped me be able to solve the questions. On the quiz I messed up by forgetting a negative when I was finding the derivative of sec(x) :((. The derivative of sec(x) was given to us in our notes but I forgot to put a few of those on my cardstock paper which would have made that question a piece of cake. I think I did pretty good on the rest of the quiz, I felt confident about it and was able to finish it pretty quickly and look over my answers (and I still missed my negative mistake). Cresswell said that each question was worth multiple points so hopefully me missing that question doesn't result in a 75%.
This week we learned how to take the derivative of functions that had multiplication or division in them, antiderivatives, and the derivatives of trig functions. It was fairly easy, probably the worst part about it was the length of my work for certain problems. I understood the concept and feel confident about the quiz but also know the exact question I messed up on and we haven't even gotten our scores. The question I messed up was the product rule for one of the trig functions. I forgot to apply the product rule for some reason, so I didn't do the (u)(v')+(v)(u'). Depending on how many points I'll get docked for that I may retake this quiz because I know that I could easily score 100% on it. I've been able to calculate the problems without the calculator. I will occasionally double check it by graphing if I am unsure about that problem but I felt pretty good about the homework problems for the most part and felt no need to do that. I think next week I need to improve on staying on top of the homework assignments, I did a little bit of cramming on Thursday night, which is probably why I messed up the trig function on the quiz.
This week we learned about how to graph derivatives without a calculator, how to know when derivatives will fail, and the different rules of derivatives that allow us to write/find the derivative without using the four step process. I really struggled with the concept of tangent and normal lines. For some reason I had a hard time figuring out how to write the equation for them. After blindly doing a homework question about tangent lines I thought I came to a an answer but all I really found was the slope. After talking with some other people they explained to me how to do this kind of problem and I thought I got it but when I tried to do it later on my own I was lost again. I had just memorized the steps instead of actually understanding the "why" behind each step. So I took a closer look and realized that once you find the slope you want to find the y-value at the given x-value and then plug all those numbers into the y=mx+b formula to find the y-intercept and then rewrite the equation. You want to find the slope at the tangent so that's why you plug it into the derivative first. But next you want to find the values at the actual equation not the derivative. This week we learned about derivatives and the first thing we did to help us understand was create graphs on gifsmos. It represented how the slope of a line got more accurate the closer the two points got together which turned it from a secant line into a tangent line. The activity at the beginning of the hour helped me to understand that concept. I struggled a little bit at first because I just didn't know where to start. Talking with my classmates definitely helped me get on the right track. I was almost finished with my first graph but was stumped on how to get the sliders to move so I asked Mr. Cresswell who helped explain things. The first graph required a stationary point with one slider number f(a) and the second graph both sliders numbers moved by adding f(b) and slope formula m=(f(a)-f(b))/(a-b), the avg rate of change. For the third graph I kept everything the same expect for the original f(x)=.5x^2 function. I changed it to f(x)=sinx^2 and it still worked because of the f(a) an f(b) slider points.
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March 2017
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