Well the week of math after spring break went pretty much how I expected it to, I struggled with just about everything we did and trying to force myself to remember old stuff wasn’t a walk in the park. Our main focus this week was on known cross sections (it’s funny because the name is “known” cross sections, yet I have no idea what these are). I don’t know if I took bad notes on Tuesday or if the hype about the T-shirts have thrown me off but I just do not understand how to solve these problems. There are different ways to solve them, examples being with rectangles, semi-circles, or equilateral triangles. Trying to make these different methods work in an integral and doing it correctly hasn’t clicked yet for me, so I’m hoping some miracle will happen by the time of the quiz on Thursday. Besides getting the correct integral, sometimes having to split integrals into two sections has given me trouble as well. In problem three in the worksheet, when you have to add integrals because you can’t take certain areas while respecting one variable (or something like that, not really sure because it doesn’t make sense to me). Rotating the areas about lines or axis’ has come better to me, as this is simply just subtracting the top line from the bottom, or shifting them up or down to be able to flip around the y axis. I am crossing my fingers that Tuesday be the day that I get some good practice in at these known cross section problems, that way when I say “known” in the name, it will actually feel relevant to me. Well also I’d like to pass the test.
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This week in calculus we dove even deeper into finding areas under curves, and we learned how to find the volume of a curve after rotating it on a certain axis, or even a line. We used three methods to do this: being the disc method, the washer method, and the shell method. Out of these methods, the disc method is probably the easiest for me, just because of the simplicity of the formula. It is just the integral from either the x axis bounds or y axis bounds (depending on whether you are taking respect to y or x) multiplied by the function squared, time pi. The washer method was a little more difficult because you have to subtract functions inside of the integral, and grasping which function to subtract from the other was a little confusing. The last method was the shell method, and the only thing I never really understood was if p(x) does not equal just x, how to find what it is. We did have an example in class but I would like more confirmation on that part of the equation. Probably the top two most confusing things were figuring out whether to take respect to y or x, and rotating it over lines rather than axis’. Members in my pod just shifted the equation up, down, left, or right so they could just treat it like they were rotating around an axis. This is a good idea, but I would like more practice in finding the volume when rotating around a line. The final struggle was frq questions from the AP exam, these did not go well and kind of ruined any sort of confidence I was trying to build up in convincing myself if I wanted to take the AP exam.
This week in calculus I felt like the blind squirrel who finally found a nut. I was basically playing catch up from the one day that I missed last week in trying to teach myself the new section, copying down notes, and getting peer help. Even though Mr. Cresswell missed a day and we had a snow day so the rest of the class probably felt like we didn’t get much done, I did a lot of work in class and I know that it paid off during the quiz on Friday. It finally makes sense on when to know whether to solve an integral using respect to y or x, why I’m doing it one way or the other, and how to solve them. When you solve using respect to y, your equation that you are subtracting should be in the form of x=, and you use the bound set on the y axis. You take the right line and subtract the left on the graph to get the correct area. When solving with respect to x, you should end up using the y= equation, and use the bounds set by the x axis and take the top line on the graph minus the bottom line. In knowing when to use respect to y or x, basically what worked for me was if it was already in the form of y= or x=, and which bounds are easier to work with (on the y axis or x axis). Unless of course directions were specific on which variable to respect, and now I am confident I can do both. There was one problem on the quiz that I did wrong, which was the last problem working with velocity and position with points on the graph. I think I did this one wrong because just after I handed my quiz in I realized the steps I took to find slope only work in a linear function… So I kind of turned back into the blind squirrel at that point in time.
This week in calculus we dealt again with definite and indefinite integrals and how to solve them. We also learned a completely new subject (for me at least), being slope fields. Solving integrals using U substitution has been a major struggle for me, and I just can’t seem to get the hang of it. It hasn’t really “clicked” yet how to make the U value end up matching the derivative of another value in the original integral. I would still like more practice with these, and maybe even taking steps backwards so I really understand the easy ones (this feels like geometric proofs all over again, I’m just falling further and further behind). As for slope fields, I understood how to draw these pretty well, they are just time consuming. I did have a little trouble on matching he slope fields with their differential equations and their own equations, and I wish we could have a few more homework problems involving these before seeing them again on a quiz or test. My participation this week I felt was pretty good, I feel like I am making a decent effort at mastering the U substitution problems, they just haven’t seemed to click yet.
In learning the fundamental theorem of calculus (and also what the words deductive and inductive mean), I think I was more of a deductive learner. I feel like this is usually the case with most subjects for me, as I seem to learn better after having the specific rules of something presented to me and then I get to actually work them out. An example involving the fundamental theorem of calculus is when you take the derivative of an integral between bounds, and the answer ends up being the same with just plugging in an x value (or whatever value the derivative pays respect to) for the t variable. When this was presented using deductive reasoning and the answer was just there in front of me it was easier to do, and I still had my basic reference formula to go back to if I needed reassurance (subtracting the derivatives of the bounds). I believe the fundamental theorem of calculus is so fundamental because it needs to be kept relatively broad and/or simple so it can be used in any situation with integrals and a function. When it is kept fundamental, it will work for more functions and be generally simpler to use.
This week in calculus we learned about how to more accurately measure the area under a curve by hand, using trapezoids. This allows for a more accurate line following the curve rather than a rectangle which has to have 90 degree angles. We also practiced more with anti-derivatives in finding the area of a definite integral by subtracting the antiderivative of the upper bound from the anti-derivative of the lower bound. Another thing we did was learn about the fundamental theorem of calculus, which I found to be pretty interesting. It basically says that when you take the derivative of an integral with the lower bound being a constant and the upper bound x, the derivative is the same as the integral. This is because the derivative of a constant is zero, and the derivative of x is 1, so essentially it is just multiplying the integral by 1, keeping it the same but replacing the variable to an x. We also learned how to go in reverse if you want to write the integral given an x and y value, you just plug in x for the lower bound constant and add the y value. I definitely think all of this stuff is starting to come together, and everything is starting to make more sense as to why we are doing this and have to learn all of these different little tricks, and I feel like I actually understand the fundamental theorem of calculus. (I can’t take that with a grain of salt). I felt like my participation was good this week again because I understand what is going on, and it is cool to be able to engage in a conversation and not feel like my peers are speaking a foreign language.
*STAR WARS SPOILER ALERT* Well this week reminded me of when Han Solo sees that Finn and Rey have the Millennium Falcon and bring it back to him, and he is overjoyed. This was the first week in a long time that I understood what we were doing and was able to do most of the problems with ease, and I am very happy about that. This week we learned about finding the area of a under the curve of a function between certain intervals. We learned how to solve it using methods of drawing rectangles and also using our calculators. We learned about LRAM, MRAM, and RRAM, which are different ways to fit in the rectangles to figure out the area. I liked RRAM the best, as for me it was easiest to draw them in accurately. We also learned about Riemann sums for functions on a certain interval [a,b]. We used integrals to set things up simpler to solve the area and net area of a function, using upper and lower bounds, the integrand, and the variable of integration. This week connects to previous weeks because we do use the limit of the function as n approaches infinity to get the area. If I had to pick one thing I struggled with it would probably be understanding what questions in the homework were asking me to do, and the problems in the homework that gave you information about an integral and then asked you to find different values between different intervals. I would definitely like to do more problems like these in notes and on the whiteboards just to keep practicing. I felt my participation was pretty good this week because I actually understood what was going on and I could explain things, rather than just smile through my teeth and nod my head like I was engaged with what was going on.
Well, I knew I would regret my statement from last week’s blog. I miss just being able to do a couple of steps of algebra and then pushing some buttons to get an answer. This week was probably one of the hardest weeks of math in my whole life, and there was basically nothing that I understood how to do. While we really haven’t learned any huge impossible formulas and most of what we did last week was applying old formulas and concepts, it was still extremely tough. Since it is in the blog expectations, I guess what I understood relatively well this week was problems like the one on the quiz about Mr. Cresswell and his dog, in finding how long they should walk in the woods vs trail given different speeds in both cases. I wrote down an example on my notecard and it helped on the quiz, and it actually made sense that I had to find the time using time=distance/rate. As for what I didn’t understand, it was literally everything else. It was just hard to envision what the questions were asking, and what I had to do to solve it. Friday’s class period was a big help from the new steps that we learned, but I am still not comfortable at all and if these type of problems are going to be a frequent enemy I would love to do more whiteboard work and example problems using the new and improved steps. Remembering and applying all of these area and volume formulas and then taking the derivatives and finding out what it means just isn’t clicking. I felt like my participation this week was good but it was a lost cause, and I am sure if you see the score on my quiz you would agree.
Well, I’m about 1/3 of the way done with AP calculus, yippee. This week we relearned lots of concepts that we knew about before, but we didn’t know how to find them without a graphing calculator, such as finding the intervals of increasing and decreasing, maximums and minimums, and concavity. We also learned about a very critical concept, called critical points. These are points that occur when f’=0 or f’ does not exist. I feel like I have a pretty good understanding of these, which has helped in finding these other concepts without a calculator. Finding these relearned terms can be found using first and second derivative tests. I think I have managed to grasp these tests now in knowing that the first derivative test finds increasing and decreasing and the second derivative test finds concavity. The biggest challenge this week was the sheet of paper on the non-calculator portion of the quiz. I think I just need more practice at doing these, especially with more complex functions. The thing that came relatively easy was finding symmetry, because you simply replace the x values with negatives and then check to see if it fits odd or even. What we did this week connects to previous weeks because pretty much all of the terms we are learning about have been taught in the past, just with a calculator in our hands. Personally I kind of like finding these without a calculator because it is actually finding the values on a function and not just pushing buttons. I have a bad feeling that these are going to get super hard and I am probably going to regret saying that I like not using a calculator, but for now I’ll stick with my gut.
This week was one of the better weeks in the last month, probably the best since the beginning of derivatives. We learned about taking the derivative of implicitly defined functions, such as x^2+y^2=25. These are solved by taking the derivative of both sides with respect to x, and we learned that the derivative of y with respect to x is simply dy/dx. Taking these derivatives to me was pretty simple because it was just solving for dy/dx, which was pretty easy especially using all of the other rules we have learned. The quiz on Wednesday kind of shot me in the foot because I felt like I did well for the most part, but ended up with 16 out of 23. The biggest problem for me still is the U substitution in antiderivatives, which I just cannot seem to grasp. After the quiz, we learned about inverse trig derivatives which was again just plugging in formulas and using rules that I have become (knock on wood) pretty comfortable with. Everything we have done so far has connected to our new stuff because we use rules and other formulas in every little new thing we are learning. My participation this week I felt was very good because I understood most everything we did, so I was able to feel a part of conversations and understand what people were talking about. I also got to prove Gavin wrong so that was pretty freakin’ awesome.
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April 2016
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