This week in AP Calculus, the new topic that we learned about seemed oddly familiar. Back in trimester one, we learned about a topic called U-Substitution which was a strategy for anti-deriving a function. Just when we thought it was gone forever, it decided to make a comeback, like Brett Favre, and this time it was not much different than the first time around, also like Brett Favre. The only difference this time was that U-Substitution was applied to the fundamental theorem of calculus. The process for integrating with U-Substitution is this, first you start by choosing a "U", then you take the derivative of the U function and try to match it with the integral. To find the bounds for the new Integral, you plug in the original bounds of the integral to the u equation and then you get the new bounds. Next, you plug in the du (derivative of u) to the integral and then the fundamental theorem of calculus takes over. You antiderive the U function and then evaluate using the new bounds of the U function. After you contemplate, calculate, and evaluate, you get the value of the integral.
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This week in AP Calculus, we learned learned about definite integrals. An definite integral is an integral expressed as the difference between the values of the integral at specified upper and lower limits of the independent variable. Under the topic of finite sums, we first learned an approximation method for interpreting the area under a curve called the Rectangular Approximation Method (RAM). The rectangular approximation method is used to approximate the area by taking the areas of rectangles on a given interval with a certain sub-interval that are touching the curve and adding up their areas to get the area under the curve. There are three types of RAM, right rectangle approximation method (RRAM), middle rectangle approximation method (MRAM), and left rectangle approximation method (LRAM). RRAM is when the rectangles right edge is touching the curve, MRAM is when the middle of the rectangle is intersecting the curve and LRAM is when the left edge of the rectangle is touching the curve. One way to define RRAM, MRAM, and LRAM is by using Riemann Sums. Riemann Sums is an easier way to define the RAM and it is used to approximate the area under the curve as well as the length of the curve. In section 5.3, we learned about the rules for definite integrals as well as how to calculate the average value of an integral. Over the past couple of weeks I have began to notice everything that we have learned up until this point in the class is starting to come together and it is making more sense because we get to see the whole reason behind learning derivatives. These past couple weeks, we have been learning about the topics of optimization and related rates. At first, it was a little bit difficult to understand these topics, but after some practice, I began comprehending the ideas of the topics more in depth and they overall became easier for me to solve. Optimization and Related Rates are very similar topics but are different in their own ways. Optimization in simplest form is the mathematical steps to figuring the most effective way to do something using the least amount of resources or the most effective use of a situation using mathematical equations like area, trigonometry, or Pythagorean Theorem. Some examples of the optimization problems that we were working on are the open top box problem, the paint can problem, the inscribed rectangle problem, and the boat problem. The boat problem consists of a boat that is a certain distance from shore and they want to reach a point up the beach. We used optimization to find the most optimum place to land on the beach to make the trip in the shortest amount of time. Related Rates deals more with a situation that is already happening and you are required to calculate how fast a certain value is changing using mathematical equations like area, trigonometry, or Pythagorean Theorem. This week in AP Calculus was mainly focused on implicit differentiation. Implicit Differentiation is under the topic of derivatives and it ties in a lot of the stuff we have been doing over the past couple weeks such as the chain rule, product rule, and the quotient rule. This topic was somewhat difficult for me to rap my head around, but after doing the homework and going over the tougher problems in class, I now mostly understand what to do. The goal for Implicit Differentiation problems is to get dy/dx by itself on on side of the equation. The steps for solving an implicit differentiation problem is first taking the derivative of both sides of the function keeping in mind that y is a function not a variable, then you move all of the terms with dy/dx in them to one side, next you factor out dy/dx, and finally you divide to get dy/dx by itself on one side of the equation. The part I struggled with the most was remembering that y was not a variable but it was a function so every time you take the derivative of y, you have to use the chain rule which gives you the term dy/dx to solve the problem.
This week in AP Calculus we worked on an assignment over secant and tangent lines. This activity helped me understand how the slope of the tangent line is related to the curve. Although I struggled a little bit at the start of the activity, I figured it out working together with classmates. The change from the first graph to the second graph was adding another slider point in and adding a slope equation to the formula for the slope line.
This week in AP Calculus we learned about section 2.2, Limits Involving Infinity, and section 2.3, Continuity; we also took a quiz over section 2.1 and 2.2. Section 2.2, Limits involving infinity, was very similar to Section 2.1 which was about limits. The only difference was we looked at the limit of a function as it approaches infinity instead of approaching a certain number. I did not have any problems with understanding most of the information in section 2.2 although there were a couple problems that made me think and I did not understand at first. After asking my classmates and our teacher, I understood how to do the problems that I struggled with for the most part. We also got a worksheet over the topic of limits that helped me further understand the topic and it also made me think more about how to solve limit problems. On Wednesday, we took a quiz over section 2.1 and 2.2 and although I thought that I understood the topic of limits, I did not score as well on the quiz as I would have liked to. After I get my quiz back, I will look over it and decide whether or not to retake the quiz to try to score better on it. On Thursday, we learned about section 2.3 which is the topic of continuity. Although I was not in class on Thursday due to a tennis match, after talking about the section on Friday with classmates and going through the assignment, I believe that I have a pretty strong hold on the subject.
This week in AP Calculus we spent the majority of the week working on a review packet of things that we should have learned in past math classes. Most of the stuff was fairly simple for me once I started to remember the past topics. After a long summer of not doing very much school work, it can sometimes be hard to get back into the right frame of mind right away. After a little while of working on the packet I began to get back into thinking in the way of math and I started to remember more of the topics. Once I started remembering things mostly everything began coming back to me and it became easier. Although, there were a couple topics that it took me a little while to understand. One topic that I really understand and remember well is functions. Functions may be a tough topic for some people, but when I learned about functions it just stuck with me and I picked it up very quickly. Reviewing this stuff this week is really helpful not only because we have a test over it but also because they are very important topics in math. Moving forward this year in calculus, the topics will definitely be tied into the things that we learn. There are still a few things that I need to study before the test, but I do believe that I will do well on the test. To help myself be prepared for the test I can watch some review videos and look over the packet.
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January 2018
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