This week in AP Calculus, we learned about volumes using integrals, position and velocity, and areas under curves. This was the beginning of chapter 7, and we will be using these more as the chapter goes along. This relates to previous topics because we are still using integrals. We are just applying them in different ways. It could be leading into more with integrals and even more in-depth topics. I understood area under the curve very well this week. It was probably the simplest topic we learned, and it was necessary to understand when we were learning about volumes using integrals. I didn't really struggle with anything this week. Everything was pretty easy once you understood it. To figure out areas under curves, you have to take the antiderivative of the function on top in relation to the axis and subtract the antiderivative of the function on bottom in relation to the axis. If there is only one function, you just take the antiderivative of that function. You have to make sure that you correctly put in the bounds. This should be the range of values on the function for which you are evaluating the area under. In addition, I exemplified participation in class this week.
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This week in AP Calculus, we finished sections 6.1 and 6.2 with a quiz. We also started learning about differential equations and separation of variables. This was important because it finishes up chapter 6 for us. Next week, we will have the test over chapter 6, so we will need to know everything that we went over this week. Everything in chapter 6 relates to each other because it all deals with solving equations using the derivative. I understood everything pretty well this week. Separating variables was a fairly simple concept for me to grasp. I didn't really struggle with anything, but I should review some more before the chapter 6 test next week. When solving a separation of variables problem, you will be given the equation of a derivative. You must separate the y's and dy's from the x's and dx's. Once you do this you should take the antiderivative of both sides and simplify. I thought that my participation was spectacular this week. I still need to look at chapter 6 as a whole and review all four sections to prepare for the test. Once I have done that, I think that I will be very well prepared.
This week in AP Calculus, we learned more about u-substitutions, integrals, and slope fields. This is important because all of these concepts can be tied together. By looking at derivatives and anti-derivatives, we can determine slope and original functions. To evaluate an integral, you can use u-substitution. We learned a little bit about u-substitution at the beginning of the year. We have also determined slope from the derivative previously. Like I said before, these concepts all tie together, and we are just taking a more in-depth look at them. I understood slope fields very well this week. Since using derivatives to determine slope is fresh in my mind, I was able to understand these extremely well. I did not really struggle because these concepts are pretty familiar to me. To determine slope field, what you do is you take a point on the dot paper and plug it into the derivative. If the point we are looking at is (1,1), we would plug in 1 for x and 1 for y, solve the equation, and make a mark on the dot of the number/slope we get. Once the entire field has a mark, the result is the slope field. My participation was very solid this week. All I really have to do is prepare for the upcoming test next week.
This week, we learned about the fundamental theorem of calculus. In my experience, deductive learning really helped solidify the concepts for me. Inductive learning was what let me discover the topics, but deductive learning told me why those things were true. Combining the two methods of learning was helpful, because it went from discovery learning to analytic learning. The two together helped with complete understanding of the concepts. The fundamental theorem of calculus is very fundamental, because it is the basis for a lot of the calculus we have been doing and will be doing. We can describe a lot of things we haven't been able to in the past. Once we understand the theorem, it will also allow us to understand previous topics and upcoming topics better. The theorem shows us the net area of a graph. This can describe the behavior and relationships that graphs share. It will help us learn more about a graph. We will think of graphs as more than just lines, and we will be able to describe them better. Using a combination of all the concepts in calculus and the fundamental theorem, we will be able to understand equations and graphs at a higher level.
This week in AP Calculus, we continued learning about how to calculate the area under the curves of functions. We got much more in-depth this week, since we had only just begun learning about this topic before break. We also learned about how to calculate the definite integrals using antiderivatives. This was taught by hand and in our calculators. Before, we had to draw the function ourselves, and look at the graph to figure out the problem. This process became much easier by learning these alternative methods. This will help us calculate area under curves much easier. The lessons we learned this week finished off the sections, so now we have to study for the upcoming test. I understood pretty much everything this week. I didn't really struggle at all, but I have to look over the chapter and study for the upcoming tests. To calculate an integral using antiderivatives, you calculate the antiderivative. You can calculate the antiderivative of something by adding one to the exponent and then dividing the whole term by the exponent you get. Then, you plug in the intervals and subtract, multiplying by the interval. My participation was miraculous this week. I still need to prepare for the test next week.
This week in AP Calculus, we started to take a look at calculating the area under a curve. By using geometric shapes such as rectangles, we can estimate the area of different things. For math, we are trying to calculate the area under the curve of a function. Using rectangles to calculate the area is not very accurate, so I am sure we will transition into finding a more accurate calculation for the area under the curve. So far, this section only uses basic math skills learned in geometry class. I am understanding this concept very easily. I have not struggled with anything so far, however the chapter just started. To calculate the area under the curve, you draw rectangles with LRAM, MRAM, or RRAM. These are where you have the rectangle intersect the function. You must multiply the height by the interval on the x-axis of the rectangles that you draw. You can calculate the area by adding the areas of the rectangles together. I thought that my participation in class was stellar this week. We are still just beginning to learn about the area under the curve, so I do not really have anything that I need to work on yet.
This week in AP Calculus, we finished up chapter four. We learned more about related rates and differentials and had a quiz over it. We did some more review and had a chapter review assignment. It is important that we went over this stuff because we have a test over chapter four next week. All of this relates to previous topics because chapter tests are over the entire chapter plus previous chapters. I assume this will be leading us into chapter 5. I can only guess what is in chapter 5. I think I understood optimization pretty well. I didn't like related rates as much, but I got most of it. I struggled with a question on the related rates quiz. It confused me a little bit, but I understand how to do it now. To figure out a related rates problem, you have to set up equations that involve what you know from the problem. You take the derivative of the equation and plug in anything that you can. Since you took the derivative, you can now solve for any missing rates. My participation was really great this week. I still need to prepare for the test next week. http://www.textbooknbeyond.com/index.php?main_page=product_info&products_id=21542
This week in AP Calc, we applied what we know about derivatives to solve story problems. We found the derivative after setting up some equations to solve the problem. By doing this, we are able to optimize the problem in whatever way it wants us to. The derivative gives us critical points. Through this, we can determine the minimum and maximum of the problem. By using this method, we can do things like maximize the volume of a box. What you must do is set up the equations for volume of a box. Set up the variables you know. Find the derivative. Solve for the unknown variable. Plug this back into the equations if needed. The quiz over this went fairly well for me, but I think derivative applications was a fairly difficult chapter. We are now looking at rate problems. In these, we must understand the problem and how far and at what rate things are traveling. We must then differentiate the function with relation to time. Plugging in what you know will give you what you need to know. This is basically just a different type of application problem. We'll see how it goes, but I feel pretty decent about it. This week, we learned about optimization. We learned about both maximizing and minimizing whatever amount we are looking for. At the beginning of the week, we did an activity on finding the shortest distance from point A to point B. In these problems, there is usually a factor that prevents full optimization. In this case, we could not travel through cars. We had to go around them. This meant we had to find the shortest route, but it could not be a direct route. Then, we started learning about other types of applications relating to optimization. By setting up two equations, we can use substitution and the derivative to calculate maximums and minimums, representing maximization and minimization. This is important because we are finally using derivatives to solve real-world problems. Before, we were just looking at derivatives, what they were, and analyzing graphs based on that information. Now, we are applying what we learned about the derivative to solve application problems. This could be leading us into solving other types of application problems. We only really learned about applications of derivatives this week. I understand derivatives pretty well so I should be able to figure out the application problems, even though I am not amazing at them yet. Application problems and story problems are not my best types of problems so we will see if I struggle with them or not. To solve an application problem, you need to understand the question, set up some equations, and substitute based on what the question is asking. My participation was breathtaking this week, to say the least. I still need to work on more application problems.
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February 2015
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