Websites:
slader.com
appcalc.com
This week we finished up with the last and final section to chapter 7. We learned another application technique involving integrals called the "shell method." It is important to remember that it is possible to use washer, disc, or shell methods on a variety of graphs. However, how do you decide which one to use? This is the question I tend to ask myself constantly when solving these kind of problems. Honestly, this is the hardest part for me with dealing with chapter 7, is deciding exactly what equation to use for the certain type of graph. But with practice and drawing in the rectangle somewhat helps me figure it all out. Overall, I would have to say the easiest method for me to use is the shell method. However, what can sometimes trip me up with the shell method is the P(x) value when it is not getting rotated exactly on the x axis or y axis. I get tripped off whether it is x+a a-x x-a ect. I hope for the chapter test I will be able to master this skill better. What I found easy in this chapter is setting up the equation and finding the values in general for the shell method. When you have to rotate the equation over horizontally you must remember it must be in f(y) unlike when you use disc or washer method you do just the opposite. Same goes for vertically, you must take f(x). Once you do this it is fairly simple after you plug in the expression multiplied by the P(x) value and integrate it by the (a vs b) or (c vs d) multiplied by 2 pi. In other words it seems complicated but after you figure out where you are rotating it across and the p(x) value it is fairly simple to just plug in. Next week we will be continuing to review this chapter and wrap up the final lessons for the class before we dive into the AP review. I will not be taking the AP test however I still need to be conscious about my ability to gather and absorb all material learned.
Websites: slader.com appcalc.com
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This week we continued to learn more about chapter 7. We dove into section 7.3 that dealt with revolutions of solids. It is very important to understand the integrals from previous sections of when to integrate either from x or y when setting up the equation formula for solving the volume. We learned two methods in particular. One method we used was called the "disk method". This was the easiest method because all you needed to find was the integral you were integrating and the function to plug into the equation. The second method we used is called the "washer method". This method is very confusing to me. I was absent the day it was taught so I have been a little fuzzy on this topic. However, I get confused of which function/value is the larger R and which one is the little r by looking at the graphs. I am a little nervous going into a test on Monday and being confused on the washer method. For next week I hope to get a better understanding of the "washer method" to solve equations. I believe next week in order to successfully do well in the class I will have to make sure I understand the revolution methods well. The quiz we had earlier in the week I felt I was a little confused on. However, after learning more about these kind of solids on graphs I believe I have a much better understanding therefore should be able to complete a mastery on that quiz pretty well. I believe my participation went well this week. I was engaged in the CCCs and contributed towards my work and helping others when we had a sub. Next week I believe we will learn other methods in solving these kinds of equations with revolutions.
These websites were helpful: slader.com solidsrevcalcclassroom.com To start off with the final (third) trimester of this course we began looking at chapter 7. Chapter 7 is the last chapter to be learned for the Calculus exam. Because of the weird schedule with cancellations of delayed exams we went at a little bit of a slower pace with the two first sections. The first section, 7.1, dealt with applications of integrals. This included interpretation between velocity, acceleration, displacement, ect. Because most of these problems were story problems, I struggled with it the most. For this section it was important to remember the standing, velocity, acceleration, and juel rules for derivatives when solving the story problems for anti deriving. However, i grasp the concept of finding the total distance well. i understood the reason for the absolute values of the integral of velocity to find the the total distance. The reason for the absolute values is because you are adding the positive and negative values not just the displacement mechanism. In 7.2 we dove deeper into integrals but ones where we had to solve for their areas within a plane. These areas varied between finding them inside bounded areas. For solving these problems it was important to remember a few specific rules. I had a little trouble with the problems that had you integrate with respect to y instead of x. I got confused on what places to look for the upper and lower bounds to plugged into the equation. However, with practice with the homework, I got more used to dealing with such complex problems and it began to get easier to grasp. I believe my participation went well this week, and I was engaged to ask questions to problems that I needed extra help on. Next week, I know we will have a quiz over 7.1/7.2 and will continue working on further sections within this chapter. I will have to try my best not to fall behind and get caught up with "senioritis" and stay focused in class. This week we finished up with the chapter by taking Chapter 6 Test. Chapter 6 works a lot with anti-substitution and derivatives to solve for equations and expressions. On Monday we continued reviewing section 6.4. Because I was not there last Friday I spent my time learning the rules for exponential growth with anti-derivatives. On Tuesday we did not have school, due to poor weather conditions. This then, moved the Chapter 6 test to Thursday. On Wednesday we continued reviewing and did a table rotations. These table rotations were a great review for the test the next day and even had some challenging complex questions involved that really made you think about the sections. Overall, I thought the test was not too hard except I had some complications with the spiral questions inside. This is something I could work on for the future, are my skills for all the sections. This will become extremely important for my exam preparation for next week. I felt my participation went well this week in group discussions and rotations. I also felt that I struggled a bit at first with the last section on 6.4 because of my absence but soon fell in to place pretty quick. Next week I am guessing we might dive a little into chapter 7 before we take the exam. I found slader.com to be very helpful with my work this week including mathhelp.com.
This week we finished up summarizing 6.1 and dove into section 6.4. As you can see we skipped section 6.3 because it is not tested in the AP calculus exam. When we first started working on section 6.4 that dealt with solving anti-derivatives with x and y values in the equation, I thought it was quite simple. However as we worked more with this section the problems started to get more complicated. The reason I thought these problems were complicated was because it included a lot of algebra skills involving expotentionals, like ln and e to be able to solve the equations. In order to help me with the steps in solving down to the solution I found slader to be a big help. Slader, was able to show me a full explanation in simplifying the algebra and trigonometric problems. This section in particular connects well with the rest of the units we have learned because it evolves us to use all our anti-derivative and integrating skills to solve. For example the only difference in section 6.4 than the other sections that dealt with integrating for the anti-derivative is simply that there are two variables. With these two variables you must get the x's and the y's on separate sides and then derive it after that, so the dx variable gets cancelled out. Next week I believe we will be reviewing more for chapter 6 before we take the test and jump into Chapter 7. For next week, I need to get more comfortable with solving problems that deal with ln or e. Overall, I thought my participation went well this week in the CCC groups even though I was absent on today, this Friday.
These were some websites that helped me: https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html This week we started working on Chapter 6. We learned section 6.2 first that dealt with anti-differentiation by substitution and started the beginning of section 6.1 that dealt with differential equalities with slope fields. These first two sections are crucial in in calculus because it describes the solving techniques with anti-differentiation for indefinite integrals. Indefinite integrals are simply all the anti-derivatives of a function. This can be described as a family of functions, which would be explained as the same F(x) term however just a different constant of integration in other words, the C value would only change. These sections refreshed my memory on u-substitution we learned in the first trimester. When I first learned the rules to u-substitution I was fairly confused. However when I relearned/refreshed my memory this time around it "clicked" faster and I was able to successfully go through the process and solve the problem a lot faster than when I did it in the first trimester. This realization reminded me in the importance each section I learn in this class plays. Each section or chapter I learn is a building block and will come back again in calculus including the AP assessment. Therefore it is important for me to focus on understanding ALL concepts in this class so I will be able to stay successful in this calculus course. I felt I understood these two sections fairly well this week. Overall I thought they were quite simple once you were able to get the process down. However, in some of the more complex problems in section 6.2 I got a little thrown off at. When there became more complexity involved in solving the solution it became more difficult to identify the u substitution value. However, I was able to have access to slader.com that was able to help me find this value when I came across a stump. Overall I felt my participation went well this week because I was engaged in the CCC conversations and made sure to fill out the logs when available. https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/USubstitution.html This week we continued working with section 5.3 that dealt with definite integrals. I 5.3 we learned the many different rules with dealing with definite integrals. Section 5.3 also contained the Mean Value Theorem for definite integrals that states when f is continuous at [a,b] then at some point c in [a,b] f(c)=1/(b-a) on the integral [a,b] f(x)dx. What this means is that if you have the equation of the function and the integral amount you can easily find the area it processes under the curve by plugging the information in to the Mean Value Theorem formula. We also learned a second strategy for solving the area under the curve by using the antidervatives. For this the formula is on integral [a,x] f(t)dt=F'(x)-F'(a). Throughout section 5.3 these formulas came very useful to know for the completeness of my homework.
I found section 5.3 to be fairly easily along with the other 2 sections in this chapter. I would say the hardest thing I learned this week was practice with AP style questions. On Wednesday we had AP style quiz format, that along with my other classmates struggled on. I felt I understood the sections but the wording messed me up on the test to where I wasn't exactly sure what the question was asking. If I decide to take the AP test at the end of the year I need to continue practicing and improving on those kind of questions so I will be prepared to take the test. We also had some practice dealing with AP style format questions. I believe my participation this week was well. I felt I engaged in group discussions and completed all the assigned homework with the CCC. Next week I believe we will continue working with Chapter 5 dealing with the properties of integrals that connects to calculus. I hope to continue getting extra practice on AP style format questions to work with also. These websites helped with my learning this week: This week we began Chapter 5. Chapter 5 consists of definite integrals. This week we explored the first two sections of it that dealt with estimations of the area under a curved graph. The previous section that dealt with derivatives comes into play in this chapter. According to the book we have not learned the way to find anti dervatives, however because we skipped ahead and learned it before it helps understanding more clear on the purpose of integral area estimates. The purpose of this week was being able to find approximate areas for a "curvy thing." This week we dove into section 4.6 that dealt with Related Rates. This section is similar to the set up in the previous section in 4.4, Optimization. Using the techniqes in 4.6 helps relate rate problems in calculus by solving for an equation's dervative down to the solving process.
To simplify, there are five steps to solving rate problems successfully. The first step is to draw a picture of the problem. Doing this it will help mentally inscribe what you are "actually" trying to solve. The second step involves finding a model to illustrate your picture. This is where you would come up with an equation made up of variables. You do NOT plug in the values yet, that is to come. The third step is then to find the dervative of the equation in step two. Step four, you organize all the data from the story problem. This is where you would come up with the variable you need to solve and the values for the other variables. The last and final step involves simply just plugging the values in from step 4 into step 3. Watch out, you do NOT want to plug the values back into the original equation (step two) but the dervative one (step 3). I found this week to be a bit challenging, however a good challenge. I was able to keep up and follow each problem. Usually, I struggle with story problems. However, I am becoming more confident in dealing with them, which I believe is helping me grasp this section's concept faster. I believe my participation went well this week. I was involved in group discussions and asked questions when needed. Next week, I am assuming we will have a chapter test. Therefore my goal is to keep working out the more difficult problems and reanalyzing the process to each story problem to do well on the test. These websites were helpful to my understanding this week: http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx https://www.khanacademy.org/math/calculus/derivative_applications/rates_of_change/v/falling-ladder-related-rates This week we dove into section 4.4 that dealt with optimization. This consisted of various topics of story problems that were all associated with finding derivatives to solve them. In other terms, everything I have learned about the process of derivatives was used to solve equations used in complicated geometry, algebra, trigonometry, ect.
At first I struggled with the story problems. However as the week went on, with more practice I became a lot more confident and started to grasps the concepts better. I understood when to find the maximum or minimum capacity of an object fairly well. The process in solving for these equations is to find an equation in where you are able to substitute one variable out to plug in to the second equation. When you do this you are now only left with one variable. From this you are able to solve the equation right away by graphing the function and finding its min or max point. If you want to challenge your mathematical skills you can also find the answer by solving for the function's derivative to interpret its zeros. From knowledge of the previous trimester you know that zeros of the derivative are the max/min of the function. To find which point is the max and which one is the min you do the testing method of increasing vs. decreasing of the number line to figure it out. I believe my participation this week was well. I enjoyed going over the story problems one by one from the assignment in those two days. I found this strategy to be helpful, and it really allowed me to understand the methods behind each story problem. I need to continue working on my basic algebra skills to help with simplifying my answers by hand when solving complicated problems. Next week, I am assuming we will continue working in chapter 4 and find new techniques and strategies in analyzing and solving graphs using derivative properties. These links I found helpful with this section: http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx https://www.khanacademy.org/math/calculus/derivative_applications |