This week we finished up the limits and continuity sections with the chapter test on Tuesday. I appreciate you giving us an extra day to go over the review before the test. I felt that the activity allowed me to participate in really understanding the concepts of limit and continuity problems. On Wednesday we started a new lab that involved the introduction of derivatives. Mathematicians use derivatives frequently in calculus. From our discussion on Wednesday in class, I learned that knowing the limits of a function will help measure the derivative of the function. Therefore of what we learned in the second chapter will come in great handy for this new chapter involving derivatives. This is because you can not use the value 0 but instead you can use numbers very close to the number zero to find the values. Right now I do not know a well enough understanding of what exactly a derivative is except the fact that it is considered a slope of a point at a certain time. Next week I am assuming I will learn more about derivatives and knowing what their purpose serves. When I looked up what a derivative was the explanation that was given to me was the measure of a function in relationship to its change in input. This would make sense in why it would involve knowing the slope when you zoom up on a point like we did in the lab. There are special formulas involving derivatives I am assuming I will learn in the future that I found on http://mathworld.wolfram.com/Derivative.html. This website helped me understand the meaning of derivatives better. This week I felt I participated well in class by working in my group during the lab and in discussions. I felt I understood the concepts in chapter 2 well and I struggled the most with derivatives. This is to be expected though because I only worked with derivatives for one day in which we didn't even get to finish the lab (since I was gone on Friday). However, my goals for next week are to understand and learn more about derivatives and their relationships within a function.
This week I continued to learn more about limits. By learning how to solve limits in forms of continuity and infinity allows you to interpret values with holes, assympotes, ect. These sections were a review of what we learned last year in pre calc. My knowledge expanded in learning how to solve limits involving infinity. I also got a better understanding in knowing how to solve limits without the use of a calculator. Learning how to solve these problems algebraically is an important skill that will come into handy with future sections when I have to find limits and understand graphs better for more complex problems. Telling whether functions were continuous or not was an easier subject for me. I understood the difference in a removable discontinuity, jumps, and infinite. Graphs that involve assymptotes will be a continuous function. How so? This is because the definition of a continuous function is that every point in the DOMAIN has to be continuous. Therefore the assymptote is not in the domain. However, this does not mean it is a function. For more information in continuous functions I checked out other methods at http://mathworld.wolfram.com/ContinuousFunction.html. The thing I struggled most this week was probably the quiz we had over limits. I thought I had a good understanding of limits, but I did not do so well on the quiz. I made errors, that I realized I had made after I took the test. In the future I need to get better at thinking what the question is really asking in conceptional thinking. In problem 69 in section 2.3 I had a hard time trying to figure out what the question was asking of me. However, by participating in peer discussions I grew to understand how simple the problem really was. All I had to remember was that you can take a log of something X to the n power and make it n log x. From this technique it made the problem into a basic algebra to solve. I believe I participated well this week, which allowed me to understand the concepts better. For next week I really want to do well on the test and if we have another quiz to make up for the previous two quizzes.
This week after we took our review quiz we learned the basics on limits. The limits lab was a good review that helped me rewind the concepts of limits from last year. I was able to grasp the concept on limits better without the relying heavily on my calculator. I found the rules of limits to be very simple, almost like common sense. These limit rules like the constant, identify, sum/difference, product, quotient, constant function, and power rules were similar to basic algebra rules. Therefore basic algebra skills can be very helpful when finding certain limits. This method comes in handy when the denomonator happens to be zero in a function where you would not be able to directly substitute in to find the limit. The expression when the limit of (sinX)/(X) as X approaches 0 equals 1 will be very helpful I believe in future sections to find limits more easily when working with cos, sin, tan, ect kinds of functions to simplify in finding limits. The problems I best understood this week were the direct substitution ways to find limits instead of the algebraic. I found it easier just to plug the values in rather than making a mistake with my algebra skills. However, I need to become better with using the algebraic method for finding limits because not only will help me get an answer write on the test but is a good way to double check my answers. One and two sided limits state that when f(x) has a limit as x approaches c if the right and left hand limits both are equal (in symbols.) Therefore for a function to be continuous it must be said that when f(x) x=x(0) is defined at x0. From understanding the relationships of limits to be either one or two sided of where the values approach from you can really get a true understanding of the graph itself. To go even more in depth with different kinds of functions including continuous you can go visit http://www.mathamazement.com/Lessons/Pre-Calculus/11_Introduction-to-Calculus/limits-of-functions.html. I also found this website http://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx helpful with the limits section.
9/3-9/6
I learned a lot of things in my first week back in math. These topics were reviews of what I have learned in previous years. However, I found it helpful to review and go over these topics to get back in the groove of math. The problems and discussions we had this week all dealt with basic calc concepts to prepare and warm us up for what we will be tackling in AP Calculus. One of these basic concepts I understood were the problems dealing with finding zeros. I found this topic to be easy because I have done lots of practice with the c-quad formula to foil and find the x values. The volume section I also found to be easy. I remembered the volume formulas for different shapes and was order to easily figure out these values. However, I learned a new way to calculate volume for a variety of prisms. Instead of breaking up a complicated shape into each of the volume components and adding the parts up, I learned that it is the same process as simply just multiplying the base of the trapezoid by the height. I also was reminded of the interval notation meanings of the different signs. The box parentheses represents that the value touches and the curved parentheses means that the value does not touch. There were some things I struggled to remember in the review. I could remember doing the problems from last year, but the set-up I was struggling with. One topic in particular I noticed this with were the similar triangle story problems. I had a difficult time separating what the question was asking and the information that was being provided. For next week or before the test I need to heavily review and study the importance of relationships between similar triangles in order to tackle those problems. Another topic that was reviewed this week were the comparisons between relations and functions. The definition for a function is for each input there is exactly one output. Therefore not every relation can be a function. If a relation has two domain factors that have different outputs it CANNOT be a function. There are also other strategies that can be used to determine if a function is not a function but perhaps a relation. One way is done by a vertical line test. If one draws a vertical line and it crosses two points then it is NOT a function. Another strategy that can be used is to solve the equation out algebraically. If the values end up not being real numbers and imaginary then it is not a function. The definition for a relation is just simply a comparison between two things, like the domain and range, it is not a specific relationship like functions are. Overall I would say I participated well in class. I frequently asked questions when needed and was not afraid to ask for help. I also worked in a group of 4 on our math packet where we all contributed and discussed solutions and ways of solving the questions. |