Average Rate of Change
Cover Letter
These past two weeks, we have done various assignments: from finding the population, to learning about paradoxes, as well as finding the speed of a falling object. One thing that all of these have in common would be that we could find the rate of these. Rate is the speed of an object, or the distance divided by time. To use rate in populations, which we used in How Many More People? and Comparative Growth, we had to determine how much the population was changing by, and dividing the change in the population by how long it took to create the change. With this information, one could have a rough estimate of how much the population would increase over a set amount of time. Although, with this information it doesn't take into account different variables that could drastically change the outcome, whether it be plagues, World War's, or simply the bad economy. Through these problems, we were able to find out that even if the slope is steeper compared to another, that doesn't mean the rate is greater, one must take into account the time that it takes to get there. If one looks at Comparative Growth, one would assume that the first graph's rate would be larger due to the steeper slope, but it ends up being the second one because of the shorter length of time it took to increase that much.
Paradoxes were used to discuss how one could find the most exact time to how fast an object is at a certain time. Or, in general, a statement which leads to an answer which seems illogical or never ending (ex. Trust me, I'm lying). The San Diego Cedar Fire problems was where there was a slight reference to a paradox in the last question on the back page. To try and find out the speed of an object before it hit the ground, one would have to keep shortening the time, that is taking into account already knowing how long the object will take before reaching the ground, until the answer become zero due to the mathematics involved. For the problem, the closest that we were able to get was .16 seconds, by finding out the height of the object off the ground divided by the time it took to get that far. This was what we had learned as a class these past two weeks.
These past two weeks, we have done various assignments: from finding the population, to learning about paradoxes, as well as finding the speed of a falling object. One thing that all of these have in common would be that we could find the rate of these. Rate is the speed of an object, or the distance divided by time. To use rate in populations, which we used in How Many More People? and Comparative Growth, we had to determine how much the population was changing by, and dividing the change in the population by how long it took to create the change. With this information, one could have a rough estimate of how much the population would increase over a set amount of time. Although, with this information it doesn't take into account different variables that could drastically change the outcome, whether it be plagues, World War's, or simply the bad economy. Through these problems, we were able to find out that even if the slope is steeper compared to another, that doesn't mean the rate is greater, one must take into account the time that it takes to get there. If one looks at Comparative Growth, one would assume that the first graph's rate would be larger due to the steeper slope, but it ends up being the second one because of the shorter length of time it took to increase that much.
Paradoxes were used to discuss how one could find the most exact time to how fast an object is at a certain time. Or, in general, a statement which leads to an answer which seems illogical or never ending (ex. Trust me, I'm lying). The San Diego Cedar Fire problems was where there was a slight reference to a paradox in the last question on the back page. To try and find out the speed of an object before it hit the ground, one would have to keep shortening the time, that is taking into account already knowing how long the object will take before reaching the ground, until the answer become zero due to the mathematics involved. For the problem, the closest that we were able to get was .16 seconds, by finding out the height of the object off the ground divided by the time it took to get that far. This was what we had learned as a class these past two weeks.
Reflection
When doing these problems, one habit of a mathematician that I feel that I embodied was generalizing. I used this habit because I easily understood the problems given to me and continuously used the equation to find the rate of an object for these problems. However, these past two weeks I feel that I have not been doing the best that I could showing and describing my work. This would mean that if someone looked at my paper to try and understand the process I did, they would have a difficult time due to the lack of work and the lack reason behind it (when I try to figure out a problem, I try multiple ways, so that whenever I am done, it is difficult to understand what had occurred). When working as a group, the effort I did, and behavior in class is taken into consideration, I feel that I excelled. I was able to understand these questions easily, so I was able to help other classmates when they needed help to do it. I participated from time to time, sharing the work that I did and giving my 'two-cents' about the question and answer. This is why I believe I deserve 10/10.
When doing these problems, one habit of a mathematician that I feel that I embodied was generalizing. I used this habit because I easily understood the problems given to me and continuously used the equation to find the rate of an object for these problems. However, these past two weeks I feel that I have not been doing the best that I could showing and describing my work. This would mean that if someone looked at my paper to try and understand the process I did, they would have a difficult time due to the lack of work and the lack reason behind it (when I try to figure out a problem, I try multiple ways, so that whenever I am done, it is difficult to understand what had occurred). When working as a group, the effort I did, and behavior in class is taken into consideration, I feel that I excelled. I was able to understand these questions easily, so I was able to help other classmates when they needed help to do it. I participated from time to time, sharing the work that I did and giving my 'two-cents' about the question and answer. This is why I believe I deserve 10/10.