First Semester Calculus Reflection
Over this semester, I have gone from pre-calculus to calculus and one of the biggest struggles I encountered during this process was catching up with a class that was already a few weeks ahead of me. The main subject that I missed was limits, and I did not even encounter this subject until late in the semester while studying for the final. At first catching up with the class was relatively easy, but the hard part came at the end of the semester when I realized I had missed an entire subject during the class. It was definitely the most challenging part of the class for me.
On the contrary, one of my accomplishments this semester was accomplishing a problem of the week on my own. The problem had a simple concept, the question was: on any given circle, three points are randomly placed on the edge of the circle, what is the probability that the triangle these points create encompass the midpoint of the circle. After some experimenting at lunch I came to a simple solution. First I decided, the farthest a point can possibly be from another is 180 degrees, and the closest these two points can be is 0 degrees. The sum of these two values is 180 and the sum of any two corresponding angles will also be 180 degrees. This is significant because the average number of degrees two opposite angles create divided by two (the amount of values) is 90/360, or ¼. Therefore the probability that the triangle that three randomly placed points create a triangle that encompasses the midpoint is ¼. The reason I find this problem solving aspect of myself as important is because I know that this is a strength I will be able to apply for the rest of my life. When I am expected to know something that I don’t I can problem solve my way out.
Another strength of mine was the process of finding a derivative, or differentiating. Differentiating is the process of finding the slope of an equation at every point on the graph. To do this one must use the chain, product, and quotient rules. I practiced this and was successful in grasping the concepts. I find these concepts useful because in the real world we can use derivatives to find the acceleration and rate of change of velocity. I am interested in majors that include physics and these are important concepts to be able to grasp in the field.
Next semester I will expand my knowledge of math even more. I know that one concept we will be covering is integrals. Next year and in college I will continue to pursue mathematics and possibly obtain a major in math or science.
On the contrary, one of my accomplishments this semester was accomplishing a problem of the week on my own. The problem had a simple concept, the question was: on any given circle, three points are randomly placed on the edge of the circle, what is the probability that the triangle these points create encompass the midpoint of the circle. After some experimenting at lunch I came to a simple solution. First I decided, the farthest a point can possibly be from another is 180 degrees, and the closest these two points can be is 0 degrees. The sum of these two values is 180 and the sum of any two corresponding angles will also be 180 degrees. This is significant because the average number of degrees two opposite angles create divided by two (the amount of values) is 90/360, or ¼. Therefore the probability that the triangle that three randomly placed points create a triangle that encompasses the midpoint is ¼. The reason I find this problem solving aspect of myself as important is because I know that this is a strength I will be able to apply for the rest of my life. When I am expected to know something that I don’t I can problem solve my way out.
Another strength of mine was the process of finding a derivative, or differentiating. Differentiating is the process of finding the slope of an equation at every point on the graph. To do this one must use the chain, product, and quotient rules. I practiced this and was successful in grasping the concepts. I find these concepts useful because in the real world we can use derivatives to find the acceleration and rate of change of velocity. I am interested in majors that include physics and these are important concepts to be able to grasp in the field.
Next semester I will expand my knowledge of math even more. I know that one concept we will be covering is integrals. Next year and in college I will continue to pursue mathematics and possibly obtain a major in math or science.
Problems of the week #7
Problem: Given a circle radius 1 select three random points along the edge of the circle. What is the probability that the center of the circle is contained within the triangle made by these points.
My Solution:
1. To solve this problem the first step is to assume that the first point down will always be at (0,1). This is because there are no other given points of reference yet on this circle.
2. The next step I took was analyzing how the closest distance that point 2 could be placed to point one is 0 degrees. The farthest points 1 and 2 can be from each other is 180 degrees. If point 1 and 2 are 180 degrees apart, that point three will result in creating a triangle that contains the midpoint is 100%. If point 1 and 2 are 0 degrees apart the probability that the triangle created by point three is zero.
3. The sum of 0 and 180 degrees is 180, as well as any set of supplementary angles. The average of these numbers is 90 degrees. If we put 90 degrees over the total number of degrees in our circle 360 we get the fraction 1/8. This is the probability that a triangle will be created by three points along the circle that include the center of the circle.
My Solution:
1. To solve this problem the first step is to assume that the first point down will always be at (0,1). This is because there are no other given points of reference yet on this circle.
2. The next step I took was analyzing how the closest distance that point 2 could be placed to point one is 0 degrees. The farthest points 1 and 2 can be from each other is 180 degrees. If point 1 and 2 are 180 degrees apart, that point three will result in creating a triangle that contains the midpoint is 100%. If point 1 and 2 are 0 degrees apart the probability that the triangle created by point three is zero.
3. The sum of 0 and 180 degrees is 180, as well as any set of supplementary angles. The average of these numbers is 90 degrees. If we put 90 degrees over the total number of degrees in our circle 360 we get the fraction 1/8. This is the probability that a triangle will be created by three points along the circle that include the center of the circle.