Second semester

I feel like I have done almost the best I could have done this semester. I was almost perfect with doing my homework, although there were a few times where I got behind. I got all my blogs done which is pretty cool. There are some tests that I feel I gave my best effort on and there were others where I feel that I could have done better. I took good notes at the beginning of the semester but then I slowly started to stop. I think I did a pretty good job on my presentation as well. Overall though, I feel that I had a pretty strong second semester in Honors Math Analysis. 

Trig review week

This week we covered some of the basics of trig. A large part of trig is slaving and proving problems using the trig identities. There are multiple identities that are used, but the cool thing about them is that they can be moved around and formed to suit the user's needs. The point of proving something using trig is not of find the answer, because it it right in front of you. The point is to show how you get to the answer and what steps must be taken. This type of work is almost like a puzzle. There is not much arithmetic involved, so it feels like you are not even doing math. 

Parametric equations

Parametric equations are equations that express a bunch of different functions. When graphed, these equations can turn out to look like things ranging from a circle to a butterfly. Different types of parametric equations consist of circles, ellipses, hyperbole, parabola, spirals, butterfly curves, and many more. When graphed, these equations are not only forms of math, but art as well. There are infinite ways in which parametric graphs can look. 

Partial Fractions

Partial fractions are fractions in which the denominator consists of a variable adding or subtracting a number. The numerator of these fractions usually contain whole numbers. To add or subtract two partial fractions, you must find the common denominator, usually leaving you with quadratic equation on the bottom. You can Lao do this process in reverse. You would accomplish this by factoring the bottom of the fraction. 

Repeating decimals

Repeating decimals in essence are decimals that go on for ever. To convert a repeating decimals into a fraction, you would take the number after the decimal that is repeating, and put it over the number nine. For example, 0.7777 is 7/9. If, however, there are two numbers being repeated, the you would put them over 99. For example, you would write 45/99. If there is a number before the decimal, you would just add it to the fraction. 

Sequences and series

A sequence is essentially an ordered set of numbers. A very simple example of a sequence is 2, 4, 6, 8... A series too is an infinite set of numbers. Series can either be geometric or arithmetic. Geometric series are numbers that are multiplied by a certain number each time. Arithmetic series are similar, but they are just numbers that add or subtract a number each time. A series can have an endpoint and a starting point.

 

Parabolas

Parabolas are essentially graphed as "U"s. Any given point on a parabola is equally as far from the focus as it is the directrix. All parabolas are symmetric about their bisector or axis of symmetry. The equation of a parabola contains a y^2 and an x, or the other way around. The vertex of a parabola is the point at the very top of the bend. The can either be the highest, lowest, farthest left, or farthest right point of the graph, depending if it's positive or nevagtive and vertical or horizontal. Parabolas are used greatly in physics, such that they display the natural tendencies of gravity. 

Tower of Hanoi

When playing the tower of Hanoi puzzle, I noticed that to finish, one had to utilize all three pillars. I did not really find any specific rhythms or strategies for finding the fasted solution, but I did notice that one had to go against their instinct to complete it. Mathematical induction is seen in this puzzle through the reoccurring steps that include using all three towers. This proves that to get all the rings on the last tower, the middle tower must always be used. This is an inevitable  pattern that occurs whether there are 2 rings or 6. 

Graphing systems of inequalities

Linear inequalities are graphed as lines. There are usually two to four sets of linear inequalities in a graph. When shading in parts of the graph, one must plug in a single x-y coordinate to each equation in the set. If the result is true, then you shade the side of the line that contains the point. If it is not true, then you shade the opposite side.  Each line must go through this shading process.  Whichever potion of the plane contains shaded areas from both the lines is the "answer". 

Cramer's rule

Cramer's rule is used to solve for the varisbles in a system of linear equations. It starts by finding the determinate of a three count set. Once the determinate is found, you replace the collumn of whatever variable you are trying to find with the answers of each equation.  You then find the determinate of the new equation. If one was solving for x, they would put the outcome of Dx/x, or the determinate of x over the determinate of the original set. To solve for all,the variables, you just repeat this process for y, z, and so on. You write the answer in an ordered pair, such as (4,5,3). 

Systems of equations

Linear equations graph as straight lines. The simplest of linear equations have two variables and two equations. These quotations can be soved in one of three ways, substitution, elimination, or cramer's rule. Each linear equation has what is called a determinate. They are tested by plugging in an x-y coordinate, and if it is true, then that side is shaded. I it is not, however, then the other side is shaded.

 

Rotating conic sections

Conic sections are second degree polynomials. The equation of an expanded conic section is Ax^2+Bxy+Cy^2+Dx+Ey+F=0. If B is zero, no rotation takes place. If, however, it is not zero, then it is rotated about the axises. The center of the graph will never change. It can rotate up to 360 degrees. Comics can be parabolas, hyperbolas, or ellipses. 

Graphs of Polar equations

Like most other types of equations, there are many graph shapes that fall under the category of polar equations. The simplest of these are the line through the orgin or just a normal circle. Others that fall under this category are spirals, cardioids, lemniscates, limacons, and rose curves. An element that all do these graphs have in common is the use of theta, with the exception of the circle at the orgin. While the appearance of circles, lines, and spirals may seem obvious, the other four are a little more complex. Cardioids are essentially heart-shaped graphs that go up and down for sin and left and right for cos. Limacons are graphs with either an interior loop, a climpled side, or just convex, depending on the value of a/b. Rose curves are basically graphs that look like flowers. Their number of pedals is dependent on the value of n. Lemniscates look like an infinity sign, and are either on the x-axis for cos or diagonal for sin. 

Picture

Polar Coordinates

Polar coordinates are essentially points points on a graph. They are not, however, plotted on a traditional x-y plane, but a circle. Also, the points are not listed as (x,y), but rather (r,ø). The R sands for radius, and the ø stands for angle. These points, however, can be converted to (x,y) terms by going through specific equations for each coordinate. 

2nd semester blog

Last semester I thought that I did well on taking notes, doing my homework, and preparing for tests. This semester, however, I would like to be more on top of my blogs, know when upcoming quizzes are, and study more even when there is no upcoming test or quiz. My favorite Christmas story was the first Monday of break me and by brother went hiking and I got poison oak all over my body and I had it for two weeks. Just kidding that was terrible, I enjoyed snowboarding almost every day though.