Sequences and Series

In sequences and series, you are given a set if numbers in which you must find a pattern that separates the numbers apart. There are two different ways we can identify these sequences. The first is the arithmetic sequence which is separated by addition or subtraction meaning there is a common difference within the pattern that separates the numbers in the set. The second way is the geometric sequence that is separated through division or multiplication meaning that within the sequence, the terms are separated by a common ratio. You find the difference or ratio and you can solve any equation by simply increasing or decreasing from the first/previous term with the term that separates the numbers from each other. 

Systems of Equations

Systems of equations are when 2 or more equations are given based on the amount of variables you have and you need to solve for the variables that are given within these equations. You do this by lineing up the two equations and change one of them through multiplecation so that you can eliminate the variable if the equations were added together. After finding the first variable you plug it in to one of the original equations and from there you can find the next variable. 

Partial Fractions

Partial Fractions can only be done when the numerator has a lesser power then the denominator. If the denominator is larger, you must divide and then use the dividend. With the denominator, you split the equation into ddifferent roots in the process making new fractions with the numerators being A, B, and so forth. You then add up your new fractions so find common denominators and then you place all terms with equal powers next to each other. You then use elimination and substitution to find A and B to place back on to the fractions made earlier in place for A and B or whatever was used. 

Parametric Equations

In Parametric equations, the goal is to sketch the graph and eliminate the parameter. The equation given tells where the starting point of the graph and the direction in which the graph is heading toward. When solving these problems, you need to eliminate the t value by substitution and elimination. When graphing you need to plug in the points for t to find where you are going to display them on the graph. Once you sketch the graph and eliminate the parameter "t", you already know how to solve parametric equations. 

Rotating Conic Sections

The equation for rotating conics, is long, complex, esoteric, and just overall not very fun. Not just the equation, but the whole concept is very advanced. The equations is, Ax^2+Bxy+Cy^2+Dx+Ey+F=0. By the number of letters I'm pretty sure you can see why I think this way. First you need to find the degree of rotation by finding cot2(beta) which will be equivalent to (A-C)/B when 0<beta<90. Then you replace x and y with xcos(beta)-ycos(beta) or xsin(beta)+ysin(beta). This turns them into actual numbers that you then have to simplify till you get a point where you have an equationthat you can graph. 

Graphs of Sine and Cosine Functions

To graph Sine and Cosine functions, you need a unit circle. For graphing the important thing to remember is that the x coordinate represents the beta. The sine graph when the points are plotted looks like a letter s rotated to the side while the Cosine graph looks like a big U curve. The normal equation given for graphing Cosine and sine functions is y = Acos/sin(Bx-C) + D. A represents the amplitude which is 1/2 the overall height. The period is 2pi/B and B represents the compression or stretch of the curve depending on if it's between 0 and 1 or greater then 1. C/B represents the phase shift meaning the left or right movement on the graph. D is the vertical shift or the movement up and down on the graph which also could be found through dividing the average of the minimum and maximum points of the curve. 

Cramer's rule

To solve Cramer's problems, you need to use matrices, and with the matrices, you find determinants. You would do all this when you have 2 linear equations. So if you had the equations 2x + 3y= 4 and 3x + y = 5. Then your first matrix would consist of 2, 3 at the top and 3,1 at the bottom. To find the determinent, you cross the top on the left to the bottom of the right and the bottom of the left to the top of the right and multiply. Then you take the product of the top left to bottom right equation and subtract the product of the other equation. You repeat this to find other determinents by making two new matrices and replacing the column for each with 4 and 5 since those are the answers to the equations. In the end you take the determinent of the 2nd and 3rd matrices and divide them by the orignal determinents. 

Tower of Hanoi

The tower of Hanoi is a puzzle but a puzzle that can be solved easily through mathematical induction. The puzzle is that you have 3 slots with a tower of blocks on the first. The tower on the first as a large base and each block above it is -1 the size of the previous block. So if the base block was 4, the next would be 3 then 2 and the top would be 1. To solve it, you must figure out the pattern it takes to transport each to the 3 spot, while only moving one at a time and placing smaller pieces on bigger pieces only. 

Repeating Decimals

Repeating decimals is a way to find out the decimal numbers that infinitely last like .3333... or .1515... To accomplish this, you use the formula, a1/1-d. a1 = the first decimal place it goes up to as a fraction, so for .3333 a1 would be 3/10 because it's .3 which is 30/100 or 3/10. then d = the difference between the first and next digit. Since .3333 repeats after every decimal, d would be 1/10 since it goes downward by 1/10 each digit. The answer would be found after you multiply the reciprical of the bottom to the fraction at the numerator and in this case it would result in 3/9.

Trig Review Week

Trigonometry to me is like a complex, aggravating, annoying, puzzle that takes constant 100% thinking to solve. In order to solve trig problems, you need to know each of the very many trig formulas. With these formulas, you attempt to algebraically solve each problem by using the trig formulas to cancel out or add onto each equation trying to make them equal. An example of a formula is sin^2 + cos^2 = 1. This is utilized frequently in order to make everything in an equation equal to sin or cos. 

2nd Semester Summary

This semester, I think I really I definitely improved as I focused on acheiving the goals I had set at the beginning of the semester. In the beginning I wanted to complete all hw, not fall asleep in class, and avoid careless errors on tests. Well to follow up on my goals, I completed most of my hw leaving behind approximately 2-3 assignments which is still an improvement compared to last semester. I also only fell asleep once in her class the whole semester, like this here is the true accomplishment since I barely get any sleep. And I also brought my average test grade up to around a B+. So overall I would definitely say I had a better semester then first semester. 

Here's an artsy pic I took a while ago :D

Polar Coordinates

Polar Coordinates are another way to graph dots like rectangular coordinates. The difference is that rectangular coordinates are plotted with x and y coordinates (x, y) while polar coordinates are done through r and beta (r, beta). Beta is represented with a symbol that looks like a zero with a horizontal line crossing the middle. r represents the circles that protrude outward while beta represents the position of the circle it is located. Each polar coordinate can be graphed in 4 different ways, (+, +), (-, +), (-, -), (+, -). To find r with rectangular coordinates, you use the formula r^2 = x^2 +y^2. This form of graph overall is almost equivalent to rectangular coordinates except the graph used for polar coordinates is circular. 

  

My Polar Equation picture

 

Parabolas

Parabolas are weird U shaped lines. They consist of the vertex, focus, directrix, and the axis of symmetry. The equation for a parabola that signifies it's a parabola is (h - x)^2 = (y -k). The x and y will switch depending on if the parabola is horizontal or vertical. The vertex is just the h and k values that would make each side equal to 0. The focus is (h, k+c), the directrix is k- c and the axis of symmetry is whatever x is. The c value comes from c^2 = a^2 - b^2. 

2nd semester goals

In the year of 2014 and the first semester of my junior year of high school, I feel that in my honors math

analysis class I achieved certain goals well. But i definitely need to improve upon a plethora of aspects that ultimately build up my overall grade. To start with, I personally believe I accomplished the hw to the best of my abilities. However on the other hand, I wasn't consistent within my accomplishment of fulfilling all my hw. Another good thing would be that I took thorough, specific notes on most of the lessons. Yet throughout these notes I fell asleep frequently due to lack of sleep the previous night. So acquire more sleep would be a reasonable goal. And finally, I feel for most tests, I was able to remember the material to the best of my understanding. But on these tests, I made careless errors and for some I didn't know the material since I was sick/absent constantly throughout the year. Mainly I need to work on my attendance. 

Off topic my favorite Christmas story over break would be celebrating with friends and family while my favorite traditional Christmas story is the Christmas Shoes.

Here's my family on New Years😋