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.