Tower of Hanoi-Mathematical Induction #10

In this project, I learned how to do mathematical induction. It was difficult to get all the numbers from the rings, because my screen kept freezing, but when I did get them, I did not realize how many moves it took to complete each step. I decided to move the first ring to the ring beside it because that was the only available place. For me, this game just seemed to be to put the ring wherever it fit, there was not rhyme or reason. Mathematical induction helped us to prove that the equation we came up with for the project was true, because when we plugged in certain variables, they worked. Step one proved that n=1 which was helpful, because we then knew we were on the right track. In step two, we proved that (k+1) was true, therefore proving that our entire equation was true and proven correct. I learned that the recursion formula can be very helpful in terms of plugging in that set for a longer part of the equation. We know that using mathematical induction can help prove something, because if we plug a 1 into each side and they come out equal, then our equation is correct. 

Cramer's Rule #7

Cramer's Rule is a lot easier to use when finding the determinant of each variable. In order to find each determinant, first find the common determinant. Then, substitute the solutions given for the first column and solve for the determinant. Next, substitute the solutions given for the second column and change the first back to its original numbers and solve for the determinant. To find the determinant of X, put the first determinant you found over the common determinant. To find the determinant of Y, out the second determinant you found over the common determinant. 

Systems of Equations #6

An inconsistent equation means there is no solution and the lines are parallel. A consistent equation means that there is an answer and if there is one solution, then it is independent, but if there are infinite solutions then it is dependent. There are two methods to solving equations; substitution of elimination. The first step for substitution is to solve one equation for one variable. Then,substitute that variable into the other equation. Next, solve for the variable. Lastly, go back and substitute to find the second variable. The first step for elimination is to interchange any two equations. Next, multiply by a constant number. Finally, add one equation to the other to eliminate a variable. 

Graphs of Polar Equations #5

To find the vertices of an ellipse or parabola, use theta=0,pi. To find the vertex of a parabola, use either theta=0 or theta=pi. These are for the horizontal axis, cos. For the vertical, sin, in order to find the vertices of an ellipse or parabola, plug in theta=pi/2 and 3pi/2. To find the vertex of a parabola, use either theta=pi/2 or theta=3pi/2. Also find x or y intercepts. 

Polar Coordinates #4

In the coordinate (r, theta), r stands for the radius. If theta>0, then you rotate counterclockwise. If theta<0, then you rotate clockwise. When converting from polar (r, theta) to rectangular (x,y), substitute x=rcostheta and y=rsintheta. When converting from rectangular to polar, substitute r^2=x^2+y^2 and tan(theta)=y/x. For both of these conversions, solve for x or r. Trig identities are also key to this process.

Rotating Conic Sections #3

The starting point for rotating conic sections is to use the formula Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Identify all parts in order to being step 1. The first step is to find the angle using cot2(theta)=(A-C)/B. Step two is to replace x and y with either x=x'cos(theta)-y'sin(theta) or y=x'sin(theta)+y'cos(theta). The final step is to use algebra and simplify! A helpful hint is to use the different trig identities used previously so it will be easier to replace things. Rotating conic sections can be tricky, but the trickiest part is to not mess up on the basic algebra steps.

Parabolas #2

Parabolas are U shaped graphs. They have a focus, which is right above the vertex and that distance is called c and is (h,k+c) . The vertex is the center of the parabola such is (h,k). The directrix is y=k-c and is the line below the vertex. The parts described are for a graph that has a c>0, or positive. If the c<0, or negative, then the parts are reversed. The basic equation for a positive parabola is (x-h)^2=4c(y-k) and the basic equation for a negative parabola is (y-k)^2=4c(x-h).