2nd Semester Summary

This semester I decided to work a lot harder at math so that I could get my C up to a B. I started listening a lot closer in class and asking other people for help. I studied a lot longer for the tests and actually did my homework. This showed in my grades and I was able to get it up. I hope to keep my grade at a b till the end of the school year. I have spent a lot of time on math this semester and I think I have learned a lot. 

Trig Review Week

This week we reviewed verifying, trigonometric equations and the trig identities. Verifying is when you try and get one side of a trig equation to look exactly like the other. To do this you work with one side only and use trig identities to make it the same. It is always best to start with the more complicated side of the equation. We also looked at solving trigonometric equations such as sinx=0.5. The easiest way to solve these types of equations is by making everything into sine's and cosine's. We also reviewed the Pythagorean, double angle, and other trig identities. Here is an example of a trigonometric equation.

Repeating Decimals

Repeating decimals are decimals that go on forever and ever such as .33333333 or .6666666666667. These decimals are also fractions such as 1/3 and 2/3 and there is a way to convert them. The first thing you do is write the decimals as a geometric sequence. For example 0.3 +0.03+0.003+0.0003 etc. Once you do this you can find the rate which is 10 and use that for r. After that you use the equation a1/1-r and you plug in the corresponding numbers. For a1 you would plug in the first number of the sequence which is 0.3 and for r you would use the rate you found so 1/10. Once you simplify this equation you get 1/3. Here is an example of repeating decimals. 

Parametric Equations

Parametric equations express the coordinates of the points of a curve as functions of a parameter. To graph a parametric equation you make a table of values according to the set parameters and plug those into x and y. After you plug them in you can use those points to graph the parametric equation. Another part to parametric equations is eliminating the parameter which is usually t. To do this you can use either substitution or elimination to solve it like a system of equations. Here is an example of a parametric equation. 

Partial Fractions

Partial fraction decomposition is used to break up a simplified fraction. For example it would turn 3x+2/x^2+x back into its more complicated form 2/x + 1/x+1.  To do this you first check if the fraction is improper and if it is you divide it (more on that later). Then you factor the denominator and if it is a linear equation you set up fractions with coefficients a, b, c etc over the factored parts and if its quadratic you use the coefficients ax+b cx+d etc.  Once you get the fractions you can multiply the equation by the lcd and then group equations by powers of x. After grouping them you equate the coefficients and then solve the systems of equations and substitute the answers you get for the a, b, c coefficients in the set up fraction. Here is an example of partial fraction decomposition. 

Tower of Hanoi

On Thursday and Friday I worked on a Tower of Hanoi puzzle.  The tower of Hanoi is a mathematical puzzle that consists of three rods and a certain number of disks that are in ascending order on one rod.  The goal of this puzzle is to move all the disks to the third rod without placing a larger disk on top of a smaller disk.  For example if you have three disks on the first rod the goal would be to rearrange those disks on the third rod.  Although it may sound easy it gets very complicated very fast.  In our group we determined that to move 2 disks it would take 3 moves, 3 disks: 7 moves, 4 disks: 15 moves, 5 disks: 31 moves, 6 disks: 63 moves...etc.  We also found the general formula for this puzzle which is Tn=2n-1 which means that however many disks there are you subtract that by one and raise 2 to the power of that.  For the recursive formula we found that the formula is Tn-1=(Tn(2))+1 which means that you take the number of disks and subtract that by one and then multiply that by 2 and add 1 which will get you the number of moves.  We then used mathematical induction to prove the general formula by assuming true for n=k which made the formula Tk=2k-1 and then we showed true for k+1 which made the formula Tk+1=2(k+1)-1.  When you do the algebra the equation works out to equal 2k+1-1 on both sides.  Because they equal each other this shows that mathematical induction works.  We know that mathematical induction works to prove something when you can get both sides to equal the same thing.  

Sequences and series

Sequences are shown as things like a1, a2 a3 a4......an with n being the last term.  An arithmetic sequences is a sequence that is made by adding or subtracting. A geometric sequence is a sequence that is made by multiplying or dividing.  The general equation of an arithmetic sequence is an=a1+(n-1)d and the recursive equation is an+1=an+d where d is the common difference between an+1 and an.  The general equation of a geometric equation is an=a1rn-1 and the recursive equation is an+1=anr where r is the common ration of an+1/an.  Another thing we looked at is series which uses the symbol ∑.  This is used for summation notation and on top of the symbol is n (where you end) to the right of the symbol is the equation and the bottom of the equation is where you start.  To use summation notation you start at the number it tells you to and plug the numbers into the equation until you reach the number at the top.  Here is an example of summation notation.