Autism prevalence in the USA.

The symptoms of autism are drastic and devastating. These can be remediated to some degree, but treatment is imperfect. In addition to this, autism and other mental disorders are becoming more common in American as well as global society. Unfortunately, the causes of this terrible epidemic are unknown. Perhaps there is a genetic gene that spreads around. Or maybe the air is so polluted with chemicals, it may affect the development of a child by damaging the brain. Nevertheless, the percentage of children diagnosed with autism has substantially increased. According to the Centers for Disease Control and Prevention (CDC), in 2000, Autism Spectrum Disorder of all levels was prevalent in about 1 in 150 children of all ethnicities and gender in the United States. In a recent statistic, CDC, on March 27, 2014, discovered that autism was prevalent in 1 in 68 children in the United States. However, ASD was 5 times as common in male children (1 in 42) than it was in female children (1 in 189).  ASD is extremely prevalent in other countries as well. In Asia and Europe, ASD is prevalent in 1% of the population. In South Korea, it is reported that ASD is prevalent in 2.6% of individuals (CDC.gov). As seen through the studies throughout the decade, autism prevalence has increased by 45% in 10 years. Today in the United States, about 1.5% of children are diagnosed with autism, which means that a large minority of the population requires heavy special education. The prevalence of autistic kids is larger than the number of doctors in the United States. According a census in 2004, .29% of the population (1 in 300 people) are doctors. Therefore, not only does their need to be more quantity and better quality of special education, but there is also a need for prevention methods.

10.4: Rotating Conics

Hello Mathland!!Let's review the difficult... wait cross that out.... simple concept of Rotating Conics!
Basically all we are doing is rotating a conic by a certain angle.

Here are some Main equations for rotating conics
Ax^2 + Bxy + Cy^2 + Dx +Ey + F = 0
A'(x')^2 + C'(y')^2 + D'x' + E'y' + F' = 0
cos 2 theta = A-C/B
x = x'cos theta - y'sin theta
y = x'sin theta + y' cos theta.

Here are the steps to rotate a conic
1. Find the angle using cos 2theta = A-C/B
2. substitute the angle measure value from the unit circle to the equations in terms of x and y in order to get x' and y'
3. substitute these values into the original equation, and pray that the xy term is eliminated.

10.6 polar coordinates

Wussup mathletes in the wonderful mathland, here is another review of polar coordinates
Remember that they are denoted by (r, theta), where r is the distance from the center and theta is the angle of the line from a polar axis.
Now, how do we convert between rectangular equations to polar equations.
Here are some conversion formulas
x=r cos theta
y = r sin theta

tan theta = y/x
r^2 = x^2 + y^2

here's an example:
3x - 2y = 6
3(r cos theta) - 2(r sin theta) = 6
(r( 3 cos theta - 2 sin theta) = 6)/ (3cos theta - 2 sin theta)
r = 6/(3 cos theta - 2 sin theta)

Effect of age on athletic performance

Obviously at first glance, athletes tend to deterioate as they reach the age of 30. However, why do they do so? Can a trained 50 year old past athlete beat a nonathletic 25 year old to a race? Well first of all, at age 25, the muscles and mind of an individual have completely developed, and they have reached their peak. Additionally, their maximum oxygen consumption, VO2max is also at its greatest potential. For athletes at 25 especially, their VO2 max is extremely high, since they train their body to do so. However, after the age of 30, VO2max decreases approximately by 10%, thus causing muscles to deterriorate. It continues to drop for the remainder of an individual's life. THis comes to the next question, can a trained 50 year old beat a nonathletic 25 year old? Although it sounds unlikely, the answer is yes. For athletes that continue to train, their VO2max does not drop as fast for that of nonathletes. This is why it is extremely important to exercise, since it usually results in a healthier body. Therefore, if an athlete continues to train until he's 50, he can potentially beat a nonathletic 25 year old in a race.

works cited
http://www.svl.ch/SportsAge.html

12.4: limits at infinity and limits of sequences

Limits of infinity: as the limit of f(x) as x approaches -infinity is L1. The limit of f(x) as x approaches infinity is L2.

Here are some rules regarding limits of infinity 

The limit of 1/x^r as x approaches infinity is 0. The limit leans towards the right 

The limit of 1/x^r as x approaches -infinity is 0. Limits leans towards left.

For rational functions... Say a^n/b^m 

If n<m, the limit is 0

If n=m the limit is a/b

If n> m, there is no limit

For example

An = 2n + 1/n+4.... 

N=m 

Limit = 2/1=2

Notes for 8.1 presentation

8.1: matrices and systems of equations
Vocabulary
Matrix: a rectangular array that displays series of terms through m rows and n columns.
Augmented matrix: matrix derived from a system of linear equations. Elementary row options: the means in which we can rearrange matrices. 1. Interchange two equations
2 multiply an equation by a nonzero equation
3. Add a multiple of an equation to another equation.
Row-echelon form: the necessary form we need for augmented matrices and system of equations.
1. Rows consisting mainly of zeroes belong at the bottom
2. First nonzero has a 1
3. For each row the leading 1 in the higher row is to the left of the lower one.

How to solve system of equations through gAussian elimination with back substitution.
1. Get the matrix in row-echelon form using elementary row operations
2. Use back substitution to solve for each variable.

Gauss-Jordan elimination
1. Obtain the reduced row-echelon form using elementary row operations. 2. Variables are equal to the coefficients on the right. 

Techniques for evaluating limits

Limits of polynomial and rational functions

1. If p is a polynomial function and c is a real number,

Lim p(x) = p(c)

2. If r is a rational function given by r(x) = p(x)/q(x), and c is a real number such that q(c) doesn't = 0 

Lim r(x) = r(c) = p(c)/q(c)

Since we cannot have a limit that doesn't have a solution, we must rearrange the function to find the limit. We can do this in two ways

1. Cancellation (factoring) 

2. Rationalizing (multiplying. By conjugate)

Example of cancellation 

15 is cancellation. 17 is rationalizing

Fun Post: Getting kicked in the nuts

Ladies, I understand that you have it hard, but don't think we don't have it hard either. Our sensitive area that contains our babies are extremely precious. There is no greater physical pain than having this space violated.
Here's a video that has both math and nut shots in it. I know you want to see it. Btw, this guy is like a beast.
https://www.youtube.com/watch?v=pB7gnB31NnI
https://www.youtube.com/watch?v=a86cQobU-n4

Introduction to limits

Definition of limit:

If f(x) becomes close to a number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

Lim.  F(x) = L

X->c

Another way to define a limit is a point where an approaching line or graph ceases to pass.

How do numerically estimate a limit 

As seen through this example, we simply substitute values extremely close to c into f(x). After finding these various results, we should be able to estimate the value that resembles the limit. 

Through graphs we can tell if the function has a limit or not.

No limit: lines will never intersect. (|x|/x)

1. F(x) approaches a different number from the right side of c

2. F(x) increases or decreases without bound as x approaches c

3. F(x) oscillates between two fixed values as x approaches c.

Limit: lines intersect

Professions that use math

What's up mathland!?
As a fellow high schooler, I also wonder what is the relevance of using math for the future. Doesn't calculus seem a little bit pointless for many professional occupations? That is why Im giving a list of professional jobs using math.

The main sections of jobs that use math are...
income - economists, bankers
future outlook - statistics, actuaries, researcher.
physical demands - physicians,
job security
stress
work environment.

In general, we may need to know more math math skill than we think we do. Therefore, we should take math more serriously than we should believe it or not.

11.2-vectors In space

R,emeber that a vector is a line with direction and magnitude. It is denoted by v=<v1,v2,v3>

To find the points of a vector we usually subtract the terminal point by the initial points

Here a few things regarding vectors in space

Let's say we had to give the component form, the length, and unit vector of v with initial point (3,4,2) and terminal point (3,6,4).

V= <3-3,6-4,4-2> = <0,2,2> 

||v|| = [2^2+2^2]^1/2 = 8^1/2

U = <0,2,2>/8^1/2 = <0,1/2^1/2, 1/2^1/2>

Angle between two vectors 

Cos theta = (u * v)/||u||||v||

11.1-the three dimensional coordinate system

X= distance from yz-plane to P

Y=distance from xz-plane to P

Z=distance from xy-plane to P

Distant formula

D=[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]^1/2

Let's say we had to find the distance between (2,1,3) and (-1,2,1)

D= [(-1 - 2)^2 + (2 - 1)^2 + (1 - 3)^2]^1/2

  = [ 9 + 1 + 4]^1/2

  = (14)^1/2

Midpoint 

((X1 + x2))/2, ((y1 + y2))/2, ((z1 + z2))/2

Let's say we had to find the midpoint between (5,-2,3) and (0,4,4)

(5+0)/2, (-2+4)/2, (3+4)/2) = (5/2,1,7/2)

Standard equation of a sphere

[(x-h)^2 + (y-k)^2 + (z-j)^2] = r^2

Cool 3d graphs

Z=-1/(x^2+y^2)

Z = cos (abs x + abs y) * (abs x + abs y)

Z = -x*y*e^(x^2-y^2)

Application of polar coordinates. Batteship

In Mathland, we used our knowledge of polar coordinates to play a game if battleship. It was actually fun

Rules to battle ship

1. Set up your four ships on the coordinate graph. The points have to be adjacent to each other not diagonal. Don't show the opponent 

2. Each player gets one guess per turn. He declares a specific point on a graph (r,theta) and the other player will tell whether you hit or not. He must also tell you if the ship sank. A shirt sinks when it has been hit according to the length of it.

Highlighted would mean my ships or hits

Graphs of polar equations

Graphing polar equations is much different than graphing regular (x,y) equations. 

Let's say we were given the equation: r=4sin theta

Now remember that for sin, the value of theta is between 0<theta<2pi

By substituting various values of pi into the equation, we can also figure out the value of r for each calculated value of theta. Remember to plot the points according to polar coordinate system. So if theta is pi/6, r would be 2. That particular point is (2, pi/6)

Symmetry

1. Line theta = pi/2: replace (r, theta) with (r, pi - theta) 

2. The polar axis:    Replace (r,theta ) with (r, -theta)

3. The pole:            Replace (r, theta) with (r, pi + theta)

Let's try to see if the equation: r=3+2cos theta has any symmetry

2. R = 3 + 2cos(-theta)

    R = 3 + 2cos theta. 

So it is symmetrical to the polar axis. 

To find the maximums of |r| we have to plug in different values of theta according to the range and see which value of r is the highest.

Shapes: 

10.6 polar coordinates

In addition to graphing points in terms of (x,y), we can also do so through (r,theta) . When we do this, his is called the polar coordinate system. R is the directed distance while theta is the directed angle. To express r, we generally draw circular rings around the origin. If the point is located on a further ring, we have a higher value if |r|. For instance of r =3, then the point will be located on the third ring. The exact location on the ring is determined by the angle. Theta is expressed through radians. We can use the angles on the unit circle to express the angles in the polar coordinate system. 

Now. How do we even convert (x,y) to (r, theta) and vice Versa?

To convert to (x,y)

X = r cos theta

Y= r sin theta

 To convert to (r,theta)

Tan theta = y/x

r^2=x^2+y^2

Now let's try to convert (3,pi)

X=3 cos pi = 3

Y = 3 sin pi = 0

(3,0) 

Congrats this is the main gist of section 6

Rotation and systems of quadratic equations

Okay... Let me be frank. This section is going to make you cry, because it definitely made me. BUT! With my guidance you are going to understand this so easy! 

Let's start with the equation of an xy-plane.

Ax^2+Bx^2+Cy^2+Dx+Ey+F=0

The objective of rotation is to eliminate the xy term because it completely messes up our graphing process! We do so in e following steps.

1. Find the angle in which we rotate the graph through the equation..

    Cot 2 theta = A - C

                            B

2. Use the angle to find x and y, in the equations

    X= x' cos theta - y' sin theta

    Y= x'sin theta + y' cos theta

3. Substitute x and y into the original equation. ( we will also try to rearrange this into the equation of a parabola, ellipse, or hyperbola). 

Remember that the general second degree equation is 

A'(x')^2 + C'(y')^2 + D'x' + E'y' + F' = 0

4. We graph this new second degree equation. The angle of rotation is the angle we found earlier.

Here's an example of how to do a rotation

Hyperbolas

 Hyperbola is defined as a set of all points (x,y) the difference of whose distances from two distinct fixed points (foci) is constant. 

The equation is given by so (standard form)

Going horizontal: (x - h)^2  - (y - k)^2

                               a^2.           b^2 

Going vertical:     (y - k)^2  -. (x - h)^2

                               a^2.            b^2 

Foci: c^2 = a^2 + b^2

Notice how, unlike ellipses, a doesn't denote the major axis. This is due to the fact that such things are not used here, a and b stay in the same place regardless of anything. The direction in which a hyperbola goes (horizontal or vertical) depends on whether x or y is the first term. If y is the first term, the hyperbolas go along the y axis. Opposite is true if x is the leading term, 

Center (h,k) 

Vertices:

 if hyperbola is going horizontal (x leading term): (h+a,k) (h-a,k)

If hyperbola is going vertical (y is the leading term): (h, k+a)(h, k-a) 

Remember for the box that we add b to the h or k to find the boundaries.

As for the foci

If hyperbola going horizontal: (h + c, k) (h - c, k) 

If hyperbola going vertical: (h, k + c) (h, k - c)

Asymptotes

Horizontal: y=k +/- b/a (x - h)

Vertical: y = k +/- a/b (x - h)

Here's an example of his to make a hyperbola.

My application of math in a spiritual way

What's up mathland!? I serriously could not find any topic to do a math post on, but I suddenly remembered that we did an extra credit assignment last semester, that did apply math to Christianity.
Now of course, remember than no virtue or element of Christianity can be calculated in terms of math, because they are all aspects of the heart, not the mind.

Even though we cannot describe God or the universe with mathematically calculate functions or equations, we can truly create analogies that may be of more value. These analogies could possible be how our life works. The function y=mx+b describes the equation of a line. Let y be our continuous line. It is important to know that even though our life ends at a certain point, it will always continue in the after-life, just like how a line has no beginning and no end. Now, let mx+b represent the factors that affect y. Life has its  “ups” and “downs” similarly how a slope can be positive or negative in our basic equation. Now, let x be the decisions we make in our life. The degree of a slope depends on the value of x. Similarly, the degree of our “ups” and “downs” in our life is totally dependent on the decisions we make in our life. Therefore, our general life is completely dependent on our actions. However, allow b represent a divine intervention or influence. In a line, the value of b either lowers or raises the y-intercept. Similarly, if we allow divine intervention enter our lives, our lives can be much better or positive on the graph. However, even with divine intervention, our decisions may be negative and our lives will become even more terrible despite intervention. Therefore, humans must make positive decisions in order to live out the best of their capabilities.
So how about this philosophy!? Please comment on this opinion. It's a little bit out of the blue.

Ellipses

An ellipse is defined as a set of all points, the sum of whose distances from two foci is constant. It is also important to identify the different parts of a parabola. First of all there is a major axis that can go either horizontally or vertically. The smaller axis is the minor axis. 

The equation of an ellipse is given by:

The major axis is "a" and is always bigger than b. If it's under x, then x is the major axis. The same is true for y. 

C^2 = a^2+ b^2

The center is (h,k) 

We can find the vertices by knowing the value of a. Once we know it, we add the value of a to the y or x points of the center. 

For the foci: we add and subtract the value of c to the coordinates of the center.

Here's an example of an ellipse

Parabolas

A parabola is defined as a set of all points that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. 

The point between the midpoint and the focus is called the parabola. Therefore we could calculate the vertex if we knew the focus and directrix. 

The equation of a parabola is:

For a parabola going up the vertices axis- (x-h)^2 = 4p (y-k) 

For a parabola going across the horizontal axis - (y-k)^2 = 4p (x-h) 

The vertex is (h,k) and the focus is always found by using p as well as the directrix.

Here's an example of how to find p.

If we wanted to find the focus, we know that the vertex is (-1,1). Since the parabola is going up, the value of x will remain the same and we will add the value of p to the y of the vertex. The directrix is subtracting it from the vertex.

Monte hall problem

The montee hall show, Let's make a deal, offers an activity that awards prizes. The contestants would choose one of three of doors and one of them would have a prize in it. Once the contestant chose a door, montee hall would open another door that didn't have the prize in it, and he would offer the contestant if he or she wanted to switch doors. The competitor would choose which door from there. 

For several years during that era, it was noticeable that when people switched doors, they got the answer 2/3 of the time while those who didn't got the award 1/3 of the time. Why?

There is actually a bit of math involved. Let's start simple. Picking one door door would give you 1/3 of a chance of getting a prize. Thus you would have 2/3 of a chance picking the wrong door. 1/3+2/3=1. From this you would know that two doors together would have a probability of 2/3 getting the prize. Here's where it gets interesting when he reveals one of the wrong doors. Remember that we already established that the probability of two doors together is 2/3. Staying with your initial choice. Additionally, your initial choice will always be 1/3 of success. If montee hall didn't remove a door, and you changed your mind there are 2 doors to choose from and there is a 2/3 success rate of two doors. Thus, (1/2)(2/3)=2/6=1/3 just like your initial choice. Since he did remove one, you only have one other choice to choose from and remember that two doors has a success rate of 2/3. Therefore 1(2/3)=2/3 probability of success if you switch. Same concept works for four doors. If he didn't remove a door, you have 3 other doors to choose from and a 3/4 success rate of three doors. Thus (1/3)(3/4)=3/12 probability which is just like an initial choice of 1/4 success rate. If he does remove a door there are only two doors to choose from. Thus (1/2)(3/4)=3/8 which is greater than 1/4. Therefore you should always switch doors.

http://montyhallproblem.com

Probability

Probability is defined as

If an event E has n(E) equally likely outcomes and it's sample space S has n(S) equally likely outcomes, the probability is

P(E)=n(E)/n(S)

The probability of an event ranges from 0 _< P(E). _< 1

Must occur at 1 but cannot occur at 0

Let's say we were to determine the the probability of drawing a card less than 6 In a 52 card deck. 

To solve this we know that there are 5 cards less than six and 4 differente suites for each card.

Therefore the n(E) = 5x4 =20

The total sample space is 52.

So P(of drawing a card less than six) = 20/52=5/13= .38

We can also implement count unprincipled in probability

Let's say we were drawing two marbles from a bag of one green, two yellow, and three red marbles. What is the probability of drawing two yellow marbles?

To determine the sample space, we can use combinations to find the number of outcomes

6C2= 6!/(6-2)!2!=15 total outcomes.

The total number of times we can pick two yellow marbles is 1 because there are only two yellow marbles

Thus

P(E)=.07

Ray Lewis vs a batering Ram

Hello mathland, I know it has been a while since the football season ended, but I am doing another post about football. In this sports science video, we see the general idea of how hard a NFL lineback hits as we compare it with a battering ram. This is overall a generally interesting video to watch, and we also learn how much force it takes to break through a door.
https://www.youtube.com/watch?v=k8krW5z-MO4

Counting principles

Fundamental counting principle

E1 and E2 are two different events. The first event can happen in m1 different ways and the second event can happen in m2 different ways. M1 x m2 is the number ways that two events can occur. 

For example if we were to find the total different combinations of telephone numbers for each area code, we see that there are eight different events. (Important to note that the first digit cannot be 0 or 1, thus giving us eight ways event one can occur). 

8 x 10 x 10 x10x10x10x10. = 8,000,000. 

Permutation (order matters)

A permutation of x number of elements is an ordering of elements. 

For instance, if 4 people were in a line, we could be asked how many different ways could they line up.

In this case order matters.

4 different people can take the first spot, then 3 different people can take the second part since somebody already took the first spot. And so on.

4! = 24 different ways. 

Let's also say that 8 track runners ran in a race. How many different combinations of runners can get 1st, second, and third.

In this instance, we use the equation nPr= n!/(n-r)!

Thus 8P3=8!/(8-3)! =336

Combinations (order doesn't matter)

The number of combinations taken r at a time is denoted by 

nCr=n!/(n-r)!r!

Let's say we were choosing 4 marbles from a bag of 7 marbles. How many different combinations of 4 can we get?

7C4=7!/(7-4)!4! = 35 different ways.

Binomial expansion

In a binomial raised to a certain power, it can be difficult to calculate the coefficients and exponents of the seperate terms. However there are patterns that are noticeable when expanding a binomial.

1. In each expansion, there are n+1 terms.

2. X and y symmetric roles. The powers of x decrease while the powers of y increases by 1 successively.

3. The sum of powers of each term is n. 

4. Coefficients increase and decrease

Binomial coefficients 

nCr = n!/(n-r)!r! 

r is the specific term in the expansion. The fourth term would be r =3 which means the number of term is r+1. 

For instance...

The third term for (x+y)^4 is 6

4C2 = 4!/(4-2)!2! =6

The pascal Triangle is also a means of determining the coefficients since it already states the coefficients for an (x+y)^n. 

Here's an example of an expansion...

Blog 20: mathematical induction

One of the only reasons why the mathematical induction works is beacause of the well ordering principle. The well ordering principle states that every non empty su stet of a natural number has a least element. We use mathematical induction in order to prove an equation. 

Let's say we were tryin to prove the equation that

(2n-1)=n^2

Step 1: we have to prove the statement is true at the starting point (n=1)

Step 2: assume tha the statement is true for n. Prove the statement is true for n+1

Enginerring in Half-Pipe

Hey yall, since its almost the end of the Sochi WInter Olympics, I found it appropriate to do this math blog on one of the events. My favorite event is the snowboard half-pipe, which involves performing several airborne tricks on a litteral snow half-pipe. However, in order for one to perform such daring tricks, the half-pipe also must be designed in a certain way. One factor for a snowboard is acceleration. This is defined as the rate of change in velocity. Therefore, if one wanted to gain more height on the halfpipe, that individual would have to increase his acceleration, since it gives a greater force on the body. Gravity also contributes to the performance. The walls of a half pipe are 22 feet and the width is approximately 65 feet wide. This fits a greater radius with the pipe, and thus greater room for speed.
works cited
http://www.nsf.gov/news/special_reports/winterolympics/halfpipe.jsp

Post 19: arithmetic sequences

Arithmetic: a sequence whose common terms have a common difference 

For instance

7,11,14,18

There is a difference of four between each term

We can find the An term of a sequence in the equation

An=A1+(n-1)d

D is the common difference

Arithmetic sequence

Sn=n/2(A1+An)

This is the sum of a finite sequence

Say we were given the sequence

1,3,5,7

Sn=(4/2)(1+7) = 16

Post 18: sequences and summation notation

How do we find the terms of a sequence?

Say we were the sequence An = 3n - 2

A1 = 3(1) - 2=1

A2 =3(2)-2=4

....

We must substitute the variable n with the specific term number to find the value of that term.

A recursive sequence

In order to find the terms we have to be given the value of a few terms and also be given the recursive formula

For example

A0=1,A1=1 Ak=Ak-2+Ak-1

 This means that one term is equivalent to the sum of the two before it. 

Therefore

A2=A1+A0=1+1=2

A3=A2+A1=2+1=3

Summation notation

This involved the uppercase of Greek Letter sigma 

Basically it gives the number of terms in a sequence and the pattern. After finding the value for each term, we add them together. We are also given the term we start on.

Post #17 a review of matrices

Here's a link to a google site that reviews matrices

https://sites.google.com/a/maranathastudents.org/group-5/home

blog #15: using matrices to find an equation of the line

February 7, 2014
here is a link for solving the equation of a line using matrices.

Post #13- sample problems on determinants

So basically I solved the 2x2 matrices by first multiplying the left corner by the bottom right corner. I subtracted this by the product of the bottom left on the top right corner. That's the determinant for a 2x2

Post #14: encryption matrices

So apparently in math land we can actually create codes using the codes. They are very hard, but it can be very interesting to see what kind of message can be created. 

The way to do this is to start creating the inverse of your matrix (decryption matrix). After we find this inverse (say mxm), we will multiply this by another matrix (1xm). We will continue to multiply the original matrix by as many types of 1xm matrices depending on the length of the code. From the results we have, we will discover the code. We also have a scale that translates the numbers into letters. For example 0=space, 1= a, 2=b..... 

Post #12 inverse matrices

Here are two sample problems that I did in order to find the inverse matrix.

So basically in problem number one there was no inverse. I was able to figure this out because the third row only contained zeroes which meant I could go no further. 

Remember to also apply the elementary row operations for the other matrix as well. 

Remember that A x the inverse or A is always the leading one matrix.

Blog #11: Concussions and math

One of the most common injuries within sports such as Football, Soccer, and boxing, are concussions. For those who don't know the exact definition of a concussion, it is a type of neurological impairment to the brain caused by some type of truama. Concussions generally happen to 5-10% of athletes in any sport. 75% of concussions for males occur in football. This is due to the fact that an average professional football player will receive 900 to 1500 direct blows to the head which raises the probability that they will get a concussion. Additionally, the average professional athlete will tackle at a rate of 25 miles per hour. As for females, 50% of concussions occur from Soccer from flying soccer balls.
 However, Age is also a factor to consider. For a high school athlete, his or her frontal lobe is still developing until the age of 25. Therefore, there chances of receiving a concussion is much higher, and damage is more peramanent for that of an adult. 53% of high school athletes have had a concussion before participating in a high school sport while 36% of collegiate athletes have received multiple concussions.

Works Cited
 http://www.concussiontreatment.com/concussionfacts.html#sfaq9

Blog #10: Gaussian with back substitution vs gaus Jordan

We can also solve systems of linear equations with matrices. The key to solving any system with matrices is forming an augmented matrix, which we can find by using the coefficients of the variables. Remember that the columns are the variables and the rows are the values of the coefficients for each equation. From there we can solve using the Gaussian elimination method or the Gaus Jordan method.

Elementary Row Operations

1. We can interchange 2 rows

2. Multiply a row by a constant

3. Add two rows together.

Steps to Gaussian elimination

1. Create an augmented matrix

2. Use ERO's to rewrite in row echelon form

3. Write a system of equations and use back substitution.

Make sure that everything below your leading ones are zero. The top values don't have to be zero.

Be sure to work column by column.

Steps to Gaus Jordan Method: essentially we are using the same steps as the other elimination method. however, it is important to note that we should only have leading ones and zeroes. Therefore we don't need any new systems of equations. 

Sample problem #1 is  using the Gaus Jordan method. Sample problem #2 is solved using Gaussian elimination. 

Remember that whenever we multiply an equation by a constant, we have to multiply everything in that row. This is shown through the error of another student.

blog #9: prezi on vocabulary regarding matrices.

http://prezi.com/pxvzhbfuuv8i/?utm_campaign=share&utm_medium=copy

post #8: How to strength relates to speed

As a fellow athlete, I have always wanted to improve my athletic performance in strength and speed. However, I have never been so sure what the relationship between the two were. As it turns out, the magnitude of speed is directly proportional with the magnitude of strength if we were to eliminate the factors of weight, endurance, and efficiency. Basically, if an athlete were able to dead lift 500 pounds with perfect form, he or she should potentially be able to apply that maximum amount of force on each step. This is simple physics, if we were to apply more force on the ground, we would increase the distance of our stride and therefore go faster.
However, we must always consider technique. Running with poor technique such as: landing on the back of your heel, leaning too much backwards, or opening up the knee angle too much can lower our running efficiency to 74%. This means that, each of these poor techniques decrease the amount of force we apply on the ground.

Works Cited
http://www.athletesacceleration.com/the-simple-math-behind-running-faster/

Post #7: Chapter 7 review

This is a study guide basically for what my class has learned on Systems of Equations and inequalities.
Systems of Equations (2 variables, 2 equations)
In a system  of linear equations, we have to be able to find the values of both variables. This can be found in the following ways:
1) substitution: solving for one variable in one equation, and plugging it in the variable in the other equation.
2)elimination: basically adding two equations together to get rid of a variable.

We can apply our knowledge of Systems of linear equations in something known as Break Even. This is the point where the total Revenue equals the total cost.
Total Cost = (cost per unit)(number of units) + initial cost
Total Revenue = (Price per unit)(number of units)
since total cost = total Revenue, we can basically set the equations equal to each other using substitution and easily find our number of units, and later on our cost and revenue price.

Systems of Equations (3 variables, 2 or 3 equations)
For a system with 3 equations and 3 variables, we practically do the same thing as we would for a system with 2 variables and 2 equations. The plan is to use our knowledge of Substitution and Elimination to gradually eliminate more and more variables.
As for a system with only 2 equations, follow the simple steps:
1. Eliminate x and solve for y
2. Eliminate y and solve for x.
In this way, we will be solving in terms of Z. Don't forget that Z is a constant in this case, and we will have to replace it with a constant a instead.

Partial Fractions
Remember to do the following steps

1. Multiply by the LCD.

2. Distribute

3. Bring the terms together

4. Factor out the variable.

5. Equate

6. Solve

7. Write as a partial fraction.

Linear programming 

Basically for this we always have to find our constraints in order to find the feasible region. This is the only hard part, after finding the region we must find the vertices of it. Using the coordinates of the vertices, we plug them into our z equation in order to find our maximum and minimum.

Blog #6 Sports Science, NFL Defensive Tackle vs Average Joe

Basically, I was watching this very funny, interesting sports science video yesterday, and realized that football includes physics in it. In this particular video, scientists are determining how strong the NFL really is by matching up Defensive lineman, Kris Jenkins, with the sports science host. I honestly think that this was really funny seeing the lab rat getting destroyed every single time. Basically don't mess with an NFL athlete.
http://www.youtube.com/watch?v=2QOEIQ3_Kuo

Blog #5 systems of inequalities

Sketching the graph of an inequality:

1. Replace the inequality sign with an equal sign, and sketch the graph of the resulting equation.

A. Remember that dashed (less than, greater than) or solid lines ( less than or equal to or greater than or equal to) 

2. Test one point in each of the regions formed by the graph In step 1. If the point satisfies the inequality, shade the entire region to denote that every point in the region satisfies the inequality.

3. A solution of a system of inequalities in x and y is a point (x,y) that satisfies each inequality in the system.

4. For a system of inequalities, it is helpful to find the vertices of the solution region.

Tips

-be sure to set everything equal to y since y=mx+b.

- use different colors to graph everything

-be sure or shade every region covered by a certain equation in order to find the solution region more easily

1. Less than goes left or down from the equation line.

2. Greater than goes right of the equation line or up.

Here's an example.

Blog #4 partial fractions

Welcome to another day of Mathland!! Today In the Mathland I was practicing problems on a difficult process known as partial fractions. It's basically finding two smaller fractions that make up a bigger rational fraction. But through this blog post, I hope I can make your life easier with these tough thingies. 

To solve any partial fractions we must:

1. Multiply by the LCD

2. Distribute

3. Bring terms again

4. Factor the variables out

5. Equate with other side

6. Solve a system of equations.

7. Write as a partial fraction.

As seen in number nine I factored the denominator into two terms and used terms A and B in order to represent the coefficients. Then I found a common denominator which was x^2 + x and distributed the terms. I rearranged the terms together and factored the variables in order to help me equate. This equation was 0x+1 = x(A+B) + A. In this way (a+b) would have to equal 0 and A =1 . If we were to solve the system of equations we would find that b=-1 . Using substitution I would right this as a partial fraction. 

Of course this is just a basic lesson on how to do these things but these are the mains steps for more difficult problems.