Teddy's Presentation

Once again our good friend teddy has imparted a special gift upon us. I don't know how to get rid of this underscore but whatever. Here we go teddy! 

Elementary row options

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 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.

Here are some examples of matrices in row-echelon form:

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.

Darron's Presentation

Today My good friend D graced the class with an excellent and superb presentation. Let's take a look at what it entailed. 

Dot Product of Two Vectors

-The dot product of u = <a,b> and v = <c,d> is u • v =  ac + bd 

-The properties of the dot products of two vectors are as follows: Let u, v, and w be vectors in a plane or in space and let c be a scalar.

1. u • v = v • u 2. 0 • v = 0 3. u • (v + w) = u • v + u • w 4. v • v = ll v ll ²  5. c(u • v) = cu • v = u • c


The Angle Between Two Vectors 

If ø is the angle between two nonzero vectors u and v, then cos ø = u • v / ll u ll ll v ll

Vectors u and v  are orthogonal if u • v = 0 

Vector Components
Let u and v be nonzero vectors such that u = w + twhere w and t are orthogonal and w is parallel to v. The vectors w and t are called vector components of u. 

The vector w is the projection of u onto v and is denoted byw = proj v uThe vector t is given by t = u - w
Let u and v be nonzero vectors. The projection of u onto v isproj v u = (u • v/ ll v ll²)v

Becoming a beast at basketball

1. Here are 7 steps to becoming a great basketball player like me lol. 
   
   
   Get to know the court's geometry: You have to know all the measures from where you play, so you can have a better spacial notion. The hoop diameter (18 in), the length from hoop to hoop (94/84 ft), the ball itself (9.4 in diameter), the field wideness (50 ft), and the length from the three point line to the hoop (19 ft).
   
   

   
2. 2
   Understand shooting implications: When you shoot you are involving three factors: angles, the impulse, and the position of your arms. You gotta apply a greater angle (understanding the angle as a perpendicular line from your hips, and the extension of your arms) if you make a field shoot, but a smaller one when you shoot from inside the free throw zone. You may shoot higher when you have some defenders in front of you, also I recommend to shoot from a 45 or greater angle, cause that helps the ball to enter softer and cleaner to the basket. Your elbow should be the closest possible to your face so the ball goes in straight line and extend your arm as far as you can so you have greater force.
   
   

   
3. 3
   Recognize the math in bounding: The ball is a semi-sphere that according to Newton, will have a reaction depending on the force you apply it. When bad pumped, it will not bound as good as when full of air, the same as when you don't it bounce correctly. You have to apply the ball a relative amount of force, depending if you want to give a long bound pass, if you are running, or if you are dribbling. My advice is that when dribbling, apply considerable force to the ball, and bounce it close to the floor, so that you can have better control. Use a straighter angle on bound passes if you want them to get further, and when running keep it at the height of your hips so you maintain your speed.
   
   

   
4. 4
   Get an assistant to record percentages: You have to know how many rebounds, shoots, steals, and counter attacks you are making, so you can improve on the areas were you have low performance. Record the attempted and made shoots. The offensive and defensive rebounds. The total steals and counter attacks on the game. The second chance points you made and the team's shooting tendencies.
   
   

   
5. 5
   Understand the parabola in shoots: The parabola is the downside curve that's made in every shoot. As you are shooting you may realize that as higher the parabola the cleaner or easier the ball gets inside the hoop, and as lower, the more chances there are the ball hits the rim. For the parabola effect to be completed you should apply the "follow through" with your wrist, that means you shall give the ball a back flip effect at the end of your shot using your hand.
   
   

   
6. 6
   Apply geometry in rebounding: Whenever the ball is shot from one side, from a field shoot, it will end on the other side, the most of the times; when shot from the free throw zone it will mostly rebound on the same side. When more force is applied the rebound will get further, when little force implied, the ball will fall in the same place it hit. So familiarize yourself with rebounding implications so you can guess every time where the ball's going.
   
   

   
7. 7
   Understand defensive implications: Every defense is got to know how geometry is implied, so they can know how they can steal more balls, avoid being tricked and stop their man from scoring. The first implication are the defensive angles at which you stand your man, when you are half-sided, you have an advantage, as he cannot get pass you as easy as if you faced him directly with no body angle. The same is applied to the angle of you legs on defensive position, the more folded your legs the faster you are. Finally realize that when you displace with the higher front part of your feet (tips), you also are faster, as there is faster contact with the floor.
   
   

   

Limits in Polynomials

Today in class things got a little crazy. As we learned more about limits I was overwhelmed by a sense of scaredness and fright as I was not sure if I was for sure going to graduate. I am still not sure but with the help of Miss V all things remain possible. Anyways, let's get to it.

Limits of Polynomial and Rational Functions:

1. If p is a polynomial function and c is a real number, then lim x-->c 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) does not equal 0, then lim x--> c r(x) = r(c) = p(c) / q(c), q(c) does not equal 0.

Methods of Evaluating Limits:

1. Direct substitution (plug-in).

2. Cancellation technique (factor/cancel).

3. Rationalization technique (multiply radicals by conjugate).

Once you reach the intermediate form (0/0), use either the cancellation or rationalization method to evaluate the limit. If it contains a radical, use the rationalization technique. Finally, plug the value of x 

in the equation using direct substitution. This will give you the value of the limit. 

Limits in Polynomials

Today in class things got a little crazy. As we learned more about limits I was overwhelmed by a sense of scaredness and fright as I was not sure if I was for sure going to graduate. I am still not sure but with the help of Miss V all things remain possible. Anyways, let's get to it.

Limits of Polynomial and Rational Functions:

1. If p is a polynomial function and c is a real number, then lim x-->c 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) does not equal 0, then lim x--> c r(x) = r(c) = p(c) / q(c), q(c) does not equal 0.

Methods of Evaluating Limits:

1. Direct substitution (plug-in).

2. Cancellation technique (factor/cancel).

3. Rationalization technique (multiply radicals by conjugate).

Once you reach the intermediate form (0/0), use either the cancellation or rationalization method to evaluate the limit. If it contains a radical, use the rationalization technique. Finally, plug the value of x 

in the equation using direct substitution. This will give you the value of the limit. 

Limits

Example 1: Finding a Rectangle of Maximum Area

You are given 24 inches of wire and are asked to form a rectangle 

whose area is as large as possible. What dimensions should the rectangle have?

Let w represent the width of the rectangle and let l represent the length of the rectangle. 

Because 2w + 2l = 24 (perimeter is 24), it follows that l = 12-w and that the area of the rectangle is 

A = wl --> Formula for area

A = w(12 - w) --> Substitute 12 - w for l

A = 12w - w^2 --> Simplify

Using this model for area, you can experiment with difficult values of w to see how to obtain the maximum area. After trying several values, it appears that the maximum area occurs when w = 6.

Example 2 : Finding a Limit That Can Be Reached

Use a table to estimate numerically the limit lim x--> 2 (3x - 2)

Let f(x) = 3x - 2. Then construct a table that shows values of f(x) when x is close to 2.

From the table, it appears that the closer x gets to 2, the closer f(x) gets to 4. Thus, you can estimate the limit to be 4. For this particular function, you can obtain the limit simply by substituting 2 for x to then obtain lim x--> 2 (3x - 2) = 3(2) - 2 = 4. 

Conditions Under Which Limits Do Not Exist:

The limit of f(x) as x--> c does not exist if any of the following conditions is true.

1. f(x) approaches a different a different number from the right side of c than from the left 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.

Example 4: Oscillating Behavior

Discuss the existence of the limit.

lim x--> 0 sin (1/x) 

Let f(x) = sin(1/x). In the diagram above, you can see that as x approaches 0, f(x) oscillates between -1 and 1. Therefore, the limit does not exist because no matter how close you are to 0, it is possible to choose values of x(1) and x(2) such that sin (1/x(1)) = 1 and sin (1/x(2)) = -1, as indicated in the table. 

Math in Music Waves

Music has always been a means of cooling me down, pumping me up, or just simply giving me and opportunity to relax. It would be so ever interesting if there was somehow MATH INVOLVED IN THE MUSIC I LISTEN TO ! 

Music appears to be transmitted by magic, escaping from your expensive stereo - or a loudly passing car radio, or a guitar-strumming maestro - and accosting your eardrums in one fell swoop. In fact, sound progresses as a wave through the air, and sound cannot be produced without an atmosphere. (Or, as the horror movies would say: in space no one can hear you scream.)

A sound wave creates minute pockets of higher and lower air pressure, and all the sounds we hear are caused by these pressure changes. With music, the frequency at which these pockets strike your ear controls the pitch that you hear.

For example, consider the note called "Middle C" (usually the first note learned in piano lessons). This note has a frequency of about 262 Hertz. That means that when Middle C is played, 262 pockets of higher air pressure pound against your ear each second. Equivalently, the pockets of air arrive so quickly that one pocket strikes your ear every 0.00382 seconds. We can draw a graph by putting an X at every time when a pocket of air arrives:

This graph provides a sort of "picture" of Middle C. By itself, it does not tell us much. However, such graphs provide a new perspective on the relationships between different musical notes.

A basic rule is that higher-pitched notes have a higher frequency, corresponding to more frequent air pocket arrivals. For example, the note Middle G (seven semi-tones higher than Middle C) has a frequency of about 392 Hertz, corresponding to 392 air pockets per second, or a time period of 0.00255 of a second between arrivals:

With the higher note (Middle G), the air pockets arrive more frequently - corresponding to a higher frequency, and thus to more X's in the graph.

So how does this help us? Well, by using knowledge of sound frequencies carefully, such musical mysteries as octaves and chords can be unraveled.If you listen carefully to an ambulance siren or a train whistle, you will notice that the noise sounds higher while the vehicle is approaching, and lower after the vehicle has passed by. This is because the approaching movement compresses the X's together, making them arrive more frequently and produce a higher pitch, while the departing movement stretches out the X's and produces a lower pitch. This is musical frequency in action.

Cross Product of Two Vectors

Today we learned about one of the hardest things ever to cross a papers face. I can assure you this is quite a bit of information so hold ok to your seats folks. Also much of this is from the book so my apologies if you cannot understand the written out verbage. Hopefully the pictures help! 

 Let u = u1i + u2j + u3kand v = v1i + v2j + v3k  be vectors in space. The cross product of u and v is the vector. u x v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k. A convenient way to calculate u x v is to use the following determinant form with cofactor expansion. (This 3 x 3 determinant form is used simply to help remember the formula for the cross product--it is technically not a determinant because not all the entries of the corresponding matrix are real numbers). 

Example 1: Finding Cross Products

Given u = i + 2j + k and v = 3i + j + 2k, find the following:

a) u x v 

b) v x u

c) v x v

Algebraic Properties of the Cross Product

1) u x v = -(v x u)

2) u x (v + w) = (u x v) + (u x w)

3) c (u x v) = (cu) x v = u x (cv)

4) u x 0 = 0 x u = 0

5) u x u = 0

6) u x (v x w) = (u x v) x w

Geometric Properties of the Cross Product

This property indicates that the vectors u x v and v x u have equal lengths but opposite directions. 

Let u and v be nonzero vectors in space, and let theta be the angle between u andv. 

1) u x v is orthogonal to both u and v.

2) ||u x v|| = ||u|| ||v|| sin theta.

3) u x v = 0 if and only if u and v are scalar multiples.

4) ||u x v|| = area of parallelogram havingu and v as adjacent sides.

Example 2: Using the Cross Product

Find a unit vector that is orthogonal to both

u = 3i - 4j + k and v = -3i + 6j

The Triple Scalar Product

For vectors u, v, and w in space, the dot product of u and v x w 

is called the triple scalar product of u, v,and w. 

If the vectors u, v, and w do not lie in the same plane, the triple scalar product u x (v x w) can be used to determine the volume of the parallelpiped with u, v, andw as adjacent edges. 

Geometric Property of Triple Scalar Product:

Example 4: Volume by the Triple Scalar Product

Find the volume of the parallelpiped having

u = 3i - 5j + k, v = 2j - 2k, and w = 3i + j +k as adjacent edges. 

In class 3D

I missed this day in class so I used some of my friends pretty 3D models! Enjoy lol.

3D Vectors

Hello, as you may or may not have noticed I have not been in class for quite some time. So... I have borrowed some nuggets from some of my blogger friends to put this wonderful post together. Enjoy! 

Component form: v=<v1, v2, v3>

Unit vector form: v=v1i+v2j+v3k

Length formula: 

Length can also be called tension or magnitude. 

Unit Vector:

Vector addition and dot product:

Angle between two vectors formula:

Orthogonal if...

-The dot product between the two vectors is 0

-The angle between the vectors is 90 degrees

Parallel if...

u=cv

Vector 1=(constant)(vector 2) 

-the points are collinear 

3D Graphs

Today we learned about 3-D graphs and me oh my was it fun.. Jk I probs wasn't in class you never know though. Well anyways here are the important things to know about this subject ;)

With the points (x1, y1, z1) and (x2, y2, z2) 

The distance and midpoint formula:

The eqaution of a sphere:

(X-H)^2 + (Y-K)^2 + (Z-J)^2 = r^2

r=radius

center=(h, k, j)