Letter To Future Math Analysis Student

Dear Future Math Analysis Student, 

 To have the most successful year possible in your future math analysis class be ready to adapt to change. You have to go into the classroom expecting things to be different and keep a positive attitude. You should no give up simply because you do not like the flipped classroom method. Start the year off right by doing your work on time and take advantage of your class time to complete assignments like student problems or WPP's. With such a filled agenda of work for the class, you should plan accordingly to complete all your assignments on time. It can be tough to manage if you have sports and have late practice, so you should complete some assignments early when you have free time. Also, if you work very efficiently in class you should never have any extra work besides watching the next concept. Make sure to come to class prepared every day by watching the videos and completing the wsqs. I think that if you do all your work, and plan out your busy weeks, you should survive math analysis. 

Adjusting to the flipped classroom is essential. You will need to take on responsibility for your own learning. So you must watch every video lesson everyday before class and pause and rewind during a lesson when you do not understand. It is not like a traditional classroom and you will not be able to get help right away, you should ask questions in class the next day. Also, ask classmates around you, everyone is there to help you. For the flipped classroom, you are required to film yourself solving a problem for student videos or WPP's. You should get familiar with using a basic camera or an ipod that will be available to you in class.  Also, if you do not have a filming device at home, make sure to plan to use one during class. Do not be shy when making your videos and make sure to speak clearly. The first video will most likely be the hardest to do because you may find it a bit weird, but soon you will have a blog filled with almost 20 self created videos of you solving problems. Make sure that you have a reliable internet connection or plan to use the schools internet when you need to submit something online. 

The flipped classroom will be very different from your previous math classes because you are responsible for your own learning. You will not sit in class and take notes on a concept, you will have guided SSS packets that you will complete every night while watching the video. Also, you will most likely never have homework for this class besides watching the lesson video. Opposed to traditional classrooms, you will do most of your work during class. Also, this class has quite a bit more work than previous math classes so be ready to work hard. You will definitely go deeper into math concepts and understand where things come from and why we do certain things in math. Also, you will create your own blog that is a online portfolio of blog posts and self created videos that explain math concepts.

Unit V Big Question

Where does the formula from the difference quotient come form? 
1. A derivative is the slope of a tangent line to a graph. A tangent line is a line that touches the graph at one point. A function can have different derivatives at different values of x. To try and find the slope of the tangent line we first have to find the slope of a secant line similar to the tangent line. A secant line is a line that touches the graph at 2 points. We find the slope of the secant line using the slope formula using the 2 points on the secant lines. This is shown in the picture below. When we plug in our points we get m= f(x+h)-f(x)/ x+h-x. So when we simplify the denominator we are left with 
m=f(x+h)-f(x)/ h. So this is what we know as the difference quotient and is also the slope of the secant line. But remember we want to find the slope of the tangent line. So what we will notice is that as you make h(which is the change in x or the difference between the two x points) smaller, the secant line begins to resemble the tangent line better. So we want h(or delta x) to be as small as possible. So we use a limit as h approaches 0. We cannot just say that h=0 because then we will have a 0 in the denominator, so this is why we use a limit. By using a limit of 0, we are now finding the slope of the tangent line. 

Image: http://en.wikipedia.org/wiki/Numerical_differentiation

Unit U Big Questions

1. What is continuity? What is discontinuity?
Continuity demonstrates that a function is predictable and has no breaks, holes, or jumps. A continuous function can be drawn with a single, unbroken stroke of a pencil. You can tell if a function is continuous by imagining that you are driving a car on the line. If the imaginary car meets anything that makes the car fall off the line such as a hole or jump, the graph is discontinuous. Discontinuity or a discontinuous function has either a jump discontinuity, oscillating behavior, or a vertical asymptote. In reference to the imaginary car, a discontinuity would make the car fall off the line. 

2. What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value? 
A limit is the intended height of a function.  A limit exists when the graph reaches the same height from the left and right side. This can be tested by placing two fingers on the graph and tracing the line towards a certain point to see if your fingers meet. If they meet, the limit exists at that point. A limit doesn't exist when your two fingers do not meet on the line or in other words when the left and right behavior of a graph do not match and when their is unbounded or oscillating behavior. The value is the actual height of a graph, while a limit is the intended height. 

3. How do we evaluate limits graphically, algebraically, and numerically? 
We evaluate a limit graphically by determining left and right behavior. We do this by using two fingers to trace to a certain point and of our fingers meet at the same point, the limit exists. If they do not, then that is most likely due to a hole or jump and the limit doesn't exist. We evaluate limits algebraically by using either direct substitution, dividing out/factoring method, or rationalizing/conjugate method. These methods can all result in indeterminate form which is 0/0 and so we have to figure out which  method works best. Direct substitution is substituting the limit into the equation.  Dividing/factoring out method is factoring the numerator and denominator and canceling terms then substituting in the limit. Rationalizing/conjugate method is multiplying by the conjugate of the term with the radical in the function. We evaluate limits numerically by creating a table with values that get closer and closer to a certain value on the x axis and notice that the graph of f(x) get closer to a value on the y axis. Our table consists of x values that approach the limit from the right and left and after plugging the function into our graphing calculator we can trace to the x values and record the y values. With our table we can determine if the limit can be reached or not. If the f(x )value doesn't match the limit on the table, then the limit cannot be reached. 

Why is a normal tangent graph uphill, but a normal cotangent graph downhill?

 This has to do with the shift in the asymptotes between tangent and cotangent and the pattern for cotangent and tangent. Cotangent has asymptotes where sine is equal to zero. So its asymptotes are located at 180 degrees(pi) and 360 degrees(2pi). Tangent has asymptotes where cosine is zero. So its asymptotes are located at 90 degrees(pi/2) and 270 degrees(3pi/2). Now we pay close attention to tangent and cotangent's pattern + - + - dealing with ASTC. Once we draw our asymptotes our pattern guides us to whether our graph should be above or below the x axis. If it is positive, it is above the x axis and if it is negative then it is below the x axis. When we draw our graphs, we will see how the placement of the asymptotes affects the shape of the graph, even though they follow the same pattern of + - + -

Why do sine and cosine not have asympotes, but the other four trig graphs do?

Sine and cosine will never have asymptotes because of their trig ratios. The ratio for sin= y/r
and the ratio for cos= x/r. According to the unit circle, r will always equal 1. Asymptotes exist when it is undefined, meaning it has a 0 in the denominator. So this means that sine and cosine will always be over 1, making it impossible for them to have an asymptotes. Tangent,cotangent,cosecant, and secant can have asymptotes due to their trig ratios. Their ratios are as follows:
 tan= y/x
cot= x/y
csc= r/y
sec= r/x
Because they have an x or a y value in their denominator this opens up the possibility to be undefined and therefore have an asymptote.  For example, tangent and secant will have an asymptote wherever the x value equals zero, which is at 90 degrees and 270 degrees. Cotangent and cosecant will have asymptotes where the y value is zero so this is at 0 and 180 degrees.The image below displays the trig functions according to the unit circle.

picture:  http://htmartin.myweb.uga.edu/6190/resources/unitcircletrig.gif

How do the graphs of sine and cosine relate to each of the others?

Tangent and cotangent have asymptotes where their respective ratios are equal to zero. We know that Tangent=sine/cosine. So when cosine is 0, tangent will have asymptotes. So this means that tangents asymptotes will be at pi/2 (90 degrees) and 3pi/2 (270 degrees). These are the points where cosine, or the x value, is equal to zero. This can also be thought of as cosecant is the reciprocal of sine.  Cosecant is 1/sinx and we know based on the unit circle that sine is 0 at 0 and 180 degrees, so its asymptotes lie there. Whenever we have a zero in the denominator, it is undefined, and undefined means asymptote!  Secant is also the reciprocal of cosine  and its ratio is 1/cosx. Cosine equals 0 at 90 degrees and 270 degrees, so its asymptotes are at pi/2 and and 3pi/2. We know that cotangent=cosine/sine so when sine is 0, cotangent will have asymptotes. So cotangents asymptotes are at 0 and pi(180 degrees), the points where sine is equal to zero. The images below show that cotangents asymptote is in the same place at pi for cosecant.
photos:  http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg94.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg2.gif

How do trig graphs relate to the unit circle?

Why is the period for sine and cosine 2pi,whereas the period for tangent and cotangent is pi?
 Trigonometric graphs are cynical which means they repeat themselves over and over again. One time through their cycle is called a period. The period for sine and cosine is 2pi while the period for tangent and cotangent is pi.  The pattern for sine is ++-- (meaning it is positive in the first quadrant and second quadrant while being negative in the third and fourth quadrant) This cycle is one period. It takes one full revolution around the unit circle for this cycle to repeat, so one revolution is 360 degrees or 2pi. This is the same for cosine because its pattern is +--+(meaning it is positive in the first quadrant, negative in the second and third quadrant and positive in the fourth quadrant) and it would take one revolution or 2pi for the cycle to repeat. The period for tangent and cotangent is pi. Tangent's cycle is +-+- . This means it is positive in the first quadrant , negative in the second quadrant, positive in the third quadrant, and negative in the fourth quadrant.  Notice that it only takes half the unit circle for its cycle to repeat from positive to negative. Therefore it only takes 180 degrees or pi units to repeat its cycle, making its period pi. The image below gives you a visual of what quadrants the trig functions are positive or negative.

How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the unit circle? 

 The  amplitude is half the distance between the highest and lowest points on the graph.  Sine and cosine have amplitudes of one because based on our knowledge of the unit circle it only goes 1 unit in every direction. So the furthest points it can go to is 1 or -1. So the distance from 1 or -1 to the x axis is 1 unit, making our amplitude 1. Cotangent, tangent, cosecant, and secant do not have amplitudes, however they do have asymptotes. They have asymptotes where sine and cosine equal zero. 

Photo:  http://www.oojih.com/show/trigonometry/generalangle/