Letter to Future Students

1. What do you want to say to them to help them have the most successful year possible?  Success is viewed differently in people's eyes, but to have a successful year of learning something new in Math Analysis, remember to do your WSQ daily and consistently. At first they will be a pain in the rear end to do but you need to do them in order to do well in class and in that specific math unit. Each WSQ teaches a new thing required for future math references and these WSQs are not there for Mrs.Kirch to torture you and laugh at your suffering but it is there for her to gauge how well you are learning the concept and see if you need help on a particular math concept.
2. How can they best adjust to the flipped classroom and learn to work with all the technology I require of them?  (do you have any specific tips for them or experiences you could share?)The best way to adjust to this new setting is to go with the flow. I'm the kind of person that doesn't care what happens unless it will affect me negatively. Follow Mrs.Kirch instructions and do exactly what she recommends you to do and that is the only way you will be able to adjust to this flipped classroom. Since you will be working with technology, remember to close out all distractions such as Facebook, AIM, and other messaging system so you can focus on math and learn something new.
3. What can they expect to be different from their previous math classes?Obviously this will feel extremely different from the usually teacher in front of class explaining and droning out a lesson. Some new things implemented in this learning system is that you will be engaged in technology a lot so make sure you have a way to access a computer or internet. Instead of the teacher explaining the lesson, you will be learning the lesson at your own pace but at home therefore fulfilling the real definition of homework. In class, you will have a lot of free time to work ahead and do practice problems or ask Mrs. Kirch for a 1 on 1 lesson if you didn't understand the concept from last night.

Unit V Big Question

The difference quotient is the slope of a secant line.to derive it, you would find the slope of a secant line.for the slope formula, to find the value of m, you use y-y1/x-x1. Start with the y axis to find the coordinates.first point is x and the distance between that point from another point is h.the second point is x+h.to find the value of y,, if you start at x, the value going up would be f(x) because its a function.then you plug in the expression x+h into f(x) to find the second point.so in the end the coordinates are (x,f(x)) and(x+h,f(x+h)).plug these coordinates into the slope formula and you will get the derivative formula.

Unit U Big Questions

1) a continuity is a continuous function but a discontinuity is when there is a break in the graph or a hole.
2)a limit is the intended height of a function.it exists everywhere except at points of non-removable discontinuity.a limit is the intended height of that point but a value is the actual height.

 3) to solve numerically, we use a take that is 0.1,0.01,0.001 value away from the original point.graphically, we are already given a graph so we just look and label where the limit exists.algebraically we either plug it in through substitution, factoring, rationalize, or infinite.

Unit T Big Question 4

Sine and cosine don't have asymptotes because there are no restrictions to them and any # plugged in will produce a real value answer that can be drawn on a graph. The other trig functions have asymptotes because they're ratios are fractions such as secant=1/cos and tangent=sine/cosine. Fractions can never have a 0 at the bottom or it will become undefined and lets take tangent(y/x) for instance. It has asymptotes only when (0.1) or (0,-1) is chosen on the unit circles. secant also has asymptotes at those same coordinates because secant=1/cos and by plugging in the coordinates sec=1/0 which makes it undefined.

Unit T Big Question #3

To start off, we need to know that 0 to pi/2 is 1st quadrant, pi/2 to pi is 2nd, pi to 3pi/2 is third and 3pi/2 to 2r is fourth quadrant. According to tangent and cotangent 1st quad is +, 2nd is -, 3rd is +, and 4th is -.

On the left is a cotangent graph and it goes downhill because cot=x/y and it has asymptotes when y=0 which is (1,0) and (-1,0). In degrees, those coordinates are 0 and 180 so there are asymptotes are at 0 and pi or -pi. So from 0 to pi/2, cotangent is positive in the 1st quadrant so that part of the graph hugs up the asymptotes. From pi/2 to r, cotangent is negative in the 2nd quadrant so that part of the graph hugs down the asymptote. after connecting the two, we see that cotangent graph goes downhill.

On the left again is a tangent graph and it goes uphill because tan=y/x and it has asymptotes when x=0 which is (0,1) and (0,-1). In degrees, those coordinates are 90° and 270° so their asymptotes are at +/-pi/2 or +/-3pi/2. So from 0 to pi/2,tangent is positive in the 1st quadrant so that part of the graph hugs up the asymptotes. From -pi/2 to 0, cotangent is negative in the 4th quadrant so that part of the graph hugs down the asymptote. After connecting the two, we see that cotangent graph goes uphill.

Unit T Big Question #2

   Sine and cosine graphs are related to each other in which their amplitudes values restrict their range but the lack of asymptotes allow them to continuous travel across the x-axis.
   They are related to tangents because whenever cosine is 0, an asymptote is created and that will later give the tangent its distinctive uphill curve characteristics.
    Sine and Cosine are related to cotangents because whenver sin is 0, an asymptote is created that that shapes the downhill curve of a cotangent.
    A cosine graph is related to secant because it is sketched out and used to help draw out the secant graph. Also whenever cos=0, asymptotes form and that restricts the domain and how far the parabolas of a secant graph will stretch out too.

    A sine graph is related to cosecant because it is sketched out and used to help draw out the cosecant graph. Also whenever sin=0, asymptotes form and that restricts the domain and how far the parabolas of a secant graph will stretch out.

Unit T Big Question #1

Because unit circles are split into 4 sections/quadrants, each trig graph has its own look. Looking at the sin and cos graphs, we see that each x-value marker separates different quadrants and are degree angles after being converted. Quadrant 1, the first sin value is 1/2, next is rad2/2, and last is rad3/2. As you can see on the sin graph, the curve is going upwards from 0 to pi/2(Quadrant 1), then it goes down because the y-value is decreasing all the way down to -rad3/2 and comes back up at quadrant 4. For the cos graph and on the unit circle at 0, the beginning coordinates are (0,1) and in one rotation cos will end at (0,1) again or its amplitude. The curve for cos looks like that because it goes from 1 to rad3/2, rad2/2, 1/2, -1/2,-rad2/2, and -rad3/2. Afterwards it starts to go back up as we go counter clockwise on the unit circle from pi to 0.

The period for sine and cosine is 2pi because in one cycle it travels 2pi units on the x-axis, but tangent and cotangent is pi because in one cycle, it travels only pi units on the x-axis.

By having amplitudes, we figure out that sine and cosine have a restriction on the range values because on a unit circle, the sin and cos values can only up to 1 or -1 before it starts dropping down values. The other trig functions dont have amplitudes because they are restricted by asymptotes instead and these graphs are parabolas that continuously travel upwards or downwards which means no amplitudes to obstruct its path.