BQ #6 - UNIT U

1. What is continuity? What is discontinuity?
A continuous graph is a predictable, the graph will go where its intended to go. The graph has no jumps, holes or brakes. You can also draw this graph without lifting your pencil. As seen in the picture below.
   
A discontinuity is something that stops the graph from being continuous. There are four types of discontinuities. There are point discontinuities that when graphed it still reaches the intended point but it has no value(this is a removable discontinuity). These next three are non removable discontinuities. There are jump discontinuities which is where the limit reached two different points leaving a gap that looks like a jump. There are infinite discontinuities. These occur when there is a vertical asymptote which results in unbounded behavior, making the graph go towards infinity. and the last one is oscillating discontinuity, this graph just looks very wiggly. The reason why this is a discontinuity is because the graph doesn't know where it wants to go.

here we see how a jump discontinuity looks  then we see two holes and finally we see and infinite discontinuity




This is an oscillating discontinuity 

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 
the definition of a limit is the intended height of a function. The limit exists whenever both the left side of the function and the right side of the function meet. a limit does not exist in the non removable discontinuities because both sides of the graph do not get to the intended height. 
like said before, the limit is the intended height of the function, that does not mean that the point will always be there. the value is the actual number or the actual height that the graph reached.

3. How do we evaluate limits numerically, graphically, and algebraically?
You can evaluate limits numerically using and using the closes values to the number.
You evaluate the limits graphically by using the graph and with two different points you make those two points meet. if they meet then that's the limit, if they do not meet then the limit does not exist.
algebraically. there are different methods to evaluate a graph algebraically. The firs one is substitution, on here all you have to do is plug in the number and solve. this can give you four different answers. a number, 0, undefined, or indeterminate. if you get that last one then you have to try the other methods, if not, you're done with the problem. The other method is the factoring/divide method. Here you look to factor the function in order to find something to cancel. then after canceling then you try substitution again. the last method is the rationalizing/conjugate method. Here you use the factor with the radical sign and you rationalize it in order to be able to cancel some stuff out and then you can try substitution again.

Works cited
http://images.tutorcircle.com/cms/images/tcimages/types-of-discontinuity-in-calculus-functions-1330067235.jpg
http://www.milefoot.com/math/calculus/limits/images/oscdisc.gif
http://www.milefoot.com/math/calculus/limits/images/oscdisc.gif

BQ#4 – Unit T Concept 3

The way that a tangent and a cotangent graph go has to do a lot with the asymptotes. The position of the asymptote describes where the graph has to go. You also have to take into account how the way that the sign of the graph has been determined already. For that reason the "normal" tangent graph goes uphill and the "normal" tangent graph goes downhill. 

BQ#3 – Unit T Concepts 1-3

These two graphs when seen without a background the graphs look just the same. What differs these two are where the graph starts. Other than that the graphs are very similar. Both of them have two parts in the negative section and two in the positive one. The period for both of these are 2pi. They both have amplitude, for that reason, these two typed of graphs are very similar.

BQ#5 – Unit T Concepts 1-3: Asymptotes

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
First of all, we should be familiar that asymptotes happen whenever we get undefined. The only way to get undefined is to be divided by zero. If we refer to the circle ratios of sine and cosine the ratios would be sin=y/r and cos=x/r. We know that r will always equal one so there is no way that sine and cosine will randomly appear with a zero on the bottom. However, every other trig ratio have some value in which the bottom can equal zero, making it possible for them to have an asymptote. 

BQ#2 – Unit T Concept Intro: How do the trig graphs relate to the Unit Circle?

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The reason why sine and cosine have a period of 2pi is because that's how long it takes for a sine/cosine graph to repeat itself. The repetitions are based on the signs of the trig function and the unit circle. Whereas tangent, tangent repeats the sign pattern in the unit circle, so for it to complete a repeated cycle. So in 2pi there would be 2 graphs looking exactly the same for that reason only 1 pi is needed for the graph. 

 
Amplitude? – 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?
We have to remember that in the unit circle, a value greater than 1 and less than -1 would result as an error, because the values cannot be greater than that on the unit circle. for that reason those are used as amplitudes while graphing a sine/cosine graph. 


Works Cited
http://edunettech.blogspot.com/2013/10/the-graphs-of-six-trigonometric.html 

Reflection# 1: Unit Q, Trigonometric Identities

What does it actually mean to verify a trig identity?

When we are asked to verify an identity what we have to do is make sure that both sides equal each other exactly. we can do this by substituting trig functions with each other. No matter what, both sides equal the same thing. we just have to check that its true. 

What tips and tricks have you found helpful? 

Well the best tip for this unit is memorize the identities. If you know your identities you can look at a problem and be ahead of the problem. There were times where i knew what i was going to do in the problem two steps before i actually did them. This was thanks to me memorizing the identities. 

Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you. 

The first thing that i would see is what are the terms in which we have to end up with. Is it going to be a fraction? Do they cancel at the end? these are some questions that go through my mind in the beginning. then if i have a fraction i try to get rid of the bottom part if possible. then i try to set everything to sin and cosine. so that later i can just move those two around to turn them into the trig function it has to equal to.  

ID#3 Unit Q: Concept 1 : Pythagorean Identites

Where does where sin2x+cos2x=1 come from?
First we have to remember that the Pythagorean theorem is an identity, and what this means is that no matter what the formula will always be true. When we use x, y, and r means that we are basing things on the unit circle. for this reason the identity is equal to 1, because we are basing ourselves on how the unit circle always equals to one. we know that the ratio of cosine in the unit circle is x/r and the ratio of sine in the unit circle is y/r. What i notice is that everything of this is related to the unit circle and the ratios in it. for that reason is that the identity of sin2x+cos2x=1. 

Show and explain how to derive the two remaining Pythagorean Identities from sin2x+cos2x=1
One of the first things that you should do is to decide which trig function you want to derive. if it's sin firs you have to move sine to the other side and do the same with the one ending up with 1 - cos2x = sin2x. and you do the same thing for cosine ending up with 1 - sin2x = cos2x. 


 The connections that I see between Units N, O, P, and Q so far are that all the trigonometric functions in the end connect with each other. They all come back to its origins which for us was the unit circle. 


If I had to describe trigonometry in THREE words, they would be... Relationship, Angles, Confusing