BQ #7 - Unit V Difference Quotient

The difference quotient is derived by using the slope formula: (y2-y1/x2-x1). Looking at the video above, from the graph that he drew, the first point is called "x1" and the "h" is the distance from one point to the other. That point would then be "x+h" ( this would be x2). The y values arefound by looking at the function, f(x). We plug in the x1 value "x" and we would get f(x). For our x2, we plug in "x+h" into our function and we would get f(x+h).

y2= f(x+h)
y1= f(x)
x2= "x+h"
x1= "x"

From there, we simply plug in those into the slope formula.

We would get f(x+h) - f(x) over x+h-x. The x's on the bottom would cancel each other out. And we are left with f(x+h)-f(x)/h which is the difference quotient.

BQ #6 - Unit U Limits

A continuity is predicable, has no jumps/breaks/holes, and can be drawn without lifting up the pencil. A discontinuity has jumps, breaks and holes and is drawn having to lift up the pencil. There are two families of discontinuities: removable and non-removable. Removable discontinuity is a point discontinuity, it looks like a hole in the graph (similar to a previous unit of when we looked at asymptotes). In a non-removable discontinuity, there is a jump discontinuity (where one graph ends and then jumps to a different location and begins again), oscillating discontinuity (a wiggly graph), and infinite discontinuity (vertical asymptotes).

Point Discontinuity

Jump Discontinuity

Oscillating Discontinuity

Infinite Discontinuity

A limit is the intended height of a function. It exists when both the left and the right meet at the same point. A limit does not exist when the left and the right do not meet at the same point, when there is unbounded behavior, and when there is oscillating behavior. A limit is the intended height of the function while the value is the actual height of the function.

As you can see in this picture, the limit is the intended height. The white circle in this function is a point discontinuity, the function intends to go to the white circle. Also if you use your left and right fingers to trace the left and right sides of the graph, you will meet at the same place, that is the limit. However, in this graph, the value is at a different place, the actual height of the function, the blue circle. In some cases, the limit and the value can be the same, like if the white circle had been a blue circle instead.

When evaluating limits numerically, we set up a table. When evaluating a limit graphically, we use a graph and use our fingers to trace the graph from the left and the right of the number we are approaching. When evaluating a graph algebraically, we are using the methods of substitution (plugging in a number, the number that x is approaching), factoring/dividing out (factoring out and canceling, plugging in answer into function), and rationalizing/conjugate (rationalizing/multiplying by conjugate, plugging back in answer into function).

Numerically

Graphically

Substitution Method

Factoring/Dividing Method

Rationalizing/Conjugate Method

Credit:

http://www.conservapedia.com/images/2/2f/Br-cont-function.png

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/4a69dec7-03e0-492f-ac16-4dcd555579c9.gif
http://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png
http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg
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http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/12cf828c-12be-4ace-a420-21bd21aeb8c8.gif
https://finitemathematics.wikispaces.hcpss.org/file/view/limit_table.PNG/239144381/575x194/limit_table.PNG
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCUbZ0eJareDU5yWUxAZMRubLi5PC81ZW3WVwF2aXfRHXfEWh3DjZD2e-_DGoW8kPbJplcMPVCYkCmk8Jr4s1Ons-PCsyQ1B-ExYnitDf2V-zX-wJHweCxVIttUGGOi61Kn2ma0rf_uZSZ/s400/LimitsGraphically011.jpg
http://evaluationoflimits.weebly.com/uploads/1/3/9/2/13920575/891412.jpg?424
http://00.edu-cdn.com/files/static/mcgrawhillprof/9780071624756/EVALUATING_LIMITS_09.GIF
http://00.edu-cdn.com/files/static/mcgrawhillprof/9780071624756/EVALUATING_LIMITS_11.GIF

BQ#5 – Unit T Concepts 1-3

Sine and cosine graphs do not have asymptotes. The reason being is that when looking back at Unit Circle ratios, sine is y/r and cosine is x/r. "r" was always one no matter what and for an asymptote to occur, the answer must be undefined. However because "r" is always one and never zero, sine and cosine graphs do not have asymptotes.

In cosecant and cotangent graphs, the respective ratios are 1/sine and cosine/sine. Their graphs have asymptotes because sine can equal to zero at 0 and pi. Because sine can be zero, when using the ratios, we can never divide by zero because it leads to undefined answers. This is the reason why cosecant and tangent graphs have asymptotes.

In secant and tangent graphs, the respective ratios are 1/cosine and sine/cosine. Their graphs have asymptotes because cosine can equal to zero at 90 (pi/2) and 270 (3pi/2). Because cosine can be zero, when using the ratios, we can never divide by a zero because that is undefined. Because it is undefined, that means there is an asymptote, which is why those graphs have asymptotes.

BQ#4 - Unit T - Concept 3

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

In the Unit Circle ratios of a tangent graph, it is sin(x) over cosin(x) which is the same as y/x. On the graph for tangent, when cosine or x equals 0, which is 90 and 270, there is an asymptote. Because there is an asymptote in those areas, the graph is uphill.

In the Unit Circle ratios for a cotangent graph, it is the reciprocal of tangent, meaning the raio is x/y. On the graph for cotangent, when sine or y equals 0, which is 0 and 180, there is an asymptote. Because there is an asymptote in those areas, the graph is downhill.

BQ#3 – Unit T Concepts 1-3

Tangent: The proportion for tangent is sin(x)/cos(x). In the first quad, sine and cosine are positive and in that ratio, that also means that tangent is positive. In the second quad, sine is positive and cosine is negative and in that ratio, tangent is negative. In the third quad, sine is negative and so is cosine and that means that tangent is negative. In the fourth quad, sine is negative and cosine is positive and that means that tangent is negative. When cosine is equal to zero, that makes the ratio undefined which also means that there will be an asymptote. The asymptotes for tangent are based on sine and cosine, cosine equals to zero on 90* (pi/2) and 270* (3pi/2). 

Cotangent: The ratio for cotangent is cos(x)/sin(x). Based on ASTC, the first quad is positive, the second is negative, the third is positive, and the fourth is negative. For this ratio, when sine is equaled to zero, there is an asymptote. The asymptotes for cotangent exist on 0 and 180 (pi). 

Secant: The ratio for secant is 1/cos(x). Using the ASTC, cosine in the first quad is positive so secant is positive. In the second quad, cosine is negative so secant is negative. In the third quad, cosine is negative so secant is negative. In the fourth quad, cosine is positive so secant is positive. The asymptotes of secant are based on 90* (pi/2) and 270* (3pi/2) because asymptotes exist when the ratio is undefined which is when cosine is equaled to zero.

Cosecant: The ratio is 1/sin(x). Using the ASTC, sine is positive in the first quad so cosecant is positive. In the second quad, sine is positive so cosecant is also positive. In the third quad, sine is negative so cosecant is also negative. In the fourth quad, sine is negative and so is cosecant. The asymptote is based on the sine so when sine is equaled to zero, that means there is an asymptote. The asymptotes for cosecant exist in 0 and 180 (pi).

BQ#2 – Unit T Concept Intro

In this unit, there is a relationship between the Unit Circle and the period for the trig functions. For sine and cosine, their periods are 2pi. The reason for this is because a period is a repeating pattern. Using ASTC, sine is positive, positive, negative, negative. The cycle of that is pi. However, there is no repeating pattern so in order for there to be a period, we must go around the Unit Circle again, which is why it is 2pi. It is the same idea as AABB, that is not a period because there is not repeating pattern. So we must then repeat that entire sequence again for it to be a period: AABB AABB.


It is the same idea for cosine. There is not repeating pattern in cosine: positive, negative, negative positive. Because there is not repeating pattern, we must go around the circle again in order for there to be a period thus a full period for cosine is 2pi. Take ABBA for example, there is not repeating pattern so we must repeat the cycle again in order for there to be a period: ABBA ABBA.

Tangent and cotangent's period however, is not 2pi. It is just pi. The reason for that is because there is a repeating pattern in the first rotation using ASTC: positive, negative, positive, negative. There is a repeating pattern of positive and negative thus only having to go half the cycle compared to sine and cosine, thus the full period of tangent/cotangent is only pi.

Sine and cosine both have amplitudes of one in their graphs. The reason for that is think back to the Unit Circle, the largest number on the Unit Circle was 1 and the smallest number was -1. Those are the largest and smallest that the ratios of sine (y/r) and cosine (x/r) can be, thus the amplitudes of their graphs are restricted to only 1 and -1. However, the other trig functions do not have these restrictions which is why their amplitudes can surpass 1 and -1.

Reflection #1 - Unit Q: Verifying Trig Identities

1. To verify a trig identity is to be able to know how to manipulate what is given to you and find a way to create what is given to the answer. To be able to use knowledge of identities to substitute, cancel, or factor/multiply to get to the answer. For example, being able to see that tanx is sinx/cosx, or seeing that sin^2=cos^2-1. 
2. Some tips and tricks is just changing everything I can to sin and cos, by doing that I get a clearer image of what I can substitute with or cancel. Another important tip is to memorize the identities, to be able to look at a problem and be able to know what the identity to use without having to second-guess. I have found that looking at the problem in separate pieces to help a lot because I am not overwhelmed by the information. 
3. My thought process begins with me trying to find what I can substitute to best help me cancel things out or will just overall make the problem easier to look at. If not, then I would then try to see if moving things to one side would make an identity or would help me see what I can do to find the answer. I would try to see if I can divide/multiply by certain things like a fraction or a conjugate denominator to cancel things out or factor. I would also look for a GCF because that could probably lead to an identity. The last resort is to square things and I try to avoid doing that so I don't have to worry about having to remember to check for extraneous answers, but if I have to, then I would try to see if squaring both sides would bring me to the answer.