BQ #7: Unit V: Derivatives and the Area Problem

1.  Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG

                                   

To find our derivative, which is the slope of all tangent lines, we use the difference quotient. How is the difference quotient derived? The first point is (x,f(x)). The second point is (x+h), f(x+h). The distance across is h. We then plug it into our slope formula and get (f(x+h)-f(x)/h). The h cancels out when we simplify so it concludes to be the difference quotient, which is f(x+h)-f(x)/h. H is also known as delta x. The difference quotient is used to find the possible slopes on any graph. A tangent line touches the graph once. On the other hand, a secant line touches the graph twice. To find the slope, we find the limit as h approaches 0. We do this by simply plugging in 0 for h. We then get a derivative, which we then can use to find values and the slope.

http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm

http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm

https://www.youtube.com/watch?v=iMaJDAV7as0
Here is an application of my explanation. 



Sources: 
https://www.youtube.com/watch?v=iMaJDAV7as0
http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm
http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG
                                                                   

BQ #6: Unit U

1. What is a continuity? What is a discontinuity?
A continuous function is predictable. It contains no breaks, no jumps, and no holes. It can be drawn without lifting a pencil. The intended height and the actual height of the graph are equal. On the other hand, a discontinuity has breaks, jumps, and holes. The intended height (limit) and actual height (value) are not equal to each other. The two families of discontinuity are removable and non-removable. The removable discontinuity is point discontinuity, in which there is a hole. The non-removable discontinuities are known as jump discontinuity, oscillating behavior, and infinite discontinuity. The limits don't exist in non-removable discontinuities. For jump discontinuity, the left and right limits are different, thus they don't exist. For oscillating behavior, it is very wiggly, so a limit cannot be determined because it does not approach a value. For a infinite asymptote, there is a vertical asymptote which means unbounded behavior.

Removable and non-removable discontinuities
(Please forgive my coffee stain, I think it's a sign that I need to cut back on coffee...)

2. What is a limit? When does a limit exist? When does a limit does not exist? What is the difference between a limit and a value?

A limit is the intended height of a function. It exists at point discontinuities. Why? Because limits exists from when the left and right side matches, and meet, therefore they have the same intended height. An actual height is the value, while the intended height is it limit. A limit does not exist, however, at the three non-removable discontinuities. The right and left limits aren't the same. In an oscillating is wiggle and does not reach a single value, therefore it does not have a limit. An infinite discontinuity does not have a limit because it has unbounded behavior due to a vertical asymptote.

http://www-rohan.sdsu.edu/~jmahaffy/courses/s00a/math121/lectures/limits_cont_deriv/limits.html

Infinite discontinuity; unbounded behavior because of vertical asymptote
Limit DNE

http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx
Oscillating behavior; no limit because does not go to single value
Limit DNE 

http://www.mathsisfun.com/calculus/limits.html
Jump discontinuty, limits from left and right don't match
LIMIT DNE 

3. How do we evaluate limits numerically, graphically, and algebraically?

Numerically
For evaluating limits numerically, we use a table. The number that X is approaching in the middle is what we are using to find the limit. First, we need to subtract .1 from the given number and that would be in the first end on the table (beginning). We add .1 to the given number to get the end on the table (last). Then, basically the numbers will get smaller as we go towards the middle. Soon, it will be clear the numbers approach a certain value that will be our limit. The limit may or not be reached depending on the function though.

Graphically 
To find a limit graphically, we basically use our fingers to see if the right and left side meet. If they don't meet, the limit does not exist. Below is a video that will show my explanation with a real graph for a visual. He also explains more into depth!

https://www.youtube.com/watch?v=aVcqrDFcaCA


Algebraically 
To find limits algebraically, there are three different ways. First is direct substitution. In direct substitution, you plug in the given number and see what you get. If you get 0, a numerical answer, undefined (limit DNE), then you are done. If you get 0/0, that is an indeterminate form, and you have to use another method! The next method is factoring method. You have to basically factor the numerator and denominator. After, cancel common terms, then plug in the given into the equation that is left. The next method is conjugate method, where you rationalize depending where the radical is in the numerator or denominator. You take the conjugate of what has a radical- either in the numerator or denominator. Then you multiply the entire fraction by the conjugate. Remember, you want to try direct substitution first because you don't want to go through a huge hassle if you can already get an answer using it. That is just a cheese bucket move, and no one wants to be called a cheese bucket because of this silly mistake!




Sources
https://www.youtube.com/watch?v=aVcqrDFcaCA
http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx
http://www.mathsisfun.com/calculus/limits.html
http://www.rohan.sdsu.edu/~jmahaffy/courses/s00a/math121/lectures/limits_cont_deriv/limits.html
SSS packet

BQ #4: Unit T Concept 3

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

Basically, a normal tangent graph is uphill because it is based on its asymptotes. They are located at pi/2 and 3pi/2. Furthermore, the quadrants are shown as positive, negative, positive, and negative (based on sin/cos). Cotangent is downhill because its asymptotes are located at 0 and pi. Because of where the asymptotes are located, it is downhill. The quadrants are positive, negative, positive, negative. The asymptotes are where sine and cosine are equal to 0 (making the ratio undefined).


                          Visual of Tangent 

(https://www.desmos.com/calculator)

As you can see, the colors represent the different quadrants (I,II,III,IV)

http://mathsm1m0838.edublogs.org/2009/08/01/the-tangent-graph/

Asymptotes at pi over 2 and 3pi over 2

  Visual of Cotangent

(https://www.desmos.com/calculator)

                                           

http://www.oocities.org/mathnuts/TrigAGModule1-TrigFunctions/graphs-general-tan-cot-sec-csc-sec6p8/graphs-general-tan-cot-sec-csc.htm

Asymptotes at 0 radians and pi

BQ#3 – Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?  Emphasize asymptotes in your response.

Tangent
Sine and cosine is related to tangent because the ratio identity for tangent is (sin/cos). The quadrants in the Unit Circle is shown in the tangent graph. Using our previous knowledge of the unit circle, we know where sine and cos is positive or negative in the quadrants.  If sine or cosine is negative, then tangent would be negative. If sine or cosine was positive or negative, tangent would be positive. In the first quadrant, sine and cosine are positive, so tangent is positive. In the second quadrant, cosine is negative, but sine is positive. But, overall tan is negative because cosine is negative. In the third quadrant, both sine and cosine are negative, making tangent positive (the negatives cancel out). The fourth quadrant for tan is negative because cosine is positive, but sine is negative, making tan negative. Tangent has asymptotes when cosine is equal to 0. Therefore, at pi/2 and 3pi/2, there are asymptotes. The asmymptotes repeat itself over and over again, as the domain goes on.

Visual of Tangent

(https://www.desmos.com/calculator)

Cotangent

In cotangent, the ratio identity is cosine/sine. It is very similar to tangent. Cosine and sine are positive in the first quadrant, so cotangent is positive. In the second section, the sine is positive and cosine is negative, so cotangent is negative. For the third quadrant, cosine and sine are negative, making cotangent positive (cancels out).  For the fourth quadrant, cos is positive and sine is negative, making cotangent negative. Sine must equal to 0 to find the asymptotes. Remember, asymptotes are undefined values. Therefore, we know that sine is equal to 0 are (1,0) and (-1,0). That is at 0 and pi. Cotangent is different from tangent because of the asymptotes. 

Visual of Cotangent

https://www.desmos.com/calculator)

Secant 

The reciprocal of secant is cosine. So the graph is just cosine flipped. So since cosine is positive in the first and last quadrant, and negative in the second and third, so is secant.  If cosine is positive, so is secant. Sine has no effect on the graph because it is not part of the ratio. The asymptotes are (0,1) which is pi over 2 and (0,-1) and 3pi over 2. This is where cosine is 0, making the ratio undefined. This is similar to the tangent graph, which had cosine as the denominator. 

Visual of Secant

(https://www.desmos.com/calculator)

Cosecant

Cosecant is just secant reversed. The ratio for cosecant is 1/sin. So basically, since sin is positive. so is cosecant in the first quadrant. In the second quadrant, cosecant is positive as well. In the third and fourth, cosecant is negative. The asympytotes are where sine is equal to 0 which is at (1,0) and (-1,0). Therefore, it is at 0 and pi. Remember, the graphs don't EVER touch the asymptote. 

Visual of Cosecant 

(https://www.desmos.com/calculator

BQ#5 – Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
          Sine and cosine do not have asymptotes because they are never undefined. Remember, sine means y/r and cosine means x/r. R stands for the hypotenuse, and in the unit circle the radius is always 1, which is the hypotenuse. The other four trig graphs do have asmpytotes because they do not have 1 (r) as their denominator. Instead, their denominators can be any numbers, according to their trig ratio. For example, tangent's ratio y/x. X can be any number; there are no restrictions.
        The denominator, however, can be 0 (from the coordinates (0,1) or (-1,0), which makes the ratio undefined once in the according trig function. In cotangent, the ratio x/y also can assume asymptotes. If the coordinate is at (-1,0) or (1,0), when plugged into the ratio, it will be either positive or negative 1 over 0, which is undefined. Also, this applies to secant. Secant is r/y. Y is 0 at (1,0) or (-1,0). This results to be 1 over 0, which makes it undefined. In cosecant, the ratio is r/x. X is 0 as (0,1) and (0,-1). Thus, when plugged in, the ratio is 1/0, making is undefined as well. An asymptote is basically when the ratio is divided by 0, meaning its undefined. 

BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?
So, you may ask, how do trig graphs relate to the unit circle? Well, it uses the positive and negative signs of the quadrants of the trig function! The trig graphs are basically the unit circle unwrapped or uncoiled to a line. The points on the graph are relevant to the points in the unit circle (radians). FOR EXAMPLE, in the unit circle, sine is positive in the first two quadrants, and negative in the third and fourth concepts (we know this from our previous units).  Thus, on a trig graph, the graph will show a pattern of going uphill, uphill, downhill, downhill. This represents the quadrants. It is positive in the first two quadrants and negative in the other two. Lets use another example. For cosine, in the unit circle it is positive in the first quadrant, negative in the second, negative in the third, and positive in the fourth. So on the trig graph, the period is positive and then goes negative, and eventually goes back positive according to the marks. For tangent and cotangent, it will have positive values first, then negative, then positive, then negative. The trig graph has a shorter period and this is because in the first two quadrants, it goes positive then negative, and then it is basically repeated in the other two quadrants. 

Beautiful visuals of the trig graphs and the patterns that I talked about in my earlier paragraph.

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

The period for sine and cosine is 2pi because basically when you look at the unit circle, sine is positive, positive, negative, negative on the quadrants. In cosine, it is positive, negative. negative, positive. The period is 2pi and it takes all the quadrants (one whole revolution) to finish the pattern. But, for tangent and cotangent the pattern is positive negative. If it is positive and negative in the first two quadrants, and it repeats after that, then basically it only takes the period of pi in a revolution, in which it repeats itself twice.

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?

Sine and cosine have amplitudes of one. Why? Well think about what sine and cosine consists of. Sine is y/r and cosine is x/r. R is the hypotenuse, which is 1. Sine and cosine cannot be greater than one or less than one. On the unit circle, y and x are either 1 or -1. When you divide using the given ratios, it still equals to 1 or -1. So that clarifies that the amplitude is always one or negative one because of this rule. But, for other trig functions it is different. There are boundaries to what values it must equal to because x and y can vary in different numbers. For example, cosecant, it is r/y. Your y CAN be 1 or -1, OR it can be any other number like 2 or 3. Same goes to the other trig functions. So there are no amplitudes. 

Reflection #1: Unit Q-Verifying Trig Identities

1. What does it mean to verify a trig identity?
To me, verifying an identity basically means to prove that an equation is true by showing that both sides are equal. I think of it as taking steps to simplify one side of the equation to make it equal to the other. Verifying a trig identity takes a gradual process of manipulation, alterations, and logical thinking! It may be difficult at first, but all you have to do is to make both sides equal and there are multiple steps so it is not as difficult as other math units.
2. What tips and tricks have you found helpful? 
I found that remembering the identities (ratio, reciprocal, Pythagorean) make verifying a trig identity much faster. Also, I learned that it's a little different than other units we have went through, because there are many ways to verify an identity and it's very ambiguous. Usually, we are usually used to using a formula and getting our answer right away, but with this we basically we have trial and error. Another tip is to not touch the other side! You simply cannot, no matter how tempting it is. 
3. Explain your thought process and steps you take in verifying a trig identity.
First, I look at the problem and I try to convert anything to a trig identity of sine or cosine.  Next, I usually look to see if I can take out a LCD (least common denominator). I factor if I need it. Then, I usually check if I can use any identities, especially Pythagorean identities. I also find that powering up can lead me to the Pythagorean identity. I sometimes multiply by conjugate if there is any fraction. If I get confused at all, I usually go to three steps back and review what I did. If I can't make sense of it, I usually start from scratch because I don't want to over complicate the equation and continue if I accidentally made an error.