BQ#7: Unit V: Derivatives

Where does the difference quotient come from and what do you know about it?

          The difference quotient has a nickname called the derivative which is derived from the slope formula of (Y^2-y^1)/(x^2-x^1). The reason why we use the slope formula is because the derivative is the slope of all the tangent lines. A tangent line touches the graph one time. Looking at our formula for the difference quotient is f(x+h)-f(x)/h. We have two points that we use in our slope formula. Our first x point is simply just x. While our first y value is f(x) As we move along the x axis we have our letter H which can  be referred to as delta x. Our second points are  used from the original ones but including h because of the shift. So our points would be (x+h) and then our second point would be f(x+h) We then plug this into our slope formula and get f(x+h)-f(x)/f(x+h)-x we then simply our x values in the denominator and whats left is our difference quotient  f(x+h)-f(x)/h.




 


 
 

 

By looking at the picture above our delta x or our h has to be moved closer to be able to be touching which is the secant line that touches two points. In the SSS videos our delta x has to be moved closer and closer in order for the points to be directly touching.


Reference:
http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg
http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG
http://images.tutorvista.com/cms/images/39/difference-quotient-formula.png

Unit U: BQ#6

#1: What is continuity? What is discontinuity?
A continuity is a graph that has no jumps, no breaks, and no holes. The graph itself is predictable so you can draw it without lifting your pencil off the paper. A discontinuity has two families which include a removable discontinuity  and non-removable discontinuities . Removable discontinuity is consistent of the point discontinuity. This discontinuity has a hole. Non-removable discontinuities include jump. oscillating, and infinite discontinuity.  Jump discontinuities have different left and right because they never meet up. Oscilatting is a wiggly line. Infinite discontinuity has unbounded behavior and vertical asymptotes.

#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 limit is the intended height of a function. A limit exists on a continuous graph or the removable discontinuity. With the left and right going towards the limit, it is able to exist. The limit doesn't exist at non-removable continuities, which are stated above as jump, oscilatting and infinite. A limit doesn't exist on jump because of different left and right, they never meet up so there is no limit. On oscillating theres no single point because its wiggly and keeps on moving. A limit doesn't exist as infinite discontinuity because of unbounded behavior and its vertical asymptote.

The limit is the intended height of a function while the value is the actual height of a function. Taking the first graph as an example there is a limit ,but theres no value since theres a hole. In some cases, for example the graph below the first one there will be a different limit and a different value somewhere else. The jump discontinuity has no limit because it jumps, it has no meeting place, however the value still exists at the closed circle. Oscillating and infinite discontinuity are never at an exact point.

#3: How do we evaluate limits numerically, graphically, and algebraically?
To evaluate limits graphically you can create a table, you choose your base number, then add and subtract by a tenth. So for example we can take the number 5 if we subtract a tenth from 5 it becomes 4.9. Once we add a tenth to 5 it becomes 5.1. Basically you want to get as close to your base number of 5 So to get closer from 4.9 would be 4.99 and 4.999. Closer to 5.1 would be 5.01, and 5.001.

To evaluate graphically we take our fingers from left and right and work our way towards the center.

Algebraically we used direct substitution in which we plug it in and solve, If that doesnt work we go to the factoring method in which we factor out the numerator and denominator and cancel terms to remove the zero in the denominator. Then we can use the rationalizing method in which we change the sign and multiply by our square root problems.
 
Reference: SSS Packet

BQ#4 Unit T: Tan and Cot

Why is a "normal" tangent graph uphill, but a "normal cotangent graph is downhill? Use unit circle ratios to explain.

We know that our Tangent value is sin/cos or y/x. Our cotangent value is x/y or cos/sin. In order to for tan and cot to go their directions we must have our asymptotes in which for tan the x value has to be 0 while for cot our y must be 0. Tangent is 0 at the degrees of 90 and 270 because we take our coordinates of our quadrant angles. 90 degrees has a coordinate of (0,1) and 270 has a point of (0,-1). Once we plug this into our equation for both we get 1/0 and -1/0 which is our asymptote. These degrees also have radians in which 90 is pi/2 and 270 is 3pi/2. These serve as our asymptotes on our graph and is represented by a dotted line. The same thing applies to cotangent in which our value is x/y in which the degrees of 0 and 180 are our asymptotes. Plugging in our values we get 1/0 and -1/0. 0 has no asymptote on the plane while 180 lies on pi.

 




 



 
 
  By looking at our graph our asymptote for tan is placed at pi/2 and 3pi/2. These serve us as our walls which tan goes upwards. Tan also goes upward between 0 and pi/2. Cot goes downhill from its first asymptote  at zero. Cot also goes down at 180 which lies on pi.

Bq#3 Unit T: Relating graphs

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

A)Tangent

  Sin and cos are related to tan by looking at our trig values. Since Tan=Sin/cos or y/x By looking at our value of cosine it never touches our origin which is zero because it would cause the value to become an asymptote and become undefined. However our tan graph is going by certain amounts in which our asymptotes can act like a wall which can guide and lead the tan graph to go to that certain amount. Cosine never intersects with any of the other graphs.


 B) Cotangent

 





Cotangent has the value of cos/sin or x/y. We can see that sin goes from 0 and touches every one value of pi going to pi 2pi 3pi etc. We can see our cot  starts from the positive and transcends into negative within 0 and pi and continues which is our wall that restrains the graph and keeps it contained. Tan and cot are mirrors of the direction of the graph going up or down.
 
C) Secant

 








Secant has the value of r/x or r/cos. When cos is 0 it lies on pi/2 3pi/2 and 5pi/2. In these areas sec is split by parabolas and are split by the asymptotes into their own sections. Starting by going up then down and repeats the pattern of the direction of sec. Sin and cos act as a measurement between the graphs of secant.
 
D) Cosecant

 


Cosecant has the value of r/y or r/sin. The asymptotes lie on every pi such as 2pi 3pi etcwhich results in seperate graphs according to csc. Sin is 0 on the values of pi which is 1/0 which is an asymptote because its undefined. The pattern repeats of up and down.

Bq#5 Unit T: 6 Trig Graphs

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain.
     
        Sin and cosine don't have asymptotes simply by the reason of that their circle ratios are composed of having a denominator of r which is equal to one. Sin has a circle ratio is y/r and cosine has a value of x/r. Referring back to our unit circle, r has a value of one, so therefore whatever the value of x and y will be that value because the denominator is automatically one. In order to obtain an asymptote, the value must be undefined in which leads to having zero on the bottom which creates our asymptote. Other trig functions are able to have asymptotes because of their natural unit circle ratios.  Csc is r/y in which y can equal to 0 with the 90 degree angle which leads to 1/0 and is an asymptote. Secant is r/x in which you can use 270 to get the value of 1/0 which results in undefined.  Tangent and cotangent are the same in that you can mix up the values, as long as you have 0 on the bottom you can achieve an asymptote since dividing by 0 is undefined.

BQ#2 Unit T Unit Circle Introductions

How do the trig graphs relate to the Unit Circle?
a. Period? Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
          Trig graphs relate to a Unit circle in that it contains quadrants from the unit circle and can be created with the use of a computer cable.

Using all students take calculus Sin is positive in the first and second quadrant while sin is negative in the third and fourth quadrant. For cosine the first quadrant is positive, the second is negative and then it repeats itself by being negative then positive in the third and fourth quadrant. Tangent and Cotangent is positive, negative, positive and then negative. Our unit circle is also consistent of having quadrant angles split by 4 sections on the axis. For sine the first quadrant is to pi/2 then to the other half becomes pi. which is half of the unit circle, going into the 270 quadrant angle is 3pi/4 and completing at 360 is 2pi. Here if you look at the picture the quadrants are divided along the line by the quadrants and their values according to the trig function. When you look at the lines going up and down on the line, this is based on our sin value of the quadrants both being positive and then negative in the 3rd and 4th quadrant. We can see this by my lines indicated on the line, after the second one there's a "valley" in which the sin values are negative so they go below. Sin and cosine are 2pi because they don't have a complete pattern like tan and cot. Sin is positive positive then negative negative, so it has to go through a whole go around the unit circle, same thing applies to cosine in which it doesn't complete its pattern. While tan and cot complete its pattern of being positive negative then positive negative.

b) Amplitude?-How does the fact that sin and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about our unit circle?
Sin and cosine have restrictions on their values as -1<x<1 Since we know the unit circle has an r value of one hence our trig values for sin is y/r and x/r belongs to cos. If we get x or y values greater then -1 or less than -1 we get error and it doesn't work, meanwhile the other trig functions you can mix and match the values of the unit circle.

Reflection:#1: Unit Q verifying trig functions

1.What does it actually mean to verify a trig identity?
To verify a trig identity is to start off with what you're given and turn it around and adjust the given values to make it equal to what the trig identity is giving us. With verifying we can use our reciprocal trig values or sub in with identities. We're open to numerous different ways to verify the different identities.


2.What tips and tricks have you found helpful?
Tips and tricks I've used are basically try every single way possible and test things that you've never done before. Other tips include just knowing your identities and what values they are and make sure to not mistake for subtraction signs and addition signs.

3.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.
My thought process just include depending on the concept. For concept 4 I wonder if I'm able to factor out anything such as a trig function or if I'm able to use the zero product property. I also look for any functions I can convert and see if it cancels with anything or it can add too my current problem. I also check if I should square both sides and check for extraneous solutions. For problems like concept 5 I include the same thought process, except i put the right side as a no no touch boundary and only convert the left side if possible.