Dear Math Analysis Student

Dear Math Analysis Student,

1. What do you want to say to them to help them have the most successful year possible?
   In Mrs, Kirch's class, the best thing for you to do to be successful is just work hard. Basically, everything outlined in the KIRCH guidelines for success will help you to succeed. Put great effort into all the work you do in her class. Meaning, don't half-_ _ _ things in her class. A tip that I wish I would have stuck with the entire year is BE ORGANIZED. Every paper, every PQ, every SSS packet you will need to organize and keep track of it all. So I suggest start organizing from the very first day until your very last.
   Another tip is do your math work neatly. You'd be surprised how just your handwriting makes a difference in how many mistakes you make and it helps you get done faster. Pay special attention to the R the Kirch's guidelines for success. Remain honest. It is one important thing all students will have to learn and you decide if you want to learn the easy way or the hard way.
   
   
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 the flipped classroom is be sure you have a good camera, fast internet connection, and a working computer. Literally, those three things are your lifelines in Mrs. Kirch's class. A specific tip when working with technology and videos is to film your video a day before instead of the night of because, trust me, you will be up all night uploading that thing and it can save you from A LOT of stress. Also, I used youtube to upload my videos and i found that a few times i had to wait over 200 minutes on youtube to upload my video. Keep them short and sweet but hit all the points. Other than that, having a blog and having so much freedom to present the math is way more exciting.
   
3. What can they expect to be different from their previous math classes?
   You'll find that the flipped classroom teaches you MORE in LESS the amount of time. The flipped classroom is basically where you will be watching videos at home in which Mrs. Kirch will teach you the math concepts and you submit a response to the video, proving that you watched it, but also proving that you understood the concept and all key points. Then, you go to class the next day and talk about what you watched the night before and work out problems on your own. Yes, it is different from your traditional classrooms but it does not compare in how you come to understand the information in a flipped classroom. In my own opinion, Math Analysis was my favorite class because Mrs. Kirch has you doing new assignments youve never done before including a blog and making videos
   
   ~If there is a will, there is a way.~

Unit V Big Questions Blog Post

The video above explains in great detail where the formula for the difference quotient comes from. Basically, the difference quotient represents a formula that tells us the slope of a line that is secant to the graph, meaning it touches the graph twice. However, when we EVALUATE THE LIMIT of the difference quotient we are finding the slope of a line tangent to the graph. This is also known as the derivative. Additionally, this is called using the limit definition of the derivative. So, we are basically using the slope formula to derive the difference quotient. As you can see in the picture below, the secant line has two points that touch the graph. If we want to find the slope of these two points we easily use the slope formula derived from delta y over delta x (also known as rise over run) y sub 2 -y sub 1/x sub 2 - x sub 1. From there, we get our difference quotient and if we want to take it to the next level................

....WE FIND THE DERIVATIVE! In the picture below, you can see how as the h approaches 0, the secant line approaches the tangent line and that is why we evaluate the limit as h->0. As you can see, the tangent line only touches the graph ONCE. This new derivative is called f'(x) and is read as "f prime of x".

Works Cited:
"Deriving the Difference Quotient ... MathWOEs" on youtube.com
Pictures taken from Mrs. Kirch's SSS Packet Unit V

Unit U Big Question #1

1. What is continuity? What is discontinuity?

A graph that is continuous means we can draw the entire thing without lifting our pencil from the paper. It has no holes, no breaks, no jumps in the graph. The continuous graph is predictable and makes a good bridge because a person "driving" over it would not fall through or fall off.

On the other hand....a discontinuous graph is the exact opposite. We have four types of discontinuities divided into two families.

• Removable Discontinuities
• Wherever there is a HOLE in the graph, it is called a point discontinuity.
  
• Non-Removable Discontinuities
• Jump discontinuities includes a break in the graph. 
• Oscillating behavior occurs when the graph is "wiggly" and unpredictable.
• Infinite discontinuity occurs whenever there is a vertical asymptote and is also known as unbounded behavior.
  

(Images from education.com, wyzant.com, and en.wikipedia.org)

Unit U Big Question #2

2. What is a limit? 

A limit is the intended height of a function at a certain value. Basically what this means is it is the y-value at which a function is approaching on the x-axis but may or may not reach it but it is intended to reach it.

 When does a limit exist? When does a limit not exist?

A limit exists as long at the same height (y-value) is reached from both the left and the right. In other words, the limit exists if both the right hand limit and the left hand limit are the same. If the graph does not break at a certain x-value, then the limit exists. For instance, at a hole, the limit exists because the intended height was still reached, even if the "diner" (in the car analogy) is not there.

 

   

 

(Image from curvebank.calstatela.edu)

What is the difference between a limit and a value? 

 

(Image from Unit U SSS packet)  

 The limit is the intended height and the value is the real or actual height. As you can see in the picture above, the limit and value are two different things. We can have the limit equal the value on a continuous graph because the limit is always reached and there are not breaks OR holes in the graph.

Unit U Big Question #3

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

NUMERICALLY

When we evaluate limits numerically, it is on a table. We are finding the intended height of the function, which is the y-value as the graph, and it is referred to as the LIMIT. We use a table similar to the one in the picture above. As x is approaching a certain value, in this case 3, we write numbers approaching the limit from the left and from the right. This way, we ALWAYS find what the limit is intending to reach even if the graph never actually reaches that point. (Image from Unit U SSS Packet)

GRAPHICALLY

(Images from www.education.com)

When we evaluate limits graphically, we are looking at a graph and using our fingers to approach the limit from the left and right to see if they meet.

ALGEBRAICALLY

There are three ways of evaluating limits algebraically: direct substitution method, dividing out/factoring method, and rationalizing/conjugate method.   

1.  Direct substitution is the method used first. We take the limit and plug in that number into all the x's in the equation to find an answer. We can get a numerical answer which is what we hope for and we are finished. We can get 0/# and this means our limit is 0 and we are finished. We can get #/0 which means the limit is undefined and DNE because of unbounded behavior. And lastly, we can get 0/0 which means we are not finished. 0/0 is indeterminate and we need to use one of the other methods to evaluate it.

2. Dividing out/factoring method is used after we get indeterminate form (0/0).  We factor the numerator and denominator and cancel any common terms on top and bottom. When we do this, we are removing the zero in the denominator. Then, we use direct substitution with the simplified expression. This is similar to finding a hole in a past unit.

3. Rationalizing/conjugate method is used when the fraction has a radical in it. We use the conjugate of either the numerator or the denominator depending on where the radical is. We foil the conjugates and leave the non-conjugate factored. Then, we use direct substitution.

Unit T Big Question #4

Why DON’T sine and cosine have asymptotes, but the other four trig graphs do? 

As you can see, sine and cosine have ratios with "r" in the denominator. We know that from the Unit Circle, "r" IS ALWAYS ONE! Therefore, we can never have "r" being zero and we can never have sine or cosine being undefined. For the other graphs, however, we have either y or x as the denominator. This means that we have possibilities of having a denominator being zero (at the four quadrant angles) and having undefined answers. And we know that undefined= asymptotes. On the other hand, since "r" is always 1 we know that sine and cosine can't be undefined and will not have asymptotes restricting the graph.

Unit T Big Question #3

This video goes over the question: why is a "normal" tangent graph is uphill, but a "normal cotangent graph downhill?
Well, we need to consider two major reasons: 1) the asymptotes of a cotangent and tangent graph. The asymptotes of these two graphs is in different places sine cotangent asymptotes correspond with whenever SINE is zero. While tangent's asymptotes correspond for when COSINE is zero. And 2) We must keep in mind the rules for ASTC and the pattern for cotangent and tangent. The pattern on + - + - and whenever the asymptotes are placed will affect each graph (tangent and cotangent) differently. Pay close attention to how tangent is shaped and how cotangent is shaped once we consider all the restrictions including where the asymptotes cut us off and when tangent/cotangent needs to be above the x-axis (positive) or below the x-axis (negative).

Unit T Big Questions #2

How do the graphs of sine and cosine relate to each of the others? 

Tangent? Cotangent? 

 

The graphs of sine and cosine affect secant and cosecant in almost the same way. Since we have ratios that have sinx and cosx on the bottom we take into consideration any possibilities of it being undefined. Because undefined=asymptotes. So, when we have cosine being zero (pi/2 and 3pi/2) we have asymptotes affecting the graph of secant. When sine is 0 (0 and pi) we have asymptotes for cosecant that affect its graph.

Unit T Big Question #1

How do the trig graphs relate to the unit circle?

Part 1a:
Why is the period for sine and cosine 2pi, wheras the period for tangent and cotangent is pi? 

•  For sine and cosine graphs, it takes 4 parts in order for its "pattern" to repeat. For instance, the picture shows sine's pattern: ++--. That is one period. The period is marked at the point that the pattern is about to repeat. This basically means that for sine and cosine, it will take a whole rotation of the unit circle in order for the graph to repeat itself. (Hence, 4 parts and 4 quadrants). This is one period. Therefore, for sine and cosine, a period goes through 2pi units on the x-axis (also 360* and one revolution of the unit circle.)

• Tangent and cotangent are different. While their pattern goes + - + -. It only takes two parts for the graph to repeat itself: + - and then we go again with + -. We do not need the other two "parts" or we would be going into a SECOND period of the graph. Therefore, we only need half of the unit circle( pi and also 180*) to cover a cycle. One period for tangent and cotangent would cover only two "parts" and only pi units on the x-axis.

  Part 1b: How does the fact that sine and cosine have amplitudes of 1 relate to what we know about the Unit Circle?
  *First of all, we must remember that amplitude is HALF the distance between the highest and lowest points on the graph. *
  

  If we look closely, the Unit Circle resembles the graph greatly. That is because the Unit Circle in "line" form would be that graph and it is like we are stretching it out. If we notice, sine has two amplitudes here. One at pi/2 and another at 3pi/2. We notice that on the Unit Circle at those exact values the sine is 1. That is where we get our amplitude from.  
  
  
  
  As you can see, cosine has three amplitudes. At 0, at pi, and at 2pi. If we look at the Unit Circle, cosine begins at 0, goes halfway to pi and then ends at 2pi. If we notice, the cosine of those values is 1 and -1. Which correspond with the graph. The Unit Circle is a great reference to make sure your graphs are on point as the graphs are simply just the Unit Circle laid out flat.
  
  

  
  
  

  Since cotangent, tangent, cosecant, and secant all have asymptotes, they do not have amplitudes like sine and cosine. As you can see, where sine and cosine equal zero, there will not be amplitudes but instead....ASYMPTOTES.
  

Student Video #5: Unit S Concept 7-Solving equations with half-angle formulas

• What is this problem about?
• This problem goes over solving equations with half-angle formulas. We are given an equation and we must use our half-angle formulas to substitute in and also we need to reference our identities from Unit Q. Then, we solve for the angles using the Unit Circle. This is similar to past units and we are trying to get our equation down to where we have only one trig function or down to the point where all we need to do is Zero Product Property.

• What does the viewer need to pay special attention to? 
• The viewer must pay special attention to taking the square root of a fraction because we must remember to include both the positive and the negative answers. So, we would take the angle from all four quadrants since we have both positive and negative answers. Also, if you square both sides of an equation you have to remember to check for extraneous solutions.

Assessment #2: Unit S Concept 4

Solving exact value of an angle using

Half-angle
formulas

Solving exact values of an angle using the

Sum formula!

How do we know these are both the same answers?? 

We know that the answers from both pictures are the same because we can check it by plugging it correctly into a calculator. By plugging it in and then plugging in the sin, cos, and tan of 75° we can see that they are the exact same answer. In fact, the tangent of 75° for both the sum and difference formulas and the half-angle formulas gave us the exact same value for both. The sin of both answers plugged in the calculator gives us a value of .9659258....... The cosine gives us a value of .258819.......... The tangent gives us a value of 3.73205...... You can even try it in your calculator to see that these values are what you get if you plug in those crazy radical answers from above.