Unit V Big Question

Difference Quotient

1. Explain in detail where the formula for the difference quotient comes from. Include all appropriate terminology.

f(x+h)-f(x)/h 

This formula is derived to find the slope, or derivative, of a tangent, or secant line, that touches a point on an original graph. A tangent lint will touch at a point once, while a secant line will touch twice. When we use the difference quotient to find the derivative of the slope, we plug in the point it crosses into "h". We do this using limit notation, which is the limit as h approaches a number is f(x).

Letter to 2013-2014 Math Anlysis Students

Dear Math Analysis Students,
         Get ready for a year of not just understanding trigonometric theorems,  limits, or piecewise functions, but also  understanding what it means to be a responsible student in various degrees. In order for you to reach your fullest potential in this class, there are some tips you may want to take into consideration. Do your work in class and at home on time! it will be fairly difficult if you fall behind. Also, be patient with this new concept of learning. Just like any new class, with a new subject, the concepts may seem confusing, and the way the class is ran may be hard to adjust to, but in time, you will be professionals in math and adjust to the new class setting comfortably. Furthermore, take time to study the material that you will be exposed to, there is a lot of new math concepts and a lot of review material that will be taught, and you are expected to master this material like pros! Study hard, and you will see positive results at the end of every test! Lastly, always have respect towards you peers and your teachers. This goes for all classes you may be enrolled in. The teachers have dedicated, and still are dedicating, a tremendous amount of their time just for you guys to be successful. They do not want to fail their class, they want you succeed and go beyond their expectations. By taking these suggestion into consideration, you will be a successful Math Analysis student! ***When times get tough, and trust me, it is not always easy to be on top of things due to outside interference, remember to breath, and don't freak out. The complicated math variables, matrices,  and derivatives will soon be the last of your worries  as long as you take every step one by one, and ALWAYS ask for help! Teachers are there to support us! 
        The special scenario you may be part taking in this year is the flipped classroom. By now, you may have be explained how this classroom is ran. It is no doubt, a big adjust every student has to go through when taking this class. Probably the most important thing to do while adjusting is to simply be calm, and go with the flow. Have a positive attitude throughout the year. When you just have a negative outlook for the class, you will have it a little more difficult time not only adjusting, but also enjoying the class overall. Change may be scary, but change can also help us mature and grow as a student. Since you will be working with a lot of technology, it is fairly important that you have axis to a computer. Since there will be a majority learning through videos from Mrs. Kirch, online websites, and Blogging, internet axis is necessary. You don't have a computer or internet axis? Well like a good neighbor, Mrs. Kirch is there with a class room full of computers! Any resources such as computers, calculators, Ipods, and video files of the lessons is provided by Mrs. Kirch. Just don't be afraid to ask for those resources!
       This math class, as may already see, is definitely not the same as other math classes. First of all, don't expect to be flipping through you math text book and working on practice problems. Our work and learning are done in lovely packets called Student Success Sheets (SSS). However the textbook is there for extra practice! Secondly, be prepared to sing a lot of songs derived from Mrs. Kirch herself. Yes, we not only get to learn math and technology, but we learn how to sing! What a strange math class!
        There is a great year ahead you, and this class is just a portion of it. Just be honest with everything you do, and you will see progress in your work and success in this school year. To all the Seniors, end this year strong, and don't give up! This is only the end of the beginning of your journey. As for the rest, stay committed and honest! May God Bless you in this next school year!
        

                                                                                     With all respect,  

                                  Mauricio Ivan Chavez
                                  
                                  SFHS 2013 Graduate

Uniti U Big Questions

 

1. What is Continuity? What is Discontinuity?

Continuity is the state of being continuous. Graphically, it occurs when the graph is go on forever to infinity. These graphs include strait lines and parabolas. Discontinuity is the state when there is an end or limit. This separates to removable discontinuity , and non-removable discontinuity. Removable discontinuity includes point discontinuity, where there is a hole on the graph. Non-removable discontinuity includes jump discontinuity, oscillating behavior, and infinite 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?
A limit is the intended height of a function. It exist as long is the same height is reached between both sides of the graph, the left side and the right side. If the same height is not reached by both the left direction and the right direction of the function, then the limit does not exist. The main difference between a value and a limit is that a value is the actual height of a function, while the limit is the intended height. For example, in a jump discontinuity function, the limit does not exist, but there is a value because there is a point on the graph that does exist.





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


Numerically: We can evaluate a limit numerically using a table. we plot the values of the numbers for the limits, as x approaches a number.
 Graphically: We can simply plug in any function on the calculator ans LOOK where the limit exist or not, looking for and discontinuities.

Algebraically: to evaluate a limit algebraically, we plug in the number that the limit is approaching into the function that is given. there will be a variety of possible solutions. If it comes out to be 0/#, then the limit is 0. if it is #/0, then the limit does not exist. if it it 0/0, we must use a different method besides substitution to find the limit of the function, either getting the conjugate or dividing method. 













http://www.calculus-help.com/tutorials
http://www.sagemath.org/calctut/onesided.html
http://www.cliffsnotes.com/math/calculus/calculus/limits/evaluating-limits
http://www.showme.com/search/?q=1.3%20evaluating%20limits%20analytically

Student Video #1-Unit S Concept 3 Assessment #3

.
What is This Concept About?
This concept goes over Unit S Concept 3, which deals with using power reducing formulas to simplify trig equations. This concept is used when the trig function is at  high power, and need to be powered down in ordered to be simplified.

What is important to remember about this concept?
It is important to understand the process if substituting the formulas withing the equation in order to power down the trig functions. Also, when taking the inverse of the denominators of already complex fractions, you must be careful to multiply it correctly to the denimator of the numerator.

Student Video #2- Unit S Concept 7 Assessment #4

 
What is this concept about?
This Concept goes over solving equations with half angle formulas. there will be a trig function within the equation that is a half angle, and in order to simplify the equation, you must substitute the half angle with the corresponding formula. (Unit S Concept 7)

What is Important to remember about this concept?
What is really important about this concept is to substitute the right formula in the equation. Also, in this problem in articular, it is important to realize that in order to solve for "x", you will need to do alot of substituting, and distribution. If you do not distribute correctly, you will not be able to use the Zero Product Property correctly. Remember to keep the equation set to zero when solving for "x".

Unit S Assessment #2

 Solving Using Sum/Difference Formulas------------------------>

Solving Using Half Angle Formula             
<---------------










These two ways of finding the values of the trig functions give the same answers they are both simplifying through radical other formulas, just in a different process. The multiples within the trig functions are similar, and when you plug each of them in the calculator, they will give the same answers for each of them. Also, since we are using the half angle within the sum/difference formula, it is similar to using the "u" , but taking the square root of the values. Therefore, since each values come out the same, in know that there are two ways of finding the values of these trig function and angles.

Student Problem #10 Unit R Concept 3

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Student Problem #9 Unit R Concept 2

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Student Problem #8 Unit R Concept 1

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Concept 4 Problems Solving Trig Functions

Level 1

In this level, we simply take the trig function we have and find out how we get the given value through the unit circle. since we know that tan is x/y, we know that we can get the ratio rad3/3 with 1/2/rad3/3. the given points are found in the 60 degree, and also in 240 degree, since it is a positive function.








Level 2

The second level, we deal with simple problem solving. we must get the sin (x) by itself. In this case, we are getting no solution since sin must be -1<x<1.












Level 3
In this level, we deal with more complicated solving. we must get a common trig function, and since we know from the ratio identity, csc=1/sin, and cot=cos/sin. there is a common denominator, so we combine the equation. next, we must multiply sin to both sides. since we are now stuck, we must square both sides. we know that sin^2(x) is 1-cos^2. finally, we take what is given, and combine like terms, having a zero on the right side. we take out the common factor which is 2cos. then, we solve for both terms.
Don't forget that there may be extraneous solutions!


Level 4

In this level of problem, we are looking for half/double angles. We need to replace the function with "m", then solve for m. when we do, we take what is solve, 360 and 180, and equal it to x/4. put it as "x" and get the final angles.

Solving Trigonometric problems using Identites and Right Trianlges

 When you are solving to find all trigonometric functions, there are two possible way to do so. One way is using right triangles to find all the function with given one we know. Using SOH CAH TOA , we are able to to find the trig functions of any other angle. O=opposite, H=hypotenuse, A=adjacent.

Since we are given sin=1/2 and cos=rad3/2, we all the required information we need.   **problem worked out here------------->















The next way to find all possible trig function is using the identities. we must first know where the the functions will be located on the unit circle. since cos and sin are both in in the 1st quadrant, we now know that all possible answers will be positive. Now, here are the identities that we use:

tan=sin/cos
cot=cos/sin                 The work is shown here----->
csc=1/sin
sec=1/cos

Deriving the Pythagorean Identities

We are given the main identity of cos^2+sin^2=1. This identity is derived from the ratios of cos and sine in a right triangle. Cos is equal to x/1, that will be the first part of the equation. Sin is equal to y/1, that will be the second part of the equation. R=1. So our equation of the pythagoreon is x^2+y^2=r^2. By substituting the ratios we get cos^2+sin^2=1. 

When we divide cos^2 to the entire eqution, we get the dervation of the next Pythagoreon identity: 1+tan^2=sec^2. *This occurs because cos/cos=1. We know that sin/cos=tan through the ratio identities. And finally, 1/cos=sec. 

Our next derivation is derived when we dived sin to the entire equation: cot^2+1=csc^2. *Dividing sin will do this: cos/sin=cot, which is what we know from the ratio identity. sin/sin=1. And finally, 1/sin=csc. 

 

Reflective Unit O & P

1. How have you performed on the Unit O and P tests?  What evidence do you have from your work in the unit that supports your test grade (good or bad)?  Be specific and include a minimum of three pieces of evidence.

RESPOND HERE: Overall, i did well on the unit O and P Tests. Submitted all my WSQs, watching all the videos and practicing along at home. I took my time to finish the Practice Quizzes, and checking if the the answers are correct. For the most part, i managed to take my quizzes on time, except for the last ones, I had to rush through them. Also, i did not retake any of my quizzes, which might suggest why i did not the better grade i wanted. What is interesting is that I had gotten a better on my Unit P test then i did on my Unit O Test, and i has a disadvantage by not having internet at my home to study. I did my best to not fall completely behind by finishing my PQs and doing all my assessments. The Practice Test did give me the extra push to understand the material.

2. You are able to learn material in a variety of ways in Math Analysis.  It generally follows this pattern:

→Your initial source of information is generally the video lessons and SSS packets followed by a processing and reflection activity via the WSQ → individual supplemental research online or in the textbook before class
→ reviewing and accessing supplementary resources provided by Mrs. Kirch on the blog
→  discussion with classmates about key concepts
→ practice of math concepts through PQs
→ formatively assessing your progress through concept quizzes
→ cumulatively reviewing material through PTs
→ Final Assessment via Unit Test.

Talk through each of the steps given in the following terms:
a. How seriously do you take this step for your learning?  What evidence do you have to support your claim?  Make sure to make reference to all 8 steps.
b. How could you improve your focus and attention on this step to improve your mastery of the material?  What specific next steps would this entail?  Make sure to make reference to all 8 steps.

RESPOND HERE:
a. At my time at home, i gave my full attention to the the online videos and the SSS packets. i took all the necessary notes and usually went ahead to do any other ones that were not required. The online summary i thought was important to help me explain what i had learn. I did not take advantage of any other online research or look up the concepts in the textbooks because i felt i was already understanding the material. The only resources i would use on Mrs. Kirch's Blog is the she would post herself. During class, I personally worked batter on my own, due to distractions and time. Although, when i did get stuck on some concepts, the discussions were helpful for me to get clarification. i would also do my best to ask questions to others who new the concept i has trouble with. The PQs i took the most time in. I felt that practicing these problems on my own will really help me understand. Sometimes, when the concept is already simply, yet there is so many of them, i tend to not finish them. Also, i felt the quizzes were important, but i did not manage my time to finish as good i i should have. i usually don't take my time to retake the quizzes. The Practice Tests are sometimes tedious, but are indeed important. The are intended to be done consecutively during the unit, be i do not do that. I usually do them in in large portions at a time, and when i do, i am even more confident on the day of the test. The actual assessment is greatly beneficial to my learning. In every test day, i make sure i have all the necessary things to use and remember everything from what i practice. In every test i have taken, not gotten lower than a 72%. (not counting Unit A-D). 

b. One important thing i must need to improve on is to complete the assignment on time and not at the last minute. By doing so, i believe i would be able to learn at my full potential. For the most part, i do well when managing my work with the online videos and WSQ, along with the notes for the SSS. when i do get stuck on problems at home, i will do some search online to help me understand my mistakes and learn. During lass, i shall improve with my work ethic and remain focus the entire class period. I will separate myself from people if i have to, but also really take class time serious by getting help from Mrs. Kirch and my peers. With the PQs, i will make sure i finish the required ones during class. The biggest step i will do for the Quizzes is to finish them on time and to retakes when i need to. As for the PTs, i will need to finish them by concepts in order to stay on top of my work and make sure i look for the correction to see if any mistakes were made. All this i feel is perfect preparation for the assessment. More importantly, i will take necessary preparation on test day by doing PMI for my quiz packet.


3. Reflect on your learning this year thus far by considering the following questions:
a.  How confident do you generally feel on the day of a Unit Test?  Give evidence and specifics to back up your answer.
b.  How well do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.
c.  How DEEPLY do you feel you have learned the math material this year as compared to your previous years in math?  Give evidence to support your claim.
d.  Do you normally feel like you understand the WHY behind the math and not just the WHAT/HOW?  Meaning, do you understand why things work, how they are connected to each other, etc, and not just the procedures?  Explain your answer in detail and cite specific evidence from this year.
e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life?

RESPOND HERE: a.I generally feel confident on my test day. I have my work finished, and have understood the concepts of the Unit. the times i did not feel confident is when there was incomplete work with my practice test or the notes of the SSS.
b.Compared with previous years in math class, i felt as though it has been a little tougher to learn, but much more effective. for example, In Collage Algebra, i would learn know all the concepts during the test, but will in class, it was difficult for me to fully practice.
c. Deeply, i felt like i have learn a lot of the math material this year compared to last year. The concepts are much more in depth, and the process of learning is even greater depth. While i was getting "As" on tests in my Algebra 2 class, i basically had to look up some information on the text book and learn the concepts on my own. in this class, i still manged to get high grades, and i am still responsible for my own learning.
d. Normally, i do understand the "why" behind the math. there are time when i have no clue as to why some certain concepts are derived from, for example, the Law of Cosines. I do not know why it is the formula a^2=b^2+c^2-2(a)(c)cosA, i do remember the formula. But as soon as i do, i remember them always. The procedure, in my perspective, are only the process of understanding the "why" behind the math.
e. The value of taking your time to do your work and to do it honestly is fairly important. doing a lousy job at things can lead to a bigger mess in the end. My success and performance always relates to the time i have committed on the subject or work. When i sow a little, i will only reap a little. when i sow alot, the reaping will be immense.
 

WPP #14

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WPP #13

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WPP #11

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