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

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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             
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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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