Future Math Analysis Student

Dear Future Math Analysis Student,

Perhaps you have heard these rumors about math analysis,but I can assure you that they absolutely not true! Mrs. Kirch's math analysis is the best class that you will have in your life! Not only will you simply learn math, and memorize the what to do like the past years in math, but you will be able to connect them with the things that you have previously learned before. To be successful in this class, you must be able to make sure to never procrastinate for any of the work. Another thing that will have to do is ask as many questions as you can, especially when you are confused! If you are quiet as I was I will assure that is not good you must interact with your peers whenever there is any type of confusion. When you are introduced to the flipped classroom there will be major changes from your previous math class. The flipped classroom will require to watch videos as your lesson (very helpful indeed!) everyday, don't worry you will adjust, but remember you do not want to procrastinate because if you try to watch them during school, you will not learn effectively. Something new that you probably have not been exposed to would probably be blogposts and WPP'S. Do not worry they are very helpful, and will actually help you learn better from the concept. If you follow directions and deadlines then you will be successful in the flipped classroom. The flipped classroom is different method of learning from previous years, but it helps so much! I personally learned very much in the flipped classroom ( not very good at math overall in my high school experience), but this classroom "flipped" my world upside down  (in a good way!) and I finally understood math! At first it my be kind of stressful in the beginning of the year, but if you put hard work into it; it will pay off! Now you step into the world of math, good luck!

Unit V Big Question

Explain in detail where the formula for the difference quotient comes from now that you know.(include secant, tangent, h delta/ x)

•  The difference quotient allows us to see the slope from the tangent line. This also known as the derivative in calculus. On the picture shown above we are able to see that the tangent line only touches  the graph once and is known to be (x). But the image does not only show the line touching one point, but two points. When the line touches two points we are able to identify it to be a secant line. 

• Since we have two points, now we focus on finding the slope, which is y1-y2/ x1-x2 .  As shown in the image the ywo given points are (x1,Y1) and (x1+h, y2), and between these two given points there is a change in h, also known as delta h. This leads us to the form of ( x, f(x1)) and (x+h, f(x+h)). (if any confusion: Picture is different , instead of y2 it should be replaced in of f(x). And by using the slope formula we plug in the points, and solve it a slope formula problem. Thats how we get the difference quotient, this mainly only focuses on secant line. To find the secant line we could just plug in the lim. of zero into the difference quotient and find secant line. 

Source of info: 

http://cis.stvincent.edu/carlsond/ma109/diffquot.html 

Big Question #1,2,3

1. What is Continuity? What is discontinuity?

Continuity: Continuity means that the graph may go on and will have no interruptions. Continuity graphs will have no holes, jumps or any type of breaks. 

Example: As shown the picture is continuous and has absolutely no interruptions.

Discontinuity: A discontinuity graph means that there has to be interruptions in the graph such as a jump, hole or breaks.

Example: This image shows that there is a jump discontinuities and holes present in the graph.

Pictures from : http://www.zweigmedia.com

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

• What is a limit? 

       - The limit is the intended height of the function. It is the y-value where the x axis lies. 

• When does a limit exist? 

      - The limit will exist when the right and the left reach the intended height. It has to be the same when reaching the left or right. The limit may also exist at a hole (not actual value).

• When does the limit not exist? 

     - The limit will not exsist if there is an interruption such as jump, oscillating, or infinite discontinuity. 

• What is the difference between a limit and value?

     - The limit is the intended height, while the value is the real height of the given function. 

Example: Image shows the jump discontinuity, 

example: This picture shows the jump continuity and the left or right approach for the limits (which DNE because of Jump) 

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

• Numerically: To solve numerically, we first have to set a table, which in hte first row it will show the limit from both the right and the left. On the second row it should contain f(x) and the value of the limit is being approached. Then state the limit statemnt, mathematically and verbally. 

example: Shows a table in which how the limit approaches its value.

• Graphically: To solve graphically we use the method of two fingers(left and right side of graph) in which we place on the graph. If the fingers meet at each other then the limit exists  If the fingers are do not meet each other then limit does not exist. If the limit does not exist then it must be because of jump, oscillating, and infinite.
• Algebraically: To solve it algebraically we use the substitution method. To use the substitution method we simply plug in the numbers into the equation, while using this method we might get a numerical answer, 0/# which will be zero, a #/0 means the limit is undefined. And if the answer is 0/0 then it is indeterminate. If it is indeterminate then we use the dividing and factoring out method, which we factor both the numerator and denominator and cancel out similar terms, then use direct substitution. The other method is the rationalizing and conjugate method by using this method we multiply the conjugate change the sign, foil, and simplify. 

Extra Credit Math Mistakes: Adding and Subtracting Complex Numbers

Compare

http://mathmistakes.org/?p=1070

What did the student do wrong?

• I believe the student complicated the problem more than what he was supposed to do. I think what he did was to take each number and distribute the negative (only to the second side) and subtract that to the first numbers, but individually.

What should have the student done?

• The student should had just viewed as a regular subtraction equation and solved the it (as depicted in the picture)

Extra credit : Completing the Square Math mistakes

Compare 

http://mathmistakes.org/?p=1104

What did the student do wrong?

• The student seemed to have taken the eight and square root it and it resulted in 2.8.
• He left x alone and ignored the square root of ten and added 2.8 which then would be 12.8.
• He did not properly follow the steps in order to complete the square.

What should have the student done instead?

• The student should have seen the step where you are supposed to factor, and from there square root to take take away the ^2.
• Then use +/- to have to answers. 

Extra Credit Math Mistake (Adding Complex Numbers)

http://mathmistakes.org/?p=261

What did they do wrong?

• The student student seemed to only add together the ones in in parenthesis first  and then added then added them resulting in 14i.

What they should have done instead?

• The student should have instead seen this as any regular adding equation, with the exception of the presence of a complex number. 
• If he should have seen this as a regular addition equation then the numbers would have canceled and it should have only been 14. 

Unit T Big Question #4

4. Why do sine and cosine NOT have asymptotes, but other four trig. graphs do? Use unit circle ratios to explain. 
Explanation: Sine will never be able to have asymptotes because sine= y/r ,and r will always be 1. For Cosine it will be x/r and r will always be one and it will not have the need to be an asymptote. For the other trig. functions they will always be undefined and will need asymptotes.