Dear Incoming Math Analysis Students,
My name is Josefina and I took Math Analysis during my senior year.
First of all, I want to wish you all luck on the upcoming year! I hope you are all prepared and excited for Mrs. Kirch's class! My experience with Math Analysis was difficult but it was also fun. The most difficult part of Math Analysis was getting used to watching videos instead of having an in-class lecture. The best advice I can give you is to not slack off on your homework, especially the summer assignment! It was hard for me to keep up with the homework for this class because I had other classes and AP's to worry about, but I had to make time for this class so that my grade wouldn't go down. It is also important to do the summer assignment because if you don't do it you will fall behind the rest of your class. I also want you guys to go into Mrs. Kirch's class with a positive attitude and an open mind. The most important thing in this class is to be able to accept change and the different rules that Mrs. Kirch has set up.
The best way to adjust to the flipped classroom is to keep an open mind. At first, you guys will feel overwhelmed with all the new things that are you are exposed to, but pretty soon you will get used to it. In the flipped classroom, Mrs. Kirch required us to make our own videos as assignments. The first video that I recorded was in Mrs. Kirch's class and it was difficult for me the first time. It was difficult because I had never recorded a video before but by the end of the year, we had recorded a lot of videos and I felt that I got better at it. The most difficult part about recording videos was getting the right equipment to record the videos, but if this this a problem for you, Mrs. Kirch has the supplies for you to record your video. I would also advise you to keep up with your work in class! This is important because you will regret it if you slack off. Mrs. Kirch sets up recommended deadlines for work and it is a good idea to follow them.
The flipped classroom is very different from a normal classroom.The biggest change is the video lessons. In Math Analysis, we need to watch videos at home to learn the material and in class, we do practice problems that would normally be assigned homework. Another difference in the flipped classroom is that we do not use our textbook at all. Mrs. Kirch creates her own packets (the SSS) that we used instead of the textbook, so don't expect to be using your Math Analysis book frequently. As you can see, Mrs. Kirch had us set up our own blog and we used this blog to do a majority of our homework assignments. This blog was hard to keep up with because some of the assignments we had were difficult. The biggest difference between a flipped classroom and a normal classroom was the homework we were assigned because we don't have the usual textbook assignment, we have videos and blog posts for homework.
Please keep an open mind towards the flipped classroom, it well help a lot!
Have fun in Math Analysis next year, Mrs. Kirch is an awesome teacher!
Wednesday, June 5, 2013
Monday, June 3, 2013
Unit V Big Question
Sources:
http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/24-the-difference-quotient-01.htm
Sunday, May 19, 2013
Unit U Big Questions
1. What is continuity? What is discontinuity?
Continuity is the unbroken and consistent existence of something over a period of time. This means that something continues forever. An example of this would be a linear function. Discontinuity is a point at which a graph or a function is discontinuous. This means that something happens in the graph that causes it to become discontinuous, or broken. An example of a discontinuity is a point discontinuity which is a hole, jump discontinuity which is when the left and right side are different, unbounded behavior which is an asymptote, or oscillating behavior which is when the graph is wiggly.
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 the function. A limit exists when you reach the same height from both the left and right side of the graph. Both the left and the right side must be the same in order for the limit to exist. A limit does not exist when there are discontinuities in the graph such as asymptotes and jump discontinuities. The limit still exists when there is a point discontinuity (hole) because even though there is a hole at that point, the graph still reached its limit or its intended height. The difference between a limit and a value is that a limit is the intended height of a function and the value is the actual height of a function.
3.How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically by using a table. When we do this, we list the values from the left and the values from the right and we determine the limit based on the number that the values from the left and the right approach. We evaluate limits graphically by using a graphing calculator or by looking at a picture of the graph. When we use our graphing calculator, we simply need to graph the function and trace to an x-value to see what the limit is. When we are given a picture of the graph, we can just look at the x-values and see what the limit is. Lastly, we can evaluate limits algebraically by using the direct substitution method. The direct substitution method is when you take the number the limit is approaching and plug it in anytime you see "x'.
Works Cited:
http://mathworld.wolfram.com/Discontinuity.html
Continuity is the unbroken and consistent existence of something over a period of time. This means that something continues forever. An example of this would be a linear function. Discontinuity is a point at which a graph or a function is discontinuous. This means that something happens in the graph that causes it to become discontinuous, or broken. An example of a discontinuity is a point discontinuity which is a hole, jump discontinuity which is when the left and right side are different, unbounded behavior which is an asymptote, or oscillating behavior which is when the graph is wiggly.
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 the function. A limit exists when you reach the same height from both the left and right side of the graph. Both the left and the right side must be the same in order for the limit to exist. A limit does not exist when there are discontinuities in the graph such as asymptotes and jump discontinuities. The limit still exists when there is a point discontinuity (hole) because even though there is a hole at that point, the graph still reached its limit or its intended height. The difference between a limit and a value is that a limit is the intended height of a function and the value is the actual height of a function.
3.How do we evaluate limits numerically, graphically, and algebraically?
We evaluate limits numerically by using a table. When we do this, we list the values from the left and the values from the right and we determine the limit based on the number that the values from the left and the right approach. We evaluate limits graphically by using a graphing calculator or by looking at a picture of the graph. When we use our graphing calculator, we simply need to graph the function and trace to an x-value to see what the limit is. When we are given a picture of the graph, we can just look at the x-values and see what the limit is. Lastly, we can evaluate limits algebraically by using the direct substitution method. The direct substitution method is when you take the number the limit is approaching and plug it in anytime you see "x'.
Works Cited:
http://mathworld.wolfram.com/Discontinuity.html
Thursday, April 25, 2013
Big Question Blog Post # 3
Why is a normal tangent graph uphill, but a normal cotangent graph downhill. Use the unit circle ratios to explain.
The way that the tangent or cotangent graph faces has to do with the asymptotes and with the quadrants where tan/cot is positive or negative in. A normal tangent graph has asymptotes where it is undefined or where x is equal to zero, this will be at 90 degrees and at 270 degrees. After we figure out where our asymptotes are, we need to look at the positive and negative signs and we can draw the curves from there. The pattern for tangent is + - + - so first we draw the graph going upward, then downward, then upward,and downward again.
A normal cotangent graph goes downhill because we know that cot= x/y. There will be asymptotes where y equals zero and that will be at zero degrees and at 180 degrees. Next, we need to look at the pattern again, first it is positive, then negative. When we draw the positive side, we need to draw it above the x-axis and when we draw the negative side, we need to draw it below the x-axis.We need to start drawing our graph at the nearest asymptote so that is can go downhill That is how we know that cotangent goes downhill.
The way that the tangent or cotangent graph faces has to do with the asymptotes and with the quadrants where tan/cot is positive or negative in. A normal tangent graph has asymptotes where it is undefined or where x is equal to zero, this will be at 90 degrees and at 270 degrees. After we figure out where our asymptotes are, we need to look at the positive and negative signs and we can draw the curves from there. The pattern for tangent is + - + - so first we draw the graph going upward, then downward, then upward,and downward again.
A normal cotangent graph goes downhill because we know that cot= x/y. There will be asymptotes where y equals zero and that will be at zero degrees and at 180 degrees. Next, we need to look at the pattern again, first it is positive, then negative. When we draw the positive side, we need to draw it above the x-axis and when we draw the negative side, we need to draw it below the x-axis.We need to start drawing our graph at the nearest asymptote so that is can go downhill That is how we know that cotangent goes downhill.
Big Question Blog Post # 4
Why do sine and cosine NOT have asymptotes, but the other for trig graphs do? Use unit circle ratios to explain.
Sine and cosine will never have asymptotes. Sine does not have asymptotes because sine is equal to y/r. We know that "r" can never equal zero and in order for us to have asymptotes we need to have zero in the denominator because asymptotes only have when something is undefined. Cosine is equal to y/r and since we know that "r" can't be equal to zero, it can't have any asymptotes.
Tangent and Cotangent have asymptotes because tangent equals y/x and cotangent equals x/y. Tangent and cotangent can have asymptotes because based on the unit circle, there are some angles there where x is zero or y is zero. Remember, asymptotes are formed when we have an undefined value or any number divided by zero. We can have asymptotes in tangent and cotangent because the quadrant angles have zeros in their ordered pairs and those zeroes give us asymptotes.
Wednesday, April 24, 2013
Big Question Blog Post #2
How do the graphs of sine and cosine relate to the others? Emphasize asymptotes in your response.
a. Tangent: The sine and cosine graphs relate to the tangent graphs because we need to go through the same process to graph these trig functions. The sine and cosine graphs relate to the tangent graphs because tangent equal sine/cosine, so if we were to graph sine and cosine on the same graph, we would know where to graph tangent. We also know that since tangent equals sine/cosine we will have an asymptote where cosine is equal to zero.
b. Cotangent: Sine and cosine graphs relate to tangent graphs because we know that cotangent equals cosine/sine. When we graph cotangent, we know that we will have asymptotes where sine is equal to zero. The only reason that we have asymptotes is because we have a value divided by zero so it makes it undefined and we get an asymptote.
c. Secant: The sine and cosine graphs relate to secant because secant is the reciprocal of cosine. When we graph secant, we sketch the cosine graph first. The reason that we sketch the cosine graph first id so that we can now where we need to draw the asymptotes and the little parabolas that go on the graphs. Secant is related to sine and cosine because secant is the reciprocal of cosine.
d. Cosecant: Cosecant is related to sine and cosine because cosecant is the reciprocal of sine. When we graph cosecant, we start with the same process for graphing sine. We need to lightly sketch the sine graph because it will be our guide to draw our cosecant graph. Then we draw the shifts and draw the parabolas where those shifts are located. Cosecant is related to the seine and cosine graphs because cosecant is the reciprocal of sine.
a. Tangent: The sine and cosine graphs relate to the tangent graphs because we need to go through the same process to graph these trig functions. The sine and cosine graphs relate to the tangent graphs because tangent equal sine/cosine, so if we were to graph sine and cosine on the same graph, we would know where to graph tangent. We also know that since tangent equals sine/cosine we will have an asymptote where cosine is equal to zero.
b. Cotangent: Sine and cosine graphs relate to tangent graphs because we know that cotangent equals cosine/sine. When we graph cotangent, we know that we will have asymptotes where sine is equal to zero. The only reason that we have asymptotes is because we have a value divided by zero so it makes it undefined and we get an asymptote.
c. Secant: The sine and cosine graphs relate to secant because secant is the reciprocal of cosine. When we graph secant, we sketch the cosine graph first. The reason that we sketch the cosine graph first id so that we can now where we need to draw the asymptotes and the little parabolas that go on the graphs. Secant is related to sine and cosine because secant is the reciprocal of cosine.
d. Cosecant: Cosecant is related to sine and cosine because cosecant is the reciprocal of sine. When we graph cosecant, we start with the same process for graphing sine. We need to lightly sketch the sine graph because it will be our guide to draw our cosecant graph. Then we draw the shifts and draw the parabolas where those shifts are located. Cosecant is related to the seine and cosine graphs because cosecant is the reciprocal of sine.
Big Question Blog Post #1
How do the trig graphs relate to the Unit Circle?
The trig graphs relate to the unit circle because when we graph a trig function, we are basically taking the unit circle and stretching it into a line. We also need to look at ASTC to know how the trig graphs relate to the unit circle. Sine's pattern is positive, positive, negative, negative. That means that in the first two quadrants sine is positive and in the next two quadrants, sine is negative. When we graph sine, the first two sections are above the x-axis because they are positive. The next two sections of the sine graph are below the x-axis because sine is negative in quadrants three and four. In the picture below, we see how the sine graph relates to the unit circle because we know that the sine of 90 is one and that is shown on the graph. The cosine and tangent graphs also relate to the unit circle based on the pattern they have. For example, cosine's pattern is positive, negative, negative, positive. Tangent's pattern is positive, negative, positive, negative.
a. Period: The period for sine and cosine is 2 pi and the period of tangent and cotangent is pi. The period for sine is 2 pi because the pattern for sine and cosine takes two periods to to repeat. The pattern for sine is + + - - and the period for cosine is + - - +. The period for sine and cosine because their respective patterns take two periods to repeat. The period for tangent and cotangent is only one period long because the pattern only takes one period to repeat itself. The pattern for tangent is + - + -.
b. Amplitude: Amplitude is the distance between the lowest point and the highest point on the graph. We know that sine and cosine have amplitudes because of the values on the unit circle. Tangent and cotangent don't have amplitudes because those graphs go uphill and downhill. Since these graphs go uphill and downhill, it is not necessary to have an amplitude.
The trig graphs relate to the unit circle because when we graph a trig function, we are basically taking the unit circle and stretching it into a line. We also need to look at ASTC to know how the trig graphs relate to the unit circle. Sine's pattern is positive, positive, negative, negative. That means that in the first two quadrants sine is positive and in the next two quadrants, sine is negative. When we graph sine, the first two sections are above the x-axis because they are positive. The next two sections of the sine graph are below the x-axis because sine is negative in quadrants three and four. In the picture below, we see how the sine graph relates to the unit circle because we know that the sine of 90 is one and that is shown on the graph. The cosine and tangent graphs also relate to the unit circle based on the pattern they have. For example, cosine's pattern is positive, negative, negative, positive. Tangent's pattern is positive, negative, positive, negative.
b. Amplitude: Amplitude is the distance between the lowest point and the highest point on the graph. We know that sine and cosine have amplitudes because of the values on the unit circle. Tangent and cotangent don't have amplitudes because those graphs go uphill and downhill. Since these graphs go uphill and downhill, it is not necessary to have an amplitude.
Subscribe to:
Posts (Atom)