BQ #6: Unit U

1. What is continuity? And what is discontinuity?

A continuity is a function in which on a graph is predictable, has no breaks, no holes, and no jumps, and can be drawn without lifting your pencil. While a discontinuity is the opposite, it isn't predictable, has breaks, holes, and jumps, and you must lift your pencil to draw these. A discontinuity are set in two families: removable and non removable. These are different and it will be mentioned later in this post.  In the removable discontinuities there is only one and that is the point discontinuity, this is usually known as a hole. In the non removable discontinuities there are three; the jump, oscillating, and infinite discontinuities. The jump discontinuity is when it goes from one point on a graph to another without the graph touching. An oscillating discontinuity is when the graph just becomes a bunch of wiggly lines. The infinite discontinuity is a result of a vertical asymptote.

2. What is a limit?

A limit is the intended height of a graph. The limit exists when we reach the same height from the left and right. This can only be reached if the graph is a continuity or part of the removable discontinuities. There will be no limit if the non removable discontinuities are set. With this, the limit will not exist when the graph is different from the left and the right, when there is unbounded behavior, and/or there is oscillating behavior. A limit is different from a value because the limit is only the predicted point of a function, while the value is the actual point of the function.

3. How do we evaluate limits? 

To evaluate the limit numerically we must set up a table. In this table we have the x we approach as our middle number and we get as close to that number as possible from the left and right, usually with .1, .01, .001 and .9, .99, .999. You plug the function into your calculator and trace the graph plugging in the numbers other than the middle number. To evaluate the limit graphically we must slide our fingers from the left and the right and when they get to the same height that approaches the x that is the limit. There are several ways to evaluate the limit algebraically with direct substitution, dividing out/factoring, and rationalizing/conjugate methods. In direct substitution we take the limit and plug it straight into the function. If the answer turns out to be a numerical value, 0/#, #/0 then those are the answers but if there is a 0/0 then you must look to use a different method. In factoring method, you factor both the numerator and the denominator and cancel the common terms. Once we cancel the terms we use direct substitution again for what is remaining. In the conjugate method we multiply the top and bottom by the conjugate of where usually the radical will be. When we multiply this it is just a different form of 1. So we multiply the conjugate and simplify it by foiling, but we do not multiply out the non conjugate. We do this because this is what will cancel out. After canceling our we go back to the substitution method. 

Conjugate method

http://hsmathelearnings4.wiki.hci.edu.sg/file/view/conjugate_example.jpg/204976568/328x342/conjugate_example.jpg

http://choosgs4math.wiki.hci.edu.sg/file/viewlimit_example_2.jpg/205241252/560x356/limit_example_2.jpg

http://tutorial.math.lamar.edu/problems/CalcI/Continuity_files/image002.gif

I/D1: Unit N

How do SRT and UC relate?

Inquiry Activity Summary:
In this activity we were asked to find and simplify the lengths of the special right triangles with the hypotenuse equalling 1. We must always draw the fortunate plane for each because the coordinate is derived from the unit circle.

30-60-90 SRT

Here we have the special right triangle of 30-60-90. This triangle has an original hypotenuse of (2x) a height of (x) and the length of (x rad 3). To make the hypotenuse of (2x) to equal 1 we must divide all sides of the triangle by (2x).

We divided all sides by 2x to reduce the hypotenuse equal to 1. The effect of doing this cause the length of the triangle to become rad 3/2. The height then becomes 1/2.

45-45-90 SRT

In this special right triangle we have the 45-45-90. The rules for this triangle we have the hypotenuse equal x rad 2 and both the height and length equal x.

To make the hypotenuse equal 1 we have to divide it by x rad 2. And whatever we do to one side we must do to all to keep it proportional. When we do this the height and length become rad 2/2.

60-30-90 SRT

This triangle is similar to the 30-60-90 triangle but it is in a different position. The hypotenuse is 2x the length is x and the height is x rad 3. We must do what we did to the first triangle and divide the sides by 2x.

When we divide by 2x the hypotenuse becomes 1, the length becomes 1/2, and the height is rad 3/2.

How does this activity help you to derive the Unit Circle?

This activity helps derive the unit circle because it helps us understand and know where the points of the unit circle come from. This activity also showed the way the triangles are proportioned so the hypotenuse is equal to one. 

5. 

The quadrants of the triangles shown are in quadrant one. The quadrants change the value of it is either positive or negative. In quadrant 2 x is negative and y is positive, in quadrant 3 x and y are negative, quadrant 4 x is positive and y is negative.

This triangle is in quadrant 2 and all x-values are negative.

In this it shows a triangle in quadrant and all values are negative.

This triangle is located in quadrant and all y values are negative in this quadrant.

INQUIRY ACTIVITY REFLECTION

The coolest thing I learned from this activity was that these triangles are the only ones that have these special rules.

This activity will help me in this unit because it will help me understand the unit circle and how the quadrants can affect the triangle and which points are positive and negative.

Something I never realized before about special right triangles and the unit circle is that the quadrant affects the negative and positive of a point. And that the triangles have special rules that only apply to these specific triangles.