Basically in these steps we are finding the slope of the secant line by dividing it between the different points and intersections of the graph.

# Math Analysis

## Wednesday, June 4, 2014

## Sunday, May 18, 2014

### BQ ##6 : Unit U

1. What is a continuity?

A continuity is a continuous functions with no breaks, holes, nor jumps. So in this case it is a regular graph that you can draw without having your pencil come off the paper.

What is a discontinuity?

On the other hand a discontinuity does have these breaks, holes, and jumps. In this case there are two families of discontinuities, Removable Discontinuities and Non- Removable Discontinuities.

A Limit is the intended height of a function.

When does a limit exist?

The limit exists when the function reaches or intends to reach a certain height in the graph. so even when there is a hole the limit still exists because you still end in one spot on your graph.

When does a limit not exist?

A limit does not exist in 3 different situations. The first is in the jump discontinuities because the graph breaks and you have two different points on your left and on your right. So the reason is stated as "different left and right"

The next is an infinite discontinuity in which case the lines never interact because of the vertical asymptote. They go in one direction infinitely and never intersect. The reason for this is called "unbounded behavior." Either side reaches either negative or positive infinity.

Finally, there is oscillating behavior where the y value jumps around and it doesn't approach any single value. The reason given for this is just "oscillating."

What is the major difference between a limit and a value?

A limit is the intended height of the graph and a value is the actual height or an actual point.

3. How do you evaluate limits numerically?

When you evaluate a limit numerically you place a point on a graph and you plot points on either side of it as close as they can get to the original point. From there you just graph and calculate your y value. This helps you see if you can actually reach this point and if not, how close you can get to it.

Graphically?

As for graphically, you place your fingers on either side of a graph and you slide towards a specific point. If your fingers meet then the limit exists and if they don't then the limit does not exist. Either way it leads you to the points you are looking for.

Algebraically?

There are three methods for algebraically. The first one to use in all cases is substitution where you take the limit that it is reaching and you substitute it in any x you see in your equation. You can get several different answers that help you determine whether this method worked. If you get a numerical answer, 0/#, #/0 then this method worked. One thing to take note of is that you get #/0 then it is undefined and that means the limit does not exist. Finally, if you get 0/0 then this method did not work and you have to utilize another one.

The next is the factoring out method which is when you factor both the numerator and denominator. You cancel out the common terms and when that is done you plug in the limit you are reaching into the x.

The following is where you multiply by the conjugate. In this case you multiply the conjugate of the radical and eventually terms will cancel out. The side where there is no radical you don't multiply it out because eventually things will cancel. Then you preform the same steps by plugging stuff in.

Finally we deal with limits at infinity which can be done by the prior method with bigger on top nada, same degree ratio, bigger on bottom 0. Or we can take the largest degree in the denominator and divide everything by it. Anything left being divided by x will automatically turn into zero.

A continuity is a continuous functions with no breaks, holes, nor jumps. So in this case it is a regular graph that you can draw without having your pencil come off the paper.

What is a discontinuity?

On the other hand a discontinuity does have these breaks, holes, and jumps. In this case there are two families of discontinuities, Removable Discontinuities and Non- Removable Discontinuities.

The point discontinuity is a part of the removable discontinuities.

In the jump discontinuity the graph breaks, the oscillating is basically a very wiggly graph, and infinite is separated by a vertical asymptote. The reason they are separated is because the limit does not exist with non removable discontinuities.

2. What is a limit? A Limit is the intended height of a function.

When does a limit exist?

The limit exists when the function reaches or intends to reach a certain height in the graph. so even when there is a hole the limit still exists because you still end in one spot on your graph.

When does a limit not exist?

A limit does not exist in 3 different situations. The first is in the jump discontinuities because the graph breaks and you have two different points on your left and on your right. So the reason is stated as "different left and right"

The next is an infinite discontinuity in which case the lines never interact because of the vertical asymptote. They go in one direction infinitely and never intersect. The reason for this is called "unbounded behavior." Either side reaches either negative or positive infinity.

Finally, there is oscillating behavior where the y value jumps around and it doesn't approach any single value. The reason given for this is just "oscillating."

What is the major difference between a limit and a value?

A limit is the intended height of the graph and a value is the actual height or an actual point.

3. How do you evaluate limits numerically?

When you evaluate a limit numerically you place a point on a graph and you plot points on either side of it as close as they can get to the original point. From there you just graph and calculate your y value. This helps you see if you can actually reach this point and if not, how close you can get to it.

Graphically?

As for graphically, you place your fingers on either side of a graph and you slide towards a specific point. If your fingers meet then the limit exists and if they don't then the limit does not exist. Either way it leads you to the points you are looking for.

Algebraically?

There are three methods for algebraically. The first one to use in all cases is substitution where you take the limit that it is reaching and you substitute it in any x you see in your equation. You can get several different answers that help you determine whether this method worked. If you get a numerical answer, 0/#, #/0 then this method worked. One thing to take note of is that you get #/0 then it is undefined and that means the limit does not exist. Finally, if you get 0/0 then this method did not work and you have to utilize another one.

The next is the factoring out method which is when you factor both the numerator and denominator. You cancel out the common terms and when that is done you plug in the limit you are reaching into the x.

The following is where you multiply by the conjugate. In this case you multiply the conjugate of the radical and eventually terms will cancel out. The side where there is no radical you don't multiply it out because eventually things will cancel. Then you preform the same steps by plugging stuff in.

Finally we deal with limits at infinity which can be done by the prior method with bigger on top nada, same degree ratio, bigger on bottom 0. Or we can take the largest degree in the denominator and divide everything by it. Anything left being divided by x will automatically turn into zero.

## Sunday, April 20, 2014

### BQ #3: Unit T

**How do the graphs of sine and cosine relate to each of the others?**

In the tangent graph the lines relate to the ratios of sin/cos. There places above or below the x axis determine whether the line is positve or negative which helps you see if it will be above or below the c axis. Also, to continue it is cos that determines where the asymptote of tangent is which is when it equal zero. That divides it up and arranges it in locations of positives and negatives to determines the uphill.

The same applies to cot but sin determines the asymptotes. This causes the graph to start on the positive section and go downhill toward the negative.

For secant, there are asymptotes where ever cosine is equal to zero. Because secant is the reciprocal of cos the zero coordinates will cause an undefined. But wherever cos is positive so is sec and same applies to negative so therefore it goes up in some cases close to the asymptote and down in others.

As for cosecant, it is pretty much the same but is influenced by sin. It also has asymptotes where sin equals zero and goes up where sin is positive and down where it's negative.

### BQ #4: Unit T

**Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?**

To begin with, the location of the slopes all depends on the location of the slopes and weather the section is supposed to be above the x axis or below it. (That is defined through the pattern of it being positive in the first and third quadrant and negative in the second and fourth)

Asymptotes are caused whenever the the denominator of the ratio is zero which causes it to be undefined. For tangent the ratio is undefined in the 90 degree and 270 degree mark which is pi over two and 3pi over two on your graph. Then you must abide by the positive and negative slopes in accordance to the pattern of the unit circle. That is why in this case it starts downhill and goes uphill.

## Thursday, April 17, 2014

### BQ #5: Unit T

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?

Well to begin with we know that in order to get an asymptote we have to have a ratio that is undefined, and in this case that only happens when the bottom number is zero. The ratio for sin is y/r and for cos it is x/r. In the unit circle we know that the r is basically the hypotenuse in our unit circle and it ALWAYS equals one. Therefore if the bottom number is always one for sin and cos then it will never have the ability of being undefined. The other ratios either have y or x on the bottom which have the possibility of being zero therefore they have asymptotes.

## Wednesday, April 16, 2014

### BQ #2 Unit T: Concept 1

1.How do Trig Graphs relate to the Unit circle?

a) Period? - Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?

a) Period? - Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?

To begin with we know that sin is positive in the firsts two quadrants on the unit circle and negative in the last two and for this entire pattern to continue we must go through all the quadrants. This plays a role when we plot our period. All four quadrants makes up two pi as does one period in the graph and if we divide that in four evenly we see the pattern of positive positive negative and negative seen on our period. The coordinates and degrees (or radians) also connect. When we have sin of all those degrees it makes up the points in out graph.

For cos, when it is on the unit circle it takes 2 pi for the pattern to complete which is positive negative negative and positive. So then when we plot it on the graph and divide it evenly with the quadrants we see the lines found on either the positive or negative side in accordance to the pattern found on the unit circle. Finding the cos of the various quadrant degrees (0 90 180 270 and 360) using the coordinates on the unit circle gives us the coordinates on our graphs.

As for tan the pattern seems to be completed in the span of 180 or pi which is why the period is always completed within the span of pi instead of two pi like the rest.

b) How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

As we know from the unit circle lessons sin and cos could never be greater than one. And when we use the unit circle there are restrictions as to how wide and long it was. The coordinates only went as far as one and negative one. So this why the greatest and lowest points for sin and cos are always going to be one and negative one.

## Wednesday, April 2, 2014

### Reflection #1: Unit Q: Verifying Trig Identities

**1. What does it mean to verify a trig identity?**

Well lets begin with the definition of of an identity. In the wise words of Kirch they are "proven facts and formulas that are always true. So in this case we are given a problem comprised with many identities that can configure and order themselves to different answers. Our goal when verifying a trig identity is to take the left and make it equal the right. We do this by using the identities to alter the problem which then leads to things being canceled, replaced, or combined. In the end when we have simplified the left side (*WITHOUT HAVING TO TOUCH THE RIGHT*) and it is equal to the right then we have verified that a complicated equation makes a specific (and simple) answer.

**2. What tips and tricks have you found helpful?**

One of the very first things that I came up with (with the help of Helena) is that when we are verifying an identity "IF YOU TOUCH THE RIGHT YOUR WRONG!" Which reminds me never to touch the right of course. Another tip is to look out for identities with he power of TWO because that tells me that I can replace it with a Pythagorean Identity. Another tip is if you know you can do something other than squaring it then try it because that's the last thing you want to do. Finally, while it seems that writing your steps and thought process is useless it actually comes in handy when you look back at the problem to review, because if you don't you'll have no idea what you did.

**3. My Thought Process**

The first step when starting my problems is by giving it a good long stare. While I stare I try to envision the relationships between the trig functions and if I could convert them into any other identity that may help me combine terms or cancel them out. After that if it is a fraction I try to find a least common denominator or even multiply by the conjugate. But if it isn't then I look for methods of either factoring it or foiling it. Sometimes, if I'm lucky, I can finish the problem right away by just cancelling but yeah that doesn't work. OH! and I like to separate my fractions so I can convert them individually, and sometimes turn the fraction into 1!

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