BQ #7: Unit V - Derivatives and the Area Problem

The formula for the difference quotient comes from a secant line intersecting twice, through a parabola. As it intersects through a parabola, two points are shared by the two lines. In order to work with tangent lines (lines that touch other lines upon only one point, and which we plot out with derivatives), we want the space between those two lines to be literally 0, meaning ideally, that we want to make them the same.

If we label them exactly as we were told to by a higher entity, then we label the further point as (x+h), and the closer point as just 'x', making the space in between them, 'h'. Their respective y-values, would be f (x+h) and f (x), which basically mean that they are plugged in and calculated into whatever equation we have, to get the y-value. Below is an example of what we have just done.

Somehow, we find it necessary to find the slope of the secant line; so, we utilize the slope formula of (y2 - y1) / (x2 - x1), which, if we plug in the values of our previous graph into, we get ( f (x+h) - f (x) ) / ( (x+h) - x) ) , which simplified, would be (f (x+h) - f (x) / h). And as we all know, that is our formula for the difference quotient, which is inextricably tied to derivatives, somehow.

To make the points (nearly) equal to each other, we can set the limit of h to 0, creating a situation where the points are not equal to each other, but extremely close to one another. After working with the difference quotient, using any sort of equation, we would arrive at an end where the variable 'h' probably exists. By making the limit of h equal to 0, we replace the secant line (two intersections), with the tangent line (one intersection), which are tied to derivatives by the fact that derivatives plot out the slope of every tangent line of any graph. Something like that.

BQ #6: Unit U

1) What is a continuity? What is a discontinuity?

A continuity is when a function proceeds without interruption, meaning that there are no discontinuities. Continuous functions are predictable, and one would be able to draw the function in its entirety without lifting their writing utensil. To extrapolate on my earlier point of "no discontinuities", I meant that the function contains no holes, breaks, or jumps - respectively, a missing point in the graph, a vertical asymptote which invalidates a specific x-value, or a random jump up or down in the function. A continuous function looks like this:

A discontinuity is when a disruption in the continuation of a function occurs. This may occur through either removable discontinuities - holes, or non-removable discontinuities - jumps, oscillating behavior, or unbounded behavior. Holes are when there is a missing point in the graph, jumps are when a graph drops/jumps, oscillating behavior is when there are wiggles, and unbounded behavior is when there is a vertical asymptote. The reason why we separate these into two families, is because removable discontinuities have limits, because their left/right limit values are the same, as we can see in the picture below. Therefore, the non-removable discontinuities must not have limits - due to different left/right limit values, no exact point, or unbounded behavior.

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 an intended value of a function. It exists when a function is continuous, or when there is a jump discontinuity (a removable discontinuity). This is because as long as you can trace along the function from both sides and end up at a single, same point on the graph, there is an intended value. It doesn't exist when there is a non-removable discontinuity within the function - namely, jumps, oscillation, or unbounded behavior. This is because you can't end up at a point by tracing from left/right of the point, because of a sudden change in the y-value, no specific coordinate, or a restriction upon a value, respectively. The difference between a limit and a value is that a limit is intended, while the value is what is actually there. In the bottom example, the limit of the function is -2, but the actual value of the function is 2.

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

To evaluate a limit numerically, one would set up a table with values very near the point which you are trying to reach, and what they end up being after you plug them into the function. In the example below, you would reason that whatever is between the values which are very close to the intended point, is between the left/right close values, just as you would with any other numerical evaluation of a limit.

To evaluate a limit graphically, you would simply trace from left/right of the point, in order to find the limit. In the previous example, you would trace left/right of the x-value of 1, and end up ending up at 1 for your limit.

To evaluate a limit algebraically, you would utilize algebra skills in order to find a way in which you can plug in the number which x approaches to get something other than indeterminate form, or infinity of some sort, which are the only two restrictions for evaluating limits algebraically. The very first method in which you would try to evaluate a limit algebraically, would be plugging in the number which x approaches directly into the function. If that results in indeterminate form, or infinity, then you must move on to factoring, which would remedy the issue of indeterminate form. When you can't factor to prevent indeterminate form, you have to multiply by conjugates, usually for radicals, in order to get something other than indeterminate form. Lastly, if you have infinity of some sort, you would have to divide the entire function by the highest power variable in order to somehow fix the problem, and get an answer with substance - without the vagueness of  infinity.

BQ #4: Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" COtangent graph downhill? Use unit circle ratios to explain.

Before we delve into the material, there are two important precursors to the logic that shall be put into explaining just why tangent graphs are typically uphill, while cotangent graphs are typically downhill.

Firstly, it's crucial to know that the ratios for tangent and cotangent, in terms of sine and cosine, are sin/cos, and cos/sin, respectively. From that, we would know that whenever cosine is zero, tangent shall be undefined, resulting in an asymptote, and the same goes for cotangent whenever sine is zero as well.

Secondly, we should know that the x-axis for these graphs is in terms of the unit circle - of π and all that. And, we know that in certain areas of the unit circle, tangent/cotangent are positive/negative. They're both positive in Quadrants I and III, and therefore, are negative in Quadrants II and IV. Since every π/2 on the x-axis is a quadrant, the areas where tangent/cotangent are positive/negative can be organized neatly into certain portions of the x-axis.

Basically, asymptotes divide these portions into repeating portions of either positive to negative (going from left to right), or negative to positive. The reason for which tangent and cotangent differ in where they are allowed to be positive/negative, is that their asymptotes are located in different places - where sine or cosine are zero.

In the above picture, there are asymptotes at π/2 and 3π/2, since that is where cosine is equal to zero. In the sections where tangent is allowed to continually exist for two sections of the graph, it goes from negative to positive, from left to right.

In the above picture, there are asymptotes at π and 2π, since that is where sine is equal to zero. In the sections where cotangent is allowed to continually exist for two sections of the graph, it goes from positive to negative, from left to right.

BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.

Most simply, the graphs of sine and cosine relate to each of the others, in how the others' asymptotes would be where either sine or cosine are equal to zero. This is because sine is the denominator for cosecant and cotangent, and cosine, the denominator for secant and tangent.

Sine is equal to zero, when on the x-axis, it is at zero, and π, as illustrated by the graph below. This is so, because sine is y/r on the unit circle, and therefore, shall only be equal to zero when y is equal to zero. And, y, on the unit circle, is equal to zero, at those places - 0° (zero) and 180° (π). Therefore, there shall be asymptotes for cosecant and cotangent when sine is equal to zero. Asymptotes are important if you want to find out why cotangent goes down. It basically goes down, because the two areas where uninterrupted, consecutive points can continue from left to right, dictate that the left area shall always be positive, and that the right area shall be negative. This is because of the unit circle's ASTC thing. So, since cotangent has to go from positive to negative, it has to go downhill. It's the only way. The only way that I could possibly see sine relating to cosecant, is that they share the same positive/negative areas of the graph. In the areas where sine is positive, cosecant is positive, and vice-versa.

Cotangent is the orange line, and cosecant is the blue line.

Cosine is equal to zero, when on the x-axis, it is at π/2 and 3π/2, as illustrated by the graph below. This is so, because cosine is x/r on the unit circle, and therefore, shall only be equal to zero when x is equal to zero. And, x, on the unit circle, is equal to zero at those places - 90° (π/2) and 270° (3π/2). Therefore, there shall be asymptotes for secant and tangent when cosine is equal to zero. Asymptotes are important if you want to find out why tangent goes up. It basically goes up, because the two areas where interrupted, consecutive points can continue from left to right, dictate that the left area shall always be negative, and that the right area shall always be positive. This is because of the unit circle's ASTC thing. So, since tangent has to go from negative to positive, it has to go downhill. The only way that I could possibly see cosine relating to secant, is that they share the same positive/negative areas of the graph. In the areas where cosine is positive, secant is positive, and vice-versa.

Tangent is the orange line, and secant is the blue line.

BQ #5: Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain?
Sine and cosine do NOT have asymptotes, because their ratios on the unit circle are x (for cosine) and y (for sine), over r. And, r shall always be equal to 1, being the length of the radius of the circle. The logic behind that explaining how they don't have asymptotes, is that for there to be an asymptote, the denominator of a trigonometric function's ratio, has to be able to be equal to 0. Asymptotes only exist when there is a denominator that could possibly be equal to 0, therefore restricting a certain part of the graph. It is from that same requirement for there to be an asymptote, that we can reason that secant/cosecant/tangent/cotangent have asymptotes. Their ratios on the unit circle should be, respectively: r/x. r/y, y/x, and x/y. Since their denominators are not restricted towards being solely 1 as sine and cosine are, 0 is a possibility for the denominators. The end.

BQ #2: Unit T Intro

How do the trig graphs relate to the unit circle?
Trig graphs relate to the unit circle in how the unit circle can be utilized to find where a certain trig function is positive or negative on a graph. For example, sine is positive in the first and second quadrants - between 0° and 180°, or between 0 and π. If we utilize radians on our x-axis, and have some fraction of π as our markings, then we may isolate the area of the graph between 0 and π as the area in which sine would be positive. Finding these positive/negative areas works with every trigonometric function, though you would have to account for asymptotes as well for denominators that could possibly be equal to 0.

Period? 
Why is the period for sine and cosine 2π, whereas the period for tangent and cotangent is π?
The periods for sine and cosine are 2π because it takes a full rotation of the unit circle, which is 2π, for sine and cosine to fulfill the requirements to be periods. And, a period is a portion of any trig graph that is repeated over and over again without change. To illustrate this example, the values of sine, going through the unit circle's four different quadrants, should be +, +, -, -. Having anything less than those four repeating themselves would create inconsistency, so there you have it. On the other hand, the periods for tangent and cotangent are π because it takes half the rotation of the unit circle. π, for tangent and cotangent to continually repeat themselves consistently. The values of tangent/cotangent, going through the unit circle's four different quadrants, should be +, -, +, -. And having just two repeat themselves continually should be enough to constitute an acceptable pattern.

Amplitude?
How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
Sine and cosine have amplitudes, while the other trig functions don't have amplitudes, because their ranges are restricted by how both of their trigonometric ratios have 'r' as the sole value in their denominators. And, on the unit circle, 'r' is equal to 1, being the length of the radius. The other trig functions don't have amplitudes because restrictions such as 'r' aren't in their ratios.

BQ #1: Unit P Concepts 1-5: Non-right Triangle Trigonometry

1. Law of Sines: The Law of Sines is indispensable in sustaining our cushy, privileged lives, because it maintains the space-time continuum in our universe, and because you need it in order to solve for all of the values of any triangle that exists. Of course, you could use the Pythagorean Theorem and other simpler trigonometric functions - but only for right triangles, though, for some apparent reason.

We could prove this law by utilizing our previous knowledge of trigonometry, by creating a line perpendicular to side b and touching the vertex of angle B in a non-right triangle. We shall label this side, h, for the sake of convenience for something else we shall do in just a moment.

Now, we have something that we can work with - two right triangles. And, if we use trigonometry, we will be able to prove the Law of Sines using our prior knowledge. Basically, sine of A, would be h/c, and sine of C, would be h/a. If we multiply both sides of sin A = h/c by c, then h = sin A x c. And if we multiply both sides of sin C = h/a, then h = sin C x a.

With h being equal and all, we will be able to set them equal to each other. This makes sin A x c = sin C x a. If we divide both sides by c x a, then sin A / a = sin C / c, which proves the Law of Sines (or at least 2/3 of it). Just kidding. Since we only use two at a time (you couldn't possibly have = twice in an equation, right?), all we would have to do is switch around the letters, and everything would be fine.

4. Area of An Oblique Triangle: The area of any triangle should be 1/2 b x h (b is base, which would be the side of b anyways, and A, area in this case), which roughly translates to: half of double of the triangle. We would use it normally for our immediate needs - but, we lack the value of h, which can be substantiated, by trigonometry. Of course, this only occurs when we don't have the value of h, but do have, or can get the values of, the rest of the triangle.

So, we can utilize trigonometry, to get h, starting with the fact that sin C = h / a. If we multiply both sides by a, then h = sin C x a. And, if we substitute that into the equation for the area of a triangle, as the value of h, then it'd simply be A = 1/2 b x (sin C x a).

This is related to the area formula that we are familiar with, because it is the area formula that we are familiar with - the only thing changed, is that h is a different value. Lastly, these letter can change around a bit, as such:

[Math Is Fun]

Works Cited: Math Is Fun