BQ# 7: Unit V

 The origins of the the difference quotient comes from a graph and using an old equation from early this year (Unit A).

Here is the graph f(x) and the line that's barely touching it at x is the tangent line. We can write the coordinates for the graph as ( x, f(x) ). 

This is another graph that has a secant line going through it. It still has the original point as the first one but it has another point in it. Since we deviated from the original to the new point, then it's a change in the graph. That change can be written as delta x or as i put it h. So the new coordinates for this graph would be ( x, f(x) ), ( x+h, f(x+h) ).

Here is an image with the one of the equations, the difference quotient, we would use to find the equation. First we take the coordinates from the graph, then we use the slope formula to ind the slope,Then we plug in the denominator we see that the xs' cancel so there's only an  h left in the denominator. The last one in purple is the difference quotient and that is how we get the equation.

BQ#6 Unit U

#1 What is a continuity? 
A continuity is a continuous function that is predictable and can be drawn continuously,without lifting your pencil and has no breaks, no holes, and or jumps.If in a function the limit and the value are the same, it is continuous.

Here is an example of a continuous function that has no breaks no holes, no jumps, and can be drawn without removing the pencil.

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 discontinuity happens when the intended height of a function and the actual height of a function are different. This can result in one of three different discontinuities, either jump, infinite, or oscillating. A jump discontinuity is when there is a jump between two points that has caused a difference between left/right values. Secondly, is the oscillating discontinuity,which is a function whose graph continuously switches between increasing and decreasing causing the graph to have a series of local maxima and minima resembling waves in water or a vibrating string. The last discontinuity is an infinite discontinuity which is caused by a vertical asymptote which then results in unbounded behavior.

Jump Ex.

(Example of a jump discontinuity, as you can see there is a jump between 2 points and comes in from one on the left but is different right)

Oscillating Ex:

(Example of an oscillating discontinuity)

 Infinite Ex:

(Example of an infinite discontinuity which as you can see has a vertical asymptote which results in 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 value? 

A limit is the intended height of a function. A limit will exist as long as you can reach the same height from both the left and the right. A limit does not exist if the left and right hand limits are not equal, this can usually be caused by breaks, jumps, or holes in the graph and function(hence discontinuities). A limit is the intended height of a function while a value is the actual height of a function.

(Example of an existing limit, since both the value and the limit are the same from left and right.)
Ex of a function without a limit

This is an example of a limit NOT existing, as you can see we start from different sides but unlike the previous picture, we don't end up at the same place in the middle, instead there is a "jump discontinuity")

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

There are many ways to evaluate limits, I will be focusing on three specific ways of evaluating them, those three being numerically, graphically, and algebraically. Numerically, we evaluate limits using tables which help us to determine the "x" and the "f(x)", which are the intended and actual height of a function.

Evaluating limit algebraically:

The first way of evaluating the limit is by doing it algebraically. To do this you start by plugging the limit that is given into the function itself, substituting the x with the limit. All you would need then is the limit and the function to evaluate the limit this way. The following picture shows an example of this being done in a step by step process.

(This picture shows a limit being evaluated algebraically)

Evaluating a limit Numerically:

While multiple ways to evaluate limits, we will be focusing on three specific ways of evaluating them, those three being numerically, graphically, and algebraically. Numerically, we evaluate limits using tables which help us to determine the "x" and the "f(x)", which are the intended and actual height of a function. An example can be shown in the below picture.

(This picture shows a limit being evaluated numerically)

Evaluating a limit Graphically:

Evaluating limits can also be done graphically. There are many ways to do it graphically but there are two main ways, those being having a picture of a graph and using your fingers, the other way is using a graphing calculator by plugging in the equation of the function. The following picture shows doing it by plugging in the equation of the function into your graphing calculator.

(this image shows the process to graphing limits)

BQ#4: Unit T Concepts 1-3

Why is a "normal" tangent graph uphill, but a "normal" tangent graph downhill? Use unit circle ratios to explain.
 The reason as to why these graphs are completely different is because of their asymptotes and their location. The graphs themselves are based on their Unit Circle ratios, and since the graphs can't touch the asymptotes because we have to remember that the asymptotes themselves are based on the Unit Circle.

Both tangent and cotangent have their asymptotes based on their Unit Circle ratios. Remember the ratio for tangent is y/x and in order to show an asymptote x has to be divided by 0 so it can be undefined. If we look at the graph, we see that 90 degrees and 270 degrees has the ratio to be undefined. 90 and 270 degrees in radians are pi/2 and 3pi/2, and those are the asymptotes. The same thing applies for cotangent except that the ratios are switched, it's x/y. So in order for cotangent to have asymptotes, y must be divided by 0 and 180 and 360 degrees have those points. The asymptotes for cotangent, in radians, is pi and 2pi.

Tangent:

If you observe a tangent graph already graphed we can see that 2pi and 3pi/2 are the asymptotes and the graphs can't touch them. Also, based on the Unit Circle, the four quadrants are present and if they are positive and negative. Both quadrants 1 and 3 for tangent are positive so it's above the x-axis. Quadrants 2 and 4 are negative so the graph is below the x-axis.

Cotangent:

The asymptotes for cotangent are pi and 2pi and just like Tangent, remember the graphs CAN"T touch them. The Unit Circle and it's four quadrants are just like tangent, 1 and 3 is positive and 2 and 4 are negative. Except that the graphs of cotangent will not be the same as Tangent because of the asymptotes make the graph go downhill to follow the positive- negative rule of the quadrants.

BQ# 3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each trig graph?
Sine will be in pink with a dotted black line.
Cosine will be in Pink.
Tangent will be in Blue.
Cotangent will be Yellow.
Cosecant will be in Purple.
Secant will be in gray.

Tangent:

To observe where the tangent graph will be we must first know about it's ratio. Tangent's ratio is y/x or sin/cos, cosine will determine where tangent will be. Where sine and cosine are both positive or negative then tangent will just be positive. If one of them is negative then tangent is negative. Tangent's asymptotes is determined by cosine, if cosine is 0 on the x-axis then that is one of tangent's asymptote. 

Cotangent:

Same thing goes for cotangent except that it's sine that needs to be 0. If both sine and cosine are positive or negative then cotangent will be positive. If one of them is negative then cotangent is negative. The reason why cotangent is downhill is because of the asymptotes and where they are found. Since it has to be positive in quadrant 1, the cotangent graph has to be above the x-axis. Same thing for quadrant 2 but it's negative so it's below the x-axis.

Cosecant:

Where sine is 0, the asymptotes for secant will be those points. If sine is positive in two quadrants then cosecant will be positive, if it's negative, like in quadrants 3 and 4, then it's negative. Cosecant relates to the graphs by it's shape, the shape is between two asymptotes. So the asymptotes basically determine the shape and the asymptotes is where sine is 0.

Secant:

Where cosine is 0 the asymptotes of secant can be found. The way the secant graph looks like, positive or negative, is the same as the cosine graph. The shape of this is also determined by the location of the asymptotes and the cosine graph. As you can see the graphs are quite set up, (sorry for the color mixup) v.v

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 graphs don't have asymptotes because asymptotes occur when the denominator of the ratios is 0 (undefined). The other trig graphs can be divided by 0 and it can be undefined while sine and cosine can't be divided by 0. They can only be divided by 1 because 1 is their restriction on the Unit Circle and on the Unit Circle itself it goes from 1 to -1 on both axis. They both can't be divided by 0 because it's not undefined but no solution. The other four graphs especially tangent and cotangent have no restrictions and if their ratios equal undefined then it's their asymptote.

BQ#2: Unit T Concept Intro

A)   When you think about sine and cosine, we have to remember their periods are 2pi and it's like that due to their similarity to the Unit Circle.



In the Unit circle Sine is positive in the first and second quadrant and negative in the third and fourth quadrant. If we start from zero and do one complete rotation to get it to be positive again, then that is a 360 degree or 2pi rotation. So when we stretch the Unit Circle into a line, the cycles will be stretched out too because the first two quadrants will be above the x-axis and the last two will be below the x-axis. 

 

 

 





This also applies for cosine since quadrant I and IV are positive and quadrant II and III are negative. We need to start at 0 degrees and as we move around in the Unit Circle well reach the positive when we get to 360 degrees. When it reaches 360 degrees the it made one complete rotation. If we again stretch out the Unit Circle to a straight line for cosine then we will start above the x-axis since quad. I is positive and for quads II and III it will be below the x-axis. As we approach quad IV to make a cyclical then it go back up above the x-axis.

 

Tangent is different because it's only half the trip to get a cyclical. Quadrants I is positive and quadrant II is a negative so we already have our cyclical. On the Unit Circle, it will start at 0 degrees and when we reach a positive again it will be at 180 degrees and the radian value is pi.



BQ #1: Unit P Concepts 1 and 4: Law of Sines and Area of Obliques

Concept 1: LAW OF SINES 

 The Law of Sines is crucial when we are working on a triangle that is not a right triangle. When we have that we use the Law of Sines to solve it but only if the the triangle is an AAS or an ASA. How to derive the Law of Sines will be shown in these following pictures:

Here we are given a triangle the A,B, and C as their angles and a, b, and c as their sides we can just split the triangle in half to form two triangles.

Lets focus on angle A, angle C, side a, side c and the height, h, I can show you how they can equal each other. If we take the sine of angle A, it will be h/c but since this is a part of a triangle we don't know what h is. So c is then multiplied to both sides and it will give us h = c(sin of angle A). The same thing will happen if we do the sine of C, except it's h/a and we multiply a to both sides and we'll get h= a(sine of angle C). We can get the height in two ways but in the end the height will be the same for the both of them. Since c(sine of angle A) = a(sine of angle C), and we cross multiply, then ( sine of angle A)/a =  (sine of angle C/c.

By staying with the same angles and side but with a different height then we see how sine of B can be the same as the other two.

In this triangle we'll use the sine of angle B and it is k/c but we don't know what k is so we multiply c to both side and we'll get k= c(sine of angle B ). On the other triangle it's the sine of angle C and that's k/ b. Just like the other one we multiply b to both side and get k= b(sine of angle C). But if  c(sine of angle B ) = k and b(sine of angle C)= k then they are both the same. Which mean that if we cross multiply them then (sine of angle B )/b = (sine of angle C)/c and if that is true then (sine of angle A)/a will be the same and all three will work and we'll get the same answer.
Concept 4: AREA OF OBLIQUES.
     The "area of obliques" is derived from the area of a triangle which is A = 1/2 bh. The base is b and the height is h in a right triangle. But if we don't have a right triangle then it will almost be the same but that depends on the side and angle given to us. If we have angle A, side b, and side c given to us then the area formula will be A = 1/2 b(c(sine of angle A). The formula can be rewritten to work with the two other angle. The following images will show which formula to work with for that problem.