BQ #7: Unit V Derivatives and the Area Problem

Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc.) 


 For starters the difference quotient is crucial in calculus since it is like a part of the semester and it is also known as the derivative. The derivative is the slope of all tangent lines on the graph. When using it, we can determine the possible slopes on various graphs. Secant lines are lines that will touch the graph twice but tangent lines are lines that will touch the graph at only one point. H and delta x are variables for the derivative and on a graph, y-value is your f(x) and the x-value is your h or delta x. For derivatives, the thing to add on is that we aren't only finding the difference quotient but we're also determining h as it approaches 0. From here we can do various things like find the tangent slope or the tangent line or any specific values. The thing to remember the the difference quotient is f(x) but the derivative is f'(x) (f prime of x). 

Source 
Derivative
Difference Quotient
Secant & Tangent

BQ #6 Unit U - Introduction to Limits

What is a continuity? What is a discontinuity?

Well for starters, a continuity is when the function is predictable (meaning you know where it's going), no jumps, breaks, or holes (essentially the non-removable discontinuities that we will discuss later on), and can be drawn without lifting your hand up. A discontinuity is divided into two families: removable discontinuities and non-removable discontinuities. The removable discontinuities consist of a point discontinuity. The point discontinuity is where a hole exist and it is also where the limit can also exist at the hole. As for the non-removable discontinuities, they consist of jump discontinuity which is a results from different left/right behaviors, oscillating behavior (which is a wiggly line where there is no real value), and the infinite discontinuity which has a vertical asymptote that results in unbounded behavior. A non-removable discontinuity is where the limit does not exist and the removable discontinuity is where the limit does exist.

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 simply the intended height of the function and it exist at removable discontinuities that are also known as point discontinuities. The limit does not exist at non-removable discontinuities that are also known as jump, oscillating, and infinite discontinuities. Limits do not exist at non-removable discontinuities because of different left right behavior where the lines do not meet together at one point, or when there is no real value in the oscillating discontinuity where it does not approach a single value, or when the line reaches towards infinity because infinity is not defined as a real number. Limits do not also exist at the infinite discontinuity because of the vertical asymptotes that result in unbounded behavior.
The main difference between a limit and a value is that the limit is the intended height of the function while the graph is the actual height of the function.

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

Algebraically

By using direct substitution, factoring/dividing, or the conjugate method, we can solve for these limits algebraically. For direct substitution, it is simply plugging in whatever x approaches into the function. However, there are occasions where the result is an indeterminate form which is 0/0. To resolve this problem, we use the factoring/dividing method which removes the hole that causes the indeterminate form. Basically,  the factoring/dividing method is where we will factor out the function as best as we can and then divide/cancel out any like terms to get rid of the hole. As for the conjugate form, this is when there is a radical and we use the conjugate method to make our lives easier and get rid of the radical. However, it is important to remember that we should always use direct substitution method first.

Numerically

For finding limits numerically, we use a table. Usually, we will plug the function into the graphing calculator and hit trace to where the limit supposedly is. Since we can't actually reach the limit, we approximate as close as we can. For example the we can't reach the number 2 but we can get really really close to it so we write it as 1.9, 1.99, 1.999 or 2.1, 2.01, 2.001

Graphically

Graphically is probably the simplest method. To do this we basically just put our fingers on the graph to the left and right of where the limit is. From there we move our fingers closer to the limit. If our fingers touch then that means that we a limit> However, if our fingers do not touch for whatever reason, then that means we do not have a limit so we write DNE (does not exist).


Reference Unit U SSS - Online (Kirch's)

BQ#4 - Unit T Concept 3

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

To start off, the ratio for tangent is sine/cosine. So whenever cosine equals 0, we have an asymptote because anything with a denominator of 0 is undefined which is where asymptotes come from. Therefore, the asymptotes for tangent are pi/2 and 3pi/2 which is also at (0,1) and (0,-1). From here, looking at the unit circle, it says that tangent is only positive in the first and third quadrant and negative in the second and fourth quadrant. Therefore, when we graph tangent with its asymptotes, the only possible way to draw tangent and follow its rules is with an uphill line. For cotangent, the ratio is cosine/sine, so whenever sine is equal to 0, there will be an asymptote which is 0 and pi. On the unit circle the asymptotes would be (1,0) and (-1,0). Also on the unit circle, cotangent is only positive on the first and third quadrant. So when plotting the cotangent on the trig graph with its asymptotes, it can only be in an downhill line within its period because of the rules it has to follow.

BQ#3 - Unit T Concept 1-3

How do the graph of sine and cosine relate to each other? 

Tangent 

The ratio for tangent is sine/cosine. From this, we can determine how the graph will look based on if sine/cosine is positive or negative in the quadrants I-IV. Therefore that is why the order for tangent is +_+_ on the unit circle. Another thing is that there are asymptotes in tangent which means that there basically where tangent can't exist because of the undefined trig ratio. With the asymptotes in tangent (an example of the asymptotes are pi/2 and 3pi/2) and the order of positive and negative, it results in the shape of the line which is a uphill line. The uphill line comes from quadrant II with the start being in the negative region because of cosine and then going to positive in quadrant III also because of cosine. 

Cotangent 

The ratio for cotangent is cosine/sine. It is essentially like tangent but its relationship is inverse. For starters, the asymptotes are 0 and pi. This is when sine = 0. The order is once again +_+_ but with the asymptotes in place, the graph will change and it will result in a downhill graph because in Quadrant I, cotangent will be positive but moving to Quadrant II, cotangent will become negative and since the asymptotes restrict the graph from continuing forever, our period shows us a downhill graph. 

Secant 

The ratio for secant is r/x. Since secant is the inverse of cosine, if cosine is positive or negative, secant will also be positive or negative. Going on with asymptotes, since secant is 1/cosine, it is similar to tangent which is sine/cosine meaning that secant will have the same asymptotes as tangent which are pi/2 and 3pi/2. Therefore, when drawing it out, the graph will look similar to the sine/cosine graph but the thing to notice is that the secant graph has almost parabola looking graph. 

Cosecant

The Ratio for cosecant is r/y. Since cosecant is the inverse of sine, this means that where sine is positive or negative on the unit circle also goes along with cosecant. The asymptoes for cosecant also go along with cotangent because the denominator for cosecant and cotangent are sine so the asymptotes are 0 and pi. Once again, the graph will look similar to cos/sine except for the asymptotes which cut off the graph and also the parabolas that are within the periods. 

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 are y/r and x/r. Relating to the unit circle, r is a constant that always equals one. For there to be an asymptote, the trig ratio should result in a undefined but since r is always equal to one in the unit circle, that means that sine and cosine can't have asymptotes since its trig ratio never results in undefined. As for the four other trig graphs, there are asymptotes because for cosecant and secant, r is not a denominator so the possibility of having an undefined makes it possible for there to be a asymptote. For tangent and cotangent, there is no r, it's just y/x or x/y so having a denominator of 0 is possible so a undefined solution can result in asymptotes in the graphing. Therefore, only sine and cosine do not have asymptotes while the other four trig graphs do.

BQ#2 - Unit T Concept Intro

How do the trig graphs relate to the unit circle?

Trig graphs relate to the unit circle due to trig graphs are just unit circles that are unwrapped. Therefore, each section of the trig graph comes from the quadrants on the unit circle and this results in the distinct pattern of sin/cos/sec/csc/tan/cot on the trig graph. 

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

A period is one time a trig function goes through its cycle on the cyclical graph. The period for sine and cosine are 2pi because the trigonometric graph is just the unit circle unraveled and for sin and cosine to make the full rotation it takes 2pi. The pattern for sine is +,-,-,+ and the this whole pattern has to occur first for it to be counted as a period. For tangent and cotangent, the period is pi simply because that is the distance for each period. The pattern is only +,-,+,- for tan and since the pattern repeats itself twice in the unit circle, the period would only be half of 2pi which is just pi.

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?

Amplitude is 1/2 the distance between the highest and lowest points on the graph. They are found by looking at the value of a. Sine and cosine have amplitudes because they are the only trig functions that have restrictions. Sin has the ratio y/r and cos has the ratio x/r. Since R is a constant and in the unit circle it equals 1. Therefore the largest and smallest values can only be 1 and -1. Since sin and cosine have these restrictions while the other trig functions do not have this restriction. Therefore, sin and cos have amplitudes but the other functions do not have amplitudes. 

BQ# 1 Unit P Concept 1 & 4 Law of Sines Derivation and Area of an Oblique Triangle

1. Law of Sines - Law of Sines is important because life isn't so perfect that it will always give right triangles. Normal trig functions that we use for special right triangles don't apply to non-right triangles. Therefore we use the law of Sines to solve non-right triangles. Essentially, what we do is set a perpendicular from angle B to create two right triangles and label the line as H. From here we can determine the relationships and use the transitive property
c Sin A = a Sin C to solve for what is needed. From this, we can solve for any side or angle depending on it opposite angle or side. This video below will help you fully understand the concept of Law of Sines.

 

4. Area Formulas - Sometimes you might not be able to get the value of H from the law of Sines. Not to fear because the area of an oblique triangle is very similar to the area of a triangle (A=1/2bh) but since we don't know H, we use (A=1/2b(asinc). This will help us find the value of H so we can use either law of sines or cosines to determine the rest of the triangle if needed. Remember, "The area of an oblique (all sides different lengths!) triangle is one half of the product of two sides and the sine of their included angle." This video below will help you understand this situation better.




Sources
Law of Sines Deriviation
Area of an Oblique Triangle Deriviation