Unit V BQ # 7

The formula from the difference quotient comes from the graph we draw and the two sections that are f of x and x plus h. We put this all over h and that is how we get our difference quotient formula. Fromthe graph we pull the equation f of x plus x plus h all over h. This is the formula that we use in order to find the difference quotient.

Unit U BQ # 6

1.) A continuity is a predictable line, that contains absolutely no jumps, breaks, or holes in it. The line can also has to be able to be drawn without lifting one's pen/pencil off of the paper. On the other hand a Discontinuity is that of a point, jump, infinite, or oscillating. At a point discontinuity there is a difference in the limit and value of the function. When there is a jump discontinuity there is literally a jump in the graph, the graph stops at one point and then starts anew at another point. Infinite discontinuity occurs when there are vertical asymptotes in the function, this leads to unbounded behavior. Oscillating discontinuity occurs plain and simply because it is oscillating.

2.) A limit is the intended height that a function is intending on reaching, where as a value is the actual height that the function reaches. The limit exists at a point discontinuity because the line can still meet at the given point even if there is nothing there. The limit does not exist at the following: jump, infinite and oscillating discontinuity.

3.) One evaluates limits numerically through the use of a table. Graphically, it is as simple as making sure the graph meets at a given point. As far as solving the limit algebraically there is a number of ways to solve which include: direct substitution, dividing out, and rationalizing.  

BQ #4 Unit T Concept 3

Why is a normal Tangent graph uphill and a Cotangent graph downhill? Use the unit circle to explain.

The reason that tangent goes uphill when graphed and cotangent goes downhill when graphed is due to that fact that that they are reciprocals of each other. What this means is that when graphed what will happen to tangent the opposite will have to happen to cotangent. The other thing is that it is also based on its unit circle ratios, Tangent and Cotangent are +-+-, but Tangent is drawn from the Sine graph and Cotangent is graphed from the cosine graph line.

BQ#4 – Unit T Concepts 3 Why is a “normal” tangent graph uphill, but a “normal” Cotangent graph downhill?

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

The reason that tangent goes uphill when graphed and cotangent goes downhill when graphed is due to that fact that that they are reciprocals of each other. What this means is that when graphed what will happen to tangent the opposite will have to happen to cotangent. The other thing is that it is also based on its unit circle ratios, Tangent and Cotangent are +-+-, but Tangent is drawn from the Sine graph and Cotangent is graphed from the cosine graph line.

BQ#5 – Unit T Concepts 1-3

Sine and Cosine do not have asymptotes due to that fact that they simply move left and right and get smaller and bigger. The other four trig functions have asymptotes because they rely on the asymptotes to mark where they can start and end with their lines. This is because they move up and down, they also move left and right.
But most important ly is due to that fact that Sine and Cosine become undefined and thus create an asymptote when they are being used with the four other trig graphs.

BQ# 3: Unit T Concept 1-3 , How do the graphs of Sine and Cosine relate to each other?

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

Sine and Cosine relate to each of the other graphs due to that fact that they serve as a stepping stone in a way. All the other graphs are based on these two lines that come from Sine and Cosine. Asymptotes are formed at the mark where cosine is equal to zero and sine is equal to one. All of the lines drawn are always close to but never touching the asymptote.

Tangent?

The lines are facing upwards in the graph they are going off by the points that are created by Sin and Cosine.

Cotangent? 

These lines are the exact opposite of Tangent. The reason for this is due to that fact that Cotangent is the reciprocal of Tangent so that is why the lines are facing in the opposite direction.

Secant?

Secant is a weird graph it is a set of parabolas in each period. Each asymptote has a part of the line that makes up the parabola in the end.

Cosecant?

These lines are the exact opposite of Secant. The reason for this is due to thatfact that it is the reciprocal of Secant, so that is why the lines are facing in the opposite direction.

Reflection#1 Verifying Trig Functions

1. To verify a trig function means to  fully simplify a trig function so that a said side is equal to another. 
2. The tips that I have found useful in this unit is to work with Sin and Cos as much as possible. The reason for this id due to the fact that these two trig function are the easiest to work with, that and sec squared and tan squared.
3. When verifying a trigonometric function look for what to do first: pull out a gcf, lcd, foil, etc. After this can the identities be replaced with sin and cos? Check for this. After this simply take it step by step and organize your work so that everything doesn't get mixed up.