Unit V: BQ#7: Concepts 1-5

How is the difference quotient derived? 

The difference quotient is derived from the slope of a secant line. The difference quotient is of course f of x plus h minus f of x divided by the letter h. We know that the difference quotient is derived from the slope of a secant line because we would have to find the different quotient in order to then find the slope of the secant line. To find the derivative of the difference quotient we do all of this work but with the left over h's we have to plug in zero for them. 

Once we do this then we set our equation equal to zero, eventually finding our slope intercept form. 

BQ#6: Unit U: Concepts 1-8

1.What is continuity? What is discontinuity?

Continuity is when there are no jumps, breaks, or holes. You can draw the line without having to raise your pencil. The limit and value have to be the same. A discontinuity is when there is a jump, break, or hole. There are different types of them jump, point, infinite, and oscillating. 

2. What is a limit? 

A limit is the intended height of the function. 

When does the limit exist? 

The limit exists when you reach the same height from the left and the right. 

When does the limit not exist? 

The limit does not exist when you don't reach the same intended height from both the left and the right or when you have a jump discontinuity and have the closed circle above the open circle. 

What is the difference between a limit and a value? 

The difference is that a limit is the intended height of the function the value is the actual height of the function. 

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

Numerically is when we use a table chart to help us evaluate the limit we pick a number and get closer to it from the left and the right. Graphically is when we use a graph as a visual and use our fingers to see if the limit has the same left and right. Algebraically is when we use substitution, division, factoring,and rationalizing to help us evaluate the limit. 

BQ#4 – Unit T Concept 3

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

The difference between these two graphs is their asymptotes, when you plot the sine graphs you locate the asymptotes by looking where the graph touches the x-axis. When you do locate the asymptotes then you will see that for tangent and cotangent they are different. One will start positive and end negative whereas the other will start negative and end positive. We know that the order of the quadrants go positive,negative,postive,negative and when you find the asymptotes both graphs will be different because they either shift right or left which changes how they are placed and drawn on the graph. 

BQ#3 – Unit T Concepts 1-3

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

Remember that sine and cosine have asymptotes because they are undefined when a zero is placed as a denominator, you can't divide by zero. To relate these to all the other trig functions we know that sine and cosine are involved with their ratios. 

For tangent we know that it is equal to sine over cosine. We can see that tangent will have asymptotes at 3pi/2 and pi/2. 

Cotangent is the exact opposite of tangent instead of being sine/cosine it is cosine/sine. 

Secant will always be like the graph for cosine but the only thing that is different is the asymptotes, secant is related to cosine because it is the the reciprocal 1/cos is equal to secant. 

Cosecant follows the graph of sine and it is the reciprocal of sine,1/sin equals Cosecant. The difference again is the asymptotes. 

BQ#5 – Unit T Concepts 1-3

Why do sine and cosine not have asymptotes, but the other four trig functions do?

Sine has a ratio of opposite over hypotenuse and cosine has a ratio of adjacent over hypotenuse. In the unit circle the hypotenuse is always going to be one. You can't have any asymptotes because in order to get asymptotes on the graph you would need a zero to be divided and you can't divide by zero if you do zero is undefined. When a trig function is undefined it means it has asymptotes and since sine and cosine are not undefined they do not have asymptotes. 

BQ#2: Unit T: Concept Intro

How do the trig graphs relate to the Unit Circle? 

The period for sine and cosine is 2pi because that's how much the distance is for the pattern of sine and cosine to repeat they have to go a full revolution in the unit circle whereas the period of cotangent and tangent is pi because it only takes a half a revolution for these two to repeat their patterns. 

The unit circle only reaches a radius of one, we all know that, and sine and cosine have one as their amplitude because of the unit circles radius. Other trig functions don't have restrictions which is why they don't have amplitudes

Reflection #1: Unit Q: Concept 1-5

What does it actually mean to verify a trig function?

To verify a trig function is when you have one of the problems like in one of the concepts 1 or 5 and then just equaling the left side to the right side. It is like you are given a puzzle and you have a picture of the puzzle completed but you only have puzzle pieces. You have to put the pieces together to get it to look like the picture. It's as simple as that, but contains more math and is more challenging. 

What tips and tricks have you found helpful? 

Some tips that I recommend are to look at the problem and look for simple things to do or change don't over think the problem because the answer will fly by your head. Some tricks that I use is to actually memorize the identities that way you don't have to keep referring to your SSS packet. 

Explain your thought process and steps you take in verifying a trig identity.

My thought process first is to recognize what I have and look for simple things to do. I always try a few things like looking for a GCF, Substituting an identity, The conjugate, CLT, Separating fractions, and factoring. I try looking to see if any of these will work most of the time the problems are simple, but I overthink everything and do way more than I have to. Then make sure to write the steps that you have taken in order to receive credit, but to also help you remember what you had done in order to get that answer. Personally I think verifying is easier than simplifying because verifying already gives you the answer and when you simplify you are not sure whether you are right or not.