BQ #7: Unit V Concept- Derivatives and The Area Problem

1. 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.) Your post must include text and some sort of media to support your writing.

-The difference quotient is used in higher level math, aka Calculus. In Calculus, the difference quotient is used when we talk about the concept: derivatives. We find the difference quotient by finding f(x+h), the simplifying f(x+h)-f(x) and at the end divide everything by h. When we divide by h we may need to take out a h from the numerator. The difference quotient to us was just this formula in the beginning but now we know that the difference quotient is also known as finding the slope of the tangent line to a graph. So the answer we got from the difference quotient formula is actually the slope of a tangent line aka the derivative. But before we can call it a tangent line we must change it from a secant line to tangent line. The picture below is an example of a secant line because it touches the graph twice.

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We change it from a secant line to a tangent line by plugging in zero whereever there is an h. So all the h's cancel out and makes it a tangent line. Once it is a tangent line we notice in the picture below that a tangent line only tocuhes the graph as we sae above that a secant line touches it twice. Now we have the derivative. 

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BQ #6: Unit U Concepts 1-7

1. What is continuity? What is discontinuity?

-Continuity has four different meanings. One of those meanings is that it is a predictable function which means it goes where it should go. Another thing is that it is a function that has no breaks, no jumps, or no holes in it, it must be continuous. The third thing is that we must be able to draw it without lifting our pencial. Lastly, continuity means that the value and limit are the same.

Discontinuity on the other hand is different. This means that the graph is not continuous due to the different types of discontinuities. (In order from left to right) Point, Jump, oscillating, and infinite discontinuity.

The first one on this picture is called Point discontinuity which is a removable discontinuiy. This one is also known as a hole in the function. The second on in the picture is called jump discontinuity. This means that there is different left and right numbers. The third one is called infinite disconinity which means there is a vertical asymptote in the graph which causes unbounded behavior. The last one in the picture is called oscillating behavior which is just a wiggly graph which menas there is not one single set point. 

2. What is a limit? When does a limit exisit? When does a limit not exsist? What is the difference between  a limit and a value?

- A Limit is the intended height of a function. The limit exisit when there is a continuous function. That means at all the discontunitues it only exisits at the removable one which is the point disconintuites. The limit of a function does not exisit when it is a nonremovable discontinuty due to the fact that there is different left and rights, unbounded behavior, or oscillating behavior. The difference between the limit and the value is that the limit is the intended height of the function where as the value is the actual height of the function. They do not always have to be the same but they can.

3. How do we evaluate limits numerically, graphically, amd algebracially?

- We evaluate limits numerically by setting up a table with the correct numbers. We plug in the function in our calculator and we just use the trace buttom to find the missing values. While doing so, we can see that the numbers are approaching closer and closer to some number but sometimes it can not be reaches. We write it out as " The limit as x approaches 'a number' of f(x) is equal to L'".

We evaluate limits graphically by getting two fingers or pencials and we put one of the left side of the function and one on the right, and then we just see if our fingers meet. If they met, then that means the limit exisits. If they did not meet, we must explain why and give a reason behind that. The reason must come from the disconintiuities.

We evaluate limits algebracially with three different ways. The first way we always try is direct substitution. This means we basically just plug the number into x and see what we get. If we get a numerical answer, we are done. If we get 0/# it is zero and we are done. If we get #/0 it is undefined and we are done. But if we get 0/0 this is indeterminate form which means we must try another way. The  next way we must try is the dividing out/ factoring method. This means we factor both the numerator and denominator and cancel terms to remove the zero in the denominator. After that we use direct substitution. If we cannot see anything that might factor out, we must use the last way which is rationalizing/ conjugate method. If it is a fraction we multiply the top and bottom by a conjugate and it should help us out. After that we are able to use direct substitution.

BQ #4 Unit T- Concept 3

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

From Mrs. Kirch's SSS packet

tangent: Quadrant 1 is postive, 2 is negative, 3 is postive, and 4 is negative. The ratio for tangent is Tan(x)= y/x. So as you can see when cosine equal zero that means that there is an asymptote. There is an asymptote because when it is zero, it is undefined which means there is an asymptote where ever x equals zero. Tangent has asymptotes at pi/2 and 3pi/2 because that is where x equals zero. So as you can see in the picture above pi/2 is before quadrant 2 which is negative, so the graph will start at the bottom and work its way up because the quadrant before 3pi/2 is postive.

cotangent: The quadrant have the same signs as tangent but the ratio for cotangent is cot(x)=x/y. So as you can see, now it is when sine equals zero where their is an asymptote. There is an asymptote at zero and pi for cotangent because that is where y equals zero. So as you can see in the picture above, the quadrant after 0 is postive and the quadrant before pi is negative. So the graph will start on the top and then go downhill.

So tangent goes uphill and cotangent goes downhill because of the location of their asymptotes.

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.

a. Tangent?

-Sine and cosine relate to cosine because of their signs on the unit cicrle. The ratio for tangent is tan(x)=Sin(x0/cos(x). So since the tangent ratio includes sine and cosine, their signs affect the tangent graph. So if sine is postive and cosine is negative, then the tangent group will be negative and go downhill. If sine and cosine is postive then tangent will be postive and going uphill.

b. Cotangent?

-Cotangent is just the reciporcal of tangent. Cotangent's ratio is cot(x)=cos(x)/sin(x). So cosine and sine's signs depend on what way the graph is going. So if sin is negative and cosine is postive then the graph will go downhill because it will make cotangent negative. We get these signs from the unit circle.  Plus it has diffferent asymptotes than tangent. 

c. Secant? 

-For secant, sine does not affect this graph at all. The only one that affects it is cosine. Cosine affects this graph because the ratio for secant is sec(x)=1/cos(x). So this means that cosine determine where the asymptotes go because cosine can equal 0. If cosine is 0 then it is undefined which means there are asymptotes. So cosine affectets secant because of the asymptotes.

d. Cosecant?

                            

                                     

-For cosecant, cosine does not affect this graph at all. The only one that affects it is sine. Sine affects this graph because the ratio for cosecant is csc(x)=1/sin(x). So this means that sine determines where the asymptotes go because sine can equal 0. If sine is 0 then it is undefined which means there are asymptotes. So sine affects cosecant because of the asymptotes. 

BQ #5: Unit T Concepts 1-3

Mrs. Kirch's SSS packet

5. Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use Unit Circle ratios to explain. 

-Sine and Cosine are the only two trig functions that do NOT have asymptotes. The reason behind this is because asymptotes happen when you get undefined. The only way you can get undefined is when you divide by zero. According the the trig ratios, sine is y/r and cosine is x/r. So as you can see, they do not divide by zero because r is equal to one for the Unit Cicrle. Cosecant is r/y and cotangent is x/y so they always have the same asymptotes because sine can equal zero. Secant is r/x and tangent is y/x so they have the same asymptote because they both divide by cosine, and cosine can be zero so that means there is an asymptote present. 

BQ #2- Unit T Concept Intro

From Mrs. Kirch's awesome SSS packet 

1. How do trig graphs relate to the Unit Circle?

-Trig Graphs relate to the Unit Circle because they are basically the same thing. We just unwrap the unit cirlce and make it a line and it turns into a trig graph. It has the same pie values in the same four quadrants. Another reason why they relate is by the signs. The four quadrants stay the same as well. All the signs for each trig functions are the same, so we must remember ALL STUDENTS TAKE CALCULUS. As seen in the picture above you can see how the signs correlate with the unit circle.

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

-The period for sine and cosine is 2pie. The reason behind this is because the pattern for sine is postive postive negative negative. The pattern for cosine is postive negative negative postive. It takes 2pie for this pattern to repeat. So this makes it the period because these graphs go on forever and to repeat the period it takes 2pie. You can see a visual of the sine cosine trig graphs below.

The period for tangent and cotangent is pie. The pattern is postive negative postive negative. As you can see, the pattern repeats half way through the graph/ unit circle so that means that it is only pie and not 2pie because it repeats half way through the revelation. You can see a visial of the tangent trig graph below.

b. Amplitude? How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

-The fact that sine and cosine have amplitudes of one relate to the unit circle big time. On the Unit Circle we remember that sine and cosine could not be smaller than -1 and larger than 1. It relates to this because their amplitudes must be 1 as well. If it is greater than 1 or less than 1 it is considered undefined just like it was on the Unit Circle.

Reflection #1: Unit Q- Verifying Trig Functions

1. In this Unit, Unit Q, we learned how to verify a trig function. This actually means that we must prove that both sides are equal to each other because we know our answer already. So we must use our identities, which we all memorized to make the left side equal to the right side. WE NEVER TOUCH THE RIGHT SIDE. We are just verifying that is proven to be right. 

2. There are many tips and tricks that I have found helpful in this unit to help me understand it way more. One huge tip that I learned that if you can, change EVERYTHING into sine and cosine. By doing this it will make our lives easier because they will either cancel out or be equal to one. Another tip is to always work on the complicated side first that has a lot going on. By doing this first we are able to make it easier and get the hard side out of the way. A third tip is if you can, look for the greatest common factor. This will allow us to make our problems simplier and closer to our answer. The biggest tip I can give is to ALWAYS identify what you did in each step and make it neat. This will help you see where you went wrong if you did not get the answer right. It will also look nicer as well. 

3. The first step in my thought process is to see which identies I have in my problem. I always look to see if I have sine and cosine together, secant and tangent together, or cosecants and cotangents together because they pair up well with their idenities. Next I look for to see if anything cancel or if I have a greatest common factor so I can simplify it even more. If none of that works, the next step I do is look for a common denominator. Once I have done one or all of those steps, usually the identies work out the way I want it to and give me the same answer as the right side. Concepts 1 and 5 are tricky, but if you practice it will all make sense. Just memorize the identities.