BQ#7 - The Difference Quotient

The Difference Quotient

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

A: The formula for the difference quotient comes from way back in our middle school ages. We have learned about it in our rise over run method. The slope formula. Now it's time to talk about how to evaluate the difference quotient formula. A common expression from calculus is the difference quotient. It is used to when introducing a concept called the derivative. The expression for the difference quotient is commonly given by: f(x+h)-f(x)/h. Plug wherever there's an "x" and plug in "x+h" then you want to subtract the original equation then divide everything by "h." Evaluating the limit of the difference quotient is also known as finding the slope of the tangent line to a graph (the derivative!) We also learn about secant lines and tangent lines (these aren't things that you may think you know, they do not correspond to geometry). Secant lines: touches the graph twice. Tangent lines: touches the graph once. The only thing we do differently now is evaluate the limit as "h" approaches "0" to get the slope of the tangent line instead of the secant line (this is called the derivative)! In Math Analysis, we find the derivatives of linear, quadratic, higher powered binomials, radical and rational functions using the "limit definition of the derivative". We have been doing this for quite some time now or all year - now all we have to do is evaluate the limit as "h" approaches "0" (using direct substitution) to get the "derivative." In concepts 2,3,4, we took the derivatives and applied more things to it. We are not just doing the difference quotient, we are taking the limit as "h" approaches "0" and we call this f'(x), and we read it as "f prime of x". (http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/sld002.htm)

Limits & Functions - BQ#6

Limits & Functions

1. What is continuity? What is discontinuity? 

(http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx)A: A continuous function is predictable. It has no breaks in the graph, no holes, and no jumps. A continuous function can be drawn with a single, unbroken pencil stroke. A continuous function makes a good bridge. You can't fall thru or fall off. A function is continuous if lim_x->c f(x) = f(c). A discontinuous function is the opposite of what a continuous function is. It has breaks in the graph,  holes, and  jumps. A discontinuous function cannot be drawn with a single, unbroken pencil stroke. There are 3 types of discontinuities [in two families]:  Removable discontinuities: which is know as the point discontinuity. Then there's the Non-Removable discontinuities: Jump (different L/R , Oscillating Behavior (wiggly), and Infinite discontinuity due to 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 a value? 

A: A limit is the intended height of a function. A limit exists as long as you reach the same height from both the left and the right (example: As long as you and your friend both arrive at the same restaurant when coming from different directions.)  (http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html)  In order for a limit to exist, both the right hand limit and left hand limit must be the same. If a graph does break at a given x-value, a limit exists there. A limit can exist if your ultimate destination is a hole in the graph! (i.e. You and your friend both drive to the restaurant, but the restaurant isn't there. If you both arrived at the same "parking lot", the limit still exists!") The limits also don't exist when we're comparing left and right behavior, unbounded behavior, and oscillating behavior. The difference between a limit and a value is that a limit is the intended height, while a value is the actual height. 

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

a: In order to evaluate we must know what the term evaluating means. Evaluate means you must give a numerical answer. There are three techniques for evaluating: substitution, factoring, and the conjugate method. To evaluate limits graphically we can go to our calculator and hit the Graph button, 2nd -> hit trace, and lastly trace to the value that you're looking for. Then there's also the method of direct substitution which means you take the number the limit is approaching and plug it in anytime you see "x" and this later converts to evaluating the limits algebraically. Numerically, is a chart that helps us and supports us to find the answer that we're looking for since we're a using a chart. 

BQ#5 – Unit T Concepts 1-3

Q: Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.

A: The values of sine and cosine can only go to the range of [-1, 1]. The domain can remain infinite. The domain is also infinite as well as for the cosine. It is only their inverse properties because since cosecant and secant have asymptotes where sine are cosine are equal to zero. This is because they are recirprocals of sine and cosine, like I mentioned before. Thus, if the value of sine is 0, the value of cosecant is 1/0., which is undefined. Undefine=Asymptotes. Sine are Cosine are very much alike.

BQ#4 – Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” COtangent graph downhill? Use unit circle ratios to explain. Now let me tell you why...

A: During class our teacher Mrs. Kirch stood up explaining why a "normal" tangent graph is uphill and that's because tangent has asymptotes where cosine is zero. So its asymptotes are located at 90 degrees(π/2) and 270 degrees(3π/2). Okay, so the graph on the left of the screen is a tangent graph. What you see is that it starts off as -, +, -, + and it just repeats on forever. 

A: Cotangent have asymptotes where their respective ratios are equal to zero. Cotangent's parent asymptotes are thus at 0 and π (the two points on the unit circle where the "y" value is 0). It's also just the reciprocal of tangent so the graph literally just switches upside down. 

BQ#3 – Unit T Concepts 1-3

3. How do the graphs of sine and cosine relate to each of the others?  Emphasize asymptotes in your response.
A: Cosecant and Secant:

Cosecant and Secant have asymptotes where sine and cosine are equal to zero. This is because they are the reciprocals of sine and cosine. Thus, if the value of sine is 0, the value of cosecant is 1/0, which undefined. Undefined=asymptotes. 

Tangent and Cotangent:

Tangent and Cotangent have asymptotes where their respective ratios are equal to zero. Tangent=sine/cosine, so when cosine=0, tangent has asymptotes. Tangent's parent asymptotes are thus at π/2 and 3π/2 (the two points on the unit circle where the "x" value is 0). Cotangent= cosine/sine, so when sine=0, tangent has asymptotes. Cotangent's parent asymptotes are thus at 0 and π (the two points on the unit circle where the "y" value is 0). 

BQ#2 – Unit T Concept Intro

Unit T Big Questions

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

A: The trig graphs are related to the Unit Circle because when we work with the Unit Circle we see that in the graphs we have Sine, Cosine, and Tangent. We also see their reciprocals which is co-secant, secant, and cotangent. 

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

A: The period for sine, cosine, co-secant, and secant is 2pi. This means they go through once cycle while covering 2pi units on the x-axis. The period for tangent and cotangent is pi. This means they go through once cycle while covering pi units on the x-axis. 

b. 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? 

(http://www.regentsprep.org/Regents/math/algtrig/ATT7/graphvocab.htm)            (

http://www.icoachmath.com/math_dictionary_html5/Amplitude.html)
A: Sine and cosine graphs have an amplitude. Amplitudes are half the distance between the highest and lowest points on the graph. Sine and Cosine can't be greater than 1, while the other trig functions can. 

Reflection# 1Unit Q: Verifying Trig Identities

Reflection

1. What does it actually mean to verify a trig identity? 

A: To verify a trig identity is knowing what you already know and plugging it in and making sure it equals to the original. So, for example, lets say you have tan(theta)sin(theta)=sin(theta) we know that we cannot touch the right side because we've seen what Mrs. Kirch did in her videos. So, we work out the problem and we get the left side equal to the right side. So, basically it's just having two sides equal to each other and knowing that whatever you did in your problem makes sense.

2.  What tips and tricks have you found helpful?

A: I've learned that it helps if you do all of the Practice Quizzes in the SSS packet because if you don't do them, then you get totally lost if you don't do them. I also found that watching the SSS packet clearly and having your brain function properly while watching it as well helps out a lot! Another tips and tricks is that if we're doing a test Mrs. Kirch will probably have the same problems in the Practice Quizzes or in the class activities and you can use those in the tests. 

3. Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you

A: Okay. Well, in order to understand the concepts for any type of problem as you're trying to verify, a very helpful tip would be that you make a T-Chart and on the left side you should be able to write down your work and on the right side explain the steps that you did. Next, we have to make sure to use the Pythagorean identities, Ratio identities, and Reciprocal identities because sometimes when we're explaining or proving that your verification is true then at times you'll get stuck and it's better that you remember the identities because you as you keep on working on the problem you'll figure out that you'll have to use one of them. And make sure that you don't touch the right side.