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. 

SP7: Unit Q Concept 2

“Please see my SP7, made in collaboration with Jesus Huerta, by visiting their blog here.  Also be sure to check out the other awesome posts on their blog,”

 

I/D3: Unit Q - Pythagorean Identities

INQUIRY ACTIVITY SUMMARY

1. Where does where sin2x+cos2x=1 come from to begin with (think Unit Circle!). You should be referring to Unit Circle ratios and the Pythagorean Theorem in your explanation.

A: Well first we should ask ourselves what is an identity? And why the Pythagorean Theorem is also an "identity"? An identity are proven facts and formulas that are always true. Then, we should ask ourselves what is the Pythagorean Theorem using x, y, and r? Well it's x^2+y^2=r^2 divided by r^2, which then makes the operation equal to 1. Remember what the ratios are for cosines and sines are in the unit circle. Cosine is (x/r) and Sine is (y/r) and you're able to see that cos^2ø+sin^2ø=1. It's referred to an identity because it's a formula that's always true. For example, (√3/2,1/2) cos30º = (√3/2)^2 = 3/4 and sin30º= (1/2)^2 = 1/4. Cos^2ø+Sin^2ø=1. 

2. Show and explain how to derive the two remaining Pythagorean Identities from sin2x+cos2x=1.  Be sure to show step by step.

A: So first off with we know that sin^2x +cos^2x=1, then you have to divide cos^2x to both sides and that gets you tan^2x+1=sec^2x. And then for the other one we know that sin^2x +cos^2x=1 and now all we have to do is divide sin^2x and we get csc^2x.

INQUIRY ACTIVITY REFLECTION

“The connections that I see between Units N, O, P, and Q so far are…” is that the problems that we work with happen to be similar. For example, the Pythagorean theorem somehow always manages to come out. 

“If I had to describe trigonometry in THREE words, they would be…” awe-struck, mind-blowing, and fun.

WPP #13 & 14: Unit P Concept 6 & 7

“This WPP13-14 was made in collaboration with Rodolfo Rodriguez. Please visit the other awesome posts on their blog by going here

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BQ# 1: Unit P Concept P: Law of Sines, Law of Cosines, Area formulas

BQ #1: Unit P Concept 1-5: Law of Sines, Law of Cosines, Area formulas

3. Law of Cosines - Why do we need it? How is it derived from what we already know?  

(http://mathworld.wolfram.com/LawofCosines.html)

Answer: The Law of Cosines is a very helpful law that helps us find answers to the triangle we're working on. The Law of Cosines can be used in any triangle. The formula for the Law of Cosines is    but the law itself can be arranged in two other ways. The Law of Cosines is useful for finding: The third side of a triangle when you know two sides and the angle between them. The angles of a triangle when you know all three sides. Also, you can rewrite the c² = a² + b² - 2ab cos(C) formula into "a²=" and "b²=" form.

4. Area formulas - How is the “area of an oblique” triangle derived?  How does it relate to the area formula that you are familiar with?

Answer: What if you wanted to find out the area of the triangle on your left, but did not know the value of h? Could we do it? So, the area of a triangle is A=1/2bh, where b is the base and h is the perpendicular height of the triangle. In the given triangle on the left, we know that sinC= h/a, and therefore h=asicC. Substituting in our regular area equation for h, we get A=1/2b(asinC). 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 (the angle in between them). Depending on which sides and angles we know, the formula can be written in three ways:

Either: Area = 1 ab sin C 2

Or: Area = 1 bc sin A 2

Or: Area = 1 ca sin B 2

There are different ways to find an area of a triangle. You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been know for nearly 2000 years. :O 
References/Citations:
http://www.mathsisfun.com/algebra/trig-cosine-law.html
http://mathworld.wolfram.com/LawofCosines.html
http://www.mathsisfun.com/geometry/herons-formula.html
http://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html

I/D# 1: Unit N Concept 7

INQUIRY ACTIVITY SUMMARY

(http://www.wyzant.com/resources/lessons/math/trigonometry/unit-circle)

1. Describe the 30* triangle

The 30 degree triangle is one of the triangle that is introduced in the Unit Circle and a triangle that we worked on during class. The properties of the 30 degree triangle has several things to it, which are the sides when you use sohcahtoa, or in other words sine, cosine, and tangent. The angle itself is already given. The triangle itself also happens to land on the first (I) quadrant and everything in the first quadrant have positive angles and coordinate points. Another interesting fact about the unit circle is that when we want to find the radians for the 30 degree triangle all we have to do is multiply 30* times Pi and divide it by 180, which then gets you pi/6. To get Rad 3 / 2 and 1/2 is that you have to take in consideration that there's a chart that'll be on your bottom explaining to you why those are the numbers you want to have instead of some other numbers.

(http://www.mathsisfun.com/geometry/unit-circle.html)

2. Describe the 45* triangle

The 45 degree triangle is one of the other triangles that is introduced in the Unit Circle and a triangle that we worked on during class. The properties of the 45 degree triangles also has several things to it, which are the sides you use sohcahtoa, or in other words sine, cosine, and tangent. Just like the 30 degree triangle the 45 degree angle itself is already given. Just like the 30 degree triangle the 45 degree also happens to land on the first quadrant and everything that you work around with it is positive. Another interesting fact about the unit circle is that when we want to find the radians for the 45 degree triangle all we have to do is multiply 45* times Pi and divide it by 180, which then gets you pi/4. To get the Rad 2/2 and rad 2/2 is that you have to take in consideration that there's a chart that'll be on your bottom explaining to you why those are the numbers you want to have instead of some other numbers.

(http://www.wyzant.com/resources/lessons/math/trigonometry/unit-circle)

3. Describe the 60* triangle

The 60 degree triangle is one of the other triangles that is introduced in the Unit Circle and a triangle that we worked on during class. The properties of the 60 degree triangle also have different things to it , which are the sides when you use sohcahtoa, or in other words sine, cosine, and tangent. Just like the two above 30 degree and 45 degree angle triangles, the 60 degree (angle) has already been given. Just like the 30 degree and 45 degree angle(s), the 60 degree angle also lies on the first quadrant and everything that you work around with it is positive. Another interesting fact about the unit circle is that when we want to find the radians for the 60 degree triangle all we have to do is multiply 60* times Pi and divide it by 180, which then gets you pi/3. To get 1/2 and Rad 3/2 is that you have to take in consideration that there's a chart that'll be on your left also explaining to you why those are the numbers you want to have instead of some other number.

4. How does this activity help you to derive the Unit Circle? This activity helps me derive the Unit Circle in so many ways that I wish I could just show all of the upcoming sophomores who are about to take Math Analysis Honors next year. But mathematically speaking the Unit Circle helps you out a lot when it comes to the tests because if you don't know what something is right away when you're doing a problem and you have to refer back to the Unit Circle like everything is all there. Jesus is there with you no lie. But if you had to memorize the Unit Circle it would be somewhat difficult because you don't know what it would mean or where it came from but since we have an excellent math teacher then you know exactly why the answers are just like that. Our teacher helps explain the tree map of the Unit Circle like why those numbers are there. She helps us explain the quadrants, functions, ratios, and all of that good stuff. And all of that helps me and benefits me for future classes that I will hopefully take in college. You get the idea on why the Unit Circle is the way it is.

http://www.futurebird.com/wp-content/uploads/qq1.jpg

 5. What quadrant does the triangle drawn in this activity lie in? How do the values change if you draw the triangles in Quadrant II, III, or IV?  Re-draw the three triangles, but this time put one of the triangles in Quadrant II, one in Quadrant III, and one in Quadrant IV.  Label them as you did in the activity and describe the changes that occur.


http://yorkporc.files.wordpress.com/2011/10/image65.png
So, the triangles that you see on your left all have something in similar and those are the sides, angles, radians, and the points. They all play something significant to the making of the unit circle and if you see on the second picture to the left you can see that the lines go down from one point to the other point because those points happen to be the same but different areas in which they lie in, which in the quadrants it would be: ALL, Students, Take, Calculus.

INQUIRY ACTIVITY REFLECTION

1. “The coolest thing I learned from this activity was…” that I found out a lot about the Unit Circle there's so much things that goes along with it and your just like whoa!
2. “This activity will help me in this unit because…” it'll strengthen my knowledge of the unit circle.

WPP #12: Unit O Concept 10- PICTURE

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I/D# 2: Unit O Concept 7-8

Inquiry Activity Summary

(http://upload.wikimedia.org/wikipedia/commons/6/68/30-60-90.svg)30-60-90º Triangle

1. In this 30-60-90º triangle the ratios that make up this triangle is literally 
1:2:3 that respectively measure 30-60-90º angles in this special right 
triangle. So, in this special right triangle we are given that that this is a 30-
60-90º and so basically what that means is how and why do we get the 
numbers that are on the left of you? Well, segment AD crosses down to segment CB and so what that does is that it creates a right triangle given 
that the angles are 30º, 60º, and 90º. But now how do we find the sides for 
each segment? In order to find √3 we must use the pythagorean theorem, a^2+b^2=c^2, and so what we plug in the pythagorean theorem is(segment DB) (1/2)^2, segment CD & CB were (1/2) so square it and you get 1. Next, we leave b^2 because we don't know what that is *hint hint (check out the pictures). Finally, c^2 means the hypotenuse (segment AB) is 2 because when we multiply "n" by 2 we manage to get the segments as you see on the right. "n" as you see in the 2nd picture is a variable that'll represent different types of problems when being applied to that number.

45-45-90º

2. "In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem." - Wikipedia. So first off what we should is cut the square diagonally and label your 45-45-90º since we can assume they are since the numbers are being given to us. To prove that the sides are 1, which also equal to "n" because "n" can be any number and basically it is a variable, we have to use the pythagorean theorem, which then later unlocks the true meaning of the hypotenuse. C=√2 because of the Pythagorean Theorem. The 45-45-90º is an easier triangle to decipher. 

Inquiry Activity Reflection1. “Something I never noticed before about special right triangles is…” that I could apply this to my mathematics test on Friday because the concept is somewhat complex. 2. “Being able to derive these patterns myself aids in my learning because…” I will be able to pass the test this Friday.

Real World Application #1 Ellipses

Ellipses 

1. Mathmatical Definiton : "Set of all points such that the sum of the distance from two points is a constant."

•  2.  Algebraically Definition:  An ellipse can either be "fat" or "skinny". The equation for a fat graph is  (x-h)^2/a^2 +  (y-k)^2/b^2 =1, the bigger number "a" being on the bottom of x. If the graph is skinny then  the equation of the graph is (x-h)^2/b^2 +  (y-k)^2/a^2 =1, the bigger number  "a" being below the y in this  instance.So what you have to do is plot given information and find out which points are vertices and which  ones are co-vertices.
•  From this, you should be able to identify a and b. Remember that the vertices lie on a  long the major axis, which is the longer one. If given the focus and either the vertices or co-vertices, use  equation c^2=a^2-b^2 to find missing value. From given information, find center (h,k). Put a^2 and b^2  into general equation. Make sure they are underneath the correct term (if major axis vertical, a goes under  the y^2 term; if major axis is horizontal, a goes under the x^2 term). (Kirch) 

The eccentricity is a measure of how much the conic section deviates from being circular. The eccentricity for an ellipse is 0<e<1. "An ellipse is defined in part by the location of the foci. However if you have an ellipse with known major and minor axis lengths, you can find the location of the foci using the formula below. The major and minor axis lengths are the width and height of the ellipse." (http://www.mathopenref.com/ellipsefoci.html) 

  3. Real World Application 

www.mathwarehouse.com

As we create the ellipse we have to take in consideration that we can't skip the equation of the ellipses c^2=a^2-b^2 to find the missing value. Remember that "c" is the focus of the ellipses.

The Video Is Here: www.youtube.com/watch?v=6pDh42E2bbA :) 

Here in this picture we see that that Earth actually revolves around the sun in an ellipse motion and it isn't perfectly circular. "The radius of this orbit is 150 million km (which is, of course, the distance to the Sun.) and it takes a YEAR (365¼ days), for the Earth to complete ONE orbit, to fit this into the calendar we have 365 days for three years and then 366 days for a leap year." (www.telescope.org/nuffield/pas/earth/earth5.html) 

4. Citations
(www.telescope.org/nuffield/pas/earth/earth5.html)

www.youtube.com/watch?v=6pDh42E2bbA

http://www.mathopenref.com/ellipsefoci.html

http://en.wikipedia.org/wiki/Ellipse