BQ#7: Unit V: Derivatives

Where does the difference quotient come from and what do you know about it?

          The difference quotient has a nickname called the derivative which is derived from the slope formula of (Y^2-y^1)/(x^2-x^1). The reason why we use the slope formula is because the derivative is the slope of all the tangent lines. A tangent line touches the graph one time. Looking at our formula for the difference quotient is f(x+h)-f(x)/h. We have two points that we use in our slope formula. Our first x point is simply just x. While our first y value is f(x) As we move along the x axis we have our letter H which can  be referred to as delta x. Our second points are  used from the original ones but including h because of the shift. So our points would be (x+h) and then our second point would be f(x+h) We then plug this into our slope formula and get f(x+h)-f(x)/f(x+h)-x we then simply our x values in the denominator and whats left is our difference quotient  f(x+h)-f(x)/h.




 


 
 

 

By looking at the picture above our delta x or our h has to be moved closer to be able to be touching which is the secant line that touches two points. In the SSS videos our delta x has to be moved closer and closer in order for the points to be directly touching.


Reference:
http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg
http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG
http://images.tutorvista.com/cms/images/39/difference-quotient-formula.png

Unit U: BQ#6

#1: What is continuity? What is discontinuity?
A continuity is a graph that has no jumps, no breaks, and no holes. The graph itself is predictable so you can draw it without lifting your pencil off the paper. A discontinuity has two families which include a removable discontinuity  and non-removable discontinuities . Removable discontinuity is consistent of the point discontinuity. This discontinuity has a hole. Non-removable discontinuities include jump. oscillating, and infinite discontinuity.  Jump discontinuities have different left and right because they never meet up. Oscilatting is a wiggly line. Infinite discontinuity has unbounded behavior and vertical asymptotes.

#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?

    The limit is the intended height of a function. A limit exists on a continuous graph or the removable discontinuity. With the left and right going towards the limit, it is able to exist. The limit doesn't exist at non-removable continuities, which are stated above as jump, oscilatting and infinite. A limit doesn't exist on jump because of different left and right, they never meet up so there is no limit. On oscillating theres no single point because its wiggly and keeps on moving. A limit doesn't exist as infinite discontinuity because of unbounded behavior and its vertical asymptote.

The limit is the intended height of a function while the value is the actual height of a function. Taking the first graph as an example there is a limit ,but theres no value since theres a hole. In some cases, for example the graph below the first one there will be a different limit and a different value somewhere else. The jump discontinuity has no limit because it jumps, it has no meeting place, however the value still exists at the closed circle. Oscillating and infinite discontinuity are never at an exact point.

#3: How do we evaluate limits numerically, graphically, and algebraically?
To evaluate limits graphically you can create a table, you choose your base number, then add and subtract by a tenth. So for example we can take the number 5 if we subtract a tenth from 5 it becomes 4.9. Once we add a tenth to 5 it becomes 5.1. Basically you want to get as close to your base number of 5 So to get closer from 4.9 would be 4.99 and 4.999. Closer to 5.1 would be 5.01, and 5.001.

To evaluate graphically we take our fingers from left and right and work our way towards the center.

Algebraically we used direct substitution in which we plug it in and solve, If that doesnt work we go to the factoring method in which we factor out the numerator and denominator and cancel terms to remove the zero in the denominator. Then we can use the rationalizing method in which we change the sign and multiply by our square root problems.
 
Reference: SSS Packet

BQ#4 Unit T: Tan and Cot

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

We know that our Tangent value is sin/cos or y/x. Our cotangent value is x/y or cos/sin. In order to for tan and cot to go their directions we must have our asymptotes in which for tan the x value has to be 0 while for cot our y must be 0. Tangent is 0 at the degrees of 90 and 270 because we take our coordinates of our quadrant angles. 90 degrees has a coordinate of (0,1) and 270 has a point of (0,-1). Once we plug this into our equation for both we get 1/0 and -1/0 which is our asymptote. These degrees also have radians in which 90 is pi/2 and 270 is 3pi/2. These serve as our asymptotes on our graph and is represented by a dotted line. The same thing applies to cotangent in which our value is x/y in which the degrees of 0 and 180 are our asymptotes. Plugging in our values we get 1/0 and -1/0. 0 has no asymptote on the plane while 180 lies on pi.

 




 



 
 
  By looking at our graph our asymptote for tan is placed at pi/2 and 3pi/2. These serve us as our walls which tan goes upwards. Tan also goes upward between 0 and pi/2. Cot goes downhill from its first asymptote  at zero. Cot also goes down at 180 which lies on pi.

Bq#3 Unit T: Relating graphs

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

A)Tangent

  Sin and cos are related to tan by looking at our trig values. Since Tan=Sin/cos or y/x By looking at our value of cosine it never touches our origin which is zero because it would cause the value to become an asymptote and become undefined. However our tan graph is going by certain amounts in which our asymptotes can act like a wall which can guide and lead the tan graph to go to that certain amount. Cosine never intersects with any of the other graphs.


 B) Cotangent

 





Cotangent has the value of cos/sin or x/y. We can see that sin goes from 0 and touches every one value of pi going to pi 2pi 3pi etc. We can see our cot  starts from the positive and transcends into negative within 0 and pi and continues which is our wall that restrains the graph and keeps it contained. Tan and cot are mirrors of the direction of the graph going up or down.
 
C) Secant

 








Secant has the value of r/x or r/cos. When cos is 0 it lies on pi/2 3pi/2 and 5pi/2. In these areas sec is split by parabolas and are split by the asymptotes into their own sections. Starting by going up then down and repeats the pattern of the direction of sec. Sin and cos act as a measurement between the graphs of secant.
 
D) Cosecant

 


Cosecant has the value of r/y or r/sin. The asymptotes lie on every pi such as 2pi 3pi etcwhich results in seperate graphs according to csc. Sin is 0 on the values of pi which is 1/0 which is an asymptote because its undefined. The pattern repeats of up and down.

Bq#5 Unit T: 6 Trig Graphs

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain.
     
        Sin and cosine don't have asymptotes simply by the reason of that their circle ratios are composed of having a denominator of r which is equal to one. Sin has a circle ratio is y/r and cosine has a value of x/r. Referring back to our unit circle, r has a value of one, so therefore whatever the value of x and y will be that value because the denominator is automatically one. In order to obtain an asymptote, the value must be undefined in which leads to having zero on the bottom which creates our asymptote. Other trig functions are able to have asymptotes because of their natural unit circle ratios.  Csc is r/y in which y can equal to 0 with the 90 degree angle which leads to 1/0 and is an asymptote. Secant is r/x in which you can use 270 to get the value of 1/0 which results in undefined.  Tangent and cotangent are the same in that you can mix up the values, as long as you have 0 on the bottom you can achieve an asymptote since dividing by 0 is undefined.

BQ#2 Unit T Unit Circle Introductions

How do the trig graphs relate to the Unit Circle?
a. Period? Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
          Trig graphs relate to a Unit circle in that it contains quadrants from the unit circle and can be created with the use of a computer cable.

Using all students take calculus Sin is positive in the first and second quadrant while sin is negative in the third and fourth quadrant. For cosine the first quadrant is positive, the second is negative and then it repeats itself by being negative then positive in the third and fourth quadrant. Tangent and Cotangent is positive, negative, positive and then negative. Our unit circle is also consistent of having quadrant angles split by 4 sections on the axis. For sine the first quadrant is to pi/2 then to the other half becomes pi. which is half of the unit circle, going into the 270 quadrant angle is 3pi/4 and completing at 360 is 2pi. Here if you look at the picture the quadrants are divided along the line by the quadrants and their values according to the trig function. When you look at the lines going up and down on the line, this is based on our sin value of the quadrants both being positive and then negative in the 3rd and 4th quadrant. We can see this by my lines indicated on the line, after the second one there's a "valley" in which the sin values are negative so they go below. Sin and cosine are 2pi because they don't have a complete pattern like tan and cot. Sin is positive positive then negative negative, so it has to go through a whole go around the unit circle, same thing applies to cosine in which it doesn't complete its pattern. While tan and cot complete its pattern of being positive negative then positive negative.

b) Amplitude?-How does the fact that sin and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about our unit circle?
Sin and cosine have restrictions on their values as -1<x<1 Since we know the unit circle has an r value of one hence our trig values for sin is y/r and x/r belongs to cos. If we get x or y values greater then -1 or less than -1 we get error and it doesn't work, meanwhile the other trig functions you can mix and match the values of the unit circle.

Reflection:#1: Unit Q verifying trig functions

1.What does it actually mean to verify a trig identity?
To verify a trig identity is to start off with what you're given and turn it around and adjust the given values to make it equal to what the trig identity is giving us. With verifying we can use our reciprocal trig values or sub in with identities. We're open to numerous different ways to verify the different identities.


2.What tips and tricks have you found helpful?
Tips and tricks I've used are basically try every single way possible and test things that you've never done before. Other tips include just knowing your identities and what values they are and make sure to not mistake for subtraction signs and addition signs.

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.
My thought process just include depending on the concept. For concept 4 I wonder if I'm able to factor out anything such as a trig function or if I'm able to use the zero product property. I also look for any functions I can convert and see if it cancels with anything or it can add too my current problem. I also check if I should square both sides and check for extraneous solutions. For problems like concept 5 I include the same thought process, except i put the right side as a no no touch boundary and only convert the left side if possible.

Sp7 Unit Q Concept 2

My amazing partner Ana posted this SP on her blog here

I/D3: Unit Q - Pythagorean Identities

INQUIRY ACTIVITY SUMMARY:  

1)What is an identity? Why is the Pythagorean Theorem an identity.? 

An identity is a proven fact or formula which is always true. The Pythagorean theorem is an identity in that whatever numbers you plug in you're able to get the correct numbers.

b) What is the Pythagorean Theorem using x y and r?

Since our original theorem used a^2+b^2=c^2 With triangles being inside our unit circle we can replace a with x, b with y and c with r which results in x^2+y^2=r^2

c)

Here we divide by r to set our equation equal to one

d) What is the ratio for cosine on the unit circle?

x/r

e) What is the ratio for sine on the unit circle?

y/r

f, g and h)

What do you notice about part c in relation to parts D and e? What can you conclude?

Looking at our very top equation in this picture we take our value of the ratio for cosine and sine of the unit circle and plug it in and what we get is that our equation consists of is taken from the unit circle. Its also derived from our new transformed formula. Our sinx^2+cos^2x=1 is referred to as an identity because whatever number you plug in will result in one respectively. Using 45 45 degrees I plugged the numbers into our equation and got 1. 

2) Perform a single operation fairly to derive the identity with secant and tangent. 

Here we take our equation and divide it by cos in which results the first part results in one and our second part results in tan (because we memorized it) and our 1 over cos is a reciprocal which results in it equal to sec.

b ) Perform a single operation fairly to derive the identity with Cosecant and Cotangent.

Here we divide by the value of sin in which results it in being one for our first part. Our second part it results in being cot, and our third part which is a reciprocal leads it to being csc.

INQUIRY ACTIVITY REFLECTION:

1)The connections I see between units N, O, P, and Q are how we use triangles within the unit circle, and the use and value of trig functions such as y/r= sin and etc and how the values can be taken the root and lead into the unit circle.

2) If I had to describe trigonometry in three words, they would be : headaches are present.

WPP#13-14: Unit P Concepts 6-7: Solving law of Sin and Cosine word Problems: The Skydiving Pizzeria Evening

This WPP was made with the amazing Ana who is fabulous so its on her blog so check the wpp here

BQ#1: Unit:P Concept 3 Law of Cosines SSS or SAS

3. Law of Cosines - Why do we need it? How is it derived from what we already know?  The derivation must be shown either in a video or in multiple sequential pictures and inshould include descriptions and information beyond what you can find in the SSS.


 This video derives the law of cosin starting off by starting off with the use of the trig value functions such as sin= opp/hyp We start off by cutting or triangle and dividing line is labeled as h. Since SinC=h/a we multiply a on both side and get h=asinc. He then speaks of how the sides are achieved. Using the Pythagorean theorem A^2+B^2=C^2 we take the squared values of the legs of a triangle and set it equal to our hypotenuse. Leading into identities sinC and cosC are equal to one.


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?

 
The area of an oblique triangle is taking from the original area equation of a triangle. We take the SinC=h/a we multiply each side by the value of a and get h=asinC. We then take our our original area of a triangle formula and replace the h with our h value in which results in a=1/2absinC. We can do this with our a and b values. For example if we use  Sin A= h/c multiple each side by c and get h=csinA and plug it into our orignial equation of 1/2bh replacing the h. So our new equation would be sinA=1/2bcsinA.


 Reference:
http://www.youtube.com/watch?v=_DR0BfWh5Jk&list=WLy-6lkQIkegNnF6iwmkqxSLjHvc-2mXVZ

http://www.youtube.com/watch?v=pyftYzmOwr4

WPP#10 Unit O Concept 10: Elevation and Depression

 




 


A) Read the problem and solve for what is asked. Show all works and steps clearly. 
Nancy is about to go skiing, she measure the angle of elevation to the top of the slope to be 80. She is 30 feet away from the slope, what is the height of the slope? 

  





  

  

 B) What is the angle of depression 
 Once Nancy reaches the bottom of the slope, she estimates the angle of depression from the top to the end to be 40. Nancy knows shes 140 feet higher then the base of the course. How long is the path?

I/D2: Unit O - How can we derive the patterns for our special right triangles?

Inquiry activity summary:

 
1. 30-60-90 We begin by taking a regular triangle which is 60 in each of its corners because the sum of a triangle is 180 degrees. We split a line down the triangle in order to achieve our special right triangle which is a 30 60 90 triangle. When we split the triangle we make a 90 degree angle and split the 60 in half which changes the value to 30 and our 60 in the corner remains. Since we split the triangle in half our one becomes 1/2 for each side because 1divided by 2 is one half.  Our base of our triangle is labeled as A while the side is labeled B and the hypotenuse as C.We then take Pythagorean theorem which is a^2+b^2=c^2. Our base is labeled as A, our side is labeled as B, and our hypotenuse is labeled as C. We then plug in and square our values.  The product results in b= radical 3 over 4 in which we square the 4 and leave the radical 3. This value of radical 3 over 2 is across from our 60 degree angle. Since we have fractions, and a majority of people despise fractions we multiply each value by 2 so radical 3 over 2 becomes simply radical 3 across from our 60 degree angle. Our 30 degree angle which was once 1/2 becomes simply one. Our hypotenuse becomes then 2.  When our values vary we use a variable like n for extended values so we can use them for the numerous amount of numbers.
 

 



 
2. We are given a square with equal sides of one which has the angles of 90 degrees in 4 spots. Taking our square, we split  it diagonally so we can get our 45 45 90 triangle. With two 90 degree angles cut in half  we get the result of 2 45 degree angles and have a 90 degree angle remaining.We start off using the Pythagorean theorem  with a^2+b^2=c^2.  Our A is labeled at the base our B is the side and our hypotenuse is C. We plug in our value of one which results in c equals radical 2 which is our hypotenuse. We put N on the side just in case our numbers change.

Inquiry activity reflection:
1. Something I never noticed before about right triangles is how you can derive them from other shapes like a whole square or a whole triangle.
2. Being able to derive these patterns myself aid in my learning because I can use our knowledge of shapes and break down and use for knowledge later on.

I/D#1 Unit: N Concept 7 Unit Circle

The activity we did in class was a worksheet where we were given three different triangles of a 30 60 90 a 45 45  and a 60 30 90. We set the hypotenuse to one and solved for the opposing sides according to 30, 60, and 90. We then labeled what was r x and y for all three triangles. We then drew a coordinate plan so the triangle lies on t
he origin. We then labeled the three points of the triangle accordingly.











 

  Inquiry activity summary:
1.The 30 degree triangle consist of radical 3 over 2 as the x value and 1/2 as the y value when the hypotenuse is one. The reference angle is 30 degrees because its 30 degrees from the x axis.

2.The 45 degree triangle consist of equal value  sides in which has the value of radical 2 over 2 when the hypotenuse is one. The 45 degree angle has a 45 degree reference angle since its 45 degrees away from the x axis.

3. The 60 degree triangle has the x value of 1/2 and the y value is radical 3 over 2.  The 60 degree has a reference angle of 60 because its 60 degrees away from the x axis.

4. This activity helps me understand the unit circle in that the first quadrant consists of these angles and the whole circle is just full of these different triangles just with the addition of the quadrant angles  simply adding 30 45 and 60 to the numbers of 0 90 180 and 270. This activity showed me what was on the outside of the triangle which was in reality the outside of the circle which helped me find the coordinates since the radical 3 over 2 value is always near the x axis.

5. The quadrant that this activity lies in is quadrant one. The values that change if drawn in the different quadrant are the degrees, the value of pie and if the different coordinates are either the x value is negative, the y is negative, or both values are negative. These values change according to the saying of All Students Take Calculus  The A is in the first quadrant which means all values are positive. S is the second quadrant which means sin and csc are positive while everything else is negative. T is in the third quadrant which means tan and cot are positive while everything else is negative. C lies on the fourth quadrant in which cos and secant are positive while everything else is negative.
Inquiry activity reflection
The coolest thing I learned from this activity was that the triangles actually are inside the unit circle.
This activity will help me in this unit because knowing the coordinates on point is really helpful so you don't have to get brain  scattered everywhere.
Something I've never realized about special right triangles and the unit circle is  they connect with each other and relate like a family.

RWA #1: Unit M Concept 4: Parabolas

This picture demonstrates the use of the parabola and how its properties can be used to create monuments

        This video helps in understanding a parabola by first going over the definition. Then by further going into depth in the equal value of the distance from the focus to the vertex. The video then explains the bouncing of distance.

http://www.education.com/study-help/article/pre-calculus-help-conic-sections/

1) The set of all points equal distant to a focus and a special line called the directix (Taken from the video attached to this post)

2)  Key features include the standard form, the direction, the vertex, directrix, focus, axis of symmetry, and value of P

       The formula for parabola is (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h)
          The parabola will vary in direction depending on which variable is squared and the value of P. If x is squared and p is positive the parabola will direct upwards. If P is negative parabola goes down. When y is squared it will go either left or right. While p is positive the parabola goes to the right. While P is negative the parabola goes to the left.
            To find the standard form we can take it from the center value which is the vertex. We can also put together certain pieces if we're not given the vertex. The vertex is found between the focus and directrix on the graph and can be taken from our standard form. The direction is determined by the variables and the value of P. To find our directrix we use fvd and go down or up according to the direction and determined by the value of P. The focus is found the same way. The axis of symmetry is found through when using fvd what value that does change when using fvd such as going dvf  on the graph the y will stay the same because its moving horizontally. Value of P is found through setting the value in front of the non-squared term equal to 4p.
                   The focus of a parabola affects the shape in that if the focus is closer to the parabola its smaller. The focus is also the same distance to any point on the parabola from straight to the directrix is the same value. A parabola's eccentricity is equal to one.
                  The focus x or y value will change accordingly to what other value changes. Such as if the x values change the focus x value will chance. The focus is also in the middle between the vertex on the axis of symmetry. The focus will change in location according to what direction the parabola is facing.

3) Conics of parabolas could be used in many different way in finding the length of bridges, or monuments. Parabolas can be used to determine the waves of satellite dishes in how they the signals are able to bounce back and forth as shown in the video and these signals then bounce towards satellites which receive this signals and project images on television. (Cited from Mrs. Kirch explanation in class)


4) Works cited

• Mrs Kirch lecture
• .  http://www.education.com/study-help/article/pre-calculus-help-conic-sections/
• http://www.mathsisfun.com/geometry/parabola.html
• http://www.mathwords.com/f/focus_parabola.htm 
• https://www.youtube.com/watch?v=INYUob0_wjo