Thursday, June 5, 2014

BQ 7: Unit V

The difference quotient, which is shown in the link below, comes from the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition the derivative. The difference quotient allows you to find the slope of any curve or line at any single point. The difference quotient is the change in "y" divided by the change in "x." The only difference is that in the slope formula, y is used as the y-axis, but in the difference quotient, the change in the y-axis is described by f(x).

Difference quotient

Wednesday, June 4, 2014

WPP#13-14: Unit P concept 6-7: Applications with Law of Sines and Cosines

1.) Ricky and Ana went a a family bbq to watch the game.  However, no one could find the remote to turn on the tv so they quickly all tried to look for it before the game started. Ana looked under the couch when it was north of the couch and at Ricky was looking in the cabinets which was 25 yards away from Ana found it. Ricky saw the remote  N28E from a north-south line through where was standing and Ana saw the ball's path  N39W from a north-south line through where she is standing. What is the distance between Amarie and the football?

2) Ricky and Ana were playing baseball and they both ran to catch the ball. Ricky ran 12 yards at a bearing on 56 degrees and Ana ran 14 yards at a bearing of 315 degrees. how far apart were they before starting to run?




Monday, May 19, 2014

BQ#6: Unit U



1. What is a continuity? What is a discontinuity?
A continuity is a predictable graph in which you can draw without lifting your pencil and does not have jumps, breaks, or holes. To see an example of a continuous graph look at the first image below. A discontinuity on the other hand consists of non removable and removable. A non removable is non removable for the fact that there are different left and right heights on both sides of the graph, the graph is oscillating meaning it is extremely wiggly, or there is an unbounded behavior meaning that there is a vertical asymptote.  At vertical asymptote keeps going and going closer and closer to the graph. When saying there are different left and rights, it simply means that if you put your hand on the left side and the right side of the graph and slide them together, they do not meet.  A removable discontinuity is a point discontinuity meaning that there is a hole in the graph, as seen in the second image below.

http://www.analyzemath.com/calculus/continuity/continuous_1.gif



https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTfGXTPwPhR-q7-0XVjAsFKgDYLVYvmiIoGlO2I_UFAHoiBrlgl
2. What is a limit? What is a discontinuity? 
A limit is the intended height of a function, meaning its where two graphs are to meet up with each other. It is not able to exist when there is an unbounded behavior meaning there is a vertical asymptote, it is oscillating, and there are different left and right heights. An example of each is shown below. 

Unbounded behavior 

http://shell.cas.usf.edu/~ssuen/2233.2013s/c/chapter1.diagrams/1.6.18.png


Different left and right


http://image.tutorvista.com/content/feed/u364/discontin.GIF

Oscillating 




http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg


3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically means to set up a table. You are to have the limit in the middle and three boxes on each side. On far right side box you add 1/10 and the far left you subtract 1/10. From each sides you are to see clearly that the numbers keep getting a little closer and closer. To see an example look below. To evaluate a limit graphically you simply put a hand on the left and right side of the graph and slide them together. To evaluate it algebraically you may use substitution, which is simply plugging in the x value and solving. nalizing/conjugate method which is multiplying the conjugate and canceling. A conjugate is taking the radical expression and changing its sign.

Works Cited
http://www.analyzemath.com/calculus/continuity/continuous_1.gif
https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTfGXTPwPhR-q7-0XVjAsFKgDYLVYvmiIoGlO2I_UFAHoiBrlgl
http://shell.cas.usf.edu/~ssuen/2233.2013s/c/chapter1.diagrams/1.6.18.png
http://image.tutorvista.com/content/feed/u364/discontin.GIF
http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg

Wednesday, April 16, 2014

BQ#2: Uit T Concept Intro

The trig functions when graphed relate to the unit circle because when their values graphed and on the unit circle are the same there simple extended when there graphed.

Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
It takes up to all four quadrants to repeat the same pattern. tangent and cotangent have a period of pi because they already have a positive and negative value on the first two quadrants. 
Sine and cosine can not be bigger than one because of their ratios. 

Friday, April 4, 2014

Reflection#1: Unit Q Concept 5: Verifying Trig Identities

1. What does it actually mean to verify a trig function?
It means to solve each side of the equation so that they are equal to each other. You are proving that the identity is always true.You must simplify one of the sides by  finding the GCF, substitution for an identity, multiplying by the conjugate, combining fractions with a binomial denominator, separating fractions with monomial denominators, or factoring as you break things down.The zero product property (ZPP) may apply which means your equation is equal to zero.

2. What tips and tricks have you found helpful?
The process of verifying trig functions a bit quicker when having the identities memorized so that you won't have to keep looking back and checking what they are. Another way to make things a bit easier is to be fully aware of what you are able to do with the trig functions such as, substituting an identity, multiplying by the conjugate, separating fractions with monomial denominators etc.Always try something even if you think your unsure of what to do. Once you start the problem you start to notice things that you can either cancel, substitute, factor etc.RRemember that you can never divide by a trig function.

3. Explain your thought process and steps you take in verifying a trig function. Do not use a specific example but speak in general terms in what you do no matter what they would live you. 
It is better to first work with the one side of the equation that is more complicated and then you can move onto the other. You must look if there is a GCF, substitution for an identity, you are able to multiply by the conjugate, combine fractions with a binomial denominator, separate fractions with monomial denominators, or factor. In order to fully solve you must continue to look for these opportunities. Keep in mind that you may convert some terms to sines and cosines but not all. You cannot touch the right side.There is not only a one way process however it is possible.

Wednesday, March 26, 2014

I/D#3: Unit Q: Pythagoreom Identities

Inquiry Activity Summary: 
1.Sin2x+cos2x=1 comes from the identity, Pythagorean theorem, which is a  factor or formula that is proven.The Pythagorean theorem, a^2+b^2=c^2 is replaced x^2+y^2=r^2 because of the fact that  they hold the same values.

2. To derive the two remaining Pythagorean Identities from sin2x+cos2x=1 you must first, divide everything by r^2 and after doing that we get x^2/r^2+y^2/r^2=1. You can then notice that they x/ r is cosine and y/ r is sine meaning that you can put sin2x+cos2x=1 in place.

Inquiry Activity Reflection:

1. The connections that I see between Unit N,O,P, and Q so far are all related to the unit circle. In order to solve and understand most of them, you must fully understand and keep in  mind the unit circle. Having memorized the unit circle is key. 

2. If i had to describe trigonometry in THREE words, they would be difficult, challenging, and memorization. Trigonometry really requires a lot of practice and memorization of the unit circle. You can not solve the problems if you do not know the unit circle, formulas,  sin, cos, tan, and so much more.