Friday, June 6, 2014

WPP #12: Unit O Concept 10: Angles of Elevation and Depression

Steven is playing football and is given the chance to kick the ball into a the field goal.

He is at the 20 yd line and is getting ready to kick it up the 15 yd post. When he kicks it, it hits it exactly at the post. At what angle does he have to kick it to be able to hit the post?

When the ball comes back off the pole it falls 15 yds down and lands on the 30 yd line. At what angle is the ball coming at to the floor?

Thursday, June 5, 2014

BQ#7: Unit V Concept 1: The Difference Quotients

When you are given a secant line on a graph with ONE point , you can find the other point by using the difference quotients...
How do you find the difference quotients?
First look at the function.

When we give this function a point it will look like this 

We don't know exact numbers so we will give the x value x and the y value f(x) because it is a function. As you can see this is the point for the first point of the secant line. We need to be able to find the slope for this line in order to move on with anything else.

We need to know the point on the graph to be able to use this formula. We can get the other ordered pair on the graph by adding in another variable.

H stands for the length that we are missing. We use H to show that x has changed. When we write the next x value it will look like this...

We have to add x+h because that shows that we are adding both segments. The second point isn't just H. It is both x and h combined together and that is why we add them. The same thing for the y value. Now that we have our points, we are able to plug it in the slope formula. 

Next we are able to cancel variables like x's at the bottom

As a result, we have our difference quotients. Taking the derivative of the quotients formula will get you the slope from the secant line. 

f'(x)=2x will be the slope. From here you can use y=mx+b the find the tangent to any point on the function.

Tuesday, May 20, 2014

BQ# 6: Unit U Concept 1-8: Defining Limits

1. Continuity is when a function can be drawn without lifting your pencil from the paper. A continuous function also has no breaks, jumps, or holes.

 Although there is a hole in the function, the limit is there. A limit is an intended height, and this function is continuing on going through that intended height. The function is continuing on with it's pattern. The function is continuous. 

A discontinuity is a point in which the function stops and has a change in behavior. A jump discontinuity is when the function from the left and the right meet but have different y values, it changes and so you see two different segments. An infinite discontinuity is when there is a vertical asmyptote which leads to unbounded behavior and this causes the function to be discontinuous. An oscillating discontinuity with when the function has a very wiggly behavior and is very inconsistent, this means there is no definite point so the limit does no exist which means it is a discontinuity. 

2. A limit is the intended height in a function. A limit will always exist unless there is a jump discontinuity or infinite discontinuity or an oscillating graph. Besides these few points, their should be a limit at every single point you are on. If you happen to land on one of these points, the limit will not exist. The differences between a limit and a value is that a limit is the INTENDED height and the value is the ACTUAL height of the function.

3. When we evaluate a function numerically, we take a look at the table. We use tables to identify discontinuities. Evaluating a function graphically is the easier way you can look for limits/discontinuities. You can just take you fingers and bring them together to see what's going on on the graph. You will be able to see right away whether it is continuous or if there is a discontinuity. There are several steps in which you would have to do in order to evaluate a limit algebraically. You must first do substitution. You will substitute in the limit it is asking for and if you have a regular fraction or 0/#  or #/0 than that will be your limit. If it is 0/0 than it is indeterminate and you have to continue on by either factoring the equation or rationalizing/conjugate. From there you have to cancel out to remove the hole in the equation. Once things have canceled then you are able to use substitution again to find the limit, and that's how you evaluate a limit algebraically.

Sunday, April 20, 2014

BQ#4: Unit T: Concept 3

In the Unit Circle, tangent and cotangent are both positive in the first quadrant and negative in the second. Then it is positive in the third and then negative in the fourth quadrant. The identity for tangent is tan(x)=sin(x)/cos(x) and the identity for cotangent is cot(x)= cos(x)/sin(x). This causes the asmyptotes to be place in different positions. The asymptotes for tangent are in different places than the asmyptotes for cotangent and this is what makes the trig graph go uphill or downhill.

The reason why tangent is going uphill and it is normal is because the first asymptote is place at pi/2 which is the end of the first quadrant. In the first quadrant the function needs to be going up, in order to go on to the next part of the function it needs to start down without touching the asymptote. That is why it starts over and starts down then goes up this is why it is going uphill.

Cotangent is different. The first asymptote  is mark at pi. This means the function can start up in quadrant one and progress down to quadrant two. There is no need for a new function to start because there is no asymptote separating those two quadrants. This is why it is normal for the function to be going down hill.

BQ#3: Unit T: Concepts 1-3

Sine and cosine are a part of every graph. When we go back to our identities we know that
a. tan(x)=sin(x)/cos(x)
b. cot(x)=cos(x)/sin(x)
c. sec(x)=1/cos(x)
d. csc(x)=1/sin(x)

Asymptotes from sine and cosine cause these graphs to differ. Sine and cosine are the only functions without a x or y in its denominator. All other trig functions do. Whenever sine (y) is 0 there will be an asymptote and that is where cotangent and cosecant have their asmyptote. The same goes for cosine (x), whenever it is 0 there will be an asymptote. This is what causes the trig graphs to change and differ.

BQ#5: Unit T Concepts 1-3

Whenever you have 0 in your denominator it will be undefined. You can only have an asymptote if you have an undefined number. All the trig functions have a ratio:

Sine               Cosine                  Cosecant                   Secant                  Tangent                Cotangent
      y/r                    x/r                           r/y                           r/x                         y/x                         x/y

When we are talking about the Unit Circle "r" matter what it equal to 1. When we look at the ratios, sine and cosine are the only trig functions where r is the denominator. This means no matter what the numerator is, the function will never be undefined because the denominator will always be 1. All the other functions have the possibility of being undefined because x or y can equal to 1 or 0 and that all depends on what point on the graph you are on. If you are on 0 or 180 or 360 degrees, cosecant and cotangent will be 0 because the ordered pairs on that point are (1,0) and 0 is y. If you are on 90 or 270 degrees, secant and tangent will be 0 because the ordered pairs are (0,1) and x is 0. 

BQ#2: Unit T Intro

1. The Unit Circle is a trig graph unraveled. It follows the same patterens as it does on the Unit Circle. When you have sine on the Unit Circle, it is going to be + + - -, on the trig graph it is the same thing it is up up then goes down down and that ends that period of the graph and on the Unit Circle it marks then end of a whole circle.

The period of sine and consine is 2pi because then you look at the negatives and positive from the Unit Circle, it takes one whole circle for the pattern of positives and negatives to start again. When you go around the Unit Circle completely that is 2 pi, that is why the period is 2pi. The reason why tangent and cotangent is pi is because it takes half the circle for the pattern to start over.

Sine and cosine are the only ones that have an amplitude, they are also the only two graphs that do not have that do not have asymptotes. All the other graphs have asymptotes because they certrain points where the graph is undefined and does not touch those points. With sine and cosine they graph will touch any points as long as they are within the amplitude. With all the other trig functions they have asymptotes. Amplitude and asymptotes are restrictions for each graph.