Today was our first day of the new semester.  In pre-calculus class we watched a few videos of epic domino sequences. Below is another cool one:

We then split up into groups to create our own mini versions of these domino sequences.  My group is going to create a sinusoid shape out of the 58 dominos we were given.  We considered using the textbooks to create stairs for the dominos to climb and descend, but decided that it would be too difficult to do with so few dominos and in such a short period of time.  I am excited to present our domino sequence and interested to see what the other groups came up with.  It would be really cool if we all our ideas together to create a mega domino set up like they show in the videos we watched.  I’m interested to see how this domino activity ties into pre-calculus.  I’m thinking maybe because each concept leads to another which leads to another, and so on.  I guess we will have to see where this takes us.

Equation to look of graphs:

• Even exponent: Both ends point the same direction
• Odd exponent: Each end points a different direction
• Positive coefficient: as x??, p(x) ??
• Negative coefficient: as x??, p(x) ?-?
• Turns = degree – 1
• Branches = zeros =      degree
• A zero of multiplicity bigger than 2 flattens the graph out
• Greater the exponent the more it flattens out
• Multiplicity of root=1 will go through x-axis
• Multiplicity of root = 2 will touch x-axis
• Find the y intercept for a more accurate graph

To show the different details we learned I’ll use an example.

f(x) = -(x + 2)(2x – 1)³(x – 3)²(x +6)

degree = 7 (greatest exponent or for multiplying factors- how many factors there are)

multiplicity: -(x + 2)(2x – 1)(2x – 1)(2x – 1)(x – 3)(x – 3)(x + 6)

zeros: -2, ½, ½, ½, 3, 3, -6 (what makes each factor = 0)

y-intercept: 108 (found by putting 0 in for each x and then solving)

Local behavior- How a function behaves when the graph is zoomed in enough to see the x and y intercepts and the discontinuities/vertical asymptote

Global behavior- How the function behaves when the graph is zoomed way out, and horizontal asymptote

How do we know if we have a good model?

1.      Interpolation: Put points in your model and check how well they match the true point

2.      Extrapolate: Go outside the data set and check how well it fits the model

3.      See how close the “r” value is to 1 (the closer to 1, the better)

4.      Look at it, does the model fit a majority of the points

5.      Look at endpoint behavior

6.      Look at the residual plot and if the points are random or in a pattern (random is good, meaning that the model accounts for all sources of error)

How to do the math using your calculator:

For reach point above, you can input it into your calculator under the stat plot lists as follows—

L₃ = deviations² = (average (L₂) – L₂)²

L₄ = residuals² = (y₁(L₁) – L₂)²

r² = SL₃ – SL₄ (S represents the sum of that list)

SL₃

Now I’ll do the bluegill problem off of an assessment we did.

(Enter this data into your calculator to produce a graph.  Then using the graph and calculations you can analyze the model.  My analysis of the model is below)

As seen on the graph, the logistics function is the best fitting model for the data. Although ln could also be a good model, logistics is better for a number of reasons.  First off, logistics has a r value of .9997, while ln has an r value of .9966; also, the other regressions r values were worse than both of these (power-.9848 and linear-.9775).  Another reason to rule out power and linear is that the model misses a majority of the points while the other two hit a majority of the points; the data is more curved then the power and linear functions.  So after that step I decided to further look into only logistics and ln. When looking at the graph of the residuals both regressions are random and have no pattern, showing that both models account for all sources of error.  For interpolation, I randomly chose the point (68,95). Logistics gave me the point (68, 95.5), and ln gave me (68, 98.1); so logistics is better for interpolation.  Then I look at extrapolation and can see that logistics converges to 177.9, while ln continues to grow.  Converging makes since because bluegills don’t grow forever, so although converging is still growing a tiny amount, that is better then ln which continues to grow more rapidly for the whole graph.  Also, ln starts off with a negative total length for any initial length less then 21.83, and at about 21.84 initial length, there is no growth total growth.  Where as if you look at the logistics graph, there is no negative lengths, and it starts off at 0 initial length with a total growth after one year of 24.86; which makes since fish start off as an egg and grow.  After making these evaluations, I can say logistics the best model for this set of data.

I thought the most interesting problem in that packet we were working on for claims was number 46. I’m not exactly sure what it has to do with limits, but piecewise functions were always kinda fun. This problem dealt with transforming the signum funtion. This piecewise function only has 3 outputs. If x is less than zero, the output is negative one. If x is zero, the output is zero. If x is greater than zero, the output is positive one. This function is useful in computing for testing a value ov x to see what sign it has. The graph of the signum function looks like this:

The question first asked about the limit and continutity of the function if r(x) = |sgnx|. Because absolute value makes all imputs positive, it means all values of the function except for zero would be one. So yes, the funtion does have a limit because from both sides as x approaches 1, both outputs are at 1. We don’t care about the value at zero. It does have a function value at zero, which is zero. It is not continuous at zero because the defined point does not line up with the rest of the function.

Next, the question asked to sketch a graph of a transformed piecewise function. It looked like this:

The 3 causes a vertical dilation while the subtraction of two from the imputs causes  a horizontal transformation to the right by 2 units.

Then the question asked to sketch a graph of g(x) = x^2 – sgnx. The graph looks like this:

For part d, the problem asked you to prove that the function a(x) = |x|/x is equal to x for all x except zero. To show this, I did three calculations:

a(1) = |1|/1 = 1 and sgn(1) = 1 because according to the function, sgn of any x greater that one is 1

a(-1) = |-1|/-1 = -1 and sgn(-1) = -1 because according to the function, sgn of any x less than one is -1

a(0) = |0|/0 which is undefined and sgn(0) = 0 so just as the problem said, this doesn’t match. The above calculation will also work for every number because any number divided by itself will be one and the absolute value divided by itself will always be negative one.

Finally, the problem asked you to sketch the graph of f(x) = cosx + sgnx, which looked like this:

We know what cosine looks like without any transformations. When the signum function is added to it, 1 is added to all of the points with x values greater than zero. -1 is added to all of the points with x values less than zero. This causes a vertical translation to the function. It looks like the function has changed at zero because the signum function is piecewise. So on the right, it moves 1 up and on the left it moves 1 down. There is still a point at (0,1), which is the same zero for the cosine function.

That’s all there was to the problem. It has incorporated knowledge we’ve aquired throughout this entire year. This was one of the exploration problems so though it didn’t have much to do with limits, this still taught us something. For more practice on these types of problems, see the packet this came out of and other related worksheets.

For awhile we’ve been working on limits…and all associated whatnot. But today, we refined our definition and brought our understanding to the next level. We started class out by looking at some graphs that were discontinuous or continuous and discussed why it was either choice.

One of the new continuous graphs we learned about were graphs with cusps, i.e. where a graph suddenly changes direction while still being continuous. We related it to a hiker on a mountain. A man is hiking to the tip of the point, maybe hesitates, but then falls off the other side, onto the graph. There is no removable discontinuity or step discontinuity when a cusp is present.

We previously knew what a discontinuity was, but now we have defined continuity further. Two ideas we assume that must be present for a continuous graph are the limit of f(x) as x approaches c must exist and also that f(c) must exist. Basically, in a continuous graph limf(x) as x appoaches c = f(c).

Next, we talked about notation for “one-sided limits”. This is the same thing that we’ve previously been doing, just notated different. When we first learned limits, we did check both sides of the function. We’ll still use that notation, but we expanded it. For the limit that approaches point c on the negative, or left, side, we use the notation x?c-. For the limit that approaches point c on the positive, or right, side, we use the notation x?c+. If both of those limits match, then we can say that the limit as x?c is that number.

Which brings me to the next topic: using this notation, what is a limit?

A limit (L) equals the limit f(x) if and only if the limits from both the negative and positive branches match. If the limits of the branches do not match, then we say that the limit does not exist, abbreviated by dne.

Again, L=lim f(x) as x? c  iff  L=lim f(x) as x?c-  and  L=lim f(x) as x?c+ 

For the rest of class we worked on problems to apply our newly aquired knowledge. For more information on limits see:

http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx

http://www.mathnstuff.com/math/spoken/here/2class/420/limit.htm

For the record, the ? are right arrows that are used in limit notation. I’m not too sure why they don’t show up.

Watch the following videos and take any notes that will help you understand the concepts. Pause the videos when you need to, and rewind them so you come away with a good idea of what the presenter is talking about.
Law of Sines
Law of Cosines

You have a quiz tomorrow so make sure and study the sheets we have been doing all week. Finish them if you need to and see Mrs. Sarnow or Mrs. Hojnacki after school if you have questions.

Lastly then, print a copy of the following problems and work on them if you have time left today and then after the quiz.
2004trig-01
2004trig-02
2004ANSWERKEY

For this new section that we have started with exponential and power graphs, stat plots, regression, and other things there is a lot of definitions, equations and mathematical models that must be remembered to be successful. For this blog post I am going to hopefully cover a lot of that stuff.

First off: the goal of this section is to find a mathematical model the will fit a given set of data and decided what type will fit it the best. There are all kinds of equations that could work but the question is which one fits the data the best. The way to do this, if you’re allowed to use your calculator, is to plug the data into your calculator and try to fit a line to it. You do this by going to stat, calculate and then find the equation that gives an “r” value that is closest to 1. This number, “r”, is called the correlation coefficient. It tells how close the model fits the data. The goal is to find the one with the r value that is closest to 1. The r^2 value that some regression equations give is called the coefficient of determination. This is used because its units are the squares of the y-units.

The process behind finding the r value is very complicated (in my opinion.) first you must find the residuals or the vertical distance that each point is from the regression line. This is tedious work but is necessary if your not able to use a calculator (which does it all for you.) then you square each value to make sure they are all wither positive, or zero. Then you add them all together to get the SSres, or the sum of the squares of the residuals. Another process is to find the mean value (add all up and divide by number of terms) and then find the deviations of the mean, then square them all. When you add these up you get the SSdev, or the sum of the squares of the deviations. Most of the time the SSres will be much smaller because the line tends to fit the data a lot better than the horizontal line given by the SSdev.

The coefficient of determination is complicated as well. It is: r^2 = (SSdev-SSres)/SSdev.

*all of this information can be found between pages 325 – 330 in the book. It also gives diagrams that help in the explanation*

Section 8-3 – regression for nonlinear data

Over that past few weeks we have talked a lot about different types of functions, such as power and exponential. To me, these two functions seem a lot alike. I didn’t really understand the difference until I looked up videos on bright storm and explanations on the internet. First off, the general pattern for the two is different. For an exponential function it is add – multiply but for a power function it is multiply – multiply.

The equations for the two are obviously different as well.

POWER: y = AX^n

EXPONENTIAL: y = AB^x

• Notice the placement of the x variable in each of these equations.

I liked how one website said the difference so I am going to paste that below:

In a power function, x is brought to the power of the variable. In an exponential function, the variable is brought to the power x.

Read more: http://wiki.answers.com/Q/What_is_the_difference_between_power_functions_and_exponential_functions#ixzz1HoinQRqC

The books definition is also helpful for both. They are listed below:

Exponential function: a function in which the independent variable appears as an exponent.

Power function: a function of the form y = ax^n where a cannot equal 0.

Terms that come in handy when talking about these are concave up and concave down. These were confusing to me at first but they made sense after looking in the book. The graphs that depict these definitions are on page 272 of the book.

Concave up/down book definition: the graph of a function (or a portion of the graph between two asymptotes or two points of inflection) is called concave upward when its “hallowed out” side faces upward. It is called concave downward when its “hollowed out” side faces downward.

Examples:

   Concave downward

Concave upward