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

Today in math, we learned about imiginary numbers and how they are utilized when finding unreal values in such instances as the quadratic formula.

The fist thing we learned was that the square root of a negative number is an imaginary number. The units for imaginary numbers “i” has a value of “the square root of -1.” Futhermore, when a solution is a “complex number,” that simply means the sum of it is composed of a real number and an imaginary number.

Take for example the quadractic equation “x^2 + -4x + 18=0″. If we plug the a, b, and c values for this equation into our quadractic formula programs on our calculators, an error message would appear because none of the possible “x” values are real numbers.

Thus to solve, we would have to figure the possible “x” values manually, using the quadractic formula model “y= -b (+ or -) ‘the square root of ‘ b^2 – 4ac, ‘all over’ 2a”. After plugging the a, b, and c values into the formula manually you recive “y=-6 (+or -) ‘the square root of ‘ -12 ‘all over’ 2″. But since -12 cannot be squared this is were imaginary numbers substitute the negitive sign and the new equation is “y=-6 (+or -) ‘the square root of ‘ 12 i ‘all over’ 2″. Then through futher simplicfication we would recieve a complex number for the solutions; simplifying to “y+-3 (+or-) i ‘to the square root of ‘ 3″.

All done !

Logarithms are the opposite of exponents, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs undo exponentials. Technically speaking, logs are the inverses of exponentials.

They generally have this relationship. log_a x = N means that a^N = x. (generally when there is no a: for example log(10), the log is base 10, which means a = 10)

Once you recognize that then youre a step ahead.

What is a log, anyway?

Let’s start with the very basics. First we learned to add:

5+5+5+5 = 20

Then we learned to multiply..

5 x 4 = 20

Then exponents..

5 x 5 x 5 x 5 became 5 to the fourth power = 625

Then we get into logarithms, where the log to the base 5 of 625 = 4

if 5 to the fourth = 625 the log:base 5(625) = 4. Also if 3 to the ninth power = 729 the log:base 3(729) = 9.

bringing us back to what I stated before. log_a x = N means that a^N = x

A log scale measures exponential values rather than unit values as a linear scale does. This is useful when values change more rapid than is convenient to represent on a linear scale. Our senses of sight and hearing function on a logarithmic scale such that doubling the actual intensity of the input increases its real intensity by a constant amount.

Reading a log scale.

Step 1) Create a log. For the equation x = b^y, y is the logarithm of x to the base b. Therefore, if x = b^y, then y = logb(x). The most common base for a logarithmic scale is 10.

Step 2) Make values on a linear scale. The unit markings on a linear scale are values such like the first unit marking is 1, the second unit marking is 2, the third unit marking is 3 and so on.

Step 3) Look at a logarithmic scale. The unit markings on a logarithmic scale show the values given by raising the logarithm’s base to the power of each integer.

Step 4) Read the values on a logarithmic scale. The first unit marking shows 10^1 = 10, the second unit marking is 10^2 = 100, the third unit marking is 10^3 = 1,000 and so on.

Step 5) Look at negative values on a logarithmic scale. On this scale, the unit -1 shows 10^-1 = 0.1, -2 shows 10^-2 = 0.001, -3 shows 10^-3 = 0.001 and so on. Also, realize that negative numbers on a logarithmic scale are numbers between 0 and 1 instead of negative numbers.

This is how I learned logs.

Hopefully you can benefit.

- Brian.

Series are the sum of all the terms of a sequence.  Just like sequences, they can be infinite or finite.  There are two types of series: arithmetic and geometric.

An arithmetic series is the sum of an arithmetic sequence.  And this sum is called the partial sum (a portion of a sequence that is added together).  The equation to figure this out is: Sn (or the sum of the first n terms in this series) is equal to n over 2 (the number of terms), which is then multiplied by  a1 (the first term) plus an (the last term).  But if the series is infinite and we don’t know the last term, we simply take a1 (first term) plus n minus 1 times the common difference, and substitute that for an (last term).

Sn=(n/2)(a1+an)        an=a1+(n-1)d

Example: 3+7+11+15+…  Find S40

s40=(40/2)(2(3)+39(4))

=20(162)=3240

A Geometric Series is the sum of a Geometric Sequence.  This has two different equations, one for infinite and the other for finite.  The difference between them is when looking at the finite one, you’re finding the sum of a certain number of n terms. For the infinite series, there is no n. Not having an n tells us that we don’t have a specified term number and we are summing everything.

Finite:

Sn=(a1(1-(r^n))/(1-r)

Infinite:

There is one restriction: the absolute value of our rate has to be less than 1; meaning that our terms have to be getting smaller.

S=a1/(1-r)

Example:

1/6 + 1/2 +3/2+…     Find S7.

Rule (rate): *3

Sn= ((1/6)(1-(3^7)))/(1-3)

=(1/-12)(1-(3^7))