Blog post 4

Good morning, I am professor Matt and we are going to talk about the number "e."

e is about 2.7182818284590452353602874713527 and it goes on forever.

e is the base of all natural logs.

e is mainly used for calculating compound interest or other things that need to be compounded at a higher and higher rate.

e is calculated by (1+1/n)^n.

Here are some of the calculations done out and as you can see they get closer and closer to "e"

1 2.00000 2 2.25000 5 2.48832 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827

Why don't you try some of your own?  What would happen if you invested 10,000 at an interest rate of 5% compounded continuously for five years, use e!

The equation is A=Pe^rt

When you calculate it it comes out to 12,840.25, I hope everyone got it right.

Blog 4 - Be the Professor: Domain and Range

Hello everyone,
my name is professor Maragkos and today ill be teaching you what the domain and range of a function is.
To begin with, we need to know the domain and range of a relationship in order to determine whether it is a function or not, and what values this function can have.
So, lets say that we have a few relationships of x and y values.
(2,4), (5,-1), (-1,2), (-4, 6) and (2, 3)
The x-values of the above relationship are its domain, while the y-values its range.
So, the domain of the above relationship is {-4, -1, 2, 5}
and the range {-1, 2, 3, 4,6}
The above set of values does represent a relationship but since there are two sets with the same x-values, (2,4) and (2,3), the relationship does not represent a function. In that case, we can say that the relationship does not pass the horizontal line test, where every x-value must have only one y-value, so it is not a function.
So, in order to determine whether a relationship is a function, we need to look for x-duplicates at the given relationships.

To get deeper into the concept of domain and range, lets say that we are given a fraction, for example 2x+5/x-7
First step is to make sure that the expression is not divided by 0, in other words that x is never 0.
So, taking the denominator and solving for 0
x-7=0 => x=7
We can conclude that the domain of the fraction is all x-values except for x=7
This can be written in many different ways.

First, there is the set builder notation which has the form of {x ε R/ -∞ < 7 < ∞}
And second, there is the Interval notation which has the form of (-∞,7) U (7, ∞)

Determining the range is a little bit trickier, and usually we are asked to sketch or given a graph, to make the process easier.
Lets say the we are given a polynomial function, y= -x^4 + 4 

The domain of this function is all x-values or given in interval notation (-∞, ∞) 

To determine the range of the above function, we need to take a look at the graph

As we can see, the y-values of the function go up to 4 and down to infinity. 

So, the range of the function would be [4, ∞)
or {y ε R/ y>=4}

Blog Post 4 Be the professor Alex Freedgood

Hi Class my name is Professor Freedgood and today we will learning about Symmetry. Symmetry is a very common topic in many forms of math as well as life. You see it in art, buildings, shapes, images, and even functions! Symmetry is when an object, image, shape, function, or graph looks the same when flipped upon itself. It's as if one side is mirrored onto the other. Here are some simple images to display symmetry.

These shapes all display vertical symmetry the most simple and common form of symmetry. This is because the line dividing the shapes is a vertical line. Similar to a y-axis! But, we'll get back axis in a little bit.


Shapes, images, and functions can be divided/symmetrical in other manners as well. Horizontally and diagonally. Horizontally(x-axis) those axis again! For example a shape like a triangle can only be vertically symmetrical, unless turned on it's side. But, a shape like a circle, rectangle, or square can be horizontally symmetrical.




Symmetry is used in math when discussing functions. Using symmetry one can deduct whether a function is odd or even. When graphing a function the y-axis, x-axis, or the origin are usually the point of symmetry. See those axis are back, y-axis being vertical, x-axis being horizontal, and the origin being diagonal forms of symmetry. Here are some rules related to symmetry.
1. When a function is symmetric along the y-axis and f(x) and f(-x) are equal. then the function is even. 
2. When a function is symmetric along the origin and f(-x) and -f(x) are equal, then the function is odd.

Laura Romero Blog #4

 Domain and Range

Hi Class! Today we are going to learn about domain and range.

Domain is the set of all input values of a function. This is the collection of the x-values
Range is the set of all output values of a function. This is the collection of the y-values

In physics, for every action, there is an equal and opposite reaction. In math, for every input, there is an output. Let's look at some examples:

Table A shows the profit generated by a made up company over the course of 5 years.

Year	2009	2010	2011	2012	2013
Profit	10,000	13,000	15,000	18,000	21,000

The years represent the domain, or input, and the profit is the range, or output.

Table B shows a linear function.


The x values of this function represent the domain or input, and the y values represent the range or output.

There are different ways to write domain and range.

Set Builder Notation lists the rules that determine a set that an object is part of, rather than listing the object itself. Ex: D:{x∈ℝl x>0}
Interval Notation. Ex: D: [0,4]

Let's now apply the notations to the previous tables.

Table A
Interval Notation
D: [2009,2013]
 R:[10000,21000]
Set Builder Notation
D{x∈ℝl 2009<x<2013} 
R{y∈ℝl 10000<y<21000}
Table B
Interval Notation 
D: {-∞,∞} 
R: {-∞,∞}
Set builder Notation 
D{x∈ℝl -∞<x<∞} 
R{y∈ℝl -∞<y<∞}

Now you know more about Domain and Range! 

Be the Professor- Blog Post 4

Completing the Square:

Hello Class! Today we will learn how to "Complete the Square". The reason we do this is because it helps put the quadratic equation into a neat format that can help you solve.

Lets do an example:

x² + 6x + 10 = 0

1. First, move the number thats alone (10) to the other side of the equal sign. 

                                x² + 6x = – 10

2. Next, Take the coefficient on the x-term and divide it by two(keeping the sign). Then square it and add it to both sides of the equation. 
    
                                                   x² + 6x = – 10

                                                       +3-->  +9
                                              x² + 6x+9 = -10 + 9

3. Then, convert the left side to a squared form (x + 3)². On the right side simplify (-1). 

                                                      (x + 3)² = –1
     
^{4. Take the square root of both sides and don't forget the} "±" . 
                                            x+3 =  ± square root of -1 = ± i



5. Solve for x and simplify.

                                  x = –3 ± i

Overall, if you use "Completing the Square" this technique will help you solve for quadratic equations. Make sure to do each step because each one helps you get one step closer to the answer. 

My name is Professor Brennock and today I am going to teach you about the letter “e.” The letter “e” is one of the more famous irrational numbers and is very important. The letter is often called Eulers Number and stands for the numeric value of 2.718. The number was first used by John Napier who used it with logarithms in the1600’s. It acts as a natural constant just like pi would.

The e constant is defined as the limit:

The value of e is also equal to 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

As you can see by the graph, given the equation the larger number you substitute in for “n” the closer you get to the actual value of  “e”.

A few special things about “e” is that the derivative of e^x is special because it is equal to itself and it is the base for continuous, natural decay and growth

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

Professor Aissi: Lesson On Functions (Nathalie Aissi)

Hello class, my name is Professor Aissi and today, we are going to learn about Functions! Yay!

So what is a function class?

  A function is a rule that takes particular numbers as inputs and for each input number, there is an output number. Basically, one output for every input is a function.  

Below is an example of having an output value for every input value. In this case, you can see that for the different temperatures (output), there is a time (input) for each temperature value. This is what a function is ladies and gentlemen!

                              

Time (input)	Temperature (output)
0 (hour)	85 (degrees Fahrenheit)
1 (hour)	86 (degrees Fahrenheit)
2 (hours)	88 (degrees Fahrenheit)
3 (hours)	89 (degrees Fahrenheit)
4 (hours)	88 (degrees Fahrenheit)
5 (hours)	87 (degrees Fahrenheit)

One thing about a function is that it can be represented in many different ways! For example, a function can be represented in a table form (like the example above), in a graph form, in a formula or even in words.

Example of a function represented in a formula:  Y=3x + 12 . To write this formula, you use the slope intercept form, y=mx+b, where “m” is the slope of the function, “b” is the initial value, “x” is the input value and “y” is the output value.

Example of a function represented in Function Notation:  Output= f(input)

Ex: T=f(h),  which is temperature= f(time).

Example of a function represented in words: In one hour, the temperature was 86 degrees Fahrenheit. In two hours, the temperature was 88 degrees Fahrenheit. In three hours, the temperature was 89 degrees Fahrenheit.

Example of a function represented in a graph is below:

 (Every input value graphed has an output value because without every input having an output, a function would not exist. )

                                

Now class, another important thing about a function is that it has to pass the vertical time test!

The vertical line test is the idea of single valued means that no vertical line ever crosses more than one value. If a graph does not pass the vertical line test, then the graph is not a function!! For example, the graph above passes the vertical line test because if a vertical line is placed on any part of the graph, the line would not cross more than one value.

Below is an example of a graph that is not a function because it doesn’t pass the vertical line test:

                                       

You can see on that for the graph above, if a vertical line was placed on the graph, the vertical line would cross more than one value and a function can only cross one value.

When you use a function to describe an actual situation, the function is called a mathematical model. A mathematical model is a function in which the output values depend on the input values.

An example of a mathematical model: T=1/4R+40 is a mathematical model of the relationship between the temperature and the cricket’s chirp rate, This means that when the cricket’s chirp rate is, lets say6, 60 chirps per minute, the temperature is 70 degrees Fahrenheit, which means that the chirp rate depends on the temperature.

An example of a non- mathematical model:

           

                                     Passing Yards for Tom Brady

Year	Passing Yards
2010	3000
2011	5114
2012	2681
2013	6381

The graph above represents how many passing yards Tom Brady threw each year. But even though this is a function, this is not a mathematical model function because the output (passing yards) do not depend on the input (years). In other words, the number of passing yards done by Tom Brady is not dependent on the year he did it.

Now class, here is all there is about functions! Congratulations, you guys are experts now!