Glencoe Algebra 2, Chapter 2, 2.6, Section 2.6, Problem 9

f(x) = |3x-2| , the the domain and range is given as follows
(i)Domain definition:
The domain of a function is the set of the input or argument values for which the function is real and defined.
In this function, The function has no undefined points, so the domain is
-oo (ii)Range definition
It is the set of values of the dependent variable for which a function is defined.
For this function the interval has a minimum point at x= 2/3 with value f(x) = 0
so the range of |3x-2| is f(x) >= 0
It can also be observed from the graph below:

Why are scientists are developing alternative energy resources?

There are a lot of possibilities:  
We are going to (someday) run out of our current energy resources.  Much of the world's energy depends on fossil fuels, and those are finite resources.  Once they are used up, that's it.  An alternative energy source will have to be used.  Scientists know this, and they are trying to work ahead of the coming problem.  
They are curious.  Scientists are inquisitive people by nature, so looking in to new technology and energy resources is satisfying their curiosity.  
They are hoping to discover something amazing, patent it, and get rich.  A person that figured out an almost unlimited fuel source that is cheap and easy to use and maintain would likely become one of the richest people in history.  
They are environmentally conscious.  This is similar to #1, but it could be different in a key area -- pollution.  Fossil fuels and their burning pollutes a great deal.  Many new energy resources could be renewable and clean.  
https://www.eia.gov/energyexplained/renewable-sources/

Why does everybody fear Capulet's nephew Tybalt?

Tybalt has the deserved reputation of a hothead, a young man with a violent temper who is quick to take offense and forever getting into fights and scrapes. In the very first scene of the play we see him living up to his nickname of "The Prince of Cats," so-called because he's always looking for a fight. He shamelessly fuels the fire of the skirmish between the servants, goading the Montagues into further acts of violence. The dominant moral code permits young men to fight to the death if needs be to defend their honor and the honor of their families. But even by the standards of the time, Tybalt's taste for violence is excessive. He doesn't fight because he has to; he fights because he wants to. This, more than anything else, is what makes him so widely feared.

f(x)=1/x ,c=1 Use the definition of Taylor series to find the Taylor series, centered at c for the function.

Taylor series is an example of infinite series derived from the expansion of f(x) about a single point. It is represented by infinite sum of f^n(x) centered at x=c . The general formula for Taylor series is:
f(x) = sum_(n=0)^oo (f^n(c))/(n!) (x-c)^n
or
f(x) =f(c)+f'(c)(x-c) +(f^2(c))/(2!)(x-c)^2 +(f^3(c))/(3!)(x-c)^3 +(f^4(c))/(4!)(x-c)^4 +...
To apply the definition of Taylor series for the given function f(x) = 1/x centered at c=1 , we list f^n(x) using the  Power rule for differentiation: d/(dx) x^n= n *x^(n-1) and basic differentiation property: d/(dx) c* f(x)= c * d/(dx) f(x) .
f(x) =1/x
f'(x) = d/(dx) 1/x
       = d/(dx) x^(-1)
       =-1 *x^(-1-1)
       =-x^(-2) or -1/x^2
f^2(x)= d/(dx) -x^(-2)
            =-1 *d/(dx) x^(-2)
            =-1 *(-2x^(-2-1))
           =2x^(-3) or 2/x^3
f^3(x)= d/(dx) 2x^(-3)
           =2 *d/(dx) x^(-3)
          =2 *(-3x^(-3-1))
          =-6x^(-4) or -6/x^4
f^4(x)= d/(dx) -6x^(-4)
            =-6 *d/(dx) x^(-4)
            =-6 *(-4x^(-4-1))
           =24x^(-5) or 24/x^5
Plug-in x=1 , we get:
f(1)=1/1 =1
f'(1)=-1/1^2 = -1
f^2(1)=2/1^3 =2
f^3(1)=-6/1^4 = -6
f^4(1)=24/1^5 = 24
Plug-in the values on the formula for Taylor series, we get:
1/x =sum_(n=0)^oo (f^n(1))/(n!) (x-1)^n
=f(1)+f'(1)(x-1) +(f^2(1))/(2!)(x-1)^2 +(f^3(1))/(3!)(x-1)^3 +(f^4(1))/(4!)(x-1)^4 +...
=1+(-1)*(x-1) +2/(2!)(x-1)^2 +(-6)/(3!)(x-1)^3 +24/(4!)(x-1)^4 +...
=1-(x-1) +2/2(x-1)^2 -6/6(x-1)^3 +24/24(x-1)^4 +...
= 1-(x-1)+ (x-1)^2 -(x-1)^3 + (x-1)^4 +...
The Taylor series for the given function f(x)=1/x centered at c=1 will be:
1/x=1-(x-1)+ (x-1)^2 -(x-1)^3 + (x-1)^4 +...
or
1/x =sum_(n=0)^oo(-1)^n(x-1)^n

Beginning Algebra With Applications, Chapter 1, 1.2, Section 1.2, Problem 172

Determine which statement is true for all real numbers.

a.) $||x| - |y|| \leq |x| - |y|$

b.) $||x| - |y|| = |x| - |y|$

c.) $||x| - |y|| \geq |x| - |y|$


If we let $x = -2$ and $y = 3$, then we substitute this to the given statement. We have

a.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| & \leq |-2|-|3|
\\
|2-3| & \leq 2-3
\\
|-1| & \leq -1
\\
1 & \leq -1

\end{aligned}
\end{equation}
$


The statement is false.

b.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| =& |-2| - |3|
\\
|2-3| =& 2-3
\\
|-1| =& -1
\\
1 =& -1

\end{aligned}
\end{equation}
$


The statement is false.

c.)


$
\begin{equation}
\begin{aligned}

||-2| - |3|| \geq & |-2| - |3|
\\
|2 -3| =& 2-3
\\
|-1| =& -1
\\
1 \geq & -1

\end{aligned}
\end{equation}
$


The statement is true.

So the statement that is true for all real numbers is $||x| - |y|| \geq |x| - |y|$

Describe the personality of the Artful Dodger.

The Artful Dodger (or, by his true name, Jack Dawkins) is a character from the Charles Dickens novel Oliver Twist. For generations of readers, he has become a beloved character despite being a disloyal thief and, in a way, the true villain of the book, as it is the Artful Dodger who persuades young Oliver to join Fagin's gang of child criminals.
As the leader of that operation of pickpockets, the Artful Dodger is portrayed somewhat as a man in a child's body. We read that he "was of a rather saturnine disposition, and seldom gave way to merriment when it interfered with business." In order to command the respect of the other young thieves—and of his insidious boss, Fagin—he tries very hard to come across as confident, mature, and grown up, even dressing in men's clothing (which end up looking quite comical on him). The impression he makes is well described in this sentence: "He was, altogether, as roistering and swaggering a young gentleman as ever stood four feet six, or something less."
However, despite his show of maturity, Jack Dawkins is in reality just a boy, and a rather cocky, audacious one at that. Our last glimpse of the Artful Dodger is when he is being kicked out of the country (presumably to Australia) for having been caught with a stolen silver snuff box. In the courtroom he cries out with indignation: "I am an Englishman; where are my privileges?" and generally makes as inconvenient a commotion as he can. Yet we read that, once out in the yard, he is now grinning with “great glee and self-approval," supremely pleased with his dramatic performance.
As the best of Fagin's gang of pickpockets, the Artful Dodger is very intelligent and quite cunning, and we know he is not a heartless boy. One example is his relationship with Oliver. Though their friendship ends badly, he is for a time very close to Oliver, and the Artful Dodger tries to teach the boy how to be a good thief (though with poor results). He also respects Fagin a great deal and shows implicit trust in him, giving Fagin everything he manages to steal.

What was the most difficult part to read in “The Cold Equations” by Tom Godwin? How did you get through this part?

This is a matter of opinion, but personally, the most difficult part for me (assuming you mean emotionally difficult rather than challenging to read) was when the girl says goodbye to her brother. In their conversation, she tells her brother,

Maybe I’ll come to you in your dreams with my hair in braids and crying because the kitten in my arms is dead; maybe I’ll be the touch of a breeze that whispers to you as it goes by; maybe I’ll be one of those gold-winged larks you told me about, singing my silly head off to you; maybe, at times, I’ll be nothing you can see, but you will know I’m there beside you.

She wants her brother and her parents to remember her as she was in her life rather than “the other way,” which refers to the way her body will be affected when it is out in space. Though the entire story is quite sad, this part is the most filled with emotion and, for me, the most difficult part to get through.
What helps me get through that part is to remind myself that here on Earth, we have options. This type of situation in which only one option exists does not typically arise in real life. The setting of the story—a space ship on a mission to deliver medicine—is unrealistic if not impossible today. Though there are real life situations in which a person’s life has to be sacrificed for the greater good (such as war), there probably cannot be, and will never be, a situation where a young girl has to die in such a cold and calculated manner. On Earth, we have learned to come up with creative solutions to problems so that if a situation like this were to occur, we would find a way out of it. I like to believe that, as humans, we do everything we can to spare the lives of the innocent at any costs.