Beginning Algebra With Applications, Chapter 3, 3.2, Section 3.2, Problem 176

The pressure at a certain depth in the ocean can be approximated by the equation $\displaystyle P = \frac{1}{2} D + 15$, where $P$ is the pressure in pounds per square inch and $D$ is the depth in feet.

Find the depth of a diver when the pressure on the diver is $45$ lb/in$^2$.

We solve for $D$ (depth),


$
\begin{equation}
\begin{aligned}

P =& \frac{1}{2} D + 15
&& \text{Given equation}
\\
\\
P - 15 =& \frac{1}{2} D
&& \text{Subtract } 15
\\
\\
2(P - 15) =& D
&& \text{Multiply both sides by } 2
\\
\\
2P - 30 =& D
&& \text{Apply Distributive Property}
\\
\\
2(45) - 30 =& D
&& \text{Substitute } P = 45
\\
\\
90-30 =& D
&& \text{Simplify}
\\
\\
D =& 60 \text{ ft}
&&

\end{aligned}
\end{equation}
$


The depth of the diver is $60$ ft.

Calculus: Early Transcendentals, Chapter 3, 3.6, Section 3.6, Problem 12

In order to find h'(x) we need to know a few things:
The derivative of log(f(x)), and of sqrt(x^2-1).
We need to know the Chain rule. Given h(x)=log(x+sqrt(x^2-1)), we start by differentiating log(f(x)), where f(x) = x+sqrt(x^2-1):
d/dx (log(f(x))) = 1/f(x) * d/dx(f(x)).
Now find the derivative of f(x), that is
d/dx (x+sqrt(g(x))), where g(x) = x^2-1
We have d/dx(x+sqrt(g(x))) = 1 + 1/(2*sqrt(g(x)) * d/dx(g(x))
Finally we are left with finding the derivative of g(x), simple enough.
d/dx (x^2-1) = 2*x
Back substituting we have
d/dx (h(x)) = 1/(x+sqrt(x^2-1)) * (1 + 1/(2*sqrt(x^2-1))*2*x)
Simplifying, h'(x) = 1/(x+sqrt(x^2-1)) * (1 + x/(sqrt(x^2-1))

What does Jesus understand about Daniel when he goes to his house?

Daniel goes to Jesus's house when he loses all hope. When his difficulties are at their worst, he goes to Jesus to seek solace and comfort. At this time his sister's health is at its worst. Leah, whose mental health had been severely suffering because of the extreme stress she'd faced in losing family and friends in the political unrest, was made even worse by her brother's refusal to admit one of her only friends into their home. This was because her friend, Marcus, was a Roman soldier.
When Jesus sees Daniel, he immediately understands his suffering and pain. He knows that Daniel is holding onto his anger and bitterness, refusing to be freed by love. Daniel does not yet understand how powerful love is. But Jesus doesn't lose hope for Daniel. Even though Daniel has rejected Jesus before, hearing his teachings but not yet believing them, Jesus accepts Daniel into his home. Daniel still believes that violence and rebellion are the answer. In fact, he's waiting for the coming of a Messiah who he thinks will be a violent warrior king who will save the Jewish people. What he doesn't understand is that the Messiah is standing right in front of him, preaching love and grace, and dressed in simple clothes rather than royal garb.
What does Jesus understand about Daniel? Everything. He understands his immense suffering, the depths of his anger, and the fear he has for his sister's well-being. He knows that Daniel doesn't agree with him (yet). Nonetheless, Jesus doesn't lose hope for Daniel.

College Algebra, Chapter 5, 5.2, Section 5.2, Problem 60

Graph the function $y = 1 + \ln(-x)$, not by plotting points, but by starting from the graphs of family of Logarithmic functions. State the domain, range and asymptote.







We start with the graph of $y = \ln (x)$ and reflect it to the $y$-axis then shift $1$ unit upward to have the graph of $y = 1 + \ln (-x)$. Its domain is $(- \infty, 0)$ and its range is all reals. Also the vertical asymptote is $x = 0$.

What is an easy and smart sounding research topic for a high school student to write a 10 page paper about?

This is the kind of assignment that the college freshmen I teach in First-Year English Composition and Rhetoric are given three times per semester. The advice that I like to give my students is to choose a subject that you are passionate about or have an interest in learning more about. Select a topic that personally affects you, and you will have no trouble sounding "smart" because the act of researching will be an engaging process for you. Do NOT pick a topic that you have no interest in simply because it seems easy to write about; this will often lead to the classic teenage disease known as "procrastination," which can lead to the even more fatal condition known as "writing your paper the night before it is due."
Here is what I like to ask my students when they are searching for a topic: Look at the world around you and your life. Are you happy with it? Would you want to be living this exact life in this exact world's conditions in fifteen years? The answer is likely no. The average human being can find issue with something in their life. So what is it? What would you like to change? 
Personally, a topic that I care a lot about right now is the private prison industrial complex and the incentivizing of locking people up. What this means is that non-violent offenders are being given longer and longer prison sentences for drug-related offenses, all for the financial benefits of huge corporate investors who profit when prisons are full. This has resulted in huge personal consequences for inmates who will likely not see the light of day for years (if ever) and has decreased efforts toward limiting recidivism (the likelihood that one re-commits a crime) and rehabilitating criminals. A vast majority of these sentences are also reflective of a sinister judicial system which is predicated on racial bias and prejudice, with 70% of imprisoned individuals being people of color. This topic impacts me because I spent a year teaching Creative Writing to maximum security students in a state prison; I saw firsthand how damaging imprisonment was to the psyche of my students, many who came from impoverished and underprivileged backgrounds. 
So, to reiterate: think hard about what troubles you about the world around you and choose something to advocate on the behalf of. Pick up any newspaper and you'll find some cause worth fighting for. 

"Easy and smart sounding" is going to differ from student to student and reader to reader.  There is not going to be a single topic that I could recommend that would fit those two characteristics for every person.  My recommendation is that you pick a topic that you are interested in.  That will likely make the research feel easier because you are personally interested and invested in the topic.  For "smart sounding," I recommend picking a topic that hasn't been done so many times that your reader is likely to be bored with it before he/she even begins reading the paper.  
I'd like to suggest a possible topic for you.  More than likely, you and your classmates were not happy with this assignment.  First, it's homework.  Second, it's a big homework assignment.  I think an interesting research topic for you is to research whether or not homework is actually beneficial to student learners.  I did my master's thesis on this very topic, and a flood of new research has come out since then.  You shouldn't struggle to find information about how and when homework is and/or isn't beneficial to students.  You could even guide your research toward finding homework types that are most beneficial to students.  
If that doesn't sound interesting, then perhaps you could research on the benefits and the pitfalls of the "gamification of education" that is happening across the nation.  In my experience, students like games and video games.  Gamification of education seeks to bring those elements into the learning process.  Your paper could explain what the general idea of gamification is, and how it can benefit and hurt student learning. 
https://inside.rotman.utoronto.ca/behaviouraleconomicsinaction/files/2013/09/GuideGamificationEducationDec2013.pdf

What did Auggie compare his friendship with Summer to?

Wonder, written by R.J. Palacio, is a story about August "Auggie" Pullman, a young boy with a facial deformity. Auggie has never attended a real school and is worried about starting school as a fifth-grader.
Auggie faces many issues on his first day of school, especially finding a place to sit in the cafeteria during lunch. Auggie decides to eat by himself and sits alone at a table. Summer, a girl at the school, asks Auggie if the seats by him are taken and sits down.
While Auggie and Summer are eating, a friendship begins to form. Because of their names, August and Summer, they decide that only people with "summer" names should be allowed to sit at their table. After school, Auggie tells his mom that his friendship with Summer is sort of like Beauty and the Beast. He makes this comparison because he sees himself as ugly because of his face and because of how he must eat due to his deformity. He sees Summer as beautiful because she is one of the more popular girls and also has a great personality.

On Auggie's first day of school, he faces the uncomfortable predicament of finding a place to sit at lunch. Although a teacher specifically said not to save seats, it seems as if many students are saving seats for their friends. Finally, Auggie finds a table and begins to eat lunch by himself. Already self-conscious about the way he looks, Auggie also hates the way he eats. He has a hole in the top of his mouth which causes crumbs to fly out as he chews. As he begins eating, Summer approaches him and says, "Hey, is this seat taken?"
Auggie and Summer proceed to have a conversation about a "summer only" lunch table. They make a list of kids with summer names that could potentially sit at their table. When his mom picks him up from school, she notices Summer and asks about her. Auggie explains that they ate lunch together. Auggie then makes a comparison about his friendship with Summer. He says, "We're kind of like Beauty and the Beast." Auggie makes that comparison because he feels he is ugly and Summer is pretty.

Calculus and Its Applications, Chapter 1, 1.6, Section 1.6, Problem 18

Take the derivative of $\displaystyle F(x) = \frac{x^3 + 27}{x + 3}$: first, use the Quotient Rule;
then, by dividing the expression before differentiating. Compare your results as a check.
By using Quotient Rule,

$
\begin{equation}
\begin{aligned}
F'(x) &= \frac{(x + 3) \cdot \frac{d}{dx} (x^3 + 27) - (x^3 + 27) \cdot \frac{d}{dx} (x + 3)}{(x + 3)^2}\\
\\
&= \frac{(x + 3)(3x^2) - (x^3 + 27) (1)}{x^2 + 6x + 9}\\
\\
&= \frac{3x^3 + 9x^2 - x^3 - 27}{x^2 + 6x + 9}\\
\\
&= \frac{2x^3 + 9x^2 - 27}{x^2 + 6x + 9}
\end{aligned}
\end{equation}
$


By further simplifying, we get
$F'(x) = 2x - 3$




By dividing the expression first,



Thus,
$\displaystyle F(x) = \frac{x^3 + 27}{x + 3} = x^2 - 3x + 9$

Hence,
$\displaystyle F'(x) = \frac{d}{dx} (x^2 - 3x + 9) = 2x - 3$
Both results agree.