How many ways are there to arrange four people into two pairs? Write out all permutations of [4] and then group them into equivalence classes. What is the size of each equivalence class and what then is the answer to the original question?

Suppose that you have a unit cube whose center is at (cx, cy, cz).

(a) Let us rotate it, along the direction of the x-axis, about its centre for angle θ. Derive the general matrix for doing this rotation.

(b) Suppose that we rotate it, centering at (cx, 0, 0), along the direction of the x-axis. Show the rotation matrix and explain why it is so simple.

1. Use the ę notation to prove the following limits:

lim n→∞ [n^2+ 3ncos(2n+1)+2] / [n^2−nsin(4n+3)+4] = 1

2. Let {an} a sequence converging to L > 0. Show ∃N ∈ N, ∀n ∈ N, n ≥ N, an > 0

3.Let {an} a sequence converging to L. Let {bn} a sequence such that ∃Nb ∈ N, ∀n ∈ N, n ≥ Nb, an = bn. Show that {bn} converges to L as well.

Thank you. Please complete proofs fully.

Each Friday evening in August 2011, each episode of Family Guy was typically watched by 1.3 million viewers, while each episode of American Dad was typically watched by 0.9 million viewers.† Your marketing services firm has been hired to promote Gauss Jordan Sneakers by buying at least 20 commercial spots during episodes of Family Guy and American Dad. You have been quoted a price of $2,000 per spot for Family Guy and $3,000 per spot for American Dad. Gauss Jordan Inc.'s advertising budget for TV commercials is $50,000, and it would like at least 75% of the total number of spots to appear on Family Guy. How many spots should you purchase on each show to maximize exposure? Hint [Calculate exposure as Number of ads × Number of viewers.]

NO case number

Write a short paragraph answering to both of these replies:

Question #3

3. What did Alex Rogo finally determine to be "The Goal" for his plant (Chapters 4-5)? Do you agree or disagree with his conclusion? Why do you think it took him so long to come to that conclusion? How do the other goals that he thought of relating to it? Explain your answers.

Alex Rogo determined the goal for to company to be making money. I disagree with this conclusion, making money to me seems too broad of a goal. Jonah had mentioned how until you get the goal, you are just playing with words and numbers which leads me to believe that the numbers are a tool of the goal instead of the actual goal.

Alex arrived at this conclusion by thinking of the definition of productivity which says that anything benefiting towards the goal is productive, and anything not beneficial towards the goal is not productive. As he thought about goals, he came up with ideas such as technology, constant labor, sales, quality, and efficiencies. These are all tools used to accomplish the goal but are not to goal themselves.

Write answer to this:

2. s a result of Bill Peach personally expediting completion of Burnside’s order (Chapters 1-2), what additional costs did the plant and its operation incur for the order. What were the direct costs of Peach's actions? What were the indirect costs of his actions? Explain your answers.

Bill Peach determined that the finishing of a product for another company was more important than the morality of his own employees. Some of the direct costs of Peach’s actions were that an employee quit as well as a broke a machine. All other production was at a standstill while making this specific product which made employees inefficient. This can be directly traced because employees did something different from their usual routines which slows down the rest of production. Another large direct cost of this is the overtime cost of the hourly employees to be able to get the product out the door.

Some of the indirect costs include upsetting the manager of the department, other employees may have felt unappreciated or unimportant. The machine was broken and it may affect is depreciation in the future. Indirectly other employees could feel less appreciated and more like pawns in the corporate structure. Employees that were not directly working on the production of Peach’s order request were less productive which was shown in the book when they were sitting and talking during work. This scenario proves that getting results is more important to the company than their employees. Employee morale arguably was affected the most. A company can recover from lost materials or money but loyal employees are nearly impossible to recover from. Due to the small city atmosphere it will be difficult to find new employees because if a person had a bad experience then they’re very likely to tell others about it.

Write an answer to this:

a. Calculate the tax owed by the couple if they delay their marriage until next year so they can each file a tax return at the single tax rate this year.

The couple owes

​$nothing.

​(Simplify your answer. Round to the nearest dollar as​ needed.)

b. Calculate the tax owed by the couple if they marry before the end of the year and file a joint return.

The couple owes

​$nothing.

​(Simplify your answer. Round to the nearest dollar as​ needed.)

c. Does the couple face a​ "marriage penalty" if they marry before the end of the​ year?

NoNo

Yes

An application of Linear Algebra to things other than lines and planes. Here’s the scenario:
You are given four points on the (x,y)-plane and you need to find a curve that “fits”. In
other words, you will need to find an equation to a curve that goes through all the four
points given. You will be asked to turn in a screenshot of your graph for this example.
I recommend you use Grapher which you can have access to on any Mac computer on
campus, or Matlab which we all have a license for now. Follow the steps below.
(a) Graph the following four points: (0,2),(1,3),(2,0),(3,8)
(b) We are going to find a degree three polynomial that fits all of these data points.
Recall that a degree three polynomial can be written as
p(x) = a + bx + cx 2 + dx 3 .
Since we want the polynomial to pass through the point (3,8), for example, we require
that
p(3) = a + 3b + 9c + 27d = 8.
Do the same for the three other points.
(c) Write the four linear equations you found above as a system of four equations and
four unknowns.
(d) Write the augmented matrix corresponding to the system of equations and solve the
system. (Recall to keep up the practice of writing the variables you are solving for
on the top of the augmented matrix.)
(e) Write the polynomial p(x) that you found. Graph the polynomial and verify that it
passes through all four points. Turn in all the work above and a screenshot of your
graph from Grapher, or matlab, etc.

Solve:

y'''−7y''+8y'+16y=0

y(0)=1,  y'(0)=−3,  y''(0)=10y(0)=1,  y′(0)=-3,  y′′(0)=10

y(t)= ?

Inference: The differing meanings of "valid inference" and "warranted inference" are closely related to the differing purposes of deductive and inductive arguments – the purpose of deductive being to prove; the purpose of inductive to make the conclusion most probable.

• Look up the words "valid" and "warranted." Each of these words, you will find, has what is known as a lexical definition – that is just the dictionary definition of the word. Words also have a certain connotations - meanings that go beyond their lexical definitions; associated ideas and concepts – think of terms such a "fur baby" as the name for a pet.
• Briefly discuss how the lexical definitions and connotations of "valid" and "warranted" can help us understand the differing purposes of deductive and inductive arguments.

Consider the following various differential equations that describe the mass-

spring systems.

A. y''+y=0

B. y''+9y=3sin(9t)

C. y''+y=5cos(t)

D. y''+400y=sin(19t)

E. y''+4y'+5y=0

F. y''+6y'+9y=0

(I) Which system is undergoing resonance?

(II) Which system is critically damped?

(III) Which system gives damped oscillation as the solution?

(IV) Which system describes undamped free oscillation?

(V) What is the frequency and period of the case (A)?

(VI) What is the quasi-frequency and quasi-period of the case (E)?

Hello, I ran into a conceptual question in a lecture and it said the following was possible: That simple interest and compound interest can be the same over several periods. How is this so?

On a particularly strange railway line, there is just one infinitely long track, so overtaking
is impossible. Any time a train catches up to the one in front of it, they link up to form a
single train moving at the speed of the slower train. At first, there are three equally spaced
trains, each moving at a different speed. After all the linking that will happen has happened,
how many trains are there? What would have happened if the three equally spaced trains
had started in a different order, but each train kept its same starting speed? On average
(where we are averaging over all possible orderings of the three trains), how many trains will there be after a long time has elapsed? What if at the start there are 4 trains (all moving
at different speeds)? Or 5? Or n? (Assume the Earth is flat and extends infinitely far in all
directions.)

Consider the vector field F(x,y,z)=〈 4x^(2) , 7(x+y)^2 , −4(x+y+z)^(2) 〉.

Find the divergence and curl of F.

div(F)=∇⋅F= ?

curl(F)=∇×F= ?

Using the definition of Compact Set, prove that the union of two compact sets is compact. Use this result to show that the union of a finite collection of compact sets is compact. Is the union of any collection of compact sets compact?

a. Find the x-perpendicular of the projection of (1,1,1,1,1) onto (1,2,3,4,1).

b. What is the projection of the projection of (1,1,1,1,1) onto (1,2,3,4,1)? Make a sketch.