This problem is an example of over-damped harmonic motion.
A mass m=4kg is attached to both a spring with spring constant k=72N/m and a dash-pot with damping constant c=36N⋅s/m

The ball is started in motion with initial position x0=−3m and initial velocity v0=4m/s

Determine the position function x(t) in meters.

Each of the following 13 statements is either (always) True (T) or (sometimes) False (F). If your answer is sometimes False (F), please provide a counterexample; that is, give an example where the statement is not true. If it is always true, you need not give a proof; merely answer T.

1. ?(? + h) = ?(?) + ?(h)

Answer: _____

2. sin⁡(? + ?) = sin(?) + sin⁡(?)

Answer: _____

3. sin(? ∗ ?) = sin(?) ∗ sin⁡(?)

Answer: _____

4. √((X)^2+(y)^2) =?+?

Answer:_____

5. (?+?)^2 =?^2 +?^2

Answer:_____

6. ?^? ∗ ?^?=?^(?+?)

Answer: _____

7. (?^?)^? = ?^(?+?)

Answer: _____

8. √?^2 = ?

Answer: _____

9. ?+?/b+x = ?/?

Answer: _____

10. −x represents a negative number

Answer: __F___

11. A square is a rectangle.

Answer: _____

12. If?,?≠0and?<?then1/b <1/a.

Answer:_____

3|Page

13. If |? − 7| > 0 then ? > 7. Answer: _____

Describe a problem that can be solved by using the shortest-route model. Give a detailed mathematical example.

Explain what it means for an assignment model to be balanced. Give detailed mathematical examples.

In a job shop operation, four jobs may be performed on any of four machines. The hours required for each job on each machine are presented in the following table. The plant supervisor would like to assign jobs so that total time is minimized. Find the best solution. Which assignments should be made?

Explanation of the problem chapter 3.12 problem 49 of the book introduction to the mathematical programming 4th edition

Solution: Step # 1
Let Xij be the amount of money invested at the beginning of month i, for a period of j month.
Objective function: Step # 2
The objective is to maximize the available cash at the beginning of month 5.
X14 = collect the money invested at the beginning of month 1 of 4 months.
X23 = collect the money invested at the beginning of month 2 of 3 months.
X32 = collect the money invested at the beginning of month 3 of 2 months.
X41 = collect the money invested at the beginning of month 4 of 1 month.
Therefore the objective function is:
Maximize Z = 1.08 X14 + 1.03 X23 + 1.01 X32 + 1.001 X41
Restriction 1
It would be the money invested at the beginning of month 1 plus bills paid in month 1 which would be equal to the money available at month 1.

                       X11 + X12 + X13 + X14 + 600 = 400 + 400 (MONTH 1)
Restriction 2
It would be the money invested at the beginning of month 2 plus bills paid in month 2 which would be equal to the money available at month 2.

X21 + X22 + X23 + 500 = 1.001 X11 + 800 (MONTH 2)
Restriction 3 and 4
In the same way for the rest of the month we have:

X31 + X32 + 500 = 1.001 X12 + 1.001 X21 + 300 (MONTH 3)
X41 + 250 = 1.001 X13 + 1.001 X22 + 1.001 X31 + 300 (MONTH 4)

What I need is an explanation of the problem of how the data was extracted at each step

Now that we have seen Euler in action, let's return to examining the content of the M-File Euler.m. We have already explained the first line, where we defined the parameters our function takes. To see the meaning behind the third and fourth lines, type:

>> x = zeros(10,1);
y = zeros(10,1);
[x,y]

Thus we can see that the third and fourth lines of our M-File zero out the contents of our arrays x and y before we begin. Now, you'll continue examining our code.

Question: In your Word document, briefly explain what is happening in each remaining line of the M-File Euler.m.

The M-File is:

function [x,y] = Euler(h, x0, y0, interval_length, func)
nsteps = floor(interval_length/h) + 1;
x = zeros(nsteps,1);
y = zeros(nsteps,1);
x(1) = x0;
y(1) = y0;
for i=2:nsteps
    y(i) = y(i-1) + h* func(x(i-1), y(i-1));
    x(i) = x(i-1) + h;
end

Show all work

A ball is projected upward from the top of a 90 foot building at a velocity of 64 feet per second. The ball's height above the ground below the building is described by the function h(t)=-16t^2 + 64t + 90 , with t being the time in seconds after the ball is projected upward. a.) Determine the amount of the vertical intercept, and interpret what this means in the context of the problem (in terms of seconds and feet above the ground). b.) Determine the amount of all horizontal intercepts (if any) , and interpret what they mean in teh context of the problem (in terms of seconds and feet above the ground). c. Write the coordinates of the vertex. Interpret what these numbers mean in the context of the problem. d. If nothing stops the ball before, then how much time elapses until the ball hits the ground below the building?

In Exercises 3–6, find (a) the maximum value of Q(x) subject to
the constraint x^Tx = 1, (b) a unit vector u where this maximum is
attained, and (c) the maximum of Q(x) subject to the constraints
x^Tx = 1 and x^Tu = 0. Q(x) = 3x²₁ + 9x²₂ + 8x₁x₂

consider f(x) = ln(x)

a) Approximate f(0.9) and f(1.1)

b) Use Taylor remainder to find an error formula for Taylor polynomial.

Give error bounds for each of the two approximations in (a).

Which of the two approximations in part (a) is closer to correct value?

c) Compare an actual error in each case with error bound in part (b).

consider f(x) = ln(x)

a) Approximate f(0.9) and f(1.1)

b) Use Taylor remainder to find an error formula for Taylor polynomial.

Give error bounds for each of the two approximations in (a).

Which of the two approximations in part (a) is closer to correct value?

c) Compare an actual error in each case with error bound in part (b).

given the logistics equation y'=9(1-y)y
on the interval [0,5], w/ initial condition y(0)=1.25
1. what are the equilibrium Solutions? What is the long-term behavior of the solution?
2. solve the ode exactly and plot the solution on [0,5].

given the ivp y'=-y^2 on the interval [0,2] with initial condition y(0)=-1/2
1. solve the ode. what is the behavior of the true solution as x approaches 2?

Determine the solution of the following initial boundary-value problem using the method of separation of Variables

Uxx=4Utt 0<x<Pi t>0

U(x,0)=sinx 0<=x<Pi

Ut(x,0)=x 0<=x<Pi

U(0,t)=0 t>=0

U(pi,t)=0 t>=0