Superior Consulting is a firm that specializes in developing computerized decision support systems for manufacturing companies. They currently operate offices in Chicago, Charlotte, Pittsburgh and Houston. They are considering opening new offices in one or more cities including: Atlanta, Boston, Denver, Detroit, Miami, St. Louis and Washington DC. They have $14 million available for this purpose. The executive team ranked the prospective cities from 7 to 1, with 7 being the highest preference.

Due to the specialized nature of their work, they must staff any new offices with a minimum number of its employees from its existing offices. However, it has a limited number of employees available to transfer to any new offices. See the tables below for the costs for opening a new office, the start-up staffing needs, preference and available number of employees from existing offices.

The HR team developed the following cost per employee (in 1,000’s) to transfer them from an existing office to a prospective new office.

In addition, the company would like at least one new office in the Midwest (Detroit and/or St. Louis) and one new office in the Southeast (Atlanta and/or Miami.)

Formulate and solve their problem using Excel and Solver.

Show that the map T(a,b,c) = (bx^2 + cx + a) is an isomoprhism between R^3 and P_2

Use Euler's method with the given step size to estimate y(1.4) where y(x) is the solution of the initial-value problem y′=x−xy ,y(1)=3.

Let A ⊆ R, let f : A → R be a function, and let c be a limit point of A. Suppose that a student copied down the following definition of the limit of f at c: “we say that limx→c f(x) = L provided that, for all ε > 0, there exists a δ ≥ 0 such that if 0 < |x − c| < δ and x ∈ A, then |f(x) − L| < ε”. What was the student’s mistake? If this were the correct definition of a limit, which functions would have which limits, and at which points?

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation

y′′−6y′+9y=12te3t−6e3t+18t−21y″−6y′+9y=12te3t−6e3t+18t−21

with initial values y(0)=−3andy′(0)=−2.y(0)=−3andy′(0)=−2.

A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)

B. Write the fundamental solutions for the associated homogeneous equation.

C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.

D. Write the general solution. (Use c1 and c2 for c1c1 and c2c2 ).

E. Plug in the initial values and solve for c1c1 and c2c2 to find the solution to the initial value problem.

2. (a) Let p be a prime. Determine the number of elements of order p in Zp^2 ⊕ Zp^2 .

(b) Determine the number of subgroups of of Zp^2 ⊕ Zp^2 which are isomorphic to Zp^2 .

Let u be a unit vector, and P = Identity vector − u⊗u. Compute P^2 and P^(−1).

Hint: Rank one Matrix.

Consider the following functions.
f1(x) = cos(2x), f2(x) = 1, f3(x) = cos2(x)
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?

Determine whether f1, f2, f3 are linearly independent on the interval (−∞, ∞).
linearly dependent or linearly independent?  

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

A mass that weight 15lb15lb stretches a spring 8in8in. The system is acted on by an external force 9sin(43–√t)lb9sin⁡(43t)lb.If the mass is pulled down 3in3in and then released, determine the position of the mass at any time tt. Use 32ft/s232ft/s2 as the acceleration due to gravity. Pay close attention to the units. Answer must be in inches

5. Provide a counterexample to a false statement. (The statement might be a “for all,” “there
exists,” or P ==> Q.)
6. Compute the product of two sets.
7. Given a relation (as ordered pairs or as a diagram), determine the domain, range, and target
of the relation.
8. Given a relation (as ordered pairs or as a diagram), determine if a pair is in the relation.
9. Convert a relation from a list of ordered pairs to a mapping diagram.
10. Convert a relation from a mapping diagram to a list of ordered pairs.

1. Write the following complex numbers in polar form.

a. 6 + ?2

b. 10 + ?0

c. -1/j

d. 1/6 (18 + ?24)

e. sqrt(-j)

1. Locate a root of sin(x)=x2 where x is in radians. Use a graphical technique and bisection with
the initial interval from 0.5 to 1. Perform the computation until ea is less than es=2%. Also
perform an error check by substituting your final answer into the original equation.
2. Determine the positive real root of ln(x
2
)=0.7 using three iterations of the bisection method,
with initial interval of [0.5:2].
3. Determine the lowest positive root of f(x)=7sin(x)e
-x
-1 using (a) the Newton-Raphson
method (three iterations, xi=0.3). (b) the secant method (five iterations, xi-1=0.5 and xi=0.4). (c)
the modified secant method (three iterations, xi=0.3, δ=0.01).
4. Use the Newton-Raphson method to find the root of f(x)=e
-0.5x
(4-x)-2. Employ initial guesses
of (a) 2, (b) 6, and (c) 8. Explain your results.

2.48. Is it true that {ax + by + cz|x, y, z ∈ Z} = {n · gcd(a, b, c)|n ∈ Z}?

2.49. What are all the integer values of e for which the Diophantine equation 18x + 14y + 63z = e has an integer solution. Find a solution for each such e.

2.50. For integers a, b and k > 0, is it true that a | b iff a^k | b^k ?

Question 5.

Use MATLAB to solve for and plot the response of the following models for 0≤t ≤1.5, where the input is f (t) =5t and the initial conditions are zero

a. 3¨ x +21˙ x +30x = f (t) b. 5¨

A (Turn in the MATLAB script and answers from MATLAB, .m file, screen shots if needed)

B (Turn in the MATLAB plot with t being time in SI units)

C Comment on the response the analytical solution compared with the MATLAB and the plots. (Do the calculations and MATLAB agree ? Why and Why not.? Do the plots make sense?)