Let (X, d) be a metric space.

Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Consider the following initial value problem: ?? − 2?? = √? − 2? + 3 ?? ?(0) = 6

1. Write the equation in the form ?? ?? = ?(?? + ?? + ? ), where ?, ?, ??? ? are constants and ? is a function.

2. Use the substitution ? = ?? + ?? + ? to transfer the equation into the variables ? and ? only.

3. Solve the equation in (2).

4. Re-substitute ? = ?? + ?? + ? to write your solution in terms of ? and ?.

5. Use the initial condition to write the solution for this initial value problem.

6. Discussion and conclusion.

please share the solution with explanation as it mentioned in part 6.

(PDE)

Find the series soln to Ut=Uxx on -2<x<2, T>0

with Dirichlet boundary { U(t,-2)=0=U(t, 2)

initial condition { U(0,x) = { x, IxI <1

Use the method of steepest ascent to approximate the optimal solution to the following problem: max⁡ z=-(x1-2)^2-x1-(x2)^2 . Begin at the point(2.5,1.5)

(p.s. The answer already exists on the Chegg.Study website is incorrect)

You have agreed to pay off an $8,000 loan in 30 monthly payments of $298.79 per month. The annual interest rate is 9% on the unpaid balance.

(a) How much of the first month’s payment will apply towards reducing the principal of $8,000?

(b) What is the unpaid balance (on the principal) after 12 monthly payments have been made?

x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

Suppose that over a certain region of space the electrical potential V is given by the following equation.

V(x, y, z) = 4x² − 4xy + xyz

(a) Find the rate of change of the potential at P(3, 6, 6) in the direction of the vector v = i + j − k.
???
(b) In which direction does V change most rapidly at P?

???
(c) What is the maximum rate of change at P?

???

1.Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

ln(x) = 1/(x-3)

2. Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

6e^−x2 sin(x) = x² − x + 1

Hey.
Given C = {z | z = z(t) = 10*e^(it), 0 <= t <= 2 pi}.

How do I solve the following two integrals. Is there a way to do it with residues?

a) f(z) = (cos(z) -1)/ z^3
b) f(z) = (sin(pi*z))/(z^3-1)

Thank you!

Recall the following theorem, phrased in terms of least upper bounds.
Theorem (The Least Upper Bound Property of R). Every nonempty subset of R that
has an upper bound has a least upper bound.
A consequence of the Least Upper Bound Property of R is the Archimedean Property.
Theorem (Archimedean Property of R). For any x; y 2 R, if x > 0, then there exists
n 2 N so that nx > y.
Prove the following statements by using the above theorems.
(a) For any two real numbers a; b 2 R, if a < b, then there exists a real number r 2 R
such that a < r < b.
(b) Prove that for any two rational numbers a; b 2 Q, if a < b, then there exists an
irrational number r 2 R, r =2 Q, such that a < r < b.
(c) For any two real irrational numbers a; b 2 R, a; b =2 Q, if a < b, then there exists
a rational number q 2 Q such that a < q < b.
(d) Prove that the Least Upper Bound Property is equivalent to the Greatest Lower
Bound Property: \Every nonempty subset of R that has a lower bound has a
greatest lower bound."

Let T denote the counterclockwise rotation through 60 degrees, followed by reflection in the line y=x

(i) Show that T is a linear transformation.

(ii) Write it as a composition of two linear transformations.

(iii) Find the standard matrix of T.

The Verbrugge Publishing Company’s 2019 balance sheet and income statement are as follows (in millions of dollars):

Balance Sheet

Current assets Net fixed assets

Total assets

Income Statement

Net sales Operating expense

$300 200

$500

Current liabilities
Advance payments by customers

Noncallable preferred stock, $6 coupon, $110 par value (1,000,000 shares)
Callable preferred stock, $10 coupon, no par, $100 call price (200,000 shares)

Common stock, $2 par value (5,000,000 shares)

Retained earnings Total liabilities & equity

$ 40 80 110

200

10

60 $500

$540 516 $24 4 $28 7 $21 6 2 $13

Net operating income Other income
EBT
Taxes (25%)

Net income
Dividends on $6 preferred
Dividends on $10 preferred
Income available to common stockholders

(24-3)

Liquidation

Verbrugge and its creditors have agreed upon a voluntary reorganization plan. In this plan, each share of the noncallable preferred will be exchanged for 1 share of $2.40 preferred with a par value of $35 plus one 8% subordinated income debenture with a par value of $75. The callable preferred issue will be retired with cash generated by reducing current assets.

1. Assume that the reorganization takes place and construct the projected balance. Show the new preferred stock at its par value. What is the value for total assets? For debt? For preferred stock? 

2. Construct the projected income statement. What is the income available to common shareholders in the proposed recapitalization? 

3. What were the total cash flows received by the noncallable preferred stockholders prior to the reorganization? What were the total cash flows to the original noncall- able preferred stockholders after the reorganization? What was the net income to common stockholders before the reorganization? After the reorganization. 

4. Required pre-tax earnings are defined as the amount that is just large enough to meet fixed charges (debenture interest and/or preferred dividends). What are the required pre-tax earnings before and after the recapitalization? 

5. How is the debt ratio (i.e., liabilities/total assets) affected by the reorganization? Suppose you treated preferred stock as debt and calculated the resulting debt ratios. How are these ratios affected? If you were a holder of Verbrugge’s common stock, would you vote in favor of the reorganization? Why or why not? 


Consider again the Ohio Trust bank location problem discussed in Section 7.3. The file OhioTrustFull contains data for all of Ohio’s 88 counties. The file contains an 88 X 88 matrix with the rows and columns each being the 88 counties. The entries in the matrix are zeros and ones and indicate if the county of the row shares a border with the county of the column (1 = yes and 0 = no).

a. Create a model to find the location of required principal places of business (PPBs) to minimize the number of PPBs needed to open all counties to branches.

b. Solve the model constructed in part (a). What is the minimum number PPBs needed to open up the entire state to Ohio Trust branches?

2. The following data represent Salary, years of experience, and rank (A, B, or C). Salary is the dependent variable. For class and the interaction effect between class and years of experience, do the following: a. Indicate a scheme (method) of coding “Rank”. How many variables are needed? b. Using the coding method given in Part a, give the data (the numbers) that you would input for the following cases including the interaction term between rank and experience: Absenteeism (Hrs/Yr) Y Experience (Years) X1 Rank 36.5 4 B 27.7 1 A 32.2 5 C In the heading, give the variable. In the body, give the numbers. You might not need all columns.