Let A and B be sets, and let R be a relation from A to B. Prove that Rng(R^-1) = Dom(R)

(1 point) Find y as a function of t if y′′+25y=0, y(0)=4,y′(0)=6. y= ?

Problem 6-09 (Algorithmic)

The Ace Manufacturing Company has orders for three similar products:

Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are as follows:

Use the transportation model to develop the minimum cost production schedule for the products and machines. Show the linear programming formulation. If the constant is "1" it must be entered in the box. If your answer is zero enter "0".

The linear programming formulation and optimal solution are shown.

Let x_ij = Units of product j on machine i.

If required, round your answers to the nearest whole number.

Suppose you want your daughter’s college fund to contain $125,000 after 14 years. If you can get an APR of 7.8%, compounded monthly, how much should you deposit at the end of each month?

a. $398.54

b. $406.64

c. $412.50

d. $476.83

A cake is shaped like a hemisphere of radius 2 with its base on the​ xy-plane. A wedge of the cake is removed by making two slices from the center of the cake​ outward, perpendicular to the​ xy-plane and separated by an angle of phi.

Consider the parabolas y=x^2 and y=a(x-b)^2+c, where a,b,c are all real numbers

(a) Derive an equation for a line tangent to both of these parabolas (show all steps, assuming that such a line exists)

(b) Assume that the doubly-tangent line has an equation y+Ax+B. Find an example of values of a,b,c (other than the ones given here or in class) such that A,B ∈ Z

(c) Consider the case a = 1. Does your algebra accurately reflect what we can expect geometrically? Explain.

given y1 find y2 (Differential Equations)

(3x-1)y''-(3x+2)y'-(6x-8)y=0 and y1=e^(2x)

For each proposition, either give a counterexample showing it is false, or write a proof.
(a) For all a, b, c ∈ Z, if ab divides c then a divides c and b divides c.
(b) For all a, b, c ∈ Z, if a divides bc, then a divides b or a divides c.

Egor, a United States citizen, is engaged in numerous, diverse operations and pays U.S. income tax at a rate of 37%. Egor owns MY LLC, a disregarded entity for U.S. tax purposes. MY LLC manufactures the ubiquitous product, widgets. U.S. sales result in $100,000 of taxable U.S.-source income. Egor projects that he could earn approximately $100,000 of net income in the United Kingdom (the "U.K."), where the corporate income tax rate is 20%. To further limit his liability (widgets being a very dangerous product); Egor’s MY LLC forms a private limited company in the United Kingdom. The private limited company in the U.K. is not a "per se" entity and, therefore, Egor (via the MY LLC) would consider checking-the-box to treat the private limited company in the U.K. as a disregarded entity. Assume that both the withholding tax rate on any dividends from a U.K. private limited company to the United States is 15% and that the title on all widget sales passes in the U.K.

1. What is Egor's foreign tax credit position in Year 1 if the U.K. private limited company is not "checked" as a disregarded entity and pays a dividend of $80,000 to the MY LLC?
2. What is Egor's foreign tax credit position in Year 1 if the U.K. private limited company is not "checked" as a disregarded entity and does not pay a dividend?

Linear Algebra

Carefully prove the following statement:

Let A be an n×n matrix. Assume that there exists an integer k ≥ 1 such that A^k = I . Prove that A is invertible.

Determine the reasonable form of the paticular solution for each nonhomogeneous differential equation. DO NOT SOLVE IT.

a) y''-y'+y=e^(-x) + xcos(2x) + (e^(x))sin(2x)

b) D^2[y] + 4y = 1 + x^2 + xsin(2x)

3.Consider the system x*=4x-y, y*=2x+y

(a) Write the system in matrix form  and find the eigenvalues and eigenvectors of the matrix A.

(b) Classify the fixed point at the origin

(c) Find the  general solution of the system

(d) Solve the system subject to the initial condition

Given two graphs G = [V ; E] and G0 = [V 0 ; E0 ], and an isomorphism, f : V → V 0 , and making direct use of the formal definition for isomorphism:

(a) Explain why G and G0 must have the same number of vertices.

(b) Explain why G and G0 must have the same number of edges.

(c) Explain why G and G0 must have the same degree sequences.

(d) Given two vertices, u, v ∈ V explain why: u is connected to v → f(u) is connected to f(v)

Note: Problems that ask you to “explain” are asking for responses that can be less formal than problems that ask you to “prove”. Nonetheless, responses need to be sufficiently precise and based on definitions and theorems as given in the text. Explanations should be concise, but care must be taken to ensure that explanations are at an appropriate level of detail and will be clear to the intended reader.

On an island inhabited by kings and peasants, where the former always tell the truth and the latter always lie, you meet three individuals: Erin, William, and Sam. Erin says that William is a king. William says that Erin is a king but Sam is a peasant. Sam says that both Erin and William are kings. Determine who is a king and who is a peasant by constructing a truth table.