4-Consider the following problem:

max − 3x1 + 2x2 − x3 + x4

s.t.

2x1 − 3x2 − x3 + x4 ≤ 0

− x1 + 2x2 + 2x3 − 3x4 ≤ 1

− x1 + x2 − 4x3 + x4 ≤ 8

x1, x2, x3, x4 ≥ 0

Use the Simplex method to verify that the optimal objective value is unbounded. Make use of the final tableau to construct an unbounded direction..

y′′+12y′+35y=8cos(8t+0) y(0) = 14; y'(0) = 3

find the particular, complimentary, and total solution.

I have found the complimentary to be. C1e^-5t + C2e^-7t

I'm having trouble finding the particular, so I can't get to the total solution

Give the Laplace transform of the solution to y"+2y'+3y=0 y(0)=-5 y'(0)=4

USING BISECTION METHOD, FIND THE ROOT OF 0.5e^x - 5x + 2 = 0 ON THE INTERVAL [ 0 , 1 ] UP TO 3 DECIMAL PLACES.

USE NEWTON'S METHOD TO APPROXIMATE THE ROOT OF f(x)=x^2-5    IN THE INTERVAL  [ 2 , 3 ] UP TO 4 DECIMAL PLACES.

short paragraphs

1) When factoring a trinomial, is the factored form unique or can there be more than one factored form?  

2) When factoring, what does it mean that a polynomial is prime?

USE NEWTON'S METHOD TO APPROXIMATE THE SOLUTION TO   2COSX = 3X .  LET X₀ =pi/6 .  ANSWER UP TO 3 DECIMAL PLACES.

USING REGULA FALSI METHOD, SOLVE THE EQUATION x^3 - 4x + 1 = 0 UP TO 3 DECIMAL PLACES.

Let A = {a, b, c, d} and B = {b, d, e}. Write out all of the elements of the following sets.

(a) B ∩ ∅

(b) A ∪ B

(c) (A ∩ B) × B

(d) P(A\B)

(e) {X ∈ P(A) | |X| ≤ 3}

2. Firm I has variable cost VCi = yi^2/10 and fixed cost FCi = 2000.

(1) Find total cost Ci(yi), average cost ACi, marginal cost Mci and the firm supply function Si(p)

(2) There are n=50 firms identical to firm I, facing a market demand of D(p) = 1000-250p. Find the market supply function S(p), the market equilibrium price p*, the market equilibrium quantity Y*.

(3) Given price p* you found in part b, what is the profit maximising yi* that firm i produces? How much profit does firm i make?

(4) The government introduces a tax on demand so that D'(p ) = 1000-250(p+t), where t=8. What is the new equilibrium price p? What is the new market equilibrium quantity Y'?

(5) At the new market price p', and assuming that in the short run the number of firms remains n=50, how much will firm I produce and how much will profit be?

(6) Given what you found in part e, will firms enter or exit? What is the long-run equilibrium number of firms n? What is the long run equilibrium price?

Looking at the first 2z integers 1, 2, . . . , 2z. Choose z + 1 of them. Then demonstrate that there is at least one pair of integers from the selection that are relatively prime. (A way to approach this problem is to show that at least two integers from the selection are consecutive.)

Two square matrices A and B are called similar if there exists an invertible matrixV suchthatB=V−1AV.

(a) Explain in detail how similarity is related to the change of basis for- mula. What does similarity imply about the geometric relationship between B and A?

Decide, with justification, on the truth of the following propositions, both when the Universe of discourse is the set of all positive integers, and when the Universe of discourse is the set of all real numbers.

1. ∃x∀y, x < x·y

2. ∀y∃x, x < x·y

3. ∃x∀y, x = x·y

4. ∀y∃x, x = x·y

5. ∀x∃y,∃z, y2 − z2 = 4x

6. ∀x∀y∃z, z < x2 + y2

7. ∀x∃y∃z, x > yz.

A group of six friends play some games of ping-pong with these results: Amy beats Bob Bob beats Carl Frank beats Bob Amy beats Elise Carl beats Dave Elise beats Carl Elise beats Dave Frank beats Elise Frank beats Amy Consider the relation R = {hx, yi : x has beaten y}. (a) Draw the directed graph G representing R. (b) Is R reflexive? Irreflexive? Symmetric? Asymmetric? Antisymmetric? Transitive? An equivalence? An order? (c) The players want to rank themselves. Find every possible topological order of G. (d) In order to have a definitive ranking, the players want there to be only one possible topological order. Which two players should face each other? (e) The transitive closure of R (R+), is an order. Is it partial or total?

Prove or disprove: Between any n-dimensional vector space V and Rⁿ there is exactly one isomorphism T : V → Rⁿ .

dy/dx + y/x \ x³y²