1) Use the simplex method to solve the linear programming problem. Maximize P = 6x + 5y subject to 3x + 4y ≤ 34 x + y ≤ 10 3x + y ≤ 28 x ≥ 0, y ≥ 0   The maximum is P = at (x, y) =

2) Use the simplex method to solve the linear programming problem. Maximize P = x + 2y + 3z subject to 2x + y + z ≤ 21 3x + 2y + 4z ≤ 36 2x + 5y − 2z ≤ 15 x ≥ 0, y ≥ 0, z ≥ 0   The maximum is P = at (x, y, z) =

Which of the following vectors does not belong to span{ (2,0,3), (3,0,2) } ?

a) (5,0,5)

b) (1,0,-1)

c) (3,2,0)

d) (0,0,5)

1. Let R be the relation on A = {1, 2, 3, 4, 5} given by R = {(1, 1),(1, 3),(2, 2),(2, 4),(2, 5),(3, 1),(3, 3),(4, 2),(4, 4),(4, 5),(5, 2),(5, 4),(5, 5)}.

(a) Draw the digraph which represents R.

(b) Give the 0 -1 matrix of R with respect to the natural ordering.

(c) Which of the five properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) does R have? Give a brief reason why or why not each property holds.

2. Let A = {1, 2, 3, 4}, B = {α, β, γ}, and C = {x, y, z}. Further suppose S = {(1, γ),(2, α),(2, γ),(3, β),(3, γ)} and R = {(α, x),(α, y),(β, z)}.

(a) Compute the composition relation R ◦ S. Hint: It may be helpful to draw bipartite graphs.

(b) Is the relation R ◦ S a function from A to B? Why or why not?

Let G be a finite group and p be a prime number.Suppose that pr divides the order of G. Show that G has a proper subgroup of order pr.

Use Theorem 3.5.1 to find the general solution to each of the following systems. Then find a specific solution satisfying the given boundary condition.

a. f1′=2f1+4f2,f1(0)=0 f 2′ = 3 f 1 + 3 f 2 , f 2 ( 0 ) = 1

c. f1′= 4f2+4f3 f2′= f1+f2−2f3 f 3′ = − f 1 + f 2 + 4 f 3 f1(0) = f2(0) = f3(0) = 1

Prove the following more general version of the Chinese Remainder Theorem: Theorem. Let m1, . . . , mN ∈ N, and let M = lcm(m1, . . . , mN ) be their least common multiple. Let a1, . . . , aN ∈ Z, and consider the system of simultaneous congruence equations    x ≡ a1 mod m1 . . . x ≡ aN mod mN This system is solvable for x ∈ Z if and only if gcd(mi , mj )| ai − aj for all i 6= j, and the solutions are precisely given by one congruence class x ≡ b mod M. Hint: Regarding existence: For x ≡ ai mod mi and x ≡ aj mod mj , argue by reducing further modulo gcd(mi , mj ) that gcd(mi , mj )| ai − aj is a necessary condition for existence. To prove sufficiency of this condition, first treat the case N = 2. In that case, reduce the problem by the prime factors of m1 and m2 and thereby consolidate to a single system of congruence equations with coprime moduli to which the standard Chinese Remainder Theorem can be applied. This establishes existence for N = 2. Then proceed to treat the general case N > 2 by induction with respect to N. At some point, you will probably have to apply the identity lcm(gcd(m1, mN+1), . . . , gcd(mN , mN+1)) = gcd(lcm(m1, . . . , mN ), mN+1) which is valid in view of Problem 2 (this identity can be proved based on Problem 2 by induction, but you may just use it in your proof).

. Let U be a non-empty set. For A and B subsets of U, define the relation A R B if an only if A is a proper subest of B. a. Is R reflexive? Prove or explain why not. b. Is R symmetric? Prove or explain why not c. Is R transitive? Prove or explain why not. d. Is R antisymmetric? Prove or explain why not. e. Is R an equivalence relation? Prove or explain why no

Solve these first-order Differential Equations using an integrating factor.

1. dy/dx+2xy=0

2. dy/dx-y=5

3. dy/dx+y=x

4. (x)dy/dx+(x+1)y=3/x

5. (x^2)dy/dx=e^x-2xy

Is the following argument p& ∼ p,(p ∨ q) ≡ (s ≡ (t ∨ p)) ∴ r valid? If not provide a counterexample.

A new type of fueling truck is under consideration for an airport on a resort island. The company has been testing one alternative and feel it has the potential to significantly reduce fueling time while airplanes are at the airport’s one gate. With current equipment refueling requires 20 minutes. They estimate that the first fueling operation with this equipment will require 30 minutes. They hope, by spending the money for the new equipment, that within two weeks they will achieve a refueling time of 15 minutes – a 25% improvement on the current time. Assume 6 planes per day are schedule to arrive/depart from the island (7 days/week). All flights to and from the island are scheduled to arrive and depart between 6:00 am and 9:00 am.

a. Determine the learning rate required to achieve their objective.

b. If their learning rate is actually 92%, how many days will it take to get the refueling time below the 20 minutes required with the old system?

c. After 10 weeks with a learning rate of 92%, what would be the expected time to refuel planes with the new system?

d. What impact (if any) will this change have on the demand for fueling systems and the way they are scheduled?

e. What impact might this have on the scheduling of flights to/from this destination?

f. Assuming there’s demand for up to six additional flights in the scheduling window, what is the business case for purchasing this equipment?

solve the initial values:

if Y(3)-4Y"+20Y'=51e^3x

Y"(0)=41, Y'(0)= 11. Y(0)= 7 > solution is Y(x)= e^3x+2 e^2x sin(4x)+6

so, what is the solution for:

Y(3)-8Y"+17Y'=12e^3x

Y"(0)=26, Y'(0)= 7. Y(0)= 6

Y(x)=???

Consider the initial-value problem y' = 2x − 3y + 1, y(1) = 7. The analytic solution is y(x) = 1/9 + 2/3 x + (56/9) e^(−3(x − 1)).

(a) Find a formula involving c and h for the local truncation error in the nth step if Euler's method is used.

(b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5). (Proceed as in this example.)

(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler's method. (Round your answers to four decimal places.)

h = 0.1          y(1.5) ≈______

h = 0.05         y(1.5) ≈______

(d) Calculate the errors in part (c) and verify that the global truncation error of Euler's method is O(h). (Round your answers to four decimal places.) Since y(1.5) =______, the error for h = 0.1 is E_0.1 = ______, while the error for h = 0.05 is E_0.05 = ______. With global truncation error O(h) we expect E_0.1/E_0.05 ≈ 2. We actually have E_0.1/E_0.05 = _______ (rounded to two decimal places).

1. Find the first four terms in each portion of the series solution around x₀ = 0 for the following differential equation. x² y^// - 5x y^/ + 6y = 0