3-24 Today’s Electronics specializes in manufacturing modern electronic components. It also builds the equipment that produces the components. Phyllis Weinberger, who is responsible for advising the president of Today’s Electronics on electronic manufacturing equipment, has developed the following table concerning a proposed facility:

a. develop an opportunity loss table.

b. what is the minimax regret decision?

The minimax regret decision rule states the best alternative is the medium-sized facility.

Just need help with making an excel spreadsheet showing formulas for these answers .

Find the general solution of the following equations:

y′′ −4y′ +4y=0; y′′ −5y′ +6y=0; y′′ − 2y′ = 0

Solve the differential equation y'' − y' − 2y = 9e^2t , with initial conditions y(0) = 3, y' (0) = −2, using two different methods. Indicate clearly which methods you are using. First method:

Second method:

Obtain the general solution of the following equation 4Uxx+12Uxy+9Uyy-9U = 9

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V = W = R^n to simplify things. Hint: show that the “preimage of” operation plays nice with set-complements, and then use the fact that every closed set is the complement of some open set. Note that R^n is both open and closed as a subset of itself.

A solution of hydrochloric acid accidentally leaks into a spring fed freshwater lake, which in turn feeds a single stream. the acid leaks into the lake at a rate of 2 gallons per hour and has a concentration if 3 pounds per gallon. The lake initially contains 4000 gallons of water. Water flows into the lake from the spring at a rate of 100 gallons per hour and fluid flows out of the lake into the stream at a rate of 100 gallons per hour.
a) Set up the initial value problem that models the dynamics if acid in the lake.
b) Solve the initial value problem from part a.
c) compute the amount of acid in the lake after 1 hour

By computing both sides, show that for an m × n matrix A, vectors u and v ∈ Rn , and a scalar s ∈ R, we have (a) A(sv) = s(Av); (b) A(u + v) = Au + Av; (c) A(0) = 0. Here 0 denotes the zero vector. Is the meaning of 0 on the two sides identical? Why or why not? Hint: Let x = (x1, . . . , xn) and y = (y1, . . . , yn) be vectors in Rn . The definition of vector equality is that x = y if and only if, for each i between 1 and n, we have xi = yi . So verify this property for the vectors in (a), (b), (c).

How do you determine convexity/concavity of a function f(x,y)? How about a function f(x,y,z)?

1.Determine whether S spans V . Justify your answers.

V = C ^0 [−1, 1] (the vector space of continuous functions on [−1, 1]) and S = {1, t, t2 , t3 , . . . }.

2.Let S be a set in a vector space V and v any vector. Prove that span(S) = span(S ∪ {v}) if and only if v ∈ span(S).

PROOFS:

1. State the prove The Density Theorem for Rational Numbers

2. Prove that irrational numbers are dense in the set of real numbers

3. Prove that rational numbers are countable

4. Prove that real numbers are uncountable

5. Prove that square root of 2 is irrational

You have entered a model rocket contest with your friend, Tiffany. You have been working on a pressurized rocket filled with nitrous oxide. Tiffany has determined the minimum atmospheric pressure at which the rocket fuel is stable. Based not hat value, and the equations given below, your task is to determine the optimum launch angle and initial velocity to maximize flight time. The goal is to re-use your rocket capsule, so you really want to avoid a fuel explosion.

The atmospheric pressure varies with elevation according to the equation: P(h)= 14.7e−h/10. where p is the pressure in psi and h, is an elevation in miles above sea levels. The height (in feet) of a rocket launched at an angle α degrees with the horizontal and an initial velocity, vo in feet/second, t seconds after launch is given by the equation h(t)=-16t^2+vo*t*sin(α).

1) If Tiffany has determined that the minimum safe pressure is 11 pounds per square inch, at what altitude will the rocket explode? Report your result in feet. Round to the nearest foot.

2) If the angle of launch is33o, with an initial velocity of 1,648 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

3) If the angle of launch is32o, with an initial velocity of 1,908 what is the minimum atmospheric pressure exerted on the rocket during its flight? Report your answer to one decimal place. Under these conditions, will the rocket explode during its flight?

4) Tiffany has revisited her calculation and has now concluded that the minimum safe pressure for the fuel is 9 psi. What is the maximum height your rocket can achieve without exploding in flight? Report your answer in feet to the nearest foot.

5) Tiffany has (once again) checked her calculations, and you have verified with her that the safe pressure for your fuel is 9 and the fuel capsule holds enough fuel to produce an initial velocity of 2,169 feet per second. What launch angle will you use so that your rocket achieves the maximum safe altitude? Round your answer to the nearest tenth of a degree.

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value of y(2) where y is the solution to the initial value problem

y′=y−2x             y(0) = 1

Recall that Euler’s method says: Approximate values for the solution of the initial value problem

y′=F(x, y),y(x0) =y0 with step size h, at xn=xn−1+h, are

yn=yn−1+hF(xn−1, yn−1)

Fill in the table for steps of size h= 0.2.

Graph the portion of the approximate solution curve you found above. It should look like a lot of line segments. The first segment has been given on the grid below:

(c) Suppose f(x) is an exact solution to the initial value problem above. Describe, with justification, the behavior off(x) as x→∞. Hint: Graphing a slope field may be helpful for this.

Find a base of solutions. Try to identify the series as expansions of known functions. ( show details of your work)

xy''+2y'+xy=0 INTENTIONALLY +XY