x"(t)- 4x'(t)+4x(t)=4e^2t ; x(0)= -1, x'(0)= -4

Graph theory

In a certain school, there are 39 clubs and 194 students. Some students do not belong to any clubs but every club has at least one member and no two clubs have the same number of members. Prove that there is a way to choose one student from each club ending up with 39 different students.

find the general solution of the given differential equation.

1. y''+2y'−3y=0

2.  6y''−y'−y=0

3.  y''+5y' =0

4.  y''−9y'+9y=0

Given that a particular solution of 2y′′ +3y′ +y = x^2 +7x+8 is yp1=x^2+x+1 and that a particular solution of 2y′′ + 3y′ + y = 2sinx+4cosx is yp2=sinx-cosx, find a particular solution for 2y" +3y' +y =3x^2 + 21x + 24 -sinx -2cosx

can a vector with dimensions R^N and a vector with dimensions R^N+1 be a matrix? and if so what would it dimension size be?

In a manufacturing company at present, the pattern shop and the maintenance department are located in the same room. One large counter is used by both maintenance personnel to get tools and parts and by sand molders that need various patterns for the molding operation. Peter and Bob, who work behind the counter, are able to service a total of 10 people per hour (or about 5 per hour each). On the average, 4 people from casting/maintenance and 3 people from the molding/pattern department arrive at the counter per hour. People from the molding department and from casting dept arrive randomly, and to be served they form a single line. Pete and Bob have always had a policy of first come, first served. This is a poisson arrival pattern exponential service time model. Because of the location of the pattern shop and casting department, it takes about 6 minutes for a person from the casting/maintenance department to walk to the counter, and it takes about 2 minute for a person to walk from the molding. Separating the maintenance shop from the pattern shop had a number of advantages. It would take people from their department only 2 minutes now to get to the department counter. Using time and motion studies, George was also able to determine that improving the layout of the maintenance department would allow Bob to serve 6 people from the maintenance department per hour, and improving the layout of the pattern department would allow Pete to serve 7 people from the molding shop per hour. This would act as a single server system for both the departments separately. In the present system how much minutes are spent in casting ? Give only numeric answer

QUESTION1 In the present system how much minutes are spent in casting ? Give only numeric answer

QUESTION 2 In the present system how many minutes are spent in molding ? Give only numeric answer

QUESTION 3 How much time in minutes would the new layout save in total for both the departments? Give only numeric answer

Describe all elements of (Z10xZ15) / <(2,3)> and their respective orders.

A theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $17,100. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $14,600. How many balcony seats are there?

1. During a​ one-month promotional​ campaign, Fran's Flix gave either a free DVD rental or a​ 12-serving box of microwave popcorn to new members. It cost the store $1

for each free rental and $2 or each box of popcorn. In​ all, 49 new members were signed up and the​ store's cost for the incentives was $75. How many of each incentive were given​ away?

There were            free rentals given away to new members.

2. The point at which a​ company's costs equal its revenues is the​ break-even point. C represents the​ cost, in​ dollars, of x units of a product and R represents the​ revenue, in​ dollars, from the sale of x units. Find the number of units that must be produced and sold in order to break even. That​ is, find the value of x for which C=R.

C= 16x+450 and R= 18.5x.

How many units must be produced and sold in order to break​ even?

3. Solve the following differential equation

x^2y’’ − 2xy’ + 5y = 0.

4. A coil spring is suspended from the ceiling, a 16-lb weight is attached to the end of it, and the weight then comes to rest in its equilibrium position. The mass is in a medium that exerts a viscous resistance of 8 lb when the mass has a velocity of 1 ft/s. It is then pulled down 12 in. below its equilibrium position and released with an initial velocity of 2 ft/sec, directed upward.

(a)   Use the Laplace transform to determine the resulting displacement of the weight as a function of time; the solution of the initial value problem

                                          1/2y’’ + 8y’ + 50y= 0;        y(0) = 1, y^’(0) = −2.

(b)   Write the solution in the form y(t) = Re^−µt cos(ωt − φ). Please leave the solution in exact form.

Please solve both with steps

The Main Street Entrepreneurship Kauffman Index is an indicator of main street business activity, presenting trends in small business activity and ownership over the past two decades. Their 2015 report stated that “reversing a six-year downward trend, activity in established small businesses increased last year.”   Using the calculus analysis you have just performed and any additional math you may wish to do, is their statement justified? Why or why not?

2

(a) Differentiate and integrate the following equations with respect to x: x2 + x – 12 = 0

(b) cos(x) + sin(x) + 15x3 = 0 3%

(c) e2x + 5x2 = 0 3%

(d) Differentiate the following with respect to x: x + x2ex – 15x = 0 3% (e) x3 + 4x2 cos(x)

^{The number of new businesses established in the US since 1990 can be modeled by the function Nx=110.8x^3-5305.5x^2+76,701x+332,892 where x = 0 represents 1990 and the domain is [0, 25].}   

1. According to the model, about how many new businesses were established in 2011? Don’t forget to label and interpret the answer.   

2. What was the average rate of change in the number of new businesses established between 2000 and 2010? Don’t forget to label and interpret the answers.  

3. What is the instantaneous rate of change in the number of new businesses established in 2011? Remember to label and interpret the answer.   

4. Consider the year 2015 (x = 25). Evaluate D(25), D’(25), and D’’(25). Label and interpret the meaning of each answer.

5. Use the First Derivative Test to determine the extrema of the function on the given domain. Classify as max/min and absolute/relative. For what intervals does the function increase/decrease? Please label and interpret each answer in the context of this problem.

6. Use the Second Derivative Test to discuss the concavity of this function on the given interval. What is the POI? Interpret the meaning of the POI in the context of this problem.           

Let f(x, y) = xy3 − x 2 + 2y − 1. (a) Find the gradient vector of f(x, y) at the point (2, 1).

(b) Find the directional derivative of f(x, y) at the point (2, 1) in the direction of ~u = 1 √ 10 (3i + j).

(c) Find the directional derivative of f(x, y) at point (2, 1) in the direction of ~v = 3i + 2j.

The questions below are about 2×2 autonomous systems of ODE. Parts (a) and (b) are unrelated. (a) Find the equilibrium solutions (critical points) of the system dx/dt = 2x−xy, dy/dt =−y+xy.(b) Consider the autonomous system dx/d t= −y+xy^3, dy/dt = x−x^3. The system has an isolated critical point at (x,y) = (1,1). Find the associated linear system at(1,1). Name and classify (as stable or unstable) the critical point of the linear system.