This question revisits 2.4.7 from Abbott. Remind yourself of the definition of lim sup. Let (a_n) be a bounded sequence. Let S = {s ∈ R : ∃ a subsequence (a_nk ) converging to s}. This is called the set of subsequential limits. Bolzano-Weierstrass theorem implies there is at least one convergent subsequence, so S cannot equal ∅. Show S is bounded and lim sup a_n=sup(S).

A mass of 9kg stretches a spring 17cm. The mass is acted on by an external force of 7sin(t/3)N and moves in a medium that imparts a viscous force of 4N when the speed of the mass is 12cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 7cm/s, determine the position u of the mass at any time t. Use 9.8m/s2 as the acceleration due to gravity. Pay close attention to the units.

Prove that the product of any three consecutive integers is divisible by 6.

Hint: See corollary 2 to theorem 2.4 of the Elementary Number Theory Book:

If a divides c and b divides c, with gcd(a,b)=1, then a*b divides c.

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5

As a mathematical function, does U have a maximum or minimum value? What values of Ci correspond to the minimum value of U? What values of Ci correspond to the maximum value of U? Do these values of Ci make sense from an economic standpoint?

Now let us connect the idea of economic utility to actual dollar values. To keep the values more manageable, we will use household income rather than the entire state budget, and retail costs and measures rather than industrial ones. Find the Median Household Income for Mesa, AZ for the most recent year possible. Then find the dollar cost in Mesa, AZ for a Penny, a pound of Ground Beef, a pair of Jeans, fresh Orange Juice, and a Movie Ticket. (Entertainment is often used as a stand-in for Climate.) A Cost-of- Living Index is a good place to find much of this data. Record these prices as P1, P2, P3, P4, and P5 respectively.

Construct an equation using Median Income, the Ci and Pi values that illustrates how much of each resource the Median Household can afford to purchase. Given this restriction, do the maximum or minimum values of U change? Do the values of Ci that give the maximum or minimum values change? What are these new values? How should the Median Household budget its Income so as to maximize its Economic Utility?

Write up your findings in a paper that you could turn in to an employer. Be sure to show all your work. Include any appropriate references as well as any computational devices used.

* Combine all necessary information - Discussion, Work, Conclusions, Graphs, Tables, and Sources

Find the general solution using ONLY the variation of parameters method. (1+x)y'+2y=sinx/(1+x)

A college student owes $2000 to a credit card company, which charges interest at an annual rate of 10%. The student makes payments continuously at a constant rate of $25/month ($300/year).

a. set up the initial value problem describing the situation.

b. solve the initial value problem from part (a)

c. find the time T it will take to pay off the debit

Other answer is wrong.

Solve the initial value problem (first state the type of the differential equation)

y'(x) = (2xy) / (x^2−y^2) &

y(1) = 1

Use the method of reduction of order to find a second independent solution of the given differential equation.

t²y'' + 4ty' − 4y = 0,    t > 0;    y₁(t) = t

1. Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:

   Rental Class
   Room		Super Saver	Deluxe	Business
   	Type I	$31	$35	—
   	Type II	$16	$33	$42

   Type I rooms do not have wireless Internet access and are not available for the Business rental class.

   Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 50 rentals in the Deluxe class, and 45 rentals in the Business class. Round Tree has 105 Type I rooms and 145 Type II rooms.

   1. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types.
      

      Variable	# of reservations
      SuperSaver rentals allocated to room type I	???
      SuperSaver rentals allocated to room type II	???
      Deluxe rentals allocated to room type I	???
      Deluxe rentals allocated to room type II	???
      Business rentals allocated to room type II	???

      
      Is the demand by any rental class not satisfied? Yes or No?
      Explain.
   2. How many reservations can be accommodated in each rental class?
      

      Rental Class	# of reservations
      SuperSaver	???
      Deluxe	???
      Business	???

   3. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered? Yes or No?
   4. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why? Type 1 or Type 2?
   5. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change? Yes or No?

Please be descriptive and show all work step-by-step.

Decide which special case applies. case 1) no feasible region 2) unboundedness for maximizing the objective function 3)Redundant constraint 4)More than one optimal solution

Problem - A candidate for mayor is planning an advertising campaign. Radio Ads cost $200 and reach 3,000 voters and TV ads cost $500 and reach 7,000 voters.The campaign manager

has stipulated that at least 10 ads of each type must be used. Also, the manager insists that the number of radio ads be at least as great as the number of TV ads. How many ads of each type should be purchased in order to reach the maximum number of voters?

Theorem 1 (Mean Value Theorem). Suppose ? ⊂ R is open and ? : ? → R. If ? is differentiable on the open interval (?, ?) ∈ ? then there exists ? ∈ (?, ?) such that ?(?) − ?(?) ? − ? = ? ′ (?).

4. An alternative interpretation of the Mean Value Theorem is that if ? is differentiable on (?, ? + ℎ), then there exists ? ≤ ? ≤ ? + ℎ such that ?(? + ℎ) = ?(?) + ??(?)(ℎ) (1)

a. The Mean Value Theorem fails if ? : R^? → R^? even with ? = 1 and ? = 2. Let ? : [0, 1] → R^2 be given by ?(?) = (? − ?^2 , ? − ?^3 ). Show that Equation (1) fails for ? when ? = 0 and ℎ = 1.

b. On the other hand, let ?(?, ?, ?) = ?? + ?^2 , a = (0, 0, 0) and b = (2, 1, 2). Let ? : [0, 1] → R^3 be the parameterization of the line segment going from a to b. Find (? ∘ ?)(?) and (? ∘ ?) ′ (?).

c. Find a value ?0 ∈ (0, 1) such that ?(b) − ?(a) = (? ∘ ?) ′ (?0).

WeHaul Trucking is planning its truck purchases for the coming year. It allocated $600,000 for the purchase of additional trucks, of which three sizes are available. A large truck costs $150,000 and will return the equivalent of $15,000 per year to profit. A medium-sized truck costs $90,000 and will return the equivalent of $12,000 per year. A small truck costs $50,000 and will return the equivalent of $9,000 per year. WeHaul has maintenance capacity to service either four large trucks, five medium-sized trucks, or eight small trucks, or some equivalent combination. WeHaul believes that it will be able to hire a maximum of seven new drivers for these added trucks. The company cannot spend more than one/half of the total funds it actually spends to purchase medium-sized trucks. (Hint: this is not necessarily one half of the total funds it has allocated for the purchase of additional trucks).   

You must submit your linear programming formulations and show the linear programming software solution to this problem to receive credit. If you solve this using another linear programming approach, you may submit that instead of the software solution.   

a) Formulate a linear programming model to be used for determining how many of each size of truck to purchase if the company wants to maximize its profit. Ignore the time value of money. Provide the linear programming variables, the objective function, and the constraints for the problem.

b) At optimality, how much profit will result and what is the optimal combination of trucks? You must submit your linear programming formulations and show the linear programming software solution to this problem to receive credit. If your answer is in fractional units of trucks that is acceptable – do not round to whole number of trucks.

c) Using your sensitivity analysis output, provide two sensitivity analysis interpretations. One must be for the objective function and one must be for one of the constraints. You must provide the source of your answers from the sensitivity analysis output.

d) Now suppose that there is a requirement that WeHaul must purchase at least two small trucks for each medium size truck. Also, the number of larger trucks cannot be more than the total number of medium and small trucks. Write the constraint(s) for this requirement. However, you do not need to resolve the problem.

Let ? : R^2 → R be given by ?(?, ?) = √︀ |??|.

a. Show ? is continuous at (0, 0).

b. Show ? does not have a directional derivative at (0, 0) along (1, 1).

c. Is ? differentiable at (0, 0)?

Find the maximum point between x = -1 to 0 Find the minimum point between x = 0 to 1

Use Golden Search section method

Use 10 iterations