Study these definitions and prove or disprove the claims. (In all cases, n ∈ N.)

Definition. f(n)→∞ifforanyC>0,thereisnC suchthatforalln≥nC,f(n)≥C.Definition. f(n)→aifforanyε>0,thereisnε suchthatforalln≥nε,|f(n)−a|≤ε.

(a) f(n)=(2n2 +3)/(n+1). (i)f(n)→∞. (ii)f(n)→1. (iii)f(n)→2.

(b) f(n)=(n+3)/(n+1). (i)f(n)→∞. (ii)f(n)→1. (iii)f(n)→2.

(c) f(n) = nsin2(1nπ). (i) f(n) → ∞. (ii) f(n) → 1. (iii) f(n) → 2.

Scenario

Office Equipment, Inc. (OEI) leases automatic mailing machines to business customers in Fort Wayne, Indiana. The company built its success on a reputation of providing timely maintenance and repair service. Each OEI service contract states that a service technician will arrive at a customer’s business site within an average of 3 hours from the time that the customer notifies OEI of an equipment problem.

Currently, OEI has 10 customers with service contracts. One service technician is responsible for handling all service calls. A statistical analysis of historical service records indicates that a customer requests a service call at an average rate of one call per 50 hours of operation. If the service technician is available when a customer calls for service, it takes the technician an average of 1 hour of travel time to reach the customer’s office and an average of 1.5 hours to complete the repair service. However, if the service technician is busy with another customer when a new customer calls for service, the technician completes the current service call and any other waiting service calls before responding to the new service call. In such cases, after the technician is free from all existing service commitments, the technician takes an average of 1 hour of travel time to reach the new customer’s office and an average of 1.5 hours to complete the repair service. The cost of the service technician is $80 per hour. The downtime cost (wait time and service time) for customers is $100 per hour.

OEI is planning to expand its business. Within 1 year, OEI projects that it will have 20 customers, and within 2 years, OEI projects that it will have 30 customers. Although OEI is satisfied that one service technician can handle the 10 existing customers, management is concerned about the ability of one technician to meet the average 3-hour service call guarantee when the OEI customer base expands. In a recent planning meeting, the marketing manager made a proposal to add a second service technician when OEI reaches 20 customers and to add a third service technician when OEI reaches 30 customers. Before making a final decision, management would like an analysis of OEI service capabilities. OEI is particularly interested in meeting the average 3-hour waiting time guarantee at the lowest possible total cost.

Managerial Report

Develop a managerial report (1,000-1,250 words) summarizing your analysis of the OEI service capabilities. Make recommendations regarding the number of technicians to be used when OEI reaches 20 and then 30 customers, and justify your response. Include a discussion of the following issues in your report:

1. What is the arrival rate for each customer?
2. What is the service rate in terms of the number of customers per hour? (Remember that the average travel time of 1 hour is counted as service time because the time that the service technician is busy handling a service call includes the travel time in addition to the time required to complete the repair.)
3. Waiting line models generally assume that the arriving customers are in the same location as the service facility. Consider how OEI is different in this regard, given that a service technician travels an average of 1 hour to reach each customer. How should the travel time and the waiting time predicted by the waiting line model be combined to determine the total customer waiting time? Explain.
4. OEI is satisfied that one service technician can handle the 10 existing customers. Use a waiting line model to determine the following information: (a) probability that no customers are in the system, (b) average number of customers in the waiting line, (c) average number of customers in the system, (d) average time a customer waits until the service technician arrives, (e) average time a customer waits until the machine is back in operation, (f) probability that a customer will have to wait more than one hour for the service technician to arrive, and (g) the total cost per hour for the service operation.
5. Do you agree with OEI management that one technician can meet the average 3-hour service call guarantee? Why or why not?
6. What is your recommendation for the number of service technicians to hire when OEI expands to 20 customers? Use the information that you developed in Question 4 (above) to justify your answer.
7. What is your recommendation for the number of service technicians to hire when OEI expands to 30 customers? Use the information that you developed in Question 4 (above) to justify your answer.
8. What are the annual savings of your recommendation in Question 6 (above) compared to the planning committee's proposal that 30 customers will require three service technicians? (Assume 250 days of operation per year.) How was this determination reached?

5. In math class, a student has written down a sequence of 16 numbers on the blackboard. Below each number, a second student writes down how many times that number occurs in the se‐ quence. This results in a second sequence of 16 numbers. Below each number of the second se‐ quence, a third student writes down how many times that number occurs in the second se‐ quence. This results in a third sequence of numbers. In the same way, a fourth, fifth, sixth, and seventh student each construct a sequence from the previous one. Afterward, it turns out that the first six sequences are all different. The seventh sequence, however, turns out to be equal to the sixth sequence.

   Give one sequence that could have been the sequence written down by the first student. Explain which solution strategy or algorithm you have used.

find the bessel function J-7/2(X)  in term of sinx ,cosx, and powers of x

Find s shuffle of a deck of 13 cards that requires 42 repeats to return to the original order. How about one requiring 30 repeats?

Bunker makes two types of briefcase, fabric and leather. The company is currently using a traditional costing system with labor hours as the cost driver but is considering switching to an activity-based costing system. In preparation for the possible switch, Bunker has identified two activity cost pools: materials handling and setup. Pertinent data follow:     

Total estimated overhead costs are $367,500, of which $300,000 is assigned to the materials handling cost pool and $67,500 is assigned to the setup cost pool.

Required:

1. Calculate the overhead assigned to the fabric case using the traditional costing system based on direct labor hours. (Do not round intermediate calculations and round your final answer to the nearest whole dollar amount.)



2. Calculate the overhead assigned to the fabric case using ABC. (Round activity rates or activity proportions, and intermediate calculations to four decimal places.)



3. Was the fabric case over- or undercosted by the traditional cost system compared to ABC?

Bunker makes two types of briefcase, fabric and leather. The company is currently using a traditional costing system with labor hours as the cost driver but is considering switching to an activity-based costing system. In preparation for the possible switch, Bunker has identified two activity cost pools: materials handling and setup. Pertinent data follow:

Total estimated overhead costs are $367,500, of which $300,000 is assigned to the materials handling cost pool and $67,500 is assigned to the setup cost pool.

Required:
1. Calculate the overhead assigned to the leather case line using the traditional costing system based on direct labor hours. (Do not round intermediate calculations.)



2. Calculate the overhead assigned to the leather case line using ABC.



3. Was the leather case over- or undercosted by the traditional cost system compared to ABC?

Let a and b be rational numbers. As always, prove your answers. (a) For which choices of a, b is there a rational number x such that ax = b? (b) For which choices of a, b is there exactly one rational number x such that ax = b?

Bisection search

1. In MATLAB, implement a function that performs a bisection search. It should take the following parameters:

• F: A function (assumed to be continuous) whose roots you want to find,

• a: A floating-point number giving the left endpoint of the initial interval in which you want to search for a root of F.

• b: A floating-point number giving the right endpoint of the initial interval.

• delta: A non-negative floating-point number giving the acceptable proximity of the output to a root of F.

Your function should first check a and b to determine whether or not they satisfy the condition given by the Intermediate Value Theorem that allows us to conclude that [a, b] contains a root of F. If this condition is not satisfied, return NaN (Matlab for “Not a number”). If the condition is satisfied, your function should perform a bisection search until it finds a number z that it can guarantee satisfies |x−x∗| < delta, for some real-valued root x∗ of F. It should return z.

2. Use the MATLAB function that you wrote to find a real-valued root of the function F(x) = x 5 +x+ 1, with accuracy to 4 decimal places (this last requirement will determine your choice of delta).

3. Suppose that you use bisection search to find a root of F(x) = sin x, with a = −π/2, b = 5π/2. To which root will the bisection search converge?

Prove That For All Natural Numbers A > 1 And B > 1, If A Divides B Then A Does Not Divide B+1 (prove by contradiction)

Consider the following linear program. Maximize z= 5x1+ 3x2

subject to 3x1+ 5x2≤15

5x1+ 2x2≤10

– x1+ x2≤2

x2≤2.5

x1≥0, x2≥0

a. Show the equality form of the model.

b. Sketch the graph of the feasible region and identify the extreme point solutions. From this representation find the optimal solution.

c. Analytically determine all solutions that derive from the intersection of two constraints or nonnegativity restrictions. Identify whether or not these solutions are feasible, and indicate the corresponding objective function values. Which one is optimal?

d.Let the slack variables for the first two constraints, x3and x4, be the axes of the graph, and sketch the geometric representation of the model. Show an iso-objective line in these variables, and from it determine the optimal solution.

Topology question:

Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

Use induction to prove that 8^n - 3^n is divisible by 5 for all integers n>=1.

1. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide Vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03.

1. Formulate a linear programming model for this problem.
2. Solve the model using graphical analysis.
3. What would be the effect on the optimal solution in the above problem if the cost of rice increased from $0.03 to $0.06 per ounce?