Functions and Limits

Functions and Limits

For each question, be sure to show all of your work details for full credit. Round all of the numerical answers to 3 decimal places.

1.Research and define the concepts of maximum theoretical throughput, channel capacity, and bandwidth. Next, explain the similarities and differences between Cat5e and Cat6 Ethernet cables. What specific speeds can each of these cables handle? Listing credible cited resources, answer these below.

2.In the following table, based on the first letter of your last name, select one number from each of the ranges in the x and y columns.

First Letter of Your Last Name

(in seconds)

(in gigabits [Gb])

A–D

2 to 3.99

2 to 3.99

E–H

4 to 7.99

4 to 7.99

I–L

8 to 15.99

8 to 15.99

M–P

16 to 31.99

16 to 31.99

Q–T

32 to 63.99

32 to 63.99

U–Z

64 to 79.99

64 to 79.99

Table: Estimates of Bit Transmissions and Times

Given that the throughput is the table’s gigabits per seconds (Gbps), calculate and record this transfer rate for your chosen values using the following formula:

3.Your school’s Cat5e network has a maximum bandwidth of 1 Gbps. Does your answer from above exceed that or not? Based strictly on your value, do you think that the maximum bandwidth should be raised? If you think that the network needs to be upgraded, will the network infrastructure need to be upgraded to support Cat6 cables? How did you arrive at this conclusion? Include the factors that influenced your decision.

NOTE: The questions below do not specifically use the calculations in Question 2.

4.It has been found that the school’s network throughput speed R(t) in gigabits per second, with respect to time in seconds, is best modeled with the following rational function:

Generate a graph of this function using Excel or another graphing utility. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0, online utilities like this site, and many others.)

Insert both the function and the graph into the Word document containing your answers and work details. Be sure to label and number the axes appropriately.

5.Peak measured throughput (PT): Also known as instantaneous throughputs, PTs measure values that are useful for systems that rely on bursts of data in a Cat5e cable. Unlike continuous streaming, information travels in short bursts. For example, during in-class demonstrations, a computer lab’s user experiences high traffic usage spikes on a very high bandwidth, which are transmitted over a relatively short period of time. Activity is important during these in-use peak times. Peak rates are measured limits taken with respect to throughput as time approaches zero. Thus, for the above function, use algebraic techniques and limit theorems to show the work details for calculating the following limit:

Based strictly on this result, do you think that your school should raise its current network’s bandwidth of 1 Gbps? How did you arrive at this conclusion?

6.Maximum sustained throughput (ST):One of the most accurate indicators of system performance for high-duty-cycle networks is the maximum ST averaged over a long period of time. This value measures the network capacity that is fully utilized over its entire existence. Essentially, high volumes of continuously streamed transmissions max out the amounts of data being transferred in because the network is busy processing the current data and is unable to efficiently enter the cable. This builds up the delivery time, causing latency instabilities. In this case, sustained rates are measured limits taken with respect to throughput as time increases toward infinity. Thus, for the above function, use algebraic techniques and limit theorems to show the work details for calculating the following limit:

Based strictly on this result, do you think that your school should raise its current network’s bandwidth of 1 Gbps? How did you arrive at this conclusion?

7.In general, the limit of a function f(x) is where the function f(x) gets close to a certain limit as x becomes closer to a certain value.

The following equation represents the concentration of a drug in milligrams per liter (mg/L) in the blood stream over time t (hours). The equation is:

8.Discuss the continuity of this function.

9.How do you think the concept of limits is related to the continuity of a function? Provide an example to support your thoughts.

10.Find a specific example of how the limit could be applied in your practical life.