Suppose that our thermometer has a minimum sensitivity of 0.05∘C0.05∘C — in other words, it can’t detect a difference in temperature that is less than this amount. How long will it take until our measurements of the coffee’s temperature are indistinguishable from room temperature?

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Calculus

Background: When a function f(t) represents some real-world quantity, its limit as 𝑡→∞t→∞ represents the “long-term” behavior. Often, this kind of limit can be evaluated through algebraic methods. However, in more difficult cases, limits can be evaluated in terms of their individual parts by applying limit laws (see p. 95 from Section 2.3). As we now know, limits involving infinity can be evaluated with analytic techniques (see Section 2.6), without the need for graphs or numerical calculations. By combining these methods, we can analyze some interesting physical problems.

The Application: A cup of coffee is brought into a room with constant temperature 20∘C20∘C. The temperature of the coffee after t minutes is given by

𝑇(𝑡)=20+60𝑒𝑡/5.T(t)=20+60et/5.

Answer each of the questions about this scenario, being sure to show all your work. Use exact values whenever possible, though you may give final answers in decimal form (use at least 3 places after the decimal point).

Your Task. Your solution needs to include all of the following steps. For each one, show all the math that is required in order to get your answer.

  1. Limit Laws. Use the limit laws from Section 2.3 to evaluate lim𝑡→∞𝑇(𝑡)limt→∞T(t), being sure to justify each step with a limit law (see pages 95-96). Hint: you will need to evaluate the limit lim𝑡→∞𝑒−𝑡limt→∞e−t at some point — use what you know from Pre-Calculus to get the answer here.
  2. Interpretation. In 2-3 complete sentences, explain the physical meaning of the calculation you did in Step 1. What happens to the coffee in this scenario? Why?
  3. Equation-Solving. Suppose that our thermometer has a minimum sensitivity of 0.05∘C0.05∘C — in other words, it can’t detect a difference in temperature that is less than this amount. How long will it take until our measurements of the coffee’s temperature are indistinguishable from room temperature?

 

Choose ONE of the videos as the focus of your Observation assignment. Use the notes you took during your observation, to complete your reflection.

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Observation

– Use the Reflection Format Guide to guide your observation and take notes about each video.
– Choose ONE of the videos as the focus of your Observation assignment.
– Use the notes you took during your observation, to complete your reflection. Your work must be typed and should be 2-3 pages in length. Follow the format on the Observation Questions and Reflection Format Guide (pgs. 13-14 on the syllabus). Your response to each item must be in complete sentences and equal to 1-2 paragraph(s)

 

Summarize the case scenario of the Regional Call Center’s Washington, D.C. facility. Develop bar charts showing the mean and median current account balance. Construct a scatter diagram showing current balance on the horizontal axis and past due amount on the vertical axis.

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Week 4 Assignment – Case Study: Transforming Data Into Information

Overview

You are a supervisor at Regional Call Center’s Washington, DC, facility. Regional provides contract call center services for a number of companies, including banks and major retail companies. You have been with the company for slightly more than seven years, having joined Regional right after graduating with a master’s degree in business administration from Strayer University. After the monthly staff meeting, you were handed a new assignment by the company CEO. The assignment came out of a discussion at the meeting in which one of Regional’s clients wanted a report describing the calls being handled for them by Regional. The CEO had asked you to describe the data in a file called Regional Call Center and produce a report that would both graphically and numerically analyze the data. The data are for a sample of 57 calls and for the following variables:

  • Account Number.
  • Past Due Amount.
  • Current Account Balance.
  • Nature of Call (Billing Question or Other).

Instructions

Summarize the case scenario of the Regional Call Center’s Washington, D.C. facility.
Develop bar charts showing the mean and median current account balance.
Construct a scatter diagram showing current balance on the horizontal axis and past due amount on the vertical axis.

Compute the key descriptive statistics for current and past due amount.
Repeat task 4 but compute the statistics for the past due balances.

Compute the coefficient of variation for current account balances.

Write a 4–5-page report (including a cover page and a source list page) to National’s client that contains the results of the completed tasks along with a discussion of the statistics and graphs

Watch the Math120 Chapter 7 (part1) Toner Videos and complete Toner’s Lecture notes.

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Math120 Chapter 7 (part1) Toner Videos

Watch the Math120 Chapter 7 (part1) Toner Videos and complete Toner’s Lecture notes (paper assignment)

Here is the link to the video

 

Compute the gross income, adjusted gross income, and taxable income in the following situation. Use the exemptions and deductions in the table to the right. Explain how it was decided whether to itemize deductions or use the standard deduction.

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Compute the gross income, adjusted gross income, and taxable income in the following situation. Use the exemptions and deductions in the table to the right. Explain how it was decided whether to itemize deductions or use the standard deduction.

A woman is single and earned wages of $61,100. She received $260 in interest from a savings account. She contributed $500 to a tax-deferred retirement plan. She had $2540 in itemized deductions from charitable contributions.

Her gross income is $0. (Simplify your answer.)

Tax Rate Single 10% up to $9325 15% up to $37,950 25% up to $91,900 28% up to $191,650 33% up to $416,700 35% up to $418,400 39.6% above $418,400 Standard deduction $6350 Exemption (per person) $4050

Use the marginal tax rates in the table below to compute the tax owed in the following situation.

Paul is a head of household with a taxable income of $82,000. He is entitled to an $8000 tax credit.

Tax Rate Head of Household 10% up to $13,350 15% up to $50,800 25% up to $131,200 28% up to $212,500 33% up to $416,700 35% up to $444,550 39.6% above $444,550 Standard deduction $9350 Exemption (per person) $4050

The tax owed is $ (Type an integer or a decimal.)

A woman is in the 10% tax bracket and claims the standard deduction. How much will her tax bill be reduced if she qualifies for a $1000 tax credit?

  • • •

Her tax bill will be reduced by $1 . (Simplify your answer.)

Determine how much the following individual will save in taxes with the specified tax credits or deductions.

Rosa is in the 25% tax bracket and claims the standard deduction. How much will her tax bill be reduced if she makes a $700 contribution to charity?

Her tax bill will be reduced by $ . (Simplify your answer.)

Use the marginal tax rates in the table below to compute the tax owed in the following situation.

Marco is married filing separately with a taxable income of $67,500.

Tax Rate Married Filing Separately 10% up to $9325 15% up to $37,950 25% up to $76,550 28% up to $116,675 33% up to $208,350 35% up to $235,350 39.6% above $235,350 Standard deduction $6350 Exemption (per person) $4050

—

The tax owed is $11. (Simplify your answer. Round to the nearest dollar as needed.)

 

Explain why we need to find multiplicative inverses modulo a number. Find 122-1mod 761, if possible, without using an inverse calculator. If not possible, explain what goes wrong in your calculations. If it is possible, make sure you show all of your calculations and justification.

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Applied Mathematics

1.1 Inverses

  1. Explain why we need to find multiplicative inverses modulo a number.
  2. Find 122-1mod 761, if possible, without using an inverse calculator. If not possible, explain what goes wrong in your calculations. If it is possible, make sure you show all of your calculations and justification.
  3. Find 122-1mod 760, if possible, without using an inverse calculator. If not possible, explain what goes wrong in your calculations. If it is possible, make sure you show all of your calculations and justification.

1.2 Building Blocks

a) Suppose that a and b are relatively prime natural numbers such that ab is a perfect square. Show that a and b are each perfect squares.*

Show that the converse to (a) holds, and does not require the condition that a and b are relatively

(Uses some Week 6 Information) Classify the integers numbers with rational and/or complex square roots (no proof is necessary). Draw a visual diagram or other picture that demonstrates the classification clearly.

2.1 Check Digits You have likely had the experience of entering a number in an internet form and it immediately coming back with an error message saying something like “number not valid”. How does it know that the number isn’t valid, even before connecting with its servers? It turns out that most long, multi-digit numbers — from drivers license numbers, library book numbers, transit pass numbers, tracking numbers, and lottery ticket numbers — employ check digits – extra digits that check to ensure that there are no errors in the previous ones. Usually these digits are obtained by adding or multiplying the other digits, and then taking the result modulo a number to end-up with a one-digit result. Check-Digit Scheme 1: if the ID number without the check-digit consists of the digits al, a2, • • • , ak we take Ei ai mod n or IL a, mod n for some small number n. Since we are dealing with large numbers, we’ll assume that k > 3.

  • a) Take a recent number that you were given (choose something not-too-secretive, like a tracking number) and speculate about whether the last digit could be a check-digit. If so, what could the operation and modulo be? Note that sometimes first or last digits are stripped from the calculation.
  • b) What is the greatest possible value of n so that the result of the calculation in check Digit Scheme 1 is a one-digit number?
  • c) Now we introduce a new scheme Check-Digit Scheme 2: This scheme uses a weighing vector, (wi, w2, wk). If the ID number without the check-digits consists of al, a2, , ak, we compute
    (ai,a2, • • • ak) • (w17w27• • •mod n (al • wi,a2 • w2,…,ak • wk) mod n to obtain the check digit. The ISBN system that catalogues all books uses a check-digit system. Find the weighting vector of the ISBN-10 system and check that the ISBN of a book that you have checks-out.It Throughout this problem we are using the same notation that we used in class and in the reading for RSA. tHow many of you made a mistake transcribing the number the first time?? This proves its usefulness!
  • d) We generally talk about two types of errors in transcribing numbers: either of the type where you write down the wrong number or where you swap the positions of the digits. What is the error-catching capability of Check-Digit Scheme 2? Can it catch ALL possible errors? The following theorem tells. us exactly what errors can be caught. Theorem 2.1. Suppose a number (tor • • a/ satisfies the condition (al, az • • • ak) • (wi, ta21. • • wk) = 0 mod n The single error obtained by substituting a’ for a is undetectable if and only (di — ai)wi is divisible by n. A single swapping error (where ei and aj are swapped) is undetectable if and only if (ai — ai)(wi — wi) is divisible by rt. e) At one point in Querbec, the weighting vector (12,11,10,…,2,1) was used for all drivers’ licenses while Newfoundlanders had the weighting vector (1, 2, 3, 4, 5, 6, 7, 8, 1) applied to their licenses. Which types of errors will the check-digit schemes avoid? (An easier way to answer might be: which ones will they miss? ) What about the ISBN system?

2.2 RSA, Uncovered In the following question we use standard notation to refer to RSA cryptography. a) Why do you need to make the numeric message m into smaller. messages if m > n? Answer this question by giving an example showing what happens when it does not get decomposed into smaller pieces; that is, give an example showing the case where m < n. You should choose a 2-digit n here. b) Demonstrate the importance of using very large numbers when generating keys by deducing the fol-lowing private keys, given the public keys below:

Person 71 e Amen 98662273 1313 Briana 99633329 2791 Cheng 222561187 52107

  1. c) You have to compute a lot of powers when using RSA. How can you more efficiently compute powers like m8 by first computing powers like m2 (called the “square and multiply” technique). Using this technique, how many multiplications do you need to computer m100? Is there a larger power that requires the same number of multiplications?§

littps://www.overleal.com/project/63420ccbb3d93698dff26b40

3.1 Key Exchange The Diffie-Hellman Key Exchange System provides a way for two people – not just the sender of the message – to determine the keyword. Here is the Key Exchange system procedure: 1. Beth and Stephanie agree (Not privately – perhaps over the phone) on a prime number p and some arbitrary integer q with q < p. Since p is prime, gcd(p, q) = 1. 2. Beth and Stephanie privately choose integers b and s respectively with b < p and s < p. 3. Beth computes B = q”( mod p) and Stephanie computes S = q8( mod p). 4. Beth sends B to Stephanie and Stephanie sends S to Beth. 5. Beth computes Sb mod p and Stephanie computes 138 mod p. 6. Beth and Stephanie necessarily come up with the same answer. Call this answer K. 7. By some mutually agreed upon algorithm, the integer K is interpreted as a string of letters, the keyword for the Vigenere encipherment or some other system. I a) Prove that Beth and Stephanie come up with the same answer in step (6). b) In order to determine a key in this system, is it necessary for both Beth and Stephanie to determine primes p, q and integers I), s? Or, can they come up with a key with less information than this? c) How is this system secure? Would you recommend primes or numbers of a certain length? d) You receive the following message:

yasnblifkinvlbaxfj1

Yikes!! BUT… you know that the keyword is generated by a Diffie-Hellman key exchange with p = 4576384930643, q = 64783087731, and the private keys are 476388475629 and 2243552788. Suppose that the mutually agreed upon algorithm to interpret the integer K as a string of letters is the following: • If K contains an odd number of digits, add a 0 to the end • Decompose K into groups of two digits • Compute the value of each two-digit block mod 26 • Determine the letter occupying each alphabetic position in this string of letters (with A=1). Use this information to find the plaintext message.

!This explanation comes from Lewand

 

Post an alternative solution. Explain what is different and how it works. Compare the ArrayList approach with the alternative solution, evaluate which one is more effective, and explain why.

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Array List object

Revise this code example C# for Finding the duplicates and follow the discussion question below. Apart from code, the writing part is just 150 words

 

//function to display the duplicate values in an Array  public void Display Array(ArrayList ary)

{ //loop through all the elements  for (int i = 0; i < ary.Count; i++)

{Console.Write(ary[i]+” “); }  Console.WriteLine(); }

//function to find the duplicate values in an Array public void Find Duplicate(ArrayList ary)

{ //Array list to store all the duplicate values  ArrayList dup = new ArrayList(); for (int i = 0; i < ary.Count;i++)

{ for (int j =i+1; j < ary.Count; j++) { //compare each value with following remaining values  if (ary[i].Equals(ary[j]))

{ //When duplicate value is found, check //whether the value not contained in the dup array list   if(!dup.Contains(ary[i]))

{ //if not contains, then add the value to dup array list  dup.Add(ary[i]);   }}} }

Console.WriteLine(“The numbers which duplicates are”);    DisplayArray(dup); } //Input Arraylist values: 4,5,2,5,4,7 Output: 4 5 2 5 4 7 The numbers which are duplicates: 4 5

DISCUSSION TOPICS

After reading week 2 required materials and conducting independent research as needed, discuss with your peers the following:

  • The solution uses an ArrayList object. Modify it and make it run with code that uses a different approach to solve the problem with identical results:
  1. Post an alternative solution(it must run on Visual Studio)
  2. Explain what is different and how it works
  3. Compare the ArrayList approach with the alternative solution, evaluate which one is more effective, and explain why.

 

Solve the following quadratic equation by the factorisation method: x ^ 2 + 2√2x – 6 = 0

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Algebra

Solve the following quadratic equation by the factorisation method: x ^ 2 + 2√2x – 6 = 0

 

Create a fractal image. When developing your fractal image, you may create any type of fractal art, including Mandelbrot Sets, Julia Sets, fractal landscapes, or nature pictures made of self-replicating fractals.

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LBS 330 Project 2 fractal art

When was the last time you had an art project in your math class? For Project 2, use a fractal art program to create a fractal image. The objective is to develop the most inspiring image as voted on by your classmates (no, you cannot vote for your own fractal image!). When developing your fractal image, you may create any type of fractal art, including Mandelbrot Sets, Julia Sets, fractal landscapes, or nature pictures made of self-replicating fractals. The option is yours. Be creative! Be sure to include the following elements in your submission:

1) An explanation of the mathematical algorithms, values, and software utilized to generate the fractal

2) Saint Leo color scheme, logos, or landscape In addition, any comments or recommendations on this project for future students are greatly appreciated.

You will post this project to the Project 2 Discussion Board no later than Sunday 11:59 EST/EDT. Since you are creating this image, you will own the copyright for the image you create. Digital watermarks on the image are acceptable (this allows evaluation versions of several programs to avoid purchase). Be sure to review the rubric for this project before submission. Once submitted, please vote for your favorite fractal using the discussion tool. To help you get started, the links below include some software that might help.

Fractal Arts Opens in new tab icon – They have many images and also a page with software available on the internet. The gallery might give you some ideas, and the software page has links to several programs that could be useful.

Fractal Foundation Opens in new tab icon – This site has some award-winning fractal art for inspiration. Also, it lists the software used. Some of this software is powerful but has a high learning curve.

Ultra Fractal Opens in new tab icon – This appears to be a popular application for artwork. There is an evaluation version that includes a watermark, which is ugly but acceptable.

Download.com Opens in new tab icon – This site has links to Fractal DRAW Pro and will perform self-replicating fractals.

You can also find links to Genesis ll software that will create fractal landscapes. Do not wait until the last minute to create your fractal. This project can be an enjoyable way to discover math if you give yourself some time to use the fractal image applications proficiently.

Solve the following IVP: (3x2y + 2xy + y3)dx + (x2 + y2)dy = 0, y(0) = 1

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Differential Equations

Problem

  1. Consider a general linear equation:

dy

dx + P (x)y = Q(x)

  • i) Write it in the form M (x, y)dx+N (x, y)dy by taking Q(x) to the LHS and then multiplying the ODE by dx.
  • ii) Use the above form to check when it will be exact.
  • iii) In general, find an integrating factor μ to make this ODE exact.

Problem 2. Solve the following IVP:

(3x2y + 2xy + y3)dx + (x2 + y2)dy = 0, y(0) = 1

Problem 3. Solve the following IVP:

6y′′ − 5y′ + y = 0, y(0) = 1, y′(0) = 1

Problem 4. Solve the following IVP:

y′′ + 4y′ + 4y = 0, y(0) = 1, y′(0) = 3

Problem 5. Find the general solution to the following ODEs:

  1. i) y′′ + y = 0
  2. ii) 2y′′ + 2y′ + y = 0

1Problem 6. Consider the ODE:

y′′′ − y′′ − y′ + y = 0

  • i) Write the characteristic equation associated to this ODE and find the solution(s) to that equation.
  • ii) Analogous to the 2nd order situation, try to find 3 distinct non-zero solutions to the ODE and check that all three are solutions.
  • iii) Guess what the general solution should be