Midterm Practice (Not to be turned in)
Exam 2: May 11, 2011

• The exam covers all the material from the class since the first exam. (See detail on class notes page.)
• In preparation for the midterm, I have compiled some practice problems.  Try them out and perhaps you will have questions about them that we will discuss during Q&A day on Monday, 5/9/11. 
  • Book:  1.5.2, 4.2.1, 4.2.2, 4.2.3, 4.2.7 (setup only)
  • Q1.  Create the linear program that will answer question 4.2.13 (p. 262). If you feel inspired, use Mathematica to solve the linear program once you have found it.
  • Q2.  Answer the questions from Section 4 of Mathematica Tutorial 5.
  • Q3.  Explain what types of errors may occur when your calculator calculates e¹⁰.
  • Q4.  Give the definitions of state space and random variable.
  • Q5.  Perform a sensitivity analysis on the fertilizer example on page 253 of the book and page 112 of the notes. That is, determine the equilibrium cost of both a unit of phosphate and a unit of nitrate. (Hint: Increase the amount of phosphates available and see how much the profit increases.)
• Here are some Mathematica questions that would be fair game:
  • Write an If statement that outputs "Heads" with probabilty one third, and "Tails" with probability two thirds.
  • How would you generate a list of 100 trials of Heads and Tails?
  • What does % represent in Mathematica? What would happen if you evaluated it by itself?
• Here is a copy of last year's second exam. No similarities are to be implied for this year's second exam. (You did not see Markov Chains in this class so ignore Question 1.)
• Also to keep in mind: Project Presentations start Monday, May 16.

• Read Sections 1.7, 4.1, and 4.2. Then complete the following problems, worth 10 points each.
• 5-1. (a)Modify the waiting room algorithm from the notes and tutorial in order to simulate that patients arrive between 9:00am and 3:00pm, and that the expected number of patients that arrive in any day is 13.5. (Make sure that you justify that your model ensures this restriction.) (b) Run your simulation 1000 times to determine if your modification increases, decreases, or keeps the same the expected number of patients in the waiting room at the end of the day. (c) Discuss whether the answer you find is what you expected to find.
• 5-2. (a) Write down the linear program that will answer the following optimization problem.

  A farmer has 24 acres on which to grow corn and tomatoes and 80 pounds of fertlizer with which to do it. The corn needs 2 pounds of fertilizer per acre and the tomatoes need 5 pounds of fertilizer per acre. If an acre of corn sells for 600 dollars and an acre of tomatoes sells for 400 dollars, determine how many acres of corn and tomatoes should be planted to maximize profits.

  (b) Next, use Mathematica to plot the feasible region. [Hint: use RegionPlot]
  (c) Last, use Mathematica to solve the linear program.  [Hint: You may find Tutorial 6 helpful.]
  
• Remember: Your revised project is due Wednesday, May 4, 2011.  Bring in one paper copy per group AND ALSO submit your project electronically on Blackboard (there is no need to submit figures electronically if this is a problem).

• Read Sections 2.1, 5.3A (pp.377–385), and the first bit of 5.1. Then complete the following problems.
• 4-1. Determine and justify the category of error involved in each of the two Sidelight stories in Section 2.1 (page 76 and then page 77). After this, write a paragraph discussing the difference between the two errors encountered. Last, write a few sentences explaining the morals of these errors and how it relates in general to the modeling process in real-life.
• 4-2. Suppose that you are modeling the arrival of patients at a doctor's office in the first five minutes, starting at 9am. Suppose that at every minute (9:00, 9:01, 9:02, 9:03, 9:04), there is a probability of 0.075 that a patient arrives.
  (a) What is the probability that the first patient arrives at 9:02?
  (b) What is the probability that no patients arrive in these first five minutes?
  (c) What is the probability that exactly two patients will arrive in these first five minutes?
  (d) What is the expected number of patients that arrive in these first five minutes?
• 4-3. (a) Determine the system reliability of the following communication system. There are two different methods of communicating; at least one method must succeed in order for the system to succeed. The first method is by an FM radio, which has reliability 75%. The second method is by a satellite radio; both the physical radio must transmit the signal (with 90% reliability) AND the satellite must retransmit the signal (with 95% reliability) in order for the satellite radio to succeed.
  (b) Give a pseudocode description of how you might use a computer to simulate the situation in part (a) in order to calculate the expected reliability of the system.
• 4-4. This question involves Mathematica.
  (a) Use Mathematica's RandomInteger command to simulate rolling a six-sided die 1000 times and calculate how many ones appear.
  (b) Use a Table command to repeat this experiment 500 times. The result will be a list of 500 numbers, each representing the number of ones that appeared out of 1000. Take the average of this list by using the command Mean.
  (c) Input the list from part (b) into the Histogram command to see a visualization of the 500 trials.
  (d) Discuss what you see in the answers to part (b) and (c) and how they relate to the real-life die-rolling situation.
• Remember: The final draft of your project is due Wednesday, April 13. Bring three paper copies of your final draft to class on April 13. You will participate in a peer review day. Attendance is mandatory; non-attendance will decrease your personal project grade. If you have any questions about what is expected in your final draft, see Prof. Chris as soon as possible.

• Prepare your project organizational statement in accordance with the details on the project page.
• Note: This is two days earlier than originally scheduled.
• We will be in the computer lab in Kiely 236 on Wednesday, March 30  (NOT Monday 3/28).

• The exam covers all the material from the class up through Monday, 3/7/11. (See detail on class notes page.)
• In preparation for the midterm, I have compiled some practice problems.  Try them out and perhaps you will have questions about them that we will discuss during Q&A day on 3/10/11. 
  • Book: 3.1.13 (p. 149), 3.2.3, 3.2.5 (p. 167–168), 3.3.1, 3.3.3 (p. 179),
  • P-1.  Question 3.1.2 (p149). Explain some advantages and disadvantages in complete sentences; give at least four total (advantages + disadvantages).
  • P-2.  Question 3.1.7. DO give examples of such models and explain why they satisfy the conditions you claim.
  • P-3.  What are the steps in the modeling process? What is done at each step?
  • P-4.  The following data is assumed to fit a logarithmic model, y=a+b*ln[x]. Determine the best values for a and b using the least squares criterion. [You may use Mathematica, but that is not required.]

    x	1.1	1.9	2.8	3.8	5.1	6.2	7.2	8.0	8.7	9.8
    y	3.2	5.2	6.4	7.3	8.0	8.6	9.1	9.5	9.9	10.2

• Here are some Mathematica questions that would be fair game:
  • Explain the difference between the following two lines of Mathematica code:
    a=Table[3i,{i,1,5}];
    a=Table[3i,{i,1,5}]
  • Write down the Mathematica code which would determine the numerical value of the expression Sqrt[10]+Log[2].
  • What are the differences between (), [], and {}?
  • What do you expect when you evaluate the code Table[2^k,{k,1,10}]?
  • When would you use the TableForm command?
  • What is the syntax of the command which fits datapoints to a curve of the form y=ax^3+bx^4+cx+d? 
  • You need to know how to plot a list, how to plot a function, and how to incorporate two plots together.
• Here is a copy of last year's first exam. No similarities are to be implied for this year's first exam.
• Also to keep in mind: Your project topic is due on Wednesday, March 16.

• Read Sections 3.2 and 3.4. Then complete the following problems. There are only four problems; they will be worth five points each.
• 3-1.  Find by hand the least squares best fit line to the data {(1, 2), (4, 15), (6, 11)}.
  
• 3-2.  Read http://eagereyes.org/criticism/anscombes-quartet.  This question is to understand the set of four figures in the middle of the page.  Write three paragraphs explaining this blog post in the context of this class.  Explain what makes the four data sets similar; why are they grouped together?  Discuss the differences in the figures; does the "line of best fit" fit one of the sets of data better than another?  What does "best fit" mean in this context?  Last, given these four figures and the lines of best fit, how would you modify your modeling approach to find a better fit to each data set (if necessary).
• 3-3.  Create and print out a Mathematica file (also called a notebook) with your name, the date, and which solves the following questions.
  (a) Use the Table command to create a list of 10 pairs of integers of the form {x,x+3} where each x is one more than a multiple of five; then print out the list in a nice way using the TableForm command.
  (b) Generate three three individual plots and combine them in a fourth plot, as follows. First, create a plot of the sine function from 0 to 2pi that has a thick green line. Next, create a plot of the cosine function from 0 to 2pi that has a thin dashed line and is not green. Third, create a plot of ten to twenty large bright fluorescent points of the form (x,x²-2x) where x ranges from 0 to 2pi. Last, combine all three graphs on one set of axes.
• 3-4.  Use Mathematica to solve problem 3.4.9 on page 196.
• We will be in the computer lab in Kiely 236 on Monday, February 28.
• Reminder: The first exam is scheduled for Monday, March 14.

• Read Section 1.3 to page 25, all of Section 1.4, and Section 3.2 to the middle of page 154. Then complete the following problems. There are only four problems; they will be worth five points each.
• 2-1.  On pages 42–43, complete problems (a) 1.4.2 (b) 1.4.4 (c) 1.4.9a (do not do b or c)
• 2-2.  Perhaps the data in Problem 1-5 (see below) better fits a curve of the form y=Ce^kx.  Transform the data appropriately in order to determine visually approximate values for the constants C and k. Which model (this or 1-5) is better in your opinion? Justify your answer.
• 2-3.   (a) Include recent data on population from the years 1980, 1990, 2000, and 2009 to update the chart on page 42.
  (b) Does the original model appear to still be accurate? Explain.
  (c) If there are any discrepancies in the model, try to explain them (you may have to do some research).
  (d) Modify the model to try to model the population of the United States for the years up through 2050, and explain your reasoning.
• 2-4.  Go to Mathematica's Documentation Center (One way to get there is through the help menu), and enter some mathematical phrase that is of interest to you.  This will search through Mathematica's documentation and give you relevant results (except in the case you give a precise command; in that case, it will take you directly to that command).  Find a command that is related to your phrase of interest.  You will know that it is a command because it will start with a capital letter, and in the search results it will say "Built-in Mathematica symbol".  Read through the provided help page on the command and make sure that the command does something interesting. Here is what you need to do. 
  (a) Copy down the command on paper, including the syntax of the command.  (For example, I picked the following one out at random (don't use it)):

  CylindricalDecomposition[ineqs,{x₁,x₂,...}] 

  (b) Explain in your own words what the command does.
  (c) Give three examples of how to use the command, different from those given on the help page (although you can use the given examples as a guide); print out the output from these commands and attach it to your homework.
  

• Thoroughly read the class web page including the syllabus and schedule. This should answer all the questions that you may have about the class. Read Sections 1.1 and 1.2 to page 18.
• Problem 1-1 must be completed online before class on Monday 2/7 for credit.
  • 1-1.  (a) Email me at chanusa@qc.cuny.edu with the email address where you are best contacted. (Make sure to include your name in the message as well!)
    (b) Take the syllabus quiz on Blackboard. Retake the quiz as necessary to earn a score of 100%.
    (c) Update your email address on the CUNY First system and on Blackboard to be the one where you are best contacted.  
• Problems 1-2 through 1-5 should be written up and handed in in class before class starts on Monday:
  • 1-2. Read Section 1.1 from the textbook (pp. 1-11); it gives six different models that describe real-life situations. Write two to three paragraphs explaining where you have experience with mathematical models in real life. (Do not use the examples from the book unless you feel you have a unique perspective.) In these paragraphs, explain how the real-life situation and the model differ. (You may want to address the following questions: Are there simplifying assumptions? Does the model truly describe what happens in real life or are there limits to the model's effectiveness? ... )
  • 1-3.   Here are three different vague scenarios. Choose one of the three and identify a precise problem statement that you would like to study. Determine eight variables that affect your proposed problem.  Then of those eight variables, choose those that are the most important and explain why you think they are more important than the rest.
    1. The MTA is considering implementing a fast bus corridor between Flushing and Jamaica.
    2. A local toy merchant needs to determine how many toys to stock and at what price to sell them.
    3. Queens College would like to convince more students to play intramural sports.
  • 1-4.   Suppose that a ball is thrown downward at 8 ft/sec.  Determine the new assumption that should replace Galileo's Assumption 1* on page 17.  Use the new assumption to determine the formula for the position of the ball (x(t)), and determine the position and velocity of the ball at time t=1.
  • 1-5.   Here is some data that should fit a polynomial of the form y=Cx^k for some constants C and k, where x is an independent variable and y is a dependent variable. Use the method involving logarithms from class on Wednesday to approximate C and k.  Explain the steps you applied and how you found your answer.

    x	1	3	4	6	7	8	9	10	12
    y	2	9	6	35	58	87	122	165	290

• Remember what is expected:
  • Write your solutions using complete sentences.
  • Follow the guidelines for turning in homework.
• We will be in the computer lab in Kiely 236 on Monday, February 7.
  • If you want to help getting Mathematica installed on your laptop computer, bring it to class.