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Course: MAC 2233, Spring 2011
School: University of Florida
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1 MAC Project 2233 1. What do we call the set of all points whose distance from the point ( h, k ) is r? Derive an equation for the graph using distance formula. Rewrite the following equations of circles by completing squares, draw the graphs, and label the points of intersection A and B . What is the distance between the centers of the circles? What is the slope of the line through the two centers? x2 + y 2 2...

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1 MAC Project 2233 1. What do we call the set of all points whose distance from the point ( h, k ) is r? Derive an equation for the graph using distance formula. Rewrite the following equations of circles by completing squares, draw the graphs, and label the points of intersection A and B . What is the distance between the centers of the circles? What is the slope of the line through the two centers? x2 + y 2 2 x + 2 y 8 = 0 x2 + y 2 6 x 4 y + 8 = 0 2. When the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula 40 + 24x2/5 R= 1 + 4x2/5 has been used to model the of dependence R on x, where R is measured in square millimeters and x 0 is measured in appropriate units of brightness. Should the pupil have a larger area when x = 0 or when x = 32 ? How may times larger is it? The equation above can be solved for the variable x2/5 . Show that x = R 40 24 4R 5/2 . For what x-value will R be 10 square millimeters (round to the nearest tenth)? Note that x is not a real number for some R values because of the square root. Solve an inequality in order to determine the values of R for which there is a real solution ( x, R ). What seem to be the smallest and largest possible areas R for the pupil?
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UCSB - ECON - 171
Econ 171 Spring 2010Notes for Lecture 2April 1Note: we will begin class by playing some in-class games and by continuing to go through theHandout on Normal Form GamesOverviewIn these notes we will continue looking at games in the normal (or strategi
University of Florida - MAC - 2233
Project 2MAC 22331. Suppose that at the base of a mountain range, the diversity of an insect species (measured inaverage number of species N per acre) depends linearly upon the elevation E in hundreds offeet below sea level according to the equation N
UCSB - ECON - 171
Lecture 4: Nash EquilibriumEcon 171 Spring 2010OverviewNash equilibrium Mixed-strategy Nash equilibrium1Nash EquilibriumBegin with threeACX4, 04, 2games that are not dominance solvable:22Y0, 24, 0O2, 10, 0OBB0, 01, 2H1, 11, 1H
University of Florida - MAC - 2233
Project 3MAC 22331. Examine the function f (x) = 4 | 2 x | .Write f (x) as a piecewise function without absolute value bars, using the denition of absolute value, and graph the function on the axes below:f (x) =Graph the function f (x) again, this ti
UCSB - ECON - 171
Econ 171 Spring 2010Notes on Applications of Nash Equilibrium1April 16OverviewThese notes explore some applications of Nash equilibrium and are intended to supplementChapter 10 of Watson. The rst section presents an extension of the Cournot and Bertr
University of Florida - MAC - 2233
Project 3MAC 22331. Examine the function.Writeas a piecewise function without absolute value bars, using the denition of absolute value, and graph the function on the axes below:Graph the functionagain, this time in a series of steps, beginning wit
University of Florida - MAC - 2233
Project 4MAC 22331. Consider the functions:f ( x) = x2g (x) =1 x2h(x) =xWrite a simplied formula for the function k = f + g g give its domain in interval notation.h(x) = 1; Domain: [ 1, 1 ]Write a simplied formula for the function k = (f + g ) h
UCSB - ECON - 171
Econ 171 Fall 2008Practice Questions &amp; Solutions - Part III: Perfect Bayesian EquilibriumPerfect Bayesian equilibrium practice questions with solutions below:28.2, 28.3, 29.1, 29.3, 29.428.2 (a) No.(b) Yes. The strategy prole is (AA , Y ), with belie
University of Florida - MAC - 2233
Project 5MAC 22331. An apartment complex contains 50 one-bedroom apartments. Whern rent is set at \$550 permonth, all the units will be occupied. Each \$30 increase in rent results in an average of twoless units occupied. Each occupied unit requires an
UCSB - ECON - 171
Econ 171 Spring 2010Some student questions about PS1 and my responsesProblem 4)Student question omitted. Here is the response, though:You are right that there are no PSNE, but there still can be MSNE. It is a good idea tolook for dominated strategies
University of Florida - MAC - 2233
Project 6MAC 22331. Examine the piecewise-dened function below, and sketch the graph carefully on the axes:g (x) =2 1+x0 2 1x1x&lt;0x=00&lt;x&lt;1x&gt;1Evaluate the following limits:lim g (x) = lim g (x) = 1x0x0+lim g (x) = 1lim g (x) = 1x1x1+2
UCSB - ECON - 171
Econ 171 Spring 2010Problem Set 1Due April 13Important: hand in only the two-star problems. (Read the syllabus for the meaning of onestar, two-star, etc.) The notation a.b denotes problem number b from Chapter a inWatson.* Problem 1(a) Create a larg
University of Florida - MAC - 2233
Project 7MAC 22336x1x.1. Write the domain of f (x) =4xxRewrite f (x) as a simplied rational function in lowest terms. Include the restrictions on thedomain. Then, using a number line, nd the intervals on which f (x) is positive and thoseinterva
University of Florida - MAC - 2233
Project 7MAC 22336x1x.1. Write the domain of f (x) =4xxx = 0, 2Rewrite f (x) as a simplied rational function in lowest terms. Include the restrictions on thedomain. Then, using a number line, nd the intervals on which f (x) is positive and thos
UCSB - ECON - 171
Econ 171 Spring 2010Problem Set 1Solutions to the two-star ProblemsIn some cases Ive provided more explanation than was asked of you.* Problem 3Find the rationalizable strategies for this game. For each iteration, list the eliminatedABCW5, 64,
University of Florida - MAC - 2233
Project 8MAC 2233x2 x 24x2 2x31. Examine the function: f (x) =a) What type of function is this? What is the domain of the function? What are its discontinuities?rational function; discontinuities at x = 0, 2b) Evaluate the right and left limits at
UCSB - ECON - 171
Econ 171 Spring 2010Problem Set 2Due May 6Important: hand in only the two-star problems. The notation a.b denotes problem numberb from Chapter a in Watson.* Problem 11LR2ab0, 0Figure 1: An extensive gameIn the game in Figure 1, solve for the
University of Florida - MAC - 2233
Project 9MAC 22331. Consider the following piecewise dened function for various choices of constants p and q .Sketch its graph in the case that p = q = 1. px &lt; 1x2x = 1f (x) = 2x1 &lt; x &lt; 2x+qx2Determine whether f (x) is continuous at x = 1, x
University of Florida - MAC - 2233
Project 10MAC 22331. A farmer keeps a small private orchard with apple trees. Suppose the total yield Y (numberof healthy apples) is a function of the number of trees planted p given by Y (p) = 120p 7p2 .Graph Y (p) below.Use the denition of the deri
UCSB - ECON - 171
Econ 171 Spring 2010Problem Set 3Due Tuesday, June 1Important: hand in only the two-star problems. There are no one-star problems on thisproblem set. The notation a.b denotes problem number b from Chapter a in Watson.* Problem 1Consider a two-player
University of Florida - MAC - 2233
Project 11MAC 2233In this project, we explore the average and instantaneous rates of change of total cost, total revenue, and total prot with respect to the number x of items produced/sold. In particular, note theinterpretations of the calculations, an
UCSB - ECON - 171
Econ 171 Spring 2010Problem Set 3 - SolutionsDue Tuesday, June 1Important: hand in only the two-star problems. There are no one-star problems on thisproblem set. The notation a.b denotes problem number b from Chapter a in Watson.* Problem 1Consider
University of Florida - MAC - 2233
Project 11MAC 2233In this project, we explore the average and instantaneous rates of change of total cost, total revenue, and total prot with respect to the number x of items produced/sold. In particular, note theinterpretations of the calculations, an
University of Florida - MAC - 2233
Project 12MAC 22331. Examine the function f (x) = x3 3x. From precalculus, what should be the two distinctpossibilities for the number of turning points of any cubic (degree 3) polynomial function?In order for a smooth graph to turn around, what value
UCSB - ECON - 171
Econ 171 (Grossman) Spring 2010Exam 1April 20You have 75 minutes to take this exam. Please answer all 6 questions, each of which is worth5 points. Point subtotals are indicated. Show your work to obtain full credit.Part 1 4 questions, 20 points (5 ea
University of Florida - MAC - 2233
Project 13/14MAC 22331. An underwater pipe begins to leak oil that spreads outward in the shape of an expandingsphere. Recall that the volume of a sphere is V = 4 r3 , where r is the length of its radius.3Suppose the company that owns the pipe takes
UCSB - ECON - 171
Econ 171 (Grossman) Spring 2010Exam 1April 20You have 75 minutes to take this exam. Please answer all 6 questions, each of which is worth5 points. Point subtotals are indicated. Show your work to obtain full credit.Part 1 4 questions, 20 points (5 ea
University of Florida - MAC - 2233
Project 15MAC 22331. One model that is used for an agricultural yield Y bushels (in thousands) as a function of soilNnitrogen level N points is Y (N ) =.1 + N2Calculate Y (N ) and Y (N ). Give the units for each.Y (N ) =Y (N ) =1 N2(1 + N 2 )2
University of Florida - MAC - 2233
Project 16MAC 22331. Suppose the daily supply x of a certain product in an open-air market is related to the unitprice p (in dollars) by the equation 100p2 + 10xp = x2 + 500.Calculatedximplicitly. What rate of change does it represent?dpFind all t
UCSB - ECON - 171
NAME:Econ 171 (Grossman) Spring 2010Exam 2 SolutionsMay 13You have 75 minutes to take this exam. Please answer all 4 questions, each of which is worth15 points. Point subtotals are indicated. Show your work to obtain full credit.1. Consider the exte
University of Florida - MAC - 2233
Project 17/18MAC 22331. The idea of a differential is pretty simple if you can get past the notation. Consider the following problem: Suppose the value (in dollars) of some electronic equipment t years from1600now is given by V (t) = 400 +.1 + t2Fi
UCSB - ECON - 171
NAME:Econ 171 (Grossman) Spring 2010Exam 2May 13You have 75 minutes to take this exam. Please answer all 4 questions, each of which is worth15 points. Point subtotals are indicated. Show your work to obtain full credit.1. Consider the extensive-form
University of Florida - MAC - 2233
Project 17/18MAC 22331. The idea of a differential is pretty simple if you can get past the notation. Consider the following problem: Suppose the value (in dollars) of some electronic equipment t years from1600now is given by V (t) = 400 +.1 + t2Fi
UCSB - ECON - 171
Econ 171 Spring 2010Final ExamJune 8You have three hours to take this exam. Please answer all 5 questions, for a maximum of100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you mustprovide arguments or work to suppor
University of Florida - MAC - 2233
Project 19MAC 223321. Examine the function h(x) = x 3x 3 . What is the domain of h(x) ? What are its intercepts?Calculate h (x), and use a number line to determine the interval(s) on which the function isincreasing/decreasing.What are the local extr
University of Florida - MAC - 2233
Project 20/21MAC 22331. Note that the information learned in this lecture should prepare you for curve sketching!Recall this function from a prior project:2f (x) = 3x( 7 x2 ) 3f (x) =7(3 x2 )(7 x2)13f (x) =28x( 9 x2 )43( 7 x2 ) 3(Again, i
UCSB - ECON - 171
Econ 171 Spring 2010Final ExamJune 8You have three hours to take this exam. Please answer all 5 questions, for a maximum of100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you mustprovide arguments or work to suppor
University of Florida - MAC - 2233
Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ?x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?What are the in
UCSB - ECON - 171
Class InfoGame TheoryNormal Form GamesEcon 171 Spring 2010Class 1 - March 30, 2010Index cards: Name, pronunciation, perm no.,major, nativelanguage, Math background, Reason for interest in class,Economics is. . .Web: http:/econ.ucsb.edu/~grossman/E
University of Florida - MAC - 2233
Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ? x = 0, 1x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?27xan
UCSB - ECON - 171
Introduction to Game TheoryEconomics 171 - Spring 2010SyllabusInstructor:Zack Grossmangrossman[at]econ.ucsb.eduOffice: NH 3049Office Hours: R 12:30-1:30Course Homepage: http:/econ.ucsb.edu/~grossman/Econ171S10Lectures: TR 2 3:15, PHELPS 1260Cour
University of Florida - MAC - 2233
Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?2. Sketch the graph of the funct
University of Florida - MAC - 2233
Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?Max volume when width is 2, leng
UCSB - ECON - 177
Professor GarrattEconomics 177AuctionsMW 11:00-12:15, 387 103www.econ.ucsb.edu/~garratt/Econ177Auctions have been used to allocate goods for thousands of years, but in the past 25 years there hasbeen a surge in popularity. Auctions are now routinely
University of Florida - MAC - 2233
Project 25/26MAC 22331. Carefully sketch the graph of f (x) = 4 + e(x2) below , including any intercepts and asymptotal lines, AND state its domain and range in interval notation.Domain:Range:Formula: g (x)=Now suppose g (x) is the function obtained
UCSB - ECON - 177
Cummulative Distribution Function of Bids for All Pay AuctionSpring 2011100.00%90.00%80.00%70.00%60.00%2 bidders5 bidders10 bidders50.00%40.00%30.00%20.00%10.00%0.00%05101520253035404550Bid556065707580859095100
University of Florida - MAC - 2233
Project 28MAC 22331. A population of wolves increases so that the number of wolves x years from now is given bythe function W (x) = 200(1 ex ). Calculate W (x).At what rate does the population change after 1 year? 3 years? (Include units and round to
UCSB - ECON - 177
Assignment 1: Econ 177Professor GarrattYour assignment is to estimate your own private bid functions that characterize your biddingbehavior in the 2-bidder and 5-bidder first- and second price auctions. We will only consider thecase of no entry fee an
University of Florida - MAC - 2233
Project 29MAC 22331. Examine the function h(x) =3 ln(x). What are its domain and intercepts ?xCalculate h (x), and h (x) . Use a number line to determine the interval(s) on which thefunction is increasing/decreasing, concave up/down. Are there any
UCSB - ECON - 177
Assignment 2: Econ 177Professor GarrattConsider the following two payment options:Scenario A: A gamble that pays \$100 with probability p and \$0 with probability (1-p).Scenario B: A payment of \$x with probability 1.In the table below, for each cash am
University of Florida - MAC - 2233
Project 29MAC 22331. Examine the function h(x) =3 ln(x). What are its domain and intercepts ?xDomain x &gt; 0;intercept ( 1, 0 )Calculate h (x), and h (x) . Use a number line to determine the interval(s) on which thefunction is increasing/decreasing
UCSB - ECON - 177
Suboptimal Bidding and Profits in 2-Bidder First-Price Auction with \$25 Entry Fee4030Dollars2010012345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40-10-20RoundAverage deviation from equ
University of Florida - MAC - 2233
Project 30MAC 22331. Suppose that the number of worldwide users of a certain internet service is growing exponentially. Suppose that, after assessing the data, there were 100 million users originally, and140 million users at the end of the second year.
UCSB - ECON - 177
Chapter 12AppendixBelow are brief descriptions and instructions for the experiments used in thiscourse. In each instance I use the following notation:v = your valueb = your bidB = the highest bid among everyone elser = reserve pricec = entry feeE
University of Florida - MAC - 2233
Project 31MAC 22331. Recall the formula and graph for the logistic function; note that one example is the function100P (t) =. Where does the inection point for the graph occur?1 + 2e0.2tNow sketch the function P (t) on the axes above. Include the i
UCSB - ECON - 177
Auction Theory With ExperimentsbyRodney J. Garratt1April 7, 20111 Departmentof Economics, University of California at Santa Barbara, garratt@econ.ucsb.edu.2OverviewThe goal of this text, its supplement, and software is to provide a meansof teachi
University of Florida - MAC - 2233
Project 32/33MAC 22331. Evaluate each of the following integrals using a basic substitution.8x3dx4 3x43x2 e2x dx1exdx4x21 + 2 ln(x)dx4x2. Demonstrate that F (u) = (u 1)eu is an antiderivative for the function f (u) = ueu .Use the substitut
UCSB - ECON - 177
Entry Decisions in 2-Bidder First-Price Auction with \$25 Entry Fee10.90.80.7Percent0.60.50.40.30.20.1012345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40RoundPercent correct entryP
University of Florida - MAC - 2233
Project 32/33MAC 22331. Evaluate each of the following integrals using a basic substitution.8x3dx4 3x43x2 e2x dxeach of these can be fairly easilyby taking the derivative of your answer1exdx4x21 + 2 ln(x)dx4x2. Demonstrate that F (u) = (u
UCSB - ECON - 177
FPA 2-bidder Spring 2011100Bids outside target range: 13Percent outside target: 0.056522Intercept 0.2717 s.e.: 1.3539Slope: 0.7812 s.e.: 0.012349080Average revenue:Average earnings:7062.92565Highest earnings: 124.84 Subject: 7560Bids52.4174
University of Florida - MAC - 2233
Project 34/35MAC 22331. Examine the parabolic arch formed by the function f (x) = 3x 1 x2 on [ 0, 6 ] . We would2like to approximate its area using rectangles.Sketch the parabola below on [ 0, 6 ] and shade the area beneath it.Now for some approxima