#$&* course Mth 272 07/28 10 pm Question: `qQuery problem 7.3.38 level curves of z = e^(xy), c = 1, 2, 3, 4, 1/2, 1/3, 1/4.
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Given Solution: `a The z = c level curve of e^(xy) occurs when e^(xy) = c. We solve e^(xy) = c for y in terms of x. We first take the natural log of both sides: ln(e^(xy)) = ln(c), or xy = ln(c). We then divide both sides by x: y = ln(c) / x. For c = 1 we get y = ln(1) / x = 0 / x = 0. Thus the c = 1 level curve is the x axis y = 0. For c = 2 we get y = ln(2) / x = .7 / x, approximately. This curve passes through the points (1,.7) and (-1, -.7), and is asymptotic to both the x and y axes. For c = 3 we get y = ln(3) / x = 1.1 / x, approximately. This curve passes through the points (1,1.1) and (-1, -1.1), and is asymptotic to both the x and y axes. For c = 4 we get y = ln(4) / x = 1.39 / x, approximately. This curve passes through the points (1,1.39) and (-1, -1.39), and is asymptotic to both the x and y axes. For c = 1/2 we get y = ln(1/2) / x = -.7 / x, approximately. This curve passes through the points (-1,.7) and -1, -.7), and is asymptotic to both the x and y axes. For c = 1/3 we get y = ln(1/3) / x = -1.1 / x, approximately. This curve passes through the points (-1,1.1) and -1, -1.1), and is asymptotic to both the x and y axes. For c = 1/4 we get y = ln(1/4) / x = -1.39 / x, approximately. This curve passes through the points (-1,1.39) and -1, -1.39), and is asymptotic to both the x and y axes. The c = 2, 3, 4 level curves form similar hyperbolas in the first and third quadrant which progressively 'bunch up' closer and closer together. Similar behavior is observed for the c = 1/2, 1/3, 1/4 hyperbolas, which occur in the second and fourth quadrants. STUDENT COMMENT I had a little bit of trouble applying my graphing here given dimension, but clear pertaining to concept, making a bit more progress. INSTRUCTOR RESPONSE You should be very familiar with the graph of y = 1 / x. Focus on the point (1, 1) and the fact that the graph in the right half-plane has asymptotes with the positive y and positive x axes. If you multiply this function by ln(c) the point becomes (1, ln(c)). For c = 1, 2, 3, 4 the points rise higher and higher, but with less space between successive points. The level curves pass through these points, still with asymptotes at the x and y axes. Each curve is a bit 'higher' than the preceding. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ok ********************************************* Question: `qQuery problem 7.3.46 queuing model W(x,y) = 1 / (x-y), y < x (y = ave arrival rate, x = aver service rate). What are the values of W at (15, 10), (12, 9), (12, 6) and (4,2)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: plug in values 1/5 1/3 1/6 1/2 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a 1/5, 1/3, 1/6, 1/2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Ok ********************************************* Question: `qYou may take extra time with the following: What is the nature of the worst combination of x and y, and why is this bad--both in terms of the behavior of the function and in terms of the real-world situation? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x= number of people, y= amount of food The worst combination would be if x is greater than y because there would be more people than amount of food; therefore, people would starve confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aGood answer by student: x=service, y=arrival rate therefore the worst senerio would be when the arrival rate was almost as long as the service rate...its worse to have a customer waiting a long time than having a customer being served for a long time &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Ok ********************************************* Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment." Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qQuery problem 7.3.46 queuing model W(x,y) = 1 / (x-y), y < x (y = ave arrival rate, x = aver service rate). What are the values of W at (15, 10), (12, 9), (12, 6) and (4,2)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: plug in values 1/5 1/3 1/6 1/2 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a 1/5, 1/3, 1/6, 1/2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Ok ********************************************* Question: `qYou may take extra time with the following: What is the nature of the worst combination of x and y, and why is this bad--both in terms of the behavior of the function and in terms of the real-world situation? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x= number of people, y= amount of food The worst combination would be if x is greater than y because there would be more people than amount of food; therefore, people would starve confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aGood answer by student: x=service, y=arrival rate therefore the worst senerio would be when the arrival rate was almost as long as the service rate...its worse to have a customer waiting a long time than having a customer being served for a long time &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Ok ********************************************* Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment." Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!