course Mth 272
b~˿X`y`»assignment #026
026.
Applied Calculus II
11-17-2008
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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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12:32:26
z=c
ln(z)/x
The level curves are hyperbolas
Close. Rather than ln(z) / x should be y = ln(c) / x.
A bit more detail:
** 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.
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22:52:35
What is the level curve z = c for the given function?
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RESPONSE -->
z=c
ln(z)/x
The level curves are hyperbolas
confidence assessment: 3
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22:55:22
Describe how the level curves look for the given values of c, and how they change from one value of c to another.
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RESPONSE -->
There are hyperbolas in all four quadrants of the graph. The whole number values of c have hyperbolas that are located in quadrant II and quadrant III. The fractional values of c are hyperbolas in quadrant 1 and quadrant IV. As the values of the whole numbers for c increase, the corresponding hyperbola is graphed farther and farther away from the origin. The opposite trens is present with the fractional values for c.
confidence assessment: 2
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22:56:37
What are the values of W at (15, 10), (12, 9), (12, 6) and (4,2)?
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RESPONSE -->
1/(15-10)= 1/5
1/(12-9)= 1/3
1/(12-6)= 1/6
1/(4-2)= 1/2
confidence assessment: 3
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22:58:33
You 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?
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RESPONSE -->
For a function that necesitates that y
Good. Compare with the following 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
confidence assessment: 2
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