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December 7th, 2012, 07:55 PM   #1
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Implicit and critical numbers?

the function is f(x) = x^2+xy +y^2-3=0 and you are told to differentiate it. I got dy/dx = (-2x-y)/(x+2y). I believe this is correct but the second part asks you to find where the function is "rising and falling", which I take to mean where it is increasing and decreasing. I know that youw ould usually set the derivated equal to zero and solve, but how would you do this for this problem since it has two variables?
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December 7th, 2012, 08:10 PM   #2
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Re: Implicit and critical numbers?

Here is a plot of the curve along with the lines we get when we equate the numerator and denominator of the derivative to zero:

[attachment=0:3jmbyee6]slantedellipse.jpg[/attachment:3jmbyee6]

What does this tell us?
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File Type: jpg slantedellipse.jpg (24.4 KB, 82 views)
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December 8th, 2012, 08:36 AM   #3
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Re: Implicit and critical numbers?

You know that and you know the function will be increasing when that is positive, decreasing when it is negative. That is, this is a matter of determining when and when . You should also know two things: a continuous function can only change sign when it is 0, and a rational function like this is 0 only when the numerator is 0 and discontinuous only when the denominator is 0. That means that this function can change sign only when 2x+y= 0, which gives y= -2x so that you can put into and solve for x, or when x+ 2y= 0 which occurs on the line y= -x/2, which can, again, put into and solve for x. Once you have found the values of x that satisfy those equations, you can check a single point in each interval between them to determine if the derivative is positive or negative on that interval.

Fortunately, those give the same x-values as those where MarkFL's line crosses the ellipse!
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