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Unformatted text preview: 3.4. LEVEL CURVES 259 This process breaks down if f/y = 0: either there are no solutions, if f/x negationslash = 0, or, if f/x = 0, the equation tells us nothing about the slope. Of course, as we have seen, even when f/y is zero, all is not lost, for if f/x is nonzero, then we can interchange the roles of y and x , solving for the derivative of x as a function of y . So the issue seems to be: is at least one of the partials nonzero? If so, we seem to have a perfectly reasonable way to calculate the direction of a line tangent to the level curve at that point. All that remains is to establish our original assumptionthat one of the variables can be expressed as a function of the otheras valid. This is the purpose of the Implicit Function Theorem. The Implicit Function Theorem in the Plane We want to single out points for which at least one partial is nonzero, or what is the same, at which the gradient is a nonzero vector. Note that to even talk about the gradient or partials, we need to assume that...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Slope

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