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CAF123
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Homework Statement
The temperature T of a plate lying in the (x,y) plane is given by [tex] T(x,y) = 50 - x^2 - 2y^2. [/tex] A bug on the plate is intially at the point (2,1). What is the equation of the curve the bug should follow so as to ensure that the temperature decreases as rapidly as possible?
The Attempt at a Solution
So, [tex] \vec{∇}T(2,1) = <-4,-4> [/tex] If we want the temperature to decrease as rapdily as possible we are looking for when the directional derivative is minimal. So the direction is [itex] -\vec{∇}T(2,1) = <4,4>. [/itex] My professor gave a hint to find dy/dx and then solve for y. Using the Implicit Function theorem this gives; [tex] \frac{dy}{dx} = -\frac{-2x}{-4y}. [/tex] I don't see how this method takes into account that we are considering the minimal temperature? (In fact, the gradient vector is not used at all)?