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Temperature T of a plate lying in xy plane is defined T(x,y)=50-(x^2)-(2y^2). An ant, which is initially at (2,1) moves along a curve ensuring the temperature is decreasing as rapidly as possible. I need to find the equation of this curve.

The gradient vector is <-2x, -4y>, but I need to go to the decreasing side, therefore the direction is <2x, 4y>.

Having this information, how can I find the equation of the curve?

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    Should it not be dy/dx = -(-4y/-2x)?2012-10-07

1 Answers 1

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Translate the condition to the differential equation $\frac{dy}{dx}=\frac{-4y}{-2x},$ which is separable, and easily solved. (Remember to use the initial condition to evaluate the constant of integration.)