Finding the Rate of Change of Temperature

The temperature \(T\) (in degrees Celsius) of a metal plate, located in the xy-plane, at any point \((x,y)\) is given by \(T=T(x,y) =24(x^{2}+y^{2})^{2}\).

1. Find the rate of change of \(T\) in the direction of the positive \(x\)-axis at the point \((1,-2)\).
2. Find the rate of change of \(T\) in the direction of the positive \(y\)-axis at the point \((1,-2)\).

Solution  (a) The rate of change of temperature in the direction of the positive \(x\)-axis is given by \[ \begin{equation*} T_{x}(x, y)=2(24)(x^{2}+y^{2})(2x)=96\;x(x^2+y^2) \end{equation*} \]

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At the point \((1,-2)\), \(T_{x}(1,-2)=96(1)(5) = 480\). This means that as one moves in a horizontal direction to the right away from the point \((1,-2)\), the temperature of the plate increases at the rate of \(480 ^\circ{\rm C}\) per unit of distance.

(b) The rate of change of temperature in the direction of the positive \(y\)-axis is given by \[ \begin{equation*} T_{y}(x, y)=2(24)(x^{2}+y^{2})(2y)=96\;y(x^2+y^2) \end{equation*} \]

At the point \((1,-2)\), \(T_{y}(1,-2)=96(-2) (5) =-960\). This means that as one moves in a vertical direction up from the point \((1,-2)\), the temperature of the plate decreases at the rate of \(960 {^\circ{\rm C}}\) per unit of distance.